Properties

Label 324.3.f.f.55.1
Level $324$
Weight $3$
Character 324.55
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 55.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 324.55
Dual form 324.3.f.f.271.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.00000 - 6.92820i) q^{5} -8.00000 q^{8} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.00000 - 6.92820i) q^{5} -8.00000 q^{8} -16.0000 q^{10} +(5.00000 + 8.66025i) q^{13} +(-8.00000 + 13.8564i) q^{16} -16.0000 q^{17} +(-16.0000 + 27.7128i) q^{20} +(-19.5000 + 33.7750i) q^{25} +20.0000 q^{26} +(20.0000 - 34.6410i) q^{29} +(16.0000 + 27.7128i) q^{32} +(-16.0000 + 27.7128i) q^{34} -70.0000 q^{37} +(32.0000 + 55.4256i) q^{40} +(-40.0000 - 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(39.0000 + 67.5500i) q^{50} +(20.0000 - 34.6410i) q^{52} +56.0000 q^{53} +(-40.0000 - 69.2820i) q^{58} +(11.0000 - 19.0526i) q^{61} +64.0000 q^{64} +(40.0000 - 69.2820i) q^{65} +(32.0000 + 55.4256i) q^{68} +110.000 q^{73} +(-70.0000 + 121.244i) q^{74} +128.000 q^{80} -160.000 q^{82} +(64.0000 + 110.851i) q^{85} -160.000 q^{89} +(65.0000 - 112.583i) q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} - 8q^{5} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} - 8q^{5} - 16q^{8} - 32q^{10} + 10q^{13} - 16q^{16} - 32q^{17} - 32q^{20} - 39q^{25} + 40q^{26} + 40q^{29} + 32q^{32} - 32q^{34} - 140q^{37} + 64q^{40} - 80q^{41} - 49q^{49} + 78q^{50} + 40q^{52} + 112q^{53} - 80q^{58} + 22q^{61} + 128q^{64} + 80q^{65} + 64q^{68} + 220q^{73} - 140q^{74} + 256q^{80} - 320q^{82} + 128q^{85} - 320q^{89} + 130q^{97} - 196q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 1.73205i 0.500000 0.866025i
\(3\) 0 0
\(4\) −2.00000 3.46410i −0.500000 0.866025i
\(5\) −4.00000 6.92820i −0.800000 1.38564i −0.919615 0.392820i \(-0.871499\pi\)
0.119615 0.992820i \(-0.461834\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −8.00000 −1.00000
\(9\) 0 0
\(10\) −16.0000 −1.60000
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 5.00000 + 8.66025i 0.384615 + 0.666173i 0.991716 0.128452i \(-0.0410008\pi\)
−0.607100 + 0.794625i \(0.707667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(17\) −16.0000 −0.941176 −0.470588 0.882353i \(-0.655958\pi\)
−0.470588 + 0.882353i \(0.655958\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −16.0000 + 27.7128i −0.800000 + 1.38564i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −19.5000 + 33.7750i −0.780000 + 1.35100i
\(26\) 20.0000 0.769231
\(27\) 0 0
\(28\) 0 0
\(29\) 20.0000 34.6410i 0.689655 1.19452i −0.282294 0.959328i \(-0.591095\pi\)
0.971949 0.235190i \(-0.0755712\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 16.0000 + 27.7128i 0.500000 + 0.866025i
\(33\) 0 0
\(34\) −16.0000 + 27.7128i −0.470588 + 0.815083i
\(35\) 0 0
\(36\) 0 0
\(37\) −70.0000 −1.89189 −0.945946 0.324324i \(-0.894863\pi\)
−0.945946 + 0.324324i \(0.894863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 32.0000 + 55.4256i 0.800000 + 1.38564i
\(41\) −40.0000 69.2820i −0.975610 1.68981i −0.677908 0.735147i \(-0.737113\pi\)
−0.297702 0.954659i \(-0.596220\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −24.5000 42.4352i −0.500000 0.866025i
\(50\) 39.0000 + 67.5500i 0.780000 + 1.35100i
\(51\) 0 0
\(52\) 20.0000 34.6410i 0.384615 0.666173i
\(53\) 56.0000 1.05660 0.528302 0.849057i \(-0.322829\pi\)
0.528302 + 0.849057i \(0.322829\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −40.0000 69.2820i −0.689655 1.19452i
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 11.0000 19.0526i 0.180328 0.312337i −0.761664 0.647972i \(-0.775617\pi\)
0.941992 + 0.335635i \(0.108951\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 40.0000 69.2820i 0.615385 1.06588i
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 32.0000 + 55.4256i 0.470588 + 0.815083i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 110.000 1.50685 0.753425 0.657534i \(-0.228401\pi\)
0.753425 + 0.657534i \(0.228401\pi\)
\(74\) −70.0000 + 121.244i −0.945946 + 1.63843i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 128.000 1.60000
\(81\) 0 0
\(82\) −160.000 −1.95122
\(83\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) 64.0000 + 110.851i 0.752941 + 1.30413i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −160.000 −1.79775 −0.898876 0.438202i \(-0.855615\pi\)
−0.898876 + 0.438202i \(0.855615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 65.0000 112.583i 0.670103 1.16065i −0.307771 0.951460i \(-0.599583\pi\)
0.977875 0.209192i \(-0.0670835\pi\)
\(98\) −98.0000 −1.00000
\(99\) 0 0
\(100\) 156.000 1.56000
\(101\) 20.0000 34.6410i 0.198020 0.342980i −0.749866 0.661589i \(-0.769882\pi\)
0.947886 + 0.318609i \(0.103216\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) −40.0000 69.2820i −0.384615 0.666173i
\(105\) 0 0
\(106\) 56.0000 96.9948i 0.528302 0.915046i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 182.000 1.66972 0.834862 0.550459i \(-0.185547\pi\)
0.834862 + 0.550459i \(0.185547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −112.000 193.990i −0.991150 1.71672i −0.610534 0.791990i \(-0.709045\pi\)
−0.380616 0.924733i \(-0.624288\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −160.000 −1.37931
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −60.5000 104.789i −0.500000 0.866025i
\(122\) −22.0000 38.1051i −0.180328 0.312337i
\(123\) 0 0
\(124\) 0 0
\(125\) 112.000 0.896000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 64.0000 110.851i 0.500000 0.866025i
\(129\) 0 0
\(130\) −80.0000 138.564i −0.615385 1.06588i
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 128.000 0.941176
\(137\) −88.0000 + 152.420i −0.642336 + 1.11256i 0.342574 + 0.939491i \(0.388701\pi\)
−0.984910 + 0.173067i \(0.944632\pi\)
\(138\) 0 0
\(139\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −320.000 −2.20690
\(146\) 110.000 190.526i 0.753425 1.30497i
\(147\) 0 0
\(148\) 140.000 + 242.487i 0.945946 + 1.63843i
\(149\) 140.000 + 242.487i 0.939597 + 1.62743i 0.766223 + 0.642574i \(0.222134\pi\)
0.173374 + 0.984856i \(0.444533\pi\)
\(150\) 0 0
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −85.0000 147.224i −0.541401 0.937735i −0.998824 0.0484851i \(-0.984561\pi\)
0.457423 0.889249i \(-0.348773\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 128.000 221.703i 0.800000 1.38564i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −160.000 + 277.128i −0.975610 + 1.68981i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0 0
\(169\) 34.5000 59.7558i 0.204142 0.353584i
\(170\) 256.000 1.50588
\(171\) 0 0
\(172\) 0 0
\(173\) −52.0000 + 90.0666i −0.300578 + 0.520616i −0.976267 0.216570i \(-0.930513\pi\)
0.675689 + 0.737187i \(0.263846\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −160.000 + 277.128i −0.898876 + 1.55690i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 38.0000 0.209945 0.104972 0.994475i \(-0.466525\pi\)
0.104972 + 0.994475i \(0.466525\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 280.000 + 484.974i 1.51351 + 2.62148i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 95.0000 + 164.545i 0.492228 + 0.852564i 0.999960 0.00895123i \(-0.00284930\pi\)
−0.507732 + 0.861515i \(0.669516\pi\)
\(194\) −130.000 225.167i −0.670103 1.16065i
\(195\) 0 0
\(196\) −98.0000 + 169.741i −0.500000 + 0.866025i
\(197\) 56.0000 0.284264 0.142132 0.989848i \(-0.454604\pi\)
0.142132 + 0.989848i \(0.454604\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 156.000 270.200i 0.780000 1.35100i
\(201\) 0 0
\(202\) −40.0000 69.2820i −0.198020 0.342980i
\(203\) 0 0
\(204\) 0 0
\(205\) −320.000 + 554.256i −1.56098 + 2.70369i
\(206\) 0 0
\(207\) 0 0
\(208\) −160.000 −0.769231
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) −112.000 193.990i −0.528302 0.915046i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 182.000 315.233i 0.834862 1.44602i
\(219\) 0 0
\(220\) 0 0
\(221\) −80.0000 138.564i −0.361991 0.626987i
\(222\) 0 0
\(223\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −448.000 −1.98230
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) 221.000 + 382.783i 0.965066 + 1.67154i 0.709439 + 0.704767i \(0.248948\pi\)
0.255627 + 0.966776i \(0.417718\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −160.000 + 277.128i −0.689655 + 1.19452i
\(233\) 416.000 1.78541 0.892704 0.450644i \(-0.148806\pi\)
0.892704 + 0.450644i \(0.148806\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 209.000 361.999i 0.867220 1.50207i 0.00239399 0.999997i \(-0.499238\pi\)
0.864826 0.502072i \(-0.167429\pi\)
\(242\) −242.000 −1.00000
\(243\) 0 0
\(244\) −88.0000 −0.360656
\(245\) −196.000 + 339.482i −0.800000 + 1.38564i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 112.000 193.990i 0.448000 0.775959i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −128.000 221.703i −0.500000 0.866025i
\(257\) 32.0000 + 55.4256i 0.124514 + 0.215664i 0.921543 0.388277i \(-0.126930\pi\)
−0.797029 + 0.603941i \(0.793596\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −320.000 −1.23077
\(261\) 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
\(265\) −224.000 387.979i −0.845283 1.46407i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −520.000 −1.93309 −0.966543 0.256506i \(-0.917429\pi\)
−0.966543 + 0.256506i \(0.917429\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 128.000 221.703i 0.470588 0.815083i
\(273\) 0 0
\(274\) 176.000 + 304.841i 0.642336 + 1.11256i
\(275\) 0 0
\(276\) 0 0
\(277\) −115.000 + 199.186i −0.415162 + 0.719082i −0.995445 0.0953324i \(-0.969609\pi\)
0.580283 + 0.814415i \(0.302942\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −160.000 + 277.128i −0.569395 + 0.986221i 0.427231 + 0.904143i \(0.359489\pi\)
−0.996626 + 0.0820785i \(0.973844\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −33.0000 −0.114187
\(290\) −320.000 + 554.256i −1.10345 + 1.91123i
\(291\) 0 0
\(292\) −220.000 381.051i −0.753425 1.30497i
\(293\) 68.0000 + 117.779i 0.232082 + 0.401978i 0.958421 0.285359i \(-0.0921129\pi\)
−0.726339 + 0.687337i \(0.758780\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 560.000 1.89189
\(297\) 0 0
\(298\) 560.000 1.87919
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −176.000 −0.577049
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) −25.0000 + 43.3013i −0.0798722 + 0.138343i −0.903195 0.429231i \(-0.858785\pi\)
0.823322 + 0.567574i \(0.192118\pi\)
\(314\) −340.000 −1.08280
\(315\) 0 0
\(316\) 0 0
\(317\) 308.000 533.472i 0.971609 1.68288i 0.280909 0.959734i \(-0.409364\pi\)
0.690700 0.723141i \(-0.257303\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −256.000 443.405i −0.800000 1.38564i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −390.000 −1.20000
\(326\) 0 0
\(327\) 0 0
\(328\) 320.000 + 554.256i 0.975610 + 1.68981i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −175.000 303.109i −0.519288 0.899433i −0.999749 0.0224168i \(-0.992864\pi\)
0.480461 0.877016i \(-0.340469\pi\)
\(338\) −69.0000 119.512i −0.204142 0.353584i
\(339\) 0 0
\(340\) 256.000 443.405i 0.752941 1.30413i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 104.000 + 180.133i 0.300578 + 0.520616i
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 299.000 517.883i 0.856734 1.48391i −0.0182939 0.999833i \(-0.505823\pi\)
0.875027 0.484073i \(-0.160843\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 272.000 471.118i 0.770538 1.33461i −0.166730 0.986003i \(-0.553321\pi\)
0.937268 0.348609i \(-0.113346\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 320.000 + 554.256i 0.898876 + 1.55690i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 38.0000 65.8179i 0.104972 0.181817i
\(363\) 0 0
\(364\) 0 0
\(365\) −440.000 762.102i −1.20548 2.08795i
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 1120.00 3.02703
\(371\) 0 0
\(372\) 0 0
\(373\) 275.000 + 476.314i 0.737265 + 1.27698i 0.953722 + 0.300689i \(0.0972166\pi\)
−0.216457 + 0.976292i \(0.569450\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 400.000 1.06101
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 380.000 0.984456
\(387\) 0 0
\(388\) −520.000 −1.34021
\(389\) −340.000 + 588.897i −0.874036 + 1.51387i −0.0162499 + 0.999868i \(0.505173\pi\)
−0.857786 + 0.514007i \(0.828161\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 196.000 + 339.482i 0.500000 + 0.866025i
\(393\) 0 0
\(394\) 56.0000 96.9948i 0.142132 0.246180i
\(395\) 0 0
\(396\) 0 0
\(397\) 650.000 1.63728 0.818640 0.574307i \(-0.194729\pi\)
0.818640 + 0.574307i \(0.194729\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −312.000 540.400i −0.780000 1.35100i
\(401\) −40.0000 69.2820i −0.0997506 0.172773i 0.811831 0.583893i \(-0.198471\pi\)
−0.911581 + 0.411120i \(0.865138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −160.000 −0.396040
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −391.000 677.232i −0.955990 1.65582i −0.732086 0.681213i \(-0.761453\pi\)
−0.223905 0.974611i \(-0.571880\pi\)
\(410\) 640.000 + 1108.51i 1.56098 + 2.70369i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −160.000 + 277.128i −0.384615 + 0.666173i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 29.0000 50.2295i 0.0688836 0.119310i −0.829527 0.558467i \(-0.811390\pi\)
0.898410 + 0.439157i \(0.144723\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −448.000 −1.05660
\(425\) 312.000 540.400i 0.734118 1.27153i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 290.000 0.669746 0.334873 0.942263i \(-0.391307\pi\)
0.334873 + 0.942263i \(0.391307\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −364.000 630.466i −0.834862 1.44602i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −320.000 −0.723982
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 640.000 + 1108.51i 1.43820 + 2.49104i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 560.000 1.24722 0.623608 0.781737i \(-0.285666\pi\)
0.623608 + 0.781737i \(0.285666\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −448.000 + 775.959i −0.991150 + 1.71672i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 425.000 736.122i 0.929978 1.61077i 0.146625 0.989192i \(-0.453159\pi\)
0.783353 0.621577i \(-0.213508\pi\)
\(458\) 884.000 1.93013
\(459\) 0 0
\(460\) 0 0
\(461\) 380.000 658.179i 0.824295 1.42772i −0.0781619 0.996941i \(-0.524905\pi\)
0.902457 0.430780i \(-0.141762\pi\)
\(462\) 0 0
\(463\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 320.000 + 554.256i 0.689655 + 1.19452i
\(465\) 0 0
\(466\) 416.000 720.533i 0.892704 1.54621i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) −350.000 606.218i −0.727651 1.26033i
\(482\) −418.000 723.997i −0.867220 1.50207i
\(483\) 0 0
\(484\) −242.000 + 419.156i −0.500000 + 0.866025i
\(485\) −1040.00 −2.14433
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −88.0000 + 152.420i −0.180328 + 0.312337i
\(489\) 0 0
\(490\) 392.000 + 678.964i 0.800000 + 1.38564i
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) −320.000 + 554.256i −0.649087 + 1.12425i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) −224.000 387.979i −0.448000 0.775959i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −320.000 −0.633663
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −220.000 381.051i −0.432220 0.748627i 0.564844 0.825198i \(-0.308936\pi\)
−0.997064 + 0.0765706i \(0.975603\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −1.00000
\(513\) 0 0
\(514\) 128.000 0.249027
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −320.000 + 554.256i −0.615385 + 1.06588i
\(521\) −880.000 −1.68906 −0.844530 0.535509i \(-0.820120\pi\)
−0.844530 + 0.535509i \(0.820120\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −264.500 + 458.127i −0.500000 + 0.866025i
\(530\) −896.000 −1.69057
\(531\) 0 0
\(532\) 0 0
\(533\) 400.000 692.820i 0.750469 1.29985i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −520.000 + 900.666i −0.966543 + 1.67410i
\(539\) 0 0
\(540\) 0 0
\(541\) −682.000 −1.26063 −0.630314 0.776340i \(-0.717074\pi\)
−0.630314 + 0.776340i \(0.717074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −256.000 443.405i −0.470588 0.815083i
\(545\) −728.000 1260.93i −1.33578 2.31364i
\(546\) 0 0
\(547\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 704.000 1.28467
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 230.000 + 398.372i 0.415162 + 0.719082i
\(555\) 0 0
\(556\) 0 0
\(557\) 1064.00 1.91023 0.955117 0.296230i \(-0.0957294\pi\)
0.955117 + 0.296230i \(0.0957294\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 320.000 + 554.256i 0.569395 + 0.986221i
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) −896.000 + 1551.92i −1.58584 + 2.74676i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −520.000 + 900.666i −0.913884 + 1.58289i −0.105357 + 0.994434i \(0.533599\pi\)
−0.808527 + 0.588459i \(0.799735\pi\)
\(570\) 0 0
\(571\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1150.00 −1.99307 −0.996534 0.0831889i \(-0.973490\pi\)
−0.996534 + 0.0831889i \(0.973490\pi\)
\(578\) −33.0000 + 57.1577i −0.0570934 + 0.0988887i
\(579\) 0 0
\(580\) 640.000 + 1108.51i 1.10345 + 1.91123i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −880.000 −1.50685
\(585\) 0 0
\(586\) 272.000 0.464164
\(587\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 560.000 969.948i 0.945946 1.63843i
\(593\) −736.000 −1.24115 −0.620573 0.784148i \(-0.713100\pi\)
−0.620573 + 0.784148i \(0.713100\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 560.000 969.948i 0.939597 1.62743i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 551.000 954.360i 0.916805 1.58795i 0.112569 0.993644i \(-0.464092\pi\)
0.804236 0.594309i \(-0.202575\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −484.000 + 838.313i −0.800000 + 1.38564i
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −176.000 + 304.841i −0.288525 + 0.499739i
\(611\) 0 0
\(612\) 0 0
\(613\) −70.0000 −0.114192 −0.0570962 0.998369i \(-0.518184\pi\)
−0.0570962 + 0.998369i \(0.518184\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 608.000 + 1053.09i 0.985413 + 1.70679i 0.640085 + 0.768304i \(0.278899\pi\)
0.345328 + 0.938482i \(0.387768\pi\)
\(618\) 0 0
\(619\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 39.5000 + 68.4160i 0.0632000 + 0.109466i
\(626\) 50.0000 + 86.6025i 0.0798722 + 0.138343i
\(627\) 0 0
\(628\) −340.000 + 588.897i −0.541401 + 0.937735i
\(629\) 1120.00 1.78060
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −616.000 1066.94i −0.971609 1.68288i
\(635\) 0 0
\(636\) 0 0
\(637\) 245.000 424.352i 0.384615 0.666173i
\(638\) 0 0
\(639\) 0 0
\(640\) −1024.00 −1.60000
\(641\) 200.000 346.410i 0.312012 0.540421i −0.666785 0.745250i \(-0.732330\pi\)
0.978798 + 0.204828i \(0.0656636\pi\)
\(642\) 0 0
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −390.000 + 675.500i −0.600000 + 1.03923i
\(651\) 0 0
\(652\) 0 0
\(653\) 572.000 + 990.733i 0.875957 + 1.51720i 0.855740 + 0.517407i \(0.173103\pi\)
0.0202175 + 0.999796i \(0.493564\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1280.00 1.95122
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) −589.000 1020.18i −0.891074 1.54339i −0.838589 0.544764i \(-0.816619\pi\)
−0.0524847 0.998622i \(-0.516714\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −385.000 + 666.840i −0.572065 + 0.990846i 0.424288 + 0.905527i \(0.360524\pi\)
−0.996354 + 0.0853191i \(0.972809\pi\)
\(674\) −700.000 −1.03858
\(675\) 0 0
\(676\) −276.000 −0.408284
\(677\) −52.0000 + 90.0666i −0.0768095 + 0.133038i −0.901872 0.432004i \(-0.857807\pi\)
0.825062 + 0.565042i \(0.191140\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −512.000 886.810i −0.752941 1.30413i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 1408.00 2.05547
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 280.000 + 484.974i 0.406386 + 0.703881i
\(690\) 0 0
\(691\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 416.000 0.601156
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 640.000 + 1108.51i 0.918221 + 1.59041i
\(698\) −598.000 1035.77i −0.856734 1.48391i
\(699\) 0 0
\(700\) 0 0
\(701\) −520.000 −0.741797 −0.370899 0.928673i \(-0.620950\pi\)
−0.370899 + 0.928673i \(0.620950\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −544.000 942.236i −0.770538 1.33461i
\(707\) 0 0
\(708\) 0 0
\(709\) −259.000 + 448.601i −0.365303 + 0.632724i −0.988825 0.149082i \(-0.952368\pi\)
0.623522 + 0.781806i \(0.285701\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1280.00 1.79775
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 361.000 625.270i 0.500000 0.866025i
\(723\) 0 0
\(724\) −76.0000 131.636i −0.104972 0.181817i
\(725\) 780.000 + 1351.00i 1.07586 + 1.86345i
\(726\) 0 0
\(727\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1760.00 −2.41096
\(731\) 0 0
\(732\) 0 0
\(733\) 725.000 + 1255.74i 0.989086 + 1.71315i 0.622143 + 0.782904i \(0.286262\pi\)
0.366943 + 0.930243i \(0.380404\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1120.00 1939.90i 1.51351 2.62148i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 1120.00 1939.90i 1.50336 2.60389i
\(746\) 1100.00 1.47453
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 400.000 692.820i 0.530504 0.918860i
\(755\) 0 0
\(756\) 0 0
\(757\) 1190.00 1.57199 0.785997 0.618230i \(-0.212150\pi\)
0.785997 + 0.618230i \(0.212150\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −760.000 1316.36i −0.998686 1.72977i −0.543725 0.839263i \(-0.682987\pi\)
−0.454961 0.890512i \(-0.650347\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −481.000 833.116i −0.625488 1.08338i −0.988446 0.151571i \(-0.951567\pi\)
0.362959 0.931805i \(-0.381767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 380.000 658.179i 0.492228 0.852564i
\(773\) 1496.00 1.93532 0.967658 0.252264i \(-0.0811751\pi\)
0.967658 + 0.252264i \(0.0811751\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −520.000 + 900.666i −0.670103 + 1.16065i
\(777\) 0 0
\(778\) 680.000 + 1177.79i 0.874036 + 1.51387i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) −680.000 + 1177.79i −0.866242 + 1.50038i
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) −112.000 193.990i −0.142132 0.246180i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 220.000 0.277427
\(794\) 650.000 1125.83i 0.818640 1.41793i
\(795\) 0 0
\(796\) 0 0
\(797\) 572.000 + 990.733i 0.717691 + 1.24308i 0.961912 + 0.273358i \(0.0881344\pi\)
−0.244221 + 0.969720i \(0.578532\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1248.00 −1.56000
\(801\) 0 0
\(802\) −160.000 −0.199501