Properties

Label 648.2.i.h.217.1
Level $648$
Weight $2$
Character 648.217
Analytic conductor $5.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 217.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 648.217
Dual form 648.2.i.h.433.1

$q$-expansion

\(f(q)\) \(=\) \(q+(2.00000 + 3.46410i) q^{5} +(1.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(2.00000 + 3.46410i) q^{5} +(1.50000 - 2.59808i) q^{7} +(2.00000 - 3.46410i) q^{11} +(-0.500000 - 0.866025i) q^{13} +4.00000 q^{17} -1.00000 q^{19} +(2.00000 + 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(2.00000 + 3.46410i) q^{31} +12.0000 q^{35} -9.00000 q^{37} +(4.00000 - 6.92820i) q^{43} +(-6.00000 + 10.3923i) q^{47} +(-1.00000 - 1.73205i) q^{49} +8.00000 q^{53} +16.0000 q^{55} +(2.00000 + 3.46410i) q^{59} +(2.50000 - 4.33013i) q^{61} +(2.00000 - 3.46410i) q^{65} +(-5.50000 - 9.52628i) q^{67} -8.00000 q^{71} +1.00000 q^{73} +(-6.00000 - 10.3923i) q^{77} +(2.50000 - 4.33013i) q^{79} +(4.00000 - 6.92820i) q^{83} +(8.00000 + 13.8564i) q^{85} -12.0000 q^{89} -3.00000 q^{91} +(-2.00000 - 3.46410i) q^{95} +(-2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 3 q^{7} + O(q^{10}) \) \( 2 q + 4 q^{5} + 3 q^{7} + 4 q^{11} - q^{13} + 8 q^{17} - 2 q^{19} + 4 q^{23} - 11 q^{25} + 4 q^{31} + 24 q^{35} - 18 q^{37} + 8 q^{43} - 12 q^{47} - 2 q^{49} + 16 q^{53} + 32 q^{55} + 4 q^{59} + 5 q^{61} + 4 q^{65} - 11 q^{67} - 16 q^{71} + 2 q^{73} - 12 q^{77} + 5 q^{79} + 8 q^{83} + 16 q^{85} - 24 q^{89} - 6 q^{91} - 4 q^{95} - 5 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 + 3.46410i 0.894427 + 1.54919i 0.834512 + 0.550990i \(0.185750\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) 1.50000 2.59808i 0.566947 0.981981i −0.429919 0.902867i \(-0.641458\pi\)
0.996866 0.0791130i \(-0.0252088\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 3.46410i 0.603023 1.04447i −0.389338 0.921095i \(-0.627296\pi\)
0.992361 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) −0.500000 0.866025i −0.138675 0.240192i 0.788320 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 4.00000 6.92820i 0.609994 1.05654i −0.381246 0.924473i \(-0.624505\pi\)
0.991241 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 + 10.3923i −0.875190 + 1.51587i −0.0186297 + 0.999826i \(0.505930\pi\)
−0.856560 + 0.516047i \(0.827403\pi\)
\(48\) 0 0
\(49\) −1.00000 1.73205i −0.142857 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 16.0000 2.15744
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) 2.50000 4.33013i 0.320092 0.554416i −0.660415 0.750901i \(-0.729619\pi\)
0.980507 + 0.196485i \(0.0629528\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 3.46410i 0.248069 0.429669i
\(66\) 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
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 10.3923i −0.683763 1.18431i
\(78\) 0 0
\(79\) 2.50000 4.33013i 0.281272 0.487177i −0.690426 0.723403i \(-0.742577\pi\)
0.971698 + 0.236225i \(0.0759104\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 6.92820i 0.439057 0.760469i −0.558560 0.829464i \(-0.688646\pi\)
0.997617 + 0.0689950i \(0.0219793\pi\)
\(84\) 0 0
\(85\) 8.00000 + 13.8564i 0.867722 + 1.50294i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 3.46410i −0.205196 0.355409i
\(96\) 0 0
\(97\) −2.50000 + 4.33013i −0.253837 + 0.439658i −0.964579 0.263795i \(-0.915026\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) −0.500000 0.866025i −0.0492665 0.0853320i 0.840341 0.542059i \(-0.182355\pi\)
−0.889607 + 0.456727i \(0.849022\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 + 10.3923i 0.564433 + 0.977626i 0.997102 + 0.0760733i \(0.0242383\pi\)
−0.432670 + 0.901553i \(0.642428\pi\)
\(114\) 0 0
\(115\) −8.00000 + 13.8564i −0.746004 + 1.29212i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 10.3923i 0.550019 0.952661i
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 13.8564i −0.698963 1.21064i −0.968826 0.247741i \(-0.920312\pi\)
0.269863 0.962899i \(-0.413022\pi\)
\(132\) 0 0
\(133\) −1.50000 + 2.59808i −0.130066 + 0.225282i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) −4.50000 7.79423i −0.381685 0.661098i 0.609618 0.792695i \(-0.291323\pi\)
−0.991303 + 0.131597i \(0.957989\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.00000 + 6.92820i 0.327693 + 0.567581i 0.982054 0.188602i \(-0.0603956\pi\)
−0.654361 + 0.756182i \(0.727062\pi\)
\(150\) 0 0
\(151\) 0.500000 0.866025i 0.0406894 0.0704761i −0.844963 0.534824i \(-0.820378\pi\)
0.885653 + 0.464348i \(0.153711\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 + 13.8564i −0.642575 + 1.11297i
\(156\) 0 0
\(157\) 1.00000 + 1.73205i 0.0798087 + 0.138233i 0.903167 0.429289i \(-0.141236\pi\)
−0.823359 + 0.567521i \(0.807902\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −15.0000 −1.17489 −0.587445 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 10.3923i −0.464294 0.804181i 0.534875 0.844931i \(-0.320359\pi\)
−0.999169 + 0.0407502i \(0.987025\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 16.5000 + 28.5788i 1.24728 + 2.16036i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 21.0000 1.56092 0.780459 0.625207i \(-0.214986\pi\)
0.780459 + 0.625207i \(0.214986\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −18.0000 31.1769i −1.32339 2.29217i
\(186\) 0 0
\(187\) 8.00000 13.8564i 0.585018 1.01328i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 10.3923i 0.434145 0.751961i −0.563081 0.826402i \(-0.690384\pi\)
0.997225 + 0.0744412i \(0.0237173\pi\)
\(192\) 0 0
\(193\) −11.5000 19.9186i −0.827788 1.43377i −0.899770 0.436365i \(-0.856266\pi\)
0.0719816 0.997406i \(-0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 25.0000 1.77220 0.886102 0.463491i \(-0.153403\pi\)
0.886102 + 0.463491i \(0.153403\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.00000 + 3.46410i −0.138343 + 0.239617i
\(210\) 0 0
\(211\) −6.50000 11.2583i −0.447478 0.775055i 0.550743 0.834675i \(-0.314345\pi\)
−0.998221 + 0.0596196i \(0.981011\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 32.0000 2.18238
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 3.46410i −0.134535 0.233021i
\(222\) 0 0
\(223\) −2.00000 + 3.46410i −0.133930 + 0.231973i −0.925188 0.379509i \(-0.876093\pi\)
0.791258 + 0.611482i \(0.209426\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 + 20.7846i −0.796468 + 1.37952i 0.125435 + 0.992102i \(0.459967\pi\)
−0.921903 + 0.387421i \(0.873366\pi\)
\(228\) 0 0
\(229\) −5.00000 8.66025i −0.330409 0.572286i 0.652183 0.758062i \(-0.273853\pi\)
−0.982592 + 0.185776i \(0.940520\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) −48.0000 −3.13117
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000 + 6.92820i 0.258738 + 0.448148i 0.965904 0.258900i \(-0.0833599\pi\)
−0.707166 + 0.707048i \(0.750027\pi\)
\(240\) 0 0
\(241\) 7.50000 12.9904i 0.483117 0.836784i −0.516695 0.856170i \(-0.672838\pi\)
0.999812 + 0.0193858i \(0.00617107\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.00000 6.92820i 0.255551 0.442627i
\(246\) 0 0
\(247\) 0.500000 + 0.866025i 0.0318142 + 0.0551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.00000 + 6.92820i 0.249513 + 0.432169i 0.963391 0.268101i \(-0.0863961\pi\)
−0.713878 + 0.700270i \(0.753063\pi\)
\(258\) 0 0
\(259\) −13.5000 + 23.3827i −0.838849 + 1.45293i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.00000 + 6.92820i −0.246651 + 0.427211i −0.962594 0.270947i \(-0.912663\pi\)
0.715944 + 0.698158i \(0.245997\pi\)
\(264\) 0 0
\(265\) 16.0000 + 27.7128i 0.982872 + 1.70238i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.0000 + 38.1051i 1.32665 + 2.29783i
\(276\) 0 0
\(277\) −5.00000 + 8.66025i −0.300421 + 0.520344i −0.976231 0.216731i \(-0.930460\pi\)
0.675810 + 0.737075i \(0.263794\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 27.7128i 0.954480 1.65321i 0.218926 0.975741i \(-0.429745\pi\)
0.735554 0.677466i \(-0.236922\pi\)
\(282\) 0 0
\(283\) 8.00000 + 13.8564i 0.475551 + 0.823678i 0.999608 0.0280052i \(-0.00891551\pi\)
−0.524057 + 0.851683i \(0.675582\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000 + 10.3923i 0.350524 + 0.607125i 0.986341 0.164714i \(-0.0526703\pi\)
−0.635818 + 0.771839i \(0.719337\pi\)
\(294\) 0 0
\(295\) −8.00000 + 13.8564i −0.465778 + 0.806751i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000 3.46410i 0.115663 0.200334i
\(300\) 0 0
\(301\) −12.0000 20.7846i −0.691669 1.19800i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.00000 + 10.3923i 0.340229 + 0.589294i 0.984475 0.175525i \(-0.0561621\pi\)
−0.644246 + 0.764818i \(0.722829\pi\)
\(312\) 0 0
\(313\) −1.50000 + 2.59808i −0.0847850 + 0.146852i −0.905300 0.424774i \(-0.860354\pi\)
0.820515 + 0.571626i \(0.193687\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 + 20.7846i −0.673987 + 1.16738i 0.302777 + 0.953062i \(0.402086\pi\)
−0.976764 + 0.214318i \(0.931247\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 18.0000 + 31.1769i 0.992372 + 1.71884i
\(330\) 0 0
\(331\) −8.50000 + 14.7224i −0.467202 + 0.809218i −0.999298 0.0374662i \(-0.988071\pi\)
0.532096 + 0.846684i \(0.321405\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 22.0000 38.1051i 1.20199 2.08190i
\(336\) 0 0
\(337\) 1.50000 + 2.59808i 0.0817102 + 0.141526i 0.903985 0.427565i \(-0.140628\pi\)
−0.822274 + 0.569091i \(0.807295\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 20.7846i −0.644194 1.11578i −0.984487 0.175457i \(-0.943860\pi\)
0.340293 0.940319i \(-0.389474\pi\)
\(348\) 0 0
\(349\) 16.5000 28.5788i 0.883225 1.52979i 0.0354898 0.999370i \(-0.488701\pi\)
0.847735 0.530420i \(-0.177966\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.00000 6.92820i 0.212899 0.368751i −0.739722 0.672913i \(-0.765043\pi\)
0.952620 + 0.304162i \(0.0983763\pi\)
\(354\) 0 0
\(355\) −16.0000 27.7128i −0.849192 1.47084i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 + 3.46410i 0.104685 + 0.181319i
\(366\) 0 0
\(367\) 11.5000 19.9186i 0.600295 1.03974i −0.392481 0.919760i \(-0.628383\pi\)
0.992776 0.119982i \(-0.0382835\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 20.7846i 0.623009 1.07908i
\(372\) 0 0
\(373\) 0.500000 + 0.866025i 0.0258890 + 0.0448411i 0.878680 0.477412i \(-0.158425\pi\)
−0.852791 + 0.522253i \(0.825092\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 24.0000 41.5692i 1.22315 2.11856i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.00000 3.46410i 0.101404 0.175637i −0.810859 0.585241i \(-0.801000\pi\)
0.912263 + 0.409604i \(0.134333\pi\)
\(390\) 0 0
\(391\) 8.00000 + 13.8564i 0.404577 + 0.700749i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 20.0000 1.00631
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 + 20.7846i 0.599251 + 1.03793i 0.992932 + 0.118686i \(0.0378683\pi\)
−0.393680 + 0.919247i \(0.628798\pi\)
\(402\) 0 0
\(403\) 2.00000 3.46410i 0.0996271 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.0000 + 31.1769i −0.892227 + 1.54538i
\(408\) 0 0
\(409\) 19.5000 + 33.7750i 0.964213 + 1.67007i 0.711715 + 0.702468i \(0.247919\pi\)
0.252498 + 0.967597i \(0.418748\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 32.0000 1.57082
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 10.3923i −0.293119 0.507697i 0.681426 0.731887i \(-0.261360\pi\)
−0.974546 + 0.224189i \(0.928027\pi\)
\(420\) 0 0
\(421\) −8.50000 + 14.7224i −0.414265 + 0.717527i −0.995351 0.0963145i \(-0.969295\pi\)
0.581086 + 0.813842i \(0.302628\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −22.0000 + 38.1051i −1.06716 + 1.84837i
\(426\) 0 0
\(427\) −7.50000 12.9904i −0.362950 0.628649i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.00000 3.46410i −0.0956730 0.165710i
\(438\) 0 0
\(439\) −18.0000 + 31.1769i −0.859093 + 1.48799i 0.0137020 + 0.999906i \(0.495638\pi\)
−0.872795 + 0.488087i \(0.837695\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.00000 6.92820i 0.190046 0.329169i −0.755219 0.655472i \(-0.772470\pi\)
0.945265 + 0.326303i \(0.105803\pi\)
\(444\) 0 0
\(445\) −24.0000 41.5692i −1.13771 1.97057i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.00000 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.00000 10.3923i −0.281284 0.487199i
\(456\) 0 0
\(457\) −19.0000 + 32.9090i −0.888783 + 1.53942i −0.0474665 + 0.998873i \(0.515115\pi\)
−0.841316 + 0.540544i \(0.818219\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 24.2487i 0.652045 1.12938i −0.330581 0.943778i \(-0.607245\pi\)
0.982626 0.185597i \(-0.0594220\pi\)
\(462\) 0 0
\(463\) 9.50000 + 16.4545i 0.441502 + 0.764705i 0.997801 0.0662777i \(-0.0211123\pi\)
−0.556299 + 0.830982i \(0.687779\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −33.0000 −1.52380
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −16.0000 27.7128i −0.735681 1.27424i
\(474\) 0 0
\(475\) 5.50000 9.52628i 0.252357 0.437096i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.0000 + 34.6410i −0.913823 + 1.58279i −0.105208 + 0.994450i \(0.533551\pi\)
−0.808615 + 0.588338i \(0.799782\pi\)
\(480\) 0 0
\(481\) 4.50000 + 7.79423i 0.205182 + 0.355386i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) −11.0000 −0.498458 −0.249229 0.968445i \(-0.580177\pi\)
−0.249229 + 0.968445i \(0.580177\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.00000 + 10.3923i 0.270776 + 0.468998i 0.969061 0.246822i \(-0.0793863\pi\)
−0.698285 + 0.715820i \(0.746053\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 + 20.7846i −0.538274 + 0.932317i
\(498\) 0 0
\(499\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 10.3923i −0.265945 0.460631i 0.701866 0.712309i \(-0.252351\pi\)
−0.967811 + 0.251679i \(0.919017\pi\)
\(510\) 0 0
\(511\) 1.50000 2.59808i 0.0663561 0.114932i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.00000 3.46410i 0.0881305 0.152647i
\(516\) 0 0
\(517\) 24.0000 + 41.5692i 1.05552 + 1.82821i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 0 0
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 + 13.8564i 0.348485 + 0.603595i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −24.0000 41.5692i −1.03761 1.79719i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.00000 −0.344584
\(540\) 0 0
\(541\) 9.00000 0.386940 0.193470 0.981106i \(-0.438026\pi\)
0.193470 + 0.981106i \(0.438026\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −28.0000 48.4974i −1.19939 2.07740i
\(546\) 0 0
\(547\) 3.50000 6.06218i 0.149649 0.259200i −0.781449 0.623970i \(-0.785519\pi\)
0.931098 + 0.364770i \(0.118852\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −7.50000 12.9904i −0.318932 0.552407i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.0000 27.7128i −0.674320 1.16796i −0.976667 0.214758i \(-0.931104\pi\)
0.302348 0.953198i \(-0.402230\pi\)
\(564\) 0 0
\(565\) −24.0000 + 41.5692i −1.00969 + 1.74883i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.0000 24.2487i 0.586911 1.01656i −0.407724 0.913105i \(-0.633677\pi\)
0.994634 0.103454i \(-0.0329893\pi\)
\(570\) 0 0
\(571\) 16.5000 + 28.5788i 0.690504 + 1.19599i 0.971673 + 0.236329i \(0.0759443\pi\)
−0.281170 + 0.959658i \(0.590722\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −44.0000 −1.83493
\(576\) 0 0
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 20.7846i −0.497844 0.862291i
\(582\) 0 0
\(583\) 16.0000 27.7128i 0.662652 1.14775i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.0000 24.2487i 0.577842 1.00085i −0.417885 0.908500i \(-0.637228\pi\)
0.995726 0.0923513i \(-0.0294383\pi\)
\(588\) 0 0
\(589\) −2.00000 3.46410i −0.0824086 0.142736i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −40.0000 −1.64260 −0.821302 0.570494i \(-0.806752\pi\)
−0.821302 + 0.570494i \(0.806752\pi\)
\(594\) 0 0
\(595\) 48.0000 1.96781
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −5.00000 + 8.66025i −0.203954 + 0.353259i −0.949799 0.312861i \(-0.898713\pi\)
0.745845 + 0.666120i \(0.232046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0000 17.3205i 0.406558 0.704179i
\(606\) 0 0
\(607\) −2.50000 4.33013i −0.101472 0.175754i 0.810819 0.585296i \(-0.199022\pi\)
−0.912291 + 0.409542i \(0.865689\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 35.0000 1.41364 0.706818 0.707395i \(-0.250130\pi\)
0.706818 + 0.707395i \(0.250130\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 + 31.1769i 0.724653 + 1.25514i 0.959117 + 0.283011i \(0.0913331\pi\)
−0.234464 + 0.972125i \(0.575334\pi\)
\(618\) 0 0
\(619\) 8.50000 14.7224i 0.341644 0.591744i −0.643094 0.765787i \(-0.722350\pi\)
0.984738 + 0.174042i \(0.0556830\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −18.0000 + 31.1769i −0.721155 + 1.24908i
\(624\) 0 0
\(625\) −20.5000 35.5070i −0.820000 1.42028i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 + 13.8564i 0.317470 + 0.549875i
\(636\) 0 0
\(637\) −1.00000 + 1.73205i −0.0396214 + 0.0686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.0000 34.6410i 0.789953 1.36824i −0.136043 0.990703i \(-0.543438\pi\)
0.925995 0.377535i \(-0.123228\pi\)
\(642\) 0 0
\(643\) 12.0000 + 20.7846i 0.473234 + 0.819665i 0.999531 0.0306359i \(-0.00975325\pi\)
−0.526297 + 0.850301i \(0.676420\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.0000 + 20.7846i 0.469596 + 0.813365i 0.999396 0.0347583i \(-0.0110661\pi\)
−0.529799 + 0.848123i \(0.677733\pi\)
\(654\) 0 0
\(655\) 32.0000 55.4256i 1.25034 2.16566i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 + 20.7846i −0.467454 + 0.809653i −0.999309 0.0371821i \(-0.988162\pi\)
0.531855 + 0.846836i \(0.321495\pi\)
\(660\) 0 0
\(661\) 6.50000 + 11.2583i 0.252821 + 0.437898i 0.964301 0.264807i \(-0.0853084\pi\)
−0.711481 + 0.702706i \(0.751975\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 17.3205i −0.386046 0.668651i
\(672\) 0 0
\(673\) −9.50000 + 16.4545i −0.366198 + 0.634274i −0.988968 0.148132i \(-0.952674\pi\)
0.622770 + 0.782405i \(0.286007\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.0000 + 38.1051i −0.845529 + 1.46450i 0.0396326 + 0.999214i \(0.487381\pi\)
−0.885161 + 0.465284i \(0.845952\pi\)
\(678\) 0 0
\(679\) 7.50000 + 12.9904i 0.287824 + 0.498525i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 0 0
\(685\) −48.0000 −1.83399
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.00000 6.92820i −0.152388 0.263944i
\(690\) 0 0
\(691\) 20.0000 34.6410i 0.760836 1.31781i −0.181584 0.983375i \(-0.558123\pi\)
0.942420 0.334431i \(-0.108544\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.0000 31.1769i 0.682779 1.18261i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 9.00000 0.339441
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.50000 6.06218i 0.131445 0.227670i −0.792789 0.609497i \(-0.791372\pi\)
0.924234 + 0.381827i \(0.124705\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 + 13.8564i −0.299602 + 0.518927i
\(714\) 0 0
\(715\) −8.00000 13.8564i −0.299183 0.518200i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.00000 + 10.3923i −0.222528 + 0.385429i −0.955575 0.294749i \(-0.904764\pi\)
0.733047 + 0.680178i \(0.238097\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.0000 27.7128i 0.591781 1.02500i
\(732\) 0 0
\(733\) −1.00000 1.73205i −0.0369358 0.0639748i 0.846967 0.531646i \(-0.178426\pi\)
−0.883902 + 0.467671i \(0.845093\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −44.0000 −1.62076
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.00000 + 3.46410i 0.0733729 + 0.127086i 0.900378 0.435110i \(-0.143290\pi\)
−0.827005 + 0.562195i \(0.809957\pi\)
\(744\) 0 0
\(745\) −16.0000 + 27.7128i −0.586195 + 1.01532i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −18.0000 + 31.1769i −0.657706 + 1.13918i
\(750\) 0 0
\(751\) −1.50000 2.59808i −0.0547358 0.0948051i 0.837359 0.546653i \(-0.184098\pi\)
−0.892095 + 0.451848i \(0.850765\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) −31.0000 −1.12671 −0.563357 0.826214i \(-0.690490\pi\)
−0.563357 + 0.826214i \(0.690490\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.00000 3.46410i −0.0724999 0.125574i 0.827496 0.561471i \(-0.189764\pi\)
−0.899996 + 0.435897i \(0.856431\pi\)
\(762\) 0 0
\(763\) −21.0000 + 36.3731i −0.760251 + 1.31679i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.00000 3.46410i 0.0722158 0.125081i
\(768\) 0 0
\(769\) −23.5000 40.7032i −0.847432 1.46779i −0.883493 0.468445i \(-0.844814\pi\)
0.0360609 0.999350i \(-0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) −44.0000 −1.58053
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −16.0000 + 27.7128i −0.572525 + 0.991642i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.00000 + 6.92820i −0.142766 + 0.247278i
\(786\) 0 0
\(787\) −3.50000 6.06218i −0.124762 0.216093i 0.796878 0.604140i \(-0.206483\pi\)
−0.921640 + 0.388047i \(0.873150\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000 + 13.8564i 0.283375 + 0.490819i 0.972214 0.234095i \(-0.0752127\pi\)
−0.688839 + 0.724914i \(0.741879\pi\)
\(798\) 0 0
\(799\) −24.0000 + 41.5692i −0.849059 + 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.00000 3.46410i 0.0705785 0.122245i
\(804\)