Properties

Label 648.2.i.a.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.a.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 \(-0.814250\pi\)
−0.0599153 0.998203i \(-0.519083\pi\)
\(6\) 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 \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 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 \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 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 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 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 \(-0.494070\pi\)
0.856560 0.516047i \(-0.172597\pi\)
\(48\) 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 \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 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 0.705965 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 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 \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 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 −0.997617 0.0689950i \(-0.978021\pi\)
0.558560 + 0.829464i \(0.311354\pi\)
\(84\) 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 \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 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 \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 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 \(-0.975762\pi\)
0.432670 0.901553i \(-0.357572\pi\)
\(114\) 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 \(0.0796882\pi\)
−0.269863 + 0.962899i \(0.586978\pi\)
\(132\) 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 \(-0.662010\pi\)
0.999893 0.0146279i \(-0.00465636\pi\)
\(138\) 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 0.654361 0.756182i \(-0.272938\pi\)
−0.982054 + 0.188602i \(0.939604\pi\)
\(150\) 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 0.999169 0.0407502i \(-0.0129748\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(168\) 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 \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 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 −0.997225 0.0744412i \(-0.976283\pi\)
0.563081 + 0.826402i \(0.309616\pi\)
\(192\) 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 \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 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 \(-0.540033\pi\)
0.921903 0.387421i \(-0.126634\pi\)
\(228\) 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 \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 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 0.707166 0.707048i \(-0.249973\pi\)
−0.965904 + 0.258900i \(0.916640\pi\)
\(240\) 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 \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 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 0.713878 0.700270i \(-0.246937\pi\)
−0.963391 + 0.268101i \(0.913604\pi\)
\(258\) 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 −0.715944 0.698158i \(-0.754003\pi\)
0.962594 + 0.270947i \(0.0873367\pi\)
\(264\) 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 \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 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 \(0.570255\pi\)
−0.735554 + 0.677466i \(0.763078\pi\)
\(282\) 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 0.635818 0.771839i \(-0.280663\pi\)
−0.986341 + 0.164714i \(0.947330\pi\)
\(294\) 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 0.644246 0.764818i \(-0.277171\pi\)
−0.984475 + 0.175525i \(0.943838\pi\)
\(312\) 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 \(-0.597914\pi\)
0.976764 0.214318i \(-0.0687530\pi\)
\(318\) 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 \(0.0561403\pi\)
−0.340293 + 0.940319i \(0.610526\pi\)
\(348\) 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 −0.952620 0.304162i \(-0.901624\pi\)
0.739722 + 0.672913i \(0.234957\pi\)
\(354\) 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 \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 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 −0.912263 0.409604i \(-0.865667\pi\)
0.810859 + 0.585241i \(0.199000\pi\)
\(390\) 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 \(-0.962132\pi\)
0.393680 0.919247i \(-0.371202\pi\)
\(402\) 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 0.974546 0.224189i \(-0.0719734\pi\)
−0.681426 + 0.731887i \(0.738640\pi\)
\(420\) 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 \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 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 −0.945265 0.326303i \(-0.894197\pi\)
0.755219 + 0.655472i \(0.227530\pi\)
\(444\) 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 \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 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 \(0.392755\pi\)
−0.982626 + 0.185597i \(0.940578\pi\)
\(462\) 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 \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 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 \(-0.466449\pi\)
0.808615 0.588338i \(-0.200218\pi\)
\(480\) 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 0.698285 0.715820i \(-0.253947\pi\)
−0.969061 + 0.246822i \(0.920614\pi\)
\(492\) 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 \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 + 10.3923i 0.265945 + 0.460631i 0.967811 0.251679i \(-0.0809826\pi\)
−0.701866 + 0.712309i \(0.747649\pi\)
\(510\) 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 \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 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 \(-0.297865\pi\)
0.593199 + 0.805056i \(0.297865\pi\)
\(558\) 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 \(0.0688963\pi\)
−0.302348 + 0.953198i \(0.597770\pi\)
\(564\) 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 \(0.366323\pi\)
−0.994634 + 0.103454i \(0.967011\pi\)
\(570\) 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 \(0.362772\pi\)
−0.995726 + 0.0923513i \(0.970562\pi\)
\(588\) 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 \(-0.193248\pi\)
0.821302 + 0.570494i \(0.193248\pi\)
\(594\) 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 \(-0.908667\pi\)
0.234464 0.972125i \(-0.424666\pi\)
\(618\) 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 \(0.456562\pi\)
−0.925995 + 0.377535i \(0.876772\pi\)
\(642\) 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 \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 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 0.529799 0.848123i \(-0.322267\pi\)
−0.999396 + 0.0347583i \(0.988934\pi\)
\(654\) 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 −0.531855 0.846836i \(-0.678505\pi\)
0.999309 + 0.0371821i \(0.0118382\pi\)
\(660\) 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 \(-0.512619\pi\)
0.885161 0.465284i \(-0.154048\pi\)
\(678\) 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 \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 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 \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 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 \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 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 0.827005 0.562195i \(-0.190043\pi\)
−0.900378 + 0.435110i \(0.856710\pi\)
\(744\) 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 0.899996 0.435897i \(-0.143569\pi\)
−0.827496 + 0.561471i \(0.810236\pi\)
\(762\) 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 \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 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 0.688839 0.724914i \(-0.258121\pi\)
−0.972214 + 0.234095i \(0.924787\pi\)
\(798\) 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