Properties

Label 1296.2.i.q.433.1
Level $1296$
Weight $2$
Character 1296.433
Analytic conductor $10.349$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3486121020\)
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 433.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1296.433
Dual form 1296.2.i.q.865.1

$q$-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 \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0.0599153 0.998203i \(-0.480917\pi\)
0.834512 0.550990i \(-0.185750\pi\)
\(6\) 0 0
\(7\) −1.50000 2.59808i −0.566947 0.981981i −0.996866 0.0791130i \(-0.974791\pi\)
0.429919 0.902867i \(-0.358542\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 3.46410i −0.603023 1.04447i −0.992361 0.123371i \(-0.960630\pi\)
0.389338 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 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 \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −4.00000 6.92820i −0.609994 1.05654i −0.991241 0.132068i \(-0.957838\pi\)
0.381246 0.924473i \(-0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 + 10.3923i 0.875190 + 1.51587i 0.856560 + 0.516047i \(0.172597\pi\)
0.0186297 + 0.999826i \(0.494070\pi\)
\(48\) 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 \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 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 −0.305424 0.952217i \(-0.598798\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 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 \(-0.257423\pi\)
−0.971698 + 0.236225i \(0.924090\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 6.92820i −0.439057 0.760469i 0.558560 0.829464i \(-0.311354\pi\)
−0.997617 + 0.0689950i \(0.978021\pi\)
\(84\) 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 0.710742 0.703452i \(-0.248359\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0.500000 0.866025i 0.0492665 0.0853320i −0.840341 0.542059i \(-0.817645\pi\)
0.889607 + 0.456727i \(0.150978\pi\)
\(104\) 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 −0.432670 0.901553i \(-0.642428\pi\)
0.997102 0.0760733i \(-0.0242383\pi\)
\(114\) 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 \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 13.8564i 0.698963 1.21064i −0.269863 0.962899i \(-0.586978\pi\)
0.968826 0.247741i \(-0.0796882\pi\)
\(132\) 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 −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) 4.50000 7.79423i 0.381685 0.661098i −0.609618 0.792695i \(-0.708677\pi\)
0.991303 + 0.131597i \(0.0420106\pi\)
\(140\) 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 \(-0.727062\pi\)
0.982054 + 0.188602i \(0.0603956\pi\)
\(150\) 0 0
\(151\) −0.500000 0.866025i −0.0406894 0.0704761i 0.844963 0.534824i \(-0.179622\pi\)
−0.885653 + 0.464348i \(0.846289\pi\)
\(152\) 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 −0.823359 0.567521i \(-0.807902\pi\)
0.903167 + 0.429289i \(0.141236\pi\)
\(158\) 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 \(-0.300134\pi\)
0.587445 + 0.809264i \(0.300134\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 10.3923i 0.464294 0.804181i −0.534875 0.844931i \(-0.679641\pi\)
0.999169 + 0.0407502i \(0.0129748\pi\)
\(168\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 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 0.563081 0.826402i \(-0.309616\pi\)
−0.997225 + 0.0744412i \(0.976283\pi\)
\(192\) 0 0
\(193\) −11.5000 + 19.9186i −0.827788 + 1.43377i 0.0719816 + 0.997406i \(0.477068\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 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 \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 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 \(-0.685655\pi\)
0.998221 + 0.0596196i \(0.0189888\pi\)
\(212\) 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 \(-0.123907\pi\)
−0.791258 + 0.611482i \(0.790574\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 + 20.7846i 0.796468 + 1.37952i 0.921903 + 0.387421i \(0.126634\pi\)
−0.125435 + 0.992102i \(0.540033\pi\)
\(228\) 0 0
\(229\) −5.00000 + 8.66025i −0.330409 + 0.572286i −0.982592 0.185776i \(-0.940520\pi\)
0.652183 + 0.758062i \(0.273853\pi\)
\(230\) 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 \(-0.916640\pi\)
0.707166 + 0.707048i \(0.249973\pi\)
\(240\) 0 0
\(241\) 7.50000 + 12.9904i 0.483117 + 0.836784i 0.999812 0.0193858i \(-0.00617107\pi\)
−0.516695 + 0.856170i \(0.672838\pi\)
\(242\) 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 \(-0.753063\pi\)
0.963391 + 0.268101i \(0.0863961\pi\)
\(258\) 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 \(-0.0873367\pi\)
−0.715944 + 0.698158i \(0.754003\pi\)
\(264\) 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 \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) 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 0.675810 0.737075i \(-0.263794\pi\)
−0.976231 + 0.216731i \(0.930460\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 + 27.7128i 0.954480 + 1.65321i 0.735554 + 0.677466i \(0.236922\pi\)
0.218926 + 0.975741i \(0.429745\pi\)
\(282\) 0 0
\(283\) −8.00000 + 13.8564i −0.475551 + 0.823678i −0.999608 0.0280052i \(-0.991084\pi\)
0.524057 + 0.851683i \(0.324418\pi\)
\(284\) 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 \(-0.719337\pi\)
0.986341 + 0.164714i \(0.0526703\pi\)
\(294\) 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 \(-0.943838\pi\)
0.644246 + 0.764818i \(0.277171\pi\)
\(312\) 0 0
\(313\) −1.50000 2.59808i −0.0847850 0.146852i 0.820515 0.571626i \(-0.193687\pi\)
−0.905300 + 0.424774i \(0.860354\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 20.7846i −0.673987 1.16738i −0.976764 0.214318i \(-0.931247\pi\)
0.302777 0.953062i \(-0.402086\pi\)
\(318\) 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 \(-0.0119287\pi\)
−0.532096 + 0.846684i \(0.678595\pi\)
\(332\) 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 −0.822274 0.569091i \(-0.807295\pi\)
0.903985 + 0.427565i \(0.140628\pi\)
\(338\) 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 −0.340293 0.940319i \(-0.610526\pi\)
0.984487 0.175457i \(-0.0561403\pi\)
\(348\) 0 0
\(349\) 16.5000 + 28.5788i 0.883225 + 1.52979i 0.847735 + 0.530420i \(0.177966\pi\)
0.0354898 + 0.999370i \(0.488701\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.00000 + 6.92820i 0.212899 + 0.368751i 0.952620 0.304162i \(-0.0983763\pi\)
−0.739722 + 0.672913i \(0.765043\pi\)
\(354\) 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 −0.992776 0.119982i \(-0.961716\pi\)
0.392481 0.919760i \(-0.371617\pi\)
\(368\) 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 −0.852791 0.522253i \(-0.825092\pi\)
0.878680 + 0.477412i \(0.158425\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 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 \(-0.134333\pi\)
−0.810859 + 0.585241i \(0.801000\pi\)
\(390\) 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 −0.393680 0.919247i \(-0.628798\pi\)
0.992932 0.118686i \(-0.0378683\pi\)
\(402\) 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 0.252498 0.967597i \(-0.418748\pi\)
0.711715 0.702468i \(-0.247919\pi\)
\(410\) 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 \(-0.738640\pi\)
0.974546 + 0.224189i \(0.0719734\pi\)
\(420\) 0 0
\(421\) −8.50000 14.7224i −0.414265 0.717527i 0.581086 0.813842i \(-0.302628\pi\)
−0.995351 + 0.0963145i \(0.969295\pi\)
\(422\) 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 0.872795 + 0.488087i \(0.162305\pi\)
−0.0137020 + 0.999906i \(0.504362\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.00000 6.92820i −0.190046 0.329169i 0.755219 0.655472i \(-0.227530\pi\)
−0.945265 + 0.326303i \(0.894197\pi\)
\(444\) 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 −0.841316 0.540544i \(-0.818219\pi\)
−0.0474665 0.998873i \(-0.515115\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 + 24.2487i 0.652045 + 1.12938i 0.982626 + 0.185597i \(0.0594220\pi\)
−0.330581 + 0.943778i \(0.607245\pi\)
\(462\) 0 0
\(463\) −9.50000 + 16.4545i −0.441502 + 0.764705i −0.997801 0.0662777i \(-0.978888\pi\)
0.556299 + 0.830982i \(0.312221\pi\)
\(464\) 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 0.808615 + 0.588338i \(0.200218\pi\)
0.105208 + 0.994450i \(0.466449\pi\)
\(480\) 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 \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 + 10.3923i −0.270776 + 0.468998i −0.969061 0.246822i \(-0.920614\pi\)
0.698285 + 0.715820i \(0.253947\pi\)
\(492\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(500\) 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 \(-0.919017\pi\)
0.701866 + 0.712309i \(0.252351\pi\)
\(510\) 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 \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 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 \(-0.214481\pi\)
−0.931098 + 0.364770i \(0.881148\pi\)
\(548\) 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 −0.302348 0.953198i \(-0.597770\pi\)
0.976667 0.214758i \(-0.0688963\pi\)
\(564\) 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 0.994634 + 0.103454i \(0.0329893\pi\)
−0.407724 + 0.913105i \(0.633677\pi\)
\(570\) 0 0
\(571\) −16.5000 + 28.5788i −0.690504 + 1.19599i 0.281170 + 0.959658i \(0.409278\pi\)
−0.971673 + 0.236329i \(0.924056\pi\)
\(572\) 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 −0.995726 0.0923513i \(-0.970562\pi\)
0.417885 0.908500i \(-0.362772\pi\)
\(588\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) −5.00000 8.66025i −0.203954 0.353259i 0.745845 0.666120i \(-0.232046\pi\)
−0.949799 + 0.312861i \(0.898713\pi\)
\(602\) 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 \(-0.800978\pi\)
0.912291 + 0.409542i \(0.134311\pi\)
\(608\) 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 −0.234464 0.972125i \(-0.575334\pi\)
0.959117 0.283011i \(-0.0913331\pi\)
\(618\) 0 0
\(619\) −8.50000 14.7224i −0.341644 0.591744i 0.643094 0.765787i \(-0.277650\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 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 \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 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 0.925995 + 0.377535i \(0.123228\pi\)
−0.136043 + 0.990703i \(0.543438\pi\)
\(642\) 0 0
\(643\) −12.0000 + 20.7846i −0.473234 + 0.819665i −0.999531 0.0306359i \(-0.990247\pi\)
0.526297 + 0.850301i \(0.323580\pi\)
\(644\) 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 \(-0.677733\pi\)
0.999396 + 0.0347583i \(0.0110661\pi\)
\(654\) 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 \(-0.0118382\pi\)
−0.531855 + 0.846836i \(0.678505\pi\)
\(660\) 0 0
\(661\) 6.50000 11.2583i 0.252821 0.437898i −0.711481 0.702706i \(-0.751975\pi\)
0.964301 + 0.264807i \(0.0853084\pi\)
\(662\) 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 0.622770 0.782405i \(-0.286007\pi\)
−0.988968 + 0.148132i \(0.952674\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.0000 38.1051i −0.845529 1.46450i −0.885161 0.465284i \(-0.845952\pi\)
0.0396326 0.999214i \(-0.487381\pi\)
\(678\) 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 −0.942420 0.334431i \(-0.891456\pi\)
0.181584 0.983375i \(-0.441877\pi\)
\(692\) 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 0.924234 0.381827i \(-0.124705\pi\)
−0.792789 + 0.609497i \(0.791372\pi\)
\(710\) 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 \(-0.0952359\pi\)
−0.733047 + 0.680178i \(0.761903\pi\)
\(728\) 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 −0.883902 0.467671i \(-0.845093\pi\)
0.846967 + 0.531646i \(0.178426\pi\)
\(734\) 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 \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.00000 + 3.46410i −0.0733729 + 0.127086i −0.900378 0.435110i \(-0.856710\pi\)
0.827005 + 0.562195i \(0.190043\pi\)
\(744\) 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 \(-0.815902\pi\)
0.892095 + 0.451848i \(0.149235\pi\)
\(752\) 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 \(-0.856431\pi\)
0.827496 + 0.561471i \(0.189764\pi\)
\(762\) 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 0.0360609 + 0.999350i \(0.488519\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 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 \(-0.793517\pi\)
0.921640 + 0.388047i \(0.126850\pi\)
\(788\) 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 \(-0.741879\pi\)
0.972214 + 0.234095i \(0.0752127\pi\)
\(798\) 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