Properties

Label 2592.2.i.v.865.1
Level $2592$
Weight $2$
Character 2592.865
Analytic conductor $20.697$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6972242039\)
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 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2592.865
Dual form 2592.2.i.v.1729.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(1.00000 + 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{11} +(1.50000 + 2.59808i) q^{13} +2.00000 q^{17} +3.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{29} +6.00000 q^{35} +7.00000 q^{37} +(4.00000 + 6.92820i) q^{41} +(-6.00000 + 10.3923i) q^{43} +(-3.00000 + 5.19615i) q^{47} +(-1.00000 - 1.73205i) q^{49} -4.00000 q^{53} -12.0000 q^{55} +(-3.00000 - 5.19615i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(-1.50000 - 2.59808i) q^{67} +12.0000 q^{71} -15.0000 q^{73} +(9.00000 + 15.5885i) q^{77} +(4.50000 - 7.79423i) q^{79} +(6.00000 - 10.3923i) q^{83} +(2.00000 + 3.46410i) q^{85} +10.0000 q^{89} +9.00000 q^{91} +(3.00000 + 5.19615i) q^{95} +(-4.50000 + 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 3 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} + 4 q^{17} + 6 q^{19} - 6 q^{23} + q^{25} - 8 q^{29} + 12 q^{35} + 14 q^{37} + 8 q^{41} - 12 q^{43} - 6 q^{47} - 2 q^{49} - 8 q^{53} - 24 q^{55} - 6 q^{59} + q^{61} - 6 q^{65} - 3 q^{67} + 24 q^{71} - 30 q^{73} + 18 q^{77} + 9 q^{79} + 12 q^{83} + 4 q^{85} + 20 q^{89} + 18 q^{91} + 6 q^{95} - 9 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 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
\(11\) −3.00000 + 5.19615i −0.904534 + 1.56670i −0.0829925 + 0.996550i \(0.526448\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) 0 0
\(13\) 1.50000 + 2.59808i 0.416025 + 0.720577i 0.995535 0.0943882i \(-0.0300895\pi\)
−0.579510 + 0.814965i \(0.696756\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 + 6.92820i −0.742781 + 1.28654i 0.208443 + 0.978035i \(0.433160\pi\)
−0.951224 + 0.308500i \(0.900173\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 + 6.92820i 0.624695 + 1.08200i 0.988600 + 0.150567i \(0.0481100\pi\)
−0.363905 + 0.931436i \(0.618557\pi\)
\(42\) 0 0
\(43\) −6.00000 + 10.3923i −0.914991 + 1.58481i −0.108078 + 0.994142i \(0.534469\pi\)
−0.806914 + 0.590669i \(0.798864\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) −1.00000 1.73205i −0.142857 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) 0.500000 0.866025i 0.0640184 0.110883i −0.832240 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) −1.50000 2.59808i −0.183254 0.317406i 0.759733 0.650236i \(-0.225330\pi\)
−0.942987 + 0.332830i \(0.891996\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −15.0000 −1.75562 −0.877809 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.00000 + 15.5885i 1.02565 + 1.77647i
\(78\) 0 0
\(79\) 4.50000 7.79423i 0.506290 0.876919i −0.493684 0.869641i \(-0.664350\pi\)
0.999974 0.00727784i \(-0.00231663\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 10.3923i 0.658586 1.14070i −0.322396 0.946605i \(-0.604488\pi\)
0.980982 0.194099i \(-0.0621783\pi\)
\(84\) 0 0
\(85\) 2.00000 + 3.46410i 0.216930 + 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 9.00000 0.943456
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) −4.50000 + 7.79423i −0.456906 + 0.791384i −0.998796 0.0490655i \(-0.984376\pi\)
0.541890 + 0.840450i \(0.317709\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.00000 + 13.8564i −0.796030 + 1.37876i 0.126153 + 0.992011i \(0.459737\pi\)
−0.922183 + 0.386753i \(0.873597\pi\)
\(102\) 0 0
\(103\) 7.50000 + 12.9904i 0.738997 + 1.27998i 0.952947 + 0.303136i \(0.0980336\pi\)
−0.213950 + 0.976845i \(0.568633\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.00000 + 12.1244i 0.658505 + 1.14056i 0.981003 + 0.193993i \(0.0621440\pi\)
−0.322498 + 0.946570i \(0.604523\pi\)
\(114\) 0 0
\(115\) 6.00000 10.3923i 0.559503 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 5.19615i 0.275010 0.476331i
\(120\) 0 0
\(121\) −12.5000 21.6506i −1.13636 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 4.50000 7.79423i 0.390199 0.675845i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.00000 + 12.1244i −0.598050 + 1.03585i 0.395058 + 0.918656i \(0.370724\pi\)
−0.993109 + 0.117198i \(0.962609\pi\)
\(138\) 0 0
\(139\) −10.5000 18.1865i −0.890598 1.54256i −0.839159 0.543885i \(-0.816953\pi\)
−0.0514389 0.998676i \(-0.516381\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −18.0000 −1.50524
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.00000 + 3.46410i 0.163846 + 0.283790i 0.936245 0.351348i \(-0.114277\pi\)
−0.772399 + 0.635138i \(0.780943\pi\)
\(150\) 0 0
\(151\) −1.50000 + 2.59808i −0.122068 + 0.211428i −0.920583 0.390547i \(-0.872286\pi\)
0.798515 + 0.601975i \(0.205619\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000 + 12.1244i 0.558661 + 0.967629i 0.997609 + 0.0691164i \(0.0220180\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −18.0000 −1.41860
\(162\) 0 0
\(163\) −3.00000 −0.234978 −0.117489 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 + 15.5885i 0.696441 + 1.20627i 0.969693 + 0.244328i \(0.0785675\pi\)
−0.273252 + 0.961943i \(0.588099\pi\)
\(168\) 0 0
\(169\) 2.00000 3.46410i 0.153846 0.266469i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.00000 6.92820i 0.304114 0.526742i −0.672949 0.739689i \(-0.734973\pi\)
0.977064 + 0.212947i \(0.0683062\pi\)
\(174\) 0 0
\(175\) −1.50000 2.59808i −0.113389 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 9.00000 0.668965 0.334482 0.942402i \(-0.391439\pi\)
0.334482 + 0.942402i \(0.391439\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.00000 + 12.1244i 0.514650 + 0.891400i
\(186\) 0 0
\(187\) −6.00000 + 10.3923i −0.438763 + 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 5.19615i 0.217072 0.375980i −0.736839 0.676068i \(-0.763683\pi\)
0.953912 + 0.300088i \(0.0970159\pi\)
\(192\) 0 0
\(193\) 2.50000 + 4.33013i 0.179954 + 0.311689i 0.941865 0.335993i \(-0.109072\pi\)
−0.761911 + 0.647682i \(0.775738\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −15.0000 −1.06332 −0.531661 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 + 20.7846i 0.842235 + 1.45879i
\(204\) 0 0
\(205\) −8.00000 + 13.8564i −0.558744 + 0.967773i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.00000 + 15.5885i −0.622543 + 1.07828i
\(210\) 0 0
\(211\) 7.50000 + 12.9904i 0.516321 + 0.894295i 0.999820 + 0.0189499i \(0.00603229\pi\)
−0.483499 + 0.875345i \(0.660634\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 + 5.19615i 0.201802 + 0.349531i
\(222\) 0 0
\(223\) −12.0000 + 20.7846i −0.803579 + 1.39184i 0.113666 + 0.993519i \(0.463740\pi\)
−0.917246 + 0.398321i \(0.869593\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.00000 + 10.3923i −0.398234 + 0.689761i −0.993508 0.113761i \(-0.963710\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(228\) 0 0
\(229\) 3.00000 + 5.19615i 0.198246 + 0.343371i 0.947960 0.318390i \(-0.103142\pi\)
−0.749714 + 0.661762i \(0.769809\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 10.3923i −0.388108 0.672222i 0.604087 0.796918i \(-0.293538\pi\)
−0.992195 + 0.124696i \(0.960204\pi\)
\(240\) 0 0
\(241\) 1.50000 2.59808i 0.0966235 0.167357i −0.813662 0.581339i \(-0.802529\pi\)
0.910285 + 0.413982i \(0.135862\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 3.46410i 0.127775 0.221313i
\(246\) 0 0
\(247\) 4.50000 + 7.79423i 0.286328 + 0.495935i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0000 17.3205i −0.623783 1.08042i −0.988775 0.149413i \(-0.952262\pi\)
0.364992 0.931011i \(-0.381072\pi\)
\(258\) 0 0
\(259\) 10.5000 18.1865i 0.652438 1.13006i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 10.3923i 0.369976 0.640817i −0.619586 0.784929i \(-0.712699\pi\)
0.989561 + 0.144112i \(0.0460326\pi\)
\(264\) 0 0
\(265\) −4.00000 6.92820i −0.245718 0.425596i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 15.0000 0.911185 0.455593 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 + 5.19615i 0.180907 + 0.313340i
\(276\) 0 0
\(277\) 9.00000 15.5885i 0.540758 0.936620i −0.458103 0.888899i \(-0.651471\pi\)
0.998861 0.0477206i \(-0.0151957\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.00000 6.92820i 0.238620 0.413302i −0.721699 0.692207i \(-0.756638\pi\)
0.960319 + 0.278906i \(0.0899716\pi\)
\(282\) 0 0
\(283\) 6.00000 + 10.3923i 0.356663 + 0.617758i 0.987401 0.158237i \(-0.0505811\pi\)
−0.630738 + 0.775996i \(0.717248\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00000 1.73205i −0.0584206 0.101187i 0.835336 0.549740i \(-0.185273\pi\)
−0.893757 + 0.448552i \(0.851940\pi\)
\(294\) 0 0
\(295\) 6.00000 10.3923i 0.349334 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.00000 15.5885i 0.520483 0.901504i
\(300\) 0 0
\(301\) 18.0000 + 31.1769i 1.03750 + 1.79701i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.00000 15.5885i −0.510343 0.883940i −0.999928 0.0119847i \(-0.996185\pi\)
0.489585 0.871956i \(-0.337148\pi\)
\(312\) 0 0
\(313\) 0.500000 0.866025i 0.0282617 0.0489506i −0.851549 0.524276i \(-0.824336\pi\)
0.879810 + 0.475325i \(0.157669\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 + 3.46410i −0.112331 + 0.194563i −0.916710 0.399554i \(-0.869165\pi\)
0.804379 + 0.594117i \(0.202498\pi\)
\(318\) 0 0
\(319\) −24.0000 41.5692i −1.34374 2.32743i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.00000 + 15.5885i 0.496186 + 0.859419i
\(330\) 0 0
\(331\) 1.50000 2.59808i 0.0824475 0.142803i −0.821853 0.569699i \(-0.807060\pi\)
0.904301 + 0.426896i \(0.140393\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 5.19615i 0.163908 0.283896i
\(336\) 0 0
\(337\) −4.50000 7.79423i −0.245131 0.424579i 0.717038 0.697034i \(-0.245498\pi\)
−0.962168 + 0.272456i \(0.912164\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 10.3923i −0.322097 0.557888i 0.658824 0.752297i \(-0.271054\pi\)
−0.980921 + 0.194409i \(0.937721\pi\)
\(348\) 0 0
\(349\) 0.500000 0.866025i 0.0267644 0.0463573i −0.852333 0.523000i \(-0.824813\pi\)
0.879097 + 0.476642i \(0.158146\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.0000 17.3205i 0.532246 0.921878i −0.467045 0.884234i \(-0.654681\pi\)
0.999291 0.0376440i \(-0.0119853\pi\)
\(354\) 0 0
\(355\) 12.0000 + 20.7846i 0.636894 + 1.10313i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.0000 25.9808i −0.785136 1.35990i
\(366\) 0 0
\(367\) 1.50000 2.59808i 0.0782994 0.135618i −0.824217 0.566274i \(-0.808384\pi\)
0.902516 + 0.430656i \(0.141718\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 + 10.3923i −0.311504 + 0.539542i
\(372\) 0 0
\(373\) −11.5000 19.9186i −0.595447 1.03135i −0.993484 0.113975i \(-0.963641\pi\)
0.398036 0.917370i \(-0.369692\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 27.0000 1.38690 0.693448 0.720506i \(-0.256091\pi\)
0.693448 + 0.720506i \(0.256091\pi\)
\(380\) 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
\(385\) −18.0000 + 31.1769i −0.917365 + 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.0000 29.4449i 0.861934 1.49291i −0.00812520 0.999967i \(-0.502586\pi\)
0.870059 0.492947i \(-0.164080\pi\)
\(390\) 0 0
\(391\) −6.00000 10.3923i −0.303433 0.525561i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 18.0000 0.905678
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 17.3205i −0.499376 0.864945i 0.500624 0.865665i \(-0.333104\pi\)
−1.00000 0.000720188i \(0.999771\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.0000 + 36.3731i −1.04093 + 1.80295i
\(408\) 0 0
\(409\) −10.5000 18.1865i −0.519192 0.899266i −0.999751 0.0223042i \(-0.992900\pi\)
0.480560 0.876962i \(-0.340434\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.0000 −0.885722
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.0000 25.9808i −0.732798 1.26924i −0.955683 0.294398i \(-0.904881\pi\)
0.222885 0.974845i \(-0.428453\pi\)
\(420\) 0 0
\(421\) 7.50000 12.9904i 0.365528 0.633112i −0.623333 0.781956i \(-0.714222\pi\)
0.988861 + 0.148844i \(0.0475552\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000 1.73205i 0.0485071 0.0840168i
\(426\) 0 0
\(427\) −1.50000 2.59808i −0.0725901 0.125730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.00000 15.5885i −0.430528 0.745697i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00000 10.3923i 0.285069 0.493753i −0.687557 0.726130i \(-0.741317\pi\)
0.972626 + 0.232377i \(0.0746503\pi\)
\(444\) 0 0
\(445\) 10.0000 + 17.3205i 0.474045 + 0.821071i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) −48.0000 −2.26023
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.00000 + 15.5885i 0.421927 + 0.730798i
\(456\) 0 0
\(457\) 15.0000 25.9808i 0.701670 1.21533i −0.266209 0.963915i \(-0.585771\pi\)
0.967880 0.251414i \(-0.0808954\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.00000 + 1.73205i −0.0465746 + 0.0806696i −0.888373 0.459123i \(-0.848164\pi\)
0.841798 + 0.539792i \(0.181497\pi\)
\(462\) 0 0
\(463\) 7.50000 + 12.9904i 0.348555 + 0.603714i 0.985993 0.166787i \(-0.0533393\pi\)
−0.637438 + 0.770501i \(0.720006\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −36.0000 62.3538i −1.65528 2.86703i
\(474\) 0 0
\(475\) 1.50000 2.59808i 0.0688247 0.119208i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.00000 + 10.3923i −0.274147 + 0.474837i −0.969920 0.243426i \(-0.921729\pi\)
0.695773 + 0.718262i \(0.255062\pi\)
\(480\) 0 0
\(481\) 10.5000 + 18.1865i 0.478759 + 0.829235i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) −15.0000 −0.679715 −0.339857 0.940477i \(-0.610379\pi\)
−0.339857 + 0.940477i \(0.610379\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −21.0000 36.3731i −0.947717 1.64149i −0.750218 0.661190i \(-0.770052\pi\)
−0.197499 0.980303i \(-0.563282\pi\)
\(492\) 0 0
\(493\) −8.00000 + 13.8564i −0.360302 + 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.0000 31.1769i 0.807410 1.39848i
\(498\) 0 0
\(499\) 12.0000 + 20.7846i 0.537194 + 0.930447i 0.999054 + 0.0434940i \(0.0138489\pi\)
−0.461860 + 0.886953i \(0.652818\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.00000 + 8.66025i 0.221621 + 0.383859i 0.955300 0.295637i \(-0.0955319\pi\)
−0.733679 + 0.679496i \(0.762199\pi\)
\(510\) 0 0
\(511\) −22.5000 + 38.9711i −0.995341 + 1.72398i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −15.0000 + 25.9808i −0.660979 + 1.14485i
\(516\) 0 0
\(517\) −18.0000 31.1769i −0.791639 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) −9.00000 −0.393543 −0.196771 0.980449i \(-0.563046\pi\)
−0.196771 + 0.980449i \(0.563046\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 + 20.7846i −0.519778 + 0.900281i
\(534\) 0 0
\(535\) −6.00000 10.3923i −0.259403 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −15.0000 −0.644900 −0.322450 0.946586i \(-0.604506\pi\)
−0.322450 + 0.946586i \(0.604506\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.00000 + 10.3923i 0.257012 + 0.445157i
\(546\) 0 0
\(547\) −4.50000 + 7.79423i −0.192406 + 0.333257i −0.946047 0.324029i \(-0.894962\pi\)
0.753641 + 0.657286i \(0.228296\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 + 20.7846i −0.511217 + 0.885454i
\(552\) 0 0
\(553\) −13.5000 23.3827i −0.574078 0.994333i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 + 20.7846i 0.505740 + 0.875967i 0.999978 + 0.00664037i \(0.00211371\pi\)
−0.494238 + 0.869326i \(0.664553\pi\)
\(564\) 0 0
\(565\) −14.0000 + 24.2487i −0.588984 + 1.02015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.0000 39.8372i 0.964210 1.67006i 0.252488 0.967600i \(-0.418751\pi\)
0.711722 0.702461i \(-0.247915\pi\)
\(570\) 0 0
\(571\) −1.50000 2.59808i −0.0627730 0.108726i 0.832931 0.553377i \(-0.186661\pi\)
−0.895704 + 0.444651i \(0.853328\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −18.0000 31.1769i −0.746766 1.29344i
\(582\) 0 0
\(583\) 12.0000 20.7846i 0.496989 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.00000 + 15.5885i −0.371470 + 0.643404i −0.989792 0.142520i \(-0.954479\pi\)
0.618322 + 0.785925i \(0.287813\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −28.0000 −1.14982 −0.574911 0.818216i \(-0.694963\pi\)
−0.574911 + 0.818216i \(0.694963\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 20.7846i −0.490307 0.849236i 0.509631 0.860393i \(-0.329782\pi\)
−0.999938 + 0.0111569i \(0.996449\pi\)
\(600\) 0 0
\(601\) 13.0000 22.5167i 0.530281 0.918474i −0.469095 0.883148i \(-0.655420\pi\)
0.999376 0.0353259i \(-0.0112469\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25.0000 43.3013i 1.01639 1.76045i
\(606\) 0 0
\(607\) 1.50000 + 2.59808i 0.0608831 + 0.105453i 0.894860 0.446346i \(-0.147275\pi\)
−0.833977 + 0.551799i \(0.813942\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00000 + 1.73205i 0.0402585 + 0.0697297i 0.885453 0.464730i \(-0.153849\pi\)
−0.845194 + 0.534460i \(0.820515\pi\)
\(618\) 0 0
\(619\) 4.50000 7.79423i 0.180870 0.313276i −0.761307 0.648392i \(-0.775442\pi\)
0.942177 + 0.335115i \(0.108775\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.0000 25.9808i 0.600962 1.04090i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.0000 0.558217
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000 + 20.7846i 0.476205 + 0.824812i
\(636\) 0 0
\(637\) 3.00000 5.19615i 0.118864 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 3.46410i 0.0789953 0.136824i −0.823821 0.566849i \(-0.808162\pi\)
0.902817 + 0.430026i \(0.141495\pi\)
\(642\) 0 0
\(643\) −18.0000 31.1769i −0.709851 1.22950i −0.964912 0.262573i \(-0.915429\pi\)
0.255062 0.966925i \(-0.417904\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14.0000 24.2487i −0.547862 0.948925i −0.998421 0.0561784i \(-0.982108\pi\)
0.450558 0.892747i \(-0.351225\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 10.3923i 0.233727 0.404827i −0.725175 0.688565i \(-0.758241\pi\)
0.958902 + 0.283738i \(0.0915745\pi\)
\(660\) 0 0
\(661\) 2.50000 + 4.33013i 0.0972387 + 0.168422i 0.910541 0.413419i \(-0.135666\pi\)
−0.813302 + 0.581842i \(0.802332\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 18.0000 0.698010
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 + 5.19615i 0.115814 + 0.200595i
\(672\) 0 0
\(673\) 0.500000 0.866025i 0.0192736 0.0333828i −0.856228 0.516599i \(-0.827198\pi\)
0.875501 + 0.483216i \(0.160531\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.00000 1.73205i 0.0384331 0.0665681i −0.846169 0.532915i \(-0.821097\pi\)
0.884602 + 0.466347i \(0.154430\pi\)
\(678\) 0 0
\(679\) 13.5000 + 23.3827i 0.518082 + 0.897345i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.00000 10.3923i −0.228582 0.395915i
\(690\) 0 0
\(691\) 6.00000 10.3923i 0.228251 0.395342i −0.729039 0.684472i \(-0.760033\pi\)
0.957290 + 0.289130i \(0.0933661\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.0000 36.3731i 0.796575 1.37971i
\(696\) 0 0
\(697\) 8.00000 + 13.8564i 0.303022 + 0.524849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −46.0000 −1.73740 −0.868698 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) 0 0
\(703\) 21.0000 0.792030
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 + 41.5692i 0.902613 + 1.56337i
\(708\) 0 0
\(709\) 25.5000 44.1673i 0.957673 1.65874i 0.229543 0.973299i \(-0.426277\pi\)
0.728130 0.685439i \(-0.240390\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −18.0000 31.1769i −0.673162 1.16595i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 45.0000 1.67589
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.00000 + 6.92820i 0.148556 + 0.257307i
\(726\) 0 0
\(727\) 12.0000 20.7846i 0.445055 0.770859i −0.553001 0.833181i \(-0.686517\pi\)
0.998056 + 0.0623223i \(0.0198507\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 + 20.7846i −0.443836 + 0.768747i
\(732\) 0 0
\(733\) 9.00000 + 15.5885i 0.332423 + 0.575773i 0.982986 0.183679i \(-0.0588007\pi\)
−0.650564 + 0.759452i \(0.725467\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.0000 0.663039
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.00000 5.19615i −0.110059 0.190628i 0.805735 0.592277i \(-0.201771\pi\)
−0.915794 + 0.401648i \(0.868437\pi\)
\(744\) 0 0
\(745\) −4.00000 + 6.92820i −0.146549 + 0.253830i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.00000 + 15.5885i −0.328853 + 0.569590i
\(750\) 0 0
\(751\) 22.5000 + 38.9711i 0.821037 + 1.42208i 0.904911 + 0.425601i \(0.139937\pi\)
−0.0838743 + 0.996476i \(0.526729\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −27.0000 −0.981332 −0.490666 0.871348i \(-0.663246\pi\)
−0.490666 + 0.871348i \(0.663246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17.0000 29.4449i −0.616250 1.06738i −0.990164 0.139912i \(-0.955318\pi\)
0.373914 0.927463i \(-0.378015\pi\)
\(762\) 0 0
\(763\) 9.00000 15.5885i 0.325822 0.564340i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.00000 15.5885i 0.324971 0.562867i
\(768\) 0 0
\(769\) 24.5000 + 42.4352i 0.883493 + 1.53025i 0.847432 + 0.530904i \(0.178148\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 + 20.7846i 0.429945 + 0.744686i
\(780\) 0 0
\(781\) −36.0000 + 62.3538i −1.28818 + 2.23120i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −14.0000 + 24.2487i −0.499681 + 0.865474i
\(786\) 0 0
\(787\) −13.5000 23.3827i −0.481223 0.833503i 0.518545 0.855050i \(-0.326474\pi\)
−0.999768 + 0.0215477i \(0.993141\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) 3.00000 0.106533
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −20.0000 34.6410i −0.708436 1.22705i −0.965437 0.260637i \(-0.916068\pi\)
0.257001 0.966411i \(-0.417266\pi\)
\(798\) 0 0
\(799\) −6.00000 + 10.3923i −0.212265 + 0.367653i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 45.0000 77.9423i 1.58802 2.75052i