Properties

Label 324.2.e.d.109.1
Level $324$
Weight $2$
Character 324.109
Analytic conductor $2.587$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 324.109
Dual form 324.2.e.d.217.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{5} +(-1.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+(1.50000 - 2.59808i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-2.50000 + 4.33013i) q^{13} +3.00000 q^{17} +2.00000 q^{19} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(1.50000 + 2.59808i) q^{29} +(2.00000 - 3.46410i) q^{31} -6.00000 q^{35} +5.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(5.00000 + 8.66025i) q^{43} +(1.50000 - 2.59808i) q^{49} +6.00000 q^{53} -18.0000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(7.50000 + 12.9904i) q^{65} +(-1.00000 + 1.73205i) q^{67} -6.00000 q^{71} -1.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(5.00000 + 8.66025i) q^{79} +(4.50000 - 7.79423i) q^{85} +3.00000 q^{89} +10.0000 q^{91} +(3.00000 - 5.19615i) q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - 2 q^{7} - 6 q^{11} - 5 q^{13} + 6 q^{17} + 4 q^{19} + 6 q^{23} - 4 q^{25} + 3 q^{29} + 4 q^{31} - 12 q^{35} + 10 q^{37} - 6 q^{41} + 10 q^{43} + 3 q^{49} + 12 q^{53} - 36 q^{55} - 12 q^{59} - 5 q^{61} + 15 q^{65} - 2 q^{67} - 12 q^{71} - 2 q^{73} - 12 q^{77} + 10 q^{79} + 9 q^{85} + 6 q^{89} + 20 q^{91} + 6 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 1.50000 2.59808i 0.670820 1.16190i −0.306851 0.951757i \(-0.599275\pi\)
0.977672 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) −1.00000 1.73205i −0.377964 0.654654i 0.612801 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i \(-0.806886\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) 2.00000 3.46410i 0.359211 0.622171i −0.628619 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) 5.00000 + 8.66025i 0.762493 + 1.32068i 0.941562 + 0.336840i \(0.109358\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) −2.50000 4.33013i −0.320092 0.554416i 0.660415 0.750901i \(-0.270381\pi\)
−0.980507 + 0.196485i \(0.937047\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.50000 + 12.9904i 0.930261 + 1.61126i
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 + 10.3923i −0.683763 + 1.18431i
\(78\) 0 0
\(79\) 5.00000 + 8.66025i 0.562544 + 0.974355i 0.997274 + 0.0737937i \(0.0235106\pi\)
−0.434730 + 0.900561i \(0.643156\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 4.50000 7.79423i 0.488094 0.845403i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 5.19615i 0.307794 0.533114i
\(96\) 0 0
\(97\) 5.00000 + 8.66025i 0.507673 + 0.879316i 0.999961 + 0.00888289i \(0.00282755\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 + 5.19615i 0.298511 + 0.517036i 0.975796 0.218685i \(-0.0701767\pi\)
−0.677284 + 0.735721i \(0.736843\pi\)
\(102\) 0 0
\(103\) 8.00000 13.8564i 0.788263 1.36531i −0.138767 0.990325i \(-0.544314\pi\)
0.927030 0.374987i \(-0.122353\pi\)
\(104\) 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
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.50000 7.79423i 0.423324 0.733219i −0.572938 0.819599i \(-0.694196\pi\)
0.996262 + 0.0863794i \(0.0275297\pi\)
\(114\) 0 0
\(115\) −9.00000 15.5885i −0.839254 1.45363i
\(116\) 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
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.00000 + 5.19615i −0.262111 + 0.453990i −0.966803 0.255524i \(-0.917752\pi\)
0.704692 + 0.709514i \(0.251085\pi\)
\(132\) 0 0
\(133\) −2.00000 3.46410i −0.173422 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.50000 12.9904i −0.640768 1.10984i −0.985262 0.171054i \(-0.945283\pi\)
0.344493 0.938789i \(-0.388051\pi\)
\(138\) 0 0
\(139\) −4.00000 + 6.92820i −0.339276 + 0.587643i −0.984297 0.176522i \(-0.943515\pi\)
0.645021 + 0.764165i \(0.276849\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 30.0000 2.50873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.50000 12.9904i 0.614424 1.06421i −0.376061 0.926595i \(-0.622722\pi\)
0.990485 0.137619i \(-0.0439449\pi\)
\(150\) 0 0
\(151\) 2.00000 + 3.46410i 0.162758 + 0.281905i 0.935857 0.352381i \(-0.114628\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 10.3923i −0.481932 0.834730i
\(156\) 0 0
\(157\) −2.50000 + 4.33013i −0.199522 + 0.345582i −0.948373 0.317156i \(-0.897272\pi\)
0.748852 + 0.662738i \(0.230606\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 15.5885i 0.696441 1.20627i −0.273252 0.961943i \(-0.588099\pi\)
0.969693 0.244328i \(-0.0785675\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.50000 7.79423i −0.342129 0.592584i 0.642699 0.766119i \(-0.277815\pi\)
−0.984828 + 0.173534i \(0.944481\pi\)
\(174\) 0 0
\(175\) −4.00000 + 6.92820i −0.302372 + 0.523723i
\(176\) 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
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.50000 12.9904i 0.551411 0.955072i
\(186\) 0 0
\(187\) −9.00000 15.5885i −0.658145 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i \(-0.0970159\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(192\) 0 0
\(193\) 12.5000 21.6506i 0.899770 1.55845i 0.0719816 0.997406i \(-0.477068\pi\)
0.827788 0.561041i \(-0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.00000 5.19615i 0.210559 0.364698i
\(204\) 0 0
\(205\) 9.00000 + 15.5885i 0.628587 + 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 10.3923i −0.415029 0.718851i
\(210\) 0 0
\(211\) −7.00000 + 12.1244i −0.481900 + 0.834675i −0.999784 0.0207756i \(-0.993386\pi\)
0.517884 + 0.855451i \(0.326720\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 30.0000 2.04598
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.50000 + 12.9904i −0.504505 + 0.873828i
\(222\) 0 0
\(223\) 5.00000 + 8.66025i 0.334825 + 0.579934i 0.983451 0.181173i \(-0.0579895\pi\)
−0.648626 + 0.761107i \(0.724656\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.00000 + 15.5885i 0.597351 + 1.03464i 0.993210 + 0.116331i \(0.0371134\pi\)
−0.395860 + 0.918311i \(0.629553\pi\)
\(228\) 0 0
\(229\) −2.50000 + 4.33013i −0.165205 + 0.286143i −0.936728 0.350058i \(-0.886162\pi\)
0.771523 + 0.636201i \(0.219495\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0000 0.982683 0.491341 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) −5.50000 9.52628i −0.354286 0.613642i 0.632709 0.774389i \(-0.281943\pi\)
−0.986996 + 0.160748i \(0.948609\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.50000 7.79423i −0.287494 0.497955i
\(246\) 0 0
\(247\) −5.00000 + 8.66025i −0.318142 + 0.551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.5000 + 23.3827i −0.842107 + 1.45857i 0.0460033 + 0.998941i \(0.485352\pi\)
−0.888110 + 0.459631i \(0.847982\pi\)
\(258\) 0 0
\(259\) −5.00000 8.66025i −0.310685 0.538122i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 + 20.7846i 0.739952 + 1.28163i 0.952517 + 0.304487i \(0.0984850\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(264\) 0 0
\(265\) 9.00000 15.5885i 0.552866 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 + 20.7846i −0.723627 + 1.25336i
\(276\) 0 0
\(277\) 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i \(-0.0695395\pi\)
−0.675810 + 0.737075i \(0.736206\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.50000 12.9904i −0.447412 0.774941i 0.550804 0.834634i \(-0.314321\pi\)
−0.998217 + 0.0596933i \(0.980988\pi\)
\(282\) 0 0
\(283\) 2.00000 3.46410i 0.118888 0.205919i −0.800439 0.599414i \(-0.795400\pi\)
0.919327 + 0.393494i \(0.128734\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.50000 12.9904i 0.438155 0.758906i −0.559393 0.828903i \(-0.688966\pi\)
0.997547 + 0.0699967i \(0.0222989\pi\)
\(294\) 0 0
\(295\) 18.0000 + 31.1769i 1.04800 + 1.81519i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.0000 + 25.9808i 0.867472 + 1.50251i
\(300\) 0 0
\(301\) 10.0000 17.3205i 0.576390 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.0000 −0.858898
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 + 20.7846i −0.680458 + 1.17859i 0.294384 + 0.955687i \(0.404886\pi\)
−0.974841 + 0.222900i \(0.928448\pi\)
\(312\) 0 0
\(313\) 12.5000 + 21.6506i 0.706542 + 1.22377i 0.966132 + 0.258047i \(0.0830791\pi\)
−0.259590 + 0.965719i \(0.583588\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.50000 + 12.9904i 0.421242 + 0.729612i 0.996061 0.0886679i \(-0.0282610\pi\)
−0.574819 + 0.818280i \(0.694928\pi\)
\(318\) 0 0
\(319\) 9.00000 15.5885i 0.503903 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 22.5167i −0.714545 1.23763i −0.963135 0.269019i \(-0.913301\pi\)
0.248590 0.968609i \(-0.420033\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 + 5.19615i 0.163908 + 0.283896i
\(336\) 0 0
\(337\) 5.00000 8.66025i 0.272367 0.471754i −0.697100 0.716974i \(-0.745527\pi\)
0.969468 + 0.245220i \(0.0788601\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 + 15.5885i −0.483145 + 0.836832i −0.999813 0.0193540i \(-0.993839\pi\)
0.516667 + 0.856186i \(0.327172\pi\)
\(348\) 0 0
\(349\) −1.00000 1.73205i −0.0535288 0.0927146i 0.838019 0.545640i \(-0.183714\pi\)
−0.891548 + 0.452926i \(0.850380\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.0000 + 25.9808i 0.798369 + 1.38282i 0.920677 + 0.390324i \(0.127637\pi\)
−0.122308 + 0.992492i \(0.539030\pi\)
\(354\) 0 0
\(355\) −9.00000 + 15.5885i −0.477670 + 0.827349i
\(356\) 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
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.50000 + 2.59808i −0.0785136 + 0.135990i
\(366\) 0 0
\(367\) −16.0000 27.7128i −0.835193 1.44660i −0.893873 0.448320i \(-0.852022\pi\)
0.0586798 0.998277i \(-0.481311\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 10.3923i −0.311504 0.539542i
\(372\) 0 0
\(373\) −7.00000 + 12.1244i −0.362446 + 0.627775i −0.988363 0.152115i \(-0.951392\pi\)
0.625917 + 0.779890i \(0.284725\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.0000 25.9808i 0.766464 1.32755i −0.173005 0.984921i \(-0.555348\pi\)
0.939469 0.342634i \(-0.111319\pi\)
\(384\) 0 0
\(385\) 18.0000 + 31.1769i 0.917365 + 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.0000 25.9808i −0.760530 1.31728i −0.942578 0.333987i \(-0.891606\pi\)
0.182047 0.983290i \(-0.441728\pi\)
\(390\) 0 0
\(391\) 9.00000 15.5885i 0.455150 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.50000 + 12.9904i −0.374532 + 0.648709i −0.990257 0.139253i \(-0.955530\pi\)
0.615725 + 0.787961i \(0.288863\pi\)
\(402\) 0 0
\(403\) 10.0000 + 17.3205i 0.498135 + 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.0000 25.9808i −0.743522 1.28782i
\(408\) 0 0
\(409\) −17.5000 + 30.3109i −0.865319 + 1.49878i 0.00141047 + 0.999999i \(0.499551\pi\)
−0.866730 + 0.498778i \(0.833782\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) −14.5000 25.1147i −0.706687 1.22402i −0.966079 0.258245i \(-0.916856\pi\)
0.259393 0.965772i \(-0.416478\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 10.3923i −0.291043 0.504101i
\(426\) 0 0
\(427\) −5.00000 + 8.66025i −0.241967 + 0.419099i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.00000 10.3923i 0.287019 0.497131i
\(438\) 0 0
\(439\) −4.00000 6.92820i −0.190910 0.330665i 0.754642 0.656136i \(-0.227810\pi\)
−0.945552 + 0.325471i \(0.894477\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 0 0
\(445\) 4.50000 7.79423i 0.213320 0.369482i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.0000 25.9808i 0.703211 1.21800i
\(456\) 0 0
\(457\) −17.5000 30.3109i −0.818615 1.41788i −0.906702 0.421771i \(-0.861409\pi\)
0.0880870 0.996113i \(-0.471925\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 5.19615i −0.139724 0.242009i 0.787668 0.616100i \(-0.211288\pi\)
−0.927392 + 0.374091i \(0.877955\pi\)
\(462\) 0 0
\(463\) 2.00000 3.46410i 0.0929479 0.160990i −0.815802 0.578331i \(-0.803704\pi\)
0.908750 + 0.417340i \(0.137038\pi\)
\(464\) 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
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 30.0000 51.9615i 1.37940 2.38919i
\(474\) 0 0
\(475\) −4.00000 6.92820i −0.183533 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.00000 + 15.5885i 0.411220 + 0.712255i 0.995023 0.0996406i \(-0.0317693\pi\)
−0.583803 + 0.811895i \(0.698436\pi\)
\(480\) 0 0
\(481\) −12.5000 + 21.6506i −0.569951 + 0.987184i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 30.0000 1.36223
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.00000 5.19615i 0.135388 0.234499i −0.790358 0.612646i \(-0.790105\pi\)
0.925746 + 0.378147i \(0.123439\pi\)
\(492\) 0 0
\(493\) 4.50000 + 7.79423i 0.202670 + 0.351034i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 + 10.3923i 0.269137 + 0.466159i
\(498\) 0 0
\(499\) 11.0000 19.0526i 0.492428 0.852910i −0.507534 0.861632i \(-0.669443\pi\)
0.999962 + 0.00872186i \(0.00277629\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.00000 + 5.19615i −0.132973 + 0.230315i −0.924821 0.380402i \(-0.875786\pi\)
0.791849 + 0.610718i \(0.209119\pi\)
\(510\) 0 0
\(511\) 1.00000 + 1.73205i 0.0442374 + 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.0000 41.5692i −1.05757 1.83176i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.0000 25.9808i −0.649722 1.12535i
\(534\) 0 0
\(535\) −18.0000 + 31.1769i −0.778208 + 1.34790i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 41.0000 1.76273 0.881364 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.5000 + 18.1865i −0.449771 + 0.779026i
\(546\) 0 0
\(547\) 20.0000 + 34.6410i 0.855138 + 1.48114i 0.876517 + 0.481371i \(0.159861\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.00000 + 5.19615i 0.127804 + 0.221364i
\(552\) 0 0
\(553\) 10.0000 17.3205i 0.425243 0.736543i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15.0000 −0.635570 −0.317785 0.948163i \(-0.602939\pi\)
−0.317785 + 0.948163i \(0.602939\pi\)
\(558\) 0 0
\(559\) −50.0000 −2.11477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) −13.5000 23.3827i −0.567949 0.983717i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.50000 2.59808i −0.0628833 0.108917i 0.832870 0.553469i \(-0.186696\pi\)
−0.895753 + 0.444552i \(0.853363\pi\)
\(570\) 0 0
\(571\) −22.0000 + 38.1051i −0.920671 + 1.59465i −0.122292 + 0.992494i \(0.539025\pi\)
−0.798379 + 0.602155i \(0.794309\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.0000 31.1769i −0.745484 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.0000 25.9808i −0.619116 1.07234i −0.989647 0.143521i \(-0.954158\pi\)
0.370531 0.928820i \(-0.379176\pi\)
\(588\) 0 0
\(589\) 4.00000 6.92820i 0.164817 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) −18.0000 −0.737928
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 + 10.3923i −0.245153 + 0.424618i −0.962175 0.272433i \(-0.912172\pi\)
0.717021 + 0.697051i \(0.245505\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.866025i 0.0203954 + 0.0353259i 0.876043 0.482233i \(-0.160174\pi\)
−0.855648 + 0.517559i \(0.826841\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 37.5000 + 64.9519i 1.52459 + 2.64067i
\(606\) 0 0
\(607\) 5.00000 8.66025i 0.202944 0.351509i −0.746532 0.665350i \(-0.768282\pi\)
0.949476 + 0.313841i \(0.101616\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.50000 + 12.9904i −0.301939 + 0.522973i −0.976575 0.215177i \(-0.930967\pi\)
0.674636 + 0.738150i \(0.264300\pi\)
\(618\) 0 0
\(619\) 20.0000 + 34.6410i 0.803868 + 1.39234i 0.917053 + 0.398766i \(0.130561\pi\)
−0.113185 + 0.993574i \(0.536105\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.00000 5.19615i −0.120192 0.208179i
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.00000 5.19615i 0.119051 0.206203i
\(636\) 0 0
\(637\) 7.50000 + 12.9904i 0.297161 + 0.514698i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.5000 + 18.1865i 0.414725 + 0.718325i 0.995400 0.0958109i \(-0.0305444\pi\)
−0.580674 + 0.814136i \(0.697211\pi\)
\(642\) 0 0
\(643\) 20.0000 34.6410i 0.788723 1.36611i −0.138027 0.990429i \(-0.544076\pi\)
0.926750 0.375680i \(-0.122591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 72.0000 2.82625
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.0000 + 25.9808i −0.586995 + 1.01671i 0.407628 + 0.913148i \(0.366356\pi\)
−0.994623 + 0.103558i \(0.966977\pi\)
\(654\) 0 0
\(655\) 9.00000 + 15.5885i 0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.00000 + 5.19615i 0.116863 + 0.202413i 0.918523 0.395367i \(-0.129383\pi\)
−0.801660 + 0.597781i \(0.796049\pi\)
\(660\) 0 0
\(661\) −14.5000 + 25.1147i −0.563985 + 0.976850i 0.433159 + 0.901318i \(0.357399\pi\)
−0.997143 + 0.0755324i \(0.975934\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0000 + 25.9808i −0.579069 + 1.00298i
\(672\) 0 0
\(673\) 0.500000 + 0.866025i 0.0192736 + 0.0333828i 0.875501 0.483216i \(-0.160531\pi\)
−0.856228 + 0.516599i \(0.827198\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.0000 + 36.3731i 0.807096 + 1.39793i 0.914867 + 0.403755i \(0.132295\pi\)
−0.107772 + 0.994176i \(0.534372\pi\)
\(678\) 0 0
\(679\) 10.0000 17.3205i 0.383765 0.664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −45.0000 −1.71936
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −15.0000 + 25.9808i −0.571454 + 0.989788i
\(690\) 0 0
\(691\) 5.00000 + 8.66025i 0.190209 + 0.329452i 0.945319 0.326146i \(-0.105750\pi\)
−0.755110 + 0.655598i \(0.772417\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 + 20.7846i 0.455186 + 0.788405i
\(696\) 0 0
\(697\) −9.00000 + 15.5885i −0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 10.3923i 0.225653 0.390843i
\(708\) 0 0
\(709\) 9.50000 + 16.4545i 0.356780 + 0.617961i 0.987421 0.158114i \(-0.0505412\pi\)
−0.630641 + 0.776075i \(0.717208\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.0000 20.7846i −0.449404 0.778390i
\(714\) 0 0
\(715\) 45.0000 77.9423i 1.68290 2.91488i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.00000 10.3923i 0.222834 0.385961i
\(726\) 0 0
\(727\) 5.00000 + 8.66025i 0.185440 + 0.321191i 0.943725 0.330732i \(-0.107296\pi\)
−0.758285 + 0.651923i \(0.773962\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.0000 + 25.9808i 0.554795 + 0.960933i
\(732\) 0 0
\(733\) −7.00000 + 12.1244i −0.258551 + 0.447823i −0.965854 0.259087i \(-0.916578\pi\)
0.707303 + 0.706910i \(0.249912\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18.0000 + 31.1769i −0.660356 + 1.14377i 0.320166 + 0.947361i \(0.396261\pi\)
−0.980522 + 0.196409i \(0.937072\pi\)
\(744\) 0 0
\(745\) −22.5000 38.9711i −0.824336 1.42779i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 + 20.7846i 0.438470 + 0.759453i
\(750\) 0 0
\(751\) 5.00000 8.66025i 0.182453 0.316017i −0.760263 0.649616i \(-0.774930\pi\)
0.942715 + 0.333599i \(0.108263\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.50000 7.79423i 0.163125 0.282541i −0.772863 0.634573i \(-0.781176\pi\)
0.935988 + 0.352032i \(0.114509\pi\)
\(762\) 0 0
\(763\) 7.00000 + 12.1244i 0.253417 + 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −30.0000 51.9615i −1.08324 1.87622i
\(768\) 0 0
\(769\) −11.5000 + 19.9186i −0.414701 + 0.718283i −0.995397 0.0958377i \(-0.969447\pi\)
0.580696 + 0.814120i \(0.302780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 45.0000 1.61854 0.809269 0.587439i \(-0.199864\pi\)
0.809269 + 0.587439i \(0.199864\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 + 10.3923i −0.214972 + 0.372343i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.50000 + 12.9904i 0.267686 + 0.463647i
\(786\) 0 0
\(787\) −19.0000 + 32.9090i −0.677277 + 1.17308i 0.298521 + 0.954403i \(0.403507\pi\)
−0.975798 + 0.218675i \(0.929827\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 25.0000 0.887776
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22.5000 + 38.9711i −0.796991 + 1.38043i 0.124576 + 0.992210i \(0.460243\pi\)
−0.921567 + 0.388219i \(0.873091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0