Properties

Label 1620.2.r.d.109.1
Level $1620$
Weight $2$
Character 1620.109
Analytic conductor $12.936$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 109.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1620.109
Dual form 1620.2.r.d.1189.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.23205 - 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +O(q^{10})\) \(q+(-1.23205 - 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +(-2.00000 - 3.46410i) q^{11} -4.00000i q^{17} +(-3.46410 - 2.00000i) q^{23} +(-1.96410 + 4.59808i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(-8.00000 - 4.00000i) q^{35} -8.00000i q^{37} +(-5.00000 + 8.66025i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(3.46410 - 2.00000i) q^{47} +(4.50000 - 7.79423i) q^{49} -12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +(-2.00000 + 3.46410i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(3.46410 + 2.00000i) q^{67} +8.00000i q^{73} +(-13.8564 - 8.00000i) q^{77} +(-6.00000 - 10.3923i) q^{79} +(3.46410 - 2.00000i) q^{83} +(-7.46410 + 4.92820i) q^{85} -10.0000 q^{89} +(6.92820 - 4.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + O(q^{10}) \) \( 4q + 2q^{5} - 8q^{11} + 6q^{25} + 12q^{29} - 8q^{31} - 32q^{35} - 20q^{41} + 18q^{49} - 16q^{55} - 8q^{59} - 4q^{61} - 24q^{79} - 16q^{85} - 40q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −1.23205 1.86603i −0.550990 0.834512i
\(6\) 0 0
\(7\) 3.46410 2.00000i 1.30931 0.755929i 0.327327 0.944911i \(-0.393852\pi\)
0.981981 + 0.188982i \(0.0605189\pi\)
\(8\) 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 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 2.00000i −0.722315 0.417029i 0.0932891 0.995639i \(-0.470262\pi\)
−0.815604 + 0.578610i \(0.803595\pi\)
\(24\) 0 0
\(25\) −1.96410 + 4.59808i −0.392820 + 0.919615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 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
\(35\) −8.00000 4.00000i −1.35225 0.676123i
\(36\) 0 0
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 + 8.66025i −0.780869 + 1.35250i 0.150567 + 0.988600i \(0.451890\pi\)
−0.931436 + 0.363905i \(0.881443\pi\)
\(42\) 0 0
\(43\) −3.46410 + 2.00000i −0.528271 + 0.304997i −0.740312 0.672264i \(-0.765322\pi\)
0.212041 + 0.977261i \(0.431989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 2.00000i 0.505291 0.291730i −0.225605 0.974219i \(-0.572436\pi\)
0.730896 + 0.682489i \(0.239102\pi\)
\(48\) 0 0
\(49\) 4.50000 7.79423i 0.642857 1.11346i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) −4.00000 + 8.00000i −0.539360 + 1.07872i
\(56\) 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
\(61\) −1.00000 1.73205i −0.128037 0.221766i 0.794879 0.606768i \(-0.207534\pi\)
−0.922916 + 0.385002i \(0.874201\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410 + 2.00000i 0.423207 + 0.244339i 0.696449 0.717607i \(-0.254762\pi\)
−0.273241 + 0.961946i \(0.588096\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.8564 8.00000i −1.57908 0.911685i
\(78\) 0 0
\(79\) −6.00000 10.3923i −0.675053 1.16923i −0.976453 0.215728i \(-0.930788\pi\)
0.301401 0.953498i \(-0.402546\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410 2.00000i 0.380235 0.219529i −0.297686 0.954664i \(-0.596215\pi\)
0.677920 + 0.735135i \(0.262881\pi\)
\(84\) 0 0
\(85\) −7.46410 + 4.92820i −0.809595 + 0.534539i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.92820 4.00000i 0.703452 0.406138i −0.105180 0.994453i \(-0.533542\pi\)
0.808632 + 0.588315i \(0.200208\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.00000 1.73205i −0.0995037 0.172345i 0.811976 0.583691i \(-0.198392\pi\)
−0.911479 + 0.411346i \(0.865059\pi\)
\(102\) 0 0
\(103\) −3.46410 2.00000i −0.341328 0.197066i 0.319531 0.947576i \(-0.396475\pi\)
−0.660859 + 0.750510i \(0.729808\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.3923 6.00000i −0.977626 0.564433i −0.0760733 0.997102i \(-0.524238\pi\)
−0.901553 + 0.432670i \(0.857572\pi\)
\(114\) 0 0
\(115\) 0.535898 + 8.92820i 0.0499728 + 0.832559i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.00000 13.8564i −0.733359 1.27021i
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.3923 + 6.00000i −0.887875 + 0.512615i −0.873247 0.487278i \(-0.837990\pi\)
−0.0146279 + 0.999893i \(0.504656\pi\)
\(138\) 0 0
\(139\) 8.00000 13.8564i 0.678551 1.17529i −0.296866 0.954919i \(-0.595942\pi\)
0.975417 0.220366i \(-0.0707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 12.0000i 0.498273 0.996546i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000 1.73205i 0.0819232 0.141895i −0.822153 0.569267i \(-0.807227\pi\)
0.904076 + 0.427372i \(0.140560\pi\)
\(150\) 0 0
\(151\) 10.0000 + 17.3205i 0.813788 + 1.40952i 0.910195 + 0.414181i \(0.135932\pi\)
−0.0964061 + 0.995342i \(0.530735\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.92820 0.535898i 0.717131 0.0430444i
\(156\) 0 0
\(157\) −6.92820 4.00000i −0.552931 0.319235i 0.197372 0.980329i \(-0.436759\pi\)
−0.750303 + 0.661094i \(0.770093\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 20.0000i 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3923 + 6.00000i 0.804181 + 0.464294i 0.844931 0.534875i \(-0.179641\pi\)
−0.0407502 + 0.999169i \(0.512975\pi\)
\(168\) 0 0
\(169\) −6.50000 11.2583i −0.500000 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.46410 + 2.00000i −0.263371 + 0.152057i −0.625871 0.779926i \(-0.715256\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(174\) 0 0
\(175\) 2.39230 + 19.8564i 0.180841 + 1.50100i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.9282 + 9.85641i −1.09754 + 0.724657i
\(186\) 0 0
\(187\) −13.8564 + 8.00000i −1.01328 + 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 + 20.7846i 0.868290 + 1.50392i 0.863743 + 0.503932i \(0.168114\pi\)
0.00454614 + 0.999990i \(0.498553\pi\)
\(192\) 0 0
\(193\) 13.8564 + 8.00000i 0.997406 + 0.575853i 0.907480 0.420096i \(-0.138004\pi\)
0.0899262 + 0.995948i \(0.471337\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.00000i 0.284988i 0.989796 + 0.142494i \(0.0455122\pi\)
−0.989796 + 0.142494i \(0.954488\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.7846 + 12.0000i 1.45879 + 0.842235i
\(204\) 0 0
\(205\) 22.3205 1.33975i 1.55893 0.0935719i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 + 4.00000i 0.545595 + 0.272798i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.3205 10.0000i 1.15987 0.669650i 0.208595 0.978002i \(-0.433111\pi\)
0.951272 + 0.308353i \(0.0997777\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.3205 10.0000i 1.14960 0.663723i 0.200812 0.979630i \(-0.435642\pi\)
0.948790 + 0.315906i \(0.102309\pi\)
\(228\) 0 0
\(229\) 13.0000 22.5167i 0.859064 1.48794i −0.0137585 0.999905i \(-0.504380\pi\)
0.872823 0.488037i \(-0.162287\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.0000i 1.31024i 0.755523 + 0.655122i \(0.227383\pi\)
−0.755523 + 0.655122i \(0.772617\pi\)
\(234\) 0 0
\(235\) −8.00000 4.00000i −0.521862 0.260931i
\(236\) 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
\(241\) 1.00000 + 1.73205i 0.0644157 + 0.111571i 0.896435 0.443176i \(-0.146148\pi\)
−0.832019 + 0.554747i \(0.812815\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −20.0885 + 1.20577i −1.28340 + 0.0770339i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.3923 + 6.00000i 0.648254 + 0.374270i 0.787787 0.615948i \(-0.211227\pi\)
−0.139533 + 0.990217i \(0.544560\pi\)
\(258\) 0 0
\(259\) −16.0000 27.7128i −0.994192 1.72199i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.46410 2.00000i 0.213606 0.123325i −0.389380 0.921077i \(-0.627311\pi\)
0.602986 + 0.797752i \(0.293977\pi\)
\(264\) 0 0
\(265\) −22.3923 + 14.7846i −1.37555 + 0.908211i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.8564 2.39230i 1.19739 0.144261i
\(276\) 0 0
\(277\) −27.7128 + 16.0000i −1.66510 + 0.961347i −0.694881 + 0.719125i \(0.744543\pi\)
−0.970221 + 0.242222i \(0.922124\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 + 5.19615i 0.178965 + 0.309976i 0.941526 0.336939i \(-0.109392\pi\)
−0.762561 + 0.646916i \(0.776058\pi\)
\(282\) 0 0
\(283\) −24.2487 14.0000i −1.44144 0.832214i −0.443491 0.896279i \(-0.646260\pi\)
−0.997946 + 0.0640654i \(0.979593\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000i 2.36113i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.3923 6.00000i −0.607125 0.350524i 0.164714 0.986341i \(-0.447330\pi\)
−0.771839 + 0.635818i \(0.780663\pi\)
\(294\) 0 0
\(295\) 8.92820 0.535898i 0.519820 0.0312012i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 + 13.8564i −0.461112 + 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 + 4.00000i −0.114520 + 0.229039i
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 20.7846i 0.680458 1.17859i −0.294384 0.955687i \(-0.595114\pi\)
0.974841 0.222900i \(-0.0715523\pi\)
\(312\) 0 0
\(313\) −13.8564 + 8.00000i −0.783210 + 0.452187i −0.837567 0.546335i \(-0.816023\pi\)
0.0543564 + 0.998522i \(0.482689\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.46410 + 2.00000i −0.194563 + 0.112331i −0.594117 0.804379i \(-0.702498\pi\)
0.399554 + 0.916710i \(0.369165\pi\)
\(318\) 0 0
\(319\) 12.0000 20.7846i 0.671871 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 13.8564i 0.441054 0.763928i
\(330\) 0 0
\(331\) 4.00000 + 6.92820i 0.219860 + 0.380808i 0.954765 0.297361i \(-0.0961066\pi\)
−0.734905 + 0.678170i \(0.762773\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.535898 8.92820i −0.0292793 0.487800i
\(336\) 0 0
\(337\) 6.92820 + 4.00000i 0.377403 + 0.217894i 0.676688 0.736270i \(-0.263415\pi\)
−0.299285 + 0.954164i \(0.596748\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.3923 + 6.00000i 0.557888 + 0.322097i 0.752297 0.658824i \(-0.228946\pi\)
−0.194409 + 0.980921i \(0.562279\pi\)
\(348\) 0 0
\(349\) −7.00000 12.1244i −0.374701 0.649002i 0.615581 0.788074i \(-0.288921\pi\)
−0.990282 + 0.139072i \(0.955588\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.2487 + 14.0000i −1.29063 + 0.745145i −0.978766 0.204982i \(-0.934286\pi\)
−0.311863 + 0.950127i \(0.600953\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.9282 9.85641i 0.781378 0.515908i
\(366\) 0 0
\(367\) 3.46410 2.00000i 0.180825 0.104399i −0.406855 0.913493i \(-0.633375\pi\)
0.587680 + 0.809093i \(0.300041\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −24.0000 41.5692i −1.24602 2.15817i
\(372\) 0 0
\(373\) 20.7846 + 12.0000i 1.07619 + 0.621336i 0.929865 0.367901i \(-0.119923\pi\)
0.146321 + 0.989237i \(0.453257\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.2487 + 14.0000i 1.23905 + 0.715367i 0.968900 0.247451i \(-0.0795931\pi\)
0.270151 + 0.962818i \(0.412926\pi\)
\(384\) 0 0
\(385\) 2.14359 + 35.7128i 0.109248 + 1.82009i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.0000 + 29.4449i 0.861934 + 1.49291i 0.870059 + 0.492947i \(0.164080\pi\)
−0.00812520 + 0.999967i \(0.502586\pi\)
\(390\) 0 0
\(391\) −8.00000 + 13.8564i −0.404577 + 0.700749i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 + 24.0000i −0.603786 + 1.20757i
\(396\) 0 0
\(397\) 8.00000i 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.00000 + 12.1244i −0.349563 + 0.605461i −0.986172 0.165726i \(-0.947003\pi\)
0.636609 + 0.771187i \(0.280337\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −27.7128 + 16.0000i −1.37367 + 0.793091i
\(408\) 0 0
\(409\) 13.0000 22.5167i 0.642809 1.11338i −0.341994 0.939702i \(-0.611102\pi\)
0.984803 0.173675i \(-0.0555643\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.0000i 0.787309i
\(414\) 0 0
\(415\) −8.00000 4.00000i −0.392705 0.196352i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0000 24.2487i 0.683945 1.18463i −0.289822 0.957080i \(-0.593596\pi\)
0.973767 0.227547i \(-0.0730704\pi\)
\(420\) 0 0
\(421\) −5.00000 8.66025i −0.243685 0.422075i 0.718076 0.695965i \(-0.245023\pi\)
−0.961761 + 0.273890i \(0.911690\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.3923 + 7.85641i 0.892158 + 0.381092i
\(426\) 0 0
\(427\) −6.92820 4.00000i −0.335279 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −6.00000 10.3923i −0.286364 0.495998i 0.686575 0.727059i \(-0.259113\pi\)
−0.972939 + 0.231062i \(0.925780\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.1769 18.0000i 1.48126 0.855206i 0.481486 0.876454i \(-0.340097\pi\)
0.999774 + 0.0212481i \(0.00676401\pi\)
\(444\) 0 0
\(445\) 12.3205 + 18.6603i 0.584048 + 0.884581i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 34.6410 20.0000i 1.62044 0.935561i 0.633636 0.773631i \(-0.281562\pi\)
0.986802 0.161929i \(-0.0517716\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 + 5.19615i 0.139724 + 0.242009i 0.927392 0.374091i \(-0.122045\pi\)
−0.787668 + 0.616100i \(0.788712\pi\)
\(462\) 0 0
\(463\) 10.3923 + 6.00000i 0.482971 + 0.278844i 0.721654 0.692254i \(-0.243382\pi\)
−0.238683 + 0.971098i \(0.576716\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.8564 + 8.00000i 0.637118 + 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.00000 + 13.8564i 0.365529 + 0.633115i 0.988861 0.148842i \(-0.0475547\pi\)
−0.623332 + 0.781958i \(0.714221\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.0000 8.00000i −0.726523 0.363261i
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −18.0000 + 31.1769i −0.812329 + 1.40699i 0.0989017 + 0.995097i \(0.468467\pi\)
−0.911230 + 0.411897i \(0.864866\pi\)
\(492\) 0 0
\(493\) 20.7846 12.0000i 0.936092 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0000 20.7846i 0.537194 0.930447i −0.461860 0.886953i \(-0.652818\pi\)
0.999054 0.0434940i \(-0.0138489\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) −2.00000 + 4.00000i −0.0889988 + 0.177998i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.0000 + 36.3731i −0.930809 + 1.61221i −0.148866 + 0.988857i \(0.547562\pi\)
−0.781943 + 0.623350i \(0.785771\pi\)
\(510\) 0 0
\(511\) 16.0000 + 27.7128i 0.707798 + 1.22594i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.535898 + 8.92820i 0.0236145 + 0.393424i
\(516\) 0 0
\(517\) −13.8564 8.00000i −0.609404 0.351840i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.8564 + 8.00000i 0.603595 + 0.348485i
\(528\) 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
\(535\) −22.3923 + 14.7846i −0.968104 + 0.639194i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.46410 3.73205i −0.105551 0.159863i
\(546\) 0 0
\(547\) 24.2487 14.0000i 1.03680 0.598597i 0.117875 0.993028i \(-0.462392\pi\)
0.918925 + 0.394432i \(0.129059\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −41.5692 24.0000i −1.76770 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.3205 10.0000i −0.729972 0.421450i 0.0884397 0.996082i \(-0.471812\pi\)
−0.818412 + 0.574632i \(0.805145\pi\)
\(564\) 0 0
\(565\) 1.60770 + 26.7846i 0.0676362 + 1.12684i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.0000 22.5167i −0.544988 0.943948i −0.998608 0.0527519i \(-0.983201\pi\)
0.453619 0.891196i \(-0.350133\pi\)
\(570\) 0 0
\(571\) −20.0000 + 34.6410i −0.836974 + 1.44968i 0.0554391 + 0.998462i \(0.482344\pi\)
−0.892413 + 0.451219i \(0.850989\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.0000 12.0000i 0.667246 0.500435i
\(576\) 0 0
\(577\) 32.0000i 1.33218i −0.745873 0.666089i \(-0.767967\pi\)
0.745873 0.666089i \(-0.232033\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 13.8564i 0.331896 0.574861i
\(582\) 0 0
\(583\) −41.5692 + 24.0000i −1.72162 + 0.993978i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.1769 18.0000i 1.28681 0.742940i 0.308725 0.951151i \(-0.400098\pi\)
0.978084 + 0.208212i \(0.0667643\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 36.0000i 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) −16.0000 + 32.0000i −0.655936 + 1.31187i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 20.7846i 0.490307 0.849236i −0.509631 0.860393i \(-0.670218\pi\)
0.999938 + 0.0111569i \(0.00355143\pi\)
\(600\) 0 0
\(601\) −19.0000 32.9090i −0.775026 1.34238i −0.934780 0.355228i \(-0.884403\pi\)
0.159754 0.987157i \(-0.448930\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.1603 0.669873i 0.453729 0.0272342i
\(606\) 0 0
\(607\) 24.2487 + 14.0000i 0.984225 + 0.568242i 0.903543 0.428497i \(-0.140957\pi\)
0.0806818 + 0.996740i \(0.474290\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000i 0.323117i 0.986863 + 0.161558i \(0.0516520\pi\)
−0.986863 + 0.161558i \(0.948348\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.1051 22.0000i −1.53405 0.885687i −0.999169 0.0407620i \(-0.987021\pi\)
−0.534885 0.844925i \(-0.679645\pi\)
\(618\) 0 0
\(619\) −8.00000 13.8564i −0.321547 0.556936i 0.659260 0.751915i \(-0.270870\pi\)
−0.980807 + 0.194979i \(0.937536\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −34.6410 + 20.0000i −1.38786 + 0.801283i
\(624\) 0 0
\(625\) −17.2846 18.0622i −0.691384 0.722487i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.46410 4.92820i 0.296204 0.195570i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.00000 + 12.1244i 0.276483 + 0.478883i 0.970508 0.241068i \(-0.0774976\pi\)
−0.694025 + 0.719951i \(0.744164\pi\)
\(642\) 0 0
\(643\) 31.1769 + 18.0000i 1.22950 + 0.709851i 0.966925 0.255062i \(-0.0820957\pi\)
0.262573 + 0.964912i \(0.415429\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.46410 2.00000i −0.135561 0.0782660i 0.430686 0.902502i \(-0.358272\pi\)
−0.566247 + 0.824236i \(0.691605\pi\)
\(654\) 0 0
\(655\) −26.7846 + 1.60770i −1.04656 + 0.0628178i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14.0000 24.2487i −0.545363 0.944596i −0.998584 0.0531977i \(-0.983059\pi\)
0.453221 0.891398i \(-0.350275\pi\)
\(660\) 0 0
\(661\) 11.0000 19.0526i 0.427850 0.741059i −0.568831 0.822454i \(-0.692604\pi\)
0.996682 + 0.0813955i \(0.0259377\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 + 6.92820i −0.154418 + 0.267460i
\(672\) 0 0
\(673\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.46410 + 2.00000i −0.133136 + 0.0768662i −0.565089 0.825030i \(-0.691158\pi\)
0.431953 + 0.901896i \(0.357825\pi\)
\(678\) 0 0
\(679\) 16.0000 27.7128i 0.614024 1.06352i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0000i 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) 0 0
\(685\) 24.0000 + 12.0000i 0.916993 + 0.458496i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 16.0000 + 27.7128i 0.608669 + 1.05425i 0.991460 + 0.130410i \(0.0416295\pi\)
−0.382791 + 0.923835i \(0.625037\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −35.7128 + 2.14359i −1.35466 + 0.0813111i
\(696\) 0 0
\(697\) 34.6410 + 20.0000i 1.31212 + 0.757554i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.92820 4.00000i −0.260562 0.150435i
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.8564 8.00000i 0.518927 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −29.7846 + 3.58846i −1.10617 + 0.133272i
\(726\) 0 0
\(727\) −10.3923 + 6.00000i −0.385429 + 0.222528i −0.680178 0.733047i \(-0.738097\pi\)
0.294749 + 0.955575i \(0.404764\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 + 13.8564i 0.295891 + 0.512498i
\(732\) 0 0
\(733\) −13.8564 8.00000i −0.511798 0.295487i 0.221774 0.975098i \(-0.428815\pi\)
−0.733572 + 0.679611i \(0.762148\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1769 18.0000i −1.14377 0.660356i −0.196409 0.980522i \(-0.562928\pi\)
−0.947361 + 0.320166i \(0.896261\pi\)
\(744\) 0 0
\(745\) −4.46410 + 0.267949i −0.163552 + 0.00981690i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 41.5692i −0.876941 1.51891i
\(750\) 0 0
\(751\) 14.0000 24.2487i 0.510867 0.884848i −0.489053 0.872254i \(-0.662658\pi\)
0.999921 0.0125942i \(-0.00400897\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.0000 40.0000i 0.727875 1.45575i
\(756\) 0 0
\(757\) 16.0000i 0.581530i 0.956795 + 0.290765i \(0.0939098\pi\)
−0.956795 + 0.290765i \(0.906090\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 36.3731i 0.761249 1.31852i −0.180957 0.983491i \(-0.557920\pi\)
0.942207 0.335032i \(-0.108747\pi\)
\(762\) 0 0
\(763\) 6.92820 4.00000i 0.250818 0.144810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −9.00000 + 15.5885i −0.324548 + 0.562134i −0.981421 0.191867i \(-0.938546\pi\)
0.656873 + 0.754002i \(0.271879\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.0000i 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 0 0
\(775\) −12.0000 16.0000i −0.431053 0.574737i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.07180 + 17.8564i 0.0382541 + 0.637322i
\(786\) 0 0
\(787\) −24.2487 14.0000i −0.864373 0.499046i 0.00110111 0.999999i \(-0.499650\pi\)
−0.865474 + 0.500953i \(0.832983\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.2487 + 14.0000i 0.858933 + 0.495905i 0.863655 0.504083i \(-0.168170\pi\)
−0.00472155 + 0.999989i \(0.501503\pi\)
\(798\) 0 0
\(799\) −8.00000 13.8564i −0.283020 0.490204i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 27.7128 16.0000i 0.977964 0.564628i
\(804\) 0 0
\(805\) 19.7128 +