Properties

Label 1620.2.r.c.1189.2
Level $1620$
Weight $2$
Character 1620.1189
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 1189.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1620.1189
Dual form 1620.2.r.c.109.2

$q$-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 \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.23205 1.86603i 0.550990 0.834512i
\(6\) 0 0
\(7\) 3.46410 + 2.00000i 1.30931 + 0.755929i 0.981981 0.188982i \(-0.0605189\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 3.46410i 0.603023 1.04447i −0.389338 0.921095i \(-0.627296\pi\)
0.992361 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 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 \(-0.529738\pi\)
0.815604 + 0.578610i \(0.196405\pi\)
\(24\) 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 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 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 \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 + 8.66025i 0.780869 + 1.35250i 0.931436 + 0.363905i \(0.118557\pi\)
−0.150567 + 0.988600i \(0.548110\pi\)
\(42\) 0 0
\(43\) −3.46410 2.00000i −0.528271 0.304997i 0.212041 0.977261i \(-0.431989\pi\)
−0.740312 + 0.672264i \(0.765322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.46410 2.00000i −0.505291 0.291730i 0.225605 0.974219i \(-0.427564\pi\)
−0.730896 + 0.682489i \(0.760898\pi\)
\(48\) 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 \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410 2.00000i 0.423207 0.244339i −0.273241 0.961946i \(-0.588096\pi\)
0.696449 + 0.717607i \(0.254762\pi\)
\(68\) 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 \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(74\) 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 0.301401 + 0.953498i \(0.402546\pi\)
−0.976453 + 0.215728i \(0.930788\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.46410 2.00000i −0.380235 0.219529i 0.297686 0.954664i \(-0.403785\pi\)
−0.677920 + 0.735135i \(0.737119\pi\)
\(84\) 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 \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 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 0.808632 0.588315i \(-0.200208\pi\)
−0.105180 + 0.994453i \(0.533542\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.00000 1.73205i 0.0995037 0.172345i −0.811976 0.583691i \(-0.801608\pi\)
0.911479 + 0.411346i \(0.134941\pi\)
\(102\) 0 0
\(103\) −3.46410 + 2.00000i −0.341328 + 0.197066i −0.660859 0.750510i \(-0.729808\pi\)
0.319531 + 0.947576i \(0.396475\pi\)
\(104\) 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 \(-0.475762\pi\)
0.901553 + 0.432670i \(0.142428\pi\)
\(114\) 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 \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 10.3923i −0.524222 0.907980i −0.999602 0.0281993i \(-0.991023\pi\)
0.475380 0.879781i \(-0.342311\pi\)
\(132\) 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 \(-0.162010\pi\)
0.0146279 + 0.999893i \(0.495344\pi\)
\(138\) 0 0
\(139\) 8.00000 + 13.8564i 0.678551 + 1.17529i 0.975417 + 0.220366i \(0.0707252\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 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 \(-0.192773\pi\)
−0.904076 + 0.427372i \(0.859440\pi\)
\(150\) 0 0
\(151\) 10.0000 17.3205i 0.813788 1.40952i −0.0964061 0.995342i \(-0.530735\pi\)
0.910195 0.414181i \(-0.135932\pi\)
\(152\) 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 −0.750303 0.661094i \(-0.770093\pi\)
0.197372 + 0.980329i \(0.436759\pi\)
\(158\) 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 \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.3923 + 6.00000i −0.804181 + 0.464294i −0.844931 0.534875i \(-0.820359\pi\)
0.0407502 + 0.999169i \(0.487025\pi\)
\(168\) 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 \(-0.284744\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(174\) 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 \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 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 −0.00454614 + 0.999990i \(0.501447\pi\)
−0.863743 + 0.503932i \(0.831886\pi\)
\(192\) 0 0
\(193\) 13.8564 8.00000i 0.997406 0.575853i 0.0899262 0.995948i \(-0.471337\pi\)
0.907480 + 0.420096i \(0.138004\pi\)
\(194\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(212\) 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 0.951272 0.308353i \(-0.0997777\pi\)
0.208595 + 0.978002i \(0.433111\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.3205 10.0000i −1.14960 0.663723i −0.200812 0.979630i \(-0.564358\pi\)
−0.948790 + 0.315906i \(0.897691\pi\)
\(228\) 0 0
\(229\) 13.0000 + 22.5167i 0.859064 + 1.48794i 0.872823 + 0.488037i \(0.162287\pi\)
−0.0137585 + 0.999905i \(0.504380\pi\)
\(230\) 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 \(-0.0833599\pi\)
−0.707166 + 0.707048i \(0.750027\pi\)
\(240\) 0 0
\(241\) 1.00000 1.73205i 0.0644157 0.111571i −0.832019 0.554747i \(-0.812815\pi\)
0.896435 + 0.443176i \(0.146148\pi\)
\(242\) 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 \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 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 \(-0.788773\pi\)
0.139533 + 0.990217i \(0.455440\pi\)
\(258\) 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 \(-0.372689\pi\)
−0.602986 + 0.797752i \(0.706023\pi\)
\(264\) 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 \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 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 −0.970221 0.242222i \(-0.922124\pi\)
−0.694881 0.719125i \(-0.744543\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.00000 + 5.19615i −0.178965 + 0.309976i −0.941526 0.336939i \(-0.890608\pi\)
0.762561 + 0.646916i \(0.223942\pi\)
\(282\) 0 0
\(283\) −24.2487 + 14.0000i −1.44144 + 0.832214i −0.997946 0.0640654i \(-0.979593\pi\)
−0.443491 + 0.896279i \(0.646260\pi\)
\(284\) 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 \(-0.552670\pi\)
0.771839 + 0.635818i \(0.219337\pi\)
\(294\) 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 \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −13.8564 8.00000i −0.783210 0.452187i 0.0543564 0.998522i \(-0.482689\pi\)
−0.837567 + 0.546335i \(0.816023\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.46410 + 2.00000i 0.194563 + 0.112331i 0.594117 0.804379i \(-0.297502\pi\)
−0.399554 + 0.916710i \(0.630835\pi\)
\(318\) 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 −0.734905 0.678170i \(-0.762773\pi\)
0.954765 + 0.297361i \(0.0961066\pi\)
\(332\) 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 −0.299285 0.954164i \(-0.596748\pi\)
0.676688 + 0.736270i \(0.263415\pi\)
\(338\) 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 \(-0.771054\pi\)
0.194409 + 0.980921i \(0.437721\pi\)
\(348\) 0 0
\(349\) −7.00000 + 12.1244i −0.374701 + 0.649002i −0.990282 0.139072i \(-0.955588\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.2487 + 14.0000i 1.29063 + 0.745145i 0.978766 0.204982i \(-0.0657137\pi\)
0.311863 + 0.950127i \(0.399047\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 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 0.587680 0.809093i \(-0.300041\pi\)
−0.406855 + 0.913493i \(0.633375\pi\)
\(368\) 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 0.146321 0.989237i \(-0.453257\pi\)
0.929865 + 0.367901i \(0.119923\pi\)
\(374\) 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 \(-0.920407\pi\)
−0.270151 + 0.962818i \(0.587074\pi\)
\(384\) 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 0.00812520 + 0.999967i \(0.497414\pi\)
−0.870059 + 0.492947i \(0.835920\pi\)
\(390\) 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 \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.00000 + 12.1244i 0.349563 + 0.605461i 0.986172 0.165726i \(-0.0529966\pi\)
−0.636609 + 0.771187i \(0.719663\pi\)
\(402\) 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 0.984803 + 0.173675i \(0.0555643\pi\)
−0.341994 + 0.939702i \(0.611102\pi\)
\(410\) 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 −0.973767 0.227547i \(-0.926930\pi\)
0.289822 0.957080i \(-0.406404\pi\)
\(420\) 0 0
\(421\) −5.00000 + 8.66025i −0.243685 + 0.422075i −0.961761 0.273890i \(-0.911690\pi\)
0.718076 + 0.695965i \(0.245023\pi\)
\(422\) 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 \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −6.00000 + 10.3923i −0.286364 + 0.495998i −0.972939 0.231062i \(-0.925780\pi\)
0.686575 + 0.727059i \(0.259113\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.1769 18.0000i −1.48126 0.855206i −0.481486 0.876454i \(-0.659903\pi\)
−0.999774 + 0.0212481i \(0.993236\pi\)
\(444\) 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 \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 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 0.986802 + 0.161929i \(0.0517716\pi\)
0.633636 + 0.773631i \(0.281562\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 + 5.19615i −0.139724 + 0.242009i −0.927392 0.374091i \(-0.877955\pi\)
0.787668 + 0.616100i \(0.211288\pi\)
\(462\) 0 0
\(463\) 10.3923 6.00000i 0.482971 0.278844i −0.238683 0.971098i \(-0.576716\pi\)
0.721654 + 0.692254i \(0.243382\pi\)
\(464\) 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 \(-0.952445\pi\)
0.623332 + 0.781958i \(0.285779\pi\)
\(480\) 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 \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 18.0000 + 31.1769i 0.812329 + 1.40699i 0.911230 + 0.411897i \(0.135134\pi\)
−0.0989017 + 0.995097i \(0.531533\pi\)
\(492\) 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 0.999054 + 0.0434940i \(0.0138489\pi\)
−0.461860 + 0.886953i \(0.652818\pi\)
\(500\) 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 0.781943 + 0.623350i \(0.214229\pi\)
0.148866 + 0.988857i \(0.452438\pi\)
\(510\) 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 \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 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 0.918925 0.394432i \(-0.129059\pi\)
0.117875 + 0.993028i \(0.462392\pi\)
\(548\) 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 \(-0.528188\pi\)
0.818412 + 0.574632i \(0.194855\pi\)
\(564\) 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 −0.453619 0.891196i \(-0.649867\pi\)
0.998608 0.0527519i \(-0.0167993\pi\)
\(570\) 0 0
\(571\) −20.0000 34.6410i −0.836974 1.44968i −0.892413 0.451219i \(-0.850989\pi\)
0.0554391 0.998462i \(-0.482344\pi\)
\(572\) 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 \(0.232033\pi\)
−0.745873 + 0.666089i \(0.767967\pi\)
\(578\) 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 \(-0.599902\pi\)
−0.978084 + 0.208212i \(0.933236\pi\)
\(588\) 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 \(-0.329782\pi\)
−0.999938 + 0.0111569i \(0.996449\pi\)
\(600\) 0 0
\(601\) −19.0000 + 32.9090i −0.775026 + 1.34238i 0.159754 + 0.987157i \(0.448930\pi\)
−0.934780 + 0.355228i \(0.884403\pi\)
\(602\) 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 0.0806818 0.996740i \(-0.474290\pi\)
0.903543 + 0.428497i \(0.140957\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.00000i 0.323117i −0.986863 0.161558i \(-0.948348\pi\)
0.986863 0.161558i \(-0.0516520\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.1051 22.0000i 1.53405 0.885687i 0.534885 0.844925i \(-0.320355\pi\)
0.999169 0.0407620i \(-0.0129786\pi\)
\(618\) 0 0
\(619\) −8.00000 + 13.8564i −0.321547 + 0.556936i −0.980807 0.194979i \(-0.937536\pi\)
0.659260 + 0.751915i \(0.270870\pi\)
\(620\) 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 \(-0.922502\pi\)
0.694025 + 0.719951i \(0.255836\pi\)
\(642\) 0 0
\(643\) 31.1769 18.0000i 1.22950 0.709851i 0.262573 0.964912i \(-0.415429\pi\)
0.966925 + 0.255062i \(0.0820957\pi\)
\(644\) 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 \(-0.641728\pi\)
0.566247 + 0.824236i \(0.308395\pi\)
\(654\) 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 −0.453221 0.891398i \(-0.649725\pi\)
0.998584 0.0531977i \(-0.0169414\pi\)
\(660\) 0 0
\(661\) 11.0000 + 19.0526i 0.427850 + 0.741059i 0.996682 0.0813955i \(-0.0259377\pi\)
−0.568831 + 0.822454i \(0.692604\pi\)
\(662\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.46410 + 2.00000i 0.133136 + 0.0768662i 0.565089 0.825030i \(-0.308842\pi\)
−0.431953 + 0.901896i \(0.642175\pi\)
\(678\) 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 −0.382791 0.923835i \(-0.625037\pi\)
0.991460 0.130410i \(-0.0416295\pi\)
\(692\) 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 \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 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 −0.916845 0.399244i \(-0.869273\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(710\) 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 \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 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 0.294749 0.955575i \(-0.404764\pi\)
−0.680178 + 0.733047i \(0.738097\pi\)
\(728\) 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 −0.733572 0.679611i \(-0.762148\pi\)
0.221774 + 0.975098i \(0.428815\pi\)
\(734\) 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 \(-0.437072\pi\)
0.947361 + 0.320166i \(0.103739\pi\)
\(744\) 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 0.999921 + 0.0125942i \(0.00400897\pi\)
−0.489053 + 0.872254i \(0.662658\pi\)
\(752\) 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 \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.0000 36.3731i −0.761249 1.31852i −0.942207 0.335032i \(-0.891253\pi\)
0.180957 0.983491i \(-0.442080\pi\)
\(762\) 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 0.656873 0.754002i \(-0.271879\pi\)
−0.981421 + 0.191867i \(0.938546\pi\)
\(770\) 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 −0.865474 0.500953i \(-0.832983\pi\)
0.00110111 + 0.999999i \(0.499650\pi\)
\(788\) 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 \(-0.831830\pi\)
0.00472155 + 0.999989i \(0.498497\pi\)
\(798\) 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