Properties

Label 1764.2.k.b.361.1
Level $1764$
Weight $2$
Character 1764.361
Analytic conductor $14.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.2.k.b.1549.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{5} +O(q^{10})\) \(q+(-1.50000 - 2.59808i) q^{5} +(-1.50000 + 2.59808i) q^{11} -2.00000 q^{13} +(-1.50000 + 2.59808i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +6.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(0.500000 + 0.866025i) q^{37} +6.00000 q^{41} -4.00000 q^{43} +(4.50000 + 7.79423i) q^{47} +(1.50000 - 2.59808i) q^{53} +9.00000 q^{55} +(-4.50000 + 7.79423i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(3.00000 + 5.19615i) q^{65} +(3.50000 - 6.06218i) q^{67} +(-0.500000 + 0.866025i) q^{73} +(6.50000 + 11.2583i) q^{79} +12.0000 q^{83} +9.00000 q^{85} +(-7.50000 - 12.9904i) q^{89} +(-1.50000 + 2.59808i) q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + O(q^{10}) \) \( 2q - 3q^{5} - 3q^{11} - 4q^{13} - 3q^{17} - q^{19} + 3q^{23} - 4q^{25} + 12q^{29} - 7q^{31} + q^{37} + 12q^{41} - 8q^{43} + 9q^{47} + 3q^{53} + 18q^{55} - 9q^{59} - q^{61} + 6q^{65} + 7q^{67} - q^{73} + 13q^{79} + 24q^{83} + 18q^{85} - 15q^{89} - 3q^{95} + 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −1.50000 2.59808i −0.670820 1.16190i −0.977672 0.210138i \(-0.932609\pi\)
0.306851 0.951757i \(-0.400725\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) −0.500000 0.866025i −0.114708 0.198680i 0.802955 0.596040i \(-0.203260\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 + 2.59808i 0.312772 + 0.541736i 0.978961 0.204046i \(-0.0654092\pi\)
−0.666190 + 0.745782i \(0.732076\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.50000 + 7.79423i 0.656392 + 1.13691i 0.981543 + 0.191243i \(0.0612518\pi\)
−0.325150 + 0.945662i \(0.605415\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 2.59808i 0.206041 0.356873i −0.744423 0.667708i \(-0.767275\pi\)
0.950464 + 0.310835i \(0.100609\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.50000 + 7.79423i −0.585850 + 1.01472i 0.408919 + 0.912571i \(0.365906\pi\)
−0.994769 + 0.102151i \(0.967427\pi\)
\(60\) 0 0
\(61\) −0.500000 0.866025i −0.0640184 0.110883i 0.832240 0.554416i \(-0.187058\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −0.500000 + 0.866025i −0.0585206 + 0.101361i −0.893801 0.448463i \(-0.851972\pi\)
0.835281 + 0.549823i \(0.185305\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 + 11.2583i 0.731307 + 1.26666i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.50000 12.9904i −0.794998 1.37698i −0.922840 0.385183i \(-0.874138\pi\)
0.127842 0.991795i \(-0.459195\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.50000 + 2.59808i −0.153897 + 0.266557i
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.50000 + 12.9904i −0.746278 + 1.29259i 0.203317 + 0.979113i \(0.434828\pi\)
−0.949595 + 0.313478i \(0.898506\pi\)
\(102\) 0 0
\(103\) 5.50000 + 9.52628i 0.541931 + 0.938652i 0.998793 + 0.0491146i \(0.0156400\pi\)
−0.456862 + 0.889538i \(0.651027\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.50000 + 12.9904i 0.725052 + 1.25583i 0.958952 + 0.283567i \(0.0915178\pi\)
−0.233900 + 0.972261i \(0.575149\pi\)
\(108\) 0 0
\(109\) 0.500000 0.866025i 0.0478913 0.0829502i −0.841086 0.540901i \(-0.818083\pi\)
0.888977 + 0.457951i \(0.151417\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 4.50000 7.79423i 0.419627 0.726816i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.50000 2.59808i −0.131056 0.226995i 0.793028 0.609185i \(-0.208503\pi\)
−0.924084 + 0.382190i \(0.875170\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.5000 + 18.1865i −0.897076 + 1.55378i −0.0658609 + 0.997829i \(0.520979\pi\)
−0.831215 + 0.555952i \(0.812354\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.00000 5.19615i 0.250873 0.434524i
\(144\) 0 0
\(145\) −9.00000 15.5885i −0.747409 1.29455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.50000 + 2.59808i 0.122885 + 0.212843i 0.920904 0.389789i \(-0.127452\pi\)
−0.798019 + 0.602632i \(0.794119\pi\)
\(150\) 0 0
\(151\) −8.50000 + 14.7224i −0.691720 + 1.19809i 0.279554 + 0.960130i \(0.409814\pi\)
−0.971274 + 0.237964i \(0.923520\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 21.0000 1.68676
\(156\) 0 0
\(157\) −6.50000 + 11.2583i −0.518756 + 0.898513i 0.481006 + 0.876717i \(0.340272\pi\)
−0.999762 + 0.0217953i \(0.993062\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.50000 9.52628i −0.430793 0.746156i 0.566149 0.824303i \(-0.308433\pi\)
−0.996942 + 0.0781474i \(0.975100\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.50000 + 7.79423i 0.342129 + 0.592584i 0.984828 0.173534i \(-0.0555188\pi\)
−0.642699 + 0.766119i \(0.722185\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.5000 18.1865i 0.784807 1.35933i −0.144308 0.989533i \(-0.546095\pi\)
0.929114 0.369792i \(-0.120571\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.50000 2.59808i 0.110282 0.191014i
\(186\) 0 0
\(187\) −4.50000 7.79423i −0.329073 0.569970i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.50000 7.79423i −0.325609 0.563971i 0.656027 0.754738i \(-0.272236\pi\)
−0.981635 + 0.190767i \(0.938902\pi\)
\(192\) 0 0
\(193\) −5.50000 + 9.52628i −0.395899 + 0.685717i −0.993215 0.116289i \(-0.962900\pi\)
0.597317 + 0.802005i \(0.296234\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −3.50000 + 6.06218i −0.248108 + 0.429736i −0.963001 0.269498i \(-0.913142\pi\)
0.714893 + 0.699234i \(0.246476\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 15.5885i −0.628587 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.00000 + 10.3923i 0.409197 + 0.708749i
\(216\) 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
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.50000 2.59808i 0.0995585 0.172440i −0.811943 0.583736i \(-0.801590\pi\)
0.911502 + 0.411296i \(0.134924\pi\)
\(228\) 0 0
\(229\) 5.50000 + 9.52628i 0.363450 + 0.629514i 0.988526 0.151050i \(-0.0482653\pi\)
−0.625076 + 0.780564i \(0.714932\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.5000 18.1865i −0.687878 1.19144i −0.972523 0.232806i \(-0.925209\pi\)
0.284645 0.958633i \(-0.408124\pi\)
\(234\) 0 0
\(235\) 13.5000 23.3827i 0.880643 1.52532i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −0.500000 + 0.866025i −0.0322078 + 0.0557856i −0.881680 0.471848i \(-0.843587\pi\)
0.849472 + 0.527633i \(0.176921\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 + 1.73205i 0.0636285 + 0.110208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.50000 2.59808i −0.0935674 0.162064i 0.815442 0.578838i \(-0.196494\pi\)
−0.909010 + 0.416775i \(0.863160\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.50000 + 2.59808i −0.0924940 + 0.160204i −0.908560 0.417755i \(-0.862817\pi\)
0.816066 + 0.577959i \(0.196151\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.50000 + 2.59808i −0.0914566 + 0.158408i −0.908124 0.418701i \(-0.862486\pi\)
0.816668 + 0.577108i \(0.195819\pi\)
\(270\) 0 0
\(271\) 5.50000 + 9.52628i 0.334101 + 0.578680i 0.983312 0.181928i \(-0.0582339\pi\)
−0.649211 + 0.760609i \(0.724901\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 10.3923i −0.361814 0.626680i
\(276\) 0 0
\(277\) 6.50000 11.2583i 0.390547 0.676448i −0.601975 0.798515i \(-0.705619\pi\)
0.992522 + 0.122068i \(0.0389525\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 14.5000 25.1147i 0.861936 1.49292i −0.00812260 0.999967i \(-0.502586\pi\)
0.870058 0.492949i \(-0.164081\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 27.0000 1.57200
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 5.19615i −0.173494 0.300501i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.50000 + 2.59808i −0.0858898 + 0.148765i
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.5000 23.3827i 0.765515 1.32591i −0.174459 0.984664i \(-0.555818\pi\)
0.939974 0.341246i \(-0.110849\pi\)
\(312\) 0 0
\(313\) 11.5000 + 19.9186i 0.650018 + 1.12586i 0.983118 + 0.182973i \(0.0585722\pi\)
−0.333099 + 0.942892i \(0.608094\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.50000 7.79423i −0.252745 0.437767i 0.711535 0.702650i \(-0.248000\pi\)
−0.964281 + 0.264883i \(0.914667\pi\)
\(318\) 0 0
\(319\) −9.00000 + 15.5885i −0.503903 + 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 4.00000 6.92820i 0.221880 0.384308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.50000 + 11.2583i 0.357272 + 0.618814i 0.987504 0.157593i \(-0.0503735\pi\)
−0.630232 + 0.776407i \(0.717040\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −21.0000 −1.14735
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.5000 18.1865i −0.568607 0.984856i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.50000 7.79423i 0.241573 0.418416i −0.719590 0.694399i \(-0.755670\pi\)
0.961162 + 0.275983i \(0.0890035\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.5000 18.1865i 0.558859 0.967972i −0.438733 0.898617i \(-0.644573\pi\)
0.997592 0.0693543i \(-0.0220939\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.50000 + 12.9904i 0.395835 + 0.685606i 0.993207 0.116358i \(-0.0371219\pi\)
−0.597372 + 0.801964i \(0.703789\pi\)
\(360\) 0 0
\(361\) 9.00000 15.5885i 0.473684 0.820445i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) 2.50000 4.33013i 0.130499 0.226031i −0.793370 0.608740i \(-0.791675\pi\)
0.923869 + 0.382709i \(0.125009\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.5000 + 21.6506i 0.647225 + 1.12103i 0.983783 + 0.179364i \(0.0574041\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.5000 + 28.5788i 0.843111 + 1.46031i 0.887252 + 0.461285i \(0.152611\pi\)
−0.0441413 + 0.999025i \(0.514055\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.50000 12.9904i 0.380265 0.658638i −0.610835 0.791758i \(-0.709166\pi\)
0.991100 + 0.133120i \(0.0424994\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 19.5000 33.7750i 0.981151 1.69940i
\(396\) 0 0
\(397\) −18.5000 32.0429i −0.928488 1.60819i −0.785853 0.618414i \(-0.787776\pi\)
−0.142636 0.989775i \(-0.545558\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 + 2.59808i 0.0749064 + 0.129742i 0.901046 0.433724i \(-0.142801\pi\)
−0.826139 + 0.563466i \(0.809468\pi\)
\(402\) 0 0
\(403\) 7.00000 12.1244i 0.348695 0.603957i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) 5.50000 9.52628i 0.271957 0.471044i −0.697406 0.716677i \(-0.745662\pi\)
0.969363 + 0.245633i \(0.0789957\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −18.0000 31.1769i −0.883585 1.53041i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 10.3923i −0.291043 0.504101i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.50000 + 12.9904i −0.361262 + 0.625725i −0.988169 0.153370i \(-0.950987\pi\)
0.626907 + 0.779094i \(0.284321\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.50000 2.59808i 0.0717547 0.124283i
\(438\) 0 0
\(439\) −0.500000 0.866025i −0.0238637 0.0413331i 0.853847 0.520524i \(-0.174263\pi\)
−0.877711 + 0.479191i \(0.840930\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.50000 7.79423i −0.213801 0.370315i 0.739100 0.673596i \(-0.235251\pi\)
−0.952901 + 0.303281i \(0.901918\pi\)
\(444\) 0 0
\(445\) −22.5000 + 38.9711i −1.06660 + 1.84741i
\(446\) 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
\(451\) −9.00000 + 15.5885i −0.423793 + 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.5000 19.9186i −0.537947 0.931752i −0.999014 0.0443868i \(-0.985867\pi\)
0.461067 0.887365i \(-0.347467\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.5000 + 18.1865i 0.485882 + 0.841572i 0.999868 0.0162260i \(-0.00516512\pi\)
−0.513986 + 0.857798i \(0.671832\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 10.3923i 0.275880 0.477839i
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.50000 2.59808i 0.0685367 0.118709i −0.829721 0.558179i \(-0.811500\pi\)
0.898257 + 0.439470i \(0.144834\pi\)
\(480\) 0 0
\(481\) −1.00000 1.73205i −0.0455961 0.0789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.0000 25.9808i −0.681115 1.17973i
\(486\) 0 0
\(487\) 9.50000 16.4545i 0.430486 0.745624i −0.566429 0.824110i \(-0.691675\pi\)
0.996915 + 0.0784867i \(0.0250088\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) −9.00000 + 15.5885i −0.405340 + 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −5.50000 9.52628i −0.246214 0.426455i 0.716258 0.697835i \(-0.245853\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.50000 2.59808i −0.0664863 0.115158i 0.830866 0.556473i \(-0.187846\pi\)
−0.897352 + 0.441315i \(0.854512\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.5000 28.5788i 0.727077 1.25933i
\(516\) 0 0
\(517\) −27.0000 −1.18746
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −19.5000 + 33.7750i −0.854311 + 1.47971i 0.0229727 + 0.999736i \(0.492687\pi\)
−0.877283 + 0.479973i \(0.840646\pi\)
\(522\) 0 0
\(523\) −0.500000 0.866025i −0.0218635 0.0378686i 0.854887 0.518815i \(-0.173627\pi\)
−0.876750 + 0.480946i \(0.840293\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.5000 18.1865i −0.457387 0.792218i
\(528\) 0 0
\(529\) 7.00000 12.1244i 0.304348 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 22.5000 38.9711i 0.972760 1.68487i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.5000 30.3109i −0.752384 1.30317i −0.946664 0.322221i \(-0.895571\pi\)
0.194281 0.980946i \(-0.437763\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.00000 5.19615i −0.127804 0.221364i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.5000 + 28.5788i −0.699127 + 1.21092i 0.269642 + 0.962961i \(0.413095\pi\)
−0.968769 + 0.247964i \(0.920239\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.50000 + 7.79423i −0.189652 + 0.328488i −0.945134 0.326682i \(-0.894069\pi\)
0.755482 + 0.655169i \(0.227403\pi\)
\(564\) 0 0
\(565\) 9.00000 + 15.5885i 0.378633 + 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.50000 7.79423i −0.188650 0.326751i 0.756151 0.654398i \(-0.227078\pi\)
−0.944800 + 0.327647i \(0.893744\pi\)
\(570\) 0 0
\(571\) −14.5000 + 25.1147i −0.606806 + 1.05102i 0.384957 + 0.922934i \(0.374216\pi\)
−0.991763 + 0.128085i \(0.959117\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −0.500000 + 0.866025i −0.0208153 + 0.0360531i −0.876245 0.481865i \(-0.839960\pi\)
0.855430 + 0.517918i \(0.173293\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.50000 + 7.79423i 0.186371 + 0.322804i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 7.00000 0.288430
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.5000 + 18.1865i 0.431183 + 0.746831i 0.996976 0.0777165i \(-0.0247629\pi\)
−0.565792 + 0.824548i \(0.691430\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −13.5000 + 23.3827i −0.551595 + 0.955391i 0.446565 + 0.894751i \(0.352647\pi\)
−0.998160 + 0.0606393i \(0.980686\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.00000 5.19615i 0.121967 0.211254i
\(606\) 0 0
\(607\) 23.5000 + 40.7032i 0.953836 + 1.65209i 0.737011 + 0.675881i \(0.236237\pi\)
0.216825 + 0.976210i \(0.430430\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 15.5885i −0.364101 0.630641i
\(612\) 0 0
\(613\) 12.5000 21.6506i 0.504870 0.874461i −0.495114 0.868828i \(-0.664874\pi\)
0.999984 0.00563283i \(-0.00179300\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −15.5000 + 26.8468i −0.622998 + 1.07906i 0.365927 + 0.930644i \(0.380752\pi\)
−0.988924 + 0.148420i \(0.952581\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.0000 20.7846i −0.476205 0.824812i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.50000 12.9904i 0.296232 0.513089i −0.679039 0.734103i \(-0.737603\pi\)
0.975271 + 0.221013i \(0.0709364\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.5000 + 18.1865i −0.412798 + 0.714986i −0.995194 0.0979182i \(-0.968782\pi\)
0.582397 + 0.812905i \(0.302115\pi\)
\(648\) 0 0
\(649\) −13.5000 23.3827i −0.529921 0.917851i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.5000 + 33.7750i 0.763094 + 1.32172i 0.941248 + 0.337715i \(0.109654\pi\)
−0.178154 + 0.984003i \(0.557013\pi\)
\(654\) 0 0
\(655\) −4.50000 + 7.79423i −0.175830 + 0.304546i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 5.50000 9.52628i 0.213925 0.370529i −0.739014 0.673690i \(-0.764708\pi\)
0.952940 + 0.303160i \(0.0980418\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000 + 15.5885i 0.348481 + 0.603587i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.5000 23.3827i −0.518847 0.898670i −0.999760 0.0219013i \(-0.993028\pi\)
0.480913 0.876768i \(-0.340305\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.5000 18.1865i 0.401771 0.695888i −0.592168 0.805814i \(-0.701728\pi\)
0.993940 + 0.109926i \(0.0350613\pi\)
\(684\) 0 0
\(685\) 63.0000 2.40711
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.00000 + 5.19615i −0.114291 + 0.197958i
\(690\) 0 0
\(691\) −6.50000 11.2583i −0.247272 0.428287i 0.715496 0.698617i \(-0.246201\pi\)
−0.962768 + 0.270330i \(0.912867\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 30.0000 + 51.9615i 1.13796 + 1.97101i
\(696\) 0 0
\(697\) −9.00000 + 15.5885i −0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 0.500000 0.866025i 0.0188579 0.0326628i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.500000 + 0.866025i 0.0187779 + 0.0325243i 0.875262 0.483650i \(-0.160689\pi\)
−0.856484 + 0.516174i \(0.827356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −21.0000 −0.786456
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 10.5000 + 18.1865i 0.391584 + 0.678243i 0.992659 0.120950i \(-0.0385939\pi\)
−0.601075 + 0.799193i \(0.705261\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.0000 + 20.7846i −0.445669 + 0.771921i
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.00000 10.3923i 0.221918 0.384373i
\(732\) 0 0
\(733\) −12.5000 21.6506i −0.461698 0.799684i 0.537348 0.843361i \(-0.319426\pi\)
−0.999046 + 0.0436764i \(0.986093\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.5000 + 18.1865i 0.386772 + 0.669910i
\(738\) 0 0
\(739\) 9.50000 16.4545i 0.349463 0.605288i −0.636691 0.771119i \(-0.719697\pi\)
0.986154 + 0.165831i \(0.0530307\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 4.50000 7.79423i 0.164867 0.285558i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.5000 + 21.6506i 0.456131 + 0.790043i 0.998752 0.0499348i \(-0.0159013\pi\)
−0.542621 + 0.839978i \(0.682568\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.50000 2.59808i −0.0543750 0.0941802i 0.837557 0.546350i \(-0.183983\pi\)
−0.891932 + 0.452170i \(0.850650\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.00000 15.5885i 0.324971 0.562867i
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16.5000 28.5788i 0.593464 1.02791i −0.400298 0.916385i \(-0.631093\pi\)
0.993762 0.111524i \(-0.0355733\pi\)
\(774\) 0 0
\(775\) −14.0000 24.2487i −0.502895 0.871039i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.00000 5.19615i −0.107486 0.186171i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 39.0000 1.39197
\(786\) 0 0
\(787\) −15.5000 + 26.8468i −0.552515 + 0.956985i 0.445577 + 0.895244i \(0.352999\pi\)
−0.998092 + 0.0617409i \(0.980335\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.00000 + 1.73205i 0.0355110 + 0.0615069i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.50000 2.59808i −0.0529339 0.0916841i
\(804\) 0 0
\(805\) 0 0