Properties

Label 1764.2.k.l.361.2
Level $1764$
Weight $2$
Character 1764.361
Analytic conductor $14.086$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.2.k.l.1549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 + 1.22474i) q^{5} +O(q^{10})\) \(q+(0.707107 + 1.22474i) q^{5} +(2.00000 - 3.46410i) q^{11} +4.24264 q^{13} +(0.707107 - 1.22474i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(1.50000 - 2.59808i) q^{25} -8.00000 q^{29} +(4.00000 + 6.92820i) q^{37} +7.07107 q^{41} -4.00000 q^{43} +(2.82843 + 4.89898i) q^{47} +(5.00000 - 8.66025i) q^{53} +5.65685 q^{55} +(7.07107 - 12.2474i) q^{59} +(3.53553 + 6.12372i) q^{61} +(3.00000 + 5.19615i) q^{65} +(3.53553 - 6.12372i) q^{73} +(-4.00000 - 6.92820i) q^{79} +14.1421 q^{83} +2.00000 q^{85} +(3.53553 + 6.12372i) q^{89} +(2.00000 - 3.46410i) q^{95} -1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{11} - 8 q^{23} + 6 q^{25} - 32 q^{29} + 16 q^{37} - 16 q^{43} + 20 q^{53} + 12 q^{65} - 16 q^{79} + 8 q^{85} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 0.707107 + 1.22474i 0.316228 + 0.547723i 0.979698 0.200480i \(-0.0642503\pi\)
−0.663470 + 0.748203i \(0.730917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 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
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.707107 1.22474i 0.171499 0.297044i −0.767445 0.641114i \(-0.778472\pi\)
0.938944 + 0.344070i \(0.111806\pi\)
\(18\) 0 0
\(19\) −1.41421 2.44949i −0.324443 0.561951i 0.656957 0.753928i \(-0.271843\pi\)
−0.981399 + 0.191977i \(0.938510\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 3.46410i −0.417029 0.722315i 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) 1.50000 2.59808i 0.300000 0.519615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 + 6.92820i 0.657596 + 1.13899i 0.981236 + 0.192809i \(0.0617599\pi\)
−0.323640 + 0.946180i \(0.604907\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.07107 1.10432 0.552158 0.833740i \(-0.313805\pi\)
0.552158 + 0.833740i \(0.313805\pi\)
\(42\) 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
\(47\) 2.82843 + 4.89898i 0.412568 + 0.714590i 0.995170 0.0981685i \(-0.0312984\pi\)
−0.582601 + 0.812758i \(0.697965\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.00000 8.66025i 0.686803 1.18958i −0.286064 0.958211i \(-0.592347\pi\)
0.972867 0.231367i \(-0.0743197\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.07107 12.2474i 0.920575 1.59448i 0.122047 0.992524i \(-0.461054\pi\)
0.798528 0.601958i \(-0.205612\pi\)
\(60\) 0 0
\(61\) 3.53553 + 6.12372i 0.452679 + 0.784063i 0.998551 0.0538056i \(-0.0171351\pi\)
−0.545873 + 0.837868i \(0.683802\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.53553 6.12372i 0.413803 0.716728i −0.581499 0.813547i \(-0.697534\pi\)
0.995302 + 0.0968194i \(0.0308669\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 6.92820i −0.450035 0.779484i 0.548352 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.1421 1.55230 0.776151 0.630548i \(-0.217170\pi\)
0.776151 + 0.630548i \(0.217170\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.53553 + 6.12372i 0.374766 + 0.649113i 0.990292 0.139003i \(-0.0443898\pi\)
−0.615526 + 0.788116i \(0.711056\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 3.46410i 0.205196 0.355409i
\(96\) 0 0
\(97\) −1.41421 −0.143592 −0.0717958 0.997419i \(-0.522873\pi\)
−0.0717958 + 0.997419i \(0.522873\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.36396 + 11.0227i −0.633238 + 1.09680i 0.353648 + 0.935379i \(0.384941\pi\)
−0.986886 + 0.161421i \(0.948392\pi\)
\(102\) 0 0
\(103\) 5.65685 + 9.79796i 0.557386 + 0.965422i 0.997714 + 0.0675842i \(0.0215291\pi\)
−0.440327 + 0.897837i \(0.645138\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 + 6.92820i 0.386695 + 0.669775i 0.992003 0.126217i \(-0.0402834\pi\)
−0.605308 + 0.795991i \(0.706950\pi\)
\(108\) 0 0
\(109\) 4.00000 6.92820i 0.383131 0.663602i −0.608377 0.793648i \(-0.708179\pi\)
0.991508 + 0.130046i \(0.0415126\pi\)
\(110\) 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
\(115\) 2.82843 4.89898i 0.263752 0.456832i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.24264 7.34847i −0.370681 0.642039i 0.618989 0.785399i \(-0.287542\pi\)
−0.989671 + 0.143361i \(0.954209\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0 0
\(139\) 2.82843 0.239904 0.119952 0.992780i \(-0.461726\pi\)
0.119952 + 0.992780i \(0.461726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.48528 14.6969i 0.709575 1.22902i
\(144\) 0 0
\(145\) −5.65685 9.79796i −0.469776 0.813676i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 + 8.66025i 0.409616 + 0.709476i 0.994847 0.101391i \(-0.0323294\pi\)
−0.585231 + 0.810867i \(0.698996\pi\)
\(150\) 0 0
\(151\) 2.00000 3.46410i 0.162758 0.281905i −0.773099 0.634285i \(-0.781294\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.53553 + 6.12372i −0.282166 + 0.488726i −0.971918 0.235320i \(-0.924386\pi\)
0.689752 + 0.724046i \(0.257720\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.00000 3.46410i −0.156652 0.271329i 0.777007 0.629492i \(-0.216737\pi\)
−0.933659 + 0.358162i \(0.883403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.65685 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.12132 3.67423i −0.161281 0.279347i 0.774047 0.633128i \(-0.218229\pi\)
−0.935328 + 0.353781i \(0.884896\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) −21.2132 −1.57676 −0.788382 0.615185i \(-0.789081\pi\)
−0.788382 + 0.615185i \(0.789081\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.65685 + 9.79796i −0.415900 + 0.720360i
\(186\) 0 0
\(187\) −2.82843 4.89898i −0.206835 0.358249i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 13.8564i −0.578860 1.00261i −0.995610 0.0935936i \(-0.970165\pi\)
0.416751 0.909021i \(-0.363169\pi\)
\(192\) 0 0
\(193\) 5.00000 8.66025i 0.359908 0.623379i −0.628037 0.778183i \(-0.716141\pi\)
0.987945 + 0.154805i \(0.0494748\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.00000 + 8.66025i 0.349215 + 0.604858i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.82843 4.89898i −0.192897 0.334108i
\(216\) 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
\(223\) 16.9706 1.13643 0.568216 0.822879i \(-0.307634\pi\)
0.568216 + 0.822879i \(0.307634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.24264 7.34847i 0.281594 0.487735i −0.690184 0.723634i \(-0.742470\pi\)
0.971778 + 0.235899i \(0.0758036\pi\)
\(228\) 0 0
\(229\) 10.6066 + 18.3712i 0.700904 + 1.21400i 0.968149 + 0.250373i \(0.0805532\pi\)
−0.267246 + 0.963628i \(0.586113\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) −4.00000 + 6.92820i −0.260931 + 0.451946i
\(236\) 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
\(241\) −6.36396 + 11.0227i −0.409939 + 0.710035i −0.994882 0.101039i \(-0.967783\pi\)
0.584944 + 0.811074i \(0.301117\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 10.3923i −0.381771 0.661247i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −19.7990 −1.24970 −0.624851 0.780744i \(-0.714840\pi\)
−0.624851 + 0.780744i \(0.714840\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.6066 + 18.3712i 0.661622 + 1.14596i 0.980189 + 0.198062i \(0.0634648\pi\)
−0.318568 + 0.947900i \(0.603202\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 + 20.7846i −0.739952 + 1.28163i 0.212565 + 0.977147i \(0.431818\pi\)
−0.952517 + 0.304487i \(0.901515\pi\)
\(264\) 0 0
\(265\) 14.1421 0.868744
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.19239 + 15.9217i −0.560470 + 0.970762i 0.436986 + 0.899469i \(0.356046\pi\)
−0.997455 + 0.0712937i \(0.977287\pi\)
\(270\) 0 0
\(271\) −14.1421 24.4949i −0.859074 1.48796i −0.872814 0.488053i \(-0.837707\pi\)
0.0137402 0.999906i \(-0.495626\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 10.3923i −0.361814 0.626680i
\(276\) 0 0
\(277\) −11.0000 + 19.0526i −0.660926 + 1.14476i 0.319447 + 0.947604i \(0.396503\pi\)
−0.980373 + 0.197153i \(0.936830\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) 1.41421 2.44949i 0.0840663 0.145607i −0.820927 0.571034i \(-0.806543\pi\)
0.904993 + 0.425427i \(0.139876\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.50000 + 12.9904i 0.441176 + 0.764140i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.5269 −1.90024 −0.950121 0.311881i \(-0.899041\pi\)
−0.950121 + 0.311881i \(0.899041\pi\)
\(294\) 0 0
\(295\) 20.0000 1.16445
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.48528 14.6969i −0.490716 0.849946i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.00000 + 8.66025i −0.286299 + 0.495885i
\(306\) 0 0
\(307\) −19.7990 −1.12999 −0.564994 0.825095i \(-0.691122\pi\)
−0.564994 + 0.825095i \(0.691122\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.3137 + 19.5959i −0.641542 + 1.11118i 0.343547 + 0.939135i \(0.388371\pi\)
−0.985089 + 0.172047i \(0.944962\pi\)
\(312\) 0 0
\(313\) −2.12132 3.67423i −0.119904 0.207680i 0.799825 0.600233i \(-0.204925\pi\)
−0.919730 + 0.392553i \(0.871592\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.00000 1.73205i −0.0561656 0.0972817i 0.836576 0.547852i \(-0.184554\pi\)
−0.892741 + 0.450570i \(0.851221\pi\)
\(318\) 0 0
\(319\) −16.0000 + 27.7128i −0.895828 + 1.55162i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 6.36396 11.0227i 0.353009 0.611430i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 + 17.3205i 0.549650 + 0.952021i 0.998298 + 0.0583130i \(0.0185721\pi\)
−0.448649 + 0.893708i \(0.648095\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 + 10.3923i −0.322097 + 0.557888i −0.980921 0.194409i \(-0.937721\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(348\) 0 0
\(349\) −4.24264 −0.227103 −0.113552 0.993532i \(-0.536223\pi\)
−0.113552 + 0.993532i \(0.536223\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.94975 8.57321i 0.263448 0.456306i −0.703707 0.710490i \(-0.748473\pi\)
0.967156 + 0.254184i \(0.0818068\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.0000 17.3205i −0.527780 0.914141i −0.999476 0.0323801i \(-0.989691\pi\)
0.471696 0.881761i \(-0.343642\pi\)
\(360\) 0 0
\(361\) 5.50000 9.52628i 0.289474 0.501383i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −2.82843 + 4.89898i −0.147643 + 0.255725i −0.930356 0.366658i \(-0.880502\pi\)
0.782713 + 0.622383i \(0.213835\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.00000 8.66025i −0.258890 0.448411i 0.707055 0.707159i \(-0.250023\pi\)
−0.965945 + 0.258748i \(0.916690\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −33.9411 −1.74806
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.82843 4.89898i −0.144526 0.250326i 0.784670 0.619914i \(-0.212832\pi\)
−0.929196 + 0.369587i \(0.879499\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.00000 6.92820i 0.202808 0.351274i −0.746624 0.665246i \(-0.768327\pi\)
0.949432 + 0.313972i \(0.101660\pi\)
\(390\) 0 0
\(391\) −5.65685 −0.286079
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.65685 9.79796i 0.284627 0.492989i
\(396\) 0 0
\(397\) −7.77817 13.4722i −0.390375 0.676150i 0.602124 0.798403i \(-0.294321\pi\)
−0.992499 + 0.122253i \(0.960988\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 + 20.7846i 0.599251 + 1.03793i 0.992932 + 0.118686i \(0.0378683\pi\)
−0.393680 + 0.919247i \(0.628798\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0000 1.58618
\(408\) 0 0
\(409\) −19.0919 + 33.0681i −0.944033 + 1.63511i −0.186357 + 0.982482i \(0.559668\pi\)
−0.757676 + 0.652631i \(0.773665\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.0000 + 17.3205i 0.490881 + 0.850230i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.1421 −0.690889 −0.345444 0.938439i \(-0.612272\pi\)
−0.345444 + 0.938439i \(0.612272\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.12132 3.67423i −0.102899 0.178227i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 + 31.1769i −0.867029 + 1.50174i −0.00201168 + 0.999998i \(0.500640\pi\)
−0.865018 + 0.501741i \(0.832693\pi\)
\(432\) 0 0
\(433\) −21.2132 −1.01944 −0.509721 0.860340i \(-0.670251\pi\)
−0.509721 + 0.860340i \(0.670251\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.65685 + 9.79796i −0.270604 + 0.468700i
\(438\) 0 0
\(439\) 8.48528 + 14.6969i 0.404980 + 0.701447i 0.994319 0.106439i \(-0.0339450\pi\)
−0.589339 + 0.807886i \(0.700612\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.00000 13.8564i −0.380091 0.658338i 0.610984 0.791643i \(-0.290774\pi\)
−0.991075 + 0.133306i \(0.957441\pi\)
\(444\) 0 0
\(445\) −5.00000 + 8.66025i −0.237023 + 0.410535i
\(446\) 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
\(451\) 14.1421 24.4949i 0.665927 1.15342i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0000 25.9808i −0.701670 1.21533i −0.967880 0.251414i \(-0.919105\pi\)
0.266209 0.963915i \(-0.414229\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7.07107 −0.329332 −0.164666 0.986349i \(-0.552655\pi\)
−0.164666 + 0.986349i \(0.552655\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.89949 17.1464i −0.458094 0.793442i 0.540766 0.841173i \(-0.318134\pi\)
−0.998860 + 0.0477308i \(0.984801\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 + 13.8564i −0.367840 + 0.637118i
\(474\) 0 0
\(475\) −8.48528 −0.389331
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.65685 + 9.79796i −0.258468 + 0.447680i −0.965832 0.259170i \(-0.916551\pi\)
0.707364 + 0.706850i \(0.249884\pi\)
\(480\) 0 0
\(481\) 16.9706 + 29.3939i 0.773791 + 1.34025i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.00000 1.73205i −0.0454077 0.0786484i
\(486\) 0 0
\(487\) 6.00000 10.3923i 0.271886 0.470920i −0.697459 0.716625i \(-0.745686\pi\)
0.969345 + 0.245705i \(0.0790193\pi\)
\(488\) 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
\(493\) −5.65685 + 9.79796i −0.254772 + 0.441278i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 27.7128i −0.716258 1.24060i −0.962472 0.271380i \(-0.912520\pi\)
0.246214 0.969216i \(-0.420813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 39.5980 1.76559 0.882793 0.469762i \(-0.155660\pi\)
0.882793 + 0.469762i \(0.155660\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.19239 15.9217i −0.407445 0.705716i 0.587157 0.809473i \(-0.300247\pi\)
−0.994603 + 0.103757i \(0.966914\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.00000 + 13.8564i −0.352522 + 0.610586i
\(516\) 0 0
\(517\) 22.6274 0.995153
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.5061 + 35.5176i −0.898388 + 1.55605i −0.0688342 + 0.997628i \(0.521928\pi\)
−0.829554 + 0.558426i \(0.811405\pi\)
\(522\) 0 0
\(523\) −21.2132 36.7423i −0.927589 1.60663i −0.787344 0.616514i \(-0.788544\pi\)
−0.140244 0.990117i \(-0.544789\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 0 0
\(535\) −5.65685 + 9.79796i −0.244567 + 0.423603i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.00000 + 12.1244i 0.300954 + 0.521267i 0.976352 0.216186i \(-0.0693618\pi\)
−0.675399 + 0.737453i \(0.736028\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11.3137 0.484626
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.3137 + 19.5959i 0.481980 + 0.834814i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 25.9808i 0.635570 1.10084i −0.350824 0.936442i \(-0.614098\pi\)
0.986394 0.164399i \(-0.0525683\pi\)
\(558\) 0 0
\(559\) −16.9706 −0.717778
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.07107 12.2474i 0.298010 0.516168i −0.677671 0.735366i \(-0.737010\pi\)
0.975681 + 0.219197i \(0.0703438\pi\)
\(564\) 0 0
\(565\) −4.24264 7.34847i −0.178489 0.309152i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0000 + 34.6410i 0.838444 + 1.45223i 0.891196 + 0.453619i \(0.149867\pi\)
−0.0527519 + 0.998608i \(0.516799\pi\)
\(570\) 0 0
\(571\) 10.0000 17.3205i 0.418487 0.724841i −0.577301 0.816532i \(-0.695894\pi\)
0.995788 + 0.0916910i \(0.0292272\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −6.36396 + 11.0227i −0.264935 + 0.458881i −0.967547 0.252693i \(-0.918684\pi\)
0.702611 + 0.711574i \(0.252017\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −20.0000 34.6410i −0.828315 1.43468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25.4558 −1.05068 −0.525338 0.850894i \(-0.676061\pi\)
−0.525338 + 0.850894i \(0.676061\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.94975 8.57321i −0.203262 0.352060i 0.746316 0.665592i \(-0.231821\pi\)
−0.949578 + 0.313532i \(0.898488\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.00000 6.92820i 0.163436 0.283079i −0.772663 0.634816i \(-0.781076\pi\)
0.936099 + 0.351738i \(0.114409\pi\)
\(600\) 0 0
\(601\) −29.6985 −1.21143 −0.605713 0.795683i \(-0.707112\pi\)
−0.605713 + 0.795683i \(0.707112\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.53553 6.12372i 0.143740 0.248965i
\(606\) 0 0
\(607\) 16.9706 + 29.3939i 0.688814 + 1.19306i 0.972222 + 0.234061i \(0.0752016\pi\)
−0.283408 + 0.958999i \(0.591465\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 + 20.7846i 0.485468 + 0.840855i
\(612\) 0 0
\(613\) −12.0000 + 20.7846i −0.484675 + 0.839482i −0.999845 0.0176058i \(-0.994396\pi\)
0.515170 + 0.857088i \(0.327729\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) 15.5563 26.9444i 0.625262 1.08299i −0.363228 0.931700i \(-0.618325\pi\)
0.988490 0.151286i \(-0.0483414\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 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
\(635\) −14.1421 24.4949i −0.561214 0.972050i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.00000 6.92820i 0.157991 0.273648i −0.776153 0.630544i \(-0.782832\pi\)
0.934144 + 0.356897i \(0.116165\pi\)
\(642\) 0 0
\(643\) 2.82843 0.111542 0.0557711 0.998444i \(-0.482238\pi\)
0.0557711 + 0.998444i \(0.482238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) −28.2843 48.9898i −1.11025 1.92302i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12.0000 20.7846i −0.469596 0.813365i 0.529799 0.848123i \(-0.322267\pi\)
−0.999396 + 0.0347583i \(0.988934\pi\)
\(654\) 0 0
\(655\) 6.00000 10.3923i 0.234439 0.406061i
\(656\) 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
\(661\) 10.6066 18.3712i 0.412549 0.714556i −0.582619 0.812746i \(-0.697972\pi\)
0.995168 + 0.0981898i \(0.0313052\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 + 27.7128i 0.619522 + 1.07304i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.2843 1.09190
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.36396 + 11.0227i 0.244587 + 0.423637i 0.962015 0.272995i \(-0.0880143\pi\)
−0.717428 + 0.696632i \(0.754681\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 21.2132 36.7423i 0.808159 1.39977i
\(690\) 0 0
\(691\) 21.2132 + 36.7423i 0.806988 + 1.39774i 0.914941 + 0.403589i \(0.132237\pi\)
−0.107952 + 0.994156i \(0.534429\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 + 3.46410i 0.0758643 + 0.131401i
\(696\) 0 0
\(697\) 5.00000 8.66025i 0.189389 0.328031i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) 11.3137 19.5959i 0.426705 0.739074i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 + 6.92820i 0.150223 + 0.260194i 0.931309 0.364229i \(-0.118667\pi\)
−0.781086 + 0.624423i \(0.785334\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 19.7990 + 34.2929i 0.738378 + 1.27891i 0.953225 + 0.302260i \(0.0977411\pi\)
−0.214848 + 0.976648i \(0.568926\pi\)
\(720\) 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
\(727\) 28.2843 1.04901 0.524503 0.851409i \(-0.324251\pi\)
0.524503 + 0.851409i \(0.324251\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.82843 + 4.89898i −0.104613 + 0.181195i
\(732\) 0 0
\(733\) 19.0919 + 33.0681i 0.705175 + 1.22140i 0.966628 + 0.256183i \(0.0824648\pi\)
−0.261454 + 0.965216i \(0.584202\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 6.00000 10.3923i 0.220714 0.382287i −0.734311 0.678813i \(-0.762495\pi\)
0.955025 + 0.296526i \(0.0958281\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) −7.07107 + 12.2474i −0.259064 + 0.448712i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000 + 3.46410i 0.0729810 + 0.126407i 0.900207 0.435463i \(-0.143415\pi\)
−0.827225 + 0.561870i \(0.810082\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.65685 0.205874
\(756\) 0 0
\(757\) −40.0000 −1.45382 −0.726912 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.5061 + 35.5176i 0.743345 + 1.28751i 0.950964 + 0.309302i \(0.100095\pi\)
−0.207618 + 0.978210i \(0.566571\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.0000 51.9615i 1.08324 1.87622i
\(768\) 0 0
\(769\) 46.6690 1.68293 0.841464 0.540312i \(-0.181694\pi\)
0.841464 + 0.540312i \(0.181694\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.0208 20.8207i 0.432359 0.748867i −0.564717 0.825284i \(-0.691015\pi\)
0.997076 + 0.0764173i \(0.0243481\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.0000 17.3205i −0.358287 0.620572i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −24.0416 + 41.6413i −0.856992 + 1.48435i 0.0177934 + 0.999842i \(0.494336\pi\)
−0.874785 + 0.484511i \(0.838997\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 15.0000 + 25.9808i 0.532666 + 0.922604i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.6985 1.05197 0.525987 0.850493i \(-0.323696\pi\)
0.525987 + 0.850493i \(0.323696\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0