Properties

Label 380.2.l.a.113.4
Level $380$
Weight $2$
Character 380.113
Analytic conductor $3.034$
Analytic rank $0$
Dimension $8$
CM discriminant -19
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,2,Mod(37,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.l (of order \(4\), degree \(2\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 113.4
Root \(-0.656712 - 2.13746i\) of defining polynomial
Character \(\chi\) \(=\) 380.113
Dual form 380.2.l.a.37.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.63746 + 1.52274i) q^{5} +(-2.84677 + 2.84677i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(1.63746 + 1.52274i) q^{5} +(-2.84677 + 2.84677i) q^{7} +3.00000i q^{9} -6.50958 q^{11} +(1.69609 - 1.69609i) q^{17} -4.35890i q^{19} +(6.35890 + 6.35890i) q^{23} +(0.362541 + 4.98684i) q^{25} +(-8.99636 + 0.326585i) q^{35} +(8.74854 + 8.74854i) q^{43} +(-4.56821 + 4.91238i) q^{45} +(5.35635 - 5.35635i) q^{47} -9.20822i q^{49} +(-10.6592 - 9.91238i) q^{55} +10.8109 q^{61} +(-8.54032 - 8.54032i) q^{63} +(5.11994 + 5.11994i) q^{73} +(18.5313 - 18.5313i) q^{77} -9.00000 q^{81} +(-3.64110 - 3.64110i) q^{83} +(5.35999 - 0.194578i) q^{85} +(6.63746 - 7.13752i) q^{95} -19.5287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 6 q^{7} + 14 q^{17} + 16 q^{23} + 18 q^{25} - 22 q^{35} + 2 q^{43} - 26 q^{47} - 18 q^{63} + 22 q^{73} + 26 q^{77} - 72 q^{81} - 64 q^{83} + 24 q^{85} + 38 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) 1.63746 + 1.52274i 0.732294 + 0.680989i
\(6\) 0 0
\(7\) −2.84677 + 2.84677i −1.07598 + 1.07598i −0.0791130 + 0.996866i \(0.525209\pi\)
−0.996866 + 0.0791130i \(0.974791\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) −6.50958 −1.96271 −0.981356 0.192201i \(-0.938437\pi\)
−0.981356 + 0.192201i \(0.938437\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.69609 1.69609i 0.411363 0.411363i −0.470850 0.882213i \(-0.656053\pi\)
0.882213 + 0.470850i \(0.156053\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.35890 + 6.35890i 1.32592 + 1.32592i 0.908893 + 0.417029i \(0.136929\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0.362541 + 4.98684i 0.0725083 + 0.997368i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.99636 + 0.326585i −1.52066 + 0.0552029i
\(36\) 0 0
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 8.74854 + 8.74854i 1.33414 + 1.33414i 0.901629 + 0.432511i \(0.142372\pi\)
0.432511 + 0.901629i \(0.357628\pi\)
\(44\) 0 0
\(45\) −4.56821 + 4.91238i −0.680989 + 0.732294i
\(46\) 0 0
\(47\) 5.35635 5.35635i 0.781304 0.781304i −0.198747 0.980051i \(-0.563687\pi\)
0.980051 + 0.198747i \(0.0636872\pi\)
\(48\) 0 0
\(49\) 9.20822i 1.31546i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) −10.6592 9.91238i −1.43728 1.33658i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.8109 1.38420 0.692099 0.721803i \(-0.256686\pi\)
0.692099 + 0.721803i \(0.256686\pi\)
\(62\) 0 0
\(63\) −8.54032 8.54032i −1.07598 1.07598i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.11994 + 5.11994i 0.599243 + 0.599243i 0.940111 0.340868i \(-0.110721\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 18.5313 18.5313i 2.11184 2.11184i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −3.64110 3.64110i −0.399663 0.399663i 0.478451 0.878114i \(-0.341198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 5.35999 0.194578i 0.581373 0.0211049i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.63746 7.13752i 0.680989 0.732294i
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0 0
\(99\) 19.5287i 1.96271i
\(100\) 0 0
\(101\) −17.4356 −1.73491 −0.867453 0.497519i \(-0.834245\pi\)
−0.867453 + 0.497519i \(0.834245\pi\)
\(102\) 0 0
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0.729500 + 20.0954i 0.0680263 + 1.87390i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.65679i 0.885236i
\(120\) 0 0
\(121\) 31.3746 2.85224
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.00000 + 8.71780i −0.626099 + 0.779744i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −22.3746 −1.95488 −0.977438 0.211221i \(-0.932256\pi\)
−0.977438 + 0.211221i \(0.932256\pi\)
\(132\) 0 0
\(133\) 12.4088 + 12.4088i 1.07598 + 1.07598i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.5070 + 16.5070i −1.41029 + 1.41029i −0.652512 + 0.757778i \(0.726285\pi\)
−0.757778 + 0.652512i \(0.773715\pi\)
\(138\) 0 0
\(139\) 14.3746i 1.21924i −0.792695 0.609618i \(-0.791323\pi\)
0.792695 0.609618i \(-0.208677\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 24.3746i 1.99684i −0.0561570 0.998422i \(-0.517885\pi\)
0.0561570 0.998422i \(-0.482115\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 5.08828 + 5.08828i 0.411363 + 0.411363i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.7178 17.7178i 1.41403 1.41403i 0.695756 0.718278i \(-0.255069\pi\)
0.718278 0.695756i \(-0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −36.2047 −2.85333
\(162\) 0 0
\(163\) 16.3589 + 16.3589i 1.28133 + 1.28133i 0.939913 + 0.341415i \(0.110906\pi\)
0.341415 + 0.939913i \(0.389094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 13.0767 1.00000
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) −15.2285 13.1643i −1.15116 0.995129i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −11.0409 + 11.0409i −0.807387 + 0.807387i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.3746 0.750679 0.375339 0.926887i \(-0.377526\pi\)
0.375339 + 0.926887i \(0.377526\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.28220 + 2.28220i −0.162600 + 0.162600i −0.783718 0.621117i \(-0.786679\pi\)
0.621117 + 0.783718i \(0.286679\pi\)
\(198\) 0 0
\(199\) 15.1123i 1.07128i 0.844446 + 0.535641i \(0.179930\pi\)
−0.844446 + 0.535641i \(0.820070\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −19.0767 + 19.0767i −1.32592 + 1.32592i
\(208\) 0 0
\(209\) 28.3746i 1.96271i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00364 + 27.6471i 0.0684478 + 1.88552i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) −14.9605 + 1.08762i −0.997368 + 0.0725083i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 8.37459i 0.553408i 0.960955 + 0.276704i \(0.0892422\pi\)
−0.960955 + 0.276704i \(0.910758\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.26715 + 6.26715i 0.410575 + 0.410575i 0.881939 0.471364i \(-0.156238\pi\)
−0.471364 + 0.881939i \(0.656238\pi\)
\(234\) 0 0
\(235\) 16.9271 0.614487i 1.10420 0.0400847i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.9260i 0.706745i 0.935483 + 0.353373i \(0.114965\pi\)
−0.935483 + 0.353373i \(0.885035\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.0217 15.0781i 0.895814 0.963303i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.3746 1.91723 0.958613 0.284711i \(-0.0918976\pi\)
0.958613 + 0.284711i \(0.0918976\pi\)
\(252\) 0 0
\(253\) −41.3937 41.3937i −2.60240 2.60240i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −22.8534 22.8534i −1.40920 1.40920i −0.764075 0.645128i \(-0.776804\pi\)
−0.645128 0.764075i \(-0.723196\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 26.1534 1.58871 0.794353 0.607457i \(-0.207810\pi\)
0.794353 + 0.607457i \(0.207810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.35999 32.4622i −0.142313 1.95754i
\(276\) 0 0
\(277\) 9.01660 9.01660i 0.541755 0.541755i −0.382288 0.924043i \(-0.624864\pi\)
0.924043 + 0.382288i \(0.124864\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 18.7802 + 18.7802i 1.11637 + 1.11637i 0.992270 + 0.124096i \(0.0396032\pi\)
0.124096 + 0.992270i \(0.460397\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 11.2465i 0.661560i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −49.8102 −2.87101
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 17.7025 + 16.4622i 1.01364 + 0.942623i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.1314 1.59519 0.797594 0.603195i \(-0.206106\pi\)
0.797594 + 0.603195i \(0.206106\pi\)
\(312\) 0 0
\(313\) −20.4356 20.4356i −1.15509 1.15509i −0.985518 0.169570i \(-0.945762\pi\)
−0.169570 0.985518i \(-0.554238\pi\)
\(314\) 0 0
\(315\) −0.979755 26.9891i −0.0552029 1.52066i
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.39310 7.39310i −0.411363 0.411363i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 30.4966i 1.68133i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.28630 + 6.28630i 0.339428 + 0.339428i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.4737 24.4737i 1.31382 1.31382i 0.395242 0.918577i \(-0.370661\pi\)
0.918577 0.395242i \(-0.129339\pi\)
\(348\) 0 0
\(349\) 36.8492i 1.97249i −0.165277 0.986247i \(-0.552852\pi\)
0.165277 0.986247i \(-0.447148\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.4356 10.4356i −0.555431 0.555431i 0.372572 0.928003i \(-0.378476\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 34.3746i 1.81422i −0.420892 0.907111i \(-0.638283\pi\)
0.420892 0.907111i \(-0.361717\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.587365 + 16.1800i 0.0307441 + 0.846900i
\(366\) 0 0
\(367\) 0.923303 0.923303i 0.0481960 0.0481960i −0.682598 0.730794i \(-0.739150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 58.5625 2.12593i 2.98462 0.108347i
\(386\) 0 0
\(387\) −26.2456 + 26.2456i −1.33414 + 1.33414i
\(388\) 0 0
\(389\) 6.39449i 0.324213i −0.986773 0.162107i \(-0.948171\pi\)
0.986773 0.162107i \(-0.0518289\pi\)
\(390\) 0 0
\(391\) 21.5706 1.09087
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.9607 11.9607i 0.600290 0.600290i −0.340099 0.940389i \(-0.610461\pi\)
0.940389 + 0.340099i \(0.110461\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −14.7371 13.7046i −0.732294 0.680989i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −0.417711 11.5066i −0.0205046 0.564836i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i −0.977064 0.212946i \(-0.931694\pi\)
0.977064 0.212946i \(-0.0683059\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 16.0690 + 16.0690i 0.781304 + 0.781304i
\(424\) 0 0
\(425\) 9.07305 + 7.84325i 0.440108 + 0.380453i
\(426\) 0 0
\(427\) −30.7763 + 30.7763i −1.48937 + 1.48937i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.7178 27.7178i 1.32592 1.32592i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 27.6247 1.31546
\(442\) 0 0
\(443\) −15.5329 15.5329i −0.737991 0.737991i 0.234198 0.972189i \(-0.424754\pi\)
−0.972189 + 0.234198i \(0.924754\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.9059 + 29.9059i −1.39894 + 1.39894i −0.595818 + 0.803120i \(0.703172\pi\)
−0.803120 + 0.595818i \(0.796828\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.374586 0.0174462 0.00872311 0.999962i \(-0.497223\pi\)
0.00872311 + 0.999962i \(0.497223\pi\)
\(462\) 0 0
\(463\) 1.42803 + 1.42803i 0.0663661 + 0.0663661i 0.739511 0.673145i \(-0.235057\pi\)
−0.673145 + 0.739511i \(0.735057\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.1673 + 30.1673i −1.39598 + 1.39598i −0.584792 + 0.811183i \(0.698824\pi\)
−0.811183 + 0.584792i \(0.801176\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −56.9493 56.9493i −2.61853 2.61853i
\(474\) 0 0
\(475\) 21.7371 1.58028i 0.997368 0.0725083i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.00000i 0.182765i −0.995816 0.0913823i \(-0.970871\pi\)
0.995816 0.0913823i \(-0.0291285\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 29.7371 31.9775i 1.33658 1.43728i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.3746i 1.71788i 0.512074 + 0.858941i \(0.328877\pi\)
−0.512074 + 0.858941i \(0.671123\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.3589 + 26.3589i 1.17529 + 1.17529i 0.980932 + 0.194354i \(0.0622609\pi\)
0.194354 + 0.980932i \(0.437739\pi\)
\(504\) 0 0
\(505\) −28.5501 26.5498i −1.27046 1.18145i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −29.1506 −1.28955
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −34.8676 + 34.8676i −1.53347 + 1.53347i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 57.8712i 2.51614i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 59.9416i 2.58187i
\(540\) 0 0
\(541\) 41.2657 1.77415 0.887075 0.461625i \(-0.152733\pi\)
0.887075 + 0.461625i \(0.152733\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 32.4328i 1.38420i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.62441 + 5.62441i −0.238314 + 0.238314i −0.816152 0.577838i \(-0.803897\pi\)
0.577838 + 0.816152i \(0.303897\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 25.6209 25.6209i 1.07598 1.07598i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 26.1534 1.09449 0.547243 0.836974i \(-0.315677\pi\)
0.547243 + 0.836974i \(0.315677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −29.4054 + 34.0162i −1.22629 + 1.41857i
\(576\) 0 0
\(577\) 33.2981 33.2981i 1.38622 1.38622i 0.553104 0.833112i \(-0.313443\pi\)
0.833112 0.553104i \(-0.186557\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.7308 0.860057
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.99398 + 3.99398i −0.164849 + 0.164849i −0.784711 0.619862i \(-0.787189\pi\)
0.619862 + 0.784711i \(0.287189\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.435596 0.435596i −0.0178878 0.0178878i 0.698106 0.715994i \(-0.254026\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 0 0
\(595\) −14.7048 + 15.8126i −0.602836 + 0.648253i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 51.3746 + 47.7753i 2.08867 + 1.94234i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −21.0534 21.0534i −0.850337 0.850337i 0.139837 0.990174i \(-0.455342\pi\)
−0.990174 + 0.139837i \(0.955342\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.6542 + 17.6542i −0.710733 + 0.710733i −0.966689 0.255956i \(-0.917610\pi\)
0.255956 + 0.966689i \(0.417610\pi\)
\(618\) 0 0
\(619\) 24.0000i 0.964641i −0.875995 0.482321i \(-0.839794\pi\)
0.875995 0.482321i \(-0.160206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −24.7371 + 3.61587i −0.989485 + 0.144635i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −36.9643 −1.47153 −0.735763 0.677239i \(-0.763176\pi\)
−0.735763 + 0.677239i \(0.763176\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −35.8608 35.8608i −1.41421 1.41421i −0.707818 0.706395i \(-0.750320\pi\)
−0.706395 0.707818i \(-0.749680\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.6378 29.6378i 1.16518 1.16518i 0.181857 0.983325i \(-0.441789\pi\)
0.983325 0.181857i \(-0.0582109\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.5877 + 33.5877i 1.31439 + 1.31439i 0.918147 + 0.396239i \(0.129685\pi\)
0.396239 + 0.918147i \(0.370315\pi\)
\(654\) 0 0
\(655\) −36.6375 34.0706i −1.43154 1.33125i
\(656\) 0 0
\(657\) −15.3598 + 15.3598i −0.599243 + 0.599243i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.42355 + 39.2142i 0.0552029 + 1.52066i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −70.3746 −2.71678
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) −52.1654 + 1.89371i −1.99314 + 0.0723548i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −10.6958 −0.406889 −0.203445 0.979086i \(-0.565214\pi\)
−0.203445 + 0.979086i \(0.565214\pi\)
\(692\) 0 0
\(693\) 55.5938 + 55.5938i 2.11184 + 2.11184i
\(694\) 0 0
\(695\) 21.8887 23.5378i 0.830286 0.892839i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17.4356 −0.658533 −0.329267 0.944237i \(-0.606802\pi\)
−0.329267 + 0.944237i \(0.606802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 49.6352 49.6352i 1.86672 1.86672i
\(708\) 0 0
\(709\) 52.3068i 1.96442i −0.187779 0.982211i \(-0.560129\pi\)
0.187779 0.982211i \(-0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 5.62541i 0.209793i 0.994483 + 0.104896i \(0.0334511\pi\)
−0.994483 + 0.104896i \(0.966549\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −33.5661 + 33.5661i −1.24490 + 1.24490i −0.286954 + 0.957944i \(0.592643\pi\)
−0.957944 + 0.286954i \(0.907357\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 29.6767 1.09763
\(732\) 0 0
\(733\) 33.1534 + 33.1534i 1.22455 + 1.22455i 0.965998 + 0.258551i \(0.0832450\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.7150i 0.872370i −0.899857 0.436185i \(-0.856329\pi\)
0.899857 0.436185i \(-0.143671\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 37.1161 39.9124i 1.35983 1.46228i
\(746\) 0 0
\(747\) 10.9233 10.9233i 0.399663 0.399663i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.66627 9.66627i 0.351327 0.351327i −0.509276 0.860603i \(-0.670087\pi\)
0.860603 + 0.509276i \(0.170087\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.4519 1.64763 0.823816 0.566857i \(-0.191841\pi\)
0.823816 + 0.566857i \(0.191841\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.583734 + 16.0800i 0.0211049 + 0.581373i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 44.3746i 1.60019i −0.599874 0.800094i \(-0.704783\pi\)
0.599874 0.800094i \(-0.295217\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 55.9917 2.03261i 1.99843 0.0725468i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 18.1697i 0.642799i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −33.3286 33.3286i −1.17614 1.17614i
\(804\) 0 0