Properties

Label 380.2.l.a.113.3
Level $380$
Weight $2$
Character 380.113
Analytic conductor $3.034$
Analytic rank $0$
Dimension $8$
CM discriminant -19
Inner twists $4$

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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 113.3
Root \(0.656712 - 2.13746i\) of defining polynomial
Character \(\chi\) \(=\) 380.113
Dual form 380.2.l.a.37.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.63746 - 1.52274i) q^{5} +(-2.42815 + 2.42815i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(1.63746 - 1.52274i) q^{5} +(-2.42815 + 2.42815i) q^{7} +3.00000i q^{9} +6.50958 q^{11} +(5.57882 - 5.57882i) q^{17} +4.35890i q^{19} +(-2.35890 - 2.35890i) q^{23} +(0.362541 - 4.98684i) q^{25} +(-0.278560 + 7.67341i) q^{35} +(3.07621 + 3.07621i) q^{43} +(4.56821 + 4.91238i) q^{45} +(-8.08143 + 8.08143i) q^{47} -4.79178i q^{49} +(10.6592 - 9.91238i) q^{55} -10.8109 q^{61} +(-7.28444 - 7.28444i) q^{63} +(-10.9447 - 10.9447i) q^{73} +(-15.8062 + 15.8062i) q^{77} -9.00000 q^{81} +(-12.3589 - 12.3589i) q^{83} +(0.640009 - 17.6302i) q^{85} +(6.63746 + 7.13752i) q^{95} +19.5287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 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
\(7\) −2.42815 + 2.42815i −0.917753 + 0.917753i −0.996866 0.0791130i \(-0.974791\pi\)
0.0791130 + 0.996866i \(0.474791\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 6.50958 1.96271 0.981356 0.192201i \(-0.0615626\pi\)
0.981356 + 0.192201i \(0.0615626\pi\)
\(12\) 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
\(17\) 5.57882 5.57882i 1.35306 1.35306i 0.470850 0.882213i \(-0.343947\pi\)
0.882213 0.470850i \(-0.156053\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.35890 2.35890i −0.491864 0.491864i 0.417029 0.908893i \(-0.363071\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 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
\(35\) −0.278560 + 7.67341i −0.0470852 + 1.29704i
\(36\) 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
\(43\) 3.07621 + 3.07621i 0.469118 + 0.469118i 0.901629 0.432511i \(-0.142372\pi\)
−0.432511 + 0.901629i \(0.642372\pi\)
\(44\) 0 0
\(45\) 4.56821 + 4.91238i 0.680989 + 0.732294i
\(46\) 0 0
\(47\) −8.08143 + 8.08143i −1.17880 + 1.17880i −0.198747 + 0.980051i \(0.563687\pi\)
−0.980051 + 0.198747i \(0.936313\pi\)
\(48\) 0 0
\(49\) 4.79178i 0.684540i
\(50\) 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 \(-0.743314\pi\)
−0.692099 + 0.721803i \(0.743314\pi\)
\(62\) 0 0
\(63\) −7.28444 7.28444i −0.917753 0.917753i
\(64\) 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
\(73\) −10.9447 10.9447i −1.28098 1.28098i −0.940111 0.340868i \(-0.889279\pi\)
−0.340868 0.940111i \(-0.610721\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.8062 + 15.8062i −1.80128 + 1.80128i
\(78\) 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
\(83\) −12.3589 12.3589i −1.35657 1.35657i −0.878114 0.478451i \(-0.841198\pi\)
−0.478451 0.878114i \(-0.658802\pi\)
\(84\) 0 0
\(85\) 0.640009 17.6302i 0.0694187 1.91226i
\(86\) 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 \(-0.165755\pi\)
0.867453 + 0.497519i \(0.165755\pi\)
\(102\) 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
\(115\) −7.45458 0.270616i −0.695143 0.0252350i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 27.0924i 2.48355i
\(120\) 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
\(133\) −10.5840 10.5840i −0.917753 0.917753i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.23211 1.23211i 0.105266 0.105266i −0.652512 0.757778i \(-0.726285\pi\)
0.757778 + 0.652512i \(0.226285\pi\)
\(138\) 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
\(153\) 16.7365 + 16.7365i 1.35306 + 1.35306i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.282202 0.282202i 0.0225222 0.0225222i −0.695756 0.718278i \(-0.744931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.4555 0.902820
\(162\) 0 0
\(163\) 7.64110 + 7.64110i 0.598497 + 0.598497i 0.939913 0.341415i \(-0.110906\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 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
\(175\) 11.2285 + 12.9891i 0.848792 + 0.981882i
\(176\) 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
\(187\) 36.3158 36.3158i 2.65567 2.65567i
\(188\) 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
\(197\) −19.7178 + 19.7178i −1.40483 + 1.40483i −0.621117 + 0.783718i \(0.713321\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 15.1123i 1.07128i −0.844446 0.535641i \(-0.820070\pi\)
0.844446 0.535641i \(-0.179930\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.07670 7.07670i 0.491864 0.491864i
\(208\) 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
\(215\) 9.72144 + 0.352907i 0.662997 + 0.0240681i
\(216\) 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
\(233\) 20.6573 + 20.6573i 1.35330 + 1.35330i 0.881939 + 0.471364i \(0.156238\pi\)
0.471364 + 0.881939i \(0.343762\pi\)
\(234\) 0 0
\(235\) −0.927111 + 25.5389i −0.0604781 + 1.66597i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.9260i 0.706745i −0.935483 0.353373i \(-0.885035\pi\)
0.935483 0.353373i \(-0.114965\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.29662 7.84634i −0.466164 0.501284i
\(246\) 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
\(253\) −15.3554 15.3554i −0.965388 0.965388i
\(254\) 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
\(263\) 1.92900 + 1.92900i 0.118947 + 0.118947i 0.764075 0.645128i \(-0.223196\pi\)
−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 \(-0.792190\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.35999 32.4622i 0.142313 1.95754i
\(276\) 0 0
\(277\) −21.7417 + 21.7417i −1.30633 + 1.30633i −0.382288 + 0.924043i \(0.624864\pi\)
−0.924043 + 0.382288i \(0.875136\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −14.6049 14.6049i −0.868174 0.868174i 0.124096 0.992270i \(-0.460397\pi\)
−0.992270 + 0.124096i \(0.960397\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 45.2465i 2.66156i
\(290\) 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
\(301\) −14.9390 −0.861069
\(302\) 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 \(-0.793894\pi\)
−0.797594 + 0.603195i \(0.793894\pi\)
\(312\) 0 0
\(313\) 14.4356 + 14.4356i 0.815948 + 0.815948i 0.985518 0.169570i \(-0.0542379\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −23.0202 0.835679i −1.29704 0.0470852i
\(316\) 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
\(323\) 24.3175 + 24.3175i 1.35306 + 1.35306i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 39.2458i 2.16369i
\(330\) 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
\(343\) −5.36188 5.36188i −0.289514 0.289514i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.74865 + 9.74865i −0.523335 + 0.523335i −0.918577 0.395242i \(-0.870661\pi\)
0.395242 + 0.918577i \(0.370661\pi\)
\(348\) 0 0
\(349\) 36.8492i 1.97249i 0.165277 + 0.986247i \(0.447148\pi\)
−0.165277 + 0.986247i \(0.552852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.4356 + 24.4356i 1.30058 + 1.30058i 0.928003 + 0.372572i \(0.121524\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 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
\(365\) −34.5874 1.25559i −1.81039 0.0657205i
\(366\) 0 0
\(367\) 27.0767 27.0767i 1.41339 1.41339i 0.682598 0.730794i \(-0.260850\pi\)
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
\(385\) −1.81331 + 49.9507i −0.0924146 + 2.54572i
\(386\) 0 0
\(387\) −9.22864 + 9.22864i −0.469118 + 0.469118i
\(388\) 0 0
\(389\) 6.39449i 0.324213i 0.986773 + 0.162107i \(0.0518289\pi\)
−0.986773 + 0.162107i \(0.948171\pi\)
\(390\) 0 0
\(391\) −26.3198 −1.33105
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.5136 25.5136i 1.28049 1.28049i 0.340099 0.940389i \(-0.389539\pi\)
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
\(415\) −39.0565 1.41783i −1.91721 0.0695984i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i 0.977064 + 0.212946i \(0.0683059\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −24.2443 24.2443i −1.17880 1.17880i
\(424\) 0 0
\(425\) −25.7981 29.8432i −1.25139 1.44761i
\(426\) 0 0
\(427\) 26.2505 26.2505i 1.27035 1.27035i
\(428\) 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
\(437\) 10.2822 10.2822i 0.491864 0.491864i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 14.3753 0.684540
\(442\) 0 0
\(443\) −25.3915 25.3915i −1.20639 1.20639i −0.972189 0.234198i \(-0.924754\pi\)
−0.234198 0.972189i \(-0.575246\pi\)
\(444\) 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
\(457\) 4.43161 4.43161i 0.207302 0.207302i −0.595818 0.803120i \(-0.703172\pi\)
0.803120 + 0.595818i \(0.203172\pi\)
\(458\) 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
\(463\) 30.3967 + 30.3967i 1.41266 + 1.41266i 0.739511 + 0.673145i \(0.235057\pi\)
0.673145 + 0.739511i \(0.264943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.89236 4.89236i 0.226392 0.226392i −0.584792 0.811183i \(-0.698824\pi\)
0.811183 + 0.584792i \(0.198824\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20.0249 + 20.0249i 0.920744 + 0.920744i
\(474\) 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
\(503\) 17.6411 + 17.6411i 0.786578 + 0.786578i 0.980932 0.194354i \(-0.0622609\pi\)
−0.194354 + 0.980932i \(0.562261\pi\)
\(504\) 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
\(511\) 53.1506 2.35124
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −52.6067 + 52.6067i −2.31364 + 2.31364i
\(518\) 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
\(529\) 11.8712i 0.516139i
\(530\) 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
\(539\) 31.1924i 1.34355i
\(540\) 0 0
\(541\) −41.2657 −1.77415 −0.887075 0.461625i \(-0.847267\pi\)
−0.887075 + 0.461625i \(0.847267\pi\)
\(542\) 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
\(557\) 32.8993 32.8993i 1.39399 1.39399i 0.577838 0.816152i \(-0.303897\pi\)
0.816152 0.577838i \(-0.196103\pi\)
\(558\) 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
\(567\) 21.8533 21.8533i 0.917753 0.917753i
\(568\) 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 \(-0.684323\pi\)
−0.547243 + 0.836974i \(0.684323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.6186 + 10.9083i −0.526234 + 0.454906i
\(576\) 0 0
\(577\) 6.72603 6.72603i 0.280008 0.280008i −0.553104 0.833112i \(-0.686557\pi\)
0.833112 + 0.553104i \(0.186557\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 60.0184 2.48998
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −34.0301 + 34.0301i −1.40457 + 1.40457i −0.619862 + 0.784711i \(0.712811\pi\)
−0.784711 + 0.619862i \(0.787189\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.4356 + 34.4356i 1.41410 + 1.41410i 0.715994 + 0.698106i \(0.245974\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 41.2546 + 44.3627i 1.69127 + 1.81869i
\(596\) 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
\(613\) 27.9778 + 27.9778i 1.13001 + 1.13001i 0.990174 + 0.139837i \(0.0446580\pi\)
0.139837 + 0.990174i \(0.455342\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.3699 + 30.3699i −1.22264 + 1.22264i −0.255956 + 0.966689i \(0.582390\pi\)
−0.966689 + 0.255956i \(0.917610\pi\)
\(618\) 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 \(-0.236824\pi\)
0.735763 + 0.677239i \(0.236824\pi\)
\(632\) 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
\(643\) 0.0360724 + 0.0360724i 0.00142255 + 0.00142255i 0.707818 0.706395i \(-0.249680\pi\)
−0.706395 + 0.707818i \(0.749680\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.3863 20.3863i 0.801468 0.801468i −0.181857 0.983325i \(-0.558211\pi\)
0.983325 + 0.181857i \(0.0582109\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.3368 + 13.3368i 0.521908 + 0.521908i 0.918147 0.396239i \(-0.129685\pi\)
−0.396239 + 0.918147i \(0.629685\pi\)
\(654\) 0 0
\(655\) −36.6375 + 34.0706i −1.43154 + 1.33125i
\(656\) 0 0
\(657\) 32.8341 32.8341i 1.28098 1.28098i
\(658\) 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
\(665\) −33.4476 1.21421i −1.29704 0.0470852i
\(666\) 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
\(685\) 0.141349 3.89371i 0.00540067 0.148771i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.6958 0.406889 0.203445 0.979086i \(-0.434786\pi\)
0.203445 + 0.979086i \(0.434786\pi\)
\(692\) 0 0
\(693\) −47.4186 47.4186i −1.80128 1.80128i
\(694\) 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 \(-0.393198\pi\)
0.329267 + 0.944237i \(0.393198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −42.3362 + 42.3362i −1.59222 + 1.59222i
\(708\) 0 0
\(709\) 52.3068i 1.96442i 0.187779 + 0.982211i \(0.439871\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 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
\(727\) 18.0919 18.0919i 0.670990 0.670990i −0.286954 0.957944i \(-0.592643\pi\)
0.957944 + 0.286954i \(0.0926427\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 34.3233 1.26949
\(732\) 0 0
\(733\) −19.1534 19.1534i −0.707447 0.707447i 0.258551 0.965998i \(-0.416755\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.7150i 0.872370i 0.899857 + 0.436185i \(0.143671\pi\)
−0.899857 + 0.436185i \(0.856329\pi\)
\(740\) 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
\(747\) 37.0767 37.0767i 1.35657 1.35657i
\(748\) 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
\(757\) −37.6904 + 37.6904i −1.36988 + 1.36988i −0.509276 + 0.860603i \(0.670087\pi\)
−0.860603 + 0.509276i \(0.829913\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −45.4519 −1.64763 −0.823816 0.566857i \(-0.808159\pi\)
−0.823816 + 0.566857i \(0.808159\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 52.8905 + 1.92003i 1.91226 + 0.0694187i
\(766\) 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
\(785\) 0.0323746 0.891814i 0.00115550 0.0318302i
\(786\) 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
\(799\) 90.1697i 3.18998i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −71.2453 71.2453i −2.51419 2.51419i
\(804\) 0 0