Properties

Label 380.2.l.a.113.2
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 113.2
Root \(-1.52274 + 1.63746i\) of defining polynomial
Character \(\chi\) \(=\) 380.113
Dual form 380.2.l.a.37.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.13746 + 0.656712i) q^{5} +(3.52622 - 3.52622i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(-2.13746 + 0.656712i) q^{5} +(3.52622 - 3.52622i) q^{7} +3.00000i q^{9} +2.15068 q^{11} +(3.98336 - 3.98336i) q^{17} -4.35890i q^{19} +(6.35890 + 6.35890i) q^{23} +(4.13746 - 2.80739i) q^{25} +(-5.22144 + 9.85286i) q^{35} +(-1.71019 - 1.71019i) q^{43} +(-1.97014 - 6.41238i) q^{45} +(-9.67690 + 9.67690i) q^{47} -17.8685i q^{49} +(-4.59698 + 1.41238i) q^{55} -15.1698 q^{61} +(10.5787 + 10.5787i) q^{63} +(6.91841 + 6.91841i) q^{73} +(7.58377 - 7.58377i) q^{77} -9.00000 q^{81} +(-3.64110 - 3.64110i) q^{83} +(-5.89834 + 11.1302i) q^{85} +(2.86254 + 9.31697i) q^{95} +6.45203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 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
\(5\) −2.13746 + 0.656712i −0.955901 + 0.293691i
\(6\) 0 0
\(7\) 3.52622 3.52622i 1.33279 1.33279i 0.429919 0.902867i \(-0.358542\pi\)
0.902867 0.429919i \(-0.141458\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 2.15068 0.648454 0.324227 0.945979i \(-0.394896\pi\)
0.324227 + 0.945979i \(0.394896\pi\)
\(12\) 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
\(17\) 3.98336 3.98336i 0.966106 0.966106i −0.0333386 0.999444i \(-0.510614\pi\)
0.999444 + 0.0333386i \(0.0106140\pi\)
\(18\) 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
\(25\) 4.13746 2.80739i 0.827492 0.561478i
\(26\) 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
\(35\) −5.22144 + 9.85286i −0.882585 + 1.66544i
\(36\) 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
\(43\) −1.71019 1.71019i −0.260801 0.260801i 0.564578 0.825380i \(-0.309039\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) 0 0
\(45\) −1.97014 6.41238i −0.293691 0.955901i
\(46\) 0 0
\(47\) −9.67690 + 9.67690i −1.41152 + 1.41152i −0.662145 + 0.749375i \(0.730354\pi\)
−0.749375 + 0.662145i \(0.769646\pi\)
\(48\) 0 0
\(49\) 17.8685i 2.55264i
\(50\) 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
\(55\) −4.59698 + 1.41238i −0.619857 + 0.190445i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.1698 −1.94230 −0.971149 0.238474i \(-0.923353\pi\)
−0.971149 + 0.238474i \(0.923353\pi\)
\(62\) 0 0
\(63\) 10.5787 + 10.5787i 1.33279 + 1.33279i
\(64\) 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
\(73\) 6.91841 + 6.91841i 0.809739 + 0.809739i 0.984594 0.174855i \(-0.0559458\pi\)
−0.174855 + 0.984594i \(0.555946\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.58377 7.58377i 0.864250 0.864250i
\(78\) 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
\(85\) −5.89834 + 11.1302i −0.639765 + 1.20724i
\(86\) 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
\(95\) 2.86254 + 9.31697i 0.293691 + 0.955901i
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0 0
\(99\) 6.45203i 0.648454i
\(100\) 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
\(115\) −17.7678 9.41592i −1.65686 0.878039i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 28.0924i 2.57522i
\(120\) 0 0
\(121\) −6.37459 −0.579508
\(122\) 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
\(131\) 15.3746 1.34328 0.671642 0.740876i \(-0.265589\pi\)
0.671642 + 0.740876i \(0.265589\pi\)
\(132\) 0 0
\(133\) −15.3704 15.3704i −1.33279 1.33279i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.18648 7.18648i 0.613982 0.613982i −0.329999 0.943981i \(-0.607048\pi\)
0.943981 + 0.329999i \(0.107048\pi\)
\(138\) 0 0
\(139\) 23.3746i 1.98261i 0.131597 + 0.991303i \(0.457989\pi\)
−0.131597 + 0.991303i \(0.542011\pi\)
\(140\) 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
\(149\) 13.3746i 1.09569i 0.836580 + 0.547844i \(0.184551\pi\)
−0.836580 + 0.547844i \(0.815449\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 11.9501 + 11.9501i 0.966106 + 0.966106i
\(154\) 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
\(161\) 44.8458 3.53434
\(162\) 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
\(175\) 4.69012 24.4891i 0.354540 1.85120i
\(176\) 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
\(187\) 8.56691 8.56691i 0.626475 0.626475i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −27.3746 −1.98076 −0.990378 0.138390i \(-0.955807\pi\)
−0.990378 + 0.138390i \(0.955807\pi\)
\(192\) 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
\(199\) 28.1890i 1.99826i −0.0416556 0.999132i \(-0.513263\pi\)
0.0416556 0.999132i \(-0.486737\pi\)
\(200\) 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
\(209\) 9.37459i 0.648454i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.77856 + 2.53236i 0.325895 + 0.172705i
\(216\) 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
\(225\) 8.42217 + 12.4124i 0.561478 + 0.827492i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 29.3746i 1.94113i −0.240845 0.970564i \(-0.577424\pi\)
0.240845 0.970564i \(-0.422576\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.0233 21.0233i −1.37728 1.37728i −0.849183 0.528099i \(-0.822905\pi\)
−0.528099 0.849183i \(-0.677095\pi\)
\(234\) 0 0
\(235\) 14.3290 27.0389i 0.934723 1.76382i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.5863i 1.26693i 0.773771 + 0.633465i \(0.218368\pi\)
−0.773771 + 0.633465i \(0.781632\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.7344 + 38.1931i 0.749686 + 2.44007i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.37459 −0.465480 −0.232740 0.972539i \(-0.574769\pi\)
−0.232740 + 0.972539i \(0.574769\pi\)
\(252\) 0 0
\(253\) 13.6759 + 13.6759i 0.859799 + 0.859799i
\(254\) 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
\(263\) 13.0973 + 13.0973i 0.807613 + 0.807613i 0.984272 0.176659i \(-0.0565291\pi\)
−0.176659 + 0.984272i \(0.556529\pi\)
\(264\) 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
\(275\) 8.89834 6.03779i 0.536590 0.364092i
\(276\) 0 0
\(277\) −23.3372 + 23.3372i −1.40219 + 1.40219i −0.609101 + 0.793093i \(0.708470\pi\)
−0.793093 + 0.609101i \(0.791530\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 3.25816 + 3.25816i 0.193677 + 0.193677i 0.797283 0.603606i \(-0.206270\pi\)
−0.603606 + 0.797283i \(0.706270\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 14.7342i 0.866720i
\(290\) 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
\(301\) −12.0610 −0.695185
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 32.4249 9.96221i 1.85664 0.570435i
\(306\) 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
\(311\) −32.4903 −1.84236 −0.921179 0.389139i \(-0.872773\pi\)
−0.921179 + 0.389139i \(0.872773\pi\)
\(312\) 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
\(315\) −29.5586 15.6643i −1.66544 0.882585i
\(316\) 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
\(323\) −17.3630 17.3630i −0.966106 0.966106i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 68.2458i 3.76251i
\(330\) 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
\(343\) −38.3247 38.3247i −2.06934 2.06934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.79429 + 3.79429i −0.203688 + 0.203688i −0.801578 0.597890i \(-0.796006\pi\)
0.597890 + 0.801578i \(0.296006\pi\)
\(348\) 0 0
\(349\) 23.7725i 1.27251i 0.771477 + 0.636257i \(0.219518\pi\)
−0.771477 + 0.636257i \(0.780482\pi\)
\(350\) 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
\(359\) 3.37459i 0.178104i 0.996027 + 0.0890519i \(0.0283837\pi\)
−0.996027 + 0.0890519i \(0.971616\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −19.3312 10.2444i −1.01184 0.536217i
\(366\) 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
\(385\) −11.2296 + 21.1903i −0.572315 + 1.07996i
\(386\) 0 0
\(387\) 5.13057 5.13057i 0.260801 0.260801i
\(388\) 0 0
\(389\) 36.9068i 1.87125i 0.352998 + 0.935624i \(0.385162\pi\)
−0.352998 + 0.935624i \(0.614838\pi\)
\(390\) 0 0
\(391\) 50.6595 2.56196
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −28.0757 + 28.0757i −1.40908 + 1.40908i −0.644351 + 0.764730i \(0.722873\pi\)
−0.764730 + 0.644351i \(0.777127\pi\)
\(398\) 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
\(405\) 19.2371 5.91041i 0.955901 0.293691i
\(406\) 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
\(415\) 10.1739 + 5.39155i 0.499415 + 0.264661i
\(416\) 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
\(423\) −29.0307 29.0307i −1.41152 1.41152i
\(424\) 0 0
\(425\) 5.29814 27.6638i 0.256997 1.34189i
\(426\) 0 0
\(427\) −53.4922 + 53.4922i −2.58867 + 2.58867i
\(428\) 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
\(441\) 53.6054 2.55264
\(442\) 0 0
\(443\) −14.2232 14.2232i −0.675766 0.675766i 0.283273 0.959039i \(-0.408580\pi\)
−0.959039 + 0.283273i \(0.908580\pi\)
\(444\) 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
\(457\) 18.7908 18.7908i 0.878998 0.878998i −0.114433 0.993431i \(-0.536505\pi\)
0.993431 + 0.114433i \(0.0365053\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −37.3746 −1.74071 −0.870354 0.492427i \(-0.836110\pi\)
−0.870354 + 0.492427i \(0.836110\pi\)
\(462\) 0 0
\(463\) 25.6103 + 25.6103i 1.19021 + 1.19021i 0.977007 + 0.213205i \(0.0683902\pi\)
0.213205 + 0.977007i \(0.431610\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.8467 10.8467i 0.501927 0.501927i −0.410110 0.912036i \(-0.634510\pi\)
0.912036 + 0.410110i \(0.134510\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.67806 3.67806i −0.169118 0.169118i
\(474\) 0 0
\(475\) −12.2371 18.0348i −0.561478 0.827492i
\(476\) 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
\(495\) −4.23713 13.7910i −0.190445 0.619857i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.625414i 0.0279974i 0.999902 + 0.0139987i \(0.00445607\pi\)
−0.999902 + 0.0139987i \(0.995544\pi\)
\(500\) 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
\(505\) 37.2679 11.4502i 1.65840 0.509526i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 48.7917 2.15842
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −20.8119 + 20.8119i −0.915306 + 0.915306i
\(518\) 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
\(539\) 38.4293i 1.65527i
\(540\) 0 0
\(541\) −2.03559 −0.0875168 −0.0437584 0.999042i \(-0.513933\pi\)
−0.0437584 + 0.999042i \(0.513933\pi\)
\(542\) 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
\(549\) 45.5095i 1.94230i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.3039 31.3039i 1.32639 1.32639i 0.417889 0.908498i \(-0.362770\pi\)
0.908498 0.417889i \(-0.137230\pi\)
\(558\) 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
\(567\) −31.7360 + 31.7360i −1.33279 + 1.33279i
\(568\) 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
\(575\) 44.1616 + 8.45777i 1.84167 + 0.352714i
\(576\) 0 0
\(577\) −10.8241 + 10.8241i −0.450614 + 0.450614i −0.895558 0.444945i \(-0.853223\pi\)
0.444945 + 0.895558i \(0.353223\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −25.6787 −1.06533
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.4679 31.4679i 1.29882 1.29882i 0.369649 0.929172i \(-0.379478\pi\)
0.929172 0.369649i \(-0.120522\pi\)
\(588\) 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
\(595\) 18.4486 + 60.0463i 0.756319 + 2.46166i
\(596\) 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
\(605\) 13.6254 4.18627i 0.553952 0.170196i
\(606\) 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
\(613\) −13.7028 13.7028i −0.553450 0.553450i 0.373985 0.927435i \(-0.377991\pi\)
−0.927435 + 0.373985i \(0.877991\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.1282 35.1282i 1.41421 1.41421i 0.705008 0.709199i \(-0.250943\pi\)
0.709199 0.705008i \(-0.249057\pi\)
\(618\) 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
\(625\) 9.23713 23.2309i 0.369485 0.929237i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −10.9836 −0.437249 −0.218624 0.975809i \(-0.570157\pi\)
−0.218624 + 0.975809i \(0.570157\pi\)
\(632\) 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
\(643\) 17.8992 + 17.8992i 0.705874 + 0.705874i 0.965665 0.259791i \(-0.0836535\pi\)
−0.259791 + 0.965665i \(0.583654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.83614 2.83614i 0.111500 0.111500i −0.649155 0.760656i \(-0.724878\pi\)
0.760656 + 0.649155i \(0.224878\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28.3438 28.3438i −1.10918 1.10918i −0.993259 0.115920i \(-0.963018\pi\)
−0.115920 0.993259i \(-0.536982\pi\)
\(654\) 0 0
\(655\) −32.8625 + 10.0967i −1.28405 + 0.394510i
\(656\) 0 0
\(657\) −20.7552 + 20.7552i −0.809739 + 0.809739i
\(658\) 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
\(665\) 42.9476 + 22.7597i 1.66544 + 0.882585i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −32.6254 −1.25949
\(672\) 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
\(685\) −10.6413 + 20.0802i −0.406585 + 0.767226i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 49.9259 1.89927 0.949636 0.313355i \(-0.101453\pi\)
0.949636 + 0.313355i \(0.101453\pi\)
\(692\) 0 0
\(693\) 22.7513 + 22.7513i 0.864250 + 0.864250i
\(694\) 0 0
\(695\) −15.3504 49.9622i −0.582273 1.89517i
\(696\) 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
\(707\) −61.4818 + 61.4818i −2.31226 + 2.31226i
\(708\) 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
\(719\) 43.3746i 1.61760i 0.588084 + 0.808800i \(0.299882\pi\)
−0.588084 + 0.808800i \(0.700118\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 32.4511 32.4511i 1.20354 1.20354i 0.230463 0.973081i \(-0.425976\pi\)
0.973081 0.230463i \(-0.0740239\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) −13.6246 −0.503923
\(732\) 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
\(739\) 54.2273i 1.99478i 0.0721811 + 0.997392i \(0.477004\pi\)
−0.0721811 + 0.997392i \(0.522996\pi\)
\(740\) 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
\(745\) −8.78325 28.5876i −0.321793 1.04737i
\(746\) 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
\(757\) 27.8077 27.8077i 1.01069 1.01069i 0.0107448 0.999942i \(-0.496580\pi\)
0.999942 0.0107448i \(-0.00342025\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −49.8108 −1.80564 −0.902821 0.430017i \(-0.858508\pi\)
−0.902821 + 0.430017i \(0.858508\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −33.3905 17.6950i −1.20724 0.639765i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.62541i 0.238919i −0.992839 0.119459i \(-0.961884\pi\)
0.992839 0.119459i \(-0.0381161\pi\)
\(770\) 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
\(785\) −26.2356 + 49.5066i −0.936388 + 1.76696i
\(786\) 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
\(799\) 77.0930i 2.72736i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.8793 + 14.8793i 0.525078 + 0.525078i
\(804\) 0 0