Properties

Label 360.4.f.d.289.1
Level $360$
Weight $4$
Character 360.289
Analytic conductor $21.241$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,4,Mod(289,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.f (of order \(2\), degree \(1\), 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: \(21.2406876021\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{129})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 65x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.1
Root \(5.17891i\) of defining polynomial
Character \(\chi\) \(=\) 360.289
Dual form 360.4.f.d.289.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-11.1789 - 0.178908i) q^{5} -33.0735i q^{7} +O(q^{10})\) \(q+(-11.1789 - 0.178908i) q^{5} -33.0735i q^{7} -48.3578 q^{11} +60.3578i q^{13} +17.7891i q^{17} +130.863 q^{19} +70.8625i q^{23} +(124.936 + 4.00000i) q^{25} +104.505 q^{29} -210.441 q^{31} +(-5.91712 + 369.725i) q^{35} +300.945i q^{37} -240.147 q^{41} +108.000i q^{43} +278.991i q^{47} -750.853 q^{49} +328.358i q^{53} +(540.588 + 8.65162i) q^{55} +889.533 q^{59} -241.450 q^{61} +(10.7985 - 674.735i) q^{65} +103.834i q^{67} +277.597 q^{71} -274.403i q^{73} +1599.36i q^{77} -366.991 q^{79} +57.7251i q^{83} +(3.18262 - 198.863i) q^{85} -203.175 q^{89} +1996.24 q^{91} +(-1462.90 - 23.4124i) q^{95} +1283.45i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 22 q^{5} - 148 q^{11} + 160 q^{19} + 100 q^{29} - 24 q^{31} - 796 q^{35} - 688 q^{41} - 3276 q^{49} + 1072 q^{55} + 1332 q^{59} + 488 q^{61} - 820 q^{65} - 616 q^{71} - 2104 q^{79} + 2148 q^{85} + 1368 q^{89} + 1352 q^{91} - 2944 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(4\) 0 0
\(5\) −11.1789 0.178908i −0.999872 0.0160020i
\(6\) 0 0
\(7\) 33.0735i 1.78580i −0.450257 0.892899i \(-0.648667\pi\)
0.450257 0.892899i \(-0.351333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −48.3578 −1.32549 −0.662747 0.748844i \(-0.730609\pi\)
−0.662747 + 0.748844i \(0.730609\pi\)
\(12\) 0 0
\(13\) 60.3578i 1.28771i 0.765147 + 0.643856i \(0.222666\pi\)
−0.765147 + 0.643856i \(0.777334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 17.7891i 0.253793i 0.991916 + 0.126897i \(0.0405017\pi\)
−0.991916 + 0.126897i \(0.959498\pi\)
\(18\) 0 0
\(19\) 130.863 1.58010 0.790051 0.613042i \(-0.210054\pi\)
0.790051 + 0.613042i \(0.210054\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 70.8625i 0.642429i 0.947007 + 0.321214i \(0.104091\pi\)
−0.947007 + 0.321214i \(0.895909\pi\)
\(24\) 0 0
\(25\) 124.936 + 4.00000i 0.999488 + 0.0320000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 104.505 0.669174 0.334587 0.942365i \(-0.391403\pi\)
0.334587 + 0.942365i \(0.391403\pi\)
\(30\) 0 0
\(31\) −210.441 −1.21923 −0.609617 0.792696i \(-0.708677\pi\)
−0.609617 + 0.792696i \(0.708677\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.91712 + 369.725i −0.0285764 + 1.78557i
\(36\) 0 0
\(37\) 300.945i 1.33717i 0.743638 + 0.668583i \(0.233099\pi\)
−0.743638 + 0.668583i \(0.766901\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −240.147 −0.914747 −0.457374 0.889275i \(-0.651210\pi\)
−0.457374 + 0.889275i \(0.651210\pi\)
\(42\) 0 0
\(43\) 108.000i 0.383020i 0.981491 + 0.191510i \(0.0613384\pi\)
−0.981491 + 0.191510i \(0.938662\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 278.991i 0.865850i 0.901430 + 0.432925i \(0.142519\pi\)
−0.901430 + 0.432925i \(0.857481\pi\)
\(48\) 0 0
\(49\) −750.853 −2.18908
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 328.358i 0.851008i 0.904957 + 0.425504i \(0.139903\pi\)
−0.904957 + 0.425504i \(0.860097\pi\)
\(54\) 0 0
\(55\) 540.588 + 8.65162i 1.32532 + 0.0212106i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 889.533 1.96284 0.981418 0.191882i \(-0.0614589\pi\)
0.981418 + 0.191882i \(0.0614589\pi\)
\(60\) 0 0
\(61\) −241.450 −0.506795 −0.253398 0.967362i \(-0.581548\pi\)
−0.253398 + 0.967362i \(0.581548\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.7985 674.735i 0.0206060 1.28755i
\(66\) 0 0
\(67\) 103.834i 0.189334i 0.995509 + 0.0946669i \(0.0301786\pi\)
−0.995509 + 0.0946669i \(0.969821\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 277.597 0.464010 0.232005 0.972715i \(-0.425471\pi\)
0.232005 + 0.972715i \(0.425471\pi\)
\(72\) 0 0
\(73\) 274.403i 0.439951i −0.975505 0.219976i \(-0.929402\pi\)
0.975505 0.219976i \(-0.0705978\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1599.36i 2.36706i
\(78\) 0 0
\(79\) −366.991 −0.522654 −0.261327 0.965250i \(-0.584160\pi\)
−0.261327 + 0.965250i \(0.584160\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 57.7251i 0.0763391i 0.999271 + 0.0381696i \(0.0121527\pi\)
−0.999271 + 0.0381696i \(0.987847\pi\)
\(84\) 0 0
\(85\) 3.18262 198.863i 0.00406121 0.253761i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −203.175 −0.241983 −0.120992 0.992654i \(-0.538607\pi\)
−0.120992 + 0.992654i \(0.538607\pi\)
\(90\) 0 0
\(91\) 1996.24 2.29959
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1462.90 23.4124i −1.57990 0.0252849i
\(96\) 0 0
\(97\) 1283.45i 1.34345i 0.740801 + 0.671725i \(0.234446\pi\)
−0.740801 + 0.671725i \(0.765554\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −886.908 −0.873768 −0.436884 0.899518i \(-0.643918\pi\)
−0.436884 + 0.899518i \(0.643918\pi\)
\(102\) 0 0
\(103\) 783.055i 0.749094i 0.927208 + 0.374547i \(0.122202\pi\)
−0.927208 + 0.374547i \(0.877798\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1301.51i 1.17590i −0.808897 0.587950i \(-0.799935\pi\)
0.808897 0.587950i \(-0.200065\pi\)
\(108\) 0 0
\(109\) −1161.91 −1.02102 −0.510508 0.859873i \(-0.670543\pi\)
−0.510508 + 0.859873i \(0.670543\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 507.808i 0.422748i 0.977405 + 0.211374i \(0.0677938\pi\)
−0.977405 + 0.211374i \(0.932206\pi\)
\(114\) 0 0
\(115\) 12.6779 792.166i 0.0102802 0.642346i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 588.346 0.453224
\(120\) 0 0
\(121\) 1007.48 0.756933
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1395.93 67.0677i −0.998848 0.0479898i
\(126\) 0 0
\(127\) 796.064i 0.556215i 0.960550 + 0.278107i \(0.0897071\pi\)
−0.960550 + 0.278107i \(0.910293\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 91.4764 0.0610102 0.0305051 0.999535i \(-0.490288\pi\)
0.0305051 + 0.999535i \(0.490288\pi\)
\(132\) 0 0
\(133\) 4328.08i 2.82174i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2273.28i 1.41766i −0.705380 0.708829i \(-0.749224\pi\)
0.705380 0.708829i \(-0.250776\pi\)
\(138\) 0 0
\(139\) 738.735 0.450782 0.225391 0.974268i \(-0.427634\pi\)
0.225391 + 0.974268i \(0.427634\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2918.77i 1.70685i
\(144\) 0 0
\(145\) −1168.25 18.6968i −0.669088 0.0107082i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1507.15 0.828661 0.414330 0.910127i \(-0.364016\pi\)
0.414330 + 0.910127i \(0.364016\pi\)
\(150\) 0 0
\(151\) 154.365 0.0831925 0.0415962 0.999135i \(-0.486756\pi\)
0.0415962 + 0.999135i \(0.486756\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2352.50 + 37.6496i 1.21908 + 0.0195102i
\(156\) 0 0
\(157\) 315.514i 0.160387i −0.996779 0.0801935i \(-0.974446\pi\)
0.996779 0.0801935i \(-0.0255538\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2343.67 1.14725
\(162\) 0 0
\(163\) 1457.85i 0.700539i 0.936649 + 0.350270i \(0.113910\pi\)
−0.936649 + 0.350270i \(0.886090\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2198.61i 1.01876i 0.860541 + 0.509381i \(0.170126\pi\)
−0.860541 + 0.509381i \(0.829874\pi\)
\(168\) 0 0
\(169\) −1446.07 −0.658200
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2030.49i 0.892343i −0.894948 0.446171i \(-0.852787\pi\)
0.894948 0.446171i \(-0.147213\pi\)
\(174\) 0 0
\(175\) 132.294 4132.06i 0.0571456 1.78488i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 427.220 0.178391 0.0891954 0.996014i \(-0.471570\pi\)
0.0891954 + 0.996014i \(0.471570\pi\)
\(180\) 0 0
\(181\) −3779.97 −1.55228 −0.776140 0.630561i \(-0.782825\pi\)
−0.776140 + 0.630561i \(0.782825\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 53.8416 3364.24i 0.0213974 1.33699i
\(186\) 0 0
\(187\) 860.241i 0.336401i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1565.60 −0.593103 −0.296551 0.955017i \(-0.595837\pi\)
−0.296551 + 0.955017i \(0.595837\pi\)
\(192\) 0 0
\(193\) 2642.63i 0.985599i 0.870143 + 0.492800i \(0.164026\pi\)
−0.870143 + 0.492800i \(0.835974\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 98.4522i 0.0356062i −0.999842 0.0178031i \(-0.994333\pi\)
0.999842 0.0178031i \(-0.00566721\pi\)
\(198\) 0 0
\(199\) −1131.62 −0.403106 −0.201553 0.979478i \(-0.564599\pi\)
−0.201553 + 0.979478i \(0.564599\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3456.33i 1.19501i
\(204\) 0 0
\(205\) 2684.58 + 42.9643i 0.914630 + 0.0146378i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6328.23 −2.09441
\(210\) 0 0
\(211\) −1902.85 −0.620841 −0.310420 0.950599i \(-0.600470\pi\)
−0.310420 + 0.950599i \(0.600470\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 19.3221 1207.32i 0.00612910 0.382971i
\(216\) 0 0
\(217\) 6960.00i 2.17731i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1073.71 −0.326813
\(222\) 0 0
\(223\) 4855.59i 1.45809i −0.684465 0.729046i \(-0.739964\pi\)
0.684465 0.729046i \(-0.260036\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6536.19i 1.91111i 0.294812 + 0.955555i \(0.404743\pi\)
−0.294812 + 0.955555i \(0.595257\pi\)
\(228\) 0 0
\(229\) 5510.35 1.59011 0.795053 0.606539i \(-0.207443\pi\)
0.795053 + 0.606539i \(0.207443\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5915.65i 1.66329i 0.555306 + 0.831646i \(0.312601\pi\)
−0.555306 + 0.831646i \(0.687399\pi\)
\(234\) 0 0
\(235\) 49.9137 3118.81i 0.0138554 0.865739i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2263.25 0.612543 0.306272 0.951944i \(-0.400918\pi\)
0.306272 + 0.951944i \(0.400918\pi\)
\(240\) 0 0
\(241\) 772.493 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8393.72 + 134.334i 2.18880 + 0.0350297i
\(246\) 0 0
\(247\) 7898.58i 2.03471i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −596.192 −0.149926 −0.0749628 0.997186i \(-0.523884\pi\)
−0.0749628 + 0.997186i \(0.523884\pi\)
\(252\) 0 0
\(253\) 3426.76i 0.851535i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4139.19i 1.00465i −0.864678 0.502326i \(-0.832478\pi\)
0.864678 0.502326i \(-0.167522\pi\)
\(258\) 0 0
\(259\) 9953.30 2.38791
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1611.34i 0.377792i −0.981997 0.188896i \(-0.939509\pi\)
0.981997 0.188896i \(-0.0604909\pi\)
\(264\) 0 0
\(265\) 58.7460 3670.68i 0.0136179 0.850899i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7031.33 −1.59371 −0.796855 0.604171i \(-0.793504\pi\)
−0.796855 + 0.604171i \(0.793504\pi\)
\(270\) 0 0
\(271\) −5441.27 −1.21968 −0.609840 0.792524i \(-0.708766\pi\)
−0.609840 + 0.792524i \(0.708766\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6041.63 193.431i −1.32481 0.0424158i
\(276\) 0 0
\(277\) 1080.43i 0.234355i 0.993111 + 0.117178i \(0.0373847\pi\)
−0.993111 + 0.117178i \(0.962615\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1602.99 0.340308 0.170154 0.985417i \(-0.445573\pi\)
0.170154 + 0.985417i \(0.445573\pi\)
\(282\) 0 0
\(283\) 334.810i 0.0703265i −0.999382 0.0351632i \(-0.988805\pi\)
0.999382 0.0351632i \(-0.0111951\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7942.49i 1.63355i
\(288\) 0 0
\(289\) 4596.55 0.935589
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 539.250i 0.107520i 0.998554 + 0.0537599i \(0.0171206\pi\)
−0.998554 + 0.0537599i \(0.982879\pi\)
\(294\) 0 0
\(295\) −9944.01 159.145i −1.96258 0.0314094i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4277.11 −0.827263
\(300\) 0 0
\(301\) 3571.93 0.683996
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2699.15 + 43.1974i 0.506731 + 0.00810977i
\(306\) 0 0
\(307\) 8477.58i 1.57603i 0.615656 + 0.788015i \(0.288891\pi\)
−0.615656 + 0.788015i \(0.711109\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3646.92 −0.664945 −0.332473 0.943113i \(-0.607883\pi\)
−0.332473 + 0.943113i \(0.607883\pi\)
\(312\) 0 0
\(313\) 7537.05i 1.36108i −0.732709 0.680542i \(-0.761745\pi\)
0.732709 0.680542i \(-0.238255\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10721.2i 1.89956i 0.312916 + 0.949781i \(0.398694\pi\)
−0.312916 + 0.949781i \(0.601306\pi\)
\(318\) 0 0
\(319\) −5053.62 −0.886986
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2327.92i 0.401019i
\(324\) 0 0
\(325\) −241.431 + 7540.86i −0.0412068 + 1.28705i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9227.18 1.54623
\(330\) 0 0
\(331\) −4405.14 −0.731505 −0.365753 0.930712i \(-0.619188\pi\)
−0.365753 + 0.930712i \(0.619188\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.5768 1160.75i 0.00302973 0.189310i
\(336\) 0 0
\(337\) 186.117i 0.0300844i −0.999887 0.0150422i \(-0.995212\pi\)
0.999887 0.0150422i \(-0.00478827\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10176.5 1.61609
\(342\) 0 0
\(343\) 13489.1i 2.12345i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5547.38i 0.858211i −0.903254 0.429105i \(-0.858829\pi\)
0.903254 0.429105i \(-0.141171\pi\)
\(348\) 0 0
\(349\) −9078.82 −1.39249 −0.696244 0.717805i \(-0.745147\pi\)
−0.696244 + 0.717805i \(0.745147\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10678.1i 1.61002i 0.593262 + 0.805009i \(0.297840\pi\)
−0.593262 + 0.805009i \(0.702160\pi\)
\(354\) 0 0
\(355\) −3103.23 49.6644i −0.463951 0.00742511i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −265.733 −0.0390665 −0.0195332 0.999809i \(-0.506218\pi\)
−0.0195332 + 0.999809i \(0.506218\pi\)
\(360\) 0 0
\(361\) 10266.0 1.49672
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −49.0930 + 3067.53i −0.00704012 + 0.439895i
\(366\) 0 0
\(367\) 5854.21i 0.832663i −0.909213 0.416332i \(-0.863316\pi\)
0.909213 0.416332i \(-0.136684\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10859.9 1.51973
\(372\) 0 0
\(373\) 10134.5i 1.40682i 0.710787 + 0.703408i \(0.248339\pi\)
−0.710787 + 0.703408i \(0.751661\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6307.68i 0.861703i
\(378\) 0 0
\(379\) 4235.70 0.574072 0.287036 0.957920i \(-0.407330\pi\)
0.287036 + 0.957920i \(0.407330\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8100.84i 1.08077i 0.841419 + 0.540383i \(0.181721\pi\)
−0.841419 + 0.540383i \(0.818279\pi\)
\(384\) 0 0
\(385\) 286.139 17879.1i 0.0378779 2.36676i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13820.2 −1.80131 −0.900656 0.434532i \(-0.856914\pi\)
−0.900656 + 0.434532i \(0.856914\pi\)
\(390\) 0 0
\(391\) −1260.58 −0.163044
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4102.55 + 65.6577i 0.522587 + 0.00836353i
\(396\) 0 0
\(397\) 11523.9i 1.45685i 0.685125 + 0.728425i \(0.259748\pi\)
−0.685125 + 0.728425i \(0.740252\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −700.915 −0.0872869 −0.0436434 0.999047i \(-0.513897\pi\)
−0.0436434 + 0.999047i \(0.513897\pi\)
\(402\) 0 0
\(403\) 12701.7i 1.57002i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14553.1i 1.77240i
\(408\) 0 0
\(409\) −6650.69 −0.804047 −0.402024 0.915629i \(-0.631693\pi\)
−0.402024 + 0.915629i \(0.631693\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 29419.9i 3.50523i
\(414\) 0 0
\(415\) 10.3275 645.303i 0.00122158 0.0763294i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13844.4 1.61418 0.807091 0.590426i \(-0.201040\pi\)
0.807091 + 0.590426i \(0.201040\pi\)
\(420\) 0 0
\(421\) 12576.3 1.45590 0.727949 0.685631i \(-0.240474\pi\)
0.727949 + 0.685631i \(0.240474\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −71.1563 + 2222.50i −0.00812139 + 0.253663i
\(426\) 0 0
\(427\) 7985.59i 0.905035i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13440.6 −1.50212 −0.751059 0.660236i \(-0.770456\pi\)
−0.751059 + 0.660236i \(0.770456\pi\)
\(432\) 0 0
\(433\) 3012.87i 0.334387i −0.985924 0.167193i \(-0.946530\pi\)
0.985924 0.167193i \(-0.0534704\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9273.25i 1.01510i
\(438\) 0 0
\(439\) −10675.0 −1.16057 −0.580283 0.814415i \(-0.697058\pi\)
−0.580283 + 0.814415i \(0.697058\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 125.868i 0.0134992i 0.999977 + 0.00674962i \(0.00214849\pi\)
−0.999977 + 0.00674962i \(0.997852\pi\)
\(444\) 0 0
\(445\) 2271.28 + 36.3497i 0.241952 + 0.00387223i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9707.21 −1.02029 −0.510146 0.860088i \(-0.670409\pi\)
−0.510146 + 0.860088i \(0.670409\pi\)
\(450\) 0 0
\(451\) 11613.0 1.21249
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −22315.8 357.144i −2.29930 0.0367982i
\(456\) 0 0
\(457\) 1279.97i 0.131016i −0.997852 0.0655082i \(-0.979133\pi\)
0.997852 0.0655082i \(-0.0208668\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3080.48 −0.311220 −0.155610 0.987819i \(-0.549734\pi\)
−0.155610 + 0.987819i \(0.549734\pi\)
\(462\) 0 0
\(463\) 18017.3i 1.80850i −0.427008 0.904248i \(-0.640432\pi\)
0.427008 0.904248i \(-0.359568\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7236.77i 0.717083i 0.933514 + 0.358541i \(0.116726\pi\)
−0.933514 + 0.358541i \(0.883274\pi\)
\(468\) 0 0
\(469\) 3434.16 0.338112
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5222.64i 0.507690i
\(474\) 0 0
\(475\) 16349.4 + 523.450i 1.57929 + 0.0505632i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4932.78 0.470532 0.235266 0.971931i \(-0.424404\pi\)
0.235266 + 0.971931i \(0.424404\pi\)
\(480\) 0 0
\(481\) −18164.4 −1.72188
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 229.620 14347.6i 0.0214979 1.34328i
\(486\) 0 0
\(487\) 5937.91i 0.552510i −0.961084 0.276255i \(-0.910907\pi\)
0.961084 0.276255i \(-0.0890934\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15703.5 −1.44336 −0.721678 0.692229i \(-0.756629\pi\)
−0.721678 + 0.692229i \(0.756629\pi\)
\(492\) 0 0
\(493\) 1859.04i 0.169832i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9181.09i 0.828628i
\(498\) 0 0
\(499\) 5656.77 0.507478 0.253739 0.967273i \(-0.418340\pi\)
0.253739 + 0.967273i \(0.418340\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 290.441i 0.0257457i 0.999917 + 0.0128729i \(0.00409768\pi\)
−0.999917 + 0.0128729i \(0.995902\pi\)
\(504\) 0 0
\(505\) 9914.66 + 158.675i 0.873657 + 0.0139821i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17330.6 1.50916 0.754582 0.656206i \(-0.227840\pi\)
0.754582 + 0.656206i \(0.227840\pi\)
\(510\) 0 0
\(511\) −9075.45 −0.785664
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 140.095 8753.70i 0.0119870 0.748998i
\(516\) 0 0
\(517\) 13491.4i 1.14768i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6174.36 −0.519201 −0.259601 0.965716i \(-0.583591\pi\)
−0.259601 + 0.965716i \(0.583591\pi\)
\(522\) 0 0
\(523\) 13389.4i 1.11946i 0.828675 + 0.559730i \(0.189095\pi\)
−0.828675 + 0.559730i \(0.810905\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3743.55i 0.309434i
\(528\) 0 0
\(529\) 7145.50 0.587285
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14494.7i 1.17793i
\(534\) 0 0
\(535\) −232.850 + 14549.4i −0.0188168 + 1.17575i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36309.6 2.90161
\(540\) 0 0
\(541\) −14355.5 −1.14084 −0.570418 0.821354i \(-0.693219\pi\)
−0.570418 + 0.821354i \(0.693219\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12988.9 + 207.875i 1.02089 + 0.0163384i
\(546\) 0 0
\(547\) 21133.9i 1.65195i −0.563704 0.825977i \(-0.690624\pi\)
0.563704 0.825977i \(-0.309376\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 13675.8 1.05736
\(552\) 0 0
\(553\) 12137.6i 0.933355i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3098.60i 0.235713i −0.993031 0.117856i \(-0.962398\pi\)
0.993031 0.117856i \(-0.0376023\pi\)
\(558\) 0 0
\(559\) −6518.64 −0.493219
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7908.77i 0.592033i 0.955183 + 0.296017i \(0.0956584\pi\)
−0.955183 + 0.296017i \(0.904342\pi\)
\(564\) 0 0
\(565\) 90.8511 5676.74i 0.00676484 0.422694i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10740.7 −0.791345 −0.395673 0.918392i \(-0.629488\pi\)
−0.395673 + 0.918392i \(0.629488\pi\)
\(570\) 0 0
\(571\) 14701.2 1.07745 0.538725 0.842482i \(-0.318906\pi\)
0.538725 + 0.842482i \(0.318906\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −283.450 + 8853.28i −0.0205577 + 0.642100i
\(576\) 0 0
\(577\) 15788.1i 1.13911i 0.821954 + 0.569554i \(0.192884\pi\)
−0.821954 + 0.569554i \(0.807116\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1909.17 0.136326
\(582\) 0 0
\(583\) 15878.7i 1.12801i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14778.5i 1.03913i −0.854430 0.519567i \(-0.826093\pi\)
0.854430 0.519567i \(-0.173907\pi\)
\(588\) 0 0
\(589\) −27538.8 −1.92651
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 11143.9i 0.771713i 0.922559 + 0.385857i \(0.126094\pi\)
−0.922559 + 0.385857i \(0.873906\pi\)
\(594\) 0 0
\(595\) −6577.07 105.260i −0.453166 0.00725251i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7707.87 −0.525768 −0.262884 0.964827i \(-0.584674\pi\)
−0.262884 + 0.964827i \(0.584674\pi\)
\(600\) 0 0
\(601\) −13681.4 −0.928580 −0.464290 0.885683i \(-0.653691\pi\)
−0.464290 + 0.885683i \(0.653691\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11262.5 180.246i −0.756837 0.0121125i
\(606\) 0 0
\(607\) 11552.4i 0.772481i 0.922398 + 0.386241i \(0.126227\pi\)
−0.922398 + 0.386241i \(0.873773\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16839.3 −1.11496
\(612\) 0 0
\(613\) 9904.66i 0.652603i 0.945266 + 0.326301i \(0.105802\pi\)
−0.945266 + 0.326301i \(0.894198\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20323.5i 1.32608i −0.748582 0.663042i \(-0.769265\pi\)
0.748582 0.663042i \(-0.230735\pi\)
\(618\) 0 0
\(619\) 9223.99 0.598940 0.299470 0.954106i \(-0.403190\pi\)
0.299470 + 0.954106i \(0.403190\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6719.70i 0.432134i
\(624\) 0 0
\(625\) 15593.0 + 999.488i 0.997952 + 0.0639672i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5353.54 −0.339364
\(630\) 0 0
\(631\) −16916.0 −1.06722 −0.533610 0.845730i \(-0.679165\pi\)
−0.533610 + 0.845730i \(0.679165\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 142.422 8899.13i 0.00890057 0.556143i
\(636\) 0 0
\(637\) 45319.9i 2.81890i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5811.35 0.358088 0.179044 0.983841i \(-0.442700\pi\)
0.179044 + 0.983841i \(0.442700\pi\)
\(642\) 0 0
\(643\) 27931.7i 1.71309i −0.516069 0.856547i \(-0.672605\pi\)
0.516069 0.856547i \(-0.327395\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25437.8i 1.54569i −0.634595 0.772845i \(-0.718833\pi\)
0.634595 0.772845i \(-0.281167\pi\)
\(648\) 0 0
\(649\) −43015.9 −2.60173
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22647.7i 1.35723i 0.734493 + 0.678617i \(0.237420\pi\)
−0.734493 + 0.678617i \(0.762580\pi\)
\(654\) 0 0
\(655\) −1022.61 16.3659i −0.0610024 0.000976288i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 25158.5 1.48716 0.743579 0.668649i \(-0.233127\pi\)
0.743579 + 0.668649i \(0.233127\pi\)
\(660\) 0 0
\(661\) 23441.0 1.37935 0.689675 0.724119i \(-0.257753\pi\)
0.689675 + 0.724119i \(0.257753\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −774.329 + 48383.2i −0.0451537 + 2.82138i
\(666\) 0 0
\(667\) 7405.47i 0.429896i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11676.0 0.671754
\(672\) 0 0
\(673\) 3145.33i 0.180154i 0.995935 + 0.0900770i \(0.0287113\pi\)
−0.995935 + 0.0900770i \(0.971289\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10606.2i 0.602109i 0.953607 + 0.301054i \(0.0973386\pi\)
−0.953607 + 0.301054i \(0.902661\pi\)
\(678\) 0 0
\(679\) 42448.1 2.39913
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7825.06i 0.438386i −0.975682 0.219193i \(-0.929658\pi\)
0.975682 0.219193i \(-0.0703424\pi\)
\(684\) 0 0
\(685\) −406.708 + 25412.8i −0.0226854 + 1.41748i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −19819.0 −1.09585
\(690\) 0 0
\(691\) 22750.9 1.25251 0.626256 0.779618i \(-0.284587\pi\)
0.626256 + 0.779618i \(0.284587\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8258.25 132.166i −0.450724 0.00721343i
\(696\) 0 0
\(697\) 4271.99i 0.232157i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 133.598 0.00719817 0.00359908 0.999994i \(-0.498854\pi\)
0.00359908 + 0.999994i \(0.498854\pi\)
\(702\) 0 0
\(703\) 39382.5i 2.11286i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29333.1i 1.56037i
\(708\) 0 0
\(709\) −6886.26 −0.364766 −0.182383 0.983228i \(-0.558381\pi\)
−0.182383 + 0.983228i \(0.558381\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14912.4i 0.783271i
\(714\) 0 0
\(715\) −522.193 + 32628.7i −0.0273131 + 1.70663i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1570.18 −0.0814436 −0.0407218 0.999171i \(-0.512966\pi\)
−0.0407218 + 0.999171i \(0.512966\pi\)
\(720\) 0 0
\(721\) 25898.3 1.33773
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13056.4 + 418.019i 0.668831 + 0.0214136i
\(726\) 0 0
\(727\) 3399.18i 0.173409i −0.996234 0.0867047i \(-0.972366\pi\)
0.996234 0.0867047i \(-0.0276337\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1921.22 −0.0972078
\(732\) 0 0
\(733\) 14152.1i 0.713125i 0.934272 + 0.356562i \(0.116051\pi\)
−0.934272 + 0.356562i \(0.883949\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5021.20i 0.250961i
\(738\) 0 0
\(739\) 14919.5 0.742655 0.371328 0.928502i \(-0.378903\pi\)
0.371328 + 0.928502i \(0.378903\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7287.29i 0.359818i 0.983683 + 0.179909i \(0.0575803\pi\)
−0.983683 + 0.179909i \(0.942420\pi\)
\(744\) 0 0
\(745\) −16848.3 269.642i −0.828555 0.0132603i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −43045.3 −2.09992
\(750\) 0 0
\(751\) 18783.4 0.912670 0.456335 0.889808i \(-0.349162\pi\)
0.456335 + 0.889808i \(0.349162\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1725.63 27.6172i −0.0831818 0.00133125i
\(756\) 0 0
\(757\) 30614.3i 1.46988i −0.678134 0.734939i \(-0.737211\pi\)
0.678134 0.734939i \(-0.262789\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15277.2 0.727723 0.363861 0.931453i \(-0.381458\pi\)
0.363861 + 0.931453i \(0.381458\pi\)
\(762\) 0 0
\(763\) 38428.4i 1.82333i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 53690.3i 2.52757i
\(768\) 0 0
\(769\) 17700.1 0.830016 0.415008 0.909818i \(-0.363779\pi\)
0.415008 + 0.909818i \(0.363779\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 29362.5i 1.36623i 0.730310 + 0.683116i \(0.239375\pi\)
−0.730310 + 0.683116i \(0.760625\pi\)
\(774\) 0 0
\(775\) −26291.6 841.763i −1.21861 0.0390155i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −31426.2 −1.44539
\(780\) 0 0
\(781\) −13424.0 −0.615042
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −56.4481 + 3527.10i −0.00256652 + 0.160367i
\(786\) 0 0
\(787\) 2816.92i 0.127589i −0.997963 0.0637943i \(-0.979680\pi\)
0.997963 0.0637943i \(-0.0203202\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 16795.0 0.754943
\(792\) 0 0
\(793\) 14573.4i 0.652606i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0