Properties

Label 800.4.c.f.449.1
Level $800$
Weight $4$
Character 800.449
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(449,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("800.449");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.4.c.f.449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} -6.00000i q^{7} +23.0000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} -6.00000i q^{7} +23.0000 q^{9} +60.0000 q^{11} +50.0000i q^{13} +30.0000i q^{17} -40.0000 q^{19} -12.0000 q^{21} +178.000i q^{23} -100.000i q^{27} -166.000 q^{29} +20.0000 q^{31} -120.000i q^{33} -10.0000i q^{37} +100.000 q^{39} -250.000 q^{41} +142.000i q^{43} -214.000i q^{47} +307.000 q^{49} +60.0000 q^{51} +490.000i q^{53} +80.0000i q^{57} +800.000 q^{59} +250.000 q^{61} -138.000i q^{63} +774.000i q^{67} +356.000 q^{69} +100.000 q^{71} -230.000i q^{73} -360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} +982.000i q^{83} +332.000i q^{87} -874.000 q^{89} +300.000 q^{91} -40.0000i q^{93} +310.000i q^{97} +1380.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 46 q^{9} + 120 q^{11} - 80 q^{19} - 24 q^{21} - 332 q^{29} + 40 q^{31} + 200 q^{39} - 500 q^{41} + 614 q^{49} + 120 q^{51} + 1600 q^{59} + 500 q^{61} + 712 q^{69} + 200 q^{71} + 2640 q^{79} + 842 q^{81} - 1748 q^{89} + 600 q^{91} + 2760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.00000i − 0.384900i −0.981307 0.192450i \(-0.938357\pi\)
0.981307 0.192450i \(-0.0616434\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 6.00000i − 0.323970i −0.986793 0.161985i \(-0.948210\pi\)
0.986793 0.161985i \(-0.0517895\pi\)
\(8\) 0 0
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) 60.0000 1.64461 0.822304 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) 0 0
\(13\) 50.0000i 1.06673i 0.845885 + 0.533366i \(0.179073\pi\)
−0.845885 + 0.533366i \(0.820927\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 30.0000i 0.428004i 0.976833 + 0.214002i \(0.0686499\pi\)
−0.976833 + 0.214002i \(0.931350\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) 178.000i 1.61372i 0.590743 + 0.806860i \(0.298835\pi\)
−0.590743 + 0.806860i \(0.701165\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 100.000i − 0.712778i
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) − 120.000i − 0.633010i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 0.0444322i −0.999753 0.0222161i \(-0.992928\pi\)
0.999753 0.0222161i \(-0.00707218\pi\)
\(38\) 0 0
\(39\) 100.000 0.410585
\(40\) 0 0
\(41\) −250.000 −0.952279 −0.476140 0.879370i \(-0.657964\pi\)
−0.476140 + 0.879370i \(0.657964\pi\)
\(42\) 0 0
\(43\) 142.000i 0.503600i 0.967779 + 0.251800i \(0.0810225\pi\)
−0.967779 + 0.251800i \(0.918977\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 214.000i − 0.664151i −0.943253 0.332076i \(-0.892251\pi\)
0.943253 0.332076i \(-0.107749\pi\)
\(48\) 0 0
\(49\) 307.000 0.895044
\(50\) 0 0
\(51\) 60.0000 0.164739
\(52\) 0 0
\(53\) 490.000i 1.26994i 0.772538 + 0.634969i \(0.218987\pi\)
−0.772538 + 0.634969i \(0.781013\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 80.0000i 0.185899i
\(58\) 0 0
\(59\) 800.000 1.76527 0.882637 0.470056i \(-0.155766\pi\)
0.882637 + 0.470056i \(0.155766\pi\)
\(60\) 0 0
\(61\) 250.000 0.524741 0.262371 0.964967i \(-0.415496\pi\)
0.262371 + 0.964967i \(0.415496\pi\)
\(62\) 0 0
\(63\) − 138.000i − 0.275974i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 774.000i 1.41133i 0.708545 + 0.705665i \(0.249352\pi\)
−0.708545 + 0.705665i \(0.750648\pi\)
\(68\) 0 0
\(69\) 356.000 0.621121
\(70\) 0 0
\(71\) 100.000 0.167152 0.0835762 0.996501i \(-0.473366\pi\)
0.0835762 + 0.996501i \(0.473366\pi\)
\(72\) 0 0
\(73\) − 230.000i − 0.368760i −0.982855 0.184380i \(-0.940972\pi\)
0.982855 0.184380i \(-0.0590277\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 360.000i − 0.532803i
\(78\) 0 0
\(79\) 1320.00 1.87989 0.939947 0.341321i \(-0.110874\pi\)
0.939947 + 0.341321i \(0.110874\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 982.000i 1.29866i 0.760508 + 0.649328i \(0.224950\pi\)
−0.760508 + 0.649328i \(0.775050\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 332.000i 0.409128i
\(88\) 0 0
\(89\) −874.000 −1.04094 −0.520471 0.853879i \(-0.674244\pi\)
−0.520471 + 0.853879i \(0.674244\pi\)
\(90\) 0 0
\(91\) 300.000 0.345588
\(92\) 0 0
\(93\) − 40.0000i − 0.0446001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 310.000i 0.324492i 0.986750 + 0.162246i \(0.0518738\pi\)
−0.986750 + 0.162246i \(0.948126\pi\)
\(98\) 0 0
\(99\) 1380.00 1.40096
\(100\) 0 0
\(101\) −1498.00 −1.47581 −0.737904 0.674906i \(-0.764184\pi\)
−0.737904 + 0.674906i \(0.764184\pi\)
\(102\) 0 0
\(103\) 1402.00i 1.34120i 0.741821 + 0.670598i \(0.233962\pi\)
−0.741821 + 0.670598i \(0.766038\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1194.00i − 1.07877i −0.842059 0.539385i \(-0.818657\pi\)
0.842059 0.539385i \(-0.181343\pi\)
\(108\) 0 0
\(109\) −650.000 −0.571181 −0.285590 0.958352i \(-0.592190\pi\)
−0.285590 + 0.958352i \(0.592190\pi\)
\(110\) 0 0
\(111\) −20.0000 −0.0171019
\(112\) 0 0
\(113\) − 1510.00i − 1.25707i −0.777782 0.628535i \(-0.783655\pi\)
0.777782 0.628535i \(-0.216345\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1150.00i 0.908697i
\(118\) 0 0
\(119\) 180.000 0.138660
\(120\) 0 0
\(121\) 2269.00 1.70473
\(122\) 0 0
\(123\) 500.000i 0.366532i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1246.00i − 0.870588i −0.900288 0.435294i \(-0.856645\pi\)
0.900288 0.435294i \(-0.143355\pi\)
\(128\) 0 0
\(129\) 284.000 0.193836
\(130\) 0 0
\(131\) 2660.00 1.77409 0.887043 0.461687i \(-0.152756\pi\)
0.887043 + 0.461687i \(0.152756\pi\)
\(132\) 0 0
\(133\) 240.000i 0.156471i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2770.00i − 1.72742i −0.503986 0.863712i \(-0.668134\pi\)
0.503986 0.863712i \(-0.331866\pi\)
\(138\) 0 0
\(139\) 560.000 0.341716 0.170858 0.985296i \(-0.445346\pi\)
0.170858 + 0.985296i \(0.445346\pi\)
\(140\) 0 0
\(141\) −428.000 −0.255632
\(142\) 0 0
\(143\) 3000.00i 1.75435i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 614.000i − 0.344502i
\(148\) 0 0
\(149\) 2350.00 1.29208 0.646039 0.763305i \(-0.276424\pi\)
0.646039 + 0.763305i \(0.276424\pi\)
\(150\) 0 0
\(151\) 580.000 0.312581 0.156290 0.987711i \(-0.450046\pi\)
0.156290 + 0.987711i \(0.450046\pi\)
\(152\) 0 0
\(153\) 690.000i 0.364596i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1310.00i 0.665920i 0.942941 + 0.332960i \(0.108047\pi\)
−0.942941 + 0.332960i \(0.891953\pi\)
\(158\) 0 0
\(159\) 980.000 0.488799
\(160\) 0 0
\(161\) 1068.00 0.522796
\(162\) 0 0
\(163\) 1862.00i 0.894743i 0.894348 + 0.447371i \(0.147640\pi\)
−0.894348 + 0.447371i \(0.852360\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 726.000i − 0.336405i −0.985752 0.168202i \(-0.946204\pi\)
0.985752 0.168202i \(-0.0537962\pi\)
\(168\) 0 0
\(169\) −303.000 −0.137915
\(170\) 0 0
\(171\) −920.000 −0.411428
\(172\) 0 0
\(173\) 3250.00i 1.42828i 0.700001 + 0.714141i \(0.253183\pi\)
−0.700001 + 0.714141i \(0.746817\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1600.00i − 0.679454i
\(178\) 0 0
\(179\) 1120.00 0.467669 0.233834 0.972276i \(-0.424873\pi\)
0.233834 + 0.972276i \(0.424873\pi\)
\(180\) 0 0
\(181\) −2842.00 −1.16710 −0.583548 0.812079i \(-0.698336\pi\)
−0.583548 + 0.812079i \(0.698336\pi\)
\(182\) 0 0
\(183\) − 500.000i − 0.201973i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1800.00i 0.703899i
\(188\) 0 0
\(189\) −600.000 −0.230918
\(190\) 0 0
\(191\) 3180.00 1.20469 0.602347 0.798234i \(-0.294232\pi\)
0.602347 + 0.798234i \(0.294232\pi\)
\(192\) 0 0
\(193\) − 4670.00i − 1.74173i −0.491522 0.870865i \(-0.663559\pi\)
0.491522 0.870865i \(-0.336441\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2990.00i 1.08136i 0.841227 + 0.540682i \(0.181834\pi\)
−0.841227 + 0.540682i \(0.818166\pi\)
\(198\) 0 0
\(199\) 4240.00 1.51038 0.755190 0.655506i \(-0.227545\pi\)
0.755190 + 0.655506i \(0.227545\pi\)
\(200\) 0 0
\(201\) 1548.00 0.543221
\(202\) 0 0
\(203\) 996.000i 0.344362i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4094.00i 1.37465i
\(208\) 0 0
\(209\) −2400.00 −0.794313
\(210\) 0 0
\(211\) −4060.00 −1.32465 −0.662327 0.749215i \(-0.730431\pi\)
−0.662327 + 0.749215i \(0.730431\pi\)
\(212\) 0 0
\(213\) − 200.000i − 0.0643370i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 120.000i − 0.0375398i
\(218\) 0 0
\(219\) −460.000 −0.141936
\(220\) 0 0
\(221\) −1500.00 −0.456565
\(222\) 0 0
\(223\) − 5622.00i − 1.68824i −0.536156 0.844119i \(-0.680124\pi\)
0.536156 0.844119i \(-0.319876\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1554.00i − 0.454373i −0.973851 0.227186i \(-0.927047\pi\)
0.973851 0.227186i \(-0.0729526\pi\)
\(228\) 0 0
\(229\) −1134.00 −0.327235 −0.163618 0.986524i \(-0.552316\pi\)
−0.163618 + 0.986524i \(0.552316\pi\)
\(230\) 0 0
\(231\) −720.000 −0.205076
\(232\) 0 0
\(233\) − 1710.00i − 0.480798i −0.970674 0.240399i \(-0.922722\pi\)
0.970674 0.240399i \(-0.0772782\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 2640.00i − 0.723571i
\(238\) 0 0
\(239\) −4440.00 −1.20167 −0.600836 0.799372i \(-0.705166\pi\)
−0.600836 + 0.799372i \(0.705166\pi\)
\(240\) 0 0
\(241\) −850.000 −0.227192 −0.113596 0.993527i \(-0.536237\pi\)
−0.113596 + 0.993527i \(0.536237\pi\)
\(242\) 0 0
\(243\) − 3542.00i − 0.935059i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 2000.00i − 0.515210i
\(248\) 0 0
\(249\) 1964.00 0.499853
\(250\) 0 0
\(251\) −660.000 −0.165971 −0.0829857 0.996551i \(-0.526446\pi\)
−0.0829857 + 0.996551i \(0.526446\pi\)
\(252\) 0 0
\(253\) 10680.0i 2.65394i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7590.00i 1.84222i 0.389299 + 0.921111i \(0.372717\pi\)
−0.389299 + 0.921111i \(0.627283\pi\)
\(258\) 0 0
\(259\) −60.0000 −0.0143947
\(260\) 0 0
\(261\) −3818.00 −0.905472
\(262\) 0 0
\(263\) 762.000i 0.178658i 0.996002 + 0.0893288i \(0.0284722\pi\)
−0.996002 + 0.0893288i \(0.971528\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1748.00i 0.400659i
\(268\) 0 0
\(269\) 150.000 0.0339987 0.0169994 0.999856i \(-0.494589\pi\)
0.0169994 + 0.999856i \(0.494589\pi\)
\(270\) 0 0
\(271\) −6580.00 −1.47493 −0.737466 0.675384i \(-0.763978\pi\)
−0.737466 + 0.675384i \(0.763978\pi\)
\(272\) 0 0
\(273\) − 600.000i − 0.133017i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4530.00i − 0.982604i −0.870989 0.491302i \(-0.836521\pi\)
0.870989 0.491302i \(-0.163479\pi\)
\(278\) 0 0
\(279\) 460.000 0.0987078
\(280\) 0 0
\(281\) 6950.00 1.47545 0.737726 0.675100i \(-0.235899\pi\)
0.737726 + 0.675100i \(0.235899\pi\)
\(282\) 0 0
\(283\) − 3882.00i − 0.815410i −0.913114 0.407705i \(-0.866329\pi\)
0.913114 0.407705i \(-0.133671\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1500.00i 0.308509i
\(288\) 0 0
\(289\) 4013.00 0.816813
\(290\) 0 0
\(291\) 620.000 0.124897
\(292\) 0 0
\(293\) 1370.00i 0.273161i 0.990629 + 0.136581i \(0.0436113\pi\)
−0.990629 + 0.136581i \(0.956389\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 6000.00i − 1.17224i
\(298\) 0 0
\(299\) −8900.00 −1.72141
\(300\) 0 0
\(301\) 852.000 0.163151
\(302\) 0 0
\(303\) 2996.00i 0.568039i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 4106.00i − 0.763328i −0.924301 0.381664i \(-0.875351\pi\)
0.924301 0.381664i \(-0.124649\pi\)
\(308\) 0 0
\(309\) 2804.00 0.516226
\(310\) 0 0
\(311\) 2220.00 0.404774 0.202387 0.979306i \(-0.435130\pi\)
0.202387 + 0.979306i \(0.435130\pi\)
\(312\) 0 0
\(313\) − 9430.00i − 1.70292i −0.524417 0.851462i \(-0.675717\pi\)
0.524417 0.851462i \(-0.324283\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6470.00i 1.14635i 0.819435 + 0.573173i \(0.194288\pi\)
−0.819435 + 0.573173i \(0.805712\pi\)
\(318\) 0 0
\(319\) −9960.00 −1.74813
\(320\) 0 0
\(321\) −2388.00 −0.415219
\(322\) 0 0
\(323\) − 1200.00i − 0.206718i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1300.00i 0.219848i
\(328\) 0 0
\(329\) −1284.00 −0.215165
\(330\) 0 0
\(331\) 900.000 0.149452 0.0747258 0.997204i \(-0.476192\pi\)
0.0747258 + 0.997204i \(0.476192\pi\)
\(332\) 0 0
\(333\) − 230.000i − 0.0378496i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 530.000i − 0.0856704i −0.999082 0.0428352i \(-0.986361\pi\)
0.999082 0.0428352i \(-0.0136390\pi\)
\(338\) 0 0
\(339\) −3020.00 −0.483846
\(340\) 0 0
\(341\) 1200.00 0.190568
\(342\) 0 0
\(343\) − 3900.00i − 0.613936i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 414.000i 0.0640481i 0.999487 + 0.0320240i \(0.0101953\pi\)
−0.999487 + 0.0320240i \(0.989805\pi\)
\(348\) 0 0
\(349\) −8614.00 −1.32119 −0.660597 0.750741i \(-0.729697\pi\)
−0.660597 + 0.750741i \(0.729697\pi\)
\(350\) 0 0
\(351\) 5000.00 0.760343
\(352\) 0 0
\(353\) − 2270.00i − 0.342266i −0.985248 0.171133i \(-0.945257\pi\)
0.985248 0.171133i \(-0.0547428\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 360.000i − 0.0533704i
\(358\) 0 0
\(359\) −8080.00 −1.18787 −0.593936 0.804512i \(-0.702427\pi\)
−0.593936 + 0.804512i \(0.702427\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 0 0
\(363\) − 4538.00i − 0.656152i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2374.00i − 0.337662i −0.985645 0.168831i \(-0.946001\pi\)
0.985645 0.168831i \(-0.0539991\pi\)
\(368\) 0 0
\(369\) −5750.00 −0.811201
\(370\) 0 0
\(371\) 2940.00 0.411421
\(372\) 0 0
\(373\) 1810.00i 0.251255i 0.992077 + 0.125628i \(0.0400945\pi\)
−0.992077 + 0.125628i \(0.959906\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 8300.00i − 1.13388i
\(378\) 0 0
\(379\) −8120.00 −1.10052 −0.550259 0.834994i \(-0.685471\pi\)
−0.550259 + 0.834994i \(0.685471\pi\)
\(380\) 0 0
\(381\) −2492.00 −0.335089
\(382\) 0 0
\(383\) − 11782.0i − 1.57189i −0.618299 0.785943i \(-0.712178\pi\)
0.618299 0.785943i \(-0.287822\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3266.00i 0.428993i
\(388\) 0 0
\(389\) 4350.00 0.566976 0.283488 0.958976i \(-0.408508\pi\)
0.283488 + 0.958976i \(0.408508\pi\)
\(390\) 0 0
\(391\) −5340.00 −0.690679
\(392\) 0 0
\(393\) − 5320.00i − 0.682846i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7470.00i 0.944354i 0.881504 + 0.472177i \(0.156532\pi\)
−0.881504 + 0.472177i \(0.843468\pi\)
\(398\) 0 0
\(399\) 480.000 0.0602257
\(400\) 0 0
\(401\) 11698.0 1.45678 0.728392 0.685161i \(-0.240268\pi\)
0.728392 + 0.685161i \(0.240268\pi\)
\(402\) 0 0
\(403\) 1000.00i 0.123607i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 600.000i − 0.0730735i
\(408\) 0 0
\(409\) 3650.00 0.441274 0.220637 0.975356i \(-0.429186\pi\)
0.220637 + 0.975356i \(0.429186\pi\)
\(410\) 0 0
\(411\) −5540.00 −0.664886
\(412\) 0 0
\(413\) − 4800.00i − 0.571895i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 1120.00i − 0.131527i
\(418\) 0 0
\(419\) 1120.00 0.130586 0.0652931 0.997866i \(-0.479202\pi\)
0.0652931 + 0.997866i \(0.479202\pi\)
\(420\) 0 0
\(421\) 4850.00 0.561460 0.280730 0.959787i \(-0.409424\pi\)
0.280730 + 0.959787i \(0.409424\pi\)
\(422\) 0 0
\(423\) − 4922.00i − 0.565758i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1500.00i − 0.170000i
\(428\) 0 0
\(429\) 6000.00 0.675251
\(430\) 0 0
\(431\) −12580.0 −1.40593 −0.702967 0.711223i \(-0.748142\pi\)
−0.702967 + 0.711223i \(0.748142\pi\)
\(432\) 0 0
\(433\) 13130.0i 1.45725i 0.684915 + 0.728623i \(0.259839\pi\)
−0.684915 + 0.728623i \(0.740161\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 7120.00i − 0.779395i
\(438\) 0 0
\(439\) −8560.00 −0.930630 −0.465315 0.885145i \(-0.654059\pi\)
−0.465315 + 0.885145i \(0.654059\pi\)
\(440\) 0 0
\(441\) 7061.00 0.762445
\(442\) 0 0
\(443\) − 4258.00i − 0.456667i −0.973583 0.228334i \(-0.926672\pi\)
0.973583 0.228334i \(-0.0733277\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 4700.00i − 0.497321i
\(448\) 0 0
\(449\) −2550.00 −0.268022 −0.134011 0.990980i \(-0.542786\pi\)
−0.134011 + 0.990980i \(0.542786\pi\)
\(450\) 0 0
\(451\) −15000.0 −1.56613
\(452\) 0 0
\(453\) − 1160.00i − 0.120312i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6710.00i 0.686828i 0.939184 + 0.343414i \(0.111583\pi\)
−0.939184 + 0.343414i \(0.888417\pi\)
\(458\) 0 0
\(459\) 3000.00 0.305072
\(460\) 0 0
\(461\) −14482.0 −1.46311 −0.731555 0.681782i \(-0.761205\pi\)
−0.731555 + 0.681782i \(0.761205\pi\)
\(462\) 0 0
\(463\) 162.000i 0.0162609i 0.999967 + 0.00813043i \(0.00258802\pi\)
−0.999967 + 0.00813043i \(0.997412\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15974.0i 1.58284i 0.611270 + 0.791422i \(0.290659\pi\)
−0.611270 + 0.791422i \(0.709341\pi\)
\(468\) 0 0
\(469\) 4644.00 0.457228
\(470\) 0 0
\(471\) 2620.00 0.256313
\(472\) 0 0
\(473\) 8520.00i 0.828224i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11270.0i 1.08180i
\(478\) 0 0
\(479\) 10760.0 1.02638 0.513191 0.858274i \(-0.328463\pi\)
0.513191 + 0.858274i \(0.328463\pi\)
\(480\) 0 0
\(481\) 500.000 0.0473972
\(482\) 0 0
\(483\) − 2136.00i − 0.201224i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9266.00i 0.862182i 0.902309 + 0.431091i \(0.141871\pi\)
−0.902309 + 0.431091i \(0.858129\pi\)
\(488\) 0 0
\(489\) 3724.00 0.344387
\(490\) 0 0
\(491\) 2860.00 0.262872 0.131436 0.991325i \(-0.458041\pi\)
0.131436 + 0.991325i \(0.458041\pi\)
\(492\) 0 0
\(493\) − 4980.00i − 0.454945i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 600.000i − 0.0541523i
\(498\) 0 0
\(499\) −7160.00 −0.642336 −0.321168 0.947022i \(-0.604075\pi\)
−0.321168 + 0.947022i \(0.604075\pi\)
\(500\) 0 0
\(501\) −1452.00 −0.129482
\(502\) 0 0
\(503\) − 1398.00i − 0.123924i −0.998079 0.0619620i \(-0.980264\pi\)
0.998079 0.0619620i \(-0.0197357\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 606.000i 0.0530836i
\(508\) 0 0
\(509\) −7446.00 −0.648405 −0.324203 0.945988i \(-0.605096\pi\)
−0.324203 + 0.945988i \(0.605096\pi\)
\(510\) 0 0
\(511\) −1380.00 −0.119467
\(512\) 0 0
\(513\) 4000.00i 0.344258i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 12840.0i − 1.09227i
\(518\) 0 0
\(519\) 6500.00 0.549746
\(520\) 0 0
\(521\) −16438.0 −1.38227 −0.691134 0.722726i \(-0.742889\pi\)
−0.691134 + 0.722726i \(0.742889\pi\)
\(522\) 0 0
\(523\) − 7322.00i − 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 600.000i 0.0495947i
\(528\) 0 0
\(529\) −19517.0 −1.60409
\(530\) 0 0
\(531\) 18400.0 1.50375
\(532\) 0 0
\(533\) − 12500.0i − 1.01583i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2240.00i − 0.180006i
\(538\) 0 0
\(539\) 18420.0 1.47200
\(540\) 0 0
\(541\) 10878.0 0.864476 0.432238 0.901759i \(-0.357724\pi\)
0.432238 + 0.901759i \(0.357724\pi\)
\(542\) 0 0
\(543\) 5684.00i 0.449215i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 16114.0i − 1.25957i −0.776769 0.629785i \(-0.783143\pi\)
0.776769 0.629785i \(-0.216857\pi\)
\(548\) 0 0
\(549\) 5750.00 0.447002
\(550\) 0 0
\(551\) 6640.00 0.513382
\(552\) 0 0
\(553\) − 7920.00i − 0.609028i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 3690.00i − 0.280701i −0.990102 0.140350i \(-0.955177\pi\)
0.990102 0.140350i \(-0.0448229\pi\)
\(558\) 0 0
\(559\) −7100.00 −0.537206
\(560\) 0 0
\(561\) 3600.00 0.270931
\(562\) 0 0
\(563\) − 2562.00i − 0.191786i −0.995392 0.0958929i \(-0.969429\pi\)
0.995392 0.0958929i \(-0.0305706\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2526.00i − 0.187094i
\(568\) 0 0
\(569\) 6050.00 0.445746 0.222873 0.974848i \(-0.428457\pi\)
0.222873 + 0.974848i \(0.428457\pi\)
\(570\) 0 0
\(571\) 8260.00 0.605377 0.302688 0.953090i \(-0.402116\pi\)
0.302688 + 0.953090i \(0.402116\pi\)
\(572\) 0 0
\(573\) − 6360.00i − 0.463687i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 16870.0i 1.21717i 0.793489 + 0.608585i \(0.208263\pi\)
−0.793489 + 0.608585i \(0.791737\pi\)
\(578\) 0 0
\(579\) −9340.00 −0.670392
\(580\) 0 0
\(581\) 5892.00 0.420725
\(582\) 0 0
\(583\) 29400.0i 2.08855i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 966.000i 0.0679235i 0.999423 + 0.0339617i \(0.0108124\pi\)
−0.999423 + 0.0339617i \(0.989188\pi\)
\(588\) 0 0
\(589\) −800.000 −0.0559651
\(590\) 0 0
\(591\) 5980.00 0.416217
\(592\) 0 0
\(593\) 26290.0i 1.82057i 0.413977 + 0.910287i \(0.364139\pi\)
−0.413977 + 0.910287i \(0.635861\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 8480.00i − 0.581346i
\(598\) 0 0
\(599\) 11640.0 0.793986 0.396993 0.917822i \(-0.370054\pi\)
0.396993 + 0.917822i \(0.370054\pi\)
\(600\) 0 0
\(601\) −25450.0 −1.72733 −0.863667 0.504064i \(-0.831838\pi\)
−0.863667 + 0.504064i \(0.831838\pi\)
\(602\) 0 0
\(603\) 17802.0i 1.20224i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 16694.0i − 1.11629i −0.829743 0.558145i \(-0.811513\pi\)
0.829743 0.558145i \(-0.188487\pi\)
\(608\) 0 0
\(609\) 1992.00 0.132545
\(610\) 0 0
\(611\) 10700.0 0.708471
\(612\) 0 0
\(613\) 15890.0i 1.04697i 0.852036 + 0.523484i \(0.175368\pi\)
−0.852036 + 0.523484i \(0.824632\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1230.00i 0.0802560i 0.999195 + 0.0401280i \(0.0127766\pi\)
−0.999195 + 0.0401280i \(0.987223\pi\)
\(618\) 0 0
\(619\) 10840.0 0.703871 0.351936 0.936024i \(-0.385524\pi\)
0.351936 + 0.936024i \(0.385524\pi\)
\(620\) 0 0
\(621\) 17800.0 1.15022
\(622\) 0 0
\(623\) 5244.00i 0.337233i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4800.00i 0.305731i
\(628\) 0 0
\(629\) 300.000 0.0190171
\(630\) 0 0
\(631\) −14060.0 −0.887036 −0.443518 0.896265i \(-0.646270\pi\)
−0.443518 + 0.896265i \(0.646270\pi\)
\(632\) 0 0
\(633\) 8120.00i 0.509859i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15350.0i 0.954771i
\(638\) 0 0
\(639\) 2300.00 0.142389
\(640\) 0 0
\(641\) −17650.0 −1.08757 −0.543786 0.839224i \(-0.683010\pi\)
−0.543786 + 0.839224i \(0.683010\pi\)
\(642\) 0 0
\(643\) 27358.0i 1.67791i 0.544203 + 0.838953i \(0.316832\pi\)
−0.544203 + 0.838953i \(0.683168\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6786.00i 0.412342i 0.978516 + 0.206171i \(0.0661003\pi\)
−0.978516 + 0.206171i \(0.933900\pi\)
\(648\) 0 0
\(649\) 48000.0 2.90318
\(650\) 0 0
\(651\) −240.000 −0.0144491
\(652\) 0 0
\(653\) − 9030.00i − 0.541150i −0.962699 0.270575i \(-0.912786\pi\)
0.962699 0.270575i \(-0.0872139\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 5290.00i − 0.314129i
\(658\) 0 0
\(659\) 15600.0 0.922139 0.461070 0.887364i \(-0.347466\pi\)
0.461070 + 0.887364i \(0.347466\pi\)
\(660\) 0 0
\(661\) 16850.0 0.991511 0.495756 0.868462i \(-0.334891\pi\)
0.495756 + 0.868462i \(0.334891\pi\)
\(662\) 0 0
\(663\) 3000.00i 0.175732i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 29548.0i − 1.71530i
\(668\) 0 0
\(669\) −11244.0 −0.649803
\(670\) 0 0
\(671\) 15000.0 0.862993
\(672\) 0 0
\(673\) − 7990.00i − 0.457640i −0.973469 0.228820i \(-0.926513\pi\)
0.973469 0.228820i \(-0.0734868\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18690.0i − 1.06103i −0.847677 0.530513i \(-0.821999\pi\)
0.847677 0.530513i \(-0.178001\pi\)
\(678\) 0 0
\(679\) 1860.00 0.105126
\(680\) 0 0
\(681\) −3108.00 −0.174888
\(682\) 0 0
\(683\) 19182.0i 1.07464i 0.843379 + 0.537320i \(0.180563\pi\)
−0.843379 + 0.537320i \(0.819437\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2268.00i 0.125953i
\(688\) 0 0
\(689\) −24500.0 −1.35468
\(690\) 0 0
\(691\) −23380.0 −1.28714 −0.643572 0.765385i \(-0.722548\pi\)
−0.643572 + 0.765385i \(0.722548\pi\)
\(692\) 0 0
\(693\) − 8280.00i − 0.453869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 7500.00i − 0.407579i
\(698\) 0 0
\(699\) −3420.00 −0.185059
\(700\) 0 0
\(701\) 11850.0 0.638471 0.319236 0.947675i \(-0.396574\pi\)
0.319236 + 0.947675i \(0.396574\pi\)
\(702\) 0 0
\(703\) 400.000i 0.0214599i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8988.00i 0.478117i
\(708\) 0 0
\(709\) −25646.0 −1.35847 −0.679235 0.733921i \(-0.737688\pi\)
−0.679235 + 0.733921i \(0.737688\pi\)
\(710\) 0 0
\(711\) 30360.0 1.60139
\(712\) 0 0
\(713\) 3560.00i 0.186989i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8880.00i 0.462524i
\(718\) 0 0
\(719\) 30280.0 1.57059 0.785294 0.619122i \(-0.212512\pi\)
0.785294 + 0.619122i \(0.212512\pi\)
\(720\) 0 0
\(721\) 8412.00 0.434507
\(722\) 0 0
\(723\) 1700.00i 0.0874463i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 17446.0i − 0.890009i −0.895528 0.445004i \(-0.853202\pi\)
0.895528 0.445004i \(-0.146798\pi\)
\(728\) 0 0
\(729\) 4283.00 0.217599
\(730\) 0 0
\(731\) −4260.00 −0.215543
\(732\) 0 0
\(733\) − 16750.0i − 0.844032i −0.906588 0.422016i \(-0.861323\pi\)
0.906588 0.422016i \(-0.138677\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 46440.0i 2.32108i
\(738\) 0 0
\(739\) 36560.0 1.81987 0.909933 0.414755i \(-0.136133\pi\)
0.909933 + 0.414755i \(0.136133\pi\)
\(740\) 0 0
\(741\) −4000.00 −0.198305
\(742\) 0 0
\(743\) − 30142.0i − 1.48829i −0.668016 0.744147i \(-0.732856\pi\)
0.668016 0.744147i \(-0.267144\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22586.0i 1.10626i
\(748\) 0 0
\(749\) −7164.00 −0.349488
\(750\) 0 0
\(751\) 11860.0 0.576268 0.288134 0.957590i \(-0.406965\pi\)
0.288134 + 0.957590i \(0.406965\pi\)
\(752\) 0 0
\(753\) 1320.00i 0.0638824i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 37010.0i − 1.77695i −0.458925 0.888475i \(-0.651765\pi\)
0.458925 0.888475i \(-0.348235\pi\)
\(758\) 0 0
\(759\) 21360.0 1.02150
\(760\) 0 0
\(761\) −11718.0 −0.558183 −0.279091 0.960265i \(-0.590033\pi\)
−0.279091 + 0.960265i \(0.590033\pi\)
\(762\) 0 0
\(763\) 3900.00i 0.185045i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40000.0i 1.88307i
\(768\) 0 0
\(769\) −4706.00 −0.220680 −0.110340 0.993894i \(-0.535194\pi\)
−0.110340 + 0.993894i \(0.535194\pi\)
\(770\) 0 0
\(771\) 15180.0 0.709072
\(772\) 0 0
\(773\) − 28670.0i − 1.33401i −0.745054 0.667004i \(-0.767576\pi\)
0.745054 0.667004i \(-0.232424\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 120.000i 0.00554051i
\(778\) 0 0
\(779\) 10000.0 0.459932
\(780\) 0 0
\(781\) 6000.00 0.274900
\(782\) 0 0
\(783\) 16600.0i 0.757644i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 20434.0i − 0.925532i −0.886481 0.462766i \(-0.846857\pi\)
0.886481 0.462766i \(-0.153143\pi\)
\(788\) 0 0
\(789\) 1524.00 0.0687653
\(790\) 0 0
\(791\) −9060.00 −0.407252
\(792\) 0 0
\(793\) 12500.0i 0.559758i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 3930.00i − 0.174665i −0.996179 0.0873323i \(-0.972166\pi\)
0.996179 0.0873323i \(-0.0278342\pi\)
\(798\) 0 0
\(799\) 6420.00 0.284259
\(800\) 0 0
\(801\) −20102.0 −0.886728
\(802\) 0 0
\(803\) − 13800.0i − 0.606465i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 300.000i − 0.0130861i
\(808\) 0 0
\(809\) 4854.00 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −13140.0 −0.568937 −0.284468 0.958685i \(-0.591817\pi\)
−0.284468 + 0.958685i \(0.591817\pi\)
\(812\) 0 0
\(813\) 13160.0i 0.567702i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 5680.00i − 0.243229i
\(818\) 0 0
\(819\) 6900.00 0.294390
\(820\) 0 0
\(821\) 22050.0 0.937333 0.468666 0.883375i \(-0.344735\pi\)
0.468666 + 0.883375i \(0.344735\pi\)
\(822\) 0 0
\(823\) 14578.0i 0.617445i 0.951152 + 0.308722i \(0.0999014\pi\)
−0.951152 + 0.308722i \(0.900099\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37054.0i 1.55803i 0.627003 + 0.779017i \(0.284281\pi\)
−0.627003 + 0.779017i \(0.715719\pi\)
\(828\) 0 0
\(829\) 6150.00 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(830\) 0 0
\(831\) −9060.00 −0.378204
\(832\) 0 0
\(833\) 9210.00i 0.383082i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 2000.00i − 0.0825927i
\(838\) 0 0
\(839\) 8200.00 0.337420 0.168710 0.985666i \(-0.446040\pi\)
0.168710 + 0.985666i \(0.446040\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) 0 0
\(843\) − 13900.0i − 0.567902i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 13614.0i − 0.552282i
\(848\) 0 0
\(849\) −7764.00 −0.313851
\(850\) 0 0
\(851\) 1780.00 0.0717011
\(852\) 0 0
\(853\) − 42990.0i − 1.72561i −0.505533 0.862807i \(-0.668704\pi\)
0.505533 0.862807i \(-0.331296\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 32130.0i − 1.28068i −0.768093 0.640338i \(-0.778794\pi\)
0.768093 0.640338i \(-0.221206\pi\)
\(858\) 0 0
\(859\) −15440.0 −0.613278 −0.306639 0.951826i \(-0.599204\pi\)
−0.306639 + 0.951826i \(0.599204\pi\)
\(860\) 0 0
\(861\) 3000.00 0.118745
\(862\) 0 0
\(863\) 46938.0i 1.85143i 0.378216 + 0.925717i \(0.376538\pi\)
−0.378216 + 0.925717i \(0.623462\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 8026.00i − 0.314391i
\(868\) 0 0
\(869\) 79200.0 3.09169
\(870\) 0 0
\(871\) −38700.0 −1.50551
\(872\) 0 0
\(873\) 7130.00i 0.276419i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31230.0i 1.20247i 0.799074 + 0.601233i \(0.205324\pi\)
−0.799074 + 0.601233i \(0.794676\pi\)
\(878\) 0 0
\(879\) 2740.00 0.105140
\(880\) 0 0
\(881\) 25550.0 0.977073 0.488537 0.872543i \(-0.337531\pi\)
0.488537 + 0.872543i \(0.337531\pi\)
\(882\) 0 0
\(883\) 4318.00i 0.164567i 0.996609 + 0.0822833i \(0.0262212\pi\)
−0.996609 + 0.0822833i \(0.973779\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1766.00i − 0.0668506i −0.999441 0.0334253i \(-0.989358\pi\)
0.999441 0.0334253i \(-0.0106416\pi\)
\(888\) 0 0
\(889\) −7476.00 −0.282044
\(890\) 0 0
\(891\) 25260.0 0.949766
\(892\) 0 0
\(893\) 8560.00i 0.320772i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 17800.0i 0.662569i
\(898\) 0 0
\(899\) −3320.00 −0.123168
\(900\) 0 0
\(901\) −14700.0 −0.543538
\(902\) 0 0
\(903\) − 1704.00i − 0.0627969i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 41906.0i − 1.53414i −0.641563 0.767071i \(-0.721714\pi\)
0.641563 0.767071i \(-0.278286\pi\)
\(908\) 0 0
\(909\) −34454.0 −1.25717
\(910\) 0 0
\(911\) 25140.0 0.914298 0.457149 0.889390i \(-0.348871\pi\)
0.457149 + 0.889390i \(0.348871\pi\)
\(912\) 0 0
\(913\) 58920.0i 2.13578i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 15960.0i − 0.574750i
\(918\) 0 0
\(919\) −32920.0 −1.18164 −0.590822 0.806802i \(-0.701196\pi\)
−0.590822 + 0.806802i \(0.701196\pi\)
\(920\) 0 0
\(921\) −8212.00 −0.293805
\(922\) 0 0
\(923\) 5000.00i 0.178307i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 32246.0i 1.14250i
\(928\) 0 0
\(929\) −10150.0 −0.358461 −0.179231 0.983807i \(-0.557361\pi\)
−0.179231 + 0.983807i \(0.557361\pi\)
\(930\) 0 0
\(931\) −12280.0 −0.432289
\(932\) 0 0
\(933\) − 4440.00i − 0.155798i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 28530.0i − 0.994701i −0.867550 0.497350i \(-0.834306\pi\)
0.867550 0.497350i \(-0.165694\pi\)
\(938\) 0 0
\(939\) −18860.0 −0.655456
\(940\) 0 0
\(941\) 9678.00 0.335275 0.167638 0.985849i \(-0.446386\pi\)
0.167638 + 0.985849i \(0.446386\pi\)
\(942\) 0 0
\(943\) − 44500.0i − 1.53671i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 36986.0i − 1.26915i −0.772862 0.634574i \(-0.781176\pi\)
0.772862 0.634574i \(-0.218824\pi\)
\(948\) 0 0
\(949\) 11500.0 0.393368
\(950\) 0 0
\(951\) 12940.0 0.441228
\(952\) 0 0
\(953\) − 3350.00i − 0.113869i −0.998378 0.0569345i \(-0.981867\pi\)
0.998378 0.0569345i \(-0.0181326\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 19920.0i 0.672855i
\(958\) 0 0
\(959\) −16620.0 −0.559633
\(960\) 0 0
\(961\) −29391.0 −0.986573
\(962\) 0 0
\(963\) − 27462.0i − 0.918952i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 43774.0i − 1.45572i −0.685728 0.727858i \(-0.740516\pi\)
0.685728 0.727858i \(-0.259484\pi\)
\(968\) 0 0
\(969\) −2400.00 −0.0795656
\(970\) 0 0
\(971\) 8740.00 0.288857 0.144428 0.989515i \(-0.453866\pi\)
0.144428 + 0.989515i \(0.453866\pi\)
\(972\) 0 0
\(973\) − 3360.00i − 0.110706i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48310.0i 1.58196i 0.611843 + 0.790979i \(0.290429\pi\)
−0.611843 + 0.790979i \(0.709571\pi\)
\(978\) 0 0
\(979\) −52440.0 −1.71194
\(980\) 0 0
\(981\) −14950.0 −0.486561
\(982\) 0 0
\(983\) 2282.00i 0.0740432i 0.999314 + 0.0370216i \(0.0117870\pi\)
−0.999314 + 0.0370216i \(0.988213\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2568.00i 0.0828170i
\(988\) 0 0
\(989\) −25276.0 −0.812669
\(990\) 0 0
\(991\) −31580.0 −1.01228 −0.506141 0.862451i \(-0.668929\pi\)
−0.506141 + 0.862451i \(0.668929\pi\)
\(992\) 0 0
\(993\) − 1800.00i − 0.0575239i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2790.00i 0.0886261i 0.999018 + 0.0443130i \(0.0141099\pi\)
−0.999018 + 0.0443130i \(0.985890\pi\)
\(998\) 0 0
\(999\) −1000.00 −0.0316703
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 800.4.c.f.449.1 2
4.3 odd 2 800.4.c.e.449.2 2
5.2 odd 4 160.4.a.a.1.1 1
5.3 odd 4 800.4.a.h.1.1 1
5.4 even 2 inner 800.4.c.f.449.2 2
15.2 even 4 1440.4.a.o.1.1 1
20.3 even 4 800.4.a.d.1.1 1
20.7 even 4 160.4.a.b.1.1 yes 1
20.19 odd 2 800.4.c.e.449.1 2
40.3 even 4 1600.4.a.bj.1.1 1
40.13 odd 4 1600.4.a.r.1.1 1
40.27 even 4 320.4.a.f.1.1 1
40.37 odd 4 320.4.a.i.1.1 1
60.47 odd 4 1440.4.a.n.1.1 1
80.27 even 4 1280.4.d.k.641.1 2
80.37 odd 4 1280.4.d.f.641.2 2
80.67 even 4 1280.4.d.k.641.2 2
80.77 odd 4 1280.4.d.f.641.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.a.1.1 1 5.2 odd 4
160.4.a.b.1.1 yes 1 20.7 even 4
320.4.a.f.1.1 1 40.27 even 4
320.4.a.i.1.1 1 40.37 odd 4
800.4.a.d.1.1 1 20.3 even 4
800.4.a.h.1.1 1 5.3 odd 4
800.4.c.e.449.1 2 20.19 odd 2
800.4.c.e.449.2 2 4.3 odd 2
800.4.c.f.449.1 2 1.1 even 1 trivial
800.4.c.f.449.2 2 5.4 even 2 inner
1280.4.d.f.641.1 2 80.77 odd 4
1280.4.d.f.641.2 2 80.37 odd 4
1280.4.d.k.641.1 2 80.27 even 4
1280.4.d.k.641.2 2 80.67 even 4
1440.4.a.n.1.1 1 60.47 odd 4
1440.4.a.o.1.1 1 15.2 even 4
1600.4.a.r.1.1 1 40.13 odd 4
1600.4.a.bj.1.1 1 40.3 even 4