Properties

Label 800.4.c.e.449.1
Level $800$
Weight $4$
Character 800.449
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
CM no
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.e.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 \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) 0 0
\(13\) − 50.0000i − 1.06673i −0.845885 0.533366i \(-0.820927\pi\)
0.845885 0.533366i \(-0.179073\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 30.0000i − 0.428004i −0.976833 0.214002i \(-0.931350\pi\)
0.976833 0.214002i \(-0.0686499\pi\)
\(18\) 0 0
\(19\) 40.0000 0.482980 0.241490 0.970403i \(-0.422364\pi\)
0.241490 + 0.970403i \(0.422364\pi\)
\(20\) 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 \(-0.518452\pi\)
−0.0579372 + 0.998320i \(0.518452\pi\)
\(32\) 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 \(0.00707218\pi\)
−0.999753 + 0.0222161i \(0.992928\pi\)
\(38\) 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 \(-0.781013\pi\)
0.772538 0.634969i \(-0.218987\pi\)
\(54\) 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 \(-0.844234\pi\)
−0.882637 + 0.470056i \(0.844234\pi\)
\(60\) 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 \(-0.526634\pi\)
−0.0835762 + 0.996501i \(0.526634\pi\)
\(72\) 0 0
\(73\) 230.000i 0.368760i 0.982855 + 0.184380i \(0.0590277\pi\)
−0.982855 + 0.184380i \(0.940972\pi\)
\(74\) 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 \(-0.889126\pi\)
−0.939947 + 0.341321i \(0.889126\pi\)
\(80\) 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 \(-0.948126\pi\)
0.986750 0.162246i \(-0.0518738\pi\)
\(98\) 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 \(0.216345\pi\)
−0.777782 + 0.628535i \(0.783655\pi\)
\(114\) 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 \(-0.847244\pi\)
−0.887043 + 0.461687i \(0.847244\pi\)
\(132\) 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 \(0.331866\pi\)
−0.503986 + 0.863712i \(0.668134\pi\)
\(138\) 0 0
\(139\) −560.000 −0.341716 −0.170858 0.985296i \(-0.554654\pi\)
−0.170858 + 0.985296i \(0.554654\pi\)
\(140\) 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 \(-0.549954\pi\)
−0.156290 + 0.987711i \(0.549954\pi\)
\(152\) 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 \(-0.891953\pi\)
0.942941 0.332960i \(-0.108047\pi\)
\(158\) 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 \(-0.746817\pi\)
0.700001 0.714141i \(-0.253183\pi\)
\(174\) 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 \(-0.575127\pi\)
−0.233834 + 0.972276i \(0.575127\pi\)
\(180\) 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 \(-0.705768\pi\)
−0.602347 + 0.798234i \(0.705768\pi\)
\(192\) 0 0
\(193\) 4670.00i 1.74173i 0.491522 + 0.870865i \(0.336441\pi\)
−0.491522 + 0.870865i \(0.663559\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2990.00i − 1.08136i −0.841227 0.540682i \(-0.818166\pi\)
0.841227 0.540682i \(-0.181834\pi\)
\(198\) 0 0
\(199\) −4240.00 −1.51038 −0.755190 0.655506i \(-0.772455\pi\)
−0.755190 + 0.655506i \(0.772455\pi\)
\(200\) 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 \(-0.269569\pi\)
0.662327 + 0.749215i \(0.269569\pi\)
\(212\) 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 \(0.0772782\pi\)
−0.970674 + 0.240399i \(0.922722\pi\)
\(234\) 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 \(-0.294834\pi\)
0.600836 + 0.799372i \(0.294834\pi\)
\(240\) 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 \(-0.473554\pi\)
0.0829857 + 0.996551i \(0.473554\pi\)
\(252\) 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 \(-0.627283\pi\)
0.389299 0.921111i \(-0.372717\pi\)
\(258\) 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 \(-0.236022\pi\)
0.737466 + 0.675384i \(0.236022\pi\)
\(272\) 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 \(0.163479\pi\)
−0.870989 + 0.491302i \(0.836521\pi\)
\(278\) 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 \(-0.956389\pi\)
0.990629 0.136581i \(-0.0436113\pi\)
\(294\) 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 \(-0.564870\pi\)
−0.202387 + 0.979306i \(0.564870\pi\)
\(312\) 0 0
\(313\) 9430.00i 1.70292i 0.524417 + 0.851462i \(0.324283\pi\)
−0.524417 + 0.851462i \(0.675717\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6470.00i − 1.14635i −0.819435 0.573173i \(-0.805712\pi\)
0.819435 0.573173i \(-0.194288\pi\)
\(318\) 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 \(-0.523808\pi\)
−0.0747258 + 0.997204i \(0.523808\pi\)
\(332\) 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 \(0.0136390\pi\)
−0.999082 + 0.0428352i \(0.986361\pi\)
\(338\) 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 \(0.0547428\pi\)
−0.985248 + 0.171133i \(0.945257\pi\)
\(354\) 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 \(-0.297573\pi\)
0.593936 + 0.804512i \(0.297573\pi\)
\(360\) 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 \(-0.959906\pi\)
0.992077 0.125628i \(-0.0400945\pi\)
\(374\) 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 \(-0.314529\pi\)
0.550259 + 0.834994i \(0.314529\pi\)
\(380\) 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 \(-0.843468\pi\)
0.881504 0.472177i \(-0.156532\pi\)
\(398\) 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 \(-0.520798\pi\)
−0.0652931 + 0.997866i \(0.520798\pi\)
\(420\) 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 \(-0.251858\pi\)
0.702967 + 0.711223i \(0.251858\pi\)
\(432\) 0 0
\(433\) − 13130.0i − 1.45725i −0.684915 0.728623i \(-0.740161\pi\)
0.684915 0.728623i \(-0.259839\pi\)
\(434\) 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 \(-0.345941\pi\)
0.465315 + 0.885145i \(0.345941\pi\)
\(440\) 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 \(-0.888417\pi\)
0.939184 0.343414i \(-0.111583\pi\)
\(458\) 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 \(-0.671537\pi\)
−0.513191 + 0.858274i \(0.671537\pi\)
\(480\) 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 \(-0.541959\pi\)
−0.131436 + 0.991325i \(0.541959\pi\)
\(492\) 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 \(-0.395925\pi\)
0.321168 + 0.947022i \(0.395925\pi\)
\(500\) 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 \(0.0448229\pi\)
−0.990102 + 0.140350i \(0.955177\pi\)
\(558\) 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 \(-0.597884\pi\)
−0.302688 + 0.953090i \(0.597884\pi\)
\(572\) 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 \(-0.791737\pi\)
0.793489 0.608585i \(-0.208263\pi\)
\(578\) 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 \(-0.635861\pi\)
0.413977 0.910287i \(-0.364139\pi\)
\(594\) 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 \(-0.629946\pi\)
−0.396993 + 0.917822i \(0.629946\pi\)
\(600\) 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 \(-0.824632\pi\)
0.852036 0.523484i \(-0.175368\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1230.00i − 0.0802560i −0.999195 0.0401280i \(-0.987223\pi\)
0.999195 0.0401280i \(-0.0127766\pi\)
\(618\) 0 0
\(619\) −10840.0 −0.703871 −0.351936 0.936024i \(-0.614476\pi\)
−0.351936 + 0.936024i \(0.614476\pi\)
\(620\) 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 \(-0.353730\pi\)
0.443518 + 0.896265i \(0.353730\pi\)
\(632\) 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 \(0.0872139\pi\)
−0.962699 + 0.270575i \(0.912786\pi\)
\(654\) 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 \(-0.652534\pi\)
−0.461070 + 0.887364i \(0.652534\pi\)
\(660\) 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 \(0.0734868\pi\)
−0.973469 + 0.228820i \(0.926513\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18690.0i 1.06103i 0.847677 + 0.530513i \(0.178001\pi\)
−0.847677 + 0.530513i \(0.821999\pi\)
\(678\) 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 \(-0.277452\pi\)
0.643572 + 0.765385i \(0.277452\pi\)
\(692\) 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 \(-0.787488\pi\)
−0.785294 + 0.619122i \(0.787488\pi\)
\(720\) 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 \(0.138677\pi\)
−0.906588 + 0.422016i \(0.861323\pi\)
\(734\) 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 \(-0.863867\pi\)
−0.909933 + 0.414755i \(0.863867\pi\)
\(740\) 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 \(-0.593035\pi\)
−0.288134 + 0.957590i \(0.593035\pi\)
\(752\) 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 \(0.348235\pi\)
−0.458925 + 0.888475i \(0.651765\pi\)
\(758\) 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 \(0.232424\pi\)
−0.745054 + 0.667004i \(0.767576\pi\)
\(774\) 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 \(0.0278342\pi\)
−0.996179 + 0.0873323i \(0.972166\pi\)
\(798\) 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 \(-0.408183\pi\)
0.284468 + 0.958685i \(0.408183\pi\)
\(812\) 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 \(-0.553960\pi\)
−0.168710 + 0.985666i \(0.553960\pi\)
\(840\) 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 \(0.331296\pi\)
−0.505533 + 0.862807i \(0.668704\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32130.0i 1.28068i 0.768093 + 0.640338i \(0.221206\pi\)
−0.768093 + 0.640338i \(0.778794\pi\)
\(858\) 0 0
\(859\) 15440.0 0.613278 0.306639 0.951826i \(-0.400796\pi\)
0.306639 + 0.951826i \(0.400796\pi\)
\(860\) 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 \(-0.794676\pi\)
0.799074 0.601233i \(-0.205324\pi\)
\(878\) 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 \(-0.651129\pi\)
−0.457149 + 0.889390i \(0.651129\pi\)
\(912\) 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 \(-0.298804\pi\)
0.590822 + 0.806802i \(0.298804\pi\)
\(920\) 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 \(0.165694\pi\)
−0.867550 + 0.497350i \(0.834306\pi\)
\(938\) 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 \(0.0181326\pi\)
−0.998378 + 0.0569345i \(0.981867\pi\)
\(954\) 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 \(-0.546134\pi\)
−0.144428 + 0.989515i \(0.546134\pi\)
\(972\) 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 \(-0.709571\pi\)
0.611843 0.790979i \(-0.290429\pi\)
\(978\) 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 \(-0.331071\pi\)
0.506141 + 0.862451i \(0.331071\pi\)
\(992\) 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 \(-0.985890\pi\)
0.999018 0.0443130i \(-0.0141099\pi\)
\(998\) 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.e.449.1 2
4.3 odd 2 800.4.c.f.449.2 2
5.2 odd 4 800.4.a.d.1.1 1
5.3 odd 4 160.4.a.b.1.1 yes 1
5.4 even 2 inner 800.4.c.e.449.2 2
15.8 even 4 1440.4.a.n.1.1 1
20.3 even 4 160.4.a.a.1.1 1
20.7 even 4 800.4.a.h.1.1 1
20.19 odd 2 800.4.c.f.449.1 2
40.3 even 4 320.4.a.i.1.1 1
40.13 odd 4 320.4.a.f.1.1 1
40.27 even 4 1600.4.a.r.1.1 1
40.37 odd 4 1600.4.a.bj.1.1 1
60.23 odd 4 1440.4.a.o.1.1 1
80.3 even 4 1280.4.d.f.641.1 2
80.13 odd 4 1280.4.d.k.641.2 2
80.43 even 4 1280.4.d.f.641.2 2
80.53 odd 4 1280.4.d.k.641.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.a.1.1 1 20.3 even 4
160.4.a.b.1.1 yes 1 5.3 odd 4
320.4.a.f.1.1 1 40.13 odd 4
320.4.a.i.1.1 1 40.3 even 4
800.4.a.d.1.1 1 5.2 odd 4
800.4.a.h.1.1 1 20.7 even 4
800.4.c.e.449.1 2 1.1 even 1 trivial
800.4.c.e.449.2 2 5.4 even 2 inner
800.4.c.f.449.1 2 20.19 odd 2
800.4.c.f.449.2 2 4.3 odd 2
1280.4.d.f.641.1 2 80.3 even 4
1280.4.d.f.641.2 2 80.43 even 4
1280.4.d.k.641.1 2 80.53 odd 4
1280.4.d.k.641.2 2 80.13 odd 4
1440.4.a.n.1.1 1 15.8 even 4
1440.4.a.o.1.1 1 60.23 odd 4
1600.4.a.r.1.1 1 40.27 even 4
1600.4.a.bj.1.1 1 40.37 odd 4