Properties

Label 1280.4.d.k.641.1
Level $1280$
Weight $4$
Character 1280.641
Analytic conductor $75.522$
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] = [1280,4,Mod(641,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.641");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (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: \(75.5224448073\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.k.641.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} -5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} -5.00000i q^{5} +6.00000 q^{7} +23.0000 q^{9} -60.0000i q^{11} -50.0000i q^{13} -10.0000 q^{15} -30.0000 q^{17} +40.0000i q^{19} -12.0000i q^{21} +178.000 q^{23} -25.0000 q^{25} -100.000i q^{27} -166.000i q^{29} -20.0000 q^{31} -120.000 q^{33} -30.0000i q^{35} +10.0000i q^{37} -100.000 q^{39} +250.000 q^{41} -142.000i q^{43} -115.000i q^{45} -214.000 q^{47} -307.000 q^{49} +60.0000i q^{51} +490.000i q^{53} -300.000 q^{55} +80.0000 q^{57} +800.000i q^{59} -250.000i q^{61} +138.000 q^{63} -250.000 q^{65} -774.000i q^{67} -356.000i q^{69} +100.000 q^{71} +230.000 q^{73} +50.0000i q^{75} -360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} +982.000i q^{83} +150.000i q^{85} -332.000 q^{87} -874.000 q^{89} -300.000i q^{91} +40.0000i q^{93} +200.000 q^{95} -310.000 q^{97} -1380.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{7} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{7} + 46 q^{9} - 20 q^{15} - 60 q^{17} + 356 q^{23} - 50 q^{25} - 40 q^{31} - 240 q^{33} - 200 q^{39} + 500 q^{41} - 428 q^{47} - 614 q^{49} - 600 q^{55} + 160 q^{57} + 276 q^{63} - 500 q^{65} + 200 q^{71} + 460 q^{73} + 2640 q^{79} + 842 q^{81} - 664 q^{87} - 1748 q^{89} + 400 q^{95} - 620 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) 6.00000 0.323970 0.161985 0.986793i \(-0.448210\pi\)
0.161985 + 0.986793i \(0.448210\pi\)
\(8\) 0 0
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) − 60.0000i − 1.64461i −0.569049 0.822304i \(-0.692689\pi\)
0.569049 0.822304i \(-0.307311\pi\)
\(12\) 0 0
\(13\) − 50.0000i − 1.06673i −0.845885 0.533366i \(-0.820927\pi\)
0.845885 0.533366i \(-0.179073\pi\)
\(14\) 0 0
\(15\) −10.0000 −0.172133
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) 40.0000i 0.482980i 0.970403 + 0.241490i \(0.0776362\pi\)
−0.970403 + 0.241490i \(0.922364\pi\)
\(20\) 0 0
\(21\) − 12.0000i − 0.124696i
\(22\) 0 0
\(23\) 178.000 1.61372 0.806860 0.590743i \(-0.201165\pi\)
0.806860 + 0.590743i \(0.201165\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) − 100.000i − 0.712778i
\(28\) 0 0
\(29\) − 166.000i − 1.06295i −0.847075 0.531473i \(-0.821639\pi\)
0.847075 0.531473i \(-0.178361\pi\)
\(30\) 0 0
\(31\) −20.0000 −0.115874 −0.0579372 0.998320i \(-0.518452\pi\)
−0.0579372 + 0.998320i \(0.518452\pi\)
\(32\) 0 0
\(33\) −120.000 −0.633010
\(34\) 0 0
\(35\) − 30.0000i − 0.144884i
\(36\) 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 \(-0.342036\pi\)
0.476140 + 0.879370i \(0.342036\pi\)
\(42\) 0 0
\(43\) − 142.000i − 0.503600i −0.967779 0.251800i \(-0.918977\pi\)
0.967779 0.251800i \(-0.0810225\pi\)
\(44\) 0 0
\(45\) − 115.000i − 0.380960i
\(46\) 0 0
\(47\) −214.000 −0.664151 −0.332076 0.943253i \(-0.607749\pi\)
−0.332076 + 0.943253i \(0.607749\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) 60.0000i 0.164739i
\(52\) 0 0
\(53\) 490.000i 1.26994i 0.772538 + 0.634969i \(0.218987\pi\)
−0.772538 + 0.634969i \(0.781013\pi\)
\(54\) 0 0
\(55\) −300.000 −0.735491
\(56\) 0 0
\(57\) 80.0000 0.185899
\(58\) 0 0
\(59\) 800.000i 1.76527i 0.470056 + 0.882637i \(0.344234\pi\)
−0.470056 + 0.882637i \(0.655766\pi\)
\(60\) 0 0
\(61\) − 250.000i − 0.524741i −0.964967 0.262371i \(-0.915496\pi\)
0.964967 0.262371i \(-0.0845043\pi\)
\(62\) 0 0
\(63\) 138.000 0.275974
\(64\) 0 0
\(65\) −250.000 −0.477057
\(66\) 0 0
\(67\) − 774.000i − 1.41133i −0.708545 0.705665i \(-0.750648\pi\)
0.708545 0.705665i \(-0.249352\pi\)
\(68\) 0 0
\(69\) − 356.000i − 0.621121i
\(70\) 0 0
\(71\) 100.000 0.167152 0.0835762 0.996501i \(-0.473366\pi\)
0.0835762 + 0.996501i \(0.473366\pi\)
\(72\) 0 0
\(73\) 230.000 0.368760 0.184380 0.982855i \(-0.440972\pi\)
0.184380 + 0.982855i \(0.440972\pi\)
\(74\) 0 0
\(75\) 50.0000i 0.0769800i
\(76\) 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
\(85\) 150.000i 0.191409i
\(86\) 0 0
\(87\) −332.000 −0.409128
\(88\) 0 0
\(89\) −874.000 −1.04094 −0.520471 0.853879i \(-0.674244\pi\)
−0.520471 + 0.853879i \(0.674244\pi\)
\(90\) 0 0
\(91\) − 300.000i − 0.345588i
\(92\) 0 0
\(93\) 40.0000i 0.0446001i
\(94\) 0 0
\(95\) 200.000 0.215995
\(96\) 0 0
\(97\) −310.000 −0.324492 −0.162246 0.986750i \(-0.551874\pi\)
−0.162246 + 0.986750i \(0.551874\pi\)
\(98\) 0 0
\(99\) − 1380.00i − 1.40096i
\(100\) 0 0
\(101\) − 1498.00i − 1.47581i −0.674906 0.737904i \(-0.735816\pi\)
0.674906 0.737904i \(-0.264184\pi\)
\(102\) 0 0
\(103\) 1402.00 1.34120 0.670598 0.741821i \(-0.266038\pi\)
0.670598 + 0.741821i \(0.266038\pi\)
\(104\) 0 0
\(105\) −60.0000 −0.0557657
\(106\) 0 0
\(107\) − 1194.00i − 1.07877i −0.842059 0.539385i \(-0.818657\pi\)
0.842059 0.539385i \(-0.181343\pi\)
\(108\) 0 0
\(109\) − 650.000i − 0.571181i −0.958352 0.285590i \(-0.907810\pi\)
0.958352 0.285590i \(-0.0921897\pi\)
\(110\) 0 0
\(111\) 20.0000 0.0171019
\(112\) 0 0
\(113\) −1510.00 −1.25707 −0.628535 0.777782i \(-0.716345\pi\)
−0.628535 + 0.777782i \(0.716345\pi\)
\(114\) 0 0
\(115\) − 890.000i − 0.721678i
\(116\) 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
\(125\) 125.000i 0.0894427i
\(126\) 0 0
\(127\) −1246.00 −0.870588 −0.435294 0.900288i \(-0.643355\pi\)
−0.435294 + 0.900288i \(0.643355\pi\)
\(128\) 0 0
\(129\) −284.000 −0.193836
\(130\) 0 0
\(131\) 2660.00i 1.77409i 0.461687 + 0.887043i \(0.347244\pi\)
−0.461687 + 0.887043i \(0.652756\pi\)
\(132\) 0 0
\(133\) 240.000i 0.156471i
\(134\) 0 0
\(135\) −500.000 −0.318764
\(136\) 0 0
\(137\) −2770.00 −1.72742 −0.863712 0.503986i \(-0.831866\pi\)
−0.863712 + 0.503986i \(0.831866\pi\)
\(138\) 0 0
\(139\) 560.000i 0.341716i 0.985296 + 0.170858i \(0.0546540\pi\)
−0.985296 + 0.170858i \(0.945346\pi\)
\(140\) 0 0
\(141\) 428.000i 0.255632i
\(142\) 0 0
\(143\) −3000.00 −1.75435
\(144\) 0 0
\(145\) −830.000 −0.475364
\(146\) 0 0
\(147\) 614.000i 0.344502i
\(148\) 0 0
\(149\) − 2350.00i − 1.29208i −0.763305 0.646039i \(-0.776424\pi\)
0.763305 0.646039i \(-0.223576\pi\)
\(150\) 0 0
\(151\) 580.000 0.312581 0.156290 0.987711i \(-0.450046\pi\)
0.156290 + 0.987711i \(0.450046\pi\)
\(152\) 0 0
\(153\) −690.000 −0.364596
\(154\) 0 0
\(155\) 100.000i 0.0518206i
\(156\) 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
\(165\) 600.000i 0.283091i
\(166\) 0 0
\(167\) 726.000 0.336405 0.168202 0.985752i \(-0.446204\pi\)
0.168202 + 0.985752i \(0.446204\pi\)
\(168\) 0 0
\(169\) −303.000 −0.137915
\(170\) 0 0
\(171\) 920.000i 0.411428i
\(172\) 0 0
\(173\) − 3250.00i − 1.42828i −0.700001 0.714141i \(-0.746817\pi\)
0.700001 0.714141i \(-0.253183\pi\)
\(174\) 0 0
\(175\) −150.000 −0.0647939
\(176\) 0 0
\(177\) 1600.00 0.679454
\(178\) 0 0
\(179\) − 1120.00i − 0.467669i −0.972276 0.233834i \(-0.924873\pi\)
0.972276 0.233834i \(-0.0751274\pi\)
\(180\) 0 0
\(181\) − 2842.00i − 1.16710i −0.812079 0.583548i \(-0.801664\pi\)
0.812079 0.583548i \(-0.198336\pi\)
\(182\) 0 0
\(183\) −500.000 −0.201973
\(184\) 0 0
\(185\) 50.0000 0.0198707
\(186\) 0 0
\(187\) 1800.00i 0.703899i
\(188\) 0 0
\(189\) − 600.000i − 0.230918i
\(190\) 0 0
\(191\) −3180.00 −1.20469 −0.602347 0.798234i \(-0.705768\pi\)
−0.602347 + 0.798234i \(0.705768\pi\)
\(192\) 0 0
\(193\) −4670.00 −1.74173 −0.870865 0.491522i \(-0.836441\pi\)
−0.870865 + 0.491522i \(0.836441\pi\)
\(194\) 0 0
\(195\) 500.000i 0.183619i
\(196\) 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
\(205\) − 1250.00i − 0.425872i
\(206\) 0 0
\(207\) 4094.00 1.37465
\(208\) 0 0
\(209\) 2400.00 0.794313
\(210\) 0 0
\(211\) − 4060.00i − 1.32465i −0.749215 0.662327i \(-0.769569\pi\)
0.749215 0.662327i \(-0.230431\pi\)
\(212\) 0 0
\(213\) − 200.000i − 0.0643370i
\(214\) 0 0
\(215\) −710.000 −0.225217
\(216\) 0 0
\(217\) −120.000 −0.0375398
\(218\) 0 0
\(219\) − 460.000i − 0.141936i
\(220\) 0 0
\(221\) 1500.00i 0.456565i
\(222\) 0 0
\(223\) 5622.00 1.68824 0.844119 0.536156i \(-0.180124\pi\)
0.844119 + 0.536156i \(0.180124\pi\)
\(224\) 0 0
\(225\) −575.000 −0.170370
\(226\) 0 0
\(227\) 1554.00i 0.454373i 0.973851 + 0.227186i \(0.0729526\pi\)
−0.973851 + 0.227186i \(0.927047\pi\)
\(228\) 0 0
\(229\) 1134.00i 0.327235i 0.986524 + 0.163618i \(0.0523163\pi\)
−0.986524 + 0.163618i \(0.947684\pi\)
\(230\) 0 0
\(231\) −720.000 −0.205076
\(232\) 0 0
\(233\) 1710.00 0.480798 0.240399 0.970674i \(-0.422722\pi\)
0.240399 + 0.970674i \(0.422722\pi\)
\(234\) 0 0
\(235\) 1070.00i 0.297017i
\(236\) 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
\(245\) 1535.00i 0.400276i
\(246\) 0 0
\(247\) 2000.00 0.515210
\(248\) 0 0
\(249\) 1964.00 0.499853
\(250\) 0 0
\(251\) 660.000i 0.165971i 0.996551 + 0.0829857i \(0.0264456\pi\)
−0.996551 + 0.0829857i \(0.973554\pi\)
\(252\) 0 0
\(253\) − 10680.0i − 2.65394i
\(254\) 0 0
\(255\) 300.000 0.0736734
\(256\) 0 0
\(257\) −7590.00 −1.84222 −0.921111 0.389299i \(-0.872717\pi\)
−0.921111 + 0.389299i \(0.872717\pi\)
\(258\) 0 0
\(259\) 60.0000i 0.0143947i
\(260\) 0 0
\(261\) − 3818.00i − 0.905472i
\(262\) 0 0
\(263\) 762.000 0.178658 0.0893288 0.996002i \(-0.471528\pi\)
0.0893288 + 0.996002i \(0.471528\pi\)
\(264\) 0 0
\(265\) 2450.00 0.567933
\(266\) 0 0
\(267\) 1748.00i 0.400659i
\(268\) 0 0
\(269\) 150.000i 0.0339987i 0.999856 + 0.0169994i \(0.00541133\pi\)
−0.999856 + 0.0169994i \(0.994589\pi\)
\(270\) 0 0
\(271\) 6580.00 1.47493 0.737466 0.675384i \(-0.236022\pi\)
0.737466 + 0.675384i \(0.236022\pi\)
\(272\) 0 0
\(273\) −600.000 −0.133017
\(274\) 0 0
\(275\) 1500.00i 0.328921i
\(276\) 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 \(-0.764101\pi\)
−0.737726 + 0.675100i \(0.764101\pi\)
\(282\) 0 0
\(283\) 3882.00i 0.815410i 0.913114 + 0.407705i \(0.133671\pi\)
−0.913114 + 0.407705i \(0.866329\pi\)
\(284\) 0 0
\(285\) − 400.000i − 0.0831367i
\(286\) 0 0
\(287\) 1500.00 0.308509
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) 620.000i 0.124897i
\(292\) 0 0
\(293\) 1370.00i 0.273161i 0.990629 + 0.136581i \(0.0436113\pi\)
−0.990629 + 0.136581i \(0.956389\pi\)
\(294\) 0 0
\(295\) 4000.00 0.789454
\(296\) 0 0
\(297\) −6000.00 −1.17224
\(298\) 0 0
\(299\) − 8900.00i − 1.72141i
\(300\) 0 0
\(301\) − 852.000i − 0.163151i
\(302\) 0 0
\(303\) −2996.00 −0.568039
\(304\) 0 0
\(305\) −1250.00 −0.234671
\(306\) 0 0
\(307\) 4106.00i 0.763328i 0.924301 + 0.381664i \(0.124649\pi\)
−0.924301 + 0.381664i \(0.875351\pi\)
\(308\) 0 0
\(309\) − 2804.00i − 0.516226i
\(310\) 0 0
\(311\) 2220.00 0.404774 0.202387 0.979306i \(-0.435130\pi\)
0.202387 + 0.979306i \(0.435130\pi\)
\(312\) 0 0
\(313\) 9430.00 1.70292 0.851462 0.524417i \(-0.175717\pi\)
0.851462 + 0.524417i \(0.175717\pi\)
\(314\) 0 0
\(315\) − 690.000i − 0.123419i
\(316\) 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
\(325\) 1250.00i 0.213346i
\(326\) 0 0
\(327\) −1300.00 −0.219848
\(328\) 0 0
\(329\) −1284.00 −0.215165
\(330\) 0 0
\(331\) − 900.000i − 0.149452i −0.997204 0.0747258i \(-0.976192\pi\)
0.997204 0.0747258i \(-0.0238082\pi\)
\(332\) 0 0
\(333\) 230.000i 0.0378496i
\(334\) 0 0
\(335\) −3870.00 −0.631166
\(336\) 0 0
\(337\) 530.000 0.0856704 0.0428352 0.999082i \(-0.486361\pi\)
0.0428352 + 0.999082i \(0.486361\pi\)
\(338\) 0 0
\(339\) 3020.00i 0.483846i
\(340\) 0 0
\(341\) 1200.00i 0.190568i
\(342\) 0 0
\(343\) −3900.00 −0.613936
\(344\) 0 0
\(345\) −1780.00 −0.277774
\(346\) 0 0
\(347\) 414.000i 0.0640481i 0.999487 + 0.0320240i \(0.0101953\pi\)
−0.999487 + 0.0320240i \(0.989805\pi\)
\(348\) 0 0
\(349\) − 8614.00i − 1.32119i −0.750741 0.660597i \(-0.770303\pi\)
0.750741 0.660597i \(-0.229697\pi\)
\(350\) 0 0
\(351\) −5000.00 −0.760343
\(352\) 0 0
\(353\) −2270.00 −0.342266 −0.171133 0.985248i \(-0.554743\pi\)
−0.171133 + 0.985248i \(0.554743\pi\)
\(354\) 0 0
\(355\) − 500.000i − 0.0747528i
\(356\) 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
\(365\) − 1150.00i − 0.164914i
\(366\) 0 0
\(367\) −2374.00 −0.337662 −0.168831 0.985645i \(-0.553999\pi\)
−0.168831 + 0.985645i \(0.553999\pi\)
\(368\) 0 0
\(369\) 5750.00 0.811201
\(370\) 0 0
\(371\) 2940.00i 0.411421i
\(372\) 0 0
\(373\) 1810.00i 0.251255i 0.992077 + 0.125628i \(0.0400945\pi\)
−0.992077 + 0.125628i \(0.959906\pi\)
\(374\) 0 0
\(375\) 250.000 0.0344265
\(376\) 0 0
\(377\) −8300.00 −1.13388
\(378\) 0 0
\(379\) − 8120.00i − 1.10052i −0.834994 0.550259i \(-0.814529\pi\)
0.834994 0.550259i \(-0.185471\pi\)
\(380\) 0 0
\(381\) 2492.00i 0.335089i
\(382\) 0 0
\(383\) 11782.0 1.57189 0.785943 0.618299i \(-0.212178\pi\)
0.785943 + 0.618299i \(0.212178\pi\)
\(384\) 0 0
\(385\) −1800.00 −0.238277
\(386\) 0 0
\(387\) − 3266.00i − 0.428993i
\(388\) 0 0
\(389\) − 4350.00i − 0.566976i −0.958976 0.283488i \(-0.908508\pi\)
0.958976 0.283488i \(-0.0914917\pi\)
\(390\) 0 0
\(391\) −5340.00 −0.690679
\(392\) 0 0
\(393\) 5320.00 0.682846
\(394\) 0 0
\(395\) − 6600.00i − 0.840714i
\(396\) 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
\(405\) − 2105.00i − 0.258267i
\(406\) 0 0
\(407\) 600.000 0.0730735
\(408\) 0 0
\(409\) 3650.00 0.441274 0.220637 0.975356i \(-0.429186\pi\)
0.220637 + 0.975356i \(0.429186\pi\)
\(410\) 0 0
\(411\) 5540.00i 0.664886i
\(412\) 0 0
\(413\) 4800.00i 0.571895i
\(414\) 0 0
\(415\) 4910.00 0.580777
\(416\) 0 0
\(417\) 1120.00 0.131527
\(418\) 0 0
\(419\) − 1120.00i − 0.130586i −0.997866 0.0652931i \(-0.979202\pi\)
0.997866 0.0652931i \(-0.0207982\pi\)
\(420\) 0 0
\(421\) 4850.00i 0.561460i 0.959787 + 0.280730i \(0.0905765\pi\)
−0.959787 + 0.280730i \(0.909424\pi\)
\(422\) 0 0
\(423\) −4922.00 −0.565758
\(424\) 0 0
\(425\) 750.000 0.0856008
\(426\) 0 0
\(427\) − 1500.00i − 0.170000i
\(428\) 0 0
\(429\) 6000.00i 0.675251i
\(430\) 0 0
\(431\) 12580.0 1.40593 0.702967 0.711223i \(-0.251858\pi\)
0.702967 + 0.711223i \(0.251858\pi\)
\(432\) 0 0
\(433\) 13130.0 1.45725 0.728623 0.684915i \(-0.240161\pi\)
0.728623 + 0.684915i \(0.240161\pi\)
\(434\) 0 0
\(435\) 1660.00i 0.182968i
\(436\) 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 \(0.0733277\pi\)
−0.973583 + 0.228334i \(0.926672\pi\)
\(444\) 0 0
\(445\) 4370.00i 0.465523i
\(446\) 0 0
\(447\) −4700.00 −0.497321
\(448\) 0 0
\(449\) 2550.00 0.268022 0.134011 0.990980i \(-0.457214\pi\)
0.134011 + 0.990980i \(0.457214\pi\)
\(450\) 0 0
\(451\) − 15000.0i − 1.56613i
\(452\) 0 0
\(453\) − 1160.00i − 0.120312i
\(454\) 0 0
\(455\) −1500.00 −0.154552
\(456\) 0 0
\(457\) 6710.00 0.686828 0.343414 0.939184i \(-0.388417\pi\)
0.343414 + 0.939184i \(0.388417\pi\)
\(458\) 0 0
\(459\) 3000.00i 0.305072i
\(460\) 0 0
\(461\) 14482.0i 1.46311i 0.681782 + 0.731555i \(0.261205\pi\)
−0.681782 + 0.731555i \(0.738795\pi\)
\(462\) 0 0
\(463\) −162.000 −0.0162609 −0.00813043 0.999967i \(-0.502588\pi\)
−0.00813043 + 0.999967i \(0.502588\pi\)
\(464\) 0 0
\(465\) 200.000 0.0199458
\(466\) 0 0
\(467\) − 15974.0i − 1.58284i −0.611270 0.791422i \(-0.709341\pi\)
0.611270 0.791422i \(-0.290659\pi\)
\(468\) 0 0
\(469\) − 4644.00i − 0.457228i
\(470\) 0 0
\(471\) 2620.00 0.256313
\(472\) 0 0
\(473\) −8520.00 −0.828224
\(474\) 0 0
\(475\) − 1000.00i − 0.0965961i
\(476\) 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
\(485\) 1550.00i 0.145117i
\(486\) 0 0
\(487\) −9266.00 −0.862182 −0.431091 0.902309i \(-0.641871\pi\)
−0.431091 + 0.902309i \(0.641871\pi\)
\(488\) 0 0
\(489\) 3724.00 0.344387
\(490\) 0 0
\(491\) − 2860.00i − 0.262872i −0.991325 0.131436i \(-0.958041\pi\)
0.991325 0.131436i \(-0.0419587\pi\)
\(492\) 0 0
\(493\) 4980.00i 0.454945i
\(494\) 0 0
\(495\) −6900.00 −0.626529
\(496\) 0 0
\(497\) 600.000 0.0541523
\(498\) 0 0
\(499\) 7160.00i 0.642336i 0.947022 + 0.321168i \(0.104075\pi\)
−0.947022 + 0.321168i \(0.895925\pi\)
\(500\) 0 0
\(501\) − 1452.00i − 0.129482i
\(502\) 0 0
\(503\) −1398.00 −0.123924 −0.0619620 0.998079i \(-0.519736\pi\)
−0.0619620 + 0.998079i \(0.519736\pi\)
\(504\) 0 0
\(505\) −7490.00 −0.660001
\(506\) 0 0
\(507\) 606.000i 0.0530836i
\(508\) 0 0
\(509\) − 7446.00i − 0.648405i −0.945988 0.324203i \(-0.894904\pi\)
0.945988 0.324203i \(-0.105096\pi\)
\(510\) 0 0
\(511\) 1380.00 0.119467
\(512\) 0 0
\(513\) 4000.00 0.344258
\(514\) 0 0
\(515\) − 7010.00i − 0.599801i
\(516\) 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 \(-0.257111\pi\)
0.691134 + 0.722726i \(0.257111\pi\)
\(522\) 0 0
\(523\) 7322.00i 0.612177i 0.952003 + 0.306089i \(0.0990204\pi\)
−0.952003 + 0.306089i \(0.900980\pi\)
\(524\) 0 0
\(525\) 300.000i 0.0249392i
\(526\) 0 0
\(527\) 600.000 0.0495947
\(528\) 0 0
\(529\) 19517.0 1.60409
\(530\) 0 0
\(531\) 18400.0i 1.50375i
\(532\) 0 0
\(533\) − 12500.0i − 1.01583i
\(534\) 0 0
\(535\) −5970.00 −0.482440
\(536\) 0 0
\(537\) −2240.00 −0.180006
\(538\) 0 0
\(539\) 18420.0i 1.47200i
\(540\) 0 0
\(541\) − 10878.0i − 0.864476i −0.901759 0.432238i \(-0.857724\pi\)
0.901759 0.432238i \(-0.142276\pi\)
\(542\) 0 0
\(543\) −5684.00 −0.449215
\(544\) 0 0
\(545\) −3250.00 −0.255440
\(546\) 0 0
\(547\) 16114.0i 1.25957i 0.776769 + 0.629785i \(0.216857\pi\)
−0.776769 + 0.629785i \(0.783143\pi\)
\(548\) 0 0
\(549\) − 5750.00i − 0.447002i
\(550\) 0 0
\(551\) 6640.00 0.513382
\(552\) 0 0
\(553\) 7920.00 0.609028
\(554\) 0 0
\(555\) − 100.000i − 0.00764822i
\(556\) 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
\(565\) 7550.00i 0.562179i
\(566\) 0 0
\(567\) 2526.00 0.187094
\(568\) 0 0
\(569\) 6050.00 0.445746 0.222873 0.974848i \(-0.428457\pi\)
0.222873 + 0.974848i \(0.428457\pi\)
\(570\) 0 0
\(571\) − 8260.00i − 0.605377i −0.953090 0.302688i \(-0.902116\pi\)
0.953090 0.302688i \(-0.0978842\pi\)
\(572\) 0 0
\(573\) 6360.00i 0.463687i
\(574\) 0 0
\(575\) −4450.00 −0.322744
\(576\) 0 0
\(577\) −16870.0 −1.21717 −0.608585 0.793489i \(-0.708263\pi\)
−0.608585 + 0.793489i \(0.708263\pi\)
\(578\) 0 0
\(579\) 9340.00i 0.670392i
\(580\) 0 0
\(581\) 5892.00i 0.420725i
\(582\) 0 0
\(583\) 29400.0 2.08855
\(584\) 0 0
\(585\) −5750.00 −0.406382
\(586\) 0 0
\(587\) 966.000i 0.0679235i 0.999423 + 0.0339617i \(0.0108124\pi\)
−0.999423 + 0.0339617i \(0.989188\pi\)
\(588\) 0 0
\(589\) − 800.000i − 0.0559651i
\(590\) 0 0
\(591\) −5980.00 −0.416217
\(592\) 0 0
\(593\) 26290.0 1.82057 0.910287 0.413977i \(-0.135861\pi\)
0.910287 + 0.413977i \(0.135861\pi\)
\(594\) 0 0
\(595\) 900.000i 0.0620108i
\(596\) 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 \(-0.168162\pi\)
0.863667 + 0.504064i \(0.168162\pi\)
\(602\) 0 0
\(603\) − 17802.0i − 1.20224i
\(604\) 0 0
\(605\) 11345.0i 0.762380i
\(606\) 0 0
\(607\) −16694.0 −1.11629 −0.558145 0.829743i \(-0.688487\pi\)
−0.558145 + 0.829743i \(0.688487\pi\)
\(608\) 0 0
\(609\) −1992.00 −0.132545
\(610\) 0 0
\(611\) 10700.0i 0.708471i
\(612\) 0 0
\(613\) 15890.0i 1.04697i 0.852036 + 0.523484i \(0.175368\pi\)
−0.852036 + 0.523484i \(0.824632\pi\)
\(614\) 0 0
\(615\) −2500.00 −0.163918
\(616\) 0 0
\(617\) 1230.00 0.0802560 0.0401280 0.999195i \(-0.487223\pi\)
0.0401280 + 0.999195i \(0.487223\pi\)
\(618\) 0 0
\(619\) 10840.0i 0.703871i 0.936024 + 0.351936i \(0.114476\pi\)
−0.936024 + 0.351936i \(0.885524\pi\)
\(620\) 0 0
\(621\) − 17800.0i − 1.15022i
\(622\) 0 0
\(623\) −5244.00 −0.337233
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) − 4800.00i − 0.305731i
\(628\) 0 0
\(629\) − 300.000i − 0.0190171i
\(630\) 0 0
\(631\) −14060.0 −0.887036 −0.443518 0.896265i \(-0.646270\pi\)
−0.443518 + 0.896265i \(0.646270\pi\)
\(632\) 0 0
\(633\) −8120.00 −0.509859
\(634\) 0 0
\(635\) 6230.00i 0.389339i
\(636\) 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
\(645\) 1420.00i 0.0866860i
\(646\) 0 0
\(647\) −6786.00 −0.412342 −0.206171 0.978516i \(-0.566100\pi\)
−0.206171 + 0.978516i \(0.566100\pi\)
\(648\) 0 0
\(649\) 48000.0 2.90318
\(650\) 0 0
\(651\) 240.000i 0.0144491i
\(652\) 0 0
\(653\) 9030.00i 0.541150i 0.962699 + 0.270575i \(0.0872139\pi\)
−0.962699 + 0.270575i \(0.912786\pi\)
\(654\) 0 0
\(655\) 13300.0 0.793395
\(656\) 0 0
\(657\) 5290.00 0.314129
\(658\) 0 0
\(659\) − 15600.0i − 0.922139i −0.887364 0.461070i \(-0.847466\pi\)
0.887364 0.461070i \(-0.152534\pi\)
\(660\) 0 0
\(661\) 16850.0i 0.991511i 0.868462 + 0.495756i \(0.165109\pi\)
−0.868462 + 0.495756i \(0.834891\pi\)
\(662\) 0 0
\(663\) 3000.00 0.175732
\(664\) 0 0
\(665\) 1200.00 0.0699759
\(666\) 0 0
\(667\) − 29548.0i − 1.71530i
\(668\) 0 0
\(669\) − 11244.0i − 0.649803i
\(670\) 0 0
\(671\) −15000.0 −0.862993
\(672\) 0 0
\(673\) −7990.00 −0.457640 −0.228820 0.973469i \(-0.573487\pi\)
−0.228820 + 0.973469i \(0.573487\pi\)
\(674\) 0 0
\(675\) 2500.00i 0.142556i
\(676\) 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 \(-0.819437\pi\)
0.843379 0.537320i \(-0.180563\pi\)
\(684\) 0 0
\(685\) 13850.0i 0.772527i
\(686\) 0 0
\(687\) 2268.00 0.125953
\(688\) 0 0
\(689\) 24500.0 1.35468
\(690\) 0 0
\(691\) − 23380.0i − 1.28714i −0.765385 0.643572i \(-0.777452\pi\)
0.765385 0.643572i \(-0.222548\pi\)
\(692\) 0 0
\(693\) − 8280.00i − 0.453869i
\(694\) 0 0
\(695\) 2800.00 0.152820
\(696\) 0 0
\(697\) −7500.00 −0.407579
\(698\) 0 0
\(699\) − 3420.00i − 0.185059i
\(700\) 0 0
\(701\) − 11850.0i − 0.638471i −0.947675 0.319236i \(-0.896574\pi\)
0.947675 0.319236i \(-0.103426\pi\)
\(702\) 0 0
\(703\) −400.000 −0.0214599
\(704\) 0 0
\(705\) 2140.00 0.114322
\(706\) 0 0
\(707\) − 8988.00i − 0.478117i
\(708\) 0 0
\(709\) 25646.0i 1.35847i 0.733921 + 0.679235i \(0.237688\pi\)
−0.733921 + 0.679235i \(0.762312\pi\)
\(710\) 0 0
\(711\) 30360.0 1.60139
\(712\) 0 0
\(713\) −3560.00 −0.186989
\(714\) 0 0
\(715\) 15000.0i 0.784571i
\(716\) 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
\(725\) 4150.00i 0.212589i
\(726\) 0 0
\(727\) 17446.0 0.890009 0.445004 0.895528i \(-0.353202\pi\)
0.445004 + 0.895528i \(0.353202\pi\)
\(728\) 0 0
\(729\) 4283.00 0.217599
\(730\) 0 0
\(731\) 4260.00i 0.215543i
\(732\) 0 0
\(733\) 16750.0i 0.844032i 0.906588 + 0.422016i \(0.138677\pi\)
−0.906588 + 0.422016i \(0.861323\pi\)
\(734\) 0 0
\(735\) 3070.00 0.154066
\(736\) 0 0
\(737\) −46440.0 −2.32108
\(738\) 0 0
\(739\) − 36560.0i − 1.81987i −0.414755 0.909933i \(-0.636133\pi\)
0.414755 0.909933i \(-0.363867\pi\)
\(740\) 0 0
\(741\) − 4000.00i − 0.198305i
\(742\) 0 0
\(743\) −30142.0 −1.48829 −0.744147 0.668016i \(-0.767144\pi\)
−0.744147 + 0.668016i \(0.767144\pi\)
\(744\) 0 0
\(745\) −11750.0 −0.577834
\(746\) 0 0
\(747\) 22586.0i 1.10626i
\(748\) 0 0
\(749\) − 7164.00i − 0.349488i
\(750\) 0 0
\(751\) −11860.0 −0.576268 −0.288134 0.957590i \(-0.593035\pi\)
−0.288134 + 0.957590i \(0.593035\pi\)
\(752\) 0 0
\(753\) 1320.00 0.0638824
\(754\) 0 0
\(755\) − 2900.00i − 0.139790i
\(756\) 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 \(-0.409967\pi\)
0.279091 + 0.960265i \(0.409967\pi\)
\(762\) 0 0
\(763\) − 3900.00i − 0.185045i
\(764\) 0 0
\(765\) 3450.00i 0.163052i
\(766\) 0 0
\(767\) 40000.0 1.88307
\(768\) 0 0
\(769\) 4706.00 0.220680 0.110340 0.993894i \(-0.464806\pi\)
0.110340 + 0.993894i \(0.464806\pi\)
\(770\) 0 0
\(771\) 15180.0i 0.709072i
\(772\) 0 0
\(773\) − 28670.0i − 1.33401i −0.745054 0.667004i \(-0.767576\pi\)
0.745054 0.667004i \(-0.232424\pi\)
\(774\) 0 0
\(775\) 500.000 0.0231749
\(776\) 0 0
\(777\) 120.000 0.00554051
\(778\) 0 0
\(779\) 10000.0i 0.459932i
\(780\) 0 0
\(781\) − 6000.00i − 0.274900i
\(782\) 0 0
\(783\) −16600.0 −0.757644
\(784\) 0 0
\(785\) 6550.00 0.297808
\(786\) 0 0
\(787\) 20434.0i 0.925532i 0.886481 + 0.462766i \(0.153143\pi\)
−0.886481 + 0.462766i \(0.846857\pi\)
\(788\) 0 0
\(789\) − 1524.00i − 0.0687653i
\(790\) 0 0
\(791\) −9060.00 −0.407252
\(792\) 0 0
\(793\) −12500.0 −0.559758
\(794\) 0 0
\(795\) − 4900.00i − 0.218598i
\(796\) 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
\(805\) − 5340.00i − 0.233802i
\(806\) 0 0
\(807\) 300.000 0.0130861
\(808\) 0 0
\(809\) 4854.00 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 13140.0i 0.568937i 0.958685 + 0.284468i \(0.0918171\pi\)
−0.958685 + 0.284468i \(0.908183\pi\)
\(812\) 0 0
\(813\) − 13160.0i − 0.567702i
\(814\) 0 0
\(815\) 9310.00 0.400141
\(816\) 0 0
\(817\) 5680.00 0.243229
\(818\) 0 0
\(819\) − 6900.00i − 0.294390i
\(820\) 0 0
\(821\) 22050.0i 0.937333i 0.883375 + 0.468666i \(0.155265\pi\)
−0.883375 + 0.468666i \(0.844735\pi\)
\(822\) 0 0
\(823\) 14578.0 0.617445 0.308722 0.951152i \(-0.400099\pi\)
0.308722 + 0.951152i \(0.400099\pi\)
\(824\) 0 0
\(825\) 3000.00 0.126602
\(826\) 0 0
\(827\) 37054.0i 1.55803i 0.627003 + 0.779017i \(0.284281\pi\)
−0.627003 + 0.779017i \(0.715719\pi\)
\(828\) 0 0
\(829\) 6150.00i 0.257658i 0.991667 + 0.128829i \(0.0411218\pi\)
−0.991667 + 0.128829i \(0.958878\pi\)
\(830\) 0 0
\(831\) 9060.00 0.378204
\(832\) 0 0
\(833\) 9210.00 0.383082
\(834\) 0 0
\(835\) − 3630.00i − 0.150445i
\(836\) 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
\(845\) 1515.00i 0.0616776i
\(846\) 0 0
\(847\) −13614.0 −0.552282
\(848\) 0 0
\(849\) 7764.00 0.313851
\(850\) 0 0
\(851\) 1780.00i 0.0717011i
\(852\) 0 0
\(853\) − 42990.0i − 1.72561i −0.505533 0.862807i \(-0.668704\pi\)
0.505533 0.862807i \(-0.331296\pi\)
\(854\) 0 0
\(855\) 4600.00 0.183996
\(856\) 0 0
\(857\) −32130.0 −1.28068 −0.640338 0.768093i \(-0.721206\pi\)
−0.640338 + 0.768093i \(0.721206\pi\)
\(858\) 0 0
\(859\) − 15440.0i − 0.613278i −0.951826 0.306639i \(-0.900796\pi\)
0.951826 0.306639i \(-0.0992045\pi\)
\(860\) 0 0
\(861\) − 3000.00i − 0.118745i
\(862\) 0 0
\(863\) −46938.0 −1.85143 −0.925717 0.378216i \(-0.876538\pi\)
−0.925717 + 0.378216i \(0.876538\pi\)
\(864\) 0 0
\(865\) −16250.0 −0.638747
\(866\) 0 0
\(867\) 8026.00i 0.314391i
\(868\) 0 0
\(869\) − 79200.0i − 3.09169i
\(870\) 0 0
\(871\) −38700.0 −1.50551
\(872\) 0 0
\(873\) −7130.00 −0.276419
\(874\) 0 0
\(875\) 750.000i 0.0289767i
\(876\) 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
\(885\) − 8000.00i − 0.303861i
\(886\) 0 0
\(887\) 1766.00 0.0668506 0.0334253 0.999441i \(-0.489358\pi\)
0.0334253 + 0.999441i \(0.489358\pi\)
\(888\) 0 0
\(889\) −7476.00 −0.282044
\(890\) 0 0
\(891\) − 25260.0i − 0.949766i
\(892\) 0 0
\(893\) − 8560.00i − 0.320772i
\(894\) 0 0
\(895\) −5600.00 −0.209148
\(896\) 0 0
\(897\) −17800.0 −0.662569
\(898\) 0 0
\(899\) 3320.00i 0.123168i
\(900\) 0 0
\(901\) − 14700.0i − 0.543538i
\(902\) 0 0
\(903\) −1704.00 −0.0627969
\(904\) 0 0
\(905\) −14210.0 −0.521941
\(906\) 0 0
\(907\) − 41906.0i − 1.53414i −0.641563 0.767071i \(-0.721714\pi\)
0.641563 0.767071i \(-0.278286\pi\)
\(908\) 0 0
\(909\) − 34454.0i − 1.25717i
\(910\) 0 0
\(911\) −25140.0 −0.914298 −0.457149 0.889390i \(-0.651129\pi\)
−0.457149 + 0.889390i \(0.651129\pi\)
\(912\) 0 0
\(913\) 58920.0 2.13578
\(914\) 0 0
\(915\) 2500.00i 0.0903251i
\(916\) 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
\(925\) − 250.000i − 0.00888643i
\(926\) 0 0
\(927\) 32246.0 1.14250
\(928\) 0 0
\(929\) 10150.0 0.358461 0.179231 0.983807i \(-0.442639\pi\)
0.179231 + 0.983807i \(0.442639\pi\)
\(930\) 0 0
\(931\) − 12280.0i − 0.432289i
\(932\) 0 0
\(933\) − 4440.00i − 0.155798i
\(934\) 0 0
\(935\) 9000.00 0.314793
\(936\) 0 0
\(937\) −28530.0 −0.994701 −0.497350 0.867550i \(-0.665694\pi\)
−0.497350 + 0.867550i \(0.665694\pi\)
\(938\) 0 0
\(939\) − 18860.0i − 0.655456i
\(940\) 0 0
\(941\) − 9678.00i − 0.335275i −0.985849 0.167638i \(-0.946386\pi\)
0.985849 0.167638i \(-0.0536138\pi\)
\(942\) 0 0
\(943\) 44500.0 1.53671
\(944\) 0 0
\(945\) −3000.00 −0.103270
\(946\) 0 0
\(947\) 36986.0i 1.26915i 0.772862 + 0.634574i \(0.218824\pi\)
−0.772862 + 0.634574i \(0.781176\pi\)
\(948\) 0 0
\(949\) − 11500.0i − 0.393368i
\(950\) 0 0
\(951\) 12940.0 0.441228
\(952\) 0 0
\(953\) 3350.00 0.113869 0.0569345 0.998378i \(-0.481867\pi\)
0.0569345 + 0.998378i \(0.481867\pi\)
\(954\) 0 0
\(955\) 15900.0i 0.538756i
\(956\) 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
\(965\) 23350.0i 0.778925i
\(966\) 0 0
\(967\) 43774.0 1.45572 0.727858 0.685728i \(-0.240516\pi\)
0.727858 + 0.685728i \(0.240516\pi\)
\(968\) 0 0
\(969\) −2400.00 −0.0795656
\(970\) 0 0
\(971\) − 8740.00i − 0.288857i −0.989515 0.144428i \(-0.953866\pi\)
0.989515 0.144428i \(-0.0461343\pi\)
\(972\) 0 0
\(973\) 3360.00i 0.110706i
\(974\) 0 0
\(975\) 2500.00 0.0821170
\(976\) 0 0
\(977\) −48310.0 −1.58196 −0.790979 0.611843i \(-0.790429\pi\)
−0.790979 + 0.611843i \(0.790429\pi\)
\(978\) 0 0
\(979\) 52440.0i 1.71194i
\(980\) 0 0
\(981\) − 14950.0i − 0.486561i
\(982\) 0 0
\(983\) 2282.00 0.0740432 0.0370216 0.999314i \(-0.488213\pi\)
0.0370216 + 0.999314i \(0.488213\pi\)
\(984\) 0 0
\(985\) −14950.0 −0.483601
\(986\) 0 0
\(987\) 2568.00i 0.0828170i
\(988\) 0 0
\(989\) − 25276.0i − 0.812669i
\(990\) 0 0
\(991\) 31580.0 1.01228 0.506141 0.862451i \(-0.331071\pi\)
0.506141 + 0.862451i \(0.331071\pi\)
\(992\) 0 0
\(993\) −1800.00 −0.0575239
\(994\) 0 0
\(995\) 21200.0i 0.675462i
\(996\) 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 1280.4.d.k.641.1 2
4.3 odd 2 1280.4.d.f.641.2 2
8.3 odd 2 1280.4.d.f.641.1 2
8.5 even 2 inner 1280.4.d.k.641.2 2
16.3 odd 4 160.4.a.a.1.1 1
16.5 even 4 320.4.a.f.1.1 1
16.11 odd 4 320.4.a.i.1.1 1
16.13 even 4 160.4.a.b.1.1 yes 1
48.29 odd 4 1440.4.a.n.1.1 1
48.35 even 4 1440.4.a.o.1.1 1
80.3 even 4 800.4.c.f.449.1 2
80.13 odd 4 800.4.c.e.449.2 2
80.19 odd 4 800.4.a.h.1.1 1
80.29 even 4 800.4.a.d.1.1 1
80.59 odd 4 1600.4.a.r.1.1 1
80.67 even 4 800.4.c.f.449.2 2
80.69 even 4 1600.4.a.bj.1.1 1
80.77 odd 4 800.4.c.e.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.a.1.1 1 16.3 odd 4
160.4.a.b.1.1 yes 1 16.13 even 4
320.4.a.f.1.1 1 16.5 even 4
320.4.a.i.1.1 1 16.11 odd 4
800.4.a.d.1.1 1 80.29 even 4
800.4.a.h.1.1 1 80.19 odd 4
800.4.c.e.449.1 2 80.77 odd 4
800.4.c.e.449.2 2 80.13 odd 4
800.4.c.f.449.1 2 80.3 even 4
800.4.c.f.449.2 2 80.67 even 4
1280.4.d.f.641.1 2 8.3 odd 2
1280.4.d.f.641.2 2 4.3 odd 2
1280.4.d.k.641.1 2 1.1 even 1 trivial
1280.4.d.k.641.2 2 8.5 even 2 inner
1440.4.a.n.1.1 1 48.29 odd 4
1440.4.a.o.1.1 1 48.35 even 4
1600.4.a.r.1.1 1 80.59 odd 4
1600.4.a.bj.1.1 1 80.69 even 4