Properties

Label 800.4.c.i.449.1
Level $800$
Weight $4$
Character 800.449
Analytic conductor $47.202$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.4.c.i.449.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.89898i q^{3} -20.4949i q^{7} -52.1918 q^{9} +O(q^{10})\) \(q-8.89898i q^{3} -20.4949i q^{7} -52.1918 q^{9} -61.3939 q^{11} +45.1918i q^{13} +115.576i q^{17} +64.8082 q^{19} -182.384 q^{21} -6.11123i q^{23} +224.182i q^{27} +224.384 q^{29} -58.6061 q^{31} +546.343i q^{33} +99.6163i q^{37} +402.161 q^{39} -145.959 q^{41} +6.73776i q^{43} -203.687i q^{47} -77.0408 q^{49} +1028.50 q^{51} +275.576i q^{53} -576.727i q^{57} +262.020 q^{59} -790.767 q^{61} +1069.67i q^{63} -141.748i q^{67} -54.3837 q^{69} -1043.98 q^{71} -1057.88i q^{73} +1258.26i q^{77} -826.624 q^{79} +585.808 q^{81} +1026.25i q^{83} -1996.79i q^{87} +154.849 q^{89} +926.202 q^{91} +521.535i q^{93} -414.041i q^{97} +3204.26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81} + 1560 q^{89} + 3744 q^{91} + 6272 q^{99}+O(q^{100}) \) 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
\(3\) − 8.89898i − 1.71261i −0.516471 0.856305i \(-0.672755\pi\)
0.516471 0.856305i \(-0.327245\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 20.4949i − 1.10662i −0.832975 0.553310i \(-0.813364\pi\)
0.832975 0.553310i \(-0.186636\pi\)
\(8\) 0 0
\(9\) −52.1918 −1.93303
\(10\) 0 0
\(11\) −61.3939 −1.68281 −0.841407 0.540402i \(-0.818272\pi\)
−0.841407 + 0.540402i \(0.818272\pi\)
\(12\) 0 0
\(13\) 45.1918i 0.964151i 0.876130 + 0.482075i \(0.160117\pi\)
−0.876130 + 0.482075i \(0.839883\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 115.576i 1.64889i 0.565940 + 0.824446i \(0.308513\pi\)
−0.565940 + 0.824446i \(0.691487\pi\)
\(18\) 0 0
\(19\) 64.8082 0.782527 0.391263 0.920279i \(-0.372038\pi\)
0.391263 + 0.920279i \(0.372038\pi\)
\(20\) 0 0
\(21\) −182.384 −1.89521
\(22\) 0 0
\(23\) − 6.11123i − 0.0554034i −0.999616 0.0277017i \(-0.991181\pi\)
0.999616 0.0277017i \(-0.00881886\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 224.182i 1.59792i
\(28\) 0 0
\(29\) 224.384 1.43679 0.718397 0.695634i \(-0.244876\pi\)
0.718397 + 0.695634i \(0.244876\pi\)
\(30\) 0 0
\(31\) −58.6061 −0.339547 −0.169774 0.985483i \(-0.554304\pi\)
−0.169774 + 0.985483i \(0.554304\pi\)
\(32\) 0 0
\(33\) 546.343i 2.88200i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 99.6163i 0.442617i 0.975204 + 0.221308i \(0.0710327\pi\)
−0.975204 + 0.221308i \(0.928967\pi\)
\(38\) 0 0
\(39\) 402.161 1.65121
\(40\) 0 0
\(41\) −145.959 −0.555975 −0.277988 0.960585i \(-0.589667\pi\)
−0.277988 + 0.960585i \(0.589667\pi\)
\(42\) 0 0
\(43\) 6.73776i 0.0238953i 0.999929 + 0.0119477i \(0.00380315\pi\)
−0.999929 + 0.0119477i \(0.996197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 203.687i − 0.632144i −0.948735 0.316072i \(-0.897636\pi\)
0.948735 0.316072i \(-0.102364\pi\)
\(48\) 0 0
\(49\) −77.0408 −0.224609
\(50\) 0 0
\(51\) 1028.50 2.82391
\(52\) 0 0
\(53\) 275.576i 0.714211i 0.934064 + 0.357106i \(0.116236\pi\)
−0.934064 + 0.357106i \(0.883764\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 576.727i − 1.34016i
\(58\) 0 0
\(59\) 262.020 0.578172 0.289086 0.957303i \(-0.406649\pi\)
0.289086 + 0.957303i \(0.406649\pi\)
\(60\) 0 0
\(61\) −790.767 −1.65979 −0.829897 0.557917i \(-0.811601\pi\)
−0.829897 + 0.557917i \(0.811601\pi\)
\(62\) 0 0
\(63\) 1069.67i 2.13913i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 141.748i − 0.258467i −0.991614 0.129233i \(-0.958748\pi\)
0.991614 0.129233i \(-0.0412516\pi\)
\(68\) 0 0
\(69\) −54.3837 −0.0948844
\(70\) 0 0
\(71\) −1043.98 −1.74503 −0.872516 0.488585i \(-0.837513\pi\)
−0.872516 + 0.488585i \(0.837513\pi\)
\(72\) 0 0
\(73\) − 1057.88i − 1.69610i −0.529917 0.848049i \(-0.677777\pi\)
0.529917 0.848049i \(-0.322223\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1258.26i 1.86224i
\(78\) 0 0
\(79\) −826.624 −1.17725 −0.588624 0.808407i \(-0.700330\pi\)
−0.588624 + 0.808407i \(0.700330\pi\)
\(80\) 0 0
\(81\) 585.808 0.803578
\(82\) 0 0
\(83\) 1026.25i 1.35718i 0.734518 + 0.678589i \(0.237408\pi\)
−0.734518 + 0.678589i \(0.762592\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 1996.79i − 2.46067i
\(88\) 0 0
\(89\) 154.849 0.184427 0.0922133 0.995739i \(-0.470606\pi\)
0.0922133 + 0.995739i \(0.470606\pi\)
\(90\) 0 0
\(91\) 926.202 1.06695
\(92\) 0 0
\(93\) 521.535i 0.581512i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 414.041i − 0.433397i −0.976239 0.216698i \(-0.930471\pi\)
0.976239 0.216698i \(-0.0695288\pi\)
\(98\) 0 0
\(99\) 3204.26 3.25293
\(100\) 0 0
\(101\) 1776.99 1.75066 0.875331 0.483524i \(-0.160643\pi\)
0.875331 + 0.483524i \(0.160643\pi\)
\(102\) 0 0
\(103\) 767.991i 0.734683i 0.930086 + 0.367342i \(0.119732\pi\)
−0.930086 + 0.367342i \(0.880268\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1649.00i 1.48986i 0.667144 + 0.744929i \(0.267517\pi\)
−0.667144 + 0.744929i \(0.732483\pi\)
\(108\) 0 0
\(109\) −884.220 −0.777000 −0.388500 0.921449i \(-0.627007\pi\)
−0.388500 + 0.921449i \(0.627007\pi\)
\(110\) 0 0
\(111\) 886.484 0.758030
\(112\) 0 0
\(113\) − 1138.85i − 0.948088i −0.880501 0.474044i \(-0.842794\pi\)
0.880501 0.474044i \(-0.157206\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2358.64i − 1.86373i
\(118\) 0 0
\(119\) 2368.71 1.82470
\(120\) 0 0
\(121\) 2438.21 1.83186
\(122\) 0 0
\(123\) 1298.89i 0.952169i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1701.60i 1.18892i 0.804125 + 0.594460i \(0.202634\pi\)
−0.804125 + 0.594460i \(0.797366\pi\)
\(128\) 0 0
\(129\) 59.9592 0.0409233
\(130\) 0 0
\(131\) 1540.99 1.02776 0.513880 0.857862i \(-0.328208\pi\)
0.513880 + 0.857862i \(0.328208\pi\)
\(132\) 0 0
\(133\) − 1328.24i − 0.865960i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1391.53i 0.867787i 0.900964 + 0.433894i \(0.142861\pi\)
−0.900964 + 0.433894i \(0.857139\pi\)
\(138\) 0 0
\(139\) 156.649 0.0955885 0.0477942 0.998857i \(-0.484781\pi\)
0.0477942 + 0.998857i \(0.484781\pi\)
\(140\) 0 0
\(141\) −1812.60 −1.08262
\(142\) 0 0
\(143\) − 2774.50i − 1.62249i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 685.585i 0.384667i
\(148\) 0 0
\(149\) −214.245 −0.117796 −0.0588981 0.998264i \(-0.518759\pi\)
−0.0588981 + 0.998264i \(0.518759\pi\)
\(150\) 0 0
\(151\) 35.0510 0.0188901 0.00944507 0.999955i \(-0.496993\pi\)
0.00944507 + 0.999955i \(0.496993\pi\)
\(152\) 0 0
\(153\) − 6032.10i − 3.18736i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 349.592i − 0.177710i −0.996045 0.0888550i \(-0.971679\pi\)
0.996045 0.0888550i \(-0.0283208\pi\)
\(158\) 0 0
\(159\) 2452.34 1.22317
\(160\) 0 0
\(161\) −125.249 −0.0613106
\(162\) 0 0
\(163\) 2029.04i 0.975010i 0.873120 + 0.487505i \(0.162093\pi\)
−0.873120 + 0.487505i \(0.837907\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 214.619i 0.0994476i 0.998763 + 0.0497238i \(0.0158341\pi\)
−0.998763 + 0.0497238i \(0.984166\pi\)
\(168\) 0 0
\(169\) 154.698 0.0704133
\(170\) 0 0
\(171\) −3382.46 −1.51265
\(172\) 0 0
\(173\) 3424.99i 1.50518i 0.658487 + 0.752592i \(0.271197\pi\)
−0.658487 + 0.752592i \(0.728803\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2331.71i − 0.990183i
\(178\) 0 0
\(179\) 2030.10 0.847692 0.423846 0.905734i \(-0.360680\pi\)
0.423846 + 0.905734i \(0.360680\pi\)
\(180\) 0 0
\(181\) 204.139 0.0838316 0.0419158 0.999121i \(-0.486654\pi\)
0.0419158 + 0.999121i \(0.486654\pi\)
\(182\) 0 0
\(183\) 7037.02i 2.84258i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 7095.63i − 2.77478i
\(188\) 0 0
\(189\) 4594.58 1.76829
\(190\) 0 0
\(191\) −617.353 −0.233875 −0.116937 0.993139i \(-0.537308\pi\)
−0.116937 + 0.993139i \(0.537308\pi\)
\(192\) 0 0
\(193\) 3071.33i 1.14549i 0.819734 + 0.572744i \(0.194121\pi\)
−0.819734 + 0.572744i \(0.805879\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 812.098i 0.293703i 0.989159 + 0.146852i \(0.0469140\pi\)
−0.989159 + 0.146852i \(0.953086\pi\)
\(198\) 0 0
\(199\) 1065.93 0.379709 0.189855 0.981812i \(-0.439198\pi\)
0.189855 + 0.981812i \(0.439198\pi\)
\(200\) 0 0
\(201\) −1261.41 −0.442653
\(202\) 0 0
\(203\) − 4598.72i − 1.58998i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 318.956i 0.107097i
\(208\) 0 0
\(209\) −3978.82 −1.31685
\(210\) 0 0
\(211\) −1222.24 −0.398779 −0.199390 0.979920i \(-0.563896\pi\)
−0.199390 + 0.979920i \(0.563896\pi\)
\(212\) 0 0
\(213\) 9290.33i 2.98856i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1201.13i 0.375750i
\(218\) 0 0
\(219\) −9414.03 −2.90475
\(220\) 0 0
\(221\) −5223.07 −1.58978
\(222\) 0 0
\(223\) 414.374i 0.124433i 0.998063 + 0.0622165i \(0.0198169\pi\)
−0.998063 + 0.0622165i \(0.980183\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2293.88i 0.670707i 0.942092 + 0.335353i \(0.108856\pi\)
−0.942092 + 0.335353i \(0.891144\pi\)
\(228\) 0 0
\(229\) −3214.72 −0.927662 −0.463831 0.885924i \(-0.653525\pi\)
−0.463831 + 0.885924i \(0.653525\pi\)
\(230\) 0 0
\(231\) 11197.2 3.18928
\(232\) 0 0
\(233\) 3598.60i 1.01181i 0.862589 + 0.505905i \(0.168841\pi\)
−0.862589 + 0.505905i \(0.831159\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7356.11i 2.01616i
\(238\) 0 0
\(239\) −2462.58 −0.666491 −0.333245 0.942840i \(-0.608144\pi\)
−0.333245 + 0.942840i \(0.608144\pi\)
\(240\) 0 0
\(241\) 4153.27 1.11011 0.555054 0.831815i \(-0.312698\pi\)
0.555054 + 0.831815i \(0.312698\pi\)
\(242\) 0 0
\(243\) 839.809i 0.221703i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2928.80i 0.754474i
\(248\) 0 0
\(249\) 9132.60 2.32432
\(250\) 0 0
\(251\) −1335.26 −0.335779 −0.167890 0.985806i \(-0.553695\pi\)
−0.167890 + 0.985806i \(0.553695\pi\)
\(252\) 0 0
\(253\) 375.192i 0.0932336i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 530.359i 0.128727i 0.997927 + 0.0643636i \(0.0205017\pi\)
−0.997927 + 0.0643636i \(0.979498\pi\)
\(258\) 0 0
\(259\) 2041.63 0.489809
\(260\) 0 0
\(261\) −11711.0 −2.77737
\(262\) 0 0
\(263\) − 2234.23i − 0.523836i −0.965090 0.261918i \(-0.915645\pi\)
0.965090 0.261918i \(-0.0843549\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 1378.00i − 0.315851i
\(268\) 0 0
\(269\) −5096.09 −1.15507 −0.577535 0.816366i \(-0.695985\pi\)
−0.577535 + 0.816366i \(0.695985\pi\)
\(270\) 0 0
\(271\) 1353.39 0.303367 0.151684 0.988429i \(-0.451530\pi\)
0.151684 + 0.988429i \(0.451530\pi\)
\(272\) 0 0
\(273\) − 8242.25i − 1.82727i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8879.57i 1.92607i 0.269376 + 0.963035i \(0.413183\pi\)
−0.269376 + 0.963035i \(0.586817\pi\)
\(278\) 0 0
\(279\) 3058.76 0.656356
\(280\) 0 0
\(281\) −1145.55 −0.243195 −0.121598 0.992579i \(-0.538802\pi\)
−0.121598 + 0.992579i \(0.538802\pi\)
\(282\) 0 0
\(283\) 385.399i 0.0809526i 0.999180 + 0.0404763i \(0.0128875\pi\)
−0.999180 + 0.0404763i \(0.987112\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2991.42i 0.615254i
\(288\) 0 0
\(289\) −8444.70 −1.71885
\(290\) 0 0
\(291\) −3684.54 −0.742239
\(292\) 0 0
\(293\) 3812.30i 0.760126i 0.924961 + 0.380063i \(0.124098\pi\)
−0.924961 + 0.380063i \(0.875902\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 13763.4i − 2.68900i
\(298\) 0 0
\(299\) 276.178 0.0534172
\(300\) 0 0
\(301\) 138.090 0.0264430
\(302\) 0 0
\(303\) − 15813.4i − 2.99820i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 6897.45i − 1.28228i −0.767426 0.641138i \(-0.778463\pi\)
0.767426 0.641138i \(-0.221537\pi\)
\(308\) 0 0
\(309\) 6834.33 1.25823
\(310\) 0 0
\(311\) −8125.39 −1.48151 −0.740753 0.671778i \(-0.765531\pi\)
−0.740753 + 0.671778i \(0.765531\pi\)
\(312\) 0 0
\(313\) 2052.66i 0.370681i 0.982674 + 0.185341i \(0.0593389\pi\)
−0.982674 + 0.185341i \(0.940661\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 4175.12i − 0.739741i −0.929083 0.369871i \(-0.879402\pi\)
0.929083 0.369871i \(-0.120598\pi\)
\(318\) 0 0
\(319\) −13775.8 −2.41786
\(320\) 0 0
\(321\) 14674.4 2.55154
\(322\) 0 0
\(323\) 7490.24i 1.29030i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7868.66i 1.33070i
\(328\) 0 0
\(329\) −4174.54 −0.699543
\(330\) 0 0
\(331\) −11738.2 −1.94921 −0.974607 0.223923i \(-0.928114\pi\)
−0.974607 + 0.223923i \(0.928114\pi\)
\(332\) 0 0
\(333\) − 5199.16i − 0.855592i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1698.85i 0.274606i 0.990529 + 0.137303i \(0.0438434\pi\)
−0.990529 + 0.137303i \(0.956157\pi\)
\(338\) 0 0
\(339\) −10134.6 −1.62370
\(340\) 0 0
\(341\) 3598.06 0.571395
\(342\) 0 0
\(343\) − 5450.81i − 0.858064i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9682.53i 1.49794i 0.662604 + 0.748970i \(0.269451\pi\)
−0.662604 + 0.748970i \(0.730549\pi\)
\(348\) 0 0
\(349\) 10755.1 1.64959 0.824796 0.565430i \(-0.191290\pi\)
0.824796 + 0.565430i \(0.191290\pi\)
\(350\) 0 0
\(351\) −10131.2 −1.54063
\(352\) 0 0
\(353\) − 7044.94i − 1.06222i −0.847302 0.531111i \(-0.821775\pi\)
0.847302 0.531111i \(-0.178225\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 21079.1i − 3.12500i
\(358\) 0 0
\(359\) 805.576 0.118431 0.0592154 0.998245i \(-0.481140\pi\)
0.0592154 + 0.998245i \(0.481140\pi\)
\(360\) 0 0
\(361\) −2658.90 −0.387652
\(362\) 0 0
\(363\) − 21697.6i − 3.13726i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 11157.2i − 1.58692i −0.608623 0.793460i \(-0.708278\pi\)
0.608623 0.793460i \(-0.291722\pi\)
\(368\) 0 0
\(369\) 7617.88 1.07472
\(370\) 0 0
\(371\) 5647.89 0.790361
\(372\) 0 0
\(373\) − 4634.24i − 0.643303i −0.946858 0.321652i \(-0.895762\pi\)
0.946858 0.321652i \(-0.104238\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10140.3i 1.38529i
\(378\) 0 0
\(379\) −754.678 −0.102283 −0.0511414 0.998691i \(-0.516286\pi\)
−0.0511414 + 0.998691i \(0.516286\pi\)
\(380\) 0 0
\(381\) 15142.5 2.03616
\(382\) 0 0
\(383\) 984.709i 0.131374i 0.997840 + 0.0656871i \(0.0209239\pi\)
−0.997840 + 0.0656871i \(0.979076\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 351.656i − 0.0461904i
\(388\) 0 0
\(389\) 3259.21 0.424803 0.212402 0.977182i \(-0.431872\pi\)
0.212402 + 0.977182i \(0.431872\pi\)
\(390\) 0 0
\(391\) 706.308 0.0913543
\(392\) 0 0
\(393\) − 13713.2i − 1.76015i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13713.2i 1.73361i 0.498644 + 0.866807i \(0.333831\pi\)
−0.498644 + 0.866807i \(0.666169\pi\)
\(398\) 0 0
\(399\) −11820.0 −1.48305
\(400\) 0 0
\(401\) −671.306 −0.0835996 −0.0417998 0.999126i \(-0.513309\pi\)
−0.0417998 + 0.999126i \(0.513309\pi\)
\(402\) 0 0
\(403\) − 2648.52i − 0.327375i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 6115.83i − 0.744842i
\(408\) 0 0
\(409\) −12957.4 −1.56651 −0.783253 0.621703i \(-0.786441\pi\)
−0.783253 + 0.621703i \(0.786441\pi\)
\(410\) 0 0
\(411\) 12383.2 1.48618
\(412\) 0 0
\(413\) − 5370.08i − 0.639817i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 1394.02i − 0.163706i
\(418\) 0 0
\(419\) 14023.2 1.63503 0.817516 0.575906i \(-0.195351\pi\)
0.817516 + 0.575906i \(0.195351\pi\)
\(420\) 0 0
\(421\) −4825.76 −0.558653 −0.279326 0.960196i \(-0.590111\pi\)
−0.279326 + 0.960196i \(0.590111\pi\)
\(422\) 0 0
\(423\) 10630.8i 1.22195i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16206.7i 1.83676i
\(428\) 0 0
\(429\) −24690.2 −2.77868
\(430\) 0 0
\(431\) −3901.30 −0.436007 −0.218003 0.975948i \(-0.569954\pi\)
−0.218003 + 0.975948i \(0.569954\pi\)
\(432\) 0 0
\(433\) 1027.04i 0.113987i 0.998375 + 0.0569933i \(0.0181514\pi\)
−0.998375 + 0.0569933i \(0.981849\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 396.057i − 0.0433547i
\(438\) 0 0
\(439\) −255.290 −0.0277547 −0.0138774 0.999904i \(-0.504417\pi\)
−0.0138774 + 0.999904i \(0.504417\pi\)
\(440\) 0 0
\(441\) 4020.90 0.434176
\(442\) 0 0
\(443\) − 6551.19i − 0.702611i −0.936261 0.351305i \(-0.885738\pi\)
0.936261 0.351305i \(-0.114262\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1906.56i 0.201739i
\(448\) 0 0
\(449\) 2364.38 0.248512 0.124256 0.992250i \(-0.460346\pi\)
0.124256 + 0.992250i \(0.460346\pi\)
\(450\) 0 0
\(451\) 8961.00 0.935603
\(452\) 0 0
\(453\) − 311.918i − 0.0323514i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 4440.07i − 0.454481i −0.973839 0.227241i \(-0.927030\pi\)
0.973839 0.227241i \(-0.0729704\pi\)
\(458\) 0 0
\(459\) −25909.9 −2.63479
\(460\) 0 0
\(461\) −891.518 −0.0900697 −0.0450349 0.998985i \(-0.514340\pi\)
−0.0450349 + 0.998985i \(0.514340\pi\)
\(462\) 0 0
\(463\) 3294.85i 0.330723i 0.986233 + 0.165361i \(0.0528790\pi\)
−0.986233 + 0.165361i \(0.947121\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1488.22i − 0.147466i −0.997278 0.0737329i \(-0.976509\pi\)
0.997278 0.0737329i \(-0.0234912\pi\)
\(468\) 0 0
\(469\) −2905.11 −0.286025
\(470\) 0 0
\(471\) −3111.01 −0.304348
\(472\) 0 0
\(473\) − 413.657i − 0.0402114i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 14382.8i − 1.38059i
\(478\) 0 0
\(479\) −17534.7 −1.67261 −0.836304 0.548266i \(-0.815288\pi\)
−0.836304 + 0.548266i \(0.815288\pi\)
\(480\) 0 0
\(481\) −4501.84 −0.426749
\(482\) 0 0
\(483\) 1114.59i 0.105001i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 10113.8i − 0.941067i −0.882382 0.470533i \(-0.844062\pi\)
0.882382 0.470533i \(-0.155938\pi\)
\(488\) 0 0
\(489\) 18056.4 1.66981
\(490\) 0 0
\(491\) 17855.8 1.64119 0.820594 0.571512i \(-0.193643\pi\)
0.820594 + 0.571512i \(0.193643\pi\)
\(492\) 0 0
\(493\) 25933.3i 2.36912i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21396.2i 1.93109i
\(498\) 0 0
\(499\) −2985.84 −0.267865 −0.133932 0.990990i \(-0.542760\pi\)
−0.133932 + 0.990990i \(0.542760\pi\)
\(500\) 0 0
\(501\) 1909.89 0.170315
\(502\) 0 0
\(503\) 2159.75i 0.191449i 0.995408 + 0.0957243i \(0.0305167\pi\)
−0.995408 + 0.0957243i \(0.969483\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1376.65i − 0.120590i
\(508\) 0 0
\(509\) −16880.6 −1.46998 −0.734989 0.678079i \(-0.762813\pi\)
−0.734989 + 0.678079i \(0.762813\pi\)
\(510\) 0 0
\(511\) −21681.1 −1.87694
\(512\) 0 0
\(513\) 14528.8i 1.25041i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12505.1i 1.06378i
\(518\) 0 0
\(519\) 30478.9 2.57779
\(520\) 0 0
\(521\) 3915.76 0.329276 0.164638 0.986354i \(-0.447354\pi\)
0.164638 + 0.986354i \(0.447354\pi\)
\(522\) 0 0
\(523\) 14478.8i 1.21054i 0.796019 + 0.605271i \(0.206935\pi\)
−0.796019 + 0.605271i \(0.793065\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 6773.43i − 0.559877i
\(528\) 0 0
\(529\) 12129.7 0.996930
\(530\) 0 0
\(531\) −13675.3 −1.11762
\(532\) 0 0
\(533\) − 6596.16i − 0.536044i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 18065.8i − 1.45177i
\(538\) 0 0
\(539\) 4729.83 0.377975
\(540\) 0 0
\(541\) −11827.6 −0.939940 −0.469970 0.882682i \(-0.655735\pi\)
−0.469970 + 0.882682i \(0.655735\pi\)
\(542\) 0 0
\(543\) − 1816.63i − 0.143571i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 6446.22i − 0.503877i −0.967743 0.251938i \(-0.918932\pi\)
0.967743 0.251938i \(-0.0810680\pi\)
\(548\) 0 0
\(549\) 41271.6 3.20843
\(550\) 0 0
\(551\) 14541.9 1.12433
\(552\) 0 0
\(553\) 16941.6i 1.30277i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14419.5i 1.09690i 0.836183 + 0.548451i \(0.184782\pi\)
−0.836183 + 0.548451i \(0.815218\pi\)
\(558\) 0 0
\(559\) −304.492 −0.0230387
\(560\) 0 0
\(561\) −63143.9 −4.75211
\(562\) 0 0
\(563\) − 817.038i − 0.0611617i −0.999532 0.0305808i \(-0.990264\pi\)
0.999532 0.0305808i \(-0.00973570\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 12006.1i − 0.889256i
\(568\) 0 0
\(569\) 16795.5 1.23744 0.618720 0.785612i \(-0.287652\pi\)
0.618720 + 0.785612i \(0.287652\pi\)
\(570\) 0 0
\(571\) −15342.6 −1.12447 −0.562233 0.826979i \(-0.690057\pi\)
−0.562233 + 0.826979i \(0.690057\pi\)
\(572\) 0 0
\(573\) 5493.81i 0.400536i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1287.65i − 0.0929039i −0.998921 0.0464519i \(-0.985209\pi\)
0.998921 0.0464519i \(-0.0147914\pi\)
\(578\) 0 0
\(579\) 27331.7 1.96177
\(580\) 0 0
\(581\) 21032.9 1.50188
\(582\) 0 0
\(583\) − 16918.6i − 1.20188i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 2074.66i − 0.145878i −0.997336 0.0729391i \(-0.976762\pi\)
0.997336 0.0729391i \(-0.0232379\pi\)
\(588\) 0 0
\(589\) −3798.16 −0.265705
\(590\) 0 0
\(591\) 7226.84 0.502999
\(592\) 0 0
\(593\) 16263.0i 1.12621i 0.826386 + 0.563104i \(0.190393\pi\)
−0.826386 + 0.563104i \(0.809607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 9485.73i − 0.650293i
\(598\) 0 0
\(599\) −22423.7 −1.52956 −0.764780 0.644292i \(-0.777152\pi\)
−0.764780 + 0.644292i \(0.777152\pi\)
\(600\) 0 0
\(601\) −2915.58 −0.197885 −0.0989424 0.995093i \(-0.531546\pi\)
−0.0989424 + 0.995093i \(0.531546\pi\)
\(602\) 0 0
\(603\) 7398.09i 0.499624i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11897.3i 0.795546i 0.917484 + 0.397773i \(0.130217\pi\)
−0.917484 + 0.397773i \(0.869783\pi\)
\(608\) 0 0
\(609\) −40923.9 −2.72302
\(610\) 0 0
\(611\) 9204.98 0.609482
\(612\) 0 0
\(613\) − 22791.7i − 1.50171i −0.660466 0.750856i \(-0.729641\pi\)
0.660466 0.750856i \(-0.270359\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 11894.0i − 0.776071i −0.921644 0.388036i \(-0.873154\pi\)
0.921644 0.388036i \(-0.126846\pi\)
\(618\) 0 0
\(619\) 15567.8 1.01086 0.505431 0.862867i \(-0.331333\pi\)
0.505431 + 0.862867i \(0.331333\pi\)
\(620\) 0 0
\(621\) 1370.02 0.0885301
\(622\) 0 0
\(623\) − 3173.61i − 0.204090i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 35407.5i 2.25524i
\(628\) 0 0
\(629\) −11513.2 −0.729828
\(630\) 0 0
\(631\) −23681.8 −1.49407 −0.747034 0.664786i \(-0.768523\pi\)
−0.747034 + 0.664786i \(0.768523\pi\)
\(632\) 0 0
\(633\) 10876.7i 0.682953i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3481.62i − 0.216557i
\(638\) 0 0
\(639\) 54487.1 3.37320
\(640\) 0 0
\(641\) 21351.6 1.31566 0.657831 0.753166i \(-0.271474\pi\)
0.657831 + 0.753166i \(0.271474\pi\)
\(642\) 0 0
\(643\) 7196.04i 0.441344i 0.975348 + 0.220672i \(0.0708250\pi\)
−0.975348 + 0.220672i \(0.929175\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20897.9i 1.26983i 0.772582 + 0.634915i \(0.218965\pi\)
−0.772582 + 0.634915i \(0.781035\pi\)
\(648\) 0 0
\(649\) −16086.4 −0.972956
\(650\) 0 0
\(651\) 10688.8 0.643513
\(652\) 0 0
\(653\) 8655.43i 0.518703i 0.965783 + 0.259352i \(0.0835088\pi\)
−0.965783 + 0.259352i \(0.916491\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 55212.6i 3.27861i
\(658\) 0 0
\(659\) 13004.3 0.768705 0.384352 0.923186i \(-0.374425\pi\)
0.384352 + 0.923186i \(0.374425\pi\)
\(660\) 0 0
\(661\) −19135.2 −1.12598 −0.562991 0.826463i \(-0.690349\pi\)
−0.562991 + 0.826463i \(0.690349\pi\)
\(662\) 0 0
\(663\) 46480.0i 2.72267i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 1371.26i − 0.0796033i
\(668\) 0 0
\(669\) 3687.51 0.213105
\(670\) 0 0
\(671\) 48548.3 2.79312
\(672\) 0 0
\(673\) 24101.3i 1.38044i 0.723599 + 0.690221i \(0.242487\pi\)
−0.723599 + 0.690221i \(0.757513\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 17892.9i − 1.01577i −0.861424 0.507887i \(-0.830427\pi\)
0.861424 0.507887i \(-0.169573\pi\)
\(678\) 0 0
\(679\) −8485.72 −0.479606
\(680\) 0 0
\(681\) 20413.2 1.14866
\(682\) 0 0
\(683\) 27545.4i 1.54319i 0.636117 + 0.771593i \(0.280540\pi\)
−0.636117 + 0.771593i \(0.719460\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 28607.7i 1.58872i
\(688\) 0 0
\(689\) −12453.8 −0.688608
\(690\) 0 0
\(691\) −11360.1 −0.625408 −0.312704 0.949851i \(-0.601235\pi\)
−0.312704 + 0.949851i \(0.601235\pi\)
\(692\) 0 0
\(693\) − 65671.0i − 3.59976i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 16869.3i − 0.916744i
\(698\) 0 0
\(699\) 32023.8 1.73284
\(700\) 0 0
\(701\) −148.008 −0.00797459 −0.00398729 0.999992i \(-0.501269\pi\)
−0.00398729 + 0.999992i \(0.501269\pi\)
\(702\) 0 0
\(703\) 6455.95i 0.346360i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 36419.2i − 1.93732i
\(708\) 0 0
\(709\) −2401.61 −0.127213 −0.0636067 0.997975i \(-0.520260\pi\)
−0.0636067 + 0.997975i \(0.520260\pi\)
\(710\) 0 0
\(711\) 43143.0 2.27566
\(712\) 0 0
\(713\) 358.155i 0.0188121i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21914.5i 1.14144i
\(718\) 0 0
\(719\) 13003.4 0.674472 0.337236 0.941420i \(-0.390508\pi\)
0.337236 + 0.941420i \(0.390508\pi\)
\(720\) 0 0
\(721\) 15739.9 0.813016
\(722\) 0 0
\(723\) − 36959.9i − 1.90118i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 36506.5i − 1.86238i −0.364533 0.931190i \(-0.618771\pi\)
0.364533 0.931190i \(-0.381229\pi\)
\(728\) 0 0
\(729\) 23290.3 1.18327
\(730\) 0 0
\(731\) −778.720 −0.0394008
\(732\) 0 0
\(733\) − 18661.5i − 0.940350i −0.882573 0.470175i \(-0.844191\pi\)
0.882573 0.470175i \(-0.155809\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8702.46i 0.434951i
\(738\) 0 0
\(739\) −27675.0 −1.37759 −0.688797 0.724954i \(-0.741861\pi\)
−0.688797 + 0.724954i \(0.741861\pi\)
\(740\) 0 0
\(741\) 26063.3 1.29212
\(742\) 0 0
\(743\) − 17622.0i − 0.870103i −0.900406 0.435052i \(-0.856730\pi\)
0.900406 0.435052i \(-0.143270\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 53562.0i − 2.62347i
\(748\) 0 0
\(749\) 33796.1 1.64871
\(750\) 0 0
\(751\) −34581.0 −1.68027 −0.840133 0.542381i \(-0.817523\pi\)
−0.840133 + 0.542381i \(0.817523\pi\)
\(752\) 0 0
\(753\) 11882.4i 0.575058i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 33994.4i − 1.63216i −0.577936 0.816082i \(-0.696142\pi\)
0.577936 0.816082i \(-0.303858\pi\)
\(758\) 0 0
\(759\) 3338.82 0.159673
\(760\) 0 0
\(761\) −34917.8 −1.66330 −0.831649 0.555302i \(-0.812603\pi\)
−0.831649 + 0.555302i \(0.812603\pi\)
\(762\) 0 0
\(763\) 18122.0i 0.859844i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11841.2i 0.557445i
\(768\) 0 0
\(769\) 20714.9 0.971388 0.485694 0.874129i \(-0.338567\pi\)
0.485694 + 0.874129i \(0.338567\pi\)
\(770\) 0 0
\(771\) 4719.66 0.220459
\(772\) 0 0
\(773\) − 12461.4i − 0.579826i −0.957053 0.289913i \(-0.906374\pi\)
0.957053 0.289913i \(-0.0936264\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 18168.4i − 0.838851i
\(778\) 0 0
\(779\) −9459.35 −0.435066
\(780\) 0 0
\(781\) 64093.8 2.93657
\(782\) 0 0
\(783\) 50302.7i 2.29588i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 32821.0i − 1.48658i −0.668967 0.743292i \(-0.733263\pi\)
0.668967 0.743292i \(-0.266737\pi\)
\(788\) 0 0
\(789\) −19882.4 −0.897126
\(790\) 0 0
\(791\) −23340.6 −1.04917
\(792\) 0 0
\(793\) − 35736.2i − 1.60029i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 7777.06i − 0.345643i −0.984953 0.172822i \(-0.944712\pi\)
0.984953 0.172822i \(-0.0552884\pi\)
\(798\) 0 0
\(799\) 23541.2 1.04234
\(800\) 0 0
\(801\) −8081.85 −0.356502
\(802\) 0 0
\(803\) 64947.2i 2.85422i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 45350.0i 1.97819i
\(808\) 0 0
\(809\) −20407.4 −0.886880 −0.443440 0.896304i \(-0.646242\pi\)
−0.443440 + 0.896304i \(0.646242\pi\)
\(810\) 0 0
\(811\) −25876.6 −1.12041 −0.560205 0.828354i \(-0.689277\pi\)
−0.560205 + 0.828354i \(0.689277\pi\)
\(812\) 0 0
\(813\) − 12043.8i − 0.519550i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 436.662i 0.0186987i
\(818\) 0 0
\(819\) −48340.2 −2.06245
\(820\) 0 0
\(821\) 1100.58 0.0467850 0.0233925 0.999726i \(-0.492553\pi\)
0.0233925 + 0.999726i \(0.492553\pi\)
\(822\) 0 0
\(823\) − 36245.7i − 1.53517i −0.640948 0.767585i \(-0.721458\pi\)
0.640948 0.767585i \(-0.278542\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1996.72i − 0.0839572i −0.999119 0.0419786i \(-0.986634\pi\)
0.999119 0.0419786i \(-0.0133661\pi\)
\(828\) 0 0
\(829\) −43393.8 −1.81801 −0.909004 0.416788i \(-0.863156\pi\)
−0.909004 + 0.416788i \(0.863156\pi\)
\(830\) 0 0
\(831\) 79019.1 3.29861
\(832\) 0 0
\(833\) − 8904.03i − 0.370356i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 13138.4i − 0.542569i
\(838\) 0 0
\(839\) 3424.53 0.140915 0.0704576 0.997515i \(-0.477554\pi\)
0.0704576 + 0.997515i \(0.477554\pi\)
\(840\) 0 0
\(841\) 25959.0 1.06437
\(842\) 0 0
\(843\) 10194.2i 0.416498i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 49970.8i − 2.02718i
\(848\) 0 0
\(849\) 3429.66 0.138640
\(850\) 0 0
\(851\) 608.778 0.0245225
\(852\) 0 0
\(853\) − 5173.63i − 0.207669i −0.994595 0.103835i \(-0.966889\pi\)
0.994595 0.103835i \(-0.0331113\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 28323.8i − 1.12896i −0.825445 0.564482i \(-0.809076\pi\)
0.825445 0.564482i \(-0.190924\pi\)
\(858\) 0 0
\(859\) 42970.7 1.70680 0.853401 0.521255i \(-0.174536\pi\)
0.853401 + 0.521255i \(0.174536\pi\)
\(860\) 0 0
\(861\) 26620.6 1.05369
\(862\) 0 0
\(863\) 12664.1i 0.499527i 0.968307 + 0.249764i \(0.0803529\pi\)
−0.968307 + 0.249764i \(0.919647\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 75149.2i 2.94371i
\(868\) 0 0
\(869\) 50749.7 1.98109
\(870\) 0 0
\(871\) 6405.85 0.249201
\(872\) 0 0
\(873\) 21609.6i 0.837769i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7403.14i 0.285047i 0.989791 + 0.142524i \(0.0455217\pi\)
−0.989791 + 0.142524i \(0.954478\pi\)
\(878\) 0 0
\(879\) 33925.6 1.30180
\(880\) 0 0
\(881\) −5691.74 −0.217661 −0.108831 0.994060i \(-0.534711\pi\)
−0.108831 + 0.994060i \(0.534711\pi\)
\(882\) 0 0
\(883\) − 13915.1i − 0.530329i −0.964203 0.265164i \(-0.914574\pi\)
0.964203 0.265164i \(-0.0854263\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35631.2i 1.34879i 0.738370 + 0.674396i \(0.235596\pi\)
−0.738370 + 0.674396i \(0.764404\pi\)
\(888\) 0 0
\(889\) 34874.2 1.31568
\(890\) 0 0
\(891\) −35965.0 −1.35227
\(892\) 0 0
\(893\) − 13200.6i − 0.494670i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2457.70i − 0.0914829i
\(898\) 0 0
\(899\) −13150.3 −0.487859
\(900\) 0 0
\(901\) −31849.8 −1.17766
\(902\) 0 0
\(903\) − 1228.86i − 0.0452866i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 22734.4i − 0.832285i −0.909299 0.416143i \(-0.863382\pi\)
0.909299 0.416143i \(-0.136618\pi\)
\(908\) 0 0
\(909\) −92744.3 −3.38408
\(910\) 0 0
\(911\) 32714.8 1.18978 0.594890 0.803807i \(-0.297195\pi\)
0.594890 + 0.803807i \(0.297195\pi\)
\(912\) 0 0
\(913\) − 63005.6i − 2.28388i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 31582.3i − 1.13734i
\(918\) 0 0
\(919\) −41077.0 −1.47443 −0.737217 0.675656i \(-0.763861\pi\)
−0.737217 + 0.675656i \(0.763861\pi\)
\(920\) 0 0
\(921\) −61380.3 −2.19604
\(922\) 0 0
\(923\) − 47179.3i − 1.68247i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 40082.9i − 1.42017i
\(928\) 0 0
\(929\) −27678.8 −0.977515 −0.488758 0.872420i \(-0.662550\pi\)
−0.488758 + 0.872420i \(0.662550\pi\)
\(930\) 0 0
\(931\) −4992.87 −0.175762
\(932\) 0 0
\(933\) 72307.6i 2.53724i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17120.7i 0.596915i 0.954423 + 0.298457i \(0.0964720\pi\)
−0.954423 + 0.298457i \(0.903528\pi\)
\(938\) 0 0
\(939\) 18266.6 0.634832
\(940\) 0 0
\(941\) −30687.2 −1.06310 −0.531549 0.847028i \(-0.678390\pi\)
−0.531549 + 0.847028i \(0.678390\pi\)
\(942\) 0 0
\(943\) 891.989i 0.0308029i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36649.9i 1.25761i 0.777561 + 0.628807i \(0.216456\pi\)
−0.777561 + 0.628807i \(0.783544\pi\)
\(948\) 0 0
\(949\) 47807.4 1.63529
\(950\) 0 0
\(951\) −37154.3 −1.26689
\(952\) 0 0
\(953\) 23768.8i 0.807919i 0.914777 + 0.403959i \(0.132366\pi\)
−0.914777 + 0.403959i \(0.867634\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 122590.i 4.14084i
\(958\) 0 0
\(959\) 28519.4 0.960311
\(960\) 0 0
\(961\) −26356.3 −0.884708
\(962\) 0 0
\(963\) − 86064.3i − 2.87994i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 36241.8i 1.20523i 0.798032 + 0.602615i \(0.205875\pi\)
−0.798032 + 0.602615i \(0.794125\pi\)
\(968\) 0 0
\(969\) 66655.5 2.20979
\(970\) 0 0
\(971\) 16837.4 0.556476 0.278238 0.960512i \(-0.410250\pi\)
0.278238 + 0.960512i \(0.410250\pi\)
\(972\) 0 0
\(973\) − 3210.51i − 0.105780i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26322.4i 0.861954i 0.902363 + 0.430977i \(0.141831\pi\)
−0.902363 + 0.430977i \(0.858169\pi\)
\(978\) 0 0
\(979\) −9506.78 −0.310355
\(980\) 0 0
\(981\) 46149.1 1.50196
\(982\) 0 0
\(983\) − 42438.7i − 1.37699i −0.725240 0.688497i \(-0.758271\pi\)
0.725240 0.688497i \(-0.241729\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 37149.1i 1.19804i
\(988\) 0 0
\(989\) 41.1760 0.00132388
\(990\) 0 0
\(991\) −17890.3 −0.573464 −0.286732 0.958011i \(-0.592569\pi\)
−0.286732 + 0.958011i \(0.592569\pi\)
\(992\) 0 0
\(993\) 104458.i 3.33824i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 16189.9i − 0.514283i −0.966374 0.257142i \(-0.917219\pi\)
0.966374 0.257142i \(-0.0827807\pi\)
\(998\) 0 0
\(999\) −22332.2 −0.707265
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.i.449.1 4
4.3 odd 2 800.4.c.k.449.4 4
5.2 odd 4 160.4.a.c.1.1 2
5.3 odd 4 800.4.a.s.1.2 2
5.4 even 2 inner 800.4.c.i.449.4 4
15.2 even 4 1440.4.a.t.1.2 2
20.3 even 4 800.4.a.m.1.1 2
20.7 even 4 160.4.a.g.1.2 yes 2
20.19 odd 2 800.4.c.k.449.1 4
40.3 even 4 1600.4.a.cn.1.2 2
40.13 odd 4 1600.4.a.cd.1.1 2
40.27 even 4 320.4.a.o.1.1 2
40.37 odd 4 320.4.a.s.1.2 2
60.47 odd 4 1440.4.a.x.1.1 2
80.27 even 4 1280.4.d.q.641.1 4
80.37 odd 4 1280.4.d.x.641.4 4
80.67 even 4 1280.4.d.q.641.4 4
80.77 odd 4 1280.4.d.x.641.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.c.1.1 2 5.2 odd 4
160.4.a.g.1.2 yes 2 20.7 even 4
320.4.a.o.1.1 2 40.27 even 4
320.4.a.s.1.2 2 40.37 odd 4
800.4.a.m.1.1 2 20.3 even 4
800.4.a.s.1.2 2 5.3 odd 4
800.4.c.i.449.1 4 1.1 even 1 trivial
800.4.c.i.449.4 4 5.4 even 2 inner
800.4.c.k.449.1 4 20.19 odd 2
800.4.c.k.449.4 4 4.3 odd 2
1280.4.d.q.641.1 4 80.27 even 4
1280.4.d.q.641.4 4 80.67 even 4
1280.4.d.x.641.1 4 80.77 odd 4
1280.4.d.x.641.4 4 80.37 odd 4
1440.4.a.t.1.2 2 15.2 even 4
1440.4.a.x.1.1 2 60.47 odd 4
1600.4.a.cd.1.1 2 40.13 odd 4
1600.4.a.cn.1.2 2 40.3 even 4