Properties

Label 1568.3.h.a.881.11
Level $1568$
Weight $3$
Character 1568.881
Analytic conductor $42.725$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,3,Mod(881,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.881");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.h (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: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.11
Character \(\chi\) \(=\) 1568.881
Dual form 1568.3.h.a.881.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.910863 q^{3} -6.34503 q^{5} -8.17033 q^{9} +O(q^{10})\) \(q-0.910863 q^{3} -6.34503 q^{5} -8.17033 q^{9} +13.2146i q^{11} -19.4243 q^{13} +5.77945 q^{15} -15.9268i q^{17} -16.4545 q^{19} -23.9215 q^{23} +15.2593 q^{25} +15.6398 q^{27} -16.6618i q^{29} -12.8588i q^{31} -12.0367i q^{33} +47.5556i q^{37} +17.6929 q^{39} +6.49499i q^{41} +33.2928i q^{43} +51.8409 q^{45} +21.9062i q^{47} +14.5071i q^{51} +37.1846i q^{53} -83.8473i q^{55} +14.9878 q^{57} -54.6855 q^{59} +10.2468 q^{61} +123.248 q^{65} -17.1341i q^{67} +21.7892 q^{69} -32.0568 q^{71} -107.166i q^{73} -13.8992 q^{75} +58.3083 q^{79} +59.2872 q^{81} +36.3441 q^{83} +101.056i q^{85} +15.1766i q^{87} +1.07373i q^{89} +11.7126i q^{93} +104.404 q^{95} -169.517i q^{97} -107.968i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −0.910863 −0.303621 −0.151810 0.988410i \(-0.548510\pi\)
−0.151810 + 0.988410i \(0.548510\pi\)
\(4\) 0 0
\(5\) −6.34503 −1.26901 −0.634503 0.772921i \(-0.718795\pi\)
−0.634503 + 0.772921i \(0.718795\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −8.17033 −0.907814
\(10\) 0 0
\(11\) 13.2146i 1.20133i 0.799500 + 0.600666i \(0.205098\pi\)
−0.799500 + 0.600666i \(0.794902\pi\)
\(12\) 0 0
\(13\) −19.4243 −1.49418 −0.747090 0.664723i \(-0.768550\pi\)
−0.747090 + 0.664723i \(0.768550\pi\)
\(14\) 0 0
\(15\) 5.77945 0.385296
\(16\) 0 0
\(17\) − 15.9268i − 0.936869i −0.883498 0.468434i \(-0.844818\pi\)
0.883498 0.468434i \(-0.155182\pi\)
\(18\) 0 0
\(19\) −16.4545 −0.866026 −0.433013 0.901388i \(-0.642550\pi\)
−0.433013 + 0.901388i \(0.642550\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −23.9215 −1.04006 −0.520032 0.854147i \(-0.674080\pi\)
−0.520032 + 0.854147i \(0.674080\pi\)
\(24\) 0 0
\(25\) 15.2593 0.610374
\(26\) 0 0
\(27\) 15.6398 0.579252
\(28\) 0 0
\(29\) − 16.6618i − 0.574544i −0.957849 0.287272i \(-0.907252\pi\)
0.957849 0.287272i \(-0.0927483\pi\)
\(30\) 0 0
\(31\) − 12.8588i − 0.414799i −0.978256 0.207400i \(-0.933500\pi\)
0.978256 0.207400i \(-0.0665000\pi\)
\(32\) 0 0
\(33\) − 12.0367i − 0.364749i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 47.5556i 1.28529i 0.766165 + 0.642644i \(0.222162\pi\)
−0.766165 + 0.642644i \(0.777838\pi\)
\(38\) 0 0
\(39\) 17.6929 0.453664
\(40\) 0 0
\(41\) 6.49499i 0.158415i 0.996858 + 0.0792073i \(0.0252389\pi\)
−0.996858 + 0.0792073i \(0.974761\pi\)
\(42\) 0 0
\(43\) 33.2928i 0.774252i 0.922027 + 0.387126i \(0.126532\pi\)
−0.922027 + 0.387126i \(0.873468\pi\)
\(44\) 0 0
\(45\) 51.8409 1.15202
\(46\) 0 0
\(47\) 21.9062i 0.466089i 0.972466 + 0.233045i \(0.0748688\pi\)
−0.972466 + 0.233045i \(0.925131\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 14.5071i 0.284453i
\(52\) 0 0
\(53\) 37.1846i 0.701597i 0.936451 + 0.350798i \(0.114090\pi\)
−0.936451 + 0.350798i \(0.885910\pi\)
\(54\) 0 0
\(55\) − 83.8473i − 1.52450i
\(56\) 0 0
\(57\) 14.9878 0.262944
\(58\) 0 0
\(59\) −54.6855 −0.926874 −0.463437 0.886130i \(-0.653384\pi\)
−0.463437 + 0.886130i \(0.653384\pi\)
\(60\) 0 0
\(61\) 10.2468 0.167980 0.0839902 0.996467i \(-0.473234\pi\)
0.0839902 + 0.996467i \(0.473234\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 123.248 1.89612
\(66\) 0 0
\(67\) − 17.1341i − 0.255733i −0.991791 0.127867i \(-0.959187\pi\)
0.991791 0.127867i \(-0.0408129\pi\)
\(68\) 0 0
\(69\) 21.7892 0.315785
\(70\) 0 0
\(71\) −32.0568 −0.451505 −0.225752 0.974185i \(-0.572484\pi\)
−0.225752 + 0.974185i \(0.572484\pi\)
\(72\) 0 0
\(73\) − 107.166i − 1.46802i −0.679137 0.734011i \(-0.737646\pi\)
0.679137 0.734011i \(-0.262354\pi\)
\(74\) 0 0
\(75\) −13.8992 −0.185322
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 58.3083 0.738080 0.369040 0.929414i \(-0.379687\pi\)
0.369040 + 0.929414i \(0.379687\pi\)
\(80\) 0 0
\(81\) 59.2872 0.731941
\(82\) 0 0
\(83\) 36.3441 0.437880 0.218940 0.975738i \(-0.429740\pi\)
0.218940 + 0.975738i \(0.429740\pi\)
\(84\) 0 0
\(85\) 101.056i 1.18889i
\(86\) 0 0
\(87\) 15.1766i 0.174443i
\(88\) 0 0
\(89\) 1.07373i 0.0120644i 0.999982 + 0.00603222i \(0.00192013\pi\)
−0.999982 + 0.00603222i \(0.998080\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 11.7126i 0.125942i
\(94\) 0 0
\(95\) 104.404 1.09899
\(96\) 0 0
\(97\) − 169.517i − 1.74760i −0.486286 0.873799i \(-0.661649\pi\)
0.486286 0.873799i \(-0.338351\pi\)
\(98\) 0 0
\(99\) − 107.968i − 1.09059i
\(100\) 0 0
\(101\) 28.1260 0.278476 0.139238 0.990259i \(-0.455535\pi\)
0.139238 + 0.990259i \(0.455535\pi\)
\(102\) 0 0
\(103\) 166.310i 1.61466i 0.590098 + 0.807331i \(0.299089\pi\)
−0.590098 + 0.807331i \(0.700911\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 197.584i − 1.84658i −0.384109 0.923288i \(-0.625491\pi\)
0.384109 0.923288i \(-0.374509\pi\)
\(108\) 0 0
\(109\) − 11.5198i − 0.105686i −0.998603 0.0528431i \(-0.983172\pi\)
0.998603 0.0528431i \(-0.0168283\pi\)
\(110\) 0 0
\(111\) − 43.3167i − 0.390240i
\(112\) 0 0
\(113\) −14.7908 −0.130892 −0.0654460 0.997856i \(-0.520847\pi\)
−0.0654460 + 0.997856i \(0.520847\pi\)
\(114\) 0 0
\(115\) 151.782 1.31985
\(116\) 0 0
\(117\) 158.703 1.35644
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −53.6270 −0.443198
\(122\) 0 0
\(123\) − 5.91605i − 0.0480980i
\(124\) 0 0
\(125\) 61.8047 0.494438
\(126\) 0 0
\(127\) 70.2656 0.553272 0.276636 0.960975i \(-0.410780\pi\)
0.276636 + 0.960975i \(0.410780\pi\)
\(128\) 0 0
\(129\) − 30.3252i − 0.235079i
\(130\) 0 0
\(131\) −142.129 −1.08496 −0.542478 0.840070i \(-0.682514\pi\)
−0.542478 + 0.840070i \(0.682514\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −99.2350 −0.735074
\(136\) 0 0
\(137\) 253.074 1.84725 0.923626 0.383295i \(-0.125211\pi\)
0.923626 + 0.383295i \(0.125211\pi\)
\(138\) 0 0
\(139\) 49.1909 0.353892 0.176946 0.984221i \(-0.443378\pi\)
0.176946 + 0.984221i \(0.443378\pi\)
\(140\) 0 0
\(141\) − 19.9535i − 0.141514i
\(142\) 0 0
\(143\) − 256.686i − 1.79501i
\(144\) 0 0
\(145\) 105.719i 0.729099i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 41.6936i − 0.279823i −0.990164 0.139911i \(-0.955318\pi\)
0.990164 0.139911i \(-0.0446818\pi\)
\(150\) 0 0
\(151\) 97.6290 0.646549 0.323275 0.946305i \(-0.395216\pi\)
0.323275 + 0.946305i \(0.395216\pi\)
\(152\) 0 0
\(153\) 130.127i 0.850503i
\(154\) 0 0
\(155\) 81.5892i 0.526382i
\(156\) 0 0
\(157\) 28.1654 0.179397 0.0896986 0.995969i \(-0.471410\pi\)
0.0896986 + 0.995969i \(0.471410\pi\)
\(158\) 0 0
\(159\) − 33.8701i − 0.213019i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 242.432i 1.48731i 0.668563 + 0.743655i \(0.266910\pi\)
−0.668563 + 0.743655i \(0.733090\pi\)
\(164\) 0 0
\(165\) 76.3734i 0.462869i
\(166\) 0 0
\(167\) 60.1108i 0.359945i 0.983672 + 0.179972i \(0.0576008\pi\)
−0.983672 + 0.179972i \(0.942399\pi\)
\(168\) 0 0
\(169\) 208.305 1.23257
\(170\) 0 0
\(171\) 134.439 0.786191
\(172\) 0 0
\(173\) −139.364 −0.805573 −0.402786 0.915294i \(-0.631958\pi\)
−0.402786 + 0.915294i \(0.631958\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 49.8110 0.281418
\(178\) 0 0
\(179\) 291.727i 1.62976i 0.579631 + 0.814879i \(0.303197\pi\)
−0.579631 + 0.814879i \(0.696803\pi\)
\(180\) 0 0
\(181\) 166.844 0.921791 0.460895 0.887455i \(-0.347528\pi\)
0.460895 + 0.887455i \(0.347528\pi\)
\(182\) 0 0
\(183\) −9.33343 −0.0510024
\(184\) 0 0
\(185\) − 301.742i − 1.63104i
\(186\) 0 0
\(187\) 210.467 1.12549
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 131.356 0.687728 0.343864 0.939019i \(-0.388264\pi\)
0.343864 + 0.939019i \(0.388264\pi\)
\(192\) 0 0
\(193\) −81.4392 −0.421965 −0.210982 0.977490i \(-0.567666\pi\)
−0.210982 + 0.977490i \(0.567666\pi\)
\(194\) 0 0
\(195\) −112.262 −0.575702
\(196\) 0 0
\(197\) − 2.09549i − 0.0106370i −0.999986 0.00531851i \(-0.998307\pi\)
0.999986 0.00531851i \(-0.00169294\pi\)
\(198\) 0 0
\(199\) − 126.944i − 0.637911i −0.947770 0.318955i \(-0.896668\pi\)
0.947770 0.318955i \(-0.103332\pi\)
\(200\) 0 0
\(201\) 15.6068i 0.0776459i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) − 41.2109i − 0.201029i
\(206\) 0 0
\(207\) 195.446 0.944184
\(208\) 0 0
\(209\) − 217.440i − 1.04038i
\(210\) 0 0
\(211\) − 7.16822i − 0.0339726i −0.999856 0.0169863i \(-0.994593\pi\)
0.999856 0.0169863i \(-0.00540717\pi\)
\(212\) 0 0
\(213\) 29.1994 0.137086
\(214\) 0 0
\(215\) − 211.244i − 0.982530i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 97.6132i 0.445722i
\(220\) 0 0
\(221\) 309.367i 1.39985i
\(222\) 0 0
\(223\) − 279.720i − 1.25435i −0.778878 0.627175i \(-0.784211\pi\)
0.778878 0.627175i \(-0.215789\pi\)
\(224\) 0 0
\(225\) −124.674 −0.554106
\(226\) 0 0
\(227\) −304.784 −1.34266 −0.671330 0.741159i \(-0.734276\pi\)
−0.671330 + 0.741159i \(0.734276\pi\)
\(228\) 0 0
\(229\) −414.688 −1.81087 −0.905433 0.424490i \(-0.860453\pi\)
−0.905433 + 0.424490i \(0.860453\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −164.926 −0.707836 −0.353918 0.935277i \(-0.615151\pi\)
−0.353918 + 0.935277i \(0.615151\pi\)
\(234\) 0 0
\(235\) − 138.995i − 0.591469i
\(236\) 0 0
\(237\) −53.1109 −0.224097
\(238\) 0 0
\(239\) 19.1182 0.0799926 0.0399963 0.999200i \(-0.487265\pi\)
0.0399963 + 0.999200i \(0.487265\pi\)
\(240\) 0 0
\(241\) 350.308i 1.45356i 0.686870 + 0.726780i \(0.258984\pi\)
−0.686870 + 0.726780i \(0.741016\pi\)
\(242\) 0 0
\(243\) −194.761 −0.801485
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 319.618 1.29400
\(248\) 0 0
\(249\) −33.1045 −0.132950
\(250\) 0 0
\(251\) 88.3204 0.351874 0.175937 0.984401i \(-0.443704\pi\)
0.175937 + 0.984401i \(0.443704\pi\)
\(252\) 0 0
\(253\) − 316.114i − 1.24946i
\(254\) 0 0
\(255\) − 92.0479i − 0.360972i
\(256\) 0 0
\(257\) 86.0828i 0.334952i 0.985876 + 0.167476i \(0.0535617\pi\)
−0.985876 + 0.167476i \(0.946438\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 136.132i 0.521579i
\(262\) 0 0
\(263\) −319.210 −1.21373 −0.606863 0.794806i \(-0.707572\pi\)
−0.606863 + 0.794806i \(0.707572\pi\)
\(264\) 0 0
\(265\) − 235.937i − 0.890330i
\(266\) 0 0
\(267\) − 0.978025i − 0.00366302i
\(268\) 0 0
\(269\) −57.4680 −0.213636 −0.106818 0.994279i \(-0.534066\pi\)
−0.106818 + 0.994279i \(0.534066\pi\)
\(270\) 0 0
\(271\) − 30.8765i − 0.113935i −0.998376 0.0569677i \(-0.981857\pi\)
0.998376 0.0569677i \(-0.0181432\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 201.647i 0.733261i
\(276\) 0 0
\(277\) − 356.185i − 1.28587i −0.765923 0.642933i \(-0.777718\pi\)
0.765923 0.642933i \(-0.222282\pi\)
\(278\) 0 0
\(279\) 105.060i 0.376561i
\(280\) 0 0
\(281\) −294.160 −1.04683 −0.523416 0.852077i \(-0.675343\pi\)
−0.523416 + 0.852077i \(0.675343\pi\)
\(282\) 0 0
\(283\) 415.002 1.46644 0.733219 0.679993i \(-0.238017\pi\)
0.733219 + 0.679993i \(0.238017\pi\)
\(284\) 0 0
\(285\) −95.0979 −0.333677
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 35.3381 0.122277
\(290\) 0 0
\(291\) 154.407i 0.530608i
\(292\) 0 0
\(293\) −370.564 −1.26472 −0.632362 0.774673i \(-0.717915\pi\)
−0.632362 + 0.774673i \(0.717915\pi\)
\(294\) 0 0
\(295\) 346.981 1.17621
\(296\) 0 0
\(297\) 206.675i 0.695874i
\(298\) 0 0
\(299\) 464.658 1.55404
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −25.6190 −0.0845510
\(304\) 0 0
\(305\) −65.0162 −0.213168
\(306\) 0 0
\(307\) 160.327 0.522239 0.261120 0.965306i \(-0.415908\pi\)
0.261120 + 0.965306i \(0.415908\pi\)
\(308\) 0 0
\(309\) − 151.486i − 0.490245i
\(310\) 0 0
\(311\) 472.839i 1.52038i 0.649700 + 0.760191i \(0.274895\pi\)
−0.649700 + 0.760191i \(0.725105\pi\)
\(312\) 0 0
\(313\) 231.013i 0.738060i 0.929417 + 0.369030i \(0.120310\pi\)
−0.929417 + 0.369030i \(0.879690\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 225.319i − 0.710784i −0.934717 0.355392i \(-0.884347\pi\)
0.934717 0.355392i \(-0.115653\pi\)
\(318\) 0 0
\(319\) 220.179 0.690218
\(320\) 0 0
\(321\) 179.972i 0.560659i
\(322\) 0 0
\(323\) 262.067i 0.811353i
\(324\) 0 0
\(325\) −296.403 −0.912008
\(326\) 0 0
\(327\) 10.4930i 0.0320885i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 20.6988i − 0.0625341i −0.999511 0.0312671i \(-0.990046\pi\)
0.999511 0.0312671i \(-0.00995424\pi\)
\(332\) 0 0
\(333\) − 388.545i − 1.16680i
\(334\) 0 0
\(335\) 108.716i 0.324527i
\(336\) 0 0
\(337\) 34.9645 0.103752 0.0518762 0.998654i \(-0.483480\pi\)
0.0518762 + 0.998654i \(0.483480\pi\)
\(338\) 0 0
\(339\) 13.4724 0.0397415
\(340\) 0 0
\(341\) 169.924 0.498311
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −138.253 −0.400733
\(346\) 0 0
\(347\) − 438.738i − 1.26438i −0.774815 0.632188i \(-0.782157\pi\)
0.774815 0.632188i \(-0.217843\pi\)
\(348\) 0 0
\(349\) 435.121 1.24677 0.623383 0.781917i \(-0.285758\pi\)
0.623383 + 0.781917i \(0.285758\pi\)
\(350\) 0 0
\(351\) −303.793 −0.865507
\(352\) 0 0
\(353\) − 281.109i − 0.796343i −0.917311 0.398171i \(-0.869645\pi\)
0.917311 0.398171i \(-0.130355\pi\)
\(354\) 0 0
\(355\) 203.401 0.572962
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −263.930 −0.735180 −0.367590 0.929988i \(-0.619817\pi\)
−0.367590 + 0.929988i \(0.619817\pi\)
\(360\) 0 0
\(361\) −90.2495 −0.249999
\(362\) 0 0
\(363\) 48.8468 0.134564
\(364\) 0 0
\(365\) 679.969i 1.86293i
\(366\) 0 0
\(367\) − 154.939i − 0.422176i −0.977467 0.211088i \(-0.932299\pi\)
0.977467 0.211088i \(-0.0677007\pi\)
\(368\) 0 0
\(369\) − 53.0662i − 0.143811i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 584.862i − 1.56799i −0.620765 0.783997i \(-0.713178\pi\)
0.620765 0.783997i \(-0.286822\pi\)
\(374\) 0 0
\(375\) −56.2956 −0.150122
\(376\) 0 0
\(377\) 323.644i 0.858472i
\(378\) 0 0
\(379\) 128.176i 0.338195i 0.985599 + 0.169098i \(0.0540853\pi\)
−0.985599 + 0.169098i \(0.945915\pi\)
\(380\) 0 0
\(381\) −64.0023 −0.167985
\(382\) 0 0
\(383\) 249.920i 0.652532i 0.945278 + 0.326266i \(0.105791\pi\)
−0.945278 + 0.326266i \(0.894209\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 272.013i − 0.702877i
\(388\) 0 0
\(389\) 216.423i 0.556358i 0.960529 + 0.278179i \(0.0897309\pi\)
−0.960529 + 0.278179i \(0.910269\pi\)
\(390\) 0 0
\(391\) 380.991i 0.974402i
\(392\) 0 0
\(393\) 129.460 0.329415
\(394\) 0 0
\(395\) −369.968 −0.936627
\(396\) 0 0
\(397\) 699.882 1.76293 0.881463 0.472252i \(-0.156559\pi\)
0.881463 + 0.472252i \(0.156559\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −180.981 −0.451323 −0.225662 0.974206i \(-0.572454\pi\)
−0.225662 + 0.974206i \(0.572454\pi\)
\(402\) 0 0
\(403\) 249.773i 0.619785i
\(404\) 0 0
\(405\) −376.179 −0.928837
\(406\) 0 0
\(407\) −628.431 −1.54406
\(408\) 0 0
\(409\) 358.843i 0.877366i 0.898642 + 0.438683i \(0.144555\pi\)
−0.898642 + 0.438683i \(0.855445\pi\)
\(410\) 0 0
\(411\) −230.515 −0.560864
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −230.604 −0.555672
\(416\) 0 0
\(417\) −44.8062 −0.107449
\(418\) 0 0
\(419\) −780.890 −1.86370 −0.931849 0.362846i \(-0.881805\pi\)
−0.931849 + 0.362846i \(0.881805\pi\)
\(420\) 0 0
\(421\) 114.961i 0.273068i 0.990635 + 0.136534i \(0.0435962\pi\)
−0.990635 + 0.136534i \(0.956404\pi\)
\(422\) 0 0
\(423\) − 178.981i − 0.423122i
\(424\) 0 0
\(425\) − 243.032i − 0.571840i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 233.806i 0.545001i
\(430\) 0 0
\(431\) −309.712 −0.718590 −0.359295 0.933224i \(-0.616983\pi\)
−0.359295 + 0.933224i \(0.616983\pi\)
\(432\) 0 0
\(433\) 595.775i 1.37592i 0.725747 + 0.687962i \(0.241494\pi\)
−0.725747 + 0.687962i \(0.758506\pi\)
\(434\) 0 0
\(435\) − 96.2958i − 0.221370i
\(436\) 0 0
\(437\) 393.615 0.900722
\(438\) 0 0
\(439\) − 806.900i − 1.83804i −0.394211 0.919020i \(-0.628982\pi\)
0.394211 0.919020i \(-0.371018\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 444.807i − 1.00408i −0.864845 0.502039i \(-0.832583\pi\)
0.864845 0.502039i \(-0.167417\pi\)
\(444\) 0 0
\(445\) − 6.81287i − 0.0153098i
\(446\) 0 0
\(447\) 37.9772i 0.0849601i
\(448\) 0 0
\(449\) 262.420 0.584455 0.292228 0.956349i \(-0.405604\pi\)
0.292228 + 0.956349i \(0.405604\pi\)
\(450\) 0 0
\(451\) −85.8291 −0.190308
\(452\) 0 0
\(453\) −88.9266 −0.196306
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 389.475 0.852244 0.426122 0.904666i \(-0.359879\pi\)
0.426122 + 0.904666i \(0.359879\pi\)
\(458\) 0 0
\(459\) − 249.092i − 0.542683i
\(460\) 0 0
\(461\) 158.714 0.344283 0.172141 0.985072i \(-0.444931\pi\)
0.172141 + 0.985072i \(0.444931\pi\)
\(462\) 0 0
\(463\) 528.844 1.14221 0.571106 0.820877i \(-0.306515\pi\)
0.571106 + 0.820877i \(0.306515\pi\)
\(464\) 0 0
\(465\) − 74.3166i − 0.159821i
\(466\) 0 0
\(467\) −436.898 −0.935542 −0.467771 0.883850i \(-0.654943\pi\)
−0.467771 + 0.883850i \(0.654943\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −25.6548 −0.0544688
\(472\) 0 0
\(473\) −439.953 −0.930134
\(474\) 0 0
\(475\) −251.085 −0.528600
\(476\) 0 0
\(477\) − 303.811i − 0.636920i
\(478\) 0 0
\(479\) 545.870i 1.13960i 0.821782 + 0.569802i \(0.192980\pi\)
−0.821782 + 0.569802i \(0.807020\pi\)
\(480\) 0 0
\(481\) − 923.737i − 1.92045i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1075.59i 2.21771i
\(486\) 0 0
\(487\) −648.230 −1.33107 −0.665534 0.746368i \(-0.731796\pi\)
−0.665534 + 0.746368i \(0.731796\pi\)
\(488\) 0 0
\(489\) − 220.822i − 0.451579i
\(490\) 0 0
\(491\) − 732.074i − 1.49098i −0.666514 0.745492i \(-0.732214\pi\)
0.666514 0.745492i \(-0.267786\pi\)
\(492\) 0 0
\(493\) −265.368 −0.538272
\(494\) 0 0
\(495\) 685.060i 1.38396i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 26.7040i − 0.0535151i −0.999642 0.0267575i \(-0.991482\pi\)
0.999642 0.0267575i \(-0.00851820\pi\)
\(500\) 0 0
\(501\) − 54.7527i − 0.109287i
\(502\) 0 0
\(503\) 616.414i 1.22548i 0.790286 + 0.612738i \(0.209932\pi\)
−0.790286 + 0.612738i \(0.790068\pi\)
\(504\) 0 0
\(505\) −178.460 −0.353387
\(506\) 0 0
\(507\) −189.737 −0.374235
\(508\) 0 0
\(509\) 132.753 0.260811 0.130405 0.991461i \(-0.458372\pi\)
0.130405 + 0.991461i \(0.458372\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −257.345 −0.501648
\(514\) 0 0
\(515\) − 1055.24i − 2.04902i
\(516\) 0 0
\(517\) −289.483 −0.559928
\(518\) 0 0
\(519\) 126.942 0.244589
\(520\) 0 0
\(521\) 676.054i 1.29761i 0.760955 + 0.648804i \(0.224731\pi\)
−0.760955 + 0.648804i \(0.775269\pi\)
\(522\) 0 0
\(523\) −372.449 −0.712139 −0.356069 0.934460i \(-0.615883\pi\)
−0.356069 + 0.934460i \(0.615883\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −204.799 −0.388612
\(528\) 0 0
\(529\) 43.2358 0.0817313
\(530\) 0 0
\(531\) 446.799 0.841429
\(532\) 0 0
\(533\) − 126.161i − 0.236700i
\(534\) 0 0
\(535\) 1253.67i 2.34331i
\(536\) 0 0
\(537\) − 265.723i − 0.494829i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 69.6715i − 0.128783i −0.997925 0.0643914i \(-0.979489\pi\)
0.997925 0.0643914i \(-0.0205106\pi\)
\(542\) 0 0
\(543\) −151.972 −0.279875
\(544\) 0 0
\(545\) 73.0934i 0.134116i
\(546\) 0 0
\(547\) − 466.463i − 0.852765i −0.904543 0.426383i \(-0.859788\pi\)
0.904543 0.426383i \(-0.140212\pi\)
\(548\) 0 0
\(549\) −83.7197 −0.152495
\(550\) 0 0
\(551\) 274.161i 0.497570i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 274.845i 0.495217i
\(556\) 0 0
\(557\) 137.219i 0.246353i 0.992385 + 0.123177i \(0.0393082\pi\)
−0.992385 + 0.123177i \(0.960692\pi\)
\(558\) 0 0
\(559\) − 646.692i − 1.15687i
\(560\) 0 0
\(561\) −191.706 −0.341722
\(562\) 0 0
\(563\) −169.126 −0.300402 −0.150201 0.988655i \(-0.547992\pi\)
−0.150201 + 0.988655i \(0.547992\pi\)
\(564\) 0 0
\(565\) 93.8479 0.166103
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 744.931 1.30919 0.654597 0.755978i \(-0.272838\pi\)
0.654597 + 0.755978i \(0.272838\pi\)
\(570\) 0 0
\(571\) 886.611i 1.55273i 0.630281 + 0.776367i \(0.282940\pi\)
−0.630281 + 0.776367i \(0.717060\pi\)
\(572\) 0 0
\(573\) −119.647 −0.208809
\(574\) 0 0
\(575\) −365.026 −0.634827
\(576\) 0 0
\(577\) 240.062i 0.416052i 0.978123 + 0.208026i \(0.0667038\pi\)
−0.978123 + 0.208026i \(0.933296\pi\)
\(578\) 0 0
\(579\) 74.1799 0.128117
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −491.382 −0.842850
\(584\) 0 0
\(585\) −1006.98 −1.72133
\(586\) 0 0
\(587\) −190.873 −0.325168 −0.162584 0.986695i \(-0.551983\pi\)
−0.162584 + 0.986695i \(0.551983\pi\)
\(588\) 0 0
\(589\) 211.585i 0.359227i
\(590\) 0 0
\(591\) 1.90871i 0.00322962i
\(592\) 0 0
\(593\) 736.177i 1.24145i 0.784030 + 0.620723i \(0.213161\pi\)
−0.784030 + 0.620723i \(0.786839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 115.629i 0.193683i
\(598\) 0 0
\(599\) −1116.66 −1.86421 −0.932104 0.362190i \(-0.882029\pi\)
−0.932104 + 0.362190i \(0.882029\pi\)
\(600\) 0 0
\(601\) − 183.100i − 0.304659i −0.988330 0.152329i \(-0.951323\pi\)
0.988330 0.152329i \(-0.0486774\pi\)
\(602\) 0 0
\(603\) 139.991i 0.232158i
\(604\) 0 0
\(605\) 340.264 0.562421
\(606\) 0 0
\(607\) − 454.982i − 0.749558i −0.927114 0.374779i \(-0.877719\pi\)
0.927114 0.374779i \(-0.122281\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 425.513i − 0.696421i
\(612\) 0 0
\(613\) − 268.876i − 0.438623i −0.975655 0.219312i \(-0.929619\pi\)
0.975655 0.219312i \(-0.0703811\pi\)
\(614\) 0 0
\(615\) 37.5375i 0.0610366i
\(616\) 0 0
\(617\) −184.934 −0.299731 −0.149866 0.988706i \(-0.547884\pi\)
−0.149866 + 0.988706i \(0.547884\pi\)
\(618\) 0 0
\(619\) −993.618 −1.60520 −0.802599 0.596519i \(-0.796550\pi\)
−0.802599 + 0.596519i \(0.796550\pi\)
\(620\) 0 0
\(621\) −374.127 −0.602459
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −773.636 −1.23782
\(626\) 0 0
\(627\) 198.058i 0.315883i
\(628\) 0 0
\(629\) 757.408 1.20415
\(630\) 0 0
\(631\) −805.857 −1.27711 −0.638555 0.769576i \(-0.720468\pi\)
−0.638555 + 0.769576i \(0.720468\pi\)
\(632\) 0 0
\(633\) 6.52926i 0.0103148i
\(634\) 0 0
\(635\) −445.837 −0.702105
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 261.915 0.409883
\(640\) 0 0
\(641\) 5.50441 0.00858723 0.00429361 0.999991i \(-0.498633\pi\)
0.00429361 + 0.999991i \(0.498633\pi\)
\(642\) 0 0
\(643\) 1024.08 1.59266 0.796331 0.604861i \(-0.206771\pi\)
0.796331 + 0.604861i \(0.206771\pi\)
\(644\) 0 0
\(645\) 192.414i 0.298317i
\(646\) 0 0
\(647\) 456.573i 0.705677i 0.935684 + 0.352839i \(0.114784\pi\)
−0.935684 + 0.352839i \(0.885216\pi\)
\(648\) 0 0
\(649\) − 722.650i − 1.11348i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 28.2443i − 0.0432532i −0.999766 0.0216266i \(-0.993116\pi\)
0.999766 0.0216266i \(-0.00688450\pi\)
\(654\) 0 0
\(655\) 901.814 1.37681
\(656\) 0 0
\(657\) 875.579i 1.33269i
\(658\) 0 0
\(659\) 132.188i 0.200589i 0.994958 + 0.100295i \(0.0319785\pi\)
−0.994958 + 0.100295i \(0.968021\pi\)
\(660\) 0 0
\(661\) −693.847 −1.04969 −0.524847 0.851197i \(-0.675877\pi\)
−0.524847 + 0.851197i \(0.675877\pi\)
\(662\) 0 0
\(663\) − 281.791i − 0.425024i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 398.574i 0.597562i
\(668\) 0 0
\(669\) 254.787i 0.380847i
\(670\) 0 0
\(671\) 135.408i 0.201800i
\(672\) 0 0
\(673\) 532.137 0.790694 0.395347 0.918532i \(-0.370624\pi\)
0.395347 + 0.918532i \(0.370624\pi\)
\(674\) 0 0
\(675\) 238.653 0.353560
\(676\) 0 0
\(677\) 286.230 0.422792 0.211396 0.977401i \(-0.432199\pi\)
0.211396 + 0.977401i \(0.432199\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 277.616 0.407659
\(682\) 0 0
\(683\) 447.836i 0.655690i 0.944731 + 0.327845i \(0.106322\pi\)
−0.944731 + 0.327845i \(0.893678\pi\)
\(684\) 0 0
\(685\) −1605.76 −2.34417
\(686\) 0 0
\(687\) 377.724 0.549817
\(688\) 0 0
\(689\) − 722.287i − 1.04831i
\(690\) 0 0
\(691\) 1020.73 1.47718 0.738591 0.674154i \(-0.235492\pi\)
0.738591 + 0.674154i \(0.235492\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −312.118 −0.449090
\(696\) 0 0
\(697\) 103.444 0.148414
\(698\) 0 0
\(699\) 150.225 0.214914
\(700\) 0 0
\(701\) − 1311.02i − 1.87021i −0.354369 0.935106i \(-0.615304\pi\)
0.354369 0.935106i \(-0.384696\pi\)
\(702\) 0 0
\(703\) − 782.504i − 1.11309i
\(704\) 0 0
\(705\) 126.606i 0.179582i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 537.508i − 0.758121i −0.925372 0.379061i \(-0.876247\pi\)
0.925372 0.379061i \(-0.123753\pi\)
\(710\) 0 0
\(711\) −476.398 −0.670040
\(712\) 0 0
\(713\) 307.601i 0.431417i
\(714\) 0 0
\(715\) 1628.68i 2.27787i
\(716\) 0 0
\(717\) −17.4141 −0.0242874
\(718\) 0 0
\(719\) 269.363i 0.374635i 0.982299 + 0.187318i \(0.0599793\pi\)
−0.982299 + 0.187318i \(0.940021\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 319.083i − 0.441331i
\(724\) 0 0
\(725\) − 254.248i − 0.350686i
\(726\) 0 0
\(727\) 460.316i 0.633172i 0.948564 + 0.316586i \(0.102537\pi\)
−0.948564 + 0.316586i \(0.897463\pi\)
\(728\) 0 0
\(729\) −356.185 −0.488594
\(730\) 0 0
\(731\) 530.247 0.725372
\(732\) 0 0
\(733\) 66.6821 0.0909715 0.0454857 0.998965i \(-0.485516\pi\)
0.0454857 + 0.998965i \(0.485516\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 226.421 0.307220
\(738\) 0 0
\(739\) − 933.889i − 1.26372i −0.775083 0.631860i \(-0.782292\pi\)
0.775083 0.631860i \(-0.217708\pi\)
\(740\) 0 0
\(741\) −291.128 −0.392885
\(742\) 0 0
\(743\) 1198.23 1.61269 0.806345 0.591446i \(-0.201443\pi\)
0.806345 + 0.591446i \(0.201443\pi\)
\(744\) 0 0
\(745\) 264.547i 0.355097i
\(746\) 0 0
\(747\) −296.943 −0.397514
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 168.599 0.224499 0.112249 0.993680i \(-0.464194\pi\)
0.112249 + 0.993680i \(0.464194\pi\)
\(752\) 0 0
\(753\) −80.4478 −0.106836
\(754\) 0 0
\(755\) −619.458 −0.820474
\(756\) 0 0
\(757\) 209.207i 0.276364i 0.990407 + 0.138182i \(0.0441259\pi\)
−0.990407 + 0.138182i \(0.955874\pi\)
\(758\) 0 0
\(759\) 287.936i 0.379362i
\(760\) 0 0
\(761\) 553.248i 0.727002i 0.931594 + 0.363501i \(0.118419\pi\)
−0.931594 + 0.363501i \(0.881581\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) − 825.659i − 1.07929i
\(766\) 0 0
\(767\) 1062.23 1.38492
\(768\) 0 0
\(769\) 219.524i 0.285467i 0.989761 + 0.142734i \(0.0455892\pi\)
−0.989761 + 0.142734i \(0.954411\pi\)
\(770\) 0 0
\(771\) − 78.4096i − 0.101699i
\(772\) 0 0
\(773\) 666.674 0.862451 0.431225 0.902244i \(-0.358081\pi\)
0.431225 + 0.902244i \(0.358081\pi\)
\(774\) 0 0
\(775\) − 196.216i − 0.253182i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 106.872i − 0.137191i
\(780\) 0 0
\(781\) − 423.620i − 0.542407i
\(782\) 0 0
\(783\) − 260.587i − 0.332806i
\(784\) 0 0
\(785\) −178.710 −0.227656
\(786\) 0 0
\(787\) 919.865 1.16882 0.584412 0.811457i \(-0.301325\pi\)
0.584412 + 0.811457i \(0.301325\pi\)
\(788\) 0 0
\(789\) 290.757 0.368513
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −199.037 −0.250993
\(794\) 0 0
\(795\) 214.907i 0.270323i
\(796\) 0 0
\(797\) 1016.13 1.27494 0.637470 0.770476i \(-0.279981\pi\)
0.637470 + 0.770476i \(0.279981\pi\)
\(798\) 0 0
\(799\) 348.895 0.436664
\(800\) 0 0
\(801\) − 8.77277i − 0.0109523i
\(802\) 0 0
\(803\) 1416.16 1.76358
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 52.3454 0.0648642
\(808\) 0 0
\(809\) 1131.90 1.39913 0.699567 0.714567i \(-0.253376\pi\)
0.699567 + 0.714567i \(0.253376\pi\)
\(810\) 0 0
\(811\) −481.066 −0.593176 −0.296588 0.955006i \(-0.595849\pi\)
−0.296588 + 0.955006i \(0.595849\pi\)
\(812\) 0 0
\(813\) 28.1242i 0.0345931i
\(814\) 0 0
\(815\) − 1538.23i − 1.88740i
\(816\) 0 0
\(817\) − 547.817i − 0.670523i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 727.675i − 0.886328i −0.896441 0.443164i \(-0.853856\pi\)
0.896441 0.443164i \(-0.146144\pi\)
\(822\) 0 0
\(823\) 626.646 0.761417 0.380709 0.924695i \(-0.375680\pi\)
0.380709 + 0.924695i \(0.375680\pi\)
\(824\) 0 0
\(825\) − 183.673i − 0.222633i
\(826\) 0 0
\(827\) − 1468.52i − 1.77572i −0.460116 0.887859i \(-0.652192\pi\)
0.460116 0.887859i \(-0.347808\pi\)
\(828\) 0 0
\(829\) 818.704 0.987580 0.493790 0.869581i \(-0.335611\pi\)
0.493790 + 0.869581i \(0.335611\pi\)
\(830\) 0 0
\(831\) 324.435i 0.390416i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 381.404i − 0.456772i
\(836\) 0 0
\(837\) − 201.109i − 0.240273i
\(838\) 0 0
\(839\) 1108.84i 1.32162i 0.750555 + 0.660808i \(0.229786\pi\)
−0.750555 + 0.660808i \(0.770214\pi\)
\(840\) 0 0
\(841\) 563.386 0.669900
\(842\) 0 0
\(843\) 267.939 0.317840
\(844\) 0 0
\(845\) −1321.70 −1.56414
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −378.010 −0.445241
\(850\) 0 0
\(851\) − 1137.60i − 1.33678i
\(852\) 0 0
\(853\) 610.400 0.715592 0.357796 0.933800i \(-0.383528\pi\)
0.357796 + 0.933800i \(0.383528\pi\)
\(854\) 0 0
\(855\) −853.017 −0.997680
\(856\) 0 0
\(857\) − 444.114i − 0.518219i −0.965848 0.259110i \(-0.916571\pi\)
0.965848 0.259110i \(-0.0834291\pi\)
\(858\) 0 0
\(859\) 81.5094 0.0948887 0.0474443 0.998874i \(-0.484892\pi\)
0.0474443 + 0.998874i \(0.484892\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1051.46 1.21838 0.609189 0.793025i \(-0.291495\pi\)
0.609189 + 0.793025i \(0.291495\pi\)
\(864\) 0 0
\(865\) 884.268 1.02228
\(866\) 0 0
\(867\) −32.1882 −0.0371259
\(868\) 0 0
\(869\) 770.524i 0.886679i
\(870\) 0 0
\(871\) 332.819i 0.382111i
\(872\) 0 0
\(873\) 1385.01i 1.58650i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1559.63i 1.77837i 0.457551 + 0.889183i \(0.348727\pi\)
−0.457551 + 0.889183i \(0.651273\pi\)
\(878\) 0 0
\(879\) 337.533 0.383997
\(880\) 0 0
\(881\) − 1515.22i − 1.71989i −0.510389 0.859944i \(-0.670499\pi\)
0.510389 0.859944i \(-0.329501\pi\)
\(882\) 0 0
\(883\) 763.828i 0.865037i 0.901625 + 0.432519i \(0.142375\pi\)
−0.901625 + 0.432519i \(0.857625\pi\)
\(884\) 0 0
\(885\) −316.052 −0.357121
\(886\) 0 0
\(887\) − 573.371i − 0.646416i −0.946328 0.323208i \(-0.895239\pi\)
0.946328 0.323208i \(-0.104761\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 783.460i 0.879304i
\(892\) 0 0
\(893\) − 360.455i − 0.403645i
\(894\) 0 0
\(895\) − 1851.01i − 2.06817i
\(896\) 0 0
\(897\) −423.240 −0.471840
\(898\) 0 0
\(899\) −214.250 −0.238320
\(900\) 0 0
\(901\) 592.231 0.657304
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1058.63 −1.16976
\(906\) 0 0
\(907\) 1021.95i 1.12674i 0.826205 + 0.563369i \(0.190495\pi\)
−0.826205 + 0.563369i \(0.809505\pi\)
\(908\) 0 0
\(909\) −229.799 −0.252804
\(910\) 0 0
\(911\) −630.111 −0.691669 −0.345835 0.938295i \(-0.612404\pi\)
−0.345835 + 0.938295i \(0.612404\pi\)
\(912\) 0 0
\(913\) 480.274i 0.526040i
\(914\) 0 0
\(915\) 59.2208 0.0647222
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 842.979 0.917278 0.458639 0.888623i \(-0.348337\pi\)
0.458639 + 0.888623i \(0.348337\pi\)
\(920\) 0 0
\(921\) −146.036 −0.158563
\(922\) 0 0
\(923\) 622.683 0.674630
\(924\) 0 0
\(925\) 725.668i 0.784506i
\(926\) 0 0
\(927\) − 1358.81i − 1.46581i
\(928\) 0 0
\(929\) 774.651i 0.833855i 0.908940 + 0.416927i \(0.136893\pi\)
−0.908940 + 0.416927i \(0.863107\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 430.691i − 0.461620i
\(934\) 0 0
\(935\) −1335.42 −1.42825
\(936\) 0 0
\(937\) − 1586.27i − 1.69293i −0.532447 0.846463i \(-0.678727\pi\)
0.532447 0.846463i \(-0.321273\pi\)
\(938\) 0 0
\(939\) − 210.421i − 0.224090i
\(940\) 0 0
\(941\) −820.046 −0.871462 −0.435731 0.900077i \(-0.643510\pi\)
−0.435731 + 0.900077i \(0.643510\pi\)
\(942\) 0 0
\(943\) − 155.370i − 0.164761i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 637.335i 0.673005i 0.941683 + 0.336502i \(0.109244\pi\)
−0.941683 + 0.336502i \(0.890756\pi\)
\(948\) 0 0
\(949\) 2081.62i 2.19349i
\(950\) 0 0
\(951\) 205.234i 0.215809i
\(952\) 0 0
\(953\) −350.626 −0.367918 −0.183959 0.982934i \(-0.558891\pi\)
−0.183959 + 0.982934i \(0.558891\pi\)
\(954\) 0 0
\(955\) −833.458 −0.872731
\(956\) 0 0
\(957\) −200.553 −0.209564
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 795.652 0.827942
\(962\) 0 0
\(963\) 1614.32i 1.67635i
\(964\) 0 0
\(965\) 516.734 0.535475
\(966\) 0 0
\(967\) 649.816 0.671992 0.335996 0.941863i \(-0.390927\pi\)
0.335996 + 0.941863i \(0.390927\pi\)
\(968\) 0 0
\(969\) − 238.707i − 0.246344i
\(970\) 0 0
\(971\) −970.610 −0.999598 −0.499799 0.866141i \(-0.666593\pi\)
−0.499799 + 0.866141i \(0.666593\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 269.982 0.276905
\(976\) 0 0
\(977\) −600.875 −0.615020 −0.307510 0.951545i \(-0.599496\pi\)
−0.307510 + 0.951545i \(0.599496\pi\)
\(978\) 0 0
\(979\) −14.1890 −0.0144934
\(980\) 0 0
\(981\) 94.1205i 0.0959434i
\(982\) 0 0
\(983\) 1268.62i 1.29056i 0.763945 + 0.645281i \(0.223260\pi\)
−0.763945 + 0.645281i \(0.776740\pi\)
\(984\) 0 0
\(985\) 13.2959i 0.0134984i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 796.413i − 0.805271i
\(990\) 0 0
\(991\) 1549.11 1.56318 0.781590 0.623793i \(-0.214409\pi\)
0.781590 + 0.623793i \(0.214409\pi\)
\(992\) 0 0
\(993\) 18.8538i 0.0189867i
\(994\) 0 0
\(995\) 805.464i 0.809512i
\(996\) 0 0
\(997\) −940.938 −0.943769 −0.471885 0.881660i \(-0.656426\pi\)
−0.471885 + 0.881660i \(0.656426\pi\)
\(998\) 0 0
\(999\) 743.761i 0.744506i
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 1568.3.h.a.881.11 28
4.3 odd 2 392.3.h.a.293.26 28
7.2 even 3 224.3.n.a.17.9 28
7.3 odd 6 224.3.n.a.145.6 28
7.6 odd 2 inner 1568.3.h.a.881.17 28
8.3 odd 2 392.3.h.a.293.27 28
8.5 even 2 inner 1568.3.h.a.881.18 28
28.3 even 6 56.3.j.a.5.6 yes 28
28.11 odd 6 392.3.j.e.117.6 28
28.19 even 6 392.3.j.e.325.4 28
28.23 odd 6 56.3.j.a.45.4 yes 28
28.27 even 2 392.3.h.a.293.25 28
56.3 even 6 56.3.j.a.5.4 28
56.11 odd 6 392.3.j.e.117.4 28
56.13 odd 2 inner 1568.3.h.a.881.12 28
56.19 even 6 392.3.j.e.325.6 28
56.27 even 2 392.3.h.a.293.28 28
56.37 even 6 224.3.n.a.17.6 28
56.45 odd 6 224.3.n.a.145.9 28
56.51 odd 6 56.3.j.a.45.6 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.j.a.5.4 28 56.3 even 6
56.3.j.a.5.6 yes 28 28.3 even 6
56.3.j.a.45.4 yes 28 28.23 odd 6
56.3.j.a.45.6 yes 28 56.51 odd 6
224.3.n.a.17.6 28 56.37 even 6
224.3.n.a.17.9 28 7.2 even 3
224.3.n.a.145.6 28 7.3 odd 6
224.3.n.a.145.9 28 56.45 odd 6
392.3.h.a.293.25 28 28.27 even 2
392.3.h.a.293.26 28 4.3 odd 2
392.3.h.a.293.27 28 8.3 odd 2
392.3.h.a.293.28 28 56.27 even 2
392.3.j.e.117.4 28 56.11 odd 6
392.3.j.e.117.6 28 28.11 odd 6
392.3.j.e.325.4 28 28.19 even 6
392.3.j.e.325.6 28 56.19 even 6
1568.3.h.a.881.11 28 1.1 even 1 trivial
1568.3.h.a.881.12 28 56.13 odd 2 inner
1568.3.h.a.881.17 28 7.6 odd 2 inner
1568.3.h.a.881.18 28 8.5 even 2 inner