Properties

Label 2070.4.a.bg.1.1
Level $2070$
Weight $4$
Character 2070.1
Self dual yes
Analytic conductor $122.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2070,4,Mod(1,2070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2070.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-8,0,16,-20,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(122.133953712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.12571\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -24.6431 q^{7} -8.00000 q^{8} +10.0000 q^{10} +17.3867 q^{11} +4.00699 q^{13} +49.2862 q^{14} +16.0000 q^{16} -48.2740 q^{17} +79.3172 q^{19} -20.0000 q^{20} -34.7734 q^{22} +23.0000 q^{23} +25.0000 q^{25} -8.01398 q^{26} -98.5723 q^{28} +254.267 q^{29} -220.696 q^{31} -32.0000 q^{32} +96.5480 q^{34} +123.215 q^{35} -422.904 q^{37} -158.634 q^{38} +40.0000 q^{40} +170.251 q^{41} -228.920 q^{43} +69.5468 q^{44} -46.0000 q^{46} -580.087 q^{47} +264.282 q^{49} -50.0000 q^{50} +16.0280 q^{52} +260.354 q^{53} -86.9335 q^{55} +197.145 q^{56} -508.535 q^{58} -353.130 q^{59} -80.6108 q^{61} +441.392 q^{62} +64.0000 q^{64} -20.0349 q^{65} -820.011 q^{67} -193.096 q^{68} -246.431 q^{70} -614.845 q^{71} +511.586 q^{73} +845.808 q^{74} +317.269 q^{76} -428.462 q^{77} +160.464 q^{79} -80.0000 q^{80} -340.502 q^{82} +32.5646 q^{83} +241.370 q^{85} +457.840 q^{86} -139.094 q^{88} +25.0375 q^{89} -98.7445 q^{91} +92.0000 q^{92} +1160.17 q^{94} -396.586 q^{95} -249.798 q^{97} -528.563 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −24.6431 −1.33060 −0.665301 0.746575i \(-0.731697\pi\)
−0.665301 + 0.746575i \(0.731697\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 10.0000 0.316228
\(11\) 17.3867 0.476572 0.238286 0.971195i \(-0.423414\pi\)
0.238286 + 0.971195i \(0.423414\pi\)
\(12\) 0 0
\(13\) 4.00699 0.0854876 0.0427438 0.999086i \(-0.486390\pi\)
0.0427438 + 0.999086i \(0.486390\pi\)
\(14\) 49.2862 0.940877
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −48.2740 −0.688716 −0.344358 0.938838i \(-0.611903\pi\)
−0.344358 + 0.938838i \(0.611903\pi\)
\(18\) 0 0
\(19\) 79.3172 0.957717 0.478859 0.877892i \(-0.341051\pi\)
0.478859 + 0.877892i \(0.341051\pi\)
\(20\) −20.0000 −0.223607
\(21\) 0 0
\(22\) −34.7734 −0.336987
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −8.01398 −0.0604488
\(27\) 0 0
\(28\) −98.5723 −0.665301
\(29\) 254.267 1.62815 0.814074 0.580761i \(-0.197245\pi\)
0.814074 + 0.580761i \(0.197245\pi\)
\(30\) 0 0
\(31\) −220.696 −1.27865 −0.639325 0.768937i \(-0.720786\pi\)
−0.639325 + 0.768937i \(0.720786\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 96.5480 0.486996
\(35\) 123.215 0.595063
\(36\) 0 0
\(37\) −422.904 −1.87905 −0.939527 0.342476i \(-0.888735\pi\)
−0.939527 + 0.342476i \(0.888735\pi\)
\(38\) −158.634 −0.677208
\(39\) 0 0
\(40\) 40.0000 0.158114
\(41\) 170.251 0.648505 0.324252 0.945971i \(-0.394887\pi\)
0.324252 + 0.945971i \(0.394887\pi\)
\(42\) 0 0
\(43\) −228.920 −0.811860 −0.405930 0.913904i \(-0.633052\pi\)
−0.405930 + 0.913904i \(0.633052\pi\)
\(44\) 69.5468 0.238286
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −580.087 −1.80031 −0.900154 0.435572i \(-0.856546\pi\)
−0.900154 + 0.435572i \(0.856546\pi\)
\(48\) 0 0
\(49\) 264.282 0.770500
\(50\) −50.0000 −0.141421
\(51\) 0 0
\(52\) 16.0280 0.0427438
\(53\) 260.354 0.674762 0.337381 0.941368i \(-0.390459\pi\)
0.337381 + 0.941368i \(0.390459\pi\)
\(54\) 0 0
\(55\) −86.9335 −0.213129
\(56\) 197.145 0.470439
\(57\) 0 0
\(58\) −508.535 −1.15127
\(59\) −353.130 −0.779215 −0.389607 0.920981i \(-0.627389\pi\)
−0.389607 + 0.920981i \(0.627389\pi\)
\(60\) 0 0
\(61\) −80.6108 −0.169199 −0.0845996 0.996415i \(-0.526961\pi\)
−0.0845996 + 0.996415i \(0.526961\pi\)
\(62\) 441.392 0.904142
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −20.0349 −0.0382312
\(66\) 0 0
\(67\) −820.011 −1.49523 −0.747614 0.664133i \(-0.768801\pi\)
−0.747614 + 0.664133i \(0.768801\pi\)
\(68\) −193.096 −0.344358
\(69\) 0 0
\(70\) −246.431 −0.420773
\(71\) −614.845 −1.02773 −0.513864 0.857872i \(-0.671786\pi\)
−0.513864 + 0.857872i \(0.671786\pi\)
\(72\) 0 0
\(73\) 511.586 0.820228 0.410114 0.912034i \(-0.365489\pi\)
0.410114 + 0.912034i \(0.365489\pi\)
\(74\) 845.808 1.32869
\(75\) 0 0
\(76\) 317.269 0.478859
\(77\) −428.462 −0.634127
\(78\) 0 0
\(79\) 160.464 0.228526 0.114263 0.993451i \(-0.463549\pi\)
0.114263 + 0.993451i \(0.463549\pi\)
\(80\) −80.0000 −0.111803
\(81\) 0 0
\(82\) −340.502 −0.458562
\(83\) 32.5646 0.0430654 0.0215327 0.999768i \(-0.493145\pi\)
0.0215327 + 0.999768i \(0.493145\pi\)
\(84\) 0 0
\(85\) 241.370 0.308003
\(86\) 457.840 0.574072
\(87\) 0 0
\(88\) −139.094 −0.168494
\(89\) 25.0375 0.0298199 0.0149100 0.999889i \(-0.495254\pi\)
0.0149100 + 0.999889i \(0.495254\pi\)
\(90\) 0 0
\(91\) −98.7445 −0.113750
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 1160.17 1.27301
\(95\) −396.586 −0.428304
\(96\) 0 0
\(97\) −249.798 −0.261476 −0.130738 0.991417i \(-0.541735\pi\)
−0.130738 + 0.991417i \(0.541735\pi\)
\(98\) −528.563 −0.544826
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −620.493 −0.611300 −0.305650 0.952144i \(-0.598874\pi\)
−0.305650 + 0.952144i \(0.598874\pi\)
\(102\) 0 0
\(103\) 1473.24 1.40935 0.704673 0.709532i \(-0.251094\pi\)
0.704673 + 0.709532i \(0.251094\pi\)
\(104\) −32.0559 −0.0302244
\(105\) 0 0
\(106\) −520.708 −0.477129
\(107\) 940.141 0.849410 0.424705 0.905332i \(-0.360378\pi\)
0.424705 + 0.905332i \(0.360378\pi\)
\(108\) 0 0
\(109\) −636.264 −0.559111 −0.279555 0.960130i \(-0.590187\pi\)
−0.279555 + 0.960130i \(0.590187\pi\)
\(110\) 173.867 0.150705
\(111\) 0 0
\(112\) −394.289 −0.332650
\(113\) −832.451 −0.693013 −0.346506 0.938048i \(-0.612632\pi\)
−0.346506 + 0.938048i \(0.612632\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 1017.07 0.814074
\(117\) 0 0
\(118\) 706.261 0.550988
\(119\) 1189.62 0.916406
\(120\) 0 0
\(121\) −1028.70 −0.772879
\(122\) 161.222 0.119642
\(123\) 0 0
\(124\) −882.783 −0.639325
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1614.60 1.12813 0.564067 0.825729i \(-0.309236\pi\)
0.564067 + 0.825729i \(0.309236\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 40.0699 0.0270335
\(131\) 1974.56 1.31693 0.658464 0.752612i \(-0.271206\pi\)
0.658464 + 0.752612i \(0.271206\pi\)
\(132\) 0 0
\(133\) −1954.62 −1.27434
\(134\) 1640.02 1.05729
\(135\) 0 0
\(136\) 386.192 0.243498
\(137\) −753.805 −0.470087 −0.235044 0.971985i \(-0.575523\pi\)
−0.235044 + 0.971985i \(0.575523\pi\)
\(138\) 0 0
\(139\) 1014.10 0.618810 0.309405 0.950930i \(-0.399870\pi\)
0.309405 + 0.950930i \(0.399870\pi\)
\(140\) 492.862 0.297532
\(141\) 0 0
\(142\) 1229.69 0.726714
\(143\) 69.6683 0.0407410
\(144\) 0 0
\(145\) −1271.34 −0.728130
\(146\) −1023.17 −0.579989
\(147\) 0 0
\(148\) −1691.62 −0.939527
\(149\) 2771.80 1.52399 0.761995 0.647583i \(-0.224220\pi\)
0.761995 + 0.647583i \(0.224220\pi\)
\(150\) 0 0
\(151\) 3108.62 1.67534 0.837668 0.546180i \(-0.183918\pi\)
0.837668 + 0.546180i \(0.183918\pi\)
\(152\) −634.538 −0.338604
\(153\) 0 0
\(154\) 856.924 0.448396
\(155\) 1103.48 0.571830
\(156\) 0 0
\(157\) 3712.87 1.88739 0.943693 0.330822i \(-0.107326\pi\)
0.943693 + 0.330822i \(0.107326\pi\)
\(158\) −320.928 −0.161593
\(159\) 0 0
\(160\) 160.000 0.0790569
\(161\) −566.791 −0.277450
\(162\) 0 0
\(163\) 915.791 0.440063 0.220032 0.975493i \(-0.429384\pi\)
0.220032 + 0.975493i \(0.429384\pi\)
\(164\) 681.003 0.324252
\(165\) 0 0
\(166\) −65.1292 −0.0304518
\(167\) −1432.32 −0.663688 −0.331844 0.943334i \(-0.607671\pi\)
−0.331844 + 0.943334i \(0.607671\pi\)
\(168\) 0 0
\(169\) −2180.94 −0.992692
\(170\) −482.740 −0.217791
\(171\) 0 0
\(172\) −915.680 −0.405930
\(173\) 3479.54 1.52916 0.764580 0.644529i \(-0.222946\pi\)
0.764580 + 0.644529i \(0.222946\pi\)
\(174\) 0 0
\(175\) −616.077 −0.266120
\(176\) 278.187 0.119143
\(177\) 0 0
\(178\) −50.0751 −0.0210859
\(179\) −3642.71 −1.52105 −0.760527 0.649307i \(-0.775059\pi\)
−0.760527 + 0.649307i \(0.775059\pi\)
\(180\) 0 0
\(181\) 2409.15 0.989342 0.494671 0.869080i \(-0.335289\pi\)
0.494671 + 0.869080i \(0.335289\pi\)
\(182\) 197.489 0.0804333
\(183\) 0 0
\(184\) −184.000 −0.0737210
\(185\) 2114.52 0.840338
\(186\) 0 0
\(187\) −839.326 −0.328223
\(188\) −2320.35 −0.900154
\(189\) 0 0
\(190\) 793.172 0.302857
\(191\) 189.608 0.0718299 0.0359150 0.999355i \(-0.488565\pi\)
0.0359150 + 0.999355i \(0.488565\pi\)
\(192\) 0 0
\(193\) −1855.45 −0.692012 −0.346006 0.938232i \(-0.612462\pi\)
−0.346006 + 0.938232i \(0.612462\pi\)
\(194\) 499.597 0.184891
\(195\) 0 0
\(196\) 1057.13 0.385250
\(197\) −2429.85 −0.878779 −0.439390 0.898297i \(-0.644805\pi\)
−0.439390 + 0.898297i \(0.644805\pi\)
\(198\) 0 0
\(199\) 4333.49 1.54368 0.771842 0.635815i \(-0.219336\pi\)
0.771842 + 0.635815i \(0.219336\pi\)
\(200\) −200.000 −0.0707107
\(201\) 0 0
\(202\) 1240.99 0.432255
\(203\) −6265.94 −2.16642
\(204\) 0 0
\(205\) −851.254 −0.290020
\(206\) −2946.48 −0.996558
\(207\) 0 0
\(208\) 64.1118 0.0213719
\(209\) 1379.07 0.456421
\(210\) 0 0
\(211\) −816.788 −0.266493 −0.133247 0.991083i \(-0.542540\pi\)
−0.133247 + 0.991083i \(0.542540\pi\)
\(212\) 1041.42 0.337381
\(213\) 0 0
\(214\) −1880.28 −0.600624
\(215\) 1144.60 0.363075
\(216\) 0 0
\(217\) 5438.63 1.70137
\(218\) 1272.53 0.395351
\(219\) 0 0
\(220\) −347.734 −0.106565
\(221\) −193.433 −0.0588767
\(222\) 0 0
\(223\) −4513.80 −1.35546 −0.677728 0.735313i \(-0.737035\pi\)
−0.677728 + 0.735313i \(0.737035\pi\)
\(224\) 788.579 0.235219
\(225\) 0 0
\(226\) 1664.90 0.490034
\(227\) −2792.85 −0.816599 −0.408300 0.912848i \(-0.633878\pi\)
−0.408300 + 0.912848i \(0.633878\pi\)
\(228\) 0 0
\(229\) 1404.50 0.405292 0.202646 0.979252i \(-0.435046\pi\)
0.202646 + 0.979252i \(0.435046\pi\)
\(230\) 230.000 0.0659380
\(231\) 0 0
\(232\) −2034.14 −0.575637
\(233\) −1073.79 −0.301916 −0.150958 0.988540i \(-0.548236\pi\)
−0.150958 + 0.988540i \(0.548236\pi\)
\(234\) 0 0
\(235\) 2900.44 0.805122
\(236\) −1412.52 −0.389607
\(237\) 0 0
\(238\) −2379.24 −0.647997
\(239\) 2573.18 0.696423 0.348212 0.937416i \(-0.386789\pi\)
0.348212 + 0.937416i \(0.386789\pi\)
\(240\) 0 0
\(241\) −696.127 −0.186064 −0.0930321 0.995663i \(-0.529656\pi\)
−0.0930321 + 0.995663i \(0.529656\pi\)
\(242\) 2057.40 0.546508
\(243\) 0 0
\(244\) −322.443 −0.0845996
\(245\) −1321.41 −0.344578
\(246\) 0 0
\(247\) 317.823 0.0818729
\(248\) 1765.57 0.452071
\(249\) 0 0
\(250\) 250.000 0.0632456
\(251\) −6467.81 −1.62647 −0.813236 0.581934i \(-0.802296\pi\)
−0.813236 + 0.581934i \(0.802296\pi\)
\(252\) 0 0
\(253\) 399.894 0.0993721
\(254\) −3229.21 −0.797711
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −5332.72 −1.29434 −0.647172 0.762344i \(-0.724048\pi\)
−0.647172 + 0.762344i \(0.724048\pi\)
\(258\) 0 0
\(259\) 10421.7 2.50027
\(260\) −80.1398 −0.0191156
\(261\) 0 0
\(262\) −3949.11 −0.931209
\(263\) 6872.85 1.61140 0.805699 0.592325i \(-0.201790\pi\)
0.805699 + 0.592325i \(0.201790\pi\)
\(264\) 0 0
\(265\) −1301.77 −0.301763
\(266\) 3909.24 0.901094
\(267\) 0 0
\(268\) −3280.04 −0.747614
\(269\) 1926.13 0.436572 0.218286 0.975885i \(-0.429953\pi\)
0.218286 + 0.975885i \(0.429953\pi\)
\(270\) 0 0
\(271\) −3653.06 −0.818847 −0.409423 0.912344i \(-0.634270\pi\)
−0.409423 + 0.912344i \(0.634270\pi\)
\(272\) −772.384 −0.172179
\(273\) 0 0
\(274\) 1507.61 0.332402
\(275\) 434.668 0.0953144
\(276\) 0 0
\(277\) −1047.37 −0.227185 −0.113592 0.993527i \(-0.536236\pi\)
−0.113592 + 0.993527i \(0.536236\pi\)
\(278\) −2028.19 −0.437565
\(279\) 0 0
\(280\) −985.723 −0.210387
\(281\) 2758.90 0.585701 0.292851 0.956158i \(-0.405396\pi\)
0.292851 + 0.956158i \(0.405396\pi\)
\(282\) 0 0
\(283\) 3355.58 0.704837 0.352418 0.935843i \(-0.385359\pi\)
0.352418 + 0.935843i \(0.385359\pi\)
\(284\) −2459.38 −0.513864
\(285\) 0 0
\(286\) −139.337 −0.0288082
\(287\) −4195.50 −0.862902
\(288\) 0 0
\(289\) −2582.62 −0.525670
\(290\) 2542.67 0.514866
\(291\) 0 0
\(292\) 2046.35 0.410114
\(293\) −419.625 −0.0836681 −0.0418341 0.999125i \(-0.513320\pi\)
−0.0418341 + 0.999125i \(0.513320\pi\)
\(294\) 0 0
\(295\) 1765.65 0.348475
\(296\) 3383.23 0.664346
\(297\) 0 0
\(298\) −5543.59 −1.07762
\(299\) 92.1607 0.0178254
\(300\) 0 0
\(301\) 5641.30 1.08026
\(302\) −6217.24 −1.18464
\(303\) 0 0
\(304\) 1269.08 0.239429
\(305\) 403.054 0.0756682
\(306\) 0 0
\(307\) −4133.41 −0.768425 −0.384212 0.923245i \(-0.625527\pi\)
−0.384212 + 0.923245i \(0.625527\pi\)
\(308\) −1713.85 −0.317064
\(309\) 0 0
\(310\) −2206.96 −0.404345
\(311\) 6991.03 1.27468 0.637339 0.770584i \(-0.280035\pi\)
0.637339 + 0.770584i \(0.280035\pi\)
\(312\) 0 0
\(313\) 9380.53 1.69399 0.846996 0.531600i \(-0.178409\pi\)
0.846996 + 0.531600i \(0.178409\pi\)
\(314\) −7425.75 −1.33458
\(315\) 0 0
\(316\) 641.855 0.114263
\(317\) 7995.35 1.41660 0.708302 0.705910i \(-0.249462\pi\)
0.708302 + 0.705910i \(0.249462\pi\)
\(318\) 0 0
\(319\) 4420.87 0.775929
\(320\) −320.000 −0.0559017
\(321\) 0 0
\(322\) 1133.58 0.196186
\(323\) −3828.96 −0.659595
\(324\) 0 0
\(325\) 100.175 0.0170975
\(326\) −1831.58 −0.311172
\(327\) 0 0
\(328\) −1362.01 −0.229281
\(329\) 14295.1 2.39549
\(330\) 0 0
\(331\) −4798.86 −0.796886 −0.398443 0.917193i \(-0.630449\pi\)
−0.398443 + 0.917193i \(0.630449\pi\)
\(332\) 130.258 0.0215327
\(333\) 0 0
\(334\) 2864.63 0.469298
\(335\) 4100.06 0.668687
\(336\) 0 0
\(337\) 6895.51 1.11461 0.557303 0.830309i \(-0.311836\pi\)
0.557303 + 0.830309i \(0.311836\pi\)
\(338\) 4361.89 0.701939
\(339\) 0 0
\(340\) 965.480 0.154002
\(341\) −3837.17 −0.609368
\(342\) 0 0
\(343\) 1939.86 0.305372
\(344\) 1831.36 0.287036
\(345\) 0 0
\(346\) −6959.08 −1.08128
\(347\) 8744.17 1.35277 0.676386 0.736548i \(-0.263545\pi\)
0.676386 + 0.736548i \(0.263545\pi\)
\(348\) 0 0
\(349\) −6387.08 −0.979635 −0.489818 0.871825i \(-0.662937\pi\)
−0.489818 + 0.871825i \(0.662937\pi\)
\(350\) 1232.15 0.188175
\(351\) 0 0
\(352\) −556.375 −0.0842468
\(353\) 589.384 0.0888661 0.0444331 0.999012i \(-0.485852\pi\)
0.0444331 + 0.999012i \(0.485852\pi\)
\(354\) 0 0
\(355\) 3074.23 0.459614
\(356\) 100.150 0.0149100
\(357\) 0 0
\(358\) 7285.41 1.07555
\(359\) −7214.13 −1.06058 −0.530289 0.847817i \(-0.677917\pi\)
−0.530289 + 0.847817i \(0.677917\pi\)
\(360\) 0 0
\(361\) −567.774 −0.0827780
\(362\) −4818.30 −0.699570
\(363\) 0 0
\(364\) −394.978 −0.0568749
\(365\) −2557.93 −0.366817
\(366\) 0 0
\(367\) 12356.4 1.75750 0.878748 0.477286i \(-0.158379\pi\)
0.878748 + 0.477286i \(0.158379\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) −4229.04 −0.594209
\(371\) −6415.92 −0.897839
\(372\) 0 0
\(373\) 7145.98 0.991969 0.495985 0.868331i \(-0.334807\pi\)
0.495985 + 0.868331i \(0.334807\pi\)
\(374\) 1678.65 0.232088
\(375\) 0 0
\(376\) 4640.70 0.636505
\(377\) 1018.85 0.139186
\(378\) 0 0
\(379\) 2170.69 0.294197 0.147099 0.989122i \(-0.453007\pi\)
0.147099 + 0.989122i \(0.453007\pi\)
\(380\) −1586.34 −0.214152
\(381\) 0 0
\(382\) −379.215 −0.0507914
\(383\) 7967.21 1.06294 0.531469 0.847078i \(-0.321640\pi\)
0.531469 + 0.847078i \(0.321640\pi\)
\(384\) 0 0
\(385\) 2142.31 0.283590
\(386\) 3710.90 0.489326
\(387\) 0 0
\(388\) −999.193 −0.130738
\(389\) 568.951 0.0741567 0.0370783 0.999312i \(-0.488195\pi\)
0.0370783 + 0.999312i \(0.488195\pi\)
\(390\) 0 0
\(391\) −1110.30 −0.143607
\(392\) −2114.25 −0.272413
\(393\) 0 0
\(394\) 4859.70 0.621391
\(395\) −802.319 −0.102200
\(396\) 0 0
\(397\) 8564.88 1.08277 0.541384 0.840775i \(-0.317900\pi\)
0.541384 + 0.840775i \(0.317900\pi\)
\(398\) −8666.98 −1.09155
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) 12455.6 1.55113 0.775563 0.631270i \(-0.217466\pi\)
0.775563 + 0.631270i \(0.217466\pi\)
\(402\) 0 0
\(403\) −884.325 −0.109309
\(404\) −2481.97 −0.305650
\(405\) 0 0
\(406\) 12531.9 1.53189
\(407\) −7352.91 −0.895504
\(408\) 0 0
\(409\) 11838.9 1.43129 0.715645 0.698465i \(-0.246133\pi\)
0.715645 + 0.698465i \(0.246133\pi\)
\(410\) 1702.51 0.205075
\(411\) 0 0
\(412\) 5892.96 0.704673
\(413\) 8702.22 1.03682
\(414\) 0 0
\(415\) −162.823 −0.0192594
\(416\) −128.224 −0.0151122
\(417\) 0 0
\(418\) −2758.13 −0.322738
\(419\) 1531.77 0.178596 0.0892981 0.996005i \(-0.471538\pi\)
0.0892981 + 0.996005i \(0.471538\pi\)
\(420\) 0 0
\(421\) −9985.06 −1.15592 −0.577959 0.816066i \(-0.696151\pi\)
−0.577959 + 0.816066i \(0.696151\pi\)
\(422\) 1633.58 0.188439
\(423\) 0 0
\(424\) −2082.83 −0.238564
\(425\) −1206.85 −0.137743
\(426\) 0 0
\(427\) 1986.50 0.225137
\(428\) 3760.56 0.424705
\(429\) 0 0
\(430\) −2289.20 −0.256733
\(431\) 13786.7 1.54079 0.770396 0.637566i \(-0.220059\pi\)
0.770396 + 0.637566i \(0.220059\pi\)
\(432\) 0 0
\(433\) −2621.92 −0.290996 −0.145498 0.989359i \(-0.546478\pi\)
−0.145498 + 0.989359i \(0.546478\pi\)
\(434\) −10877.3 −1.20305
\(435\) 0 0
\(436\) −2545.06 −0.279555
\(437\) 1824.30 0.199698
\(438\) 0 0
\(439\) 12062.7 1.31143 0.655717 0.755007i \(-0.272367\pi\)
0.655717 + 0.755007i \(0.272367\pi\)
\(440\) 695.468 0.0753526
\(441\) 0 0
\(442\) 386.867 0.0416321
\(443\) −3659.26 −0.392453 −0.196227 0.980559i \(-0.562869\pi\)
−0.196227 + 0.980559i \(0.562869\pi\)
\(444\) 0 0
\(445\) −125.188 −0.0133359
\(446\) 9027.60 0.958452
\(447\) 0 0
\(448\) −1577.16 −0.166325
\(449\) 10529.6 1.10674 0.553368 0.832937i \(-0.313342\pi\)
0.553368 + 0.832937i \(0.313342\pi\)
\(450\) 0 0
\(451\) 2960.10 0.309059
\(452\) −3329.81 −0.346506
\(453\) 0 0
\(454\) 5585.70 0.577423
\(455\) 493.723 0.0508705
\(456\) 0 0
\(457\) 6443.23 0.659522 0.329761 0.944064i \(-0.393032\pi\)
0.329761 + 0.944064i \(0.393032\pi\)
\(458\) −2809.00 −0.286585
\(459\) 0 0
\(460\) −460.000 −0.0466252
\(461\) 3263.86 0.329747 0.164873 0.986315i \(-0.447278\pi\)
0.164873 + 0.986315i \(0.447278\pi\)
\(462\) 0 0
\(463\) 9518.12 0.955388 0.477694 0.878526i \(-0.341473\pi\)
0.477694 + 0.878526i \(0.341473\pi\)
\(464\) 4068.28 0.407037
\(465\) 0 0
\(466\) 2147.59 0.213487
\(467\) −19092.6 −1.89187 −0.945934 0.324360i \(-0.894851\pi\)
−0.945934 + 0.324360i \(0.894851\pi\)
\(468\) 0 0
\(469\) 20207.6 1.98955
\(470\) −5800.87 −0.569307
\(471\) 0 0
\(472\) 2825.04 0.275494
\(473\) −3980.17 −0.386910
\(474\) 0 0
\(475\) 1982.93 0.191543
\(476\) 4758.48 0.458203
\(477\) 0 0
\(478\) −5146.36 −0.492446
\(479\) 6324.81 0.603315 0.301658 0.953416i \(-0.402460\pi\)
0.301658 + 0.953416i \(0.402460\pi\)
\(480\) 0 0
\(481\) −1694.57 −0.160636
\(482\) 1392.25 0.131567
\(483\) 0 0
\(484\) −4114.81 −0.386440
\(485\) 1248.99 0.116936
\(486\) 0 0
\(487\) −7873.07 −0.732573 −0.366286 0.930502i \(-0.619371\pi\)
−0.366286 + 0.930502i \(0.619371\pi\)
\(488\) 644.886 0.0598209
\(489\) 0 0
\(490\) 2642.82 0.243654
\(491\) 3556.82 0.326918 0.163459 0.986550i \(-0.447735\pi\)
0.163459 + 0.986550i \(0.447735\pi\)
\(492\) 0 0
\(493\) −12274.5 −1.12133
\(494\) −635.646 −0.0578929
\(495\) 0 0
\(496\) −3531.13 −0.319662
\(497\) 15151.7 1.36750
\(498\) 0 0
\(499\) −1933.37 −0.173446 −0.0867231 0.996232i \(-0.527640\pi\)
−0.0867231 + 0.996232i \(0.527640\pi\)
\(500\) −500.000 −0.0447214
\(501\) 0 0
\(502\) 12935.6 1.15009
\(503\) −2114.36 −0.187425 −0.0937123 0.995599i \(-0.529873\pi\)
−0.0937123 + 0.995599i \(0.529873\pi\)
\(504\) 0 0
\(505\) 3102.46 0.273382
\(506\) −799.789 −0.0702667
\(507\) 0 0
\(508\) 6458.42 0.564067
\(509\) 316.452 0.0275570 0.0137785 0.999905i \(-0.495614\pi\)
0.0137785 + 0.999905i \(0.495614\pi\)
\(510\) 0 0
\(511\) −12607.1 −1.09140
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 10665.4 0.915239
\(515\) −7366.20 −0.630279
\(516\) 0 0
\(517\) −10085.8 −0.857976
\(518\) −20843.3 −1.76796
\(519\) 0 0
\(520\) 160.280 0.0135168
\(521\) 309.041 0.0259872 0.0129936 0.999916i \(-0.495864\pi\)
0.0129936 + 0.999916i \(0.495864\pi\)
\(522\) 0 0
\(523\) −5892.46 −0.492656 −0.246328 0.969186i \(-0.579224\pi\)
−0.246328 + 0.969186i \(0.579224\pi\)
\(524\) 7898.22 0.658464
\(525\) 0 0
\(526\) −13745.7 −1.13943
\(527\) 10653.9 0.880626
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 2603.54 0.213378
\(531\) 0 0
\(532\) −7818.49 −0.637170
\(533\) 682.193 0.0554391
\(534\) 0 0
\(535\) −4700.71 −0.379868
\(536\) 6560.09 0.528643
\(537\) 0 0
\(538\) −3852.25 −0.308703
\(539\) 4594.99 0.367199
\(540\) 0 0
\(541\) 1560.55 0.124017 0.0620085 0.998076i \(-0.480249\pi\)
0.0620085 + 0.998076i \(0.480249\pi\)
\(542\) 7306.12 0.579012
\(543\) 0 0
\(544\) 1544.77 0.121749
\(545\) 3181.32 0.250042
\(546\) 0 0
\(547\) 17756.3 1.38795 0.693973 0.720001i \(-0.255859\pi\)
0.693973 + 0.720001i \(0.255859\pi\)
\(548\) −3015.22 −0.235044
\(549\) 0 0
\(550\) −869.335 −0.0673974
\(551\) 20167.8 1.55930
\(552\) 0 0
\(553\) −3954.32 −0.304078
\(554\) 2094.73 0.160644
\(555\) 0 0
\(556\) 4056.39 0.309405
\(557\) −1212.77 −0.0922559 −0.0461280 0.998936i \(-0.514688\pi\)
−0.0461280 + 0.998936i \(0.514688\pi\)
\(558\) 0 0
\(559\) −917.280 −0.0694040
\(560\) 1971.45 0.148766
\(561\) 0 0
\(562\) −5517.80 −0.414153
\(563\) −12558.2 −0.940078 −0.470039 0.882646i \(-0.655760\pi\)
−0.470039 + 0.882646i \(0.655760\pi\)
\(564\) 0 0
\(565\) 4162.26 0.309925
\(566\) −6711.17 −0.498395
\(567\) 0 0
\(568\) 4918.76 0.363357
\(569\) −9776.72 −0.720319 −0.360159 0.932891i \(-0.617278\pi\)
−0.360159 + 0.932891i \(0.617278\pi\)
\(570\) 0 0
\(571\) −18733.9 −1.37301 −0.686505 0.727125i \(-0.740856\pi\)
−0.686505 + 0.727125i \(0.740856\pi\)
\(572\) 278.673 0.0203705
\(573\) 0 0
\(574\) 8391.01 0.610164
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) 5113.58 0.368944 0.184472 0.982838i \(-0.440942\pi\)
0.184472 + 0.982838i \(0.440942\pi\)
\(578\) 5165.24 0.371705
\(579\) 0 0
\(580\) −5085.35 −0.364065
\(581\) −802.492 −0.0573029
\(582\) 0 0
\(583\) 4526.70 0.321572
\(584\) −4092.69 −0.289995
\(585\) 0 0
\(586\) 839.250 0.0591623
\(587\) 5379.95 0.378287 0.189143 0.981949i \(-0.439429\pi\)
0.189143 + 0.981949i \(0.439429\pi\)
\(588\) 0 0
\(589\) −17505.0 −1.22458
\(590\) −3531.30 −0.246409
\(591\) 0 0
\(592\) −6766.46 −0.469763
\(593\) 15060.3 1.04292 0.521461 0.853275i \(-0.325387\pi\)
0.521461 + 0.853275i \(0.325387\pi\)
\(594\) 0 0
\(595\) −5948.10 −0.409829
\(596\) 11087.2 0.761995
\(597\) 0 0
\(598\) −184.321 −0.0126045
\(599\) 3772.04 0.257298 0.128649 0.991690i \(-0.458936\pi\)
0.128649 + 0.991690i \(0.458936\pi\)
\(600\) 0 0
\(601\) −14663.9 −0.995261 −0.497631 0.867389i \(-0.665796\pi\)
−0.497631 + 0.867389i \(0.665796\pi\)
\(602\) −11282.6 −0.763861
\(603\) 0 0
\(604\) 12434.5 0.837668
\(605\) 5143.51 0.345642
\(606\) 0 0
\(607\) −24514.5 −1.63923 −0.819614 0.572915i \(-0.805812\pi\)
−0.819614 + 0.572915i \(0.805812\pi\)
\(608\) −2538.15 −0.169302
\(609\) 0 0
\(610\) −806.108 −0.0535055
\(611\) −2324.40 −0.153904
\(612\) 0 0
\(613\) 14451.2 0.952167 0.476084 0.879400i \(-0.342056\pi\)
0.476084 + 0.879400i \(0.342056\pi\)
\(614\) 8266.83 0.543358
\(615\) 0 0
\(616\) 3427.70 0.224198
\(617\) −3292.19 −0.214811 −0.107406 0.994215i \(-0.534254\pi\)
−0.107406 + 0.994215i \(0.534254\pi\)
\(618\) 0 0
\(619\) −12595.5 −0.817858 −0.408929 0.912566i \(-0.634098\pi\)
−0.408929 + 0.912566i \(0.634098\pi\)
\(620\) 4413.92 0.285915
\(621\) 0 0
\(622\) −13982.1 −0.901333
\(623\) −617.002 −0.0396785
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −18761.1 −1.19783
\(627\) 0 0
\(628\) 14851.5 0.943693
\(629\) 20415.3 1.29413
\(630\) 0 0
\(631\) 19889.3 1.25480 0.627401 0.778697i \(-0.284119\pi\)
0.627401 + 0.778697i \(0.284119\pi\)
\(632\) −1283.71 −0.0807963
\(633\) 0 0
\(634\) −15990.7 −1.00169
\(635\) −8073.02 −0.504517
\(636\) 0 0
\(637\) 1058.97 0.0658682
\(638\) −8841.75 −0.548665
\(639\) 0 0
\(640\) 640.000 0.0395285
\(641\) 19276.0 1.18776 0.593880 0.804553i \(-0.297595\pi\)
0.593880 + 0.804553i \(0.297595\pi\)
\(642\) 0 0
\(643\) −10219.2 −0.626758 −0.313379 0.949628i \(-0.601461\pi\)
−0.313379 + 0.949628i \(0.601461\pi\)
\(644\) −2267.16 −0.138725
\(645\) 0 0
\(646\) 7657.93 0.466404
\(647\) −20818.4 −1.26500 −0.632500 0.774560i \(-0.717971\pi\)
−0.632500 + 0.774560i \(0.717971\pi\)
\(648\) 0 0
\(649\) −6139.77 −0.371352
\(650\) −200.349 −0.0120898
\(651\) 0 0
\(652\) 3663.17 0.220032
\(653\) 15135.3 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(654\) 0 0
\(655\) −9872.78 −0.588949
\(656\) 2724.01 0.162126
\(657\) 0 0
\(658\) −28590.3 −1.69387
\(659\) 13207.9 0.780737 0.390369 0.920659i \(-0.372348\pi\)
0.390369 + 0.920659i \(0.372348\pi\)
\(660\) 0 0
\(661\) 6671.34 0.392564 0.196282 0.980547i \(-0.437113\pi\)
0.196282 + 0.980547i \(0.437113\pi\)
\(662\) 9597.72 0.563484
\(663\) 0 0
\(664\) −260.517 −0.0152259
\(665\) 9773.11 0.569902
\(666\) 0 0
\(667\) 5848.15 0.339492
\(668\) −5729.26 −0.331844
\(669\) 0 0
\(670\) −8200.11 −0.472833
\(671\) −1401.56 −0.0806355
\(672\) 0 0
\(673\) −9534.41 −0.546099 −0.273049 0.962000i \(-0.588032\pi\)
−0.273049 + 0.962000i \(0.588032\pi\)
\(674\) −13791.0 −0.788146
\(675\) 0 0
\(676\) −8723.78 −0.496346
\(677\) −17748.4 −1.00757 −0.503787 0.863828i \(-0.668060\pi\)
−0.503787 + 0.863828i \(0.668060\pi\)
\(678\) 0 0
\(679\) 6155.80 0.347920
\(680\) −1930.96 −0.108896
\(681\) 0 0
\(682\) 7674.35 0.430888
\(683\) −9583.57 −0.536904 −0.268452 0.963293i \(-0.586512\pi\)
−0.268452 + 0.963293i \(0.586512\pi\)
\(684\) 0 0
\(685\) 3769.03 0.210229
\(686\) −3879.73 −0.215931
\(687\) 0 0
\(688\) −3662.72 −0.202965
\(689\) 1043.24 0.0576837
\(690\) 0 0
\(691\) 9410.74 0.518092 0.259046 0.965865i \(-0.416592\pi\)
0.259046 + 0.965865i \(0.416592\pi\)
\(692\) 13918.2 0.764580
\(693\) 0 0
\(694\) −17488.3 −0.956554
\(695\) −5070.48 −0.276740
\(696\) 0 0
\(697\) −8218.69 −0.446636
\(698\) 12774.2 0.692707
\(699\) 0 0
\(700\) −2464.31 −0.133060
\(701\) 19517.9 1.05161 0.525806 0.850605i \(-0.323764\pi\)
0.525806 + 0.850605i \(0.323764\pi\)
\(702\) 0 0
\(703\) −33543.6 −1.79960
\(704\) 1112.75 0.0595715
\(705\) 0 0
\(706\) −1178.77 −0.0628378
\(707\) 15290.9 0.813397
\(708\) 0 0
\(709\) −28430.7 −1.50598 −0.752990 0.658033i \(-0.771389\pi\)
−0.752990 + 0.658033i \(0.771389\pi\)
\(710\) −6148.45 −0.324996
\(711\) 0 0
\(712\) −200.300 −0.0105429
\(713\) −5076.00 −0.266617
\(714\) 0 0
\(715\) −348.342 −0.0182199
\(716\) −14570.8 −0.760527
\(717\) 0 0
\(718\) 14428.3 0.749942
\(719\) −18716.0 −0.970779 −0.485390 0.874298i \(-0.661322\pi\)
−0.485390 + 0.874298i \(0.661322\pi\)
\(720\) 0 0
\(721\) −36305.2 −1.87528
\(722\) 1135.55 0.0585329
\(723\) 0 0
\(724\) 9636.61 0.494671
\(725\) 6356.69 0.325630
\(726\) 0 0
\(727\) 18419.5 0.939670 0.469835 0.882754i \(-0.344313\pi\)
0.469835 + 0.882754i \(0.344313\pi\)
\(728\) 789.956 0.0402167
\(729\) 0 0
\(730\) 5115.86 0.259379
\(731\) 11050.9 0.559141
\(732\) 0 0
\(733\) −21548.4 −1.08582 −0.542912 0.839790i \(-0.682678\pi\)
−0.542912 + 0.839790i \(0.682678\pi\)
\(734\) −24712.9 −1.24274
\(735\) 0 0
\(736\) −736.000 −0.0368605
\(737\) −14257.3 −0.712584
\(738\) 0 0
\(739\) 12066.4 0.600636 0.300318 0.953839i \(-0.402907\pi\)
0.300318 + 0.953839i \(0.402907\pi\)
\(740\) 8458.08 0.420169
\(741\) 0 0
\(742\) 12831.8 0.634868
\(743\) 21951.0 1.08385 0.541926 0.840426i \(-0.317695\pi\)
0.541926 + 0.840426i \(0.317695\pi\)
\(744\) 0 0
\(745\) −13859.0 −0.681549
\(746\) −14292.0 −0.701428
\(747\) 0 0
\(748\) −3357.31 −0.164111
\(749\) −23168.0 −1.13023
\(750\) 0 0
\(751\) −3112.43 −0.151230 −0.0756152 0.997137i \(-0.524092\pi\)
−0.0756152 + 0.997137i \(0.524092\pi\)
\(752\) −9281.40 −0.450077
\(753\) 0 0
\(754\) −2037.69 −0.0984197
\(755\) −15543.1 −0.749233
\(756\) 0 0
\(757\) −7684.64 −0.368960 −0.184480 0.982836i \(-0.559060\pi\)
−0.184480 + 0.982836i \(0.559060\pi\)
\(758\) −4341.38 −0.208029
\(759\) 0 0
\(760\) 3172.69 0.151428
\(761\) 36484.6 1.73793 0.868965 0.494874i \(-0.164786\pi\)
0.868965 + 0.494874i \(0.164786\pi\)
\(762\) 0 0
\(763\) 15679.5 0.743954
\(764\) 758.430 0.0359150
\(765\) 0 0
\(766\) −15934.4 −0.751611
\(767\) −1414.99 −0.0666132
\(768\) 0 0
\(769\) 2004.39 0.0939925 0.0469962 0.998895i \(-0.485035\pi\)
0.0469962 + 0.998895i \(0.485035\pi\)
\(770\) −4284.62 −0.200529
\(771\) 0 0
\(772\) −7421.81 −0.346006
\(773\) −7716.17 −0.359031 −0.179516 0.983755i \(-0.557453\pi\)
−0.179516 + 0.983755i \(0.557453\pi\)
\(774\) 0 0
\(775\) −5517.40 −0.255730
\(776\) 1998.39 0.0924457
\(777\) 0 0
\(778\) −1137.90 −0.0524367
\(779\) 13503.8 0.621084
\(780\) 0 0
\(781\) −10690.1 −0.489786
\(782\) 2220.61 0.101546
\(783\) 0 0
\(784\) 4228.51 0.192625
\(785\) −18564.4 −0.844065
\(786\) 0 0
\(787\) −57.0149 −0.00258241 −0.00129121 0.999999i \(-0.500411\pi\)
−0.00129121 + 0.999999i \(0.500411\pi\)
\(788\) −9719.39 −0.439390
\(789\) 0 0
\(790\) 1604.64 0.0722664
\(791\) 20514.2 0.922124
\(792\) 0 0
\(793\) −323.006 −0.0144644
\(794\) −17129.8 −0.765633
\(795\) 0 0
\(796\) 17334.0 0.771842
\(797\) 15184.7 0.674870 0.337435 0.941349i \(-0.390441\pi\)
0.337435 + 0.941349i \(0.390441\pi\)
\(798\) 0 0
\(799\) 28003.2 1.23990
\(800\) −800.000 −0.0353553
\(801\) 0 0
\(802\) −24911.1 −1.09681
\(803\) 8894.80 0.390898
\(804\) 0 0
\(805\) 2833.95 0.124079
\(806\) 1768.65 0.0772929
\(807\) 0 0
\(808\) 4963.94 0.216127
\(809\) 30286.4 1.31621 0.658105 0.752926i \(-0.271358\pi\)
0.658105 + 0.752926i \(0.271358\pi\)
\(810\) 0 0
\(811\) −43936.2 −1.90235 −0.951176 0.308650i \(-0.900123\pi\)
−0.951176 + 0.308650i \(0.900123\pi\)
\(812\) −25063.7 −1.08321
\(813\) 0 0
\(814\) 14705.8 0.633217
\(815\) −4578.96 −0.196802
\(816\) 0 0
\(817\) −18157.3 −0.777532
\(818\) −23677.9 −1.01207
\(819\) 0 0
\(820\) −3405.02 −0.145010
\(821\) 5245.69 0.222991 0.111496 0.993765i \(-0.464436\pi\)
0.111496 + 0.993765i \(0.464436\pi\)
\(822\) 0 0
\(823\) 10678.0 0.452260 0.226130 0.974097i \(-0.427393\pi\)
0.226130 + 0.974097i \(0.427393\pi\)
\(824\) −11785.9 −0.498279
\(825\) 0 0
\(826\) −17404.4 −0.733145
\(827\) −3393.69 −0.142697 −0.0713484 0.997451i \(-0.522730\pi\)
−0.0713484 + 0.997451i \(0.522730\pi\)
\(828\) 0 0
\(829\) 9601.74 0.402270 0.201135 0.979564i \(-0.435537\pi\)
0.201135 + 0.979564i \(0.435537\pi\)
\(830\) 325.646 0.0136185
\(831\) 0 0
\(832\) 256.447 0.0106859
\(833\) −12757.9 −0.530656
\(834\) 0 0
\(835\) 7161.58 0.296810
\(836\) 5516.26 0.228210
\(837\) 0 0
\(838\) −3063.54 −0.126287
\(839\) −11992.8 −0.493489 −0.246744 0.969081i \(-0.579361\pi\)
−0.246744 + 0.969081i \(0.579361\pi\)
\(840\) 0 0
\(841\) 40263.0 1.65087
\(842\) 19970.1 0.817358
\(843\) 0 0
\(844\) −3267.15 −0.133247
\(845\) 10904.7 0.443945
\(846\) 0 0
\(847\) 25350.4 1.02839
\(848\) 4165.66 0.168690
\(849\) 0 0
\(850\) 2413.70 0.0973991
\(851\) −9726.79 −0.391810
\(852\) 0 0
\(853\) −15441.5 −0.619821 −0.309911 0.950766i \(-0.600299\pi\)
−0.309911 + 0.950766i \(0.600299\pi\)
\(854\) −3973.00 −0.159196
\(855\) 0 0
\(856\) −7521.13 −0.300312
\(857\) 44572.4 1.77662 0.888310 0.459244i \(-0.151880\pi\)
0.888310 + 0.459244i \(0.151880\pi\)
\(858\) 0 0
\(859\) 2519.56 0.100077 0.0500386 0.998747i \(-0.484066\pi\)
0.0500386 + 0.998747i \(0.484066\pi\)
\(860\) 4578.40 0.181537
\(861\) 0 0
\(862\) −27573.4 −1.08950
\(863\) −28980.1 −1.14310 −0.571548 0.820568i \(-0.693657\pi\)
−0.571548 + 0.820568i \(0.693657\pi\)
\(864\) 0 0
\(865\) −17397.7 −0.683861
\(866\) 5243.83 0.205765
\(867\) 0 0
\(868\) 21754.5 0.850687
\(869\) 2789.94 0.108909
\(870\) 0 0
\(871\) −3285.77 −0.127823
\(872\) 5090.11 0.197675
\(873\) 0 0
\(874\) −3648.59 −0.141208
\(875\) 3080.39 0.119013
\(876\) 0 0
\(877\) −7955.28 −0.306307 −0.153153 0.988202i \(-0.548943\pi\)
−0.153153 + 0.988202i \(0.548943\pi\)
\(878\) −24125.3 −0.927323
\(879\) 0 0
\(880\) −1390.94 −0.0532823
\(881\) −29722.0 −1.13662 −0.568309 0.822815i \(-0.692402\pi\)
−0.568309 + 0.822815i \(0.692402\pi\)
\(882\) 0 0
\(883\) −19379.7 −0.738596 −0.369298 0.929311i \(-0.620402\pi\)
−0.369298 + 0.929311i \(0.620402\pi\)
\(884\) −773.734 −0.0294383
\(885\) 0 0
\(886\) 7318.53 0.277507
\(887\) −22901.7 −0.866925 −0.433463 0.901172i \(-0.642708\pi\)
−0.433463 + 0.901172i \(0.642708\pi\)
\(888\) 0 0
\(889\) −39788.8 −1.50110
\(890\) 250.375 0.00942989
\(891\) 0 0
\(892\) −18055.2 −0.677728
\(893\) −46010.9 −1.72419
\(894\) 0 0
\(895\) 18213.5 0.680236
\(896\) 3154.31 0.117610
\(897\) 0 0
\(898\) −21059.3 −0.782581
\(899\) −56115.8 −2.08183
\(900\) 0 0
\(901\) −12568.3 −0.464719
\(902\) −5920.20 −0.218538
\(903\) 0 0
\(904\) 6659.61 0.245017
\(905\) −12045.8 −0.442447
\(906\) 0 0
\(907\) 42542.2 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(908\) −11171.4 −0.408300
\(909\) 0 0
\(910\) −987.445 −0.0359709
\(911\) −220.864 −0.00803243 −0.00401622 0.999992i \(-0.501278\pi\)
−0.00401622 + 0.999992i \(0.501278\pi\)
\(912\) 0 0
\(913\) 566.191 0.0205237
\(914\) −12886.5 −0.466352
\(915\) 0 0
\(916\) 5617.99 0.202646
\(917\) −48659.1 −1.75231
\(918\) 0 0
\(919\) 27835.4 0.999135 0.499567 0.866275i \(-0.333492\pi\)
0.499567 + 0.866275i \(0.333492\pi\)
\(920\) 920.000 0.0329690
\(921\) 0 0
\(922\) −6527.72 −0.233166
\(923\) −2463.68 −0.0878580
\(924\) 0 0
\(925\) −10572.6 −0.375811
\(926\) −19036.2 −0.675562
\(927\) 0 0
\(928\) −8136.56 −0.287819
\(929\) 2172.71 0.0767323 0.0383661 0.999264i \(-0.487785\pi\)
0.0383661 + 0.999264i \(0.487785\pi\)
\(930\) 0 0
\(931\) 20962.1 0.737921
\(932\) −4295.17 −0.150958
\(933\) 0 0
\(934\) 38185.3 1.33775
\(935\) 4196.63 0.146786
\(936\) 0 0
\(937\) 54906.5 1.91432 0.957160 0.289560i \(-0.0935090\pi\)
0.957160 + 0.289560i \(0.0935090\pi\)
\(938\) −40415.2 −1.40683
\(939\) 0 0
\(940\) 11601.7 0.402561
\(941\) −5980.06 −0.207167 −0.103584 0.994621i \(-0.533031\pi\)
−0.103584 + 0.994621i \(0.533031\pi\)
\(942\) 0 0
\(943\) 3915.77 0.135223
\(944\) −5650.09 −0.194804
\(945\) 0 0
\(946\) 7960.33 0.273586
\(947\) 20268.6 0.695504 0.347752 0.937587i \(-0.386945\pi\)
0.347752 + 0.937587i \(0.386945\pi\)
\(948\) 0 0
\(949\) 2049.92 0.0701193
\(950\) −3965.86 −0.135442
\(951\) 0 0
\(952\) −9516.97 −0.323999
\(953\) 21797.9 0.740925 0.370463 0.928847i \(-0.379199\pi\)
0.370463 + 0.928847i \(0.379199\pi\)
\(954\) 0 0
\(955\) −948.038 −0.0321233
\(956\) 10292.7 0.348212
\(957\) 0 0
\(958\) −12649.6 −0.426608
\(959\) 18576.1 0.625499
\(960\) 0 0
\(961\) 18915.6 0.634945
\(962\) 3389.14 0.113587
\(963\) 0 0
\(964\) −2784.51 −0.0930321
\(965\) 9277.26 0.309477
\(966\) 0 0
\(967\) 48950.6 1.62786 0.813932 0.580960i \(-0.197323\pi\)
0.813932 + 0.580960i \(0.197323\pi\)
\(968\) 8229.62 0.273254
\(969\) 0 0
\(970\) −2497.98 −0.0826860
\(971\) 26426.2 0.873386 0.436693 0.899611i \(-0.356150\pi\)
0.436693 + 0.899611i \(0.356150\pi\)
\(972\) 0 0
\(973\) −24990.5 −0.823389
\(974\) 15746.1 0.518007
\(975\) 0 0
\(976\) −1289.77 −0.0422998
\(977\) −5770.09 −0.188947 −0.0944736 0.995527i \(-0.530117\pi\)
−0.0944736 + 0.995527i \(0.530117\pi\)
\(978\) 0 0
\(979\) 435.320 0.0142113
\(980\) −5285.63 −0.172289
\(981\) 0 0
\(982\) −7113.64 −0.231166
\(983\) 484.532 0.0157214 0.00786071 0.999969i \(-0.497498\pi\)
0.00786071 + 0.999969i \(0.497498\pi\)
\(984\) 0 0
\(985\) 12149.2 0.393002
\(986\) 24549.0 0.792901
\(987\) 0 0
\(988\) 1271.29 0.0409365
\(989\) −5265.16 −0.169285
\(990\) 0 0
\(991\) −26511.5 −0.849813 −0.424907 0.905237i \(-0.639693\pi\)
−0.424907 + 0.905237i \(0.639693\pi\)
\(992\) 7062.27 0.226035
\(993\) 0 0
\(994\) −30303.4 −0.966966
\(995\) −21667.4 −0.690356
\(996\) 0 0
\(997\) −4628.20 −0.147018 −0.0735088 0.997295i \(-0.523420\pi\)
−0.0735088 + 0.997295i \(0.523420\pi\)
\(998\) 3866.74 0.122645
\(999\) 0 0
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 2070.4.a.bg.1.1 4
3.2 odd 2 230.4.a.j.1.4 4
12.11 even 2 1840.4.a.k.1.1 4
15.2 even 4 1150.4.b.o.599.5 8
15.8 even 4 1150.4.b.o.599.4 8
15.14 odd 2 1150.4.a.n.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.j.1.4 4 3.2 odd 2
1150.4.a.n.1.1 4 15.14 odd 2
1150.4.b.o.599.4 8 15.8 even 4
1150.4.b.o.599.5 8 15.2 even 4
1840.4.a.k.1.1 4 12.11 even 2
2070.4.a.bg.1.1 4 1.1 even 1 trivial