Properties

Label 1617.4.a.be.1.6
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

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: yes
Analytic conductor: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.86830\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.86830 q^{2} -3.00000 q^{3} -4.50945 q^{4} +9.74429 q^{5} +5.60490 q^{6} +23.3714 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.86830 q^{2} -3.00000 q^{3} -4.50945 q^{4} +9.74429 q^{5} +5.60490 q^{6} +23.3714 q^{8} +9.00000 q^{9} -18.2053 q^{10} +11.0000 q^{11} +13.5284 q^{12} +20.2969 q^{13} -29.2329 q^{15} -7.58919 q^{16} -15.3274 q^{17} -16.8147 q^{18} -26.0950 q^{19} -43.9414 q^{20} -20.5513 q^{22} +106.341 q^{23} -70.1143 q^{24} -30.0489 q^{25} -37.9208 q^{26} -27.0000 q^{27} -286.202 q^{29} +54.6158 q^{30} -120.333 q^{31} -172.792 q^{32} -33.0000 q^{33} +28.6363 q^{34} -40.5851 q^{36} +132.619 q^{37} +48.7534 q^{38} -60.8908 q^{39} +227.738 q^{40} +130.355 q^{41} +4.42351 q^{43} -49.6040 q^{44} +87.6986 q^{45} -198.676 q^{46} -41.8993 q^{47} +22.7676 q^{48} +56.1404 q^{50} +45.9823 q^{51} -91.5281 q^{52} +80.9450 q^{53} +50.4441 q^{54} +107.187 q^{55} +78.2851 q^{57} +534.710 q^{58} -379.793 q^{59} +131.824 q^{60} +25.0427 q^{61} +224.819 q^{62} +383.542 q^{64} +197.779 q^{65} +61.6539 q^{66} -137.485 q^{67} +69.1184 q^{68} -319.022 q^{69} -1087.84 q^{71} +210.343 q^{72} +885.713 q^{73} -247.772 q^{74} +90.1467 q^{75} +117.674 q^{76} +113.762 q^{78} -65.3091 q^{79} -73.9513 q^{80} +81.0000 q^{81} -243.542 q^{82} +823.837 q^{83} -149.355 q^{85} -8.26444 q^{86} +858.605 q^{87} +257.086 q^{88} +150.752 q^{89} -163.847 q^{90} -479.538 q^{92} +361.000 q^{93} +78.2805 q^{94} -254.278 q^{95} +518.377 q^{96} +53.8790 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86830 −0.660544 −0.330272 0.943886i \(-0.607140\pi\)
−0.330272 + 0.943886i \(0.607140\pi\)
\(3\) −3.00000 −0.577350
\(4\) −4.50945 −0.563682
\(5\) 9.74429 0.871555 0.435778 0.900054i \(-0.356473\pi\)
0.435778 + 0.900054i \(0.356473\pi\)
\(6\) 5.60490 0.381365
\(7\) 0 0
\(8\) 23.3714 1.03288
\(9\) 9.00000 0.333333
\(10\) −18.2053 −0.575701
\(11\) 11.0000 0.301511
\(12\) 13.5284 0.325442
\(13\) 20.2969 0.433027 0.216514 0.976280i \(-0.430531\pi\)
0.216514 + 0.976280i \(0.430531\pi\)
\(14\) 0 0
\(15\) −29.2329 −0.503193
\(16\) −7.58919 −0.118581
\(17\) −15.3274 −0.218673 −0.109337 0.994005i \(-0.534873\pi\)
−0.109337 + 0.994005i \(0.534873\pi\)
\(18\) −16.8147 −0.220181
\(19\) −26.0950 −0.315085 −0.157542 0.987512i \(-0.550357\pi\)
−0.157542 + 0.987512i \(0.550357\pi\)
\(20\) −43.9414 −0.491280
\(21\) 0 0
\(22\) −20.5513 −0.199161
\(23\) 106.341 0.964068 0.482034 0.876153i \(-0.339898\pi\)
0.482034 + 0.876153i \(0.339898\pi\)
\(24\) −70.1143 −0.596334
\(25\) −30.0489 −0.240391
\(26\) −37.9208 −0.286034
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −286.202 −1.83263 −0.916315 0.400457i \(-0.868851\pi\)
−0.916315 + 0.400457i \(0.868851\pi\)
\(30\) 54.6158 0.332381
\(31\) −120.333 −0.697177 −0.348588 0.937276i \(-0.613339\pi\)
−0.348588 + 0.937276i \(0.613339\pi\)
\(32\) −172.792 −0.954552
\(33\) −33.0000 −0.174078
\(34\) 28.6363 0.144443
\(35\) 0 0
\(36\) −40.5851 −0.187894
\(37\) 132.619 0.589254 0.294627 0.955612i \(-0.404805\pi\)
0.294627 + 0.955612i \(0.404805\pi\)
\(38\) 48.7534 0.208127
\(39\) −60.8908 −0.250008
\(40\) 227.738 0.900213
\(41\) 130.355 0.496537 0.248268 0.968691i \(-0.420138\pi\)
0.248268 + 0.968691i \(0.420138\pi\)
\(42\) 0 0
\(43\) 4.42351 0.0156879 0.00784394 0.999969i \(-0.497503\pi\)
0.00784394 + 0.999969i \(0.497503\pi\)
\(44\) −49.6040 −0.169956
\(45\) 87.6986 0.290518
\(46\) −198.676 −0.636809
\(47\) −41.8993 −0.130035 −0.0650175 0.997884i \(-0.520710\pi\)
−0.0650175 + 0.997884i \(0.520710\pi\)
\(48\) 22.7676 0.0684629
\(49\) 0 0
\(50\) 56.1404 0.158789
\(51\) 45.9823 0.126251
\(52\) −91.5281 −0.244090
\(53\) 80.9450 0.209786 0.104893 0.994484i \(-0.466550\pi\)
0.104893 + 0.994484i \(0.466550\pi\)
\(54\) 50.4441 0.127122
\(55\) 107.187 0.262784
\(56\) 0 0
\(57\) 78.2851 0.181914
\(58\) 534.710 1.21053
\(59\) −379.793 −0.838049 −0.419025 0.907975i \(-0.637628\pi\)
−0.419025 + 0.907975i \(0.637628\pi\)
\(60\) 131.824 0.283641
\(61\) 25.0427 0.0525638 0.0262819 0.999655i \(-0.491633\pi\)
0.0262819 + 0.999655i \(0.491633\pi\)
\(62\) 224.819 0.460516
\(63\) 0 0
\(64\) 383.542 0.749105
\(65\) 197.779 0.377407
\(66\) 61.6539 0.114986
\(67\) −137.485 −0.250694 −0.125347 0.992113i \(-0.540004\pi\)
−0.125347 + 0.992113i \(0.540004\pi\)
\(68\) 69.1184 0.123262
\(69\) −319.022 −0.556605
\(70\) 0 0
\(71\) −1087.84 −1.81835 −0.909173 0.416418i \(-0.863285\pi\)
−0.909173 + 0.416418i \(0.863285\pi\)
\(72\) 210.343 0.344293
\(73\) 885.713 1.42007 0.710033 0.704168i \(-0.248680\pi\)
0.710033 + 0.704168i \(0.248680\pi\)
\(74\) −247.772 −0.389228
\(75\) 90.1467 0.138790
\(76\) 117.674 0.177608
\(77\) 0 0
\(78\) 113.762 0.165142
\(79\) −65.3091 −0.0930107 −0.0465054 0.998918i \(-0.514808\pi\)
−0.0465054 + 0.998918i \(0.514808\pi\)
\(80\) −73.9513 −0.103350
\(81\) 81.0000 0.111111
\(82\) −243.542 −0.327984
\(83\) 823.837 1.08949 0.544746 0.838601i \(-0.316626\pi\)
0.544746 + 0.838601i \(0.316626\pi\)
\(84\) 0 0
\(85\) −149.355 −0.190586
\(86\) −8.26444 −0.0103625
\(87\) 858.605 1.05807
\(88\) 257.086 0.311425
\(89\) 150.752 0.179547 0.0897736 0.995962i \(-0.471386\pi\)
0.0897736 + 0.995962i \(0.471386\pi\)
\(90\) −163.847 −0.191900
\(91\) 0 0
\(92\) −479.538 −0.543427
\(93\) 361.000 0.402515
\(94\) 78.2805 0.0858938
\(95\) −254.278 −0.274614
\(96\) 518.377 0.551111
\(97\) 53.8790 0.0563977 0.0281989 0.999602i \(-0.491023\pi\)
0.0281989 + 0.999602i \(0.491023\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 135.504 0.135504
\(101\) −1409.55 −1.38867 −0.694335 0.719652i \(-0.744301\pi\)
−0.694335 + 0.719652i \(0.744301\pi\)
\(102\) −85.9088 −0.0833945
\(103\) 372.287 0.356141 0.178071 0.984018i \(-0.443014\pi\)
0.178071 + 0.984018i \(0.443014\pi\)
\(104\) 474.368 0.447265
\(105\) 0 0
\(106\) −151.230 −0.138573
\(107\) 998.277 0.901935 0.450968 0.892540i \(-0.351079\pi\)
0.450968 + 0.892540i \(0.351079\pi\)
\(108\) 121.755 0.108481
\(109\) 1192.90 1.04825 0.524123 0.851642i \(-0.324393\pi\)
0.524123 + 0.851642i \(0.324393\pi\)
\(110\) −200.258 −0.173580
\(111\) −397.856 −0.340206
\(112\) 0 0
\(113\) −77.2678 −0.0643252 −0.0321626 0.999483i \(-0.510239\pi\)
−0.0321626 + 0.999483i \(0.510239\pi\)
\(114\) −146.260 −0.120162
\(115\) 1036.21 0.840238
\(116\) 1290.61 1.03302
\(117\) 182.672 0.144342
\(118\) 709.568 0.553568
\(119\) 0 0
\(120\) −683.213 −0.519738
\(121\) 121.000 0.0909091
\(122\) −46.7873 −0.0347207
\(123\) −391.065 −0.286676
\(124\) 542.637 0.392986
\(125\) −1510.84 −1.08107
\(126\) 0 0
\(127\) −381.319 −0.266430 −0.133215 0.991087i \(-0.542530\pi\)
−0.133215 + 0.991087i \(0.542530\pi\)
\(128\) 665.768 0.459736
\(129\) −13.2705 −0.00905740
\(130\) −369.511 −0.249294
\(131\) −1102.49 −0.735306 −0.367653 0.929963i \(-0.619839\pi\)
−0.367653 + 0.929963i \(0.619839\pi\)
\(132\) 148.812 0.0981244
\(133\) 0 0
\(134\) 256.864 0.165594
\(135\) −263.096 −0.167731
\(136\) −358.224 −0.225864
\(137\) 1671.10 1.04213 0.521066 0.853517i \(-0.325535\pi\)
0.521066 + 0.853517i \(0.325535\pi\)
\(138\) 596.029 0.367662
\(139\) −323.898 −0.197645 −0.0988225 0.995105i \(-0.531508\pi\)
−0.0988225 + 0.995105i \(0.531508\pi\)
\(140\) 0 0
\(141\) 125.698 0.0750757
\(142\) 2032.41 1.20110
\(143\) 223.266 0.130563
\(144\) −68.3028 −0.0395271
\(145\) −2788.83 −1.59724
\(146\) −1654.78 −0.938016
\(147\) 0 0
\(148\) −598.038 −0.332152
\(149\) 1948.10 1.07110 0.535551 0.844503i \(-0.320104\pi\)
0.535551 + 0.844503i \(0.320104\pi\)
\(150\) −168.421 −0.0916768
\(151\) −1586.19 −0.854848 −0.427424 0.904051i \(-0.640579\pi\)
−0.427424 + 0.904051i \(0.640579\pi\)
\(152\) −609.878 −0.325445
\(153\) −137.947 −0.0728912
\(154\) 0 0
\(155\) −1172.56 −0.607628
\(156\) 274.584 0.140925
\(157\) −595.179 −0.302551 −0.151275 0.988492i \(-0.548338\pi\)
−0.151275 + 0.988492i \(0.548338\pi\)
\(158\) 122.017 0.0614377
\(159\) −242.835 −0.121120
\(160\) −1683.74 −0.831945
\(161\) 0 0
\(162\) −151.332 −0.0733938
\(163\) 1523.75 0.732205 0.366102 0.930575i \(-0.380692\pi\)
0.366102 + 0.930575i \(0.380692\pi\)
\(164\) −587.829 −0.279889
\(165\) −321.561 −0.151718
\(166\) −1539.18 −0.719658
\(167\) −3149.59 −1.45942 −0.729708 0.683759i \(-0.760344\pi\)
−0.729708 + 0.683759i \(0.760344\pi\)
\(168\) 0 0
\(169\) −1785.03 −0.812487
\(170\) 279.040 0.125890
\(171\) −234.855 −0.105028
\(172\) −19.9476 −0.00884297
\(173\) −1552.57 −0.682311 −0.341156 0.940007i \(-0.610818\pi\)
−0.341156 + 0.940007i \(0.610818\pi\)
\(174\) −1604.13 −0.698902
\(175\) 0 0
\(176\) −83.4811 −0.0357536
\(177\) 1139.38 0.483848
\(178\) −281.650 −0.118599
\(179\) 2159.62 0.901774 0.450887 0.892581i \(-0.351108\pi\)
0.450887 + 0.892581i \(0.351108\pi\)
\(180\) −395.473 −0.163760
\(181\) −2184.64 −0.897144 −0.448572 0.893747i \(-0.648067\pi\)
−0.448572 + 0.893747i \(0.648067\pi\)
\(182\) 0 0
\(183\) −75.1282 −0.0303477
\(184\) 2485.33 0.995767
\(185\) 1292.27 0.513567
\(186\) −674.456 −0.265879
\(187\) −168.602 −0.0659325
\(188\) 188.943 0.0732983
\(189\) 0 0
\(190\) 475.067 0.181395
\(191\) 344.648 0.130565 0.0652824 0.997867i \(-0.479205\pi\)
0.0652824 + 0.997867i \(0.479205\pi\)
\(192\) −1150.63 −0.432496
\(193\) −1154.67 −0.430647 −0.215323 0.976543i \(-0.569081\pi\)
−0.215323 + 0.976543i \(0.569081\pi\)
\(194\) −100.662 −0.0372532
\(195\) −593.337 −0.217896
\(196\) 0 0
\(197\) −2121.12 −0.767126 −0.383563 0.923515i \(-0.625303\pi\)
−0.383563 + 0.923515i \(0.625303\pi\)
\(198\) −184.962 −0.0663872
\(199\) 5279.18 1.88056 0.940279 0.340406i \(-0.110564\pi\)
0.940279 + 0.340406i \(0.110564\pi\)
\(200\) −702.285 −0.248295
\(201\) 412.456 0.144738
\(202\) 2633.47 0.917278
\(203\) 0 0
\(204\) −207.355 −0.0711655
\(205\) 1270.22 0.432759
\(206\) −695.544 −0.235247
\(207\) 957.066 0.321356
\(208\) −154.037 −0.0513489
\(209\) −287.045 −0.0950017
\(210\) 0 0
\(211\) 1408.61 0.459587 0.229793 0.973239i \(-0.426195\pi\)
0.229793 + 0.973239i \(0.426195\pi\)
\(212\) −365.018 −0.118252
\(213\) 3263.51 1.04982
\(214\) −1865.08 −0.595768
\(215\) 43.1039 0.0136729
\(216\) −631.028 −0.198778
\(217\) 0 0
\(218\) −2228.69 −0.692413
\(219\) −2657.14 −0.819875
\(220\) −483.355 −0.148126
\(221\) −311.100 −0.0946916
\(222\) 743.315 0.224721
\(223\) −3578.02 −1.07445 −0.537224 0.843439i \(-0.680527\pi\)
−0.537224 + 0.843439i \(0.680527\pi\)
\(224\) 0 0
\(225\) −270.440 −0.0801304
\(226\) 144.359 0.0424896
\(227\) −429.439 −0.125563 −0.0627816 0.998027i \(-0.519997\pi\)
−0.0627816 + 0.998027i \(0.519997\pi\)
\(228\) −353.023 −0.102542
\(229\) −6313.08 −1.82175 −0.910874 0.412685i \(-0.864591\pi\)
−0.910874 + 0.412685i \(0.864591\pi\)
\(230\) −1935.96 −0.555014
\(231\) 0 0
\(232\) −6688.94 −1.89289
\(233\) 193.119 0.0542990 0.0271495 0.999631i \(-0.491357\pi\)
0.0271495 + 0.999631i \(0.491357\pi\)
\(234\) −341.287 −0.0953445
\(235\) −408.279 −0.113333
\(236\) 1712.66 0.472393
\(237\) 195.927 0.0536998
\(238\) 0 0
\(239\) −1899.51 −0.514097 −0.257049 0.966399i \(-0.582750\pi\)
−0.257049 + 0.966399i \(0.582750\pi\)
\(240\) 221.854 0.0596692
\(241\) −4767.31 −1.27423 −0.637115 0.770769i \(-0.719872\pi\)
−0.637115 + 0.770769i \(0.719872\pi\)
\(242\) −226.064 −0.0600494
\(243\) −243.000 −0.0641500
\(244\) −112.929 −0.0296293
\(245\) 0 0
\(246\) 730.626 0.189362
\(247\) −529.649 −0.136440
\(248\) −2812.36 −0.720100
\(249\) −2471.51 −0.629019
\(250\) 2822.70 0.714094
\(251\) −868.810 −0.218481 −0.109241 0.994015i \(-0.534842\pi\)
−0.109241 + 0.994015i \(0.534842\pi\)
\(252\) 0 0
\(253\) 1169.75 0.290677
\(254\) 712.419 0.175989
\(255\) 448.065 0.110035
\(256\) −4312.19 −1.05278
\(257\) −2121.02 −0.514807 −0.257403 0.966304i \(-0.582867\pi\)
−0.257403 + 0.966304i \(0.582867\pi\)
\(258\) 24.7933 0.00598281
\(259\) 0 0
\(260\) −891.876 −0.212738
\(261\) −2575.81 −0.610877
\(262\) 2059.79 0.485702
\(263\) −2095.52 −0.491314 −0.245657 0.969357i \(-0.579004\pi\)
−0.245657 + 0.969357i \(0.579004\pi\)
\(264\) −771.257 −0.179801
\(265\) 788.751 0.182840
\(266\) 0 0
\(267\) −452.257 −0.103662
\(268\) 619.983 0.141312
\(269\) −1833.43 −0.415563 −0.207782 0.978175i \(-0.566624\pi\)
−0.207782 + 0.978175i \(0.566624\pi\)
\(270\) 491.542 0.110794
\(271\) −6698.54 −1.50150 −0.750752 0.660584i \(-0.770309\pi\)
−0.750752 + 0.660584i \(0.770309\pi\)
\(272\) 116.323 0.0259306
\(273\) 0 0
\(274\) −3122.12 −0.688374
\(275\) −330.538 −0.0724807
\(276\) 1438.61 0.313748
\(277\) −7675.02 −1.66479 −0.832396 0.554182i \(-0.813031\pi\)
−0.832396 + 0.554182i \(0.813031\pi\)
\(278\) 605.139 0.130553
\(279\) −1083.00 −0.232392
\(280\) 0 0
\(281\) 6334.43 1.34477 0.672385 0.740202i \(-0.265270\pi\)
0.672385 + 0.740202i \(0.265270\pi\)
\(282\) −234.842 −0.0495908
\(283\) −4424.14 −0.929285 −0.464643 0.885498i \(-0.653817\pi\)
−0.464643 + 0.885498i \(0.653817\pi\)
\(284\) 4905.56 1.02497
\(285\) 762.833 0.158548
\(286\) −417.128 −0.0862424
\(287\) 0 0
\(288\) −1555.13 −0.318184
\(289\) −4678.07 −0.952182
\(290\) 5210.37 1.05505
\(291\) −161.637 −0.0325612
\(292\) −3994.08 −0.800465
\(293\) −4530.68 −0.903363 −0.451681 0.892179i \(-0.649176\pi\)
−0.451681 + 0.892179i \(0.649176\pi\)
\(294\) 0 0
\(295\) −3700.82 −0.730406
\(296\) 3099.49 0.608629
\(297\) −297.000 −0.0580259
\(298\) −3639.63 −0.707510
\(299\) 2158.39 0.417468
\(300\) −406.512 −0.0782333
\(301\) 0 0
\(302\) 2963.47 0.564665
\(303\) 4228.66 0.801749
\(304\) 198.040 0.0373631
\(305\) 244.023 0.0458123
\(306\) 257.726 0.0481478
\(307\) 2305.20 0.428549 0.214274 0.976774i \(-0.431261\pi\)
0.214274 + 0.976774i \(0.431261\pi\)
\(308\) 0 0
\(309\) −1116.86 −0.205618
\(310\) 2190.70 0.401365
\(311\) −6888.07 −1.25591 −0.627953 0.778251i \(-0.716107\pi\)
−0.627953 + 0.778251i \(0.716107\pi\)
\(312\) −1423.10 −0.258229
\(313\) 445.020 0.0803642 0.0401821 0.999192i \(-0.487206\pi\)
0.0401821 + 0.999192i \(0.487206\pi\)
\(314\) 1111.97 0.199848
\(315\) 0 0
\(316\) 294.508 0.0524285
\(317\) 9779.25 1.73267 0.866337 0.499461i \(-0.166468\pi\)
0.866337 + 0.499461i \(0.166468\pi\)
\(318\) 453.689 0.0800050
\(319\) −3148.22 −0.552559
\(320\) 3737.34 0.652886
\(321\) −2994.83 −0.520732
\(322\) 0 0
\(323\) 399.970 0.0689007
\(324\) −365.266 −0.0626313
\(325\) −609.900 −0.104096
\(326\) −2846.82 −0.483654
\(327\) −3578.69 −0.605206
\(328\) 3046.58 0.512863
\(329\) 0 0
\(330\) 600.773 0.100217
\(331\) −6913.76 −1.14808 −0.574040 0.818827i \(-0.694625\pi\)
−0.574040 + 0.818827i \(0.694625\pi\)
\(332\) −3715.06 −0.614127
\(333\) 1193.57 0.196418
\(334\) 5884.38 0.964009
\(335\) −1339.70 −0.218494
\(336\) 0 0
\(337\) 7592.36 1.22725 0.613623 0.789599i \(-0.289711\pi\)
0.613623 + 0.789599i \(0.289711\pi\)
\(338\) 3334.98 0.536684
\(339\) 231.803 0.0371382
\(340\) 673.509 0.107430
\(341\) −1323.67 −0.210207
\(342\) 438.780 0.0693758
\(343\) 0 0
\(344\) 103.384 0.0162037
\(345\) −3108.64 −0.485112
\(346\) 2900.67 0.450697
\(347\) 3489.20 0.539798 0.269899 0.962889i \(-0.413010\pi\)
0.269899 + 0.962889i \(0.413010\pi\)
\(348\) −3871.84 −0.596415
\(349\) 5517.66 0.846286 0.423143 0.906063i \(-0.360927\pi\)
0.423143 + 0.906063i \(0.360927\pi\)
\(350\) 0 0
\(351\) −548.017 −0.0833361
\(352\) −1900.72 −0.287808
\(353\) 4543.49 0.685058 0.342529 0.939507i \(-0.388716\pi\)
0.342529 + 0.939507i \(0.388716\pi\)
\(354\) −2128.70 −0.319603
\(355\) −10600.2 −1.58479
\(356\) −679.810 −0.101208
\(357\) 0 0
\(358\) −4034.82 −0.595662
\(359\) 1814.37 0.266738 0.133369 0.991066i \(-0.457420\pi\)
0.133369 + 0.991066i \(0.457420\pi\)
\(360\) 2049.64 0.300071
\(361\) −6178.05 −0.900721
\(362\) 4081.57 0.592603
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 8630.64 1.23767
\(366\) 140.362 0.0200460
\(367\) −12177.6 −1.73206 −0.866029 0.499994i \(-0.833336\pi\)
−0.866029 + 0.499994i \(0.833336\pi\)
\(368\) −807.040 −0.114320
\(369\) 1173.19 0.165512
\(370\) −2414.36 −0.339234
\(371\) 0 0
\(372\) −1627.91 −0.226890
\(373\) 5051.76 0.701260 0.350630 0.936514i \(-0.385967\pi\)
0.350630 + 0.936514i \(0.385967\pi\)
\(374\) 314.999 0.0435513
\(375\) 4532.52 0.624156
\(376\) −979.247 −0.134311
\(377\) −5809.01 −0.793579
\(378\) 0 0
\(379\) 1113.44 0.150906 0.0754532 0.997149i \(-0.475960\pi\)
0.0754532 + 0.997149i \(0.475960\pi\)
\(380\) 1146.65 0.154795
\(381\) 1143.96 0.153824
\(382\) −643.907 −0.0862438
\(383\) 529.579 0.0706533 0.0353267 0.999376i \(-0.488753\pi\)
0.0353267 + 0.999376i \(0.488753\pi\)
\(384\) −1997.31 −0.265429
\(385\) 0 0
\(386\) 2157.27 0.284461
\(387\) 39.8116 0.00522929
\(388\) −242.965 −0.0317904
\(389\) −1318.67 −0.171875 −0.0859375 0.996301i \(-0.527389\pi\)
−0.0859375 + 0.996301i \(0.527389\pi\)
\(390\) 1108.53 0.143930
\(391\) −1629.93 −0.210816
\(392\) 0 0
\(393\) 3307.47 0.424529
\(394\) 3962.90 0.506720
\(395\) −636.391 −0.0810640
\(396\) −446.436 −0.0566521
\(397\) 292.462 0.0369729 0.0184865 0.999829i \(-0.494115\pi\)
0.0184865 + 0.999829i \(0.494115\pi\)
\(398\) −9863.09 −1.24219
\(399\) 0 0
\(400\) 228.047 0.0285059
\(401\) 14223.6 1.77130 0.885649 0.464356i \(-0.153714\pi\)
0.885649 + 0.464356i \(0.153714\pi\)
\(402\) −770.591 −0.0956059
\(403\) −2442.39 −0.301897
\(404\) 6356.31 0.782768
\(405\) 789.287 0.0968395
\(406\) 0 0
\(407\) 1458.81 0.177667
\(408\) 1074.67 0.130402
\(409\) −4121.13 −0.498232 −0.249116 0.968474i \(-0.580140\pi\)
−0.249116 + 0.968474i \(0.580140\pi\)
\(410\) −2373.14 −0.285857
\(411\) −5013.31 −0.601675
\(412\) −1678.81 −0.200750
\(413\) 0 0
\(414\) −1788.09 −0.212270
\(415\) 8027.70 0.949553
\(416\) −3507.16 −0.413347
\(417\) 971.694 0.114110
\(418\) 536.287 0.0627528
\(419\) −7499.67 −0.874422 −0.437211 0.899359i \(-0.644034\pi\)
−0.437211 + 0.899359i \(0.644034\pi\)
\(420\) 0 0
\(421\) −249.778 −0.0289155 −0.0144577 0.999895i \(-0.504602\pi\)
−0.0144577 + 0.999895i \(0.504602\pi\)
\(422\) −2631.71 −0.303577
\(423\) −377.094 −0.0433450
\(424\) 1891.80 0.216684
\(425\) 460.573 0.0525672
\(426\) −6097.22 −0.693454
\(427\) 0 0
\(428\) −4501.68 −0.508404
\(429\) −669.799 −0.0753804
\(430\) −80.5311 −0.00903152
\(431\) −7821.39 −0.874114 −0.437057 0.899434i \(-0.643979\pi\)
−0.437057 + 0.899434i \(0.643979\pi\)
\(432\) 204.908 0.0228210
\(433\) 2414.70 0.267998 0.133999 0.990981i \(-0.457218\pi\)
0.133999 + 0.990981i \(0.457218\pi\)
\(434\) 0 0
\(435\) 8366.49 0.922166
\(436\) −5379.32 −0.590878
\(437\) −2774.96 −0.303763
\(438\) 4964.33 0.541564
\(439\) −3062.39 −0.332938 −0.166469 0.986047i \(-0.553237\pi\)
−0.166469 + 0.986047i \(0.553237\pi\)
\(440\) 2505.12 0.271424
\(441\) 0 0
\(442\) 581.228 0.0625480
\(443\) −923.369 −0.0990306 −0.0495153 0.998773i \(-0.515768\pi\)
−0.0495153 + 0.998773i \(0.515768\pi\)
\(444\) 1794.11 0.191768
\(445\) 1468.97 0.156485
\(446\) 6684.82 0.709721
\(447\) −5844.29 −0.618401
\(448\) 0 0
\(449\) 13717.4 1.44179 0.720893 0.693046i \(-0.243732\pi\)
0.720893 + 0.693046i \(0.243732\pi\)
\(450\) 505.263 0.0529296
\(451\) 1433.90 0.149712
\(452\) 348.436 0.0362589
\(453\) 4758.56 0.493547
\(454\) 802.320 0.0829400
\(455\) 0 0
\(456\) 1829.63 0.187896
\(457\) −11597.7 −1.18713 −0.593565 0.804786i \(-0.702280\pi\)
−0.593565 + 0.804786i \(0.702280\pi\)
\(458\) 11794.7 1.20334
\(459\) 413.841 0.0420837
\(460\) −4672.76 −0.473627
\(461\) 7553.59 0.763136 0.381568 0.924341i \(-0.375384\pi\)
0.381568 + 0.924341i \(0.375384\pi\)
\(462\) 0 0
\(463\) −3552.83 −0.356618 −0.178309 0.983975i \(-0.557063\pi\)
−0.178309 + 0.983975i \(0.557063\pi\)
\(464\) 2172.04 0.217315
\(465\) 3517.68 0.350814
\(466\) −360.805 −0.0358669
\(467\) 3997.91 0.396148 0.198074 0.980187i \(-0.436531\pi\)
0.198074 + 0.980187i \(0.436531\pi\)
\(468\) −823.753 −0.0813632
\(469\) 0 0
\(470\) 762.788 0.0748612
\(471\) 1785.54 0.174678
\(472\) −8876.31 −0.865604
\(473\) 48.6586 0.00473007
\(474\) −366.051 −0.0354711
\(475\) 784.127 0.0757436
\(476\) 0 0
\(477\) 728.505 0.0699286
\(478\) 3548.86 0.339584
\(479\) −16294.5 −1.55431 −0.777157 0.629307i \(-0.783339\pi\)
−0.777157 + 0.629307i \(0.783339\pi\)
\(480\) 5051.22 0.480324
\(481\) 2691.75 0.255163
\(482\) 8906.76 0.841684
\(483\) 0 0
\(484\) −545.644 −0.0512438
\(485\) 525.012 0.0491537
\(486\) 453.997 0.0423739
\(487\) 3631.92 0.337943 0.168971 0.985621i \(-0.445955\pi\)
0.168971 + 0.985621i \(0.445955\pi\)
\(488\) 585.284 0.0542921
\(489\) −4571.25 −0.422739
\(490\) 0 0
\(491\) −16706.9 −1.53558 −0.767792 0.640699i \(-0.778645\pi\)
−0.767792 + 0.640699i \(0.778645\pi\)
\(492\) 1763.49 0.161594
\(493\) 4386.74 0.400748
\(494\) 989.544 0.0901249
\(495\) 964.684 0.0875946
\(496\) 913.232 0.0826720
\(497\) 0 0
\(498\) 4617.53 0.415494
\(499\) −16741.8 −1.50194 −0.750968 0.660338i \(-0.770413\pi\)
−0.750968 + 0.660338i \(0.770413\pi\)
\(500\) 6813.07 0.609379
\(501\) 9448.77 0.842595
\(502\) 1623.20 0.144317
\(503\) −1763.45 −0.156319 −0.0781594 0.996941i \(-0.524904\pi\)
−0.0781594 + 0.996941i \(0.524904\pi\)
\(504\) 0 0
\(505\) −13735.1 −1.21030
\(506\) −2185.44 −0.192005
\(507\) 5355.10 0.469090
\(508\) 1719.54 0.150182
\(509\) −11700.9 −1.01892 −0.509461 0.860494i \(-0.670155\pi\)
−0.509461 + 0.860494i \(0.670155\pi\)
\(510\) −837.119 −0.0726829
\(511\) 0 0
\(512\) 2730.32 0.235672
\(513\) 704.566 0.0606381
\(514\) 3962.70 0.340053
\(515\) 3627.67 0.310397
\(516\) 59.8428 0.00510549
\(517\) −460.893 −0.0392070
\(518\) 0 0
\(519\) 4657.72 0.393933
\(520\) 4622.38 0.389817
\(521\) −14099.3 −1.18560 −0.592802 0.805348i \(-0.701978\pi\)
−0.592802 + 0.805348i \(0.701978\pi\)
\(522\) 4812.39 0.403511
\(523\) 10857.4 0.907763 0.453882 0.891062i \(-0.350039\pi\)
0.453882 + 0.891062i \(0.350039\pi\)
\(524\) 4971.63 0.414479
\(525\) 0 0
\(526\) 3915.07 0.324534
\(527\) 1844.40 0.152454
\(528\) 250.443 0.0206423
\(529\) −858.667 −0.0705734
\(530\) −1473.62 −0.120774
\(531\) −3418.14 −0.279350
\(532\) 0 0
\(533\) 2645.80 0.215014
\(534\) 844.951 0.0684731
\(535\) 9727.49 0.786086
\(536\) −3213.22 −0.258937
\(537\) −6478.86 −0.520640
\(538\) 3425.41 0.274498
\(539\) 0 0
\(540\) 1186.42 0.0945469
\(541\) −8501.46 −0.675613 −0.337806 0.941216i \(-0.609685\pi\)
−0.337806 + 0.941216i \(0.609685\pi\)
\(542\) 12514.9 0.991809
\(543\) 6553.92 0.517966
\(544\) 2648.47 0.208735
\(545\) 11623.9 0.913605
\(546\) 0 0
\(547\) 4228.35 0.330514 0.165257 0.986251i \(-0.447155\pi\)
0.165257 + 0.986251i \(0.447155\pi\)
\(548\) −7535.76 −0.587430
\(549\) 225.385 0.0175213
\(550\) 617.544 0.0478767
\(551\) 7468.44 0.577434
\(552\) −7455.99 −0.574906
\(553\) 0 0
\(554\) 14339.2 1.09967
\(555\) −3876.82 −0.296508
\(556\) 1460.60 0.111409
\(557\) 16870.9 1.28338 0.641689 0.766965i \(-0.278234\pi\)
0.641689 + 0.766965i \(0.278234\pi\)
\(558\) 2023.37 0.153505
\(559\) 89.7836 0.00679328
\(560\) 0 0
\(561\) 505.805 0.0380662
\(562\) −11834.6 −0.888279
\(563\) −13108.5 −0.981271 −0.490635 0.871365i \(-0.663235\pi\)
−0.490635 + 0.871365i \(0.663235\pi\)
\(564\) −566.829 −0.0423188
\(565\) −752.920 −0.0560629
\(566\) 8265.62 0.613834
\(567\) 0 0
\(568\) −25424.3 −1.87813
\(569\) 5912.77 0.435635 0.217817 0.975990i \(-0.430106\pi\)
0.217817 + 0.975990i \(0.430106\pi\)
\(570\) −1425.20 −0.104728
\(571\) −4877.34 −0.357461 −0.178731 0.983898i \(-0.557199\pi\)
−0.178731 + 0.983898i \(0.557199\pi\)
\(572\) −1006.81 −0.0735958
\(573\) −1033.94 −0.0753816
\(574\) 0 0
\(575\) −3195.42 −0.231753
\(576\) 3451.88 0.249702
\(577\) −25952.7 −1.87248 −0.936242 0.351355i \(-0.885721\pi\)
−0.936242 + 0.351355i \(0.885721\pi\)
\(578\) 8740.04 0.628958
\(579\) 3464.00 0.248634
\(580\) 12576.1 0.900335
\(581\) 0 0
\(582\) 301.986 0.0215081
\(583\) 890.395 0.0632528
\(584\) 20700.4 1.46676
\(585\) 1780.01 0.125802
\(586\) 8464.67 0.596711
\(587\) 4206.57 0.295782 0.147891 0.989004i \(-0.452752\pi\)
0.147891 + 0.989004i \(0.452752\pi\)
\(588\) 0 0
\(589\) 3140.10 0.219670
\(590\) 6914.23 0.482465
\(591\) 6363.37 0.442900
\(592\) −1006.47 −0.0698744
\(593\) 17777.5 1.23109 0.615543 0.788103i \(-0.288937\pi\)
0.615543 + 0.788103i \(0.288937\pi\)
\(594\) 554.885 0.0383286
\(595\) 0 0
\(596\) −8784.85 −0.603761
\(597\) −15837.5 −1.08574
\(598\) −4032.52 −0.275756
\(599\) 15790.9 1.07713 0.538565 0.842584i \(-0.318967\pi\)
0.538565 + 0.842584i \(0.318967\pi\)
\(600\) 2106.86 0.143353
\(601\) 13984.4 0.949144 0.474572 0.880217i \(-0.342603\pi\)
0.474572 + 0.880217i \(0.342603\pi\)
\(602\) 0 0
\(603\) −1237.37 −0.0835646
\(604\) 7152.83 0.481862
\(605\) 1179.06 0.0792323
\(606\) −7900.40 −0.529591
\(607\) −1960.66 −0.131105 −0.0655525 0.997849i \(-0.520881\pi\)
−0.0655525 + 0.997849i \(0.520881\pi\)
\(608\) 4509.03 0.300765
\(609\) 0 0
\(610\) −455.909 −0.0302610
\(611\) −850.428 −0.0563087
\(612\) 622.065 0.0410874
\(613\) 3005.10 0.198001 0.0990006 0.995087i \(-0.468435\pi\)
0.0990006 + 0.995087i \(0.468435\pi\)
\(614\) −4306.80 −0.283075
\(615\) −3810.65 −0.249854
\(616\) 0 0
\(617\) 18241.5 1.19023 0.595117 0.803639i \(-0.297106\pi\)
0.595117 + 0.803639i \(0.297106\pi\)
\(618\) 2086.63 0.135820
\(619\) −16968.2 −1.10179 −0.550897 0.834573i \(-0.685714\pi\)
−0.550897 + 0.834573i \(0.685714\pi\)
\(620\) 5287.61 0.342509
\(621\) −2871.20 −0.185535
\(622\) 12869.0 0.829581
\(623\) 0 0
\(624\) 462.112 0.0296463
\(625\) −10966.0 −0.701821
\(626\) −831.430 −0.0530841
\(627\) 861.136 0.0548492
\(628\) 2683.93 0.170542
\(629\) −2032.71 −0.128854
\(630\) 0 0
\(631\) −5615.52 −0.354279 −0.177140 0.984186i \(-0.556684\pi\)
−0.177140 + 0.984186i \(0.556684\pi\)
\(632\) −1526.37 −0.0960690
\(633\) −4225.83 −0.265343
\(634\) −18270.6 −1.14451
\(635\) −3715.69 −0.232209
\(636\) 1095.05 0.0682731
\(637\) 0 0
\(638\) 5881.81 0.364989
\(639\) −9790.54 −0.606116
\(640\) 6487.44 0.400685
\(641\) 11526.4 0.710241 0.355120 0.934821i \(-0.384440\pi\)
0.355120 + 0.934821i \(0.384440\pi\)
\(642\) 5595.24 0.343967
\(643\) 30977.5 1.89990 0.949949 0.312404i \(-0.101134\pi\)
0.949949 + 0.312404i \(0.101134\pi\)
\(644\) 0 0
\(645\) −129.312 −0.00789402
\(646\) −747.264 −0.0455119
\(647\) −14906.8 −0.905792 −0.452896 0.891563i \(-0.649609\pi\)
−0.452896 + 0.891563i \(0.649609\pi\)
\(648\) 1893.08 0.114764
\(649\) −4177.73 −0.252681
\(650\) 1139.48 0.0687600
\(651\) 0 0
\(652\) −6871.29 −0.412731
\(653\) −27319.8 −1.63722 −0.818612 0.574347i \(-0.805256\pi\)
−0.818612 + 0.574347i \(0.805256\pi\)
\(654\) 6686.07 0.399765
\(655\) −10743.0 −0.640860
\(656\) −989.289 −0.0588799
\(657\) 7971.41 0.473355
\(658\) 0 0
\(659\) 3178.49 0.187885 0.0939427 0.995578i \(-0.470053\pi\)
0.0939427 + 0.995578i \(0.470053\pi\)
\(660\) 1450.07 0.0855208
\(661\) 3304.67 0.194458 0.0972291 0.995262i \(-0.469002\pi\)
0.0972291 + 0.995262i \(0.469002\pi\)
\(662\) 12917.0 0.758357
\(663\) 933.300 0.0546702
\(664\) 19254.2 1.12532
\(665\) 0 0
\(666\) −2229.94 −0.129743
\(667\) −30434.9 −1.76678
\(668\) 14202.9 0.822646
\(669\) 10734.1 0.620333
\(670\) 2502.95 0.144325
\(671\) 275.470 0.0158486
\(672\) 0 0
\(673\) −1150.52 −0.0658982 −0.0329491 0.999457i \(-0.510490\pi\)
−0.0329491 + 0.999457i \(0.510490\pi\)
\(674\) −14184.8 −0.810650
\(675\) 811.320 0.0462633
\(676\) 8049.53 0.457984
\(677\) 22127.5 1.25617 0.628085 0.778144i \(-0.283839\pi\)
0.628085 + 0.778144i \(0.283839\pi\)
\(678\) −433.078 −0.0245314
\(679\) 0 0
\(680\) −3490.64 −0.196853
\(681\) 1288.32 0.0724939
\(682\) 2473.00 0.138851
\(683\) −16339.5 −0.915392 −0.457696 0.889109i \(-0.651325\pi\)
−0.457696 + 0.889109i \(0.651325\pi\)
\(684\) 1059.07 0.0592025
\(685\) 16283.7 0.908275
\(686\) 0 0
\(687\) 18939.2 1.05179
\(688\) −33.5709 −0.00186029
\(689\) 1642.93 0.0908430
\(690\) 5807.87 0.320438
\(691\) −11585.2 −0.637804 −0.318902 0.947788i \(-0.603314\pi\)
−0.318902 + 0.947788i \(0.603314\pi\)
\(692\) 7001.25 0.384606
\(693\) 0 0
\(694\) −6518.86 −0.356560
\(695\) −3156.15 −0.172259
\(696\) 20066.8 1.09286
\(697\) −1998.01 −0.108579
\(698\) −10308.7 −0.559009
\(699\) −579.358 −0.0313495
\(700\) 0 0
\(701\) 22786.2 1.22771 0.613855 0.789419i \(-0.289618\pi\)
0.613855 + 0.789419i \(0.289618\pi\)
\(702\) 1023.86 0.0550472
\(703\) −3460.69 −0.185665
\(704\) 4218.96 0.225864
\(705\) 1224.84 0.0654327
\(706\) −8488.60 −0.452511
\(707\) 0 0
\(708\) −5137.98 −0.272736
\(709\) 1634.00 0.0865529 0.0432764 0.999063i \(-0.486220\pi\)
0.0432764 + 0.999063i \(0.486220\pi\)
\(710\) 19804.4 1.04682
\(711\) −587.782 −0.0310036
\(712\) 3523.29 0.185451
\(713\) −12796.3 −0.672126
\(714\) 0 0
\(715\) 2175.57 0.113793
\(716\) −9738.71 −0.508314
\(717\) 5698.53 0.296814
\(718\) −3389.79 −0.176192
\(719\) −2413.14 −0.125167 −0.0625833 0.998040i \(-0.519934\pi\)
−0.0625833 + 0.998040i \(0.519934\pi\)
\(720\) −665.562 −0.0344500
\(721\) 0 0
\(722\) 11542.5 0.594966
\(723\) 14301.9 0.735677
\(724\) 9851.54 0.505704
\(725\) 8600.04 0.440548
\(726\) 678.193 0.0346696
\(727\) 2446.28 0.124797 0.0623986 0.998051i \(-0.480125\pi\)
0.0623986 + 0.998051i \(0.480125\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −16124.6 −0.817533
\(731\) −67.8010 −0.00343052
\(732\) 338.787 0.0171065
\(733\) −29166.8 −1.46971 −0.734856 0.678223i \(-0.762750\pi\)
−0.734856 + 0.678223i \(0.762750\pi\)
\(734\) 22751.4 1.14410
\(735\) 0 0
\(736\) −18374.9 −0.920253
\(737\) −1512.34 −0.0755871
\(738\) −2191.88 −0.109328
\(739\) −13859.6 −0.689895 −0.344947 0.938622i \(-0.612103\pi\)
−0.344947 + 0.938622i \(0.612103\pi\)
\(740\) −5827.45 −0.289488
\(741\) 1588.95 0.0787739
\(742\) 0 0
\(743\) −32907.2 −1.62483 −0.812415 0.583079i \(-0.801848\pi\)
−0.812415 + 0.583079i \(0.801848\pi\)
\(744\) 8437.07 0.415750
\(745\) 18982.8 0.933525
\(746\) −9438.20 −0.463213
\(747\) 7414.53 0.363164
\(748\) 760.302 0.0371650
\(749\) 0 0
\(750\) −8468.11 −0.412282
\(751\) 31534.5 1.53224 0.766118 0.642700i \(-0.222186\pi\)
0.766118 + 0.642700i \(0.222186\pi\)
\(752\) 317.982 0.0154197
\(753\) 2606.43 0.126140
\(754\) 10853.0 0.524194
\(755\) −15456.3 −0.745047
\(756\) 0 0
\(757\) −21693.1 −1.04155 −0.520773 0.853695i \(-0.674356\pi\)
−0.520773 + 0.853695i \(0.674356\pi\)
\(758\) −2080.24 −0.0996803
\(759\) −3509.24 −0.167823
\(760\) −5942.83 −0.283643
\(761\) 2567.63 0.122308 0.0611541 0.998128i \(-0.480522\pi\)
0.0611541 + 0.998128i \(0.480522\pi\)
\(762\) −2137.26 −0.101607
\(763\) 0 0
\(764\) −1554.18 −0.0735970
\(765\) −1344.19 −0.0635287
\(766\) −989.412 −0.0466696
\(767\) −7708.64 −0.362898
\(768\) 12936.6 0.607823
\(769\) −11645.5 −0.546096 −0.273048 0.962000i \(-0.588032\pi\)
−0.273048 + 0.962000i \(0.588032\pi\)
\(770\) 0 0
\(771\) 6363.05 0.297224
\(772\) 5206.92 0.242748
\(773\) 38312.4 1.78267 0.891333 0.453350i \(-0.149771\pi\)
0.891333 + 0.453350i \(0.149771\pi\)
\(774\) −74.3800 −0.00345418
\(775\) 3615.88 0.167595
\(776\) 1259.23 0.0582521
\(777\) 0 0
\(778\) 2463.68 0.113531
\(779\) −3401.62 −0.156451
\(780\) 2675.63 0.122824
\(781\) −11966.2 −0.548252
\(782\) 3045.20 0.139253
\(783\) 7727.44 0.352690
\(784\) 0 0
\(785\) −5799.59 −0.263690
\(786\) −6179.36 −0.280420
\(787\) −2812.42 −0.127385 −0.0636924 0.997970i \(-0.520288\pi\)
−0.0636924 + 0.997970i \(0.520288\pi\)
\(788\) 9565.11 0.432415
\(789\) 6286.57 0.283660
\(790\) 1188.97 0.0535463
\(791\) 0 0
\(792\) 2313.77 0.103808
\(793\) 508.290 0.0227616
\(794\) −546.407 −0.0244222
\(795\) −2366.25 −0.105563
\(796\) −23806.2 −1.06004
\(797\) −28166.9 −1.25185 −0.625924 0.779884i \(-0.715278\pi\)
−0.625924 + 0.779884i \(0.715278\pi\)
\(798\) 0 0
\(799\) 642.209 0.0284352
\(800\) 5192.22 0.229466
\(801\) 1356.77 0.0598491
\(802\) −26573.9 −1.17002
\(803\) 9742.84 0.428166
\(804\) −1859.95 −0.0815863
\(805\) 0 0
\(806\) 4563.13 0.199416
\(807\) 5500.30 0.239925
\(808\) −32943.2 −1.43433
\(809\) 21316.7 0.926396 0.463198 0.886255i \(-0.346702\pi\)
0.463198 + 0.886255i \(0.346702\pi\)
\(810\) −1474.63 −0.0639667
\(811\) 40821.2 1.76748 0.883739 0.467980i \(-0.155018\pi\)
0.883739 + 0.467980i \(0.155018\pi\)
\(812\) 0 0
\(813\) 20095.6 0.866894
\(814\) −2725.49 −0.117357
\(815\) 14847.9 0.638157
\(816\) −348.969 −0.0149710
\(817\) −115.432 −0.00494301
\(818\) 7699.51 0.329104
\(819\) 0 0
\(820\) −5727.98 −0.243939
\(821\) −4198.28 −0.178466 −0.0892332 0.996011i \(-0.528442\pi\)
−0.0892332 + 0.996011i \(0.528442\pi\)
\(822\) 9366.37 0.397433
\(823\) −2217.32 −0.0939138 −0.0469569 0.998897i \(-0.514952\pi\)
−0.0469569 + 0.998897i \(0.514952\pi\)
\(824\) 8700.88 0.367851
\(825\) 991.614 0.0418467
\(826\) 0 0
\(827\) −13530.1 −0.568909 −0.284454 0.958690i \(-0.591812\pi\)
−0.284454 + 0.958690i \(0.591812\pi\)
\(828\) −4315.84 −0.181142
\(829\) 6331.37 0.265257 0.132628 0.991166i \(-0.457658\pi\)
0.132628 + 0.991166i \(0.457658\pi\)
\(830\) −14998.2 −0.627221
\(831\) 23025.1 0.961168
\(832\) 7784.72 0.324383
\(833\) 0 0
\(834\) −1815.42 −0.0753750
\(835\) −30690.5 −1.27196
\(836\) 1294.42 0.0535507
\(837\) 3249.00 0.134172
\(838\) 14011.6 0.577594
\(839\) 1252.75 0.0515490 0.0257745 0.999668i \(-0.491795\pi\)
0.0257745 + 0.999668i \(0.491795\pi\)
\(840\) 0 0
\(841\) 57522.3 2.35854
\(842\) 466.660 0.0190999
\(843\) −19003.3 −0.776403
\(844\) −6352.07 −0.259061
\(845\) −17393.9 −0.708128
\(846\) 704.525 0.0286313
\(847\) 0 0
\(848\) −614.307 −0.0248766
\(849\) 13272.4 0.536523
\(850\) −860.488 −0.0347229
\(851\) 14102.8 0.568080
\(852\) −14716.7 −0.591766
\(853\) 7295.29 0.292832 0.146416 0.989223i \(-0.453226\pi\)
0.146416 + 0.989223i \(0.453226\pi\)
\(854\) 0 0
\(855\) −2288.50 −0.0915380
\(856\) 23331.1 0.931591
\(857\) −5812.12 −0.231666 −0.115833 0.993269i \(-0.536954\pi\)
−0.115833 + 0.993269i \(0.536954\pi\)
\(858\) 1251.39 0.0497921
\(859\) 36906.8 1.46594 0.732971 0.680260i \(-0.238133\pi\)
0.732971 + 0.680260i \(0.238133\pi\)
\(860\) −194.375 −0.00770714
\(861\) 0 0
\(862\) 14612.7 0.577391
\(863\) −38625.5 −1.52355 −0.761777 0.647840i \(-0.775673\pi\)
−0.761777 + 0.647840i \(0.775673\pi\)
\(864\) 4665.40 0.183704
\(865\) −15128.7 −0.594672
\(866\) −4511.38 −0.177024
\(867\) 14034.2 0.549742
\(868\) 0 0
\(869\) −718.400 −0.0280438
\(870\) −15631.1 −0.609131
\(871\) −2790.53 −0.108557
\(872\) 27879.7 1.08271
\(873\) 484.911 0.0187992
\(874\) 5184.47 0.200649
\(875\) 0 0
\(876\) 11982.2 0.462149
\(877\) 10534.7 0.405622 0.202811 0.979218i \(-0.434992\pi\)
0.202811 + 0.979218i \(0.434992\pi\)
\(878\) 5721.46 0.219920
\(879\) 13592.0 0.521557
\(880\) −813.464 −0.0311612
\(881\) −22503.3 −0.860561 −0.430281 0.902695i \(-0.641585\pi\)
−0.430281 + 0.902695i \(0.641585\pi\)
\(882\) 0 0
\(883\) −20666.2 −0.787626 −0.393813 0.919191i \(-0.628844\pi\)
−0.393813 + 0.919191i \(0.628844\pi\)
\(884\) 1402.89 0.0533759
\(885\) 11102.4 0.421700
\(886\) 1725.13 0.0654141
\(887\) −17557.9 −0.664640 −0.332320 0.943167i \(-0.607831\pi\)
−0.332320 + 0.943167i \(0.607831\pi\)
\(888\) −9298.46 −0.351392
\(889\) 0 0
\(890\) −2744.48 −0.103365
\(891\) 891.000 0.0335013
\(892\) 16134.9 0.605647
\(893\) 1093.36 0.0409721
\(894\) 10918.9 0.408481
\(895\) 21044.0 0.785946
\(896\) 0 0
\(897\) −6475.17 −0.241025
\(898\) −25628.1 −0.952363
\(899\) 34439.5 1.27767
\(900\) 1219.54 0.0451680
\(901\) −1240.68 −0.0458746
\(902\) −2678.96 −0.0988910
\(903\) 0 0
\(904\) −1805.86 −0.0664402
\(905\) −21287.8 −0.781911
\(906\) −8890.42 −0.326009
\(907\) −8826.67 −0.323137 −0.161568 0.986862i \(-0.551655\pi\)
−0.161568 + 0.986862i \(0.551655\pi\)
\(908\) 1936.53 0.0707777
\(909\) −12686.0 −0.462890
\(910\) 0 0
\(911\) 44980.2 1.63585 0.817926 0.575323i \(-0.195124\pi\)
0.817926 + 0.575323i \(0.195124\pi\)
\(912\) −594.121 −0.0215716
\(913\) 9062.21 0.328494
\(914\) 21668.0 0.784152
\(915\) −732.070 −0.0264497
\(916\) 28468.6 1.02689
\(917\) 0 0
\(918\) −773.179 −0.0277982
\(919\) −23795.8 −0.854136 −0.427068 0.904220i \(-0.640453\pi\)
−0.427068 + 0.904220i \(0.640453\pi\)
\(920\) 24217.8 0.867866
\(921\) −6915.59 −0.247423
\(922\) −14112.4 −0.504085
\(923\) −22079.8 −0.787394
\(924\) 0 0
\(925\) −3985.05 −0.141651
\(926\) 6637.76 0.235562
\(927\) 3350.58 0.118714
\(928\) 49453.5 1.74934
\(929\) 7564.29 0.267144 0.133572 0.991039i \(-0.457355\pi\)
0.133572 + 0.991039i \(0.457355\pi\)
\(930\) −6572.09 −0.231728
\(931\) 0 0
\(932\) −870.862 −0.0306073
\(933\) 20664.2 0.725098
\(934\) −7469.30 −0.261673
\(935\) −1642.90 −0.0574639
\(936\) 4269.31 0.149088
\(937\) 38741.6 1.35073 0.675365 0.737484i \(-0.263986\pi\)
0.675365 + 0.737484i \(0.263986\pi\)
\(938\) 0 0
\(939\) −1335.06 −0.0463983
\(940\) 1841.12 0.0638836
\(941\) −13893.4 −0.481308 −0.240654 0.970611i \(-0.577362\pi\)
−0.240654 + 0.970611i \(0.577362\pi\)
\(942\) −3335.92 −0.115382
\(943\) 13862.0 0.478695
\(944\) 2882.33 0.0993768
\(945\) 0 0
\(946\) −90.9088 −0.00312442
\(947\) −27554.3 −0.945508 −0.472754 0.881195i \(-0.656740\pi\)
−0.472754 + 0.881195i \(0.656740\pi\)
\(948\) −883.525 −0.0302696
\(949\) 17977.2 0.614927
\(950\) −1464.99 −0.0500320
\(951\) −29337.8 −1.00036
\(952\) 0 0
\(953\) −14453.6 −0.491288 −0.245644 0.969360i \(-0.578999\pi\)
−0.245644 + 0.969360i \(0.578999\pi\)
\(954\) −1361.07 −0.0461909
\(955\) 3358.35 0.113794
\(956\) 8565.76 0.289787
\(957\) 9444.65 0.319020
\(958\) 30443.1 1.02669
\(959\) 0 0
\(960\) −11212.0 −0.376944
\(961\) −15310.9 −0.513944
\(962\) −5029.00 −0.168546
\(963\) 8984.49 0.300645
\(964\) 21498.0 0.718260
\(965\) −11251.4 −0.375332
\(966\) 0 0
\(967\) 35948.8 1.19549 0.597743 0.801688i \(-0.296064\pi\)
0.597743 + 0.801688i \(0.296064\pi\)
\(968\) 2827.94 0.0938982
\(969\) −1199.91 −0.0397798
\(970\) −980.880 −0.0324682
\(971\) 13802.8 0.456182 0.228091 0.973640i \(-0.426752\pi\)
0.228091 + 0.973640i \(0.426752\pi\)
\(972\) 1095.80 0.0361602
\(973\) 0 0
\(974\) −6785.52 −0.223226
\(975\) 1829.70 0.0600998
\(976\) −190.054 −0.00623308
\(977\) 25188.5 0.824823 0.412411 0.910998i \(-0.364687\pi\)
0.412411 + 0.910998i \(0.364687\pi\)
\(978\) 8540.47 0.279238
\(979\) 1658.27 0.0541355
\(980\) 0 0
\(981\) 10736.1 0.349416
\(982\) 31213.5 1.01432
\(983\) 3378.02 0.109606 0.0548028 0.998497i \(-0.482547\pi\)
0.0548028 + 0.998497i \(0.482547\pi\)
\(984\) −9139.74 −0.296102
\(985\) −20668.8 −0.668593
\(986\) −8195.74 −0.264711
\(987\) 0 0
\(988\) 2388.43 0.0769089
\(989\) 470.399 0.0151242
\(990\) −1802.32 −0.0578601
\(991\) −52396.1 −1.67953 −0.839766 0.542949i \(-0.817308\pi\)
−0.839766 + 0.542949i \(0.817308\pi\)
\(992\) 20792.7 0.665492
\(993\) 20741.3 0.662844
\(994\) 0 0
\(995\) 51441.8 1.63901
\(996\) 11145.2 0.354566
\(997\) −26498.8 −0.841750 −0.420875 0.907119i \(-0.638277\pi\)
−0.420875 + 0.907119i \(0.638277\pi\)
\(998\) 31278.7 0.992095
\(999\) −3580.71 −0.113402
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 1617.4.a.be.1.6 16
7.6 odd 2 1617.4.a.bf.1.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.6 16 1.1 even 1 trivial
1617.4.a.bf.1.6 yes 16 7.6 odd 2