Properties

Label 1617.4.a.be.1.8
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 1.8
Root \(-0.347888\) 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)
 
\(f(q)\) \(=\) \(q-0.347888 q^{2} -3.00000 q^{3} -7.87897 q^{4} -10.0796 q^{5} +1.04366 q^{6} +5.52411 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.347888 q^{2} -3.00000 q^{3} -7.87897 q^{4} -10.0796 q^{5} +1.04366 q^{6} +5.52411 q^{8} +9.00000 q^{9} +3.50656 q^{10} +11.0000 q^{11} +23.6369 q^{12} -57.3186 q^{13} +30.2387 q^{15} +61.1100 q^{16} -75.9020 q^{17} -3.13099 q^{18} -28.3006 q^{19} +79.4166 q^{20} -3.82677 q^{22} +4.21125 q^{23} -16.5723 q^{24} -23.4023 q^{25} +19.9405 q^{26} -27.0000 q^{27} +176.852 q^{29} -10.5197 q^{30} +214.157 q^{31} -65.4523 q^{32} -33.0000 q^{33} +26.4054 q^{34} -70.9108 q^{36} +212.216 q^{37} +9.84544 q^{38} +171.956 q^{39} -55.6806 q^{40} -221.402 q^{41} -30.9339 q^{43} -86.6687 q^{44} -90.7161 q^{45} -1.46504 q^{46} +570.914 q^{47} -183.330 q^{48} +8.14140 q^{50} +227.706 q^{51} +451.612 q^{52} -278.018 q^{53} +9.39298 q^{54} -110.875 q^{55} +84.9018 q^{57} -61.5248 q^{58} +129.884 q^{59} -238.250 q^{60} +496.722 q^{61} -74.5027 q^{62} -466.110 q^{64} +577.747 q^{65} +11.4803 q^{66} +507.308 q^{67} +598.030 q^{68} -12.6337 q^{69} +291.686 q^{71} +49.7170 q^{72} -376.607 q^{73} -73.8275 q^{74} +70.2070 q^{75} +222.980 q^{76} -59.8214 q^{78} +1222.16 q^{79} -615.962 q^{80} +81.0000 q^{81} +77.0231 q^{82} -306.268 q^{83} +765.059 q^{85} +10.7616 q^{86} -530.556 q^{87} +60.7652 q^{88} +16.7779 q^{89} +31.5591 q^{90} -33.1803 q^{92} -642.471 q^{93} -198.614 q^{94} +285.258 q^{95} +196.357 q^{96} +501.813 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.347888 −0.122997 −0.0614985 0.998107i \(-0.519588\pi\)
−0.0614985 + 0.998107i \(0.519588\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.87897 −0.984872
\(5\) −10.0796 −0.901544 −0.450772 0.892639i \(-0.648851\pi\)
−0.450772 + 0.892639i \(0.648851\pi\)
\(6\) 1.04366 0.0710124
\(7\) 0 0
\(8\) 5.52411 0.244133
\(9\) 9.00000 0.333333
\(10\) 3.50656 0.110887
\(11\) 11.0000 0.301511
\(12\) 23.6369 0.568616
\(13\) −57.3186 −1.22287 −0.611436 0.791294i \(-0.709408\pi\)
−0.611436 + 0.791294i \(0.709408\pi\)
\(14\) 0 0
\(15\) 30.2387 0.520507
\(16\) 61.1100 0.954844
\(17\) −75.9020 −1.08288 −0.541439 0.840740i \(-0.682120\pi\)
−0.541439 + 0.840740i \(0.682120\pi\)
\(18\) −3.13099 −0.0409990
\(19\) −28.3006 −0.341716 −0.170858 0.985296i \(-0.554654\pi\)
−0.170858 + 0.985296i \(0.554654\pi\)
\(20\) 79.4166 0.887905
\(21\) 0 0
\(22\) −3.82677 −0.0370850
\(23\) 4.21125 0.0381785 0.0190893 0.999818i \(-0.493923\pi\)
0.0190893 + 0.999818i \(0.493923\pi\)
\(24\) −16.5723 −0.140951
\(25\) −23.4023 −0.187219
\(26\) 19.9405 0.150410
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 176.852 1.13243 0.566217 0.824256i \(-0.308406\pi\)
0.566217 + 0.824256i \(0.308406\pi\)
\(30\) −10.5197 −0.0640208
\(31\) 214.157 1.24077 0.620383 0.784299i \(-0.286977\pi\)
0.620383 + 0.784299i \(0.286977\pi\)
\(32\) −65.4523 −0.361576
\(33\) −33.0000 −0.174078
\(34\) 26.4054 0.133191
\(35\) 0 0
\(36\) −70.9108 −0.328291
\(37\) 212.216 0.942922 0.471461 0.881887i \(-0.343727\pi\)
0.471461 + 0.881887i \(0.343727\pi\)
\(38\) 9.84544 0.0420301
\(39\) 171.956 0.706025
\(40\) −55.6806 −0.220097
\(41\) −221.402 −0.843345 −0.421672 0.906748i \(-0.638557\pi\)
−0.421672 + 0.906748i \(0.638557\pi\)
\(42\) 0 0
\(43\) −30.9339 −0.109707 −0.0548533 0.998494i \(-0.517469\pi\)
−0.0548533 + 0.998494i \(0.517469\pi\)
\(44\) −86.6687 −0.296950
\(45\) −90.7161 −0.300515
\(46\) −1.46504 −0.00469585
\(47\) 570.914 1.77184 0.885919 0.463840i \(-0.153529\pi\)
0.885919 + 0.463840i \(0.153529\pi\)
\(48\) −183.330 −0.551279
\(49\) 0 0
\(50\) 8.14140 0.0230274
\(51\) 227.706 0.625200
\(52\) 451.612 1.20437
\(53\) −278.018 −0.720542 −0.360271 0.932848i \(-0.617316\pi\)
−0.360271 + 0.932848i \(0.617316\pi\)
\(54\) 9.39298 0.0236708
\(55\) −110.875 −0.271826
\(56\) 0 0
\(57\) 84.9018 0.197290
\(58\) −61.5248 −0.139286
\(59\) 129.884 0.286601 0.143300 0.989679i \(-0.454229\pi\)
0.143300 + 0.989679i \(0.454229\pi\)
\(60\) −238.250 −0.512632
\(61\) 496.722 1.04260 0.521301 0.853373i \(-0.325447\pi\)
0.521301 + 0.853373i \(0.325447\pi\)
\(62\) −74.5027 −0.152611
\(63\) 0 0
\(64\) −466.110 −0.910371
\(65\) 577.747 1.10247
\(66\) 11.4803 0.0214110
\(67\) 507.308 0.925038 0.462519 0.886609i \(-0.346946\pi\)
0.462519 + 0.886609i \(0.346946\pi\)
\(68\) 598.030 1.06650
\(69\) −12.6337 −0.0220424
\(70\) 0 0
\(71\) 291.686 0.487560 0.243780 0.969831i \(-0.421613\pi\)
0.243780 + 0.969831i \(0.421613\pi\)
\(72\) 49.7170 0.0813778
\(73\) −376.607 −0.603816 −0.301908 0.953337i \(-0.597624\pi\)
−0.301908 + 0.953337i \(0.597624\pi\)
\(74\) −73.8275 −0.115977
\(75\) 70.2070 0.108091
\(76\) 222.980 0.336546
\(77\) 0 0
\(78\) −59.8214 −0.0868390
\(79\) 1222.16 1.74055 0.870275 0.492567i \(-0.163941\pi\)
0.870275 + 0.492567i \(0.163941\pi\)
\(80\) −615.962 −0.860834
\(81\) 81.0000 0.111111
\(82\) 77.0231 0.103729
\(83\) −306.268 −0.405027 −0.202513 0.979279i \(-0.564911\pi\)
−0.202513 + 0.979279i \(0.564911\pi\)
\(84\) 0 0
\(85\) 765.059 0.976262
\(86\) 10.7616 0.0134936
\(87\) −530.556 −0.653812
\(88\) 60.7652 0.0736090
\(89\) 16.7779 0.0199826 0.00999132 0.999950i \(-0.496820\pi\)
0.00999132 + 0.999950i \(0.496820\pi\)
\(90\) 31.5591 0.0369624
\(91\) 0 0
\(92\) −33.1803 −0.0376009
\(93\) −642.471 −0.716356
\(94\) −198.614 −0.217931
\(95\) 285.258 0.308072
\(96\) 196.357 0.208756
\(97\) 501.813 0.525272 0.262636 0.964895i \(-0.415408\pi\)
0.262636 + 0.964895i \(0.415408\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 184.386 0.184386
\(101\) 898.860 0.885544 0.442772 0.896634i \(-0.353995\pi\)
0.442772 + 0.896634i \(0.353995\pi\)
\(102\) −79.2162 −0.0768978
\(103\) −1775.27 −1.69828 −0.849140 0.528167i \(-0.822879\pi\)
−0.849140 + 0.528167i \(0.822879\pi\)
\(104\) −316.634 −0.298544
\(105\) 0 0
\(106\) 96.7193 0.0886246
\(107\) 2060.09 1.86127 0.930636 0.365945i \(-0.119254\pi\)
0.930636 + 0.365945i \(0.119254\pi\)
\(108\) 212.732 0.189539
\(109\) −1255.52 −1.10327 −0.551636 0.834085i \(-0.685996\pi\)
−0.551636 + 0.834085i \(0.685996\pi\)
\(110\) 38.5722 0.0334338
\(111\) −636.648 −0.544396
\(112\) 0 0
\(113\) −1848.64 −1.53899 −0.769495 0.638653i \(-0.779492\pi\)
−0.769495 + 0.638653i \(0.779492\pi\)
\(114\) −29.5363 −0.0242661
\(115\) −42.4475 −0.0344196
\(116\) −1393.41 −1.11530
\(117\) −515.868 −0.407624
\(118\) −45.1851 −0.0352510
\(119\) 0 0
\(120\) 167.042 0.127073
\(121\) 121.000 0.0909091
\(122\) −172.804 −0.128237
\(123\) 664.205 0.486905
\(124\) −1687.34 −1.22200
\(125\) 1495.83 1.07033
\(126\) 0 0
\(127\) 314.312 0.219612 0.109806 0.993953i \(-0.464977\pi\)
0.109806 + 0.993953i \(0.464977\pi\)
\(128\) 685.773 0.473549
\(129\) 92.8018 0.0633391
\(130\) −200.991 −0.135601
\(131\) −798.517 −0.532571 −0.266285 0.963894i \(-0.585796\pi\)
−0.266285 + 0.963894i \(0.585796\pi\)
\(132\) 260.006 0.171444
\(133\) 0 0
\(134\) −176.487 −0.113777
\(135\) 272.148 0.173502
\(136\) −419.291 −0.264367
\(137\) −1317.28 −0.821479 −0.410739 0.911753i \(-0.634729\pi\)
−0.410739 + 0.911753i \(0.634729\pi\)
\(138\) 4.39513 0.00271115
\(139\) −126.458 −0.0771655 −0.0385827 0.999255i \(-0.512284\pi\)
−0.0385827 + 0.999255i \(0.512284\pi\)
\(140\) 0 0
\(141\) −1712.74 −1.02297
\(142\) −101.474 −0.0599685
\(143\) −630.505 −0.368710
\(144\) 549.990 0.318281
\(145\) −1782.59 −1.02094
\(146\) 131.017 0.0742676
\(147\) 0 0
\(148\) −1672.05 −0.928657
\(149\) −416.554 −0.229030 −0.114515 0.993422i \(-0.536531\pi\)
−0.114515 + 0.993422i \(0.536531\pi\)
\(150\) −24.4242 −0.0132949
\(151\) 2417.60 1.30292 0.651461 0.758682i \(-0.274157\pi\)
0.651461 + 0.758682i \(0.274157\pi\)
\(152\) −156.336 −0.0834243
\(153\) −683.118 −0.360959
\(154\) 0 0
\(155\) −2158.61 −1.11860
\(156\) −1354.84 −0.695344
\(157\) −3256.08 −1.65518 −0.827591 0.561332i \(-0.810289\pi\)
−0.827591 + 0.561332i \(0.810289\pi\)
\(158\) −425.174 −0.214083
\(159\) 834.055 0.416005
\(160\) 659.731 0.325977
\(161\) 0 0
\(162\) −28.1790 −0.0136663
\(163\) 1307.20 0.628145 0.314073 0.949399i \(-0.398306\pi\)
0.314073 + 0.949399i \(0.398306\pi\)
\(164\) 1744.42 0.830586
\(165\) 332.626 0.156939
\(166\) 106.547 0.0498171
\(167\) −2434.25 −1.12795 −0.563975 0.825792i \(-0.690729\pi\)
−0.563975 + 0.825792i \(0.690729\pi\)
\(168\) 0 0
\(169\) 1088.42 0.495414
\(170\) −266.155 −0.120077
\(171\) −254.705 −0.113905
\(172\) 243.728 0.108047
\(173\) 275.909 0.121254 0.0606270 0.998160i \(-0.480690\pi\)
0.0606270 + 0.998160i \(0.480690\pi\)
\(174\) 184.574 0.0804169
\(175\) 0 0
\(176\) 672.210 0.287896
\(177\) −389.651 −0.165469
\(178\) −5.83684 −0.00245781
\(179\) −1834.83 −0.766156 −0.383078 0.923716i \(-0.625136\pi\)
−0.383078 + 0.923716i \(0.625136\pi\)
\(180\) 714.750 0.295968
\(181\) −2893.38 −1.18820 −0.594098 0.804393i \(-0.702491\pi\)
−0.594098 + 0.804393i \(0.702491\pi\)
\(182\) 0 0
\(183\) −1490.16 −0.601946
\(184\) 23.2634 0.00932065
\(185\) −2139.05 −0.850086
\(186\) 223.508 0.0881098
\(187\) −834.922 −0.326500
\(188\) −4498.22 −1.74503
\(189\) 0 0
\(190\) −99.2378 −0.0378919
\(191\) 223.671 0.0847344 0.0423672 0.999102i \(-0.486510\pi\)
0.0423672 + 0.999102i \(0.486510\pi\)
\(192\) 1398.33 0.525603
\(193\) −2403.77 −0.896513 −0.448257 0.893905i \(-0.647955\pi\)
−0.448257 + 0.893905i \(0.647955\pi\)
\(194\) −174.575 −0.0646069
\(195\) −1733.24 −0.636512
\(196\) 0 0
\(197\) −2497.97 −0.903418 −0.451709 0.892165i \(-0.649185\pi\)
−0.451709 + 0.892165i \(0.649185\pi\)
\(198\) −34.4409 −0.0123617
\(199\) −1163.62 −0.414506 −0.207253 0.978287i \(-0.566452\pi\)
−0.207253 + 0.978287i \(0.566452\pi\)
\(200\) −129.277 −0.0457064
\(201\) −1521.92 −0.534071
\(202\) −312.703 −0.108919
\(203\) 0 0
\(204\) −1794.09 −0.615742
\(205\) 2231.63 0.760312
\(206\) 617.597 0.208884
\(207\) 37.9012 0.0127262
\(208\) −3502.74 −1.16765
\(209\) −311.307 −0.103031
\(210\) 0 0
\(211\) −2652.81 −0.865531 −0.432765 0.901507i \(-0.642462\pi\)
−0.432765 + 0.901507i \(0.642462\pi\)
\(212\) 2190.50 0.709642
\(213\) −875.058 −0.281493
\(214\) −716.680 −0.228931
\(215\) 311.801 0.0989052
\(216\) −149.151 −0.0469835
\(217\) 0 0
\(218\) 436.779 0.135699
\(219\) 1129.82 0.348613
\(220\) 873.583 0.267713
\(221\) 4350.60 1.32422
\(222\) 221.483 0.0669592
\(223\) 2993.32 0.898869 0.449434 0.893313i \(-0.351626\pi\)
0.449434 + 0.893313i \(0.351626\pi\)
\(224\) 0 0
\(225\) −210.621 −0.0624062
\(226\) 643.122 0.189291
\(227\) −2722.91 −0.796148 −0.398074 0.917353i \(-0.630321\pi\)
−0.398074 + 0.917353i \(0.630321\pi\)
\(228\) −668.939 −0.194305
\(229\) −5969.62 −1.72264 −0.861318 0.508066i \(-0.830360\pi\)
−0.861318 + 0.508066i \(0.830360\pi\)
\(230\) 14.7670 0.00423351
\(231\) 0 0
\(232\) 976.950 0.276465
\(233\) 2733.48 0.768569 0.384284 0.923215i \(-0.374448\pi\)
0.384284 + 0.923215i \(0.374448\pi\)
\(234\) 179.464 0.0501365
\(235\) −5754.57 −1.59739
\(236\) −1023.35 −0.282265
\(237\) −3666.47 −1.00491
\(238\) 0 0
\(239\) 6063.77 1.64114 0.820570 0.571546i \(-0.193656\pi\)
0.820570 + 0.571546i \(0.193656\pi\)
\(240\) 1847.89 0.497003
\(241\) 4652.12 1.24344 0.621721 0.783239i \(-0.286434\pi\)
0.621721 + 0.783239i \(0.286434\pi\)
\(242\) −42.0945 −0.0111816
\(243\) −243.000 −0.0641500
\(244\) −3913.66 −1.02683
\(245\) 0 0
\(246\) −231.069 −0.0598879
\(247\) 1622.15 0.417874
\(248\) 1183.03 0.302912
\(249\) 918.803 0.233842
\(250\) −520.382 −0.131647
\(251\) −3238.56 −0.814407 −0.407203 0.913337i \(-0.633496\pi\)
−0.407203 + 0.913337i \(0.633496\pi\)
\(252\) 0 0
\(253\) 46.3237 0.0115113
\(254\) −109.345 −0.0270116
\(255\) −2295.18 −0.563645
\(256\) 3490.31 0.852126
\(257\) −165.054 −0.0400615 −0.0200307 0.999799i \(-0.506376\pi\)
−0.0200307 + 0.999799i \(0.506376\pi\)
\(258\) −32.2847 −0.00779052
\(259\) 0 0
\(260\) −4552.05 −1.08579
\(261\) 1591.67 0.377478
\(262\) 277.795 0.0655046
\(263\) 3216.42 0.754117 0.377058 0.926189i \(-0.376936\pi\)
0.377058 + 0.926189i \(0.376936\pi\)
\(264\) −182.296 −0.0424982
\(265\) 2802.30 0.649601
\(266\) 0 0
\(267\) −50.3337 −0.0115370
\(268\) −3997.07 −0.911044
\(269\) 5036.31 1.14152 0.570760 0.821117i \(-0.306649\pi\)
0.570760 + 0.821117i \(0.306649\pi\)
\(270\) −94.6772 −0.0213403
\(271\) −1757.16 −0.393874 −0.196937 0.980416i \(-0.563099\pi\)
−0.196937 + 0.980416i \(0.563099\pi\)
\(272\) −4638.37 −1.03398
\(273\) 0 0
\(274\) 458.265 0.101039
\(275\) −257.426 −0.0564486
\(276\) 99.5409 0.0217089
\(277\) −5141.78 −1.11531 −0.557653 0.830074i \(-0.688298\pi\)
−0.557653 + 0.830074i \(0.688298\pi\)
\(278\) 43.9931 0.00949113
\(279\) 1927.41 0.413589
\(280\) 0 0
\(281\) 188.028 0.0399176 0.0199588 0.999801i \(-0.493646\pi\)
0.0199588 + 0.999801i \(0.493646\pi\)
\(282\) 595.843 0.125822
\(283\) 4720.29 0.991492 0.495746 0.868468i \(-0.334895\pi\)
0.495746 + 0.868468i \(0.334895\pi\)
\(284\) −2298.19 −0.480184
\(285\) −855.773 −0.177865
\(286\) 219.345 0.0453502
\(287\) 0 0
\(288\) −589.071 −0.120525
\(289\) 848.110 0.172626
\(290\) 620.143 0.125573
\(291\) −1505.44 −0.303266
\(292\) 2967.28 0.594681
\(293\) 733.866 0.146324 0.0731620 0.997320i \(-0.476691\pi\)
0.0731620 + 0.997320i \(0.476691\pi\)
\(294\) 0 0
\(295\) −1309.17 −0.258383
\(296\) 1172.30 0.230199
\(297\) −297.000 −0.0580259
\(298\) 144.914 0.0281700
\(299\) −241.383 −0.0466874
\(300\) −553.159 −0.106456
\(301\) 0 0
\(302\) −841.053 −0.160256
\(303\) −2696.58 −0.511269
\(304\) −1729.45 −0.326285
\(305\) −5006.74 −0.939951
\(306\) 237.649 0.0443970
\(307\) −3178.13 −0.590831 −0.295416 0.955369i \(-0.595458\pi\)
−0.295416 + 0.955369i \(0.595458\pi\)
\(308\) 0 0
\(309\) 5325.82 0.980503
\(310\) 750.955 0.137585
\(311\) 386.879 0.0705398 0.0352699 0.999378i \(-0.488771\pi\)
0.0352699 + 0.999378i \(0.488771\pi\)
\(312\) 949.903 0.172364
\(313\) −4179.63 −0.754781 −0.377391 0.926054i \(-0.623179\pi\)
−0.377391 + 0.926054i \(0.623179\pi\)
\(314\) 1132.75 0.203583
\(315\) 0 0
\(316\) −9629.34 −1.71422
\(317\) 355.365 0.0629631 0.0314816 0.999504i \(-0.489977\pi\)
0.0314816 + 0.999504i \(0.489977\pi\)
\(318\) −290.158 −0.0511674
\(319\) 1945.37 0.341442
\(320\) 4698.19 0.820739
\(321\) −6180.26 −1.07461
\(322\) 0 0
\(323\) 2148.07 0.370037
\(324\) −638.197 −0.109430
\(325\) 1341.39 0.228944
\(326\) −454.759 −0.0772600
\(327\) 3766.55 0.636974
\(328\) −1223.05 −0.205889
\(329\) 0 0
\(330\) −115.717 −0.0193030
\(331\) 10564.8 1.75436 0.877178 0.480165i \(-0.159423\pi\)
0.877178 + 0.480165i \(0.159423\pi\)
\(332\) 2413.07 0.398900
\(333\) 1909.95 0.314307
\(334\) 846.846 0.138735
\(335\) −5113.45 −0.833963
\(336\) 0 0
\(337\) −4615.35 −0.746036 −0.373018 0.927824i \(-0.621677\pi\)
−0.373018 + 0.927824i \(0.621677\pi\)
\(338\) −378.650 −0.0609345
\(339\) 5545.93 0.888536
\(340\) −6027.88 −0.961493
\(341\) 2355.73 0.374105
\(342\) 88.6090 0.0140100
\(343\) 0 0
\(344\) −170.882 −0.0267830
\(345\) 127.343 0.0198722
\(346\) −95.9854 −0.0149139
\(347\) 4346.07 0.672361 0.336181 0.941798i \(-0.390865\pi\)
0.336181 + 0.941798i \(0.390865\pi\)
\(348\) 4180.24 0.643921
\(349\) 846.492 0.129833 0.0649164 0.997891i \(-0.479322\pi\)
0.0649164 + 0.997891i \(0.479322\pi\)
\(350\) 0 0
\(351\) 1547.60 0.235342
\(352\) −719.976 −0.109019
\(353\) −6847.40 −1.03244 −0.516218 0.856457i \(-0.672661\pi\)
−0.516218 + 0.856457i \(0.672661\pi\)
\(354\) 135.555 0.0203522
\(355\) −2940.07 −0.439557
\(356\) −132.193 −0.0196803
\(357\) 0 0
\(358\) 638.317 0.0942349
\(359\) 10573.8 1.55450 0.777251 0.629191i \(-0.216614\pi\)
0.777251 + 0.629191i \(0.216614\pi\)
\(360\) −501.126 −0.0733657
\(361\) −6058.08 −0.883230
\(362\) 1006.57 0.146145
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 3796.04 0.544366
\(366\) 518.411 0.0740376
\(367\) −6295.05 −0.895365 −0.447683 0.894193i \(-0.647751\pi\)
−0.447683 + 0.894193i \(0.647751\pi\)
\(368\) 257.349 0.0364545
\(369\) −1992.62 −0.281115
\(370\) 744.149 0.104558
\(371\) 0 0
\(372\) 5062.01 0.705519
\(373\) 1381.14 0.191723 0.0958617 0.995395i \(-0.469439\pi\)
0.0958617 + 0.995395i \(0.469439\pi\)
\(374\) 290.460 0.0401586
\(375\) −4487.49 −0.617955
\(376\) 3153.79 0.432565
\(377\) −10136.9 −1.38482
\(378\) 0 0
\(379\) 4860.66 0.658774 0.329387 0.944195i \(-0.393158\pi\)
0.329387 + 0.944195i \(0.393158\pi\)
\(380\) −2247.54 −0.303411
\(381\) −942.936 −0.126793
\(382\) −77.8125 −0.0104221
\(383\) −6167.56 −0.822839 −0.411420 0.911446i \(-0.634967\pi\)
−0.411420 + 0.911446i \(0.634967\pi\)
\(384\) −2057.32 −0.273404
\(385\) 0 0
\(386\) 836.243 0.110269
\(387\) −278.405 −0.0365688
\(388\) −3953.77 −0.517325
\(389\) 5637.12 0.734739 0.367370 0.930075i \(-0.380258\pi\)
0.367370 + 0.930075i \(0.380258\pi\)
\(390\) 602.974 0.0782892
\(391\) −319.642 −0.0413427
\(392\) 0 0
\(393\) 2395.55 0.307480
\(394\) 869.016 0.111118
\(395\) −12318.8 −1.56918
\(396\) −780.018 −0.0989833
\(397\) 9281.83 1.17341 0.586703 0.809803i \(-0.300426\pi\)
0.586703 + 0.809803i \(0.300426\pi\)
\(398\) 404.809 0.0509830
\(399\) 0 0
\(400\) −1430.12 −0.178765
\(401\) −5809.36 −0.723455 −0.361727 0.932284i \(-0.617813\pi\)
−0.361727 + 0.932284i \(0.617813\pi\)
\(402\) 529.460 0.0656892
\(403\) −12275.2 −1.51730
\(404\) −7082.10 −0.872147
\(405\) −816.445 −0.100172
\(406\) 0 0
\(407\) 2334.38 0.284302
\(408\) 1257.87 0.152632
\(409\) −5878.30 −0.710669 −0.355334 0.934739i \(-0.615633\pi\)
−0.355334 + 0.934739i \(0.615633\pi\)
\(410\) −776.359 −0.0935162
\(411\) 3951.83 0.474281
\(412\) 13987.3 1.67259
\(413\) 0 0
\(414\) −13.1854 −0.00156528
\(415\) 3087.04 0.365150
\(416\) 3751.64 0.442161
\(417\) 379.373 0.0445515
\(418\) 108.300 0.0126725
\(419\) 8934.42 1.04171 0.520853 0.853646i \(-0.325614\pi\)
0.520853 + 0.853646i \(0.325614\pi\)
\(420\) 0 0
\(421\) 8004.64 0.926656 0.463328 0.886187i \(-0.346655\pi\)
0.463328 + 0.886187i \(0.346655\pi\)
\(422\) 922.882 0.106458
\(423\) 5138.23 0.590613
\(424\) −1535.80 −0.175908
\(425\) 1776.28 0.202735
\(426\) 304.422 0.0346228
\(427\) 0 0
\(428\) −16231.4 −1.83311
\(429\) 1891.51 0.212875
\(430\) −108.472 −0.0121651
\(431\) −12239.7 −1.36791 −0.683953 0.729526i \(-0.739741\pi\)
−0.683953 + 0.729526i \(0.739741\pi\)
\(432\) −1649.97 −0.183760
\(433\) 8347.05 0.926405 0.463203 0.886252i \(-0.346700\pi\)
0.463203 + 0.886252i \(0.346700\pi\)
\(434\) 0 0
\(435\) 5347.78 0.589440
\(436\) 9892.18 1.08658
\(437\) −119.181 −0.0130462
\(438\) −393.052 −0.0428784
\(439\) 9684.79 1.05292 0.526458 0.850201i \(-0.323520\pi\)
0.526458 + 0.850201i \(0.323520\pi\)
\(440\) −612.487 −0.0663617
\(441\) 0 0
\(442\) −1513.52 −0.162875
\(443\) 5765.29 0.618323 0.309162 0.951009i \(-0.399952\pi\)
0.309162 + 0.951009i \(0.399952\pi\)
\(444\) 5016.14 0.536160
\(445\) −169.114 −0.0180152
\(446\) −1041.34 −0.110558
\(447\) 1249.66 0.132230
\(448\) 0 0
\(449\) −5179.91 −0.544443 −0.272222 0.962235i \(-0.587758\pi\)
−0.272222 + 0.962235i \(0.587758\pi\)
\(450\) 73.2726 0.00767579
\(451\) −2435.42 −0.254278
\(452\) 14565.4 1.51571
\(453\) −7252.79 −0.752242
\(454\) 947.268 0.0979239
\(455\) 0 0
\(456\) 469.007 0.0481650
\(457\) −18949.6 −1.93966 −0.969832 0.243773i \(-0.921615\pi\)
−0.969832 + 0.243773i \(0.921615\pi\)
\(458\) 2076.76 0.211879
\(459\) 2049.35 0.208400
\(460\) 334.443 0.0338989
\(461\) −19026.5 −1.92224 −0.961121 0.276126i \(-0.910949\pi\)
−0.961121 + 0.276126i \(0.910949\pi\)
\(462\) 0 0
\(463\) −10895.6 −1.09365 −0.546827 0.837246i \(-0.684164\pi\)
−0.546827 + 0.837246i \(0.684164\pi\)
\(464\) 10807.4 1.08130
\(465\) 6475.83 0.645827
\(466\) −950.947 −0.0945317
\(467\) −15831.9 −1.56876 −0.784381 0.620280i \(-0.787019\pi\)
−0.784381 + 0.620280i \(0.787019\pi\)
\(468\) 4064.51 0.401457
\(469\) 0 0
\(470\) 2001.95 0.196474
\(471\) 9768.24 0.955620
\(472\) 717.492 0.0699688
\(473\) −340.273 −0.0330778
\(474\) 1275.52 0.123601
\(475\) 662.300 0.0639756
\(476\) 0 0
\(477\) −2502.16 −0.240181
\(478\) −2109.51 −0.201856
\(479\) 14727.9 1.40487 0.702437 0.711746i \(-0.252095\pi\)
0.702437 + 0.711746i \(0.252095\pi\)
\(480\) −1979.19 −0.188203
\(481\) −12163.9 −1.15307
\(482\) −1618.42 −0.152940
\(483\) 0 0
\(484\) −953.356 −0.0895338
\(485\) −5058.05 −0.473556
\(486\) 84.5369 0.00789027
\(487\) −1200.48 −0.111702 −0.0558509 0.998439i \(-0.517787\pi\)
−0.0558509 + 0.998439i \(0.517787\pi\)
\(488\) 2743.94 0.254534
\(489\) −3921.59 −0.362660
\(490\) 0 0
\(491\) 18870.7 1.73446 0.867232 0.497904i \(-0.165897\pi\)
0.867232 + 0.497904i \(0.165897\pi\)
\(492\) −5233.25 −0.479539
\(493\) −13423.4 −1.22629
\(494\) −564.327 −0.0513973
\(495\) −997.877 −0.0906086
\(496\) 13087.1 1.18474
\(497\) 0 0
\(498\) −319.641 −0.0287619
\(499\) −13583.8 −1.21863 −0.609315 0.792929i \(-0.708555\pi\)
−0.609315 + 0.792929i \(0.708555\pi\)
\(500\) −11785.6 −1.05414
\(501\) 7302.74 0.651222
\(502\) 1126.66 0.100170
\(503\) 16992.7 1.50629 0.753146 0.657853i \(-0.228535\pi\)
0.753146 + 0.657853i \(0.228535\pi\)
\(504\) 0 0
\(505\) −9060.12 −0.798357
\(506\) −16.1155 −0.00141585
\(507\) −3265.27 −0.286027
\(508\) −2476.46 −0.216289
\(509\) 15348.3 1.33654 0.668270 0.743918i \(-0.267035\pi\)
0.668270 + 0.743918i \(0.267035\pi\)
\(510\) 798.465 0.0693267
\(511\) 0 0
\(512\) −6700.42 −0.578358
\(513\) 764.116 0.0657632
\(514\) 57.4204 0.00492745
\(515\) 17894.0 1.53107
\(516\) −731.183 −0.0623809
\(517\) 6280.05 0.534229
\(518\) 0 0
\(519\) −827.726 −0.0700060
\(520\) 3191.54 0.269150
\(521\) 680.526 0.0572253 0.0286127 0.999591i \(-0.490891\pi\)
0.0286127 + 0.999591i \(0.490891\pi\)
\(522\) −553.723 −0.0464287
\(523\) −12211.6 −1.02099 −0.510494 0.859881i \(-0.670537\pi\)
−0.510494 + 0.859881i \(0.670537\pi\)
\(524\) 6291.49 0.524514
\(525\) 0 0
\(526\) −1118.95 −0.0927542
\(527\) −16254.9 −1.34360
\(528\) −2016.63 −0.166217
\(529\) −12149.3 −0.998542
\(530\) −974.889 −0.0798990
\(531\) 1168.95 0.0955335
\(532\) 0 0
\(533\) 12690.4 1.03130
\(534\) 17.5105 0.00141902
\(535\) −20764.8 −1.67802
\(536\) 2802.43 0.225833
\(537\) 5504.50 0.442340
\(538\) −1752.07 −0.140404
\(539\) 0 0
\(540\) −2144.25 −0.170877
\(541\) −3227.43 −0.256484 −0.128242 0.991743i \(-0.540933\pi\)
−0.128242 + 0.991743i \(0.540933\pi\)
\(542\) 611.295 0.0484453
\(543\) 8680.14 0.686005
\(544\) 4967.96 0.391543
\(545\) 12655.1 0.994648
\(546\) 0 0
\(547\) −14060.4 −1.09905 −0.549525 0.835478i \(-0.685191\pi\)
−0.549525 + 0.835478i \(0.685191\pi\)
\(548\) 10378.8 0.809051
\(549\) 4470.49 0.347534
\(550\) 89.5554 0.00694301
\(551\) −5005.02 −0.386971
\(552\) −69.7902 −0.00538128
\(553\) 0 0
\(554\) 1788.77 0.137179
\(555\) 6417.14 0.490797
\(556\) 996.357 0.0759981
\(557\) 19281.7 1.46677 0.733387 0.679811i \(-0.237938\pi\)
0.733387 + 0.679811i \(0.237938\pi\)
\(558\) −670.525 −0.0508702
\(559\) 1773.09 0.134157
\(560\) 0 0
\(561\) 2504.77 0.188505
\(562\) −65.4129 −0.00490975
\(563\) −9162.10 −0.685856 −0.342928 0.939362i \(-0.611419\pi\)
−0.342928 + 0.939362i \(0.611419\pi\)
\(564\) 13494.7 1.00750
\(565\) 18633.5 1.38747
\(566\) −1642.13 −0.121951
\(567\) 0 0
\(568\) 1611.31 0.119030
\(569\) −7422.13 −0.546840 −0.273420 0.961895i \(-0.588155\pi\)
−0.273420 + 0.961895i \(0.588155\pi\)
\(570\) 297.713 0.0218769
\(571\) −9768.97 −0.715970 −0.357985 0.933727i \(-0.616536\pi\)
−0.357985 + 0.933727i \(0.616536\pi\)
\(572\) 4967.73 0.363132
\(573\) −671.013 −0.0489214
\(574\) 0 0
\(575\) −98.5531 −0.00714773
\(576\) −4194.99 −0.303457
\(577\) −1112.36 −0.0802570 −0.0401285 0.999195i \(-0.512777\pi\)
−0.0401285 + 0.999195i \(0.512777\pi\)
\(578\) −295.047 −0.0212325
\(579\) 7211.31 0.517602
\(580\) 14045.0 1.00549
\(581\) 0 0
\(582\) 523.724 0.0373008
\(583\) −3058.20 −0.217252
\(584\) −2080.42 −0.147412
\(585\) 5199.72 0.367491
\(586\) −255.303 −0.0179974
\(587\) 11714.8 0.823719 0.411860 0.911247i \(-0.364879\pi\)
0.411860 + 0.911247i \(0.364879\pi\)
\(588\) 0 0
\(589\) −6060.77 −0.423989
\(590\) 455.446 0.0317804
\(591\) 7493.92 0.521588
\(592\) 12968.5 0.900343
\(593\) −4776.42 −0.330766 −0.165383 0.986229i \(-0.552886\pi\)
−0.165383 + 0.986229i \(0.552886\pi\)
\(594\) 103.323 0.00713701
\(595\) 0 0
\(596\) 3282.02 0.225565
\(597\) 3490.85 0.239315
\(598\) 83.9743 0.00574241
\(599\) 8522.36 0.581326 0.290663 0.956825i \(-0.406124\pi\)
0.290663 + 0.956825i \(0.406124\pi\)
\(600\) 387.831 0.0263886
\(601\) −18817.2 −1.27715 −0.638576 0.769559i \(-0.720476\pi\)
−0.638576 + 0.769559i \(0.720476\pi\)
\(602\) 0 0
\(603\) 4565.77 0.308346
\(604\) −19048.2 −1.28321
\(605\) −1219.63 −0.0819585
\(606\) 938.109 0.0628846
\(607\) −13226.6 −0.884434 −0.442217 0.896908i \(-0.645808\pi\)
−0.442217 + 0.896908i \(0.645808\pi\)
\(608\) 1852.34 0.123556
\(609\) 0 0
\(610\) 1741.79 0.115611
\(611\) −32724.0 −2.16673
\(612\) 5382.27 0.355499
\(613\) −15169.6 −0.999500 −0.499750 0.866170i \(-0.666575\pi\)
−0.499750 + 0.866170i \(0.666575\pi\)
\(614\) 1105.63 0.0726705
\(615\) −6694.90 −0.438967
\(616\) 0 0
\(617\) 17958.9 1.17179 0.585896 0.810386i \(-0.300743\pi\)
0.585896 + 0.810386i \(0.300743\pi\)
\(618\) −1852.79 −0.120599
\(619\) −2029.16 −0.131759 −0.0658794 0.997828i \(-0.520985\pi\)
−0.0658794 + 0.997828i \(0.520985\pi\)
\(620\) 17007.6 1.10168
\(621\) −113.704 −0.00734746
\(622\) −134.591 −0.00867619
\(623\) 0 0
\(624\) 10508.2 0.674144
\(625\) −12152.0 −0.777730
\(626\) 1454.04 0.0928359
\(627\) 933.920 0.0594851
\(628\) 25654.6 1.63014
\(629\) −16107.6 −1.02107
\(630\) 0 0
\(631\) 14750.2 0.930581 0.465290 0.885158i \(-0.345950\pi\)
0.465290 + 0.885158i \(0.345950\pi\)
\(632\) 6751.33 0.424926
\(633\) 7958.43 0.499714
\(634\) −123.627 −0.00774428
\(635\) −3168.13 −0.197990
\(636\) −6571.50 −0.409712
\(637\) 0 0
\(638\) −676.773 −0.0419964
\(639\) 2625.17 0.162520
\(640\) −6912.29 −0.426926
\(641\) 5087.09 0.313460 0.156730 0.987641i \(-0.449905\pi\)
0.156730 + 0.987641i \(0.449905\pi\)
\(642\) 2150.04 0.132173
\(643\) −4855.47 −0.297793 −0.148897 0.988853i \(-0.547572\pi\)
−0.148897 + 0.988853i \(0.547572\pi\)
\(644\) 0 0
\(645\) −935.402 −0.0571030
\(646\) −747.289 −0.0455134
\(647\) 6395.76 0.388629 0.194315 0.980939i \(-0.437752\pi\)
0.194315 + 0.980939i \(0.437752\pi\)
\(648\) 447.453 0.0271259
\(649\) 1428.72 0.0864133
\(650\) −466.654 −0.0281595
\(651\) 0 0
\(652\) −10299.4 −0.618642
\(653\) −15360.1 −0.920500 −0.460250 0.887789i \(-0.652240\pi\)
−0.460250 + 0.887789i \(0.652240\pi\)
\(654\) −1310.34 −0.0783460
\(655\) 8048.70 0.480136
\(656\) −13529.9 −0.805263
\(657\) −3389.47 −0.201272
\(658\) 0 0
\(659\) −2697.45 −0.159450 −0.0797252 0.996817i \(-0.525404\pi\)
−0.0797252 + 0.996817i \(0.525404\pi\)
\(660\) −2620.75 −0.154564
\(661\) 1290.65 0.0759464 0.0379732 0.999279i \(-0.487910\pi\)
0.0379732 + 0.999279i \(0.487910\pi\)
\(662\) −3675.36 −0.215781
\(663\) −13051.8 −0.764539
\(664\) −1691.86 −0.0988806
\(665\) 0 0
\(666\) −664.448 −0.0386589
\(667\) 744.768 0.0432347
\(668\) 19179.4 1.11089
\(669\) −8979.97 −0.518962
\(670\) 1778.91 0.102575
\(671\) 5463.94 0.314356
\(672\) 0 0
\(673\) −24211.0 −1.38673 −0.693363 0.720589i \(-0.743872\pi\)
−0.693363 + 0.720589i \(0.743872\pi\)
\(674\) 1605.63 0.0917603
\(675\) 631.863 0.0360303
\(676\) −8575.66 −0.487919
\(677\) 23924.5 1.35819 0.679094 0.734051i \(-0.262373\pi\)
0.679094 + 0.734051i \(0.262373\pi\)
\(678\) −1929.37 −0.109287
\(679\) 0 0
\(680\) 4226.27 0.238338
\(681\) 8168.72 0.459657
\(682\) −819.530 −0.0460138
\(683\) 7909.54 0.443119 0.221559 0.975147i \(-0.428885\pi\)
0.221559 + 0.975147i \(0.428885\pi\)
\(684\) 2006.82 0.112182
\(685\) 13277.6 0.740599
\(686\) 0 0
\(687\) 17908.9 0.994564
\(688\) −1890.37 −0.104753
\(689\) 15935.6 0.881130
\(690\) −44.3010 −0.00244422
\(691\) −31763.7 −1.74870 −0.874348 0.485299i \(-0.838711\pi\)
−0.874348 + 0.485299i \(0.838711\pi\)
\(692\) −2173.88 −0.119420
\(693\) 0 0
\(694\) −1511.95 −0.0826985
\(695\) 1274.64 0.0695681
\(696\) −2930.85 −0.159617
\(697\) 16804.8 0.913240
\(698\) −294.485 −0.0159691
\(699\) −8200.45 −0.443733
\(700\) 0 0
\(701\) −772.079 −0.0415992 −0.0207996 0.999784i \(-0.506621\pi\)
−0.0207996 + 0.999784i \(0.506621\pi\)
\(702\) −538.393 −0.0289463
\(703\) −6005.84 −0.322211
\(704\) −5127.21 −0.274487
\(705\) 17263.7 0.922253
\(706\) 2382.13 0.126987
\(707\) 0 0
\(708\) 3070.05 0.162966
\(709\) −35890.8 −1.90114 −0.950569 0.310512i \(-0.899499\pi\)
−0.950569 + 0.310512i \(0.899499\pi\)
\(710\) 1022.82 0.0540642
\(711\) 10999.4 0.580183
\(712\) 92.6830 0.00487843
\(713\) 901.868 0.0473706
\(714\) 0 0
\(715\) 6355.22 0.332408
\(716\) 14456.6 0.754565
\(717\) −18191.3 −0.947513
\(718\) −3678.52 −0.191199
\(719\) 15491.6 0.803530 0.401765 0.915743i \(-0.368397\pi\)
0.401765 + 0.915743i \(0.368397\pi\)
\(720\) −5543.66 −0.286945
\(721\) 0 0
\(722\) 2107.53 0.108635
\(723\) −13956.4 −0.717901
\(724\) 22796.9 1.17022
\(725\) −4138.75 −0.212013
\(726\) 126.283 0.00645567
\(727\) 38315.5 1.95467 0.977333 0.211706i \(-0.0679020\pi\)
0.977333 + 0.211706i \(0.0679020\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −1320.60 −0.0669555
\(731\) 2347.95 0.118799
\(732\) 11741.0 0.592840
\(733\) 4666.39 0.235139 0.117570 0.993065i \(-0.462490\pi\)
0.117570 + 0.993065i \(0.462490\pi\)
\(734\) 2189.97 0.110127
\(735\) 0 0
\(736\) −275.636 −0.0138045
\(737\) 5580.39 0.278910
\(738\) 693.208 0.0345763
\(739\) −8887.99 −0.442422 −0.221211 0.975226i \(-0.571001\pi\)
−0.221211 + 0.975226i \(0.571001\pi\)
\(740\) 16853.5 0.837225
\(741\) −4866.45 −0.241260
\(742\) 0 0
\(743\) 12880.1 0.635971 0.317986 0.948096i \(-0.396994\pi\)
0.317986 + 0.948096i \(0.396994\pi\)
\(744\) −3549.08 −0.174887
\(745\) 4198.68 0.206480
\(746\) −480.483 −0.0235814
\(747\) −2756.41 −0.135009
\(748\) 6578.33 0.321561
\(749\) 0 0
\(750\) 1561.15 0.0760067
\(751\) −24404.5 −1.18580 −0.592898 0.805278i \(-0.702016\pi\)
−0.592898 + 0.805278i \(0.702016\pi\)
\(752\) 34888.6 1.69183
\(753\) 9715.68 0.470198
\(754\) 3526.52 0.170329
\(755\) −24368.3 −1.17464
\(756\) 0 0
\(757\) 39568.5 1.89979 0.949895 0.312569i \(-0.101190\pi\)
0.949895 + 0.312569i \(0.101190\pi\)
\(758\) −1690.97 −0.0810273
\(759\) −138.971 −0.00664603
\(760\) 1575.79 0.0752106
\(761\) 33434.3 1.59263 0.796317 0.604880i \(-0.206779\pi\)
0.796317 + 0.604880i \(0.206779\pi\)
\(762\) 328.036 0.0155952
\(763\) 0 0
\(764\) −1762.30 −0.0834525
\(765\) 6885.53 0.325421
\(766\) 2145.62 0.101207
\(767\) −7444.76 −0.350476
\(768\) −10470.9 −0.491975
\(769\) 38008.7 1.78235 0.891176 0.453657i \(-0.149881\pi\)
0.891176 + 0.453657i \(0.149881\pi\)
\(770\) 0 0
\(771\) 495.163 0.0231295
\(772\) 18939.2 0.882951
\(773\) 22171.3 1.03163 0.515813 0.856701i \(-0.327490\pi\)
0.515813 + 0.856701i \(0.327490\pi\)
\(774\) 96.8540 0.00449786
\(775\) −5011.78 −0.232295
\(776\) 2772.07 0.128236
\(777\) 0 0
\(778\) −1961.09 −0.0903708
\(779\) 6265.80 0.288184
\(780\) 13656.2 0.626883
\(781\) 3208.55 0.147005
\(782\) 111.200 0.00508503
\(783\) −4775.01 −0.217937
\(784\) 0 0
\(785\) 32819.9 1.49222
\(786\) −833.384 −0.0378191
\(787\) 36159.2 1.63778 0.818892 0.573947i \(-0.194589\pi\)
0.818892 + 0.573947i \(0.194589\pi\)
\(788\) 19681.5 0.889750
\(789\) −9649.25 −0.435390
\(790\) 4285.57 0.193005
\(791\) 0 0
\(792\) 546.887 0.0245363
\(793\) −28471.4 −1.27497
\(794\) −3229.04 −0.144325
\(795\) −8406.91 −0.375047
\(796\) 9168.12 0.408235
\(797\) 16643.7 0.739710 0.369855 0.929089i \(-0.379407\pi\)
0.369855 + 0.929089i \(0.379407\pi\)
\(798\) 0 0
\(799\) −43333.5 −1.91868
\(800\) 1531.74 0.0676939
\(801\) 151.001 0.00666088
\(802\) 2021.01 0.0889828
\(803\) −4142.68 −0.182057
\(804\) 11991.2 0.525992
\(805\) 0 0
\(806\) 4270.39 0.186623
\(807\) −15108.9 −0.659057
\(808\) 4965.40 0.216191
\(809\) −34965.3 −1.51955 −0.759774 0.650187i \(-0.774691\pi\)
−0.759774 + 0.650187i \(0.774691\pi\)
\(810\) 284.032 0.0123208
\(811\) 12291.0 0.532175 0.266087 0.963949i \(-0.414269\pi\)
0.266087 + 0.963949i \(0.414269\pi\)
\(812\) 0 0
\(813\) 5271.48 0.227403
\(814\) −812.103 −0.0349683
\(815\) −13176.0 −0.566300
\(816\) 13915.1 0.596969
\(817\) 875.449 0.0374885
\(818\) 2044.99 0.0874102
\(819\) 0 0
\(820\) −17583.0 −0.748810
\(821\) 20687.3 0.879404 0.439702 0.898144i \(-0.355084\pi\)
0.439702 + 0.898144i \(0.355084\pi\)
\(822\) −1374.80 −0.0583352
\(823\) −41610.7 −1.76240 −0.881201 0.472742i \(-0.843264\pi\)
−0.881201 + 0.472742i \(0.843264\pi\)
\(824\) −9806.80 −0.414607
\(825\) 772.277 0.0325906
\(826\) 0 0
\(827\) 24355.7 1.02410 0.512050 0.858955i \(-0.328886\pi\)
0.512050 + 0.858955i \(0.328886\pi\)
\(828\) −298.623 −0.0125336
\(829\) −35382.4 −1.48237 −0.741183 0.671303i \(-0.765735\pi\)
−0.741183 + 0.671303i \(0.765735\pi\)
\(830\) −1073.95 −0.0449123
\(831\) 15425.3 0.643922
\(832\) 26716.8 1.11327
\(833\) 0 0
\(834\) −131.979 −0.00547971
\(835\) 24536.1 1.01690
\(836\) 2452.78 0.101473
\(837\) −5782.24 −0.238785
\(838\) −3108.18 −0.128127
\(839\) −3280.27 −0.134979 −0.0674897 0.997720i \(-0.521499\pi\)
−0.0674897 + 0.997720i \(0.521499\pi\)
\(840\) 0 0
\(841\) 6887.67 0.282409
\(842\) −2784.72 −0.113976
\(843\) −564.085 −0.0230464
\(844\) 20901.4 0.852437
\(845\) −10970.8 −0.446637
\(846\) −1787.53 −0.0726436
\(847\) 0 0
\(848\) −16989.7 −0.688006
\(849\) −14160.9 −0.572438
\(850\) −617.948 −0.0249358
\(851\) 893.695 0.0359994
\(852\) 6894.56 0.277234
\(853\) 29009.7 1.16445 0.582224 0.813028i \(-0.302183\pi\)
0.582224 + 0.813028i \(0.302183\pi\)
\(854\) 0 0
\(855\) 2567.32 0.102691
\(856\) 11380.1 0.454399
\(857\) −13538.6 −0.539640 −0.269820 0.962911i \(-0.586964\pi\)
−0.269820 + 0.962911i \(0.586964\pi\)
\(858\) −658.036 −0.0261829
\(859\) −20023.0 −0.795316 −0.397658 0.917534i \(-0.630177\pi\)
−0.397658 + 0.917534i \(0.630177\pi\)
\(860\) −2456.67 −0.0974090
\(861\) 0 0
\(862\) 4258.06 0.168249
\(863\) −40818.4 −1.61005 −0.805025 0.593241i \(-0.797848\pi\)
−0.805025 + 0.593241i \(0.797848\pi\)
\(864\) 1767.21 0.0695854
\(865\) −2781.04 −0.109316
\(866\) −2903.84 −0.113945
\(867\) −2544.33 −0.0996655
\(868\) 0 0
\(869\) 13443.7 0.524795
\(870\) −1860.43 −0.0724994
\(871\) −29078.2 −1.13120
\(872\) −6935.61 −0.269346
\(873\) 4516.31 0.175091
\(874\) 41.4616 0.00160464
\(875\) 0 0
\(876\) −8901.84 −0.343339
\(877\) 5277.12 0.203188 0.101594 0.994826i \(-0.467606\pi\)
0.101594 + 0.994826i \(0.467606\pi\)
\(878\) −3369.22 −0.129506
\(879\) −2201.60 −0.0844802
\(880\) −6775.59 −0.259551
\(881\) −41475.1 −1.58607 −0.793036 0.609174i \(-0.791501\pi\)
−0.793036 + 0.609174i \(0.791501\pi\)
\(882\) 0 0
\(883\) 5610.40 0.213822 0.106911 0.994269i \(-0.465904\pi\)
0.106911 + 0.994269i \(0.465904\pi\)
\(884\) −34278.2 −1.30419
\(885\) 3927.52 0.149177
\(886\) −2005.68 −0.0760520
\(887\) 28484.5 1.07826 0.539130 0.842222i \(-0.318753\pi\)
0.539130 + 0.842222i \(0.318753\pi\)
\(888\) −3516.91 −0.132905
\(889\) 0 0
\(890\) 58.8328 0.00221582
\(891\) 891.000 0.0335013
\(892\) −23584.3 −0.885270
\(893\) −16157.2 −0.605465
\(894\) −434.743 −0.0162640
\(895\) 18494.3 0.690723
\(896\) 0 0
\(897\) 724.149 0.0269550
\(898\) 1802.03 0.0669649
\(899\) 37874.1 1.40509
\(900\) 1659.48 0.0614621
\(901\) 21102.1 0.780260
\(902\) 847.254 0.0312755
\(903\) 0 0
\(904\) −10212.1 −0.375719
\(905\) 29164.0 1.07121
\(906\) 2523.16 0.0925236
\(907\) −34526.8 −1.26400 −0.631998 0.774970i \(-0.717765\pi\)
−0.631998 + 0.774970i \(0.717765\pi\)
\(908\) 21453.7 0.784104
\(909\) 8089.74 0.295181
\(910\) 0 0
\(911\) −4049.17 −0.147261 −0.0736307 0.997286i \(-0.523459\pi\)
−0.0736307 + 0.997286i \(0.523459\pi\)
\(912\) 5188.35 0.188381
\(913\) −3368.94 −0.122120
\(914\) 6592.36 0.238573
\(915\) 15020.2 0.542681
\(916\) 47034.5 1.69658
\(917\) 0 0
\(918\) −712.946 −0.0256326
\(919\) −42686.0 −1.53219 −0.766095 0.642727i \(-0.777803\pi\)
−0.766095 + 0.642727i \(0.777803\pi\)
\(920\) −234.485 −0.00840298
\(921\) 9534.38 0.341117
\(922\) 6619.11 0.236430
\(923\) −16719.0 −0.596223
\(924\) 0 0
\(925\) −4966.35 −0.176533
\(926\) 3790.45 0.134516
\(927\) −15977.5 −0.566094
\(928\) −11575.4 −0.409462
\(929\) 25572.5 0.903128 0.451564 0.892239i \(-0.350866\pi\)
0.451564 + 0.892239i \(0.350866\pi\)
\(930\) −2252.87 −0.0794348
\(931\) 0 0
\(932\) −21537.1 −0.756942
\(933\) −1160.64 −0.0407262
\(934\) 5507.72 0.192953
\(935\) 8415.65 0.294354
\(936\) −2849.71 −0.0995146
\(937\) −39541.9 −1.37863 −0.689316 0.724461i \(-0.742089\pi\)
−0.689316 + 0.724461i \(0.742089\pi\)
\(938\) 0 0
\(939\) 12538.9 0.435773
\(940\) 45340.1 1.57322
\(941\) −12094.7 −0.418997 −0.209499 0.977809i \(-0.567183\pi\)
−0.209499 + 0.977809i \(0.567183\pi\)
\(942\) −3398.26 −0.117538
\(943\) −932.377 −0.0321977
\(944\) 7937.20 0.273659
\(945\) 0 0
\(946\) 118.377 0.00406847
\(947\) −47504.8 −1.63009 −0.815047 0.579394i \(-0.803289\pi\)
−0.815047 + 0.579394i \(0.803289\pi\)
\(948\) 28888.0 0.989704
\(949\) 21586.6 0.738389
\(950\) −230.406 −0.00786881
\(951\) −1066.10 −0.0363518
\(952\) 0 0
\(953\) −1746.32 −0.0593588 −0.0296794 0.999559i \(-0.509449\pi\)
−0.0296794 + 0.999559i \(0.509449\pi\)
\(954\) 870.474 0.0295415
\(955\) −2254.51 −0.0763918
\(956\) −47776.3 −1.61631
\(957\) −5836.12 −0.197132
\(958\) −5123.66 −0.172795
\(959\) 0 0
\(960\) −14094.6 −0.473854
\(961\) 16072.2 0.539500
\(962\) 4231.69 0.141825
\(963\) 18540.8 0.620424
\(964\) −36653.9 −1.22463
\(965\) 24228.9 0.808246
\(966\) 0 0
\(967\) 3694.27 0.122854 0.0614269 0.998112i \(-0.480435\pi\)
0.0614269 + 0.998112i \(0.480435\pi\)
\(968\) 668.417 0.0221939
\(969\) −6444.21 −0.213641
\(970\) 1759.64 0.0582459
\(971\) −10587.6 −0.349919 −0.174959 0.984576i \(-0.555979\pi\)
−0.174959 + 0.984576i \(0.555979\pi\)
\(972\) 1914.59 0.0631796
\(973\) 0 0
\(974\) 417.632 0.0137390
\(975\) −4024.17 −0.132181
\(976\) 30354.7 0.995522
\(977\) −4040.14 −0.132299 −0.0661493 0.997810i \(-0.521071\pi\)
−0.0661493 + 0.997810i \(0.521071\pi\)
\(978\) 1364.28 0.0446061
\(979\) 184.557 0.00602499
\(980\) 0 0
\(981\) −11299.6 −0.367757
\(982\) −6564.89 −0.213334
\(983\) −39320.9 −1.27583 −0.637916 0.770106i \(-0.720203\pi\)
−0.637916 + 0.770106i \(0.720203\pi\)
\(984\) 3669.14 0.118870
\(985\) 25178.5 0.814471
\(986\) 4669.85 0.150830
\(987\) 0 0
\(988\) −12780.9 −0.411553
\(989\) −130.270 −0.00418843
\(990\) 347.150 0.0111446
\(991\) −17420.2 −0.558397 −0.279199 0.960233i \(-0.590069\pi\)
−0.279199 + 0.960233i \(0.590069\pi\)
\(992\) −14017.1 −0.448632
\(993\) −31694.3 −1.01288
\(994\) 0 0
\(995\) 11728.8 0.373695
\(996\) −7239.22 −0.230305
\(997\) 4364.82 0.138651 0.0693256 0.997594i \(-0.477915\pi\)
0.0693256 + 0.997594i \(0.477915\pi\)
\(998\) 4725.66 0.149888
\(999\) −5729.84 −0.181465
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.1.8 16
7.6 odd 2 1617.4.a.bf.1.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.8 16 1.1 even 1 trivial
1617.4.a.bf.1.8 yes 16 7.6 odd 2