Properties

Label 169.4.b.e.168.3
Level $169$
Weight $4$
Character 169.168
Analytic conductor $9.971$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.3
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.4.b.e.168.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} +17.8078i q^{5} -1.61553i q^{6} +5.43845i q^{7} +6.93087i q^{8} -13.4233 q^{9} +O(q^{10})\) \(q+0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} +17.8078i q^{5} -1.61553i q^{6} +5.43845i q^{7} +6.93087i q^{8} -13.4233 q^{9} -7.80776 q^{10} -22.4233i q^{11} -28.7689 q^{12} -2.38447 q^{14} -65.6155i q^{15} +59.4233 q^{16} -67.9848 q^{17} -5.88540i q^{18} +80.8078i q^{19} +139.039i q^{20} -20.0388i q^{21} +9.83143 q^{22} -140.531 q^{23} -25.5379i q^{24} -192.116 q^{25} +148.946 q^{27} +42.4621i q^{28} -106.693 q^{29} +28.7689 q^{30} +276.155i q^{31} +81.5009i q^{32} +82.6222i q^{33} -29.8078i q^{34} -96.8466 q^{35} -104.806 q^{36} -4.29168i q^{37} -35.4299 q^{38} -123.423 q^{40} -227.769i q^{41} +8.78596 q^{42} -27.5294 q^{43} -175.076i q^{44} -239.039i q^{45} -61.6155i q^{46} +318.617i q^{47} -218.955 q^{48} +313.423 q^{49} -84.2329i q^{50} +250.501 q^{51} -67.6562 q^{53} +65.3050i q^{54} +399.309 q^{55} -37.6932 q^{56} -297.749i q^{57} -46.7793i q^{58} -291.115i q^{59} -512.311i q^{60} +663.311 q^{61} -121.080 q^{62} -73.0019i q^{63} +439.652 q^{64} -36.2255 q^{66} +425.101i q^{67} -530.810 q^{68} +517.810 q^{69} -42.4621i q^{70} +152.963i q^{71} -93.0351i q^{72} +117.268i q^{73} +1.88167 q^{74} +707.884 q^{75} +630.928i q^{76} +121.948 q^{77} +202.462 q^{79} +1058.20i q^{80} -186.386 q^{81} +99.8647 q^{82} -336.155i q^{83} -156.458i q^{84} -1210.66i q^{85} -12.0702i q^{86} +393.128 q^{87} +155.413 q^{88} +718.194i q^{89} +104.806 q^{90} -1097.23 q^{92} -1017.54i q^{93} -139.697 q^{94} -1439.01 q^{95} -300.303i q^{96} -759.368i q^{97} +137.420i q^{98} +300.994i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.438447i 0.155014i 0.996992 + 0.0775072i \(0.0246961\pi\)
−0.996992 + 0.0775072i \(0.975304\pi\)
\(3\) −3.68466 −0.709113 −0.354556 0.935035i \(-0.615368\pi\)
−0.354556 + 0.935035i \(0.615368\pi\)
\(4\) 7.80776 0.975971
\(5\) 17.8078i 1.59277i 0.604787 + 0.796387i \(0.293258\pi\)
−0.604787 + 0.796387i \(0.706742\pi\)
\(6\) − 1.61553i − 0.109923i
\(7\) 5.43845i 0.293649i 0.989163 + 0.146824i \(0.0469052\pi\)
−0.989163 + 0.146824i \(0.953095\pi\)
\(8\) 6.93087i 0.306304i
\(9\) −13.4233 −0.497159
\(10\) −7.80776 −0.246903
\(11\) − 22.4233i − 0.614625i −0.951609 0.307313i \(-0.900570\pi\)
0.951609 0.307313i \(-0.0994297\pi\)
\(12\) −28.7689 −0.692073
\(13\) 0 0
\(14\) −2.38447 −0.0455198
\(15\) − 65.6155i − 1.12946i
\(16\) 59.4233 0.928489
\(17\) −67.9848 −0.969926 −0.484963 0.874535i \(-0.661167\pi\)
−0.484963 + 0.874535i \(0.661167\pi\)
\(18\) − 5.88540i − 0.0770668i
\(19\) 80.8078i 0.975714i 0.872923 + 0.487857i \(0.162221\pi\)
−0.872923 + 0.487857i \(0.837779\pi\)
\(20\) 139.039i 1.55450i
\(21\) − 20.0388i − 0.208230i
\(22\) 9.83143 0.0952758
\(23\) −140.531 −1.27403 −0.637017 0.770850i \(-0.719832\pi\)
−0.637017 + 0.770850i \(0.719832\pi\)
\(24\) − 25.5379i − 0.217204i
\(25\) −192.116 −1.53693
\(26\) 0 0
\(27\) 148.946 1.06165
\(28\) 42.4621i 0.286592i
\(29\) −106.693 −0.683187 −0.341594 0.939848i \(-0.610967\pi\)
−0.341594 + 0.939848i \(0.610967\pi\)
\(30\) 28.7689 0.175082
\(31\) 276.155i 1.59997i 0.600023 + 0.799983i \(0.295158\pi\)
−0.600023 + 0.799983i \(0.704842\pi\)
\(32\) 81.5009i 0.450233i
\(33\) 82.6222i 0.435839i
\(34\) − 29.8078i − 0.150353i
\(35\) −96.8466 −0.467716
\(36\) −104.806 −0.485212
\(37\) − 4.29168i − 0.0190688i −0.999955 0.00953442i \(-0.996965\pi\)
0.999955 0.00953442i \(-0.00303495\pi\)
\(38\) −35.4299 −0.151250
\(39\) 0 0
\(40\) −123.423 −0.487873
\(41\) − 227.769i − 0.867598i −0.901010 0.433799i \(-0.857173\pi\)
0.901010 0.433799i \(-0.142827\pi\)
\(42\) 8.78596 0.0322787
\(43\) −27.5294 −0.0976323 −0.0488162 0.998808i \(-0.515545\pi\)
−0.0488162 + 0.998808i \(0.515545\pi\)
\(44\) − 175.076i − 0.599856i
\(45\) − 239.039i − 0.791862i
\(46\) − 61.6155i − 0.197494i
\(47\) 318.617i 0.988832i 0.869225 + 0.494416i \(0.164618\pi\)
−0.869225 + 0.494416i \(0.835382\pi\)
\(48\) −218.955 −0.658403
\(49\) 313.423 0.913771
\(50\) − 84.2329i − 0.238247i
\(51\) 250.501 0.687787
\(52\) 0 0
\(53\) −67.6562 −0.175345 −0.0876726 0.996149i \(-0.527943\pi\)
−0.0876726 + 0.996149i \(0.527943\pi\)
\(54\) 65.3050i 0.164572i
\(55\) 399.309 0.978960
\(56\) −37.6932 −0.0899457
\(57\) − 297.749i − 0.691892i
\(58\) − 46.7793i − 0.105904i
\(59\) − 291.115i − 0.642371i −0.947016 0.321186i \(-0.895919\pi\)
0.947016 0.321186i \(-0.104081\pi\)
\(60\) − 512.311i − 1.10232i
\(61\) 663.311 1.39227 0.696133 0.717913i \(-0.254902\pi\)
0.696133 + 0.717913i \(0.254902\pi\)
\(62\) −121.080 −0.248018
\(63\) − 73.0019i − 0.145990i
\(64\) 439.652 0.858696
\(65\) 0 0
\(66\) −36.2255 −0.0675613
\(67\) 425.101i 0.775140i 0.921840 + 0.387570i \(0.126685\pi\)
−0.921840 + 0.387570i \(0.873315\pi\)
\(68\) −530.810 −0.946619
\(69\) 517.810 0.903434
\(70\) − 42.4621i − 0.0725028i
\(71\) 152.963i 0.255681i 0.991795 + 0.127841i \(0.0408046\pi\)
−0.991795 + 0.127841i \(0.959195\pi\)
\(72\) − 93.0351i − 0.152282i
\(73\) 117.268i 0.188016i 0.995571 + 0.0940081i \(0.0299680\pi\)
−0.995571 + 0.0940081i \(0.970032\pi\)
\(74\) 1.88167 0.00295595
\(75\) 707.884 1.08986
\(76\) 630.928i 0.952268i
\(77\) 121.948 0.180484
\(78\) 0 0
\(79\) 202.462 0.288339 0.144169 0.989553i \(-0.453949\pi\)
0.144169 + 0.989553i \(0.453949\pi\)
\(80\) 1058.20i 1.47887i
\(81\) −186.386 −0.255674
\(82\) 99.8647 0.134490
\(83\) − 336.155i − 0.444552i −0.974984 0.222276i \(-0.928651\pi\)
0.974984 0.222276i \(-0.0713486\pi\)
\(84\) − 156.458i − 0.203226i
\(85\) − 1210.66i − 1.54487i
\(86\) − 12.0702i − 0.0151344i
\(87\) 393.128 0.484457
\(88\) 155.413 0.188262
\(89\) 718.194i 0.855376i 0.903927 + 0.427688i \(0.140672\pi\)
−0.903927 + 0.427688i \(0.859328\pi\)
\(90\) 104.806 0.122750
\(91\) 0 0
\(92\) −1097.23 −1.24342
\(93\) − 1017.54i − 1.13456i
\(94\) −139.697 −0.153283
\(95\) −1439.01 −1.55409
\(96\) − 300.303i − 0.319266i
\(97\) − 759.368i − 0.794868i −0.917631 0.397434i \(-0.869901\pi\)
0.917631 0.397434i \(-0.130099\pi\)
\(98\) 137.420i 0.141648i
\(99\) 300.994i 0.305566i
\(100\) −1500.00 −1.50000
\(101\) 348.697 0.343531 0.171766 0.985138i \(-0.445053\pi\)
0.171766 + 0.985138i \(0.445053\pi\)
\(102\) 109.831i 0.106617i
\(103\) 580.303 0.555136 0.277568 0.960706i \(-0.410472\pi\)
0.277568 + 0.960706i \(0.410472\pi\)
\(104\) 0 0
\(105\) 356.847 0.331663
\(106\) − 29.6637i − 0.0271810i
\(107\) 571.493 0.516340 0.258170 0.966100i \(-0.416881\pi\)
0.258170 + 0.966100i \(0.416881\pi\)
\(108\) 1162.94 1.03614
\(109\) − 176.004i − 0.154661i −0.997005 0.0773307i \(-0.975360\pi\)
0.997005 0.0773307i \(-0.0246397\pi\)
\(110\) 175.076i 0.151753i
\(111\) 15.8134i 0.0135220i
\(112\) 323.170i 0.272649i
\(113\) 1264.88 1.05301 0.526505 0.850172i \(-0.323502\pi\)
0.526505 + 0.850172i \(0.323502\pi\)
\(114\) 130.547 0.107253
\(115\) − 2502.55i − 2.02925i
\(116\) −833.035 −0.666770
\(117\) 0 0
\(118\) 127.638 0.0995768
\(119\) − 369.732i − 0.284817i
\(120\) 454.773 0.345957
\(121\) 828.196 0.622236
\(122\) 290.827i 0.215821i
\(123\) 839.251i 0.615225i
\(124\) 2156.16i 1.56152i
\(125\) − 1195.19i − 0.855211i
\(126\) 32.0075 0.0226306
\(127\) −2604.11 −1.81950 −0.909752 0.415151i \(-0.863729\pi\)
−0.909752 + 0.415151i \(0.863729\pi\)
\(128\) 844.772i 0.583344i
\(129\) 101.436 0.0692323
\(130\) 0 0
\(131\) 2131.70 1.42174 0.710870 0.703324i \(-0.248302\pi\)
0.710870 + 0.703324i \(0.248302\pi\)
\(132\) 645.094i 0.425366i
\(133\) −439.469 −0.286517
\(134\) −186.384 −0.120158
\(135\) 2652.40i 1.69098i
\(136\) − 471.194i − 0.297092i
\(137\) 687.985i 0.429040i 0.976720 + 0.214520i \(0.0688188\pi\)
−0.976720 + 0.214520i \(0.931181\pi\)
\(138\) 227.032i 0.140045i
\(139\) −679.580 −0.414685 −0.207343 0.978268i \(-0.566482\pi\)
−0.207343 + 0.978268i \(0.566482\pi\)
\(140\) −756.155 −0.456477
\(141\) − 1174.00i − 0.701194i
\(142\) −67.0662 −0.0396343
\(143\) 0 0
\(144\) −797.656 −0.461607
\(145\) − 1899.97i − 1.08816i
\(146\) −51.4158 −0.0291452
\(147\) −1154.86 −0.647966
\(148\) − 33.5084i − 0.0186106i
\(149\) 1975.46i 1.08615i 0.839685 + 0.543074i \(0.182740\pi\)
−0.839685 + 0.543074i \(0.817260\pi\)
\(150\) 310.370i 0.168944i
\(151\) 1803.24i 0.971824i 0.874008 + 0.485912i \(0.161513\pi\)
−0.874008 + 0.485912i \(0.838487\pi\)
\(152\) −560.068 −0.298865
\(153\) 912.580 0.482208
\(154\) 53.4677i 0.0279776i
\(155\) −4917.71 −2.54839
\(156\) 0 0
\(157\) −397.168 −0.201894 −0.100947 0.994892i \(-0.532187\pi\)
−0.100947 + 0.994892i \(0.532187\pi\)
\(158\) 88.7689i 0.0446967i
\(159\) 249.290 0.124340
\(160\) −1451.35 −0.717120
\(161\) − 764.272i − 0.374118i
\(162\) − 81.7206i − 0.0396332i
\(163\) 941.393i 0.452365i 0.974085 + 0.226183i \(0.0726246\pi\)
−0.974085 + 0.226183i \(0.927375\pi\)
\(164\) − 1778.37i − 0.846750i
\(165\) −1471.32 −0.694193
\(166\) 147.386 0.0689120
\(167\) 3680.43i 1.70539i 0.522408 + 0.852696i \(0.325034\pi\)
−0.522408 + 0.852696i \(0.674966\pi\)
\(168\) 138.886 0.0637817
\(169\) 0 0
\(170\) 530.810 0.239478
\(171\) − 1084.71i − 0.485085i
\(172\) −214.943 −0.0952863
\(173\) −1422.77 −0.625269 −0.312634 0.949874i \(-0.601211\pi\)
−0.312634 + 0.949874i \(0.601211\pi\)
\(174\) 172.366i 0.0750978i
\(175\) − 1044.82i − 0.451318i
\(176\) − 1332.47i − 0.570673i
\(177\) 1072.66i 0.455514i
\(178\) −314.890 −0.132596
\(179\) −1167.89 −0.487666 −0.243833 0.969817i \(-0.578405\pi\)
−0.243833 + 0.969817i \(0.578405\pi\)
\(180\) − 1866.36i − 0.772834i
\(181\) 1133.96 0.465673 0.232836 0.972516i \(-0.425199\pi\)
0.232836 + 0.972516i \(0.425199\pi\)
\(182\) 0 0
\(183\) −2444.07 −0.987274
\(184\) − 974.004i − 0.390242i
\(185\) 76.4252 0.0303724
\(186\) 446.137 0.175873
\(187\) 1524.44i 0.596141i
\(188\) 2487.69i 0.965071i
\(189\) 810.035i 0.311753i
\(190\) − 630.928i − 0.240907i
\(191\) 2682.12 1.01608 0.508040 0.861333i \(-0.330370\pi\)
0.508040 + 0.861333i \(0.330370\pi\)
\(192\) −1619.97 −0.608913
\(193\) − 1970.67i − 0.734983i −0.930027 0.367491i \(-0.880217\pi\)
0.930027 0.367491i \(-0.119783\pi\)
\(194\) 332.943 0.123216
\(195\) 0 0
\(196\) 2447.14 0.891813
\(197\) 4016.05i 1.45244i 0.687460 + 0.726222i \(0.258726\pi\)
−0.687460 + 0.726222i \(0.741274\pi\)
\(198\) −131.970 −0.0473672
\(199\) 4226.06 1.50541 0.752707 0.658356i \(-0.228748\pi\)
0.752707 + 0.658356i \(0.228748\pi\)
\(200\) − 1331.53i − 0.470768i
\(201\) − 1566.35i − 0.549662i
\(202\) 152.885i 0.0532523i
\(203\) − 580.245i − 0.200617i
\(204\) 1955.85 0.671260
\(205\) 4056.06 1.38189
\(206\) 254.432i 0.0860541i
\(207\) 1886.39 0.633398
\(208\) 0 0
\(209\) 1811.98 0.599699
\(210\) 156.458i 0.0514126i
\(211\) 1364.67 0.445249 0.222625 0.974904i \(-0.428538\pi\)
0.222625 + 0.974904i \(0.428538\pi\)
\(212\) −528.244 −0.171132
\(213\) − 563.617i − 0.181307i
\(214\) 250.570i 0.0800401i
\(215\) − 490.237i − 0.155506i
\(216\) 1032.33i 0.325189i
\(217\) −1501.86 −0.469828
\(218\) 77.1683 0.0239748
\(219\) − 432.093i − 0.133325i
\(220\) 3117.71 0.955436
\(221\) 0 0
\(222\) −6.93332 −0.00209610
\(223\) 1059.47i 0.318149i 0.987267 + 0.159075i \(0.0508510\pi\)
−0.987267 + 0.159075i \(0.949149\pi\)
\(224\) −443.239 −0.132210
\(225\) 2578.84 0.764099
\(226\) 554.584i 0.163232i
\(227\) − 3464.19i − 1.01289i −0.862272 0.506446i \(-0.830959\pi\)
0.862272 0.506446i \(-0.169041\pi\)
\(228\) − 2324.75i − 0.675266i
\(229\) − 2324.64i − 0.670815i −0.942073 0.335407i \(-0.891126\pi\)
0.942073 0.335407i \(-0.108874\pi\)
\(230\) 1097.23 0.314563
\(231\) −449.336 −0.127983
\(232\) − 739.476i − 0.209263i
\(233\) 3731.01 1.04904 0.524521 0.851398i \(-0.324245\pi\)
0.524521 + 0.851398i \(0.324245\pi\)
\(234\) 0 0
\(235\) −5673.86 −1.57499
\(236\) − 2272.95i − 0.626935i
\(237\) −746.004 −0.204465
\(238\) 162.108 0.0441508
\(239\) − 6044.47i − 1.63592i −0.575278 0.817958i \(-0.695106\pi\)
0.575278 0.817958i \(-0.304894\pi\)
\(240\) − 3899.09i − 1.04869i
\(241\) − 5173.96i − 1.38292i −0.722414 0.691461i \(-0.756967\pi\)
0.722414 0.691461i \(-0.243033\pi\)
\(242\) 363.120i 0.0964556i
\(243\) −3334.77 −0.880353
\(244\) 5178.97 1.35881
\(245\) 5581.37i 1.45543i
\(246\) −367.967 −0.0953688
\(247\) 0 0
\(248\) −1914.00 −0.490076
\(249\) 1238.62i 0.315238i
\(250\) 524.029 0.132570
\(251\) −5620.73 −1.41346 −0.706728 0.707486i \(-0.749829\pi\)
−0.706728 + 0.707486i \(0.749829\pi\)
\(252\) − 569.981i − 0.142482i
\(253\) 3151.17i 0.783054i
\(254\) − 1141.76i − 0.282050i
\(255\) 4460.86i 1.09549i
\(256\) 3146.83 0.768270
\(257\) 1674.14 0.406342 0.203171 0.979143i \(-0.434875\pi\)
0.203171 + 0.979143i \(0.434875\pi\)
\(258\) 44.4745i 0.0107320i
\(259\) 23.3401 0.00559954
\(260\) 0 0
\(261\) 1432.17 0.339653
\(262\) 934.640i 0.220390i
\(263\) −6309.18 −1.47924 −0.739622 0.673023i \(-0.764996\pi\)
−0.739622 + 0.673023i \(0.764996\pi\)
\(264\) −572.644 −0.133499
\(265\) − 1204.81i − 0.279285i
\(266\) − 192.684i − 0.0444143i
\(267\) − 2646.30i − 0.606558i
\(268\) 3319.09i 0.756514i
\(269\) −2482.73 −0.562731 −0.281366 0.959601i \(-0.590787\pi\)
−0.281366 + 0.959601i \(0.590787\pi\)
\(270\) −1162.94 −0.262126
\(271\) 2835.72i 0.635638i 0.948151 + 0.317819i \(0.102950\pi\)
−0.948151 + 0.317819i \(0.897050\pi\)
\(272\) −4039.88 −0.900566
\(273\) 0 0
\(274\) −301.645 −0.0665075
\(275\) 4307.88i 0.944637i
\(276\) 4042.94 0.881725
\(277\) 3837.51 0.832396 0.416198 0.909274i \(-0.363362\pi\)
0.416198 + 0.909274i \(0.363362\pi\)
\(278\) − 297.960i − 0.0642822i
\(279\) − 3706.91i − 0.795438i
\(280\) − 671.231i − 0.143263i
\(281\) − 9122.13i − 1.93659i −0.249819 0.968293i \(-0.580371\pi\)
0.249819 0.968293i \(-0.419629\pi\)
\(282\) 514.735 0.108695
\(283\) −2127.85 −0.446952 −0.223476 0.974709i \(-0.571740\pi\)
−0.223476 + 0.974709i \(0.571740\pi\)
\(284\) 1194.30i 0.249537i
\(285\) 5302.24 1.10203
\(286\) 0 0
\(287\) 1238.71 0.254769
\(288\) − 1094.01i − 0.223838i
\(289\) −291.061 −0.0592430
\(290\) 833.035 0.168681
\(291\) 2798.01i 0.563651i
\(292\) 915.601i 0.183498i
\(293\) 8274.77i 1.64989i 0.565215 + 0.824944i \(0.308793\pi\)
−0.565215 + 0.824944i \(0.691207\pi\)
\(294\) − 506.344i − 0.100444i
\(295\) 5184.10 1.02315
\(296\) 29.7450 0.00584086
\(297\) − 3339.86i − 0.652520i
\(298\) −866.136 −0.168369
\(299\) 0 0
\(300\) 5526.99 1.06367
\(301\) − 149.717i − 0.0286696i
\(302\) −790.625 −0.150647
\(303\) −1284.83 −0.243602
\(304\) 4801.86i 0.905940i
\(305\) 11812.1i 2.21757i
\(306\) 400.118i 0.0747492i
\(307\) − 3610.49i − 0.671211i −0.942003 0.335605i \(-0.891059\pi\)
0.942003 0.335605i \(-0.108941\pi\)
\(308\) 952.140 0.176147
\(309\) −2138.22 −0.393654
\(310\) − 2156.16i − 0.395037i
\(311\) −3331.06 −0.607354 −0.303677 0.952775i \(-0.598214\pi\)
−0.303677 + 0.952775i \(0.598214\pi\)
\(312\) 0 0
\(313\) −358.125 −0.0646724 −0.0323362 0.999477i \(-0.510295\pi\)
−0.0323362 + 0.999477i \(0.510295\pi\)
\(314\) − 174.137i − 0.0312966i
\(315\) 1300.00 0.232529
\(316\) 1580.78 0.281410
\(317\) − 3047.46i − 0.539944i −0.962868 0.269972i \(-0.912986\pi\)
0.962868 0.269972i \(-0.0870144\pi\)
\(318\) 109.301i 0.0192744i
\(319\) 2392.41i 0.419904i
\(320\) 7829.23i 1.36771i
\(321\) −2105.76 −0.366143
\(322\) 335.093 0.0579938
\(323\) − 5493.70i − 0.946371i
\(324\) −1455.26 −0.249530
\(325\) 0 0
\(326\) −412.751 −0.0701232
\(327\) 648.514i 0.109672i
\(328\) 1578.64 0.265749
\(329\) −1732.78 −0.290369
\(330\) − 645.094i − 0.107610i
\(331\) − 7694.77i − 1.27777i −0.769301 0.638887i \(-0.779395\pi\)
0.769301 0.638887i \(-0.220605\pi\)
\(332\) − 2624.62i − 0.433870i
\(333\) 57.6084i 0.00948025i
\(334\) −1613.68 −0.264360
\(335\) −7570.10 −1.23462
\(336\) − 1190.77i − 0.193339i
\(337\) −4712.21 −0.761693 −0.380846 0.924638i \(-0.624367\pi\)
−0.380846 + 0.924638i \(0.624367\pi\)
\(338\) 0 0
\(339\) −4660.66 −0.746703
\(340\) − 9452.53i − 1.50775i
\(341\) 6192.31 0.983380
\(342\) 475.586 0.0751952
\(343\) 3569.92i 0.561976i
\(344\) − 190.803i − 0.0299052i
\(345\) 9221.03i 1.43897i
\(346\) − 623.811i − 0.0969257i
\(347\) −5261.98 −0.814058 −0.407029 0.913415i \(-0.633435\pi\)
−0.407029 + 0.913415i \(0.633435\pi\)
\(348\) 3069.45 0.472815
\(349\) 50.3345i 0.00772018i 0.999993 + 0.00386009i \(0.00122871\pi\)
−0.999993 + 0.00386009i \(0.998771\pi\)
\(350\) 458.096 0.0699608
\(351\) 0 0
\(352\) 1827.52 0.276725
\(353\) − 9057.64i − 1.36569i −0.730562 0.682846i \(-0.760742\pi\)
0.730562 0.682846i \(-0.239258\pi\)
\(354\) −470.304 −0.0706112
\(355\) −2723.93 −0.407243
\(356\) 5607.49i 0.834821i
\(357\) 1362.34i 0.201968i
\(358\) − 512.059i − 0.0755953i
\(359\) 7177.86i 1.05525i 0.849479 + 0.527623i \(0.176917\pi\)
−0.849479 + 0.527623i \(0.823083\pi\)
\(360\) 1656.75 0.242551
\(361\) 329.105 0.0479815
\(362\) 497.183i 0.0721861i
\(363\) −3051.62 −0.441235
\(364\) 0 0
\(365\) −2088.28 −0.299467
\(366\) − 1071.60i − 0.153042i
\(367\) 4004.14 0.569522 0.284761 0.958599i \(-0.408086\pi\)
0.284761 + 0.958599i \(0.408086\pi\)
\(368\) −8350.83 −1.18293
\(369\) 3057.41i 0.431334i
\(370\) 33.5084i 0.00470816i
\(371\) − 367.945i − 0.0514899i
\(372\) − 7944.70i − 1.10729i
\(373\) −10014.2 −1.39012 −0.695060 0.718952i \(-0.744622\pi\)
−0.695060 + 0.718952i \(0.744622\pi\)
\(374\) −668.388 −0.0924105
\(375\) 4403.88i 0.606441i
\(376\) −2208.30 −0.302883
\(377\) 0 0
\(378\) −355.158 −0.0483263
\(379\) 8169.12i 1.10717i 0.832791 + 0.553587i \(0.186742\pi\)
−0.832791 + 0.553587i \(0.813258\pi\)
\(380\) −11235.4 −1.51675
\(381\) 9595.24 1.29023
\(382\) 1175.97i 0.157507i
\(383\) 7310.25i 0.975290i 0.873042 + 0.487645i \(0.162144\pi\)
−0.873042 + 0.487645i \(0.837856\pi\)
\(384\) − 3112.70i − 0.413656i
\(385\) 2171.62i 0.287470i
\(386\) 864.033 0.113933
\(387\) 369.535 0.0485388
\(388\) − 5928.97i − 0.775767i
\(389\) −8785.47 −1.14509 −0.572546 0.819872i \(-0.694044\pi\)
−0.572546 + 0.819872i \(0.694044\pi\)
\(390\) 0 0
\(391\) 9553.99 1.23572
\(392\) 2172.30i 0.279892i
\(393\) −7854.60 −1.00817
\(394\) −1760.82 −0.225150
\(395\) 3605.40i 0.459259i
\(396\) 2350.09i 0.298224i
\(397\) 11266.8i 1.42434i 0.702006 + 0.712171i \(0.252288\pi\)
−0.702006 + 0.712171i \(0.747712\pi\)
\(398\) 1852.90i 0.233361i
\(399\) 1619.29 0.203173
\(400\) −11416.2 −1.42702
\(401\) 1576.23i 0.196293i 0.995172 + 0.0981464i \(0.0312913\pi\)
−0.995172 + 0.0981464i \(0.968709\pi\)
\(402\) 686.763 0.0852055
\(403\) 0 0
\(404\) 2722.54 0.335276
\(405\) − 3319.12i − 0.407231i
\(406\) 254.407 0.0310985
\(407\) −96.2335 −0.0117202
\(408\) 1736.19i 0.210672i
\(409\) − 6755.78i − 0.816753i −0.912814 0.408377i \(-0.866095\pi\)
0.912814 0.408377i \(-0.133905\pi\)
\(410\) 1778.37i 0.214213i
\(411\) − 2534.99i − 0.304238i
\(412\) 4530.87 0.541796
\(413\) 1583.21 0.188631
\(414\) 827.083i 0.0981858i
\(415\) 5986.17 0.708072
\(416\) 0 0
\(417\) 2504.02 0.294059
\(418\) 794.456i 0.0929620i
\(419\) 10756.2 1.25411 0.627057 0.778973i \(-0.284259\pi\)
0.627057 + 0.778973i \(0.284259\pi\)
\(420\) 2786.17 0.323694
\(421\) − 7886.03i − 0.912925i −0.889743 0.456463i \(-0.849116\pi\)
0.889743 0.456463i \(-0.150884\pi\)
\(422\) 598.335i 0.0690201i
\(423\) − 4276.89i − 0.491607i
\(424\) − 468.916i − 0.0537089i
\(425\) 13061.0 1.49071
\(426\) 247.116 0.0281052
\(427\) 3607.38i 0.408837i
\(428\) 4462.08 0.503932
\(429\) 0 0
\(430\) 214.943 0.0241057
\(431\) 14084.6i 1.57409i 0.616897 + 0.787044i \(0.288390\pi\)
−0.616897 + 0.787044i \(0.711610\pi\)
\(432\) 8850.86 0.985735
\(433\) −1864.14 −0.206894 −0.103447 0.994635i \(-0.532987\pi\)
−0.103447 + 0.994635i \(0.532987\pi\)
\(434\) − 658.485i − 0.0728301i
\(435\) 7000.73i 0.771630i
\(436\) − 1374.20i − 0.150945i
\(437\) − 11356.0i − 1.24309i
\(438\) 189.450 0.0206673
\(439\) −6154.49 −0.669106 −0.334553 0.942377i \(-0.608585\pi\)
−0.334553 + 0.942377i \(0.608585\pi\)
\(440\) 2767.56i 0.299859i
\(441\) −4207.17 −0.454289
\(442\) 0 0
\(443\) −14539.3 −1.55933 −0.779663 0.626200i \(-0.784609\pi\)
−0.779663 + 0.626200i \(0.784609\pi\)
\(444\) 123.467i 0.0131970i
\(445\) −12789.4 −1.36242
\(446\) −464.521 −0.0493177
\(447\) − 7278.90i − 0.770202i
\(448\) 2391.03i 0.252155i
\(449\) − 7043.87i − 0.740358i −0.928960 0.370179i \(-0.879296\pi\)
0.928960 0.370179i \(-0.120704\pi\)
\(450\) 1130.68i 0.118446i
\(451\) −5107.33 −0.533248
\(452\) 9875.90 1.02771
\(453\) − 6644.32i − 0.689133i
\(454\) 1518.87 0.157013
\(455\) 0 0
\(456\) 2063.66 0.211929
\(457\) − 14098.9i − 1.44314i −0.692340 0.721572i \(-0.743420\pi\)
0.692340 0.721572i \(-0.256580\pi\)
\(458\) 1019.23 0.103986
\(459\) −10126.1 −1.02973
\(460\) − 19539.3i − 1.98049i
\(461\) − 14449.7i − 1.45985i −0.683529 0.729924i \(-0.739556\pi\)
0.683529 0.729924i \(-0.260444\pi\)
\(462\) − 197.010i − 0.0198393i
\(463\) 15806.5i 1.58659i 0.608840 + 0.793293i \(0.291635\pi\)
−0.608840 + 0.793293i \(0.708365\pi\)
\(464\) −6340.06 −0.634332
\(465\) 18120.1 1.80709
\(466\) 1635.85i 0.162617i
\(467\) 15071.3 1.49340 0.746699 0.665162i \(-0.231638\pi\)
0.746699 + 0.665162i \(0.231638\pi\)
\(468\) 0 0
\(469\) −2311.89 −0.227619
\(470\) − 2487.69i − 0.244146i
\(471\) 1463.43 0.143166
\(472\) 2017.68 0.196761
\(473\) 617.299i 0.0600073i
\(474\) − 327.083i − 0.0316950i
\(475\) − 15524.5i − 1.49961i
\(476\) − 2886.78i − 0.277973i
\(477\) 908.169 0.0871744
\(478\) 2650.18 0.253591
\(479\) − 392.545i − 0.0374443i −0.999825 0.0187222i \(-0.994040\pi\)
0.999825 0.0187222i \(-0.00595980\pi\)
\(480\) 5347.73 0.508519
\(481\) 0 0
\(482\) 2268.51 0.214373
\(483\) 2816.08i 0.265292i
\(484\) 6466.36 0.607284
\(485\) 13522.7 1.26605
\(486\) − 1462.12i − 0.136467i
\(487\) − 9497.89i − 0.883758i −0.897075 0.441879i \(-0.854312\pi\)
0.897075 0.441879i \(-0.145688\pi\)
\(488\) 4597.32i 0.426457i
\(489\) − 3468.71i − 0.320778i
\(490\) −2447.14 −0.225613
\(491\) 1893.82 0.174067 0.0870337 0.996205i \(-0.472261\pi\)
0.0870337 + 0.996205i \(0.472261\pi\)
\(492\) 6552.67i 0.600442i
\(493\) 7253.52 0.662641
\(494\) 0 0
\(495\) −5360.04 −0.486699
\(496\) 16410.1i 1.48555i
\(497\) −831.881 −0.0750804
\(498\) −543.068 −0.0488664
\(499\) 13370.1i 1.19945i 0.800205 + 0.599727i \(0.204724\pi\)
−0.800205 + 0.599727i \(0.795276\pi\)
\(500\) − 9331.79i − 0.834661i
\(501\) − 13561.1i − 1.20932i
\(502\) − 2464.39i − 0.219106i
\(503\) 5554.71 0.492391 0.246195 0.969220i \(-0.420820\pi\)
0.246195 + 0.969220i \(0.420820\pi\)
\(504\) 505.966 0.0447173
\(505\) 6209.51i 0.547168i
\(506\) −1381.62 −0.121385
\(507\) 0 0
\(508\) −20332.3 −1.77578
\(509\) − 2197.55i − 0.191365i −0.995412 0.0956824i \(-0.969497\pi\)
0.995412 0.0956824i \(-0.0305033\pi\)
\(510\) −1955.85 −0.169817
\(511\) −637.756 −0.0552107
\(512\) 8137.89i 0.702437i
\(513\) 12036.0i 1.03587i
\(514\) 734.022i 0.0629890i
\(515\) 10333.9i 0.884206i
\(516\) 791.991 0.0675687
\(517\) 7144.45 0.607761
\(518\) 10.2334i 0 0.000868010i
\(519\) 5242.44 0.443386
\(520\) 0 0
\(521\) 17005.2 1.42997 0.714983 0.699142i \(-0.246435\pi\)
0.714983 + 0.699142i \(0.246435\pi\)
\(522\) 627.932i 0.0526511i
\(523\) −14486.2 −1.21116 −0.605581 0.795783i \(-0.707059\pi\)
−0.605581 + 0.795783i \(0.707059\pi\)
\(524\) 16643.8 1.38758
\(525\) 3849.79i 0.320035i
\(526\) − 2766.24i − 0.229304i
\(527\) − 18774.4i − 1.55185i
\(528\) 4909.68i 0.404671i
\(529\) 7582.03 0.623163
\(530\) 528.244 0.0432933
\(531\) 3907.72i 0.319361i
\(532\) −3431.27 −0.279632
\(533\) 0 0
\(534\) 1160.26 0.0940252
\(535\) 10177.0i 0.822413i
\(536\) −2946.32 −0.237429
\(537\) 4303.28 0.345810
\(538\) − 1088.55i − 0.0872315i
\(539\) − 7027.98i − 0.561626i
\(540\) 20709.3i 1.65034i
\(541\) − 15266.7i − 1.21325i −0.794990 0.606623i \(-0.792524\pi\)
0.794990 0.606623i \(-0.207476\pi\)
\(542\) −1243.31 −0.0985330
\(543\) −4178.27 −0.330215
\(544\) − 5540.83i − 0.436693i
\(545\) 3134.23 0.246341
\(546\) 0 0
\(547\) 15260.5 1.19286 0.596430 0.802665i \(-0.296586\pi\)
0.596430 + 0.802665i \(0.296586\pi\)
\(548\) 5371.62i 0.418731i
\(549\) −8903.81 −0.692177
\(550\) −1888.78 −0.146432
\(551\) − 8621.64i − 0.666595i
\(552\) 3588.87i 0.276726i
\(553\) 1101.08i 0.0846703i
\(554\) 1682.55i 0.129033i
\(555\) −281.601 −0.0215374
\(556\) −5306.00 −0.404721
\(557\) − 10442.1i − 0.794337i −0.917746 0.397169i \(-0.869993\pi\)
0.917746 0.397169i \(-0.130007\pi\)
\(558\) 1625.29 0.123304
\(559\) 0 0
\(560\) −5754.94 −0.434269
\(561\) − 5617.06i − 0.422731i
\(562\) 3999.57 0.300199
\(563\) −7145.26 −0.534879 −0.267440 0.963575i \(-0.586178\pi\)
−0.267440 + 0.963575i \(0.586178\pi\)
\(564\) − 9166.29i − 0.684344i
\(565\) 22524.7i 1.67721i
\(566\) − 932.950i − 0.0692841i
\(567\) − 1013.65i − 0.0750783i
\(568\) −1060.17 −0.0783162
\(569\) −4438.86 −0.327042 −0.163521 0.986540i \(-0.552285\pi\)
−0.163521 + 0.986540i \(0.552285\pi\)
\(570\) 2324.75i 0.170830i
\(571\) −10117.3 −0.741497 −0.370748 0.928733i \(-0.620899\pi\)
−0.370748 + 0.928733i \(0.620899\pi\)
\(572\) 0 0
\(573\) −9882.70 −0.720516
\(574\) 543.109i 0.0394929i
\(575\) 26998.4 1.95810
\(576\) −5901.58 −0.426909
\(577\) − 3105.60i − 0.224069i −0.993704 0.112035i \(-0.964263\pi\)
0.993704 0.112035i \(-0.0357368\pi\)
\(578\) − 127.615i − 0.00918352i
\(579\) 7261.23i 0.521186i
\(580\) − 14834.5i − 1.06202i
\(581\) 1828.16 0.130542
\(582\) −1226.78 −0.0873741
\(583\) 1517.08i 0.107772i
\(584\) −812.769 −0.0575901
\(585\) 0 0
\(586\) −3628.05 −0.255757
\(587\) − 19662.3i − 1.38254i −0.722597 0.691270i \(-0.757052\pi\)
0.722597 0.691270i \(-0.242948\pi\)
\(588\) −9016.86 −0.632396
\(589\) −22315.5 −1.56111
\(590\) 2272.95i 0.158603i
\(591\) − 14797.8i − 1.02995i
\(592\) − 255.026i − 0.0177052i
\(593\) 6395.51i 0.442888i 0.975173 + 0.221444i \(0.0710769\pi\)
−0.975173 + 0.221444i \(0.928923\pi\)
\(594\) 1464.35 0.101150
\(595\) 6584.10 0.453650
\(596\) 15423.9i 1.06005i
\(597\) −15571.6 −1.06751
\(598\) 0 0
\(599\) 8878.48 0.605618 0.302809 0.953051i \(-0.402076\pi\)
0.302809 + 0.953051i \(0.402076\pi\)
\(600\) 4906.25i 0.333828i
\(601\) 19100.6 1.29639 0.648194 0.761475i \(-0.275525\pi\)
0.648194 + 0.761475i \(0.275525\pi\)
\(602\) 65.6430 0.00444420
\(603\) − 5706.26i − 0.385368i
\(604\) 14079.3i 0.948472i
\(605\) 14748.3i 0.991082i
\(606\) − 563.330i − 0.0377619i
\(607\) 16595.8 1.10972 0.554861 0.831943i \(-0.312771\pi\)
0.554861 + 0.831943i \(0.312771\pi\)
\(608\) −6585.91 −0.439299
\(609\) 2138.01i 0.142260i
\(610\) −5178.97 −0.343755
\(611\) 0 0
\(612\) 7125.21 0.470620
\(613\) − 16469.2i − 1.08513i −0.840015 0.542564i \(-0.817454\pi\)
0.840015 0.542564i \(-0.182546\pi\)
\(614\) 1583.01 0.104047
\(615\) −14945.2 −0.979915
\(616\) 845.205i 0.0552829i
\(617\) − 10116.0i − 0.660055i −0.943971 0.330027i \(-0.892942\pi\)
0.943971 0.330027i \(-0.107058\pi\)
\(618\) − 937.496i − 0.0610220i
\(619\) 18854.8i 1.22430i 0.790743 + 0.612148i \(0.209694\pi\)
−0.790743 + 0.612148i \(0.790306\pi\)
\(620\) −38396.3 −2.48715
\(621\) −20931.6 −1.35258
\(622\) − 1460.49i − 0.0941487i
\(623\) −3905.86 −0.251180
\(624\) 0 0
\(625\) −2730.82 −0.174773
\(626\) − 157.019i − 0.0100252i
\(627\) −6676.51 −0.425254
\(628\) −3100.99 −0.197043
\(629\) 291.769i 0.0184954i
\(630\) 569.981i 0.0360454i
\(631\) 18946.2i 1.19531i 0.801755 + 0.597653i \(0.203900\pi\)
−0.801755 + 0.597653i \(0.796100\pi\)
\(632\) 1403.24i 0.0883194i
\(633\) −5028.33 −0.315732
\(634\) 1336.15 0.0836991
\(635\) − 46373.3i − 2.89806i
\(636\) 1946.40 0.121352
\(637\) 0 0
\(638\) −1048.95 −0.0650912
\(639\) − 2053.27i − 0.127114i
\(640\) −15043.5 −0.929135
\(641\) 23586.9 1.45340 0.726698 0.686957i \(-0.241054\pi\)
0.726698 + 0.686957i \(0.241054\pi\)
\(642\) − 923.263i − 0.0567575i
\(643\) 27153.0i 1.66534i 0.553772 + 0.832669i \(0.313188\pi\)
−0.553772 + 0.832669i \(0.686812\pi\)
\(644\) − 5967.25i − 0.365128i
\(645\) 1806.35i 0.110272i
\(646\) 2408.70 0.146701
\(647\) −6856.72 −0.416639 −0.208319 0.978061i \(-0.566799\pi\)
−0.208319 + 0.978061i \(0.566799\pi\)
\(648\) − 1291.82i − 0.0783140i
\(649\) −6527.75 −0.394817
\(650\) 0 0
\(651\) 5533.83 0.333161
\(652\) 7350.17i 0.441495i
\(653\) −8073.89 −0.483853 −0.241926 0.970295i \(-0.577779\pi\)
−0.241926 + 0.970295i \(0.577779\pi\)
\(654\) −284.339 −0.0170008
\(655\) 37960.9i 2.26451i
\(656\) − 13534.8i − 0.805555i
\(657\) − 1574.12i − 0.0934739i
\(658\) − 759.734i − 0.0450114i
\(659\) 5305.73 0.313629 0.156815 0.987628i \(-0.449877\pi\)
0.156815 + 0.987628i \(0.449877\pi\)
\(660\) −11487.7 −0.677512
\(661\) 25848.3i 1.52100i 0.649336 + 0.760502i \(0.275047\pi\)
−0.649336 + 0.760502i \(0.724953\pi\)
\(662\) 3373.75 0.198073
\(663\) 0 0
\(664\) 2329.85 0.136168
\(665\) − 7825.96i − 0.456357i
\(666\) −25.2583 −0.00146958
\(667\) 14993.7 0.870404
\(668\) 28735.9i 1.66441i
\(669\) − 3903.78i − 0.225604i
\(670\) − 3319.09i − 0.191385i
\(671\) − 14873.6i − 0.855722i
\(672\) 1633.18 0.0937521
\(673\) 14529.1 0.832177 0.416089 0.909324i \(-0.363401\pi\)
0.416089 + 0.909324i \(0.363401\pi\)
\(674\) − 2066.06i − 0.118073i
\(675\) −28615.0 −1.63169
\(676\) 0 0
\(677\) 12058.1 0.684535 0.342267 0.939603i \(-0.388805\pi\)
0.342267 + 0.939603i \(0.388805\pi\)
\(678\) − 2043.45i − 0.115750i
\(679\) 4129.78 0.233412
\(680\) 8390.91 0.473201
\(681\) 12764.4i 0.718255i
\(682\) 2715.00i 0.152438i
\(683\) 30028.8i 1.68231i 0.540792 + 0.841156i \(0.318125\pi\)
−0.540792 + 0.841156i \(0.681875\pi\)
\(684\) − 8469.13i − 0.473429i
\(685\) −12251.5 −0.683364
\(686\) −1565.22 −0.0871144
\(687\) 8565.50i 0.475683i
\(688\) −1635.89 −0.0906505
\(689\) 0 0
\(690\) −4042.94 −0.223061
\(691\) − 449.696i − 0.0247572i −0.999923 0.0123786i \(-0.996060\pi\)
0.999923 0.0123786i \(-0.00394034\pi\)
\(692\) −11108.7 −0.610244
\(693\) −1636.94 −0.0897291
\(694\) − 2307.10i − 0.126191i
\(695\) − 12101.8i − 0.660500i
\(696\) 2724.72i 0.148391i
\(697\) 15484.8i 0.841506i
\(698\) −22.0690 −0.00119674
\(699\) −13747.5 −0.743889
\(700\) − 8157.67i − 0.440473i
\(701\) 26986.0 1.45399 0.726994 0.686644i \(-0.240917\pi\)
0.726994 + 0.686644i \(0.240917\pi\)
\(702\) 0 0
\(703\) 346.801 0.0186057
\(704\) − 9858.46i − 0.527776i
\(705\) 20906.2 1.11684
\(706\) 3971.29 0.211702
\(707\) 1896.37i 0.100877i
\(708\) 8375.06i 0.444568i
\(709\) − 9098.87i − 0.481968i −0.970529 0.240984i \(-0.922530\pi\)
0.970529 0.240984i \(-0.0774701\pi\)
\(710\) − 1194.30i − 0.0631285i
\(711\) −2717.71 −0.143350
\(712\) −4977.71 −0.262005
\(713\) − 38808.4i − 2.03841i
\(714\) −597.312 −0.0313079
\(715\) 0 0
\(716\) −9118.62 −0.475948
\(717\) 22271.8i 1.16005i
\(718\) −3147.11 −0.163578
\(719\) 6293.55 0.326439 0.163220 0.986590i \(-0.447812\pi\)
0.163220 + 0.986590i \(0.447812\pi\)
\(720\) − 14204.5i − 0.735235i
\(721\) 3155.95i 0.163015i
\(722\) 144.295i 0.00743783i
\(723\) 19064.3i 0.980648i
\(724\) 8853.72 0.454483
\(725\) 20497.5 1.05001
\(726\) − 1337.97i − 0.0683979i
\(727\) −18070.7 −0.921878 −0.460939 0.887432i \(-0.652487\pi\)
−0.460939 + 0.887432i \(0.652487\pi\)
\(728\) 0 0
\(729\) 17319.9 0.879944
\(730\) − 915.601i − 0.0464218i
\(731\) 1871.58 0.0946962
\(732\) −19082.7 −0.963550
\(733\) − 34771.5i − 1.75214i −0.482188 0.876068i \(-0.660158\pi\)
0.482188 0.876068i \(-0.339842\pi\)
\(734\) 1755.61i 0.0882842i
\(735\) − 20565.4i − 1.03206i
\(736\) − 11453.4i − 0.573613i
\(737\) 9532.17 0.476421
\(738\) −1340.51 −0.0668631
\(739\) 23631.5i 1.17632i 0.808746 + 0.588158i \(0.200147\pi\)
−0.808746 + 0.588158i \(0.799853\pi\)
\(740\) 596.710 0.0296425
\(741\) 0 0
\(742\) 161.324 0.00798167
\(743\) − 32502.8i − 1.60486i −0.596745 0.802431i \(-0.703540\pi\)
0.596745 0.802431i \(-0.296460\pi\)
\(744\) 7052.42 0.347519
\(745\) −35178.6 −1.72999
\(746\) − 4390.69i − 0.215489i
\(747\) 4512.31i 0.221013i
\(748\) 11902.5i 0.581816i
\(749\) 3108.04i 0.151622i
\(750\) −1930.87 −0.0940072
\(751\) −2020.86 −0.0981920 −0.0490960 0.998794i \(-0.515634\pi\)
−0.0490960 + 0.998794i \(0.515634\pi\)
\(752\) 18933.3i 0.918120i
\(753\) 20710.5 1.00230
\(754\) 0 0
\(755\) −32111.6 −1.54790
\(756\) 6324.56i 0.304262i
\(757\) 12568.2 0.603434 0.301717 0.953398i \(-0.402440\pi\)
0.301717 + 0.953398i \(0.402440\pi\)
\(758\) −3581.73 −0.171628
\(759\) − 11611.0i − 0.555273i
\(760\) − 9973.56i − 0.476025i
\(761\) − 8704.81i − 0.414651i −0.978272 0.207325i \(-0.933524\pi\)
0.978272 0.207325i \(-0.0664758\pi\)
\(762\) 4207.01i 0.200005i
\(763\) 957.187 0.0454161
\(764\) 20941.4 0.991665
\(765\) 16251.0i 0.768048i
\(766\) −3205.16 −0.151184
\(767\) 0 0
\(768\) −11595.0 −0.544790
\(769\) 21915.9i 1.02771i 0.857878 + 0.513853i \(0.171782\pi\)
−0.857878 + 0.513853i \(0.828218\pi\)
\(770\) −952.140 −0.0445620
\(771\) −6168.63 −0.288143
\(772\) − 15386.5i − 0.717322i
\(773\) 23077.5i 1.07379i 0.843649 + 0.536896i \(0.180403\pi\)
−0.843649 + 0.536896i \(0.819597\pi\)
\(774\) 162.022i 0.00752422i
\(775\) − 53054.0i − 2.45904i
\(776\) 5263.08 0.243471
\(777\) −86.0001 −0.00397070
\(778\) − 3851.96i − 0.177506i
\(779\) 18405.5 0.846528
\(780\) 0 0
\(781\) 3429.94 0.157148
\(782\) 4188.92i 0.191554i
\(783\) −15891.5 −0.725309
\(784\) 18624.6 0.848426
\(785\) − 7072.67i − 0.321572i
\(786\) − 3443.83i − 0.156282i
\(787\) − 16522.4i − 0.748362i −0.927356 0.374181i \(-0.877924\pi\)
0.927356 0.374181i \(-0.122076\pi\)
\(788\) 31356.4i 1.41754i
\(789\) 23247.2 1.04895
\(790\) −1580.78 −0.0711918
\(791\) 6879.00i 0.309215i
\(792\) −2086.15 −0.0935962
\(793\) 0 0
\(794\) −4939.89 −0.220794
\(795\) 4439.30i 0.198045i
\(796\) 32996.1 1.46924
\(797\) 11719.4 0.520855 0.260427 0.965493i \(-0.416137\pi\)
0.260427 + 0.965493i \(0.416137\pi\)
\(798\) 709.974i 0.0314948i
\(799\) − 21661.2i − 0.959095i
\(800\) − 15657.7i − 0.691978i
\(801\) − 9640.53i − 0.425258i
\(802\) −691.096 −0.0304282
\(803\) 2629.53 0.115559
\(804\) − 12229.7i − 0.536454i
\(805\) 13610.0 0.595886
\(806\) 0 0
\(807\) 9148.01 0.399040
\(808\) 2416.77i 0.105225i
\(809\) −24096.0 −1.04718 −0.523592 0.851969i \(-0.675408\pi\)
−0.523592 + 0.851969i \(0.675408\pi\)
\(810\) 1455.26 0.0631267
\(811\) − 16622.6i − 0.719729i −0.933005 0.359864i \(-0.882823\pi\)
0.933005 0.359864i \(-0.117177\pi\)
\(812\) − 4530.42i − 0.195796i
\(813\) − 10448.7i − 0.450739i
\(814\) − 42.1933i − 0.00181680i
\(815\) −16764.1 −0.720516
\(816\) 14885.6 0.638603
\(817\) − 2224.59i − 0.0952613i
\(818\) 2962.05 0.126609
\(819\) 0 0
\(820\) 31668.7 1.34868
\(821\) 38005.5i 1.61559i 0.589461 + 0.807797i \(0.299340\pi\)
−0.589461 + 0.807797i \(0.700660\pi\)
\(822\) 1111.46 0.0471613
\(823\) 15859.5 0.671722 0.335861 0.941912i \(-0.390973\pi\)
0.335861 + 0.941912i \(0.390973\pi\)
\(824\) 4022.01i 0.170040i
\(825\) − 15873.1i − 0.669854i
\(826\) 694.155i 0.0292406i
\(827\) 12201.0i 0.513023i 0.966541 + 0.256512i \(0.0825732\pi\)
−0.966541 + 0.256512i \(0.917427\pi\)
\(828\) 14728.5 0.618177
\(829\) 5431.41 0.227552 0.113776 0.993506i \(-0.463705\pi\)
0.113776 + 0.993506i \(0.463705\pi\)
\(830\) 2624.62i 0.109761i
\(831\) −14139.9 −0.590263
\(832\) 0 0
\(833\) −21308.0 −0.886290
\(834\) 1097.88i 0.0455834i
\(835\) −65540.3 −2.71630
\(836\) 14147.5 0.585288
\(837\) 41132.2i 1.69861i
\(838\) 4716.02i 0.194406i
\(839\) − 7960.90i − 0.327582i −0.986495 0.163791i \(-0.947628\pi\)
0.986495 0.163791i \(-0.0523722\pi\)
\(840\) 2473.26i 0.101590i
\(841\) −13005.6 −0.533255
\(842\) 3457.61 0.141517
\(843\) 33611.9i 1.37326i
\(844\) 10655.0 0.434550
\(845\) 0 0
\(846\) 1875.19 0.0762062
\(847\) 4504.10i 0.182719i
\(848\) −4020.35 −0.162806
\(849\) 7840.40 0.316940
\(850\) 5726.56i 0.231082i
\(851\) 603.115i 0.0242944i
\(852\) − 4400.59i − 0.176950i
\(853\) 13576.7i 0.544969i 0.962160 + 0.272485i \(0.0878454\pi\)
−0.962160 + 0.272485i \(0.912155\pi\)
\(854\) −1581.65 −0.0633756
\(855\) 19316.2 0.772631
\(856\) 3960.95i 0.158157i
\(857\) 31223.9 1.24456 0.622281 0.782794i \(-0.286206\pi\)
0.622281 + 0.782794i \(0.286206\pi\)
\(858\) 0 0
\(859\) −11815.8 −0.469323 −0.234661 0.972077i \(-0.575398\pi\)
−0.234661 + 0.972077i \(0.575398\pi\)
\(860\) − 3827.65i − 0.151770i
\(861\) −4564.22 −0.180660
\(862\) −6175.36 −0.244006
\(863\) 1790.84i 0.0706384i 0.999376 + 0.0353192i \(0.0112448\pi\)
−0.999376 + 0.0353192i \(0.988755\pi\)
\(864\) 12139.2i 0.477992i
\(865\) − 25336.4i − 0.995912i
\(866\) − 817.328i − 0.0320715i
\(867\) 1072.46 0.0420100
\(868\) −11726.1 −0.458538
\(869\) − 4539.87i − 0.177220i
\(870\) −3069.45 −0.119614
\(871\) 0 0
\(872\) 1219.86 0.0473734
\(873\) 10193.2i 0.395176i
\(874\) 4979.01 0.192698
\(875\) 6500.00 0.251132
\(876\) − 3373.68i − 0.130121i
\(877\) − 43542.5i − 1.67654i −0.545255 0.838270i \(-0.683567\pi\)
0.545255 0.838270i \(-0.316433\pi\)
\(878\) − 2698.42i − 0.103721i
\(879\) − 30489.7i − 1.16996i
\(880\) 23728.2 0.908953
\(881\) 1020.04 0.0390080 0.0195040 0.999810i \(-0.493791\pi\)
0.0195040 + 0.999810i \(0.493791\pi\)
\(882\) − 1844.62i − 0.0704214i
\(883\) 34781.9 1.32560 0.662800 0.748797i \(-0.269368\pi\)
0.662800 + 0.748797i \(0.269368\pi\)
\(884\) 0 0
\(885\) −19101.6 −0.725531
\(886\) − 6374.70i − 0.241718i
\(887\) 49785.1 1.88458 0.942288 0.334802i \(-0.108670\pi\)
0.942288 + 0.334802i \(0.108670\pi\)
\(888\) −109.600 −0.00414183
\(889\) − 14162.3i − 0.534295i
\(890\) − 5607.49i − 0.211195i
\(891\) 4179.40i 0.157144i
\(892\) 8272.08i 0.310504i
\(893\) −25746.8 −0.964818
\(894\) 3191.41 0.119392
\(895\) − 20797.5i − 0.776743i
\(896\) −4594.25 −0.171298
\(897\) 0 0
\(898\) 3088.36 0.114766
\(899\) − 29463.9i − 1.09308i
\(900\) 20134.9 0.745738
\(901\) 4599.60 0.170072
\(902\) − 2239.29i − 0.0826611i
\(903\) 551.656i 0.0203300i
\(904\) 8766.74i 0.322541i
\(905\) 20193.3i 0.741712i
\(906\) 2913.18 0.106826
\(907\) −17389.9 −0.636627 −0.318314 0.947985i \(-0.603116\pi\)
−0.318314 + 0.947985i \(0.603116\pi\)
\(908\) − 27047.6i − 0.988553i
\(909\) −4680.66 −0.170790
\(910\) 0 0
\(911\) 20419.5 0.742621 0.371311 0.928509i \(-0.378909\pi\)
0.371311 + 0.928509i \(0.378909\pi\)
\(912\) − 17693.2i − 0.642414i
\(913\) −7537.71 −0.273233
\(914\) 6181.60 0.223708
\(915\) − 43523.5i − 1.57250i
\(916\) − 18150.2i − 0.654695i
\(917\) 11593.2i 0.417492i
\(918\) − 4439.75i − 0.159623i
\(919\) −33231.8 −1.19283 −0.596417 0.802674i \(-0.703410\pi\)
−0.596417 + 0.802674i \(0.703410\pi\)
\(920\) 17344.8 0.621567
\(921\) 13303.4i 0.475964i
\(922\) 6335.43 0.226297
\(923\) 0 0
\(924\) −3508.31 −0.124908
\(925\) 824.502i 0.0293075i
\(926\) −6930.31 −0.245944
\(927\) −7789.58 −0.275991
\(928\) − 8695.59i − 0.307594i
\(929\) 25222.8i 0.890780i 0.895337 + 0.445390i \(0.146935\pi\)
−0.895337 + 0.445390i \(0.853065\pi\)
\(930\) 7944.70i 0.280126i
\(931\) 25327.0i 0.891579i
\(932\) 29130.9 1.02383
\(933\) 12273.8 0.430683
\(934\) 6607.97i 0.231498i
\(935\) −27146.9 −0.949519
\(936\) 0 0
\(937\) −26979.4 −0.940639 −0.470319 0.882496i \(-0.655861\pi\)
−0.470319 + 0.882496i \(0.655861\pi\)
\(938\) − 1013.64i − 0.0352842i
\(939\) 1319.57 0.0458600
\(940\) −44300.2 −1.53714
\(941\) − 7641.67i − 0.264730i −0.991201 0.132365i \(-0.957743\pi\)
0.991201 0.132365i \(-0.0422572\pi\)
\(942\) 641.635i 0.0221928i
\(943\) 32008.7i 1.10535i
\(944\) − 17299.0i − 0.596434i
\(945\) −14424.9 −0.496553
\(946\) −270.653 −0.00930200
\(947\) − 2869.32i − 0.0984587i −0.998788 0.0492293i \(-0.984323\pi\)
0.998788 0.0492293i \(-0.0156765\pi\)
\(948\) −5824.62 −0.199552
\(949\) 0 0
\(950\) 6806.67 0.232461
\(951\) 11228.8i 0.382881i
\(952\) 2562.56 0.0872407
\(953\) −12313.6 −0.418548 −0.209274 0.977857i \(-0.567110\pi\)
−0.209274 + 0.977857i \(0.567110\pi\)
\(954\) 398.184i 0.0135133i
\(955\) 47762.6i 1.61839i
\(956\) − 47193.8i − 1.59661i
\(957\) − 8815.22i − 0.297759i
\(958\) 172.110 0.00580442
\(959\) −3741.57 −0.125987
\(960\) − 28848.0i − 0.969861i
\(961\) −46470.7 −1.55989
\(962\) 0 0
\(963\) −7671.32 −0.256703
\(964\) − 40397.1i − 1.34969i
\(965\) 35093.2 1.17066
\(966\) −1234.70 −0.0411241
\(967\) 17838.0i 0.593207i 0.955001 + 0.296603i \(0.0958540\pi\)
−0.955001 + 0.296603i \(0.904146\pi\)
\(968\) 5740.12i 0.190593i
\(969\) 20242.4i 0.671084i
\(970\) 5928.97i 0.196255i
\(971\) 41525.3 1.37241 0.686206 0.727408i \(-0.259275\pi\)
0.686206 + 0.727408i \(0.259275\pi\)
\(972\) −26037.1 −0.859199
\(973\) − 3695.86i − 0.121772i
\(974\) 4164.32 0.136995
\(975\) 0 0
\(976\) 39416.1 1.29270
\(977\) 31654.4i 1.03655i 0.855213 + 0.518277i \(0.173426\pi\)
−0.855213 + 0.518277i \(0.826574\pi\)
\(978\) 1520.85 0.0497253
\(979\) 16104.3 0.525735
\(980\) 43578.0i 1.42046i
\(981\) 2362.55i 0.0768913i
\(982\) 830.342i 0.0269830i
\(983\) − 39913.2i − 1.29505i −0.762045 0.647525i \(-0.775804\pi\)
0.762045 0.647525i \(-0.224196\pi\)
\(984\) −5816.74 −0.188446
\(985\) −71516.8 −2.31342
\(986\) 3180.28i 0.102719i
\(987\) 6384.72 0.205905
\(988\) 0 0
\(989\) 3868.74 0.124387
\(990\) − 2350.09i − 0.0754453i
\(991\) −2700.94 −0.0865773 −0.0432887 0.999063i \(-0.513784\pi\)
−0.0432887 + 0.999063i \(0.513784\pi\)
\(992\) −22506.9 −0.720358
\(993\) 28352.6i 0.906085i
\(994\) − 364.736i − 0.0116386i
\(995\) 75256.6i 2.39778i
\(996\) 9670.83i 0.307663i
\(997\) −9729.08 −0.309050 −0.154525 0.987989i \(-0.549385\pi\)
−0.154525 + 0.987989i \(0.549385\pi\)
\(998\) −5862.08 −0.185933
\(999\) − 639.228i − 0.0202445i
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 169.4.b.e.168.3 4
13.2 odd 12 169.4.c.f.22.2 4
13.3 even 3 169.4.e.g.147.2 8
13.4 even 6 169.4.e.g.23.2 8
13.5 odd 4 169.4.a.j.1.1 2
13.6 odd 12 169.4.c.f.146.2 4
13.7 odd 12 13.4.c.b.3.1 4
13.8 odd 4 169.4.a.f.1.2 2
13.9 even 3 169.4.e.g.23.3 8
13.10 even 6 169.4.e.g.147.3 8
13.11 odd 12 13.4.c.b.9.1 yes 4
13.12 even 2 inner 169.4.b.e.168.2 4
39.5 even 4 1521.4.a.l.1.2 2
39.8 even 4 1521.4.a.t.1.1 2
39.11 even 12 117.4.g.d.100.2 4
39.20 even 12 117.4.g.d.55.2 4
52.7 even 12 208.4.i.e.81.1 4
52.11 even 12 208.4.i.e.113.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 13.7 odd 12
13.4.c.b.9.1 yes 4 13.11 odd 12
117.4.g.d.55.2 4 39.20 even 12
117.4.g.d.100.2 4 39.11 even 12
169.4.a.f.1.2 2 13.8 odd 4
169.4.a.j.1.1 2 13.5 odd 4
169.4.b.e.168.2 4 13.12 even 2 inner
169.4.b.e.168.3 4 1.1 even 1 trivial
169.4.c.f.22.2 4 13.2 odd 12
169.4.c.f.146.2 4 13.6 odd 12
169.4.e.g.23.2 8 13.4 even 6
169.4.e.g.23.3 8 13.9 even 3
169.4.e.g.147.2 8 13.3 even 3
169.4.e.g.147.3 8 13.10 even 6
208.4.i.e.81.1 4 52.7 even 12
208.4.i.e.113.1 4 52.11 even 12
1521.4.a.l.1.2 2 39.5 even 4
1521.4.a.t.1.1 2 39.8 even 4