Properties

Label 684.3.s.a.445.40
Level $684$
Weight $3$
Character 684.445
Analytic conductor $18.638$
Analytic rank $0$
Dimension $80$
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] = [684,3,Mod(445,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.445");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.s (of order \(6\), degree \(2\), 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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 445.40
Character \(\chi\) \(=\) 684.445
Dual form 684.3.s.a.601.40

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.99550 + 0.164289i) q^{3} +(-4.18843 - 7.25458i) q^{5} +(-4.50755 - 7.80731i) q^{7} +(8.94602 + 0.984254i) q^{9} +O(q^{10})\) \(q+(2.99550 + 0.164289i) q^{3} +(-4.18843 - 7.25458i) q^{5} +(-4.50755 - 7.80731i) q^{7} +(8.94602 + 0.984254i) q^{9} +(-1.48869 - 2.57849i) q^{11} +5.56875i q^{13} +(-11.3546 - 22.4192i) q^{15} +(-13.0392 + 22.5845i) q^{17} +(-10.0502 - 16.1243i) q^{19} +(-12.2197 - 24.1273i) q^{21} +18.9366 q^{23} +(-22.5860 + 39.1201i) q^{25} +(26.6361 + 4.41806i) q^{27} +(-13.6152 - 7.86072i) q^{29} +(-16.0762 - 9.28158i) q^{31} +(-4.03576 - 7.96844i) q^{33} +(-37.7592 + 65.4008i) q^{35} -32.1993i q^{37} +(-0.914885 + 16.6812i) q^{39} +(-62.9342 + 36.3351i) q^{41} -63.1803 q^{43} +(-30.3295 - 69.0221i) q^{45} +(-20.3688 + 35.2799i) q^{47} +(-16.1361 + 27.9485i) q^{49} +(-42.7692 + 65.5097i) q^{51} +(73.4198 - 42.3890i) q^{53} +(-12.4706 + 21.5997i) q^{55} +(-27.4562 - 49.9515i) q^{57} +(13.4480 - 7.76418i) q^{59} +(-8.32790 + 14.4244i) q^{61} +(-32.6403 - 74.2809i) q^{63} +(40.3990 - 23.3244i) q^{65} -63.1854i q^{67} +(56.7247 + 3.11108i) q^{69} +(84.2994 + 48.6703i) q^{71} +(20.4220 - 35.3719i) q^{73} +(-74.0832 + 113.473i) q^{75} +(-13.4207 + 23.2454i) q^{77} -107.749i q^{79} +(79.0625 + 17.6103i) q^{81} +(-57.3231 - 99.2866i) q^{83} +218.455 q^{85} +(-39.4928 - 25.7836i) q^{87} +(2.70957 - 1.56437i) q^{89} +(43.4770 - 25.1015i) q^{91} +(-46.6313 - 30.4441i) q^{93} +(-74.8808 + 140.446i) q^{95} -120.628i q^{97} +(-10.7800 - 24.5325i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 80 q - q^{7} + 4 q^{9} - 6 q^{11} + 33 q^{15} - 21 q^{17} - 20 q^{19} - 48 q^{23} - 200 q^{25} - 63 q^{27} - 27 q^{29} - 24 q^{31} + 27 q^{33} - 54 q^{35} - 81 q^{39} - 18 q^{41} - 152 q^{43} + 188 q^{45} - 12 q^{47} - 267 q^{49} - 126 q^{51} - 36 q^{53} + 126 q^{57} - 135 q^{59} - 7 q^{61} - 190 q^{63} - 288 q^{65} + 48 q^{69} - 81 q^{71} + 55 q^{73} + 165 q^{75} + 30 q^{77} + 28 q^{81} - 93 q^{83} + 306 q^{87} + 216 q^{89} + 96 q^{91} + 24 q^{93} + 288 q^{95} - 241 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.99550 + 0.164289i 0.998499 + 0.0547630i
\(4\) 0 0
\(5\) −4.18843 7.25458i −0.837687 1.45092i −0.891824 0.452383i \(-0.850574\pi\)
0.0541368 0.998534i \(-0.482759\pi\)
\(6\) 0 0
\(7\) −4.50755 7.80731i −0.643936 1.11533i −0.984546 0.175125i \(-0.943967\pi\)
0.340610 0.940205i \(-0.389366\pi\)
\(8\) 0 0
\(9\) 8.94602 + 0.984254i 0.994002 + 0.109362i
\(10\) 0 0
\(11\) −1.48869 2.57849i −0.135336 0.234408i 0.790390 0.612604i \(-0.209878\pi\)
−0.925726 + 0.378196i \(0.876545\pi\)
\(12\) 0 0
\(13\) 5.56875i 0.428366i 0.976794 + 0.214183i \(0.0687088\pi\)
−0.976794 + 0.214183i \(0.931291\pi\)
\(14\) 0 0
\(15\) −11.3546 22.4192i −0.756973 1.49461i
\(16\) 0 0
\(17\) −13.0392 + 22.5845i −0.767010 + 1.32850i 0.172167 + 0.985068i \(0.444923\pi\)
−0.939177 + 0.343433i \(0.888410\pi\)
\(18\) 0 0
\(19\) −10.0502 16.1243i −0.528957 0.848649i
\(20\) 0 0
\(21\) −12.2197 24.1273i −0.581891 1.14892i
\(22\) 0 0
\(23\) 18.9366 0.823332 0.411666 0.911335i \(-0.364947\pi\)
0.411666 + 0.911335i \(0.364947\pi\)
\(24\) 0 0
\(25\) −22.5860 + 39.1201i −0.903439 + 1.56480i
\(26\) 0 0
\(27\) 26.6361 + 4.41806i 0.986521 + 0.163632i
\(28\) 0 0
\(29\) −13.6152 7.86072i −0.469489 0.271059i 0.246537 0.969133i \(-0.420707\pi\)
−0.716026 + 0.698074i \(0.754041\pi\)
\(30\) 0 0
\(31\) −16.0762 9.28158i −0.518586 0.299406i 0.217770 0.976000i \(-0.430122\pi\)
−0.736356 + 0.676594i \(0.763455\pi\)
\(32\) 0 0
\(33\) −4.03576 7.96844i −0.122296 0.241468i
\(34\) 0 0
\(35\) −37.7592 + 65.4008i −1.07883 + 1.86859i
\(36\) 0 0
\(37\) 32.1993i 0.870251i −0.900370 0.435126i \(-0.856704\pi\)
0.900370 0.435126i \(-0.143296\pi\)
\(38\) 0 0
\(39\) −0.914885 + 16.6812i −0.0234586 + 0.427723i
\(40\) 0 0
\(41\) −62.9342 + 36.3351i −1.53498 + 0.886222i −0.535860 + 0.844307i \(0.680012\pi\)
−0.999121 + 0.0419144i \(0.986654\pi\)
\(42\) 0 0
\(43\) −63.1803 −1.46931 −0.734655 0.678441i \(-0.762656\pi\)
−0.734655 + 0.678441i \(0.762656\pi\)
\(44\) 0 0
\(45\) −30.3295 69.0221i −0.673988 1.53382i
\(46\) 0 0
\(47\) −20.3688 + 35.2799i −0.433379 + 0.750635i −0.997162 0.0752882i \(-0.976012\pi\)
0.563782 + 0.825923i \(0.309346\pi\)
\(48\) 0 0
\(49\) −16.1361 + 27.9485i −0.329307 + 0.570377i
\(50\) 0 0
\(51\) −42.7692 + 65.5097i −0.838612 + 1.28450i
\(52\) 0 0
\(53\) 73.4198 42.3890i 1.38528 0.799792i 0.392501 0.919752i \(-0.371610\pi\)
0.992779 + 0.119960i \(0.0382767\pi\)
\(54\) 0 0
\(55\) −12.4706 + 21.5997i −0.226738 + 0.392721i
\(56\) 0 0
\(57\) −27.4562 49.9515i −0.481688 0.876343i
\(58\) 0 0
\(59\) 13.4480 7.76418i 0.227931 0.131596i −0.381686 0.924292i \(-0.624656\pi\)
0.609617 + 0.792696i \(0.291323\pi\)
\(60\) 0 0
\(61\) −8.32790 + 14.4244i −0.136523 + 0.236465i −0.926178 0.377086i \(-0.876926\pi\)
0.789655 + 0.613551i \(0.210259\pi\)
\(62\) 0 0
\(63\) −32.6403 74.2809i −0.518100 1.17906i
\(64\) 0 0
\(65\) 40.3990 23.3244i 0.621523 0.358836i
\(66\) 0 0
\(67\) 63.1854i 0.943065i −0.881849 0.471533i \(-0.843701\pi\)
0.881849 0.471533i \(-0.156299\pi\)
\(68\) 0 0
\(69\) 56.7247 + 3.11108i 0.822097 + 0.0450881i
\(70\) 0 0
\(71\) 84.2994 + 48.6703i 1.18732 + 0.685497i 0.957695 0.287784i \(-0.0929184\pi\)
0.229620 + 0.973280i \(0.426252\pi\)
\(72\) 0 0
\(73\) 20.4220 35.3719i 0.279753 0.484547i −0.691570 0.722309i \(-0.743081\pi\)
0.971323 + 0.237763i \(0.0764140\pi\)
\(74\) 0 0
\(75\) −74.0832 + 113.473i −0.987776 + 1.51298i
\(76\) 0 0
\(77\) −13.4207 + 23.2454i −0.174295 + 0.301888i
\(78\) 0 0
\(79\) 107.749i 1.36391i −0.731394 0.681955i \(-0.761130\pi\)
0.731394 0.681955i \(-0.238870\pi\)
\(80\) 0 0
\(81\) 79.0625 + 17.6103i 0.976080 + 0.217411i
\(82\) 0 0
\(83\) −57.3231 99.2866i −0.690640 1.19622i −0.971628 0.236512i \(-0.923996\pi\)
0.280988 0.959711i \(-0.409338\pi\)
\(84\) 0 0
\(85\) 218.455 2.57006
\(86\) 0 0
\(87\) −39.4928 25.7836i −0.453940 0.296363i
\(88\) 0 0
\(89\) 2.70957 1.56437i 0.0304446 0.0175772i −0.484700 0.874680i \(-0.661071\pi\)
0.515145 + 0.857103i \(0.327738\pi\)
\(90\) 0 0
\(91\) 43.4770 25.1015i 0.477769 0.275840i
\(92\) 0 0
\(93\) −46.6313 30.4441i −0.501411 0.327356i
\(94\) 0 0
\(95\) −74.8808 + 140.446i −0.788219 + 1.47837i
\(96\) 0 0
\(97\) 120.628i 1.24359i −0.783181 0.621794i \(-0.786404\pi\)
0.783181 0.621794i \(-0.213596\pi\)
\(98\) 0 0
\(99\) −10.7800 24.5325i −0.108889 0.247803i
\(100\) 0 0
\(101\) −8.81674 + 15.2710i −0.0872944 + 0.151198i −0.906367 0.422492i \(-0.861155\pi\)
0.819072 + 0.573690i \(0.194489\pi\)
\(102\) 0 0
\(103\) 100.731 + 58.1573i 0.977976 + 0.564634i 0.901658 0.432449i \(-0.142351\pi\)
0.0763172 + 0.997084i \(0.475684\pi\)
\(104\) 0 0
\(105\) −123.852 + 189.705i −1.17954 + 1.80671i
\(106\) 0 0
\(107\) 5.14531i 0.0480870i 0.999711 + 0.0240435i \(0.00765403\pi\)
−0.999711 + 0.0240435i \(0.992346\pi\)
\(108\) 0 0
\(109\) −9.07991 5.24229i −0.0833019 0.0480944i 0.457770 0.889070i \(-0.348648\pi\)
−0.541072 + 0.840976i \(0.681981\pi\)
\(110\) 0 0
\(111\) 5.28999 96.4530i 0.0476576 0.868946i
\(112\) 0 0
\(113\) −177.083 102.239i −1.56711 0.904772i −0.996504 0.0835482i \(-0.973375\pi\)
−0.570607 0.821223i \(-0.693292\pi\)
\(114\) 0 0
\(115\) −79.3149 137.377i −0.689695 1.19459i
\(116\) 0 0
\(117\) −5.48107 + 49.8182i −0.0468468 + 0.425796i
\(118\) 0 0
\(119\) 235.099 1.97562
\(120\) 0 0
\(121\) 56.0676 97.1119i 0.463369 0.802578i
\(122\) 0 0
\(123\) −194.489 + 98.5023i −1.58121 + 0.800832i
\(124\) 0 0
\(125\) 168.978 1.35182
\(126\) 0 0
\(127\) 36.6506 21.1602i 0.288587 0.166616i −0.348717 0.937228i \(-0.613383\pi\)
0.637305 + 0.770612i \(0.280049\pi\)
\(128\) 0 0
\(129\) −189.257 10.3798i −1.46710 0.0804638i
\(130\) 0 0
\(131\) 108.540 + 187.997i 0.828552 + 1.43509i 0.899174 + 0.437591i \(0.144168\pi\)
−0.0706218 + 0.997503i \(0.522498\pi\)
\(132\) 0 0
\(133\) −80.5859 + 151.146i −0.605909 + 1.13644i
\(134\) 0 0
\(135\) −79.5123 211.738i −0.588980 1.56843i
\(136\) 0 0
\(137\) −29.4674 + 51.0391i −0.215091 + 0.372548i −0.953301 0.302023i \(-0.902338\pi\)
0.738210 + 0.674571i \(0.235671\pi\)
\(138\) 0 0
\(139\) 2.16605 0.0155831 0.00779155 0.999970i \(-0.497520\pi\)
0.00779155 + 0.999970i \(0.497520\pi\)
\(140\) 0 0
\(141\) −66.8109 + 102.334i −0.473836 + 0.725776i
\(142\) 0 0
\(143\) 14.3590 8.29016i 0.100412 0.0579731i
\(144\) 0 0
\(145\) 131.697i 0.908252i
\(146\) 0 0
\(147\) −52.9272 + 81.0686i −0.360049 + 0.551487i
\(148\) 0 0
\(149\) −71.1507 123.237i −0.477522 0.827092i 0.522146 0.852856i \(-0.325131\pi\)
−0.999668 + 0.0257641i \(0.991798\pi\)
\(150\) 0 0
\(151\) −247.779 + 143.055i −1.64092 + 0.947387i −0.660416 + 0.750900i \(0.729620\pi\)
−0.980506 + 0.196487i \(0.937047\pi\)
\(152\) 0 0
\(153\) −138.878 + 189.208i −0.907697 + 1.23665i
\(154\) 0 0
\(155\) 155.501i 1.00323i
\(156\) 0 0
\(157\) −87.4600 151.485i −0.557070 0.964874i −0.997739 0.0672042i \(-0.978592\pi\)
0.440669 0.897670i \(-0.354741\pi\)
\(158\) 0 0
\(159\) 226.893 114.914i 1.42700 0.722729i
\(160\) 0 0
\(161\) −85.3579 147.844i −0.530173 0.918287i
\(162\) 0 0
\(163\) 280.563 1.72124 0.860622 0.509244i \(-0.170075\pi\)
0.860622 + 0.509244i \(0.170075\pi\)
\(164\) 0 0
\(165\) −40.9042 + 62.6530i −0.247904 + 0.379715i
\(166\) 0 0
\(167\) 5.68711i 0.0340545i −0.999855 0.0170273i \(-0.994580\pi\)
0.999855 0.0170273i \(-0.00542021\pi\)
\(168\) 0 0
\(169\) 137.989 0.816503
\(170\) 0 0
\(171\) −74.0386 154.140i −0.432974 0.901406i
\(172\) 0 0
\(173\) 1.03532i 0.00598452i 0.999996 + 0.00299226i \(0.000952468\pi\)
−0.999996 + 0.00299226i \(0.999048\pi\)
\(174\) 0 0
\(175\) 407.230 2.32703
\(176\) 0 0
\(177\) 41.5589 21.0482i 0.234796 0.118917i
\(178\) 0 0
\(179\) 185.084i 1.03399i −0.855988 0.516996i \(-0.827050\pi\)
0.855988 0.516996i \(-0.172950\pi\)
\(180\) 0 0
\(181\) 77.7940 44.9144i 0.429801 0.248146i −0.269461 0.963011i \(-0.586845\pi\)
0.699262 + 0.714866i \(0.253512\pi\)
\(182\) 0 0
\(183\) −27.3160 + 41.8399i −0.149268 + 0.228634i
\(184\) 0 0
\(185\) −233.592 + 134.865i −1.26266 + 0.728998i
\(186\) 0 0
\(187\) 77.6452 0.415215
\(188\) 0 0
\(189\) −85.5703 227.871i −0.452753 1.20567i
\(190\) 0 0
\(191\) 107.195 + 185.668i 0.561233 + 0.972083i 0.997389 + 0.0722127i \(0.0230060\pi\)
−0.436157 + 0.899871i \(0.643661\pi\)
\(192\) 0 0
\(193\) 7.35082 4.24400i 0.0380871 0.0219896i −0.480836 0.876811i \(-0.659667\pi\)
0.518923 + 0.854821i \(0.326333\pi\)
\(194\) 0 0
\(195\) 124.847 63.2310i 0.640241 0.324261i
\(196\) 0 0
\(197\) −182.684 −0.927331 −0.463666 0.886010i \(-0.653466\pi\)
−0.463666 + 0.886010i \(0.653466\pi\)
\(198\) 0 0
\(199\) −117.295 203.161i −0.589422 1.02091i −0.994308 0.106541i \(-0.966022\pi\)
0.404886 0.914367i \(-0.367311\pi\)
\(200\) 0 0
\(201\) 10.3807 189.272i 0.0516451 0.941650i
\(202\) 0 0
\(203\) 141.731i 0.698180i
\(204\) 0 0
\(205\) 527.192 + 304.374i 2.57167 + 1.48475i
\(206\) 0 0
\(207\) 169.408 + 18.6385i 0.818394 + 0.0900409i
\(208\) 0 0
\(209\) −26.6148 + 49.9184i −0.127344 + 0.238844i
\(210\) 0 0
\(211\) −57.8006 + 33.3712i −0.273936 + 0.158157i −0.630675 0.776047i \(-0.717222\pi\)
0.356739 + 0.934204i \(0.383889\pi\)
\(212\) 0 0
\(213\) 244.523 + 159.641i 1.14799 + 0.749489i
\(214\) 0 0
\(215\) 264.627 + 458.347i 1.23082 + 2.13185i
\(216\) 0 0
\(217\) 167.349i 0.771193i
\(218\) 0 0
\(219\) 66.9852 102.601i 0.305869 0.468500i
\(220\) 0 0
\(221\) −125.768 72.6120i −0.569084 0.328561i
\(222\) 0 0
\(223\) 230.654i 1.03432i −0.855888 0.517161i \(-0.826989\pi\)
0.855888 0.517161i \(-0.173011\pi\)
\(224\) 0 0
\(225\) −240.559 + 327.738i −1.06915 + 1.45661i
\(226\) 0 0
\(227\) −76.8934 + 44.3944i −0.338737 + 0.195570i −0.659713 0.751517i \(-0.729322\pi\)
0.320976 + 0.947087i \(0.395989\pi\)
\(228\) 0 0
\(229\) 67.4283 116.789i 0.294447 0.509997i −0.680409 0.732832i \(-0.738198\pi\)
0.974856 + 0.222836i \(0.0715313\pi\)
\(230\) 0 0
\(231\) −44.0207 + 67.4265i −0.190566 + 0.291890i
\(232\) 0 0
\(233\) 33.1778 57.4656i 0.142394 0.246633i −0.786004 0.618222i \(-0.787853\pi\)
0.928398 + 0.371588i \(0.121187\pi\)
\(234\) 0 0
\(235\) 341.254 1.45215
\(236\) 0 0
\(237\) 17.7019 322.762i 0.0746918 1.36186i
\(238\) 0 0
\(239\) 157.173 272.231i 0.657626 1.13904i −0.323602 0.946193i \(-0.604894\pi\)
0.981228 0.192849i \(-0.0617728\pi\)
\(240\) 0 0
\(241\) −43.9174 25.3557i −0.182230 0.105210i 0.406110 0.913824i \(-0.366885\pi\)
−0.588340 + 0.808614i \(0.700218\pi\)
\(242\) 0 0
\(243\) 233.938 + 65.7408i 0.962709 + 0.270538i
\(244\) 0 0
\(245\) 270.339 1.10343
\(246\) 0 0
\(247\) 89.7924 55.9669i 0.363532 0.226587i
\(248\) 0 0
\(249\) −155.400 306.830i −0.624095 1.23225i
\(250\) 0 0
\(251\) 10.9851 + 19.0267i 0.0437653 + 0.0758038i 0.887078 0.461619i \(-0.152731\pi\)
−0.843313 + 0.537423i \(0.819398\pi\)
\(252\) 0 0
\(253\) −28.1908 48.8279i −0.111426 0.192996i
\(254\) 0 0
\(255\) 654.381 + 35.8897i 2.56620 + 0.140744i
\(256\) 0 0
\(257\) 410.241i 1.59627i 0.602479 + 0.798135i \(0.294180\pi\)
−0.602479 + 0.798135i \(0.705820\pi\)
\(258\) 0 0
\(259\) −251.390 + 145.140i −0.970618 + 0.560386i
\(260\) 0 0
\(261\) −114.065 83.7230i −0.437029 0.320778i
\(262\) 0 0
\(263\) 254.216 0.966600 0.483300 0.875455i \(-0.339438\pi\)
0.483300 + 0.875455i \(0.339438\pi\)
\(264\) 0 0
\(265\) −615.028 355.087i −2.32086 1.33995i
\(266\) 0 0
\(267\) 8.37352 4.24092i 0.0313615 0.0158836i
\(268\) 0 0
\(269\) −0.296857 0.171391i −0.00110356 0.000637140i 0.499448 0.866344i \(-0.333536\pi\)
−0.500552 + 0.865707i \(0.666869\pi\)
\(270\) 0 0
\(271\) 177.957 308.230i 0.656666 1.13738i −0.324807 0.945780i \(-0.605299\pi\)
0.981473 0.191599i \(-0.0613673\pi\)
\(272\) 0 0
\(273\) 134.359 68.0486i 0.492158 0.249262i
\(274\) 0 0
\(275\) 134.494 0.489070
\(276\) 0 0
\(277\) −146.001 252.882i −0.527081 0.912931i −0.999502 0.0315577i \(-0.989953\pi\)
0.472421 0.881373i \(-0.343380\pi\)
\(278\) 0 0
\(279\) −134.682 98.8562i −0.482732 0.354323i
\(280\) 0 0
\(281\) 293.470 + 169.435i 1.04438 + 0.602972i 0.921070 0.389396i \(-0.127316\pi\)
0.123308 + 0.992368i \(0.460650\pi\)
\(282\) 0 0
\(283\) 38.4339 + 66.5694i 0.135809 + 0.235228i 0.925906 0.377754i \(-0.123303\pi\)
−0.790097 + 0.612981i \(0.789970\pi\)
\(284\) 0 0
\(285\) −247.379 + 408.402i −0.867996 + 1.43299i
\(286\) 0 0
\(287\) 567.359 + 327.565i 1.97686 + 1.14134i
\(288\) 0 0
\(289\) −195.540 338.685i −0.676610 1.17192i
\(290\) 0 0
\(291\) 19.8178 361.341i 0.0681025 1.24172i
\(292\) 0 0
\(293\) −13.8270 7.98304i −0.0471913 0.0272459i 0.476219 0.879327i \(-0.342007\pi\)
−0.523410 + 0.852081i \(0.675340\pi\)
\(294\) 0 0
\(295\) −112.652 65.0395i −0.381870 0.220473i
\(296\) 0 0
\(297\) −28.2610 75.2580i −0.0951548 0.253394i
\(298\) 0 0
\(299\) 105.454i 0.352687i
\(300\) 0 0
\(301\) 284.789 + 493.268i 0.946141 + 1.63877i
\(302\) 0 0
\(303\) −28.9194 + 44.2959i −0.0954435 + 0.146191i
\(304\) 0 0
\(305\) 139.524 0.457454
\(306\) 0 0
\(307\) 392.315 + 226.503i 1.27790 + 0.737795i 0.976461 0.215692i \(-0.0692006\pi\)
0.301436 + 0.953486i \(0.402534\pi\)
\(308\) 0 0
\(309\) 292.186 + 190.759i 0.945587 + 0.617344i
\(310\) 0 0
\(311\) −179.030 + 310.089i −0.575659 + 0.997071i 0.420311 + 0.907380i \(0.361921\pi\)
−0.995970 + 0.0896903i \(0.971412\pi\)
\(312\) 0 0
\(313\) −289.612 + 501.622i −0.925277 + 1.60263i −0.134161 + 0.990959i \(0.542834\pi\)
−0.791115 + 0.611667i \(0.790499\pi\)
\(314\) 0 0
\(315\) −402.165 + 547.912i −1.27672 + 1.73940i
\(316\) 0 0
\(317\) −276.153 159.437i −0.871146 0.502956i −0.00341721 0.999994i \(-0.501088\pi\)
−0.867729 + 0.497038i \(0.834421\pi\)
\(318\) 0 0
\(319\) 46.8088i 0.146736i
\(320\) 0 0
\(321\) −0.845318 + 15.4128i −0.00263339 + 0.0480149i
\(322\) 0 0
\(323\) 495.206 16.7304i 1.53315 0.0517968i
\(324\) 0 0
\(325\) −217.850 125.776i −0.670308 0.387002i
\(326\) 0 0
\(327\) −26.3376 17.1950i −0.0805431 0.0525841i
\(328\) 0 0
\(329\) 367.254 1.11627
\(330\) 0 0
\(331\) 454.593 262.459i 1.37339 0.792928i 0.382038 0.924147i \(-0.375222\pi\)
0.991353 + 0.131219i \(0.0418890\pi\)
\(332\) 0 0
\(333\) 31.6923 288.056i 0.0951721 0.865032i
\(334\) 0 0
\(335\) −458.383 + 264.648i −1.36831 + 0.789993i
\(336\) 0 0
\(337\) −183.075 + 105.698i −0.543249 + 0.313645i −0.746395 0.665504i \(-0.768217\pi\)
0.203146 + 0.979149i \(0.434884\pi\)
\(338\) 0 0
\(339\) −513.656 335.350i −1.51521 0.989234i
\(340\) 0 0
\(341\) 55.2696i 0.162081i
\(342\) 0 0
\(343\) −150.804 −0.439661
\(344\) 0 0
\(345\) −215.018 424.544i −0.623241 1.23056i
\(346\) 0 0
\(347\) −137.369 237.930i −0.395875 0.685676i 0.597337 0.801990i \(-0.296225\pi\)
−0.993213 + 0.116314i \(0.962892\pi\)
\(348\) 0 0
\(349\) −50.0768 86.7355i −0.143486 0.248526i 0.785321 0.619089i \(-0.212498\pi\)
−0.928807 + 0.370563i \(0.879165\pi\)
\(350\) 0 0
\(351\) −24.6031 + 148.330i −0.0700943 + 0.422592i
\(352\) 0 0
\(353\) 285.912 + 495.215i 0.809950 + 1.40287i 0.912898 + 0.408187i \(0.133839\pi\)
−0.102948 + 0.994687i \(0.532828\pi\)
\(354\) 0 0
\(355\) 815.409i 2.29693i
\(356\) 0 0
\(357\) 704.239 + 38.6242i 1.97266 + 0.108191i
\(358\) 0 0
\(359\) 30.1543 52.2287i 0.0839951 0.145484i −0.820967 0.570975i \(-0.806565\pi\)
0.904963 + 0.425491i \(0.139899\pi\)
\(360\) 0 0
\(361\) −158.988 + 324.105i −0.440410 + 0.897797i
\(362\) 0 0
\(363\) 183.905 281.687i 0.506625 0.775998i
\(364\) 0 0
\(365\) −342.145 −0.937382
\(366\) 0 0
\(367\) 262.298 454.314i 0.714709 1.23791i −0.248363 0.968667i \(-0.579893\pi\)
0.963072 0.269245i \(-0.0867741\pi\)
\(368\) 0 0
\(369\) −598.774 + 263.111i −1.62269 + 0.713038i
\(370\) 0 0
\(371\) −661.887 382.141i −1.78406 1.03003i
\(372\) 0 0
\(373\) −525.452 303.370i −1.40872 0.813324i −0.413454 0.910525i \(-0.635678\pi\)
−0.995265 + 0.0972011i \(0.969011\pi\)
\(374\) 0 0
\(375\) 506.173 + 27.7612i 1.34979 + 0.0740298i
\(376\) 0 0
\(377\) 43.7744 75.8195i 0.116113 0.201113i
\(378\) 0 0
\(379\) 488.868i 1.28989i 0.764229 + 0.644945i \(0.223120\pi\)
−0.764229 + 0.644945i \(0.776880\pi\)
\(380\) 0 0
\(381\) 113.263 57.3641i 0.297279 0.150562i
\(382\) 0 0
\(383\) −153.679 + 88.7264i −0.401250 + 0.231662i −0.687023 0.726636i \(-0.741083\pi\)
0.285773 + 0.958297i \(0.407750\pi\)
\(384\) 0 0
\(385\) 224.847 0.584018
\(386\) 0 0
\(387\) −565.212 62.1855i −1.46050 0.160686i
\(388\) 0 0
\(389\) 335.243 580.659i 0.861808 1.49270i −0.00837339 0.999965i \(-0.502665\pi\)
0.870182 0.492731i \(-0.164001\pi\)
\(390\) 0 0
\(391\) −246.918 + 427.675i −0.631504 + 1.09380i
\(392\) 0 0
\(393\) 294.246 + 580.978i 0.748719 + 1.47831i
\(394\) 0 0
\(395\) −781.673 + 451.299i −1.97892 + 1.14253i
\(396\) 0 0
\(397\) −143.711 + 248.915i −0.361992 + 0.626989i −0.988289 0.152596i \(-0.951237\pi\)
0.626296 + 0.779585i \(0.284570\pi\)
\(398\) 0 0
\(399\) −266.227 + 439.518i −0.667235 + 1.10155i
\(400\) 0 0
\(401\) −116.457 + 67.2365i −0.290417 + 0.167672i −0.638130 0.769929i \(-0.720292\pi\)
0.347713 + 0.937601i \(0.386958\pi\)
\(402\) 0 0
\(403\) 51.6868 89.5242i 0.128255 0.222144i
\(404\) 0 0
\(405\) −203.393 647.325i −0.502204 1.59833i
\(406\) 0 0
\(407\) −83.0256 + 47.9348i −0.203994 + 0.117776i
\(408\) 0 0
\(409\) 599.099i 1.46479i −0.680880 0.732395i \(-0.738402\pi\)
0.680880 0.732395i \(-0.261598\pi\)
\(410\) 0 0
\(411\) −96.6548 + 148.046i −0.235170 + 0.360210i
\(412\) 0 0
\(413\) −121.235 69.9949i −0.293547 0.169479i
\(414\) 0 0
\(415\) −480.188 + 831.711i −1.15708 + 2.00412i
\(416\) 0 0
\(417\) 6.48840 + 0.355858i 0.0155597 + 0.000853376i
\(418\) 0 0
\(419\) −161.894 + 280.408i −0.386381 + 0.669232i −0.991960 0.126553i \(-0.959609\pi\)
0.605578 + 0.795786i \(0.292942\pi\)
\(420\) 0 0
\(421\) 652.095i 1.54892i −0.632624 0.774459i \(-0.718022\pi\)
0.632624 0.774459i \(-0.281978\pi\)
\(422\) 0 0
\(423\) −216.944 + 295.566i −0.512871 + 0.698738i
\(424\) 0 0
\(425\) −589.005 1020.19i −1.38589 2.40044i
\(426\) 0 0
\(427\) 150.154 0.351648
\(428\) 0 0
\(429\) 44.3743 22.4741i 0.103436 0.0523872i
\(430\) 0 0
\(431\) 99.9113 57.6838i 0.231813 0.133837i −0.379595 0.925153i \(-0.623937\pi\)
0.611408 + 0.791315i \(0.290603\pi\)
\(432\) 0 0
\(433\) −60.5946 + 34.9843i −0.139941 + 0.0807952i −0.568336 0.822797i \(-0.692413\pi\)
0.428395 + 0.903592i \(0.359079\pi\)
\(434\) 0 0
\(435\) −21.6363 + 394.497i −0.0497386 + 0.906889i
\(436\) 0 0
\(437\) −190.317 305.341i −0.435507 0.698720i
\(438\) 0 0
\(439\) 449.021i 1.02283i 0.859335 + 0.511414i \(0.170878\pi\)
−0.859335 + 0.511414i \(0.829122\pi\)
\(440\) 0 0
\(441\) −171.862 + 234.146i −0.389710 + 0.530942i
\(442\) 0 0
\(443\) −96.3807 + 166.936i −0.217564 + 0.376831i −0.954063 0.299607i \(-0.903144\pi\)
0.736499 + 0.676439i \(0.236478\pi\)
\(444\) 0 0
\(445\) −22.6977 13.1045i −0.0510061 0.0294484i
\(446\) 0 0
\(447\) −192.885 380.845i −0.431511 0.852001i
\(448\) 0 0
\(449\) 264.055i 0.588096i −0.955791 0.294048i \(-0.904997\pi\)
0.955791 0.294048i \(-0.0950025\pi\)
\(450\) 0 0
\(451\) 187.379 + 108.183i 0.415475 + 0.239875i
\(452\) 0 0
\(453\) −765.725 + 387.815i −1.69034 + 0.856104i
\(454\) 0 0
\(455\) −364.201 210.272i −0.800442 0.462135i
\(456\) 0 0
\(457\) −225.519 390.610i −0.493476 0.854726i 0.506495 0.862243i \(-0.330941\pi\)
−0.999972 + 0.00751646i \(0.997607\pi\)
\(458\) 0 0
\(459\) −447.092 + 543.955i −0.974057 + 1.18509i
\(460\) 0 0
\(461\) 599.782 1.30104 0.650522 0.759487i \(-0.274550\pi\)
0.650522 + 0.759487i \(0.274550\pi\)
\(462\) 0 0
\(463\) −39.3790 + 68.2064i −0.0850518 + 0.147314i −0.905413 0.424531i \(-0.860439\pi\)
0.820361 + 0.571845i \(0.193772\pi\)
\(464\) 0 0
\(465\) −25.5471 + 465.803i −0.0549400 + 1.00173i
\(466\) 0 0
\(467\) 570.793 1.22225 0.611127 0.791532i \(-0.290716\pi\)
0.611127 + 0.791532i \(0.290716\pi\)
\(468\) 0 0
\(469\) −493.308 + 284.811i −1.05183 + 0.607274i
\(470\) 0 0
\(471\) −237.099 468.142i −0.503395 0.993933i
\(472\) 0 0
\(473\) 94.0560 + 162.910i 0.198850 + 0.344418i
\(474\) 0 0
\(475\) 857.778 28.9797i 1.80585 0.0610099i
\(476\) 0 0
\(477\) 698.537 306.949i 1.46444 0.643498i
\(478\) 0 0
\(479\) 462.338 800.793i 0.965215 1.67180i 0.256179 0.966629i \(-0.417536\pi\)
0.709036 0.705172i \(-0.249130\pi\)
\(480\) 0 0
\(481\) 179.310 0.372786
\(482\) 0 0
\(483\) −231.400 456.891i −0.479090 0.945943i
\(484\) 0 0
\(485\) −875.105 + 505.242i −1.80434 + 1.04174i
\(486\) 0 0
\(487\) 236.919i 0.486486i 0.969965 + 0.243243i \(0.0782113\pi\)
−0.969965 + 0.243243i \(0.921789\pi\)
\(488\) 0 0
\(489\) 840.425 + 46.0934i 1.71866 + 0.0942605i
\(490\) 0 0
\(491\) −276.148 478.302i −0.562420 0.974139i −0.997285 0.0736437i \(-0.976537\pi\)
0.434865 0.900496i \(-0.356796\pi\)
\(492\) 0 0
\(493\) 355.061 204.995i 0.720205 0.415811i
\(494\) 0 0
\(495\) −132.822 + 180.957i −0.268326 + 0.365569i
\(496\) 0 0
\(497\) 877.535i 1.76566i
\(498\) 0 0
\(499\) 185.062 + 320.536i 0.370865 + 0.642357i 0.989699 0.143165i \(-0.0457278\pi\)
−0.618834 + 0.785522i \(0.712395\pi\)
\(500\) 0 0
\(501\) 0.934328 17.0357i 0.00186493 0.0340034i
\(502\) 0 0
\(503\) −410.372 710.785i −0.815848 1.41309i −0.908717 0.417412i \(-0.862937\pi\)
0.0928687 0.995678i \(-0.470396\pi\)
\(504\) 0 0
\(505\) 147.713 0.292502
\(506\) 0 0
\(507\) 413.346 + 22.6701i 0.815278 + 0.0447141i
\(508\) 0 0
\(509\) 37.8132i 0.0742891i −0.999310 0.0371446i \(-0.988174\pi\)
0.999310 0.0371446i \(-0.0118262\pi\)
\(510\) 0 0
\(511\) −368.213 −0.720573
\(512\) 0 0
\(513\) −196.459 473.891i −0.382961 0.923765i
\(514\) 0 0
\(515\) 974.353i 1.89195i
\(516\) 0 0
\(517\) 121.292 0.234607
\(518\) 0 0
\(519\) −0.170092 + 3.10131i −0.000327730 + 0.00597554i
\(520\) 0 0
\(521\) 190.170i 0.365009i 0.983205 + 0.182504i \(0.0584204\pi\)
−0.983205 + 0.182504i \(0.941580\pi\)
\(522\) 0 0
\(523\) 86.4225 49.8961i 0.165244 0.0954035i −0.415097 0.909777i \(-0.636253\pi\)
0.580341 + 0.814373i \(0.302919\pi\)
\(524\) 0 0
\(525\) 1219.86 + 66.9034i 2.32354 + 0.127435i
\(526\) 0 0
\(527\) 419.240 242.048i 0.795522 0.459295i
\(528\) 0 0
\(529\) −170.403 −0.322124
\(530\) 0 0
\(531\) 127.948 56.2223i 0.240956 0.105880i
\(532\) 0 0
\(533\) −202.341 350.465i −0.379627 0.657533i
\(534\) 0 0
\(535\) 37.3271 21.5508i 0.0697703 0.0402819i
\(536\) 0 0
\(537\) 30.4073 554.420i 0.0566245 1.03244i
\(538\) 0 0
\(539\) 96.0865 0.178268
\(540\) 0 0
\(541\) 151.059 + 261.642i 0.279222 + 0.483627i 0.971192 0.238300i \(-0.0765901\pi\)
−0.691970 + 0.721927i \(0.743257\pi\)
\(542\) 0 0
\(543\) 240.411 121.760i 0.442745 0.224236i
\(544\) 0 0
\(545\) 87.8279i 0.161152i
\(546\) 0 0
\(547\) 313.718 + 181.125i 0.573525 + 0.331125i 0.758556 0.651608i \(-0.225905\pi\)
−0.185031 + 0.982733i \(0.559239\pi\)
\(548\) 0 0
\(549\) −88.6988 + 120.844i −0.161564 + 0.220116i
\(550\) 0 0
\(551\) 10.0860 + 298.537i 0.0183049 + 0.541810i
\(552\) 0 0
\(553\) −841.229 + 485.684i −1.52121 + 0.878271i
\(554\) 0 0
\(555\) −721.883 + 365.610i −1.30069 + 0.658757i
\(556\) 0 0
\(557\) 198.669 + 344.104i 0.356676 + 0.617781i 0.987403 0.158223i \(-0.0505766\pi\)
−0.630727 + 0.776005i \(0.717243\pi\)
\(558\) 0 0
\(559\) 351.836i 0.629402i
\(560\) 0 0
\(561\) 232.586 + 12.7563i 0.414592 + 0.0227384i
\(562\) 0 0
\(563\) 250.075 + 144.381i 0.444182 + 0.256449i 0.705370 0.708839i \(-0.250781\pi\)
−0.261188 + 0.965288i \(0.584114\pi\)
\(564\) 0 0
\(565\) 1712.89i 3.03166i
\(566\) 0 0
\(567\) −218.889 696.645i −0.386048 1.22865i
\(568\) 0 0
\(569\) 506.239 292.277i 0.889700 0.513668i 0.0158554 0.999874i \(-0.494953\pi\)
0.873844 + 0.486206i \(0.161620\pi\)
\(570\) 0 0
\(571\) −523.855 + 907.343i −0.917434 + 1.58904i −0.114135 + 0.993465i \(0.536410\pi\)
−0.803298 + 0.595577i \(0.796924\pi\)
\(572\) 0 0
\(573\) 290.601 + 573.779i 0.507156 + 1.00136i
\(574\) 0 0
\(575\) −427.703 + 740.803i −0.743831 + 1.28835i
\(576\) 0 0
\(577\) 514.036 0.890877 0.445438 0.895313i \(-0.353048\pi\)
0.445438 + 0.895313i \(0.353048\pi\)
\(578\) 0 0
\(579\) 22.7166 11.5052i 0.0392342 0.0198708i
\(580\) 0 0
\(581\) −516.774 + 895.079i −0.889456 + 1.54058i
\(582\) 0 0
\(583\) −218.599 126.208i −0.374955 0.216481i
\(584\) 0 0
\(585\) 384.367 168.897i 0.657038 0.288713i
\(586\) 0 0
\(587\) −506.000 −0.862010 −0.431005 0.902350i \(-0.641841\pi\)
−0.431005 + 0.902350i \(0.641841\pi\)
\(588\) 0 0
\(589\) 11.9091 + 352.499i 0.0202191 + 0.598470i
\(590\) 0 0
\(591\) −547.230 30.0130i −0.925940 0.0507834i
\(592\) 0 0
\(593\) −110.508 191.405i −0.186353 0.322774i 0.757678 0.652628i \(-0.226334\pi\)
−0.944032 + 0.329855i \(0.893000\pi\)
\(594\) 0 0
\(595\) −984.697 1705.55i −1.65495 2.86646i
\(596\) 0 0
\(597\) −317.980 627.838i −0.532629 1.05165i
\(598\) 0 0
\(599\) 149.926i 0.250294i −0.992138 0.125147i \(-0.960060\pi\)
0.992138 0.125147i \(-0.0399402\pi\)
\(600\) 0 0
\(601\) −933.044 + 538.693i −1.55249 + 0.896328i −0.554547 + 0.832153i \(0.687108\pi\)
−0.997939 + 0.0641750i \(0.979558\pi\)
\(602\) 0 0
\(603\) 62.1905 565.257i 0.103135 0.937409i
\(604\) 0 0
\(605\) −939.342 −1.55263
\(606\) 0 0
\(607\) 990.463 + 571.844i 1.63174 + 0.942083i 0.983558 + 0.180594i \(0.0578019\pi\)
0.648178 + 0.761489i \(0.275531\pi\)
\(608\) 0 0
\(609\) −23.2848 + 424.553i −0.0382344 + 0.697132i
\(610\) 0 0
\(611\) −196.465 113.429i −0.321546 0.185645i
\(612\) 0 0
\(613\) −357.940 + 619.969i −0.583914 + 1.01137i 0.411095 + 0.911592i \(0.365146\pi\)
−0.995010 + 0.0997770i \(0.968187\pi\)
\(614\) 0 0
\(615\) 1529.20 + 998.364i 2.48650 + 1.62336i
\(616\) 0 0
\(617\) −1147.16 −1.85925 −0.929625 0.368506i \(-0.879869\pi\)
−0.929625 + 0.368506i \(0.879869\pi\)
\(618\) 0 0
\(619\) 194.222 + 336.402i 0.313767 + 0.543460i 0.979175 0.203020i \(-0.0650757\pi\)
−0.665408 + 0.746480i \(0.731742\pi\)
\(620\) 0 0
\(621\) 504.398 + 83.6633i 0.812235 + 0.134724i
\(622\) 0 0
\(623\) −24.4271 14.1030i −0.0392088 0.0226372i
\(624\) 0 0
\(625\) −143.103 247.862i −0.228965 0.396579i
\(626\) 0 0
\(627\) −87.9256 + 145.158i −0.140232 + 0.231512i
\(628\) 0 0
\(629\) 727.206 + 419.852i 1.15613 + 0.667492i
\(630\) 0 0
\(631\) −398.613 690.418i −0.631717 1.09417i −0.987201 0.159483i \(-0.949017\pi\)
0.355484 0.934682i \(-0.384316\pi\)
\(632\) 0 0
\(633\) −178.624 + 90.4673i −0.282186 + 0.142918i
\(634\) 0 0
\(635\) −307.017 177.256i −0.483492 0.279144i
\(636\) 0 0
\(637\) −155.638 89.8578i −0.244330 0.141064i
\(638\) 0 0
\(639\) 706.240 + 518.377i 1.10523 + 0.811232i
\(640\) 0 0
\(641\) 769.355i 1.20024i −0.799909 0.600121i \(-0.795119\pi\)
0.799909 0.600121i \(-0.204881\pi\)
\(642\) 0 0
\(643\) 159.471 + 276.211i 0.248010 + 0.429567i 0.962974 0.269595i \(-0.0868899\pi\)
−0.714963 + 0.699162i \(0.753557\pi\)
\(644\) 0 0
\(645\) 717.387 + 1416.45i 1.11223 + 2.19605i
\(646\) 0 0
\(647\) 585.103 0.904333 0.452166 0.891934i \(-0.350651\pi\)
0.452166 + 0.891934i \(0.350651\pi\)
\(648\) 0 0
\(649\) −40.0397 23.1169i −0.0616945 0.0356193i
\(650\) 0 0
\(651\) −27.4936 + 501.293i −0.0422328 + 0.770035i
\(652\) 0 0
\(653\) −89.3580 + 154.773i −0.136842 + 0.237018i −0.926300 0.376788i \(-0.877029\pi\)
0.789457 + 0.613805i \(0.210362\pi\)
\(654\) 0 0
\(655\) 909.228 1574.83i 1.38813 2.40432i
\(656\) 0 0
\(657\) 217.510 296.337i 0.331066 0.451046i
\(658\) 0 0
\(659\) −118.419 68.3690i −0.179694 0.103747i 0.407455 0.913225i \(-0.366416\pi\)
−0.587149 + 0.809479i \(0.699750\pi\)
\(660\) 0 0
\(661\) 633.064i 0.957737i 0.877887 + 0.478869i \(0.158953\pi\)
−0.877887 + 0.478869i \(0.841047\pi\)
\(662\) 0 0
\(663\) −364.807 238.171i −0.550237 0.359233i
\(664\) 0 0
\(665\) 1434.03 48.4482i 2.15644 0.0728545i
\(666\) 0 0
\(667\) −257.826 148.856i −0.386545 0.223172i
\(668\) 0 0
\(669\) 37.8939 690.923i 0.0566425 1.03277i
\(670\) 0 0
\(671\) 49.5907 0.0739057
\(672\) 0 0
\(673\) −293.528 + 169.468i −0.436149 + 0.251811i −0.701963 0.712214i \(-0.747693\pi\)
0.265814 + 0.964024i \(0.414359\pi\)
\(674\) 0 0
\(675\) −774.437 + 942.219i −1.14731 + 1.39588i
\(676\) 0 0
\(677\) 90.2123 52.0841i 0.133253 0.0769337i −0.431892 0.901926i \(-0.642154\pi\)
0.565145 + 0.824992i \(0.308820\pi\)
\(678\) 0 0
\(679\) −941.780 + 543.737i −1.38701 + 0.800791i
\(680\) 0 0
\(681\) −237.627 + 120.351i −0.348939 + 0.176726i
\(682\) 0 0
\(683\) 517.968i 0.758371i −0.925321 0.379186i \(-0.876204\pi\)
0.925321 0.379186i \(-0.123796\pi\)
\(684\) 0 0
\(685\) 493.690 0.720715
\(686\) 0 0
\(687\) 221.169 338.764i 0.321934 0.493107i
\(688\) 0 0
\(689\) 236.054 + 408.857i 0.342603 + 0.593406i
\(690\) 0 0
\(691\) −44.7261 77.4679i −0.0647266 0.112110i 0.831846 0.555006i \(-0.187284\pi\)
−0.896573 + 0.442897i \(0.853951\pi\)
\(692\) 0 0
\(693\) −142.941 + 194.744i −0.206264 + 0.281016i
\(694\) 0 0
\(695\) −9.07236 15.7138i −0.0130538 0.0226098i
\(696\) 0 0
\(697\) 1895.12i 2.71896i
\(698\) 0 0
\(699\) 108.825 166.687i 0.155687 0.238465i
\(700\) 0 0
\(701\) 291.669 505.186i 0.416076 0.720665i −0.579465 0.814997i \(-0.696738\pi\)
0.995541 + 0.0943325i \(0.0300717\pi\)
\(702\) 0 0
\(703\) −519.192 + 323.609i −0.738538 + 0.460325i
\(704\) 0 0
\(705\) 1022.23 + 56.0643i 1.44997 + 0.0795238i
\(706\) 0 0
\(707\) 158.968 0.224848
\(708\) 0 0
\(709\) −426.312 + 738.395i −0.601287 + 1.04146i 0.391340 + 0.920246i \(0.372012\pi\)
−0.992627 + 0.121213i \(0.961322\pi\)
\(710\) 0 0
\(711\) 106.052 963.924i 0.149159 1.35573i
\(712\) 0 0
\(713\) −304.429 175.762i −0.426969 0.246510i
\(714\) 0 0
\(715\) −120.283 69.4456i −0.168228 0.0971267i
\(716\) 0 0
\(717\) 515.535 789.646i 0.719017 1.10132i
\(718\) 0 0
\(719\) −443.865 + 768.797i −0.617337 + 1.06926i 0.372633 + 0.927979i \(0.378455\pi\)
−0.989970 + 0.141280i \(0.954878\pi\)
\(720\) 0 0
\(721\) 1048.59i 1.45435i
\(722\) 0 0
\(723\) −127.389 83.1681i −0.176195 0.115032i
\(724\) 0 0
\(725\) 615.024 355.084i 0.848309 0.489771i
\(726\) 0 0
\(727\) −331.033 −0.455341 −0.227671 0.973738i \(-0.573111\pi\)
−0.227671 + 0.973738i \(0.573111\pi\)
\(728\) 0 0
\(729\) 689.961 + 235.360i 0.946449 + 0.322853i
\(730\) 0 0
\(731\) 823.819 1426.90i 1.12698 1.95198i
\(732\) 0 0
\(733\) −411.381 + 712.533i −0.561229 + 0.972078i 0.436160 + 0.899869i \(0.356338\pi\)
−0.997390 + 0.0722087i \(0.976995\pi\)
\(734\) 0 0
\(735\) 809.801 + 44.4138i 1.10177 + 0.0604269i
\(736\) 0 0
\(737\) −162.923 + 94.0635i −0.221062 + 0.127630i
\(738\) 0 0
\(739\) 351.505 608.824i 0.475649 0.823849i −0.523962 0.851742i \(-0.675546\pi\)
0.999611 + 0.0278931i \(0.00887980\pi\)
\(740\) 0 0
\(741\) 278.168 152.897i 0.375395 0.206339i
\(742\) 0 0
\(743\) −928.485 + 536.061i −1.24964 + 0.721482i −0.971038 0.238924i \(-0.923205\pi\)
−0.278605 + 0.960406i \(0.589872\pi\)
\(744\) 0 0
\(745\) −596.020 + 1032.34i −0.800027 + 1.38569i
\(746\) 0 0
\(747\) −415.090 944.640i −0.555677 1.26458i
\(748\) 0 0
\(749\) 40.1711 23.1928i 0.0536329 0.0309650i
\(750\) 0 0
\(751\) 120.921i 0.161014i −0.996754 0.0805068i \(-0.974346\pi\)
0.996754 0.0805068i \(-0.0256539\pi\)
\(752\) 0 0
\(753\) 29.7800 + 58.7993i 0.0395484 + 0.0780867i
\(754\) 0 0
\(755\) 2075.62 + 1198.36i 2.74916 + 1.58723i
\(756\) 0 0
\(757\) −500.057 + 866.125i −0.660578 + 1.14415i 0.319887 + 0.947456i \(0.396355\pi\)
−0.980464 + 0.196698i \(0.936978\pi\)
\(758\) 0 0
\(759\) −76.4237 150.895i −0.100690 0.198808i
\(760\) 0 0
\(761\) 710.262 1230.21i 0.933327 1.61657i 0.155736 0.987799i \(-0.450225\pi\)
0.777591 0.628771i \(-0.216442\pi\)
\(762\) 0 0
\(763\) 94.5195i 0.123879i
\(764\) 0 0
\(765\) 1954.30 + 215.015i 2.55464 + 0.281066i
\(766\) 0 0
\(767\) 43.2368 + 74.8884i 0.0563713 + 0.0976380i
\(768\) 0 0
\(769\) 452.651 0.588623 0.294312 0.955709i \(-0.404910\pi\)
0.294312 + 0.955709i \(0.404910\pi\)
\(770\) 0 0
\(771\) −67.3981 + 1228.88i −0.0874164 + 1.59387i
\(772\) 0 0
\(773\) −158.416 + 91.4618i −0.204937 + 0.118321i −0.598956 0.800782i \(-0.704418\pi\)
0.394019 + 0.919102i \(0.371084\pi\)
\(774\) 0 0
\(775\) 726.192 419.267i 0.937022 0.540990i
\(776\) 0 0
\(777\) −776.883 + 393.466i −0.999850 + 0.506392i
\(778\) 0 0
\(779\) 1218.38 + 649.598i 1.56403 + 0.833887i
\(780\) 0 0
\(781\) 289.820i 0.371088i
\(782\) 0 0
\(783\) −327.926 269.532i −0.418807 0.344229i
\(784\) 0 0
\(785\) −732.641 + 1268.97i −0.933301 + 1.61652i
\(786\) 0 0
\(787\) 355.648 + 205.334i 0.451904 + 0.260907i 0.708634 0.705576i \(-0.249312\pi\)
−0.256730 + 0.966483i \(0.582645\pi\)
\(788\) 0 0
\(789\) 761.503 + 41.7648i 0.965149 + 0.0529339i
\(790\) 0 0
\(791\) 1843.39i 2.33046i
\(792\) 0 0
\(793\) −80.3257 46.3760i −0.101293 0.0584818i
\(794\) 0 0
\(795\) −1783.98 1164.70i −2.24400 1.46504i
\(796\) 0 0
\(797\) 1016.85 + 587.077i 1.27584 + 0.736609i 0.976081 0.217405i \(-0.0697594\pi\)
0.299762 + 0.954014i \(0.403093\pi\)
\(798\) 0 0
\(799\) −531.186 920.040i −0.664813 1.15149i
\(800\) 0 0
\(801\) 25.7796 11.3280i 0.0321843 0.0141423i
\(802\) 0 0
\(803\) −121.608 −0.151442
\(804\) 0 0
\(805\) −715.032 + 1238.47i −0.888239 + 1.53847i
\(806\) 0 0
\(807\) −0.861078 0.562171i −0.00106701 0.000696618i
\(808\) 0 0
\(809\) 500.592 0.618778 0.309389 0.950936i \(-0.399875\pi\)
0.309389 + 0.950936i \(0.399875\pi\)
\(810\) 0 0
\(811\) 671.823 387.877i 0.828389 0.478270i −0.0249120 0.999690i \(-0.507931\pi\)
0.853301 + 0.521419i \(0.174597\pi\)
\(812\) 0 0
\(813\) 583.707 894.065i 0.717967 1.09971i
\(814\) 0 0
\(815\) −1175.12 2035.37i −1.44186 2.49738i
\(816\) 0 0
\(817\) 634.973 + 1018.74i 0.777201 + 1.24693i
\(818\) 0 0
\(819\) 413.652 181.766i 0.505070 0.221936i
\(820\) 0 0
\(821\) 724.841 1255.46i 0.882875 1.52918i 0.0347454 0.999396i \(-0.488938\pi\)
0.848130 0.529789i \(-0.177729\pi\)
\(822\) 0 0
\(823\) 22.0138 0.0267482 0.0133741 0.999911i \(-0.495743\pi\)
0.0133741 + 0.999911i \(0.495743\pi\)
\(824\) 0 0
\(825\) 402.877 + 22.0959i 0.488336 + 0.0267829i
\(826\) 0 0
\(827\) −417.259 + 240.905i −0.504546 + 0.291300i −0.730589 0.682818i \(-0.760754\pi\)
0.226043 + 0.974117i \(0.427421\pi\)
\(828\) 0 0
\(829\) 1578.05i 1.90355i −0.306791 0.951777i \(-0.599255\pi\)
0.306791 0.951777i \(-0.400745\pi\)
\(830\) 0 0
\(831\) −395.801 781.493i −0.476295 0.940425i
\(832\) 0 0
\(833\) −420.802 728.850i −0.505164 0.874970i
\(834\) 0 0
\(835\) −41.2576 + 23.8201i −0.0494103 + 0.0285270i
\(836\) 0 0
\(837\) −387.199 318.250i −0.462604 0.380227i
\(838\) 0 0
\(839\) 934.080i 1.11333i 0.830739 + 0.556663i \(0.187918\pi\)
−0.830739 + 0.556663i \(0.812082\pi\)
\(840\) 0 0
\(841\) −296.918 514.277i −0.353054 0.611507i
\(842\) 0 0
\(843\) 851.254 + 555.757i 1.00979 + 0.659261i
\(844\) 0 0
\(845\) −577.958 1001.05i −0.683974 1.18468i
\(846\) 0 0
\(847\) −1010.91 −1.19352
\(848\) 0 0
\(849\) 104.192 + 205.723i 0.122723 + 0.242312i
\(850\) 0 0
\(851\) 609.747i 0.716506i
\(852\) 0 0
\(853\) 293.146 0.343665 0.171832 0.985126i \(-0.445031\pi\)
0.171832 + 0.985126i \(0.445031\pi\)
\(854\) 0 0
\(855\) −808.119 + 1182.73i −0.945168 + 1.38331i
\(856\) 0 0
\(857\) 801.284i 0.934987i 0.883996 + 0.467494i \(0.154843\pi\)
−0.883996 + 0.467494i \(0.845157\pi\)
\(858\) 0 0
\(859\) −850.247 −0.989810 −0.494905 0.868947i \(-0.664797\pi\)
−0.494905 + 0.868947i \(0.664797\pi\)
\(860\) 0 0
\(861\) 1645.71 + 1074.43i 1.91139 + 1.24789i
\(862\) 0 0
\(863\) 57.6973i 0.0668566i 0.999441 + 0.0334283i \(0.0106425\pi\)
−0.999441 + 0.0334283i \(0.989357\pi\)
\(864\) 0 0
\(865\) 7.51083 4.33638i 0.00868304 0.00501316i
\(866\) 0 0
\(867\) −530.098 1046.66i −0.611416 1.20722i
\(868\) 0 0
\(869\) −277.829 + 160.405i −0.319712 + 0.184586i
\(870\) 0 0
\(871\) 351.864 0.403977
\(872\) 0 0
\(873\) 118.729 1079.14i 0.136001 1.23613i
\(874\) 0 0
\(875\) −761.676 1319.26i −0.870487 1.50773i
\(876\) 0 0
\(877\) 1377.56 795.333i 1.57076 0.906879i 0.574684 0.818375i \(-0.305125\pi\)
0.996076 0.0885036i \(-0.0282085\pi\)
\(878\) 0 0
\(879\) −40.1073 26.1848i −0.0456284 0.0297893i
\(880\) 0 0
\(881\) −1154.00 −1.30987 −0.654936 0.755684i \(-0.727304\pi\)
−0.654936 + 0.755684i \(0.727304\pi\)
\(882\) 0 0
\(883\) 184.480 + 319.529i 0.208924 + 0.361867i 0.951376 0.308032i \(-0.0996705\pi\)
−0.742452 + 0.669899i \(0.766337\pi\)
\(884\) 0 0
\(885\) −326.763 213.333i −0.369224 0.241055i
\(886\) 0 0
\(887\) 1079.34i 1.21685i 0.793613 + 0.608423i \(0.208198\pi\)
−0.793613 + 0.608423i \(0.791802\pi\)
\(888\) 0 0
\(889\) −330.409 190.762i −0.371664 0.214580i
\(890\) 0 0
\(891\) −72.2916 230.078i −0.0811354 0.258225i
\(892\) 0 0
\(893\) 773.574 26.1349i 0.866265 0.0292664i
\(894\) 0 0
\(895\) −1342.71 + 775.214i −1.50024 + 0.866161i
\(896\) 0 0
\(897\) −17.3248 + 315.886i −0.0193142 + 0.352158i
\(898\) 0 0
\(899\) 145.920 + 252.741i 0.162314 + 0.281135i
\(900\) 0 0
\(901\) 2210.87i 2.45379i
\(902\) 0 0
\(903\) 772.045 + 1524.37i 0.854978 + 1.68812i
\(904\) 0 0
\(905\) −651.670 376.242i −0.720078 0.415737i
\(906\) 0 0
\(907\) 1159.72i 1.27863i 0.768945 + 0.639315i \(0.220782\pi\)
−0.768945 + 0.639315i \(0.779218\pi\)
\(908\) 0 0
\(909\) −93.9053 + 127.937i −0.103306 + 0.140745i
\(910\) 0 0
\(911\) 228.498 131.923i 0.250821 0.144812i −0.369319 0.929303i \(-0.620409\pi\)
0.620140 + 0.784491i \(0.287076\pi\)
\(912\) 0 0
\(913\) −170.673 + 295.614i −0.186936 + 0.323783i
\(914\) 0 0
\(915\) 417.943 + 22.9222i 0.456768 + 0.0250516i
\(916\) 0 0
\(917\) 978.503 1694.82i 1.06707 1.84822i
\(918\) 0 0
\(919\) 197.600 0.215016 0.107508 0.994204i \(-0.465713\pi\)
0.107508 + 0.994204i \(0.465713\pi\)
\(920\) 0 0
\(921\) 1137.97 + 742.942i 1.23558 + 0.806669i
\(922\) 0 0
\(923\) −271.033 + 469.442i −0.293643 + 0.508605i
\(924\) 0 0
\(925\) 1259.64 + 727.253i 1.36177 + 0.786219i
\(926\) 0 0
\(927\) 843.904 + 619.422i 0.910360 + 0.668201i
\(928\) 0 0
\(929\) 16.2727 0.0175164 0.00875820 0.999962i \(-0.497212\pi\)
0.00875820 + 0.999962i \(0.497212\pi\)
\(930\) 0 0
\(931\) 612.821 20.7039i 0.658239 0.0222384i
\(932\) 0 0
\(933\) −587.228 + 899.458i −0.629398 + 0.964050i
\(934\) 0 0
\(935\) −325.212 563.284i −0.347820 0.602442i
\(936\) 0 0
\(937\) 136.677 + 236.732i 0.145867 + 0.252649i 0.929696 0.368328i \(-0.120070\pi\)
−0.783829 + 0.620976i \(0.786736\pi\)
\(938\) 0 0
\(939\) −949.942 + 1455.03i −1.01165 + 1.54955i
\(940\) 0 0
\(941\) 1205.09i 1.28065i 0.768104 + 0.640325i \(0.221200\pi\)
−0.768104 + 0.640325i \(0.778800\pi\)
\(942\) 0 0
\(943\) −1191.76 + 688.065i −1.26380 + 0.729655i
\(944\) 0 0
\(945\) −1294.70 + 1575.20i −1.37005 + 1.66688i
\(946\) 0 0
\(947\) 1029.74 1.08737 0.543686 0.839289i \(-0.317028\pi\)
0.543686 + 0.839289i \(0.317028\pi\)
\(948\) 0 0
\(949\) 196.977 + 113.725i 0.207563 + 0.119837i
\(950\) 0 0
\(951\) −801.023 522.963i −0.842296 0.549908i
\(952\) 0 0
\(953\) −618.547 357.118i −0.649052 0.374730i 0.139041 0.990287i \(-0.455598\pi\)
−0.788093 + 0.615556i \(0.788931\pi\)
\(954\) 0 0
\(955\) 897.962 1555.32i 0.940275 1.62860i
\(956\) 0 0
\(957\) −7.69016 + 140.216i −0.00803570 + 0.146516i
\(958\) 0 0
\(959\) 531.304 0.554019
\(960\) 0 0
\(961\) −308.205 533.826i −0.320712 0.555490i
\(962\) 0 0
\(963\) −5.06430 + 46.0301i −0.00525888 + 0.0477986i
\(964\) 0 0
\(965\) −61.5768 35.5514i −0.0638102 0.0368408i
\(966\) 0 0
\(967\) 545.797 + 945.348i 0.564423 + 0.977609i 0.997103 + 0.0760618i \(0.0242346\pi\)
−0.432680 + 0.901548i \(0.642432\pi\)
\(968\) 0 0
\(969\) 1486.14 + 31.2411i 1.53368 + 0.0322406i
\(970\) 0 0
\(971\) 997.067 + 575.657i 1.02685 + 0.592850i 0.916079 0.400997i \(-0.131336\pi\)
0.110766 + 0.993847i \(0.464670\pi\)
\(972\) 0 0
\(973\) −9.76358 16.9110i −0.0100345 0.0173803i
\(974\) 0 0
\(975\) −631.906 412.551i −0.648108 0.423130i
\(976\) 0 0
\(977\) −742.979 428.959i −0.760470 0.439057i 0.0689947 0.997617i \(-0.478021\pi\)
−0.829464 + 0.558560i \(0.811354\pi\)
\(978\) 0 0
\(979\) −8.06743 4.65773i −0.00824048 0.00475764i
\(980\) 0 0
\(981\) −76.0693 55.8345i −0.0775426 0.0569159i
\(982\) 0 0
\(983\) 877.350i 0.892523i 0.894903 + 0.446261i \(0.147245\pi\)
−0.894903 + 0.446261i \(0.852755\pi\)
\(984\) 0 0
\(985\) 765.161 + 1325.30i 0.776813 + 1.34548i
\(986\) 0 0
\(987\) 1100.11 + 60.3358i 1.11460 + 0.0611305i
\(988\) 0 0
\(989\) −1196.42 −1.20973
\(990\) 0 0
\(991\) −452.536 261.272i −0.456646 0.263645i 0.253987 0.967208i \(-0.418258\pi\)
−0.710633 + 0.703563i \(0.751591\pi\)
\(992\) 0 0
\(993\) 1404.85 711.511i 1.41475 0.716527i
\(994\) 0 0
\(995\) −982.564 + 1701.85i −0.987502 + 1.71040i
\(996\) 0 0
\(997\) 577.092 999.553i 0.578829 1.00256i −0.416785 0.909005i \(-0.636843\pi\)
0.995614 0.0935557i \(-0.0298233\pi\)
\(998\) 0 0
\(999\) 142.259 857.663i 0.142401 0.858522i
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 684.3.s.a.445.40 80
3.2 odd 2 2052.3.s.a.901.39 80
9.2 odd 6 2052.3.bl.a.1585.2 80
9.7 even 3 684.3.bl.a.673.29 yes 80
19.12 odd 6 684.3.bl.a.373.29 yes 80
57.50 even 6 2052.3.bl.a.145.2 80
171.88 odd 6 inner 684.3.s.a.601.40 yes 80
171.164 even 6 2052.3.s.a.829.39 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.s.a.445.40 80 1.1 even 1 trivial
684.3.s.a.601.40 yes 80 171.88 odd 6 inner
684.3.bl.a.373.29 yes 80 19.12 odd 6
684.3.bl.a.673.29 yes 80 9.7 even 3
2052.3.s.a.829.39 80 171.164 even 6
2052.3.s.a.901.39 80 3.2 odd 2
2052.3.bl.a.145.2 80 57.50 even 6
2052.3.bl.a.1585.2 80 9.2 odd 6