Properties

Label 201.5.b.a.133.17
Level $201$
Weight $5$
Character 201.133
Analytic conductor $20.777$
Analytic rank $0$
Dimension $46$
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] = [201,5,Mod(133,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.133");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 201.b (of order \(2\), degree \(1\), 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: \(20.7773625799\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 133.17
Character \(\chi\) \(=\) 201.133
Dual form 201.5.b.a.133.30

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.53609i q^{2} +5.19615i q^{3} +9.56827 q^{4} -8.16007i q^{5} +13.1779 q^{6} -69.7059i q^{7} -64.8433i q^{8} -27.0000 q^{9} +O(q^{10})\) \(q-2.53609i q^{2} +5.19615i q^{3} +9.56827 q^{4} -8.16007i q^{5} +13.1779 q^{6} -69.7059i q^{7} -64.8433i q^{8} -27.0000 q^{9} -20.6946 q^{10} +42.4935i q^{11} +49.7182i q^{12} -8.13871i q^{13} -176.780 q^{14} +42.4009 q^{15} -11.3560 q^{16} -51.4498 q^{17} +68.4743i q^{18} -552.939 q^{19} -78.0777i q^{20} +362.203 q^{21} +107.767 q^{22} -661.237 q^{23} +336.936 q^{24} +558.413 q^{25} -20.6405 q^{26} -140.296i q^{27} -666.965i q^{28} +952.351 q^{29} -107.532i q^{30} -1572.07i q^{31} -1008.69i q^{32} -220.803 q^{33} +130.481i q^{34} -568.805 q^{35} -258.343 q^{36} -824.332 q^{37} +1402.30i q^{38} +42.2900 q^{39} -529.126 q^{40} -1049.72i q^{41} -918.577i q^{42} -2489.51i q^{43} +406.589i q^{44} +220.322i q^{45} +1676.95i q^{46} -3338.19 q^{47} -59.0076i q^{48} -2457.92 q^{49} -1416.18i q^{50} -267.341i q^{51} -77.8734i q^{52} +5321.62i q^{53} -355.803 q^{54} +346.750 q^{55} -4519.96 q^{56} -2873.16i q^{57} -2415.24i q^{58} +6339.83 q^{59} +405.704 q^{60} -5490.98i q^{61} -3986.90 q^{62} +1882.06i q^{63} -2739.83 q^{64} -66.4124 q^{65} +559.975i q^{66} +(3106.61 + 3240.39i) q^{67} -492.285 q^{68} -3435.89i q^{69} +1442.54i q^{70} -238.544 q^{71} +1750.77i q^{72} -3348.79 q^{73} +2090.58i q^{74} +2901.60i q^{75} -5290.67 q^{76} +2962.05 q^{77} -107.251i q^{78} -1448.51i q^{79} +92.6659i q^{80} +729.000 q^{81} -2662.19 q^{82} +1344.13 q^{83} +3465.65 q^{84} +419.833i q^{85} -6313.62 q^{86} +4948.56i q^{87} +2755.42 q^{88} +4416.22 q^{89} +558.755 q^{90} -567.316 q^{91} -6326.89 q^{92} +8168.70 q^{93} +8465.95i q^{94} +4512.02i q^{95} +5241.32 q^{96} -4826.09i q^{97} +6233.49i q^{98} -1147.32i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 396 q^{4} - 1242 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q - 396 q^{4} - 1242 q^{9} + 396 q^{10} + 792 q^{14} - 252 q^{15} + 3396 q^{16} + 462 q^{17} - 590 q^{19} - 936 q^{21} + 3184 q^{22} - 1446 q^{23} - 1404 q^{24} - 6278 q^{25} + 2700 q^{26} - 1014 q^{29} + 540 q^{33} + 9924 q^{35} + 10692 q^{36} - 386 q^{37} + 4968 q^{39} - 9988 q^{40} - 2754 q^{47} - 19062 q^{49} - 2320 q^{55} - 3396 q^{56} - 7098 q^{59} + 72 q^{60} - 21180 q^{62} - 75644 q^{64} + 18396 q^{65} + 8574 q^{67} + 9084 q^{68} - 23040 q^{71} - 22338 q^{73} + 28016 q^{76} + 45084 q^{77} + 33534 q^{81} + 17564 q^{82} + 35856 q^{83} + 40176 q^{84} + 31764 q^{86} - 19448 q^{88} - 14538 q^{89} - 10692 q^{90} + 13792 q^{91} - 67692 q^{92} + 22464 q^{93} + 22464 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.53609i 0.634022i −0.948422 0.317011i \(-0.897321\pi\)
0.948422 0.317011i \(-0.102679\pi\)
\(3\) 5.19615i 0.577350i
\(4\) 9.56827 0.598017
\(5\) 8.16007i 0.326403i −0.986593 0.163201i \(-0.947818\pi\)
0.986593 0.163201i \(-0.0521820\pi\)
\(6\) 13.1779 0.366053
\(7\) 69.7059i 1.42257i −0.702904 0.711285i \(-0.748114\pi\)
0.702904 0.711285i \(-0.251886\pi\)
\(8\) 64.8433i 1.01318i
\(9\) −27.0000 −0.333333
\(10\) −20.6946 −0.206946
\(11\) 42.4935i 0.351186i 0.984463 + 0.175593i \(0.0561843\pi\)
−0.984463 + 0.175593i \(0.943816\pi\)
\(12\) 49.7182i 0.345265i
\(13\) 8.13871i 0.0481581i −0.999710 0.0240790i \(-0.992335\pi\)
0.999710 0.0240790i \(-0.00766533\pi\)
\(14\) −176.780 −0.901940
\(15\) 42.4009 0.188449
\(16\) −11.3560 −0.0443595
\(17\) −51.4498 −0.178027 −0.0890134 0.996030i \(-0.528371\pi\)
−0.0890134 + 0.996030i \(0.528371\pi\)
\(18\) 68.4743i 0.211341i
\(19\) −552.939 −1.53169 −0.765844 0.643027i \(-0.777678\pi\)
−0.765844 + 0.643027i \(0.777678\pi\)
\(20\) 78.0777i 0.195194i
\(21\) 362.203 0.821321
\(22\) 107.767 0.222659
\(23\) −661.237 −1.24998 −0.624988 0.780635i \(-0.714896\pi\)
−0.624988 + 0.780635i \(0.714896\pi\)
\(24\) 336.936 0.584958
\(25\) 558.413 0.893461
\(26\) −20.6405 −0.0305332
\(27\) 140.296i 0.192450i
\(28\) 666.965i 0.850721i
\(29\) 952.351 1.13240 0.566202 0.824267i \(-0.308412\pi\)
0.566202 + 0.824267i \(0.308412\pi\)
\(30\) 107.532i 0.119481i
\(31\) 1572.07i 1.63587i −0.575314 0.817933i \(-0.695120\pi\)
0.575314 0.817933i \(-0.304880\pi\)
\(32\) 1008.69i 0.985052i
\(33\) −220.803 −0.202757
\(34\) 130.481i 0.112873i
\(35\) −568.805 −0.464331
\(36\) −258.343 −0.199339
\(37\) −824.332 −0.602142 −0.301071 0.953602i \(-0.597344\pi\)
−0.301071 + 0.953602i \(0.597344\pi\)
\(38\) 1402.30i 0.971123i
\(39\) 42.2900 0.0278041
\(40\) −529.126 −0.330704
\(41\) 1049.72i 0.624463i −0.950006 0.312231i \(-0.898924\pi\)
0.950006 0.312231i \(-0.101076\pi\)
\(42\) 918.577i 0.520735i
\(43\) 2489.51i 1.34641i −0.739456 0.673205i \(-0.764917\pi\)
0.739456 0.673205i \(-0.235083\pi\)
\(44\) 406.589i 0.210015i
\(45\) 220.322i 0.108801i
\(46\) 1676.95i 0.792511i
\(47\) −3338.19 −1.51118 −0.755589 0.655046i \(-0.772649\pi\)
−0.755589 + 0.655046i \(0.772649\pi\)
\(48\) 59.0076i 0.0256109i
\(49\) −2457.92 −1.02371
\(50\) 1416.18i 0.566474i
\(51\) 267.341i 0.102784i
\(52\) 77.8734i 0.0287993i
\(53\) 5321.62i 1.89449i 0.320512 + 0.947245i \(0.396145\pi\)
−0.320512 + 0.947245i \(0.603855\pi\)
\(54\) −355.803 −0.122018
\(55\) 346.750 0.114628
\(56\) −4519.96 −1.44132
\(57\) 2873.16i 0.884320i
\(58\) 2415.24i 0.717968i
\(59\) 6339.83 1.82127 0.910634 0.413214i \(-0.135594\pi\)
0.910634 + 0.413214i \(0.135594\pi\)
\(60\) 405.704 0.112695
\(61\) 5490.98i 1.47567i −0.674979 0.737837i \(-0.735847\pi\)
0.674979 0.737837i \(-0.264153\pi\)
\(62\) −3986.90 −1.03717
\(63\) 1882.06i 0.474190i
\(64\) −2739.83 −0.668904
\(65\) −66.4124 −0.0157189
\(66\) 559.975i 0.128552i
\(67\) 3106.61 + 3240.39i 0.692049 + 0.721851i
\(68\) −492.285 −0.106463
\(69\) 3435.89i 0.721674i
\(70\) 1442.54i 0.294396i
\(71\) −238.544 −0.0473208 −0.0236604 0.999720i \(-0.507532\pi\)
−0.0236604 + 0.999720i \(0.507532\pi\)
\(72\) 1750.77i 0.337726i
\(73\) −3348.79 −0.628408 −0.314204 0.949355i \(-0.601738\pi\)
−0.314204 + 0.949355i \(0.601738\pi\)
\(74\) 2090.58i 0.381771i
\(75\) 2901.60i 0.515840i
\(76\) −5290.67 −0.915974
\(77\) 2962.05 0.499587
\(78\) 107.251i 0.0176284i
\(79\) 1448.51i 0.232096i −0.993244 0.116048i \(-0.962977\pi\)
0.993244 0.116048i \(-0.0370226\pi\)
\(80\) 92.6659i 0.0144790i
\(81\) 729.000 0.111111
\(82\) −2662.19 −0.395923
\(83\) 1344.13 0.195113 0.0975563 0.995230i \(-0.468897\pi\)
0.0975563 + 0.995230i \(0.468897\pi\)
\(84\) 3465.65 0.491164
\(85\) 419.833i 0.0581084i
\(86\) −6313.62 −0.853653
\(87\) 4948.56i 0.653793i
\(88\) 2755.42 0.355813
\(89\) 4416.22 0.557533 0.278766 0.960359i \(-0.410075\pi\)
0.278766 + 0.960359i \(0.410075\pi\)
\(90\) 558.755 0.0689821
\(91\) −567.316 −0.0685082
\(92\) −6326.89 −0.747506
\(93\) 8168.70 0.944467
\(94\) 8465.95i 0.958120i
\(95\) 4512.02i 0.499947i
\(96\) 5241.32 0.568720
\(97\) 4826.09i 0.512922i −0.966555 0.256461i \(-0.917443\pi\)
0.966555 0.256461i \(-0.0825566\pi\)
\(98\) 6233.49i 0.649051i
\(99\) 1147.32i 0.117062i
\(100\) 5343.05 0.534305
\(101\) 3271.88i 0.320741i −0.987057 0.160371i \(-0.948731\pi\)
0.987057 0.160371i \(-0.0512690\pi\)
\(102\) −677.999 −0.0651672
\(103\) 18834.7 1.77536 0.887678 0.460465i \(-0.152317\pi\)
0.887678 + 0.460465i \(0.152317\pi\)
\(104\) −527.741 −0.0487926
\(105\) 2955.60i 0.268081i
\(106\) 13496.1 1.20115
\(107\) 1334.21 0.116535 0.0582674 0.998301i \(-0.481442\pi\)
0.0582674 + 0.998301i \(0.481442\pi\)
\(108\) 1342.39i 0.115088i
\(109\) 16139.4i 1.35842i −0.733943 0.679211i \(-0.762322\pi\)
0.733943 0.679211i \(-0.237678\pi\)
\(110\) 879.387i 0.0726766i
\(111\) 4283.35i 0.347647i
\(112\) 791.582i 0.0631044i
\(113\) 21501.8i 1.68390i 0.539553 + 0.841951i \(0.318593\pi\)
−0.539553 + 0.841951i \(0.681407\pi\)
\(114\) −7286.57 −0.560678
\(115\) 5395.74i 0.407995i
\(116\) 9112.35 0.677196
\(117\) 219.745i 0.0160527i
\(118\) 16078.4i 1.15472i
\(119\) 3586.35i 0.253256i
\(120\) 2749.42i 0.190932i
\(121\) 12835.3 0.876668
\(122\) −13925.6 −0.935609
\(123\) 5454.52 0.360534
\(124\) 15042.0i 0.978275i
\(125\) 9656.73i 0.618031i
\(126\) 4773.07 0.300647
\(127\) −1460.37 −0.0905431 −0.0452715 0.998975i \(-0.514415\pi\)
−0.0452715 + 0.998975i \(0.514415\pi\)
\(128\) 9190.65i 0.560953i
\(129\) 12935.9 0.777350
\(130\) 168.428i 0.00996613i
\(131\) 6483.08 0.377780 0.188890 0.981998i \(-0.439511\pi\)
0.188890 + 0.981998i \(0.439511\pi\)
\(132\) −2112.70 −0.121252
\(133\) 38543.1i 2.17893i
\(134\) 8217.90 7878.62i 0.457669 0.438774i
\(135\) −1144.83 −0.0628162
\(136\) 3336.17i 0.180373i
\(137\) 32095.2i 1.71001i 0.518619 + 0.855006i \(0.326446\pi\)
−0.518619 + 0.855006i \(0.673554\pi\)
\(138\) −8713.71 −0.457557
\(139\) 28368.8i 1.46829i 0.678994 + 0.734144i \(0.262416\pi\)
−0.678994 + 0.734144i \(0.737584\pi\)
\(140\) −5442.48 −0.277677
\(141\) 17345.8i 0.872479i
\(142\) 604.969i 0.0300024i
\(143\) 345.842 0.0169124
\(144\) 306.613 0.0147865
\(145\) 7771.25i 0.369619i
\(146\) 8492.81i 0.398424i
\(147\) 12771.7i 0.591037i
\(148\) −7887.43 −0.360091
\(149\) −9734.03 −0.438450 −0.219225 0.975674i \(-0.570353\pi\)
−0.219225 + 0.975674i \(0.570353\pi\)
\(150\) 7358.71 0.327054
\(151\) 791.752 0.0347245 0.0173622 0.999849i \(-0.494473\pi\)
0.0173622 + 0.999849i \(0.494473\pi\)
\(152\) 35854.4i 1.55187i
\(153\) 1389.14 0.0593423
\(154\) 7512.01i 0.316749i
\(155\) −12828.2 −0.533951
\(156\) 404.642 0.0166273
\(157\) 28652.3 1.16241 0.581206 0.813757i \(-0.302581\pi\)
0.581206 + 0.813757i \(0.302581\pi\)
\(158\) −3673.55 −0.147154
\(159\) −27651.9 −1.09378
\(160\) −8231.01 −0.321524
\(161\) 46092.1i 1.77818i
\(162\) 1848.81i 0.0704468i
\(163\) −33278.1 −1.25252 −0.626258 0.779615i \(-0.715415\pi\)
−0.626258 + 0.779615i \(0.715415\pi\)
\(164\) 10044.0i 0.373439i
\(165\) 1801.76i 0.0661805i
\(166\) 3408.83i 0.123706i
\(167\) −19217.5 −0.689072 −0.344536 0.938773i \(-0.611964\pi\)
−0.344536 + 0.938773i \(0.611964\pi\)
\(168\) 23486.4i 0.832144i
\(169\) 28494.8 0.997681
\(170\) 1064.73 0.0368420
\(171\) 14929.4 0.510562
\(172\) 23820.3i 0.805176i
\(173\) −36614.7 −1.22338 −0.611692 0.791096i \(-0.709511\pi\)
−0.611692 + 0.791096i \(0.709511\pi\)
\(174\) 12550.0 0.414519
\(175\) 38924.7i 1.27101i
\(176\) 482.557i 0.0155784i
\(177\) 32942.7i 1.05151i
\(178\) 11199.9i 0.353488i
\(179\) 49518.1i 1.54546i 0.634734 + 0.772730i \(0.281110\pi\)
−0.634734 + 0.772730i \(0.718890\pi\)
\(180\) 2108.10i 0.0650647i
\(181\) 51803.6 1.58126 0.790629 0.612296i \(-0.209754\pi\)
0.790629 + 0.612296i \(0.209754\pi\)
\(182\) 1438.76i 0.0434357i
\(183\) 28532.0 0.851980
\(184\) 42876.8i 1.26645i
\(185\) 6726.60i 0.196541i
\(186\) 20716.5i 0.598813i
\(187\) 2186.28i 0.0625205i
\(188\) −31940.7 −0.903710
\(189\) −9779.47 −0.273774
\(190\) 11442.9 0.316977
\(191\) 29898.4i 0.819562i −0.912184 0.409781i \(-0.865605\pi\)
0.912184 0.409781i \(-0.134395\pi\)
\(192\) 14236.6i 0.386192i
\(193\) −39039.5 −1.04807 −0.524035 0.851697i \(-0.675574\pi\)
−0.524035 + 0.851697i \(0.675574\pi\)
\(194\) −12239.4 −0.325204
\(195\) 345.089i 0.00907532i
\(196\) −23518.0 −0.612193
\(197\) 17800.8i 0.458676i −0.973347 0.229338i \(-0.926344\pi\)
0.973347 0.229338i \(-0.0736561\pi\)
\(198\) −2909.71 −0.0742198
\(199\) 11121.0 0.280827 0.140413 0.990093i \(-0.455157\pi\)
0.140413 + 0.990093i \(0.455157\pi\)
\(200\) 36209.4i 0.905234i
\(201\) −16837.6 + 16142.4i −0.416761 + 0.399554i
\(202\) −8297.78 −0.203357
\(203\) 66384.5i 1.61092i
\(204\) 2557.99i 0.0614664i
\(205\) −8565.80 −0.203826
\(206\) 47766.5i 1.12561i
\(207\) 17853.4 0.416658
\(208\) 92.4234i 0.00213627i
\(209\) 23496.3i 0.537907i
\(210\) −7495.65 −0.169969
\(211\) 24317.1 0.546194 0.273097 0.961987i \(-0.411952\pi\)
0.273097 + 0.961987i \(0.411952\pi\)
\(212\) 50918.7i 1.13294i
\(213\) 1239.51i 0.0273207i
\(214\) 3383.66i 0.0738855i
\(215\) −20314.6 −0.439472
\(216\) −9097.27 −0.194986
\(217\) −109582. −2.32713
\(218\) −40930.9 −0.861269
\(219\) 17400.8i 0.362812i
\(220\) 3317.79 0.0685495
\(221\) 418.735i 0.00857343i
\(222\) −10863.0 −0.220415
\(223\) 44620.8 0.897278 0.448639 0.893713i \(-0.351909\pi\)
0.448639 + 0.893713i \(0.351909\pi\)
\(224\) −70311.9 −1.40131
\(225\) −15077.2 −0.297820
\(226\) 54530.3 1.06763
\(227\) 33993.9 0.659705 0.329853 0.944032i \(-0.393001\pi\)
0.329853 + 0.944032i \(0.393001\pi\)
\(228\) 27491.1i 0.528838i
\(229\) 36241.4i 0.691089i 0.938402 + 0.345544i \(0.112306\pi\)
−0.938402 + 0.345544i \(0.887694\pi\)
\(230\) 13684.1 0.258678
\(231\) 15391.3i 0.288436i
\(232\) 61753.6i 1.14732i
\(233\) 33280.3i 0.613021i −0.951867 0.306510i \(-0.900839\pi\)
0.951867 0.306510i \(-0.0991614\pi\)
\(234\) 557.293 0.0101777
\(235\) 27239.9i 0.493253i
\(236\) 60661.2 1.08915
\(237\) 7526.68 0.134001
\(238\) 9095.30 0.160570
\(239\) 10500.7i 0.183832i −0.995767 0.0919161i \(-0.970701\pi\)
0.995767 0.0919161i \(-0.0292992\pi\)
\(240\) −481.506 −0.00835948
\(241\) 25198.2 0.433846 0.216923 0.976189i \(-0.430398\pi\)
0.216923 + 0.976189i \(0.430398\pi\)
\(242\) 32551.4i 0.555827i
\(243\) 3788.00i 0.0641500i
\(244\) 52539.2i 0.882477i
\(245\) 20056.8i 0.334140i
\(246\) 13833.1i 0.228586i
\(247\) 4500.21i 0.0737631i
\(248\) −101938. −1.65742
\(249\) 6984.31i 0.112648i
\(250\) −24490.3 −0.391845
\(251\) 104168.i 1.65344i −0.562616 0.826719i \(-0.690205\pi\)
0.562616 0.826719i \(-0.309795\pi\)
\(252\) 18008.1i 0.283574i
\(253\) 28098.3i 0.438974i
\(254\) 3703.62i 0.0574063i
\(255\) −2181.52 −0.0335489
\(256\) −67145.6 −1.02456
\(257\) 47433.0 0.718149 0.359074 0.933309i \(-0.383092\pi\)
0.359074 + 0.933309i \(0.383092\pi\)
\(258\) 32806.5i 0.492857i
\(259\) 57460.8i 0.856589i
\(260\) −635.452 −0.00940017
\(261\) −25713.5 −0.377468
\(262\) 16441.7i 0.239521i
\(263\) 2241.07 0.0323999 0.0161999 0.999869i \(-0.494843\pi\)
0.0161999 + 0.999869i \(0.494843\pi\)
\(264\) 14317.6i 0.205429i
\(265\) 43424.8 0.618366
\(266\) 97748.7 1.38149
\(267\) 22947.3i 0.321892i
\(268\) 29724.8 + 31004.9i 0.413857 + 0.431679i
\(269\) 130716. 1.80644 0.903218 0.429182i \(-0.141198\pi\)
0.903218 + 0.429182i \(0.141198\pi\)
\(270\) 2903.38i 0.0398268i
\(271\) 23195.9i 0.315843i −0.987452 0.157922i \(-0.949521\pi\)
0.987452 0.157922i \(-0.0504794\pi\)
\(272\) 584.264 0.00789717
\(273\) 2947.86i 0.0395532i
\(274\) 81396.2 1.08418
\(275\) 23728.9i 0.313771i
\(276\) 32875.5i 0.431573i
\(277\) −122354. −1.59462 −0.797312 0.603568i \(-0.793745\pi\)
−0.797312 + 0.603568i \(0.793745\pi\)
\(278\) 71945.7 0.930926
\(279\) 42445.8i 0.545288i
\(280\) 36883.2i 0.470449i
\(281\) 103622.i 1.31232i 0.754621 + 0.656161i \(0.227821\pi\)
−0.754621 + 0.656161i \(0.772179\pi\)
\(282\) −43990.4 −0.553171
\(283\) 69630.9 0.869419 0.434710 0.900571i \(-0.356851\pi\)
0.434710 + 0.900571i \(0.356851\pi\)
\(284\) −2282.45 −0.0282986
\(285\) −23445.1 −0.288644
\(286\) 877.086i 0.0107228i
\(287\) −73171.9 −0.888342
\(288\) 27234.7i 0.328351i
\(289\) −80873.9 −0.968306
\(290\) −19708.6 −0.234347
\(291\) 25077.1 0.296136
\(292\) −32042.1 −0.375799
\(293\) 131459. 1.53129 0.765643 0.643266i \(-0.222421\pi\)
0.765643 + 0.643266i \(0.222421\pi\)
\(294\) −32390.2 −0.374730
\(295\) 51733.5i 0.594467i
\(296\) 53452.4i 0.610076i
\(297\) 5961.67 0.0675858
\(298\) 24686.3i 0.277987i
\(299\) 5381.62i 0.0601964i
\(300\) 27763.3i 0.308481i
\(301\) −173534. −1.91536
\(302\) 2007.95i 0.0220161i
\(303\) 17001.2 0.185180
\(304\) 6279.19 0.0679448
\(305\) −44806.8 −0.481664
\(306\) 3522.99i 0.0376243i
\(307\) −115161. −1.22188 −0.610942 0.791675i \(-0.709209\pi\)
−0.610942 + 0.791675i \(0.709209\pi\)
\(308\) 28341.7 0.298761
\(309\) 97868.2i 1.02500i
\(310\) 32533.3i 0.338536i
\(311\) 36460.2i 0.376962i −0.982077 0.188481i \(-0.939644\pi\)
0.982077 0.188481i \(-0.0603564\pi\)
\(312\) 2742.22i 0.0281704i
\(313\) 3411.80i 0.0348253i 0.999848 + 0.0174127i \(0.00554290\pi\)
−0.999848 + 0.0174127i \(0.994457\pi\)
\(314\) 72664.6i 0.736994i
\(315\) 15357.7 0.154777
\(316\) 13859.7i 0.138797i
\(317\) 143439. 1.42741 0.713706 0.700445i \(-0.247015\pi\)
0.713706 + 0.700445i \(0.247015\pi\)
\(318\) 70127.7i 0.693483i
\(319\) 40468.7i 0.397684i
\(320\) 22357.2i 0.218332i
\(321\) 6932.74i 0.0672813i
\(322\) 116894. 1.12740
\(323\) 28448.6 0.272681
\(324\) 6975.27 0.0664463
\(325\) 4544.76i 0.0430274i
\(326\) 84396.2i 0.794123i
\(327\) 83862.9 0.784285
\(328\) −68067.5 −0.632691
\(329\) 232692.i 2.14976i
\(330\) 4569.43 0.0419599
\(331\) 174114.i 1.58920i −0.607135 0.794599i \(-0.707681\pi\)
0.607135 0.794599i \(-0.292319\pi\)
\(332\) 12861.0 0.116681
\(333\) 22257.0 0.200714
\(334\) 48737.3i 0.436887i
\(335\) 26441.8 25350.1i 0.235614 0.225887i
\(336\) −4113.18 −0.0364334
\(337\) 100618.i 0.885964i 0.896531 + 0.442982i \(0.146079\pi\)
−0.896531 + 0.442982i \(0.853921\pi\)
\(338\) 72265.2i 0.632551i
\(339\) −111726. −0.972202
\(340\) 4017.08i 0.0347498i
\(341\) 66802.6 0.574493
\(342\) 37862.1i 0.323708i
\(343\) 3967.46i 0.0337228i
\(344\) −161428. −1.36415
\(345\) −28037.1 −0.235556
\(346\) 92857.9i 0.775652i
\(347\) 149814.i 1.24421i −0.782935 0.622104i \(-0.786278\pi\)
0.782935 0.622104i \(-0.213722\pi\)
\(348\) 47349.1i 0.390979i
\(349\) −128819. −1.05762 −0.528808 0.848742i \(-0.677361\pi\)
−0.528808 + 0.848742i \(0.677361\pi\)
\(350\) −98716.5 −0.805849
\(351\) −1141.83 −0.00926802
\(352\) 42862.9 0.345936
\(353\) 46449.1i 0.372759i −0.982478 0.186379i \(-0.940325\pi\)
0.982478 0.186379i \(-0.0596754\pi\)
\(354\) 83545.6 0.666680
\(355\) 1946.54i 0.0154456i
\(356\) 42255.5 0.333414
\(357\) −18635.2 −0.146217
\(358\) 125582. 0.979856
\(359\) −55914.3 −0.433844 −0.216922 0.976189i \(-0.569602\pi\)
−0.216922 + 0.976189i \(0.569602\pi\)
\(360\) 14286.4 0.110235
\(361\) 175421. 1.34607
\(362\) 131378.i 1.00255i
\(363\) 66694.2i 0.506145i
\(364\) −5428.24 −0.0409690
\(365\) 27326.3i 0.205114i
\(366\) 72359.5i 0.540174i
\(367\) 45255.2i 0.335997i 0.985787 + 0.167999i \(0.0537304\pi\)
−0.985787 + 0.167999i \(0.946270\pi\)
\(368\) 7509.02 0.0554482
\(369\) 28342.5i 0.208154i
\(370\) 17059.2 0.124611
\(371\) 370948. 2.69504
\(372\) 78160.3 0.564807
\(373\) 31935.1i 0.229536i −0.993392 0.114768i \(-0.963388\pi\)
0.993392 0.114768i \(-0.0366124\pi\)
\(374\) −5544.59 −0.0396394
\(375\) 50177.8 0.356820
\(376\) 216460.i 1.53109i
\(377\) 7750.91i 0.0545343i
\(378\) 24801.6i 0.173578i
\(379\) 52284.2i 0.363992i 0.983299 + 0.181996i \(0.0582558\pi\)
−0.983299 + 0.181996i \(0.941744\pi\)
\(380\) 43172.2i 0.298976i
\(381\) 7588.30i 0.0522751i
\(382\) −75825.0 −0.519620
\(383\) 140143.i 0.955377i −0.878529 0.477689i \(-0.841475\pi\)
0.878529 0.477689i \(-0.158525\pi\)
\(384\) 47756.0 0.323866
\(385\) 24170.5i 0.163066i
\(386\) 99007.7i 0.664499i
\(387\) 67216.8i 0.448803i
\(388\) 46177.3i 0.306736i
\(389\) 100857. 0.666513 0.333257 0.942836i \(-0.391852\pi\)
0.333257 + 0.942836i \(0.391852\pi\)
\(390\) −875.176 −0.00575395
\(391\) 34020.5 0.222529
\(392\) 159380.i 1.03719i
\(393\) 33687.1i 0.218111i
\(394\) −45144.3 −0.290811
\(395\) −11819.9 −0.0757567
\(396\) 10977.9i 0.0700050i
\(397\) 259406. 1.64588 0.822942 0.568126i \(-0.192331\pi\)
0.822942 + 0.568126i \(0.192331\pi\)
\(398\) 28203.9i 0.178050i
\(399\) −200276. −1.25801
\(400\) −6341.35 −0.0396335
\(401\) 62793.9i 0.390507i −0.980753 0.195253i \(-0.937447\pi\)
0.980753 0.195253i \(-0.0625529\pi\)
\(402\) 40938.5 + 42701.5i 0.253326 + 0.264235i
\(403\) −12794.6 −0.0787801
\(404\) 31306.3i 0.191809i
\(405\) 5948.69i 0.0362670i
\(406\) −168357. −1.02136
\(407\) 35028.7i 0.211464i
\(408\) −17335.3 −0.104138
\(409\) 73604.2i 0.440003i −0.975500 0.220002i \(-0.929394\pi\)
0.975500 0.220002i \(-0.0706063\pi\)
\(410\) 21723.6i 0.129230i
\(411\) −166772. −0.987275
\(412\) 180216. 1.06169
\(413\) 441924.i 2.59088i
\(414\) 45277.8i 0.264170i
\(415\) 10968.2i 0.0636853i
\(416\) −8209.47 −0.0474382
\(417\) −147409. −0.847717
\(418\) −59588.7 −0.341045
\(419\) −182512. −1.03959 −0.519797 0.854290i \(-0.673992\pi\)
−0.519797 + 0.854290i \(0.673992\pi\)
\(420\) 28279.9i 0.160317i
\(421\) 25338.0 0.142958 0.0714790 0.997442i \(-0.477228\pi\)
0.0714790 + 0.997442i \(0.477228\pi\)
\(422\) 61670.2i 0.346299i
\(423\) 90131.2 0.503726
\(424\) 345072. 1.91945
\(425\) −28730.2 −0.159060
\(426\) −3143.51 −0.0173219
\(427\) −382754. −2.09925
\(428\) 12766.0 0.0696897
\(429\) 1797.05i 0.00976440i
\(430\) 51519.5i 0.278635i
\(431\) −146413. −0.788179 −0.394090 0.919072i \(-0.628940\pi\)
−0.394090 + 0.919072i \(0.628940\pi\)
\(432\) 1593.21i 0.00853698i
\(433\) 307516.i 1.64018i 0.572232 + 0.820092i \(0.306078\pi\)
−0.572232 + 0.820092i \(0.693922\pi\)
\(434\) 277910.i 1.47545i
\(435\) 40380.6 0.213400
\(436\) 154426.i 0.812359i
\(437\) 365624. 1.91457
\(438\) −44130.0 −0.230030
\(439\) −88851.9 −0.461039 −0.230520 0.973068i \(-0.574043\pi\)
−0.230520 + 0.973068i \(0.574043\pi\)
\(440\) 22484.4i 0.116138i
\(441\) 66363.8 0.341235
\(442\) 1061.95 0.00543574
\(443\) 145343.i 0.740606i −0.928911 0.370303i \(-0.879254\pi\)
0.928911 0.370303i \(-0.120746\pi\)
\(444\) 40984.3i 0.207899i
\(445\) 36036.6i 0.181980i
\(446\) 113162.i 0.568894i
\(447\) 50579.5i 0.253139i
\(448\) 190982.i 0.951562i
\(449\) −216267. −1.07275 −0.536373 0.843981i \(-0.680206\pi\)
−0.536373 + 0.843981i \(0.680206\pi\)
\(450\) 38237.0i 0.188825i
\(451\) 44606.4 0.219303
\(452\) 205735.i 1.00700i
\(453\) 4114.07i 0.0200482i
\(454\) 86211.6i 0.418267i
\(455\) 4629.34i 0.0223613i
\(456\) −186305. −0.895973
\(457\) −413963. −1.98212 −0.991058 0.133428i \(-0.957401\pi\)
−0.991058 + 0.133428i \(0.957401\pi\)
\(458\) 91911.3 0.438165
\(459\) 7218.20i 0.0342613i
\(460\) 51627.9i 0.243988i
\(461\) 115022. 0.541226 0.270613 0.962688i \(-0.412774\pi\)
0.270613 + 0.962688i \(0.412774\pi\)
\(462\) 39033.6 0.182875
\(463\) 172721.i 0.805719i −0.915262 0.402859i \(-0.868016\pi\)
0.915262 0.402859i \(-0.131984\pi\)
\(464\) −10814.9 −0.0502328
\(465\) 66657.1i 0.308277i
\(466\) −84401.6 −0.388668
\(467\) −126068. −0.578058 −0.289029 0.957320i \(-0.593332\pi\)
−0.289029 + 0.957320i \(0.593332\pi\)
\(468\) 2102.58i 0.00959977i
\(469\) 225874. 216549.i 1.02688 0.984488i
\(470\) 69082.7 0.312733
\(471\) 148882.i 0.671118i
\(472\) 411096.i 1.84527i
\(473\) 105788. 0.472840
\(474\) 19088.3i 0.0849593i
\(475\) −308769. −1.36850
\(476\) 34315.2i 0.151451i
\(477\) 143684.i 0.631496i
\(478\) −26630.6 −0.116554
\(479\) 137760. 0.600417 0.300208 0.953874i \(-0.402944\pi\)
0.300208 + 0.953874i \(0.402944\pi\)
\(480\) 42769.6i 0.185632i
\(481\) 6709.00i 0.0289980i
\(482\) 63904.9i 0.275068i
\(483\) −239502. −1.02663
\(484\) 122812. 0.524262
\(485\) −39381.2 −0.167419
\(486\) 9606.68 0.0406725
\(487\) 134028.i 0.565115i −0.959250 0.282558i \(-0.908817\pi\)
0.959250 0.282558i \(-0.0911829\pi\)
\(488\) −356053. −1.49512
\(489\) 172918.i 0.723141i
\(490\) 50865.7 0.211852
\(491\) −59774.2 −0.247942 −0.123971 0.992286i \(-0.539563\pi\)
−0.123971 + 0.992286i \(0.539563\pi\)
\(492\) 52190.3 0.215605
\(493\) −48998.2 −0.201598
\(494\) 11412.9 0.0467674
\(495\) −9362.24 −0.0382093
\(496\) 17852.4i 0.0725661i
\(497\) 16627.9i 0.0673172i
\(498\) 17712.8 0.0714215
\(499\) 382461.i 1.53598i 0.640460 + 0.767992i \(0.278744\pi\)
−0.640460 + 0.767992i \(0.721256\pi\)
\(500\) 92398.2i 0.369593i
\(501\) 99857.3i 0.397836i
\(502\) −264180. −1.04831
\(503\) 394723.i 1.56011i −0.625708 0.780057i \(-0.715190\pi\)
0.625708 0.780057i \(-0.284810\pi\)
\(504\) 122039. 0.480438
\(505\) −26698.8 −0.104691
\(506\) −71259.6 −0.278319
\(507\) 148063.i 0.576011i
\(508\) −13973.2 −0.0541463
\(509\) −49356.1 −0.190504 −0.0952522 0.995453i \(-0.530366\pi\)
−0.0952522 + 0.995453i \(0.530366\pi\)
\(510\) 5532.52i 0.0212707i
\(511\) 233430.i 0.893955i
\(512\) 23236.5i 0.0886404i
\(513\) 77575.2i 0.294773i
\(514\) 120294.i 0.455322i
\(515\) 153693.i 0.579481i
\(516\) 123774. 0.464868
\(517\) 141852.i 0.530705i
\(518\) 145726. 0.543096
\(519\) 190255.i 0.706321i
\(520\) 4306.40i 0.0159260i
\(521\) 164403.i 0.605668i 0.953043 + 0.302834i \(0.0979328\pi\)
−0.953043 + 0.302834i \(0.902067\pi\)
\(522\) 65211.6i 0.239323i
\(523\) 12914.4 0.0472142 0.0236071 0.999721i \(-0.492485\pi\)
0.0236071 + 0.999721i \(0.492485\pi\)
\(524\) 62031.9 0.225919
\(525\) 202259. 0.733819
\(526\) 5683.54i 0.0205422i
\(527\) 80882.4i 0.291228i
\(528\) 2507.44 0.00899420
\(529\) 157393. 0.562438
\(530\) 110129.i 0.392058i
\(531\) −171175. −0.607089
\(532\) 368791.i 1.30304i
\(533\) −8543.39 −0.0300729
\(534\) 58196.4 0.204086
\(535\) 10887.2i 0.0380372i
\(536\) 210118. 201443.i 0.731363 0.701168i
\(537\) −257304. −0.892272
\(538\) 331506.i 1.14532i
\(539\) 104445.i 0.359511i
\(540\) −10954.0 −0.0375651
\(541\) 173516.i 0.592849i −0.955056 0.296424i \(-0.904206\pi\)
0.955056 0.296424i \(-0.0957943\pi\)
\(542\) −58826.7 −0.200252
\(543\) 269179.i 0.912939i
\(544\) 51897.0i 0.175366i
\(545\) −131699. −0.443393
\(546\) −7476.03 −0.0250776
\(547\) 184590.i 0.616925i −0.951236 0.308463i \(-0.900186\pi\)
0.951236 0.308463i \(-0.0998145\pi\)
\(548\) 307095.i 1.02262i
\(549\) 148256.i 0.491891i
\(550\) 60178.6 0.198938
\(551\) −526592. −1.73449
\(552\) −222794. −0.731183
\(553\) −100970. −0.330173
\(554\) 310300.i 1.01103i
\(555\) −34952.5 −0.113473
\(556\) 271440.i 0.878061i
\(557\) −132778. −0.427971 −0.213986 0.976837i \(-0.568645\pi\)
−0.213986 + 0.976837i \(0.568645\pi\)
\(558\) 107646. 0.345725
\(559\) −20261.4 −0.0648405
\(560\) 6459.36 0.0205975
\(561\) 11360.2 0.0360962
\(562\) 262795. 0.832041
\(563\) 221405.i 0.698506i 0.937028 + 0.349253i \(0.113565\pi\)
−0.937028 + 0.349253i \(0.886435\pi\)
\(564\) 165969.i 0.521757i
\(565\) 175456. 0.549630
\(566\) 176590.i 0.551231i
\(567\) 50815.6i 0.158063i
\(568\) 15468.0i 0.0479444i
\(569\) −244081. −0.753891 −0.376946 0.926235i \(-0.623026\pi\)
−0.376946 + 0.926235i \(0.623026\pi\)
\(570\) 59458.9i 0.183007i
\(571\) 453087. 1.38966 0.694831 0.719173i \(-0.255479\pi\)
0.694831 + 0.719173i \(0.255479\pi\)
\(572\) 3309.11 0.0101139
\(573\) 155357. 0.473174
\(574\) 185570.i 0.563228i
\(575\) −369244. −1.11680
\(576\) 73975.4 0.222968
\(577\) 123405.i 0.370665i −0.982676 0.185333i \(-0.940664\pi\)
0.982676 0.185333i \(-0.0593362\pi\)
\(578\) 205103.i 0.613927i
\(579\) 202855.i 0.605103i
\(580\) 74357.4i 0.221039i
\(581\) 93693.9i 0.277561i
\(582\) 63597.6i 0.187756i
\(583\) −226134. −0.665318
\(584\) 217147.i 0.636689i
\(585\) 1793.14 0.00523964
\(586\) 333392.i 0.970868i
\(587\) 192782.i 0.559488i −0.960075 0.279744i \(-0.909750\pi\)
0.960075 0.279744i \(-0.0902496\pi\)
\(588\) 122203.i 0.353450i
\(589\) 869257.i 2.50563i
\(590\) −131201. −0.376905
\(591\) 92495.5 0.264817
\(592\) 9361.13 0.0267107
\(593\) 381805.i 1.08576i −0.839812 0.542878i \(-0.817335\pi\)
0.839812 0.542878i \(-0.182665\pi\)
\(594\) 15119.3i 0.0428508i
\(595\) 29264.9 0.0826633
\(596\) −93137.8 −0.262200
\(597\) 57786.5i 0.162135i
\(598\) 13648.2 0.0381658
\(599\) 63839.1i 0.177923i −0.996035 0.0889617i \(-0.971645\pi\)
0.996035 0.0889617i \(-0.0283549\pi\)
\(600\) 188149. 0.522637
\(601\) 454809. 1.25916 0.629579 0.776936i \(-0.283227\pi\)
0.629579 + 0.776936i \(0.283227\pi\)
\(602\) 440097.i 1.21438i
\(603\) −83878.4 87490.5i −0.230683 0.240617i
\(604\) 7575.70 0.0207658
\(605\) 104737.i 0.286147i
\(606\) 43116.5i 0.117408i
\(607\) 56775.6 0.154093 0.0770467 0.997027i \(-0.475451\pi\)
0.0770467 + 0.997027i \(0.475451\pi\)
\(608\) 557746.i 1.50879i
\(609\) 344944. 0.930067
\(610\) 113634.i 0.305385i
\(611\) 27168.6i 0.0727754i
\(612\) 13291.7 0.0354877
\(613\) 433945. 1.15482 0.577409 0.816455i \(-0.304064\pi\)
0.577409 + 0.816455i \(0.304064\pi\)
\(614\) 292059.i 0.774701i
\(615\) 44509.2i 0.117679i
\(616\) 192069.i 0.506170i
\(617\) 4914.68 0.0129100 0.00645499 0.999979i \(-0.497945\pi\)
0.00645499 + 0.999979i \(0.497945\pi\)
\(618\) 248202. 0.649873
\(619\) −539171. −1.40717 −0.703583 0.710613i \(-0.748418\pi\)
−0.703583 + 0.710613i \(0.748418\pi\)
\(620\) −122743. −0.319311
\(621\) 92769.0i 0.240558i
\(622\) −92466.1 −0.239002
\(623\) 307836.i 0.793129i
\(624\) −480.246 −0.00123337
\(625\) 270209. 0.691734
\(626\) 8652.63 0.0220800
\(627\) 122090. 0.310561
\(628\) 274153. 0.695141
\(629\) 42411.7 0.107197
\(630\) 38948.5i 0.0981319i
\(631\) 211466.i 0.531106i 0.964096 + 0.265553i \(0.0855545\pi\)
−0.964096 + 0.265553i \(0.914445\pi\)
\(632\) −93926.2 −0.235154
\(633\) 126355.i 0.315345i
\(634\) 363774.i 0.905011i
\(635\) 11916.7i 0.0295535i
\(636\) −264581. −0.654101
\(637\) 20004.3i 0.0492997i
\(638\) 102632. 0.252140
\(639\) 6440.69 0.0157736
\(640\) −74996.3 −0.183096
\(641\) 62879.9i 0.153037i −0.997068 0.0765184i \(-0.975620\pi\)
0.997068 0.0765184i \(-0.0243804\pi\)
\(642\) 17582.0 0.0426578
\(643\) 337262. 0.815729 0.407864 0.913043i \(-0.366274\pi\)
0.407864 + 0.913043i \(0.366274\pi\)
\(644\) 441022.i 1.06338i
\(645\) 105558.i 0.253729i
\(646\) 72148.0i 0.172886i
\(647\) 307016.i 0.733420i 0.930335 + 0.366710i \(0.119516\pi\)
−0.930335 + 0.366710i \(0.880484\pi\)
\(648\) 47270.8i 0.112575i
\(649\) 269402.i 0.639604i
\(650\) −11525.9 −0.0272803
\(651\) 569407.i 1.34357i
\(652\) −318414. −0.749026
\(653\) 574452.i 1.34718i 0.739103 + 0.673592i \(0.235250\pi\)
−0.739103 + 0.673592i \(0.764750\pi\)
\(654\) 212683.i 0.497254i
\(655\) 52902.4i 0.123308i
\(656\) 11920.7i 0.0277008i
\(657\) 90417.3 0.209469
\(658\) 590127. 1.36299
\(659\) 636830. 1.46640 0.733201 0.680012i \(-0.238025\pi\)
0.733201 + 0.680012i \(0.238025\pi\)
\(660\) 17239.8i 0.0395770i
\(661\) 761413.i 1.74268i −0.490680 0.871340i \(-0.663252\pi\)
0.490680 0.871340i \(-0.336748\pi\)
\(662\) −441568. −1.00759
\(663\) −2175.81 −0.00494987
\(664\) 87157.9i 0.197684i
\(665\) 314515. 0.711209
\(666\) 56445.6i 0.127257i
\(667\) −629730. −1.41548
\(668\) −183879. −0.412077
\(669\) 231856.i 0.518044i
\(670\) −64290.1 67058.6i −0.143217 0.149384i
\(671\) 233331. 0.518236
\(672\) 365351.i 0.809044i
\(673\) 694522.i 1.53340i 0.642004 + 0.766701i \(0.278103\pi\)
−0.642004 + 0.766701i \(0.721897\pi\)
\(674\) 255176. 0.561720
\(675\) 78343.2i 0.171947i
\(676\) 272645. 0.596630
\(677\) 335951.i 0.732990i 0.930420 + 0.366495i \(0.119442\pi\)
−0.930420 + 0.366495i \(0.880558\pi\)
\(678\) 283348.i 0.616397i
\(679\) −336407. −0.729668
\(680\) 27223.4 0.0588741
\(681\) 176638.i 0.380881i
\(682\) 169417.i 0.364241i
\(683\) 268779.i 0.576175i −0.957604 0.288087i \(-0.906981\pi\)
0.957604 0.288087i \(-0.0930193\pi\)
\(684\) 142848. 0.305325
\(685\) 261899. 0.558152
\(686\) 10061.8 0.0213810
\(687\) −188316. −0.399000
\(688\) 28271.0i 0.0597260i
\(689\) 43311.1 0.0912349
\(690\) 71104.4i 0.149348i
\(691\) −20490.6 −0.0429139 −0.0214569 0.999770i \(-0.506830\pi\)
−0.0214569 + 0.999770i \(0.506830\pi\)
\(692\) −350339. −0.731604
\(693\) −79975.3 −0.166529
\(694\) −379941. −0.788854
\(695\) 231491. 0.479253
\(696\) 320881. 0.662408
\(697\) 54007.9i 0.111171i
\(698\) 326695.i 0.670551i
\(699\) 172929. 0.353928
\(700\) 372442.i 0.760086i
\(701\) 343916.i 0.699869i 0.936774 + 0.349935i \(0.113796\pi\)
−0.936774 + 0.349935i \(0.886204\pi\)
\(702\) 2895.78i 0.00587613i
\(703\) 455805. 0.922293
\(704\) 116425.i 0.234910i
\(705\) −141543. −0.284780
\(706\) −117799. −0.236337
\(707\) −228070. −0.456277
\(708\) 315205.i 0.628820i
\(709\) 392130. 0.780076 0.390038 0.920799i \(-0.372462\pi\)
0.390038 + 0.920799i \(0.372462\pi\)
\(710\) 4936.58 0.00979287
\(711\) 39109.8i 0.0773653i
\(712\) 286362.i 0.564879i
\(713\) 1.03951e6i 2.04479i
\(714\) 47260.6i 0.0927049i
\(715\) 2822.10i 0.00552026i
\(716\) 473802.i 0.924211i
\(717\) 54563.1 0.106136
\(718\) 141803.i 0.275066i
\(719\) −122930. −0.237794 −0.118897 0.992907i \(-0.537936\pi\)
−0.118897 + 0.992907i \(0.537936\pi\)
\(720\) 2501.98i 0.00482635i
\(721\) 1.31289e6i 2.52557i
\(722\) 444882.i 0.853434i
\(723\) 130934.i 0.250481i
\(724\) 495670. 0.945618
\(725\) 531805. 1.01176
\(726\) 169142. 0.320907
\(727\) 81782.9i 0.154737i −0.997003 0.0773684i \(-0.975348\pi\)
0.997003 0.0773684i \(-0.0246518\pi\)
\(728\) 36786.7i 0.0694109i
\(729\) −19683.0 −0.0370370
\(730\) 69301.9 0.130047
\(731\) 128085.i 0.239697i
\(732\) 273001. 0.509498
\(733\) 358817.i 0.667828i 0.942603 + 0.333914i \(0.108370\pi\)
−0.942603 + 0.333914i \(0.891630\pi\)
\(734\) 114771. 0.213030
\(735\) −104218. −0.192916
\(736\) 666985.i 1.23129i
\(737\) −137695. + 132011.i −0.253504 + 0.243038i
\(738\) 71879.0 0.131974
\(739\) 286254.i 0.524158i −0.965046 0.262079i \(-0.915592\pi\)
0.965046 0.262079i \(-0.0844081\pi\)
\(740\) 64361.9i 0.117535i
\(741\) −23383.8 −0.0425871
\(742\) 940757.i 1.70872i
\(743\) −187651. −0.339917 −0.169958 0.985451i \(-0.554363\pi\)
−0.169958 + 0.985451i \(0.554363\pi\)
\(744\) 529686.i 0.956913i
\(745\) 79430.3i 0.143111i
\(746\) −80990.1 −0.145531
\(747\) −36291.5 −0.0650375
\(748\) 20918.9i 0.0373883i
\(749\) 93002.1i 0.165779i
\(750\) 127255.i 0.226232i
\(751\) −11651.4 −0.0206585 −0.0103292 0.999947i \(-0.503288\pi\)
−0.0103292 + 0.999947i \(0.503288\pi\)
\(752\) 37908.6 0.0670351
\(753\) 541274. 0.954612
\(754\) −19657.0 −0.0345759
\(755\) 6460.75i 0.0113342i
\(756\) −93572.6 −0.163721
\(757\) 507606.i 0.885799i 0.896571 + 0.442899i \(0.146050\pi\)
−0.896571 + 0.442899i \(0.853950\pi\)
\(758\) 132597. 0.230779
\(759\) 146003. 0.253442
\(760\) 292574. 0.506535
\(761\) 946519. 1.63441 0.817203 0.576350i \(-0.195523\pi\)
0.817203 + 0.576350i \(0.195523\pi\)
\(762\) −19244.6 −0.0331435
\(763\) −1.12501e6 −1.93245
\(764\) 286076.i 0.490112i
\(765\) 11335.5i 0.0193695i
\(766\) −355416. −0.605730
\(767\) 51598.1i 0.0877087i
\(768\) 348899.i 0.591530i
\(769\) 651018.i 1.10088i 0.834874 + 0.550441i \(0.185540\pi\)
−0.834874 + 0.550441i \(0.814460\pi\)
\(770\) −61298.5 −0.103388
\(771\) 246469.i 0.414623i
\(772\) −373541. −0.626763
\(773\) −1.09679e6 −1.83554 −0.917771 0.397109i \(-0.870013\pi\)
−0.917771 + 0.397109i \(0.870013\pi\)
\(774\) 170468. 0.284551
\(775\) 877863.i 1.46158i
\(776\) −312939. −0.519681
\(777\) −298575. −0.494552
\(778\) 255783.i 0.422584i
\(779\) 580432.i 0.956482i
\(780\) 3301.90i 0.00542719i
\(781\) 10136.6i 0.0166184i
\(782\) 86278.9i 0.141088i
\(783\) 133611.i 0.217931i
\(784\) 27912.2 0.0454110
\(785\) 233804.i 0.379414i
\(786\) 85433.4 0.138287
\(787\) 283075.i 0.457037i 0.973540 + 0.228519i \(0.0733881\pi\)
−0.973540 + 0.228519i \(0.926612\pi\)
\(788\) 170322.i 0.274296i
\(789\) 11644.9i 0.0187061i
\(790\) 29976.4i 0.0480314i
\(791\) 1.49880e6 2.39547
\(792\) −74396.3 −0.118604
\(793\) −44689.5 −0.0710656
\(794\) 657876.i 1.04353i
\(795\) 225642.i 0.357014i
\(796\) 106409. 0.167939
\(797\) −731479. −1.15156 −0.575778 0.817606i \(-0.695301\pi\)
−0.575778 + 0.817606i \(0.695301\pi\)
\(798\) 507917.i 0.797604i
\(799\) 171749. 0.269030
\(800\) 563268.i 0.880106i
\(801\) −119238. −0.185844
\(802\) −159251. −0.247590
\(803\) 142302.i 0.220688i
\(804\) −161106. + 154455.i −0.249230 + 0.238940i
\(805\) 376115. 0.580402
\(806\) 32448.2i 0.0499483i
\(807\) 679218.i 1.04295i
\(808\) −212160. −0.324968
\(809\) 460289.i 0.703289i 0.936134 + 0.351645i \(0.114377\pi\)
−0.936134 + 0.351645i \(0.885623\pi\)
\(810\) −15086.4 −0.0229940
\(811\) 85621.2i 0.130179i −0.997879 0.0650893i \(-0.979267\pi\)
0.997879 0.0650893i \(-0.0207332\pi\)
\(812\) 635185.i 0.963359i
\(813\) 120529. 0.182352
\(814\) −88835.9 −0.134073
\(815\) 271552.i 0.408825i
\(816\) 3035.93i 0.00455943i
\(817\) 1.37655e6i 2.06228i
\(818\) −186667. −0.278972
\(819\) 15317.5 0.0228361
\(820\) −81959.9 −0.121892
\(821\) −1.12464e6 −1.66850 −0.834249 0.551388i \(-0.814098\pi\)
−0.834249 + 0.551388i \(0.814098\pi\)
\(822\) 422947.i 0.625954i
\(823\) −88269.9 −0.130321 −0.0651603 0.997875i \(-0.520756\pi\)
−0.0651603 + 0.997875i \(0.520756\pi\)
\(824\) 1.22131e6i 1.79875i
\(825\) −123299. −0.181156
\(826\) −1.12076e6 −1.64267
\(827\) −762169. −1.11440 −0.557199 0.830379i \(-0.688124\pi\)
−0.557199 + 0.830379i \(0.688124\pi\)
\(828\) 170826. 0.249169
\(829\) −832615. −1.21153 −0.605767 0.795642i \(-0.707133\pi\)
−0.605767 + 0.795642i \(0.707133\pi\)
\(830\) −27816.3 −0.0403778
\(831\) 635769.i 0.920656i
\(832\) 22298.7i 0.0322131i
\(833\) 126459. 0.182247
\(834\) 373841.i 0.537471i
\(835\) 156816.i 0.224915i
\(836\) 224819.i 0.321677i
\(837\) −220555. −0.314822
\(838\) 462867.i 0.659125i
\(839\) −847059. −1.20334 −0.601672 0.798743i \(-0.705499\pi\)
−0.601672 + 0.798743i \(0.705499\pi\)
\(840\) −191651. −0.271614
\(841\) 199691. 0.282337
\(842\) 64259.4i 0.0906384i
\(843\) −538437. −0.757670
\(844\) 232672. 0.326633
\(845\) 232519.i 0.325646i
\(846\) 228581.i 0.319373i
\(847\) 894697.i 1.24712i
\(848\) 60432.4i 0.0840385i
\(849\) 361813.i 0.501960i
\(850\) 72862.3i 0.100848i
\(851\) 545079. 0.752662
\(852\) 11860.0i 0.0163382i
\(853\) −556392. −0.764686 −0.382343 0.924020i \(-0.624883\pi\)
−0.382343 + 0.924020i \(0.624883\pi\)
\(854\) 970697.i 1.33097i
\(855\) 121825.i 0.166649i
\(856\) 86514.4i 0.118070i
\(857\) 892287.i 1.21491i −0.794355 0.607454i \(-0.792191\pi\)
0.794355 0.607454i \(-0.207809\pi\)
\(858\) 4557.47 0.00619084
\(859\) 970247. 1.31491 0.657455 0.753494i \(-0.271633\pi\)
0.657455 + 0.753494i \(0.271633\pi\)
\(860\) −194375. −0.262811
\(861\) 380212.i 0.512885i
\(862\) 371316.i 0.499723i
\(863\) 115316. 0.154834 0.0774170 0.996999i \(-0.475333\pi\)
0.0774170 + 0.996999i \(0.475333\pi\)
\(864\) −141516. −0.189573
\(865\) 298778.i 0.399316i
\(866\) 779888. 1.03991
\(867\) 420233.i 0.559052i
\(868\) −1.04851e6 −1.39166
\(869\) 61552.2 0.0815088
\(870\) 102409.i 0.135300i
\(871\) 26372.6 25283.8i 0.0347629 0.0333277i
\(872\) −1.04653e6 −1.37632
\(873\) 130304.i 0.170974i
\(874\) 927253.i 1.21388i
\(875\) −673131. −0.879192
\(876\) 166496.i 0.216967i
\(877\) 87113.3 0.113262 0.0566311 0.998395i \(-0.481964\pi\)
0.0566311 + 0.998395i \(0.481964\pi\)
\(878\) 225336.i 0.292309i
\(879\) 683083.i 0.884088i
\(880\) −3937.70 −0.00508484
\(881\) 10742.7 0.0138408 0.00692038 0.999976i \(-0.497797\pi\)
0.00692038 + 0.999976i \(0.497797\pi\)
\(882\) 168304.i 0.216350i
\(883\) 945802.i 1.21305i −0.795064 0.606525i \(-0.792563\pi\)
0.795064 0.606525i \(-0.207437\pi\)
\(884\) 4006.57i 0.00512705i
\(885\) 268815. 0.343215
\(886\) −368603. −0.469560
\(887\) −48296.9 −0.0613864 −0.0306932 0.999529i \(-0.509771\pi\)
−0.0306932 + 0.999529i \(0.509771\pi\)
\(888\) −277747. −0.352228
\(889\) 101796.i 0.128804i
\(890\) −91392.0 −0.115379
\(891\) 30977.8i 0.0390207i
\(892\) 426943. 0.536587
\(893\) 1.84582e6 2.31465
\(894\) −128274. −0.160496
\(895\) 404071. 0.504443
\(896\) −640643. −0.797995
\(897\) −27963.7 −0.0347544
\(898\) 548471.i 0.680144i
\(899\) 1.49716e6i 1.85246i
\(900\) −144262. −0.178102
\(901\) 273796.i 0.337270i
\(902\) 113126.i 0.139043i
\(903\) 901708.i 1.10584i
\(904\) 1.39425e6 1.70609
\(905\) 422721.i 0.516127i
\(906\) 10433.6 0.0127110
\(907\) −1.18879e6 −1.44508 −0.722540 0.691329i \(-0.757025\pi\)
−0.722540 + 0.691329i \(0.757025\pi\)
\(908\) 325263. 0.394515
\(909\) 88340.9i 0.106914i
\(910\) 11740.4 0.0141775
\(911\) −135791. −0.163619 −0.0818095 0.996648i \(-0.526070\pi\)
−0.0818095 + 0.996648i \(0.526070\pi\)
\(912\) 32627.6i 0.0392279i
\(913\) 57116.8i 0.0685208i
\(914\) 1.04985e6i 1.25670i
\(915\) 232823.i 0.278089i
\(916\) 346767.i 0.413283i
\(917\) 451909.i 0.537419i
\(918\) 18306.0 0.0217224
\(919\) 440505.i 0.521578i 0.965396 + 0.260789i \(0.0839827\pi\)
−0.965396 + 0.260789i \(0.916017\pi\)
\(920\) 349878. 0.413371
\(921\) 598396.i 0.705455i
\(922\) 291705.i 0.343149i
\(923\) 1941.44i 0.00227888i
\(924\) 147268.i 0.172490i
\(925\) −460318. −0.537990
\(926\) −438036. −0.510843
\(927\) −508538. −0.591785
\(928\) 960630.i 1.11548i
\(929\) 1.16404e6i 1.34876i 0.738384 + 0.674381i \(0.235589\pi\)
−0.738384 + 0.674381i \(0.764411\pi\)
\(930\) −169048. −0.195454
\(931\) 1.35908e6 1.56800
\(932\) 318435.i 0.366597i
\(933\) 189453. 0.217639
\(934\) 319719.i 0.366501i
\(935\) −17840.2 −0.0204069
\(936\) 14249.0 0.0162642
\(937\) 391873.i 0.446341i −0.974779 0.223170i \(-0.928359\pi\)
0.974779 0.223170i \(-0.0716407\pi\)
\(938\) −549187. 572837.i −0.624186 0.651066i
\(939\) −17728.2 −0.0201064
\(940\) 260638.i 0.294973i
\(941\) 1.56608e6i 1.76863i −0.466895 0.884313i \(-0.654627\pi\)
0.466895 0.884313i \(-0.345373\pi\)
\(942\) 377577. 0.425504
\(943\) 694115.i 0.780563i
\(944\) −71995.3 −0.0807904
\(945\) 79801.1i 0.0893605i
\(946\) 268288.i 0.299791i
\(947\) 573431. 0.639412 0.319706 0.947517i \(-0.396416\pi\)
0.319706 + 0.947517i \(0.396416\pi\)
\(948\) 72017.3 0.0801346
\(949\) 27254.8i 0.0302629i
\(950\) 783064.i 0.867661i
\(951\) 745332.i 0.824117i
\(952\) 232551. 0.256593
\(953\) 1.50854e6 1.66100 0.830502 0.557016i \(-0.188054\pi\)
0.830502 + 0.557016i \(0.188054\pi\)
\(954\) −364394. −0.400382
\(955\) −243973. −0.267507
\(956\) 100473.i 0.109935i
\(957\) −210282. −0.229603
\(958\) 349372.i 0.380677i
\(959\) 2.23723e6 2.43261
\(960\) −116171. −0.126054
\(961\) −1.54787e6 −1.67606
\(962\) 17014.6 0.0183853
\(963\) −36023.6 −0.0388449
\(964\) 241103. 0.259447
\(965\) 318565.i 0.342093i
\(966\) 607397.i 0.650906i
\(967\) −318251. −0.340343 −0.170171 0.985414i \(-0.554432\pi\)
−0.170171 + 0.985414i \(0.554432\pi\)
\(968\) 832284.i 0.888220i
\(969\) 147823.i 0.157433i
\(970\) 99874.1i 0.106147i
\(971\) 562745. 0.596861 0.298431 0.954431i \(-0.403537\pi\)
0.298431 + 0.954431i \(0.403537\pi\)
\(972\) 36244.5i 0.0383628i
\(973\) 1.97747e6 2.08874
\(974\) −339906. −0.358295
\(975\) 23615.3 0.0248419
\(976\) 62355.7i 0.0654601i
\(977\) 585418. 0.613306 0.306653 0.951821i \(-0.400791\pi\)
0.306653 + 0.951821i \(0.400791\pi\)
\(978\) −438535. −0.458487
\(979\) 187660.i 0.195798i
\(980\) 191908.i 0.199821i
\(981\) 435764.i 0.452807i
\(982\) 151592.i 0.157201i
\(983\) 1.91764e6i 1.98454i 0.124093 + 0.992271i \(0.460398\pi\)
−0.124093 + 0.992271i \(0.539602\pi\)
\(984\) 353689.i 0.365285i
\(985\) −145255. −0.149713
\(986\) 124264.i 0.127818i
\(987\) −1.20910e6 −1.24116
\(988\) 43059.2i 0.0441115i
\(989\) 1.64616e6i 1.68298i
\(990\) 23743.5i 0.0242255i
\(991\) 512865.i 0.522223i 0.965309 + 0.261111i \(0.0840890\pi\)
−0.965309 + 0.261111i \(0.915911\pi\)
\(992\) −1.58573e6 −1.61141
\(993\) 904723. 0.917524
\(994\) 42169.9 0.0426805
\(995\) 90748.2i 0.0916625i
\(996\) 66827.7i 0.0673656i
\(997\) 169477. 0.170499 0.0852493 0.996360i \(-0.472831\pi\)
0.0852493 + 0.996360i \(0.472831\pi\)
\(998\) 969955. 0.973847
\(999\) 115651.i 0.115882i
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 201.5.b.a.133.17 46
67.66 odd 2 inner 201.5.b.a.133.30 yes 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.5.b.a.133.17 46 1.1 even 1 trivial
201.5.b.a.133.30 yes 46 67.66 odd 2 inner