Properties

Label 483.3.f.a.22.36
Level $483$
Weight $3$
Character 483.22
Analytic conductor $13.161$
Analytic rank $0$
Dimension $48$
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] = [483,3,Mod(22,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.22");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 483.f (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: \(13.1607967686\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 22.36
Character \(\chi\) \(=\) 483.22
Dual form 483.3.f.a.22.35

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.39960 q^{2} -1.73205 q^{3} +1.75809 q^{4} +1.89961i q^{5} -4.15623 q^{6} -2.64575i q^{7} -5.37969 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+2.39960 q^{2} -1.73205 q^{3} +1.75809 q^{4} +1.89961i q^{5} -4.15623 q^{6} -2.64575i q^{7} -5.37969 q^{8} +3.00000 q^{9} +4.55832i q^{10} -11.7870i q^{11} -3.04511 q^{12} -9.50541 q^{13} -6.34875i q^{14} -3.29023i q^{15} -19.9415 q^{16} -23.2236i q^{17} +7.19881 q^{18} -20.1037i q^{19} +3.33970i q^{20} +4.58258i q^{21} -28.2841i q^{22} +(21.2213 + 8.86874i) q^{23} +9.31789 q^{24} +21.3915 q^{25} -22.8092 q^{26} -5.19615 q^{27} -4.65148i q^{28} -22.9320 q^{29} -7.89524i q^{30} -27.0399 q^{31} -26.3329 q^{32} +20.4157i q^{33} -55.7275i q^{34} +5.02591 q^{35} +5.27428 q^{36} -36.2524i q^{37} -48.2409i q^{38} +16.4639 q^{39} -10.2193i q^{40} -50.4211 q^{41} +10.9964i q^{42} -12.8026i q^{43} -20.7226i q^{44} +5.69884i q^{45} +(50.9228 + 21.2815i) q^{46} -35.2976 q^{47} +34.5397 q^{48} -7.00000 q^{49} +51.3310 q^{50} +40.2245i q^{51} -16.7114 q^{52} -43.6619i q^{53} -12.4687 q^{54} +22.3908 q^{55} +14.2333i q^{56} +34.8206i q^{57} -55.0276 q^{58} +21.5090 q^{59} -5.78452i q^{60} +88.3762i q^{61} -64.8851 q^{62} -7.93725i q^{63} +16.5775 q^{64} -18.0566i q^{65} +48.9896i q^{66} -19.2677i q^{67} -40.8293i q^{68} +(-36.7564 - 15.3611i) q^{69} +12.0602 q^{70} +116.730 q^{71} -16.1391 q^{72} -24.8834 q^{73} -86.9914i q^{74} -37.0511 q^{75} -35.3441i q^{76} -31.1855 q^{77} +39.5067 q^{78} +26.1733i q^{79} -37.8811i q^{80} +9.00000 q^{81} -120.991 q^{82} +64.1905i q^{83} +8.05659i q^{84} +44.1159 q^{85} -30.7211i q^{86} +39.7193 q^{87} +63.4104i q^{88} +99.9201i q^{89} +13.6750i q^{90} +25.1490i q^{91} +(37.3091 + 15.5921i) q^{92} +46.8345 q^{93} -84.7003 q^{94} +38.1892 q^{95} +45.6099 q^{96} +49.1664i q^{97} -16.7972 q^{98} -35.3610i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{2} + 116 q^{4} - 24 q^{6} - 4 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 4 q^{2} + 116 q^{4} - 24 q^{6} - 4 q^{8} + 144 q^{9} + 16 q^{13} + 324 q^{16} + 12 q^{18} - 4 q^{23} - 24 q^{24} - 176 q^{25} + 136 q^{26} - 128 q^{29} - 8 q^{31} - 252 q^{32} - 56 q^{35} + 348 q^{36} + 96 q^{39} - 24 q^{41} - 148 q^{46} - 408 q^{47} - 96 q^{48} - 336 q^{49} + 236 q^{50} - 32 q^{52} - 72 q^{54} - 24 q^{55} - 56 q^{58} + 136 q^{59} + 184 q^{62} + 716 q^{64} - 48 q^{69} - 112 q^{70} + 48 q^{71} - 12 q^{72} - 224 q^{73} - 48 q^{75} + 224 q^{77} - 96 q^{78} + 432 q^{81} + 640 q^{82} - 424 q^{85} + 312 q^{87} + 1060 q^{92} + 192 q^{93} + 216 q^{94} + 624 q^{95} + 48 q^{96} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.39960 1.19980 0.599901 0.800075i \(-0.295207\pi\)
0.599901 + 0.800075i \(0.295207\pi\)
\(3\) −1.73205 −0.577350
\(4\) 1.75809 0.439523
\(5\) 1.89961i 0.379923i 0.981792 + 0.189961i \(0.0608363\pi\)
−0.981792 + 0.189961i \(0.939164\pi\)
\(6\) −4.15623 −0.692706
\(7\) 2.64575i 0.377964i
\(8\) −5.37969 −0.672461
\(9\) 3.00000 0.333333
\(10\) 4.55832i 0.455832i
\(11\) 11.7870i 1.07155i −0.844362 0.535773i \(-0.820020\pi\)
0.844362 0.535773i \(-0.179980\pi\)
\(12\) −3.04511 −0.253759
\(13\) −9.50541 −0.731185 −0.365593 0.930775i \(-0.619134\pi\)
−0.365593 + 0.930775i \(0.619134\pi\)
\(14\) 6.34875i 0.453482i
\(15\) 3.29023i 0.219349i
\(16\) −19.9415 −1.24634
\(17\) 23.2236i 1.36610i −0.730374 0.683048i \(-0.760654\pi\)
0.730374 0.683048i \(-0.239346\pi\)
\(18\) 7.19881 0.399934
\(19\) 20.1037i 1.05809i −0.848594 0.529044i \(-0.822550\pi\)
0.848594 0.529044i \(-0.177450\pi\)
\(20\) 3.33970i 0.166985i
\(21\) 4.58258i 0.218218i
\(22\) 28.2841i 1.28564i
\(23\) 21.2213 + 8.86874i 0.922667 + 0.385597i
\(24\) 9.31789 0.388245
\(25\) 21.3915 0.855659
\(26\) −22.8092 −0.877277
\(27\) −5.19615 −0.192450
\(28\) 4.65148i 0.166124i
\(29\) −22.9320 −0.790757 −0.395379 0.918518i \(-0.629387\pi\)
−0.395379 + 0.918518i \(0.629387\pi\)
\(30\) 7.89524i 0.263175i
\(31\) −27.0399 −0.872256 −0.436128 0.899885i \(-0.643651\pi\)
−0.436128 + 0.899885i \(0.643651\pi\)
\(32\) −26.3329 −0.822902
\(33\) 20.4157i 0.618657i
\(34\) 55.7275i 1.63904i
\(35\) 5.02591 0.143597
\(36\) 5.27428 0.146508
\(37\) 36.2524i 0.979795i −0.871780 0.489897i \(-0.837034\pi\)
0.871780 0.489897i \(-0.162966\pi\)
\(38\) 48.2409i 1.26950i
\(39\) 16.4639 0.422150
\(40\) 10.2193i 0.255483i
\(41\) −50.4211 −1.22978 −0.614891 0.788612i \(-0.710800\pi\)
−0.614891 + 0.788612i \(0.710800\pi\)
\(42\) 10.9964i 0.261818i
\(43\) 12.8026i 0.297734i −0.988857 0.148867i \(-0.952437\pi\)
0.988857 0.148867i \(-0.0475626\pi\)
\(44\) 20.7226i 0.470969i
\(45\) 5.69884i 0.126641i
\(46\) 50.9228 + 21.2815i 1.10702 + 0.462640i
\(47\) −35.2976 −0.751013 −0.375507 0.926820i \(-0.622531\pi\)
−0.375507 + 0.926820i \(0.622531\pi\)
\(48\) 34.5397 0.719576
\(49\) −7.00000 −0.142857
\(50\) 51.3310 1.02662
\(51\) 40.2245i 0.788716i
\(52\) −16.7114 −0.321373
\(53\) 43.6619i 0.823810i −0.911227 0.411905i \(-0.864863\pi\)
0.911227 0.411905i \(-0.135137\pi\)
\(54\) −12.4687 −0.230902
\(55\) 22.3908 0.407105
\(56\) 14.2333i 0.254166i
\(57\) 34.8206i 0.610888i
\(58\) −55.0276 −0.948752
\(59\) 21.5090 0.364559 0.182279 0.983247i \(-0.441652\pi\)
0.182279 + 0.983247i \(0.441652\pi\)
\(60\) 5.78452i 0.0964087i
\(61\) 88.3762i 1.44879i 0.689385 + 0.724395i \(0.257881\pi\)
−0.689385 + 0.724395i \(0.742119\pi\)
\(62\) −64.8851 −1.04653
\(63\) 7.93725i 0.125988i
\(64\) 16.5775 0.259023
\(65\) 18.0566i 0.277794i
\(66\) 48.9896i 0.742266i
\(67\) 19.2677i 0.287577i −0.989608 0.143788i \(-0.954072\pi\)
0.989608 0.143788i \(-0.0459285\pi\)
\(68\) 40.8293i 0.600431i
\(69\) −36.7564 15.3611i −0.532702 0.222625i
\(70\) 12.0602 0.172288
\(71\) 116.730 1.64409 0.822044 0.569424i \(-0.192834\pi\)
0.822044 + 0.569424i \(0.192834\pi\)
\(72\) −16.1391 −0.224154
\(73\) −24.8834 −0.340868 −0.170434 0.985369i \(-0.554517\pi\)
−0.170434 + 0.985369i \(0.554517\pi\)
\(74\) 86.9914i 1.17556i
\(75\) −37.0511 −0.494015
\(76\) 35.3441i 0.465054i
\(77\) −31.1855 −0.405006
\(78\) 39.5067 0.506496
\(79\) 26.1733i 0.331308i 0.986184 + 0.165654i \(0.0529735\pi\)
−0.986184 + 0.165654i \(0.947027\pi\)
\(80\) 37.8811i 0.473514i
\(81\) 9.00000 0.111111
\(82\) −120.991 −1.47549
\(83\) 64.1905i 0.773380i 0.922210 + 0.386690i \(0.126382\pi\)
−0.922210 + 0.386690i \(0.873618\pi\)
\(84\) 8.05659i 0.0959118i
\(85\) 44.1159 0.519011
\(86\) 30.7211i 0.357222i
\(87\) 39.7193 0.456544
\(88\) 63.4104i 0.720573i
\(89\) 99.9201i 1.12270i 0.827579 + 0.561349i \(0.189717\pi\)
−0.827579 + 0.561349i \(0.810283\pi\)
\(90\) 13.6750i 0.151944i
\(91\) 25.1490i 0.276362i
\(92\) 37.3091 + 15.5921i 0.405534 + 0.169479i
\(93\) 46.8345 0.503597
\(94\) −84.7003 −0.901067
\(95\) 38.1892 0.401992
\(96\) 45.6099 0.475103
\(97\) 49.1664i 0.506870i 0.967352 + 0.253435i \(0.0815605\pi\)
−0.967352 + 0.253435i \(0.918440\pi\)
\(98\) −16.7972 −0.171400
\(99\) 35.3610i 0.357182i
\(100\) 37.6082 0.376082
\(101\) −6.46798 −0.0640394 −0.0320197 0.999487i \(-0.510194\pi\)
−0.0320197 + 0.999487i \(0.510194\pi\)
\(102\) 96.5228i 0.946302i
\(103\) 67.3537i 0.653919i −0.945038 0.326960i \(-0.893976\pi\)
0.945038 0.326960i \(-0.106024\pi\)
\(104\) 51.1361 0.491694
\(105\) −8.70512 −0.0829060
\(106\) 104.771i 0.988409i
\(107\) 97.3074i 0.909415i 0.890641 + 0.454708i \(0.150256\pi\)
−0.890641 + 0.454708i \(0.849744\pi\)
\(108\) −9.13532 −0.0845863
\(109\) 149.211i 1.36891i −0.729056 0.684454i \(-0.760041\pi\)
0.729056 0.684454i \(-0.239959\pi\)
\(110\) 53.7289 0.488445
\(111\) 62.7910i 0.565685i
\(112\) 52.7602i 0.471073i
\(113\) 16.3063i 0.144304i −0.997394 0.0721519i \(-0.977013\pi\)
0.997394 0.0721519i \(-0.0229866\pi\)
\(114\) 83.5556i 0.732944i
\(115\) −16.8472 + 40.3124i −0.146497 + 0.350542i
\(116\) −40.3165 −0.347556
\(117\) −28.5162 −0.243728
\(118\) 51.6130 0.437398
\(119\) −61.4439 −0.516336
\(120\) 17.7004i 0.147503i
\(121\) −17.9335 −0.148211
\(122\) 212.068i 1.73826i
\(123\) 87.3319 0.710015
\(124\) −47.5387 −0.383377
\(125\) 88.1259i 0.705007i
\(126\) 19.0463i 0.151161i
\(127\) 148.405 1.16854 0.584270 0.811559i \(-0.301381\pi\)
0.584270 + 0.811559i \(0.301381\pi\)
\(128\) 145.111 1.13368
\(129\) 22.1747i 0.171897i
\(130\) 43.3287i 0.333298i
\(131\) −36.7507 −0.280540 −0.140270 0.990113i \(-0.544797\pi\)
−0.140270 + 0.990113i \(0.544797\pi\)
\(132\) 35.8927i 0.271914i
\(133\) −53.1894 −0.399920
\(134\) 46.2347i 0.345035i
\(135\) 9.87068i 0.0731162i
\(136\) 124.936i 0.918646i
\(137\) 78.8141i 0.575286i 0.957738 + 0.287643i \(0.0928716\pi\)
−0.957738 + 0.287643i \(0.907128\pi\)
\(138\) −88.2009 36.8606i −0.639137 0.267106i
\(139\) 72.4952 0.521548 0.260774 0.965400i \(-0.416022\pi\)
0.260774 + 0.965400i \(0.416022\pi\)
\(140\) 8.83601 0.0631143
\(141\) 61.1373 0.433598
\(142\) 280.106 1.97258
\(143\) 112.040i 0.783499i
\(144\) −59.8244 −0.415448
\(145\) 43.5619i 0.300427i
\(146\) −59.7102 −0.408974
\(147\) 12.1244 0.0824786
\(148\) 63.7351i 0.430642i
\(149\) 208.161i 1.39706i −0.715583 0.698528i \(-0.753839\pi\)
0.715583 0.698528i \(-0.246161\pi\)
\(150\) −88.9079 −0.592720
\(151\) −5.35118 −0.0354383 −0.0177191 0.999843i \(-0.505640\pi\)
−0.0177191 + 0.999843i \(0.505640\pi\)
\(152\) 108.152i 0.711523i
\(153\) 69.6709i 0.455365i
\(154\) −74.8328 −0.485927
\(155\) 51.3654i 0.331390i
\(156\) 28.9450 0.185545
\(157\) 160.794i 1.02416i −0.858937 0.512082i \(-0.828874\pi\)
0.858937 0.512082i \(-0.171126\pi\)
\(158\) 62.8056i 0.397504i
\(159\) 75.6247i 0.475627i
\(160\) 50.0223i 0.312639i
\(161\) 23.4645 56.1464i 0.145742 0.348735i
\(162\) 21.5964 0.133311
\(163\) −219.889 −1.34901 −0.674506 0.738270i \(-0.735643\pi\)
−0.674506 + 0.738270i \(0.735643\pi\)
\(164\) −88.6449 −0.540518
\(165\) −38.7819 −0.235042
\(166\) 154.032i 0.927902i
\(167\) 165.497 0.990997 0.495499 0.868609i \(-0.334985\pi\)
0.495499 + 0.868609i \(0.334985\pi\)
\(168\) 24.6528i 0.146743i
\(169\) −78.6472 −0.465368
\(170\) 105.861 0.622710
\(171\) 60.3111i 0.352696i
\(172\) 22.5081i 0.130861i
\(173\) −192.854 −1.11476 −0.557382 0.830256i \(-0.688194\pi\)
−0.557382 + 0.830256i \(0.688194\pi\)
\(174\) 95.3106 0.547762
\(175\) 56.5965i 0.323409i
\(176\) 235.050i 1.33551i
\(177\) −37.2546 −0.210478
\(178\) 239.769i 1.34701i
\(179\) −59.6812 −0.333415 −0.166707 0.986006i \(-0.553314\pi\)
−0.166707 + 0.986006i \(0.553314\pi\)
\(180\) 10.0191i 0.0556616i
\(181\) 357.591i 1.97564i −0.155605 0.987819i \(-0.549733\pi\)
0.155605 0.987819i \(-0.450267\pi\)
\(182\) 60.3475i 0.331580i
\(183\) 153.072i 0.836459i
\(184\) −114.164 47.7111i −0.620458 0.259299i
\(185\) 68.8656 0.372246
\(186\) 112.384 0.604217
\(187\) −273.737 −1.46383
\(188\) −62.0565 −0.330088
\(189\) 13.7477i 0.0727393i
\(190\) 91.6390 0.482311
\(191\) 83.7639i 0.438555i 0.975663 + 0.219277i \(0.0703700\pi\)
−0.975663 + 0.219277i \(0.929630\pi\)
\(192\) −28.7130 −0.149547
\(193\) −203.616 −1.05500 −0.527501 0.849554i \(-0.676871\pi\)
−0.527501 + 0.849554i \(0.676871\pi\)
\(194\) 117.980i 0.608144i
\(195\) 31.2750i 0.160384i
\(196\) −12.3066 −0.0627890
\(197\) 308.447 1.56572 0.782861 0.622196i \(-0.213759\pi\)
0.782861 + 0.622196i \(0.213759\pi\)
\(198\) 84.8524i 0.428547i
\(199\) 336.923i 1.69308i −0.532326 0.846539i \(-0.678682\pi\)
0.532326 0.846539i \(-0.321318\pi\)
\(200\) −115.079 −0.575397
\(201\) 33.3726i 0.166033i
\(202\) −15.5206 −0.0768346
\(203\) 60.6723i 0.298878i
\(204\) 70.7184i 0.346659i
\(205\) 95.7806i 0.467222i
\(206\) 161.622i 0.784573i
\(207\) 63.6640 + 26.6062i 0.307556 + 0.128532i
\(208\) 189.552 0.911307
\(209\) −236.962 −1.13379
\(210\) −20.8888 −0.0994707
\(211\) 214.467 1.01643 0.508215 0.861230i \(-0.330305\pi\)
0.508215 + 0.861230i \(0.330305\pi\)
\(212\) 76.7617i 0.362084i
\(213\) −202.183 −0.949214
\(214\) 233.499i 1.09112i
\(215\) 24.3199 0.113116
\(216\) 27.9537 0.129415
\(217\) 71.5410i 0.329682i
\(218\) 358.047i 1.64242i
\(219\) 43.0992 0.196800
\(220\) 39.3650 0.178932
\(221\) 220.750i 0.998869i
\(222\) 150.673i 0.678709i
\(223\) 22.7824 0.102163 0.0510816 0.998694i \(-0.483733\pi\)
0.0510816 + 0.998694i \(0.483733\pi\)
\(224\) 69.6702i 0.311028i
\(225\) 64.1744 0.285220
\(226\) 39.1287i 0.173136i
\(227\) 57.0590i 0.251361i 0.992071 + 0.125681i \(0.0401114\pi\)
−0.992071 + 0.125681i \(0.959889\pi\)
\(228\) 61.2178i 0.268499i
\(229\) 350.946i 1.53252i −0.642533 0.766258i \(-0.722117\pi\)
0.642533 0.766258i \(-0.277883\pi\)
\(230\) −40.4265 + 96.7336i −0.175768 + 0.420581i
\(231\) 54.0149 0.233831
\(232\) 123.367 0.531753
\(233\) 197.772 0.848805 0.424402 0.905474i \(-0.360484\pi\)
0.424402 + 0.905474i \(0.360484\pi\)
\(234\) −68.4276 −0.292426
\(235\) 67.0519i 0.285327i
\(236\) 37.8147 0.160232
\(237\) 45.3335i 0.191281i
\(238\) −147.441 −0.619500
\(239\) 16.7669 0.0701542 0.0350771 0.999385i \(-0.488832\pi\)
0.0350771 + 0.999385i \(0.488832\pi\)
\(240\) 65.6120i 0.273383i
\(241\) 300.467i 1.24675i −0.781922 0.623376i \(-0.785760\pi\)
0.781922 0.623376i \(-0.214240\pi\)
\(242\) −43.0334 −0.177824
\(243\) −15.5885 −0.0641500
\(244\) 155.373i 0.636777i
\(245\) 13.2973i 0.0542747i
\(246\) 209.562 0.851877
\(247\) 191.094i 0.773659i
\(248\) 145.466 0.586558
\(249\) 111.181i 0.446511i
\(250\) 211.467i 0.845868i
\(251\) 175.805i 0.700418i 0.936672 + 0.350209i \(0.113889\pi\)
−0.936672 + 0.350209i \(0.886111\pi\)
\(252\) 13.9544i 0.0553747i
\(253\) 104.536 250.136i 0.413185 0.988680i
\(254\) 356.112 1.40202
\(255\) −76.4110 −0.299651
\(256\) 281.899 1.10117
\(257\) −258.428 −1.00556 −0.502779 0.864415i \(-0.667689\pi\)
−0.502779 + 0.864415i \(0.667689\pi\)
\(258\) 53.2105i 0.206242i
\(259\) −95.9149 −0.370328
\(260\) 31.7452i 0.122097i
\(261\) −68.7959 −0.263586
\(262\) −88.1871 −0.336592
\(263\) 513.701i 1.95324i 0.214981 + 0.976618i \(0.431031\pi\)
−0.214981 + 0.976618i \(0.568969\pi\)
\(264\) 109.830i 0.416023i
\(265\) 82.9408 0.312984
\(266\) −127.633 −0.479825
\(267\) 173.067i 0.648190i
\(268\) 33.8743i 0.126397i
\(269\) 328.407 1.22085 0.610423 0.792076i \(-0.291000\pi\)
0.610423 + 0.792076i \(0.291000\pi\)
\(270\) 23.6857i 0.0877249i
\(271\) −524.797 −1.93652 −0.968260 0.249946i \(-0.919587\pi\)
−0.968260 + 0.249946i \(0.919587\pi\)
\(272\) 463.114i 1.70262i
\(273\) 43.5593i 0.159558i
\(274\) 189.123i 0.690228i
\(275\) 252.141i 0.916878i
\(276\) −64.6212 27.0063i −0.234135 0.0978487i
\(277\) −258.121 −0.931846 −0.465923 0.884825i \(-0.654278\pi\)
−0.465923 + 0.884825i \(0.654278\pi\)
\(278\) 173.960 0.625754
\(279\) −81.1198 −0.290752
\(280\) −27.0378 −0.0965636
\(281\) 362.403i 1.28969i 0.764313 + 0.644845i \(0.223078\pi\)
−0.764313 + 0.644845i \(0.776922\pi\)
\(282\) 146.705 0.520231
\(283\) 33.3103i 0.117704i 0.998267 + 0.0588521i \(0.0187440\pi\)
−0.998267 + 0.0588521i \(0.981256\pi\)
\(284\) 205.222 0.722614
\(285\) −66.1457 −0.232090
\(286\) 268.852i 0.940043i
\(287\) 133.402i 0.464814i
\(288\) −78.9986 −0.274301
\(289\) −250.337 −0.866218
\(290\) 104.531i 0.360452i
\(291\) 85.1588i 0.292642i
\(292\) −43.7472 −0.149819
\(293\) 413.973i 1.41288i −0.707774 0.706439i \(-0.750300\pi\)
0.707774 0.706439i \(-0.249700\pi\)
\(294\) 29.0936 0.0989579
\(295\) 40.8587i 0.138504i
\(296\) 195.027i 0.658874i
\(297\) 61.2471i 0.206219i
\(298\) 499.504i 1.67619i
\(299\) −201.718 84.3010i −0.674641 0.281943i
\(300\) −65.1393 −0.217131
\(301\) −33.8724 −0.112533
\(302\) −12.8407 −0.0425189
\(303\) 11.2029 0.0369732
\(304\) 400.897i 1.31874i
\(305\) −167.881 −0.550428
\(306\) 167.182i 0.546348i
\(307\) 322.114 1.04923 0.524616 0.851339i \(-0.324209\pi\)
0.524616 + 0.851339i \(0.324209\pi\)
\(308\) −54.8270 −0.178010
\(309\) 116.660i 0.377540i
\(310\) 123.257i 0.397602i
\(311\) 546.043 1.75577 0.877883 0.478876i \(-0.158955\pi\)
0.877883 + 0.478876i \(0.158955\pi\)
\(312\) −88.5704 −0.283879
\(313\) 159.579i 0.509836i 0.966963 + 0.254918i \(0.0820485\pi\)
−0.966963 + 0.254918i \(0.917952\pi\)
\(314\) 385.841i 1.22879i
\(315\) 15.0777 0.0478658
\(316\) 46.0151i 0.145617i
\(317\) 498.849 1.57366 0.786829 0.617172i \(-0.211722\pi\)
0.786829 + 0.617172i \(0.211722\pi\)
\(318\) 181.469i 0.570658i
\(319\) 270.299i 0.847333i
\(320\) 31.4908i 0.0984088i
\(321\) 168.541i 0.525051i
\(322\) 56.3054 134.729i 0.174862 0.418413i
\(323\) −466.881 −1.44545
\(324\) 15.8228 0.0488359
\(325\) −203.335 −0.625645
\(326\) −527.646 −1.61855
\(327\) 258.441i 0.790340i
\(328\) 271.250 0.826981
\(329\) 93.3887i 0.283856i
\(330\) −93.0612 −0.282004
\(331\) 83.6420 0.252695 0.126347 0.991986i \(-0.459675\pi\)
0.126347 + 0.991986i \(0.459675\pi\)
\(332\) 112.853i 0.339918i
\(333\) 108.757i 0.326598i
\(334\) 397.126 1.18900
\(335\) 36.6011 0.109257
\(336\) 91.3833i 0.271974i
\(337\) 382.659i 1.13549i −0.823205 0.567744i \(-0.807816\pi\)
0.823205 0.567744i \(-0.192184\pi\)
\(338\) −188.722 −0.558349
\(339\) 28.2434i 0.0833139i
\(340\) 77.5599 0.228117
\(341\) 318.720i 0.934663i
\(342\) 144.723i 0.423165i
\(343\) 18.5203i 0.0539949i
\(344\) 68.8738i 0.200215i
\(345\) 29.1802 69.8231i 0.0845802 0.202386i
\(346\) −462.774 −1.33750
\(347\) −297.380 −0.857003 −0.428502 0.903541i \(-0.640958\pi\)
−0.428502 + 0.903541i \(0.640958\pi\)
\(348\) 69.8302 0.200662
\(349\) 330.611 0.947309 0.473654 0.880711i \(-0.342935\pi\)
0.473654 + 0.880711i \(0.342935\pi\)
\(350\) 135.809i 0.388026i
\(351\) 49.3916 0.140717
\(352\) 310.386i 0.881778i
\(353\) 435.318 1.23320 0.616598 0.787278i \(-0.288510\pi\)
0.616598 + 0.787278i \(0.288510\pi\)
\(354\) −89.3963 −0.252532
\(355\) 221.742i 0.624626i
\(356\) 175.669i 0.493452i
\(357\) 106.424 0.298107
\(358\) −143.211 −0.400031
\(359\) 455.565i 1.26898i −0.772930 0.634491i \(-0.781210\pi\)
0.772930 0.634491i \(-0.218790\pi\)
\(360\) 30.6580i 0.0851611i
\(361\) −43.1583 −0.119552
\(362\) 858.075i 2.37037i
\(363\) 31.0618 0.0855697
\(364\) 44.2142i 0.121468i
\(365\) 47.2688i 0.129503i
\(366\) 367.312i 1.00358i
\(367\) 64.4735i 0.175677i −0.996135 0.0878385i \(-0.972004\pi\)
0.996135 0.0878385i \(-0.0279960\pi\)
\(368\) −423.185 176.856i −1.14996 0.480586i
\(369\) −151.263 −0.409928
\(370\) 165.250 0.446622
\(371\) −115.519 −0.311371
\(372\) 82.3395 0.221343
\(373\) 317.319i 0.850721i −0.905024 0.425360i \(-0.860147\pi\)
0.905024 0.425360i \(-0.139853\pi\)
\(374\) −656.860 −1.75631
\(375\) 152.638i 0.407036i
\(376\) 189.890 0.505027
\(377\) 217.978 0.578190
\(378\) 32.9891i 0.0872727i
\(379\) 463.183i 1.22212i −0.791585 0.611059i \(-0.790744\pi\)
0.791585 0.611059i \(-0.209256\pi\)
\(380\) 67.1402 0.176685
\(381\) −257.044 −0.674657
\(382\) 201.000i 0.526178i
\(383\) 94.3791i 0.246421i −0.992381 0.123210i \(-0.960681\pi\)
0.992381 0.123210i \(-0.0393190\pi\)
\(384\) −251.339 −0.654530
\(385\) 59.2404i 0.153871i
\(386\) −488.596 −1.26579
\(387\) 38.4077i 0.0992447i
\(388\) 86.4391i 0.222781i
\(389\) 609.706i 1.56737i −0.621161 0.783683i \(-0.713339\pi\)
0.621161 0.783683i \(-0.286661\pi\)
\(390\) 75.0475i 0.192429i
\(391\) 205.964 492.837i 0.526763 1.26045i
\(392\) 37.6578 0.0960658
\(393\) 63.6541 0.161970
\(394\) 740.151 1.87856
\(395\) −49.7192 −0.125871
\(396\) 62.1679i 0.156990i
\(397\) −197.643 −0.497842 −0.248921 0.968524i \(-0.580076\pi\)
−0.248921 + 0.968524i \(0.580076\pi\)
\(398\) 808.480i 2.03136i
\(399\) 92.1267 0.230894
\(400\) −426.578 −1.06644
\(401\) 145.916i 0.363880i −0.983310 0.181940i \(-0.941762\pi\)
0.983310 0.181940i \(-0.0582377\pi\)
\(402\) 80.0809i 0.199206i
\(403\) 257.026 0.637781
\(404\) −11.3713 −0.0281468
\(405\) 17.0965i 0.0422136i
\(406\) 145.589i 0.358594i
\(407\) −427.307 −1.04990
\(408\) 216.395i 0.530381i
\(409\) −210.872 −0.515578 −0.257789 0.966201i \(-0.582994\pi\)
−0.257789 + 0.966201i \(0.582994\pi\)
\(410\) 229.835i 0.560574i
\(411\) 136.510i 0.332141i
\(412\) 118.414i 0.287413i
\(413\) 56.9074i 0.137790i
\(414\) 152.768 + 63.8444i 0.369006 + 0.154213i
\(415\) −121.937 −0.293825
\(416\) 250.305 0.601694
\(417\) −125.565 −0.301116
\(418\) −568.615 −1.36032
\(419\) 509.278i 1.21546i 0.794143 + 0.607731i \(0.207920\pi\)
−0.794143 + 0.607731i \(0.792080\pi\)
\(420\) −15.3044 −0.0364391
\(421\) 642.060i 1.52508i 0.646940 + 0.762541i \(0.276048\pi\)
−0.646940 + 0.762541i \(0.723952\pi\)
\(422\) 514.635 1.21951
\(423\) −105.893 −0.250338
\(424\) 234.888i 0.553980i
\(425\) 496.788i 1.16891i
\(426\) −485.158 −1.13887
\(427\) 233.821 0.547591
\(428\) 171.075i 0.399709i
\(429\) 194.060i 0.452353i
\(430\) 58.3582 0.135717
\(431\) 203.279i 0.471645i −0.971796 0.235822i \(-0.924222\pi\)
0.971796 0.235822i \(-0.0757784\pi\)
\(432\) 103.619 0.239859
\(433\) 279.424i 0.645321i −0.946515 0.322661i \(-0.895423\pi\)
0.946515 0.322661i \(-0.104577\pi\)
\(434\) 171.670i 0.395553i
\(435\) 75.4514i 0.173451i
\(436\) 262.327i 0.601667i
\(437\) 178.294 426.627i 0.407996 0.976264i
\(438\) 103.421 0.236121
\(439\) 63.9956 0.145776 0.0728880 0.997340i \(-0.476778\pi\)
0.0728880 + 0.997340i \(0.476778\pi\)
\(440\) −120.455 −0.273762
\(441\) −21.0000 −0.0476190
\(442\) 529.713i 1.19844i
\(443\) 602.496 1.36004 0.680018 0.733195i \(-0.261972\pi\)
0.680018 + 0.733195i \(0.261972\pi\)
\(444\) 110.392i 0.248632i
\(445\) −189.810 −0.426539
\(446\) 54.6687 0.122576
\(447\) 360.546i 0.806590i
\(448\) 43.8599i 0.0979015i
\(449\) −757.082 −1.68615 −0.843075 0.537796i \(-0.819257\pi\)
−0.843075 + 0.537796i \(0.819257\pi\)
\(450\) 153.993 0.342207
\(451\) 594.314i 1.31777i
\(452\) 28.6680i 0.0634249i
\(453\) 9.26851 0.0204603
\(454\) 136.919i 0.301583i
\(455\) −47.7733 −0.104996
\(456\) 187.324i 0.410798i
\(457\) 466.080i 1.01987i 0.860213 + 0.509935i \(0.170330\pi\)
−0.860213 + 0.509935i \(0.829670\pi\)
\(458\) 842.131i 1.83871i
\(459\) 120.674i 0.262905i
\(460\) −29.6189 + 70.8729i −0.0643889 + 0.154071i
\(461\) −512.022 −1.11068 −0.555339 0.831624i \(-0.687411\pi\)
−0.555339 + 0.831624i \(0.687411\pi\)
\(462\) 129.614 0.280550
\(463\) −236.851 −0.511557 −0.255779 0.966735i \(-0.582332\pi\)
−0.255779 + 0.966735i \(0.582332\pi\)
\(464\) 457.297 0.985554
\(465\) 88.9676i 0.191328i
\(466\) 474.573 1.01840
\(467\) 455.178i 0.974685i 0.873211 + 0.487343i \(0.162034\pi\)
−0.873211 + 0.487343i \(0.837966\pi\)
\(468\) −50.1342 −0.107124
\(469\) −50.9774 −0.108694
\(470\) 160.898i 0.342336i
\(471\) 278.503i 0.591301i
\(472\) −115.712 −0.245152
\(473\) −150.904 −0.319036
\(474\) 108.782i 0.229499i
\(475\) 430.047i 0.905363i
\(476\) −108.024 −0.226941
\(477\) 130.986i 0.274603i
\(478\) 40.2338 0.0841711
\(479\) 5.78512i 0.0120775i −0.999982 0.00603874i \(-0.998078\pi\)
0.999982 0.00603874i \(-0.00192220\pi\)
\(480\) 86.6412i 0.180502i
\(481\) 344.594i 0.716412i
\(482\) 721.002i 1.49586i
\(483\) −40.6417 + 97.2484i −0.0841443 + 0.201342i
\(484\) −31.5288 −0.0651422
\(485\) −93.3972 −0.192572
\(486\) −37.4061 −0.0769673
\(487\) −286.687 −0.588680 −0.294340 0.955701i \(-0.595100\pi\)
−0.294340 + 0.955701i \(0.595100\pi\)
\(488\) 475.436i 0.974254i
\(489\) 380.859 0.778852
\(490\) 31.9082i 0.0651188i
\(491\) −81.5009 −0.165990 −0.0829948 0.996550i \(-0.526448\pi\)
−0.0829948 + 0.996550i \(0.526448\pi\)
\(492\) 153.538 0.312068
\(493\) 532.563i 1.08025i
\(494\) 458.549i 0.928237i
\(495\) 67.1723 0.135702
\(496\) 539.216 1.08713
\(497\) 308.839i 0.621407i
\(498\) 266.791i 0.535724i
\(499\) 821.224 1.64574 0.822870 0.568230i \(-0.192372\pi\)
0.822870 + 0.568230i \(0.192372\pi\)
\(500\) 154.933i 0.309867i
\(501\) −286.648 −0.572152
\(502\) 421.862i 0.840362i
\(503\) 596.272i 1.18543i 0.805412 + 0.592716i \(0.201944\pi\)
−0.805412 + 0.592716i \(0.798056\pi\)
\(504\) 42.6999i 0.0847221i
\(505\) 12.2867i 0.0243300i
\(506\) 250.845 600.227i 0.495740 1.18622i
\(507\) 136.221 0.268680
\(508\) 260.909 0.513600
\(509\) 145.511 0.285876 0.142938 0.989732i \(-0.454345\pi\)
0.142938 + 0.989732i \(0.454345\pi\)
\(510\) −183.356 −0.359522
\(511\) 65.8352i 0.128836i
\(512\) 96.0009 0.187502
\(513\) 104.462i 0.203629i
\(514\) −620.125 −1.20647
\(515\) 127.946 0.248439
\(516\) 38.9852i 0.0755526i
\(517\) 416.053i 0.804745i
\(518\) −230.158 −0.444320
\(519\) 334.033 0.643610
\(520\) 97.1389i 0.186806i
\(521\) 757.208i 1.45337i −0.686968 0.726687i \(-0.741059\pi\)
0.686968 0.726687i \(-0.258941\pi\)
\(522\) −165.083 −0.316251
\(523\) 291.346i 0.557066i 0.960427 + 0.278533i \(0.0898482\pi\)
−0.960427 + 0.278533i \(0.910152\pi\)
\(524\) −64.6111 −0.123304
\(525\) 98.0280i 0.186720i
\(526\) 1232.68i 2.34350i
\(527\) 627.966i 1.19159i
\(528\) 407.119i 0.771059i
\(529\) 371.691 + 376.413i 0.702629 + 0.711556i
\(530\) 199.025 0.375519
\(531\) 64.5269 0.121520
\(532\) −93.5118 −0.175774
\(533\) 479.273 0.899199
\(534\) 415.291i 0.777699i
\(535\) −184.847 −0.345508
\(536\) 103.654i 0.193384i
\(537\) 103.371 0.192497
\(538\) 788.047 1.46477
\(539\) 82.5090i 0.153078i
\(540\) 17.3536i 0.0321362i
\(541\) −361.351 −0.667931 −0.333966 0.942585i \(-0.608387\pi\)
−0.333966 + 0.942585i \(0.608387\pi\)
\(542\) −1259.30 −2.32344
\(543\) 619.365i 1.14064i
\(544\) 611.545i 1.12416i
\(545\) 283.443 0.520080
\(546\) 104.525i 0.191438i
\(547\) 536.912 0.981558 0.490779 0.871284i \(-0.336712\pi\)
0.490779 + 0.871284i \(0.336712\pi\)
\(548\) 138.562i 0.252851i
\(549\) 265.129i 0.482930i
\(550\) 605.039i 1.10007i
\(551\) 461.017i 0.836691i
\(552\) 197.738 + 82.6380i 0.358221 + 0.149706i
\(553\) 69.2481 0.125223
\(554\) −619.389 −1.11803
\(555\) −119.279 −0.214917
\(556\) 127.453 0.229233
\(557\) 915.897i 1.64434i 0.569242 + 0.822170i \(0.307237\pi\)
−0.569242 + 0.822170i \(0.692763\pi\)
\(558\) −194.655 −0.348845
\(559\) 121.694i 0.217699i
\(560\) −100.224 −0.178971
\(561\) 474.127 0.845145
\(562\) 869.623i 1.54737i
\(563\) 676.872i 1.20226i 0.799152 + 0.601130i \(0.205282\pi\)
−0.799152 + 0.601130i \(0.794718\pi\)
\(564\) 107.485 0.190576
\(565\) 30.9757 0.0548243
\(566\) 79.9314i 0.141222i
\(567\) 23.8118i 0.0419961i
\(568\) −627.972 −1.10558
\(569\) 726.413i 1.27665i −0.769767 0.638325i \(-0.779628\pi\)
0.769767 0.638325i \(-0.220372\pi\)
\(570\) −158.723 −0.278462
\(571\) 281.192i 0.492455i 0.969212 + 0.246227i \(0.0791910\pi\)
−0.969212 + 0.246227i \(0.920809\pi\)
\(572\) 196.977i 0.344366i
\(573\) 145.083i 0.253200i
\(574\) 320.111i 0.557685i
\(575\) 453.956 + 189.715i 0.789488 + 0.329940i
\(576\) 49.7324 0.0863410
\(577\) 962.280 1.66773 0.833865 0.551969i \(-0.186123\pi\)
0.833865 + 0.551969i \(0.186123\pi\)
\(578\) −600.709 −1.03929
\(579\) 352.673 0.609106
\(580\) 76.5858i 0.132044i
\(581\) 169.832 0.292310
\(582\) 204.347i 0.351112i
\(583\) −514.644 −0.882751
\(584\) 133.865 0.229220
\(585\) 54.1698i 0.0925980i
\(586\) 993.372i 1.69517i
\(587\) −105.339 −0.179452 −0.0897262 0.995966i \(-0.528599\pi\)
−0.0897262 + 0.995966i \(0.528599\pi\)
\(588\) 21.3157 0.0362513
\(589\) 543.603i 0.922925i
\(590\) 98.0447i 0.166177i
\(591\) −534.247 −0.903970
\(592\) 722.927i 1.22116i
\(593\) 28.3074 0.0477359 0.0238680 0.999715i \(-0.492402\pi\)
0.0238680 + 0.999715i \(0.492402\pi\)
\(594\) 146.969i 0.247422i
\(595\) 116.720i 0.196168i
\(596\) 365.967i 0.614038i
\(597\) 583.567i 0.977499i
\(598\) −484.042 202.289i −0.809435 0.338276i
\(599\) −175.103 −0.292325 −0.146162 0.989261i \(-0.546692\pi\)
−0.146162 + 0.989261i \(0.546692\pi\)
\(600\) 199.323 0.332206
\(601\) 155.499 0.258734 0.129367 0.991597i \(-0.458705\pi\)
0.129367 + 0.991597i \(0.458705\pi\)
\(602\) −81.2803 −0.135017
\(603\) 57.8030i 0.0958590i
\(604\) −9.40786 −0.0155759
\(605\) 34.0668i 0.0563087i
\(606\) 26.8824 0.0443605
\(607\) 562.737 0.927079 0.463540 0.886076i \(-0.346579\pi\)
0.463540 + 0.886076i \(0.346579\pi\)
\(608\) 529.388i 0.870704i
\(609\) 105.087i 0.172557i
\(610\) −402.847 −0.660404
\(611\) 335.518 0.549130
\(612\) 122.488i 0.200144i
\(613\) 1112.46i 1.81478i −0.420288 0.907391i \(-0.638071\pi\)
0.420288 0.907391i \(-0.361929\pi\)
\(614\) 772.947 1.25887
\(615\) 165.897i 0.269751i
\(616\) 167.768 0.272351
\(617\) 670.161i 1.08616i 0.839681 + 0.543080i \(0.182742\pi\)
−0.839681 + 0.543080i \(0.817258\pi\)
\(618\) 279.938i 0.452974i
\(619\) 631.709i 1.02053i 0.860017 + 0.510266i \(0.170453\pi\)
−0.860017 + 0.510266i \(0.829547\pi\)
\(620\) 90.3052i 0.145654i
\(621\) −110.269 46.0833i −0.177567 0.0742083i
\(622\) 1310.29 2.10657
\(623\) 264.364 0.424340
\(624\) −328.314 −0.526144
\(625\) 367.382 0.587810
\(626\) 382.925i 0.611702i
\(627\) 410.431 0.654595
\(628\) 282.690i 0.450144i
\(629\) −841.913 −1.33849
\(630\) 36.1805 0.0574294
\(631\) 7.16238i 0.0113508i 0.999984 + 0.00567542i \(0.00180655\pi\)
−0.999984 + 0.00567542i \(0.998193\pi\)
\(632\) 140.804i 0.222792i
\(633\) −371.467 −0.586836
\(634\) 1197.04 1.88808
\(635\) 281.911i 0.443955i
\(636\) 132.955i 0.209049i
\(637\) 66.5379 0.104455
\(638\) 648.611i 1.01663i
\(639\) 350.191 0.548029
\(640\) 275.655i 0.430710i
\(641\) 857.091i 1.33711i 0.743661 + 0.668557i \(0.233088\pi\)
−0.743661 + 0.668557i \(0.766912\pi\)
\(642\) 404.432i 0.629957i
\(643\) 582.061i 0.905227i −0.891707 0.452613i \(-0.850492\pi\)
0.891707 0.452613i \(-0.149508\pi\)
\(644\) 41.2527 98.7106i 0.0640570 0.153277i
\(645\) −42.1234 −0.0653075
\(646\) −1120.33 −1.73425
\(647\) 798.806 1.23463 0.617315 0.786716i \(-0.288220\pi\)
0.617315 + 0.786716i \(0.288220\pi\)
\(648\) −48.4172 −0.0747179
\(649\) 253.526i 0.390641i
\(650\) −487.922 −0.750650
\(651\) 123.913i 0.190342i
\(652\) −386.585 −0.592921
\(653\) −429.160 −0.657213 −0.328607 0.944467i \(-0.606579\pi\)
−0.328607 + 0.944467i \(0.606579\pi\)
\(654\) 620.156i 0.948251i
\(655\) 69.8121i 0.106583i
\(656\) 1005.47 1.53273
\(657\) −74.6501 −0.113623
\(658\) 224.096i 0.340571i
\(659\) 799.463i 1.21315i −0.795028 0.606573i \(-0.792544\pi\)
0.795028 0.606573i \(-0.207456\pi\)
\(660\) −68.1822 −0.103306
\(661\) 112.845i 0.170718i −0.996350 0.0853590i \(-0.972796\pi\)
0.996350 0.0853590i \(-0.0272037\pi\)
\(662\) 200.707 0.303183
\(663\) 382.350i 0.576698i
\(664\) 345.325i 0.520068i
\(665\) 101.039i 0.151939i
\(666\) 260.974i 0.391853i
\(667\) −486.647 203.378i −0.729606 0.304914i
\(668\) 290.958 0.435566
\(669\) −39.4603 −0.0589839
\(670\) 87.8281 0.131087
\(671\) 1041.69 1.55244
\(672\) 120.672i 0.179572i
\(673\) −32.4266 −0.0481822 −0.0240911 0.999710i \(-0.507669\pi\)
−0.0240911 + 0.999710i \(0.507669\pi\)
\(674\) 918.230i 1.36236i
\(675\) −111.153 −0.164672
\(676\) −138.269 −0.204540
\(677\) 621.675i 0.918279i 0.888364 + 0.459139i \(0.151842\pi\)
−0.888364 + 0.459139i \(0.848158\pi\)
\(678\) 67.7729i 0.0999601i
\(679\) 130.082 0.191579
\(680\) −237.330 −0.349015
\(681\) 98.8290i 0.145123i
\(682\) 764.801i 1.12141i
\(683\) 111.256 0.162894 0.0814468 0.996678i \(-0.474046\pi\)
0.0814468 + 0.996678i \(0.474046\pi\)
\(684\) 106.032i 0.155018i
\(685\) −149.716 −0.218564
\(686\) 44.4413i 0.0647832i
\(687\) 607.856i 0.884798i
\(688\) 255.302i 0.371079i
\(689\) 415.025i 0.602358i
\(690\) 70.0208 167.548i 0.101479 0.242823i
\(691\) 123.165 0.178242 0.0891209 0.996021i \(-0.471594\pi\)
0.0891209 + 0.996021i \(0.471594\pi\)
\(692\) −339.056 −0.489965
\(693\) −93.5565 −0.135002
\(694\) −713.594 −1.02823
\(695\) 137.713i 0.198148i
\(696\) −213.678 −0.307008
\(697\) 1170.96i 1.68000i
\(698\) 793.334 1.13658
\(699\) −342.550 −0.490058
\(700\) 99.5019i 0.142146i
\(701\) 1207.42i 1.72242i −0.508247 0.861211i \(-0.669706\pi\)
0.508247 0.861211i \(-0.330294\pi\)
\(702\) 118.520 0.168832
\(703\) −728.807 −1.03671
\(704\) 195.399i 0.277555i
\(705\) 116.137i 0.164734i
\(706\) 1044.59 1.47959
\(707\) 17.1127i 0.0242046i
\(708\) −65.4971 −0.0925100
\(709\) 1165.00i 1.64315i 0.570098 + 0.821576i \(0.306905\pi\)
−0.570098 + 0.821576i \(0.693095\pi\)
\(710\) 532.093i 0.749427i
\(711\) 78.5200i 0.110436i
\(712\) 537.539i 0.754970i
\(713\) −573.824 239.810i −0.804802 0.336340i
\(714\) 255.375 0.357669
\(715\) −212.833 −0.297669
\(716\) −104.925 −0.146543
\(717\) −29.0410 −0.0405035
\(718\) 1093.17i 1.52253i
\(719\) −341.047 −0.474335 −0.237167 0.971469i \(-0.576219\pi\)
−0.237167 + 0.971469i \(0.576219\pi\)
\(720\) 113.643i 0.157838i
\(721\) −178.201 −0.247158
\(722\) −103.563 −0.143439
\(723\) 520.425i 0.719813i
\(724\) 628.677i 0.868339i
\(725\) −490.548 −0.676618
\(726\) 74.5360 0.102667
\(727\) 970.051i 1.33432i 0.744914 + 0.667160i \(0.232490\pi\)
−0.744914 + 0.667160i \(0.767510\pi\)
\(728\) 135.293i 0.185843i
\(729\) 27.0000 0.0370370
\(730\) 113.426i 0.155378i
\(731\) −297.322 −0.406733
\(732\) 269.115i 0.367643i
\(733\) 399.314i 0.544766i −0.962189 0.272383i \(-0.912188\pi\)
0.962189 0.272383i \(-0.0878118\pi\)
\(734\) 154.711i 0.210778i
\(735\) 23.0316i 0.0313355i
\(736\) −558.819 233.539i −0.759265 0.317309i
\(737\) −227.108 −0.308152
\(738\) −362.972 −0.491832
\(739\) −352.600 −0.477131 −0.238565 0.971126i \(-0.576677\pi\)
−0.238565 + 0.971126i \(0.576677\pi\)
\(740\) 121.072 0.163611
\(741\) 330.984i 0.446672i
\(742\) −277.199 −0.373583
\(743\) 1220.21i 1.64228i 0.570727 + 0.821140i \(0.306661\pi\)
−0.570727 + 0.821140i \(0.693339\pi\)
\(744\) −251.955 −0.338650
\(745\) 395.426 0.530773
\(746\) 761.439i 1.02070i
\(747\) 192.572i 0.257793i
\(748\) −481.255 −0.643389
\(749\) 257.451 0.343727
\(750\) 366.272i 0.488362i
\(751\) 966.244i 1.28661i −0.765610 0.643305i \(-0.777563\pi\)
0.765610 0.643305i \(-0.222437\pi\)
\(752\) 703.887 0.936020
\(753\) 304.503i 0.404386i
\(754\) 523.060 0.693713
\(755\) 10.1652i 0.0134638i
\(756\) 24.1698i 0.0319706i
\(757\) 426.679i 0.563644i −0.959467 0.281822i \(-0.909061\pi\)
0.959467 0.281822i \(-0.0909388\pi\)
\(758\) 1111.45i 1.46630i
\(759\) −181.062 + 433.248i −0.238553 + 0.570815i
\(760\) −205.446 −0.270324
\(761\) 10.0740 0.0132379 0.00661894 0.999978i \(-0.497893\pi\)
0.00661894 + 0.999978i \(0.497893\pi\)
\(762\) −616.804 −0.809454
\(763\) −394.775 −0.517399
\(764\) 147.265i 0.192755i
\(765\) 132.348 0.173004
\(766\) 226.472i 0.295656i
\(767\) −204.452 −0.266560
\(768\) −488.263 −0.635759
\(769\) 242.409i 0.315226i −0.987501 0.157613i \(-0.949620\pi\)
0.987501 0.157613i \(-0.0503799\pi\)
\(770\) 142.153i 0.184615i
\(771\) 447.611 0.580559
\(772\) −357.975 −0.463698
\(773\) 826.460i 1.06916i −0.845118 0.534579i \(-0.820470\pi\)
0.845118 0.534579i \(-0.179530\pi\)
\(774\) 92.1632i 0.119074i
\(775\) −578.424 −0.746354
\(776\) 264.500i 0.340851i
\(777\) 166.129 0.213809
\(778\) 1463.05i 1.88053i
\(779\) 1013.65i 1.30122i
\(780\) 54.9843i 0.0704927i
\(781\) 1375.90i 1.76172i
\(782\) 494.233 1182.61i 0.632011 1.51229i
\(783\) 119.158 0.152181
\(784\) 139.590 0.178049
\(785\) 305.446 0.389103
\(786\) 152.744 0.194331
\(787\) 1271.91i 1.61614i −0.589084 0.808072i \(-0.700511\pi\)
0.589084 0.808072i \(-0.299489\pi\)
\(788\) 542.279 0.688171
\(789\) 889.757i 1.12770i
\(790\) −119.306 −0.151021
\(791\) −43.1425 −0.0545417
\(792\) 190.231i 0.240191i
\(793\) 840.052i 1.05933i
\(794\) −474.266 −0.597312
\(795\) −143.658 −0.180702
\(796\) 592.341i 0.744147i
\(797\) 322.055i 0.404084i 0.979377 + 0.202042i \(0.0647578\pi\)
−0.979377 + 0.202042i \(0.935242\pi\)
\(798\) 221.067 0.277027
\(799\) 819.739i 1.02596i
\(800\) −563.299 −0.704124
\(801\) 299.760i 0.374233i
\(802\) 350.140i 0.436584i
\(803\) 293.300i 0.365256i
\(804\) 58.6720i 0.0729752i
\(805\) 106.656 + 44.5735i 0.132493 + 0.0553708i
\(806\) 616.760 0.765210
\(807\) −568.818 −0.704855
\(808\) 34.7957 0.0430640
\(809\) −906.479 −1.12049 −0.560246 0.828326i \(-0.689294\pi\)
−0.560246 + 0.828326i \(0.689294\pi\)
\(810\) 41.0249i 0.0506480i
\(811\) −37.9719 −0.0468211 −0.0234106 0.999726i \(-0.507452\pi\)
−0.0234106 + 0.999726i \(0.507452\pi\)
\(812\) 106.667i 0.131364i
\(813\) 908.975 1.11805
\(814\) −1025.37 −1.25967
\(815\) 417.704i 0.512520i
\(816\) 802.136i 0.983010i
\(817\) −257.379 −0.315029
\(818\) −506.008 −0.618592
\(819\) 75.4469i 0.0921207i
\(820\) 168.391i 0.205355i
\(821\) −401.596 −0.489154 −0.244577 0.969630i \(-0.578649\pi\)
−0.244577 + 0.969630i \(0.578649\pi\)
\(822\) 327.570i 0.398503i
\(823\) 92.9808 0.112978 0.0564889 0.998403i \(-0.482009\pi\)
0.0564889 + 0.998403i \(0.482009\pi\)
\(824\) 362.342i 0.439735i
\(825\) 436.722i 0.529360i
\(826\) 136.555i 0.165321i
\(827\) 23.9802i 0.0289966i −0.999895 0.0144983i \(-0.995385\pi\)
0.999895 0.0144983i \(-0.00461512\pi\)
\(828\) 111.927 + 46.7762i 0.135178 + 0.0564930i
\(829\) 984.055 1.18704 0.593519 0.804820i \(-0.297738\pi\)
0.593519 + 0.804820i \(0.297738\pi\)
\(830\) −292.601 −0.352531
\(831\) 447.079 0.538002
\(832\) −157.576 −0.189394
\(833\) 162.565i 0.195157i
\(834\) −301.307 −0.361279
\(835\) 314.379i 0.376502i
\(836\) −416.602 −0.498327
\(837\) 140.504 0.167866
\(838\) 1222.07i 1.45831i
\(839\) 640.165i 0.763010i −0.924367 0.381505i \(-0.875406\pi\)
0.924367 0.381505i \(-0.124594\pi\)
\(840\) 46.8308 0.0557510
\(841\) −315.125 −0.374703
\(842\) 1540.69i 1.82980i
\(843\) 627.700i 0.744603i
\(844\) 377.053 0.446745
\(845\) 149.399i 0.176804i
\(846\) −254.101 −0.300356
\(847\) 47.4477i 0.0560185i
\(848\) 870.684i 1.02675i
\(849\) 57.6951i 0.0679565i
\(850\) 1192.09i 1.40246i
\(851\) 321.513 769.325i 0.377806 0.904025i
\(852\) −355.456 −0.417202
\(853\) 677.711 0.794503 0.397252 0.917710i \(-0.369964\pi\)
0.397252 + 0.917710i \(0.369964\pi\)
\(854\) 561.078 0.657000
\(855\) 114.568 0.133997
\(856\) 523.484i 0.611546i
\(857\) 194.493 0.226946 0.113473 0.993541i \(-0.463802\pi\)
0.113473 + 0.993541i \(0.463802\pi\)
\(858\) 465.666i 0.542734i
\(859\) −894.039 −1.04079 −0.520395 0.853926i \(-0.674215\pi\)
−0.520395 + 0.853926i \(0.674215\pi\)
\(860\) 42.7567 0.0497171
\(861\) 231.058i 0.268361i
\(862\) 487.789i 0.565880i
\(863\) 190.049 0.220219 0.110109 0.993919i \(-0.464880\pi\)
0.110109 + 0.993919i \(0.464880\pi\)
\(864\) 136.830 0.158368
\(865\) 366.349i 0.423524i
\(866\) 670.507i 0.774257i
\(867\) 433.596 0.500111
\(868\) 125.776i 0.144903i
\(869\) 308.505 0.355012
\(870\) 181.053i 0.208107i
\(871\) 183.147i 0.210272i
\(872\) 802.709i 0.920538i
\(873\) 147.499i 0.168957i
\(874\) 427.836 1023.74i 0.489515 1.17132i
\(875\) 233.159 0.266468
\(876\) 75.7724 0.0864982
\(877\) −849.024 −0.968101 −0.484050 0.875040i \(-0.660835\pi\)
−0.484050 + 0.875040i \(0.660835\pi\)
\(878\) 153.564 0.174902
\(879\) 717.023i 0.815726i
\(880\) −446.505 −0.507392
\(881\) 1068.08i 1.21235i 0.795330 + 0.606176i \(0.207297\pi\)
−0.795330 + 0.606176i \(0.792703\pi\)
\(882\) −50.3917 −0.0571334
\(883\) −14.2405 −0.0161274 −0.00806368 0.999967i \(-0.502567\pi\)
−0.00806368 + 0.999967i \(0.502567\pi\)
\(884\) 388.099i 0.439026i
\(885\) 70.7694i 0.0799654i
\(886\) 1445.75 1.63177
\(887\) 608.195 0.685676 0.342838 0.939395i \(-0.388612\pi\)
0.342838 + 0.939395i \(0.388612\pi\)
\(888\) 337.796i 0.380401i
\(889\) 392.642i 0.441667i
\(890\) −455.468 −0.511761
\(891\) 106.083i 0.119061i
\(892\) 40.0535 0.0449031
\(893\) 709.613i 0.794639i
\(894\) 865.167i 0.967748i
\(895\) 113.371i 0.126672i
\(896\) 383.927i 0.428490i
\(897\) 349.385 + 146.014i 0.389504 + 0.162780i
\(898\) −1816.69 −2.02305
\(899\) 620.079 0.689743
\(900\) 112.825 0.125361
\(901\) −1013.99 −1.12540
\(902\) 1426.12i 1.58106i
\(903\) 58.6687 0.0649709
\(904\) 87.7230i 0.0970387i
\(905\) 679.284 0.750590
\(906\) 22.2407 0.0245483
\(907\) 256.082i 0.282339i −0.989985 0.141170i \(-0.954914\pi\)
0.989985 0.141170i \(-0.0450863\pi\)
\(908\) 100.315i 0.110479i
\(909\) −19.4039 −0.0213465
\(910\) −114.637 −0.125975
\(911\) 358.680i 0.393721i −0.980431 0.196861i \(-0.936925\pi\)
0.980431 0.196861i \(-0.0630746\pi\)
\(912\) 694.375i 0.761376i
\(913\) 756.614 0.828712
\(914\) 1118.41i 1.22364i
\(915\) 290.778 0.317790
\(916\) 616.996i 0.673576i
\(917\) 97.2332i 0.106034i
\(918\) 289.568i 0.315434i
\(919\) 852.534i 0.927676i 0.885920 + 0.463838i \(0.153528\pi\)
−0.885920 + 0.463838i \(0.846472\pi\)
\(920\) 90.6326 216.868i 0.0985137 0.235726i
\(921\) −557.919 −0.605775
\(922\) −1228.65 −1.33259
\(923\) −1109.57 −1.20213
\(924\) 94.9631 0.102774
\(925\) 775.492i 0.838370i
\(926\) −568.348 −0.613767
\(927\) 202.061i 0.217973i
\(928\) 603.864 0.650716
\(929\) −1365.22 −1.46956 −0.734781 0.678304i \(-0.762715\pi\)
−0.734781 + 0.678304i \(0.762715\pi\)
\(930\) 213.487i 0.229556i
\(931\) 140.726i 0.151156i
\(932\) 347.701 0.373069
\(933\) −945.774 −1.01369
\(934\) 1092.25i 1.16943i
\(935\) 519.995i 0.556144i
\(936\) 153.408 0.163898
\(937\) 849.124i 0.906215i 0.891456 + 0.453108i \(0.149685\pi\)
−0.891456 + 0.453108i \(0.850315\pi\)
\(938\) −122.326 −0.130411
\(939\) 276.398i 0.294354i
\(940\) 117.883i 0.125408i
\(941\) 1017.42i 1.08122i −0.841275 0.540608i \(-0.818194\pi\)
0.841275 0.540608i \(-0.181806\pi\)
\(942\) 668.296i 0.709444i
\(943\) −1070.00 447.172i −1.13468 0.474201i
\(944\) −428.921 −0.454365
\(945\) −26.1154 −0.0276353
\(946\) −362.109 −0.382780
\(947\) 139.135 0.146922 0.0734610 0.997298i \(-0.476596\pi\)
0.0734610 + 0.997298i \(0.476596\pi\)
\(948\) 79.7005i 0.0840723i
\(949\) 236.527 0.249238
\(950\) 1031.94i 1.08626i
\(951\) −864.032 −0.908551
\(952\) 330.549 0.347216
\(953\) 1583.35i 1.66143i −0.556694 0.830717i \(-0.687931\pi\)
0.556694 0.830717i \(-0.312069\pi\)
\(954\) 314.314i 0.329470i
\(955\) −159.119 −0.166617
\(956\) 29.4777 0.0308344
\(957\) 468.172i 0.489208i
\(958\) 13.8820i 0.0144906i
\(959\) 208.523 0.217437
\(960\) 54.5437i 0.0568163i
\(961\) −229.842 −0.239169
\(962\) 826.889i 0.859552i
\(963\) 291.922i 0.303138i
\(964\) 528.249i 0.547977i
\(965\) 386.791i 0.400820i
\(966\) −97.5239 + 233.358i −0.100956 + 0.241571i
\(967\) 1209.98 1.25127 0.625636 0.780115i \(-0.284840\pi\)
0.625636 + 0.780115i \(0.284840\pi\)
\(968\) 96.4768 0.0996661
\(969\) 808.661 0.834531
\(970\) −224.116 −0.231048
\(971\) 361.766i 0.372570i −0.982496 0.186285i \(-0.940355\pi\)
0.982496 0.186285i \(-0.0596448\pi\)
\(972\) −27.4059 −0.0281954
\(973\) 191.804i 0.197127i
\(974\) −687.935 −0.706299
\(975\) 352.186 0.361216
\(976\) 1762.35i 1.80569i
\(977\) 467.002i 0.477995i 0.971020 + 0.238998i \(0.0768189\pi\)
−0.971020 + 0.238998i \(0.923181\pi\)
\(978\) 913.909 0.934467
\(979\) 1177.76 1.20302
\(980\) 23.3779i 0.0238550i
\(981\) 447.633i 0.456303i
\(982\) −195.570 −0.199155
\(983\) 1055.40i 1.07365i 0.843695 + 0.536824i \(0.180376\pi\)
−0.843695 + 0.536824i \(0.819624\pi\)
\(984\) −469.818 −0.477458
\(985\) 585.931i 0.594854i
\(986\) 1277.94i 1.29609i
\(987\) 161.754i 0.163885i
\(988\) 335.961i 0.340041i
\(989\) 113.543 271.688i 0.114806 0.274709i
\(990\) 161.187 0.162815
\(991\) 1049.02 1.05855 0.529276 0.848450i \(-0.322464\pi\)
0.529276 + 0.848450i \(0.322464\pi\)
\(992\) 712.039 0.717782
\(993\) −144.872 −0.145893
\(994\) 741.091i 0.745564i
\(995\) 640.023 0.643239
\(996\) 195.467i 0.196252i
\(997\) 1636.68 1.64160 0.820802 0.571212i \(-0.193527\pi\)
0.820802 + 0.571212i \(0.193527\pi\)
\(998\) 1970.61 1.97456
\(999\) 188.373i 0.188562i
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 483.3.f.a.22.36 yes 48
23.22 odd 2 inner 483.3.f.a.22.35 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.3.f.a.22.35 48 23.22 odd 2 inner
483.3.f.a.22.36 yes 48 1.1 even 1 trivial