Properties

Label 19.5.b.b.18.3
Level $19$
Weight $5$
Character 19.18
Analytic conductor $1.964$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,5,Mod(18,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.18");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 19.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: \(1.96402929859\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.12107488.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 35x^{2} + 142 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 18.3
Root \(2.16425i\) of defining polynomial
Character \(\chi\) \(=\) 19.18
Dual form 19.5.b.b.18.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.16425i q^{2} -16.8206i q^{3} +11.3160 q^{4} -14.7720 q^{5} +36.4040 q^{6} +59.6320 q^{7} +59.1188i q^{8} -201.932 q^{9} +O(q^{10})\) \(q+2.16425i q^{2} -16.8206i q^{3} +11.3160 q^{4} -14.7720 q^{5} +36.4040 q^{6} +59.6320 q^{7} +59.1188i q^{8} -201.932 q^{9} -31.9704i q^{10} -17.1240 q^{11} -190.342i q^{12} +163.384i q^{13} +129.059i q^{14} +248.474i q^{15} +53.1081 q^{16} +177.528 q^{17} -437.032i q^{18} +(272.004 + 237.349i) q^{19} -167.160 q^{20} -1003.05i q^{21} -37.0608i q^{22} -502.236 q^{23} +994.412 q^{24} -406.788 q^{25} -353.604 q^{26} +2034.15i q^{27} +674.796 q^{28} -500.627i q^{29} -537.760 q^{30} -1465.70i q^{31} +1060.84i q^{32} +288.036i q^{33} +384.216i q^{34} -880.884 q^{35} -2285.06 q^{36} -600.763i q^{37} +(-513.684 + 588.686i) q^{38} +2748.21 q^{39} -873.303i q^{40} +325.088i q^{41} +2170.84 q^{42} -752.973 q^{43} -193.776 q^{44} +2982.94 q^{45} -1086.97i q^{46} -2257.62 q^{47} -893.309i q^{48} +1154.98 q^{49} -880.392i q^{50} -2986.13i q^{51} +1848.85i q^{52} +1501.20i q^{53} -4402.41 q^{54} +252.956 q^{55} +3525.37i q^{56} +(3992.36 - 4575.27i) q^{57} +1083.48 q^{58} +4555.57i q^{59} +2811.73i q^{60} +770.012 q^{61} +3172.15 q^{62} -12041.6 q^{63} -1446.20 q^{64} -2413.51i q^{65} -623.383 q^{66} +1613.61i q^{67} +2008.91 q^{68} +8447.91i q^{69} -1906.46i q^{70} -7676.78i q^{71} -11938.0i q^{72} -2330.98 q^{73} +1300.20 q^{74} +6842.41i q^{75} +(3078.00 + 2685.85i) q^{76} -1021.14 q^{77} +5947.83i q^{78} +53.3968i q^{79} -784.513 q^{80} +17859.1 q^{81} -703.573 q^{82} +9438.26 q^{83} -11350.5i q^{84} -2622.44 q^{85} -1629.62i q^{86} -8420.83 q^{87} -1012.35i q^{88} -9238.06i q^{89} +6455.84i q^{90} +9742.91i q^{91} -5683.31 q^{92} -24654.0 q^{93} -4886.06i q^{94} +(-4018.05 - 3506.13i) q^{95} +17843.9 q^{96} -2562.56i q^{97} +2499.66i q^{98} +3457.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 42 q^{5} + 26 q^{6} + 136 q^{7} - 278 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 42 q^{5} + 26 q^{6} + 136 q^{7} - 278 q^{9} + 222 q^{11} - 454 q^{16} + 300 q^{17} + 114 q^{19} - 156 q^{20} - 78 q^{23} + 1910 q^{24} - 1986 q^{25} + 414 q^{26} + 1110 q^{28} - 784 q^{30} - 1866 q^{35} - 6372 q^{36} + 2166 q^{38} + 6362 q^{39} + 3950 q^{42} + 2986 q^{43} - 4056 q^{44} + 5182 q^{45} - 7578 q^{47} - 2352 q^{49} - 15542 q^{54} - 1090 q^{55} + 10450 q^{57} + 14638 q^{58} + 158 q^{61} + 18396 q^{62} - 23030 q^{63} + 3494 q^{64} - 7244 q^{66} + 4806 q^{68} - 15168 q^{73} - 17868 q^{74} + 12312 q^{76} + 102 q^{77} + 1920 q^{80} + 40712 q^{81} - 29164 q^{82} + 33276 q^{83} - 4902 q^{85} - 7214 q^{87} - 24630 q^{92} - 40004 q^{93} - 5358 q^{95} + 47538 q^{96} + 23042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16425i 0.541063i 0.962711 + 0.270532i \(0.0871995\pi\)
−0.962711 + 0.270532i \(0.912801\pi\)
\(3\) 16.8206i 1.86895i −0.356024 0.934477i \(-0.615868\pi\)
0.356024 0.934477i \(-0.384132\pi\)
\(4\) 11.3160 0.707250
\(5\) −14.7720 −0.590880 −0.295440 0.955361i \(-0.595466\pi\)
−0.295440 + 0.955361i \(0.595466\pi\)
\(6\) 36.4040 1.01122
\(7\) 59.6320 1.21698 0.608490 0.793562i \(-0.291776\pi\)
0.608490 + 0.793562i \(0.291776\pi\)
\(8\) 59.1188i 0.923731i
\(9\) −201.932 −2.49299
\(10\) 31.9704i 0.319704i
\(11\) −17.1240 −0.141521 −0.0707605 0.997493i \(-0.522543\pi\)
−0.0707605 + 0.997493i \(0.522543\pi\)
\(12\) 190.342i 1.32182i
\(13\) 163.384i 0.966769i 0.875408 + 0.483384i \(0.160593\pi\)
−0.875408 + 0.483384i \(0.839407\pi\)
\(14\) 129.059i 0.658463i
\(15\) 248.474i 1.10433i
\(16\) 53.1081 0.207453
\(17\) 177.528 0.614284 0.307142 0.951664i \(-0.400627\pi\)
0.307142 + 0.951664i \(0.400627\pi\)
\(18\) 437.032i 1.34886i
\(19\) 272.004 + 237.349i 0.753474 + 0.657478i
\(20\) −167.160 −0.417900
\(21\) 1003.05i 2.27448i
\(22\) 37.0608i 0.0765718i
\(23\) −502.236 −0.949407 −0.474703 0.880146i \(-0.657445\pi\)
−0.474703 + 0.880146i \(0.657445\pi\)
\(24\) 994.412 1.72641
\(25\) −406.788 −0.650861
\(26\) −353.604 −0.523083
\(27\) 2034.15i 2.79033i
\(28\) 674.796 0.860709
\(29\) 500.627i 0.595275i −0.954679 0.297638i \(-0.903801\pi\)
0.954679 0.297638i \(-0.0961987\pi\)
\(30\) −537.760 −0.597511
\(31\) 1465.70i 1.52518i −0.646880 0.762592i \(-0.723926\pi\)
0.646880 0.762592i \(-0.276074\pi\)
\(32\) 1060.84i 1.03598i
\(33\) 288.036i 0.264496i
\(34\) 384.216i 0.332367i
\(35\) −880.884 −0.719089
\(36\) −2285.06 −1.76317
\(37\) 600.763i 0.438833i −0.975631 0.219417i \(-0.929585\pi\)
0.975631 0.219417i \(-0.0704154\pi\)
\(38\) −513.684 + 588.686i −0.355737 + 0.407677i
\(39\) 2748.21 1.80685
\(40\) 873.303i 0.545814i
\(41\) 325.088i 0.193390i 0.995314 + 0.0966948i \(0.0308271\pi\)
−0.995314 + 0.0966948i \(0.969173\pi\)
\(42\) 2170.84 1.23064
\(43\) −752.973 −0.407232 −0.203616 0.979051i \(-0.565269\pi\)
−0.203616 + 0.979051i \(0.565269\pi\)
\(44\) −193.776 −0.100091
\(45\) 2982.94 1.47306
\(46\) 1086.97i 0.513689i
\(47\) −2257.62 −1.02201 −0.511005 0.859578i \(-0.670727\pi\)
−0.511005 + 0.859578i \(0.670727\pi\)
\(48\) 893.309i 0.387721i
\(49\) 1154.98 0.481040
\(50\) 880.392i 0.352157i
\(51\) 2986.13i 1.14807i
\(52\) 1848.85i 0.683747i
\(53\) 1501.20i 0.534427i 0.963637 + 0.267213i \(0.0861028\pi\)
−0.963637 + 0.267213i \(0.913897\pi\)
\(54\) −4402.41 −1.50974
\(55\) 252.956 0.0836219
\(56\) 3525.37i 1.12416i
\(57\) 3992.36 4575.27i 1.22880 1.40821i
\(58\) 1083.48 0.322082
\(59\) 4555.57i 1.30870i 0.756193 + 0.654348i \(0.227057\pi\)
−0.756193 + 0.654348i \(0.772943\pi\)
\(60\) 2811.73i 0.781036i
\(61\) 770.012 0.206937 0.103468 0.994633i \(-0.467006\pi\)
0.103468 + 0.994633i \(0.467006\pi\)
\(62\) 3172.15 0.825221
\(63\) −12041.6 −3.03392
\(64\) −1446.20 −0.353075
\(65\) 2413.51i 0.571244i
\(66\) −623.383 −0.143109
\(67\) 1613.61i 0.359458i 0.983716 + 0.179729i \(0.0575221\pi\)
−0.983716 + 0.179729i \(0.942478\pi\)
\(68\) 2008.91 0.434452
\(69\) 8447.91i 1.77440i
\(70\) 1906.46i 0.389073i
\(71\) 7676.78i 1.52287i −0.648242 0.761435i \(-0.724495\pi\)
0.648242 0.761435i \(-0.275505\pi\)
\(72\) 11938.0i 2.30285i
\(73\) −2330.98 −0.437413 −0.218707 0.975791i \(-0.570184\pi\)
−0.218707 + 0.975791i \(0.570184\pi\)
\(74\) 1300.20 0.237437
\(75\) 6842.41i 1.21643i
\(76\) 3078.00 + 2685.85i 0.532895 + 0.465001i
\(77\) −1021.14 −0.172228
\(78\) 5947.83i 0.977618i
\(79\) 53.3968i 0.00855580i 0.999991 + 0.00427790i \(0.00136170\pi\)
−0.999991 + 0.00427790i \(0.998638\pi\)
\(80\) −784.513 −0.122580
\(81\) 17859.1 2.72200
\(82\) −703.573 −0.104636
\(83\) 9438.26 1.37005 0.685024 0.728520i \(-0.259792\pi\)
0.685024 + 0.728520i \(0.259792\pi\)
\(84\) 11350.5i 1.60863i
\(85\) −2622.44 −0.362968
\(86\) 1629.62i 0.220339i
\(87\) −8420.83 −1.11254
\(88\) 1012.35i 0.130727i
\(89\) 9238.06i 1.16627i −0.812374 0.583137i \(-0.801825\pi\)
0.812374 0.583137i \(-0.198175\pi\)
\(90\) 6455.84i 0.797017i
\(91\) 9742.91i 1.17654i
\(92\) −5683.31 −0.671468
\(93\) −24654.0 −2.85050
\(94\) 4886.06i 0.552972i
\(95\) −4018.05 3506.13i −0.445213 0.388490i
\(96\) 17843.9 1.93619
\(97\) 2562.56i 0.272352i −0.990685 0.136176i \(-0.956519\pi\)
0.990685 0.136176i \(-0.0434813\pi\)
\(98\) 2499.66i 0.260273i
\(99\) 3457.89 0.352810
\(100\) −4603.21 −0.460321
\(101\) 7050.24 0.691133 0.345566 0.938394i \(-0.387687\pi\)
0.345566 + 0.938394i \(0.387687\pi\)
\(102\) 6462.73 0.621178
\(103\) 8275.94i 0.780086i 0.920797 + 0.390043i \(0.127540\pi\)
−0.920797 + 0.390043i \(0.872460\pi\)
\(104\) −9659.05 −0.893034
\(105\) 14817.0i 1.34394i
\(106\) −3248.99 −0.289159
\(107\) 6473.72i 0.565440i 0.959202 + 0.282720i \(0.0912367\pi\)
−0.959202 + 0.282720i \(0.908763\pi\)
\(108\) 23018.4i 1.97346i
\(109\) 5991.18i 0.504266i −0.967693 0.252133i \(-0.918868\pi\)
0.967693 0.252133i \(-0.0811320\pi\)
\(110\) 547.461i 0.0452448i
\(111\) −10105.2 −0.820159
\(112\) 3166.94 0.252467
\(113\) 20337.5i 1.59273i −0.604817 0.796364i \(-0.706754\pi\)
0.604817 0.796364i \(-0.293246\pi\)
\(114\) 9902.04 + 8640.47i 0.761930 + 0.664856i
\(115\) 7419.03 0.560986
\(116\) 5665.09i 0.421009i
\(117\) 32992.4i 2.41014i
\(118\) −9859.42 −0.708088
\(119\) 10586.4 0.747571
\(120\) −14689.5 −1.02010
\(121\) −14347.8 −0.979972
\(122\) 1666.50i 0.111966i
\(123\) 5468.17 0.361436
\(124\) 16585.9i 1.07869i
\(125\) 15241.6 0.975461
\(126\) 26061.1i 1.64154i
\(127\) 15199.0i 0.942342i 0.882042 + 0.471171i \(0.156169\pi\)
−0.882042 + 0.471171i \(0.843831\pi\)
\(128\) 13843.5i 0.844940i
\(129\) 12665.4i 0.761098i
\(130\) 5223.44 0.309079
\(131\) 29704.4 1.73092 0.865461 0.500976i \(-0.167025\pi\)
0.865461 + 0.500976i \(0.167025\pi\)
\(132\) 3259.42i 0.187065i
\(133\) 16220.2 + 14153.6i 0.916963 + 0.800137i
\(134\) −3492.26 −0.194490
\(135\) 30048.4i 1.64875i
\(136\) 10495.2i 0.567433i
\(137\) 7080.63 0.377251 0.188626 0.982049i \(-0.439597\pi\)
0.188626 + 0.982049i \(0.439597\pi\)
\(138\) −18283.4 −0.960062
\(139\) −16337.3 −0.845574 −0.422787 0.906229i \(-0.638948\pi\)
−0.422787 + 0.906229i \(0.638948\pi\)
\(140\) −9968.09 −0.508576
\(141\) 37974.5i 1.91009i
\(142\) 16614.5 0.823969
\(143\) 2797.79i 0.136818i
\(144\) −10724.2 −0.517179
\(145\) 7395.26i 0.351736i
\(146\) 5044.82i 0.236668i
\(147\) 19427.4i 0.899041i
\(148\) 6798.23i 0.310365i
\(149\) −12128.3 −0.546294 −0.273147 0.961972i \(-0.588065\pi\)
−0.273147 + 0.961972i \(0.588065\pi\)
\(150\) −14808.7 −0.658165
\(151\) 26542.8i 1.16411i −0.813151 0.582053i \(-0.802250\pi\)
0.813151 0.582053i \(-0.197750\pi\)
\(152\) −14031.8 + 16080.5i −0.607332 + 0.696007i
\(153\) −35848.6 −1.53140
\(154\) 2210.01i 0.0931863i
\(155\) 21651.4i 0.901201i
\(156\) 31098.8 1.27789
\(157\) −8292.18 −0.336411 −0.168205 0.985752i \(-0.553797\pi\)
−0.168205 + 0.985752i \(0.553797\pi\)
\(158\) −115.564 −0.00462923
\(159\) 25251.1 0.998819
\(160\) 15670.7i 0.612138i
\(161\) −29949.4 −1.15541
\(162\) 38651.5i 1.47278i
\(163\) 16703.6 0.628688 0.314344 0.949309i \(-0.398215\pi\)
0.314344 + 0.949309i \(0.398215\pi\)
\(164\) 3678.70i 0.136775i
\(165\) 4254.87i 0.156285i
\(166\) 20426.8i 0.741283i
\(167\) 20319.3i 0.728576i 0.931286 + 0.364288i \(0.118688\pi\)
−0.931286 + 0.364288i \(0.881312\pi\)
\(168\) 59298.8 2.10101
\(169\) 1866.70 0.0653585
\(170\) 5675.64i 0.196389i
\(171\) −54926.3 47928.5i −1.87840 1.63908i
\(172\) −8520.64 −0.288015
\(173\) 19570.3i 0.653893i 0.945043 + 0.326946i \(0.106020\pi\)
−0.945043 + 0.326946i \(0.893980\pi\)
\(174\) 18224.8i 0.601956i
\(175\) −24257.6 −0.792084
\(176\) −909.424 −0.0293590
\(177\) 76627.4 2.44589
\(178\) 19993.5 0.631028
\(179\) 23233.3i 0.725110i −0.931962 0.362555i \(-0.881904\pi\)
0.931962 0.362555i \(-0.118096\pi\)
\(180\) 33755.0 1.04182
\(181\) 53293.1i 1.62672i −0.581759 0.813361i \(-0.697635\pi\)
0.581759 0.813361i \(-0.302365\pi\)
\(182\) −21086.1 −0.636582
\(183\) 12952.1i 0.386756i
\(184\) 29691.6i 0.876996i
\(185\) 8874.46i 0.259298i
\(186\) 53357.4i 1.54230i
\(187\) −3040.00 −0.0869340
\(188\) −25547.2 −0.722817
\(189\) 121300.i 3.39577i
\(190\) 7588.15 8696.07i 0.210198 0.240888i
\(191\) 5420.24 0.148577 0.0742886 0.997237i \(-0.476331\pi\)
0.0742886 + 0.997237i \(0.476331\pi\)
\(192\) 24325.9i 0.659882i
\(193\) 38918.6i 1.04482i 0.852694 + 0.522411i \(0.174967\pi\)
−0.852694 + 0.522411i \(0.825033\pi\)
\(194\) 5546.04 0.147360
\(195\) −40596.6 −1.06763
\(196\) 13069.7 0.340216
\(197\) −41122.0 −1.05960 −0.529800 0.848122i \(-0.677733\pi\)
−0.529800 + 0.848122i \(0.677733\pi\)
\(198\) 7483.75i 0.190893i
\(199\) 22778.9 0.575211 0.287606 0.957749i \(-0.407141\pi\)
0.287606 + 0.957749i \(0.407141\pi\)
\(200\) 24048.8i 0.601220i
\(201\) 27141.8 0.671811
\(202\) 15258.5i 0.373947i
\(203\) 29853.4i 0.724438i
\(204\) 33791.0i 0.811972i
\(205\) 4802.20i 0.114270i
\(206\) −17911.2 −0.422076
\(207\) 101418. 2.36686
\(208\) 8677.00i 0.200559i
\(209\) −4657.81 4064.38i −0.106632 0.0930469i
\(210\) −32067.7 −0.727159
\(211\) 80470.1i 1.80746i 0.428099 + 0.903732i \(0.359183\pi\)
−0.428099 + 0.903732i \(0.640817\pi\)
\(212\) 16987.6i 0.377973i
\(213\) −129128. −2.84617
\(214\) −14010.8 −0.305939
\(215\) 11122.9 0.240625
\(216\) −120256. −2.57751
\(217\) 87402.8i 1.85612i
\(218\) 12966.4 0.272840
\(219\) 39208.4i 0.817505i
\(220\) 2862.45 0.0591416
\(221\) 29005.2i 0.593870i
\(222\) 21870.2i 0.443758i
\(223\) 8564.16i 0.172217i 0.996286 + 0.0861083i \(0.0274431\pi\)
−0.996286 + 0.0861083i \(0.972557\pi\)
\(224\) 63260.0i 1.26076i
\(225\) 82143.5 1.62259
\(226\) 44015.6 0.861767
\(227\) 37739.2i 0.732387i −0.930539 0.366193i \(-0.880661\pi\)
0.930539 0.366193i \(-0.119339\pi\)
\(228\) 45177.5 51773.8i 0.869066 0.995956i
\(229\) −44028.8 −0.839587 −0.419793 0.907620i \(-0.637897\pi\)
−0.419793 + 0.907620i \(0.637897\pi\)
\(230\) 16056.7i 0.303529i
\(231\) 17176.2i 0.321886i
\(232\) 29596.4 0.549874
\(233\) −5412.82 −0.0997038 −0.0498519 0.998757i \(-0.515875\pi\)
−0.0498519 + 0.998757i \(0.515875\pi\)
\(234\) 71404.0 1.30404
\(235\) 33349.6 0.603885
\(236\) 51550.9i 0.925576i
\(237\) 898.165 0.0159904
\(238\) 22911.6i 0.404483i
\(239\) −24069.2 −0.421372 −0.210686 0.977554i \(-0.567570\pi\)
−0.210686 + 0.977554i \(0.567570\pi\)
\(240\) 13196.0i 0.229097i
\(241\) 24651.4i 0.424432i −0.977223 0.212216i \(-0.931932\pi\)
0.977223 0.212216i \(-0.0680680\pi\)
\(242\) 31052.2i 0.530227i
\(243\) 135634.i 2.29697i
\(244\) 8713.46 0.146356
\(245\) −17061.3 −0.284237
\(246\) 11834.5i 0.195560i
\(247\) −38779.1 + 44441.1i −0.635629 + 0.728435i
\(248\) 86650.5 1.40886
\(249\) 158757.i 2.56056i
\(250\) 32986.6i 0.527786i
\(251\) 91859.2 1.45806 0.729030 0.684482i \(-0.239972\pi\)
0.729030 + 0.684482i \(0.239972\pi\)
\(252\) −136263. −2.14574
\(253\) 8600.31 0.134361
\(254\) −32894.6 −0.509867
\(255\) 44111.1i 0.678371i
\(256\) −53100.0 −0.810242
\(257\) 63651.3i 0.963698i −0.876254 0.481849i \(-0.839965\pi\)
0.876254 0.481849i \(-0.160035\pi\)
\(258\) −27411.2 −0.411803
\(259\) 35824.7i 0.534051i
\(260\) 27311.3i 0.404013i
\(261\) 101093.i 1.48401i
\(262\) 64287.8i 0.936539i
\(263\) 113072. 1.63472 0.817360 0.576127i \(-0.195437\pi\)
0.817360 + 0.576127i \(0.195437\pi\)
\(264\) −17028.3 −0.244323
\(265\) 22175.8i 0.315782i
\(266\) −30632.0 + 35104.5i −0.432925 + 0.496135i
\(267\) −155390. −2.17971
\(268\) 18259.6i 0.254227i
\(269\) 54854.5i 0.758067i −0.925383 0.379034i \(-0.876257\pi\)
0.925383 0.379034i \(-0.123743\pi\)
\(270\) 65032.4 0.892077
\(271\) 61064.9 0.831482 0.415741 0.909483i \(-0.363522\pi\)
0.415741 + 0.909483i \(0.363522\pi\)
\(272\) 9428.17 0.127435
\(273\) 163881. 2.19889
\(274\) 15324.3i 0.204117i
\(275\) 6965.85 0.0921104
\(276\) 95596.6i 1.25494i
\(277\) −31384.3 −0.409027 −0.204514 0.978864i \(-0.565561\pi\)
−0.204514 + 0.978864i \(0.565561\pi\)
\(278\) 35358.1i 0.457509i
\(279\) 295972.i 3.80227i
\(280\) 52076.8i 0.664245i
\(281\) 79910.3i 1.01202i 0.862527 + 0.506011i \(0.168880\pi\)
−0.862527 + 0.506011i \(0.831120\pi\)
\(282\) −82186.4 −1.03348
\(283\) −99909.8 −1.24749 −0.623743 0.781630i \(-0.714389\pi\)
−0.623743 + 0.781630i \(0.714389\pi\)
\(284\) 86870.5i 1.07705i
\(285\) −58975.1 + 67585.9i −0.726071 + 0.832082i
\(286\) 6055.13 0.0740272
\(287\) 19385.7i 0.235351i
\(288\) 214218.i 2.58268i
\(289\) −52004.8 −0.622655
\(290\) −16005.2 −0.190312
\(291\) −43103.8 −0.509014
\(292\) −26377.3 −0.309361
\(293\) 1641.27i 0.0191182i −0.999954 0.00955908i \(-0.996957\pi\)
0.999954 0.00955908i \(-0.00304279\pi\)
\(294\) 42045.8 0.486438
\(295\) 67294.9i 0.773283i
\(296\) 35516.3 0.405364
\(297\) 34832.8i 0.394890i
\(298\) 26248.7i 0.295580i
\(299\) 82057.3i 0.917857i
\(300\) 77428.8i 0.860320i
\(301\) −44901.3 −0.495594
\(302\) 57445.3 0.629855
\(303\) 118589.i 1.29170i
\(304\) 14445.6 + 12605.2i 0.156311 + 0.136396i
\(305\) −11374.6 −0.122275
\(306\) 77585.5i 0.828586i
\(307\) 72395.1i 0.768126i 0.923307 + 0.384063i \(0.125476\pi\)
−0.923307 + 0.384063i \(0.874524\pi\)
\(308\) −11555.2 −0.121808
\(309\) 139206. 1.45795
\(310\) −46859.0 −0.487607
\(311\) −124301. −1.28515 −0.642573 0.766225i \(-0.722133\pi\)
−0.642573 + 0.766225i \(0.722133\pi\)
\(312\) 162471.i 1.66904i
\(313\) 87747.3 0.895664 0.447832 0.894118i \(-0.352196\pi\)
0.447832 + 0.894118i \(0.352196\pi\)
\(314\) 17946.4i 0.182019i
\(315\) 177879. 1.79268
\(316\) 604.238i 0.00605110i
\(317\) 40640.3i 0.404426i −0.979342 0.202213i \(-0.935187\pi\)
0.979342 0.202213i \(-0.0648133\pi\)
\(318\) 54649.9i 0.540424i
\(319\) 8572.74i 0.0842439i
\(320\) 21363.2 0.208625
\(321\) 108892. 1.05678
\(322\) 64818.0i 0.625150i
\(323\) 48288.4 + 42136.2i 0.462847 + 0.403878i
\(324\) 202093. 1.92514
\(325\) 66462.6i 0.629232i
\(326\) 36150.9i 0.340160i
\(327\) −100775. −0.942450
\(328\) −19218.8 −0.178640
\(329\) −134626. −1.24377
\(330\) 9208.62 0.0845603
\(331\) 146431.i 1.33652i −0.743927 0.668260i \(-0.767039\pi\)
0.743927 0.668260i \(-0.232961\pi\)
\(332\) 106803. 0.968967
\(333\) 121313.i 1.09401i
\(334\) −43976.0 −0.394206
\(335\) 23836.2i 0.212397i
\(336\) 53269.8i 0.471848i
\(337\) 191079.i 1.68249i 0.540654 + 0.841245i \(0.318177\pi\)
−0.540654 + 0.841245i \(0.681823\pi\)
\(338\) 4040.02i 0.0353631i
\(339\) −342089. −2.97674
\(340\) −29675.6 −0.256709
\(341\) 25098.7i 0.215845i
\(342\) 103729. 118875.i 0.886849 1.01633i
\(343\) −74302.9 −0.631564
\(344\) 44514.8i 0.376173i
\(345\) 124792.i 1.04846i
\(346\) −42355.2 −0.353797
\(347\) 75833.7 0.629801 0.314900 0.949125i \(-0.398029\pi\)
0.314900 + 0.949125i \(0.398029\pi\)
\(348\) −95290.2 −0.786846
\(349\) 167272. 1.37332 0.686659 0.726980i \(-0.259077\pi\)
0.686659 + 0.726980i \(0.259077\pi\)
\(350\) 52499.6i 0.428568i
\(351\) −332347. −2.69760
\(352\) 18165.9i 0.146612i
\(353\) 71568.7 0.574346 0.287173 0.957879i \(-0.407285\pi\)
0.287173 + 0.957879i \(0.407285\pi\)
\(354\) 165841.i 1.32338i
\(355\) 113401.i 0.899833i
\(356\) 104538.i 0.824848i
\(357\) 178069.i 1.39718i
\(358\) 50282.7 0.392331
\(359\) −121681. −0.944132 −0.472066 0.881563i \(-0.656492\pi\)
−0.472066 + 0.881563i \(0.656492\pi\)
\(360\) 176348.i 1.36071i
\(361\) 17651.5 + 129120.i 0.135446 + 0.990785i
\(362\) 115340. 0.880160
\(363\) 241338.i 1.83152i
\(364\) 110251.i 0.832107i
\(365\) 34433.2 0.258459
\(366\) 28031.5 0.209259
\(367\) 233608. 1.73443 0.867214 0.497935i \(-0.165908\pi\)
0.867214 + 0.497935i \(0.165908\pi\)
\(368\) −26672.8 −0.196958
\(369\) 65645.7i 0.482118i
\(370\) −19206.6 −0.140297
\(371\) 89519.8i 0.650387i
\(372\) −278984. −2.01602
\(373\) 42576.8i 0.306024i 0.988224 + 0.153012i \(0.0488973\pi\)
−0.988224 + 0.153012i \(0.951103\pi\)
\(374\) 6579.32i 0.0470368i
\(375\) 256372.i 1.82309i
\(376\) 133468.i 0.944062i
\(377\) 81794.3 0.575493
\(378\) −262525. −1.83733
\(379\) 99133.0i 0.690144i 0.938576 + 0.345072i \(0.112146\pi\)
−0.938576 + 0.345072i \(0.887854\pi\)
\(380\) −45468.2 39675.3i −0.314877 0.274760i
\(381\) 255657. 1.76119
\(382\) 11730.8i 0.0803897i
\(383\) 198667.i 1.35434i 0.735826 + 0.677171i \(0.236794\pi\)
−0.735826 + 0.677171i \(0.763206\pi\)
\(384\) 232856. 1.57915
\(385\) 15084.3 0.101766
\(386\) −84229.6 −0.565315
\(387\) 152049. 1.01523
\(388\) 28998.0i 0.192621i
\(389\) −252659. −1.66969 −0.834845 0.550485i \(-0.814443\pi\)
−0.834845 + 0.550485i \(0.814443\pi\)
\(390\) 87861.3i 0.577655i
\(391\) −89161.0 −0.583205
\(392\) 68280.8i 0.444351i
\(393\) 499645.i 3.23501i
\(394\) 88998.6i 0.573311i
\(395\) 788.777i 0.00505545i
\(396\) 39129.5 0.249525
\(397\) 37586.2 0.238478 0.119239 0.992866i \(-0.461955\pi\)
0.119239 + 0.992866i \(0.461955\pi\)
\(398\) 49299.4i 0.311226i
\(399\) 238072. 272832.i 1.49542 1.71376i
\(400\) −21603.7 −0.135023
\(401\) 175688.i 1.09258i 0.837596 + 0.546290i \(0.183960\pi\)
−0.837596 + 0.546290i \(0.816040\pi\)
\(402\) 58741.8i 0.363492i
\(403\) 239472. 1.47450
\(404\) 79780.6 0.488804
\(405\) −263814. −1.60838
\(406\) 64610.3 0.391967
\(407\) 10287.5i 0.0621041i
\(408\) 176536. 1.06051
\(409\) 276594.i 1.65347i −0.562591 0.826735i \(-0.690195\pi\)
0.562591 0.826735i \(-0.309805\pi\)
\(410\) 10393.2 0.0618274
\(411\) 119100.i 0.705066i
\(412\) 93650.5i 0.551716i
\(413\) 271658.i 1.59266i
\(414\) 219493.i 1.28062i
\(415\) −139422. −0.809534
\(416\) −173324. −1.00155
\(417\) 274804.i 1.58034i
\(418\) 8796.35 10080.7i 0.0503443 0.0576949i
\(419\) −91901.2 −0.523471 −0.261736 0.965140i \(-0.584295\pi\)
−0.261736 + 0.965140i \(0.584295\pi\)
\(420\) 167669.i 0.950505i
\(421\) 154865.i 0.873753i −0.899521 0.436877i \(-0.856085\pi\)
0.899521 0.436877i \(-0.143915\pi\)
\(422\) −174158. −0.977952
\(423\) 455886. 2.54786
\(424\) −88749.4 −0.493666
\(425\) −72216.3 −0.399813
\(426\) 279466.i 1.53996i
\(427\) 45917.4 0.251838
\(428\) 73256.7i 0.399908i
\(429\) −47060.5 −0.255706
\(430\) 24072.8i 0.130194i
\(431\) 110424.i 0.594443i −0.954809 0.297221i \(-0.903940\pi\)
0.954809 0.297221i \(-0.0960600\pi\)
\(432\) 108030.i 0.578863i
\(433\) 34749.0i 0.185339i −0.995697 0.0926694i \(-0.970460\pi\)
0.995697 0.0926694i \(-0.0295400\pi\)
\(434\) 189162. 1.00428
\(435\) 124393. 0.657379
\(436\) 67796.3i 0.356642i
\(437\) −136610. 119205.i −0.715353 0.624214i
\(438\) −84856.9 −0.442322
\(439\) 82718.3i 0.429213i −0.976701 0.214606i \(-0.931153\pi\)
0.976701 0.214606i \(-0.0688468\pi\)
\(440\) 14954.5i 0.0772441i
\(441\) −233227. −1.19923
\(442\) −62774.7 −0.321322
\(443\) −214703. −1.09404 −0.547018 0.837121i \(-0.684237\pi\)
−0.547018 + 0.837121i \(0.684237\pi\)
\(444\) −114350. −0.580058
\(445\) 136465.i 0.689128i
\(446\) −18535.0 −0.0931801
\(447\) 204005.i 1.02100i
\(448\) −86239.6 −0.429686
\(449\) 200750.i 0.995781i 0.867240 + 0.497891i \(0.165892\pi\)
−0.867240 + 0.497891i \(0.834108\pi\)
\(450\) 177779.i 0.877923i
\(451\) 5566.82i 0.0273687i
\(452\) 230140.i 1.12646i
\(453\) −446465. −2.17566
\(454\) 81677.1 0.396268
\(455\) 143922.i 0.695193i
\(456\) 270484. + 236023.i 1.30081 + 1.13508i
\(457\) 130986. 0.627183 0.313591 0.949558i \(-0.398468\pi\)
0.313591 + 0.949558i \(0.398468\pi\)
\(458\) 95289.4i 0.454270i
\(459\) 361118.i 1.71405i
\(460\) 83953.8 0.396757
\(461\) 35097.5 0.165149 0.0825743 0.996585i \(-0.473686\pi\)
0.0825743 + 0.996585i \(0.473686\pi\)
\(462\) −37173.6 −0.174161
\(463\) −28532.2 −0.133099 −0.0665493 0.997783i \(-0.521199\pi\)
−0.0665493 + 0.997783i \(0.521199\pi\)
\(464\) 26587.3i 0.123492i
\(465\) 364188. 1.68430
\(466\) 11714.7i 0.0539461i
\(467\) 402160. 1.84402 0.922008 0.387170i \(-0.126547\pi\)
0.922008 + 0.387170i \(0.126547\pi\)
\(468\) 373343.i 1.70457i
\(469\) 96222.7i 0.437453i
\(470\) 72176.9i 0.326740i
\(471\) 139479.i 0.628736i
\(472\) −269320. −1.20888
\(473\) 12893.9 0.0576319
\(474\) 1943.86i 0.00865182i
\(475\) −110648. 96550.9i −0.490407 0.427926i
\(476\) 119795. 0.528720
\(477\) 303141.i 1.33232i
\(478\) 52091.9i 0.227989i
\(479\) −272020. −1.18558 −0.592788 0.805359i \(-0.701973\pi\)
−0.592788 + 0.805359i \(0.701973\pi\)
\(480\) −263591. −1.14406
\(481\) 98154.9 0.424250
\(482\) 53351.9 0.229644
\(483\) 503766.i 2.15941i
\(484\) −162359. −0.693085
\(485\) 37854.2i 0.160928i
\(486\) 293546. 1.24281
\(487\) 412092.i 1.73754i −0.495212 0.868772i \(-0.664910\pi\)
0.495212 0.868772i \(-0.335090\pi\)
\(488\) 45522.2i 0.191154i
\(489\) 280965.i 1.17499i
\(490\) 36925.0i 0.153790i
\(491\) 343889. 1.42645 0.713223 0.700938i \(-0.247235\pi\)
0.713223 + 0.700938i \(0.247235\pi\)
\(492\) 61877.9 0.255626
\(493\) 88875.3i 0.365668i
\(494\) −96181.8 83927.8i −0.394130 0.343916i
\(495\) −51080.0 −0.208468
\(496\) 77840.6i 0.316405i
\(497\) 457782.i 1.85330i
\(498\) 343591. 1.38542
\(499\) −166687. −0.669425 −0.334712 0.942320i \(-0.608639\pi\)
−0.334712 + 0.942320i \(0.608639\pi\)
\(500\) 172474. 0.689895
\(501\) 341782. 1.36168
\(502\) 198807.i 0.788903i
\(503\) −341395. −1.34934 −0.674669 0.738121i \(-0.735714\pi\)
−0.674669 + 0.738121i \(0.735714\pi\)
\(504\) 711885.i 2.80252i
\(505\) −104146. −0.408377
\(506\) 18613.3i 0.0726978i
\(507\) 31399.1i 0.122152i
\(508\) 171992.i 0.666472i
\(509\) 390009.i 1.50535i 0.658390 + 0.752677i \(0.271238\pi\)
−0.658390 + 0.752677i \(0.728762\pi\)
\(510\) −95467.5 −0.367042
\(511\) −139001. −0.532323
\(512\) 106574.i 0.406548i
\(513\) −482804. + 553297.i −1.83458 + 2.10244i
\(514\) 137758. 0.521422
\(515\) 122252.i 0.460938i
\(516\) 143322.i 0.538287i
\(517\) 38659.6 0.144636
\(518\) 77533.7 0.288955
\(519\) 329185. 1.22209
\(520\) 142684. 0.527676
\(521\) 385501.i 1.42020i 0.704099 + 0.710102i \(0.251351\pi\)
−0.704099 + 0.710102i \(0.748649\pi\)
\(522\) −218790. −0.802946
\(523\) 480249.i 1.75575i 0.478889 + 0.877875i \(0.341039\pi\)
−0.478889 + 0.877875i \(0.658961\pi\)
\(524\) 336135. 1.22420
\(525\) 408027.i 1.48037i
\(526\) 244716.i 0.884487i
\(527\) 260203.i 0.936896i
\(528\) 15297.0i 0.0548706i
\(529\) −27599.8 −0.0986267
\(530\) 47994.1 0.170858
\(531\) 919916.i 3.26257i
\(532\) 183547. + 160163.i 0.648522 + 0.565897i
\(533\) −53114.2 −0.186963
\(534\) 336302.i 1.17936i
\(535\) 95629.9i 0.334107i
\(536\) −95394.5 −0.332043
\(537\) −390797. −1.35520
\(538\) 118719. 0.410162
\(539\) −19777.9 −0.0680772
\(540\) 340028.i 1.16608i
\(541\) 398980. 1.36319 0.681595 0.731730i \(-0.261287\pi\)
0.681595 + 0.731730i \(0.261287\pi\)
\(542\) 132160.i 0.449885i
\(543\) −896420. −3.04027
\(544\) 188329.i 0.636383i
\(545\) 88501.8i 0.297961i
\(546\) 354681.i 1.18974i
\(547\) 4309.05i 0.0144015i −0.999974 0.00720073i \(-0.997708\pi\)
0.999974 0.00720073i \(-0.00229208\pi\)
\(548\) 80124.5 0.266811
\(549\) −155490. −0.515891
\(550\) 15075.9i 0.0498376i
\(551\) 118823. 136172.i 0.391380 0.448524i
\(552\) −499430. −1.63907
\(553\) 3184.16i 0.0104122i
\(554\) 67923.5i 0.221310i
\(555\) 149274. 0.484615
\(556\) −184873. −0.598033
\(557\) −208935. −0.673444 −0.336722 0.941604i \(-0.609318\pi\)
−0.336722 + 0.941604i \(0.609318\pi\)
\(558\) −640559. −2.05727
\(559\) 123024.i 0.393699i
\(560\) −46782.1 −0.149177
\(561\) 51134.5i 0.162476i
\(562\) −172946. −0.547568
\(563\) 552442.i 1.74289i −0.490493 0.871445i \(-0.663183\pi\)
0.490493 0.871445i \(-0.336817\pi\)
\(564\) 429720.i 1.35091i
\(565\) 300426.i 0.941111i
\(566\) 216230.i 0.674969i
\(567\) 1.06497e6 3.31262
\(568\) 453842. 1.40672
\(569\) 376174.i 1.16189i 0.813943 + 0.580944i \(0.197317\pi\)
−0.813943 + 0.580944i \(0.802683\pi\)
\(570\) −146273. 127637.i −0.450209 0.392850i
\(571\) −390679. −1.19825 −0.599125 0.800656i \(-0.704485\pi\)
−0.599125 + 0.800656i \(0.704485\pi\)
\(572\) 31659.8i 0.0967646i
\(573\) 91171.7i 0.277684i
\(574\) −41955.5 −0.127340
\(575\) 204304. 0.617932
\(576\) 292034. 0.880213
\(577\) −374980. −1.12631 −0.563153 0.826352i \(-0.690412\pi\)
−0.563153 + 0.826352i \(0.690412\pi\)
\(578\) 112552.i 0.336896i
\(579\) 654633. 1.95272
\(580\) 83684.8i 0.248766i
\(581\) 562823. 1.66732
\(582\) 93287.6i 0.275409i
\(583\) 25706.7i 0.0756326i
\(584\) 137804.i 0.404052i
\(585\) 487364.i 1.42411i
\(586\) 3552.13 0.0103441
\(587\) 269098. 0.780970 0.390485 0.920609i \(-0.372307\pi\)
0.390485 + 0.920609i \(0.372307\pi\)
\(588\) 219840.i 0.635847i
\(589\) 347884. 398677.i 1.00277 1.14919i
\(590\) 145643. 0.418395
\(591\) 691697.i 1.98035i
\(592\) 31905.3i 0.0910374i
\(593\) 56198.5 0.159814 0.0799071 0.996802i \(-0.474538\pi\)
0.0799071 + 0.996802i \(0.474538\pi\)
\(594\) 75387.0 0.213660
\(595\) −156382. −0.441725
\(596\) −137244. −0.386367
\(597\) 383155.i 1.07504i
\(598\) 177593. 0.496619
\(599\) 149473.i 0.416592i −0.978066 0.208296i \(-0.933208\pi\)
0.978066 0.208296i \(-0.0667917\pi\)
\(600\) −404515. −1.12365
\(601\) 111386.i 0.308377i 0.988041 + 0.154188i \(0.0492763\pi\)
−0.988041 + 0.154188i \(0.950724\pi\)
\(602\) 97177.8i 0.268148i
\(603\) 325839.i 0.896125i
\(604\) 300358.i 0.823314i
\(605\) 211945. 0.579046
\(606\) 256657. 0.698889
\(607\) 496593.i 1.34779i 0.738826 + 0.673896i \(0.235381\pi\)
−0.738826 + 0.673896i \(0.764619\pi\)
\(608\) −251790. + 288553.i −0.681131 + 0.780581i
\(609\) −502151. −1.35394
\(610\) 24617.6i 0.0661585i
\(611\) 368859.i 0.988047i
\(612\) −405663. −1.08309
\(613\) 21019.3 0.0559369 0.0279684 0.999609i \(-0.491096\pi\)
0.0279684 + 0.999609i \(0.491096\pi\)
\(614\) −156681. −0.415605
\(615\) −80775.8 −0.213566
\(616\) 60368.6i 0.159092i
\(617\) −415170. −1.09058 −0.545288 0.838249i \(-0.683580\pi\)
−0.545288 + 0.838249i \(0.683580\pi\)
\(618\) 301277.i 0.788841i
\(619\) −40823.3 −0.106543 −0.0532717 0.998580i \(-0.516965\pi\)
−0.0532717 + 0.998580i \(0.516965\pi\)
\(620\) 245007.i 0.637375i
\(621\) 1.02162e6i 2.64915i
\(622\) 269018.i 0.695345i
\(623\) 550884.i 1.41933i
\(624\) 145952. 0.374836
\(625\) 29093.9 0.0744804
\(626\) 189907.i 0.484611i
\(627\) −68365.2 + 78347.0i −0.173900 + 0.199291i
\(628\) −93834.4 −0.237926
\(629\) 106652.i 0.269568i
\(630\) 384975.i 0.969954i
\(631\) −515651. −1.29508 −0.647541 0.762030i \(-0.724203\pi\)
−0.647541 + 0.762030i \(0.724203\pi\)
\(632\) −3156.75 −0.00790326
\(633\) 1.35355e6 3.37807
\(634\) 87956.0 0.218820
\(635\) 224520.i 0.556811i
\(636\) 285742. 0.706415
\(637\) 188705.i 0.465054i
\(638\) −18553.6 −0.0455813
\(639\) 1.55019e6i 3.79649i
\(640\) 204496.i 0.499258i
\(641\) 100947.i 0.245685i −0.992426 0.122843i \(-0.960799\pi\)
0.992426 0.122843i \(-0.0392010\pi\)
\(642\) 235670.i 0.571786i
\(643\) −256498. −0.620386 −0.310193 0.950674i \(-0.600394\pi\)
−0.310193 + 0.950674i \(0.600394\pi\)
\(644\) −338907. −0.817163
\(645\) 187094.i 0.449718i
\(646\) −91193.4 + 104508.i −0.218524 + 0.250430i
\(647\) 752626. 1.79792 0.898961 0.438029i \(-0.144323\pi\)
0.898961 + 0.438029i \(0.144323\pi\)
\(648\) 1.05581e6i 2.51440i
\(649\) 78009.8i 0.185208i
\(650\) 143842. 0.340454
\(651\) −1.47017e6 −3.46900
\(652\) 189018. 0.444640
\(653\) −703535. −1.64991 −0.824953 0.565201i \(-0.808798\pi\)
−0.824953 + 0.565201i \(0.808798\pi\)
\(654\) 218103.i 0.509925i
\(655\) −438793. −1.02277
\(656\) 17264.8i 0.0401194i
\(657\) 470699. 1.09047
\(658\) 291366.i 0.672956i
\(659\) 813907.i 1.87415i 0.349130 + 0.937074i \(0.386477\pi\)
−0.349130 + 0.937074i \(0.613523\pi\)
\(660\) 48148.2i 0.110533i
\(661\) 426236.i 0.975545i −0.872971 0.487772i \(-0.837810\pi\)
0.872971 0.487772i \(-0.162190\pi\)
\(662\) 316913. 0.723143
\(663\) 487885. 1.10992
\(664\) 557979.i 1.26556i
\(665\) −239604. 209077.i −0.541815 0.472785i
\(666\) −262553. −0.591927
\(667\) 251433.i 0.565158i
\(668\) 229933.i 0.515286i
\(669\) 144054. 0.321865
\(670\) 51587.6 0.114920
\(671\) −13185.7 −0.0292859
\(672\) 1.06407e6 2.35631
\(673\) 513214.i 1.13310i −0.824027 0.566550i \(-0.808278\pi\)
0.824027 0.566550i \(-0.191722\pi\)
\(674\) −413543. −0.910334
\(675\) 827467.i 1.81611i
\(676\) 21123.6 0.0462248
\(677\) 139181.i 0.303671i −0.988406 0.151836i \(-0.951482\pi\)
0.988406 0.151836i \(-0.0485184\pi\)
\(678\) 740368.i 1.61060i
\(679\) 152811.i 0.331447i
\(680\) 155036.i 0.335285i
\(681\) −634795. −1.36880
\(682\) −54320.0 −0.116786
\(683\) 340635.i 0.730211i 0.930966 + 0.365105i \(0.118967\pi\)
−0.930966 + 0.365105i \(0.881033\pi\)
\(684\) −621547. 542359.i −1.32850 1.15924i
\(685\) −104595. −0.222910
\(686\) 160810.i 0.341716i
\(687\) 740590.i 1.56915i
\(688\) −39988.9 −0.0844817
\(689\) −245273. −0.516667
\(690\) 270083. 0.567281
\(691\) 665671. 1.39413 0.697065 0.717008i \(-0.254489\pi\)
0.697065 + 0.717008i \(0.254489\pi\)
\(692\) 221458.i 0.462466i
\(693\) 206201. 0.429363
\(694\) 164123.i 0.340762i
\(695\) 241335. 0.499633
\(696\) 497829.i 1.02769i
\(697\) 57712.3i 0.118796i
\(698\) 362018.i 0.743052i
\(699\) 91046.8i 0.186342i
\(700\) −274499. −0.560202
\(701\) −287944. −0.585965 −0.292982 0.956118i \(-0.594648\pi\)
−0.292982 + 0.956118i \(0.594648\pi\)
\(702\) 719283.i 1.45957i
\(703\) 142591. 163410.i 0.288523 0.330649i
\(704\) 24764.7 0.0499676
\(705\) 560959.i 1.12863i
\(706\) 154893.i 0.310758i
\(707\) 420420. 0.841095
\(708\) 867116. 1.72986
\(709\) 461125. 0.917331 0.458665 0.888609i \(-0.348328\pi\)
0.458665 + 0.888609i \(0.348328\pi\)
\(710\) −245430. −0.486867
\(711\) 10782.5i 0.0213295i
\(712\) 546143. 1.07732
\(713\) 736129.i 1.44802i
\(714\) 385386. 0.755961
\(715\) 41329.0i 0.0808430i
\(716\) 262908.i 0.512834i
\(717\) 404858.i 0.787525i
\(718\) 263348.i 0.510835i
\(719\) 680993. 1.31730 0.658650 0.752449i \(-0.271128\pi\)
0.658650 + 0.752449i \(0.271128\pi\)
\(720\) 158418. 0.305591
\(721\) 493511.i 0.949349i
\(722\) −279449. + 38202.3i −0.536077 + 0.0732849i
\(723\) −414651. −0.793243
\(724\) 603065.i 1.15050i
\(725\) 203649.i 0.387441i
\(726\) −522316. −0.990970
\(727\) 11537.8 0.0218301 0.0109150 0.999940i \(-0.496526\pi\)
0.0109150 + 0.999940i \(0.496526\pi\)
\(728\) −575989. −1.08680
\(729\) −834857. −1.57093
\(730\) 74522.1i 0.139843i
\(731\) −133674. −0.250156
\(732\) 146566.i 0.273533i
\(733\) 27216.9 0.0506560 0.0253280 0.999679i \(-0.491937\pi\)
0.0253280 + 0.999679i \(0.491937\pi\)
\(734\) 505588.i 0.938436i
\(735\) 286981.i 0.531226i
\(736\) 532792.i 0.983563i
\(737\) 27631.5i 0.0508708i
\(738\) 142074. 0.260857
\(739\) 488729. 0.894909 0.447455 0.894307i \(-0.352331\pi\)
0.447455 + 0.894307i \(0.352331\pi\)
\(740\) 100423.i 0.183388i
\(741\) 747525. + 652287.i 1.36141 + 1.18796i
\(742\) −193744. −0.351900
\(743\) 4601.59i 0.00833548i −0.999991 0.00416774i \(-0.998673\pi\)
0.999991 0.00416774i \(-0.00132664\pi\)
\(744\) 1.45751e6i 2.63309i
\(745\) 179159. 0.322794
\(746\) −92146.9 −0.165578
\(747\) −1.90589e6 −3.41552
\(748\) −34400.6 −0.0614841
\(749\) 386041.i 0.688129i
\(750\) 554854. 0.986408
\(751\) 62141.6i 0.110180i −0.998481 0.0550900i \(-0.982455\pi\)
0.998481 0.0550900i \(-0.0175446\pi\)
\(752\) −119898. −0.212019
\(753\) 1.54513e6i 2.72505i
\(754\) 177024.i 0.311378i
\(755\) 392090.i 0.687847i
\(756\) 1.37264e6i 2.40166i
\(757\) −295159. −0.515067 −0.257533 0.966269i \(-0.582910\pi\)
−0.257533 + 0.966269i \(0.582910\pi\)
\(758\) −214549. −0.373412
\(759\) 144662.i 0.251114i
\(760\) 207278. 237542.i 0.358861 0.411257i
\(761\) 327534. 0.565571 0.282785 0.959183i \(-0.408742\pi\)
0.282785 + 0.959183i \(0.408742\pi\)
\(762\) 553306.i 0.952918i
\(763\) 357266.i 0.613681i
\(764\) 61335.5 0.105081
\(765\) 529556. 0.904875
\(766\) −429966. −0.732785
\(767\) −744307. −1.26521
\(768\) 893173.i 1.51430i
\(769\) 805330. 1.36182 0.680912 0.732365i \(-0.261583\pi\)
0.680912 + 0.732365i \(0.261583\pi\)
\(770\) 32646.2i 0.0550619i
\(771\) −1.07065e6 −1.80111
\(772\) 440403.i 0.738950i
\(773\) 585854.i 0.980462i 0.871593 + 0.490231i \(0.163088\pi\)
−0.871593 + 0.490231i \(0.836912\pi\)
\(774\) 329073.i 0.549301i
\(775\) 596230.i 0.992683i
\(776\) 151496. 0.251580
\(777\) −602592. −0.998117
\(778\) 546819.i 0.903409i
\(779\) −77159.5 + 88425.3i −0.127149 + 0.145714i
\(780\) −459391. −0.755081
\(781\) 131457.i 0.215518i
\(782\) 192967.i 0.315551i
\(783\) 1.01835e6 1.66101
\(784\) 61338.6 0.0997934
\(785\) 122492. 0.198778
\(786\) 1.08136e6 1.75035
\(787\) 762631.i 1.23130i 0.788018 + 0.615652i \(0.211107\pi\)
−0.788018 + 0.615652i \(0.788893\pi\)
\(788\) −465337. −0.749403
\(789\) 1.90194e6i 3.05522i
\(790\) 1707.11 0.00273532
\(791\) 1.21277e6i 1.93832i
\(792\) 204426.i 0.325901i
\(793\) 125808.i 0.200060i
\(794\) 81346.2i 0.129032i
\(795\) −373010. −0.590182
\(796\) 257767. 0.406818
\(797\) 739146.i 1.16363i −0.813322 0.581813i \(-0.802343\pi\)
0.813322 0.581813i \(-0.197657\pi\)
\(798\) 590479. + 515249.i 0.927253 + 0.809117i
\(799\) −400791. −0.627804
\(800\) 431537.i 0.674276i
\(801\) 1.86546e6i 2.90751i
\(802\) −380233. −0.591155
\(803\) 39915.7 0.0619031
\(804\) 307137. 0.475138
\(805\) 442412. 0.682708
\(806\) 518278.i 0.797798i
\(807\) −922685. −1.41679
\(808\) 416802.i 0.638421i
\(809\) −283400. −0.433015 −0.216508 0.976281i \(-0.569467\pi\)
−0.216508 + 0.976281i \(0.569467\pi\)
\(810\) 570961.i 0.870234i
\(811\) 14690.1i 0.0223348i −0.999938 0.0111674i \(-0.996445\pi\)
0.999938 0.0111674i \(-0.00355477\pi\)
\(812\) 337821.i 0.512359i
\(813\) 1.02715e6i 1.55400i
\(814\) −22264.7 −0.0336022
\(815\) −246746. −0.371479
\(816\) 158587.i 0.238171i
\(817\) −204812. 178718.i −0.306839 0.267746i
\(818\) 598620. 0.894632
\(819\) 1.96741e6i 2.93310i
\(820\) 54341.7i 0.0808176i
\(821\) 668244. 0.991399 0.495699 0.868494i \(-0.334912\pi\)
0.495699 + 0.868494i \(0.334912\pi\)
\(822\) 257763. 0.381485
\(823\) 6475.45 0.00956028 0.00478014 0.999989i \(-0.498478\pi\)
0.00478014 + 0.999989i \(0.498478\pi\)
\(824\) −489263. −0.720590
\(825\) 117170.i 0.172150i
\(826\) −587937. −0.861729
\(827\) 55503.2i 0.0811534i 0.999176 + 0.0405767i \(0.0129195\pi\)
−0.999176 + 0.0405767i \(0.987080\pi\)
\(828\) 1.14764e6 1.67396
\(829\) 475443.i 0.691813i 0.938269 + 0.345907i \(0.112429\pi\)
−0.938269 + 0.345907i \(0.887571\pi\)
\(830\) 301745.i 0.438009i
\(831\) 527902.i 0.764453i
\(832\) 236285.i 0.341342i
\(833\) 205041. 0.295495
\(834\) −594745. −0.855064
\(835\) 300156.i 0.430501i
\(836\) −52707.8 45992.5i −0.0754158 0.0658074i
\(837\) 2.98145e6 4.25576
\(838\) 198897.i 0.283231i
\(839\) 389106.i 0.552770i −0.961047 0.276385i \(-0.910864\pi\)
0.961047 0.276385i \(-0.0891364\pi\)
\(840\) −875962. −1.24144
\(841\) 456654. 0.645647
\(842\) 335167. 0.472756
\(843\) 1.34414e6 1.89142
\(844\) 910600.i 1.27833i
\(845\) −27575.0 −0.0386190
\(846\) 986653.i 1.37855i
\(847\) −855586. −1.19261
\(848\) 79726.1i 0.110869i
\(849\) 1.68054e6i 2.33149i
\(850\) 156294.i 0.216324i
\(851\) 301725.i 0.416631i
\(852\) −1.46121e6 −2.01296
\(853\) −1.17337e6 −1.61264 −0.806322 0.591476i \(-0.798545\pi\)
−0.806322 + 0.591476i \(0.798545\pi\)
\(854\) 99376.9i 0.136260i
\(855\) 811372. + 707999.i 1.10991 + 0.968502i
\(856\) −382719. −0.522314
\(857\) 170943.i 0.232750i −0.993205 0.116375i \(-0.962873\pi\)
0.993205 0.116375i \(-0.0371274\pi\)
\(858\) 101851.i 0.138353i
\(859\) 494381. 0.670002 0.335001 0.942218i \(-0.391263\pi\)
0.335001 + 0.942218i \(0.391263\pi\)
\(860\) 125867. 0.170182
\(861\) 326078. 0.439861
\(862\) 238986. 0.321631
\(863\) 1.38223e6i 1.85591i −0.372691 0.927956i \(-0.621565\pi\)
0.372691 0.927956i \(-0.378435\pi\)
\(864\) −2.15790e6 −2.89071
\(865\) 289093.i 0.386372i
\(866\) 75205.6 0.100280
\(867\) 874751.i 1.16371i
\(868\) 989050.i 1.31274i
\(869\) 914.368i 0.00121083i
\(870\) 269217.i 0.355684i
\(871\) −263637. −0.347513
\(872\) 354191. 0.465806
\(873\) 517464.i 0.678971i
\(874\) 257991. 295659.i 0.337739 0.387052i
\(875\) 908886. 1.18712
\(876\) 443682.i 0.578181i
\(877\) 1.30994e6i 1.70315i −0.524235 0.851574i \(-0.675649\pi\)
0.524235 0.851574i \(-0.324351\pi\)
\(878\) 179023. 0.232231
\(879\) −27607.2 −0.0357309
\(880\) 13434.0 0.0173476
\(881\) −1.16822e6 −1.50513 −0.752565 0.658517i \(-0.771184\pi\)
−0.752565 + 0.658517i \(0.771184\pi\)
\(882\) 504762.i 0.648858i
\(883\) 761544. 0.976728 0.488364 0.872640i \(-0.337594\pi\)
0.488364 + 0.872640i \(0.337594\pi\)
\(884\) 328223.i 0.420015i
\(885\) −1.13194e6 −1.44523
\(886\) 464672.i 0.591942i
\(887\) 347725.i 0.441966i −0.975278 0.220983i \(-0.929074\pi\)
0.975278 0.220983i \(-0.0709265\pi\)
\(888\) 597406.i 0.757606i
\(889\) 906349.i 1.14681i
\(890\) −295344. −0.372862
\(891\) −305819. −0.385220
\(892\) 96912.1i 0.121800i
\(893\) −614082. 535845.i −0.770058 0.671949i
\(894\) −441518. −0.552425
\(895\) 343202.i 0.428453i
\(896\) 825516.i 1.02827i
\(897\) −1.38025e6 −1.71543
\(898\) −434475. −0.538781
\(899\) −733769. −0.907905
\(900\) 929537. 1.14758
\(901\) 266506.i 0.328290i
\(902\) 12048.0 0.0148082
\(903\) 755266.i 0.926242i
\(904\) 1.20233e6 1.47125
\(905\) 787245.i 0.961198i
\(906\) 966264.i 1.17717i
\(907\) 687599.i 0.835835i 0.908485 + 0.417917i \(0.137240\pi\)
−0.908485 + 0.417917i \(0.862760\pi\)
\(908\) 427057.i 0.517981i
\(909\) −1.42367e6 −1.72299
\(910\) 311484. 0.376143
\(911\) 449372.i 0.541464i 0.962655 + 0.270732i \(0.0872657\pi\)
−0.962655 + 0.270732i \(0.912734\pi\)
\(912\) 212026. 242984.i 0.254918 0.292138i
\(913\) −161621. −0.193891
\(914\) 283488.i 0.339346i
\(915\) 191328.i 0.228526i
\(916\) −498230. −0.593798
\(917\) 1.77133e6 2.10650
\(918\) −781552. −0.927411
\(919\) 74612.4 0.0883446 0.0441723 0.999024i \(-0.485935\pi\)
0.0441723 + 0.999024i \(0.485935\pi\)
\(920\) 438604.i 0.518200i
\(921\) 1.21773e6 1.43559
\(922\) 75960.0i 0.0893559i
\(923\) 1.25426e6 1.47226
\(924\) 194366.i 0.227654i
\(925\) 244383.i 0.285619i
\(926\) 61751.0i 0.0720148i
\(927\) 1.67118e6i 1.94475i
\(928\) 531084. 0.616691
\(929\) 178101. 0.206365 0.103182 0.994662i \(-0.467097\pi\)
0.103182 + 0.994662i \(0.467097\pi\)
\(930\) 788196.i 0.911315i
\(931\) 314158. + 274133.i 0.362451 + 0.316273i
\(932\) −61251.5 −0.0705156
\(933\) 2.09081e6i 2.40188i
\(934\) 870376.i 0.997730i
\(935\) 44906.8 0.0513676
\(936\) 1.95047e6 2.22632
\(937\) −1.24353e6 −1.41637 −0.708186 0.706026i \(-0.750486\pi\)
−0.708186 + 0.706026i \(0.750486\pi\)
\(938\) −208250. −0.236690
\(939\) 1.47596e6i 1.67395i
\(940\) 377384. 0.427098
\(941\) 1.31010e6i 1.47954i −0.672861 0.739769i \(-0.734935\pi\)
0.672861 0.739769i \(-0.265065\pi\)
\(942\) −301869. −0.340186
\(943\) 163271.i 0.183606i
\(944\) 241938.i 0.271494i
\(945\) 1.79185e6i 2.00649i
\(946\) 27905.7i 0.0311825i
\(947\) 660926. 0.736975 0.368488 0.929633i \(-0.379876\pi\)
0.368488 + 0.929633i \(0.379876\pi\)
\(948\) 10163.6 0.0113092
\(949\) 380844.i 0.422877i
\(950\) 208961. 239470.i 0.231535 0.265341i
\(951\) −683594. −0.755853
\(952\) 625852.i 0.690554i
\(953\) 61065.5i 0.0672373i 0.999435 + 0.0336187i \(0.0107032\pi\)
−0.999435 + 0.0336187i \(0.989297\pi\)
\(954\) 656075. 0.720869
\(955\) −80067.8 −0.0877913
\(956\) −272367. −0.298016
\(957\) 144199. 0.157448
\(958\) 588719.i 0.641471i
\(959\) 422232. 0.459107
\(960\) 359342.i 0.389911i
\(961\) −1.22476e6 −1.32619
\(962\) 212432.i 0.229546i
\(963\) 1.30725e6i 1.40964i
\(964\) 278956.i 0.300179i
\(965\) 574905.i 0.617364i
\(966\) −1.09028e6 −1.16838
\(967\) −269267. −0.287959 −0.143979 0.989581i \(-0.545990\pi\)
−0.143979 + 0.989581i \(0.545990\pi\)
\(968\) 848222.i 0.905230i
\(969\) 708755. 812238.i 0.754829 0.865040i
\(970\) −81926.1 −0.0870720
\(971\) 1.49753e6i 1.58832i 0.607709 + 0.794160i \(0.292089\pi\)
−0.607709 + 0.794160i \(0.707911\pi\)
\(972\) 1.53483e6i 1.62453i
\(973\) −974228. −1.02905
\(974\) 891871. 0.940122
\(975\) −1.11794e6 −1.17601
\(976\) 40893.9 0.0429298
\(977\) 1.59483e6i 1.67080i −0.549639 0.835402i \(-0.685235\pi\)
0.549639 0.835402i \(-0.314765\pi\)
\(978\) 608079. 0.635744
\(979\) 158193.i 0.165052i
\(980\) −193066. −0.201027
\(981\) 1.20981e6i 1.25713i
\(982\) 744263.i 0.771797i
\(983\) 1.32566e6i 1.37191i 0.727642 + 0.685957i \(0.240616\pi\)
−0.727642 + 0.685957i \(0.759384\pi\)
\(984\) 323272.i 0.333870i
\(985\) 607455. 0.626097
\(986\) 192349. 0.197850
\(987\) 2.26450e6i 2.32454i
\(988\) −438824. + 502896.i −0.449549 + 0.515186i
\(989\) 378170. 0.386629
\(990\) 110550.i 0.112795i
\(991\) 666235.i 0.678391i −0.940716 0.339196i \(-0.889845\pi\)
0.940716 0.339196i \(-0.110155\pi\)
\(992\) 1.55487e6 1.58005
\(993\) −2.46305e6 −2.49790
\(994\) 990756. 1.00275
\(995\) −336490. −0.339881
\(996\) 1.79650e6i 1.81096i
\(997\) −823511. −0.828475 −0.414237 0.910169i \(-0.635952\pi\)
−0.414237 + 0.910169i \(0.635952\pi\)
\(998\) 360754.i 0.362201i
\(999\) 1.22204e6 1.22449
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 19.5.b.b.18.3 yes 4
3.2 odd 2 171.5.c.c.37.2 4
4.3 odd 2 304.5.e.c.113.4 4
19.18 odd 2 inner 19.5.b.b.18.2 4
57.56 even 2 171.5.c.c.37.3 4
76.75 even 2 304.5.e.c.113.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.5.b.b.18.2 4 19.18 odd 2 inner
19.5.b.b.18.3 yes 4 1.1 even 1 trivial
171.5.c.c.37.2 4 3.2 odd 2
171.5.c.c.37.3 4 57.56 even 2
304.5.e.c.113.1 4 76.75 even 2
304.5.e.c.113.4 4 4.3 odd 2