Properties

Label 531.8.a.c.1.15
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,8,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

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: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(-14.3045\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+14.3045 q^{2} +76.6177 q^{4} -348.885 q^{5} -1316.66 q^{7} -734.996 q^{8} +O(q^{10})\) \(q+14.3045 q^{2} +76.6177 q^{4} -348.885 q^{5} -1316.66 q^{7} -734.996 q^{8} -4990.62 q^{10} -4994.84 q^{11} +3397.75 q^{13} -18834.2 q^{14} -20320.8 q^{16} -13080.3 q^{17} +8796.50 q^{19} -26730.8 q^{20} -71448.5 q^{22} -107491. q^{23} +43596.1 q^{25} +48603.0 q^{26} -100880. q^{28} -31901.3 q^{29} -154292. q^{31} -196599. q^{32} -187107. q^{34} +459365. q^{35} -528856. q^{37} +125829. q^{38} +256430. q^{40} +480637. q^{41} -438138. q^{43} -382693. q^{44} -1.53760e6 q^{46} -1.23501e6 q^{47} +910062. q^{49} +623619. q^{50} +260328. q^{52} +1.18221e6 q^{53} +1.74263e6 q^{55} +967743. q^{56} -456331. q^{58} +205379. q^{59} +873606. q^{61} -2.20706e6 q^{62} -211175. q^{64} -1.18542e6 q^{65} -832263. q^{67} -1.00219e6 q^{68} +6.57097e6 q^{70} -3.15545e6 q^{71} +2.69697e6 q^{73} -7.56500e6 q^{74} +673968. q^{76} +6.57652e6 q^{77} +6.93335e6 q^{79} +7.08963e6 q^{80} +6.87525e6 q^{82} +3.14360e6 q^{83} +4.56354e6 q^{85} -6.26733e6 q^{86} +3.67119e6 q^{88} +470189. q^{89} -4.47369e6 q^{91} -8.23572e6 q^{92} -1.76661e7 q^{94} -3.06897e6 q^{95} +1.03272e7 q^{97} +1.30179e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8} + 6521 q^{10} + 1764 q^{11} + 18192 q^{13} + 7827 q^{14} + 139226 q^{16} + 15507 q^{17} + 52083 q^{19} - 721 q^{20} - 234434 q^{22} - 63823 q^{23} + 202153 q^{25} + 367956 q^{26} + 182306 q^{28} + 502955 q^{29} + 347531 q^{31} + 243908 q^{32} - 330872 q^{34} - 92641 q^{35} + 447615 q^{37} - 775669 q^{38} + 2203270 q^{40} - 940335 q^{41} + 478562 q^{43} + 596924 q^{44} - 3078663 q^{46} - 703121 q^{47} + 1895082 q^{49} + 876967 q^{50} + 6278296 q^{52} + 1005974 q^{53} + 5212846 q^{55} - 3425294 q^{56} + 6710166 q^{58} + 3491443 q^{59} + 11510749 q^{61} - 5996234 q^{62} + 29496941 q^{64} - 11094180 q^{65} + 14007144 q^{67} - 19688159 q^{68} + 30909708 q^{70} - 5229074 q^{71} + 5452211 q^{73} - 12819662 q^{74} + 41929340 q^{76} - 9930777 q^{77} + 15275654 q^{79} - 36576105 q^{80} + 32025935 q^{82} - 7826609 q^{83} + 11836945 q^{85} - 51649136 q^{86} + 30223741 q^{88} + 6436185 q^{89} + 11633535 q^{91} - 43357972 q^{92} - 4494252 q^{94} - 23741055 q^{95} + 26377540 q^{97} - 26517816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.3045 1.26435 0.632174 0.774826i \(-0.282163\pi\)
0.632174 + 0.774826i \(0.282163\pi\)
\(3\) 0 0
\(4\) 76.6177 0.598576
\(5\) −348.885 −1.24821 −0.624105 0.781340i \(-0.714536\pi\)
−0.624105 + 0.781340i \(0.714536\pi\)
\(6\) 0 0
\(7\) −1316.66 −1.45088 −0.725441 0.688285i \(-0.758364\pi\)
−0.725441 + 0.688285i \(0.758364\pi\)
\(8\) −734.996 −0.507540
\(9\) 0 0
\(10\) −4990.62 −1.57817
\(11\) −4994.84 −1.13148 −0.565740 0.824584i \(-0.691409\pi\)
−0.565740 + 0.824584i \(0.691409\pi\)
\(12\) 0 0
\(13\) 3397.75 0.428933 0.214466 0.976731i \(-0.431199\pi\)
0.214466 + 0.976731i \(0.431199\pi\)
\(14\) −18834.2 −1.83442
\(15\) 0 0
\(16\) −20320.8 −1.24028
\(17\) −13080.3 −0.645725 −0.322863 0.946446i \(-0.604645\pi\)
−0.322863 + 0.946446i \(0.604645\pi\)
\(18\) 0 0
\(19\) 8796.50 0.294220 0.147110 0.989120i \(-0.453003\pi\)
0.147110 + 0.989120i \(0.453003\pi\)
\(20\) −26730.8 −0.747149
\(21\) 0 0
\(22\) −71448.5 −1.43058
\(23\) −107491. −1.84215 −0.921075 0.389385i \(-0.872687\pi\)
−0.921075 + 0.389385i \(0.872687\pi\)
\(24\) 0 0
\(25\) 43596.1 0.558030
\(26\) 48603.0 0.542320
\(27\) 0 0
\(28\) −100880. −0.868462
\(29\) −31901.3 −0.242893 −0.121447 0.992598i \(-0.538753\pi\)
−0.121447 + 0.992598i \(0.538753\pi\)
\(30\) 0 0
\(31\) −154292. −0.930200 −0.465100 0.885258i \(-0.653982\pi\)
−0.465100 + 0.885258i \(0.653982\pi\)
\(32\) −196599. −1.06061
\(33\) 0 0
\(34\) −187107. −0.816422
\(35\) 459365. 1.81101
\(36\) 0 0
\(37\) −528856. −1.71645 −0.858225 0.513274i \(-0.828432\pi\)
−0.858225 + 0.513274i \(0.828432\pi\)
\(38\) 125829. 0.371996
\(39\) 0 0
\(40\) 256430. 0.633517
\(41\) 480637. 1.08911 0.544557 0.838724i \(-0.316698\pi\)
0.544557 + 0.838724i \(0.316698\pi\)
\(42\) 0 0
\(43\) −438138. −0.840371 −0.420186 0.907438i \(-0.638035\pi\)
−0.420186 + 0.907438i \(0.638035\pi\)
\(44\) −382693. −0.677277
\(45\) 0 0
\(46\) −1.53760e6 −2.32912
\(47\) −1.23501e6 −1.73511 −0.867555 0.497341i \(-0.834310\pi\)
−0.867555 + 0.497341i \(0.834310\pi\)
\(48\) 0 0
\(49\) 910062. 1.10506
\(50\) 623619. 0.705544
\(51\) 0 0
\(52\) 260328. 0.256749
\(53\) 1.18221e6 1.09076 0.545379 0.838190i \(-0.316386\pi\)
0.545379 + 0.838190i \(0.316386\pi\)
\(54\) 0 0
\(55\) 1.74263e6 1.41233
\(56\) 967743. 0.736380
\(57\) 0 0
\(58\) −456331. −0.307102
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 873606. 0.492789 0.246395 0.969170i \(-0.420754\pi\)
0.246395 + 0.969170i \(0.420754\pi\)
\(62\) −2.20706e6 −1.17610
\(63\) 0 0
\(64\) −211175. −0.100696
\(65\) −1.18542e6 −0.535399
\(66\) 0 0
\(67\) −832263. −0.338064 −0.169032 0.985611i \(-0.554064\pi\)
−0.169032 + 0.985611i \(0.554064\pi\)
\(68\) −1.00219e6 −0.386516
\(69\) 0 0
\(70\) 6.57097e6 2.28974
\(71\) −3.15545e6 −1.04630 −0.523152 0.852240i \(-0.675244\pi\)
−0.523152 + 0.852240i \(0.675244\pi\)
\(72\) 0 0
\(73\) 2.69697e6 0.811420 0.405710 0.914002i \(-0.367024\pi\)
0.405710 + 0.914002i \(0.367024\pi\)
\(74\) −7.56500e6 −2.17019
\(75\) 0 0
\(76\) 673968. 0.176113
\(77\) 6.57652e6 1.64164
\(78\) 0 0
\(79\) 6.93335e6 1.58215 0.791076 0.611718i \(-0.209521\pi\)
0.791076 + 0.611718i \(0.209521\pi\)
\(80\) 7.08963e6 1.54813
\(81\) 0 0
\(82\) 6.87525e6 1.37702
\(83\) 3.14360e6 0.603468 0.301734 0.953392i \(-0.402435\pi\)
0.301734 + 0.953392i \(0.402435\pi\)
\(84\) 0 0
\(85\) 4.56354e6 0.806001
\(86\) −6.26733e6 −1.06252
\(87\) 0 0
\(88\) 3.67119e6 0.574271
\(89\) 470189. 0.0706980 0.0353490 0.999375i \(-0.488746\pi\)
0.0353490 + 0.999375i \(0.488746\pi\)
\(90\) 0 0
\(91\) −4.47369e6 −0.622331
\(92\) −8.23572e6 −1.10267
\(93\) 0 0
\(94\) −1.76661e7 −2.19378
\(95\) −3.06897e6 −0.367249
\(96\) 0 0
\(97\) 1.03272e7 1.14890 0.574448 0.818541i \(-0.305217\pi\)
0.574448 + 0.818541i \(0.305217\pi\)
\(98\) 1.30179e7 1.39718
\(99\) 0 0
\(100\) 3.34023e6 0.334023
\(101\) 1.36297e7 1.31632 0.658159 0.752879i \(-0.271336\pi\)
0.658159 + 0.752879i \(0.271336\pi\)
\(102\) 0 0
\(103\) −1.58162e6 −0.142618 −0.0713088 0.997454i \(-0.522718\pi\)
−0.0713088 + 0.997454i \(0.522718\pi\)
\(104\) −2.49733e6 −0.217701
\(105\) 0 0
\(106\) 1.69109e7 1.37910
\(107\) −1.70106e7 −1.34238 −0.671191 0.741284i \(-0.734217\pi\)
−0.671191 + 0.741284i \(0.734217\pi\)
\(108\) 0 0
\(109\) −5.78972e6 −0.428218 −0.214109 0.976810i \(-0.568685\pi\)
−0.214109 + 0.976810i \(0.568685\pi\)
\(110\) 2.49273e7 1.78567
\(111\) 0 0
\(112\) 2.67557e7 1.79950
\(113\) −2.12637e7 −1.38632 −0.693161 0.720783i \(-0.743783\pi\)
−0.693161 + 0.720783i \(0.743783\pi\)
\(114\) 0 0
\(115\) 3.75021e7 2.29939
\(116\) −2.44421e6 −0.145390
\(117\) 0 0
\(118\) 2.93784e6 0.164604
\(119\) 1.72224e7 0.936871
\(120\) 0 0
\(121\) 5.46122e6 0.280247
\(122\) 1.24965e7 0.623057
\(123\) 0 0
\(124\) −1.18215e7 −0.556795
\(125\) 1.20466e7 0.551672
\(126\) 0 0
\(127\) 1.24454e7 0.539132 0.269566 0.962982i \(-0.413120\pi\)
0.269566 + 0.962982i \(0.413120\pi\)
\(128\) 2.21439e7 0.933294
\(129\) 0 0
\(130\) −1.69569e7 −0.676930
\(131\) −8.16356e6 −0.317271 −0.158635 0.987337i \(-0.550709\pi\)
−0.158635 + 0.987337i \(0.550709\pi\)
\(132\) 0 0
\(133\) −1.15820e7 −0.426878
\(134\) −1.19051e7 −0.427430
\(135\) 0 0
\(136\) 9.61400e6 0.327731
\(137\) −4.91255e7 −1.63224 −0.816122 0.577880i \(-0.803880\pi\)
−0.816122 + 0.577880i \(0.803880\pi\)
\(138\) 0 0
\(139\) −2.62822e6 −0.0830060 −0.0415030 0.999138i \(-0.513215\pi\)
−0.0415030 + 0.999138i \(0.513215\pi\)
\(140\) 3.51955e7 1.08402
\(141\) 0 0
\(142\) −4.51371e7 −1.32289
\(143\) −1.69712e7 −0.485329
\(144\) 0 0
\(145\) 1.11299e7 0.303182
\(146\) 3.85787e7 1.02592
\(147\) 0 0
\(148\) −4.05197e7 −1.02743
\(149\) 1.97981e7 0.490311 0.245156 0.969484i \(-0.421161\pi\)
0.245156 + 0.969484i \(0.421161\pi\)
\(150\) 0 0
\(151\) 1.33779e7 0.316205 0.158103 0.987423i \(-0.449462\pi\)
0.158103 + 0.987423i \(0.449462\pi\)
\(152\) −6.46539e6 −0.149328
\(153\) 0 0
\(154\) 9.40736e7 2.07561
\(155\) 5.38301e7 1.16109
\(156\) 0 0
\(157\) −2.96493e7 −0.611457 −0.305728 0.952119i \(-0.598900\pi\)
−0.305728 + 0.952119i \(0.598900\pi\)
\(158\) 9.91778e7 2.00039
\(159\) 0 0
\(160\) 6.85904e7 1.32386
\(161\) 1.41530e8 2.67274
\(162\) 0 0
\(163\) −5.91942e7 −1.07059 −0.535294 0.844666i \(-0.679799\pi\)
−0.535294 + 0.844666i \(0.679799\pi\)
\(164\) 3.68253e7 0.651918
\(165\) 0 0
\(166\) 4.49675e7 0.762993
\(167\) −1.03566e8 −1.72071 −0.860357 0.509692i \(-0.829759\pi\)
−0.860357 + 0.509692i \(0.829759\pi\)
\(168\) 0 0
\(169\) −5.12038e7 −0.816017
\(170\) 6.52790e7 1.01907
\(171\) 0 0
\(172\) −3.35691e7 −0.503026
\(173\) −7.49114e7 −1.09998 −0.549992 0.835170i \(-0.685369\pi\)
−0.549992 + 0.835170i \(0.685369\pi\)
\(174\) 0 0
\(175\) −5.74014e7 −0.809635
\(176\) 1.01499e8 1.40336
\(177\) 0 0
\(178\) 6.72580e6 0.0893869
\(179\) −1.39539e8 −1.81849 −0.909244 0.416263i \(-0.863339\pi\)
−0.909244 + 0.416263i \(0.863339\pi\)
\(180\) 0 0
\(181\) 1.39686e7 0.175096 0.0875480 0.996160i \(-0.472097\pi\)
0.0875480 + 0.996160i \(0.472097\pi\)
\(182\) −6.39938e7 −0.786843
\(183\) 0 0
\(184\) 7.90055e7 0.934965
\(185\) 1.84510e8 2.14249
\(186\) 0 0
\(187\) 6.53342e7 0.730625
\(188\) −9.46235e7 −1.03860
\(189\) 0 0
\(190\) −4.39000e7 −0.464330
\(191\) 3.81975e7 0.396660 0.198330 0.980135i \(-0.436448\pi\)
0.198330 + 0.980135i \(0.436448\pi\)
\(192\) 0 0
\(193\) −3.69244e7 −0.369711 −0.184856 0.982766i \(-0.559182\pi\)
−0.184856 + 0.982766i \(0.559182\pi\)
\(194\) 1.47725e8 1.45260
\(195\) 0 0
\(196\) 6.97268e7 0.661460
\(197\) −1.04700e8 −0.975695 −0.487847 0.872929i \(-0.662218\pi\)
−0.487847 + 0.872929i \(0.662218\pi\)
\(198\) 0 0
\(199\) −1.24096e8 −1.11628 −0.558138 0.829748i \(-0.688484\pi\)
−0.558138 + 0.829748i \(0.688484\pi\)
\(200\) −3.20430e7 −0.283222
\(201\) 0 0
\(202\) 1.94965e8 1.66428
\(203\) 4.20033e7 0.352409
\(204\) 0 0
\(205\) −1.67687e8 −1.35944
\(206\) −2.26243e7 −0.180318
\(207\) 0 0
\(208\) −6.90449e7 −0.531998
\(209\) −4.39371e7 −0.332904
\(210\) 0 0
\(211\) −8.39750e7 −0.615406 −0.307703 0.951483i \(-0.599560\pi\)
−0.307703 + 0.951483i \(0.599560\pi\)
\(212\) 9.05781e7 0.652901
\(213\) 0 0
\(214\) −2.43327e8 −1.69724
\(215\) 1.52860e8 1.04896
\(216\) 0 0
\(217\) 2.03150e8 1.34961
\(218\) −8.28188e7 −0.541416
\(219\) 0 0
\(220\) 1.33516e8 0.845384
\(221\) −4.44437e7 −0.276973
\(222\) 0 0
\(223\) 2.55405e8 1.54227 0.771137 0.636669i \(-0.219688\pi\)
0.771137 + 0.636669i \(0.219688\pi\)
\(224\) 2.58854e8 1.53882
\(225\) 0 0
\(226\) −3.04166e8 −1.75279
\(227\) −2.94977e8 −1.67377 −0.836887 0.547375i \(-0.815627\pi\)
−0.836887 + 0.547375i \(0.815627\pi\)
\(228\) 0 0
\(229\) 3.95841e7 0.217819 0.108910 0.994052i \(-0.465264\pi\)
0.108910 + 0.994052i \(0.465264\pi\)
\(230\) 5.36447e8 2.90723
\(231\) 0 0
\(232\) 2.34473e7 0.123278
\(233\) 1.72426e7 0.0893009 0.0446505 0.999003i \(-0.485783\pi\)
0.0446505 + 0.999003i \(0.485783\pi\)
\(234\) 0 0
\(235\) 4.30876e8 2.16578
\(236\) 1.57357e7 0.0779279
\(237\) 0 0
\(238\) 2.46357e8 1.18453
\(239\) −3.47868e6 −0.0164824 −0.00824121 0.999966i \(-0.502623\pi\)
−0.00824121 + 0.999966i \(0.502623\pi\)
\(240\) 0 0
\(241\) 2.24809e8 1.03456 0.517279 0.855817i \(-0.326945\pi\)
0.517279 + 0.855817i \(0.326945\pi\)
\(242\) 7.81198e7 0.354330
\(243\) 0 0
\(244\) 6.69337e7 0.294972
\(245\) −3.17507e8 −1.37934
\(246\) 0 0
\(247\) 2.98883e7 0.126201
\(248\) 1.13404e8 0.472113
\(249\) 0 0
\(250\) 1.72321e8 0.697505
\(251\) −4.20064e8 −1.67671 −0.838354 0.545127i \(-0.816481\pi\)
−0.838354 + 0.545127i \(0.816481\pi\)
\(252\) 0 0
\(253\) 5.36900e8 2.08436
\(254\) 1.78024e8 0.681650
\(255\) 0 0
\(256\) 3.43787e8 1.28070
\(257\) −1.99799e8 −0.734221 −0.367111 0.930177i \(-0.619653\pi\)
−0.367111 + 0.930177i \(0.619653\pi\)
\(258\) 0 0
\(259\) 6.96325e8 2.49036
\(260\) −9.08245e7 −0.320477
\(261\) 0 0
\(262\) −1.16775e8 −0.401141
\(263\) −5.10505e8 −1.73043 −0.865216 0.501399i \(-0.832819\pi\)
−0.865216 + 0.501399i \(0.832819\pi\)
\(264\) 0 0
\(265\) −4.12455e8 −1.36150
\(266\) −1.65675e8 −0.539723
\(267\) 0 0
\(268\) −6.37661e7 −0.202357
\(269\) 6.12082e8 1.91724 0.958621 0.284686i \(-0.0918893\pi\)
0.958621 + 0.284686i \(0.0918893\pi\)
\(270\) 0 0
\(271\) 1.95076e8 0.595404 0.297702 0.954659i \(-0.403780\pi\)
0.297702 + 0.954659i \(0.403780\pi\)
\(272\) 2.65803e8 0.800882
\(273\) 0 0
\(274\) −7.02714e8 −2.06372
\(275\) −2.17755e8 −0.631400
\(276\) 0 0
\(277\) −4.11429e8 −1.16310 −0.581548 0.813512i \(-0.697553\pi\)
−0.581548 + 0.813512i \(0.697553\pi\)
\(278\) −3.75953e7 −0.104949
\(279\) 0 0
\(280\) −3.37632e8 −0.919157
\(281\) 1.45356e8 0.390805 0.195403 0.980723i \(-0.437399\pi\)
0.195403 + 0.980723i \(0.437399\pi\)
\(282\) 0 0
\(283\) −7.33769e8 −1.92445 −0.962226 0.272252i \(-0.912232\pi\)
−0.962226 + 0.272252i \(0.912232\pi\)
\(284\) −2.41764e8 −0.626292
\(285\) 0 0
\(286\) −2.42764e8 −0.613625
\(287\) −6.32837e8 −1.58018
\(288\) 0 0
\(289\) −2.39243e8 −0.583039
\(290\) 1.59207e8 0.383328
\(291\) 0 0
\(292\) 2.06635e8 0.485696
\(293\) 3.59570e8 0.835117 0.417558 0.908650i \(-0.362886\pi\)
0.417558 + 0.908650i \(0.362886\pi\)
\(294\) 0 0
\(295\) −7.16538e7 −0.162503
\(296\) 3.88707e8 0.871167
\(297\) 0 0
\(298\) 2.83202e8 0.619924
\(299\) −3.65228e8 −0.790159
\(300\) 0 0
\(301\) 5.76880e8 1.21928
\(302\) 1.91364e8 0.399794
\(303\) 0 0
\(304\) −1.78752e8 −0.364916
\(305\) −3.04788e8 −0.615105
\(306\) 0 0
\(307\) −2.59711e8 −0.512278 −0.256139 0.966640i \(-0.582450\pi\)
−0.256139 + 0.966640i \(0.582450\pi\)
\(308\) 5.03878e8 0.982648
\(309\) 0 0
\(310\) 7.70010e8 1.46802
\(311\) −4.42793e8 −0.834717 −0.417359 0.908742i \(-0.637044\pi\)
−0.417359 + 0.908742i \(0.637044\pi\)
\(312\) 0 0
\(313\) 4.37731e7 0.0806867 0.0403433 0.999186i \(-0.487155\pi\)
0.0403433 + 0.999186i \(0.487155\pi\)
\(314\) −4.24117e8 −0.773094
\(315\) 0 0
\(316\) 5.31217e8 0.947038
\(317\) −1.74353e8 −0.307413 −0.153706 0.988117i \(-0.549121\pi\)
−0.153706 + 0.988117i \(0.549121\pi\)
\(318\) 0 0
\(319\) 1.59342e8 0.274829
\(320\) 7.36761e7 0.125690
\(321\) 0 0
\(322\) 2.02451e9 3.37927
\(323\) −1.15061e8 −0.189985
\(324\) 0 0
\(325\) 1.48128e8 0.239357
\(326\) −8.46741e8 −1.35360
\(327\) 0 0
\(328\) −3.53266e8 −0.552769
\(329\) 1.62609e9 2.51744
\(330\) 0 0
\(331\) 4.19111e8 0.635231 0.317615 0.948220i \(-0.397118\pi\)
0.317615 + 0.948220i \(0.397118\pi\)
\(332\) 2.40856e8 0.361221
\(333\) 0 0
\(334\) −1.48145e9 −2.17558
\(335\) 2.90364e8 0.421975
\(336\) 0 0
\(337\) −5.81583e8 −0.827765 −0.413882 0.910330i \(-0.635827\pi\)
−0.413882 + 0.910330i \(0.635827\pi\)
\(338\) −7.32443e8 −1.03173
\(339\) 0 0
\(340\) 3.49648e8 0.482453
\(341\) 7.70661e8 1.05250
\(342\) 0 0
\(343\) −1.13916e8 −0.152425
\(344\) 3.22030e8 0.426522
\(345\) 0 0
\(346\) −1.07157e9 −1.39076
\(347\) −6.12801e8 −0.787347 −0.393673 0.919250i \(-0.628796\pi\)
−0.393673 + 0.919250i \(0.628796\pi\)
\(348\) 0 0
\(349\) 5.26220e8 0.662642 0.331321 0.943518i \(-0.392506\pi\)
0.331321 + 0.943518i \(0.392506\pi\)
\(350\) −8.21096e8 −1.02366
\(351\) 0 0
\(352\) 9.81978e8 1.20006
\(353\) −7.42142e8 −0.897999 −0.448999 0.893532i \(-0.648219\pi\)
−0.448999 + 0.893532i \(0.648219\pi\)
\(354\) 0 0
\(355\) 1.10089e9 1.30601
\(356\) 3.60248e7 0.0423181
\(357\) 0 0
\(358\) −1.99603e9 −2.29920
\(359\) −4.68258e8 −0.534139 −0.267070 0.963677i \(-0.586055\pi\)
−0.267070 + 0.963677i \(0.586055\pi\)
\(360\) 0 0
\(361\) −8.16493e8 −0.913435
\(362\) 1.99813e8 0.221382
\(363\) 0 0
\(364\) −3.42764e8 −0.372512
\(365\) −9.40933e8 −1.01282
\(366\) 0 0
\(367\) 4.07678e8 0.430513 0.215257 0.976558i \(-0.430941\pi\)
0.215257 + 0.976558i \(0.430941\pi\)
\(368\) 2.18430e9 2.28479
\(369\) 0 0
\(370\) 2.63932e9 2.70885
\(371\) −1.55657e9 −1.58256
\(372\) 0 0
\(373\) 1.76997e9 1.76598 0.882989 0.469393i \(-0.155527\pi\)
0.882989 + 0.469393i \(0.155527\pi\)
\(374\) 9.34570e8 0.923765
\(375\) 0 0
\(376\) 9.07726e8 0.880638
\(377\) −1.08393e8 −0.104185
\(378\) 0 0
\(379\) −2.03436e8 −0.191951 −0.0959757 0.995384i \(-0.530597\pi\)
−0.0959757 + 0.995384i \(0.530597\pi\)
\(380\) −2.35137e8 −0.219826
\(381\) 0 0
\(382\) 5.46395e8 0.501516
\(383\) 1.58998e9 1.44609 0.723044 0.690802i \(-0.242742\pi\)
0.723044 + 0.690802i \(0.242742\pi\)
\(384\) 0 0
\(385\) −2.29445e9 −2.04912
\(386\) −5.28184e8 −0.467444
\(387\) 0 0
\(388\) 7.91244e8 0.687701
\(389\) 6.10709e8 0.526030 0.263015 0.964792i \(-0.415283\pi\)
0.263015 + 0.964792i \(0.415283\pi\)
\(390\) 0 0
\(391\) 1.40602e9 1.18952
\(392\) −6.68892e8 −0.560860
\(393\) 0 0
\(394\) −1.49767e9 −1.23362
\(395\) −2.41894e9 −1.97486
\(396\) 0 0
\(397\) 1.32027e8 0.105900 0.0529501 0.998597i \(-0.483138\pi\)
0.0529501 + 0.998597i \(0.483138\pi\)
\(398\) −1.77513e9 −1.41136
\(399\) 0 0
\(400\) −8.85907e8 −0.692115
\(401\) −6.34337e8 −0.491263 −0.245632 0.969363i \(-0.578995\pi\)
−0.245632 + 0.969363i \(0.578995\pi\)
\(402\) 0 0
\(403\) −5.24244e8 −0.398993
\(404\) 1.04427e9 0.787916
\(405\) 0 0
\(406\) 6.00835e8 0.445568
\(407\) 2.64155e9 1.94213
\(408\) 0 0
\(409\) 9.45727e8 0.683493 0.341746 0.939792i \(-0.388982\pi\)
0.341746 + 0.939792i \(0.388982\pi\)
\(410\) −2.39868e9 −1.71881
\(411\) 0 0
\(412\) −1.21180e8 −0.0853675
\(413\) −2.70415e8 −0.188889
\(414\) 0 0
\(415\) −1.09676e9 −0.753255
\(416\) −6.67992e8 −0.454930
\(417\) 0 0
\(418\) −6.28496e8 −0.420907
\(419\) 1.99014e9 1.32170 0.660852 0.750517i \(-0.270195\pi\)
0.660852 + 0.750517i \(0.270195\pi\)
\(420\) 0 0
\(421\) 2.39417e8 0.156375 0.0781875 0.996939i \(-0.475087\pi\)
0.0781875 + 0.996939i \(0.475087\pi\)
\(422\) −1.20122e9 −0.778087
\(423\) 0 0
\(424\) −8.68918e8 −0.553603
\(425\) −5.70252e8 −0.360334
\(426\) 0 0
\(427\) −1.15025e9 −0.714979
\(428\) −1.30331e9 −0.803518
\(429\) 0 0
\(430\) 2.18658e9 1.32625
\(431\) 1.41644e9 0.852171 0.426086 0.904683i \(-0.359892\pi\)
0.426086 + 0.904683i \(0.359892\pi\)
\(432\) 0 0
\(433\) 2.49096e9 1.47455 0.737273 0.675595i \(-0.236113\pi\)
0.737273 + 0.675595i \(0.236113\pi\)
\(434\) 2.90595e9 1.70638
\(435\) 0 0
\(436\) −4.43595e8 −0.256321
\(437\) −9.45545e8 −0.541997
\(438\) 0 0
\(439\) 1.29699e9 0.731661 0.365831 0.930681i \(-0.380785\pi\)
0.365831 + 0.930681i \(0.380785\pi\)
\(440\) −1.28082e9 −0.716811
\(441\) 0 0
\(442\) −6.35743e8 −0.350190
\(443\) 4.26015e8 0.232815 0.116408 0.993202i \(-0.462862\pi\)
0.116408 + 0.993202i \(0.462862\pi\)
\(444\) 0 0
\(445\) −1.64042e8 −0.0882460
\(446\) 3.65343e9 1.94997
\(447\) 0 0
\(448\) 2.78047e8 0.146098
\(449\) −3.29234e9 −1.71649 −0.858247 0.513236i \(-0.828446\pi\)
−0.858247 + 0.513236i \(0.828446\pi\)
\(450\) 0 0
\(451\) −2.40070e9 −1.23231
\(452\) −1.62918e9 −0.829819
\(453\) 0 0
\(454\) −4.21948e9 −2.11623
\(455\) 1.56081e9 0.776800
\(456\) 0 0
\(457\) −4.28524e8 −0.210024 −0.105012 0.994471i \(-0.533488\pi\)
−0.105012 + 0.994471i \(0.533488\pi\)
\(458\) 5.66229e8 0.275399
\(459\) 0 0
\(460\) 2.87332e9 1.37636
\(461\) 2.18183e9 1.03721 0.518605 0.855014i \(-0.326452\pi\)
0.518605 + 0.855014i \(0.326452\pi\)
\(462\) 0 0
\(463\) −3.49857e9 −1.63816 −0.819082 0.573677i \(-0.805517\pi\)
−0.819082 + 0.573677i \(0.805517\pi\)
\(464\) 6.48260e8 0.301256
\(465\) 0 0
\(466\) 2.46646e8 0.112907
\(467\) −5.99769e8 −0.272505 −0.136253 0.990674i \(-0.543506\pi\)
−0.136253 + 0.990674i \(0.543506\pi\)
\(468\) 0 0
\(469\) 1.09581e9 0.490490
\(470\) 6.16345e9 2.73830
\(471\) 0 0
\(472\) −1.50953e8 −0.0660761
\(473\) 2.18843e9 0.950863
\(474\) 0 0
\(475\) 3.83493e8 0.164184
\(476\) 1.31954e9 0.560788
\(477\) 0 0
\(478\) −4.97606e7 −0.0208395
\(479\) −1.04242e9 −0.433381 −0.216691 0.976240i \(-0.569526\pi\)
−0.216691 + 0.976240i \(0.569526\pi\)
\(480\) 0 0
\(481\) −1.79692e9 −0.736242
\(482\) 3.21578e9 1.30804
\(483\) 0 0
\(484\) 4.18426e8 0.167749
\(485\) −3.60300e9 −1.43406
\(486\) 0 0
\(487\) 4.49609e9 1.76394 0.881970 0.471305i \(-0.156217\pi\)
0.881970 + 0.471305i \(0.156217\pi\)
\(488\) −6.42097e8 −0.250110
\(489\) 0 0
\(490\) −4.54177e9 −1.74397
\(491\) 3.34941e9 1.27698 0.638489 0.769631i \(-0.279560\pi\)
0.638489 + 0.769631i \(0.279560\pi\)
\(492\) 0 0
\(493\) 4.17280e8 0.156842
\(494\) 4.27536e8 0.159562
\(495\) 0 0
\(496\) 3.13533e9 1.15371
\(497\) 4.15467e9 1.51806
\(498\) 0 0
\(499\) −2.53248e9 −0.912419 −0.456209 0.889872i \(-0.650793\pi\)
−0.456209 + 0.889872i \(0.650793\pi\)
\(500\) 9.22986e8 0.330217
\(501\) 0 0
\(502\) −6.00879e9 −2.11994
\(503\) 2.14316e9 0.750874 0.375437 0.926848i \(-0.377493\pi\)
0.375437 + 0.926848i \(0.377493\pi\)
\(504\) 0 0
\(505\) −4.75520e9 −1.64304
\(506\) 7.68007e9 2.63535
\(507\) 0 0
\(508\) 9.53536e8 0.322711
\(509\) −3.85176e9 −1.29463 −0.647316 0.762222i \(-0.724109\pi\)
−0.647316 + 0.762222i \(0.724109\pi\)
\(510\) 0 0
\(511\) −3.55100e9 −1.17727
\(512\) 2.08327e9 0.685962
\(513\) 0 0
\(514\) −2.85801e9 −0.928311
\(515\) 5.51806e8 0.178017
\(516\) 0 0
\(517\) 6.16866e9 1.96324
\(518\) 9.96056e9 3.14869
\(519\) 0 0
\(520\) 8.71283e8 0.271736
\(521\) 4.67938e9 1.44963 0.724814 0.688945i \(-0.241926\pi\)
0.724814 + 0.688945i \(0.241926\pi\)
\(522\) 0 0
\(523\) 3.37502e9 1.03162 0.515810 0.856703i \(-0.327491\pi\)
0.515810 + 0.856703i \(0.327491\pi\)
\(524\) −6.25474e8 −0.189911
\(525\) 0 0
\(526\) −7.30250e9 −2.18787
\(527\) 2.01819e9 0.600654
\(528\) 0 0
\(529\) 8.14951e9 2.39352
\(530\) −5.89995e9 −1.72140
\(531\) 0 0
\(532\) −8.87389e8 −0.255519
\(533\) 1.63308e9 0.467157
\(534\) 0 0
\(535\) 5.93475e9 1.67558
\(536\) 6.11710e8 0.171581
\(537\) 0 0
\(538\) 8.75551e9 2.42406
\(539\) −4.54561e9 −1.25035
\(540\) 0 0
\(541\) 3.20625e9 0.870576 0.435288 0.900291i \(-0.356647\pi\)
0.435288 + 0.900291i \(0.356647\pi\)
\(542\) 2.79046e9 0.752798
\(543\) 0 0
\(544\) 2.57158e9 0.684862
\(545\) 2.01995e9 0.534506
\(546\) 0 0
\(547\) −4.82907e8 −0.126156 −0.0630779 0.998009i \(-0.520092\pi\)
−0.0630779 + 0.998009i \(0.520092\pi\)
\(548\) −3.76388e9 −0.977022
\(549\) 0 0
\(550\) −3.11487e9 −0.798309
\(551\) −2.80620e8 −0.0714641
\(552\) 0 0
\(553\) −9.12889e9 −2.29551
\(554\) −5.88527e9 −1.47056
\(555\) 0 0
\(556\) −2.01368e8 −0.0496854
\(557\) 3.29285e9 0.807380 0.403690 0.914896i \(-0.367727\pi\)
0.403690 + 0.914896i \(0.367727\pi\)
\(558\) 0 0
\(559\) −1.48868e9 −0.360463
\(560\) −9.33466e9 −2.24616
\(561\) 0 0
\(562\) 2.07924e9 0.494114
\(563\) −1.67347e9 −0.395220 −0.197610 0.980281i \(-0.563318\pi\)
−0.197610 + 0.980281i \(0.563318\pi\)
\(564\) 0 0
\(565\) 7.41859e9 1.73042
\(566\) −1.04962e10 −2.43318
\(567\) 0 0
\(568\) 2.31925e9 0.531041
\(569\) −5.59667e9 −1.27361 −0.636805 0.771025i \(-0.719745\pi\)
−0.636805 + 0.771025i \(0.719745\pi\)
\(570\) 0 0
\(571\) −6.50267e9 −1.46172 −0.730862 0.682525i \(-0.760882\pi\)
−0.730862 + 0.682525i \(0.760882\pi\)
\(572\) −1.30029e9 −0.290506
\(573\) 0 0
\(574\) −9.05240e9 −1.99789
\(575\) −4.68619e9 −1.02797
\(576\) 0 0
\(577\) −4.50802e9 −0.976945 −0.488473 0.872579i \(-0.662446\pi\)
−0.488473 + 0.872579i \(0.662446\pi\)
\(578\) −3.42225e9 −0.737164
\(579\) 0 0
\(580\) 8.52748e8 0.181477
\(581\) −4.13907e9 −0.875560
\(582\) 0 0
\(583\) −5.90494e9 −1.23417
\(584\) −1.98226e9 −0.411828
\(585\) 0 0
\(586\) 5.14346e9 1.05588
\(587\) −6.96930e9 −1.42218 −0.711092 0.703099i \(-0.751799\pi\)
−0.711092 + 0.703099i \(0.751799\pi\)
\(588\) 0 0
\(589\) −1.35723e9 −0.273683
\(590\) −1.02497e9 −0.205461
\(591\) 0 0
\(592\) 1.07468e10 2.12888
\(593\) −1.72102e9 −0.338917 −0.169459 0.985537i \(-0.554202\pi\)
−0.169459 + 0.985537i \(0.554202\pi\)
\(594\) 0 0
\(595\) −6.00865e9 −1.16941
\(596\) 1.51689e9 0.293489
\(597\) 0 0
\(598\) −5.22438e9 −0.999036
\(599\) −4.56819e9 −0.868460 −0.434230 0.900802i \(-0.642979\pi\)
−0.434230 + 0.900802i \(0.642979\pi\)
\(600\) 0 0
\(601\) −2.34532e9 −0.440698 −0.220349 0.975421i \(-0.570720\pi\)
−0.220349 + 0.975421i \(0.570720\pi\)
\(602\) 8.25197e9 1.54159
\(603\) 0 0
\(604\) 1.02499e9 0.189273
\(605\) −1.90534e9 −0.349807
\(606\) 0 0
\(607\) 3.67032e9 0.666107 0.333053 0.942908i \(-0.391921\pi\)
0.333053 + 0.942908i \(0.391921\pi\)
\(608\) −1.72938e9 −0.312052
\(609\) 0 0
\(610\) −4.35984e9 −0.777706
\(611\) −4.19624e9 −0.744246
\(612\) 0 0
\(613\) 3.62973e9 0.636447 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(614\) −3.71502e9 −0.647697
\(615\) 0 0
\(616\) −4.83372e9 −0.833199
\(617\) −4.89208e9 −0.838486 −0.419243 0.907874i \(-0.637704\pi\)
−0.419243 + 0.907874i \(0.637704\pi\)
\(618\) 0 0
\(619\) −8.57865e9 −1.45379 −0.726895 0.686749i \(-0.759037\pi\)
−0.726895 + 0.686749i \(0.759037\pi\)
\(620\) 4.12434e9 0.694997
\(621\) 0 0
\(622\) −6.33392e9 −1.05537
\(623\) −6.19080e8 −0.102574
\(624\) 0 0
\(625\) −7.60884e9 −1.24663
\(626\) 6.26150e8 0.102016
\(627\) 0 0
\(628\) −2.27166e9 −0.366003
\(629\) 6.91761e9 1.10836
\(630\) 0 0
\(631\) 5.81549e9 0.921475 0.460737 0.887537i \(-0.347585\pi\)
0.460737 + 0.887537i \(0.347585\pi\)
\(632\) −5.09598e9 −0.803005
\(633\) 0 0
\(634\) −2.49403e9 −0.388677
\(635\) −4.34201e9 −0.672950
\(636\) 0 0
\(637\) 3.09216e9 0.473995
\(638\) 2.27930e9 0.347479
\(639\) 0 0
\(640\) −7.72567e9 −1.16495
\(641\) 1.97168e6 0.000295688 0 0.000147844 1.00000i \(-0.499953\pi\)
0.000147844 1.00000i \(0.499953\pi\)
\(642\) 0 0
\(643\) −5.68675e9 −0.843580 −0.421790 0.906694i \(-0.638598\pi\)
−0.421790 + 0.906694i \(0.638598\pi\)
\(644\) 1.08437e10 1.59984
\(645\) 0 0
\(646\) −1.64589e9 −0.240208
\(647\) −8.00900e9 −1.16256 −0.581278 0.813705i \(-0.697447\pi\)
−0.581278 + 0.813705i \(0.697447\pi\)
\(648\) 0 0
\(649\) −1.02583e9 −0.147306
\(650\) 2.11890e9 0.302631
\(651\) 0 0
\(652\) −4.53532e9 −0.640828
\(653\) −4.57134e9 −0.642462 −0.321231 0.947001i \(-0.604097\pi\)
−0.321231 + 0.947001i \(0.604097\pi\)
\(654\) 0 0
\(655\) 2.84815e9 0.396021
\(656\) −9.76692e9 −1.35081
\(657\) 0 0
\(658\) 2.32604e10 3.18292
\(659\) −1.31699e10 −1.79260 −0.896298 0.443452i \(-0.853754\pi\)
−0.896298 + 0.443452i \(0.853754\pi\)
\(660\) 0 0
\(661\) −7.04155e9 −0.948338 −0.474169 0.880434i \(-0.657251\pi\)
−0.474169 + 0.880434i \(0.657251\pi\)
\(662\) 5.99517e9 0.803153
\(663\) 0 0
\(664\) −2.31053e9 −0.306284
\(665\) 4.04080e9 0.532834
\(666\) 0 0
\(667\) 3.42911e9 0.447446
\(668\) −7.93497e9 −1.02998
\(669\) 0 0
\(670\) 4.15351e9 0.533523
\(671\) −4.36352e9 −0.557581
\(672\) 0 0
\(673\) 8.33165e8 0.105361 0.0526803 0.998611i \(-0.483224\pi\)
0.0526803 + 0.998611i \(0.483224\pi\)
\(674\) −8.31923e9 −1.04658
\(675\) 0 0
\(676\) −3.92312e9 −0.488448
\(677\) −1.49936e10 −1.85715 −0.928573 0.371150i \(-0.878964\pi\)
−0.928573 + 0.371150i \(0.878964\pi\)
\(678\) 0 0
\(679\) −1.35974e10 −1.66691
\(680\) −3.35419e9 −0.409078
\(681\) 0 0
\(682\) 1.10239e10 1.33073
\(683\) −6.61866e9 −0.794873 −0.397436 0.917630i \(-0.630100\pi\)
−0.397436 + 0.917630i \(0.630100\pi\)
\(684\) 0 0
\(685\) 1.71392e10 2.03738
\(686\) −1.62951e9 −0.192718
\(687\) 0 0
\(688\) 8.90331e9 1.04230
\(689\) 4.01684e9 0.467862
\(690\) 0 0
\(691\) 7.24311e9 0.835125 0.417563 0.908648i \(-0.362884\pi\)
0.417563 + 0.908648i \(0.362884\pi\)
\(692\) −5.73954e9 −0.658424
\(693\) 0 0
\(694\) −8.76579e9 −0.995480
\(695\) 9.16947e8 0.103609
\(696\) 0 0
\(697\) −6.28689e9 −0.703269
\(698\) 7.52730e9 0.837810
\(699\) 0 0
\(700\) −4.39796e9 −0.484628
\(701\) 2.33354e9 0.255860 0.127930 0.991783i \(-0.459167\pi\)
0.127930 + 0.991783i \(0.459167\pi\)
\(702\) 0 0
\(703\) −4.65208e9 −0.505014
\(704\) 1.05479e9 0.113936
\(705\) 0 0
\(706\) −1.06160e10 −1.13538
\(707\) −1.79457e10 −1.90982
\(708\) 0 0
\(709\) 4.23075e9 0.445816 0.222908 0.974839i \(-0.428445\pi\)
0.222908 + 0.974839i \(0.428445\pi\)
\(710\) 1.57477e10 1.65125
\(711\) 0 0
\(712\) −3.45587e8 −0.0358820
\(713\) 1.65850e10 1.71357
\(714\) 0 0
\(715\) 5.92100e9 0.605793
\(716\) −1.06912e10 −1.08850
\(717\) 0 0
\(718\) −6.69818e9 −0.675338
\(719\) 1.25609e10 1.26029 0.630143 0.776479i \(-0.282996\pi\)
0.630143 + 0.776479i \(0.282996\pi\)
\(720\) 0 0
\(721\) 2.08247e9 0.206921
\(722\) −1.16795e10 −1.15490
\(723\) 0 0
\(724\) 1.07024e9 0.104808
\(725\) −1.39077e9 −0.135542
\(726\) 0 0
\(727\) 9.77361e9 0.943375 0.471688 0.881766i \(-0.343645\pi\)
0.471688 + 0.881766i \(0.343645\pi\)
\(728\) 3.28815e9 0.315858
\(729\) 0 0
\(730\) −1.34595e10 −1.28056
\(731\) 5.73099e9 0.542649
\(732\) 0 0
\(733\) −4.89324e9 −0.458915 −0.229458 0.973319i \(-0.573695\pi\)
−0.229458 + 0.973319i \(0.573695\pi\)
\(734\) 5.83162e9 0.544318
\(735\) 0 0
\(736\) 2.11326e10 1.95380
\(737\) 4.15702e9 0.382512
\(738\) 0 0
\(739\) −1.16838e10 −1.06495 −0.532473 0.846447i \(-0.678737\pi\)
−0.532473 + 0.846447i \(0.678737\pi\)
\(740\) 1.41367e10 1.28244
\(741\) 0 0
\(742\) −2.22659e10 −2.00091
\(743\) 2.02352e9 0.180987 0.0904934 0.995897i \(-0.471156\pi\)
0.0904934 + 0.995897i \(0.471156\pi\)
\(744\) 0 0
\(745\) −6.90728e9 −0.612012
\(746\) 2.53185e10 2.23281
\(747\) 0 0
\(748\) 5.00575e9 0.437335
\(749\) 2.23972e10 1.94764
\(750\) 0 0
\(751\) −5.88364e8 −0.0506881 −0.0253441 0.999679i \(-0.508068\pi\)
−0.0253441 + 0.999679i \(0.508068\pi\)
\(752\) 2.50963e10 2.15203
\(753\) 0 0
\(754\) −1.55050e9 −0.131726
\(755\) −4.66736e9 −0.394691
\(756\) 0 0
\(757\) −1.84565e10 −1.54637 −0.773185 0.634181i \(-0.781337\pi\)
−0.773185 + 0.634181i \(0.781337\pi\)
\(758\) −2.91005e9 −0.242693
\(759\) 0 0
\(760\) 2.25568e9 0.186393
\(761\) −1.56129e10 −1.28421 −0.642105 0.766617i \(-0.721939\pi\)
−0.642105 + 0.766617i \(0.721939\pi\)
\(762\) 0 0
\(763\) 7.62311e9 0.621293
\(764\) 2.92660e9 0.237431
\(765\) 0 0
\(766\) 2.27438e10 1.82836
\(767\) 6.97826e8 0.0558423
\(768\) 0 0
\(769\) 2.12983e10 1.68889 0.844446 0.535641i \(-0.179930\pi\)
0.844446 + 0.535641i \(0.179930\pi\)
\(770\) −3.28209e10 −2.59080
\(771\) 0 0
\(772\) −2.82906e9 −0.221300
\(773\) −1.58105e10 −1.23116 −0.615582 0.788073i \(-0.711079\pi\)
−0.615582 + 0.788073i \(0.711079\pi\)
\(774\) 0 0
\(775\) −6.72651e9 −0.519079
\(776\) −7.59043e9 −0.583110
\(777\) 0 0
\(778\) 8.73587e9 0.665086
\(779\) 4.22792e9 0.320439
\(780\) 0 0
\(781\) 1.57610e10 1.18387
\(782\) 2.01124e10 1.50397
\(783\) 0 0
\(784\) −1.84932e10 −1.37058
\(785\) 1.03442e10 0.763227
\(786\) 0 0
\(787\) 2.69394e9 0.197005 0.0985024 0.995137i \(-0.468595\pi\)
0.0985024 + 0.995137i \(0.468595\pi\)
\(788\) −8.02186e9 −0.584027
\(789\) 0 0
\(790\) −3.46017e10 −2.49691
\(791\) 2.79971e10 2.01139
\(792\) 0 0
\(793\) 2.96829e9 0.211374
\(794\) 1.88858e9 0.133895
\(795\) 0 0
\(796\) −9.50795e9 −0.668176
\(797\) −5.54182e9 −0.387747 −0.193873 0.981027i \(-0.562105\pi\)
−0.193873 + 0.981027i \(0.562105\pi\)
\(798\) 0 0
\(799\) 1.61543e10 1.12040
\(800\) −8.57093e9 −0.591852
\(801\) 0 0
\(802\) −9.07385e9 −0.621128
\(803\) −1.34709e10 −0.918106
\(804\) 0 0
\(805\) −4.93776e10 −3.33614
\(806\) −7.49903e9 −0.504466
\(807\) 0 0
\(808\) −1.00178e10 −0.668084
\(809\) −1.56002e10 −1.03589 −0.517943 0.855415i \(-0.673302\pi\)
−0.517943 + 0.855415i \(0.673302\pi\)
\(810\) 0 0
\(811\) −7.79846e9 −0.513377 −0.256688 0.966494i \(-0.582631\pi\)
−0.256688 + 0.966494i \(0.582631\pi\)
\(812\) 3.21820e9 0.210944
\(813\) 0 0
\(814\) 3.77859e10 2.45553
\(815\) 2.06520e10 1.33632
\(816\) 0 0
\(817\) −3.85408e9 −0.247254
\(818\) 1.35281e10 0.864173
\(819\) 0 0
\(820\) −1.28478e10 −0.813731
\(821\) −5.49348e9 −0.346455 −0.173227 0.984882i \(-0.555420\pi\)
−0.173227 + 0.984882i \(0.555420\pi\)
\(822\) 0 0
\(823\) 2.49423e10 1.55969 0.779843 0.625976i \(-0.215299\pi\)
0.779843 + 0.625976i \(0.215299\pi\)
\(824\) 1.16249e9 0.0723841
\(825\) 0 0
\(826\) −3.86814e9 −0.238821
\(827\) 3.09149e10 1.90063 0.950317 0.311285i \(-0.100759\pi\)
0.950317 + 0.311285i \(0.100759\pi\)
\(828\) 0 0
\(829\) −7.17903e8 −0.0437648 −0.0218824 0.999761i \(-0.506966\pi\)
−0.0218824 + 0.999761i \(0.506966\pi\)
\(830\) −1.56885e10 −0.952376
\(831\) 0 0
\(832\) −7.17521e8 −0.0431920
\(833\) −1.19039e10 −0.713563
\(834\) 0 0
\(835\) 3.61326e10 2.14781
\(836\) −3.36636e9 −0.199268
\(837\) 0 0
\(838\) 2.84678e10 1.67109
\(839\) 1.00495e10 0.587460 0.293730 0.955888i \(-0.405103\pi\)
0.293730 + 0.955888i \(0.405103\pi\)
\(840\) 0 0
\(841\) −1.62322e10 −0.941003
\(842\) 3.42473e9 0.197713
\(843\) 0 0
\(844\) −6.43397e9 −0.368367
\(845\) 1.78643e10 1.01856
\(846\) 0 0
\(847\) −7.19059e9 −0.406605
\(848\) −2.40234e10 −1.35285
\(849\) 0 0
\(850\) −8.15714e9 −0.455588
\(851\) 5.68473e10 3.16196
\(852\) 0 0
\(853\) 7.41016e9 0.408795 0.204398 0.978888i \(-0.434476\pi\)
0.204398 + 0.978888i \(0.434476\pi\)
\(854\) −1.64536e10 −0.903982
\(855\) 0 0
\(856\) 1.25027e10 0.681313
\(857\) −1.81207e10 −0.983429 −0.491714 0.870757i \(-0.663630\pi\)
−0.491714 + 0.870757i \(0.663630\pi\)
\(858\) 0 0
\(859\) 1.04208e10 0.560953 0.280477 0.959861i \(-0.409508\pi\)
0.280477 + 0.959861i \(0.409508\pi\)
\(860\) 1.17118e10 0.627882
\(861\) 0 0
\(862\) 2.02614e10 1.07744
\(863\) 2.49924e10 1.32364 0.661821 0.749662i \(-0.269784\pi\)
0.661821 + 0.749662i \(0.269784\pi\)
\(864\) 0 0
\(865\) 2.61355e10 1.37301
\(866\) 3.56318e10 1.86434
\(867\) 0 0
\(868\) 1.55649e10 0.807843
\(869\) −3.46309e10 −1.79017
\(870\) 0 0
\(871\) −2.82782e9 −0.145007
\(872\) 4.25542e9 0.217337
\(873\) 0 0
\(874\) −1.35255e10 −0.685273
\(875\) −1.58614e10 −0.800410
\(876\) 0 0
\(877\) −1.00470e10 −0.502964 −0.251482 0.967862i \(-0.580918\pi\)
−0.251482 + 0.967862i \(0.580918\pi\)
\(878\) 1.85527e10 0.925075
\(879\) 0 0
\(880\) −3.54115e10 −1.75168
\(881\) 3.06184e10 1.50858 0.754288 0.656544i \(-0.227982\pi\)
0.754288 + 0.656544i \(0.227982\pi\)
\(882\) 0 0
\(883\) 5.86608e9 0.286738 0.143369 0.989669i \(-0.454206\pi\)
0.143369 + 0.989669i \(0.454206\pi\)
\(884\) −3.40517e9 −0.165789
\(885\) 0 0
\(886\) 6.09391e9 0.294359
\(887\) −1.17203e9 −0.0563905 −0.0281953 0.999602i \(-0.508976\pi\)
−0.0281953 + 0.999602i \(0.508976\pi\)
\(888\) 0 0
\(889\) −1.63864e10 −0.782217
\(890\) −2.34653e9 −0.111574
\(891\) 0 0
\(892\) 1.95685e10 0.923168
\(893\) −1.08637e10 −0.510504
\(894\) 0 0
\(895\) 4.86832e10 2.26986
\(896\) −2.91560e10 −1.35410
\(897\) 0 0
\(898\) −4.70952e10 −2.17025
\(899\) 4.92210e9 0.225939
\(900\) 0 0
\(901\) −1.54637e10 −0.704330
\(902\) −3.43408e10 −1.55807
\(903\) 0 0
\(904\) 1.56287e10 0.703614
\(905\) −4.87342e9 −0.218557
\(906\) 0 0
\(907\) 1.61777e10 0.719932 0.359966 0.932965i \(-0.382788\pi\)
0.359966 + 0.932965i \(0.382788\pi\)
\(908\) −2.26004e10 −1.00188
\(909\) 0 0
\(910\) 2.23265e10 0.982145
\(911\) 1.77685e10 0.778637 0.389319 0.921103i \(-0.372710\pi\)
0.389319 + 0.921103i \(0.372710\pi\)
\(912\) 0 0
\(913\) −1.57018e10 −0.682812
\(914\) −6.12981e9 −0.265543
\(915\) 0 0
\(916\) 3.03284e9 0.130381
\(917\) 1.07487e10 0.460322
\(918\) 0 0
\(919\) 2.24473e10 0.954024 0.477012 0.878897i \(-0.341720\pi\)
0.477012 + 0.878897i \(0.341720\pi\)
\(920\) −2.75639e10 −1.16703
\(921\) 0 0
\(922\) 3.12098e10 1.31139
\(923\) −1.07214e10 −0.448794
\(924\) 0 0
\(925\) −2.30560e10 −0.957830
\(926\) −5.00452e10 −2.07121
\(927\) 0 0
\(928\) 6.27175e9 0.257615
\(929\) 2.97565e10 1.21766 0.608831 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(930\) 0 0
\(931\) 8.00535e9 0.325130
\(932\) 1.32109e9 0.0534534
\(933\) 0 0
\(934\) −8.57937e9 −0.344541
\(935\) −2.27941e10 −0.911974
\(936\) 0 0
\(937\) 2.60896e9 0.103605 0.0518023 0.998657i \(-0.483503\pi\)
0.0518023 + 0.998657i \(0.483503\pi\)
\(938\) 1.56750e10 0.620151
\(939\) 0 0
\(940\) 3.30127e10 1.29639
\(941\) −3.46382e10 −1.35516 −0.677582 0.735447i \(-0.736972\pi\)
−0.677582 + 0.735447i \(0.736972\pi\)
\(942\) 0 0
\(943\) −5.16642e10 −2.00631
\(944\) −4.17346e9 −0.161471
\(945\) 0 0
\(946\) 3.13043e10 1.20222
\(947\) 4.81487e10 1.84230 0.921149 0.389209i \(-0.127252\pi\)
0.921149 + 0.389209i \(0.127252\pi\)
\(948\) 0 0
\(949\) 9.16362e9 0.348045
\(950\) 5.48566e9 0.207585
\(951\) 0 0
\(952\) −1.26584e10 −0.475499
\(953\) 1.99375e9 0.0746182 0.0373091 0.999304i \(-0.488121\pi\)
0.0373091 + 0.999304i \(0.488121\pi\)
\(954\) 0 0
\(955\) −1.33266e10 −0.495115
\(956\) −2.66528e8 −0.00986598
\(957\) 0 0
\(958\) −1.49113e10 −0.547945
\(959\) 6.46818e10 2.36819
\(960\) 0 0
\(961\) −3.70674e9 −0.134729
\(962\) −2.57039e10 −0.930866
\(963\) 0 0
\(964\) 1.72244e10 0.619261
\(965\) 1.28824e10 0.461478
\(966\) 0 0
\(967\) −7.83920e9 −0.278791 −0.139396 0.990237i \(-0.544516\pi\)
−0.139396 + 0.990237i \(0.544516\pi\)
\(968\) −4.01398e9 −0.142237
\(969\) 0 0
\(970\) −5.15390e10 −1.81315
\(971\) 4.89013e10 1.71417 0.857084 0.515177i \(-0.172274\pi\)
0.857084 + 0.515177i \(0.172274\pi\)
\(972\) 0 0
\(973\) 3.46048e9 0.120432
\(974\) 6.43142e10 2.23023
\(975\) 0 0
\(976\) −1.77524e10 −0.611198
\(977\) 2.82603e10 0.969494 0.484747 0.874654i \(-0.338912\pi\)
0.484747 + 0.874654i \(0.338912\pi\)
\(978\) 0 0
\(979\) −2.34852e9 −0.0799934
\(980\) −2.43267e10 −0.825642
\(981\) 0 0
\(982\) 4.79115e10 1.61454
\(983\) −4.62408e10 −1.55270 −0.776351 0.630301i \(-0.782932\pi\)
−0.776351 + 0.630301i \(0.782932\pi\)
\(984\) 0 0
\(985\) 3.65282e10 1.21787
\(986\) 5.96897e9 0.198303
\(987\) 0 0
\(988\) 2.28997e9 0.0755407
\(989\) 4.70959e10 1.54809
\(990\) 0 0
\(991\) 1.15771e10 0.377870 0.188935 0.981990i \(-0.439497\pi\)
0.188935 + 0.981990i \(0.439497\pi\)
\(992\) 3.03335e10 0.986578
\(993\) 0 0
\(994\) 5.94304e10 1.91936
\(995\) 4.32953e10 1.39335
\(996\) 0 0
\(997\) 4.71011e9 0.150521 0.0752607 0.997164i \(-0.476021\pi\)
0.0752607 + 0.997164i \(0.476021\pi\)
\(998\) −3.62258e10 −1.15361
\(999\) 0 0
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 531.8.a.c.1.15 17
3.2 odd 2 177.8.a.c.1.3 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.3 17 3.2 odd 2
531.8.a.c.1.15 17 1.1 even 1 trivial