Properties

Label 256.8.a.q.1.3
Level $256$
Weight $8$
Character 256.1
Self dual yes
Analytic conductor $79.971$
Analytic rank $1$
Dimension $6$
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] = [256,8,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, 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("256.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 256.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: \(79.9705665239\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 163x^{4} + 4820x^{2} - 15296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{27} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(11.2068\) of defining polynomial
Character \(\chi\) \(=\) 256.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-21.9408 q^{3} -184.916 q^{5} +1051.96 q^{7} -1705.60 q^{9} +O(q^{10})\) \(q-21.9408 q^{3} -184.916 q^{5} +1051.96 q^{7} -1705.60 q^{9} -4324.35 q^{11} +11253.2 q^{13} +4057.20 q^{15} -21746.4 q^{17} +45466.5 q^{19} -23080.8 q^{21} +4414.37 q^{23} -43931.2 q^{25} +85406.8 q^{27} -23687.1 q^{29} -72941.9 q^{31} +94879.8 q^{33} -194523. q^{35} +483338. q^{37} -246905. q^{39} +411040. q^{41} +96165.3 q^{43} +315392. q^{45} +156171. q^{47} +283070. q^{49} +477134. q^{51} -686962. q^{53} +799641. q^{55} -997572. q^{57} -1.79961e6 q^{59} +1.36394e6 q^{61} -1.79422e6 q^{63} -2.08090e6 q^{65} -1.08853e6 q^{67} -96854.8 q^{69} -5.60830e6 q^{71} -21698.7 q^{73} +963886. q^{75} -4.54903e6 q^{77} -2.34010e6 q^{79} +1.85626e6 q^{81} -882169. q^{83} +4.02125e6 q^{85} +519714. q^{87} +1.34738e6 q^{89} +1.18379e7 q^{91} +1.60040e6 q^{93} -8.40746e6 q^{95} +7.32798e6 q^{97} +7.37562e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 688 q^{7} + 2918 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 688 q^{7} + 2918 q^{9} - 17872 q^{15} + 1452 q^{17} - 1296 q^{23} + 39314 q^{25} + 89280 q^{31} + 53880 q^{33} - 328208 q^{39} - 521244 q^{41} - 1566432 q^{47} - 511050 q^{49} - 3270256 q^{55} + 1889896 q^{57} - 5776816 q^{63} + 1416480 q^{65} - 7597104 q^{71} - 2089564 q^{73} - 16015904 q^{79} - 723058 q^{81} - 37453776 q^{87} - 2169084 q^{89} - 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −21.9408 −0.469168 −0.234584 0.972096i \(-0.575373\pi\)
−0.234584 + 0.972096i \(0.575373\pi\)
\(4\) 0 0
\(5\) −184.916 −0.661574 −0.330787 0.943705i \(-0.607314\pi\)
−0.330787 + 0.943705i \(0.607314\pi\)
\(6\) 0 0
\(7\) 1051.96 1.15919 0.579595 0.814905i \(-0.303211\pi\)
0.579595 + 0.814905i \(0.303211\pi\)
\(8\) 0 0
\(9\) −1705.60 −0.779882
\(10\) 0 0
\(11\) −4324.35 −0.979596 −0.489798 0.871836i \(-0.662929\pi\)
−0.489798 + 0.871836i \(0.662929\pi\)
\(12\) 0 0
\(13\) 11253.2 1.42061 0.710307 0.703892i \(-0.248556\pi\)
0.710307 + 0.703892i \(0.248556\pi\)
\(14\) 0 0
\(15\) 4057.20 0.310389
\(16\) 0 0
\(17\) −21746.4 −1.07354 −0.536768 0.843730i \(-0.680355\pi\)
−0.536768 + 0.843730i \(0.680355\pi\)
\(18\) 0 0
\(19\) 45466.5 1.52074 0.760368 0.649492i \(-0.225019\pi\)
0.760368 + 0.649492i \(0.225019\pi\)
\(20\) 0 0
\(21\) −23080.8 −0.543855
\(22\) 0 0
\(23\) 4414.37 0.0756522 0.0378261 0.999284i \(-0.487957\pi\)
0.0378261 + 0.999284i \(0.487957\pi\)
\(24\) 0 0
\(25\) −43931.2 −0.562320
\(26\) 0 0
\(27\) 85406.8 0.835063
\(28\) 0 0
\(29\) −23687.1 −0.180351 −0.0901756 0.995926i \(-0.528743\pi\)
−0.0901756 + 0.995926i \(0.528743\pi\)
\(30\) 0 0
\(31\) −72941.9 −0.439755 −0.219878 0.975527i \(-0.570566\pi\)
−0.219878 + 0.975527i \(0.570566\pi\)
\(32\) 0 0
\(33\) 94879.8 0.459595
\(34\) 0 0
\(35\) −194523. −0.766890
\(36\) 0 0
\(37\) 483338. 1.56872 0.784359 0.620307i \(-0.212992\pi\)
0.784359 + 0.620307i \(0.212992\pi\)
\(38\) 0 0
\(39\) −246905. −0.666506
\(40\) 0 0
\(41\) 411040. 0.931410 0.465705 0.884940i \(-0.345801\pi\)
0.465705 + 0.884940i \(0.345801\pi\)
\(42\) 0 0
\(43\) 96165.3 0.184450 0.0922250 0.995738i \(-0.470602\pi\)
0.0922250 + 0.995738i \(0.470602\pi\)
\(44\) 0 0
\(45\) 315392. 0.515949
\(46\) 0 0
\(47\) 156171. 0.219411 0.109705 0.993964i \(-0.465009\pi\)
0.109705 + 0.993964i \(0.465009\pi\)
\(48\) 0 0
\(49\) 283070. 0.343722
\(50\) 0 0
\(51\) 477134. 0.503668
\(52\) 0 0
\(53\) −686962. −0.633822 −0.316911 0.948455i \(-0.602646\pi\)
−0.316911 + 0.948455i \(0.602646\pi\)
\(54\) 0 0
\(55\) 799641. 0.648075
\(56\) 0 0
\(57\) −997572. −0.713480
\(58\) 0 0
\(59\) −1.79961e6 −1.14077 −0.570384 0.821378i \(-0.693206\pi\)
−0.570384 + 0.821378i \(0.693206\pi\)
\(60\) 0 0
\(61\) 1.36394e6 0.769379 0.384690 0.923046i \(-0.374308\pi\)
0.384690 + 0.923046i \(0.374308\pi\)
\(62\) 0 0
\(63\) −1.79422e6 −0.904031
\(64\) 0 0
\(65\) −2.08090e6 −0.939841
\(66\) 0 0
\(67\) −1.08853e6 −0.442160 −0.221080 0.975256i \(-0.570958\pi\)
−0.221080 + 0.975256i \(0.570958\pi\)
\(68\) 0 0
\(69\) −96854.8 −0.0354936
\(70\) 0 0
\(71\) −5.60830e6 −1.85963 −0.929816 0.368025i \(-0.880034\pi\)
−0.929816 + 0.368025i \(0.880034\pi\)
\(72\) 0 0
\(73\) −21698.7 −0.00652837 −0.00326418 0.999995i \(-0.501039\pi\)
−0.00326418 + 0.999995i \(0.501039\pi\)
\(74\) 0 0
\(75\) 963886. 0.263822
\(76\) 0 0
\(77\) −4.54903e6 −1.13554
\(78\) 0 0
\(79\) −2.34010e6 −0.533998 −0.266999 0.963697i \(-0.586032\pi\)
−0.266999 + 0.963697i \(0.586032\pi\)
\(80\) 0 0
\(81\) 1.85626e6 0.388097
\(82\) 0 0
\(83\) −882169. −0.169347 −0.0846736 0.996409i \(-0.526985\pi\)
−0.0846736 + 0.996409i \(0.526985\pi\)
\(84\) 0 0
\(85\) 4.02125e6 0.710223
\(86\) 0 0
\(87\) 519714. 0.0846150
\(88\) 0 0
\(89\) 1.34738e6 0.202594 0.101297 0.994856i \(-0.467701\pi\)
0.101297 + 0.994856i \(0.467701\pi\)
\(90\) 0 0
\(91\) 1.18379e7 1.64676
\(92\) 0 0
\(93\) 1.60040e6 0.206319
\(94\) 0 0
\(95\) −8.40746e6 −1.00608
\(96\) 0 0
\(97\) 7.32798e6 0.815236 0.407618 0.913153i \(-0.366360\pi\)
0.407618 + 0.913153i \(0.366360\pi\)
\(98\) 0 0
\(99\) 7.37562e6 0.763969
\(100\) 0 0
\(101\) −9.66027e6 −0.932963 −0.466482 0.884531i \(-0.654479\pi\)
−0.466482 + 0.884531i \(0.654479\pi\)
\(102\) 0 0
\(103\) −1.38659e7 −1.25031 −0.625154 0.780501i \(-0.714964\pi\)
−0.625154 + 0.780501i \(0.714964\pi\)
\(104\) 0 0
\(105\) 4.26800e6 0.359800
\(106\) 0 0
\(107\) −1.88354e7 −1.48638 −0.743192 0.669078i \(-0.766689\pi\)
−0.743192 + 0.669078i \(0.766689\pi\)
\(108\) 0 0
\(109\) −8.60619e6 −0.636529 −0.318265 0.948002i \(-0.603100\pi\)
−0.318265 + 0.948002i \(0.603100\pi\)
\(110\) 0 0
\(111\) −1.06048e7 −0.735992
\(112\) 0 0
\(113\) −1.31340e7 −0.856292 −0.428146 0.903709i \(-0.640833\pi\)
−0.428146 + 0.903709i \(0.640833\pi\)
\(114\) 0 0
\(115\) −816286. −0.0500495
\(116\) 0 0
\(117\) −1.91935e7 −1.10791
\(118\) 0 0
\(119\) −2.28763e7 −1.24443
\(120\) 0 0
\(121\) −787128. −0.0403921
\(122\) 0 0
\(123\) −9.01856e6 −0.436988
\(124\) 0 0
\(125\) 2.25701e7 1.03359
\(126\) 0 0
\(127\) −3.18768e7 −1.38090 −0.690449 0.723381i \(-0.742587\pi\)
−0.690449 + 0.723381i \(0.742587\pi\)
\(128\) 0 0
\(129\) −2.10994e6 −0.0865380
\(130\) 0 0
\(131\) −1.17050e7 −0.454905 −0.227453 0.973789i \(-0.573040\pi\)
−0.227453 + 0.973789i \(0.573040\pi\)
\(132\) 0 0
\(133\) 4.78288e7 1.76282
\(134\) 0 0
\(135\) −1.57931e7 −0.552456
\(136\) 0 0
\(137\) −3.79174e7 −1.25984 −0.629921 0.776659i \(-0.716913\pi\)
−0.629921 + 0.776659i \(0.716913\pi\)
\(138\) 0 0
\(139\) 2.80300e7 0.885260 0.442630 0.896704i \(-0.354045\pi\)
0.442630 + 0.896704i \(0.354045\pi\)
\(140\) 0 0
\(141\) −3.42652e6 −0.102941
\(142\) 0 0
\(143\) −4.86630e7 −1.39163
\(144\) 0 0
\(145\) 4.38012e6 0.119316
\(146\) 0 0
\(147\) −6.21078e6 −0.161263
\(148\) 0 0
\(149\) 6.82083e6 0.168922 0.0844608 0.996427i \(-0.473083\pi\)
0.0844608 + 0.996427i \(0.473083\pi\)
\(150\) 0 0
\(151\) −7.62068e7 −1.80125 −0.900626 0.434595i \(-0.856891\pi\)
−0.900626 + 0.434595i \(0.856891\pi\)
\(152\) 0 0
\(153\) 3.70907e7 0.837230
\(154\) 0 0
\(155\) 1.34881e7 0.290931
\(156\) 0 0
\(157\) −4.28737e7 −0.884183 −0.442092 0.896970i \(-0.645763\pi\)
−0.442092 + 0.896970i \(0.645763\pi\)
\(158\) 0 0
\(159\) 1.50725e7 0.297369
\(160\) 0 0
\(161\) 4.64373e6 0.0876952
\(162\) 0 0
\(163\) −4.12127e7 −0.745374 −0.372687 0.927957i \(-0.621563\pi\)
−0.372687 + 0.927957i \(0.621563\pi\)
\(164\) 0 0
\(165\) −1.75448e7 −0.304056
\(166\) 0 0
\(167\) 2.41703e7 0.401582 0.200791 0.979634i \(-0.435649\pi\)
0.200791 + 0.979634i \(0.435649\pi\)
\(168\) 0 0
\(169\) 6.38869e7 1.01814
\(170\) 0 0
\(171\) −7.75477e7 −1.18599
\(172\) 0 0
\(173\) 5.90454e7 0.867011 0.433505 0.901151i \(-0.357277\pi\)
0.433505 + 0.901151i \(0.357277\pi\)
\(174\) 0 0
\(175\) −4.62137e7 −0.651835
\(176\) 0 0
\(177\) 3.94850e7 0.535212
\(178\) 0 0
\(179\) 1.20311e8 1.56791 0.783954 0.620819i \(-0.213200\pi\)
0.783954 + 0.620819i \(0.213200\pi\)
\(180\) 0 0
\(181\) 3.21539e6 0.0403049 0.0201525 0.999797i \(-0.493585\pi\)
0.0201525 + 0.999797i \(0.493585\pi\)
\(182\) 0 0
\(183\) −2.99259e7 −0.360968
\(184\) 0 0
\(185\) −8.93768e7 −1.03782
\(186\) 0 0
\(187\) 9.40392e7 1.05163
\(188\) 0 0
\(189\) 8.98443e7 0.967997
\(190\) 0 0
\(191\) −7.04122e7 −0.731191 −0.365596 0.930774i \(-0.619135\pi\)
−0.365596 + 0.930774i \(0.619135\pi\)
\(192\) 0 0
\(193\) 1.17861e7 0.118010 0.0590050 0.998258i \(-0.481207\pi\)
0.0590050 + 0.998258i \(0.481207\pi\)
\(194\) 0 0
\(195\) 4.56566e7 0.440943
\(196\) 0 0
\(197\) −8.97008e7 −0.835920 −0.417960 0.908465i \(-0.637255\pi\)
−0.417960 + 0.908465i \(0.637255\pi\)
\(198\) 0 0
\(199\) −3.98955e7 −0.358871 −0.179436 0.983770i \(-0.557427\pi\)
−0.179436 + 0.983770i \(0.557427\pi\)
\(200\) 0 0
\(201\) 2.38833e7 0.207447
\(202\) 0 0
\(203\) −2.49178e7 −0.209061
\(204\) 0 0
\(205\) −7.60078e7 −0.616197
\(206\) 0 0
\(207\) −7.52915e6 −0.0589997
\(208\) 0 0
\(209\) −1.96613e8 −1.48971
\(210\) 0 0
\(211\) −6.63604e7 −0.486318 −0.243159 0.969986i \(-0.578184\pi\)
−0.243159 + 0.969986i \(0.578184\pi\)
\(212\) 0 0
\(213\) 1.23051e8 0.872479
\(214\) 0 0
\(215\) −1.77825e7 −0.122027
\(216\) 0 0
\(217\) −7.67317e7 −0.509760
\(218\) 0 0
\(219\) 476088. 0.00306290
\(220\) 0 0
\(221\) −2.44718e8 −1.52508
\(222\) 0 0
\(223\) 1.57269e8 0.949675 0.474837 0.880074i \(-0.342507\pi\)
0.474837 + 0.880074i \(0.342507\pi\)
\(224\) 0 0
\(225\) 7.49291e7 0.438543
\(226\) 0 0
\(227\) −1.95129e8 −1.10721 −0.553607 0.832778i \(-0.686749\pi\)
−0.553607 + 0.832778i \(0.686749\pi\)
\(228\) 0 0
\(229\) −6.95136e7 −0.382512 −0.191256 0.981540i \(-0.561256\pi\)
−0.191256 + 0.981540i \(0.561256\pi\)
\(230\) 0 0
\(231\) 9.98095e7 0.532758
\(232\) 0 0
\(233\) 3.81872e7 0.197776 0.0988878 0.995099i \(-0.468472\pi\)
0.0988878 + 0.995099i \(0.468472\pi\)
\(234\) 0 0
\(235\) −2.88785e7 −0.145157
\(236\) 0 0
\(237\) 5.13437e7 0.250535
\(238\) 0 0
\(239\) 2.63549e8 1.24873 0.624366 0.781132i \(-0.285357\pi\)
0.624366 + 0.781132i \(0.285357\pi\)
\(240\) 0 0
\(241\) 1.21337e8 0.558386 0.279193 0.960235i \(-0.409933\pi\)
0.279193 + 0.960235i \(0.409933\pi\)
\(242\) 0 0
\(243\) −2.27512e8 −1.01715
\(244\) 0 0
\(245\) −5.23441e7 −0.227398
\(246\) 0 0
\(247\) 5.11645e8 2.16038
\(248\) 0 0
\(249\) 1.93555e7 0.0794523
\(250\) 0 0
\(251\) −3.62042e7 −0.144511 −0.0722555 0.997386i \(-0.523020\pi\)
−0.0722555 + 0.997386i \(0.523020\pi\)
\(252\) 0 0
\(253\) −1.90893e7 −0.0741085
\(254\) 0 0
\(255\) −8.82295e7 −0.333214
\(256\) 0 0
\(257\) 1.75550e8 0.645111 0.322555 0.946551i \(-0.395458\pi\)
0.322555 + 0.946551i \(0.395458\pi\)
\(258\) 0 0
\(259\) 5.08451e8 1.81844
\(260\) 0 0
\(261\) 4.04008e7 0.140653
\(262\) 0 0
\(263\) −1.10733e8 −0.375348 −0.187674 0.982231i \(-0.560095\pi\)
−0.187674 + 0.982231i \(0.560095\pi\)
\(264\) 0 0
\(265\) 1.27030e8 0.419320
\(266\) 0 0
\(267\) −2.95627e7 −0.0950505
\(268\) 0 0
\(269\) 3.07561e8 0.963382 0.481691 0.876341i \(-0.340023\pi\)
0.481691 + 0.876341i \(0.340023\pi\)
\(270\) 0 0
\(271\) 2.86036e8 0.873027 0.436513 0.899698i \(-0.356213\pi\)
0.436513 + 0.899698i \(0.356213\pi\)
\(272\) 0 0
\(273\) −2.59734e8 −0.772607
\(274\) 0 0
\(275\) 1.89974e8 0.550846
\(276\) 0 0
\(277\) −3.15642e8 −0.892309 −0.446155 0.894956i \(-0.647207\pi\)
−0.446155 + 0.894956i \(0.647207\pi\)
\(278\) 0 0
\(279\) 1.24410e8 0.342957
\(280\) 0 0
\(281\) 2.50451e8 0.673366 0.336683 0.941618i \(-0.390695\pi\)
0.336683 + 0.941618i \(0.390695\pi\)
\(282\) 0 0
\(283\) −3.47387e7 −0.0911088 −0.0455544 0.998962i \(-0.514505\pi\)
−0.0455544 + 0.998962i \(0.514505\pi\)
\(284\) 0 0
\(285\) 1.84467e8 0.472020
\(286\) 0 0
\(287\) 4.32397e8 1.07968
\(288\) 0 0
\(289\) 6.25676e7 0.152478
\(290\) 0 0
\(291\) −1.60782e8 −0.382482
\(292\) 0 0
\(293\) 5.34336e8 1.24102 0.620508 0.784200i \(-0.286926\pi\)
0.620508 + 0.784200i \(0.286926\pi\)
\(294\) 0 0
\(295\) 3.32777e8 0.754703
\(296\) 0 0
\(297\) −3.69329e8 −0.818024
\(298\) 0 0
\(299\) 4.96760e7 0.107472
\(300\) 0 0
\(301\) 1.01162e8 0.213813
\(302\) 0 0
\(303\) 2.11954e8 0.437716
\(304\) 0 0
\(305\) −2.52214e8 −0.509002
\(306\) 0 0
\(307\) 1.68851e8 0.333057 0.166529 0.986037i \(-0.446744\pi\)
0.166529 + 0.986037i \(0.446744\pi\)
\(308\) 0 0
\(309\) 3.04229e8 0.586605
\(310\) 0 0
\(311\) 6.99677e8 1.31897 0.659487 0.751716i \(-0.270774\pi\)
0.659487 + 0.751716i \(0.270774\pi\)
\(312\) 0 0
\(313\) −5.06345e8 −0.933343 −0.466671 0.884431i \(-0.654547\pi\)
−0.466671 + 0.884431i \(0.654547\pi\)
\(314\) 0 0
\(315\) 3.31779e8 0.598084
\(316\) 0 0
\(317\) −4.78996e8 −0.844548 −0.422274 0.906468i \(-0.638768\pi\)
−0.422274 + 0.906468i \(0.638768\pi\)
\(318\) 0 0
\(319\) 1.02431e8 0.176671
\(320\) 0 0
\(321\) 4.13263e8 0.697364
\(322\) 0 0
\(323\) −9.88733e8 −1.63256
\(324\) 0 0
\(325\) −4.94369e8 −0.798839
\(326\) 0 0
\(327\) 1.88827e8 0.298639
\(328\) 0 0
\(329\) 1.64285e8 0.254339
\(330\) 0 0
\(331\) 4.20825e8 0.637828 0.318914 0.947784i \(-0.396682\pi\)
0.318914 + 0.947784i \(0.396682\pi\)
\(332\) 0 0
\(333\) −8.24382e8 −1.22341
\(334\) 0 0
\(335\) 2.01287e8 0.292522
\(336\) 0 0
\(337\) −1.02987e9 −1.46580 −0.732902 0.680334i \(-0.761835\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(338\) 0 0
\(339\) 2.88170e8 0.401745
\(340\) 0 0
\(341\) 3.15427e8 0.430782
\(342\) 0 0
\(343\) −5.68554e8 −0.760751
\(344\) 0 0
\(345\) 1.79100e7 0.0234816
\(346\) 0 0
\(347\) −5.31496e8 −0.682883 −0.341442 0.939903i \(-0.610915\pi\)
−0.341442 + 0.939903i \(0.610915\pi\)
\(348\) 0 0
\(349\) 1.31436e9 1.65511 0.827555 0.561385i \(-0.189731\pi\)
0.827555 + 0.561385i \(0.189731\pi\)
\(350\) 0 0
\(351\) 9.61103e8 1.18630
\(352\) 0 0
\(353\) 1.42031e9 1.71858 0.859291 0.511486i \(-0.170905\pi\)
0.859291 + 0.511486i \(0.170905\pi\)
\(354\) 0 0
\(355\) 1.03706e9 1.23028
\(356\) 0 0
\(357\) 5.01924e8 0.583847
\(358\) 0 0
\(359\) 7.59524e8 0.866385 0.433193 0.901301i \(-0.357387\pi\)
0.433193 + 0.901301i \(0.357387\pi\)
\(360\) 0 0
\(361\) 1.17333e9 1.31264
\(362\) 0 0
\(363\) 1.72702e7 0.0189507
\(364\) 0 0
\(365\) 4.01244e6 0.00431900
\(366\) 0 0
\(367\) −3.76159e8 −0.397228 −0.198614 0.980078i \(-0.563644\pi\)
−0.198614 + 0.980078i \(0.563644\pi\)
\(368\) 0 0
\(369\) −7.01071e8 −0.726390
\(370\) 0 0
\(371\) −7.22655e8 −0.734720
\(372\) 0 0
\(373\) −4.57130e8 −0.456099 −0.228049 0.973650i \(-0.573235\pi\)
−0.228049 + 0.973650i \(0.573235\pi\)
\(374\) 0 0
\(375\) −4.95206e8 −0.484927
\(376\) 0 0
\(377\) −2.66557e8 −0.256209
\(378\) 0 0
\(379\) −1.91565e9 −1.80750 −0.903752 0.428056i \(-0.859199\pi\)
−0.903752 + 0.428056i \(0.859199\pi\)
\(380\) 0 0
\(381\) 6.99402e8 0.647873
\(382\) 0 0
\(383\) −1.57853e9 −1.43568 −0.717839 0.696209i \(-0.754869\pi\)
−0.717839 + 0.696209i \(0.754869\pi\)
\(384\) 0 0
\(385\) 8.41187e8 0.751243
\(386\) 0 0
\(387\) −1.64020e8 −0.143849
\(388\) 0 0
\(389\) −2.21214e9 −1.90541 −0.952707 0.303890i \(-0.901715\pi\)
−0.952707 + 0.303890i \(0.901715\pi\)
\(390\) 0 0
\(391\) −9.59967e7 −0.0812153
\(392\) 0 0
\(393\) 2.56817e8 0.213427
\(394\) 0 0
\(395\) 4.32721e8 0.353279
\(396\) 0 0
\(397\) −1.30879e9 −1.04979 −0.524896 0.851167i \(-0.675896\pi\)
−0.524896 + 0.851167i \(0.675896\pi\)
\(398\) 0 0
\(399\) −1.04940e9 −0.827060
\(400\) 0 0
\(401\) 1.04514e8 0.0809413 0.0404706 0.999181i \(-0.487114\pi\)
0.0404706 + 0.999181i \(0.487114\pi\)
\(402\) 0 0
\(403\) −8.20832e8 −0.624722
\(404\) 0 0
\(405\) −3.43251e8 −0.256755
\(406\) 0 0
\(407\) −2.09013e9 −1.53671
\(408\) 0 0
\(409\) −1.94107e9 −1.40284 −0.701421 0.712747i \(-0.747451\pi\)
−0.701421 + 0.712747i \(0.747451\pi\)
\(410\) 0 0
\(411\) 8.31937e8 0.591077
\(412\) 0 0
\(413\) −1.89312e9 −1.32237
\(414\) 0 0
\(415\) 1.63127e8 0.112036
\(416\) 0 0
\(417\) −6.15000e8 −0.415336
\(418\) 0 0
\(419\) 1.96429e9 1.30454 0.652268 0.757989i \(-0.273818\pi\)
0.652268 + 0.757989i \(0.273818\pi\)
\(420\) 0 0
\(421\) 2.33176e9 1.52299 0.761493 0.648174i \(-0.224467\pi\)
0.761493 + 0.648174i \(0.224467\pi\)
\(422\) 0 0
\(423\) −2.66366e8 −0.171115
\(424\) 0 0
\(425\) 9.55346e8 0.603670
\(426\) 0 0
\(427\) 1.43480e9 0.891857
\(428\) 0 0
\(429\) 1.06771e9 0.652907
\(430\) 0 0
\(431\) −1.72745e8 −0.103929 −0.0519643 0.998649i \(-0.516548\pi\)
−0.0519643 + 0.998649i \(0.516548\pi\)
\(432\) 0 0
\(433\) −2.71381e9 −1.60647 −0.803233 0.595665i \(-0.796889\pi\)
−0.803233 + 0.595665i \(0.796889\pi\)
\(434\) 0 0
\(435\) −9.61033e7 −0.0559791
\(436\) 0 0
\(437\) 2.00706e8 0.115047
\(438\) 0 0
\(439\) −2.73808e8 −0.154461 −0.0772307 0.997013i \(-0.524608\pi\)
−0.0772307 + 0.997013i \(0.524608\pi\)
\(440\) 0 0
\(441\) −4.82804e8 −0.268063
\(442\) 0 0
\(443\) 2.72048e9 1.48673 0.743365 0.668886i \(-0.233229\pi\)
0.743365 + 0.668886i \(0.233229\pi\)
\(444\) 0 0
\(445\) −2.49152e8 −0.134031
\(446\) 0 0
\(447\) −1.49654e8 −0.0792526
\(448\) 0 0
\(449\) 1.69111e9 0.881675 0.440837 0.897587i \(-0.354682\pi\)
0.440837 + 0.897587i \(0.354682\pi\)
\(450\) 0 0
\(451\) −1.77748e9 −0.912406
\(452\) 0 0
\(453\) 1.67204e9 0.845089
\(454\) 0 0
\(455\) −2.18902e9 −1.08945
\(456\) 0 0
\(457\) 1.15773e8 0.0567413 0.0283706 0.999597i \(-0.490968\pi\)
0.0283706 + 0.999597i \(0.490968\pi\)
\(458\) 0 0
\(459\) −1.85729e9 −0.896470
\(460\) 0 0
\(461\) 9.34713e8 0.444350 0.222175 0.975007i \(-0.428684\pi\)
0.222175 + 0.975007i \(0.428684\pi\)
\(462\) 0 0
\(463\) −1.91738e9 −0.897790 −0.448895 0.893584i \(-0.648182\pi\)
−0.448895 + 0.893584i \(0.648182\pi\)
\(464\) 0 0
\(465\) −2.95940e8 −0.136495
\(466\) 0 0
\(467\) 1.65838e9 0.753485 0.376742 0.926318i \(-0.377044\pi\)
0.376742 + 0.926318i \(0.377044\pi\)
\(468\) 0 0
\(469\) −1.14509e9 −0.512548
\(470\) 0 0
\(471\) 9.40684e8 0.414830
\(472\) 0 0
\(473\) −4.15853e8 −0.180686
\(474\) 0 0
\(475\) −1.99740e9 −0.855140
\(476\) 0 0
\(477\) 1.17168e9 0.494306
\(478\) 0 0
\(479\) −4.09755e9 −1.70353 −0.851766 0.523923i \(-0.824468\pi\)
−0.851766 + 0.523923i \(0.824468\pi\)
\(480\) 0 0
\(481\) 5.43912e9 2.22854
\(482\) 0 0
\(483\) −1.01887e8 −0.0411438
\(484\) 0 0
\(485\) −1.35506e9 −0.539339
\(486\) 0 0
\(487\) −3.24167e9 −1.27180 −0.635898 0.771773i \(-0.719370\pi\)
−0.635898 + 0.771773i \(0.719370\pi\)
\(488\) 0 0
\(489\) 9.04239e8 0.349705
\(490\) 0 0
\(491\) −2.16351e9 −0.824847 −0.412423 0.910992i \(-0.635318\pi\)
−0.412423 + 0.910992i \(0.635318\pi\)
\(492\) 0 0
\(493\) 5.15110e8 0.193613
\(494\) 0 0
\(495\) −1.36387e9 −0.505422
\(496\) 0 0
\(497\) −5.89969e9 −2.15567
\(498\) 0 0
\(499\) −2.16713e9 −0.780787 −0.390394 0.920648i \(-0.627661\pi\)
−0.390394 + 0.920648i \(0.627661\pi\)
\(500\) 0 0
\(501\) −5.30316e8 −0.188409
\(502\) 0 0
\(503\) −2.58480e9 −0.905606 −0.452803 0.891611i \(-0.649576\pi\)
−0.452803 + 0.891611i \(0.649576\pi\)
\(504\) 0 0
\(505\) 1.78633e9 0.617224
\(506\) 0 0
\(507\) −1.40173e9 −0.477680
\(508\) 0 0
\(509\) 2.11747e9 0.711712 0.355856 0.934541i \(-0.384189\pi\)
0.355856 + 0.934541i \(0.384189\pi\)
\(510\) 0 0
\(511\) −2.28261e7 −0.00756762
\(512\) 0 0
\(513\) 3.88315e9 1.26991
\(514\) 0 0
\(515\) 2.56402e9 0.827172
\(516\) 0 0
\(517\) −6.75339e8 −0.214934
\(518\) 0 0
\(519\) −1.29550e9 −0.406773
\(520\) 0 0
\(521\) 2.34354e9 0.726006 0.363003 0.931788i \(-0.381751\pi\)
0.363003 + 0.931788i \(0.381751\pi\)
\(522\) 0 0
\(523\) 3.26155e9 0.996939 0.498470 0.866907i \(-0.333896\pi\)
0.498470 + 0.866907i \(0.333896\pi\)
\(524\) 0 0
\(525\) 1.01397e9 0.305820
\(526\) 0 0
\(527\) 1.58622e9 0.472093
\(528\) 0 0
\(529\) −3.38534e9 −0.994277
\(530\) 0 0
\(531\) 3.06942e9 0.889664
\(532\) 0 0
\(533\) 4.62554e9 1.32317
\(534\) 0 0
\(535\) 3.48296e9 0.983354
\(536\) 0 0
\(537\) −2.63973e9 −0.735612
\(538\) 0 0
\(539\) −1.22409e9 −0.336709
\(540\) 0 0
\(541\) −7.24638e9 −1.96757 −0.983787 0.179344i \(-0.942603\pi\)
−0.983787 + 0.179344i \(0.942603\pi\)
\(542\) 0 0
\(543\) −7.05482e7 −0.0189098
\(544\) 0 0
\(545\) 1.59142e9 0.421111
\(546\) 0 0
\(547\) −6.41460e9 −1.67577 −0.837884 0.545848i \(-0.816207\pi\)
−0.837884 + 0.545848i \(0.816207\pi\)
\(548\) 0 0
\(549\) −2.32634e9 −0.600025
\(550\) 0 0
\(551\) −1.07697e9 −0.274267
\(552\) 0 0
\(553\) −2.46168e9 −0.619005
\(554\) 0 0
\(555\) 1.96100e9 0.486914
\(556\) 0 0
\(557\) 1.76223e9 0.432085 0.216042 0.976384i \(-0.430685\pi\)
0.216042 + 0.976384i \(0.430685\pi\)
\(558\) 0 0
\(559\) 1.08217e9 0.262032
\(560\) 0 0
\(561\) −2.06330e9 −0.493391
\(562\) 0 0
\(563\) −2.33270e9 −0.550909 −0.275454 0.961314i \(-0.588828\pi\)
−0.275454 + 0.961314i \(0.588828\pi\)
\(564\) 0 0
\(565\) 2.42868e9 0.566501
\(566\) 0 0
\(567\) 1.95270e9 0.449878
\(568\) 0 0
\(569\) −5.53591e9 −1.25978 −0.629892 0.776683i \(-0.716901\pi\)
−0.629892 + 0.776683i \(0.716901\pi\)
\(570\) 0 0
\(571\) 3.70890e9 0.833717 0.416858 0.908971i \(-0.363131\pi\)
0.416858 + 0.908971i \(0.363131\pi\)
\(572\) 0 0
\(573\) 1.54490e9 0.343051
\(574\) 0 0
\(575\) −1.93929e8 −0.0425407
\(576\) 0 0
\(577\) 3.63713e9 0.788212 0.394106 0.919065i \(-0.371054\pi\)
0.394106 + 0.919065i \(0.371054\pi\)
\(578\) 0 0
\(579\) −2.58596e8 −0.0553665
\(580\) 0 0
\(581\) −9.28003e8 −0.196306
\(582\) 0 0
\(583\) 2.97067e9 0.620889
\(584\) 0 0
\(585\) 3.54919e9 0.732965
\(586\) 0 0
\(587\) −2.04311e9 −0.416926 −0.208463 0.978030i \(-0.566846\pi\)
−0.208463 + 0.978030i \(0.566846\pi\)
\(588\) 0 0
\(589\) −3.31641e9 −0.668752
\(590\) 0 0
\(591\) 1.96811e9 0.392187
\(592\) 0 0
\(593\) −5.74764e9 −1.13187 −0.565937 0.824448i \(-0.691486\pi\)
−0.565937 + 0.824448i \(0.691486\pi\)
\(594\) 0 0
\(595\) 4.23018e9 0.823284
\(596\) 0 0
\(597\) 8.75341e8 0.168371
\(598\) 0 0
\(599\) 3.14564e8 0.0598020 0.0299010 0.999553i \(-0.490481\pi\)
0.0299010 + 0.999553i \(0.490481\pi\)
\(600\) 0 0
\(601\) 2.49586e9 0.468985 0.234493 0.972118i \(-0.424657\pi\)
0.234493 + 0.972118i \(0.424657\pi\)
\(602\) 0 0
\(603\) 1.85660e9 0.344833
\(604\) 0 0
\(605\) 1.45552e8 0.0267224
\(606\) 0 0
\(607\) 2.81126e9 0.510200 0.255100 0.966915i \(-0.417892\pi\)
0.255100 + 0.966915i \(0.417892\pi\)
\(608\) 0 0
\(609\) 5.46717e8 0.0980849
\(610\) 0 0
\(611\) 1.75743e9 0.311698
\(612\) 0 0
\(613\) −2.34343e9 −0.410904 −0.205452 0.978667i \(-0.565866\pi\)
−0.205452 + 0.978667i \(0.565866\pi\)
\(614\) 0 0
\(615\) 1.66767e9 0.289100
\(616\) 0 0
\(617\) 6.26737e8 0.107420 0.0537102 0.998557i \(-0.482895\pi\)
0.0537102 + 0.998557i \(0.482895\pi\)
\(618\) 0 0
\(619\) −8.90879e9 −1.50974 −0.754869 0.655876i \(-0.772300\pi\)
−0.754869 + 0.655876i \(0.772300\pi\)
\(620\) 0 0
\(621\) 3.77017e8 0.0631743
\(622\) 0 0
\(623\) 1.41739e9 0.234845
\(624\) 0 0
\(625\) −7.41437e8 −0.121477
\(626\) 0 0
\(627\) 4.31385e9 0.698922
\(628\) 0 0
\(629\) −1.05109e10 −1.68407
\(630\) 0 0
\(631\) −2.21942e9 −0.351671 −0.175835 0.984420i \(-0.556263\pi\)
−0.175835 + 0.984420i \(0.556263\pi\)
\(632\) 0 0
\(633\) 1.45600e9 0.228165
\(634\) 0 0
\(635\) 5.89451e9 0.913566
\(636\) 0 0
\(637\) 3.18545e9 0.488296
\(638\) 0 0
\(639\) 9.56552e9 1.45029
\(640\) 0 0
\(641\) 8.58104e9 1.28688 0.643438 0.765498i \(-0.277507\pi\)
0.643438 + 0.765498i \(0.277507\pi\)
\(642\) 0 0
\(643\) 7.71279e9 1.14412 0.572062 0.820210i \(-0.306144\pi\)
0.572062 + 0.820210i \(0.306144\pi\)
\(644\) 0 0
\(645\) 3.90161e8 0.0572513
\(646\) 0 0
\(647\) 1.18898e10 1.72587 0.862935 0.505316i \(-0.168624\pi\)
0.862935 + 0.505316i \(0.168624\pi\)
\(648\) 0 0
\(649\) 7.78217e9 1.11749
\(650\) 0 0
\(651\) 1.68356e9 0.239163
\(652\) 0 0
\(653\) −2.00345e9 −0.281567 −0.140784 0.990040i \(-0.544962\pi\)
−0.140784 + 0.990040i \(0.544962\pi\)
\(654\) 0 0
\(655\) 2.16443e9 0.300954
\(656\) 0 0
\(657\) 3.70094e7 0.00509135
\(658\) 0 0
\(659\) −7.66944e9 −1.04391 −0.521957 0.852972i \(-0.674798\pi\)
−0.521957 + 0.852972i \(0.674798\pi\)
\(660\) 0 0
\(661\) −4.04041e9 −0.544152 −0.272076 0.962276i \(-0.587710\pi\)
−0.272076 + 0.962276i \(0.587710\pi\)
\(662\) 0 0
\(663\) 5.36930e9 0.715518
\(664\) 0 0
\(665\) −8.84429e9 −1.16624
\(666\) 0 0
\(667\) −1.04564e8 −0.0136440
\(668\) 0 0
\(669\) −3.45060e9 −0.445557
\(670\) 0 0
\(671\) −5.89816e9 −0.753681
\(672\) 0 0
\(673\) 4.11138e9 0.519918 0.259959 0.965620i \(-0.416291\pi\)
0.259959 + 0.965620i \(0.416291\pi\)
\(674\) 0 0
\(675\) −3.75203e9 −0.469572
\(676\) 0 0
\(677\) 1.12671e10 1.39557 0.697787 0.716305i \(-0.254168\pi\)
0.697787 + 0.716305i \(0.254168\pi\)
\(678\) 0 0
\(679\) 7.70872e9 0.945014
\(680\) 0 0
\(681\) 4.28129e9 0.519469
\(682\) 0 0
\(683\) −8.00228e9 −0.961039 −0.480520 0.876984i \(-0.659552\pi\)
−0.480520 + 0.876984i \(0.659552\pi\)
\(684\) 0 0
\(685\) 7.01151e9 0.833479
\(686\) 0 0
\(687\) 1.52518e9 0.179462
\(688\) 0 0
\(689\) −7.73055e9 −0.900416
\(690\) 0 0
\(691\) −7.24716e9 −0.835593 −0.417797 0.908541i \(-0.637197\pi\)
−0.417797 + 0.908541i \(0.637197\pi\)
\(692\) 0 0
\(693\) 7.75884e9 0.885585
\(694\) 0 0
\(695\) −5.18318e9 −0.585665
\(696\) 0 0
\(697\) −8.93865e9 −0.999902
\(698\) 0 0
\(699\) −8.37859e8 −0.0927899
\(700\) 0 0
\(701\) −4.68293e8 −0.0513458 −0.0256729 0.999670i \(-0.508173\pi\)
−0.0256729 + 0.999670i \(0.508173\pi\)
\(702\) 0 0
\(703\) 2.19757e10 2.38561
\(704\) 0 0
\(705\) 6.33617e8 0.0681028
\(706\) 0 0
\(707\) −1.01622e10 −1.08148
\(708\) 0 0
\(709\) 1.08926e10 1.14781 0.573903 0.818923i \(-0.305429\pi\)
0.573903 + 0.818923i \(0.305429\pi\)
\(710\) 0 0
\(711\) 3.99128e9 0.416455
\(712\) 0 0
\(713\) −3.21992e8 −0.0332684
\(714\) 0 0
\(715\) 8.99855e9 0.920664
\(716\) 0 0
\(717\) −5.78249e9 −0.585865
\(718\) 0 0
\(719\) 7.40284e9 0.742758 0.371379 0.928481i \(-0.378885\pi\)
0.371379 + 0.928481i \(0.378885\pi\)
\(720\) 0 0
\(721\) −1.45863e10 −1.44935
\(722\) 0 0
\(723\) −2.66224e9 −0.261977
\(724\) 0 0
\(725\) 1.04060e9 0.101415
\(726\) 0 0
\(727\) 1.04295e10 1.00668 0.503342 0.864087i \(-0.332103\pi\)
0.503342 + 0.864087i \(0.332103\pi\)
\(728\) 0 0
\(729\) 9.32176e8 0.0891152
\(730\) 0 0
\(731\) −2.09125e9 −0.198014
\(732\) 0 0
\(733\) 1.55538e10 1.45873 0.729363 0.684127i \(-0.239816\pi\)
0.729363 + 0.684127i \(0.239816\pi\)
\(734\) 0 0
\(735\) 1.14847e9 0.106688
\(736\) 0 0
\(737\) 4.70720e9 0.433138
\(738\) 0 0
\(739\) 1.07140e10 0.976554 0.488277 0.872689i \(-0.337626\pi\)
0.488277 + 0.872689i \(0.337626\pi\)
\(740\) 0 0
\(741\) −1.12259e10 −1.01358
\(742\) 0 0
\(743\) −5.46982e9 −0.489229 −0.244615 0.969620i \(-0.578661\pi\)
−0.244615 + 0.969620i \(0.578661\pi\)
\(744\) 0 0
\(745\) −1.26128e9 −0.111754
\(746\) 0 0
\(747\) 1.50463e9 0.132071
\(748\) 0 0
\(749\) −1.98140e10 −1.72300
\(750\) 0 0
\(751\) −5.68185e9 −0.489497 −0.244749 0.969587i \(-0.578705\pi\)
−0.244749 + 0.969587i \(0.578705\pi\)
\(752\) 0 0
\(753\) 7.94349e8 0.0677999
\(754\) 0 0
\(755\) 1.40918e10 1.19166
\(756\) 0 0
\(757\) −5.38304e9 −0.451016 −0.225508 0.974241i \(-0.572404\pi\)
−0.225508 + 0.974241i \(0.572404\pi\)
\(758\) 0 0
\(759\) 4.18835e8 0.0347693
\(760\) 0 0
\(761\) 3.28593e9 0.270279 0.135139 0.990827i \(-0.456852\pi\)
0.135139 + 0.990827i \(0.456852\pi\)
\(762\) 0 0
\(763\) −9.05334e9 −0.737858
\(764\) 0 0
\(765\) −6.85865e9 −0.553890
\(766\) 0 0
\(767\) −2.02515e10 −1.62059
\(768\) 0 0
\(769\) 7.10914e9 0.563735 0.281868 0.959453i \(-0.409046\pi\)
0.281868 + 0.959453i \(0.409046\pi\)
\(770\) 0 0
\(771\) −3.85170e9 −0.302665
\(772\) 0 0
\(773\) −1.08690e10 −0.846370 −0.423185 0.906043i \(-0.639088\pi\)
−0.423185 + 0.906043i \(0.639088\pi\)
\(774\) 0 0
\(775\) 3.20443e9 0.247283
\(776\) 0 0
\(777\) −1.11558e10 −0.853155
\(778\) 0 0
\(779\) 1.86886e10 1.41643
\(780\) 0 0
\(781\) 2.42523e10 1.82169
\(782\) 0 0
\(783\) −2.02304e9 −0.150605
\(784\) 0 0
\(785\) 7.92802e9 0.584953
\(786\) 0 0
\(787\) 1.94246e10 1.42050 0.710248 0.703951i \(-0.248583\pi\)
0.710248 + 0.703951i \(0.248583\pi\)
\(788\) 0 0
\(789\) 2.42958e9 0.176101
\(790\) 0 0
\(791\) −1.38164e10 −0.992606
\(792\) 0 0
\(793\) 1.53487e10 1.09299
\(794\) 0 0
\(795\) −2.78714e9 −0.196732
\(796\) 0 0
\(797\) 2.30309e10 1.61141 0.805707 0.592314i \(-0.201786\pi\)
0.805707 + 0.592314i \(0.201786\pi\)
\(798\) 0 0
\(799\) −3.39616e9 −0.235545
\(800\) 0 0
\(801\) −2.29810e9 −0.157999
\(802\) 0 0
\(803\) 9.38331e7 0.00639516
\(804\) 0 0
\(805\) −8.58697e8 −0.0580169
\(806\) 0 0
\(807\) −6.74814e9 −0.451988
\(808\) 0 0
\(809\) 1.50546e10 0.999656 0.499828 0.866125i \(-0.333397\pi\)
0.499828 + 0.866125i \(0.333397\pi\)
\(810\) 0 0
\(811\) −1.98821e10 −1.30885 −0.654424 0.756128i \(-0.727089\pi\)
−0.654424 + 0.756128i \(0.727089\pi\)
\(812\) 0 0
\(813\) −6.27585e9 −0.409596
\(814\) 0 0
\(815\) 7.62086e9 0.493120
\(816\) 0 0
\(817\) 4.37230e9 0.280500
\(818\) 0 0
\(819\) −2.01908e10 −1.28428
\(820\) 0 0
\(821\) 2.29870e10 1.44971 0.724855 0.688901i \(-0.241907\pi\)
0.724855 + 0.688901i \(0.241907\pi\)
\(822\) 0 0
\(823\) 1.46696e10 0.917316 0.458658 0.888613i \(-0.348330\pi\)
0.458658 + 0.888613i \(0.348330\pi\)
\(824\) 0 0
\(825\) −4.16819e9 −0.258439
\(826\) 0 0
\(827\) 1.26591e9 0.0778277 0.0389138 0.999243i \(-0.487610\pi\)
0.0389138 + 0.999243i \(0.487610\pi\)
\(828\) 0 0
\(829\) −2.49778e10 −1.52270 −0.761349 0.648342i \(-0.775463\pi\)
−0.761349 + 0.648342i \(0.775463\pi\)
\(830\) 0 0
\(831\) 6.92544e9 0.418643
\(832\) 0 0
\(833\) −6.15576e9 −0.368998
\(834\) 0 0
\(835\) −4.46947e9 −0.265676
\(836\) 0 0
\(837\) −6.22973e9 −0.367223
\(838\) 0 0
\(839\) −3.09477e10 −1.80909 −0.904547 0.426373i \(-0.859791\pi\)
−0.904547 + 0.426373i \(0.859791\pi\)
\(840\) 0 0
\(841\) −1.66888e10 −0.967473
\(842\) 0 0
\(843\) −5.49510e9 −0.315921
\(844\) 0 0
\(845\) −1.18137e10 −0.673577
\(846\) 0 0
\(847\) −8.28024e8 −0.0468221
\(848\) 0 0
\(849\) 7.62194e8 0.0427453
\(850\) 0 0
\(851\) 2.13363e9 0.118677
\(852\) 0 0
\(853\) −1.86385e10 −1.02823 −0.514113 0.857723i \(-0.671879\pi\)
−0.514113 + 0.857723i \(0.671879\pi\)
\(854\) 0 0
\(855\) 1.43398e10 0.784623
\(856\) 0 0
\(857\) 2.37366e10 1.28821 0.644103 0.764939i \(-0.277231\pi\)
0.644103 + 0.764939i \(0.277231\pi\)
\(858\) 0 0
\(859\) 2.46268e10 1.32566 0.662830 0.748770i \(-0.269355\pi\)
0.662830 + 0.748770i \(0.269355\pi\)
\(860\) 0 0
\(861\) −9.48713e9 −0.506552
\(862\) 0 0
\(863\) −2.50051e10 −1.32431 −0.662157 0.749365i \(-0.730359\pi\)
−0.662157 + 0.749365i \(0.730359\pi\)
\(864\) 0 0
\(865\) −1.09184e10 −0.573592
\(866\) 0 0
\(867\) −1.37278e9 −0.0715378
\(868\) 0 0
\(869\) 1.01194e10 0.523102
\(870\) 0 0
\(871\) −1.22495e10 −0.628139
\(872\) 0 0
\(873\) −1.24986e10 −0.635788
\(874\) 0 0
\(875\) 2.37428e10 1.19813
\(876\) 0 0
\(877\) −1.07280e10 −0.537057 −0.268528 0.963272i \(-0.586537\pi\)
−0.268528 + 0.963272i \(0.586537\pi\)
\(878\) 0 0
\(879\) −1.17238e10 −0.582245
\(880\) 0 0
\(881\) 5.94499e9 0.292911 0.146455 0.989217i \(-0.453214\pi\)
0.146455 + 0.989217i \(0.453214\pi\)
\(882\) 0 0
\(883\) −1.60870e10 −0.786342 −0.393171 0.919465i \(-0.628622\pi\)
−0.393171 + 0.919465i \(0.628622\pi\)
\(884\) 0 0
\(885\) −7.30139e9 −0.354082
\(886\) 0 0
\(887\) −2.61884e10 −1.26001 −0.630007 0.776589i \(-0.716948\pi\)
−0.630007 + 0.776589i \(0.716948\pi\)
\(888\) 0 0
\(889\) −3.35330e10 −1.60072
\(890\) 0 0
\(891\) −8.02711e9 −0.380178
\(892\) 0 0
\(893\) 7.10056e9 0.333666
\(894\) 0 0
\(895\) −2.22474e10 −1.03729
\(896\) 0 0
\(897\) −1.08993e9 −0.0504226
\(898\) 0 0
\(899\) 1.72778e9 0.0793104
\(900\) 0 0
\(901\) 1.49390e10 0.680430
\(902\) 0 0
\(903\) −2.21957e9 −0.100314
\(904\) 0 0
\(905\) −5.94575e8 −0.0266647
\(906\) 0 0
\(907\) −1.79753e9 −0.0799929 −0.0399964 0.999200i \(-0.512735\pi\)
−0.0399964 + 0.999200i \(0.512735\pi\)
\(908\) 0 0
\(909\) 1.64766e10 0.727601
\(910\) 0 0
\(911\) −3.70021e10 −1.62148 −0.810740 0.585406i \(-0.800935\pi\)
−0.810740 + 0.585406i \(0.800935\pi\)
\(912\) 0 0
\(913\) 3.81481e9 0.165892
\(914\) 0 0
\(915\) 5.53377e9 0.238807
\(916\) 0 0
\(917\) −1.23131e10 −0.527322
\(918\) 0 0
\(919\) −2.44812e10 −1.04047 −0.520234 0.854024i \(-0.674155\pi\)
−0.520234 + 0.854024i \(0.674155\pi\)
\(920\) 0 0
\(921\) −3.70472e9 −0.156260
\(922\) 0 0
\(923\) −6.31115e10 −2.64182
\(924\) 0 0
\(925\) −2.12336e10 −0.882121
\(926\) 0 0
\(927\) 2.36497e10 0.975093
\(928\) 0 0
\(929\) −6.19981e9 −0.253702 −0.126851 0.991922i \(-0.540487\pi\)
−0.126851 + 0.991922i \(0.540487\pi\)
\(930\) 0 0
\(931\) 1.28702e10 0.522711
\(932\) 0 0
\(933\) −1.53515e10 −0.618820
\(934\) 0 0
\(935\) −1.73893e10 −0.695732
\(936\) 0 0
\(937\) 2.78747e10 1.10693 0.553466 0.832871i \(-0.313305\pi\)
0.553466 + 0.832871i \(0.313305\pi\)
\(938\) 0 0
\(939\) 1.11096e10 0.437894
\(940\) 0 0
\(941\) 3.45921e10 1.35336 0.676680 0.736277i \(-0.263418\pi\)
0.676680 + 0.736277i \(0.263418\pi\)
\(942\) 0 0
\(943\) 1.81448e9 0.0704632
\(944\) 0 0
\(945\) −1.66136e10 −0.640402
\(946\) 0 0
\(947\) −2.20475e10 −0.843598 −0.421799 0.906689i \(-0.638601\pi\)
−0.421799 + 0.906689i \(0.638601\pi\)
\(948\) 0 0
\(949\) −2.44181e8 −0.00927429
\(950\) 0 0
\(951\) 1.05096e10 0.396235
\(952\) 0 0
\(953\) 3.95876e9 0.148161 0.0740805 0.997252i \(-0.476398\pi\)
0.0740805 + 0.997252i \(0.476398\pi\)
\(954\) 0 0
\(955\) 1.30203e10 0.483737
\(956\) 0 0
\(957\) −2.24743e9 −0.0828885
\(958\) 0 0
\(959\) −3.98874e10 −1.46040
\(960\) 0 0
\(961\) −2.21921e10 −0.806615
\(962\) 0 0
\(963\) 3.21256e10 1.15920
\(964\) 0 0
\(965\) −2.17943e9 −0.0780724
\(966\) 0 0
\(967\) −9.76598e9 −0.347315 −0.173657 0.984806i \(-0.555559\pi\)
−0.173657 + 0.984806i \(0.555559\pi\)
\(968\) 0 0
\(969\) 2.16936e10 0.765946
\(970\) 0 0
\(971\) −2.15553e10 −0.755589 −0.377795 0.925889i \(-0.623317\pi\)
−0.377795 + 0.925889i \(0.623317\pi\)
\(972\) 0 0
\(973\) 2.94863e10 1.02619
\(974\) 0 0
\(975\) 1.08468e10 0.374789
\(976\) 0 0
\(977\) 5.24340e10 1.79880 0.899398 0.437131i \(-0.144005\pi\)
0.899398 + 0.437131i \(0.144005\pi\)
\(978\) 0 0
\(979\) −5.82657e9 −0.198460
\(980\) 0 0
\(981\) 1.46787e10 0.496417
\(982\) 0 0
\(983\) −2.85116e10 −0.957380 −0.478690 0.877984i \(-0.658888\pi\)
−0.478690 + 0.877984i \(0.658888\pi\)
\(984\) 0 0
\(985\) 1.65871e10 0.553023
\(986\) 0 0
\(987\) −3.60455e9 −0.119328
\(988\) 0 0
\(989\) 4.24509e8 0.0139540
\(990\) 0 0
\(991\) 2.72290e10 0.888738 0.444369 0.895844i \(-0.353428\pi\)
0.444369 + 0.895844i \(0.353428\pi\)
\(992\) 0 0
\(993\) −9.23324e9 −0.299248
\(994\) 0 0
\(995\) 7.37731e9 0.237420
\(996\) 0 0
\(997\) 1.30057e10 0.415624 0.207812 0.978169i \(-0.433366\pi\)
0.207812 + 0.978169i \(0.433366\pi\)
\(998\) 0 0
\(999\) 4.12804e10 1.30998
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 256.8.a.q.1.3 6
4.3 odd 2 256.8.a.r.1.4 6
8.3 odd 2 256.8.a.r.1.3 6
8.5 even 2 inner 256.8.a.q.1.4 6
16.3 odd 4 8.8.b.a.5.4 yes 6
16.5 even 4 32.8.b.a.17.4 6
16.11 odd 4 8.8.b.a.5.3 6
16.13 even 4 32.8.b.a.17.3 6
48.5 odd 4 288.8.d.b.145.4 6
48.11 even 4 72.8.d.b.37.4 6
48.29 odd 4 288.8.d.b.145.3 6
48.35 even 4 72.8.d.b.37.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.3 6 16.11 odd 4
8.8.b.a.5.4 yes 6 16.3 odd 4
32.8.b.a.17.3 6 16.13 even 4
32.8.b.a.17.4 6 16.5 even 4
72.8.d.b.37.3 6 48.35 even 4
72.8.d.b.37.4 6 48.11 even 4
256.8.a.q.1.3 6 1.1 even 1 trivial
256.8.a.q.1.4 6 8.5 even 2 inner
256.8.a.r.1.3 6 8.3 odd 2
256.8.a.r.1.4 6 4.3 odd 2
288.8.d.b.145.3 6 48.29 odd 4
288.8.d.b.145.4 6 48.5 odd 4