Properties

Label 256.9.d.e.127.2
Level $256$
Weight $9$
Character 256.127
Analytic conductor $104.289$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,9,Mod(127,256)]
 
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([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.d (of order \(2\), degree \(1\), not 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: \(104.288924176\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(3.12250 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 256.127
Dual form 256.9.d.e.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-99.9200 q^{3} +610.000i q^{5} +1398.88i q^{7} +3423.00 q^{9} +O(q^{10})\) \(q-99.9200 q^{3} +610.000i q^{5} +1398.88i q^{7} +3423.00 q^{9} -18485.2 q^{11} +5470.00i q^{13} -60951.2i q^{15} +73090.0 q^{17} +19484.4 q^{19} -139776. i q^{21} -237210. i q^{23} +18525.0 q^{25} +313549. q^{27} +128222. i q^{29} +67945.6i q^{31} +1.84704e6 q^{33} -853317. q^{35} -3.47203e6i q^{37} -546562. i q^{39} -2.14688e6 q^{41} -5.92815e6 q^{43} +2.08803e6i q^{45} -7.62629e6i q^{47} +3.80794e6 q^{49} -7.30315e6 q^{51} +824290. i q^{53} -1.12760e7i q^{55} -1.94688e6 q^{57} -3.72552e6 q^{59} +1.47461e7i q^{61} +4.78836e6i q^{63} -3.33670e6 q^{65} -1.52567e7 q^{67} +2.37020e7i q^{69} -1.19604e6i q^{71} +5.72563e6 q^{73} -1.85102e6 q^{75} -2.58586e7i q^{77} -3.59132e7i q^{79} -5.37881e7 q^{81} +5.19603e7 q^{83} +4.45849e7i q^{85} -1.28119e7i q^{87} +8.33242e7 q^{89} -7.65187e6 q^{91} -6.78912e6i q^{93} +1.18855e7i q^{95} +1.20619e8 q^{97} -6.32748e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13692 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 13692 q^{9} + 292360 q^{17} + 74100 q^{25} + 7388160 q^{33} - 8587528 q^{41} + 15231748 q^{49} - 7787520 q^{57} - 13346800 q^{65} + 22902520 q^{73} - 215152380 q^{81} + 333296888 q^{89} + 482476040 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −99.9200 −1.23358 −0.616790 0.787128i \(-0.711567\pi\)
−0.616790 + 0.787128i \(0.711567\pi\)
\(4\) 0 0
\(5\) 610.000i 0.976000i 0.872844 + 0.488000i \(0.162273\pi\)
−0.872844 + 0.488000i \(0.837727\pi\)
\(6\) 0 0
\(7\) 1398.88i 0.582624i 0.956628 + 0.291312i \(0.0940917\pi\)
−0.956628 + 0.291312i \(0.905908\pi\)
\(8\) 0 0
\(9\) 3423.00 0.521719
\(10\) 0 0
\(11\) −18485.2 −1.26256 −0.631282 0.775554i \(-0.717471\pi\)
−0.631282 + 0.775554i \(0.717471\pi\)
\(12\) 0 0
\(13\) 5470.00i 0.191520i 0.995404 + 0.0957600i \(0.0305281\pi\)
−0.995404 + 0.0957600i \(0.969472\pi\)
\(14\) 0 0
\(15\) − 60951.2i − 1.20397i
\(16\) 0 0
\(17\) 73090.0 0.875109 0.437555 0.899192i \(-0.355845\pi\)
0.437555 + 0.899192i \(0.355845\pi\)
\(18\) 0 0
\(19\) 19484.4 0.149511 0.0747554 0.997202i \(-0.476182\pi\)
0.0747554 + 0.997202i \(0.476182\pi\)
\(20\) 0 0
\(21\) − 139776.i − 0.718713i
\(22\) 0 0
\(23\) − 237210.i − 0.847660i −0.905742 0.423830i \(-0.860685\pi\)
0.905742 0.423830i \(-0.139315\pi\)
\(24\) 0 0
\(25\) 18525.0 0.0474240
\(26\) 0 0
\(27\) 313549. 0.589997
\(28\) 0 0
\(29\) 128222.i 0.181289i 0.995883 + 0.0906443i \(0.0288926\pi\)
−0.995883 + 0.0906443i \(0.971107\pi\)
\(30\) 0 0
\(31\) 67945.6i 0.0735723i 0.999323 + 0.0367862i \(0.0117120\pi\)
−0.999323 + 0.0367862i \(0.988288\pi\)
\(32\) 0 0
\(33\) 1.84704e6 1.55747
\(34\) 0 0
\(35\) −853317. −0.568641
\(36\) 0 0
\(37\) − 3.47203e6i − 1.85258i −0.376813 0.926289i \(-0.622980\pi\)
0.376813 0.926289i \(-0.377020\pi\)
\(38\) 0 0
\(39\) − 546562.i − 0.236255i
\(40\) 0 0
\(41\) −2.14688e6 −0.759754 −0.379877 0.925037i \(-0.624034\pi\)
−0.379877 + 0.925037i \(0.624034\pi\)
\(42\) 0 0
\(43\) −5.92815e6 −1.73399 −0.866993 0.498321i \(-0.833950\pi\)
−0.866993 + 0.498321i \(0.833950\pi\)
\(44\) 0 0
\(45\) 2.08803e6i 0.509198i
\(46\) 0 0
\(47\) − 7.62629e6i − 1.56287i −0.623989 0.781433i \(-0.714489\pi\)
0.623989 0.781433i \(-0.285511\pi\)
\(48\) 0 0
\(49\) 3.80794e6 0.660550
\(50\) 0 0
\(51\) −7.30315e6 −1.07952
\(52\) 0 0
\(53\) 824290.i 0.104466i 0.998635 + 0.0522332i \(0.0166339\pi\)
−0.998635 + 0.0522332i \(0.983366\pi\)
\(54\) 0 0
\(55\) − 1.12760e7i − 1.23226i
\(56\) 0 0
\(57\) −1.94688e6 −0.184433
\(58\) 0 0
\(59\) −3.72552e6 −0.307453 −0.153726 0.988113i \(-0.549127\pi\)
−0.153726 + 0.988113i \(0.549127\pi\)
\(60\) 0 0
\(61\) 1.47461e7i 1.06502i 0.846424 + 0.532509i \(0.178751\pi\)
−0.846424 + 0.532509i \(0.821249\pi\)
\(62\) 0 0
\(63\) 4.78836e6i 0.303966i
\(64\) 0 0
\(65\) −3.33670e6 −0.186923
\(66\) 0 0
\(67\) −1.52567e7 −0.757113 −0.378557 0.925578i \(-0.623579\pi\)
−0.378557 + 0.925578i \(0.623579\pi\)
\(68\) 0 0
\(69\) 2.37020e7i 1.04566i
\(70\) 0 0
\(71\) − 1.19604e6i − 0.0470666i −0.999723 0.0235333i \(-0.992508\pi\)
0.999723 0.0235333i \(-0.00749158\pi\)
\(72\) 0 0
\(73\) 5.72563e6 0.201619 0.100810 0.994906i \(-0.467857\pi\)
0.100810 + 0.994906i \(0.467857\pi\)
\(74\) 0 0
\(75\) −1.85102e6 −0.0585013
\(76\) 0 0
\(77\) − 2.58586e7i − 0.735600i
\(78\) 0 0
\(79\) − 3.59132e7i − 0.922032i −0.887392 0.461016i \(-0.847485\pi\)
0.887392 0.461016i \(-0.152515\pi\)
\(80\) 0 0
\(81\) −5.37881e7 −1.24953
\(82\) 0 0
\(83\) 5.19603e7 1.09486 0.547431 0.836851i \(-0.315606\pi\)
0.547431 + 0.836851i \(0.315606\pi\)
\(84\) 0 0
\(85\) 4.45849e7i 0.854107i
\(86\) 0 0
\(87\) − 1.28119e7i − 0.223634i
\(88\) 0 0
\(89\) 8.33242e7 1.32804 0.664020 0.747715i \(-0.268849\pi\)
0.664020 + 0.747715i \(0.268849\pi\)
\(90\) 0 0
\(91\) −7.65187e6 −0.111584
\(92\) 0 0
\(93\) − 6.78912e6i − 0.0907573i
\(94\) 0 0
\(95\) 1.18855e7i 0.145923i
\(96\) 0 0
\(97\) 1.20619e8 1.36248 0.681238 0.732062i \(-0.261442\pi\)
0.681238 + 0.732062i \(0.261442\pi\)
\(98\) 0 0
\(99\) −6.32748e7 −0.658704
\(100\) 0 0
\(101\) 2.77246e7i 0.266428i 0.991087 + 0.133214i \(0.0425298\pi\)
−0.991087 + 0.133214i \(0.957470\pi\)
\(102\) 0 0
\(103\) 1.04501e8i 0.928477i 0.885710 + 0.464238i \(0.153672\pi\)
−0.885710 + 0.464238i \(0.846328\pi\)
\(104\) 0 0
\(105\) 8.52634e7 0.701464
\(106\) 0 0
\(107\) 1.00328e8 0.765394 0.382697 0.923874i \(-0.374995\pi\)
0.382697 + 0.923874i \(0.374995\pi\)
\(108\) 0 0
\(109\) 5.90716e7i 0.418478i 0.977865 + 0.209239i \(0.0670987\pi\)
−0.977865 + 0.209239i \(0.932901\pi\)
\(110\) 0 0
\(111\) 3.46925e8i 2.28530i
\(112\) 0 0
\(113\) 5.50849e7 0.337846 0.168923 0.985629i \(-0.445971\pi\)
0.168923 + 0.985629i \(0.445971\pi\)
\(114\) 0 0
\(115\) 1.44698e8 0.827316
\(116\) 0 0
\(117\) 1.87238e7i 0.0999196i
\(118\) 0 0
\(119\) 1.02244e8i 0.509859i
\(120\) 0 0
\(121\) 1.27344e8 0.594067
\(122\) 0 0
\(123\) 2.14516e8 0.937217
\(124\) 0 0
\(125\) 2.49581e8i 1.02229i
\(126\) 0 0
\(127\) − 2.57160e8i − 0.988529i −0.869312 0.494264i \(-0.835438\pi\)
0.869312 0.494264i \(-0.164562\pi\)
\(128\) 0 0
\(129\) 5.92341e8 2.13901
\(130\) 0 0
\(131\) 3.12175e8 1.06002 0.530009 0.847992i \(-0.322188\pi\)
0.530009 + 0.847992i \(0.322188\pi\)
\(132\) 0 0
\(133\) 2.72563e7i 0.0871085i
\(134\) 0 0
\(135\) 1.91265e8i 0.575838i
\(136\) 0 0
\(137\) −2.21980e8 −0.630132 −0.315066 0.949070i \(-0.602027\pi\)
−0.315066 + 0.949070i \(0.602027\pi\)
\(138\) 0 0
\(139\) −2.95030e8 −0.790328 −0.395164 0.918611i \(-0.629312\pi\)
−0.395164 + 0.918611i \(0.629312\pi\)
\(140\) 0 0
\(141\) 7.62019e8i 1.92792i
\(142\) 0 0
\(143\) − 1.01114e8i − 0.241806i
\(144\) 0 0
\(145\) −7.82154e7 −0.176938
\(146\) 0 0
\(147\) −3.80489e8 −0.814841
\(148\) 0 0
\(149\) 4.03603e8i 0.818859i 0.912342 + 0.409429i \(0.134272\pi\)
−0.912342 + 0.409429i \(0.865728\pi\)
\(150\) 0 0
\(151\) − 8.36985e8i − 1.60994i −0.593316 0.804970i \(-0.702181\pi\)
0.593316 0.804970i \(-0.297819\pi\)
\(152\) 0 0
\(153\) 2.50187e8 0.456561
\(154\) 0 0
\(155\) −4.14468e7 −0.0718066
\(156\) 0 0
\(157\) 2.71319e8i 0.446561i 0.974754 + 0.223281i \(0.0716767\pi\)
−0.974754 + 0.223281i \(0.928323\pi\)
\(158\) 0 0
\(159\) − 8.23630e7i − 0.128868i
\(160\) 0 0
\(161\) 3.31828e8 0.493867
\(162\) 0 0
\(163\) −5.78509e8 −0.819520 −0.409760 0.912193i \(-0.634388\pi\)
−0.409760 + 0.912193i \(0.634388\pi\)
\(164\) 0 0
\(165\) 1.12669e9i 1.52009i
\(166\) 0 0
\(167\) 4.68118e8i 0.601852i 0.953647 + 0.300926i \(0.0972958\pi\)
−0.953647 + 0.300926i \(0.902704\pi\)
\(168\) 0 0
\(169\) 7.85810e8 0.963320
\(170\) 0 0
\(171\) 6.66951e7 0.0780026
\(172\) 0 0
\(173\) 2.06197e8i 0.230196i 0.993354 + 0.115098i \(0.0367182\pi\)
−0.993354 + 0.115098i \(0.963282\pi\)
\(174\) 0 0
\(175\) 2.59142e7i 0.0276303i
\(176\) 0 0
\(177\) 3.72253e8 0.379268
\(178\) 0 0
\(179\) −1.41911e8 −0.138230 −0.0691152 0.997609i \(-0.522018\pi\)
−0.0691152 + 0.997609i \(0.522018\pi\)
\(180\) 0 0
\(181\) 4.82566e8i 0.449616i 0.974403 + 0.224808i \(0.0721755\pi\)
−0.974403 + 0.224808i \(0.927824\pi\)
\(182\) 0 0
\(183\) − 1.47343e9i − 1.31379i
\(184\) 0 0
\(185\) 2.11794e9 1.80812
\(186\) 0 0
\(187\) −1.35108e9 −1.10488
\(188\) 0 0
\(189\) 4.38617e8i 0.343747i
\(190\) 0 0
\(191\) − 9.92461e8i − 0.745727i −0.927886 0.372864i \(-0.878376\pi\)
0.927886 0.372864i \(-0.121624\pi\)
\(192\) 0 0
\(193\) 1.17593e9 0.847526 0.423763 0.905773i \(-0.360709\pi\)
0.423763 + 0.905773i \(0.360709\pi\)
\(194\) 0 0
\(195\) 3.33403e8 0.230585
\(196\) 0 0
\(197\) 1.70538e9i 1.13229i 0.824306 + 0.566144i \(0.191565\pi\)
−0.824306 + 0.566144i \(0.808435\pi\)
\(198\) 0 0
\(199\) 2.49036e9i 1.58800i 0.607919 + 0.793999i \(0.292004\pi\)
−0.607919 + 0.793999i \(0.707996\pi\)
\(200\) 0 0
\(201\) 1.52445e9 0.933960
\(202\) 0 0
\(203\) −1.79367e8 −0.105623
\(204\) 0 0
\(205\) − 1.30960e9i − 0.741519i
\(206\) 0 0
\(207\) − 8.11970e8i − 0.442241i
\(208\) 0 0
\(209\) −3.60173e8 −0.188767
\(210\) 0 0
\(211\) −1.46774e9 −0.740491 −0.370245 0.928934i \(-0.620726\pi\)
−0.370245 + 0.928934i \(0.620726\pi\)
\(212\) 0 0
\(213\) 1.19508e8i 0.0580604i
\(214\) 0 0
\(215\) − 3.61617e9i − 1.69237i
\(216\) 0 0
\(217\) −9.50477e7 −0.0428650
\(218\) 0 0
\(219\) −5.72105e8 −0.248713
\(220\) 0 0
\(221\) 3.99802e8i 0.167601i
\(222\) 0 0
\(223\) 1.47920e9i 0.598147i 0.954230 + 0.299073i \(0.0966776\pi\)
−0.954230 + 0.299073i \(0.903322\pi\)
\(224\) 0 0
\(225\) 6.34111e7 0.0247420
\(226\) 0 0
\(227\) 7.50054e8 0.282481 0.141241 0.989975i \(-0.454891\pi\)
0.141241 + 0.989975i \(0.454891\pi\)
\(228\) 0 0
\(229\) − 2.84784e9i − 1.03556i −0.855515 0.517778i \(-0.826759\pi\)
0.855515 0.517778i \(-0.173241\pi\)
\(230\) 0 0
\(231\) 2.58379e9i 0.907421i
\(232\) 0 0
\(233\) −2.20621e8 −0.0748553 −0.0374276 0.999299i \(-0.511916\pi\)
−0.0374276 + 0.999299i \(0.511916\pi\)
\(234\) 0 0
\(235\) 4.65204e9 1.52536
\(236\) 0 0
\(237\) 3.58845e9i 1.13740i
\(238\) 0 0
\(239\) 4.04493e9i 1.23971i 0.784717 + 0.619855i \(0.212808\pi\)
−0.784717 + 0.619855i \(0.787192\pi\)
\(240\) 0 0
\(241\) 6.17983e9 1.83193 0.915964 0.401260i \(-0.131427\pi\)
0.915964 + 0.401260i \(0.131427\pi\)
\(242\) 0 0
\(243\) 3.31731e9 0.951395
\(244\) 0 0
\(245\) 2.32284e9i 0.644696i
\(246\) 0 0
\(247\) 1.06580e8i 0.0286343i
\(248\) 0 0
\(249\) −5.19187e9 −1.35060
\(250\) 0 0
\(251\) 5.21367e9 1.31356 0.656778 0.754084i \(-0.271919\pi\)
0.656778 + 0.754084i \(0.271919\pi\)
\(252\) 0 0
\(253\) 4.38487e9i 1.07022i
\(254\) 0 0
\(255\) − 4.45492e9i − 1.05361i
\(256\) 0 0
\(257\) −6.13693e9 −1.40676 −0.703378 0.710816i \(-0.748326\pi\)
−0.703378 + 0.710816i \(0.748326\pi\)
\(258\) 0 0
\(259\) 4.85695e9 1.07936
\(260\) 0 0
\(261\) 4.38904e8i 0.0945818i
\(262\) 0 0
\(263\) 6.96916e9i 1.45666i 0.685228 + 0.728329i \(0.259703\pi\)
−0.685228 + 0.728329i \(0.740297\pi\)
\(264\) 0 0
\(265\) −5.02817e8 −0.101959
\(266\) 0 0
\(267\) −8.32575e9 −1.63824
\(268\) 0 0
\(269\) − 2.70720e9i − 0.517025i −0.966008 0.258513i \(-0.916768\pi\)
0.966008 0.258513i \(-0.0832324\pi\)
\(270\) 0 0
\(271\) 7.99032e9i 1.48145i 0.671808 + 0.740725i \(0.265518\pi\)
−0.671808 + 0.740725i \(0.734482\pi\)
\(272\) 0 0
\(273\) 7.64575e8 0.137648
\(274\) 0 0
\(275\) −3.42438e8 −0.0598758
\(276\) 0 0
\(277\) − 8.22965e9i − 1.39786i −0.715192 0.698928i \(-0.753661\pi\)
0.715192 0.698928i \(-0.246339\pi\)
\(278\) 0 0
\(279\) 2.32578e8i 0.0383841i
\(280\) 0 0
\(281\) −3.08105e9 −0.494167 −0.247083 0.968994i \(-0.579472\pi\)
−0.247083 + 0.968994i \(0.579472\pi\)
\(282\) 0 0
\(283\) 1.17112e9 0.182582 0.0912908 0.995824i \(-0.470901\pi\)
0.0912908 + 0.995824i \(0.470901\pi\)
\(284\) 0 0
\(285\) − 1.18760e9i − 0.180007i
\(286\) 0 0
\(287\) − 3.00323e9i − 0.442650i
\(288\) 0 0
\(289\) −1.63361e9 −0.234184
\(290\) 0 0
\(291\) −1.20522e10 −1.68072
\(292\) 0 0
\(293\) 4.80980e9i 0.652614i 0.945264 + 0.326307i \(0.105804\pi\)
−0.945264 + 0.326307i \(0.894196\pi\)
\(294\) 0 0
\(295\) − 2.27256e9i − 0.300074i
\(296\) 0 0
\(297\) −5.79601e9 −0.744909
\(298\) 0 0
\(299\) 1.29754e9 0.162344
\(300\) 0 0
\(301\) − 8.29277e9i − 1.01026i
\(302\) 0 0
\(303\) − 2.77025e9i − 0.328661i
\(304\) 0 0
\(305\) −8.99511e9 −1.03946
\(306\) 0 0
\(307\) 3.49176e9 0.393089 0.196545 0.980495i \(-0.437028\pi\)
0.196545 + 0.980495i \(0.437028\pi\)
\(308\) 0 0
\(309\) − 1.04417e10i − 1.14535i
\(310\) 0 0
\(311\) − 1.29807e10i − 1.38757i −0.720182 0.693785i \(-0.755942\pi\)
0.720182 0.693785i \(-0.244058\pi\)
\(312\) 0 0
\(313\) 6.31165e9 0.657606 0.328803 0.944399i \(-0.393355\pi\)
0.328803 + 0.944399i \(0.393355\pi\)
\(314\) 0 0
\(315\) −2.92090e9 −0.296671
\(316\) 0 0
\(317\) − 1.65902e10i − 1.64291i −0.570273 0.821455i \(-0.693163\pi\)
0.570273 0.821455i \(-0.306837\pi\)
\(318\) 0 0
\(319\) − 2.37021e9i − 0.228888i
\(320\) 0 0
\(321\) −1.00247e10 −0.944175
\(322\) 0 0
\(323\) 1.42411e9 0.130838
\(324\) 0 0
\(325\) 1.01332e8i 0.00908264i
\(326\) 0 0
\(327\) − 5.90243e9i − 0.516226i
\(328\) 0 0
\(329\) 1.06683e10 0.910563
\(330\) 0 0
\(331\) 5.48640e9 0.457062 0.228531 0.973537i \(-0.426608\pi\)
0.228531 + 0.973537i \(0.426608\pi\)
\(332\) 0 0
\(333\) − 1.18848e10i − 0.966526i
\(334\) 0 0
\(335\) − 9.30657e9i − 0.738942i
\(336\) 0 0
\(337\) −3.56226e8 −0.0276189 −0.0138095 0.999905i \(-0.504396\pi\)
−0.0138095 + 0.999905i \(0.504396\pi\)
\(338\) 0 0
\(339\) −5.50408e9 −0.416760
\(340\) 0 0
\(341\) − 1.25599e9i − 0.0928897i
\(342\) 0 0
\(343\) 1.33911e10i 0.967476i
\(344\) 0 0
\(345\) −1.44582e10 −1.02056
\(346\) 0 0
\(347\) −1.59731e10 −1.10172 −0.550859 0.834599i \(-0.685700\pi\)
−0.550859 + 0.834599i \(0.685700\pi\)
\(348\) 0 0
\(349\) − 1.03634e10i − 0.698553i −0.937020 0.349277i \(-0.886427\pi\)
0.937020 0.349277i \(-0.113573\pi\)
\(350\) 0 0
\(351\) 1.71511e9i 0.112996i
\(352\) 0 0
\(353\) −1.30979e10 −0.843536 −0.421768 0.906704i \(-0.638590\pi\)
−0.421768 + 0.906704i \(0.638590\pi\)
\(354\) 0 0
\(355\) 7.29586e8 0.0459370
\(356\) 0 0
\(357\) − 1.02162e10i − 0.628952i
\(358\) 0 0
\(359\) 3.31454e9i 0.199547i 0.995010 + 0.0997737i \(0.0318119\pi\)
−0.995010 + 0.0997737i \(0.968188\pi\)
\(360\) 0 0
\(361\) −1.66039e10 −0.977647
\(362\) 0 0
\(363\) −1.27242e10 −0.732829
\(364\) 0 0
\(365\) 3.49263e9i 0.196780i
\(366\) 0 0
\(367\) − 1.96628e10i − 1.08388i −0.840418 0.541939i \(-0.817691\pi\)
0.840418 0.541939i \(-0.182309\pi\)
\(368\) 0 0
\(369\) −7.34878e9 −0.396378
\(370\) 0 0
\(371\) −1.15308e9 −0.0608646
\(372\) 0 0
\(373\) − 2.10063e10i − 1.08521i −0.839987 0.542606i \(-0.817438\pi\)
0.839987 0.542606i \(-0.182562\pi\)
\(374\) 0 0
\(375\) − 2.49382e10i − 1.26107i
\(376\) 0 0
\(377\) −7.01374e8 −0.0347204
\(378\) 0 0
\(379\) 3.04816e9 0.147734 0.0738670 0.997268i \(-0.476466\pi\)
0.0738670 + 0.997268i \(0.476466\pi\)
\(380\) 0 0
\(381\) 2.56955e10i 1.21943i
\(382\) 0 0
\(383\) 2.23357e10i 1.03802i 0.854770 + 0.519008i \(0.173698\pi\)
−0.854770 + 0.519008i \(0.826302\pi\)
\(384\) 0 0
\(385\) 1.57737e10 0.717945
\(386\) 0 0
\(387\) −2.02921e10 −0.904654
\(388\) 0 0
\(389\) 3.13680e10i 1.36990i 0.728592 + 0.684948i \(0.240175\pi\)
−0.728592 + 0.684948i \(0.759825\pi\)
\(390\) 0 0
\(391\) − 1.73377e10i − 0.741795i
\(392\) 0 0
\(393\) −3.11926e10 −1.30762
\(394\) 0 0
\(395\) 2.19071e10 0.899904
\(396\) 0 0
\(397\) − 7.65788e9i − 0.308281i −0.988049 0.154140i \(-0.950739\pi\)
0.988049 0.154140i \(-0.0492608\pi\)
\(398\) 0 0
\(399\) − 2.72345e9i − 0.107455i
\(400\) 0 0
\(401\) −3.26120e10 −1.26125 −0.630623 0.776089i \(-0.717201\pi\)
−0.630623 + 0.776089i \(0.717201\pi\)
\(402\) 0 0
\(403\) −3.71662e8 −0.0140906
\(404\) 0 0
\(405\) − 3.28107e10i − 1.21954i
\(406\) 0 0
\(407\) 6.41811e10i 2.33900i
\(408\) 0 0
\(409\) −2.26168e10 −0.808236 −0.404118 0.914707i \(-0.632422\pi\)
−0.404118 + 0.914707i \(0.632422\pi\)
\(410\) 0 0
\(411\) 2.21802e10 0.777318
\(412\) 0 0
\(413\) − 5.21155e9i − 0.179129i
\(414\) 0 0
\(415\) 3.16958e10i 1.06858i
\(416\) 0 0
\(417\) 2.94794e10 0.974932
\(418\) 0 0
\(419\) 4.94503e10 1.60440 0.802201 0.597054i \(-0.203662\pi\)
0.802201 + 0.597054i \(0.203662\pi\)
\(420\) 0 0
\(421\) − 3.34077e10i − 1.06345i −0.846916 0.531726i \(-0.821543\pi\)
0.846916 0.531726i \(-0.178457\pi\)
\(422\) 0 0
\(423\) − 2.61048e10i − 0.815378i
\(424\) 0 0
\(425\) 1.35399e9 0.0415012
\(426\) 0 0
\(427\) −2.06280e10 −0.620505
\(428\) 0 0
\(429\) 1.01033e10i 0.298287i
\(430\) 0 0
\(431\) 3.06956e10i 0.889544i 0.895644 + 0.444772i \(0.146715\pi\)
−0.895644 + 0.444772i \(0.853285\pi\)
\(432\) 0 0
\(433\) 2.88433e9 0.0820529 0.0410265 0.999158i \(-0.486937\pi\)
0.0410265 + 0.999158i \(0.486937\pi\)
\(434\) 0 0
\(435\) 7.81528e9 0.218267
\(436\) 0 0
\(437\) − 4.62189e9i − 0.126734i
\(438\) 0 0
\(439\) 6.92422e10i 1.86429i 0.362088 + 0.932144i \(0.382064\pi\)
−0.362088 + 0.932144i \(0.617936\pi\)
\(440\) 0 0
\(441\) 1.30346e10 0.344621
\(442\) 0 0
\(443\) 2.06609e10 0.536455 0.268228 0.963356i \(-0.413562\pi\)
0.268228 + 0.963356i \(0.413562\pi\)
\(444\) 0 0
\(445\) 5.08278e10i 1.29617i
\(446\) 0 0
\(447\) − 4.03280e10i − 1.01013i
\(448\) 0 0
\(449\) 2.11092e10 0.519382 0.259691 0.965692i \(-0.416379\pi\)
0.259691 + 0.965692i \(0.416379\pi\)
\(450\) 0 0
\(451\) 3.96855e10 0.959237
\(452\) 0 0
\(453\) 8.36315e10i 1.98599i
\(454\) 0 0
\(455\) − 4.66764e9i − 0.108906i
\(456\) 0 0
\(457\) 2.06831e10 0.474188 0.237094 0.971487i \(-0.423805\pi\)
0.237094 + 0.971487i \(0.423805\pi\)
\(458\) 0 0
\(459\) 2.29173e10 0.516312
\(460\) 0 0
\(461\) − 7.65072e10i − 1.69394i −0.531640 0.846971i \(-0.678424\pi\)
0.531640 0.846971i \(-0.321576\pi\)
\(462\) 0 0
\(463\) − 3.41303e9i − 0.0742704i −0.999310 0.0371352i \(-0.988177\pi\)
0.999310 0.0371352i \(-0.0118232\pi\)
\(464\) 0 0
\(465\) 4.14136e9 0.0885791
\(466\) 0 0
\(467\) −1.92903e10 −0.405576 −0.202788 0.979223i \(-0.565000\pi\)
−0.202788 + 0.979223i \(0.565000\pi\)
\(468\) 0 0
\(469\) − 2.13423e10i − 0.441112i
\(470\) 0 0
\(471\) − 2.71102e10i − 0.550869i
\(472\) 0 0
\(473\) 1.09583e11 2.18927
\(474\) 0 0
\(475\) 3.60948e8 0.00709040
\(476\) 0 0
\(477\) 2.82154e9i 0.0545021i
\(478\) 0 0
\(479\) 2.43887e10i 0.463282i 0.972801 + 0.231641i \(0.0744095\pi\)
−0.972801 + 0.231641i \(0.925590\pi\)
\(480\) 0 0
\(481\) 1.89920e10 0.354806
\(482\) 0 0
\(483\) −3.31563e10 −0.609224
\(484\) 0 0
\(485\) 7.35776e10i 1.32978i
\(486\) 0 0
\(487\) − 9.30801e10i − 1.65478i −0.561626 0.827391i \(-0.689824\pi\)
0.561626 0.827391i \(-0.310176\pi\)
\(488\) 0 0
\(489\) 5.78046e10 1.01094
\(490\) 0 0
\(491\) −2.12850e9 −0.0366225 −0.0183113 0.999832i \(-0.505829\pi\)
−0.0183113 + 0.999832i \(0.505829\pi\)
\(492\) 0 0
\(493\) 9.37175e9i 0.158647i
\(494\) 0 0
\(495\) − 3.85976e10i − 0.642895i
\(496\) 0 0
\(497\) 1.67312e9 0.0274221
\(498\) 0 0
\(499\) −1.04101e10 −0.167901 −0.0839503 0.996470i \(-0.526754\pi\)
−0.0839503 + 0.996470i \(0.526754\pi\)
\(500\) 0 0
\(501\) − 4.67744e10i − 0.742433i
\(502\) 0 0
\(503\) 3.93019e10i 0.613962i 0.951716 + 0.306981i \(0.0993188\pi\)
−0.951716 + 0.306981i \(0.900681\pi\)
\(504\) 0 0
\(505\) −1.69120e10 −0.260034
\(506\) 0 0
\(507\) −7.85181e10 −1.18833
\(508\) 0 0
\(509\) 3.25113e10i 0.484354i 0.970232 + 0.242177i \(0.0778615\pi\)
−0.970232 + 0.242177i \(0.922139\pi\)
\(510\) 0 0
\(511\) 8.00947e9i 0.117468i
\(512\) 0 0
\(513\) 6.10931e9 0.0882110
\(514\) 0 0
\(515\) −6.37455e10 −0.906194
\(516\) 0 0
\(517\) 1.40973e11i 1.97322i
\(518\) 0 0
\(519\) − 2.06032e10i − 0.283965i
\(520\) 0 0
\(521\) −1.84550e9 −0.0250475 −0.0125237 0.999922i \(-0.503987\pi\)
−0.0125237 + 0.999922i \(0.503987\pi\)
\(522\) 0 0
\(523\) 6.23770e10 0.833715 0.416858 0.908972i \(-0.363131\pi\)
0.416858 + 0.908972i \(0.363131\pi\)
\(524\) 0 0
\(525\) − 2.58935e9i − 0.0340842i
\(526\) 0 0
\(527\) 4.96614e9i 0.0643838i
\(528\) 0 0
\(529\) 2.20424e10 0.281473
\(530\) 0 0
\(531\) −1.27524e10 −0.160404
\(532\) 0 0
\(533\) − 1.17434e10i − 0.145508i
\(534\) 0 0
\(535\) 6.11998e10i 0.747025i
\(536\) 0 0
\(537\) 1.41797e10 0.170518
\(538\) 0 0
\(539\) −7.03905e10 −0.833986
\(540\) 0 0
\(541\) 7.45917e10i 0.870766i 0.900245 + 0.435383i \(0.143387\pi\)
−0.900245 + 0.435383i \(0.856613\pi\)
\(542\) 0 0
\(543\) − 4.82179e10i − 0.554638i
\(544\) 0 0
\(545\) −3.60337e10 −0.408435
\(546\) 0 0
\(547\) −1.41531e9 −0.0158089 −0.00790445 0.999969i \(-0.502516\pi\)
−0.00790445 + 0.999969i \(0.502516\pi\)
\(548\) 0 0
\(549\) 5.04758e10i 0.555641i
\(550\) 0 0
\(551\) 2.49833e9i 0.0271046i
\(552\) 0 0
\(553\) 5.02383e10 0.537198
\(554\) 0 0
\(555\) −2.11624e11 −2.23046
\(556\) 0 0
\(557\) 1.37543e11i 1.42895i 0.699661 + 0.714475i \(0.253334\pi\)
−0.699661 + 0.714475i \(0.746666\pi\)
\(558\) 0 0
\(559\) − 3.24270e10i − 0.332093i
\(560\) 0 0
\(561\) 1.35000e11 1.36296
\(562\) 0 0
\(563\) 1.06415e11 1.05918 0.529589 0.848255i \(-0.322346\pi\)
0.529589 + 0.848255i \(0.322346\pi\)
\(564\) 0 0
\(565\) 3.36018e10i 0.329738i
\(566\) 0 0
\(567\) − 7.52431e10i − 0.728005i
\(568\) 0 0
\(569\) −4.02429e10 −0.383919 −0.191960 0.981403i \(-0.561484\pi\)
−0.191960 + 0.981403i \(0.561484\pi\)
\(570\) 0 0
\(571\) 1.50341e11 1.41427 0.707137 0.707077i \(-0.249986\pi\)
0.707137 + 0.707077i \(0.249986\pi\)
\(572\) 0 0
\(573\) 9.91667e10i 0.919914i
\(574\) 0 0
\(575\) − 4.39432e9i − 0.0401994i
\(576\) 0 0
\(577\) 4.96477e9 0.0447915 0.0223958 0.999749i \(-0.492871\pi\)
0.0223958 + 0.999749i \(0.492871\pi\)
\(578\) 0 0
\(579\) −1.17499e11 −1.04549
\(580\) 0 0
\(581\) 7.26862e10i 0.637892i
\(582\) 0 0
\(583\) − 1.52372e10i − 0.131895i
\(584\) 0 0
\(585\) −1.14215e10 −0.0975216
\(586\) 0 0
\(587\) 1.53440e11 1.29237 0.646185 0.763181i \(-0.276363\pi\)
0.646185 + 0.763181i \(0.276363\pi\)
\(588\) 0 0
\(589\) 1.32388e9i 0.0109999i
\(590\) 0 0
\(591\) − 1.70402e11i − 1.39677i
\(592\) 0 0
\(593\) 2.06036e11 1.66619 0.833094 0.553131i \(-0.186567\pi\)
0.833094 + 0.553131i \(0.186567\pi\)
\(594\) 0 0
\(595\) −6.23689e10 −0.497623
\(596\) 0 0
\(597\) − 2.48837e11i − 1.95892i
\(598\) 0 0
\(599\) − 2.30634e11i − 1.79150i −0.444558 0.895750i \(-0.646639\pi\)
0.444558 0.895750i \(-0.353361\pi\)
\(600\) 0 0
\(601\) −1.01422e11 −0.777382 −0.388691 0.921368i \(-0.627073\pi\)
−0.388691 + 0.921368i \(0.627073\pi\)
\(602\) 0 0
\(603\) −5.22236e10 −0.395001
\(604\) 0 0
\(605\) 7.76795e10i 0.579809i
\(606\) 0 0
\(607\) 1.97883e11i 1.45765i 0.684700 + 0.728825i \(0.259933\pi\)
−0.684700 + 0.728825i \(0.740067\pi\)
\(608\) 0 0
\(609\) 1.79224e10 0.130294
\(610\) 0 0
\(611\) 4.17158e10 0.299320
\(612\) 0 0
\(613\) 1.27158e11i 0.900538i 0.892893 + 0.450269i \(0.148672\pi\)
−0.892893 + 0.450269i \(0.851328\pi\)
\(614\) 0 0
\(615\) 1.30855e11i 0.914723i
\(616\) 0 0
\(617\) 5.06702e10 0.349632 0.174816 0.984601i \(-0.444067\pi\)
0.174816 + 0.984601i \(0.444067\pi\)
\(618\) 0 0
\(619\) −7.06748e10 −0.481395 −0.240698 0.970600i \(-0.577376\pi\)
−0.240698 + 0.970600i \(0.577376\pi\)
\(620\) 0 0
\(621\) − 7.43769e10i − 0.500117i
\(622\) 0 0
\(623\) 1.16561e11i 0.773748i
\(624\) 0 0
\(625\) −1.45008e11 −0.950327
\(626\) 0 0
\(627\) 3.59885e10 0.232859
\(628\) 0 0
\(629\) − 2.53771e11i − 1.62121i
\(630\) 0 0
\(631\) 1.65273e11i 1.04252i 0.853399 + 0.521259i \(0.174537\pi\)
−0.853399 + 0.521259i \(0.825463\pi\)
\(632\) 0 0
\(633\) 1.46657e11 0.913454
\(634\) 0 0
\(635\) 1.56868e11 0.964804
\(636\) 0 0
\(637\) 2.08294e10i 0.126508i
\(638\) 0 0
\(639\) − 4.09405e9i − 0.0245556i
\(640\) 0 0
\(641\) 1.12013e11 0.663490 0.331745 0.943369i \(-0.392363\pi\)
0.331745 + 0.943369i \(0.392363\pi\)
\(642\) 0 0
\(643\) 2.65913e11 1.55559 0.777795 0.628518i \(-0.216338\pi\)
0.777795 + 0.628518i \(0.216338\pi\)
\(644\) 0 0
\(645\) 3.61328e11i 2.08767i
\(646\) 0 0
\(647\) 2.71996e11i 1.55219i 0.630614 + 0.776097i \(0.282803\pi\)
−0.630614 + 0.776097i \(0.717197\pi\)
\(648\) 0 0
\(649\) 6.88669e10 0.388179
\(650\) 0 0
\(651\) 9.49716e9 0.0528774
\(652\) 0 0
\(653\) − 3.03789e11i − 1.67078i −0.549656 0.835391i \(-0.685241\pi\)
0.549656 0.835391i \(-0.314759\pi\)
\(654\) 0 0
\(655\) 1.90427e11i 1.03458i
\(656\) 0 0
\(657\) 1.95988e10 0.105189
\(658\) 0 0
\(659\) −4.18575e10 −0.221938 −0.110969 0.993824i \(-0.535395\pi\)
−0.110969 + 0.993824i \(0.535395\pi\)
\(660\) 0 0
\(661\) − 2.46529e11i − 1.29141i −0.763589 0.645703i \(-0.776565\pi\)
0.763589 0.645703i \(-0.223435\pi\)
\(662\) 0 0
\(663\) − 3.99482e10i − 0.206749i
\(664\) 0 0
\(665\) −1.66264e10 −0.0850179
\(666\) 0 0
\(667\) 3.04155e10 0.153671
\(668\) 0 0
\(669\) − 1.47802e11i − 0.737862i
\(670\) 0 0
\(671\) − 2.72584e11i − 1.34465i
\(672\) 0 0
\(673\) −3.15336e11 −1.53714 −0.768569 0.639767i \(-0.779031\pi\)
−0.768569 + 0.639767i \(0.779031\pi\)
\(674\) 0 0
\(675\) 5.80849e9 0.0279800
\(676\) 0 0
\(677\) − 2.47236e10i − 0.117695i −0.998267 0.0588475i \(-0.981257\pi\)
0.998267 0.0588475i \(-0.0187426\pi\)
\(678\) 0 0
\(679\) 1.68731e11i 0.793811i
\(680\) 0 0
\(681\) −7.49454e10 −0.348463
\(682\) 0 0
\(683\) 7.20843e10 0.331251 0.165626 0.986189i \(-0.447036\pi\)
0.165626 + 0.986189i \(0.447036\pi\)
\(684\) 0 0
\(685\) − 1.35408e11i − 0.615009i
\(686\) 0 0
\(687\) 2.84556e11i 1.27744i
\(688\) 0 0
\(689\) −4.50887e9 −0.0200074
\(690\) 0 0
\(691\) 2.95424e11 1.29578 0.647892 0.761732i \(-0.275651\pi\)
0.647892 + 0.761732i \(0.275651\pi\)
\(692\) 0 0
\(693\) − 8.85139e10i − 0.383776i
\(694\) 0 0
\(695\) − 1.79968e11i − 0.771360i
\(696\) 0 0
\(697\) −1.56916e11 −0.664867
\(698\) 0 0
\(699\) 2.20444e10 0.0923400
\(700\) 0 0
\(701\) 2.87925e11i 1.19236i 0.802851 + 0.596180i \(0.203315\pi\)
−0.802851 + 0.596180i \(0.796685\pi\)
\(702\) 0 0
\(703\) − 6.76504e10i − 0.276980i
\(704\) 0 0
\(705\) −4.64831e11 −1.88165
\(706\) 0 0
\(707\) −3.87834e10 −0.155227
\(708\) 0 0
\(709\) 2.51685e11i 0.996030i 0.867168 + 0.498015i \(0.165938\pi\)
−0.867168 + 0.498015i \(0.834062\pi\)
\(710\) 0 0
\(711\) − 1.22931e11i − 0.481042i
\(712\) 0 0
\(713\) 1.61174e10 0.0623643
\(714\) 0 0
\(715\) 6.16795e10 0.236003
\(716\) 0 0
\(717\) − 4.04170e11i − 1.52928i
\(718\) 0 0
\(719\) 1.38856e11i 0.519574i 0.965666 + 0.259787i \(0.0836524\pi\)
−0.965666 + 0.259787i \(0.916348\pi\)
\(720\) 0 0
\(721\) −1.46184e11 −0.540953
\(722\) 0 0
\(723\) −6.17489e11 −2.25983
\(724\) 0 0
\(725\) 2.37531e9i 0.00859743i
\(726\) 0 0
\(727\) − 1.79083e11i − 0.641088i −0.947234 0.320544i \(-0.896134\pi\)
0.947234 0.320544i \(-0.103866\pi\)
\(728\) 0 0
\(729\) 2.14381e10 0.0759061
\(730\) 0 0
\(731\) −4.33289e11 −1.51743
\(732\) 0 0
\(733\) − 2.17618e11i − 0.753839i −0.926246 0.376920i \(-0.876983\pi\)
0.926246 0.376920i \(-0.123017\pi\)
\(734\) 0 0
\(735\) − 2.32098e11i − 0.795285i
\(736\) 0 0
\(737\) 2.82023e11 0.955904
\(738\) 0 0
\(739\) −4.84950e11 −1.62599 −0.812997 0.582268i \(-0.802166\pi\)
−0.812997 + 0.582268i \(0.802166\pi\)
\(740\) 0 0
\(741\) − 1.06494e10i − 0.0353227i
\(742\) 0 0
\(743\) 2.03509e11i 0.667771i 0.942614 + 0.333886i \(0.108360\pi\)
−0.942614 + 0.333886i \(0.891640\pi\)
\(744\) 0 0
\(745\) −2.46198e11 −0.799206
\(746\) 0 0
\(747\) 1.77860e11 0.571210
\(748\) 0 0
\(749\) 1.40346e11i 0.445937i
\(750\) 0 0
\(751\) − 2.34693e11i − 0.737804i −0.929468 0.368902i \(-0.879734\pi\)
0.929468 0.368902i \(-0.120266\pi\)
\(752\) 0 0
\(753\) −5.20950e11 −1.62038
\(754\) 0 0
\(755\) 5.10561e11 1.57130
\(756\) 0 0
\(757\) − 3.84882e11i − 1.17204i −0.810295 0.586022i \(-0.800693\pi\)
0.810295 0.586022i \(-0.199307\pi\)
\(758\) 0 0
\(759\) − 4.38136e11i − 1.32021i
\(760\) 0 0
\(761\) −2.39209e11 −0.713244 −0.356622 0.934249i \(-0.616072\pi\)
−0.356622 + 0.934249i \(0.616072\pi\)
\(762\) 0 0
\(763\) −8.26340e10 −0.243815
\(764\) 0 0
\(765\) 1.52614e11i 0.445604i
\(766\) 0 0
\(767\) − 2.03786e10i − 0.0588833i
\(768\) 0 0
\(769\) 2.08457e11 0.596089 0.298045 0.954552i \(-0.403666\pi\)
0.298045 + 0.954552i \(0.403666\pi\)
\(770\) 0 0
\(771\) 6.13202e11 1.73535
\(772\) 0 0
\(773\) − 5.54469e10i − 0.155296i −0.996981 0.0776478i \(-0.975259\pi\)
0.996981 0.0776478i \(-0.0247410\pi\)
\(774\) 0 0
\(775\) 1.25869e9i 0.00348909i
\(776\) 0 0
\(777\) −4.85306e11 −1.33147
\(778\) 0 0
\(779\) −4.18307e10 −0.113591
\(780\) 0 0
\(781\) 2.21091e10i 0.0594246i
\(782\) 0 0
\(783\) 4.02039e10i 0.106960i
\(784\) 0 0
\(785\) −1.65504e11 −0.435844
\(786\) 0 0
\(787\) −4.05908e11 −1.05811 −0.529053 0.848589i \(-0.677453\pi\)
−0.529053 + 0.848589i \(0.677453\pi\)
\(788\) 0 0
\(789\) − 6.96358e11i − 1.79690i
\(790\) 0 0
\(791\) 7.70572e10i 0.196837i
\(792\) 0 0
\(793\) −8.06610e10 −0.203972
\(794\) 0 0
\(795\) 5.02414e10 0.125775
\(796\) 0 0
\(797\) 3.09015e11i 0.765855i 0.923778 + 0.382927i \(0.125084\pi\)
−0.923778 + 0.382927i \(0.874916\pi\)
\(798\) 0 0
\(799\) − 5.57406e11i − 1.36768i
\(800\) 0 0
\(801\) 2.85219e11 0.692864
\(802\) 0 0
\(803\) −1.05839e11 −0.254557
\(804\) 0 0
\(805\) 2.02415e11i 0.482014i
\(806\) 0 0
\(807\) 2.70504e11i 0.637792i
\(808\) 0 0
\(809\) −4.77958e11 −1.11582 −0.557912 0.829900i \(-0.688397\pi\)
−0.557912 + 0.829900i \(0.688397\pi\)
\(810\) 0 0
\(811\) −6.37503e11 −1.47366 −0.736832 0.676075i \(-0.763679\pi\)
−0.736832 + 0.676075i \(0.763679\pi\)
\(812\) 0 0
\(813\) − 7.98393e11i − 1.82749i
\(814\) 0 0
\(815\) − 3.52890e11i − 0.799851i
\(816\) 0 0
\(817\) −1.15506e11 −0.259250
\(818\) 0 0
\(819\) −2.61924e10 −0.0582155
\(820\) 0 0
\(821\) − 8.43824e11i − 1.85729i −0.370973 0.928644i \(-0.620976\pi\)
0.370973 0.928644i \(-0.379024\pi\)
\(822\) 0 0
\(823\) − 2.60916e11i − 0.568723i −0.958717 0.284362i \(-0.908218\pi\)
0.958717 0.284362i \(-0.0917817\pi\)
\(824\) 0 0
\(825\) 3.42164e10 0.0738616
\(826\) 0 0
\(827\) 2.78675e11 0.595765 0.297883 0.954602i \(-0.403720\pi\)
0.297883 + 0.954602i \(0.403720\pi\)
\(828\) 0 0
\(829\) − 4.75156e11i − 1.00605i −0.864273 0.503023i \(-0.832221\pi\)
0.864273 0.503023i \(-0.167779\pi\)
\(830\) 0 0
\(831\) 8.22306e11i 1.72437i
\(832\) 0 0
\(833\) 2.78322e11 0.578053
\(834\) 0 0
\(835\) −2.85552e11 −0.587408
\(836\) 0 0
\(837\) 2.13043e10i 0.0434075i
\(838\) 0 0
\(839\) − 3.35440e11i − 0.676966i −0.940973 0.338483i \(-0.890086\pi\)
0.940973 0.338483i \(-0.109914\pi\)
\(840\) 0 0
\(841\) 4.83806e11 0.967134
\(842\) 0 0
\(843\) 3.07858e11 0.609594
\(844\) 0 0
\(845\) 4.79344e11i 0.940200i
\(846\) 0 0
\(847\) 1.78138e11i 0.346117i
\(848\) 0 0
\(849\) −1.17019e11 −0.225229
\(850\) 0 0
\(851\) −8.23600e11 −1.57036
\(852\) 0 0
\(853\) − 9.75408e10i − 0.184243i −0.995748 0.0921213i \(-0.970635\pi\)
0.995748 0.0921213i \(-0.0293648\pi\)
\(854\) 0 0
\(855\) 4.06840e10i 0.0761306i
\(856\) 0 0
\(857\) 7.94769e10 0.147339 0.0736695 0.997283i \(-0.476529\pi\)
0.0736695 + 0.997283i \(0.476529\pi\)
\(858\) 0 0
\(859\) −4.15618e11 −0.763347 −0.381673 0.924297i \(-0.624652\pi\)
−0.381673 + 0.924297i \(0.624652\pi\)
\(860\) 0 0
\(861\) 3.00083e11i 0.546045i
\(862\) 0 0
\(863\) 4.80012e11i 0.865383i 0.901542 + 0.432692i \(0.142436\pi\)
−0.901542 + 0.432692i \(0.857564\pi\)
\(864\) 0 0
\(865\) −1.25780e11 −0.224671
\(866\) 0 0
\(867\) 1.63230e11 0.288884
\(868\) 0 0
\(869\) 6.63863e11i 1.16412i
\(870\) 0 0
\(871\) − 8.34540e10i − 0.145002i
\(872\) 0 0
\(873\) 4.12879e11 0.710830
\(874\) 0 0
\(875\) −3.49134e11 −0.595608
\(876\) 0 0
\(877\) − 2.74155e11i − 0.463444i −0.972782 0.231722i \(-0.925564\pi\)
0.972782 0.231722i \(-0.0744360\pi\)
\(878\) 0 0
\(879\) − 4.80595e11i − 0.805051i
\(880\) 0 0
\(881\) 8.01838e11 1.33101 0.665507 0.746391i \(-0.268215\pi\)
0.665507 + 0.746391i \(0.268215\pi\)
\(882\) 0 0
\(883\) 9.95008e11 1.63676 0.818378 0.574681i \(-0.194874\pi\)
0.818378 + 0.574681i \(0.194874\pi\)
\(884\) 0 0
\(885\) 2.27075e11i 0.370165i
\(886\) 0 0
\(887\) 5.46038e11i 0.882122i 0.897477 + 0.441061i \(0.145398\pi\)
−0.897477 + 0.441061i \(0.854602\pi\)
\(888\) 0 0
\(889\) 3.59736e11 0.575940
\(890\) 0 0
\(891\) 9.94283e11 1.57761
\(892\) 0 0
\(893\) − 1.48594e11i − 0.233665i
\(894\) 0 0
\(895\) − 8.65656e10i − 0.134913i
\(896\) 0 0
\(897\) −1.29650e11 −0.200264
\(898\) 0 0
\(899\) −8.71212e9 −0.0133378
\(900\) 0 0
\(901\) 6.02474e10i 0.0914195i
\(902\) 0 0
\(903\) 8.28613e11i 1.24624i
\(904\) 0 0
\(905\) −2.94365e11 −0.438826
\(906\) 0 0
\(907\) 6.55018e11 0.967886 0.483943 0.875100i \(-0.339204\pi\)
0.483943 + 0.875100i \(0.339204\pi\)
\(908\) 0 0
\(909\) 9.49014e10i 0.139001i
\(910\) 0 0
\(911\) 1.19425e11i 0.173389i 0.996235 + 0.0866943i \(0.0276304\pi\)
−0.996235 + 0.0866943i \(0.972370\pi\)
\(912\) 0 0
\(913\) −9.60496e11 −1.38233
\(914\) 0 0
\(915\) 8.98791e11 1.28225
\(916\) 0 0
\(917\) 4.36696e11i 0.617592i
\(918\) 0 0
\(919\) 5.33989e10i 0.0748635i 0.999299 + 0.0374318i \(0.0119177\pi\)
−0.999299 + 0.0374318i \(0.988082\pi\)
\(920\) 0 0
\(921\) −3.48897e11 −0.484907
\(922\) 0 0
\(923\) 6.54235e9 0.00901420
\(924\) 0 0
\(925\) − 6.43194e10i − 0.0878567i
\(926\) 0 0
\(927\) 3.57707e11i 0.484404i
\(928\) 0 0
\(929\) −6.30991e11 −0.847150 −0.423575 0.905861i \(-0.639225\pi\)
−0.423575 + 0.905861i \(0.639225\pi\)
\(930\) 0 0
\(931\) 7.41953e10 0.0987593
\(932\) 0 0
\(933\) 1.29703e12i 1.71168i
\(934\) 0 0
\(935\) − 8.24161e11i − 1.07836i
\(936\) 0 0
\(937\) 8.41436e11 1.09160 0.545799 0.837916i \(-0.316226\pi\)
0.545799 + 0.837916i \(0.316226\pi\)
\(938\) 0 0
\(939\) −6.30660e11 −0.811209
\(940\) 0 0
\(941\) 4.52935e11i 0.577666i 0.957379 + 0.288833i \(0.0932673\pi\)
−0.957379 + 0.288833i \(0.906733\pi\)
\(942\) 0 0
\(943\) 5.09262e11i 0.644013i
\(944\) 0 0
\(945\) −2.67556e11 −0.335497
\(946\) 0 0
\(947\) 1.00309e12 1.24721 0.623605 0.781739i \(-0.285667\pi\)
0.623605 + 0.781739i \(0.285667\pi\)
\(948\) 0 0
\(949\) 3.13192e10i 0.0386141i
\(950\) 0 0
\(951\) 1.65769e12i 2.02666i
\(952\) 0 0
\(953\) 1.39040e12 1.68565 0.842826 0.538186i \(-0.180890\pi\)
0.842826 + 0.538186i \(0.180890\pi\)
\(954\) 0 0
\(955\) 6.05401e11 0.727830
\(956\) 0 0
\(957\) 2.36831e11i 0.282352i
\(958\) 0 0
\(959\) − 3.10523e11i − 0.367130i
\(960\) 0 0
\(961\) 8.48274e11 0.994587
\(962\) 0 0
\(963\) 3.43421e11 0.399321
\(964\) 0 0
\(965\) 7.17318e11i 0.827185i
\(966\) 0 0
\(967\) − 9.01843e11i − 1.03140i −0.856771 0.515698i \(-0.827533\pi\)
0.856771 0.515698i \(-0.172467\pi\)
\(968\) 0 0
\(969\) −1.42297e11 −0.161399
\(970\) 0 0
\(971\) 1.28411e12 1.44452 0.722261 0.691621i \(-0.243103\pi\)
0.722261 + 0.691621i \(0.243103\pi\)
\(972\) 0 0
\(973\) − 4.12712e11i − 0.460464i
\(974\) 0 0
\(975\) − 1.01251e10i − 0.0112042i
\(976\) 0 0
\(977\) 1.20120e12 1.31837 0.659183 0.751983i \(-0.270902\pi\)
0.659183 + 0.751983i \(0.270902\pi\)
\(978\) 0 0
\(979\) −1.54026e12 −1.67674
\(980\) 0 0
\(981\) 2.02202e11i 0.218328i
\(982\) 0 0
\(983\) 3.77388e11i 0.404179i 0.979367 + 0.202090i \(0.0647733\pi\)
−0.979367 + 0.202090i \(0.935227\pi\)
\(984\) 0 0
\(985\) −1.04028e12 −1.10511
\(986\) 0 0
\(987\) −1.06597e12 −1.12325
\(988\) 0 0
\(989\) 1.40622e12i 1.46983i
\(990\) 0 0
\(991\) 3.35563e11i 0.347920i 0.984753 + 0.173960i \(0.0556563\pi\)
−0.984753 + 0.173960i \(0.944344\pi\)
\(992\) 0 0
\(993\) −5.48200e11 −0.563822
\(994\) 0 0
\(995\) −1.51912e12 −1.54989
\(996\) 0 0
\(997\) − 1.03163e12i − 1.04410i −0.852915 0.522050i \(-0.825167\pi\)
0.852915 0.522050i \(-0.174833\pi\)
\(998\) 0 0
\(999\) − 1.08865e12i − 1.09302i
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 256.9.d.e.127.2 4
4.3 odd 2 inner 256.9.d.e.127.4 4
8.3 odd 2 inner 256.9.d.e.127.1 4
8.5 even 2 inner 256.9.d.e.127.3 4
16.3 odd 4 4.9.b.b.3.2 yes 2
16.5 even 4 64.9.c.b.63.2 2
16.11 odd 4 64.9.c.b.63.1 2
16.13 even 4 4.9.b.b.3.1 2
48.29 odd 4 36.9.d.b.19.2 2
48.35 even 4 36.9.d.b.19.1 2
80.3 even 4 100.9.d.b.99.4 4
80.13 odd 4 100.9.d.b.99.2 4
80.19 odd 4 100.9.b.c.51.1 2
80.29 even 4 100.9.b.c.51.2 2
80.67 even 4 100.9.d.b.99.1 4
80.77 odd 4 100.9.d.b.99.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.9.b.b.3.1 2 16.13 even 4
4.9.b.b.3.2 yes 2 16.3 odd 4
36.9.d.b.19.1 2 48.35 even 4
36.9.d.b.19.2 2 48.29 odd 4
64.9.c.b.63.1 2 16.11 odd 4
64.9.c.b.63.2 2 16.5 even 4
100.9.b.c.51.1 2 80.19 odd 4
100.9.b.c.51.2 2 80.29 even 4
100.9.d.b.99.1 4 80.67 even 4
100.9.d.b.99.2 4 80.13 odd 4
100.9.d.b.99.3 4 80.77 odd 4
100.9.d.b.99.4 4 80.3 even 4
256.9.d.e.127.1 4 8.3 odd 2 inner
256.9.d.e.127.2 4 1.1 even 1 trivial
256.9.d.e.127.3 4 8.5 even 2 inner
256.9.d.e.127.4 4 4.3 odd 2 inner