Properties

Label 64.9.c.b.63.2
Level $64$
Weight $9$
Character 64.63
Analytic conductor $26.072$
Analytic rank $0$
Dimension $2$
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] = [64,9,Mod(63,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 64.c (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: \(26.0722310439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-39}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 63.2
Root \(0.500000 - 3.12250i\) of defining polynomial
Character \(\chi\) \(=\) 64.63
Dual form 64.9.c.b.63.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+99.9200i q^{3} -610.000 q^{5} -1398.88i q^{7} -3423.00 q^{9} +O(q^{10})\) \(q+99.9200i q^{3} -610.000 q^{5} -1398.88i q^{7} -3423.00 q^{9} -18485.2i q^{11} +5470.00 q^{13} -60951.2i q^{15} +73090.0 q^{17} -19484.4i q^{19} +139776. q^{21} +237210. i q^{23} -18525.0 q^{25} +313549. i q^{27} +128222. q^{29} +67945.6i q^{31} +1.84704e6 q^{33} +853317. i q^{35} +3.47203e6 q^{37} +546562. i q^{39} +2.14688e6 q^{41} -5.92815e6i q^{43} +2.08803e6 q^{45} -7.62629e6i q^{47} +3.80794e6 q^{49} +7.30315e6i q^{51} -824290. q^{53} +1.12760e7i q^{55} +1.94688e6 q^{57} -3.72552e6i q^{59} +1.47461e7 q^{61} +4.78836e6i q^{63} -3.33670e6 q^{65} +1.52567e7i q^{67} -2.37020e7 q^{69} +1.19604e6i q^{71} -5.72563e6 q^{73} -1.85102e6i q^{75} -2.58586e7 q^{77} -3.59132e7i q^{79} -5.37881e7 q^{81} -5.19603e7i q^{83} -4.45849e7 q^{85} +1.28119e7i q^{87} -8.33242e7 q^{89} -7.65187e6i q^{91} -6.78912e6 q^{93} +1.18855e7i q^{95} +1.20619e8 q^{97} +6.32748e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1220 q^{5} - 6846 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1220 q^{5} - 6846 q^{9} + 10940 q^{13} + 146180 q^{17} + 279552 q^{21} - 37050 q^{25} + 256444 q^{29} + 3694080 q^{33} + 6944060 q^{37} + 4293764 q^{41} + 4176060 q^{45} + 7615874 q^{49} - 1648580 q^{53} + 3893760 q^{57} + 29492156 q^{61} - 6673400 q^{65} - 47404032 q^{69} - 11451260 q^{73} - 51717120 q^{77} - 107576190 q^{81} - 89169800 q^{85} - 166648444 q^{89} - 13578240 q^{93} + 241238020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 99.9200i 1.23358i 0.787128 + 0.616790i \(0.211567\pi\)
−0.787128 + 0.616790i \(0.788433\pi\)
\(4\) 0 0
\(5\) −610.000 −0.976000 −0.488000 0.872844i \(-0.662273\pi\)
−0.488000 + 0.872844i \(0.662273\pi\)
\(6\) 0 0
\(7\) − 1398.88i − 0.582624i −0.956628 0.291312i \(-0.905908\pi\)
0.956628 0.291312i \(-0.0940917\pi\)
\(8\) 0 0
\(9\) −3423.00 −0.521719
\(10\) 0 0
\(11\) − 18485.2i − 1.26256i −0.775554 0.631282i \(-0.782529\pi\)
0.775554 0.631282i \(-0.217471\pi\)
\(12\) 0 0
\(13\) 5470.00 0.191520 0.0957600 0.995404i \(-0.469472\pi\)
0.0957600 + 0.995404i \(0.469472\pi\)
\(14\) 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
\(19\) − 19484.4i − 0.149511i −0.997202 0.0747554i \(-0.976182\pi\)
0.997202 0.0747554i \(-0.0238176\pi\)
\(20\) 0 0
\(21\) 139776. 0.718713
\(22\) 0 0
\(23\) 237210.i 0.847660i 0.905742 + 0.423830i \(0.139315\pi\)
−0.905742 + 0.423830i \(0.860685\pi\)
\(24\) 0 0
\(25\) −18525.0 −0.0474240
\(26\) 0 0
\(27\) 313549.i 0.589997i
\(28\) 0 0
\(29\) 128222. 0.181289 0.0906443 0.995883i \(-0.471107\pi\)
0.0906443 + 0.995883i \(0.471107\pi\)
\(30\) 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.i 0.568641i
\(36\) 0 0
\(37\) 3.47203e6 1.85258 0.926289 0.376813i \(-0.122980\pi\)
0.926289 + 0.376813i \(0.122980\pi\)
\(38\) 0 0
\(39\) 546562.i 0.236255i
\(40\) 0 0
\(41\) 2.14688e6 0.759754 0.379877 0.925037i \(-0.375966\pi\)
0.379877 + 0.925037i \(0.375966\pi\)
\(42\) 0 0
\(43\) − 5.92815e6i − 1.73399i −0.498321 0.866993i \(-0.666050\pi\)
0.498321 0.866993i \(-0.333950\pi\)
\(44\) 0 0
\(45\) 2.08803e6 0.509198
\(46\) 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
\(51\) 7.30315e6i 1.07952i
\(52\) 0 0
\(53\) −824290. −0.104466 −0.0522332 0.998635i \(-0.516634\pi\)
−0.0522332 + 0.998635i \(0.516634\pi\)
\(54\) 0 0
\(55\) 1.12760e7i 1.23226i
\(56\) 0 0
\(57\) 1.94688e6 0.184433
\(58\) 0 0
\(59\) − 3.72552e6i − 0.307453i −0.988113 0.153726i \(-0.950873\pi\)
0.988113 0.153726i \(-0.0491274\pi\)
\(60\) 0 0
\(61\) 1.47461e7 1.06502 0.532509 0.846424i \(-0.321249\pi\)
0.532509 + 0.846424i \(0.321249\pi\)
\(62\) 0 0
\(63\) 4.78836e6i 0.303966i
\(64\) 0 0
\(65\) −3.33670e6 −0.186923
\(66\) 0 0
\(67\) 1.52567e7i 0.757113i 0.925578 + 0.378557i \(0.123579\pi\)
−0.925578 + 0.378557i \(0.876421\pi\)
\(68\) 0 0
\(69\) −2.37020e7 −1.04566
\(70\) 0 0
\(71\) 1.19604e6i 0.0470666i 0.999723 + 0.0235333i \(0.00749158\pi\)
−0.999723 + 0.0235333i \(0.992508\pi\)
\(72\) 0 0
\(73\) −5.72563e6 −0.201619 −0.100810 0.994906i \(-0.532143\pi\)
−0.100810 + 0.994906i \(0.532143\pi\)
\(74\) 0 0
\(75\) − 1.85102e6i − 0.0585013i
\(76\) 0 0
\(77\) −2.58586e7 −0.735600
\(78\) 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
\(83\) − 5.19603e7i − 1.09486i −0.836851 0.547431i \(-0.815606\pi\)
0.836851 0.547431i \(-0.184394\pi\)
\(84\) 0 0
\(85\) −4.45849e7 −0.854107
\(86\) 0 0
\(87\) 1.28119e7i 0.223634i
\(88\) 0 0
\(89\) −8.33242e7 −1.32804 −0.664020 0.747715i \(-0.731151\pi\)
−0.664020 + 0.747715i \(0.731151\pi\)
\(90\) 0 0
\(91\) − 7.65187e6i − 0.111584i
\(92\) 0 0
\(93\) −6.78912e6 −0.0907573
\(94\) 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
\(99\) 6.32748e7i 0.658704i
\(100\) 0 0
\(101\) −2.77246e7 −0.266428 −0.133214 0.991087i \(-0.542530\pi\)
−0.133214 + 0.991087i \(0.542530\pi\)
\(102\) 0 0
\(103\) − 1.04501e8i − 0.928477i −0.885710 0.464238i \(-0.846328\pi\)
0.885710 0.464238i \(-0.153672\pi\)
\(104\) 0 0
\(105\) −8.52634e7 −0.701464
\(106\) 0 0
\(107\) 1.00328e8i 0.765394i 0.923874 + 0.382697i \(0.125005\pi\)
−0.923874 + 0.382697i \(0.874995\pi\)
\(108\) 0 0
\(109\) 5.90716e7 0.418478 0.209239 0.977865i \(-0.432901\pi\)
0.209239 + 0.977865i \(0.432901\pi\)
\(110\) 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
\(115\) − 1.44698e8i − 0.827316i
\(116\) 0 0
\(117\) −1.87238e7 −0.0999196
\(118\) 0 0
\(119\) − 1.02244e8i − 0.509859i
\(120\) 0 0
\(121\) −1.27344e8 −0.594067
\(122\) 0 0
\(123\) 2.14516e8i 0.937217i
\(124\) 0 0
\(125\) 2.49581e8 1.02229
\(126\) 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
\(131\) − 3.12175e8i − 1.06002i −0.847992 0.530009i \(-0.822188\pi\)
0.847992 0.530009i \(-0.177812\pi\)
\(132\) 0 0
\(133\) −2.72563e7 −0.0871085
\(134\) 0 0
\(135\) − 1.91265e8i − 0.575838i
\(136\) 0 0
\(137\) 2.21980e8 0.630132 0.315066 0.949070i \(-0.397973\pi\)
0.315066 + 0.949070i \(0.397973\pi\)
\(138\) 0 0
\(139\) − 2.95030e8i − 0.790328i −0.918611 0.395164i \(-0.870688\pi\)
0.918611 0.395164i \(-0.129312\pi\)
\(140\) 0 0
\(141\) 7.62019e8 1.92792
\(142\) 0 0
\(143\) − 1.01114e8i − 0.241806i
\(144\) 0 0
\(145\) −7.82154e7 −0.176938
\(146\) 0 0
\(147\) 3.80489e8i 0.814841i
\(148\) 0 0
\(149\) −4.03603e8 −0.818859 −0.409429 0.912342i \(-0.634272\pi\)
−0.409429 + 0.912342i \(0.634272\pi\)
\(150\) 0 0
\(151\) 8.36985e8i 1.60994i 0.593316 + 0.804970i \(0.297819\pi\)
−0.593316 + 0.804970i \(0.702181\pi\)
\(152\) 0 0
\(153\) −2.50187e8 −0.456561
\(154\) 0 0
\(155\) − 4.14468e7i − 0.0718066i
\(156\) 0 0
\(157\) 2.71319e8 0.446561 0.223281 0.974754i \(-0.428323\pi\)
0.223281 + 0.974754i \(0.428323\pi\)
\(158\) 0 0
\(159\) − 8.23630e7i − 0.128868i
\(160\) 0 0
\(161\) 3.31828e8 0.493867
\(162\) 0 0
\(163\) 5.78509e8i 0.819520i 0.912193 + 0.409760i \(0.134388\pi\)
−0.912193 + 0.409760i \(0.865612\pi\)
\(164\) 0 0
\(165\) −1.12669e9 −1.52009
\(166\) 0 0
\(167\) − 4.68118e8i − 0.601852i −0.953647 0.300926i \(-0.902704\pi\)
0.953647 0.300926i \(-0.0972958\pi\)
\(168\) 0 0
\(169\) −7.85810e8 −0.963320
\(170\) 0 0
\(171\) 6.66951e7i 0.0780026i
\(172\) 0 0
\(173\) 2.06197e8 0.230196 0.115098 0.993354i \(-0.463282\pi\)
0.115098 + 0.993354i \(0.463282\pi\)
\(174\) 0 0
\(175\) 2.59142e7i 0.0276303i
\(176\) 0 0
\(177\) 3.72253e8 0.379268
\(178\) 0 0
\(179\) 1.41911e8i 0.138230i 0.997609 + 0.0691152i \(0.0220176\pi\)
−0.997609 + 0.0691152i \(0.977982\pi\)
\(180\) 0 0
\(181\) −4.82566e8 −0.449616 −0.224808 0.974403i \(-0.572176\pi\)
−0.224808 + 0.974403i \(0.572176\pi\)
\(182\) 0 0
\(183\) 1.47343e9i 1.31379i
\(184\) 0 0
\(185\) −2.11794e9 −1.80812
\(186\) 0 0
\(187\) − 1.35108e9i − 1.10488i
\(188\) 0 0
\(189\) 4.38617e8 0.343747
\(190\) 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
\(195\) − 3.33403e8i − 0.230585i
\(196\) 0 0
\(197\) −1.70538e9 −1.13229 −0.566144 0.824306i \(-0.691565\pi\)
−0.566144 + 0.824306i \(0.691565\pi\)
\(198\) 0 0
\(199\) − 2.49036e9i − 1.58800i −0.607919 0.793999i \(-0.707996\pi\)
0.607919 0.793999i \(-0.292004\pi\)
\(200\) 0 0
\(201\) −1.52445e9 −0.933960
\(202\) 0 0
\(203\) − 1.79367e8i − 0.105623i
\(204\) 0 0
\(205\) −1.30960e9 −0.741519
\(206\) 0 0
\(207\) − 8.11970e8i − 0.442241i
\(208\) 0 0
\(209\) −3.60173e8 −0.188767
\(210\) 0 0
\(211\) 1.46774e9i 0.740491i 0.928934 + 0.370245i \(0.120726\pi\)
−0.928934 + 0.370245i \(0.879274\pi\)
\(212\) 0 0
\(213\) −1.19508e8 −0.0580604
\(214\) 0 0
\(215\) 3.61617e9i 1.69237i
\(216\) 0 0
\(217\) 9.50477e7 0.0428650
\(218\) 0 0
\(219\) − 5.72105e8i − 0.248713i
\(220\) 0 0
\(221\) 3.99802e8 0.167601
\(222\) 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
\(227\) − 7.50054e8i − 0.282481i −0.989975 0.141241i \(-0.954891\pi\)
0.989975 0.141241i \(-0.0451091\pi\)
\(228\) 0 0
\(229\) 2.84784e9 1.03556 0.517778 0.855515i \(-0.326759\pi\)
0.517778 + 0.855515i \(0.326759\pi\)
\(230\) 0 0
\(231\) − 2.58379e9i − 0.907421i
\(232\) 0 0
\(233\) 2.20621e8 0.0748553 0.0374276 0.999299i \(-0.488084\pi\)
0.0374276 + 0.999299i \(0.488084\pi\)
\(234\) 0 0
\(235\) 4.65204e9i 1.52536i
\(236\) 0 0
\(237\) 3.58845e9 1.13740
\(238\) 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
\(243\) − 3.31731e9i − 0.951395i
\(244\) 0 0
\(245\) −2.32284e9 −0.644696
\(246\) 0 0
\(247\) − 1.06580e8i − 0.0286343i
\(248\) 0 0
\(249\) 5.19187e9 1.35060
\(250\) 0 0
\(251\) 5.21367e9i 1.31356i 0.754084 + 0.656778i \(0.228081\pi\)
−0.754084 + 0.656778i \(0.771919\pi\)
\(252\) 0 0
\(253\) 4.38487e9 1.07022
\(254\) 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
\(259\) − 4.85695e9i − 1.07936i
\(260\) 0 0
\(261\) −4.38904e8 −0.0945818
\(262\) 0 0
\(263\) − 6.96916e9i − 1.45666i −0.685228 0.728329i \(-0.740297\pi\)
0.685228 0.728329i \(-0.259703\pi\)
\(264\) 0 0
\(265\) 5.02817e8 0.101959
\(266\) 0 0
\(267\) − 8.32575e9i − 1.63824i
\(268\) 0 0
\(269\) −2.70720e9 −0.517025 −0.258513 0.966008i \(-0.583232\pi\)
−0.258513 + 0.966008i \(0.583232\pi\)
\(270\) 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
\(275\) 3.42438e8i 0.0598758i
\(276\) 0 0
\(277\) 8.22965e9 1.39786 0.698928 0.715192i \(-0.253661\pi\)
0.698928 + 0.715192i \(0.253661\pi\)
\(278\) 0 0
\(279\) − 2.32578e8i − 0.0383841i
\(280\) 0 0
\(281\) 3.08105e9 0.494167 0.247083 0.968994i \(-0.420528\pi\)
0.247083 + 0.968994i \(0.420528\pi\)
\(282\) 0 0
\(283\) 1.17112e9i 0.182582i 0.995824 + 0.0912908i \(0.0290993\pi\)
−0.995824 + 0.0912908i \(0.970901\pi\)
\(284\) 0 0
\(285\) −1.18760e9 −0.180007
\(286\) 0 0
\(287\) − 3.00323e9i − 0.442650i
\(288\) 0 0
\(289\) −1.63361e9 −0.234184
\(290\) 0 0
\(291\) 1.20522e10i 1.68072i
\(292\) 0 0
\(293\) −4.80980e9 −0.652614 −0.326307 0.945264i \(-0.605804\pi\)
−0.326307 + 0.945264i \(0.605804\pi\)
\(294\) 0 0
\(295\) 2.27256e9i 0.300074i
\(296\) 0 0
\(297\) 5.79601e9 0.744909
\(298\) 0 0
\(299\) 1.29754e9i 0.162344i
\(300\) 0 0
\(301\) −8.29277e9 −1.01026
\(302\) 0 0
\(303\) − 2.77025e9i − 0.328661i
\(304\) 0 0
\(305\) −8.99511e9 −1.03946
\(306\) 0 0
\(307\) − 3.49176e9i − 0.393089i −0.980495 0.196545i \(-0.937028\pi\)
0.980495 0.196545i \(-0.0629721\pi\)
\(308\) 0 0
\(309\) 1.04417e10 1.14535
\(310\) 0 0
\(311\) 1.29807e10i 1.38757i 0.720182 + 0.693785i \(0.244058\pi\)
−0.720182 + 0.693785i \(0.755942\pi\)
\(312\) 0 0
\(313\) −6.31165e9 −0.657606 −0.328803 0.944399i \(-0.606645\pi\)
−0.328803 + 0.944399i \(0.606645\pi\)
\(314\) 0 0
\(315\) − 2.92090e9i − 0.296671i
\(316\) 0 0
\(317\) −1.65902e10 −1.64291 −0.821455 0.570273i \(-0.806837\pi\)
−0.821455 + 0.570273i \(0.806837\pi\)
\(318\) 0 0
\(319\) − 2.37021e9i − 0.228888i
\(320\) 0 0
\(321\) −1.00247e10 −0.944175
\(322\) 0 0
\(323\) − 1.42411e9i − 0.130838i
\(324\) 0 0
\(325\) −1.01332e8 −0.00908264
\(326\) 0 0
\(327\) 5.90243e9i 0.516226i
\(328\) 0 0
\(329\) −1.06683e10 −0.910563
\(330\) 0 0
\(331\) 5.48640e9i 0.457062i 0.973537 + 0.228531i \(0.0733922\pi\)
−0.973537 + 0.228531i \(0.926608\pi\)
\(332\) 0 0
\(333\) −1.18848e10 −0.966526
\(334\) 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
\(339\) 5.50408e9i 0.416760i
\(340\) 0 0
\(341\) 1.25599e9 0.0928897
\(342\) 0 0
\(343\) − 1.33911e10i − 0.967476i
\(344\) 0 0
\(345\) 1.44582e10 1.02056
\(346\) 0 0
\(347\) − 1.59731e10i − 1.10172i −0.834599 0.550859i \(-0.814300\pi\)
0.834599 0.550859i \(-0.185700\pi\)
\(348\) 0 0
\(349\) −1.03634e10 −0.698553 −0.349277 0.937020i \(-0.613573\pi\)
−0.349277 + 0.937020i \(0.613573\pi\)
\(350\) 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
\(355\) − 7.29586e8i − 0.0459370i
\(356\) 0 0
\(357\) 1.02162e10 0.628952
\(358\) 0 0
\(359\) − 3.31454e9i − 0.199547i −0.995010 0.0997737i \(-0.968188\pi\)
0.995010 0.0997737i \(-0.0318119\pi\)
\(360\) 0 0
\(361\) 1.66039e10 0.977647
\(362\) 0 0
\(363\) − 1.27242e10i − 0.732829i
\(364\) 0 0
\(365\) 3.49263e9 0.196780
\(366\) 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
\(371\) 1.15308e9i 0.0608646i
\(372\) 0 0
\(373\) 2.10063e10 1.08521 0.542606 0.839987i \(-0.317438\pi\)
0.542606 + 0.839987i \(0.317438\pi\)
\(374\) 0 0
\(375\) 2.49382e10i 1.26107i
\(376\) 0 0
\(377\) 7.01374e8 0.0347204
\(378\) 0 0
\(379\) 3.04816e9i 0.147734i 0.997268 + 0.0738670i \(0.0235340\pi\)
−0.997268 + 0.0738670i \(0.976466\pi\)
\(380\) 0 0
\(381\) 2.56955e10 1.21943
\(382\) 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
\(387\) 2.02921e10i 0.904654i
\(388\) 0 0
\(389\) −3.13680e10 −1.36990 −0.684948 0.728592i \(-0.740175\pi\)
−0.684948 + 0.728592i \(0.740175\pi\)
\(390\) 0 0
\(391\) 1.73377e10i 0.741795i
\(392\) 0 0
\(393\) 3.11926e10 1.30762
\(394\) 0 0
\(395\) 2.19071e10i 0.899904i
\(396\) 0 0
\(397\) −7.65788e9 −0.308281 −0.154140 0.988049i \(-0.549261\pi\)
−0.154140 + 0.988049i \(0.549261\pi\)
\(398\) 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
\(403\) 3.71662e8i 0.0140906i
\(404\) 0 0
\(405\) 3.28107e10 1.21954
\(406\) 0 0
\(407\) − 6.41811e10i − 2.33900i
\(408\) 0 0
\(409\) 2.26168e10 0.808236 0.404118 0.914707i \(-0.367578\pi\)
0.404118 + 0.914707i \(0.367578\pi\)
\(410\) 0 0
\(411\) 2.21802e10i 0.777318i
\(412\) 0 0
\(413\) −5.21155e9 −0.179129
\(414\) 0 0
\(415\) 3.16958e10i 1.06858i
\(416\) 0 0
\(417\) 2.94794e10 0.974932
\(418\) 0 0
\(419\) − 4.94503e10i − 1.60440i −0.597054 0.802201i \(-0.703662\pi\)
0.597054 0.802201i \(-0.296338\pi\)
\(420\) 0 0
\(421\) 3.34077e10 1.06345 0.531726 0.846916i \(-0.321543\pi\)
0.531726 + 0.846916i \(0.321543\pi\)
\(422\) 0 0
\(423\) 2.61048e10i 0.815378i
\(424\) 0 0
\(425\) −1.35399e9 −0.0415012
\(426\) 0 0
\(427\) − 2.06280e10i − 0.620505i
\(428\) 0 0
\(429\) 1.01033e10 0.298287
\(430\) 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
\(435\) − 7.81528e9i − 0.218267i
\(436\) 0 0
\(437\) 4.62189e9 0.126734
\(438\) 0 0
\(439\) − 6.92422e10i − 1.86429i −0.362088 0.932144i \(-0.617936\pi\)
0.362088 0.932144i \(-0.382064\pi\)
\(440\) 0 0
\(441\) −1.30346e10 −0.344621
\(442\) 0 0
\(443\) 2.06609e10i 0.536455i 0.963356 + 0.268228i \(0.0864379\pi\)
−0.963356 + 0.268228i \(0.913562\pi\)
\(444\) 0 0
\(445\) 5.08278e10 1.29617
\(446\) 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
\(451\) − 3.96855e10i − 0.959237i
\(452\) 0 0
\(453\) −8.36315e10 −1.98599
\(454\) 0 0
\(455\) 4.66764e9i 0.108906i
\(456\) 0 0
\(457\) −2.06831e10 −0.474188 −0.237094 0.971487i \(-0.576195\pi\)
−0.237094 + 0.971487i \(0.576195\pi\)
\(458\) 0 0
\(459\) 2.29173e10i 0.516312i
\(460\) 0 0
\(461\) −7.65072e10 −1.69394 −0.846971 0.531640i \(-0.821576\pi\)
−0.846971 + 0.531640i \(0.821576\pi\)
\(462\) 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
\(467\) 1.92903e10i 0.405576i 0.979223 + 0.202788i \(0.0650002\pi\)
−0.979223 + 0.202788i \(0.935000\pi\)
\(468\) 0 0
\(469\) 2.13423e10 0.441112
\(470\) 0 0
\(471\) 2.71102e10i 0.550869i
\(472\) 0 0
\(473\) −1.09583e11 −2.18927
\(474\) 0 0
\(475\) 3.60948e8i 0.00709040i
\(476\) 0 0
\(477\) 2.82154e9 0.0545021
\(478\) 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
\(483\) 3.31563e10i 0.609224i
\(484\) 0 0
\(485\) −7.35776e10 −1.32978
\(486\) 0 0
\(487\) 9.30801e10i 1.65478i 0.561626 + 0.827391i \(0.310176\pi\)
−0.561626 + 0.827391i \(0.689824\pi\)
\(488\) 0 0
\(489\) −5.78046e10 −1.01094
\(490\) 0 0
\(491\) − 2.12850e9i − 0.0366225i −0.999832 0.0183113i \(-0.994171\pi\)
0.999832 0.0183113i \(-0.00582898\pi\)
\(492\) 0 0
\(493\) 9.37175e9 0.158647
\(494\) 0 0
\(495\) − 3.85976e10i − 0.642895i
\(496\) 0 0
\(497\) 1.67312e9 0.0274221
\(498\) 0 0
\(499\) 1.04101e10i 0.167901i 0.996470 + 0.0839503i \(0.0267537\pi\)
−0.996470 + 0.0839503i \(0.973246\pi\)
\(500\) 0 0
\(501\) 4.67744e10 0.742433
\(502\) 0 0
\(503\) − 3.93019e10i − 0.613962i −0.951716 0.306981i \(-0.900681\pi\)
0.951716 0.306981i \(-0.0993188\pi\)
\(504\) 0 0
\(505\) 1.69120e10 0.260034
\(506\) 0 0
\(507\) − 7.85181e10i − 1.18833i
\(508\) 0 0
\(509\) 3.25113e10 0.484354 0.242177 0.970232i \(-0.422139\pi\)
0.242177 + 0.970232i \(0.422139\pi\)
\(510\) 0 0
\(511\) 8.00947e9i 0.117468i
\(512\) 0 0
\(513\) 6.10931e9 0.0882110
\(514\) 0 0
\(515\) 6.37455e10i 0.906194i
\(516\) 0 0
\(517\) −1.40973e11 −1.97322
\(518\) 0 0
\(519\) 2.06032e10i 0.283965i
\(520\) 0 0
\(521\) 1.84550e9 0.0250475 0.0125237 0.999922i \(-0.496013\pi\)
0.0125237 + 0.999922i \(0.496013\pi\)
\(522\) 0 0
\(523\) 6.23770e10i 0.833715i 0.908972 + 0.416858i \(0.136869\pi\)
−0.908972 + 0.416858i \(0.863131\pi\)
\(524\) 0 0
\(525\) −2.58935e9 −0.0340842
\(526\) 0 0
\(527\) 4.96614e9i 0.0643838i
\(528\) 0 0
\(529\) 2.20424e10 0.281473
\(530\) 0 0
\(531\) 1.27524e10i 0.160404i
\(532\) 0 0
\(533\) 1.17434e10 0.145508
\(534\) 0 0
\(535\) − 6.11998e10i − 0.747025i
\(536\) 0 0
\(537\) −1.41797e10 −0.170518
\(538\) 0 0
\(539\) − 7.03905e10i − 0.833986i
\(540\) 0 0
\(541\) 7.45917e10 0.870766 0.435383 0.900245i \(-0.356613\pi\)
0.435383 + 0.900245i \(0.356613\pi\)
\(542\) 0 0
\(543\) − 4.82179e10i − 0.554638i
\(544\) 0 0
\(545\) −3.60337e10 −0.408435
\(546\) 0 0
\(547\) 1.41531e9i 0.0158089i 0.999969 + 0.00790445i \(0.00251609\pi\)
−0.999969 + 0.00790445i \(0.997484\pi\)
\(548\) 0 0
\(549\) −5.04758e10 −0.555641
\(550\) 0 0
\(551\) − 2.49833e9i − 0.0271046i
\(552\) 0 0
\(553\) −5.02383e10 −0.537198
\(554\) 0 0
\(555\) − 2.11624e11i − 2.23046i
\(556\) 0 0
\(557\) 1.37543e11 1.42895 0.714475 0.699661i \(-0.246666\pi\)
0.714475 + 0.699661i \(0.246666\pi\)
\(558\) 0 0
\(559\) − 3.24270e10i − 0.332093i
\(560\) 0 0
\(561\) 1.35000e11 1.36296
\(562\) 0 0
\(563\) − 1.06415e11i − 1.05918i −0.848255 0.529589i \(-0.822346\pi\)
0.848255 0.529589i \(-0.177654\pi\)
\(564\) 0 0
\(565\) −3.36018e10 −0.329738
\(566\) 0 0
\(567\) 7.52431e10i 0.728005i
\(568\) 0 0
\(569\) 4.02429e10 0.383919 0.191960 0.981403i \(-0.438516\pi\)
0.191960 + 0.981403i \(0.438516\pi\)
\(570\) 0 0
\(571\) 1.50341e11i 1.41427i 0.707077 + 0.707137i \(0.250014\pi\)
−0.707077 + 0.707137i \(0.749986\pi\)
\(572\) 0 0
\(573\) 9.91667e10 0.919914
\(574\) 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
\(579\) 1.17499e11i 1.04549i
\(580\) 0 0
\(581\) −7.26862e10 −0.637892
\(582\) 0 0
\(583\) 1.52372e10i 0.131895i
\(584\) 0 0
\(585\) 1.14215e10 0.0975216
\(586\) 0 0
\(587\) 1.53440e11i 1.29237i 0.763181 + 0.646185i \(0.223637\pi\)
−0.763181 + 0.646185i \(0.776363\pi\)
\(588\) 0 0
\(589\) 1.32388e9 0.0109999
\(590\) 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
\(595\) 6.23689e10i 0.497623i
\(596\) 0 0
\(597\) 2.48837e11 1.95892
\(598\) 0 0
\(599\) 2.30634e11i 1.79150i 0.444558 + 0.895750i \(0.353361\pi\)
−0.444558 + 0.895750i \(0.646639\pi\)
\(600\) 0 0
\(601\) 1.01422e11 0.777382 0.388691 0.921368i \(-0.372927\pi\)
0.388691 + 0.921368i \(0.372927\pi\)
\(602\) 0 0
\(603\) − 5.22236e10i − 0.395001i
\(604\) 0 0
\(605\) 7.76795e10 0.579809
\(606\) 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
\(611\) − 4.17158e10i − 0.299320i
\(612\) 0 0
\(613\) −1.27158e11 −0.900538 −0.450269 0.892893i \(-0.648672\pi\)
−0.450269 + 0.892893i \(0.648672\pi\)
\(614\) 0 0
\(615\) − 1.30855e11i − 0.914723i
\(616\) 0 0
\(617\) −5.06702e10 −0.349632 −0.174816 0.984601i \(-0.555933\pi\)
−0.174816 + 0.984601i \(0.555933\pi\)
\(618\) 0 0
\(619\) − 7.06748e10i − 0.481395i −0.970600 0.240698i \(-0.922624\pi\)
0.970600 0.240698i \(-0.0773762\pi\)
\(620\) 0 0
\(621\) −7.43769e10 −0.500117
\(622\) 0 0
\(623\) 1.16561e11i 0.773748i
\(624\) 0 0
\(625\) −1.45008e11 −0.950327
\(626\) 0 0
\(627\) − 3.59885e10i − 0.232859i
\(628\) 0 0
\(629\) 2.53771e11 1.62121
\(630\) 0 0
\(631\) − 1.65273e11i − 1.04252i −0.853399 0.521259i \(-0.825463\pi\)
0.853399 0.521259i \(-0.174537\pi\)
\(632\) 0 0
\(633\) −1.46657e11 −0.913454
\(634\) 0 0
\(635\) 1.56868e11i 0.964804i
\(636\) 0 0
\(637\) 2.08294e10 0.126508
\(638\) 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
\(643\) − 2.65913e11i − 1.55559i −0.628518 0.777795i \(-0.716338\pi\)
0.628518 0.777795i \(-0.283662\pi\)
\(644\) 0 0
\(645\) −3.61328e11 −2.08767
\(646\) 0 0
\(647\) − 2.71996e11i − 1.55219i −0.630614 0.776097i \(-0.717197\pi\)
0.630614 0.776097i \(-0.282803\pi\)
\(648\) 0 0
\(649\) −6.88669e10 −0.388179
\(650\) 0 0
\(651\) 9.49716e9i 0.0528774i
\(652\) 0 0
\(653\) −3.03789e11 −1.67078 −0.835391 0.549656i \(-0.814759\pi\)
−0.835391 + 0.549656i \(0.814759\pi\)
\(654\) 0 0
\(655\) 1.90427e11i 1.03458i
\(656\) 0 0
\(657\) 1.95988e10 0.105189
\(658\) 0 0
\(659\) 4.18575e10i 0.221938i 0.993824 + 0.110969i \(0.0353954\pi\)
−0.993824 + 0.110969i \(0.964605\pi\)
\(660\) 0 0
\(661\) 2.46529e11 1.29141 0.645703 0.763589i \(-0.276565\pi\)
0.645703 + 0.763589i \(0.276565\pi\)
\(662\) 0 0
\(663\) 3.99482e10i 0.206749i
\(664\) 0 0
\(665\) 1.66264e10 0.0850179
\(666\) 0 0
\(667\) 3.04155e10i 0.153671i
\(668\) 0 0
\(669\) −1.47802e11 −0.737862
\(670\) 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
\(675\) − 5.80849e9i − 0.0279800i
\(676\) 0 0
\(677\) 2.47236e10 0.117695 0.0588475 0.998267i \(-0.481257\pi\)
0.0588475 + 0.998267i \(0.481257\pi\)
\(678\) 0 0
\(679\) − 1.68731e11i − 0.793811i
\(680\) 0 0
\(681\) 7.49454e10 0.348463
\(682\) 0 0
\(683\) 7.20843e10i 0.331251i 0.986189 + 0.165626i \(0.0529644\pi\)
−0.986189 + 0.165626i \(0.947036\pi\)
\(684\) 0 0
\(685\) −1.35408e11 −0.615009
\(686\) 0 0
\(687\) 2.84556e11i 1.27744i
\(688\) 0 0
\(689\) −4.50887e9 −0.0200074
\(690\) 0 0
\(691\) − 2.95424e11i − 1.29578i −0.761732 0.647892i \(-0.775651\pi\)
0.761732 0.647892i \(-0.224349\pi\)
\(692\) 0 0
\(693\) 8.85139e10 0.383776
\(694\) 0 0
\(695\) 1.79968e11i 0.771360i
\(696\) 0 0
\(697\) 1.56916e11 0.664867
\(698\) 0 0
\(699\) 2.20444e10i 0.0923400i
\(700\) 0 0
\(701\) 2.87925e11 1.19236 0.596180 0.802851i \(-0.296685\pi\)
0.596180 + 0.802851i \(0.296685\pi\)
\(702\) 0 0
\(703\) − 6.76504e10i − 0.276980i
\(704\) 0 0
\(705\) −4.64831e11 −1.88165
\(706\) 0 0
\(707\) 3.87834e10i 0.155227i
\(708\) 0 0
\(709\) −2.51685e11 −0.996030 −0.498015 0.867168i \(-0.665938\pi\)
−0.498015 + 0.867168i \(0.665938\pi\)
\(710\) 0 0
\(711\) 1.22931e11i 0.481042i
\(712\) 0 0
\(713\) −1.61174e10 −0.0623643
\(714\) 0 0
\(715\) 6.16795e10i 0.236003i
\(716\) 0 0
\(717\) −4.04170e11 −1.52928
\(718\) 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
\(723\) 6.17489e11i 2.25983i
\(724\) 0 0
\(725\) −2.37531e9 −0.00859743
\(726\) 0 0
\(727\) 1.79083e11i 0.641088i 0.947234 + 0.320544i \(0.103866\pi\)
−0.947234 + 0.320544i \(0.896134\pi\)
\(728\) 0 0
\(729\) −2.14381e10 −0.0759061
\(730\) 0 0
\(731\) − 4.33289e11i − 1.51743i
\(732\) 0 0
\(733\) −2.17618e11 −0.753839 −0.376920 0.926246i \(-0.623017\pi\)
−0.376920 + 0.926246i \(0.623017\pi\)
\(734\) 0 0
\(735\) − 2.32098e11i − 0.795285i
\(736\) 0 0
\(737\) 2.82023e11 0.955904
\(738\) 0 0
\(739\) 4.84950e11i 1.62599i 0.582268 + 0.812997i \(0.302166\pi\)
−0.582268 + 0.812997i \(0.697834\pi\)
\(740\) 0 0
\(741\) 1.06494e10 0.0353227
\(742\) 0 0
\(743\) − 2.03509e11i − 0.667771i −0.942614 0.333886i \(-0.891640\pi\)
0.942614 0.333886i \(-0.108360\pi\)
\(744\) 0 0
\(745\) 2.46198e11 0.799206
\(746\) 0 0
\(747\) 1.77860e11i 0.571210i
\(748\) 0 0
\(749\) 1.40346e11 0.445937
\(750\) 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
\(755\) − 5.10561e11i − 1.57130i
\(756\) 0 0
\(757\) 3.84882e11 1.17204 0.586022 0.810295i \(-0.300693\pi\)
0.586022 + 0.810295i \(0.300693\pi\)
\(758\) 0 0
\(759\) 4.38136e11i 1.32021i
\(760\) 0 0
\(761\) 2.39209e11 0.713244 0.356622 0.934249i \(-0.383928\pi\)
0.356622 + 0.934249i \(0.383928\pi\)
\(762\) 0 0
\(763\) − 8.26340e10i − 0.243815i
\(764\) 0 0
\(765\) 1.52614e11 0.445604
\(766\) 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
\(771\) − 6.13202e11i − 1.73535i
\(772\) 0 0
\(773\) 5.54469e10 0.155296 0.0776478 0.996981i \(-0.475259\pi\)
0.0776478 + 0.996981i \(0.475259\pi\)
\(774\) 0 0
\(775\) − 1.25869e9i − 0.00348909i
\(776\) 0 0
\(777\) 4.85306e11 1.33147
\(778\) 0 0
\(779\) − 4.18307e10i − 0.113591i
\(780\) 0 0
\(781\) 2.21091e10 0.0594246
\(782\) 0 0
\(783\) 4.02039e10i 0.106960i
\(784\) 0 0
\(785\) −1.65504e11 −0.435844
\(786\) 0 0
\(787\) 4.05908e11i 1.05811i 0.848589 + 0.529053i \(0.177453\pi\)
−0.848589 + 0.529053i \(0.822547\pi\)
\(788\) 0 0
\(789\) 6.96358e11 1.79690
\(790\) 0 0
\(791\) − 7.70572e10i − 0.196837i
\(792\) 0 0
\(793\) 8.06610e10 0.203972
\(794\) 0 0
\(795\) 5.02414e10i 0.125775i
\(796\) 0 0
\(797\) 3.09015e11 0.765855 0.382927 0.923778i \(-0.374916\pi\)
0.382927 + 0.923778i \(0.374916\pi\)
\(798\) 0 0
\(799\) − 5.57406e11i − 1.36768i
\(800\) 0 0
\(801\) 2.85219e11 0.692864
\(802\) 0 0
\(803\) 1.05839e11i 0.254557i
\(804\) 0 0
\(805\) −2.02415e11 −0.482014
\(806\) 0 0
\(807\) − 2.70504e11i − 0.637792i
\(808\) 0 0
\(809\) 4.77958e11 1.11582 0.557912 0.829900i \(-0.311603\pi\)
0.557912 + 0.829900i \(0.311603\pi\)
\(810\) 0 0
\(811\) − 6.37503e11i − 1.47366i −0.676075 0.736832i \(-0.736321\pi\)
0.676075 0.736832i \(-0.263679\pi\)
\(812\) 0 0
\(813\) −7.98393e11 −1.82749
\(814\) 0 0
\(815\) − 3.52890e11i − 0.799851i
\(816\) 0 0
\(817\) −1.15506e11 −0.259250
\(818\) 0 0
\(819\) 2.61924e10i 0.0582155i
\(820\) 0 0
\(821\) 8.43824e11 1.85729 0.928644 0.370973i \(-0.120976\pi\)
0.928644 + 0.370973i \(0.120976\pi\)
\(822\) 0 0
\(823\) 2.60916e11i 0.568723i 0.958717 + 0.284362i \(0.0917817\pi\)
−0.958717 + 0.284362i \(0.908218\pi\)
\(824\) 0 0
\(825\) −3.42164e10 −0.0738616
\(826\) 0 0
\(827\) 2.78675e11i 0.595765i 0.954602 + 0.297883i \(0.0962804\pi\)
−0.954602 + 0.297883i \(0.903720\pi\)
\(828\) 0 0
\(829\) −4.75156e11 −1.00605 −0.503023 0.864273i \(-0.667779\pi\)
−0.503023 + 0.864273i \(0.667779\pi\)
\(830\) 0 0
\(831\) 8.22306e11i 1.72437i
\(832\) 0 0
\(833\) 2.78322e11 0.578053
\(834\) 0 0
\(835\) 2.85552e11i 0.587408i
\(836\) 0 0
\(837\) −2.13043e10 −0.0434075
\(838\) 0 0
\(839\) 3.35440e11i 0.676966i 0.940973 + 0.338483i \(0.109914\pi\)
−0.940973 + 0.338483i \(0.890086\pi\)
\(840\) 0 0
\(841\) −4.83806e11 −0.967134
\(842\) 0 0
\(843\) 3.07858e11i 0.609594i
\(844\) 0 0
\(845\) 4.79344e11 0.940200
\(846\) 0 0
\(847\) 1.78138e11i 0.346117i
\(848\) 0 0
\(849\) −1.17019e11 −0.225229
\(850\) 0 0
\(851\) 8.23600e11i 1.57036i
\(852\) 0 0
\(853\) 9.75408e10 0.184243 0.0921213 0.995748i \(-0.470635\pi\)
0.0921213 + 0.995748i \(0.470635\pi\)
\(854\) 0 0
\(855\) − 4.06840e10i − 0.0761306i
\(856\) 0 0
\(857\) −7.94769e10 −0.147339 −0.0736695 0.997283i \(-0.523471\pi\)
−0.0736695 + 0.997283i \(0.523471\pi\)
\(858\) 0 0
\(859\) − 4.15618e11i − 0.763347i −0.924297 0.381673i \(-0.875348\pi\)
0.924297 0.381673i \(-0.124652\pi\)
\(860\) 0 0
\(861\) 3.00083e11 0.546045
\(862\) 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
\(867\) − 1.63230e11i − 0.288884i
\(868\) 0 0
\(869\) −6.63863e11 −1.16412
\(870\) 0 0
\(871\) 8.34540e10i 0.145002i
\(872\) 0 0
\(873\) −4.12879e11 −0.710830
\(874\) 0 0
\(875\) − 3.49134e11i − 0.595608i
\(876\) 0 0
\(877\) −2.74155e11 −0.463444 −0.231722 0.972782i \(-0.574436\pi\)
−0.231722 + 0.972782i \(0.574436\pi\)
\(878\) 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
\(883\) − 9.95008e11i − 1.63676i −0.574681 0.818378i \(-0.694874\pi\)
0.574681 0.818378i \(-0.305126\pi\)
\(884\) 0 0
\(885\) −2.27075e11 −0.370165
\(886\) 0 0
\(887\) − 5.46038e11i − 0.882122i −0.897477 0.441061i \(-0.854602\pi\)
0.897477 0.441061i \(-0.145398\pi\)
\(888\) 0 0
\(889\) −3.59736e11 −0.575940
\(890\) 0 0
\(891\) 9.94283e11i 1.57761i
\(892\) 0 0
\(893\) −1.48594e11 −0.233665
\(894\) 0 0
\(895\) − 8.65656e10i − 0.134913i
\(896\) 0 0
\(897\) −1.29650e11 −0.200264
\(898\) 0 0
\(899\) 8.71212e9i 0.0133378i
\(900\) 0 0
\(901\) −6.02474e10 −0.0914195
\(902\) 0 0
\(903\) − 8.28613e11i − 1.24624i
\(904\) 0 0
\(905\) 2.94365e11 0.438826
\(906\) 0 0
\(907\) 6.55018e11i 0.967886i 0.875100 + 0.483943i \(0.160796\pi\)
−0.875100 + 0.483943i \(0.839204\pi\)
\(908\) 0 0
\(909\) 9.49014e10 0.139001
\(910\) 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
\(915\) − 8.98791e11i − 1.28225i
\(916\) 0 0
\(917\) −4.36696e11 −0.617592
\(918\) 0 0
\(919\) − 5.33989e10i − 0.0748635i −0.999299 0.0374318i \(-0.988082\pi\)
0.999299 0.0374318i \(-0.0119177\pi\)
\(920\) 0 0
\(921\) 3.48897e11 0.484907
\(922\) 0 0
\(923\) 6.54235e9i 0.00901420i
\(924\) 0 0
\(925\) −6.43194e10 −0.0878567
\(926\) 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
\(931\) − 7.41953e10i − 0.0987593i
\(932\) 0 0
\(933\) −1.29703e12 −1.71168
\(934\) 0 0
\(935\) 8.24161e11i 1.07836i
\(936\) 0 0
\(937\) −8.41436e11 −1.09160 −0.545799 0.837916i \(-0.683774\pi\)
−0.545799 + 0.837916i \(0.683774\pi\)
\(938\) 0 0
\(939\) − 6.30660e11i − 0.811209i
\(940\) 0 0
\(941\) 4.52935e11 0.577666 0.288833 0.957379i \(-0.406733\pi\)
0.288833 + 0.957379i \(0.406733\pi\)
\(942\) 0 0
\(943\) 5.09262e11i 0.644013i
\(944\) 0 0
\(945\) −2.67556e11 −0.335497
\(946\) 0 0
\(947\) − 1.00309e12i − 1.24721i −0.781739 0.623605i \(-0.785667\pi\)
0.781739 0.623605i \(-0.214333\pi\)
\(948\) 0 0
\(949\) −3.13192e10 −0.0386141
\(950\) 0 0
\(951\) − 1.65769e12i − 2.02666i
\(952\) 0 0
\(953\) −1.39040e12 −1.68565 −0.842826 0.538186i \(-0.819110\pi\)
−0.842826 + 0.538186i \(0.819110\pi\)
\(954\) 0 0
\(955\) 6.05401e11i 0.727830i
\(956\) 0 0
\(957\) 2.36831e11 0.282352
\(958\) 0 0
\(959\) − 3.10523e11i − 0.367130i
\(960\) 0 0
\(961\) 8.48274e11 0.994587
\(962\) 0 0
\(963\) − 3.43421e11i − 0.399321i
\(964\) 0 0
\(965\) −7.17318e11 −0.827185
\(966\) 0 0
\(967\) 9.01843e11i 1.03140i 0.856771 + 0.515698i \(0.172467\pi\)
−0.856771 + 0.515698i \(0.827533\pi\)
\(968\) 0 0
\(969\) 1.42297e11 0.161399
\(970\) 0 0
\(971\) 1.28411e12i 1.44452i 0.691621 + 0.722261i \(0.256897\pi\)
−0.691621 + 0.722261i \(0.743103\pi\)
\(972\) 0 0
\(973\) −4.12712e11 −0.460464
\(974\) 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
\(979\) 1.54026e12i 1.67674i
\(980\) 0 0
\(981\) −2.02202e11 −0.218328
\(982\) 0 0
\(983\) − 3.77388e11i − 0.404179i −0.979367 0.202090i \(-0.935227\pi\)
0.979367 0.202090i \(-0.0647733\pi\)
\(984\) 0 0
\(985\) 1.04028e12 1.10511
\(986\) 0 0
\(987\) − 1.06597e12i − 1.12325i
\(988\) 0 0
\(989\) 1.40622e12 1.46983
\(990\) 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
\(995\) 1.51912e12i 1.54989i
\(996\) 0 0
\(997\) 1.03163e12 1.04410 0.522050 0.852915i \(-0.325167\pi\)
0.522050 + 0.852915i \(0.325167\pi\)
\(998\) 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 64.9.c.b.63.2 2
4.3 odd 2 inner 64.9.c.b.63.1 2
8.3 odd 2 4.9.b.b.3.2 yes 2
8.5 even 2 4.9.b.b.3.1 2
16.3 odd 4 256.9.d.e.127.4 4
16.5 even 4 256.9.d.e.127.3 4
16.11 odd 4 256.9.d.e.127.1 4
16.13 even 4 256.9.d.e.127.2 4
24.5 odd 2 36.9.d.b.19.2 2
24.11 even 2 36.9.d.b.19.1 2
40.3 even 4 100.9.d.b.99.4 4
40.13 odd 4 100.9.d.b.99.2 4
40.19 odd 2 100.9.b.c.51.1 2
40.27 even 4 100.9.d.b.99.1 4
40.29 even 2 100.9.b.c.51.2 2
40.37 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 8.5 even 2
4.9.b.b.3.2 yes 2 8.3 odd 2
36.9.d.b.19.1 2 24.11 even 2
36.9.d.b.19.2 2 24.5 odd 2
64.9.c.b.63.1 2 4.3 odd 2 inner
64.9.c.b.63.2 2 1.1 even 1 trivial
100.9.b.c.51.1 2 40.19 odd 2
100.9.b.c.51.2 2 40.29 even 2
100.9.d.b.99.1 4 40.27 even 4
100.9.d.b.99.2 4 40.13 odd 4
100.9.d.b.99.3 4 40.37 odd 4
100.9.d.b.99.4 4 40.3 even 4
256.9.d.e.127.1 4 16.11 odd 4
256.9.d.e.127.2 4 16.13 even 4
256.9.d.e.127.3 4 16.5 even 4
256.9.d.e.127.4 4 16.3 odd 4