Properties

Label 48.22.c.b.47.6
Level $48$
Weight $22$
Character 48.47
Analytic conductor $134.149$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,22,Mod(47,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(134.149125258\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{68}\cdot 3^{56}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 47.6
Root \(0.500000 + 3.90105e7i\) of defining polynomial
Character \(\chi\) \(=\) 48.47
Dual form 48.22.c.b.47.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-30363.0 + 97665.0i) q^{3} -3.55520e7i q^{5} +1.36883e9i q^{7} +(-8.61653e9 - 5.93080e9i) q^{9} +O(q^{10})\) \(q+(-30363.0 + 97665.0i) q^{3} -3.55520e7i q^{5} +1.36883e9i q^{7} +(-8.61653e9 - 5.93080e9i) q^{9} -8.13058e10 q^{11} -1.01527e11 q^{13} +(3.47219e12 + 1.07947e12i) q^{15} +1.39503e13i q^{17} +9.45210e12i q^{19} +(-1.33686e14 - 4.15616e13i) q^{21} +1.97092e14 q^{23} -7.87110e14 q^{25} +(8.40855e14 - 6.61456e14i) q^{27} +3.11066e15i q^{29} -3.21136e15i q^{31} +(2.46869e15 - 7.94072e15i) q^{33} +4.86645e16 q^{35} +2.51240e16 q^{37} +(3.08267e15 - 9.91566e15i) q^{39} -1.94790e16i q^{41} +2.27069e16i q^{43} +(-2.10852e17 + 3.06335e17i) q^{45} -2.64317e17 q^{47} -1.31514e18 q^{49} +(-1.36245e18 - 4.23573e17i) q^{51} +1.40537e18i q^{53} +2.89059e18i q^{55} +(-9.23139e17 - 2.86994e17i) q^{57} +1.50456e17 q^{59} -1.64227e17 q^{61} +(8.11823e18 - 1.17945e19i) q^{63} +3.60950e18i q^{65} +1.03110e19i q^{67} +(-5.98430e18 + 1.92490e19i) q^{69} +1.34577e19 q^{71} -4.29171e19 q^{73} +(2.38990e19 - 7.68731e19i) q^{75} -1.11293e20i q^{77} +1.61084e20i q^{79} +(3.90703e19 + 1.02206e20i) q^{81} -1.59123e20 q^{83} +4.95961e20 q^{85} +(-3.03802e20 - 9.44489e19i) q^{87} -4.43760e20i q^{89} -1.38973e20i q^{91} +(3.13637e20 + 9.75065e19i) q^{93} +3.36042e20 q^{95} -1.06077e21 q^{97} +(7.00574e20 + 4.82208e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6223178268 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6223178268 q^{9} - 1558099630680 q^{13} - 467668605656952 q^{21} - 23\!\cdots\!04 q^{25}+ \cdots - 47\!\cdots\!00 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −30363.0 + 97665.0i −0.296873 + 0.954917i
\(4\) 0 0
\(5\) 3.55520e7i 1.62809i −0.580799 0.814047i \(-0.697260\pi\)
0.580799 0.814047i \(-0.302740\pi\)
\(6\) 0 0
\(7\) 1.36883e9i 1.83155i 0.401692 + 0.915775i \(0.368422\pi\)
−0.401692 + 0.915775i \(0.631578\pi\)
\(8\) 0 0
\(9\) −8.61653e9 5.93080e9i −0.823732 0.566979i
\(10\) 0 0
\(11\) −8.13058e10 −0.945144 −0.472572 0.881292i \(-0.656674\pi\)
−0.472572 + 0.881292i \(0.656674\pi\)
\(12\) 0 0
\(13\) −1.01527e11 −0.204257 −0.102129 0.994771i \(-0.532565\pi\)
−0.102129 + 0.994771i \(0.532565\pi\)
\(14\) 0 0
\(15\) 3.47219e12 + 1.07947e12i 1.55469 + 0.483338i
\(16\) 0 0
\(17\) 1.39503e13i 1.67830i 0.543900 + 0.839150i \(0.316947\pi\)
−0.543900 + 0.839150i \(0.683053\pi\)
\(18\) 0 0
\(19\) 9.45210e12i 0.353684i 0.984239 + 0.176842i \(0.0565882\pi\)
−0.984239 + 0.176842i \(0.943412\pi\)
\(20\) 0 0
\(21\) −1.33686e14 4.15616e13i −1.74898 0.543738i
\(22\) 0 0
\(23\) 1.97092e14 0.992035 0.496018 0.868313i \(-0.334795\pi\)
0.496018 + 0.868313i \(0.334795\pi\)
\(24\) 0 0
\(25\) −7.87110e14 −1.65069
\(26\) 0 0
\(27\) 8.40855e14 6.61456e14i 0.785962 0.618275i
\(28\) 0 0
\(29\) 3.11066e15i 1.37301i 0.727126 + 0.686504i \(0.240856\pi\)
−0.727126 + 0.686504i \(0.759144\pi\)
\(30\) 0 0
\(31\) 3.21136e15i 0.703706i −0.936055 0.351853i \(-0.885552\pi\)
0.936055 0.351853i \(-0.114448\pi\)
\(32\) 0 0
\(33\) 2.46869e15 7.94072e15i 0.280588 0.902534i
\(34\) 0 0
\(35\) 4.86645e16 2.98193
\(36\) 0 0
\(37\) 2.51240e16 0.858954 0.429477 0.903078i \(-0.358698\pi\)
0.429477 + 0.903078i \(0.358698\pi\)
\(38\) 0 0
\(39\) 3.08267e15 9.91566e15i 0.0606386 0.195049i
\(40\) 0 0
\(41\) 1.94790e16i 0.226639i −0.993559 0.113320i \(-0.963852\pi\)
0.993559 0.113320i \(-0.0361484\pi\)
\(42\) 0 0
\(43\) 2.27069e16i 0.160228i 0.996786 + 0.0801141i \(0.0255285\pi\)
−0.996786 + 0.0801141i \(0.974472\pi\)
\(44\) 0 0
\(45\) −2.10852e17 + 3.06335e17i −0.923095 + 1.34111i
\(46\) 0 0
\(47\) −2.64317e17 −0.732990 −0.366495 0.930420i \(-0.619442\pi\)
−0.366495 + 0.930420i \(0.619442\pi\)
\(48\) 0 0
\(49\) −1.31514e18 −2.35457
\(50\) 0 0
\(51\) −1.36245e18 4.23573e17i −1.60264 0.498243i
\(52\) 0 0
\(53\) 1.40537e18i 1.10381i 0.833908 + 0.551903i \(0.186098\pi\)
−0.833908 + 0.551903i \(0.813902\pi\)
\(54\) 0 0
\(55\) 2.89059e18i 1.53878i
\(56\) 0 0
\(57\) −9.23139e17 2.86994e17i −0.337739 0.104999i
\(58\) 0 0
\(59\) 1.50456e17 0.0383234 0.0191617 0.999816i \(-0.493900\pi\)
0.0191617 + 0.999816i \(0.493900\pi\)
\(60\) 0 0
\(61\) −1.64227e17 −0.0294769 −0.0147384 0.999891i \(-0.504692\pi\)
−0.0147384 + 0.999891i \(0.504692\pi\)
\(62\) 0 0
\(63\) 8.11823e18 1.17945e19i 1.03845 1.50871i
\(64\) 0 0
\(65\) 3.60950e18i 0.332550i
\(66\) 0 0
\(67\) 1.03110e19i 0.691059i 0.938408 + 0.345529i \(0.112301\pi\)
−0.938408 + 0.345529i \(0.887699\pi\)
\(68\) 0 0
\(69\) −5.98430e18 + 1.92490e19i −0.294509 + 0.947311i
\(70\) 0 0
\(71\) 1.34577e19 0.490636 0.245318 0.969443i \(-0.421108\pi\)
0.245318 + 0.969443i \(0.421108\pi\)
\(72\) 0 0
\(73\) −4.29171e19 −1.16880 −0.584401 0.811465i \(-0.698670\pi\)
−0.584401 + 0.811465i \(0.698670\pi\)
\(74\) 0 0
\(75\) 2.38990e19 7.68731e19i 0.490046 1.57627i
\(76\) 0 0
\(77\) 1.11293e20i 1.73108i
\(78\) 0 0
\(79\) 1.61084e20i 1.91411i 0.289905 + 0.957055i \(0.406376\pi\)
−0.289905 + 0.957055i \(0.593624\pi\)
\(80\) 0 0
\(81\) 3.90703e19 + 1.02206e20i 0.357070 + 0.934078i
\(82\) 0 0
\(83\) −1.59123e20 −1.12568 −0.562838 0.826567i \(-0.690291\pi\)
−0.562838 + 0.826567i \(0.690291\pi\)
\(84\) 0 0
\(85\) 4.95961e20 2.73243
\(86\) 0 0
\(87\) −3.03802e20 9.44489e19i −1.31111 0.407610i
\(88\) 0 0
\(89\) 4.43760e20i 1.50853i −0.656571 0.754264i \(-0.727994\pi\)
0.656571 0.754264i \(-0.272006\pi\)
\(90\) 0 0
\(91\) 1.38973e20i 0.374107i
\(92\) 0 0
\(93\) 3.13637e20 + 9.75065e19i 0.671981 + 0.208912i
\(94\) 0 0
\(95\) 3.36042e20 0.575831
\(96\) 0 0
\(97\) −1.06077e21 −1.46056 −0.730278 0.683150i \(-0.760610\pi\)
−0.730278 + 0.683150i \(0.760610\pi\)
\(98\) 0 0
\(99\) 7.00574e20 + 4.82208e20i 0.778546 + 0.535877i
\(100\) 0 0
\(101\) 1.15660e21i 1.04185i −0.853601 0.520927i \(-0.825586\pi\)
0.853601 0.520927i \(-0.174414\pi\)
\(102\) 0 0
\(103\) 1.98586e21i 1.45598i −0.685585 0.727992i \(-0.740454\pi\)
0.685585 0.727992i \(-0.259546\pi\)
\(104\) 0 0
\(105\) −1.47760e21 + 4.75282e21i −0.885257 + 2.84750i
\(106\) 0 0
\(107\) 2.46370e21 1.21076 0.605380 0.795937i \(-0.293021\pi\)
0.605380 + 0.795937i \(0.293021\pi\)
\(108\) 0 0
\(109\) −1.35594e20 −0.0548609 −0.0274304 0.999624i \(-0.508732\pi\)
−0.0274304 + 0.999624i \(0.508732\pi\)
\(110\) 0 0
\(111\) −7.62838e20 + 2.45373e21i −0.255001 + 0.820230i
\(112\) 0 0
\(113\) 4.56440e21i 1.26491i −0.774596 0.632456i \(-0.782047\pi\)
0.774596 0.632456i \(-0.217953\pi\)
\(114\) 0 0
\(115\) 7.00703e21i 1.61513i
\(116\) 0 0
\(117\) 8.74813e20 + 6.02138e20i 0.168253 + 0.115810i
\(118\) 0 0
\(119\) −1.90955e22 −3.07389
\(120\) 0 0
\(121\) −7.89623e20 −0.106702
\(122\) 0 0
\(123\) 1.90241e21 + 5.91440e20i 0.216422 + 0.0672832i
\(124\) 0 0
\(125\) 1.10308e22i 1.05938i
\(126\) 0 0
\(127\) 3.26959e21i 0.265800i −0.991129 0.132900i \(-0.957571\pi\)
0.991129 0.132900i \(-0.0424289\pi\)
\(128\) 0 0
\(129\) −2.21767e21 6.89449e20i −0.153005 0.0475675i
\(130\) 0 0
\(131\) 2.36405e22 1.38774 0.693870 0.720100i \(-0.255904\pi\)
0.693870 + 0.720100i \(0.255904\pi\)
\(132\) 0 0
\(133\) −1.29383e22 −0.647790
\(134\) 0 0
\(135\) −2.35161e22 2.98941e22i −1.00661 1.27962i
\(136\) 0 0
\(137\) 2.16152e22i 0.792854i −0.918066 0.396427i \(-0.870250\pi\)
0.918066 0.396427i \(-0.129750\pi\)
\(138\) 0 0
\(139\) 3.40010e22i 1.07111i −0.844499 0.535557i \(-0.820102\pi\)
0.844499 0.535557i \(-0.179898\pi\)
\(140\) 0 0
\(141\) 8.02546e21 2.58145e22i 0.217605 0.699944i
\(142\) 0 0
\(143\) 8.25476e21 0.193053
\(144\) 0 0
\(145\) 1.10590e23 2.23539
\(146\) 0 0
\(147\) 3.99315e22 1.28443e23i 0.699010 2.24842i
\(148\) 0 0
\(149\) 4.03375e22i 0.612708i 0.951918 + 0.306354i \(0.0991091\pi\)
−0.951918 + 0.306354i \(0.900891\pi\)
\(150\) 0 0
\(151\) 2.84800e22i 0.376082i −0.982161 0.188041i \(-0.939786\pi\)
0.982161 0.188041i \(-0.0602138\pi\)
\(152\) 0 0
\(153\) 8.27364e22 1.20203e23i 0.951561 1.38247i
\(154\) 0 0
\(155\) −1.14170e23 −1.14570
\(156\) 0 0
\(157\) −1.30679e23 −1.14619 −0.573097 0.819488i \(-0.694258\pi\)
−0.573097 + 0.819488i \(0.694258\pi\)
\(158\) 0 0
\(159\) −1.37255e23 4.26711e22i −1.05404 0.327691i
\(160\) 0 0
\(161\) 2.69785e23i 1.81696i
\(162\) 0 0
\(163\) 2.56766e23i 1.51903i −0.650487 0.759517i \(-0.725435\pi\)
0.650487 0.759517i \(-0.274565\pi\)
\(164\) 0 0
\(165\) −2.82309e23 8.77668e22i −1.46941 0.456824i
\(166\) 0 0
\(167\) −3.27675e23 −1.50286 −0.751432 0.659810i \(-0.770637\pi\)
−0.751432 + 0.659810i \(0.770637\pi\)
\(168\) 0 0
\(169\) −2.36757e23 −0.958279
\(170\) 0 0
\(171\) 5.60585e22 8.14444e22i 0.200531 0.291341i
\(172\) 0 0
\(173\) 2.54074e23i 0.804408i 0.915550 + 0.402204i \(0.131756\pi\)
−0.915550 + 0.402204i \(0.868244\pi\)
\(174\) 0 0
\(175\) 1.07742e24i 3.02332i
\(176\) 0 0
\(177\) −4.56829e21 + 1.46943e22i −0.0113772 + 0.0365956i
\(178\) 0 0
\(179\) −4.15928e22 −0.0920579 −0.0460290 0.998940i \(-0.514657\pi\)
−0.0460290 + 0.998940i \(0.514657\pi\)
\(180\) 0 0
\(181\) 1.99467e23 0.392867 0.196433 0.980517i \(-0.437064\pi\)
0.196433 + 0.980517i \(0.437064\pi\)
\(182\) 0 0
\(183\) 4.98642e21 1.60392e22i 0.00875090 0.0281480i
\(184\) 0 0
\(185\) 8.93208e23i 1.39846i
\(186\) 0 0
\(187\) 1.13424e24i 1.58624i
\(188\) 0 0
\(189\) 9.05418e23 + 1.15098e24i 1.13240 + 1.43953i
\(190\) 0 0
\(191\) 1.63158e24 1.82708 0.913542 0.406745i \(-0.133336\pi\)
0.913542 + 0.406745i \(0.133336\pi\)
\(192\) 0 0
\(193\) 5.17016e23 0.518982 0.259491 0.965746i \(-0.416445\pi\)
0.259491 + 0.965746i \(0.416445\pi\)
\(194\) 0 0
\(195\) −3.52522e23 1.09595e23i −0.317558 0.0987253i
\(196\) 0 0
\(197\) 1.35824e24i 1.09921i −0.835424 0.549606i \(-0.814778\pi\)
0.835424 0.549606i \(-0.185222\pi\)
\(198\) 0 0
\(199\) 1.58981e24i 1.15715i −0.815631 0.578573i \(-0.803610\pi\)
0.815631 0.578573i \(-0.196390\pi\)
\(200\) 0 0
\(201\) −1.00702e24 3.13072e23i −0.659903 0.205157i
\(202\) 0 0
\(203\) −4.25795e24 −2.51473
\(204\) 0 0
\(205\) −6.92517e23 −0.368990
\(206\) 0 0
\(207\) −1.69825e24 1.16891e24i −0.817171 0.562463i
\(208\) 0 0
\(209\) 7.68511e23i 0.334283i
\(210\) 0 0
\(211\) 3.51376e24i 1.38295i 0.722400 + 0.691476i \(0.243039\pi\)
−0.722400 + 0.691476i \(0.756961\pi\)
\(212\) 0 0
\(213\) −4.08617e23 + 1.31435e24i −0.145657 + 0.468516i
\(214\) 0 0
\(215\) 8.07277e23 0.260867
\(216\) 0 0
\(217\) 4.39579e24 1.28887
\(218\) 0 0
\(219\) 1.30309e24 4.19150e24i 0.346986 1.11611i
\(220\) 0 0
\(221\) 1.41634e24i 0.342805i
\(222\) 0 0
\(223\) 3.91484e24i 0.862012i 0.902349 + 0.431006i \(0.141841\pi\)
−0.902349 + 0.431006i \(0.858159\pi\)
\(224\) 0 0
\(225\) 6.78216e24 + 4.66819e24i 1.35973 + 0.935906i
\(226\) 0 0
\(227\) 2.66708e24 0.487264 0.243632 0.969868i \(-0.421661\pi\)
0.243632 + 0.969868i \(0.421661\pi\)
\(228\) 0 0
\(229\) −8.18167e24 −1.36323 −0.681616 0.731711i \(-0.738722\pi\)
−0.681616 + 0.731711i \(0.738722\pi\)
\(230\) 0 0
\(231\) 1.08695e25 + 3.37920e24i 1.65304 + 0.513911i
\(232\) 0 0
\(233\) 6.36272e24i 0.883904i −0.897039 0.441952i \(-0.854286\pi\)
0.897039 0.441952i \(-0.145714\pi\)
\(234\) 0 0
\(235\) 9.39701e24i 1.19338i
\(236\) 0 0
\(237\) −1.57322e25 4.89098e24i −1.82782 0.568249i
\(238\) 0 0
\(239\) 7.64315e24 0.813007 0.406504 0.913649i \(-0.366748\pi\)
0.406504 + 0.913649i \(0.366748\pi\)
\(240\) 0 0
\(241\) −2.50959e24 −0.244581 −0.122291 0.992494i \(-0.539024\pi\)
−0.122291 + 0.992494i \(0.539024\pi\)
\(242\) 0 0
\(243\) −1.11682e25 + 7.12520e23i −0.997971 + 0.0636695i
\(244\) 0 0
\(245\) 4.67558e25i 3.83347i
\(246\) 0 0
\(247\) 9.59647e23i 0.0722426i
\(248\) 0 0
\(249\) 4.83145e24 1.55408e25i 0.334183 1.07493i
\(250\) 0 0
\(251\) 1.37900e25 0.876984 0.438492 0.898735i \(-0.355513\pi\)
0.438492 + 0.898735i \(0.355513\pi\)
\(252\) 0 0
\(253\) −1.60247e25 −0.937616
\(254\) 0 0
\(255\) −1.50589e25 + 4.84380e25i −0.811186 + 2.60924i
\(256\) 0 0
\(257\) 1.53289e25i 0.760701i −0.924842 0.380350i \(-0.875803\pi\)
0.924842 0.380350i \(-0.124197\pi\)
\(258\) 0 0
\(259\) 3.43903e25i 1.57322i
\(260\) 0 0
\(261\) 1.84487e25 2.68031e25i 0.778467 1.13099i
\(262\) 0 0
\(263\) −4.28430e25 −1.66857 −0.834286 0.551332i \(-0.814120\pi\)
−0.834286 + 0.551332i \(0.814120\pi\)
\(264\) 0 0
\(265\) 4.99636e25 1.79710
\(266\) 0 0
\(267\) 4.33398e25 + 1.34739e25i 1.44052 + 0.447842i
\(268\) 0 0
\(269\) 2.46839e25i 0.758602i −0.925273 0.379301i \(-0.876164\pi\)
0.925273 0.379301i \(-0.123836\pi\)
\(270\) 0 0
\(271\) 4.10139e24i 0.116615i −0.998299 0.0583073i \(-0.981430\pi\)
0.998299 0.0583073i \(-0.0185703\pi\)
\(272\) 0 0
\(273\) 1.35728e25 + 4.21964e24i 0.357241 + 0.111063i
\(274\) 0 0
\(275\) 6.39966e25 1.56014
\(276\) 0 0
\(277\) 3.49763e25 0.790199 0.395100 0.918638i \(-0.370710\pi\)
0.395100 + 0.918638i \(0.370710\pi\)
\(278\) 0 0
\(279\) −1.90459e25 + 2.76708e25i −0.398986 + 0.579666i
\(280\) 0 0
\(281\) 2.73843e25i 0.532212i 0.963944 + 0.266106i \(0.0857371\pi\)
−0.963944 + 0.266106i \(0.914263\pi\)
\(282\) 0 0
\(283\) 4.89255e25i 0.882629i 0.897353 + 0.441314i \(0.145488\pi\)
−0.897353 + 0.441314i \(0.854512\pi\)
\(284\) 0 0
\(285\) −1.02032e25 + 3.28195e25i −0.170949 + 0.549871i
\(286\) 0 0
\(287\) 2.66633e25 0.415101
\(288\) 0 0
\(289\) −1.25519e26 −1.81669
\(290\) 0 0
\(291\) 3.22082e25 1.03600e26i 0.433600 1.39471i
\(292\) 0 0
\(293\) 7.20642e25i 0.902838i −0.892312 0.451419i \(-0.850918\pi\)
0.892312 0.451419i \(-0.149082\pi\)
\(294\) 0 0
\(295\) 5.34902e24i 0.0623940i
\(296\) 0 0
\(297\) −6.83663e25 + 5.37802e25i −0.742847 + 0.584359i
\(298\) 0 0
\(299\) −2.00102e25 −0.202630
\(300\) 0 0
\(301\) −3.10818e25 −0.293466
\(302\) 0 0
\(303\) 1.12959e26 + 3.51177e25i 0.994884 + 0.309299i
\(304\) 0 0
\(305\) 5.83860e24i 0.0479911i
\(306\) 0 0
\(307\) 1.74780e26i 1.34134i 0.741755 + 0.670671i \(0.233994\pi\)
−0.741755 + 0.670671i \(0.766006\pi\)
\(308\) 0 0
\(309\) 1.93948e26 + 6.02965e25i 1.39034 + 0.432243i
\(310\) 0 0
\(311\) −1.44780e26 −0.969894 −0.484947 0.874544i \(-0.661161\pi\)
−0.484947 + 0.874544i \(0.661161\pi\)
\(312\) 0 0
\(313\) −3.52618e25 −0.220846 −0.110423 0.993885i \(-0.535221\pi\)
−0.110423 + 0.993885i \(0.535221\pi\)
\(314\) 0 0
\(315\) −4.19319e26 2.88620e26i −2.45632 1.69069i
\(316\) 0 0
\(317\) 1.26009e25i 0.0690686i −0.999404 0.0345343i \(-0.989005\pi\)
0.999404 0.0345343i \(-0.0109948\pi\)
\(318\) 0 0
\(319\) 2.52914e26i 1.29769i
\(320\) 0 0
\(321\) −7.48052e25 + 2.40617e26i −0.359442 + 1.15617i
\(322\) 0 0
\(323\) −1.31860e26 −0.593588
\(324\) 0 0
\(325\) 7.99132e25 0.337165
\(326\) 0 0
\(327\) 4.11704e24 1.32428e25i 0.0162867 0.0523876i
\(328\) 0 0
\(329\) 3.61804e26i 1.34251i
\(330\) 0 0
\(331\) 3.75401e26i 1.30708i −0.756892 0.653540i \(-0.773283\pi\)
0.756892 0.653540i \(-0.226717\pi\)
\(332\) 0 0
\(333\) −2.16481e26 1.49005e26i −0.707548 0.487009i
\(334\) 0 0
\(335\) 3.66577e26 1.12511
\(336\) 0 0
\(337\) −2.88457e26 −0.831702 −0.415851 0.909433i \(-0.636516\pi\)
−0.415851 + 0.909433i \(0.636516\pi\)
\(338\) 0 0
\(339\) 4.45782e26 + 1.38589e26i 1.20789 + 0.375519i
\(340\) 0 0
\(341\) 2.61102e26i 0.665104i
\(342\) 0 0
\(343\) 1.03564e27i 2.48097i
\(344\) 0 0
\(345\) 6.84341e26 + 2.12754e26i 1.54231 + 0.479488i
\(346\) 0 0
\(347\) −3.55460e26 −0.753931 −0.376965 0.926227i \(-0.623032\pi\)
−0.376965 + 0.926227i \(0.623032\pi\)
\(348\) 0 0
\(349\) −6.03239e25 −0.120454 −0.0602272 0.998185i \(-0.519183\pi\)
−0.0602272 + 0.998185i \(0.519183\pi\)
\(350\) 0 0
\(351\) −8.53697e25 + 6.71559e25i −0.160538 + 0.126287i
\(352\) 0 0
\(353\) 1.42270e25i 0.0252046i 0.999921 + 0.0126023i \(0.00401155\pi\)
−0.999921 + 0.0126023i \(0.995988\pi\)
\(354\) 0 0
\(355\) 4.78450e26i 0.798801i
\(356\) 0 0
\(357\) 5.79797e26 1.86496e27i 0.912556 2.93531i
\(358\) 0 0
\(359\) 1.85469e26 0.275283 0.137641 0.990482i \(-0.456048\pi\)
0.137641 + 0.990482i \(0.456048\pi\)
\(360\) 0 0
\(361\) 6.24867e26 0.874907
\(362\) 0 0
\(363\) 2.39753e25 7.71185e25i 0.0316770 0.101892i
\(364\) 0 0
\(365\) 1.52579e27i 1.90292i
\(366\) 0 0
\(367\) 5.69151e26i 0.670246i −0.942174 0.335123i \(-0.891222\pi\)
0.942174 0.335123i \(-0.108778\pi\)
\(368\) 0 0
\(369\) −1.15526e26 + 1.67841e26i −0.128500 + 0.186690i
\(370\) 0 0
\(371\) −1.92370e27 −2.02167
\(372\) 0 0
\(373\) −1.39733e26 −0.138789 −0.0693945 0.997589i \(-0.522107\pi\)
−0.0693945 + 0.997589i \(0.522107\pi\)
\(374\) 0 0
\(375\) −1.07733e27 3.34929e26i −1.01162 0.314503i
\(376\) 0 0
\(377\) 3.15817e26i 0.280447i
\(378\) 0 0
\(379\) 1.28970e27i 1.08337i 0.840581 + 0.541686i \(0.182214\pi\)
−0.840581 + 0.541686i \(0.817786\pi\)
\(380\) 0 0
\(381\) 3.19325e26 + 9.92746e25i 0.253817 + 0.0789089i
\(382\) 0 0
\(383\) −1.08252e25 −0.00814420 −0.00407210 0.999992i \(-0.501296\pi\)
−0.00407210 + 0.999992i \(0.501296\pi\)
\(384\) 0 0
\(385\) −3.95671e27 −2.81836
\(386\) 0 0
\(387\) 1.34670e26 1.95655e26i 0.0908460 0.131985i
\(388\) 0 0
\(389\) 7.32643e25i 0.0468190i 0.999726 + 0.0234095i \(0.00745215\pi\)
−0.999726 + 0.0234095i \(0.992548\pi\)
\(390\) 0 0
\(391\) 2.74949e27i 1.66493i
\(392\) 0 0
\(393\) −7.17796e26 + 2.30885e27i −0.411983 + 1.32518i
\(394\) 0 0
\(395\) 5.72685e27 3.11635
\(396\) 0 0
\(397\) −2.26195e27 −1.16730 −0.583650 0.812006i \(-0.698376\pi\)
−0.583650 + 0.812006i \(0.698376\pi\)
\(398\) 0 0
\(399\) 3.92845e26 1.26362e27i 0.192312 0.618586i
\(400\) 0 0
\(401\) 5.07907e26i 0.235922i 0.993018 + 0.117961i \(0.0376358\pi\)
−0.993018 + 0.117961i \(0.962364\pi\)
\(402\) 0 0
\(403\) 3.26041e26i 0.143737i
\(404\) 0 0
\(405\) 3.63363e27 1.38903e27i 1.52077 0.581344i
\(406\) 0 0
\(407\) −2.04272e27 −0.811835
\(408\) 0 0
\(409\) 1.42237e27 0.536928 0.268464 0.963290i \(-0.413484\pi\)
0.268464 + 0.963290i \(0.413484\pi\)
\(410\) 0 0
\(411\) 2.11105e27 + 6.56302e26i 0.757109 + 0.235377i
\(412\) 0 0
\(413\) 2.05948e26i 0.0701911i
\(414\) 0 0
\(415\) 5.65715e27i 1.83271i
\(416\) 0 0
\(417\) 3.32070e27 + 1.03237e27i 1.02283 + 0.317986i
\(418\) 0 0
\(419\) −5.74722e26 −0.168349 −0.0841745 0.996451i \(-0.526825\pi\)
−0.0841745 + 0.996451i \(0.526825\pi\)
\(420\) 0 0
\(421\) 3.15296e27 0.878531 0.439265 0.898357i \(-0.355239\pi\)
0.439265 + 0.898357i \(0.355239\pi\)
\(422\) 0 0
\(423\) 2.27750e27 + 1.56761e27i 0.603787 + 0.415590i
\(424\) 0 0
\(425\) 1.09804e28i 2.77035i
\(426\) 0 0
\(427\) 2.24798e26i 0.0539883i
\(428\) 0 0
\(429\) −2.50639e26 + 8.06200e26i −0.0573122 + 0.184349i
\(430\) 0 0
\(431\) 5.60602e26 0.122080 0.0610398 0.998135i \(-0.480558\pi\)
0.0610398 + 0.998135i \(0.480558\pi\)
\(432\) 0 0
\(433\) −4.72416e27 −0.979945 −0.489973 0.871738i \(-0.662993\pi\)
−0.489973 + 0.871738i \(0.662993\pi\)
\(434\) 0 0
\(435\) −3.35785e27 + 1.08008e28i −0.663627 + 2.13461i
\(436\) 0 0
\(437\) 1.86294e27i 0.350867i
\(438\) 0 0
\(439\) 4.42156e27i 0.793776i 0.917867 + 0.396888i \(0.129910\pi\)
−0.917867 + 0.396888i \(0.870090\pi\)
\(440\) 0 0
\(441\) 1.13319e28 + 7.79981e27i 1.93954 + 1.33499i
\(442\) 0 0
\(443\) −5.39591e26 −0.0880695 −0.0440347 0.999030i \(-0.514021\pi\)
−0.0440347 + 0.999030i \(0.514021\pi\)
\(444\) 0 0
\(445\) −1.57766e28 −2.45603
\(446\) 0 0
\(447\) −3.93956e27 1.22477e27i −0.585085 0.181897i
\(448\) 0 0
\(449\) 2.75637e27i 0.390617i −0.980742 0.195309i \(-0.937429\pi\)
0.980742 0.195309i \(-0.0625709\pi\)
\(450\) 0 0
\(451\) 1.58375e27i 0.214207i
\(452\) 0 0
\(453\) 2.78150e27 + 8.64739e26i 0.359127 + 0.111649i
\(454\) 0 0
\(455\) −4.94078e27 −0.609082
\(456\) 0 0
\(457\) 3.15103e27 0.370964 0.185482 0.982648i \(-0.440615\pi\)
0.185482 + 0.982648i \(0.440615\pi\)
\(458\) 0 0
\(459\) 9.22751e27 + 1.17302e28i 1.03765 + 1.31908i
\(460\) 0 0
\(461\) 7.89127e27i 0.847788i 0.905712 + 0.423894i \(0.139337\pi\)
−0.905712 + 0.423894i \(0.860663\pi\)
\(462\) 0 0
\(463\) 1.39692e28i 1.43407i −0.697035 0.717037i \(-0.745498\pi\)
0.697035 0.717037i \(-0.254502\pi\)
\(464\) 0 0
\(465\) 3.46655e27 1.11504e28i 0.340128 1.09405i
\(466\) 0 0
\(467\) 9.34461e27 0.876463 0.438232 0.898862i \(-0.355605\pi\)
0.438232 + 0.898862i \(0.355605\pi\)
\(468\) 0 0
\(469\) −1.41139e28 −1.26571
\(470\) 0 0
\(471\) 3.96779e27 1.27627e28i 0.340274 1.09452i
\(472\) 0 0
\(473\) 1.84620e27i 0.151439i
\(474\) 0 0
\(475\) 7.43985e27i 0.583823i
\(476\) 0 0
\(477\) 8.33494e27 1.21094e28i 0.625834 0.909241i
\(478\) 0 0
\(479\) 2.21401e28 1.59095 0.795477 0.605984i \(-0.207220\pi\)
0.795477 + 0.605984i \(0.207220\pi\)
\(480\) 0 0
\(481\) −2.55077e27 −0.175448
\(482\) 0 0
\(483\) −2.63485e28 8.19147e27i −1.73505 0.539407i
\(484\) 0 0
\(485\) 3.77126e28i 2.37792i
\(486\) 0 0
\(487\) 1.55360e28i 0.938180i 0.883150 + 0.469090i \(0.155418\pi\)
−0.883150 + 0.469090i \(0.844582\pi\)
\(488\) 0 0
\(489\) 2.50770e28 + 7.79618e27i 1.45055 + 0.450961i
\(490\) 0 0
\(491\) −8.88611e27 −0.492443 −0.246222 0.969214i \(-0.579189\pi\)
−0.246222 + 0.969214i \(0.579189\pi\)
\(492\) 0 0
\(493\) −4.33946e28 −2.30432
\(494\) 0 0
\(495\) 1.71435e28 2.49068e28i 0.872458 1.26755i
\(496\) 0 0
\(497\) 1.84213e28i 0.898624i
\(498\) 0 0
\(499\) 8.00716e27i 0.374475i 0.982315 + 0.187237i \(0.0599534\pi\)
−0.982315 + 0.187237i \(0.940047\pi\)
\(500\) 0 0
\(501\) 9.94918e27 3.20023e28i 0.446160 1.43511i
\(502\) 0 0
\(503\) −4.50080e28 −1.93565 −0.967824 0.251629i \(-0.919034\pi\)
−0.967824 + 0.251629i \(0.919034\pi\)
\(504\) 0 0
\(505\) −4.11193e28 −1.69624
\(506\) 0 0
\(507\) 7.18864e27 2.31228e28i 0.284488 0.915077i
\(508\) 0 0
\(509\) 3.13277e28i 1.18957i −0.803883 0.594787i \(-0.797236\pi\)
0.803883 0.594787i \(-0.202764\pi\)
\(510\) 0 0
\(511\) 5.87461e28i 2.14072i
\(512\) 0 0
\(513\) 6.25216e27 + 7.94785e27i 0.218674 + 0.277982i
\(514\) 0 0
\(515\) −7.06012e28 −2.37048
\(516\) 0 0
\(517\) 2.14905e28 0.692781
\(518\) 0 0
\(519\) −2.48142e28 7.71445e27i −0.768143 0.238807i
\(520\) 0 0
\(521\) 4.52720e28i 1.34597i −0.739658 0.672983i \(-0.765013\pi\)
0.739658 0.672983i \(-0.234987\pi\)
\(522\) 0 0
\(523\) 2.14432e28i 0.612382i 0.951970 + 0.306191i \(0.0990547\pi\)
−0.951970 + 0.306191i \(0.900945\pi\)
\(524\) 0 0
\(525\) 1.05226e29 + 3.27136e28i 2.88702 + 0.897543i
\(526\) 0 0
\(527\) 4.47994e28 1.18103
\(528\) 0 0
\(529\) −6.26278e26 −0.0158665
\(530\) 0 0
\(531\) −1.29641e27 8.92324e26i −0.0315682 0.0217285i
\(532\) 0 0
\(533\) 1.97765e27i 0.0462928i
\(534\) 0 0
\(535\) 8.75895e28i 1.97123i
\(536\) 0 0
\(537\) 1.26288e27 4.06216e27i 0.0273295 0.0879076i
\(538\) 0 0
\(539\) 1.06928e29 2.22541
\(540\) 0 0
\(541\) 6.03725e28 1.20856 0.604279 0.796773i \(-0.293461\pi\)
0.604279 + 0.796773i \(0.293461\pi\)
\(542\) 0 0
\(543\) −6.05640e27 + 1.94809e28i −0.116632 + 0.375155i
\(544\) 0 0
\(545\) 4.82065e27i 0.0893186i
\(546\) 0 0
\(547\) 4.25089e28i 0.757901i −0.925417 0.378951i \(-0.876285\pi\)
0.925417 0.378951i \(-0.123715\pi\)
\(548\) 0 0
\(549\) 1.41507e27 + 9.73997e26i 0.0242810 + 0.0167128i
\(550\) 0 0
\(551\) −2.94023e28 −0.485612
\(552\) 0 0
\(553\) −2.20495e29 −3.50579
\(554\) 0 0
\(555\) 8.72351e28 + 2.71204e28i 1.33541 + 0.415165i
\(556\) 0 0
\(557\) 8.71218e28i 1.28424i −0.766603 0.642122i \(-0.778054\pi\)
0.766603 0.642122i \(-0.221946\pi\)
\(558\) 0 0
\(559\) 2.30537e27i 0.0327278i
\(560\) 0 0
\(561\) 1.10775e29 + 3.44389e28i 1.51472 + 0.470911i
\(562\) 0 0
\(563\) 1.63734e28 0.215676 0.107838 0.994168i \(-0.465607\pi\)
0.107838 + 0.994168i \(0.465607\pi\)
\(564\) 0 0
\(565\) −1.62274e29 −2.05940
\(566\) 0 0
\(567\) −1.39902e29 + 5.34804e28i −1.71081 + 0.653992i
\(568\) 0 0
\(569\) 1.60930e29i 1.89653i 0.317487 + 0.948263i \(0.397161\pi\)
−0.317487 + 0.948263i \(0.602839\pi\)
\(570\) 0 0
\(571\) 5.06590e28i 0.575411i 0.957719 + 0.287705i \(0.0928924\pi\)
−0.957719 + 0.287705i \(0.907108\pi\)
\(572\) 0 0
\(573\) −4.95397e28 + 1.59348e29i −0.542413 + 1.74471i
\(574\) 0 0
\(575\) −1.55133e29 −1.63754
\(576\) 0 0
\(577\) 2.67417e27 0.0272171 0.0136086 0.999907i \(-0.495668\pi\)
0.0136086 + 0.999907i \(0.495668\pi\)
\(578\) 0 0
\(579\) −1.56981e28 + 5.04943e28i −0.154072 + 0.495585i
\(580\) 0 0
\(581\) 2.17812e29i 2.06173i
\(582\) 0 0
\(583\) 1.14264e29i 1.04326i
\(584\) 0 0
\(585\) 2.14072e28 3.11014e28i 0.188549 0.273932i
\(586\) 0 0
\(587\) 1.08450e29 0.921575 0.460787 0.887511i \(-0.347567\pi\)
0.460787 + 0.887511i \(0.347567\pi\)
\(588\) 0 0
\(589\) 3.03541e28 0.248890
\(590\) 0 0
\(591\) 1.32653e29 + 4.12403e28i 1.04966 + 0.326327i
\(592\) 0 0
\(593\) 7.92155e28i 0.604973i 0.953154 + 0.302486i \(0.0978166\pi\)
−0.953154 + 0.302486i \(0.902183\pi\)
\(594\) 0 0
\(595\) 6.78885e29i 5.00458i
\(596\) 0 0
\(597\) 1.55269e29 + 4.82714e28i 1.10498 + 0.343526i
\(598\) 0 0
\(599\) −8.02774e28 −0.551584 −0.275792 0.961217i \(-0.588940\pi\)
−0.275792 + 0.961217i \(0.588940\pi\)
\(600\) 0 0
\(601\) 1.12347e29 0.745383 0.372692 0.927955i \(-0.378435\pi\)
0.372692 + 0.927955i \(0.378435\pi\)
\(602\) 0 0
\(603\) 6.11524e28 8.88450e28i 0.391816 0.569247i
\(604\) 0 0
\(605\) 2.80727e28i 0.173721i
\(606\) 0 0
\(607\) 7.77330e28i 0.464648i −0.972638 0.232324i \(-0.925367\pi\)
0.972638 0.232324i \(-0.0746330\pi\)
\(608\) 0 0
\(609\) 1.29284e29 4.15852e29i 0.746558 2.40136i
\(610\) 0 0
\(611\) 2.68354e28 0.149719
\(612\) 0 0
\(613\) −2.58302e29 −1.39249 −0.696247 0.717802i \(-0.745148\pi\)
−0.696247 + 0.717802i \(0.745148\pi\)
\(614\) 0 0
\(615\) 2.10269e28 6.76346e28i 0.109543 0.352355i
\(616\) 0 0
\(617\) 1.88675e29i 0.949995i 0.879987 + 0.474997i \(0.157551\pi\)
−0.879987 + 0.474997i \(0.842449\pi\)
\(618\) 0 0
\(619\) 7.62056e28i 0.370881i 0.982655 + 0.185441i \(0.0593713\pi\)
−0.982655 + 0.185441i \(0.940629\pi\)
\(620\) 0 0
\(621\) 1.65726e29 1.30368e29i 0.779702 0.613350i
\(622\) 0 0
\(623\) 6.07431e29 2.76294
\(624\) 0 0
\(625\) 1.68452e28 0.0740860
\(626\) 0 0
\(627\) 7.50565e28 + 2.33343e28i 0.319212 + 0.0992396i
\(628\) 0 0
\(629\) 3.50487e29i 1.44158i
\(630\) 0 0
\(631\) 3.16492e29i 1.25909i 0.776966 + 0.629543i \(0.216758\pi\)
−0.776966 + 0.629543i \(0.783242\pi\)
\(632\) 0 0
\(633\) −3.43172e29 1.06688e29i −1.32060 0.410561i
\(634\) 0 0
\(635\) −1.16241e29 −0.432747
\(636\) 0 0
\(637\) 1.33522e29 0.480939
\(638\) 0 0
\(639\) −1.15959e29 7.98151e28i −0.404153 0.278180i
\(640\) 0 0
\(641\) 3.12846e29i 1.05517i −0.849504 0.527583i \(-0.823098\pi\)
0.849504 0.527583i \(-0.176902\pi\)
\(642\) 0 0
\(643\) 2.05138e29i 0.669624i 0.942285 + 0.334812i \(0.108673\pi\)
−0.942285 + 0.334812i \(0.891327\pi\)
\(644\) 0 0
\(645\) −2.45113e28 + 7.88426e28i −0.0774444 + 0.249106i
\(646\) 0 0
\(647\) −3.82322e29 −1.16932 −0.584661 0.811278i \(-0.698772\pi\)
−0.584661 + 0.811278i \(0.698772\pi\)
\(648\) 0 0
\(649\) −1.22329e28 −0.0362211
\(650\) 0 0
\(651\) −1.33469e29 + 4.29315e29i −0.382632 + 1.23077i
\(652\) 0 0
\(653\) 1.78353e29i 0.495099i −0.968875 0.247549i \(-0.920375\pi\)
0.968875 0.247549i \(-0.0796252\pi\)
\(654\) 0 0
\(655\) 8.40468e29i 2.25937i
\(656\) 0 0
\(657\) 3.69797e29 + 2.54533e29i 0.962779 + 0.662685i
\(658\) 0 0
\(659\) −6.40298e28 −0.161467 −0.0807337 0.996736i \(-0.525726\pi\)
−0.0807337 + 0.996736i \(0.525726\pi\)
\(660\) 0 0
\(661\) 4.51347e29 1.10254 0.551271 0.834327i \(-0.314143\pi\)
0.551271 + 0.834327i \(0.314143\pi\)
\(662\) 0 0
\(663\) 1.38326e29 + 4.30042e28i 0.327350 + 0.101770i
\(664\) 0 0
\(665\) 4.59982e29i 1.05466i
\(666\) 0 0
\(667\) 6.13086e29i 1.36207i
\(668\) 0 0
\(669\) −3.82343e29 1.18866e29i −0.823150 0.255909i
\(670\) 0 0
\(671\) 1.33526e28 0.0278599
\(672\) 0 0
\(673\) −3.16425e29 −0.639901 −0.319951 0.947434i \(-0.603666\pi\)
−0.319951 + 0.947434i \(0.603666\pi\)
\(674\) 0 0
\(675\) −6.61845e29 + 5.20639e29i −1.29738 + 1.02058i
\(676\) 0 0
\(677\) 5.55817e29i 1.05621i −0.849178 0.528106i \(-0.822902\pi\)
0.849178 0.528106i \(-0.177098\pi\)
\(678\) 0 0
\(679\) 1.45201e30i 2.67508i
\(680\) 0 0
\(681\) −8.09805e28 + 2.60480e29i −0.144656 + 0.465297i
\(682\) 0 0
\(683\) −6.76496e26 −0.00117179 −0.000585893 1.00000i \(-0.500186\pi\)
−0.000585893 1.00000i \(0.500186\pi\)
\(684\) 0 0
\(685\) −7.68464e29 −1.29084
\(686\) 0 0
\(687\) 2.48420e29 7.99063e29i 0.404707 1.30177i
\(688\) 0 0
\(689\) 1.42683e29i 0.225460i
\(690\) 0 0
\(691\) 1.14106e30i 1.74900i −0.485023 0.874501i \(-0.661189\pi\)
0.485023 0.874501i \(-0.338811\pi\)
\(692\) 0 0
\(693\) −6.60059e29 + 9.58963e29i −0.981485 + 1.42595i
\(694\) 0 0
\(695\) −1.20880e30 −1.74388
\(696\) 0 0
\(697\) 2.71737e29 0.380369
\(698\) 0 0
\(699\) 6.21414e29 + 1.93191e29i 0.844055 + 0.262408i
\(700\) 0 0
\(701\) 6.98936e29i 0.921294i 0.887584 + 0.460647i \(0.152383\pi\)
−0.887584 + 0.460647i \(0.847617\pi\)
\(702\) 0 0
\(703\) 2.37474e29i 0.303798i
\(704\) 0 0
\(705\) −9.17759e29 2.85321e29i −1.13957 0.354282i
\(706\) 0 0
\(707\) 1.58318e30 1.90821
\(708\) 0 0
\(709\) 8.53396e29 0.998539 0.499270 0.866447i \(-0.333602\pi\)
0.499270 + 0.866447i \(0.333602\pi\)
\(710\) 0 0
\(711\) 9.55355e29 1.38798e30i 1.08526 1.57672i
\(712\) 0 0
\(713\) 6.32934e29i 0.698101i
\(714\) 0 0
\(715\) 2.93473e29i 0.314308i
\(716\) 0 0
\(717\) −2.32069e29 + 7.46468e29i −0.241360 + 0.776354i
\(718\) 0 0
\(719\) −4.51890e29 −0.456435 −0.228218 0.973610i \(-0.573290\pi\)
−0.228218 + 0.973610i \(0.573290\pi\)
\(720\) 0 0
\(721\) 2.71829e30 2.66671
\(722\) 0 0
\(723\) 7.61986e28 2.45099e29i 0.0726097 0.233555i
\(724\) 0 0
\(725\) 2.44843e30i 2.26641i
\(726\) 0 0
\(727\) 5.79335e29i 0.520977i −0.965477 0.260488i \(-0.916116\pi\)
0.965477 0.260488i \(-0.0838835\pi\)
\(728\) 0 0
\(729\) 2.69512e29 1.11238e30i 0.235472 0.971881i
\(730\) 0 0
\(731\) −3.16768e29 −0.268911
\(732\) 0 0
\(733\) 1.47112e29 0.121354 0.0606772 0.998157i \(-0.480674\pi\)
0.0606772 + 0.998157i \(0.480674\pi\)
\(734\) 0 0
\(735\) −4.56640e30 1.41965e30i −3.66064 1.13805i
\(736\) 0 0
\(737\) 8.38343e29i 0.653150i
\(738\) 0 0
\(739\) 1.59263e30i 1.20600i 0.797741 + 0.603000i \(0.206028\pi\)
−0.797741 + 0.603000i \(0.793972\pi\)
\(740\) 0 0
\(741\) 9.37238e28 + 2.91377e28i 0.0689857 + 0.0214469i
\(742\) 0 0
\(743\) −5.91119e29 −0.422953 −0.211477 0.977383i \(-0.567827\pi\)
−0.211477 + 0.977383i \(0.567827\pi\)
\(744\) 0 0
\(745\) 1.43408e30 0.997546
\(746\) 0 0
\(747\) 1.37109e30 + 9.43727e29i 0.927256 + 0.638234i
\(748\) 0 0
\(749\) 3.37237e30i 2.21757i
\(750\) 0 0
\(751\) 3.06438e29i 0.195940i −0.995189 0.0979701i \(-0.968765\pi\)
0.995189 0.0979701i \(-0.0312349\pi\)
\(752\) 0 0
\(753\) −4.18707e29 + 1.34680e30i −0.260353 + 0.837447i
\(754\) 0 0
\(755\) −1.01252e30 −0.612297
\(756\) 0 0
\(757\) 1.30557e30 0.767881 0.383940 0.923358i \(-0.374567\pi\)
0.383940 + 0.923358i \(0.374567\pi\)
\(758\) 0 0
\(759\) 4.86558e29 1.56505e30i 0.278353 0.895346i
\(760\) 0 0
\(761\) 5.41042e29i 0.301087i −0.988603 0.150544i \(-0.951898\pi\)
0.988603 0.150544i \(-0.0481024\pi\)
\(762\) 0 0
\(763\) 1.85605e29i 0.100480i
\(764\) 0 0
\(765\) −4.27347e30 2.94145e30i −2.25079 1.54923i
\(766\) 0 0
\(767\) −1.52754e28 −0.00782783
\(768\) 0 0
\(769\) −9.95470e29 −0.496366 −0.248183 0.968713i \(-0.579833\pi\)
−0.248183 + 0.968713i \(0.579833\pi\)
\(770\) 0 0
\(771\) 1.49710e30 + 4.65432e29i 0.726406 + 0.225832i
\(772\) 0 0
\(773\) 2.95905e30i 1.39723i 0.715498 + 0.698615i \(0.246200\pi\)
−0.715498 + 0.698615i \(0.753800\pi\)
\(774\) 0 0
\(775\) 2.52769e30i 1.16160i
\(776\) 0 0
\(777\) −3.35873e30 1.04419e30i −1.50229 0.467046i
\(778\) 0 0
\(779\) 1.84117e29 0.0801588
\(780\) 0 0
\(781\) −1.09419e30 −0.463722
\(782\) 0 0
\(783\) 2.05757e30 + 2.61561e30i 0.848897 + 1.07913i
\(784\) 0 0
\(785\) 4.64589e30i 1.86611i
\(786\) 0 0
\(787\) 1.17587e29i 0.0459859i 0.999736 + 0.0229930i \(0.00731953\pi\)
−0.999736 + 0.0229930i \(0.992680\pi\)
\(788\) 0 0
\(789\) 1.30084e30 4.18426e30i 0.495354 1.59335i
\(790\) 0 0
\(791\) 6.24787e30 2.31675
\(792\) 0 0
\(793\) 1.66735e28 0.00602087
\(794\) 0 0
\(795\) −1.51704e30 + 4.87969e30i −0.533511 + 1.71608i
\(796\) 0 0
\(797\) 2.73219e30i 0.935835i −0.883772 0.467917i \(-0.845004\pi\)
0.883772 0.467917i \(-0.154996\pi\)
\(798\) 0 0
\(799\) 3.68730e30i 1.23018i
\(800\) 0 0
\(801\) −2.63185e30 + 3.82368e30i −0.855304 + 1.24262i
\(802\) 0 0
\(803\) 3.48941e30 1.10469
\(804\) 0 0
\(805\) 9.59140e30 2.95818
\(806\) 0 0
\(807\) 2.41075e30 + 7.49476e29i 0.724402 + 0.225209i
\(808\) 0 0
\(809\) 1.64561e30i 0.481801i 0.970550 + 0.240901i \(0.0774428\pi\)
−0.970550 + 0.240901i \(0.922557\pi\)
\(810\) 0 0
\(811\) 3.93458e30i 1.12248i 0.827652 + 0.561241i \(0.189676\pi\)
−0.827652 + 0.561241i \(0.810324\pi\)
\(812\) 0 0
\(813\) 4.00562e29 + 1.24530e29i 0.111357 + 0.0346198i
\(814\) 0 0
\(815\) −9.12855e30 −2.47313
\(816\) 0 0
\(817\) −2.14628e29 −0.0566702
\(818\) 0 0
\(819\) −8.24222e29 + 1.19747e30i −0.212111 + 0.308164i
\(820\) 0 0
\(821\) 1.85981e28i 0.00466515i −0.999997 0.00233257i \(-0.999258\pi\)
0.999997 0.00233257i \(-0.000742482\pi\)
\(822\) 0 0
\(823\) 2.35040e29i 0.0574702i −0.999587 0.0287351i \(-0.990852\pi\)
0.999587 0.0287351i \(-0.00914792\pi\)
\(824\) 0 0
\(825\) −1.94313e30 + 6.25022e30i −0.463164 + 1.48980i
\(826\) 0 0
\(827\) −2.97803e30 −0.692025 −0.346012 0.938230i \(-0.612464\pi\)
−0.346012 + 0.938230i \(0.612464\pi\)
\(828\) 0 0
\(829\) 1.27916e30 0.289803 0.144902 0.989446i \(-0.453713\pi\)
0.144902 + 0.989446i \(0.453713\pi\)
\(830\) 0 0
\(831\) −1.06199e30 + 3.41596e30i −0.234589 + 0.754574i
\(832\) 0 0
\(833\) 1.83466e31i 3.95168i
\(834\) 0 0
\(835\) 1.16495e31i 2.44680i
\(836\) 0 0
\(837\) −2.12418e30 2.70029e30i −0.435084 0.553086i
\(838\) 0 0
\(839\) 3.30738e30 0.660668 0.330334 0.943864i \(-0.392839\pi\)
0.330334 + 0.943864i \(0.392839\pi\)
\(840\) 0 0
\(841\) −4.54335e30 −0.885154
\(842\) 0 0
\(843\) −2.67448e30 8.31468e29i −0.508218 0.158000i
\(844\) 0 0
\(845\) 8.41718e30i 1.56017i
\(846\) 0 0
\(847\) 1.08086e30i 0.195430i
\(848\) 0 0
\(849\) −4.77831e30 1.48553e30i −0.842837 0.262029i
\(850\) 0 0
\(851\) 4.95173e30 0.852112
\(852\) 0 0
\(853\) 9.54603e30 1.60272 0.801361 0.598182i \(-0.204110\pi\)
0.801361 + 0.598182i \(0.204110\pi\)
\(854\) 0 0
\(855\) −2.89551e30 1.99299e30i −0.474331 0.326484i
\(856\) 0 0
\(857\) 6.28873e30i 1.00523i 0.864511 + 0.502614i \(0.167628\pi\)
−0.864511 + 0.502614i \(0.832372\pi\)
\(858\) 0 0
\(859\) 6.97554e30i 1.08805i 0.839069 + 0.544026i \(0.183101\pi\)
−0.839069 + 0.544026i \(0.816899\pi\)
\(860\) 0 0
\(861\) −8.09578e29 + 2.60407e30i −0.123233 + 0.396387i
\(862\) 0 0
\(863\) 1.06325e31 1.57951 0.789755 0.613422i \(-0.210207\pi\)
0.789755 + 0.613422i \(0.210207\pi\)
\(864\) 0 0
\(865\) 9.03286e30 1.30965
\(866\) 0 0
\(867\) 3.81113e30 1.22588e31i 0.539328 1.73479i
\(868\) 0 0
\(869\) 1.30970e31i 1.80911i
\(870\) 0 0
\(871\) 1.04685e30i 0.141154i
\(872\) 0 0
\(873\) 9.14016e30 + 6.29122e30i 1.20311 + 0.828104i
\(874\) 0 0
\(875\) −1.50993e31 −1.94031
\(876\) 0 0
\(877\) −9.26733e30 −1.16268 −0.581338 0.813662i \(-0.697471\pi\)
−0.581338 + 0.813662i \(0.697471\pi\)
\(878\) 0 0
\(879\) 7.03815e30 + 2.18809e30i 0.862135 + 0.268029i
\(880\) 0 0
\(881\) 5.29699e29i 0.0633552i −0.999498 0.0316776i \(-0.989915\pi\)
0.999498 0.0316776i \(-0.0100850\pi\)
\(882\) 0 0
\(883\) 1.17325e31i 1.37026i −0.728421 0.685130i \(-0.759746\pi\)
0.728421 0.685130i \(-0.240254\pi\)
\(884\) 0 0
\(885\) 5.22412e29 + 1.62412e29i 0.0595811 + 0.0185231i
\(886\) 0 0
\(887\) 3.21463e30 0.358041 0.179020 0.983845i \(-0.442707\pi\)
0.179020 + 0.983845i \(0.442707\pi\)
\(888\) 0 0
\(889\) 4.47550e30 0.486826
\(890\) 0 0
\(891\) −3.17664e30 8.30992e30i −0.337483 0.882838i
\(892\) 0 0
\(893\) 2.49835e30i 0.259247i
\(894\) 0 0
\(895\) 1.47871e30i 0.149879i
\(896\) 0 0
\(897\) 6.07570e29 1.95430e30i 0.0601556 0.193495i
\(898\) 0 0
\(899\) 9.98945e30 0.966195
\(900\) 0 0
\(901\) −1.96053e31 −1.85252
\(902\) 0 0
\(903\) 9.43736e29 3.03560e30i 0.0871222 0.280236i
\(904\) 0 0
\(905\) 7.09144e30i 0.639624i
\(906\) 0 0
\(907\) 1.03172e31i 0.909251i −0.890683 0.454626i \(-0.849773\pi\)
0.890683 0.454626i \(-0.150227\pi\)
\(908\) 0 0
\(909\) −6.85953e30 + 9.96584e30i −0.590709 + 0.858209i
\(910\) 0 0
\(911\) −1.92471e31 −1.61966 −0.809829 0.586666i \(-0.800440\pi\)
−0.809829 + 0.586666i \(0.800440\pi\)
\(912\) 0 0
\(913\) 1.29376e31 1.06393
\(914\) 0 0
\(915\) −5.70227e29 1.77277e29i −0.0458275 0.0142473i
\(916\) 0 0
\(917\) 3.23597e31i 2.54172i
\(918\) 0 0
\(919\) 1.58189e31i 1.21441i −0.794546 0.607204i \(-0.792291\pi\)
0.794546 0.607204i \(-0.207709\pi\)
\(920\) 0 0
\(921\) −1.70699e31 5.30685e30i −1.28087 0.398208i
\(922\) 0 0
\(923\) −1.36633e30 −0.100216
\(924\) 0 0
\(925\) −1.97753e31 −1.41787
\(926\) 0 0
\(927\) −1.17777e31 + 1.71112e31i −0.825512 + 1.19934i
\(928\) 0 0
\(929\) 1.61527e31i 1.10683i 0.832906 + 0.553414i \(0.186675\pi\)
−0.832906 + 0.553414i \(0.813325\pi\)
\(930\) 0 0
\(931\) 1.24308e31i 0.832776i
\(932\) 0 0
\(933\) 4.39595e30 1.41399e31i 0.287936 0.926168i
\(934\) 0 0
\(935\) −4.03245e31 −2.58254
\(936\) 0 0
\(937\) −2.33084e31 −1.45964 −0.729822 0.683637i \(-0.760397\pi\)
−0.729822 + 0.683637i \(0.760397\pi\)
\(938\) 0 0
\(939\) 1.07065e30 3.44384e30i 0.0655632 0.210889i
\(940\) 0 0
\(941\) 9.65228e30i 0.578015i −0.957327 0.289007i \(-0.906675\pi\)
0.957327 0.289007i \(-0.0933252\pi\)
\(942\) 0 0
\(943\) 3.83915e30i 0.224834i
\(944\) 0 0
\(945\) 4.09198e31 3.21895e31i 2.34369 1.84366i
\(946\) 0 0
\(947\) 1.47142e31 0.824258 0.412129 0.911125i \(-0.364785\pi\)
0.412129 + 0.911125i \(0.364785\pi\)
\(948\) 0 0
\(949\) 4.35726e30 0.238736
\(950\) 0 0
\(951\) 1.23067e30 + 3.82602e29i 0.0659548 + 0.0205046i
\(952\) 0 0
\(953\) 2.06433e31i 1.08219i 0.840961 + 0.541096i \(0.181990\pi\)
−0.840961 + 0.541096i \(0.818010\pi\)
\(954\) 0 0
\(955\) 5.80061e31i 2.97466i
\(956\) 0 0
\(957\) 2.47009e31 + 7.67924e30i 1.23919 + 0.385250i
\(958\) 0 0
\(959\) 2.95874e31 1.45215
\(960\) 0 0
\(961\) 1.05127e31 0.504798
\(962\) 0 0
\(963\) −2.12285e31 1.46117e31i −0.997342 0.686475i
\(964\) 0 0
\(965\) 1.83810e31i 0.844951i
\(966\) 0 0
\(967\) 1.75379e31i 0.788860i 0.918926 + 0.394430i \(0.129058\pi\)
−0.918926 + 0.394430i \(0.870942\pi\)
\(968\) 0 0
\(969\) 4.00365e30 1.28781e31i 0.176221 0.566828i
\(970\) 0 0
\(971\) 1.39045e31 0.598899 0.299449 0.954112i \(-0.403197\pi\)
0.299449 + 0.954112i \(0.403197\pi\)
\(972\) 0 0
\(973\) 4.65414e31 1.96180
\(974\) 0 0
\(975\) −2.42640e30 + 7.80471e30i −0.100095 + 0.321965i
\(976\) 0 0
\(977\) 2.26862e31i 0.915942i −0.888967 0.457971i \(-0.848576\pi\)
0.888967 0.457971i \(-0.151424\pi\)
\(978\) 0 0
\(979\) 3.60803e31i 1.42578i
\(980\) 0 0
\(981\) 1.16835e30 + 8.04181e29i 0.0451907 + 0.0311049i
\(982\) 0 0
\(983\) 4.70993e31 1.78321 0.891606 0.452813i \(-0.149579\pi\)
0.891606 + 0.452813i \(0.149579\pi\)
\(984\) 0 0
\(985\) −4.82883e31 −1.78962
\(986\) 0 0
\(987\) 3.53356e31 + 1.09855e31i 1.28198 + 0.398555i
\(988\) 0 0
\(989\) 4.47535e30i 0.158952i
\(990\) 0 0
\(991\) 1.02957e31i 0.357999i −0.983849 0.179000i \(-0.942714\pi\)
0.983849 0.179000i \(-0.0572861\pi\)
\(992\) 0 0
\(993\) 3.66636e31 + 1.13983e31i 1.24815 + 0.388037i
\(994\) 0 0
\(995\) −5.65210e31 −1.88394
\(996\) 0 0
\(997\) −4.94089e31 −1.61252 −0.806262 0.591558i \(-0.798513\pi\)
−0.806262 + 0.591558i \(0.798513\pi\)
\(998\) 0 0
\(999\) 2.11256e31 1.66184e31i 0.675105 0.531070i
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 48.22.c.b.47.6 yes 12
3.2 odd 2 inner 48.22.c.b.47.8 yes 12
4.3 odd 2 inner 48.22.c.b.47.7 yes 12
12.11 even 2 inner 48.22.c.b.47.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.22.c.b.47.5 12 12.11 even 2 inner
48.22.c.b.47.6 yes 12 1.1 even 1 trivial
48.22.c.b.47.7 yes 12 4.3 odd 2 inner
48.22.c.b.47.8 yes 12 3.2 odd 2 inner