Properties

Label 64.24.a.h.1.1
Level $64$
Weight $24$
Character 64.1
Self dual yes
Analytic conductor $214.531$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,24,Mod(1,64)]
 
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([0, 0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(214.530583901\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 166408x - 10560732 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(436.575\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-574251. q^{3} -8.41392e7 q^{5} +5.95738e9 q^{7} +2.35621e11 q^{9} +O(q^{10})\) \(q-574251. q^{3} -8.41392e7 q^{5} +5.95738e9 q^{7} +2.35621e11 q^{9} +1.24326e12 q^{11} -7.21479e12 q^{13} +4.83170e13 q^{15} -6.50936e13 q^{17} -5.56139e14 q^{19} -3.42103e15 q^{21} +3.08126e15 q^{23} -4.84153e15 q^{25} -8.12435e16 q^{27} +4.69935e16 q^{29} -2.40563e17 q^{31} -7.13941e17 q^{33} -5.01249e17 q^{35} +5.28193e17 q^{37} +4.14310e18 q^{39} +1.59837e18 q^{41} +7.35994e18 q^{43} -1.98249e19 q^{45} +1.63997e19 q^{47} +8.12160e18 q^{49} +3.73800e19 q^{51} +2.64089e19 q^{53} -1.04607e20 q^{55} +3.19363e20 q^{57} -3.24947e20 q^{59} +3.22520e20 q^{61} +1.40368e21 q^{63} +6.07047e20 q^{65} -1.09896e21 q^{67} -1.76941e21 q^{69} +2.30177e20 q^{71} +8.33706e20 q^{73} +2.78025e21 q^{75} +7.40655e21 q^{77} -2.59992e21 q^{79} +2.44721e22 q^{81} -1.05807e22 q^{83} +5.47692e21 q^{85} -2.69861e22 q^{87} +4.07434e22 q^{89} -4.29812e22 q^{91} +1.38143e23 q^{93} +4.67930e22 q^{95} -9.40029e22 q^{97} +2.92937e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 213948 q^{3} - 95628618 q^{5} + 8647912920 q^{7} + 174509823951 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 213948 q^{3} - 95628618 q^{5} + 8647912920 q^{7} + 174509823951 q^{9} + 35420906796 q^{11} - 3164858452338 q^{13} + 19825526344392 q^{15} - 30233487828906 q^{17} - 382754784400236 q^{19} - 27788918984928 q^{21} + 37\!\cdots\!20 q^{23}+ \cdots + 19\!\cdots\!00 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −574251. −1.87157 −0.935787 0.352565i \(-0.885309\pi\)
−0.935787 + 0.352565i \(0.885309\pi\)
\(4\) 0 0
\(5\) −8.41392e7 −0.770625 −0.385312 0.922786i \(-0.625906\pi\)
−0.385312 + 0.922786i \(0.625906\pi\)
\(6\) 0 0
\(7\) 5.95738e9 1.13875 0.569374 0.822079i \(-0.307186\pi\)
0.569374 + 0.822079i \(0.307186\pi\)
\(8\) 0 0
\(9\) 2.35621e11 2.50279
\(10\) 0 0
\(11\) 1.24326e12 1.31385 0.656924 0.753957i \(-0.271857\pi\)
0.656924 + 0.753957i \(0.271857\pi\)
\(12\) 0 0
\(13\) −7.21479e12 −1.11654 −0.558271 0.829658i \(-0.688535\pi\)
−0.558271 + 0.829658i \(0.688535\pi\)
\(14\) 0 0
\(15\) 4.83170e13 1.44228
\(16\) 0 0
\(17\) −6.50936e13 −0.460655 −0.230327 0.973113i \(-0.573980\pi\)
−0.230327 + 0.973113i \(0.573980\pi\)
\(18\) 0 0
\(19\) −5.56139e14 −1.09526 −0.547629 0.836721i \(-0.684470\pi\)
−0.547629 + 0.836721i \(0.684470\pi\)
\(20\) 0 0
\(21\) −3.42103e15 −2.13125
\(22\) 0 0
\(23\) 3.08126e15 0.674307 0.337154 0.941450i \(-0.390536\pi\)
0.337154 + 0.941450i \(0.390536\pi\)
\(24\) 0 0
\(25\) −4.84153e15 −0.406137
\(26\) 0 0
\(27\) −8.12435e16 −2.81258
\(28\) 0 0
\(29\) 4.69935e16 0.715255 0.357628 0.933864i \(-0.383586\pi\)
0.357628 + 0.933864i \(0.383586\pi\)
\(30\) 0 0
\(31\) −2.40563e17 −1.70047 −0.850234 0.526405i \(-0.823540\pi\)
−0.850234 + 0.526405i \(0.823540\pi\)
\(32\) 0 0
\(33\) −7.13941e17 −2.45896
\(34\) 0 0
\(35\) −5.01249e17 −0.877548
\(36\) 0 0
\(37\) 5.28193e17 0.488059 0.244030 0.969768i \(-0.421531\pi\)
0.244030 + 0.969768i \(0.421531\pi\)
\(38\) 0 0
\(39\) 4.14310e18 2.08969
\(40\) 0 0
\(41\) 1.59837e18 0.453589 0.226795 0.973943i \(-0.427175\pi\)
0.226795 + 0.973943i \(0.427175\pi\)
\(42\) 0 0
\(43\) 7.35994e18 1.20778 0.603889 0.797068i \(-0.293617\pi\)
0.603889 + 0.797068i \(0.293617\pi\)
\(44\) 0 0
\(45\) −1.98249e19 −1.92871
\(46\) 0 0
\(47\) 1.63997e19 0.967631 0.483815 0.875170i \(-0.339251\pi\)
0.483815 + 0.875170i \(0.339251\pi\)
\(48\) 0 0
\(49\) 8.12160e18 0.296747
\(50\) 0 0
\(51\) 3.73800e19 0.862149
\(52\) 0 0
\(53\) 2.64089e19 0.391361 0.195681 0.980668i \(-0.437308\pi\)
0.195681 + 0.980668i \(0.437308\pi\)
\(54\) 0 0
\(55\) −1.04607e20 −1.01248
\(56\) 0 0
\(57\) 3.19363e20 2.04986
\(58\) 0 0
\(59\) −3.24947e20 −1.40286 −0.701429 0.712739i \(-0.747454\pi\)
−0.701429 + 0.712739i \(0.747454\pi\)
\(60\) 0 0
\(61\) 3.22520e20 0.948995 0.474498 0.880257i \(-0.342630\pi\)
0.474498 + 0.880257i \(0.342630\pi\)
\(62\) 0 0
\(63\) 1.40368e21 2.85005
\(64\) 0 0
\(65\) 6.07047e20 0.860436
\(66\) 0 0
\(67\) −1.09896e21 −1.09932 −0.549658 0.835390i \(-0.685242\pi\)
−0.549658 + 0.835390i \(0.685242\pi\)
\(68\) 0 0
\(69\) −1.76941e21 −1.26202
\(70\) 0 0
\(71\) 2.30177e20 0.118193 0.0590964 0.998252i \(-0.481178\pi\)
0.0590964 + 0.998252i \(0.481178\pi\)
\(72\) 0 0
\(73\) 8.33706e20 0.311028 0.155514 0.987834i \(-0.450297\pi\)
0.155514 + 0.987834i \(0.450297\pi\)
\(74\) 0 0
\(75\) 2.78025e21 0.760116
\(76\) 0 0
\(77\) 7.40655e21 1.49614
\(78\) 0 0
\(79\) −2.59992e21 −0.391064 −0.195532 0.980697i \(-0.562643\pi\)
−0.195532 + 0.980697i \(0.562643\pi\)
\(80\) 0 0
\(81\) 2.44721e22 2.76117
\(82\) 0 0
\(83\) −1.05807e22 −0.901817 −0.450908 0.892570i \(-0.648900\pi\)
−0.450908 + 0.892570i \(0.648900\pi\)
\(84\) 0 0
\(85\) 5.47692e21 0.354992
\(86\) 0 0
\(87\) −2.69861e22 −1.33865
\(88\) 0 0
\(89\) 4.07434e22 1.55622 0.778112 0.628126i \(-0.216178\pi\)
0.778112 + 0.628126i \(0.216178\pi\)
\(90\) 0 0
\(91\) −4.29812e22 −1.27146
\(92\) 0 0
\(93\) 1.38143e23 3.18255
\(94\) 0 0
\(95\) 4.67930e22 0.844034
\(96\) 0 0
\(97\) −9.40029e22 −1.33434 −0.667169 0.744906i \(-0.732494\pi\)
−0.667169 + 0.744906i \(0.732494\pi\)
\(98\) 0 0
\(99\) 2.92937e23 3.28828
\(100\) 0 0
\(101\) 1.31018e23 1.16851 0.584257 0.811568i \(-0.301386\pi\)
0.584257 + 0.811568i \(0.301386\pi\)
\(102\) 0 0
\(103\) 5.91184e22 0.420819 0.210409 0.977613i \(-0.432520\pi\)
0.210409 + 0.977613i \(0.432520\pi\)
\(104\) 0 0
\(105\) 2.87842e23 1.64240
\(106\) 0 0
\(107\) −7.78257e22 −0.357445 −0.178723 0.983899i \(-0.557196\pi\)
−0.178723 + 0.983899i \(0.557196\pi\)
\(108\) 0 0
\(109\) 6.81626e22 0.253012 0.126506 0.991966i \(-0.459624\pi\)
0.126506 + 0.991966i \(0.459624\pi\)
\(110\) 0 0
\(111\) −3.03315e23 −0.913439
\(112\) 0 0
\(113\) −4.27400e23 −1.04817 −0.524086 0.851665i \(-0.675593\pi\)
−0.524086 + 0.851665i \(0.675593\pi\)
\(114\) 0 0
\(115\) −2.59254e23 −0.519638
\(116\) 0 0
\(117\) −1.69995e24 −2.79447
\(118\) 0 0
\(119\) −3.87787e23 −0.524570
\(120\) 0 0
\(121\) 6.50256e23 0.726194
\(122\) 0 0
\(123\) −9.17865e23 −0.848926
\(124\) 0 0
\(125\) 1.41038e24 1.08360
\(126\) 0 0
\(127\) 6.80149e23 0.435373 0.217686 0.976019i \(-0.430149\pi\)
0.217686 + 0.976019i \(0.430149\pi\)
\(128\) 0 0
\(129\) −4.22645e24 −2.26045
\(130\) 0 0
\(131\) 6.38677e22 0.0286195 0.0143097 0.999898i \(-0.495445\pi\)
0.0143097 + 0.999898i \(0.495445\pi\)
\(132\) 0 0
\(133\) −3.31313e24 −1.24722
\(134\) 0 0
\(135\) 6.83576e24 2.16745
\(136\) 0 0
\(137\) 3.16090e24 0.846300 0.423150 0.906060i \(-0.360924\pi\)
0.423150 + 0.906060i \(0.360924\pi\)
\(138\) 0 0
\(139\) −1.51803e24 −0.344040 −0.172020 0.985093i \(-0.555029\pi\)
−0.172020 + 0.985093i \(0.555029\pi\)
\(140\) 0 0
\(141\) −9.41752e24 −1.81099
\(142\) 0 0
\(143\) −8.96984e24 −1.46697
\(144\) 0 0
\(145\) −3.95400e24 −0.551194
\(146\) 0 0
\(147\) −4.66384e24 −0.555385
\(148\) 0 0
\(149\) 5.94136e23 0.0605680 0.0302840 0.999541i \(-0.490359\pi\)
0.0302840 + 0.999541i \(0.490359\pi\)
\(150\) 0 0
\(151\) −1.03677e25 −0.906664 −0.453332 0.891342i \(-0.649765\pi\)
−0.453332 + 0.891342i \(0.649765\pi\)
\(152\) 0 0
\(153\) −1.53374e25 −1.15292
\(154\) 0 0
\(155\) 2.02407e25 1.31042
\(156\) 0 0
\(157\) 1.27647e25 0.713125 0.356563 0.934271i \(-0.383949\pi\)
0.356563 + 0.934271i \(0.383949\pi\)
\(158\) 0 0
\(159\) −1.51653e25 −0.732461
\(160\) 0 0
\(161\) 1.83562e25 0.767866
\(162\) 0 0
\(163\) 4.28936e25 1.55681 0.778404 0.627764i \(-0.216030\pi\)
0.778404 + 0.627764i \(0.216030\pi\)
\(164\) 0 0
\(165\) 6.00704e25 1.89494
\(166\) 0 0
\(167\) 6.74074e25 1.85126 0.925631 0.378427i \(-0.123535\pi\)
0.925631 + 0.378427i \(0.123535\pi\)
\(168\) 0 0
\(169\) 1.02993e25 0.246668
\(170\) 0 0
\(171\) −1.31038e26 −2.74120
\(172\) 0 0
\(173\) 5.33848e25 0.976984 0.488492 0.872569i \(-0.337547\pi\)
0.488492 + 0.872569i \(0.337547\pi\)
\(174\) 0 0
\(175\) −2.88428e25 −0.462488
\(176\) 0 0
\(177\) 1.86601e26 2.62555
\(178\) 0 0
\(179\) 5.76759e25 0.713156 0.356578 0.934266i \(-0.383943\pi\)
0.356578 + 0.934266i \(0.383943\pi\)
\(180\) 0 0
\(181\) 1.31592e26 1.43194 0.715971 0.698130i \(-0.245984\pi\)
0.715971 + 0.698130i \(0.245984\pi\)
\(182\) 0 0
\(183\) −1.85208e26 −1.77612
\(184\) 0 0
\(185\) −4.44417e25 −0.376111
\(186\) 0 0
\(187\) −8.09280e25 −0.605230
\(188\) 0 0
\(189\) −4.83999e26 −3.20283
\(190\) 0 0
\(191\) −4.07437e25 −0.238878 −0.119439 0.992842i \(-0.538110\pi\)
−0.119439 + 0.992842i \(0.538110\pi\)
\(192\) 0 0
\(193\) −3.00984e26 −1.56543 −0.782717 0.622378i \(-0.786167\pi\)
−0.782717 + 0.622378i \(0.786167\pi\)
\(194\) 0 0
\(195\) −3.48597e26 −1.61037
\(196\) 0 0
\(197\) 4.17423e25 0.171481 0.0857403 0.996318i \(-0.472674\pi\)
0.0857403 + 0.996318i \(0.472674\pi\)
\(198\) 0 0
\(199\) 3.13013e25 0.114486 0.0572430 0.998360i \(-0.481769\pi\)
0.0572430 + 0.998360i \(0.481769\pi\)
\(200\) 0 0
\(201\) 6.31080e26 2.05745
\(202\) 0 0
\(203\) 2.79958e26 0.814496
\(204\) 0 0
\(205\) −1.34485e26 −0.349547
\(206\) 0 0
\(207\) 7.26008e26 1.68765
\(208\) 0 0
\(209\) −6.91423e26 −1.43900
\(210\) 0 0
\(211\) −2.37199e26 −0.442451 −0.221226 0.975223i \(-0.571006\pi\)
−0.221226 + 0.975223i \(0.571006\pi\)
\(212\) 0 0
\(213\) −1.32179e26 −0.221207
\(214\) 0 0
\(215\) −6.19259e26 −0.930744
\(216\) 0 0
\(217\) −1.43312e27 −1.93640
\(218\) 0 0
\(219\) −4.78756e26 −0.582113
\(220\) 0 0
\(221\) 4.69637e26 0.514341
\(222\) 0 0
\(223\) −1.49431e27 −1.47549 −0.737744 0.675080i \(-0.764109\pi\)
−0.737744 + 0.675080i \(0.764109\pi\)
\(224\) 0 0
\(225\) −1.14076e27 −1.01648
\(226\) 0 0
\(227\) 4.09710e26 0.329746 0.164873 0.986315i \(-0.447279\pi\)
0.164873 + 0.986315i \(0.447279\pi\)
\(228\) 0 0
\(229\) −1.57268e27 −1.14428 −0.572141 0.820155i \(-0.693887\pi\)
−0.572141 + 0.820155i \(0.693887\pi\)
\(230\) 0 0
\(231\) −4.25322e27 −2.80014
\(232\) 0 0
\(233\) 2.46809e27 1.47152 0.735762 0.677240i \(-0.236824\pi\)
0.735762 + 0.677240i \(0.236824\pi\)
\(234\) 0 0
\(235\) −1.37985e27 −0.745681
\(236\) 0 0
\(237\) 1.49300e27 0.731906
\(238\) 0 0
\(239\) −1.15361e27 −0.513432 −0.256716 0.966487i \(-0.582641\pi\)
−0.256716 + 0.966487i \(0.582641\pi\)
\(240\) 0 0
\(241\) −6.38713e26 −0.258291 −0.129146 0.991626i \(-0.541223\pi\)
−0.129146 + 0.991626i \(0.541223\pi\)
\(242\) 0 0
\(243\) −6.40458e27 −2.35515
\(244\) 0 0
\(245\) −6.83345e26 −0.228681
\(246\) 0 0
\(247\) 4.01243e27 1.22290
\(248\) 0 0
\(249\) 6.07600e27 1.68782
\(250\) 0 0
\(251\) 5.85944e26 0.148460 0.0742298 0.997241i \(-0.476350\pi\)
0.0742298 + 0.997241i \(0.476350\pi\)
\(252\) 0 0
\(253\) 3.83079e27 0.885937
\(254\) 0 0
\(255\) −3.14512e27 −0.664394
\(256\) 0 0
\(257\) −5.28864e26 −0.102121 −0.0510603 0.998696i \(-0.516260\pi\)
−0.0510603 + 0.998696i \(0.516260\pi\)
\(258\) 0 0
\(259\) 3.14664e27 0.555777
\(260\) 0 0
\(261\) 1.10726e28 1.79013
\(262\) 0 0
\(263\) −5.59316e27 −0.828259 −0.414129 0.910218i \(-0.635914\pi\)
−0.414129 + 0.910218i \(0.635914\pi\)
\(264\) 0 0
\(265\) −2.22202e27 −0.301593
\(266\) 0 0
\(267\) −2.33969e28 −2.91259
\(268\) 0 0
\(269\) −1.11747e28 −1.27668 −0.638341 0.769753i \(-0.720379\pi\)
−0.638341 + 0.769753i \(0.720379\pi\)
\(270\) 0 0
\(271\) 1.14584e28 1.20220 0.601100 0.799174i \(-0.294729\pi\)
0.601100 + 0.799174i \(0.294729\pi\)
\(272\) 0 0
\(273\) 2.46820e28 2.37963
\(274\) 0 0
\(275\) −6.01927e27 −0.533602
\(276\) 0 0
\(277\) 1.04295e28 0.850638 0.425319 0.905044i \(-0.360162\pi\)
0.425319 + 0.905044i \(0.360162\pi\)
\(278\) 0 0
\(279\) −5.66815e28 −4.25592
\(280\) 0 0
\(281\) 9.50359e25 0.00657302 0.00328651 0.999995i \(-0.498954\pi\)
0.00328651 + 0.999995i \(0.498954\pi\)
\(282\) 0 0
\(283\) −2.19749e28 −1.40082 −0.700412 0.713738i \(-0.747001\pi\)
−0.700412 + 0.713738i \(0.747001\pi\)
\(284\) 0 0
\(285\) −2.68709e28 −1.57967
\(286\) 0 0
\(287\) 9.52209e27 0.516524
\(288\) 0 0
\(289\) −1.57304e28 −0.787797
\(290\) 0 0
\(291\) 5.39812e28 2.49731
\(292\) 0 0
\(293\) −2.29758e28 −0.982410 −0.491205 0.871044i \(-0.663443\pi\)
−0.491205 + 0.871044i \(0.663443\pi\)
\(294\) 0 0
\(295\) 2.73407e28 1.08108
\(296\) 0 0
\(297\) −1.01007e29 −3.69531
\(298\) 0 0
\(299\) −2.22306e28 −0.752893
\(300\) 0 0
\(301\) 4.38459e28 1.37535
\(302\) 0 0
\(303\) −7.52370e28 −2.18696
\(304\) 0 0
\(305\) −2.71366e28 −0.731320
\(306\) 0 0
\(307\) −3.40762e28 −0.851842 −0.425921 0.904760i \(-0.640050\pi\)
−0.425921 + 0.904760i \(0.640050\pi\)
\(308\) 0 0
\(309\) −3.39488e28 −0.787593
\(310\) 0 0
\(311\) −5.66460e28 −1.22018 −0.610091 0.792331i \(-0.708867\pi\)
−0.610091 + 0.792331i \(0.708867\pi\)
\(312\) 0 0
\(313\) 3.78832e28 0.758031 0.379016 0.925390i \(-0.376263\pi\)
0.379016 + 0.925390i \(0.376263\pi\)
\(314\) 0 0
\(315\) −1.18105e29 −2.19632
\(316\) 0 0
\(317\) −3.72097e28 −0.643390 −0.321695 0.946843i \(-0.604253\pi\)
−0.321695 + 0.946843i \(0.604253\pi\)
\(318\) 0 0
\(319\) 5.84250e28 0.939736
\(320\) 0 0
\(321\) 4.46915e28 0.668985
\(322\) 0 0
\(323\) 3.62010e28 0.504536
\(324\) 0 0
\(325\) 3.49306e28 0.453469
\(326\) 0 0
\(327\) −3.91424e28 −0.473531
\(328\) 0 0
\(329\) 9.76990e28 1.10189
\(330\) 0 0
\(331\) −5.62405e28 −0.591599 −0.295799 0.955250i \(-0.595586\pi\)
−0.295799 + 0.955250i \(0.595586\pi\)
\(332\) 0 0
\(333\) 1.24453e29 1.22151
\(334\) 0 0
\(335\) 9.24657e28 0.847160
\(336\) 0 0
\(337\) 9.79219e28 0.837792 0.418896 0.908034i \(-0.362417\pi\)
0.418896 + 0.908034i \(0.362417\pi\)
\(338\) 0 0
\(339\) 2.45435e29 1.96173
\(340\) 0 0
\(341\) −2.99081e29 −2.23416
\(342\) 0 0
\(343\) −1.14663e29 −0.800828
\(344\) 0 0
\(345\) 1.48877e29 0.972541
\(346\) 0 0
\(347\) 7.68930e28 0.470000 0.235000 0.971995i \(-0.424491\pi\)
0.235000 + 0.971995i \(0.424491\pi\)
\(348\) 0 0
\(349\) 3.15740e29 1.80650 0.903249 0.429116i \(-0.141175\pi\)
0.903249 + 0.429116i \(0.141175\pi\)
\(350\) 0 0
\(351\) 5.86155e29 3.14037
\(352\) 0 0
\(353\) −2.41820e29 −1.21362 −0.606809 0.794847i \(-0.707551\pi\)
−0.606809 + 0.794847i \(0.707551\pi\)
\(354\) 0 0
\(355\) −1.93669e28 −0.0910824
\(356\) 0 0
\(357\) 2.22687e29 0.981771
\(358\) 0 0
\(359\) −2.43466e29 −1.00659 −0.503294 0.864115i \(-0.667879\pi\)
−0.503294 + 0.864115i \(0.667879\pi\)
\(360\) 0 0
\(361\) 5.14606e28 0.199592
\(362\) 0 0
\(363\) −3.73410e29 −1.35913
\(364\) 0 0
\(365\) −7.01473e28 −0.239686
\(366\) 0 0
\(367\) 2.79274e29 0.896128 0.448064 0.894001i \(-0.352114\pi\)
0.448064 + 0.894001i \(0.352114\pi\)
\(368\) 0 0
\(369\) 3.76609e29 1.13524
\(370\) 0 0
\(371\) 1.57328e29 0.445662
\(372\) 0 0
\(373\) 1.37422e29 0.365937 0.182968 0.983119i \(-0.441429\pi\)
0.182968 + 0.983119i \(0.441429\pi\)
\(374\) 0 0
\(375\) −8.09911e29 −2.02805
\(376\) 0 0
\(377\) −3.39049e29 −0.798613
\(378\) 0 0
\(379\) 1.24160e29 0.275189 0.137594 0.990489i \(-0.456063\pi\)
0.137594 + 0.990489i \(0.456063\pi\)
\(380\) 0 0
\(381\) −3.90576e29 −0.814832
\(382\) 0 0
\(383\) 3.88270e29 0.762690 0.381345 0.924433i \(-0.375461\pi\)
0.381345 + 0.924433i \(0.375461\pi\)
\(384\) 0 0
\(385\) −6.23181e29 −1.15296
\(386\) 0 0
\(387\) 1.73415e30 3.02282
\(388\) 0 0
\(389\) 1.17387e30 1.92841 0.964203 0.265165i \(-0.0854265\pi\)
0.964203 + 0.265165i \(0.0854265\pi\)
\(390\) 0 0
\(391\) −2.00570e29 −0.310623
\(392\) 0 0
\(393\) −3.66761e28 −0.0535634
\(394\) 0 0
\(395\) 2.18755e29 0.301364
\(396\) 0 0
\(397\) −1.47771e30 −1.92087 −0.960433 0.278511i \(-0.910159\pi\)
−0.960433 + 0.278511i \(0.910159\pi\)
\(398\) 0 0
\(399\) 1.90257e30 2.33427
\(400\) 0 0
\(401\) −4.26558e29 −0.494103 −0.247052 0.969002i \(-0.579462\pi\)
−0.247052 + 0.969002i \(0.579462\pi\)
\(402\) 0 0
\(403\) 1.73561e30 1.89865
\(404\) 0 0
\(405\) −2.05906e30 −2.12783
\(406\) 0 0
\(407\) 6.56679e29 0.641235
\(408\) 0 0
\(409\) −1.42830e30 −1.31826 −0.659129 0.752030i \(-0.729075\pi\)
−0.659129 + 0.752030i \(0.729075\pi\)
\(410\) 0 0
\(411\) −1.81515e30 −1.58391
\(412\) 0 0
\(413\) −1.93583e30 −1.59750
\(414\) 0 0
\(415\) 8.90255e29 0.694963
\(416\) 0 0
\(417\) 8.71728e29 0.643897
\(418\) 0 0
\(419\) −1.76509e30 −1.23397 −0.616985 0.786975i \(-0.711646\pi\)
−0.616985 + 0.786975i \(0.711646\pi\)
\(420\) 0 0
\(421\) 2.02002e30 1.33694 0.668471 0.743739i \(-0.266949\pi\)
0.668471 + 0.743739i \(0.266949\pi\)
\(422\) 0 0
\(423\) 3.86410e30 2.42178
\(424\) 0 0
\(425\) 3.15153e29 0.187089
\(426\) 0 0
\(427\) 1.92138e30 1.08067
\(428\) 0 0
\(429\) 5.15094e30 2.74554
\(430\) 0 0
\(431\) −2.37148e30 −1.19820 −0.599102 0.800673i \(-0.704476\pi\)
−0.599102 + 0.800673i \(0.704476\pi\)
\(432\) 0 0
\(433\) 2.23333e30 1.06990 0.534949 0.844884i \(-0.320331\pi\)
0.534949 + 0.844884i \(0.320331\pi\)
\(434\) 0 0
\(435\) 2.27059e30 1.03160
\(436\) 0 0
\(437\) −1.71361e30 −0.738541
\(438\) 0 0
\(439\) −3.90666e30 −1.59758 −0.798792 0.601607i \(-0.794527\pi\)
−0.798792 + 0.601607i \(0.794527\pi\)
\(440\) 0 0
\(441\) 1.91362e30 0.742697
\(442\) 0 0
\(443\) −3.27739e30 −1.20749 −0.603747 0.797176i \(-0.706326\pi\)
−0.603747 + 0.797176i \(0.706326\pi\)
\(444\) 0 0
\(445\) −3.42811e30 −1.19927
\(446\) 0 0
\(447\) −3.41183e29 −0.113358
\(448\) 0 0
\(449\) 8.52929e29 0.269203 0.134602 0.990900i \(-0.457025\pi\)
0.134602 + 0.990900i \(0.457025\pi\)
\(450\) 0 0
\(451\) 1.98718e30 0.595947
\(452\) 0 0
\(453\) 5.95364e30 1.69689
\(454\) 0 0
\(455\) 3.61641e30 0.979820
\(456\) 0 0
\(457\) 7.07801e30 1.82337 0.911685 0.410890i \(-0.134782\pi\)
0.911685 + 0.410890i \(0.134782\pi\)
\(458\) 0 0
\(459\) 5.28843e30 1.29563
\(460\) 0 0
\(461\) 5.36478e30 1.25023 0.625117 0.780531i \(-0.285051\pi\)
0.625117 + 0.780531i \(0.285051\pi\)
\(462\) 0 0
\(463\) 2.58238e30 0.572584 0.286292 0.958142i \(-0.407577\pi\)
0.286292 + 0.958142i \(0.407577\pi\)
\(464\) 0 0
\(465\) −1.16233e31 −2.45255
\(466\) 0 0
\(467\) −3.99016e30 −0.801394 −0.400697 0.916211i \(-0.631232\pi\)
−0.400697 + 0.916211i \(0.631232\pi\)
\(468\) 0 0
\(469\) −6.54693e30 −1.25184
\(470\) 0 0
\(471\) −7.33016e30 −1.33467
\(472\) 0 0
\(473\) 9.15029e30 1.58684
\(474\) 0 0
\(475\) 2.69256e30 0.444825
\(476\) 0 0
\(477\) 6.22248e30 0.979495
\(478\) 0 0
\(479\) −5.56629e30 −0.835041 −0.417520 0.908668i \(-0.637101\pi\)
−0.417520 + 0.908668i \(0.637101\pi\)
\(480\) 0 0
\(481\) −3.81080e30 −0.544939
\(482\) 0 0
\(483\) −1.05411e31 −1.43712
\(484\) 0 0
\(485\) 7.90932e30 1.02827
\(486\) 0 0
\(487\) 3.57505e30 0.443301 0.221651 0.975126i \(-0.428856\pi\)
0.221651 + 0.975126i \(0.428856\pi\)
\(488\) 0 0
\(489\) −2.46317e31 −2.91368
\(490\) 0 0
\(491\) −6.34262e30 −0.715865 −0.357933 0.933747i \(-0.616518\pi\)
−0.357933 + 0.933747i \(0.616518\pi\)
\(492\) 0 0
\(493\) −3.05898e30 −0.329486
\(494\) 0 0
\(495\) −2.46475e31 −2.53403
\(496\) 0 0
\(497\) 1.37125e30 0.134592
\(498\) 0 0
\(499\) −3.30165e30 −0.309439 −0.154720 0.987958i \(-0.549447\pi\)
−0.154720 + 0.987958i \(0.549447\pi\)
\(500\) 0 0
\(501\) −3.87087e31 −3.46477
\(502\) 0 0
\(503\) 2.32758e31 1.99009 0.995044 0.0994338i \(-0.0317031\pi\)
0.995044 + 0.0994338i \(0.0317031\pi\)
\(504\) 0 0
\(505\) −1.10237e31 −0.900487
\(506\) 0 0
\(507\) −5.91440e30 −0.461657
\(508\) 0 0
\(509\) −2.41190e31 −1.79930 −0.899652 0.436607i \(-0.856180\pi\)
−0.899652 + 0.436607i \(0.856180\pi\)
\(510\) 0 0
\(511\) 4.96670e30 0.354183
\(512\) 0 0
\(513\) 4.51827e31 3.08051
\(514\) 0 0
\(515\) −4.97418e30 −0.324293
\(516\) 0 0
\(517\) 2.03890e31 1.27132
\(518\) 0 0
\(519\) −3.06562e31 −1.82850
\(520\) 0 0
\(521\) 6.94775e30 0.396470 0.198235 0.980155i \(-0.436479\pi\)
0.198235 + 0.980155i \(0.436479\pi\)
\(522\) 0 0
\(523\) 2.69454e31 1.47135 0.735673 0.677337i \(-0.236866\pi\)
0.735673 + 0.677337i \(0.236866\pi\)
\(524\) 0 0
\(525\) 1.65630e31 0.865580
\(526\) 0 0
\(527\) 1.56591e31 0.783328
\(528\) 0 0
\(529\) −1.13863e31 −0.545310
\(530\) 0 0
\(531\) −7.65641e31 −3.51106
\(532\) 0 0
\(533\) −1.15319e31 −0.506452
\(534\) 0 0
\(535\) 6.54819e30 0.275456
\(536\) 0 0
\(537\) −3.31204e31 −1.33472
\(538\) 0 0
\(539\) 1.00972e31 0.389881
\(540\) 0 0
\(541\) −7.63627e30 −0.282561 −0.141281 0.989970i \(-0.545122\pi\)
−0.141281 + 0.989970i \(0.545122\pi\)
\(542\) 0 0
\(543\) −7.55667e31 −2.67999
\(544\) 0 0
\(545\) −5.73515e30 −0.194978
\(546\) 0 0
\(547\) −2.26514e31 −0.738312 −0.369156 0.929367i \(-0.620353\pi\)
−0.369156 + 0.929367i \(0.620353\pi\)
\(548\) 0 0
\(549\) 7.59925e31 2.37514
\(550\) 0 0
\(551\) −2.61349e31 −0.783390
\(552\) 0 0
\(553\) −1.54887e31 −0.445324
\(554\) 0 0
\(555\) 2.55207e31 0.703919
\(556\) 0 0
\(557\) −1.58983e31 −0.420741 −0.210371 0.977622i \(-0.567467\pi\)
−0.210371 + 0.977622i \(0.567467\pi\)
\(558\) 0 0
\(559\) −5.31005e31 −1.34854
\(560\) 0 0
\(561\) 4.64730e31 1.13273
\(562\) 0 0
\(563\) −7.95857e31 −1.86204 −0.931019 0.364972i \(-0.881079\pi\)
−0.931019 + 0.364972i \(0.881079\pi\)
\(564\) 0 0
\(565\) 3.59611e31 0.807748
\(566\) 0 0
\(567\) 1.45789e32 3.14428
\(568\) 0 0
\(569\) 7.26713e31 1.50512 0.752561 0.658522i \(-0.228818\pi\)
0.752561 + 0.658522i \(0.228818\pi\)
\(570\) 0 0
\(571\) 3.78373e31 0.752670 0.376335 0.926484i \(-0.377184\pi\)
0.376335 + 0.926484i \(0.377184\pi\)
\(572\) 0 0
\(573\) 2.33971e31 0.447078
\(574\) 0 0
\(575\) −1.49180e31 −0.273861
\(576\) 0 0
\(577\) −6.81473e31 −1.20206 −0.601031 0.799225i \(-0.705243\pi\)
−0.601031 + 0.799225i \(0.705243\pi\)
\(578\) 0 0
\(579\) 1.72840e32 2.92982
\(580\) 0 0
\(581\) −6.30335e31 −1.02694
\(582\) 0 0
\(583\) 3.28330e31 0.514189
\(584\) 0 0
\(585\) 1.43033e32 2.15349
\(586\) 0 0
\(587\) 1.10842e32 1.60459 0.802296 0.596927i \(-0.203612\pi\)
0.802296 + 0.596927i \(0.203612\pi\)
\(588\) 0 0
\(589\) 1.33786e32 1.86245
\(590\) 0 0
\(591\) −2.39706e31 −0.320939
\(592\) 0 0
\(593\) 1.12848e31 0.145333 0.0726666 0.997356i \(-0.476849\pi\)
0.0726666 + 0.997356i \(0.476849\pi\)
\(594\) 0 0
\(595\) 3.26281e31 0.404246
\(596\) 0 0
\(597\) −1.79748e31 −0.214269
\(598\) 0 0
\(599\) −7.82635e31 −0.897741 −0.448870 0.893597i \(-0.648174\pi\)
−0.448870 + 0.893597i \(0.648174\pi\)
\(600\) 0 0
\(601\) −8.73347e31 −0.964119 −0.482059 0.876139i \(-0.660111\pi\)
−0.482059 + 0.876139i \(0.660111\pi\)
\(602\) 0 0
\(603\) −2.58938e32 −2.75136
\(604\) 0 0
\(605\) −5.47120e31 −0.559623
\(606\) 0 0
\(607\) 9.31517e31 0.917320 0.458660 0.888612i \(-0.348330\pi\)
0.458660 + 0.888612i \(0.348330\pi\)
\(608\) 0 0
\(609\) −1.60766e32 −1.52439
\(610\) 0 0
\(611\) −1.18320e32 −1.08040
\(612\) 0 0
\(613\) −1.93566e32 −1.70229 −0.851145 0.524930i \(-0.824091\pi\)
−0.851145 + 0.524930i \(0.824091\pi\)
\(614\) 0 0
\(615\) 7.72284e31 0.654204
\(616\) 0 0
\(617\) −7.76155e31 −0.633387 −0.316694 0.948528i \(-0.602573\pi\)
−0.316694 + 0.948528i \(0.602573\pi\)
\(618\) 0 0
\(619\) −1.48903e32 −1.17074 −0.585372 0.810765i \(-0.699051\pi\)
−0.585372 + 0.810765i \(0.699051\pi\)
\(620\) 0 0
\(621\) −2.50332e32 −1.89655
\(622\) 0 0
\(623\) 2.42724e32 1.77215
\(624\) 0 0
\(625\) −6.09526e31 −0.428916
\(626\) 0 0
\(627\) 3.97050e32 2.69320
\(628\) 0 0
\(629\) −3.43819e31 −0.224827
\(630\) 0 0
\(631\) −2.00939e31 −0.126685 −0.0633426 0.997992i \(-0.520176\pi\)
−0.0633426 + 0.997992i \(0.520176\pi\)
\(632\) 0 0
\(633\) 1.36212e32 0.828080
\(634\) 0 0
\(635\) −5.72271e31 −0.335509
\(636\) 0 0
\(637\) −5.85957e31 −0.331331
\(638\) 0 0
\(639\) 5.42345e31 0.295812
\(640\) 0 0
\(641\) 2.52315e31 0.132762 0.0663812 0.997794i \(-0.478855\pi\)
0.0663812 + 0.997794i \(0.478855\pi\)
\(642\) 0 0
\(643\) 7.18001e31 0.364501 0.182251 0.983252i \(-0.441662\pi\)
0.182251 + 0.983252i \(0.441662\pi\)
\(644\) 0 0
\(645\) 3.55610e32 1.74196
\(646\) 0 0
\(647\) 1.17766e32 0.556700 0.278350 0.960480i \(-0.410212\pi\)
0.278350 + 0.960480i \(0.410212\pi\)
\(648\) 0 0
\(649\) −4.03992e32 −1.84314
\(650\) 0 0
\(651\) 8.22971e32 3.62413
\(652\) 0 0
\(653\) −2.88510e32 −1.22648 −0.613238 0.789898i \(-0.710133\pi\)
−0.613238 + 0.789898i \(0.710133\pi\)
\(654\) 0 0
\(655\) −5.37377e30 −0.0220549
\(656\) 0 0
\(657\) 1.96438e32 0.778439
\(658\) 0 0
\(659\) −1.55020e32 −0.593208 −0.296604 0.955001i \(-0.595854\pi\)
−0.296604 + 0.955001i \(0.595854\pi\)
\(660\) 0 0
\(661\) 2.65192e32 0.980039 0.490020 0.871711i \(-0.336990\pi\)
0.490020 + 0.871711i \(0.336990\pi\)
\(662\) 0 0
\(663\) −2.69689e32 −0.962627
\(664\) 0 0
\(665\) 2.78764e32 0.961142
\(666\) 0 0
\(667\) 1.44799e32 0.482302
\(668\) 0 0
\(669\) 8.58111e32 2.76149
\(670\) 0 0
\(671\) 4.00976e32 1.24683
\(672\) 0 0
\(673\) 5.25147e32 1.57800 0.789001 0.614392i \(-0.210599\pi\)
0.789001 + 0.614392i \(0.210599\pi\)
\(674\) 0 0
\(675\) 3.93343e32 1.14229
\(676\) 0 0
\(677\) −4.13293e32 −1.16008 −0.580041 0.814587i \(-0.696963\pi\)
−0.580041 + 0.814587i \(0.696963\pi\)
\(678\) 0 0
\(679\) −5.60011e32 −1.51948
\(680\) 0 0
\(681\) −2.35276e32 −0.617143
\(682\) 0 0
\(683\) −5.82971e32 −1.47846 −0.739229 0.673454i \(-0.764810\pi\)
−0.739229 + 0.673454i \(0.764810\pi\)
\(684\) 0 0
\(685\) −2.65956e32 −0.652180
\(686\) 0 0
\(687\) 9.03114e32 2.14161
\(688\) 0 0
\(689\) −1.90535e32 −0.436971
\(690\) 0 0
\(691\) −7.25689e30 −0.0160973 −0.00804865 0.999968i \(-0.502562\pi\)
−0.00804865 + 0.999968i \(0.502562\pi\)
\(692\) 0 0
\(693\) 1.74514e33 3.74453
\(694\) 0 0
\(695\) 1.27725e32 0.265126
\(696\) 0 0
\(697\) −1.04044e32 −0.208948
\(698\) 0 0
\(699\) −1.41730e33 −2.75407
\(700\) 0 0
\(701\) −8.94327e32 −1.68166 −0.840832 0.541297i \(-0.817934\pi\)
−0.840832 + 0.541297i \(0.817934\pi\)
\(702\) 0 0
\(703\) −2.93748e32 −0.534551
\(704\) 0 0
\(705\) 7.92382e32 1.39560
\(706\) 0 0
\(707\) 7.80522e32 1.33064
\(708\) 0 0
\(709\) −1.06809e33 −1.76269 −0.881346 0.472472i \(-0.843362\pi\)
−0.881346 + 0.472472i \(0.843362\pi\)
\(710\) 0 0
\(711\) −6.12594e32 −0.978752
\(712\) 0 0
\(713\) −7.41235e32 −1.14664
\(714\) 0 0
\(715\) 7.54715e32 1.13048
\(716\) 0 0
\(717\) 6.62461e32 0.960927
\(718\) 0 0
\(719\) 1.23737e32 0.173827 0.0869133 0.996216i \(-0.472300\pi\)
0.0869133 + 0.996216i \(0.472300\pi\)
\(720\) 0 0
\(721\) 3.52191e32 0.479206
\(722\) 0 0
\(723\) 3.66781e32 0.483411
\(724\) 0 0
\(725\) −2.27521e32 −0.290492
\(726\) 0 0
\(727\) −1.92923e32 −0.238637 −0.119319 0.992856i \(-0.538071\pi\)
−0.119319 + 0.992856i \(0.538071\pi\)
\(728\) 0 0
\(729\) 1.37396e33 1.64667
\(730\) 0 0
\(731\) −4.79085e32 −0.556368
\(732\) 0 0
\(733\) −5.14954e31 −0.0579526 −0.0289763 0.999580i \(-0.509225\pi\)
−0.0289763 + 0.999580i \(0.509225\pi\)
\(734\) 0 0
\(735\) 3.92411e32 0.427993
\(736\) 0 0
\(737\) −1.36629e33 −1.44433
\(738\) 0 0
\(739\) −1.78518e33 −1.82924 −0.914621 0.404312i \(-0.867511\pi\)
−0.914621 + 0.404312i \(0.867511\pi\)
\(740\) 0 0
\(741\) −2.30414e33 −2.28875
\(742\) 0 0
\(743\) −1.03301e32 −0.0994792 −0.0497396 0.998762i \(-0.515839\pi\)
−0.0497396 + 0.998762i \(0.515839\pi\)
\(744\) 0 0
\(745\) −4.99901e31 −0.0466752
\(746\) 0 0
\(747\) −2.49304e33 −2.25706
\(748\) 0 0
\(749\) −4.63637e32 −0.407040
\(750\) 0 0
\(751\) −2.00354e33 −1.70584 −0.852922 0.522038i \(-0.825172\pi\)
−0.852922 + 0.522038i \(0.825172\pi\)
\(752\) 0 0
\(753\) −3.36479e32 −0.277853
\(754\) 0 0
\(755\) 8.72327e32 0.698698
\(756\) 0 0
\(757\) −2.90156e32 −0.225439 −0.112719 0.993627i \(-0.535956\pi\)
−0.112719 + 0.993627i \(0.535956\pi\)
\(758\) 0 0
\(759\) −2.19984e33 −1.65810
\(760\) 0 0
\(761\) −1.59845e33 −1.16889 −0.584446 0.811433i \(-0.698688\pi\)
−0.584446 + 0.811433i \(0.698688\pi\)
\(762\) 0 0
\(763\) 4.06071e32 0.288117
\(764\) 0 0
\(765\) 1.29047e33 0.888471
\(766\) 0 0
\(767\) 2.34442e33 1.56635
\(768\) 0 0
\(769\) −8.79322e32 −0.570158 −0.285079 0.958504i \(-0.592020\pi\)
−0.285079 + 0.958504i \(0.592020\pi\)
\(770\) 0 0
\(771\) 3.03701e32 0.191126
\(772\) 0 0
\(773\) 8.24155e32 0.503437 0.251718 0.967801i \(-0.419004\pi\)
0.251718 + 0.967801i \(0.419004\pi\)
\(774\) 0 0
\(775\) 1.16469e33 0.690623
\(776\) 0 0
\(777\) −1.80696e33 −1.04018
\(778\) 0 0
\(779\) −8.88915e32 −0.496798
\(780\) 0 0
\(781\) 2.86169e32 0.155287
\(782\) 0 0
\(783\) −3.81792e33 −2.01172
\(784\) 0 0
\(785\) −1.07401e33 −0.549552
\(786\) 0 0
\(787\) −1.89467e31 −0.00941509 −0.00470755 0.999989i \(-0.501498\pi\)
−0.00470755 + 0.999989i \(0.501498\pi\)
\(788\) 0 0
\(789\) 3.21187e33 1.55015
\(790\) 0 0
\(791\) −2.54619e33 −1.19360
\(792\) 0 0
\(793\) −2.32692e33 −1.05959
\(794\) 0 0
\(795\) 1.27600e33 0.564453
\(796\) 0 0
\(797\) 5.78084e32 0.248439 0.124220 0.992255i \(-0.460357\pi\)
0.124220 + 0.992255i \(0.460357\pi\)
\(798\) 0 0
\(799\) −1.06751e33 −0.445744
\(800\) 0 0
\(801\) 9.59998e33 3.89490
\(802\) 0 0
\(803\) 1.03651e33 0.408644
\(804\) 0 0
\(805\) −1.54448e33 −0.591737
\(806\) 0 0
\(807\) 6.41706e33 2.38941
\(808\) 0 0
\(809\) 4.46172e33 1.61471 0.807355 0.590066i \(-0.200898\pi\)
0.807355 + 0.590066i \(0.200898\pi\)
\(810\) 0 0
\(811\) 2.69737e33 0.948857 0.474429 0.880294i \(-0.342655\pi\)
0.474429 + 0.880294i \(0.342655\pi\)
\(812\) 0 0
\(813\) −6.57999e33 −2.25001
\(814\) 0 0
\(815\) −3.60903e33 −1.19971
\(816\) 0 0
\(817\) −4.09315e33 −1.32283
\(818\) 0 0
\(819\) −1.01273e34 −3.18220
\(820\) 0 0
\(821\) 3.28224e33 1.00282 0.501412 0.865209i \(-0.332814\pi\)
0.501412 + 0.865209i \(0.332814\pi\)
\(822\) 0 0
\(823\) 2.64872e33 0.786933 0.393466 0.919339i \(-0.371276\pi\)
0.393466 + 0.919339i \(0.371276\pi\)
\(824\) 0 0
\(825\) 3.45657e33 0.998676
\(826\) 0 0
\(827\) 1.65312e33 0.464505 0.232253 0.972655i \(-0.425390\pi\)
0.232253 + 0.972655i \(0.425390\pi\)
\(828\) 0 0
\(829\) −3.48612e33 −0.952719 −0.476360 0.879251i \(-0.658044\pi\)
−0.476360 + 0.879251i \(0.658044\pi\)
\(830\) 0 0
\(831\) −5.98913e33 −1.59203
\(832\) 0 0
\(833\) −5.28664e32 −0.136698
\(834\) 0 0
\(835\) −5.67160e33 −1.42663
\(836\) 0 0
\(837\) 1.95442e34 4.78271
\(838\) 0 0
\(839\) −4.22678e33 −1.00634 −0.503172 0.864186i \(-0.667834\pi\)
−0.503172 + 0.864186i \(0.667834\pi\)
\(840\) 0 0
\(841\) −2.10833e33 −0.488410
\(842\) 0 0
\(843\) −5.45745e31 −0.0123019
\(844\) 0 0
\(845\) −8.66577e32 −0.190088
\(846\) 0 0
\(847\) 3.87382e33 0.826952
\(848\) 0 0
\(849\) 1.26191e34 2.62175
\(850\) 0 0
\(851\) 1.62750e33 0.329102
\(852\) 0 0
\(853\) −4.60865e33 −0.907111 −0.453555 0.891228i \(-0.649845\pi\)
−0.453555 + 0.891228i \(0.649845\pi\)
\(854\) 0 0
\(855\) 1.10254e34 2.11244
\(856\) 0 0
\(857\) −1.09714e33 −0.204637 −0.102318 0.994752i \(-0.532626\pi\)
−0.102318 + 0.994752i \(0.532626\pi\)
\(858\) 0 0
\(859\) 2.95181e33 0.536003 0.268001 0.963419i \(-0.413637\pi\)
0.268001 + 0.963419i \(0.413637\pi\)
\(860\) 0 0
\(861\) −5.46807e33 −0.966713
\(862\) 0 0
\(863\) 1.04670e33 0.180177 0.0900884 0.995934i \(-0.471285\pi\)
0.0900884 + 0.995934i \(0.471285\pi\)
\(864\) 0 0
\(865\) −4.49175e33 −0.752888
\(866\) 0 0
\(867\) 9.03319e33 1.47442
\(868\) 0 0
\(869\) −3.23237e33 −0.513799
\(870\) 0 0
\(871\) 7.92878e33 1.22743
\(872\) 0 0
\(873\) −2.21490e34 −3.33957
\(874\) 0 0
\(875\) 8.40216e33 1.23395
\(876\) 0 0
\(877\) −3.86117e33 −0.552362 −0.276181 0.961106i \(-0.589069\pi\)
−0.276181 + 0.961106i \(0.589069\pi\)
\(878\) 0 0
\(879\) 1.31939e34 1.83865
\(880\) 0 0
\(881\) −7.78654e33 −1.05711 −0.528557 0.848898i \(-0.677267\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(882\) 0 0
\(883\) −4.55914e33 −0.603024 −0.301512 0.953462i \(-0.597491\pi\)
−0.301512 + 0.953462i \(0.597491\pi\)
\(884\) 0 0
\(885\) −1.57004e34 −2.02332
\(886\) 0 0
\(887\) −3.58889e32 −0.0450649 −0.0225325 0.999746i \(-0.507173\pi\)
−0.0225325 + 0.999746i \(0.507173\pi\)
\(888\) 0 0
\(889\) 4.05190e33 0.495780
\(890\) 0 0
\(891\) 3.04251e34 3.62776
\(892\) 0 0
\(893\) −9.12049e33 −1.05981
\(894\) 0 0
\(895\) −4.85280e33 −0.549576
\(896\) 0 0
\(897\) 1.27660e34 1.40909
\(898\) 0 0
\(899\) −1.13049e34 −1.21627
\(900\) 0 0
\(901\) −1.71905e33 −0.180282
\(902\) 0 0
\(903\) −2.51786e34 −2.57408
\(904\) 0 0
\(905\) −1.10720e34 −1.10349
\(906\) 0 0
\(907\) −1.09184e34 −1.06090 −0.530450 0.847716i \(-0.677977\pi\)
−0.530450 + 0.847716i \(0.677977\pi\)
\(908\) 0 0
\(909\) 3.08705e34 2.92455
\(910\) 0 0
\(911\) −1.42933e34 −1.32029 −0.660146 0.751137i \(-0.729506\pi\)
−0.660146 + 0.751137i \(0.729506\pi\)
\(912\) 0 0
\(913\) −1.31546e34 −1.18485
\(914\) 0 0
\(915\) 1.55832e34 1.36872
\(916\) 0 0
\(917\) 3.80484e32 0.0325904
\(918\) 0 0
\(919\) 2.15002e34 1.79604 0.898018 0.439959i \(-0.145007\pi\)
0.898018 + 0.439959i \(0.145007\pi\)
\(920\) 0 0
\(921\) 1.95683e34 1.59429
\(922\) 0 0
\(923\) −1.66068e33 −0.131967
\(924\) 0 0
\(925\) −2.55726e33 −0.198219
\(926\) 0 0
\(927\) 1.39295e34 1.05322
\(928\) 0 0
\(929\) −1.53709e34 −1.13376 −0.566878 0.823801i \(-0.691849\pi\)
−0.566878 + 0.823801i \(0.691849\pi\)
\(930\) 0 0
\(931\) −4.51674e33 −0.325015
\(932\) 0 0
\(933\) 3.25290e34 2.28366
\(934\) 0 0
\(935\) 6.80921e33 0.466405
\(936\) 0 0
\(937\) −3.02985e33 −0.202496 −0.101248 0.994861i \(-0.532284\pi\)
−0.101248 + 0.994861i \(0.532284\pi\)
\(938\) 0 0
\(939\) −2.17545e34 −1.41871
\(940\) 0 0
\(941\) −1.40737e34 −0.895631 −0.447815 0.894126i \(-0.647798\pi\)
−0.447815 + 0.894126i \(0.647798\pi\)
\(942\) 0 0
\(943\) 4.92499e33 0.305859
\(944\) 0 0
\(945\) 4.07232e34 2.46818
\(946\) 0 0
\(947\) 2.84567e34 1.68329 0.841647 0.540028i \(-0.181586\pi\)
0.841647 + 0.540028i \(0.181586\pi\)
\(948\) 0 0
\(949\) −6.01502e33 −0.347276
\(950\) 0 0
\(951\) 2.13677e34 1.20415
\(952\) 0 0
\(953\) −3.91513e33 −0.215366 −0.107683 0.994185i \(-0.534343\pi\)
−0.107683 + 0.994185i \(0.534343\pi\)
\(954\) 0 0
\(955\) 3.42814e33 0.184085
\(956\) 0 0
\(957\) −3.35506e34 −1.75879
\(958\) 0 0
\(959\) 1.88307e34 0.963723
\(960\) 0 0
\(961\) 3.78570e34 1.89159
\(962\) 0 0
\(963\) −1.83373e34 −0.894611
\(964\) 0 0
\(965\) 2.53245e34 1.20636
\(966\) 0 0
\(967\) 1.18860e34 0.552882 0.276441 0.961031i \(-0.410845\pi\)
0.276441 + 0.961031i \(0.410845\pi\)
\(968\) 0 0
\(969\) −2.07885e34 −0.944277
\(970\) 0 0
\(971\) −2.66114e33 −0.118045 −0.0590224 0.998257i \(-0.518798\pi\)
−0.0590224 + 0.998257i \(0.518798\pi\)
\(972\) 0 0
\(973\) −9.04346e33 −0.391775
\(974\) 0 0
\(975\) −2.00589e34 −0.848702
\(976\) 0 0
\(977\) 1.53142e34 0.632860 0.316430 0.948616i \(-0.397516\pi\)
0.316430 + 0.948616i \(0.397516\pi\)
\(978\) 0 0
\(979\) 5.06545e34 2.04464
\(980\) 0 0
\(981\) 1.60605e34 0.633237
\(982\) 0 0
\(983\) −1.82163e34 −0.701610 −0.350805 0.936449i \(-0.614092\pi\)
−0.350805 + 0.936449i \(0.614092\pi\)
\(984\) 0 0
\(985\) −3.51217e33 −0.132147
\(986\) 0 0
\(987\) −5.61037e34 −2.06227
\(988\) 0 0
\(989\) 2.26779e34 0.814413
\(990\) 0 0
\(991\) −4.71132e34 −1.65309 −0.826543 0.562874i \(-0.809696\pi\)
−0.826543 + 0.562874i \(0.809696\pi\)
\(992\) 0 0
\(993\) 3.22961e34 1.10722
\(994\) 0 0
\(995\) −2.63367e33 −0.0882257
\(996\) 0 0
\(997\) 2.34856e34 0.768791 0.384395 0.923169i \(-0.374410\pi\)
0.384395 + 0.923169i \(0.374410\pi\)
\(998\) 0 0
\(999\) −4.29122e34 −1.37271
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 64.24.a.h.1.1 3
4.3 odd 2 64.24.a.k.1.3 3
8.3 odd 2 8.24.a.a.1.1 3
8.5 even 2 16.24.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.24.a.a.1.1 3 8.3 odd 2
16.24.a.e.1.3 3 8.5 even 2
64.24.a.h.1.1 3 1.1 even 1 trivial
64.24.a.k.1.3 3 4.3 odd 2