Properties

Label 54.9.b.a.53.2
Level $54$
Weight $9$
Character 54.53
Analytic conductor $21.998$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,9,Mod(53,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 54.b (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: \(21.9984449433\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 54.53
Dual form 54.9.b.a.53.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+11.3137i q^{2} -128.000 q^{4} +678.823i q^{5} -2065.00 q^{7} -1448.15i q^{8} +O(q^{10})\) \(q+11.3137i q^{2} -128.000 q^{4} +678.823i q^{5} -2065.00 q^{7} -1448.15i q^{8} -7680.00 q^{10} +6652.46i q^{11} +8063.00 q^{13} -23362.8i q^{14} +16384.0 q^{16} -21586.6i q^{17} -226609. q^{19} -86889.3i q^{20} -75264.0 q^{22} -368329. i q^{23} -70175.0 q^{25} +91222.4i q^{26} +264320. q^{28} -937047. i q^{29} +826370. q^{31} +185364. i q^{32} +244224. q^{34} -1.40177e6i q^{35} +1.34458e6 q^{37} -2.56379e6i q^{38} +983040. q^{40} -5.19191e6i q^{41} -6.14774e6 q^{43} -851515. i q^{44} +4.16717e6 q^{46} +5.91078e6i q^{47} -1.50058e6 q^{49} -793939. i q^{50} -1.03206e6 q^{52} -768156. i q^{53} -4.51584e6 q^{55} +2.99044e6i q^{56} +1.06015e7 q^{58} +473954. i q^{59} -1.49857e7 q^{61} +9.34931e6i q^{62} -2.09715e6 q^{64} +5.47335e6i q^{65} -1.00237e7 q^{67} +2.76308e6i q^{68} +1.58592e7 q^{70} +4.54849e7i q^{71} -2.32616e7 q^{73} +1.52121e7i q^{74} +2.90060e7 q^{76} -1.37373e7i q^{77} +1.42672e7 q^{79} +1.11218e7i q^{80} +5.87397e7 q^{82} +3.61918e7i q^{83} +1.46534e7 q^{85} -6.95538e7i q^{86} +9.63379e6 q^{88} +1.15088e8i q^{89} -1.66501e7 q^{91} +4.71461e7i q^{92} -6.68728e7 q^{94} -1.53827e8i q^{95} -4.05716e7 q^{97} -1.69771e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} - 4130 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} - 4130 q^{7} - 15360 q^{10} + 16126 q^{13} + 32768 q^{16} - 453218 q^{19} - 150528 q^{22} - 140350 q^{25} + 528640 q^{28} + 1652740 q^{31} + 488448 q^{34} + 2689150 q^{37} + 1966080 q^{40} - 12295484 q^{43} + 8334336 q^{46} - 3001152 q^{49} - 2064128 q^{52} - 9031680 q^{55} + 21202944 q^{58} - 29971394 q^{61} - 4194304 q^{64} - 20047394 q^{67} + 31718400 q^{70} - 46523138 q^{73} + 58011904 q^{76} + 28534366 q^{79} + 117479424 q^{82} + 29306880 q^{85} + 19267584 q^{88} - 33300190 q^{91} - 133745664 q^{94} - 81143234 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.3137i 0.707107i
\(3\) 0 0
\(4\) −128.000 −0.500000
\(5\) 678.823i 1.08612i 0.839695 + 0.543058i \(0.182734\pi\)
−0.839695 + 0.543058i \(0.817266\pi\)
\(6\) 0 0
\(7\) −2065.00 −0.860058 −0.430029 0.902815i \(-0.641497\pi\)
−0.430029 + 0.902815i \(0.641497\pi\)
\(8\) − 1448.15i − 0.353553i
\(9\) 0 0
\(10\) −7680.00 −0.768000
\(11\) 6652.46i 0.454372i 0.973851 + 0.227186i \(0.0729525\pi\)
−0.973851 + 0.227186i \(0.927047\pi\)
\(12\) 0 0
\(13\) 8063.00 0.282308 0.141154 0.989988i \(-0.454919\pi\)
0.141154 + 0.989988i \(0.454919\pi\)
\(14\) − 23362.8i − 0.608153i
\(15\) 0 0
\(16\) 16384.0 0.250000
\(17\) − 21586.6i − 0.258457i −0.991615 0.129228i \(-0.958750\pi\)
0.991615 0.129228i \(-0.0412500\pi\)
\(18\) 0 0
\(19\) −226609. −1.73885 −0.869426 0.494063i \(-0.835511\pi\)
−0.869426 + 0.494063i \(0.835511\pi\)
\(20\) − 86889.3i − 0.543058i
\(21\) 0 0
\(22\) −75264.0 −0.321290
\(23\) − 368329.i − 1.31621i −0.752927 0.658104i \(-0.771359\pi\)
0.752927 0.658104i \(-0.228641\pi\)
\(24\) 0 0
\(25\) −70175.0 −0.179648
\(26\) 91222.4i 0.199622i
\(27\) 0 0
\(28\) 264320. 0.430029
\(29\) − 937047.i − 1.32486i −0.749125 0.662429i \(-0.769526\pi\)
0.749125 0.662429i \(-0.230474\pi\)
\(30\) 0 0
\(31\) 826370. 0.894804 0.447402 0.894333i \(-0.352349\pi\)
0.447402 + 0.894333i \(0.352349\pi\)
\(32\) 185364.i 0.176777i
\(33\) 0 0
\(34\) 244224. 0.182756
\(35\) − 1.40177e6i − 0.934123i
\(36\) 0 0
\(37\) 1.34458e6 0.717428 0.358714 0.933448i \(-0.383215\pi\)
0.358714 + 0.933448i \(0.383215\pi\)
\(38\) − 2.56379e6i − 1.22955i
\(39\) 0 0
\(40\) 983040. 0.384000
\(41\) − 5.19191e6i − 1.83735i −0.395017 0.918674i \(-0.629261\pi\)
0.395017 0.918674i \(-0.370739\pi\)
\(42\) 0 0
\(43\) −6.14774e6 −1.79822 −0.899108 0.437727i \(-0.855784\pi\)
−0.899108 + 0.437727i \(0.855784\pi\)
\(44\) − 851515.i − 0.227186i
\(45\) 0 0
\(46\) 4.16717e6 0.930700
\(47\) 5.91078e6i 1.21130i 0.795729 + 0.605652i \(0.207088\pi\)
−0.795729 + 0.605652i \(0.792912\pi\)
\(48\) 0 0
\(49\) −1.50058e6 −0.260300
\(50\) − 793939.i − 0.127030i
\(51\) 0 0
\(52\) −1.03206e6 −0.141154
\(53\) − 768156.i − 0.0973522i −0.998815 0.0486761i \(-0.984500\pi\)
0.998815 0.0486761i \(-0.0155002\pi\)
\(54\) 0 0
\(55\) −4.51584e6 −0.493501
\(56\) 2.99044e6i 0.304077i
\(57\) 0 0
\(58\) 1.06015e7 0.936816
\(59\) 473954.i 0.0391136i 0.999809 + 0.0195568i \(0.00622552\pi\)
−0.999809 + 0.0195568i \(0.993774\pi\)
\(60\) 0 0
\(61\) −1.49857e7 −1.08232 −0.541162 0.840918i \(-0.682016\pi\)
−0.541162 + 0.840918i \(0.682016\pi\)
\(62\) 9.34931e6i 0.632722i
\(63\) 0 0
\(64\) −2.09715e6 −0.125000
\(65\) 5.47335e6i 0.306619i
\(66\) 0 0
\(67\) −1.00237e7 −0.497426 −0.248713 0.968577i \(-0.580008\pi\)
−0.248713 + 0.968577i \(0.580008\pi\)
\(68\) 2.76308e6i 0.129228i
\(69\) 0 0
\(70\) 1.58592e7 0.660525
\(71\) 4.54849e7i 1.78992i 0.446145 + 0.894961i \(0.352797\pi\)
−0.446145 + 0.894961i \(0.647203\pi\)
\(72\) 0 0
\(73\) −2.32616e7 −0.819120 −0.409560 0.912283i \(-0.634318\pi\)
−0.409560 + 0.912283i \(0.634318\pi\)
\(74\) 1.52121e7i 0.507298i
\(75\) 0 0
\(76\) 2.90060e7 0.869426
\(77\) − 1.37373e7i − 0.390786i
\(78\) 0 0
\(79\) 1.42672e7 0.366294 0.183147 0.983086i \(-0.441372\pi\)
0.183147 + 0.983086i \(0.441372\pi\)
\(80\) 1.11218e7i 0.271529i
\(81\) 0 0
\(82\) 5.87397e7 1.29920
\(83\) 3.61918e7i 0.762602i 0.924451 + 0.381301i \(0.124524\pi\)
−0.924451 + 0.381301i \(0.875476\pi\)
\(84\) 0 0
\(85\) 1.46534e7 0.280714
\(86\) − 6.95538e7i − 1.27153i
\(87\) 0 0
\(88\) 9.63379e6 0.160645
\(89\) 1.15088e8i 1.83429i 0.398549 + 0.917147i \(0.369514\pi\)
−0.398549 + 0.917147i \(0.630486\pi\)
\(90\) 0 0
\(91\) −1.66501e7 −0.242801
\(92\) 4.71461e7i 0.658104i
\(93\) 0 0
\(94\) −6.68728e7 −0.856522
\(95\) − 1.53827e8i − 1.88860i
\(96\) 0 0
\(97\) −4.05716e7 −0.458285 −0.229142 0.973393i \(-0.573592\pi\)
−0.229142 + 0.973393i \(0.573592\pi\)
\(98\) − 1.69771e7i − 0.184060i
\(99\) 0 0
\(100\) 8.98240e6 0.0898240
\(101\) − 6.49465e7i − 0.624123i −0.950062 0.312061i \(-0.898981\pi\)
0.950062 0.312061i \(-0.101019\pi\)
\(102\) 0 0
\(103\) −1.37263e8 −1.21956 −0.609782 0.792569i \(-0.708743\pi\)
−0.609782 + 0.792569i \(0.708743\pi\)
\(104\) − 1.16765e7i − 0.0998110i
\(105\) 0 0
\(106\) 8.69069e6 0.0688384
\(107\) − 1.39108e8i − 1.06125i −0.847607 0.530625i \(-0.821957\pi\)
0.847607 0.530625i \(-0.178043\pi\)
\(108\) 0 0
\(109\) −4.29417e7 −0.304210 −0.152105 0.988364i \(-0.548605\pi\)
−0.152105 + 0.988364i \(0.548605\pi\)
\(110\) − 5.10909e7i − 0.348958i
\(111\) 0 0
\(112\) −3.38330e7 −0.215015
\(113\) − 2.91649e8i − 1.78874i −0.447329 0.894369i \(-0.647625\pi\)
0.447329 0.894369i \(-0.352375\pi\)
\(114\) 0 0
\(115\) 2.50030e8 1.42956
\(116\) 1.19942e8i 0.662429i
\(117\) 0 0
\(118\) −5.36218e6 −0.0276575
\(119\) 4.45762e7i 0.222288i
\(120\) 0 0
\(121\) 1.70104e8 0.793546
\(122\) − 1.69544e8i − 0.765319i
\(123\) 0 0
\(124\) −1.05775e8 −0.447402
\(125\) 2.17529e8i 0.890997i
\(126\) 0 0
\(127\) −3.39515e8 −1.30510 −0.652551 0.757745i \(-0.726301\pi\)
−0.652551 + 0.757745i \(0.726301\pi\)
\(128\) − 2.37266e7i − 0.0883883i
\(129\) 0 0
\(130\) −6.19238e7 −0.216813
\(131\) − 2.24537e8i − 0.762436i −0.924485 0.381218i \(-0.875505\pi\)
0.924485 0.381218i \(-0.124495\pi\)
\(132\) 0 0
\(133\) 4.67948e8 1.49551
\(134\) − 1.13405e8i − 0.351733i
\(135\) 0 0
\(136\) −3.12607e7 −0.0913782
\(137\) 4.01263e8i 1.13906i 0.821970 + 0.569531i \(0.192875\pi\)
−0.821970 + 0.569531i \(0.807125\pi\)
\(138\) 0 0
\(139\) 2.69764e8 0.722645 0.361322 0.932441i \(-0.382325\pi\)
0.361322 + 0.932441i \(0.382325\pi\)
\(140\) 1.79426e8i 0.467062i
\(141\) 0 0
\(142\) −5.14603e8 −1.26567
\(143\) 5.36388e7i 0.128273i
\(144\) 0 0
\(145\) 6.36088e8 1.43895
\(146\) − 2.63175e8i − 0.579205i
\(147\) 0 0
\(148\) −1.72106e8 −0.358714
\(149\) − 1.94198e8i − 0.394003i −0.980403 0.197002i \(-0.936880\pi\)
0.980403 0.197002i \(-0.0631204\pi\)
\(150\) 0 0
\(151\) −8.75100e7 −0.168325 −0.0841627 0.996452i \(-0.526822\pi\)
−0.0841627 + 0.996452i \(0.526822\pi\)
\(152\) 3.28165e8i 0.614777i
\(153\) 0 0
\(154\) 1.55420e8 0.276328
\(155\) 5.60959e8i 0.971861i
\(156\) 0 0
\(157\) 2.84655e8 0.468512 0.234256 0.972175i \(-0.424735\pi\)
0.234256 + 0.972175i \(0.424735\pi\)
\(158\) 1.61415e8i 0.259009i
\(159\) 0 0
\(160\) −1.25829e8 −0.192000
\(161\) 7.60600e8i 1.13202i
\(162\) 0 0
\(163\) −2.63153e8 −0.372785 −0.186393 0.982475i \(-0.559680\pi\)
−0.186393 + 0.982475i \(0.559680\pi\)
\(164\) 6.64564e8i 0.918674i
\(165\) 0 0
\(166\) −4.09464e8 −0.539241
\(167\) 1.16384e9i 1.49633i 0.663511 + 0.748167i \(0.269066\pi\)
−0.663511 + 0.748167i \(0.730934\pi\)
\(168\) 0 0
\(169\) −7.50719e8 −0.920302
\(170\) 1.65785e8i 0.198495i
\(171\) 0 0
\(172\) 7.86911e8 0.899108
\(173\) − 1.32417e9i − 1.47829i −0.673547 0.739145i \(-0.735230\pi\)
0.673547 0.739145i \(-0.264770\pi\)
\(174\) 0 0
\(175\) 1.44911e8 0.154508
\(176\) 1.08994e8i 0.113593i
\(177\) 0 0
\(178\) −1.30207e9 −1.29704
\(179\) − 4.06696e8i − 0.396148i −0.980187 0.198074i \(-0.936531\pi\)
0.980187 0.198074i \(-0.0634686\pi\)
\(180\) 0 0
\(181\) −1.29071e9 −1.20258 −0.601289 0.799032i \(-0.705346\pi\)
−0.601289 + 0.799032i \(0.705346\pi\)
\(182\) − 1.88374e8i − 0.171687i
\(183\) 0 0
\(184\) −5.33398e8 −0.465350
\(185\) 9.12728e8i 0.779210i
\(186\) 0 0
\(187\) 1.43604e8 0.117435
\(188\) − 7.56580e8i − 0.605652i
\(189\) 0 0
\(190\) 1.74036e9 1.33544
\(191\) − 2.97714e8i − 0.223700i −0.993725 0.111850i \(-0.964322\pi\)
0.993725 0.111850i \(-0.0356776\pi\)
\(192\) 0 0
\(193\) 2.22004e9 1.60004 0.800021 0.599972i \(-0.204822\pi\)
0.800021 + 0.599972i \(0.204822\pi\)
\(194\) − 4.59015e8i − 0.324056i
\(195\) 0 0
\(196\) 1.92074e8 0.130150
\(197\) − 1.91580e9i − 1.27199i −0.771693 0.635996i \(-0.780590\pi\)
0.771693 0.635996i \(-0.219410\pi\)
\(198\) 0 0
\(199\) −1.75472e9 −1.11891 −0.559457 0.828859i \(-0.688990\pi\)
−0.559457 + 0.828859i \(0.688990\pi\)
\(200\) 1.01624e8i 0.0635152i
\(201\) 0 0
\(202\) 7.34786e8 0.441322
\(203\) 1.93500e9i 1.13945i
\(204\) 0 0
\(205\) 3.52438e9 1.99557
\(206\) − 1.55295e9i − 0.862362i
\(207\) 0 0
\(208\) 1.32104e8 0.0705770
\(209\) − 1.50751e9i − 0.790086i
\(210\) 0 0
\(211\) −2.15389e9 −1.08666 −0.543331 0.839519i \(-0.682837\pi\)
−0.543331 + 0.839519i \(0.682837\pi\)
\(212\) 9.83239e7i 0.0486761i
\(213\) 0 0
\(214\) 1.57383e9 0.750417
\(215\) − 4.17323e9i − 1.95307i
\(216\) 0 0
\(217\) −1.70645e9 −0.769583
\(218\) − 4.85829e8i − 0.215109i
\(219\) 0 0
\(220\) 5.78028e8 0.246750
\(221\) − 1.74052e8i − 0.0729644i
\(222\) 0 0
\(223\) −2.39374e9 −0.967959 −0.483980 0.875079i \(-0.660809\pi\)
−0.483980 + 0.875079i \(0.660809\pi\)
\(224\) − 3.82776e8i − 0.152038i
\(225\) 0 0
\(226\) 3.29963e9 1.26483
\(227\) − 9.92657e8i − 0.373849i −0.982374 0.186924i \(-0.940148\pi\)
0.982374 0.186924i \(-0.0598519\pi\)
\(228\) 0 0
\(229\) −2.22150e9 −0.807800 −0.403900 0.914803i \(-0.632346\pi\)
−0.403900 + 0.914803i \(0.632346\pi\)
\(230\) 2.82877e9i 1.01085i
\(231\) 0 0
\(232\) −1.35699e9 −0.468408
\(233\) 1.24042e9i 0.420867i 0.977608 + 0.210433i \(0.0674875\pi\)
−0.977608 + 0.210433i \(0.932512\pi\)
\(234\) 0 0
\(235\) −4.01237e9 −1.31562
\(236\) − 6.06661e7i − 0.0195568i
\(237\) 0 0
\(238\) −5.04323e8 −0.157181
\(239\) − 7.59290e8i − 0.232711i −0.993208 0.116355i \(-0.962879\pi\)
0.993208 0.116355i \(-0.0371211\pi\)
\(240\) 0 0
\(241\) 4.47467e9 1.32646 0.663228 0.748417i \(-0.269186\pi\)
0.663228 + 0.748417i \(0.269186\pi\)
\(242\) 1.92450e9i 0.561122i
\(243\) 0 0
\(244\) 1.91817e9 0.541162
\(245\) − 1.01862e9i − 0.282716i
\(246\) 0 0
\(247\) −1.82715e9 −0.490892
\(248\) − 1.19671e9i − 0.316361i
\(249\) 0 0
\(250\) −2.46106e9 −0.630030
\(251\) − 1.07356e9i − 0.270477i −0.990813 0.135239i \(-0.956820\pi\)
0.990813 0.135239i \(-0.0431801\pi\)
\(252\) 0 0
\(253\) 2.45029e9 0.598048
\(254\) − 3.84118e9i − 0.922846i
\(255\) 0 0
\(256\) 2.68435e8 0.0625000
\(257\) 7.58666e9i 1.73907i 0.493867 + 0.869537i \(0.335583\pi\)
−0.493867 + 0.869537i \(0.664417\pi\)
\(258\) 0 0
\(259\) −2.77655e9 −0.617030
\(260\) − 7.00588e8i − 0.153310i
\(261\) 0 0
\(262\) 2.54035e9 0.539124
\(263\) 2.81146e9i 0.587637i 0.955861 + 0.293819i \(0.0949262\pi\)
−0.955861 + 0.293819i \(0.905074\pi\)
\(264\) 0 0
\(265\) 5.21441e8 0.105736
\(266\) 5.29422e9i 1.05749i
\(267\) 0 0
\(268\) 1.28303e9 0.248713
\(269\) 2.96285e9i 0.565849i 0.959142 + 0.282924i \(0.0913045\pi\)
−0.959142 + 0.282924i \(0.908695\pi\)
\(270\) 0 0
\(271\) −8.40415e9 −1.55818 −0.779089 0.626914i \(-0.784318\pi\)
−0.779089 + 0.626914i \(0.784318\pi\)
\(272\) − 3.53674e8i − 0.0646142i
\(273\) 0 0
\(274\) −4.53978e9 −0.805438
\(275\) − 4.66836e8i − 0.0816270i
\(276\) 0 0
\(277\) 5.98162e9 1.01601 0.508007 0.861353i \(-0.330382\pi\)
0.508007 + 0.861353i \(0.330382\pi\)
\(278\) 3.05203e9i 0.510987i
\(279\) 0 0
\(280\) −2.02998e9 −0.330262
\(281\) 1.08341e10i 1.73767i 0.495105 + 0.868833i \(0.335130\pi\)
−0.495105 + 0.868833i \(0.664870\pi\)
\(282\) 0 0
\(283\) 3.78670e9 0.590357 0.295179 0.955442i \(-0.404621\pi\)
0.295179 + 0.955442i \(0.404621\pi\)
\(284\) − 5.82207e9i − 0.894961i
\(285\) 0 0
\(286\) −6.06854e8 −0.0907026
\(287\) 1.07213e10i 1.58023i
\(288\) 0 0
\(289\) 6.50978e9 0.933200
\(290\) 7.19652e9i 1.01749i
\(291\) 0 0
\(292\) 2.97748e9 0.409560
\(293\) − 7.32039e9i − 0.993262i −0.867962 0.496631i \(-0.834570\pi\)
0.867962 0.496631i \(-0.165430\pi\)
\(294\) 0 0
\(295\) −3.21731e8 −0.0424819
\(296\) − 1.94715e9i − 0.253649i
\(297\) 0 0
\(298\) 2.19710e9 0.278602
\(299\) − 2.96984e9i − 0.371576i
\(300\) 0 0
\(301\) 1.26951e10 1.54657
\(302\) − 9.90063e8i − 0.119024i
\(303\) 0 0
\(304\) −3.71276e9 −0.434713
\(305\) − 1.01726e10i − 1.17553i
\(306\) 0 0
\(307\) −2.21986e8 −0.0249903 −0.0124951 0.999922i \(-0.503977\pi\)
−0.0124951 + 0.999922i \(0.503977\pi\)
\(308\) 1.75838e9i 0.195393i
\(309\) 0 0
\(310\) −6.34652e9 −0.687209
\(311\) − 7.19555e9i − 0.769171i −0.923089 0.384585i \(-0.874344\pi\)
0.923089 0.384585i \(-0.125656\pi\)
\(312\) 0 0
\(313\) 4.23341e9 0.441076 0.220538 0.975378i \(-0.429219\pi\)
0.220538 + 0.975378i \(0.429219\pi\)
\(314\) 3.22051e9i 0.331288i
\(315\) 0 0
\(316\) −1.82620e9 −0.183147
\(317\) − 5.40044e9i − 0.534801i −0.963586 0.267400i \(-0.913835\pi\)
0.963586 0.267400i \(-0.0861646\pi\)
\(318\) 0 0
\(319\) 6.23367e9 0.601978
\(320\) − 1.42359e9i − 0.135765i
\(321\) 0 0
\(322\) −8.60520e9 −0.800456
\(323\) 4.89171e9i 0.449418i
\(324\) 0 0
\(325\) −5.65821e8 −0.0507161
\(326\) − 2.97724e9i − 0.263599i
\(327\) 0 0
\(328\) −7.51868e9 −0.649601
\(329\) − 1.22058e10i − 1.04179i
\(330\) 0 0
\(331\) −1.35941e10 −1.13250 −0.566252 0.824232i \(-0.691607\pi\)
−0.566252 + 0.824232i \(0.691607\pi\)
\(332\) − 4.63255e9i − 0.381301i
\(333\) 0 0
\(334\) −1.31674e10 −1.05807
\(335\) − 6.80431e9i − 0.540263i
\(336\) 0 0
\(337\) −2.35433e9 −0.182536 −0.0912680 0.995826i \(-0.529092\pi\)
−0.0912680 + 0.995826i \(0.529092\pi\)
\(338\) − 8.49341e9i − 0.650752i
\(339\) 0 0
\(340\) −1.87564e9 −0.140357
\(341\) 5.49739e9i 0.406574i
\(342\) 0 0
\(343\) 1.50030e10 1.08393
\(344\) 8.90288e9i 0.635765i
\(345\) 0 0
\(346\) 1.49813e10 1.04531
\(347\) − 8.33815e9i − 0.575111i −0.957764 0.287555i \(-0.907157\pi\)
0.957764 0.287555i \(-0.0928426\pi\)
\(348\) 0 0
\(349\) −1.26463e10 −0.852434 −0.426217 0.904621i \(-0.640154\pi\)
−0.426217 + 0.904621i \(0.640154\pi\)
\(350\) 1.63949e9i 0.109253i
\(351\) 0 0
\(352\) −1.23313e9 −0.0803224
\(353\) − 3.76139e9i − 0.242242i −0.992638 0.121121i \(-0.961351\pi\)
0.992638 0.121121i \(-0.0386490\pi\)
\(354\) 0 0
\(355\) −3.08762e10 −1.94406
\(356\) − 1.47312e10i − 0.917147i
\(357\) 0 0
\(358\) 4.60124e9 0.280119
\(359\) 1.64726e10i 0.991707i 0.868406 + 0.495853i \(0.165145\pi\)
−0.868406 + 0.495853i \(0.834855\pi\)
\(360\) 0 0
\(361\) 3.43681e10 2.02361
\(362\) − 1.46027e10i − 0.850351i
\(363\) 0 0
\(364\) 2.13121e9 0.121401
\(365\) − 1.57905e10i − 0.889659i
\(366\) 0 0
\(367\) 7.38723e9 0.407209 0.203605 0.979053i \(-0.434734\pi\)
0.203605 + 0.979053i \(0.434734\pi\)
\(368\) − 6.03470e9i − 0.329052i
\(369\) 0 0
\(370\) −1.03263e10 −0.550984
\(371\) 1.58624e9i 0.0837286i
\(372\) 0 0
\(373\) 2.30025e10 1.18834 0.594168 0.804341i \(-0.297482\pi\)
0.594168 + 0.804341i \(0.297482\pi\)
\(374\) 1.62469e9i 0.0830394i
\(375\) 0 0
\(376\) 8.55972e9 0.428261
\(377\) − 7.55541e9i − 0.374018i
\(378\) 0 0
\(379\) −2.05891e10 −0.997883 −0.498942 0.866636i \(-0.666278\pi\)
−0.498942 + 0.866636i \(0.666278\pi\)
\(380\) 1.96899e10i 0.944298i
\(381\) 0 0
\(382\) 3.36825e9 0.158180
\(383\) − 1.43645e10i − 0.667569i −0.942649 0.333785i \(-0.891674\pi\)
0.942649 0.333785i \(-0.108326\pi\)
\(384\) 0 0
\(385\) 9.32521e9 0.424439
\(386\) 2.51169e10i 1.13140i
\(387\) 0 0
\(388\) 5.19317e9 0.229142
\(389\) 3.33296e10i 1.45557i 0.685808 + 0.727783i \(0.259449\pi\)
−0.685808 + 0.727783i \(0.740551\pi\)
\(390\) 0 0
\(391\) −7.95096e9 −0.340183
\(392\) 2.17307e9i 0.0920298i
\(393\) 0 0
\(394\) 2.16747e10 0.899434
\(395\) 9.68488e9i 0.397838i
\(396\) 0 0
\(397\) −4.32631e9 −0.174163 −0.0870814 0.996201i \(-0.527754\pi\)
−0.0870814 + 0.996201i \(0.527754\pi\)
\(398\) − 1.98524e10i − 0.791192i
\(399\) 0 0
\(400\) −1.14975e9 −0.0449120
\(401\) 1.20770e10i 0.467069i 0.972348 + 0.233535i \(0.0750293\pi\)
−0.972348 + 0.233535i \(0.924971\pi\)
\(402\) 0 0
\(403\) 6.66302e9 0.252610
\(404\) 8.31315e9i 0.312061i
\(405\) 0 0
\(406\) −2.18920e10 −0.805716
\(407\) 8.94473e9i 0.325979i
\(408\) 0 0
\(409\) −3.81221e10 −1.36234 −0.681168 0.732127i \(-0.738528\pi\)
−0.681168 + 0.732127i \(0.738528\pi\)
\(410\) 3.98738e10i 1.41108i
\(411\) 0 0
\(412\) 1.75697e10 0.609782
\(413\) − 9.78715e8i − 0.0336400i
\(414\) 0 0
\(415\) −2.45678e10 −0.828275
\(416\) 1.49459e9i 0.0499055i
\(417\) 0 0
\(418\) 1.70555e10 0.558675
\(419\) 2.52616e10i 0.819605i 0.912174 + 0.409803i \(0.134402\pi\)
−0.912174 + 0.409803i \(0.865598\pi\)
\(420\) 0 0
\(421\) −3.33965e9 −0.106310 −0.0531548 0.998586i \(-0.516928\pi\)
−0.0531548 + 0.998586i \(0.516928\pi\)
\(422\) − 2.43685e10i − 0.768386i
\(423\) 0 0
\(424\) −1.11241e9 −0.0344192
\(425\) 1.51484e9i 0.0464312i
\(426\) 0 0
\(427\) 3.09455e10 0.930862
\(428\) 1.78059e10i 0.530625i
\(429\) 0 0
\(430\) 4.72147e10 1.38103
\(431\) − 5.55244e9i − 0.160907i −0.996758 0.0804534i \(-0.974363\pi\)
0.996758 0.0804534i \(-0.0256368\pi\)
\(432\) 0 0
\(433\) 1.14713e10 0.326334 0.163167 0.986598i \(-0.447829\pi\)
0.163167 + 0.986598i \(0.447829\pi\)
\(434\) − 1.93063e10i − 0.544178i
\(435\) 0 0
\(436\) 5.49653e9 0.152105
\(437\) 8.34667e10i 2.28869i
\(438\) 0 0
\(439\) −5.98486e10 −1.61137 −0.805686 0.592343i \(-0.798203\pi\)
−0.805686 + 0.592343i \(0.798203\pi\)
\(440\) 6.53963e9i 0.174479i
\(441\) 0 0
\(442\) 1.96918e9 0.0515936
\(443\) − 3.61609e9i − 0.0938910i −0.998897 0.0469455i \(-0.985051\pi\)
0.998897 0.0469455i \(-0.0149487\pi\)
\(444\) 0 0
\(445\) −7.81241e10 −1.99226
\(446\) − 2.70821e10i − 0.684451i
\(447\) 0 0
\(448\) 4.33062e9 0.107507
\(449\) 2.39980e9i 0.0590459i 0.999564 + 0.0295230i \(0.00939882\pi\)
−0.999564 + 0.0295230i \(0.990601\pi\)
\(450\) 0 0
\(451\) 3.45390e10 0.834839
\(452\) 3.73311e10i 0.894369i
\(453\) 0 0
\(454\) 1.12306e10 0.264351
\(455\) − 1.13025e10i − 0.263710i
\(456\) 0 0
\(457\) −4.11731e10 −0.943948 −0.471974 0.881612i \(-0.656458\pi\)
−0.471974 + 0.881612i \(0.656458\pi\)
\(458\) − 2.51334e10i − 0.571201i
\(459\) 0 0
\(460\) −3.20039e10 −0.714778
\(461\) − 2.07400e10i − 0.459203i −0.973285 0.229601i \(-0.926258\pi\)
0.973285 0.229601i \(-0.0737422\pi\)
\(462\) 0 0
\(463\) −7.35380e9 −0.160025 −0.0800125 0.996794i \(-0.525496\pi\)
−0.0800125 + 0.996794i \(0.525496\pi\)
\(464\) − 1.53526e10i − 0.331214i
\(465\) 0 0
\(466\) −1.40337e10 −0.297598
\(467\) − 9.09281e10i − 1.91175i −0.293776 0.955874i \(-0.594912\pi\)
0.293776 0.955874i \(-0.405088\pi\)
\(468\) 0 0
\(469\) 2.06989e10 0.427816
\(470\) − 4.53948e10i − 0.930282i
\(471\) 0 0
\(472\) 6.86359e8 0.0138288
\(473\) − 4.08976e10i − 0.817059i
\(474\) 0 0
\(475\) 1.59023e10 0.312381
\(476\) − 5.70576e9i − 0.111144i
\(477\) 0 0
\(478\) 8.59039e9 0.164551
\(479\) − 1.91197e10i − 0.363194i −0.983373 0.181597i \(-0.941873\pi\)
0.983373 0.181597i \(-0.0581266\pi\)
\(480\) 0 0
\(481\) 1.08413e10 0.202536
\(482\) 5.06251e10i 0.937946i
\(483\) 0 0
\(484\) −2.17733e10 −0.396773
\(485\) − 2.75409e10i − 0.497750i
\(486\) 0 0
\(487\) 5.28737e10 0.939992 0.469996 0.882669i \(-0.344255\pi\)
0.469996 + 0.882669i \(0.344255\pi\)
\(488\) 2.17016e10i 0.382660i
\(489\) 0 0
\(490\) 1.15244e10 0.199910
\(491\) 9.78032e10i 1.68278i 0.540429 + 0.841389i \(0.318262\pi\)
−0.540429 + 0.841389i \(0.681738\pi\)
\(492\) 0 0
\(493\) −2.02276e10 −0.342418
\(494\) − 2.06718e10i − 0.347113i
\(495\) 0 0
\(496\) 1.35392e10 0.223701
\(497\) − 9.39263e10i − 1.53944i
\(498\) 0 0
\(499\) 8.89351e10 1.43440 0.717201 0.696866i \(-0.245423\pi\)
0.717201 + 0.696866i \(0.245423\pi\)
\(500\) − 2.78437e10i − 0.445499i
\(501\) 0 0
\(502\) 1.21459e10 0.191256
\(503\) − 2.97791e10i − 0.465200i −0.972572 0.232600i \(-0.925277\pi\)
0.972572 0.232600i \(-0.0747233\pi\)
\(504\) 0 0
\(505\) 4.40871e10 0.677870
\(506\) 2.77219e10i 0.422884i
\(507\) 0 0
\(508\) 4.34579e10 0.652551
\(509\) 2.67186e10i 0.398054i 0.979994 + 0.199027i \(0.0637781\pi\)
−0.979994 + 0.199027i \(0.936222\pi\)
\(510\) 0 0
\(511\) 4.80351e10 0.704491
\(512\) 3.03700e9i 0.0441942i
\(513\) 0 0
\(514\) −8.58333e10 −1.22971
\(515\) − 9.31772e10i − 1.32459i
\(516\) 0 0
\(517\) −3.93212e10 −0.550383
\(518\) − 3.14130e10i − 0.436306i
\(519\) 0 0
\(520\) 7.92625e9 0.108406
\(521\) 1.06779e10i 0.144922i 0.997371 + 0.0724610i \(0.0230853\pi\)
−0.997371 + 0.0724610i \(0.976915\pi\)
\(522\) 0 0
\(523\) 4.63928e10 0.620075 0.310037 0.950724i \(-0.399658\pi\)
0.310037 + 0.950724i \(0.399658\pi\)
\(524\) 2.87408e10i 0.381218i
\(525\) 0 0
\(526\) −3.18081e10 −0.415522
\(527\) − 1.78385e10i − 0.231268i
\(528\) 0 0
\(529\) −5.73553e10 −0.732405
\(530\) 5.89943e9i 0.0747665i
\(531\) 0 0
\(532\) −5.98973e10 −0.747757
\(533\) − 4.18623e10i − 0.518698i
\(534\) 0 0
\(535\) 9.44298e10 1.15264
\(536\) 1.45159e10i 0.175867i
\(537\) 0 0
\(538\) −3.35208e10 −0.400115
\(539\) − 9.98252e9i − 0.118273i
\(540\) 0 0
\(541\) −1.22420e11 −1.42911 −0.714553 0.699581i \(-0.753370\pi\)
−0.714553 + 0.699581i \(0.753370\pi\)
\(542\) − 9.50822e10i − 1.10180i
\(543\) 0 0
\(544\) 4.00137e9 0.0456891
\(545\) − 2.91498e10i − 0.330407i
\(546\) 0 0
\(547\) −5.53975e10 −0.618786 −0.309393 0.950934i \(-0.600126\pi\)
−0.309393 + 0.950934i \(0.600126\pi\)
\(548\) − 5.13617e10i − 0.569531i
\(549\) 0 0
\(550\) 5.28165e9 0.0577190
\(551\) 2.12343e11i 2.30373i
\(552\) 0 0
\(553\) −2.94617e10 −0.315034
\(554\) 6.76743e10i 0.718431i
\(555\) 0 0
\(556\) −3.45298e10 −0.361322
\(557\) 1.17293e11i 1.21857i 0.792951 + 0.609285i \(0.208544\pi\)
−0.792951 + 0.609285i \(0.791456\pi\)
\(558\) 0 0
\(559\) −4.95692e10 −0.507651
\(560\) − 2.29666e10i − 0.233531i
\(561\) 0 0
\(562\) −1.22574e11 −1.22872
\(563\) 1.03752e11i 1.03267i 0.856385 + 0.516337i \(0.172705\pi\)
−0.856385 + 0.516337i \(0.827295\pi\)
\(564\) 0 0
\(565\) 1.97978e11 1.94278
\(566\) 4.28416e10i 0.417445i
\(567\) 0 0
\(568\) 6.58692e10 0.632833
\(569\) 2.67515e10i 0.255211i 0.991825 + 0.127605i \(0.0407291\pi\)
−0.991825 + 0.127605i \(0.959271\pi\)
\(570\) 0 0
\(571\) 1.14161e11 1.07393 0.536963 0.843606i \(-0.319572\pi\)
0.536963 + 0.843606i \(0.319572\pi\)
\(572\) − 6.86577e9i − 0.0641364i
\(573\) 0 0
\(574\) −1.21298e11 −1.11739
\(575\) 2.58475e10i 0.236454i
\(576\) 0 0
\(577\) 1.43414e11 1.29386 0.646930 0.762549i \(-0.276052\pi\)
0.646930 + 0.762549i \(0.276052\pi\)
\(578\) 7.36497e10i 0.659872i
\(579\) 0 0
\(580\) −8.14193e10 −0.719475
\(581\) − 7.47361e10i − 0.655883i
\(582\) 0 0
\(583\) 5.11012e9 0.0442341
\(584\) 3.36864e10i 0.289603i
\(585\) 0 0
\(586\) 8.28207e10 0.702342
\(587\) 1.93256e11i 1.62773i 0.581057 + 0.813863i \(0.302639\pi\)
−0.581057 + 0.813863i \(0.697361\pi\)
\(588\) 0 0
\(589\) −1.87263e11 −1.55593
\(590\) − 3.63997e9i − 0.0300393i
\(591\) 0 0
\(592\) 2.20295e10 0.179357
\(593\) − 2.21385e10i − 0.179031i −0.995985 0.0895156i \(-0.971468\pi\)
0.995985 0.0895156i \(-0.0285319\pi\)
\(594\) 0 0
\(595\) −3.02594e10 −0.241430
\(596\) 2.48574e10i 0.197002i
\(597\) 0 0
\(598\) 3.35999e10 0.262744
\(599\) − 1.32315e11i − 1.02778i −0.857856 0.513891i \(-0.828204\pi\)
0.857856 0.513891i \(-0.171796\pi\)
\(600\) 0 0
\(601\) −2.29562e10 −0.175955 −0.0879775 0.996122i \(-0.528040\pi\)
−0.0879775 + 0.996122i \(0.528040\pi\)
\(602\) 1.43629e11i 1.09359i
\(603\) 0 0
\(604\) 1.12013e10 0.0841627
\(605\) 1.15470e11i 0.861883i
\(606\) 0 0
\(607\) 1.56978e11 1.15634 0.578169 0.815917i \(-0.303767\pi\)
0.578169 + 0.815917i \(0.303767\pi\)
\(608\) − 4.20051e10i − 0.307389i
\(609\) 0 0
\(610\) 1.15090e11 0.831225
\(611\) 4.76586e10i 0.341961i
\(612\) 0 0
\(613\) −1.72931e11 −1.22470 −0.612352 0.790586i \(-0.709776\pi\)
−0.612352 + 0.790586i \(0.709776\pi\)
\(614\) − 2.51148e9i − 0.0176708i
\(615\) 0 0
\(616\) −1.98938e10 −0.138164
\(617\) 1.42425e10i 0.0982758i 0.998792 + 0.0491379i \(0.0156474\pi\)
−0.998792 + 0.0491379i \(0.984353\pi\)
\(618\) 0 0
\(619\) 1.29983e11 0.885367 0.442684 0.896678i \(-0.354027\pi\)
0.442684 + 0.896678i \(0.354027\pi\)
\(620\) − 7.18027e10i − 0.485930i
\(621\) 0 0
\(622\) 8.14084e10 0.543886
\(623\) − 2.37656e11i − 1.57760i
\(624\) 0 0
\(625\) −1.75075e11 −1.14737
\(626\) 4.78956e10i 0.311888i
\(627\) 0 0
\(628\) −3.64359e10 −0.234256
\(629\) − 2.90247e10i − 0.185424i
\(630\) 0 0
\(631\) −5.79110e9 −0.0365295 −0.0182648 0.999833i \(-0.505814\pi\)
−0.0182648 + 0.999833i \(0.505814\pi\)
\(632\) − 2.06611e10i − 0.129505i
\(633\) 0 0
\(634\) 6.10990e10 0.378161
\(635\) − 2.30471e11i − 1.41749i
\(636\) 0 0
\(637\) −1.20991e10 −0.0734847
\(638\) 7.05259e10i 0.425663i
\(639\) 0 0
\(640\) 1.61061e10 0.0960000
\(641\) 2.18083e11i 1.29178i 0.763429 + 0.645892i \(0.223514\pi\)
−0.763429 + 0.645892i \(0.776486\pi\)
\(642\) 0 0
\(643\) 1.74278e11 1.01953 0.509764 0.860314i \(-0.329733\pi\)
0.509764 + 0.860314i \(0.329733\pi\)
\(644\) − 9.73567e10i − 0.566008i
\(645\) 0 0
\(646\) −5.53434e10 −0.317786
\(647\) 2.22876e11i 1.27188i 0.771738 + 0.635941i \(0.219388\pi\)
−0.771738 + 0.635941i \(0.780612\pi\)
\(648\) 0 0
\(649\) −3.15296e9 −0.0177721
\(650\) − 6.40153e9i − 0.0358617i
\(651\) 0 0
\(652\) 3.36836e10 0.186393
\(653\) − 2.56934e11i − 1.41308i −0.707671 0.706542i \(-0.750254\pi\)
0.707671 0.706542i \(-0.249746\pi\)
\(654\) 0 0
\(655\) 1.52421e11 0.828094
\(656\) − 8.50642e10i − 0.459337i
\(657\) 0 0
\(658\) 1.38092e11 0.736658
\(659\) − 2.69559e11i − 1.42926i −0.699500 0.714632i \(-0.746594\pi\)
0.699500 0.714632i \(-0.253406\pi\)
\(660\) 0 0
\(661\) −2.56807e11 −1.34525 −0.672623 0.739986i \(-0.734832\pi\)
−0.672623 + 0.739986i \(0.734832\pi\)
\(662\) − 1.53800e11i − 0.800802i
\(663\) 0 0
\(664\) 5.24114e10 0.269621
\(665\) 3.17653e11i 1.62430i
\(666\) 0 0
\(667\) −3.45142e11 −1.74379
\(668\) − 1.48972e11i − 0.748167i
\(669\) 0 0
\(670\) 7.69820e10 0.382023
\(671\) − 9.96918e10i − 0.491778i
\(672\) 0 0
\(673\) 1.67895e11 0.818421 0.409210 0.912440i \(-0.365804\pi\)
0.409210 + 0.912440i \(0.365804\pi\)
\(674\) − 2.66363e10i − 0.129072i
\(675\) 0 0
\(676\) 9.60920e10 0.460151
\(677\) − 1.70201e11i − 0.810230i −0.914266 0.405115i \(-0.867231\pi\)
0.914266 0.405115i \(-0.132769\pi\)
\(678\) 0 0
\(679\) 8.37804e10 0.394152
\(680\) − 2.12204e10i − 0.0992473i
\(681\) 0 0
\(682\) −6.21959e10 −0.287491
\(683\) − 3.04016e11i − 1.39706i −0.715582 0.698529i \(-0.753838\pi\)
0.715582 0.698529i \(-0.246162\pi\)
\(684\) 0 0
\(685\) −2.72387e11 −1.23715
\(686\) 1.69740e11i 0.766455i
\(687\) 0 0
\(688\) −1.00725e11 −0.449554
\(689\) − 6.19364e9i − 0.0274833i
\(690\) 0 0
\(691\) −4.17969e10 −0.183329 −0.0916646 0.995790i \(-0.529219\pi\)
−0.0916646 + 0.995790i \(0.529219\pi\)
\(692\) 1.69494e11i 0.739145i
\(693\) 0 0
\(694\) 9.43354e10 0.406665
\(695\) 1.83122e11i 0.784876i
\(696\) 0 0
\(697\) −1.12075e11 −0.474875
\(698\) − 1.43076e11i − 0.602762i
\(699\) 0 0
\(700\) −1.85487e10 −0.0772539
\(701\) − 2.07821e11i − 0.860631i −0.902678 0.430316i \(-0.858402\pi\)
0.902678 0.430316i \(-0.141598\pi\)
\(702\) 0 0
\(703\) −3.04693e11 −1.24750
\(704\) − 1.39512e10i − 0.0567965i
\(705\) 0 0
\(706\) 4.25553e10 0.171291
\(707\) 1.34114e11i 0.536782i
\(708\) 0 0
\(709\) −3.47189e11 −1.37398 −0.686990 0.726667i \(-0.741068\pi\)
−0.686990 + 0.726667i \(0.741068\pi\)
\(710\) − 3.49324e11i − 1.37466i
\(711\) 0 0
\(712\) 1.66665e11 0.648521
\(713\) − 3.04376e11i − 1.17775i
\(714\) 0 0
\(715\) −3.64112e10 −0.139319
\(716\) 5.20570e10i 0.198074i
\(717\) 0 0
\(718\) −1.86366e11 −0.701242
\(719\) 1.56177e11i 0.584387i 0.956359 + 0.292194i \(0.0943851\pi\)
−0.956359 + 0.292194i \(0.905615\pi\)
\(720\) 0 0
\(721\) 2.83448e11 1.04890
\(722\) 3.88830e11i 1.43091i
\(723\) 0 0
\(724\) 1.65210e11 0.601289
\(725\) 6.57572e10i 0.238008i
\(726\) 0 0
\(727\) −2.70116e11 −0.966969 −0.483485 0.875353i \(-0.660629\pi\)
−0.483485 + 0.875353i \(0.660629\pi\)
\(728\) 2.41119e10i 0.0858433i
\(729\) 0 0
\(730\) 1.78649e11 0.629084
\(731\) 1.32709e11i 0.464761i
\(732\) 0 0
\(733\) 1.53459e11 0.531589 0.265795 0.964030i \(-0.414366\pi\)
0.265795 + 0.964030i \(0.414366\pi\)
\(734\) 8.35770e10i 0.287940i
\(735\) 0 0
\(736\) 6.82749e10 0.232675
\(737\) − 6.66822e10i − 0.226017i
\(738\) 0 0
\(739\) 7.60266e10 0.254911 0.127455 0.991844i \(-0.459319\pi\)
0.127455 + 0.991844i \(0.459319\pi\)
\(740\) − 1.16829e11i − 0.389605i
\(741\) 0 0
\(742\) −1.79463e10 −0.0592050
\(743\) − 4.18589e11i − 1.37351i −0.726888 0.686756i \(-0.759034\pi\)
0.726888 0.686756i \(-0.240966\pi\)
\(744\) 0 0
\(745\) 1.31826e11 0.427933
\(746\) 2.60243e11i 0.840280i
\(747\) 0 0
\(748\) −1.83813e10 −0.0587177
\(749\) 2.87259e11i 0.912737i
\(750\) 0 0
\(751\) 3.73788e11 1.17507 0.587537 0.809197i \(-0.300098\pi\)
0.587537 + 0.809197i \(0.300098\pi\)
\(752\) 9.68422e10i 0.302826i
\(753\) 0 0
\(754\) 8.54797e10 0.264471
\(755\) − 5.94038e10i − 0.182821i
\(756\) 0 0
\(757\) 4.74806e11 1.44588 0.722941 0.690910i \(-0.242790\pi\)
0.722941 + 0.690910i \(0.242790\pi\)
\(758\) − 2.32939e11i − 0.705610i
\(759\) 0 0
\(760\) −2.22766e11 −0.667719
\(761\) 5.91673e11i 1.76418i 0.471079 + 0.882091i \(0.343865\pi\)
−0.471079 + 0.882091i \(0.656135\pi\)
\(762\) 0 0
\(763\) 8.86745e10 0.261638
\(764\) 3.81074e10i 0.111850i
\(765\) 0 0
\(766\) 1.62516e11 0.472043
\(767\) 3.82149e9i 0.0110421i
\(768\) 0 0
\(769\) 4.29184e11 1.22727 0.613633 0.789592i \(-0.289708\pi\)
0.613633 + 0.789592i \(0.289708\pi\)
\(770\) 1.05503e11i 0.300124i
\(771\) 0 0
\(772\) −2.84165e11 −0.800021
\(773\) − 1.95537e11i − 0.547660i −0.961778 0.273830i \(-0.911709\pi\)
0.961778 0.273830i \(-0.0882906\pi\)
\(774\) 0 0
\(775\) −5.79905e10 −0.160750
\(776\) 5.87540e10i 0.162028i
\(777\) 0 0
\(778\) −3.77082e11 −1.02924
\(779\) 1.17653e12i 3.19488i
\(780\) 0 0
\(781\) −3.02587e11 −0.813290
\(782\) − 8.99548e10i − 0.240546i
\(783\) 0 0
\(784\) −2.45854e10 −0.0650749
\(785\) 1.93230e11i 0.508858i
\(786\) 0 0
\(787\) −1.30204e10 −0.0339409 −0.0169705 0.999856i \(-0.505402\pi\)
−0.0169705 + 0.999856i \(0.505402\pi\)
\(788\) 2.45222e11i 0.635996i
\(789\) 0 0
\(790\) −1.09572e11 −0.281314
\(791\) 6.02256e11i 1.53842i
\(792\) 0 0
\(793\) −1.20830e11 −0.305549
\(794\) − 4.89466e10i − 0.123152i
\(795\) 0 0
\(796\) 2.24605e11 0.559457
\(797\) 1.33234e11i 0.330204i 0.986277 + 0.165102i \(0.0527953\pi\)
−0.986277 + 0.165102i \(0.947205\pi\)
\(798\) 0 0
\(799\) 1.27593e11 0.313070
\(800\) − 1.30079e10i − 0.0317576i
\(801\) 0 0
\(802\) −1.36636e11 −0.330268
\(803\) − 1.54747e11i − 0.372185i
\(804\) 0 0
\(805\) −5.16312e11 −1.22950
\(806\) 7.53835e10i 0.178622i
\(807\) 0 0
\(808\) −9.40525e10 −0.220661
\(809\) − 7.75707e11i − 1.81094i −0.424413 0.905469i \(-0.639519\pi\)
0.424413 0.905469i \(-0.360481\pi\)
\(810\) 0 0
\(811\) −7.76807e11 −1.79568 −0.897842 0.440318i \(-0.854866\pi\)
−0.897842 + 0.440318i \(0.854866\pi\)
\(812\) − 2.47680e11i − 0.569727i
\(813\) 0 0
\(814\) −1.01198e11 −0.230502
\(815\) − 1.78634e11i − 0.404888i
\(816\) 0 0
\(817\) 1.39313e12 3.12683
\(818\) − 4.31303e11i − 0.963317i
\(819\) 0 0
\(820\) −4.51121e11 −0.997786
\(821\) − 2.75649e10i − 0.0606713i −0.999540 0.0303356i \(-0.990342\pi\)
0.999540 0.0303356i \(-0.00965761\pi\)
\(822\) 0 0
\(823\) 2.84608e11 0.620366 0.310183 0.950677i \(-0.399610\pi\)
0.310183 + 0.950677i \(0.399610\pi\)
\(824\) 1.98778e11i 0.431181i
\(825\) 0 0
\(826\) 1.10729e10 0.0237871
\(827\) 2.82378e11i 0.603683i 0.953358 + 0.301842i \(0.0976014\pi\)
−0.953358 + 0.301842i \(0.902399\pi\)
\(828\) 0 0
\(829\) 8.40257e11 1.77907 0.889537 0.456863i \(-0.151027\pi\)
0.889537 + 0.456863i \(0.151027\pi\)
\(830\) − 2.77953e11i − 0.585679i
\(831\) 0 0
\(832\) −1.69093e10 −0.0352885
\(833\) 3.23923e10i 0.0672762i
\(834\) 0 0
\(835\) −7.90043e11 −1.62519
\(836\) 1.92961e11i 0.395043i
\(837\) 0 0
\(838\) −2.85802e11 −0.579548
\(839\) − 3.34946e11i − 0.675968i −0.941152 0.337984i \(-0.890255\pi\)
0.941152 0.337984i \(-0.109745\pi\)
\(840\) 0 0
\(841\) −3.77810e11 −0.755248
\(842\) − 3.77838e10i − 0.0751722i
\(843\) 0 0
\(844\) 2.75698e11 0.543331
\(845\) − 5.09605e11i − 0.999555i
\(846\) 0 0
\(847\) −3.51264e11 −0.682496
\(848\) − 1.25855e10i − 0.0243380i
\(849\) 0 0
\(850\) −1.71384e10 −0.0328318
\(851\) − 4.95246e11i − 0.944284i
\(852\) 0 0
\(853\) 6.09757e10 0.115176 0.0575878 0.998340i \(-0.481659\pi\)
0.0575878 + 0.998340i \(0.481659\pi\)
\(854\) 3.50108e11i 0.658219i
\(855\) 0 0
\(856\) −2.01450e11 −0.375209
\(857\) 4.12820e11i 0.765311i 0.923891 + 0.382655i \(0.124990\pi\)
−0.923891 + 0.382655i \(0.875010\pi\)
\(858\) 0 0
\(859\) 4.28958e11 0.787847 0.393924 0.919143i \(-0.371117\pi\)
0.393924 + 0.919143i \(0.371117\pi\)
\(860\) 5.34173e11i 0.976535i
\(861\) 0 0
\(862\) 6.28186e10 0.113778
\(863\) − 9.65828e11i − 1.74123i −0.491963 0.870616i \(-0.663720\pi\)
0.491963 0.870616i \(-0.336280\pi\)
\(864\) 0 0
\(865\) 8.98876e11 1.60559
\(866\) 1.29783e11i 0.230753i
\(867\) 0 0
\(868\) 2.18426e11 0.384792
\(869\) 9.49119e10i 0.166434i
\(870\) 0 0
\(871\) −8.08211e10 −0.140427
\(872\) 6.21862e10i 0.107554i
\(873\) 0 0
\(874\) −9.44318e11 −1.61835
\(875\) − 4.49197e11i − 0.766310i
\(876\) 0 0
\(877\) −1.01992e12 −1.72412 −0.862059 0.506808i \(-0.830825\pi\)
−0.862059 + 0.506808i \(0.830825\pi\)
\(878\) − 6.77110e11i − 1.13941i
\(879\) 0 0
\(880\) −7.39875e10 −0.123375
\(881\) 7.44050e11i 1.23509i 0.786535 + 0.617545i \(0.211873\pi\)
−0.786535 + 0.617545i \(0.788127\pi\)
\(882\) 0 0
\(883\) 1.35286e11 0.222540 0.111270 0.993790i \(-0.464508\pi\)
0.111270 + 0.993790i \(0.464508\pi\)
\(884\) 2.22787e10i 0.0364822i
\(885\) 0 0
\(886\) 4.09113e10 0.0663909
\(887\) 5.56566e11i 0.899129i 0.893248 + 0.449565i \(0.148421\pi\)
−0.893248 + 0.449565i \(0.851579\pi\)
\(888\) 0 0
\(889\) 7.01099e11 1.12246
\(890\) − 8.83874e11i − 1.40874i
\(891\) 0 0
\(892\) 3.06398e11 0.483980
\(893\) − 1.33944e12i − 2.10628i
\(894\) 0 0
\(895\) 2.76074e11 0.430263
\(896\) 4.89954e10i 0.0760191i
\(897\) 0 0
\(898\) −2.71507e10 −0.0417518
\(899\) − 7.74347e11i − 1.18549i
\(900\) 0 0
\(901\) −1.65818e10 −0.0251613
\(902\) 3.90764e11i 0.590321i
\(903\) 0 0
\(904\) −4.22353e11 −0.632415
\(905\) − 8.76161e11i − 1.30614i
\(906\) 0 0
\(907\) 3.75761e10 0.0555243 0.0277621 0.999615i \(-0.491162\pi\)
0.0277621 + 0.999615i \(0.491162\pi\)
\(908\) 1.27060e11i 0.186924i
\(909\) 0 0
\(910\) 1.27873e11 0.186471
\(911\) 5.43769e11i 0.789479i 0.918793 + 0.394740i \(0.129165\pi\)
−0.918793 + 0.394740i \(0.870835\pi\)
\(912\) 0 0
\(913\) −2.40765e11 −0.346505
\(914\) − 4.65820e11i − 0.667472i
\(915\) 0 0
\(916\) 2.84352e11 0.403900
\(917\) 4.63670e11i 0.655740i
\(918\) 0 0
\(919\) 9.08317e10 0.127343 0.0636715 0.997971i \(-0.479719\pi\)
0.0636715 + 0.997971i \(0.479719\pi\)
\(920\) − 3.62082e11i − 0.505424i
\(921\) 0 0
\(922\) 2.34646e11 0.324705
\(923\) 3.66745e11i 0.505309i
\(924\) 0 0
\(925\) −9.43556e10 −0.128884
\(926\) − 8.31987e10i − 0.113155i
\(927\) 0 0
\(928\) 1.73695e11 0.234204
\(929\) − 1.38421e11i − 0.185841i −0.995674 0.0929203i \(-0.970380\pi\)
0.995674 0.0929203i \(-0.0296202\pi\)
\(930\) 0 0
\(931\) 3.40044e11 0.452623
\(932\) − 1.58774e11i − 0.210433i
\(933\) 0 0
\(934\) 1.02873e12 1.35181
\(935\) 9.74814e10i 0.127549i
\(936\) 0 0
\(937\) 1.28993e12 1.67343 0.836714 0.547640i \(-0.184474\pi\)
0.836714 + 0.547640i \(0.184474\pi\)
\(938\) 2.34182e11i 0.302511i
\(939\) 0 0
\(940\) 5.13583e11 0.657809
\(941\) 1.44651e12i 1.84486i 0.386163 + 0.922431i \(0.373800\pi\)
−0.386163 + 0.922431i \(0.626200\pi\)
\(942\) 0 0
\(943\) −1.91233e12 −2.41833
\(944\) 7.76526e9i 0.00977841i
\(945\) 0 0
\(946\) 4.62704e11 0.577748
\(947\) − 6.92009e11i − 0.860423i −0.902728 0.430212i \(-0.858439\pi\)
0.902728 0.430212i \(-0.141561\pi\)
\(948\) 0 0
\(949\) −1.87558e11 −0.231244
\(950\) 1.79914e11i 0.220887i
\(951\) 0 0
\(952\) 6.45533e10 0.0785906
\(953\) − 3.33259e11i − 0.404027i −0.979383 0.202013i \(-0.935252\pi\)
0.979383 0.202013i \(-0.0647484\pi\)
\(954\) 0 0
\(955\) 2.02095e11 0.242964
\(956\) 9.71891e10i 0.116355i
\(957\) 0 0
\(958\) 2.16314e11 0.256817
\(959\) − 8.28609e11i − 0.979659i
\(960\) 0 0
\(961\) −1.70004e11 −0.199326
\(962\) 1.22655e11i 0.143214i
\(963\) 0 0
\(964\) −5.72758e11 −0.663228
\(965\) 1.50701e12i 1.73783i
\(966\) 0 0
\(967\) −1.25085e12 −1.43054 −0.715270 0.698848i \(-0.753696\pi\)
−0.715270 + 0.698848i \(0.753696\pi\)
\(968\) − 2.46336e11i − 0.280561i
\(969\) 0 0
\(970\) 3.11590e11 0.351963
\(971\) 7.48637e11i 0.842160i 0.907024 + 0.421080i \(0.138349\pi\)
−0.907024 + 0.421080i \(0.861651\pi\)
\(972\) 0 0
\(973\) −5.57063e11 −0.621516
\(974\) 5.98198e11i 0.664675i
\(975\) 0 0
\(976\) −2.45526e11 −0.270581
\(977\) − 6.85254e11i − 0.752096i −0.926600 0.376048i \(-0.877283\pi\)
0.926600 0.376048i \(-0.122717\pi\)
\(978\) 0 0
\(979\) −7.65616e11 −0.833452
\(980\) 1.30384e11i 0.141358i
\(981\) 0 0
\(982\) −1.10652e12 −1.18990
\(983\) 1.15973e12i 1.24206i 0.783785 + 0.621032i \(0.213286\pi\)
−0.783785 + 0.621032i \(0.786714\pi\)
\(984\) 0 0
\(985\) 1.30048e12 1.38153
\(986\) − 2.28849e11i − 0.242126i
\(987\) 0 0
\(988\) 2.33875e11 0.245446
\(989\) 2.26439e12i 2.36683i
\(990\) 0 0
\(991\) −1.55112e12 −1.60824 −0.804121 0.594465i \(-0.797364\pi\)
−0.804121 + 0.594465i \(0.797364\pi\)
\(992\) 1.53179e11i 0.158180i
\(993\) 0 0
\(994\) 1.06266e12 1.08855
\(995\) − 1.19115e12i − 1.21527i
\(996\) 0 0
\(997\) −3.61247e11 −0.365615 −0.182807 0.983149i \(-0.558518\pi\)
−0.182807 + 0.983149i \(0.558518\pi\)
\(998\) 1.00619e12i 1.01428i
\(999\) 0 0
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 54.9.b.a.53.2 yes 2
3.2 odd 2 inner 54.9.b.a.53.1 2
4.3 odd 2 432.9.e.g.161.2 2
9.2 odd 6 162.9.d.c.53.1 4
9.4 even 3 162.9.d.c.107.1 4
9.5 odd 6 162.9.d.c.107.2 4
9.7 even 3 162.9.d.c.53.2 4
12.11 even 2 432.9.e.g.161.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.9.b.a.53.1 2 3.2 odd 2 inner
54.9.b.a.53.2 yes 2 1.1 even 1 trivial
162.9.d.c.53.1 4 9.2 odd 6
162.9.d.c.53.2 4 9.7 even 3
162.9.d.c.107.1 4 9.4 even 3
162.9.d.c.107.2 4 9.5 odd 6
432.9.e.g.161.1 2 12.11 even 2
432.9.e.g.161.2 2 4.3 odd 2