Properties

Label 75.22.a.j.1.3
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(89.2274\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-340.910 q^{2} -59049.0 q^{3} -1.98093e6 q^{4} +2.01304e7 q^{6} +6.73159e8 q^{7} +1.39026e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-340.910 q^{2} -59049.0 q^{3} -1.98093e6 q^{4} +2.01304e7 q^{6} +6.73159e8 q^{7} +1.39026e9 q^{8} +3.48678e9 q^{9} +8.81884e10 q^{11} +1.16972e11 q^{12} -4.35956e11 q^{13} -2.29486e11 q^{14} +3.68036e12 q^{16} +3.38833e12 q^{17} -1.18868e12 q^{18} -4.68698e13 q^{19} -3.97494e13 q^{21} -3.00643e13 q^{22} +2.77663e13 q^{23} -8.20934e13 q^{24} +1.48622e14 q^{26} -2.05891e14 q^{27} -1.33348e15 q^{28} -4.93794e14 q^{29} -1.99343e15 q^{31} -4.17025e15 q^{32} -5.20744e15 q^{33} -1.15512e15 q^{34} -6.90709e15 q^{36} -2.48383e16 q^{37} +1.59784e16 q^{38} +2.57428e16 q^{39} +1.53977e17 q^{41} +1.35509e16 q^{42} -8.38852e16 q^{43} -1.74695e17 q^{44} -9.46580e15 q^{46} +1.06134e16 q^{47} -2.17322e17 q^{48} -1.05402e17 q^{49} -2.00078e17 q^{51} +8.63600e17 q^{52} +2.83801e17 q^{53} +7.01903e16 q^{54} +9.35865e17 q^{56} +2.76761e18 q^{57} +1.68339e17 q^{58} +7.02994e18 q^{59} +6.54670e18 q^{61} +6.79580e17 q^{62} +2.34716e18 q^{63} -6.29660e18 q^{64} +1.77527e18 q^{66} +1.27152e18 q^{67} -6.71206e18 q^{68} -1.63957e18 q^{69} +9.18512e18 q^{71} +4.84753e18 q^{72} -3.67454e19 q^{73} +8.46760e18 q^{74} +9.28459e19 q^{76} +5.93649e19 q^{77} -8.77597e18 q^{78} -6.42331e19 q^{79} +1.21577e19 q^{81} -5.24921e19 q^{82} +2.62194e20 q^{83} +7.87409e19 q^{84} +2.85973e19 q^{86} +2.91580e19 q^{87} +1.22605e20 q^{88} -2.98217e20 q^{89} -2.93468e20 q^{91} -5.50032e19 q^{92} +1.17710e20 q^{93} -3.61820e18 q^{94} +2.46249e20 q^{96} +6.86712e20 q^{97} +3.59327e19 q^{98} +3.07494e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −340.910 −0.235410 −0.117705 0.993049i \(-0.537554\pi\)
−0.117705 + 0.993049i \(0.537554\pi\)
\(3\) −59049.0 −0.577350
\(4\) −1.98093e6 −0.944582
\(5\) 0 0
\(6\) 2.01304e7 0.135914
\(7\) 6.73159e8 0.900717 0.450359 0.892848i \(-0.351296\pi\)
0.450359 + 0.892848i \(0.351296\pi\)
\(8\) 1.39026e9 0.457773
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 8.81884e10 1.02515 0.512576 0.858642i \(-0.328691\pi\)
0.512576 + 0.858642i \(0.328691\pi\)
\(12\) 1.16972e11 0.545355
\(13\) −4.35956e11 −0.877077 −0.438539 0.898712i \(-0.644504\pi\)
−0.438539 + 0.898712i \(0.644504\pi\)
\(14\) −2.29486e11 −0.212038
\(15\) 0 0
\(16\) 3.68036e12 0.836818
\(17\) 3.38833e12 0.407636 0.203818 0.979009i \(-0.434665\pi\)
0.203818 + 0.979009i \(0.434665\pi\)
\(18\) −1.18868e12 −0.0784699
\(19\) −4.68698e13 −1.75380 −0.876900 0.480672i \(-0.840393\pi\)
−0.876900 + 0.480672i \(0.840393\pi\)
\(20\) 0 0
\(21\) −3.97494e13 −0.520029
\(22\) −3.00643e13 −0.241331
\(23\) 2.77663e13 0.139758 0.0698788 0.997555i \(-0.477739\pi\)
0.0698788 + 0.997555i \(0.477739\pi\)
\(24\) −8.20934e13 −0.264296
\(25\) 0 0
\(26\) 1.48622e14 0.206472
\(27\) −2.05891e14 −0.192450
\(28\) −1.33348e15 −0.850802
\(29\) −4.93794e14 −0.217955 −0.108977 0.994044i \(-0.534758\pi\)
−0.108977 + 0.994044i \(0.534758\pi\)
\(30\) 0 0
\(31\) −1.99343e15 −0.436821 −0.218410 0.975857i \(-0.570087\pi\)
−0.218410 + 0.975857i \(0.570087\pi\)
\(32\) −4.17025e15 −0.654768
\(33\) −5.20744e15 −0.591872
\(34\) −1.15512e15 −0.0959614
\(35\) 0 0
\(36\) −6.90709e15 −0.314861
\(37\) −2.48383e16 −0.849186 −0.424593 0.905384i \(-0.639583\pi\)
−0.424593 + 0.905384i \(0.639583\pi\)
\(38\) 1.59784e16 0.412862
\(39\) 2.57428e16 0.506381
\(40\) 0 0
\(41\) 1.53977e17 1.79153 0.895766 0.444526i \(-0.146628\pi\)
0.895766 + 0.444526i \(0.146628\pi\)
\(42\) 1.35509e16 0.122420
\(43\) −8.38852e16 −0.591925 −0.295962 0.955200i \(-0.595640\pi\)
−0.295962 + 0.955200i \(0.595640\pi\)
\(44\) −1.74695e17 −0.968341
\(45\) 0 0
\(46\) −9.46580e15 −0.0329003
\(47\) 1.06134e16 0.0294324 0.0147162 0.999892i \(-0.495316\pi\)
0.0147162 + 0.999892i \(0.495316\pi\)
\(48\) −2.17322e17 −0.483137
\(49\) −1.05402e17 −0.188708
\(50\) 0 0
\(51\) −2.00078e17 −0.235349
\(52\) 8.63600e17 0.828472
\(53\) 2.83801e17 0.222904 0.111452 0.993770i \(-0.464450\pi\)
0.111452 + 0.993770i \(0.464450\pi\)
\(54\) 7.01903e16 0.0453046
\(55\) 0 0
\(56\) 9.35865e17 0.412324
\(57\) 2.76761e18 1.01256
\(58\) 1.68339e17 0.0513087
\(59\) 7.02994e18 1.79063 0.895314 0.445436i \(-0.146951\pi\)
0.895314 + 0.445436i \(0.146951\pi\)
\(60\) 0 0
\(61\) 6.54670e18 1.17506 0.587529 0.809203i \(-0.300101\pi\)
0.587529 + 0.809203i \(0.300101\pi\)
\(62\) 6.79580e17 0.102832
\(63\) 2.34716e18 0.300239
\(64\) −6.29660e18 −0.682679
\(65\) 0 0
\(66\) 1.77527e18 0.139332
\(67\) 1.27152e18 0.0852192 0.0426096 0.999092i \(-0.486433\pi\)
0.0426096 + 0.999092i \(0.486433\pi\)
\(68\) −6.71206e18 −0.385046
\(69\) −1.63957e18 −0.0806891
\(70\) 0 0
\(71\) 9.18512e18 0.334867 0.167433 0.985883i \(-0.446452\pi\)
0.167433 + 0.985883i \(0.446452\pi\)
\(72\) 4.84753e18 0.152591
\(73\) −3.67454e19 −1.00072 −0.500361 0.865817i \(-0.666799\pi\)
−0.500361 + 0.865817i \(0.666799\pi\)
\(74\) 8.46760e18 0.199907
\(75\) 0 0
\(76\) 9.28459e19 1.65661
\(77\) 5.93649e19 0.923373
\(78\) −8.77597e18 −0.119207
\(79\) −6.42331e19 −0.763263 −0.381632 0.924315i \(-0.624638\pi\)
−0.381632 + 0.924315i \(0.624638\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −5.24921e19 −0.421744
\(83\) 2.62194e20 1.85482 0.927412 0.374042i \(-0.122028\pi\)
0.927412 + 0.374042i \(0.122028\pi\)
\(84\) 7.87409e19 0.491211
\(85\) 0 0
\(86\) 2.85973e19 0.139345
\(87\) 2.91580e19 0.125836
\(88\) 1.22605e20 0.469288
\(89\) −2.98217e20 −1.01377 −0.506883 0.862015i \(-0.669202\pi\)
−0.506883 + 0.862015i \(0.669202\pi\)
\(90\) 0 0
\(91\) −2.93468e20 −0.789999
\(92\) −5.50032e19 −0.132013
\(93\) 1.17710e20 0.252199
\(94\) −3.61820e18 −0.00692867
\(95\) 0 0
\(96\) 2.46249e20 0.378031
\(97\) 6.86712e20 0.945522 0.472761 0.881191i \(-0.343257\pi\)
0.472761 + 0.881191i \(0.343257\pi\)
\(98\) 3.59327e19 0.0444238
\(99\) 3.07494e20 0.341718
\(100\) 0 0
\(101\) −1.72493e21 −1.55381 −0.776904 0.629619i \(-0.783211\pi\)
−0.776904 + 0.629619i \(0.783211\pi\)
\(102\) 6.82084e19 0.0554034
\(103\) 7.93905e20 0.582073 0.291037 0.956712i \(-0.406000\pi\)
0.291037 + 0.956712i \(0.406000\pi\)
\(104\) −6.06092e20 −0.401503
\(105\) 0 0
\(106\) −9.67505e19 −0.0524737
\(107\) −2.41854e21 −1.18856 −0.594282 0.804257i \(-0.702564\pi\)
−0.594282 + 0.804257i \(0.702564\pi\)
\(108\) 4.07856e20 0.181785
\(109\) −4.56052e20 −0.184517 −0.0922584 0.995735i \(-0.529409\pi\)
−0.0922584 + 0.995735i \(0.529409\pi\)
\(110\) 0 0
\(111\) 1.46667e21 0.490278
\(112\) 2.47747e21 0.753736
\(113\) 2.66253e21 0.737854 0.368927 0.929458i \(-0.379725\pi\)
0.368927 + 0.929458i \(0.379725\pi\)
\(114\) −9.43506e20 −0.238366
\(115\) 0 0
\(116\) 9.78172e20 0.205876
\(117\) −1.52009e21 −0.292359
\(118\) −2.39657e21 −0.421531
\(119\) 2.28089e21 0.367165
\(120\) 0 0
\(121\) 3.76952e20 0.0509378
\(122\) −2.23183e21 −0.276620
\(123\) −9.09217e21 −1.03434
\(124\) 3.94885e21 0.412613
\(125\) 0 0
\(126\) −8.00170e20 −0.0706792
\(127\) 3.90399e21 0.317372 0.158686 0.987329i \(-0.449274\pi\)
0.158686 + 0.987329i \(0.449274\pi\)
\(128\) 1.08922e22 0.815478
\(129\) 4.95334e21 0.341748
\(130\) 0 0
\(131\) −1.19004e22 −0.698575 −0.349288 0.937016i \(-0.613576\pi\)
−0.349288 + 0.937016i \(0.613576\pi\)
\(132\) 1.03156e22 0.559072
\(133\) −3.15508e22 −1.57968
\(134\) −4.33473e20 −0.0200614
\(135\) 0 0
\(136\) 4.71066e21 0.186605
\(137\) 4.39883e22 1.61351 0.806754 0.590887i \(-0.201222\pi\)
0.806754 + 0.590887i \(0.201222\pi\)
\(138\) 5.58946e20 0.0189950
\(139\) 2.38509e22 0.751362 0.375681 0.926749i \(-0.377409\pi\)
0.375681 + 0.926749i \(0.377409\pi\)
\(140\) 0 0
\(141\) −6.26709e20 −0.0169928
\(142\) −3.13130e21 −0.0788309
\(143\) −3.84463e22 −0.899138
\(144\) 1.28326e22 0.278939
\(145\) 0 0
\(146\) 1.25269e22 0.235579
\(147\) 6.22390e21 0.108951
\(148\) 4.92029e22 0.802126
\(149\) 9.91268e21 0.150569 0.0752844 0.997162i \(-0.476014\pi\)
0.0752844 + 0.997162i \(0.476014\pi\)
\(150\) 0 0
\(151\) −1.80864e22 −0.238833 −0.119416 0.992844i \(-0.538102\pi\)
−0.119416 + 0.992844i \(0.538102\pi\)
\(152\) −6.51611e22 −0.802843
\(153\) 1.18144e22 0.135879
\(154\) −2.02381e22 −0.217371
\(155\) 0 0
\(156\) −5.09947e22 −0.478318
\(157\) −1.17597e23 −1.03146 −0.515729 0.856752i \(-0.672479\pi\)
−0.515729 + 0.856752i \(0.672479\pi\)
\(158\) 2.18977e22 0.179679
\(159\) −1.67582e22 −0.128694
\(160\) 0 0
\(161\) 1.86911e22 0.125882
\(162\) −4.14466e21 −0.0261566
\(163\) −3.82771e22 −0.226448 −0.113224 0.993569i \(-0.536118\pi\)
−0.113224 + 0.993569i \(0.536118\pi\)
\(164\) −3.05018e23 −1.69225
\(165\) 0 0
\(166\) −8.93844e22 −0.436643
\(167\) 3.13595e23 1.43829 0.719145 0.694860i \(-0.244534\pi\)
0.719145 + 0.694860i \(0.244534\pi\)
\(168\) −5.52619e22 −0.238056
\(169\) −5.70065e22 −0.230735
\(170\) 0 0
\(171\) −1.63425e23 −0.584600
\(172\) 1.66171e23 0.559121
\(173\) 4.81111e23 1.52322 0.761608 0.648039i \(-0.224410\pi\)
0.761608 + 0.648039i \(0.224410\pi\)
\(174\) −9.94025e21 −0.0296231
\(175\) 0 0
\(176\) 3.24566e23 0.857866
\(177\) −4.15111e23 −1.03382
\(178\) 1.01665e23 0.238650
\(179\) −3.28060e22 −0.0726099 −0.0363050 0.999341i \(-0.511559\pi\)
−0.0363050 + 0.999341i \(0.511559\pi\)
\(180\) 0 0
\(181\) −5.50948e23 −1.08514 −0.542569 0.840011i \(-0.682548\pi\)
−0.542569 + 0.840011i \(0.682548\pi\)
\(182\) 1.00046e23 0.185973
\(183\) −3.86576e23 −0.678420
\(184\) 3.86023e22 0.0639773
\(185\) 0 0
\(186\) −4.01285e22 −0.0593700
\(187\) 2.98812e23 0.417889
\(188\) −2.10244e22 −0.0278013
\(189\) −1.38598e23 −0.173343
\(190\) 0 0
\(191\) 1.18267e24 1.32438 0.662191 0.749335i \(-0.269627\pi\)
0.662191 + 0.749335i \(0.269627\pi\)
\(192\) 3.71808e23 0.394145
\(193\) −1.17187e24 −1.17632 −0.588161 0.808744i \(-0.700148\pi\)
−0.588161 + 0.808744i \(0.700148\pi\)
\(194\) −2.34107e23 −0.222585
\(195\) 0 0
\(196\) 2.08795e23 0.178251
\(197\) 6.31695e23 0.511225 0.255613 0.966779i \(-0.417723\pi\)
0.255613 + 0.966779i \(0.417723\pi\)
\(198\) −1.04828e23 −0.0804436
\(199\) −1.80597e24 −1.31448 −0.657239 0.753682i \(-0.728276\pi\)
−0.657239 + 0.753682i \(0.728276\pi\)
\(200\) 0 0
\(201\) −7.50820e22 −0.0492014
\(202\) 5.88046e23 0.365781
\(203\) −3.32402e23 −0.196316
\(204\) 3.96340e23 0.222306
\(205\) 0 0
\(206\) −2.70650e23 −0.137026
\(207\) 9.68151e22 0.0465859
\(208\) −1.60448e24 −0.733954
\(209\) −4.13338e24 −1.79791
\(210\) 0 0
\(211\) 2.94629e23 0.115961 0.0579803 0.998318i \(-0.481534\pi\)
0.0579803 + 0.998318i \(0.481534\pi\)
\(212\) −5.62191e23 −0.210551
\(213\) −5.42372e23 −0.193336
\(214\) 8.24502e23 0.279800
\(215\) 0 0
\(216\) −2.86242e23 −0.0880985
\(217\) −1.34190e24 −0.393452
\(218\) 1.55472e23 0.0434371
\(219\) 2.16978e24 0.577767
\(220\) 0 0
\(221\) −1.47717e24 −0.357528
\(222\) −5.00003e23 −0.115416
\(223\) −7.94458e24 −1.74932 −0.874662 0.484734i \(-0.838916\pi\)
−0.874662 + 0.484734i \(0.838916\pi\)
\(224\) −2.80725e24 −0.589761
\(225\) 0 0
\(226\) −9.07681e23 −0.173698
\(227\) 6.99177e24 1.27737 0.638684 0.769469i \(-0.279479\pi\)
0.638684 + 0.769469i \(0.279479\pi\)
\(228\) −5.48246e24 −0.956444
\(229\) −1.97790e24 −0.329558 −0.164779 0.986330i \(-0.552691\pi\)
−0.164779 + 0.986330i \(0.552691\pi\)
\(230\) 0 0
\(231\) −3.50544e24 −0.533109
\(232\) −6.86501e23 −0.0997740
\(233\) 5.40517e24 0.750883 0.375441 0.926846i \(-0.377491\pi\)
0.375441 + 0.926846i \(0.377491\pi\)
\(234\) 5.18212e23 0.0688242
\(235\) 0 0
\(236\) −1.39258e25 −1.69140
\(237\) 3.79290e24 0.440670
\(238\) −7.77577e23 −0.0864341
\(239\) −1.53053e25 −1.62804 −0.814018 0.580840i \(-0.802724\pi\)
−0.814018 + 0.580840i \(0.802724\pi\)
\(240\) 0 0
\(241\) −1.34112e25 −1.30704 −0.653519 0.756910i \(-0.726708\pi\)
−0.653519 + 0.756910i \(0.726708\pi\)
\(242\) −1.28507e23 −0.0119912
\(243\) −7.17898e23 −0.0641500
\(244\) −1.29686e25 −1.10994
\(245\) 0 0
\(246\) 3.09961e24 0.243494
\(247\) 2.04332e25 1.53822
\(248\) −2.77138e24 −0.199965
\(249\) −1.54823e25 −1.07088
\(250\) 0 0
\(251\) 1.69729e24 0.107940 0.0539699 0.998543i \(-0.482812\pi\)
0.0539699 + 0.998543i \(0.482812\pi\)
\(252\) −4.64957e24 −0.283601
\(253\) 2.44867e24 0.143273
\(254\) −1.33091e24 −0.0747125
\(255\) 0 0
\(256\) 9.49167e24 0.490708
\(257\) 7.35666e24 0.365076 0.182538 0.983199i \(-0.441569\pi\)
0.182538 + 0.983199i \(0.441569\pi\)
\(258\) −1.68864e24 −0.0804507
\(259\) −1.67201e25 −0.764877
\(260\) 0 0
\(261\) −1.72175e24 −0.0726516
\(262\) 4.05696e24 0.164451
\(263\) −4.57145e25 −1.78040 −0.890202 0.455566i \(-0.849437\pi\)
−0.890202 + 0.455566i \(0.849437\pi\)
\(264\) −7.23969e24 −0.270943
\(265\) 0 0
\(266\) 1.07560e25 0.371872
\(267\) 1.76094e25 0.585298
\(268\) −2.51880e24 −0.0804966
\(269\) −3.25178e25 −0.999360 −0.499680 0.866210i \(-0.666549\pi\)
−0.499680 + 0.866210i \(0.666549\pi\)
\(270\) 0 0
\(271\) −1.68503e25 −0.479104 −0.239552 0.970884i \(-0.577001\pi\)
−0.239552 + 0.970884i \(0.577001\pi\)
\(272\) 1.24703e25 0.341117
\(273\) 1.73290e25 0.456106
\(274\) −1.49960e25 −0.379835
\(275\) 0 0
\(276\) 3.24788e24 0.0762175
\(277\) 6.04936e25 1.36670 0.683348 0.730093i \(-0.260523\pi\)
0.683348 + 0.730093i \(0.260523\pi\)
\(278\) −8.13100e24 −0.176878
\(279\) −6.95067e24 −0.145607
\(280\) 0 0
\(281\) 9.79283e23 0.0190323 0.00951616 0.999955i \(-0.496971\pi\)
0.00951616 + 0.999955i \(0.496971\pi\)
\(282\) 2.13651e23 0.00400027
\(283\) −4.41068e25 −0.795697 −0.397849 0.917451i \(-0.630243\pi\)
−0.397849 + 0.917451i \(0.630243\pi\)
\(284\) −1.81951e25 −0.316309
\(285\) 0 0
\(286\) 1.31067e25 0.211666
\(287\) 1.03651e26 1.61366
\(288\) −1.45408e25 −0.218256
\(289\) −5.76111e25 −0.833833
\(290\) 0 0
\(291\) −4.05497e25 −0.545897
\(292\) 7.27902e25 0.945263
\(293\) −8.77281e25 −1.09908 −0.549539 0.835468i \(-0.685197\pi\)
−0.549539 + 0.835468i \(0.685197\pi\)
\(294\) −2.12179e24 −0.0256481
\(295\) 0 0
\(296\) −3.45316e25 −0.388735
\(297\) −1.81572e25 −0.197291
\(298\) −3.37933e24 −0.0354454
\(299\) −1.21049e25 −0.122578
\(300\) 0 0
\(301\) −5.64681e25 −0.533157
\(302\) 6.16582e24 0.0562235
\(303\) 1.01855e26 0.897091
\(304\) −1.72498e26 −1.46761
\(305\) 0 0
\(306\) −4.02764e24 −0.0319871
\(307\) −1.76055e26 −1.35113 −0.675563 0.737303i \(-0.736099\pi\)
−0.675563 + 0.737303i \(0.736099\pi\)
\(308\) −1.17598e26 −0.872201
\(309\) −4.68793e25 −0.336060
\(310\) 0 0
\(311\) −2.72398e26 −1.82482 −0.912410 0.409278i \(-0.865781\pi\)
−0.912410 + 0.409278i \(0.865781\pi\)
\(312\) 3.57891e25 0.231808
\(313\) −9.00000e25 −0.563672 −0.281836 0.959463i \(-0.590943\pi\)
−0.281836 + 0.959463i \(0.590943\pi\)
\(314\) 4.00901e25 0.242815
\(315\) 0 0
\(316\) 1.27241e26 0.720965
\(317\) −8.30387e25 −0.455154 −0.227577 0.973760i \(-0.573080\pi\)
−0.227577 + 0.973760i \(0.573080\pi\)
\(318\) 5.71302e24 0.0302957
\(319\) −4.35469e25 −0.223437
\(320\) 0 0
\(321\) 1.42812e26 0.686218
\(322\) −6.37199e24 −0.0296339
\(323\) −1.58811e26 −0.714912
\(324\) −2.40835e25 −0.104954
\(325\) 0 0
\(326\) 1.30490e25 0.0533081
\(327\) 2.69294e25 0.106531
\(328\) 2.14067e26 0.820116
\(329\) 7.14449e24 0.0265103
\(330\) 0 0
\(331\) −9.84201e25 −0.342681 −0.171340 0.985212i \(-0.554810\pi\)
−0.171340 + 0.985212i \(0.554810\pi\)
\(332\) −5.19389e26 −1.75203
\(333\) −8.66056e25 −0.283062
\(334\) −1.06908e26 −0.338587
\(335\) 0 0
\(336\) −1.46292e26 −0.435170
\(337\) 6.17471e26 1.78034 0.890169 0.455631i \(-0.150586\pi\)
0.890169 + 0.455631i \(0.150586\pi\)
\(338\) 1.94340e25 0.0543173
\(339\) −1.57220e26 −0.426000
\(340\) 0 0
\(341\) −1.75798e26 −0.447808
\(342\) 5.57131e25 0.137621
\(343\) −4.46943e26 −1.07069
\(344\) −1.16622e26 −0.270967
\(345\) 0 0
\(346\) −1.64015e26 −0.358580
\(347\) 1.24054e26 0.263118 0.131559 0.991308i \(-0.458002\pi\)
0.131559 + 0.991308i \(0.458002\pi\)
\(348\) −5.77601e25 −0.118863
\(349\) 8.43585e26 1.68446 0.842232 0.539115i \(-0.181241\pi\)
0.842232 + 0.539115i \(0.181241\pi\)
\(350\) 0 0
\(351\) 8.97596e25 0.168794
\(352\) −3.67768e26 −0.671238
\(353\) 8.92103e26 1.58045 0.790224 0.612818i \(-0.209964\pi\)
0.790224 + 0.612818i \(0.209964\pi\)
\(354\) 1.41515e26 0.243371
\(355\) 0 0
\(356\) 5.90748e26 0.957585
\(357\) −1.34684e26 −0.211983
\(358\) 1.11839e25 0.0170931
\(359\) −1.31171e26 −0.194692 −0.0973458 0.995251i \(-0.531035\pi\)
−0.0973458 + 0.995251i \(0.531035\pi\)
\(360\) 0 0
\(361\) 1.48257e27 2.07582
\(362\) 1.87823e26 0.255452
\(363\) −2.22587e25 −0.0294090
\(364\) 5.81341e26 0.746219
\(365\) 0 0
\(366\) 1.31788e26 0.159707
\(367\) 6.63211e25 0.0781012 0.0390506 0.999237i \(-0.487567\pi\)
0.0390506 + 0.999237i \(0.487567\pi\)
\(368\) 1.02190e26 0.116952
\(369\) 5.36884e26 0.597177
\(370\) 0 0
\(371\) 1.91043e26 0.200773
\(372\) −2.33176e26 −0.238222
\(373\) 1.49862e27 1.48850 0.744252 0.667899i \(-0.232806\pi\)
0.744252 + 0.667899i \(0.232806\pi\)
\(374\) −1.01868e26 −0.0983751
\(375\) 0 0
\(376\) 1.47553e25 0.0134734
\(377\) 2.15273e26 0.191163
\(378\) 4.72492e25 0.0408066
\(379\) 2.60984e26 0.219231 0.109615 0.993974i \(-0.465038\pi\)
0.109615 + 0.993974i \(0.465038\pi\)
\(380\) 0 0
\(381\) −2.30527e26 −0.183235
\(382\) −4.03184e26 −0.311772
\(383\) 6.73245e26 0.506508 0.253254 0.967400i \(-0.418499\pi\)
0.253254 + 0.967400i \(0.418499\pi\)
\(384\) −6.43175e26 −0.470816
\(385\) 0 0
\(386\) 3.99501e26 0.276918
\(387\) −2.92489e26 −0.197308
\(388\) −1.36033e27 −0.893123
\(389\) 9.85802e26 0.629969 0.314984 0.949097i \(-0.398001\pi\)
0.314984 + 0.949097i \(0.398001\pi\)
\(390\) 0 0
\(391\) 9.40815e25 0.0569703
\(392\) −1.46536e26 −0.0863857
\(393\) 7.02707e26 0.403323
\(394\) −2.15351e26 −0.120347
\(395\) 0 0
\(396\) −6.09125e26 −0.322780
\(397\) −2.43807e27 −1.25819 −0.629093 0.777330i \(-0.716574\pi\)
−0.629093 + 0.777330i \(0.716574\pi\)
\(398\) 6.15673e26 0.309441
\(399\) 1.86305e27 0.912028
\(400\) 0 0
\(401\) 3.26497e26 0.151657 0.0758286 0.997121i \(-0.475840\pi\)
0.0758286 + 0.997121i \(0.475840\pi\)
\(402\) 2.55962e25 0.0115825
\(403\) 8.69049e26 0.383126
\(404\) 3.41697e27 1.46770
\(405\) 0 0
\(406\) 1.13319e26 0.0462146
\(407\) −2.19045e27 −0.870546
\(408\) −2.78160e26 −0.107736
\(409\) 4.25757e27 1.60719 0.803596 0.595176i \(-0.202918\pi\)
0.803596 + 0.595176i \(0.202918\pi\)
\(410\) 0 0
\(411\) −2.59746e27 −0.931559
\(412\) −1.57267e27 −0.549816
\(413\) 4.73227e27 1.61285
\(414\) −3.30052e25 −0.0109668
\(415\) 0 0
\(416\) 1.81805e27 0.574283
\(417\) −1.40837e27 −0.433799
\(418\) 1.40911e27 0.423246
\(419\) −5.00076e26 −0.146483 −0.0732417 0.997314i \(-0.523334\pi\)
−0.0732417 + 0.997314i \(0.523334\pi\)
\(420\) 0 0
\(421\) −4.63434e27 −1.29130 −0.645648 0.763635i \(-0.723413\pi\)
−0.645648 + 0.763635i \(0.723413\pi\)
\(422\) −1.00442e26 −0.0272982
\(423\) 3.70065e25 0.00981080
\(424\) 3.94557e26 0.102039
\(425\) 0 0
\(426\) 1.84900e26 0.0455130
\(427\) 4.40697e27 1.05840
\(428\) 4.79096e27 1.12270
\(429\) 2.27022e27 0.519118
\(430\) 0 0
\(431\) 3.30727e27 0.720209 0.360105 0.932912i \(-0.382741\pi\)
0.360105 + 0.932912i \(0.382741\pi\)
\(432\) −7.57754e26 −0.161046
\(433\) −2.58162e27 −0.535512 −0.267756 0.963487i \(-0.586282\pi\)
−0.267756 + 0.963487i \(0.586282\pi\)
\(434\) 4.57466e26 0.0926224
\(435\) 0 0
\(436\) 9.03408e26 0.174291
\(437\) −1.30140e27 −0.245107
\(438\) −7.39699e26 −0.136012
\(439\) 6.51723e27 1.17000 0.585000 0.811034i \(-0.301095\pi\)
0.585000 + 0.811034i \(0.301095\pi\)
\(440\) 0 0
\(441\) −3.67515e26 −0.0629028
\(442\) 5.03580e26 0.0841656
\(443\) −1.02894e28 −1.67939 −0.839694 0.543060i \(-0.817266\pi\)
−0.839694 + 0.543060i \(0.817266\pi\)
\(444\) −2.90538e27 −0.463108
\(445\) 0 0
\(446\) 2.70838e27 0.411808
\(447\) −5.85334e26 −0.0869310
\(448\) −4.23862e27 −0.614901
\(449\) 2.52487e27 0.357810 0.178905 0.983866i \(-0.442745\pi\)
0.178905 + 0.983866i \(0.442745\pi\)
\(450\) 0 0
\(451\) 1.35790e28 1.83659
\(452\) −5.27429e27 −0.696964
\(453\) 1.06798e27 0.137890
\(454\) −2.38356e27 −0.300705
\(455\) 0 0
\(456\) 3.84770e27 0.463522
\(457\) −4.85166e27 −0.571176 −0.285588 0.958352i \(-0.592189\pi\)
−0.285588 + 0.958352i \(0.592189\pi\)
\(458\) 6.74286e26 0.0775812
\(459\) −6.97628e26 −0.0784496
\(460\) 0 0
\(461\) −1.01408e28 −1.08946 −0.544731 0.838611i \(-0.683368\pi\)
−0.544731 + 0.838611i \(0.683368\pi\)
\(462\) 1.19504e27 0.125499
\(463\) 9.15460e27 0.939808 0.469904 0.882718i \(-0.344289\pi\)
0.469904 + 0.882718i \(0.344289\pi\)
\(464\) −1.81734e27 −0.182389
\(465\) 0 0
\(466\) −1.84267e27 −0.176765
\(467\) −3.58291e27 −0.336054 −0.168027 0.985782i \(-0.553740\pi\)
−0.168027 + 0.985782i \(0.553740\pi\)
\(468\) 3.01119e27 0.276157
\(469\) 8.55935e26 0.0767584
\(470\) 0 0
\(471\) 6.94401e27 0.595512
\(472\) 9.77343e27 0.819702
\(473\) −7.39770e27 −0.606813
\(474\) −1.29304e27 −0.103738
\(475\) 0 0
\(476\) −4.51829e27 −0.346817
\(477\) 9.89554e26 0.0743013
\(478\) 5.21772e27 0.383255
\(479\) −9.63095e27 −0.692064 −0.346032 0.938223i \(-0.612471\pi\)
−0.346032 + 0.938223i \(0.612471\pi\)
\(480\) 0 0
\(481\) 1.08284e28 0.744802
\(482\) 4.57200e27 0.307690
\(483\) −1.10369e27 −0.0726781
\(484\) −7.46717e26 −0.0481149
\(485\) 0 0
\(486\) 2.44738e26 0.0151015
\(487\) 1.93192e27 0.116664 0.0583318 0.998297i \(-0.481422\pi\)
0.0583318 + 0.998297i \(0.481422\pi\)
\(488\) 9.10161e27 0.537910
\(489\) 2.26022e27 0.130740
\(490\) 0 0
\(491\) −8.62908e27 −0.478199 −0.239100 0.970995i \(-0.576852\pi\)
−0.239100 + 0.970995i \(0.576852\pi\)
\(492\) 1.80110e28 0.977021
\(493\) −1.67314e27 −0.0888463
\(494\) −6.96587e27 −0.362112
\(495\) 0 0
\(496\) −7.33655e27 −0.365540
\(497\) 6.18305e27 0.301620
\(498\) 5.27806e27 0.252096
\(499\) −3.45711e28 −1.61681 −0.808403 0.588629i \(-0.799668\pi\)
−0.808403 + 0.588629i \(0.799668\pi\)
\(500\) 0 0
\(501\) −1.85175e28 −0.830397
\(502\) −5.78622e26 −0.0254101
\(503\) −2.28934e28 −0.984569 −0.492285 0.870434i \(-0.663838\pi\)
−0.492285 + 0.870434i \(0.663838\pi\)
\(504\) 3.26316e27 0.137441
\(505\) 0 0
\(506\) −8.34774e26 −0.0337278
\(507\) 3.36617e27 0.133215
\(508\) −7.73353e27 −0.299784
\(509\) −3.01380e28 −1.14440 −0.572199 0.820115i \(-0.693910\pi\)
−0.572199 + 0.820115i \(0.693910\pi\)
\(510\) 0 0
\(511\) −2.47355e28 −0.901367
\(512\) −2.60785e28 −0.930995
\(513\) 9.65008e27 0.337519
\(514\) −2.50796e27 −0.0859423
\(515\) 0 0
\(516\) −9.81222e27 −0.322809
\(517\) 9.35977e26 0.0301727
\(518\) 5.70004e27 0.180059
\(519\) −2.84091e28 −0.879429
\(520\) 0 0
\(521\) −7.99529e27 −0.237705 −0.118853 0.992912i \(-0.537922\pi\)
−0.118853 + 0.992912i \(0.537922\pi\)
\(522\) 5.86962e26 0.0171029
\(523\) −2.08661e28 −0.595901 −0.297950 0.954581i \(-0.596303\pi\)
−0.297950 + 0.954581i \(0.596303\pi\)
\(524\) 2.35739e28 0.659862
\(525\) 0 0
\(526\) 1.55845e28 0.419124
\(527\) −6.75441e27 −0.178064
\(528\) −1.91653e28 −0.495289
\(529\) −3.87006e28 −0.980468
\(530\) 0 0
\(531\) 2.45119e28 0.596876
\(532\) 6.25001e28 1.49214
\(533\) −6.71272e28 −1.57131
\(534\) −6.00322e27 −0.137785
\(535\) 0 0
\(536\) 1.76774e27 0.0390111
\(537\) 1.93716e27 0.0419214
\(538\) 1.10856e28 0.235259
\(539\) −9.29526e27 −0.193455
\(540\) 0 0
\(541\) −6.81219e28 −1.36369 −0.681844 0.731497i \(-0.738822\pi\)
−0.681844 + 0.731497i \(0.738822\pi\)
\(542\) 5.74442e27 0.112786
\(543\) 3.25329e28 0.626505
\(544\) −1.41302e28 −0.266907
\(545\) 0 0
\(546\) −5.90762e27 −0.107372
\(547\) 5.16259e27 0.0920450 0.0460225 0.998940i \(-0.485345\pi\)
0.0460225 + 0.998940i \(0.485345\pi\)
\(548\) −8.71379e28 −1.52409
\(549\) 2.28269e28 0.391686
\(550\) 0 0
\(551\) 2.31440e28 0.382250
\(552\) −2.27943e27 −0.0369373
\(553\) −4.32391e28 −0.687484
\(554\) −2.06228e28 −0.321733
\(555\) 0 0
\(556\) −4.72470e28 −0.709724
\(557\) 8.97767e28 1.32338 0.661689 0.749778i \(-0.269840\pi\)
0.661689 + 0.749778i \(0.269840\pi\)
\(558\) 2.36955e27 0.0342773
\(559\) 3.65703e28 0.519164
\(560\) 0 0
\(561\) −1.76445e28 −0.241268
\(562\) −3.33847e26 −0.00448039
\(563\) −7.85492e26 −0.0103467 −0.00517337 0.999987i \(-0.501647\pi\)
−0.00517337 + 0.999987i \(0.501647\pi\)
\(564\) 1.24147e27 0.0160511
\(565\) 0 0
\(566\) 1.50364e28 0.187315
\(567\) 8.18405e27 0.100080
\(568\) 1.27697e28 0.153293
\(569\) −9.58086e28 −1.12908 −0.564541 0.825405i \(-0.690947\pi\)
−0.564541 + 0.825405i \(0.690947\pi\)
\(570\) 0 0
\(571\) −6.50183e26 −0.00738511 −0.00369255 0.999993i \(-0.501175\pi\)
−0.00369255 + 0.999993i \(0.501175\pi\)
\(572\) 7.61596e28 0.849310
\(573\) −6.98355e28 −0.764632
\(574\) −3.53356e28 −0.379872
\(575\) 0 0
\(576\) −2.19549e28 −0.227560
\(577\) 1.67471e29 1.70448 0.852242 0.523148i \(-0.175242\pi\)
0.852242 + 0.523148i \(0.175242\pi\)
\(578\) 1.96402e28 0.196292
\(579\) 6.91975e28 0.679150
\(580\) 0 0
\(581\) 1.76498e29 1.67067
\(582\) 1.38238e28 0.128510
\(583\) 2.50280e28 0.228510
\(584\) −5.10856e28 −0.458104
\(585\) 0 0
\(586\) 2.99074e28 0.258734
\(587\) −2.23014e29 −1.89510 −0.947551 0.319603i \(-0.896450\pi\)
−0.947551 + 0.319603i \(0.896450\pi\)
\(588\) −1.23291e28 −0.102913
\(589\) 9.34317e28 0.766097
\(590\) 0 0
\(591\) −3.73010e28 −0.295156
\(592\) −9.14138e28 −0.710614
\(593\) 1.71015e29 1.30605 0.653024 0.757337i \(-0.273500\pi\)
0.653024 + 0.757337i \(0.273500\pi\)
\(594\) 6.18997e27 0.0464441
\(595\) 0 0
\(596\) −1.96363e28 −0.142225
\(597\) 1.06641e29 0.758914
\(598\) 4.12668e27 0.0288561
\(599\) −1.56747e29 −1.07701 −0.538504 0.842623i \(-0.681010\pi\)
−0.538504 + 0.842623i \(0.681010\pi\)
\(600\) 0 0
\(601\) −1.53561e29 −1.01883 −0.509413 0.860522i \(-0.670137\pi\)
−0.509413 + 0.860522i \(0.670137\pi\)
\(602\) 1.92505e28 0.125510
\(603\) 4.43352e27 0.0284064
\(604\) 3.58279e28 0.225597
\(605\) 0 0
\(606\) −3.47235e28 −0.211184
\(607\) −2.86065e29 −1.70995 −0.854976 0.518667i \(-0.826428\pi\)
−0.854976 + 0.518667i \(0.826428\pi\)
\(608\) 1.95459e29 1.14833
\(609\) 1.96280e28 0.113343
\(610\) 0 0
\(611\) −4.62697e27 −0.0258145
\(612\) −2.34035e28 −0.128349
\(613\) −4.90169e28 −0.264247 −0.132124 0.991233i \(-0.542180\pi\)
−0.132124 + 0.991233i \(0.542180\pi\)
\(614\) 6.00189e28 0.318068
\(615\) 0 0
\(616\) 8.25325e28 0.422695
\(617\) −2.77126e29 −1.39535 −0.697676 0.716413i \(-0.745783\pi\)
−0.697676 + 0.716413i \(0.745783\pi\)
\(618\) 1.59816e28 0.0791118
\(619\) −8.92512e28 −0.434373 −0.217186 0.976130i \(-0.569688\pi\)
−0.217186 + 0.976130i \(0.569688\pi\)
\(620\) 0 0
\(621\) −5.71683e27 −0.0268964
\(622\) 9.28631e28 0.429580
\(623\) −2.00748e29 −0.913116
\(624\) 9.47429e28 0.423749
\(625\) 0 0
\(626\) 3.06818e28 0.132694
\(627\) 2.44072e29 1.03803
\(628\) 2.32952e29 0.974296
\(629\) −8.41603e28 −0.346159
\(630\) 0 0
\(631\) 7.08364e28 0.281805 0.140902 0.990024i \(-0.455000\pi\)
0.140902 + 0.990024i \(0.455000\pi\)
\(632\) −8.93006e28 −0.349402
\(633\) −1.73976e28 −0.0669499
\(634\) 2.83087e28 0.107148
\(635\) 0 0
\(636\) 3.31968e28 0.121562
\(637\) 4.59508e28 0.165512
\(638\) 1.48456e28 0.0525992
\(639\) 3.20265e28 0.111622
\(640\) 0 0
\(641\) 2.59855e28 0.0876440 0.0438220 0.999039i \(-0.486047\pi\)
0.0438220 + 0.999039i \(0.486047\pi\)
\(642\) −4.86860e28 −0.161542
\(643\) 1.38976e29 0.453652 0.226826 0.973935i \(-0.427165\pi\)
0.226826 + 0.973935i \(0.427165\pi\)
\(644\) −3.70259e28 −0.118906
\(645\) 0 0
\(646\) 5.41400e28 0.168297
\(647\) −4.60493e29 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(648\) 1.69023e28 0.0508637
\(649\) 6.19959e29 1.83567
\(650\) 0 0
\(651\) 7.92377e28 0.227160
\(652\) 7.58243e28 0.213899
\(653\) 6.22853e29 1.72901 0.864504 0.502625i \(-0.167632\pi\)
0.864504 + 0.502625i \(0.167632\pi\)
\(654\) −9.18049e27 −0.0250784
\(655\) 0 0
\(656\) 5.66691e29 1.49919
\(657\) −1.28123e29 −0.333574
\(658\) −2.43562e27 −0.00624077
\(659\) 6.30181e29 1.58916 0.794582 0.607157i \(-0.207690\pi\)
0.794582 + 0.607157i \(0.207690\pi\)
\(660\) 0 0
\(661\) −3.89533e29 −0.951545 −0.475773 0.879568i \(-0.657832\pi\)
−0.475773 + 0.879568i \(0.657832\pi\)
\(662\) 3.35524e28 0.0806704
\(663\) 8.72252e28 0.206419
\(664\) 3.64517e29 0.849089
\(665\) 0 0
\(666\) 2.95247e28 0.0666356
\(667\) −1.37108e28 −0.0304609
\(668\) −6.21211e29 −1.35858
\(669\) 4.69120e29 1.00997
\(670\) 0 0
\(671\) 5.77344e29 1.20461
\(672\) 1.65765e29 0.340499
\(673\) 2.70633e29 0.547297 0.273648 0.961830i \(-0.411770\pi\)
0.273648 + 0.961830i \(0.411770\pi\)
\(674\) −2.10502e29 −0.419109
\(675\) 0 0
\(676\) 1.12926e29 0.217948
\(677\) −4.23921e29 −0.805572 −0.402786 0.915294i \(-0.631958\pi\)
−0.402786 + 0.915294i \(0.631958\pi\)
\(678\) 5.35977e28 0.100285
\(679\) 4.62267e29 0.851648
\(680\) 0 0
\(681\) −4.12857e29 −0.737488
\(682\) 5.99311e28 0.105418
\(683\) −9.87881e29 −1.71115 −0.855573 0.517682i \(-0.826795\pi\)
−0.855573 + 0.517682i \(0.826795\pi\)
\(684\) 3.23734e29 0.552203
\(685\) 0 0
\(686\) 1.52367e29 0.252051
\(687\) 1.16793e29 0.190271
\(688\) −3.08728e29 −0.495333
\(689\) −1.23725e29 −0.195504
\(690\) 0 0
\(691\) −1.20248e29 −0.184314 −0.0921569 0.995744i \(-0.529376\pi\)
−0.0921569 + 0.995744i \(0.529376\pi\)
\(692\) −9.53049e29 −1.43880
\(693\) 2.06993e29 0.307791
\(694\) −4.22911e28 −0.0619405
\(695\) 0 0
\(696\) 4.05372e28 0.0576045
\(697\) 5.21725e29 0.730293
\(698\) −2.87586e29 −0.396539
\(699\) −3.19170e29 −0.433522
\(700\) 0 0
\(701\) 3.01199e29 0.397022 0.198511 0.980099i \(-0.436389\pi\)
0.198511 + 0.980099i \(0.436389\pi\)
\(702\) −3.05999e28 −0.0397356
\(703\) 1.16416e30 1.48930
\(704\) −5.55288e29 −0.699850
\(705\) 0 0
\(706\) −3.04126e29 −0.372053
\(707\) −1.16115e30 −1.39954
\(708\) 8.22306e29 0.976527
\(709\) −4.42994e29 −0.518337 −0.259168 0.965832i \(-0.583448\pi\)
−0.259168 + 0.965832i \(0.583448\pi\)
\(710\) 0 0
\(711\) −2.23967e29 −0.254421
\(712\) −4.14599e29 −0.464075
\(713\) −5.53502e28 −0.0610491
\(714\) 4.59151e28 0.0499028
\(715\) 0 0
\(716\) 6.49864e28 0.0685861
\(717\) 9.03763e29 0.939947
\(718\) 4.47176e28 0.0458323
\(719\) 1.48484e30 1.49978 0.749888 0.661565i \(-0.230108\pi\)
0.749888 + 0.661565i \(0.230108\pi\)
\(720\) 0 0
\(721\) 5.34425e29 0.524283
\(722\) −5.05422e29 −0.488668
\(723\) 7.91918e29 0.754619
\(724\) 1.09139e30 1.02500
\(725\) 0 0
\(726\) 7.58819e27 0.00692315
\(727\) 1.41765e30 1.27484 0.637421 0.770516i \(-0.280001\pi\)
0.637421 + 0.770516i \(0.280001\pi\)
\(728\) −4.07997e29 −0.361640
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) −2.84231e29 −0.241290
\(732\) 7.65782e29 0.640824
\(733\) 1.26744e30 1.04553 0.522764 0.852477i \(-0.324901\pi\)
0.522764 + 0.852477i \(0.324901\pi\)
\(734\) −2.26095e28 −0.0183858
\(735\) 0 0
\(736\) −1.15793e29 −0.0915089
\(737\) 1.12133e29 0.0873627
\(738\) −1.83029e29 −0.140581
\(739\) −1.86110e30 −1.40930 −0.704650 0.709555i \(-0.748896\pi\)
−0.704650 + 0.709555i \(0.748896\pi\)
\(740\) 0 0
\(741\) −1.20656e30 −0.888091
\(742\) −6.51285e28 −0.0472640
\(743\) 5.15013e29 0.368499 0.184249 0.982880i \(-0.441015\pi\)
0.184249 + 0.982880i \(0.441015\pi\)
\(744\) 1.63647e29 0.115450
\(745\) 0 0
\(746\) −5.10895e29 −0.350408
\(747\) 9.14214e29 0.618275
\(748\) −5.91926e29 −0.394731
\(749\) −1.62806e30 −1.07056
\(750\) 0 0
\(751\) −1.60839e30 −1.02842 −0.514211 0.857663i \(-0.671915\pi\)
−0.514211 + 0.857663i \(0.671915\pi\)
\(752\) 3.90611e28 0.0246296
\(753\) −1.00223e29 −0.0623191
\(754\) −7.33885e28 −0.0450017
\(755\) 0 0
\(756\) 2.74552e29 0.163737
\(757\) −2.14104e29 −0.125927 −0.0629634 0.998016i \(-0.520055\pi\)
−0.0629634 + 0.998016i \(0.520055\pi\)
\(758\) −8.89718e28 −0.0516090
\(759\) −1.44591e29 −0.0827187
\(760\) 0 0
\(761\) 1.25293e30 0.697250 0.348625 0.937262i \(-0.386649\pi\)
0.348625 + 0.937262i \(0.386649\pi\)
\(762\) 7.85887e28 0.0431353
\(763\) −3.06996e29 −0.166198
\(764\) −2.34279e30 −1.25099
\(765\) 0 0
\(766\) −2.29516e29 −0.119237
\(767\) −3.06475e30 −1.57052
\(768\) −5.60474e29 −0.283310
\(769\) 2.53517e30 1.26410 0.632048 0.774929i \(-0.282215\pi\)
0.632048 + 0.774929i \(0.282215\pi\)
\(770\) 0 0
\(771\) −4.34403e29 −0.210777
\(772\) 2.32139e30 1.11113
\(773\) −1.70099e30 −0.803187 −0.401594 0.915818i \(-0.631544\pi\)
−0.401594 + 0.915818i \(0.631544\pi\)
\(774\) 9.97125e28 0.0464483
\(775\) 0 0
\(776\) 9.54707e29 0.432835
\(777\) 9.87306e29 0.441602
\(778\) −3.36069e29 −0.148301
\(779\) −7.21686e30 −3.14199
\(780\) 0 0
\(781\) 8.10022e29 0.343290
\(782\) −3.20733e28 −0.0134113
\(783\) 1.01668e29 0.0419454
\(784\) −3.87919e29 −0.157915
\(785\) 0 0
\(786\) −2.39560e29 −0.0949460
\(787\) 4.98615e30 1.94998 0.974991 0.222246i \(-0.0713388\pi\)
0.974991 + 0.222246i \(0.0713388\pi\)
\(788\) −1.25135e30 −0.482894
\(789\) 2.69940e30 1.02792
\(790\) 0 0
\(791\) 1.79231e30 0.664598
\(792\) 4.27496e29 0.156429
\(793\) −2.85408e30 −1.03062
\(794\) 8.31160e29 0.296189
\(795\) 0 0
\(796\) 3.57751e30 1.24163
\(797\) −4.48685e30 −1.53684 −0.768420 0.639946i \(-0.778957\pi\)
−0.768420 + 0.639946i \(0.778957\pi\)
\(798\) −6.35130e29 −0.214700
\(799\) 3.59616e28 0.0119977
\(800\) 0 0
\(801\) −1.03982e30 −0.337922
\(802\) −1.11306e29 −0.0357016
\(803\) −3.24052e30 −1.02589
\(804\) 1.48732e29 0.0464747
\(805\) 0 0
\(806\) −2.96267e29 −0.0901915
\(807\) 1.92014e30 0.576981
\(808\) −2.39810e30 −0.711292
\(809\) −5.22233e30 −1.52899 −0.764496 0.644629i \(-0.777012\pi\)
−0.764496 + 0.644629i \(0.777012\pi\)
\(810\) 0 0
\(811\) −6.36541e30 −1.81597 −0.907983 0.419007i \(-0.862378\pi\)
−0.907983 + 0.419007i \(0.862378\pi\)
\(812\) 6.58466e29 0.185436
\(813\) 9.94992e29 0.276611
\(814\) 7.46744e29 0.204935
\(815\) 0 0
\(816\) −7.36359e29 −0.196944
\(817\) 3.93168e30 1.03812
\(818\) −1.45145e30 −0.378348
\(819\) −1.02326e30 −0.263333
\(820\) 0 0
\(821\) −7.95104e29 −0.199444 −0.0997220 0.995015i \(-0.531795\pi\)
−0.0997220 + 0.995015i \(0.531795\pi\)
\(822\) 8.85501e29 0.219298
\(823\) 1.00191e30 0.244980 0.122490 0.992470i \(-0.460912\pi\)
0.122490 + 0.992470i \(0.460912\pi\)
\(824\) 1.10373e30 0.266458
\(825\) 0 0
\(826\) −1.61328e30 −0.379680
\(827\) −4.61386e30 −1.07215 −0.536076 0.844170i \(-0.680094\pi\)
−0.536076 + 0.844170i \(0.680094\pi\)
\(828\) −1.91784e29 −0.0440042
\(829\) −2.84827e30 −0.645294 −0.322647 0.946519i \(-0.604573\pi\)
−0.322647 + 0.946519i \(0.604573\pi\)
\(830\) 0 0
\(831\) −3.57209e30 −0.789062
\(832\) 2.74505e30 0.598763
\(833\) −3.57138e29 −0.0769243
\(834\) 4.80127e29 0.102121
\(835\) 0 0
\(836\) 8.18794e30 1.69828
\(837\) 4.10430e29 0.0840662
\(838\) 1.70481e29 0.0344836
\(839\) −6.35489e30 −1.26943 −0.634713 0.772748i \(-0.718882\pi\)
−0.634713 + 0.772748i \(0.718882\pi\)
\(840\) 0 0
\(841\) −4.88901e30 −0.952496
\(842\) 1.57989e30 0.303984
\(843\) −5.78257e28 −0.0109883
\(844\) −5.83641e29 −0.109534
\(845\) 0 0
\(846\) −1.26159e28 −0.00230956
\(847\) 2.53749e29 0.0458806
\(848\) 1.04449e30 0.186530
\(849\) 2.60446e30 0.459396
\(850\) 0 0
\(851\) −6.89666e29 −0.118680
\(852\) 1.07440e30 0.182621
\(853\) −7.05513e30 −1.18451 −0.592257 0.805749i \(-0.701763\pi\)
−0.592257 + 0.805749i \(0.701763\pi\)
\(854\) −1.50238e30 −0.249156
\(855\) 0 0
\(856\) −3.36239e30 −0.544093
\(857\) −3.70327e30 −0.591953 −0.295976 0.955195i \(-0.595645\pi\)
−0.295976 + 0.955195i \(0.595645\pi\)
\(858\) −7.73939e29 −0.122205
\(859\) −5.45289e29 −0.0850548 −0.0425274 0.999095i \(-0.513541\pi\)
−0.0425274 + 0.999095i \(0.513541\pi\)
\(860\) 0 0
\(861\) −6.12048e30 −0.931649
\(862\) −1.12748e30 −0.169544
\(863\) 7.07308e30 1.05074 0.525370 0.850874i \(-0.323927\pi\)
0.525370 + 0.850874i \(0.323927\pi\)
\(864\) 8.58618e29 0.126010
\(865\) 0 0
\(866\) 8.80098e29 0.126065
\(867\) 3.40188e30 0.481414
\(868\) 2.65821e30 0.371648
\(869\) −5.66462e30 −0.782461
\(870\) 0 0
\(871\) −5.54327e29 −0.0747439
\(872\) −6.34030e29 −0.0844669
\(873\) 2.39442e30 0.315174
\(874\) 4.43660e29 0.0577006
\(875\) 0 0
\(876\) −4.29819e30 −0.545748
\(877\) −1.58002e31 −1.98229 −0.991146 0.132779i \(-0.957610\pi\)
−0.991146 + 0.132779i \(0.957610\pi\)
\(878\) −2.22179e30 −0.275429
\(879\) 5.18026e30 0.634553
\(880\) 0 0
\(881\) 9.64321e30 1.15339 0.576693 0.816961i \(-0.304343\pi\)
0.576693 + 0.816961i \(0.304343\pi\)
\(882\) 1.25289e29 0.0148079
\(883\) 7.91697e30 0.924638 0.462319 0.886714i \(-0.347018\pi\)
0.462319 + 0.886714i \(0.347018\pi\)
\(884\) 2.92617e30 0.337715
\(885\) 0 0
\(886\) 3.50776e30 0.395344
\(887\) −7.76439e30 −0.864787 −0.432394 0.901685i \(-0.642331\pi\)
−0.432394 + 0.901685i \(0.642331\pi\)
\(888\) 2.03906e30 0.224436
\(889\) 2.62801e30 0.285863
\(890\) 0 0
\(891\) 1.07217e30 0.113906
\(892\) 1.57377e31 1.65238
\(893\) −4.97447e29 −0.0516186
\(894\) 1.99546e29 0.0204644
\(895\) 0 0
\(896\) 7.33221e30 0.734515
\(897\) 7.14782e29 0.0707706
\(898\) −8.60752e29 −0.0842318
\(899\) 9.84344e29 0.0952073
\(900\) 0 0
\(901\) 9.61613e29 0.0908636
\(902\) −4.62920e30 −0.432352
\(903\) 3.33438e30 0.307818
\(904\) 3.70160e30 0.337770
\(905\) 0 0
\(906\) −3.64086e29 −0.0324607
\(907\) 1.25327e31 1.10451 0.552253 0.833677i \(-0.313768\pi\)
0.552253 + 0.833677i \(0.313768\pi\)
\(908\) −1.38502e31 −1.20658
\(909\) −6.01446e30 −0.517936
\(910\) 0 0
\(911\) 9.93009e30 0.835623 0.417811 0.908534i \(-0.362797\pi\)
0.417811 + 0.908534i \(0.362797\pi\)
\(912\) 1.01858e31 0.847326
\(913\) 2.31225e31 1.90148
\(914\) 1.65398e30 0.134460
\(915\) 0 0
\(916\) 3.91809e30 0.311295
\(917\) −8.01087e30 −0.629219
\(918\) 2.37828e29 0.0184678
\(919\) 2.22319e31 1.70673 0.853364 0.521316i \(-0.174559\pi\)
0.853364 + 0.521316i \(0.174559\pi\)
\(920\) 0 0
\(921\) 1.03959e31 0.780072
\(922\) 3.45709e30 0.256470
\(923\) −4.00431e30 −0.293704
\(924\) 6.94403e30 0.503566
\(925\) 0 0
\(926\) −3.12089e30 −0.221240
\(927\) 2.76818e30 0.194024
\(928\) 2.05925e30 0.142710
\(929\) −1.62508e31 −1.11355 −0.556773 0.830664i \(-0.687961\pi\)
−0.556773 + 0.830664i \(0.687961\pi\)
\(930\) 0 0
\(931\) 4.94018e30 0.330957
\(932\) −1.07073e31 −0.709271
\(933\) 1.60848e31 1.05356
\(934\) 1.22145e30 0.0791103
\(935\) 0 0
\(936\) −2.11331e30 −0.133834
\(937\) 1.81129e30 0.113428 0.0567142 0.998390i \(-0.481938\pi\)
0.0567142 + 0.998390i \(0.481938\pi\)
\(938\) −2.91797e29 −0.0180697
\(939\) 5.31441e30 0.325436
\(940\) 0 0
\(941\) −3.25431e31 −1.94880 −0.974402 0.224812i \(-0.927823\pi\)
−0.974402 + 0.224812i \(0.927823\pi\)
\(942\) −2.36728e30 −0.140189
\(943\) 4.27536e30 0.250380
\(944\) 2.58727e31 1.49843
\(945\) 0 0
\(946\) 2.52195e30 0.142850
\(947\) −1.36434e31 −0.764270 −0.382135 0.924107i \(-0.624811\pi\)
−0.382135 + 0.924107i \(0.624811\pi\)
\(948\) −7.51348e30 −0.416249
\(949\) 1.60194e31 0.877710
\(950\) 0 0
\(951\) 4.90335e30 0.262783
\(952\) 3.17102e30 0.168078
\(953\) −1.03136e31 −0.540673 −0.270337 0.962766i \(-0.587135\pi\)
−0.270337 + 0.962766i \(0.587135\pi\)
\(954\) −3.37348e29 −0.0174912
\(955\) 0 0
\(956\) 3.03188e31 1.53781
\(957\) 2.57140e30 0.129001
\(958\) 3.28328e30 0.162918
\(959\) 2.96111e31 1.45331
\(960\) 0 0
\(961\) −1.68517e31 −0.809187
\(962\) −3.69151e30 −0.175334
\(963\) −8.43291e30 −0.396188
\(964\) 2.65667e31 1.23461
\(965\) 0 0
\(966\) 3.76260e29 0.0171091
\(967\) −3.33709e31 −1.50103 −0.750517 0.660851i \(-0.770196\pi\)
−0.750517 + 0.660851i \(0.770196\pi\)
\(968\) 5.24061e29 0.0233180
\(969\) 9.37760e30 0.412755
\(970\) 0 0
\(971\) 2.13037e30 0.0917598 0.0458799 0.998947i \(-0.485391\pi\)
0.0458799 + 0.998947i \(0.485391\pi\)
\(972\) 1.42211e30 0.0605950
\(973\) 1.60555e31 0.676765
\(974\) −6.58611e29 −0.0274637
\(975\) 0 0
\(976\) 2.40943e31 0.983310
\(977\) −4.34834e31 −1.75562 −0.877811 0.479008i \(-0.840997\pi\)
−0.877811 + 0.479008i \(0.840997\pi\)
\(978\) −7.70532e29 −0.0307774
\(979\) −2.62993e31 −1.03926
\(980\) 0 0
\(981\) −1.59015e30 −0.0615056
\(982\) 2.94174e30 0.112573
\(983\) 4.41457e31 1.67138 0.835692 0.549198i \(-0.185067\pi\)
0.835692 + 0.549198i \(0.185067\pi\)
\(984\) −1.26405e31 −0.473494
\(985\) 0 0
\(986\) 5.70389e29 0.0209153
\(987\) −4.21875e29 −0.0153057
\(988\) −4.04768e31 −1.45297
\(989\) −2.32918e30 −0.0827260
\(990\) 0 0
\(991\) −3.77164e31 −1.31147 −0.655734 0.754992i \(-0.727641\pi\)
−0.655734 + 0.754992i \(0.727641\pi\)
\(992\) 8.31311e30 0.286017
\(993\) 5.81161e30 0.197847
\(994\) −2.10786e30 −0.0710044
\(995\) 0 0
\(996\) 3.06694e31 1.01154
\(997\) −2.89594e31 −0.945126 −0.472563 0.881297i \(-0.656671\pi\)
−0.472563 + 0.881297i \(0.656671\pi\)
\(998\) 1.17856e31 0.380612
\(999\) 5.11398e30 0.163426
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 75.22.a.j.1.3 yes 6
5.2 odd 4 75.22.b.i.49.6 12
5.3 odd 4 75.22.b.i.49.7 12
5.4 even 2 75.22.a.i.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.22.a.i.1.4 6 5.4 even 2
75.22.a.j.1.3 yes 6 1.1 even 1 trivial
75.22.b.i.49.6 12 5.2 odd 4
75.22.b.i.49.7 12 5.3 odd 4