Properties

Label 198.10.a.n.1.1
Level $198$
Weight $10$
Character 198.1
Self dual yes
Analytic conductor $101.977$
Analytic rank $1$
Dimension $2$
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] = [198,10,Mod(1,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("198.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 198.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: \(101.977095560\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{889}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(15.4081\) of defining polynomial
Character \(\chi\) \(=\) 198.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +256.000 q^{4} -1185.58 q^{5} +1174.66 q^{7} +4096.00 q^{8} +O(q^{10})\) \(q+16.0000 q^{2} +256.000 q^{4} -1185.58 q^{5} +1174.66 q^{7} +4096.00 q^{8} -18969.3 q^{10} -14641.0 q^{11} +82461.9 q^{13} +18794.5 q^{14} +65536.0 q^{16} -440826. q^{17} +804401. q^{19} -303509. q^{20} -234256. q^{22} +228403. q^{23} -547523. q^{25} +1.31939e6 q^{26} +300712. q^{28} -2.28497e6 q^{29} +2.56508e6 q^{31} +1.04858e6 q^{32} -7.05322e6 q^{34} -1.39265e6 q^{35} -4.68709e6 q^{37} +1.28704e7 q^{38} -4.85614e6 q^{40} -3.30235e6 q^{41} +7.81398e6 q^{43} -3.74810e6 q^{44} +3.65445e6 q^{46} +2.85515e6 q^{47} -3.89738e7 q^{49} -8.76036e6 q^{50} +2.11103e7 q^{52} -9.97047e6 q^{53} +1.73581e7 q^{55} +4.81140e6 q^{56} -3.65595e7 q^{58} -6.31658e7 q^{59} -2.12529e8 q^{61} +4.10412e7 q^{62} +1.67772e7 q^{64} -9.77653e7 q^{65} -2.27892e8 q^{67} -1.12852e8 q^{68} -2.22824e7 q^{70} +1.55605e8 q^{71} -1.04177e8 q^{73} -7.49935e7 q^{74} +2.05927e8 q^{76} -1.71982e7 q^{77} -2.96624e8 q^{79} -7.76982e7 q^{80} -5.28376e7 q^{82} +7.38793e8 q^{83} +5.22635e8 q^{85} +1.25024e8 q^{86} -5.99695e7 q^{88} -9.00049e8 q^{89} +9.68645e7 q^{91} +5.84712e7 q^{92} +4.56823e7 q^{94} -9.53683e8 q^{95} +1.29157e9 q^{97} -6.23581e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} + 512 q^{4} + 521 q^{5} - 7490 q^{7} + 8192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{2} + 512 q^{4} + 521 q^{5} - 7490 q^{7} + 8192 q^{8} + 8336 q^{10} - 29282 q^{11} + 150314 q^{13} - 119840 q^{14} + 131072 q^{16} - 690472 q^{17} + 511212 q^{19} + 133376 q^{20} - 468512 q^{22} - 874751 q^{23} + 411771 q^{25} + 2405024 q^{26} - 1917440 q^{28} + 2951058 q^{29} - 5818705 q^{31} + 2097152 q^{32} - 11047552 q^{34} - 16179590 q^{35} + 2658905 q^{37} + 8179392 q^{38} + 2134016 q^{40} - 13427994 q^{41} - 17820762 q^{43} - 7496192 q^{44} - 13996016 q^{46} - 56044104 q^{47} - 4251114 q^{49} + 6588336 q^{50} + 38480384 q^{52} - 96842752 q^{53} - 7627961 q^{55} - 30679040 q^{56} + 47216928 q^{58} + 119136183 q^{59} - 90424326 q^{61} - 93099280 q^{62} + 33554432 q^{64} + 18029712 q^{65} - 295944891 q^{67} - 176760832 q^{68} - 258873440 q^{70} + 322953267 q^{71} - 255975514 q^{73} + 42542480 q^{74} + 130870272 q^{76} + 109661090 q^{77} - 889658 q^{79} + 34144256 q^{80} - 214847904 q^{82} + 277699042 q^{83} + 96595042 q^{85} - 285132192 q^{86} - 119939072 q^{88} - 1363672217 q^{89} - 491050280 q^{91} - 223936256 q^{92} - 896705664 q^{94} - 1454033872 q^{95} + 1398434043 q^{97} - 68017824 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 0 0
\(4\) 256.000 0.500000
\(5\) −1185.58 −0.848333 −0.424166 0.905584i \(-0.639433\pi\)
−0.424166 + 0.905584i \(0.639433\pi\)
\(6\) 0 0
\(7\) 1174.66 0.184914 0.0924570 0.995717i \(-0.470528\pi\)
0.0924570 + 0.995717i \(0.470528\pi\)
\(8\) 4096.00 0.353553
\(9\) 0 0
\(10\) −18969.3 −0.599862
\(11\) −14641.0 −0.301511
\(12\) 0 0
\(13\) 82461.9 0.800771 0.400386 0.916347i \(-0.368876\pi\)
0.400386 + 0.916347i \(0.368876\pi\)
\(14\) 18794.5 0.130754
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) −440826. −1.28011 −0.640055 0.768329i \(-0.721089\pi\)
−0.640055 + 0.768329i \(0.721089\pi\)
\(18\) 0 0
\(19\) 804401. 1.41606 0.708030 0.706183i \(-0.249584\pi\)
0.708030 + 0.706183i \(0.249584\pi\)
\(20\) −303509. −0.424166
\(21\) 0 0
\(22\) −234256. −0.213201
\(23\) 228403. 0.170187 0.0850936 0.996373i \(-0.472881\pi\)
0.0850936 + 0.996373i \(0.472881\pi\)
\(24\) 0 0
\(25\) −547523. −0.280332
\(26\) 1.31939e6 0.566231
\(27\) 0 0
\(28\) 300712. 0.0924570
\(29\) −2.28497e6 −0.599914 −0.299957 0.953953i \(-0.596972\pi\)
−0.299957 + 0.953953i \(0.596972\pi\)
\(30\) 0 0
\(31\) 2.56508e6 0.498853 0.249427 0.968394i \(-0.419758\pi\)
0.249427 + 0.968394i \(0.419758\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 0 0
\(34\) −7.05322e6 −0.905175
\(35\) −1.39265e6 −0.156869
\(36\) 0 0
\(37\) −4.68709e6 −0.411146 −0.205573 0.978642i \(-0.565906\pi\)
−0.205573 + 0.978642i \(0.565906\pi\)
\(38\) 1.28704e7 1.00131
\(39\) 0 0
\(40\) −4.85614e6 −0.299931
\(41\) −3.30235e6 −0.182514 −0.0912569 0.995827i \(-0.529088\pi\)
−0.0912569 + 0.995827i \(0.529088\pi\)
\(42\) 0 0
\(43\) 7.81398e6 0.348549 0.174275 0.984697i \(-0.444242\pi\)
0.174275 + 0.984697i \(0.444242\pi\)
\(44\) −3.74810e6 −0.150756
\(45\) 0 0
\(46\) 3.65445e6 0.120341
\(47\) 2.85515e6 0.0853470 0.0426735 0.999089i \(-0.486412\pi\)
0.0426735 + 0.999089i \(0.486412\pi\)
\(48\) 0 0
\(49\) −3.89738e7 −0.965807
\(50\) −8.76036e6 −0.198224
\(51\) 0 0
\(52\) 2.11103e7 0.400386
\(53\) −9.97047e6 −0.173570 −0.0867849 0.996227i \(-0.527659\pi\)
−0.0867849 + 0.996227i \(0.527659\pi\)
\(54\) 0 0
\(55\) 1.73581e7 0.255782
\(56\) 4.81140e6 0.0653770
\(57\) 0 0
\(58\) −3.65595e7 −0.424203
\(59\) −6.31658e7 −0.678653 −0.339326 0.940669i \(-0.610199\pi\)
−0.339326 + 0.940669i \(0.610199\pi\)
\(60\) 0 0
\(61\) −2.12529e8 −1.96532 −0.982660 0.185414i \(-0.940637\pi\)
−0.982660 + 0.185414i \(0.940637\pi\)
\(62\) 4.10412e7 0.352743
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) −9.77653e7 −0.679320
\(66\) 0 0
\(67\) −2.27892e8 −1.38163 −0.690815 0.723031i \(-0.742748\pi\)
−0.690815 + 0.723031i \(0.742748\pi\)
\(68\) −1.12852e8 −0.640055
\(69\) 0 0
\(70\) −2.22824e7 −0.110923
\(71\) 1.55605e8 0.726708 0.363354 0.931651i \(-0.381632\pi\)
0.363354 + 0.931651i \(0.381632\pi\)
\(72\) 0 0
\(73\) −1.04177e8 −0.429357 −0.214679 0.976685i \(-0.568870\pi\)
−0.214679 + 0.976685i \(0.568870\pi\)
\(74\) −7.49935e7 −0.290724
\(75\) 0 0
\(76\) 2.05927e8 0.708030
\(77\) −1.71982e7 −0.0557537
\(78\) 0 0
\(79\) −2.96624e8 −0.856811 −0.428405 0.903587i \(-0.640924\pi\)
−0.428405 + 0.903587i \(0.640924\pi\)
\(80\) −7.76982e7 −0.212083
\(81\) 0 0
\(82\) −5.28376e7 −0.129057
\(83\) 7.38793e8 1.70872 0.854361 0.519680i \(-0.173949\pi\)
0.854361 + 0.519680i \(0.173949\pi\)
\(84\) 0 0
\(85\) 5.22635e8 1.08596
\(86\) 1.25024e8 0.246462
\(87\) 0 0
\(88\) −5.99695e7 −0.106600
\(89\) −9.00049e8 −1.52059 −0.760293 0.649580i \(-0.774945\pi\)
−0.760293 + 0.649580i \(0.774945\pi\)
\(90\) 0 0
\(91\) 9.68645e7 0.148074
\(92\) 5.84712e7 0.0850936
\(93\) 0 0
\(94\) 4.56823e7 0.0603494
\(95\) −9.53683e8 −1.20129
\(96\) 0 0
\(97\) 1.29157e9 1.48131 0.740653 0.671888i \(-0.234516\pi\)
0.740653 + 0.671888i \(0.234516\pi\)
\(98\) −6.23581e8 −0.682929
\(99\) 0 0
\(100\) −1.40166e8 −0.140166
\(101\) −1.45652e9 −1.39275 −0.696373 0.717681i \(-0.745204\pi\)
−0.696373 + 0.717681i \(0.745204\pi\)
\(102\) 0 0
\(103\) −1.95971e9 −1.71563 −0.857814 0.513960i \(-0.828178\pi\)
−0.857814 + 0.513960i \(0.828178\pi\)
\(104\) 3.37764e8 0.283115
\(105\) 0 0
\(106\) −1.59528e8 −0.122732
\(107\) −6.07627e8 −0.448136 −0.224068 0.974573i \(-0.571934\pi\)
−0.224068 + 0.974573i \(0.571934\pi\)
\(108\) 0 0
\(109\) −1.14761e9 −0.778708 −0.389354 0.921088i \(-0.627302\pi\)
−0.389354 + 0.921088i \(0.627302\pi\)
\(110\) 2.77729e8 0.180865
\(111\) 0 0
\(112\) 7.69823e7 0.0462285
\(113\) 2.54053e8 0.146579 0.0732895 0.997311i \(-0.476650\pi\)
0.0732895 + 0.997311i \(0.476650\pi\)
\(114\) 0 0
\(115\) −2.70790e8 −0.144375
\(116\) −5.84952e8 −0.299957
\(117\) 0 0
\(118\) −1.01065e9 −0.479880
\(119\) −5.17820e8 −0.236710
\(120\) 0 0
\(121\) 2.14359e8 0.0909091
\(122\) −3.40046e9 −1.38969
\(123\) 0 0
\(124\) 6.56660e8 0.249427
\(125\) 2.96472e9 1.08615
\(126\) 0 0
\(127\) −1.26227e8 −0.0430561 −0.0215280 0.999768i \(-0.506853\pi\)
−0.0215280 + 0.999768i \(0.506853\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 0 0
\(130\) −1.56425e9 −0.480352
\(131\) 1.87859e9 0.557328 0.278664 0.960389i \(-0.410108\pi\)
0.278664 + 0.960389i \(0.410108\pi\)
\(132\) 0 0
\(133\) 9.44895e8 0.261849
\(134\) −3.64627e9 −0.976960
\(135\) 0 0
\(136\) −1.80563e9 −0.452587
\(137\) −6.32338e9 −1.53358 −0.766791 0.641897i \(-0.778148\pi\)
−0.766791 + 0.641897i \(0.778148\pi\)
\(138\) 0 0
\(139\) −7.82379e9 −1.77767 −0.888833 0.458230i \(-0.848483\pi\)
−0.888833 + 0.458230i \(0.848483\pi\)
\(140\) −3.56519e8 −0.0784343
\(141\) 0 0
\(142\) 2.48967e9 0.513860
\(143\) −1.20733e9 −0.241442
\(144\) 0 0
\(145\) 2.70901e9 0.508927
\(146\) −1.66683e9 −0.303602
\(147\) 0 0
\(148\) −1.19990e9 −0.205573
\(149\) 2.69524e8 0.0447981 0.0223991 0.999749i \(-0.492870\pi\)
0.0223991 + 0.999749i \(0.492870\pi\)
\(150\) 0 0
\(151\) 1.18223e10 1.85057 0.925286 0.379270i \(-0.123825\pi\)
0.925286 + 0.379270i \(0.123825\pi\)
\(152\) 3.29483e9 0.500653
\(153\) 0 0
\(154\) −2.75170e8 −0.0394238
\(155\) −3.04111e9 −0.423194
\(156\) 0 0
\(157\) 3.66990e9 0.482065 0.241032 0.970517i \(-0.422514\pi\)
0.241032 + 0.970517i \(0.422514\pi\)
\(158\) −4.74599e9 −0.605857
\(159\) 0 0
\(160\) −1.24317e9 −0.149965
\(161\) 2.68295e8 0.0314700
\(162\) 0 0
\(163\) −1.30610e10 −1.44921 −0.724605 0.689165i \(-0.757978\pi\)
−0.724605 + 0.689165i \(0.757978\pi\)
\(164\) −8.45401e8 −0.0912569
\(165\) 0 0
\(166\) 1.18207e10 1.20825
\(167\) 1.16911e9 0.116314 0.0581571 0.998307i \(-0.481478\pi\)
0.0581571 + 0.998307i \(0.481478\pi\)
\(168\) 0 0
\(169\) −3.80453e9 −0.358765
\(170\) 8.36217e9 0.767889
\(171\) 0 0
\(172\) 2.00038e9 0.174275
\(173\) −1.90893e10 −1.62025 −0.810124 0.586258i \(-0.800601\pi\)
−0.810124 + 0.586258i \(0.800601\pi\)
\(174\) 0 0
\(175\) −6.43151e8 −0.0518373
\(176\) −9.59513e8 −0.0753778
\(177\) 0 0
\(178\) −1.44008e10 −1.07522
\(179\) 1.99470e10 1.45224 0.726122 0.687566i \(-0.241321\pi\)
0.726122 + 0.687566i \(0.241321\pi\)
\(180\) 0 0
\(181\) −8.20947e9 −0.568541 −0.284270 0.958744i \(-0.591751\pi\)
−0.284270 + 0.958744i \(0.591751\pi\)
\(182\) 1.54983e9 0.104704
\(183\) 0 0
\(184\) 9.35539e8 0.0601703
\(185\) 5.55693e9 0.348788
\(186\) 0 0
\(187\) 6.45414e9 0.385968
\(188\) 7.30918e8 0.0426735
\(189\) 0 0
\(190\) −1.52589e10 −0.849440
\(191\) −1.24525e10 −0.677027 −0.338513 0.940962i \(-0.609924\pi\)
−0.338513 + 0.940962i \(0.609924\pi\)
\(192\) 0 0
\(193\) 1.60846e9 0.0834456 0.0417228 0.999129i \(-0.486715\pi\)
0.0417228 + 0.999129i \(0.486715\pi\)
\(194\) 2.06651e10 1.04744
\(195\) 0 0
\(196\) −9.97729e9 −0.482903
\(197\) −4.02780e10 −1.90533 −0.952664 0.304024i \(-0.901670\pi\)
−0.952664 + 0.304024i \(0.901670\pi\)
\(198\) 0 0
\(199\) 3.22083e10 1.45589 0.727945 0.685636i \(-0.240476\pi\)
0.727945 + 0.685636i \(0.240476\pi\)
\(200\) −2.24265e9 −0.0991122
\(201\) 0 0
\(202\) −2.33044e10 −0.984819
\(203\) −2.68405e9 −0.110933
\(204\) 0 0
\(205\) 3.91520e9 0.154832
\(206\) −3.13553e10 −1.21313
\(207\) 0 0
\(208\) 5.40423e9 0.200193
\(209\) −1.17772e10 −0.426958
\(210\) 0 0
\(211\) 1.06111e10 0.368544 0.184272 0.982875i \(-0.441007\pi\)
0.184272 + 0.982875i \(0.441007\pi\)
\(212\) −2.55244e9 −0.0867849
\(213\) 0 0
\(214\) −9.72203e9 −0.316880
\(215\) −9.26410e9 −0.295686
\(216\) 0 0
\(217\) 3.01309e9 0.0922450
\(218\) −1.83618e10 −0.550630
\(219\) 0 0
\(220\) 4.44367e9 0.127891
\(221\) −3.63514e10 −1.02508
\(222\) 0 0
\(223\) −2.11175e10 −0.571835 −0.285918 0.958254i \(-0.592298\pi\)
−0.285918 + 0.958254i \(0.592298\pi\)
\(224\) 1.23172e9 0.0326885
\(225\) 0 0
\(226\) 4.06485e9 0.103647
\(227\) 5.33286e9 0.133304 0.0666521 0.997776i \(-0.478768\pi\)
0.0666521 + 0.997776i \(0.478768\pi\)
\(228\) 0 0
\(229\) −2.62407e10 −0.630545 −0.315272 0.949001i \(-0.602096\pi\)
−0.315272 + 0.949001i \(0.602096\pi\)
\(230\) −4.33265e9 −0.102089
\(231\) 0 0
\(232\) −9.35923e9 −0.212102
\(233\) −4.77174e10 −1.06066 −0.530329 0.847792i \(-0.677932\pi\)
−0.530329 + 0.847792i \(0.677932\pi\)
\(234\) 0 0
\(235\) −3.38501e9 −0.0724026
\(236\) −1.61704e10 −0.339326
\(237\) 0 0
\(238\) −8.28512e9 −0.167380
\(239\) 2.64340e10 0.524050 0.262025 0.965061i \(-0.415610\pi\)
0.262025 + 0.965061i \(0.415610\pi\)
\(240\) 0 0
\(241\) −4.95287e10 −0.945758 −0.472879 0.881127i \(-0.656785\pi\)
−0.472879 + 0.881127i \(0.656785\pi\)
\(242\) 3.42974e9 0.0642824
\(243\) 0 0
\(244\) −5.44074e10 −0.982660
\(245\) 4.62066e10 0.819325
\(246\) 0 0
\(247\) 6.63325e10 1.13394
\(248\) 1.05066e10 0.176371
\(249\) 0 0
\(250\) 4.74355e10 0.768022
\(251\) 8.76978e10 1.39462 0.697311 0.716768i \(-0.254380\pi\)
0.697311 + 0.716768i \(0.254380\pi\)
\(252\) 0 0
\(253\) −3.34405e9 −0.0513134
\(254\) −2.01963e9 −0.0304453
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) 1.09539e11 1.56628 0.783140 0.621846i \(-0.213617\pi\)
0.783140 + 0.621846i \(0.213617\pi\)
\(258\) 0 0
\(259\) −5.50573e9 −0.0760266
\(260\) −2.50279e10 −0.339660
\(261\) 0 0
\(262\) 3.00574e10 0.394091
\(263\) −4.08592e9 −0.0526610 −0.0263305 0.999653i \(-0.508382\pi\)
−0.0263305 + 0.999653i \(0.508382\pi\)
\(264\) 0 0
\(265\) 1.18208e10 0.147245
\(266\) 1.51183e10 0.185155
\(267\) 0 0
\(268\) −5.83403e10 −0.690815
\(269\) 8.48247e10 0.987727 0.493864 0.869539i \(-0.335584\pi\)
0.493864 + 0.869539i \(0.335584\pi\)
\(270\) 0 0
\(271\) 6.76455e10 0.761864 0.380932 0.924603i \(-0.375603\pi\)
0.380932 + 0.924603i \(0.375603\pi\)
\(272\) −2.88900e10 −0.320028
\(273\) 0 0
\(274\) −1.01174e11 −1.08441
\(275\) 8.01628e9 0.0845232
\(276\) 0 0
\(277\) 1.22944e11 1.25472 0.627361 0.778728i \(-0.284135\pi\)
0.627361 + 0.778728i \(0.284135\pi\)
\(278\) −1.25181e11 −1.25700
\(279\) 0 0
\(280\) −5.70430e9 −0.0554614
\(281\) 1.28901e11 1.23333 0.616664 0.787226i \(-0.288484\pi\)
0.616664 + 0.787226i \(0.288484\pi\)
\(282\) 0 0
\(283\) 2.08828e10 0.193530 0.0967651 0.995307i \(-0.469150\pi\)
0.0967651 + 0.995307i \(0.469150\pi\)
\(284\) 3.98348e10 0.363354
\(285\) 0 0
\(286\) −1.93172e10 −0.170725
\(287\) −3.87913e9 −0.0337494
\(288\) 0 0
\(289\) 7.57401e10 0.638683
\(290\) 4.33442e10 0.359866
\(291\) 0 0
\(292\) −2.66693e10 −0.214679
\(293\) −3.85968e9 −0.0305948 −0.0152974 0.999883i \(-0.504869\pi\)
−0.0152974 + 0.999883i \(0.504869\pi\)
\(294\) 0 0
\(295\) 7.48881e10 0.575723
\(296\) −1.91983e10 −0.145362
\(297\) 0 0
\(298\) 4.31239e9 0.0316770
\(299\) 1.88346e10 0.136281
\(300\) 0 0
\(301\) 9.17874e9 0.0644516
\(302\) 1.89157e11 1.30855
\(303\) 0 0
\(304\) 5.27172e10 0.354015
\(305\) 2.51970e11 1.66725
\(306\) 0 0
\(307\) 4.11326e10 0.264279 0.132140 0.991231i \(-0.457815\pi\)
0.132140 + 0.991231i \(0.457815\pi\)
\(308\) −4.40273e9 −0.0278768
\(309\) 0 0
\(310\) −4.86577e10 −0.299243
\(311\) 2.75638e11 1.67078 0.835388 0.549661i \(-0.185243\pi\)
0.835388 + 0.549661i \(0.185243\pi\)
\(312\) 0 0
\(313\) 3.14160e11 1.85013 0.925064 0.379810i \(-0.124011\pi\)
0.925064 + 0.379810i \(0.124011\pi\)
\(314\) 5.87183e10 0.340871
\(315\) 0 0
\(316\) −7.59358e10 −0.428405
\(317\) −2.58171e11 −1.43595 −0.717977 0.696067i \(-0.754932\pi\)
−0.717977 + 0.696067i \(0.754932\pi\)
\(318\) 0 0
\(319\) 3.34542e10 0.180881
\(320\) −1.98907e10 −0.106042
\(321\) 0 0
\(322\) 4.29273e9 0.0222527
\(323\) −3.54601e11 −1.81271
\(324\) 0 0
\(325\) −4.51498e10 −0.224482
\(326\) −2.08975e11 −1.02475
\(327\) 0 0
\(328\) −1.35264e10 −0.0645284
\(329\) 3.35382e9 0.0157819
\(330\) 0 0
\(331\) −5.94811e10 −0.272366 −0.136183 0.990684i \(-0.543484\pi\)
−0.136183 + 0.990684i \(0.543484\pi\)
\(332\) 1.89131e11 0.854361
\(333\) 0 0
\(334\) 1.87058e10 0.0822466
\(335\) 2.70184e11 1.17208
\(336\) 0 0
\(337\) 9.61669e10 0.406154 0.203077 0.979163i \(-0.434906\pi\)
0.203077 + 0.979163i \(0.434906\pi\)
\(338\) −6.08724e10 −0.253685
\(339\) 0 0
\(340\) 1.33795e11 0.542980
\(341\) −3.75553e10 −0.150410
\(342\) 0 0
\(343\) −9.31825e10 −0.363505
\(344\) 3.20061e10 0.123231
\(345\) 0 0
\(346\) −3.05428e11 −1.14569
\(347\) 3.65974e11 1.35509 0.677543 0.735483i \(-0.263045\pi\)
0.677543 + 0.735483i \(0.263045\pi\)
\(348\) 0 0
\(349\) −4.32258e11 −1.55966 −0.779828 0.625993i \(-0.784694\pi\)
−0.779828 + 0.625993i \(0.784694\pi\)
\(350\) −1.02904e10 −0.0366545
\(351\) 0 0
\(352\) −1.53522e10 −0.0533002
\(353\) 3.67882e11 1.26102 0.630511 0.776180i \(-0.282845\pi\)
0.630511 + 0.776180i \(0.282845\pi\)
\(354\) 0 0
\(355\) −1.84482e11 −0.616490
\(356\) −2.30413e11 −0.760293
\(357\) 0 0
\(358\) 3.19152e11 1.02689
\(359\) 1.25751e11 0.399564 0.199782 0.979840i \(-0.435977\pi\)
0.199782 + 0.979840i \(0.435977\pi\)
\(360\) 0 0
\(361\) 3.24374e11 1.00522
\(362\) −1.31352e11 −0.402019
\(363\) 0 0
\(364\) 2.47973e10 0.0740369
\(365\) 1.23510e11 0.364238
\(366\) 0 0
\(367\) 5.19603e11 1.49511 0.747557 0.664197i \(-0.231227\pi\)
0.747557 + 0.664197i \(0.231227\pi\)
\(368\) 1.49686e10 0.0425468
\(369\) 0 0
\(370\) 8.89109e10 0.246631
\(371\) −1.17119e10 −0.0320955
\(372\) 0 0
\(373\) 4.65238e11 1.24447 0.622237 0.782829i \(-0.286224\pi\)
0.622237 + 0.782829i \(0.286224\pi\)
\(374\) 1.03266e11 0.272920
\(375\) 0 0
\(376\) 1.16947e10 0.0301747
\(377\) −1.88423e11 −0.480394
\(378\) 0 0
\(379\) 4.48793e11 1.11730 0.558650 0.829404i \(-0.311320\pi\)
0.558650 + 0.829404i \(0.311320\pi\)
\(380\) −2.44143e11 −0.600645
\(381\) 0 0
\(382\) −1.99240e11 −0.478730
\(383\) −2.75704e11 −0.654710 −0.327355 0.944901i \(-0.606157\pi\)
−0.327355 + 0.944901i \(0.606157\pi\)
\(384\) 0 0
\(385\) 2.03898e10 0.0472977
\(386\) 2.57354e10 0.0590049
\(387\) 0 0
\(388\) 3.30641e11 0.740653
\(389\) −8.29663e11 −1.83708 −0.918540 0.395327i \(-0.870631\pi\)
−0.918540 + 0.395327i \(0.870631\pi\)
\(390\) 0 0
\(391\) −1.00686e11 −0.217858
\(392\) −1.59637e11 −0.341464
\(393\) 0 0
\(394\) −6.44448e11 −1.34727
\(395\) 3.51672e11 0.726861
\(396\) 0 0
\(397\) 1.53152e11 0.309432 0.154716 0.987959i \(-0.450554\pi\)
0.154716 + 0.987959i \(0.450554\pi\)
\(398\) 5.15332e11 1.02947
\(399\) 0 0
\(400\) −3.58824e10 −0.0700829
\(401\) −2.05461e11 −0.396807 −0.198404 0.980120i \(-0.563576\pi\)
−0.198404 + 0.980120i \(0.563576\pi\)
\(402\) 0 0
\(403\) 2.11521e11 0.399467
\(404\) −3.72870e11 −0.696373
\(405\) 0 0
\(406\) −4.29448e10 −0.0784411
\(407\) 6.86237e10 0.123965
\(408\) 0 0
\(409\) −6.57635e11 −1.16206 −0.581032 0.813880i \(-0.697351\pi\)
−0.581032 + 0.813880i \(0.697351\pi\)
\(410\) 6.26432e10 0.109483
\(411\) 0 0
\(412\) −5.01685e11 −0.857814
\(413\) −7.41981e10 −0.125492
\(414\) 0 0
\(415\) −8.75899e11 −1.44956
\(416\) 8.64676e10 0.141558
\(417\) 0 0
\(418\) −1.88436e11 −0.301905
\(419\) 3.79470e11 0.601471 0.300735 0.953708i \(-0.402768\pi\)
0.300735 + 0.953708i \(0.402768\pi\)
\(420\) 0 0
\(421\) 9.15367e11 1.42012 0.710061 0.704140i \(-0.248667\pi\)
0.710061 + 0.704140i \(0.248667\pi\)
\(422\) 1.69778e11 0.260600
\(423\) 0 0
\(424\) −4.08390e10 −0.0613662
\(425\) 2.41362e11 0.358855
\(426\) 0 0
\(427\) −2.49648e11 −0.363415
\(428\) −1.55553e11 −0.224068
\(429\) 0 0
\(430\) −1.48226e11 −0.209081
\(431\) −4.90485e11 −0.684664 −0.342332 0.939579i \(-0.611217\pi\)
−0.342332 + 0.939579i \(0.611217\pi\)
\(432\) 0 0
\(433\) −5.67227e11 −0.775464 −0.387732 0.921772i \(-0.626741\pi\)
−0.387732 + 0.921772i \(0.626741\pi\)
\(434\) 4.82094e10 0.0652271
\(435\) 0 0
\(436\) −2.93788e11 −0.389354
\(437\) 1.83728e11 0.240995
\(438\) 0 0
\(439\) −1.15068e12 −1.47865 −0.739323 0.673351i \(-0.764854\pi\)
−0.739323 + 0.673351i \(0.764854\pi\)
\(440\) 7.10987e10 0.0904326
\(441\) 0 0
\(442\) −5.81622e11 −0.724838
\(443\) 4.44419e11 0.548247 0.274123 0.961695i \(-0.411612\pi\)
0.274123 + 0.961695i \(0.411612\pi\)
\(444\) 0 0
\(445\) 1.06708e12 1.28996
\(446\) −3.37880e11 −0.404349
\(447\) 0 0
\(448\) 1.97075e10 0.0231143
\(449\) −6.03886e11 −0.701208 −0.350604 0.936524i \(-0.614024\pi\)
−0.350604 + 0.936524i \(0.614024\pi\)
\(450\) 0 0
\(451\) 4.83497e10 0.0550300
\(452\) 6.50377e10 0.0732895
\(453\) 0 0
\(454\) 8.53258e10 0.0942603
\(455\) −1.14841e11 −0.125616
\(456\) 0 0
\(457\) −2.80219e11 −0.300521 −0.150260 0.988646i \(-0.548011\pi\)
−0.150260 + 0.988646i \(0.548011\pi\)
\(458\) −4.19851e11 −0.445862
\(459\) 0 0
\(460\) −6.93224e10 −0.0721877
\(461\) −1.25259e12 −1.29168 −0.645838 0.763475i \(-0.723492\pi\)
−0.645838 + 0.763475i \(0.723492\pi\)
\(462\) 0 0
\(463\) 1.46886e11 0.148548 0.0742739 0.997238i \(-0.476336\pi\)
0.0742739 + 0.997238i \(0.476336\pi\)
\(464\) −1.49748e11 −0.149979
\(465\) 0 0
\(466\) −7.63479e11 −0.749999
\(467\) −1.49431e12 −1.45383 −0.726916 0.686726i \(-0.759047\pi\)
−0.726916 + 0.686726i \(0.759047\pi\)
\(468\) 0 0
\(469\) −2.67695e11 −0.255483
\(470\) −5.41601e10 −0.0511964
\(471\) 0 0
\(472\) −2.58727e11 −0.239940
\(473\) −1.14404e11 −0.105092
\(474\) 0 0
\(475\) −4.40428e11 −0.396966
\(476\) −1.32562e11 −0.118355
\(477\) 0 0
\(478\) 4.22944e11 0.370559
\(479\) 1.58007e11 0.137140 0.0685702 0.997646i \(-0.478156\pi\)
0.0685702 + 0.997646i \(0.478156\pi\)
\(480\) 0 0
\(481\) −3.86507e11 −0.329234
\(482\) −7.92459e11 −0.668752
\(483\) 0 0
\(484\) 5.48759e10 0.0454545
\(485\) −1.53126e12 −1.25664
\(486\) 0 0
\(487\) −1.99896e12 −1.61036 −0.805181 0.593029i \(-0.797932\pi\)
−0.805181 + 0.593029i \(0.797932\pi\)
\(488\) −8.70518e11 −0.694846
\(489\) 0 0
\(490\) 7.39305e11 0.579351
\(491\) −9.33204e11 −0.724619 −0.362310 0.932058i \(-0.618012\pi\)
−0.362310 + 0.932058i \(0.618012\pi\)
\(492\) 0 0
\(493\) 1.00727e12 0.767956
\(494\) 1.06132e12 0.801817
\(495\) 0 0
\(496\) 1.68105e11 0.124713
\(497\) 1.82782e11 0.134379
\(498\) 0 0
\(499\) −1.93508e12 −1.39716 −0.698580 0.715532i \(-0.746184\pi\)
−0.698580 + 0.715532i \(0.746184\pi\)
\(500\) 7.58968e11 0.543074
\(501\) 0 0
\(502\) 1.40316e12 0.986147
\(503\) −1.50533e12 −1.04852 −0.524260 0.851558i \(-0.675658\pi\)
−0.524260 + 0.851558i \(0.675658\pi\)
\(504\) 0 0
\(505\) 1.72683e12 1.18151
\(506\) −5.35048e10 −0.0362840
\(507\) 0 0
\(508\) −3.23140e10 −0.0215280
\(509\) 1.56888e12 1.03600 0.518001 0.855380i \(-0.326676\pi\)
0.518001 + 0.855380i \(0.326676\pi\)
\(510\) 0 0
\(511\) −1.22372e11 −0.0793942
\(512\) 6.87195e10 0.0441942
\(513\) 0 0
\(514\) 1.75262e12 1.10753
\(515\) 2.32339e12 1.45542
\(516\) 0 0
\(517\) −4.18022e10 −0.0257331
\(518\) −8.80916e10 −0.0537590
\(519\) 0 0
\(520\) −4.00447e11 −0.240176
\(521\) 8.36119e11 0.497163 0.248581 0.968611i \(-0.420036\pi\)
0.248581 + 0.968611i \(0.420036\pi\)
\(522\) 0 0
\(523\) −1.04380e12 −0.610043 −0.305021 0.952345i \(-0.598664\pi\)
−0.305021 + 0.952345i \(0.598664\pi\)
\(524\) 4.80919e11 0.278664
\(525\) 0 0
\(526\) −6.53748e10 −0.0372370
\(527\) −1.13075e12 −0.638587
\(528\) 0 0
\(529\) −1.74898e12 −0.971036
\(530\) 1.89133e11 0.104118
\(531\) 0 0
\(532\) 2.41893e11 0.130925
\(533\) −2.72318e11 −0.146152
\(534\) 0 0
\(535\) 7.20391e11 0.380169
\(536\) −9.33444e11 −0.488480
\(537\) 0 0
\(538\) 1.35719e12 0.698428
\(539\) 5.70615e11 0.291202
\(540\) 0 0
\(541\) −2.61498e11 −0.131245 −0.0656223 0.997845i \(-0.520903\pi\)
−0.0656223 + 0.997845i \(0.520903\pi\)
\(542\) 1.08233e12 0.538719
\(543\) 0 0
\(544\) −4.62240e11 −0.226294
\(545\) 1.36058e12 0.660604
\(546\) 0 0
\(547\) 3.97347e12 1.89770 0.948848 0.315734i \(-0.102251\pi\)
0.948848 + 0.315734i \(0.102251\pi\)
\(548\) −1.61879e12 −0.766791
\(549\) 0 0
\(550\) 1.28260e11 0.0597669
\(551\) −1.83803e12 −0.849514
\(552\) 0 0
\(553\) −3.48432e11 −0.158436
\(554\) 1.96710e12 0.887223
\(555\) 0 0
\(556\) −2.00289e12 −0.888833
\(557\) 1.26345e12 0.556175 0.278087 0.960556i \(-0.410300\pi\)
0.278087 + 0.960556i \(0.410300\pi\)
\(558\) 0 0
\(559\) 6.44356e11 0.279108
\(560\) −9.12688e10 −0.0392172
\(561\) 0 0
\(562\) 2.06242e12 0.872095
\(563\) −2.98705e12 −1.25301 −0.626505 0.779417i \(-0.715515\pi\)
−0.626505 + 0.779417i \(0.715515\pi\)
\(564\) 0 0
\(565\) −3.01201e11 −0.124348
\(566\) 3.34124e11 0.136847
\(567\) 0 0
\(568\) 6.37357e11 0.256930
\(569\) 4.48345e11 0.179311 0.0896555 0.995973i \(-0.471423\pi\)
0.0896555 + 0.995973i \(0.471423\pi\)
\(570\) 0 0
\(571\) 1.93797e12 0.762931 0.381466 0.924383i \(-0.375420\pi\)
0.381466 + 0.924383i \(0.375420\pi\)
\(572\) −3.09075e11 −0.120721
\(573\) 0 0
\(574\) −6.20660e10 −0.0238644
\(575\) −1.25056e11 −0.0477089
\(576\) 0 0
\(577\) −1.54518e12 −0.580347 −0.290174 0.956974i \(-0.593713\pi\)
−0.290174 + 0.956974i \(0.593713\pi\)
\(578\) 1.21184e12 0.451617
\(579\) 0 0
\(580\) 6.93508e11 0.254463
\(581\) 8.67828e11 0.315967
\(582\) 0 0
\(583\) 1.45978e11 0.0523333
\(584\) −4.26709e11 −0.151801
\(585\) 0 0
\(586\) −6.17549e10 −0.0216338
\(587\) 1.73985e10 0.00604841 0.00302421 0.999995i \(-0.499037\pi\)
0.00302421 + 0.999995i \(0.499037\pi\)
\(588\) 0 0
\(589\) 2.06335e12 0.706406
\(590\) 1.19821e12 0.407098
\(591\) 0 0
\(592\) −3.07173e11 −0.102786
\(593\) −8.05756e10 −0.0267582 −0.0133791 0.999910i \(-0.504259\pi\)
−0.0133791 + 0.999910i \(0.504259\pi\)
\(594\) 0 0
\(595\) 6.13917e11 0.200809
\(596\) 6.89982e10 0.0223991
\(597\) 0 0
\(598\) 3.01353e11 0.0963652
\(599\) −4.11009e11 −0.130446 −0.0652229 0.997871i \(-0.520776\pi\)
−0.0652229 + 0.997871i \(0.520776\pi\)
\(600\) 0 0
\(601\) −3.07127e12 −0.960246 −0.480123 0.877201i \(-0.659408\pi\)
−0.480123 + 0.877201i \(0.659408\pi\)
\(602\) 1.46860e11 0.0455742
\(603\) 0 0
\(604\) 3.02651e12 0.925286
\(605\) −2.54140e11 −0.0771212
\(606\) 0 0
\(607\) 1.67239e12 0.500021 0.250010 0.968243i \(-0.419566\pi\)
0.250010 + 0.968243i \(0.419566\pi\)
\(608\) 8.43476e11 0.250326
\(609\) 0 0
\(610\) 4.03152e12 1.17892
\(611\) 2.35441e11 0.0683434
\(612\) 0 0
\(613\) −2.38637e11 −0.0682598 −0.0341299 0.999417i \(-0.510866\pi\)
−0.0341299 + 0.999417i \(0.510866\pi\)
\(614\) 6.58121e11 0.186874
\(615\) 0 0
\(616\) −7.04436e10 −0.0197119
\(617\) −2.55970e12 −0.711060 −0.355530 0.934665i \(-0.615700\pi\)
−0.355530 + 0.934665i \(0.615700\pi\)
\(618\) 0 0
\(619\) −4.65350e12 −1.27401 −0.637004 0.770860i \(-0.719827\pi\)
−0.637004 + 0.770860i \(0.719827\pi\)
\(620\) −7.78524e11 −0.211597
\(621\) 0 0
\(622\) 4.41022e12 1.18142
\(623\) −1.05725e12 −0.281178
\(624\) 0 0
\(625\) −2.44554e12 −0.641083
\(626\) 5.02657e12 1.30824
\(627\) 0 0
\(628\) 9.39493e11 0.241032
\(629\) 2.06619e12 0.526312
\(630\) 0 0
\(631\) −2.86910e12 −0.720466 −0.360233 0.932862i \(-0.617303\pi\)
−0.360233 + 0.932862i \(0.617303\pi\)
\(632\) −1.21497e12 −0.302928
\(633\) 0 0
\(634\) −4.13073e12 −1.01537
\(635\) 1.49652e11 0.0365259
\(636\) 0 0
\(637\) −3.21385e12 −0.773390
\(638\) 5.35267e11 0.127902
\(639\) 0 0
\(640\) −3.18252e11 −0.0749827
\(641\) 6.45940e12 1.51123 0.755615 0.655016i \(-0.227338\pi\)
0.755615 + 0.655016i \(0.227338\pi\)
\(642\) 0 0
\(643\) −2.55823e12 −0.590188 −0.295094 0.955468i \(-0.595351\pi\)
−0.295094 + 0.955468i \(0.595351\pi\)
\(644\) 6.86836e10 0.0157350
\(645\) 0 0
\(646\) −5.67362e12 −1.28178
\(647\) 9.97781e11 0.223855 0.111927 0.993716i \(-0.464298\pi\)
0.111927 + 0.993716i \(0.464298\pi\)
\(648\) 0 0
\(649\) 9.24810e11 0.204622
\(650\) −7.22397e11 −0.158732
\(651\) 0 0
\(652\) −3.34361e12 −0.724605
\(653\) 7.53538e12 1.62179 0.810897 0.585188i \(-0.198979\pi\)
0.810897 + 0.585188i \(0.198979\pi\)
\(654\) 0 0
\(655\) −2.22722e12 −0.472800
\(656\) −2.16423e11 −0.0456284
\(657\) 0 0
\(658\) 5.36611e10 0.0111595
\(659\) 8.67293e12 1.79135 0.895677 0.444706i \(-0.146692\pi\)
0.895677 + 0.444706i \(0.146692\pi\)
\(660\) 0 0
\(661\) 4.77055e12 0.971991 0.485995 0.873961i \(-0.338457\pi\)
0.485995 + 0.873961i \(0.338457\pi\)
\(662\) −9.51697e11 −0.192592
\(663\) 0 0
\(664\) 3.02610e12 0.604124
\(665\) −1.12025e12 −0.222135
\(666\) 0 0
\(667\) −5.21894e11 −0.102098
\(668\) 2.99293e11 0.0581571
\(669\) 0 0
\(670\) 4.32294e12 0.828787
\(671\) 3.11163e12 0.592567
\(672\) 0 0
\(673\) −7.71533e12 −1.44973 −0.724864 0.688892i \(-0.758098\pi\)
−0.724864 + 0.688892i \(0.758098\pi\)
\(674\) 1.53867e12 0.287194
\(675\) 0 0
\(676\) −9.73959e11 −0.179383
\(677\) −9.14395e12 −1.67296 −0.836479 0.548000i \(-0.815389\pi\)
−0.836479 + 0.548000i \(0.815389\pi\)
\(678\) 0 0
\(679\) 1.51715e12 0.273914
\(680\) 2.14071e12 0.383945
\(681\) 0 0
\(682\) −6.00885e11 −0.106356
\(683\) −8.83407e11 −0.155334 −0.0776672 0.996979i \(-0.524747\pi\)
−0.0776672 + 0.996979i \(0.524747\pi\)
\(684\) 0 0
\(685\) 7.49688e12 1.30099
\(686\) −1.49092e12 −0.257037
\(687\) 0 0
\(688\) 5.12097e11 0.0871373
\(689\) −8.22184e11 −0.138990
\(690\) 0 0
\(691\) 6.58731e12 1.09915 0.549575 0.835445i \(-0.314790\pi\)
0.549575 + 0.835445i \(0.314790\pi\)
\(692\) −4.88685e12 −0.810124
\(693\) 0 0
\(694\) 5.85558e12 0.958191
\(695\) 9.27573e12 1.50805
\(696\) 0 0
\(697\) 1.45576e12 0.233638
\(698\) −6.91613e12 −1.10284
\(699\) 0 0
\(700\) −1.64647e11 −0.0259186
\(701\) −6.57894e12 −1.02902 −0.514512 0.857483i \(-0.672027\pi\)
−0.514512 + 0.857483i \(0.672027\pi\)
\(702\) 0 0
\(703\) −3.77030e12 −0.582207
\(704\) −2.45635e11 −0.0376889
\(705\) 0 0
\(706\) 5.88612e12 0.891678
\(707\) −1.71092e12 −0.257538
\(708\) 0 0
\(709\) 8.74938e11 0.130038 0.0650188 0.997884i \(-0.479289\pi\)
0.0650188 + 0.997884i \(0.479289\pi\)
\(710\) −2.95171e12 −0.435924
\(711\) 0 0
\(712\) −3.68660e12 −0.537609
\(713\) 5.85872e11 0.0848985
\(714\) 0 0
\(715\) 1.43138e12 0.204823
\(716\) 5.10644e12 0.726122
\(717\) 0 0
\(718\) 2.01202e12 0.282535
\(719\) −8.46541e12 −1.18132 −0.590660 0.806920i \(-0.701133\pi\)
−0.590660 + 0.806920i \(0.701133\pi\)
\(720\) 0 0
\(721\) −2.30198e12 −0.317244
\(722\) 5.18998e12 0.710801
\(723\) 0 0
\(724\) −2.10163e12 −0.284270
\(725\) 1.25107e12 0.168175
\(726\) 0 0
\(727\) −4.57971e12 −0.608042 −0.304021 0.952665i \(-0.598329\pi\)
−0.304021 + 0.952665i \(0.598329\pi\)
\(728\) 3.96757e11 0.0523520
\(729\) 0 0
\(730\) 1.97616e12 0.257555
\(731\) −3.44461e12 −0.446182
\(732\) 0 0
\(733\) 2.48940e12 0.318512 0.159256 0.987237i \(-0.449090\pi\)
0.159256 + 0.987237i \(0.449090\pi\)
\(734\) 8.31365e12 1.05721
\(735\) 0 0
\(736\) 2.39498e11 0.0300851
\(737\) 3.33656e12 0.416577
\(738\) 0 0
\(739\) −8.85822e12 −1.09256 −0.546282 0.837602i \(-0.683957\pi\)
−0.546282 + 0.837602i \(0.683957\pi\)
\(740\) 1.42257e12 0.174394
\(741\) 0 0
\(742\) −1.87390e11 −0.0226949
\(743\) 1.02362e13 1.23222 0.616112 0.787659i \(-0.288707\pi\)
0.616112 + 0.787659i \(0.288707\pi\)
\(744\) 0 0
\(745\) −3.19543e11 −0.0380037
\(746\) 7.44381e12 0.879975
\(747\) 0 0
\(748\) 1.65226e12 0.192984
\(749\) −7.13753e11 −0.0828667
\(750\) 0 0
\(751\) 3.19150e12 0.366113 0.183056 0.983102i \(-0.441401\pi\)
0.183056 + 0.983102i \(0.441401\pi\)
\(752\) 1.87115e11 0.0213367
\(753\) 0 0
\(754\) −3.01477e12 −0.339690
\(755\) −1.40163e13 −1.56990
\(756\) 0 0
\(757\) 1.65793e13 1.83499 0.917496 0.397746i \(-0.130207\pi\)
0.917496 + 0.397746i \(0.130207\pi\)
\(758\) 7.18069e12 0.790050
\(759\) 0 0
\(760\) −3.90628e12 −0.424720
\(761\) −8.76788e12 −0.947685 −0.473842 0.880610i \(-0.657133\pi\)
−0.473842 + 0.880610i \(0.657133\pi\)
\(762\) 0 0
\(763\) −1.34805e12 −0.143994
\(764\) −3.18784e12 −0.338513
\(765\) 0 0
\(766\) −4.41127e12 −0.462950
\(767\) −5.20877e12 −0.543446
\(768\) 0 0
\(769\) 1.02168e13 1.05353 0.526764 0.850011i \(-0.323405\pi\)
0.526764 + 0.850011i \(0.323405\pi\)
\(770\) 3.26237e11 0.0334445
\(771\) 0 0
\(772\) 4.11767e11 0.0417228
\(773\) 4.77737e12 0.481262 0.240631 0.970617i \(-0.422646\pi\)
0.240631 + 0.970617i \(0.422646\pi\)
\(774\) 0 0
\(775\) −1.40444e12 −0.139844
\(776\) 5.29026e12 0.523720
\(777\) 0 0
\(778\) −1.32746e13 −1.29901
\(779\) −2.65641e12 −0.258450
\(780\) 0 0
\(781\) −2.27821e12 −0.219111
\(782\) −1.61098e12 −0.154049
\(783\) 0 0
\(784\) −2.55419e12 −0.241452
\(785\) −4.35096e12 −0.408951
\(786\) 0 0
\(787\) −1.03319e11 −0.00960053 −0.00480026 0.999988i \(-0.501528\pi\)
−0.00480026 + 0.999988i \(0.501528\pi\)
\(788\) −1.03112e13 −0.952664
\(789\) 0 0
\(790\) 5.62676e12 0.513968
\(791\) 2.98426e11 0.0271045
\(792\) 0 0
\(793\) −1.75255e13 −1.57377
\(794\) 2.45043e12 0.218801
\(795\) 0 0
\(796\) 8.24531e12 0.727945
\(797\) 1.18659e13 1.04169 0.520844 0.853652i \(-0.325617\pi\)
0.520844 + 0.853652i \(0.325617\pi\)
\(798\) 0 0
\(799\) −1.25862e12 −0.109254
\(800\) −5.74119e11 −0.0495561
\(801\) 0 0
\(802\) −3.28738e12 −0.280585
\(803\) 1.52526e12 0.129456
\(804\) 0 0
\(805\) −3.18086e11 −0.0266970
\(806\) 3.38434e12 0.282466
\(807\) 0 0
\(808\) −5.96592e12 −0.492410
\(809\) −1.31694e13 −1.08093 −0.540465 0.841367i \(-0.681751\pi\)
−0.540465 + 0.841367i \(0.681751\pi\)
\(810\) 0 0
\(811\) −3.03385e12 −0.246263 −0.123132 0.992390i \(-0.539294\pi\)
−0.123132 + 0.992390i \(0.539294\pi\)
\(812\) −6.87118e11 −0.0554663
\(813\) 0 0
\(814\) 1.09798e12 0.0876566
\(815\) 1.54848e13 1.22941
\(816\) 0 0
\(817\) 6.28557e12 0.493566
\(818\) −1.05222e13 −0.821704
\(819\) 0 0
\(820\) 1.00229e12 0.0774162
\(821\) −1.60816e13 −1.23534 −0.617670 0.786438i \(-0.711923\pi\)
−0.617670 + 0.786438i \(0.711923\pi\)
\(822\) 0 0
\(823\) 9.77833e12 0.742960 0.371480 0.928441i \(-0.378850\pi\)
0.371480 + 0.928441i \(0.378850\pi\)
\(824\) −8.02695e12 −0.606566
\(825\) 0 0
\(826\) −1.18717e12 −0.0887366
\(827\) −1.19955e13 −0.891749 −0.445874 0.895096i \(-0.647107\pi\)
−0.445874 + 0.895096i \(0.647107\pi\)
\(828\) 0 0
\(829\) 1.25391e13 0.922085 0.461043 0.887378i \(-0.347475\pi\)
0.461043 + 0.887378i \(0.347475\pi\)
\(830\) −1.40144e13 −1.02500
\(831\) 0 0
\(832\) 1.38348e12 0.100096
\(833\) 1.71807e13 1.23634
\(834\) 0 0
\(835\) −1.38608e12 −0.0986732
\(836\) −3.01497e12 −0.213479
\(837\) 0 0
\(838\) 6.07152e12 0.425304
\(839\) −1.36095e13 −0.948233 −0.474116 0.880462i \(-0.657232\pi\)
−0.474116 + 0.880462i \(0.657232\pi\)
\(840\) 0 0
\(841\) −9.28607e12 −0.640103
\(842\) 1.46459e13 1.00418
\(843\) 0 0
\(844\) 2.71644e12 0.184272
\(845\) 4.51057e12 0.304352
\(846\) 0 0
\(847\) 2.51798e11 0.0168104
\(848\) −6.53425e11 −0.0433925
\(849\) 0 0
\(850\) 3.86180e12 0.253749
\(851\) −1.07055e12 −0.0699718
\(852\) 0 0
\(853\) −5.86443e12 −0.379276 −0.189638 0.981854i \(-0.560731\pi\)
−0.189638 + 0.981854i \(0.560731\pi\)
\(854\) −3.99437e12 −0.256974
\(855\) 0 0
\(856\) −2.48884e12 −0.158440
\(857\) 1.19243e12 0.0755128 0.0377564 0.999287i \(-0.487979\pi\)
0.0377564 + 0.999287i \(0.487979\pi\)
\(858\) 0 0
\(859\) 1.96633e13 1.23222 0.616108 0.787662i \(-0.288709\pi\)
0.616108 + 0.787662i \(0.288709\pi\)
\(860\) −2.37161e12 −0.147843
\(861\) 0 0
\(862\) −7.84775e12 −0.484131
\(863\) 1.56800e13 0.962271 0.481135 0.876646i \(-0.340225\pi\)
0.481135 + 0.876646i \(0.340225\pi\)
\(864\) 0 0
\(865\) 2.26319e13 1.37451
\(866\) −9.07564e12 −0.548336
\(867\) 0 0
\(868\) 7.71350e11 0.0461225
\(869\) 4.34288e12 0.258338
\(870\) 0 0
\(871\) −1.87924e13 −1.10637
\(872\) −4.70061e12 −0.275315
\(873\) 0 0
\(874\) 2.93964e12 0.170409
\(875\) 3.48253e12 0.200844
\(876\) 0 0
\(877\) −1.56274e13 −0.892047 −0.446023 0.895021i \(-0.647160\pi\)
−0.446023 + 0.895021i \(0.647160\pi\)
\(878\) −1.84109e13 −1.04556
\(879\) 0 0
\(880\) 1.13758e12 0.0639455
\(881\) −7.19215e12 −0.402223 −0.201112 0.979568i \(-0.564455\pi\)
−0.201112 + 0.979568i \(0.564455\pi\)
\(882\) 0 0
\(883\) 8.94240e11 0.0495029 0.0247515 0.999694i \(-0.492121\pi\)
0.0247515 + 0.999694i \(0.492121\pi\)
\(884\) −9.30596e12 −0.512538
\(885\) 0 0
\(886\) 7.11071e12 0.387669
\(887\) −1.98339e13 −1.07585 −0.537926 0.842992i \(-0.680792\pi\)
−0.537926 + 0.842992i \(0.680792\pi\)
\(888\) 0 0
\(889\) −1.48273e11 −0.00796168
\(890\) 1.70733e13 0.912142
\(891\) 0 0
\(892\) −5.40609e12 −0.285918
\(893\) 2.29668e12 0.120856
\(894\) 0 0
\(895\) −2.36488e13 −1.23199
\(896\) 3.15320e11 0.0163442
\(897\) 0 0
\(898\) −9.66218e12 −0.495829
\(899\) −5.86112e12 −0.299269
\(900\) 0 0
\(901\) 4.39525e12 0.222189
\(902\) 7.73595e11 0.0389121
\(903\) 0 0
\(904\) 1.04060e12 0.0518235
\(905\) 9.73300e12 0.482312
\(906\) 0 0
\(907\) −1.02442e13 −0.502625 −0.251313 0.967906i \(-0.580862\pi\)
−0.251313 + 0.967906i \(0.580862\pi\)
\(908\) 1.36521e12 0.0666521
\(909\) 0 0
\(910\) −1.83745e12 −0.0888239
\(911\) 7.68070e12 0.369461 0.184730 0.982789i \(-0.440859\pi\)
0.184730 + 0.982789i \(0.440859\pi\)
\(912\) 0 0
\(913\) −1.08167e13 −0.515199
\(914\) −4.48350e12 −0.212500
\(915\) 0 0
\(916\) −6.71762e12 −0.315272
\(917\) 2.20670e12 0.103058
\(918\) 0 0
\(919\) 1.89989e13 0.878633 0.439317 0.898332i \(-0.355221\pi\)
0.439317 + 0.898332i \(0.355221\pi\)
\(920\) −1.10916e12 −0.0510444
\(921\) 0 0
\(922\) −2.00414e13 −0.913352
\(923\) 1.28315e13 0.581927
\(924\) 0 0
\(925\) 2.56629e12 0.115257
\(926\) 2.35018e12 0.105039
\(927\) 0 0
\(928\) −2.39596e12 −0.106051
\(929\) −2.17473e13 −0.957931 −0.478966 0.877834i \(-0.658988\pi\)
−0.478966 + 0.877834i \(0.658988\pi\)
\(930\) 0 0
\(931\) −3.13506e13 −1.36764
\(932\) −1.22157e13 −0.530329
\(933\) 0 0
\(934\) −2.39089e13 −1.02802
\(935\) −7.65191e12 −0.327429
\(936\) 0 0
\(937\) 3.95881e13 1.67779 0.838893 0.544297i \(-0.183203\pi\)
0.838893 + 0.544297i \(0.183203\pi\)
\(938\) −4.28311e12 −0.180654
\(939\) 0 0
\(940\) −8.66562e11 −0.0362013
\(941\) 4.13659e13 1.71985 0.859923 0.510424i \(-0.170512\pi\)
0.859923 + 0.510424i \(0.170512\pi\)
\(942\) 0 0
\(943\) −7.54267e11 −0.0310615
\(944\) −4.13963e12 −0.169663
\(945\) 0 0
\(946\) −1.83047e12 −0.0743109
\(947\) −7.78696e12 −0.314625 −0.157313 0.987549i \(-0.550283\pi\)
−0.157313 + 0.987549i \(0.550283\pi\)
\(948\) 0 0
\(949\) −8.59064e12 −0.343817
\(950\) −7.04685e12 −0.280698
\(951\) 0 0
\(952\) −2.12099e12 −0.0836898
\(953\) −1.26501e13 −0.496794 −0.248397 0.968658i \(-0.579904\pi\)
−0.248397 + 0.968658i \(0.579904\pi\)
\(954\) 0 0
\(955\) 1.47634e13 0.574344
\(956\) 6.76710e12 0.262025
\(957\) 0 0
\(958\) 2.52810e12 0.0969729
\(959\) −7.42781e12 −0.283581
\(960\) 0 0
\(961\) −1.98600e13 −0.751145
\(962\) −6.18411e12 −0.232803
\(963\) 0 0
\(964\) −1.26793e13 −0.472879
\(965\) −1.90696e12 −0.0707896
\(966\) 0 0
\(967\) −6.51628e12 −0.239652 −0.119826 0.992795i \(-0.538234\pi\)
−0.119826 + 0.992795i \(0.538234\pi\)
\(968\) 8.78014e11 0.0321412
\(969\) 0 0
\(970\) −2.45001e13 −0.888578
\(971\) −2.06099e13 −0.744028 −0.372014 0.928227i \(-0.621333\pi\)
−0.372014 + 0.928227i \(0.621333\pi\)
\(972\) 0 0
\(973\) −9.19027e12 −0.328716
\(974\) −3.19833e13 −1.13870
\(975\) 0 0
\(976\) −1.39283e13 −0.491330
\(977\) 3.23219e13 1.13494 0.567468 0.823395i \(-0.307923\pi\)
0.567468 + 0.823395i \(0.307923\pi\)
\(978\) 0 0
\(979\) 1.31776e13 0.458474
\(980\) 1.18289e13 0.409663
\(981\) 0 0
\(982\) −1.49313e13 −0.512383
\(983\) −2.94079e13 −1.00455 −0.502277 0.864707i \(-0.667504\pi\)
−0.502277 + 0.864707i \(0.667504\pi\)
\(984\) 0 0
\(985\) 4.77528e13 1.61635
\(986\) 1.61164e13 0.543027
\(987\) 0 0
\(988\) 1.69811e13 0.566970
\(989\) 1.78474e12 0.0593186
\(990\) 0 0
\(991\) 3.67690e13 1.21102 0.605508 0.795839i \(-0.292970\pi\)
0.605508 + 0.795839i \(0.292970\pi\)
\(992\) 2.68968e12 0.0881857
\(993\) 0 0
\(994\) 2.92451e12 0.0950200
\(995\) −3.81855e13 −1.23508
\(996\) 0 0
\(997\) −7.67620e12 −0.246047 −0.123024 0.992404i \(-0.539259\pi\)
−0.123024 + 0.992404i \(0.539259\pi\)
\(998\) −3.09613e13 −0.987941
\(999\) 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 198.10.a.n.1.1 2
3.2 odd 2 22.10.a.d.1.2 2
12.11 even 2 176.10.a.e.1.1 2
33.32 even 2 242.10.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.10.a.d.1.2 2 3.2 odd 2
176.10.a.e.1.1 2 12.11 even 2
198.10.a.n.1.1 2 1.1 even 1 trivial
242.10.a.e.1.2 2 33.32 even 2