Properties

Label 6.20.a.a.1.1
Level $6$
Weight $20$
Character 6.1
Self dual yes
Analytic conductor $13.729$
Analytic rank $0$
Dimension $1$
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] = [6,20,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(13.7290017934\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-512.000 q^{2} -19683.0 q^{3} +262144. q^{4} -3.73247e6 q^{5} +1.00777e7 q^{6} -1.49673e8 q^{7} -1.34218e8 q^{8} +3.87420e8 q^{9} +O(q^{10})\) \(q-512.000 q^{2} -19683.0 q^{3} +262144. q^{4} -3.73247e6 q^{5} +1.00777e7 q^{6} -1.49673e8 q^{7} -1.34218e8 q^{8} +3.87420e8 q^{9} +1.91103e9 q^{10} -7.45967e9 q^{11} -5.15978e9 q^{12} +5.92385e10 q^{13} +7.66324e10 q^{14} +7.34663e10 q^{15} +6.87195e10 q^{16} +5.23110e11 q^{17} -1.98359e11 q^{18} +9.69502e11 q^{19} -9.78446e11 q^{20} +2.94601e12 q^{21} +3.81935e12 q^{22} -1.36837e12 q^{23} +2.64181e12 q^{24} -5.14212e12 q^{25} -3.03301e13 q^{26} -7.62560e12 q^{27} -3.92358e13 q^{28} -9.86429e13 q^{29} -3.76147e13 q^{30} +1.94952e14 q^{31} -3.51844e13 q^{32} +1.46829e14 q^{33} -2.67833e14 q^{34} +5.58649e14 q^{35} +1.01560e14 q^{36} +1.18732e15 q^{37} -4.96385e14 q^{38} -1.16599e15 q^{39} +5.00964e14 q^{40} +1.87020e15 q^{41} -1.50836e15 q^{42} -1.48369e14 q^{43} -1.95551e15 q^{44} -1.44604e15 q^{45} +7.00608e14 q^{46} -1.05723e16 q^{47} -1.35261e15 q^{48} +1.10030e16 q^{49} +2.63277e15 q^{50} -1.02964e16 q^{51} +1.55290e16 q^{52} -3.63439e16 q^{53} +3.90431e15 q^{54} +2.78430e16 q^{55} +2.00887e16 q^{56} -1.90827e16 q^{57} +5.05052e16 q^{58} +4.14701e16 q^{59} +1.92587e16 q^{60} +1.47498e17 q^{61} -9.98154e16 q^{62} -5.79863e16 q^{63} +1.80144e16 q^{64} -2.21106e17 q^{65} -7.51763e16 q^{66} -3.73864e17 q^{67} +1.37130e17 q^{68} +2.69337e16 q^{69} -2.86028e17 q^{70} +6.82838e17 q^{71} -5.19987e16 q^{72} -6.94764e16 q^{73} -6.07907e17 q^{74} +1.01212e17 q^{75} +2.54149e17 q^{76} +1.11651e18 q^{77} +5.96987e17 q^{78} +4.08505e17 q^{79} -2.56494e17 q^{80} +1.50095e17 q^{81} -9.57542e17 q^{82} -1.99775e17 q^{83} +7.72278e17 q^{84} -1.95250e18 q^{85} +7.59648e16 q^{86} +1.94159e18 q^{87} +1.00122e18 q^{88} +1.27068e18 q^{89} +7.40371e17 q^{90} -8.86638e18 q^{91} -3.58711e17 q^{92} -3.83724e18 q^{93} +5.41302e18 q^{94} -3.61864e18 q^{95} +6.92534e17 q^{96} +2.51426e18 q^{97} -5.63354e18 q^{98} -2.89003e18 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −512.000 −0.707107
\(3\) −19683.0 −0.577350
\(4\) 262144. 0.500000
\(5\) −3.73247e6 −0.854637 −0.427319 0.904101i \(-0.640542\pi\)
−0.427319 + 0.904101i \(0.640542\pi\)
\(6\) 1.00777e7 0.408248
\(7\) −1.49673e8 −1.40188 −0.700940 0.713220i \(-0.747236\pi\)
−0.700940 + 0.713220i \(0.747236\pi\)
\(8\) −1.34218e8 −0.353553
\(9\) 3.87420e8 0.333333
\(10\) 1.91103e9 0.604320
\(11\) −7.45967e9 −0.953870 −0.476935 0.878939i \(-0.658252\pi\)
−0.476935 + 0.878939i \(0.658252\pi\)
\(12\) −5.15978e9 −0.288675
\(13\) 5.92385e10 1.54932 0.774661 0.632376i \(-0.217920\pi\)
0.774661 + 0.632376i \(0.217920\pi\)
\(14\) 7.66324e10 0.991279
\(15\) 7.34663e10 0.493425
\(16\) 6.87195e10 0.250000
\(17\) 5.23110e11 1.06986 0.534932 0.844895i \(-0.320337\pi\)
0.534932 + 0.844895i \(0.320337\pi\)
\(18\) −1.98359e11 −0.235702
\(19\) 9.69502e11 0.689270 0.344635 0.938737i \(-0.388003\pi\)
0.344635 + 0.938737i \(0.388003\pi\)
\(20\) −9.78446e11 −0.427319
\(21\) 2.94601e12 0.809376
\(22\) 3.81935e12 0.674488
\(23\) −1.36837e12 −0.158413 −0.0792064 0.996858i \(-0.525239\pi\)
−0.0792064 + 0.996858i \(0.525239\pi\)
\(24\) 2.64181e12 0.204124
\(25\) −5.14212e12 −0.269595
\(26\) −3.03301e13 −1.09554
\(27\) −7.62560e12 −0.192450
\(28\) −3.92358e13 −0.700940
\(29\) −9.86429e13 −1.26266 −0.631328 0.775516i \(-0.717490\pi\)
−0.631328 + 0.775516i \(0.717490\pi\)
\(30\) −3.76147e13 −0.348904
\(31\) 1.94952e14 1.32432 0.662158 0.749364i \(-0.269641\pi\)
0.662158 + 0.749364i \(0.269641\pi\)
\(32\) −3.51844e13 −0.176777
\(33\) 1.46829e14 0.550717
\(34\) −2.67833e14 −0.756508
\(35\) 5.58649e14 1.19810
\(36\) 1.01560e14 0.166667
\(37\) 1.18732e15 1.50193 0.750967 0.660340i \(-0.229588\pi\)
0.750967 + 0.660340i \(0.229588\pi\)
\(38\) −4.96385e14 −0.487388
\(39\) −1.16599e15 −0.894502
\(40\) 5.00964e14 0.302160
\(41\) 1.87020e15 0.892157 0.446078 0.894994i \(-0.352820\pi\)
0.446078 + 0.894994i \(0.352820\pi\)
\(42\) −1.50836e15 −0.572315
\(43\) −1.48369e14 −0.0450186 −0.0225093 0.999747i \(-0.507166\pi\)
−0.0225093 + 0.999747i \(0.507166\pi\)
\(44\) −1.95551e15 −0.476935
\(45\) −1.44604e15 −0.284879
\(46\) 7.00608e14 0.112015
\(47\) −1.05723e16 −1.37797 −0.688985 0.724775i \(-0.741944\pi\)
−0.688985 + 0.724775i \(0.741944\pi\)
\(48\) −1.35261e15 −0.144338
\(49\) 1.10030e16 0.965270
\(50\) 2.63277e15 0.190633
\(51\) −1.02964e16 −0.617686
\(52\) 1.55290e16 0.774661
\(53\) −3.63439e16 −1.51290 −0.756451 0.654051i \(-0.773068\pi\)
−0.756451 + 0.654051i \(0.773068\pi\)
\(54\) 3.90431e15 0.136083
\(55\) 2.78430e16 0.815213
\(56\) 2.00887e16 0.495640
\(57\) −1.90827e16 −0.397950
\(58\) 5.05052e16 0.892832
\(59\) 4.14701e16 0.623220 0.311610 0.950210i \(-0.399132\pi\)
0.311610 + 0.950210i \(0.399132\pi\)
\(60\) 1.92587e16 0.246712
\(61\) 1.47498e17 1.61492 0.807461 0.589921i \(-0.200841\pi\)
0.807461 + 0.589921i \(0.200841\pi\)
\(62\) −9.98154e16 −0.936433
\(63\) −5.79863e16 −0.467294
\(64\) 1.80144e16 0.125000
\(65\) −2.21106e17 −1.32411
\(66\) −7.51763e16 −0.389416
\(67\) −3.73864e17 −1.67881 −0.839407 0.543504i \(-0.817097\pi\)
−0.839407 + 0.543504i \(0.817097\pi\)
\(68\) 1.37130e17 0.534932
\(69\) 2.69337e16 0.0914597
\(70\) −2.86028e17 −0.847184
\(71\) 6.82838e17 1.76752 0.883758 0.467944i \(-0.155005\pi\)
0.883758 + 0.467944i \(0.155005\pi\)
\(72\) −5.19987e16 −0.117851
\(73\) −6.94764e16 −0.138124 −0.0690622 0.997612i \(-0.522001\pi\)
−0.0690622 + 0.997612i \(0.522001\pi\)
\(74\) −6.07907e17 −1.06203
\(75\) 1.01212e17 0.155651
\(76\) 2.54149e17 0.344635
\(77\) 1.11651e18 1.33721
\(78\) 5.96987e17 0.632508
\(79\) 4.08505e17 0.383477 0.191739 0.981446i \(-0.438587\pi\)
0.191739 + 0.981446i \(0.438587\pi\)
\(80\) −2.56494e17 −0.213659
\(81\) 1.50095e17 0.111111
\(82\) −9.57542e17 −0.630850
\(83\) −1.99775e17 −0.117301 −0.0586503 0.998279i \(-0.518680\pi\)
−0.0586503 + 0.998279i \(0.518680\pi\)
\(84\) 7.72278e17 0.404688
\(85\) −1.95250e18 −0.914346
\(86\) 7.59648e16 0.0318330
\(87\) 1.94159e18 0.728995
\(88\) 1.00122e18 0.337244
\(89\) 1.27068e18 0.384443 0.192221 0.981352i \(-0.438431\pi\)
0.192221 + 0.981352i \(0.438431\pi\)
\(90\) 7.40371e17 0.201440
\(91\) −8.86638e18 −2.17197
\(92\) −3.58711e17 −0.0792064
\(93\) −3.83724e18 −0.764594
\(94\) 5.41302e18 0.974372
\(95\) −3.61864e18 −0.589076
\(96\) 6.92534e17 0.102062
\(97\) 2.51426e18 0.335798 0.167899 0.985804i \(-0.446302\pi\)
0.167899 + 0.985804i \(0.446302\pi\)
\(98\) −5.63354e18 −0.682549
\(99\) −2.89003e18 −0.317957
\(100\) −1.34798e18 −0.134798
\(101\) 1.60915e19 1.46401 0.732004 0.681300i \(-0.238585\pi\)
0.732004 + 0.681300i \(0.238585\pi\)
\(102\) 5.27175e18 0.436770
\(103\) 1.50360e17 0.0113548 0.00567739 0.999984i \(-0.498193\pi\)
0.00567739 + 0.999984i \(0.498193\pi\)
\(104\) −7.95085e18 −0.547768
\(105\) −1.09959e19 −0.691723
\(106\) 1.86081e19 1.06978
\(107\) 2.03779e19 1.07155 0.535776 0.844360i \(-0.320019\pi\)
0.535776 + 0.844360i \(0.320019\pi\)
\(108\) −1.99900e18 −0.0962250
\(109\) 3.58279e19 1.58005 0.790023 0.613077i \(-0.210069\pi\)
0.790023 + 0.613077i \(0.210069\pi\)
\(110\) −1.42556e19 −0.576443
\(111\) −2.33700e19 −0.867142
\(112\) −1.02854e19 −0.350470
\(113\) −1.67259e18 −0.0523776 −0.0261888 0.999657i \(-0.508337\pi\)
−0.0261888 + 0.999657i \(0.508337\pi\)
\(114\) 9.77035e18 0.281393
\(115\) 5.10742e18 0.135385
\(116\) −2.58586e19 −0.631328
\(117\) 2.29502e19 0.516441
\(118\) −2.12327e19 −0.440683
\(119\) −7.82953e19 −1.49982
\(120\) −9.86048e18 −0.174452
\(121\) −5.51238e18 −0.0901318
\(122\) −7.55188e19 −1.14192
\(123\) −3.68111e19 −0.515087
\(124\) 5.11055e19 0.662158
\(125\) 9.03841e19 1.08504
\(126\) 2.96890e19 0.330426
\(127\) −3.02978e19 −0.312806 −0.156403 0.987693i \(-0.549990\pi\)
−0.156403 + 0.987693i \(0.549990\pi\)
\(128\) −9.22337e18 −0.0883883
\(129\) 2.92034e18 0.0259915
\(130\) 1.13206e20 0.936286
\(131\) 1.61202e20 1.23964 0.619818 0.784746i \(-0.287206\pi\)
0.619818 + 0.784746i \(0.287206\pi\)
\(132\) 3.84903e19 0.275359
\(133\) −1.45108e20 −0.966275
\(134\) 1.91418e20 1.18710
\(135\) 2.84623e19 0.164475
\(136\) −7.02107e19 −0.378254
\(137\) 5.03120e19 0.252829 0.126414 0.991978i \(-0.459653\pi\)
0.126414 + 0.991978i \(0.459653\pi\)
\(138\) −1.37901e19 −0.0646718
\(139\) −3.17871e20 −1.39191 −0.695955 0.718086i \(-0.745019\pi\)
−0.695955 + 0.718086i \(0.745019\pi\)
\(140\) 1.46447e20 0.599050
\(141\) 2.08095e20 0.795572
\(142\) −3.49613e20 −1.24982
\(143\) −4.41899e20 −1.47785
\(144\) 2.66233e19 0.0833333
\(145\) 3.68182e20 1.07911
\(146\) 3.55719e19 0.0976687
\(147\) −2.16572e20 −0.557299
\(148\) 3.11248e20 0.750967
\(149\) 4.24814e20 0.961455 0.480728 0.876870i \(-0.340373\pi\)
0.480728 + 0.876870i \(0.340373\pi\)
\(150\) −5.18208e19 −0.110062
\(151\) −5.76835e20 −1.15019 −0.575097 0.818086i \(-0.695036\pi\)
−0.575097 + 0.818086i \(0.695036\pi\)
\(152\) −1.30124e20 −0.243694
\(153\) 2.02664e20 0.356621
\(154\) −5.71653e20 −0.945552
\(155\) −7.27653e20 −1.13181
\(156\) −3.05657e20 −0.447251
\(157\) 5.56405e20 0.766204 0.383102 0.923706i \(-0.374856\pi\)
0.383102 + 0.923706i \(0.374856\pi\)
\(158\) −2.09154e20 −0.271159
\(159\) 7.15357e20 0.873474
\(160\) 1.31325e20 0.151080
\(161\) 2.04808e20 0.222076
\(162\) −7.68485e19 −0.0785674
\(163\) 1.49613e21 1.44273 0.721367 0.692553i \(-0.243514\pi\)
0.721367 + 0.692553i \(0.243514\pi\)
\(164\) 4.90261e20 0.446078
\(165\) −5.48034e20 −0.470663
\(166\) 1.02285e20 0.0829440
\(167\) −9.59191e20 −0.734681 −0.367340 0.930087i \(-0.619732\pi\)
−0.367340 + 0.930087i \(0.619732\pi\)
\(168\) −3.95406e20 −0.286158
\(169\) 2.04727e21 1.40040
\(170\) 9.99678e20 0.646540
\(171\) 3.75605e20 0.229757
\(172\) −3.88940e19 −0.0225093
\(173\) −5.51701e20 −0.302180 −0.151090 0.988520i \(-0.548278\pi\)
−0.151090 + 0.988520i \(0.548278\pi\)
\(174\) −9.94093e20 −0.515477
\(175\) 7.69635e20 0.377941
\(176\) −5.12625e20 −0.238468
\(177\) −8.16257e20 −0.359816
\(178\) −6.50589e20 −0.271842
\(179\) −1.32712e21 −0.525782 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(180\) −3.79070e20 −0.142440
\(181\) −2.95073e21 −1.05192 −0.525961 0.850509i \(-0.676294\pi\)
−0.525961 + 0.850509i \(0.676294\pi\)
\(182\) 4.53959e21 1.53581
\(183\) −2.90319e21 −0.932375
\(184\) 1.83660e20 0.0560074
\(185\) −4.43163e21 −1.28361
\(186\) 1.96467e21 0.540650
\(187\) −3.90223e21 −1.02051
\(188\) −2.77147e21 −0.688985
\(189\) 1.14134e21 0.269792
\(190\) 1.85274e21 0.416540
\(191\) 4.41459e21 0.944220 0.472110 0.881540i \(-0.343493\pi\)
0.472110 + 0.881540i \(0.343493\pi\)
\(192\) −3.54577e20 −0.0721688
\(193\) 5.13561e21 0.994941 0.497470 0.867481i \(-0.334262\pi\)
0.497470 + 0.867481i \(0.334262\pi\)
\(194\) −1.28730e21 −0.237445
\(195\) 4.35203e21 0.764475
\(196\) 2.88437e21 0.482635
\(197\) 8.08427e20 0.128888 0.0644439 0.997921i \(-0.479473\pi\)
0.0644439 + 0.997921i \(0.479473\pi\)
\(198\) 1.47970e21 0.224829
\(199\) −4.98582e21 −0.722159 −0.361079 0.932535i \(-0.617592\pi\)
−0.361079 + 0.932535i \(0.617592\pi\)
\(200\) 6.90164e20 0.0953164
\(201\) 7.35876e21 0.969264
\(202\) −8.23885e21 −1.03521
\(203\) 1.47641e22 1.77009
\(204\) −2.69913e21 −0.308843
\(205\) −6.98047e21 −0.762470
\(206\) −7.69844e19 −0.00802905
\(207\) −5.30136e20 −0.0528043
\(208\) 4.07084e21 0.387331
\(209\) −7.23217e21 −0.657474
\(210\) 5.62990e21 0.489122
\(211\) 1.35801e22 1.12777 0.563883 0.825854i \(-0.309307\pi\)
0.563883 + 0.825854i \(0.309307\pi\)
\(212\) −9.52733e21 −0.756451
\(213\) −1.34403e22 −1.02048
\(214\) −1.04335e22 −0.757701
\(215\) 5.53782e20 0.0384746
\(216\) 1.02349e21 0.0680414
\(217\) −2.91790e22 −1.85653
\(218\) −1.83439e22 −1.11726
\(219\) 1.36750e21 0.0797462
\(220\) 7.29888e21 0.407606
\(221\) 3.09883e22 1.65756
\(222\) 1.19654e22 0.613162
\(223\) 2.50315e21 0.122911 0.0614554 0.998110i \(-0.480426\pi\)
0.0614554 + 0.998110i \(0.480426\pi\)
\(224\) 5.26614e21 0.247820
\(225\) −1.99216e21 −0.0898651
\(226\) 8.56368e20 0.0370365
\(227\) 8.31116e19 0.00344680 0.00172340 0.999999i \(-0.499451\pi\)
0.00172340 + 0.999999i \(0.499451\pi\)
\(228\) −5.00242e21 −0.198975
\(229\) 3.24628e20 0.0123865 0.00619325 0.999981i \(-0.498029\pi\)
0.00619325 + 0.999981i \(0.498029\pi\)
\(230\) −2.61500e21 −0.0957320
\(231\) −2.19762e22 −0.772040
\(232\) 1.32396e22 0.446416
\(233\) −1.38992e22 −0.449892 −0.224946 0.974371i \(-0.572221\pi\)
−0.224946 + 0.974371i \(0.572221\pi\)
\(234\) −1.17505e22 −0.365179
\(235\) 3.94608e22 1.17766
\(236\) 1.08711e22 0.311610
\(237\) −8.04060e21 −0.221401
\(238\) 4.00872e22 1.06053
\(239\) −6.81749e21 −0.173318 −0.0866592 0.996238i \(-0.527619\pi\)
−0.0866592 + 0.996238i \(0.527619\pi\)
\(240\) 5.04856e21 0.123356
\(241\) −1.10203e22 −0.258841 −0.129420 0.991590i \(-0.541312\pi\)
−0.129420 + 0.991590i \(0.541312\pi\)
\(242\) 2.82234e21 0.0637328
\(243\) −2.95431e21 −0.0641500
\(244\) 3.86656e22 0.807461
\(245\) −4.10684e22 −0.824955
\(246\) 1.88473e22 0.364221
\(247\) 5.74318e22 1.06790
\(248\) −2.61660e22 −0.468216
\(249\) 3.93218e21 0.0677235
\(250\) −4.62767e22 −0.767242
\(251\) 1.28693e22 0.205426 0.102713 0.994711i \(-0.467248\pi\)
0.102713 + 0.994711i \(0.467248\pi\)
\(252\) −1.52007e22 −0.233647
\(253\) 1.02076e22 0.151105
\(254\) 1.55125e22 0.221187
\(255\) 3.84310e22 0.527898
\(256\) 4.72237e21 0.0625000
\(257\) −1.11082e23 −1.41671 −0.708354 0.705857i \(-0.750562\pi\)
−0.708354 + 0.705857i \(0.750562\pi\)
\(258\) −1.49521e21 −0.0183788
\(259\) −1.77709e23 −2.10553
\(260\) −5.79616e22 −0.662054
\(261\) −3.82163e22 −0.420885
\(262\) −8.25356e22 −0.876555
\(263\) 8.23499e22 0.843497 0.421748 0.906713i \(-0.361417\pi\)
0.421748 + 0.906713i \(0.361417\pi\)
\(264\) −1.97070e22 −0.194708
\(265\) 1.35653e23 1.29298
\(266\) 7.42953e22 0.683259
\(267\) −2.50108e22 −0.221958
\(268\) −9.80061e22 −0.839407
\(269\) 1.28354e23 1.06111 0.530556 0.847650i \(-0.321983\pi\)
0.530556 + 0.847650i \(0.321983\pi\)
\(270\) −1.45727e22 −0.116301
\(271\) 2.82596e22 0.217750 0.108875 0.994055i \(-0.465275\pi\)
0.108875 + 0.994055i \(0.465275\pi\)
\(272\) 3.59479e22 0.267466
\(273\) 1.74517e23 1.25399
\(274\) −2.57597e22 −0.178777
\(275\) 3.83586e22 0.257159
\(276\) 7.06051e21 0.0457298
\(277\) 4.69743e22 0.293969 0.146985 0.989139i \(-0.453043\pi\)
0.146985 + 0.989139i \(0.453043\pi\)
\(278\) 1.62750e23 0.984228
\(279\) 7.55284e22 0.441439
\(280\) −7.49806e22 −0.423592
\(281\) 1.44399e22 0.0788593 0.0394296 0.999222i \(-0.487446\pi\)
0.0394296 + 0.999222i \(0.487446\pi\)
\(282\) −1.06544e23 −0.562554
\(283\) −2.54603e23 −1.29985 −0.649924 0.759999i \(-0.725199\pi\)
−0.649924 + 0.759999i \(0.725199\pi\)
\(284\) 1.79002e23 0.883758
\(285\) 7.12257e22 0.340103
\(286\) 2.26253e23 1.04500
\(287\) −2.79918e23 −1.25070
\(288\) −1.36311e22 −0.0589256
\(289\) 3.45721e22 0.144609
\(290\) −1.88509e23 −0.763048
\(291\) −4.94882e22 −0.193873
\(292\) −1.82128e22 −0.0690622
\(293\) 1.20485e21 0.00442274 0.00221137 0.999998i \(-0.499296\pi\)
0.00221137 + 0.999998i \(0.499296\pi\)
\(294\) 1.10885e23 0.394070
\(295\) −1.54786e23 −0.532627
\(296\) −1.59359e23 −0.531014
\(297\) 5.68845e22 0.183572
\(298\) −2.17505e23 −0.679852
\(299\) −8.10604e22 −0.245433
\(300\) 2.65322e22 0.0778255
\(301\) 2.22067e22 0.0631107
\(302\) 2.95340e23 0.813309
\(303\) −3.16729e23 −0.845245
\(304\) 6.66237e22 0.172318
\(305\) −5.50531e23 −1.38017
\(306\) −1.03764e23 −0.252169
\(307\) 4.80032e22 0.113098 0.0565490 0.998400i \(-0.481990\pi\)
0.0565490 + 0.998400i \(0.481990\pi\)
\(308\) 2.92686e23 0.668606
\(309\) −2.95954e21 −0.00655569
\(310\) 3.72558e23 0.800310
\(311\) −2.92996e23 −0.610434 −0.305217 0.952283i \(-0.598729\pi\)
−0.305217 + 0.952283i \(0.598729\pi\)
\(312\) 1.56497e23 0.316254
\(313\) 9.11835e23 1.78750 0.893748 0.448569i \(-0.148066\pi\)
0.893748 + 0.448569i \(0.148066\pi\)
\(314\) −2.84880e23 −0.541788
\(315\) 2.16432e23 0.399366
\(316\) 1.07087e23 0.191739
\(317\) 1.82852e23 0.317714 0.158857 0.987302i \(-0.449219\pi\)
0.158857 + 0.987302i \(0.449219\pi\)
\(318\) −3.66263e23 −0.617639
\(319\) 7.35844e23 1.20441
\(320\) −6.72383e22 −0.106830
\(321\) −4.01098e23 −0.618661
\(322\) −1.04862e23 −0.157031
\(323\) 5.07157e23 0.737425
\(324\) 3.93464e22 0.0555556
\(325\) −3.04612e23 −0.417690
\(326\) −7.66017e23 −1.02017
\(327\) −7.05201e23 −0.912240
\(328\) −2.51014e23 −0.315425
\(329\) 1.58238e24 1.93175
\(330\) 2.80594e23 0.332809
\(331\) −3.98958e23 −0.459792 −0.229896 0.973215i \(-0.573839\pi\)
−0.229896 + 0.973215i \(0.573839\pi\)
\(332\) −5.23699e22 −0.0586503
\(333\) 4.59991e23 0.500644
\(334\) 4.91106e23 0.519498
\(335\) 1.39544e24 1.43478
\(336\) 2.02448e23 0.202344
\(337\) −1.41916e24 −1.37895 −0.689474 0.724311i \(-0.742158\pi\)
−0.689474 + 0.724311i \(0.742158\pi\)
\(338\) −1.04820e24 −0.990233
\(339\) 3.29217e22 0.0302402
\(340\) −5.11835e23 −0.457173
\(341\) −1.45428e24 −1.26323
\(342\) −1.92310e23 −0.162463
\(343\) 5.92534e22 0.0486877
\(344\) 1.99137e22 0.0159165
\(345\) −1.00529e23 −0.0781649
\(346\) 2.82471e23 0.213674
\(347\) −1.47004e23 −0.108193 −0.0540964 0.998536i \(-0.517228\pi\)
−0.0540964 + 0.998536i \(0.517228\pi\)
\(348\) 5.08976e23 0.364497
\(349\) 7.67378e23 0.534771 0.267385 0.963590i \(-0.413840\pi\)
0.267385 + 0.963590i \(0.413840\pi\)
\(350\) −3.94053e23 −0.267244
\(351\) −4.51729e23 −0.298167
\(352\) 2.62464e23 0.168622
\(353\) −1.87042e24 −1.16971 −0.584857 0.811137i \(-0.698849\pi\)
−0.584857 + 0.811137i \(0.698849\pi\)
\(354\) 4.17923e23 0.254429
\(355\) −2.54868e24 −1.51059
\(356\) 3.33102e23 0.192221
\(357\) 1.54109e24 0.865923
\(358\) 6.79484e23 0.371784
\(359\) 1.62492e24 0.865835 0.432918 0.901434i \(-0.357484\pi\)
0.432918 + 0.901434i \(0.357484\pi\)
\(360\) 1.94084e23 0.100720
\(361\) −1.03849e24 −0.524907
\(362\) 1.51077e24 0.743821
\(363\) 1.08500e23 0.0520376
\(364\) −2.32427e24 −1.08598
\(365\) 2.59319e23 0.118046
\(366\) 1.48644e24 0.659289
\(367\) −3.00365e23 −0.129814 −0.0649071 0.997891i \(-0.520675\pi\)
−0.0649071 + 0.997891i \(0.520675\pi\)
\(368\) −9.40340e22 −0.0396032
\(369\) 7.24553e23 0.297386
\(370\) 2.26900e24 0.907648
\(371\) 5.43969e24 2.12091
\(372\) −1.00591e24 −0.382297
\(373\) 2.68937e24 0.996359 0.498180 0.867074i \(-0.334002\pi\)
0.498180 + 0.867074i \(0.334002\pi\)
\(374\) 1.99794e24 0.721611
\(375\) −1.77903e24 −0.626450
\(376\) 1.41899e24 0.487186
\(377\) −5.84345e24 −1.95626
\(378\) −5.84368e23 −0.190772
\(379\) −2.46626e22 −0.00785176 −0.00392588 0.999992i \(-0.501250\pi\)
−0.00392588 + 0.999992i \(0.501250\pi\)
\(380\) −9.48605e23 −0.294538
\(381\) 5.96351e23 0.180599
\(382\) −2.26027e24 −0.667664
\(383\) 4.45159e24 1.28271 0.641353 0.767246i \(-0.278373\pi\)
0.641353 + 0.767246i \(0.278373\pi\)
\(384\) 1.81544e23 0.0510310
\(385\) −4.16734e24 −1.14283
\(386\) −2.62943e24 −0.703529
\(387\) −5.74811e22 −0.0150062
\(388\) 6.59098e23 0.167899
\(389\) −7.02195e24 −1.74557 −0.872784 0.488107i \(-0.837688\pi\)
−0.872784 + 0.488107i \(0.837688\pi\)
\(390\) −2.22824e24 −0.540565
\(391\) −7.15811e23 −0.169480
\(392\) −1.47680e24 −0.341274
\(393\) −3.17295e24 −0.715704
\(394\) −4.13915e23 −0.0911374
\(395\) −1.52473e24 −0.327734
\(396\) −7.57604e23 −0.158978
\(397\) 5.45409e24 1.11741 0.558705 0.829366i \(-0.311298\pi\)
0.558705 + 0.829366i \(0.311298\pi\)
\(398\) 2.55274e24 0.510643
\(399\) 2.85616e24 0.557879
\(400\) −3.53364e23 −0.0673988
\(401\) 3.85979e24 0.718940 0.359470 0.933157i \(-0.382958\pi\)
0.359470 + 0.933157i \(0.382958\pi\)
\(402\) −3.76768e24 −0.685373
\(403\) 1.15487e25 2.05179
\(404\) 4.21829e24 0.732004
\(405\) −5.60224e23 −0.0949597
\(406\) −7.55924e24 −1.25164
\(407\) −8.85700e24 −1.43265
\(408\) 1.38196e24 0.218385
\(409\) 7.54727e24 1.16525 0.582624 0.812742i \(-0.302026\pi\)
0.582624 + 0.812742i \(0.302026\pi\)
\(410\) 3.57400e24 0.539148
\(411\) −9.90291e23 −0.145971
\(412\) 3.94160e22 0.00567739
\(413\) −6.20695e24 −0.873680
\(414\) 2.71430e23 0.0373383
\(415\) 7.45656e23 0.100249
\(416\) −2.08427e24 −0.273884
\(417\) 6.25666e24 0.803619
\(418\) 3.70287e24 0.464905
\(419\) 1.19281e25 1.46400 0.731998 0.681307i \(-0.238588\pi\)
0.731998 + 0.681307i \(0.238588\pi\)
\(420\) −2.88251e24 −0.345861
\(421\) 6.32070e24 0.741456 0.370728 0.928742i \(-0.379108\pi\)
0.370728 + 0.928742i \(0.379108\pi\)
\(422\) −6.95300e24 −0.797451
\(423\) −4.09593e24 −0.459324
\(424\) 4.87799e24 0.534891
\(425\) −2.68990e24 −0.288430
\(426\) 6.88143e24 0.721586
\(427\) −2.20764e25 −2.26393
\(428\) 5.34194e24 0.535776
\(429\) 8.69791e24 0.853239
\(430\) −2.83537e23 −0.0272056
\(431\) −1.40410e25 −1.31784 −0.658921 0.752212i \(-0.728987\pi\)
−0.658921 + 0.752212i \(0.728987\pi\)
\(432\) −5.24027e23 −0.0481125
\(433\) −6.29547e24 −0.565448 −0.282724 0.959201i \(-0.591238\pi\)
−0.282724 + 0.959201i \(0.591238\pi\)
\(434\) 1.49396e25 1.31277
\(435\) −7.24693e24 −0.623026
\(436\) 9.39207e24 0.790023
\(437\) −1.32664e24 −0.109189
\(438\) −7.00162e23 −0.0563890
\(439\) −3.14113e24 −0.247556 −0.123778 0.992310i \(-0.539501\pi\)
−0.123778 + 0.992310i \(0.539501\pi\)
\(440\) −3.73703e24 −0.288221
\(441\) 4.26279e24 0.321757
\(442\) −1.58660e25 −1.17208
\(443\) 1.93668e25 1.40030 0.700152 0.713993i \(-0.253115\pi\)
0.700152 + 0.713993i \(0.253115\pi\)
\(444\) −6.12630e24 −0.433571
\(445\) −4.74279e24 −0.328559
\(446\) −1.28161e24 −0.0869111
\(447\) −8.36161e24 −0.555096
\(448\) −2.69626e24 −0.175235
\(449\) −2.12764e25 −1.35381 −0.676906 0.736070i \(-0.736679\pi\)
−0.676906 + 0.736070i \(0.736679\pi\)
\(450\) 1.01999e24 0.0635442
\(451\) −1.39511e25 −0.851002
\(452\) −4.38460e23 −0.0261888
\(453\) 1.13538e25 0.664064
\(454\) −4.25531e22 −0.00243726
\(455\) 3.30935e25 1.85624
\(456\) 2.56124e24 0.140697
\(457\) −2.08995e25 −1.12443 −0.562214 0.826992i \(-0.690050\pi\)
−0.562214 + 0.826992i \(0.690050\pi\)
\(458\) −1.66209e23 −0.00875858
\(459\) −3.98903e24 −0.205895
\(460\) 1.33888e24 0.0676927
\(461\) 3.78803e25 1.87609 0.938046 0.346510i \(-0.112633\pi\)
0.938046 + 0.346510i \(0.112633\pi\)
\(462\) 1.12518e25 0.545915
\(463\) −1.09015e25 −0.518163 −0.259082 0.965855i \(-0.583420\pi\)
−0.259082 + 0.965855i \(0.583420\pi\)
\(464\) −6.77869e24 −0.315664
\(465\) 1.43224e25 0.653450
\(466\) 7.11638e24 0.318121
\(467\) −3.52044e25 −1.54201 −0.771004 0.636830i \(-0.780245\pi\)
−0.771004 + 0.636830i \(0.780245\pi\)
\(468\) 6.01626e24 0.258220
\(469\) 5.59572e25 2.35350
\(470\) −2.02040e25 −0.832735
\(471\) −1.09517e25 −0.442368
\(472\) −5.56603e24 −0.220342
\(473\) 1.10678e24 0.0429419
\(474\) 4.11679e24 0.156554
\(475\) −4.98530e24 −0.185824
\(476\) −2.05247e25 −0.749911
\(477\) −1.40804e25 −0.504301
\(478\) 3.49056e24 0.122555
\(479\) 4.63983e25 1.59704 0.798518 0.601971i \(-0.205618\pi\)
0.798518 + 0.601971i \(0.205618\pi\)
\(480\) −2.58487e24 −0.0872260
\(481\) 7.03349e25 2.32698
\(482\) 5.64241e24 0.183028
\(483\) −4.03124e24 −0.128216
\(484\) −1.44504e24 −0.0450659
\(485\) −9.38441e24 −0.286986
\(486\) 1.51261e24 0.0453609
\(487\) −2.50165e25 −0.735701 −0.367851 0.929885i \(-0.619906\pi\)
−0.367851 + 0.929885i \(0.619906\pi\)
\(488\) −1.97968e25 −0.570961
\(489\) −2.94483e25 −0.832963
\(490\) 2.10270e25 0.583332
\(491\) 1.16110e24 0.0315934 0.0157967 0.999875i \(-0.494972\pi\)
0.0157967 + 0.999875i \(0.494972\pi\)
\(492\) −9.64982e24 −0.257543
\(493\) −5.16011e25 −1.35087
\(494\) −2.94051e25 −0.755121
\(495\) 1.07870e25 0.271738
\(496\) 1.33970e25 0.331079
\(497\) −1.02202e26 −2.47785
\(498\) −2.01328e24 −0.0478878
\(499\) 3.16323e25 0.738202 0.369101 0.929389i \(-0.379666\pi\)
0.369101 + 0.929389i \(0.379666\pi\)
\(500\) 2.36937e25 0.542522
\(501\) 1.88798e25 0.424168
\(502\) −6.58909e24 −0.145258
\(503\) −7.15936e23 −0.0154874 −0.00774369 0.999970i \(-0.502465\pi\)
−0.00774369 + 0.999970i \(0.502465\pi\)
\(504\) 7.78278e24 0.165213
\(505\) −6.00611e25 −1.25120
\(506\) −5.22630e24 −0.106848
\(507\) −4.02965e25 −0.808522
\(508\) −7.94238e24 −0.156403
\(509\) 7.34242e25 1.41912 0.709562 0.704643i \(-0.248893\pi\)
0.709562 + 0.704643i \(0.248893\pi\)
\(510\) −1.96767e25 −0.373280
\(511\) 1.03987e25 0.193634
\(512\) −2.41785e24 −0.0441942
\(513\) −7.39303e24 −0.132650
\(514\) 5.68742e25 1.00176
\(515\) −5.61215e23 −0.00970422
\(516\) 7.65550e23 0.0129957
\(517\) 7.88659e25 1.31440
\(518\) 9.09870e25 1.48884
\(519\) 1.08591e25 0.174464
\(520\) 2.96763e25 0.468143
\(521\) −8.40278e25 −1.30156 −0.650780 0.759266i \(-0.725558\pi\)
−0.650780 + 0.759266i \(0.725558\pi\)
\(522\) 1.95667e25 0.297611
\(523\) 2.62032e25 0.391371 0.195685 0.980667i \(-0.437307\pi\)
0.195685 + 0.980667i \(0.437307\pi\)
\(524\) 4.22582e25 0.619818
\(525\) −1.51487e25 −0.218204
\(526\) −4.21631e25 −0.596442
\(527\) 1.01981e26 1.41684
\(528\) 1.00900e25 0.137679
\(529\) −7.27430e25 −0.974905
\(530\) −6.94541e25 −0.914276
\(531\) 1.60664e25 0.207740
\(532\) −3.80392e25 −0.483137
\(533\) 1.10788e26 1.38224
\(534\) 1.28055e25 0.156948
\(535\) −7.60599e25 −0.915788
\(536\) 5.01791e25 0.593550
\(537\) 2.61217e25 0.303560
\(538\) −6.57171e25 −0.750320
\(539\) −8.20788e25 −0.920742
\(540\) 7.46123e24 0.0822375
\(541\) 8.43002e24 0.0912966 0.0456483 0.998958i \(-0.485465\pi\)
0.0456483 + 0.998958i \(0.485465\pi\)
\(542\) −1.44689e25 −0.153972
\(543\) 5.80793e25 0.607327
\(544\) −1.84053e25 −0.189127
\(545\) −1.33727e26 −1.35037
\(546\) −8.93527e25 −0.886701
\(547\) 9.35298e25 0.912158 0.456079 0.889939i \(-0.349253\pi\)
0.456079 + 0.889939i \(0.349253\pi\)
\(548\) 1.31890e25 0.126414
\(549\) 5.71436e25 0.538307
\(550\) −1.96396e25 −0.181839
\(551\) −9.56345e25 −0.870311
\(552\) −3.61498e24 −0.0323359
\(553\) −6.11420e25 −0.537589
\(554\) −2.40508e25 −0.207868
\(555\) 8.72278e25 0.741091
\(556\) −8.33281e25 −0.695955
\(557\) −6.72734e25 −0.552356 −0.276178 0.961107i \(-0.589068\pi\)
−0.276178 + 0.961107i \(0.589068\pi\)
\(558\) −3.86705e25 −0.312144
\(559\) −8.78913e24 −0.0697483
\(560\) 3.83901e25 0.299525
\(561\) 7.68076e25 0.589193
\(562\) −7.39320e24 −0.0557619
\(563\) 1.94848e26 1.44500 0.722499 0.691372i \(-0.242994\pi\)
0.722499 + 0.691372i \(0.242994\pi\)
\(564\) 5.45508e25 0.397786
\(565\) 6.24291e24 0.0447638
\(566\) 1.30357e26 0.919131
\(567\) −2.24651e25 −0.155765
\(568\) −9.16490e25 −0.624911
\(569\) −2.11021e25 −0.141501 −0.0707504 0.997494i \(-0.522539\pi\)
−0.0707504 + 0.997494i \(0.522539\pi\)
\(570\) −3.64676e25 −0.240489
\(571\) −3.02789e25 −0.196380 −0.0981900 0.995168i \(-0.531305\pi\)
−0.0981900 + 0.995168i \(0.531305\pi\)
\(572\) −1.15841e26 −0.738926
\(573\) −8.68923e25 −0.545146
\(574\) 1.43318e26 0.884377
\(575\) 7.03635e24 0.0427074
\(576\) 6.97915e24 0.0416667
\(577\) −1.45327e26 −0.853446 −0.426723 0.904382i \(-0.640332\pi\)
−0.426723 + 0.904382i \(0.640332\pi\)
\(578\) −1.77009e25 −0.102254
\(579\) −1.01084e26 −0.574429
\(580\) 9.65167e25 0.539556
\(581\) 2.99009e25 0.164441
\(582\) 2.53379e25 0.137089
\(583\) 2.71114e26 1.44311
\(584\) 9.32497e24 0.0488343
\(585\) −8.56610e25 −0.441370
\(586\) −6.16885e23 −0.00312735
\(587\) 1.44980e26 0.723178 0.361589 0.932338i \(-0.382234\pi\)
0.361589 + 0.932338i \(0.382234\pi\)
\(588\) −5.67731e25 −0.278649
\(589\) 1.89006e26 0.912811
\(590\) 7.92505e25 0.376624
\(591\) −1.59123e25 −0.0744134
\(592\) 8.15919e25 0.375483
\(593\) −9.52077e24 −0.0431174 −0.0215587 0.999768i \(-0.506863\pi\)
−0.0215587 + 0.999768i \(0.506863\pi\)
\(594\) −2.91248e25 −0.129805
\(595\) 2.92235e26 1.28180
\(596\) 1.11362e26 0.480728
\(597\) 9.81360e25 0.416939
\(598\) 4.15029e25 0.173547
\(599\) −3.73628e26 −1.53775 −0.768873 0.639402i \(-0.779182\pi\)
−0.768873 + 0.639402i \(0.779182\pi\)
\(600\) −1.35845e25 −0.0550309
\(601\) −2.04542e26 −0.815597 −0.407798 0.913072i \(-0.633703\pi\)
−0.407798 + 0.913072i \(0.633703\pi\)
\(602\) −1.13698e25 −0.0446260
\(603\) −1.44842e26 −0.559605
\(604\) −1.51214e26 −0.575097
\(605\) 2.05748e25 0.0770300
\(606\) 1.62165e26 0.597679
\(607\) −2.78691e26 −1.01118 −0.505591 0.862773i \(-0.668726\pi\)
−0.505591 + 0.862773i \(0.668726\pi\)
\(608\) −3.41113e25 −0.121847
\(609\) −2.90603e26 −1.02196
\(610\) 2.81872e26 0.975929
\(611\) −6.26287e26 −2.13492
\(612\) 5.31271e25 0.178311
\(613\) 1.83909e26 0.607754 0.303877 0.952711i \(-0.401719\pi\)
0.303877 + 0.952711i \(0.401719\pi\)
\(614\) −2.45776e25 −0.0799724
\(615\) 1.37397e26 0.440212
\(616\) −1.49855e26 −0.472776
\(617\) 5.72893e26 1.77977 0.889886 0.456183i \(-0.150784\pi\)
0.889886 + 0.456183i \(0.150784\pi\)
\(618\) 1.51528e24 0.00463557
\(619\) −4.19679e26 −1.26432 −0.632159 0.774838i \(-0.717831\pi\)
−0.632159 + 0.774838i \(0.717831\pi\)
\(620\) −1.90750e26 −0.565905
\(621\) 1.04347e25 0.0304866
\(622\) 1.50014e26 0.431642
\(623\) −1.90186e26 −0.538943
\(624\) −8.01263e25 −0.223625
\(625\) −2.39278e26 −0.657723
\(626\) −4.66860e26 −1.26395
\(627\) 1.42351e26 0.379593
\(628\) 1.45858e26 0.383102
\(629\) 6.21098e26 1.60686
\(630\) −1.10813e26 −0.282395
\(631\) 3.87489e26 0.972704 0.486352 0.873763i \(-0.338327\pi\)
0.486352 + 0.873763i \(0.338327\pi\)
\(632\) −5.48286e25 −0.135580
\(633\) −2.67297e26 −0.651116
\(634\) −9.36202e25 −0.224658
\(635\) 1.13086e26 0.267336
\(636\) 1.87526e26 0.436737
\(637\) 6.51801e26 1.49551
\(638\) −3.76752e26 −0.851646
\(639\) 2.64545e26 0.589172
\(640\) 3.44260e25 0.0755400
\(641\) 2.30520e26 0.498376 0.249188 0.968455i \(-0.419836\pi\)
0.249188 + 0.968455i \(0.419836\pi\)
\(642\) 2.05362e26 0.437459
\(643\) −3.49990e26 −0.734600 −0.367300 0.930103i \(-0.619718\pi\)
−0.367300 + 0.930103i \(0.619718\pi\)
\(644\) 5.36892e25 0.111038
\(645\) −1.09001e25 −0.0222133
\(646\) −2.59664e26 −0.521439
\(647\) 3.77903e26 0.747808 0.373904 0.927467i \(-0.378019\pi\)
0.373904 + 0.927467i \(0.378019\pi\)
\(648\) −2.01454e25 −0.0392837
\(649\) −3.09354e26 −0.594471
\(650\) 1.55961e26 0.295352
\(651\) 5.74330e26 1.07187
\(652\) 3.92201e26 0.721367
\(653\) 8.44532e26 1.53088 0.765439 0.643508i \(-0.222522\pi\)
0.765439 + 0.643508i \(0.222522\pi\)
\(654\) 3.61063e26 0.645051
\(655\) −6.01684e26 −1.05944
\(656\) 1.28519e26 0.223039
\(657\) −2.69166e25 −0.0460415
\(658\) −8.10181e26 −1.36595
\(659\) −2.79794e26 −0.464972 −0.232486 0.972600i \(-0.574686\pi\)
−0.232486 + 0.972600i \(0.574686\pi\)
\(660\) −1.43664e26 −0.235332
\(661\) −1.04973e26 −0.169498 −0.0847491 0.996402i \(-0.527009\pi\)
−0.0847491 + 0.996402i \(0.527009\pi\)
\(662\) 2.04267e26 0.325122
\(663\) −6.09942e26 −0.956996
\(664\) 2.68134e25 0.0414720
\(665\) 5.41612e26 0.825814
\(666\) −2.35516e26 −0.354009
\(667\) 1.34980e26 0.200021
\(668\) −2.51446e26 −0.367340
\(669\) −4.92694e25 −0.0709626
\(670\) −7.14463e26 −1.01454
\(671\) −1.10028e27 −1.54043
\(672\) −1.03653e26 −0.143079
\(673\) −8.12980e26 −1.10646 −0.553231 0.833028i \(-0.686605\pi\)
−0.553231 + 0.833028i \(0.686605\pi\)
\(674\) 7.26611e26 0.975063
\(675\) 3.92118e25 0.0518837
\(676\) 5.36681e26 0.700201
\(677\) −7.93130e26 −1.02036 −0.510178 0.860069i \(-0.670421\pi\)
−0.510178 + 0.860069i \(0.670421\pi\)
\(678\) −1.68559e25 −0.0213830
\(679\) −3.76316e26 −0.470749
\(680\) 2.62060e26 0.323270
\(681\) −1.63589e24 −0.00199001
\(682\) 7.44590e26 0.893235
\(683\) 5.15218e26 0.609528 0.304764 0.952428i \(-0.401422\pi\)
0.304764 + 0.952428i \(0.401422\pi\)
\(684\) 9.84626e25 0.114878
\(685\) −1.87788e26 −0.216077
\(686\) −3.03377e25 −0.0344274
\(687\) −6.38965e24 −0.00715135
\(688\) −1.01958e25 −0.0112546
\(689\) −2.15296e27 −2.34397
\(690\) 5.14710e25 0.0552709
\(691\) 9.85717e26 1.04402 0.522012 0.852938i \(-0.325182\pi\)
0.522012 + 0.852938i \(0.325182\pi\)
\(692\) −1.44625e26 −0.151090
\(693\) 4.32558e26 0.445737
\(694\) 7.52659e25 0.0765039
\(695\) 1.18645e27 1.18958
\(696\) −2.60596e26 −0.257739
\(697\) 9.78321e26 0.954487
\(698\) −3.92898e26 −0.378140
\(699\) 2.73578e26 0.259745
\(700\) 2.01755e26 0.188970
\(701\) 1.32660e27 1.22580 0.612899 0.790161i \(-0.290003\pi\)
0.612899 + 0.790161i \(0.290003\pi\)
\(702\) 2.31285e26 0.210836
\(703\) 1.15111e27 1.03524
\(704\) −1.34382e26 −0.119234
\(705\) −7.76708e26 −0.679925
\(706\) 9.57655e26 0.827112
\(707\) −2.40846e27 −2.05236
\(708\) −2.13977e26 −0.179908
\(709\) −5.63780e26 −0.467703 −0.233851 0.972272i \(-0.575133\pi\)
−0.233851 + 0.972272i \(0.575133\pi\)
\(710\) 1.30492e27 1.06814
\(711\) 1.58263e26 0.127826
\(712\) −1.70548e26 −0.135921
\(713\) −2.66767e26 −0.209789
\(714\) −7.89037e26 −0.612300
\(715\) 1.64938e27 1.26303
\(716\) −3.47896e26 −0.262891
\(717\) 1.34189e26 0.100065
\(718\) −8.31960e26 −0.612238
\(719\) −1.69545e27 −1.23129 −0.615645 0.788024i \(-0.711104\pi\)
−0.615645 + 0.788024i \(0.711104\pi\)
\(720\) −9.93709e25 −0.0712198
\(721\) −2.25048e25 −0.0159181
\(722\) 5.31705e26 0.371165
\(723\) 2.16913e26 0.149442
\(724\) −7.73517e26 −0.525961
\(725\) 5.07234e26 0.340406
\(726\) −5.55521e25 −0.0367962
\(727\) 2.57739e27 1.68501 0.842507 0.538685i \(-0.181079\pi\)
0.842507 + 0.538685i \(0.181079\pi\)
\(728\) 1.19003e27 0.767906
\(729\) 5.81497e25 0.0370370
\(730\) −1.32771e26 −0.0834713
\(731\) −7.76132e25 −0.0481638
\(732\) −7.61055e26 −0.466188
\(733\) −7.00439e26 −0.423528 −0.211764 0.977321i \(-0.567921\pi\)
−0.211764 + 0.977321i \(0.567921\pi\)
\(734\) 1.53787e26 0.0917925
\(735\) 8.08350e26 0.476288
\(736\) 4.81454e25 0.0280037
\(737\) 2.78890e27 1.60137
\(738\) −3.70971e26 −0.210283
\(739\) −2.24718e27 −1.25752 −0.628760 0.777599i \(-0.716437\pi\)
−0.628760 + 0.777599i \(0.716437\pi\)
\(740\) −1.16173e27 −0.641804
\(741\) −1.13043e27 −0.616554
\(742\) −2.78512e27 −1.49971
\(743\) −2.12060e26 −0.112737 −0.0563683 0.998410i \(-0.517952\pi\)
−0.0563683 + 0.998410i \(0.517952\pi\)
\(744\) 5.15026e26 0.270325
\(745\) −1.58561e27 −0.821695
\(746\) −1.37696e27 −0.704532
\(747\) −7.73971e25 −0.0391002
\(748\) −1.02295e27 −0.510256
\(749\) −3.05001e27 −1.50219
\(750\) 9.10864e26 0.442967
\(751\) 3.31814e27 1.59337 0.796684 0.604397i \(-0.206586\pi\)
0.796684 + 0.604397i \(0.206586\pi\)
\(752\) −7.26523e26 −0.344493
\(753\) −2.53307e26 −0.118603
\(754\) 2.99185e27 1.38329
\(755\) 2.15302e27 0.982998
\(756\) 2.99196e26 0.134896
\(757\) 2.98175e26 0.132758 0.0663789 0.997794i \(-0.478855\pi\)
0.0663789 + 0.997794i \(0.478855\pi\)
\(758\) 1.26273e25 0.00555203
\(759\) −2.00917e26 −0.0872407
\(760\) 4.85686e26 0.208270
\(761\) −7.82078e26 −0.331204 −0.165602 0.986193i \(-0.552957\pi\)
−0.165602 + 0.986193i \(0.552957\pi\)
\(762\) −3.05332e26 −0.127703
\(763\) −5.36246e27 −2.21504
\(764\) 1.15726e27 0.472110
\(765\) −7.56437e26 −0.304782
\(766\) −2.27922e27 −0.907010
\(767\) 2.45663e27 0.965569
\(768\) −9.29503e25 −0.0360844
\(769\) −1.07371e27 −0.411706 −0.205853 0.978583i \(-0.565997\pi\)
−0.205853 + 0.978583i \(0.565997\pi\)
\(770\) 2.13368e27 0.808104
\(771\) 2.18643e27 0.817937
\(772\) 1.34627e27 0.497470
\(773\) 3.25274e27 1.18725 0.593627 0.804740i \(-0.297696\pi\)
0.593627 + 0.804740i \(0.297696\pi\)
\(774\) 2.94303e25 0.0106110
\(775\) −1.00247e27 −0.357029
\(776\) −3.37458e26 −0.118723
\(777\) 3.49785e27 1.21563
\(778\) 3.59524e27 1.23430
\(779\) 1.81316e27 0.614937
\(780\) 1.14086e27 0.382237
\(781\) −5.09375e27 −1.68598
\(782\) 3.66495e26 0.119841
\(783\) 7.52211e26 0.242998
\(784\) 7.56121e26 0.241317
\(785\) −2.07677e27 −0.654827
\(786\) 1.62455e27 0.506079
\(787\) 1.32360e27 0.407378 0.203689 0.979036i \(-0.434707\pi\)
0.203689 + 0.979036i \(0.434707\pi\)
\(788\) 2.11924e26 0.0644439
\(789\) −1.62089e27 −0.486993
\(790\) 7.80663e26 0.231743
\(791\) 2.50342e26 0.0734271
\(792\) 3.87893e26 0.112415
\(793\) 8.73753e27 2.50204
\(794\) −2.79250e27 −0.790129
\(795\) −2.67005e27 −0.746503
\(796\) −1.30700e27 −0.361079
\(797\) −5.29249e27 −1.44480 −0.722398 0.691478i \(-0.756960\pi\)
−0.722398 + 0.691478i \(0.756960\pi\)
\(798\) −1.46235e27 −0.394480
\(799\) −5.53048e27 −1.47424
\(800\) 1.80922e26 0.0476582
\(801\) 4.92288e26 0.128148
\(802\) −1.97621e27 −0.508367
\(803\) 5.18271e26 0.131753
\(804\) 1.92905e27 0.484632
\(805\) −7.64441e26 −0.189794
\(806\) −5.91291e27 −1.45084
\(807\) −2.52639e27 −0.612634
\(808\) −2.15976e27 −0.517605
\(809\) 8.43529e26 0.199797 0.0998986 0.994998i \(-0.468148\pi\)
0.0998986 + 0.994998i \(0.468148\pi\)
\(810\) 2.86835e26 0.0671466
\(811\) 2.58962e27 0.599153 0.299576 0.954072i \(-0.403155\pi\)
0.299576 + 0.954072i \(0.403155\pi\)
\(812\) 3.87033e27 0.885046
\(813\) −5.56234e26 −0.125718
\(814\) 4.53479e27 1.01304
\(815\) −5.58426e27 −1.23301
\(816\) −7.07562e26 −0.154422
\(817\) −1.43844e26 −0.0310300
\(818\) −3.86420e27 −0.823955
\(819\) −3.43502e27 −0.723989
\(820\) −1.82989e27 −0.381235
\(821\) 6.26683e27 1.29059 0.645294 0.763934i \(-0.276735\pi\)
0.645294 + 0.763934i \(0.276735\pi\)
\(822\) 5.07029e26 0.103217
\(823\) −4.11316e27 −0.827709 −0.413854 0.910343i \(-0.635818\pi\)
−0.413854 + 0.910343i \(0.635818\pi\)
\(824\) −2.01810e25 −0.00401452
\(825\) −7.55012e26 −0.148471
\(826\) 3.17796e27 0.617785
\(827\) 5.78083e27 1.11093 0.555466 0.831539i \(-0.312540\pi\)
0.555466 + 0.831539i \(0.312540\pi\)
\(828\) −1.38972e26 −0.0264021
\(829\) 4.03871e27 0.758534 0.379267 0.925287i \(-0.376176\pi\)
0.379267 + 0.925287i \(0.376176\pi\)
\(830\) −3.81776e26 −0.0708870
\(831\) −9.24595e26 −0.169723
\(832\) 1.06715e27 0.193665
\(833\) 5.75579e27 1.03271
\(834\) −3.20341e27 −0.568245
\(835\) 3.58016e27 0.627886
\(836\) −1.89587e27 −0.328737
\(837\) −1.48663e27 −0.254865
\(838\) −6.10721e27 −1.03520
\(839\) 7.16187e27 1.20030 0.600148 0.799889i \(-0.295108\pi\)
0.600148 + 0.799889i \(0.295108\pi\)
\(840\) 1.47584e27 0.244561
\(841\) 3.62716e27 0.594299
\(842\) −3.23620e27 −0.524289
\(843\) −2.84220e26 −0.0455294
\(844\) 3.55994e27 0.563883
\(845\) −7.64140e27 −1.19683
\(846\) 2.09711e27 0.324791
\(847\) 8.25052e26 0.126354
\(848\) −2.49753e27 −0.378225
\(849\) 5.01135e27 0.750468
\(850\) 1.37723e27 0.203951
\(851\) −1.62470e27 −0.237926
\(852\) −3.52329e27 −0.510238
\(853\) 1.17165e28 1.67796 0.838982 0.544159i \(-0.183151\pi\)
0.838982 + 0.544159i \(0.183151\pi\)
\(854\) 1.13031e28 1.60084
\(855\) −1.40194e27 −0.196359
\(856\) −2.73507e27 −0.378851
\(857\) −1.10156e28 −1.50901 −0.754505 0.656295i \(-0.772123\pi\)
−0.754505 + 0.656295i \(0.772123\pi\)
\(858\) −4.45333e27 −0.603331
\(859\) 3.04449e27 0.407924 0.203962 0.978979i \(-0.434618\pi\)
0.203962 + 0.978979i \(0.434618\pi\)
\(860\) 1.45171e26 0.0192373
\(861\) 5.50962e27 0.722090
\(862\) 7.18898e27 0.931855
\(863\) −2.46553e26 −0.0316088 −0.0158044 0.999875i \(-0.505031\pi\)
−0.0158044 + 0.999875i \(0.505031\pi\)
\(864\) 2.68302e26 0.0340207
\(865\) 2.05921e27 0.258254
\(866\) 3.22328e27 0.399832
\(867\) −6.80482e26 −0.0834902
\(868\) −7.64909e27 −0.928266
\(869\) −3.04731e27 −0.365788
\(870\) 3.71043e27 0.440546
\(871\) −2.21471e28 −2.60102
\(872\) −4.80874e27 −0.558631
\(873\) 9.74076e26 0.111933
\(874\) 6.79240e26 0.0772085
\(875\) −1.35280e28 −1.52110
\(876\) 3.58483e26 0.0398731
\(877\) −1.08028e28 −1.18861 −0.594303 0.804241i \(-0.702572\pi\)
−0.594303 + 0.804241i \(0.702572\pi\)
\(878\) 1.60826e27 0.175048
\(879\) −2.37151e25 −0.00255347
\(880\) 1.91336e27 0.203803
\(881\) 1.28087e28 1.34969 0.674847 0.737958i \(-0.264210\pi\)
0.674847 + 0.737958i \(0.264210\pi\)
\(882\) −2.18255e27 −0.227516
\(883\) 9.95748e26 0.102689 0.0513443 0.998681i \(-0.483649\pi\)
0.0513443 + 0.998681i \(0.483649\pi\)
\(884\) 8.12339e27 0.828782
\(885\) 3.04666e27 0.307512
\(886\) −9.91580e27 −0.990165
\(887\) 4.91403e27 0.485471 0.242736 0.970092i \(-0.421955\pi\)
0.242736 + 0.970092i \(0.421955\pi\)
\(888\) 3.13667e27 0.306581
\(889\) 4.53475e27 0.438517
\(890\) 2.42831e27 0.232326
\(891\) −1.11966e27 −0.105986
\(892\) 6.56185e26 0.0614554
\(893\) −1.02499e28 −0.949794
\(894\) 4.28114e27 0.392512
\(895\) 4.95343e27 0.449353
\(896\) 1.38049e27 0.123910
\(897\) 1.59551e27 0.141701
\(898\) 1.08935e28 0.957289
\(899\) −1.92306e28 −1.67215
\(900\) −5.22234e26 −0.0449326
\(901\) −1.90119e28 −1.61860
\(902\) 7.14295e27 0.601749
\(903\) −4.37095e26 −0.0364370
\(904\) 2.24492e26 0.0185183
\(905\) 1.10135e28 0.899011
\(906\) −5.81317e27 −0.469564
\(907\) −4.38931e27 −0.350854 −0.175427 0.984492i \(-0.556131\pi\)
−0.175427 + 0.984492i \(0.556131\pi\)
\(908\) 2.17872e25 0.00172340
\(909\) 6.23418e27 0.488003
\(910\) −1.69439e28 −1.31256
\(911\) 1.44663e28 1.10900 0.554500 0.832183i \(-0.312909\pi\)
0.554500 + 0.832183i \(0.312909\pi\)
\(912\) −1.31135e27 −0.0994876
\(913\) 1.49026e27 0.111890
\(914\) 1.07005e28 0.795091
\(915\) 1.08361e28 0.796843
\(916\) 8.50992e25 0.00619325
\(917\) −2.41276e28 −1.73782
\(918\) 2.04238e27 0.145590
\(919\) 2.02703e28 1.43009 0.715044 0.699079i \(-0.246406\pi\)
0.715044 + 0.699079i \(0.246406\pi\)
\(920\) −6.85506e26 −0.0478660
\(921\) −9.44846e26 −0.0652972
\(922\) −1.93947e28 −1.32660
\(923\) 4.04503e28 2.73845
\(924\) −5.76094e27 −0.386020
\(925\) −6.10534e27 −0.404914
\(926\) 5.58157e27 0.366397
\(927\) 5.82526e25 0.00378493
\(928\) 3.47069e27 0.223208
\(929\) 1.69685e27 0.108017 0.0540087 0.998540i \(-0.482800\pi\)
0.0540087 + 0.998540i \(0.482800\pi\)
\(930\) −7.33307e27 −0.462059
\(931\) 1.06674e28 0.665332
\(932\) −3.64359e27 −0.224946
\(933\) 5.76705e27 0.352434
\(934\) 1.80247e28 1.09036
\(935\) 1.45650e28 0.872167
\(936\) −3.08032e27 −0.182589
\(937\) 1.88106e28 1.10377 0.551883 0.833922i \(-0.313910\pi\)
0.551883 + 0.833922i \(0.313910\pi\)
\(938\) −2.86501e28 −1.66417
\(939\) −1.79477e28 −1.03201
\(940\) 1.03444e28 0.588832
\(941\) −2.39592e28 −1.35012 −0.675058 0.737765i \(-0.735881\pi\)
−0.675058 + 0.737765i \(0.735881\pi\)
\(942\) 5.60728e27 0.312802
\(943\) −2.55913e27 −0.141329
\(944\) 2.84981e27 0.155805
\(945\) −4.26003e27 −0.230574
\(946\) −5.66672e26 −0.0303645
\(947\) 1.03960e28 0.551495 0.275748 0.961230i \(-0.411075\pi\)
0.275748 + 0.961230i \(0.411075\pi\)
\(948\) −2.10779e27 −0.110700
\(949\) −4.11568e27 −0.213999
\(950\) 2.55247e27 0.131397
\(951\) −3.59907e27 −0.183432
\(952\) 1.05086e28 0.530267
\(953\) −1.88398e28 −0.941227 −0.470614 0.882339i \(-0.655967\pi\)
−0.470614 + 0.882339i \(0.655967\pi\)
\(954\) 7.20915e27 0.356594
\(955\) −1.64773e28 −0.806965
\(956\) −1.78716e27 −0.0866592
\(957\) −1.44836e28 −0.695366
\(958\) −2.37559e28 −1.12927
\(959\) −7.53033e27 −0.354436
\(960\) 1.32345e27 0.0616781
\(961\) 1.63356e28 0.753812
\(962\) −3.60115e28 −1.64542
\(963\) 7.89481e27 0.357184
\(964\) −2.88891e27 −0.129420
\(965\) −1.91685e28 −0.850313
\(966\) 2.06399e27 0.0906621
\(967\) −2.13424e28 −0.928308 −0.464154 0.885754i \(-0.653642\pi\)
−0.464154 + 0.885754i \(0.653642\pi\)
\(968\) 7.39859e26 0.0318664
\(969\) −9.98236e27 −0.425753
\(970\) 4.80482e27 0.202930
\(971\) −1.04800e28 −0.438306 −0.219153 0.975691i \(-0.570329\pi\)
−0.219153 + 0.975691i \(0.570329\pi\)
\(972\) −7.74455e26 −0.0320750
\(973\) 4.75767e28 1.95129
\(974\) 1.28084e28 0.520219
\(975\) 5.99567e27 0.241154
\(976\) 1.01360e28 0.403730
\(977\) −3.66937e27 −0.144742 −0.0723708 0.997378i \(-0.523056\pi\)
−0.0723708 + 0.997378i \(0.523056\pi\)
\(978\) 1.50775e28 0.588994
\(979\) −9.47887e27 −0.366708
\(980\) −1.07658e28 −0.412478
\(981\) 1.38805e28 0.526682
\(982\) −5.94485e26 −0.0223399
\(983\) 2.27156e28 0.845407 0.422703 0.906268i \(-0.361081\pi\)
0.422703 + 0.906268i \(0.361081\pi\)
\(984\) 4.94071e27 0.182111
\(985\) −3.01743e27 −0.110152
\(986\) 2.64198e28 0.955209
\(987\) −3.11461e28 −1.11530
\(988\) 1.50554e28 0.533951
\(989\) 2.03024e26 0.00713152
\(990\) −5.52292e27 −0.192148
\(991\) 4.05166e28 1.39615 0.698077 0.716022i \(-0.254039\pi\)
0.698077 + 0.716022i \(0.254039\pi\)
\(992\) −6.85926e27 −0.234108
\(993\) 7.85269e27 0.265461
\(994\) 5.23275e28 1.75210
\(995\) 1.86095e28 0.617184
\(996\) 1.03080e27 0.0338618
\(997\) −4.63139e28 −1.50698 −0.753489 0.657460i \(-0.771631\pi\)
−0.753489 + 0.657460i \(0.771631\pi\)
\(998\) −1.61957e28 −0.521987
\(999\) −9.05401e27 −0.289047
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 6.20.a.a.1.1 1
3.2 odd 2 18.20.a.f.1.1 1
4.3 odd 2 48.20.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.20.a.a.1.1 1 1.1 even 1 trivial
18.20.a.f.1.1 1 3.2 odd 2
48.20.a.d.1.1 1 4.3 odd 2