Properties

Label 10.32.a.b.1.1
Level $10$
Weight $32$
Character 10.1
Self dual yes
Analytic conductor $60.877$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,32,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 32, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 32);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(60.8771328190\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{337159}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 337159 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-580.654\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32768.0 q^{2} +1.07495e7 q^{3} +1.07374e9 q^{4} -3.05176e10 q^{5} -3.52239e11 q^{6} +2.26000e13 q^{7} -3.51844e13 q^{8} -5.02122e14 q^{9} +1.00000e15 q^{10} +2.45408e16 q^{11} +1.15422e16 q^{12} -1.48420e17 q^{13} -7.40557e17 q^{14} -3.28048e17 q^{15} +1.15292e18 q^{16} +1.16191e19 q^{17} +1.64535e19 q^{18} +3.78068e19 q^{19} -3.27680e19 q^{20} +2.42938e20 q^{21} -8.04154e20 q^{22} +8.75576e20 q^{23} -3.78214e20 q^{24} +9.31323e20 q^{25} +4.86342e21 q^{26} -1.20372e22 q^{27} +2.42666e22 q^{28} -8.57580e22 q^{29} +1.07495e22 q^{30} +2.14345e22 q^{31} -3.77789e22 q^{32} +2.63801e23 q^{33} -3.80734e23 q^{34} -6.89698e23 q^{35} -5.39149e23 q^{36} -2.44667e24 q^{37} -1.23885e24 q^{38} -1.59544e24 q^{39} +1.07374e24 q^{40} +1.07003e25 q^{41} -7.96061e24 q^{42} +6.78689e24 q^{43} +2.63505e25 q^{44} +1.53235e25 q^{45} -2.86909e25 q^{46} -1.41828e26 q^{47} +1.23933e25 q^{48} +3.52985e26 q^{49} -3.05176e25 q^{50} +1.24899e26 q^{51} -1.59365e26 q^{52} +5.25772e26 q^{53} +3.94436e26 q^{54} -7.48926e26 q^{55} -7.95167e26 q^{56} +4.06404e26 q^{57} +2.81012e27 q^{58} +2.03548e27 q^{59} -3.52239e26 q^{60} +6.56027e27 q^{61} -7.02366e26 q^{62} -1.13480e28 q^{63} +1.23794e27 q^{64} +4.52941e27 q^{65} -8.64424e27 q^{66} +2.16532e28 q^{67} +1.24759e28 q^{68} +9.41199e27 q^{69} +2.26000e28 q^{70} +2.83140e28 q^{71} +1.76668e28 q^{72} +2.94533e28 q^{73} +8.01724e28 q^{74} +1.00112e28 q^{75} +4.05948e28 q^{76} +5.54623e29 q^{77} +5.22793e28 q^{78} -2.36152e29 q^{79} -3.51844e28 q^{80} +1.80753e29 q^{81} -3.50626e29 q^{82} +7.97795e29 q^{83} +2.60853e29 q^{84} -3.54586e29 q^{85} -2.22393e29 q^{86} -9.21854e29 q^{87} -8.63453e29 q^{88} +1.17539e30 q^{89} -5.02122e29 q^{90} -3.35429e30 q^{91} +9.40142e29 q^{92} +2.30410e29 q^{93} +4.64740e30 q^{94} -1.15377e30 q^{95} -4.06104e29 q^{96} -3.02840e30 q^{97} -1.15666e31 q^{98} -1.23225e31 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 65536 q^{2} + 29024244 q^{3} + 2147483648 q^{4} - 61035156250 q^{5} - 951066427392 q^{6} + 12795158105212 q^{7} - 70368744177664 q^{8} - 785828540134926 q^{9} + 20\!\cdots\!00 q^{10} - 500906012122176 q^{11}+ \cdots - 52\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32768.0 −0.707107
\(3\) 1.07495e7 0.432522 0.216261 0.976336i \(-0.430614\pi\)
0.216261 + 0.976336i \(0.430614\pi\)
\(4\) 1.07374e9 0.500000
\(5\) −3.05176e10 −0.447214
\(6\) −3.52239e11 −0.305839
\(7\) 2.26000e13 1.79924 0.899620 0.436674i \(-0.143844\pi\)
0.899620 + 0.436674i \(0.143844\pi\)
\(8\) −3.51844e13 −0.353553
\(9\) −5.02122e14 −0.812925
\(10\) 1.00000e15 0.316228
\(11\) 2.45408e16 1.77134 0.885671 0.464313i \(-0.153699\pi\)
0.885671 + 0.464313i \(0.153699\pi\)
\(12\) 1.15422e16 0.216261
\(13\) −1.48420e17 −0.804211 −0.402105 0.915593i \(-0.631721\pi\)
−0.402105 + 0.915593i \(0.631721\pi\)
\(14\) −7.40557e17 −1.27225
\(15\) −3.28048e17 −0.193430
\(16\) 1.15292e18 0.250000
\(17\) 1.16191e19 0.984496 0.492248 0.870455i \(-0.336175\pi\)
0.492248 + 0.870455i \(0.336175\pi\)
\(18\) 1.64535e19 0.574825
\(19\) 3.78068e19 0.571333 0.285667 0.958329i \(-0.407785\pi\)
0.285667 + 0.958329i \(0.407785\pi\)
\(20\) −3.27680e19 −0.223607
\(21\) 2.42938e20 0.778211
\(22\) −8.04154e20 −1.25253
\(23\) 8.75576e20 0.684719 0.342359 0.939569i \(-0.388774\pi\)
0.342359 + 0.939569i \(0.388774\pi\)
\(24\) −3.78214e20 −0.152920
\(25\) 9.31323e20 0.200000
\(26\) 4.86342e21 0.568663
\(27\) −1.20372e22 −0.784130
\(28\) 2.42666e22 0.899620
\(29\) −8.57580e22 −1.84546 −0.922732 0.385443i \(-0.874049\pi\)
−0.922732 + 0.385443i \(0.874049\pi\)
\(30\) 1.07495e22 0.136775
\(31\) 2.14345e22 0.164062 0.0820308 0.996630i \(-0.473859\pi\)
0.0820308 + 0.996630i \(0.473859\pi\)
\(32\) −3.77789e22 −0.176777
\(33\) 2.63801e23 0.766144
\(34\) −3.80734e23 −0.696143
\(35\) −6.89698e23 −0.804645
\(36\) −5.39149e23 −0.406462
\(37\) −2.44667e24 −1.20628 −0.603140 0.797635i \(-0.706084\pi\)
−0.603140 + 0.797635i \(0.706084\pi\)
\(38\) −1.23885e24 −0.403994
\(39\) −1.59544e24 −0.347839
\(40\) 1.07374e24 0.158114
\(41\) 1.07003e25 1.07459 0.537297 0.843393i \(-0.319445\pi\)
0.537297 + 0.843393i \(0.319445\pi\)
\(42\) −7.96061e24 −0.550278
\(43\) 6.78689e24 0.325769 0.162884 0.986645i \(-0.447920\pi\)
0.162884 + 0.986645i \(0.447920\pi\)
\(44\) 2.63505e25 0.885671
\(45\) 1.53235e25 0.363551
\(46\) −2.86909e25 −0.484169
\(47\) −1.41828e26 −1.71492 −0.857460 0.514551i \(-0.827959\pi\)
−0.857460 + 0.514551i \(0.827959\pi\)
\(48\) 1.23933e25 0.108130
\(49\) 3.52985e26 2.23726
\(50\) −3.05176e25 −0.141421
\(51\) 1.24899e26 0.425816
\(52\) −1.59365e26 −0.402105
\(53\) 5.25772e26 0.987464 0.493732 0.869614i \(-0.335632\pi\)
0.493732 + 0.869614i \(0.335632\pi\)
\(54\) 3.94436e26 0.554464
\(55\) −7.48926e26 −0.792168
\(56\) −7.95167e26 −0.636127
\(57\) 4.06404e26 0.247114
\(58\) 2.81012e27 1.30494
\(59\) 2.03548e27 0.725203 0.362602 0.931944i \(-0.381889\pi\)
0.362602 + 0.931944i \(0.381889\pi\)
\(60\) −3.52239e26 −0.0967149
\(61\) 6.56027e27 1.39415 0.697074 0.716999i \(-0.254485\pi\)
0.697074 + 0.716999i \(0.254485\pi\)
\(62\) −7.02366e26 −0.116009
\(63\) −1.13480e28 −1.46265
\(64\) 1.23794e27 0.125000
\(65\) 4.52941e27 0.359654
\(66\) −8.64424e27 −0.541746
\(67\) 2.16532e28 1.07489 0.537444 0.843299i \(-0.319390\pi\)
0.537444 + 0.843299i \(0.319390\pi\)
\(68\) 1.24759e28 0.492248
\(69\) 9.41199e27 0.296156
\(70\) 2.26000e28 0.568970
\(71\) 2.83140e28 0.572133 0.286066 0.958210i \(-0.407652\pi\)
0.286066 + 0.958210i \(0.407652\pi\)
\(72\) 1.76668e28 0.287412
\(73\) 2.94533e28 0.386928 0.193464 0.981107i \(-0.438028\pi\)
0.193464 + 0.981107i \(0.438028\pi\)
\(74\) 8.01724e28 0.852969
\(75\) 1.00112e28 0.0865044
\(76\) 4.05948e28 0.285667
\(77\) 5.54623e29 3.18707
\(78\) 5.22793e28 0.245959
\(79\) −2.36152e29 −0.911951 −0.455976 0.889992i \(-0.650710\pi\)
−0.455976 + 0.889992i \(0.650710\pi\)
\(80\) −3.51844e28 −0.111803
\(81\) 1.80753e29 0.473771
\(82\) −3.50626e29 −0.759853
\(83\) 7.97795e29 1.43279 0.716393 0.697697i \(-0.245792\pi\)
0.716393 + 0.697697i \(0.245792\pi\)
\(84\) 2.60853e29 0.389105
\(85\) −3.54586e29 −0.440280
\(86\) −2.22393e29 −0.230353
\(87\) −9.21854e29 −0.798204
\(88\) −8.63453e29 −0.626264
\(89\) 1.17539e30 0.715542 0.357771 0.933809i \(-0.383537\pi\)
0.357771 + 0.933809i \(0.383537\pi\)
\(90\) −5.02122e29 −0.257069
\(91\) −3.35429e30 −1.44697
\(92\) 9.40142e29 0.342359
\(93\) 2.30410e29 0.0709603
\(94\) 4.64740e30 1.21263
\(95\) −1.15377e30 −0.255508
\(96\) −4.06104e29 −0.0764598
\(97\) −3.02840e30 −0.485570 −0.242785 0.970080i \(-0.578061\pi\)
−0.242785 + 0.970080i \(0.578061\pi\)
\(98\) −1.15666e31 −1.58198
\(99\) −1.23225e31 −1.43997
\(100\) 1.00000e30 0.100000
\(101\) 1.24493e31 1.06699 0.533497 0.845802i \(-0.320877\pi\)
0.533497 + 0.845802i \(0.320877\pi\)
\(102\) −4.09270e30 −0.301097
\(103\) 5.50424e30 0.348113 0.174056 0.984736i \(-0.444312\pi\)
0.174056 + 0.984736i \(0.444312\pi\)
\(104\) 5.22206e30 0.284331
\(105\) −7.41389e30 −0.348026
\(106\) −1.72285e31 −0.698242
\(107\) 2.87060e31 1.00583 0.502914 0.864336i \(-0.332261\pi\)
0.502914 + 0.864336i \(0.332261\pi\)
\(108\) −1.29249e31 −0.392065
\(109\) −6.39739e31 −1.68226 −0.841128 0.540837i \(-0.818108\pi\)
−0.841128 + 0.540837i \(0.818108\pi\)
\(110\) 2.45408e31 0.560148
\(111\) −2.63004e31 −0.521743
\(112\) 2.60560e31 0.449810
\(113\) −5.00616e31 −0.752990 −0.376495 0.926419i \(-0.622871\pi\)
−0.376495 + 0.926419i \(0.622871\pi\)
\(114\) −1.33170e31 −0.174736
\(115\) −2.67204e31 −0.306216
\(116\) −9.20819e31 −0.922732
\(117\) 7.45249e31 0.653763
\(118\) −6.66985e31 −0.512796
\(119\) 2.62592e32 1.77134
\(120\) 1.15422e31 0.0683877
\(121\) 4.10309e32 2.13765
\(122\) −2.14967e32 −0.985812
\(123\) 1.15022e32 0.464786
\(124\) 2.30151e31 0.0820308
\(125\) −2.84217e31 −0.0894427
\(126\) 3.71850e32 1.03425
\(127\) −1.96018e32 −0.482324 −0.241162 0.970485i \(-0.577528\pi\)
−0.241162 + 0.970485i \(0.577528\pi\)
\(128\) −4.05648e31 −0.0883883
\(129\) 7.29555e31 0.140902
\(130\) −1.48420e32 −0.254314
\(131\) 8.78188e32 1.33623 0.668117 0.744056i \(-0.267101\pi\)
0.668117 + 0.744056i \(0.267101\pi\)
\(132\) 2.83254e32 0.383072
\(133\) 8.54435e32 1.02797
\(134\) −7.09533e32 −0.760061
\(135\) 3.67347e32 0.350674
\(136\) −4.08810e32 −0.348072
\(137\) 1.58360e33 1.20359 0.601794 0.798651i \(-0.294453\pi\)
0.601794 + 0.798651i \(0.294453\pi\)
\(138\) −3.08412e32 −0.209414
\(139\) 1.84495e31 0.0112009 0.00560047 0.999984i \(-0.498217\pi\)
0.00560047 + 0.999984i \(0.498217\pi\)
\(140\) −7.40557e32 −0.402322
\(141\) −1.52457e33 −0.741740
\(142\) −9.27793e32 −0.404559
\(143\) −3.64234e33 −1.42453
\(144\) −5.78907e32 −0.203231
\(145\) 2.61713e33 0.825316
\(146\) −9.65126e32 −0.273599
\(147\) 3.79441e33 0.967666
\(148\) −2.62709e33 −0.603140
\(149\) 5.96260e32 0.123324 0.0616621 0.998097i \(-0.480360\pi\)
0.0616621 + 0.998097i \(0.480360\pi\)
\(150\) −3.28048e32 −0.0611678
\(151\) −8.01584e33 −1.34836 −0.674180 0.738567i \(-0.735503\pi\)
−0.674180 + 0.738567i \(0.735503\pi\)
\(152\) −1.33021e33 −0.201997
\(153\) −5.83420e33 −0.800321
\(154\) −1.81739e34 −2.25360
\(155\) −6.54129e32 −0.0733706
\(156\) −1.71309e33 −0.173919
\(157\) 5.03735e33 0.463189 0.231594 0.972812i \(-0.425606\pi\)
0.231594 + 0.972812i \(0.425606\pi\)
\(158\) 7.73823e33 0.644847
\(159\) 5.65177e33 0.427100
\(160\) 1.15292e33 0.0790569
\(161\) 1.97880e34 1.23197
\(162\) −5.92293e33 −0.335007
\(163\) 2.60494e34 1.33933 0.669667 0.742661i \(-0.266437\pi\)
0.669667 + 0.742661i \(0.266437\pi\)
\(164\) 1.14893e34 0.537297
\(165\) −8.05057e33 −0.342630
\(166\) −2.61422e34 −1.01313
\(167\) 1.50501e34 0.531415 0.265708 0.964054i \(-0.414394\pi\)
0.265708 + 0.964054i \(0.414394\pi\)
\(168\) −8.54764e33 −0.275139
\(169\) −1.20315e34 −0.353245
\(170\) 1.16191e34 0.311325
\(171\) −1.89836e34 −0.464451
\(172\) 7.28737e33 0.162884
\(173\) 2.58953e33 0.0529062 0.0264531 0.999650i \(-0.491579\pi\)
0.0264531 + 0.999650i \(0.491579\pi\)
\(174\) 3.02073e34 0.564415
\(175\) 2.10479e34 0.359848
\(176\) 2.82936e34 0.442836
\(177\) 2.18803e34 0.313666
\(178\) −3.85150e34 −0.505965
\(179\) −8.17306e34 −0.984380 −0.492190 0.870488i \(-0.663803\pi\)
−0.492190 + 0.870488i \(0.663803\pi\)
\(180\) 1.64535e34 0.181775
\(181\) 2.40793e34 0.244133 0.122066 0.992522i \(-0.461048\pi\)
0.122066 + 0.992522i \(0.461048\pi\)
\(182\) 1.09913e35 1.02316
\(183\) 7.05195e34 0.603000
\(184\) −3.08066e34 −0.242085
\(185\) 7.46664e34 0.539465
\(186\) −7.55007e33 −0.0501765
\(187\) 2.85142e35 1.74388
\(188\) −1.52286e35 −0.857460
\(189\) −2.72041e35 −1.41084
\(190\) 3.78068e34 0.180671
\(191\) −1.66142e35 −0.731916 −0.365958 0.930631i \(-0.619259\pi\)
−0.365958 + 0.930631i \(0.619259\pi\)
\(192\) 1.33072e34 0.0540652
\(193\) 1.35705e35 0.508694 0.254347 0.967113i \(-0.418140\pi\)
0.254347 + 0.967113i \(0.418140\pi\)
\(194\) 9.92347e34 0.343350
\(195\) 4.86889e34 0.155558
\(196\) 3.79015e35 1.11863
\(197\) −1.30234e35 −0.355220 −0.177610 0.984101i \(-0.556837\pi\)
−0.177610 + 0.984101i \(0.556837\pi\)
\(198\) 4.03783e35 1.01821
\(199\) −5.17445e35 −1.20682 −0.603409 0.797432i \(-0.706191\pi\)
−0.603409 + 0.797432i \(0.706191\pi\)
\(200\) −3.27680e34 −0.0707107
\(201\) 2.32761e35 0.464913
\(202\) −4.07937e35 −0.754479
\(203\) −1.93813e36 −3.32043
\(204\) 1.34110e35 0.212908
\(205\) −3.26546e35 −0.480573
\(206\) −1.80363e35 −0.246153
\(207\) −4.39646e35 −0.556625
\(208\) −1.71116e35 −0.201053
\(209\) 9.27811e35 1.01203
\(210\) 2.42938e35 0.246092
\(211\) −1.18735e36 −1.11738 −0.558689 0.829377i \(-0.688695\pi\)
−0.558689 + 0.829377i \(0.688695\pi\)
\(212\) 5.64543e35 0.493732
\(213\) 3.04361e35 0.247460
\(214\) −9.40638e35 −0.711228
\(215\) −2.07119e35 −0.145688
\(216\) 4.23522e35 0.277232
\(217\) 4.84420e35 0.295186
\(218\) 2.09630e36 1.18953
\(219\) 3.16608e35 0.167355
\(220\) −8.04154e35 −0.396084
\(221\) −1.72450e36 −0.791742
\(222\) 8.61812e35 0.368928
\(223\) 2.87825e35 0.114922 0.0574611 0.998348i \(-0.481699\pi\)
0.0574611 + 0.998348i \(0.481699\pi\)
\(224\) −8.53804e35 −0.318064
\(225\) −4.67638e35 −0.162585
\(226\) 1.64042e36 0.532444
\(227\) 3.90915e36 1.18490 0.592450 0.805607i \(-0.298161\pi\)
0.592450 + 0.805607i \(0.298161\pi\)
\(228\) 4.36373e35 0.123557
\(229\) −9.35252e35 −0.247445 −0.123723 0.992317i \(-0.539483\pi\)
−0.123723 + 0.992317i \(0.539483\pi\)
\(230\) 8.75576e35 0.216527
\(231\) 5.96191e36 1.37848
\(232\) 3.01734e36 0.652470
\(233\) 2.81203e36 0.568856 0.284428 0.958697i \(-0.408196\pi\)
0.284428 + 0.958697i \(0.408196\pi\)
\(234\) −2.44203e36 −0.462280
\(235\) 4.32823e36 0.766935
\(236\) 2.18558e36 0.362602
\(237\) −2.53851e36 −0.394439
\(238\) −8.60460e36 −1.25253
\(239\) 9.22821e36 1.25878 0.629391 0.777089i \(-0.283305\pi\)
0.629391 + 0.777089i \(0.283305\pi\)
\(240\) −3.78214e35 −0.0483574
\(241\) −2.25483e36 −0.270303 −0.135151 0.990825i \(-0.543152\pi\)
−0.135151 + 0.990825i \(0.543152\pi\)
\(242\) −1.34450e37 −1.51155
\(243\) 9.37808e36 0.989046
\(244\) 7.04403e36 0.697074
\(245\) −1.07723e37 −1.00053
\(246\) −3.76905e36 −0.328653
\(247\) −5.61128e36 −0.459472
\(248\) −7.54159e35 −0.0580046
\(249\) 8.57589e36 0.619711
\(250\) 9.31323e35 0.0632456
\(251\) −1.87833e37 −1.19903 −0.599515 0.800364i \(-0.704640\pi\)
−0.599515 + 0.800364i \(0.704640\pi\)
\(252\) −1.21848e37 −0.731323
\(253\) 2.14873e37 1.21287
\(254\) 6.42312e36 0.341054
\(255\) −3.81162e36 −0.190431
\(256\) 1.32923e36 0.0625000
\(257\) −1.96277e37 −0.868773 −0.434386 0.900727i \(-0.643035\pi\)
−0.434386 + 0.900727i \(0.643035\pi\)
\(258\) −2.39061e36 −0.0996329
\(259\) −5.52947e37 −2.17039
\(260\) 4.86342e36 0.179827
\(261\) 4.30610e37 1.50022
\(262\) −2.87765e37 −0.944860
\(263\) 1.77659e37 0.549886 0.274943 0.961461i \(-0.411341\pi\)
0.274943 + 0.961461i \(0.411341\pi\)
\(264\) −9.28168e36 −0.270873
\(265\) −1.60453e37 −0.441607
\(266\) −2.79981e37 −0.726881
\(267\) 1.26348e37 0.309488
\(268\) 2.32500e37 0.537444
\(269\) 4.11974e37 0.898897 0.449449 0.893306i \(-0.351621\pi\)
0.449449 + 0.893306i \(0.351621\pi\)
\(270\) −1.20372e37 −0.247964
\(271\) 2.67128e37 0.519631 0.259816 0.965658i \(-0.416338\pi\)
0.259816 + 0.965658i \(0.416338\pi\)
\(272\) 1.33959e37 0.246124
\(273\) −3.60569e37 −0.625846
\(274\) −5.18915e37 −0.851065
\(275\) 2.28554e37 0.354268
\(276\) 1.01060e37 0.148078
\(277\) −1.71079e37 −0.237007 −0.118503 0.992954i \(-0.537810\pi\)
−0.118503 + 0.992954i \(0.537810\pi\)
\(278\) −6.04552e35 −0.00792025
\(279\) −1.07627e37 −0.133370
\(280\) 2.42666e37 0.284485
\(281\) −8.17662e37 −0.907038 −0.453519 0.891247i \(-0.649832\pi\)
−0.453519 + 0.891247i \(0.649832\pi\)
\(282\) 4.99572e37 0.524490
\(283\) 3.56430e36 0.0354229 0.0177115 0.999843i \(-0.494362\pi\)
0.0177115 + 0.999843i \(0.494362\pi\)
\(284\) 3.04019e37 0.286066
\(285\) −1.24025e37 −0.110513
\(286\) 1.19352e38 1.00730
\(287\) 2.41826e38 1.93345
\(288\) 1.89696e37 0.143706
\(289\) −4.28571e36 −0.0307685
\(290\) −8.57580e37 −0.583587
\(291\) −3.25538e37 −0.210020
\(292\) 3.16253e37 0.193464
\(293\) 1.72791e38 1.00248 0.501238 0.865310i \(-0.332878\pi\)
0.501238 + 0.865310i \(0.332878\pi\)
\(294\) −1.24335e38 −0.684243
\(295\) −6.21178e37 −0.324321
\(296\) 8.60844e37 0.426484
\(297\) −2.95403e38 −1.38896
\(298\) −1.95383e37 −0.0872034
\(299\) −1.29953e38 −0.550658
\(300\) 1.07495e37 0.0432522
\(301\) 1.53384e38 0.586136
\(302\) 2.62663e38 0.953435
\(303\) 1.33823e38 0.461499
\(304\) 4.35883e37 0.142833
\(305\) −2.00203e38 −0.623482
\(306\) 1.91175e38 0.565912
\(307\) −1.39185e38 −0.391694 −0.195847 0.980634i \(-0.562746\pi\)
−0.195847 + 0.980634i \(0.562746\pi\)
\(308\) 5.95522e38 1.59353
\(309\) 5.91677e37 0.150567
\(310\) 2.14345e37 0.0518809
\(311\) −7.23579e37 −0.166610 −0.0833048 0.996524i \(-0.526548\pi\)
−0.0833048 + 0.996524i \(0.526548\pi\)
\(312\) 5.61344e37 0.122980
\(313\) −7.16835e38 −1.49445 −0.747226 0.664570i \(-0.768615\pi\)
−0.747226 + 0.664570i \(0.768615\pi\)
\(314\) −1.65064e38 −0.327524
\(315\) 3.46312e38 0.654115
\(316\) −2.53566e38 −0.455976
\(317\) 1.01778e39 1.74275 0.871373 0.490620i \(-0.163230\pi\)
0.871373 + 0.490620i \(0.163230\pi\)
\(318\) −1.85197e38 −0.302005
\(319\) −2.10457e39 −3.26895
\(320\) −3.77789e37 −0.0559017
\(321\) 3.08575e38 0.435043
\(322\) −6.48414e38 −0.871137
\(323\) 4.39281e38 0.562475
\(324\) 1.94083e38 0.236886
\(325\) −1.38227e38 −0.160842
\(326\) −8.53585e38 −0.947053
\(327\) −6.87686e38 −0.727612
\(328\) −3.76482e38 −0.379926
\(329\) −3.20530e39 −3.08555
\(330\) 2.63801e38 0.242276
\(331\) 1.30666e39 1.14506 0.572529 0.819885i \(-0.305963\pi\)
0.572529 + 0.819885i \(0.305963\pi\)
\(332\) 8.56626e38 0.716393
\(333\) 1.22853e39 0.980615
\(334\) −4.93161e38 −0.375767
\(335\) −6.60805e38 −0.480705
\(336\) 2.80089e38 0.194553
\(337\) 1.56627e39 1.03897 0.519486 0.854479i \(-0.326123\pi\)
0.519486 + 0.854479i \(0.326123\pi\)
\(338\) 3.94248e38 0.249782
\(339\) −5.38137e38 −0.325685
\(340\) −3.80734e38 −0.220140
\(341\) 5.26020e38 0.290609
\(342\) 6.22056e38 0.328416
\(343\) 4.41174e39 2.22614
\(344\) −2.38792e38 −0.115177
\(345\) −2.87231e38 −0.132445
\(346\) −8.48537e37 −0.0374103
\(347\) −1.53164e39 −0.645726 −0.322863 0.946446i \(-0.604645\pi\)
−0.322863 + 0.946446i \(0.604645\pi\)
\(348\) −9.89833e38 −0.399102
\(349\) 2.63344e39 1.01562 0.507808 0.861470i \(-0.330456\pi\)
0.507808 + 0.861470i \(0.330456\pi\)
\(350\) −6.89698e38 −0.254451
\(351\) 1.78656e39 0.630606
\(352\) −9.27126e38 −0.313132
\(353\) 4.56209e39 1.47454 0.737269 0.675600i \(-0.236115\pi\)
0.737269 + 0.675600i \(0.236115\pi\)
\(354\) −7.16974e38 −0.221796
\(355\) −8.64075e38 −0.255866
\(356\) 1.26206e39 0.357771
\(357\) 2.82272e39 0.766145
\(358\) 2.67815e39 0.696062
\(359\) 2.02315e39 0.503575 0.251787 0.967783i \(-0.418982\pi\)
0.251787 + 0.967783i \(0.418982\pi\)
\(360\) −5.39149e38 −0.128535
\(361\) −2.94951e39 −0.673578
\(362\) −7.89031e38 −0.172628
\(363\) 4.41061e39 0.924582
\(364\) −3.60164e39 −0.723484
\(365\) −8.98844e38 −0.173039
\(366\) −2.31078e39 −0.426385
\(367\) −1.03029e40 −1.82237 −0.911185 0.411998i \(-0.864831\pi\)
−0.911185 + 0.411998i \(0.864831\pi\)
\(368\) 1.00947e39 0.171180
\(369\) −5.37284e39 −0.873564
\(370\) −2.44667e39 −0.381459
\(371\) 1.18824e40 1.77668
\(372\) 2.47401e38 0.0354801
\(373\) −4.89843e39 −0.673860 −0.336930 0.941530i \(-0.609389\pi\)
−0.336930 + 0.941530i \(0.609389\pi\)
\(374\) −9.34353e39 −1.23311
\(375\) −3.05519e38 −0.0386859
\(376\) 4.99011e39 0.606316
\(377\) 1.27282e40 1.48414
\(378\) 8.91425e39 0.997613
\(379\) −1.32326e40 −1.42147 −0.710737 0.703458i \(-0.751638\pi\)
−0.710737 + 0.703458i \(0.751638\pi\)
\(380\) −1.23885e39 −0.127754
\(381\) −2.10709e39 −0.208616
\(382\) 5.44413e39 0.517543
\(383\) 2.21210e39 0.201940 0.100970 0.994889i \(-0.467805\pi\)
0.100970 + 0.994889i \(0.467805\pi\)
\(384\) −4.36051e38 −0.0382299
\(385\) −1.69257e40 −1.42530
\(386\) −4.44677e39 −0.359701
\(387\) −3.40785e39 −0.264826
\(388\) −3.25172e39 −0.242785
\(389\) −1.98512e40 −1.42419 −0.712096 0.702082i \(-0.752254\pi\)
−0.712096 + 0.702082i \(0.752254\pi\)
\(390\) −1.59544e39 −0.109996
\(391\) 1.01734e40 0.674103
\(392\) −1.24196e40 −0.790992
\(393\) 9.44007e39 0.577950
\(394\) 4.26750e39 0.251178
\(395\) 7.20679e39 0.407837
\(396\) −1.32312e40 −0.719984
\(397\) 4.66363e39 0.244046 0.122023 0.992527i \(-0.461062\pi\)
0.122023 + 0.992527i \(0.461062\pi\)
\(398\) 1.69556e40 0.853349
\(399\) 9.18474e39 0.444618
\(400\) 1.07374e39 0.0500000
\(401\) −1.07728e40 −0.482605 −0.241303 0.970450i \(-0.577575\pi\)
−0.241303 + 0.970450i \(0.577575\pi\)
\(402\) −7.62712e39 −0.328743
\(403\) −3.18130e39 −0.131940
\(404\) 1.33673e40 0.533497
\(405\) −5.51616e39 −0.211877
\(406\) 6.35087e40 2.34790
\(407\) −6.00432e40 −2.13674
\(408\) −4.39450e39 −0.150549
\(409\) −1.59543e40 −0.526219 −0.263110 0.964766i \(-0.584748\pi\)
−0.263110 + 0.964766i \(0.584748\pi\)
\(410\) 1.07003e40 0.339817
\(411\) 1.70229e40 0.520578
\(412\) 5.91013e39 0.174056
\(413\) 4.60018e40 1.30481
\(414\) 1.44063e40 0.393593
\(415\) −2.43468e40 −0.640761
\(416\) 5.60714e39 0.142166
\(417\) 1.98322e38 0.00484465
\(418\) −3.04025e40 −0.715611
\(419\) −5.46661e40 −1.23994 −0.619971 0.784625i \(-0.712855\pi\)
−0.619971 + 0.784625i \(0.712855\pi\)
\(420\) −7.96061e39 −0.174013
\(421\) −2.46945e40 −0.520270 −0.260135 0.965572i \(-0.583767\pi\)
−0.260135 + 0.965572i \(0.583767\pi\)
\(422\) 3.89070e40 0.790105
\(423\) 7.12147e40 1.39410
\(424\) −1.84989e40 −0.349121
\(425\) 1.08211e40 0.196899
\(426\) −9.97330e39 −0.174981
\(427\) 1.48262e41 2.50841
\(428\) 3.08228e40 0.502914
\(429\) −3.91533e40 −0.616142
\(430\) 6.78689e39 0.103017
\(431\) 4.73716e40 0.693618 0.346809 0.937936i \(-0.387265\pi\)
0.346809 + 0.937936i \(0.387265\pi\)
\(432\) −1.38780e40 −0.196032
\(433\) −3.30096e38 −0.00449860 −0.00224930 0.999997i \(-0.500716\pi\)
−0.00224930 + 0.999997i \(0.500716\pi\)
\(434\) −1.58735e40 −0.208728
\(435\) 2.81327e40 0.356967
\(436\) −6.86914e40 −0.841128
\(437\) 3.31027e40 0.391203
\(438\) −1.03746e40 −0.118338
\(439\) −1.10291e39 −0.0121434 −0.00607170 0.999982i \(-0.501933\pi\)
−0.00607170 + 0.999982i \(0.501933\pi\)
\(440\) 2.63505e40 0.280074
\(441\) −1.77242e41 −1.81873
\(442\) 5.65085e40 0.559846
\(443\) −1.21843e41 −1.16558 −0.582791 0.812622i \(-0.698039\pi\)
−0.582791 + 0.812622i \(0.698039\pi\)
\(444\) −2.82399e40 −0.260871
\(445\) −3.58699e40 −0.320000
\(446\) −9.43146e39 −0.0812622
\(447\) 6.40949e39 0.0533405
\(448\) 2.79775e40 0.224905
\(449\) 1.34618e41 1.04540 0.522702 0.852515i \(-0.324924\pi\)
0.522702 + 0.852515i \(0.324924\pi\)
\(450\) 1.53235e40 0.114965
\(451\) 2.62593e41 1.90347
\(452\) −5.37533e40 −0.376495
\(453\) −8.61661e40 −0.583196
\(454\) −1.28095e41 −0.837851
\(455\) 1.02365e41 0.647104
\(456\) −1.42991e40 −0.0873681
\(457\) 6.73134e40 0.397558 0.198779 0.980044i \(-0.436302\pi\)
0.198779 + 0.980044i \(0.436302\pi\)
\(458\) 3.06463e40 0.174970
\(459\) −1.39862e41 −0.771972
\(460\) −2.86909e40 −0.153108
\(461\) −3.76769e41 −1.94407 −0.972033 0.234842i \(-0.924543\pi\)
−0.972033 + 0.234842i \(0.924543\pi\)
\(462\) −1.95360e41 −0.974731
\(463\) −3.37703e41 −1.62941 −0.814704 0.579877i \(-0.803100\pi\)
−0.814704 + 0.579877i \(0.803100\pi\)
\(464\) −9.88722e40 −0.461366
\(465\) −7.03155e39 −0.0317344
\(466\) −9.21445e40 −0.402242
\(467\) −1.12227e41 −0.473900 −0.236950 0.971522i \(-0.576148\pi\)
−0.236950 + 0.971522i \(0.576148\pi\)
\(468\) 8.00205e40 0.326881
\(469\) 4.89364e41 1.93398
\(470\) −1.41828e41 −0.542305
\(471\) 5.41490e40 0.200339
\(472\) −7.16169e40 −0.256398
\(473\) 1.66556e41 0.577048
\(474\) 8.31820e40 0.278911
\(475\) 3.52104e40 0.114267
\(476\) 2.81956e41 0.885672
\(477\) −2.64001e41 −0.802734
\(478\) −3.02390e41 −0.890093
\(479\) 9.96622e40 0.284008 0.142004 0.989866i \(-0.454645\pi\)
0.142004 + 0.989866i \(0.454645\pi\)
\(480\) 1.23933e40 0.0341939
\(481\) 3.63134e41 0.970104
\(482\) 7.38864e40 0.191133
\(483\) 2.12711e41 0.532856
\(484\) 4.40565e41 1.06883
\(485\) 9.24195e40 0.217153
\(486\) −3.07301e41 −0.699361
\(487\) 1.24135e41 0.273650 0.136825 0.990595i \(-0.456310\pi\)
0.136825 + 0.990595i \(0.456310\pi\)
\(488\) −2.30819e41 −0.492906
\(489\) 2.80017e41 0.579292
\(490\) 3.52985e41 0.707485
\(491\) 2.86826e40 0.0557000 0.0278500 0.999612i \(-0.491134\pi\)
0.0278500 + 0.999612i \(0.491134\pi\)
\(492\) 1.23504e41 0.232393
\(493\) −9.96429e41 −1.81685
\(494\) 1.83871e41 0.324896
\(495\) 3.76052e41 0.643973
\(496\) 2.47123e40 0.0410154
\(497\) 6.39897e41 1.02940
\(498\) −2.81015e41 −0.438202
\(499\) 9.27513e41 1.40204 0.701022 0.713140i \(-0.252728\pi\)
0.701022 + 0.713140i \(0.252728\pi\)
\(500\) −3.05176e40 −0.0447214
\(501\) 1.61781e41 0.229849
\(502\) 6.15492e41 0.847842
\(503\) 2.27661e41 0.304078 0.152039 0.988375i \(-0.451416\pi\)
0.152039 + 0.988375i \(0.451416\pi\)
\(504\) 3.99271e41 0.517124
\(505\) −3.79921e41 −0.477175
\(506\) −7.04097e41 −0.857630
\(507\) −1.29332e41 −0.152786
\(508\) −2.10473e41 −0.241162
\(509\) −4.37667e40 −0.0486427 −0.0243214 0.999704i \(-0.507742\pi\)
−0.0243214 + 0.999704i \(0.507742\pi\)
\(510\) 1.24899e41 0.134655
\(511\) 6.65645e41 0.696176
\(512\) −4.35561e40 −0.0441942
\(513\) −4.55089e41 −0.447999
\(514\) 6.43161e41 0.614315
\(515\) −1.67976e41 −0.155681
\(516\) 7.83354e40 0.0704511
\(517\) −3.48056e42 −3.03771
\(518\) 1.81190e42 1.53470
\(519\) 2.78361e40 0.0228831
\(520\) −1.59365e41 −0.127157
\(521\) −1.96331e42 −1.52057 −0.760283 0.649592i \(-0.774940\pi\)
−0.760283 + 0.649592i \(0.774940\pi\)
\(522\) −1.41102e42 −1.06082
\(523\) −1.34397e42 −0.980874 −0.490437 0.871477i \(-0.663163\pi\)
−0.490437 + 0.871477i \(0.663163\pi\)
\(524\) 9.42947e41 0.668117
\(525\) 2.26254e41 0.155642
\(526\) −5.82152e41 −0.388828
\(527\) 2.49049e41 0.161518
\(528\) 3.04142e41 0.191536
\(529\) −8.68537e41 −0.531160
\(530\) 5.25772e41 0.312263
\(531\) −1.02206e42 −0.589536
\(532\) 9.17443e41 0.513983
\(533\) −1.58813e42 −0.864200
\(534\) −4.14017e41 −0.218841
\(535\) −8.76038e41 −0.449820
\(536\) −7.61856e41 −0.380031
\(537\) −8.78562e41 −0.425766
\(538\) −1.34996e42 −0.635616
\(539\) 8.66255e42 3.96296
\(540\) 3.94436e41 0.175337
\(541\) −5.23970e40 −0.0226333 −0.0113167 0.999936i \(-0.503602\pi\)
−0.0113167 + 0.999936i \(0.503602\pi\)
\(542\) −8.75325e41 −0.367435
\(543\) 2.58840e41 0.105593
\(544\) −4.38957e41 −0.174036
\(545\) 1.95233e42 0.752327
\(546\) 1.18151e42 0.442540
\(547\) 6.69118e41 0.243612 0.121806 0.992554i \(-0.461131\pi\)
0.121806 + 0.992554i \(0.461131\pi\)
\(548\) 1.70038e42 0.601794
\(549\) −3.29405e42 −1.13334
\(550\) −7.48926e41 −0.250506
\(551\) −3.24224e42 −1.05437
\(552\) −3.31155e41 −0.104707
\(553\) −5.33704e42 −1.64082
\(554\) 5.60593e41 0.167589
\(555\) 8.02625e41 0.233330
\(556\) 1.98099e40 0.00560047
\(557\) 6.32573e42 1.73922 0.869612 0.493736i \(-0.164369\pi\)
0.869612 + 0.493736i \(0.164369\pi\)
\(558\) 3.52673e41 0.0943067
\(559\) −1.00731e42 −0.261987
\(560\) −7.95167e41 −0.201161
\(561\) 3.06513e42 0.754266
\(562\) 2.67931e42 0.641373
\(563\) 2.31235e42 0.538485 0.269242 0.963072i \(-0.413227\pi\)
0.269242 + 0.963072i \(0.413227\pi\)
\(564\) −1.63700e42 −0.370870
\(565\) 1.52776e42 0.336747
\(566\) −1.16795e41 −0.0250478
\(567\) 4.08503e42 0.852428
\(568\) −9.96210e41 −0.202279
\(569\) −4.99520e42 −0.986990 −0.493495 0.869749i \(-0.664281\pi\)
−0.493495 + 0.869749i \(0.664281\pi\)
\(570\) 4.06404e41 0.0781444
\(571\) 2.08977e42 0.391056 0.195528 0.980698i \(-0.437358\pi\)
0.195528 + 0.980698i \(0.437358\pi\)
\(572\) −3.91094e42 −0.712266
\(573\) −1.78594e42 −0.316570
\(574\) −7.92416e42 −1.36716
\(575\) 8.15443e41 0.136944
\(576\) −6.21597e41 −0.101616
\(577\) −3.85855e42 −0.614044 −0.307022 0.951702i \(-0.599333\pi\)
−0.307022 + 0.951702i \(0.599333\pi\)
\(578\) 1.40434e41 0.0217566
\(579\) 1.45875e42 0.220021
\(580\) 2.81012e42 0.412658
\(581\) 1.80302e43 2.57792
\(582\) 1.06672e42 0.148506
\(583\) 1.29029e43 1.74914
\(584\) −1.03630e42 −0.136800
\(585\) −2.27432e42 −0.292372
\(586\) −5.66203e42 −0.708857
\(587\) 9.46507e42 1.15407 0.577037 0.816718i \(-0.304209\pi\)
0.577037 + 0.816718i \(0.304209\pi\)
\(588\) 4.07422e42 0.483833
\(589\) 8.10371e41 0.0937339
\(590\) 2.03548e42 0.229329
\(591\) −1.39995e42 −0.153640
\(592\) −2.82082e42 −0.301570
\(593\) −1.21956e43 −1.27015 −0.635075 0.772450i \(-0.719031\pi\)
−0.635075 + 0.772450i \(0.719031\pi\)
\(594\) 9.67978e42 0.982145
\(595\) −8.01366e42 −0.792169
\(596\) 6.40229e41 0.0616621
\(597\) −5.56227e42 −0.521975
\(598\) 4.25829e42 0.389374
\(599\) 4.86657e42 0.433618 0.216809 0.976214i \(-0.430435\pi\)
0.216809 + 0.976214i \(0.430435\pi\)
\(600\) −3.52239e41 −0.0305839
\(601\) −1.25867e43 −1.06502 −0.532508 0.846425i \(-0.678750\pi\)
−0.532508 + 0.846425i \(0.678750\pi\)
\(602\) −5.02608e42 −0.414461
\(603\) −1.08726e43 −0.873804
\(604\) −8.60694e42 −0.674180
\(605\) −1.25216e43 −0.955988
\(606\) −4.38512e42 −0.326329
\(607\) 1.81008e43 1.31303 0.656513 0.754315i \(-0.272031\pi\)
0.656513 + 0.754315i \(0.272031\pi\)
\(608\) −1.42830e42 −0.100998
\(609\) −2.08339e43 −1.43616
\(610\) 6.56027e42 0.440868
\(611\) 2.10500e43 1.37916
\(612\) −6.26442e42 −0.400160
\(613\) 1.34234e43 0.836037 0.418018 0.908439i \(-0.362725\pi\)
0.418018 + 0.908439i \(0.362725\pi\)
\(614\) 4.56081e42 0.276970
\(615\) −3.51020e42 −0.207858
\(616\) −1.95141e43 −1.12680
\(617\) −3.23957e42 −0.182418 −0.0912089 0.995832i \(-0.529073\pi\)
−0.0912089 + 0.995832i \(0.529073\pi\)
\(618\) −1.93881e42 −0.106467
\(619\) −2.85690e43 −1.52999 −0.764997 0.644034i \(-0.777260\pi\)
−0.764997 + 0.644034i \(0.777260\pi\)
\(620\) −7.02366e41 −0.0366853
\(621\) −1.05395e43 −0.536908
\(622\) 2.37102e42 0.117811
\(623\) 2.65637e43 1.28743
\(624\) −1.83941e42 −0.0869597
\(625\) 8.67362e41 0.0400000
\(626\) 2.34893e43 1.05674
\(627\) 9.97349e42 0.437724
\(628\) 5.40882e42 0.231594
\(629\) −2.84280e43 −1.18758
\(630\) −1.13480e43 −0.462529
\(631\) −8.68109e42 −0.345239 −0.172619 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(632\) 8.30886e42 0.322424
\(633\) −1.27634e43 −0.483290
\(634\) −3.33505e43 −1.23231
\(635\) 5.98200e42 0.215702
\(636\) 6.06855e42 0.213550
\(637\) −5.23900e43 −1.79923
\(638\) 6.89626e43 2.31150
\(639\) −1.42171e43 −0.465101
\(640\) 1.23794e42 0.0395285
\(641\) 4.01231e43 1.25053 0.625267 0.780411i \(-0.284990\pi\)
0.625267 + 0.780411i \(0.284990\pi\)
\(642\) −1.01114e43 −0.307622
\(643\) −1.38043e43 −0.409964 −0.204982 0.978766i \(-0.565714\pi\)
−0.204982 + 0.978766i \(0.565714\pi\)
\(644\) 2.12472e43 0.615987
\(645\) −2.22643e42 −0.0630134
\(646\) −1.43944e43 −0.397730
\(647\) 4.95630e43 1.33703 0.668515 0.743699i \(-0.266930\pi\)
0.668515 + 0.743699i \(0.266930\pi\)
\(648\) −6.35970e42 −0.167503
\(649\) 4.99522e43 1.28458
\(650\) 4.52941e42 0.113733
\(651\) 5.20727e42 0.127675
\(652\) 2.79703e43 0.669667
\(653\) 8.53813e42 0.199622 0.0998110 0.995006i \(-0.468176\pi\)
0.0998110 + 0.995006i \(0.468176\pi\)
\(654\) 2.25341e43 0.514500
\(655\) −2.68002e43 −0.597582
\(656\) 1.23366e43 0.268649
\(657\) −1.47892e43 −0.314543
\(658\) 1.05031e44 2.18181
\(659\) 3.89436e43 0.790154 0.395077 0.918648i \(-0.370718\pi\)
0.395077 + 0.918648i \(0.370718\pi\)
\(660\) −8.64424e42 −0.171315
\(661\) −8.37311e42 −0.162093 −0.0810465 0.996710i \(-0.525826\pi\)
−0.0810465 + 0.996710i \(0.525826\pi\)
\(662\) −4.28165e43 −0.809678
\(663\) −1.85375e43 −0.342446
\(664\) −2.80699e43 −0.506566
\(665\) −2.60753e43 −0.459720
\(666\) −4.02563e43 −0.693399
\(667\) −7.50876e43 −1.26362
\(668\) 1.61599e43 0.265708
\(669\) 3.09398e42 0.0497064
\(670\) 2.16532e43 0.339910
\(671\) 1.60994e44 2.46951
\(672\) −9.17796e42 −0.137570
\(673\) 9.42152e43 1.38003 0.690013 0.723797i \(-0.257605\pi\)
0.690013 + 0.723797i \(0.257605\pi\)
\(674\) −5.13236e43 −0.734664
\(675\) −1.12105e43 −0.156826
\(676\) −1.29187e43 −0.176622
\(677\) −1.26241e42 −0.0168685 −0.00843423 0.999964i \(-0.502685\pi\)
−0.00843423 + 0.999964i \(0.502685\pi\)
\(678\) 1.76337e43 0.230294
\(679\) −6.84419e43 −0.873657
\(680\) 1.24759e43 0.155662
\(681\) 4.20213e43 0.512495
\(682\) −1.72366e43 −0.205492
\(683\) −5.02289e43 −0.585373 −0.292686 0.956209i \(-0.594549\pi\)
−0.292686 + 0.956209i \(0.594549\pi\)
\(684\) −2.03835e43 −0.232225
\(685\) −4.83278e43 −0.538261
\(686\) −1.44564e44 −1.57412
\(687\) −1.00535e43 −0.107025
\(688\) 7.82475e42 0.0814422
\(689\) −7.80349e43 −0.794129
\(690\) 9.41199e42 0.0936527
\(691\) 5.98132e43 0.581952 0.290976 0.956730i \(-0.406020\pi\)
0.290976 + 0.956730i \(0.406020\pi\)
\(692\) 2.78049e42 0.0264531
\(693\) −2.78488e44 −2.59085
\(694\) 5.01886e43 0.456597
\(695\) −5.63033e41 −0.00500921
\(696\) 3.24348e43 0.282208
\(697\) 1.24327e44 1.05793
\(698\) −8.62926e43 −0.718149
\(699\) 3.02278e43 0.246043
\(700\) 2.26000e43 0.179924
\(701\) −1.56022e44 −1.21495 −0.607473 0.794340i \(-0.707817\pi\)
−0.607473 + 0.794340i \(0.707817\pi\)
\(702\) −5.85421e43 −0.445906
\(703\) −9.25007e43 −0.689188
\(704\) 3.03801e43 0.221418
\(705\) 4.65263e43 0.331716
\(706\) −1.49490e44 −1.04266
\(707\) 2.81353e44 1.91978
\(708\) 2.34938e43 0.156833
\(709\) −8.54271e43 −0.557929 −0.278964 0.960301i \(-0.589991\pi\)
−0.278964 + 0.960301i \(0.589991\pi\)
\(710\) 2.83140e43 0.180924
\(711\) 1.18577e44 0.741348
\(712\) −4.13552e43 −0.252982
\(713\) 1.87675e43 0.112336
\(714\) −9.24950e43 −0.541746
\(715\) 1.11156e44 0.637070
\(716\) −8.77575e43 −0.492190
\(717\) 9.91985e43 0.544450
\(718\) −6.62945e43 −0.356081
\(719\) 2.30590e44 1.21211 0.606057 0.795421i \(-0.292750\pi\)
0.606057 + 0.795421i \(0.292750\pi\)
\(720\) 1.76668e43 0.0908877
\(721\) 1.24396e44 0.626339
\(722\) 9.66495e43 0.476292
\(723\) −2.42383e43 −0.116912
\(724\) 2.58550e43 0.122066
\(725\) −7.98683e43 −0.369093
\(726\) −1.44527e44 −0.653778
\(727\) −3.33530e44 −1.47690 −0.738450 0.674308i \(-0.764442\pi\)
−0.738450 + 0.674308i \(0.764442\pi\)
\(728\) 1.18019e44 0.511581
\(729\) −1.08371e43 −0.0459871
\(730\) 2.94533e43 0.122357
\(731\) 7.88574e43 0.320718
\(732\) 7.57197e43 0.301500
\(733\) 6.64546e43 0.259068 0.129534 0.991575i \(-0.458652\pi\)
0.129534 + 0.991575i \(0.458652\pi\)
\(734\) 3.37606e44 1.28861
\(735\) −1.15796e44 −0.432753
\(736\) −3.30783e43 −0.121042
\(737\) 5.31388e44 1.90400
\(738\) 1.76057e44 0.617703
\(739\) 5.13394e43 0.176385 0.0881926 0.996103i \(-0.471891\pi\)
0.0881926 + 0.996103i \(0.471891\pi\)
\(740\) 8.01724e43 0.269732
\(741\) −6.03184e43 −0.198732
\(742\) −3.89364e44 −1.25631
\(743\) −5.18614e44 −1.63877 −0.819385 0.573244i \(-0.805685\pi\)
−0.819385 + 0.573244i \(0.805685\pi\)
\(744\) −8.10682e42 −0.0250882
\(745\) −1.81964e43 −0.0551523
\(746\) 1.60512e44 0.476491
\(747\) −4.00591e44 −1.16475
\(748\) 3.06169e44 0.871939
\(749\) 6.48756e44 1.80973
\(750\) 1.00112e43 0.0273551
\(751\) 1.16699e44 0.312356 0.156178 0.987729i \(-0.450083\pi\)
0.156178 + 0.987729i \(0.450083\pi\)
\(752\) −1.63516e44 −0.428730
\(753\) −2.01911e44 −0.518607
\(754\) −4.17077e44 −1.04945
\(755\) 2.44624e44 0.603005
\(756\) −2.92102e44 −0.705419
\(757\) 7.15124e44 1.69198 0.845990 0.533199i \(-0.179010\pi\)
0.845990 + 0.533199i \(0.179010\pi\)
\(758\) 4.33607e44 1.00513
\(759\) 2.30978e44 0.524593
\(760\) 4.05948e43 0.0903357
\(761\) −2.08933e44 −0.455559 −0.227779 0.973713i \(-0.573146\pi\)
−0.227779 + 0.973713i \(0.573146\pi\)
\(762\) 6.90453e43 0.147514
\(763\) −1.44581e45 −3.02678
\(764\) −1.78393e44 −0.365958
\(765\) 1.78046e44 0.357914
\(766\) −7.24860e43 −0.142793
\(767\) −3.02105e44 −0.583216
\(768\) 1.42885e43 0.0270326
\(769\) 4.48666e44 0.831889 0.415944 0.909390i \(-0.363451\pi\)
0.415944 + 0.909390i \(0.363451\pi\)
\(770\) 5.54623e44 1.00784
\(771\) −2.10988e44 −0.375763
\(772\) 1.45712e44 0.254347
\(773\) −9.87774e44 −1.68996 −0.844978 0.534800i \(-0.820387\pi\)
−0.844978 + 0.534800i \(0.820387\pi\)
\(774\) 1.11668e44 0.187260
\(775\) 1.99624e43 0.0328123
\(776\) 1.06552e44 0.171675
\(777\) −5.94390e44 −0.938740
\(778\) 6.50485e44 1.00706
\(779\) 4.04543e44 0.613952
\(780\) 5.22793e43 0.0777791
\(781\) 6.94849e44 1.01344
\(782\) −3.33362e44 −0.476662
\(783\) 1.03229e45 1.44708
\(784\) 4.06964e44 0.559316
\(785\) −1.53728e44 −0.207144
\(786\) −3.09332e44 −0.408673
\(787\) −8.12711e44 −1.05276 −0.526378 0.850251i \(-0.676450\pi\)
−0.526378 + 0.850251i \(0.676450\pi\)
\(788\) −1.39838e44 −0.177610
\(789\) 1.90974e44 0.237838
\(790\) −2.36152e44 −0.288384
\(791\) −1.13139e45 −1.35481
\(792\) 4.33559e44 0.509106
\(793\) −9.73673e44 −1.12119
\(794\) −1.52818e44 −0.172566
\(795\) −1.72478e44 −0.191005
\(796\) −5.55603e44 −0.603409
\(797\) 2.32602e44 0.247748 0.123874 0.992298i \(-0.460468\pi\)
0.123874 + 0.992298i \(0.460468\pi\)
\(798\) −3.00965e44 −0.314392
\(799\) −1.64791e45 −1.68833
\(800\) −3.51844e43 −0.0353553
\(801\) −5.90187e44 −0.581682
\(802\) 3.53004e44 0.341253
\(803\) 7.22808e44 0.685381
\(804\) 2.49925e44 0.232456
\(805\) −6.03882e44 −0.550955
\(806\) 1.04245e44 0.0932958
\(807\) 4.42851e44 0.388793
\(808\) −4.38019e44 −0.377240
\(809\) 1.01189e45 0.854931 0.427466 0.904032i \(-0.359407\pi\)
0.427466 + 0.904032i \(0.359407\pi\)
\(810\) 1.80753e44 0.149820
\(811\) 7.77255e44 0.632034 0.316017 0.948754i \(-0.397654\pi\)
0.316017 + 0.948754i \(0.397654\pi\)
\(812\) −2.08105e45 −1.66022
\(813\) 2.87149e44 0.224752
\(814\) 1.96750e45 1.51090
\(815\) −7.94963e44 −0.598969
\(816\) 1.43999e44 0.106454
\(817\) 2.56591e44 0.186123
\(818\) 5.22791e44 0.372093
\(819\) 1.68426e45 1.17628
\(820\) −3.50626e44 −0.240287
\(821\) 2.49289e45 1.67642 0.838212 0.545344i \(-0.183601\pi\)
0.838212 + 0.545344i \(0.183601\pi\)
\(822\) −5.57807e44 −0.368104
\(823\) −2.55800e45 −1.65654 −0.828272 0.560326i \(-0.810676\pi\)
−0.828272 + 0.560326i \(0.810676\pi\)
\(824\) −1.93663e44 −0.123077
\(825\) 2.45684e44 0.153229
\(826\) −1.50739e45 −0.922643
\(827\) −1.08245e45 −0.650240 −0.325120 0.945673i \(-0.605405\pi\)
−0.325120 + 0.945673i \(0.605405\pi\)
\(828\) −4.72066e44 −0.278312
\(829\) 8.46679e43 0.0489918 0.0244959 0.999700i \(-0.492202\pi\)
0.0244959 + 0.999700i \(0.492202\pi\)
\(830\) 7.97795e44 0.453086
\(831\) −1.83901e44 −0.102511
\(832\) −1.83735e44 −0.100526
\(833\) 4.10137e45 2.20258
\(834\) −6.49862e42 −0.00342568
\(835\) −4.59292e44 −0.237656
\(836\) 9.96229e44 0.506013
\(837\) −2.58012e44 −0.128646
\(838\) 1.79130e45 0.876771
\(839\) −6.17907e43 −0.0296902 −0.0148451 0.999890i \(-0.504726\pi\)
−0.0148451 + 0.999890i \(0.504726\pi\)
\(840\) 2.60853e44 0.123046
\(841\) 5.19500e45 2.40574
\(842\) 8.09191e44 0.367886
\(843\) −8.78944e44 −0.392314
\(844\) −1.27490e45 −0.558689
\(845\) 3.67172e44 0.157976
\(846\) −2.33356e45 −0.985778
\(847\) 9.27298e45 3.84615
\(848\) 6.06173e44 0.246866
\(849\) 3.83144e43 0.0153212
\(850\) −3.54586e44 −0.139229
\(851\) −2.14224e45 −0.825963
\(852\) 3.26805e44 0.123730
\(853\) 1.05224e45 0.391205 0.195602 0.980683i \(-0.437334\pi\)
0.195602 + 0.980683i \(0.437334\pi\)
\(854\) −4.85825e45 −1.77371
\(855\) 5.79335e44 0.207709
\(856\) −1.01000e45 −0.355614
\(857\) −2.23665e45 −0.773383 −0.386692 0.922209i \(-0.626382\pi\)
−0.386692 + 0.922209i \(0.626382\pi\)
\(858\) 1.28298e45 0.435678
\(859\) −4.60157e45 −1.53466 −0.767329 0.641254i \(-0.778415\pi\)
−0.767329 + 0.641254i \(0.778415\pi\)
\(860\) −2.22393e44 −0.0728441
\(861\) 2.59951e45 0.836261
\(862\) −1.55227e45 −0.490462
\(863\) −4.42805e45 −1.37419 −0.687093 0.726569i \(-0.741114\pi\)
−0.687093 + 0.726569i \(0.741114\pi\)
\(864\) 4.54753e44 0.138616
\(865\) −7.90262e43 −0.0236604
\(866\) 1.08166e43 0.00318099
\(867\) −4.60692e43 −0.0133080
\(868\) 5.20142e44 0.147593
\(869\) −5.79536e45 −1.61538
\(870\) −9.21854e44 −0.252414
\(871\) −3.21377e45 −0.864437
\(872\) 2.25088e45 0.594767
\(873\) 1.52063e45 0.394732
\(874\) −1.08471e45 −0.276622
\(875\) −6.42331e44 −0.160929
\(876\) 3.39955e44 0.0836774
\(877\) −5.41361e45 −1.30916 −0.654581 0.755992i \(-0.727155\pi\)
−0.654581 + 0.755992i \(0.727155\pi\)
\(878\) 3.61402e43 0.00858668
\(879\) 1.85742e45 0.433593
\(880\) −8.63453e44 −0.198042
\(881\) −4.09130e45 −0.922008 −0.461004 0.887398i \(-0.652511\pi\)
−0.461004 + 0.887398i \(0.652511\pi\)
\(882\) 5.80785e45 1.28603
\(883\) 7.61955e45 1.65782 0.828912 0.559379i \(-0.188960\pi\)
0.828912 + 0.559379i \(0.188960\pi\)
\(884\) −1.85167e45 −0.395871
\(885\) −6.67734e44 −0.140276
\(886\) 3.99255e45 0.824190
\(887\) −4.92537e45 −0.999131 −0.499566 0.866276i \(-0.666507\pi\)
−0.499566 + 0.866276i \(0.666507\pi\)
\(888\) 9.25363e44 0.184464
\(889\) −4.43001e45 −0.867816
\(890\) 1.17539e45 0.226274
\(891\) 4.43584e45 0.839211
\(892\) 3.09050e44 0.0574611
\(893\) −5.36205e45 −0.979791
\(894\) −2.10026e44 −0.0377174
\(895\) 2.49422e45 0.440228
\(896\) −9.16765e44 −0.159032
\(897\) −1.39693e45 −0.238172
\(898\) −4.41116e45 −0.739213
\(899\) −1.83818e45 −0.302770
\(900\) −5.02122e44 −0.0812925
\(901\) 6.10899e45 0.972154
\(902\) −8.60465e45 −1.34596
\(903\) 1.64880e45 0.253517
\(904\) 1.76139e45 0.266222
\(905\) −7.34842e44 −0.109180
\(906\) 2.82349e45 0.412382
\(907\) −2.39988e45 −0.344570 −0.172285 0.985047i \(-0.555115\pi\)
−0.172285 + 0.985047i \(0.555115\pi\)
\(908\) 4.19742e45 0.592450
\(909\) −6.25105e45 −0.867386
\(910\) −3.35429e45 −0.457572
\(911\) −1.27477e46 −1.70961 −0.854805 0.518949i \(-0.826324\pi\)
−0.854805 + 0.518949i \(0.826324\pi\)
\(912\) 4.68552e44 0.0617786
\(913\) 1.95786e46 2.53795
\(914\) −2.20573e45 −0.281116
\(915\) −2.15208e45 −0.269670
\(916\) −1.00422e45 −0.123723
\(917\) 1.98471e46 2.40420
\(918\) 4.58298e45 0.545867
\(919\) −8.11185e44 −0.0950013 −0.0475007 0.998871i \(-0.515126\pi\)
−0.0475007 + 0.998871i \(0.515126\pi\)
\(920\) 9.40142e44 0.108264
\(921\) −1.49617e45 −0.169416
\(922\) 1.23460e46 1.37466
\(923\) −4.20236e45 −0.460115
\(924\) 6.40155e45 0.689239
\(925\) −2.27864e45 −0.241256
\(926\) 1.10659e46 1.15217
\(927\) −2.76380e45 −0.282990
\(928\) 3.23984e45 0.326235
\(929\) −8.34797e45 −0.826680 −0.413340 0.910577i \(-0.635638\pi\)
−0.413340 + 0.910577i \(0.635638\pi\)
\(930\) 2.30410e44 0.0224396
\(931\) 1.33453e46 1.27822
\(932\) 3.01939e45 0.284428
\(933\) −7.77810e44 −0.0720623
\(934\) 3.67746e45 0.335098
\(935\) −8.70184e45 −0.779886
\(936\) −2.62211e45 −0.231140
\(937\) 7.08971e45 0.614703 0.307351 0.951596i \(-0.400557\pi\)
0.307351 + 0.951596i \(0.400557\pi\)
\(938\) −1.60355e46 −1.36753
\(939\) −7.70561e45 −0.646384
\(940\) 4.64740e45 0.383468
\(941\) −1.27325e46 −1.03341 −0.516706 0.856163i \(-0.672842\pi\)
−0.516706 + 0.856163i \(0.672842\pi\)
\(942\) −1.77435e45 −0.141661
\(943\) 9.36889e45 0.735795
\(944\) 2.34674e45 0.181301
\(945\) 8.30204e45 0.630946
\(946\) −5.45770e45 −0.408035
\(947\) 8.84487e45 0.650529 0.325265 0.945623i \(-0.394547\pi\)
0.325265 + 0.945623i \(0.394547\pi\)
\(948\) −2.72571e45 −0.197220
\(949\) −4.37145e45 −0.311171
\(950\) −1.15377e45 −0.0807987
\(951\) 1.09406e46 0.753776
\(952\) −9.23912e45 −0.626265
\(953\) 9.38297e44 0.0625749 0.0312875 0.999510i \(-0.490039\pi\)
0.0312875 + 0.999510i \(0.490039\pi\)
\(954\) 8.65080e45 0.567618
\(955\) 5.07024e45 0.327323
\(956\) 9.90872e45 0.629391
\(957\) −2.26231e46 −1.41389
\(958\) −3.26573e45 −0.200824
\(959\) 3.57895e46 2.16554
\(960\) −4.06104e44 −0.0241787
\(961\) −1.66097e46 −0.973084
\(962\) −1.18992e46 −0.685967
\(963\) −1.44139e46 −0.817663
\(964\) −2.42111e45 −0.135151
\(965\) −4.14138e45 −0.227495
\(966\) −6.97011e45 −0.376786
\(967\) −1.81981e46 −0.968092 −0.484046 0.875043i \(-0.660833\pi\)
−0.484046 + 0.875043i \(0.660833\pi\)
\(968\) −1.44364e46 −0.755775
\(969\) 4.72204e45 0.243283
\(970\) −3.02840e45 −0.153551
\(971\) −1.52145e46 −0.759205 −0.379602 0.925150i \(-0.623939\pi\)
−0.379602 + 0.925150i \(0.623939\pi\)
\(972\) 1.00696e46 0.494523
\(973\) 4.16958e44 0.0201532
\(974\) −4.06766e45 −0.193500
\(975\) −1.48587e45 −0.0695678
\(976\) 7.56347e45 0.348537
\(977\) −1.63297e46 −0.740650 −0.370325 0.928902i \(-0.620754\pi\)
−0.370325 + 0.928902i \(0.620754\pi\)
\(978\) −9.17560e45 −0.409621
\(979\) 2.88449e46 1.26747
\(980\) −1.15666e46 −0.500267
\(981\) 3.21227e46 1.36755
\(982\) −9.39872e44 −0.0393859
\(983\) 2.28872e46 0.944089 0.472044 0.881575i \(-0.343516\pi\)
0.472044 + 0.881575i \(0.343516\pi\)
\(984\) −4.04699e45 −0.164327
\(985\) 3.97442e45 0.158859
\(986\) 3.26510e46 1.28471
\(987\) −3.44554e46 −1.33457
\(988\) −6.02507e45 −0.229736
\(989\) 5.94243e45 0.223060
\(990\) −1.23225e46 −0.455358
\(991\) −1.29262e46 −0.470250 −0.235125 0.971965i \(-0.575550\pi\)
−0.235125 + 0.971965i \(0.575550\pi\)
\(992\) −8.09773e44 −0.0290023
\(993\) 1.40459e46 0.495262
\(994\) −2.09681e46 −0.727898
\(995\) 1.57912e46 0.539705
\(996\) 9.20829e45 0.309856
\(997\) −7.75877e45 −0.257050 −0.128525 0.991706i \(-0.541024\pi\)
−0.128525 + 0.991706i \(0.541024\pi\)
\(998\) −3.03927e46 −0.991394
\(999\) 2.94511e46 0.945880
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 10.32.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.32.a.b.1.1 2 1.1 even 1 trivial