Properties

Label 1.66.a.a.1.2
Level $1$
Weight $66$
Character 1.1
Self dual yes
Analytic conductor $26.757$
Analytic rank $1$
Dimension $5$
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] = [1,66,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 66, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 66);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(26.7572356472\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.39707e8\) of defining polynomial
Character \(\chi\) \(=\) 1.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.49786e9 q^{2} -4.87294e15 q^{3} +1.93244e19 q^{4} -6.28165e22 q^{5} +3.65366e25 q^{6} -4.41462e27 q^{7} +1.31731e29 q^{8} +1.34445e31 q^{9} +O(q^{10})\) \(q-7.49786e9 q^{2} -4.87294e15 q^{3} +1.93244e19 q^{4} -6.28165e22 q^{5} +3.65366e25 q^{6} -4.41462e27 q^{7} +1.31731e29 q^{8} +1.34445e31 q^{9} +4.70989e32 q^{10} +2.47205e33 q^{11} -9.41665e34 q^{12} +8.18765e35 q^{13} +3.31002e37 q^{14} +3.06101e38 q^{15} -1.70064e39 q^{16} +1.52850e40 q^{17} -1.00805e41 q^{18} -1.62754e41 q^{19} -1.21389e42 q^{20} +2.15121e43 q^{21} -1.85351e43 q^{22} -1.77529e44 q^{23} -6.41916e44 q^{24} +1.23540e45 q^{25} -6.13898e45 q^{26} -1.53177e46 q^{27} -8.53097e46 q^{28} -4.19469e47 q^{29} -2.29510e48 q^{30} -6.78523e47 q^{31} +7.89116e48 q^{32} -1.20462e49 q^{33} -1.14605e50 q^{34} +2.77311e50 q^{35} +2.59806e50 q^{36} +1.36074e51 q^{37} +1.22030e51 q^{38} -3.98979e51 q^{39} -8.27486e51 q^{40} +3.18837e52 q^{41} -1.61295e53 q^{42} +1.25486e53 q^{43} +4.77709e52 q^{44} -8.44534e53 q^{45} +1.33108e54 q^{46} -9.94260e53 q^{47} +8.28712e54 q^{48} +1.09505e55 q^{49} -9.26288e54 q^{50} -7.44827e55 q^{51} +1.58221e55 q^{52} -1.30085e55 q^{53} +1.14850e56 q^{54} -1.55286e56 q^{55} -5.81540e56 q^{56} +7.93088e56 q^{57} +3.14512e57 q^{58} -1.98182e57 q^{59} +5.91521e57 q^{60} -5.55785e57 q^{61} +5.08747e57 q^{62} -5.93522e58 q^{63} +3.57579e57 q^{64} -5.14319e58 q^{65} +9.03203e58 q^{66} +1.23347e59 q^{67} +2.95372e59 q^{68} +8.65086e59 q^{69} -2.07923e60 q^{70} -5.36951e59 q^{71} +1.77105e60 q^{72} +5.21560e60 q^{73} -1.02026e61 q^{74} -6.02004e60 q^{75} -3.14511e60 q^{76} -1.09132e61 q^{77} +2.99149e61 q^{78} +5.19565e61 q^{79} +1.06828e62 q^{80} -6.38500e61 q^{81} -2.39059e62 q^{82} +8.74678e60 q^{83} +4.15709e62 q^{84} -9.60148e62 q^{85} -9.40879e62 q^{86} +2.04405e63 q^{87} +3.25645e62 q^{88} -1.81776e63 q^{89} +6.33220e63 q^{90} -3.61453e63 q^{91} -3.43063e63 q^{92} +3.30640e63 q^{93} +7.45482e63 q^{94} +1.02236e64 q^{95} -3.84531e64 q^{96} -1.86066e63 q^{97} -8.21053e64 q^{98} +3.32354e64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3959709648 q^{2} - 22\!\cdots\!04 q^{3}+ \cdots + 31\!\cdots\!65 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3959709648 q^{2} - 22\!\cdots\!04 q^{3}+ \cdots - 36\!\cdots\!20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.49786e9 −1.23442 −0.617209 0.786799i \(-0.711737\pi\)
−0.617209 + 0.786799i \(0.711737\pi\)
\(3\) −4.87294e15 −1.51827 −0.759137 0.650931i \(-0.774379\pi\)
−0.759137 + 0.650931i \(0.774379\pi\)
\(4\) 1.93244e19 0.523788
\(5\) −6.28165e22 −1.20656 −0.603279 0.797530i \(-0.706140\pi\)
−0.603279 + 0.797530i \(0.706140\pi\)
\(6\) 3.65366e25 1.87418
\(7\) −4.41462e27 −1.51080 −0.755399 0.655265i \(-0.772557\pi\)
−0.755399 + 0.655265i \(0.772557\pi\)
\(8\) 1.31731e29 0.587845
\(9\) 1.34445e31 1.30516
\(10\) 4.70989e32 1.48940
\(11\) 2.47205e33 0.353017 0.176508 0.984299i \(-0.443520\pi\)
0.176508 + 0.984299i \(0.443520\pi\)
\(12\) −9.41665e34 −0.795254
\(13\) 8.18765e35 0.512862 0.256431 0.966563i \(-0.417453\pi\)
0.256431 + 0.966563i \(0.417453\pi\)
\(14\) 3.31002e37 1.86496
\(15\) 3.06101e38 1.83189
\(16\) −1.70064e39 −1.24943
\(17\) 1.52850e40 1.56558 0.782789 0.622287i \(-0.213796\pi\)
0.782789 + 0.622287i \(0.213796\pi\)
\(18\) −1.00805e41 −1.61111
\(19\) −1.62754e41 −0.448785 −0.224393 0.974499i \(-0.572040\pi\)
−0.224393 + 0.974499i \(0.572040\pi\)
\(20\) −1.21389e42 −0.631981
\(21\) 2.15121e43 2.29381
\(22\) −1.85351e43 −0.435770
\(23\) −1.77529e44 −0.984269 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(24\) −6.41916e44 −0.892509
\(25\) 1.23540e45 0.455783
\(26\) −6.13898e45 −0.633086
\(27\) −1.53177e46 −0.463309
\(28\) −8.53097e46 −0.791338
\(29\) −4.19469e47 −1.24384 −0.621921 0.783080i \(-0.713647\pi\)
−0.621921 + 0.783080i \(0.713647\pi\)
\(30\) −2.29510e48 −2.26131
\(31\) −6.78523e47 −0.230308 −0.115154 0.993348i \(-0.536736\pi\)
−0.115154 + 0.993348i \(0.536736\pi\)
\(32\) 7.89116e48 0.954480
\(33\) −1.20462e49 −0.535976
\(34\) −1.14605e50 −1.93258
\(35\) 2.77311e50 1.82287
\(36\) 2.59806e50 0.683625
\(37\) 1.36074e51 1.46967 0.734833 0.678248i \(-0.237260\pi\)
0.734833 + 0.678248i \(0.237260\pi\)
\(38\) 1.22030e51 0.553988
\(39\) −3.98979e51 −0.778665
\(40\) −8.27486e51 −0.709269
\(41\) 3.18837e52 1.22488 0.612440 0.790517i \(-0.290188\pi\)
0.612440 + 0.790517i \(0.290188\pi\)
\(42\) −1.61295e53 −2.83152
\(43\) 1.25486e53 1.02535 0.512674 0.858583i \(-0.328655\pi\)
0.512674 + 0.858583i \(0.328655\pi\)
\(44\) 4.77709e52 0.184906
\(45\) −8.44534e53 −1.57475
\(46\) 1.33108e54 1.21500
\(47\) −9.94260e53 −0.451149 −0.225574 0.974226i \(-0.572426\pi\)
−0.225574 + 0.974226i \(0.572426\pi\)
\(48\) 8.28712e54 1.89698
\(49\) 1.09505e55 1.28251
\(50\) −9.26288e54 −0.562627
\(51\) −7.44827e55 −2.37698
\(52\) 1.58221e55 0.268631
\(53\) −1.30085e55 −0.118922 −0.0594608 0.998231i \(-0.518938\pi\)
−0.0594608 + 0.998231i \(0.518938\pi\)
\(54\) 1.14850e56 0.571917
\(55\) −1.55286e56 −0.425935
\(56\) −5.81540e56 −0.888115
\(57\) 7.93088e56 0.681379
\(58\) 3.14512e57 1.53542
\(59\) −1.98182e57 −0.555103 −0.277552 0.960711i \(-0.589523\pi\)
−0.277552 + 0.960711i \(0.589523\pi\)
\(60\) 5.91521e57 0.959520
\(61\) −5.55785e57 −0.526850 −0.263425 0.964680i \(-0.584852\pi\)
−0.263425 + 0.964680i \(0.584852\pi\)
\(62\) 5.08747e57 0.284297
\(63\) −5.93522e58 −1.97183
\(64\) 3.57579e57 0.0712071
\(65\) −5.14319e58 −0.618798
\(66\) 9.03203e58 0.661619
\(67\) 1.23347e59 0.554239 0.277119 0.960835i \(-0.410620\pi\)
0.277119 + 0.960835i \(0.410620\pi\)
\(68\) 2.95372e59 0.820031
\(69\) 8.65086e59 1.49439
\(70\) −2.07923e60 −2.25018
\(71\) −5.36951e59 −0.366470 −0.183235 0.983069i \(-0.558657\pi\)
−0.183235 + 0.983069i \(0.558657\pi\)
\(72\) 1.77105e60 0.767228
\(73\) 5.21560e60 1.44315 0.721574 0.692337i \(-0.243419\pi\)
0.721574 + 0.692337i \(0.243419\pi\)
\(74\) −1.02026e61 −1.81418
\(75\) −6.02004e60 −0.692004
\(76\) −3.14511e60 −0.235068
\(77\) −1.09132e61 −0.533337
\(78\) 2.99149e61 0.961198
\(79\) 5.19565e61 1.10347 0.551734 0.834020i \(-0.313967\pi\)
0.551734 + 0.834020i \(0.313967\pi\)
\(80\) 1.06828e62 1.50752
\(81\) −6.38500e61 −0.601725
\(82\) −2.39059e62 −1.51201
\(83\) 8.74678e60 0.0373087 0.0186544 0.999826i \(-0.494062\pi\)
0.0186544 + 0.999826i \(0.494062\pi\)
\(84\) 4.15709e62 1.20147
\(85\) −9.60148e62 −1.88896
\(86\) −9.40879e62 −1.26571
\(87\) 2.04405e63 1.88849
\(88\) 3.25645e62 0.207519
\(89\) −1.81776e63 −0.802344 −0.401172 0.916003i \(-0.631397\pi\)
−0.401172 + 0.916003i \(0.631397\pi\)
\(90\) 6.33220e63 1.94390
\(91\) −3.61453e63 −0.774831
\(92\) −3.43063e63 −0.515548
\(93\) 3.30640e63 0.349671
\(94\) 7.45482e63 0.556906
\(95\) 1.02236e64 0.541485
\(96\) −3.84531e64 −1.44916
\(97\) −1.86066e63 −0.0500709 −0.0250355 0.999687i \(-0.507970\pi\)
−0.0250355 + 0.999687i \(0.507970\pi\)
\(98\) −8.21053e64 −1.58316
\(99\) 3.32354e64 0.460742
\(100\) 2.38734e64 0.238734
\(101\) 2.25012e64 0.162840 0.0814200 0.996680i \(-0.474054\pi\)
0.0814200 + 0.996680i \(0.474054\pi\)
\(102\) 5.58461e65 2.93418
\(103\) −1.05413e65 −0.403352 −0.201676 0.979452i \(-0.564639\pi\)
−0.201676 + 0.979452i \(0.564639\pi\)
\(104\) 1.07857e65 0.301483
\(105\) −1.35132e66 −2.76761
\(106\) 9.75357e64 0.146799
\(107\) 6.77694e65 0.751729 0.375864 0.926675i \(-0.377346\pi\)
0.375864 + 0.926675i \(0.377346\pi\)
\(108\) −2.96005e65 −0.242676
\(109\) −2.43834e65 −0.148161 −0.0740805 0.997252i \(-0.523602\pi\)
−0.0740805 + 0.997252i \(0.523602\pi\)
\(110\) 1.16431e66 0.525782
\(111\) −6.63080e66 −2.23136
\(112\) 7.50768e66 1.88764
\(113\) 3.56635e66 0.671699 0.335850 0.941916i \(-0.390977\pi\)
0.335850 + 0.941916i \(0.390977\pi\)
\(114\) −5.94646e66 −0.841106
\(115\) 1.11517e67 1.18758
\(116\) −8.10597e66 −0.651509
\(117\) 1.10079e67 0.669364
\(118\) 1.48594e67 0.685230
\(119\) −6.74773e67 −2.36527
\(120\) 4.03229e67 1.07686
\(121\) −4.29260e67 −0.875379
\(122\) 4.16719e67 0.650353
\(123\) −1.55367e68 −1.85970
\(124\) −1.31120e67 −0.120633
\(125\) 9.26607e67 0.656629
\(126\) 4.45014e68 2.43406
\(127\) 3.29671e68 1.39463 0.697317 0.716763i \(-0.254377\pi\)
0.697317 + 0.716763i \(0.254377\pi\)
\(128\) −3.17943e68 −1.04238
\(129\) −6.11487e68 −1.55676
\(130\) 3.85629e68 0.763855
\(131\) −8.62701e68 −1.33212 −0.666060 0.745898i \(-0.732021\pi\)
−0.666060 + 0.745898i \(0.732021\pi\)
\(132\) −2.32784e68 −0.280738
\(133\) 7.18494e68 0.678024
\(134\) −9.24837e68 −0.684163
\(135\) 9.62203e68 0.559010
\(136\) 2.01350e69 0.920316
\(137\) −8.40720e68 −0.302854 −0.151427 0.988468i \(-0.548387\pi\)
−0.151427 + 0.988468i \(0.548387\pi\)
\(138\) −6.48629e69 −1.84470
\(139\) 3.55176e69 0.798845 0.399422 0.916767i \(-0.369211\pi\)
0.399422 + 0.916767i \(0.369211\pi\)
\(140\) 5.35885e69 0.954796
\(141\) 4.84497e69 0.684967
\(142\) 4.02598e69 0.452377
\(143\) 2.02403e69 0.181049
\(144\) −2.28642e70 −1.63071
\(145\) 2.63496e70 1.50077
\(146\) −3.91058e70 −1.78145
\(147\) −5.33611e70 −1.94720
\(148\) 2.62954e70 0.769794
\(149\) 5.65978e70 1.33121 0.665604 0.746305i \(-0.268174\pi\)
0.665604 + 0.746305i \(0.268174\pi\)
\(150\) 4.51374e70 0.854222
\(151\) −1.02555e71 −1.56390 −0.781948 0.623343i \(-0.785774\pi\)
−0.781948 + 0.623343i \(0.785774\pi\)
\(152\) −2.14396e70 −0.263816
\(153\) 2.05498e71 2.04332
\(154\) 8.18253e70 0.658361
\(155\) 4.26224e70 0.277880
\(156\) −7.71002e70 −0.407855
\(157\) −2.01202e71 −0.864755 −0.432377 0.901693i \(-0.642325\pi\)
−0.432377 + 0.901693i \(0.642325\pi\)
\(158\) −3.89562e71 −1.36214
\(159\) 6.33895e70 0.180555
\(160\) −4.95695e71 −1.15164
\(161\) 7.83721e71 1.48703
\(162\) 4.78739e71 0.742780
\(163\) −8.87967e71 −1.12798 −0.563988 0.825783i \(-0.690734\pi\)
−0.563988 + 0.825783i \(0.690734\pi\)
\(164\) 6.16133e71 0.641578
\(165\) 7.56697e71 0.646686
\(166\) −6.55821e70 −0.0460546
\(167\) −2.94357e72 −1.70055 −0.850276 0.526338i \(-0.823565\pi\)
−0.850276 + 0.526338i \(0.823565\pi\)
\(168\) 2.83381e72 1.34840
\(169\) −1.87832e72 −0.736973
\(170\) 7.19905e72 2.33177
\(171\) −2.18814e72 −0.585734
\(172\) 2.42494e72 0.537066
\(173\) −2.44850e72 −0.449160 −0.224580 0.974456i \(-0.572101\pi\)
−0.224580 + 0.974456i \(0.572101\pi\)
\(174\) −1.53260e73 −2.33119
\(175\) −5.45383e72 −0.688597
\(176\) −4.20407e72 −0.441071
\(177\) 9.65728e72 0.842799
\(178\) 1.36293e73 0.990428
\(179\) 1.53155e73 0.927701 0.463850 0.885913i \(-0.346467\pi\)
0.463850 + 0.885913i \(0.346467\pi\)
\(180\) −1.63201e73 −0.824833
\(181\) 1.32295e73 0.558459 0.279229 0.960224i \(-0.409921\pi\)
0.279229 + 0.960224i \(0.409921\pi\)
\(182\) 2.71013e73 0.956465
\(183\) 2.70830e73 0.799902
\(184\) −2.33860e73 −0.578597
\(185\) −8.54768e73 −1.77324
\(186\) −2.47909e73 −0.431640
\(187\) 3.77852e73 0.552675
\(188\) −1.92135e73 −0.236306
\(189\) 6.76217e73 0.699967
\(190\) −7.66551e73 −0.668419
\(191\) −1.32871e74 −0.976889 −0.488444 0.872595i \(-0.662435\pi\)
−0.488444 + 0.872595i \(0.662435\pi\)
\(192\) −1.74246e73 −0.108112
\(193\) 2.53755e74 1.32985 0.664925 0.746910i \(-0.268464\pi\)
0.664925 + 0.746910i \(0.268464\pi\)
\(194\) 1.39510e73 0.0618085
\(195\) 2.50625e74 0.939505
\(196\) 2.11612e74 0.671765
\(197\) −5.80730e74 −1.56251 −0.781253 0.624214i \(-0.785419\pi\)
−0.781253 + 0.624214i \(0.785419\pi\)
\(198\) −2.49195e74 −0.568748
\(199\) 6.30427e74 1.22155 0.610773 0.791806i \(-0.290859\pi\)
0.610773 + 0.791806i \(0.290859\pi\)
\(200\) 1.62741e74 0.267930
\(201\) −6.01062e74 −0.841486
\(202\) −1.68711e74 −0.201013
\(203\) 1.85179e75 1.87919
\(204\) −1.43933e75 −1.24503
\(205\) −2.00282e75 −1.47789
\(206\) 7.90371e74 0.497905
\(207\) −2.38678e75 −1.28462
\(208\) −1.39243e75 −0.640787
\(209\) −4.02335e74 −0.158429
\(210\) 1.01320e76 3.41639
\(211\) 3.20088e75 0.924890 0.462445 0.886648i \(-0.346972\pi\)
0.462445 + 0.886648i \(0.346972\pi\)
\(212\) −2.51381e74 −0.0622897
\(213\) 2.61653e75 0.556402
\(214\) −5.08126e75 −0.927948
\(215\) −7.88261e75 −1.23714
\(216\) −2.01781e75 −0.272354
\(217\) 2.99542e75 0.347949
\(218\) 1.82824e75 0.182893
\(219\) −2.54153e76 −2.19110
\(220\) −3.00080e75 −0.223100
\(221\) 1.25148e76 0.802925
\(222\) 4.97168e76 2.75443
\(223\) −1.64082e76 −0.785514 −0.392757 0.919642i \(-0.628479\pi\)
−0.392757 + 0.919642i \(0.628479\pi\)
\(224\) −3.48364e76 −1.44203
\(225\) 1.66093e76 0.594868
\(226\) −2.67400e76 −0.829158
\(227\) 3.31692e76 0.891036 0.445518 0.895273i \(-0.353020\pi\)
0.445518 + 0.895273i \(0.353020\pi\)
\(228\) 1.53259e76 0.356898
\(229\) 7.56824e76 1.52877 0.764384 0.644761i \(-0.223043\pi\)
0.764384 + 0.644761i \(0.223043\pi\)
\(230\) −8.36140e76 −1.46597
\(231\) 5.31791e76 0.809752
\(232\) −5.52569e76 −0.731185
\(233\) −7.85150e74 −0.00903410 −0.00451705 0.999990i \(-0.501438\pi\)
−0.00451705 + 0.999990i \(0.501438\pi\)
\(234\) −8.25354e76 −0.826276
\(235\) 6.24559e76 0.544337
\(236\) −3.82974e76 −0.290757
\(237\) −2.53181e77 −1.67537
\(238\) 5.05935e77 2.91974
\(239\) −3.24479e77 −1.63401 −0.817006 0.576629i \(-0.804368\pi\)
−0.817006 + 0.576629i \(0.804368\pi\)
\(240\) −5.20568e77 −2.28882
\(241\) 4.35995e77 1.67466 0.837332 0.546694i \(-0.184114\pi\)
0.837332 + 0.546694i \(0.184114\pi\)
\(242\) 3.21853e77 1.08058
\(243\) 4.68926e77 1.37689
\(244\) −1.07402e77 −0.275958
\(245\) −6.87872e77 −1.54743
\(246\) 1.16492e78 2.29565
\(247\) −1.33257e77 −0.230165
\(248\) −8.93823e76 −0.135385
\(249\) −4.26225e76 −0.0566448
\(250\) −6.94757e77 −0.810555
\(251\) −1.07974e78 −1.10642 −0.553212 0.833040i \(-0.686598\pi\)
−0.553212 + 0.833040i \(0.686598\pi\)
\(252\) −1.14694e78 −1.03282
\(253\) −4.38860e77 −0.347463
\(254\) −2.47183e78 −1.72156
\(255\) 4.67874e78 2.86796
\(256\) 2.25197e78 1.21552
\(257\) 8.71232e77 0.414293 0.207146 0.978310i \(-0.433582\pi\)
0.207146 + 0.978310i \(0.433582\pi\)
\(258\) 4.58484e78 1.92169
\(259\) −6.00714e78 −2.22037
\(260\) −9.93890e77 −0.324119
\(261\) −5.63954e78 −1.62341
\(262\) 6.46841e78 1.64439
\(263\) 2.13751e78 0.480116 0.240058 0.970759i \(-0.422834\pi\)
0.240058 + 0.970759i \(0.422834\pi\)
\(264\) −1.58685e78 −0.315071
\(265\) 8.17146e77 0.143486
\(266\) −5.38717e78 −0.836965
\(267\) 8.85781e78 1.21818
\(268\) 2.38360e78 0.290304
\(269\) −7.76436e78 −0.837831 −0.418916 0.908025i \(-0.637590\pi\)
−0.418916 + 0.908025i \(0.637590\pi\)
\(270\) −7.21446e78 −0.690052
\(271\) −2.75336e78 −0.233540 −0.116770 0.993159i \(-0.537254\pi\)
−0.116770 + 0.993159i \(0.537254\pi\)
\(272\) −2.59943e79 −1.95609
\(273\) 1.76134e79 1.17641
\(274\) 6.30360e78 0.373848
\(275\) 3.05398e78 0.160899
\(276\) 1.67172e79 0.782744
\(277\) 3.41655e79 1.42231 0.711157 0.703033i \(-0.248171\pi\)
0.711157 + 0.703033i \(0.248171\pi\)
\(278\) −2.66306e79 −0.986108
\(279\) −9.12238e78 −0.300588
\(280\) 3.65303e79 1.07156
\(281\) 2.92641e79 0.764503 0.382252 0.924058i \(-0.375149\pi\)
0.382252 + 0.924058i \(0.375149\pi\)
\(282\) −3.63269e79 −0.845536
\(283\) −1.48186e79 −0.307431 −0.153715 0.988115i \(-0.549124\pi\)
−0.153715 + 0.988115i \(0.549124\pi\)
\(284\) −1.03762e79 −0.191953
\(285\) −4.98190e79 −0.822123
\(286\) −1.51759e79 −0.223490
\(287\) −1.40754e80 −1.85055
\(288\) 1.06093e80 1.24574
\(289\) 1.38311e80 1.45103
\(290\) −1.97565e80 −1.85257
\(291\) 9.06688e78 0.0760214
\(292\) 1.00788e80 0.755904
\(293\) −1.37981e80 −0.926021 −0.463010 0.886353i \(-0.653231\pi\)
−0.463010 + 0.886353i \(0.653231\pi\)
\(294\) 4.00094e80 2.40366
\(295\) 1.24491e80 0.669765
\(296\) 1.79251e80 0.863936
\(297\) −3.78661e79 −0.163556
\(298\) −4.24362e80 −1.64327
\(299\) −1.45354e80 −0.504794
\(300\) −1.16334e80 −0.362464
\(301\) −5.53974e80 −1.54910
\(302\) 7.68946e80 1.93050
\(303\) −1.09647e80 −0.247236
\(304\) 2.76785e80 0.560727
\(305\) 3.49124e80 0.635675
\(306\) −1.54080e81 −2.52231
\(307\) 8.32934e80 1.22635 0.613174 0.789948i \(-0.289893\pi\)
0.613174 + 0.789948i \(0.289893\pi\)
\(308\) −2.10890e80 −0.279356
\(309\) 5.13671e80 0.612399
\(310\) −3.19577e80 −0.343020
\(311\) −2.02227e79 −0.0195491 −0.00977455 0.999952i \(-0.503111\pi\)
−0.00977455 + 0.999952i \(0.503111\pi\)
\(312\) −5.25578e80 −0.457734
\(313\) −2.05669e81 −1.61428 −0.807139 0.590361i \(-0.798985\pi\)
−0.807139 + 0.590361i \(0.798985\pi\)
\(314\) 1.50858e81 1.06747
\(315\) 3.72829e81 2.37912
\(316\) 1.00403e81 0.577983
\(317\) 1.39661e81 0.725517 0.362758 0.931883i \(-0.381835\pi\)
0.362758 + 0.931883i \(0.381835\pi\)
\(318\) −4.75285e80 −0.222881
\(319\) −1.03695e81 −0.439097
\(320\) −2.24619e80 −0.0859155
\(321\) −3.30236e81 −1.14133
\(322\) −5.87623e81 −1.83562
\(323\) −2.48768e81 −0.702608
\(324\) −1.23386e81 −0.315176
\(325\) 1.01151e81 0.233754
\(326\) 6.65785e81 1.39239
\(327\) 1.18819e81 0.224949
\(328\) 4.20006e81 0.720039
\(329\) 4.38928e81 0.681595
\(330\) −5.67361e81 −0.798282
\(331\) −8.98103e81 −1.14529 −0.572646 0.819803i \(-0.694083\pi\)
−0.572646 + 0.819803i \(0.694083\pi\)
\(332\) 1.69026e80 0.0195419
\(333\) 1.82944e82 1.91814
\(334\) 2.20705e82 2.09919
\(335\) −7.74821e81 −0.668722
\(336\) −3.65844e82 −2.86596
\(337\) 1.09804e82 0.780990 0.390495 0.920605i \(-0.372304\pi\)
0.390495 + 0.920605i \(0.372304\pi\)
\(338\) 1.40834e82 0.909732
\(339\) −1.73786e82 −1.01982
\(340\) −1.85543e82 −0.989415
\(341\) −1.67734e81 −0.0813026
\(342\) 1.64063e82 0.723041
\(343\) −1.06488e82 −0.426819
\(344\) 1.65304e82 0.602746
\(345\) −5.43416e82 −1.80307
\(346\) 1.83585e82 0.554452
\(347\) 5.19171e82 1.42759 0.713793 0.700357i \(-0.246976\pi\)
0.713793 + 0.700357i \(0.246976\pi\)
\(348\) 3.94999e82 0.989169
\(349\) −7.56370e82 −1.72547 −0.862735 0.505656i \(-0.831251\pi\)
−0.862735 + 0.505656i \(0.831251\pi\)
\(350\) 4.08920e82 0.850017
\(351\) −1.25416e82 −0.237614
\(352\) 1.95074e82 0.336947
\(353\) 5.18406e81 0.0816567 0.0408284 0.999166i \(-0.487000\pi\)
0.0408284 + 0.999166i \(0.487000\pi\)
\(354\) −7.24089e82 −1.04037
\(355\) 3.37294e82 0.442167
\(356\) −3.51270e82 −0.420258
\(357\) 3.28812e83 3.59113
\(358\) −1.14834e83 −1.14517
\(359\) −6.01244e82 −0.547622 −0.273811 0.961784i \(-0.588284\pi\)
−0.273811 + 0.961784i \(0.588284\pi\)
\(360\) −1.11251e83 −0.925706
\(361\) −1.05029e83 −0.798592
\(362\) −9.91932e82 −0.689372
\(363\) 2.09176e83 1.32907
\(364\) −6.98486e82 −0.405847
\(365\) −3.27625e83 −1.74124
\(366\) −2.03065e83 −0.987414
\(367\) −5.84299e82 −0.260009 −0.130004 0.991513i \(-0.541499\pi\)
−0.130004 + 0.991513i \(0.541499\pi\)
\(368\) 3.01913e83 1.22978
\(369\) 4.28660e83 1.59866
\(370\) 6.40893e83 2.18892
\(371\) 5.74274e82 0.179666
\(372\) 6.38941e82 0.183153
\(373\) 3.39596e82 0.0892124 0.0446062 0.999005i \(-0.485797\pi\)
0.0446062 + 0.999005i \(0.485797\pi\)
\(374\) −2.83308e83 −0.682232
\(375\) −4.51530e83 −0.996943
\(376\) −1.30975e83 −0.265205
\(377\) −3.43447e83 −0.637919
\(378\) −5.07018e83 −0.864052
\(379\) 8.27047e83 1.29347 0.646734 0.762715i \(-0.276134\pi\)
0.646734 + 0.762715i \(0.276134\pi\)
\(380\) 1.97565e83 0.283624
\(381\) −1.60647e84 −2.11744
\(382\) 9.96246e83 1.20589
\(383\) −1.10487e84 −1.22844 −0.614218 0.789136i \(-0.710528\pi\)
−0.614218 + 0.789136i \(0.710528\pi\)
\(384\) 1.54932e84 1.58262
\(385\) 6.85526e83 0.643502
\(386\) −1.90262e84 −1.64159
\(387\) 1.68710e84 1.33824
\(388\) −3.59561e82 −0.0262266
\(389\) 6.97054e83 0.467632 0.233816 0.972281i \(-0.424879\pi\)
0.233816 + 0.972281i \(0.424879\pi\)
\(390\) −1.87915e84 −1.15974
\(391\) −2.71352e84 −1.54095
\(392\) 1.44252e84 0.753918
\(393\) 4.20389e84 2.02252
\(394\) 4.35423e84 1.92879
\(395\) −3.26372e84 −1.33140
\(396\) 6.42254e83 0.241331
\(397\) 6.86328e83 0.237596 0.118798 0.992918i \(-0.462096\pi\)
0.118798 + 0.992918i \(0.462096\pi\)
\(398\) −4.72685e84 −1.50790
\(399\) −3.50118e84 −1.02943
\(400\) −2.10098e84 −0.569471
\(401\) 1.98012e84 0.494879 0.247439 0.968903i \(-0.420411\pi\)
0.247439 + 0.968903i \(0.420411\pi\)
\(402\) 4.50667e84 1.03875
\(403\) −5.55551e83 −0.118116
\(404\) 4.34822e83 0.0852937
\(405\) 4.01083e84 0.726016
\(406\) −1.38845e85 −2.31971
\(407\) 3.36382e84 0.518817
\(408\) −9.81166e84 −1.39729
\(409\) −1.24288e84 −0.163463 −0.0817316 0.996654i \(-0.526045\pi\)
−0.0817316 + 0.996654i \(0.526045\pi\)
\(410\) 1.50169e85 1.82433
\(411\) 4.09678e84 0.459815
\(412\) −2.03704e84 −0.211271
\(413\) 8.74897e84 0.838649
\(414\) 1.78957e85 1.58576
\(415\) −5.49442e83 −0.0450151
\(416\) 6.46101e84 0.489516
\(417\) −1.73075e85 −1.21286
\(418\) 3.01665e84 0.195567
\(419\) −2.85264e84 −0.171116 −0.0855580 0.996333i \(-0.527267\pi\)
−0.0855580 + 0.996333i \(0.527267\pi\)
\(420\) −2.61134e85 −1.44964
\(421\) 6.27821e83 0.0322602 0.0161301 0.999870i \(-0.494865\pi\)
0.0161301 + 0.999870i \(0.494865\pi\)
\(422\) −2.39998e85 −1.14170
\(423\) −1.33673e85 −0.588819
\(424\) −1.71362e84 −0.0699074
\(425\) 1.88831e85 0.713564
\(426\) −1.96184e85 −0.686832
\(427\) 2.45357e85 0.795964
\(428\) 1.30960e85 0.393747
\(429\) −9.86297e84 −0.274882
\(430\) 5.91027e85 1.52715
\(431\) 5.37189e85 1.28711 0.643555 0.765400i \(-0.277459\pi\)
0.643555 + 0.765400i \(0.277459\pi\)
\(432\) 2.60499e85 0.578874
\(433\) −5.31909e85 −1.09643 −0.548215 0.836338i \(-0.684692\pi\)
−0.548215 + 0.836338i \(0.684692\pi\)
\(434\) −2.24592e85 −0.429515
\(435\) −1.28400e86 −2.27858
\(436\) −4.71195e84 −0.0776050
\(437\) 2.88934e85 0.441725
\(438\) 1.90560e86 2.70473
\(439\) −1.02401e86 −1.34961 −0.674806 0.737995i \(-0.735773\pi\)
−0.674806 + 0.737995i \(0.735773\pi\)
\(440\) −2.04559e85 −0.250384
\(441\) 1.47224e86 1.67388
\(442\) −9.38342e85 −0.991145
\(443\) −3.98853e85 −0.391464 −0.195732 0.980657i \(-0.562708\pi\)
−0.195732 + 0.980657i \(0.562708\pi\)
\(444\) −1.28136e86 −1.16876
\(445\) 1.14185e86 0.968075
\(446\) 1.23027e86 0.969653
\(447\) −2.75798e86 −2.02114
\(448\) −1.57858e85 −0.107580
\(449\) −1.07852e86 −0.683634 −0.341817 0.939766i \(-0.611042\pi\)
−0.341817 + 0.939766i \(0.611042\pi\)
\(450\) −1.24535e86 −0.734316
\(451\) 7.88182e85 0.432403
\(452\) 6.89175e85 0.351828
\(453\) 4.99746e86 2.37442
\(454\) −2.48698e86 −1.09991
\(455\) 2.27052e86 0.934879
\(456\) 1.04474e86 0.400545
\(457\) 4.87620e86 1.74103 0.870513 0.492146i \(-0.163787\pi\)
0.870513 + 0.492146i \(0.163787\pi\)
\(458\) −5.67456e86 −1.88714
\(459\) −2.34130e86 −0.725347
\(460\) 2.15500e86 0.622039
\(461\) 4.13101e86 1.11116 0.555579 0.831464i \(-0.312497\pi\)
0.555579 + 0.831464i \(0.312497\pi\)
\(462\) −3.98730e86 −0.999572
\(463\) −6.31593e86 −1.47589 −0.737947 0.674859i \(-0.764204\pi\)
−0.737947 + 0.674859i \(0.764204\pi\)
\(464\) 7.13366e86 1.55410
\(465\) −2.07696e86 −0.421898
\(466\) 5.88694e84 0.0111519
\(467\) −1.06959e87 −1.88981 −0.944904 0.327348i \(-0.893845\pi\)
−0.944904 + 0.327348i \(0.893845\pi\)
\(468\) 2.12720e86 0.350605
\(469\) −5.44529e86 −0.837343
\(470\) −4.68286e86 −0.671940
\(471\) 9.80443e86 1.31293
\(472\) −2.61067e86 −0.326314
\(473\) 3.10209e86 0.361965
\(474\) 1.89831e87 2.06810
\(475\) −2.01066e86 −0.204549
\(476\) −1.30396e87 −1.23890
\(477\) −1.74892e86 −0.155211
\(478\) 2.43290e87 2.01705
\(479\) −7.36945e85 −0.0570862 −0.0285431 0.999593i \(-0.509087\pi\)
−0.0285431 + 0.999593i \(0.509087\pi\)
\(480\) 2.41549e87 1.74850
\(481\) 1.11413e87 0.753736
\(482\) −3.26903e87 −2.06724
\(483\) −3.81902e87 −2.25772
\(484\) −8.29519e86 −0.458513
\(485\) 1.16880e86 0.0604135
\(486\) −3.51594e87 −1.69966
\(487\) 3.08739e87 1.39605 0.698024 0.716074i \(-0.254063\pi\)
0.698024 + 0.716074i \(0.254063\pi\)
\(488\) −7.32139e86 −0.309706
\(489\) 4.32701e87 1.71258
\(490\) 5.15757e87 1.91017
\(491\) −7.58213e86 −0.262810 −0.131405 0.991329i \(-0.541949\pi\)
−0.131405 + 0.991329i \(0.541949\pi\)
\(492\) −3.00238e87 −0.974090
\(493\) −6.41157e87 −1.94733
\(494\) 9.99141e86 0.284120
\(495\) −2.08773e87 −0.555912
\(496\) 1.15392e87 0.287755
\(497\) 2.37043e87 0.553662
\(498\) 3.19578e86 0.0699234
\(499\) −7.04288e87 −1.44372 −0.721859 0.692040i \(-0.756712\pi\)
−0.721859 + 0.692040i \(0.756712\pi\)
\(500\) 1.79061e87 0.343935
\(501\) 1.43439e88 2.58190
\(502\) 8.09571e87 1.36579
\(503\) −5.74552e87 −0.908594 −0.454297 0.890850i \(-0.650110\pi\)
−0.454297 + 0.890850i \(0.650110\pi\)
\(504\) −7.81850e87 −1.15913
\(505\) −1.41345e87 −0.196476
\(506\) 3.29051e87 0.428915
\(507\) 9.15293e87 1.11893
\(508\) 6.37069e87 0.730493
\(509\) 1.76018e88 1.89334 0.946671 0.322202i \(-0.104423\pi\)
0.946671 + 0.322202i \(0.104423\pi\)
\(510\) −3.50805e88 −3.54026
\(511\) −2.30249e88 −2.18031
\(512\) −5.15491e87 −0.458086
\(513\) 2.49301e87 0.207926
\(514\) −6.53237e87 −0.511410
\(515\) 6.62167e87 0.486668
\(516\) −1.18166e88 −0.815413
\(517\) −2.45786e87 −0.159263
\(518\) 4.50407e88 2.74087
\(519\) 1.19314e88 0.681948
\(520\) −6.77517e87 −0.363757
\(521\) −1.77119e88 −0.893386 −0.446693 0.894687i \(-0.647398\pi\)
−0.446693 + 0.894687i \(0.647398\pi\)
\(522\) 4.22845e88 2.00396
\(523\) −1.01782e88 −0.453281 −0.226640 0.973978i \(-0.572774\pi\)
−0.226640 + 0.973978i \(0.572774\pi\)
\(524\) −1.66712e88 −0.697749
\(525\) 2.65762e88 1.04548
\(526\) −1.60267e88 −0.592663
\(527\) −1.03712e88 −0.360565
\(528\) 2.04862e88 0.669667
\(529\) −1.01548e87 −0.0312148
\(530\) −6.12685e87 −0.177121
\(531\) −2.66445e88 −0.724496
\(532\) 1.38844e88 0.355141
\(533\) 2.61053e88 0.628194
\(534\) −6.64146e88 −1.50374
\(535\) −4.25704e88 −0.907005
\(536\) 1.62486e88 0.325806
\(537\) −7.46315e88 −1.40850
\(538\) 5.82161e88 1.03423
\(539\) 2.70702e88 0.452748
\(540\) 1.85940e88 0.292803
\(541\) 2.29885e88 0.340879 0.170440 0.985368i \(-0.445481\pi\)
0.170440 + 0.985368i \(0.445481\pi\)
\(542\) 2.06443e88 0.288286
\(543\) −6.44667e88 −0.847893
\(544\) 1.20616e89 1.49431
\(545\) 1.53168e88 0.178765
\(546\) −1.32063e89 −1.45218
\(547\) 1.50750e89 1.56195 0.780976 0.624561i \(-0.214722\pi\)
0.780976 + 0.624561i \(0.214722\pi\)
\(548\) −1.62464e88 −0.158631
\(549\) −7.47223e88 −0.687621
\(550\) −2.28983e88 −0.198617
\(551\) 6.82700e88 0.558217
\(552\) 1.13958e89 0.878469
\(553\) −2.29368e89 −1.66712
\(554\) −2.56168e89 −1.75573
\(555\) 4.16523e89 2.69226
\(556\) 6.86355e88 0.418425
\(557\) 1.01681e89 0.584715 0.292357 0.956309i \(-0.405560\pi\)
0.292357 + 0.956309i \(0.405560\pi\)
\(558\) 6.83983e88 0.371051
\(559\) 1.02744e89 0.525862
\(560\) −4.71606e89 −2.27755
\(561\) −1.84125e89 −0.839112
\(562\) −2.19418e89 −0.943717
\(563\) −1.75906e89 −0.714097 −0.357048 0.934086i \(-0.616217\pi\)
−0.357048 + 0.934086i \(0.616217\pi\)
\(564\) 9.36260e88 0.358778
\(565\) −2.24026e89 −0.810444
\(566\) 1.11108e89 0.379498
\(567\) 2.81873e89 0.909085
\(568\) −7.07329e88 −0.215427
\(569\) 4.73652e89 1.36242 0.681209 0.732089i \(-0.261454\pi\)
0.681209 + 0.732089i \(0.261454\pi\)
\(570\) 3.73536e89 1.01484
\(571\) −3.10204e89 −0.796113 −0.398056 0.917361i \(-0.630315\pi\)
−0.398056 + 0.917361i \(0.630315\pi\)
\(572\) 3.91131e88 0.0948312
\(573\) 6.47471e89 1.48318
\(574\) 1.05536e90 2.28435
\(575\) −2.19319e89 −0.448613
\(576\) 4.80747e88 0.0929363
\(577\) −8.08841e89 −1.47791 −0.738957 0.673752i \(-0.764681\pi\)
−0.738957 + 0.673752i \(0.764681\pi\)
\(578\) −1.03704e90 −1.79118
\(579\) −1.23653e90 −2.01908
\(580\) 5.09189e89 0.786084
\(581\) −3.86137e88 −0.0563659
\(582\) −6.79822e88 −0.0938422
\(583\) −3.21576e88 −0.0419813
\(584\) 6.87054e89 0.848347
\(585\) −6.91475e89 −0.807627
\(586\) 1.03456e90 1.14310
\(587\) 2.04373e89 0.213641 0.106820 0.994278i \(-0.465933\pi\)
0.106820 + 0.994278i \(0.465933\pi\)
\(588\) −1.03117e90 −1.01992
\(589\) 1.10432e89 0.103359
\(590\) −9.33415e89 −0.826770
\(591\) 2.82986e90 2.37231
\(592\) −2.31413e90 −1.83625
\(593\) −3.01234e89 −0.226270 −0.113135 0.993580i \(-0.536089\pi\)
−0.113135 + 0.993580i \(0.536089\pi\)
\(594\) 2.83915e89 0.201896
\(595\) 4.23868e90 2.85384
\(596\) 1.09372e90 0.697271
\(597\) −3.07203e90 −1.85464
\(598\) 1.08985e90 0.623127
\(599\) −2.20682e90 −1.19507 −0.597537 0.801841i \(-0.703854\pi\)
−0.597537 + 0.801841i \(0.703854\pi\)
\(600\) −7.93025e89 −0.406791
\(601\) −2.25821e90 −1.09735 −0.548673 0.836037i \(-0.684867\pi\)
−0.548673 + 0.836037i \(0.684867\pi\)
\(602\) 4.15362e90 1.91223
\(603\) 1.65833e90 0.723368
\(604\) −1.98182e90 −0.819151
\(605\) 2.69646e90 1.05620
\(606\) 8.22118e89 0.305192
\(607\) −1.94001e89 −0.0682610 −0.0341305 0.999417i \(-0.510866\pi\)
−0.0341305 + 0.999417i \(0.510866\pi\)
\(608\) −1.28431e90 −0.428356
\(609\) −9.02368e90 −2.85313
\(610\) −2.61768e90 −0.784689
\(611\) −8.14066e89 −0.231377
\(612\) 3.97113e90 1.07027
\(613\) 6.13649e90 1.56839 0.784196 0.620514i \(-0.213076\pi\)
0.784196 + 0.620514i \(0.213076\pi\)
\(614\) −6.24522e90 −1.51383
\(615\) 9.75963e90 2.24384
\(616\) −1.43760e90 −0.313519
\(617\) −5.08036e89 −0.105106 −0.0525529 0.998618i \(-0.516736\pi\)
−0.0525529 + 0.998618i \(0.516736\pi\)
\(618\) −3.85143e90 −0.755956
\(619\) −5.67982e90 −1.05776 −0.528882 0.848696i \(-0.677389\pi\)
−0.528882 + 0.848696i \(0.677389\pi\)
\(620\) 8.23651e89 0.145550
\(621\) 2.71933e90 0.456021
\(622\) 1.51627e89 0.0241318
\(623\) 8.02469e90 1.21218
\(624\) 6.78521e90 0.972890
\(625\) −9.16919e90 −1.24804
\(626\) 1.54208e91 1.99270
\(627\) 1.96055e90 0.240538
\(628\) −3.88810e90 −0.452948
\(629\) 2.07989e91 2.30088
\(630\) −2.79542e91 −2.93683
\(631\) −7.67702e90 −0.766017 −0.383009 0.923745i \(-0.625112\pi\)
−0.383009 + 0.923745i \(0.625112\pi\)
\(632\) 6.84427e90 0.648667
\(633\) −1.55977e91 −1.40424
\(634\) −1.04715e91 −0.895591
\(635\) −2.07088e91 −1.68271
\(636\) 1.22496e90 0.0945728
\(637\) 8.96589e90 0.657752
\(638\) 7.77490e90 0.542029
\(639\) −7.21902e90 −0.478300
\(640\) 1.99721e91 1.25769
\(641\) −2.19458e91 −1.31361 −0.656804 0.754061i \(-0.728092\pi\)
−0.656804 + 0.754061i \(0.728092\pi\)
\(642\) 2.47606e91 1.40888
\(643\) 7.55485e90 0.408667 0.204333 0.978901i \(-0.434497\pi\)
0.204333 + 0.978901i \(0.434497\pi\)
\(644\) 1.51449e91 0.778890
\(645\) 3.84115e91 1.87832
\(646\) 1.86523e91 0.867312
\(647\) −3.77147e91 −1.66771 −0.833856 0.551982i \(-0.813872\pi\)
−0.833856 + 0.551982i \(0.813872\pi\)
\(648\) −8.41101e90 −0.353721
\(649\) −4.89916e90 −0.195961
\(650\) −7.58412e90 −0.288550
\(651\) −1.45965e91 −0.528282
\(652\) −1.71594e91 −0.590820
\(653\) −8.24655e90 −0.270143 −0.135072 0.990836i \(-0.543126\pi\)
−0.135072 + 0.990836i \(0.543126\pi\)
\(654\) −8.90888e90 −0.277681
\(655\) 5.41919e91 1.60728
\(656\) −5.42228e91 −1.53041
\(657\) 7.01210e91 1.88353
\(658\) −3.29102e91 −0.841373
\(659\) −1.36580e91 −0.332361 −0.166180 0.986095i \(-0.553143\pi\)
−0.166180 + 0.986095i \(0.553143\pi\)
\(660\) 1.46227e91 0.338727
\(661\) −5.25207e90 −0.115820 −0.0579099 0.998322i \(-0.518444\pi\)
−0.0579099 + 0.998322i \(0.518444\pi\)
\(662\) 6.73385e91 1.41377
\(663\) −6.09838e91 −1.21906
\(664\) 1.15222e90 0.0219317
\(665\) −4.51333e91 −0.818075
\(666\) −1.37169e92 −2.36779
\(667\) 7.44677e91 1.22427
\(668\) −5.68827e91 −0.890729
\(669\) 7.99563e91 1.19263
\(670\) 5.80950e91 0.825482
\(671\) −1.37393e91 −0.185987
\(672\) 1.69756e92 2.18939
\(673\) −2.81696e91 −0.346171 −0.173086 0.984907i \(-0.555374\pi\)
−0.173086 + 0.984907i \(0.555374\pi\)
\(674\) −8.23293e91 −0.964068
\(675\) −1.89235e91 −0.211169
\(676\) −3.62973e91 −0.386018
\(677\) 1.70699e92 1.73021 0.865106 0.501589i \(-0.167251\pi\)
0.865106 + 0.501589i \(0.167251\pi\)
\(678\) 1.30302e92 1.25889
\(679\) 8.21410e90 0.0756471
\(680\) −1.26481e92 −1.11042
\(681\) −1.61632e92 −1.35284
\(682\) 1.25765e91 0.100361
\(683\) 5.51499e91 0.419636 0.209818 0.977741i \(-0.432713\pi\)
0.209818 + 0.977741i \(0.432713\pi\)
\(684\) −4.22843e91 −0.306801
\(685\) 5.28111e91 0.365411
\(686\) 7.98435e91 0.526873
\(687\) −3.68796e92 −2.32109
\(688\) −2.13407e92 −1.28111
\(689\) −1.06509e91 −0.0609903
\(690\) 4.07446e92 2.22574
\(691\) 3.73079e91 0.194430 0.0972152 0.995263i \(-0.469006\pi\)
0.0972152 + 0.995263i \(0.469006\pi\)
\(692\) −4.73157e91 −0.235265
\(693\) −1.46722e92 −0.696088
\(694\) −3.89267e92 −1.76224
\(695\) −2.23109e92 −0.963853
\(696\) 2.69264e92 1.11014
\(697\) 4.87341e92 1.91764
\(698\) 5.67115e92 2.12995
\(699\) 3.82599e90 0.0137162
\(700\) −1.05392e92 −0.360679
\(701\) 2.66904e92 0.872005 0.436003 0.899945i \(-0.356394\pi\)
0.436003 + 0.899945i \(0.356394\pi\)
\(702\) 9.40351e91 0.293315
\(703\) −2.21465e92 −0.659564
\(704\) 8.83955e90 0.0251373
\(705\) −3.04344e92 −0.826453
\(706\) −3.88693e91 −0.100799
\(707\) −9.93342e91 −0.246019
\(708\) 1.86621e92 0.441448
\(709\) −3.84583e92 −0.868936 −0.434468 0.900687i \(-0.643064\pi\)
−0.434468 + 0.900687i \(0.643064\pi\)
\(710\) −2.52898e92 −0.545819
\(711\) 6.98528e92 1.44020
\(712\) −2.39454e92 −0.471653
\(713\) 1.20457e92 0.226685
\(714\) −2.46539e93 −4.43296
\(715\) −1.27142e92 −0.218446
\(716\) 2.95963e92 0.485919
\(717\) 1.58117e93 2.48088
\(718\) 4.50804e92 0.675995
\(719\) −2.82885e92 −0.405434 −0.202717 0.979237i \(-0.564977\pi\)
−0.202717 + 0.979237i \(0.564977\pi\)
\(720\) 1.43625e93 1.96754
\(721\) 4.65358e92 0.609383
\(722\) 7.87492e92 0.985796
\(723\) −2.12458e93 −2.54260
\(724\) 2.55653e92 0.292514
\(725\) −5.18213e92 −0.566922
\(726\) −1.56837e93 −1.64062
\(727\) 5.75312e92 0.575487 0.287743 0.957708i \(-0.407095\pi\)
0.287743 + 0.957708i \(0.407095\pi\)
\(728\) −4.76145e92 −0.455480
\(729\) −1.62732e93 −1.48877
\(730\) 2.45649e93 2.14942
\(731\) 1.91805e93 1.60526
\(732\) 5.23363e92 0.418979
\(733\) −4.37341e92 −0.334920 −0.167460 0.985879i \(-0.553556\pi\)
−0.167460 + 0.985879i \(0.553556\pi\)
\(734\) 4.38099e92 0.320959
\(735\) 3.35196e93 2.34942
\(736\) −1.40091e93 −0.939465
\(737\) 3.04920e92 0.195656
\(738\) −3.21403e93 −1.97341
\(739\) −1.48307e93 −0.871399 −0.435699 0.900092i \(-0.643499\pi\)
−0.435699 + 0.900092i \(0.643499\pi\)
\(740\) −1.65179e93 −0.928802
\(741\) 6.49353e92 0.349453
\(742\) −4.30582e92 −0.221784
\(743\) 7.24601e92 0.357242 0.178621 0.983918i \(-0.442836\pi\)
0.178621 + 0.983918i \(0.442836\pi\)
\(744\) 4.35554e92 0.205552
\(745\) −3.55528e93 −1.60618
\(746\) −2.54624e92 −0.110125
\(747\) 1.17596e92 0.0486937
\(748\) 7.30176e92 0.289485
\(749\) −2.99176e93 −1.13571
\(750\) 3.38551e93 1.23064
\(751\) −4.54486e92 −0.158206 −0.0791030 0.996866i \(-0.525206\pi\)
−0.0791030 + 0.996866i \(0.525206\pi\)
\(752\) 1.69088e93 0.563681
\(753\) 5.26149e93 1.67985
\(754\) 2.57511e93 0.787458
\(755\) 6.44217e93 1.88693
\(756\) 1.30675e93 0.366634
\(757\) −7.28700e93 −1.95854 −0.979270 0.202559i \(-0.935074\pi\)
−0.979270 + 0.202559i \(0.935074\pi\)
\(758\) −6.20108e93 −1.59668
\(759\) 2.13854e93 0.527545
\(760\) 1.34676e93 0.318309
\(761\) −7.53622e93 −1.70668 −0.853341 0.521354i \(-0.825427\pi\)
−0.853341 + 0.521354i \(0.825427\pi\)
\(762\) 1.20451e94 2.61380
\(763\) 1.07644e93 0.223841
\(764\) −2.56764e93 −0.511683
\(765\) −1.29087e94 −2.46539
\(766\) 8.28419e93 1.51640
\(767\) −1.62265e93 −0.284691
\(768\) −1.09737e94 −1.84550
\(769\) 1.01482e94 1.63600 0.818001 0.575217i \(-0.195082\pi\)
0.818001 + 0.575217i \(0.195082\pi\)
\(770\) −5.13998e93 −0.794351
\(771\) −4.24546e93 −0.629010
\(772\) 4.90366e93 0.696560
\(773\) −1.13392e93 −0.154436 −0.0772179 0.997014i \(-0.524604\pi\)
−0.0772179 + 0.997014i \(0.524604\pi\)
\(774\) −1.26496e94 −1.65195
\(775\) −8.38249e92 −0.104971
\(776\) −2.45106e92 −0.0294339
\(777\) 2.92724e94 3.37113
\(778\) −5.22641e93 −0.577254
\(779\) −5.18919e93 −0.549708
\(780\) 4.84316e93 0.492101
\(781\) −1.32737e93 −0.129370
\(782\) 2.03456e94 1.90218
\(783\) 6.42529e93 0.576283
\(784\) −1.86229e94 −1.60241
\(785\) 1.26388e94 1.04338
\(786\) −3.15202e94 −2.49664
\(787\) 2.00701e94 1.52535 0.762676 0.646781i \(-0.223885\pi\)
0.762676 + 0.646781i \(0.223885\pi\)
\(788\) −1.12222e94 −0.818423
\(789\) −1.04159e94 −0.728947
\(790\) 2.44709e94 1.64350
\(791\) −1.57441e94 −1.01480
\(792\) 4.37813e93 0.270844
\(793\) −4.55057e93 −0.270201
\(794\) −5.14599e93 −0.293293
\(795\) −3.98190e93 −0.217851
\(796\) 1.21826e94 0.639832
\(797\) −4.77501e93 −0.240757 −0.120378 0.992728i \(-0.538411\pi\)
−0.120378 + 0.992728i \(0.538411\pi\)
\(798\) 2.62513e94 1.27074
\(799\) −1.51972e94 −0.706308
\(800\) 9.74877e93 0.435036
\(801\) −2.44388e94 −1.04718
\(802\) −1.48466e94 −0.610887
\(803\) 1.28932e94 0.509456
\(804\) −1.16151e94 −0.440761
\(805\) −4.92306e94 −1.79419
\(806\) 4.16544e93 0.145805
\(807\) 3.78352e94 1.27206
\(808\) 2.96410e93 0.0957246
\(809\) 5.87211e93 0.182166 0.0910829 0.995843i \(-0.470967\pi\)
0.0910829 + 0.995843i \(0.470967\pi\)
\(810\) −3.00727e94 −0.896208
\(811\) 1.43057e94 0.409573 0.204786 0.978807i \(-0.434350\pi\)
0.204786 + 0.978807i \(0.434350\pi\)
\(812\) 3.57848e94 0.984299
\(813\) 1.34170e94 0.354578
\(814\) −2.52214e94 −0.640437
\(815\) 5.57790e94 1.36097
\(816\) 1.26668e95 2.96987
\(817\) −2.04233e94 −0.460161
\(818\) 9.31890e93 0.201782
\(819\) −4.85955e94 −1.01127
\(820\) −3.87033e94 −0.774101
\(821\) 8.34931e93 0.160508 0.0802542 0.996774i \(-0.474427\pi\)
0.0802542 + 0.996774i \(0.474427\pi\)
\(822\) −3.07170e94 −0.567604
\(823\) −7.51066e94 −1.33408 −0.667042 0.745020i \(-0.732440\pi\)
−0.667042 + 0.745020i \(0.732440\pi\)
\(824\) −1.38861e94 −0.237108
\(825\) −1.48819e94 −0.244289
\(826\) −6.55985e94 −1.03524
\(827\) −3.46296e94 −0.525434 −0.262717 0.964873i \(-0.584618\pi\)
−0.262717 + 0.964873i \(0.584618\pi\)
\(828\) −4.61230e94 −0.672871
\(829\) 8.40749e94 1.17935 0.589677 0.807639i \(-0.299255\pi\)
0.589677 + 0.807639i \(0.299255\pi\)
\(830\) 4.11964e93 0.0555675
\(831\) −1.66487e95 −2.15946
\(832\) 2.92774e93 0.0365194
\(833\) 1.67378e95 2.00787
\(834\) 1.29769e95 1.49718
\(835\) 1.84905e95 2.05181
\(836\) −7.77487e93 −0.0829830
\(837\) 1.03934e94 0.106704
\(838\) 2.13886e94 0.211229
\(839\) −1.27529e95 −1.21156 −0.605780 0.795632i \(-0.707139\pi\)
−0.605780 + 0.795632i \(0.707139\pi\)
\(840\) −1.78010e95 −1.62692
\(841\) 6.22255e94 0.547140
\(842\) −4.70731e93 −0.0398226
\(843\) −1.42602e95 −1.16073
\(844\) 6.18551e94 0.484447
\(845\) 1.17989e95 0.889201
\(846\) 1.00226e95 0.726849
\(847\) 1.89502e95 1.32252
\(848\) 2.21227e94 0.148585
\(849\) 7.22100e94 0.466764
\(850\) −1.41583e95 −0.880837
\(851\) −2.41570e95 −1.44655
\(852\) 5.05628e94 0.291437
\(853\) −2.73020e95 −1.51479 −0.757394 0.652959i \(-0.773528\pi\)
−0.757394 + 0.652959i \(0.773528\pi\)
\(854\) −1.83966e95 −0.982552
\(855\) 1.37451e95 0.706723
\(856\) 8.92732e94 0.441900
\(857\) −2.34418e95 −1.11716 −0.558578 0.829452i \(-0.688653\pi\)
−0.558578 + 0.829452i \(0.688653\pi\)
\(858\) 7.39512e94 0.339319
\(859\) 3.16987e95 1.40044 0.700218 0.713930i \(-0.253086\pi\)
0.700218 + 0.713930i \(0.253086\pi\)
\(860\) −1.52326e95 −0.648001
\(861\) 6.85887e95 2.80964
\(862\) −4.02777e95 −1.58883
\(863\) −4.68316e95 −1.77905 −0.889525 0.456886i \(-0.848965\pi\)
−0.889525 + 0.456886i \(0.848965\pi\)
\(864\) −1.20874e95 −0.442219
\(865\) 1.53806e95 0.541938
\(866\) 3.98818e95 1.35345
\(867\) −6.73982e95 −2.20307
\(868\) 5.78845e94 0.182252
\(869\) 1.28439e95 0.389542
\(870\) 9.62723e95 2.81271
\(871\) 1.00992e95 0.284248
\(872\) −3.21205e94 −0.0870956
\(873\) −2.50156e94 −0.0653503
\(874\) −2.16639e95 −0.545274
\(875\) −4.09061e95 −0.992034
\(876\) −4.91134e95 −1.14767
\(877\) −1.75364e94 −0.0394870 −0.0197435 0.999805i \(-0.506285\pi\)
−0.0197435 + 0.999805i \(0.506285\pi\)
\(878\) 7.67791e95 1.66599
\(879\) 6.72371e95 1.40595
\(880\) 2.64085e95 0.532178
\(881\) −7.68400e94 −0.149235 −0.0746174 0.997212i \(-0.523774\pi\)
−0.0746174 + 0.997212i \(0.523774\pi\)
\(882\) −1.10386e96 −2.06626
\(883\) 6.26629e95 1.13054 0.565272 0.824904i \(-0.308771\pi\)
0.565272 + 0.824904i \(0.308771\pi\)
\(884\) 2.41841e95 0.420563
\(885\) −6.06637e95 −1.01689
\(886\) 2.99055e95 0.483231
\(887\) 5.78744e94 0.0901507 0.0450753 0.998984i \(-0.485647\pi\)
0.0450753 + 0.998984i \(0.485647\pi\)
\(888\) −8.73480e95 −1.31169
\(889\) −1.45537e96 −2.10701
\(890\) −8.56143e95 −1.19501
\(891\) −1.57841e95 −0.212419
\(892\) −3.17079e95 −0.411443
\(893\) 1.61819e95 0.202469
\(894\) 2.06789e96 2.49493
\(895\) −9.62066e95 −1.11933
\(896\) 1.40360e96 1.57482
\(897\) 7.08302e95 0.766415
\(898\) 8.08662e95 0.843891
\(899\) 2.84619e95 0.286467
\(900\) 3.20965e95 0.311585
\(901\) −1.98834e95 −0.186181
\(902\) −5.90967e95 −0.533766
\(903\) 2.69948e96 2.35195
\(904\) 4.69798e95 0.394855
\(905\) −8.31033e95 −0.673813
\(906\) −3.74703e96 −2.93103
\(907\) 1.32052e96 0.996572 0.498286 0.867013i \(-0.333963\pi\)
0.498286 + 0.867013i \(0.333963\pi\)
\(908\) 6.40975e95 0.466714
\(909\) 3.02517e95 0.212532
\(910\) −1.70241e96 −1.15403
\(911\) 1.65330e95 0.108144 0.0540721 0.998537i \(-0.482780\pi\)
0.0540721 + 0.998537i \(0.482780\pi\)
\(912\) −1.34876e96 −0.851338
\(913\) 2.16225e94 0.0131706
\(914\) −3.65611e96 −2.14915
\(915\) −1.70126e96 −0.965129
\(916\) 1.46251e96 0.800751
\(917\) 3.80849e96 2.01257
\(918\) 1.75548e96 0.895381
\(919\) 2.04368e96 1.00614 0.503069 0.864246i \(-0.332204\pi\)
0.503069 + 0.864246i \(0.332204\pi\)
\(920\) 1.46902e96 0.698111
\(921\) −4.05884e96 −1.86193
\(922\) −3.09737e96 −1.37163
\(923\) −4.39637e95 −0.187948
\(924\) 1.02765e96 0.424138
\(925\) 1.68106e96 0.669850
\(926\) 4.73559e96 1.82187
\(927\) −1.41722e96 −0.526437
\(928\) −3.31010e96 −1.18722
\(929\) −2.46989e96 −0.855396 −0.427698 0.903922i \(-0.640675\pi\)
−0.427698 + 0.903922i \(0.640675\pi\)
\(930\) 1.55728e96 0.520799
\(931\) −1.78223e96 −0.575572
\(932\) −1.51725e94 −0.00473196
\(933\) 9.85441e94 0.0296809
\(934\) 8.01961e96 2.33281
\(935\) −2.37354e96 −0.666835
\(936\) 1.45007e96 0.393482
\(937\) −4.67372e96 −1.22497 −0.612486 0.790481i \(-0.709831\pi\)
−0.612486 + 0.790481i \(0.709831\pi\)
\(938\) 4.08280e96 1.03363
\(939\) 1.00221e97 2.45092
\(940\) 1.20692e96 0.285117
\(941\) −2.71979e96 −0.620688 −0.310344 0.950624i \(-0.600444\pi\)
−0.310344 + 0.950624i \(0.600444\pi\)
\(942\) −7.35122e96 −1.62071
\(943\) −5.66027e96 −1.20561
\(944\) 3.37037e96 0.693565
\(945\) −4.24776e96 −0.844551
\(946\) −2.32590e96 −0.446817
\(947\) 1.54092e96 0.286026 0.143013 0.989721i \(-0.454321\pi\)
0.143013 + 0.989721i \(0.454321\pi\)
\(948\) −4.89256e96 −0.877537
\(949\) 4.27035e96 0.740136
\(950\) 1.50757e96 0.252499
\(951\) −6.80557e96 −1.10153
\(952\) −8.88883e96 −1.39041
\(953\) 7.16977e96 1.08389 0.541946 0.840413i \(-0.317688\pi\)
0.541946 + 0.840413i \(0.317688\pi\)
\(954\) 1.31132e96 0.191595
\(955\) 8.34647e96 1.17867
\(956\) −6.27036e96 −0.855876
\(957\) 5.05299e96 0.666669
\(958\) 5.52551e95 0.0704682
\(959\) 3.71146e96 0.457551
\(960\) 1.09455e96 0.130443
\(961\) −8.21941e96 −0.946958
\(962\) −8.35356e96 −0.930425
\(963\) 9.11124e96 0.981123
\(964\) 8.42534e96 0.877170
\(965\) −1.59400e97 −1.60454
\(966\) 2.86345e97 2.78697
\(967\) 1.00835e97 0.948962 0.474481 0.880266i \(-0.342636\pi\)
0.474481 + 0.880266i \(0.342636\pi\)
\(968\) −5.65468e96 −0.514587
\(969\) 1.21223e97 1.06675
\(970\) −8.76350e95 −0.0745755
\(971\) −3.00015e96 −0.246898 −0.123449 0.992351i \(-0.539396\pi\)
−0.123449 + 0.992351i \(0.539396\pi\)
\(972\) 9.06169e96 0.721200
\(973\) −1.56796e97 −1.20689
\(974\) −2.31488e97 −1.72331
\(975\) −4.92900e96 −0.354903
\(976\) 9.45190e96 0.658264
\(977\) −7.90349e96 −0.532409 −0.266205 0.963917i \(-0.585770\pi\)
−0.266205 + 0.963917i \(0.585770\pi\)
\(978\) −3.24433e97 −2.11403
\(979\) −4.49359e96 −0.283241
\(980\) −1.32927e97 −0.810523
\(981\) −3.27823e96 −0.193373
\(982\) 5.68497e96 0.324418
\(983\) 2.35570e97 1.30056 0.650281 0.759694i \(-0.274651\pi\)
0.650281 + 0.759694i \(0.274651\pi\)
\(984\) −2.04666e97 −1.09322
\(985\) 3.64794e97 1.88526
\(986\) 4.80730e97 2.40382
\(987\) −2.13887e97 −1.03485
\(988\) −2.57511e96 −0.120558
\(989\) −2.22774e97 −1.00922
\(990\) 1.56535e97 0.686228
\(991\) 8.80715e96 0.373630 0.186815 0.982395i \(-0.440184\pi\)
0.186815 + 0.982395i \(0.440184\pi\)
\(992\) −5.35433e96 −0.219824
\(993\) 4.37640e97 1.73887
\(994\) −1.77732e97 −0.683451
\(995\) −3.96012e97 −1.47387
\(996\) −8.23653e95 −0.0296699
\(997\) 2.37020e97 0.826402 0.413201 0.910640i \(-0.364411\pi\)
0.413201 + 0.910640i \(0.364411\pi\)
\(998\) 5.28065e97 1.78215
\(999\) −2.08434e97 −0.680910
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 1.66.a.a.1.2 5
3.2 odd 2 9.66.a.b.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.66.a.a.1.2 5 1.1 even 1 trivial
9.66.a.b.1.4 5 3.2 odd 2