Properties

Label 21.30.a.d.1.5
Level $21$
Weight $30$
Character 21.1
Self dual yes
Analytic conductor $111.884$
Analytic rank $0$
Dimension $8$
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] = [21,30,Mod(1,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.1");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(111.883889004\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3 x^{7} - 2857317313 x^{6} - 1405020216555 x^{5} + \cdots - 19\!\cdots\!94 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{16}\cdot 5^{2}\cdot 7^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-239.721\) of defining polynomial
Character \(\chi\) \(=\) 21.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2672.72 q^{2} +4.78297e6 q^{3} -5.29727e8 q^{4} +1.94732e10 q^{5} -1.27835e10 q^{6} -6.78223e11 q^{7} +2.85072e12 q^{8} +2.28768e13 q^{9} +O(q^{10})\) \(q-2672.72 q^{2} +4.78297e6 q^{3} -5.29727e8 q^{4} +1.94732e10 q^{5} -1.27835e10 q^{6} -6.78223e11 q^{7} +2.85072e12 q^{8} +2.28768e13 q^{9} -5.20466e13 q^{10} -2.34539e15 q^{11} -2.53367e15 q^{12} -1.06675e16 q^{13} +1.81270e15 q^{14} +9.31399e16 q^{15} +2.76776e17 q^{16} -1.80413e17 q^{17} -6.11433e16 q^{18} +3.63408e18 q^{19} -1.03155e19 q^{20} -3.24392e18 q^{21} +6.26858e18 q^{22} +1.42481e19 q^{23} +1.36349e19 q^{24} +1.92943e20 q^{25} +2.85113e19 q^{26} +1.09419e20 q^{27} +3.59273e20 q^{28} +1.38009e21 q^{29} -2.48937e20 q^{30} -1.39623e21 q^{31} -2.27021e21 q^{32} -1.12179e22 q^{33} +4.82194e20 q^{34} -1.32072e22 q^{35} -1.21185e22 q^{36} +1.35650e22 q^{37} -9.71289e21 q^{38} -5.10223e22 q^{39} +5.55128e22 q^{40} -4.23543e23 q^{41} +8.67009e21 q^{42} +3.71448e23 q^{43} +1.24242e24 q^{44} +4.45485e23 q^{45} -3.80811e22 q^{46} +2.04365e24 q^{47} +1.32381e24 q^{48} +4.59987e23 q^{49} -5.15682e23 q^{50} -8.62910e23 q^{51} +5.65087e24 q^{52} +1.15964e25 q^{53} -2.92446e23 q^{54} -4.56724e25 q^{55} -1.93342e24 q^{56} +1.73817e25 q^{57} -3.68861e24 q^{58} -2.23571e25 q^{59} -4.93388e25 q^{60} -4.97521e25 q^{61} +3.73174e24 q^{62} -1.55156e25 q^{63} -1.42525e26 q^{64} -2.07731e26 q^{65} +2.99824e25 q^{66} +5.48772e26 q^{67} +9.55698e25 q^{68} +6.81480e25 q^{69} +3.52992e25 q^{70} +1.06838e27 q^{71} +6.52153e25 q^{72} -1.06623e27 q^{73} -3.62554e25 q^{74} +9.22839e26 q^{75} -1.92507e27 q^{76} +1.59070e27 q^{77} +1.36368e26 q^{78} -5.67748e27 q^{79} +5.38973e27 q^{80} +5.23348e26 q^{81} +1.13201e27 q^{82} -5.28365e27 q^{83} +1.71839e27 q^{84} -3.51323e27 q^{85} -9.92776e26 q^{86} +6.60095e27 q^{87} -6.68605e27 q^{88} +2.36536e28 q^{89} -1.19066e27 q^{90} +7.23494e27 q^{91} -7.54759e27 q^{92} -6.67813e27 q^{93} -5.46211e27 q^{94} +7.07674e28 q^{95} -1.08584e28 q^{96} +2.86997e28 q^{97} -1.22942e27 q^{98} -5.36550e28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 19461 q^{2} + 38263752 q^{3} + 1467008653 q^{4} - 8498112672 q^{5} - 93081359709 q^{6} - 5425784582792 q^{7} - 16689415716987 q^{8} + 183014339639688 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 19461 q^{2} + 38263752 q^{3} + 1467008653 q^{4} - 8498112672 q^{5} - 93081359709 q^{6} - 5425784582792 q^{7} - 16689415716987 q^{8} + 183014339639688 q^{9} + 636942348029770 q^{10} + 395708686257744 q^{11} + 70\!\cdots\!57 q^{12}+ \cdots + 90\!\cdots\!84 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2672.72 −0.115350 −0.0576751 0.998335i \(-0.518369\pi\)
−0.0576751 + 0.998335i \(0.518369\pi\)
\(3\) 4.78297e6 0.577350
\(4\) −5.29727e8 −0.986694
\(5\) 1.94732e10 1.42683 0.713417 0.700740i \(-0.247147\pi\)
0.713417 + 0.700740i \(0.247147\pi\)
\(6\) −1.27835e10 −0.0665975
\(7\) −6.78223e11 −0.377964
\(8\) 2.85072e12 0.229166
\(9\) 2.28768e13 0.333333
\(10\) −5.20466e13 −0.164586
\(11\) −2.34539e15 −1.86218 −0.931089 0.364791i \(-0.881140\pi\)
−0.931089 + 0.364791i \(0.881140\pi\)
\(12\) −2.53367e15 −0.569668
\(13\) −1.06675e16 −0.751422 −0.375711 0.926737i \(-0.622602\pi\)
−0.375711 + 0.926737i \(0.622602\pi\)
\(14\) 1.81270e15 0.0435983
\(15\) 9.31399e16 0.823783
\(16\) 2.76776e17 0.960260
\(17\) −1.80413e17 −0.259872 −0.129936 0.991522i \(-0.541477\pi\)
−0.129936 + 0.991522i \(0.541477\pi\)
\(18\) −6.11433e16 −0.0384501
\(19\) 3.63408e18 1.04344 0.521720 0.853117i \(-0.325291\pi\)
0.521720 + 0.853117i \(0.325291\pi\)
\(20\) −1.03155e19 −1.40785
\(21\) −3.24392e18 −0.218218
\(22\) 6.26858e18 0.214803
\(23\) 1.42481e19 0.256273 0.128136 0.991757i \(-0.459101\pi\)
0.128136 + 0.991757i \(0.459101\pi\)
\(24\) 1.36349e19 0.132309
\(25\) 1.92943e20 1.03585
\(26\) 2.85113e19 0.0866768
\(27\) 1.09419e20 0.192450
\(28\) 3.59273e20 0.372935
\(29\) 1.38009e21 0.861267 0.430634 0.902527i \(-0.358290\pi\)
0.430634 + 0.902527i \(0.358290\pi\)
\(30\) −2.48937e20 −0.0950236
\(31\) −1.39623e21 −0.331293 −0.165647 0.986185i \(-0.552971\pi\)
−0.165647 + 0.986185i \(0.552971\pi\)
\(32\) −2.27021e21 −0.339932
\(33\) −1.12179e22 −1.07513
\(34\) 4.82194e20 0.0299763
\(35\) −1.32072e22 −0.539292
\(36\) −1.21185e22 −0.328898
\(37\) 1.35650e22 0.247454 0.123727 0.992316i \(-0.460515\pi\)
0.123727 + 0.992316i \(0.460515\pi\)
\(38\) −9.71289e21 −0.120361
\(39\) −5.10223e22 −0.433834
\(40\) 5.55128e22 0.326981
\(41\) −4.23543e23 −1.74394 −0.871969 0.489561i \(-0.837157\pi\)
−0.871969 + 0.489561i \(0.837157\pi\)
\(42\) 8.67009e21 0.0251715
\(43\) 3.71448e23 0.766664 0.383332 0.923611i \(-0.374776\pi\)
0.383332 + 0.923611i \(0.374776\pi\)
\(44\) 1.24242e24 1.83740
\(45\) 4.45485e23 0.475611
\(46\) −3.80811e22 −0.0295611
\(47\) 2.04365e24 1.16142 0.580708 0.814112i \(-0.302776\pi\)
0.580708 + 0.814112i \(0.302776\pi\)
\(48\) 1.32381e24 0.554406
\(49\) 4.59987e23 0.142857
\(50\) −5.15682e23 −0.119486
\(51\) −8.62910e23 −0.150037
\(52\) 5.65087e24 0.741424
\(53\) 1.15964e25 1.15431 0.577157 0.816633i \(-0.304162\pi\)
0.577157 + 0.816633i \(0.304162\pi\)
\(54\) −2.92446e23 −0.0221992
\(55\) −4.56724e25 −2.65702
\(56\) −1.93342e24 −0.0866165
\(57\) 1.73817e25 0.602430
\(58\) −3.68861e24 −0.0993474
\(59\) −2.23571e25 −0.469961 −0.234981 0.972000i \(-0.575503\pi\)
−0.234981 + 0.972000i \(0.575503\pi\)
\(60\) −4.93388e25 −0.812822
\(61\) −4.97521e25 −0.644954 −0.322477 0.946577i \(-0.604516\pi\)
−0.322477 + 0.946577i \(0.604516\pi\)
\(62\) 3.73174e24 0.0382148
\(63\) −1.55156e25 −0.125988
\(64\) −1.42525e26 −0.921049
\(65\) −2.07731e26 −1.07215
\(66\) 2.99824e25 0.124016
\(67\) 5.48772e26 1.82519 0.912593 0.408869i \(-0.134077\pi\)
0.912593 + 0.408869i \(0.134077\pi\)
\(68\) 9.55698e25 0.256414
\(69\) 6.81480e25 0.147959
\(70\) 3.52992e25 0.0622075
\(71\) 1.06838e27 1.53279 0.766393 0.642372i \(-0.222050\pi\)
0.766393 + 0.642372i \(0.222050\pi\)
\(72\) 6.52153e25 0.0763886
\(73\) −1.06623e27 −1.02251 −0.511257 0.859428i \(-0.670820\pi\)
−0.511257 + 0.859428i \(0.670820\pi\)
\(74\) −3.62554e25 −0.0285439
\(75\) 9.22839e26 0.598050
\(76\) −1.92507e27 −1.02956
\(77\) 1.59070e27 0.703837
\(78\) 1.36368e26 0.0500429
\(79\) −5.67748e27 −1.73206 −0.866029 0.499994i \(-0.833336\pi\)
−0.866029 + 0.499994i \(0.833336\pi\)
\(80\) 5.38973e27 1.37013
\(81\) 5.23348e26 0.111111
\(82\) 1.13201e27 0.201164
\(83\) −5.28365e27 −0.787592 −0.393796 0.919198i \(-0.628838\pi\)
−0.393796 + 0.919198i \(0.628838\pi\)
\(84\) 1.71839e27 0.215314
\(85\) −3.51323e27 −0.370793
\(86\) −9.92776e26 −0.0884349
\(87\) 6.60095e27 0.497253
\(88\) −6.68605e27 −0.426748
\(89\) 2.36536e28 1.28157 0.640784 0.767721i \(-0.278609\pi\)
0.640784 + 0.767721i \(0.278609\pi\)
\(90\) −1.19066e27 −0.0548619
\(91\) 7.23494e27 0.284011
\(92\) −7.54759e27 −0.252863
\(93\) −6.67813e27 −0.191272
\(94\) −5.46211e27 −0.133970
\(95\) 7.07674e28 1.48882
\(96\) −1.08584e28 −0.196260
\(97\) 2.86997e28 0.446362 0.223181 0.974777i \(-0.428356\pi\)
0.223181 + 0.974777i \(0.428356\pi\)
\(98\) −1.22942e27 −0.0164786
\(99\) −5.36550e28 −0.620726
\(100\) −1.02207e29 −1.02207
\(101\) 1.00238e29 0.867704 0.433852 0.900984i \(-0.357154\pi\)
0.433852 + 0.900984i \(0.357154\pi\)
\(102\) 2.30632e27 0.0173068
\(103\) −2.20235e28 −0.143465 −0.0717327 0.997424i \(-0.522853\pi\)
−0.0717327 + 0.997424i \(0.522853\pi\)
\(104\) −3.04101e28 −0.172200
\(105\) −6.31696e28 −0.311361
\(106\) −3.09940e28 −0.133150
\(107\) 2.29131e29 0.859053 0.429526 0.903054i \(-0.358681\pi\)
0.429526 + 0.903054i \(0.358681\pi\)
\(108\) −5.79622e28 −0.189889
\(109\) 6.79956e29 1.94893 0.974466 0.224534i \(-0.0720861\pi\)
0.974466 + 0.224534i \(0.0720861\pi\)
\(110\) 1.22070e29 0.306488
\(111\) 6.48809e28 0.142868
\(112\) −1.87716e29 −0.362944
\(113\) 8.64483e29 1.46933 0.734664 0.678431i \(-0.237340\pi\)
0.734664 + 0.678431i \(0.237340\pi\)
\(114\) −4.64564e28 −0.0694905
\(115\) 2.77456e29 0.365658
\(116\) −7.31074e29 −0.849808
\(117\) −2.44038e29 −0.250474
\(118\) 5.97544e28 0.0542101
\(119\) 1.22360e29 0.0982222
\(120\) 2.65516e29 0.188783
\(121\) 3.91455e30 2.46771
\(122\) 1.32973e29 0.0743956
\(123\) −2.02579e30 −1.00686
\(124\) 7.39622e29 0.326885
\(125\) 1.30047e29 0.0511572
\(126\) 4.14688e28 0.0145328
\(127\) 1.93309e30 0.604084 0.302042 0.953295i \(-0.402332\pi\)
0.302042 + 0.953295i \(0.402332\pi\)
\(128\) 1.59974e30 0.446175
\(129\) 1.77662e30 0.442634
\(130\) 5.55207e29 0.123673
\(131\) 1.56740e30 0.312425 0.156212 0.987723i \(-0.450072\pi\)
0.156212 + 0.987723i \(0.450072\pi\)
\(132\) 5.94245e30 1.06082
\(133\) −2.46472e30 −0.394383
\(134\) −1.46671e30 −0.210536
\(135\) 2.13074e30 0.274594
\(136\) −5.14307e29 −0.0595537
\(137\) 5.59686e30 0.582768 0.291384 0.956606i \(-0.405884\pi\)
0.291384 + 0.956606i \(0.405884\pi\)
\(138\) −1.82141e29 −0.0170671
\(139\) −9.85701e29 −0.0831822 −0.0415911 0.999135i \(-0.513243\pi\)
−0.0415911 + 0.999135i \(0.513243\pi\)
\(140\) 6.99622e30 0.532117
\(141\) 9.77471e30 0.670543
\(142\) −2.85549e30 −0.176807
\(143\) 2.50195e31 1.39928
\(144\) 6.33175e30 0.320087
\(145\) 2.68749e31 1.22889
\(146\) 2.84973e30 0.117947
\(147\) 2.20010e30 0.0824786
\(148\) −7.18575e30 −0.244161
\(149\) −2.84172e31 −0.875752 −0.437876 0.899035i \(-0.644269\pi\)
−0.437876 + 0.899035i \(0.644269\pi\)
\(150\) −2.46649e30 −0.0689853
\(151\) 2.81089e31 0.713967 0.356984 0.934111i \(-0.383805\pi\)
0.356984 + 0.934111i \(0.383805\pi\)
\(152\) 1.03598e31 0.239121
\(153\) −4.12727e30 −0.0866239
\(154\) −4.25149e30 −0.0811878
\(155\) −2.71891e31 −0.472700
\(156\) 2.70279e31 0.428061
\(157\) 9.89253e31 1.42811 0.714056 0.700088i \(-0.246856\pi\)
0.714056 + 0.700088i \(0.246856\pi\)
\(158\) 1.51743e31 0.199793
\(159\) 5.54653e31 0.666443
\(160\) −4.42084e31 −0.485026
\(161\) −9.66336e30 −0.0968619
\(162\) −1.39876e30 −0.0128167
\(163\) 4.69173e31 0.393199 0.196600 0.980484i \(-0.437010\pi\)
0.196600 + 0.980484i \(0.437010\pi\)
\(164\) 2.24362e32 1.72073
\(165\) −2.18450e32 −1.53403
\(166\) 1.41217e31 0.0908490
\(167\) −1.40088e32 −0.826058 −0.413029 0.910718i \(-0.635529\pi\)
−0.413029 + 0.910718i \(0.635529\pi\)
\(168\) −9.24751e30 −0.0500081
\(169\) −8.77426e31 −0.435365
\(170\) 9.38988e30 0.0427711
\(171\) 8.31362e31 0.347813
\(172\) −1.96766e32 −0.756463
\(173\) −3.03186e31 −0.107162 −0.0535810 0.998564i \(-0.517064\pi\)
−0.0535810 + 0.998564i \(0.517064\pi\)
\(174\) −1.76425e31 −0.0573583
\(175\) −1.30858e32 −0.391516
\(176\) −6.49148e32 −1.78818
\(177\) −1.06933e32 −0.271332
\(178\) −6.32194e31 −0.147829
\(179\) −8.79328e31 −0.189575 −0.0947877 0.995498i \(-0.530217\pi\)
−0.0947877 + 0.995498i \(0.530217\pi\)
\(180\) −2.35986e32 −0.469283
\(181\) 1.39286e32 0.255605 0.127802 0.991800i \(-0.459208\pi\)
0.127802 + 0.991800i \(0.459208\pi\)
\(182\) −1.93370e31 −0.0327607
\(183\) −2.37963e32 −0.372364
\(184\) 4.06172e31 0.0587289
\(185\) 2.64154e32 0.353076
\(186\) 1.78488e31 0.0220633
\(187\) 4.23139e32 0.483927
\(188\) −1.08258e33 −1.14596
\(189\) −7.42105e31 −0.0727393
\(190\) −1.89141e32 −0.171735
\(191\) 2.33825e33 1.96746 0.983732 0.179644i \(-0.0574947\pi\)
0.983732 + 0.179644i \(0.0574947\pi\)
\(192\) −6.81694e32 −0.531768
\(193\) −1.92217e33 −1.39063 −0.695315 0.718705i \(-0.744735\pi\)
−0.695315 + 0.718705i \(0.744735\pi\)
\(194\) −7.67062e31 −0.0514879
\(195\) −9.93570e32 −0.619009
\(196\) −2.43668e32 −0.140956
\(197\) −8.31047e32 −0.446545 −0.223272 0.974756i \(-0.571674\pi\)
−0.223272 + 0.974756i \(0.571674\pi\)
\(198\) 1.43405e32 0.0716010
\(199\) 3.47494e33 1.61279 0.806394 0.591378i \(-0.201416\pi\)
0.806394 + 0.591378i \(0.201416\pi\)
\(200\) 5.50026e32 0.237382
\(201\) 2.62476e33 1.05377
\(202\) −2.67908e32 −0.100090
\(203\) −9.36012e32 −0.325528
\(204\) 4.57107e32 0.148041
\(205\) −8.24775e33 −2.48831
\(206\) 5.88627e31 0.0165488
\(207\) 3.25950e32 0.0854242
\(208\) −2.95251e33 −0.721561
\(209\) −8.52334e33 −1.94307
\(210\) 1.68835e32 0.0359155
\(211\) −8.10783e33 −1.60994 −0.804969 0.593317i \(-0.797818\pi\)
−0.804969 + 0.593317i \(0.797818\pi\)
\(212\) −6.14294e33 −1.13895
\(213\) 5.11005e33 0.884954
\(214\) −6.12404e32 −0.0990920
\(215\) 7.23329e33 1.09390
\(216\) 3.11923e32 0.0441030
\(217\) 9.46956e32 0.125217
\(218\) −1.81733e33 −0.224810
\(219\) −5.09974e33 −0.590348
\(220\) 2.41939e34 2.62167
\(221\) 1.92456e33 0.195273
\(222\) −1.73409e32 −0.0164798
\(223\) 1.50914e34 1.34372 0.671858 0.740680i \(-0.265496\pi\)
0.671858 + 0.740680i \(0.265496\pi\)
\(224\) 1.53971e33 0.128482
\(225\) 4.41391e33 0.345285
\(226\) −2.31052e33 −0.169487
\(227\) 5.78815e33 0.398258 0.199129 0.979973i \(-0.436189\pi\)
0.199129 + 0.979973i \(0.436189\pi\)
\(228\) −9.20757e33 −0.594415
\(229\) 1.93852e34 1.17451 0.587253 0.809404i \(-0.300209\pi\)
0.587253 + 0.809404i \(0.300209\pi\)
\(230\) −7.41562e32 −0.0421788
\(231\) 7.60826e33 0.406361
\(232\) 3.93426e33 0.197373
\(233\) −6.44258e33 −0.303668 −0.151834 0.988406i \(-0.548518\pi\)
−0.151834 + 0.988406i \(0.548518\pi\)
\(234\) 6.52246e32 0.0288923
\(235\) 3.97965e34 1.65715
\(236\) 1.18432e34 0.463708
\(237\) −2.71552e34 −1.00000
\(238\) −3.27035e32 −0.0113300
\(239\) 2.07853e34 0.677620 0.338810 0.940855i \(-0.389976\pi\)
0.338810 + 0.940855i \(0.389976\pi\)
\(240\) 2.57789e34 0.791046
\(241\) −4.38919e34 −1.26805 −0.634026 0.773312i \(-0.718599\pi\)
−0.634026 + 0.773312i \(0.718599\pi\)
\(242\) −1.04625e34 −0.284651
\(243\) 2.50316e33 0.0641500
\(244\) 2.63551e34 0.636373
\(245\) 8.95743e33 0.203833
\(246\) 5.41437e33 0.116142
\(247\) −3.87666e34 −0.784064
\(248\) −3.98026e33 −0.0759211
\(249\) −2.52715e34 −0.454717
\(250\) −3.47580e32 −0.00590099
\(251\) −2.43361e34 −0.389925 −0.194963 0.980811i \(-0.562459\pi\)
−0.194963 + 0.980811i \(0.562459\pi\)
\(252\) 8.21902e33 0.124312
\(253\) −3.34173e34 −0.477225
\(254\) −5.16660e33 −0.0696813
\(255\) −1.68037e34 −0.214078
\(256\) 7.22421e34 0.869582
\(257\) 1.41213e35 1.60637 0.803184 0.595731i \(-0.203138\pi\)
0.803184 + 0.595731i \(0.203138\pi\)
\(258\) −4.74842e33 −0.0510579
\(259\) −9.20009e33 −0.0935288
\(260\) 1.10041e35 1.05789
\(261\) 3.15721e34 0.287089
\(262\) −4.18922e33 −0.0360383
\(263\) 1.27035e35 1.03411 0.517054 0.855953i \(-0.327028\pi\)
0.517054 + 0.855953i \(0.327028\pi\)
\(264\) −3.19792e34 −0.246383
\(265\) 2.25820e35 1.64701
\(266\) 6.58751e33 0.0454922
\(267\) 1.13134e35 0.739913
\(268\) −2.90700e35 −1.80090
\(269\) −1.61544e35 −0.948160 −0.474080 0.880482i \(-0.657219\pi\)
−0.474080 + 0.880482i \(0.657219\pi\)
\(270\) −5.69488e33 −0.0316745
\(271\) −1.92714e35 −1.01592 −0.507960 0.861381i \(-0.669600\pi\)
−0.507960 + 0.861381i \(0.669600\pi\)
\(272\) −4.99340e34 −0.249544
\(273\) 3.46045e34 0.163974
\(274\) −1.49588e34 −0.0672225
\(275\) −4.52526e35 −1.92894
\(276\) −3.60999e34 −0.145990
\(277\) 1.24537e35 0.477906 0.238953 0.971031i \(-0.423196\pi\)
0.238953 + 0.971031i \(0.423196\pi\)
\(278\) 2.63450e33 0.00959509
\(279\) −3.19413e34 −0.110431
\(280\) −3.76500e34 −0.123587
\(281\) 9.35141e34 0.291498 0.145749 0.989322i \(-0.453441\pi\)
0.145749 + 0.989322i \(0.453441\pi\)
\(282\) −2.61251e34 −0.0773474
\(283\) −3.30375e35 −0.929188 −0.464594 0.885524i \(-0.653800\pi\)
−0.464594 + 0.885524i \(0.653800\pi\)
\(284\) −5.65953e35 −1.51239
\(285\) 3.38478e35 0.859568
\(286\) −6.68700e34 −0.161408
\(287\) 2.87256e35 0.659147
\(288\) −5.19352e34 −0.113311
\(289\) −4.49420e35 −0.932467
\(290\) −7.18292e34 −0.141752
\(291\) 1.37270e35 0.257707
\(292\) 5.64811e35 1.00891
\(293\) −7.63998e35 −1.29871 −0.649355 0.760486i \(-0.724961\pi\)
−0.649355 + 0.760486i \(0.724961\pi\)
\(294\) −5.88026e33 −0.00951393
\(295\) −4.35366e35 −0.670556
\(296\) 3.86700e34 0.0567080
\(297\) −2.56630e35 −0.358376
\(298\) 7.59513e34 0.101018
\(299\) −1.51991e35 −0.192569
\(300\) −4.88853e35 −0.590093
\(301\) −2.51924e35 −0.289772
\(302\) −7.51273e34 −0.0823563
\(303\) 4.79434e35 0.500969
\(304\) 1.00583e36 1.00197
\(305\) −9.68835e35 −0.920242
\(306\) 1.10311e34 0.00999209
\(307\) 1.73065e36 1.49521 0.747604 0.664144i \(-0.231204\pi\)
0.747604 + 0.664144i \(0.231204\pi\)
\(308\) −8.42637e35 −0.694472
\(309\) −1.05338e35 −0.0828297
\(310\) 7.26690e34 0.0545261
\(311\) −1.05243e36 −0.753648 −0.376824 0.926285i \(-0.622984\pi\)
−0.376824 + 0.926285i \(0.622984\pi\)
\(312\) −1.45450e35 −0.0994199
\(313\) 5.43426e34 0.0354607 0.0177304 0.999843i \(-0.494356\pi\)
0.0177304 + 0.999843i \(0.494356\pi\)
\(314\) −2.64400e35 −0.164733
\(315\) −3.02138e35 −0.179764
\(316\) 3.00751e36 1.70901
\(317\) 2.67754e36 1.45337 0.726686 0.686970i \(-0.241059\pi\)
0.726686 + 0.686970i \(0.241059\pi\)
\(318\) −1.48243e35 −0.0768744
\(319\) −3.23686e36 −1.60383
\(320\) −2.77543e36 −1.31418
\(321\) 1.09593e36 0.495974
\(322\) 2.58275e34 0.0111730
\(323\) −6.55636e35 −0.271160
\(324\) −2.77232e35 −0.109633
\(325\) −2.05822e36 −0.778363
\(326\) −1.25397e35 −0.0453556
\(327\) 3.25221e36 1.12522
\(328\) −1.20740e36 −0.399651
\(329\) −1.38605e36 −0.438974
\(330\) 5.83855e35 0.176951
\(331\) 5.14201e36 1.49151 0.745756 0.666219i \(-0.232089\pi\)
0.745756 + 0.666219i \(0.232089\pi\)
\(332\) 2.79889e36 0.777113
\(333\) 3.10323e35 0.0824847
\(334\) 3.74415e35 0.0952860
\(335\) 1.06864e37 2.60424
\(336\) −8.97839e35 −0.209546
\(337\) −8.50236e35 −0.190067 −0.0950333 0.995474i \(-0.530296\pi\)
−0.0950333 + 0.995474i \(0.530296\pi\)
\(338\) 2.34511e35 0.0502194
\(339\) 4.13479e36 0.848317
\(340\) 1.86105e36 0.365860
\(341\) 3.27471e36 0.616927
\(342\) −2.22200e35 −0.0401204
\(343\) −3.11973e35 −0.0539949
\(344\) 1.05889e36 0.175693
\(345\) 1.32706e36 0.211113
\(346\) 8.10331e34 0.0123612
\(347\) −9.07198e36 −1.32717 −0.663583 0.748103i \(-0.730965\pi\)
−0.663583 + 0.748103i \(0.730965\pi\)
\(348\) −3.49671e36 −0.490637
\(349\) −7.98276e36 −1.07445 −0.537223 0.843440i \(-0.680527\pi\)
−0.537223 + 0.843440i \(0.680527\pi\)
\(350\) 3.49748e35 0.0451615
\(351\) −1.16723e36 −0.144611
\(352\) 5.32454e36 0.633014
\(353\) −2.38372e35 −0.0271970 −0.0135985 0.999908i \(-0.504329\pi\)
−0.0135985 + 0.999908i \(0.504329\pi\)
\(354\) 2.85803e35 0.0312982
\(355\) 2.08049e37 2.18703
\(356\) −1.25299e37 −1.26452
\(357\) 5.85246e35 0.0567086
\(358\) 2.35020e35 0.0218676
\(359\) −6.70211e36 −0.598883 −0.299442 0.954115i \(-0.596800\pi\)
−0.299442 + 0.954115i \(0.596800\pi\)
\(360\) 1.26995e36 0.108994
\(361\) 1.07673e36 0.0887675
\(362\) −3.72273e35 −0.0294841
\(363\) 1.87232e37 1.42473
\(364\) −3.83255e36 −0.280232
\(365\) −2.07629e37 −1.45896
\(366\) 6.36008e35 0.0429523
\(367\) 6.00718e36 0.389953 0.194977 0.980808i \(-0.437537\pi\)
0.194977 + 0.980808i \(0.437537\pi\)
\(368\) 3.94352e36 0.246088
\(369\) −9.68930e36 −0.581313
\(370\) −7.06011e35 −0.0407274
\(371\) −7.86496e36 −0.436289
\(372\) 3.53759e36 0.188727
\(373\) 6.08471e36 0.312221 0.156110 0.987740i \(-0.450104\pi\)
0.156110 + 0.987740i \(0.450104\pi\)
\(374\) −1.13093e36 −0.0558212
\(375\) 6.22013e35 0.0295356
\(376\) 5.82587e36 0.266157
\(377\) −1.47222e37 −0.647175
\(378\) 1.98344e35 0.00839050
\(379\) 3.06661e37 1.24850 0.624252 0.781223i \(-0.285404\pi\)
0.624252 + 0.781223i \(0.285404\pi\)
\(380\) −3.74874e37 −1.46901
\(381\) 9.24590e36 0.348768
\(382\) −6.24948e36 −0.226947
\(383\) −1.06910e37 −0.373798 −0.186899 0.982379i \(-0.559844\pi\)
−0.186899 + 0.982379i \(0.559844\pi\)
\(384\) 7.65152e36 0.257599
\(385\) 3.09761e37 1.00426
\(386\) 5.13743e36 0.160409
\(387\) 8.49753e36 0.255555
\(388\) −1.52030e37 −0.440422
\(389\) −3.13590e37 −0.875172 −0.437586 0.899176i \(-0.644166\pi\)
−0.437586 + 0.899176i \(0.644166\pi\)
\(390\) 2.65554e36 0.0714028
\(391\) −2.57054e36 −0.0665979
\(392\) 1.31129e36 0.0327380
\(393\) 7.49682e36 0.180379
\(394\) 2.22116e36 0.0515091
\(395\) −1.10559e38 −2.47136
\(396\) 2.84225e37 0.612467
\(397\) −8.36014e37 −1.73681 −0.868403 0.495859i \(-0.834853\pi\)
−0.868403 + 0.495859i \(0.834853\pi\)
\(398\) −9.28754e36 −0.186036
\(399\) −1.17887e37 −0.227697
\(400\) 5.34019e37 0.994689
\(401\) −4.15145e37 −0.745772 −0.372886 0.927877i \(-0.621632\pi\)
−0.372886 + 0.927877i \(0.621632\pi\)
\(402\) −7.01525e36 −0.121553
\(403\) 1.48943e37 0.248941
\(404\) −5.30987e37 −0.856158
\(405\) 1.01913e37 0.158537
\(406\) 2.50170e36 0.0375498
\(407\) −3.18152e37 −0.460803
\(408\) −2.45992e36 −0.0343833
\(409\) 1.92537e37 0.259733 0.129866 0.991532i \(-0.458545\pi\)
0.129866 + 0.991532i \(0.458545\pi\)
\(410\) 2.20439e37 0.287027
\(411\) 2.67696e37 0.336462
\(412\) 1.16665e37 0.141556
\(413\) 1.51631e37 0.177629
\(414\) −8.71173e35 −0.00985370
\(415\) −1.02890e38 −1.12376
\(416\) 2.42175e37 0.255433
\(417\) −4.71458e36 −0.0480253
\(418\) 2.27805e37 0.224134
\(419\) 3.02161e37 0.287167 0.143584 0.989638i \(-0.454137\pi\)
0.143584 + 0.989638i \(0.454137\pi\)
\(420\) 3.34627e37 0.307218
\(421\) −1.00326e38 −0.889863 −0.444931 0.895565i \(-0.646772\pi\)
−0.444931 + 0.895565i \(0.646772\pi\)
\(422\) 2.16700e37 0.185707
\(423\) 4.67521e37 0.387138
\(424\) 3.30581e37 0.264529
\(425\) −3.48094e37 −0.269189
\(426\) −1.36577e37 −0.102080
\(427\) 3.37430e37 0.243770
\(428\) −1.21377e38 −0.847622
\(429\) 1.19667e38 0.807876
\(430\) −1.93326e37 −0.126182
\(431\) 1.97532e38 1.24657 0.623287 0.781994i \(-0.285797\pi\)
0.623287 + 0.781994i \(0.285797\pi\)
\(432\) 3.02846e37 0.184802
\(433\) 2.49009e37 0.146940 0.0734700 0.997297i \(-0.476593\pi\)
0.0734700 + 0.997297i \(0.476593\pi\)
\(434\) −2.53095e36 −0.0144438
\(435\) 1.28542e38 0.709497
\(436\) −3.60192e38 −1.92300
\(437\) 5.17786e37 0.267405
\(438\) 1.36302e37 0.0680969
\(439\) 3.60571e38 1.74283 0.871416 0.490545i \(-0.163202\pi\)
0.871416 + 0.490545i \(0.163202\pi\)
\(440\) −1.30199e38 −0.608898
\(441\) 1.05230e37 0.0476190
\(442\) −5.14380e36 −0.0225248
\(443\) 7.19114e36 0.0304750 0.0152375 0.999884i \(-0.495150\pi\)
0.0152375 + 0.999884i \(0.495150\pi\)
\(444\) −3.43692e37 −0.140967
\(445\) 4.60612e38 1.82858
\(446\) −4.03350e37 −0.154998
\(447\) −1.35919e38 −0.505615
\(448\) 9.66640e37 0.348124
\(449\) 6.39610e37 0.223020 0.111510 0.993763i \(-0.464431\pi\)
0.111510 + 0.993763i \(0.464431\pi\)
\(450\) −1.17972e37 −0.0398287
\(451\) 9.93373e38 3.24752
\(452\) −4.57940e38 −1.44978
\(453\) 1.34444e38 0.412209
\(454\) −1.54701e37 −0.0459392
\(455\) 1.40888e38 0.405236
\(456\) 4.95504e37 0.138056
\(457\) 3.36298e38 0.907695 0.453847 0.891079i \(-0.350051\pi\)
0.453847 + 0.891079i \(0.350051\pi\)
\(458\) −5.18112e37 −0.135480
\(459\) −1.97406e37 −0.0500123
\(460\) −1.46976e38 −0.360793
\(461\) −3.00963e38 −0.715896 −0.357948 0.933742i \(-0.616523\pi\)
−0.357948 + 0.933742i \(0.616523\pi\)
\(462\) −2.03348e37 −0.0468738
\(463\) −2.02310e37 −0.0451952 −0.0225976 0.999745i \(-0.507194\pi\)
−0.0225976 + 0.999745i \(0.507194\pi\)
\(464\) 3.81977e38 0.827041
\(465\) −1.30045e38 −0.272914
\(466\) 1.72192e37 0.0350282
\(467\) −9.23292e38 −1.82072 −0.910362 0.413812i \(-0.864197\pi\)
−0.910362 + 0.413812i \(0.864197\pi\)
\(468\) 1.29274e38 0.247141
\(469\) −3.72190e38 −0.689856
\(470\) −1.06365e38 −0.191152
\(471\) 4.73157e38 0.824521
\(472\) −6.37339e37 −0.107699
\(473\) −8.71190e38 −1.42767
\(474\) 7.25782e37 0.115351
\(475\) 7.01170e38 1.08085
\(476\) −6.48176e37 −0.0969153
\(477\) 2.65289e38 0.384771
\(478\) −5.55532e37 −0.0781636
\(479\) 3.71292e38 0.506817 0.253408 0.967359i \(-0.418448\pi\)
0.253408 + 0.967359i \(0.418448\pi\)
\(480\) −2.11448e38 −0.280030
\(481\) −1.44705e38 −0.185942
\(482\) 1.17311e38 0.146270
\(483\) −4.62196e37 −0.0559232
\(484\) −2.07364e39 −2.43487
\(485\) 5.58876e38 0.636884
\(486\) −6.69024e36 −0.00739972
\(487\) 3.31078e38 0.355435 0.177718 0.984082i \(-0.443129\pi\)
0.177718 + 0.984082i \(0.443129\pi\)
\(488\) −1.41829e38 −0.147801
\(489\) 2.24404e38 0.227014
\(490\) −2.39407e37 −0.0235122
\(491\) 4.67267e38 0.445536 0.222768 0.974871i \(-0.428491\pi\)
0.222768 + 0.974871i \(0.428491\pi\)
\(492\) 1.07312e39 0.993466
\(493\) −2.48987e38 −0.223819
\(494\) 1.03612e38 0.0904420
\(495\) −1.04484e39 −0.885673
\(496\) −3.86443e38 −0.318128
\(497\) −7.24603e38 −0.579339
\(498\) 6.75437e37 0.0524517
\(499\) −8.77872e38 −0.662176 −0.331088 0.943600i \(-0.607416\pi\)
−0.331088 + 0.943600i \(0.607416\pi\)
\(500\) −6.88897e37 −0.0504765
\(501\) −6.70034e38 −0.476925
\(502\) 6.50435e37 0.0449780
\(503\) 1.58017e39 1.06161 0.530807 0.847492i \(-0.321889\pi\)
0.530807 + 0.847492i \(0.321889\pi\)
\(504\) −4.42305e37 −0.0288722
\(505\) 1.95195e39 1.23807
\(506\) 8.93150e37 0.0550481
\(507\) −4.19670e38 −0.251358
\(508\) −1.02401e39 −0.596047
\(509\) 5.38247e38 0.304490 0.152245 0.988343i \(-0.451350\pi\)
0.152245 + 0.988343i \(0.451350\pi\)
\(510\) 4.49115e37 0.0246939
\(511\) 7.23141e38 0.386474
\(512\) −1.05194e39 −0.546482
\(513\) 3.97638e38 0.200810
\(514\) −3.77424e38 −0.185295
\(515\) −4.28869e38 −0.204701
\(516\) −9.41126e38 −0.436744
\(517\) −4.79316e39 −2.16276
\(518\) 2.45893e37 0.0107886
\(519\) −1.45013e38 −0.0618700
\(520\) −5.92182e38 −0.245701
\(521\) −2.42785e39 −0.979661 −0.489830 0.871818i \(-0.662941\pi\)
−0.489830 + 0.871818i \(0.662941\pi\)
\(522\) −8.43835e37 −0.0331158
\(523\) 8.33744e38 0.318242 0.159121 0.987259i \(-0.449134\pi\)
0.159121 + 0.987259i \(0.449134\pi\)
\(524\) −8.30294e38 −0.308268
\(525\) −6.25891e38 −0.226042
\(526\) −3.39530e38 −0.119285
\(527\) 2.51898e38 0.0860937
\(528\) −3.10486e39 −1.03240
\(529\) −2.88805e39 −0.934324
\(530\) −6.03554e38 −0.189983
\(531\) −5.11459e38 −0.156654
\(532\) 1.30563e39 0.389136
\(533\) 4.51814e39 1.31043
\(534\) −3.02376e38 −0.0853492
\(535\) 4.46193e39 1.22573
\(536\) 1.56439e39 0.418270
\(537\) −4.20580e38 −0.109451
\(538\) 4.31761e38 0.109371
\(539\) −1.07885e39 −0.266026
\(540\) −1.12871e39 −0.270941
\(541\) −5.64635e39 −1.31949 −0.659747 0.751488i \(-0.729337\pi\)
−0.659747 + 0.751488i \(0.729337\pi\)
\(542\) 5.15071e38 0.117187
\(543\) 6.66202e38 0.147573
\(544\) 4.09576e38 0.0883387
\(545\) 1.32410e40 2.78080
\(546\) −9.24882e37 −0.0189144
\(547\) 4.46905e39 0.890019 0.445010 0.895526i \(-0.353200\pi\)
0.445010 + 0.895526i \(0.353200\pi\)
\(548\) −2.96481e39 −0.575014
\(549\) −1.13817e39 −0.214985
\(550\) 1.20948e39 0.222504
\(551\) 5.01538e39 0.898681
\(552\) 1.94271e38 0.0339071
\(553\) 3.85059e39 0.654656
\(554\) −3.32853e38 −0.0551265
\(555\) 1.26344e39 0.203848
\(556\) 5.22153e38 0.0820754
\(557\) 1.23118e40 1.88547 0.942736 0.333540i \(-0.108243\pi\)
0.942736 + 0.333540i \(0.108243\pi\)
\(558\) 8.53701e37 0.0127383
\(559\) −3.96242e39 −0.576089
\(560\) −3.65544e39 −0.517861
\(561\) 2.02386e39 0.279396
\(562\) −2.49937e38 −0.0336244
\(563\) −9.45065e39 −1.23905 −0.619527 0.784976i \(-0.712675\pi\)
−0.619527 + 0.784976i \(0.712675\pi\)
\(564\) −5.17793e39 −0.661621
\(565\) 1.68343e40 2.09649
\(566\) 8.83000e38 0.107182
\(567\) −3.54946e38 −0.0419961
\(568\) 3.04566e39 0.351262
\(569\) 1.58490e40 1.78187 0.890934 0.454133i \(-0.150051\pi\)
0.890934 + 0.454133i \(0.150051\pi\)
\(570\) −9.04658e38 −0.0991514
\(571\) 2.09571e39 0.223928 0.111964 0.993712i \(-0.464286\pi\)
0.111964 + 0.993712i \(0.464286\pi\)
\(572\) −1.32535e40 −1.38066
\(573\) 1.11838e40 1.13592
\(574\) −7.67756e38 −0.0760328
\(575\) 2.74906e39 0.265461
\(576\) −3.26052e39 −0.307016
\(577\) −8.37574e39 −0.769084 −0.384542 0.923108i \(-0.625641\pi\)
−0.384542 + 0.923108i \(0.625641\pi\)
\(578\) 1.20117e39 0.107560
\(579\) −9.19369e39 −0.802880
\(580\) −1.42364e40 −1.21253
\(581\) 3.58349e39 0.297682
\(582\) −3.66884e38 −0.0297266
\(583\) −2.71981e40 −2.14954
\(584\) −3.03952e39 −0.234325
\(585\) −4.75222e39 −0.357385
\(586\) 2.04195e39 0.149806
\(587\) 2.19066e40 1.56791 0.783956 0.620817i \(-0.213199\pi\)
0.783956 + 0.620817i \(0.213199\pi\)
\(588\) −1.16545e39 −0.0813812
\(589\) −5.07402e39 −0.345685
\(590\) 1.16361e39 0.0773489
\(591\) −3.97487e39 −0.257813
\(592\) 3.75446e39 0.237620
\(593\) 8.92945e39 0.551482 0.275741 0.961232i \(-0.411077\pi\)
0.275741 + 0.961232i \(0.411077\pi\)
\(594\) 6.85901e38 0.0413388
\(595\) 2.38275e39 0.140147
\(596\) 1.50534e40 0.864099
\(597\) 1.66205e40 0.931144
\(598\) 4.06230e38 0.0222129
\(599\) −1.76630e40 −0.942706 −0.471353 0.881945i \(-0.656234\pi\)
−0.471353 + 0.881945i \(0.656234\pi\)
\(600\) 2.63076e39 0.137053
\(601\) −3.56607e40 −1.81347 −0.906735 0.421702i \(-0.861433\pi\)
−0.906735 + 0.421702i \(0.861433\pi\)
\(602\) 6.73324e38 0.0334253
\(603\) 1.25541e40 0.608395
\(604\) −1.48901e40 −0.704467
\(605\) 7.62290e40 3.52101
\(606\) −1.28139e39 −0.0577869
\(607\) 2.55200e40 1.12368 0.561842 0.827245i \(-0.310093\pi\)
0.561842 + 0.827245i \(0.310093\pi\)
\(608\) −8.25014e39 −0.354699
\(609\) −4.47692e39 −0.187944
\(610\) 2.58943e39 0.106150
\(611\) −2.18006e40 −0.872713
\(612\) 2.18633e39 0.0854713
\(613\) 9.39524e39 0.358700 0.179350 0.983785i \(-0.442601\pi\)
0.179350 + 0.983785i \(0.442601\pi\)
\(614\) −4.62553e39 −0.172473
\(615\) −3.94487e40 −1.43663
\(616\) 4.53464e39 0.161295
\(617\) −4.84847e40 −1.68450 −0.842248 0.539091i \(-0.818768\pi\)
−0.842248 + 0.539091i \(0.818768\pi\)
\(618\) 2.81539e38 0.00955444
\(619\) 5.52968e40 1.83310 0.916549 0.399923i \(-0.130963\pi\)
0.916549 + 0.399923i \(0.130963\pi\)
\(620\) 1.44028e40 0.466411
\(621\) 1.55901e39 0.0493197
\(622\) 2.81286e39 0.0869335
\(623\) −1.60424e40 −0.484387
\(624\) −1.41218e40 −0.416593
\(625\) −3.34059e40 −0.962861
\(626\) −1.45243e38 −0.00409041
\(627\) −4.07669e40 −1.12183
\(628\) −5.24035e40 −1.40911
\(629\) −2.44730e39 −0.0643062
\(630\) 8.07532e38 0.0207358
\(631\) −5.95042e40 −1.49321 −0.746606 0.665266i \(-0.768318\pi\)
−0.746606 + 0.665266i \(0.768318\pi\)
\(632\) −1.61849e40 −0.396928
\(633\) −3.87795e40 −0.929498
\(634\) −7.15632e39 −0.167647
\(635\) 3.76435e40 0.861928
\(636\) −2.93815e40 −0.657576
\(637\) −4.90691e39 −0.107346
\(638\) 8.65123e39 0.185003
\(639\) 2.44412e40 0.510929
\(640\) 3.11522e40 0.636618
\(641\) 2.12073e40 0.423686 0.211843 0.977304i \(-0.432053\pi\)
0.211843 + 0.977304i \(0.432053\pi\)
\(642\) −2.92911e39 −0.0572108
\(643\) −1.69538e39 −0.0323748 −0.0161874 0.999869i \(-0.505153\pi\)
−0.0161874 + 0.999869i \(0.505153\pi\)
\(644\) 5.11895e39 0.0955731
\(645\) 3.45966e40 0.631565
\(646\) 1.75233e39 0.0312784
\(647\) −9.52815e40 −1.66302 −0.831508 0.555513i \(-0.812522\pi\)
−0.831508 + 0.555513i \(0.812522\pi\)
\(648\) 1.49192e39 0.0254629
\(649\) 5.24362e40 0.875151
\(650\) 5.50104e39 0.0897845
\(651\) 4.52926e39 0.0722941
\(652\) −2.48534e40 −0.387967
\(653\) 5.05290e40 0.771436 0.385718 0.922617i \(-0.373954\pi\)
0.385718 + 0.922617i \(0.373954\pi\)
\(654\) −8.69225e39 −0.129794
\(655\) 3.05223e40 0.445778
\(656\) −1.17226e41 −1.67463
\(657\) −2.43919e40 −0.340838
\(658\) 3.70453e39 0.0506357
\(659\) 1.12152e40 0.149958 0.0749788 0.997185i \(-0.476111\pi\)
0.0749788 + 0.997185i \(0.476111\pi\)
\(660\) 1.15719e41 1.51362
\(661\) −3.04968e40 −0.390242 −0.195121 0.980779i \(-0.562510\pi\)
−0.195121 + 0.980779i \(0.562510\pi\)
\(662\) −1.37431e40 −0.172046
\(663\) 9.20510e39 0.112741
\(664\) −1.50622e40 −0.180489
\(665\) −4.79961e40 −0.562719
\(666\) −8.29408e38 −0.00951463
\(667\) 1.96637e40 0.220719
\(668\) 7.42082e40 0.815067
\(669\) 7.21815e40 0.775795
\(670\) −2.85617e40 −0.300399
\(671\) 1.16688e41 1.20102
\(672\) 7.36439e39 0.0741793
\(673\) 1.41854e41 1.39838 0.699188 0.714938i \(-0.253545\pi\)
0.699188 + 0.714938i \(0.253545\pi\)
\(674\) 2.27244e39 0.0219242
\(675\) 2.11116e40 0.199350
\(676\) 4.64796e40 0.429572
\(677\) −1.27467e41 −1.15308 −0.576542 0.817068i \(-0.695598\pi\)
−0.576542 + 0.817068i \(0.695598\pi\)
\(678\) −1.10511e40 −0.0978536
\(679\) −1.94648e40 −0.168709
\(680\) −1.00152e40 −0.0849732
\(681\) 2.76845e40 0.229934
\(682\) −8.75238e39 −0.0711627
\(683\) −1.29019e41 −1.02696 −0.513482 0.858101i \(-0.671645\pi\)
−0.513482 + 0.858101i \(0.671645\pi\)
\(684\) −4.40395e40 −0.343186
\(685\) 1.08989e41 0.831514
\(686\) 8.33818e38 0.00622833
\(687\) 9.27187e40 0.678101
\(688\) 1.02808e41 0.736197
\(689\) −1.23705e41 −0.867377
\(690\) −3.54687e39 −0.0243519
\(691\) −2.21087e41 −1.48638 −0.743192 0.669079i \(-0.766689\pi\)
−0.743192 + 0.669079i \(0.766689\pi\)
\(692\) 1.60606e40 0.105736
\(693\) 3.63901e40 0.234612
\(694\) 2.42469e40 0.153089
\(695\) −1.91948e40 −0.118687
\(696\) 1.88175e40 0.113953
\(697\) 7.64126e40 0.453200
\(698\) 2.13357e40 0.123938
\(699\) −3.08147e40 −0.175323
\(700\) 6.93192e40 0.386306
\(701\) 4.29431e40 0.234413 0.117207 0.993108i \(-0.462606\pi\)
0.117207 + 0.993108i \(0.462606\pi\)
\(702\) 3.11967e39 0.0166810
\(703\) 4.92963e40 0.258203
\(704\) 3.34278e41 1.71516
\(705\) 1.90345e41 0.956754
\(706\) 6.37101e38 0.00313719
\(707\) −6.79836e40 −0.327961
\(708\) 5.66456e40 0.267722
\(709\) 1.08342e41 0.501680 0.250840 0.968029i \(-0.419293\pi\)
0.250840 + 0.968029i \(0.419293\pi\)
\(710\) −5.56057e40 −0.252275
\(711\) −1.29882e41 −0.577353
\(712\) 6.74297e40 0.293691
\(713\) −1.98936e40 −0.0849014
\(714\) −1.56420e39 −0.00654136
\(715\) 4.87210e41 1.99654
\(716\) 4.65804e40 0.187053
\(717\) 9.94153e40 0.391224
\(718\) 1.79129e40 0.0690814
\(719\) −3.17789e41 −1.20108 −0.600538 0.799596i \(-0.705047\pi\)
−0.600538 + 0.799596i \(0.705047\pi\)
\(720\) 1.23300e41 0.456710
\(721\) 1.49369e40 0.0542248
\(722\) −2.87781e39 −0.0102394
\(723\) −2.09934e41 −0.732110
\(724\) −7.37837e40 −0.252204
\(725\) 2.66279e41 0.892147
\(726\) −5.00418e40 −0.164343
\(727\) −4.95180e41 −1.59410 −0.797048 0.603916i \(-0.793606\pi\)
−0.797048 + 0.603916i \(0.793606\pi\)
\(728\) 2.06248e40 0.0650856
\(729\) 1.19725e40 0.0370370
\(730\) 5.54936e40 0.168291
\(731\) −6.70141e40 −0.199234
\(732\) 1.26055e41 0.367410
\(733\) 3.31979e41 0.948643 0.474321 0.880352i \(-0.342693\pi\)
0.474321 + 0.880352i \(0.342693\pi\)
\(734\) −1.60555e40 −0.0449812
\(735\) 4.28431e40 0.117683
\(736\) −3.23461e40 −0.0871152
\(737\) −1.28708e42 −3.39882
\(738\) 2.58968e40 0.0670546
\(739\) −2.21021e41 −0.561163 −0.280581 0.959830i \(-0.590527\pi\)
−0.280581 + 0.959830i \(0.590527\pi\)
\(740\) −1.39930e41 −0.348378
\(741\) −1.85419e41 −0.452680
\(742\) 2.10208e40 0.0503261
\(743\) −2.82062e41 −0.662227 −0.331113 0.943591i \(-0.607424\pi\)
−0.331113 + 0.943591i \(0.607424\pi\)
\(744\) −1.90375e40 −0.0438331
\(745\) −5.53376e41 −1.24955
\(746\) −1.62627e40 −0.0360148
\(747\) −1.20873e41 −0.262531
\(748\) −2.24149e41 −0.477488
\(749\) −1.55402e41 −0.324691
\(750\) −1.66247e39 −0.00340694
\(751\) 3.22452e41 0.648166 0.324083 0.946029i \(-0.394944\pi\)
0.324083 + 0.946029i \(0.394944\pi\)
\(752\) 5.65633e41 1.11526
\(753\) −1.16399e41 −0.225123
\(754\) 3.93482e40 0.0746519
\(755\) 5.47372e41 1.01871
\(756\) 3.93113e40 0.0717715
\(757\) 7.91119e41 1.41694 0.708470 0.705741i \(-0.249386\pi\)
0.708470 + 0.705741i \(0.249386\pi\)
\(758\) −8.19619e40 −0.144015
\(759\) −1.59834e41 −0.275526
\(760\) 2.01738e41 0.341186
\(761\) −7.23142e41 −1.19990 −0.599951 0.800037i \(-0.704813\pi\)
−0.599951 + 0.800037i \(0.704813\pi\)
\(762\) −2.47117e40 −0.0402305
\(763\) −4.61162e41 −0.736627
\(764\) −1.23863e42 −1.94128
\(765\) −8.03714e40 −0.123598
\(766\) 2.85741e40 0.0431177
\(767\) 2.38495e41 0.353139
\(768\) 3.45532e41 0.502054
\(769\) −7.92719e41 −1.13028 −0.565142 0.824994i \(-0.691179\pi\)
−0.565142 + 0.824994i \(0.691179\pi\)
\(770\) −8.27904e40 −0.115842
\(771\) 6.75419e41 0.927437
\(772\) 1.01823e42 1.37213
\(773\) 8.00140e41 1.05819 0.529095 0.848563i \(-0.322532\pi\)
0.529095 + 0.848563i \(0.322532\pi\)
\(774\) −2.27115e40 −0.0294783
\(775\) −2.69393e41 −0.343171
\(776\) 8.18147e40 0.102291
\(777\) −4.40037e40 −0.0539989
\(778\) 8.38140e40 0.100951
\(779\) −1.53919e42 −1.81970
\(780\) 5.26321e41 0.610772
\(781\) −2.50578e42 −2.85432
\(782\) 6.87033e39 0.00768209
\(783\) 1.51009e41 0.165751
\(784\) 1.27313e41 0.137180
\(785\) 1.92640e42 2.03768
\(786\) −2.00369e40 −0.0208067
\(787\) 9.37568e41 0.955804 0.477902 0.878413i \(-0.341397\pi\)
0.477902 + 0.878413i \(0.341397\pi\)
\(788\) 4.40228e41 0.440603
\(789\) 6.07606e41 0.597043
\(790\) 2.95493e41 0.285072
\(791\) −5.86312e41 −0.555354
\(792\) −1.52955e41 −0.142249
\(793\) 5.30731e41 0.484633
\(794\) 2.23443e41 0.200341
\(795\) 1.08009e42 0.950903
\(796\) −1.84077e42 −1.59133
\(797\) −1.97737e42 −1.67858 −0.839289 0.543686i \(-0.817028\pi\)
−0.839289 + 0.543686i \(0.817028\pi\)
\(798\) 3.15078e40 0.0262650
\(799\) −3.68701e41 −0.301819
\(800\) −4.38021e41 −0.352120
\(801\) 5.41118e41 0.427189
\(802\) 1.10957e41 0.0860251
\(803\) 2.50072e42 1.90410
\(804\) −1.39041e42 −1.03975
\(805\) −1.88177e41 −0.138206
\(806\) −3.98083e40 −0.0287154
\(807\) −7.72658e41 −0.547420
\(808\) 2.85750e41 0.198848
\(809\) −6.85121e41 −0.468289 −0.234144 0.972202i \(-0.575229\pi\)
−0.234144 + 0.972202i \(0.575229\pi\)
\(810\) −2.72384e40 −0.0182873
\(811\) −6.85487e41 −0.452061 −0.226030 0.974120i \(-0.572575\pi\)
−0.226030 + 0.974120i \(0.572575\pi\)
\(812\) 4.95831e41 0.321197
\(813\) −9.21746e41 −0.586541
\(814\) 8.50332e40 0.0531538
\(815\) 9.13632e41 0.561030
\(816\) −2.38833e41 −0.144074
\(817\) 1.34987e42 0.799968
\(818\) −5.14598e40 −0.0299602
\(819\) 1.65512e41 0.0946703
\(820\) 4.36906e42 2.45520
\(821\) −7.85106e41 −0.433463 −0.216732 0.976231i \(-0.569540\pi\)
−0.216732 + 0.976231i \(0.569540\pi\)
\(822\) −7.15477e40 −0.0388109
\(823\) 6.55490e41 0.349356 0.174678 0.984626i \(-0.444112\pi\)
0.174678 + 0.984626i \(0.444112\pi\)
\(824\) −6.27829e40 −0.0328773
\(825\) −2.16442e42 −1.11368
\(826\) −4.05268e40 −0.0204895
\(827\) 2.86888e41 0.142522 0.0712612 0.997458i \(-0.477298\pi\)
0.0712612 + 0.997458i \(0.477298\pi\)
\(828\) −1.72665e41 −0.0842875
\(829\) 8.78790e41 0.421545 0.210773 0.977535i \(-0.432402\pi\)
0.210773 + 0.977535i \(0.432402\pi\)
\(830\) 2.74996e41 0.129626
\(831\) 5.95657e41 0.275919
\(832\) 1.52039e42 0.692096
\(833\) −8.29876e40 −0.0371245
\(834\) 1.26007e40 0.00553973
\(835\) −2.72796e42 −1.17865
\(836\) 4.51505e42 1.91722
\(837\) −1.52774e41 −0.0637574
\(838\) −8.07592e40 −0.0331248
\(839\) 4.67265e42 1.88371 0.941856 0.336016i \(-0.109080\pi\)
0.941856 + 0.336016i \(0.109080\pi\)
\(840\) −1.80079e41 −0.0713532
\(841\) −6.63024e41 −0.258219
\(842\) 2.68143e41 0.102646
\(843\) 4.47275e41 0.168296
\(844\) 4.29494e42 1.58852
\(845\) −1.70863e42 −0.621193
\(846\) −1.24955e41 −0.0446565
\(847\) −2.65494e42 −0.932706
\(848\) 3.20961e42 1.10844
\(849\) −1.58017e42 −0.536467
\(850\) 9.30358e40 0.0310510
\(851\) 1.93275e41 0.0634156
\(852\) −2.70693e42 −0.873179
\(853\) 1.89245e42 0.600156 0.300078 0.953915i \(-0.402987\pi\)
0.300078 + 0.953915i \(0.402987\pi\)
\(854\) −9.01857e40 −0.0281189
\(855\) 1.61893e42 0.496272
\(856\) 6.53189e41 0.196865
\(857\) −1.78935e42 −0.530240 −0.265120 0.964215i \(-0.585412\pi\)
−0.265120 + 0.964215i \(0.585412\pi\)
\(858\) −3.19837e41 −0.0931887
\(859\) −5.36533e42 −1.53708 −0.768538 0.639804i \(-0.779016\pi\)
−0.768538 + 0.639804i \(0.779016\pi\)
\(860\) −3.83167e42 −1.07935
\(861\) 1.37394e42 0.380559
\(862\) −5.27949e41 −0.143793
\(863\) 2.68416e42 0.718871 0.359436 0.933170i \(-0.382969\pi\)
0.359436 + 0.933170i \(0.382969\pi\)
\(864\) −2.48405e41 −0.0654200
\(865\) −5.90401e41 −0.152902
\(866\) −6.65531e40 −0.0169496
\(867\) −2.14956e42 −0.538360
\(868\) −5.01629e41 −0.123551
\(869\) 1.33159e43 3.22540
\(870\) −3.43557e41 −0.0818407
\(871\) −5.85402e42 −1.37149
\(872\) 1.93837e42 0.446629
\(873\) 6.56557e41 0.148787
\(874\) −1.38390e41 −0.0308453
\(875\) −8.82011e40 −0.0193356
\(876\) 2.70147e42 0.582493
\(877\) 2.15557e41 0.0457160 0.0228580 0.999739i \(-0.492723\pi\)
0.0228580 + 0.999739i \(0.492723\pi\)
\(878\) −9.63706e41 −0.201036
\(879\) −3.65418e42 −0.749810
\(880\) −1.26410e43 −2.55143
\(881\) −1.18085e42 −0.234446 −0.117223 0.993106i \(-0.537399\pi\)
−0.117223 + 0.993106i \(0.537399\pi\)
\(882\) −2.81251e40 −0.00549287
\(883\) 8.24508e42 1.58404 0.792018 0.610498i \(-0.209031\pi\)
0.792018 + 0.610498i \(0.209031\pi\)
\(884\) −1.01949e42 −0.192675
\(885\) −2.08234e42 −0.387146
\(886\) −1.92199e40 −0.00351530
\(887\) −6.80014e42 −1.22356 −0.611780 0.791028i \(-0.709546\pi\)
−0.611780 + 0.791028i \(0.709546\pi\)
\(888\) 1.84957e41 0.0327404
\(889\) −1.31106e42 −0.228322
\(890\) −1.23109e42 −0.210928
\(891\) −1.22745e42 −0.206909
\(892\) −7.99430e42 −1.32584
\(893\) 7.42679e42 1.21187
\(894\) 3.63273e41 0.0583229
\(895\) −1.71234e42 −0.270493
\(896\) −1.08498e42 −0.168638
\(897\) −7.26969e41 −0.111180
\(898\) −1.70950e41 −0.0257254
\(899\) −1.92693e42 −0.285332
\(900\) −2.33817e42 −0.340690
\(901\) −2.09215e42 −0.299973
\(902\) −2.65501e42 −0.374603
\(903\) −1.20495e42 −0.167300
\(904\) 2.46440e42 0.336720
\(905\) 2.71236e42 0.364705
\(906\) −3.59331e41 −0.0475484
\(907\) −1.57296e42 −0.204839 −0.102420 0.994741i \(-0.532658\pi\)
−0.102420 + 0.994741i \(0.532658\pi\)
\(908\) −3.06614e42 −0.392959
\(909\) 2.29312e42 0.289235
\(910\) −3.76554e41 −0.0467441
\(911\) −1.50764e42 −0.184196 −0.0920981 0.995750i \(-0.529357\pi\)
−0.0920981 + 0.995750i \(0.529357\pi\)
\(912\) 4.81084e42 0.578490
\(913\) 1.23922e43 1.46664
\(914\) −8.98832e41 −0.104703
\(915\) −4.63391e42 −0.531302
\(916\) −1.02689e43 −1.15888
\(917\) −1.06305e42 −0.118086
\(918\) 5.27612e40 0.00576893
\(919\) 1.99622e42 0.214849 0.107424 0.994213i \(-0.465740\pi\)
0.107424 + 0.994213i \(0.465740\pi\)
\(920\) 7.90949e41 0.0837963
\(921\) 8.27763e42 0.863259
\(922\) 8.04390e41 0.0825788
\(923\) −1.13970e43 −1.15177
\(924\) −4.03030e42 −0.400954
\(925\) 2.61727e42 0.256326
\(926\) 5.40717e40 0.00521328
\(927\) −5.03827e41 −0.0478218
\(928\) −3.13311e42 −0.292772
\(929\) −9.33009e42 −0.858338 −0.429169 0.903224i \(-0.641193\pi\)
−0.429169 + 0.903224i \(0.641193\pi\)
\(930\) 3.47574e41 0.0314807
\(931\) 1.67163e42 0.149063
\(932\) 3.41281e42 0.299628
\(933\) −5.03375e42 −0.435119
\(934\) 2.46770e42 0.210021
\(935\) 8.23990e42 0.690484
\(936\) −6.95685e41 −0.0574001
\(937\) 3.45465e42 0.280660 0.140330 0.990105i \(-0.455184\pi\)
0.140330 + 0.990105i \(0.455184\pi\)
\(938\) 9.94759e41 0.0795750
\(939\) 2.59919e41 0.0204733
\(940\) −2.10813e43 −1.63510
\(941\) 2.35450e43 1.79825 0.899124 0.437694i \(-0.144205\pi\)
0.899124 + 0.437694i \(0.144205\pi\)
\(942\) −1.26462e42 −0.0951088
\(943\) −6.03466e42 −0.446923
\(944\) −6.18792e42 −0.451285
\(945\) −1.44512e42 −0.103787
\(946\) 2.32845e42 0.164682
\(947\) −5.81953e42 −0.405334 −0.202667 0.979248i \(-0.564961\pi\)
−0.202667 + 0.979248i \(0.564961\pi\)
\(948\) 1.43848e43 0.986698
\(949\) 1.13740e43 0.768339
\(950\) −1.87403e42 −0.124677
\(951\) 1.28066e43 0.839105
\(952\) 3.48815e41 0.0225092
\(953\) 2.14066e43 1.36051 0.680253 0.732977i \(-0.261870\pi\)
0.680253 + 0.732977i \(0.261870\pi\)
\(954\) −7.09043e41 −0.0443835
\(955\) 4.55333e43 2.80724
\(956\) −1.10105e43 −0.668604
\(957\) −1.54818e43 −0.925974
\(958\) −9.92361e41 −0.0584615
\(959\) −3.79592e42 −0.220266
\(960\) −1.32748e43 −0.758744
\(961\) −1.58124e43 −0.890245
\(962\) 3.86755e41 0.0214485
\(963\) 5.24179e42 0.286351
\(964\) 2.32507e43 1.25118
\(965\) −3.74309e43 −1.98420
\(966\) 1.23532e41 0.00645076
\(967\) 2.74428e43 1.41171 0.705853 0.708358i \(-0.250564\pi\)
0.705853 + 0.708358i \(0.250564\pi\)
\(968\) 1.11593e43 0.565514
\(969\) −3.13589e42 −0.156555
\(970\) −1.49372e42 −0.0734647
\(971\) 1.57511e42 0.0763189 0.0381594 0.999272i \(-0.487851\pi\)
0.0381594 + 0.999272i \(0.487851\pi\)
\(972\) −1.32599e42 −0.0632965
\(973\) 6.68525e41 0.0314399
\(974\) −8.84879e41 −0.0409996
\(975\) −9.84439e42 −0.449388
\(976\) −1.37702e43 −0.619324
\(977\) 3.21646e42 0.142530 0.0712652 0.997457i \(-0.477296\pi\)
0.0712652 + 0.997457i \(0.477296\pi\)
\(978\) −5.99769e41 −0.0261861
\(979\) −5.54769e43 −2.38651
\(980\) −4.74500e42 −0.201121
\(981\) 1.55552e43 0.649644
\(982\) −1.24887e42 −0.0513928
\(983\) 6.59024e42 0.267224 0.133612 0.991034i \(-0.457342\pi\)
0.133612 + 0.991034i \(0.457342\pi\)
\(984\) −5.77496e42 −0.230739
\(985\) −1.61832e43 −0.637145
\(986\) 6.65473e41 0.0258176
\(987\) −6.62944e42 −0.253442
\(988\) 2.05357e43 0.773632
\(989\) 5.29241e42 0.196475
\(990\) 2.79256e42 0.102163
\(991\) 1.96287e43 0.707659 0.353830 0.935310i \(-0.384879\pi\)
0.353830 + 0.935310i \(0.384879\pi\)
\(992\) 3.16974e42 0.112617
\(993\) 2.45941e43 0.861125
\(994\) 1.93666e42 0.0668269
\(995\) 6.76683e43 2.30118
\(996\) 1.33870e43 0.448666
\(997\) 5.40710e43 1.78601 0.893007 0.450043i \(-0.148591\pi\)
0.893007 + 0.450043i \(0.148591\pi\)
\(998\) 2.34631e42 0.0763822
\(999\) 1.48427e42 0.0476225
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 21.30.a.d.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.30.a.d.1.5 8 1.1 even 1 trivial