Properties

Label 10.20.a.a.1.1
Level $10$
Weight $20$
Character 10.1
Self dual yes
Analytic conductor $22.882$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,20,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.8816696556\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-512.000 q^{2} -26622.0 q^{3} +262144. q^{4} -1.95312e6 q^{5} +1.36305e7 q^{6} -3.98840e7 q^{7} -1.34218e8 q^{8} -4.53531e8 q^{9} +O(q^{10})\) \(q-512.000 q^{2} -26622.0 q^{3} +262144. q^{4} -1.95312e6 q^{5} +1.36305e7 q^{6} -3.98840e7 q^{7} -1.34218e8 q^{8} -4.53531e8 q^{9} +1.00000e9 q^{10} -1.01616e10 q^{11} -6.97880e9 q^{12} -2.69706e10 q^{13} +2.04206e10 q^{14} +5.19961e10 q^{15} +6.87195e10 q^{16} -8.01548e10 q^{17} +2.32208e11 q^{18} -1.16977e12 q^{19} -5.12000e11 q^{20} +1.06179e12 q^{21} +5.20273e12 q^{22} +1.37959e13 q^{23} +3.57314e12 q^{24} +3.81470e12 q^{25} +1.38090e13 q^{26} +4.30156e13 q^{27} -1.04554e13 q^{28} +6.53248e13 q^{29} -2.66220e13 q^{30} -8.92654e12 q^{31} -3.51844e13 q^{32} +2.70522e14 q^{33} +4.10392e13 q^{34} +7.78985e13 q^{35} -1.18890e14 q^{36} +5.25455e14 q^{37} +5.98923e14 q^{38} +7.18013e14 q^{39} +2.62144e14 q^{40} -2.63523e15 q^{41} -5.43638e14 q^{42} -1.50171e15 q^{43} -2.66380e15 q^{44} +8.85802e14 q^{45} -7.06349e15 q^{46} -3.65161e15 q^{47} -1.82945e15 q^{48} -9.80816e15 q^{49} -1.95312e15 q^{50} +2.13388e15 q^{51} -7.07019e15 q^{52} +4.33068e16 q^{53} -2.20240e16 q^{54} +1.98468e16 q^{55} +5.35314e15 q^{56} +3.11417e16 q^{57} -3.34463e16 q^{58} +5.16521e16 q^{59} +1.36305e16 q^{60} +4.52000e16 q^{61} +4.57039e15 q^{62} +1.80886e16 q^{63} +1.80144e16 q^{64} +5.26771e16 q^{65} -1.38507e17 q^{66} +3.22077e17 q^{67} -2.10121e16 q^{68} -3.67274e17 q^{69} -3.98840e16 q^{70} +3.93293e17 q^{71} +6.08718e16 q^{72} -6.72470e17 q^{73} -2.69033e17 q^{74} -1.01555e17 q^{75} -3.06649e17 q^{76} +4.05285e17 q^{77} -3.67622e17 q^{78} -4.82639e17 q^{79} -1.34218e17 q^{80} -6.18041e17 q^{81} +1.34924e18 q^{82} -3.13265e17 q^{83} +2.78343e17 q^{84} +1.56552e17 q^{85} +7.68875e17 q^{86} -1.73908e18 q^{87} +1.36386e18 q^{88} -4.23010e18 q^{89} -4.53531e17 q^{90} +1.07570e18 q^{91} +3.61651e18 q^{92} +2.37642e17 q^{93} +1.86962e18 q^{94} +2.28471e18 q^{95} +9.36678e17 q^{96} +3.54705e17 q^{97} +5.02178e18 q^{98} +4.60859e18 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −512.000 −0.707107
\(3\) −26622.0 −0.780888 −0.390444 0.920627i \(-0.627678\pi\)
−0.390444 + 0.920627i \(0.627678\pi\)
\(4\) 262144. 0.500000
\(5\) −1.95312e6 −0.447214
\(6\) 1.36305e7 0.552171
\(7\) −3.98840e7 −0.373566 −0.186783 0.982401i \(-0.559806\pi\)
−0.186783 + 0.982401i \(0.559806\pi\)
\(8\) −1.34218e8 −0.353553
\(9\) −4.53531e8 −0.390214
\(10\) 1.00000e9 0.316228
\(11\) −1.01616e10 −1.29936 −0.649682 0.760206i \(-0.725098\pi\)
−0.649682 + 0.760206i \(0.725098\pi\)
\(12\) −6.97880e9 −0.390444
\(13\) −2.69706e10 −0.705390 −0.352695 0.935738i \(-0.614735\pi\)
−0.352695 + 0.935738i \(0.614735\pi\)
\(14\) 2.04206e10 0.264151
\(15\) 5.19961e10 0.349224
\(16\) 6.87195e10 0.250000
\(17\) −8.01548e10 −0.163932 −0.0819661 0.996635i \(-0.526120\pi\)
−0.0819661 + 0.996635i \(0.526120\pi\)
\(18\) 2.32208e11 0.275923
\(19\) −1.16977e12 −0.831653 −0.415826 0.909444i \(-0.636508\pi\)
−0.415826 + 0.909444i \(0.636508\pi\)
\(20\) −5.12000e11 −0.223607
\(21\) 1.06179e12 0.291713
\(22\) 5.20273e12 0.918789
\(23\) 1.37959e13 1.59711 0.798555 0.601921i \(-0.205598\pi\)
0.798555 + 0.601921i \(0.205598\pi\)
\(24\) 3.57314e12 0.276086
\(25\) 3.81470e12 0.200000
\(26\) 1.38090e13 0.498786
\(27\) 4.30156e13 1.08560
\(28\) −1.04554e13 −0.186783
\(29\) 6.53248e13 0.836174 0.418087 0.908407i \(-0.362701\pi\)
0.418087 + 0.908407i \(0.362701\pi\)
\(30\) −2.66220e13 −0.246938
\(31\) −8.92654e12 −0.0606383 −0.0303192 0.999540i \(-0.509652\pi\)
−0.0303192 + 0.999540i \(0.509652\pi\)
\(32\) −3.51844e13 −0.176777
\(33\) 2.70522e14 1.01466
\(34\) 4.10392e13 0.115918
\(35\) 7.78985e13 0.167064
\(36\) −1.18890e14 −0.195107
\(37\) 5.25455e14 0.664690 0.332345 0.943158i \(-0.392160\pi\)
0.332345 + 0.943158i \(0.392160\pi\)
\(38\) 5.98923e14 0.588067
\(39\) 7.18013e14 0.550831
\(40\) 2.62144e14 0.158114
\(41\) −2.63523e15 −1.25710 −0.628552 0.777767i \(-0.716352\pi\)
−0.628552 + 0.777767i \(0.716352\pi\)
\(42\) −5.43638e14 −0.206273
\(43\) −1.50171e15 −0.455654 −0.227827 0.973702i \(-0.573162\pi\)
−0.227827 + 0.973702i \(0.573162\pi\)
\(44\) −2.66380e15 −0.649682
\(45\) 8.85802e14 0.174509
\(46\) −7.06349e15 −1.12933
\(47\) −3.65161e15 −0.475943 −0.237971 0.971272i \(-0.576482\pi\)
−0.237971 + 0.971272i \(0.576482\pi\)
\(48\) −1.82945e15 −0.195222
\(49\) −9.80816e15 −0.860448
\(50\) −1.95312e15 −0.141421
\(51\) 2.13388e15 0.128013
\(52\) −7.07019e15 −0.352695
\(53\) 4.33068e16 1.80275 0.901375 0.433040i \(-0.142559\pi\)
0.901375 + 0.433040i \(0.142559\pi\)
\(54\) −2.20240e16 −0.767636
\(55\) 1.98468e16 0.581093
\(56\) 5.35314e15 0.132076
\(57\) 3.11417e16 0.649428
\(58\) −3.34463e16 −0.591265
\(59\) 5.16521e16 0.776236 0.388118 0.921610i \(-0.373125\pi\)
0.388118 + 0.921610i \(0.373125\pi\)
\(60\) 1.36305e16 0.174612
\(61\) 4.52000e16 0.494886 0.247443 0.968902i \(-0.420410\pi\)
0.247443 + 0.968902i \(0.420410\pi\)
\(62\) 4.57039e15 0.0428778
\(63\) 1.80886e16 0.145771
\(64\) 1.80144e16 0.125000
\(65\) 5.26771e16 0.315460
\(66\) −1.38507e17 −0.717471
\(67\) 3.22077e17 1.44627 0.723135 0.690707i \(-0.242700\pi\)
0.723135 + 0.690707i \(0.242700\pi\)
\(68\) −2.10121e16 −0.0819661
\(69\) −3.67274e17 −1.24716
\(70\) −3.98840e16 −0.118132
\(71\) 3.93293e17 1.01803 0.509017 0.860756i \(-0.330009\pi\)
0.509017 + 0.860756i \(0.330009\pi\)
\(72\) 6.08718e16 0.137961
\(73\) −6.72470e17 −1.33692 −0.668460 0.743748i \(-0.733046\pi\)
−0.668460 + 0.743748i \(0.733046\pi\)
\(74\) −2.69033e17 −0.470006
\(75\) −1.01555e17 −0.156178
\(76\) −3.06649e17 −0.415826
\(77\) 4.05285e17 0.485398
\(78\) −3.67622e17 −0.389496
\(79\) −4.82639e17 −0.453070 −0.226535 0.974003i \(-0.572740\pi\)
−0.226535 + 0.974003i \(0.572740\pi\)
\(80\) −1.34218e17 −0.111803
\(81\) −6.18041e17 −0.457519
\(82\) 1.34924e18 0.888907
\(83\) −3.13265e17 −0.183938 −0.0919688 0.995762i \(-0.529316\pi\)
−0.0919688 + 0.995762i \(0.529316\pi\)
\(84\) 2.78343e17 0.145857
\(85\) 1.56552e17 0.0733128
\(86\) 7.68875e17 0.322196
\(87\) −1.73908e18 −0.652959
\(88\) 1.36386e18 0.459394
\(89\) −4.23010e18 −1.27981 −0.639905 0.768454i \(-0.721026\pi\)
−0.639905 + 0.768454i \(0.721026\pi\)
\(90\) −4.53531e17 −0.123396
\(91\) 1.07570e18 0.263510
\(92\) 3.61651e18 0.798555
\(93\) 2.37642e17 0.0473517
\(94\) 1.86962e18 0.336542
\(95\) 2.28471e18 0.371926
\(96\) 9.36678e17 0.138043
\(97\) 3.54705e17 0.0473736 0.0236868 0.999719i \(-0.492460\pi\)
0.0236868 + 0.999719i \(0.492460\pi\)
\(98\) 5.02178e18 0.608429
\(99\) 4.60859e18 0.507030
\(100\) 1.00000e18 0.100000
\(101\) 1.29200e19 1.17546 0.587730 0.809057i \(-0.300022\pi\)
0.587730 + 0.809057i \(0.300022\pi\)
\(102\) −1.09255e18 −0.0905187
\(103\) 1.35900e19 1.02628 0.513139 0.858305i \(-0.328482\pi\)
0.513139 + 0.858305i \(0.328482\pi\)
\(104\) 3.61994e18 0.249393
\(105\) −2.07381e18 −0.130458
\(106\) −2.21731e19 −1.27474
\(107\) −3.28307e19 −1.72637 −0.863185 0.504887i \(-0.831534\pi\)
−0.863185 + 0.504887i \(0.831534\pi\)
\(108\) 1.12763e19 0.542801
\(109\) −3.70765e19 −1.63511 −0.817554 0.575851i \(-0.804671\pi\)
−0.817554 + 0.575851i \(0.804671\pi\)
\(110\) −1.01616e19 −0.410895
\(111\) −1.39887e19 −0.519048
\(112\) −2.74081e18 −0.0933916
\(113\) 2.45337e19 0.768277 0.384139 0.923275i \(-0.374498\pi\)
0.384139 + 0.923275i \(0.374498\pi\)
\(114\) −1.59445e19 −0.459215
\(115\) −2.69451e19 −0.714250
\(116\) 1.71245e19 0.418087
\(117\) 1.22320e19 0.275253
\(118\) −2.64459e19 −0.548882
\(119\) 3.19689e18 0.0612396
\(120\) −6.97880e18 −0.123469
\(121\) 4.20986e19 0.688346
\(122\) −2.31424e19 −0.349937
\(123\) 7.01550e19 0.981658
\(124\) −2.34004e18 −0.0303192
\(125\) −7.45058e18 −0.0894427
\(126\) −9.26138e18 −0.103075
\(127\) 3.49512e19 0.360850 0.180425 0.983589i \(-0.442253\pi\)
0.180425 + 0.983589i \(0.442253\pi\)
\(128\) −9.22337e18 −0.0883883
\(129\) 3.99785e19 0.355815
\(130\) −2.69706e19 −0.223064
\(131\) 2.02167e20 1.55465 0.777325 0.629099i \(-0.216576\pi\)
0.777325 + 0.629099i \(0.216576\pi\)
\(132\) 7.09156e19 0.507329
\(133\) 4.66552e19 0.310677
\(134\) −1.64904e20 −1.02267
\(135\) −8.40149e19 −0.485496
\(136\) 1.07582e19 0.0579588
\(137\) 1.09975e20 0.552647 0.276323 0.961065i \(-0.410884\pi\)
0.276323 + 0.961065i \(0.410884\pi\)
\(138\) 1.88044e20 0.881879
\(139\) −3.97605e20 −1.74105 −0.870526 0.492123i \(-0.836221\pi\)
−0.870526 + 0.492123i \(0.836221\pi\)
\(140\) 2.04206e19 0.0835319
\(141\) 9.72131e19 0.371658
\(142\) −2.01366e20 −0.719859
\(143\) 2.74064e20 0.916559
\(144\) −3.11664e19 −0.0975535
\(145\) −1.27587e20 −0.373949
\(146\) 3.44304e20 0.945345
\(147\) 2.61113e20 0.671914
\(148\) 1.37745e20 0.332345
\(149\) −8.57609e20 −1.94097 −0.970487 0.241153i \(-0.922475\pi\)
−0.970487 + 0.241153i \(0.922475\pi\)
\(150\) 5.19961e19 0.110434
\(151\) −5.47594e20 −1.09189 −0.545944 0.837822i \(-0.683829\pi\)
−0.545944 + 0.837822i \(0.683829\pi\)
\(152\) 1.57004e20 0.294034
\(153\) 3.63526e19 0.0639687
\(154\) −2.07506e20 −0.343228
\(155\) 1.74346e19 0.0271183
\(156\) 1.88223e20 0.275415
\(157\) −4.22122e20 −0.581288 −0.290644 0.956831i \(-0.593869\pi\)
−0.290644 + 0.956831i \(0.593869\pi\)
\(158\) 2.47111e20 0.320369
\(159\) −1.15291e21 −1.40775
\(160\) 6.87195e19 0.0790569
\(161\) −5.50235e20 −0.596627
\(162\) 3.16437e20 0.323515
\(163\) 1.28823e21 1.24226 0.621128 0.783710i \(-0.286675\pi\)
0.621128 + 0.783710i \(0.286675\pi\)
\(164\) −6.90809e20 −0.628552
\(165\) −5.28362e20 −0.453769
\(166\) 1.60392e20 0.130064
\(167\) 2.10976e21 1.61594 0.807972 0.589221i \(-0.200565\pi\)
0.807972 + 0.589221i \(0.200565\pi\)
\(168\) −1.42511e20 −0.103136
\(169\) −7.34504e20 −0.502424
\(170\) −8.01548e19 −0.0518399
\(171\) 5.30527e20 0.324522
\(172\) −3.93664e20 −0.227827
\(173\) 1.13537e21 0.621868 0.310934 0.950431i \(-0.399358\pi\)
0.310934 + 0.950431i \(0.399358\pi\)
\(174\) 8.90407e20 0.461711
\(175\) −1.52145e20 −0.0747132
\(176\) −6.98298e20 −0.324841
\(177\) −1.37508e21 −0.606153
\(178\) 2.16581e21 0.904963
\(179\) 2.71859e21 1.07706 0.538530 0.842607i \(-0.318980\pi\)
0.538530 + 0.842607i \(0.318980\pi\)
\(180\) 2.32208e20 0.0872545
\(181\) 3.93952e21 1.40442 0.702209 0.711971i \(-0.252197\pi\)
0.702209 + 0.711971i \(0.252197\pi\)
\(182\) −5.50757e20 −0.186330
\(183\) −1.20332e21 −0.386451
\(184\) −1.85165e21 −0.564664
\(185\) −1.02628e21 −0.297258
\(186\) −1.21673e20 −0.0334827
\(187\) 8.14499e20 0.213008
\(188\) −9.57247e20 −0.237971
\(189\) −1.71564e21 −0.405544
\(190\) −1.16977e21 −0.262992
\(191\) 6.11189e21 1.30725 0.653625 0.756819i \(-0.273248\pi\)
0.653625 + 0.756819i \(0.273248\pi\)
\(192\) −4.79579e20 −0.0976110
\(193\) −3.21787e21 −0.623410 −0.311705 0.950179i \(-0.600900\pi\)
−0.311705 + 0.950179i \(0.600900\pi\)
\(194\) −1.81609e20 −0.0334982
\(195\) −1.40237e21 −0.246339
\(196\) −2.57115e21 −0.430224
\(197\) −4.85425e21 −0.773915 −0.386958 0.922098i \(-0.626474\pi\)
−0.386958 + 0.922098i \(0.626474\pi\)
\(198\) −2.35960e21 −0.358524
\(199\) 2.76811e21 0.400939 0.200470 0.979700i \(-0.435753\pi\)
0.200470 + 0.979700i \(0.435753\pi\)
\(200\) −5.12000e20 −0.0707107
\(201\) −8.57434e21 −1.12937
\(202\) −6.61501e21 −0.831176
\(203\) −2.60541e21 −0.312366
\(204\) 5.59384e20 0.0640064
\(205\) 5.14693e21 0.562194
\(206\) −6.95807e21 −0.725689
\(207\) −6.25686e21 −0.623215
\(208\) −1.85341e21 −0.176348
\(209\) 1.18867e22 1.08062
\(210\) 1.06179e21 0.0922479
\(211\) 3.92985e21 0.326356 0.163178 0.986597i \(-0.447825\pi\)
0.163178 + 0.986597i \(0.447825\pi\)
\(212\) 1.13526e22 0.901375
\(213\) −1.04703e22 −0.794970
\(214\) 1.68093e22 1.22073
\(215\) 2.93302e21 0.203775
\(216\) −5.77346e21 −0.383818
\(217\) 3.56026e20 0.0226524
\(218\) 1.89832e22 1.15620
\(219\) 1.79025e22 1.04399
\(220\) 5.20273e21 0.290547
\(221\) 2.16183e21 0.115636
\(222\) 7.16219e21 0.367022
\(223\) −3.09507e21 −0.151976 −0.0759880 0.997109i \(-0.524211\pi\)
−0.0759880 + 0.997109i \(0.524211\pi\)
\(224\) 1.40329e21 0.0660378
\(225\) −1.73008e21 −0.0780428
\(226\) −1.25613e22 −0.543254
\(227\) 1.39820e22 0.579859 0.289929 0.957048i \(-0.406368\pi\)
0.289929 + 0.957048i \(0.406368\pi\)
\(228\) 8.16360e21 0.324714
\(229\) −2.64165e22 −1.00795 −0.503975 0.863719i \(-0.668129\pi\)
−0.503975 + 0.863719i \(0.668129\pi\)
\(230\) 1.37959e22 0.505051
\(231\) −1.07895e22 −0.379042
\(232\) −8.76774e21 −0.295632
\(233\) −5.72660e22 −1.85360 −0.926800 0.375556i \(-0.877452\pi\)
−0.926800 + 0.375556i \(0.877452\pi\)
\(234\) −6.26279e21 −0.194633
\(235\) 7.13205e21 0.212848
\(236\) 1.35403e22 0.388118
\(237\) 1.28488e22 0.353797
\(238\) −1.63681e21 −0.0433029
\(239\) 5.61325e22 1.42703 0.713517 0.700638i \(-0.247101\pi\)
0.713517 + 0.700638i \(0.247101\pi\)
\(240\) 3.57314e21 0.0873059
\(241\) −4.77482e22 −1.12149 −0.560744 0.827989i \(-0.689485\pi\)
−0.560744 + 0.827989i \(0.689485\pi\)
\(242\) −2.15545e22 −0.486734
\(243\) −3.35419e22 −0.728330
\(244\) 1.18489e22 0.247443
\(245\) 1.91566e22 0.384804
\(246\) −3.59194e22 −0.694137
\(247\) 3.15495e22 0.586640
\(248\) 1.19810e21 0.0214389
\(249\) 8.33975e21 0.143635
\(250\) 3.81470e21 0.0632456
\(251\) 1.05736e23 1.68781 0.843907 0.536490i \(-0.180250\pi\)
0.843907 + 0.536490i \(0.180250\pi\)
\(252\) 4.74182e21 0.0728854
\(253\) −1.40188e23 −2.07523
\(254\) −1.78950e22 −0.255160
\(255\) −4.16773e21 −0.0572491
\(256\) 4.72237e21 0.0625000
\(257\) −3.19013e22 −0.406859 −0.203429 0.979090i \(-0.565209\pi\)
−0.203429 + 0.979090i \(0.565209\pi\)
\(258\) −2.04690e22 −0.251599
\(259\) −2.09572e22 −0.248306
\(260\) 1.38090e22 0.157730
\(261\) −2.96268e22 −0.326287
\(262\) −1.03509e23 −1.09930
\(263\) 5.93491e22 0.607903 0.303952 0.952687i \(-0.401694\pi\)
0.303952 + 0.952687i \(0.401694\pi\)
\(264\) −3.63088e22 −0.358736
\(265\) −8.45836e22 −0.806214
\(266\) −2.38875e22 −0.219682
\(267\) 1.12614e23 0.999389
\(268\) 8.44306e22 0.723135
\(269\) 1.96711e23 1.62623 0.813115 0.582103i \(-0.197770\pi\)
0.813115 + 0.582103i \(0.197770\pi\)
\(270\) 4.30156e22 0.343297
\(271\) 1.06648e23 0.821760 0.410880 0.911689i \(-0.365221\pi\)
0.410880 + 0.911689i \(0.365221\pi\)
\(272\) −5.50819e21 −0.0409831
\(273\) −2.86372e22 −0.205772
\(274\) −5.63071e22 −0.390780
\(275\) −3.87633e22 −0.259873
\(276\) −9.62787e22 −0.623582
\(277\) 6.44307e22 0.403213 0.201607 0.979467i \(-0.435384\pi\)
0.201607 + 0.979467i \(0.435384\pi\)
\(278\) 2.03574e23 1.23111
\(279\) 4.04846e21 0.0236619
\(280\) −1.04554e22 −0.0590660
\(281\) 1.73023e23 0.944918 0.472459 0.881353i \(-0.343367\pi\)
0.472459 + 0.881353i \(0.343367\pi\)
\(282\) −4.97731e22 −0.262802
\(283\) 7.83542e22 0.400029 0.200014 0.979793i \(-0.435901\pi\)
0.200014 + 0.979793i \(0.435901\pi\)
\(284\) 1.03099e23 0.509017
\(285\) −6.08236e22 −0.290433
\(286\) −1.40321e23 −0.648105
\(287\) 1.05103e23 0.469612
\(288\) 1.59572e22 0.0689807
\(289\) −2.32648e23 −0.973126
\(290\) 6.53248e22 0.264422
\(291\) −9.44296e21 −0.0369934
\(292\) −1.76284e23 −0.668460
\(293\) 3.32561e23 1.22076 0.610379 0.792110i \(-0.291017\pi\)
0.610379 + 0.792110i \(0.291017\pi\)
\(294\) −1.33690e23 −0.475115
\(295\) −1.00883e23 −0.347143
\(296\) −7.05253e22 −0.235003
\(297\) −4.37107e23 −1.41059
\(298\) 4.39096e23 1.37248
\(299\) −3.72084e23 −1.12659
\(300\) −2.66220e22 −0.0780888
\(301\) 5.98942e22 0.170217
\(302\) 2.80368e23 0.772081
\(303\) −3.43955e23 −0.917903
\(304\) −8.03861e22 −0.207913
\(305\) −8.82813e22 −0.221320
\(306\) −1.86125e22 −0.0452327
\(307\) −4.74187e23 −1.11721 −0.558605 0.829434i \(-0.688663\pi\)
−0.558605 + 0.829434i \(0.688663\pi\)
\(308\) 1.06243e23 0.242699
\(309\) −3.61793e23 −0.801409
\(310\) −8.92654e21 −0.0191755
\(311\) 2.15965e23 0.449944 0.224972 0.974365i \(-0.427771\pi\)
0.224972 + 0.974365i \(0.427771\pi\)
\(312\) −9.63700e22 −0.194748
\(313\) −5.79666e23 −1.13634 −0.568168 0.822913i \(-0.692348\pi\)
−0.568168 + 0.822913i \(0.692348\pi\)
\(314\) 2.16126e23 0.411032
\(315\) −3.53293e22 −0.0651907
\(316\) −1.26521e23 −0.226535
\(317\) 6.42977e23 1.11720 0.558602 0.829436i \(-0.311338\pi\)
0.558602 + 0.829436i \(0.311338\pi\)
\(318\) 5.90292e23 0.995426
\(319\) −6.63803e23 −1.08649
\(320\) −3.51844e22 −0.0559017
\(321\) 8.74018e23 1.34810
\(322\) 2.81721e23 0.421879
\(323\) 9.37628e22 0.136335
\(324\) −1.62016e23 −0.228760
\(325\) −1.02885e23 −0.141078
\(326\) −6.59573e23 −0.878407
\(327\) 9.87050e23 1.27684
\(328\) 3.53694e23 0.444454
\(329\) 1.45641e23 0.177796
\(330\) 2.70522e23 0.320863
\(331\) 1.44955e24 1.67058 0.835288 0.549813i \(-0.185301\pi\)
0.835288 + 0.549813i \(0.185301\pi\)
\(332\) −8.21206e22 −0.0919688
\(333\) −2.38310e23 −0.259371
\(334\) −1.08020e24 −1.14265
\(335\) −6.29057e23 −0.646792
\(336\) 7.29658e22 0.0729283
\(337\) 1.33044e24 1.29274 0.646370 0.763024i \(-0.276286\pi\)
0.646370 + 0.763024i \(0.276286\pi\)
\(338\) 3.76066e23 0.355268
\(339\) −6.53137e23 −0.599939
\(340\) 4.10392e22 0.0366564
\(341\) 9.07077e22 0.0787912
\(342\) −2.71630e23 −0.229472
\(343\) 8.45823e23 0.695001
\(344\) 2.01556e23 0.161098
\(345\) 7.17332e23 0.557749
\(346\) −5.81308e23 −0.439727
\(347\) −8.08452e23 −0.595010 −0.297505 0.954720i \(-0.596154\pi\)
−0.297505 + 0.954720i \(0.596154\pi\)
\(348\) −4.55888e23 −0.326479
\(349\) 1.87880e24 1.30930 0.654648 0.755933i \(-0.272817\pi\)
0.654648 + 0.755933i \(0.272817\pi\)
\(350\) 7.78985e22 0.0528302
\(351\) −1.16016e24 −0.765773
\(352\) 3.57529e23 0.229697
\(353\) −2.51702e24 −1.57408 −0.787039 0.616903i \(-0.788387\pi\)
−0.787039 + 0.616903i \(0.788387\pi\)
\(354\) 7.04042e23 0.428615
\(355\) −7.68151e23 −0.455279
\(356\) −1.10890e24 −0.639905
\(357\) −8.51077e22 −0.0478212
\(358\) −1.39192e24 −0.761596
\(359\) −2.04732e23 −0.109091 −0.0545453 0.998511i \(-0.517371\pi\)
−0.0545453 + 0.998511i \(0.517371\pi\)
\(360\) −1.18890e23 −0.0616982
\(361\) −6.10053e23 −0.308354
\(362\) −2.01703e24 −0.993074
\(363\) −1.12075e24 −0.537521
\(364\) 2.81988e23 0.131755
\(365\) 1.31342e24 0.597889
\(366\) 6.16098e23 0.273262
\(367\) −5.80738e23 −0.250988 −0.125494 0.992094i \(-0.540052\pi\)
−0.125494 + 0.992094i \(0.540052\pi\)
\(368\) 9.48046e23 0.399278
\(369\) 1.19516e24 0.490540
\(370\) 5.25455e23 0.210193
\(371\) −1.72725e24 −0.673446
\(372\) 6.22965e22 0.0236759
\(373\) 4.35467e24 1.61332 0.806662 0.591012i \(-0.201272\pi\)
0.806662 + 0.591012i \(0.201272\pi\)
\(374\) −4.17023e23 −0.150619
\(375\) 1.98349e23 0.0698447
\(376\) 4.90111e23 0.168271
\(377\) −1.76185e24 −0.589829
\(378\) 8.78406e23 0.286763
\(379\) 2.38731e24 0.760041 0.380021 0.924978i \(-0.375917\pi\)
0.380021 + 0.924978i \(0.375917\pi\)
\(380\) 5.98923e23 0.185963
\(381\) −9.30471e23 −0.281784
\(382\) −3.12929e24 −0.924365
\(383\) −3.07728e24 −0.886704 −0.443352 0.896348i \(-0.646211\pi\)
−0.443352 + 0.896348i \(0.646211\pi\)
\(384\) 2.45545e23 0.0690214
\(385\) −7.91572e23 −0.217077
\(386\) 1.64755e24 0.440818
\(387\) 6.81071e23 0.177803
\(388\) 9.29838e22 0.0236868
\(389\) 5.97646e24 1.48567 0.742836 0.669474i \(-0.233480\pi\)
0.742836 + 0.669474i \(0.233480\pi\)
\(390\) 7.18013e23 0.174188
\(391\) −1.10581e24 −0.261818
\(392\) 1.31643e24 0.304214
\(393\) −5.38209e24 −1.21401
\(394\) 2.48538e24 0.547241
\(395\) 9.42654e23 0.202619
\(396\) 1.20811e24 0.253515
\(397\) −2.25921e24 −0.462856 −0.231428 0.972852i \(-0.574340\pi\)
−0.231428 + 0.972852i \(0.574340\pi\)
\(398\) −1.41727e24 −0.283507
\(399\) −1.24206e24 −0.242604
\(400\) 2.62144e23 0.0500000
\(401\) 7.37768e22 0.0137419 0.00687097 0.999976i \(-0.497813\pi\)
0.00687097 + 0.999976i \(0.497813\pi\)
\(402\) 4.39006e24 0.798589
\(403\) 2.40755e23 0.0427737
\(404\) 3.38689e24 0.587730
\(405\) 1.20711e24 0.204609
\(406\) 1.33397e24 0.220876
\(407\) −5.33945e24 −0.863673
\(408\) −2.86405e23 −0.0452593
\(409\) 1.18987e25 1.83708 0.918539 0.395330i \(-0.129370\pi\)
0.918539 + 0.395330i \(0.129370\pi\)
\(410\) −2.63523e24 −0.397531
\(411\) −2.92775e24 −0.431555
\(412\) 3.56253e24 0.513139
\(413\) −2.06009e24 −0.289976
\(414\) 3.20351e24 0.440679
\(415\) 6.11846e23 0.0822594
\(416\) 9.48945e23 0.124697
\(417\) 1.05850e25 1.35957
\(418\) −6.08601e24 −0.764113
\(419\) −9.99966e24 −1.22730 −0.613652 0.789577i \(-0.710300\pi\)
−0.613652 + 0.789577i \(0.710300\pi\)
\(420\) −5.43638e23 −0.0652291
\(421\) −3.46749e24 −0.406757 −0.203379 0.979100i \(-0.565192\pi\)
−0.203379 + 0.979100i \(0.565192\pi\)
\(422\) −2.01208e24 −0.230769
\(423\) 1.65612e24 0.185719
\(424\) −5.81254e24 −0.637368
\(425\) −3.05766e23 −0.0327865
\(426\) 5.36077e24 0.562129
\(427\) −1.80276e24 −0.184873
\(428\) −8.60637e24 −0.863185
\(429\) −7.29614e24 −0.715730
\(430\) −1.50171e24 −0.144091
\(431\) −7.34068e24 −0.688973 −0.344487 0.938791i \(-0.611947\pi\)
−0.344487 + 0.938791i \(0.611947\pi\)
\(432\) 2.95601e24 0.271400
\(433\) −1.93970e24 −0.174221 −0.0871105 0.996199i \(-0.527763\pi\)
−0.0871105 + 0.996199i \(0.527763\pi\)
\(434\) −1.82285e23 −0.0160177
\(435\) 3.39663e24 0.292012
\(436\) −9.71937e24 −0.817554
\(437\) −1.61380e25 −1.32824
\(438\) −9.16607e24 −0.738209
\(439\) −1.07757e25 −0.849241 −0.424621 0.905371i \(-0.639593\pi\)
−0.424621 + 0.905371i \(0.639593\pi\)
\(440\) −2.66380e24 −0.205447
\(441\) 4.44830e24 0.335759
\(442\) −1.10685e24 −0.0817672
\(443\) 2.13229e25 1.54174 0.770871 0.636992i \(-0.219822\pi\)
0.770871 + 0.636992i \(0.219822\pi\)
\(444\) −3.66704e24 −0.259524
\(445\) 8.26192e24 0.572349
\(446\) 1.58468e24 0.107463
\(447\) 2.28313e25 1.51568
\(448\) −7.18487e23 −0.0466958
\(449\) −8.14780e24 −0.518442 −0.259221 0.965818i \(-0.583466\pi\)
−0.259221 + 0.965818i \(0.583466\pi\)
\(450\) 8.85802e23 0.0551846
\(451\) 2.67781e25 1.63344
\(452\) 6.43137e24 0.384139
\(453\) 1.45781e25 0.852642
\(454\) −7.15876e24 −0.410022
\(455\) −2.10097e24 −0.117845
\(456\) −4.17976e24 −0.229607
\(457\) 7.48065e24 0.402472 0.201236 0.979543i \(-0.435504\pi\)
0.201236 + 0.979543i \(0.435504\pi\)
\(458\) 1.35253e25 0.712728
\(459\) −3.44791e24 −0.177965
\(460\) −7.06349e24 −0.357125
\(461\) −2.41292e25 −1.19504 −0.597522 0.801852i \(-0.703848\pi\)
−0.597522 + 0.801852i \(0.703848\pi\)
\(462\) 5.52422e24 0.268023
\(463\) 3.66162e25 1.74042 0.870209 0.492683i \(-0.163984\pi\)
0.870209 + 0.492683i \(0.163984\pi\)
\(464\) 4.48908e24 0.209044
\(465\) −4.64145e23 −0.0211763
\(466\) 2.93202e25 1.31069
\(467\) −2.16793e25 −0.949589 −0.474794 0.880097i \(-0.657478\pi\)
−0.474794 + 0.880097i \(0.657478\pi\)
\(468\) 3.20655e24 0.137627
\(469\) −1.28457e25 −0.540278
\(470\) −3.65161e24 −0.150506
\(471\) 1.12377e25 0.453921
\(472\) −6.93263e24 −0.274441
\(473\) 1.52597e25 0.592060
\(474\) −6.57859e24 −0.250172
\(475\) −4.46233e24 −0.166331
\(476\) 8.38047e23 0.0306198
\(477\) −1.96410e25 −0.703458
\(478\) −2.87398e25 −1.00906
\(479\) 3.20019e24 0.110151 0.0550754 0.998482i \(-0.482460\pi\)
0.0550754 + 0.998482i \(0.482460\pi\)
\(480\) −1.82945e24 −0.0617346
\(481\) −1.41719e25 −0.468866
\(482\) 2.44471e25 0.793012
\(483\) 1.46484e25 0.465899
\(484\) 1.10359e25 0.344173
\(485\) −6.92783e23 −0.0211861
\(486\) 1.71735e25 0.515007
\(487\) −1.18018e25 −0.347076 −0.173538 0.984827i \(-0.555520\pi\)
−0.173538 + 0.984827i \(0.555520\pi\)
\(488\) −6.06665e24 −0.174969
\(489\) −3.42952e25 −0.970062
\(490\) −9.80816e24 −0.272098
\(491\) −2.40797e25 −0.655203 −0.327602 0.944816i \(-0.606240\pi\)
−0.327602 + 0.944816i \(0.606240\pi\)
\(492\) 1.83907e25 0.490829
\(493\) −5.23609e24 −0.137076
\(494\) −1.61534e25 −0.414817
\(495\) −9.00115e24 −0.226751
\(496\) −6.13427e23 −0.0151596
\(497\) −1.56861e25 −0.380303
\(498\) −4.26995e24 −0.101565
\(499\) 2.32722e25 0.543102 0.271551 0.962424i \(-0.412463\pi\)
0.271551 + 0.962424i \(0.412463\pi\)
\(500\) −1.95313e24 −0.0447214
\(501\) −5.61660e25 −1.26187
\(502\) −5.41371e25 −1.19346
\(503\) 3.42180e25 0.740216 0.370108 0.928989i \(-0.379321\pi\)
0.370108 + 0.928989i \(0.379321\pi\)
\(504\) −2.42781e24 −0.0515377
\(505\) −2.52343e25 −0.525682
\(506\) 7.17762e25 1.46741
\(507\) 1.95540e25 0.392337
\(508\) 9.16225e24 0.180425
\(509\) −4.64751e25 −0.898259 −0.449129 0.893467i \(-0.648266\pi\)
−0.449129 + 0.893467i \(0.648266\pi\)
\(510\) 2.13388e24 0.0404812
\(511\) 2.68208e25 0.499428
\(512\) −2.41785e24 −0.0441942
\(513\) −5.03185e25 −0.902843
\(514\) 1.63335e25 0.287693
\(515\) −2.65429e25 −0.458966
\(516\) 1.04801e25 0.177907
\(517\) 3.71061e25 0.618422
\(518\) 1.07301e25 0.175579
\(519\) −3.02257e25 −0.485609
\(520\) −7.07019e24 −0.111532
\(521\) 6.75250e25 1.04594 0.522969 0.852351i \(-0.324824\pi\)
0.522969 + 0.852351i \(0.324824\pi\)
\(522\) 1.51689e25 0.230720
\(523\) −1.63639e25 −0.244411 −0.122206 0.992505i \(-0.538997\pi\)
−0.122206 + 0.992505i \(0.538997\pi\)
\(524\) 5.29969e25 0.777325
\(525\) 4.05042e24 0.0583427
\(526\) −3.03867e25 −0.429853
\(527\) 7.15505e23 0.00994058
\(528\) 1.85901e25 0.253664
\(529\) 1.15711e26 1.55076
\(530\) 4.33068e25 0.570079
\(531\) −2.34258e25 −0.302898
\(532\) 1.22304e25 0.155339
\(533\) 7.10738e25 0.886749
\(534\) −5.76582e25 −0.706675
\(535\) 6.41224e25 0.772056
\(536\) −4.32285e25 −0.511334
\(537\) −7.23742e25 −0.841063
\(538\) −1.00716e26 −1.14992
\(539\) 9.96664e25 1.11804
\(540\) −2.20240e25 −0.242748
\(541\) 2.04269e25 0.221222 0.110611 0.993864i \(-0.464719\pi\)
0.110611 + 0.993864i \(0.464719\pi\)
\(542\) −5.46039e25 −0.581072
\(543\) −1.04878e26 −1.09669
\(544\) 2.82019e24 0.0289794
\(545\) 7.24150e25 0.731243
\(546\) 1.46623e25 0.145503
\(547\) 8.46896e25 0.825943 0.412972 0.910744i \(-0.364491\pi\)
0.412972 + 0.910744i \(0.364491\pi\)
\(548\) 2.88292e25 0.276323
\(549\) −2.04996e25 −0.193112
\(550\) 1.98468e25 0.183758
\(551\) −7.64151e25 −0.695407
\(552\) 4.92947e25 0.440939
\(553\) 1.92496e25 0.169252
\(554\) −3.29885e25 −0.285115
\(555\) 2.73216e25 0.232125
\(556\) −1.04230e26 −0.870526
\(557\) 8.29899e25 0.681398 0.340699 0.940172i \(-0.389336\pi\)
0.340699 + 0.940172i \(0.389336\pi\)
\(558\) −2.07281e24 −0.0167315
\(559\) 4.05021e25 0.321414
\(560\) 5.35314e24 0.0417660
\(561\) −2.16836e25 −0.166335
\(562\) −8.85878e25 −0.668158
\(563\) 9.14845e24 0.0678450 0.0339225 0.999424i \(-0.489200\pi\)
0.0339225 + 0.999424i \(0.489200\pi\)
\(564\) 2.54838e25 0.185829
\(565\) −4.79174e25 −0.343584
\(566\) −4.01174e25 −0.282863
\(567\) 2.46499e25 0.170914
\(568\) −5.27869e25 −0.359929
\(569\) −5.94829e24 −0.0398865 −0.0199432 0.999801i \(-0.506349\pi\)
−0.0199432 + 0.999801i \(0.506349\pi\)
\(570\) 3.11417e25 0.205367
\(571\) −6.17752e25 −0.400655 −0.200328 0.979729i \(-0.564201\pi\)
−0.200328 + 0.979729i \(0.564201\pi\)
\(572\) 7.18443e25 0.458279
\(573\) −1.62711e26 −1.02082
\(574\) −5.38130e25 −0.332066
\(575\) 5.26271e25 0.319422
\(576\) −8.17008e24 −0.0487767
\(577\) −5.21343e25 −0.306163 −0.153082 0.988214i \(-0.548920\pi\)
−0.153082 + 0.988214i \(0.548920\pi\)
\(578\) 1.19116e26 0.688104
\(579\) 8.56661e25 0.486814
\(580\) −3.34463e25 −0.186974
\(581\) 1.24943e25 0.0687129
\(582\) 4.83480e24 0.0261583
\(583\) −4.40065e26 −2.34243
\(584\) 9.02574e25 0.472673
\(585\) −2.38907e25 −0.123097
\(586\) −1.70271e26 −0.863206
\(587\) −7.85282e25 −0.391709 −0.195855 0.980633i \(-0.562748\pi\)
−0.195855 + 0.980633i \(0.562748\pi\)
\(588\) 6.84492e25 0.335957
\(589\) 1.04420e25 0.0504300
\(590\) 5.16521e25 0.245467
\(591\) 1.29230e26 0.604341
\(592\) 3.61090e25 0.166172
\(593\) −7.16513e25 −0.324492 −0.162246 0.986750i \(-0.551874\pi\)
−0.162246 + 0.986750i \(0.551874\pi\)
\(594\) 2.23799e26 0.997438
\(595\) −6.24393e24 −0.0273872
\(596\) −2.24817e26 −0.970487
\(597\) −7.36925e25 −0.313089
\(598\) 1.90507e26 0.796617
\(599\) 6.18751e25 0.254660 0.127330 0.991860i \(-0.459359\pi\)
0.127330 + 0.991860i \(0.459359\pi\)
\(600\) 1.36305e25 0.0552171
\(601\) 2.47224e26 0.985789 0.492894 0.870089i \(-0.335939\pi\)
0.492894 + 0.870089i \(0.335939\pi\)
\(602\) −3.06658e25 −0.120362
\(603\) −1.46072e26 −0.564355
\(604\) −1.43549e26 −0.545944
\(605\) −8.22238e25 −0.307838
\(606\) 1.76105e26 0.649055
\(607\) −9.24185e25 −0.335325 −0.167663 0.985844i \(-0.553622\pi\)
−0.167663 + 0.985844i \(0.553622\pi\)
\(608\) 4.11577e25 0.147017
\(609\) 6.93613e25 0.243923
\(610\) 4.52000e25 0.156497
\(611\) 9.84863e25 0.335725
\(612\) 9.52962e24 0.0319843
\(613\) 7.50279e25 0.247941 0.123970 0.992286i \(-0.460437\pi\)
0.123970 + 0.992286i \(0.460437\pi\)
\(614\) 2.42784e26 0.789987
\(615\) −1.37022e26 −0.439011
\(616\) −5.43964e25 −0.171614
\(617\) 3.84695e26 1.19511 0.597554 0.801828i \(-0.296139\pi\)
0.597554 + 0.801828i \(0.296139\pi\)
\(618\) 1.85238e26 0.566682
\(619\) 1.19514e26 0.360044 0.180022 0.983663i \(-0.442383\pi\)
0.180022 + 0.983663i \(0.442383\pi\)
\(620\) 4.57039e24 0.0135591
\(621\) 5.93438e26 1.73383
\(622\) −1.10574e26 −0.318159
\(623\) 1.68713e26 0.478094
\(624\) 4.93415e25 0.137708
\(625\) 1.45519e25 0.0400000
\(626\) 2.96789e26 0.803511
\(627\) −3.16449e26 −0.843843
\(628\) −1.10657e26 −0.290644
\(629\) −4.21177e25 −0.108964
\(630\) 1.80886e25 0.0460968
\(631\) −8.74692e25 −0.219572 −0.109786 0.993955i \(-0.535016\pi\)
−0.109786 + 0.993955i \(0.535016\pi\)
\(632\) 6.47787e25 0.160184
\(633\) −1.04620e26 −0.254848
\(634\) −3.29204e26 −0.789982
\(635\) −6.82641e25 −0.161377
\(636\) −3.02229e26 −0.703873
\(637\) 2.64532e26 0.606952
\(638\) 3.39867e26 0.768268
\(639\) −1.78371e26 −0.397251
\(640\) 1.80144e25 0.0395285
\(641\) −7.90635e26 −1.70933 −0.854663 0.519183i \(-0.826236\pi\)
−0.854663 + 0.519183i \(0.826236\pi\)
\(642\) −4.47497e26 −0.953252
\(643\) 7.69562e25 0.161525 0.0807625 0.996733i \(-0.474264\pi\)
0.0807625 + 0.996733i \(0.474264\pi\)
\(644\) −1.44241e26 −0.298313
\(645\) −7.80830e25 −0.159125
\(646\) −4.80065e25 −0.0964032
\(647\) −7.71883e26 −1.52743 −0.763714 0.645555i \(-0.776626\pi\)
−0.763714 + 0.645555i \(0.776626\pi\)
\(648\) 8.29520e25 0.161757
\(649\) −5.24867e26 −1.00861
\(650\) 5.26771e25 0.0997573
\(651\) −9.47813e24 −0.0176890
\(652\) 3.37701e26 0.621128
\(653\) 3.79913e26 0.688666 0.344333 0.938848i \(-0.388105\pi\)
0.344333 + 0.938848i \(0.388105\pi\)
\(654\) −5.05369e26 −0.902860
\(655\) −3.94857e26 −0.695261
\(656\) −1.81091e26 −0.314276
\(657\) 3.04986e26 0.521685
\(658\) −7.45681e25 −0.125721
\(659\) −4.70607e26 −0.782072 −0.391036 0.920375i \(-0.627883\pi\)
−0.391036 + 0.920375i \(0.627883\pi\)
\(660\) −1.38507e26 −0.226884
\(661\) 1.07871e27 1.74177 0.870885 0.491486i \(-0.163546\pi\)
0.870885 + 0.491486i \(0.163546\pi\)
\(662\) −7.42168e26 −1.18128
\(663\) −5.75521e25 −0.0902990
\(664\) 4.20458e25 0.0650318
\(665\) −9.11235e25 −0.138939
\(666\) 1.22015e26 0.183403
\(667\) 9.01213e26 1.33546
\(668\) 5.53061e26 0.807972
\(669\) 8.23970e25 0.118676
\(670\) 3.22077e26 0.457351
\(671\) −4.59304e26 −0.643037
\(672\) −3.73585e25 −0.0515681
\(673\) 1.37764e27 1.87497 0.937484 0.348029i \(-0.113149\pi\)
0.937484 + 0.348029i \(0.113149\pi\)
\(674\) −6.81185e26 −0.914105
\(675\) 1.64092e26 0.217120
\(676\) −1.92546e26 −0.251212
\(677\) 8.45165e26 1.08730 0.543650 0.839312i \(-0.317042\pi\)
0.543650 + 0.839312i \(0.317042\pi\)
\(678\) 3.34406e26 0.424221
\(679\) −1.41471e25 −0.0176972
\(680\) −2.10121e25 −0.0259200
\(681\) −3.72228e26 −0.452805
\(682\) −4.64424e25 −0.0557138
\(683\) −4.20176e26 −0.497090 −0.248545 0.968620i \(-0.579952\pi\)
−0.248545 + 0.968620i \(0.579952\pi\)
\(684\) 1.39075e26 0.162261
\(685\) −2.14795e26 −0.247151
\(686\) −4.33061e26 −0.491440
\(687\) 7.03261e26 0.787095
\(688\) −1.03197e26 −0.113914
\(689\) −1.16801e27 −1.27164
\(690\) −3.67274e26 −0.394388
\(691\) 9.77912e26 1.03576 0.517879 0.855454i \(-0.326722\pi\)
0.517879 + 0.855454i \(0.326722\pi\)
\(692\) 2.97630e26 0.310934
\(693\) −1.83809e26 −0.189409
\(694\) 4.13927e26 0.420735
\(695\) 7.76573e26 0.778622
\(696\) 2.33415e26 0.230856
\(697\) 2.11226e26 0.206080
\(698\) −9.61944e26 −0.925813
\(699\) 1.52454e27 1.44745
\(700\) −3.98840e25 −0.0373566
\(701\) −8.91378e25 −0.0823646 −0.0411823 0.999152i \(-0.513112\pi\)
−0.0411823 + 0.999152i \(0.513112\pi\)
\(702\) 5.94001e26 0.541483
\(703\) −6.14662e26 −0.552791
\(704\) −1.83055e26 −0.162420
\(705\) −1.89869e26 −0.166210
\(706\) 1.28871e27 1.11304
\(707\) −5.15300e26 −0.439112
\(708\) −3.60469e26 −0.303077
\(709\) −1.16549e27 −0.966869 −0.483435 0.875380i \(-0.660611\pi\)
−0.483435 + 0.875380i \(0.660611\pi\)
\(710\) 3.93293e26 0.321931
\(711\) 2.18892e26 0.176794
\(712\) 5.67755e26 0.452481
\(713\) −1.23150e26 −0.0968461
\(714\) 4.35752e25 0.0338147
\(715\) −5.35282e26 −0.409897
\(716\) 7.12662e26 0.538530
\(717\) −1.49436e27 −1.11435
\(718\) 1.04823e26 0.0771387
\(719\) −1.66537e27 −1.20945 −0.604723 0.796436i \(-0.706716\pi\)
−0.604723 + 0.796436i \(0.706716\pi\)
\(720\) 6.08718e25 0.0436272
\(721\) −5.42023e26 −0.383383
\(722\) 3.12347e26 0.218039
\(723\) 1.27115e27 0.875757
\(724\) 1.03272e27 0.702209
\(725\) 2.49194e26 0.167235
\(726\) 5.73823e26 0.380085
\(727\) 1.92104e26 0.125591 0.0627955 0.998026i \(-0.479998\pi\)
0.0627955 + 0.998026i \(0.479998\pi\)
\(728\) −1.44378e26 −0.0931649
\(729\) 1.61128e27 1.02626
\(730\) −6.72470e26 −0.422771
\(731\) 1.20369e26 0.0746964
\(732\) −3.15442e26 −0.193225
\(733\) −1.95377e27 −1.18137 −0.590684 0.806903i \(-0.701142\pi\)
−0.590684 + 0.806903i \(0.701142\pi\)
\(734\) 2.97338e26 0.177475
\(735\) −5.09986e26 −0.300489
\(736\) −4.85400e26 −0.282332
\(737\) −3.27281e27 −1.87923
\(738\) −6.11920e26 −0.346864
\(739\) −4.82280e26 −0.269884 −0.134942 0.990854i \(-0.543085\pi\)
−0.134942 + 0.990854i \(0.543085\pi\)
\(740\) −2.69033e26 −0.148629
\(741\) −8.39911e26 −0.458100
\(742\) 8.84352e26 0.476198
\(743\) 1.08282e27 0.575657 0.287828 0.957682i \(-0.407067\pi\)
0.287828 + 0.957682i \(0.407067\pi\)
\(744\) −3.18958e25 −0.0167414
\(745\) 1.67502e27 0.868030
\(746\) −2.22959e27 −1.14079
\(747\) 1.42075e26 0.0717750
\(748\) 2.13516e26 0.106504
\(749\) 1.30942e27 0.644914
\(750\) −1.01555e26 −0.0493877
\(751\) −1.90953e27 −0.916951 −0.458475 0.888707i \(-0.651604\pi\)
−0.458475 + 0.888707i \(0.651604\pi\)
\(752\) −2.50937e26 −0.118986
\(753\) −2.81492e27 −1.31799
\(754\) 9.02068e26 0.417072
\(755\) 1.06952e27 0.488307
\(756\) −4.49744e26 −0.202772
\(757\) −2.96368e26 −0.131953 −0.0659767 0.997821i \(-0.521016\pi\)
−0.0659767 + 0.997821i \(0.521016\pi\)
\(758\) −1.22231e27 −0.537430
\(759\) 3.73208e27 1.62052
\(760\) −3.06649e26 −0.131496
\(761\) −1.09087e27 −0.461975 −0.230988 0.972957i \(-0.574196\pi\)
−0.230988 + 0.972957i \(0.574196\pi\)
\(762\) 4.76401e26 0.199251
\(763\) 1.47876e27 0.610821
\(764\) 1.60219e27 0.653625
\(765\) −7.10012e25 −0.0286077
\(766\) 1.57557e27 0.626995
\(767\) −1.39309e27 −0.547549
\(768\) −1.25719e26 −0.0488055
\(769\) 2.20592e27 0.845842 0.422921 0.906166i \(-0.361005\pi\)
0.422921 + 0.906166i \(0.361005\pi\)
\(770\) 4.05285e26 0.153496
\(771\) 8.49277e26 0.317711
\(772\) −8.43545e26 −0.311705
\(773\) 1.83711e27 0.670550 0.335275 0.942120i \(-0.391171\pi\)
0.335275 + 0.942120i \(0.391171\pi\)
\(774\) −3.48708e26 −0.125725
\(775\) −3.40520e25 −0.0121277
\(776\) −4.76077e25 −0.0167491
\(777\) 5.57924e26 0.193899
\(778\) −3.05995e27 −1.05053
\(779\) 3.08261e27 1.04547
\(780\) −3.67622e26 −0.123170
\(781\) −3.99648e27 −1.32280
\(782\) 5.66173e26 0.185133
\(783\) 2.80998e27 0.907752
\(784\) −6.74012e26 −0.215112
\(785\) 8.24457e26 0.259960
\(786\) 2.75563e27 0.858433
\(787\) −2.08651e26 −0.0642187 −0.0321093 0.999484i \(-0.510222\pi\)
−0.0321093 + 0.999484i \(0.510222\pi\)
\(788\) −1.27251e27 −0.386958
\(789\) −1.57999e27 −0.474704
\(790\) −4.82639e26 −0.143273
\(791\) −9.78503e26 −0.287003
\(792\) −6.18554e26 −0.179262
\(793\) −1.21907e27 −0.349088
\(794\) 1.15671e27 0.327289
\(795\) 2.25178e27 0.629563
\(796\) 7.25642e26 0.200470
\(797\) 4.57657e27 1.24936 0.624678 0.780883i \(-0.285230\pi\)
0.624678 + 0.780883i \(0.285230\pi\)
\(798\) 6.35932e26 0.171547
\(799\) 2.92694e26 0.0780224
\(800\) −1.34218e26 −0.0353553
\(801\) 1.91848e27 0.499400
\(802\) −3.77737e25 −0.00971702
\(803\) 6.83335e27 1.73715
\(804\) −2.24771e27 −0.564687
\(805\) 1.07468e27 0.266820
\(806\) −1.23266e26 −0.0302456
\(807\) −5.23684e27 −1.26990
\(808\) −1.73409e27 −0.415588
\(809\) 1.09533e27 0.259439 0.129720 0.991551i \(-0.458592\pi\)
0.129720 + 0.991551i \(0.458592\pi\)
\(810\) −6.18041e26 −0.144680
\(811\) 1.55516e26 0.0359812 0.0179906 0.999838i \(-0.494273\pi\)
0.0179906 + 0.999838i \(0.494273\pi\)
\(812\) −6.82994e26 −0.156183
\(813\) −2.83919e27 −0.641703
\(814\) 2.73380e27 0.610709
\(815\) −2.51607e27 −0.555553
\(816\) 1.46639e26 0.0320032
\(817\) 1.75666e27 0.378946
\(818\) −6.09213e27 −1.29901
\(819\) −4.87862e26 −0.102825
\(820\) 1.34924e27 0.281097
\(821\) −1.23074e27 −0.253458 −0.126729 0.991937i \(-0.540448\pi\)
−0.126729 + 0.991937i \(0.540448\pi\)
\(822\) 1.49901e27 0.305156
\(823\) −7.12638e27 −1.43407 −0.717035 0.697037i \(-0.754501\pi\)
−0.717035 + 0.697037i \(0.754501\pi\)
\(824\) −1.82402e27 −0.362844
\(825\) 1.03196e27 0.202931
\(826\) 1.05477e27 0.205044
\(827\) 3.28596e27 0.631480 0.315740 0.948846i \(-0.397747\pi\)
0.315740 + 0.948846i \(0.397747\pi\)
\(828\) −1.64020e27 −0.311607
\(829\) −7.47757e27 −1.40441 −0.702203 0.711977i \(-0.747800\pi\)
−0.702203 + 0.711977i \(0.747800\pi\)
\(830\) −3.13265e26 −0.0581662
\(831\) −1.71527e27 −0.314864
\(832\) −4.85860e26 −0.0881738
\(833\) 7.86171e26 0.141055
\(834\) −5.41954e27 −0.961358
\(835\) −4.12062e27 −0.722672
\(836\) 3.11604e27 0.540310
\(837\) −3.83981e26 −0.0658290
\(838\) 5.11983e27 0.867835
\(839\) −6.82604e27 −1.14401 −0.572006 0.820250i \(-0.693834\pi\)
−0.572006 + 0.820250i \(0.693834\pi\)
\(840\) 2.78343e26 0.0461239
\(841\) −1.83594e27 −0.300812
\(842\) 1.77536e27 0.287621
\(843\) −4.60622e27 −0.737875
\(844\) 1.03019e27 0.163178
\(845\) 1.43458e27 0.224691
\(846\) −8.47931e26 −0.131323
\(847\) −1.67906e27 −0.257143
\(848\) 2.97602e27 0.450687
\(849\) −2.08595e27 −0.312378
\(850\) 1.56552e26 0.0231835
\(851\) 7.24911e27 1.06158
\(852\) −2.74471e27 −0.397485
\(853\) 3.69477e27 0.529141 0.264571 0.964366i \(-0.414770\pi\)
0.264571 + 0.964366i \(0.414770\pi\)
\(854\) 9.23013e26 0.130725
\(855\) −1.03619e27 −0.145131
\(856\) 4.40646e27 0.610364
\(857\) 8.05314e27 1.10318 0.551592 0.834114i \(-0.314021\pi\)
0.551592 + 0.834114i \(0.314021\pi\)
\(858\) 3.73562e27 0.506097
\(859\) −6.31890e26 −0.0846655 −0.0423328 0.999104i \(-0.513479\pi\)
−0.0423328 + 0.999104i \(0.513479\pi\)
\(860\) 7.68875e26 0.101887
\(861\) −2.79806e27 −0.366714
\(862\) 3.75843e27 0.487178
\(863\) 7.45376e27 0.955592 0.477796 0.878471i \(-0.341436\pi\)
0.477796 + 0.878471i \(0.341436\pi\)
\(864\) −1.51348e27 −0.191909
\(865\) −2.21751e27 −0.278108
\(866\) 9.93128e26 0.123193
\(867\) 6.19355e27 0.759903
\(868\) 9.33302e25 0.0113262
\(869\) 4.90438e27 0.588702
\(870\) −1.73908e27 −0.206484
\(871\) −8.68663e27 −1.02019
\(872\) 4.97632e27 0.578098
\(873\) −1.60870e26 −0.0184858
\(874\) 8.26268e27 0.939209
\(875\) 2.97159e26 0.0334128
\(876\) 4.69303e27 0.521993
\(877\) −2.72071e27 −0.299355 −0.149678 0.988735i \(-0.547824\pi\)
−0.149678 + 0.988735i \(0.547824\pi\)
\(878\) 5.51714e27 0.600504
\(879\) −8.85345e27 −0.953275
\(880\) 1.36386e27 0.145273
\(881\) 8.26420e27 0.870822 0.435411 0.900232i \(-0.356603\pi\)
0.435411 + 0.900232i \(0.356603\pi\)
\(882\) −2.27753e27 −0.237417
\(883\) 1.67843e28 1.73092 0.865459 0.500979i \(-0.167027\pi\)
0.865459 + 0.500979i \(0.167027\pi\)
\(884\) 5.66710e26 0.0578181
\(885\) 2.68571e27 0.271080
\(886\) −1.09173e28 −1.09018
\(887\) −1.79212e28 −1.77049 −0.885245 0.465126i \(-0.846009\pi\)
−0.885245 + 0.465126i \(0.846009\pi\)
\(888\) 1.87753e27 0.183511
\(889\) −1.39400e27 −0.134801
\(890\) −4.23010e27 −0.404712
\(891\) 6.28027e27 0.594484
\(892\) −8.11355e26 −0.0759880
\(893\) 4.27155e27 0.395819
\(894\) −1.16896e28 −1.07175
\(895\) −5.30974e27 −0.481676
\(896\) 3.67865e26 0.0330189
\(897\) 9.90562e27 0.879738
\(898\) 4.17167e27 0.366594
\(899\) −5.83124e26 −0.0507042
\(900\) −4.53531e26 −0.0390214
\(901\) −3.47125e27 −0.295529
\(902\) −1.37104e28 −1.15501
\(903\) −1.59450e27 −0.132920
\(904\) −3.29286e27 −0.271627
\(905\) −7.69437e27 −0.628075
\(906\) −7.46396e27 −0.602909
\(907\) −8.20682e27 −0.656003 −0.328001 0.944677i \(-0.606375\pi\)
−0.328001 + 0.944677i \(0.606375\pi\)
\(908\) 3.66529e27 0.289929
\(909\) −5.85959e27 −0.458681
\(910\) 1.07570e27 0.0833292
\(911\) 1.30357e28 0.999331 0.499666 0.866218i \(-0.333456\pi\)
0.499666 + 0.866218i \(0.333456\pi\)
\(912\) 2.14004e27 0.162357
\(913\) 3.18327e27 0.239002
\(914\) −3.83009e27 −0.284591
\(915\) 2.35023e27 0.172826
\(916\) −6.92493e27 −0.503975
\(917\) −8.06323e27 −0.580765
\(918\) 1.76533e27 0.125840
\(919\) 1.03919e27 0.0733155 0.0366578 0.999328i \(-0.488329\pi\)
0.0366578 + 0.999328i \(0.488329\pi\)
\(920\) 3.61651e27 0.252525
\(921\) 1.26238e28 0.872416
\(922\) 1.23542e28 0.845024
\(923\) −1.06074e28 −0.718111
\(924\) −2.82840e27 −0.189521
\(925\) 2.00445e27 0.132938
\(926\) −1.87475e28 −1.23066
\(927\) −6.16347e27 −0.400468
\(928\) −2.29841e27 −0.147816
\(929\) −3.13678e27 −0.199680 −0.0998399 0.995004i \(-0.531833\pi\)
−0.0998399 + 0.995004i \(0.531833\pi\)
\(930\) 2.37642e26 0.0149739
\(931\) 1.14733e28 0.715594
\(932\) −1.50120e28 −0.926800
\(933\) −5.74941e27 −0.351356
\(934\) 1.10998e28 0.671460
\(935\) −1.59082e27 −0.0952599
\(936\) −1.64175e27 −0.0973167
\(937\) 9.68011e27 0.568008 0.284004 0.958823i \(-0.408337\pi\)
0.284004 + 0.958823i \(0.408337\pi\)
\(938\) 6.57702e27 0.382034
\(939\) 1.54319e28 0.887351
\(940\) 1.86962e27 0.106424
\(941\) 2.78537e28 1.56957 0.784785 0.619768i \(-0.212773\pi\)
0.784785 + 0.619768i \(0.212773\pi\)
\(942\) −5.75372e27 −0.320970
\(943\) −3.63553e28 −2.00774
\(944\) 3.54950e27 0.194059
\(945\) 3.35085e27 0.181365
\(946\) −7.81298e27 −0.418650
\(947\) 6.85631e27 0.363719 0.181859 0.983325i \(-0.441788\pi\)
0.181859 + 0.983325i \(0.441788\pi\)
\(948\) 3.36824e27 0.176898
\(949\) 1.81369e28 0.943051
\(950\) 2.28471e27 0.117613
\(951\) −1.71173e28 −0.872411
\(952\) −4.29080e26 −0.0216515
\(953\) −2.64880e28 −1.32332 −0.661662 0.749802i \(-0.730149\pi\)
−0.661662 + 0.749802i \(0.730149\pi\)
\(954\) 1.00562e28 0.497420
\(955\) −1.19373e28 −0.584620
\(956\) 1.47148e28 0.713517
\(957\) 1.76718e28 0.848431
\(958\) −1.63849e27 −0.0778884
\(959\) −4.38624e27 −0.206450
\(960\) 9.36678e26 0.0436530
\(961\) −2.15910e28 −0.996323
\(962\) 7.25599e27 0.331538
\(963\) 1.48897e28 0.673654
\(964\) −1.25169e28 −0.560744
\(965\) 6.28490e27 0.278798
\(966\) −7.49996e27 −0.329440
\(967\) −8.38952e27 −0.364910 −0.182455 0.983214i \(-0.558404\pi\)
−0.182455 + 0.983214i \(0.558404\pi\)
\(968\) −5.65038e27 −0.243367
\(969\) −2.49615e27 −0.106462
\(970\) 3.54705e26 0.0149808
\(971\) −1.23262e28 −0.515520 −0.257760 0.966209i \(-0.582984\pi\)
−0.257760 + 0.966209i \(0.582984\pi\)
\(972\) −8.79281e27 −0.364165
\(973\) 1.58581e28 0.650398
\(974\) 6.04254e27 0.245420
\(975\) 2.73900e27 0.110166
\(976\) 3.10612e27 0.123722
\(977\) 3.70018e28 1.45957 0.729784 0.683678i \(-0.239621\pi\)
0.729784 + 0.683678i \(0.239621\pi\)
\(978\) 1.75592e28 0.685937
\(979\) 4.29845e28 1.66294
\(980\) 5.02178e27 0.192402
\(981\) 1.68153e28 0.638042
\(982\) 1.23288e28 0.463299
\(983\) −2.36780e28 −0.881223 −0.440612 0.897698i \(-0.645238\pi\)
−0.440612 + 0.897698i \(0.645238\pi\)
\(984\) −9.41605e27 −0.347068
\(985\) 9.48096e27 0.346105
\(986\) 2.68088e27 0.0969274
\(987\) −3.87725e27 −0.138839
\(988\) 8.27052e27 0.293320
\(989\) −2.07174e28 −0.727730
\(990\) 4.60859e27 0.160337
\(991\) 1.70262e28 0.586704 0.293352 0.956005i \(-0.405229\pi\)
0.293352 + 0.956005i \(0.405229\pi\)
\(992\) 3.14075e26 0.0107194
\(993\) −3.85898e28 −1.30453
\(994\) 8.03129e27 0.268915
\(995\) −5.40646e27 −0.179305
\(996\) 2.18622e27 0.0718173
\(997\) 3.10765e28 1.01118 0.505590 0.862774i \(-0.331275\pi\)
0.505590 + 0.862774i \(0.331275\pi\)
\(998\) −1.19153e28 −0.384031
\(999\) 2.26028e28 0.721588
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 10.20.a.a.1.1 1
3.2 odd 2 90.20.a.e.1.1 1
4.3 odd 2 80.20.a.c.1.1 1
5.2 odd 4 50.20.b.e.49.1 2
5.3 odd 4 50.20.b.e.49.2 2
5.4 even 2 50.20.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.20.a.a.1.1 1 1.1 even 1 trivial
50.20.a.e.1.1 1 5.4 even 2
50.20.b.e.49.1 2 5.2 odd 4
50.20.b.e.49.2 2 5.3 odd 4
80.20.a.c.1.1 1 4.3 odd 2
90.20.a.e.1.1 1 3.2 odd 2