Properties

Label 50.18.b.d.49.1
Level $50$
Weight $18$
Character 50.49
Analytic conductor $91.611$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,18,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(91.6110436723\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 41641x^{2} + 433472400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-143.792i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.18.b.d.49.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-256.000i q^{2} -8003.53i q^{3} -65536.0 q^{4} -2.04890e6 q^{6} +2.54694e7i q^{7} +1.67772e7i q^{8} +6.50836e7 q^{9} -2.81089e8 q^{11} +5.24520e8i q^{12} +1.52610e9i q^{13} +6.52016e9 q^{14} +4.29497e9 q^{16} +5.46901e10i q^{17} -1.66614e10i q^{18} -6.88947e8 q^{19} +2.03845e11 q^{21} +7.19587e10i q^{22} -3.91035e11i q^{23} +1.34277e11 q^{24} +3.90681e11 q^{26} -1.55448e12i q^{27} -1.66916e12i q^{28} -5.12259e12 q^{29} -7.31204e10 q^{31} -1.09951e12i q^{32} +2.24970e12i q^{33} +1.40007e13 q^{34} -4.26532e12 q^{36} -6.81637e12i q^{37} +1.76370e11i q^{38} +1.22142e13 q^{39} -5.76386e13 q^{41} -5.21843e13i q^{42} -7.57081e13i q^{43} +1.84214e13 q^{44} -1.00105e14 q^{46} -4.60351e13i q^{47} -3.43749e13i q^{48} -4.16059e14 q^{49} +4.37714e14 q^{51} -1.00014e14i q^{52} -6.58261e14i q^{53} -3.97946e14 q^{54} -4.27305e14 q^{56} +5.51401e12i q^{57} +1.31138e15i q^{58} +2.98287e14 q^{59} +8.50623e14 q^{61} +1.87188e13i q^{62} +1.65764e15i q^{63} -2.81475e14 q^{64} +5.75924e14 q^{66} -6.12967e15i q^{67} -3.58417e15i q^{68} -3.12966e15 q^{69} +5.41472e14 q^{71} +1.09192e15i q^{72} +7.16849e15i q^{73} -1.74499e15 q^{74} +4.51508e13 q^{76} -7.15916e15i q^{77} -3.12683e15i q^{78} -5.45373e15 q^{79} -4.03640e15 q^{81} +1.47555e16i q^{82} +3.64723e15i q^{83} -1.33592e16 q^{84} -1.93813e16 q^{86} +4.09988e16i q^{87} -4.71589e15i q^{88} -6.81792e14 q^{89} -3.88687e16 q^{91} +2.56269e16i q^{92} +5.85222e14i q^{93} -1.17850e16 q^{94} -8.79998e15 q^{96} -1.20373e17i q^{97} +1.06511e17i q^{98} -1.82943e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 262144 q^{4} + 669696 q^{6} + 215038188 q^{9} - 942962592 q^{11} + 309168128 q^{14} + 17179869184 q^{16} - 257345058800 q^{19} + 870762493248 q^{21} - 43889197056 q^{24} + 789419124736 q^{26} - 5086109499960 q^{29}+ \cdots - 52\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 256.000i − 0.707107i
\(3\) − 8003.53i − 0.704289i −0.935946 0.352145i \(-0.885452\pi\)
0.935946 0.352145i \(-0.114548\pi\)
\(4\) −65536.0 −0.500000
\(5\) 0 0
\(6\) −2.04890e6 −0.498008
\(7\) 2.54694e7i 1.66988i 0.550342 + 0.834939i \(0.314497\pi\)
−0.550342 + 0.834939i \(0.685503\pi\)
\(8\) 1.67772e7i 0.353553i
\(9\) 6.50836e7 0.503976
\(10\) 0 0
\(11\) −2.81089e8 −0.395372 −0.197686 0.980265i \(-0.563343\pi\)
−0.197686 + 0.980265i \(0.563343\pi\)
\(12\) 5.24520e8i 0.352145i
\(13\) 1.52610e9i 0.518876i 0.965760 + 0.259438i \(0.0835373\pi\)
−0.965760 + 0.259438i \(0.916463\pi\)
\(14\) 6.52016e9 1.18078
\(15\) 0 0
\(16\) 4.29497e9 0.250000
\(17\) 5.46901e10i 1.90148i 0.309984 + 0.950742i \(0.399676\pi\)
−0.309984 + 0.950742i \(0.600324\pi\)
\(18\) − 1.66614e10i − 0.356365i
\(19\) −6.88947e8 −0.00930637 −0.00465318 0.999989i \(-0.501481\pi\)
−0.00465318 + 0.999989i \(0.501481\pi\)
\(20\) 0 0
\(21\) 2.03845e11 1.17608
\(22\) 7.19587e10i 0.279570i
\(23\) − 3.91035e11i − 1.04119i −0.853804 0.520594i \(-0.825710\pi\)
0.853804 0.520594i \(-0.174290\pi\)
\(24\) 1.34277e11 0.249004
\(25\) 0 0
\(26\) 3.90681e11 0.366901
\(27\) − 1.55448e12i − 1.05923i
\(28\) − 1.66916e12i − 0.834939i
\(29\) −5.12259e12 −1.90154 −0.950772 0.309892i \(-0.899707\pi\)
−0.950772 + 0.309892i \(0.899707\pi\)
\(30\) 0 0
\(31\) −7.31204e10 −0.0153980 −0.00769900 0.999970i \(-0.502451\pi\)
−0.00769900 + 0.999970i \(0.502451\pi\)
\(32\) − 1.09951e12i − 0.176777i
\(33\) 2.24970e12i 0.278456i
\(34\) 1.40007e13 1.34455
\(35\) 0 0
\(36\) −4.26532e12 −0.251988
\(37\) − 6.81637e12i − 0.319035i −0.987195 0.159517i \(-0.949006\pi\)
0.987195 0.159517i \(-0.0509938\pi\)
\(38\) 1.76370e11i 0.00658059i
\(39\) 1.22142e13 0.365439
\(40\) 0 0
\(41\) −5.76386e13 −1.12733 −0.563665 0.826004i \(-0.690609\pi\)
−0.563665 + 0.826004i \(0.690609\pi\)
\(42\) − 5.21843e13i − 0.831613i
\(43\) − 7.57081e13i − 0.987781i −0.869524 0.493890i \(-0.835574\pi\)
0.869524 0.493890i \(-0.164426\pi\)
\(44\) 1.84214e13 0.197686
\(45\) 0 0
\(46\) −1.00105e14 −0.736231
\(47\) − 4.60351e13i − 0.282006i −0.990009 0.141003i \(-0.954967\pi\)
0.990009 0.141003i \(-0.0450327\pi\)
\(48\) − 3.43749e13i − 0.176072i
\(49\) −4.16059e14 −1.78849
\(50\) 0 0
\(51\) 4.37714e14 1.33919
\(52\) − 1.00014e14i − 0.259438i
\(53\) − 6.58261e14i − 1.45229i −0.687541 0.726145i \(-0.741310\pi\)
0.687541 0.726145i \(-0.258690\pi\)
\(54\) −3.97946e14 −0.748992
\(55\) 0 0
\(56\) −4.27305e14 −0.590391
\(57\) 5.51401e12i 0.00655437i
\(58\) 1.31138e15i 1.34459i
\(59\) 2.98287e14 0.264479 0.132240 0.991218i \(-0.457783\pi\)
0.132240 + 0.991218i \(0.457783\pi\)
\(60\) 0 0
\(61\) 8.50623e14 0.568111 0.284056 0.958808i \(-0.408320\pi\)
0.284056 + 0.958808i \(0.408320\pi\)
\(62\) 1.87188e13i 0.0108880i
\(63\) 1.65764e15i 0.841580i
\(64\) −2.81475e14 −0.125000
\(65\) 0 0
\(66\) 5.75924e14 0.196898
\(67\) − 6.12967e15i − 1.84417i −0.386987 0.922085i \(-0.626484\pi\)
0.386987 0.922085i \(-0.373516\pi\)
\(68\) − 3.58417e15i − 0.950742i
\(69\) −3.12966e15 −0.733298
\(70\) 0 0
\(71\) 5.41472e14 0.0995131 0.0497565 0.998761i \(-0.484155\pi\)
0.0497565 + 0.998761i \(0.484155\pi\)
\(72\) 1.09192e15i 0.178183i
\(73\) 7.16849e15i 1.04036i 0.854057 + 0.520180i \(0.174135\pi\)
−0.854057 + 0.520180i \(0.825865\pi\)
\(74\) −1.74499e15 −0.225592
\(75\) 0 0
\(76\) 4.51508e13 0.00465318
\(77\) − 7.15916e15i − 0.660223i
\(78\) − 3.12683e15i − 0.258404i
\(79\) −5.45373e15 −0.404448 −0.202224 0.979339i \(-0.564817\pi\)
−0.202224 + 0.979339i \(0.564817\pi\)
\(80\) 0 0
\(81\) −4.03640e15 −0.242031
\(82\) 1.47555e16i 0.797142i
\(83\) 3.64723e15i 0.177746i 0.996043 + 0.0888728i \(0.0283265\pi\)
−0.996043 + 0.0888728i \(0.971674\pi\)
\(84\) −1.33592e16 −0.588039
\(85\) 0 0
\(86\) −1.93813e16 −0.698467
\(87\) 4.09988e16i 1.33924i
\(88\) − 4.71589e15i − 0.139785i
\(89\) −6.81792e14 −0.0183585 −0.00917925 0.999958i \(-0.502922\pi\)
−0.00917925 + 0.999958i \(0.502922\pi\)
\(90\) 0 0
\(91\) −3.88687e16 −0.866460
\(92\) 2.56269e16i 0.520594i
\(93\) 5.85222e14i 0.0108446i
\(94\) −1.17850e16 −0.199408
\(95\) 0 0
\(96\) −8.79998e15 −0.124502
\(97\) − 1.20373e17i − 1.55944i −0.626129 0.779720i \(-0.715362\pi\)
0.626129 0.779720i \(-0.284638\pi\)
\(98\) 1.06511e17i 1.26466i
\(99\) −1.82943e16 −0.199258
\(100\) 0 0
\(101\) 7.74092e16 0.711314 0.355657 0.934617i \(-0.384257\pi\)
0.355657 + 0.934617i \(0.384257\pi\)
\(102\) − 1.12055e17i − 0.946954i
\(103\) 1.39309e17i 1.08359i 0.840512 + 0.541793i \(0.182254\pi\)
−0.840512 + 0.541793i \(0.817746\pi\)
\(104\) −2.56037e16 −0.183450
\(105\) 0 0
\(106\) −1.68515e17 −1.02692
\(107\) 2.75662e17i 1.55101i 0.631341 + 0.775505i \(0.282505\pi\)
−0.631341 + 0.775505i \(0.717495\pi\)
\(108\) 1.01874e17i 0.529617i
\(109\) −1.37550e17 −0.661202 −0.330601 0.943771i \(-0.607251\pi\)
−0.330601 + 0.943771i \(0.607251\pi\)
\(110\) 0 0
\(111\) −5.45550e16 −0.224693
\(112\) 1.09390e17i 0.417470i
\(113\) 3.90258e17i 1.38097i 0.723345 + 0.690486i \(0.242604\pi\)
−0.723345 + 0.690486i \(0.757396\pi\)
\(114\) 1.41159e15 0.00463464
\(115\) 0 0
\(116\) 3.35714e17 0.950772
\(117\) 9.93239e16i 0.261501i
\(118\) − 7.63614e16i − 0.187015i
\(119\) −1.39292e18 −3.17525
\(120\) 0 0
\(121\) −4.26436e17 −0.843681
\(122\) − 2.17759e17i − 0.401715i
\(123\) 4.61313e17i 0.793966i
\(124\) 4.79202e15 0.00769900
\(125\) 0 0
\(126\) 4.24355e17 0.595087
\(127\) − 7.37824e17i − 0.967433i −0.875225 0.483717i \(-0.839287\pi\)
0.875225 0.483717i \(-0.160713\pi\)
\(128\) 7.20576e16i 0.0883883i
\(129\) −6.05933e17 −0.695684
\(130\) 0 0
\(131\) −1.45408e18 −1.46482 −0.732409 0.680865i \(-0.761604\pi\)
−0.732409 + 0.680865i \(0.761604\pi\)
\(132\) − 1.47437e17i − 0.139228i
\(133\) − 1.75470e16i − 0.0155405i
\(134\) −1.56919e18 −1.30403
\(135\) 0 0
\(136\) −9.17547e17 −0.672276
\(137\) − 2.27534e18i − 1.56647i −0.621725 0.783236i \(-0.713568\pi\)
0.621725 0.783236i \(-0.286432\pi\)
\(138\) 8.01194e17i 0.518520i
\(139\) −1.41598e17 −0.0861849 −0.0430924 0.999071i \(-0.513721\pi\)
−0.0430924 + 0.999071i \(0.513721\pi\)
\(140\) 0 0
\(141\) −3.68444e17 −0.198614
\(142\) − 1.38617e17i − 0.0703664i
\(143\) − 4.28969e17i − 0.205149i
\(144\) 2.79532e17 0.125994
\(145\) 0 0
\(146\) 1.83513e18 0.735646
\(147\) 3.32994e18i 1.25962i
\(148\) 4.46717e17i 0.159517i
\(149\) 1.31089e18 0.442063 0.221032 0.975267i \(-0.429058\pi\)
0.221032 + 0.975267i \(0.429058\pi\)
\(150\) 0 0
\(151\) −2.76548e18 −0.832658 −0.416329 0.909214i \(-0.636684\pi\)
−0.416329 + 0.909214i \(0.636684\pi\)
\(152\) − 1.15586e16i − 0.00329030i
\(153\) 3.55943e18i 0.958303i
\(154\) −1.83274e18 −0.466848
\(155\) 0 0
\(156\) −8.00468e17 −0.182719
\(157\) − 3.81235e18i − 0.824225i −0.911133 0.412112i \(-0.864791\pi\)
0.911133 0.412112i \(-0.135209\pi\)
\(158\) 1.39615e18i 0.285988i
\(159\) −5.26842e18 −1.02283
\(160\) 0 0
\(161\) 9.95942e18 1.73866
\(162\) 1.03332e18i 0.171142i
\(163\) − 1.03773e19i − 1.63114i −0.578658 0.815570i \(-0.696423\pi\)
0.578658 0.815570i \(-0.303577\pi\)
\(164\) 3.77740e18 0.563665
\(165\) 0 0
\(166\) 9.33690e17 0.125685
\(167\) 1.21920e19i 1.55949i 0.626095 + 0.779747i \(0.284652\pi\)
−0.626095 + 0.779747i \(0.715348\pi\)
\(168\) 3.41995e18i 0.415806i
\(169\) 6.32145e18 0.730768
\(170\) 0 0
\(171\) −4.48391e16 −0.00469019
\(172\) 4.96161e18i 0.493890i
\(173\) 3.88210e18i 0.367853i 0.982940 + 0.183927i \(0.0588809\pi\)
−0.982940 + 0.183927i \(0.941119\pi\)
\(174\) 1.04957e19 0.946983
\(175\) 0 0
\(176\) −1.20727e18 −0.0988430
\(177\) − 2.38735e18i − 0.186270i
\(178\) 1.74539e17i 0.0129814i
\(179\) 1.62354e19 1.15136 0.575680 0.817675i \(-0.304737\pi\)
0.575680 + 0.817675i \(0.304737\pi\)
\(180\) 0 0
\(181\) −2.32195e19 −1.49825 −0.749127 0.662427i \(-0.769527\pi\)
−0.749127 + 0.662427i \(0.769527\pi\)
\(182\) 9.95039e18i 0.612680i
\(183\) − 6.80799e18i − 0.400115i
\(184\) 6.56048e18 0.368116
\(185\) 0 0
\(186\) 1.49817e17 0.00766832
\(187\) − 1.53728e19i − 0.751793i
\(188\) 3.01696e18i 0.141003i
\(189\) 3.95915e19 1.76879
\(190\) 0 0
\(191\) 4.47928e19 1.82989 0.914944 0.403580i \(-0.132234\pi\)
0.914944 + 0.403580i \(0.132234\pi\)
\(192\) 2.25279e18i 0.0880362i
\(193\) − 3.61154e19i − 1.35038i −0.737645 0.675189i \(-0.764062\pi\)
0.737645 0.675189i \(-0.235938\pi\)
\(194\) −3.08154e19 −1.10269
\(195\) 0 0
\(196\) 2.72668e19 0.894247
\(197\) − 1.80503e19i − 0.566920i −0.958984 0.283460i \(-0.908518\pi\)
0.958984 0.283460i \(-0.0914823\pi\)
\(198\) 4.68333e18i 0.140897i
\(199\) 6.39636e18 0.184366 0.0921832 0.995742i \(-0.470615\pi\)
0.0921832 + 0.995742i \(0.470615\pi\)
\(200\) 0 0
\(201\) −4.90590e19 −1.29883
\(202\) − 1.98168e19i − 0.502975i
\(203\) − 1.30469e20i − 3.17535i
\(204\) −2.86860e19 −0.669597
\(205\) 0 0
\(206\) 3.56631e19 0.766211
\(207\) − 2.54500e19i − 0.524734i
\(208\) 6.55453e18i 0.129719i
\(209\) 1.93655e17 0.00367948
\(210\) 0 0
\(211\) 7.72101e18 0.135292 0.0676462 0.997709i \(-0.478451\pi\)
0.0676462 + 0.997709i \(0.478451\pi\)
\(212\) 4.31398e19i 0.726145i
\(213\) − 4.33369e18i − 0.0700860i
\(214\) 7.05695e19 1.09673
\(215\) 0 0
\(216\) 2.60798e19 0.374496
\(217\) − 1.86233e18i − 0.0257128i
\(218\) 3.52127e19i 0.467541i
\(219\) 5.73733e19 0.732715
\(220\) 0 0
\(221\) −8.34623e19 −0.986634
\(222\) 1.39661e19i 0.158882i
\(223\) 1.08588e20i 1.18902i 0.804088 + 0.594510i \(0.202654\pi\)
−0.804088 + 0.594510i \(0.797346\pi\)
\(224\) 2.80039e19 0.295196
\(225\) 0 0
\(226\) 9.99061e19 0.976495
\(227\) 1.95539e19i 0.184083i 0.995755 + 0.0920415i \(0.0293392\pi\)
−0.995755 + 0.0920415i \(0.970661\pi\)
\(228\) − 3.61366e17i − 0.00327719i
\(229\) −1.20601e20 −1.05378 −0.526890 0.849933i \(-0.676642\pi\)
−0.526890 + 0.849933i \(0.676642\pi\)
\(230\) 0 0
\(231\) −5.72986e19 −0.464988
\(232\) − 8.59427e19i − 0.672297i
\(233\) − 7.20388e19i − 0.543302i −0.962396 0.271651i \(-0.912430\pi\)
0.962396 0.271651i \(-0.0875696\pi\)
\(234\) 2.54269e19 0.184909
\(235\) 0 0
\(236\) −1.95485e19 −0.132240
\(237\) 4.36491e19i 0.284849i
\(238\) 3.56588e20i 2.24524i
\(239\) −1.70425e20 −1.03550 −0.517750 0.855532i \(-0.673230\pi\)
−0.517750 + 0.855532i \(0.673230\pi\)
\(240\) 0 0
\(241\) −1.71129e20 −0.968679 −0.484340 0.874880i \(-0.660940\pi\)
−0.484340 + 0.874880i \(0.660940\pi\)
\(242\) 1.09168e20i 0.596573i
\(243\) − 1.68440e20i − 0.888775i
\(244\) −5.57464e19 −0.284056
\(245\) 0 0
\(246\) 1.18096e20 0.561419
\(247\) − 1.05140e18i − 0.00482885i
\(248\) − 1.22676e18i − 0.00544401i
\(249\) 2.91907e19 0.125184
\(250\) 0 0
\(251\) 1.85831e20 0.744546 0.372273 0.928123i \(-0.378578\pi\)
0.372273 + 0.928123i \(0.378578\pi\)
\(252\) − 1.08635e20i − 0.420790i
\(253\) 1.09916e20i 0.411657i
\(254\) −1.88883e20 −0.684078
\(255\) 0 0
\(256\) 1.84467e19 0.0625000
\(257\) 1.68273e19i 0.0551547i 0.999620 + 0.0275773i \(0.00877925\pi\)
−0.999620 + 0.0275773i \(0.991221\pi\)
\(258\) 1.55119e20i 0.491923i
\(259\) 1.73609e20 0.532750
\(260\) 0 0
\(261\) −3.33396e20 −0.958333
\(262\) 3.72245e20i 1.03578i
\(263\) − 2.77268e20i − 0.746924i −0.927645 0.373462i \(-0.878171\pi\)
0.927645 0.373462i \(-0.121829\pi\)
\(264\) −3.77438e19 −0.0984491
\(265\) 0 0
\(266\) −4.49204e18 −0.0109888
\(267\) 5.45675e18i 0.0129297i
\(268\) 4.01714e20i 0.922085i
\(269\) −9.40799e19 −0.209220 −0.104610 0.994513i \(-0.533359\pi\)
−0.104610 + 0.994513i \(0.533359\pi\)
\(270\) 0 0
\(271\) −6.17628e20 −1.28970 −0.644849 0.764310i \(-0.723080\pi\)
−0.644849 + 0.764310i \(0.723080\pi\)
\(272\) 2.34892e20i 0.475371i
\(273\) 3.11087e20i 0.610238i
\(274\) −5.82488e20 −1.10766
\(275\) 0 0
\(276\) 2.05106e20 0.366649
\(277\) − 6.67865e20i − 1.15774i −0.815420 0.578869i \(-0.803494\pi\)
0.815420 0.578869i \(-0.196506\pi\)
\(278\) 3.62491e19i 0.0609419i
\(279\) −4.75894e18 −0.00776023
\(280\) 0 0
\(281\) −1.13640e20 −0.174393 −0.0871963 0.996191i \(-0.527791\pi\)
−0.0871963 + 0.996191i \(0.527791\pi\)
\(282\) 9.43216e19i 0.140441i
\(283\) 3.76957e20i 0.544637i 0.962207 + 0.272318i \(0.0877904\pi\)
−0.962207 + 0.272318i \(0.912210\pi\)
\(284\) −3.54859e19 −0.0497565
\(285\) 0 0
\(286\) −1.09816e20 −0.145062
\(287\) − 1.46802e21i − 1.88250i
\(288\) − 7.15602e19i − 0.0890913i
\(289\) −2.16376e21 −2.61564
\(290\) 0 0
\(291\) −9.63408e20 −1.09830
\(292\) − 4.69794e20i − 0.520180i
\(293\) 3.57350e20i 0.384344i 0.981361 + 0.192172i \(0.0615531\pi\)
−0.981361 + 0.192172i \(0.938447\pi\)
\(294\) 8.52464e20 0.890684
\(295\) 0 0
\(296\) 1.14360e20 0.112796
\(297\) 4.36946e20i 0.418792i
\(298\) − 3.35589e20i − 0.312586i
\(299\) 5.96757e20 0.540247
\(300\) 0 0
\(301\) 1.92824e21 1.64947
\(302\) 7.07963e20i 0.588778i
\(303\) − 6.19547e20i − 0.500971i
\(304\) −2.95900e18 −0.00232659
\(305\) 0 0
\(306\) 9.11213e20 0.677623
\(307\) 2.25658e21i 1.63220i 0.577908 + 0.816102i \(0.303869\pi\)
−0.577908 + 0.816102i \(0.696131\pi\)
\(308\) 4.69182e20i 0.330112i
\(309\) 1.11497e21 0.763158
\(310\) 0 0
\(311\) 5.76844e20 0.373762 0.186881 0.982383i \(-0.440162\pi\)
0.186881 + 0.982383i \(0.440162\pi\)
\(312\) 2.04920e20i 0.129202i
\(313\) − 2.63218e21i − 1.61506i −0.589827 0.807530i \(-0.700804\pi\)
0.589827 0.807530i \(-0.299196\pi\)
\(314\) −9.75961e20 −0.582815
\(315\) 0 0
\(316\) 3.57415e20 0.202224
\(317\) − 9.95998e20i − 0.548599i −0.961644 0.274299i \(-0.911554\pi\)
0.961644 0.274299i \(-0.0884459\pi\)
\(318\) 1.34871e21i 0.723252i
\(319\) 1.43990e21 0.751817
\(320\) 0 0
\(321\) 2.20627e21 1.09236
\(322\) − 2.54961e21i − 1.22942i
\(323\) − 3.76786e19i − 0.0176959i
\(324\) 2.64529e20 0.121016
\(325\) 0 0
\(326\) −2.65660e21 −1.15339
\(327\) 1.10088e21i 0.465678i
\(328\) − 9.67016e20i − 0.398571i
\(329\) 1.17249e21 0.470915
\(330\) 0 0
\(331\) −3.56520e21 −1.36002 −0.680011 0.733202i \(-0.738025\pi\)
−0.680011 + 0.733202i \(0.738025\pi\)
\(332\) − 2.39025e20i − 0.0888728i
\(333\) − 4.43634e20i − 0.160786i
\(334\) 3.12114e21 1.10273
\(335\) 0 0
\(336\) 8.75508e20 0.294019
\(337\) 3.65283e20i 0.119612i 0.998210 + 0.0598060i \(0.0190482\pi\)
−0.998210 + 0.0598060i \(0.980952\pi\)
\(338\) − 1.61829e21i − 0.516731i
\(339\) 3.12344e21 0.972605
\(340\) 0 0
\(341\) 2.05533e19 0.00608793
\(342\) 1.14788e19i 0.00331646i
\(343\) − 4.67180e21i − 1.31669i
\(344\) 1.27017e21 0.349233
\(345\) 0 0
\(346\) 9.93818e20 0.260112
\(347\) − 2.06934e21i − 0.528483i −0.964457 0.264242i \(-0.914878\pi\)
0.964457 0.264242i \(-0.0851217\pi\)
\(348\) − 2.68690e21i − 0.669618i
\(349\) −3.13803e20 −0.0763204 −0.0381602 0.999272i \(-0.512150\pi\)
−0.0381602 + 0.999272i \(0.512150\pi\)
\(350\) 0 0
\(351\) 2.37228e21 0.549611
\(352\) 3.09060e20i 0.0698925i
\(353\) 6.87901e21i 1.51859i 0.650746 + 0.759296i \(0.274456\pi\)
−0.650746 + 0.759296i \(0.725544\pi\)
\(354\) −6.11161e20 −0.131713
\(355\) 0 0
\(356\) 4.46820e19 0.00917925
\(357\) 1.11483e22i 2.23629i
\(358\) − 4.15625e21i − 0.814134i
\(359\) 9.50907e21 1.81901 0.909505 0.415693i \(-0.136461\pi\)
0.909505 + 0.415693i \(0.136461\pi\)
\(360\) 0 0
\(361\) −5.47991e21 −0.999913
\(362\) 5.94420e21i 1.05943i
\(363\) 3.41300e21i 0.594196i
\(364\) 2.54730e21 0.433230
\(365\) 0 0
\(366\) −1.74285e21 −0.282924
\(367\) 1.88008e20i 0.0298205i 0.999889 + 0.0149103i \(0.00474626\pi\)
−0.999889 + 0.0149103i \(0.995254\pi\)
\(368\) − 1.67948e21i − 0.260297i
\(369\) −3.75133e21 −0.568147
\(370\) 0 0
\(371\) 1.67655e22 2.42515
\(372\) − 3.83531e19i − 0.00542232i
\(373\) − 9.66745e20i − 0.133594i −0.997767 0.0667970i \(-0.978722\pi\)
0.997767 0.0667970i \(-0.0212780\pi\)
\(374\) −3.93543e21 −0.531598
\(375\) 0 0
\(376\) 7.72342e20 0.0997040
\(377\) − 7.81756e21i − 0.986665i
\(378\) − 1.01354e22i − 1.25073i
\(379\) −1.09478e22 −1.32097 −0.660484 0.750840i \(-0.729649\pi\)
−0.660484 + 0.750840i \(0.729649\pi\)
\(380\) 0 0
\(381\) −5.90520e21 −0.681353
\(382\) − 1.14670e22i − 1.29393i
\(383\) 9.48135e21i 1.04636i 0.852223 + 0.523179i \(0.175254\pi\)
−0.852223 + 0.523179i \(0.824746\pi\)
\(384\) 5.76715e20 0.0622510
\(385\) 0 0
\(386\) −9.24554e21 −0.954861
\(387\) − 4.92736e21i − 0.497818i
\(388\) 7.88875e21i 0.779720i
\(389\) −8.68048e21 −0.839406 −0.419703 0.907661i \(-0.637866\pi\)
−0.419703 + 0.907661i \(0.637866\pi\)
\(390\) 0 0
\(391\) 2.13857e22 1.97980
\(392\) − 6.98030e21i − 0.632328i
\(393\) 1.16378e22i 1.03166i
\(394\) −4.62088e21 −0.400873
\(395\) 0 0
\(396\) 1.19893e21 0.0996291
\(397\) − 5.61938e21i − 0.457055i −0.973537 0.228528i \(-0.926609\pi\)
0.973537 0.228528i \(-0.0733911\pi\)
\(398\) − 1.63747e21i − 0.130367i
\(399\) −1.40438e20 −0.0109450
\(400\) 0 0
\(401\) −4.97857e21 −0.371858 −0.185929 0.982563i \(-0.559530\pi\)
−0.185929 + 0.982563i \(0.559530\pi\)
\(402\) 1.25591e22i 0.918411i
\(403\) − 1.11589e20i − 0.00798965i
\(404\) −5.07309e21 −0.355657
\(405\) 0 0
\(406\) −3.34001e22 −2.24531
\(407\) 1.91600e21i 0.126137i
\(408\) 7.34362e21i 0.473477i
\(409\) −2.65693e20 −0.0167777 −0.00838883 0.999965i \(-0.502670\pi\)
−0.00838883 + 0.999965i \(0.502670\pi\)
\(410\) 0 0
\(411\) −1.82108e22 −1.10325
\(412\) − 9.12977e21i − 0.541793i
\(413\) 7.59717e21i 0.441648i
\(414\) −6.51519e21 −0.371043
\(415\) 0 0
\(416\) 1.67796e21 0.0917252
\(417\) 1.13328e21i 0.0606991i
\(418\) − 4.95757e19i − 0.00260178i
\(419\) −1.71242e22 −0.880626 −0.440313 0.897844i \(-0.645133\pi\)
−0.440313 + 0.897844i \(0.645133\pi\)
\(420\) 0 0
\(421\) 2.71839e22 1.34250 0.671249 0.741231i \(-0.265758\pi\)
0.671249 + 0.741231i \(0.265758\pi\)
\(422\) − 1.97658e21i − 0.0956662i
\(423\) − 2.99613e21i − 0.142124i
\(424\) 1.10438e22 0.513462
\(425\) 0 0
\(426\) −1.10942e21 −0.0495583
\(427\) 2.16648e22i 0.948677i
\(428\) − 1.80658e22i − 0.775505i
\(429\) −3.43327e21 −0.144484
\(430\) 0 0
\(431\) −2.34759e22 −0.949656 −0.474828 0.880079i \(-0.657490\pi\)
−0.474828 + 0.880079i \(0.657490\pi\)
\(432\) − 6.67643e21i − 0.264809i
\(433\) 1.88732e22i 0.734003i 0.930220 + 0.367002i \(0.119616\pi\)
−0.930220 + 0.367002i \(0.880384\pi\)
\(434\) −4.76757e20 −0.0181817
\(435\) 0 0
\(436\) 9.01446e21 0.330601
\(437\) 2.69402e20i 0.00968968i
\(438\) − 1.46876e22i − 0.518107i
\(439\) −5.51895e22 −1.90945 −0.954724 0.297494i \(-0.903849\pi\)
−0.954724 + 0.297494i \(0.903849\pi\)
\(440\) 0 0
\(441\) −2.70786e22 −0.901359
\(442\) 2.13664e22i 0.697656i
\(443\) 1.20735e22i 0.386723i 0.981128 + 0.193362i \(0.0619391\pi\)
−0.981128 + 0.193362i \(0.938061\pi\)
\(444\) 3.57532e21 0.112346
\(445\) 0 0
\(446\) 2.77984e22 0.840764
\(447\) − 1.04918e22i − 0.311341i
\(448\) − 7.16899e21i − 0.208735i
\(449\) −2.57466e22 −0.735573 −0.367787 0.929910i \(-0.619884\pi\)
−0.367787 + 0.929910i \(0.619884\pi\)
\(450\) 0 0
\(451\) 1.62016e22 0.445714
\(452\) − 2.55760e22i − 0.690486i
\(453\) 2.21336e22i 0.586432i
\(454\) 5.00580e21 0.130166
\(455\) 0 0
\(456\) −9.25097e19 −0.00231732
\(457\) − 1.01069e22i − 0.248503i −0.992251 0.124252i \(-0.960347\pi\)
0.992251 0.124252i \(-0.0396530\pi\)
\(458\) 3.08739e22i 0.745136i
\(459\) 8.50144e22 2.01412
\(460\) 0 0
\(461\) 1.68599e21 0.0384942 0.0192471 0.999815i \(-0.493873\pi\)
0.0192471 + 0.999815i \(0.493873\pi\)
\(462\) 1.46684e22i 0.328796i
\(463\) 2.77945e22i 0.611674i 0.952084 + 0.305837i \(0.0989363\pi\)
−0.952084 + 0.305837i \(0.901064\pi\)
\(464\) −2.20013e22 −0.475386
\(465\) 0 0
\(466\) −1.84419e22 −0.384172
\(467\) 8.26643e22i 1.69093i 0.534034 + 0.845463i \(0.320676\pi\)
−0.534034 + 0.845463i \(0.679324\pi\)
\(468\) − 6.50929e21i − 0.130751i
\(469\) 1.56119e23 3.07954
\(470\) 0 0
\(471\) −3.05123e22 −0.580493
\(472\) 5.00442e21i 0.0935075i
\(473\) 2.12807e22i 0.390541i
\(474\) 1.11742e22 0.201418
\(475\) 0 0
\(476\) 9.12865e22 1.58762
\(477\) − 4.28420e22i − 0.731920i
\(478\) 4.36287e22i 0.732209i
\(479\) −2.59191e22 −0.427335 −0.213667 0.976906i \(-0.568541\pi\)
−0.213667 + 0.976906i \(0.568541\pi\)
\(480\) 0 0
\(481\) 1.04024e22 0.165540
\(482\) 4.38091e22i 0.684960i
\(483\) − 7.97106e22i − 1.22452i
\(484\) 2.79469e22 0.421841
\(485\) 0 0
\(486\) −4.31206e22 −0.628459
\(487\) − 4.14828e22i − 0.594116i −0.954859 0.297058i \(-0.903994\pi\)
0.954859 0.297058i \(-0.0960055\pi\)
\(488\) 1.42711e22i 0.200858i
\(489\) −8.30554e22 −1.14879
\(490\) 0 0
\(491\) 3.08644e22 0.412349 0.206174 0.978515i \(-0.433899\pi\)
0.206174 + 0.978515i \(0.433899\pi\)
\(492\) − 3.02326e22i − 0.396983i
\(493\) − 2.80155e23i − 3.61575i
\(494\) −2.69158e20 −0.00341451
\(495\) 0 0
\(496\) −3.14050e20 −0.00384950
\(497\) 1.37910e22i 0.166175i
\(498\) − 7.47282e21i − 0.0885187i
\(499\) −3.24182e22 −0.377516 −0.188758 0.982024i \(-0.560446\pi\)
−0.188758 + 0.982024i \(0.560446\pi\)
\(500\) 0 0
\(501\) 9.75788e22 1.09833
\(502\) − 4.75728e22i − 0.526474i
\(503\) − 6.39324e22i − 0.695654i −0.937559 0.347827i \(-0.886920\pi\)
0.937559 0.347827i \(-0.113080\pi\)
\(504\) −2.78106e22 −0.297543
\(505\) 0 0
\(506\) 2.81384e22 0.291085
\(507\) − 5.05939e22i − 0.514672i
\(508\) 4.83540e22i 0.483717i
\(509\) 9.33355e22 0.918218 0.459109 0.888380i \(-0.348169\pi\)
0.459109 + 0.888380i \(0.348169\pi\)
\(510\) 0 0
\(511\) −1.82577e23 −1.73728
\(512\) − 4.72237e21i − 0.0441942i
\(513\) 1.07095e21i 0.00985763i
\(514\) 4.30778e21 0.0390003
\(515\) 0 0
\(516\) 3.97104e22 0.347842
\(517\) 1.29400e22i 0.111497i
\(518\) − 4.44438e22i − 0.376711i
\(519\) 3.10705e22 0.259075
\(520\) 0 0
\(521\) 3.30201e22 0.266476 0.133238 0.991084i \(-0.457463\pi\)
0.133238 + 0.991084i \(0.457463\pi\)
\(522\) 8.53495e22i 0.677644i
\(523\) − 4.06002e22i − 0.317149i −0.987347 0.158575i \(-0.949310\pi\)
0.987347 0.158575i \(-0.0506898\pi\)
\(524\) 9.52948e22 0.732409
\(525\) 0 0
\(526\) −7.09807e22 −0.528155
\(527\) − 3.99896e21i − 0.0292790i
\(528\) 9.66240e21i 0.0696140i
\(529\) −1.18585e22 −0.0840729
\(530\) 0 0
\(531\) 1.94136e22 0.133291
\(532\) 1.14996e21i 0.00777025i
\(533\) − 8.79621e22i − 0.584944i
\(534\) 1.39693e21 0.00914268
\(535\) 0 0
\(536\) 1.02839e23 0.652013
\(537\) − 1.29940e23i − 0.810890i
\(538\) 2.40844e22i 0.147941i
\(539\) 1.16949e23 0.707121
\(540\) 0 0
\(541\) −4.20460e22 −0.246347 −0.123174 0.992385i \(-0.539307\pi\)
−0.123174 + 0.992385i \(0.539307\pi\)
\(542\) 1.58113e23i 0.911954i
\(543\) 1.85838e23i 1.05520i
\(544\) 6.01324e22 0.336138
\(545\) 0 0
\(546\) 7.96383e22 0.431504
\(547\) − 2.01110e23i − 1.07285i −0.843947 0.536427i \(-0.819774\pi\)
0.843947 0.536427i \(-0.180226\pi\)
\(548\) 1.49117e23i 0.783236i
\(549\) 5.53616e22 0.286315
\(550\) 0 0
\(551\) 3.52919e21 0.0176965
\(552\) − 5.25070e22i − 0.259260i
\(553\) − 1.38903e23i − 0.675380i
\(554\) −1.70973e23 −0.818645
\(555\) 0 0
\(556\) 9.27976e21 0.0430924
\(557\) 1.46727e23i 0.671030i 0.942035 + 0.335515i \(0.108910\pi\)
−0.942035 + 0.335515i \(0.891090\pi\)
\(558\) 1.21829e21i 0.00548731i
\(559\) 1.15538e23 0.512536
\(560\) 0 0
\(561\) −1.23036e23 −0.529480
\(562\) 2.90919e22i 0.123314i
\(563\) 3.39706e23i 1.41834i 0.705036 + 0.709172i \(0.250931\pi\)
−0.705036 + 0.709172i \(0.749069\pi\)
\(564\) 2.41463e22 0.0993068
\(565\) 0 0
\(566\) 9.65010e22 0.385116
\(567\) − 1.02805e23i − 0.404163i
\(568\) 9.08439e21i 0.0351832i
\(569\) −2.72602e23 −1.04010 −0.520050 0.854136i \(-0.674087\pi\)
−0.520050 + 0.854136i \(0.674087\pi\)
\(570\) 0 0
\(571\) −2.79882e22 −0.103650 −0.0518249 0.998656i \(-0.516504\pi\)
−0.0518249 + 0.998656i \(0.516504\pi\)
\(572\) 2.81129e22i 0.102574i
\(573\) − 3.58501e23i − 1.28877i
\(574\) −3.75813e23 −1.33113
\(575\) 0 0
\(576\) −1.83194e22 −0.0629971
\(577\) 3.86678e23i 1.31025i 0.755519 + 0.655127i \(0.227385\pi\)
−0.755519 + 0.655127i \(0.772615\pi\)
\(578\) 5.53923e23i 1.84954i
\(579\) −2.89051e23 −0.951056
\(580\) 0 0
\(581\) −9.28926e22 −0.296814
\(582\) 2.46632e23i 0.776613i
\(583\) 1.85030e23i 0.574195i
\(584\) −1.20267e23 −0.367823
\(585\) 0 0
\(586\) 9.14817e22 0.271772
\(587\) 3.64152e23i 1.06625i 0.846036 + 0.533125i \(0.178983\pi\)
−0.846036 + 0.533125i \(0.821017\pi\)
\(588\) − 2.18231e23i − 0.629809i
\(589\) 5.03761e19 0.000143299 0
\(590\) 0 0
\(591\) −1.44466e23 −0.399276
\(592\) − 2.92761e22i − 0.0797587i
\(593\) 6.86689e23i 1.84415i 0.387017 + 0.922073i \(0.373506\pi\)
−0.387017 + 0.922073i \(0.626494\pi\)
\(594\) 1.11858e23 0.296130
\(595\) 0 0
\(596\) −8.59108e22 −0.221032
\(597\) − 5.11935e22i − 0.129847i
\(598\) − 1.52770e23i − 0.382013i
\(599\) 3.82850e23 0.943846 0.471923 0.881640i \(-0.343560\pi\)
0.471923 + 0.881640i \(0.343560\pi\)
\(600\) 0 0
\(601\) −2.66402e23 −0.638417 −0.319209 0.947684i \(-0.603417\pi\)
−0.319209 + 0.947684i \(0.603417\pi\)
\(602\) − 4.93629e23i − 1.16635i
\(603\) − 3.98941e23i − 0.929418i
\(604\) 1.81239e23 0.416329
\(605\) 0 0
\(606\) −1.58604e23 −0.354240
\(607\) − 1.11277e23i − 0.245076i −0.992464 0.122538i \(-0.960897\pi\)
0.992464 0.122538i \(-0.0391034\pi\)
\(608\) 7.57505e20i 0.00164515i
\(609\) −1.04421e24 −2.23636
\(610\) 0 0
\(611\) 7.02541e22 0.146326
\(612\) − 2.33271e23i − 0.479152i
\(613\) 3.91229e23i 0.792532i 0.918136 + 0.396266i \(0.129694\pi\)
−0.918136 + 0.396266i \(0.870306\pi\)
\(614\) 5.77684e23 1.15414
\(615\) 0 0
\(616\) 1.20111e23 0.233424
\(617\) − 5.72626e23i − 1.09761i −0.835951 0.548804i \(-0.815083\pi\)
0.835951 0.548804i \(-0.184917\pi\)
\(618\) − 2.85431e23i − 0.539634i
\(619\) −7.86155e23 −1.46601 −0.733007 0.680222i \(-0.761884\pi\)
−0.733007 + 0.680222i \(0.761884\pi\)
\(620\) 0 0
\(621\) −6.07855e23 −1.10286
\(622\) − 1.47672e23i − 0.264290i
\(623\) − 1.73648e22i − 0.0306565i
\(624\) 5.24594e22 0.0913597
\(625\) 0 0
\(626\) −6.73838e23 −1.14202
\(627\) − 1.54993e21i − 0.00259142i
\(628\) 2.49846e23i 0.412112i
\(629\) 3.72788e23 0.606640
\(630\) 0 0
\(631\) 6.52712e23 1.03389 0.516943 0.856020i \(-0.327070\pi\)
0.516943 + 0.856020i \(0.327070\pi\)
\(632\) − 9.14983e22i − 0.142994i
\(633\) − 6.17954e22i − 0.0952850i
\(634\) −2.54975e23 −0.387918
\(635\) 0 0
\(636\) 3.45271e23 0.511416
\(637\) − 6.34945e23i − 0.928007i
\(638\) − 3.68615e23i − 0.531615i
\(639\) 3.52410e22 0.0501522
\(640\) 0 0
\(641\) 7.10734e23 0.984950 0.492475 0.870327i \(-0.336092\pi\)
0.492475 + 0.870327i \(0.336092\pi\)
\(642\) − 5.64805e23i − 0.772415i
\(643\) − 2.34278e23i − 0.316183i −0.987424 0.158092i \(-0.949466\pi\)
0.987424 0.158092i \(-0.0505341\pi\)
\(644\) −6.52701e23 −0.869329
\(645\) 0 0
\(646\) −9.64571e21 −0.0125129
\(647\) 1.50806e23i 0.193078i 0.995329 + 0.0965391i \(0.0307773\pi\)
−0.995329 + 0.0965391i \(0.969223\pi\)
\(648\) − 6.77195e22i − 0.0855710i
\(649\) −8.38450e22 −0.104568
\(650\) 0 0
\(651\) −1.49052e22 −0.0181092
\(652\) 6.80089e23i 0.815570i
\(653\) − 6.74375e22i − 0.0798251i −0.999203 0.0399126i \(-0.987292\pi\)
0.999203 0.0399126i \(-0.0127079\pi\)
\(654\) 2.81826e23 0.329284
\(655\) 0 0
\(656\) −2.47556e23 −0.281832
\(657\) 4.66551e23i 0.524317i
\(658\) − 3.00157e23i − 0.332987i
\(659\) −7.11301e22 −0.0778981 −0.0389491 0.999241i \(-0.512401\pi\)
−0.0389491 + 0.999241i \(0.512401\pi\)
\(660\) 0 0
\(661\) 1.69805e24 1.81233 0.906164 0.422926i \(-0.138997\pi\)
0.906164 + 0.422926i \(0.138997\pi\)
\(662\) 9.12691e23i 0.961680i
\(663\) 6.67994e23i 0.694876i
\(664\) −6.11903e22 −0.0628426
\(665\) 0 0
\(666\) −1.13570e23 −0.113693
\(667\) 2.00311e24i 1.97986i
\(668\) − 7.99012e23i − 0.779747i
\(669\) 8.69085e23 0.837414
\(670\) 0 0
\(671\) −2.39101e23 −0.224615
\(672\) − 2.24130e23i − 0.207903i
\(673\) 1.39848e24i 1.28094i 0.767984 + 0.640469i \(0.221260\pi\)
−0.767984 + 0.640469i \(0.778740\pi\)
\(674\) 9.35124e22 0.0845785
\(675\) 0 0
\(676\) −4.14282e23 −0.365384
\(677\) − 9.38639e23i − 0.817514i −0.912643 0.408757i \(-0.865962\pi\)
0.912643 0.408757i \(-0.134038\pi\)
\(678\) − 7.99602e23i − 0.687735i
\(679\) 3.06582e24 2.60408
\(680\) 0 0
\(681\) 1.56500e23 0.129648
\(682\) − 5.26165e21i − 0.00430482i
\(683\) − 1.23254e24i − 0.995922i −0.867199 0.497961i \(-0.834082\pi\)
0.867199 0.497961i \(-0.165918\pi\)
\(684\) 2.93858e21 0.00234509
\(685\) 0 0
\(686\) −1.19598e24 −0.931041
\(687\) 9.65236e23i 0.742167i
\(688\) − 3.25164e23i − 0.246945i
\(689\) 1.00457e24 0.753559
\(690\) 0 0
\(691\) 3.83055e23 0.280348 0.140174 0.990127i \(-0.455234\pi\)
0.140174 + 0.990127i \(0.455234\pi\)
\(692\) − 2.54417e23i − 0.183927i
\(693\) − 4.65944e23i − 0.332737i
\(694\) −5.29751e23 −0.373694
\(695\) 0 0
\(696\) −6.87845e23 −0.473492
\(697\) − 3.15226e24i − 2.14360i
\(698\) 8.03335e22i 0.0539667i
\(699\) −5.76565e23 −0.382642
\(700\) 0 0
\(701\) 1.63235e24 1.05733 0.528665 0.848831i \(-0.322693\pi\)
0.528665 + 0.848831i \(0.322693\pi\)
\(702\) − 6.07304e23i − 0.388634i
\(703\) 4.69611e21i 0.00296906i
\(704\) 7.91195e22 0.0494215
\(705\) 0 0
\(706\) 1.76103e24 1.07381
\(707\) 1.97156e24i 1.18781i
\(708\) 1.56457e23i 0.0931350i
\(709\) 6.09431e23 0.358453 0.179226 0.983808i \(-0.442641\pi\)
0.179226 + 0.983808i \(0.442641\pi\)
\(710\) 0 0
\(711\) −3.54948e23 −0.203832
\(712\) − 1.14386e22i − 0.00649071i
\(713\) 2.85926e22i 0.0160322i
\(714\) 2.85396e24 1.58130
\(715\) 0 0
\(716\) −1.06400e24 −0.575680
\(717\) 1.36400e24i 0.729292i
\(718\) − 2.43432e24i − 1.28623i
\(719\) 4.49148e23 0.234527 0.117264 0.993101i \(-0.462588\pi\)
0.117264 + 0.993101i \(0.462588\pi\)
\(720\) 0 0
\(721\) −3.54812e24 −1.80946
\(722\) 1.40286e24i 0.707046i
\(723\) 1.36964e24i 0.682230i
\(724\) 1.52171e24 0.749127
\(725\) 0 0
\(726\) 8.73727e23 0.420160
\(727\) − 5.00713e23i − 0.237983i −0.992895 0.118992i \(-0.962034\pi\)
0.992895 0.118992i \(-0.0379662\pi\)
\(728\) − 6.52109e23i − 0.306340i
\(729\) −1.86938e24 −0.867986
\(730\) 0 0
\(731\) 4.14048e24 1.87825
\(732\) 4.46168e23i 0.200057i
\(733\) − 1.20329e24i − 0.533320i −0.963791 0.266660i \(-0.914080\pi\)
0.963791 0.266660i \(-0.0859201\pi\)
\(734\) 4.81301e22 0.0210863
\(735\) 0 0
\(736\) −4.29948e23 −0.184058
\(737\) 1.72298e24i 0.729133i
\(738\) 9.60340e23i 0.401741i
\(739\) 2.19426e24 0.907426 0.453713 0.891148i \(-0.350099\pi\)
0.453713 + 0.891148i \(0.350099\pi\)
\(740\) 0 0
\(741\) −8.41491e21 −0.00340091
\(742\) − 4.29197e24i − 1.71484i
\(743\) 9.89633e22i 0.0390903i 0.999809 + 0.0195452i \(0.00622181\pi\)
−0.999809 + 0.0195452i \(0.993778\pi\)
\(744\) −9.81839e21 −0.00383416
\(745\) 0 0
\(746\) −2.47487e23 −0.0944652
\(747\) 2.37375e23i 0.0895796i
\(748\) 1.00747e24i 0.375897i
\(749\) −7.02094e24 −2.59000
\(750\) 0 0
\(751\) −1.01856e24 −0.367322 −0.183661 0.982990i \(-0.558795\pi\)
−0.183661 + 0.982990i \(0.558795\pi\)
\(752\) − 1.97719e23i − 0.0705014i
\(753\) − 1.48731e24i − 0.524376i
\(754\) −2.00130e24 −0.697677
\(755\) 0 0
\(756\) −2.59467e24 −0.884397
\(757\) − 1.43989e24i − 0.485305i −0.970113 0.242653i \(-0.921982\pi\)
0.970113 0.242653i \(-0.0780175\pi\)
\(758\) 2.80263e24i 0.934066i
\(759\) 8.79713e23 0.289925
\(760\) 0 0
\(761\) −7.15424e23 −0.230565 −0.115283 0.993333i \(-0.536777\pi\)
−0.115283 + 0.993333i \(0.536777\pi\)
\(762\) 1.51173e24i 0.481789i
\(763\) − 3.50330e24i − 1.10413i
\(764\) −2.93554e24 −0.914944
\(765\) 0 0
\(766\) 2.42722e24 0.739887
\(767\) 4.55214e23i 0.137232i
\(768\) − 1.47639e23i − 0.0440181i
\(769\) −1.87177e24 −0.551922 −0.275961 0.961169i \(-0.588996\pi\)
−0.275961 + 0.961169i \(0.588996\pi\)
\(770\) 0 0
\(771\) 1.34678e23 0.0388449
\(772\) 2.36686e24i 0.675189i
\(773\) 1.86999e24i 0.527611i 0.964576 + 0.263805i \(0.0849777\pi\)
−0.964576 + 0.263805i \(0.915022\pi\)
\(774\) −1.26140e24 −0.352011
\(775\) 0 0
\(776\) 2.01952e24 0.551345
\(777\) − 1.38948e24i − 0.375210i
\(778\) 2.22220e24i 0.593550i
\(779\) 3.97099e22 0.0104913
\(780\) 0 0
\(781\) −1.52202e23 −0.0393447
\(782\) − 5.47475e24i − 1.39993i
\(783\) 7.96294e24i 2.01418i
\(784\) −1.78696e24 −0.447124
\(785\) 0 0
\(786\) 2.97928e24 0.729490
\(787\) − 3.29304e24i − 0.797648i −0.917027 0.398824i \(-0.869418\pi\)
0.917027 0.398824i \(-0.130582\pi\)
\(788\) 1.18295e24i 0.283460i
\(789\) −2.21913e24 −0.526051
\(790\) 0 0
\(791\) −9.93963e24 −2.30606
\(792\) − 3.06927e23i − 0.0704484i
\(793\) 1.29813e24i 0.294779i
\(794\) −1.43856e24 −0.323187
\(795\) 0 0
\(796\) −4.19192e23 −0.0921832
\(797\) − 3.80089e24i − 0.826969i −0.910511 0.413485i \(-0.864312\pi\)
0.910511 0.413485i \(-0.135688\pi\)
\(798\) 3.59522e22i 0.00773929i
\(799\) 2.51767e24 0.536229
\(800\) 0 0
\(801\) −4.43735e22 −0.00925225
\(802\) 1.27451e24i 0.262944i
\(803\) − 2.01498e24i − 0.411329i
\(804\) 3.21513e24 0.649415
\(805\) 0 0
\(806\) −2.85667e22 −0.00564953
\(807\) 7.52972e23i 0.147351i
\(808\) 1.29871e24i 0.251487i
\(809\) −1.38713e24 −0.265799 −0.132900 0.991129i \(-0.542429\pi\)
−0.132900 + 0.991129i \(0.542429\pi\)
\(810\) 0 0
\(811\) 2.43418e24 0.456747 0.228374 0.973574i \(-0.426659\pi\)
0.228374 + 0.973574i \(0.426659\pi\)
\(812\) 8.55042e24i 1.58767i
\(813\) 4.94321e24i 0.908321i
\(814\) 4.90497e23 0.0891926
\(815\) 0 0
\(816\) 1.87997e24 0.334799
\(817\) 5.21589e22i 0.00919265i
\(818\) 6.80174e22i 0.0118636i
\(819\) −2.52972e24 −0.436675
\(820\) 0 0
\(821\) −7.07675e24 −1.19651 −0.598256 0.801305i \(-0.704140\pi\)
−0.598256 + 0.801305i \(0.704140\pi\)
\(822\) 4.66196e24i 0.780115i
\(823\) 5.31425e24i 0.880124i 0.897968 + 0.440062i \(0.145043\pi\)
−0.897968 + 0.440062i \(0.854957\pi\)
\(824\) −2.33722e24 −0.383105
\(825\) 0 0
\(826\) 1.94488e24 0.312292
\(827\) 5.13487e24i 0.816080i 0.912964 + 0.408040i \(0.133788\pi\)
−0.912964 + 0.408040i \(0.866212\pi\)
\(828\) 1.66789e24i 0.262367i
\(829\) −1.06738e25 −1.66191 −0.830954 0.556341i \(-0.812205\pi\)
−0.830954 + 0.556341i \(0.812205\pi\)
\(830\) 0 0
\(831\) −5.34528e24 −0.815383
\(832\) − 4.29558e23i − 0.0648595i
\(833\) − 2.27543e25i − 3.40079i
\(834\) 2.90121e23 0.0429208
\(835\) 0 0
\(836\) −1.26914e22 −0.00183974
\(837\) 1.13664e23i 0.0163101i
\(838\) 4.38380e24i 0.622697i
\(839\) 6.47302e24 0.910186 0.455093 0.890444i \(-0.349606\pi\)
0.455093 + 0.890444i \(0.349606\pi\)
\(840\) 0 0
\(841\) 1.89837e25 2.61587
\(842\) − 6.95908e24i − 0.949290i
\(843\) 9.09523e23i 0.122823i
\(844\) −5.06004e23 −0.0676462
\(845\) 0 0
\(846\) −7.67010e23 −0.100497
\(847\) − 1.08611e25i − 1.40885i
\(848\) − 2.82721e24i − 0.363073i
\(849\) 3.01699e24 0.383582
\(850\) 0 0
\(851\) −2.66544e24 −0.332175
\(852\) 2.84013e23i 0.0350430i
\(853\) − 2.69306e24i − 0.328988i −0.986378 0.164494i \(-0.947401\pi\)
0.986378 0.164494i \(-0.0525991\pi\)
\(854\) 5.54620e24 0.670816
\(855\) 0 0
\(856\) −4.62484e24 −0.548365
\(857\) − 1.86785e24i − 0.219283i −0.993971 0.109641i \(-0.965030\pi\)
0.993971 0.109641i \(-0.0349702\pi\)
\(858\) 8.78916e23i 0.102166i
\(859\) 1.13124e25 1.30200 0.651000 0.759077i \(-0.274350\pi\)
0.651000 + 0.759077i \(0.274350\pi\)
\(860\) 0 0
\(861\) −1.17493e25 −1.32583
\(862\) 6.00984e24i 0.671508i
\(863\) 2.37061e24i 0.262282i 0.991364 + 0.131141i \(0.0418640\pi\)
−0.991364 + 0.131141i \(0.958136\pi\)
\(864\) −1.70917e24 −0.187248
\(865\) 0 0
\(866\) 4.83153e24 0.519019
\(867\) 1.73178e25i 1.84217i
\(868\) 1.22050e23i 0.0128564i
\(869\) 1.53298e24 0.159907
\(870\) 0 0
\(871\) 9.35446e24 0.956895
\(872\) − 2.30770e24i − 0.233770i
\(873\) − 7.83429e24i − 0.785921i
\(874\) 6.89670e22 0.00685164
\(875\) 0 0
\(876\) −3.76002e24 −0.366357
\(877\) − 6.79534e24i − 0.655714i −0.944727 0.327857i \(-0.893674\pi\)
0.944727 0.327857i \(-0.106326\pi\)
\(878\) 1.41285e25i 1.35018i
\(879\) 2.86007e24 0.270689
\(880\) 0 0
\(881\) −1.88402e25 −1.74900 −0.874501 0.485024i \(-0.838811\pi\)
−0.874501 + 0.485024i \(0.838811\pi\)
\(882\) 6.93212e24i 0.637357i
\(883\) − 3.85491e24i − 0.351033i −0.984476 0.175517i \(-0.943840\pi\)
0.984476 0.175517i \(-0.0561596\pi\)
\(884\) 5.46979e24 0.493317
\(885\) 0 0
\(886\) 3.09081e24 0.273455
\(887\) 8.32727e24i 0.729712i 0.931064 + 0.364856i \(0.118882\pi\)
−0.931064 + 0.364856i \(0.881118\pi\)
\(888\) − 9.15281e23i − 0.0794410i
\(889\) 1.87919e25 1.61550
\(890\) 0 0
\(891\) 1.13459e24 0.0956924
\(892\) − 7.11640e24i − 0.594510i
\(893\) 3.17158e22i 0.00262445i
\(894\) −2.68590e24 −0.220151
\(895\) 0 0
\(896\) −1.83526e24 −0.147598
\(897\) − 4.77617e24i − 0.380491i
\(898\) 6.59113e24i 0.520129i
\(899\) 3.74565e23 0.0292799
\(900\) 0 0
\(901\) 3.60004e25 2.76151
\(902\) − 4.14760e24i − 0.315168i
\(903\) − 1.54327e25i − 1.16171i
\(904\) −6.54744e24 −0.488248
\(905\) 0 0
\(906\) 5.66621e24 0.414670
\(907\) 1.92435e25i 1.39515i 0.716511 + 0.697576i \(0.245738\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(908\) − 1.28148e24i − 0.0920415i
\(909\) 5.03807e24 0.358485
\(910\) 0 0
\(911\) −1.57682e25 −1.10123 −0.550613 0.834761i \(-0.685606\pi\)
−0.550613 + 0.834761i \(0.685606\pi\)
\(912\) 2.36825e22i 0.00163859i
\(913\) − 1.02519e24i − 0.0702756i
\(914\) −2.58738e24 −0.175718
\(915\) 0 0
\(916\) 7.90373e24 0.526890
\(917\) − 3.70346e25i − 2.44607i
\(918\) − 2.17637e25i − 1.42420i
\(919\) −1.03058e25 −0.668189 −0.334094 0.942540i \(-0.608430\pi\)
−0.334094 + 0.942540i \(0.608430\pi\)
\(920\) 0 0
\(921\) 1.80606e25 1.14954
\(922\) − 4.31612e23i − 0.0272195i
\(923\) 8.26339e23i 0.0516349i
\(924\) 3.75512e24 0.232494
\(925\) 0 0
\(926\) 7.11539e24 0.432519
\(927\) 9.06674e24i 0.546102i
\(928\) 5.63234e24i 0.336149i
\(929\) −9.73688e24 −0.575819 −0.287910 0.957658i \(-0.592960\pi\)
−0.287910 + 0.957658i \(0.592960\pi\)
\(930\) 0 0
\(931\) 2.86642e23 0.0166444
\(932\) 4.72113e24i 0.271651i
\(933\) − 4.61679e24i − 0.263236i
\(934\) 2.11621e25 1.19567
\(935\) 0 0
\(936\) −1.66638e24 −0.0924547
\(937\) − 3.22145e25i − 1.77119i −0.464459 0.885594i \(-0.653751\pi\)
0.464459 0.885594i \(-0.346249\pi\)
\(938\) − 3.99664e25i − 2.17756i
\(939\) −2.10667e25 −1.13747
\(940\) 0 0
\(941\) −7.90374e24 −0.419103 −0.209552 0.977798i \(-0.567200\pi\)
−0.209552 + 0.977798i \(0.567200\pi\)
\(942\) 7.81114e24i 0.410470i
\(943\) 2.25387e25i 1.17376i
\(944\) 1.28113e24 0.0661198
\(945\) 0 0
\(946\) 5.44786e24 0.276154
\(947\) 2.36682e25i 1.18902i 0.804086 + 0.594512i \(0.202655\pi\)
−0.804086 + 0.594512i \(0.797345\pi\)
\(948\) − 2.86059e24i − 0.142424i
\(949\) −1.09398e25 −0.539818
\(950\) 0 0
\(951\) −7.97150e24 −0.386372
\(952\) − 2.33693e25i − 1.12262i
\(953\) − 9.09477e24i − 0.433014i −0.976281 0.216507i \(-0.930534\pi\)
0.976281 0.216507i \(-0.0694664\pi\)
\(954\) −1.09676e25 −0.517546
\(955\) 0 0
\(956\) 1.11690e25 0.517750
\(957\) − 1.15243e25i − 0.529496i
\(958\) 6.63529e24i 0.302171i
\(959\) 5.79516e25 2.61582
\(960\) 0 0
\(961\) −2.25448e25 −0.999763
\(962\) − 2.66302e24i − 0.117054i
\(963\) 1.79411e25i 0.781673i
\(964\) 1.12151e25 0.484340
\(965\) 0 0
\(966\) −2.04059e25 −0.865865
\(967\) 3.62970e25i 1.52667i 0.646001 + 0.763337i \(0.276440\pi\)
−0.646001 + 0.763337i \(0.723560\pi\)
\(968\) − 7.15441e24i − 0.298286i
\(969\) −3.01562e23 −0.0124630
\(970\) 0 0
\(971\) −9.70396e24 −0.394081 −0.197040 0.980395i \(-0.563133\pi\)
−0.197040 + 0.980395i \(0.563133\pi\)
\(972\) 1.10389e25i 0.444387i
\(973\) − 3.60641e24i − 0.143918i
\(974\) −1.06196e25 −0.420104
\(975\) 0 0
\(976\) 3.65340e24 0.142028
\(977\) 3.70141e25i 1.42647i 0.700924 + 0.713236i \(0.252771\pi\)
−0.700924 + 0.713236i \(0.747229\pi\)
\(978\) 2.12622e25i 0.812321i
\(979\) 1.91644e23 0.00725843
\(980\) 0 0
\(981\) −8.95223e24 −0.333230
\(982\) − 7.90127e24i − 0.291574i
\(983\) 6.43184e24i 0.235305i 0.993055 + 0.117652i \(0.0375368\pi\)
−0.993055 + 0.117652i \(0.962463\pi\)
\(984\) −7.73954e24 −0.280709
\(985\) 0 0
\(986\) −7.17196e25 −2.55672
\(987\) − 9.38403e24i − 0.331661i
\(988\) 6.89045e22i 0.00241442i
\(989\) −2.96045e25 −1.02847
\(990\) 0 0
\(991\) 5.17070e25 1.76573 0.882863 0.469631i \(-0.155613\pi\)
0.882863 + 0.469631i \(0.155613\pi\)
\(992\) 8.03967e22i 0.00272201i
\(993\) 2.85342e25i 0.957848i
\(994\) 3.53048e24 0.117503
\(995\) 0 0
\(996\) −1.91304e24 −0.0625922
\(997\) − 2.77457e25i − 0.900093i −0.893005 0.450047i \(-0.851407\pi\)
0.893005 0.450047i \(-0.148593\pi\)
\(998\) 8.29907e24i 0.266944i
\(999\) −1.05959e25 −0.337933
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 50.18.b.d.49.1 4
5.2 odd 4 50.18.a.f.1.1 2
5.3 odd 4 10.18.a.c.1.2 2
5.4 even 2 inner 50.18.b.d.49.4 4
15.8 even 4 90.18.a.k.1.2 2
20.3 even 4 80.18.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.18.a.c.1.2 2 5.3 odd 4
50.18.a.f.1.1 2 5.2 odd 4
50.18.b.d.49.1 4 1.1 even 1 trivial
50.18.b.d.49.4 4 5.4 even 2 inner
80.18.a.d.1.1 2 20.3 even 4
90.18.a.k.1.2 2 15.8 even 4