LMFDB - Embedded newform 121.16.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,16,Mod(1,121)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 16, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	121.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:	$172.659141171$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	121.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-216.000 q^{2} -3348.00 q^{3} +13888.0 q^{4} +52110.0 q^{5} +723168. q^{6} -2.82246e6 q^{7} +4.07808e6 q^{8} -3.13980e6 q^{9} +O(q^{10})$

\(q-216.000 q^{2} -3348.00 q^{3} +13888.0 q^{4} +52110.0 q^{5} +723168. q^{6} -2.82246e6 q^{7} +4.07808e6 q^{8} -3.13980e6 q^{9} -1.12558e7 q^{10} -4.64970e7 q^{12} +1.90073e8 q^{13} +6.09650e8 q^{14} -1.74464e8 q^{15} -1.33595e9 q^{16} -1.64653e9 q^{17} +6.78197e8 q^{18} -1.56326e9 q^{19} +7.23704e8 q^{20} +9.44958e9 q^{21} +9.45112e9 q^{23} -1.36534e10 q^{24} -2.78021e10 q^{25} -4.10558e10 q^{26} +5.85522e10 q^{27} -3.91983e10 q^{28} +3.69026e10 q^{29} +3.76843e10 q^{30} +7.15885e10 q^{31} +1.54934e11 q^{32} +3.55650e11 q^{34} -1.47078e11 q^{35} -4.36056e10 q^{36} -1.03365e12 q^{37} +3.37664e11 q^{38} -6.36366e11 q^{39} +2.12509e11 q^{40} -1.64197e12 q^{41} -2.04111e12 q^{42} +4.92403e11 q^{43} -1.63615e11 q^{45} -2.04144e12 q^{46} -3.41068e12 q^{47} +4.47275e12 q^{48} +3.21870e12 q^{49} +6.00526e12 q^{50} +5.51258e12 q^{51} +2.63974e12 q^{52} +6.79715e12 q^{53} -1.26473e13 q^{54} -1.15102e13 q^{56} +5.23379e12 q^{57} -7.97095e12 q^{58} +9.85886e12 q^{59} -2.42296e12 q^{60} -4.93184e12 q^{61} -1.54631e13 q^{62} +8.86196e12 q^{63} +1.03106e13 q^{64} +9.90472e12 q^{65} -2.88378e13 q^{67} -2.28670e13 q^{68} -3.16423e13 q^{69} +3.17689e13 q^{70} +1.25050e14 q^{71} -1.28044e13 q^{72} +8.21715e13 q^{73} +2.23269e14 q^{74} +9.30815e13 q^{75} -2.17105e13 q^{76} +1.37455e14 q^{78} +2.54131e13 q^{79} -6.96162e13 q^{80} -1.50980e14 q^{81} +3.54666e14 q^{82} +2.81737e14 q^{83} +1.31236e14 q^{84} -8.58006e13 q^{85} -1.06359e14 q^{86} -1.23550e14 q^{87} +7.15619e14 q^{89} +3.53409e13 q^{90} -5.36474e14 q^{91} +1.31257e14 q^{92} -2.39678e14 q^{93} +7.36708e14 q^{94} -8.14613e13 q^{95} -5.18719e14 q^{96} +6.12786e14 q^{97} -6.95238e14 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−216.000	−1.19324	−0.596621	−	0.802523i	$-0.703491\pi$
$2$	−216.000	−1.19324	−0.596621	+	0.802523i	$0.703491\pi$
$3$	−3348.00	−0.883845	−0.441922	−	0.897053i	$-0.645703\pi$
$3$	−3348.00	−0.883845	−0.441922	+	0.897053i	$0.645703\pi$
$4$	13888.0	0.423828
$5$	52110.0	0.298295	0.149148	−	0.988815i	$-0.452347\pi$
$5$	52110.0	0.298295	0.149148	+	0.988815i	$0.452347\pi$
$6$	723168.	1.05464
$7$	−2.82246e6	−1.29536	−0.647682	−	0.761911i	$-0.724261\pi$
$7$	−2.82246e6	−1.29536	−0.647682	+	0.761911i	$0.724261\pi$
$8$	4.07808e6	0.687513
$9$	−3.13980e6	−0.218818
$10$	−1.12558e7	−0.355938
$11$	0	0
$11$	0	0
$12$	−4.64970e7	−0.374598
$13$	1.90073e8	0.840129	0.420065	−	0.907494i	$-0.362007\pi$
$13$	1.90073e8	0.840129	0.420065	+	0.907494i	$0.362007\pi$
$14$	6.09650e8	1.54568
$15$	−1.74464e8	−0.263647
$16$	−1.33595e9	−1.24420
$17$	−1.64653e9	−0.973200	−0.486600	−	0.873625i	$-0.661763\pi$
$17$	−1.64653e9	−0.973200	−0.486600	+	0.873625i	$0.661763\pi$
$18$	6.78197e8	0.261103
$19$	−1.56326e9	−0.401216	−0.200608	−	0.979672i	$-0.564292\pi$
$19$	−1.56326e9	−0.401216	−0.200608	+	0.979672i	$0.564292\pi$
$20$	7.23704e8	0.126426
$21$	9.44958e9	1.14490
$22$	0	0
$23$	9.45112e9	0.578794	0.289397	−	0.957209i	$-0.406545\pi$
$23$	9.45112e9	0.578794	0.289397	+	0.957209i	$0.406545\pi$
$24$	−1.36534e10	−0.607655
$25$	−2.78021e10	−0.911020
$26$	−4.10558e10	−1.00248
$27$	5.85522e10	1.07725
$28$	−3.91983e10	−0.549012
$29$	3.69026e10	0.397257	0.198629	−	0.980075i	$-0.436351\pi$
$29$	3.69026e10	0.397257	0.198629	+	0.980075i	$0.436351\pi$
$30$	3.76843e10	0.314594
$31$	7.15885e10	0.467337	0.233669	−	0.972316i	$-0.424927\pi$
$31$	7.15885e10	0.467337	0.233669	+	0.972316i	$0.424927\pi$
$32$	1.54934e11	0.797117
$33$	0	0
$34$	3.55650e11	1.16126
$35$	−1.47078e11	−0.386401
$36$	−4.36056e10	−0.0927413
$37$	−1.03365e12	−1.79003	−0.895017	−	0.446031i	$-0.852837\pi$
$37$	−1.03365e12	−1.79003	−0.895017	+	0.446031i	$0.852837\pi$
$38$	3.37664e11	0.478748
$39$	−6.36366e11	−0.742544
$40$	2.12509e11	0.205082
$41$	−1.64197e12	−1.31670	−0.658351	−	0.752711i	$-0.728746\pi$
$41$	−1.64197e12	−1.31670	−0.658351	+	0.752711i	$0.728746\pi$
$42$	−2.04111e12	−1.36614
$43$	4.92403e11	0.276253	0.138127	−	0.990415i	$-0.455892\pi$
$43$	4.92403e11	0.276253	0.138127	+	0.990415i	$0.455892\pi$
$44$	0	0
$45$	−1.63615e11	−0.0652724
$46$	−2.04144e12	−0.690642
$47$	−3.41068e12	−0.981991	−0.490996	−	0.871162i	$-0.663367\pi$
$47$	−3.41068e12	−0.981991	−0.490996	+	0.871162i	$0.663367\pi$
$48$	4.47275e12	1.09968
$49$	3.21870e12	0.677968
$50$	6.00526e12	1.08707
$51$	5.51258e12	0.860158
$52$	2.63974e12	0.356070
$53$	6.79715e12	0.794800	0.397400	−	0.917645i	$-0.369913\pi$
$53$	6.79715e12	0.794800	0.397400	+	0.917645i	$0.369913\pi$
$54$	−1.26473e13	−1.28542
$55$	0	0
$56$	−1.15102e13	−0.890580
$57$	5.23379e12	0.354613
$58$	−7.97095e12	−0.474024
$59$	9.85886e12	0.515747	0.257873	−	0.966179i	$-0.416978\pi$
$59$	9.85886e12	0.515747	0.257873	+	0.966179i	$0.416978\pi$
$60$	−2.42296e12	−0.111741
$61$	−4.93184e12	−0.200926	−0.100463	−	0.994941i	$-0.532032\pi$
$61$	−4.93184e12	−0.200926	−0.100463	+	0.994941i	$0.532032\pi$
$62$	−1.54631e13	−0.557647
$63$	8.86196e12	0.283449
$64$	1.03106e13	0.293044
$65$	9.90472e12	0.250606
$66$	0	0
$67$	−2.88378e13	−0.581302	−0.290651	−	0.956829i	$-0.593872\pi$
$67$	−2.88378e13	−0.581302	−0.290651	+	0.956829i	$0.593872\pi$
$68$	−2.28670e13	−0.412470
$69$	−3.16423e13	−0.511564
$70$	3.17689e13	0.461070
$71$	1.25050e14	1.63172	0.815862	−	0.578247i	$-0.196263\pi$
$71$	1.25050e14	1.63172	0.815862	+	0.578247i	$0.196263\pi$
$72$	−1.28044e13	−0.150440
$73$	8.21715e13	0.870562	0.435281	−	0.900295i	$-0.356649\pi$
$73$	8.21715e13	0.870562	0.435281	+	0.900295i	$0.356649\pi$
$74$	2.23269e14	2.13595
$75$	9.30815e13	0.805200
$76$	−2.17105e13	−0.170047
$77$	0	0
$78$	1.37455e14	0.886035
$79$	2.54131e13	0.148886	0.0744430	−	0.997225i	$-0.476282\pi$
$79$	2.54131e13	0.148886	0.0744430	+	0.997225i	$0.476282\pi$
$80$	−6.96162e13	−0.371138
$81$	−1.50980e14	−0.733300
$82$	3.54666e14	1.57114
$83$	2.81737e14	1.13961	0.569807	−	0.821779i	$-0.307018\pi$
$83$	2.81737e14	1.13961	0.569807	+	0.821779i	$0.307018\pi$
$84$	1.31236e14	0.485241
$85$	−8.58006e13	−0.290301
$86$	−1.06359e14	−0.329637
$87$	−1.23550e14	−0.351114
$88$	0	0
$89$	7.15619e14	1.71497	0.857485	−	0.514509i	$-0.172026\pi$
$89$	7.15619e14	1.71497	0.857485	+	0.514509i	$0.172026\pi$
$90$	3.53409e13	0.0778858
$91$	−5.36474e14	−1.08827
$92$	1.31257e14	0.245309
$93$	−2.39678e14	−0.413054
$94$	7.36708e14	1.17175
$95$	−8.14613e13	−0.119681
$96$	−5.18719e14	−0.704528
$97$	6.12786e14	0.770054	0.385027	−	0.922905i	$-0.374192\pi$
$97$	6.12786e14	0.770054	0.385027	+	0.922905i	$0.374192\pi$
$98$	−6.95238e14	−0.808981
$99$	0	0
$100$	−3.86116e14	−0.386116
$101$	8.17642e14	0.758844	0.379422	−	0.925224i	$-0.376123\pi$
$101$	8.17642e14	0.758844	0.379422	+	0.925224i	$0.376123\pi$
$102$	−1.19072e15	−1.02638
$103$	7.41115e14	0.593753	0.296877	−	0.954916i	$-0.404055\pi$
$103$	7.41115e14	0.593753	0.296877	+	0.954916i	$0.404055\pi$
$104$	7.75134e14	0.577600
$105$	4.92418e14	0.341518
$106$	−1.46818e15	−0.948389
$107$	2.51430e15	1.51370	0.756849	−	0.653590i	$-0.226738\pi$
$107$	2.51430e15	1.51370	0.756849	+	0.653590i	$0.226738\pi$
$108$	8.13173e14	0.456567
$109$	−1.26835e15	−0.664572	−0.332286	−	0.943179i	$-0.607820\pi$
$109$	−1.26835e15	−0.664572	−0.332286	+	0.943179i	$0.607820\pi$
$110$	0	0
$111$	3.46067e15	1.58211
$112$	3.77065e15	1.61169
$113$	−2.05416e15	−0.821385	−0.410692	−	0.911774i	$-0.634713\pi$
$113$	−2.05416e15	−0.821385	−0.410692	+	0.911774i	$0.634713\pi$
$114$	−1.13050e15	−0.423139
$115$	4.92498e14	0.172652
$116$	5.12503e14	0.168369
$117$	−5.96793e14	−0.183836
$118$	−2.12951e15	−0.615411
$119$	4.64725e15	1.26065
$120$	−7.11479e14	−0.181260
$121$	0	0
$122$	1.06528e15	0.239753
$123$	5.49733e15	1.16376
$124$	9.94221e14	0.198071
$125$	−3.03904e15	−0.570048
$126$	−1.91418e15	−0.338224
$127$	−2.99068e15	−0.498014	−0.249007	−	0.968502i	$-0.580104\pi$
$127$	−2.99068e15	−0.498014	−0.249007	+	0.968502i	$0.580104\pi$
$128$	−7.30396e15	−1.14679
$129$	−1.64857e15	−0.244165
$130$	−2.13942e15	−0.299034
$131$	1.62623e15	0.214608	0.107304	−	0.994226i	$-0.465778\pi$
$131$	1.62623e15	0.214608	0.107304	+	0.994226i	$0.465778\pi$
$132$	0	0
$133$	4.41222e15	0.519721
$134$	6.22897e15	0.693634
$135$	3.05116e15	0.321337
$136$	−6.71467e15	−0.669088
$137$	1.05922e16	0.999038	0.499519	−	0.866303i	$-0.333510\pi$
$137$	1.05922e16	0.999038	0.499519	+	0.866303i	$0.333510\pi$
$138$	6.83474e15	0.610421
$139$	1.86709e16	1.57963	0.789813	−	0.613347i	$-0.210177\pi$
$139$	1.86709e16	1.57963	0.789813	+	0.613347i	$0.210177\pi$
$140$	−2.04262e15	−0.163767
$141$	1.14190e16	0.867928
$142$	−2.70108e16	−1.94704
$143$	0	0
$144$	4.19461e15	0.272253
$145$	1.92299e15	0.118500
$146$	−1.77490e16	−1.03879
$147$	−1.07762e16	−0.599219
$148$	−1.43554e16	−0.758667
$149$	1.25560e16	0.630889	0.315444	−	0.948944i	$-0.397846\pi$
$149$	1.25560e16	0.630889	0.315444	+	0.948944i	$0.397846\pi$
$150$	−2.01056e16	−0.960799
$151$	−2.87588e16	−1.30751	−0.653753	−	0.756708i	$-0.726806\pi$
$151$	−2.87588e16	−1.30751	−0.653753	+	0.756708i	$0.726806\pi$
$152$	−6.37509e15	−0.275841
$153$	5.16977e15	0.212954
$154$	0	0
$155$	3.73048e15	0.139404
$156$	−8.83784e15	−0.314711
$157$	−1.45276e16	−0.493114	−0.246557	−	0.969128i	$-0.579299\pi$
$157$	−1.45276e16	−0.493114	−0.246557	+	0.969128i	$0.579299\pi$
$158$	−5.48922e15	−0.177657
$159$	−2.27569e16	−0.702480
$160$	8.07362e15	0.237776
$161$	−2.66754e16	−0.749750
$162$	3.26117e16	0.875005
$163$	1.67741e16	0.429767	0.214884	−	0.976640i	$-0.431063\pi$
$163$	1.67741e16	0.429767	0.214884	+	0.976640i	$0.431063\pi$
$164$	−2.28037e16	−0.558055
$165$	0	0
$166$	−6.08551e16	−1.35984
$167$	−6.41999e16	−1.37139	−0.685695	−	0.727889i	$-0.740502\pi$
$167$	−6.41999e16	−1.37139	−0.685695	+	0.727889i	$0.740502\pi$
$168$	3.85362e16	0.787134
$169$	−1.50580e16	−0.294183
$170$	1.85329e16	0.346399
$171$	4.90832e15	0.0877934
$172$	6.83849e15	0.117084
$173$	7.59860e16	1.24563	0.622814	−	0.782370i	$-0.285990\pi$
$173$	7.59860e16	1.24563	0.622814	+	0.782370i	$0.285990\pi$
$174$	2.66868e16	0.418964
$175$	7.84703e16	1.18010
$176$	0	0
$177$	−3.30075e16	−0.455840
$178$	−1.54574e17	−2.04638
$179$	9.33749e16	1.18531	0.592655	−	0.805456i	$-0.298080\pi$
$179$	9.33749e16	1.18531	0.592655	+	0.805456i	$0.298080\pi$
$180$	−2.27229e15	−0.0276643
$181$	7.43177e16	0.867966	0.433983	−	0.900921i	$-0.357108\pi$
$181$	7.43177e16	0.867966	0.433983	+	0.900921i	$0.357108\pi$
$182$	1.15878e17	1.29857
$183$	1.65118e16	0.177587
$184$	3.85424e16	0.397929
$185$	−5.38636e16	−0.533958
$186$	5.17705e16	0.492873
$187$	0	0
$188$	−4.73676e16	−0.416196
$189$	−1.65261e17	−1.39543
$190$	1.75956e16	0.142808
$191$	−9.86224e16	−0.769529	−0.384765	−	0.923015i	$-0.625717\pi$
$191$	−9.86224e16	−0.769529	−0.384765	+	0.923015i	$0.625717\pi$
$192$	−3.45197e16	−0.259005
$193$	8.91178e15	0.0643109	0.0321554	−	0.999483i	$-0.489763\pi$
$193$	8.91178e15	0.0643109	0.0321554	+	0.999483i	$0.489763\pi$
$194$	−1.32362e17	−0.918861
$195$	−3.31610e16	−0.221497
$196$	4.47013e16	0.287342
$197$	−3.54176e16	−0.219140	−0.109570	−	0.993979i	$-0.534947\pi$
$197$	−3.54176e16	−0.219140	−0.109570	+	0.993979i	$0.534947\pi$
$198$	0	0
$199$	−2.86461e17	−1.64311	−0.821556	−	0.570127i	$-0.806894\pi$
$199$	−2.86461e17	−1.64311	−0.821556	+	0.570127i	$0.806894\pi$
$200$	−1.13379e17	−0.626338
$201$	9.65490e16	0.513780
$202$	−1.76611e17	−0.905485
$203$	−1.04156e17	−0.514593
$204$	7.65587e16	0.364559
$205$	−8.55633e16	−0.392766
$206$	−1.60081e17	−0.708492
$207$	−2.96746e16	−0.126651
$208$	−2.53928e17	−1.04529
$209$	0	0
$210$	−1.06362e17	−0.407514
$211$	−3.75834e17	−1.38956	−0.694780	−	0.719222i	$-0.744498\pi$
$211$	−3.75834e17	−1.38956	−0.694780	+	0.719222i	$0.744498\pi$
$212$	9.43988e16	0.336859
$213$	−4.18668e17	−1.44219
$214$	−5.43089e17	−1.80621
$215$	2.56591e16	0.0824050
$216$	2.38781e17	0.740621
$217$	−2.02055e17	−0.605372
$218$	2.73964e17	0.792995
$219$	−2.75110e17	−0.769441
$220$	0	0
$221$	−3.12961e17	−0.817614
$222$	−7.47504e17	−1.88784
$223$	−2.53078e16	−0.0617970	−0.0308985	−	0.999523i	$-0.509837\pi$
$223$	−2.53078e16	−0.0617970	−0.0308985	+	0.999523i	$0.509837\pi$
$224$	−4.37295e17	−1.03256
$225$	8.72932e16	0.199348
$226$	4.43699e17	0.980111
$227$	−3.03692e17	−0.648992	−0.324496	−	0.945887i	$-0.605195\pi$
$227$	−3.03692e17	−0.648992	−0.324496	+	0.945887i	$0.605195\pi$
$228$	7.26868e16	0.150295
$229$	1.07992e17	0.216085	0.108042	−	0.994146i	$-0.465542\pi$
$229$	1.07992e17	0.216085	0.108042	+	0.994146i	$0.465542\pi$
$230$	−1.06379e17	−0.206015
$231$	0	0
$232$	1.50492e17	0.273119
$233$	7.90506e17	1.38911	0.694554	−	0.719441i	$-0.255602\pi$
$233$	7.90506e17	1.38911	0.694554	+	0.719441i	$0.255602\pi$
$234$	1.28907e17	0.219360
$235$	−1.77731e17	−0.292923
$236$	1.36920e17	0.218588
$237$	−8.50830e16	−0.131592
$238$	−1.00381e18	−1.50426
$239$	−3.52956e17	−0.512551	−0.256275	−	0.966604i	$-0.582495\pi$
$239$	−3.52956e17	−0.512551	−0.256275	+	0.966604i	$0.582495\pi$
$240$	2.33075e17	0.328028
$241$	−6.85690e16	−0.0935405	−0.0467703	−	0.998906i	$-0.514893\pi$
$241$	−6.85690e16	−0.0935405	−0.0467703	+	0.998906i	$0.514893\pi$
$242$	0	0
$243$	−3.34679e17	−0.429123
$244$	−6.84934e16	−0.0851580
$245$	1.67726e17	0.202235
$246$	−1.18742e18	−1.38865
$247$	−2.97134e17	−0.337073
$248$	2.91944e17	0.321300
$249$	−9.43255e17	−1.00724
$250$	6.56433e17	0.680205
$251$	1.58806e18	1.59703	0.798515	−	0.601975i	$-0.205619\pi$
$251$	1.58806e18	1.59703	0.798515	+	0.601975i	$0.205619\pi$
$252$	1.23075e17	0.120134
$253$	0	0
$254$	6.45986e17	0.594251
$255$	2.87260e17	0.256581
$256$	1.23980e18	1.07535
$257$	−8.28562e17	−0.697954	−0.348977	−	0.937131i	$-0.613471\pi$
$257$	−8.28562e17	−0.697954	−0.348977	+	0.937131i	$0.613471\pi$
$258$	3.56090e17	0.291348
$259$	2.91744e18	2.31875
$260$	1.37557e17	0.106214
$261$	−1.15867e17	−0.0869271
$262$	−3.51265e17	−0.256080
$263$	−1.40445e18	−0.995038	−0.497519	−	0.867453i	$-0.665756\pi$
$263$	−1.40445e18	−0.995038	−0.497519	+	0.867453i	$0.665756\pi$
$264$	0	0
$265$	3.54200e17	0.237085
$266$	−9.53041e17	−0.620153
$267$	−2.39589e18	−1.51577
$268$	−4.00500e17	−0.246372
$269$	1.43582e18	0.858930	0.429465	−	0.903083i	$-0.358702\pi$
$269$	1.43582e18	0.858930	0.429465	+	0.903083i	$0.358702\pi$
$270$	−6.59050e17	−0.383433
$271$	−5.09160e17	−0.288127	−0.144064	−	0.989568i	$-0.546017\pi$
$271$	−5.09160e17	−0.288127	−0.144064	+	0.989568i	$0.546017\pi$
$272$	2.19967e18	1.21085
$273$	1.79611e18	0.961865
$274$	−2.28792e18	−1.19209
$275$	0	0
$276$	−4.39449e17	−0.216815
$277$	−5.68946e17	−0.273195	−0.136598	−	0.990627i	$-0.543617\pi$
$277$	−5.68946e17	−0.273195	−0.136598	+	0.990627i	$0.543617\pi$
$278$	−4.03292e18	−1.88488
$279$	−2.24774e17	−0.102262
$280$	−5.99797e17	−0.265655
$281$	4.06184e18	1.75156	0.875780	−	0.482710i	$-0.160348\pi$
$281$	4.06184e18	1.75156	0.875780	+	0.482710i	$0.160348\pi$
$282$	−2.46650e18	−1.03565
$283$	−2.78506e18	−1.13877	−0.569385	−	0.822071i	$-0.692819\pi$
$283$	−2.78506e18	−1.13877	−0.569385	+	0.822071i	$0.692819\pi$
$284$	1.73670e18	0.691570
$285$	2.72733e17	0.105779
$286$	0	0
$287$	4.63440e18	1.70561
$288$	−4.86463e17	−0.174424
$289$	−1.51369e17	−0.0528813
$290$	−4.15366e17	−0.141399
$291$	−2.05161e18	−0.680608
$292$	1.14120e18	0.368969
$293$	3.63803e18	1.14646	0.573230	−	0.819395i	$-0.305690\pi$
$293$	3.63803e18	1.14646	0.573230	+	0.819395i	$0.305690\pi$
$294$	2.32766e18	0.715013
$295$	5.13745e17	0.153845
$296$	−4.21532e18	−1.23067
$297$	0	0
$298$	−2.71209e18	−0.752803
$299$	1.79641e18	0.486262
$300$	1.29272e18	0.341267
$301$	−1.38979e18	−0.357849
$302$	6.21190e18	1.56017
$303$	−2.73746e18	−0.670701
$304$	2.08843e18	0.499192
$305$	−2.56998e17	−0.0599351
$306$	−1.11667e18	−0.254106
$307$	9.75296e17	0.216570	0.108285	−	0.994120i	$-0.465464\pi$
$307$	9.75296e17	0.216570	0.108285	+	0.994120i	$0.465464\pi$
$308$	0	0
$309$	−2.48125e18	−0.524786
$310$	−8.05783e17	−0.166343
$311$	3.36692e17	0.0678468	0.0339234	−	0.999424i	$-0.489200\pi$
$311$	3.36692e17	0.0678468	0.0339234	+	0.999424i	$0.489200\pi$
$312$	−2.59515e18	−0.510508
$313$	3.65551e18	0.702046	0.351023	−	0.936367i	$-0.385834\pi$
$313$	3.65551e18	0.702046	0.351023	+	0.936367i	$0.385834\pi$
$314$	3.13797e18	0.588405
$315$	4.61797e17	0.0845515
$316$	3.52937e17	0.0631021
$317$	−7.97380e17	−0.139226	−0.0696131	−	0.997574i	$-0.522176\pi$
$317$	−7.97380e17	−0.139226	−0.0696131	+	0.997574i	$0.522176\pi$
$318$	4.91548e18	0.838229
$319$	0	0
$320$	5.37283e17	0.0874135
$321$	−8.41788e18	−1.33787
$322$	5.76188e18	0.894633
$323$	2.57395e18	0.390464
$324$	−2.09681e18	−0.310793
$325$	−5.28444e18	−0.765375
$326$	−3.62321e18	−0.512816
$327$	4.24645e18	0.587378
$328$	−6.69610e18	−0.905249
$329$	9.62651e18	1.27204
$330$	0	0
$331$	−1.01585e19	−1.28269	−0.641343	−	0.767255i	$-0.721622\pi$
$331$	−1.01585e19	−1.28269	−0.641343	+	0.767255i	$0.721622\pi$
$332$	3.91276e18	0.483000
$333$	3.24546e18	0.391692
$334$	1.38672e19	1.63640
$335$	−1.50274e18	−0.173399
$336$	−1.26241e19	−1.42448
$337$	4.81465e18	0.531301	0.265651	−	0.964069i	$-0.414413\pi$
$337$	4.81465e18	0.531301	0.265651	+	0.964069i	$0.414413\pi$
$338$	3.25253e18	0.351032
$339$	6.87734e18	0.725977
$340$	−1.19160e18	−0.123038
$341$	0	0
$342$	−1.06020e18	−0.104759
$343$	4.31515e18	0.417148
$344$	2.00806e18	0.189928
$345$	−1.64888e18	−0.152597
$346$	−1.64130e19	−1.48634
$347$	−4.50275e18	−0.399031	−0.199516	−	0.979895i	$-0.563937\pi$
$347$	−4.50275e18	−0.399031	−0.199516	+	0.979895i	$0.563937\pi$
$348$	−1.71586e18	−0.148812
$349$	−2.24323e19	−1.90407	−0.952036	−	0.305986i	$-0.901014\pi$
$349$	−2.24323e19	−1.90407	−0.952036	+	0.305986i	$0.901014\pi$
$350$	−1.69496e19	−1.40815
$351$	1.11292e19	0.905026
$352$	0	0
$353$	8.02510e18	0.625374	0.312687	−	0.949856i	$-0.398771\pi$
$353$	8.02510e18	0.625374	0.312687	+	0.949856i	$0.398771\pi$
$354$	7.12961e18	0.543928
$355$	6.51636e18	0.486735
$356$	9.93851e18	0.726853
$357$	−1.55590e19	−1.11422
$358$	−2.01690e19	−1.41436
$359$	−1.61507e18	−0.110913	−0.0554567	−	0.998461i	$-0.517661\pi$
$359$	−1.61507e18	−0.110913	−0.0554567	+	0.998461i	$0.517661\pi$
$360$	−6.67236e17	−0.0448756
$361$	−1.27374e19	−0.839026
$362$	−1.60526e19	−1.03569
$363$	0	0
$364$	−7.45055e18	−0.461241
$365$	4.28195e18	0.259684
$366$	−3.56655e18	−0.211905
$367$	−9.97799e18	−0.580828	−0.290414	−	0.956901i	$-0.593793\pi$
$367$	−9.97799e18	−0.580828	−0.290414	+	0.956901i	$0.593793\pi$
$368$	−1.26262e19	−0.720135
$369$	5.15547e18	0.288118
$370$	1.16345e19	0.637142
$371$	−1.91847e19	−1.02956
$372$	−3.32865e18	−0.175064
$373$	2.36866e19	1.22092	0.610459	−	0.792048i	$-0.290985\pi$
$373$	2.36866e19	1.22092	0.610459	+	0.792048i	$0.290985\pi$
$374$	0	0
$375$	1.01747e19	0.503834
$376$	−1.39090e19	−0.675132
$377$	7.01419e18	0.333747
$378$	3.56964e19	1.66508
$379$	1.86851e19	0.854480	0.427240	−	0.904138i	$-0.359486\pi$
$379$	1.86851e19	0.854480	0.427240	+	0.904138i	$0.359486\pi$
$380$	−1.13133e18	−0.0507241
$381$	1.00128e19	0.440167
$382$	2.13024e19	0.918235
$383$	−3.02521e19	−1.27869	−0.639343	−	0.768921i	$-0.720794\pi$
$383$	−3.02521e19	−1.27869	−0.639343	+	0.768921i	$0.720794\pi$
$384$	2.44537e19	1.01358
$385$	0	0
$386$	−1.92494e18	−0.0767385
$387$	−1.54605e18	−0.0604493
$388$	8.51037e18	0.326370
$389$	−1.00714e18	−0.0378852	−0.0189426	−	0.999821i	$-0.506030\pi$
$389$	−1.00714e18	−0.0378852	−0.0189426	+	0.999821i	$0.506030\pi$
$390$	7.16278e18	0.264300
$391$	−1.55615e19	−0.563283
$392$	1.31261e19	0.466112
$393$	−5.44461e18	−0.189680
$394$	7.65020e18	0.261487
$395$	1.32428e18	0.0444120
$396$	0	0
$397$	3.56324e19	1.15058	0.575290	−	0.817950i	$-0.304889\pi$
$397$	3.56324e19	1.15058	0.575290	+	0.817950i	$0.304889\pi$
$398$	6.18756e19	1.96063
$399$	−1.47721e19	−0.459353
$400$	3.71422e19	1.13349
$401$	3.94327e19	1.18106	0.590532	−	0.807014i	$-0.298918\pi$
$401$	3.94327e19	1.18106	0.590532	+	0.807014i	$0.298918\pi$
$402$	−2.08546e19	−0.613065
$403$	1.36071e19	0.392624
$404$	1.13554e19	0.321620
$405$	−7.86757e18	−0.218740
$406$	2.24977e19	0.614034
$407$	0	0
$408$	2.24807e19	0.591370
$409$	5.27823e19	1.36321	0.681607	−	0.731719i	$-0.261282\pi$
$409$	5.27823e19	1.36321	0.681607	+	0.731719i	$0.261282\pi$
$410$	1.84817e19	0.468665
$411$	−3.54627e19	−0.882995
$412$	1.02926e19	0.251649
$413$	−2.78262e19	−0.668080
$414$	6.40972e18	0.151125
$415$	1.46813e19	0.339941
$416$	2.94488e19	0.669681
$417$	−6.25102e19	−1.39615
$418$	0	0
$419$	8.62630e18	0.185874	0.0929372	−	0.995672i	$-0.470374\pi$
$419$	8.62630e18	0.185874	0.0929372	+	0.995672i	$0.470374\pi$
$420$	6.83870e18	0.144745
$421$	−4.29249e19	−0.892469	−0.446235	−	0.894916i	$-0.647235\pi$
$421$	−4.29249e19	−0.892469	−0.446235	+	0.894916i	$0.647235\pi$
$422$	8.11801e19	1.65808
$423$	1.07089e19	0.214878
$424$	2.77193e19	0.546435
$425$	4.57770e19	0.886605
$426$	9.04322e19	1.72088
$427$	1.39199e19	0.260272
$428$	3.49186e19	0.641547
$429$	0	0
$430$	−5.54237e18	−0.0983291
$431$	−5.04764e19	−0.880053	−0.440026	−	0.897985i	$-0.645031\pi$
$431$	−5.04764e19	−0.880053	−0.440026	+	0.897985i	$0.645031\pi$
$432$	−7.82227e19	−1.34031
$433$	5.05734e19	0.851653	0.425827	−	0.904805i	$-0.359983\pi$
$433$	5.05734e19	0.851653	0.425827	+	0.904805i	$0.359983\pi$
$434$	4.36440e19	0.722356
$435$	−6.43818e18	−0.104735
$436$	−1.76149e19	−0.281664
$437$	−1.47745e19	−0.232222
$438$	5.94238e19	0.918130
$439$	−2.47946e19	−0.376594	−0.188297	−	0.982112i	$-0.560297\pi$
$439$	−2.47946e19	−0.376594	−0.188297	+	0.982112i	$0.560297\pi$
$440$	0	0
$441$	−1.01061e19	−0.148352
$442$	6.75996e19	0.975612
$443$	−1.30654e20	−1.85394	−0.926970	−	0.375135i	$-0.877596\pi$
$443$	−1.30654e20	−1.85394	−0.926970	+	0.375135i	$0.877596\pi$
$444$	4.80617e19	0.670544
$445$	3.72909e19	0.511567
$446$	5.46648e18	0.0737389
$447$	−4.20373e19	−0.557608
$448$	−2.91011e19	−0.379598
$449$	−7.78280e19	−0.998363	−0.499181	−	0.866498i	$-0.666366\pi$
$449$	−7.78280e19	−0.998363	−0.499181	+	0.866498i	$0.666366\pi$
$450$	−1.88553e19	−0.237870
$451$	0	0
$452$	−2.85282e19	−0.348126
$453$	9.62844e19	1.15563
$454$	6.55975e19	0.774405
$455$	−2.79556e19	−0.324627
$456$	2.13438e19	0.243801
$457$	1.18451e20	1.33096	0.665482	−	0.746414i	$-0.268226\pi$
$457$	1.18451e20	1.33096	0.665482	+	0.746414i	$0.268226\pi$
$458$	−2.33262e19	−0.257841
$459$	−9.64078e19	−1.04838
$460$	6.83981e18	0.0731746
$461$	−1.38643e20	−1.45929	−0.729644	−	0.683827i	$-0.760314\pi$
$461$	−1.38643e20	−1.45929	−0.729644	+	0.683827i	$0.760314\pi$
$462$	0	0
$463$	1.75645e20	1.78969	0.894846	−	0.446375i	$-0.147285\pi$
$463$	1.75645e20	1.78969	0.894846	+	0.446375i	$0.147285\pi$
$464$	−4.92999e19	−0.494266
$465$	−1.24896e19	−0.123212
$466$	−1.70749e20	−1.65754
$467$	−1.36631e20	−1.30519	−0.652593	−	0.757708i	$-0.726319\pi$
$467$	−1.36631e20	−1.30519	−0.652593	+	0.757708i	$0.726319\pi$
$468$	−8.28826e18	−0.0779147
$469$	8.13935e19	0.752997
$470$	3.83899e19	0.349528
$471$	4.86385e19	0.435837
$472$	4.02052e19	0.354583
$473$	0	0
$474$	1.83779e19	0.157021
$475$	4.34619e19	0.365516
$476$	6.45410e19	0.534298
$477$	−2.13417e19	−0.173917
$478$	7.62386e19	0.611597
$479$	−6.41058e19	−0.506269	−0.253134	−	0.967431i	$-0.581461\pi$
$479$	−6.41058e19	−0.506269	−0.253134	+	0.967431i	$0.581461\pi$
$480$	−2.70305e19	−0.210157
$481$	−1.96470e20	−1.50386
$482$	1.48109e19	0.111617
$483$	8.93091e19	0.662662
$484$	0	0
$485$	3.19323e19	0.229703
$486$	7.22907e19	0.512047
$487$	−2.41343e19	−0.168332	−0.0841662	−	0.996452i	$-0.526823\pi$
$487$	−2.41343e19	−0.168332	−0.0841662	+	0.996452i	$0.526823\pi$
$488$	−2.01124e19	−0.138139
$489$	−5.61598e19	−0.379847
$490$	−3.62289e19	−0.241315
$491$	2.80908e19	0.184269	0.0921346	−	0.995747i	$-0.470631\pi$
$491$	2.80908e19	0.184269	0.0921346	+	0.995747i	$0.470631\pi$
$492$	7.63469e19	0.493234
$493$	−6.07611e19	−0.386611
$494$	6.41808e19	0.402210
$495$	0	0
$496$	−9.56384e19	−0.581460
$497$	−3.52948e20	−2.11368
$498$	2.03743e20	1.20188
$499$	−1.71994e20	−0.999443	−0.499722	−	0.866186i	$-0.666564\pi$
$499$	−1.71994e20	−0.999443	−0.499722	+	0.866186i	$0.666564\pi$
$500$	−4.22062e19	−0.241602
$501$	2.14941e20	1.21210
$502$	−3.43020e20	−1.90564
$503$	1.83497e20	1.00431	0.502155	−	0.864778i	$-0.332541\pi$
$503$	1.83497e20	1.00431	0.502155	+	0.864778i	$0.332541\pi$
$504$	3.61398e19	0.194875
$505$	4.26073e19	0.226359
$506$	0	0
$507$	5.04142e19	0.260012
$508$	−4.15345e19	−0.211072
$509$	2.67204e20	1.33801	0.669004	−	0.743258i	$-0.266721\pi$
$509$	2.67204e20	1.33801	0.669004	+	0.743258i	$0.266721\pi$
$510$	−6.20482e19	−0.306163
$511$	−2.31925e20	−1.12769
$512$	−2.84604e19	−0.136369
$513$	−9.15321e19	−0.432209
$514$	1.78969e20	0.832828
$515$	3.86195e19	0.177114
$516$	−2.28953e19	−0.103484
$517$	0	0
$518$	−6.30167e20	−2.76683
$519$	−2.54401e20	−1.10094
$520$	4.03922e19	0.172295
$521$	−2.01468e20	−0.847076	−0.423538	−	0.905878i	$-0.639212\pi$
$521$	−2.01468e20	−0.847076	−0.423538	+	0.905878i	$0.639212\pi$
$522$	2.50272e19	0.103725
$523$	−3.58989e20	−1.46662	−0.733311	−	0.679894i	$-0.762026\pi$
$523$	−3.58989e20	−1.46662	−0.733311	+	0.679894i	$0.762026\pi$
$524$	2.25850e19	0.0909570
$525$	−2.62718e20	−1.04303
$526$	3.03362e20	1.18732
$527$	−1.17872e20	−0.454813
$528$	0	0
$529$	−1.77312e20	−0.664997
$530$	−7.65071e19	−0.282900
$531$	−3.09549e19	−0.112855
$532$	6.12770e19	0.220272
$533$	−3.12095e20	−1.10620
$534$	5.17512e20	1.80868
$535$	1.31020e20	0.451528
$536$	−1.17603e20	−0.399652
$537$	−3.12619e20	−1.04763
$538$	−3.10137e20	−1.02491
$539$	0	0
$540$	4.23744e19	0.136192
$541$	−2.02328e20	−0.641323	−0.320662	−	0.947194i	$-0.603905\pi$
$541$	−2.02328e20	−0.641323	−0.320662	+	0.947194i	$0.603905\pi$
$542$	1.09979e20	0.343806
$543$	−2.48816e20	−0.767147
$544$	−2.55103e20	−0.775755
$545$	−6.60939e19	−0.198238
$546$	−3.87961e20	−1.14774
$547$	−7.40963e19	−0.216218	−0.108109	−	0.994139i	$-0.534480\pi$
$547$	−7.40963e19	−0.216218	−0.108109	+	0.994139i	$0.534480\pi$
$548$	1.47104e20	0.423420
$549$	1.54850e19	0.0439662
$550$	0	0
$551$	−5.76882e19	−0.159386
$552$	−1.29040e20	−0.351707
$553$	−7.17273e19	−0.192862
$554$	1.22892e20	0.325988
$555$	1.80335e20	0.471936
$556$	2.59302e20	0.669490
$557$	−2.09626e18	−0.00533987	−0.00266994	−	0.999996i	$-0.500850\pi$
$557$	−2.09626e18	−0.00533987	−0.00266994	+	0.999996i	$0.500850\pi$
$558$	4.85511e19	0.122023
$559$	9.35927e19	0.232088
$560$	1.96489e20	0.480759
$561$	0	0
$562$	−8.77357e20	−2.09004
$563$	−6.87353e20	−1.61572	−0.807861	−	0.589373i	$-0.799375\pi$
$563$	−6.87353e20	−1.61572	−0.807861	+	0.589373i	$0.799375\pi$
$564$	1.58587e20	0.367852
$565$	−1.07042e20	−0.245015
$566$	6.01573e20	1.35883
$567$	4.26134e20	0.949891
$568$	5.09964e20	1.12183
$569$	9.05218e19	0.196522	0.0982610	−	0.995161i	$-0.468672\pi$
$569$	9.05218e19	0.196522	0.0982610	+	0.995161i	$0.468672\pi$
$570$	−5.89102e19	−0.126220
$571$	−2.05774e20	−0.435130	−0.217565	−	0.976046i	$-0.569811\pi$
$571$	−2.05774e20	−0.435130	−0.217565	+	0.976046i	$0.569811\pi$
$572$	0	0
$573$	3.30188e20	0.680145
$574$	−1.00103e21	−2.03520
$575$	−2.62761e20	−0.527293
$576$	−3.23731e19	−0.0641233
$577$	5.70778e20	1.11596	0.557980	−	0.829854i	$-0.311576\pi$
$577$	5.70778e20	1.11596	0.557980	+	0.829854i	$0.311576\pi$
$578$	3.26956e19	0.0631002
$579$	−2.98366e19	−0.0568408
$580$	2.67065e19	0.0502236
$581$	−7.95190e20	−1.47622
$582$	4.43147e20	0.812130
$583$	0	0
$584$	3.35102e20	0.598522
$585$	−3.10989e19	−0.0548373
$586$	−7.85814e20	−1.36800
$587$	−9.30363e20	−1.59907	−0.799534	−	0.600621i	$-0.794920\pi$
$587$	−9.30363e20	−1.59907	−0.799534	+	0.600621i	$0.794920\pi$
$588$	−1.49660e20	−0.253966
$589$	−1.11911e20	−0.187503
$590$	−1.10969e20	−0.183574
$591$	1.18578e20	0.193686
$592$	1.38090e21	2.22716
$593$	−3.54225e20	−0.564116	−0.282058	−	0.959397i	$-0.591017\pi$
$593$	−3.54225e20	−0.564116	−0.282058	+	0.959397i	$0.591017\pi$
$594$	0	0
$595$	2.42168e20	0.376045
$596$	1.74377e20	0.267388
$597$	9.59071e20	1.45226
$598$	−3.88024e20	−0.580229
$599$	−3.30045e20	−0.487385	−0.243693	−	0.969853i	$-0.578359\pi$
$599$	−3.30045e20	−0.487385	−0.243693	+	0.969853i	$0.578359\pi$
$600$	3.79594e20	0.553586
$601$	3.35884e20	0.483761	0.241880	−	0.970306i	$-0.422236\pi$
$601$	3.35884e20	0.483761	0.241880	+	0.970306i	$0.422236\pi$
$602$	3.00194e20	0.427000
$603$	9.05451e19	0.127199
$604$	−3.99402e20	−0.554158
$605$	0	0
$606$	5.91292e20	0.800309
$607$	1.33438e21	1.78387	0.891934	−	0.452165i	$-0.149348\pi$
$607$	1.33438e21	1.78387	0.891934	+	0.452165i	$0.149348\pi$
$608$	−2.42202e20	−0.319816
$609$	3.48714e20	0.454820
$610$	5.55116e19	0.0715172
$611$	−6.48280e20	−0.825000
$612$	7.17978e19	0.0902559
$613$	−5.68844e18	−0.00706381	−0.00353191	−	0.999994i	$-0.501124\pi$
$613$	−5.68844e18	−0.00706381	−0.00353191	+	0.999994i	$0.501124\pi$
$614$	−2.10664e20	−0.258421
$615$	2.86466e20	0.347144
$616$	0	0
$617$	3.98915e20	0.471783	0.235891	−	0.971779i	$-0.424199\pi$
$617$	3.98915e20	0.471783	0.235891	+	0.971779i	$0.424199\pi$
$618$	5.35950e20	0.626197
$619$	−5.40017e20	−0.623343	−0.311672	−	0.950190i	$-0.600889\pi$
$619$	−5.40017e20	−0.623343	−0.311672	+	0.950190i	$0.600889\pi$
$620$	5.18088e19	0.0590835
$621$	5.53384e20	0.623504
$622$	−7.27254e19	−0.0809577
$623$	−2.01980e21	−2.22151
$624$	8.50151e20	0.923871
$625$	6.90089e20	0.740978
$626$	−7.89591e20	−0.837711
$627$	0	0
$628$	−2.01760e20	−0.208996
$629$	1.70194e21	1.74206
$630$	−9.97480e19	−0.100891
$631$	9.59111e20	0.958625	0.479312	−	0.877644i	$-0.340886\pi$
$631$	9.59111e20	0.958625	0.479312	+	0.877644i	$0.340886\pi$
$632$	1.03637e20	0.102361
$633$	1.25829e21	1.22816
$634$	1.72234e20	0.166131
$635$	−1.55844e20	−0.148555
$636$	−3.16047e20	−0.297731
$637$	6.11788e20	0.569581
$638$	0	0
$639$	−3.92633e20	−0.357051
$640$	−3.80609e20	−0.342082
$641$	−9.25925e20	−0.822509	−0.411255	−	0.911521i	$-0.634909\pi$
$641$	−9.25925e20	−0.822509	−0.411255	+	0.911521i	$0.634909\pi$
$642$	1.81826e21	1.59641
$643$	−7.65928e20	−0.664669	−0.332335	−	0.943162i	$-0.607836\pi$
$643$	−7.65928e20	−0.664669	−0.332335	+	0.943162i	$0.607836\pi$
$644$	−3.70467e20	−0.317765
$645$	−8.59068e19	−0.0728332
$646$	−5.55972e20	−0.465918
$647$	1.36075e21	1.12719	0.563596	−	0.826051i	$-0.309418\pi$
$647$	1.36075e21	1.12719	0.563596	+	0.826051i	$0.309418\pi$
$648$	−6.15709e20	−0.504153
$649$	0	0
$650$	1.14144e21	0.913278
$651$	6.76481e20	0.535055
$652$	2.32959e20	0.182147
$653$	−2.71809e20	−0.210094	−0.105047	−	0.994467i	$-0.533499\pi$
$653$	−2.71809e20	−0.210094	−0.105047	+	0.994467i	$0.533499\pi$
$654$	−9.17233e20	−0.700885
$655$	8.47427e19	0.0640166
$656$	2.19359e21	1.63824
$657$	−2.58002e20	−0.190495
$658$	−2.07933e21	−1.51785
$659$	−6.74316e20	−0.486657	−0.243329	−	0.969944i	$-0.578239\pi$
$659$	−6.74316e20	−0.486657	−0.243329	+	0.969944i	$0.578239\pi$
$660$	0	0
$661$	1.26727e21	0.894042	0.447021	−	0.894524i	$-0.352485\pi$
$661$	1.26727e21	0.894042	0.447021	+	0.894524i	$0.352485\pi$
$662$	2.19424e21	1.53056
$663$	1.04779e21	0.722644
$664$	1.14894e21	0.783499
$665$	2.29921e20	0.155030
$666$	−7.01020e20	−0.467384
$667$	3.48770e20	0.229930
$668$	−8.91609e20	−0.581234
$669$	8.47305e19	0.0546190
$670$	3.24592e20	0.206908
$671$	0	0
$672$	1.46406e21	0.912620
$673$	1.13945e21	0.702394	0.351197	−	0.936302i	$-0.385775\pi$
$673$	1.13945e21	0.702394	0.351197	+	0.936302i	$0.385775\pi$
$674$	−1.03997e21	−0.633971
$675$	−1.62788e21	−0.981393
$676$	−2.09126e20	−0.124683
$677$	−1.74431e21	−1.02851	−0.514256	−	0.857637i	$-0.671932\pi$
$677$	−1.74431e21	−1.02851	−0.514256	+	0.857637i	$0.671932\pi$
$678$	−1.48550e21	−0.866266
$679$	−1.72956e21	−0.997500
$680$	−3.49902e20	−0.199586
$681$	1.01676e21	0.573608
$682$	0	0
$683$	−1.43739e21	−0.793267	−0.396634	−	0.917977i	$-0.629822\pi$
$683$	−1.43739e21	−0.793267	−0.396634	+	0.917977i	$0.629822\pi$
$684$	6.81667e19	0.0372093
$685$	5.51960e20	0.298008
$686$	−9.32073e20	−0.497759
$687$	−3.61556e20	−0.190985
$688$	−6.57825e20	−0.343714
$689$	1.29196e21	0.667735
$690$	3.56159e20	0.182085
$691$	−1.77548e21	−0.897903	−0.448951	−	0.893556i	$-0.648202\pi$
$691$	−1.77548e21	−0.897903	−0.448951	+	0.893556i	$0.648202\pi$
$692$	1.05529e21	0.527932
$693$	0	0
$694$	9.72594e20	0.476141
$695$	9.72941e20	0.471195
$696$	−5.03846e20	−0.241395
$697$	2.70356e21	1.28141
$698$	4.84538e21	2.27202
$699$	−2.64661e21	−1.22776
$700$	1.08980e21	0.500161
$701$	−1.43100e21	−0.649764	−0.324882	−	0.945755i	$-0.605325\pi$
$701$	−1.43100e21	−0.649764	−0.324882	+	0.945755i	$0.605325\pi$
$702$	−2.40391e21	−1.07992
$703$	1.61586e21	0.718191
$704$	0	0
$705$	5.95043e20	0.258899
$706$	−1.73342e21	−0.746223
$707$	−2.30776e21	−0.982980
$708$	−4.58408e20	−0.193198
$709$	−2.41840e21	−1.00851	−0.504257	−	0.863554i	$-0.668234\pi$
$709$	−2.41840e21	−1.00851	−0.504257	+	0.863554i	$0.668234\pi$
$710$	−1.40753e21	−0.580793
$711$	−7.97921e19	−0.0325790
$712$	2.91835e21	1.17906
$713$	6.76591e20	0.270492
$714$	3.36074e21	1.32953
$715$	0	0
$716$	1.29679e21	0.502368
$717$	1.18170e21	0.453015
$718$	3.48856e20	0.132347
$719$	4.74444e21	1.78122	0.890611	−	0.454766i	$-0.150277\pi$
$719$	4.74444e21	1.78122	0.890611	+	0.454766i	$0.150277\pi$
$720$	2.18581e20	0.0812118
$721$	−2.09176e21	−0.769127
$722$	2.75127e21	1.00116
$723$	2.29569e20	0.0826753
$724$	1.03212e21	0.367868
$725$	−1.02597e21	−0.361909
$726$	0	0
$727$	−3.59265e21	−1.24138	−0.620692	−	0.784054i	$-0.713148\pi$
$727$	−3.59265e21	−1.24138	−0.620692	+	0.784054i	$0.713148\pi$
$728$	−2.18778e21	−0.748202
$729$	3.28690e21	1.11258
$730$	−9.24902e20	−0.309866
$731$	−8.10755e20	−0.268850
$732$	2.29316e20	0.0752664
$733$	2.76824e21	0.899339	0.449669	−	0.893195i	$-0.351542\pi$
$733$	2.76824e21	0.899339	0.449669	+	0.893195i	$0.351542\pi$
$734$	2.15525e21	0.693069
$735$	−5.61548e20	−0.178744
$736$	1.46430e21	0.461367
$737$	0	0
$738$	−1.11358e21	−0.343795
$739$	−3.55824e21	−1.08743	−0.543716	−	0.839269i	$-0.682983\pi$
$739$	−3.55824e21	−1.08743	−0.543716	+	0.839269i	$0.682983\pi$
$740$	−7.48058e20	−0.226307
$741$	9.94803e20	0.297921
$742$	4.14389e21	1.22851
$743$	1.94092e21	0.569628	0.284814	−	0.958583i	$-0.408068\pi$
$743$	1.94092e21	0.569628	0.284814	+	0.958583i	$0.408068\pi$
$744$	−9.77427e20	−0.283980
$745$	6.54291e20	0.188191
$746$	−5.11630e21	−1.45685
$747$	−8.84598e20	−0.249368
$748$	0	0
$749$	−7.09651e21	−1.96079
$750$	−2.19774e21	−0.601196
$751$	4.75565e21	1.28798	0.643992	−	0.765032i	$-0.277277\pi$
$751$	4.75565e21	1.28798	0.643992	+	0.765032i	$0.277277\pi$
$752$	4.55650e21	1.22179
$753$	−5.31681e21	−1.41153
$754$	−1.51507e21	−0.398241
$755$	−1.49862e21	−0.390022
$756$	−2.29514e21	−0.591421
$757$	3.62137e21	0.923960	0.461980	−	0.886890i	$-0.347139\pi$
$757$	3.62137e21	0.923960	0.461980	+	0.886890i	$0.347139\pi$
$758$	−4.03599e21	−1.01960
$759$	0	0
$760$	−3.32206e20	−0.0822821
$761$	3.86361e21	0.947564	0.473782	−	0.880642i	$-0.342888\pi$
$761$	3.86361e21	0.947564	0.473782	+	0.880642i	$0.342888\pi$
$762$	−2.16276e21	−0.525226
$763$	3.57987e21	0.860862
$764$	−1.36967e21	−0.326148
$765$	2.69397e20	0.0635231
$766$	6.53445e21	1.52578
$767$	1.87391e21	0.433294
$768$	−4.15085e21	−0.950446
$769$	5.39327e21	1.22294	0.611469	−	0.791268i	$-0.290579\pi$
$769$	5.39327e21	1.22294	0.611469	+	0.791268i	$0.290579\pi$
$770$	0	0
$771$	2.77403e21	0.616883
$772$	1.23767e20	0.0272568
$773$	6.57037e21	1.43299	0.716496	−	0.697591i	$-0.245745\pi$
$773$	6.57037e21	1.43299	0.716496	+	0.697591i	$0.245745\pi$
$774$	3.33947e20	0.0721306
$775$	−1.99031e21	−0.425754
$776$	2.49899e21	0.529422
$777$	−9.76758e21	−2.04941
$778$	2.17543e20	0.0452062
$779$	2.56683e21	0.528282
$780$	−4.60540e20	−0.0938767
$781$	0	0
$782$	3.36129e21	0.672133
$783$	2.16073e21	0.427944
$784$	−4.30001e21	−0.843527
$785$	−7.57035e20	−0.147094
$786$	1.17604e21	0.226335
$787$	3.72074e20	0.0709281	0.0354641	−	0.999371i	$-0.488709\pi$
$787$	3.72074e20	0.0709281	0.0354641	+	0.999371i	$0.488709\pi$
$788$	−4.91879e20	−0.0928778
$789$	4.70211e21	0.879459
$790$	−2.86044e20	−0.0529943
$791$	5.79778e21	1.06399
$792$	0	0
$793$	−9.37412e20	−0.168804
$794$	−7.69660e21	−1.37292
$795$	−1.18586e21	−0.209546
$796$	−3.97837e21	−0.696397
$797$	2.61511e21	0.453474	0.226737	−	0.973956i	$-0.427194\pi$
$797$	2.61511e21	0.453474	0.226737	+	0.973956i	$0.427194\pi$
$798$	3.19078e21	0.548119
$799$	5.61579e21	0.955674
$800$	−4.30750e21	−0.726190
$801$	−2.24690e21	−0.375267
$802$	−8.51746e21	−1.40930
$803$	0	0
$804$	1.34087e21	0.217755
$805$	−1.39005e21	−0.223647
$806$	−2.93913e21	−0.468495
$807$	−4.80713e21	−0.759161
$808$	3.33441e21	0.521715
$809$	−5.34899e21	−0.829198	−0.414599	−	0.910004i	$-0.636078\pi$
$809$	−5.34899e21	−0.829198	−0.414599	+	0.910004i	$0.636078\pi$
$810$	1.69939e21	0.261010
$811$	−8.46492e21	−1.28815	−0.644075	−	0.764962i	$-0.722758\pi$
$811$	−8.46492e21	−1.28815	−0.644075	+	0.764962i	$0.722758\pi$
$812$	−1.44652e21	−0.218099
$813$	1.70467e21	0.254660
$814$	0	0
$815$	8.74100e20	0.128197
$816$	−7.36451e21	−1.07021
$817$	−7.69753e20	−0.110837
$818$	−1.14010e22	−1.62664
$819$	1.68442e21	0.238134
$820$	−1.18830e21	−0.166465
$821$	−7.99397e21	−1.10966	−0.554829	−	0.831965i	$-0.687216\pi$
$821$	−7.99397e21	−1.10966	−0.554829	+	0.831965i	$0.687216\pi$
$822$	7.65994e21	1.05363
$823$	1.96841e21	0.268297	0.134148	−	0.990961i	$-0.457170\pi$
$823$	1.96841e21	0.268297	0.134148	+	0.990961i	$0.457170\pi$
$824$	3.02232e21	0.408213
$825$	0	0
$826$	6.01046e21	0.797181
$827$	1.43539e22	1.88659	0.943296	−	0.331954i	$-0.107708\pi$
$827$	1.43539e22	1.88659	0.943296	+	0.331954i	$0.107708\pi$
$828$	−4.12121e20	−0.0536782
$829$	−8.83327e21	−1.14015	−0.570076	−	0.821592i	$-0.693086\pi$
$829$	−8.83327e21	−1.14015	−0.570076	+	0.821592i	$0.693086\pi$
$830$	−3.17116e21	−0.405632
$831$	1.90483e21	0.241462
$832$	1.95976e21	0.246195
$833$	−5.29967e21	−0.659799
$834$	1.35022e22	1.66594
$835$	−3.34546e21	−0.409079
$836$	0	0
$837$	4.19166e21	0.503437
$838$	−1.86328e21	−0.221793
$839$	1.26696e22	1.49469	0.747343	−	0.664439i	$-0.231329\pi$
$839$	1.26696e22	1.49469	0.747343	+	0.664439i	$0.231329\pi$
$840$	2.00812e21	0.234798
$841$	−7.26739e21	−0.842187
$842$	9.27177e21	1.06493
$843$	−1.35990e22	−1.54811
$844$	−5.21958e21	−0.588935
$845$	−7.84673e20	−0.0877533
$846$	−2.31312e21	−0.256401
$847$	0	0
$848$	−9.08064e21	−0.988889
$849$	9.32438e21	1.00650
$850$	−9.88783e21	−1.05793
$851$	−9.76917e21	−1.03606
$852$	−5.81446e21	−0.611241
$853$	−6.00532e21	−0.625776	−0.312888	−	0.949790i	$-0.601296\pi$
$853$	−6.00532e21	−0.625776	−0.312888	+	0.949790i	$0.601296\pi$
$854$	−3.00670e21	−0.310568
$855$	2.55773e20	0.0261883
$856$	1.02535e22	1.04069
$857$	−1.47589e22	−1.48491	−0.742453	−	0.669898i	$-0.766338\pi$
$857$	−1.47589e22	−1.48491	−0.742453	+	0.669898i	$0.766338\pi$
$858$	0	0
$859$	9.64956e20	0.0954023	0.0477012	−	0.998862i	$-0.484810\pi$
$859$	9.64956e20	0.0954023	0.0477012	+	0.998862i	$0.484810\pi$
$860$	3.56354e20	0.0349255
$861$	−1.55160e22	−1.50749
$862$	1.09029e22	1.05012
$863$	3.44391e21	0.328829	0.164415	−	0.986391i	$-0.447426\pi$
$863$	3.44391e21	0.328829	0.164415	+	0.986391i	$0.447426\pi$
$864$	9.07173e21	0.858691
$865$	3.95963e21	0.371564
$866$	−1.09239e22	−1.01623
$867$	5.06782e20	0.0467389
$868$	−2.80614e21	−0.256574
$869$	0	0
$870$	1.39065e21	0.124975
$871$	−5.48130e21	−0.488368
$872$	−5.17245e21	−0.456901
$873$	−1.92403e21	−0.168502
$874$	3.19130e21	0.277097
$875$	8.57756e21	0.738420
$876$	−3.82073e21	−0.326111
$877$	1.09850e22	0.929617	0.464808	−	0.885411i	$-0.346123\pi$
$877$	1.09850e22	0.929617	0.464808	+	0.885411i	$0.346123\pi$
$878$	5.35564e21	0.449368
$879$	−1.21801e22	−1.01329
$880$	0	0
$881$	−7.98462e21	−0.653033	−0.326516	−	0.945192i	$-0.605875\pi$
$881$	−7.98462e21	−0.653033	−0.326516	+	0.945192i	$0.605875\pi$
$882$	2.18291e21	0.177020
$883$	5.45236e21	0.438409	0.219204	−	0.975679i	$-0.429654\pi$
$883$	5.45236e21	0.438409	0.219204	+	0.975679i	$0.429654\pi$
$884$	−4.34640e21	−0.346528
$885$	−1.72002e21	−0.135975
$886$	2.82213e22	2.21220
$887$	−1.67127e22	−1.29903	−0.649517	−	0.760347i	$-0.725029\pi$
$887$	−1.67127e22	−1.29903	−0.649517	+	0.760347i	$0.725029\pi$
$888$	1.41129e22	1.08772
$889$	8.44105e21	0.645109
$890$	−8.05483e21	−0.610424
$891$	0	0
$892$	−3.51475e20	−0.0261913
$893$	5.33178e21	0.393991
$894$	9.08007e21	0.665361
$895$	4.86576e21	0.353572
$896$	2.06151e22	1.48551
$897$	−6.01436e21	−0.429780
$898$	1.68108e22	1.19129
$899$	2.64180e21	0.185653
$900$	1.21233e21	0.0844892
$901$	−1.11917e22	−0.773500
$902$	0	0
$903$	4.65300e21	0.316283
$904$	−8.37704e21	−0.564712
$905$	3.87269e21	0.258910
$906$	−2.07974e22	−1.37895
$907$	−1.53384e22	−1.00862	−0.504309	−	0.863523i	$-0.668253\pi$
$907$	−1.53384e22	−1.00862	−0.504309	+	0.863523i	$0.668253\pi$
$908$	−4.21767e21	−0.275061
$909$	−2.56723e21	−0.166049
$910$	6.03842e21	0.387358
$911$	−1.49134e22	−0.948829	−0.474415	−	0.880302i	$-0.657340\pi$
$911$	−1.49134e22	−0.948829	−0.474415	+	0.880302i	$0.657340\pi$
$912$	−6.99206e21	−0.441209
$913$	0	0
$914$	−2.55854e22	−1.58816
$915$	8.60430e20	0.0529734
$916$	1.49979e21	0.0915827
$917$	−4.58995e21	−0.277996
$918$	2.08241e22	1.25097
$919$	5.86667e21	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$919$	5.86667e21	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$920$	2.00844e21	0.118700
$921$	−3.26529e21	−0.191414
$922$	2.99469e22	1.74129
$923$	2.37687e22	1.37086
$924$	0	0
$925$	2.87377e22	1.63076
$926$	−3.79393e22	−2.13554
$927$	−2.32695e21	−0.129924
$928$	5.71747e21	0.316660
$929$	−1.67946e22	−0.922684	−0.461342	−	0.887222i	$-0.652632\pi$
$929$	−1.67946e22	−0.922684	−0.461342	+	0.887222i	$0.652632\pi$
$930$	2.69776e21	0.147022
$931$	−5.03165e21	−0.272012
$932$	1.09786e22	0.588743
$933$	−1.12724e21	−0.0599661
$934$	2.95123e22	1.55740
$935$	0	0
$936$	−2.43377e21	−0.126389
$937$	−5.04466e21	−0.259887	−0.129944	−	0.991521i	$-0.541480\pi$
$937$	−5.04466e21	−0.259887	−0.129944	+	0.991521i	$0.541480\pi$
$938$	−1.75810e22	−0.898508
$939$	−1.22387e22	−0.620500
$940$	−2.46833e21	−0.124149
$941$	−1.65425e22	−0.825430	−0.412715	−	0.910860i	$-0.635419\pi$
$941$	−1.65425e22	−0.825430	−0.412715	+	0.910860i	$0.635419\pi$
$942$	−1.05059e22	−0.520059
$943$	−1.55185e22	−0.762100
$944$	−1.31709e22	−0.641691
$945$	−8.61175e21	−0.416249
$946$	0	0
$947$	−9.81583e21	−0.466984	−0.233492	−	0.972359i	$-0.575015\pi$
$947$	−9.81583e21	−0.466984	−0.233492	+	0.972359i	$0.575015\pi$
$948$	−1.18163e21	−0.0557725
$949$	1.56186e22	0.731384
$950$	−9.38776e21	−0.436149
$951$	2.66963e21	0.123054
$952$	1.89519e22	0.866712
$953$	5.97914e21	0.271295	0.135648	−	0.990757i	$-0.456689\pi$
$953$	5.97914e21	0.271295	0.135648	+	0.990757i	$0.456689\pi$
$954$	4.60981e21	0.207525
$955$	−5.13921e21	−0.229547
$956$	−4.90186e21	−0.217233
$957$	0	0
$958$	1.38469e22	0.604101
$959$	−2.98960e22	−1.29412
$960$	−1.79882e21	−0.0772600
$961$	−1.83404e22	−0.781596
$962$	4.24375e22	1.79447
$963$	−7.89441e21	−0.331225
$964$	−9.52286e20	−0.0396451
$965$	4.64393e20	0.0191836
$966$	−1.92908e22	−0.790717
$967$	1.44757e22	0.588764	0.294382	−	0.955688i	$-0.404886\pi$
$967$	1.44757e22	0.588764	0.294382	+	0.955688i	$0.404886\pi$
$968$	0	0
$969$	−8.61757e21	−0.345109
$970$	−6.89737e21	−0.274092
$971$	1.77921e21	0.0701590	0.0350795	−	0.999385i	$-0.488832\pi$
$971$	1.77921e21	0.0701590	0.0350795	+	0.999385i	$0.488832\pi$
$972$	−4.64802e21	−0.181874
$973$	−5.26978e22	−2.04619
$974$	5.21301e21	0.200861
$975$	1.76923e22	0.676472
$976$	6.58868e21	0.249991
$977$	1.04088e22	0.391913	0.195957	−	0.980613i	$-0.437219\pi$
$977$	1.04088e22	0.391913	0.195957	+	0.980613i	$0.437219\pi$
$978$	1.21305e22	0.453250
$979$	0	0
$980$	2.32938e21	0.0857127
$981$	3.98238e21	0.145420
$982$	−6.06761e21	−0.219878
$983$	3.26461e22	1.17403	0.587017	−	0.809575i	$-0.300302\pi$
$983$	3.26461e22	1.17403	0.587017	+	0.809575i	$0.300302\pi$
$984$	2.24185e22	0.800100
$985$	−1.84561e21	−0.0653684
$986$	1.31244e22	0.461320
$987$	−3.22295e22	−1.12428
$988$	−4.12659e21	−0.142861
$989$	4.65376e21	0.159894
$990$	0	0
$991$	−7.47327e21	−0.252906	−0.126453	−	0.991973i	$-0.540359\pi$
$991$	−7.47327e21	−0.252906	−0.126453	+	0.991973i	$0.540359\pi$
$992$	1.10915e22	0.372523
$993$	3.40107e22	1.13369
$994$	7.62369e22	2.52213
$995$	−1.49275e22	−0.490132
$996$	−1.30999e22	−0.426897
$997$	3.10809e22	1.00526	0.502632	−	0.864500i	$-0.332365\pi$
$997$	3.10809e22	1.00526	0.502632	+	0.864500i	$0.332365\pi$
$998$	3.71506e22	1.19258
$999$	−6.05226e22	−1.92831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.16.a.a.1.1		1
11.10	odd	2		1.16.a.a.1.1	✓	1
33.32	even	2		9.16.a.a.1.1		1
44.43	even	2		16.16.a.d.1.1		1
55.32	even	4		25.16.b.a.24.2		2
55.43	even	4		25.16.b.a.24.1		2
55.54	odd	2		25.16.a.a.1.1		1
77.10	even	6		49.16.c.b.30.1		2
77.32	odd	6		49.16.c.c.30.1		2
77.54	even	6		49.16.c.b.18.1		2
77.65	odd	6		49.16.c.c.18.1		2
77.76	even	2		49.16.a.a.1.1		1
88.21	odd	2		64.16.a.i.1.1		1
88.43	even	2		64.16.a.c.1.1		1
132.131	odd	2		144.16.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.16.a.a.1.1	✓	1	11.10	odd	2
9.16.a.a.1.1		1	33.32	even	2
16.16.a.d.1.1		1	44.43	even	2
25.16.a.a.1.1		1	55.54	odd	2
25.16.b.a.24.1		2	55.43	even	4
25.16.b.a.24.2		2	55.32	even	4
49.16.a.a.1.1		1	77.76	even	2
49.16.c.b.18.1		2	77.54	even	6
49.16.c.b.30.1		2	77.10	even	6
49.16.c.c.18.1		2	77.65	odd	6
49.16.c.c.30.1		2	77.32	odd	6
64.16.a.c.1.1		1	88.43	even	2
64.16.a.i.1.1		1	88.21	odd	2
121.16.a.a.1.1		1	1.1	even	1	trivial
144.16.a.f.1.1		1	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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