LMFDB - Embedded newform 121.18.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,18,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, 18, names="a")

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 18 $
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:	$221.698725687$
Analytic rank:	$0$
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)

$f(q)$

$=$

$q+528.000 q^{2} -4284.00 q^{3} +147712. q^{4} -1.02585e6 q^{5} -2.26195e6 q^{6} -3.22599e6 q^{7} +8.78592e6 q^{8} -1.10788e8 q^{9} +O(q^{10})$

\(q+528.000 q^{2} -4284.00 q^{3} +147712. q^{4} -1.02585e6 q^{5} -2.26195e6 q^{6} -3.22599e6 q^{7} +8.78592e6 q^{8} -1.10788e8 q^{9} -5.41649e8 q^{10} -6.32798e8 q^{12} -2.54106e9 q^{13} -1.70332e9 q^{14} +4.39474e9 q^{15} -1.47219e10 q^{16} +5.42974e9 q^{17} -5.84958e10 q^{18} -1.48750e9 q^{19} -1.51530e11 q^{20} +1.38201e10 q^{21} -3.17092e11 q^{23} -3.76389e10 q^{24} +2.89429e11 q^{25} -1.34168e12 q^{26} +1.02785e12 q^{27} -4.76518e11 q^{28} -2.43341e12 q^{29} +2.32042e12 q^{30} -8.84972e12 q^{31} -8.92477e12 q^{32} +2.86690e12 q^{34} +3.30938e12 q^{35} -1.63646e13 q^{36} +1.26917e13 q^{37} -7.85400e11 q^{38} +1.08859e13 q^{39} -9.01304e12 q^{40} -4.88642e13 q^{41} +7.29704e12 q^{42} +9.10200e13 q^{43} +1.13651e14 q^{45} -1.67424e14 q^{46} -4.93050e13 q^{47} +6.30688e13 q^{48} -2.22223e14 q^{49} +1.52818e14 q^{50} -2.32610e13 q^{51} -3.75346e14 q^{52} +2.29405e13 q^{53} +5.42705e14 q^{54} -2.83433e13 q^{56} +6.37245e12 q^{57} -1.28484e15 q^{58} +3.26951e13 q^{59} +6.49156e14 q^{60} +1.30829e15 q^{61} -4.67265e15 q^{62} +3.57400e14 q^{63} -2.78265e15 q^{64} +2.60675e15 q^{65} +5.19614e15 q^{67} +8.02038e14 q^{68} +1.35842e15 q^{69} +1.74735e15 q^{70} -3.70949e15 q^{71} -9.73370e14 q^{72} -3.40237e15 q^{73} +6.70119e15 q^{74} -1.23991e15 q^{75} -2.19722e14 q^{76} +5.74777e15 q^{78} -2.36653e15 q^{79} +1.51025e16 q^{80} +9.90381e15 q^{81} -2.58003e16 q^{82} +2.97668e16 q^{83} +2.04140e15 q^{84} -5.57010e15 q^{85} +4.80585e16 q^{86} +1.04247e16 q^{87} +2.91672e16 q^{89} +6.00079e16 q^{90} +8.19745e15 q^{91} -4.68383e16 q^{92} +3.79122e16 q^{93} -2.60330e16 q^{94} +1.52595e15 q^{95} +3.82337e16 q^{96} -6.37699e16 q^{97} -1.17334e17 q^{98} +O(q^{100})\)

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 $

$2$	528.000	1.45841	0.729204	−	0.684297i	$-0.239891\pi$
$2$	528.000	1.45841	0.729204	+	0.684297i	$0.239891\pi$
$3$	−4284.00	−0.376980	−0.188490	−	0.982075i	$-0.560359\pi$
$3$	−4284.00	−0.376980	−0.188490	+	0.982075i	$0.560359\pi$
$4$	147712.	1.12695
$5$	−1.02585e6	−1.17446	−0.587231	−	0.809420i	$-0.699782\pi$
$5$	−1.02585e6	−1.17446	−0.587231	+	0.809420i	$0.699782\pi$
$6$	−2.26195e6	−0.549791
$7$	−3.22599e6	−0.211510	−0.105755	−	0.994392i	$-0.533726\pi$
$7$	−3.22599e6	−0.211510	−0.105755	+	0.994392i	$0.533726\pi$
$8$	8.78592e6	0.185149
$9$	−1.10788e8	−0.857886
$10$	−5.41649e8	−1.71284
$11$	0	0
$11$	0	0
$12$	−6.32798e8	−0.424839
$13$	−2.54106e9	−0.863967	−0.431984	−	0.901881i	$-0.642186\pi$
$13$	−2.54106e9	−0.863967	−0.431984	+	0.901881i	$0.642186\pi$
$14$	−1.70332e9	−0.308467
$15$	4.39474e9	0.442749
$16$	−1.47219e10	−0.856930
$17$	5.42974e9	0.188783	0.0943916	−	0.995535i	$-0.469909\pi$
$17$	5.42974e9	0.188783	0.0943916	+	0.995535i	$0.469909\pi$
$18$	−5.84958e10	−1.25115
$19$	−1.48750e9	−0.0200933	−0.0100467	−	0.999950i	$-0.503198\pi$
$19$	−1.48750e9	−0.0200933	−0.0100467	+	0.999950i	$0.503198\pi$
$20$	−1.51530e11	−1.32356
$21$	1.38201e10	0.0797350
$22$	0	0
$23$	−3.17092e11	−0.844303	−0.422152	−	0.906525i	$-0.638725\pi$
$23$	−3.17092e11	−0.844303	−0.422152	+	0.906525i	$0.638725\pi$
$24$	−3.76389e10	−0.0697977
$25$	2.89429e11	0.379360
$26$	−1.34168e12	−1.26002
$27$	1.02785e12	0.700387
$28$	−4.76518e11	−0.238361
$29$	−2.43341e12	−0.903301	−0.451650	−	0.892195i	$-0.649165\pi$
$29$	−2.43341e12	−0.903301	−0.451650	+	0.892195i	$0.649165\pi$
$30$	2.32042e12	0.645709
$31$	−8.84972e12	−1.86361	−0.931805	−	0.362958i	$-0.881767\pi$
$31$	−8.84972e12	−1.86361	−0.931805	+	0.362958i	$0.881767\pi$
$32$	−8.92477e12	−1.43490
$33$	0	0
$34$	2.86690e12	0.275323
$35$	3.30938e12	0.248410
$36$	−1.63646e13	−0.966797
$37$	1.26917e13	0.594023	0.297012	−	0.954874i	$-0.404010\pi$
$37$	1.26917e13	0.594023	0.297012	+	0.954874i	$0.404010\pi$
$38$	−7.85400e11	−0.0293042
$39$	1.08859e13	0.325699
$40$	−9.01304e12	−0.217451
$41$	−4.88642e13	−0.955713	−0.477857	−	0.878438i	$-0.658586\pi$
$41$	−4.88642e13	−0.955713	−0.477857	+	0.878438i	$0.658586\pi$
$42$	7.29704e12	0.116286
$43$	9.10200e13	1.18756	0.593779	−	0.804628i	$-0.297635\pi$
$43$	9.10200e13	1.18756	0.593779	+	0.804628i	$0.297635\pi$
$44$	0	0
$45$	1.13651e14	1.00755
$46$	−1.67424e14	−1.23134
$47$	−4.93050e13	−0.302036	−0.151018	−	0.988531i	$-0.548255\pi$
$47$	−4.93050e13	−0.302036	−0.151018	+	0.988531i	$0.548255\pi$
$48$	6.30688e13	0.323046
$49$	−2.22223e14	−0.955264
$50$	1.52818e14	0.553262
$51$	−2.32610e13	−0.0711676
$52$	−3.75346e14	−0.973650
$53$	2.29405e13	0.0506124	0.0253062	−	0.999680i	$-0.491944\pi$
$53$	2.29405e13	0.0506124	0.0253062	+	0.999680i	$0.491944\pi$
$54$	5.42705e14	1.02145
$55$	0	0
$56$	−2.83433e13	−0.0391609
$57$	6.37245e12	0.00757478
$58$	−1.28484e15	−1.31738
$59$	3.26951e13	0.0289895	0.0144947	−	0.999895i	$-0.495386\pi$
$59$	3.26951e13	0.0289895	0.0144947	+	0.999895i	$0.495386\pi$
$60$	6.49156e14	0.498957
$61$	1.30829e15	0.873774	0.436887	−	0.899516i	$-0.356081\pi$
$61$	1.30829e15	0.873774	0.436887	+	0.899516i	$0.356081\pi$
$62$	−4.67265e15	−2.71790
$63$	3.57400e14	0.181451
$64$	−2.78265e15	−1.23574
$65$	2.60675e15	1.01470
$66$	0	0
$67$	5.19614e15	1.56331	0.781655	−	0.623711i	$-0.214376\pi$
$67$	5.19614e15	1.56331	0.781655	+	0.623711i	$0.214376\pi$
$68$	8.02038e14	0.212750
$69$	1.35842e15	0.318286
$70$	1.74735e15	0.362283
$71$	−3.70949e15	−0.681739	−0.340870	−	0.940111i	$-0.610721\pi$
$71$	−3.70949e15	−0.681739	−0.340870	+	0.940111i	$0.610721\pi$
$72$	−9.73370e14	−0.158837
$73$	−3.40237e15	−0.493785	−0.246892	−	0.969043i	$-0.579409\pi$
$73$	−3.40237e15	−0.493785	−0.246892	+	0.969043i	$0.579409\pi$
$74$	6.70119e15	0.866328
$75$	−1.23991e15	−0.143011
$76$	−2.19722e14	−0.0226442
$77$	0	0
$78$	5.74777e15	0.475001
$79$	−2.36653e15	−0.175502	−0.0877511	−	0.996142i	$-0.527968\pi$
$79$	−2.36653e15	−0.175502	−0.0877511	+	0.996142i	$0.527968\pi$
$80$	1.51025e16	1.00643
$81$	9.90381e15	0.593854
$82$	−2.58003e16	−1.39382
$83$	2.97668e16	1.45067	0.725333	−	0.688398i	$-0.241686\pi$
$83$	2.97668e16	1.45067	0.725333	+	0.688398i	$0.241686\pi$
$84$	2.04140e15	0.0898576
$85$	−5.57010e15	−0.221719
$86$	4.80585e16	1.73194
$87$	1.04247e16	0.340527
$88$	0	0
$89$	2.91672e16	0.785379	0.392690	−	0.919671i	$-0.371545\pi$
$89$	2.91672e16	0.785379	0.392690	+	0.919671i	$0.371545\pi$
$90$	6.00079e16	1.46942
$91$	8.19745e15	0.182737
$92$	−4.68383e16	−0.951490
$93$	3.79122e16	0.702545
$94$	−2.60330e16	−0.440492
$95$	1.52595e15	0.0235988
$96$	3.82337e16	0.540930
$97$	−6.37699e16	−0.826144	−0.413072	−	0.910698i	$-0.635544\pi$
$97$	−6.37699e16	−0.826144	−0.413072	+	0.910698i	$0.635544\pi$
$98$	−1.17334e17	−1.39316
$99$	0	0
$100$	4.27521e16	0.427521
$101$	1.60611e17	1.47586	0.737929	−	0.674878i	$-0.235804\pi$
$101$	1.60611e17	1.47586	0.737929	+	0.674878i	$0.235804\pi$
$102$	−1.22818e16	−0.103791
$103$	−9.07136e16	−0.705596	−0.352798	−	0.935700i	$-0.614770\pi$
$103$	−9.07136e16	−0.705596	−0.352798	+	0.935700i	$0.614770\pi$
$104$	−2.23256e16	−0.159963
$105$	−1.41774e16	−0.0936456
$106$	1.21126e16	0.0738135
$107$	−1.95549e17	−1.10026	−0.550129	−	0.835080i	$-0.685421\pi$
$107$	−1.95549e17	−1.10026	−0.550129	+	0.835080i	$0.685421\pi$
$108$	1.51826e17	0.789303
$109$	−2.13756e17	−1.02753	−0.513763	−	0.857932i	$-0.671749\pi$
$109$	−2.13756e17	−1.02753	−0.513763	+	0.857932i	$0.671749\pi$
$110$	0	0
$111$	−5.43710e16	−0.223935
$112$	4.74929e16	0.181249
$113$	−2.81383e17	−0.995706	−0.497853	−	0.867262i	$-0.665878\pi$
$113$	−2.81383e17	−0.995706	−0.497853	+	0.867262i	$0.665878\pi$
$114$	3.36465e15	0.0110471
$115$	3.25289e17	0.991602
$116$	−3.59444e17	−1.01798
$117$	2.81518e17	0.741185
$118$	1.72630e16	0.0422785
$119$	−1.75163e16	−0.0399295
$120$	3.86118e16	0.0819747
$121$	0	0
$122$	6.90775e17	1.27432
$123$	2.09334e17	0.360285
$124$	−1.30721e18	−2.10020
$125$	4.85751e17	0.728918
$126$	1.88707e17	0.264630
$127$	8.70305e17	1.14114	0.570571	−	0.821248i	$-0.306722\pi$
$127$	8.70305e17	1.14114	0.570571	+	0.821248i	$0.306722\pi$
$128$	−2.99449e17	−0.367315
$129$	−3.89930e17	−0.447686
$130$	1.37636e18	1.47984
$131$	−1.31783e17	−0.132756	−0.0663781	−	0.997795i	$-0.521144\pi$
$131$	−1.31783e17	−0.132756	−0.0663781	+	0.997795i	$0.521144\pi$
$132$	0	0
$133$	4.79866e15	0.00424993
$134$	2.74356e18	2.27994
$135$	−1.05442e18	−0.822577
$136$	4.77053e16	0.0349531
$137$	1.87128e18	1.28829	0.644147	−	0.764902i	$-0.277213\pi$
$137$	1.87128e18	1.28829	0.644147	+	0.764902i	$0.277213\pi$
$138$	7.17246e17	0.464190
$139$	−1.58706e18	−0.965980	−0.482990	−	0.875626i	$-0.660449\pi$
$139$	−1.58706e18	−0.965980	−0.482990	+	0.875626i	$0.660449\pi$
$140$	4.88836e17	0.279946
$141$	2.11223e17	0.113862
$142$	−1.95861e18	−0.994254
$143$	0	0
$144$	1.63101e18	0.735148
$145$	2.49631e18	1.06089
$146$	−1.79645e18	−0.720139
$147$	9.52005e17	0.360116
$148$	1.87471e18	0.669436
$149$	4.77240e17	0.160936	0.0804680	−	0.996757i	$-0.474359\pi$
$149$	4.77240e17	0.160936	0.0804680	+	0.996757i	$0.474359\pi$
$150$	−6.54674e17	−0.208569
$151$	3.92964e18	1.18317	0.591587	−	0.806241i	$-0.298501\pi$
$151$	3.92964e18	1.18317	0.591587	+	0.806241i	$0.298501\pi$
$152$	−1.30691e16	−0.00372026
$153$	−6.01548e17	−0.161954
$154$	0	0
$155$	9.07849e18	2.18874
$156$	1.60798e18	0.367047
$157$	−2.29453e18	−0.496075	−0.248038	−	0.968750i	$-0.579786\pi$
$157$	−2.29453e18	−0.496075	−0.248038	+	0.968750i	$0.579786\pi$
$158$	−1.24953e18	−0.255954
$159$	−9.82769e16	−0.0190799
$160$	9.15548e18	1.68524
$161$	1.02294e18	0.178578
$162$	5.22921e18	0.866081
$163$	8.01044e18	1.25910	0.629552	−	0.776958i	$-0.283238\pi$
$163$	8.01044e18	1.25910	0.629552	+	0.776958i	$0.283238\pi$
$164$	−7.21782e18	−1.07704
$165$	0	0
$166$	1.57168e19	2.11566
$167$	8.61477e18	1.10193	0.550965	−	0.834528i	$-0.314260\pi$
$167$	8.61477e18	1.10193	0.550965	+	0.834528i	$0.314260\pi$
$168$	1.21423e17	0.0147629
$169$	−2.19341e18	−0.253561
$170$	−2.94101e18	−0.323356
$171$	1.64796e17	0.0172378
$172$	1.34447e19	1.33832
$173$	−2.31430e18	−0.219294	−0.109647	−	0.993971i	$-0.534972\pi$
$173$	−2.31430e18	−0.219294	−0.109647	+	0.993971i	$0.534972\pi$
$174$	5.50426e18	0.496627
$175$	−9.33695e17	−0.0802383
$176$	0	0
$177$	−1.40066e17	−0.0109285
$178$	1.54003e19	1.14540
$179$	−7.90307e18	−0.560461	−0.280230	−	0.959933i	$-0.590411\pi$
$179$	−7.90307e18	−0.560461	−0.280230	+	0.959933i	$0.590411\pi$
$180$	1.67877e19	1.13547
$181$	1.40729e19	0.908060	0.454030	−	0.890986i	$-0.349986\pi$
$181$	1.40729e19	0.908060	0.454030	+	0.890986i	$0.349986\pi$
$182$	4.32826e18	0.266505
$183$	−5.60470e18	−0.329396
$184$	−2.78594e18	−0.156322
$185$	−1.30197e19	−0.697658
$186$	2.00176e19	1.02460
$187$	0	0
$188$	−7.28294e18	−0.340381
$189$	−3.31584e18	−0.148138
$190$	8.05703e17	0.0344167
$191$	−2.82501e19	−1.15408	−0.577040	−	0.816716i	$-0.695792\pi$
$191$	−2.82501e19	−1.15408	−0.577040	+	0.816716i	$0.695792\pi$
$192$	1.19209e19	0.465851
$193$	−4.91755e19	−1.83870	−0.919351	−	0.393437i	$-0.871286\pi$
$193$	−4.91755e19	−1.83870	−0.919351	+	0.393437i	$0.871286\pi$
$194$	−3.36705e19	−1.20486
$195$	−1.11673e19	−0.382521
$196$	−3.28251e19	−1.07654
$197$	−1.29458e19	−0.406598	−0.203299	−	0.979117i	$-0.565166\pi$
$197$	−1.29458e19	−0.406598	−0.203299	+	0.979117i	$0.565166\pi$
$198$	0	0
$199$	−5.51755e19	−1.59036	−0.795179	−	0.606375i	$-0.792623\pi$
$199$	−5.51755e19	−1.59036	−0.795179	+	0.606375i	$0.792623\pi$
$200$	2.54290e18	0.0702383
$201$	−2.22603e19	−0.589338
$202$	8.48028e19	2.15240
$203$	7.85016e18	0.191057
$204$	−3.43593e18	−0.0802025
$205$	5.01273e19	1.12245
$206$	−4.78968e19	−1.02905
$207$	3.51298e19	0.724316
$208$	3.74094e19	0.740359
$209$	0	0
$210$	−7.48567e18	−0.136574
$211$	−1.73510e19	−0.304035	−0.152017	−	0.988378i	$-0.548577\pi$
$211$	−1.73510e19	−0.304035	−0.152017	+	0.988378i	$0.548577\pi$
$212$	3.38858e18	0.0570378
$213$	1.58915e19	0.257002
$214$	−1.03250e20	−1.60462
$215$	−9.33728e19	−1.39474
$216$	9.03061e18	0.129676
$217$	2.85491e19	0.394171
$218$	−1.12863e20	−1.49855
$219$	1.45758e19	0.186147
$220$	0	0
$221$	−1.37973e19	−0.163102
$222$	−2.87079e19	−0.326589
$223$	9.48415e19	1.03850	0.519251	−	0.854622i	$-0.326211\pi$
$223$	9.48415e19	1.03850	0.519251	+	0.854622i	$0.326211\pi$
$224$	2.87912e19	0.303496
$225$	−3.20651e19	−0.325448
$226$	−1.48570e20	−1.45214
$227$	1.83782e20	1.73015	0.865073	−	0.501647i	$-0.167272\pi$
$227$	1.83782e20	1.73015	0.865073	+	0.501647i	$0.167272\pi$
$228$	9.41287e17	0.00853642
$229$	−9.14908e19	−0.799422	−0.399711	−	0.916641i	$-0.630889\pi$
$229$	−9.14908e19	−0.799422	−0.399711	+	0.916641i	$0.630889\pi$
$230$	1.71752e20	1.44616
$231$	0	0
$232$	−2.13798e19	−0.167246
$233$	−9.87497e19	−0.744750	−0.372375	−	0.928082i	$-0.621456\pi$
$233$	−9.87497e19	−0.744750	−0.372375	+	0.928082i	$0.621456\pi$
$234$	1.48642e20	1.08095
$235$	5.05795e19	0.354730
$236$	4.82946e18	0.0326698
$237$	1.01382e19	0.0661609
$238$	−9.24861e18	−0.0582334
$239$	−1.89337e20	−1.15041	−0.575206	−	0.818009i	$-0.695078\pi$
$239$	−1.89337e20	−1.15041	−0.575206	+	0.818009i	$0.695078\pi$
$240$	−6.46991e19	−0.379405
$241$	1.38762e20	0.785463	0.392731	−	0.919653i	$-0.371530\pi$
$241$	1.38762e20	0.785463	0.392731	+	0.919653i	$0.371530\pi$
$242$	0	0
$243$	−1.75165e20	−0.924258
$244$	1.93250e20	0.984702
$245$	2.27968e20	1.12192
$246$	1.10528e20	0.525443
$247$	3.77983e18	0.0173600
$248$	−7.77529e19	−0.345046
$249$	−1.27521e20	−0.546873
$250$	2.56476e20	1.06306
$251$	−3.34508e20	−1.34023	−0.670115	−	0.742257i	$-0.733755\pi$
$251$	−3.34508e20	−1.34023	−0.670115	+	0.742257i	$0.733755\pi$
$252$	5.27922e19	0.204487
$253$	0	0
$254$	4.59521e20	1.66425
$255$	2.38623e19	0.0835836
$256$	2.06618e20	0.700048
$257$	4.91382e19	0.161060	0.0805300	−	0.996752i	$-0.474339\pi$
$257$	4.91382e19	0.161060	0.0805300	+	0.996752i	$0.474339\pi$
$258$	−2.05883e20	−0.652909
$259$	−4.09432e19	−0.125642
$260$	3.85048e20	1.14352
$261$	2.69591e20	0.774929
$262$	−6.95816e19	−0.193613
$263$	−1.78845e19	−0.0481784	−0.0240892	−	0.999710i	$-0.507669\pi$
$263$	−1.78845e19	−0.0481784	−0.0240892	+	0.999710i	$0.507669\pi$
$264$	0	0
$265$	−2.35335e19	−0.0594423
$266$	2.53369e18	0.00619812
$267$	−1.24952e20	−0.296073
$268$	7.67533e20	1.76178
$269$	1.15237e20	0.256270	0.128135	−	0.991757i	$-0.459101\pi$
$269$	1.15237e20	0.256270	0.128135	+	0.991757i	$0.459101\pi$
$270$	−5.56734e20	−1.19965
$271$	1.47067e20	0.307098	0.153549	−	0.988141i	$-0.450930\pi$
$271$	1.47067e20	0.307098	0.153549	+	0.988141i	$0.450930\pi$
$272$	−7.99363e19	−0.161774
$273$	−3.51179e19	−0.0688884
$274$	9.88038e20	1.87886
$275$	0	0
$276$	2.00655e20	0.358693
$277$	1.78744e20	0.309852	0.154926	−	0.987926i	$-0.450486\pi$
$277$	1.78744e20	0.309852	0.154926	+	0.987926i	$0.450486\pi$
$278$	−8.37969e20	−1.40879
$279$	9.80439e20	1.59877
$280$	2.90760e19	0.0459929
$281$	−2.80546e20	−0.430527	−0.215263	−	0.976556i	$-0.569061\pi$
$281$	−2.80546e20	−0.430527	−0.215263	+	0.976556i	$0.569061\pi$
$282$	1.11526e20	0.166057
$283$	−3.39877e20	−0.491062	−0.245531	−	0.969389i	$-0.578962\pi$
$283$	−3.39877e20	−0.491062	−0.245531	+	0.969389i	$0.578962\pi$
$284$	−5.47936e20	−0.768288
$285$	−6.53718e18	−0.00889629
$286$	0	0
$287$	1.57635e20	0.202142
$288$	9.88753e20	1.23098
$289$	−7.97758e20	−0.964361
$290$	1.31805e21	1.54721
$291$	2.73190e20	0.311440
$292$	−5.02571e20	−0.556472
$293$	7.64887e20	0.822664	0.411332	−	0.911486i	$-0.365064\pi$
$293$	7.64887e20	0.822664	0.411332	+	0.911486i	$0.365064\pi$
$294$	5.02659e20	0.525196
$295$	−3.35403e19	−0.0340470
$296$	1.11508e20	0.109983
$297$	0	0
$298$	2.51983e20	0.234710
$299$	8.05751e20	0.729450
$300$	−1.83150e20	−0.161167
$301$	−2.93630e20	−0.251180
$302$	2.07485e21	1.72555
$303$	−6.88059e20	−0.556370
$304$	2.18989e19	0.0172185
$305$	−1.34211e21	−1.02621
$306$	−3.17617e20	−0.236196
$307$	−1.38472e21	−1.00158	−0.500789	−	0.865569i	$-0.666957\pi$
$307$	−1.38472e21	−1.00158	−0.500789	+	0.865569i	$0.666957\pi$
$308$	0	0
$309$	3.88617e20	0.265996
$310$	4.79344e21	3.19207
$311$	−2.34733e21	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$311$	−2.34733e21	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$312$	9.56428e19	0.0603029
$313$	1.58424e21	0.972065	0.486033	−	0.873941i	$-0.338444\pi$
$313$	1.58424e21	0.972065	0.486033	+	0.873941i	$0.338444\pi$
$314$	−1.21151e21	−0.723480
$315$	−3.66638e20	−0.213107
$316$	−3.49565e20	−0.197783
$317$	8.98378e20	0.494829	0.247415	−	0.968910i	$-0.420419\pi$
$317$	8.98378e20	0.494829	0.247415	+	0.968910i	$0.420419\pi$
$318$	−5.18902e19	−0.0278263
$319$	0	0
$320$	2.85458e21	1.45133
$321$	8.37734e20	0.414776
$322$	5.40110e20	0.260440
$323$	−8.07674e18	−0.00379328
$324$	1.46291e21	0.669245
$325$	−7.35457e20	−0.327755
$326$	4.22951e21	1.83629
$327$	9.15730e20	0.387357
$328$	−4.29317e20	−0.176950
$329$	1.59058e20	0.0638836
$330$	0	0
$331$	4.00469e21	1.52767	0.763837	−	0.645409i	$-0.223313\pi$
$331$	4.00469e21	1.52767	0.763837	+	0.645409i	$0.223313\pi$
$332$	4.39691e21	1.63483
$333$	−1.40608e21	−0.509604
$334$	4.54860e21	1.60706
$335$	−5.33046e21	−1.83605
$336$	−2.03459e20	−0.0683273
$337$	1.06753e21	0.349564	0.174782	−	0.984607i	$-0.444078\pi$
$337$	1.06753e21	0.349564	0.174782	+	0.984607i	$0.444078\pi$
$338$	−1.15812e21	−0.369795
$339$	1.20544e21	0.375362
$340$	−8.22771e20	−0.249867
$341$	0	0
$342$	8.70125e19	0.0251397
$343$	1.46736e21	0.413557
$344$	7.99694e20	0.219876
$345$	−1.39354e21	−0.373814
$346$	−1.22195e21	−0.319821
$347$	1.61841e21	0.413321	0.206660	−	0.978413i	$-0.433740\pi$
$347$	1.61841e21	0.413321	0.206660	+	0.978413i	$0.433740\pi$
$348$	1.53986e21	0.383758
$349$	5.60078e21	1.36217	0.681086	−	0.732203i	$-0.261508\pi$
$349$	5.60078e21	1.36217	0.681086	+	0.732203i	$0.261508\pi$
$350$	−4.92991e20	−0.117020
$351$	−2.61183e21	−0.605111
$352$	0	0
$353$	5.10774e21	1.12757	0.563785	−	0.825921i	$-0.309344\pi$
$353$	5.10774e21	1.12757	0.563785	+	0.825921i	$0.309344\pi$
$354$	−7.39547e19	−0.0159382
$355$	3.80538e21	0.800676
$356$	4.30834e21	0.885086
$357$	7.50399e19	0.0150526
$358$	−4.17282e21	−0.817380
$359$	−5.84965e21	−1.11899	−0.559496	−	0.828833i	$-0.689005\pi$
$359$	−5.84965e21	−1.11899	−0.559496	+	0.828833i	$0.689005\pi$
$360$	9.98532e20	0.186548
$361$	−5.47817e21	−0.999596
$362$	7.43047e21	1.32432
$363$	0	0
$364$	1.21086e21	0.205936
$365$	3.49032e21	0.579931
$366$	−2.95928e21	−0.480393
$367$	4.29535e20	0.0681297	0.0340649	−	0.999420i	$-0.489155\pi$
$367$	4.29535e20	0.0681297	0.0340649	+	0.999420i	$0.489155\pi$
$368$	4.66821e21	0.723509
$369$	5.41354e21	0.819893
$370$	−6.87442e21	−1.01747
$371$	−7.40057e19	−0.0107050
$372$	5.60009e21	0.791735
$373$	6.54734e21	0.904774	0.452387	−	0.891822i	$-0.350573\pi$
$373$	6.54734e21	0.904774	0.452387	+	0.891822i	$0.350573\pi$
$374$	0	0
$375$	−2.08096e21	−0.274788
$376$	−4.33190e20	−0.0559218
$377$	6.18345e21	0.780422
$378$	−1.75076e21	−0.216046
$379$	1.51850e21	0.183223	0.0916117	−	0.995795i	$-0.470798\pi$
$379$	1.51850e21	0.183223	0.0916117	+	0.995795i	$0.470798\pi$
$380$	2.25401e20	0.0265948
$381$	−3.72839e21	−0.430188
$382$	−1.49160e22	−1.68312
$383$	−8.04597e21	−0.887951	−0.443975	−	0.896039i	$-0.646432\pi$
$383$	−8.04597e21	−0.887951	−0.443975	+	0.896039i	$0.646432\pi$
$384$	1.28284e21	0.138470
$385$	0	0
$386$	−2.59647e22	−2.68158
$387$	−1.00839e22	−1.01879
$388$	−9.41958e21	−0.931026
$389$	6.84040e21	0.661470	0.330735	−	0.943724i	$-0.392703\pi$
$389$	6.84040e21	0.661470	0.330735	+	0.943724i	$0.392703\pi$
$390$	−5.89635e21	−0.557871
$391$	−1.72173e21	−0.159390
$392$	−1.95244e21	−0.176867
$393$	5.64560e20	0.0500465
$394$	−6.83536e21	−0.592985
$395$	2.42771e21	0.206121
$396$	0	0
$397$	6.23458e21	0.507093	0.253547	−	0.967323i	$-0.418403\pi$
$397$	6.23458e21	0.507093	0.253547	+	0.967323i	$0.418403\pi$
$398$	−2.91327e22	−2.31939
$399$	−2.05575e19	−0.00160214
$400$	−4.26095e21	−0.325085
$401$	−2.32048e21	−0.173320	−0.0866602	−	0.996238i	$-0.527619\pi$
$401$	−2.32048e21	−0.173320	−0.0866602	+	0.996238i	$0.527619\pi$
$402$	−1.17534e22	−0.859494
$403$	2.24877e22	1.61010
$404$	2.37242e22	1.66322
$405$	−1.01598e22	−0.697458
$406$	4.14489e21	0.278639
$407$	0	0
$408$	−2.04369e20	−0.0131766
$409$	−2.38590e22	−1.50662	−0.753312	−	0.657663i	$-0.771545\pi$
$409$	−2.38590e22	−1.50662	−0.753312	+	0.657663i	$0.771545\pi$
$410$	2.64672e22	1.63699
$411$	−8.01658e21	−0.485661
$412$	−1.33995e22	−0.795173
$413$	−1.05474e20	−0.00613155
$414$	1.85485e22	1.05635
$415$	−3.05362e22	−1.70375
$416$	2.26784e22	1.23971
$417$	6.79898e21	0.364156
$418$	0	0
$419$	−2.61839e22	−1.34653	−0.673263	−	0.739403i	$-0.735108\pi$
$419$	−2.61839e22	−1.34653	−0.673263	+	0.739403i	$0.735108\pi$
$420$	−2.09417e21	−0.105534
$421$	1.11903e22	0.552642	0.276321	−	0.961065i	$-0.410885\pi$
$421$	1.11903e22	0.552642	0.276321	+	0.961065i	$0.410885\pi$
$422$	−9.16132e21	−0.443407
$423$	5.46238e21	0.259113
$424$	2.01553e20	0.00937086
$425$	1.57152e21	0.0716168
$426$	8.39069e21	0.374814
$427$	−4.22052e21	−0.184812
$428$	−2.88850e22	−1.23994
$429$	0	0
$430$	−4.93009e22	−2.03410
$431$	−1.74296e22	−0.705068	−0.352534	−	0.935799i	$-0.614680\pi$
$431$	−1.74296e22	−0.705068	−0.352534	+	0.935799i	$0.614680\pi$
$432$	−1.51319e22	−0.600182
$433$	−4.51413e22	−1.75560	−0.877802	−	0.479024i	$-0.840991\pi$
$433$	−4.51413e22	−1.75560	−0.877802	+	0.479024i	$0.840991\pi$
$434$	1.50739e22	0.574863
$435$	−1.06942e22	−0.399936
$436$	−3.15743e22	−1.15797
$437$	4.71674e20	0.0169648
$438$	7.69600e21	0.271478
$439$	−3.85266e22	−1.33294	−0.666472	−	0.745530i	$-0.732196\pi$
$439$	−3.85266e22	−1.33294	−0.666472	+	0.745530i	$0.732196\pi$
$440$	0	0
$441$	2.46196e22	0.819507
$442$	−7.28499e21	−0.237870
$443$	−3.84966e21	−0.123308	−0.0616539	−	0.998098i	$-0.519637\pi$
$443$	−3.84966e21	−0.123308	−0.0616539	+	0.998098i	$0.519637\pi$
$444$	−8.03126e21	−0.252364
$445$	−2.99212e22	−0.922398
$446$	5.00763e22	1.51456
$447$	−2.04449e21	−0.0606697
$448$	8.97679e21	0.261371
$449$	−2.77614e21	−0.0793136	−0.0396568	−	0.999213i	$-0.512626\pi$
$449$	−2.77614e21	−0.0793136	−0.0396568	+	0.999213i	$0.512626\pi$
$450$	−1.69304e22	−0.474635
$451$	0	0
$452$	−4.15636e22	−1.12211
$453$	−1.68346e22	−0.446034
$454$	9.70368e22	2.52326
$455$	−8.40936e21	−0.214618
$456$	5.59878e19	0.00140247
$457$	1.34139e22	0.329814	0.164907	−	0.986309i	$-0.447268\pi$
$457$	1.34139e22	0.329814	0.164907	+	0.986309i	$0.447268\pi$
$458$	−4.83071e22	−1.16588
$459$	5.58096e21	0.132221
$460$	4.80490e22	1.11749
$461$	4.32728e22	0.988000	0.494000	−	0.869462i	$-0.335534\pi$
$461$	4.32728e22	0.988000	0.494000	+	0.869462i	$0.335534\pi$
$462$	0	0
$463$	3.85721e22	0.848859	0.424429	−	0.905461i	$-0.360475\pi$
$463$	3.85721e22	0.848859	0.424429	+	0.905461i	$0.360475\pi$
$464$	3.58245e22	0.774065
$465$	−3.88922e22	−0.825112
$466$	−5.21398e22	−1.08615
$467$	4.96724e22	1.01607	0.508033	−	0.861338i	$-0.330373\pi$
$467$	4.96724e22	1.01607	0.508033	+	0.861338i	$0.330373\pi$
$468$	4.15836e22	0.835281
$469$	−1.67627e22	−0.330655
$470$	2.67060e22	0.517341
$471$	9.82978e21	0.187011
$472$	2.87256e20	0.00536739
$473$	0	0
$474$	5.35299e21	0.0964895
$475$	−4.30525e20	−0.00762260
$476$	−2.58737e21	−0.0449986
$477$	−2.54152e21	−0.0434197
$478$	−9.99699e22	−1.67777
$479$	4.26513e22	0.703202	0.351601	−	0.936150i	$-0.385637\pi$
$479$	4.26513e22	0.703202	0.351601	+	0.936150i	$0.385637\pi$
$480$	−3.92221e22	−0.635302
$481$	−3.22503e22	−0.513217
$482$	7.32663e22	1.14553
$483$	−4.38226e21	−0.0673205
$484$	0	0
$485$	6.54183e22	0.970275
$486$	−9.24869e22	−1.34794
$487$	−9.28126e22	−1.32926	−0.664631	−	0.747172i	$-0.731411\pi$
$487$	−9.28126e22	−1.32926	−0.664631	+	0.747172i	$0.731411\pi$
$488$	1.14945e22	0.161779
$489$	−3.43167e22	−0.474658
$490$	1.20367e23	1.63622
$491$	−1.29877e21	−0.0173516	−0.00867579	−	0.999962i	$-0.502762\pi$
$491$	−1.29877e21	−0.0173516	−0.00867579	+	0.999962i	$0.502762\pi$
$492$	3.09211e22	0.406024
$493$	−1.32128e22	−0.170528
$494$	1.99575e21	0.0253179
$495$	0	0
$496$	1.30285e23	1.59698
$497$	1.19668e22	0.144194
$498$	−6.73310e22	−0.797564
$499$	−7.66788e22	−0.892937	−0.446468	−	0.894799i	$-0.647318\pi$
$499$	−7.66788e22	−0.892937	−0.446468	+	0.894799i	$0.647318\pi$
$500$	7.17512e22	0.821456
$501$	−3.69057e22	−0.415406
$502$	−1.76620e23	−1.95460
$503$	−6.67712e22	−0.726543	−0.363272	−	0.931683i	$-0.618340\pi$
$503$	−6.67712e22	−0.726543	−0.363272	+	0.931683i	$0.618340\pi$
$504$	3.14008e21	0.0335956
$505$	−1.64763e23	−1.73334
$506$	0	0
$507$	9.39656e21	0.0955875
$508$	1.28555e23	1.28601
$509$	−1.15659e23	−1.13783	−0.568916	−	0.822395i	$-0.692637\pi$
$509$	−1.15659e23	−1.13783	−0.568916	+	0.822395i	$0.692637\pi$
$510$	1.25993e22	0.121899
$511$	1.09760e22	0.104440
$512$	1.48344e23	1.38827
$513$	−1.52893e21	−0.0140731
$514$	2.59450e22	0.234891
$515$	9.30585e22	0.828695
$516$	−5.75973e22	−0.504521
$517$	0	0
$518$	−2.16180e22	−0.183237
$519$	9.91446e21	0.0826697
$520$	2.29027e22	0.187870
$521$	−1.79855e23	−1.45145	−0.725725	−	0.687985i	$-0.758496\pi$
$521$	−1.79855e23	−1.45145	−0.725725	+	0.687985i	$0.758496\pi$
$522$	1.42344e23	1.13016
$523$	−9.37211e22	−0.732104	−0.366052	−	0.930594i	$-0.619291\pi$
$523$	−9.37211e22	−0.732104	−0.366052	+	0.930594i	$0.619291\pi$
$524$	−1.94660e22	−0.149610
$525$	3.99995e21	0.0302483
$526$	−9.44299e21	−0.0702637
$527$	−4.80517e22	−0.351818
$528$	0	0
$529$	−4.05028e22	−0.287152
$530$	−1.24257e22	−0.0866912
$531$	−3.62221e21	−0.0248697
$532$	7.08820e20	0.00478947
$533$	1.24167e23	0.825705
$534$	−6.59748e22	−0.431795
$535$	2.00604e23	1.29221
$536$	4.56529e22	0.289446
$537$	3.38568e22	0.211283
$538$	6.08452e22	0.373746
$539$	0	0
$540$	−1.55750e23	−0.927006
$541$	1.52396e23	0.892889	0.446444	−	0.894811i	$-0.352690\pi$
$541$	1.52396e23	0.892889	0.446444	+	0.894811i	$0.352690\pi$
$542$	7.76514e22	0.447874
$543$	−6.02881e22	−0.342321
$544$	−4.84592e22	−0.270886
$545$	2.19281e23	1.20679
$546$	−1.85422e22	−0.100467
$547$	−2.04979e23	−1.09349	−0.546747	−	0.837298i	$-0.684134\pi$
$547$	−2.04979e23	−1.09349	−0.546747	+	0.837298i	$0.684134\pi$
$548$	2.76411e23	1.45185
$549$	−1.44942e23	−0.749598
$550$	0	0
$551$	3.61970e21	0.0181503
$552$	1.19350e22	0.0589304
$553$	7.63442e21	0.0371204
$554$	9.43770e22	0.451890
$555$	5.57765e22	0.263003
$556$	−2.34428e23	−1.08861
$557$	4.07359e23	1.86298	0.931490	−	0.363767i	$-0.118509\pi$
$557$	4.07359e23	1.86298	0.931490	+	0.363767i	$0.118509\pi$
$558$	5.17672e23	2.33165
$559$	−2.31288e23	−1.02601
$560$	−4.87206e22	−0.212870
$561$	0	0
$562$	−1.48128e23	−0.627884
$563$	−2.21297e23	−0.923962	−0.461981	−	0.886890i	$-0.652861\pi$
$563$	−2.21297e23	−0.923962	−0.461981	+	0.886890i	$0.652861\pi$
$564$	3.12001e22	0.128317
$565$	2.88657e23	1.16942
$566$	−1.79455e23	−0.716169
$567$	−3.19496e22	−0.125606
$568$	−3.25913e22	−0.126224
$569$	−4.88686e23	−1.86456	−0.932278	−	0.361743i	$-0.882182\pi$
$569$	−4.88686e23	−1.86456	−0.932278	+	0.361743i	$0.882182\pi$
$570$	−3.45163e21	−0.0129744
$571$	−3.42218e20	−0.00126735	−0.000633673	−	1.00000i	$-0.500202\pi$
$571$	−3.42218e20	−0.00126735	−0.000633673		1.00000i	$0.500202\pi$
$572$	0	0
$573$	1.21023e23	0.435065
$574$	8.32315e22	0.294806
$575$	−9.17755e22	−0.320295
$576$	3.08282e23	1.06013
$577$	−1.95911e23	−0.663840	−0.331920	−	0.943308i	$-0.607696\pi$
$577$	−1.95911e23	−0.663840	−0.331920	+	0.943308i	$0.607696\pi$
$578$	−4.21216e23	−1.40643
$579$	2.10668e23	0.693155
$580$	3.68736e23	1.19558
$581$	−9.60273e22	−0.306830
$582$	1.44244e23	0.454207
$583$	0	0
$584$	−2.98930e22	−0.0914239
$585$	−2.88795e23	−0.870493
$586$	4.03860e23	1.19978
$587$	−4.80676e23	−1.40744	−0.703719	−	0.710479i	$-0.748478\pi$
$587$	−4.80676e23	−1.40744	−0.703719	+	0.710479i	$0.748478\pi$
$588$	1.40623e23	0.405834
$589$	1.31640e22	0.0374461
$590$	−1.77093e22	−0.0496545
$591$	5.54596e22	0.153279
$592$	−1.86846e23	−0.509036
$593$	6.40804e23	1.72092	0.860459	−	0.509519i	$-0.170177\pi$
$593$	6.40804e23	1.72092	0.860459	+	0.509519i	$0.170177\pi$
$594$	0	0
$595$	1.79691e22	0.0468956
$596$	7.04940e22	0.181367
$597$	2.36372e23	0.599534
$598$	4.25436e23	1.06384
$599$	3.60950e23	0.889854	0.444927	−	0.895567i	$-0.353230\pi$
$599$	3.60950e23	0.889854	0.444927	+	0.895567i	$0.353230\pi$
$600$	−1.08938e22	−0.0264785
$601$	−6.53955e23	−1.56717	−0.783583	−	0.621287i	$-0.786610\pi$
$601$	−6.53955e23	−1.56717	−0.783583	+	0.621287i	$0.786610\pi$
$602$	−1.55036e23	−0.366323
$603$	−5.75668e23	−1.34114
$604$	5.80455e23	1.33338
$605$	0	0
$606$	−3.63295e23	−0.811414
$607$	−4.08715e23	−0.900154	−0.450077	−	0.892990i	$-0.648603\pi$
$607$	−4.08715e23	−0.900154	−0.450077	+	0.892990i	$0.648603\pi$
$608$	1.32756e22	0.0288319
$609$	−3.36301e22	−0.0720247
$610$	−7.08631e23	−1.49664
$611$	1.25287e23	0.260949
$612$	−8.88558e22	−0.182515
$613$	−1.80349e23	−0.365343	−0.182671	−	0.983174i	$-0.558474\pi$
$613$	−1.80349e23	−0.365343	−0.182671	+	0.983174i	$0.558474\pi$
$614$	−7.31131e23	−1.46071
$615$	−2.14745e23	−0.423141
$616$	0	0
$617$	4.42851e23	0.848856	0.424428	−	0.905462i	$-0.360475\pi$
$617$	4.42851e23	0.848856	0.424428	+	0.905462i	$0.360475\pi$
$618$	2.05190e23	0.387930
$619$	1.71098e22	0.0319062	0.0159531	−	0.999873i	$-0.494922\pi$
$619$	1.71098e22	0.0319062	0.0159531	+	0.999873i	$0.494922\pi$
$620$	1.34100e24	2.46661
$621$	−3.25923e23	−0.591339
$622$	−1.23939e24	−2.21815
$623$	−9.40931e22	−0.166115
$624$	−1.60262e23	−0.279101
$625$	−7.19124e23	−1.23545
$626$	8.36481e23	1.41767
$627$	0	0
$628$	−3.38930e23	−0.559053
$629$	6.89124e22	0.112142
$630$	−1.93585e23	−0.310797
$631$	−2.82293e23	−0.447147	−0.223574	−	0.974687i	$-0.571772\pi$
$631$	−2.82293e23	−0.447147	−0.223574	+	0.974687i	$0.571772\pi$
$632$	−2.07922e22	−0.0324941
$633$	7.43316e22	0.114615
$634$	4.74344e23	0.721663
$635$	−8.92802e23	−1.34023
$636$	−1.45167e22	−0.0215021
$637$	5.64684e23	0.825316
$638$	0	0
$639$	4.10965e23	0.584854
$640$	3.07190e23	0.431397
$641$	−3.77613e23	−0.523304	−0.261652	−	0.965162i	$-0.584267\pi$
$641$	−3.77613e23	−0.523304	−0.261652	+	0.965162i	$0.584267\pi$
$642$	4.42323e23	0.604912
$643$	−4.65536e23	−0.628289	−0.314145	−	0.949375i	$-0.601718\pi$
$643$	−4.65536e23	−0.628289	−0.314145	+	0.949375i	$0.601718\pi$
$644$	1.51100e23	0.201249
$645$	4.00009e23	0.525790
$646$	−4.26452e21	−0.00553215
$647$	1.00412e24	1.28558	0.642788	−	0.766045i	$-0.277778\pi$
$647$	1.00412e24	1.28558	0.642788	+	0.766045i	$0.277778\pi$
$648$	8.70141e22	0.109952
$649$	0	0
$650$	−3.88321e23	−0.478000
$651$	−1.22304e23	−0.148595
$652$	1.18324e24	1.41895
$653$	1.26098e23	0.149261	0.0746304	−	0.997211i	$-0.476222\pi$
$653$	1.26098e23	0.149261	0.0746304	+	0.997211i	$0.476222\pi$
$654$	4.83505e23	0.564924
$655$	1.35190e23	0.155917
$656$	7.19375e23	0.818979
$657$	3.76940e23	0.423611
$658$	8.39824e22	0.0931683
$659$	1.06105e24	1.16201	0.581005	−	0.813900i	$-0.302660\pi$
$659$	1.06105e24	1.16201	0.581005	+	0.813900i	$0.302660\pi$
$660$	0	0
$661$	1.57511e24	1.68112	0.840560	−	0.541719i	$-0.182226\pi$
$661$	1.57511e24	1.68112	0.840560	+	0.541719i	$0.182226\pi$
$662$	2.11448e24	2.22797
$663$	5.91077e22	0.0614864
$664$	2.61528e23	0.268590
$665$	−4.92271e21	−0.00499137
$666$	−7.42408e23	−0.743211
$667$	7.71615e23	0.762660
$668$	1.27251e24	1.24182
$669$	−4.06301e23	−0.391495
$670$	−2.81449e24	−2.67771
$671$	0	0
$672$	−1.23342e23	−0.114412
$673$	−2.91330e23	−0.266844	−0.133422	−	0.991059i	$-0.542597\pi$
$673$	−2.91330e23	−0.266844	−0.133422	+	0.991059i	$0.542597\pi$
$674$	5.63657e23	0.509807
$675$	2.97489e23	0.265699
$676$	−3.23993e23	−0.285751
$677$	−1.33861e24	−1.16587	−0.582937	−	0.812517i	$-0.698097\pi$
$677$	−1.33861e24	−1.16587	−0.582937	+	0.812517i	$0.698097\pi$
$678$	6.36475e23	0.547430
$679$	2.05721e23	0.174737
$680$	−4.89385e22	−0.0410511
$681$	−7.87321e23	−0.652231
$682$	0	0
$683$	−1.05295e24	−0.850812	−0.425406	−	0.905003i	$-0.639869\pi$
$683$	−1.05295e24	−0.850812	−0.425406	+	0.905003i	$0.639869\pi$
$684$	2.43424e22	0.0194261
$685$	−1.91966e24	−1.51305
$686$	7.74764e23	0.603135
$687$	3.91947e23	0.301366
$688$	−1.33999e24	−1.01765
$689$	−5.82932e22	−0.0437275
$690$	−7.35787e23	−0.545174
$691$	−5.71311e22	−0.0418128	−0.0209064	−	0.999781i	$-0.506655\pi$
$691$	−5.71311e22	−0.0418128	−0.0209064	+	0.999781i	$0.506655\pi$
$692$	−3.41850e23	−0.247134
$693$	0	0
$694$	8.54519e23	0.602790
$695$	1.62809e24	1.13451
$696$	9.15909e22	0.0630483
$697$	−2.65320e23	−0.180423
$698$	2.95721e24	1.98660
$699$	4.23044e23	0.280756
$700$	−1.37918e23	−0.0904248
$701$	3.18403e23	0.206241	0.103120	−	0.994669i	$-0.467117\pi$
$701$	3.18403e23	0.206241	0.103120	+	0.994669i	$0.467117\pi$
$702$	−1.37905e24	−0.882498
$703$	−1.88788e22	−0.0119359
$704$	0	0
$705$	−2.16683e23	−0.133726
$706$	2.69689e24	1.64446
$707$	−5.18131e23	−0.312158
$708$	−2.06894e22	−0.0123159
$709$	2.12997e24	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$709$	2.12997e24	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$710$	2.00924e24	1.16771
$711$	2.62182e23	0.150561
$712$	2.56261e23	0.145413
$713$	2.80617e24	1.57345
$714$	3.96210e22	0.0219529
$715$	0	0
$716$	−1.16738e24	−0.631613
$717$	8.11119e23	0.433683
$718$	−3.08861e24	−1.63195
$719$	−2.34234e24	−1.22308	−0.611540	−	0.791213i	$-0.709450\pi$
$719$	−2.34234e24	−1.22308	−0.611540	+	0.791213i	$0.709450\pi$
$720$	−1.67317e24	−0.863403
$721$	2.92641e23	0.149240
$722$	−2.89248e24	−1.45782
$723$	−5.94456e23	−0.296104
$724$	2.07873e24	1.02334
$725$	−7.04299e23	−0.342676
$726$	0	0
$727$	3.58185e24	1.70241	0.851207	−	0.524830i	$-0.175871\pi$
$727$	3.58185e24	1.70241	0.851207	+	0.524830i	$0.175871\pi$
$728$	7.20222e22	0.0338337
$729$	−5.28574e23	−0.245427
$730$	1.84289e24	0.845776
$731$	4.94215e23	0.224191
$732$	−8.27881e23	−0.371213
$733$	3.29013e24	1.45824	0.729119	−	0.684387i	$-0.239930\pi$
$733$	3.29013e24	1.45824	0.729119	+	0.684387i	$0.239930\pi$
$734$	2.26794e23	0.0993609
$735$	−9.76615e23	−0.422942
$736$	2.82997e24	1.21149
$737$	0	0
$738$	2.85835e24	1.19574
$739$	−1.74130e24	−0.720103	−0.360052	−	0.932932i	$-0.617241\pi$
$739$	−1.74130e24	−0.720103	−0.360052	+	0.932932i	$0.617241\pi$
$740$	−1.92317e24	−0.786227
$741$	−1.61928e22	−0.00654436
$742$	−3.90750e22	−0.0156123
$743$	4.61544e24	1.82309	0.911545	−	0.411199i	$-0.134890\pi$
$743$	4.61544e24	1.82309	0.911545	+	0.411199i	$0.134890\pi$
$744$	3.33094e23	0.130076
$745$	−4.89576e23	−0.189013
$746$	3.45700e24	1.31953
$747$	−3.29778e24	−1.24451
$748$	0	0
$749$	6.30841e23	0.232715
$750$	−1.09875e24	−0.400753
$751$	−8.19607e22	−0.0295574	−0.0147787	−	0.999891i	$-0.504704\pi$
$751$	−8.19607e22	−0.0295574	−0.0147787	+	0.999891i	$0.504704\pi$
$752$	7.25865e23	0.258824
$753$	1.43303e24	0.505241
$754$	3.26486e24	1.13817
$755$	−4.03122e24	−1.38959
$756$	−4.89789e23	−0.166945
$757$	−2.51608e24	−0.848025	−0.424013	−	0.905656i	$-0.639379\pi$
$757$	−2.51608e24	−0.848025	−0.424013	+	0.905656i	$0.639379\pi$
$758$	8.01767e23	0.267215
$759$	0	0
$760$	1.34069e22	0.00436931
$761$	5.27745e23	0.170081	0.0850403	−	0.996378i	$-0.472898\pi$
$761$	5.27745e23	0.170081	0.0850403	+	0.996378i	$0.472898\pi$
$762$	−1.96859e24	−0.627390
$763$	6.89574e23	0.217331
$764$	−4.17287e24	−1.30059
$765$	6.17098e23	0.190209
$766$	−4.24827e24	−1.29499
$767$	−8.30803e22	−0.0250460
$768$	−8.85151e23	−0.263905
$769$	7.51719e23	0.221657	0.110828	−	0.993840i	$-0.464650\pi$
$769$	7.51719e23	0.221657	0.110828	+	0.993840i	$0.464650\pi$
$770$	0	0
$771$	−2.10508e23	−0.0607165
$772$	−7.26381e24	−2.07213
$773$	2.81914e24	0.795409	0.397704	−	0.917514i	$-0.369807\pi$
$773$	2.81914e24	0.795409	0.397704	+	0.917514i	$0.369807\pi$
$774$	−5.32429e24	−1.48581
$775$	−2.56136e24	−0.706980
$776$	−5.60277e23	−0.152960
$777$	1.75401e23	0.0473644
$778$	3.61173e24	0.964693
$779$	7.26854e22	0.0192034
$780$	−1.64955e24	−0.431083
$781$	0	0
$782$	−9.09072e23	−0.232456
$783$	−2.50118e24	−0.632660
$784$	3.27156e24	0.818594
$785$	2.35385e24	0.582621
$786$	2.98088e23	0.0729882
$787$	3.20732e24	0.776887	0.388443	−	0.921473i	$-0.373013\pi$
$787$	3.20732e24	0.776887	0.388443	+	0.921473i	$0.373013\pi$
$788$	−1.91224e24	−0.458216
$789$	7.66170e22	0.0181623
$790$	1.28183e24	0.300608
$791$	9.07739e23	0.210601
$792$	0	0
$793$	−3.32444e24	−0.754912
$794$	3.29186e24	0.739549
$795$	1.00817e23	0.0224086
$796$	−8.15008e24	−1.79226
$797$	2.36328e24	0.514186	0.257093	−	0.966387i	$-0.417235\pi$
$797$	2.36328e24	0.514186	0.257093	+	0.966387i	$0.417235\pi$
$798$	−1.08543e22	−0.00233657
$799$	−2.67713e23	−0.0570194
$800$	−2.58309e24	−0.544345
$801$	−3.23136e24	−0.673766
$802$	−1.22521e24	−0.252772
$803$	0	0
$804$	−3.28811e24	−0.664156
$805$	−1.04938e24	−0.209733
$806$	1.18735e25	2.34818
$807$	−4.93676e23	−0.0966088
$808$	1.41112e24	0.273254
$809$	−9.60721e24	−1.84092	−0.920460	−	0.390836i	$-0.872186\pi$
$809$	−9.60721e24	−1.84092	−0.920460	+	0.390836i	$0.872186\pi$
$810$	−5.36439e24	−1.01718
$811$	6.64616e24	1.24708	0.623539	−	0.781792i	$-0.285694\pi$
$811$	6.64616e24	1.24708	0.623539	+	0.781792i	$0.285694\pi$
$812$	1.15956e24	0.215312
$813$	−6.30035e23	−0.115770
$814$	0	0
$815$	−8.21751e24	−1.47877
$816$	3.42447e23	0.0609856
$817$	−1.35392e23	−0.0238620
$818$	−1.25976e25	−2.19727
$819$	−9.08175e23	−0.156768
$820$	7.40440e24	1.26495
$821$	5.09458e24	0.861373	0.430687	−	0.902502i	$-0.358271\pi$
$821$	5.09458e24	0.861373	0.430687	+	0.902502i	$0.358271\pi$
$822$	−4.23275e24	−0.708292
$823$	5.78198e24	0.957586	0.478793	−	0.877928i	$-0.341074\pi$
$823$	5.78198e24	0.957586	0.478793	+	0.877928i	$0.341074\pi$
$824$	−7.97002e23	−0.130641
$825$	0	0
$826$	−5.56903e22	−0.00894230
$827$	−8.38907e24	−1.33327	−0.666633	−	0.745386i	$-0.732265\pi$
$827$	−8.38907e24	−1.33327	−0.666633	+	0.745386i	$0.732265\pi$
$828$	5.18909e24	0.816270
$829$	5.02769e24	0.782808	0.391404	−	0.920219i	$-0.371989\pi$
$829$	5.02769e24	0.782808	0.391404	+	0.920219i	$0.371989\pi$
$830$	−1.61231e25	−2.48477
$831$	−7.65741e23	−0.116808
$832$	7.07088e24	1.06764
$833$	−1.20662e24	−0.180338
$834$	3.58986e24	0.531087
$835$	−8.83747e24	−1.29417
$836$	0	0
$837$	−9.09619e24	−1.30525
$838$	−1.38251e25	−1.96378
$839$	9.38328e23	0.131940	0.0659702	−	0.997822i	$-0.478986\pi$
$839$	9.38328e23	0.131940	0.0659702	+	0.997822i	$0.478986\pi$
$840$	−1.24562e23	−0.0173384
$841$	−1.33566e24	−0.184048
$842$	5.90847e24	0.805977
$843$	1.20186e24	0.162300
$844$	−2.56295e24	−0.342633
$845$	2.25011e24	0.297798
$846$	2.88414e24	0.377892
$847$	0	0
$848$	−3.37728e23	−0.0433713
$849$	1.45603e24	0.185121
$850$	8.29764e23	0.104447
$851$	−4.02442e24	−0.501536
$852$	2.34736e24	0.289630
$853$	−2.33188e24	−0.284865	−0.142433	−	0.989805i	$-0.545492\pi$
$853$	−2.33188e24	−0.284865	−0.142433	+	0.989805i	$0.545492\pi$
$854$	−2.22843e24	−0.269531
$855$	−1.69056e23	−0.0202451
$856$	−1.71808e24	−0.203712
$857$	1.09008e25	1.27974	0.639872	−	0.768482i	$-0.278988\pi$
$857$	1.09008e25	1.27974	0.639872	+	0.768482i	$0.278988\pi$
$858$	0	0
$859$	−1.47838e25	−1.70155	−0.850773	−	0.525533i	$-0.823866\pi$
$859$	−1.47838e25	−1.70155	−0.850773	+	0.525533i	$0.823866\pi$
$860$	−1.37923e25	−1.57181
$861$	−6.75310e23	−0.0762037
$862$	−9.20284e24	−1.02828
$863$	5.86087e24	0.648441	0.324220	−	0.945982i	$-0.394898\pi$
$863$	5.86087e24	0.648441	0.324220	+	0.945982i	$0.394898\pi$
$864$	−9.17333e24	−1.00499
$865$	2.37412e24	0.257553
$866$	−2.38346e25	−2.56039
$867$	3.41760e24	0.363545
$868$	4.21705e24	0.444213
$869$	0	0
$870$	−5.64654e24	−0.583269
$871$	−1.32037e25	−1.35065
$872$	−1.87804e24	−0.190246
$873$	7.06491e24	0.708737
$874$	2.49044e23	0.0247417
$875$	−1.56703e24	−0.154173
$876$	2.15302e24	0.209779
$877$	1.08364e25	1.04565	0.522827	−	0.852439i	$-0.324878\pi$
$877$	1.08364e25	1.04565	0.522827	+	0.852439i	$0.324878\pi$
$878$	−2.03420e25	−1.94398
$879$	−3.27678e24	−0.310128
$880$	0	0
$881$	5.00054e24	0.464217	0.232109	−	0.972690i	$-0.425437\pi$
$881$	5.00054e24	0.464217	0.232109	+	0.972690i	$0.425437\pi$
$882$	1.29991e25	1.19518
$883$	−1.04108e24	−0.0948025	−0.0474012	−	0.998876i	$-0.515094\pi$
$883$	−1.04108e24	−0.0948025	−0.0474012	+	0.998876i	$0.515094\pi$
$884$	−2.03803e24	−0.183809
$885$	1.43686e23	0.0128351
$886$	−2.03262e24	−0.179833
$887$	−1.47651e25	−1.29386	−0.646929	−	0.762550i	$-0.723947\pi$
$887$	−1.47651e25	−1.29386	−0.646929	+	0.762550i	$0.723947\pi$
$888$	−4.77700e23	−0.0414615
$889$	−2.80760e24	−0.241363
$890$	−1.57984e25	−1.34523
$891$	0	0
$892$	1.40092e25	1.17034
$893$	7.33412e22	0.00606891
$894$	−1.07949e24	−0.0884812
$895$	8.10737e24	0.658240
$896$	9.66021e23	0.0776906
$897$	−3.45184e24	−0.274988
$898$	−1.46580e24	−0.115672
$899$	2.15350e25	1.68340
$900$	−4.73640e24	−0.366764
$901$	1.24561e23	0.00955478
$902$	0	0
$903$	1.25791e24	0.0946899
$904$	−2.47221e24	−0.184354
$905$	−1.44366e25	−1.06648
$906$	−8.88866e24	−0.650499
$907$	−2.99747e23	−0.0217317	−0.0108658	−	0.999941i	$-0.503459\pi$
$907$	−2.99747e23	−0.0217317	−0.0108658	+	0.999941i	$0.503459\pi$
$908$	2.71468e25	1.94979
$909$	−1.77937e25	−1.26612
$910$	−4.44014e24	−0.313000
$911$	−1.71431e24	−0.119724	−0.0598621	−	0.998207i	$-0.519066\pi$
$911$	−1.71431e24	−0.119724	−0.0598621	+	0.998207i	$0.519066\pi$
$912$	−9.38148e22	−0.00649106
$913$	0	0
$914$	7.08256e24	0.481003
$915$	5.74958e24	0.386863
$916$	−1.35143e25	−0.900911
$917$	4.25132e23	0.0280792
$918$	2.94675e24	0.192832
$919$	2.42539e25	1.57253	0.786265	−	0.617889i	$-0.212012\pi$
$919$	2.42539e25	1.57253	0.786265	+	0.617889i	$0.212012\pi$
$920$	2.85796e24	0.183594
$921$	5.93213e24	0.377575
$922$	2.28480e25	1.44091
$923$	9.42605e24	0.589000
$924$	0	0
$925$	3.67333e24	0.225349
$926$	2.03661e25	1.23798
$927$	1.00499e25	0.605320
$928$	2.17176e25	1.29615
$929$	1.40138e25	0.828745	0.414373	−	0.910107i	$-0.364001\pi$
$929$	1.40138e25	0.828745	0.414373	+	0.910107i	$0.364001\pi$
$930$	−2.05351e25	−1.20335
$931$	3.30557e23	0.0191944
$932$	−1.45865e25	−0.839298
$933$	1.00560e25	0.573364
$934$	2.62270e25	1.48184
$935$	0	0
$936$	2.47340e24	0.137230
$937$	2.43217e25	1.33724	0.668618	−	0.743606i	$-0.266886\pi$
$937$	2.43217e25	1.33724	0.668618	+	0.743606i	$0.266886\pi$
$938$	−8.85072e24	−0.482230
$939$	−6.78690e24	−0.366449
$940$	7.47120e24	0.399764
$941$	−5.98394e23	−0.0317304	−0.0158652	−	0.999874i	$-0.505050\pi$
$941$	−5.98394e23	−0.0317304	−0.0158652	+	0.999874i	$0.505050\pi$
$942$	5.19012e24	0.272738
$943$	1.54944e25	0.806912
$944$	−4.81335e23	−0.0248419
$945$	3.40155e24	0.173983
$946$	0	0
$947$	3.62477e25	1.82098	0.910491	−	0.413530i	$-0.135704\pi$
$947$	3.62477e25	1.82098	0.910491	+	0.413530i	$0.135704\pi$
$948$	1.49754e24	0.0745602
$949$	8.64565e24	0.426614
$950$	−2.27317e23	−0.0111169
$951$	−3.84865e24	−0.186541
$952$	−1.53897e23	−0.00739291
$953$	1.88157e25	0.895839	0.447919	−	0.894074i	$-0.352165\pi$
$953$	1.88157e25	0.895839	0.447919	+	0.894074i	$0.352165\pi$
$954$	−1.34192e24	−0.0633236
$955$	2.89803e25	1.35542
$956$	−2.79673e25	−1.29646
$957$	0	0
$958$	2.25199e25	1.02556
$959$	−6.03674e24	−0.272486
$960$	−1.22290e25	−0.547124
$961$	5.57675e25	2.47305
$962$	−1.70282e25	−0.748479
$963$	2.16644e25	0.943896
$964$	2.04968e25	0.885180
$965$	5.04467e25	2.15949
$966$	−2.31383e24	−0.0981807
$967$	1.81212e24	0.0762186	0.0381093	−	0.999274i	$-0.487866\pi$
$967$	1.81212e24	0.0762186	0.0381093	+	0.999274i	$0.487866\pi$
$968$	0	0
$969$	3.46008e22	0.00142999
$970$	3.45409e25	1.41506
$971$	2.03685e25	0.827170	0.413585	−	0.910465i	$-0.364276\pi$
$971$	2.03685e25	0.827170	0.413585	+	0.910465i	$0.364276\pi$
$972$	−2.58739e25	−1.04160
$973$	5.11985e24	0.204314
$974$	−4.90050e25	−1.93861
$975$	3.15070e24	0.123557
$976$	−1.92605e25	−0.748763
$977$	−2.52590e25	−0.973449	−0.486724	−	0.873556i	$-0.661808\pi$
$977$	−2.52590e25	−0.973449	−0.486724	+	0.873556i	$0.661808\pi$
$978$	−1.81192e25	−0.692244
$979$	0	0
$980$	3.36736e25	1.26435
$981$	2.36815e25	0.881499
$982$	−6.85750e23	−0.0253057
$983$	−1.51840e24	−0.0555495	−0.0277747	−	0.999614i	$-0.508842\pi$
$983$	−1.51840e24	−0.0555495	−0.0277747	+	0.999614i	$0.508842\pi$
$984$	1.83919e24	0.0667066
$985$	1.32804e25	0.477533
$986$	−6.97635e24	−0.248699
$987$	−6.81402e23	−0.0240828
$988$	5.58327e23	0.0195639
$989$	−2.88617e25	−1.00266
$990$	0	0
$991$	−6.69650e24	−0.228677	−0.114338	−	0.993442i	$-0.536475\pi$
$991$	−6.69650e24	−0.228677	−0.114338	+	0.993442i	$0.536475\pi$
$992$	7.89818e25	2.67410
$993$	−1.71561e25	−0.575903
$994$	6.31846e24	0.210294
$995$	5.66018e25	1.86781
$996$	−1.88363e25	−0.616300
$997$	3.45326e25	1.12026	0.560132	−	0.828404i	$-0.310751\pi$
$997$	3.45326e25	1.12026	0.560132	+	0.828404i	$0.310751\pi$
$998$	−4.04864e25	−1.30227
$999$	1.30451e25	0.416046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.18.a.b.1.1		1
11.10	odd	2		1.18.a.a.1.1	✓	1
33.32	even	2		9.18.a.b.1.1		1
44.43	even	2		16.18.a.b.1.1		1
55.32	even	4		25.18.b.a.24.1		2
55.43	even	4		25.18.b.a.24.2		2
55.54	odd	2		25.18.a.a.1.1		1
77.76	even	2		49.18.a.a.1.1		1
88.21	odd	2		64.18.a.d.1.1		1
88.43	even	2		64.18.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.18.a.a.1.1	✓	1	11.10	odd	2
9.18.a.b.1.1		1	33.32	even	2
16.18.a.b.1.1		1	44.43	even	2
25.18.a.a.1.1		1	55.54	odd	2
25.18.b.a.24.1		2	55.32	even	4
25.18.b.a.24.2		2	55.43	even	4
49.18.a.a.1.1		1	77.76	even	2
64.18.a.b.1.1		1	88.43	even	2
64.18.a.d.1.1		1	88.21	odd	2
121.18.a.b.1.1		1	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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