LMFDB - Embedded newform 16.18.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,18,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	16.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:	$29.3155339751$
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	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4284.00 q^{3} -1.02585e6 q^{5} -3.22599e6 q^{7} -1.10788e8 q^{9} +O(q^{10})$

$q+4284.00 q^{3} -1.02585e6 q^{5} -3.22599e6 q^{7} -1.10788e8 q^{9} +7.53618e8 q^{11} +2.54106e9 q^{13} -4.39474e9 q^{15} -5.42974e9 q^{17} -1.48750e9 q^{19} -1.38201e10 q^{21} +3.17092e11 q^{23} +2.89429e11 q^{25} -1.02785e12 q^{27} +2.43341e12 q^{29} +8.84972e12 q^{31} +3.22850e12 q^{33} +3.30938e12 q^{35} +1.26917e13 q^{37} +1.08859e13 q^{39} +4.88642e13 q^{41} +9.10200e13 q^{43} +1.13651e14 q^{45} +4.93050e13 q^{47} -2.22223e14 q^{49} -2.32610e13 q^{51} +2.29405e13 q^{53} -7.73099e14 q^{55} -6.37245e12 q^{57} -3.26951e13 q^{59} -1.30829e15 q^{61} +3.57400e14 q^{63} -2.60675e15 q^{65} -5.19614e15 q^{67} +1.35842e15 q^{69} +3.70949e15 q^{71} +3.40237e15 q^{73} +1.23991e15 q^{75} -2.43117e15 q^{77} -2.36653e15 q^{79} +9.90381e15 q^{81} +2.97668e16 q^{83} +5.57010e15 q^{85} +1.04247e16 q^{87} +2.91672e16 q^{89} -8.19745e15 q^{91} +3.79122e16 q^{93} +1.52595e15 q^{95} -6.37699e16 q^{97} -8.34915e16 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	4284.00	0.376980	0.188490	−	0.982075i	$-0.439641\pi$
$3$	4284.00	0.376980	0.188490	+	0.982075i	$0.439641\pi$
$4$	0	0
$5$	−1.02585e6	−1.17446	−0.587231	−	0.809420i	$-0.699782\pi$
$5$	−1.02585e6	−1.17446	−0.587231	+	0.809420i	$0.699782\pi$
$6$	0	0
$7$	−3.22599e6	−0.211510	−0.105755	−	0.994392i	$-0.533726\pi$
$7$	−3.22599e6	−0.211510	−0.105755	+	0.994392i	$0.533726\pi$
$8$	0	0
$9$	−1.10788e8	−0.857886
$10$	0	0
$11$	7.53618e8	1.06002	0.530009	−	0.847992i	$-0.322188\pi$
$11$	7.53618e8	1.06002	0.530009	+	0.847992i	$0.322188\pi$
$12$	0	0
$13$	2.54106e9	0.863967	0.431984	−	0.901881i	$-0.357814\pi$
$13$	2.54106e9	0.863967	0.431984	+	0.901881i	$0.357814\pi$
$14$	0	0
$15$	−4.39474e9	−0.442749
$16$	0	0
$17$	−5.42974e9	−0.188783	−0.0943916	−	0.995535i	$-0.530091\pi$
$17$	−5.42974e9	−0.188783	−0.0943916	+	0.995535i	$0.530091\pi$
$18$	0	0
$19$	−1.48750e9	−0.0200933	−0.0100467	−	0.999950i	$-0.503198\pi$
$19$	−1.48750e9	−0.0200933	−0.0100467	+	0.999950i	$0.503198\pi$
$20$	0	0
$21$	−1.38201e10	−0.0797350
$22$	0	0
$23$	3.17092e11	0.844303	0.422152	−	0.906525i	$-0.361275\pi$
$23$	3.17092e11	0.844303	0.422152	+	0.906525i	$0.361275\pi$
$24$	0	0
$25$	2.89429e11	0.379360
$26$	0	0
$27$	−1.02785e12	−0.700387
$28$	0	0
$29$	2.43341e12	0.903301	0.451650	−	0.892195i	$-0.350835\pi$
$29$	2.43341e12	0.903301	0.451650	+	0.892195i	$0.350835\pi$
$30$	0	0
$31$	8.84972e12	1.86361	0.931805	−	0.362958i	$-0.118233\pi$
$31$	8.84972e12	1.86361	0.931805	+	0.362958i	$0.118233\pi$
$32$	0	0
$33$	3.22850e12	0.399606
$34$	0	0
$35$	3.30938e12	0.248410
$36$	0	0
$37$	1.26917e13	0.594023	0.297012	−	0.954874i	$-0.404010\pi$
$37$	1.26917e13	0.594023	0.297012	+	0.954874i	$0.404010\pi$
$38$	0	0
$39$	1.08859e13	0.325699
$40$	0	0
$41$	4.88642e13	0.955713	0.477857	−	0.878438i	$-0.341414\pi$
$41$	4.88642e13	0.955713	0.477857	+	0.878438i	$0.341414\pi$
$42$	0	0
$43$	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
$46$	0	0
$47$	4.93050e13	0.302036	0.151018	−	0.988531i	$-0.451745\pi$
$47$	4.93050e13	0.302036	0.151018	+	0.988531i	$0.451745\pi$
$48$	0	0
$49$	−2.22223e14	−0.955264
$50$	0	0
$51$	−2.32610e13	−0.0711676
$52$	0	0
$53$	2.29405e13	0.0506124	0.0253062	−	0.999680i	$-0.491944\pi$
$53$	2.29405e13	0.0506124	0.0253062	+	0.999680i	$0.491944\pi$
$54$	0	0
$55$	−7.73099e14	−1.24495
$56$	0	0
$57$	−6.37245e12	−0.00757478
$58$	0	0
$59$	−3.26951e13	−0.0289895	−0.0144947	−	0.999895i	$-0.504614\pi$
$59$	−3.26951e13	−0.0289895	−0.0144947	+	0.999895i	$0.504614\pi$
$60$	0	0
$61$	−1.30829e15	−0.873774	−0.436887	−	0.899516i	$-0.643919\pi$
$61$	−1.30829e15	−0.873774	−0.436887	+	0.899516i	$0.643919\pi$
$62$	0	0
$63$	3.57400e14	0.181451
$64$	0	0
$65$	−2.60675e15	−1.01470
$66$	0	0
$67$	−5.19614e15	−1.56331	−0.781655	−	0.623711i	$-0.785624\pi$
$67$	−5.19614e15	−1.56331	−0.781655	+	0.623711i	$0.785624\pi$
$68$	0	0
$69$	1.35842e15	0.318286
$70$	0	0
$71$	3.70949e15	0.681739	0.340870	−	0.940111i	$-0.389279\pi$
$71$	3.70949e15	0.681739	0.340870	+	0.940111i	$0.389279\pi$
$72$	0	0
$73$	3.40237e15	0.493785	0.246892	−	0.969043i	$-0.420591\pi$
$73$	3.40237e15	0.493785	0.246892	+	0.969043i	$0.420591\pi$
$74$	0	0
$75$	1.23991e15	0.143011
$76$	0	0
$77$	−2.43117e15	−0.224204
$78$	0	0
$79$	−2.36653e15	−0.175502	−0.0877511	−	0.996142i	$-0.527968\pi$
$79$	−2.36653e15	−0.175502	−0.0877511	+	0.996142i	$0.527968\pi$
$80$	0	0
$81$	9.90381e15	0.593854
$82$	0	0
$83$	2.97668e16	1.45067	0.725333	−	0.688398i	$-0.241686\pi$
$83$	2.97668e16	1.45067	0.725333	+	0.688398i	$0.241686\pi$
$84$	0	0
$85$	5.57010e15	0.221719
$86$	0	0
$87$	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	$0.371545\pi$
$90$	0	0
$91$	−8.19745e15	−0.182737
$92$	0	0
$93$	3.79122e16	0.702545
$94$	0	0
$95$	1.52595e15	0.0235988
$96$	0	0
$97$	−6.37699e16	−0.826144	−0.413072	−	0.910698i	$-0.635544\pi$
$97$	−6.37699e16	−0.826144	−0.413072	+	0.910698i	$0.635544\pi$
$98$	0	0
$99$	−8.34915e16	−0.909375
$100$	0	0
$101$	−1.60611e17	−1.47586	−0.737929	−	0.674878i	$-0.764196\pi$
$101$	−1.60611e17	−1.47586	−0.737929	+	0.674878i	$0.764196\pi$
$102$	0	0
$103$	9.07136e16	0.705596	0.352798	−	0.935700i	$-0.385230\pi$
$103$	9.07136e16	0.705596	0.352798	+	0.935700i	$0.385230\pi$
$104$	0	0
$105$	1.41774e16	0.0936456
$106$	0	0
$107$	−1.95549e17	−1.10026	−0.550129	−	0.835080i	$-0.685421\pi$
$107$	−1.95549e17	−1.10026	−0.550129	+	0.835080i	$0.685421\pi$
$108$	0	0
$109$	2.13756e17	1.02753	0.513763	−	0.857932i	$-0.328251\pi$
$109$	2.13756e17	1.02753	0.513763	+	0.857932i	$0.328251\pi$
$110$	0	0
$111$	5.43710e16	0.223935
$112$	0	0
$113$	−2.81383e17	−0.995706	−0.497853	−	0.867262i	$-0.665878\pi$
$113$	−2.81383e17	−0.995706	−0.497853	+	0.867262i	$0.665878\pi$
$114$	0	0
$115$	−3.25289e17	−0.991602
$116$	0	0
$117$	−2.81518e17	−0.741185
$118$	0	0
$119$	1.75163e16	0.0399295
$120$	0	0
$121$	6.24934e16	0.123640
$122$	0	0
$123$	2.09334e17	0.360285
$124$	0	0
$125$	4.85751e17	0.728918
$126$	0	0
$127$	8.70305e17	1.14114	0.570571	−	0.821248i	$-0.306722\pi$
$127$	8.70305e17	1.14114	0.570571	+	0.821248i	$0.306722\pi$
$128$	0	0
$129$	3.89930e17	0.447686
$130$	0	0
$131$	−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
$134$	0	0
$135$	1.05442e18	0.822577
$136$	0	0
$137$	1.87128e18	1.28829	0.644147	−	0.764902i	$-0.277213\pi$
$137$	1.87128e18	1.28829	0.644147	+	0.764902i	$0.277213\pi$
$138$	0	0
$139$	−1.58706e18	−0.965980	−0.482990	−	0.875626i	$-0.660449\pi$
$139$	−1.58706e18	−0.965980	−0.482990	+	0.875626i	$0.660449\pi$
$140$	0	0
$141$	2.11223e17	0.113862
$142$	0	0
$143$	1.91499e18	0.915821
$144$	0	0
$145$	−2.49631e18	−1.06089
$146$	0	0
$147$	−9.52005e17	−0.360116
$148$	0	0
$149$	−4.77240e17	−0.160936	−0.0804680	−	0.996757i	$-0.525641\pi$
$149$	−4.77240e17	−0.160936	−0.0804680	+	0.996757i	$0.525641\pi$
$150$	0	0
$151$	3.92964e18	1.18317	0.591587	−	0.806241i	$-0.298501\pi$
$151$	3.92964e18	1.18317	0.591587	+	0.806241i	$0.298501\pi$
$152$	0	0
$153$	6.01548e17	0.161954
$154$	0	0
$155$	−9.07849e18	−2.18874
$156$	0	0
$157$	−2.29453e18	−0.496075	−0.248038	−	0.968750i	$-0.579786\pi$
$157$	−2.29453e18	−0.496075	−0.248038	+	0.968750i	$0.579786\pi$
$158$	0	0
$159$	9.82769e16	0.0190799
$160$	0	0
$161$	−1.02294e18	−0.178578
$162$	0	0
$163$	−8.01044e18	−1.25910	−0.629552	−	0.776958i	$-0.716762\pi$
$163$	−8.01044e18	−1.25910	−0.629552	+	0.776958i	$0.716762\pi$
$164$	0	0
$165$	−3.31196e18	−0.469322
$166$	0	0
$167$	8.61477e18	1.10193	0.550965	−	0.834528i	$-0.314260\pi$
$167$	8.61477e18	1.10193	0.550965	+	0.834528i	$0.314260\pi$
$168$	0	0
$169$	−2.19341e18	−0.253561
$170$	0	0
$171$	1.64796e17	0.0172378
$172$	0	0
$173$	2.31430e18	0.219294	0.109647	−	0.993971i	$-0.465028\pi$
$173$	2.31430e18	0.219294	0.109647	+	0.993971i	$0.465028\pi$
$174$	0	0
$175$	−9.33695e17	−0.0802383
$176$	0	0
$177$	−1.40066e17	−0.0109285
$178$	0	0
$179$	7.90307e18	0.560461	0.280230	−	0.959933i	$-0.409589\pi$
$179$	7.90307e18	0.560461	0.280230	+	0.959933i	$0.409589\pi$
$180$	0	0
$181$	1.40729e19	0.908060	0.454030	−	0.890986i	$-0.349986\pi$
$181$	1.40729e19	0.908060	0.454030	+	0.890986i	$0.349986\pi$
$182$	0	0
$183$	−5.60470e18	−0.329396
$184$	0	0
$185$	−1.30197e19	−0.697658
$186$	0	0
$187$	−4.09195e18	−0.200114
$188$	0	0
$189$	3.31584e18	0.148138
$190$	0	0
$191$	2.82501e19	1.15408	0.577040	−	0.816716i	$-0.304208\pi$
$191$	2.82501e19	1.15408	0.577040	+	0.816716i	$0.304208\pi$
$192$	0	0
$193$	4.91755e19	1.83870	0.919351	−	0.393437i	$-0.128714\pi$
$193$	4.91755e19	1.83870	0.919351	+	0.393437i	$0.128714\pi$
$194$	0	0
$195$	−1.11673e19	−0.382521
$196$	0	0
$197$	1.29458e19	0.406598	0.203299	−	0.979117i	$-0.434834\pi$
$197$	1.29458e19	0.406598	0.203299	+	0.979117i	$0.434834\pi$
$198$	0	0
$199$	5.51755e19	1.59036	0.795179	−	0.606375i	$-0.207377\pi$
$199$	5.51755e19	1.59036	0.795179	+	0.606375i	$0.207377\pi$
$200$	0	0
$201$	−2.22603e19	−0.589338
$202$	0	0
$203$	−7.85016e18	−0.191057
$204$	0	0
$205$	−5.01273e19	−1.12245
$206$	0	0
$207$	−3.51298e19	−0.724316
$208$	0	0
$209$	−1.12101e18	−0.0212993
$210$	0	0
$211$	−1.73510e19	−0.304035	−0.152017	−	0.988378i	$-0.548577\pi$
$211$	−1.73510e19	−0.304035	−0.152017	+	0.988378i	$0.548577\pi$
$212$	0	0
$213$	1.58915e19	0.257002
$214$	0	0
$215$	−9.33728e19	−1.39474
$216$	0	0
$217$	−2.85491e19	−0.394171
$218$	0	0
$219$	1.45758e19	0.186147
$220$	0	0
$221$	−1.37973e19	−0.163102
$222$	0	0
$223$	−9.48415e19	−1.03850	−0.519251	−	0.854622i	$-0.673789\pi$
$223$	−9.48415e19	−1.03850	−0.519251	+	0.854622i	$0.673789\pi$
$224$	0	0
$225$	−3.20651e19	−0.325448
$226$	0	0
$227$	1.83782e20	1.73015	0.865073	−	0.501647i	$-0.167272\pi$
$227$	1.83782e20	1.73015	0.865073	+	0.501647i	$0.167272\pi$
$228$	0	0
$229$	−9.14908e19	−0.799422	−0.399711	−	0.916641i	$-0.630889\pi$
$229$	−9.14908e19	−0.799422	−0.399711	+	0.916641i	$0.630889\pi$
$230$	0	0
$231$	−1.04151e19	−0.0845205
$232$	0	0
$233$	9.87497e19	0.744750	0.372375	−	0.928082i	$-0.378544\pi$
$233$	9.87497e19	0.744750	0.372375	+	0.928082i	$0.378544\pi$
$234$	0	0
$235$	−5.05795e19	−0.354730
$236$	0	0
$237$	−1.01382e19	−0.0661609
$238$	0	0
$239$	−1.89337e20	−1.15041	−0.575206	−	0.818009i	$-0.695078\pi$
$239$	−1.89337e20	−1.15041	−0.575206	+	0.818009i	$0.695078\pi$
$240$	0	0
$241$	−1.38762e20	−0.785463	−0.392731	−	0.919653i	$-0.628470\pi$
$241$	−1.38762e20	−0.785463	−0.392731	+	0.919653i	$0.628470\pi$
$242$	0	0
$243$	1.75165e20	0.924258
$244$	0	0
$245$	2.27968e20	1.12192
$246$	0	0
$247$	−3.77983e18	−0.0173600
$248$	0	0
$249$	1.27521e20	0.546873
$250$	0	0
$251$	3.34508e20	1.34023	0.670115	−	0.742257i	$-0.266245\pi$
$251$	3.34508e20	1.34023	0.670115	+	0.742257i	$0.266245\pi$
$252$	0	0
$253$	2.38966e20	0.894977
$254$	0	0
$255$	2.38623e19	0.0835836
$256$	0	0
$257$	4.91382e19	0.161060	0.0805300	−	0.996752i	$-0.474339\pi$
$257$	4.91382e19	0.161060	0.0805300	+	0.996752i	$0.474339\pi$
$258$	0	0
$259$	−4.09432e19	−0.125642
$260$	0	0
$261$	−2.69591e20	−0.774929
$262$	0	0
$263$	−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
$266$	0	0
$267$	1.24952e20	0.296073
$268$	0	0
$269$	1.15237e20	0.256270	0.128135	−	0.991757i	$-0.459101\pi$
$269$	1.15237e20	0.256270	0.128135	+	0.991757i	$0.459101\pi$
$270$	0	0
$271$	1.47067e20	0.307098	0.153549	−	0.988141i	$-0.450930\pi$
$271$	1.47067e20	0.307098	0.153549	+	0.988141i	$0.450930\pi$
$272$	0	0
$273$	−3.51179e19	−0.0688884
$274$	0	0
$275$	2.18119e20	0.402129
$276$	0	0
$277$	−1.78744e20	−0.309852	−0.154926	−	0.987926i	$-0.549514\pi$
$277$	−1.78744e20	−0.309852	−0.154926	+	0.987926i	$0.549514\pi$
$278$	0	0
$279$	−9.80439e20	−1.59877
$280$	0	0
$281$	2.80546e20	0.430527	0.215263	−	0.976556i	$-0.430939\pi$
$281$	2.80546e20	0.430527	0.215263	+	0.976556i	$0.430939\pi$
$282$	0	0
$283$	−3.39877e20	−0.491062	−0.245531	−	0.969389i	$-0.578962\pi$
$283$	−3.39877e20	−0.491062	−0.245531	+	0.969389i	$0.578962\pi$
$284$	0	0
$285$	6.53718e18	0.00889629
$286$	0	0
$287$	−1.57635e20	−0.202142
$288$	0	0
$289$	−7.97758e20	−0.964361
$290$	0	0
$291$	−2.73190e20	−0.311440
$292$	0	0
$293$	−7.64887e20	−0.822664	−0.411332	−	0.911486i	$-0.634936\pi$
$293$	−7.64887e20	−0.822664	−0.411332	+	0.911486i	$0.634936\pi$
$294$	0	0
$295$	3.35403e19	0.0340470
$296$	0	0
$297$	−7.74607e20	−0.742423
$298$	0	0
$299$	8.05751e20	0.729450
$300$	0	0
$301$	−2.93630e20	−0.251180
$302$	0	0
$303$	−6.88059e20	−0.556370
$304$	0	0
$305$	1.34211e21	1.02621
$306$	0	0
$307$	−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
$310$	0	0
$311$	2.34733e21	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$311$	2.34733e21	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$312$	0	0
$313$	1.58424e21	0.972065	0.486033	−	0.873941i	$-0.338444\pi$
$313$	1.58424e21	0.972065	0.486033	+	0.873941i	$0.338444\pi$
$314$	0	0
$315$	−3.66638e20	−0.213107
$316$	0	0
$317$	8.98378e20	0.494829	0.247415	−	0.968910i	$-0.420419\pi$
$317$	8.98378e20	0.494829	0.247415	+	0.968910i	$0.420419\pi$
$318$	0	0
$319$	1.83386e21	0.957516
$320$	0	0
$321$	−8.37734e20	−0.414776
$322$	0	0
$323$	8.07674e18	0.00379328
$324$	0	0
$325$	7.35457e20	0.327755
$326$	0	0
$327$	9.15730e20	0.387357
$328$	0	0
$329$	−1.59058e20	−0.0638836
$330$	0	0
$331$	−4.00469e21	−1.52767	−0.763837	−	0.645409i	$-0.776687\pi$
$331$	−4.00469e21	−1.52767	−0.763837	+	0.645409i	$0.776687\pi$
$332$	0	0
$333$	−1.40608e21	−0.509604
$334$	0	0
$335$	5.33046e21	1.83605
$336$	0	0
$337$	−1.06753e21	−0.349564	−0.174782	−	0.984607i	$-0.555922\pi$
$337$	−1.06753e21	−0.349564	−0.174782	+	0.984607i	$0.555922\pi$
$338$	0	0
$339$	−1.20544e21	−0.375362
$340$	0	0
$341$	6.66931e21	1.97546
$342$	0	0
$343$	1.46736e21	0.413557
$344$	0	0
$345$	−1.39354e21	−0.373814
$346$	0	0
$347$	1.61841e21	0.413321	0.206660	−	0.978413i	$-0.433740\pi$
$347$	1.61841e21	0.413321	0.206660	+	0.978413i	$0.433740\pi$
$348$	0	0
$349$	−5.60078e21	−1.36217	−0.681086	−	0.732203i	$-0.738492\pi$
$349$	−5.60078e21	−1.36217	−0.681086	+	0.732203i	$0.738492\pi$
$350$	0	0
$351$	−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	$0.309344\pi$
$354$	0	0
$355$	−3.80538e21	−0.800676
$356$	0	0
$357$	7.50399e19	0.0150526
$358$	0	0
$359$	−5.84965e21	−1.11899	−0.559496	−	0.828833i	$-0.689005\pi$
$359$	−5.84965e21	−1.11899	−0.559496	+	0.828833i	$0.689005\pi$
$360$	0	0
$361$	−5.47817e21	−0.999596
$362$	0	0
$363$	2.67722e20	0.0466098
$364$	0	0
$365$	−3.49032e21	−0.579931
$366$	0	0
$367$	−4.29535e20	−0.0681297	−0.0340649	−	0.999420i	$-0.510845\pi$
$367$	−4.29535e20	−0.0681297	−0.0340649	+	0.999420i	$0.510845\pi$
$368$	0	0
$369$	−5.41354e21	−0.819893
$370$	0	0
$371$	−7.40057e19	−0.0107050
$372$	0	0
$373$	−6.54734e21	−0.904774	−0.452387	−	0.891822i	$-0.649427\pi$
$373$	−6.54734e21	−0.904774	−0.452387	+	0.891822i	$0.649427\pi$
$374$	0	0
$375$	2.08096e21	0.274788
$376$	0	0
$377$	6.18345e21	0.780422
$378$	0	0
$379$	−1.51850e21	−0.183223	−0.0916117	−	0.995795i	$-0.529202\pi$
$379$	−1.51850e21	−0.183223	−0.0916117	+	0.995795i	$0.529202\pi$
$380$	0	0
$381$	3.72839e21	0.430188
$382$	0	0
$383$	8.04597e21	0.887951	0.443975	−	0.896039i	$-0.353568\pi$
$383$	8.04597e21	0.887951	0.443975	+	0.896039i	$0.353568\pi$
$384$	0	0
$385$	2.49401e21	0.263319
$386$	0	0
$387$	−1.00839e22	−1.01879
$388$	0	0
$389$	6.84040e21	0.661470	0.330735	−	0.943724i	$-0.392703\pi$
$389$	6.84040e21	0.661470	0.330735	+	0.943724i	$0.392703\pi$
$390$	0	0
$391$	−1.72173e21	−0.159390
$392$	0	0
$393$	−5.64560e20	−0.0500465
$394$	0	0
$395$	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	$0.418403\pi$
$398$	0	0
$399$	2.05575e19	0.00160214
$400$	0	0
$401$	−2.32048e21	−0.173320	−0.0866602	−	0.996238i	$-0.527619\pi$
$401$	−2.32048e21	−0.173320	−0.0866602	+	0.996238i	$0.527619\pi$
$402$	0	0
$403$	2.24877e22	1.61010
$404$	0	0
$405$	−1.01598e22	−0.697458
$406$	0	0
$407$	9.56466e21	0.629676
$408$	0	0
$409$	2.38590e22	1.50662	0.753312	−	0.657663i	$-0.228455\pi$
$409$	2.38590e22	1.50662	0.753312	+	0.657663i	$0.228455\pi$
$410$	0	0
$411$	8.01658e21	0.485661
$412$	0	0
$413$	1.05474e20	0.00613155
$414$	0	0
$415$	−3.05362e22	−1.70375
$416$	0	0
$417$	−6.79898e21	−0.364156
$418$	0	0
$419$	2.61839e22	1.34653	0.673263	−	0.739403i	$-0.264892\pi$
$419$	2.61839e22	1.34653	0.673263	+	0.739403i	$0.264892\pi$
$420$	0	0
$421$	1.11903e22	0.552642	0.276321	−	0.961065i	$-0.410885\pi$
$421$	1.11903e22	0.552642	0.276321	+	0.961065i	$0.410885\pi$
$422$	0	0
$423$	−5.46238e21	−0.259113
$424$	0	0
$425$	−1.57152e21	−0.0716168
$426$	0	0
$427$	4.22052e21	0.184812
$428$	0	0
$429$	8.20383e21	0.345247
$430$	0	0
$431$	−1.74296e22	−0.705068	−0.352534	−	0.935799i	$-0.614680\pi$
$431$	−1.74296e22	−0.705068	−0.352534	+	0.935799i	$0.614680\pi$
$432$	0	0
$433$	−4.51413e22	−1.75560	−0.877802	−	0.479024i	$-0.840991\pi$
$433$	−4.51413e22	−1.75560	−0.877802	+	0.479024i	$0.840991\pi$
$434$	0	0
$435$	−1.06942e22	−0.399936
$436$	0	0
$437$	−4.71674e20	−0.0169648
$438$	0	0
$439$	−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
$442$	0	0
$443$	3.84966e21	0.123308	0.0616539	−	0.998098i	$-0.480363\pi$
$443$	3.84966e21	0.123308	0.0616539	+	0.998098i	$0.480363\pi$
$444$	0	0
$445$	−2.99212e22	−0.922398
$446$	0	0
$447$	−2.04449e21	−0.0606697
$448$	0	0
$449$	−2.77614e21	−0.0793136	−0.0396568	−	0.999213i	$-0.512626\pi$
$449$	−2.77614e21	−0.0793136	−0.0396568	+	0.999213i	$0.512626\pi$
$450$	0	0
$451$	3.68249e22	1.01307
$452$	0	0
$453$	1.68346e22	0.446034
$454$	0	0
$455$	8.40936e21	0.214618
$456$	0	0
$457$	−1.34139e22	−0.329814	−0.164907	−	0.986309i	$-0.552732\pi$
$457$	−1.34139e22	−0.329814	−0.164907	+	0.986309i	$0.552732\pi$
$458$	0	0
$459$	5.58096e21	0.132221
$460$	0	0
$461$	−4.32728e22	−0.988000	−0.494000	−	0.869462i	$-0.664466\pi$
$461$	−4.32728e22	−0.988000	−0.494000	+	0.869462i	$0.664466\pi$
$462$	0	0
$463$	−3.85721e22	−0.848859	−0.424429	−	0.905461i	$-0.639525\pi$
$463$	−3.85721e22	−0.848859	−0.424429	+	0.905461i	$0.639525\pi$
$464$	0	0
$465$	−3.88922e22	−0.825112
$466$	0	0
$467$	−4.96724e22	−1.01607	−0.508033	−	0.861338i	$-0.669627\pi$
$467$	−4.96724e22	−1.01607	−0.508033	+	0.861338i	$0.669627\pi$
$468$	0	0
$469$	1.67627e22	0.330655
$470$	0	0
$471$	−9.82978e21	−0.187011
$472$	0	0
$473$	6.85943e22	1.25883
$474$	0	0
$475$	−4.30525e20	−0.00762260
$476$	0	0
$477$	−2.54152e21	−0.0434197
$478$	0	0
$479$	4.26513e22	0.703202	0.351601	−	0.936150i	$-0.385637\pi$
$479$	4.26513e22	0.703202	0.351601	+	0.936150i	$0.385637\pi$
$480$	0	0
$481$	3.22503e22	0.513217
$482$	0	0
$483$	−4.38226e21	−0.0673205
$484$	0	0
$485$	6.54183e22	0.970275
$486$	0	0
$487$	9.28126e22	1.32926	0.664631	−	0.747172i	$-0.268589\pi$
$487$	9.28126e22	1.32926	0.664631	+	0.747172i	$0.268589\pi$
$488$	0	0
$489$	−3.43167e22	−0.474658
$490$	0	0
$491$	−1.29877e21	−0.0173516	−0.00867579	−	0.999962i	$-0.502762\pi$
$491$	−1.29877e21	−0.0173516	−0.00867579	+	0.999962i	$0.502762\pi$
$492$	0	0
$493$	−1.32128e22	−0.170528
$494$	0	0
$495$	8.56497e22	1.06803
$496$	0	0
$497$	−1.19668e22	−0.144194
$498$	0	0
$499$	7.66788e22	0.892937	0.446468	−	0.894799i	$-0.352682\pi$
$499$	7.66788e22	0.892937	0.446468	+	0.894799i	$0.352682\pi$
$500$	0	0
$501$	3.69057e22	0.415406
$502$	0	0
$503$	−6.67712e22	−0.726543	−0.363272	−	0.931683i	$-0.618340\pi$
$503$	−6.67712e22	−0.726543	−0.363272	+	0.931683i	$0.618340\pi$
$504$	0	0
$505$	1.64763e23	1.73334
$506$	0	0
$507$	−9.39656e21	−0.0955875
$508$	0	0
$509$	−1.15659e23	−1.13783	−0.568916	−	0.822395i	$-0.692637\pi$
$509$	−1.15659e23	−1.13783	−0.568916	+	0.822395i	$0.692637\pi$
$510$	0	0
$511$	−1.09760e22	−0.104440
$512$	0	0
$513$	1.52893e21	0.0140731
$514$	0	0
$515$	−9.30585e22	−0.828695
$516$	0	0
$517$	3.71571e22	0.320164
$518$	0	0
$519$	9.91446e21	0.0826697
$520$	0	0
$521$	−1.79855e23	−1.45145	−0.725725	−	0.687985i	$-0.758496\pi$
$521$	−1.79855e23	−1.45145	−0.725725	+	0.687985i	$0.758496\pi$
$522$	0	0
$523$	−9.37211e22	−0.732104	−0.366052	−	0.930594i	$-0.619291\pi$
$523$	−9.37211e22	−0.732104	−0.366052	+	0.930594i	$0.619291\pi$
$524$	0	0
$525$	−3.99995e21	−0.0302483
$526$	0	0
$527$	−4.80517e22	−0.351818
$528$	0	0
$529$	−4.05028e22	−0.287152
$530$	0	0
$531$	3.62221e21	0.0248697
$532$	0	0
$533$	1.24167e23	0.825705
$534$	0	0
$535$	2.00604e23	1.29221
$536$	0	0
$537$	3.38568e22	0.211283
$538$	0	0
$539$	−1.67472e23	−1.01260
$540$	0	0
$541$	−1.52396e23	−0.892889	−0.446444	−	0.894811i	$-0.647310\pi$
$541$	−1.52396e23	−0.892889	−0.446444	+	0.894811i	$0.647310\pi$
$542$	0	0
$543$	6.02881e22	0.342321
$544$	0	0
$545$	−2.19281e23	−1.20679
$546$	0	0
$547$	−2.04979e23	−1.09349	−0.546747	−	0.837298i	$-0.684134\pi$
$547$	−2.04979e23	−1.09349	−0.546747	+	0.837298i	$0.684134\pi$
$548$	0	0
$549$	1.44942e23	0.749598
$550$	0	0
$551$	−3.61970e21	−0.0181503
$552$	0	0
$553$	7.63442e21	0.0371204
$554$	0	0
$555$	−5.57765e22	−0.263003
$556$	0	0
$557$	−4.07359e23	−1.86298	−0.931490	−	0.363767i	$-0.881491\pi$
$557$	−4.07359e23	−1.86298	−0.931490	+	0.363767i	$0.881491\pi$
$558$	0	0
$559$	2.31288e23	1.02601
$560$	0	0
$561$	−1.75299e22	−0.0754390
$562$	0	0
$563$	−2.21297e23	−0.923962	−0.461981	−	0.886890i	$-0.652861\pi$
$563$	−2.21297e23	−0.923962	−0.461981	+	0.886890i	$0.652861\pi$
$564$	0	0
$565$	2.88657e23	1.16942
$566$	0	0
$567$	−3.19496e22	−0.125606
$568$	0	0
$569$	4.88686e23	1.86456	0.932278	−	0.361743i	$-0.117818\pi$
$569$	4.88686e23	1.86456	0.932278	+	0.361743i	$0.117818\pi$
$570$	0	0
$571$	−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
$574$	0	0
$575$	9.17755e22	0.320295
$576$	0	0
$577$	−1.95911e23	−0.663840	−0.331920	−	0.943308i	$-0.607696\pi$
$577$	−1.95911e23	−0.663840	−0.331920	+	0.943308i	$0.607696\pi$
$578$	0	0
$579$	2.10668e23	0.693155
$580$	0	0
$581$	−9.60273e22	−0.306830
$582$	0	0
$583$	1.72883e22	0.0536501
$584$	0	0
$585$	2.88795e23	0.870493
$586$	0	0
$587$	4.80676e23	1.40744	0.703719	−	0.710479i	$-0.251522\pi$
$587$	4.80676e23	1.40744	0.703719	+	0.710479i	$0.251522\pi$
$588$	0	0
$589$	−1.31640e22	−0.0374461
$590$	0	0
$591$	5.54596e22	0.153279
$592$	0	0
$593$	−6.40804e23	−1.72092	−0.860459	−	0.509519i	$-0.829823\pi$
$593$	−6.40804e23	−1.72092	−0.860459	+	0.509519i	$0.829823\pi$
$594$	0	0
$595$	−1.79691e22	−0.0468956
$596$	0	0
$597$	2.36372e23	0.599534
$598$	0	0
$599$	−3.60950e23	−0.889854	−0.444927	−	0.895567i	$-0.646770\pi$
$599$	−3.60950e23	−0.889854	−0.444927	+	0.895567i	$0.646770\pi$
$600$	0	0
$601$	6.53955e23	1.56717	0.783583	−	0.621287i	$-0.213390\pi$
$601$	6.53955e23	1.56717	0.783583	+	0.621287i	$0.213390\pi$
$602$	0	0
$603$	5.75668e23	1.34114
$604$	0	0
$605$	−6.41089e22	−0.145210
$606$	0	0
$607$	−4.08715e23	−0.900154	−0.450077	−	0.892990i	$-0.648603\pi$
$607$	−4.08715e23	−0.900154	−0.450077	+	0.892990i	$0.648603\pi$
$608$	0	0
$609$	−3.36301e22	−0.0720247
$610$	0	0
$611$	1.25287e23	0.260949
$612$	0	0
$613$	1.80349e23	0.365343	0.182671	−	0.983174i	$-0.441526\pi$
$613$	1.80349e23	0.365343	0.182671	+	0.983174i	$0.441526\pi$
$614$	0	0
$615$	−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	$0.360475\pi$
$618$	0	0
$619$	−1.71098e22	−0.0319062	−0.0159531	−	0.999873i	$-0.505078\pi$
$619$	−1.71098e22	−0.0319062	−0.0159531	+	0.999873i	$0.505078\pi$
$620$	0	0
$621$	−3.25923e23	−0.591339
$622$	0	0
$623$	−9.40931e22	−0.166115
$624$	0	0
$625$	−7.19124e23	−1.23545
$626$	0	0
$627$	−4.80239e21	−0.00802941
$628$	0	0
$629$	−6.89124e22	−0.112142
$630$	0	0
$631$	2.82293e23	0.447147	0.223574	−	0.974687i	$-0.428228\pi$
$631$	2.82293e23	0.447147	0.223574	+	0.974687i	$0.428228\pi$
$632$	0	0
$633$	−7.43316e22	−0.114615
$634$	0	0
$635$	−8.92802e23	−1.34023
$636$	0	0
$637$	−5.64684e23	−0.825316
$638$	0	0
$639$	−4.10965e23	−0.584854
$640$	0	0
$641$	−3.77613e23	−0.523304	−0.261652	−	0.965162i	$-0.584267\pi$
$641$	−3.77613e23	−0.523304	−0.261652	+	0.965162i	$0.584267\pi$
$642$	0	0
$643$	4.65536e23	0.628289	0.314145	−	0.949375i	$-0.398282\pi$
$643$	4.65536e23	0.628289	0.314145	+	0.949375i	$0.398282\pi$
$644$	0	0
$645$	−4.00009e23	−0.525790
$646$	0	0
$647$	−1.00412e24	−1.28558	−0.642788	−	0.766045i	$-0.722222\pi$
$647$	−1.00412e24	−1.28558	−0.642788	+	0.766045i	$0.722222\pi$
$648$	0	0
$649$	−2.46396e22	−0.0307294
$650$	0	0
$651$	−1.22304e23	−0.148595
$652$	0	0
$653$	1.26098e23	0.149261	0.0746304	−	0.997211i	$-0.476222\pi$
$653$	1.26098e23	0.149261	0.0746304	+	0.997211i	$0.476222\pi$
$654$	0	0
$655$	1.35190e23	0.155917
$656$	0	0
$657$	−3.76940e23	−0.423611
$658$	0	0
$659$	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	$0.182226\pi$
$662$	0	0
$663$	−5.91077e22	−0.0614864
$664$	0	0
$665$	−4.92271e21	−0.00499137
$666$	0	0
$667$	7.71615e23	0.762660
$668$	0	0
$669$	−4.06301e23	−0.391495
$670$	0	0
$671$	−9.85948e23	−0.926217
$672$	0	0
$673$	2.91330e23	0.266844	0.133422	−	0.991059i	$-0.457403\pi$
$673$	2.91330e23	0.266844	0.133422	+	0.991059i	$0.457403\pi$
$674$	0	0
$675$	−2.97489e23	−0.265699
$676$	0	0
$677$	1.33861e24	1.16587	0.582937	−	0.812517i	$-0.301903\pi$
$677$	1.33861e24	1.16587	0.582937	+	0.812517i	$0.301903\pi$
$678$	0	0
$679$	2.05721e23	0.174737
$680$	0	0
$681$	7.87321e23	0.652231
$682$	0	0
$683$	1.05295e24	0.850812	0.425406	−	0.905003i	$-0.360131\pi$
$683$	1.05295e24	0.850812	0.425406	+	0.905003i	$0.360131\pi$
$684$	0	0
$685$	−1.91966e24	−1.51305
$686$	0	0
$687$	−3.91947e23	−0.301366
$688$	0	0
$689$	5.82932e22	0.0437275
$690$	0	0
$691$	5.71311e22	0.0418128	0.0209064	−	0.999781i	$-0.493345\pi$
$691$	5.71311e22	0.0418128	0.0209064	+	0.999781i	$0.493345\pi$
$692$	0	0
$693$	2.69343e23	0.192341
$694$	0	0
$695$	1.62809e24	1.13451
$696$	0	0
$697$	−2.65320e23	−0.180423
$698$	0	0
$699$	4.23044e23	0.280756
$700$	0	0
$701$	−3.18403e23	−0.206241	−0.103120	−	0.994669i	$-0.532883\pi$
$701$	−3.18403e23	−0.206241	−0.103120	+	0.994669i	$0.532883\pi$
$702$	0	0
$703$	−1.88788e22	−0.0119359
$704$	0	0
$705$	−2.16683e23	−0.133726
$706$	0	0
$707$	5.18131e23	0.312158
$708$	0	0
$709$	2.12997e24	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$709$	2.12997e24	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$710$	0	0
$711$	2.62182e23	0.150561
$712$	0	0
$713$	2.80617e24	1.57345
$714$	0	0
$715$	−1.96450e24	−1.07560
$716$	0	0
$717$	−8.11119e23	−0.433683
$718$	0	0
$719$	2.34234e24	1.22308	0.611540	−	0.791213i	$-0.290550\pi$
$719$	2.34234e24	1.22308	0.611540	+	0.791213i	$0.290550\pi$
$720$	0	0
$721$	−2.92641e23	−0.149240
$722$	0	0
$723$	−5.94456e23	−0.296104
$724$	0	0
$725$	7.04299e23	0.342676
$726$	0	0
$727$	−3.58185e24	−1.70241	−0.851207	−	0.524830i	$-0.824129\pi$
$727$	−3.58185e24	−1.70241	−0.851207	+	0.524830i	$0.824129\pi$
$728$	0	0
$729$	−5.28574e23	−0.245427
$730$	0	0
$731$	−4.94215e23	−0.224191
$732$	0	0
$733$	−3.29013e24	−1.45824	−0.729119	−	0.684387i	$-0.760070\pi$
$733$	−3.29013e24	−1.45824	−0.729119	+	0.684387i	$0.760070\pi$
$734$	0	0
$735$	9.76615e23	0.422942
$736$	0	0
$737$	−3.91591e24	−1.65714
$738$	0	0
$739$	−1.74130e24	−0.720103	−0.360052	−	0.932932i	$-0.617241\pi$
$739$	−1.74130e24	−0.720103	−0.360052	+	0.932932i	$0.617241\pi$
$740$	0	0
$741$	−1.61928e22	−0.00654436
$742$	0	0
$743$	4.61544e24	1.82309	0.911545	−	0.411199i	$-0.134890\pi$
$743$	4.61544e24	1.82309	0.911545	+	0.411199i	$0.134890\pi$
$744$	0	0
$745$	4.89576e23	0.189013
$746$	0	0
$747$	−3.29778e24	−1.24451
$748$	0	0
$749$	6.30841e23	0.232715
$750$	0	0
$751$	8.19607e22	0.0295574	0.0147787	−	0.999891i	$-0.495296\pi$
$751$	8.19607e22	0.0295574	0.0147787	+	0.999891i	$0.495296\pi$
$752$	0	0
$753$	1.43303e24	0.505241
$754$	0	0
$755$	−4.03122e24	−1.38959
$756$	0	0
$757$	−2.51608e24	−0.848025	−0.424013	−	0.905656i	$-0.639379\pi$
$757$	−2.51608e24	−0.848025	−0.424013	+	0.905656i	$0.639379\pi$
$758$	0	0
$759$	1.02373e24	0.337389
$760$	0	0
$761$	−5.27745e23	−0.170081	−0.0850403	−	0.996378i	$-0.527102\pi$
$761$	−5.27745e23	−0.170081	−0.0850403	+	0.996378i	$0.527102\pi$
$762$	0	0
$763$	−6.89574e23	−0.217331
$764$	0	0
$765$	−6.17098e23	−0.190209
$766$	0	0
$767$	−8.30803e22	−0.0250460
$768$	0	0
$769$	−7.51719e23	−0.221657	−0.110828	−	0.993840i	$-0.535350\pi$
$769$	−7.51719e23	−0.221657	−0.110828	+	0.993840i	$0.535350\pi$
$770$	0	0
$771$	2.10508e23	0.0607165
$772$	0	0
$773$	2.81914e24	0.795409	0.397704	−	0.917514i	$-0.369807\pi$
$773$	2.81914e24	0.795409	0.397704	+	0.917514i	$0.369807\pi$
$774$	0	0
$775$	2.56136e24	0.706980
$776$	0	0
$777$	−1.75401e23	−0.0473644
$778$	0	0
$779$	−7.26854e22	−0.0192034
$780$	0	0
$781$	2.79554e24	0.722656
$782$	0	0
$783$	−2.50118e24	−0.632660
$784$	0	0
$785$	2.35385e24	0.582621
$786$	0	0
$787$	3.20732e24	0.776887	0.388443	−	0.921473i	$-0.373013\pi$
$787$	3.20732e24	0.776887	0.388443	+	0.921473i	$0.373013\pi$
$788$	0	0
$789$	−7.66170e22	−0.0181623
$790$	0	0
$791$	9.07739e23	0.210601
$792$	0	0
$793$	−3.32444e24	−0.754912
$794$	0	0
$795$	−1.00817e23	−0.0224086
$796$	0	0
$797$	2.36328e24	0.514186	0.257093	−	0.966387i	$-0.417235\pi$
$797$	2.36328e24	0.514186	0.257093	+	0.966387i	$0.417235\pi$
$798$	0	0
$799$	−2.67713e23	−0.0570194
$800$	0	0
$801$	−3.23136e24	−0.673766
$802$	0	0
$803$	2.56409e24	0.523421
$804$	0	0
$805$	1.04938e24	0.209733
$806$	0	0
$807$	4.93676e23	0.0966088
$808$	0	0
$809$	9.60721e24	1.84092	0.920460	−	0.390836i	$-0.127814\pi$
$809$	9.60721e24	1.84092	0.920460	+	0.390836i	$0.127814\pi$
$810$	0	0
$811$	6.64616e24	1.24708	0.623539	−	0.781792i	$-0.285694\pi$
$811$	6.64616e24	1.24708	0.623539	+	0.781792i	$0.285694\pi$
$812$	0	0
$813$	6.30035e23	0.115770
$814$	0	0
$815$	8.21751e24	1.47877
$816$	0	0
$817$	−1.35392e23	−0.0238620
$818$	0	0
$819$	9.08175e23	0.156768
$820$	0	0
$821$	−5.09458e24	−0.861373	−0.430687	−	0.902502i	$-0.641729\pi$
$821$	−5.09458e24	−0.861373	−0.430687	+	0.902502i	$0.641729\pi$
$822$	0	0
$823$	−5.78198e24	−0.957586	−0.478793	−	0.877928i	$-0.658926\pi$
$823$	−5.78198e24	−0.957586	−0.478793	+	0.877928i	$0.658926\pi$
$824$	0	0
$825$	9.34421e23	0.151595
$826$	0	0
$827$	−8.38907e24	−1.33327	−0.666633	−	0.745386i	$-0.732265\pi$
$827$	−8.38907e24	−1.33327	−0.666633	+	0.745386i	$0.732265\pi$
$828$	0	0
$829$	5.02769e24	0.782808	0.391404	−	0.920219i	$-0.371989\pi$
$829$	5.02769e24	0.782808	0.391404	+	0.920219i	$0.371989\pi$
$830$	0	0
$831$	−7.65741e23	−0.116808
$832$	0	0
$833$	1.20662e24	0.180338
$834$	0	0
$835$	−8.83747e24	−1.29417
$836$	0	0
$837$	−9.09619e24	−1.30525
$838$	0	0
$839$	−9.38328e23	−0.131940	−0.0659702	−	0.997822i	$-0.521014\pi$
$839$	−9.38328e23	−0.131940	−0.0659702	+	0.997822i	$0.521014\pi$
$840$	0	0
$841$	−1.33566e24	−0.184048
$842$	0	0
$843$	1.20186e24	0.162300
$844$	0	0
$845$	2.25011e24	0.297798
$846$	0	0
$847$	−2.01603e23	−0.0261510
$848$	0	0
$849$	−1.45603e24	−0.185121
$850$	0	0
$851$	4.02442e24	0.501536
$852$	0	0
$853$	2.33188e24	0.284865	0.142433	−	0.989805i	$-0.454508\pi$
$853$	2.33188e24	0.284865	0.142433	+	0.989805i	$0.454508\pi$
$854$	0	0
$855$	−1.69056e23	−0.0202451
$856$	0	0
$857$	−1.09008e25	−1.27974	−0.639872	−	0.768482i	$-0.721012\pi$
$857$	−1.09008e25	−1.27974	−0.639872	+	0.768482i	$0.721012\pi$
$858$	0	0
$859$	1.47838e25	1.70155	0.850773	−	0.525533i	$-0.176134\pi$
$859$	1.47838e25	1.70155	0.850773	+	0.525533i	$0.176134\pi$
$860$	0	0
$861$	−6.75310e23	−0.0762037
$862$	0	0
$863$	−5.86087e24	−0.648441	−0.324220	−	0.945982i	$-0.605102\pi$
$863$	−5.86087e24	−0.648441	−0.324220	+	0.945982i	$0.605102\pi$
$864$	0	0
$865$	−2.37412e24	−0.257553
$866$	0	0
$867$	−3.41760e24	−0.363545
$868$	0	0
$869$	−1.78346e24	−0.186036
$870$	0	0
$871$	−1.32037e25	−1.35065
$872$	0	0
$873$	7.06491e24	0.708737
$874$	0	0
$875$	−1.56703e24	−0.154173
$876$	0	0
$877$	−1.08364e25	−1.04565	−0.522827	−	0.852439i	$-0.675122\pi$
$877$	−1.08364e25	−1.04565	−0.522827	+	0.852439i	$0.675122\pi$
$878$	0	0
$879$	−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	$0.425437\pi$
$882$	0	0
$883$	1.04108e24	0.0948025	0.0474012	−	0.998876i	$-0.484906\pi$
$883$	1.04108e24	0.0948025	0.0474012	+	0.998876i	$0.484906\pi$
$884$	0	0
$885$	1.43686e23	0.0128351
$886$	0	0
$887$	−1.47651e25	−1.29386	−0.646929	−	0.762550i	$-0.723947\pi$
$887$	−1.47651e25	−1.29386	−0.646929	+	0.762550i	$0.723947\pi$
$888$	0	0
$889$	−2.80760e24	−0.241363
$890$	0	0
$891$	7.46369e24	0.629496
$892$	0	0
$893$	−7.33412e22	−0.00606891
$894$	0	0
$895$	−8.10737e24	−0.658240
$896$	0	0
$897$	3.45184e24	0.274988
$898$	0	0
$899$	2.15350e25	1.68340
$900$	0	0
$901$	−1.24561e23	−0.00955478
$902$	0	0
$903$	−1.25791e24	−0.0946899
$904$	0	0
$905$	−1.44366e25	−1.06648
$906$	0	0
$907$	2.99747e23	0.0217317	0.0108658	−	0.999941i	$-0.496541\pi$
$907$	2.99747e23	0.0217317	0.0108658	+	0.999941i	$0.496541\pi$
$908$	0	0
$909$	1.77937e25	1.26612
$910$	0	0
$911$	1.71431e24	0.119724	0.0598621	−	0.998207i	$-0.480934\pi$
$911$	1.71431e24	0.119724	0.0598621	+	0.998207i	$0.480934\pi$
$912$	0	0
$913$	2.24328e25	1.53773
$914$	0	0
$915$	5.74958e24	0.386863
$916$	0	0
$917$	4.25132e23	0.0280792
$918$	0	0
$919$	2.42539e25	1.57253	0.786265	−	0.617889i	$-0.212012\pi$
$919$	2.42539e25	1.57253	0.786265	+	0.617889i	$0.212012\pi$
$920$	0	0
$921$	−5.93213e24	−0.377575
$922$	0	0
$923$	9.42605e24	0.589000
$924$	0	0
$925$	3.67333e24	0.225349
$926$	0	0
$927$	−1.00499e25	−0.605320
$928$	0	0
$929$	1.40138e25	0.828745	0.414373	−	0.910107i	$-0.364001\pi$
$929$	1.40138e25	0.828745	0.414373	+	0.910107i	$0.364001\pi$
$930$	0	0
$931$	3.30557e23	0.0191944
$932$	0	0
$933$	1.00560e25	0.573364
$934$	0	0
$935$	4.19773e24	0.235026
$936$	0	0
$937$	−2.43217e25	−1.33724	−0.668618	−	0.743606i	$-0.733114\pi$
$937$	−2.43217e25	−1.33724	−0.668618	+	0.743606i	$0.733114\pi$
$938$	0	0
$939$	6.78690e24	0.366449
$940$	0	0
$941$	5.98394e23	0.0317304	0.0158652	−	0.999874i	$-0.494950\pi$
$941$	5.98394e23	0.0317304	0.0158652	+	0.999874i	$0.494950\pi$
$942$	0	0
$943$	1.54944e25	0.806912
$944$	0	0
$945$	−3.40155e24	−0.173983
$946$	0	0
$947$	−3.62477e25	−1.82098	−0.910491	−	0.413530i	$-0.864296\pi$
$947$	−3.62477e25	−1.82098	−0.910491	+	0.413530i	$0.864296\pi$
$948$	0	0
$949$	8.64565e24	0.426614
$950$	0	0
$951$	3.84865e24	0.186541
$952$	0	0
$953$	−1.88157e25	−0.895839	−0.447919	−	0.894074i	$-0.647835\pi$
$953$	−1.88157e25	−0.895839	−0.447919	+	0.894074i	$0.647835\pi$
$954$	0	0
$955$	−2.89803e25	−1.35542
$956$	0	0
$957$	7.85627e24	0.360965
$958$	0	0
$959$	−6.03674e24	−0.272486
$960$	0	0
$961$	5.57675e25	2.47305
$962$	0	0
$963$	2.16644e25	0.943896
$964$	0	0
$965$	−5.04467e25	−2.15949
$966$	0	0
$967$	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
$970$	0	0
$971$	−2.03685e25	−0.827170	−0.413585	−	0.910465i	$-0.635724\pi$
$971$	−2.03685e25	−0.827170	−0.413585	+	0.910465i	$0.635724\pi$
$972$	0	0
$973$	5.11985e24	0.204314
$974$	0	0
$975$	3.15070e24	0.123557
$976$	0	0
$977$	−2.52590e25	−0.973449	−0.486724	−	0.873556i	$-0.661808\pi$
$977$	−2.52590e25	−0.973449	−0.486724	+	0.873556i	$0.661808\pi$
$978$	0	0
$979$	2.19809e25	0.832517
$980$	0	0
$981$	−2.36815e25	−0.881499
$982$	0	0
$983$	1.51840e24	0.0555495	0.0277747	−	0.999614i	$-0.491158\pi$
$983$	1.51840e24	0.0555495	0.0277747	+	0.999614i	$0.491158\pi$
$984$	0	0
$985$	−1.32804e25	−0.477533
$986$	0	0
$987$	−6.81402e23	−0.0240828
$988$	0	0
$989$	2.88617e25	1.00266
$990$	0	0
$991$	6.69650e24	0.228677	0.114338	−	0.993442i	$-0.463525\pi$
$991$	6.69650e24	0.228677	0.114338	+	0.993442i	$0.463525\pi$
$992$	0	0
$993$	−1.71561e25	−0.575903
$994$	0	0
$995$	−5.66018e25	−1.86781
$996$	0	0
$997$	−3.45326e25	−1.12026	−0.560132	−	0.828404i	$-0.689249\pi$
$997$	−3.45326e25	−1.12026	−0.560132	+	0.828404i	$0.689249\pi$
$998$	0	0
$999$	−1.30451e25	−0.416046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.18.a.b.1.1		1
4.3	odd	2		1.18.a.a.1.1	✓	1
8.3	odd	2		64.18.a.d.1.1		1
8.5	even	2		64.18.a.b.1.1		1
12.11	even	2		9.18.a.b.1.1		1
20.3	even	4		25.18.b.a.24.2		2
20.7	even	4		25.18.b.a.24.1		2
20.19	odd	2		25.18.a.a.1.1		1
28.27	even	2		49.18.a.a.1.1		1
44.43	even	2		121.18.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.18.a.a.1.1	✓	1	4.3	odd	2
9.18.a.b.1.1		1	12.11	even	2
16.18.a.b.1.1		1	1.1	even	1	trivial
25.18.a.a.1.1		1	20.19	odd	2
25.18.b.a.24.1		2	20.7	even	4
25.18.b.a.24.2		2	20.3	even	4
49.18.a.a.1.1		1	28.27	even	2
64.18.a.b.1.1		1	8.5	even	2
64.18.a.d.1.1		1	8.3	odd	2
121.18.a.b.1.1		1	44.43	even	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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