LMFDB - Embedded newform 16.22.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,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, 22, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 22 $
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:	$44.7163750859$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
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)

$f(q)$

$=$

$q-59316.0 q^{3} +4.97535e6 q^{5} -1.42743e9 q^{7} -6.94197e9 q^{9} +O(q^{10})$

$q-59316.0 q^{3} +4.97535e6 q^{5} -1.42743e9 q^{7} -6.94197e9 q^{9} +1.06768e11 q^{11} -1.50151e11 q^{13} -2.95118e11 q^{15} -1.12040e13 q^{17} -1.10241e13 q^{19} +8.46692e13 q^{21} -1.29503e14 q^{23} -4.52083e14 q^{25} +1.03224e15 q^{27} +2.38237e15 q^{29} +8.78553e14 q^{31} -6.33304e15 q^{33} -7.10194e15 q^{35} +3.11300e16 q^{37} +8.90633e15 q^{39} -2.46129e16 q^{41} +1.33386e17 q^{43} -3.45387e16 q^{45} +1.92524e17 q^{47} +1.47900e18 q^{49} +6.64575e17 q^{51} -5.94166e17 q^{53} +5.31208e17 q^{55} +6.53903e17 q^{57} +2.95595e18 q^{59} +7.98415e18 q^{61} +9.90914e18 q^{63} -7.47052e17 q^{65} -4.83704e18 q^{67} +7.68159e18 q^{69} -8.84902e18 q^{71} +3.66844e19 q^{73} +2.68158e19 q^{75} -1.52403e20 q^{77} -3.38406e19 q^{79} +1.13873e19 q^{81} -2.04215e20 q^{83} -5.57437e19 q^{85} -1.41313e20 q^{87} -4.10241e19 q^{89} +2.14329e20 q^{91} -5.21122e19 q^{93} -5.48485e19 q^{95} -7.27592e20 q^{97} -7.41179e20 q^{99} +O(q^{100})$

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
$3$	−59316.0	−0.579961	−0.289980	−	0.957033i	$-0.593649\pi$
$3$	−59316.0	−0.579961	−0.289980	+	0.957033i	$0.593649\pi$
$4$	0	0
$5$	4.97535e6	0.227845	0.113922	−	0.993490i	$-0.463659\pi$
$5$	4.97535e6	0.227845	0.113922	+	0.993490i	$0.463659\pi$
$6$	0	0
$7$	−1.42743e9	−1.90996	−0.954980	−	0.296671i	$-0.904123\pi$
$7$	−1.42743e9	−1.90996	−0.954980	+	0.296671i	$0.904123\pi$
$8$	0	0
$9$	−6.94197e9	−0.663645
$10$	0	0
$11$	1.06768e11	1.24113	0.620565	−	0.784155i	$-0.286903\pi$
$11$	1.06768e11	1.24113	0.620565	+	0.784155i	$0.286903\pi$
$12$	0	0
$13$	−1.50151e11	−0.302080	−0.151040	−	0.988528i	$-0.548262\pi$
$13$	−1.50151e11	−0.302080	−0.151040	+	0.988528i	$0.548262\pi$
$14$	0	0
$15$	−2.95118e11	−0.132141
$16$	0	0
$17$	−1.12040e13	−1.34790	−0.673952	−	0.738776i	$-0.735404\pi$
$17$	−1.12040e13	−1.34790	−0.673952	+	0.738776i	$0.735404\pi$
$18$	0	0
$19$	−1.10241e13	−0.412504	−0.206252	−	0.978499i	$-0.566127\pi$
$19$	−1.10241e13	−0.412504	−0.206252	+	0.978499i	$0.566127\pi$
$20$	0	0
$21$	8.46692e13	1.10770
$22$	0	0
$23$	−1.29503e14	−0.651834	−0.325917	−	0.945398i	$-0.605673\pi$
$23$	−1.29503e14	−0.651834	−0.325917	+	0.945398i	$0.605673\pi$
$24$	0	0
$25$	−4.52083e14	−0.948087
$26$	0	0
$27$	1.03224e15	0.964849
$28$	0	0
$29$	2.38237e15	1.05155	0.525776	−	0.850623i	$-0.323775\pi$
$29$	2.38237e15	1.05155	0.525776	+	0.850623i	$0.323775\pi$
$30$	0	0
$31$	8.78553e14	0.192517	0.0962587	−	0.995356i	$-0.469312\pi$
$31$	8.78553e14	0.192517	0.0962587	+	0.995356i	$0.469312\pi$
$32$	0	0
$33$	−6.33304e15	−0.719807
$34$	0	0
$35$	−7.10194e15	−0.435174
$36$	0	0
$37$	3.11300e16	1.06429	0.532146	−	0.846652i	$-0.321386\pi$
$37$	3.11300e16	1.06429	0.532146	+	0.846652i	$0.321386\pi$
$38$	0	0
$39$	8.90633e15	0.175194
$40$	0	0
$41$	−2.46129e16	−0.286373	−0.143187	−	0.989696i	$-0.545735\pi$
$41$	−2.46129e16	−0.286373	−0.143187	+	0.989696i	$0.545735\pi$
$42$	0	0
$43$	1.33386e17	0.941222	0.470611	−	0.882341i	$-0.344034\pi$
$43$	1.33386e17	0.941222	0.470611	+	0.882341i	$0.344034\pi$
$44$	0	0
$45$	−3.45387e16	−0.151208
$46$	0	0
$47$	1.92524e17	0.533897	0.266948	−	0.963711i	$-0.413985\pi$
$47$	1.92524e17	0.533897	0.266948	+	0.963711i	$0.413985\pi$
$48$	0	0
$49$	1.47900e18	2.64794
$50$	0	0
$51$	6.64575e17	0.781731
$52$	0	0
$53$	−5.94166e17	−0.466672	−0.233336	−	0.972396i	$-0.574964\pi$
$53$	−5.94166e17	−0.466672	−0.233336	+	0.972396i	$0.574964\pi$
$54$	0	0
$55$	5.31208e17	0.282785
$56$	0	0
$57$	6.53903e17	0.239236
$58$	0	0
$59$	2.95595e18	0.752925	0.376462	−	0.926432i	$-0.377140\pi$
$59$	2.95595e18	0.752925	0.376462	+	0.926432i	$0.377140\pi$
$60$	0	0
$61$	7.98415e18	1.43306	0.716532	−	0.697554i	$-0.245728\pi$
$61$	7.98415e18	1.43306	0.716532	+	0.697554i	$0.245728\pi$
$62$	0	0
$63$	9.90914e18	1.26754
$64$	0	0
$65$	−7.47052e17	−0.0688272
$66$	0	0
$67$	−4.83704e18	−0.324186	−0.162093	−	0.986775i	$-0.551824\pi$
$67$	−4.83704e18	−0.324186	−0.162093	+	0.986775i	$0.551824\pi$
$68$	0	0
$69$	7.68159e18	0.378038
$70$	0	0
$71$	−8.84902e18	−0.322613	−0.161307	−	0.986904i	$-0.551571\pi$
$71$	−8.84902e18	−0.322613	−0.161307	+	0.986904i	$0.551571\pi$
$72$	0	0
$73$	3.66844e19	0.999060	0.499530	−	0.866297i	$-0.333506\pi$
$73$	3.66844e19	0.999060	0.499530	+	0.866297i	$0.333506\pi$
$74$	0	0
$75$	2.68158e19	0.549853
$76$	0	0
$77$	−1.52403e20	−2.37051
$78$	0	0
$79$	−3.38406e19	−0.402118	−0.201059	−	0.979579i	$-0.564438\pi$
$79$	−3.38406e19	−0.402118	−0.201059	+	0.979579i	$0.564438\pi$
$80$	0	0
$81$	1.13873e19	0.104071
$82$	0	0
$83$	−2.04215e20	−1.44466	−0.722332	−	0.691547i	$-0.756930\pi$
$83$	−2.04215e20	−1.44466	−0.722332	+	0.691547i	$0.756930\pi$
$84$	0	0
$85$	−5.57437e19	−0.307112
$86$	0	0
$87$	−1.41313e20	−0.609858
$88$	0	0
$89$	−4.10241e19	−0.139458	−0.0697290	−	0.997566i	$-0.522213\pi$
$89$	−4.10241e19	−0.139458	−0.0697290	+	0.997566i	$0.522213\pi$
$90$	0	0
$91$	2.14329e20	0.576960
$92$	0	0
$93$	−5.21122e19	−0.111653
$94$	0	0
$95$	−5.48485e19	−0.0939869
$96$	0	0
$97$	−7.27592e20	−1.00181	−0.500905	−	0.865503i	$-0.666999\pi$
$97$	−7.27592e20	−1.00181	−0.500905	+	0.865503i	$0.666999\pi$
$98$	0	0
$99$	−7.41179e20	−0.823671
$100$	0	0
$101$	5.93965e20	0.535040	0.267520	−	0.963552i	$-0.413796\pi$
$101$	5.93965e20	0.535040	0.267520	+	0.963552i	$0.413796\pi$
$102$	0	0
$103$	6.95712e20	0.510080	0.255040	−	0.966930i	$-0.417911\pi$
$103$	6.95712e20	0.510080	0.255040	+	0.966930i	$0.417911\pi$
$104$	0	0
$105$	4.21259e20	0.252384
$106$	0	0
$107$	2.38158e21	1.17041	0.585203	−	0.810887i	$-0.301015\pi$
$107$	2.38158e21	1.17041	0.585203	+	0.810887i	$0.301015\pi$
$108$	0	0
$109$	2.01913e21	0.816933	0.408466	−	0.912773i	$-0.366064\pi$
$109$	2.01913e21	0.816933	0.408466	+	0.912773i	$0.366064\pi$
$110$	0	0
$111$	−1.84651e21	−0.617248
$112$	0	0
$113$	1.81974e21	0.504296	0.252148	−	0.967689i	$-0.418863\pi$
$113$	1.81974e21	0.504296	0.252148	+	0.967689i	$0.418863\pi$
$114$	0	0
$115$	−6.44322e20	−0.148517
$116$	0	0
$117$	1.04234e21	0.200474
$118$	0	0
$119$	1.59929e22	2.57444
$120$	0	0
$121$	3.99913e21	0.540405
$122$	0	0
$123$	1.45994e21	0.166085
$124$	0	0
$125$	−4.62170e21	−0.443861
$126$	0	0
$127$	−2.20220e22	−1.79027	−0.895134	−	0.445797i	$-0.852920\pi$
$127$	−2.20220e22	−1.79027	−0.895134	+	0.445797i	$0.852920\pi$
$128$	0	0
$129$	−7.91193e21	−0.545872
$130$	0	0
$131$	−1.44136e22	−0.846101	−0.423051	−	0.906106i	$-0.639041\pi$
$131$	−1.44136e22	−0.846101	−0.423051	+	0.906106i	$0.639041\pi$
$132$	0	0
$133$	1.57360e22	0.787867
$134$	0	0
$135$	5.13574e21	0.219836
$136$	0	0
$137$	−3.57623e22	−1.31177	−0.655887	−	0.754859i	$-0.727705\pi$
$137$	−3.57623e22	−1.31177	−0.655887	+	0.754859i	$0.727705\pi$
$138$	0	0
$139$	2.10431e22	0.662909	0.331454	−	0.943471i	$-0.392461\pi$
$139$	2.10431e22	0.662909	0.331454	+	0.943471i	$0.392461\pi$
$140$	0	0
$141$	−1.14198e22	−0.309639
$142$	0	0
$143$	−1.60313e22	−0.374921
$144$	0	0
$145$	1.18531e22	0.239590
$146$	0	0
$147$	−8.77283e22	−1.53570
$148$	0	0
$149$	−8.71910e22	−1.32439	−0.662195	−	0.749332i	$-0.730375\pi$
$149$	−8.71910e22	−1.32439	−0.662195	+	0.749332i	$0.730375\pi$
$150$	0	0
$151$	4.00667e22	0.529086	0.264543	−	0.964374i	$-0.414779\pi$
$151$	4.00667e22	0.529086	0.264543	+	0.964374i	$0.414779\pi$
$152$	0	0
$153$	7.77776e22	0.894530
$154$	0	0
$155$	4.37111e21	0.0438641
$156$	0	0
$157$	−4.60441e22	−0.403857	−0.201929	−	0.979400i	$-0.564721\pi$
$157$	−4.60441e22	−0.403857	−0.201929	+	0.979400i	$0.564721\pi$
$158$	0	0
$159$	3.52436e22	0.270651
$160$	0	0
$161$	1.84856e23	1.24498
$162$	0	0
$163$	−5.72127e22	−0.338472	−0.169236	−	0.985576i	$-0.554130\pi$
$163$	−5.72127e22	−0.338472	−0.169236	+	0.985576i	$0.554130\pi$
$164$	0	0
$165$	−3.15091e22	−0.164004
$166$	0	0
$167$	1.32913e23	0.609600	0.304800	−	0.952416i	$-0.401410\pi$
$167$	1.32913e23	0.609600	0.304800	+	0.952416i	$0.401410\pi$
$168$	0	0
$169$	−2.24519e23	−0.908748
$170$	0	0
$171$	7.65286e22	0.273757
$172$	0	0
$173$	5.53136e23	1.75125	0.875624	−	0.482994i	$-0.160451\pi$
$173$	5.53136e23	1.75125	0.875624	+	0.482994i	$0.160451\pi$
$174$	0	0
$175$	6.45315e23	1.81081
$176$	0	0
$177$	−1.75335e23	−0.436667
$178$	0	0
$179$	5.31479e22	0.117633	0.0588165	−	0.998269i	$-0.481267\pi$
$179$	5.31479e22	0.117633	0.0588165	+	0.998269i	$0.481267\pi$
$180$	0	0
$181$	7.59350e23	1.49560	0.747802	−	0.663921i	$-0.231109\pi$
$181$	7.59350e23	1.49560	0.747802	+	0.663921i	$0.231109\pi$
$182$	0	0
$183$	−4.73588e23	−0.831121
$184$	0	0
$185$	1.54883e23	0.242493
$186$	0	0
$187$	−1.19623e24	−1.67292
$188$	0	0
$189$	−1.47344e24	−1.84282
$190$	0	0
$191$	9.64674e23	1.08026	0.540132	−	0.841580i	$-0.318374\pi$
$191$	9.64674e23	1.08026	0.540132	+	0.841580i	$0.318374\pi$
$192$	0	0
$193$	−3.41192e23	−0.342489	−0.171245	−	0.985229i	$-0.554779\pi$
$193$	−3.41192e23	−0.342489	−0.171245	+	0.985229i	$0.554779\pi$
$194$	0	0
$195$	4.43121e22	0.0399171
$196$	0	0
$197$	−5.00591e23	−0.405124	−0.202562	−	0.979269i	$-0.564927\pi$
$197$	−5.00591e23	−0.405124	−0.202562	+	0.979269i	$0.564927\pi$
$198$	0	0
$199$	5.09875e23	0.371113	0.185557	−	0.982634i	$-0.440591\pi$
$199$	5.09875e23	0.371113	0.185557	+	0.982634i	$0.440591\pi$
$200$	0	0
$201$	2.86914e23	0.188015
$202$	0	0
$203$	−3.40066e24	−2.00842
$204$	0	0
$205$	−1.22458e23	−0.0652486
$206$	0	0
$207$	8.99004e23	0.432587
$208$	0	0
$209$	−1.17702e24	−0.511972
$210$	0	0
$211$	−7.29976e23	−0.287305	−0.143652	−	0.989628i	$-0.545885\pi$
$211$	−7.29976e23	−0.287305	−0.143652	+	0.989628i	$0.545885\pi$
$212$	0	0
$213$	5.24888e23	0.187103
$214$	0	0
$215$	6.63643e23	0.214452
$216$	0	0
$217$	−1.25407e24	−0.367701
$218$	0	0
$219$	−2.17597e24	−0.579416
$220$	0	0
$221$	1.68228e24	0.407174
$222$	0	0
$223$	5.87017e24	1.29256	0.646279	−	0.763101i	$-0.276324\pi$
$223$	5.87017e24	1.29256	0.646279	+	0.763101i	$0.276324\pi$
$224$	0	0
$225$	3.13834e24	0.629193
$226$	0	0
$227$	−6.13596e24	−1.12101	−0.560507	−	0.828150i	$-0.689394\pi$
$227$	−6.13596e24	−1.12101	−0.560507	+	0.828150i	$0.689394\pi$
$228$	0	0
$229$	−3.05217e24	−0.508552	−0.254276	−	0.967132i	$-0.581837\pi$
$229$	−3.05217e24	−0.508552	−0.254276	+	0.967132i	$0.581837\pi$
$230$	0	0
$231$	9.03995e24	1.37480
$232$	0	0
$233$	−7.45995e23	−0.103633	−0.0518166	−	0.998657i	$-0.516501\pi$
$233$	−7.45995e23	−0.103633	−0.0518166	+	0.998657i	$0.516501\pi$
$234$	0	0
$235$	9.57874e23	0.121645
$236$	0	0
$237$	2.00729e24	0.233213
$238$	0	0
$239$	1.02561e25	1.09095	0.545473	−	0.838128i	$-0.316350\pi$
$239$	1.02561e25	1.09095	0.545473	+	0.838128i	$0.316350\pi$
$240$	0	0
$241$	1.46660e25	1.42933	0.714663	−	0.699469i	$-0.246580\pi$
$241$	1.46660e25	1.42933	0.714663	+	0.699469i	$0.246580\pi$
$242$	0	0
$243$	−1.14730e25	−1.02521
$244$	0	0
$245$	7.35854e24	0.603320
$246$	0	0
$247$	1.65527e24	0.124609
$248$	0	0
$249$	1.21132e25	0.837848
$250$	0	0
$251$	1.52767e25	0.971526	0.485763	−	0.874090i	$-0.338542\pi$
$251$	1.52767e25	0.971526	0.485763	+	0.874090i	$0.338542\pi$
$252$	0	0
$253$	−1.38267e25	−0.809011
$254$	0	0
$255$	3.30649e24	0.178113
$256$	0	0
$257$	−3.66073e25	−1.81665	−0.908323	−	0.418270i	$-0.862636\pi$
$257$	−3.66073e25	−1.81665	−0.908323	+	0.418270i	$0.862636\pi$
$258$	0	0
$259$	−4.44358e25	−2.03276
$260$	0	0
$261$	−1.65383e25	−0.697857
$262$	0	0
$263$	1.74317e25	0.678899	0.339449	−	0.940624i	$-0.389759\pi$
$263$	1.74317e25	0.678899	0.339449	+	0.940624i	$0.389759\pi$
$264$	0	0
$265$	−2.95619e24	−0.106329
$266$	0	0
$267$	2.43338e24	0.0808802
$268$	0	0
$269$	4.63224e25	1.42361	0.711807	−	0.702375i	$-0.247877\pi$
$269$	4.63224e25	1.42361	0.711807	+	0.702375i	$0.247877\pi$
$270$	0	0
$271$	−1.79840e25	−0.511338	−0.255669	−	0.966764i	$-0.582296\pi$
$271$	−1.79840e25	−0.511338	−0.255669	+	0.966764i	$0.582296\pi$
$272$	0	0
$273$	−1.27131e25	−0.334614
$274$	0	0
$275$	−4.82680e25	−1.17670
$276$	0	0
$277$	−1.07659e25	−0.243227	−0.121614	−	0.992578i	$-0.538807\pi$
$277$	−1.07659e25	−0.243227	−0.121614	+	0.992578i	$0.538807\pi$
$278$	0	0
$279$	−6.09888e24	−0.127763
$280$	0	0
$281$	−8.52851e25	−1.65751	−0.828756	−	0.559610i	$-0.810951\pi$
$281$	−8.52851e25	−1.65751	−0.828756	+	0.559610i	$0.810951\pi$
$282$	0	0
$283$	−8.16776e23	−0.0147348	−0.00736742	−	0.999973i	$-0.502345\pi$
$283$	−8.16776e23	−0.0147348	−0.00736742	+	0.999973i	$0.502345\pi$
$284$	0	0
$285$	3.25340e24	0.0545087
$286$	0	0
$287$	3.51331e25	0.546962
$288$	0	0
$289$	5.64373e25	0.816843
$290$	0	0
$291$	4.31579e25	0.581010
$292$	0	0
$293$	5.29341e25	0.663171	0.331586	−	0.943425i	$-0.392416\pi$
$293$	5.29341e25	0.663171	0.331586	+	0.943425i	$0.392416\pi$
$294$	0	0
$295$	1.47069e25	0.171550
$296$	0	0
$297$	1.10210e26	1.19750
$298$	0	0
$299$	1.94449e25	0.196906
$300$	0	0
$301$	−1.90399e26	−1.79769
$302$	0	0
$303$	−3.52316e25	−0.310302
$304$	0	0
$305$	3.97239e25	0.326516
$306$	0	0
$307$	1.79951e26	1.38103	0.690514	−	0.723319i	$-0.257385\pi$
$307$	1.79951e26	1.38103	0.690514	+	0.723319i	$0.257385\pi$
$308$	0	0
$309$	−4.12669e25	−0.295827
$310$	0	0
$311$	−2.49771e26	−1.67324	−0.836620	−	0.547783i	$-0.815472\pi$
$311$	−2.49771e26	−1.67324	−0.836620	+	0.547783i	$0.815472\pi$
$312$	0	0
$313$	2.14796e26	1.34527	0.672637	−	0.739972i	$-0.265161\pi$
$313$	2.14796e26	1.34527	0.672637	+	0.739972i	$0.265161\pi$
$314$	0	0
$315$	4.93014e25	0.288801
$316$	0	0
$317$	3.06450e26	1.67972	0.839860	−	0.542803i	$-0.182637\pi$
$317$	3.06450e26	1.67972	0.839860	+	0.542803i	$0.182637\pi$
$318$	0	0
$319$	2.54361e26	1.30511
$320$	0	0
$321$	−1.41266e26	−0.678789
$322$	0	0
$323$	1.23513e26	0.556016
$324$	0	0
$325$	6.78805e25	0.286398
$326$	0	0
$327$	−1.19767e26	−0.473789
$328$	0	0
$329$	−2.74814e26	−1.01972
$330$	0	0
$331$	1.04905e26	0.365261	0.182631	−	0.983182i	$-0.441539\pi$
$331$	1.04905e26	0.365261	0.182631	+	0.983182i	$0.441539\pi$
$332$	0	0
$333$	−2.16103e26	−0.706313
$334$	0	0
$335$	−2.40660e25	−0.0738640
$336$	0	0
$337$	−1.95001e26	−0.562242	−0.281121	−	0.959672i	$-0.590706\pi$
$337$	−1.95001e26	−0.562242	−0.281121	+	0.959672i	$0.590706\pi$
$338$	0	0
$339$	−1.07940e26	−0.292472
$340$	0	0
$341$	9.38012e25	0.238939
$342$	0	0
$343$	−1.31388e27	−3.14751
$344$	0	0
$345$	3.82186e25	0.0861339
$346$	0	0
$347$	5.59947e26	1.18765	0.593824	−	0.804595i	$-0.297618\pi$
$347$	5.59947e26	1.18765	0.593824	+	0.804595i	$0.297618\pi$
$348$	0	0
$349$	−2.09819e26	−0.418966	−0.209483	−	0.977812i	$-0.567178\pi$
$349$	−2.09819e26	−0.418966	−0.209483	+	0.977812i	$0.567178\pi$
$350$	0	0
$351$	−1.54991e26	−0.291462
$352$	0	0
$353$	5.72422e26	1.01410	0.507051	−	0.861916i	$-0.330736\pi$
$353$	5.72422e26	1.01410	0.507051	+	0.861916i	$0.330736\pi$
$354$	0	0
$355$	−4.40270e25	−0.0735057
$356$	0	0
$357$	−9.48632e26	−1.49307
$358$	0	0
$359$	4.84990e26	0.719848	0.359924	−	0.932982i	$-0.382803\pi$
$359$	4.84990e26	0.719848	0.359924	+	0.932982i	$0.382803\pi$
$360$	0	0
$361$	−5.92680e26	−0.829840
$362$	0	0
$363$	−2.37213e26	−0.313414
$364$	0	0
$365$	1.82518e26	0.227630
$366$	0	0
$367$	−6.65755e26	−0.784008	−0.392004	−	0.919963i	$-0.628218\pi$
$367$	−6.65755e26	−0.784008	−0.392004	+	0.919963i	$0.628218\pi$
$368$	0	0
$369$	1.70862e26	0.190050
$370$	0	0
$371$	8.48128e26	0.891324
$372$	0	0
$373$	1.46537e27	1.45547	0.727735	−	0.685858i	$-0.240573\pi$
$373$	1.46537e27	1.45547	0.727735	+	0.685858i	$0.240573\pi$
$374$	0	0
$375$	2.74141e26	0.257422
$376$	0	0
$377$	−3.57714e26	−0.317652
$378$	0	0
$379$	1.80711e27	1.51800	0.759002	−	0.651088i	$-0.225687\pi$
$379$	1.80711e27	1.51800	0.759002	+	0.651088i	$0.225687\pi$
$380$	0	0
$381$	1.30626e27	1.03829
$382$	0	0
$383$	−9.69700e26	−0.729542	−0.364771	−	0.931097i	$-0.618853\pi$
$383$	−9.69700e26	−0.729542	−0.364771	+	0.931097i	$0.618853\pi$
$384$	0	0
$385$	−7.58260e26	−0.540108
$386$	0	0
$387$	−9.25962e26	−0.624637
$388$	0	0
$389$	3.19483e25	0.0204163	0.0102081	−	0.999948i	$-0.496751\pi$
$389$	3.19483e25	0.0204163	0.0102081	+	0.999948i	$0.496751\pi$
$390$	0	0
$391$	1.45095e27	0.878609
$392$	0	0
$393$	8.54954e26	0.490706
$394$	0	0
$395$	−1.68369e26	−0.0916204
$396$	0	0
$397$	−5.23867e26	−0.270346	−0.135173	−	0.990822i	$-0.543159\pi$
$397$	−5.23867e26	−0.270346	−0.135173	+	0.990822i	$0.543159\pi$
$398$	0	0
$399$	−9.33398e26	−0.456932
$400$	0	0
$401$	1.65135e27	0.767049	0.383525	−	0.923531i	$-0.374710\pi$
$401$	1.65135e27	0.767049	0.383525	+	0.923531i	$0.374710\pi$
$402$	0	0
$403$	−1.31915e26	−0.0581557
$404$	0	0
$405$	5.66558e25	0.0237119
$406$	0	0
$407$	3.32369e27	1.32093
$408$	0	0
$409$	1.41506e27	0.534172	0.267086	−	0.963673i	$-0.413939\pi$
$409$	1.41506e27	0.534172	0.267086	+	0.963673i	$0.413939\pi$
$410$	0	0
$411$	2.12127e27	0.760778
$412$	0	0
$413$	−4.21941e27	−1.43806
$414$	0	0
$415$	−1.01604e27	−0.329159
$416$	0	0
$417$	−1.24819e27	−0.384461
$418$	0	0
$419$	−1.80515e27	−0.528769	−0.264385	−	0.964417i	$-0.585169\pi$
$419$	−1.80515e27	−0.528769	−0.264385	+	0.964417i	$0.585169\pi$
$420$	0	0
$421$	−1.98386e27	−0.552775	−0.276388	−	0.961046i	$-0.589137\pi$
$421$	−1.98386e27	−0.552775	−0.276388	+	0.961046i	$0.589137\pi$
$422$	0	0
$423$	−1.33650e27	−0.354318
$424$	0	0
$425$	5.06513e27	1.27793
$426$	0	0
$427$	−1.13968e28	−2.73709
$428$	0	0
$429$	9.50910e26	0.217439
$430$	0	0
$431$	3.46591e27	0.754756	0.377378	−	0.926059i	$-0.376826\pi$
$431$	3.46591e27	0.754756	0.377378	+	0.926059i	$0.376826\pi$
$432$	0	0
$433$	−5.88060e27	−1.21983	−0.609915	−	0.792467i	$-0.708796\pi$
$433$	−5.88060e27	−1.21983	−0.609915	+	0.792467i	$0.708796\pi$
$434$	0	0
$435$	−7.03080e26	−0.138953
$436$	0	0
$437$	1.42765e27	0.268884
$438$	0	0
$439$	6.95231e26	0.124811	0.0624053	−	0.998051i	$-0.480123\pi$
$439$	6.95231e26	0.124811	0.0624053	+	0.998051i	$0.480123\pi$
$440$	0	0
$441$	−1.02672e28	−1.75730
$442$	0	0
$443$	4.77606e26	0.0779526	0.0389763	−	0.999240i	$-0.487590\pi$
$443$	4.77606e26	0.0779526	0.0389763	+	0.999240i	$0.487590\pi$
$444$	0	0
$445$	−2.04109e26	−0.0317747
$446$	0	0
$447$	5.17182e27	0.768094
$448$	0	0
$449$	6.02792e27	0.854241	0.427121	−	0.904195i	$-0.359528\pi$
$449$	6.02792e27	0.854241	0.427121	+	0.904195i	$0.359528\pi$
$450$	0	0
$451$	−2.62787e27	−0.355427
$452$	0	0
$453$	−2.37660e27	−0.306849
$454$	0	0
$455$	1.06636e27	0.131457
$456$	0	0
$457$	−3.92994e27	−0.462664	−0.231332	−	0.972875i	$-0.574308\pi$
$457$	−3.92994e27	−0.462664	−0.231332	+	0.972875i	$0.574308\pi$
$458$	0	0
$459$	−1.15652e28	−1.30052
$460$	0	0
$461$	1.19082e28	1.27934	0.639672	−	0.768648i	$-0.279070\pi$
$461$	1.19082e28	1.27934	0.639672	+	0.768648i	$0.279070\pi$
$462$	0	0
$463$	6.15811e27	0.632189	0.316095	−	0.948728i	$-0.397628\pi$
$463$	6.15811e27	0.632189	0.316095	+	0.948728i	$0.397628\pi$
$464$	0	0
$465$	−2.59277e26	−0.0254394
$466$	0	0
$467$	1.30147e28	1.22069	0.610346	−	0.792135i	$-0.291030\pi$
$467$	1.30147e28	1.22069	0.610346	+	0.792135i	$0.291030\pi$
$468$	0	0
$469$	6.90452e27	0.619182
$470$	0	0
$471$	2.73115e27	0.234221
$472$	0	0
$473$	1.42414e28	1.16818
$474$	0	0
$475$	4.98379e27	0.391090
$476$	0	0
$477$	4.12468e27	0.309705
$478$	0	0
$479$	1.70133e28	1.22255	0.611273	−	0.791420i	$-0.290658\pi$
$479$	1.70133e28	1.22255	0.611273	+	0.791420i	$0.290658\pi$
$480$	0	0
$481$	−4.67419e27	−0.321501
$482$	0	0
$483$	−1.09649e28	−0.722038
$484$	0	0
$485$	−3.62003e27	−0.228257
$486$	0	0
$487$	2.45443e27	0.148216	0.0741082	−	0.997250i	$-0.476389\pi$
$487$	2.45443e27	0.148216	0.0741082	+	0.997250i	$0.476389\pi$
$488$	0	0
$489$	3.39363e27	0.196301
$490$	0	0
$491$	−3.07994e28	−1.70682	−0.853408	−	0.521243i	$-0.825468\pi$
$491$	−3.07994e28	−1.70682	−0.853408	+	0.521243i	$0.825468\pi$
$492$	0	0
$493$	−2.66920e28	−1.41739
$494$	0	0
$495$	−3.68763e27	−0.187669
$496$	0	0
$497$	1.26313e28	0.616178
$498$	0	0
$499$	2.07463e28	0.970252	0.485126	−	0.874444i	$-0.338774\pi$
$499$	2.07463e28	0.970252	0.485126	+	0.874444i	$0.338774\pi$
$500$	0	0
$501$	−7.88388e27	−0.353544
$502$	0	0
$503$	3.54838e28	1.52604	0.763022	−	0.646373i	$-0.223715\pi$
$503$	3.54838e28	1.52604	0.763022	+	0.646373i	$0.223715\pi$
$504$	0	0
$505$	2.95518e27	0.121906
$506$	0	0
$507$	1.33176e28	0.527038
$508$	0	0
$509$	2.04071e28	0.774900	0.387450	−	0.921891i	$-0.373356\pi$
$509$	2.04071e28	0.774900	0.387450	+	0.921891i	$0.373356\pi$
$510$	0	0
$511$	−5.23643e28	−1.90816
$512$	0	0
$513$	−1.13794e28	−0.398005
$514$	0	0
$515$	3.46141e27	0.116219
$516$	0	0
$517$	2.05554e28	0.662636
$518$	0	0
$519$	−3.28098e28	−1.01566
$520$	0	0
$521$	−2.64349e28	−0.785927	−0.392963	−	0.919554i	$-0.628550\pi$
$521$	−2.64349e28	−0.785927	−0.392963	+	0.919554i	$0.628550\pi$
$522$	0	0
$523$	−3.70305e28	−1.05753	−0.528763	−	0.848769i	$-0.677344\pi$
$523$	−3.70305e28	−1.05753	−0.528763	+	0.848769i	$0.677344\pi$
$524$	0	0
$525$	−3.82775e28	−1.05020
$526$	0	0
$527$	−9.84329e27	−0.259495
$528$	0	0
$529$	−2.27006e28	−0.575112
$530$	0	0
$531$	−2.05201e28	−0.499675
$532$	0	0
$533$	3.69564e27	0.0865077
$534$	0	0
$535$	1.18492e28	0.266670
$536$	0	0
$537$	−3.15252e27	−0.0682225
$538$	0	0
$539$	1.57910e29	3.28645
$540$	0	0
$541$	−9.59401e27	−0.192056	−0.0960282	−	0.995379i	$-0.530614\pi$
$541$	−9.59401e27	−0.192056	−0.0960282	+	0.995379i	$0.530614\pi$
$542$	0	0
$543$	−4.50416e28	−0.867392
$544$	0	0
$545$	1.00459e28	0.186134
$546$	0	0
$547$	−7.60716e28	−1.35630	−0.678150	−	0.734924i	$-0.737218\pi$
$547$	−7.60716e28	−1.35630	−0.678150	+	0.734924i	$0.737218\pi$
$548$	0	0
$549$	−5.54257e28	−0.951046
$550$	0	0
$551$	−2.62634e28	−0.433769
$552$	0	0
$553$	4.83050e28	0.768029
$554$	0	0
$555$	−9.18702e27	−0.140637
$556$	0	0
$557$	4.97908e28	0.733956	0.366978	−	0.930230i	$-0.380392\pi$
$557$	4.97908e28	0.733956	0.366978	+	0.930230i	$0.380392\pi$
$558$	0	0
$559$	−2.00280e28	−0.284324
$560$	0	0
$561$	7.09553e28	0.970230
$562$	0	0
$563$	8.49629e28	1.11916	0.559579	−	0.828777i	$-0.310963\pi$
$563$	8.49629e28	1.11916	0.559579	+	0.828777i	$0.310963\pi$
$564$	0	0
$565$	9.05383e27	0.114901
$566$	0	0
$567$	−1.62545e28	−0.198771
$568$	0	0
$569$	5.57042e28	0.656461	0.328230	−	0.944598i	$-0.393548\pi$
$569$	5.57042e28	0.656461	0.328230	+	0.944598i	$0.393548\pi$
$570$	0	0
$571$	1.58127e29	1.79608	0.898041	−	0.439911i	$-0.144990\pi$
$571$	1.58127e29	1.79608	0.898041	+	0.439911i	$0.144990\pi$
$572$	0	0
$573$	−5.72206e28	−0.626511
$574$	0	0
$575$	5.85460e28	0.617995
$576$	0	0
$577$	−1.14412e29	−1.16446	−0.582229	−	0.813025i	$-0.697819\pi$
$577$	−1.14412e29	−1.16446	−0.582229	+	0.813025i	$0.697819\pi$
$578$	0	0
$579$	2.02381e28	0.198630
$580$	0	0
$581$	2.91501e29	2.75925
$582$	0	0
$583$	−6.34379e28	−0.579200
$584$	0	0
$585$	5.18601e27	0.0456769
$586$	0	0
$587$	−3.03657e28	−0.258038	−0.129019	−	0.991642i	$-0.541183\pi$
$587$	−3.03657e28	−0.258038	−0.129019	+	0.991642i	$0.541183\pi$
$588$	0	0
$589$	−9.68522e27	−0.0794143
$590$	0	0
$591$	2.96931e28	0.234956
$592$	0	0
$593$	−1.80310e29	−1.37704	−0.688519	−	0.725218i	$-0.741739\pi$
$593$	−1.80310e29	−1.37704	−0.688519	+	0.725218i	$0.741739\pi$
$594$	0	0
$595$	7.95700e28	0.586572
$596$	0	0
$597$	−3.02437e28	−0.215231
$598$	0	0
$599$	−2.43184e29	−1.67091	−0.835455	−	0.549558i	$-0.814796\pi$
$599$	−2.43184e29	−1.67091	−0.835455	+	0.549558i	$0.814796\pi$
$600$	0	0
$601$	−1.53373e29	−1.01758	−0.508789	−	0.860891i	$-0.669907\pi$
$601$	−1.53373e29	−1.01758	−0.508789	+	0.860891i	$0.669907\pi$
$602$	0	0
$603$	3.35786e28	0.215145
$604$	0	0
$605$	1.98971e28	0.123128
$606$	0	0
$607$	2.90773e28	0.173809	0.0869047	−	0.996217i	$-0.472302\pi$
$607$	2.90773e28	0.173809	0.0869047	+	0.996217i	$0.472302\pi$
$608$	0	0
$609$	2.01713e29	1.16480
$610$	0	0
$611$	−2.89076e28	−0.161280
$612$	0	0
$613$	−1.21635e29	−0.655726	−0.327863	−	0.944725i	$-0.606329\pi$
$613$	−1.21635e29	−0.655726	−0.327863	+	0.944725i	$0.606329\pi$
$614$	0	0
$615$	7.26371e27	0.0378416
$616$	0	0
$617$	8.98199e28	0.452250	0.226125	−	0.974098i	$-0.427394\pi$
$617$	8.98199e28	0.452250	0.226125	+	0.974098i	$0.427394\pi$
$618$	0	0
$619$	−3.27118e29	−1.59203	−0.796017	−	0.605275i	$-0.793063\pi$
$619$	−3.27118e29	−1.59203	−0.796017	+	0.605275i	$0.793063\pi$
$620$	0	0
$621$	−1.33677e29	−0.628922
$622$	0	0
$623$	5.85588e28	0.266359
$624$	0	0
$625$	1.92575e29	0.846956
$626$	0	0
$627$	6.98158e28	0.296924
$628$	0	0
$629$	−3.48780e29	−1.43456
$630$	0	0
$631$	−9.03181e28	−0.359308	−0.179654	−	0.983730i	$-0.557498\pi$
$631$	−9.03181e28	−0.359308	−0.179654	+	0.983730i	$0.557498\pi$
$632$	0	0
$633$	4.32993e28	0.166626
$634$	0	0
$635$	−1.09567e29	−0.407903
$636$	0	0
$637$	−2.22072e29	−0.799891
$638$	0	0
$639$	6.14296e28	0.214101
$640$	0	0
$641$	−3.16031e29	−1.06591	−0.532955	−	0.846144i	$-0.678918\pi$
$641$	−3.16031e29	−1.06591	−0.532955	+	0.846144i	$0.678918\pi$
$642$	0	0
$643$	1.84895e29	0.603546	0.301773	−	0.953380i	$-0.402422\pi$
$643$	1.84895e29	0.603546	0.301773	+	0.953380i	$0.402422\pi$
$644$	0	0
$645$	−3.93646e28	−0.124374
$646$	0	0
$647$	1.55883e29	0.476765	0.238383	−	0.971171i	$-0.423383\pi$
$647$	1.55883e29	0.476765	0.238383	+	0.971171i	$0.423383\pi$
$648$	0	0
$649$	3.15601e29	0.934478
$650$	0	0
$651$	7.43864e28	0.213252
$652$	0	0
$653$	5.45219e29	1.51350	0.756751	−	0.653704i	$-0.226786\pi$
$653$	5.45219e29	1.51350	0.756751	+	0.653704i	$0.226786\pi$
$654$	0	0
$655$	−7.17125e28	−0.192780
$656$	0	0
$657$	−2.54662e29	−0.663022
$658$	0	0
$659$	−3.88443e29	−0.979558	−0.489779	−	0.871847i	$-0.662923\pi$
$659$	−3.88443e29	−0.979558	−0.489779	+	0.871847i	$0.662923\pi$
$660$	0	0
$661$	5.22430e29	1.27618	0.638091	−	0.769961i	$-0.279724\pi$
$661$	5.22430e29	1.27618	0.638091	+	0.769961i	$0.279724\pi$
$662$	0	0
$663$	−9.97864e28	−0.236145
$664$	0	0
$665$	7.82922e28	0.179511
$666$	0	0
$667$	−3.08524e29	−0.685437
$668$	0	0
$669$	−3.48195e29	−0.749633
$670$	0	0
$671$	8.52451e29	1.77862
$672$	0	0
$673$	−3.52401e29	−0.712655	−0.356327	−	0.934361i	$-0.615971\pi$
$673$	−3.52401e29	−0.712655	−0.356327	+	0.934361i	$0.615971\pi$
$674$	0	0
$675$	−4.66656e29	−0.914761
$676$	0	0
$677$	4.45535e29	0.846644	0.423322	−	0.905979i	$-0.360864\pi$
$677$	4.45535e29	0.846644	0.423322	+	0.905979i	$0.360864\pi$
$678$	0	0
$679$	1.03858e30	1.91341
$680$	0	0
$681$	3.63961e29	0.650144
$682$	0	0
$683$	6.32620e29	1.09579	0.547893	−	0.836548i	$-0.315430\pi$
$683$	6.32620e29	1.09579	0.547893	+	0.836548i	$0.315430\pi$
$684$	0	0
$685$	−1.77930e29	−0.298881
$686$	0	0
$687$	1.81042e29	0.294940
$688$	0	0
$689$	8.92144e28	0.140972
$690$	0	0
$691$	−3.28853e29	−0.504061	−0.252030	−	0.967719i	$-0.581098\pi$
$691$	−3.28853e29	−0.504061	−0.252030	+	0.967719i	$0.581098\pi$
$692$	0	0
$693$	1.05798e30	1.57318
$694$	0	0
$695$	1.04697e29	0.151040
$696$	0	0
$697$	2.75763e29	0.386004
$698$	0	0
$699$	4.42494e28	0.0601032
$700$	0	0
$701$	−1.17358e30	−1.54694	−0.773471	−	0.633832i	$-0.781481\pi$
$701$	−1.17358e30	−1.54694	−0.773471	+	0.633832i	$0.781481\pi$
$702$	0	0
$703$	−3.43179e29	−0.439025
$704$	0	0
$705$	−5.68173e28	−0.0705496
$706$	0	0
$707$	−8.47841e29	−1.02190
$708$	0	0
$709$	4.17850e28	0.0488917	0.0244458	−	0.999701i	$-0.492218\pi$
$709$	4.17850e28	0.0488917	0.0244458	+	0.999701i	$0.492218\pi$
$710$	0	0
$711$	2.34920e29	0.266864
$712$	0	0
$713$	−1.13775e29	−0.125489
$714$	0	0
$715$	−7.97611e28	−0.0854236
$716$	0	0
$717$	−6.08350e29	−0.632706
$718$	0	0
$719$	−1.11149e30	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$719$	−1.11149e30	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$720$	0	0
$721$	−9.93077e29	−0.974233
$722$	0	0
$723$	−8.69926e29	−0.828954
$724$	0	0
$725$	−1.07703e30	−0.996962
$726$	0	0
$727$	1.37984e30	1.24084	0.620421	−	0.784269i	$-0.286962\pi$
$727$	1.37984e30	1.24084	0.620421	+	0.784269i	$0.286962\pi$
$728$	0	0
$729$	5.61417e29	0.490509
$730$	0	0
$731$	−1.49446e30	−1.26868
$732$	0	0
$733$	2.24957e30	1.85570	0.927849	−	0.372956i	$-0.121656\pi$
$733$	2.24957e30	1.85570	0.927849	+	0.372956i	$0.121656\pi$
$734$	0	0
$735$	−4.36479e29	−0.349902
$736$	0	0
$737$	−5.16441e29	−0.402357
$738$	0	0
$739$	1.72254e30	1.30438	0.652188	−	0.758057i	$-0.273851\pi$
$739$	1.72254e30	1.30438	0.652188	+	0.758057i	$0.273851\pi$
$740$	0	0
$741$	−9.81839e28	−0.0722685
$742$	0	0
$743$	−1.06638e30	−0.763006	−0.381503	−	0.924368i	$-0.624593\pi$
$743$	−1.06638e30	−0.763006	−0.381503	+	0.924368i	$0.624593\pi$
$744$	0	0
$745$	−4.33806e29	−0.301755
$746$	0	0
$747$	1.41765e30	0.958744
$748$	0	0
$749$	−3.39954e30	−2.23543
$750$	0	0
$751$	−4.90593e29	−0.313691	−0.156846	−	0.987623i	$-0.550132\pi$
$751$	−4.90593e29	−0.313691	−0.156846	+	0.987623i	$0.550132\pi$
$752$	0	0
$753$	−9.06151e29	−0.563447
$754$	0	0
$755$	1.99346e29	0.120549
$756$	0	0
$757$	1.80527e30	1.06178	0.530891	−	0.847440i	$-0.321857\pi$
$757$	1.80527e30	1.06178	0.530891	+	0.847440i	$0.321857\pi$
$758$	0	0
$759$	8.20147e29	0.469195
$760$	0	0
$761$	1.03009e30	0.573240	0.286620	−	0.958044i	$-0.407468\pi$
$761$	1.03009e30	0.573240	0.286620	+	0.958044i	$0.407468\pi$
$762$	0	0
$763$	−2.88216e30	−1.56031
$764$	0	0
$765$	3.86971e29	0.203814
$766$	0	0
$767$	−4.43838e29	−0.227443
$768$	0	0
$769$	7.98485e29	0.398144	0.199072	−	0.979985i	$-0.436207\pi$
$769$	7.98485e29	0.398144	0.199072	+	0.979985i	$0.436207\pi$
$770$	0	0
$771$	2.17140e30	1.05358
$772$	0	0
$773$	2.56868e30	1.21290	0.606450	−	0.795122i	$-0.292593\pi$
$773$	2.56868e30	1.21290	0.606450	+	0.795122i	$0.292593\pi$
$774$	0	0
$775$	−3.97179e29	−0.182523
$776$	0	0
$777$	2.63575e30	1.17892
$778$	0	0
$779$	2.71334e29	0.118130
$780$	0	0
$781$	−9.44791e29	−0.400405
$782$	0	0
$783$	2.45917e30	1.01459
$784$	0	0
$785$	−2.29086e29	−0.0920166
$786$	0	0
$787$	3.43359e30	1.34281	0.671403	−	0.741092i	$-0.265692\pi$
$787$	3.43359e30	1.34281	0.671403	+	0.741092i	$0.265692\pi$
$788$	0	0
$789$	−1.03398e30	−0.393735
$790$	0	0
$791$	−2.59754e30	−0.963184
$792$	0	0
$793$	−1.19882e30	−0.432900
$794$	0	0
$795$	1.75349e29	0.0616664
$796$	0	0
$797$	−3.29683e30	−1.12923	−0.564617	−	0.825353i	$-0.690976\pi$
$797$	−3.29683e30	−1.12923	−0.564617	+	0.825353i	$0.690976\pi$
$798$	0	0
$799$	−2.15704e30	−0.719641
$800$	0	0
$801$	2.84788e29	0.0925507
$802$	0	0
$803$	3.91672e30	1.23996
$804$	0	0
$805$	9.19722e29	0.283661
$806$	0	0
$807$	−2.74766e30	−0.825640
$808$	0	0
$809$	5.34854e30	1.56594	0.782971	−	0.622059i	$-0.213703\pi$
$809$	5.34854e30	1.56594	0.782971	+	0.622059i	$0.213703\pi$
$810$	0	0
$811$	5.96622e30	1.70208	0.851041	−	0.525099i	$-0.175972\pi$
$811$	5.96622e30	1.70208	0.851041	+	0.525099i	$0.175972\pi$
$812$	0	0
$813$	1.06674e30	0.296556
$814$	0	0
$815$	−2.84653e29	−0.0771190
$816$	0	0
$817$	−1.47046e30	−0.388258
$818$	0	0
$819$	−1.48786e30	−0.382897
$820$	0	0
$821$	6.73645e30	1.68977	0.844886	−	0.534947i	$-0.179668\pi$
$821$	6.73645e30	1.68977	0.844886	+	0.534947i	$0.179668\pi$
$822$	0	0
$823$	2.93477e29	0.0717588	0.0358794	−	0.999356i	$-0.488577\pi$
$823$	2.93477e29	0.0717588	0.0358794	+	0.999356i	$0.488577\pi$
$824$	0	0
$825$	2.86306e30	0.682440
$826$	0	0
$827$	−6.88636e30	−1.60023	−0.800114	−	0.599847i	$-0.795228\pi$
$827$	−6.88636e30	−1.60023	−0.800114	+	0.599847i	$0.795228\pi$
$828$	0	0
$829$	−4.58006e30	−1.03764	−0.518822	−	0.854882i	$-0.673629\pi$
$829$	−4.58006e30	−1.03764	−0.518822	+	0.854882i	$0.673629\pi$
$830$	0	0
$831$	6.38590e29	0.141062
$832$	0	0
$833$	−1.65707e31	−3.56917
$834$	0	0
$835$	6.61290e29	0.138894
$836$	0	0
$837$	9.06874e29	0.185750
$838$	0	0
$839$	−6.83025e30	−1.36438	−0.682191	−	0.731174i	$-0.738973\pi$
$839$	−6.83025e30	−1.36438	−0.682191	+	0.731174i	$0.738973\pi$
$840$	0	0
$841$	5.42848e29	0.105760
$842$	0	0
$843$	5.05877e30	0.961292
$844$	0	0
$845$	−1.11706e30	−0.207053
$846$	0	0
$847$	−5.70847e30	−1.03215
$848$	0	0
$849$	4.84479e28	0.00854563
$850$	0	0
$851$	−4.03142e30	−0.693742
$852$	0	0
$853$	6.29163e30	1.05633	0.528164	−	0.849143i	$-0.322881\pi$
$853$	6.29163e30	1.05633	0.528164	+	0.849143i	$0.322881\pi$
$854$	0	0
$855$	3.80757e29	0.0623740
$856$	0	0
$857$	−7.23134e30	−1.15590	−0.577950	−	0.816072i	$-0.696147\pi$
$857$	−7.23134e30	−1.15590	−0.577950	+	0.816072i	$0.696147\pi$
$858$	0	0
$859$	3.32473e30	0.518594	0.259297	−	0.965798i	$-0.416509\pi$
$859$	3.32473e30	0.518594	0.259297	+	0.965798i	$0.416509\pi$
$860$	0	0
$861$	−2.08396e30	−0.317216
$862$	0	0
$863$	3.34330e30	0.496664	0.248332	−	0.968675i	$-0.420118\pi$
$863$	3.34330e30	0.496664	0.248332	+	0.968675i	$0.420118\pi$
$864$	0	0
$865$	2.75204e30	0.399012
$866$	0	0
$867$	−3.34763e30	−0.473737
$868$	0	0
$869$	−3.61309e30	−0.499081
$870$	0	0
$871$	7.26285e29	0.0979301
$872$	0	0
$873$	5.05092e30	0.664846
$874$	0	0
$875$	6.59714e30	0.847756
$876$	0	0
$877$	−1.00355e31	−1.25905	−0.629523	−	0.776982i	$-0.716749\pi$
$877$	−1.00355e31	−1.25905	−0.629523	+	0.776982i	$0.716749\pi$
$878$	0	0
$879$	−3.13984e30	−0.384613
$880$	0	0
$881$	−2.14507e29	−0.0256563	−0.0128281	−	0.999918i	$-0.504083\pi$
$881$	−2.14507e29	−0.0256563	−0.0128281	+	0.999918i	$0.504083\pi$
$882$	0	0
$883$	−8.37555e30	−0.978196	−0.489098	−	0.872229i	$-0.662674\pi$
$883$	−8.37555e30	−0.978196	−0.489098	+	0.872229i	$0.662674\pi$
$884$	0	0
$885$	−8.72355e29	−0.0994922
$886$	0	0
$887$	6.39937e30	0.712754	0.356377	−	0.934342i	$-0.384012\pi$
$887$	6.39937e30	0.712754	0.356377	+	0.934342i	$0.384012\pi$
$888$	0	0
$889$	3.14348e31	3.41934
$890$	0	0
$891$	1.21580e30	0.129165
$892$	0	0
$893$	−2.12240e30	−0.220235
$894$	0	0
$895$	2.64429e29	0.0268020
$896$	0	0
$897$	−1.15340e30	−0.114198
$898$	0	0
$899$	2.09304e30	0.202442
$900$	0	0
$901$	6.65703e30	0.629028
$902$	0	0
$903$	1.12937e31	1.04259
$904$	0	0
$905$	3.77803e30	0.340765
$906$	0	0
$907$	−3.33701e30	−0.294091	−0.147045	−	0.989130i	$-0.546976\pi$
$907$	−3.33701e30	−0.294091	−0.147045	+	0.989130i	$0.546976\pi$
$908$	0	0
$909$	−4.12328e30	−0.355077
$910$	0	0
$911$	5.75477e30	0.484267	0.242134	−	0.970243i	$-0.422153\pi$
$911$	5.75477e30	0.484267	0.242134	+	0.970243i	$0.422153\pi$
$912$	0	0
$913$	−2.18036e31	−1.79302
$914$	0	0
$915$	−2.35627e30	−0.189366
$916$	0	0
$917$	2.05743e31	1.61602
$918$	0	0
$919$	2.25413e31	1.73048	0.865239	−	0.501359i	$-0.167166\pi$
$919$	2.25413e31	1.73048	0.865239	+	0.501359i	$0.167166\pi$
$920$	0	0
$921$	−1.06740e31	−0.800942
$922$	0	0
$923$	1.32868e30	0.0974550
$924$	0	0
$925$	−1.40733e31	−1.00904
$926$	0	0
$927$	−4.82961e30	−0.338513
$928$	0	0
$929$	2.59042e30	0.177503	0.0887513	−	0.996054i	$-0.471712\pi$
$929$	2.59042e30	0.177503	0.0887513	+	0.996054i	$0.471712\pi$
$930$	0	0
$931$	−1.63046e31	−1.09229
$932$	0	0
$933$	1.48154e31	0.970414
$934$	0	0
$935$	−5.95164e30	−0.381167
$936$	0	0
$937$	5.29691e30	0.331709	0.165854	−	0.986150i	$-0.446962\pi$
$937$	5.29691e30	0.331709	0.165854	+	0.986150i	$0.446962\pi$
$938$	0	0
$939$	−1.27408e31	−0.780207
$940$	0	0
$941$	1.38757e31	0.830930	0.415465	−	0.909609i	$-0.363619\pi$
$941$	1.38757e31	0.830930	0.415465	+	0.909609i	$0.363619\pi$
$942$	0	0
$943$	3.18744e30	0.186668
$944$	0	0
$945$	−7.33088e30	−0.419877
$946$	0	0
$947$	8.16878e30	0.457596	0.228798	−	0.973474i	$-0.426520\pi$
$947$	8.16878e30	0.457596	0.228798	+	0.973474i	$0.426520\pi$
$948$	0	0
$949$	−5.50819e30	−0.301796
$950$	0	0
$951$	−1.81774e31	−0.974172
$952$	0	0
$953$	2.02516e31	1.06166	0.530828	−	0.847479i	$-0.321881\pi$
$953$	2.02516e31	1.06166	0.530828	+	0.847479i	$0.321881\pi$
$954$	0	0
$955$	4.79959e30	0.246132
$956$	0	0
$957$	−1.50877e31	−0.756914
$958$	0	0
$959$	5.10480e31	2.50544
$960$	0	0
$961$	−2.00537e31	−0.962937
$962$	0	0
$963$	−1.65329e31	−0.776734
$964$	0	0
$965$	−1.69755e30	−0.0780343
$966$	0	0
$967$	−1.34251e31	−0.603864	−0.301932	−	0.953329i	$-0.597632\pi$
$967$	−1.34251e31	−0.603864	−0.301932	+	0.953329i	$0.597632\pi$
$968$	0	0
$969$	−7.32632e30	−0.322468
$970$	0	0
$971$	−4.11059e31	−1.77053	−0.885263	−	0.465090i	$-0.846022\pi$
$971$	−4.11059e31	−1.77053	−0.885263	+	0.465090i	$0.846022\pi$
$972$	0	0
$973$	−3.00374e31	−1.26613
$974$	0	0
$975$	−4.02640e30	−0.166100
$976$	0	0
$977$	3.60772e30	0.145660	0.0728299	−	0.997344i	$-0.476797\pi$
$977$	3.60772e30	0.145660	0.0728299	+	0.997344i	$0.476797\pi$
$978$	0	0
$979$	−4.38005e30	−0.173086
$980$	0	0
$981$	−1.40167e31	−0.542154
$982$	0	0
$983$	−3.62646e31	−1.37300	−0.686500	−	0.727130i	$-0.740854\pi$
$983$	−3.62646e31	−1.37300	−0.686500	+	0.727130i	$0.740854\pi$
$984$	0	0
$985$	−2.49062e30	−0.0923053
$986$	0	0
$987$	1.63009e31	0.591399
$988$	0	0
$989$	−1.72739e31	−0.613520
$990$	0	0
$991$	2.38846e31	0.830508	0.415254	−	0.909706i	$-0.363693\pi$
$991$	2.38846e31	0.830508	0.415254	+	0.909706i	$0.363693\pi$
$992$	0	0
$993$	−6.22256e30	−0.211837
$994$	0	0
$995$	2.53681e30	0.0845561
$996$	0	0
$997$	5.34032e31	1.74288	0.871442	−	0.490499i	$-0.163186\pi$
$997$	5.34032e31	1.74288	0.871442	+	0.490499i	$0.163186\pi$
$998$	0	0
$999$	3.21335e31	1.02688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.a.b.1.1		1
4.3	odd	2		2.22.a.b.1.1	✓	1
8.3	odd	2		64.22.a.c.1.1		1
8.5	even	2		64.22.a.e.1.1		1
12.11	even	2		18.22.a.b.1.1		1
20.3	even	4		50.22.b.c.49.1		2
20.7	even	4		50.22.b.c.49.2		2
20.19	odd	2		50.22.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.22.a.b.1.1	✓	1	4.3	odd	2
16.22.a.b.1.1		1	1.1	even	1	trivial
18.22.a.b.1.1		1	12.11	even	2
50.22.a.a.1.1		1	20.19	odd	2
50.22.b.c.49.1		2	20.3	even	4
50.22.b.c.49.2		2	20.7	even	4
64.22.a.c.1.1		1	8.3	odd	2
64.22.a.e.1.1		1	8.5	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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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