LMFDB - Embedded newform 20.25.d.b.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,25,Mod(19,20)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 25, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20.19"); S:= CuspForms(chi, 25); N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 25 $
Character orbit:	$[\chi]$	$=$	20.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,4096] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.9934304516$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			19.1
Character	$\chi$	$=$	20.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+4096.00 q^{2} -834398. q^{3} +1.67772e7 q^{4} +2.44141e8 q^{5} -3.41769e9 q^{6} -2.61398e10 q^{7} +6.87195e10 q^{8} +4.13790e11 q^{9} +1.00000e12 q^{10} -1.39989e13 q^{12} -1.07069e14 q^{14} -2.03710e14 q^{15} +2.81475e14 q^{16} +1.69489e15 q^{18} +4.09600e15 q^{20} +2.18110e16 q^{21} -4.01974e16 q^{23} -5.73394e16 q^{24} +5.96046e16 q^{25} -1.09607e17 q^{27} -4.38553e17 q^{28} -3.09695e16 q^{29} -8.34398e17 q^{30} +1.15292e18 q^{32} -6.38179e18 q^{35} +6.94225e18 q^{36} +1.67772e19 q^{40} -2.93337e19 q^{41} +8.93379e19 q^{42} +7.23521e19 q^{43} +1.01023e20 q^{45} -1.64648e20 q^{46} +2.02721e20 q^{47} -2.34862e20 q^{48} +4.91709e20 q^{49} +2.44141e20 q^{50} -4.48952e20 q^{54} -1.79631e21 q^{56} -1.26851e20 q^{58} -3.41769e21 q^{60} +5.00828e21 q^{61} -1.08164e22 q^{63} +4.72237e21 q^{64} +1.63648e22 q^{67} +3.35406e22 q^{69} -2.61398e22 q^{70} +2.84355e22 q^{72} -4.97340e22 q^{75} +6.87195e22 q^{80} -2.54105e22 q^{81} -1.20151e23 q^{82} +1.79224e23 q^{83} +3.65928e23 q^{84} +2.96354e23 q^{86} +2.58409e22 q^{87} +4.29891e22 q^{89} +4.13790e23 q^{90} -6.74400e23 q^{92} +8.30345e23 q^{94} -9.61995e23 q^{96} +2.01404e24 q^{98} +O(q^{100})\)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	4096.00	1.00000
$2$	4096.00	1.00000
$3$	−834398.	−1.57007	−0.785034	−	0.619453i	$-0.787354\pi$
$3$	−834398.	−1.57007	−0.785034	+	0.619453i	$0.787354\pi$
$4$	1.67772e7	1.00000
$5$	2.44141e8	1.00000
$5$	2.44141e8	1.00000
$6$	−3.41769e9	−1.57007
$7$	−2.61398e10	−1.88854	−0.944270	−	0.329173i	$-0.893230\pi$
$7$	−2.61398e10	−1.88854	−0.944270	+	0.329173i	$0.893230\pi$
$8$	6.87195e10	1.00000
$9$	4.13790e11	1.46511
$10$	1.00000e12	1.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.39989e13	−1.57007
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−1.07069e14	−1.88854
$15$	−2.03710e14	−1.57007
$16$	2.81475e14	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.69489e15	1.46511
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	4.09600e15	1.00000
$21$	2.18110e16	2.96513
$22$	0	0
$23$	−4.01974e16	−1.83427	−0.917136	−	0.398575i	$-0.869505\pi$
$23$	−4.01974e16	−1.83427	−0.917136	+	0.398575i	$0.869505\pi$
$24$	−5.73394e16	−1.57007
$25$	5.96046e16	1.00000
$26$	0	0
$27$	−1.09607e17	−0.730255
$28$	−4.38553e17	−1.88854
$29$	−3.09695e16	−0.0875302	−0.0437651	−	0.999042i	$-0.513935\pi$
$29$	−3.09695e16	−0.0875302	−0.0437651	+	0.999042i	$0.513935\pi$
$30$	−8.34398e17	−1.57007
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.15292e18	1.00000
$33$	0	0
$34$	0	0
$35$	−6.38179e18	−1.88854
$36$	6.94225e18	1.46511
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	1.67772e19	1.00000
$41$	−2.93337e19	−1.30005	−0.650027	−	0.759911i	$-0.725242\pi$
$41$	−2.93337e19	−1.30005	−0.650027	+	0.759911i	$0.725242\pi$
$42$	8.93379e19	2.96513
$43$	7.23521e19	1.81063	0.905314	−	0.424742i	$-0.139635\pi$
$43$	7.23521e19	1.81063	0.905314	+	0.424742i	$0.139635\pi$
$44$	0	0
$45$	1.01023e20	1.46511
$46$	−1.64648e20	−1.83427
$47$	2.02721e20	1.74471	0.872357	−	0.488869i	$-0.162590\pi$
$47$	2.02721e20	1.74471	0.872357	+	0.488869i	$0.162590\pi$
$48$	−2.34862e20	−1.57007
$49$	4.91709e20	2.56658
$50$	2.44141e20	1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−4.48952e20	−0.730255
$55$	0	0
$56$	−1.79631e21	−1.88854
$57$	0	0
$58$	−1.26851e20	−0.0875302
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−3.41769e21	−1.57007
$61$	5.00828e21	1.88682	0.943411	−	0.331627i	$-0.107597\pi$
$61$	5.00828e21	1.88682	0.943411	+	0.331627i	$0.107597\pi$
$62$	0	0
$63$	−1.08164e22	−2.76692
$64$	4.72237e21	1.00000
$65$	0	0
$66$	0	0
$67$	1.63648e22	1.99993	0.999964	−	0.00848499i	$-0.00270089\pi$
$67$	1.63648e22	1.99993	0.999964	+	0.00848499i	$0.00270089\pi$
$68$	0	0
$69$	3.35406e22	2.87993
$70$	−2.61398e22	−1.88854
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	2.84355e22	1.46511
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−4.97340e22	−1.57007
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	6.87195e22	1.00000
$81$	−2.54105e22	−0.318562
$82$	−1.20151e23	−1.30005
$83$	1.79224e23	1.67671	0.838357	−	0.545121i	$-0.183516\pi$
$83$	1.79224e23	1.67671	0.838357	+	0.545121i	$0.183516\pi$
$84$	3.65928e23	2.96513
$85$	0	0
$86$	2.96354e23	1.81063
$87$	2.58409e22	0.137428
$88$	0	0
$89$	4.29891e22	0.174052	0.0870258	−	0.996206i	$-0.472264\pi$
$89$	4.29891e22	0.174052	0.0870258	+	0.996206i	$0.472264\pi$
$90$	4.13790e23	1.46511
$91$	0	0
$92$	−6.74400e23	−1.83427
$93$	0	0
$94$	8.30345e23	1.74471
$95$	0	0
$96$	−9.61995e23	−1.57007
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	2.01404e24	2.56658
$99$	0	0
$100$	1.00000e24	1.00000
$101$	1.63137e23	0.144776	0.0723881	−	0.997377i	$-0.476938\pi$
$101$	1.63137e23	0.144776	0.0723881	+	0.997377i	$0.476938\pi$
$102$	0	0
$103$	−2.41811e24	−1.69601	−0.848006	−	0.529987i	$-0.822197\pi$
$103$	−2.41811e24	−1.69601	−0.848006	+	0.529987i	$0.822197\pi$
$104$	0	0
$105$	5.32495e24	2.96513
$106$	0	0
$107$	1.90838e24	0.847345	0.423672	−	0.905816i	$-0.360741\pi$
$107$	1.90838e24	0.847345	0.423672	+	0.905816i	$0.360741\pi$
$108$	−1.83891e24	−0.730255
$109$	−2.85198e24	−1.01398	−0.506988	−	0.861953i	$-0.669241\pi$
$109$	−2.85198e24	−1.01398	−0.506988	+	0.861953i	$0.669241\pi$
$110$	0	0
$111$	0	0
$112$	−7.35770e24	−1.88854
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−9.81381e24	−1.83427
$116$	−5.19581e23	−0.0875302
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−1.39989e25	−1.57007
$121$	9.84973e24	1.00000
$122$	2.05139e25	1.88682
$123$	2.44760e25	2.04117
$124$	0	0
$125$	1.45519e25	1.00000
$126$	−4.43040e25	−2.76692
$127$	−5.87349e24	−0.333620	−0.166810	−	0.985989i	$-0.553347\pi$
$127$	−5.87349e24	−0.333620	−0.166810	+	0.985989i	$0.553347\pi$
$128$	1.93428e25	1.00000
$129$	−6.03704e25	−2.84281
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	6.70304e25	1.99993
$135$	−2.67596e25	−0.730255
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	1.37382e26	2.87993
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−1.07069e26	−1.88854
$141$	−1.69150e26	−2.73932
$142$	0	0
$143$	0	0
$144$	1.16472e26	1.46511
$145$	−7.56090e24	−0.0875302
$146$	0	0
$147$	−4.10281e26	−4.02970
$148$	0	0
$149$	−6.86998e25	−0.573747	−0.286873	−	0.957968i	$-0.592616\pi$
$149$	−6.86998e25	−0.573747	−0.286873	+	0.957968i	$0.592616\pi$
$150$	−2.03710e26	−1.57007
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	2.81475e26	1.00000
$161$	1.05075e27	3.46409
$162$	−1.04082e26	−0.318562
$163$	7.02891e26	1.99819	0.999095	−	0.0425351i	$-0.0135434\pi$
$163$	7.02891e26	1.99819	0.999095	+	0.0425351i	$0.0135434\pi$
$164$	−4.92139e26	−1.30005
$165$	0	0
$166$	7.34102e26	1.67671
$167$	−8.65117e26	−1.83856	−0.919279	−	0.393607i	$-0.871227\pi$
$167$	−8.65117e26	−1.83856	−0.919279	+	0.393607i	$0.871227\pi$
$168$	1.49884e27	2.96513
$169$	5.42801e26	1.00000
$170$	0	0
$171$	0	0
$172$	1.21387e27	1.81063
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	1.05844e26	0.137428
$175$	−1.55805e27	−1.88854
$176$	0	0
$177$	0	0
$178$	1.76083e26	0.174052
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	1.69489e27	1.46511
$181$	−5.26957e26	−0.426219	−0.213109	−	0.977028i	$-0.568359\pi$
$181$	−5.26957e26	−0.426219	−0.213109	+	0.977028i	$0.568359\pi$
$182$	0	0
$183$	−4.17890e27	−2.96244
$184$	−2.76234e27	−1.83427
$185$	0	0
$186$	0	0
$187$	0	0
$188$	3.40109e27	1.74471
$189$	2.86511e27	1.37911
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−3.94033e27	−1.57007
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	8.24950e27	2.56658
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	4.09600e27	1.00000
$201$	−1.36548e28	−3.14002
$202$	6.68211e26	0.144776
$203$	8.09536e26	0.165304
$204$	0	0
$205$	−7.16156e27	−1.30005
$206$	−9.90456e27	−1.69601
$207$	−1.66333e28	−2.68741
$208$	0	0
$209$	0	0
$210$	2.18110e28	2.96513
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	7.81673e27	0.847345
$215$	1.76641e28	1.81063
$216$	−7.53216e27	−0.730255
$217$	0	0
$218$	−1.16817e28	−1.01398
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.50378e28	−1.65554	−0.827768	−	0.561071i	$-0.810389\pi$
$223$	−2.50378e28	−1.65554	−0.827768	+	0.561071i	$0.810389\pi$
$224$	−3.01372e28	−1.88854
$225$	2.46638e28	1.46511
$226$	0	0
$227$	−6.69727e27	−0.357756	−0.178878	−	0.983871i	$-0.557247\pi$
$227$	−6.69727e27	−0.357756	−0.178878	+	0.983871i	$0.557247\pi$
$228$	0	0
$229$	2.15967e28	1.03839	0.519196	−	0.854655i	$-0.326231\pi$
$229$	2.15967e28	1.03839	0.519196	+	0.854655i	$0.326231\pi$
$230$	−4.01974e28	−1.83427
$231$	0	0
$232$	−2.12821e27	−0.0875302
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	4.94924e28	1.74471
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−5.73394e28	−1.57007
$241$	−2.05926e28	−0.536422	−0.268211	−	0.963360i	$-0.586432\pi$
$241$	−2.05926e28	−0.536422	−0.268211	+	0.963360i	$0.586432\pi$
$242$	4.03445e28	1.00000
$243$	5.21588e28	1.23042
$244$	8.40250e28	1.88682
$245$	1.20046e29	2.56658
$246$	1.00254e29	2.04117
$247$	0	0
$248$	0	0
$249$	−1.49544e29	−2.63255
$250$	5.96046e28	1.00000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.81469e29	−2.76692
$253$	0	0
$254$	−2.40578e28	−0.333620
$255$	0	0
$256$	7.92282e28	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−2.47277e29	−2.84281
$259$	0	0
$260$	0	0
$261$	−1.28149e28	−0.128241
$262$	0	0
$263$	1.85459e29	1.69347	0.846736	−	0.532013i	$-0.178564\pi$
$263$	1.85459e29	1.69347	0.846736	+	0.532013i	$0.178564\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.58700e28	−0.273273
$268$	2.74557e29	1.99993
$269$	1.89860e29	1.32253	0.661267	−	0.750151i	$-0.270019\pi$
$269$	1.89860e29	1.32253	0.661267	+	0.750151i	$0.270019\pi$
$270$	−1.09607e29	−0.730255
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	5.62718e29	2.87993
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−4.38553e29	−1.88854
$281$	−3.88928e29	−1.60470	−0.802349	−	0.596856i	$-0.796417\pi$
$281$	−3.88928e29	−1.60470	−0.802349	+	0.596856i	$0.796417\pi$
$282$	−6.92838e29	−2.73932
$283$	3.58181e29	1.35727	0.678634	−	0.734476i	$-0.262572\pi$
$283$	3.58181e29	1.35727	0.678634	+	0.734476i	$0.262572\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.66779e29	2.45520
$288$	4.77068e29	1.46511
$289$	3.39449e29	1.00000
$290$	−3.09695e28	−0.0875302
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.68051e30	−4.02970
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−2.81394e29	−0.573747
$299$	0	0
$300$	−8.34398e29	−1.57007
$301$	−1.89127e30	−3.41944
$302$	0	0
$303$	−1.36122e29	−0.227308
$304$	0	0
$305$	1.22273e30	1.88682
$306$	0	0
$307$	−1.36529e30	−1.94790	−0.973948	−	0.226773i	$-0.927182\pi$
$307$	−1.36529e30	−1.94790	−0.973948	+	0.226773i	$0.927182\pi$
$308$	0	0
$309$	2.01766e30	2.66285
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−2.64072e30	−2.76692
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	1.15292e30	1.00000
$321$	−1.59235e30	−1.33039
$322$	4.30388e30	3.46409
$323$	0	0
$324$	−4.26318e29	−0.318562
$325$	0	0
$326$	2.87904e30	1.99819
$327$	2.37968e30	1.59201
$328$	−2.01580e30	−1.30005
$329$	−5.29909e30	−3.29496
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	3.00688e30	1.67671
$333$	0	0
$334$	−3.54352e30	−1.83856
$335$	3.99532e30	1.99993
$336$	6.13925e30	2.96513
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	2.22331e30	1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−7.84527e30	−2.95855
$344$	4.97200e30	1.81063
$345$	8.18863e30	2.87993
$346$	0	0
$347$	−2.62054e30	−0.859880	−0.429940	−	0.902857i	$-0.641465\pi$
$347$	−2.62054e30	−0.859880	−0.429940	+	0.902857i	$0.641465\pi$
$348$	4.33538e29	0.137428
$349$	6.06863e30	1.85860	0.929299	−	0.369328i	$-0.120412\pi$
$349$	6.06863e30	1.85860	0.929299	+	0.369328i	$0.120412\pi$
$350$	−6.38179e30	−1.88854
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	7.21237e29	0.174052
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	6.94225e30	1.46511
$361$	4.89876e30	1.00000
$362$	−2.15842e30	−0.426219
$363$	−8.21860e30	−1.57007
$364$	0	0
$365$	0	0
$366$	−1.71168e31	−2.96244
$367$	−4.78471e30	−0.801424	−0.400712	−	0.916204i	$-0.631237\pi$
$367$	−4.78471e30	−0.801424	−0.400712	+	0.916204i	$0.631237\pi$
$368$	−1.13146e31	−1.83427
$369$	−1.21380e31	−1.90472
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.21421e31	−1.57007
$376$	1.39309e31	1.74471
$377$	0	0
$378$	1.17355e31	1.37911
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	4.90083e30	0.523806
$382$	0	0
$383$	1.53718e31	1.54291	0.771453	−	0.636286i	$-0.219530\pi$
$383$	1.53718e31	1.54291	0.771453	+	0.636286i	$0.219530\pi$
$384$	−1.61396e31	−1.57007
$385$	0	0
$386$	0	0
$387$	2.99386e31	2.65277
$388$	0	0
$389$	−2.40106e31	−1.99990	−0.999948	−	0.0102012i	$-0.996753\pi$
$389$	−2.40106e31	−1.99990	−0.999948	+	0.0102012i	$0.996753\pi$
$390$	0	0
$391$	0	0
$392$	3.37900e31	2.56658
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.67772e31	1.00000
$401$	6.58776e30	0.381071	0.190535	−	0.981680i	$-0.438978\pi$
$401$	6.58776e30	0.381071	0.190535	+	0.981680i	$0.438978\pi$
$402$	−5.59300e31	−3.14002
$403$	0	0
$404$	2.73699e30	0.144776
$405$	−6.20374e30	−0.318562
$406$	3.31586e30	0.165304
$407$	0	0
$408$	0	0
$409$	−3.16362e31	−1.44379	−0.721895	−	0.692003i	$-0.756729\pi$
$409$	−3.16362e31	−1.44379	−0.721895	+	0.692003i	$0.756729\pi$
$410$	−2.93337e31	−1.30005
$411$	0	0
$412$	−4.05691e31	−1.69601
$413$	0	0
$414$	−6.81300e31	−2.68741
$415$	4.37559e31	1.67671
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	8.93379e31	2.96513
$421$	−2.15824e29	−0.00696168	−0.00348084	−	0.999994i	$-0.501108\pi$
$421$	−2.15824e29	−0.00696168	−0.00348084	+	0.999994i	$0.501108\pi$
$422$	0	0
$423$	8.38840e31	2.55620
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.30916e32	−3.56334
$428$	3.20173e31	0.847345
$429$	0	0
$430$	7.23521e31	1.81063
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−3.08517e31	−0.730255
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	6.30880e30	0.137428
$436$	−4.78482e31	−1.01398
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	2.03464e32	3.76032
$442$	0	0
$443$	7.73164e31	1.35340	0.676701	−	0.736258i	$-0.263409\pi$
$443$	7.73164e31	1.35340	0.676701	+	0.736258i	$0.263409\pi$
$444$	0	0
$445$	1.04954e31	0.174052
$446$	−1.02555e32	−1.65554
$447$	5.73230e31	0.900821
$448$	−1.23442e32	−1.88854
$449$	9.61046e31	1.43149	0.715745	−	0.698362i	$-0.246087\pi$
$449$	9.61046e31	1.43149	0.715745	+	0.698362i	$0.246087\pi$
$450$	1.01023e32	1.46511
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−2.74320e31	−0.357756
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	8.84601e31	1.03839
$459$	0	0
$460$	−1.64648e32	−1.83427
$461$	5.85571e31	0.635577	0.317789	−	0.948162i	$-0.397060\pi$
$461$	5.85571e31	0.635577	0.317789	+	0.948162i	$0.397060\pi$
$462$	0	0
$463$	1.17151e32	1.20718	0.603592	−	0.797293i	$-0.293736\pi$
$463$	1.17151e32	1.20718	0.603592	+	0.797293i	$0.293736\pi$
$464$	−8.71713e30	−0.0875302
$465$	0	0
$466$	0	0
$467$	−5.82441e30	−0.0541315	−0.0270657	−	0.999634i	$-0.508616\pi$
$467$	−5.82441e30	−0.0541315	−0.0270657	+	0.999634i	$0.508616\pi$
$468$	0	0
$469$	−4.27774e32	−3.77694
$470$	2.02721e32	1.74471
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−2.34862e32	−1.57007
$481$	0	0
$482$	−8.43473e31	−0.536422
$483$	−8.76745e32	−5.43886
$484$	1.65251e32	1.00000
$485$	0	0
$486$	2.13643e32	1.23042
$487$	3.55617e32	1.99818	0.999091	−	0.0426223i	$-0.0135712\pi$
$487$	3.55617e32	1.99818	0.999091	+	0.0426223i	$0.0135712\pi$
$488$	3.44167e32	1.88682
$489$	−5.86491e32	−3.13729
$490$	4.91709e32	2.56658
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	4.10639e32	2.04117
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−6.12533e32	−2.63255
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	2.44141e32	1.00000
$501$	7.21852e32	2.88666
$502$	0	0
$503$	2.05360e32	0.782888	0.391444	−	0.920202i	$-0.371976\pi$
$503$	2.05360e32	0.782888	0.391444	+	0.920202i	$0.371976\pi$
$504$	−7.43298e32	−2.76692
$505$	3.98285e31	0.144776
$506$	0	0
$507$	−4.52912e32	−1.57007
$508$	−9.85409e31	−0.333620
$509$	−6.03783e32	−1.99649	−0.998246	−	0.0591970i	$-0.981146\pi$
$509$	−6.03783e32	−1.99649	−0.998246	+	0.0591970i	$0.981146\pi$
$510$	0	0
$511$	0	0
$512$	3.24519e32	1.00000
$513$	0	0
$514$	0	0
$515$	−5.90358e32	−1.69601
$516$	−1.01285e33	−2.84281
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.77291e32	−1.94326	−0.971630	−	0.236505i	$-0.923998\pi$
$521$	−7.77291e32	−1.94326	−0.971630	+	0.236505i	$0.923998\pi$
$522$	−5.24897e31	−0.128241
$523$	−2.62258e32	−0.626193	−0.313097	−	0.949721i	$-0.601366\pi$
$523$	−2.62258e32	−0.626193	−0.313097	+	0.949721i	$0.601366\pi$
$524$	0	0
$525$	1.30004e33	2.96513
$526$	7.59639e32	1.69347
$527$	0	0
$528$	0	0
$529$	1.13558e33	2.36455
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−1.46924e32	−0.273273
$535$	4.65914e32	0.847345
$536$	1.12458e33	1.99993
$537$	0	0
$538$	7.77668e32	1.32253
$539$	0	0
$540$	−4.48952e32	−0.730255
$541$	−7.29092e32	−1.15988	−0.579942	−	0.814657i	$-0.696925\pi$
$541$	−7.29092e32	−1.15988	−0.579942	+	0.814657i	$0.696925\pi$
$542$	0	0
$543$	4.39692e32	0.669192
$544$	0	0
$545$	−6.96283e32	−1.01398
$546$	0	0
$547$	−7.59564e31	−0.105856	−0.0529281	−	0.998598i	$-0.516855\pi$
$547$	−7.59564e31	−0.105856	−0.0529281	+	0.998598i	$0.516855\pi$
$548$	0	0
$549$	2.07238e33	2.76440
$550$	0	0
$551$	0	0
$552$	2.30489e33	2.87993
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−1.79631e33	−1.88854
$561$	0	0
$562$	−1.59305e33	−1.60470
$563$	1.53111e33	1.50975	0.754877	−	0.655866i	$-0.227697\pi$
$563$	1.53111e33	1.50975	0.754877	+	0.655866i	$0.227697\pi$
$564$	−2.83787e33	−2.73932
$565$	0	0
$566$	1.46711e33	1.35727
$567$	6.64227e32	0.601616
$568$	0	0
$569$	−2.31529e32	−0.201028	−0.100514	−	0.994936i	$-0.532049\pi$
$569$	−2.31529e32	−0.201028	−0.100514	+	0.994936i	$0.532049\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	3.14073e33	2.45520
$575$	−2.39595e33	−1.83427
$576$	1.95407e33	1.46511
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.39038e33	1.00000
$579$	0	0
$580$	−1.26851e32	−0.0875302
$581$	−4.68488e33	−3.16654
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.48508e32	0.447239	0.223619	−	0.974677i	$-0.428213\pi$
$587$	7.48508e32	0.447239	0.223619	+	0.974677i	$0.428213\pi$
$588$	−6.88337e33	−4.02970
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	−1.15259e33	−0.573747
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−3.41769e33	−1.57007
$601$	−1.93306e33	−0.870468	−0.435234	−	0.900317i	$-0.643334\pi$
$601$	−1.93306e33	−0.870468	−0.435234	+	0.900317i	$0.643334\pi$
$602$	−7.74664e33	−3.41944
$603$	6.77162e33	2.93012
$604$	0	0
$605$	2.40472e33	1.00000
$606$	−5.57554e32	−0.227308
$607$	7.06595e31	0.0282427	0.0141214	−	0.999900i	$-0.495505\pi$
$607$	7.06595e31	0.0282427	0.0141214	+	0.999900i	$0.495505\pi$
$608$	0	0
$609$	−6.75475e32	−0.259539
$610$	5.00828e33	1.88682
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−5.59224e33	−1.94790
$615$	5.97559e33	2.04117
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	8.26435e33	2.66285
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	4.40593e33	1.33949
$622$	0	0
$623$	−1.12373e33	−0.328703
$624$	0	0
$625$	3.55271e33	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−1.08164e34	−2.76692
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.43396e33	−0.333620
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	4.72237e33	1.00000
$641$	8.24959e33	1.71449	0.857247	−	0.514905i	$-0.172173\pi$
$641$	8.24959e33	1.71449	0.857247	+	0.514905i	$0.172173\pi$
$642$	−6.52227e33	−1.33039
$643$	−7.68447e33	−1.53845	−0.769223	−	0.638980i	$-0.779356\pi$
$643$	−7.68447e33	−1.53845	−0.769223	+	0.638980i	$0.779356\pi$
$644$	1.76287e34	3.46409
$645$	−1.47389e34	−2.84281
$646$	0	0
$647$	−9.17137e33	−1.70444	−0.852222	−	0.523181i	$-0.824745\pi$
$647$	−9.17137e33	−1.70444	−0.852222	+	0.523181i	$0.824745\pi$
$648$	−1.74620e33	−0.318562
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1.17926e34	1.99819
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	9.74718e33	1.59201
$655$	0	0
$656$	−8.25672e33	−1.30005
$657$	0	0
$658$	−2.17051e34	−3.29496
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1.27431e34	−1.83172	−0.915860	−	0.401498i	$-0.868490\pi$
$661$	−1.27431e34	−1.83172	−0.915860	+	0.401498i	$0.868490\pi$
$662$	0	0
$663$	0	0
$664$	1.23162e34	1.67671
$665$	0	0
$666$	0	0
$667$	1.24489e33	0.160554
$668$	−1.45143e34	−1.83856
$669$	2.08915e34	2.59930
$670$	1.63648e34	1.99993
$671$	0	0
$672$	2.51464e34	2.96513
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−6.53310e33	−0.730255
$676$	9.10669e33	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.58819e33	0.561701
$682$	0	0
$683$	6.44378e33	0.625305	0.312653	−	0.949868i	$-0.398782\pi$
$683$	6.44378e33	0.625305	0.312653	+	0.949868i	$0.398782\pi$
$684$	0	0
$685$	0	0
$686$	−3.21342e34	−2.95855
$687$	−1.80202e34	−1.63035
$688$	2.03653e34	1.81063
$689$	0	0
$690$	3.35406e34	2.87993
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	−1.07338e34	−0.859880
$695$	0	0
$696$	1.77577e33	0.137428
$697$	0	0
$698$	2.48571e34	1.85860
$699$	0	0
$700$	−2.61398e34	−1.88854
$701$	−1.43011e34	−1.01567	−0.507837	−	0.861453i	$-0.669555\pi$
$701$	−1.43011e34	−1.01567	−0.507837	+	0.861453i	$0.669555\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−4.12964e34	−2.73932
$706$	0	0
$707$	−4.26438e33	−0.273415
$708$	0	0
$709$	3.20837e34	1.98852	0.994258	−	0.107009i	$-0.0341275\pi$
$709$	3.20837e34	1.98852	0.994258	+	0.107009i	$0.0341275\pi$
$710$	0	0
$711$	0	0
$712$	2.95419e33	0.174052
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	2.84355e34	1.46511
$721$	6.32088e34	3.20298
$722$	2.00653e34	1.00000
$723$	1.71824e34	0.842218
$724$	−8.84088e33	−0.426219
$725$	−1.84592e33	−0.0875302
$726$	−3.36634e34	−1.57007
$727$	−1.36875e34	−0.627931	−0.313966	−	0.949434i	$-0.601658\pi$
$727$	−1.36875e34	−0.627931	−0.313966	+	0.949434i	$0.601658\pi$
$728$	0	0
$729$	−3.63445e34	−1.61328
$730$	0	0
$731$	0	0
$732$	−7.01103e34	−2.96244
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1.95982e34	−0.801424
$735$	−1.00166e35	−4.02970
$736$	−4.63444e34	−1.83427
$737$	0	0
$738$	−4.97174e34	−1.90472
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.55181e34	1.96141	0.980706	−	0.195490i	$-0.0626298\pi$
$743$	5.55181e34	1.96141	0.980706	+	0.195490i	$0.0626298\pi$
$744$	0	0
$745$	−1.67724e34	−0.573747
$746$	0	0
$747$	7.41612e34	2.45657
$748$	0	0
$749$	−4.98848e34	−1.60024
$750$	−4.97340e34	−1.57007
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	5.70609e34	1.74471
$753$	0	0
$754$	0	0
$755$	0	0
$756$	4.80686e34	1.37911
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.55411e34	−0.677056	−0.338528	−	0.940956i	$-0.609929\pi$
$761$	−2.55411e34	−0.677056	−0.338528	+	0.940956i	$0.609929\pi$
$762$	2.00738e34	0.523806
$763$	7.45501e34	1.91493
$764$	0	0
$765$	0	0
$766$	6.29629e34	1.54291
$767$	0	0
$768$	−6.61078e34	−1.57007
$769$	8.44606e34	1.97487	0.987434	−	0.158030i	$-0.0505141\pi$
$769$	8.44606e34	1.97487	0.987434	+	0.158030i	$0.0505141\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.22628e35	2.65277
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−9.83473e34	−1.99990
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.39448e33	0.0639193
$784$	1.38404e35	2.56658
$785$	0	0
$786$	0	0
$787$	7.14886e34	1.26631	0.633155	−	0.774025i	$-0.281759\pi$
$787$	7.14886e34	1.26631	0.633155	+	0.774025i	$0.281759\pi$
$788$	0	0
$789$	−1.54746e35	−2.65887
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	6.87195e34	1.00000
$801$	1.77885e34	0.255005
$802$	2.69835e34	0.381071
$803$	0	0
$804$	−2.29089e35	−3.14002
$805$	2.56531e35	3.46409
$806$	0	0
$807$	−1.58419e35	−2.07647
$808$	1.12107e34	0.144776
$809$	2.76115e34	0.351324	0.175662	−	0.984451i	$-0.443793\pi$
$809$	2.76115e34	0.351324	0.175662	+	0.984451i	$0.443793\pi$
$810$	−2.54105e34	−0.318562
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.35818e34	0.165304
$813$	0	0
$814$	0	0
$815$	1.71604e35	1.99819
$816$	0	0
$817$	0	0
$818$	−1.29582e35	−1.44379
$819$	0	0
$820$	−1.20151e35	−1.30005
$821$	4.95423e34	0.528273	0.264137	−	0.964485i	$-0.414913\pi$
$821$	4.95423e34	0.528273	0.264137	+	0.964485i	$0.414913\pi$
$822$	0	0
$823$	1.76546e35	1.82835	0.914176	−	0.405318i	$-0.132839\pi$
$823$	1.76546e35	1.82835	0.914176	+	0.405318i	$0.132839\pi$
$824$	−1.66171e35	−1.69601
$825$	0	0
$826$	0	0
$827$	−5.82998e34	−0.569641	−0.284820	−	0.958581i	$-0.591934\pi$
$827$	−5.82998e34	−0.569641	−0.284820	+	0.958581i	$0.591934\pi$
$828$	−2.79060e35	−2.68741
$829$	1.69051e35	1.60459	0.802295	−	0.596928i	$-0.203612\pi$
$829$	1.69051e35	1.60459	0.802295	+	0.596928i	$0.203612\pi$
$830$	1.79224e35	1.67671
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.11210e35	−1.83856
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	3.65928e35	2.96513
$841$	−1.24226e35	−0.992338
$842$	−8.84014e32	−0.00696168
$843$	3.24521e35	2.51948
$844$	0	0
$845$	1.32520e35	1.00000
$846$	3.43589e35	2.55620
$847$	−2.57470e35	−1.88854
$848$	0	0
$849$	−2.98866e35	−2.13100
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	−5.36230e35	−3.56334
$855$	0	0
$856$	1.31143e35	0.847345
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	2.96354e35	1.81063
$861$	−6.39799e35	−3.85483
$862$	0	0
$863$	−2.49582e35	−1.46246	−0.731229	−	0.682132i	$-0.761053\pi$
$863$	−2.49582e35	−1.46246	−0.731229	+	0.682132i	$0.761053\pi$
$864$	−1.26369e35	−0.730255
$865$	0	0
$866$	0	0
$867$	−2.83235e35	−1.57007
$868$	0	0
$869$	0	0
$870$	2.58409e34	0.137428
$871$	0	0
$872$	−1.95986e35	−1.01398
$873$	0	0
$874$	0	0
$875$	−3.80384e35	−1.88854
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.51651e34	0.343799	0.171900	−	0.985114i	$-0.445009\pi$
$881$	7.51651e34	0.343799	0.171900	+	0.985114i	$0.445009\pi$
$882$	8.33390e35	3.76032
$883$	−4.33662e35	−1.93029	−0.965145	−	0.261715i	$-0.915712\pi$
$883$	−4.33662e35	−1.93029	−0.965145	+	0.261715i	$0.915712\pi$
$884$	0	0
$885$	0	0
$886$	3.16688e35	1.35340
$887$	3.88632e35	1.63853	0.819266	−	0.573413i	$-0.194381\pi$
$887$	3.88632e35	1.63853	0.819266	+	0.573413i	$0.194381\pi$
$888$	0	0
$889$	1.53532e35	0.630054
$890$	4.29891e34	0.174052
$891$	0	0
$892$	−4.20065e35	−1.65554
$893$	0	0
$894$	2.34795e35	0.900821
$895$	0	0
$896$	−5.05618e35	−1.88854
$897$	0	0
$898$	3.93644e35	1.43149
$899$	0	0
$900$	4.13790e35	1.46511
$901$	0	0
$902$	0	0
$903$	1.57807e36	5.36876
$904$	0	0
$905$	−1.28652e35	−0.426219
$906$	0	0
$907$	−7.48855e34	−0.241608	−0.120804	−	0.992676i	$-0.538547\pi$
$907$	−7.48855e34	−0.241608	−0.120804	+	0.992676i	$0.538547\pi$
$908$	−1.12361e35	−0.357756
$909$	6.75047e34	0.212113
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.02024e36	−2.96244
$916$	3.62332e35	1.03839
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−6.74400e35	−1.83427
$921$	1.13920e36	3.05833
$922$	2.39850e35	0.635577
$923$	0	0
$924$	0	0
$925$	0	0
$926$	4.79850e35	1.20718
$927$	−1.00059e36	−2.48484
$928$	−3.57054e34	−0.0875302
$929$	−7.06348e35	−1.70935	−0.854673	−	0.519167i	$-0.826242\pi$
$929$	−7.06348e35	−1.70935	−0.854673	+	0.519167i	$0.826242\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−2.38568e34	−0.0541315
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−1.75216e36	−3.77694
$939$	0	0
$940$	8.30345e35	1.74471
$941$	7.03496e35	1.45944	0.729719	−	0.683747i	$-0.239651\pi$
$941$	7.03496e35	1.45944	0.729719	+	0.683747i	$0.239651\pi$
$942$	0	0
$943$	1.17914e36	2.38465
$944$	0	0
$945$	6.99491e35	1.37911
$946$	0	0
$947$	−8.74520e35	−1.68101	−0.840504	−	0.541805i	$-0.817741\pi$
$947$	−8.74520e35	−1.68101	−0.840504	+	0.541805i	$0.817741\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−9.61995e35	−1.57007
$961$	6.20413e35	1.00000
$962$	0	0
$963$	7.89670e35	1.24145
$964$	−3.45486e35	−0.536422
$965$	0	0
$966$	−3.59115e36	−5.43886
$967$	7.05504e34	0.105531	0.0527657	−	0.998607i	$-0.483196\pi$
$967$	7.05504e34	0.105531	0.0527657	+	0.998607i	$0.483196\pi$
$968$	6.76868e35	1.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	8.75080e35	1.23042
$973$	0	0
$974$	1.45661e36	1.99818
$975$	0	0
$976$	1.40971e36	1.88682
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−2.40227e36	−3.13729
$979$	0	0
$980$	2.01404e36	2.56658
$981$	−1.18012e36	−1.48559
$982$	0	0
$983$	−2.73448e35	−0.335918	−0.167959	−	0.985794i	$-0.553718\pi$
$983$	−2.73448e35	−0.335918	−0.167959	+	0.985794i	$0.553718\pi$
$984$	1.68198e36	2.04117
$985$	0	0
$986$	0	0
$987$	4.42155e36	5.17331
$988$	0	0
$989$	−2.90836e36	−3.32119
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−2.50894e36	−2.63255
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.25.d.b.19.1	yes	1
4.3	odd	2		20.25.d.a.19.1	✓	1
5.4	even	2		20.25.d.a.19.1	✓	1
20.19	odd	2	CM	20.25.d.b.19.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.25.d.a.19.1	✓	1	4.3	odd	2
20.25.d.a.19.1	✓	1	5.4	even	2
20.25.d.b.19.1	yes	1	1.1	even	1	trivial
20.25.d.b.19.1	yes	1	20.19	odd	2	CM

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 25 \)
Character orbit:	\([\chi]\)	\(=\)	20.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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