LMFDB - Embedded newform 8.15.d.a.3.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8,15,Mod(3,8)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8.3"); S:= CuspForms(chi, 15); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 15, names="a")

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	8.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$9.94631745215$
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			3.1
Character	$\chi$	$=$	8.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-128.000 q^{2} +3022.00 q^{3} +16384.0 q^{4} -386816. q^{6} -2.09715e6 q^{8} +4.34952e6 q^{9} +3.87123e7 q^{11} +4.95124e7 q^{12} +2.68435e8 q^{16} -3.28636e8 q^{17} -5.56738e8 q^{18} +1.77897e9 q^{19} -4.95517e9 q^{22} -6.33759e9 q^{24} +6.10352e9 q^{25} -1.30990e9 q^{27} -3.43597e10 q^{32} +1.16988e11 q^{33} +4.20654e10 q^{34} +7.12625e10 q^{36} -2.27709e11 q^{38} -3.33394e11 q^{41} -4.95013e11 q^{43} +6.34262e11 q^{44} +8.11212e11 q^{48} +6.78223e11 q^{49} -7.81250e11 q^{50} -9.93139e11 q^{51} +1.67667e11 q^{54} +5.37606e12 q^{57} -3.91449e12 q^{59} +4.39805e12 q^{64} -1.49745e13 q^{66} +2.71110e12 q^{67} -5.38438e12 q^{68} -9.12159e12 q^{72} +1.57940e12 q^{73} +1.84448e13 q^{75} +2.91467e13 q^{76} -2.47621e13 q^{81} +4.26744e13 q^{82} -3.11463e13 q^{83} +6.33616e13 q^{86} -8.11855e13 q^{88} -3.84337e13 q^{89} -1.03835e14 q^{96} -6.28150e13 q^{97} -8.68126e13 q^{98} +1.68380e14 q^{99} +O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−128.000	−1.00000
$2$	−128.000	−1.00000
$3$	3022.00	1.38180	0.690901	−	0.722950i	$-0.257214\pi$
$3$	3022.00	1.38180	0.690901	+	0.722950i	$0.257214\pi$
$4$	16384.0	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−386816.	−1.38180
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−2.09715e6	−1.00000
$9$	4.34952e6	0.909376
$10$	0	0
$11$	3.87123e7	1.98655	0.993275	−	0.115776i	$-0.0369354\pi$
$11$	3.87123e7	1.98655	0.993275	+	0.115776i	$0.0369354\pi$
$12$	4.95124e7	1.38180
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	2.68435e8	1.00000
$17$	−3.28636e8	−0.800890	−0.400445	−	0.916321i	$-0.631144\pi$
$17$	−3.28636e8	−0.800890	−0.400445	+	0.916321i	$0.631144\pi$
$18$	−5.56738e8	−0.909376
$19$	1.77897e9	1.99019	0.995095	−	0.0989284i	$-0.0315415\pi$
$19$	1.77897e9	1.99019	0.995095	+	0.0989284i	$0.0315415\pi$
$20$	0	0
$21$	0	0
$22$	−4.95517e9	−1.98655
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−6.33759e9	−1.38180
$25$	6.10352e9	1.00000
$26$	0	0
$27$	−1.30990e9	−0.125225
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−3.43597e10	−1.00000
$33$	1.16988e11	2.74502
$34$	4.20654e10	0.800890
$35$	0	0
$36$	7.12625e10	0.909376
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−2.27709e11	−1.99019
$39$	0	0
$40$	0	0
$41$	−3.33394e11	−1.71187	−0.855934	−	0.517085i	$-0.827017\pi$
$41$	−3.33394e11	−1.71187	−0.855934	+	0.517085i	$0.827017\pi$
$42$	0	0
$43$	−4.95013e11	−1.82111	−0.910557	−	0.413384i	$-0.864347\pi$
$43$	−4.95013e11	−1.82111	−0.910557	+	0.413384i	$0.864347\pi$
$44$	6.34262e11	1.98655
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	8.11212e11	1.38180
$49$	6.78223e11	1.00000
$50$	−7.81250e11	−1.00000
$51$	−9.93139e11	−1.10667
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.67667e11	0.125225
$55$	0	0
$56$	0	0
$57$	5.37606e12	2.75005
$58$	0	0
$59$	−3.91449e12	−1.57294	−0.786469	−	0.617630i	$-0.788093\pi$
$59$	−3.91449e12	−1.57294	−0.786469	+	0.617630i	$0.788093\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	4.39805e12	1.00000
$65$	0	0
$66$	−1.49745e13	−2.74502
$67$	2.71110e12	0.447324	0.223662	−	0.974667i	$-0.428199\pi$
$67$	2.71110e12	0.447324	0.223662	+	0.974667i	$0.428199\pi$
$68$	−5.38438e12	−0.800890
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−9.12159e12	−0.909376
$73$	1.57940e12	0.142966	0.0714830	−	0.997442i	$-0.477227\pi$
$73$	1.57940e12	0.142966	0.0714830	+	0.997442i	$0.477227\pi$
$74$	0	0
$75$	1.84448e13	1.38180
$76$	2.91467e13	1.99019
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−2.47621e13	−1.08241
$82$	4.26744e13	1.71187
$83$	−3.11463e13	−1.14778	−0.573891	−	0.818932i	$-0.694567\pi$
$83$	−3.11463e13	−1.14778	−0.573891	+	0.818932i	$0.694567\pi$
$84$	0	0
$85$	0	0
$86$	6.33616e13	1.82111
$87$	0	0
$88$	−8.11855e13	−1.98655
$89$	−3.84337e13	−0.868924	−0.434462	−	0.900690i	$-0.643061\pi$
$89$	−3.84337e13	−0.868924	−0.434462	+	0.900690i	$0.643061\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.03835e14	−1.38180
$97$	−6.28150e13	−0.777430	−0.388715	−	0.921358i	$-0.627081\pi$
$97$	−6.28150e13	−0.777430	−0.388715	+	0.921358i	$0.627081\pi$
$98$	−8.68126e13	−1.00000
$99$	1.68380e14	1.80652
$100$	1.00000e14	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	1.27122e14	1.10667
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.79454e14	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$107$	1.79454e14	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$108$	−2.14614e13	−0.125225
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.33670e11	−0.00396866	−0.00198433	−	0.999998i	$-0.500632\pi$
$113$	−9.33670e11	−0.00396866	−0.00198433	+	0.999998i	$0.500632\pi$
$114$	−6.88136e14	−2.75005
$115$	0	0
$116$	0	0
$117$	0	0
$118$	5.01055e14	1.57294
$119$	0	0
$120$	0	0
$121$	1.11889e15	2.94638
$122$	0	0
$123$	−1.00752e15	−2.36546
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−5.62950e14	−1.00000
$129$	−1.49593e15	−2.51642
$130$	0	0
$131$	−1.31730e15	−1.98970	−0.994848	−	0.101381i	$-0.967674\pi$
$131$	−1.31730e15	−1.98970	−0.994848	+	0.101381i	$0.967674\pi$
$132$	1.91674e15	2.74502
$133$	0	0
$134$	−3.47021e14	−0.447324
$135$	0	0
$136$	6.89200e14	0.800890
$137$	1.60085e15	1.76729	0.883644	−	0.468159i	$-0.155083\pi$
$137$	1.60085e15	1.76729	0.883644	+	0.468159i	$0.155083\pi$
$138$	0	0
$139$	8.56226e14	0.854053	0.427027	−	0.904239i	$-0.359561\pi$
$139$	8.56226e14	0.854053	0.427027	+	0.904239i	$0.359561\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.16756e15	0.909376
$145$	0	0
$146$	−2.02164e14	−0.142966
$147$	2.04959e15	1.38180
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.36094e15	−1.38180
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−3.73078e15	−1.99019
$153$	−1.42941e15	−0.728310
$154$	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
$160$	0	0
$161$	0	0
$162$	3.16955e15	1.08241
$163$	−2.78599e15	−0.911309	−0.455655	−	0.890157i	$-0.650595\pi$
$163$	−2.78599e15	−0.911309	−0.455655	+	0.890157i	$0.650595\pi$
$164$	−5.46232e15	−1.71187
$165$	0	0
$166$	3.98672e15	1.14778
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	3.93738e15	1.00000
$170$	0	0
$171$	7.73767e15	1.80983
$172$	−8.11029e15	−1.82111
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	1.03917e16	1.98655
$177$	−1.18296e16	−2.17349
$178$	4.91951e15	0.868924
$179$	7.27403e15	1.23539	0.617695	−	0.786418i	$-0.288067\pi$
$179$	7.27403e15	1.23539	0.617695	+	0.786418i	$0.288067\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.27222e16	−1.59101
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.32909e16	1.38180
$193$	−1.94418e16	−1.94911	−0.974554	−	0.224153i	$-0.928038\pi$
$193$	−1.94418e16	−1.94911	−0.974554	+	0.224153i	$0.928038\pi$
$194$	8.04032e15	0.777430
$195$	0	0
$196$	1.11120e16	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−2.15526e16	−1.80652
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.28000e16	−1.00000
$201$	8.19296e15	0.618113
$202$	0	0
$203$	0	0
$204$	−1.62716e16	−1.10667
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.88681e16	3.95361
$210$	0	0
$211$	−2.70870e16	−1.45474	−0.727368	−	0.686248i	$-0.759256\pi$
$211$	−2.70870e16	−1.45474	−0.727368	+	0.686248i	$0.759256\pi$
$212$	0	0
$213$	0	0
$214$	−2.29701e16	−1.11755
$215$	0	0
$216$	2.74706e15	0.125225
$217$	0	0
$218$	0	0
$219$	4.77295e15	0.197551
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.65473e16	0.909376
$226$	1.19510e14	0.00396866
$227$	1.56540e16	0.504014	0.252007	−	0.967725i	$-0.418909\pi$
$227$	1.56540e16	0.504014	0.252007	+	0.967725i	$0.418909\pi$
$228$	8.80813e16	2.75005
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.42439e16	−1.72322	−0.861610	−	0.507571i	$-0.830543\pi$
$233$	−6.42439e16	−1.72322	−0.861610	+	0.507571i	$0.830543\pi$
$234$	0	0
$235$	0	0
$236$	−6.41351e16	−1.57294
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−2.26270e16	−0.479190	−0.239595	−	0.970873i	$-0.577015\pi$
$241$	−2.26270e16	−0.479190	−0.239595	+	0.970873i	$0.577015\pi$
$242$	−1.43218e17	−2.94638
$243$	−6.85659e16	−1.37045
$244$	0	0
$245$	0	0
$246$	1.28962e17	2.36546
$247$	0	0
$248$	0	0
$249$	−9.41240e16	−1.58601
$250$	0	0
$251$	1.11663e17	1.77907	0.889534	−	0.456869i	$-0.151029\pi$
$251$	1.11663e17	1.77907	0.889534	+	0.456869i	$0.151029\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	7.20576e16	1.00000
$257$	4.88939e16	0.660272	0.330136	−	0.943933i	$-0.392905\pi$
$257$	4.88939e16	0.660272	0.330136	+	0.943933i	$0.392905\pi$
$258$	1.91479e17	2.51642
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.68615e17	1.98970
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−2.45343e17	−2.74502
$265$	0	0
$266$	0	0
$267$	−1.16147e17	−1.20068
$268$	4.44187e16	0.447324
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−8.82176e16	−0.800890
$273$	0	0
$274$	−2.04909e17	−1.76729
$275$	2.36281e17	1.98655
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.09597e17	−0.854053
$279$	0	0
$280$	0	0
$281$	2.22626e16	0.160928	0.0804638	−	0.996758i	$-0.474360\pi$
$281$	2.22626e16	0.160928	0.0804638	+	0.996758i	$0.474360\pi$
$282$	0	0
$283$	2.47411e17	1.70182	0.850910	−	0.525311i	$-0.176051\pi$
$283$	2.47411e17	1.70182	0.850910	+	0.525311i	$0.176051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.49448e17	−0.909376
$289$	−6.03761e16	−0.358575
$290$	0	0
$291$	−1.89827e17	−1.07425
$292$	2.58769e16	0.142966
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−2.62348e17	−1.38180
$295$	0	0
$296$	0	0
$297$	−5.07091e16	−0.248766
$298$	0	0
$299$	0	0
$300$	3.02200e17	1.38180
$301$	0	0
$302$	0	0
$303$	0	0
$304$	4.77540e17	1.99019
$305$	0	0
$306$	1.82964e17	0.728310
$307$	−4.93680e17	−1.92078	−0.960388	−	0.278665i	$-0.910108\pi$
$307$	−4.93680e17	−1.92078	−0.960388	+	0.278665i	$0.910108\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	3.79697e17	1.29011	0.645055	−	0.764136i	$-0.276834\pi$
$313$	3.79697e17	1.29011	0.645055	+	0.764136i	$0.276834\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	5.42309e17	1.54423
$322$	0	0
$323$	−5.84635e17	−1.59392
$324$	−4.05702e17	−1.08241
$325$	0	0
$326$	3.56606e17	0.911309
$327$	0	0
$328$	6.99177e17	1.71187
$329$	0	0
$330$	0	0
$331$	−1.28422e17	−0.295014	−0.147507	−	0.989061i	$-0.547125\pi$
$331$	−1.28422e17	−0.295014	−0.147507	+	0.989061i	$0.547125\pi$
$332$	−5.10300e17	−1.14778
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.83482e17	−1.38458	−0.692290	−	0.721620i	$-0.743398\pi$
$337$	−6.83482e17	−1.38458	−0.692290	+	0.721620i	$0.743398\pi$
$338$	−5.03984e17	−1.00000
$339$	−2.82155e15	−0.00548390
$340$	0	0
$341$	0	0
$342$	−9.90422e17	−1.80983
$343$	0	0
$344$	1.03812e18	1.82111
$345$	0	0
$346$	0	0
$347$	7.74680e17	1.27884	0.639420	−	0.768858i	$-0.279174\pi$
$347$	7.74680e17	1.27884	0.639420	+	0.768858i	$0.279174\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−1.33014e18	−1.98655
$353$	−1.26968e18	−1.85897	−0.929484	−	0.368862i	$-0.879747\pi$
$353$	−1.26968e18	−1.85897	−0.929484	+	0.368862i	$0.879747\pi$
$354$	1.51419e18	2.17349
$355$	0	0
$356$	−6.29697e17	−0.868924
$357$	0	0
$358$	−9.31076e17	−1.23539
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.36574e18	2.96085
$362$	0	0
$363$	3.38128e18	4.07132
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−1.45010e18	−1.55673
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	1.62845e18	1.59101
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.24161e18	−1.99566	−0.997830	−	0.0658458i	$-0.979025\pi$
$379$	−2.24161e18	−1.99566	−0.997830	+	0.0658458i	$0.979025\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.70123e18	−1.38180
$385$	0	0
$386$	2.48855e18	1.94911
$387$	−2.15306e18	−1.65608
$388$	−1.02916e18	−0.777430
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.42234e18	−1.00000
$393$	−3.98089e18	−2.74936
$394$	0	0
$395$	0	0
$396$	2.75873e18	1.80652
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.63840e18	1.00000
$401$	1.69870e18	1.01884	0.509419	−	0.860519i	$-0.329860\pi$
$401$	1.69870e18	1.01884	0.509419	+	0.860519i	$0.329860\pi$
$402$	−1.04870e18	−0.618113
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.08276e18	1.10667
$409$	7.50795e17	0.392156	0.196078	−	0.980588i	$-0.437180\pi$
$409$	7.50795e17	0.392156	0.196078	+	0.980588i	$0.437180\pi$
$410$	0	0
$411$	4.83778e18	2.44204
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.58752e18	1.18013
$418$	−8.81511e18	−3.95361
$419$	−4.51606e18	−1.99187	−0.995936	−	0.0900663i	$-0.971292\pi$
$419$	−4.51606e18	−1.99187	−0.995936	+	0.0900663i	$0.971292\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	3.46714e18	1.45474
$423$	0	0
$424$	0	0
$425$	−2.00584e18	−0.800890
$426$	0	0
$427$	0	0
$428$	2.94017e18	1.11755
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−3.51623e17	−0.125225
$433$	5.25040e18	1.83983	0.919913	−	0.392122i	$-0.128259\pi$
$433$	5.25040e18	1.83983	0.919913	+	0.392122i	$0.128259\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−6.10938e17	−0.197551
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	2.94994e18	0.909376
$442$	0	0
$443$	3.94208e18	1.17733	0.588667	−	0.808376i	$-0.299653\pi$
$443$	3.94208e18	1.17733	0.588667	+	0.808376i	$0.299653\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.19930e18	−0.597805	−0.298902	−	0.954284i	$-0.596620\pi$
$449$	−2.19930e18	−0.597805	−0.298902	+	0.954284i	$0.596620\pi$
$450$	−3.39806e18	−0.909376
$451$	−1.29064e19	−3.40071
$452$	−1.52972e16	−0.00396866
$453$	0	0
$454$	−2.00371e18	−0.504014
$455$	0	0
$456$	−1.12744e19	−2.75005
$457$	8.01731e18	1.92582	0.962908	−	0.269828i	$-0.0869669\pi$
$457$	8.01731e18	1.92582	0.962908	+	0.269828i	$0.0869669\pi$
$458$	0	0
$459$	4.30480e17	0.100291
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	8.22322e18	1.72322
$467$	2.44265e18	0.504248	0.252124	−	0.967695i	$-0.418871\pi$
$467$	2.44265e18	0.504248	0.252124	+	0.967695i	$0.418871\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	8.20929e18	1.57294
$473$	−1.91631e19	−3.61773
$474$	0	0
$475$	1.08580e19	1.99019
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	2.89626e18	0.479190
$483$	0	0
$484$	1.83319e19	2.94638
$485$	0	0
$486$	8.77643e18	1.37045
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−8.41925e18	−1.25925
$490$	0	0
$491$	−2.26387e18	−0.329065	−0.164532	−	0.986372i	$-0.552612\pi$
$491$	−2.26387e18	−0.329065	−0.164532	+	0.986372i	$0.552612\pi$
$492$	−1.65071e19	−2.36546
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	1.20479e19	1.58601
$499$	3.81392e18	0.495071	0.247535	−	0.968879i	$-0.420379\pi$
$499$	3.81392e18	0.495071	0.247535	+	0.968879i	$0.420379\pi$
$500$	0	0
$501$	0	0
$502$	−1.42928e19	−1.77907
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.18988e19	1.38180
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−9.22337e18	−1.00000
$513$	−2.33027e18	−0.249221
$514$	−6.25842e18	−0.660272
$515$	0	0
$516$	−2.45093e19	−2.51642
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.65089e18	0.542317	0.271158	−	0.962535i	$-0.412593\pi$
$521$	5.65089e18	0.542317	0.271158	+	0.962535i	$0.412593\pi$
$522$	0	0
$523$	−8.51815e18	−0.795854	−0.397927	−	0.917417i	$-0.630270\pi$
$523$	−8.51815e18	−0.795854	−0.397927	+	0.917417i	$0.630270\pi$
$524$	−2.15827e19	−1.98970
$525$	0	0
$526$	0	0
$527$	0	0
$528$	3.14038e19	2.74502
$529$	1.15928e19	1.00000
$530$	0	0
$531$	−1.70262e19	−1.43039
$532$	0	0
$533$	0	0
$534$	1.48668e19	1.20068
$535$	0	0
$536$	−5.68560e18	−0.447324
$537$	2.19821e19	1.70706
$538$	0	0
$539$	2.62555e19	1.98655
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.12919e19	0.800890
$545$	0	0
$546$	0	0
$547$	−2.43861e19	−1.66430	−0.832150	−	0.554551i	$-0.812890\pi$
$547$	−2.43861e19	−1.66430	−0.832150	+	0.554551i	$0.812890\pi$
$548$	2.62284e19	1.76729
$549$	0	0
$550$	−3.02439e19	−1.98655
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.40284e19	0.854053
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.84466e19	−2.19846
$562$	−2.84961e18	−0.160928
$563$	3.54216e19	1.97565	0.987823	−	0.155580i	$-0.0497246\pi$
$563$	3.54216e19	1.97565	0.987823	+	0.155580i	$0.0497246\pi$
$564$	0	0
$565$	0	0
$566$	−3.16686e19	−1.70182
$567$	0	0
$568$	0	0
$569$	3.10898e19	1.61002	0.805011	−	0.593260i	$-0.202159\pi$
$569$	3.10898e19	1.61002	0.805011	+	0.593260i	$0.202159\pi$
$570$	0	0
$571$	−3.62362e19	−1.83101	−0.915503	−	0.402312i	$-0.868207\pi$
$571$	−3.62362e19	−1.83101	−0.915503	+	0.402312i	$0.868207\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.91294e19	0.909376
$577$	−5.16622e17	−0.0242628	−0.0121314	−	0.999926i	$-0.503862\pi$
$577$	−5.16622e17	−0.0242628	−0.0121314	+	0.999926i	$0.503862\pi$
$578$	7.72814e18	0.358575
$579$	−5.87532e19	−2.69328
$580$	0	0
$581$	0	0
$582$	2.42979e19	1.07425
$583$	0	0
$584$	−3.31225e18	−0.142966
$585$	0	0
$586$	0	0
$587$	7.99248e18	0.332824	0.166412	−	0.986056i	$-0.446782\pi$
$587$	7.99248e18	0.332824	0.166412	+	0.986056i	$0.446782\pi$
$588$	3.35805e19	1.38180
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.85851e19	−1.10855	−0.554277	−	0.832332i	$-0.687005\pi$
$593$	−2.85851e19	−1.10855	−0.554277	+	0.832332i	$0.687005\pi$
$594$	6.49077e18	0.248766
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−3.86816e19	−1.38180
$601$	1.32271e19	0.467030	0.233515	−	0.972353i	$-0.424977\pi$
$601$	1.32271e19	0.467030	0.233515	+	0.972353i	$0.424977\pi$
$602$	0	0
$603$	1.17920e19	0.406786
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−6.11251e19	−1.99019
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−2.34194e19	−0.728310
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	6.31910e19	1.92078
$615$	0	0
$616$	0	0
$617$	6.40050e19	1.88026	0.940129	−	0.340817i	$-0.110704\pi$
$617$	6.40050e19	1.88026	0.940129	+	0.340817i	$0.110704\pi$
$618$	0	0
$619$	−1.08767e19	−0.312365	−0.156182	−	0.987728i	$-0.549919\pi$
$619$	−1.08767e19	−0.312365	−0.156182	+	0.987728i	$0.549919\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.72529e19	1.00000
$626$	−4.86012e19	−1.29011
$627$	2.08119e20	5.46311
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−8.18570e19	−2.01016
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.81552e19	−0.858119	−0.429059	−	0.903276i	$-0.641155\pi$
$641$	−3.81552e19	−0.858119	−0.429059	+	0.903276i	$0.641155\pi$
$642$	−6.94156e19	−1.54423
$643$	−6.40914e19	−1.41034	−0.705168	−	0.709040i	$-0.749128\pi$
$643$	−6.40914e19	−1.41034	−0.705168	+	0.709040i	$0.749128\pi$
$644$	0	0
$645$	0	0
$646$	7.48333e19	1.59392
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	5.19299e19	1.08241
$649$	−1.51539e20	−3.12472
$650$	0	0
$651$	0	0
$652$	−4.56456e19	−0.911309
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−8.94947e19	−1.71187
$657$	6.86964e18	0.130010
$658$	0	0
$659$	−3.21522e19	−0.595679	−0.297839	−	0.954616i	$-0.596266\pi$
$659$	−3.21522e19	−0.595679	−0.297839	+	0.954616i	$0.596266\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.64380e19	0.295014
$663$	0	0
$664$	6.53184e19	1.14778
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.13872e20	1.82100	0.910502	−	0.413504i	$-0.135695\pi$
$673$	1.13872e20	1.82100	0.910502	+	0.413504i	$0.135695\pi$
$674$	8.74857e19	1.38458
$675$	−7.99498e18	−0.125225
$676$	6.45100e19	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	3.61158e17	0.00548390
$679$	0	0
$680$	0	0
$681$	4.73062e19	0.696448
$682$	0	0
$683$	−1.21775e20	−1.75636	−0.878180	−	0.478331i	$-0.841242\pi$
$683$	−1.21775e20	−1.75636	−0.878180	+	0.478331i	$0.841242\pi$
$684$	1.26774e20	1.80983
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.32879e20	−1.82111
$689$	0	0
$690$	0	0
$691$	1.05570e20	1.40345	0.701723	−	0.712450i	$-0.252414\pi$
$691$	1.05570e20	1.40345	0.701723	+	0.712450i	$0.252414\pi$
$692$	0	0
$693$	0	0
$694$	−9.91591e19	−1.27884
$695$	0	0
$696$	0	0
$697$	1.09565e20	1.37102
$698$	0	0
$699$	−1.94145e20	−2.38115
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.70258e20	1.98655
$705$	0	0
$706$	1.62519e20	1.85897
$707$	0	0
$708$	−1.93816e20	−2.17349
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	8.06013e19	0.868924
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.19178e20	1.23539
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−3.02815e20	−2.96085
$723$	−6.83788e19	−0.662145
$724$	0	0
$725$	0	0
$726$	−4.32804e20	−4.07132
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−8.87697e19	−0.811283
$730$	0	0
$731$	1.62679e20	1.45851
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.04953e20	0.888633
$738$	1.85613e20	1.55673
$739$	2.40601e20	1.99888	0.999441	−	0.0334271i	$-0.0106422\pi$
$739$	2.40601e20	1.99888	0.999441	+	0.0334271i	$0.0106422\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.35471e20	−1.04376
$748$	−2.08441e20	−1.59101
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	3.37445e20	2.45832
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	2.86927e20	1.99566
$759$	0	0
$760$	0	0
$761$	−2.86406e20	−1.93771	−0.968855	−	0.247628i	$-0.920349\pi$
$761$	−2.86406e20	−1.93771	−0.968855	+	0.247628i	$0.920349\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	2.17758e20	1.38180
$769$	−2.67301e20	−1.68080	−0.840401	−	0.541965i	$-0.817680\pi$
$769$	−2.67301e20	−1.68080	−0.840401	+	0.541965i	$0.817680\pi$
$770$	0	0
$771$	1.47757e20	0.912364
$772$	−3.18535e20	−1.94911
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.75592e20	1.65608
$775$	0	0
$776$	1.31733e20	0.777430
$777$	0	0
$778$	0	0
$779$	−5.93098e20	−3.40694
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.82059e20	1.00000
$785$	0	0
$786$	5.09554e20	2.74936
$787$	3.56201e20	1.90490	0.952450	−	0.304694i	$-0.0985543\pi$
$787$	3.56201e20	1.90490	0.952450	+	0.304694i	$0.0985543\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−3.53117e20	−1.80652
$793$	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
$800$	−2.09715e20	−1.00000
$801$	−1.67168e20	−0.790178
$802$	−2.17433e20	−1.01884
$803$	6.11422e19	0.284009
$804$	1.34233e20	0.618113
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.14025e19	−0.182552	−0.0912760	−	0.995826i	$-0.529095\pi$
$809$	−4.14025e19	−0.182552	−0.0912760	+	0.995826i	$0.529095\pi$
$810$	0	0
$811$	3.21183e20	1.39189	0.695947	−	0.718093i	$-0.254985\pi$
$811$	3.21183e20	1.39189	0.695947	+	0.718093i	$0.254985\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−2.66594e20	−1.10667
$817$	−8.80614e20	−3.62436
$818$	−9.61018e19	−0.392156
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−6.19235e20	−2.44204
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	7.14041e20	2.74502
$826$	0	0
$827$	1.09065e20	0.412238	0.206119	−	0.978527i	$-0.433917\pi$
$827$	1.09065e20	0.412238	0.206119	+	0.978527i	$0.433917\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.22889e20	−0.800890
$834$	−3.31202e20	−1.18013
$835$	0	0
$836$	1.12833e21	3.95361
$837$	0	0
$838$	5.78055e20	1.99187
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.97558e20	1.00000
$842$	0	0
$843$	6.72774e19	0.222370
$844$	−4.43794e20	−1.45474
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.47676e20	2.35158
$850$	2.56747e20	0.800890
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−3.76342e20	−1.11755
$857$	5.06915e20	1.49303	0.746516	−	0.665367i	$-0.231725\pi$
$857$	5.06915e20	1.49303	0.746516	+	0.665367i	$0.231725\pi$
$858$	0	0
$859$	−3.08240e20	−0.893175	−0.446587	−	0.894740i	$-0.647361\pi$
$859$	−3.08240e20	−0.893175	−0.446587	+	0.894740i	$0.647361\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	4.50078e19	0.125225
$865$	0	0
$866$	−6.72051e20	−1.83983
$867$	−1.82456e20	−0.495479
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.73215e20	−0.706976
$874$	0	0
$875$	0	0
$876$	7.82001e19	0.197551
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.20410e20	0.777812	0.388906	−	0.921277i	$-0.372853\pi$
$881$	3.20410e20	0.777812	0.388906	+	0.921277i	$0.372853\pi$
$882$	−3.77593e20	−0.909376
$883$	−7.45828e20	−1.78202	−0.891012	−	0.453979i	$-0.850004\pi$
$883$	−7.45828e20	−1.78202	−0.891012	+	0.453979i	$0.850004\pi$
$884$	0	0
$885$	0	0
$886$	−5.04587e20	−1.17733
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.58597e20	−2.15027
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	2.81510e20	0.597805
$899$	0	0
$900$	4.34952e20	0.909376
$901$	0	0
$902$	1.65202e21	3.40071
$903$	0	0
$904$	1.95805e18	0.00396866
$905$	0	0
$906$	0	0
$907$	6.98908e20	1.38410	0.692052	−	0.721848i	$-0.256707\pi$
$907$	6.98908e20	1.38410	0.692052	+	0.721848i	$0.256707\pi$
$908$	2.56474e20	0.504014
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.44312e21	2.75005
$913$	−1.20574e21	−2.28013
$914$	−1.02622e21	−1.92582
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−5.51014e19	−0.100291
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.49190e21	−2.65413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.06394e21	1.78159	0.890794	−	0.454408i	$-0.150149\pi$
$929$	1.06394e21	1.78159	0.890794	+	0.454408i	$0.150149\pi$
$930$	0	0
$931$	1.20654e21	1.99019
$932$	−1.05257e21	−1.72322
$933$	0	0
$934$	−3.12660e20	−0.504248
$935$	0	0
$936$	0	0
$937$	4.81305e20	0.759002	0.379501	−	0.925191i	$-0.376096\pi$
$937$	4.81305e20	0.759002	0.379501	+	0.925191i	$0.376096\pi$
$938$	0	0
$939$	1.14745e21	1.78268
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−1.05079e21	−1.57294
$945$	0	0
$946$	2.45287e21	3.61773
$947$	−9.06817e20	−1.32761	−0.663804	−	0.747907i	$-0.731059\pi$
$947$	−9.06817e20	−1.32761	−0.663804	+	0.747907i	$0.731059\pi$
$948$	0	0
$949$	0	0
$950$	−1.38982e21	−1.99019
$951$	0	0
$952$	0	0
$953$	7.11937e20	0.997220	0.498610	−	0.866826i	$-0.333844\pi$
$953$	7.11937e20	0.997220	0.498610	+	0.866826i	$0.333844\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.56944e20	1.00000
$962$	0	0
$963$	7.80537e20	1.01627
$964$	−3.70721e20	−0.479190
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.34648e21	−2.94638
$969$	−1.76677e21	−2.20249
$970$	0	0
$971$	8.31378e20	1.02156	0.510780	−	0.859712i	$-0.329357\pi$
$971$	8.31378e20	1.02156	0.510780	+	0.859712i	$0.329357\pi$
$972$	−1.12338e21	−1.37045
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.79186e20	−0.446263	−0.223131	−	0.974788i	$-0.571628\pi$
$977$	−3.79186e20	−0.446263	−0.223131	+	0.974788i	$0.571628\pi$
$978$	1.07766e21	1.25925
$979$	−1.48785e21	−1.72616
$980$	0	0
$981$	0	0
$982$	2.89776e20	0.329065
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	2.11291e21	2.36546
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−3.88091e20	−0.407651
$994$	0	0
$995$	0	0
$996$	−1.54213e21	−1.58601
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−4.88181e20	−0.495071
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.15.d.a.3.1	✓	1
3.2	odd	2		72.15.b.a.19.1		1
4.3	odd	2		32.15.d.a.15.1		1
8.3	odd	2	CM	8.15.d.a.3.1	✓	1
8.5	even	2		32.15.d.a.15.1		1
24.11	even	2		72.15.b.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.15.d.a.3.1	✓	1	1.1	even	1	trivial
8.15.d.a.3.1	✓	1	8.3	odd	2	CM
32.15.d.a.15.1		1	4.3	odd	2
32.15.d.a.15.1		1	8.5	even	2
72.15.b.a.19.1		1	3.2	odd	2
72.15.b.a.19.1		1	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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