LMFDB - Embedded newform 8.25.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,25,Mod(3,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 25, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 25);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 25 $
Character orbit:	$[\chi]$	$=$	8.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.1973721806$
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)

$f(q)$

$=$

$q+4096.00 q^{2} -629918. q^{3} +1.67772e7 q^{4} -2.58014e9 q^{6} +6.87195e10 q^{8} +1.14367e11 q^{9} +O(q^{10})$

$q+4096.00 q^{2} -629918. q^{3} +1.67772e7 q^{4} -2.58014e9 q^{6} +6.87195e10 q^{8} +1.14367e11 q^{9} -2.57846e12 q^{11} -1.05683e13 q^{12} +2.81475e14 q^{16} +8.86489e14 q^{17} +4.68448e14 q^{18} +3.30531e15 q^{19} -1.05614e16 q^{22} -4.32876e16 q^{24} +5.96046e16 q^{25} +1.05866e17 q^{27} +1.15292e18 q^{32} +1.62422e18 q^{33} +3.63106e18 q^{34} +1.91876e18 q^{36} +1.35385e19 q^{38} +2.93199e19 q^{41} -3.04897e19 q^{43} -4.32594e19 q^{44} -1.77306e20 q^{48} +1.91581e20 q^{49} +2.44141e20 q^{50} -5.58415e20 q^{51} +4.33625e20 q^{54} -2.08207e21 q^{57} -3.48377e21 q^{59} +4.72237e21 q^{64} +6.65279e21 q^{66} +1.42756e22 q^{67} +1.48728e22 q^{68} +7.85925e21 q^{72} -4.33894e22 q^{73} -3.75460e22 q^{75} +5.54539e22 q^{76} -9.89873e22 q^{81} +1.20094e23 q^{82} -1.76554e23 q^{83} -1.24886e23 q^{86} -1.77190e23 q^{88} +2.56541e23 q^{89} -7.26246e23 q^{96} +1.35627e24 q^{97} +7.84717e23 q^{98} -2.94891e23 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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 $

$2$	4096.00	1.00000
$2$	4096.00	1.00000
$3$	−629918.	−1.18530	−0.592651	−	0.805459i	$-0.701919\pi$
$3$	−629918.	−1.18530	−0.592651	+	0.805459i	$0.701919\pi$
$4$	1.67772e7	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−2.58014e9	−1.18530
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	6.87195e10	1.00000
$9$	1.14367e11	0.404940
$10$	0	0
$11$	−2.57846e12	−0.821576	−0.410788	−	0.911731i	$-0.634746\pi$
$11$	−2.57846e12	−0.821576	−0.410788	+	0.911731i	$0.634746\pi$
$12$	−1.05683e13	−1.18530
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	2.81475e14	1.00000
$17$	8.86489e14	1.52155	0.760775	−	0.649016i	$-0.224819\pi$
$17$	8.86489e14	1.52155	0.760775	+	0.649016i	$0.224819\pi$
$18$	4.68448e14	0.404940
$19$	3.30531e15	1.49337	0.746687	−	0.665175i	$-0.231643\pi$
$19$	3.30531e15	1.49337	0.746687	+	0.665175i	$0.231643\pi$
$20$	0	0
$21$	0	0
$22$	−1.05614e16	−0.821576
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−4.32876e16	−1.18530
$25$	5.96046e16	1.00000
$26$	0	0
$27$	1.05866e17	0.705325
$28$	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			$\pi$
$32$	1.15292e18	1.00000
$33$	1.62422e18	0.973816
$34$	3.63106e18	1.52155
$35$	0	0
$36$	1.91876e18	0.404940
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.35385e19	1.49337
$39$	0	0
$40$	0	0
$41$	2.93199e19	1.29944	0.649720	−	0.760174i	$-0.274886\pi$
$41$	2.93199e19	1.29944	0.649720	+	0.760174i	$0.274886\pi$
$42$	0	0
$43$	−3.04897e19	−0.763012	−0.381506	−	0.924366i	$-0.624594\pi$
$43$	−3.04897e19	−0.763012	−0.381506	+	0.924366i	$0.624594\pi$
$44$	−4.32594e19	−0.821576
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.77306e20	−1.18530
$49$	1.91581e20	1.00000
$50$	2.44141e20	1.00000
$51$	−5.58415e20	−1.80350
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	4.33625e20	0.705325
$55$	0	0
$56$	0	0
$57$	−2.08207e21	−1.77010
$58$	0	0
$59$	−3.48377e21	−1.95806	−0.979030	−	0.203717i	$-0.934698\pi$
$59$	−3.48377e21	−1.95806	−0.979030	+	0.203717i	$0.934698\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	4.72237e21	1.00000
$65$	0	0
$66$	6.65279e21	0.973816
$67$	1.42756e22	1.74460	0.872302	−	0.488968i	$-0.162626\pi$
$67$	1.42756e22	1.74460	0.872302	+	0.488968i	$0.162626\pi$
$68$	1.48728e22	1.52155
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	7.85925e21	0.404940
$73$	−4.33894e22	−1.89456	−0.947282	−	0.320400i	$-0.896183\pi$
$73$	−4.33894e22	−1.89456	−0.947282	+	0.320400i	$0.896183\pi$
$74$	0	0
$75$	−3.75460e22	−1.18530
$76$	5.54539e22	1.49337
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−9.89873e22	−1.24096
$82$	1.20094e23	1.29944
$83$	−1.76554e23	−1.65174	−0.825868	−	0.563863i	$-0.809314\pi$
$83$	−1.76554e23	−1.65174	−0.825868	+	0.563863i	$0.809314\pi$
$84$	0	0
$85$	0	0
$86$	−1.24886e23	−0.763012
$87$	0	0
$88$	−1.77190e23	−0.821576
$89$	2.56541e23	1.03867	0.519333	−	0.854572i	$-0.326180\pi$
$89$	2.56541e23	1.03867	0.519333	+	0.854572i	$0.326180\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−7.26246e23	−1.18530
$97$	1.35627e24	1.95472	0.977360	−	0.211581i	$-0.0678612\pi$
$97$	1.35627e24	1.95472	0.977360	+	0.211581i	$0.0678612\pi$
$98$	7.84717e23	1.00000
$99$	−2.94891e23	−0.332690
$100$	1.00000e24	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−2.28727e24	−1.80350
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.20608e24	1.42354	0.711768	−	0.702414i	$-0.247895\pi$
$107$	3.20608e24	1.42354	0.711768	+	0.702414i	$0.247895\pi$
$108$	1.77613e24	0.705325
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.38197e24	1.24165	0.620827	−	0.783948i	$-0.286797\pi$
$113$	5.38197e24	1.24165	0.620827	+	0.783948i	$0.286797\pi$
$114$	−8.52817e24	−1.77010
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.42695e25	−1.95806
$119$	0	0
$120$	0	0
$121$	−3.20128e24	−0.325012
$122$	0	0
$123$	−1.84691e25	−1.54023
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.93428e25	1.00000
$129$	1.92060e25	0.904399
$130$	0	0
$131$	−4.91687e25	−1.92501	−0.962506	−	0.271262i	$-0.912559\pi$
$131$	−4.91687e25	−1.92501	−0.962506	+	0.271262i	$0.912559\pi$
$132$	2.72498e25	0.973816
$133$	0	0
$134$	5.84729e25	1.74460
$135$	0	0
$136$	6.09190e25	1.52155
$137$	8.72632e25	1.99611	0.998055	−	0.0623463i	$-0.0198583\pi$
$137$	8.72632e25	1.99611	0.998055	+	0.0623463i	$0.0198583\pi$
$138$	0	0
$139$	−8.64496e25	−1.66183	−0.830913	−	0.556403i	$-0.812181\pi$
$139$	−8.64496e25	−1.66183	−0.830913	+	0.556403i	$0.812181\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.21915e25	0.404940
$145$	0	0
$146$	−1.77723e26	−1.89456
$147$	−1.20680e26	−1.18530
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.53789e26	−1.18530
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	2.27139e26	1.49337
$153$	1.01385e26	0.616137
$154$	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
$162$	−4.05452e26	−1.24096
$163$	−2.15258e26	−0.611938	−0.305969	−	0.952041i	$-0.598980\pi$
$163$	−2.15258e26	−0.611938	−0.305969	+	0.952041i	$0.598980\pi$
$164$	4.91906e26	1.29944
$165$	0	0
$166$	−7.23166e26	−1.65174
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	5.42801e26	1.00000
$170$	0	0
$171$	3.78019e26	0.604728
$172$	−5.11532e26	−0.763012
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−7.25772e26	−0.821576
$177$	2.19449e27	2.32089
$178$	1.05079e27	1.03867
$179$	9.00969e26	0.832671	0.416336	−	0.909211i	$-0.363314\pi$
$179$	9.00969e26	0.832671	0.416336	+	0.909211i	$0.363314\pi$
$180$	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
$187$	−2.28577e27	−1.25007
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−2.97470e27	−1.18530
$193$	−5.33215e27	−1.99625	−0.998125	−	0.0612048i	$-0.980506\pi$
$193$	−5.33215e27	−1.99625	−0.998125	+	0.0612048i	$0.980506\pi$
$194$	5.55527e27	1.95472
$195$	0	0
$196$	3.21420e27	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.20787e27	−0.332690
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	4.09600e27	1.00000
$201$	−8.99246e27	−2.06788
$202$	0	0
$203$	0	0
$204$	−9.36865e27	−1.80350
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.52260e27	−1.22692
$210$	0	0
$211$	1.36781e28	1.75646	0.878229	−	0.478241i	$-0.158725\pi$
$211$	1.36781e28	1.75646	0.878229	+	0.478241i	$0.158725\pi$
$212$	0	0
$213$	0	0
$214$	1.31321e28	1.42354
$215$	0	0
$216$	7.27502e27	0.705325
$217$	0	0
$218$	0	0
$219$	2.73318e28	2.24563
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	6.81681e27	0.404940
$226$	2.20446e28	1.24165
$227$	−2.36955e28	−1.26577	−0.632887	−	0.774244i	$-0.718130\pi$
$227$	−2.36955e28	−1.26577	−0.632887	+	0.774244i	$0.718130\pi$
$228$	−3.49314e28	−1.77010
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.21426e28	−0.474288	−0.237144	−	0.971474i	$-0.576211\pi$
$233$	−1.21426e28	−0.474288	−0.237144	+	0.971474i	$0.576211\pi$
$234$	0	0
$235$	0	0
$236$	−5.84480e28	−1.95806
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	1.96215e28	0.511125	0.255562	−	0.966793i	$-0.417739\pi$
$241$	1.96215e28	0.511125	0.255562	+	0.966793i	$0.417739\pi$
$242$	−1.31125e28	−0.325012
$243$	3.24543e28	0.765591
$244$	0	0
$245$	0	0
$246$	−7.56495e28	−1.54023
$247$	0	0
$248$	0	0
$249$	1.11215e29	1.95781
$250$	0	0
$251$	−1.74737e28	−0.279448	−0.139724	−	0.990191i	$-0.544621\pi$
$251$	−1.74737e28	−0.279448	−0.139724	+	0.990191i	$0.544621\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	7.92282e28	1.00000
$257$	−1.19249e29	−1.43634	−0.718168	−	0.695869i	$-0.755019\pi$
$257$	−1.19249e29	−1.43634	−0.718168	+	0.695869i	$0.755019\pi$
$258$	7.86677e28	0.904399
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−2.01395e29	−1.92501
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.11615e29	0.973816
$265$	0	0
$266$	0	0
$267$	−1.61600e29	−1.23113
$268$	2.39505e29	1.74460
$269$	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			$\pi$
$272$	2.49524e29	1.52155
$273$	0	0
$274$	3.57430e29	1.99611
$275$	−1.53688e29	−0.821576
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−3.54098e29	−1.66183
$279$	0	0
$280$	0	0
$281$	2.47185e29	1.01987	0.509937	−	0.860212i	$-0.329669\pi$
$281$	2.47185e29	1.01987	0.509937	+	0.860212i	$0.329669\pi$
$282$	0	0
$283$	−9.13009e28	−0.345970	−0.172985	−	0.984924i	$-0.555341\pi$
$283$	−9.13009e28	−0.345970	−0.172985	+	0.984924i	$0.555341\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.31856e29	0.404940
$289$	4.46414e29	1.31511
$290$	0	0
$291$	−8.54338e29	−2.31693
$292$	−7.27953e29	−1.89456
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−4.94307e29	−1.18530
$295$	0	0
$296$	0	0
$297$	−2.72970e29	−0.579478
$298$	0	0
$299$	0	0
$300$	−6.29918e29	−1.18530
$301$	0	0
$302$	0	0
$303$	0	0
$304$	9.30361e29	1.49337
$305$	0	0
$306$	4.15274e29	0.616137
$307$	1.28368e30	1.83146	0.915729	−	0.401796i	$-0.131614\pi$
$307$	1.28368e30	1.83146	0.915729	+	0.401796i	$0.131614\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.46610e30	1.65818	0.829091	−	0.559114i	$-0.188859\pi$
$313$	1.46610e30	1.65818	0.829091	+	0.559114i	$0.188859\pi$
$314$	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
$321$	−2.01957e30	−1.68732
$322$	0	0
$323$	2.93012e30	2.27224
$324$	−1.66073e30	−1.24096
$325$	0	0
$326$	−8.81696e29	−0.611938
$327$	0	0
$328$	2.01485e30	1.29944
$329$	0	0
$330$	0	0
$331$	3.34831e30	1.93593	0.967966	−	0.251080i	$-0.0807856\pi$
$331$	3.34831e30	1.93593	0.967966	+	0.251080i	$0.0807856\pi$
$332$	−2.96209e30	−1.65174
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.28859e30	−1.99874	−0.999368	−	0.0355482i	$-0.988682\pi$
$337$	−4.28859e30	−1.99874	−0.999368	+	0.0355482i	$0.988682\pi$
$338$	2.22331e30	1.00000
$339$	−3.39020e30	−1.47173
$340$	0	0
$341$	0	0
$342$	1.54836e30	0.604728
$343$	0	0
$344$	−2.09523e30	−0.763012
$345$	0	0
$346$	0	0
$347$	−4.49911e30	−1.47629	−0.738147	−	0.674639i	$-0.764299\pi$
$347$	−4.49911e30	−1.47629	−0.738147	+	0.674639i	$0.764299\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−2.97276e30	−0.821576
$353$	−7.33981e30	−1.96060	−0.980298	−	0.197525i	$-0.936710\pi$
$353$	−7.33981e30	−1.96060	−0.980298	+	0.197525i	$0.936710\pi$
$354$	8.98864e30	2.32089
$355$	0	0
$356$	4.30404e30	1.03867
$357$	0	0
$358$	3.69037e30	0.832671
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	6.02630e30	1.23017
$362$	0	0
$363$	2.01655e30	0.385238
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	3.35323e30	0.526196
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−9.36253e30	−1.25007
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	9.33495e30	1.06278	0.531388	−	0.847129i	$-0.321671\pi$
$379$	9.33495e30	1.06278	0.531388	+	0.847129i	$0.321671\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.21844e31	−1.18530
$385$	0	0
$386$	−2.18405e31	−1.99625
$387$	−3.48702e30	−0.308974
$388$	2.27544e31	1.95472
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.31654e31	1.00000
$393$	3.09723e31	2.28172
$394$	0	0
$395$	0	0
$396$	−4.94745e30	−0.332690
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.67772e31	1.00000
$401$	−3.08650e31	−1.78539	−0.892696	−	0.450659i	$-0.851189\pi$
$401$	−3.08650e31	−1.78539	−0.892696	+	0.450659i	$0.851189\pi$
$402$	−3.68331e31	−2.06788
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−3.83740e31	−1.80350
$409$	1.44027e31	0.657299	0.328649	−	0.944452i	$-0.393407\pi$
$409$	1.44027e31	0.657299	0.328649	+	0.944452i	$0.393407\pi$
$410$	0	0
$411$	−5.49687e31	−2.36599
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.44562e31	1.96976
$418$	−3.49086e31	−1.22692
$419$	−4.08359e30	−0.139468	−0.0697339	−	0.997566i	$-0.522215\pi$
$419$	−4.08359e30	−0.139468	−0.0697339	+	0.997566i	$0.522215\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	5.60257e31	1.75646
$423$	0	0
$424$	0	0
$425$	5.28388e31	1.52155
$426$	0	0
$427$	0	0
$428$	5.37890e31	1.42354
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	2.97985e31	0.705325
$433$	−6.88602e31	−1.58531	−0.792654	−	0.609672i	$-0.791301\pi$
$433$	−6.88602e31	−1.58531	−0.792654	+	0.609672i	$0.791301\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.11951e32	2.24563
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	2.19106e31	0.404940
$442$	0	0
$443$	−4.91271e30	−0.0859956	−0.0429978	−	0.999075i	$-0.513691\pi$
$443$	−4.91271e30	−0.0859956	−0.0429978	+	0.999075i	$0.513691\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.33929e32	−1.99488	−0.997441	−	0.0714981i	$-0.977222\pi$
$449$	−1.33929e32	−1.99488	−0.997441	+	0.0714981i	$0.977222\pi$
$450$	2.79217e31	0.404940
$451$	−7.56001e31	−1.06759
$452$	9.02946e31	1.24165
$453$	0	0
$454$	−9.70569e31	−1.26577
$455$	0	0
$456$	−1.43079e32	−1.77010
$457$	−1.65936e32	−1.99962	−0.999809	−	0.0195676i	$-0.993771\pi$
$457$	−1.65936e32	−1.99962	−0.999809	+	0.0195676i	$0.993771\pi$
$458$	0	0
$459$	9.38486e31	1.07319
$460$	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
$466$	−4.97363e31	−0.474288
$467$	1.94979e32	1.81211	0.906056	−	0.423157i	$-0.139078\pi$
$467$	1.94979e32	1.81211	0.906056	+	0.423157i	$0.139078\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−2.39403e32	−1.95806
$473$	7.86163e31	0.626872
$474$	0	0
$475$	1.97012e32	1.49337
$476$	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
$482$	8.03695e31	0.511125
$483$	0	0
$484$	−5.37086e31	−0.325012
$485$	0	0
$486$	1.32933e32	0.765591
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.35595e32	0.725332
$490$	0	0
$491$	2.31312e31	0.117820	0.0589102	−	0.998263i	$-0.481237\pi$
$491$	2.31312e31	0.117820	0.0589102	+	0.998263i	$0.481237\pi$
$492$	−3.09860e32	−1.54023
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	4.55535e32	1.95781
$499$	9.67606e31	0.405968	0.202984	−	0.979182i	$-0.434936\pi$
$499$	9.67606e31	0.405968	0.202984	+	0.979182i	$0.434936\pi$
$500$	0	0
$501$	0	0
$502$	−7.15723e31	−0.279448
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.41920e32	−1.18530
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	3.24519e32	1.00000
$513$	3.49918e32	1.05331
$514$	−4.88443e32	−1.43634
$515$	0	0
$516$	3.22223e32	0.904399
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.99794e32	−1.99952	−0.999759	−	0.0219359i	$-0.993017\pi$
$521$	−7.99794e32	−1.99952	−0.999759	+	0.0219359i	$0.993017\pi$
$522$	0	0
$523$	−2.37714e31	−0.0567591	−0.0283795	−	0.999597i	$-0.509035\pi$
$523$	−2.37714e31	−0.0567591	−0.0283795	+	0.999597i	$0.509035\pi$
$524$	−8.24914e32	−1.92501
$525$	0	0
$526$	0	0
$527$	0	0
$528$	4.57177e32	0.973816
$529$	4.80251e32	1.00000
$530$	0	0
$531$	−3.98429e32	−0.792898
$532$	0	0
$533$	0	0
$534$	−6.61912e32	−1.23113
$535$	0	0
$536$	9.81012e32	1.74460
$537$	−5.67537e32	−0.986967
$538$	0	0
$539$	−4.93984e32	−0.821576
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.02205e33	1.52155
$545$	0	0
$546$	0	0
$547$	−4.71131e32	−0.656589	−0.328295	−	0.944575i	$-0.606474\pi$
$547$	−4.71131e32	−0.656589	−0.328295	+	0.944575i	$0.606474\pi$
$548$	1.46403e33	1.99611
$549$	0	0
$550$	−6.29506e32	−0.821576
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−1.45038e33	−1.66183
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.43985e33	1.48171
$562$	1.01247e33	1.01987
$563$	−1.52943e33	−1.50809	−0.754046	−	0.656821i	$-0.771901\pi$
$563$	−1.52943e33	−1.50809	−0.754046	+	0.656821i	$0.771901\pi$
$564$	0	0
$565$	0	0
$566$	−3.73969e32	−0.345970
$567$	0	0
$568$	0	0
$569$	1.75216e33	1.52134	0.760672	−	0.649136i	$-0.224869\pi$
$569$	1.75216e33	1.52134	0.760672	+	0.649136i	$0.224869\pi$
$570$	0	0
$571$	1.12088e33	0.933097	0.466548	−	0.884496i	$-0.345497\pi$
$571$	1.12088e33	0.933097	0.466548	+	0.884496i	$0.345497\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	5.40084e32	0.404940
$577$	2.72299e33	1.99957	0.999784	−	0.0207957i	$-0.00661997\pi$
$577$	2.72299e33	1.99957	0.999784	+	0.0207957i	$0.00661997\pi$
$578$	1.82851e33	1.31511
$579$	3.35882e33	2.36616
$580$	0	0
$581$	0	0
$582$	−3.49937e33	−2.31693
$583$	0	0
$584$	−2.98170e33	−1.89456
$585$	0	0
$586$	0	0
$587$	−3.30331e33	−1.97375	−0.986876	−	0.161478i	$-0.948374\pi$
$587$	−3.30331e33	−1.97375	−0.986876	+	0.161478i	$0.948374\pi$
$588$	−2.02468e33	−1.18530
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.56614e33	−1.88600	−0.943000	−	0.332792i	$-0.892009\pi$
$593$	−3.56614e33	−1.88600	−0.943000	+	0.332792i	$0.892009\pi$
$594$	−1.11808e33	−0.579478
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−2.58014e33	−1.18530
$601$	−2.61105e33	−1.17577	−0.587884	−	0.808945i	$-0.700039\pi$
$601$	−2.61105e33	−1.17577	−0.587884	+	0.808945i	$0.700039\pi$
$602$	0	0
$603$	1.63266e33	0.706461
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	3.81076e33	1.49337
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.70096e33	0.616137
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	5.25796e33	1.83146
$615$	0	0
$616$	0	0
$617$	−6.02166e33	−1.97831	−0.989155	−	0.146873i	$-0.953079\pi$
$617$	−6.02166e33	−1.97831	−0.989155	+	0.146873i	$0.953079\pi$
$618$	0	0
$619$	5.11844e33	1.61752	0.808760	−	0.588139i	$-0.200139\pi$
$619$	5.11844e33	1.61752	0.808760	+	0.588139i	$0.200139\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.55271e33	1.00000
$626$	6.00515e33	1.65818
$627$	5.36854e33	1.45427
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−8.61611e33	−2.08193
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.36209e32	−0.173787	−0.0868937	−	0.996218i	$-0.527694\pi$
$641$	−8.36209e32	−0.173787	−0.0868937	+	0.996218i	$0.527694\pi$
$642$	−8.27214e33	−1.68732
$643$	−6.26520e33	−1.25431	−0.627153	−	0.778896i	$-0.715780\pi$
$643$	−6.26520e33	−1.25431	−0.627153	+	0.778896i	$0.715780\pi$
$644$	0	0
$645$	0	0
$646$	1.20018e34	2.27224
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−6.80235e33	−1.24096
$649$	8.98277e33	1.60870
$650$	0	0
$651$	0	0
$652$	−3.61143e33	−0.611938
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	8.25281e33	1.29944
$657$	−4.96232e33	−0.767186
$658$	0	0
$659$	−9.07510e33	−1.35278	−0.676389	−	0.736545i	$-0.736456\pi$
$659$	−9.07510e33	−1.35278	−0.676389	+	0.736545i	$0.736456\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.37147e34	1.93593
$663$	0	0
$664$	−1.21327e34	−1.65174
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−9.94033e32	−0.115138	−0.0575692	−	0.998342i	$-0.518335\pi$
$673$	−9.94033e32	−0.115138	−0.0575692	+	0.998342i	$0.518335\pi$
$674$	−1.75661e34	−1.99874
$675$	6.31008e33	0.705325
$676$	9.10669e33	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−1.38863e34	−1.47173
$679$	0	0
$680$	0	0
$681$	1.49262e34	1.50032
$682$	0	0
$683$	−1.89329e34	−1.83725	−0.918626	−	0.395127i	$-0.870700\pi$
$683$	−1.89329e34	−1.83725	−0.918626	+	0.395127i	$0.870700\pi$
$684$	6.34210e33	0.604728
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−8.58208e33	−0.763012
$689$	0	0
$690$	0	0
$691$	2.14858e34	1.81307	0.906534	−	0.422133i	$-0.138718\pi$
$691$	2.14858e34	1.81307	0.906534	+	0.422133i	$0.138718\pi$
$692$	0	0
$693$	0	0
$694$	−1.84284e34	−1.47629
$695$	0	0
$696$	0	0
$697$	2.59917e34	1.97716
$698$	0	0
$699$	7.64887e33	0.562175
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−1.21764e34	−0.821576
$705$	0	0
$706$	−3.00638e34	−1.96060
$707$	0	0
$708$	3.68175e34	2.32089
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.76293e34	1.03867
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.51158e34	0.832671
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	2.46837e34	1.23017
$723$	−1.23599e34	−0.605837
$724$	0	0
$725$	0	0
$726$	8.25977e33	0.385238
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	7.51337e33	0.333507
$730$	0	0
$731$	−2.70287e34	−1.16096
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.68090e34	−1.43333
$738$	1.37348e34	0.526196
$739$	−4.64077e34	−1.74927	−0.874635	−	0.484781i	$-0.838899\pi$
$739$	−4.64077e34	−1.74927	−0.874635	+	0.484781i	$0.838899\pi$
$740$	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
$747$	−2.01920e34	−0.668855
$748$	−3.83489e34	−1.25007
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1.10070e34	0.331230
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	3.82360e34	1.06278
$759$	0	0
$760$	0	0
$761$	1.82440e34	0.483620	0.241810	−	0.970324i	$-0.422259\pi$
$761$	1.82440e34	0.483620	0.241810	+	0.970324i	$0.422259\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−4.99072e34	−1.18530
$769$	−7.99160e34	−1.86860	−0.934302	−	0.356481i	$-0.883976\pi$
$769$	−7.99160e34	−1.86860	−0.934302	+	0.356481i	$0.883976\pi$
$770$	0	0
$771$	7.51169e34	1.70249
$772$	−8.94587e34	−1.99625
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−1.42828e34	−0.308974
$775$	0	0
$776$	9.32020e34	1.95472
$777$	0	0
$778$	0	0
$779$	9.69112e34	1.94055
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	5.39253e34	1.00000
$785$	0	0
$786$	1.26862e35	2.28172
$787$	3.40535e34	0.603206	0.301603	−	0.953434i	$-0.402478\pi$
$787$	3.40535e34	0.603206	0.301603	+	0.953434i	$0.402478\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−2.02648e34	−0.332690
$793$	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
$801$	2.93398e34	0.420598
$802$	−1.26423e35	−1.78539
$803$	1.11878e35	1.55653
$804$	−1.50868e35	−2.06788
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.29241e35	−1.64445	−0.822223	−	0.569165i	$-0.807266\pi$
$809$	−1.29241e35	−1.64445	−0.822223	+	0.569165i	$0.807266\pi$
$810$	0	0
$811$	4.02532e34	0.497222	0.248611	−	0.968603i	$-0.420026\pi$
$811$	4.02532e34	0.497222	0.248611	+	0.968603i	$0.420026\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−1.57180e35	−1.80350
$817$	−1.00778e35	−1.13946
$818$	5.89933e34	0.657299
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−2.25152e35	−2.36599
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	9.68109e34	0.973816
$826$	0	0
$827$	−1.12226e35	−1.09655	−0.548273	−	0.836299i	$-0.684714\pi$
$827$	−1.12226e35	−1.09655	−0.548273	+	0.836299i	$0.684714\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.69835e35	1.52155
$834$	2.23052e35	1.96976
$835$	0	0
$836$	−1.42985e35	−1.22692
$837$	0	0
$838$	−1.67264e34	−0.139468
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.25185e35	1.00000
$842$	0	0
$843$	−1.55706e35	−1.20886
$844$	2.29481e35	1.75646
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.75121e34	0.410079
$850$	2.16428e35	1.52155
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	2.20320e35	1.42354
$857$	−3.12387e35	−1.99032	−0.995161	−	0.0982622i	$-0.968672\pi$
$857$	−3.12387e35	−1.99032	−0.995161	+	0.0982622i	$0.968672\pi$
$858$	0	0
$859$	−3.03808e35	−1.88227	−0.941135	−	0.338032i	$-0.890239\pi$
$859$	−3.03808e35	−1.88227	−0.941135	+	0.338032i	$0.890239\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.22055e35	0.705325
$865$	0	0
$866$	−2.82051e35	−1.58531
$867$	−2.81204e35	−1.55881
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.55113e35	0.791546
$874$	0	0
$875$	0	0
$876$	4.58551e35	2.24563
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.94263e35	0.888543	0.444272	−	0.895892i	$-0.353463\pi$
$881$	1.94263e35	0.888543	0.444272	+	0.895892i	$0.353463\pi$
$882$	8.97458e34	0.404940
$883$	3.10517e35	1.38216	0.691078	−	0.722780i	$-0.257136\pi$
$883$	3.10517e35	1.38216	0.691078	+	0.722780i	$0.257136\pi$
$884$	0	0
$885$	0	0
$886$	−2.01225e34	−0.0859956
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.55235e35	1.01955
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−5.48571e35	−1.99488
$899$	0	0
$900$	1.14367e35	0.404940
$901$	0	0
$902$	−3.09658e35	−1.06759
$903$	0	0
$904$	3.69846e35	1.24165
$905$	0	0
$906$	0	0
$907$	−4.03748e35	−1.30263	−0.651317	−	0.758806i	$-0.725783\pi$
$907$	−4.03748e35	−1.30263	−0.651317	+	0.758806i	$0.725783\pi$
$908$	−3.97545e35	−1.26577
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−5.86051e35	−1.77010
$913$	4.55237e35	1.35703
$914$	−6.79675e35	−1.99962
$915$	0	0
$916$	0	0
$917$	0	0
$918$	3.84404e35	1.07319
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−8.08615e35	−2.17083
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.55845e35	1.10313	0.551567	−	0.834131i	$-0.314030\pi$
$929$	4.55845e35	1.10313	0.551567	+	0.834131i	$0.314030\pi$
$930$	0	0
$931$	6.33235e35	1.49337
$932$	−2.03720e35	−0.474288
$933$	0	0
$934$	7.98634e35	1.81211
$935$	0	0
$936$	0	0
$937$	5.40504e33	0.0118011	0.00590057	−	0.999983i	$-0.498122\pi$
$937$	5.40504e33	0.0118011	0.00590057	+	0.999983i	$0.498122\pi$
$938$	0	0
$939$	−9.23524e35	−1.96545
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−9.80595e35	−1.95806
$945$	0	0
$946$	3.22012e35	0.626872
$947$	9.78529e35	1.88094	0.940468	−	0.339881i	$-0.110387\pi$
$947$	9.78529e35	1.88094	0.940468	+	0.339881i	$0.110387\pi$
$948$	0	0
$949$	0	0
$950$	8.06960e35	1.49337
$951$	0	0
$952$	0	0
$953$	7.02211e35	1.25127	0.625637	−	0.780114i	$-0.284839\pi$
$953$	7.02211e35	1.25127	0.625637	+	0.780114i	$0.284839\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	6.20413e35	1.00000
$962$	0	0
$963$	3.66670e35	0.576447
$964$	3.29194e35	0.511125
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.19991e35	−0.325012
$969$	−1.84573e36	−2.69329
$970$	0	0
$971$	1.40463e36	1.99954	0.999771	−	0.0214148i	$-0.00681706\pi$
$971$	1.40463e36	1.99954	0.999771	+	0.0214148i	$0.00681706\pi$
$972$	5.44493e35	0.765591
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.01581e36	−1.34301	−0.671506	−	0.740999i	$-0.734352\pi$
$977$	−1.01581e36	−1.34301	−0.671506	+	0.740999i	$0.734352\pi$
$978$	5.55396e35	0.725332
$979$	−6.61480e35	−0.853344
$980$	0	0
$981$	0	0
$982$	9.47455e34	0.117820
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−1.26919e36	−1.54023
$985$	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
$993$	−2.10916e36	−2.29466
$994$	0	0
$995$	0	0
$996$	1.86587e36	1.95781
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	3.96332e35	0.405968
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.25.d.a.3.1	✓	1
4.3	odd	2		32.25.d.a.15.1		1
8.3	odd	2	CM	8.25.d.a.3.1	✓	1
8.5	even	2		32.25.d.a.15.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.25.d.a.3.1	✓	1	1.1	even	1	trivial
8.25.d.a.3.1	✓	1	8.3	odd	2	CM
32.25.d.a.15.1		1	4.3	odd	2
32.25.d.a.15.1		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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 25 \)
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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