LMFDB - Embedded newform 8.27.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,27,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, 27, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 27);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 27 $
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:	$34.2634336584$
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)

$f(q)$

$=$

$q-8192.00 q^{2} +3.05136e6 q^{3} +6.71089e7 q^{4} -2.49967e10 q^{6} -5.49756e11 q^{8} +6.76892e12 q^{9} +O(q^{10})$

$q-8192.00 q^{2} +3.05136e6 q^{3} +6.71089e7 q^{4} -2.49967e10 q^{6} -5.49756e11 q^{8} +6.76892e12 q^{9} +3.05104e13 q^{11} +2.04773e14 q^{12} +4.50360e15 q^{16} +1.37207e16 q^{17} -5.54510e16 q^{18} -8.11750e16 q^{19} -2.49941e17 q^{22} -1.67750e18 q^{24} +1.49012e18 q^{25} +1.28983e19 q^{27} -3.68935e19 q^{32} +9.30981e19 q^{33} -1.12400e20 q^{34} +4.54255e20 q^{36} +6.64986e20 q^{38} -1.83869e21 q^{41} +2.92081e21 q^{43} +2.04752e21 q^{44} +1.37421e22 q^{48} +9.38748e21 q^{49} -1.22070e22 q^{50} +4.18667e22 q^{51} -1.05663e23 q^{54} -2.47694e23 q^{57} +1.12080e23 q^{59} +3.02231e23 q^{64} -7.62660e23 q^{66} -3.27612e22 q^{67} +9.20779e23 q^{68} -3.72125e24 q^{72} +3.32970e24 q^{73} +4.54688e24 q^{75} -5.44756e24 q^{76} +2.21515e25 q^{81} +1.50625e25 q^{82} -1.08793e25 q^{83} -2.39273e25 q^{86} -1.67733e25 q^{88} -2.76449e24 q^{89} -1.12575e26 q^{96} -3.88312e25 q^{97} -7.69022e25 q^{98} +2.06522e26 q^{99} +O(q^{100})$

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 $

$2$	−8192.00	−1.00000
$2$	−8192.00	−1.00000
$3$	3.05136e6	1.91389	0.956945	−	0.290270i	$-0.0937452\pi$
$3$	3.05136e6	1.91389	0.956945	+	0.290270i	$0.0937452\pi$
$4$	6.71089e7	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−2.49967e10	−1.91389
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−5.49756e11	−1.00000
$9$	6.76892e12	2.66297
$10$	0	0
$11$	3.05104e13	0.883777	0.441889	−	0.897070i	$-0.354309\pi$
$11$	3.05104e13	0.883777	0.441889	+	0.897070i	$0.354309\pi$
$12$	2.04773e14	1.91389
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.50360e15	1.00000
$17$	1.37207e16	1.38529	0.692643	−	0.721280i	$-0.256446\pi$
$17$	1.37207e16	1.38529	0.692643	+	0.721280i	$0.256446\pi$
$18$	−5.54510e16	−2.66297
$19$	−8.11750e16	−1.93030	−0.965152	−	0.261691i	$-0.915720\pi$
$19$	−8.11750e16	−1.93030	−0.965152	+	0.261691i	$0.915720\pi$
$20$	0	0
$21$	0	0
$22$	−2.49941e17	−0.883777
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.67750e18	−1.91389
$25$	1.49012e18	1.00000
$26$	0	0
$27$	1.28983e19	3.18275
$28$	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			$\pi$
$32$	−3.68935e19	−1.00000
$33$	9.30981e19	1.69145
$34$	−1.12400e20	−1.38529
$35$	0	0
$36$	4.54255e20	2.66297
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	6.64986e20	1.93030
$39$	0	0
$40$	0	0
$41$	−1.83869e21	−1.98755	−0.993773	−	0.111420i	$-0.964460\pi$
$41$	−1.83869e21	−1.98755	−0.993773	+	0.111420i	$0.964460\pi$
$42$	0	0
$43$	2.92081e21	1.69986	0.849930	−	0.526895i	$-0.176644\pi$
$43$	2.92081e21	1.69986	0.849930	+	0.526895i	$0.176644\pi$
$44$	2.04752e21	0.883777
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.37421e22	1.91389
$49$	9.38748e21	1.00000
$50$	−1.22070e22	−1.00000
$51$	4.18667e22	2.65129
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.05663e23	−3.18275
$55$	0	0
$56$	0	0
$57$	−2.47694e23	−3.69439
$58$	0	0
$59$	1.12080e23	1.06770	0.533851	−	0.845578i	$-0.320744\pi$
$59$	1.12080e23	1.06770	0.533851	+	0.845578i	$0.320744\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	3.02231e23	1.00000
$65$	0	0
$66$	−7.62660e23	−1.69145
$67$	−3.27612e22	−0.0597568	−0.0298784	−	0.999554i	$-0.509512\pi$
$67$	−3.27612e22	−0.0597568	−0.0298784	+	0.999554i	$0.509512\pi$
$68$	9.20779e23	1.38529
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.72125e24	−2.66297
$73$	3.32970e24	1.99163	0.995814	−	0.0914042i	$-0.0291355\pi$
$73$	3.32970e24	1.99163	0.995814	+	0.0914042i	$0.0291355\pi$
$74$	0	0
$75$	4.54688e24	1.91389
$76$	−5.44756e24	−1.93030
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.21515e25	3.42845
$82$	1.50625e25	1.98755
$83$	−1.08793e25	−1.22626	−0.613132	−	0.789980i	$-0.710091\pi$
$83$	−1.08793e25	−1.22626	−0.613132	+	0.789980i	$0.710091\pi$
$84$	0	0
$85$	0	0
$86$	−2.39273e25	−1.69986
$87$	0	0
$88$	−1.67733e25	−0.883777
$89$	−2.76449e24	−0.125761	−0.0628804	−	0.998021i	$-0.520029\pi$
$89$	−2.76449e24	−0.125761	−0.0628804	+	0.998021i	$0.520029\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.12575e26	−1.91389
$97$	−3.88312e25	−0.576963	−0.288482	−	0.957485i	$-0.593150\pi$
$97$	−3.88312e25	−0.576963	−0.288482	+	0.957485i	$0.593150\pi$
$98$	−7.69022e25	−1.00000
$99$	2.06522e26	2.35348
$100$	1.00000e26	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−3.42972e26	−2.65129
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.74121e25	−0.404226	−0.202113	−	0.979362i	$-0.564781\pi$
$107$	−9.74121e25	−0.404226	−0.202113	+	0.979362i	$0.564781\pi$
$108$	8.65587e26	3.18275
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.28283e26	−0.874402	−0.437201	−	0.899364i	$-0.644030\pi$
$113$	−4.28283e26	−0.874402	−0.437201	+	0.899364i	$0.644030\pi$
$114$	2.02911e27	3.69439
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−9.18156e26	−1.06770
$119$	0	0
$120$	0	0
$121$	−2.60934e26	−0.218938
$122$	0	0
$123$	−5.61049e27	−3.80394
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−2.47588e27	−1.00000
$129$	8.91244e27	3.25335
$130$	0	0
$131$	2.39497e26	0.0715771	0.0357885	−	0.999359i	$-0.488606\pi$
$131$	2.39497e26	0.0715771	0.0357885	+	0.999359i	$0.488606\pi$
$132$	6.24771e27	1.69145
$133$	0	0
$134$	2.68380e26	0.0597568
$135$	0	0
$136$	−7.54302e27	−1.38529
$137$	−1.07544e28	−1.79563	−0.897815	−	0.440372i	$-0.854847\pi$
$137$	−1.07544e28	−1.79563	−0.897815	+	0.440372i	$0.854847\pi$
$138$	0	0
$139$	−1.43076e28	−1.97867	−0.989335	−	0.145657i	$-0.953471\pi$
$139$	−1.43076e28	−1.97867	−0.989335	+	0.145657i	$0.953471\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.04845e28	2.66297
$145$	0	0
$146$	−2.72769e28	−1.99163
$147$	2.86446e28	1.91389
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−3.72480e28	−1.91389
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	4.46264e28	1.93030
$153$	9.28742e28	3.68898
$154$	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
$162$	−1.81465e29	−3.42845
$163$	5.17065e28	0.901792	0.450896	−	0.892577i	$-0.351105\pi$
$163$	5.17065e28	0.901792	0.450896	+	0.892577i	$0.351105\pi$
$164$	−1.23392e29	−1.98755
$165$	0	0
$166$	8.91229e28	1.22626
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	9.17333e28	1.00000
$170$	0	0
$171$	−5.49467e29	−5.14035
$172$	1.96012e29	1.69986
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	1.37407e29	0.883777
$177$	3.41995e29	2.04346
$178$	2.26467e28	0.125761
$179$	−3.65920e29	−1.88928	−0.944642	−	0.328104i	$-0.893590\pi$
$179$	−3.65920e29	−1.88928	−0.944642	+	0.328104i	$0.893590\pi$
$180$	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
$187$	4.18623e29	1.22429
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	9.22216e29	1.91389
$193$	−2.00239e29	−0.388421	−0.194211	−	0.980960i	$-0.562215\pi$
$193$	−2.00239e29	−0.388421	−0.194211	+	0.980960i	$0.562215\pi$
$194$	3.18105e29	0.576963
$195$	0	0
$196$	6.29983e29	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.69183e30	−2.35348
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−8.19200e29	−1.00000
$201$	−9.99661e28	−0.114368
$202$	0	0
$203$	0	0
$204$	2.80963e30	2.65129
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.47668e30	−1.70596
$210$	0	0
$211$	−2.87288e30	−1.74842	−0.874209	−	0.485550i	$-0.838619\pi$
$211$	−2.87288e30	−1.74842	−0.874209	+	0.485550i	$0.838619\pi$
$212$	0	0
$213$	0	0
$214$	7.98000e29	0.404226
$215$	0	0
$216$	−7.09089e30	−3.18275
$217$	0	0
$218$	0	0
$219$	1.01601e31	3.81176
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	1.00865e31	2.66297
$226$	3.50850e30	0.874402
$227$	−6.51396e30	−1.53288	−0.766441	−	0.642314i	$-0.777974\pi$
$227$	−6.51396e30	−1.53288	−0.766441	+	0.642314i	$0.777974\pi$
$228$	−1.66225e31	−3.69439
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.58588e30	0.265855	0.132927	−	0.991126i	$-0.457562\pi$
$233$	1.58588e30	0.265855	0.132927	+	0.991126i	$0.457562\pi$
$234$	0	0
$235$	0	0
$236$	7.52153e30	1.06770
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−1.44727e31	−1.56433	−0.782164	−	0.623072i	$-0.785884\pi$
$241$	−1.44727e31	−1.56433	−0.782164	+	0.623072i	$0.785884\pi$
$242$	2.13757e30	0.218938
$243$	3.48065e31	3.37893
$244$	0	0
$245$	0	0
$246$	4.59611e31	3.80394
$247$	0	0
$248$	0	0
$249$	−3.31965e31	−2.34693
$250$	0	0
$251$	−7.48688e30	−0.477027	−0.238513	−	0.971139i	$-0.576660\pi$
$251$	−7.48688e30	−0.477027	−0.238513	+	0.971139i	$0.576660\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	2.02824e31	1.00000
$257$	−4.26238e31	−1.99766	−0.998831	−	0.0483487i	$-0.984604\pi$
$257$	−4.26238e31	−1.99766	−0.998831	+	0.0483487i	$0.984604\pi$
$258$	−7.30107e31	−3.25335
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−1.96196e30	−0.0715771
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−5.11812e31	−1.69145
$265$	0	0
$266$	0	0
$267$	−8.43546e30	−0.240692
$268$	−2.19856e30	−0.0597568
$269$	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			$\pi$
$272$	6.17925e31	1.38529
$273$	0	0
$274$	8.80997e31	1.79563
$275$	4.54640e31	0.883777
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.17208e32	1.97867
$279$	0	0
$280$	0	0
$281$	7.67298e31	1.12663	0.563316	−	0.826242i	$-0.309525\pi$
$281$	7.67298e31	1.12663	0.563316	+	0.826242i	$0.309525\pi$
$282$	0	0
$283$	−1.41819e32	−1.89894	−0.949472	−	0.313852i	$-0.898380\pi$
$283$	−1.41819e32	−1.89894	−0.949472	+	0.313852i	$0.898380\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−2.49729e32	−2.66297
$289$	9.01565e31	0.919020
$290$	0	0
$291$	−1.18488e32	−1.10424
$292$	2.23453e32	1.99163
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−2.34656e32	−1.91389
$295$	0	0
$296$	0	0
$297$	3.93531e32	2.81284
$298$	0	0
$299$	0	0
$300$	3.05136e32	1.91389
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−3.65580e32	−1.93030
$305$	0	0
$306$	−7.60825e32	−3.68898
$307$	4.29126e32	1.99428	0.997139	−	0.0755933i	$-0.0240851\pi$
$307$	4.29126e32	1.99428	0.997139	+	0.0755933i	$0.0240851\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−2.17797e32	−0.787003	−0.393502	−	0.919324i	$-0.628736\pi$
$313$	−2.17797e32	−0.787003	−0.393502	+	0.919324i	$0.628736\pi$
$314$	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
$321$	−2.97239e32	−0.773643
$322$	0	0
$323$	−1.11378e33	−2.67402
$324$	1.48656e33	3.42845
$325$	0	0
$326$	−4.23580e32	−0.901792
$327$	0	0
$328$	1.01083e33	1.98755
$329$	0	0
$330$	0	0
$331$	3.10852e32	0.542988	0.271494	−	0.962440i	$-0.412482\pi$
$331$	3.10852e32	0.542988	0.271494	+	0.962440i	$0.412482\pi$
$332$	−7.30094e32	−1.22626
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.06122e33	1.46762	0.733812	−	0.679352i	$-0.237739\pi$
$337$	1.06122e33	1.46762	0.733812	+	0.679352i	$0.237739\pi$
$338$	−7.51479e32	−1.00000
$339$	−1.30685e33	−1.67351
$340$	0	0
$341$	0	0
$342$	4.50124e33	5.14035
$343$	0	0
$344$	−1.60573e33	−1.69986
$345$	0	0
$346$	0	0
$347$	1.02662e33	0.970789	0.485395	−	0.874295i	$-0.338676\pi$
$347$	1.02662e33	0.970789	0.485395	+	0.874295i	$0.338676\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−1.12563e33	−0.883777
$353$	−1.21393e33	−0.918589	−0.459294	−	0.888284i	$-0.651898\pi$
$353$	−1.21393e33	−0.918589	−0.459294	+	0.888284i	$0.651898\pi$
$354$	−2.80162e33	−2.04346
$355$	0	0
$356$	−1.85522e32	−0.125761
$357$	0	0
$358$	2.99762e33	1.88928
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	4.82093e33	2.72607
$362$	0	0
$363$	−7.96202e32	−0.419022
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−1.24459e34	−5.29278
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−3.42936e33	−1.22429
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	5.60111e33	1.68254	0.841268	−	0.540619i	$-0.181810\pi$
$379$	5.60111e33	1.68254	0.841268	+	0.540619i	$0.181810\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−7.55480e33	−1.91389
$385$	0	0
$386$	1.64036e33	0.388421
$387$	1.97707e34	4.52668
$388$	−2.60592e33	−0.576963
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−5.16082e33	−1.00000
$393$	7.30792e32	0.136991
$394$	0	0
$395$	0	0
$396$	1.38595e34	2.35348
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	6.71089e33	1.00000
$401$	9.97030e33	1.43824	0.719120	−	0.694886i	$-0.244545\pi$
$401$	9.97030e33	1.43824	0.719120	+	0.694886i	$0.244545\pi$
$402$	8.18922e32	0.114368
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−2.30165e34	−2.65129
$409$	1.30476e34	1.45589	0.727944	−	0.685637i	$-0.240476\pi$
$409$	1.30476e34	1.45589	0.727944	+	0.685637i	$0.240476\pi$
$410$	0	0
$411$	−3.28154e34	−3.43664
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.36575e34	−3.78696
$418$	2.02890e34	1.70596
$419$	−1.82822e34	−1.49021	−0.745104	−	0.666948i	$-0.767600\pi$
$419$	−1.82822e34	−1.49021	−0.745104	+	0.666948i	$0.767600\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	2.35346e34	1.74842
$423$	0	0
$424$	0	0
$425$	2.04454e34	1.38529
$426$	0	0
$427$	0	0
$428$	−6.53722e33	−0.404226
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	5.80886e34	3.18275
$433$	−2.82285e33	−0.150088	−0.0750439	−	0.997180i	$-0.523910\pi$
$433$	−2.82285e33	−0.150088	−0.0750439	+	0.997180i	$0.523910\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−8.32316e34	−3.81176
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	6.35431e34	2.66297
$442$	0	0
$443$	−8.93680e33	−0.353129	−0.176565	−	0.984289i	$-0.556498\pi$
$443$	−8.93680e33	−0.353129	−0.176565	+	0.984289i	$0.556498\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.68506e34	−1.88596	−0.942981	−	0.332848i	$-0.891991\pi$
$449$	−5.68506e34	−1.88596	−0.942981	+	0.332848i	$0.891991\pi$
$450$	−8.26284e34	−2.66297
$451$	−5.60990e34	−1.75655
$452$	−2.87416e34	−0.874402
$453$	0	0
$454$	5.33624e34	1.53288
$455$	0	0
$456$	1.36171e35	3.69439
$457$	2.11794e34	0.558473	0.279237	−	0.960222i	$-0.409919\pi$
$457$	2.11794e34	0.558473	0.279237	+	0.960222i	$0.409919\pi$
$458$	0	0
$459$	1.76973e35	4.40902
$460$	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
$466$	−1.29916e34	−0.265855
$467$	−4.58122e34	−0.911720	−0.455860	−	0.890052i	$-0.650668\pi$
$467$	−4.58122e34	−0.911720	−0.455860	+	0.890052i	$0.650668\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−6.16164e34	−1.06770
$473$	8.91150e34	1.50230
$474$	0	0
$475$	−1.20960e35	−1.93030
$476$	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
$482$	1.18560e35	1.56433
$483$	0	0
$484$	−1.75110e34	−0.218938
$485$	0	0
$486$	−2.85135e35	−3.37893
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.57775e35	1.72593
$490$	0	0
$491$	−1.07365e35	−1.11379	−0.556894	−	0.830583i	$-0.688007\pi$
$491$	−1.07365e35	−1.11379	−0.556894	+	0.830583i	$0.688007\pi$
$492$	−3.76514e35	−3.80394
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	2.71946e35	2.34693
$499$	9.98217e34	0.839301	0.419650	−	0.907686i	$-0.362153\pi$
$499$	9.98217e34	0.839301	0.419650	+	0.907686i	$0.362153\pi$
$500$	0	0
$501$	0	0
$502$	6.13325e34	0.477027
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.79911e35	1.91389
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.66153e35	−1.00000
$513$	−1.04702e36	−6.14367
$514$	3.49174e35	1.99766
$515$	0	0
$516$	5.98103e35	3.25335
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.99914e35	1.91901	0.959503	−	0.281700i	$-0.0908982\pi$
$521$	3.99914e35	1.91901	0.959503	+	0.281700i	$0.0908982\pi$
$522$	0	0
$523$	−4.09695e35	−1.87042	−0.935208	−	0.354099i	$-0.884788\pi$
$523$	−4.09695e35	−1.87042	−0.935208	+	0.354099i	$0.884788\pi$
$524$	1.60724e34	0.0715771
$525$	0	0
$526$	0	0
$527$	0	0
$528$	4.19277e35	1.69145
$529$	2.54053e35	1.00000
$530$	0	0
$531$	7.58658e35	2.84326
$532$	0	0
$533$	0	0
$534$	6.91033e34	0.240692
$535$	0	0
$536$	1.80106e34	0.0597568
$537$	−1.11655e36	−3.61588
$538$	0	0
$539$	2.86416e35	0.883777
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−5.06204e35	−1.38529
$545$	0	0
$546$	0	0
$547$	2.38025e34	0.0606439	0.0303219	−	0.999540i	$-0.490347\pi$
$547$	2.38025e34	0.0606439	0.0303219	+	0.999540i	$0.490347\pi$
$548$	−7.21713e35	−1.79563
$549$	0	0
$550$	−3.72441e35	−0.883777
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−9.60165e35	−1.97867
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.27737e36	2.34315
$562$	−6.28571e35	−1.12663
$563$	−5.61955e35	−0.984221	−0.492110	−	0.870533i	$-0.663774\pi$
$563$	−5.61955e35	−0.984221	−0.492110	+	0.870533i	$0.663774\pi$
$564$	0	0
$565$	0	0
$566$	1.16178e36	1.89894
$567$	0	0
$568$	0	0
$569$	1.25893e36	1.92107	0.960535	−	0.278161i	$-0.0897248\pi$
$569$	1.25893e36	1.92107	0.960535	+	0.278161i	$0.0897248\pi$
$570$	0	0
$571$	−1.36049e36	−1.98347	−0.991734	−	0.128311i	$-0.959045\pi$
$571$	−1.36049e36	−1.98347	−0.991734	+	0.128311i	$0.959045\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	2.04578e36	2.66297
$577$	3.54035e34	0.0450569	0.0225284	−	0.999746i	$-0.492828\pi$
$577$	3.54035e34	0.0450569	0.0225284	+	0.999746i	$0.492828\pi$
$578$	−7.38562e35	−0.919020
$579$	−6.11000e35	−0.743396
$580$	0	0
$581$	0	0
$582$	9.70652e35	1.10424
$583$	0	0
$584$	−1.83052e36	−1.99163
$585$	0	0
$586$	0	0
$587$	1.95755e36	1.99259	0.996296	−	0.0859896i	$-0.0274052\pi$
$587$	1.95755e36	1.99259	0.996296	+	0.0859896i	$0.0274052\pi$
$588$	1.92230e36	1.91389
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.04959e36	1.82792	0.913958	−	0.405808i	$-0.133010\pi$
$593$	2.04959e36	1.82792	0.913958	+	0.405808i	$0.133010\pi$
$594$	−3.22380e36	−2.81284
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−2.49967e36	−1.91389
$601$	−2.59563e36	−1.94480	−0.972401	−	0.233316i	$-0.925042\pi$
$601$	−2.59563e36	−1.94480	−0.972401	+	0.233316i	$0.925042\pi$
$602$	0	0
$603$	−2.21758e35	−0.159131
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	2.99483e36	1.93030
$609$	0	0
$610$	0	0
$611$	0	0
$612$	6.23268e36	3.68898
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−3.51540e36	−1.99428
$615$	0	0
$616$	0	0
$617$	1.53670e36	0.818240	0.409120	−	0.912481i	$-0.365836\pi$
$617$	1.53670e36	0.818240	0.409120	+	0.912481i	$0.365836\pi$
$618$	0	0
$619$	6.14666e35	0.313806	0.156903	−	0.987614i	$-0.449849\pi$
$619$	6.14666e35	0.313806	0.156903	+	0.987614i	$0.449849\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.22045e36	1.00000
$626$	1.78420e36	0.787003
$627$	−7.55724e36	−3.26502
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−8.76618e36	−3.34628
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.89589e36	−1.91159	−0.955795	−	0.294035i	$-0.905002\pi$
$641$	−5.89589e36	−1.91159	−0.955795	+	0.294035i	$0.905002\pi$
$642$	2.43498e36	0.773643
$643$	−2.15235e36	−0.670148	−0.335074	−	0.942192i	$-0.608761\pi$
$643$	−2.15235e36	−0.670148	−0.335074	+	0.942192i	$0.608761\pi$
$644$	0	0
$645$	0	0
$646$	9.12406e36	2.67402
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.21779e37	−3.42845
$649$	3.41959e36	0.943611
$650$	0	0
$651$	0	0
$652$	3.46997e36	0.901792
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−8.28070e36	−1.98755
$657$	2.25385e37	5.30365
$658$	0	0
$659$	8.78684e36	1.98757	0.993785	−	0.111314i	$-0.0355060\pi$
$659$	8.78684e36	1.98757	0.993785	+	0.111314i	$0.0355060\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−2.54650e36	−0.542988
$663$	0	0
$664$	5.98093e36	1.22626
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−3.76883e36	−0.648650	−0.324325	−	0.945946i	$-0.605137\pi$
$673$	−3.76883e36	−0.648650	−0.324325	+	0.945946i	$0.605137\pi$
$674$	−8.69348e36	−1.46762
$675$	1.92199e37	3.18275
$676$	6.15612e36	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	1.07057e37	1.67351
$679$	0	0
$680$	0	0
$681$	−1.98764e37	−2.93377
$682$	0	0
$683$	−9.08839e36	−1.29127	−0.645636	−	0.763645i	$-0.723408\pi$
$683$	−9.08839e36	−1.29127	−0.645636	+	0.763645i	$0.723408\pi$
$684$	−3.68741e37	−5.14035
$685$	0	0
$686$	0	0
$687$	0	0
$688$	1.31542e37	1.69986
$689$	0	0
$690$	0	0
$691$	1.37430e37	1.67829	0.839146	−	0.543906i	$-0.183055\pi$
$691$	1.37430e37	1.67829	0.839146	+	0.543906i	$0.183055\pi$
$692$	0	0
$693$	0	0
$694$	−8.41004e36	−0.970789
$695$	0	0
$696$	0	0
$697$	−2.52280e37	−2.75332
$698$	0	0
$699$	4.83910e36	0.508817
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	9.22120e36	0.883777
$705$	0	0
$706$	9.94449e36	0.918589
$707$	0	0
$708$	2.29509e37	2.04346
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.51980e36	0.125761
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−2.45565e37	−1.88928
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−3.94931e37	−2.72607
$723$	−4.41614e37	−2.99395
$724$	0	0
$725$	0	0
$726$	6.52249e36	0.419022
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	4.99011e37	3.03845
$730$	0	0
$731$	4.00755e37	2.35479
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.99556e35	−0.0528117
$738$	1.01957e38	5.29278
$739$	−1.10805e37	−0.565174	−0.282587	−	0.959242i	$-0.591193\pi$
$739$	−1.10805e37	−0.565174	−0.282587	+	0.959242i	$0.591193\pi$
$740$	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
$747$	−7.36408e37	−3.26551
$748$	2.80933e37	1.22429
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−2.28452e37	−0.912976
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−4.58843e37	−1.68254
$759$	0	0
$760$	0	0
$761$	3.50791e37	1.22194	0.610968	−	0.791655i	$-0.290780\pi$
$761$	3.50791e37	1.22194	0.610968	+	0.791655i	$0.290780\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	6.18889e37	1.91389
$769$	2.50394e37	0.761346	0.380673	−	0.924710i	$-0.375692\pi$
$769$	2.50394e37	0.761346	0.380673	+	0.924710i	$0.375692\pi$
$770$	0	0
$771$	−1.30061e38	−3.82330
$772$	−1.34378e37	−0.388421
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−1.61962e38	−4.52668
$775$	0	0
$776$	2.13477e37	0.576963
$777$	0	0
$778$	0	0
$779$	1.49255e38	3.83657
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	4.22775e37	1.00000
$785$	0	0
$786$	−5.98665e36	−0.136991
$787$	8.42753e37	1.89683	0.948416	−	0.317028i	$-0.102685\pi$
$787$	8.42753e37	1.89683	0.948416	+	0.317028i	$0.102685\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.13537e38	−2.35348
$793$	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
$800$	−5.49756e37	−1.00000
$801$	−1.87126e37	−0.334898
$802$	−8.16767e37	−1.43824
$803$	1.01591e38	1.76016
$804$	−6.70861e36	−0.114368
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.17413e38	1.84665	0.923324	−	0.384022i	$-0.125461\pi$
$809$	1.17413e38	1.84665	0.923324	+	0.384022i	$0.125461\pi$
$810$	0	0
$811$	−1.30003e38	−1.98007	−0.990036	−	0.140814i	$-0.955028\pi$
$811$	−1.30003e38	−1.98007	−0.990036	+	0.140814i	$0.955028\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	1.88551e38	2.65129
$817$	−2.37097e38	−3.28125
$818$	−1.06886e38	−1.45589
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.68824e38	3.43664
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	1.38727e38	1.69145
$826$	0	0
$827$	−1.62324e38	−1.91783	−0.958915	−	0.283692i	$-0.908441\pi$
$827$	−1.62324e38	−1.91783	−0.958915	+	0.283692i	$0.908441\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.28803e38	1.38529
$834$	3.57642e38	3.78696
$835$	0	0
$836$	−1.66207e38	−1.70596
$837$	0	0
$838$	1.49768e38	1.49021
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.05281e38	1.00000
$842$	0	0
$843$	2.34130e38	2.15625
$844$	−1.92795e38	−1.74842
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.32742e38	−3.63437
$850$	−1.67489e38	−1.38529
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	5.35529e37	0.404226
$857$	−1.68900e38	−1.25568	−0.627840	−	0.778342i	$-0.716061\pi$
$857$	−1.68900e38	−1.25568	−0.627840	+	0.778342i	$0.716061\pi$
$858$	0	0
$859$	−2.23183e38	−1.60972	−0.804858	−	0.593467i	$-0.797759\pi$
$859$	−2.23183e38	−1.60972	−0.804858	+	0.593467i	$0.797759\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−4.75862e38	−3.18275
$865$	0	0
$866$	2.31248e37	0.150088
$867$	2.75100e38	1.75890
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.62845e38	−1.53644
$874$	0	0
$875$	0	0
$876$	6.81834e38	3.81176
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.97659e37	0.310289	0.155145	−	0.987892i	$-0.450416\pi$
$881$	5.97659e37	0.310289	0.155145	+	0.987892i	$0.450416\pi$
$882$	−5.20545e38	−2.66297
$883$	−2.54276e38	−1.28179	−0.640893	−	0.767630i	$-0.721436\pi$
$883$	−2.54276e38	−1.28179	−0.640893	+	0.767630i	$0.721436\pi$
$884$	0	0
$885$	0	0
$886$	7.32102e37	0.353129
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.75851e38	3.02999
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	4.65720e38	1.88596
$899$	0	0
$900$	6.76892e38	2.66297
$901$	0	0
$902$	4.59563e38	1.75655
$903$	0	0
$904$	2.35451e38	0.874402
$905$	0	0
$906$	0	0
$907$	2.06316e38	0.733903	0.366952	−	0.930240i	$-0.380401\pi$
$907$	2.06316e38	0.733903	0.366952	+	0.930240i	$0.380401\pi$
$908$	−4.37145e38	−1.53288
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−1.11551e39	−3.69439
$913$	−3.31930e38	−1.08374
$914$	−1.73501e38	−0.558473
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.44976e39	−4.40902
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1.30942e39	3.81683
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.67597e38	1.99953	0.999767	−	0.0215685i	$-0.00686599\pi$
$929$	7.67597e38	1.99953	0.999767	+	0.0215685i	$0.00686599\pi$
$930$	0	0
$931$	−7.62029e38	−1.93030
$932$	1.06427e38	0.265855
$933$	0	0
$934$	3.75293e38	0.911720
$935$	0	0
$936$	0	0
$937$	7.90860e38	1.84283	0.921414	−	0.388581i	$-0.127035\pi$
$937$	7.90860e38	1.84283	0.921414	+	0.388581i	$0.127035\pi$
$938$	0	0
$939$	−6.64578e38	−1.50624
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	5.04761e38	1.06770
$945$	0	0
$946$	−7.30030e38	−1.50230
$947$	−7.71396e38	−1.56577	−0.782885	−	0.622167i	$-0.786252\pi$
$947$	−7.71396e38	−1.56577	−0.782885	+	0.622167i	$0.786252\pi$
$948$	0	0
$949$	0	0
$950$	9.90906e38	1.93030
$951$	0	0
$952$	0	0
$953$	−8.81979e38	−1.64911	−0.824557	−	0.565779i	$-0.808576\pi$
$953$	−8.81979e38	−1.64911	−0.824557	+	0.565779i	$0.808576\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.96217e38	1.00000
$962$	0	0
$963$	−6.59375e38	−1.07644
$964$	−9.71247e38	−1.56433
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.43450e38	0.218938
$969$	−3.39853e39	−5.11779
$970$	0	0
$971$	7.09328e38	1.03991	0.519956	−	0.854193i	$-0.325948\pi$
$971$	7.09328e38	1.03991	0.519956	+	0.854193i	$0.325948\pi$
$972$	2.33583e39	3.37893
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.18342e39	1.60144	0.800718	−	0.599042i	$-0.204452\pi$
$977$	1.18342e39	1.60144	0.800718	+	0.599042i	$0.204452\pi$
$978$	−1.29249e39	−1.72593
$979$	−8.43458e37	−0.111145
$980$	0	0
$981$	0	0
$982$	8.79533e38	1.11379
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	3.08440e39	3.80394
$985$	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
$993$	9.48522e38	1.03922
$994$	0	0
$995$	0	0
$996$	−2.22778e39	−2.34693
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−8.17739e38	−0.839301
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.27.d.a.3.1	✓	1
4.3	odd	2		32.27.d.a.15.1		1
8.3	odd	2	CM	8.27.d.a.3.1	✓	1
8.5	even	2		32.27.d.a.15.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.27.d.a.3.1	✓	1	1.1	even	1	trivial
8.27.d.a.3.1	✓	1	8.3	odd	2	CM
32.27.d.a.15.1		1	4.3	odd	2
32.27.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:	\( k \)	\(=\)	\( 27 \)
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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