LMFDB - Embedded newform 8.39.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,39,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, 39, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 39);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 39 $
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:	$73.1707752268$
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)

$f(q)$

$=$

$q-524288. q^{2} +4.01779e8 q^{3} +2.74878e11 q^{4} -2.10648e14 q^{6} -1.44115e17 q^{8} -1.18943e18 q^{9} +O(q^{10})$

$q-524288. q^{2} +4.01779e8 q^{3} +2.74878e11 q^{4} -2.10648e14 q^{6} -1.44115e17 q^{8} -1.18943e18 q^{9} -6.28204e19 q^{11} +1.10440e20 q^{12} +7.55579e22 q^{16} -4.30022e23 q^{17} +6.23601e23 q^{18} +3.20041e24 q^{19} +3.29360e25 q^{22} -5.79024e25 q^{24} +3.63798e26 q^{25} -1.02063e27 q^{27} -3.96141e28 q^{32} -2.52399e28 q^{33} +2.25455e29 q^{34} -3.26947e29 q^{36} -1.67793e30 q^{38} -8.34214e30 q^{41} -7.54580e29 q^{43} -1.72679e31 q^{44} +3.03576e31 q^{48} +1.29935e32 q^{49} -1.90735e32 q^{50} -1.72774e32 q^{51} +5.35104e32 q^{54} +1.28586e33 q^{57} +7.93284e33 q^{59} +2.07692e34 q^{64} +1.32330e34 q^{66} -2.79189e34 q^{67} -1.18204e35 q^{68} +1.71414e35 q^{72} +1.27448e35 q^{73} +1.46166e35 q^{75} +8.79721e35 q^{76} +1.19667e36 q^{81} +4.37368e36 q^{82} +5.42827e36 q^{83} +3.95617e35 q^{86} +9.05338e36 q^{88} +1.18877e37 q^{89} -1.59161e37 q^{96} -2.07397e37 q^{97} -6.81233e37 q^{98} +7.47202e37 q^{99} +O(q^{100})$

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 $

$2$	−524288.	−1.00000
$2$	−524288.	−1.00000
$3$	4.01779e8	0.345687	0.172844	−	0.984949i	$-0.444704\pi$
$3$	4.01779e8	0.345687	0.172844	+	0.984949i	$0.444704\pi$
$4$	2.74878e11	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−2.10648e14	−0.345687
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1.44115e17	−1.00000
$9$	−1.18943e18	−0.880500
$10$	0	0
$11$	−6.28204e19	−1.02716	−0.513582	−	0.858040i	$-0.671682\pi$
$11$	−6.28204e19	−1.02716	−0.513582	+	0.858040i	$0.671682\pi$
$12$	1.10440e20	0.345687
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	7.55579e22	1.00000
$17$	−4.30022e23	−1.79871	−0.899355	−	0.437219i	$-0.855964\pi$
$17$	−4.30022e23	−1.79871	−0.899355	+	0.437219i	$0.855964\pi$
$18$	6.23601e23	0.880500
$19$	3.20041e24	1.61766	0.808829	−	0.588044i	$-0.200102\pi$
$19$	3.20041e24	1.61766	0.808829	+	0.588044i	$0.200102\pi$
$20$	0	0
$21$	0	0
$22$	3.29360e25	1.02716
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−5.79024e25	−0.345687
$25$	3.63798e26	1.00000
$26$	0	0
$27$	−1.02063e27	−0.650065
$28$	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			$\pi$
$32$	−3.96141e28	−1.00000
$33$	−2.52399e28	−0.355078
$34$	2.25455e29	1.79871
$35$	0	0
$36$	−3.26947e29	−0.880500
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.67793e30	−1.61766
$39$	0	0
$40$	0	0
$41$	−8.34214e30	−1.89838	−0.949192	−	0.314698i	$-0.898097\pi$
$41$	−8.34214e30	−1.89838	−0.949192	+	0.314698i	$0.898097\pi$
$42$	0	0
$43$	−7.54580e29	−0.0694712	−0.0347356	−	0.999397i	$-0.511059\pi$
$43$	−7.54580e29	−0.0694712	−0.0347356	+	0.999397i	$0.511059\pi$
$44$	−1.72679e31	−1.02716
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	3.03576e31	0.345687
$49$	1.29935e32	1.00000
$50$	−1.90735e32	−1.00000
$51$	−1.72774e32	−0.621791
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	5.35104e32	0.650065
$55$	0	0
$56$	0	0
$57$	1.28586e33	0.559204
$58$	0	0
$59$	7.93284e33	1.79160	0.895798	−	0.444460i	$-0.146605\pi$
$59$	7.93284e33	1.79160	0.895798	+	0.444460i	$0.146605\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	2.07692e34	1.00000
$65$	0	0
$66$	1.32330e34	0.355078
$67$	−2.79189e34	−0.562960	−0.281480	−	0.959567i	$-0.590825\pi$
$67$	−2.79189e34	−0.562960	−0.281480	+	0.959567i	$0.590825\pi$
$68$	−1.18204e35	−1.79871
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.71414e35	0.880500
$73$	1.27448e35	0.503732	0.251866	−	0.967762i	$-0.418956\pi$
$73$	1.27448e35	0.503732	0.251866	+	0.967762i	$0.418956\pi$
$74$	0	0
$75$	1.46166e35	0.345687
$76$	8.79721e35	1.61766
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.19667e36	0.655781
$82$	4.37368e36	1.89838
$83$	5.42827e36	1.87145	0.935724	−	0.352734i	$-0.114748\pi$
$83$	5.42827e36	1.87145	0.935724	+	0.352734i	$0.114748\pi$
$84$	0	0
$85$	0	0
$86$	3.95617e35	0.0694712
$87$	0	0
$88$	9.05338e36	1.02716
$89$	1.18877e37	1.08815	0.544074	−	0.839037i	$-0.316881\pi$
$89$	1.18877e37	1.08815	0.544074	+	0.839037i	$0.316881\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.59161e37	−0.345687
$97$	−2.07397e37	−0.369946	−0.184973	−	0.982744i	$-0.559220\pi$
$97$	−2.07397e37	−0.369946	−0.184973	+	0.982744i	$0.559220\pi$
$98$	−6.81233e37	−1.00000
$99$	7.47202e37	0.904419
$100$	1.00000e38	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	9.05832e37	0.621791
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.33673e38	−0.369618	−0.184809	−	0.982775i	$-0.559167\pi$
$107$	−1.33673e38	−0.369618	−0.184809	+	0.982775i	$0.559167\pi$
$108$	−2.80549e38	−0.650065
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.60136e39	−1.57036	−0.785178	−	0.619270i	$-0.787429\pi$
$113$	−1.60136e39	−1.57036	−0.785178	+	0.619270i	$0.787429\pi$
$114$	−6.74159e38	−0.559204
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−4.15909e39	−1.79160
$119$	0	0
$120$	0	0
$121$	2.05973e38	0.0550665
$122$	0	0
$123$	−3.35169e39	−0.656247
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.08890e40	−1.00000
$129$	−3.03174e38	−0.0240153
$130$	0	0
$131$	3.33151e40	1.97009	0.985047	−	0.172284i	$-0.0551147\pi$
$131$	3.33151e40	1.97009	0.985047	+	0.172284i	$0.0551147\pi$
$132$	−6.93790e39	−0.355078
$133$	0	0
$134$	1.46375e40	0.562960
$135$	0	0
$136$	6.19727e40	1.79871
$137$	7.21594e40	1.82223	0.911113	−	0.412156i	$-0.135224\pi$
$137$	7.21594e40	1.82223	0.911113	+	0.412156i	$0.135224\pi$
$138$	0	0
$139$	1.54686e40	0.296599	0.148300	−	0.988942i	$-0.452620\pi$
$139$	1.54686e40	0.296599	0.148300	+	0.988942i	$0.452620\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−8.98704e40	−0.880500
$145$	0	0
$146$	−6.68196e40	−0.503732
$147$	5.22051e40	0.345687
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−7.66332e40	−0.345687
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−4.61227e41	−1.61766
$153$	5.11479e41	1.58377
$154$	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
$162$	−6.27400e41	−0.655781
$163$	2.12256e42	1.97376	0.986882	−	0.161442i	$-0.0516143\pi$
$163$	2.12256e42	1.97376	0.986882	+	0.161442i	$0.0516143\pi$
$164$	−2.29307e42	−1.89838
$165$	0	0
$166$	−2.84598e42	−1.87145
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	2.13721e42	1.00000
$170$	0	0
$171$	−3.80665e42	−1.42435
$172$	−2.07417e41	−0.0694712
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−4.74658e42	−1.02716
$177$	3.18725e42	0.619332
$178$	−6.23258e42	−1.08815
$179$	1.23876e43	1.94438	0.972188	−	0.234201i	$-0.0752474\pi$
$179$	1.23876e43	1.94438	0.972188	+	0.234201i	$0.0752474\pi$
$180$	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
$187$	2.70142e43	1.84757
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	8.34462e42	0.345687
$193$	5.25645e43	1.97289	0.986446	−	0.164086i	$-0.0524674\pi$
$193$	5.25645e43	1.97289	0.986446	+	0.164086i	$0.0524674\pi$
$194$	1.08736e43	0.369946
$195$	0	0
$196$	3.57162e43	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−3.91749e43	−0.904419
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−5.24288e43	−1.00000
$201$	−1.12172e43	−0.194608
$202$	0	0
$203$	0	0
$204$	−4.74917e43	−0.621791
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.01051e44	−1.66160
$210$	0	0
$211$	−2.80425e44	−1.93397	−0.966987	−	0.254826i	$-0.917982\pi$
$211$	−2.80425e44	−1.93397	−0.966987	+	0.254826i	$0.917982\pi$
$212$	0	0
$213$	0	0
$214$	7.00833e43	0.369618
$215$	0	0
$216$	1.47088e44	0.650065
$217$	0	0
$218$	0	0
$219$	5.12060e43	0.174134
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−4.32710e44	−0.880500
$226$	8.39573e44	1.57036
$227$	−1.05673e45	−1.81750	−0.908748	−	0.417346i	$-0.862960\pi$
$227$	−1.05673e45	−1.81750	−0.908748	+	0.417346i	$0.862960\pi$
$228$	3.53453e44	0.559204
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.33133e45	1.39484	0.697418	−	0.716664i	$-0.254332\pi$
$233$	1.33133e45	1.39484	0.697418	+	0.716664i	$0.254332\pi$
$234$	0	0
$235$	0	0
$236$	2.18056e45	1.79160
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−3.62489e45	−1.99973	−0.999864	−	0.0164801i	$-0.994754\pi$
$241$	−3.62489e45	−1.99973	−0.999864	+	0.0164801i	$0.994754\pi$
$242$	−1.07989e44	−0.0550665
$243$	1.85952e45	0.876760
$244$	0	0
$245$	0	0
$246$	1.75725e45	0.656247
$247$	0	0
$248$	0	0
$249$	2.18096e45	0.646935
$250$	0	0
$251$	−4.52650e45	−1.15335	−0.576676	−	0.816973i	$-0.695651\pi$
$251$	−4.52650e45	−1.15335	−0.576676	+	0.816973i	$0.695651\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	5.70899e45	1.00000
$257$	5.16133e45	0.839523	0.419761	−	0.907634i	$-0.362114\pi$
$257$	5.16133e45	0.839523	0.419761	+	0.907634i	$0.362114\pi$
$258$	1.58951e44	0.0240153
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−1.74667e46	−1.97009
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	3.63746e45	0.355078
$265$	0	0
$266$	0	0
$267$	4.77623e45	0.376159
$268$	−7.67429e45	−0.562960
$269$	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			$\pi$
$272$	−3.24916e46	−1.79871
$273$	0	0
$274$	−3.78323e46	−1.82223
$275$	−2.28539e46	−1.02716
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−8.11000e45	−0.296599
$279$	0	0
$280$	0	0
$281$	6.02485e46	1.79691	0.898454	−	0.439067i	$-0.144691\pi$
$281$	6.02485e46	1.79691	0.898454	+	0.439067i	$0.144691\pi$
$282$	0	0
$283$	2.84057e46	0.740394	0.370197	−	0.928953i	$-0.379290\pi$
$283$	2.84057e46	0.740394	0.370197	+	0.928953i	$0.379290\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	4.71180e46	0.880500
$289$	1.27763e47	2.23536
$290$	0	0
$291$	−8.33275e45	−0.127886
$292$	3.50327e46	0.503732
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−2.73705e46	−0.345687
$295$	0	0
$296$	0	0
$297$	6.41164e46	0.667723
$298$	0	0
$299$	0	0
$300$	4.01779e46	0.345687
$301$	0	0
$302$	0	0
$303$	0	0
$304$	2.41816e47	1.61766
$305$	0	0
$306$	−2.68162e47	−1.58377
$307$	−3.57205e47	−1.98284	−0.991422	−	0.130698i	$-0.958278\pi$
$307$	−3.57205e47	−1.98284	−0.991422	+	0.130698i	$0.958278\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	5.59840e46	0.215140	0.107570	−	0.994197i	$-0.465693\pi$
$313$	5.59840e46	0.215140	0.107570	+	0.994197i	$0.465693\pi$
$314$	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
$321$	−5.37071e46	−0.127772
$322$	0	0
$323$	−1.37625e48	−2.90970
$324$	3.28938e47	0.655781
$325$	0	0
$326$	−1.11283e48	−1.97376
$327$	0	0
$328$	1.20223e48	1.89838
$329$	0	0
$330$	0	0
$331$	−5.88933e47	−0.782230	−0.391115	−	0.920342i	$-0.627910\pi$
$331$	−5.88933e47	−0.782230	−0.391115	+	0.920342i	$0.627910\pi$
$332$	1.49211e48	1.87145
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.41125e48	1.33240	0.666200	−	0.745773i	$-0.267920\pi$
$337$	1.41125e48	1.33240	0.666200	+	0.745773i	$0.267920\pi$
$338$	−1.12051e48	−1.00000
$339$	−6.43392e47	−0.542852
$340$	0	0
$341$	0	0
$342$	1.99578e48	1.42435
$343$	0	0
$344$	1.08746e47	0.0694712
$345$	0	0
$346$	0	0
$347$	−3.65786e48	−1.98138	−0.990688	−	0.136152i	$-0.956526\pi$
$347$	−3.65786e48	−1.98138	−0.990688	+	0.136152i	$0.956526\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	2.48857e48	1.02716
$353$	4.28702e48	1.67662	0.838312	−	0.545191i	$-0.183543\pi$
$353$	4.28702e48	1.67662	0.838312	+	0.545191i	$0.183543\pi$
$354$	−1.67103e48	−0.619332
$355$	0	0
$356$	3.26767e48	1.08815
$357$	0	0
$358$	−6.49468e48	−1.94438
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	6.32846e48	1.61682
$362$	0	0
$363$	8.27554e46	0.0190358
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	9.92235e48	1.67153
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−1.41632e49	−1.84757
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9.36203e48	−0.948910	−0.474455	−	0.880280i	$-0.657355\pi$
$379$	−9.36203e48	−0.948910	−0.474455	+	0.880280i	$0.657355\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−4.37498e48	−0.345687
$385$	0	0
$386$	−2.75589e49	−1.97289
$387$	8.97517e47	0.0611694
$388$	−5.70087e48	−0.369946
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.87256e49	−1.00000
$393$	1.33853e49	0.681036
$394$	0	0
$395$	0	0
$396$	2.05389e49	0.904419
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	2.74878e49	1.00000
$401$	−4.85705e49	−1.68511	−0.842557	−	0.538608i	$-0.818950\pi$
$401$	−4.85705e49	−1.68511	−0.842557	+	0.538608i	$0.818950\pi$
$402$	5.88106e48	0.194608
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.48993e49	0.621791
$409$	8.31100e49	1.98112	0.990559	−	0.137084i	$-0.0437729\pi$
$409$	8.31100e49	1.98112	0.990559	+	0.137084i	$0.0437729\pi$
$410$	0	0
$411$	2.89921e49	0.629920
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.21496e48	0.102530
$418$	1.05409e50	1.66160
$419$	−2.70803e48	−0.0407931	−0.0203966	−	0.999792i	$-0.506493\pi$
$419$	−2.70803e48	−0.0407931	−0.0203966	+	0.999792i	$0.506493\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.47024e50	1.93397
$423$	0	0
$424$	0	0
$425$	−1.56441e50	−1.79871
$426$	0	0
$427$	0	0
$428$	−3.67438e49	−0.369618
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−7.71166e49	−0.650065
$433$	−2.40039e50	−1.93648	−0.968239	−	0.250028i	$-0.919560\pi$
$433$	−2.40039e50	−1.93648	−0.968239	+	0.250028i	$0.919560\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−2.68467e49	−0.174134
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.54548e50	−0.880500
$442$	0	0
$443$	−3.18650e50	−1.66588	−0.832940	−	0.553364i	$-0.813344\pi$
$443$	−3.18650e50	−1.66588	−0.832940	+	0.553364i	$0.813344\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.80978e50	0.732734	0.366367	−	0.930471i	$-0.380602\pi$
$449$	1.80978e50	0.732734	0.366367	+	0.930471i	$0.380602\pi$
$450$	2.26865e50	0.880500
$451$	5.24057e50	1.94995
$452$	−4.40178e50	−1.57036
$453$	0	0
$454$	5.54032e50	1.81750
$455$	0	0
$456$	−1.85311e50	−0.559204
$457$	−6.61533e50	−1.91489	−0.957444	−	0.288618i	$-0.906804\pi$
$457$	−6.61533e50	−1.91489	−0.957444	+	0.288618i	$0.906804\pi$
$458$	0	0
$459$	4.38893e50	1.16928
$460$	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
$466$	−6.98000e50	−1.39484
$467$	6.64999e50	1.27585	0.637926	−	0.770098i	$-0.279793\pi$
$467$	6.64999e50	1.27585	0.637926	+	0.770098i	$0.279793\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.14324e51	−1.79160
$473$	4.74031e49	0.0713583
$474$	0	0
$475$	1.16430e51	1.61766
$476$	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
$482$	1.90049e51	1.99973
$483$	0	0
$484$	5.66173e49	0.0550665
$485$	0	0
$486$	−9.74922e50	−0.876760
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	8.52799e50	0.682305
$490$	0	0
$491$	2.63371e51	1.94993	0.974966	−	0.222354i	$-0.0713741\pi$
$491$	2.63371e51	1.94993	0.974966	+	0.222354i	$0.0713741\pi$
$492$	−9.21307e50	−0.656247
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−1.14345e51	−0.646935
$499$	−3.29944e51	−1.79693	−0.898463	−	0.439049i	$-0.855315\pi$
$499$	−3.29944e51	−1.79693	−0.898463	+	0.439049i	$0.855315\pi$
$500$	0	0
$501$	0	0
$502$	2.37319e51	1.15335
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.58686e50	0.345687
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−2.99316e51	−1.00000
$513$	−3.26643e51	−1.05158
$514$	−2.70603e51	−0.839523
$515$	0	0
$516$	−8.33360e49	−0.0240153
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.43578e51	−0.584414	−0.292207	−	0.956355i	$-0.594390\pi$
$521$	−2.43578e51	−0.584414	−0.292207	+	0.956355i	$0.594390\pi$
$522$	0	0
$523$	−8.12031e51	−1.81151	−0.905755	−	0.423802i	$-0.860695\pi$
$523$	−8.12031e51	−1.81151	−0.905755	+	0.423802i	$0.860695\pi$
$524$	9.15760e51	1.97009
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−1.90707e51	−0.355078
$529$	5.56747e51	1.00000
$530$	0	0
$531$	−9.43552e51	−1.57750
$532$	0	0
$533$	0	0
$534$	−2.50412e51	−0.376159
$535$	0	0
$536$	4.02354e51	0.562960
$537$	4.97708e51	0.672146
$538$	0	0
$539$	−8.16256e51	−1.02716
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.70349e52	1.79871
$545$	0	0
$546$	0	0
$547$	1.67591e52	1.59401	0.797005	−	0.603972i	$-0.206416\pi$
$547$	1.67591e52	1.59401	0.797005	+	0.603972i	$0.206416\pi$
$548$	1.98350e52	1.82223
$549$	0	0
$550$	1.19820e52	1.02716
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	4.25198e51	0.296599
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.08537e52	0.638682
$562$	−3.15876e52	−1.79691
$563$	−2.33712e52	−1.28535	−0.642676	−	0.766138i	$-0.722176\pi$
$563$	−2.33712e52	−1.28535	−0.642676	+	0.766138i	$0.722176\pi$
$564$	0	0
$565$	0	0
$566$	−1.48928e52	−0.740394
$567$	0	0
$568$	0	0
$569$	4.43667e52	1.99491	0.997456	−	0.0712849i	$-0.0227100\pi$
$569$	4.43667e52	1.99491	0.997456	+	0.0712849i	$0.0227100\pi$
$570$	0	0
$571$	−3.72272e52	−1.56594	−0.782969	−	0.622060i	$-0.786296\pi$
$571$	−3.72272e52	−1.56594	−0.782969	+	0.622060i	$0.786296\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−2.47034e52	−0.880500
$577$	−1.90928e51	−0.0658460	−0.0329230	−	0.999458i	$-0.510482\pi$
$577$	−1.90928e51	−0.0658460	−0.0329230	+	0.999458i	$0.510482\pi$
$578$	−6.69848e52	−2.23536
$579$	2.11193e52	0.682003
$580$	0	0
$581$	0	0
$582$	4.36876e51	0.127886
$583$	0	0
$584$	−1.83672e52	−0.503732
$585$	0	0
$586$	0	0
$587$	−4.02066e50	−0.0100040	−0.00500200	−	0.999987i	$-0.501592\pi$
$587$	−4.02066e50	−0.0100040	−0.00500200	+	0.999987i	$0.501592\pi$
$588$	1.43500e52	0.345687
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.39583e52	0.491380	0.245690	−	0.969348i	$-0.420985\pi$
$593$	2.39583e52	0.491380	0.245690	+	0.969348i	$0.420985\pi$
$594$	−3.36155e52	−0.667723
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−2.10648e52	−0.345687
$601$	8.16753e52	1.29860	0.649301	−	0.760532i	$-0.275062\pi$
$601$	8.16753e52	1.29860	0.649301	+	0.760532i	$0.275062\pi$
$602$	0	0
$603$	3.32075e52	0.495686
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−1.26781e53	−1.61766
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.40594e53	1.58377
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.87278e53	1.98284
$615$	0	0
$616$	0	0
$617$	−1.82335e53	−1.75975	−0.879877	−	0.475201i	$-0.842375\pi$
$617$	−1.82335e53	−1.75975	−0.879877	+	0.475201i	$0.842375\pi$
$618$	0	0
$619$	1.00182e53	0.909216	0.454608	−	0.890692i	$-0.349779\pi$
$619$	1.00182e53	0.909216	0.454608	+	0.890692i	$0.349779\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.32349e53	1.00000
$626$	−2.93518e52	−0.215140
$627$	−8.07780e52	−0.574394
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−1.12669e53	−0.668550
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.69089e53	−1.72515	−0.862577	−	0.505926i	$-0.831151\pi$
$641$	−3.69089e53	−1.72515	−0.862577	+	0.505926i	$0.831151\pi$
$642$	2.81580e52	0.127772
$643$	4.51493e53	1.98903	0.994517	−	0.104577i	$-0.0333487\pi$
$643$	4.51493e53	1.98903	0.994517	+	0.104577i	$0.0333487\pi$
$644$	0	0
$645$	0	0
$646$	7.21549e53	2.90970
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.72458e53	−0.655781
$649$	−4.98344e53	−1.84026
$650$	0	0
$651$	0	0
$652$	5.83445e53	1.97376
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−6.30314e53	−1.89838
$657$	−1.51590e53	−0.443536
$658$	0	0
$659$	−3.63299e53	−1.00332	−0.501662	−	0.865064i	$-0.667278\pi$
$659$	−3.63299e53	−1.00332	−0.501662	+	0.865064i	$0.667278\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	3.08771e53	0.782230
$663$	0	0
$664$	−7.82296e53	−1.87145
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.02330e53	−0.930471	−0.465236	−	0.885187i	$-0.654030\pi$
$673$	−5.02330e53	−0.930471	−0.465236	+	0.885187i	$0.654030\pi$
$674$	−7.39899e53	−1.33240
$675$	−3.71303e53	−0.650065
$676$	5.87472e53	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	3.37323e53	0.542852
$679$	0	0
$680$	0	0
$681$	−4.24572e53	−0.628285
$682$	0	0
$683$	8.82700e53	1.23543	0.617717	−	0.786400i	$-0.288058\pi$
$683$	8.82700e53	1.23543	0.617717	+	0.786400i	$0.288058\pi$
$684$	−1.04636e54	−1.42435
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−5.70145e52	−0.0694712
$689$	0	0
$690$	0	0
$691$	1.67032e54	1.87377	0.936884	−	0.349639i	$-0.113696\pi$
$691$	1.67032e54	1.87377	0.936884	+	0.349639i	$0.113696\pi$
$692$	0	0
$693$	0	0
$694$	1.91777e54	1.98138
$695$	0	0
$696$	0	0
$697$	3.58730e54	3.41464
$698$	0	0
$699$	5.34900e53	0.482177
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−1.30473e54	−1.02716
$705$	0	0
$706$	−2.24763e54	−1.67662
$707$	0	0
$708$	8.76104e53	0.619332
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−1.71320e54	−1.08815
$713$	0	0
$714$	0	0
$715$	0	0
$716$	3.40508e54	1.94438
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−3.31794e54	−1.61682
$723$	−1.45640e54	−0.691280
$724$	0	0
$725$	0	0
$726$	−4.33877e52	−0.0190358
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−8.69409e53	−0.352697
$730$	0	0
$731$	3.24486e53	0.124959
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.75388e54	0.578252
$738$	−5.20217e54	−1.67153
$739$	−5.47939e54	−1.71588	−0.857942	−	0.513747i	$-0.828257\pi$
$739$	−5.47939e54	−1.71588	−0.857942	+	0.513747i	$0.828257\pi$
$740$	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
$747$	−6.45652e54	−1.64781
$748$	7.42560e54	1.84757
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−1.81865e54	−0.398699
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	4.90840e54	0.948910
$759$	0	0
$760$	0	0
$761$	6.69090e52	0.0119999	0.00599993	−	0.999982i	$-0.498090\pi$
$761$	6.69090e52	0.0119999	0.00599993	+	0.999982i	$0.498090\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	2.29375e54	0.345687
$769$	1.33089e55	1.95678	0.978389	−	0.206772i	$-0.0662959\pi$
$769$	1.33089e55	1.95678	0.978389	+	0.206772i	$0.0662959\pi$
$770$	0	0
$771$	2.07371e54	0.290212
$772$	1.44488e55	1.97289
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−4.70557e53	−0.0611694
$775$	0	0
$776$	2.98890e54	0.369946
$777$	0	0
$778$	0	0
$779$	−2.66982e55	−3.07094
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.81760e54	1.00000
$785$	0	0
$786$	−7.01776e54	−0.681036
$787$	1.21984e55	1.15554	0.577768	−	0.816201i	$-0.303924\pi$
$787$	1.21984e55	1.15554	0.577768	+	0.816201i	$0.303924\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.07683e55	−0.904419
$793$	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
$800$	−1.44115e55	−1.00000
$801$	−1.41395e55	−0.958114
$802$	2.54649e55	1.68511
$803$	−8.00636e54	−0.517416
$804$	−3.08337e54	−0.194608
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.28811e55	1.28368	0.641839	−	0.766840i	$-0.278172\pi$
$809$	2.28811e55	1.28368	0.641839	+	0.766840i	$0.278172\pi$
$810$	0	0
$811$	3.24511e55	1.73714	0.868568	−	0.495571i	$-0.165041\pi$
$811$	3.24511e55	1.73714	0.868568	+	0.495571i	$0.165041\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−1.30544e55	−0.621791
$817$	−2.41496e54	−0.112381
$818$	−4.35736e55	−1.98112
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−1.52002e55	−0.629920
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−9.18223e54	−0.355078
$826$	0	0
$827$	3.81969e55	1.41066	0.705332	−	0.708877i	$-0.250798\pi$
$827$	3.81969e55	1.41066	0.705332	+	0.708877i	$0.250798\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.58748e55	−1.79871
$834$	−3.25843e54	−0.102530
$835$	0	0
$836$	−5.52645e55	−1.66160
$837$	0	0
$838$	1.41979e54	0.0407931
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	3.72498e55	1.00000
$842$	0	0
$843$	2.42066e55	0.621168
$844$	−7.70827e55	−1.93397
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.14128e55	0.255945
$850$	8.20202e55	1.79871
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	1.92643e55	0.369618
$857$	−8.61449e55	−1.61657	−0.808284	−	0.588793i	$-0.799603\pi$
$857$	−8.61449e55	−1.61657	−0.808284	+	0.588793i	$0.799603\pi$
$858$	0	0
$859$	8.05171e55	1.44550	0.722750	−	0.691110i	$-0.242878\pi$
$859$	8.05171e55	1.44550	0.722750	+	0.691110i	$0.242878\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	4.04313e55	0.650065
$865$	0	0
$866$	1.25850e56	1.93648
$867$	5.13326e55	0.772735
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.46683e55	0.325738
$874$	0	0
$875$	0	0
$876$	1.40754e55	0.174134
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.17546e56	−1.30517	−0.652585	−	0.757715i	$-0.726316\pi$
$881$	−1.17546e56	−1.30517	−0.652585	+	0.757715i	$0.726316\pi$
$882$	8.10275e55	0.880500
$883$	−5.40904e55	−0.575263	−0.287632	−	0.957741i	$-0.592868\pi$
$883$	−5.40904e55	−0.575263	−0.287632	+	0.957741i	$0.592868\pi$
$884$	0	0
$885$	0	0
$886$	1.67065e56	1.66588
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−7.51753e55	−0.673595
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−9.48848e55	−0.732734
$899$	0	0
$900$	−1.18943e56	−0.880500
$901$	0	0
$902$	−2.74757e56	−1.94995
$903$	0	0
$904$	2.30780e56	1.57036
$905$	0	0
$906$	0	0
$907$	−3.12544e56	−1.99697	−0.998487	−	0.0549811i	$-0.982490\pi$
$907$	−3.12544e56	−1.99697	−0.998487	+	0.0549811i	$0.982490\pi$
$908$	−2.90472e56	−1.81750
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	9.71565e55	0.559204
$913$	−3.41006e56	−1.92228
$914$	3.46834e56	1.91489
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−2.30106e56	−1.16928
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.43518e56	−0.685444
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.29567e56	1.74074	0.870368	−	0.492402i	$-0.163881\pi$
$929$	4.29567e56	1.74074	0.870368	+	0.492402i	$0.163881\pi$
$930$	0	0
$931$	4.15844e56	1.61766
$932$	3.65953e56	1.39484
$933$	0	0
$934$	−3.48651e56	−1.27585
$935$	0	0
$936$	0	0
$937$	5.38711e56	1.85483	0.927414	−	0.374035i	$-0.122026\pi$
$937$	5.38711e56	1.85483	0.927414	+	0.374035i	$0.122026\pi$
$938$	0	0
$939$	2.24932e55	0.0743713
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	5.99388e56	1.79160
$945$	0	0
$946$	−2.48529e55	−0.0713583
$947$	−6.24330e56	−1.75697	−0.878486	−	0.477768i	$-0.841446\pi$
$947$	−6.24330e56	−1.75697	−0.878486	+	0.477768i	$0.841446\pi$
$948$	0	0
$949$	0	0
$950$	−6.10429e56	−1.61766
$951$	0	0
$952$	0	0
$953$	7.91823e56	1.97635	0.988173	−	0.153346i	$-0.0490050\pi$
$953$	7.91823e56	1.97635	0.988173	+	0.153346i	$0.0490050\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.69618e56	1.00000
$962$	0	0
$963$	1.58994e56	0.325448
$964$	−9.96402e56	−1.99973
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.96838e55	−0.0550665
$969$	−5.52946e56	−1.00585
$970$	0	0
$971$	6.04938e56	1.05815	0.529073	−	0.848576i	$-0.322540\pi$
$971$	6.04938e56	1.05815	0.529073	+	0.848576i	$0.322540\pi$
$972$	5.11140e56	0.876760
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.12103e57	1.74430	0.872152	−	0.489236i	$-0.162724\pi$
$977$	1.12103e57	1.74430	0.872152	+	0.489236i	$0.162724\pi$
$978$	−4.47113e56	−0.682305
$979$	−7.46791e56	−1.11771
$980$	0	0
$981$	0	0
$982$	−1.38082e57	−1.94993
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	4.83030e56	0.656247
$985$	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
$993$	−2.36621e56	−0.270407
$994$	0	0
$995$	0	0
$996$	5.99499e56	0.646935
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.72986e57	1.79693
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.39.d.a.3.1	✓	1
8.3	odd	2	CM	8.39.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.39.d.a.3.1	✓	1	1.1	even	1	trivial
8.39.d.a.3.1	✓	1	8.3	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 39 \)
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

Downloads

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