LMFDB - Embedded newform 8.55.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,55,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, 55, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 55);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 55 $
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:	$147.743169528$
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)

$f(q)$

$=$

$q-1.34218e8 q^{2} +3.76111e12 q^{3} +1.80144e16 q^{4} -5.04807e20 q^{6} -2.41785e24 q^{8} -4.40038e25 q^{9} +O(q^{10})$

$q-1.34218e8 q^{2} +3.76111e12 q^{3} +1.80144e16 q^{4} -5.04807e20 q^{6} -2.41785e24 q^{8} -4.40038e25 q^{9} +5.86597e27 q^{11} +6.77541e28 q^{12} +3.24519e32 q^{16} -3.33486e33 q^{17} +5.90609e33 q^{18} -6.70523e34 q^{19} -7.87317e35 q^{22} -9.09380e36 q^{24} +5.55112e37 q^{25} -3.84211e38 q^{27} -4.35561e40 q^{32} +2.20625e40 q^{33} +4.47597e41 q^{34} -7.92702e41 q^{36} +8.99960e42 q^{38} -2.55249e43 q^{41} +2.42764e44 q^{43} +1.05672e44 q^{44} +1.22055e45 q^{48} +4.31811e45 q^{49} -7.45058e45 q^{50} -1.25428e46 q^{51} +5.15679e46 q^{54} -2.52191e47 q^{57} +1.23270e48 q^{59} +5.84601e48 q^{64} -2.96119e48 q^{66} -1.64693e49 q^{67} -6.00755e49 q^{68} +1.06395e50 q^{72} -4.07540e50 q^{73} +2.08783e50 q^{75} -1.20791e51 q^{76} +1.11375e51 q^{81} +3.42589e51 q^{82} -6.96783e51 q^{83} -3.25832e52 q^{86} -1.41830e52 q^{88} -6.38159e52 q^{89} -1.63819e53 q^{96} +7.79578e53 q^{97} -5.79568e53 q^{98} -2.58125e53 q^{99} +O(q^{100})$

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 $

$2$	−1.34218e8	−1.00000
$2$	−1.34218e8	−1.00000
$3$	3.76111e12	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$3$	3.76111e12	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$4$	1.80144e16	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−5.04807e20	−0.493221
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−2.41785e24	−1.00000
$9$	−4.40038e25	−0.756733
$10$	0	0
$11$	5.86597e27	0.447443	0.223721	−	0.974653i	$-0.428179\pi$
$11$	5.86597e27	0.447443	0.223721	+	0.974653i	$0.428179\pi$
$12$	6.77541e28	0.493221
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	3.24519e32	1.00000
$17$	−3.33486e33	−1.99966	−0.999832	−	0.0183553i	$-0.994157\pi$
$17$	−3.33486e33	−1.99966	−0.999832	+	0.0183553i	$0.994157\pi$
$18$	5.90609e33	0.756733
$19$	−6.70523e34	−1.99557	−0.997784	−	0.0665438i	$-0.978803\pi$
$19$	−6.70523e34	−1.99557	−0.997784	+	0.0665438i	$0.978803\pi$
$20$	0	0
$21$	0	0
$22$	−7.87317e35	−0.447443
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−9.09380e36	−0.493221
$25$	5.55112e37	1.00000
$26$	0	0
$27$	−3.84211e38	−0.866458
$28$	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			$\pi$
$32$	−4.35561e40	−1.00000
$33$	2.20625e40	0.220688
$34$	4.47597e41	1.99966
$35$	0	0
$36$	−7.92702e41	−0.756733
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	8.99960e42	1.99557
$39$	0	0
$40$	0	0
$41$	−2.55249e43	−0.727444	−0.363722	−	0.931508i	$-0.618494\pi$
$41$	−2.55249e43	−0.727444	−0.363722	+	0.931508i	$0.618494\pi$
$42$	0	0
$43$	2.42764e44	1.91221	0.956104	−	0.293027i	$-0.0946627\pi$
$43$	2.42764e44	1.91221	0.956104	+	0.293027i	$0.0946627\pi$
$44$	1.05672e44	0.447443
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.22055e45	0.493221
$49$	4.31811e45	1.00000
$50$	−7.45058e45	−1.00000
$51$	−1.25428e46	−0.986277
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	5.15679e46	0.866458
$55$	0	0
$56$	0	0
$57$	−2.52191e47	−0.984256
$58$	0	0
$59$	1.23270e48	1.89606	0.948032	−	0.318174i	$-0.103070\pi$
$59$	1.23270e48	1.89606	0.948032	+	0.318174i	$0.103070\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	5.84601e48	1.00000
$65$	0	0
$66$	−2.96119e48	−0.220688
$67$	−1.64693e49	−0.817817	−0.408908	−	0.912575i	$-0.634090\pi$
$67$	−1.64693e49	−0.817817	−0.408908	+	0.912575i	$0.634090\pi$
$68$	−6.00755e49	−1.99966
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.06395e50	0.756733
$73$	−4.07540e50	−1.99734	−0.998671	−	0.0515332i	$-0.983589\pi$
$73$	−4.07540e50	−1.99734	−0.998671	+	0.0515332i	$0.983589\pi$
$74$	0	0
$75$	2.08783e50	0.493221
$76$	−1.20791e51	−1.99557
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.11375e51	0.329377
$82$	3.42589e51	0.727444
$83$	−6.96783e51	−1.06657	−0.533285	−	0.845936i	$-0.679043\pi$
$83$	−6.96783e51	−1.06657	−0.533285	+	0.845936i	$0.679043\pi$
$84$	0	0
$85$	0	0
$86$	−3.25832e52	−1.91221
$87$	0	0
$88$	−1.41830e52	−0.447443
$89$	−6.38159e52	−1.48388	−0.741940	−	0.670466i	$-0.766094\pi$
$89$	−6.38159e52	−1.48388	−0.741940	+	0.670466i	$0.766094\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.63819e53	−0.493221
$97$	7.79578e53	1.77428	0.887142	−	0.461497i	$-0.152688\pi$
$97$	7.79578e53	1.77428	0.887142	+	0.461497i	$0.152688\pi$
$98$	−5.79568e53	−1.00000
$99$	−2.58125e53	−0.338594
$100$	1.00000e54	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	1.68346e54	0.986277
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.22237e55	1.96716	0.983580	−	0.180474i	$-0.0577630\pi$
$107$	1.22237e55	1.96716	0.983580	+	0.180474i	$0.0577630\pi$
$108$	−6.92132e54	−0.866458
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.38977e55	0.881534	0.440767	−	0.897622i	$-0.354707\pi$
$113$	2.38977e55	0.881534	0.440767	+	0.897622i	$0.354707\pi$
$114$	3.38485e55	0.984256
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.65450e56	−1.89606
$119$	0	0
$120$	0	0
$121$	−1.37462e56	−0.799795
$122$	0	0
$123$	−9.60018e55	−0.358791
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−7.84638e56	−1.00000
$129$	9.13060e56	0.943142
$130$	0	0
$131$	−4.88497e56	−0.333071	−0.166536	−	0.986035i	$-0.553258\pi$
$131$	−4.88497e56	−0.333071	−0.166536	+	0.986035i	$0.553258\pi$
$132$	3.97444e56	0.220688
$133$	0	0
$134$	2.21048e57	0.817817
$135$	0	0
$136$	8.06320e57	1.99966
$137$	−1.37510e57	−0.279820	−0.139910	−	0.990164i	$-0.544681\pi$
$137$	−1.37510e57	−0.279820	−0.139910	+	0.990164i	$0.544681\pi$
$138$	0	0
$139$	7.50082e57	1.03207	0.516035	−	0.856567i	$-0.327407\pi$
$139$	7.50082e57	1.03207	0.516035	+	0.856567i	$0.327407\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.42800e58	−0.756733
$145$	0	0
$146$	5.46990e58	1.99734
$147$	1.62409e58	0.493221
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.80224e58	−0.493221
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.62122e59	1.99557
$153$	1.46747e59	1.51321
$154$	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
$162$	−1.49485e59	−0.329377
$163$	4.93436e59	0.920800	0.460400	−	0.887712i	$-0.347706\pi$
$163$	4.93436e59	0.920800	0.460400	+	0.887712i	$0.347706\pi$
$164$	−4.59815e59	−0.727444
$165$	0	0
$166$	9.35206e59	1.06657
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.42214e60	1.00000
$170$	0	0
$171$	2.95056e60	1.51011
$172$	4.37324e60	1.91221
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	1.90362e60	0.447443
$177$	4.63632e60	0.935180
$178$	8.56523e60	1.48388
$179$	7.28628e60	1.08511	0.542556	−	0.840020i	$-0.317457\pi$
$179$	7.28628e60	1.08511	0.542556	+	0.840020i	$0.317457\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.95622e61	−0.894734
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	2.19875e61	0.493221
$193$	−6.18861e61	−1.20655	−0.603275	−	0.797533i	$-0.706138\pi$
$193$	−6.18861e61	−1.20655	−0.603275	+	0.797533i	$0.706138\pi$
$194$	−1.04633e62	−1.77428
$195$	0	0
$196$	7.77882e61	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	3.46449e61	0.338594
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−1.34218e62	−1.00000
$201$	−6.19429e61	−0.403365
$202$	0	0
$203$	0	0
$204$	−2.25951e62	−0.986277
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.93327e62	−0.892902
$210$	0	0
$211$	4.94379e62	0.867828	0.433914	−	0.900954i	$-0.357132\pi$
$211$	4.94379e62	0.867828	0.433914	+	0.900954i	$0.357132\pi$
$212$	0	0
$213$	0	0
$214$	−1.64063e63	−1.96716
$215$	0	0
$216$	9.28964e62	0.866458
$217$	0	0
$218$	0	0
$219$	−1.53280e63	−0.985132
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−2.44270e63	−0.756733
$226$	−3.20750e63	−0.881534
$227$	2.91702e63	0.711609	0.355805	−	0.934560i	$-0.384207\pi$
$227$	2.91702e63	0.711609	0.355805	+	0.934560i	$0.384207\pi$
$228$	−4.54307e63	−0.984256
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.33065e64	−1.60492	−0.802459	−	0.596707i	$-0.796476\pi$
$233$	−1.33065e64	−1.60492	−0.802459	+	0.596707i	$0.796476\pi$
$234$	0	0
$235$	0	0
$236$	2.22064e64	1.89606
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	4.05234e64	1.96447	0.982236	−	0.187648i	$-0.0600864\pi$
$241$	4.05234e64	1.96447	0.982236	+	0.187648i	$0.0600864\pi$
$242$	1.84499e64	0.799795
$243$	2.65307e64	1.02891
$244$	0	0
$245$	0	0
$246$	1.28851e64	0.358791
$247$	0	0
$248$	0	0
$249$	−2.62068e64	−0.526055
$250$	0	0
$251$	8.04267e64	1.30080	0.650399	−	0.759593i	$-0.274602\pi$
$251$	8.04267e64	1.30080	0.650399	+	0.759593i	$0.274602\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.05312e65	1.00000
$257$	1.89834e65	1.62248	0.811240	−	0.584714i	$-0.198793\pi$
$257$	1.89834e65	1.62248	0.811240	+	0.584714i	$0.198793\pi$
$258$	−1.22549e65	−0.943142
$259$	0	0
$260$	0	0
$261$	0	0
$262$	6.55649e64	0.333071
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−5.33440e64	−0.220688
$265$	0	0
$266$	0	0
$267$	−2.40019e65	−0.731882
$268$	−2.96685e65	−0.817817
$269$	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			$\pi$
$272$	−1.08222e66	−1.99966
$273$	0	0
$274$	1.84562e65	0.279820
$275$	3.25627e65	0.447443
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.00674e66	−1.03207
$279$	0	0
$280$	0	0
$281$	−1.79490e66	−1.37711	−0.688556	−	0.725184i	$-0.741755\pi$
$281$	−1.79490e66	−1.37711	−0.688556	+	0.725184i	$0.741755\pi$
$282$	0	0
$283$	−2.22890e66	−1.41207	−0.706034	−	0.708178i	$-0.749518\pi$
$283$	−2.22890e66	−1.41207	−0.706034	+	0.708178i	$0.749518\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.91664e66	0.756733
$289$	8.34003e66	2.99865
$290$	0	0
$291$	2.93208e66	0.875114
$292$	−7.34158e66	−1.99734
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−2.17982e66	−0.493221
$295$	0	0
$296$	0	0
$297$	−2.25377e66	−0.387690
$298$	0	0
$299$	0	0
$300$	3.76111e66	0.493221
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−2.17597e67	−1.99557
$305$	0	0
$306$	−1.96960e67	−1.51321
$307$	2.27950e67	1.60363	0.801813	−	0.597575i	$-0.203869\pi$
$307$	2.27950e67	1.60363	0.801813	+	0.597575i	$0.203869\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	4.66161e67	1.94464	0.972320	−	0.233653i	$-0.0750679\pi$
$313$	4.66161e67	1.94464	0.972320	+	0.233653i	$0.0750679\pi$
$314$	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
$321$	4.59745e67	0.970245
$322$	0	0
$323$	2.23610e68	3.99046
$324$	2.00636e67	0.329377
$325$	0	0
$326$	−6.62278e67	−0.920800
$327$	0	0
$328$	6.17154e67	0.727444
$329$	0	0
$330$	0	0
$331$	−1.17270e68	−1.08101	−0.540507	−	0.841340i	$-0.681768\pi$
$331$	−1.17270e68	−1.08101	−0.540507	+	0.841340i	$0.681768\pi$
$332$	−1.25521e68	−1.06657
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.28475e68	1.29667	0.648337	−	0.761354i	$-0.275465\pi$
$337$	2.28475e68	1.29667	0.648337	+	0.761354i	$0.275465\pi$
$338$	−1.90876e68	−1.00000
$339$	8.98820e67	0.434791
$340$	0	0
$341$	0	0
$342$	−3.96017e68	−1.51011
$343$	0	0
$344$	−5.86966e68	−1.91221
$345$	0	0
$346$	0	0
$347$	5.98514e68	1.54234	0.771168	−	0.636632i	$-0.219673\pi$
$347$	5.98514e68	1.54234	0.771168	+	0.636632i	$0.219673\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−2.55499e68	−0.447443
$353$	−1.16318e69	−1.88682	−0.943411	−	0.331626i	$-0.892403\pi$
$353$	−1.16318e69	−1.88682	−0.943411	+	0.331626i	$0.892403\pi$
$354$	−6.22276e68	−0.935180
$355$	0	0
$356$	−1.14961e69	−1.48388
$357$	0	0
$358$	−9.77948e68	−1.08511
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	3.36701e69	2.98229
$362$	0	0
$363$	−5.17011e68	−0.394476
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	1.12319e69	0.550481
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	2.62559e69	0.894734
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.40877e69	1.52586	0.762932	−	0.646478i	$-0.223759\pi$
$379$	6.40877e69	1.52586	0.762932	+	0.646478i	$0.223759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−2.95111e69	−0.493221
$385$	0	0
$386$	8.30621e69	1.20655
$387$	−1.06825e70	−1.44703
$388$	1.40436e70	1.77428
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.04406e70	−1.00000
$393$	−1.83729e69	−0.164278
$394$	0	0
$395$	0	0
$396$	−4.64997e69	−0.338594
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.80144e70	1.00000
$401$	1.01495e70	0.526681	0.263341	−	0.964703i	$-0.415176\pi$
$401$	1.01495e70	0.526681	0.263341	+	0.964703i	$0.415176\pi$
$402$	8.31384e69	0.403365
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	3.03266e70	0.986277
$409$	−6.14698e70	−1.87125	−0.935627	−	0.352991i	$-0.885165\pi$
$409$	−6.14698e70	−1.87125	−0.935627	+	0.352991i	$0.885165\pi$
$410$	0	0
$411$	−5.17189e69	−0.138013
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.82114e70	0.509039
$418$	5.27914e70	0.892902
$419$	−1.07544e71	−1.70533	−0.852667	−	0.522455i	$-0.825016\pi$
$419$	−1.07544e71	−1.70533	−0.852667	+	0.522455i	$0.825016\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−6.63544e70	−0.867828
$423$	0	0
$424$	0	0
$425$	−1.85122e71	−1.99966
$426$	0	0
$427$	0	0
$428$	2.20202e71	1.96716
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.24683e71	−0.866458
$433$	−2.36301e71	−1.54274	−0.771369	−	0.636388i	$-0.780428\pi$
$433$	−2.36301e71	−1.54274	−0.771369	+	0.636388i	$0.780428\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	2.05729e71	0.985132
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.90013e71	−0.756733
$442$	0	0
$443$	−5.00824e71	−1.76516	−0.882580	−	0.470162i	$-0.844196\pi$
$443$	−5.00824e71	−1.76516	−0.882580	+	0.470162i	$0.844196\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.77026e71	1.16921	0.584603	−	0.811320i	$-0.301250\pi$
$449$	4.77026e71	1.16921	0.584603	+	0.811320i	$0.301250\pi$
$450$	3.27854e71	0.756733
$451$	−1.49728e71	−0.325489
$452$	4.30503e71	0.881534
$453$	0	0
$454$	−3.91516e71	−0.711609
$455$	0	0
$456$	6.09760e71	0.984256
$457$	9.69518e71	1.47509	0.737545	−	0.675298i	$-0.235985\pi$
$457$	9.69518e71	1.47509	0.737545	+	0.675298i	$0.235985\pi$
$458$	0	0
$459$	1.28129e72	1.73262
$460$	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
$466$	1.78596e72	1.60492
$467$	1.96255e72	1.66443	0.832213	−	0.554456i	$-0.187074\pi$
$467$	1.96255e72	1.66443	0.832213	+	0.554456i	$0.187074\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−2.98049e72	−1.89606
$473$	1.42404e72	0.855603
$474$	0	0
$475$	−3.72215e72	−1.99557
$476$	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
$482$	−5.43896e72	−1.96447
$483$	0	0
$484$	−2.47630e72	−0.799795
$485$	0	0
$486$	−3.56089e72	−1.02891
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.85587e72	0.454158
$490$	0	0
$491$	2.34650e72	0.514305	0.257152	−	0.966371i	$-0.417216\pi$
$491$	2.34650e72	0.514305	0.257152	+	0.966371i	$0.417216\pi$
$492$	−1.72942e72	−0.358791
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	3.51741e72	0.526055
$499$	9.48111e71	0.134321	0.0671606	−	0.997742i	$-0.478606\pi$
$499$	9.48111e71	0.134321	0.0671606	+	0.997742i	$0.478606\pi$
$500$	0	0
$501$	0	0
$502$	−1.07947e73	−1.30080
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.34881e72	0.493221
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.41348e73	−1.00000
$513$	2.57622e73	1.72908
$514$	−2.54791e73	−1.62248
$515$	0	0
$516$	1.64482e73	0.943142
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.03810e73	−1.34271	−0.671357	−	0.741134i	$-0.734288\pi$
$521$	−3.03810e73	−1.34271	−0.671357	+	0.741134i	$0.734288\pi$
$522$	0	0
$523$	4.50449e73	1.79515	0.897574	−	0.440864i	$-0.145328\pi$
$523$	4.50449e73	1.79515	0.897574	+	0.440864i	$0.145328\pi$
$524$	−8.79998e72	−0.333071
$525$	0	0
$526$	0	0
$527$	0	0
$528$	7.15971e72	0.220688
$529$	3.41431e73	1.00000
$530$	0	0
$531$	−5.42435e73	−1.43481
$532$	0	0
$533$	0	0
$534$	3.22148e73	0.731882
$535$	0	0
$536$	3.98204e73	0.817817
$537$	2.74045e73	0.535200
$538$	0	0
$539$	2.53299e73	0.447443
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.45254e74	1.99966
$545$	0	0
$546$	0	0
$547$	−1.53508e74	−1.82168	−0.910842	−	0.412756i	$-0.864566\pi$
$547$	−1.53508e74	−1.82168	−0.910842	+	0.412756i	$0.864566\pi$
$548$	−2.47716e73	−0.279820
$549$	0	0
$550$	−4.37049e73	−0.447443
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.35123e74	1.03207
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.35755e73	−0.441302
$562$	2.40907e74	1.37711
$563$	−2.70118e74	−1.47173	−0.735864	−	0.677130i	$-0.763224\pi$
$563$	−2.70118e74	−1.47173	−0.735864	+	0.677130i	$0.763224\pi$
$564$	0	0
$565$	0	0
$566$	2.99157e74	1.41207
$567$	0	0
$568$	0	0
$569$	9.14447e72	0.0374220	0.0187110	−	0.999825i	$-0.494044\pi$
$569$	9.14447e72	0.0374220	0.0187110	+	0.999825i	$0.494044\pi$
$570$	0	0
$571$	−4.11141e74	−1.53044	−0.765218	−	0.643771i	$-0.777369\pi$
$571$	−4.11141e74	−1.53044	−0.765218	+	0.643771i	$0.777369\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−2.57247e74	−0.756733
$577$	−3.33278e73	−0.0935534	−0.0467767	−	0.998905i	$-0.514895\pi$
$577$	−3.33278e73	−0.0935534	−0.0467767	+	0.998905i	$0.514895\pi$
$578$	−1.11938e75	−2.99865
$579$	−2.32760e74	−0.595096
$580$	0	0
$581$	0	0
$582$	−3.93537e74	−0.875114
$583$	0	0
$584$	9.85370e74	1.99734
$585$	0	0
$586$	0	0
$587$	−1.03438e75	−1.82579	−0.912895	−	0.408195i	$-0.866158\pi$
$587$	−1.03438e75	−1.82579	−0.912895	+	0.408195i	$0.866158\pi$
$588$	2.92570e74	0.493221
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.28850e73	0.0441084	0.0220542	−	0.999757i	$-0.492979\pi$
$593$	3.28850e73	0.0441084	0.0220542	+	0.999757i	$0.492979\pi$
$594$	3.02496e74	0.387690
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−5.04807e74	−0.493221
$601$	2.08182e75	1.94461	0.972306	−	0.233713i	$-0.0750878\pi$
$601$	2.08182e75	1.94461	0.972306	+	0.233713i	$0.0750878\pi$
$602$	0	0
$603$	7.24713e74	0.618869
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	2.92054e75	1.99557
$609$	0	0
$610$	0	0
$611$	0	0
$612$	2.64355e75	1.51321
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−3.05949e75	−1.60363
$615$	0	0
$616$	0	0
$617$	3.91201e75	1.79763	0.898817	−	0.438325i	$-0.144428\pi$
$617$	3.91201e75	1.79763	0.898817	+	0.438325i	$0.144428\pi$
$618$	0	0
$619$	7.38415e74	0.310923	0.155462	−	0.987842i	$-0.450314\pi$
$619$	7.38415e74	0.310923	0.155462	+	0.987842i	$0.450314\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.08149e75	1.00000
$626$	−6.25671e75	−1.94464
$627$	−1.47934e75	−0.440398
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	1.85941e75	0.428031
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.01441e76	1.66360	0.831798	−	0.555078i	$-0.187312\pi$
$641$	1.01441e76	1.66360	0.831798	+	0.555078i	$0.187312\pi$
$642$	−6.17060e75	−0.970245
$643$	−1.24742e76	−1.88069	−0.940343	−	0.340228i	$-0.889496\pi$
$643$	−1.24742e76	−1.88069	−0.940343	+	0.340228i	$0.889496\pi$
$644$	0	0
$645$	0	0
$646$	−3.00124e76	−3.99046
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.69289e75	−0.329377
$649$	7.23098e75	0.848380
$650$	0	0
$651$	0	0
$652$	8.88895e75	0.920800
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−8.28330e75	−0.727444
$657$	1.79333e76	1.51145
$658$	0	0
$659$	−2.26881e76	−1.76154	−0.880769	−	0.473545i	$-0.842974\pi$
$659$	−2.26881e76	−1.76154	−0.880769	+	0.473545i	$0.842974\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.57397e76	1.08101
$663$	0	0
$664$	1.68472e76	1.06657
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	2.26688e76	0.997752	0.498876	−	0.866673i	$-0.333746\pi$
$673$	2.26688e76	0.997752	0.498876	+	0.866673i	$0.333746\pi$
$674$	−3.06653e76	−1.29667
$675$	−2.13280e76	−0.866458
$676$	2.56189e76	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−1.20638e76	−0.434791
$679$	0	0
$680$	0	0
$681$	1.09712e76	0.350981
$682$	0	0
$683$	−6.71104e76	−1.98349	−0.991747	−	0.128212i	$-0.959076\pi$
$683$	−6.71104e76	−1.98349	−0.991747	+	0.128212i	$0.959076\pi$
$684$	5.31525e76	1.51011
$685$	0	0
$686$	0	0
$687$	0	0
$688$	7.87813e76	1.91221
$689$	0	0
$690$	0	0
$691$	−5.15773e76	−1.11314	−0.556572	−	0.830800i	$-0.687884\pi$
$691$	−5.15773e76	−1.11314	−0.556572	+	0.830800i	$0.687884\pi$
$692$	0	0
$693$	0	0
$694$	−8.03312e76	−1.54234
$695$	0	0
$696$	0	0
$697$	8.51219e76	1.45464
$698$	0	0
$699$	−5.00471e76	−0.791580
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	3.42925e76	0.447443
$705$	0	0
$706$	1.56120e77	1.88682
$707$	0	0
$708$	8.35205e76	0.935180
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.54297e77	1.48388
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.31258e77	1.08511
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−4.51912e77	−2.98229
$723$	1.52413e77	0.968920
$724$	0	0
$725$	0	0
$726$	6.93920e76	0.394476
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	3.50204e76	0.178106
$730$	0	0
$731$	−8.09583e77	−3.82377
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.66086e76	−0.365926
$738$	−1.50752e77	−0.550481
$739$	−1.96593e77	−0.692098	−0.346049	−	0.938216i	$-0.612477\pi$
$739$	−1.96593e77	−0.692098	−0.346049	+	0.938216i	$0.612477\pi$
$740$	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
$747$	3.06611e77	0.807108
$748$	−3.52401e77	−0.894734
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	3.02494e77	0.641582
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−8.60170e77	−1.52586
$759$	0	0
$760$	0	0
$761$	1.95901e77	0.312357	0.156179	−	0.987729i	$-0.450082\pi$
$761$	1.95901e77	0.312357	0.156179	+	0.987729i	$0.450082\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	3.96091e77	0.493221
$769$	1.66257e78	1.99879	0.999397	−	0.0347179i	$-0.0110533\pi$
$769$	1.66257e78	1.99879	0.999397	+	0.0347179i	$0.0110533\pi$
$770$	0	0
$771$	7.13985e77	0.800242
$772$	−1.11484e78	−1.20655
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.43378e78	1.44703
$775$	0	0
$776$	−1.88490e78	−1.77428
$777$	0	0
$778$	0	0
$779$	1.71150e78	1.45166
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.40131e78	1.00000
$785$	0	0
$786$	2.46597e77	0.164278
$787$	2.97134e78	1.91265	0.956324	−	0.292310i	$-0.0944238\pi$
$787$	2.97134e78	1.91265	0.956324	+	0.292310i	$0.0944238\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	6.24108e77	0.338594
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−2.41785e78	−1.00000
$801$	2.80814e78	1.12290
$802$	−1.36225e78	−0.526681
$803$	−2.39061e78	−0.893696
$804$	−1.11586e78	−0.403365
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.48214e78	−1.67626	−0.838128	−	0.545474i	$-0.816350\pi$
$809$	−5.48214e78	−1.67626	−0.838128	+	0.545474i	$0.816350\pi$
$810$	0	0
$811$	1.20084e78	0.343497	0.171748	−	0.985141i	$-0.445058\pi$
$811$	1.20084e78	0.343497	0.171748	+	0.985141i	$0.445058\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−4.07036e78	−0.986277
$817$	−1.62778e79	−3.81594
$818$	8.25034e78	1.87125
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	6.94159e77	0.138013
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	1.22472e78	0.220688
$826$	0	0
$827$	1.17526e79	1.98375	0.991874	−	0.127227i	$-0.0406076\pi$
$827$	1.17526e79	1.98375	0.991874	+	0.127227i	$0.0406076\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.44003e79	−1.99966
$834$	−3.78647e78	−0.509039
$835$	0	0
$836$	−7.08554e78	−0.892902
$837$	0	0
$838$	1.44344e79	1.70533
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	9.32163e78	1.00000
$842$	0	0
$843$	−6.75080e78	−0.679221
$844$	8.90594e78	0.867828
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.38312e78	−0.696463
$850$	2.48466e79	1.99966
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−2.95550e79	−1.96716
$857$	−2.62087e79	−1.69030	−0.845150	−	0.534529i	$-0.820489\pi$
$857$	−2.62087e79	−1.69030	−0.845150	+	0.534529i	$0.820489\pi$
$858$	0	0
$859$	3.15098e77	0.0190823	0.00954114	−	0.999954i	$-0.496963\pi$
$859$	3.15098e77	0.0190823	0.00954114	+	0.999954i	$0.496963\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.67347e79	0.866458
$865$	0	0
$866$	3.17158e79	1.54274
$867$	3.13678e79	1.47900
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.43044e79	−1.34266
$874$	0	0
$875$	0	0
$876$	−2.76125e79	−0.985132
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.92018e79	1.19938	0.599690	−	0.800233i	$-0.295291\pi$
$881$	3.92018e79	1.19938	0.599690	+	0.800233i	$0.295291\pi$
$882$	2.55032e79	0.756733
$883$	1.69778e79	0.488588	0.244294	−	0.969701i	$-0.421444\pi$
$883$	1.69778e79	0.488588	0.244294	+	0.969701i	$0.421444\pi$
$884$	0	0
$885$	0	0
$886$	6.72194e79	1.76516
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.53324e78	0.147377
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−6.40253e79	−1.16921
$899$	0	0
$900$	−4.40038e79	−0.756733
$901$	0	0
$902$	2.00962e79	0.325489
$903$	0	0
$904$	−5.77812e79	−0.881534
$905$	0	0
$906$	0	0
$907$	1.31024e80	1.82791	0.913955	−	0.405815i	$-0.133012\pi$
$907$	1.31024e80	1.82791	0.913955	+	0.405815i	$0.133012\pi$
$908$	5.25484e79	0.711609
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−8.18406e79	−0.984256
$913$	−4.08731e79	−0.477229
$914$	−1.30127e80	−1.47509
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.71972e80	−1.73262
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	8.57344e79	0.790943
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.65479e80	1.20870	0.604352	−	0.796718i	$-0.293432\pi$
$929$	1.65479e80	1.20870	0.604352	+	0.796718i	$0.293432\pi$
$930$	0	0
$931$	−2.89539e80	−1.99557
$932$	−2.39708e80	−1.60492
$933$	0	0
$934$	−2.63409e80	−1.66443
$935$	0	0
$936$	0	0
$937$	−3.20596e80	−1.85776	−0.928879	−	0.370384i	$-0.879226\pi$
$937$	−3.20596e80	−1.85776	−0.928879	+	0.370384i	$0.879226\pi$
$938$	0	0
$939$	1.75328e80	0.959138
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	4.00034e80	1.89606
$945$	0	0
$946$	−1.91132e80	−0.855603
$947$	3.26720e80	1.42143	0.710717	−	0.703478i	$-0.248371\pi$
$947$	3.26720e80	1.42143	0.710717	+	0.703478i	$0.248371\pi$
$948$	0	0
$949$	0	0
$950$	4.99578e80	1.99557
$951$	0	0
$952$	0	0
$953$	4.92643e80	1.80727	0.903636	−	0.428301i	$-0.140888\pi$
$953$	4.92643e80	1.80727	0.903636	+	0.428301i	$0.140888\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.41611e80	1.00000
$962$	0	0
$963$	−5.37888e80	−1.48861
$964$	7.30005e80	1.96447
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	3.32364e80	0.799795
$969$	8.41021e80	1.96818
$970$	0	0
$971$	−9.02495e80	−1.99768	−0.998839	−	0.0481683i	$-0.984662\pi$
$971$	−9.02495e80	−1.99768	−0.998839	+	0.0481683i	$0.984662\pi$
$972$	4.77934e80	1.02891
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.91263e80	−0.920791	−0.460395	−	0.887714i	$-0.652292\pi$
$977$	−4.91263e80	−0.920791	−0.460395	+	0.887714i	$0.652292\pi$
$978$	−2.49090e80	−0.454158
$979$	−3.74342e80	−0.663952
$980$	0	0
$981$	0	0
$982$	−3.14942e80	−0.514305
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	2.32118e80	0.358791
$985$	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
$993$	−4.41065e80	−0.533179
$994$	0	0
$995$	0	0
$996$	−4.72099e80	−0.526055
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.27253e80	−0.134321
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.55.d.a.3.1	✓	1
8.3	odd	2	CM	8.55.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.55.d.a.3.1	✓	1	1.1	even	1	trivial
8.55.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:	\( k \)	\(=\)	\( 55 \)
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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