LMFDB - Embedded newform 1280.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(1,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.a (trivial)

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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} -2.82843 q^{11} +6.00000 q^{13} +1.41421 q^{15} -6.00000 q^{17} +6.00000 q^{21} -4.24264 q^{23} +1.00000 q^{25} +5.65685 q^{27} +8.48528 q^{31} +4.00000 q^{33} +4.24264 q^{35} +6.00000 q^{37} -8.48528 q^{39} -4.24264 q^{43} +1.00000 q^{45} -4.24264 q^{47} +11.0000 q^{49} +8.48528 q^{51} +6.00000 q^{53} +2.82843 q^{55} +11.3137 q^{59} +6.00000 q^{61} +4.24264 q^{63} -6.00000 q^{65} -12.7279 q^{67} +6.00000 q^{69} +8.48528 q^{71} +2.00000 q^{73} -1.41421 q^{75} +12.0000 q^{77} -5.00000 q^{81} -9.89949 q^{83} +6.00000 q^{85} +6.00000 q^{89} -25.4558 q^{91} -12.0000 q^{93} -10.0000 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^9	$ 2 q - 2 q^{5} - 2 q^{9} + 12 q^{13} - 12 q^{17} + 12 q^{21} + 2 q^{25} + 8 q^{33} + 12 q^{37} + 2 q^{45} + 22 q^{49} + 12 q^{53} + 12 q^{61} - 12 q^{65} + 12 q^{69} + 4 q^{73} + 24 q^{77} - 10 q^{81} + 12 q^{85} + 12 q^{89} - 24 q^{93} - 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 12 * q^13 - 12 * q^17 + 12 * q^21 + 2 * q^25 + 8 * q^33 + 12 * q^37 + 2 * q^45 + 22 * q^49 + 12 * q^53 + 12 * q^61 - 12 * q^65 + 12 * q^69 + 4 * q^73 + 24 * q^77 - 10 * q^81 + 12 * q^85 + 12 * q^89 - 24 * q^93 - 20 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	−4.24264	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	4.24264	0.717137
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−8.48528	−1.35873
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−4.24264	−0.646997	−0.323498	−	0.946229i	$-0.604859\pi$
$43$	−4.24264	−0.646997	−0.323498	+	0.946229i	$0.604859\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	8.48528	1.18818
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	−12.7279	−1.55496	−0.777482	−	0.628906i	$-0.783503\pi$
$67$	−12.7279	−1.55496	−0.777482	+	0.628906i	$0.783503\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−1.41421	−0.163299
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−9.89949	−1.08661	−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949	−1.08661	−0.543305	+	0.839535i	$0.682827\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−25.4558	−2.66850
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	12.7279	1.25412	0.627060	−	0.778971i	$-0.284258\pi$
$103$	12.7279	1.25412	0.627060	+	0.778971i	$0.284258\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	7.07107	0.683586	0.341793	−	0.939775i	$-0.388966\pi$
$107$	7.07107	0.683586	0.341793	+	0.939775i	$0.388966\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−8.48528	−0.805387
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.24264	0.395628
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	25.4558	2.33353
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	14.1421	1.23560	0.617802	−	0.786334i	$-0.288023\pi$
$131$	14.1421	1.23560	0.617802	+	0.786334i	$0.288023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.65685	−0.486864
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	−16.9706	−1.41915
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.5563	−1.28307
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−8.48528	−0.681554
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	−8.48528	−0.672927
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	4.24264	0.332309	0.166155	−	0.986100i	$-0.446865\pi$
$163$	4.24264	0.332309	0.166155	+	0.986100i	$0.446865\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	21.2132	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$167$	21.2132	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.24264	−0.320713
$176$	0	0
$177$	−16.0000	−1.20263
$178$	0	0
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	−8.48528	−0.627250
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	16.9706	1.24101
$188$	0	0
$189$	−24.0000	−1.74574
$190$	0	0
$191$	−25.4558	−1.84192	−0.920960	−	0.389657i	$-0.872594\pi$
$191$	−25.4558	−1.84192	−0.920960	+	0.389657i	$0.872594\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	8.48528	0.607644
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	18.0000	1.26962
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.24264	0.294884
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.48528	0.584151	0.292075	−	0.956395i	$-0.405654\pi$
$211$	8.48528	0.584151	0.292075	+	0.956395i	$0.405654\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	4.24264	0.289346
$216$	0	0
$217$	−36.0000	−2.44384
$218$	0	0
$219$	−2.82843	−0.191127
$220$	0	0
$221$	−36.0000	−2.42162
$222$	0	0
$223$	12.7279	0.852325	0.426162	−	0.904647i	$-0.359865\pi$
$223$	12.7279	0.852325	0.426162	+	0.904647i	$0.359865\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−9.89949	−0.657053	−0.328526	−	0.944495i	$-0.606552\pi$
$227$	−9.89949	−0.657053	−0.328526	+	0.944495i	$0.606552\pi$
$228$	0	0
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	−16.9706	−1.11658
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	4.24264	0.276759
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−11.0000	−0.702764
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	−8.48528	−0.531369
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−25.4558	−1.58175
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.24264	0.261612	0.130806	−	0.991408i	$-0.458243\pi$
$263$	4.24264	0.261612	0.130806	+	0.991408i	$0.458243\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−8.48528	−0.519291
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	36.0000	2.17882
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	−8.48528	−0.508001
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	4.24264	0.252199	0.126099	−	0.992018i	$-0.459754\pi$
$283$	4.24264	0.252199	0.126099	+	0.992018i	$0.459754\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	14.1421	0.829027
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−11.3137	−0.658710
$296$	0	0
$297$	−16.0000	−0.928414
$298$	0	0
$299$	−25.4558	−1.47215
$300$	0	0
$301$	18.0000	1.03750
$302$	0	0
$303$	16.9706	0.974933
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	21.2132	1.21070	0.605351	−	0.795959i	$-0.293033\pi$
$307$	21.2132	1.21070	0.605351	+	0.795959i	$0.293033\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	0	0
$311$	−8.48528	−0.481156	−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528	−0.481156	−0.240578	+	0.970630i	$0.577337\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	−4.24264	−0.239046
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	−25.4558	−1.40771
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	12.7279	0.695401
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−8.48528	−0.460857
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	26.8701	1.44246	0.721230	−	0.692696i	$-0.243577\pi$
$347$	26.8701	1.44246	0.721230	+	0.692696i	$0.243577\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	33.9411	1.81164
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	−8.48528	−0.450352
$356$	0	0
$357$	−36.0000	−1.90532
$358$	0	0
$359$	16.9706	0.895672	0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706	0.895672	0.447836	+	0.894116i	$0.352195\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	4.24264	0.222681
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.4558	−1.32160
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	1.41421	0.0730297
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.9706	0.871719	0.435860	−	0.900015i	$-0.356444\pi$
$379$	16.9706	0.871719	0.435860	+	0.900015i	$0.356444\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	0	0
$383$	−21.2132	−1.08394	−0.541972	−	0.840397i	$-0.682322\pi$
$383$	−21.2132	−1.08394	−0.541972	+	0.840397i	$0.682322\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	4.24264	0.215666
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	25.4558	1.28736
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	50.9117	2.53609
$404$	0	0
$405$	5.00000	0.248452
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	25.4558	1.25564
$412$	0	0
$413$	−48.0000	−2.36193
$414$	0	0
$415$	9.89949	0.485947
$416$	0	0
$417$	−24.0000	−1.17529
$418$	0	0
$419$	11.3137	0.552711	0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137	0.552711	0.276355	+	0.961056i	$0.410873\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	4.24264	0.206284
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−25.4558	−1.23189
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	8.48528	0.408722	0.204361	−	0.978896i	$-0.434488\pi$
$431$	8.48528	0.408722	0.204361	+	0.978896i	$0.434488\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	9.89949	0.470339	0.235170	−	0.971954i	$-0.424435\pi$
$443$	9.89949	0.470339	0.235170	+	0.971954i	$0.424435\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	−8.48528	−0.401340
$448$	0	0
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	25.4558	1.19339
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	−33.9411	−1.58424
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	21.2132	0.985861	0.492931	−	0.870069i	$-0.335926\pi$
$463$	21.2132	0.985861	0.492931	+	0.870069i	$0.335926\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	7.07107	0.327210	0.163605	−	0.986526i	$-0.447688\pi$
$467$	7.07107	0.327210	0.163605	+	0.986526i	$0.447688\pi$
$468$	0	0
$469$	54.0000	2.49349
$470$	0	0
$471$	25.4558	1.17294
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	−25.4558	−1.15828
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	12.7279	0.576757	0.288379	−	0.957516i	$-0.406884\pi$
$487$	12.7279	0.576757	0.288379	+	0.957516i	$0.406884\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	36.7696	1.65939	0.829693	−	0.558219i	$-0.188515\pi$
$491$	36.7696	1.65939	0.829693	+	0.558219i	$0.188515\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.82843	−0.127128
$496$	0	0
$497$	−36.0000	−1.61482
$498$	0	0
$499$	−16.9706	−0.759707	−0.379853	−	0.925047i	$-0.624026\pi$
$499$	−16.9706	−0.759707	−0.379853	+	0.925047i	$0.624026\pi$
$500$	0	0
$501$	−30.0000	−1.34030
$502$	0	0
$503$	12.7279	0.567510	0.283755	−	0.958897i	$-0.408420\pi$
$503$	12.7279	0.567510	0.283755	+	0.958897i	$0.408420\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	−32.5269	−1.44457
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	−8.48528	−0.375367
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.7279	−0.560859
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	−8.48528	−0.372463
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	4.24264	0.185518	0.0927589	−	0.995689i	$-0.470431\pi$
$523$	4.24264	0.185518	0.0927589	+	0.995689i	$0.470431\pi$
$524$	0	0
$525$	6.00000	0.261861
$526$	0	0
$527$	−50.9117	−2.21775
$528$	0	0
$529$	−5.00000	−0.217391
$530$	0	0
$531$	−11.3137	−0.490973
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−7.07107	−0.305709
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	−31.1127	−1.34012
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	0	0
$543$	−16.9706	−0.728277
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	12.7279	0.544207	0.272103	−	0.962268i	$-0.412281\pi$
$547$	12.7279	0.544207	0.272103	+	0.962268i	$0.412281\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.48528	0.360180
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−25.4558	−1.07667
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	−1.41421	−0.0596020	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$	−1.41421	−0.0596020	−0.0298010	+	0.999556i	$0.509487\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	21.2132	0.890871
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−8.48528	−0.355098	−0.177549	−	0.984112i	$-0.556817\pi$
$571$	−8.48528	−0.355098	−0.177549	+	0.984112i	$0.556817\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	−4.24264	−0.176930
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	19.7990	0.822818
$580$	0	0
$581$	42.0000	1.74245
$582$	0	0
$583$	−16.9706	−0.702849
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	−35.3553	−1.45927	−0.729636	−	0.683836i	$-0.760310\pi$
$587$	−35.3553	−1.45927	−0.729636	+	0.683836i	$0.760310\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	8.48528	0.349038
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−25.4558	−1.04359
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	12.7279	0.518321
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	−38.1838	−1.54983	−0.774916	−	0.632065i	$-0.782208\pi$
$607$	−38.1838	−1.54983	−0.774916	+	0.632065i	$0.782208\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25.4558	−1.02983
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	16.9706	0.682105	0.341052	−	0.940044i	$-0.389217\pi$
$619$	16.9706	0.682105	0.341052	+	0.940044i	$0.389217\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	0	0
$623$	−25.4558	−1.01987
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−36.0000	−1.43541
$630$	0	0
$631$	8.48528	0.337794	0.168897	−	0.985634i	$-0.445980\pi$
$631$	8.48528	0.337794	0.168897	+	0.985634i	$0.445980\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	−4.24264	−0.168364
$636$	0	0
$637$	66.0000	2.61502
$638$	0	0
$639$	−8.48528	−0.335673
$640$	0	0
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	29.6985	1.17119	0.585597	−	0.810602i	$-0.300860\pi$
$643$	29.6985	1.17119	0.585597	+	0.810602i	$0.300860\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−21.2132	−0.833977	−0.416989	−	0.908912i	$-0.636914\pi$
$647$	−21.2132	−0.833977	−0.416989	+	0.908912i	$0.636914\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	50.9117	1.99539
$652$	0	0
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	−14.1421	−0.552579
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	22.6274	0.881439	0.440720	−	0.897645i	$-0.354723\pi$
$659$	22.6274	0.881439	0.440720	+	0.897645i	$0.354723\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	50.9117	1.97725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−18.0000	−0.695920
$670$	0	0
$671$	−16.9706	−0.655141
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	42.4264	1.62818
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	18.3848	0.703474	0.351737	−	0.936099i	$-0.385591\pi$
$683$	18.3848	0.703474	0.351737	+	0.936099i	$0.385591\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	33.9411	1.29493
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	−42.4264	−1.61398	−0.806988	−	0.590567i	$-0.798904\pi$
$691$	−42.4264	−1.61398	−0.806988	+	0.590567i	$0.798904\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	−16.9706	−0.643730
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−8.48528	−0.320943
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	50.9117	1.91473
$708$	0	0
$709$	24.0000	0.901339	0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000	0.901339	0.450669	+	0.892691i	$0.351185\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.0000	−1.34821
$714$	0	0
$715$	16.9706	0.634663
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	−54.0000	−2.01107
$722$	0	0
$723$	11.3137	0.420761
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.24264	−0.157351	−0.0786754	−	0.996900i	$-0.525069\pi$
$727$	−4.24264	−0.157351	−0.0786754	+	0.996900i	$0.525069\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	25.4558	0.941518
$732$	0	0
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	0	0
$735$	15.5563	0.573805
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	−16.9706	−0.624272	−0.312136	−	0.950037i	$-0.601045\pi$
$739$	−16.9706	−0.624272	−0.312136	+	0.950037i	$0.601045\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−46.6690	−1.71212	−0.856061	−	0.516875i	$-0.827095\pi$
$743$	−46.6690	−1.71212	−0.856061	+	0.516875i	$0.827095\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	9.89949	0.362204
$748$	0	0
$749$	−30.0000	−1.09618
$750$	0	0
$751$	−42.4264	−1.54816	−0.774081	−	0.633087i	$-0.781788\pi$
$751$	−42.4264	−1.54816	−0.774081	+	0.633087i	$0.781788\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	8.48528	0.308811
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	0	0
$759$	−16.9706	−0.615992
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−76.3675	−2.76469
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	67.8823	2.45109
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	−8.48528	−0.305590
$772$	0	0
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	8.48528	0.304800
$776$	0	0
$777$	36.0000	1.29149
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	38.1838	1.36110	0.680552	−	0.732700i	$-0.261740\pi$
$787$	38.1838	1.36110	0.680552	+	0.732700i	$0.261740\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−25.4558	−0.905106
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	8.48528	0.300942
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	25.4558	0.900563
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−5.65685	−0.199626
$804$	0	0
$805$	−18.0000	−0.634417
$806$	0	0
$807$	25.4558	0.896088
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−8.48528	−0.297959	−0.148979	−	0.988840i	$-0.547599\pi$
$811$	−8.48528	−0.297959	−0.148979	+	0.988840i	$0.547599\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	−4.24264	−0.148613
$816$	0	0
$817$	0	0
$818$	0	0
$819$	25.4558	0.889499
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	29.6985	1.03522	0.517612	−	0.855615i	$-0.326821\pi$
$823$	29.6985	1.03522	0.517612	+	0.855615i	$0.326821\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−18.3848	−0.639301	−0.319651	−	0.947535i	$-0.603566\pi$
$827$	−18.3848	−0.639301	−0.319651	+	0.947535i	$0.603566\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	−25.4558	−0.883053
$832$	0	0
$833$	−66.0000	−2.28676
$834$	0	0
$835$	−21.2132	−0.734113
$836$	0	0
$837$	48.0000	1.65912
$838$	0	0
$839$	16.9706	0.585889	0.292944	−	0.956129i	$-0.405365\pi$
$839$	16.9706	0.585889	0.292944	+	0.956129i	$0.405365\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23.0000	−0.791224
$846$	0	0
$847$	12.7279	0.437337
$848$	0	0
$849$	−6.00000	−0.205919
$850$	0	0
$851$	−25.4558	−0.872615
$852$	0	0
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.1838	1.29979	0.649895	−	0.760024i	$-0.274813\pi$
$863$	38.1838	1.29979	0.649895	+	0.760024i	$0.274813\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−26.8701	−0.912555
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−76.3675	−2.58762
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	4.24264	0.143427
$876$	0	0
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	0	0
$879$	25.4558	0.858604
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−46.6690	−1.57054	−0.785269	−	0.619154i	$-0.787475\pi$
$883$	−46.6690	−1.57054	−0.785269	+	0.619154i	$0.787475\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	29.6985	0.997178	0.498589	−	0.866839i	$-0.333852\pi$
$887$	29.6985	0.997178	0.498589	+	0.866839i	$0.333852\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	14.1421	0.473779
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−5.65685	−0.189088
$896$	0	0
$897$	36.0000	1.20201
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	−25.4558	−0.847117
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−21.2132	−0.704373	−0.352186	−	0.935930i	$-0.614562\pi$
$907$	−21.2132	−0.704373	−0.352186	+	0.935930i	$0.614562\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	42.4264	1.40565	0.702825	−	0.711363i	$-0.251922\pi$
$911$	42.4264	1.40565	0.702825	+	0.711363i	$0.251922\pi$
$912$	0	0
$913$	28.0000	0.926665
$914$	0	0
$915$	8.48528	0.280515
$916$	0	0
$917$	−60.0000	−1.98137
$918$	0	0
$919$	−33.9411	−1.11961	−0.559807	−	0.828623i	$-0.689125\pi$
$919$	−33.9411	−1.11961	−0.559807	+	0.828623i	$0.689125\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	0	0
$923$	50.9117	1.67578
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−12.7279	−0.418040
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	−16.9706	−0.554997
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−36.7696	−1.19993
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	24.0000	0.780720
$946$	0	0
$947$	9.89949	0.321690	0.160845	−	0.986980i	$-0.448578\pi$
$947$	9.89949	0.321690	0.160845	+	0.986980i	$0.448578\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	8.48528	0.275154
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	25.4558	0.823732
$956$	0	0
$957$	0	0
$958$	0	0
$959$	76.3675	2.46604
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−7.07107	−0.227862
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	4.24264	0.136434	0.0682171	−	0.997671i	$-0.478269\pi$
$967$	4.24264	0.136434	0.0682171	+	0.997671i	$0.478269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.7990	−0.635380	−0.317690	−	0.948195i	$-0.602907\pi$
$971$	−19.7990	−0.635380	−0.317690	+	0.948195i	$0.602907\pi$
$972$	0	0
$973$	−72.0000	−2.30821
$974$	0	0
$975$	−8.48528	−0.271746
$976$	0	0
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	−16.9706	−0.542382
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	−29.6985	−0.947235	−0.473617	−	0.880731i	$-0.657052\pi$
$983$	−29.6985	−0.947235	−0.473617	+	0.880731i	$0.657052\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−25.4558	−0.810268
$988$	0	0
$989$	18.0000	0.572367
$990$	0	0
$991$	42.4264	1.34772	0.673860	−	0.738859i	$-0.264635\pi$
$991$	42.4264	1.34772	0.673860	+	0.738859i	$0.264635\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	16.9706	0.538003
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	33.9411	1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.2.a.g.1.1		2
4.3	odd	2	inner	1280.2.a.g.1.2		2
5.4	even	2		6400.2.a.bk.1.2		2
8.3	odd	2		1280.2.a.i.1.1		2
8.5	even	2		1280.2.a.i.1.2		2
16.3	odd	4		640.2.d.b.321.4	yes	4
16.5	even	4		640.2.d.b.321.3	yes	4
16.11	odd	4		640.2.d.b.321.1	✓	4
16.13	even	4		640.2.d.b.321.2	yes	4
20.19	odd	2		6400.2.a.bk.1.1		2
40.19	odd	2		6400.2.a.bp.1.2		2
40.29	even	2		6400.2.a.bp.1.1		2
48.5	odd	4		5760.2.k.v.2881.4		4
48.11	even	4		5760.2.k.v.2881.3		4
48.29	odd	4		5760.2.k.v.2881.2		4
48.35	even	4		5760.2.k.v.2881.1		4
80.3	even	4		3200.2.f.g.449.2		4
80.13	odd	4		3200.2.f.g.449.3		4
80.19	odd	4		3200.2.d.v.1601.2		4
80.27	even	4		3200.2.f.g.449.1		4
80.29	even	4		3200.2.d.v.1601.3		4
80.37	odd	4		3200.2.f.g.449.4		4
80.43	even	4		3200.2.f.l.449.4		4
80.53	odd	4		3200.2.f.l.449.1		4
80.59	odd	4		3200.2.d.v.1601.4		4
80.67	even	4		3200.2.f.l.449.3		4
80.69	even	4		3200.2.d.v.1601.1		4
80.77	odd	4		3200.2.f.l.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.b.321.1	✓	4	16.11	odd	4
640.2.d.b.321.2	yes	4	16.13	even	4
640.2.d.b.321.3	yes	4	16.5	even	4
640.2.d.b.321.4	yes	4	16.3	odd	4
1280.2.a.g.1.1		2	1.1	even	1	trivial
1280.2.a.g.1.2		2	4.3	odd	2	inner
1280.2.a.i.1.1		2	8.3	odd	2
1280.2.a.i.1.2		2	8.5	even	2
3200.2.d.v.1601.1		4	80.69	even	4
3200.2.d.v.1601.2		4	80.19	odd	4
3200.2.d.v.1601.3		4	80.29	even	4
3200.2.d.v.1601.4		4	80.59	odd	4
3200.2.f.g.449.1		4	80.27	even	4
3200.2.f.g.449.2		4	80.3	even	4
3200.2.f.g.449.3		4	80.13	odd	4
3200.2.f.g.449.4		4	80.37	odd	4
3200.2.f.l.449.1		4	80.53	odd	4
3200.2.f.l.449.2		4	80.77	odd	4
3200.2.f.l.449.3		4	80.67	even	4
3200.2.f.l.449.4		4	80.43	even	4
5760.2.k.v.2881.1		4	48.35	even	4
5760.2.k.v.2881.2		4	48.29	odd	4
5760.2.k.v.2881.3		4	48.11	even	4
5760.2.k.v.2881.4		4	48.5	odd	4
6400.2.a.bk.1.1		2	20.19	odd	2
6400.2.a.bk.1.2		2	5.4	even	2
6400.2.a.bp.1.1		2	40.29	even	2
6400.2.a.bp.1.2		2	40.19	odd	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} -2.82843 q^{11} +6.00000 q^{13} +1.41421 q^{15} -6.00000 q^{17} +6.00000 q^{21} -4.24264 q^{23} +1.00000 q^{25} +5.65685 q^{27} +8.48528 q^{31} +4.00000 q^{33} +4.24264 q^{35} +6.00000 q^{37} -8.48528 q^{39} -4.24264 q^{43} +1.00000 q^{45} -4.24264 q^{47} +11.0000 q^{49} +8.48528 q^{51} +6.00000 q^{53} +2.82843 q^{55} +11.3137 q^{59} +6.00000 q^{61} +4.24264 q^{63} -6.00000 q^{65} -12.7279 q^{67} +6.00000 q^{69} +8.48528 q^{71} +2.00000 q^{73} -1.41421 q^{75} +12.0000 q^{77} -5.00000 q^{81} -9.89949 q^{83} +6.00000 q^{85} +6.00000 q^{89} -25.4558 q^{91} -12.0000 q^{93} -10.0000 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{9} + 12 q^{13} - 12 q^{17} + 12 q^{21} + 2 q^{25} + 8 q^{33} + 12 q^{37} + 2 q^{45} + 22 q^{49} + 12 q^{53} + 12 q^{61} - 12 q^{65} + 12 q^{69} + 4 q^{73} + 24 q^{77} - 10 q^{81} + 12 q^{85} + 12 q^{89} - 24 q^{93} - 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 12 * q^13 - 12 * q^17 + 12 * q^21 + 2 * q^25 + 8 * q^33 + 12 * q^37 + 2 * q^45 + 22 * q^49 + 12 * q^53 + 12 * q^61 - 12 * q^65 + 12 * q^69 + 4 * q^73 + 24 * q^77 - 10 * q^81 + 12 * q^85 + 12 * q^89 - 24 * q^93 - 20 * q^97

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