LMFDB - Embedded newform 8280.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.00000 q^{7} +6.24264 q^{11} -2.58579 q^{13} +1.58579 q^{17} +7.07107 q^{19} +1.00000 q^{23} +1.00000 q^{25} +7.24264 q^{29} +0.656854 q^{31} +1.00000 q^{35} -3.00000 q^{37} +4.41421 q^{41} +2.00000 q^{43} -2.58579 q^{47} -6.00000 q^{49} +3.58579 q^{53} -6.24264 q^{55} -2.07107 q^{59} -13.0711 q^{61} +2.58579 q^{65} -0.656854 q^{67} +12.0711 q^{71} -15.5563 q^{73} -6.24264 q^{77} -5.65685 q^{79} -1.92893 q^{83} -1.58579 q^{85} +5.17157 q^{89} +2.58579 q^{91} -7.07107 q^{95} -6.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{23} + 2 q^{25} + 6 q^{29} - 10 q^{31} + 2 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 8 q^{47} - 12 q^{49} + 10 q^{53} - 4 q^{55} + 10 q^{59} - 12 q^{61} + 8 q^{65} + 10 q^{67} + 10 q^{71} - 4 q^{77} - 18 q^{83} - 6 q^{85} + 16 q^{89} + 8 q^{91} - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^23 + 2 * q^25 + 6 * q^29 - 10 * q^31 + 2 * q^35 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 8 * q^47 - 12 * q^49 + 10 * q^53 - 4 * q^55 + 10 * q^59 - 12 * q^61 + 8 * q^65 + 10 * q^67 + 10 * q^71 - 4 * q^77 - 18 * q^83 - 6 * q^85 + 16 * q^89 + 8 * q^91 - 8 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.24264	1.88223	0.941113	−	0.338091i	$-0.109781\pi$
$11$	6.24264	1.88223	0.941113	+	0.338091i	$0.109781\pi$
$12$	0	0
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.58579	0.384610	0.192305	−	0.981335i	$-0.438404\pi$
$17$	1.58579	0.384610	0.192305	+	0.981335i	$0.438404\pi$
$18$	0	0
$19$	7.07107	1.62221	0.811107	−	0.584898i	$-0.198865\pi$
$19$	7.07107	1.62221	0.811107	+	0.584898i	$0.198865\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.24264	1.34492	0.672462	−	0.740131i	$-0.265237\pi$
$29$	7.24264	1.34492	0.672462	+	0.740131i	$0.265237\pi$
$30$	0	0
$31$	0.656854	0.117975	0.0589873	−	0.998259i	$-0.481213\pi$
$31$	0.656854	0.117975	0.0589873	+	0.998259i	$0.481213\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.41421	0.689384	0.344692	−	0.938716i	$-0.387983\pi$
$41$	4.41421	0.689384	0.344692	+	0.938716i	$0.387983\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.58579	−0.377176	−0.188588	−	0.982056i	$-0.560391\pi$
$47$	−2.58579	−0.377176	−0.188588	+	0.982056i	$0.560391\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.58579	0.492546	0.246273	−	0.969201i	$-0.420794\pi$
$53$	3.58579	0.492546	0.246273	+	0.969201i	$0.420794\pi$
$54$	0	0
$55$	−6.24264	−0.841757
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.07107	−0.269630	−0.134815	−	0.990871i	$-0.543044\pi$
$59$	−2.07107	−0.269630	−0.134815	+	0.990871i	$0.543044\pi$
$60$	0	0
$61$	−13.0711	−1.67358	−0.836789	−	0.547525i	$-0.815570\pi$
$61$	−13.0711	−1.67358	−0.836789	+	0.547525i	$0.815570\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.58579	0.320727
$66$	0	0
$67$	−0.656854	−0.0802475	−0.0401238	−	0.999195i	$-0.512775\pi$
$67$	−0.656854	−0.0802475	−0.0401238	+	0.999195i	$0.512775\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0711	1.43257	0.716286	−	0.697807i	$-0.245841\pi$
$71$	12.0711	1.43257	0.716286	+	0.697807i	$0.245841\pi$
$72$	0	0
$73$	−15.5563	−1.82073	−0.910366	−	0.413803i	$-0.864200\pi$
$73$	−15.5563	−1.82073	−0.910366	+	0.413803i	$0.864200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.24264	−0.711415
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.92893	−0.211728	−0.105864	−	0.994381i	$-0.533761\pi$
$83$	−1.92893	−0.211728	−0.105864	+	0.994381i	$0.533761\pi$
$84$	0	0
$85$	−1.58579	−0.172003
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.17157	0.548186	0.274093	−	0.961703i	$-0.411622\pi$
$89$	5.17157	0.548186	0.274093	+	0.961703i	$0.411622\pi$
$90$	0	0
$91$	2.58579	0.271064
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.07107	−0.725476
$96$	0	0
$97$	−6.82843	−0.693322	−0.346661	−	0.937991i	$-0.612685\pi$
$97$	−6.82843	−0.693322	−0.346661	+	0.937991i	$0.612685\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.2426	−1.11868	−0.559342	−	0.828937i	$-0.688946\pi$
$101$	−11.2426	−1.11868	−0.559342	+	0.828937i	$0.688946\pi$
$102$	0	0
$103$	6.48528	0.639014	0.319507	−	0.947584i	$-0.396483\pi$
$103$	6.48528	0.639014	0.319507	+	0.947584i	$0.396483\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.24264	0.506825	0.253413	−	0.967358i	$-0.418447\pi$
$107$	5.24264	0.506825	0.253413	+	0.967358i	$0.418447\pi$
$108$	0	0
$109$	18.3848	1.76094	0.880471	−	0.474100i	$-0.157226\pi$
$109$	18.3848	1.76094	0.880471	+	0.474100i	$0.157226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.24264	0.869474	0.434737	−	0.900557i	$-0.356841\pi$
$113$	9.24264	0.869474	0.434737	+	0.900557i	$0.356841\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.58579	−0.145369
$120$	0	0
$121$	27.9706	2.54278
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.58579	0.761865	0.380933	−	0.924603i	$-0.375603\pi$
$127$	8.58579	0.761865	0.380933	+	0.924603i	$0.375603\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	−7.07107	−0.613139
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.4853	−1.40843	−0.704216	−	0.709985i	$-0.748701\pi$
$137$	−16.4853	−1.40843	−0.704216	+	0.709985i	$0.748701\pi$
$138$	0	0
$139$	19.4853	1.65272	0.826360	−	0.563142i	$-0.190408\pi$
$139$	19.4853	1.65272	0.826360	+	0.563142i	$0.190408\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.1421	−1.34987
$144$	0	0
$145$	−7.24264	−0.601469
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.4142	1.09894	0.549468	−	0.835515i	$-0.314831\pi$
$149$	13.4142	1.09894	0.549468	+	0.835515i	$0.314831\pi$
$150$	0	0
$151$	−11.6569	−0.948621	−0.474311	−	0.880358i	$-0.657303\pi$
$151$	−11.6569	−0.948621	−0.474311	+	0.880358i	$0.657303\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.656854	−0.0527598
$156$	0	0
$157$	−15.4853	−1.23586	−0.617930	−	0.786233i	$-0.712028\pi$
$157$	−15.4853	−1.23586	−0.617930	+	0.786233i	$0.712028\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−3.65685	−0.286427	−0.143213	−	0.989692i	$-0.545744\pi$
$163$	−3.65685	−0.286427	−0.143213	+	0.989692i	$0.545744\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.07107	−0.237646	−0.118823	−	0.992915i	$-0.537912\pi$
$167$	−3.07107	−0.237646	−0.118823	+	0.992915i	$0.537912\pi$
$168$	0	0
$169$	−6.31371	−0.485670
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.3137	−0.860165	−0.430083	−	0.902790i	$-0.641516\pi$
$173$	−11.3137	−0.860165	−0.430083	+	0.902790i	$0.641516\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.6569	1.76820	0.884098	−	0.467301i	$-0.154774\pi$
$179$	23.6569	1.76820	0.884098	+	0.467301i	$0.154774\pi$
$180$	0	0
$181$	22.1421	1.64581	0.822906	−	0.568178i	$-0.192351\pi$
$181$	22.1421	1.64581	0.822906	+	0.568178i	$0.192351\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	9.89949	0.723923
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.89949	−0.426872	−0.213436	−	0.976957i	$-0.568466\pi$
$191$	−5.89949	−0.426872	−0.213436	+	0.976957i	$0.568466\pi$
$192$	0	0
$193$	24.4853	1.76249	0.881245	−	0.472661i	$-0.156706\pi$
$193$	24.4853	1.76249	0.881245	+	0.472661i	$0.156706\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.7990	1.83810	0.919051	−	0.394139i	$-0.128957\pi$
$197$	25.7990	1.83810	0.919051	+	0.394139i	$0.128957\pi$
$198$	0	0
$199$	−27.4558	−1.94629	−0.973147	−	0.230186i	$-0.926066\pi$
$199$	−27.4558	−1.94629	−0.973147	+	0.230186i	$0.926066\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.24264	−0.508334
$204$	0	0
$205$	−4.41421	−0.308302
$206$	0	0
$207$	0	0
$208$	0	0
$209$	44.1421	3.05338
$210$	0	0
$211$	15.8284	1.08967	0.544837	−	0.838542i	$-0.316592\pi$
$211$	15.8284	1.08967	0.544837	+	0.838542i	$0.316592\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−0.656854	−0.0445902
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.10051	−0.275830
$222$	0	0
$223$	6.97056	0.466783	0.233392	−	0.972383i	$-0.425018\pi$
$223$	6.97056	0.466783	0.233392	+	0.972383i	$0.425018\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	1.17157	0.0774197	0.0387099	−	0.999250i	$-0.487675\pi$
$229$	1.17157	0.0774197	0.0387099	+	0.999250i	$0.487675\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.6569	1.94289	0.971443	−	0.237275i	$-0.0762542\pi$
$233$	29.6569	1.94289	0.971443	+	0.237275i	$0.0762542\pi$
$234$	0	0
$235$	2.58579	0.168678
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.07107	−0.522074	−0.261037	−	0.965329i	$-0.584064\pi$
$239$	−8.07107	−0.522074	−0.261037	+	0.965329i	$0.584064\pi$
$240$	0	0
$241$	12.2426	0.788618	0.394309	−	0.918978i	$-0.370984\pi$
$241$	12.2426	0.788618	0.394309	+	0.918978i	$0.370984\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−18.2843	−1.16340
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.6569	−0.988252	−0.494126	−	0.869390i	$-0.664512\pi$
$251$	−15.6569	−0.988252	−0.494126	+	0.869390i	$0.664512\pi$
$252$	0	0
$253$	6.24264	0.392471
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.7279	−0.793946	−0.396973	−	0.917830i	$-0.629939\pi$
$257$	−12.7279	−0.793946	−0.396973	+	0.917830i	$0.629939\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.27208	−0.263428	−0.131714	−	0.991288i	$-0.542048\pi$
$263$	−4.27208	−0.263428	−0.131714	+	0.991288i	$0.542048\pi$
$264$	0	0
$265$	−3.58579	−0.220273
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.5858	0.950282	0.475141	−	0.879910i	$-0.342397\pi$
$269$	15.5858	0.950282	0.475141	+	0.879910i	$0.342397\pi$
$270$	0	0
$271$	−9.97056	−0.605669	−0.302834	−	0.953043i	$-0.597933\pi$
$271$	−9.97056	−0.605669	−0.302834	+	0.953043i	$0.597933\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.24264	0.376445
$276$	0	0
$277$	−0.343146	−0.0206176	−0.0103088	−	0.999947i	$-0.503281\pi$
$277$	−0.343146	−0.0206176	−0.0103088	+	0.999947i	$0.503281\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.07107	0.302515	0.151257	−	0.988494i	$-0.451668\pi$
$281$	5.07107	0.302515	0.151257	+	0.988494i	$0.451668\pi$
$282$	0	0
$283$	26.4558	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$283$	26.4558	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.41421	−0.260563
$288$	0	0
$289$	−14.4853	−0.852075
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.0416	−0.878741	−0.439371	−	0.898306i	$-0.644799\pi$
$293$	−15.0416	−0.878741	−0.439371	+	0.898306i	$0.644799\pi$
$294$	0	0
$295$	2.07107	0.120582
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.58579	−0.149540
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.0711	0.748447
$306$	0	0
$307$	14.3848	0.820983	0.410491	−	0.911865i	$-0.365357\pi$
$307$	14.3848	0.820983	0.410491	+	0.911865i	$0.365357\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.9706	−0.735493	−0.367747	−	0.929926i	$-0.619871\pi$
$311$	−12.9706	−0.735493	−0.367747	+	0.929926i	$0.619871\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.5563	1.54772	0.773859	−	0.633357i	$-0.218324\pi$
$317$	27.5563	1.54772	0.773859	+	0.633357i	$0.218324\pi$
$318$	0	0
$319$	45.2132	2.53145
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.2132	0.623919
$324$	0	0
$325$	−2.58579	−0.143434
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.58579	0.142559
$330$	0	0
$331$	−8.79899	−0.483636	−0.241818	−	0.970322i	$-0.577744\pi$
$331$	−8.79899	−0.483636	−0.241818	+	0.970322i	$0.577744\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.656854	0.0358878
$336$	0	0
$337$	−3.65685	−0.199202	−0.0996008	−	0.995027i	$-0.531757\pi$
$337$	−3.65685	−0.199202	−0.0996008	+	0.995027i	$0.531757\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.10051	0.222055
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.8284	−1.01076	−0.505381	−	0.862896i	$-0.668648\pi$
$347$	−18.8284	−1.01076	−0.505381	+	0.862896i	$0.668648\pi$
$348$	0	0
$349$	−6.17157	−0.330357	−0.165178	−	0.986264i	$-0.552820\pi$
$349$	−6.17157	−0.330357	−0.165178	+	0.986264i	$0.552820\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.2426	1.29031	0.645153	−	0.764054i	$-0.276794\pi$
$353$	24.2426	1.29031	0.645153	+	0.764054i	$0.276794\pi$
$354$	0	0
$355$	−12.0711	−0.640666
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.2426	1.06837	0.534183	−	0.845369i	$-0.320619\pi$
$359$	20.2426	1.06837	0.534183	+	0.845369i	$0.320619\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.5563	0.814257
$366$	0	0
$367$	9.14214	0.477216	0.238608	−	0.971116i	$-0.423309\pi$
$367$	9.14214	0.477216	0.238608	+	0.971116i	$0.423309\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.58579	−0.186165
$372$	0	0
$373$	8.97056	0.464478	0.232239	−	0.972659i	$-0.425395\pi$
$373$	8.97056	0.464478	0.232239	+	0.972659i	$0.425395\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.7279	−0.964537
$378$	0	0
$379$	−6.34315	−0.325826	−0.162913	−	0.986640i	$-0.552089\pi$
$379$	−6.34315	−0.325826	−0.162913	+	0.986640i	$0.552089\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.9289	−0.609540	−0.304770	−	0.952426i	$-0.598580\pi$
$383$	−11.9289	−0.609540	−0.304770	+	0.952426i	$0.598580\pi$
$384$	0	0
$385$	6.24264	0.318154
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.85786	0.398410	0.199205	−	0.979958i	$-0.436164\pi$
$389$	7.85786	0.398410	0.199205	+	0.979958i	$0.436164\pi$
$390$	0	0
$391$	1.58579	0.0801967
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.65685	0.284627
$396$	0	0
$397$	−16.0000	−0.803017	−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000	−0.803017	−0.401508	+	0.915855i	$0.631514\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.48528	−0.124109	−0.0620545	−	0.998073i	$-0.519765\pi$
$401$	−2.48528	−0.124109	−0.0620545	+	0.998073i	$0.519765\pi$
$402$	0	0
$403$	−1.69848	−0.0846076
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.7279	−0.928309
$408$	0	0
$409$	22.4558	1.11037	0.555185	−	0.831727i	$-0.312647\pi$
$409$	22.4558	1.11037	0.555185	+	0.831727i	$0.312647\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.07107	0.101911
$414$	0	0
$415$	1.92893	0.0946876
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.384776	0.0187976	0.00939878	−	0.999956i	$-0.497008\pi$
$419$	0.384776	0.0187976	0.00939878	+	0.999956i	$0.497008\pi$
$420$	0	0
$421$	−38.8701	−1.89441	−0.947205	−	0.320628i	$-0.896106\pi$
$421$	−38.8701	−1.89441	−0.947205	+	0.320628i	$0.896106\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.58579	0.0769219
$426$	0	0
$427$	13.0711	0.632553
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.8284	0.521587	0.260793	−	0.965395i	$-0.416016\pi$
$431$	10.8284	0.521587	0.260793	+	0.965395i	$0.416016\pi$
$432$	0	0
$433$	29.8284	1.43346	0.716731	−	0.697349i	$-0.245637\pi$
$433$	29.8284	1.43346	0.716731	+	0.697349i	$0.245637\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.07107	0.338255
$438$	0	0
$439$	−41.7990	−1.99496	−0.997478	−	0.0709697i	$-0.977391\pi$
$439$	−41.7990	−1.99496	−0.997478	+	0.0709697i	$0.977391\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.928932	0.0441349	0.0220675	−	0.999756i	$-0.492975\pi$
$443$	0.928932	0.0441349	0.0220675	+	0.999756i	$0.492975\pi$
$444$	0	0
$445$	−5.17157	−0.245156
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.27208	0.295998	0.147999	−	0.988988i	$-0.452717\pi$
$449$	6.27208	0.295998	0.147999	+	0.988988i	$0.452717\pi$
$450$	0	0
$451$	27.5563	1.29758
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.58579	−0.121224
$456$	0	0
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.51472	−0.0705475	−0.0352737	−	0.999378i	$-0.511230\pi$
$461$	−1.51472	−0.0705475	−0.0352737	+	0.999378i	$0.511230\pi$
$462$	0	0
$463$	−20.0416	−0.931414	−0.465707	−	0.884939i	$-0.654200\pi$
$463$	−20.0416	−0.931414	−0.465707	+	0.884939i	$0.654200\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.58579	0.351028	0.175514	−	0.984477i	$-0.443841\pi$
$467$	7.58579	0.351028	0.175514	+	0.984477i	$0.443841\pi$
$468$	0	0
$469$	0.656854	0.0303307
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.4853	0.574074
$474$	0	0
$475$	7.07107	0.324443
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.89949	−0.452319	−0.226160	−	0.974090i	$-0.572617\pi$
$479$	−9.89949	−0.452319	−0.226160	+	0.974090i	$0.572617\pi$
$480$	0	0
$481$	7.75736	0.353705
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.82843	0.310063
$486$	0	0
$487$	−9.07107	−0.411049	−0.205525	−	0.978652i	$-0.565890\pi$
$487$	−9.07107	−0.411049	−0.205525	+	0.978652i	$0.565890\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.10051	−0.230183	−0.115091	−	0.993355i	$-0.536716\pi$
$491$	−5.10051	−0.230183	−0.115091	+	0.993355i	$0.536716\pi$
$492$	0	0
$493$	11.4853	0.517271
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0711	−0.541461
$498$	0	0
$499$	−4.17157	−0.186745	−0.0933726	−	0.995631i	$-0.529765\pi$
$499$	−4.17157	−0.186745	−0.0933726	+	0.995631i	$0.529765\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	34.6985	1.54713	0.773564	−	0.633718i	$-0.218472\pi$
$503$	34.6985	1.54713	0.773564	+	0.633718i	$0.218472\pi$
$504$	0	0
$505$	11.2426	0.500291
$506$	0	0
$507$	0	0
$508$	0	0
$509$	23.7990	1.05487	0.527436	−	0.849595i	$-0.323154\pi$
$509$	23.7990	1.05487	0.527436	+	0.849595i	$0.323154\pi$
$510$	0	0
$511$	15.5563	0.688172
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.48528	−0.285776
$516$	0	0
$517$	−16.1421	−0.709930
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.07107	−0.397411	−0.198705	−	0.980059i	$-0.563674\pi$
$521$	−9.07107	−0.397411	−0.198705	+	0.980059i	$0.563674\pi$
$522$	0	0
$523$	41.6569	1.82153	0.910764	−	0.412928i	$-0.135494\pi$
$523$	41.6569	1.82153	0.910764	+	0.412928i	$0.135494\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.04163	0.0453741
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.4142	−0.494404
$534$	0	0
$535$	−5.24264	−0.226659
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−37.4558	−1.61334
$540$	0	0
$541$	27.5147	1.18295	0.591475	−	0.806323i	$-0.298546\pi$
$541$	27.5147	1.18295	0.591475	+	0.806323i	$0.298546\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.3848	−0.787517
$546$	0	0
$547$	−31.3137	−1.33888	−0.669439	−	0.742867i	$-0.733465\pi$
$547$	−31.3137	−1.33888	−0.669439	+	0.742867i	$0.733465\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	51.2132	2.18176
$552$	0	0
$553$	5.65685	0.240554
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.4142	−1.20395	−0.601974	−	0.798515i	$-0.705619\pi$
$557$	−28.4142	−1.20395	−0.601974	+	0.798515i	$0.705619\pi$
$558$	0	0
$559$	−5.17157	−0.218734
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.24264	−0.220951	−0.110475	−	0.993879i	$-0.535237\pi$
$563$	−5.24264	−0.220951	−0.110475	+	0.993879i	$0.535237\pi$
$564$	0	0
$565$	−9.24264	−0.388841
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.1421	1.01209	0.506045	−	0.862507i	$-0.331107\pi$
$569$	24.1421	1.01209	0.506045	+	0.862507i	$0.331107\pi$
$570$	0	0
$571$	5.27208	0.220630	0.110315	−	0.993897i	$-0.464814\pi$
$571$	5.27208	0.220630	0.110315	+	0.993897i	$0.464814\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	45.9411	1.91255	0.956277	−	0.292462i	$-0.0944746\pi$
$577$	45.9411	1.91255	0.956277	+	0.292462i	$0.0944746\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.92893	0.0800256
$582$	0	0
$583$	22.3848	0.927083
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	4.64466	0.191380
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.8701	−1.10342	−0.551711	−	0.834036i	$-0.686025\pi$
$593$	−26.8701	−1.10342	−0.551711	+	0.834036i	$0.686025\pi$
$594$	0	0
$595$	1.58579	0.0650109
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.48528	−0.346699	−0.173350	−	0.984860i	$-0.555459\pi$
$599$	−8.48528	−0.346699	−0.173350	+	0.984860i	$0.555459\pi$
$600$	0	0
$601$	−22.3137	−0.910195	−0.455098	−	0.890442i	$-0.650396\pi$
$601$	−22.3137	−0.910195	−0.455098	+	0.890442i	$0.650396\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−27.9706	−1.13717
$606$	0	0
$607$	34.0416	1.38171	0.690854	−	0.722995i	$-0.257235\pi$
$607$	34.0416	1.38171	0.690854	+	0.722995i	$0.257235\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.68629	0.270498
$612$	0	0
$613$	3.37258	0.136217	0.0681087	−	0.997678i	$-0.478304\pi$
$613$	3.37258	0.136217	0.0681087	+	0.997678i	$0.478304\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.10051	−0.366373	−0.183186	−	0.983078i	$-0.558641\pi$
$617$	−9.10051	−0.366373	−0.183186	+	0.983078i	$0.558641\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.17157	−0.207195
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.75736	−0.189688
$630$	0	0
$631$	−12.1005	−0.481714	−0.240857	−	0.970561i	$-0.577428\pi$
$631$	−12.1005	−0.481714	−0.240857	+	0.970561i	$0.577428\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.58579	−0.340717
$636$	0	0
$637$	15.5147	0.614716
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.92893	0.273676	0.136838	−	0.990593i	$-0.456306\pi$
$641$	6.92893	0.273676	0.136838	+	0.990593i	$0.456306\pi$
$642$	0	0
$643$	26.1716	1.03211	0.516053	−	0.856557i	$-0.327401\pi$
$643$	26.1716	1.03211	0.516053	+	0.856557i	$0.327401\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.2721	0.679035	0.339518	−	0.940600i	$-0.389736\pi$
$647$	17.2721	0.679035	0.339518	+	0.940600i	$0.389736\pi$
$648$	0	0
$649$	−12.9289	−0.507505
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.8406	1.48082	0.740409	−	0.672157i	$-0.234632\pi$
$653$	37.8406	1.48082	0.740409	+	0.672157i	$0.234632\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.6985	−0.923162	−0.461581	−	0.887098i	$-0.652718\pi$
$659$	−23.6985	−0.923162	−0.461581	+	0.887098i	$0.652718\pi$
$660$	0	0
$661$	1.51472	0.0589157	0.0294579	−	0.999566i	$-0.490622\pi$
$661$	1.51472	0.0589157	0.0294579	+	0.999566i	$0.490622\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.07107	0.274204
$666$	0	0
$667$	7.24264	0.280436
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−81.5980	−3.15006
$672$	0	0
$673$	−20.8701	−0.804482	−0.402241	−	0.915534i	$-0.631769\pi$
$673$	−20.8701	−0.804482	−0.402241	+	0.915534i	$0.631769\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.8995	−0.418902	−0.209451	−	0.977819i	$-0.567168\pi$
$677$	−10.8995	−0.418902	−0.209451	+	0.977819i	$0.567168\pi$
$678$	0	0
$679$	6.82843	0.262051
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.2721	−0.737426	−0.368713	−	0.929543i	$-0.620201\pi$
$683$	−19.2721	−0.737426	−0.368713	+	0.929543i	$0.620201\pi$
$684$	0	0
$685$	16.4853	0.629870
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.27208	−0.353238
$690$	0	0
$691$	44.6274	1.69771	0.848853	−	0.528628i	$-0.177293\pi$
$691$	44.6274	1.69771	0.848853	+	0.528628i	$0.177293\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.4853	−0.739119
$696$	0	0
$697$	7.00000	0.265144
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.6985	−1.12170	−0.560848	−	0.827919i	$-0.689525\pi$
$701$	−29.6985	−1.12170	−0.560848	+	0.827919i	$0.689525\pi$
$702$	0	0
$703$	−21.2132	−0.800071
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.2426	0.422823
$708$	0	0
$709$	22.9289	0.861114	0.430557	−	0.902563i	$-0.358317\pi$
$709$	22.9289	0.861114	0.430557	+	0.902563i	$0.358317\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.656854	0.0245994
$714$	0	0
$715$	16.1421	0.603682
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.41421	0.0900350	0.0450175	−	0.998986i	$-0.485666\pi$
$719$	2.41421	0.0900350	0.0450175	+	0.998986i	$0.485666\pi$
$720$	0	0
$721$	−6.48528	−0.241524
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.24264	0.268985
$726$	0	0
$727$	19.1421	0.709943	0.354971	−	0.934877i	$-0.384491\pi$
$727$	19.1421	0.709943	0.354971	+	0.934877i	$0.384491\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.17157	0.117305
$732$	0	0
$733$	−29.8284	−1.10174	−0.550869	−	0.834592i	$-0.685704\pi$
$733$	−29.8284	−1.10174	−0.550869	+	0.834592i	$0.685704\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.10051	−0.151044
$738$	0	0
$739$	25.1421	0.924868	0.462434	−	0.886654i	$-0.346976\pi$
$739$	25.1421	0.924868	0.462434	+	0.886654i	$0.346976\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.6274	−0.683374	−0.341687	−	0.939814i	$-0.610998\pi$
$743$	−18.6274	−0.683374	−0.341687	+	0.939814i	$0.610998\pi$
$744$	0	0
$745$	−13.4142	−0.491459
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.24264	−0.191562
$750$	0	0
$751$	−11.2721	−0.411324	−0.205662	−	0.978623i	$-0.565935\pi$
$751$	−11.2721	−0.411324	−0.205662	+	0.978623i	$0.565935\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.6569	0.424236
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.8995	1.12011	0.560053	−	0.828457i	$-0.310781\pi$
$761$	30.8995	1.12011	0.560053	+	0.828457i	$0.310781\pi$
$762$	0	0
$763$	−18.3848	−0.665574
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.35534	0.193370
$768$	0	0
$769$	−19.4142	−0.700094	−0.350047	−	0.936732i	$-0.613834\pi$
$769$	−19.4142	−0.700094	−0.350047	+	0.936732i	$0.613834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.1127	−0.903241	−0.451620	−	0.892210i	$-0.649154\pi$
$773$	−25.1127	−0.903241	−0.451620	+	0.892210i	$0.649154\pi$
$774$	0	0
$775$	0.656854	0.0235949
$776$	0	0
$777$	0	0
$778$	0	0
$779$	31.2132	1.11833
$780$	0	0
$781$	75.3553	2.69643
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.4853	0.552693
$786$	0	0
$787$	−27.7696	−0.989878	−0.494939	−	0.868928i	$-0.664810\pi$
$787$	−27.7696	−0.989878	−0.494939	+	0.868928i	$0.664810\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.24264	−0.328630
$792$	0	0
$793$	33.7990	1.20024
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.5269	−1.32927	−0.664636	−	0.747168i	$-0.731413\pi$
$797$	−37.5269	−1.32927	−0.664636	+	0.747168i	$0.731413\pi$
$798$	0	0
$799$	−4.10051	−0.145065
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−97.1127	−3.42703
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.41421	−0.295828	−0.147914	−	0.989000i	$-0.547256\pi$
$809$	−8.41421	−0.295828	−0.147914	+	0.989000i	$0.547256\pi$
$810$	0	0
$811$	−29.8284	−1.04742	−0.523709	−	0.851897i	$-0.675452\pi$
$811$	−29.8284	−1.04742	−0.523709	+	0.851897i	$0.675452\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.65685	0.128094
$816$	0	0
$817$	14.1421	0.494771
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.1127	0.806639	0.403319	−	0.915059i	$-0.367856\pi$
$821$	23.1127	0.806639	0.403319	+	0.915059i	$0.367856\pi$
$822$	0	0
$823$	6.62742	0.231017	0.115509	−	0.993306i	$-0.463150\pi$
$823$	6.62742	0.231017	0.115509	+	0.993306i	$0.463150\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.0416	−0.592596	−0.296298	−	0.955096i	$-0.595752\pi$
$827$	−17.0416	−0.592596	−0.296298	+	0.955096i	$0.595752\pi$
$828$	0	0
$829$	−40.7990	−1.41701	−0.708504	−	0.705707i	$-0.750629\pi$
$829$	−40.7990	−1.41701	−0.708504	+	0.705707i	$0.750629\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.51472	−0.329665
$834$	0	0
$835$	3.07107	0.106279
$836$	0	0
$837$	0	0
$838$	0	0
$839$	51.6569	1.78339	0.891696	−	0.452634i	$-0.149516\pi$
$839$	51.6569	1.78339	0.891696	+	0.452634i	$0.149516\pi$
$840$	0	0
$841$	23.4558	0.808822
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.31371	0.217198
$846$	0	0
$847$	−27.9706	−0.961080
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	10.4853	0.359009	0.179505	−	0.983757i	$-0.442551\pi$
$853$	10.4853	0.359009	0.179505	+	0.983757i	$0.442551\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.6569	1.08138	0.540689	−	0.841223i	$-0.318164\pi$
$857$	31.6569	1.08138	0.540689	+	0.841223i	$0.318164\pi$
$858$	0	0
$859$	−16.4558	−0.561466	−0.280733	−	0.959786i	$-0.590578\pi$
$859$	−16.4558	−0.561466	−0.280733	+	0.959786i	$0.590578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.1421	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$863$	36.1421	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$864$	0	0
$865$	11.3137	0.384678
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−35.3137	−1.19794
$870$	0	0
$871$	1.69848	0.0575510
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	7.17157	0.242167	0.121083	−	0.992642i	$-0.461363\pi$
$877$	7.17157	0.242167	0.121083	+	0.992642i	$0.461363\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22.5858	−0.760934	−0.380467	−	0.924794i	$-0.624237\pi$
$881$	−22.5858	−0.760934	−0.380467	+	0.924794i	$0.624237\pi$
$882$	0	0
$883$	−20.3848	−0.686002	−0.343001	−	0.939335i	$-0.611443\pi$
$883$	−20.3848	−0.686002	−0.343001	+	0.939335i	$0.611443\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.4853	0.486368	0.243184	−	0.969980i	$-0.421808\pi$
$887$	14.4853	0.486368	0.243184	+	0.969980i	$0.421808\pi$
$888$	0	0
$889$	−8.58579	−0.287958
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−18.2843	−0.611860
$894$	0	0
$895$	−23.6569	−0.790761
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.75736	0.158667
$900$	0	0
$901$	5.68629	0.189438
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.1421	−0.736029
$906$	0	0
$907$	−31.9706	−1.06157	−0.530783	−	0.847508i	$-0.678102\pi$
$907$	−31.9706	−1.06157	−0.530783	+	0.847508i	$0.678102\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.9706	0.959838	0.479919	−	0.877313i	$-0.340666\pi$
$911$	28.9706	0.959838	0.479919	+	0.877313i	$0.340666\pi$
$912$	0	0
$913$	−12.0416	−0.398520
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.65685	−0.186806
$918$	0	0
$919$	−37.9411	−1.25156	−0.625781	−	0.779999i	$-0.715220\pi$
$919$	−37.9411	−1.25156	−0.625781	+	0.779999i	$0.715220\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−31.2132	−1.02740
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	0	0
$927$	0	0
$928$	0	0
$929$	57.3848	1.88273	0.941367	−	0.337385i	$-0.109542\pi$
$929$	57.3848	1.88273	0.941367	+	0.337385i	$0.109542\pi$
$930$	0	0
$931$	−42.4264	−1.39047
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.89949	−0.323748
$936$	0	0
$937$	50.8284	1.66049	0.830246	−	0.557397i	$-0.188200\pi$
$937$	50.8284	1.66049	0.830246	+	0.557397i	$0.188200\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.44365	−0.144859	−0.0724294	−	0.997374i	$-0.523075\pi$
$941$	−4.44365	−0.144859	−0.0724294	+	0.997374i	$0.523075\pi$
$942$	0	0
$943$	4.41421	0.143747
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.4558	−1.60710	−0.803549	−	0.595238i	$-0.797058\pi$
$947$	−49.4558	−1.60710	−0.803549	+	0.595238i	$0.797058\pi$
$948$	0	0
$949$	40.2254	1.30577
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.6863	1.12360	0.561800	−	0.827273i	$-0.310109\pi$
$953$	34.6863	1.12360	0.561800	+	0.827273i	$0.310109\pi$
$954$	0	0
$955$	5.89949	0.190903
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.4853	0.532337
$960$	0	0
$961$	−30.5685	−0.986082
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−24.4853	−0.788209
$966$	0	0
$967$	30.5269	0.981679	0.490840	−	0.871250i	$-0.336690\pi$
$967$	30.5269	0.981679	0.490840	+	0.871250i	$0.336690\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.54416	−0.0816458	−0.0408229	−	0.999166i	$-0.512998\pi$
$971$	−2.54416	−0.0816458	−0.0408229	+	0.999166i	$0.512998\pi$
$972$	0	0
$973$	−19.4853	−0.624669
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.6985	0.598218	0.299109	−	0.954219i	$-0.403311\pi$
$977$	18.6985	0.598218	0.299109	+	0.954219i	$0.403311\pi$
$978$	0	0
$979$	32.2843	1.03181
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.24264	−0.294794	−0.147397	−	0.989077i	$-0.547090\pi$
$983$	−9.24264	−0.294794	−0.147397	+	0.989077i	$0.547090\pi$
$984$	0	0
$985$	−25.7990	−0.822024
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.00000	0.0635963
$990$	0	0
$991$	−29.8284	−0.947531	−0.473766	−	0.880651i	$-0.657106\pi$
$991$	−29.8284	−0.947531	−0.473766	+	0.880651i	$0.657106\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	27.4558	0.870409
$996$	0	0
$997$	56.0833	1.77617	0.888087	−	0.459675i	$-0.152034\pi$
$997$	56.0833	1.77617	0.888087	+	0.459675i	$0.152034\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.y.1.2		2
3.2	odd	2		2760.2.a.q.1.1	✓	2
12.11	even	2		5520.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.q.1.1	✓	2	3.2	odd	2
5520.2.a.bk.1.2		2	12.11	even	2
8280.2.a.y.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.00000 q^{7} +6.24264 q^{11} -2.58579 q^{13} +1.58579 q^{17} +7.07107 q^{19} +1.00000 q^{23} +1.00000 q^{25} +7.24264 q^{29} +0.656854 q^{31} +1.00000 q^{35} -3.00000 q^{37} +4.41421 q^{41} +2.00000 q^{43} -2.58579 q^{47} -6.00000 q^{49} +3.58579 q^{53} -6.24264 q^{55} -2.07107 q^{59} -13.0711 q^{61} +2.58579 q^{65} -0.656854 q^{67} +12.0711 q^{71} -15.5563 q^{73} -6.24264 q^{77} -5.65685 q^{79} -1.92893 q^{83} -1.58579 q^{85} +5.17157 q^{89} +2.58579 q^{91} -7.07107 q^{95} -6.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{23} + 2 q^{25} + 6 q^{29} - 10 q^{31} + 2 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 8 q^{47} - 12 q^{49} + 10 q^{53} - 4 q^{55} + 10 q^{59} - 12 q^{61} + 8 q^{65} + 10 q^{67} + 10 q^{71} - 4 q^{77} - 18 q^{83} - 6 q^{85} + 16 q^{89} + 8 q^{91} - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^23 + 2 * q^25 + 6 * q^29 - 10 * q^31 + 2 * q^35 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 8 * q^47 - 12 * q^49 + 10 * q^53 - 4 * q^55 + 10 * q^59 - 12 * q^61 + 8 * q^65 + 10 * q^67 + 10 * q^71 - 4 * q^77 - 18 * q^83 - 6 * q^85 + 16 * q^89 + 8 * q^91 - 8 * q^97

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