LMFDB - Embedded newform 2736.2.a.y.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2736,2,Mod(1,2736)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2736.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-3,0,-1,0,0,0,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$21.8470699930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	2736.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.37228 q^{5} +2.37228 q^{7} -4.37228 q^{11} +2.00000 q^{13} +4.37228 q^{17} -1.00000 q^{19} +2.74456 q^{23} +14.1168 q^{25} +8.74456 q^{29} -4.74456 q^{31} -10.3723 q^{35} -6.74456 q^{37} +2.37228 q^{43} -7.62772 q^{47} -1.37228 q^{49} -8.74456 q^{53} +19.1168 q^{55} -8.37228 q^{61} -8.74456 q^{65} -13.4891 q^{67} -12.0000 q^{71} +0.372281 q^{73} -10.3723 q^{77} -8.00000 q^{79} -2.74456 q^{83} -19.1168 q^{85} -3.25544 q^{89} +4.74456 q^{91} +4.37228 q^{95} +14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} - q^{7} - 3 q^{11} + 4 q^{13} + 3 q^{17} - 2 q^{19} - 6 q^{23} + 11 q^{25} + 6 q^{29} + 2 q^{31} - 15 q^{35} - 2 q^{37} - q^{43} - 21 q^{47} + 3 q^{49} - 6 q^{53} + 21 q^{55} - 11 q^{61}+ \cdots + 28 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 - q^7 - 3 * q^11 + 4 * q^13 + 3 * q^17 - 2 * q^19 - 6 * q^23 + 11 * q^25 + 6 * q^29 + 2 * q^31 - 15 * q^35 - 2 * q^37 - q^43 - 21 * q^47 + 3 * q^49 - 6 * q^53 + 21 * q^55 - 11 * q^61 - 6 * q^65 - 4 * q^67 - 24 * q^71 - 5 * q^73 - 15 * q^77 - 16 * q^79 + 6 * q^83 - 21 * q^85 - 18 * q^89 - 2 * q^91 + 3 * q^95 + 28 * q^97

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
$5$	−4.37228	−1.95534	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−4.37228	−1.95534	−0.977672	+	0.210138i	$0.932609\pi$
$6$	0	0
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.37228	−1.31829	−0.659146	−	0.752015i	$-0.729082\pi$
$11$	−4.37228	−1.31829	−0.659146	+	0.752015i	$0.729082\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.37228	1.06043	0.530217	−	0.847862i	$-0.322110\pi$
$17$	4.37228	1.06043	0.530217	+	0.847862i	$0.322110\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.74456	0.572281	0.286140	−	0.958188i	$-0.407628\pi$
$23$	2.74456	0.572281	0.286140	+	0.958188i	$0.407628\pi$
$24$	0	0
$25$	14.1168	2.82337
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10.3723	−1.75324
$36$	0	0
$37$	−6.74456	−1.10880	−0.554400	−	0.832251i	$-0.687052\pi$
$37$	−6.74456	−1.10880	−0.554400	+	0.832251i	$0.687052\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.37228	0.361770	0.180885	−	0.983504i	$-0.442104\pi$
$43$	2.37228	0.361770	0.180885	+	0.983504i	$0.442104\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.62772	−1.11262	−0.556309	−	0.830976i	$-0.687783\pi$
$47$	−7.62772	−1.11262	−0.556309	+	0.830976i	$0.687783\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.74456	−1.20116	−0.600579	−	0.799565i	$-0.705063\pi$
$53$	−8.74456	−1.20116	−0.600579	+	0.799565i	$0.705063\pi$
$54$	0	0
$55$	19.1168	2.57771
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.37228	−1.07196	−0.535980	−	0.844230i	$-0.680058\pi$
$61$	−8.37228	−1.07196	−0.535980	+	0.844230i	$0.680058\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.74456	−1.08463
$66$	0	0
$67$	−13.4891	−1.64796	−0.823979	−	0.566620i	$-0.808251\pi$
$67$	−13.4891	−1.64796	−0.823979	+	0.566620i	$0.808251\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	0.372281	0.0435722	0.0217861	−	0.999763i	$-0.493065\pi$
$73$	0.372281	0.0435722	0.0217861	+	0.999763i	$0.493065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.3723	−1.18203
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.74456	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$84$	0	0
$85$	−19.1168	−2.07351
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.25544	−0.345076	−0.172538	−	0.985003i	$-0.555197\pi$
$89$	−3.25544	−0.345076	−0.172538	+	0.985003i	$0.555197\pi$
$90$	0	0
$91$	4.74456	0.497365
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.37228	0.448587
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.4891	1.14321	0.571605	−	0.820529i	$-0.306321\pi$
$101$	11.4891	1.14321	0.571605	+	0.820529i	$0.306321\pi$
$102$	0	0
$103$	12.7446	1.25576	0.627880	−	0.778311i	$-0.283923\pi$
$103$	12.7446	1.25576	0.627880	+	0.778311i	$0.283923\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.25544	0.314715	0.157358	−	0.987542i	$-0.449703\pi$
$107$	3.25544	0.314715	0.157358	+	0.987542i	$0.449703\pi$
$108$	0	0
$109$	16.2337	1.55491	0.777453	−	0.628941i	$-0.216511\pi$
$109$	16.2337	1.55491	0.777453	+	0.628941i	$0.216511\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.25544	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$113$	−3.25544	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.3723	0.950825
$120$	0	0
$121$	8.11684	0.737895
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−39.8614	−3.56531
$126$	0	0
$127$	−22.2337	−1.97292	−0.986460	−	0.164000i	$-0.947560\pi$
$127$	−22.2337	−1.97292	−0.986460	+	0.164000i	$0.947560\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.86141	0.861595	0.430798	−	0.902449i	$-0.358232\pi$
$131$	9.86141	0.861595	0.430798	+	0.902449i	$0.358232\pi$
$132$	0	0
$133$	−2.37228	−0.205703
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.37228	0.373549	0.186775	−	0.982403i	$-0.440197\pi$
$137$	4.37228	0.373549	0.186775	+	0.982403i	$0.440197\pi$
$138$	0	0
$139$	−9.62772	−0.816612	−0.408306	−	0.912845i	$-0.633880\pi$
$139$	−9.62772	−0.816612	−0.408306	+	0.912845i	$0.633880\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.74456	−0.731257
$144$	0	0
$145$	−38.2337	−3.17513
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.37228	0.358191	0.179096	−	0.983832i	$-0.442683\pi$
$149$	4.37228	0.358191	0.179096	+	0.983832i	$0.442683\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.7446	1.66624
$156$	0	0
$157$	−15.4891	−1.23617	−0.618083	−	0.786113i	$-0.712091\pi$
$157$	−15.4891	−1.23617	−0.618083	+	0.786113i	$0.712091\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.51087	0.513129
$162$	0	0
$163$	−13.4891	−1.05655	−0.528275	−	0.849073i	$-0.677161\pi$
$163$	−13.4891	−1.05655	−0.528275	+	0.849073i	$0.677161\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.2337	−1.10144	−0.550718	−	0.834691i	$-0.685646\pi$
$167$	−14.2337	−1.10144	−0.550718	+	0.834691i	$0.685646\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.7446	−1.57718	−0.788590	−	0.614919i	$-0.789189\pi$
$173$	−20.7446	−1.57718	−0.788590	+	0.614919i	$0.789189\pi$
$174$	0	0
$175$	33.4891	2.53154
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.25544	0.243323	0.121661	−	0.992572i	$-0.461178\pi$
$179$	3.25544	0.243323	0.121661	+	0.992572i	$0.461178\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	29.4891	2.16808
$186$	0	0
$187$	−19.1168	−1.39796
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.3723	−1.18466	−0.592328	−	0.805697i	$-0.701791\pi$
$191$	−16.3723	−1.18466	−0.592328	+	0.805697i	$0.701791\pi$
$192$	0	0
$193$	5.25544	0.378295	0.189147	−	0.981949i	$-0.439428\pi$
$193$	5.25544	0.378295	0.189147	+	0.981949i	$0.439428\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.4891	−0.818566	−0.409283	−	0.912407i	$-0.634221\pi$
$197$	−11.4891	−0.818566	−0.409283	+	0.912407i	$0.634221\pi$
$198$	0	0
$199$	−3.11684	−0.220947	−0.110474	−	0.993879i	$-0.535237\pi$
$199$	−3.11684	−0.220947	−0.110474	+	0.993879i	$0.535237\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.7446	1.45598
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.37228	0.302437
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.3723	−0.707384
$216$	0	0
$217$	−11.2554	−0.764069
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.74456	0.588223
$222$	0	0
$223$	9.48913	0.635439	0.317719	−	0.948185i	$-0.397083\pi$
$223$	9.48913	0.635439	0.317719	+	0.948185i	$0.397083\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.2337	1.74119	0.870596	−	0.491999i	$-0.163734\pi$
$227$	26.2337	1.74119	0.870596	+	0.491999i	$0.163734\pi$
$228$	0	0
$229$	0.372281	0.0246010	0.0123005	−	0.999924i	$-0.496085\pi$
$229$	0.372281	0.0246010	0.0123005	+	0.999924i	$0.496085\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.37228	−0.286438	−0.143219	−	0.989691i	$-0.545745\pi$
$233$	−4.37228	−0.286438	−0.143219	+	0.989691i	$0.545745\pi$
$234$	0	0
$235$	33.3505	2.17555
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.86141	−0.637881	−0.318941	−	0.947775i	$-0.603327\pi$
$239$	−9.86141	−0.637881	−0.318941	+	0.947775i	$0.603327\pi$
$240$	0	0
$241$	10.7446	0.692118	0.346059	−	0.938213i	$-0.387520\pi$
$241$	10.7446	0.692118	0.346059	+	0.938213i	$0.387520\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.37228	0.275976	0.137988	−	0.990434i	$-0.455936\pi$
$251$	4.37228	0.275976	0.137988	+	0.990434i	$0.455936\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.48913	−0.342402	−0.171201	−	0.985236i	$-0.554765\pi$
$257$	−5.48913	−0.342402	−0.171201	+	0.985236i	$0.554765\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.13859	−0.131871	−0.0659357	−	0.997824i	$-0.521003\pi$
$263$	−2.13859	−0.131871	−0.0659357	+	0.997824i	$0.521003\pi$
$264$	0	0
$265$	38.2337	2.34868
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.25544	−0.198488	−0.0992438	−	0.995063i	$-0.531642\pi$
$269$	−3.25544	−0.198488	−0.0992438	+	0.995063i	$0.531642\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−61.7228	−3.72203
$276$	0	0
$277$	6.88316	0.413569	0.206784	−	0.978387i	$-0.433700\pi$
$277$	6.88316	0.413569	0.206784	+	0.978387i	$0.433700\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.2554	0.910063	0.455032	−	0.890475i	$-0.349628\pi$
$281$	15.2554	0.910063	0.455032	+	0.890475i	$0.349628\pi$
$282$	0	0
$283$	8.88316	0.528049	0.264024	−	0.964516i	$-0.414950\pi$
$283$	8.88316	0.528049	0.264024	+	0.964516i	$0.414950\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.11684	0.124520
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.51087	−0.380369	−0.190185	−	0.981748i	$-0.560909\pi$
$293$	−6.51087	−0.380369	−0.190185	+	0.981748i	$0.560909\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.48913	0.317444
$300$	0	0
$301$	5.62772	0.324376
$302$	0	0
$303$	0	0
$304$	0	0
$305$	36.6060	2.09605
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.86141	0.559189	0.279595	−	0.960118i	$-0.409800\pi$
$311$	9.86141	0.559189	0.279595	+	0.960118i	$0.409800\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.51087	−0.365687	−0.182844	−	0.983142i	$-0.558530\pi$
$317$	−6.51087	−0.365687	−0.182844	+	0.983142i	$0.558530\pi$
$318$	0	0
$319$	−38.2337	−2.14068
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.37228	−0.243280
$324$	0	0
$325$	28.2337	1.56612
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−18.0951	−0.997615
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	58.9783	3.22233
$336$	0	0
$337$	−3.48913	−0.190065	−0.0950324	−	0.995474i	$-0.530295\pi$
$337$	−3.48913	−0.190065	−0.0950324	+	0.995474i	$0.530295\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.7446	1.12338
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.11684	−0.0599553	−0.0299777	−	0.999551i	$-0.509544\pi$
$347$	−1.11684	−0.0599553	−0.0299777	+	0.999551i	$0.509544\pi$
$348$	0	0
$349$	24.3723	1.30462	0.652309	−	0.757953i	$-0.273800\pi$
$349$	24.3723	1.30462	0.652309	+	0.757953i	$0.273800\pi$
$350$	0	0
$351$	0	0
$352$	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
$355$	52.4674	2.78468
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.8614	−1.78714	−0.893568	−	0.448927i	$-0.851806\pi$
$359$	−33.8614	−1.78714	−0.893568	+	0.448927i	$0.851806\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.62772	−0.0851987
$366$	0	0
$367$	−2.51087	−0.131067	−0.0655333	−	0.997850i	$-0.520875\pi$
$367$	−2.51087	−0.131067	−0.0655333	+	0.997850i	$0.520875\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.7446	−1.07700
$372$	0	0
$373$	4.23369	0.219212	0.109606	−	0.993975i	$-0.465041\pi$
$373$	4.23369	0.219212	0.109606	+	0.993975i	$0.465041\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.4891	0.900736
$378$	0	0
$379$	35.7228	1.83496	0.917479	−	0.397785i	$-0.130221\pi$
$379$	35.7228	1.83496	0.917479	+	0.397785i	$0.130221\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.2337	−1.34048	−0.670239	−	0.742145i	$-0.733809\pi$
$383$	−26.2337	−1.34048	−0.670239	+	0.742145i	$0.733809\pi$
$384$	0	0
$385$	45.3505	2.31128
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.6277	0.995165	0.497582	−	0.867417i	$-0.334221\pi$
$389$	19.6277	0.995165	0.497582	+	0.867417i	$0.334221\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	34.9783	1.75995
$396$	0	0
$397$	−10.6060	−0.532298	−0.266149	−	0.963932i	$-0.585751\pi$
$397$	−10.6060	−0.532298	−0.266149	+	0.963932i	$0.585751\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.4891	0.873365	0.436683	−	0.899616i	$-0.356153\pi$
$401$	17.4891	0.873365	0.436683	+	0.899616i	$0.356153\pi$
$402$	0	0
$403$	−9.48913	−0.472687
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.4891	1.46172
$408$	0	0
$409$	−38.4674	−1.90209	−0.951045	−	0.309053i	$-0.899988\pi$
$409$	−38.4674	−1.90209	−0.951045	+	0.309053i	$0.899988\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	37.7228	1.84288	0.921440	−	0.388521i	$-0.127014\pi$
$419$	37.7228	1.84288	0.921440	+	0.388521i	$0.127014\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	61.7228	2.99400
$426$	0	0
$427$	−19.8614	−0.961161
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.74456	0.421211	0.210605	−	0.977571i	$-0.432456\pi$
$431$	8.74456	0.421211	0.210605	+	0.977571i	$0.432456\pi$
$432$	0	0
$433$	−15.4891	−0.744360	−0.372180	−	0.928161i	$-0.621390\pi$
$433$	−15.4891	−0.744360	−0.372180	+	0.928161i	$0.621390\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.74456	−0.131290
$438$	0	0
$439$	30.2337	1.44298	0.721488	−	0.692427i	$-0.243459\pi$
$439$	30.2337	1.44298	0.721488	+	0.692427i	$0.243459\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.62772	−0.362404	−0.181202	−	0.983446i	$-0.557999\pi$
$443$	−7.62772	−0.362404	−0.181202	+	0.983446i	$0.557999\pi$
$444$	0	0
$445$	14.2337	0.674742
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.9783	−1.08441	−0.542205	−	0.840246i	$-0.682411\pi$
$449$	−22.9783	−1.08441	−0.542205	+	0.840246i	$0.682411\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−20.7446	−0.972520
$456$	0	0
$457$	−13.8614	−0.648409	−0.324205	−	0.945987i	$-0.605097\pi$
$457$	−13.8614	−0.648409	−0.324205	+	0.945987i	$0.605097\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	39.3505	1.83274	0.916368	−	0.400336i	$-0.131107\pi$
$461$	39.3505	1.83274	0.916368	+	0.400336i	$0.131107\pi$
$462$	0	0
$463$	−17.3505	−0.806348	−0.403174	−	0.915123i	$-0.632093\pi$
$463$	−17.3505	−0.806348	−0.403174	+	0.915123i	$0.632093\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.1168	−0.606975	−0.303488	−	0.952835i	$-0.598151\pi$
$467$	−13.1168	−0.606975	−0.303488	+	0.952835i	$0.598151\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.3723	−0.476918
$474$	0	0
$475$	−14.1168	−0.647725
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.76631	0.172087	0.0860436	−	0.996291i	$-0.472578\pi$
$479$	3.76631	0.172087	0.0860436	+	0.996291i	$0.472578\pi$
$480$	0	0
$481$	−13.4891	−0.615051
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−61.2119	−2.77949
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.7446	−1.74852	−0.874259	−	0.485460i	$-0.838652\pi$
$491$	−38.7446	−1.74852	−0.874259	+	0.485460i	$0.838652\pi$
$492$	0	0
$493$	38.2337	1.72196
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−28.4674	−1.27694
$498$	0	0
$499$	−17.3505	−0.776716	−0.388358	−	0.921508i	$-0.626958\pi$
$499$	−17.3505	−0.776716	−0.388358	+	0.921508i	$0.626958\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.7446	0.657428	0.328714	−	0.944430i	$-0.393385\pi$
$503$	14.7446	0.657428	0.328714	+	0.944430i	$0.393385\pi$
$504$	0	0
$505$	−50.2337	−2.23537
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.25544	0.144295	0.0721474	−	0.997394i	$-0.477015\pi$
$509$	3.25544	0.144295	0.0721474	+	0.997394i	$0.477015\pi$
$510$	0	0
$511$	0.883156	0.0390685
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−55.7228	−2.45544
$516$	0	0
$517$	33.3505	1.46675
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.4891	0.766212	0.383106	−	0.923704i	$-0.374854\pi$
$521$	17.4891	0.766212	0.383106	+	0.923704i	$0.374854\pi$
$522$	0	0
$523$	7.25544	0.317258	0.158629	−	0.987338i	$-0.449293\pi$
$523$	7.25544	0.317258	0.158629	+	0.987338i	$0.449293\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.7446	−0.903647
$528$	0	0
$529$	−15.4674	−0.672495
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−14.2337	−0.615376
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−1.86141	−0.0800281	−0.0400141	−	0.999199i	$-0.512740\pi$
$541$	−1.86141	−0.0800281	−0.0400141	+	0.999199i	$0.512740\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−70.9783	−3.04037
$546$	0	0
$547$	7.25544	0.310220	0.155110	−	0.987897i	$-0.450427\pi$
$547$	7.25544	0.310220	0.155110	+	0.987897i	$0.450427\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.74456	−0.372531
$552$	0	0
$553$	−18.9783	−0.807037
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.8614	0.926298	0.463149	−	0.886281i	$-0.346720\pi$
$557$	21.8614	0.926298	0.463149	+	0.886281i	$0.346720\pi$
$558$	0	0
$559$	4.74456	0.200674
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.02175	0.0430616	0.0215308	−	0.999768i	$-0.493146\pi$
$563$	1.02175	0.0430616	0.0215308	+	0.999768i	$0.493146\pi$
$564$	0	0
$565$	14.2337	0.598816
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.7446	1.37272	0.686362	−	0.727260i	$-0.259207\pi$
$569$	32.7446	1.37272	0.686362	+	0.727260i	$0.259207\pi$
$570$	0	0
$571$	26.9783	1.12900	0.564502	−	0.825431i	$-0.309068\pi$
$571$	26.9783	1.12900	0.564502	+	0.825431i	$0.309068\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.7446	1.61576
$576$	0	0
$577$	21.1168	0.879106	0.439553	−	0.898217i	$-0.355137\pi$
$577$	21.1168	0.879106	0.439553	+	0.898217i	$0.355137\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.51087	−0.270117
$582$	0	0
$583$	38.2337	1.58348
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.60597	−0.272658	−0.136329	−	0.990664i	$-0.543530\pi$
$587$	−6.60597	−0.272658	−0.136329	+	0.990664i	$0.543530\pi$
$588$	0	0
$589$	4.74456	0.195496
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.9783	0.697213	0.348607	−	0.937269i	$-0.386655\pi$
$593$	16.9783	0.697213	0.348607	+	0.937269i	$0.386655\pi$
$594$	0	0
$595$	−45.3505	−1.85919
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−1.25544	−0.0512104	−0.0256052	−	0.999672i	$-0.508151\pi$
$601$	−1.25544	−0.0512104	−0.0256052	+	0.999672i	$0.508151\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−35.4891	−1.44284
$606$	0	0
$607$	19.2554	0.781554	0.390777	−	0.920485i	$-0.372206\pi$
$607$	19.2554	0.781554	0.390777	+	0.920485i	$0.372206\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.2554	−0.617169
$612$	0	0
$613$	−29.1168	−1.17602	−0.588009	−	0.808854i	$-0.700088\pi$
$613$	−29.1168	−1.17602	−0.588009	+	0.808854i	$0.700088\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.8832	0.438139	0.219070	−	0.975709i	$-0.429698\pi$
$617$	10.8832	0.438139	0.219070	+	0.975709i	$0.429698\pi$
$618$	0	0
$619$	−6.97825	−0.280480	−0.140240	−	0.990118i	$-0.544787\pi$
$619$	−6.97825	−0.280480	−0.140240	+	0.990118i	$0.544787\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.72281	−0.309408
$624$	0	0
$625$	103.701	4.14804
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.4891	−1.17581
$630$	0	0
$631$	−4.13859	−0.164755	−0.0823774	−	0.996601i	$-0.526251\pi$
$631$	−4.13859	−0.164755	−0.0823774	+	0.996601i	$0.526251\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	97.2119	3.85774
$636$	0	0
$637$	−2.74456	−0.108744
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−14.2337	−0.562197	−0.281098	−	0.959679i	$-0.590699\pi$
$641$	−14.2337	−0.562197	−0.281098	+	0.959679i	$0.590699\pi$
$642$	0	0
$643$	37.3505	1.47296	0.736481	−	0.676459i	$-0.236486\pi$
$643$	37.3505	1.47296	0.736481	+	0.676459i	$0.236486\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.8614	−0.859461	−0.429730	−	0.902957i	$-0.641391\pi$
$647$	−21.8614	−0.859461	−0.429730	+	0.902957i	$0.641391\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.6060	−1.19770	−0.598852	−	0.800860i	$-0.704376\pi$
$653$	−30.6060	−1.19770	−0.598852	+	0.800860i	$0.704376\pi$
$654$	0	0
$655$	−43.1168	−1.68471
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.74456	−0.340640	−0.170320	−	0.985389i	$-0.554480\pi$
$659$	−8.74456	−0.340640	−0.170320	+	0.985389i	$0.554480\pi$
$660$	0	0
$661$	−24.2337	−0.942581	−0.471291	−	0.881978i	$-0.656212\pi$
$661$	−24.2337	−0.942581	−0.471291	+	0.881978i	$0.656212\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.3723	0.402220
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36.6060	1.41316
$672$	0	0
$673$	−36.2337	−1.39671	−0.698353	−	0.715753i	$-0.746083\pi$
$673$	−36.2337	−1.39671	−0.698353	+	0.715753i	$0.746083\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.74456	−0.336081	−0.168040	−	0.985780i	$-0.553744\pi$
$677$	−8.74456	−0.336081	−0.168040	+	0.985780i	$0.553744\pi$
$678$	0	0
$679$	33.2119	1.27456
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.7446	0.793769	0.396884	−	0.917869i	$-0.370091\pi$
$683$	20.7446	0.793769	0.396884	+	0.917869i	$0.370091\pi$
$684$	0	0
$685$	−19.1168	−0.730417
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17.4891	−0.666283
$690$	0	0
$691$	−29.3505	−1.11655	−0.558273	−	0.829657i	$-0.688536\pi$
$691$	−29.3505	−1.11655	−0.558273	+	0.829657i	$0.688536\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	42.0951	1.59676
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.5109	−0.472529	−0.236265	−	0.971689i	$-0.575923\pi$
$701$	−12.5109	−0.472529	−0.236265	+	0.971689i	$0.575923\pi$
$702$	0	0
$703$	6.74456	0.254376
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.2554	1.02505
$708$	0	0
$709$	−50.4674	−1.89534	−0.947671	−	0.319248i	$-0.896570\pi$
$709$	−50.4674	−1.89534	−0.947671	+	0.319248i	$0.896570\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.0217	−0.487668
$714$	0	0
$715$	38.2337	1.42986
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.8832	−0.405873	−0.202937	−	0.979192i	$-0.565049\pi$
$719$	−10.8832	−0.405873	−0.202937	+	0.979192i	$0.565049\pi$
$720$	0	0
$721$	30.2337	1.12596
$722$	0	0
$723$	0	0
$724$	0	0
$725$	123.446	4.58466
$726$	0	0
$727$	39.5842	1.46810	0.734049	−	0.679097i	$-0.237628\pi$
$727$	39.5842	1.46810	0.734049	+	0.679097i	$0.237628\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10.3723	0.383633
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	58.9783	2.17249
$738$	0	0
$739$	−44.6060	−1.64086	−0.820429	−	0.571749i	$-0.806265\pi$
$739$	−44.6060	−1.64086	−0.820429	+	0.571749i	$0.806265\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.4674	1.48460	0.742302	−	0.670065i	$-0.233734\pi$
$743$	40.4674	1.48460	0.742302	+	0.670065i	$0.233734\pi$
$744$	0	0
$745$	−19.1168	−0.700387
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.72281	0.282185
$750$	0	0
$751$	2.97825	0.108678	0.0543390	−	0.998523i	$-0.482695\pi$
$751$	2.97825	0.108678	0.0543390	+	0.998523i	$0.482695\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	34.9783	1.27299
$756$	0	0
$757$	32.0951	1.16652	0.583258	−	0.812287i	$-0.301778\pi$
$757$	32.0951	1.16652	0.583258	+	0.812287i	$0.301778\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41.5842	−1.50743	−0.753713	−	0.657203i	$-0.771739\pi$
$761$	−41.5842	−1.50743	−0.753713	+	0.657203i	$0.771739\pi$
$762$	0	0
$763$	38.5109	1.39419
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.1168	0.761493	0.380746	−	0.924679i	$-0.375667\pi$
$769$	21.1168	0.761493	0.380746	+	0.924679i	$0.375667\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−41.4891	−1.49226	−0.746130	−	0.665800i	$-0.768090\pi$
$773$	−41.4891	−1.49226	−0.746130	+	0.665800i	$0.768090\pi$
$774$	0	0
$775$	−66.9783	−2.40593
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	52.4674	1.87743
$782$	0	0
$783$	0	0
$784$	0	0
$785$	67.7228	2.41713
$786$	0	0
$787$	21.4891	0.766005	0.383002	−	0.923747i	$-0.374890\pi$
$787$	21.4891	0.766005	0.383002	+	0.923747i	$0.374890\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.72281	−0.274592
$792$	0	0
$793$	−16.7446	−0.594617
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.51087	0.230627	0.115314	−	0.993329i	$-0.463213\pi$
$797$	6.51087	0.230627	0.115314	+	0.993329i	$0.463213\pi$
$798$	0	0
$799$	−33.3505	−1.17986
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.62772	−0.0574409
$804$	0	0
$805$	−28.4674	−1.00334
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.6277	0.690074	0.345037	−	0.938589i	$-0.387866\pi$
$809$	19.6277	0.690074	0.345037	+	0.938589i	$0.387866\pi$
$810$	0	0
$811$	−34.2337	−1.20211	−0.601054	−	0.799209i	$-0.705252\pi$
$811$	−34.2337	−1.20211	−0.601054	+	0.799209i	$0.705252\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	58.9783	2.06592
$816$	0	0
$817$	−2.37228	−0.0829956
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41.5842	−1.45130	−0.725650	−	0.688064i	$-0.758461\pi$
$821$	−41.5842	−1.45130	−0.725650	+	0.688064i	$0.758461\pi$
$822$	0	0
$823$	37.3505	1.30196	0.650979	−	0.759096i	$-0.274359\pi$
$823$	37.3505	1.30196	0.650979	+	0.759096i	$0.274359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.4891	−0.608156	−0.304078	−	0.952647i	$-0.598348\pi$
$827$	−17.4891	−0.608156	−0.304078	+	0.952647i	$0.598348\pi$
$828$	0	0
$829$	−7.76631	−0.269735	−0.134868	−	0.990864i	$-0.543061\pi$
$829$	−7.76631	−0.269735	−0.134868	+	0.990864i	$0.543061\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	62.2337	2.15369
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.74456	0.301896	0.150948	−	0.988542i	$-0.451767\pi$
$839$	8.74456	0.301896	0.150948	+	0.988542i	$0.451767\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39.3505	1.35370
$846$	0	0
$847$	19.2554	0.661625
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.5109	−0.634545
$852$	0	0
$853$	31.4891	1.07817	0.539084	−	0.842252i	$-0.318771\pi$
$853$	31.4891	1.07817	0.539084	+	0.842252i	$0.318771\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.9783	−0.375010	−0.187505	−	0.982264i	$-0.560040\pi$
$857$	−10.9783	−0.375010	−0.187505	+	0.982264i	$0.560040\pi$
$858$	0	0
$859$	−44.6060	−1.52194	−0.760968	−	0.648789i	$-0.775276\pi$
$859$	−44.6060	−1.52194	−0.760968	+	0.648789i	$0.775276\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.02175	0.0347808	0.0173904	−	0.999849i	$-0.494464\pi$
$863$	1.02175	0.0347808	0.0173904	+	0.999849i	$0.494464\pi$
$864$	0	0
$865$	90.7011	3.08393
$866$	0	0
$867$	0	0
$868$	0	0
$869$	34.9783	1.18656
$870$	0	0
$871$	−26.9783	−0.914123
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−94.5625	−3.19679
$876$	0	0
$877$	−16.5109	−0.557533	−0.278766	−	0.960359i	$-0.589925\pi$
$877$	−16.5109	−0.557533	−0.278766	+	0.960359i	$0.589925\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	56.8397	1.91498	0.957488	−	0.288472i	$-0.0931472\pi$
$881$	56.8397	1.91498	0.957488	+	0.288472i	$0.0931472\pi$
$882$	0	0
$883$	1.35053	0.0454490	0.0227245	−	0.999742i	$-0.492766\pi$
$883$	1.35053	0.0454490	0.0227245	+	0.999742i	$0.492766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.4891	−1.39307	−0.696534	−	0.717524i	$-0.745276\pi$
$887$	−41.4891	−1.39307	−0.696534	+	0.717524i	$0.745276\pi$
$888$	0	0
$889$	−52.7446	−1.76900
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.62772	0.255252
$894$	0	0
$895$	−14.2337	−0.475780
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.4891	−1.38374
$900$	0	0
$901$	−38.2337	−1.27375
$902$	0	0
$903$	0	0
$904$	0	0
$905$	96.1902	3.19747
$906$	0	0
$907$	−5.76631	−0.191467	−0.0957336	−	0.995407i	$-0.530520\pi$
$907$	−5.76631	−0.191467	−0.0957336	+	0.995407i	$0.530520\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.2337	−1.26674	−0.633369	−	0.773850i	$-0.718329\pi$
$911$	−38.2337	−1.26674	−0.633369	+	0.773850i	$0.718329\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	23.3940	0.772539
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−95.2119	−3.13055
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11.4891	−0.376946	−0.188473	−	0.982078i	$-0.560354\pi$
$929$	−11.4891	−0.376946	−0.188473	+	0.982078i	$0.560354\pi$
$930$	0	0
$931$	1.37228	0.0449747
$932$	0	0
$933$	0	0
$934$	0	0
$935$	83.5842	2.73350
$936$	0	0
$937$	16.8397	0.550128	0.275064	−	0.961426i	$-0.411301\pi$
$937$	16.8397	0.550128	0.275064	+	0.961426i	$0.411301\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−27.2554	−0.888502	−0.444251	−	0.895902i	$-0.646530\pi$
$941$	−27.2554	−0.888502	−0.444251	+	0.895902i	$0.646530\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.7228	0.445932	0.222966	−	0.974826i	$-0.428426\pi$
$947$	13.7228	0.445932	0.222966	+	0.974826i	$0.428426\pi$
$948$	0	0
$949$	0.744563	0.0241695
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−16.4674	−0.533431	−0.266715	−	0.963775i	$-0.585938\pi$
$953$	−16.4674	−0.533431	−0.266715	+	0.963775i	$0.585938\pi$
$954$	0	0
$955$	71.5842	2.31641
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.3723	0.334938
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22.9783	−0.739696
$966$	0	0
$967$	50.9783	1.63935	0.819675	−	0.572829i	$-0.194154\pi$
$967$	50.9783	1.63935	0.819675	+	0.572829i	$0.194154\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.02175	−0.0327895	−0.0163947	−	0.999866i	$-0.505219\pi$
$971$	−1.02175	−0.0327895	−0.0163947	+	0.999866i	$0.505219\pi$
$972$	0	0
$973$	−22.8397	−0.732206
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.72281	−0.247075	−0.123537	−	0.992340i	$-0.539424\pi$
$977$	−7.72281	−0.247075	−0.123537	+	0.992340i	$0.539424\pi$
$978$	0	0
$979$	14.2337	0.454911
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.51087	−0.207665	−0.103832	−	0.994595i	$-0.533111\pi$
$983$	−6.51087	−0.207665	−0.103832	+	0.994595i	$0.533111\pi$
$984$	0	0
$985$	50.2337	1.60058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.51087	0.207034
$990$	0	0
$991$	37.9565	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$991$	37.9565	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.6277	0.432028
$996$	0	0
$997$	18.8832	0.598036	0.299018	−	0.954248i	$-0.403341\pi$
$997$	18.8832	0.598036	0.299018	+	0.954248i	$0.403341\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.a.y.1.1		2
3.2	odd	2		912.2.a.n.1.2		2
4.3	odd	2		684.2.a.d.1.1		2
12.11	even	2		228.2.a.c.1.2	✓	2
24.5	odd	2		3648.2.a.bq.1.1		2
24.11	even	2		3648.2.a.bk.1.1		2
60.23	odd	4		5700.2.f.m.3649.4		4
60.47	odd	4		5700.2.f.m.3649.1		4
60.59	even	2		5700.2.a.t.1.2		2
228.227	odd	2		4332.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.a.c.1.2	✓	2	12.11	even	2
684.2.a.d.1.1		2	4.3	odd	2
912.2.a.n.1.2		2	3.2	odd	2
2736.2.a.y.1.1		2	1.1	even	1	trivial
3648.2.a.bk.1.1		2	24.11	even	2
3648.2.a.bq.1.1		2	24.5	odd	2
4332.2.a.i.1.2		2	228.227	odd	2
5700.2.a.t.1.2		2	60.59	even	2
5700.2.f.m.3649.1		4	60.47	odd	4
5700.2.f.m.3649.4		4	60.23	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.a (trivial)

\(f(q)\)	\(=\)	\(q-4.37228 q^{5} +2.37228 q^{7} -4.37228 q^{11} +2.00000 q^{13} +4.37228 q^{17} -1.00000 q^{19} +2.74456 q^{23} +14.1168 q^{25} +8.74456 q^{29} -4.74456 q^{31} -10.3723 q^{35} -6.74456 q^{37} +2.37228 q^{43} -7.62772 q^{47} -1.37228 q^{49} -8.74456 q^{53} +19.1168 q^{55} -8.37228 q^{61} -8.74456 q^{65} -13.4891 q^{67} -12.0000 q^{71} +0.372281 q^{73} -10.3723 q^{77} -8.00000 q^{79} -2.74456 q^{83} -19.1168 q^{85} -3.25544 q^{89} +4.74456 q^{91} +4.37228 q^{95} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} - q^{7} - 3 q^{11} + 4 q^{13} + 3 q^{17} - 2 q^{19} - 6 q^{23} + 11 q^{25} + 6 q^{29} + 2 q^{31} - 15 q^{35} - 2 q^{37} - q^{43} - 21 q^{47} + 3 q^{49} - 6 q^{53} + 21 q^{55} - 11 q^{61}+ \cdots + 28 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 - q^7 - 3 * q^11 + 4 * q^13 + 3 * q^17 - 2 * q^19 - 6 * q^23 + 11 * q^25 + 6 * q^29 + 2 * q^31 - 15 * q^35 - 2 * q^37 - q^43 - 21 * q^47 + 3 * q^49 - 6 * q^53 + 21 * q^55 - 11 * q^61 - 6 * q^65 - 4 * q^67 - 24 * q^71 - 5 * q^73 - 15 * q^77 - 16 * q^79 + 6 * q^83 - 21 * q^85 - 18 * q^89 - 2 * q^91 + 3 * q^95 + 28 * q^97

Introduction

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