LMFDB - Embedded newform 5400.2.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(1,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.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:	$43.1192170915$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	5400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.60555 q^{7} +O(q^{10})$	$q+4.60555 q^{7} +1.00000 q^{11} -1.60555 q^{13} +4.60555 q^{17} +6.60555 q^{19} -1.60555 q^{23} +6.60555 q^{29} -2.00000 q^{31} +5.60555 q^{37} -6.00000 q^{41} +1.39445 q^{43} -12.8167 q^{47} +14.2111 q^{49} +7.81665 q^{53} -12.2111 q^{59} -2.39445 q^{61} -0.605551 q^{67} +11.6056 q^{71} +6.00000 q^{73} +4.60555 q^{77} +8.60555 q^{79} -9.21110 q^{83} -8.60555 q^{89} -7.39445 q^{91} -4.21110 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} + 4 q^{23} + 6 q^{29} - 4 q^{31} + 4 q^{37} - 12 q^{41} + 10 q^{43} - 4 q^{47} + 14 q^{49} - 6 q^{53} - 10 q^{59} - 12 q^{61} + 6 q^{67} + 16 q^{71} + 12 q^{73} + 2 q^{77} + 10 q^{79} - 4 q^{83} - 10 q^{89} - 22 q^{91} + 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 2 * q^11 + 4 * q^13 + 2 * q^17 + 6 * q^19 + 4 * q^23 + 6 * q^29 - 4 * q^31 + 4 * q^37 - 12 * q^41 + 10 * q^43 - 4 * q^47 + 14 * q^49 - 6 * q^53 - 10 * q^59 - 12 * q^61 + 6 * q^67 + 16 * q^71 + 12 * q^73 + 2 * q^77 + 10 * q^79 - 4 * q^83 - 10 * q^89 - 22 * q^91 + 6 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.60555	1.74073	0.870367	−	0.492403i	$-0.163881\pi$
$7$	4.60555	1.74073	0.870367	+	0.492403i	$0.163881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−1.60555	−0.445300	−0.222650	−	0.974898i	$-0.571471\pi$
$13$	−1.60555	−0.445300	−0.222650	+	0.974898i	$0.571471\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.60555	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$17$	4.60555	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$18$	0	0
$19$	6.60555	1.51542	0.757709	−	0.652593i	$-0.226319\pi$
$19$	6.60555	1.51542	0.757709	+	0.652593i	$0.226319\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.60555	−0.334781	−0.167390	−	0.985891i	$-0.553534\pi$
$23$	−1.60555	−0.334781	−0.167390	+	0.985891i	$0.553534\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.60555	1.22662	0.613310	−	0.789842i	$-0.289838\pi$
$29$	6.60555	1.22662	0.613310	+	0.789842i	$0.289838\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.60555	0.921547	0.460773	−	0.887518i	$-0.347572\pi$
$37$	5.60555	0.921547	0.460773	+	0.887518i	$0.347572\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	1.39445	0.212651	0.106326	−	0.994331i	$-0.466091\pi$
$43$	1.39445	0.212651	0.106326	+	0.994331i	$0.466091\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.8167	−1.86950	−0.934751	−	0.355305i	$-0.884377\pi$
$47$	−12.8167	−1.86950	−0.934751	+	0.355305i	$0.884377\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.81665	1.07370	0.536850	−	0.843678i	$-0.319614\pi$
$53$	7.81665	1.07370	0.536850	+	0.843678i	$0.319614\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.2111	−1.58975	−0.794875	−	0.606773i	$-0.792464\pi$
$59$	−12.2111	−1.58975	−0.794875	+	0.606773i	$0.792464\pi$
$60$	0	0
$61$	−2.39445	−0.306578	−0.153289	−	0.988181i	$-0.548986\pi$
$61$	−2.39445	−0.306578	−0.153289	+	0.988181i	$0.548986\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.605551	−0.0739799	−0.0369899	−	0.999316i	$-0.511777\pi$
$67$	−0.605551	−0.0739799	−0.0369899	+	0.999316i	$0.511777\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.6056	1.37733	0.688663	−	0.725082i	$-0.258198\pi$
$71$	11.6056	1.37733	0.688663	+	0.725082i	$0.258198\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.60555	0.524851
$78$	0	0
$79$	8.60555	0.968200	0.484100	−	0.875013i	$-0.339147\pi$
$79$	8.60555	0.968200	0.484100	+	0.875013i	$0.339147\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.21110	−1.01105	−0.505525	−	0.862812i	$-0.668701\pi$
$83$	−9.21110	−1.01105	−0.505525	+	0.862812i	$0.668701\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.60555	−0.912187	−0.456093	−	0.889932i	$-0.650752\pi$
$89$	−8.60555	−0.912187	−0.456093	+	0.889932i	$0.650752\pi$
$90$	0	0
$91$	−7.39445	−0.775149
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.21110	−0.427573	−0.213786	−	0.976880i	$-0.568580\pi$
$97$	−4.21110	−0.427573	−0.213786	+	0.976880i	$0.568580\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.8167	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$101$	−13.8167	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$102$	0	0
$103$	15.2111	1.49879	0.749397	−	0.662121i	$-0.230343\pi$
$103$	15.2111	1.49879	0.749397	+	0.662121i	$0.230343\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.4222	1.29757	0.648787	−	0.760970i	$-0.275277\pi$
$107$	13.4222	1.29757	0.648787	+	0.760970i	$0.275277\pi$
$108$	0	0
$109$	−3.21110	−0.307568	−0.153784	−	0.988105i	$-0.549146\pi$
$109$	−3.21110	−0.307568	−0.153784	+	0.988105i	$0.549146\pi$
$110$	0	0
$111$	0	0
$112$	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
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.2111	1.94442
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.788897	0.0700033	0.0350017	−	0.999387i	$-0.488856\pi$
$127$	0.788897	0.0700033	0.0350017	+	0.999387i	$0.488856\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.2111	1.24163	0.620815	−	0.783957i	$-0.286802\pi$
$131$	14.2111	1.24163	0.620815	+	0.783957i	$0.286802\pi$
$132$	0	0
$133$	30.4222	2.63794
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.81665	−0.326079	−0.163039	−	0.986620i	$-0.552130\pi$
$137$	−3.81665	−0.326079	−0.163039	+	0.986620i	$0.552130\pi$
$138$	0	0
$139$	−19.0278	−1.61391	−0.806957	−	0.590611i	$-0.798887\pi$
$139$	−19.0278	−1.61391	−0.806957	+	0.590611i	$0.798887\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.60555	−0.134263
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.78890	−0.392322	−0.196161	−	0.980572i	$-0.562847\pi$
$149$	−4.78890	−0.392322	−0.196161	+	0.980572i	$0.562847\pi$
$150$	0	0
$151$	−6.60555	−0.537552	−0.268776	−	0.963203i	$-0.586619\pi$
$151$	−6.60555	−0.537552	−0.268776	+	0.963203i	$0.586619\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2111	1.21398	0.606989	−	0.794710i	$-0.292377\pi$
$157$	15.2111	1.21398	0.606989	+	0.794710i	$0.292377\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.39445	−0.582764
$162$	0	0
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.6056	−1.82665	−0.913326	−	0.407229i	$-0.866495\pi$
$167$	−23.6056	−1.82665	−0.913326	+	0.407229i	$0.866495\pi$
$168$	0	0
$169$	−10.4222	−0.801708
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−24.6056	−1.87073	−0.935363	−	0.353690i	$-0.884927\pi$
$173$	−24.6056	−1.87073	−0.935363	+	0.353690i	$0.884927\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.211103	−0.0157785	−0.00788927	−	0.999969i	$-0.502511\pi$
$179$	−0.211103	−0.0157785	−0.00788927	+	0.999969i	$0.502511\pi$
$180$	0	0
$181$	−14.8167	−1.10131	−0.550657	−	0.834732i	$-0.685623\pi$
$181$	−14.8167	−1.10131	−0.550657	+	0.834732i	$0.685623\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.60555	0.336791
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.78890	0.201798	0.100899	−	0.994897i	$-0.467828\pi$
$191$	2.78890	0.201798	0.100899	+	0.994897i	$0.467828\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.0000	1.13995	0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000	1.13995	0.569976	+	0.821661i	$0.306952\pi$
$198$	0	0
$199$	8.78890	0.623028	0.311514	−	0.950241i	$-0.399164\pi$
$199$	8.78890	0.623028	0.311514	+	0.950241i	$0.399164\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	30.4222	2.13522
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.60555	0.456916
$210$	0	0
$211$	−9.21110	−0.634118	−0.317059	−	0.948406i	$-0.602695\pi$
$211$	−9.21110	−0.634118	−0.317059	+	0.948406i	$0.602695\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.21110	−0.625290
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.39445	−0.497404
$222$	0	0
$223$	0.183346	0.0122778	0.00613888	−	0.999981i	$-0.498046\pi$
$223$	0.183346	0.0122778	0.00613888	+	0.999981i	$0.498046\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.4222	1.28910	0.644549	−	0.764563i	$-0.277045\pi$
$227$	19.4222	1.28910	0.644549	+	0.764563i	$0.277045\pi$
$228$	0	0
$229$	15.6056	1.03124	0.515622	−	0.856816i	$-0.327561\pi$
$229$	15.6056	1.03124	0.515622	+	0.856816i	$0.327561\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.0278	−1.24655	−0.623275	−	0.782003i	$-0.714198\pi$
$233$	−19.0278	−1.24655	−0.623275	+	0.782003i	$0.714198\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.18335	−0.0765443	−0.0382722	−	0.999267i	$-0.512185\pi$
$239$	−1.18335	−0.0765443	−0.0382722	+	0.999267i	$0.512185\pi$
$240$	0	0
$241$	−29.4222	−1.89525	−0.947625	−	0.319384i	$-0.896524\pi$
$241$	−29.4222	−1.89525	−0.947625	+	0.319384i	$0.896524\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.6056	−0.674815
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.2111	0.896997	0.448498	−	0.893784i	$-0.351959\pi$
$251$	14.2111	0.896997	0.448498	+	0.893784i	$0.351959\pi$
$252$	0	0
$253$	−1.60555	−0.100940
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.60555	−0.536800	−0.268400	−	0.963308i	$-0.586495\pi$
$257$	−8.60555	−0.536800	−0.268400	+	0.963308i	$0.586495\pi$
$258$	0	0
$259$	25.8167	1.60417
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.8167	1.03696	0.518480	−	0.855090i	$-0.326498\pi$
$263$	16.8167	1.03696	0.518480	+	0.855090i	$0.326498\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.78890	0.170042	0.0850210	−	0.996379i	$-0.472904\pi$
$269$	2.78890	0.170042	0.0850210	+	0.996379i	$0.472904\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.42221	−0.506041	−0.253021	−	0.967461i	$-0.581424\pi$
$277$	−8.42221	−0.506041	−0.253021	+	0.967461i	$0.581424\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−0.183346	−0.0109375	−0.00546876	−	0.999985i	$-0.501741\pi$
$281$	−0.183346	−0.0109375	−0.00546876	+	0.999985i	$0.501741\pi$
$282$	0	0
$283$	15.8167	0.940202	0.470101	−	0.882612i	$-0.344217\pi$
$283$	15.8167	0.940202	0.470101	+	0.882612i	$0.344217\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.6333	−1.63114
$288$	0	0
$289$	4.21110	0.247712
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.605551	0.0353767	0.0176883	−	0.999844i	$-0.494369\pi$
$293$	0.605551	0.0353767	0.0176883	+	0.999844i	$0.494369\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.57779	0.149078
$300$	0	0
$301$	6.42221	0.370170
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.4222	1.16556	0.582778	−	0.812631i	$-0.301966\pi$
$307$	20.4222	1.16556	0.582778	+	0.812631i	$0.301966\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.2389	−1.43116	−0.715582	−	0.698529i	$-0.753839\pi$
$311$	−25.2389	−1.43116	−0.715582	+	0.698529i	$0.753839\pi$
$312$	0	0
$313$	−8.42221	−0.476051	−0.238026	−	0.971259i	$-0.576500\pi$
$313$	−8.42221	−0.476051	−0.238026	+	0.971259i	$0.576500\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.6056	1.49432	0.747158	−	0.664646i	$-0.231418\pi$
$317$	26.6056	1.49432	0.747158	+	0.664646i	$0.231418\pi$
$318$	0	0
$319$	6.60555	0.369840
$320$	0	0
$321$	0	0
$322$	0	0
$323$	30.4222	1.69274
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−59.0278	−3.25431
$330$	0	0
$331$	−0.788897	−0.0433617	−0.0216809	−	0.999765i	$-0.506902\pi$
$331$	−0.788897	−0.0433617	−0.0216809	+	0.999765i	$0.506902\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.4222	1.76615	0.883075	−	0.469232i	$-0.155469\pi$
$337$	32.4222	1.76615	0.883075	+	0.469232i	$0.155469\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	33.2111	1.79323
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.57779	−0.299432	−0.149716	−	0.988729i	$-0.547836\pi$
$347$	−5.57779	−0.299432	−0.149716	+	0.988729i	$0.547836\pi$
$348$	0	0
$349$	23.2111	1.24246	0.621231	−	0.783628i	$-0.286633\pi$
$349$	23.2111	1.24246	0.621231	+	0.783628i	$0.286633\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.6056	−0.990274	−0.495137	−	0.868815i	$-0.664882\pi$
$353$	−18.6056	−0.990274	−0.495137	+	0.868815i	$0.664882\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.8167	1.20422	0.602108	−	0.798414i	$-0.294327\pi$
$359$	22.8167	1.20422	0.602108	+	0.798414i	$0.294327\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.2111	1.31601	0.658004	−	0.753014i	$-0.271401\pi$
$367$	25.2111	1.31601	0.658004	+	0.753014i	$0.271401\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	36.0000	1.86903
$372$	0	0
$373$	−5.63331	−0.291682	−0.145841	−	0.989308i	$-0.546589\pi$
$373$	−5.63331	−0.291682	−0.145841	+	0.989308i	$0.546589\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.6056	−0.546214
$378$	0	0
$379$	25.6333	1.31669	0.658347	−	0.752714i	$-0.271256\pi$
$379$	25.6333	1.31669	0.658347	+	0.752714i	$0.271256\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.18335	0.367052	0.183526	−	0.983015i	$-0.441249\pi$
$383$	7.18335	0.367052	0.183526	+	0.983015i	$0.441249\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.1833	0.820528	0.410264	−	0.911967i	$-0.365437\pi$
$389$	16.1833	0.820528	0.410264	+	0.911967i	$0.365437\pi$
$390$	0	0
$391$	−7.39445	−0.373953
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.39445	−0.120174	−0.0600870	−	0.998193i	$-0.519138\pi$
$397$	−2.39445	−0.120174	−0.0600870	+	0.998193i	$0.519138\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.6056	−1.62824	−0.814122	−	0.580694i	$-0.802781\pi$
$401$	−32.6056	−1.62824	−0.814122	+	0.580694i	$0.802781\pi$
$402$	0	0
$403$	3.21110	0.159956
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.60555	0.277857
$408$	0	0
$409$	35.4222	1.75152	0.875758	−	0.482751i	$-0.160362\pi$
$409$	35.4222	1.75152	0.875758	+	0.482751i	$0.160362\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−56.2389	−2.76733
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.63331	0.177499	0.0887493	−	0.996054i	$-0.471713\pi$
$419$	3.63331	0.177499	0.0887493	+	0.996054i	$0.471713\pi$
$420$	0	0
$421$	22.0278	1.07357	0.536784	−	0.843720i	$-0.319639\pi$
$421$	22.0278	1.07357	0.536784	+	0.843720i	$0.319639\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11.0278	−0.533671
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.2389	−0.926703	−0.463352	−	0.886175i	$-0.653353\pi$
$431$	−19.2389	−0.926703	−0.463352	+	0.886175i	$0.653353\pi$
$432$	0	0
$433$	−2.57779	−0.123881	−0.0619405	−	0.998080i	$-0.519729\pi$
$433$	−2.57779	−0.123881	−0.0619405	+	0.998080i	$0.519729\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.6056	−0.507332
$438$	0	0
$439$	5.81665	0.277614	0.138807	−	0.990319i	$-0.455673\pi$
$439$	5.81665	0.277614	0.138807	+	0.990319i	$0.455673\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.42221	0.352640	0.176320	−	0.984333i	$-0.443581\pi$
$443$	7.42221	0.352640	0.176320	+	0.984333i	$0.443581\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.63331	0.454624	0.227312	−	0.973822i	$-0.427006\pi$
$449$	9.63331	0.454624	0.227312	+	0.973822i	$0.427006\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	39.0555	1.82694	0.913470	−	0.406906i	$-0.133392\pi$
$457$	39.0555	1.82694	0.913470	+	0.406906i	$0.133392\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.0555	1.77242	0.886211	−	0.463282i	$-0.153328\pi$
$461$	38.0555	1.77242	0.886211	+	0.463282i	$0.153328\pi$
$462$	0	0
$463$	−42.2389	−1.96301	−0.981503	−	0.191446i	$-0.938682\pi$
$463$	−42.2389	−1.96301	−0.981503	+	0.191446i	$0.938682\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.4222	−0.806204	−0.403102	−	0.915155i	$-0.632068\pi$
$467$	−17.4222	−0.806204	−0.403102	+	0.915155i	$0.632068\pi$
$468$	0	0
$469$	−2.78890	−0.128779
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.39445	0.0641168
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.4222	1.75556	0.877778	−	0.479068i	$-0.159025\pi$
$479$	38.4222	1.75556	0.877778	+	0.479068i	$0.159025\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	29.8167	1.35112	0.675561	−	0.737304i	$-0.263902\pi$
$487$	29.8167	1.35112	0.675561	+	0.737304i	$0.263902\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.6333	1.24707	0.623537	−	0.781794i	$-0.285695\pi$
$491$	27.6333	1.24707	0.623537	+	0.781794i	$0.285695\pi$
$492$	0	0
$493$	30.4222	1.37015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	53.4500	2.39756
$498$	0	0
$499$	1.21110	0.0542164	0.0271082	−	0.999633i	$-0.491370\pi$
$499$	1.21110	0.0542164	0.0271082	+	0.999633i	$0.491370\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.21110	−0.232352	−0.116176	−	0.993229i	$-0.537064\pi$
$503$	−5.21110	−0.232352	−0.116176	+	0.993229i	$0.537064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.2111	−0.851517	−0.425759	−	0.904837i	$-0.639993\pi$
$509$	−19.2111	−0.851517	−0.425759	+	0.904837i	$0.639993\pi$
$510$	0	0
$511$	27.6333	1.22243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.8167	−0.563676
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.6056	−1.25323	−0.626616	−	0.779328i	$-0.715561\pi$
$521$	−28.6056	−1.25323	−0.626616	+	0.779328i	$0.715561\pi$
$522$	0	0
$523$	−16.8444	−0.736555	−0.368277	−	0.929716i	$-0.620052\pi$
$523$	−16.8444	−0.736555	−0.368277	+	0.929716i	$0.620052\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.21110	−0.401242
$528$	0	0
$529$	−20.4222	−0.887922
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.63331	0.417265
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.2111	0.612116
$540$	0	0
$541$	−29.2389	−1.25708	−0.628538	−	0.777779i	$-0.716346\pi$
$541$	−29.2389	−1.25708	−0.628538	+	0.777779i	$0.716346\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−21.2111	−0.906921	−0.453461	−	0.891276i	$-0.649811\pi$
$547$	−21.2111	−0.906921	−0.453461	+	0.891276i	$0.649811\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	43.6333	1.85884
$552$	0	0
$553$	39.6333	1.68538
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.972244	0.0411953	0.0205976	−	0.999788i	$-0.493443\pi$
$557$	0.972244	0.0411953	0.0205976	+	0.999788i	$0.493443\pi$
$558$	0	0
$559$	−2.23886	−0.0946936
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.7889	1.08687	0.543436	−	0.839450i	$-0.317123\pi$
$563$	25.7889	1.08687	0.543436	+	0.839450i	$0.317123\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.8444	−1.79613	−0.898066	−	0.439862i	$-0.855027\pi$
$569$	−42.8444	−1.79613	−0.898066	+	0.439862i	$0.855027\pi$
$570$	0	0
$571$	11.5778	0.484516	0.242258	−	0.970212i	$-0.422112\pi$
$571$	11.5778	0.484516	0.242258	+	0.970212i	$0.422112\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.6333	−1.19202	−0.596010	−	0.802977i	$-0.703248\pi$
$577$	−28.6333	−1.19202	−0.596010	+	0.802977i	$0.703248\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−42.4222	−1.75997
$582$	0	0
$583$	7.81665	0.323733
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.4222	0.925463	0.462732	−	0.886498i	$-0.346869\pi$
$587$	22.4222	0.925463	0.462732	+	0.886498i	$0.346869\pi$
$588$	0	0
$589$	−13.2111	−0.544354
$590$	0	0
$591$	0	0
$592$	0	0
$593$	43.2666	1.77675	0.888373	−	0.459122i	$-0.151836\pi$
$593$	43.2666	1.77675	0.888373	+	0.459122i	$0.151836\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−44.8444	−1.83229	−0.916146	−	0.400844i	$-0.868717\pi$
$599$	−44.8444	−1.83229	−0.916146	+	0.400844i	$0.868717\pi$
$600$	0	0
$601$	−13.8444	−0.564725	−0.282363	−	0.959308i	$-0.591118\pi$
$601$	−13.8444	−0.564725	−0.282363	+	0.959308i	$0.591118\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.5778	−0.794638	−0.397319	−	0.917681i	$-0.630059\pi$
$607$	−19.5778	−0.794638	−0.397319	+	0.917681i	$0.630059\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.5778	0.832488
$612$	0	0
$613$	−46.0278	−1.85904	−0.929522	−	0.368767i	$-0.879780\pi$
$613$	−46.0278	−1.85904	−0.929522	+	0.368767i	$0.879780\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.2111	−1.41755	−0.708773	−	0.705437i	$-0.750751\pi$
$617$	−35.2111	−1.41755	−0.708773	+	0.705437i	$0.750751\pi$
$618$	0	0
$619$	−25.6333	−1.03029	−0.515145	−	0.857103i	$-0.672262\pi$
$619$	−25.6333	−1.03029	−0.515145	+	0.857103i	$0.672262\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−39.6333	−1.58787
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.8167	1.02938
$630$	0	0
$631$	−21.0278	−0.837102	−0.418551	−	0.908193i	$-0.637462\pi$
$631$	−21.0278	−0.837102	−0.418551	+	0.908193i	$0.637462\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−22.8167	−0.904029
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.4500	−1.16320	−0.581602	−	0.813474i	$-0.697574\pi$
$641$	−29.4500	−1.16320	−0.581602	+	0.813474i	$0.697574\pi$
$642$	0	0
$643$	−7.57779	−0.298839	−0.149420	−	0.988774i	$-0.547740\pi$
$643$	−7.57779	−0.298839	−0.149420	+	0.988774i	$0.547740\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.1833	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$647$	−13.1833	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$648$	0	0
$649$	−12.2111	−0.479328
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.2389	−0.635476	−0.317738	−	0.948179i	$-0.602923\pi$
$653$	−16.2389	−0.635476	−0.317738	+	0.948179i	$0.602923\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.5778	0.528916	0.264458	−	0.964397i	$-0.414807\pi$
$659$	13.5778	0.528916	0.264458	+	0.964397i	$0.414807\pi$
$660$	0	0
$661$	−44.4500	−1.72890	−0.864452	−	0.502716i	$-0.832334\pi$
$661$	−44.4500	−1.72890	−0.864452	+	0.502716i	$0.832334\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.6056	−0.410649
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.39445	−0.0924367
$672$	0	0
$673$	−29.8444	−1.15042	−0.575209	−	0.818007i	$-0.695079\pi$
$673$	−29.8444	−1.15042	−0.575209	+	0.818007i	$0.695079\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	0	0
$679$	−19.3944	−0.744291
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−31.4222	−1.20234	−0.601169	−	0.799122i	$-0.705298\pi$
$683$	−31.4222	−1.20234	−0.601169	+	0.799122i	$0.705298\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.5500	−0.478118
$690$	0	0
$691$	−32.4222	−1.23340	−0.616699	−	0.787199i	$-0.711531\pi$
$691$	−32.4222	−1.23340	−0.616699	+	0.787199i	$0.711531\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−27.6333	−1.04669
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23.0278	−0.869746	−0.434873	−	0.900492i	$-0.643207\pi$
$701$	−23.0278	−0.869746	−0.434873	+	0.900492i	$0.643207\pi$
$702$	0	0
$703$	37.0278	1.39653
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−63.6333	−2.39318
$708$	0	0
$709$	12.8167	0.481340	0.240670	−	0.970607i	$-0.422633\pi$
$709$	12.8167	0.481340	0.240670	+	0.970607i	$0.422633\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.21110	0.120257
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.39445	0.163885	0.0819426	−	0.996637i	$-0.473888\pi$
$719$	4.39445	0.163885	0.0819426	+	0.996637i	$0.473888\pi$
$720$	0	0
$721$	70.0555	2.60900
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.0555	−1.55975	−0.779876	−	0.625934i	$-0.784718\pi$
$727$	−42.0555	−1.55975	−0.779876	+	0.625934i	$0.784718\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.42221	0.237534
$732$	0	0
$733$	27.2389	1.00609	0.503045	−	0.864260i	$-0.332213\pi$
$733$	27.2389	1.00609	0.503045	+	0.864260i	$0.332213\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.605551	−0.0223058
$738$	0	0
$739$	−24.2389	−0.891641	−0.445820	−	0.895122i	$-0.647088\pi$
$739$	−24.2389	−0.891641	−0.445820	+	0.895122i	$0.647088\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.6056	1.30624	0.653120	−	0.757254i	$-0.273460\pi$
$743$	35.6056	1.30624	0.653120	+	0.757254i	$0.273460\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	61.8167	2.25873
$750$	0	0
$751$	29.2111	1.06593	0.532964	−	0.846138i	$-0.321078\pi$
$751$	29.2111	1.06593	0.532964	+	0.846138i	$0.321078\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.0278	0.655230	0.327615	−	0.944811i	$-0.393755\pi$
$757$	18.0278	0.655230	0.327615	+	0.944811i	$0.393755\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.42221	0.160305	0.0801524	−	0.996783i	$-0.474459\pi$
$761$	4.42221	0.160305	0.0801524	+	0.996783i	$0.474459\pi$
$762$	0	0
$763$	−14.7889	−0.535394
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.6056	0.707915
$768$	0	0
$769$	−40.0555	−1.44444	−0.722219	−	0.691664i	$-0.756878\pi$
$769$	−40.0555	−1.44444	−0.722219	+	0.691664i	$0.756878\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.81665	0.281145	0.140573	−	0.990070i	$-0.455106\pi$
$773$	7.81665	0.281145	0.140573	+	0.990070i	$0.455106\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−39.6333	−1.42001
$780$	0	0
$781$	11.6056	0.415279
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	52.8444	1.88370	0.941850	−	0.336034i	$-0.109086\pi$
$787$	52.8444	1.88370	0.941850	+	0.336034i	$0.109086\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	27.6333	0.982527
$792$	0	0
$793$	3.84441	0.136519
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.3944	0.757830	0.378915	−	0.925431i	$-0.376297\pi$
$797$	21.3944	0.757830	0.378915	+	0.925431i	$0.376297\pi$
$798$	0	0
$799$	−59.0278	−2.08825
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.18335	−0.147079	−0.0735393	−	0.997292i	$-0.523429\pi$
$809$	−4.18335	−0.147079	−0.0735393	+	0.997292i	$0.523429\pi$
$810$	0	0
$811$	−2.18335	−0.0766677	−0.0383338	−	0.999265i	$-0.512205\pi$
$811$	−2.18335	−0.0766677	−0.0383338	+	0.999265i	$0.512205\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.21110	0.322256
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.6333	−1.03421	−0.517105	−	0.855922i	$-0.672990\pi$
$821$	−29.6333	−1.03421	−0.517105	+	0.855922i	$0.672990\pi$
$822$	0	0
$823$	−46.6056	−1.62457	−0.812284	−	0.583263i	$-0.801776\pi$
$823$	−46.6056	−1.62457	−0.812284	+	0.583263i	$0.801776\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	−54.8167	−1.90386	−0.951931	−	0.306314i	$-0.900904\pi$
$829$	−54.8167	−1.90386	−0.951931	+	0.306314i	$0.900904\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	65.4500	2.26771
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.4500	−0.844106	−0.422053	−	0.906571i	$-0.638690\pi$
$839$	−24.4500	−0.844106	−0.422053	+	0.906571i	$0.638690\pi$
$840$	0	0
$841$	14.6333	0.504597
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−46.0555	−1.58249
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.00000	−0.308516
$852$	0	0
$853$	−17.1833	−0.588347	−0.294173	−	0.955752i	$-0.595044\pi$
$853$	−17.1833	−0.588347	−0.294173	+	0.955752i	$0.595044\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.21110	−0.178008	−0.0890039	−	0.996031i	$-0.528368\pi$
$857$	−5.21110	−0.178008	−0.0890039	+	0.996031i	$0.528368\pi$
$858$	0	0
$859$	−18.6056	−0.634813	−0.317407	−	0.948290i	$-0.602812\pi$
$859$	−18.6056	−0.634813	−0.317407	+	0.948290i	$0.602812\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.4222	0.354776	0.177388	−	0.984141i	$-0.443235\pi$
$863$	10.4222	0.354776	0.177388	+	0.984141i	$0.443235\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.60555	0.291923
$870$	0	0
$871$	0.972244	0.0329432
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.816654	−0.0275764	−0.0137882	−	0.999905i	$-0.504389\pi$
$877$	−0.816654	−0.0275764	−0.0137882	+	0.999905i	$0.504389\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.21110	−0.175567	−0.0877833	−	0.996140i	$-0.527978\pi$
$881$	−5.21110	−0.175567	−0.0877833	+	0.996140i	$0.527978\pi$
$882$	0	0
$883$	27.0278	0.909556	0.454778	−	0.890605i	$-0.349719\pi$
$883$	27.0278	0.909556	0.454778	+	0.890605i	$0.349719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.97224	−0.334835	−0.167418	−	0.985886i	$-0.553543\pi$
$887$	−9.97224	−0.334835	−0.167418	+	0.985886i	$0.553543\pi$
$888$	0	0
$889$	3.63331	0.121857
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−84.6611	−2.83307
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.2111	−0.440615
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−9.39445	−0.311938	−0.155969	−	0.987762i	$-0.549850\pi$
$907$	−9.39445	−0.311938	−0.155969	+	0.987762i	$0.549850\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.3944	0.543172	0.271586	−	0.962414i	$-0.412452\pi$
$911$	16.3944	0.543172	0.271586	+	0.962414i	$0.412452\pi$
$912$	0	0
$913$	−9.21110	−0.304843
$914$	0	0
$915$	0	0
$916$	0	0
$917$	65.4500	2.16135
$918$	0	0
$919$	31.4500	1.03744	0.518719	−	0.854945i	$-0.326409\pi$
$919$	31.4500	1.03744	0.518719	+	0.854945i	$0.326409\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.6333	−0.613323
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.4777	1.52488	0.762442	−	0.647056i	$-0.224000\pi$
$929$	46.4777	1.52488	0.762442	+	0.647056i	$0.224000\pi$
$930$	0	0
$931$	93.8722	3.07654
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−11.7889	−0.385126	−0.192563	−	0.981285i	$-0.561680\pi$
$937$	−11.7889	−0.385126	−0.192563	+	0.981285i	$0.561680\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.2389	1.31175	0.655875	−	0.754870i	$-0.272300\pi$
$941$	40.2389	1.31175	0.655875	+	0.754870i	$0.272300\pi$
$942$	0	0
$943$	9.63331	0.313704
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.0555	−1.10666	−0.553328	−	0.832964i	$-0.686642\pi$
$947$	−34.0555	−1.10666	−0.553328	+	0.832964i	$0.686642\pi$
$948$	0	0
$949$	−9.63331	−0.312710
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.4222	−0.726326	−0.363163	−	0.931726i	$-0.618303\pi$
$953$	−22.4222	−0.726326	−0.363163	+	0.931726i	$0.618303\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.5778	−0.567617
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.63331	0.116839	0.0584196	−	0.998292i	$-0.481394\pi$
$967$	3.63331	0.116839	0.0584196	+	0.998292i	$0.481394\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.42221	0.109824	0.0549119	−	0.998491i	$-0.482512\pi$
$971$	3.42221	0.109824	0.0549119	+	0.998491i	$0.482512\pi$
$972$	0	0
$973$	−87.6333	−2.80939
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.3667	−0.395646	−0.197823	−	0.980238i	$-0.563387\pi$
$977$	−12.3667	−0.395646	−0.197823	+	0.980238i	$0.563387\pi$
$978$	0	0
$979$	−8.60555	−0.275035
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.3944	−0.586692	−0.293346	−	0.956006i	$-0.594769\pi$
$983$	−18.3944	−0.586692	−0.293346	+	0.956006i	$0.594769\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.23886	−0.0711916
$990$	0	0
$991$	−26.4222	−0.839329	−0.419665	−	0.907679i	$-0.637852\pi$
$991$	−26.4222	−0.839329	−0.419665	+	0.907679i	$0.637852\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−16.8167	−0.532589	−0.266294	−	0.963892i	$-0.585799\pi$
$997$	−16.8167	−0.532589	−0.266294	+	0.963892i	$0.585799\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.a.ce.1.2	yes	2
3.2	odd	2		5400.2.a.cd.1.2	yes	2
5.2	odd	4		5400.2.f.bf.649.4		4
5.3	odd	4		5400.2.f.bf.649.1		4
5.4	even	2		5400.2.a.by.1.1	yes	2
15.2	even	4		5400.2.f.bc.649.4		4
15.8	even	4		5400.2.f.bc.649.1		4
15.14	odd	2		5400.2.a.bx.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5400.2.a.bx.1.1	✓	2	15.14	odd	2
5400.2.a.by.1.1	yes	2	5.4	even	2
5400.2.a.cd.1.2	yes	2	3.2	odd	2
5400.2.a.ce.1.2	yes	2	1.1	even	1	trivial
5400.2.f.bc.649.1		4	15.8	even	4
5400.2.f.bc.649.4		4	15.2	even	4
5400.2.f.bf.649.1		4	5.3	odd	4
5400.2.f.bf.649.4		4	5.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.a (trivial)

\(f(q)\)	\(=\)	\(q+4.60555 q^{7} +O(q^{10})\)	\(q+4.60555 q^{7} +1.00000 q^{11} -1.60555 q^{13} +4.60555 q^{17} +6.60555 q^{19} -1.60555 q^{23} +6.60555 q^{29} -2.00000 q^{31} +5.60555 q^{37} -6.00000 q^{41} +1.39445 q^{43} -12.8167 q^{47} +14.2111 q^{49} +7.81665 q^{53} -12.2111 q^{59} -2.39445 q^{61} -0.605551 q^{67} +11.6056 q^{71} +6.00000 q^{73} +4.60555 q^{77} +8.60555 q^{79} -9.21110 q^{83} -8.60555 q^{89} -7.39445 q^{91} -4.21110 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} + 4 q^{23} + 6 q^{29} - 4 q^{31} + 4 q^{37} - 12 q^{41} + 10 q^{43} - 4 q^{47} + 14 q^{49} - 6 q^{53} - 10 q^{59} - 12 q^{61} + 6 q^{67} + 16 q^{71} + 12 q^{73} + 2 q^{77} + 10 q^{79} - 4 q^{83} - 10 q^{89} - 22 q^{91} + 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 2 * q^11 + 4 * q^13 + 2 * q^17 + 6 * q^19 + 4 * q^23 + 6 * q^29 - 4 * q^31 + 4 * q^37 - 12 * q^41 + 10 * q^43 - 4 * q^47 + 14 * q^49 - 6 * q^53 - 10 * q^59 - 12 * q^61 + 6 * q^67 + 16 * q^71 + 12 * q^73 + 2 * q^77 + 10 * q^79 - 4 * q^83 - 10 * q^89 - 22 * q^91 + 6 * q^97

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