LMFDB - Embedded newform 5400.2.a.cd.1.1

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			1.1
Root			$-1.30278$ 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)

$f(q)$	$=$	$q-2.60555 q^{7} +O(q^{10})$	$q-2.60555 q^{7} -1.00000 q^{11} +5.60555 q^{13} +2.60555 q^{17} -0.605551 q^{19} -5.60555 q^{23} +0.605551 q^{29} -2.00000 q^{31} -1.60555 q^{37} +6.00000 q^{41} +8.60555 q^{43} -8.81665 q^{47} -0.211103 q^{49} +13.8167 q^{53} -2.21110 q^{59} -9.60555 q^{61} +6.60555 q^{67} -4.39445 q^{71} +6.00000 q^{73} +2.60555 q^{77} +1.39445 q^{79} -5.21110 q^{83} +1.39445 q^{89} -14.6056 q^{91} +10.2111 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.60555	−0.984806	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	−2.60555	−0.984806	−0.492403	+	0.870367i	$0.663881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	5.60555	1.55470	0.777350	−	0.629068i	$-0.216563\pi$
$13$	5.60555	1.55470	0.777350	+	0.629068i	$0.216563\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	0	0
$19$	−0.605551	−0.138923	−0.0694615	−	0.997585i	$-0.522128\pi$
$19$	−0.605551	−0.138923	−0.0694615	+	0.997585i	$0.522128\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.60555	−1.16884	−0.584419	−	0.811452i	$-0.698678\pi$
$23$	−5.60555	−1.16884	−0.584419	+	0.811452i	$0.698678\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.605551	0.112448	0.0562240	−	0.998418i	$-0.482094\pi$
$29$	0.605551	0.112448	0.0562240	+	0.998418i	$0.482094\pi$
$30$	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
$37$	−1.60555	−0.263951	−0.131976	−	0.991253i	$-0.542132\pi$
$37$	−1.60555	−0.263951	−0.131976	+	0.991253i	$0.542132\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	8.60555	1.31233	0.656167	−	0.754616i	$-0.272177\pi$
$43$	8.60555	1.31233	0.656167	+	0.754616i	$0.272177\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.81665	−1.28604	−0.643021	−	0.765849i	$-0.722319\pi$
$47$	−8.81665	−1.28604	−0.643021	+	0.765849i	$0.722319\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.8167	1.89786	0.948932	−	0.315482i	$-0.102166\pi$
$53$	13.8167	1.89786	0.948932	+	0.315482i	$0.102166\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.21110	−0.287861	−0.143931	−	0.989588i	$-0.545974\pi$
$59$	−2.21110	−0.287861	−0.143931	+	0.989588i	$0.545974\pi$
$60$	0	0
$61$	−9.60555	−1.22986	−0.614932	−	0.788580i	$-0.710817\pi$
$61$	−9.60555	−1.22986	−0.614932	+	0.788580i	$0.710817\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.60555	0.806997	0.403498	−	0.914980i	$-0.367794\pi$
$67$	6.60555	0.806997	0.403498	+	0.914980i	$0.367794\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.39445	−0.521525	−0.260763	−	0.965403i	$-0.583974\pi$
$71$	−4.39445	−0.521525	−0.260763	+	0.965403i	$0.583974\pi$
$72$	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
$77$	2.60555	0.296930
$78$	0	0
$79$	1.39445	0.156888	0.0784439	−	0.996919i	$-0.475005\pi$
$79$	1.39445	0.156888	0.0784439	+	0.996919i	$0.475005\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.21110	−0.571993	−0.285996	−	0.958231i	$-0.592325\pi$
$83$	−5.21110	−0.571993	−0.285996	+	0.958231i	$0.592325\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.39445	0.147811	0.0739056	−	0.997265i	$-0.476454\pi$
$89$	1.39445	0.147811	0.0739056	+	0.997265i	$0.476454\pi$
$90$	0	0
$91$	−14.6056	−1.53108
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.2111	1.03678	0.518390	−	0.855144i	$-0.326531\pi$
$97$	10.2111	1.03678	0.518390	+	0.855144i	$0.326531\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.81665	−0.777786	−0.388893	−	0.921283i	$-0.627142\pi$
$101$	−7.81665	−0.777786	−0.388893	+	0.921283i	$0.627142\pi$
$102$	0	0
$103$	0.788897	0.0777324	0.0388662	−	0.999244i	$-0.487625\pi$
$103$	0.788897	0.0777324	0.0388662	+	0.999244i	$0.487625\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.4222	1.49092	0.745460	−	0.666550i	$-0.232230\pi$
$107$	15.4222	1.49092	0.745460	+	0.666550i	$0.232230\pi$
$108$	0	0
$109$	11.2111	1.07383	0.536914	−	0.843637i	$-0.319590\pi$
$109$	11.2111	1.07383	0.536914	+	0.843637i	$0.319590\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.78890	−0.622337
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.2111	1.34977	0.674884	−	0.737924i	$-0.264194\pi$
$127$	15.2111	1.34977	0.674884	+	0.737924i	$0.264194\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.211103	0.0184441	0.00922206	−	0.999957i	$-0.497064\pi$
$131$	0.211103	0.0184441	0.00922206	+	0.999957i	$0.497064\pi$
$132$	0	0
$133$	1.57779	0.136812
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.8167	−1.52218	−0.761090	−	0.648647i	$-0.775335\pi$
$137$	−17.8167	−1.52218	−0.761090	+	0.648647i	$0.775335\pi$
$138$	0	0
$139$	17.0278	1.44428	0.722138	−	0.691749i	$-0.243160\pi$
$139$	17.0278	1.44428	0.722138	+	0.691749i	$0.243160\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.60555	−0.468760
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.2111	1.57383	0.786917	−	0.617058i	$-0.211676\pi$
$149$	19.2111	1.57383	0.786917	+	0.617058i	$0.211676\pi$
$150$	0	0
$151$	0.605551	0.0492791	0.0246395	−	0.999696i	$-0.492156\pi$
$151$	0.605551	0.0492791	0.0246395	+	0.999696i	$0.492156\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.788897	0.0629609	0.0314804	−	0.999504i	$-0.489978\pi$
$157$	0.788897	0.0629609	0.0314804	+	0.999504i	$0.489978\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.6056	1.15108
$162$	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
$167$	16.3944	1.26864	0.634320	−	0.773070i	$-0.281280\pi$
$167$	16.3944	1.26864	0.634320	+	0.773070i	$0.281280\pi$
$168$	0	0
$169$	18.4222	1.41709
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.3944	1.32248	0.661238	−	0.750176i	$-0.270031\pi$
$173$	17.3944	1.32248	0.661238	+	0.750176i	$0.270031\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.2111	−1.06219	−0.531094	−	0.847313i	$-0.678219\pi$
$179$	−14.2111	−1.06219	−0.531094	+	0.847313i	$0.678219\pi$
$180$	0	0
$181$	6.81665	0.506678	0.253339	−	0.967378i	$-0.418471\pi$
$181$	6.81665	0.506678	0.253339	+	0.967378i	$0.418471\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.2111	−1.24535	−0.622676	−	0.782480i	$-0.713954\pi$
$191$	−17.2111	−1.24535	−0.622676	+	0.782480i	$0.713954\pi$
$192$	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	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	23.2111	1.64539	0.822696	−	0.568482i	$-0.192469\pi$
$199$	23.2111	1.64539	0.822696	+	0.568482i	$0.192469\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.57779	−0.110739
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.605551	0.0418869
$210$	0	0
$211$	5.21110	0.358747	0.179374	−	0.983781i	$-0.442593\pi$
$211$	5.21110	0.358747	0.179374	+	0.983781i	$0.442593\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.21110	0.353753
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.6056	0.982476
$222$	0	0
$223$	21.8167	1.46095	0.730476	−	0.682939i	$-0.239298\pi$
$223$	21.8167	1.46095	0.730476	+	0.682939i	$0.239298\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.42221	0.625374	0.312687	−	0.949856i	$-0.398771\pi$
$227$	9.42221	0.625374	0.312687	+	0.949856i	$0.398771\pi$
$228$	0	0
$229$	8.39445	0.554721	0.277360	−	0.960766i	$-0.410540\pi$
$229$	8.39445	0.554721	0.277360	+	0.960766i	$0.410540\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.0278	−1.11553	−0.557763	−	0.830000i	$-0.688340\pi$
$233$	−17.0278	−1.11553	−0.557763	+	0.830000i	$0.688340\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.8167	1.47589	0.737943	−	0.674863i	$-0.235797\pi$
$239$	22.8167	1.47589	0.737943	+	0.674863i	$0.235797\pi$
$240$	0	0
$241$	−0.577795	−0.0372190	−0.0186095	−	0.999827i	$-0.505924\pi$
$241$	−0.577795	−0.0372190	−0.0186095	+	0.999827i	$0.505924\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.39445	−0.215984
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.211103	0.0133247	0.00666234	−	0.999978i	$-0.497879\pi$
$251$	0.211103	0.0133247	0.00666234	+	0.999978i	$0.497879\pi$
$252$	0	0
$253$	5.60555	0.352418
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.39445	0.0869833	0.0434917	−	0.999054i	$-0.486152\pi$
$257$	1.39445	0.0869833	0.0434917	+	0.999054i	$0.486152\pi$
$258$	0	0
$259$	4.18335	0.259940
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.81665	0.297008	0.148504	−	0.988912i	$-0.452554\pi$
$263$	4.81665	0.297008	0.148504	+	0.988912i	$0.452554\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.2111	−1.04938	−0.524690	−	0.851294i	$-0.675819\pi$
$269$	−17.2111	−1.04938	−0.524690	+	0.851294i	$0.675819\pi$
$270$	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
$277$	20.4222	1.22705	0.613526	−	0.789675i	$-0.289751\pi$
$277$	20.4222	1.22705	0.613526	+	0.789675i	$0.289751\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.8167	1.30147	0.650736	−	0.759304i	$-0.274460\pi$
$281$	21.8167	1.30147	0.650736	+	0.759304i	$0.274460\pi$
$282$	0	0
$283$	−5.81665	−0.345764	−0.172882	−	0.984943i	$-0.555308\pi$
$283$	−5.81665	−0.345764	−0.172882	+	0.984943i	$0.555308\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.6333	−0.922805
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.60555	0.385900	0.192950	−	0.981209i	$-0.438194\pi$
$293$	6.60555	0.385900	0.192950	+	0.981209i	$0.438194\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−31.4222	−1.81719
$300$	0	0
$301$	−22.4222	−1.29239
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.42221	−0.480681	−0.240340	−	0.970689i	$-0.577259\pi$
$307$	−8.42221	−0.480681	−0.240340	+	0.970689i	$0.577259\pi$
$308$	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
$313$	20.4222	1.15433	0.577166	−	0.816627i	$-0.304159\pi$
$313$	20.4222	1.15433	0.577166	+	0.816627i	$0.304159\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.3944	−1.08930	−0.544650	−	0.838663i	$-0.683338\pi$
$317$	−19.3944	−1.08930	−0.544650	+	0.838663i	$0.683338\pi$
$318$	0	0
$319$	−0.605551	−0.0339044
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.57779	−0.0877909
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	22.9722	1.26650
$330$	0	0
$331$	−15.2111	−0.836078	−0.418039	−	0.908429i	$-0.637282\pi$
$331$	−15.2111	−0.836078	−0.418039	+	0.908429i	$0.637282\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.57779	0.194895	0.0974475	−	0.995241i	$-0.468932\pi$
$337$	3.57779	0.194895	0.0974475	+	0.995241i	$0.468932\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	18.7889	1.01451
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.4222	1.84788	0.923940	−	0.382536i	$-0.124949\pi$
$347$	34.4222	1.84788	0.923940	+	0.382536i	$0.124949\pi$
$348$	0	0
$349$	8.78890	0.470459	0.235229	−	0.971940i	$-0.424416\pi$
$349$	8.78890	0.470459	0.235229	+	0.971940i	$0.424416\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.3944	0.606465	0.303233	−	0.952917i	$-0.401934\pi$
$353$	11.3944	0.606465	0.303233	+	0.952917i	$0.401934\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.18335	−0.0624546	−0.0312273	−	0.999512i	$-0.509942\pi$
$359$	−1.18335	−0.0624546	−0.0312273	+	0.999512i	$0.509942\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.7889	0.563176	0.281588	−	0.959535i	$-0.409139\pi$
$367$	10.7889	0.563176	0.281588	+	0.959535i	$0.409139\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−36.0000	−1.86903
$372$	0	0
$373$	37.6333	1.94858	0.974289	−	0.225300i	$-0.0723363\pi$
$373$	37.6333	1.94858	0.974289	+	0.225300i	$0.0723363\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.39445	0.174823
$378$	0	0
$379$	−17.6333	−0.905762	−0.452881	−	0.891571i	$-0.649604\pi$
$379$	−17.6333	−0.905762	−0.452881	+	0.891571i	$0.649604\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.8167	−1.47246	−0.736231	−	0.676730i	$-0.763396\pi$
$383$	−28.8167	−1.47246	−0.736231	+	0.676730i	$0.763396\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−37.8167	−1.91738	−0.958690	−	0.284452i	$-0.908188\pi$
$389$	−37.8167	−1.91738	−0.958690	+	0.284452i	$0.908188\pi$
$390$	0	0
$391$	−14.6056	−0.738634
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9.60555	−0.482089	−0.241044	−	0.970514i	$-0.577490\pi$
$397$	−9.60555	−0.482089	−0.241044	+	0.970514i	$0.577490\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.3944	1.26814	0.634069	−	0.773276i	$-0.281383\pi$
$401$	25.3944	1.26814	0.634069	+	0.773276i	$0.281383\pi$
$402$	0	0
$403$	−11.2111	−0.558465
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.60555	0.0795842
$408$	0	0
$409$	6.57779	0.325251	0.162626	−	0.986688i	$-0.448004\pi$
$409$	6.57779	0.325251	0.162626	+	0.986688i	$0.448004\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.76114	0.283487
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	39.6333	1.93621	0.968107	−	0.250538i	$-0.0806073\pi$
$419$	39.6333	1.93621	0.968107	+	0.250538i	$0.0806073\pi$
$420$	0	0
$421$	−14.0278	−0.683671	−0.341836	−	0.939760i	$-0.611049\pi$
$421$	−14.0278	−0.683671	−0.341836	+	0.939760i	$0.611049\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.0278	1.21118
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.2389	−1.50472	−0.752361	−	0.658751i	$-0.771085\pi$
$431$	−31.2389	−1.50472	−0.752361	+	0.658751i	$0.771085\pi$
$432$	0	0
$433$	−31.4222	−1.51005	−0.755027	−	0.655693i	$-0.772376\pi$
$433$	−31.4222	−1.51005	−0.755027	+	0.655693i	$0.772376\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.39445	0.162379
$438$	0	0
$439$	−15.8167	−0.754888	−0.377444	−	0.926032i	$-0.623197\pi$
$439$	−15.8167	−0.754888	−0.377444	+	0.926032i	$0.623197\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.4222	1.01780	0.508900	−	0.860826i	$-0.330052\pi$
$443$	21.4222	1.01780	0.508900	+	0.860826i	$0.330052\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.6333	1.58725	0.793627	−	0.608405i	$-0.208190\pi$
$449$	33.6333	1.58725	0.793627	+	0.608405i	$0.208190\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.0555	−1.54627	−0.773136	−	0.634240i	$-0.781313\pi$
$457$	−33.0555	−1.54627	−0.773136	+	0.634240i	$0.781313\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	34.0555	1.58612	0.793062	−	0.609141i	$-0.208486\pi$
$461$	34.0555	1.58612	0.793062	+	0.609141i	$0.208486\pi$
$462$	0	0
$463$	8.23886	0.382892	0.191446	−	0.981503i	$-0.438682\pi$
$463$	8.23886	0.382892	0.191446	+	0.981503i	$0.438682\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.4222	−0.528557	−0.264278	−	0.964446i	$-0.585134\pi$
$467$	−11.4222	−0.528557	−0.264278	+	0.964446i	$0.585134\pi$
$468$	0	0
$469$	−17.2111	−0.794735
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.60555	−0.395684
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.57779	−0.437621	−0.218810	−	0.975767i	$-0.570218\pi$
$479$	−9.57779	−0.437621	−0.218810	+	0.975767i	$0.570218\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.18335	0.370823	0.185411	−	0.982661i	$-0.440638\pi$
$487$	8.18335	0.370823	0.185411	+	0.982661i	$0.440638\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.6333	0.705521	0.352761	−	0.935714i	$-0.385243\pi$
$491$	15.6333	0.705521	0.352761	+	0.935714i	$0.385243\pi$
$492$	0	0
$493$	1.57779	0.0710603
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.4500	0.513601
$498$	0	0
$499$	−13.2111	−0.591410	−0.295705	−	0.955279i	$-0.595555\pi$
$499$	−13.2111	−0.591410	−0.295705	+	0.955279i	$0.595555\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.21110	−0.410703	−0.205351	−	0.978688i	$-0.565834\pi$
$503$	−9.21110	−0.410703	−0.205351	+	0.978688i	$0.565834\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.78890	0.212264	0.106132	−	0.994352i	$-0.466153\pi$
$509$	4.78890	0.212264	0.106132	+	0.994352i	$0.466153\pi$
$510$	0	0
$511$	−15.6333	−0.691577
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.81665	0.387756
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.3944	0.937308	0.468654	−	0.883382i	$-0.344739\pi$
$521$	21.3944	0.937308	0.468654	+	0.883382i	$0.344739\pi$
$522$	0	0
$523$	40.8444	1.78600	0.893001	−	0.450055i	$-0.148596\pi$
$523$	40.8444	1.78600	0.893001	+	0.450055i	$0.148596\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.21110	−0.226999
$528$	0	0
$529$	8.42221	0.366183
$530$	0	0
$531$	0	0
$532$	0	0
$533$	33.6333	1.45682
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.211103	0.00909283
$540$	0	0
$541$	21.2389	0.913130	0.456565	−	0.889690i	$-0.349080\pi$
$541$	21.2389	0.913130	0.456565	+	0.889690i	$0.349080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6.78890	−0.290272	−0.145136	−	0.989412i	$-0.546362\pi$
$547$	−6.78890	−0.290272	−0.145136	+	0.989412i	$0.546362\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.366692	−0.0156216
$552$	0	0
$553$	−3.63331	−0.154504
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−37.0278	−1.56892	−0.784458	−	0.620182i	$-0.787059\pi$
$557$	−37.0278	−1.56892	−0.784458	+	0.620182i	$0.787059\pi$
$558$	0	0
$559$	48.2389	2.04029
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−40.2111	−1.69470	−0.847348	−	0.531038i	$-0.821802\pi$
$563$	−40.2111	−1.69470	−0.847348	+	0.531038i	$0.821802\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.8444	−0.622310	−0.311155	−	0.950359i	$-0.600716\pi$
$569$	−14.8444	−0.622310	−0.311155	+	0.950359i	$0.600716\pi$
$570$	0	0
$571$	40.4222	1.69162	0.845808	−	0.533487i	$-0.179119\pi$
$571$	40.4222	1.69162	0.845808	+	0.533487i	$0.179119\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.6333	0.609193	0.304596	−	0.952482i	$-0.401478\pi$
$577$	14.6333	0.609193	0.304596	+	0.952482i	$0.401478\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.5778	0.563302
$582$	0	0
$583$	−13.8167	−0.572227
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.42221	0.265073	0.132536	−	0.991178i	$-0.457688\pi$
$587$	6.42221	0.265073	0.132536	+	0.991178i	$0.457688\pi$
$588$	0	0
$589$	1.21110	0.0499026
$590$	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
$599$	−12.8444	−0.524808	−0.262404	−	0.964958i	$-0.584515\pi$
$599$	−12.8444	−0.524808	−0.262404	+	0.964958i	$0.584515\pi$
$600$	0	0
$601$	43.8444	1.78845	0.894225	−	0.447617i	$-0.147727\pi$
$601$	43.8444	1.78845	0.894225	+	0.447617i	$0.147727\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−48.4222	−1.96540	−0.982698	−	0.185213i	$-0.940702\pi$
$607$	−48.4222	−1.96540	−0.982698	+	0.185213i	$0.940702\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−49.4222	−1.99941
$612$	0	0
$613$	−9.97224	−0.402775	−0.201388	−	0.979512i	$-0.564545\pi$
$613$	−9.97224	−0.402775	−0.201388	+	0.979512i	$0.564545\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.7889	0.836929	0.418465	−	0.908233i	$-0.362568\pi$
$617$	20.7889	0.836929	0.418465	+	0.908233i	$0.362568\pi$
$618$	0	0
$619$	17.6333	0.708742	0.354371	−	0.935105i	$-0.384695\pi$
$619$	17.6333	0.708742	0.354371	+	0.935105i	$0.384695\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.63331	−0.145565
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.18335	−0.166801
$630$	0	0
$631$	15.0278	0.598246	0.299123	−	0.954215i	$-0.403306\pi$
$631$	15.0278	0.598246	0.299123	+	0.954215i	$0.403306\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.18335	−0.0468859
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.4500	−1.40019	−0.700095	−	0.714050i	$-0.746859\pi$
$641$	−35.4500	−1.40019	−0.700095	+	0.714050i	$0.746859\pi$
$642$	0	0
$643$	−36.4222	−1.43635	−0.718176	−	0.695862i	$-0.755023\pi$
$643$	−36.4222	−1.43635	−0.718176	+	0.695862i	$0.755023\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.8167	1.36878	0.684392	−	0.729114i	$-0.260068\pi$
$647$	34.8167	1.36878	0.684392	+	0.729114i	$0.260068\pi$
$648$	0	0
$649$	2.21110	0.0867934
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.2389	−1.33987	−0.669935	−	0.742420i	$-0.733678\pi$
$653$	−34.2389	−1.33987	−0.669935	+	0.742420i	$0.733678\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−42.4222	−1.65253	−0.826267	−	0.563278i	$-0.809540\pi$
$659$	−42.4222	−1.65253	−0.826267	+	0.563278i	$0.809540\pi$
$660$	0	0
$661$	20.4500	0.795411	0.397706	−	0.917513i	$-0.369807\pi$
$661$	20.4500	0.795411	0.397706	+	0.917513i	$0.369807\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.39445	−0.131434
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.60555	0.370818
$672$	0	0
$673$	27.8444	1.07332	0.536662	−	0.843798i	$-0.319685\pi$
$673$	27.8444	1.07332	0.536662	+	0.843798i	$0.319685\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	−26.6056	−1.02103
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.57779	0.0986366	0.0493183	−	0.998783i	$-0.484295\pi$
$683$	2.57779	0.0986366	0.0493183	+	0.998783i	$0.484295\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	77.4500	2.95061
$690$	0	0
$691$	−3.57779	−0.136106	−0.0680529	−	0.997682i	$-0.521679\pi$
$691$	−3.57779	−0.136106	−0.0680529	+	0.997682i	$0.521679\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.6333	0.592154
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.0278	−0.492052	−0.246026	−	0.969263i	$-0.579125\pi$
$701$	−13.0278	−0.492052	−0.246026	+	0.969263i	$0.579125\pi$
$702$	0	0
$703$	0.972244	0.0366689
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.3667	0.765968
$708$	0	0
$709$	−8.81665	−0.331116	−0.165558	−	0.986200i	$-0.552943\pi$
$709$	−8.81665	−0.331116	−0.165558	+	0.986200i	$0.552943\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.2111	0.419859
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.6056	−0.432814	−0.216407	−	0.976303i	$-0.569434\pi$
$719$	−11.6056	−0.432814	−0.216407	+	0.976303i	$0.569434\pi$
$720$	0	0
$721$	−2.05551	−0.0765513
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30.0555	1.11470	0.557349	−	0.830279i	$-0.311819\pi$
$727$	30.0555	1.11470	0.557349	+	0.830279i	$0.311819\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22.4222	0.829315
$732$	0	0
$733$	−23.2389	−0.858347	−0.429173	−	0.903222i	$-0.641195\pi$
$733$	−23.2389	−0.858347	−0.429173	+	0.903222i	$0.641195\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.60555	−0.243319
$738$	0	0
$739$	26.2389	0.965212	0.482606	−	0.875838i	$-0.339690\pi$
$739$	26.2389	0.965212	0.482606	+	0.875838i	$0.339690\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.3944	−1.04169	−0.520846	−	0.853651i	$-0.674383\pi$
$743$	−28.3944	−1.04169	−0.520846	+	0.853651i	$0.674383\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−40.1833	−1.46827
$750$	0	0
$751$	14.7889	0.539655	0.269827	−	0.962909i	$-0.413033\pi$
$751$	14.7889	0.539655	0.269827	+	0.962909i	$0.413033\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.0278	−0.655230	−0.327615	−	0.944811i	$-0.606245\pi$
$757$	−18.0278	−0.655230	−0.327615	+	0.944811i	$0.606245\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.4222	0.885304	0.442652	−	0.896693i	$-0.354038\pi$
$761$	24.4222	0.885304	0.442652	+	0.896693i	$0.354038\pi$
$762$	0	0
$763$	−29.2111	−1.05751
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.3944	−0.447538
$768$	0	0
$769$	32.0555	1.15595	0.577976	−	0.816054i	$-0.303843\pi$
$769$	32.0555	1.15595	0.577976	+	0.816054i	$0.303843\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.8167	0.496950	0.248475	−	0.968638i	$-0.420071\pi$
$773$	13.8167	0.496950	0.248475	+	0.968638i	$0.420071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.63331	−0.130177
$780$	0	0
$781$	4.39445	0.157246
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.84441	−0.172685	−0.0863423	−	0.996266i	$-0.527518\pi$
$787$	−4.84441	−0.172685	−0.0863423	+	0.996266i	$0.527518\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.6333	0.555856
$792$	0	0
$793$	−53.8444	−1.91207
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.6056	−1.01326	−0.506630	−	0.862163i	$-0.669109\pi$
$797$	−28.6056	−1.01326	−0.506630	+	0.862163i	$0.669109\pi$
$798$	0	0
$799$	−22.9722	−0.812700
$800$	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
$809$	25.8167	0.907665	0.453833	−	0.891087i	$-0.350056\pi$
$809$	25.8167	0.907665	0.453833	+	0.891087i	$0.350056\pi$
$810$	0	0
$811$	−23.8167	−0.836316	−0.418158	−	0.908374i	$-0.637324\pi$
$811$	−23.8167	−0.836316	−0.418158	+	0.908374i	$0.637324\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.21110	−0.182313
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.6333	−0.475806	−0.237903	−	0.971289i	$-0.576460\pi$
$821$	−13.6333	−0.475806	−0.237903	+	0.971289i	$0.576460\pi$
$822$	0	0
$823$	−39.3944	−1.37320	−0.686602	−	0.727033i	$-0.740899\pi$
$823$	−39.3944	−1.37320	−0.686602	+	0.727033i	$0.740899\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	−33.1833	−1.15251	−0.576253	−	0.817272i	$-0.695486\pi$
$829$	−33.1833	−1.15251	−0.576253	+	0.817272i	$0.695486\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.550039	−0.0190577
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.4500	−1.39649	−0.698244	−	0.715860i	$-0.746035\pi$
$839$	−40.4500	−1.39649	−0.698244	+	0.715860i	$0.746035\pi$
$840$	0	0
$841$	−28.6333	−0.987355
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.0555	0.895278
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.00000	0.308516
$852$	0	0
$853$	−38.8167	−1.32906	−0.664528	−	0.747263i	$-0.731368\pi$
$853$	−38.8167	−1.32906	−0.664528	+	0.747263i	$0.731368\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.21110	−0.314645	−0.157323	−	0.987547i	$-0.550286\pi$
$857$	−9.21110	−0.314645	−0.157323	+	0.987547i	$0.550286\pi$
$858$	0	0
$859$	−11.3944	−0.388774	−0.194387	−	0.980925i	$-0.562272\pi$
$859$	−11.3944	−0.388774	−0.194387	+	0.980925i	$0.562272\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.4222	0.627099	0.313550	−	0.949572i	$-0.398482\pi$
$863$	18.4222	0.627099	0.313550	+	0.949572i	$0.398482\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.39445	−0.0473034
$870$	0	0
$871$	37.0278	1.25464
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.8167	0.702928	0.351464	−	0.936201i	$-0.385684\pi$
$877$	20.8167	0.702928	0.351464	+	0.936201i	$0.385684\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.21110	−0.310330	−0.155165	−	0.987889i	$-0.549591\pi$
$881$	−9.21110	−0.310330	−0.155165	+	0.987889i	$0.549591\pi$
$882$	0	0
$883$	−9.02776	−0.303808	−0.151904	−	0.988395i	$-0.548540\pi$
$883$	−9.02776	−0.303808	−0.151904	+	0.988395i	$0.548540\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.0278	1.54546	0.772730	−	0.634734i	$-0.218890\pi$
$887$	46.0278	1.54546	0.772730	+	0.634734i	$0.218890\pi$
$888$	0	0
$889$	−39.6333	−1.32926
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.33894	0.178661
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.21110	−0.0403925
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.6056	−0.551378	−0.275689	−	0.961247i	$-0.588906\pi$
$907$	−16.6056	−0.551378	−0.275689	+	0.961247i	$0.588906\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.6056	−0.782087	−0.391043	−	0.920372i	$-0.627886\pi$
$911$	−23.6056	−0.782087	−0.391043	+	0.920372i	$0.627886\pi$
$912$	0	0
$913$	5.21110	0.172462
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.550039	−0.0181639
$918$	0	0
$919$	−33.4500	−1.10341	−0.551706	−	0.834039i	$-0.686023\pi$
$919$	−33.4500	−1.10341	−0.551706	+	0.834039i	$0.686023\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−24.6333	−0.810815
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.4777	1.78736	0.893678	−	0.448709i	$-0.148116\pi$
$929$	54.4777	1.78736	0.893678	+	0.448709i	$0.148116\pi$
$930$	0	0
$931$	0.127833	0.00418957
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.2111	−0.856279	−0.428140	−	0.903713i	$-0.640831\pi$
$937$	−26.2111	−0.856279	−0.428140	+	0.903713i	$0.640831\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.2389	0.333777	0.166889	−	0.985976i	$-0.446628\pi$
$941$	10.2389	0.333777	0.166889	+	0.985976i	$0.446628\pi$
$942$	0	0
$943$	−33.6333	−1.09525
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.0555	−1.23664	−0.618319	−	0.785927i	$-0.712186\pi$
$947$	−38.0555	−1.23664	−0.618319	+	0.785927i	$0.712186\pi$
$948$	0	0
$949$	33.6333	1.09178
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.42221	−0.208036	−0.104018	−	0.994575i	$-0.533170\pi$
$953$	−6.42221	−0.208036	−0.104018	+	0.994575i	$0.533170\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	46.4222	1.49905
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39.6333	−1.27452	−0.637261	−	0.770648i	$-0.719933\pi$
$967$	−39.6333	−1.27452	−0.637261	+	0.770648i	$0.719933\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.4222	0.815837	0.407919	−	0.913018i	$-0.366255\pi$
$971$	25.4222	0.815837	0.407919	+	0.913018i	$0.366255\pi$
$972$	0	0
$973$	−44.3667	−1.42233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.6333	1.77987	0.889934	−	0.456090i	$-0.150751\pi$
$977$	55.6333	1.77987	0.889934	+	0.456090i	$0.150751\pi$
$978$	0	0
$979$	−1.39445	−0.0445668
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.6056	0.816690	0.408345	−	0.912828i	$-0.366106\pi$
$983$	25.6056	0.816690	0.408345	+	0.912828i	$0.366106\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.2389	−1.53391
$990$	0	0
$991$	2.42221	0.0769439	0.0384719	−	0.999260i	$-0.487751\pi$
$991$	2.42221	0.0769439	0.0384719	+	0.999260i	$0.487751\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.81665	0.152545	0.0762725	−	0.997087i	$-0.475698\pi$
$997$	4.81665	0.152545	0.0762725	+	0.997087i	$0.475698\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.a.cd.1.1	yes	2
3.2	odd	2		5400.2.a.ce.1.1	yes	2
5.2	odd	4		5400.2.f.bc.649.2		4
5.3	odd	4		5400.2.f.bc.649.3		4
5.4	even	2		5400.2.a.bx.1.2	✓	2
15.2	even	4		5400.2.f.bf.649.2		4
15.8	even	4		5400.2.f.bf.649.3		4
15.14	odd	2		5400.2.a.by.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5400.2.a.bx.1.2	✓	2	5.4	even	2
5400.2.a.by.1.2	yes	2	15.14	odd	2
5400.2.a.cd.1.1	yes	2	1.1	even	1	trivial
5400.2.a.ce.1.1	yes	2	3.2	odd	2
5400.2.f.bc.649.2		4	5.2	odd	4
5400.2.f.bc.649.3		4	5.3	odd	4
5400.2.f.bf.649.2		4	15.2	even	4
5400.2.f.bf.649.3		4	15.8	even	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)

\(f(q)\)	\(=\)	\(q-2.60555 q^{7} +O(q^{10})\)	\(q-2.60555 q^{7} -1.00000 q^{11} +5.60555 q^{13} +2.60555 q^{17} -0.605551 q^{19} -5.60555 q^{23} +0.605551 q^{29} -2.00000 q^{31} -1.60555 q^{37} +6.00000 q^{41} +8.60555 q^{43} -8.81665 q^{47} -0.211103 q^{49} +13.8167 q^{53} -2.21110 q^{59} -9.60555 q^{61} +6.60555 q^{67} -4.39445 q^{71} +6.00000 q^{73} +2.60555 q^{77} +1.39445 q^{79} -5.21110 q^{83} +1.39445 q^{89} -14.6056 q^{91} +10.2111 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 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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