LMFDB - Embedded newform 5472.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(1,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, 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("5472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.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:	$43.6941399860$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	5472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{5} -3.00000 q^{7} +O(q^{10})$	$q+2.56155 q^{5} -3.00000 q^{7} +0.561553 q^{11} -1.56155 q^{13} -0.123106 q^{17} -1.00000 q^{19} +5.56155 q^{23} +1.56155 q^{25} -4.68466 q^{29} -9.12311 q^{31} -7.68466 q^{35} +7.12311 q^{37} -4.00000 q^{41} -5.43845 q^{43} -3.68466 q^{47} +2.00000 q^{49} +4.43845 q^{53} +1.43845 q^{55} -4.43845 q^{59} +14.8078 q^{61} -4.00000 q^{65} +0.438447 q^{67} -4.24621 q^{71} -9.24621 q^{73} -1.68466 q^{77} -10.0000 q^{79} -17.3693 q^{83} -0.315342 q^{85} +15.3693 q^{89} +4.68466 q^{91} -2.56155 q^{95} +7.12311 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + q^5 - 6 * q^7	$ 2 q + q^{5} - 6 q^{7} - 3 q^{11} + q^{13} + 8 q^{17} - 2 q^{19} + 7 q^{23} - q^{25} + 3 q^{29} - 10 q^{31} - 3 q^{35} + 6 q^{37} - 8 q^{41} - 15 q^{43} + 5 q^{47} + 4 q^{49} + 13 q^{53} + 7 q^{55} - 13 q^{59} + 9 q^{61} - 8 q^{65} + 5 q^{67} + 8 q^{71} - 2 q^{73} + 9 q^{77} - 20 q^{79} - 10 q^{83} - 13 q^{85} + 6 q^{89} - 3 q^{91} - q^{95} + 6 q^{97}+O(q^{100}) $ 2 * q + q^5 - 6 * q^7 - 3 * q^11 + q^13 + 8 * q^17 - 2 * q^19 + 7 * q^23 - q^25 + 3 * q^29 - 10 * q^31 - 3 * q^35 + 6 * q^37 - 8 * q^41 - 15 * q^43 + 5 * q^47 + 4 * q^49 + 13 * q^53 + 7 * q^55 - 13 * q^59 + 9 * q^61 - 8 * q^65 + 5 * q^67 + 8 * q^71 - 2 * q^73 + 9 * q^77 - 20 * q^79 - 10 * q^83 - 13 * q^85 + 6 * q^89 - 3 * q^91 - q^95 + 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
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.561553	0.169315	0.0846573	−	0.996410i	$-0.473020\pi$
$11$	0.561553	0.169315	0.0846573	+	0.996410i	$0.473020\pi$
$12$	0	0
$13$	−1.56155	−0.433097	−0.216548	−	0.976272i	$-0.569480\pi$
$13$	−1.56155	−0.433097	−0.216548	+	0.976272i	$0.569480\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.123106	−0.0298575	−0.0149287	−	0.999889i	$-0.504752\pi$
$17$	−0.123106	−0.0298575	−0.0149287	+	0.999889i	$0.504752\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.68466	−0.869919	−0.434960	−	0.900450i	$-0.643237\pi$
$29$	−4.68466	−0.869919	−0.434960	+	0.900450i	$0.643237\pi$
$30$	0	0
$31$	−9.12311	−1.63856	−0.819279	−	0.573395i	$-0.805626\pi$
$31$	−9.12311	−1.63856	−0.819279	+	0.573395i	$0.805626\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.68466	−1.29894
$36$	0	0
$37$	7.12311	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−5.43845	−0.829355	−0.414678	−	0.909968i	$-0.636106\pi$
$43$	−5.43845	−0.829355	−0.414678	+	0.909968i	$0.636106\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.68466	−0.537463	−0.268731	−	0.963215i	$-0.586604\pi$
$47$	−3.68466	−0.537463	−0.268731	+	0.963215i	$0.586604\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.43845	0.609668	0.304834	−	0.952406i	$-0.401399\pi$
$53$	4.43845	0.609668	0.304834	+	0.952406i	$0.401399\pi$
$54$	0	0
$55$	1.43845	0.193960
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.43845	−0.577837	−0.288918	−	0.957354i	$-0.593296\pi$
$59$	−4.43845	−0.577837	−0.288918	+	0.957354i	$0.593296\pi$
$60$	0	0
$61$	14.8078	1.89594	0.947970	−	0.318360i	$-0.103132\pi$
$61$	14.8078	1.89594	0.947970	+	0.318360i	$0.103132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	0.438447	0.0535648	0.0267824	−	0.999641i	$-0.491474\pi$
$67$	0.438447	0.0535648	0.0267824	+	0.999641i	$0.491474\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.24621	−0.503933	−0.251966	−	0.967736i	$-0.581077\pi$
$71$	−4.24621	−0.503933	−0.251966	+	0.967736i	$0.581077\pi$
$72$	0	0
$73$	−9.24621	−1.08219	−0.541094	−	0.840962i	$-0.681990\pi$
$73$	−9.24621	−1.08219	−0.541094	+	0.840962i	$0.681990\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.68466	−0.191985
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.3693	−1.90653	−0.953265	−	0.302135i	$-0.902301\pi$
$83$	−17.3693	−1.90653	−0.953265	+	0.302135i	$0.902301\pi$
$84$	0	0
$85$	−0.315342	−0.0342036
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.3693	1.62914	0.814572	−	0.580062i	$-0.196972\pi$
$89$	15.3693	1.62914	0.814572	+	0.580062i	$0.196972\pi$
$90$	0	0
$91$	4.68466	0.491086
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.56155	−0.262810
$96$	0	0
$97$	7.12311	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$97$	7.12311	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	−14.4924	−1.42798	−0.713990	−	0.700155i	$-0.753114\pi$
$103$	−14.4924	−1.42798	−0.713990	+	0.700155i	$0.753114\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.9309	1.63677	0.818384	−	0.574671i	$-0.194870\pi$
$107$	16.9309	1.63677	0.818384	+	0.574671i	$0.194870\pi$
$108$	0	0
$109$	−3.56155	−0.341135	−0.170567	−	0.985346i	$-0.554560\pi$
$109$	−3.56155	−0.341135	−0.170567	+	0.985346i	$0.554560\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.87689	−0.458780	−0.229390	−	0.973335i	$-0.573673\pi$
$113$	−4.87689	−0.458780	−0.229390	+	0.973335i	$0.573673\pi$
$114$	0	0
$115$	14.2462	1.32847
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.369317	0.0338552
$120$	0	0
$121$	−10.6847	−0.971333
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.80776	−0.787790
$126$	0	0
$127$	−15.1231	−1.34196	−0.670979	−	0.741476i	$-0.734126\pi$
$127$	−15.1231	−1.34196	−0.670979	+	0.741476i	$0.734126\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.4384	−1.17412	−0.587061	−	0.809542i	$-0.699715\pi$
$131$	−13.4384	−1.17412	−0.587061	+	0.809542i	$0.699715\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.3693	−1.22765	−0.613827	−	0.789441i	$-0.710371\pi$
$137$	−14.3693	−1.22765	−0.613827	+	0.789441i	$0.710371\pi$
$138$	0	0
$139$	−5.43845	−0.461283	−0.230642	−	0.973039i	$-0.574082\pi$
$139$	−5.43845	−0.461283	−0.230642	+	0.973039i	$0.574082\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.876894	−0.0733296
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.5616	−0.865236	−0.432618	−	0.901577i	$-0.642410\pi$
$149$	−10.5616	−0.865236	−0.432618	+	0.901577i	$0.642410\pi$
$150$	0	0
$151$	22.4924	1.83041	0.915204	−	0.402992i	$-0.132030\pi$
$151$	22.4924	1.83041	0.915204	+	0.402992i	$0.132030\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−23.3693	−1.87707
$156$	0	0
$157$	−7.75379	−0.618820	−0.309410	−	0.950929i	$-0.600131\pi$
$157$	−7.75379	−0.618820	−0.309410	+	0.950929i	$0.600131\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.6847	−1.31494
$162$	0	0
$163$	−16.4924	−1.29179	−0.645893	−	0.763428i	$-0.723515\pi$
$163$	−16.4924	−1.29179	−0.645893	+	0.763428i	$0.723515\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−10.5616	−0.812427
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.2462	1.53929	0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462	1.53929	0.769645	+	0.638471i	$0.220433\pi$
$174$	0	0
$175$	−4.68466	−0.354127
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.24621	0.167890	0.0839449	−	0.996470i	$-0.473248\pi$
$179$	2.24621	0.167890	0.0839449	+	0.996470i	$0.473248\pi$
$180$	0	0
$181$	−0.246211	−0.0183007	−0.00915037	−	0.999958i	$-0.502913\pi$
$181$	−0.246211	−0.0183007	−0.00915037	+	0.999958i	$0.502913\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.2462	1.34149
$186$	0	0
$187$	−0.0691303	−0.00505531
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	0	0
$193$	18.4924	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$193$	18.4924	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.3693	−0.810030	−0.405015	−	0.914310i	$-0.632734\pi$
$197$	−11.3693	−0.810030	−0.405015	+	0.914310i	$0.632734\pi$
$198$	0	0
$199$	23.7386	1.68279	0.841394	−	0.540423i	$-0.181736\pi$
$199$	23.7386	1.68279	0.841394	+	0.540423i	$0.181736\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.0540	0.986396
$204$	0	0
$205$	−10.2462	−0.715626
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.561553	−0.0388434
$210$	0	0
$211$	−15.5616	−1.07130	−0.535651	−	0.844440i	$-0.679934\pi$
$211$	−15.5616	−1.07130	−0.535651	+	0.844440i	$0.679934\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.9309	−0.950077
$216$	0	0
$217$	27.3693	1.85795
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.192236	0.0129312
$222$	0	0
$223$	1.36932	0.0916962	0.0458481	−	0.998948i	$-0.485401\pi$
$223$	1.36932	0.0916962	0.0458481	+	0.998948i	$0.485401\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.6847	1.10740	0.553700	−	0.832716i	$-0.313216\pi$
$227$	16.6847	1.10740	0.553700	+	0.832716i	$0.313216\pi$
$228$	0	0
$229$	−12.5616	−0.830091	−0.415045	−	0.909801i	$-0.636234\pi$
$229$	−12.5616	−0.830091	−0.415045	+	0.909801i	$0.636234\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.9309	1.30571	0.652857	−	0.757481i	$-0.273570\pi$
$233$	19.9309	1.30571	0.652857	+	0.757481i	$0.273570\pi$
$234$	0	0
$235$	−9.43845	−0.615696
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−14.2462	−0.917679	−0.458840	−	0.888519i	$-0.651735\pi$
$241$	−14.2462	−0.917679	−0.458840	+	0.888519i	$0.651735\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.12311	0.327303
$246$	0	0
$247$	1.56155	0.0993592
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.68466	0.358812	0.179406	−	0.983775i	$-0.442582\pi$
$251$	5.68466	0.358812	0.179406	+	0.983775i	$0.442582\pi$
$252$	0	0
$253$	3.12311	0.196348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.2462	−1.63719	−0.818597	−	0.574369i	$-0.805248\pi$
$257$	−26.2462	−1.63719	−0.818597	+	0.574369i	$0.805248\pi$
$258$	0	0
$259$	−21.3693	−1.32782
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.0540	0.681617	0.340809	−	0.940133i	$-0.389299\pi$
$263$	11.0540	0.681617	0.340809	+	0.940133i	$0.389299\pi$
$264$	0	0
$265$	11.3693	0.698412
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.4924	1.12750	0.563751	−	0.825944i	$-0.309358\pi$
$269$	18.4924	1.12750	0.563751	+	0.825944i	$0.309358\pi$
$270$	0	0
$271$	−10.0540	−0.610736	−0.305368	−	0.952234i	$-0.598779\pi$
$271$	−10.0540	−0.610736	−0.305368	+	0.952234i	$0.598779\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.876894	0.0528787
$276$	0	0
$277$	−11.6847	−0.702063	−0.351032	−	0.936364i	$-0.614169\pi$
$277$	−11.6847	−0.702063	−0.351032	+	0.936364i	$0.614169\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.630683	0.0376234	0.0188117	−	0.999823i	$-0.494012\pi$
$281$	0.630683	0.0376234	0.0188117	+	0.999823i	$0.494012\pi$
$282$	0	0
$283$	−28.5616	−1.69781	−0.848904	−	0.528547i	$-0.822737\pi$
$283$	−28.5616	−1.69781	−0.848904	+	0.528547i	$0.822737\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−16.9848	−0.999109
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.3153	−0.894732	−0.447366	−	0.894351i	$-0.647638\pi$
$293$	−15.3153	−0.894732	−0.447366	+	0.894351i	$0.647638\pi$
$294$	0	0
$295$	−11.3693	−0.661947
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.68466	−0.502247
$300$	0	0
$301$	16.3153	0.940401
$302$	0	0
$303$	0	0
$304$	0	0
$305$	37.9309	2.17192
$306$	0	0
$307$	−4.63068	−0.264287	−0.132144	−	0.991231i	$-0.542186\pi$
$307$	−4.63068	−0.264287	−0.132144	+	0.991231i	$0.542186\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.12311	−0.347209	−0.173605	−	0.984815i	$-0.555542\pi$
$311$	−6.12311	−0.347209	−0.173605	+	0.984815i	$0.555542\pi$
$312$	0	0
$313$	−0.930870	−0.0526159	−0.0263079	−	0.999654i	$-0.508375\pi$
$313$	−0.930870	−0.0526159	−0.0263079	+	0.999654i	$0.508375\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.4384	−0.923275	−0.461638	−	0.887069i	$-0.652738\pi$
$317$	−16.4384	−0.923275	−0.461638	+	0.887069i	$0.652738\pi$
$318$	0	0
$319$	−2.63068	−0.147290
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.123106	0.00684978
$324$	0	0
$325$	−2.43845	−0.135261
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.0540	0.609425
$330$	0	0
$331$	−4.43845	−0.243959	−0.121980	−	0.992533i	$-0.538924\pi$
$331$	−4.43845	−0.243959	−0.121980	+	0.992533i	$0.538924\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.12311	0.0613618
$336$	0	0
$337$	−20.4924	−1.11629	−0.558147	−	0.829742i	$-0.688487\pi$
$337$	−20.4924	−1.11629	−0.558147	+	0.829742i	$0.688487\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.12311	−0.277432
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.56155	−0.244877	−0.122438	−	0.992476i	$-0.539071\pi$
$347$	−4.56155	−0.244877	−0.122438	+	0.992476i	$0.539071\pi$
$348$	0	0
$349$	13.6847	0.732523	0.366261	−	0.930512i	$-0.380638\pi$
$349$	13.6847	0.732523	0.366261	+	0.930512i	$0.380638\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.9309	1.53983	0.769917	−	0.638144i	$-0.220297\pi$
$353$	28.9309	1.53983	0.769917	+	0.638144i	$0.220297\pi$
$354$	0	0
$355$	−10.8769	−0.577286
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.8769	1.47129	0.735643	−	0.677369i	$-0.236880\pi$
$359$	27.8769	1.47129	0.735643	+	0.677369i	$0.236880\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−23.6847	−1.23971
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.3153	−0.691298
$372$	0	0
$373$	−14.9309	−0.773091	−0.386546	−	0.922270i	$-0.626332\pi$
$373$	−14.9309	−0.773091	−0.386546	+	0.922270i	$0.626332\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.31534	0.376759
$378$	0	0
$379$	−16.0540	−0.824637	−0.412319	−	0.911040i	$-0.635281\pi$
$379$	−16.0540	−0.824637	−0.412319	+	0.911040i	$0.635281\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.2462	1.23892	0.619462	−	0.785027i	$-0.287351\pi$
$383$	24.2462	1.23892	0.619462	+	0.785027i	$0.287351\pi$
$384$	0	0
$385$	−4.31534	−0.219930
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.8078	−1.35921	−0.679604	−	0.733579i	$-0.737848\pi$
$389$	−26.8078	−1.35921	−0.679604	+	0.733579i	$0.737848\pi$
$390$	0	0
$391$	−0.684658	−0.0346247
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−25.6155	−1.28886
$396$	0	0
$397$	−5.43845	−0.272948	−0.136474	−	0.990644i	$-0.543577\pi$
$397$	−5.43845	−0.272948	−0.136474	+	0.990644i	$0.543577\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.3693	−1.16701	−0.583504	−	0.812110i	$-0.698319\pi$
$401$	−23.3693	−1.16701	−0.583504	+	0.812110i	$0.698319\pi$
$402$	0	0
$403$	14.2462	0.709654
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	18.8769	0.933402	0.466701	−	0.884415i	$-0.345442\pi$
$409$	18.8769	0.933402	0.466701	+	0.884415i	$0.345442\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.3153	0.655205
$414$	0	0
$415$	−44.4924	−2.18405
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.2462	−1.28221	−0.641106	−	0.767453i	$-0.721524\pi$
$419$	−26.2462	−1.28221	−0.641106	+	0.767453i	$0.721524\pi$
$420$	0	0
$421$	20.0540	0.977371	0.488685	−	0.872460i	$-0.337477\pi$
$421$	20.0540	0.977371	0.488685	+	0.872460i	$0.337477\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.192236	−0.00932481
$426$	0	0
$427$	−44.4233	−2.14979
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.1231	−1.49915	−0.749574	−	0.661921i	$-0.769741\pi$
$431$	−31.1231	−1.49915	−0.749574	+	0.661921i	$0.769741\pi$
$432$	0	0
$433$	39.6155	1.90380	0.951900	−	0.306408i	$-0.0991271\pi$
$433$	39.6155	1.90380	0.951900	+	0.306408i	$0.0991271\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.56155	−0.266045
$438$	0	0
$439$	−1.75379	−0.0837038	−0.0418519	−	0.999124i	$-0.513326\pi$
$439$	−1.75379	−0.0837038	−0.0418519	+	0.999124i	$0.513326\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.68466	0.0800405	0.0400203	−	0.999199i	$-0.487258\pi$
$443$	1.68466	0.0800405	0.0400203	+	0.999199i	$0.487258\pi$
$444$	0	0
$445$	39.3693	1.86628
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.1231	0.713703	0.356852	−	0.934161i	$-0.383850\pi$
$449$	15.1231	0.713703	0.356852	+	0.934161i	$0.383850\pi$
$450$	0	0
$451$	−2.24621	−0.105770
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−36.1231	−1.68977	−0.844884	−	0.534950i	$-0.820330\pi$
$457$	−36.1231	−1.68977	−0.844884	+	0.534950i	$0.820330\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.9309	1.58032	0.790159	−	0.612902i	$-0.209998\pi$
$461$	33.9309	1.58032	0.790159	+	0.612902i	$0.209998\pi$
$462$	0	0
$463$	−15.0540	−0.699618	−0.349809	−	0.936821i	$-0.613753\pi$
$463$	−15.0540	−0.699618	−0.349809	+	0.936821i	$0.613753\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.6847	0.818348	0.409174	−	0.912456i	$-0.365817\pi$
$467$	17.6847	0.818348	0.409174	+	0.912456i	$0.365817\pi$
$468$	0	0
$469$	−1.31534	−0.0607368
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.05398	−0.140422
$474$	0	0
$475$	−1.56155	−0.0716490
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.2462	−0.650926	−0.325463	−	0.945555i	$-0.605520\pi$
$479$	−14.2462	−0.650926	−0.325463	+	0.945555i	$0.605520\pi$
$480$	0	0
$481$	−11.1231	−0.507170
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.2462	0.828518
$486$	0	0
$487$	6.49242	0.294200	0.147100	−	0.989122i	$-0.453006\pi$
$487$	6.49242	0.294200	0.147100	+	0.989122i	$0.453006\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.4924	−1.46636	−0.733181	−	0.680033i	$-0.761965\pi$
$491$	−32.4924	−1.46636	−0.733181	+	0.680033i	$0.761965\pi$
$492$	0	0
$493$	0.576708	0.0259736
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.7386	0.571406
$498$	0	0
$499$	3.68466	0.164948	0.0824740	−	0.996593i	$-0.473718\pi$
$499$	3.68466	0.164948	0.0824740	+	0.996593i	$0.473718\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.43845	−0.108725	−0.0543625	−	0.998521i	$-0.517313\pi$
$503$	−2.43845	−0.108725	−0.0543625	+	0.998521i	$0.517313\pi$
$504$	0	0
$505$	0.630683	0.0280650
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.6155	0.869443	0.434721	−	0.900565i	$-0.356847\pi$
$509$	19.6155	0.869443	0.434721	+	0.900565i	$0.356847\pi$
$510$	0	0
$511$	27.7386	1.22708
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−37.1231	−1.63584
$516$	0	0
$517$	−2.06913	−0.0910002
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−23.3693	−1.02383	−0.511914	−	0.859037i	$-0.671063\pi$
$521$	−23.3693	−1.02383	−0.511914	+	0.859037i	$0.671063\pi$
$522$	0	0
$523$	32.5464	1.42315	0.711577	−	0.702608i	$-0.247981\pi$
$523$	32.5464	1.42315	0.711577	+	0.702608i	$0.247981\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.12311	0.0489232
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.24621	0.270553
$534$	0	0
$535$	43.3693	1.87502
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.12311	0.0483756
$540$	0	0
$541$	−35.4384	−1.52362	−0.761809	−	0.647802i	$-0.775688\pi$
$541$	−35.4384	−1.52362	−0.761809	+	0.647802i	$0.775688\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.12311	−0.390791
$546$	0	0
$547$	−24.4924	−1.04722	−0.523610	−	0.851958i	$-0.675415\pi$
$547$	−24.4924	−1.04722	−0.523610	+	0.851958i	$0.675415\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.68466	0.199573
$552$	0	0
$553$	30.0000	1.27573
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.9309	0.759755	0.379878	−	0.925037i	$-0.375966\pi$
$557$	17.9309	0.759755	0.379878	+	0.925037i	$0.375966\pi$
$558$	0	0
$559$	8.49242	0.359191
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.2462	−1.27473	−0.637363	−	0.770564i	$-0.719975\pi$
$563$	−30.2462	−1.27473	−0.637363	+	0.770564i	$0.719975\pi$
$564$	0	0
$565$	−12.4924	−0.525560
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.7386	1.12094	0.560471	−	0.828174i	$-0.310620\pi$
$569$	26.7386	1.12094	0.560471	+	0.828174i	$0.310620\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.68466	0.362175
$576$	0	0
$577$	5.24621	0.218403	0.109201	−	0.994020i	$-0.465171\pi$
$577$	5.24621	0.218403	0.109201	+	0.994020i	$0.465171\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	52.1080	2.16180
$582$	0	0
$583$	2.49242	0.103226
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.9309	−1.40048	−0.700238	−	0.713909i	$-0.746923\pi$
$587$	−33.9309	−1.40048	−0.700238	+	0.713909i	$0.746923\pi$
$588$	0	0
$589$	9.12311	0.375911
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−32.7386	−1.34441	−0.672207	−	0.740363i	$-0.734654\pi$
$593$	−32.7386	−1.34441	−0.672207	+	0.740363i	$0.734654\pi$
$594$	0	0
$595$	0.946025	0.0387832
$596$	0	0
$597$	0	0
$598$	0	0
$599$	43.8617	1.79214	0.896071	−	0.443911i	$-0.146409\pi$
$599$	43.8617	1.79214	0.896071	+	0.443911i	$0.146409\pi$
$600$	0	0
$601$	−4.63068	−0.188890	−0.0944448	−	0.995530i	$-0.530108\pi$
$601$	−4.63068	−0.188890	−0.0944448	+	0.995530i	$0.530108\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−27.3693	−1.11272
$606$	0	0
$607$	12.8769	0.522657	0.261329	−	0.965250i	$-0.415839\pi$
$607$	12.8769	0.522657	0.261329	+	0.965250i	$0.415839\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.75379	0.232773
$612$	0	0
$613$	−41.5464	−1.67804	−0.839022	−	0.544098i	$-0.816872\pi$
$613$	−41.5464	−1.67804	−0.839022	+	0.544098i	$0.816872\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.6847	−0.872991	−0.436496	−	0.899706i	$-0.643781\pi$
$617$	−21.6847	−0.872991	−0.436496	+	0.899706i	$0.643781\pi$
$618$	0	0
$619$	47.1231	1.89404	0.947019	−	0.321178i	$-0.104079\pi$
$619$	47.1231	1.89404	0.947019	+	0.321178i	$0.104079\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−46.1080	−1.84728
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.876894	−0.0349641
$630$	0	0
$631$	45.3002	1.80337	0.901686	−	0.432391i	$-0.142330\pi$
$631$	45.3002	1.80337	0.901686	+	0.432391i	$0.142330\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−38.7386	−1.53730
$636$	0	0
$637$	−3.12311	−0.123742
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.1080	−0.952207	−0.476103	−	0.879389i	$-0.657951\pi$
$641$	−24.1080	−0.952207	−0.476103	+	0.879389i	$0.657951\pi$
$642$	0	0
$643$	−27.3002	−1.07661	−0.538307	−	0.842749i	$-0.680936\pi$
$643$	−27.3002	−1.07661	−0.538307	+	0.842749i	$0.680936\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.0000	1.84776	0.923880	−	0.382682i	$-0.124999\pi$
$647$	47.0000	1.84776	0.923880	+	0.382682i	$0.124999\pi$
$648$	0	0
$649$	−2.49242	−0.0978361
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.80776	−0.344674	−0.172337	−	0.985038i	$-0.555132\pi$
$653$	−8.80776	−0.344674	−0.172337	+	0.985038i	$0.555132\pi$
$654$	0	0
$655$	−34.4233	−1.34503
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−32.5464	−1.26783	−0.633914	−	0.773404i	$-0.718553\pi$
$659$	−32.5464	−1.26783	−0.633914	+	0.773404i	$0.718553\pi$
$660$	0	0
$661$	−21.1771	−0.823693	−0.411846	−	0.911253i	$-0.635116\pi$
$661$	−21.1771	−0.823693	−0.411846	+	0.911253i	$0.635116\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.68466	0.297998
$666$	0	0
$667$	−26.0540	−1.00881
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.31534	0.321010
$672$	0	0
$673$	−21.1231	−0.814236	−0.407118	−	0.913376i	$-0.633466\pi$
$673$	−21.1231	−0.814236	−0.407118	+	0.913376i	$0.633466\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.43845	0.0937171	0.0468586	−	0.998902i	$-0.485079\pi$
$677$	2.43845	0.0937171	0.0468586	+	0.998902i	$0.485079\pi$
$678$	0	0
$679$	−21.3693	−0.820079
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17.7538	−0.679330	−0.339665	−	0.940547i	$-0.610314\pi$
$683$	−17.7538	−0.679330	−0.339665	+	0.940547i	$0.610314\pi$
$684$	0	0
$685$	−36.8078	−1.40635
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.93087	−0.264045
$690$	0	0
$691$	45.0540	1.71393	0.856967	−	0.515371i	$-0.172346\pi$
$691$	45.0540	1.71393	0.856967	+	0.515371i	$0.172346\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.9309	−0.528428
$696$	0	0
$697$	0.492423	0.0186518
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.6155	−0.514251	−0.257126	−	0.966378i	$-0.582775\pi$
$701$	−13.6155	−0.514251	−0.257126	+	0.966378i	$0.582775\pi$
$702$	0	0
$703$	−7.12311	−0.268653
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.738634	−0.0277792
$708$	0	0
$709$	50.4924	1.89628	0.948141	−	0.317849i	$-0.102960\pi$
$709$	50.4924	1.89628	0.948141	+	0.317849i	$0.102960\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−50.7386	−1.90018
$714$	0	0
$715$	−2.24621	−0.0840035
$716$	0	0
$717$	0	0
$718$	0	0
$719$	23.4924	0.876120	0.438060	−	0.898946i	$-0.355666\pi$
$719$	23.4924	0.876120	0.438060	+	0.898946i	$0.355666\pi$
$720$	0	0
$721$	43.4773	1.61918
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.31534	−0.271685
$726$	0	0
$727$	−22.3693	−0.829632	−0.414816	−	0.909905i	$-0.636154\pi$
$727$	−22.3693	−0.829632	−0.414816	+	0.909905i	$0.636154\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.669503	0.0247625
$732$	0	0
$733$	3.36932	0.124449	0.0622243	−	0.998062i	$-0.480181\pi$
$733$	3.36932	0.124449	0.0622243	+	0.998062i	$0.480181\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.246211	0.00906931
$738$	0	0
$739$	−1.43845	−0.0529141	−0.0264571	−	0.999650i	$-0.508423\pi$
$739$	−1.43845	−0.0529141	−0.0264571	+	0.999650i	$0.508423\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.8769	0.839272	0.419636	−	0.907693i	$-0.362158\pi$
$743$	22.8769	0.839272	0.419636	+	0.907693i	$0.362158\pi$
$744$	0	0
$745$	−27.0540	−0.991181
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−50.7926	−1.85592
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	57.6155	2.09684
$756$	0	0
$757$	21.0540	0.765220	0.382610	−	0.923910i	$-0.375025\pi$
$757$	21.0540	0.765220	0.382610	+	0.923910i	$0.375025\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.8617	1.48124	0.740618	−	0.671926i	$-0.234533\pi$
$761$	40.8617	1.48124	0.740618	+	0.671926i	$0.234533\pi$
$762$	0	0
$763$	10.6847	0.386811
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.93087	0.250259
$768$	0	0
$769$	12.3693	0.446049	0.223024	−	0.974813i	$-0.428407\pi$
$769$	12.3693	0.446049	0.223024	+	0.974813i	$0.428407\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.0540	1.15290	0.576451	−	0.817132i	$-0.304437\pi$
$773$	32.0540	1.15290	0.576451	+	0.817132i	$0.304437\pi$
$774$	0	0
$775$	−14.2462	−0.511739
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	−2.38447	−0.0853231
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.8617	−0.708896
$786$	0	0
$787$	−22.4384	−0.799844	−0.399922	−	0.916549i	$-0.630963\pi$
$787$	−22.4384	−0.799844	−0.399922	+	0.916549i	$0.630963\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.6307	0.520207
$792$	0	0
$793$	−23.1231	−0.821126
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.68466	0.0950955	0.0475477	−	0.998869i	$-0.484859\pi$
$797$	2.68466	0.0950955	0.0475477	+	0.998869i	$0.484859\pi$
$798$	0	0
$799$	0.453602	0.0160473
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.19224	−0.183230
$804$	0	0
$805$	−42.7386	−1.50634
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.12311	−0.215277	−0.107638	−	0.994190i	$-0.534329\pi$
$809$	−6.12311	−0.215277	−0.107638	+	0.994190i	$0.534329\pi$
$810$	0	0
$811$	11.4233	0.401126	0.200563	−	0.979681i	$-0.435723\pi$
$811$	11.4233	0.401126	0.200563	+	0.979681i	$0.435723\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−42.2462	−1.47982
$816$	0	0
$817$	5.43845	0.190267
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.31534	−0.150606	−0.0753032	−	0.997161i	$-0.523992\pi$
$821$	−4.31534	−0.150606	−0.0753032	+	0.997161i	$0.523992\pi$
$822$	0	0
$823$	−20.3693	−0.710030	−0.355015	−	0.934861i	$-0.615524\pi$
$823$	−20.3693	−0.710030	−0.355015	+	0.934861i	$0.615524\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.5616	0.680222	0.340111	−	0.940385i	$-0.389535\pi$
$827$	19.5616	0.680222	0.340111	+	0.940385i	$0.389535\pi$
$828$	0	0
$829$	18.5464	0.644143	0.322072	−	0.946715i	$-0.395621\pi$
$829$	18.5464	0.644143	0.322072	+	0.946715i	$0.395621\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.246211	−0.00853071
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.6155	1.02244	0.511221	−	0.859449i	$-0.329193\pi$
$839$	29.6155	1.02244	0.511221	+	0.859449i	$0.329193\pi$
$840$	0	0
$841$	−7.05398	−0.243241
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−27.0540	−0.930685
$846$	0	0
$847$	32.0540	1.10139
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39.6155	1.35800
$852$	0	0
$853$	32.2462	1.10409	0.552045	−	0.833815i	$-0.313848\pi$
$853$	32.2462	1.10409	0.552045	+	0.833815i	$0.313848\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.6307	−0.431456	−0.215728	−	0.976454i	$-0.569212\pi$
$857$	−12.6307	−0.431456	−0.215728	+	0.976454i	$0.569212\pi$
$858$	0	0
$859$	35.9309	1.22595	0.612973	−	0.790104i	$-0.289974\pi$
$859$	35.9309	1.22595	0.612973	+	0.790104i	$0.289974\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.6307	0.770357	0.385179	−	0.922842i	$-0.374140\pi$
$863$	22.6307	0.770357	0.385179	+	0.922842i	$0.374140\pi$
$864$	0	0
$865$	51.8617	1.76335
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.61553	−0.190494
$870$	0	0
$871$	−0.684658	−0.0231988
$872$	0	0
$873$	0	0
$874$	0	0
$875$	26.4233	0.893270
$876$	0	0
$877$	2.68466	0.0906545	0.0453272	−	0.998972i	$-0.485567\pi$
$877$	2.68466	0.0906545	0.0453272	+	0.998972i	$0.485567\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−49.6847	−1.67392	−0.836959	−	0.547265i	$-0.815669\pi$
$881$	−49.6847	−1.67392	−0.836959	+	0.547265i	$0.815669\pi$
$882$	0	0
$883$	−0.0691303	−0.00232642	−0.00116321	−	0.999999i	$-0.500370\pi$
$883$	−0.0691303	−0.00232642	−0.00116321	+	0.999999i	$0.500370\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.2311	−1.25010	−0.625048	−	0.780586i	$-0.714921\pi$
$887$	−37.2311	−1.25010	−0.625048	+	0.780586i	$0.714921\pi$
$888$	0	0
$889$	45.3693	1.52164
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.68466	0.123302
$894$	0	0
$895$	5.75379	0.192328
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42.7386	1.42541
$900$	0	0
$901$	−0.546398	−0.0182032
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.630683	−0.0209646
$906$	0	0
$907$	15.5616	0.516713	0.258356	−	0.966050i	$-0.416819\pi$
$907$	15.5616	0.516713	0.258356	+	0.966050i	$0.416819\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.12311	−0.302262	−0.151131	−	0.988514i	$-0.548292\pi$
$911$	−9.12311	−0.302262	−0.151131	+	0.988514i	$0.548292\pi$
$912$	0	0
$913$	−9.75379	−0.322803
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.3153	1.33133
$918$	0	0
$919$	24.3002	0.801589	0.400795	−	0.916168i	$-0.368734\pi$
$919$	24.3002	0.801589	0.400795	+	0.916168i	$0.368734\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.63068	0.218252
$924$	0	0
$925$	11.1231	0.365725
$926$	0	0
$927$	0	0
$928$	0	0
$929$	55.1771	1.81030	0.905151	−	0.425091i	$-0.139758\pi$
$929$	55.1771	1.81030	0.905151	+	0.425091i	$0.139758\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.177081	−0.00579117
$936$	0	0
$937$	−12.7538	−0.416648	−0.208324	−	0.978060i	$-0.566801\pi$
$937$	−12.7538	−0.416648	−0.208324	+	0.978060i	$0.566801\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.3002	−0.596569	−0.298285	−	0.954477i	$-0.596414\pi$
$941$	−18.3002	−0.596569	−0.298285	+	0.954477i	$0.596414\pi$
$942$	0	0
$943$	−22.2462	−0.724436
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.9848	1.20185	0.600923	−	0.799307i	$-0.294800\pi$
$947$	36.9848	1.20185	0.600923	+	0.799307i	$0.294800\pi$
$948$	0	0
$949$	14.4384	0.468692
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.24621	0.267121	0.133560	−	0.991041i	$-0.457359\pi$
$953$	8.24621	0.267121	0.133560	+	0.991041i	$0.457359\pi$
$954$	0	0
$955$	−2.56155	−0.0828899
$956$	0	0
$957$	0	0
$958$	0	0
$959$	43.1080	1.39203
$960$	0	0
$961$	52.2311	1.68487
$962$	0	0
$963$	0	0
$964$	0	0
$965$	47.3693	1.52487
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.5076	−0.754394	−0.377197	−	0.926133i	$-0.623112\pi$
$971$	−23.5076	−0.754394	−0.377197	+	0.926133i	$0.623112\pi$
$972$	0	0
$973$	16.3153	0.523046
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.6155	1.71531	0.857656	−	0.514223i	$-0.171920\pi$
$977$	53.6155	1.71531	0.857656	+	0.514223i	$0.171920\pi$
$978$	0	0
$979$	8.63068	0.275838
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.6155	0.370478	0.185239	−	0.982694i	$-0.440694\pi$
$983$	11.6155	0.370478	0.185239	+	0.982694i	$0.440694\pi$
$984$	0	0
$985$	−29.1231	−0.927939
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.2462	−0.961774
$990$	0	0
$991$	18.2462	0.579610	0.289805	−	0.957086i	$-0.406410\pi$
$991$	18.2462	0.579610	0.289805	+	0.957086i	$0.406410\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	60.8078	1.92774
$996$	0	0
$997$	−46.9157	−1.48584	−0.742918	−	0.669383i	$-0.766559\pi$
$997$	−46.9157	−1.48584	−0.742918	+	0.669383i	$0.766559\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.a.bc.1.2		2
3.2	odd	2		608.2.a.g.1.2	✓	2
4.3	odd	2		5472.2.a.bf.1.2		2
12.11	even	2		608.2.a.h.1.1	yes	2
24.5	odd	2		1216.2.a.t.1.1		2
24.11	even	2		1216.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.g.1.2	✓	2	3.2	odd	2
608.2.a.h.1.1	yes	2	12.11	even	2
1216.2.a.s.1.2		2	24.11	even	2
1216.2.a.t.1.1		2	24.5	odd	2
5472.2.a.bc.1.2		2	1.1	even	1	trivial
5472.2.a.bf.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{5} -3.00000 q^{7} +O(q^{10})\)	\(q+2.56155 q^{5} -3.00000 q^{7} +0.561553 q^{11} -1.56155 q^{13} -0.123106 q^{17} -1.00000 q^{19} +5.56155 q^{23} +1.56155 q^{25} -4.68466 q^{29} -9.12311 q^{31} -7.68466 q^{35} +7.12311 q^{37} -4.00000 q^{41} -5.43845 q^{43} -3.68466 q^{47} +2.00000 q^{49} +4.43845 q^{53} +1.43845 q^{55} -4.43845 q^{59} +14.8078 q^{61} -4.00000 q^{65} +0.438447 q^{67} -4.24621 q^{71} -9.24621 q^{73} -1.68466 q^{77} -10.0000 q^{79} -17.3693 q^{83} -0.315342 q^{85} +15.3693 q^{89} +4.68466 q^{91} -2.56155 q^{95} +7.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + q^5 - 6 * q^7	\( 2 q + q^{5} - 6 q^{7} - 3 q^{11} + q^{13} + 8 q^{17} - 2 q^{19} + 7 q^{23} - q^{25} + 3 q^{29} - 10 q^{31} - 3 q^{35} + 6 q^{37} - 8 q^{41} - 15 q^{43} + 5 q^{47} + 4 q^{49} + 13 q^{53} + 7 q^{55} - 13 q^{59} + 9 q^{61} - 8 q^{65} + 5 q^{67} + 8 q^{71} - 2 q^{73} + 9 q^{77} - 20 q^{79} - 10 q^{83} - 13 q^{85} + 6 q^{89} - 3 q^{91} - q^{95} + 6 q^{97}+O(q^{100}) \) 2 * q + q^5 - 6 * q^7 - 3 * q^11 + q^13 + 8 * q^17 - 2 * q^19 + 7 * q^23 - q^25 + 3 * q^29 - 10 * q^31 - 3 * q^35 + 6 * q^37 - 8 * q^41 - 15 * q^43 + 5 * q^47 + 4 * q^49 + 13 * q^53 + 7 * q^55 - 13 * q^59 + 9 * q^61 - 8 * q^65 + 5 * q^67 + 8 * q^71 - 2 * q^73 + 9 * q^77 - 20 * q^79 - 10 * q^83 - 13 * q^85 + 6 * q^89 - 3 * q^91 - q^95 + 6 * q^97

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