LMFDB - Embedded newform 5472.2.a.bt.1.1

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.329727$ 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)

$f(q)$	$=$	$q-3.40617 q^{5} -2.50407 q^{7} +O(q^{10})$	$q-3.40617 q^{5} -2.50407 q^{7} -1.40617 q^{11} +6.22101 q^{13} +3.16352 q^{17} -1.00000 q^{19} -7.91023 q^{23} +6.60197 q^{25} -6.22101 q^{29} -8.13124 q^{31} +8.52927 q^{35} -6.13124 q^{37} -1.68923 q^{41} -1.71694 q^{43} +9.84818 q^{47} -0.729644 q^{49} -2.78713 q^{53} +4.78964 q^{55} +9.25078 q^{59} +6.52927 q^{61} -21.1898 q^{65} -1.87233 q^{67} -13.0081 q^{71} -6.28663 q^{73} +3.52114 q^{77} +11.2543 q^{79} +1.80420 q^{83} -10.7755 q^{85} +7.57426 q^{89} -15.5778 q^{91} +3.40617 q^{95} +4.81233 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} + q^{7}+O(q^{10}) $ 4 * q - q^5 + q^7	$ 4 q - q^{5} + q^{7} + 7 q^{11} + 10 q^{13} - 5 q^{17} - 4 q^{19} - 8 q^{23} + 17 q^{25} - 10 q^{29} + 6 q^{31} + 5 q^{35} + 14 q^{37} + 2 q^{41} - 3 q^{43} - 3 q^{47} + 7 q^{49} - 4 q^{53} + 35 q^{55} + 20 q^{59} - 3 q^{61} + 12 q^{65} - 8 q^{67} - 30 q^{71} + 9 q^{73} + 7 q^{77} - 10 q^{79} + 4 q^{83} + 19 q^{85} + 16 q^{89} - 10 q^{91} + q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q - q^5 + q^7 + 7 * q^11 + 10 * q^13 - 5 * q^17 - 4 * q^19 - 8 * q^23 + 17 * q^25 - 10 * q^29 + 6 * q^31 + 5 * q^35 + 14 * q^37 + 2 * q^41 - 3 * q^43 - 3 * q^47 + 7 * q^49 - 4 * q^53 + 35 * q^55 + 20 * q^59 - 3 * q^61 + 12 * q^65 - 8 * q^67 - 30 * q^71 + 9 * q^73 + 7 * q^77 - 10 * q^79 + 4 * q^83 + 19 * q^85 + 16 * q^89 - 10 * 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
$5$	−3.40617	−1.52328	−0.761642	−	0.647998i	$-0.775606\pi$
$5$	−3.40617	−1.52328	−0.761642	+	0.647998i	$0.775606\pi$
$6$	0	0
$7$	−2.50407	−0.946449	−0.473224	−	0.880942i	$-0.656910\pi$
$7$	−2.50407	−0.946449	−0.473224	+	0.880942i	$0.656910\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.40617	−0.423975	−0.211988	−	0.977272i	$-0.567994\pi$
$11$	−1.40617	−0.423975	−0.211988	+	0.977272i	$0.567994\pi$
$12$	0	0
$13$	6.22101	1.72540	0.862698	−	0.505719i	$-0.168773\pi$
$13$	6.22101	1.72540	0.862698	+	0.505719i	$0.168773\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.16352	0.767267	0.383633	−	0.923485i	$-0.374673\pi$
$17$	3.16352	0.767267	0.383633	+	0.923485i	$0.374673\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.91023	−1.64940	−0.824699	−	0.565572i	$-0.808655\pi$
$23$	−7.91023	−1.64940	−0.824699	+	0.565572i	$0.808655\pi$
$24$	0	0
$25$	6.60197	1.32039
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.22101	−1.15521	−0.577606	−	0.816316i	$-0.696013\pi$
$29$	−6.22101	−1.15521	−0.577606	+	0.816316i	$0.696013\pi$
$30$	0	0
$31$	−8.13124	−1.46041	−0.730207	−	0.683226i	$-0.760576\pi$
$31$	−8.13124	−1.46041	−0.730207	+	0.683226i	$0.760576\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.52927	1.44171
$36$	0	0
$37$	−6.13124	−1.00797	−0.503985	−	0.863712i	$-0.668133\pi$
$37$	−6.13124	−1.00797	−0.503985	+	0.863712i	$0.668133\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.68923	−0.263813	−0.131906	−	0.991262i	$-0.542110\pi$
$41$	−1.68923	−0.263813	−0.131906	+	0.991262i	$0.542110\pi$
$42$	0	0
$43$	−1.71694	−0.261831	−0.130915	−	0.991394i	$-0.541792\pi$
$43$	−1.71694	−0.261831	−0.130915	+	0.991394i	$0.541792\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.84818	1.43650	0.718252	−	0.695783i	$-0.244942\pi$
$47$	9.84818	1.43650	0.718252	+	0.695783i	$0.244942\pi$
$48$	0	0
$49$	−0.729644	−0.104235
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.78713	−0.382842	−0.191421	−	0.981508i	$-0.561310\pi$
$53$	−2.78713	−0.382842	−0.191421	+	0.981508i	$0.561310\pi$
$54$	0	0
$55$	4.78964	0.645834
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.25078	1.20435	0.602174	−	0.798365i	$-0.294301\pi$
$59$	9.25078	1.20435	0.602174	+	0.798365i	$0.294301\pi$
$60$	0	0
$61$	6.52927	0.835988	0.417994	−	0.908450i	$-0.362733\pi$
$61$	6.52927	0.835988	0.417994	+	0.908450i	$0.362733\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21.1898	−2.62827
$66$	0	0
$67$	−1.87233	−0.228741	−0.114370	−	0.993438i	$-0.536485\pi$
$67$	−1.87233	−0.228741	−0.114370	+	0.993438i	$0.536485\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.0081	−1.54378	−0.771891	−	0.635755i	$-0.780689\pi$
$71$	−13.0081	−1.54378	−0.771891	+	0.635755i	$0.780689\pi$
$72$	0	0
$73$	−6.28663	−0.735794	−0.367897	−	0.929867i	$-0.619922\pi$
$73$	−6.28663	−0.735794	−0.367897	+	0.929867i	$0.619922\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.52114	0.401271
$78$	0	0
$79$	11.2543	1.26621	0.633106	−	0.774065i	$-0.281780\pi$
$79$	11.2543	1.26621	0.633106	+	0.774065i	$0.281780\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.80420	0.198036	0.0990182	−	0.995086i	$-0.468430\pi$
$83$	1.80420	0.198036	0.0990182	+	0.995086i	$0.468430\pi$
$84$	0	0
$85$	−10.7755	−1.16877
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.57426	0.802870	0.401435	−	0.915888i	$-0.368512\pi$
$89$	7.57426	0.802870	0.401435	+	0.915888i	$0.368512\pi$
$90$	0	0
$91$	−15.5778	−1.63300
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.40617	0.349465
$96$	0	0
$97$	4.81233	0.488618	0.244309	−	0.969697i	$-0.421439\pi$
$97$	4.81233	0.488618	0.244309	+	0.969697i	$0.421439\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	−7.68923	−0.757642	−0.378821	−	0.925470i	$-0.623670\pi$
$103$	−7.68923	−0.757642	−0.378821	+	0.925470i	$0.623670\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.8886	−1.14931	−0.574657	−	0.818394i	$-0.694865\pi$
$107$	−11.8886	−1.14931	−0.574657	+	0.818394i	$0.694865\pi$
$108$	0	0
$109$	−5.65489	−0.541640	−0.270820	−	0.962630i	$-0.587295\pi$
$109$	−5.65489	−0.541640	−0.270820	+	0.962630i	$0.587295\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.81233	0.828995	0.414497	−	0.910051i	$-0.363957\pi$
$113$	8.81233	0.828995	0.414497	+	0.910051i	$0.363957\pi$
$114$	0	0
$115$	26.9436	2.51250
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.92167	−0.726179
$120$	0	0
$121$	−9.02270	−0.820245
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.45657	−0.488051
$126$	0	0
$127$	−12.0504	−1.06930	−0.534650	−	0.845073i	$-0.679557\pi$
$127$	−12.0504	−1.06930	−0.534650	+	0.845073i	$0.679557\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.920878	0.0804575	0.0402287	−	0.999190i	$-0.487191\pi$
$131$	0.920878	0.0804575	0.0402287	+	0.999190i	$0.487191\pi$
$132$	0	0
$133$	2.50407	0.217130
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.525705	0.0449140	0.0224570	−	0.999748i	$-0.492851\pi$
$137$	0.525705	0.0449140	0.0224570	+	0.999748i	$0.492851\pi$
$138$	0	0
$139$	4.52927	0.384168	0.192084	−	0.981379i	$-0.438475\pi$
$139$	4.52927	0.384168	0.192084	+	0.981379i	$0.438475\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.74777	−0.731525
$144$	0	0
$145$	21.1898	1.75972
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.9713	1.55419	0.777094	−	0.629384i	$-0.216693\pi$
$149$	18.9713	1.55419	0.777094	+	0.629384i	$0.216693\pi$
$150$	0	0
$151$	−6.55698	−0.533600	−0.266800	−	0.963752i	$-0.585966\pi$
$151$	−6.55698	−0.533600	−0.266800	+	0.963752i	$0.585966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	27.6964	2.22463
$156$	0	0
$157$	17.5651	1.40185	0.700925	−	0.713235i	$-0.252771\pi$
$157$	17.5651	1.40185	0.700925	+	0.713235i	$0.252771\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	19.8078	1.56107
$162$	0	0
$163$	22.9436	1.79708	0.898540	−	0.438892i	$-0.144629\pi$
$163$	22.9436	1.79708	0.898540	+	0.438892i	$0.144629\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.24621	0.483346	0.241673	−	0.970358i	$-0.422304\pi$
$167$	6.24621	0.483346	0.241673	+	0.970358i	$0.422304\pi$
$168$	0	0
$169$	25.7009	1.97699
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.9436	1.28820	0.644098	−	0.764943i	$-0.277233\pi$
$173$	16.9436	1.28820	0.644098	+	0.764943i	$0.277233\pi$
$174$	0	0
$175$	−16.5318	−1.24969
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.2462	1.36379	0.681893	−	0.731452i	$-0.261157\pi$
$179$	18.2462	1.36379	0.681893	+	0.731452i	$0.261157\pi$
$180$	0	0
$181$	10.9273	0.812220	0.406110	−	0.913824i	$-0.366885\pi$
$181$	10.9273	0.812220	0.406110	+	0.913824i	$0.366885\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.8840	1.53542
$186$	0	0
$187$	−4.44844	−0.325302
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.5122	−1.12242	−0.561212	−	0.827672i	$-0.689665\pi$
$191$	−15.5122	−1.12242	−0.561212	+	0.827672i	$0.689665\pi$
$192$	0	0
$193$	−9.56512	−0.688512	−0.344256	−	0.938876i	$-0.611869\pi$
$193$	−9.56512	−0.688512	−0.344256	+	0.938876i	$0.611869\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.1394	−1.36362	−0.681812	−	0.731527i	$-0.738808\pi$
$197$	−19.1394	−1.36362	−0.681812	+	0.731527i	$0.738808\pi$
$198$	0	0
$199$	0.133749	0.00948125	0.00474062	−	0.999989i	$-0.498491\pi$
$199$	0.133749	0.00948125	0.00474062	+	0.999989i	$0.498491\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.5778	1.09335
$204$	0	0
$205$	5.75379	0.401862
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.40617	0.0972666
$210$	0	0
$211$	23.0117	1.58419	0.792095	−	0.610397i	$-0.208990\pi$
$211$	23.0117	1.58419	0.792095	+	0.610397i	$0.208990\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.84818	0.398843
$216$	0	0
$217$	20.3612	1.38221
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.6803	1.32384
$222$	0	0
$223$	−7.51471	−0.503222	−0.251611	−	0.967828i	$-0.580960\pi$
$223$	−7.51471	−0.503222	−0.251611	+	0.967828i	$0.580960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.81690	0.386081	0.193041	−	0.981191i	$-0.438165\pi$
$227$	5.81690	0.386081	0.193041	+	0.981191i	$0.438165\pi$
$228$	0	0
$229$	−26.1702	−1.72938	−0.864688	−	0.502309i	$-0.832484\pi$
$229$	−26.1702	−1.72938	−0.864688	+	0.502309i	$0.832484\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.22352	−0.145667	−0.0728337	−	0.997344i	$-0.523204\pi$
$233$	−2.22352	−0.145667	−0.0728337	+	0.997344i	$0.523204\pi$
$234$	0	0
$235$	−33.5445	−2.18820
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.12873	0.525804	0.262902	−	0.964823i	$-0.415320\pi$
$239$	8.12873	0.525804	0.262902	+	0.964823i	$0.415320\pi$
$240$	0	0
$241$	25.9517	1.67170	0.835848	−	0.548960i	$-0.184976\pi$
$241$	25.9517	1.67170	0.835848	+	0.548960i	$0.184976\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.48529	0.158779
$246$	0	0
$247$	−6.22101	−0.395833
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.53741	−0.0970403	−0.0485202	−	0.998822i	$-0.515451\pi$
$251$	−1.53741	−0.0970403	−0.0485202	+	0.998822i	$0.515451\pi$
$252$	0	0
$253$	11.1231	0.699304
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.370318	0.0230998	0.0115499	−	0.999933i	$-0.496323\pi$
$257$	0.370318	0.0230998	0.0115499	+	0.999933i	$0.496323\pi$
$258$	0	0
$259$	15.3530	0.953992
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.9067	−1.16584	−0.582919	−	0.812530i	$-0.698090\pi$
$263$	−18.9067	−1.16584	−0.582919	+	0.812530i	$0.698090\pi$
$264$	0	0
$265$	9.49342	0.583176
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.1898	0.926138	0.463069	−	0.886322i	$-0.346748\pi$
$269$	15.1898	0.926138	0.463069	+	0.886322i	$0.346748\pi$
$270$	0	0
$271$	28.4027	1.72534	0.862669	−	0.505768i	$-0.168791\pi$
$271$	28.4027	1.72534	0.862669	+	0.505768i	$0.168791\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.28347	−0.559814
$276$	0	0
$277$	28.4647	1.71028	0.855139	−	0.518398i	$-0.173471\pi$
$277$	28.4647	1.71028	0.855139	+	0.518398i	$0.173471\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	27.1394	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$281$	27.1394	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$282$	0	0
$283$	26.9117	1.59974	0.799868	−	0.600175i	$-0.204903\pi$
$283$	26.9117	1.59974	0.799868	+	0.600175i	$0.204903\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.22994	0.249685
$288$	0	0
$289$	−6.99213	−0.411302
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.2714	−0.600062	−0.300031	−	0.953929i	$-0.596997\pi$
$293$	−10.2714	−0.600062	−0.300031	+	0.953929i	$0.596997\pi$
$294$	0	0
$295$	−31.5097	−1.83457
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−49.2096	−2.84587
$300$	0	0
$301$	4.29933	0.247809
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−22.2398	−1.27345
$306$	0	0
$307$	−5.05854	−0.288706	−0.144353	−	0.989526i	$-0.546110\pi$
$307$	−5.05854	−0.288706	−0.144353	+	0.989526i	$0.546110\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.4222	−0.761105	−0.380553	−	0.924759i	$-0.624266\pi$
$311$	−13.4222	−0.761105	−0.380553	+	0.924759i	$0.624266\pi$
$312$	0	0
$313$	6.94001	0.392272	0.196136	−	0.980577i	$-0.437161\pi$
$313$	6.94001	0.392272	0.196136	+	0.980577i	$0.437161\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.60448	−0.314779	−0.157389	−	0.987537i	$-0.550308\pi$
$317$	−5.60448	−0.314779	−0.157389	+	0.987537i	$0.550308\pi$
$318$	0	0
$319$	8.74777	0.489781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.16352	−0.176023
$324$	0	0
$325$	41.0709	2.27820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.6605	−1.35958
$330$	0	0
$331$	−4.11140	−0.225983	−0.112992	−	0.993596i	$-0.536043\pi$
$331$	−4.11140	−0.225983	−0.112992	+	0.993596i	$0.536043\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.37745	0.348437
$336$	0	0
$337$	−0.632801	−0.0344709	−0.0172354	−	0.999851i	$-0.505486\pi$
$337$	−0.632801	−0.0344709	−0.0172354	+	0.999851i	$0.505486\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.4339	0.619179
$342$	0	0
$343$	19.3556	1.04510
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.02871	−0.162590	−0.0812949	−	0.996690i	$-0.525906\pi$
$347$	−3.02871	−0.162590	−0.0812949	+	0.996690i	$0.525906\pi$
$348$	0	0
$349$	19.0954	1.02215	0.511076	−	0.859535i	$-0.329247\pi$
$349$	19.0954	1.02215	0.511076	+	0.859535i	$0.329247\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.2025	−0.809147	−0.404573	−	0.914506i	$-0.632580\pi$
$353$	−15.2025	−0.809147	−0.404573	+	0.914506i	$0.632580\pi$
$354$	0	0
$355$	44.3079	2.35162
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.9411	−1.84412	−0.922059	−	0.387048i	$-0.873495\pi$
$359$	−34.9411	−1.84412	−0.922059	+	0.387048i	$0.873495\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	21.4133	1.12082
$366$	0	0
$367$	−7.60839	−0.397155	−0.198577	−	0.980085i	$-0.563632\pi$
$367$	−7.60839	−0.397155	−0.198577	+	0.980085i	$0.563632\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.97916	0.362340
$372$	0	0
$373$	9.38739	0.486060	0.243030	−	0.970019i	$-0.421859\pi$
$373$	9.38739	0.486060	0.243030	+	0.970019i	$0.421859\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−38.7009	−1.99320
$378$	0	0
$379$	−11.0046	−0.565267	−0.282633	−	0.959228i	$-0.591208\pi$
$379$	−11.0046	−0.565267	−0.282633	+	0.959228i	$0.591208\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.6923	1.05733	0.528665	−	0.848831i	$-0.322693\pi$
$383$	20.6923	1.05733	0.528665	+	0.848831i	$0.322693\pi$
$384$	0	0
$385$	−11.9936	−0.611249
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.4306	0.731659	0.365830	−	0.930682i	$-0.380785\pi$
$389$	14.4306	0.731659	0.365830	+	0.930682i	$0.380785\pi$
$390$	0	0
$391$	−25.0242	−1.26553
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−38.3342	−1.92880
$396$	0	0
$397$	−14.2739	−0.716388	−0.358194	−	0.933647i	$-0.616607\pi$
$397$	−14.2739	−0.716388	−0.358194	+	0.933647i	$0.616607\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.38559	−0.268943	−0.134472	−	0.990917i	$-0.542934\pi$
$401$	−5.38559	−0.268943	−0.134472	+	0.990917i	$0.542934\pi$
$402$	0	0
$403$	−50.5845	−2.51979
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.62155	0.427354
$408$	0	0
$409$	5.07983	0.251182	0.125591	−	0.992082i	$-0.459917\pi$
$409$	5.07983	0.251182	0.125591	+	0.992082i	$0.459917\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−23.1646	−1.13985
$414$	0	0
$415$	−6.14539	−0.301666
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.30576	0.210350	0.105175	−	0.994454i	$-0.466460\pi$
$419$	4.30576	0.210350	0.105175	+	0.994454i	$0.466460\pi$
$420$	0	0
$421$	−17.8670	−0.870782	−0.435391	−	0.900241i	$-0.643390\pi$
$421$	−17.8670	−0.870782	−0.435391	+	0.900241i	$0.643390\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20.8855	1.01309
$426$	0	0
$427$	−16.3497	−0.791219
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.1958	−0.876461	−0.438230	−	0.898863i	$-0.644395\pi$
$431$	−18.1958	−0.876461	−0.438230	+	0.898863i	$0.644395\pi$
$432$	0	0
$433$	−16.6074	−0.798100	−0.399050	−	0.916929i	$-0.630660\pi$
$433$	−16.6074	−0.798100	−0.399050	+	0.916929i	$0.630660\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.91023	0.378398
$438$	0	0
$439$	11.6084	0.554038	0.277019	−	0.960864i	$-0.410653\pi$
$439$	11.6084	0.554038	0.277019	+	0.960864i	$0.410653\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.8140	1.13144	0.565720	−	0.824598i	$-0.308598\pi$
$443$	23.8140	1.13144	0.565720	+	0.824598i	$0.308598\pi$
$444$	0	0
$445$	−25.7992	−1.22300
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.72336	−0.175716	−0.0878582	−	0.996133i	$-0.528002\pi$
$449$	−3.72336	−0.175716	−0.0878582	+	0.996133i	$0.528002\pi$
$450$	0	0
$451$	2.37533	0.111850
$452$	0	0
$453$	0	0
$454$	0	0
$455$	53.0607	2.48752
$456$	0	0
$457$	26.7719	1.25234	0.626169	−	0.779688i	$-0.284622\pi$
$457$	26.7719	1.25234	0.626169	+	0.779688i	$0.284622\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.76735	−0.361761	−0.180881	−	0.983505i	$-0.557895\pi$
$461$	−7.76735	−0.361761	−0.180881	+	0.983505i	$0.557895\pi$
$462$	0	0
$463$	13.7928	0.641004	0.320502	−	0.947248i	$-0.396148\pi$
$463$	13.7928	0.641004	0.320502	+	0.947248i	$0.396148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.7282	1.28311	0.641554	−	0.767078i	$-0.278290\pi$
$467$	27.7282	1.28311	0.641554	+	0.767078i	$0.278290\pi$
$468$	0	0
$469$	4.68843	0.216492
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.41430	0.111010
$474$	0	0
$475$	−6.60197	−0.302919
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.0163	0.640420	0.320210	−	0.947347i	$-0.396247\pi$
$479$	14.0163	0.640420	0.320210	+	0.947347i	$0.396247\pi$
$480$	0	0
$481$	−38.1425	−1.73915
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.3916	−0.744304
$486$	0	0
$487$	37.5764	1.70275	0.851374	−	0.524559i	$-0.175770\pi$
$487$	37.5764	1.70275	0.851374	+	0.524559i	$0.175770\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.24309	0.236617	0.118309	−	0.992977i	$-0.462253\pi$
$491$	5.24309	0.236617	0.118309	+	0.992977i	$0.462253\pi$
$492$	0	0
$493$	−19.6803	−0.886356
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.5733	1.46111
$498$	0	0
$499$	−29.0542	−1.30065	−0.650323	−	0.759658i	$-0.725367\pi$
$499$	−29.0542	−1.30065	−0.650323	+	0.759658i	$0.725367\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.5512	−1.40680	−0.703399	−	0.710796i	$-0.748335\pi$
$503$	−31.5512	−1.40680	−0.703399	+	0.710796i	$0.748335\pi$
$504$	0	0
$505$	−55.3373	−2.46248
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.5097	−0.776104	−0.388052	−	0.921638i	$-0.626852\pi$
$509$	−17.5097	−0.776104	−0.388052	+	0.921638i	$0.626852\pi$
$510$	0	0
$511$	15.7421	0.696391
$512$	0	0
$513$	0	0
$514$	0	0
$515$	26.1908	1.15410
$516$	0	0
$517$	−13.8482	−0.609042
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.8780	1.65947	0.829733	−	0.558161i	$-0.188493\pi$
$521$	37.8780	1.65947	0.829733	+	0.558161i	$0.188493\pi$
$522$	0	0
$523$	−28.3739	−1.24070	−0.620352	−	0.784324i	$-0.713010\pi$
$523$	−28.3739	−1.24070	−0.620352	+	0.784324i	$0.713010\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.7234	−1.12053
$528$	0	0
$529$	39.5718	1.72051
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.5087	−0.455182
$534$	0	0
$535$	40.4945	1.75073
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.02600	0.0441930
$540$	0	0
$541$	38.5868	1.65898	0.829488	−	0.558524i	$-0.188632\pi$
$541$	38.5868	1.65898	0.829488	+	0.558524i	$0.188632\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.2615	0.825071
$546$	0	0
$547$	13.3947	0.572717	0.286359	−	0.958123i	$-0.407555\pi$
$547$	13.3947	0.572717	0.286359	+	0.958123i	$0.407555\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.22101	0.265024
$552$	0	0
$553$	−28.1816	−1.19841
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.46471	−0.189176	−0.0945879	−	0.995517i	$-0.530153\pi$
$557$	−4.46471	−0.189176	−0.0945879	+	0.995517i	$0.530153\pi$
$558$	0	0
$559$	−10.6811	−0.451762
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.1171	−0.932124	−0.466062	−	0.884752i	$-0.654328\pi$
$563$	−22.1171	−0.932124	−0.466062	+	0.884752i	$0.654328\pi$
$564$	0	0
$565$	−30.0163	−1.26279
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.70238	−0.280978	−0.140489	−	0.990082i	$-0.544868\pi$
$569$	−6.70238	−0.280978	−0.140489	+	0.990082i	$0.544868\pi$
$570$	0	0
$571$	6.37533	0.266799	0.133400	−	0.991062i	$-0.457411\pi$
$571$	6.37533	0.266799	0.133400	+	0.991062i	$0.457411\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−52.2231	−2.17785
$576$	0	0
$577$	44.5654	1.85528	0.927641	−	0.373474i	$-0.121834\pi$
$577$	44.5654	1.85528	0.927641	+	0.373474i	$0.121834\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.51783	−0.187431
$582$	0	0
$583$	3.91917	0.162315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.6311	−1.22301	−0.611503	−	0.791242i	$-0.709435\pi$
$587$	−29.6311	−1.22301	−0.611503	+	0.791242i	$0.709435\pi$
$588$	0	0
$589$	8.13124	0.335042
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−43.6572	−1.79279	−0.896393	−	0.443259i	$-0.853822\pi$
$593$	−43.6572	−1.79279	−0.896393	+	0.443259i	$0.853822\pi$
$594$	0	0
$595$	26.9825	1.10618
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.5661	−0.513438	−0.256719	−	0.966486i	$-0.582641\pi$
$599$	−12.5661	−0.513438	−0.256719	+	0.966486i	$0.582641\pi$
$600$	0	0
$601$	−37.9213	−1.54684	−0.773421	−	0.633893i	$-0.781456\pi$
$601$	−37.9213	−1.54684	−0.773421	+	0.633893i	$0.781456\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	30.7328	1.24947
$606$	0	0
$607$	−1.93332	−0.0784710	−0.0392355	−	0.999230i	$-0.512492\pi$
$607$	−1.93332	−0.0784710	−0.0392355	+	0.999230i	$0.512492\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	61.2656	2.47854
$612$	0	0
$613$	18.4306	0.744404	0.372202	−	0.928152i	$-0.378603\pi$
$613$	18.4306	0.744404	0.372202	+	0.928152i	$0.378603\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.5314	−1.22915	−0.614574	−	0.788859i	$-0.710672\pi$
$617$	−30.5314	−1.22915	−0.614574	+	0.788859i	$0.710672\pi$
$618$	0	0
$619$	−28.9094	−1.16197	−0.580984	−	0.813915i	$-0.697332\pi$
$619$	−28.9094	−1.16197	−0.580984	+	0.813915i	$0.697332\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−18.9665	−0.759875
$624$	0	0
$625$	−14.4238	−0.576954
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19.3963	−0.773382
$630$	0	0
$631$	−32.5314	−1.29505	−0.647527	−	0.762042i	$-0.724197\pi$
$631$	−32.5314	−1.29505	−0.647527	+	0.762042i	$0.724197\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	41.0457	1.62885
$636$	0	0
$637$	−4.53912	−0.179846
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.1312	1.19011	0.595056	−	0.803684i	$-0.297130\pi$
$641$	30.1312	1.19011	0.595056	+	0.803684i	$0.297130\pi$
$642$	0	0
$643$	−20.1752	−0.795633	−0.397817	−	0.917465i	$-0.630232\pi$
$643$	−20.1752	−0.795633	−0.397817	+	0.917465i	$0.630232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.2741	0.561174	0.280587	−	0.959829i	$-0.409471\pi$
$647$	14.2741	0.561174	0.280587	+	0.959829i	$0.409471\pi$
$648$	0	0
$649$	−13.0081	−0.510614
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.21538	−0.360626	−0.180313	−	0.983609i	$-0.557711\pi$
$653$	−9.21538	−0.360626	−0.180313	+	0.983609i	$0.557711\pi$
$654$	0	0
$655$	−3.13666	−0.122560
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.1511	−0.473339	−0.236669	−	0.971590i	$-0.576056\pi$
$659$	−12.1511	−0.473339	−0.236669	+	0.971590i	$0.576056\pi$
$660$	0	0
$661$	13.6924	0.532574	0.266287	−	0.963894i	$-0.414203\pi$
$661$	13.6924	0.532574	0.266287	+	0.963894i	$0.414203\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.52927	−0.330751
$666$	0	0
$667$	49.2096	1.90540
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.18124	−0.354438
$672$	0	0
$673$	40.5341	1.56247	0.781237	−	0.624234i	$-0.214589\pi$
$673$	40.5341	1.56247	0.781237	+	0.624234i	$0.214589\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.5653	−0.444492	−0.222246	−	0.974991i	$-0.571339\pi$
$677$	−11.5653	−0.444492	−0.222246	+	0.974991i	$0.571339\pi$
$678$	0	0
$679$	−12.0504	−0.462452
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.2625	1.69366	0.846828	−	0.531866i	$-0.178509\pi$
$683$	44.2625	1.69366	0.846828	+	0.531866i	$0.178509\pi$
$684$	0	0
$685$	−1.79064	−0.0684168
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17.3387	−0.660554
$690$	0	0
$691$	32.6676	1.24274	0.621368	−	0.783519i	$-0.286577\pi$
$691$	32.6676	1.24274	0.621368	+	0.783519i	$0.286577\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.4275	−0.585197
$696$	0	0
$697$	−5.34391	−0.202415
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.73150	0.329784	0.164892	−	0.986312i	$-0.447272\pi$
$701$	8.73150	0.329784	0.164892	+	0.986312i	$0.447272\pi$
$702$	0	0
$703$	6.13124	0.231244
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40.6816	−1.52999
$708$	0	0
$709$	14.3058	0.537264	0.268632	−	0.963243i	$-0.413428\pi$
$709$	14.3058	0.537264	0.268632	+	0.963243i	$0.413428\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	64.3200	2.40880
$714$	0	0
$715$	29.7964	1.11432
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.4959	−0.801663	−0.400831	−	0.916152i	$-0.631279\pi$
$719$	−21.4959	−0.801663	−0.400831	+	0.916152i	$0.631279\pi$
$720$	0	0
$721$	19.2543	0.717069
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−41.0709	−1.52533
$726$	0	0
$727$	47.6002	1.76539	0.882696	−	0.469944i	$-0.155726\pi$
$727$	47.6002	1.76539	0.882696	+	0.469944i	$0.155726\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.43158	−0.200894
$732$	0	0
$733$	−49.7239	−1.83659	−0.918297	−	0.395892i	$-0.870435\pi$
$733$	−49.7239	−1.83659	−0.918297	+	0.395892i	$0.870435\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.63280	0.0969805
$738$	0	0
$739$	−35.5283	−1.30693	−0.653464	−	0.756957i	$-0.726685\pi$
$739$	−35.5283	−1.30693	−0.653464	+	0.756957i	$0.726685\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.3612	0.746979	0.373490	−	0.927634i	$-0.378161\pi$
$743$	20.3612	0.746979	0.373490	+	0.927634i	$0.378161\pi$
$744$	0	0
$745$	−64.6194	−2.36747
$746$	0	0
$747$	0	0
$748$	0	0
$749$	29.7699	1.08777
$750$	0	0
$751$	−18.9868	−0.692840	−0.346420	−	0.938080i	$-0.612603\pi$
$751$	−18.9868	−0.692840	−0.346420	+	0.938080i	$0.612603\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.3342	0.812824
$756$	0	0
$757$	−22.4738	−0.816826	−0.408413	−	0.912797i	$-0.633918\pi$
$757$	−22.4738	−0.816826	−0.408413	+	0.912797i	$0.633918\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.40973	0.0511028	0.0255514	−	0.999674i	$-0.491866\pi$
$761$	1.40973	0.0511028	0.0255514	+	0.999674i	$0.491866\pi$
$762$	0	0
$763$	14.1602	0.512634
$764$	0	0
$765$	0	0
$766$	0	0
$767$	57.5492	2.07798
$768$	0	0
$769$	−0.114116	−0.00411512	−0.00205756	−	0.999998i	$-0.500655\pi$
$769$	−0.114116	−0.00411512	−0.00205756	+	0.999998i	$0.500655\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	46.1799	1.66097	0.830487	−	0.557038i	$-0.188062\pi$
$773$	46.1799	1.66097	0.830487	+	0.557038i	$0.188062\pi$
$774$	0	0
$775$	−53.6822	−1.92832
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.68923	0.0605228
$780$	0	0
$781$	18.2916	0.654525
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−59.8297	−2.13541
$786$	0	0
$787$	−12.6847	−0.452159	−0.226080	−	0.974109i	$-0.572591\pi$
$787$	−12.6847	−0.452159	−0.226080	+	0.974109i	$0.572591\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22.0667	−0.784601
$792$	0	0
$793$	40.6186	1.44241
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.2796	−1.39135	−0.695677	−	0.718355i	$-0.744895\pi$
$797$	−39.2796	−1.39135	−0.695677	+	0.718355i	$0.744895\pi$
$798$	0	0
$799$	31.1549	1.10218
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.84004	0.311958
$804$	0	0
$805$	−67.4685	−2.37795
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.09896	0.0386374	0.0193187	−	0.999813i	$-0.493850\pi$
$809$	1.09896	0.0386374	0.0193187	+	0.999813i	$0.493850\pi$
$810$	0	0
$811$	−36.5627	−1.28389	−0.641944	−	0.766751i	$-0.721872\pi$
$811$	−36.5627	−1.28389	−0.641944	+	0.766751i	$0.721872\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−78.1496	−2.73746
$816$	0	0
$817$	1.71694	0.0600681
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.8823	0.624097	0.312049	−	0.950066i	$-0.398985\pi$
$821$	17.8823	0.624097	0.312049	+	0.950066i	$0.398985\pi$
$822$	0	0
$823$	8.31328	0.289783	0.144891	−	0.989448i	$-0.453717\pi$
$823$	8.31328	0.289783	0.144891	+	0.989448i	$0.453717\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.280204	0.00974363	0.00487182	−	0.999988i	$-0.498449\pi$
$827$	0.280204	0.00974363	0.00487182	+	0.999988i	$0.498449\pi$
$828$	0	0
$829$	6.04148	0.209829	0.104915	−	0.994481i	$-0.466543\pi$
$829$	6.04148	0.209829	0.104915	+	0.994481i	$0.466543\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.30824	−0.0799759
$834$	0	0
$835$	−21.2756	−0.736274
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.5942	1.02171	0.510853	−	0.859668i	$-0.329330\pi$
$839$	29.5942	1.02171	0.510853	+	0.859668i	$0.329330\pi$
$840$	0	0
$841$	9.70093	0.334515
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−87.5416	−3.01152
$846$	0	0
$847$	22.5934	0.776320
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.4996	1.66254
$852$	0	0
$853$	15.2331	0.521570	0.260785	−	0.965397i	$-0.416019\pi$
$853$	15.2331	0.521570	0.260785	+	0.965397i	$0.416019\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.25535	0.0770412	0.0385206	−	0.999258i	$-0.487735\pi$
$857$	2.25535	0.0770412	0.0385206	+	0.999258i	$0.487735\pi$
$858$	0	0
$859$	1.91686	0.0654025	0.0327013	−	0.999465i	$-0.489589\pi$
$859$	1.91686	0.0654025	0.0327013	+	0.999465i	$0.489589\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.8530	−1.22045	−0.610225	−	0.792228i	$-0.708921\pi$
$863$	−35.8530	−1.22045	−0.610225	+	0.792228i	$0.708921\pi$
$864$	0	0
$865$	−57.7126	−1.96229
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.8255	−0.536843
$870$	0	0
$871$	−11.6478	−0.394669
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.6636	0.461915
$876$	0	0
$877$	48.2323	1.62869	0.814344	−	0.580383i	$-0.197097\pi$
$877$	48.2323	1.62869	0.814344	+	0.580383i	$0.197097\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.77336	−0.160819	−0.0804094	−	0.996762i	$-0.525623\pi$
$881$	−4.77336	−0.160819	−0.0804094	+	0.996762i	$0.525623\pi$
$882$	0	0
$883$	22.2144	0.747573	0.373787	−	0.927515i	$-0.378059\pi$
$883$	22.2144	0.747573	0.373787	+	0.927515i	$0.378059\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.1729	1.41603	0.708014	−	0.706198i	$-0.249591\pi$
$887$	42.1729	1.41603	0.708014	+	0.706198i	$0.249591\pi$
$888$	0	0
$889$	30.1750	1.01204
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.84818	−0.329557
$894$	0	0
$895$	−62.1496	−2.07743
$896$	0	0
$897$	0	0
$898$	0	0
$899$	50.5845	1.68709
$900$	0	0
$901$	−8.81714	−0.293742
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−37.2202	−1.23724
$906$	0	0
$907$	26.7564	0.888430	0.444215	−	0.895920i	$-0.353483\pi$
$907$	26.7564	0.888430	0.444215	+	0.895920i	$0.353483\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11.1281	−0.368691	−0.184346	−	0.982861i	$-0.559017\pi$
$911$	−11.1281	−0.368691	−0.184346	+	0.982861i	$0.559017\pi$
$912$	0	0
$913$	−2.53700	−0.0839625
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.30594	−0.0761489
$918$	0	0
$919$	5.45195	0.179843	0.0899216	−	0.995949i	$-0.471338\pi$
$919$	5.45195	0.179843	0.0899216	+	0.995949i	$0.471338\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−80.9237	−2.66364
$924$	0	0
$925$	−40.4783	−1.33092
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.94402	0.0637813	0.0318906	−	0.999491i	$-0.489847\pi$
$929$	1.94402	0.0637813	0.0318906	+	0.999491i	$0.489847\pi$
$930$	0	0
$931$	0.729644	0.0239131
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.1521	0.495527
$936$	0	0
$937$	−11.2897	−0.368820	−0.184410	−	0.982849i	$-0.559037\pi$
$937$	−11.2897	−0.368820	−0.184410	+	0.982849i	$0.559037\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−51.4229	−1.67634	−0.838170	−	0.545409i	$-0.816374\pi$
$941$	−51.4229	−1.67634	−0.838170	+	0.545409i	$0.816374\pi$
$942$	0	0
$943$	13.3622	0.435133
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.94045	−0.0630563	−0.0315281	−	0.999503i	$-0.510037\pi$
$947$	−1.94045	−0.0630563	−0.0315281	+	0.999503i	$0.510037\pi$
$948$	0	0
$949$	−39.1092	−1.26954
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.4523	0.759693	0.379847	−	0.925049i	$-0.375977\pi$
$953$	23.4523	0.759693	0.379847	+	0.925049i	$0.375977\pi$
$954$	0	0
$955$	52.8371	1.70977
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.31640	−0.0425088
$960$	0	0
$961$	35.1171	1.13281
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.5804	1.04880
$966$	0	0
$967$	−59.0194	−1.89794	−0.948968	−	0.315373i	$-0.897870\pi$
$967$	−59.0194	−1.89794	−0.948968	+	0.315373i	$0.897870\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.7743	1.27642	0.638209	−	0.769863i	$-0.279676\pi$
$971$	39.7743	1.27642	0.638209	+	0.769863i	$0.279676\pi$
$972$	0	0
$973$	−11.3416	−0.363595
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.9538	1.34222	0.671111	−	0.741357i	$-0.265817\pi$
$977$	41.9538	1.34222	0.671111	+	0.741357i	$0.265817\pi$
$978$	0	0
$979$	−10.6507	−0.340397
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.93132	0.252970	0.126485	−	0.991969i	$-0.459630\pi$
$983$	7.93132	0.252970	0.126485	+	0.991969i	$0.459630\pi$
$984$	0	0
$985$	65.1919	2.07719
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.5814	0.431863
$990$	0	0
$991$	−24.8408	−0.789093	−0.394546	−	0.918876i	$-0.629098\pi$
$991$	−24.8408	−0.789093	−0.394546	+	0.918876i	$0.629098\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.455573	−0.0144426
$996$	0	0
$997$	28.1157	0.890433	0.445216	−	0.895423i	$-0.353127\pi$
$997$	28.1157	0.890433	0.445216	+	0.895423i	$0.353127\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.a.bt.1.1		4
3.2	odd	2		608.2.a.i.1.4	✓	4
4.3	odd	2		5472.2.a.bs.1.1		4
12.11	even	2		608.2.a.j.1.2	yes	4
24.5	odd	2		1216.2.a.x.1.1		4
24.11	even	2		1216.2.a.w.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.i.1.4	✓	4	3.2	odd	2
608.2.a.j.1.2	yes	4	12.11	even	2
1216.2.a.w.1.3		4	24.11	even	2
1216.2.a.x.1.1		4	24.5	odd	2
5472.2.a.bs.1.1		4	4.3	odd	2
5472.2.a.bt.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.40617 q^{5} -2.50407 q^{7} +O(q^{10})\)	\(q-3.40617 q^{5} -2.50407 q^{7} -1.40617 q^{11} +6.22101 q^{13} +3.16352 q^{17} -1.00000 q^{19} -7.91023 q^{23} +6.60197 q^{25} -6.22101 q^{29} -8.13124 q^{31} +8.52927 q^{35} -6.13124 q^{37} -1.68923 q^{41} -1.71694 q^{43} +9.84818 q^{47} -0.729644 q^{49} -2.78713 q^{53} +4.78964 q^{55} +9.25078 q^{59} +6.52927 q^{61} -21.1898 q^{65} -1.87233 q^{67} -13.0081 q^{71} -6.28663 q^{73} +3.52114 q^{77} +11.2543 q^{79} +1.80420 q^{83} -10.7755 q^{85} +7.57426 q^{89} -15.5778 q^{91} +3.40617 q^{95} +4.81233 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} + q^{7}+O(q^{10}) \) 4 * q - q^5 + q^7	\( 4 q - q^{5} + q^{7} + 7 q^{11} + 10 q^{13} - 5 q^{17} - 4 q^{19} - 8 q^{23} + 17 q^{25} - 10 q^{29} + 6 q^{31} + 5 q^{35} + 14 q^{37} + 2 q^{41} - 3 q^{43} - 3 q^{47} + 7 q^{49} - 4 q^{53} + 35 q^{55} + 20 q^{59} - 3 q^{61} + 12 q^{65} - 8 q^{67} - 30 q^{71} + 9 q^{73} + 7 q^{77} - 10 q^{79} + 4 q^{83} + 19 q^{85} + 16 q^{89} - 10 q^{91} + q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q - q^5 + q^7 + 7 * q^11 + 10 * q^13 - 5 * q^17 - 4 * q^19 - 8 * q^23 + 17 * q^25 - 10 * q^29 + 6 * q^31 + 5 * q^35 + 14 * q^37 + 2 * q^41 - 3 * q^43 - 3 * q^47 + 7 * q^49 - 4 * q^53 + 35 * q^55 + 20 * q^59 - 3 * q^61 + 12 * q^65 - 8 * q^67 - 30 * q^71 + 9 * q^73 + 7 * q^77 - 10 * q^79 + 4 * q^83 + 19 * q^85 + 16 * q^89 - 10 * q^91 + q^95 - 6 * q^97

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