LMFDB - Embedded newform 5328.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5328,2,Mod(1,5328)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5328, 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("5328.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.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:	$42.5442941969$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 666)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	5328.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{5} +1.56155 q^{7} +O(q^{10})$	$q+2.00000 q^{5} +1.56155 q^{7} +1.56155 q^{11} +6.68466 q^{13} +3.56155 q^{17} +3.56155 q^{19} -8.68466 q^{23} -1.00000 q^{25} -1.12311 q^{29} +9.12311 q^{31} +3.12311 q^{35} -1.00000 q^{37} +11.1231 q^{41} -1.12311 q^{43} +10.2462 q^{47} -4.56155 q^{49} -1.56155 q^{53} +3.12311 q^{55} -0.876894 q^{59} -12.2462 q^{61} +13.3693 q^{65} -2.24621 q^{67} -2.24621 q^{71} +3.56155 q^{73} +2.43845 q^{77} +6.00000 q^{79} -14.9309 q^{83} +7.12311 q^{85} -12.9309 q^{89} +10.4384 q^{91} +7.12311 q^{95} +2.87689 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - q^{7}+O(q^{10}) $ 2 * q + 4 * q^5 - q^7	$ 2 q + 4 q^{5} - q^{7} - q^{11} + q^{13} + 3 q^{17} + 3 q^{19} - 5 q^{23} - 2 q^{25} + 6 q^{29} + 10 q^{31} - 2 q^{35} - 2 q^{37} + 14 q^{41} + 6 q^{43} + 4 q^{47} - 5 q^{49} + q^{53} - 2 q^{55} - 10 q^{59} - 8 q^{61} + 2 q^{65} + 12 q^{67} + 12 q^{71} + 3 q^{73} + 9 q^{77} + 12 q^{79} - q^{83} + 6 q^{85} + 3 q^{89} + 25 q^{91} + 6 q^{95} + 14 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - q^7 - q^11 + q^13 + 3 * q^17 + 3 * q^19 - 5 * q^23 - 2 * q^25 + 6 * q^29 + 10 * q^31 - 2 * q^35 - 2 * q^37 + 14 * q^41 + 6 * q^43 + 4 * q^47 - 5 * q^49 + q^53 - 2 * q^55 - 10 * q^59 - 8 * q^61 + 2 * q^65 + 12 * q^67 + 12 * q^71 + 3 * q^73 + 9 * q^77 + 12 * q^79 - q^83 + 6 * q^85 + 3 * q^89 + 25 * q^91 + 6 * q^95 + 14 * 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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	6.68466	1.85399	0.926995	−	0.375073i	$-0.122382\pi$
$13$	6.68466	1.85399	0.926995	+	0.375073i	$0.122382\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	3.56155	0.817076	0.408538	−	0.912741i	$-0.366039\pi$
$19$	3.56155	0.817076	0.408538	+	0.912741i	$0.366039\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.68466	−1.81088	−0.905438	−	0.424478i	$-0.860458\pi$
$23$	−8.68466	−1.81088	−0.905438	+	0.424478i	$0.860458\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	0	0
$31$	9.12311	1.63856	0.819279	−	0.573395i	$-0.194374\pi$
$31$	9.12311	1.63856	0.819279	+	0.573395i	$0.194374\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.12311	0.527901
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.1231	1.73714	0.868569	−	0.495569i	$-0.165040\pi$
$41$	11.1231	1.73714	0.868569	+	0.495569i	$0.165040\pi$
$42$	0	0
$43$	−1.12311	−0.171272	−0.0856360	−	0.996326i	$-0.527292\pi$
$43$	−1.12311	−0.171272	−0.0856360	+	0.996326i	$0.527292\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.56155	−0.214496	−0.107248	−	0.994232i	$-0.534204\pi$
$53$	−1.56155	−0.214496	−0.107248	+	0.994232i	$0.534204\pi$
$54$	0	0
$55$	3.12311	0.421119
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.876894	−0.114162	−0.0570810	−	0.998370i	$-0.518179\pi$
$59$	−0.876894	−0.114162	−0.0570810	+	0.998370i	$0.518179\pi$
$60$	0	0
$61$	−12.2462	−1.56797	−0.783983	−	0.620782i	$-0.786815\pi$
$61$	−12.2462	−1.56797	−0.783983	+	0.620782i	$0.786815\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.3693	1.65826
$66$	0	0
$67$	−2.24621	−0.274418	−0.137209	−	0.990542i	$-0.543813\pi$
$67$	−2.24621	−0.274418	−0.137209	+	0.990542i	$0.543813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.24621	−0.266576	−0.133288	−	0.991077i	$-0.542554\pi$
$71$	−2.24621	−0.266576	−0.133288	+	0.991077i	$0.542554\pi$
$72$	0	0
$73$	3.56155	0.416848	0.208424	−	0.978039i	$-0.433167\pi$
$73$	3.56155	0.416848	0.208424	+	0.978039i	$0.433167\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.43845	0.277887
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.9309	−1.63888	−0.819438	−	0.573168i	$-0.805714\pi$
$83$	−14.9309	−1.63888	−0.819438	+	0.573168i	$0.805714\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.9309	−1.37067	−0.685335	−	0.728228i	$-0.740344\pi$
$89$	−12.9309	−1.37067	−0.685335	+	0.728228i	$0.740344\pi$
$90$	0	0
$91$	10.4384	1.09425
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.12311	0.730815
$96$	0	0
$97$	2.87689	0.292104	0.146052	−	0.989277i	$-0.453343\pi$
$97$	2.87689	0.292104	0.146052	+	0.989277i	$0.453343\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.1231	−1.10679	−0.553395	−	0.832919i	$-0.686668\pi$
$101$	−11.1231	−1.10679	−0.553395	+	0.832919i	$0.686668\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.43845	0.622428	0.311214	−	0.950340i	$-0.399264\pi$
$107$	6.43845	0.622428	0.311214	+	0.950340i	$0.399264\pi$
$108$	0	0
$109$	3.56155	0.341135	0.170567	−	0.985346i	$-0.445440\pi$
$109$	3.56155	0.341135	0.170567	+	0.985346i	$0.445440\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.2462	−1.15203	−0.576013	−	0.817440i	$-0.695392\pi$
$113$	−12.2462	−1.15203	−0.576013	+	0.817440i	$0.695392\pi$
$114$	0	0
$115$	−17.3693	−1.61970
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.56155	0.509827
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−5.56155	−0.493508	−0.246754	−	0.969078i	$-0.579364\pi$
$127$	−5.56155	−0.493508	−0.246754	+	0.969078i	$0.579364\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.1231	−1.32131	−0.660656	−	0.750689i	$-0.729722\pi$
$131$	−15.1231	−1.32131	−0.660656	+	0.750689i	$0.729722\pi$
$132$	0	0
$133$	5.56155	0.482248
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.75379	−0.491579	−0.245790	−	0.969323i	$-0.579047\pi$
$137$	−5.75379	−0.491579	−0.245790	+	0.969323i	$0.579047\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.4384	0.872907
$144$	0	0
$145$	−2.24621	−0.186538
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.12311	0.255855	0.127927	−	0.991784i	$-0.459168\pi$
$149$	3.12311	0.255855	0.127927	+	0.991784i	$0.459168\pi$
$150$	0	0
$151$	−3.31534	−0.269799	−0.134899	−	0.990859i	$-0.543071\pi$
$151$	−3.31534	−0.269799	−0.134899	+	0.990859i	$0.543071\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.2462	1.46557
$156$	0	0
$157$	−10.4924	−0.837386	−0.418693	−	0.908128i	$-0.637512\pi$
$157$	−10.4924	−0.837386	−0.418693	+	0.908128i	$0.637512\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.5616	−1.06880
$162$	0	0
$163$	20.9309	1.63943	0.819716	−	0.572770i	$-0.194131\pi$
$163$	20.9309	1.63943	0.819716	+	0.572770i	$0.194131\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.6847	1.29110	0.645549	−	0.763719i	$-0.276629\pi$
$167$	16.6847	1.29110	0.645549	+	0.763719i	$0.276629\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.56155	0.118723	0.0593613	−	0.998237i	$-0.481094\pi$
$173$	1.56155	0.118723	0.0593613	+	0.998237i	$0.481094\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.12311	−0.532406	−0.266203	−	0.963917i	$-0.585769\pi$
$179$	−7.12311	−0.532406	−0.266203	+	0.963917i	$0.585769\pi$
$180$	0	0
$181$	14.4924	1.07721	0.538607	−	0.842557i	$-0.318951\pi$
$181$	14.4924	1.07721	0.538607	+	0.842557i	$0.318951\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	5.56155	0.406701
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.43845	−0.176440	−0.0882199	−	0.996101i	$-0.528118\pi$
$191$	−2.43845	−0.176440	−0.0882199	+	0.996101i	$0.528118\pi$
$192$	0	0
$193$	0.630683	0.0453976	0.0226988	−	0.999742i	$-0.492774\pi$
$193$	0.630683	0.0453976	0.0226988	+	0.999742i	$0.492774\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.8078	0.841268	0.420634	−	0.907230i	$-0.361808\pi$
$197$	11.8078	0.841268	0.420634	+	0.907230i	$0.361808\pi$
$198$	0	0
$199$	−13.1231	−0.930272	−0.465136	−	0.885239i	$-0.653995\pi$
$199$	−13.1231	−0.930272	−0.465136	+	0.885239i	$0.653995\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.75379	−0.123092
$204$	0	0
$205$	22.2462	1.55374
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.56155	0.384701
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.24621	−0.153190
$216$	0	0
$217$	14.2462	0.967096
$218$	0	0
$219$	0	0
$220$	0	0
$221$	23.8078	1.60148
$222$	0	0
$223$	−28.4924	−1.90799	−0.953997	−	0.299817i	$-0.903075\pi$
$223$	−28.4924	−1.90799	−0.953997	+	0.299817i	$0.903075\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.36932	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$227$	5.36932	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$228$	0	0
$229$	27.3693	1.80862	0.904308	−	0.426881i	$-0.140388\pi$
$229$	27.3693	1.80862	0.904308	+	0.426881i	$0.140388\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	20.4924	1.33678
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−8.24621	−0.531185	−0.265593	−	0.964085i	$-0.585568\pi$
$241$	−8.24621	−0.531185	−0.265593	+	0.964085i	$0.585568\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.12311	−0.582854
$246$	0	0
$247$	23.8078	1.51485
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.12311	−0.449606	−0.224803	−	0.974404i	$-0.572174\pi$
$251$	−7.12311	−0.449606	−0.224803	+	0.974404i	$0.572174\pi$
$252$	0	0
$253$	−13.5616	−0.852608
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.5616	1.46973	0.734865	−	0.678214i	$-0.237246\pi$
$257$	23.5616	1.46973	0.734865	+	0.678214i	$0.237246\pi$
$258$	0	0
$259$	−1.56155	−0.0970302
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.7538	−0.848095	−0.424047	−	0.905640i	$-0.639391\pi$
$263$	−13.7538	−0.848095	−0.424047	+	0.905640i	$0.639391\pi$
$264$	0	0
$265$	−3.12311	−0.191851
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0540	1.10077	0.550385	−	0.834911i	$-0.314481\pi$
$269$	18.0540	1.10077	0.550385	+	0.834911i	$0.314481\pi$
$270$	0	0
$271$	5.75379	0.349518	0.174759	−	0.984611i	$-0.444085\pi$
$271$	5.75379	0.349518	0.174759	+	0.984611i	$0.444085\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.56155	−0.0941652
$276$	0	0
$277$	27.5616	1.65601	0.828007	−	0.560718i	$-0.189475\pi$
$277$	27.5616	1.65601	0.828007	+	0.560718i	$0.189475\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.9309	−0.771391	−0.385696	−	0.922626i	$-0.626038\pi$
$281$	−12.9309	−0.771391	−0.385696	+	0.922626i	$0.626038\pi$
$282$	0	0
$283$	6.19224	0.368090	0.184045	−	0.982918i	$-0.441081\pi$
$283$	6.19224	0.368090	0.184045	+	0.982918i	$0.441081\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.3693	1.02528
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.3002	−1.65331	−0.826657	−	0.562706i	$-0.809760\pi$
$293$	−28.3002	−1.65331	−0.826657	+	0.562706i	$0.809760\pi$
$294$	0	0
$295$	−1.75379	−0.102110
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−58.0540	−3.35735
$300$	0	0
$301$	−1.75379	−0.101087
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.4924	−1.40243
$306$	0	0
$307$	17.3693	0.991319	0.495660	−	0.868517i	$-0.334926\pi$
$307$	17.3693	0.991319	0.495660	+	0.868517i	$0.334926\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.2462	−1.71511	−0.857553	−	0.514396i	$-0.828016\pi$
$311$	−30.2462	−1.71511	−0.857553	+	0.514396i	$0.828016\pi$
$312$	0	0
$313$	−2.87689	−0.162612	−0.0813058	−	0.996689i	$-0.525909\pi$
$313$	−2.87689	−0.162612	−0.0813058	+	0.996689i	$0.525909\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.12311	−0.400073	−0.200037	−	0.979788i	$-0.564106\pi$
$317$	−7.12311	−0.400073	−0.200037	+	0.979788i	$0.564106\pi$
$318$	0	0
$319$	−1.75379	−0.0981933
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.6847	0.705793
$324$	0	0
$325$	−6.68466	−0.370798
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	23.3693	1.28449	0.642247	−	0.766498i	$-0.278002\pi$
$331$	23.3693	1.28449	0.642247	+	0.766498i	$0.278002\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.49242	−0.245447
$336$	0	0
$337$	−4.93087	−0.268602	−0.134301	−	0.990941i	$-0.542879\pi$
$337$	−4.93087	−0.268602	−0.134301	+	0.990941i	$0.542879\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.2462	0.771476
$342$	0	0
$343$	−18.0540	−0.974823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.6155	−0.623554	−0.311777	−	0.950155i	$-0.600924\pi$
$347$	−11.6155	−0.623554	−0.311777	+	0.950155i	$0.600924\pi$
$348$	0	0
$349$	−6.49242	−0.347531	−0.173766	−	0.984787i	$-0.555594\pi$
$349$	−6.49242	−0.347531	−0.173766	+	0.984787i	$0.555594\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	−4.49242	−0.238433
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.4924	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$359$	28.4924	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$360$	0	0
$361$	−6.31534	−0.332386
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.12311	0.372840
$366$	0	0
$367$	−24.6847	−1.28853	−0.644264	−	0.764803i	$-0.722836\pi$
$367$	−24.6847	−1.28853	−0.644264	+	0.764803i	$0.722836\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.43845	−0.126598
$372$	0	0
$373$	27.3693	1.41713	0.708565	−	0.705646i	$-0.249343\pi$
$373$	27.3693	1.41713	0.708565	+	0.705646i	$0.249343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.50758	−0.386660
$378$	0	0
$379$	−14.2462	−0.731779	−0.365889	−	0.930658i	$-0.619235\pi$
$379$	−14.2462	−0.731779	−0.365889	+	0.930658i	$0.619235\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.8078	−0.603349	−0.301674	−	0.953411i	$-0.597546\pi$
$383$	−11.8078	−0.603349	−0.301674	+	0.953411i	$0.597546\pi$
$384$	0	0
$385$	4.87689	0.248550
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.7386	0.848682	0.424341	−	0.905502i	$-0.360506\pi$
$389$	16.7386	0.848682	0.424341	+	0.905502i	$0.360506\pi$
$390$	0	0
$391$	−30.9309	−1.56424
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−14.4924	−0.727354	−0.363677	−	0.931525i	$-0.618479\pi$
$397$	−14.4924	−0.727354	−0.363677	+	0.931525i	$0.618479\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.5616	0.577356	0.288678	−	0.957426i	$-0.406784\pi$
$401$	11.5616	0.577356	0.288678	+	0.957426i	$0.406784\pi$
$402$	0	0
$403$	60.9848	3.03787
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.56155	−0.0774033
$408$	0	0
$409$	20.2462	1.00111	0.500555	−	0.865705i	$-0.333129\pi$
$409$	20.2462	1.00111	0.500555	+	0.865705i	$0.333129\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.36932	−0.0673797
$414$	0	0
$415$	−29.8617	−1.46586
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.3153	−1.13903	−0.569514	−	0.821981i	$-0.692869\pi$
$419$	−23.3153	−1.13903	−0.569514	+	0.821981i	$0.692869\pi$
$420$	0	0
$421$	20.2462	0.986740	0.493370	−	0.869820i	$-0.335765\pi$
$421$	20.2462	0.986740	0.493370	+	0.869820i	$0.335765\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.56155	−0.172761
$426$	0	0
$427$	−19.1231	−0.925432
$428$	0	0
$429$	0	0
$430$	0	0
$431$	37.1771	1.79076	0.895378	−	0.445306i	$-0.146905\pi$
$431$	37.1771	1.79076	0.895378	+	0.445306i	$0.146905\pi$
$432$	0	0
$433$	20.4384	0.982209	0.491105	−	0.871101i	$-0.336593\pi$
$433$	20.4384	0.982209	0.491105	+	0.871101i	$0.336593\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−30.9309	−1.47962
$438$	0	0
$439$	10.4924	0.500776	0.250388	−	0.968146i	$-0.419442\pi$
$439$	10.4924	0.500776	0.250388	+	0.968146i	$0.419442\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.7386	0.890299	0.445150	−	0.895456i	$-0.353150\pi$
$443$	18.7386	0.890299	0.445150	+	0.895456i	$0.353150\pi$
$444$	0	0
$445$	−25.8617	−1.22596
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.2462	1.52179	0.760896	−	0.648873i	$-0.224760\pi$
$449$	32.2462	1.52179	0.760896	+	0.648873i	$0.224760\pi$
$450$	0	0
$451$	17.3693	0.817889
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.8769	0.978724
$456$	0	0
$457$	5.61553	0.262683	0.131342	−	0.991337i	$-0.458072\pi$
$457$	5.61553	0.262683	0.131342	+	0.991337i	$0.458072\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.9848	−1.25681	−0.628405	−	0.777887i	$-0.716292\pi$
$461$	−26.9848	−1.25681	−0.628405	+	0.777887i	$0.716292\pi$
$462$	0	0
$463$	11.3693	0.528377	0.264188	−	0.964471i	$-0.414896\pi$
$463$	11.3693	0.528377	0.264188	+	0.964471i	$0.414896\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.6155	0.907698	0.453849	−	0.891079i	$-0.350050\pi$
$467$	19.6155	0.907698	0.453849	+	0.891079i	$0.350050\pi$
$468$	0	0
$469$	−3.50758	−0.161965
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.75379	−0.0806393
$474$	0	0
$475$	−3.56155	−0.163415
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−42.0540	−1.92150	−0.960748	−	0.277424i	$-0.910519\pi$
$479$	−42.0540	−1.92150	−0.960748	+	0.277424i	$0.910519\pi$
$480$	0	0
$481$	−6.68466	−0.304794
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.75379	0.261266
$486$	0	0
$487$	−13.1231	−0.594665	−0.297332	−	0.954774i	$-0.596097\pi$
$487$	−13.1231	−0.594665	−0.297332	+	0.954774i	$0.596097\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.06913	0.0482492	0.0241246	−	0.999709i	$-0.492320\pi$
$491$	1.06913	0.0482492	0.0241246	+	0.999709i	$0.492320\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.50758	−0.157336
$498$	0	0
$499$	33.4233	1.49623	0.748116	−	0.663568i	$-0.230959\pi$
$499$	33.4233	1.49623	0.748116	+	0.663568i	$0.230959\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.9848	−1.47072	−0.735361	−	0.677676i	$-0.762987\pi$
$503$	−32.9848	−1.47072	−0.735361	+	0.677676i	$0.762987\pi$
$504$	0	0
$505$	−22.2462	−0.989943
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.94602	0.0862560	0.0431280	−	0.999070i	$-0.486268\pi$
$509$	1.94602	0.0862560	0.0431280	+	0.999070i	$0.486268\pi$
$510$	0	0
$511$	5.56155	0.246029
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20.0000	0.881305
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	29.1231	1.27346	0.636732	−	0.771085i	$-0.280286\pi$
$523$	29.1231	1.27346	0.636732	+	0.771085i	$0.280286\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.4924	1.41539
$528$	0	0
$529$	52.4233	2.27927
$530$	0	0
$531$	0	0
$532$	0	0
$533$	74.3542	3.22064
$534$	0	0
$535$	12.8769	0.556717
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.12311	−0.306814
$540$	0	0
$541$	−3.06913	−0.131952	−0.0659761	−	0.997821i	$-0.521016\pi$
$541$	−3.06913	−0.131952	−0.0659761	+	0.997821i	$0.521016\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.12311	0.305120
$546$	0	0
$547$	2.19224	0.0937332	0.0468666	−	0.998901i	$-0.485076\pi$
$547$	2.19224	0.0937332	0.0468666	+	0.998901i	$0.485076\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	9.36932	0.398424
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.0000	−1.44063	−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000	−1.44063	−0.720313	+	0.693649i	$0.756002\pi$
$558$	0	0
$559$	−7.50758	−0.317537
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.87689	0.374116	0.187058	−	0.982349i	$-0.440105\pi$
$563$	8.87689	0.374116	0.187058	+	0.982349i	$0.440105\pi$
$564$	0	0
$565$	−24.4924	−1.03040
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.3002	−1.60563	−0.802814	−	0.596230i	$-0.796665\pi$
$569$	−38.3002	−1.60563	−0.802814	+	0.596230i	$0.796665\pi$
$570$	0	0
$571$	−10.7386	−0.449398	−0.224699	−	0.974428i	$-0.572140\pi$
$571$	−10.7386	−0.449398	−0.224699	+	0.974428i	$0.572140\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.68466	0.362175
$576$	0	0
$577$	−9.12311	−0.379800	−0.189900	−	0.981803i	$-0.560816\pi$
$577$	−9.12311	−0.379800	−0.189900	+	0.981803i	$0.560816\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−23.3153	−0.967283
$582$	0	0
$583$	−2.43845	−0.100990
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.7538	0.567680	0.283840	−	0.958872i	$-0.408392\pi$
$587$	13.7538	0.567680	0.283840	+	0.958872i	$0.408392\pi$
$588$	0	0
$589$	32.4924	1.33883
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.3693	1.37031	0.685157	−	0.728396i	$-0.259734\pi$
$593$	33.3693	1.37031	0.685157	+	0.728396i	$0.259734\pi$
$594$	0	0
$595$	11.1231	0.456003
$596$	0	0
$597$	0	0
$598$	0	0
$599$	34.2462	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$599$	34.2462	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$600$	0	0
$601$	−9.31534	−0.379981	−0.189990	−	0.981786i	$-0.560846\pi$
$601$	−9.31534	−0.379981	−0.189990	+	0.981786i	$0.560846\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.1231	−0.696153
$606$	0	0
$607$	−35.8617	−1.45558	−0.727792	−	0.685798i	$-0.759453\pi$
$607$	−35.8617	−1.45558	−0.727792	+	0.685798i	$0.759453\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	68.4924	2.77091
$612$	0	0
$613$	−30.9848	−1.25147	−0.625733	−	0.780037i	$-0.715200\pi$
$613$	−30.9848	−1.25147	−0.625733	+	0.780037i	$0.715200\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.1231	−0.930901	−0.465451	−	0.885074i	$-0.654108\pi$
$617$	−23.1231	−0.930901	−0.465451	+	0.885074i	$0.654108\pi$
$618$	0	0
$619$	−18.7386	−0.753169	−0.376585	−	0.926382i	$-0.622902\pi$
$619$	−18.7386	−0.753169	−0.376585	+	0.926382i	$0.622902\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20.1922	−0.808985
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.56155	−0.142008
$630$	0	0
$631$	12.7386	0.507117	0.253559	−	0.967320i	$-0.418399\pi$
$631$	12.7386	0.507117	0.253559	+	0.967320i	$0.418399\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.1231	−0.441407
$636$	0	0
$637$	−30.4924	−1.20815
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.8617	−0.863487	−0.431743	−	0.901996i	$-0.642101\pi$
$641$	−21.8617	−0.863487	−0.431743	+	0.901996i	$0.642101\pi$
$642$	0	0
$643$	−2.19224	−0.0864533	−0.0432267	−	0.999065i	$-0.513764\pi$
$643$	−2.19224	−0.0864533	−0.0432267	+	0.999065i	$0.513764\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.9309	−0.586993	−0.293497	−	0.955960i	$-0.594819\pi$
$647$	−14.9309	−0.586993	−0.293497	+	0.955960i	$0.594819\pi$
$648$	0	0
$649$	−1.36932	−0.0537504
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.7386	0.655033	0.327517	−	0.944845i	$-0.393788\pi$
$653$	16.7386	0.655033	0.327517	+	0.944845i	$0.393788\pi$
$654$	0	0
$655$	−30.2462	−1.18182
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.2462	0.710771	0.355386	−	0.934720i	$-0.384350\pi$
$659$	18.2462	0.710771	0.355386	+	0.934720i	$0.384350\pi$
$660$	0	0
$661$	−19.5616	−0.760856	−0.380428	−	0.924810i	$-0.624223\pi$
$661$	−19.5616	−0.760856	−0.380428	+	0.924810i	$0.624223\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.1231	0.431336
$666$	0	0
$667$	9.75379	0.377668
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19.1231	−0.738239
$672$	0	0
$673$	0.0539753	0.00208060	0.00104030	−	0.999999i	$-0.499669\pi$
$673$	0.0539753	0.00208060	0.00104030	+	0.999999i	$0.499669\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.4384	−0.862380	−0.431190	−	0.902261i	$-0.641906\pi$
$677$	−22.4384	−0.862380	−0.431190	+	0.902261i	$0.641906\pi$
$678$	0	0
$679$	4.49242	0.172403
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.1080	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$683$	−24.1080	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$684$	0	0
$685$	−11.5076	−0.439682
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.4384	−0.397673
$690$	0	0
$691$	31.1231	1.18398	0.591989	−	0.805946i	$-0.298343\pi$
$691$	31.1231	1.18398	0.591989	+	0.805946i	$0.298343\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.0000	0.910372
$696$	0	0
$697$	39.6155	1.50055
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.6307	0.779210	0.389605	−	0.920982i	$-0.372612\pi$
$701$	20.6307	0.779210	0.389605	+	0.920982i	$0.372612\pi$
$702$	0	0
$703$	−3.56155	−0.134327
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.3693	−0.653240
$708$	0	0
$709$	−42.3002	−1.58862	−0.794308	−	0.607515i	$-0.792167\pi$
$709$	−42.3002	−1.58862	−0.794308	+	0.607515i	$0.792167\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−79.2311	−2.96723
$714$	0	0
$715$	20.8769	0.780752
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.6307	−0.843982	−0.421991	−	0.906600i	$-0.638669\pi$
$719$	−22.6307	−0.843982	−0.421991	+	0.906600i	$0.638669\pi$
$720$	0	0
$721$	15.6155	0.581553
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.12311	0.0417111
$726$	0	0
$727$	31.3693	1.16342	0.581712	−	0.813395i	$-0.302383\pi$
$727$	31.3693	1.16342	0.581712	+	0.813395i	$0.302383\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	−47.8617	−1.76781	−0.883907	−	0.467663i	$-0.845096\pi$
$733$	−47.8617	−1.76781	−0.883907	+	0.467663i	$0.845096\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.50758	−0.129203
$738$	0	0
$739$	−47.6155	−1.75157	−0.875783	−	0.482705i	$-0.839654\pi$
$739$	−47.6155	−1.75157	−0.875783	+	0.482705i	$0.839654\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.9848	0.476368	0.238184	−	0.971220i	$-0.423448\pi$
$743$	12.9848	0.476368	0.238184	+	0.971220i	$0.423448\pi$
$744$	0	0
$745$	6.24621	0.228843
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.0540	0.367364
$750$	0	0
$751$	39.2311	1.43156	0.715781	−	0.698325i	$-0.246071\pi$
$751$	39.2311	1.43156	0.715781	+	0.698325i	$0.246071\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.63068	−0.241315
$756$	0	0
$757$	−10.6847	−0.388341	−0.194170	−	0.980968i	$-0.562201\pi$
$757$	−10.6847	−0.388341	−0.194170	+	0.980968i	$0.562201\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.1080	1.45391	0.726956	−	0.686684i	$-0.240934\pi$
$761$	40.1080	1.45391	0.726956	+	0.686684i	$0.240934\pi$
$762$	0	0
$763$	5.56155	0.201342
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.86174	−0.211655
$768$	0	0
$769$	−5.61553	−0.202501	−0.101251	−	0.994861i	$-0.532284\pi$
$769$	−5.61553	−0.202501	−0.101251	+	0.994861i	$0.532284\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16.6847	−0.600105	−0.300053	−	0.953923i	$-0.597004\pi$
$773$	−16.6847	−0.600105	−0.300053	+	0.953923i	$0.597004\pi$
$774$	0	0
$775$	−9.12311	−0.327712
$776$	0	0
$777$	0	0
$778$	0	0
$779$	39.6155	1.41937
$780$	0	0
$781$	−3.50758	−0.125511
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20.9848	−0.748981
$786$	0	0
$787$	−35.2311	−1.25585	−0.627926	−	0.778273i	$-0.716096\pi$
$787$	−35.2311	−1.25585	−0.627926	+	0.778273i	$0.716096\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19.1231	−0.679939
$792$	0	0
$793$	−81.8617	−2.90700
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	36.4924	1.29101
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.56155	0.196263
$804$	0	0
$805$	−27.1231	−0.955964
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−37.4233	−1.31573	−0.657866	−	0.753135i	$-0.728541\pi$
$809$	−37.4233	−1.31573	−0.657866	+	0.753135i	$0.728541\pi$
$810$	0	0
$811$	−17.3693	−0.609919	−0.304960	−	0.952365i	$-0.598643\pi$
$811$	−17.3693	−0.609919	−0.304960	+	0.952365i	$0.598643\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	41.8617	1.46635
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.3153	−1.23251	−0.616257	−	0.787545i	$-0.711352\pi$
$821$	−35.3153	−1.23251	−0.616257	+	0.787545i	$0.711352\pi$
$822$	0	0
$823$	−19.8078	−0.690455	−0.345228	−	0.938519i	$-0.612198\pi$
$823$	−19.8078	−0.690455	−0.345228	+	0.938519i	$0.612198\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.2462	1.74723	0.873616	−	0.486616i	$-0.161769\pi$
$827$	50.2462	1.74723	0.873616	+	0.486616i	$0.161769\pi$
$828$	0	0
$829$	3.56155	0.123698	0.0618489	−	0.998086i	$-0.480300\pi$
$829$	3.56155	0.123698	0.0618489	+	0.998086i	$0.480300\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16.2462	−0.562898
$834$	0	0
$835$	33.3693	1.15479
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.1231	−1.07449	−0.537244	−	0.843427i	$-0.680535\pi$
$839$	−31.1231	−1.07449	−0.537244	+	0.843427i	$0.680535\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	0	0
$843$	0	0
$844$	0	0
$845$	63.3693	2.17997
$846$	0	0
$847$	−13.3693	−0.459375
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.68466	0.297706
$852$	0	0
$853$	36.4384	1.24763	0.623814	−	0.781573i	$-0.285582\pi$
$853$	36.4384	1.24763	0.623814	+	0.781573i	$0.285582\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.6695	1.49172	0.745861	−	0.666102i	$-0.232038\pi$
$857$	43.6695	1.49172	0.745861	+	0.666102i	$0.232038\pi$
$858$	0	0
$859$	−0.0539753	−0.00184161	−0.000920807	−	1.00000i	$-0.500293\pi$
$859$	−0.0539753	−0.00184161	−0.000920807		1.00000i	$0.500293\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−57.3693	−1.95287	−0.976437	−	0.215802i	$-0.930763\pi$
$863$	−57.3693	−1.95287	−0.976437	+	0.215802i	$0.930763\pi$
$864$	0	0
$865$	3.12311	0.106189
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.36932	0.317832
$870$	0	0
$871$	−15.0152	−0.508769
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−18.7386	−0.633481
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.4924	0.555644	0.277822	−	0.960633i	$-0.410387\pi$
$881$	16.4924	0.555644	0.277822	+	0.960633i	$0.410387\pi$
$882$	0	0
$883$	28.4384	0.957030	0.478515	−	0.878079i	$-0.341175\pi$
$883$	28.4384	0.957030	0.478515	+	0.878079i	$0.341175\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.1231	−0.776398	−0.388199	−	0.921576i	$-0.626903\pi$
$887$	−23.1231	−0.776398	−0.388199	+	0.921576i	$0.626903\pi$
$888$	0	0
$889$	−8.68466	−0.291274
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.4924	1.22117
$894$	0	0
$895$	−14.2462	−0.476198
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.2462	−0.341730
$900$	0	0
$901$	−5.56155	−0.185282
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.9848	0.963489
$906$	0	0
$907$	−38.3002	−1.27174	−0.635868	−	0.771797i	$-0.719358\pi$
$907$	−38.3002	−1.27174	−0.635868	+	0.771797i	$0.719358\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.4924	−1.20905	−0.604524	−	0.796587i	$-0.706637\pi$
$911$	−36.4924	−1.20905	−0.604524	+	0.796587i	$0.706637\pi$
$912$	0	0
$913$	−23.3153	−0.771625
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23.6155	−0.779853
$918$	0	0
$919$	−44.2462	−1.45955	−0.729774	−	0.683689i	$-0.760375\pi$
$919$	−44.2462	−1.45955	−0.729774	+	0.683689i	$0.760375\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−15.0152	−0.494230
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.2462	1.51729	0.758644	−	0.651505i	$-0.225862\pi$
$929$	46.2462	1.51729	0.758644	+	0.651505i	$0.225862\pi$
$930$	0	0
$931$	−16.2462	−0.532448
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.1231	0.363764
$936$	0	0
$937$	−32.2462	−1.05344	−0.526719	−	0.850040i	$-0.676578\pi$
$937$	−32.2462	−1.05344	−0.526719	+	0.850040i	$0.676578\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−49.8617	−1.62545	−0.812723	−	0.582650i	$-0.802016\pi$
$941$	−49.8617	−1.62545	−0.812723	+	0.582650i	$0.802016\pi$
$942$	0	0
$943$	−96.6004	−3.14574
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.36932	0.174479	0.0872397	−	0.996187i	$-0.472195\pi$
$947$	5.36932	0.174479	0.0872397	+	0.996187i	$0.472195\pi$
$948$	0	0
$949$	23.8078	0.772833
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49.8617	1.61518	0.807590	−	0.589744i	$-0.200771\pi$
$953$	49.8617	1.61518	0.807590	+	0.589744i	$0.200771\pi$
$954$	0	0
$955$	−4.87689	−0.157813
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.98485	−0.290136
$960$	0	0
$961$	52.2311	1.68487
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.26137	0.0406048
$966$	0	0
$967$	3.36932	0.108350	0.0541750	−	0.998531i	$-0.482747\pi$
$967$	3.36932	0.108350	0.0541750	+	0.998531i	$0.482747\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.75379	−0.313014	−0.156507	−	0.987677i	$-0.550023\pi$
$971$	−9.75379	−0.313014	−0.156507	+	0.987677i	$0.550023\pi$
$972$	0	0
$973$	18.7386	0.600733
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.9157	−1.08506	−0.542530	−	0.840036i	$-0.682533\pi$
$977$	−33.9157	−1.08506	−0.542530	+	0.840036i	$0.682533\pi$
$978$	0	0
$979$	−20.1922	−0.645347
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.6155	0.625638	0.312819	−	0.949813i	$-0.398727\pi$
$983$	19.6155	0.625638	0.312819	+	0.949813i	$0.398727\pi$
$984$	0	0
$985$	23.6155	0.752453
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.75379	0.310152
$990$	0	0
$991$	−60.7386	−1.92943	−0.964713	−	0.263303i	$-0.915188\pi$
$991$	−60.7386	−1.92943	−0.964713	+	0.263303i	$0.915188\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−26.2462	−0.832061
$996$	0	0
$997$	−36.9309	−1.16961	−0.584806	−	0.811173i	$-0.698829\pi$
$997$	−36.9309	−1.16961	−0.584806	+	0.811173i	$0.698829\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.a.bh.1.2		2
3.2	odd	2		5328.2.a.y.1.2		2
4.3	odd	2		666.2.a.k.1.1	yes	2
12.11	even	2		666.2.a.h.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
666.2.a.h.1.1	✓	2	12.11	even	2
666.2.a.k.1.1	yes	2	4.3	odd	2
5328.2.a.y.1.2		2	3.2	odd	2
5328.2.a.bh.1.2		2	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:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{5} +1.56155 q^{7} +O(q^{10})\)	\(q+2.00000 q^{5} +1.56155 q^{7} +1.56155 q^{11} +6.68466 q^{13} +3.56155 q^{17} +3.56155 q^{19} -8.68466 q^{23} -1.00000 q^{25} -1.12311 q^{29} +9.12311 q^{31} +3.12311 q^{35} -1.00000 q^{37} +11.1231 q^{41} -1.12311 q^{43} +10.2462 q^{47} -4.56155 q^{49} -1.56155 q^{53} +3.12311 q^{55} -0.876894 q^{59} -12.2462 q^{61} +13.3693 q^{65} -2.24621 q^{67} -2.24621 q^{71} +3.56155 q^{73} +2.43845 q^{77} +6.00000 q^{79} -14.9309 q^{83} +7.12311 q^{85} -12.9309 q^{89} +10.4384 q^{91} +7.12311 q^{95} +2.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - q^{7}+O(q^{10}) \) 2 * q + 4 * q^5 - q^7	\( 2 q + 4 q^{5} - q^{7} - q^{11} + q^{13} + 3 q^{17} + 3 q^{19} - 5 q^{23} - 2 q^{25} + 6 q^{29} + 10 q^{31} - 2 q^{35} - 2 q^{37} + 14 q^{41} + 6 q^{43} + 4 q^{47} - 5 q^{49} + q^{53} - 2 q^{55} - 10 q^{59} - 8 q^{61} + 2 q^{65} + 12 q^{67} + 12 q^{71} + 3 q^{73} + 9 q^{77} + 12 q^{79} - q^{83} + 6 q^{85} + 3 q^{89} + 25 q^{91} + 6 q^{95} + 14 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - q^7 - q^11 + q^13 + 3 * q^17 + 3 * q^19 - 5 * q^23 - 2 * q^25 + 6 * q^29 + 10 * q^31 - 2 * q^35 - 2 * q^37 + 14 * q^41 + 6 * q^43 + 4 * q^47 - 5 * q^49 + q^53 - 2 * q^55 - 10 * q^59 - 8 * q^61 + 2 * q^65 + 12 * q^67 + 12 * q^71 + 3 * q^73 + 9 * q^77 + 12 * q^79 - q^83 + 6 * q^85 + 3 * q^89 + 25 * q^91 + 6 * q^95 + 14 * q^97

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