LMFDB - Embedded newform 2268.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2268.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.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:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.11126 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+1.11126 q^{5} +1.00000 q^{7} +1.88874 q^{11} -1.00000 q^{13} +5.87636 q^{17} +7.09888 q^{19} -3.98762 q^{23} -3.76509 q^{25} -0.987620 q^{29} -0.666208 q^{31} +1.11126 q^{35} -1.33379 q^{37} +1.88874 q^{41} -10.8640 q^{43} +11.0989 q^{47} +1.00000 q^{49} +12.2101 q^{53} +2.09888 q^{55} -4.76509 q^{59} +3.76509 q^{61} -1.11126 q^{65} +4.09888 q^{67} +10.7651 q^{71} -3.09888 q^{73} +1.88874 q^{77} +6.43130 q^{79} +11.8764 q^{83} +6.53018 q^{85} -14.3090 q^{89} -1.00000 q^{91} +7.88874 q^{95} +0.765092 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 + 3 * q^7	$ 3 q + 3 q^{5} + 3 q^{7} + 6 q^{11} - 3 q^{13} + 3 q^{19} + 6 q^{23} + 6 q^{25} + 15 q^{29} - 3 q^{31} + 3 q^{35} - 3 q^{37} + 6 q^{41} + 3 q^{43} + 15 q^{47} + 3 q^{49} + 18 q^{53} - 12 q^{55} + 3 q^{59} - 6 q^{61} - 3 q^{65} - 6 q^{67} + 15 q^{71} + 9 q^{73} + 6 q^{77} + 3 q^{79} + 18 q^{83} - 15 q^{85} - 6 q^{89} - 3 q^{91} + 24 q^{95} - 15 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^7 + 6 * q^11 - 3 * q^13 + 3 * q^19 + 6 * q^23 + 6 * q^25 + 15 * q^29 - 3 * q^31 + 3 * q^35 - 3 * q^37 + 6 * q^41 + 3 * q^43 + 15 * q^47 + 3 * q^49 + 18 * q^53 - 12 * q^55 + 3 * q^59 - 6 * q^61 - 3 * q^65 - 6 * q^67 + 15 * q^71 + 9 * q^73 + 6 * q^77 + 3 * q^79 + 18 * q^83 - 15 * q^85 - 6 * q^89 - 3 * q^91 + 24 * q^95 - 15 * 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
$5$	1.11126	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$5$	1.11126	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.88874	0.569475	0.284738	−	0.958605i	$-0.408094\pi$
$11$	1.88874	0.569475	0.284738	+	0.958605i	$0.408094\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.87636	1.42523	0.712613	−	0.701557i	$-0.247512\pi$
$17$	5.87636	1.42523	0.712613	+	0.701557i	$0.247512\pi$
$18$	0	0
$19$	7.09888	1.62860	0.814298	−	0.580447i	$-0.197122\pi$
$19$	7.09888	1.62860	0.814298	+	0.580447i	$0.197122\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.98762	−0.831476	−0.415738	−	0.909484i	$-0.636477\pi$
$23$	−3.98762	−0.831476	−0.415738	+	0.909484i	$0.636477\pi$
$24$	0	0
$25$	−3.76509	−0.753018
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.987620	−0.183396	−0.0916982	−	0.995787i	$-0.529230\pi$
$29$	−0.987620	−0.183396	−0.0916982	+	0.995787i	$0.529230\pi$
$30$	0	0
$31$	−0.666208	−0.119654	−0.0598272	−	0.998209i	$-0.519055\pi$
$31$	−0.666208	−0.119654	−0.0598272	+	0.998209i	$0.519055\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.11126	0.187838
$36$	0	0
$37$	−1.33379	−0.219274	−0.109637	−	0.993972i	$-0.534969\pi$
$37$	−1.33379	−0.219274	−0.109637	+	0.993972i	$0.534969\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.88874	0.294971	0.147485	−	0.989064i	$-0.452882\pi$
$41$	1.88874	0.294971	0.147485	+	0.989064i	$0.452882\pi$
$42$	0	0
$43$	−10.8640	−1.65674	−0.828370	−	0.560181i	$-0.810732\pi$
$43$	−10.8640	−1.65674	−0.828370	+	0.560181i	$0.810732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.0989	1.61894	0.809469	−	0.587162i	$-0.199755\pi$
$47$	11.0989	1.61894	0.809469	+	0.587162i	$0.199755\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.2101	1.67719	0.838596	−	0.544753i	$-0.183377\pi$
$53$	12.2101	1.67719	0.838596	+	0.544753i	$0.183377\pi$
$54$	0	0
$55$	2.09888	0.283014
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.76509	−0.620362	−0.310181	−	0.950677i	$-0.600390\pi$
$59$	−4.76509	−0.620362	−0.310181	+	0.950677i	$0.600390\pi$
$60$	0	0
$61$	3.76509	0.482071	0.241035	−	0.970516i	$-0.422513\pi$
$61$	3.76509	0.482071	0.241035	+	0.970516i	$0.422513\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.11126	−0.137835
$66$	0	0
$67$	4.09888	0.500758	0.250379	−	0.968148i	$-0.419445\pi$
$67$	4.09888	0.500758	0.250379	+	0.968148i	$0.419445\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.7651	1.27758	0.638791	−	0.769381i	$-0.279435\pi$
$71$	10.7651	1.27758	0.638791	+	0.769381i	$0.279435\pi$
$72$	0	0
$73$	−3.09888	−0.362697	−0.181348	−	0.983419i	$-0.558046\pi$
$73$	−3.09888	−0.362697	−0.181348	+	0.983419i	$0.558046\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.88874	0.215241
$78$	0	0
$79$	6.43130	0.723578	0.361789	−	0.932260i	$-0.382166\pi$
$79$	6.43130	0.723578	0.361789	+	0.932260i	$0.382166\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.8764	1.30360	0.651800	−	0.758391i	$-0.274014\pi$
$83$	11.8764	1.30360	0.651800	+	0.758391i	$0.274014\pi$
$84$	0	0
$85$	6.53018	0.708298
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.3090	−1.51675	−0.758377	−	0.651816i	$-0.774007\pi$
$89$	−14.3090	−1.51675	−0.758377	+	0.651816i	$0.774007\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.88874	0.809367
$96$	0	0
$97$	0.765092	0.0776833	0.0388417	−	0.999245i	$-0.487633\pi$
$97$	0.765092	0.0776833	0.0388417	+	0.999245i	$0.487633\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.87636	−0.883230	−0.441615	−	0.897205i	$-0.645594\pi$
$101$	−8.87636	−0.883230	−0.441615	+	0.897205i	$0.645594\pi$
$102$	0	0
$103$	−9.96286	−0.981670	−0.490835	−	0.871253i	$-0.663308\pi$
$103$	−9.96286	−0.981670	−0.490835	+	0.871253i	$0.663308\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.22253	0.504881	0.252440	−	0.967612i	$-0.418767\pi$
$107$	5.22253	0.504881	0.252440	+	0.967612i	$0.418767\pi$
$108$	0	0
$109$	−9.09888	−0.871515	−0.435758	−	0.900064i	$-0.643519\pi$
$109$	−9.09888	−0.871515	−0.435758	+	0.900064i	$0.643519\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.4203	1.16840	0.584202	−	0.811609i	$-0.301408\pi$
$113$	12.4203	1.16840	0.584202	+	0.811609i	$0.301408\pi$
$114$	0	0
$115$	−4.43130	−0.413221
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.87636	0.538685
$120$	0	0
$121$	−7.43268	−0.675698
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.74033	−0.871202
$126$	0	0
$127$	7.66621	0.680266	0.340133	−	0.940377i	$-0.389528\pi$
$127$	7.66621	0.680266	0.340133	+	0.940377i	$0.389528\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.5426	1.27059	0.635295	−	0.772270i	$-0.280879\pi$
$131$	14.5426	1.27059	0.635295	+	0.772270i	$0.280879\pi$
$132$	0	0
$133$	7.09888	0.615551
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.64145	−0.396546	−0.198273	−	0.980147i	$-0.563533\pi$
$137$	−4.64145	−0.396546	−0.198273	+	0.980147i	$0.563533\pi$
$138$	0	0
$139$	−19.2967	−1.63672	−0.818360	−	0.574705i	$-0.805117\pi$
$139$	−19.2967	−1.63672	−0.818360	+	0.574705i	$0.805117\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.88874	−0.157944
$144$	0	0
$145$	−1.09751	−0.0911430
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.6414	1.11755	0.558775	−	0.829319i	$-0.311271\pi$
$149$	13.6414	1.11755	0.558775	+	0.829319i	$0.311271\pi$
$150$	0	0
$151$	16.9629	1.38042	0.690209	−	0.723610i	$-0.257519\pi$
$151$	16.9629	1.38042	0.690209	+	0.723610i	$0.257519\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.740333	−0.0594649
$156$	0	0
$157$	22.6291	1.80600	0.902998	−	0.429644i	$-0.141361\pi$
$157$	22.6291	1.80600	0.902998	+	0.429644i	$0.141361\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.98762	−0.314269
$162$	0	0
$163$	−18.0989	−1.41761	−0.708807	−	0.705402i	$-0.750766\pi$
$163$	−18.0989	−1.41761	−0.708807	+	0.705402i	$0.750766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.2225	1.33272	0.666360	−	0.745631i	$-0.267852\pi$
$167$	17.2225	1.33272	0.666360	+	0.745631i	$0.267852\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.863976	0.0656869	0.0328435	−	0.999461i	$-0.489544\pi$
$173$	0.863976	0.0656869	0.0328435	+	0.999461i	$0.489544\pi$
$174$	0	0
$175$	−3.76509	−0.284614
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.67859	0.349694	0.174847	−	0.984596i	$-0.444057\pi$
$179$	4.67859	0.349694	0.174847	+	0.984596i	$0.444057\pi$
$180$	0	0
$181$	−16.8640	−1.25349	−0.626745	−	0.779225i	$-0.715613\pi$
$181$	−16.8640	−1.25349	−0.626745	+	0.779225i	$0.715613\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.48220	−0.108973
$186$	0	0
$187$	11.0989	0.811631
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.0617	1.23454	0.617272	−	0.786750i	$-0.288238\pi$
$191$	17.0617	1.23454	0.617272	+	0.786750i	$0.288238\pi$
$192$	0	0
$193$	7.43268	0.535016	0.267508	−	0.963556i	$-0.413800\pi$
$193$	7.43268	0.535016	0.267508	+	0.963556i	$0.413800\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.2953	1.87346	0.936730	−	0.350052i	$-0.113836\pi$
$197$	26.2953	1.87346	0.936730	+	0.350052i	$0.113836\pi$
$198$	0	0
$199$	10.4327	0.739553	0.369776	−	0.929121i	$-0.379434\pi$
$199$	10.4327	0.739553	0.369776	+	0.929121i	$0.379434\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.987620	−0.0693174
$204$	0	0
$205$	2.09888	0.146592
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.4079	0.927445
$210$	0	0
$211$	25.3955	1.74830	0.874150	−	0.485655i	$-0.161419\pi$
$211$	25.3955	1.74830	0.874150	+	0.485655i	$0.161419\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.0727	−0.823355
$216$	0	0
$217$	−0.666208	−0.0452251
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.87636	−0.395286
$222$	0	0
$223$	14.5302	0.973013	0.486507	−	0.873677i	$-0.338271\pi$
$223$	14.5302	0.973013	0.486507	+	0.873677i	$0.338271\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.4079	−1.28815	−0.644074	−	0.764963i	$-0.722757\pi$
$227$	−19.4079	−1.28815	−0.644074	+	0.764963i	$0.722757\pi$
$228$	0	0
$229$	−10.5302	−0.695854	−0.347927	−	0.937522i	$-0.613114\pi$
$229$	−10.5302	−0.695854	−0.347927	+	0.937522i	$0.613114\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.6414	−1.48329	−0.741645	−	0.670792i	$-0.765954\pi$
$233$	−22.6414	−1.48329	−0.741645	+	0.670792i	$0.765954\pi$
$234$	0	0
$235$	12.3338	0.804568
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.97524	−0.321822	−0.160911	−	0.986969i	$-0.551443\pi$
$239$	−4.97524	−0.321822	−0.160911	+	0.986969i	$0.551443\pi$
$240$	0	0
$241$	0.234908	0.0151318	0.00756588	−	0.999971i	$-0.497592\pi$
$241$	0.234908	0.0151318	0.00756588	+	0.999971i	$0.497592\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.11126	0.0709961
$246$	0	0
$247$	−7.09888	−0.451691
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.09888	0.511197	0.255599	−	0.966783i	$-0.417727\pi$
$251$	8.09888	0.511197	0.255599	+	0.966783i	$0.417727\pi$
$252$	0	0
$253$	−7.53156	−0.473505
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.7527	−1.48165	−0.740827	−	0.671696i	$-0.765566\pi$
$257$	−23.7527	−1.48165	−0.740827	+	0.671696i	$0.765566\pi$
$258$	0	0
$259$	−1.33379	−0.0828778
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.65383	0.410293	0.205146	−	0.978731i	$-0.434233\pi$
$263$	6.65383	0.410293	0.205146	+	0.978731i	$0.434233\pi$
$264$	0	0
$265$	13.5687	0.833519
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.9876	−1.15770	−0.578848	−	0.815436i	$-0.696497\pi$
$269$	−18.9876	−1.15770	−0.578848	+	0.815436i	$0.696497\pi$
$270$	0	0
$271$	−22.3338	−1.35668	−0.678341	−	0.734748i	$-0.737301\pi$
$271$	−22.3338	−1.35668	−0.678341	+	0.734748i	$0.737301\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.11126	−0.428825
$276$	0	0
$277$	−31.8268	−1.91229	−0.956145	−	0.292895i	$-0.905381\pi$
$277$	−31.8268	−1.91229	−0.956145	+	0.292895i	$0.905381\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.8516	−1.18425	−0.592123	−	0.805847i	$-0.701710\pi$
$281$	−19.8516	−1.18425	−0.592123	+	0.805847i	$0.701710\pi$
$282$	0	0
$283$	−3.13602	−0.186417	−0.0932086	−	0.995647i	$-0.529712\pi$
$283$	−3.13602	−0.186417	−0.0932086	+	0.995647i	$0.529712\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.88874	0.111489
$288$	0	0
$289$	17.5316	1.03127
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.87773	−0.577063	−0.288532	−	0.957470i	$-0.593167\pi$
$293$	−9.87773	−0.577063	−0.288532	+	0.957470i	$0.593167\pi$
$294$	0	0
$295$	−5.29528	−0.308303
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.98762	0.230610
$300$	0	0
$301$	−10.8640	−0.626189
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.18401	0.239576
$306$	0	0
$307$	15.4313	0.880711	0.440355	−	0.897824i	$-0.354852\pi$
$307$	15.4313	0.880711	0.440355	+	0.897824i	$0.354852\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.6662	−1.00176	−0.500879	−	0.865517i	$-0.666990\pi$
$311$	−17.6662	−1.00176	−0.500879	+	0.865517i	$0.666990\pi$
$312$	0	0
$313$	13.4327	0.759260	0.379630	−	0.925138i	$-0.376051\pi$
$313$	13.4327	0.759260	0.379630	+	0.925138i	$0.376051\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.8640	−0.891010	−0.445505	−	0.895280i	$-0.646976\pi$
$317$	−15.8640	−0.891010	−0.445505	+	0.895280i	$0.646976\pi$
$318$	0	0
$319$	−1.86535	−0.104440
$320$	0	0
$321$	0	0
$322$	0	0
$323$	41.7156	2.32112
$324$	0	0
$325$	3.76509	0.208850
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.0989	0.611901
$330$	0	0
$331$	−18.0989	−0.994805	−0.497402	−	0.867520i	$-0.665713\pi$
$331$	−18.0989	−0.994805	−0.497402	+	0.867520i	$0.665713\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.55494	0.248863
$336$	0	0
$337$	−1.03714	−0.0564966	−0.0282483	−	0.999601i	$-0.508993\pi$
$337$	−1.03714	−0.0564966	−0.0282483	+	0.999601i	$0.508993\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.25829	−0.0681402
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.9642	1.17910	0.589551	−	0.807731i	$-0.299305\pi$
$347$	21.9642	1.17910	0.589551	+	0.807731i	$0.299305\pi$
$348$	0	0
$349$	23.5302	1.25954	0.629771	−	0.776781i	$-0.283149\pi$
$349$	23.5302	1.25954	0.629771	+	0.776781i	$0.283149\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.5426	−1.57239	−0.786196	−	0.617977i	$-0.787952\pi$
$353$	−29.5426	−1.57239	−0.786196	+	0.617977i	$0.787952\pi$
$354$	0	0
$355$	11.9629	0.634923
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.0741	−1.79836	−0.899182	−	0.437575i	$-0.855837\pi$
$359$	−34.0741	−1.79836	−0.899182	+	0.437575i	$0.855837\pi$
$360$	0	0
$361$	31.3942	1.65232
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.44368	−0.180250
$366$	0	0
$367$	−27.0617	−1.41261	−0.706306	−	0.707907i	$-0.749640\pi$
$367$	−27.0617	−1.41261	−0.706306	+	0.707907i	$0.749640\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.2101	0.633919
$372$	0	0
$373$	−10.8640	−0.562515	−0.281258	−	0.959632i	$-0.590752\pi$
$373$	−10.8640	−0.562515	−0.281258	+	0.959632i	$0.590752\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.987620	0.0508650
$378$	0	0
$379$	0.765092	0.0393001	0.0196501	−	0.999807i	$-0.493745\pi$
$379$	0.765092	0.0393001	0.0196501	+	0.999807i	$0.493745\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.75409	−0.498411	−0.249205	−	0.968451i	$-0.580169\pi$
$383$	−9.75409	−0.498411	−0.249205	+	0.968451i	$0.580169\pi$
$384$	0	0
$385$	2.09888	0.106969
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.6538	1.25000	0.624999	−	0.780625i	$-0.285099\pi$
$389$	24.6538	1.25000	0.624999	+	0.780625i	$0.285099\pi$
$390$	0	0
$391$	−23.4327	−1.18504
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.14687	0.359598
$396$	0	0
$397$	14.1964	0.712496	0.356248	−	0.934391i	$-0.384056\pi$
$397$	14.1964	0.712496	0.356248	+	0.934391i	$0.384056\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.9491	−1.94503	−0.972513	−	0.232850i	$-0.925195\pi$
$401$	−38.9491	−1.94503	−0.972513	+	0.232850i	$0.925195\pi$
$402$	0	0
$403$	0.666208	0.0331862
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.51918	−0.124871
$408$	0	0
$409$	−8.19777	−0.405354	−0.202677	−	0.979246i	$-0.564964\pi$
$409$	−8.19777	−0.405354	−0.202677	+	0.979246i	$0.564964\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.76509	−0.234475
$414$	0	0
$415$	13.1978	0.647853
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.457436	0.0223472	0.0111736	−	0.999938i	$-0.496443\pi$
$419$	0.457436	0.0223472	0.0111736	+	0.999938i	$0.496443\pi$
$420$	0	0
$421$	−18.6291	−0.907925	−0.453963	−	0.891021i	$-0.649990\pi$
$421$	−18.6291	−0.907925	−0.453963	+	0.891021i	$0.649990\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−22.1250	−1.07322
$426$	0	0
$427$	3.76509	0.182206
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.5178	−0.506625	−0.253312	−	0.967385i	$-0.581520\pi$
$431$	−10.5178	−0.506625	−0.253312	+	0.967385i	$0.581520\pi$
$432$	0	0
$433$	−2.80223	−0.134667	−0.0673333	−	0.997731i	$-0.521449\pi$
$433$	−2.80223	−0.134667	−0.0673333	+	0.997731i	$0.521449\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−28.3077	−1.35414
$438$	0	0
$439$	4.29528	0.205002	0.102501	−	0.994733i	$-0.467315\pi$
$439$	4.29528	0.205002	0.102501	+	0.994733i	$0.467315\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.1978	−1.05465	−0.527324	−	0.849664i	$-0.676805\pi$
$443$	−22.1978	−1.05465	−0.527324	+	0.849664i	$0.676805\pi$
$444$	0	0
$445$	−15.9011	−0.753785
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.4313	−0.633862	−0.316931	−	0.948449i	$-0.602652\pi$
$449$	−13.4313	−0.633862	−0.316931	+	0.948449i	$0.602652\pi$
$450$	0	0
$451$	3.56732	0.167979
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.11126	−0.0520969
$456$	0	0
$457$	14.9011	0.697045	0.348522	−	0.937300i	$-0.386684\pi$
$457$	14.9011	0.697045	0.348522	+	0.937300i	$0.386684\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.79123	−0.316299	−0.158150	−	0.987415i	$-0.550553\pi$
$461$	−6.79123	−0.316299	−0.158150	+	0.987415i	$0.550553\pi$
$462$	0	0
$463$	4.43268	0.206004	0.103002	−	0.994681i	$-0.467155\pi$
$463$	4.43268	0.206004	0.103002	+	0.994681i	$0.467155\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4189	1.22252	0.611261	−	0.791429i	$-0.290663\pi$
$467$	26.4189	1.22252	0.611261	+	0.791429i	$0.290663\pi$
$468$	0	0
$469$	4.09888	0.189269
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−20.5192	−0.943473
$474$	0	0
$475$	−26.7280	−1.22636
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.51918	0.252178	0.126089	−	0.992019i	$-0.459758\pi$
$479$	5.51918	0.252178	0.126089	+	0.992019i	$0.459758\pi$
$480$	0	0
$481$	1.33379	0.0608157
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.850219	0.0386065
$486$	0	0
$487$	−40.4930	−1.83492	−0.917458	−	0.397834i	$-0.869762\pi$
$487$	−40.4930	−1.83492	−0.917458	+	0.397834i	$0.869762\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.11126	−0.0501506	−0.0250753	−	0.999686i	$-0.507983\pi$
$491$	−1.11126	−0.0501506	−0.0250753	+	0.999686i	$0.507983\pi$
$492$	0	0
$493$	−5.80361	−0.261381
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.7651	0.482880
$498$	0	0
$499$	2.66758	0.119418	0.0597088	−	0.998216i	$-0.480983\pi$
$499$	2.66758	0.119418	0.0597088	+	0.998216i	$0.480983\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.86398	−0.439813	−0.219906	−	0.975521i	$-0.570575\pi$
$503$	−9.86398	−0.439813	−0.219906	+	0.975521i	$0.570575\pi$
$504$	0	0
$505$	−9.86398	−0.438941
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.98762	−0.442693	−0.221347	−	0.975195i	$-0.571045\pi$
$509$	−9.98762	−0.442693	−0.221347	+	0.975195i	$0.571045\pi$
$510$	0	0
$511$	−3.09888	−0.137087
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.0714	−0.487863
$516$	0	0
$517$	20.9629	0.921946
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−38.7527	−1.69779	−0.848894	−	0.528564i	$-0.822731\pi$
$521$	−38.7527	−1.69779	−0.848894	+	0.528564i	$0.822731\pi$
$522$	0	0
$523$	−19.6304	−0.858379	−0.429190	−	0.903214i	$-0.641201\pi$
$523$	−19.6304	−0.858379	−0.429190	+	0.903214i	$0.641201\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.91487	−0.170535
$528$	0	0
$529$	−7.09888	−0.308647
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.88874	−0.0818102
$534$	0	0
$535$	5.80361	0.250912
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.88874	0.0813536
$540$	0	0
$541$	−0.332415	−0.0142916	−0.00714582	−	0.999974i	$-0.502275\pi$
$541$	−0.332415	−0.0142916	−0.00714582	+	0.999974i	$0.502275\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.1113	−0.433119
$546$	0	0
$547$	−27.2953	−1.16706	−0.583531	−	0.812091i	$-0.698329\pi$
$547$	−27.2953	−1.16706	−0.583531	+	0.812091i	$0.698329\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.01100	−0.298679
$552$	0	0
$553$	6.43130	0.273487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.4079	1.33080	0.665398	−	0.746489i	$-0.268262\pi$
$557$	31.4079	1.33080	0.665398	+	0.746489i	$0.268262\pi$
$558$	0	0
$559$	10.8640	0.459497
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−23.6662	−0.997412	−0.498706	−	0.866771i	$-0.666191\pi$
$563$	−23.6662	−0.997412	−0.498706	+	0.866771i	$0.666191\pi$
$564$	0	0
$565$	13.8022	0.580664
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.72795	0.198206	0.0991030	−	0.995077i	$-0.468403\pi$
$569$	4.72795	0.198206	0.0991030	+	0.995077i	$0.468403\pi$
$570$	0	0
$571$	12.7651	0.534202	0.267101	−	0.963668i	$-0.413934\pi$
$571$	12.7651	0.534202	0.267101	+	0.963668i	$0.413934\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.0138	0.626117
$576$	0	0
$577$	−10.6304	−0.442551	−0.221276	−	0.975211i	$-0.571022\pi$
$577$	−10.6304	−0.442551	−0.221276	+	0.975211i	$0.571022\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.8764	0.492714
$582$	0	0
$583$	23.0617	0.955120
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.61669	−0.149277	−0.0746384	−	0.997211i	$-0.523780\pi$
$587$	−3.61669	−0.149277	−0.0746384	+	0.997211i	$0.523780\pi$
$588$	0	0
$589$	−4.72933	−0.194869
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.6786	−0.931298	−0.465649	−	0.884970i	$-0.654179\pi$
$593$	−22.6786	−0.931298	−0.465649	+	0.884970i	$0.654179\pi$
$594$	0	0
$595$	6.53018	0.267711
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.5906	1.65848	0.829242	−	0.558889i	$-0.188772\pi$
$599$	40.5906	1.65848	0.829242	+	0.558889i	$0.188772\pi$
$600$	0	0
$601$	13.1991	0.538404	0.269202	−	0.963084i	$-0.413240\pi$
$601$	13.1991	0.538404	0.269202	+	0.963084i	$0.413240\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.25967	−0.335803
$606$	0	0
$607$	−1.66758	−0.0676852	−0.0338426	−	0.999427i	$-0.510774\pi$
$607$	−1.66758	−0.0676852	−0.0338426	+	0.999427i	$0.510774\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.0989	−0.449013
$612$	0	0
$613$	22.9257	0.925961	0.462981	−	0.886368i	$-0.346780\pi$
$613$	22.9257	0.925961	0.462981	+	0.886368i	$0.346780\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.7513	0.432832	0.216416	−	0.976301i	$-0.430563\pi$
$617$	10.7513	0.432832	0.216416	+	0.976301i	$0.430563\pi$
$618$	0	0
$619$	25.9629	1.04354	0.521768	−	0.853088i	$-0.325273\pi$
$619$	25.9629	1.04354	0.521768	+	0.853088i	$0.325273\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.3090	−0.573279
$624$	0	0
$625$	8.00138	0.320055
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.83784	−0.312515
$630$	0	0
$631$	3.00138	0.119483	0.0597415	−	0.998214i	$-0.480972\pi$
$631$	3.00138	0.119483	0.0597415	+	0.998214i	$0.480972\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.51918	0.338073
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.5453	−0.535008	−0.267504	−	0.963557i	$-0.586199\pi$
$641$	−13.5453	−0.535008	−0.267504	+	0.963557i	$0.586199\pi$
$642$	0	0
$643$	34.9629	1.37880	0.689400	−	0.724381i	$-0.257874\pi$
$643$	34.9629	1.37880	0.689400	+	0.724381i	$0.257874\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.9890	−1.02173	−0.510866	−	0.859660i	$-0.670675\pi$
$647$	−25.9890	−1.02173	−0.510866	+	0.859660i	$0.670675\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.87636	0.229960	0.114980	−	0.993368i	$-0.463320\pi$
$653$	5.87636	0.229960	0.114980	+	0.993368i	$0.463320\pi$
$654$	0	0
$655$	16.1606	0.631448
$656$	0	0
$657$	0	0
$658$	0	0
$659$	41.2719	1.60772	0.803862	−	0.594815i	$-0.202775\pi$
$659$	41.2719	1.60772	0.803862	+	0.594815i	$0.202775\pi$
$660$	0	0
$661$	−36.9629	−1.43769	−0.718844	−	0.695171i	$-0.755329\pi$
$661$	−36.9629	−1.43769	−0.718844	+	0.695171i	$0.755329\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.88874	0.305912
$666$	0	0
$667$	3.93825	0.152490
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.11126	0.274527
$672$	0	0
$673$	45.3942	1.74982	0.874908	−	0.484289i	$-0.160922\pi$
$673$	45.3942	1.74982	0.874908	+	0.484289i	$0.160922\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.29528	−0.0882146	−0.0441073	−	0.999027i	$-0.514044\pi$
$677$	−2.29528	−0.0882146	−0.0441073	+	0.999027i	$0.514044\pi$
$678$	0	0
$679$	0.765092	0.0293615
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.1483	−1.23012	−0.615059	−	0.788481i	$-0.710868\pi$
$683$	−32.1483	−1.23012	−0.615059	+	0.788481i	$0.710868\pi$
$684$	0	0
$685$	−5.15787	−0.197072
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.2101	−0.465170
$690$	0	0
$691$	−6.80361	−0.258821	−0.129411	−	0.991591i	$-0.541309\pi$
$691$	−6.80361	−0.258821	−0.129411	+	0.991591i	$0.541309\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.4437	−0.813405
$696$	0	0
$697$	11.0989	0.420400
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−42.5933	−1.60873	−0.804363	−	0.594137i	$-0.797493\pi$
$701$	−42.5933	−1.60873	−0.804363	+	0.594137i	$0.797493\pi$
$702$	0	0
$703$	−9.46844	−0.357109
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.87636	−0.333830
$708$	0	0
$709$	−5.13465	−0.192836	−0.0964178	−	0.995341i	$-0.530739\pi$
$709$	−5.13465	−0.192836	−0.0964178	+	0.995341i	$0.530739\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.65658	0.0994898
$714$	0	0
$715$	−2.09888	−0.0784938
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.6057	−0.880344	−0.440172	−	0.897914i	$-0.645082\pi$
$719$	−23.6057	−0.880344	−0.440172	+	0.897914i	$0.645082\pi$
$720$	0	0
$721$	−9.96286	−0.371036
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.71848	0.138101
$726$	0	0
$727$	0.00137742	5.10856e−5 0	2.55428e−5	−	1.00000i	$-0.499992\pi$
$727$	0.00137742	5.10856e−5 0	2.55428e−5		1.00000i	$0.499992\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−63.8406	−2.36123
$732$	0	0
$733$	−21.3955	−0.790262	−0.395131	−	0.918625i	$-0.629301\pi$
$733$	−21.3955	−0.790262	−0.395131	+	0.918625i	$0.629301\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.74171	0.285170
$738$	0	0
$739$	−32.0232	−1.17799	−0.588997	−	0.808135i	$-0.700477\pi$
$739$	−32.0232	−1.17799	−0.588997	+	0.808135i	$0.700477\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	50.5192	1.85337	0.926685	−	0.375840i	$-0.122646\pi$
$743$	50.5192	1.85337	0.926685	+	0.375840i	$0.122646\pi$
$744$	0	0
$745$	15.1593	0.555392
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.22253	0.190827
$750$	0	0
$751$	40.6291	1.48258	0.741288	−	0.671187i	$-0.234215\pi$
$751$	40.6291	1.48258	0.741288	+	0.671187i	$0.234215\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.8502	0.686030
$756$	0	0
$757$	−7.90112	−0.287171	−0.143585	−	0.989638i	$-0.545863\pi$
$757$	−7.90112	−0.287171	−0.143585	+	0.989638i	$0.545863\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0248	−0.798397	−0.399198	−	0.916865i	$-0.630712\pi$
$761$	−22.0248	−0.798397	−0.399198	+	0.916865i	$0.630712\pi$
$762$	0	0
$763$	−9.09888	−0.329402
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.76509	0.172057
$768$	0	0
$769$	−48.8255	−1.76069	−0.880346	−	0.474333i	$-0.842689\pi$
$769$	−48.8255	−1.76069	−0.880346	+	0.474333i	$0.842689\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.4807	−1.24018	−0.620092	−	0.784529i	$-0.712905\pi$
$773$	−34.4807	−1.24018	−0.620092	+	0.784529i	$0.712905\pi$
$774$	0	0
$775$	2.50833	0.0901020
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.4079	0.480388
$780$	0	0
$781$	20.3324	0.727551
$782$	0	0
$783$	0	0
$784$	0	0
$785$	25.1469	0.897530
$786$	0	0
$787$	11.7266	0.418007	0.209004	−	0.977915i	$-0.432978\pi$
$787$	11.7266	0.418007	0.209004	+	0.977915i	$0.432978\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.4203	0.441615
$792$	0	0
$793$	−3.76509	−0.133702
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.2471	1.85069	0.925344	−	0.379128i	$-0.123776\pi$
$797$	52.2471	1.85069	0.925344	+	0.379128i	$0.123776\pi$
$798$	0	0
$799$	65.2210	2.30735
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.85297	−0.206547
$804$	0	0
$805$	−4.43130	−0.156183
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.5439	1.07387	0.536934	−	0.843624i	$-0.319582\pi$
$809$	30.5439	1.07387	0.536934	+	0.843624i	$0.319582\pi$
$810$	0	0
$811$	−27.9629	−0.981909	−0.490954	−	0.871185i	$-0.663352\pi$
$811$	−27.9629	−0.981909	−0.490954	+	0.871185i	$0.663352\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.1126	−0.704515
$816$	0	0
$817$	−77.1221	−2.69816
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.56870	0.264149	0.132075	−	0.991240i	$-0.457836\pi$
$821$	7.56870	0.264149	0.132075	+	0.991240i	$0.457836\pi$
$822$	0	0
$823$	−36.3955	−1.26867	−0.634334	−	0.773059i	$-0.718726\pi$
$823$	−36.3955	−1.26867	−0.634334	+	0.773059i	$0.718726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.36955	0.0823975	0.0411987	−	0.999151i	$-0.486882\pi$
$827$	2.36955	0.0823975	0.0411987	+	0.999151i	$0.486882\pi$
$828$	0	0
$829$	−22.2335	−0.772202	−0.386101	−	0.922456i	$-0.626178\pi$
$829$	−22.2335	−0.772202	−0.386101	+	0.922456i	$0.626178\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.87636	0.203604
$834$	0	0
$835$	19.1388	0.662325
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.1099	0.418080	0.209040	−	0.977907i	$-0.432966\pi$
$839$	12.1099	0.418080	0.209040	+	0.977907i	$0.432966\pi$
$840$	0	0
$841$	−28.0246	−0.966366
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−13.3352	−0.458744
$846$	0	0
$847$	−7.43268	−0.255390
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.31866	0.182321
$852$	0	0
$853$	−33.0989	−1.13328	−0.566642	−	0.823964i	$-0.691758\pi$
$853$	−33.0989	−1.13328	−0.566642	+	0.823964i	$0.691758\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.5426	−0.599243	−0.299621	−	0.954058i	$-0.596860\pi$
$857$	−17.5426	−0.599243	−0.299621	+	0.954058i	$0.596860\pi$
$858$	0	0
$859$	13.8626	0.472986	0.236493	−	0.971633i	$-0.424002\pi$
$859$	13.8626	0.472986	0.236493	+	0.971633i	$0.424002\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.9890	0.374070	0.187035	−	0.982353i	$-0.440112\pi$
$863$	10.9890	0.374070	0.187035	+	0.982353i	$0.440112\pi$
$864$	0	0
$865$	0.960106	0.0326446
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.1470	0.412060
$870$	0	0
$871$	−4.09888	−0.138885
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.74033	−0.329283
$876$	0	0
$877$	24.4944	0.827118	0.413559	−	0.910477i	$-0.364286\pi$
$877$	24.4944	0.827118	0.413559	+	0.910477i	$0.364286\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.9243	0.671268	0.335634	−	0.941992i	$-0.391049\pi$
$881$	19.9243	0.671268	0.335634	+	0.941992i	$0.391049\pi$
$882$	0	0
$883$	39.5316	1.33034	0.665171	−	0.746691i	$-0.268358\pi$
$883$	39.5316	1.33034	0.665171	+	0.746691i	$0.268358\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.0755	1.37918	0.689590	−	0.724200i	$-0.257791\pi$
$887$	41.0755	1.37918	0.689590	+	0.724200i	$0.257791\pi$
$888$	0	0
$889$	7.66621	0.257116
$890$	0	0
$891$	0	0
$892$	0	0
$893$	78.7897	2.63660
$894$	0	0
$895$	5.19915	0.173788
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.657960	0.0219442
$900$	0	0
$901$	71.7512	2.39038
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.7403	−0.622950
$906$	0	0
$907$	−11.6648	−0.387324	−0.193662	−	0.981068i	$-0.562037\pi$
$907$	−11.6648	−0.387324	−0.193662	+	0.981068i	$0.562037\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	41.6057	1.37846	0.689229	−	0.724544i	$-0.257949\pi$
$911$	41.6057	1.37846	0.689229	+	0.724544i	$0.257949\pi$
$912$	0	0
$913$	22.4313	0.742368
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.5426	0.480238
$918$	0	0
$919$	−28.3338	−0.934646	−0.467323	−	0.884087i	$-0.654781\pi$
$919$	−28.3338	−0.934646	−0.467323	+	0.884087i	$0.654781\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.7651	−0.354337
$924$	0	0
$925$	5.02185	0.165117
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.8255	−0.978542	−0.489271	−	0.872132i	$-0.662737\pi$
$929$	−29.8255	−0.978542	−0.489271	+	0.872132i	$0.662737\pi$
$930$	0	0
$931$	7.09888	0.232657
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.3338	0.403358
$936$	0	0
$937$	−2.56870	−0.0839158	−0.0419579	−	0.999119i	$-0.513360\pi$
$937$	−2.56870	−0.0839158	−0.0419579	+	0.999119i	$0.513360\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.66896	−0.152204	−0.0761019	−	0.997100i	$-0.524247\pi$
$941$	−4.66896	−0.152204	−0.0761019	+	0.997100i	$0.524247\pi$
$942$	0	0
$943$	−7.53156	−0.245261
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.6908	−1.02981	−0.514907	−	0.857246i	$-0.672173\pi$
$947$	−31.6908	−1.02981	−0.514907	+	0.857246i	$0.672173\pi$
$948$	0	0
$949$	3.09888	0.100594
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.0604	−0.423067	−0.211533	−	0.977371i	$-0.567846\pi$
$953$	−13.0604	−0.423067	−0.211533	+	0.977371i	$0.567846\pi$
$954$	0	0
$955$	18.9601	0.613535
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.64145	−0.149880
$960$	0	0
$961$	−30.5562	−0.985683
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.25967	0.265888
$966$	0	0
$967$	−8.96148	−0.288182	−0.144091	−	0.989564i	$-0.546026\pi$
$967$	−8.96148	−0.288182	−0.144091	+	0.989564i	$0.546026\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.543941	−0.0174559	−0.00872795	−	0.999962i	$-0.502778\pi$
$971$	−0.543941	−0.0174559	−0.00872795	+	0.999962i	$0.502778\pi$
$972$	0	0
$973$	−19.2967	−0.618622
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−36.4559	−1.16633	−0.583164	−	0.812355i	$-0.698185\pi$
$977$	−36.4559	−1.16633	−0.583164	+	0.812355i	$0.698185\pi$
$978$	0	0
$979$	−27.0260	−0.863754
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.3832	1.16044	0.580221	−	0.814459i	$-0.302966\pi$
$983$	36.3832	1.16044	0.580221	+	0.814459i	$0.302966\pi$
$984$	0	0
$985$	29.2210	0.931058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	43.3214	1.37754
$990$	0	0
$991$	−31.9642	−1.01538	−0.507689	−	0.861541i	$-0.669500\pi$
$991$	−31.9642	−1.01538	−0.507689	+	0.861541i	$0.669500\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.5935	0.367537
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.a.j.1.2		3
3.2	odd	2		2268.2.a.g.1.2		3
4.3	odd	2		9072.2.a.bz.1.2		3
9.2	odd	6		252.2.j.b.85.3	✓	6
9.4	even	3		756.2.j.a.505.2		6
9.5	odd	6		252.2.j.b.169.3	yes	6
9.7	even	3		756.2.j.a.253.2		6
12.11	even	2		9072.2.a.bt.1.2		3
36.7	odd	6		3024.2.r.i.1009.2		6
36.11	even	6		1008.2.r.g.337.1		6
36.23	even	6		1008.2.r.g.673.1		6
36.31	odd	6		3024.2.r.i.2017.2		6
63.2	odd	6		1764.2.l.d.949.3		6
63.4	even	3		5292.2.l.g.3313.2		6
63.5	even	6		1764.2.i.e.1537.3		6
63.11	odd	6		1764.2.i.f.373.1		6
63.13	odd	6		5292.2.j.e.3529.2		6
63.16	even	3		5292.2.l.g.361.2		6
63.20	even	6		1764.2.j.d.589.1		6
63.23	odd	6		1764.2.i.f.1537.1		6
63.25	even	3		5292.2.i.d.1549.2		6
63.31	odd	6		5292.2.l.d.3313.2		6
63.32	odd	6		1764.2.l.d.961.3		6
63.34	odd	6		5292.2.j.e.1765.2		6
63.38	even	6		1764.2.i.e.373.3		6
63.40	odd	6		5292.2.i.g.2125.2		6
63.41	even	6		1764.2.j.d.1177.1		6
63.47	even	6		1764.2.l.g.949.1		6
63.52	odd	6		5292.2.i.g.1549.2		6
63.58	even	3		5292.2.i.d.2125.2		6
63.59	even	6		1764.2.l.g.961.1		6
63.61	odd	6		5292.2.l.d.361.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.3	✓	6	9.2	odd	6
252.2.j.b.169.3	yes	6	9.5	odd	6
756.2.j.a.253.2		6	9.7	even	3
756.2.j.a.505.2		6	9.4	even	3
1008.2.r.g.337.1		6	36.11	even	6
1008.2.r.g.673.1		6	36.23	even	6
1764.2.i.e.373.3		6	63.38	even	6
1764.2.i.e.1537.3		6	63.5	even	6
1764.2.i.f.373.1		6	63.11	odd	6
1764.2.i.f.1537.1		6	63.23	odd	6
1764.2.j.d.589.1		6	63.20	even	6
1764.2.j.d.1177.1		6	63.41	even	6
1764.2.l.d.949.3		6	63.2	odd	6
1764.2.l.d.961.3		6	63.32	odd	6
1764.2.l.g.949.1		6	63.47	even	6
1764.2.l.g.961.1		6	63.59	even	6
2268.2.a.g.1.2		3	3.2	odd	2
2268.2.a.j.1.2		3	1.1	even	1	trivial
3024.2.r.i.1009.2		6	36.7	odd	6
3024.2.r.i.2017.2		6	36.31	odd	6
5292.2.i.d.1549.2		6	63.25	even	3
5292.2.i.d.2125.2		6	63.58	even	3
5292.2.i.g.1549.2		6	63.52	odd	6
5292.2.i.g.2125.2		6	63.40	odd	6
5292.2.j.e.1765.2		6	63.34	odd	6
5292.2.j.e.3529.2		6	63.13	odd	6
5292.2.l.d.361.2		6	63.61	odd	6
5292.2.l.d.3313.2		6	63.31	odd	6
5292.2.l.g.361.2		6	63.16	even	3
5292.2.l.g.3313.2		6	63.4	even	3
9072.2.a.bt.1.2		3	12.11	even	2
9072.2.a.bz.1.2		3	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q+1.11126 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+1.11126 q^{5} +1.00000 q^{7} +1.88874 q^{11} -1.00000 q^{13} +5.87636 q^{17} +7.09888 q^{19} -3.98762 q^{23} -3.76509 q^{25} -0.987620 q^{29} -0.666208 q^{31} +1.11126 q^{35} -1.33379 q^{37} +1.88874 q^{41} -10.8640 q^{43} +11.0989 q^{47} +1.00000 q^{49} +12.2101 q^{53} +2.09888 q^{55} -4.76509 q^{59} +3.76509 q^{61} -1.11126 q^{65} +4.09888 q^{67} +10.7651 q^{71} -3.09888 q^{73} +1.88874 q^{77} +6.43130 q^{79} +11.8764 q^{83} +6.53018 q^{85} -14.3090 q^{89} -1.00000 q^{91} +7.88874 q^{95} +0.765092 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 + 3 * q^7	\( 3 q + 3 q^{5} + 3 q^{7} + 6 q^{11} - 3 q^{13} + 3 q^{19} + 6 q^{23} + 6 q^{25} + 15 q^{29} - 3 q^{31} + 3 q^{35} - 3 q^{37} + 6 q^{41} + 3 q^{43} + 15 q^{47} + 3 q^{49} + 18 q^{53} - 12 q^{55} + 3 q^{59} - 6 q^{61} - 3 q^{65} - 6 q^{67} + 15 q^{71} + 9 q^{73} + 6 q^{77} + 3 q^{79} + 18 q^{83} - 15 q^{85} - 6 q^{89} - 3 q^{91} + 24 q^{95} - 15 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^7 + 6 * q^11 - 3 * q^13 + 3 * q^19 + 6 * q^23 + 6 * q^25 + 15 * q^29 - 3 * q^31 + 3 * q^35 - 3 * q^37 + 6 * q^41 + 3 * q^43 + 15 * q^47 + 3 * q^49 + 18 * q^53 - 12 * q^55 + 3 * q^59 - 6 * q^61 - 3 * q^65 - 6 * q^67 + 15 * q^71 + 9 * q^73 + 6 * q^77 + 3 * q^79 + 18 * q^83 - 15 * q^85 - 6 * q^89 - 3 * q^91 + 24 * q^95 - 15 * q^97

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