LMFDB - Embedded newform 9072.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.44949 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.44949 q^{5} +1.00000 q^{7} +2.00000 q^{11} -4.89898 q^{13} -2.00000 q^{17} -7.44949 q^{19} -1.00000 q^{23} +6.89898 q^{25} -2.89898 q^{29} -6.00000 q^{31} -3.44949 q^{35} -7.79796 q^{37} +9.79796 q^{41} +2.89898 q^{43} -9.79796 q^{47} +1.00000 q^{49} +1.10102 q^{53} -6.89898 q^{55} -2.00000 q^{59} +11.4495 q^{61} +16.8990 q^{65} +3.10102 q^{67} +9.89898 q^{71} +2.89898 q^{73} +2.00000 q^{77} -7.89898 q^{79} +2.00000 q^{83} +6.89898 q^{85} +7.10102 q^{89} -4.89898 q^{91} +25.6969 q^{95} -6.89898 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{17} - 10 q^{19} - 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{35} + 4 q^{37} - 4 q^{43} + 2 q^{49} + 12 q^{53} - 4 q^{55} - 4 q^{59} + 18 q^{61} + 24 q^{65} + 16 q^{67} + 10 q^{71} - 4 q^{73} + 4 q^{77} - 6 q^{79} + 4 q^{83} + 4 q^{85} + 24 q^{89} + 22 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 4 * q^11 - 4 * q^17 - 10 * q^19 - 2 * q^23 + 4 * q^25 + 4 * q^29 - 12 * q^31 - 2 * q^35 + 4 * q^37 - 4 * q^43 + 2 * q^49 + 12 * q^53 - 4 * q^55 - 4 * q^59 + 18 * q^61 + 24 * q^65 + 16 * q^67 + 10 * q^71 - 4 * q^73 + 4 * q^77 - 6 * q^79 + 4 * q^83 + 4 * q^85 + 24 * q^89 + 22 * q^95 - 4 * 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
$5$	−3.44949	−1.54266	−0.771329	−	0.636436i	$-0.780408\pi$
$5$	−3.44949	−1.54266	−0.771329	+	0.636436i	$0.780408\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−4.89898	−1.35873	−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898	−1.35873	−0.679366	+	0.733799i	$0.737745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−7.44949	−1.70903	−0.854515	−	0.519427i	$-0.826146\pi$
$19$	−7.44949	−1.70903	−0.854515	+	0.519427i	$0.826146\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	6.89898	1.37980
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.89898	−0.538327	−0.269163	−	0.963095i	$-0.586747\pi$
$29$	−2.89898	−0.538327	−0.269163	+	0.963095i	$0.586747\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.44949	−0.583070
$36$	0	0
$37$	−7.79796	−1.28198	−0.640988	−	0.767551i	$-0.721475\pi$
$37$	−7.79796	−1.28198	−0.640988	+	0.767551i	$0.721475\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.79796	1.53018	0.765092	−	0.643921i	$-0.222693\pi$
$41$	9.79796	1.53018	0.765092	+	0.643921i	$0.222693\pi$
$42$	0	0
$43$	2.89898	0.442090	0.221045	−	0.975264i	$-0.429053\pi$
$43$	2.89898	0.442090	0.221045	+	0.975264i	$0.429053\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.10102	0.151237	0.0756184	−	0.997137i	$-0.475907\pi$
$53$	1.10102	0.151237	0.0756184	+	0.997137i	$0.475907\pi$
$54$	0	0
$55$	−6.89898	−0.930258
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	11.4495	1.46596	0.732978	−	0.680252i	$-0.238130\pi$
$61$	11.4495	1.46596	0.732978	+	0.680252i	$0.238130\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.8990	2.09606
$66$	0	0
$67$	3.10102	0.378850	0.189425	−	0.981895i	$-0.439338\pi$
$67$	3.10102	0.378850	0.189425	+	0.981895i	$0.439338\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.89898	1.17479	0.587396	−	0.809299i	$-0.300153\pi$
$71$	9.89898	1.17479	0.587396	+	0.809299i	$0.300153\pi$
$72$	0	0
$73$	2.89898	0.339300	0.169650	−	0.985504i	$-0.445736\pi$
$73$	2.89898	0.339300	0.169650	+	0.985504i	$0.445736\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−7.89898	−0.888705	−0.444352	−	0.895852i	$-0.646566\pi$
$79$	−7.89898	−0.888705	−0.444352	+	0.895852i	$0.646566\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	6.89898	0.748299
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.10102	0.752707	0.376353	−	0.926476i	$-0.377178\pi$
$89$	7.10102	0.752707	0.376353	+	0.926476i	$0.377178\pi$
$90$	0	0
$91$	−4.89898	−0.513553
$92$	0	0
$93$	0	0
$94$	0	0
$95$	25.6969	2.63645
$96$	0	0
$97$	−6.89898	−0.700485	−0.350243	−	0.936659i	$-0.613901\pi$
$97$	−6.89898	−0.700485	−0.350243	+	0.936659i	$0.613901\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.24745	−0.721148	−0.360574	−	0.932731i	$-0.617419\pi$
$101$	−7.24745	−0.721148	−0.360574	+	0.932731i	$0.617419\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−16.6969	−1.59928	−0.799638	−	0.600482i	$-0.794975\pi$
$109$	−16.6969	−1.59928	−0.799638	+	0.600482i	$0.794975\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.8990	1.49565	0.747825	−	0.663896i	$-0.231098\pi$
$113$	15.8990	1.49565	0.747825	+	0.663896i	$0.231098\pi$
$114$	0	0
$115$	3.44949	0.321667
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.55051	−0.585895
$126$	0	0
$127$	3.00000	0.266207	0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000	0.266207	0.133103	+	0.991102i	$0.457506\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.4495	−1.17509	−0.587544	−	0.809192i	$-0.699905\pi$
$131$	−13.4495	−1.17509	−0.587544	+	0.809192i	$0.699905\pi$
$132$	0	0
$133$	−7.44949	−0.645953
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.7980	1.00797	0.503984	−	0.863713i	$-0.331867\pi$
$137$	11.7980	1.00797	0.503984	+	0.863713i	$0.331867\pi$
$138$	0	0
$139$	−9.44949	−0.801495	−0.400748	−	0.916188i	$-0.631250\pi$
$139$	−9.44949	−0.801495	−0.400748	+	0.916188i	$0.631250\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.79796	−0.819346
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.6969	1.66242
$156$	0	0
$157$	6.34847	0.506663	0.253332	−	0.967380i	$-0.418474\pi$
$157$	6.34847	0.506663	0.253332	+	0.967380i	$0.418474\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	0.202041	0.0158251	0.00791254	−	0.999969i	$-0.497481\pi$
$163$	0.202041	0.0158251	0.00791254	+	0.999969i	$0.497481\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.6969	1.44681	0.723406	−	0.690423i	$-0.242575\pi$
$167$	18.6969	1.44681	0.723406	+	0.690423i	$0.242575\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.8990	0.980691	0.490346	−	0.871528i	$-0.336871\pi$
$173$	12.8990	0.980691	0.490346	+	0.871528i	$0.336871\pi$
$174$	0	0
$175$	6.89898	0.521514
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.69694	−0.650040	−0.325020	−	0.945707i	$-0.605371\pi$
$179$	−8.69694	−0.650040	−0.325020	+	0.945707i	$0.605371\pi$
$180$	0	0
$181$	4.34847	0.323219	0.161610	−	0.986855i	$-0.448331\pi$
$181$	4.34847	0.323219	0.161610	+	0.986855i	$0.448331\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	26.8990	1.97765
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.8990	1.00569	0.502847	−	0.864375i	$-0.332286\pi$
$191$	13.8990	1.00569	0.502847	+	0.864375i	$0.332286\pi$
$192$	0	0
$193$	−8.10102	−0.583124	−0.291562	−	0.956552i	$-0.594175\pi$
$193$	−8.10102	−0.583124	−0.291562	+	0.956552i	$0.594175\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.6969	0.904619	0.452310	−	0.891861i	$-0.350600\pi$
$197$	12.6969	0.904619	0.452310	+	0.891861i	$0.350600\pi$
$198$	0	0
$199$	−6.89898	−0.489056	−0.244528	−	0.969642i	$-0.578633\pi$
$199$	−6.89898	−0.489056	−0.244528	+	0.969642i	$0.578633\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.89898	−0.203468
$204$	0	0
$205$	−33.7980	−2.36055
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.8990	−1.03058
$210$	0	0
$211$	−3.10102	−0.213483	−0.106742	−	0.994287i	$-0.534042\pi$
$211$	−3.10102	−0.213483	−0.106742	+	0.994287i	$0.534042\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.79796	0.659082
$222$	0	0
$223$	20.8990	1.39950	0.699750	−	0.714388i	$-0.253295\pi$
$223$	20.8990	1.39950	0.699750	+	0.714388i	$0.253295\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.550510	−0.0365386	−0.0182693	−	0.999833i	$-0.505816\pi$
$227$	−0.550510	−0.0365386	−0.0182693	+	0.999833i	$0.505816\pi$
$228$	0	0
$229$	−23.2474	−1.53623	−0.768117	−	0.640309i	$-0.778806\pi$
$229$	−23.2474	−1.53623	−0.768117	+	0.640309i	$0.778806\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.00000	0.458585	0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000	0.458585	0.229293	+	0.973358i	$0.426359\pi$
$234$	0	0
$235$	33.7980	2.20474
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.7980	−0.827831	−0.413916	−	0.910315i	$-0.635839\pi$
$239$	−12.7980	−0.827831	−0.413916	+	0.910315i	$0.635839\pi$
$240$	0	0
$241$	−8.89898	−0.573234	−0.286617	−	0.958045i	$-0.592531\pi$
$241$	−8.89898	−0.573234	−0.286617	+	0.958045i	$0.592531\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.44949	−0.220380
$246$	0	0
$247$	36.4949	2.32211
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.5505	0.792181	0.396091	−	0.918211i	$-0.370367\pi$
$251$	12.5505	0.792181	0.396091	+	0.918211i	$0.370367\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.7980	−1.73399	−0.866995	−	0.498318i	$-0.833951\pi$
$257$	−27.7980	−1.73399	−0.866995	+	0.498318i	$0.833951\pi$
$258$	0	0
$259$	−7.79796	−0.484542
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.1010	0.992831	0.496416	−	0.868085i	$-0.334649\pi$
$263$	16.1010	0.992831	0.496416	+	0.868085i	$0.334649\pi$
$264$	0	0
$265$	−3.79796	−0.233307
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.65153	0.222638	0.111319	−	0.993785i	$-0.464493\pi$
$269$	3.65153	0.222638	0.111319	+	0.993785i	$0.464493\pi$
$270$	0	0
$271$	−16.8990	−1.02654	−0.513270	−	0.858227i	$-0.671566\pi$
$271$	−16.8990	−1.02654	−0.513270	+	0.858227i	$0.671566\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.7980	0.832048
$276$	0	0
$277$	10.6969	0.642717	0.321358	−	0.946958i	$-0.395861\pi$
$277$	10.6969	0.642717	0.321358	+	0.946958i	$0.395861\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	20.5505	1.22160	0.610801	−	0.791785i	$-0.290848\pi$
$283$	20.5505	1.22160	0.610801	+	0.791785i	$0.290848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.79796	0.578355
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.2474	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$293$	−27.2474	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$294$	0	0
$295$	6.89898	0.401674
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.89898	0.283315
$300$	0	0
$301$	2.89898	0.167094
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−39.4949	−2.26147
$306$	0	0
$307$	−0.752551	−0.0429504	−0.0214752	−	0.999769i	$-0.506836\pi$
$307$	−0.752551	−0.0429504	−0.0214752	+	0.999769i	$0.506836\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.30306	0.0738898	0.0369449	−	0.999317i	$-0.488237\pi$
$311$	1.30306	0.0738898	0.0369449	+	0.999317i	$0.488237\pi$
$312$	0	0
$313$	24.6969	1.39595	0.697977	−	0.716120i	$-0.254084\pi$
$313$	24.6969	1.39595	0.697977	+	0.716120i	$0.254084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.69694	0.488469	0.244234	−	0.969716i	$-0.421463\pi$
$317$	8.69694	0.488469	0.244234	+	0.969716i	$0.421463\pi$
$318$	0	0
$319$	−5.79796	−0.324623
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14.8990	0.829001
$324$	0	0
$325$	−33.7980	−1.87477
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.79796	−0.540179
$330$	0	0
$331$	24.6969	1.35747	0.678733	−	0.734385i	$-0.262529\pi$
$331$	24.6969	1.35747	0.678733	+	0.734385i	$0.262529\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.6969	−0.584436
$336$	0	0
$337$	35.3939	1.92803	0.964014	−	0.265853i	$-0.0856535\pi$
$337$	35.3939	1.92803	0.964014	+	0.265853i	$0.0856535\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.5959	−1.05196	−0.525982	−	0.850496i	$-0.676302\pi$
$347$	−19.5959	−1.05196	−0.525982	+	0.850496i	$0.676302\pi$
$348$	0	0
$349$	20.8990	1.11870	0.559348	−	0.828933i	$-0.311051\pi$
$349$	20.8990	1.11870	0.559348	+	0.828933i	$0.311051\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−34.1464	−1.81230
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.7980	0.569894	0.284947	−	0.958543i	$-0.408024\pi$
$359$	10.7980	0.569894	0.284947	+	0.958543i	$0.408024\pi$
$360$	0	0
$361$	36.4949	1.92078
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−5.79796	−0.302651	−0.151325	−	0.988484i	$-0.548354\pi$
$367$	−5.79796	−0.302651	−0.151325	+	0.988484i	$0.548354\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.10102	0.0571621
$372$	0	0
$373$	2.89898	0.150103	0.0750517	−	0.997180i	$-0.476088\pi$
$373$	2.89898	0.150103	0.0750517	+	0.997180i	$0.476088\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.2020	0.731442
$378$	0	0
$379$	26.4949	1.36095	0.680476	−	0.732771i	$-0.261773\pi$
$379$	26.4949	1.36095	0.680476	+	0.732771i	$0.261773\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.89898	0.352521	0.176261	−	0.984344i	$-0.443600\pi$
$383$	6.89898	0.352521	0.176261	+	0.984344i	$0.443600\pi$
$384$	0	0
$385$	−6.89898	−0.351605
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.1010	0.765652	0.382826	−	0.923820i	$-0.374951\pi$
$389$	15.1010	0.765652	0.382826	+	0.923820i	$0.374951\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	27.2474	1.37097
$396$	0	0
$397$	9.30306	0.466907	0.233454	−	0.972368i	$-0.424997\pi$
$397$	9.30306	0.466907	0.233454	+	0.972368i	$0.424997\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.1010	0.504421	0.252210	−	0.967672i	$-0.418842\pi$
$401$	10.1010	0.504421	0.252210	+	0.967672i	$0.418842\pi$
$402$	0	0
$403$	29.3939	1.46421
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.5959	−0.773061
$408$	0	0
$409$	5.79796	0.286691	0.143345	−	0.989673i	$-0.454214\pi$
$409$	5.79796	0.286691	0.143345	+	0.989673i	$0.454214\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.00000	−0.0984136
$414$	0	0
$415$	−6.89898	−0.338658
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.5505	1.19937	0.599685	−	0.800236i	$-0.295292\pi$
$419$	24.5505	1.19937	0.599685	+	0.800236i	$0.295292\pi$
$420$	0	0
$421$	13.1010	0.638505	0.319252	−	0.947670i	$-0.396568\pi$
$421$	13.1010	0.638505	0.319252	+	0.947670i	$0.396568\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.7980	−0.669299
$426$	0	0
$427$	11.4495	0.554080
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.59592	−0.365882	−0.182941	−	0.983124i	$-0.558562\pi$
$431$	−7.59592	−0.365882	−0.182941	+	0.983124i	$0.558562\pi$
$432$	0	0
$433$	11.7980	0.566974	0.283487	−	0.958976i	$-0.408509\pi$
$433$	11.7980	0.566974	0.283487	+	0.958976i	$0.408509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.44949	0.356357
$438$	0	0
$439$	−21.7980	−1.04036	−0.520180	−	0.854057i	$-0.674135\pi$
$439$	−21.7980	−1.04036	−0.520180	+	0.854057i	$0.674135\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.10102	−0.242357	−0.121178	−	0.992631i	$-0.538667\pi$
$443$	−5.10102	−0.242357	−0.121178	+	0.992631i	$0.538667\pi$
$444$	0	0
$445$	−24.4949	−1.16117
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.5959	0.877596	0.438798	−	0.898586i	$-0.355404\pi$
$449$	18.5959	0.877596	0.438798	+	0.898586i	$0.355404\pi$
$450$	0	0
$451$	19.5959	0.922736
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.8990	0.792236
$456$	0	0
$457$	31.4949	1.47327	0.736635	−	0.676291i	$-0.236414\pi$
$457$	31.4949	1.47327	0.736635	+	0.676291i	$0.236414\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.3485	−0.947723	−0.473861	−	0.880599i	$-0.657140\pi$
$461$	−20.3485	−0.947723	−0.473861	+	0.880599i	$0.657140\pi$
$462$	0	0
$463$	25.6969	1.19424	0.597119	−	0.802153i	$-0.296312\pi$
$463$	25.6969	1.19424	0.597119	+	0.802153i	$0.296312\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	3.10102	0.143192
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.79796	0.266590
$474$	0	0
$475$	−51.3939	−2.35811
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.5959	−1.35227	−0.676136	−	0.736777i	$-0.736347\pi$
$479$	−29.5959	−1.35227	−0.676136	+	0.736777i	$0.736347\pi$
$480$	0	0
$481$	38.2020	1.74186
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.7980	1.08061
$486$	0	0
$487$	−22.3939	−1.01476	−0.507382	−	0.861721i	$-0.669387\pi$
$487$	−22.3939	−1.01476	−0.507382	+	0.861721i	$0.669387\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.79796	−0.171399	−0.0856997	−	0.996321i	$-0.527313\pi$
$491$	−3.79796	−0.171399	−0.0856997	+	0.996321i	$0.527313\pi$
$492$	0	0
$493$	5.79796	0.261127
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.89898	0.444030
$498$	0	0
$499$	−33.3939	−1.49492	−0.747458	−	0.664309i	$-0.768726\pi$
$499$	−33.3939	−1.49492	−0.747458	+	0.664309i	$0.768726\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.4949	1.09217	0.546087	−	0.837729i	$-0.316117\pi$
$503$	24.4949	1.09217	0.546087	+	0.837729i	$0.316117\pi$
$504$	0	0
$505$	25.0000	1.11249
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.8990	0.749034	0.374517	−	0.927220i	$-0.377809\pi$
$509$	16.8990	0.749034	0.374517	+	0.927220i	$0.377809\pi$
$510$	0	0
$511$	2.89898	0.128243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	48.2929	2.12804
$516$	0	0
$517$	−19.5959	−0.861827
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.6969	1.69534	0.847672	−	0.530521i	$-0.178004\pi$
$521$	38.6969	1.69534	0.847672	+	0.530521i	$0.178004\pi$
$522$	0	0
$523$	−0.348469	−0.0152375	−0.00761875	−	0.999971i	$-0.502425\pi$
$523$	−0.348469	−0.0152375	−0.00761875	+	0.999971i	$0.502425\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−48.0000	−2.07911
$534$	0	0
$535$	41.3939	1.78961
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	30.4949	1.31108	0.655539	−	0.755161i	$-0.272441\pi$
$541$	30.4949	1.31108	0.655539	+	0.755161i	$0.272441\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	57.5959	2.46714
$546$	0	0
$547$	−31.5959	−1.35094	−0.675472	−	0.737386i	$-0.736060\pi$
$547$	−31.5959	−1.35094	−0.675472	+	0.737386i	$0.736060\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.5959	0.920017
$552$	0	0
$553$	−7.89898	−0.335899
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.10102	0.131394	0.0656972	−	0.997840i	$-0.479073\pi$
$557$	3.10102	0.131394	0.0656972	+	0.997840i	$0.479073\pi$
$558$	0	0
$559$	−14.2020	−0.600682
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.9444	0.587686	0.293843	−	0.955854i	$-0.405066\pi$
$563$	13.9444	0.587686	0.293843	+	0.955854i	$0.405066\pi$
$564$	0	0
$565$	−54.8434	−2.30728
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−14.2020	−0.594337	−0.297168	−	0.954825i	$-0.596042\pi$
$571$	−14.2020	−0.594337	−0.297168	+	0.954825i	$0.596042\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.89898	−0.287707
$576$	0	0
$577$	23.5959	0.982311	0.491155	−	0.871072i	$-0.336575\pi$
$577$	23.5959	0.982311	0.491155	+	0.871072i	$0.336575\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	2.20204	0.0911992
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.1464	−0.748983	−0.374492	−	0.927230i	$-0.622183\pi$
$587$	−18.1464	−0.748983	−0.374492	+	0.927230i	$0.622183\pi$
$588$	0	0
$589$	44.6969	1.84171
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.6969	0.603531	0.301765	−	0.953382i	$-0.402424\pi$
$593$	14.6969	0.603531	0.301765	+	0.953382i	$0.402424\pi$
$594$	0	0
$595$	6.89898	0.282831
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.2020	−0.580280	−0.290140	−	0.956984i	$-0.593702\pi$
$599$	−14.2020	−0.580280	−0.290140	+	0.956984i	$0.593702\pi$
$600$	0	0
$601$	−12.6969	−0.517919	−0.258959	−	0.965888i	$-0.583380\pi$
$601$	−12.6969	−0.517919	−0.258959	+	0.965888i	$0.583380\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.1464	0.981692
$606$	0	0
$607$	8.69694	0.352998	0.176499	−	0.984301i	$-0.443523\pi$
$607$	8.69694	0.352998	0.176499	+	0.984301i	$0.443523\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	14.6969	0.593604	0.296802	−	0.954939i	$-0.404080\pi$
$613$	14.6969	0.593604	0.296802	+	0.954939i	$0.404080\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.3939	−1.74697	−0.873486	−	0.486850i	$-0.838146\pi$
$617$	−43.3939	−1.74697	−0.873486	+	0.486850i	$0.838146\pi$
$618$	0	0
$619$	4.14643	0.166659	0.0833295	−	0.996522i	$-0.473445\pi$
$619$	4.14643	0.166659	0.0833295	+	0.996522i	$0.473445\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.10102	0.284496
$624$	0	0
$625$	−11.8990	−0.475959
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.5959	0.621850
$630$	0	0
$631$	−18.1010	−0.720590	−0.360295	−	0.932838i	$-0.617324\pi$
$631$	−18.1010	−0.720590	−0.360295	+	0.932838i	$0.617324\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.3485	−0.410666
$636$	0	0
$637$	−4.89898	−0.194105
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−41.4949	−1.63895	−0.819475	−	0.573115i	$-0.805735\pi$
$641$	−41.4949	−1.63895	−0.819475	+	0.573115i	$0.805735\pi$
$642$	0	0
$643$	−19.3939	−0.764820	−0.382410	−	0.923993i	$-0.624906\pi$
$643$	−19.3939	−0.764820	−0.382410	+	0.923993i	$0.624906\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.3031	−0.837510	−0.418755	−	0.908099i	$-0.637533\pi$
$647$	−21.3031	−0.837510	−0.418755	+	0.908099i	$0.637533\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.79796	0.383424	0.191712	−	0.981451i	$-0.438596\pi$
$653$	9.79796	0.383424	0.191712	+	0.981451i	$0.438596\pi$
$654$	0	0
$655$	46.3939	1.81276
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.69694	0.182967	0.0914834	−	0.995807i	$-0.470839\pi$
$659$	4.69694	0.182967	0.0914834	+	0.995807i	$0.470839\pi$
$660$	0	0
$661$	9.44949	0.367543	0.183771	−	0.982969i	$-0.441169\pi$
$661$	9.44949	0.367543	0.183771	+	0.982969i	$0.441169\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	25.6969	0.996485
$666$	0	0
$667$	2.89898	0.112249
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.8990	0.884005
$672$	0	0
$673$	30.5959	1.17939	0.589693	−	0.807628i	$-0.299249\pi$
$673$	30.5959	1.17939	0.589693	+	0.807628i	$0.299249\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.6969	0.564849	0.282425	−	0.959289i	$-0.408861\pi$
$677$	14.6969	0.564849	0.282425	+	0.959289i	$0.408861\pi$
$678$	0	0
$679$	−6.89898	−0.264759
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.2020	1.23218	0.616088	−	0.787677i	$-0.288716\pi$
$683$	32.2020	1.23218	0.616088	+	0.787677i	$0.288716\pi$
$684$	0	0
$685$	−40.6969	−1.55495
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.39388	−0.205490
$690$	0	0
$691$	−6.95459	−0.264565	−0.132283	−	0.991212i	$-0.542231\pi$
$691$	−6.95459	−0.264565	−0.132283	+	0.991212i	$0.542231\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32.5959	1.23643
$696$	0	0
$697$	−19.5959	−0.742248
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−51.3939	−1.94112	−0.970560	−	0.240860i	$-0.922571\pi$
$701$	−51.3939	−1.94112	−0.970560	+	0.240860i	$0.922571\pi$
$702$	0	0
$703$	58.0908	2.19094
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.24745	−0.272568
$708$	0	0
$709$	−11.5959	−0.435494	−0.217747	−	0.976005i	$-0.569871\pi$
$709$	−11.5959	−0.435494	−0.217747	+	0.976005i	$0.569871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	33.7980	1.26397
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.79796	−0.365402	−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796	−0.365402	−0.182701	+	0.983169i	$0.558484\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−40.4949	−1.50187	−0.750936	−	0.660375i	$-0.770397\pi$
$727$	−40.4949	−1.50187	−0.750936	+	0.660375i	$0.770397\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.79796	−0.214445
$732$	0	0
$733$	−12.5505	−0.463564	−0.231782	−	0.972768i	$-0.574456\pi$
$733$	−12.5505	−0.463564	−0.231782	+	0.972768i	$0.574456\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.20204	0.228455
$738$	0	0
$739$	25.5959	0.941561	0.470781	−	0.882250i	$-0.343972\pi$
$739$	25.5959	0.941561	0.470781	+	0.882250i	$0.343972\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−20.6969	−0.758277
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−40.5959	−1.48137	−0.740683	−	0.671855i	$-0.765498\pi$
$751$	−40.5959	−1.48137	−0.740683	+	0.671855i	$0.765498\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.2474	−0.627699
$756$	0	0
$757$	23.3939	0.850265	0.425132	−	0.905131i	$-0.360228\pi$
$757$	23.3939	0.850265	0.425132	+	0.905131i	$0.360228\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	−16.6969	−0.604470
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.79796	0.353784
$768$	0	0
$769$	54.0908	1.95056	0.975282	−	0.220962i	$-0.0709198\pi$
$769$	54.0908	1.95056	0.975282	+	0.220962i	$0.0709198\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.9444	0.717350	0.358675	−	0.933463i	$-0.383229\pi$
$773$	19.9444	0.717350	0.358675	+	0.933463i	$0.383229\pi$
$774$	0	0
$775$	−41.3939	−1.48691
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−72.9898	−2.61513
$780$	0	0
$781$	19.7980	0.708427
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−21.8990	−0.781608
$786$	0	0
$787$	−47.3939	−1.68941	−0.844705	−	0.535233i	$-0.820224\pi$
$787$	−47.3939	−1.68941	−0.844705	+	0.535233i	$0.820224\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.8990	0.565303
$792$	0	0
$793$	−56.0908	−1.99184
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.9444	−1.27322	−0.636608	−	0.771188i	$-0.719663\pi$
$797$	−35.9444	−1.27322	−0.636608	+	0.771188i	$0.719663\pi$
$798$	0	0
$799$	19.5959	0.693254
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.79796	0.204606
$804$	0	0
$805$	3.44949	0.121579
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35.7980	1.25859	0.629295	−	0.777167i	$-0.283344\pi$
$809$	35.7980	1.25859	0.629295	+	0.777167i	$0.283344\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.696938	−0.0244127
$816$	0	0
$817$	−21.5959	−0.755546
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.5959	−1.38191	−0.690954	−	0.722899i	$-0.742809\pi$
$821$	−39.5959	−1.38191	−0.690954	+	0.722899i	$0.742809\pi$
$822$	0	0
$823$	−45.3939	−1.58233	−0.791166	−	0.611602i	$-0.790525\pi$
$823$	−45.3939	−1.58233	−0.791166	+	0.611602i	$0.790525\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.4949	0.434490	0.217245	−	0.976117i	$-0.430293\pi$
$827$	12.4949	0.434490	0.217245	+	0.976117i	$0.430293\pi$
$828$	0	0
$829$	30.6969	1.06615	0.533074	−	0.846068i	$-0.321037\pi$
$829$	30.6969	1.06615	0.533074	+	0.846068i	$0.321037\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−64.4949	−2.23194
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44.8990	−1.55008	−0.775042	−	0.631909i	$-0.782272\pi$
$839$	−44.8990	−1.55008	−0.775042	+	0.631909i	$0.782272\pi$
$840$	0	0
$841$	−20.5959	−0.710204
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−37.9444	−1.30533
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.79796	0.267311
$852$	0	0
$853$	−38.8434	−1.32997	−0.664986	−	0.746856i	$-0.731562\pi$
$853$	−38.8434	−1.32997	−0.664986	+	0.746856i	$0.731562\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.1010	0.857435	0.428717	−	0.903439i	$-0.358966\pi$
$857$	25.1010	0.857435	0.428717	+	0.903439i	$0.358966\pi$
$858$	0	0
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.10102	0.0715196	0.0357598	−	0.999360i	$-0.488615\pi$
$863$	2.10102	0.0715196	0.0357598	+	0.999360i	$0.488615\pi$
$864$	0	0
$865$	−44.4949	−1.51287
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.7980	−0.535909
$870$	0	0
$871$	−15.1918	−0.514756
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.55051	−0.221448
$876$	0	0
$877$	−26.4949	−0.894669	−0.447335	−	0.894367i	$-0.647627\pi$
$877$	−26.4949	−0.894669	−0.447335	+	0.894367i	$0.647627\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.5959	0.660203	0.330102	−	0.943945i	$-0.392917\pi$
$881$	19.5959	0.660203	0.330102	+	0.943945i	$0.392917\pi$
$882$	0	0
$883$	0.202041	0.00679922	0.00339961	−	0.999994i	$-0.498918\pi$
$883$	0.202041	0.00679922	0.00339961	+	0.999994i	$0.498918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.7980	−1.13482	−0.567412	−	0.823434i	$-0.692055\pi$
$887$	−33.7980	−1.13482	−0.567412	+	0.823434i	$0.692055\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	72.9898	2.44251
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.3939	0.580118
$900$	0	0
$901$	−2.20204	−0.0733606
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.0000	−0.498617
$906$	0	0
$907$	26.6969	0.886457	0.443229	−	0.896409i	$-0.353833\pi$
$907$	26.6969	0.886457	0.443229	+	0.896409i	$0.353833\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−45.9898	−1.52371	−0.761855	−	0.647748i	$-0.775711\pi$
$911$	−45.9898	−1.52371	−0.761855	+	0.647748i	$0.775711\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.4495	−0.444141
$918$	0	0
$919$	−3.69694	−0.121951	−0.0609754	−	0.998139i	$-0.519421\pi$
$919$	−3.69694	−0.121951	−0.0609754	+	0.998139i	$0.519421\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.4949	−1.59623
$924$	0	0
$925$	−53.7980	−1.76887
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.2929	1.12511	0.562556	−	0.826759i	$-0.309818\pi$
$929$	34.2929	1.12511	0.562556	+	0.826759i	$0.309818\pi$
$930$	0	0
$931$	−7.44949	−0.244147
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.7980	0.451242
$936$	0	0
$937$	6.40408	0.209212	0.104606	−	0.994514i	$-0.466642\pi$
$937$	6.40408	0.209212	0.104606	+	0.994514i	$0.466642\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.44949	−0.112450	−0.0562251	−	0.998418i	$-0.517906\pi$
$941$	−3.44949	−0.112450	−0.0562251	+	0.998418i	$0.517906\pi$
$942$	0	0
$943$	−9.79796	−0.319065
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.50510	0.113901	0.0569503	−	0.998377i	$-0.481862\pi$
$947$	3.50510	0.113901	0.0569503	+	0.998377i	$0.481862\pi$
$948$	0	0
$949$	−14.2020	−0.461018
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−55.3939	−1.79438	−0.897192	−	0.441641i	$-0.854396\pi$
$953$	−55.3939	−1.79438	−0.897192	+	0.441641i	$0.854396\pi$
$954$	0	0
$955$	−47.9444	−1.55144
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.7980	0.380976
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	27.9444	0.899562
$966$	0	0
$967$	14.5959	0.469373	0.234687	−	0.972071i	$-0.424594\pi$
$967$	14.5959	0.469373	0.234687	+	0.972071i	$0.424594\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53.9444	−1.73116	−0.865579	−	0.500773i	$-0.833049\pi$
$971$	−53.9444	−1.73116	−0.865579	+	0.500773i	$0.833049\pi$
$972$	0	0
$973$	−9.44949	−0.302937
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.59592	−0.0510579	−0.0255290	−	0.999674i	$-0.508127\pi$
$977$	−1.59592	−0.0510579	−0.0255290	+	0.999674i	$0.508127\pi$
$978$	0	0
$979$	14.2020	0.453899
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.1918	1.44140	0.720698	−	0.693249i	$-0.243822\pi$
$983$	45.1918	1.44140	0.720698	+	0.693249i	$0.243822\pi$
$984$	0	0
$985$	−43.7980	−1.39552
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.89898	−0.0921822
$990$	0	0
$991$	−17.7980	−0.565371	−0.282685	−	0.959213i	$-0.591225\pi$
$991$	−17.7980	−0.565371	−0.282685	+	0.959213i	$0.591225\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	23.7980	0.754446
$996$	0	0
$997$	17.8536	0.565428	0.282714	−	0.959204i	$-0.408765\pi$
$997$	17.8536	0.565428	0.282714	+	0.959204i	$0.408765\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bd.1.1		2
3.2	odd	2		9072.2.a.bk.1.2		2
4.3	odd	2		1134.2.a.i.1.1		2
9.2	odd	6		1008.2.r.e.337.1		4
9.4	even	3		3024.2.r.e.2017.2		4
9.5	odd	6		1008.2.r.e.673.2		4
9.7	even	3		3024.2.r.e.1009.2		4
12.11	even	2		1134.2.a.p.1.2		2
28.27	even	2		7938.2.a.bm.1.2		2
36.7	odd	6		378.2.f.d.253.2		4
36.11	even	6		126.2.f.c.85.2	yes	4
36.23	even	6		126.2.f.c.43.1	✓	4
36.31	odd	6		378.2.f.d.127.2		4
84.83	odd	2		7938.2.a.bn.1.1		2
252.11	even	6		882.2.e.m.373.2		4
252.23	even	6		882.2.e.m.655.2		4
252.31	even	6		2646.2.h.n.667.2		4
252.47	odd	6		882.2.h.l.67.2		4
252.59	odd	6		882.2.h.l.79.2		4
252.67	odd	6		2646.2.h.m.667.1		4
252.79	odd	6		2646.2.h.m.361.1		4
252.83	odd	6		882.2.f.j.589.1		4
252.95	even	6		882.2.h.k.79.1		4
252.103	even	6		2646.2.e.k.2125.1		4
252.115	even	6		2646.2.e.k.1549.1		4
252.131	odd	6		882.2.e.n.655.1		4
252.139	even	6		2646.2.f.k.883.1		4
252.151	odd	6		2646.2.e.l.1549.2		4
252.167	odd	6		882.2.f.j.295.2		4
252.187	even	6		2646.2.h.n.361.2		4
252.191	even	6		882.2.h.k.67.1		4
252.223	even	6		2646.2.f.k.1765.1		4
252.227	odd	6		882.2.e.n.373.1		4
252.247	odd	6		2646.2.e.l.2125.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.c.43.1	✓	4	36.23	even	6
126.2.f.c.85.2	yes	4	36.11	even	6
378.2.f.d.127.2		4	36.31	odd	6
378.2.f.d.253.2		4	36.7	odd	6
882.2.e.m.373.2		4	252.11	even	6
882.2.e.m.655.2		4	252.23	even	6
882.2.e.n.373.1		4	252.227	odd	6
882.2.e.n.655.1		4	252.131	odd	6
882.2.f.j.295.2		4	252.167	odd	6
882.2.f.j.589.1		4	252.83	odd	6
882.2.h.k.67.1		4	252.191	even	6
882.2.h.k.79.1		4	252.95	even	6
882.2.h.l.67.2		4	252.47	odd	6
882.2.h.l.79.2		4	252.59	odd	6
1008.2.r.e.337.1		4	9.2	odd	6
1008.2.r.e.673.2		4	9.5	odd	6
1134.2.a.i.1.1		2	4.3	odd	2
1134.2.a.p.1.2		2	12.11	even	2
2646.2.e.k.1549.1		4	252.115	even	6
2646.2.e.k.2125.1		4	252.103	even	6
2646.2.e.l.1549.2		4	252.151	odd	6
2646.2.e.l.2125.2		4	252.247	odd	6
2646.2.f.k.883.1		4	252.139	even	6
2646.2.f.k.1765.1		4	252.223	even	6
2646.2.h.m.361.1		4	252.79	odd	6
2646.2.h.m.667.1		4	252.67	odd	6
2646.2.h.n.361.2		4	252.187	even	6
2646.2.h.n.667.2		4	252.31	even	6
3024.2.r.e.1009.2		4	9.7	even	3
3024.2.r.e.2017.2		4	9.4	even	3
7938.2.a.bm.1.2		2	28.27	even	2
7938.2.a.bn.1.1		2	84.83	odd	2
9072.2.a.bd.1.1		2	1.1	even	1	trivial
9072.2.a.bk.1.2		2	3.2	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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-3.44949 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.44949 q^{5} +1.00000 q^{7} +2.00000 q^{11} -4.89898 q^{13} -2.00000 q^{17} -7.44949 q^{19} -1.00000 q^{23} +6.89898 q^{25} -2.89898 q^{29} -6.00000 q^{31} -3.44949 q^{35} -7.79796 q^{37} +9.79796 q^{41} +2.89898 q^{43} -9.79796 q^{47} +1.00000 q^{49} +1.10102 q^{53} -6.89898 q^{55} -2.00000 q^{59} +11.4495 q^{61} +16.8990 q^{65} +3.10102 q^{67} +9.89898 q^{71} +2.89898 q^{73} +2.00000 q^{77} -7.89898 q^{79} +2.00000 q^{83} +6.89898 q^{85} +7.10102 q^{89} -4.89898 q^{91} +25.6969 q^{95} -6.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{17} - 10 q^{19} - 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{35} + 4 q^{37} - 4 q^{43} + 2 q^{49} + 12 q^{53} - 4 q^{55} - 4 q^{59} + 18 q^{61} + 24 q^{65} + 16 q^{67} + 10 q^{71} - 4 q^{73} + 4 q^{77} - 6 q^{79} + 4 q^{83} + 4 q^{85} + 24 q^{89} + 22 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 4 * q^11 - 4 * q^17 - 10 * q^19 - 2 * q^23 + 4 * q^25 + 4 * q^29 - 12 * q^31 - 2 * q^35 + 4 * q^37 - 4 * q^43 + 2 * q^49 + 12 * q^53 - 4 * q^55 - 4 * q^59 + 18 * q^61 + 24 * q^65 + 16 * q^67 + 10 * q^71 - 4 * q^73 + 4 * q^77 - 6 * q^79 + 4 * q^83 + 4 * q^85 + 24 * q^89 + 22 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more