LMFDB - Embedded newform 9072.2.a.bk.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:	$1$
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)

$f(q)$	$=$	$q-1.44949 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.44949 q^{5} +1.00000 q^{7} -2.00000 q^{11} +4.89898 q^{13} +2.00000 q^{17} -2.55051 q^{19} +1.00000 q^{23} -2.89898 q^{25} -6.89898 q^{29} -6.00000 q^{31} -1.44949 q^{35} +11.7980 q^{37} +9.79796 q^{41} -6.89898 q^{43} -9.79796 q^{47} +1.00000 q^{49} -10.8990 q^{53} +2.89898 q^{55} +2.00000 q^{59} +6.55051 q^{61} -7.10102 q^{65} +12.8990 q^{67} -0.101021 q^{71} -6.89898 q^{73} -2.00000 q^{77} +1.89898 q^{79} -2.00000 q^{83} -2.89898 q^{85} -16.8990 q^{89} +4.89898 q^{91} +3.69694 q^{95} +2.89898 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.44949	−0.648232	−0.324116	−	0.946017i	$-0.605067\pi$
$5$	−1.44949	−0.648232	−0.324116	+	0.946017i	$0.605067\pi$
$6$	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	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.89898	1.35873	0.679366	−	0.733799i	$-0.262255\pi$
$13$	4.89898	1.35873	0.679366	+	0.733799i	$0.262255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.55051	−0.585127	−0.292564	−	0.956246i	$-0.594508\pi$
$19$	−2.55051	−0.585127	−0.292564	+	0.956246i	$0.594508\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−2.89898	−0.579796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.89898	−1.28111	−0.640554	−	0.767913i	$-0.721295\pi$
$29$	−6.89898	−1.28111	−0.640554	+	0.767913i	$0.721295\pi$
$30$	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
$35$	−1.44949	−0.245008
$36$	0	0
$37$	11.7980	1.93957	0.969786	−	0.243956i	$-0.0784453\pi$
$37$	11.7980	1.93957	0.969786	+	0.243956i	$0.0784453\pi$
$38$	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
$43$	−6.89898	−1.05208	−0.526042	−	0.850458i	$-0.676325\pi$
$43$	−6.89898	−1.05208	−0.526042	+	0.850458i	$0.676325\pi$
$44$	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
$53$	−10.8990	−1.49709	−0.748545	−	0.663084i	$-0.769247\pi$
$53$	−10.8990	−1.49709	−0.748545	+	0.663084i	$0.769247\pi$
$54$	0	0
$55$	2.89898	0.390898
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	6.55051	0.838707	0.419353	−	0.907823i	$-0.362257\pi$
$61$	6.55051	0.838707	0.419353	+	0.907823i	$0.362257\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.10102	−0.880773
$66$	0	0
$67$	12.8990	1.57586	0.787931	−	0.615764i	$-0.211153\pi$
$67$	12.8990	1.57586	0.787931	+	0.615764i	$0.211153\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.101021	−0.0119889	−0.00599446	−	0.999982i	$-0.501908\pi$
$71$	−0.101021	−0.0119889	−0.00599446	+	0.999982i	$0.501908\pi$
$72$	0	0
$73$	−6.89898	−0.807464	−0.403732	−	0.914877i	$-0.632287\pi$
$73$	−6.89898	−0.807464	−0.403732	+	0.914877i	$0.632287\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	1.89898	0.213652	0.106826	−	0.994278i	$-0.465931\pi$
$79$	1.89898	0.213652	0.106826	+	0.994278i	$0.465931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−2.89898	−0.314438
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.8990	−1.79129	−0.895644	−	0.444771i	$-0.853285\pi$
$89$	−16.8990	−1.79129	−0.895644	+	0.444771i	$0.853285\pi$
$90$	0	0
$91$	4.89898	0.513553
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.69694	0.379298
$96$	0	0
$97$	2.89898	0.294347	0.147173	−	0.989111i	$-0.452982\pi$
$97$	2.89898	0.294347	0.147173	+	0.989111i	$0.452982\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.2474	−1.71619	−0.858093	−	0.513495i	$-0.828351\pi$
$101$	−17.2474	−1.71619	−0.858093	+	0.513495i	$0.828351\pi$
$102$	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	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	12.6969	1.21615	0.608073	−	0.793881i	$-0.291943\pi$
$109$	12.6969	1.21615	0.608073	+	0.793881i	$0.291943\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.10102	−0.573936	−0.286968	−	0.957940i	$-0.592647\pi$
$113$	−6.10102	−0.573936	−0.286968	+	0.957940i	$0.592647\pi$
$114$	0	0
$115$	−1.44949	−0.135166
$116$	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
$125$	11.4495	1.02407
$126$	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
$131$	8.55051	0.747062	0.373531	−	0.927618i	$-0.378147\pi$
$131$	8.55051	0.747062	0.373531	+	0.927618i	$0.378147\pi$
$132$	0	0
$133$	−2.55051	−0.221157
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.79796	0.666225	0.333112	−	0.942887i	$-0.391901\pi$
$137$	7.79796	0.666225	0.333112	+	0.942887i	$0.391901\pi$
$138$	0	0
$139$	−4.55051	−0.385969	−0.192985	−	0.981202i	$-0.561817\pi$
$139$	−4.55051	−0.385969	−0.192985	+	0.981202i	$0.561817\pi$
$140$	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	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	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
$155$	8.69694	0.698555
$156$	0	0
$157$	−8.34847	−0.666280	−0.333140	−	0.942877i	$-0.608108\pi$
$157$	−8.34847	−0.666280	−0.333140	+	0.942877i	$0.608108\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	19.7980	1.55070	0.775348	−	0.631534i	$-0.217575\pi$
$163$	19.7980	1.55070	0.775348	+	0.631534i	$0.217575\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.6969	0.827754	0.413877	−	0.910333i	$-0.364174\pi$
$167$	10.6969	0.827754	0.413877	+	0.910333i	$0.364174\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.10102	−0.235766	−0.117883	−	0.993027i	$-0.537611\pi$
$173$	−3.10102	−0.235766	−0.117883	+	0.993027i	$0.537611\pi$
$174$	0	0
$175$	−2.89898	−0.219142
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.6969	−1.54696	−0.773481	−	0.633820i	$-0.781486\pi$
$179$	−20.6969	−1.54696	−0.773481	+	0.633820i	$0.781486\pi$
$180$	0	0
$181$	−10.3485	−0.769196	−0.384598	−	0.923084i	$-0.625660\pi$
$181$	−10.3485	−0.769196	−0.384598	+	0.923084i	$0.625660\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−17.1010	−1.25729
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.10102	−0.296739	−0.148370	−	0.988932i	$-0.547403\pi$
$191$	−4.10102	−0.296739	−0.148370	+	0.988932i	$0.547403\pi$
$192$	0	0
$193$	−17.8990	−1.28840	−0.644198	−	0.764858i	$-0.722809\pi$
$193$	−17.8990	−1.28840	−0.644198	+	0.764858i	$0.722809\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.6969	1.18961	0.594804	−	0.803871i	$-0.297230\pi$
$197$	16.6969	1.18961	0.594804	+	0.803871i	$0.297230\pi$
$198$	0	0
$199$	2.89898	0.205503	0.102752	−	0.994707i	$-0.467235\pi$
$199$	2.89898	0.205503	0.102752	+	0.994707i	$0.467235\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.89898	−0.484213
$204$	0	0
$205$	−14.2020	−0.991914
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.10102	0.352845
$210$	0	0
$211$	−12.8990	−0.888002	−0.444001	−	0.896026i	$-0.646441\pi$
$211$	−12.8990	−0.888002	−0.444001	+	0.896026i	$0.646441\pi$
$212$	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
$223$	11.1010	0.743379	0.371690	−	0.928357i	$-0.378779\pi$
$223$	11.1010	0.743379	0.371690	+	0.928357i	$0.378779\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.44949	0.361695	0.180848	−	0.983511i	$-0.442116\pi$
$227$	5.44949	0.361695	0.180848	+	0.983511i	$0.442116\pi$
$228$	0	0
$229$	1.24745	0.0824337	0.0412169	−	0.999150i	$-0.486877\pi$
$229$	1.24745	0.0824337	0.0412169	+	0.999150i	$0.486877\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.00000	−0.458585	−0.229293	−	0.973358i	$-0.573641\pi$
$233$	−7.00000	−0.458585	−0.229293	+	0.973358i	$0.573641\pi$
$234$	0	0
$235$	14.2020	0.926439
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.79796	−0.439723	−0.219862	−	0.975531i	$-0.570561\pi$
$239$	−6.79796	−0.439723	−0.219862	+	0.975531i	$0.570561\pi$
$240$	0	0
$241$	0.898979	0.0579084	0.0289542	−	0.999581i	$-0.490782\pi$
$241$	0.898979	0.0579084	0.0289542	+	0.999581i	$0.490782\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.44949	−0.0926045
$246$	0	0
$247$	−12.4949	−0.795031
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−17.4495	−1.10140	−0.550701	−	0.834703i	$-0.685640\pi$
$251$	−17.4495	−1.10140	−0.550701	+	0.834703i	$0.685640\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.20204	0.511629	0.255815	−	0.966726i	$-0.417656\pi$
$257$	8.20204	0.511629	0.255815	+	0.966726i	$0.417656\pi$
$258$	0	0
$259$	11.7980	0.733090
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.8990	−1.59700	−0.798500	−	0.601995i	$-0.794373\pi$
$263$	−25.8990	−1.59700	−0.798500	+	0.601995i	$0.794373\pi$
$264$	0	0
$265$	15.7980	0.970461
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.3485	−1.11873	−0.559363	−	0.828923i	$-0.688954\pi$
$269$	−18.3485	−1.11873	−0.559363	+	0.828923i	$0.688954\pi$
$270$	0	0
$271$	−7.10102	−0.431356	−0.215678	−	0.976465i	$-0.569196\pi$
$271$	−7.10102	−0.431356	−0.215678	+	0.976465i	$0.569196\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.79796	0.349630
$276$	0	0
$277$	−18.6969	−1.12339	−0.561695	−	0.827344i	$-0.689851\pi$
$277$	−18.6969	−1.12339	−0.561695	+	0.827344i	$0.689851\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.0000	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$281$	−19.0000	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$282$	0	0
$283$	25.4495	1.51282	0.756408	−	0.654101i	$-0.226953\pi$
$283$	25.4495	1.51282	0.756408	+	0.654101i	$0.226953\pi$
$284$	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
$293$	2.75255	0.160806	0.0804029	−	0.996762i	$-0.474379\pi$
$293$	2.75255	0.160806	0.0804029	+	0.996762i	$0.474379\pi$
$294$	0	0
$295$	−2.89898	−0.168785
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.89898	0.283315
$300$	0	0
$301$	−6.89898	−0.397651
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.49490	−0.543676
$306$	0	0
$307$	−25.2474	−1.44095	−0.720474	−	0.693482i	$-0.756076\pi$
$307$	−25.2474	−1.44095	−0.720474	+	0.693482i	$0.756076\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.6969	−1.74066	−0.870332	−	0.492466i	$-0.836096\pi$
$311$	−30.6969	−1.74066	−0.870332	+	0.492466i	$0.836096\pi$
$312$	0	0
$313$	−4.69694	−0.265487	−0.132743	−	0.991150i	$-0.542379\pi$
$313$	−4.69694	−0.265487	−0.132743	+	0.991150i	$0.542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.6969	1.16246	0.581228	−	0.813741i	$-0.302572\pi$
$317$	20.6969	1.16246	0.581228	+	0.813741i	$0.302572\pi$
$318$	0	0
$319$	13.7980	0.772537
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.10102	−0.283828
$324$	0	0
$325$	−14.2020	−0.787787
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.79796	−0.540179
$330$	0	0
$331$	−4.69694	−0.258167	−0.129084	−	0.991634i	$-0.541204\pi$
$331$	−4.69694	−0.258167	−0.129084	+	0.991634i	$0.541204\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−18.6969	−1.02152
$336$	0	0
$337$	−23.3939	−1.27435	−0.637173	−	0.770721i	$-0.719896\pi$
$337$	−23.3939	−1.27435	−0.637173	+	0.770721i	$0.719896\pi$
$338$	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
$349$	11.1010	0.594224	0.297112	−	0.954843i	$-0.403977\pi$
$349$	11.1010	0.594224	0.297112	+	0.954843i	$0.403977\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0.146428	0.00777160
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.79796	0.464339	0.232169	−	0.972675i	$-0.425418\pi$
$359$	8.79796	0.464339	0.232169	+	0.972675i	$0.425418\pi$
$360$	0	0
$361$	−12.4949	−0.657626
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	13.7980	0.720248	0.360124	−	0.932905i	$-0.382734\pi$
$367$	13.7980	0.720248	0.360124	+	0.932905i	$0.382734\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.8990	−0.565847
$372$	0	0
$373$	−6.89898	−0.357216	−0.178608	−	0.983920i	$-0.557159\pi$
$373$	−6.89898	−0.357216	−0.178608	+	0.983920i	$0.557159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−33.7980	−1.74068
$378$	0	0
$379$	−22.4949	−1.15549	−0.577743	−	0.816219i	$-0.696066\pi$
$379$	−22.4949	−1.15549	−0.577743	+	0.816219i	$0.696066\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.89898	0.148131	0.0740655	−	0.997253i	$-0.476403\pi$
$383$	2.89898	0.148131	0.0740655	+	0.997253i	$0.476403\pi$
$384$	0	0
$385$	2.89898	0.147746
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.8990	−1.26243	−0.631214	−	0.775609i	$-0.717443\pi$
$389$	−24.8990	−1.26243	−0.631214	+	0.775609i	$0.717443\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.75255	−0.138496
$396$	0	0
$397$	38.6969	1.94214	0.971072	−	0.238788i	$-0.0767500\pi$
$397$	38.6969	1.94214	0.971072	+	0.238788i	$0.0767500\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.8990	−0.993708	−0.496854	−	0.867834i	$-0.665511\pi$
$401$	−19.8990	−0.993708	−0.496854	+	0.867834i	$0.665511\pi$
$402$	0	0
$403$	−29.3939	−1.46421
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.5959	−1.16961
$408$	0	0
$409$	−13.7980	−0.682265	−0.341133	−	0.940015i	$-0.610811\pi$
$409$	−13.7980	−0.682265	−0.341133	+	0.940015i	$0.610811\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.00000	0.0984136
$414$	0	0
$415$	2.89898	0.142305
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.4495	−1.43870	−0.719351	−	0.694647i	$-0.755561\pi$
$419$	−29.4495	−1.43870	−0.719351	+	0.694647i	$0.755561\pi$
$420$	0	0
$421$	22.8990	1.11603	0.558014	−	0.829832i	$-0.311564\pi$
$421$	22.8990	1.11603	0.558014	+	0.829832i	$0.311564\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.79796	−0.281242
$426$	0	0
$427$	6.55051	0.317001
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.5959	−1.52192	−0.760961	−	0.648798i	$-0.775272\pi$
$431$	−31.5959	−1.52192	−0.760961	+	0.648798i	$0.775272\pi$
$432$	0	0
$433$	−7.79796	−0.374746	−0.187373	−	0.982289i	$-0.559997\pi$
$433$	−7.79796	−0.374746	−0.187373	+	0.982289i	$0.559997\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.55051	−0.122007
$438$	0	0
$439$	−2.20204	−0.105098	−0.0525488	−	0.998618i	$-0.516735\pi$
$439$	−2.20204	−0.105098	−0.0525488	+	0.998618i	$0.516735\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.8990	0.707872	0.353936	−	0.935270i	$-0.384843\pi$
$443$	14.8990	0.707872	0.353936	+	0.935270i	$0.384843\pi$
$444$	0	0
$445$	24.4949	1.16117
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.5959	0.971981	0.485991	−	0.873964i	$-0.338459\pi$
$449$	20.5959	0.971981	0.485991	+	0.873964i	$0.338459\pi$
$450$	0	0
$451$	−19.5959	−0.922736
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.10102	−0.332901
$456$	0	0
$457$	−17.4949	−0.818377	−0.409188	−	0.912450i	$-0.634188\pi$
$457$	−17.4949	−0.818377	−0.409188	+	0.912450i	$0.634188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.65153	0.263218	0.131609	−	0.991302i	$-0.457986\pi$
$461$	5.65153	0.263218	0.131609	+	0.991302i	$0.457986\pi$
$462$	0	0
$463$	−3.69694	−0.171811	−0.0859057	−	0.996303i	$-0.527378\pi$
$463$	−3.69694	−0.171811	−0.0859057	+	0.996303i	$0.527378\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	0	0
$469$	12.8990	0.595620
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.7980	0.634431
$474$	0	0
$475$	7.39388	0.339254
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.59592	−0.438449	−0.219224	−	0.975674i	$-0.570353\pi$
$479$	−9.59592	−0.438449	−0.219224	+	0.975674i	$0.570353\pi$
$480$	0	0
$481$	57.7980	2.63536
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.20204	−0.190805
$486$	0	0
$487$	36.3939	1.64916	0.824582	−	0.565742i	$-0.191410\pi$
$487$	36.3939	1.64916	0.824582	+	0.565742i	$0.191410\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.7980	−0.712952	−0.356476	−	0.934304i	$-0.616022\pi$
$491$	−15.7980	−0.712952	−0.356476	+	0.934304i	$0.616022\pi$
$492$	0	0
$493$	−13.7980	−0.621429
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.101021	−0.00453139
$498$	0	0
$499$	25.3939	1.13679	0.568393	−	0.822757i	$-0.307565\pi$
$499$	25.3939	1.13679	0.568393	+	0.822757i	$0.307565\pi$
$500$	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
$509$	−7.10102	−0.314747	−0.157374	−	0.987539i	$-0.550303\pi$
$509$	−7.10102	−0.314747	−0.157374	+	0.987539i	$0.550303\pi$
$510$	0	0
$511$	−6.89898	−0.305193
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20.2929	0.894210
$516$	0	0
$517$	19.5959	0.861827
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.30306	−0.407575	−0.203787	−	0.979015i	$-0.565325\pi$
$521$	−9.30306	−0.407575	−0.203787	+	0.979015i	$0.565325\pi$
$522$	0	0
$523$	14.3485	0.627415	0.313707	−	0.949520i	$-0.398429\pi$
$523$	14.3485	0.627415	0.313707	+	0.949520i	$0.398429\pi$
$524$	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
$535$	−17.3939	−0.752003
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−18.4949	−0.795158	−0.397579	−	0.917568i	$-0.630149\pi$
$541$	−18.4949	−0.795158	−0.397579	+	0.917568i	$0.630149\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.4041	−0.788344
$546$	0	0
$547$	7.59592	0.324778	0.162389	−	0.986727i	$-0.448080\pi$
$547$	7.59592	0.324778	0.162389	+	0.986727i	$0.448080\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.5959	0.749611
$552$	0	0
$553$	1.89898	0.0807528
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.8990	−0.546547	−0.273274	−	0.961936i	$-0.588106\pi$
$557$	−12.8990	−0.546547	−0.273274	+	0.961936i	$0.588106\pi$
$558$	0	0
$559$	−33.7980	−1.42950
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.9444	1.68346	0.841728	−	0.539902i	$-0.181539\pi$
$563$	39.9444	1.68346	0.841728	+	0.539902i	$0.181539\pi$
$564$	0	0
$565$	8.84337	0.372043
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−33.7980	−1.41440	−0.707200	−	0.707013i	$-0.750042\pi$
$571$	−33.7980	−1.41440	−0.707200	+	0.707013i	$0.750042\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.89898	−0.120896
$576$	0	0
$577$	−15.5959	−0.649267	−0.324633	−	0.945840i	$-0.605241\pi$
$577$	−15.5959	−0.649267	−0.324633	+	0.945840i	$0.605241\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	21.7980	0.902779
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.1464	−0.666434	−0.333217	−	0.942850i	$-0.608134\pi$
$587$	−16.1464	−0.666434	−0.333217	+	0.942850i	$0.608134\pi$
$588$	0	0
$589$	15.3031	0.630552
$590$	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
$595$	−2.89898	−0.118847
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.7980	1.38095	0.690474	−	0.723358i	$-0.257402\pi$
$599$	33.7980	1.38095	0.690474	+	0.723358i	$0.257402\pi$
$600$	0	0
$601$	16.6969	0.681082	0.340541	−	0.940230i	$-0.389390\pi$
$601$	16.6969	0.681082	0.340541	+	0.940230i	$0.389390\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.1464	0.412511
$606$	0	0
$607$	−20.6969	−0.840063	−0.420031	−	0.907510i	$-0.637981\pi$
$607$	−20.6969	−0.840063	−0.420031	+	0.907510i	$0.637981\pi$
$608$	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	$-0.595920\pi$
$613$	−14.6969	−0.593604	−0.296802	+	0.954939i	$0.595920\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.3939	−0.619734	−0.309867	−	0.950780i	$-0.600285\pi$
$617$	−15.3939	−0.619734	−0.309867	+	0.950780i	$0.600285\pi$
$618$	0	0
$619$	−30.1464	−1.21169	−0.605844	−	0.795584i	$-0.707164\pi$
$619$	−30.1464	−1.21169	−0.605844	+	0.795584i	$0.707164\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.8990	−0.677043
$624$	0	0
$625$	−2.10102	−0.0840408
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.5959	0.940831
$630$	0	0
$631$	−27.8990	−1.11064	−0.555320	−	0.831636i	$-0.687404\pi$
$631$	−27.8990	−1.11064	−0.555320	+	0.831636i	$0.687404\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.34847	−0.172564
$636$	0	0
$637$	4.89898	0.194105
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.49490	−0.296031	−0.148015	−	0.988985i	$-0.547288\pi$
$641$	−7.49490	−0.296031	−0.148015	+	0.988985i	$0.547288\pi$
$642$	0	0
$643$	39.3939	1.55354	0.776771	−	0.629783i	$-0.216856\pi$
$643$	39.3939	1.55354	0.776771	+	0.629783i	$0.216856\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	50.6969	1.99310	0.996551	−	0.0829807i	$-0.0264440\pi$
$647$	50.6969	1.99310	0.996551	+	0.0829807i	$0.0264440\pi$
$648$	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
$655$	−12.3939	−0.484269
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.6969	0.962056	0.481028	−	0.876705i	$-0.340264\pi$
$659$	24.6969	0.962056	0.481028	+	0.876705i	$0.340264\pi$
$660$	0	0
$661$	4.55051	0.176994	0.0884972	−	0.996076i	$-0.471794\pi$
$661$	4.55051	0.176994	0.0884972	+	0.996076i	$0.471794\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.69694	0.143361
$666$	0	0
$667$	−6.89898	−0.267130
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13.1010	−0.505759
$672$	0	0
$673$	−8.59592	−0.331348	−0.165674	−	0.986181i	$-0.552980\pi$
$673$	−8.59592	−0.331348	−0.165674	+	0.986181i	$0.552980\pi$
$674$	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
$679$	2.89898	0.111253
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−51.7980	−1.98199	−0.990997	−	0.133885i	$-0.957255\pi$
$683$	−51.7980	−1.98199	−0.990997	+	0.133885i	$0.957255\pi$
$684$	0	0
$685$	−11.3031	−0.431868
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−53.3939	−2.03414
$690$	0	0
$691$	−51.0454	−1.94186	−0.970929	−	0.239366i	$-0.923060\pi$
$691$	−51.0454	−1.94186	−0.970929	+	0.239366i	$0.923060\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.59592	0.250197
$696$	0	0
$697$	19.5959	0.742248
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.39388	−0.279263	−0.139631	−	0.990204i	$-0.544592\pi$
$701$	−7.39388	−0.279263	−0.139631	+	0.990204i	$0.544592\pi$
$702$	0	0
$703$	−30.0908	−1.13490
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.2474	−0.648657
$708$	0	0
$709$	27.5959	1.03639	0.518193	−	0.855264i	$-0.326605\pi$
$709$	27.5959	1.03639	0.518193	+	0.855264i	$0.326605\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	14.2020	0.531126
$716$	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
$727$	8.49490	0.315058	0.157529	−	0.987514i	$-0.449647\pi$
$727$	8.49490	0.315058	0.157529	+	0.987514i	$0.449647\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.7980	−0.510336
$732$	0	0
$733$	−17.4495	−0.644512	−0.322256	−	0.946653i	$-0.604441\pi$
$733$	−17.4495	−0.644512	−0.322256	+	0.946653i	$0.604441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.7980	−0.950280
$738$	0	0
$739$	−13.5959	−0.500134	−0.250067	−	0.968229i	$-0.580453\pi$
$739$	−13.5959	−0.500134	−0.250067	+	0.968229i	$0.580453\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	8.69694	0.318631
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−1.40408	−0.0512357	−0.0256178	−	0.999672i	$-0.508155\pi$
$751$	−1.40408	−0.0512357	−0.0256178	+	0.999672i	$0.508155\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.24745	−0.263762
$756$	0	0
$757$	−35.3939	−1.28641	−0.643206	−	0.765693i	$-0.722396\pi$
$757$	−35.3939	−1.28641	−0.643206	+	0.765693i	$0.722396\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	12.6969	0.459660
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.79796	0.353784
$768$	0	0
$769$	−34.0908	−1.22935	−0.614673	−	0.788782i	$-0.710712\pi$
$769$	−34.0908	−1.22935	−0.614673	+	0.788782i	$0.710712\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.9444	1.22089	0.610447	−	0.792057i	$-0.290990\pi$
$773$	33.9444	1.22089	0.610447	+	0.792057i	$0.290990\pi$
$774$	0	0
$775$	17.3939	0.624807
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.9898	−0.895352
$780$	0	0
$781$	0.202041	0.00722960
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.1010	0.431904
$786$	0	0
$787$	11.3939	0.406148	0.203074	−	0.979163i	$-0.434907\pi$
$787$	11.3939	0.406148	0.203074	+	0.979163i	$0.434907\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.10102	−0.216927
$792$	0	0
$793$	32.0908	1.13958
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.9444	−0.635623	−0.317811	−	0.948154i	$-0.602948\pi$
$797$	−17.9444	−0.635623	−0.317811	+	0.948154i	$0.602948\pi$
$798$	0	0
$799$	−19.5959	−0.693254
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.7980	0.486919
$804$	0	0
$805$	−1.44949	−0.0510878
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.2020	−0.569633	−0.284817	−	0.958582i	$-0.591933\pi$
$809$	−16.2020	−0.569633	−0.284817	+	0.958582i	$0.591933\pi$
$810$	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
$815$	−28.6969	−1.00521
$816$	0	0
$817$	17.5959	0.615603
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.404082	0.0141026	0.00705128	−	0.999975i	$-0.497755\pi$
$821$	0.404082	0.0141026	0.00705128	+	0.999975i	$0.497755\pi$
$822$	0	0
$823$	13.3939	0.466881	0.233441	−	0.972371i	$-0.425002\pi$
$823$	13.3939	0.466881	0.233441	+	0.972371i	$0.425002\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.4949	1.26905	0.634526	−	0.772902i	$-0.281195\pi$
$827$	36.4949	1.26905	0.634526	+	0.772902i	$0.281195\pi$
$828$	0	0
$829$	1.30306	0.0452572	0.0226286	−	0.999744i	$-0.492796\pi$
$829$	1.30306	0.0452572	0.0226286	+	0.999744i	$0.492796\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−15.5051	−0.536576
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.1010	1.21182	0.605911	−	0.795533i	$-0.292809\pi$
$839$	35.1010	1.21182	0.605911	+	0.795533i	$0.292809\pi$
$840$	0	0
$841$	18.5959	0.641239
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−15.9444	−0.548504
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.7980	0.404429
$852$	0	0
$853$	24.8434	0.850621	0.425310	−	0.905048i	$-0.360165\pi$
$853$	24.8434	0.850621	0.425310	+	0.905048i	$0.360165\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.8990	−1.19213	−0.596063	−	0.802938i	$-0.703269\pi$
$857$	−34.8990	−1.19213	−0.596063	+	0.802938i	$0.703269\pi$
$858$	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
$863$	−11.8990	−0.405046	−0.202523	−	0.979278i	$-0.564914\pi$
$863$	−11.8990	−0.405046	−0.202523	+	0.979278i	$0.564914\pi$
$864$	0	0
$865$	4.49490	0.152831
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.79796	−0.128837
$870$	0	0
$871$	63.1918	2.14117
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.4495	0.387063
$876$	0	0
$877$	22.4949	0.759599	0.379799	−	0.925069i	$-0.375993\pi$
$877$	22.4949	0.759599	0.379799	+	0.925069i	$0.375993\pi$
$878$	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
$883$	19.7980	0.666254	0.333127	−	0.942882i	$-0.391896\pi$
$883$	19.7980	0.666254	0.333127	+	0.942882i	$0.391896\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.2020	0.476858	0.238429	−	0.971160i	$-0.423368\pi$
$887$	14.2020	0.476858	0.238429	+	0.971160i	$0.423368\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.9898	0.836252
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.3939	1.38056
$900$	0	0
$901$	−21.7980	−0.726195
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.0000	0.498617
$906$	0	0
$907$	−2.69694	−0.0895504	−0.0447752	−	0.998997i	$-0.514257\pi$
$907$	−2.69694	−0.0895504	−0.0447752	+	0.998997i	$0.514257\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−51.9898	−1.72250	−0.861249	−	0.508183i	$-0.830318\pi$
$911$	−51.9898	−1.72250	−0.861249	+	0.508183i	$0.830318\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.55051	0.282363
$918$	0	0
$919$	25.6969	0.847664	0.423832	−	0.905741i	$-0.360685\pi$
$919$	25.6969	0.847664	0.423832	+	0.905741i	$0.360685\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.494897	−0.0162897
$924$	0	0
$925$	−34.2020	−1.12456
$926$	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
$931$	−2.55051	−0.0835896
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.79796	0.189614
$936$	0	0
$937$	45.5959	1.48955	0.744777	−	0.667314i	$-0.232556\pi$
$937$	45.5959	1.48955	0.744777	+	0.667314i	$0.232556\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.44949	−0.0472520	−0.0236260	−	0.999721i	$-0.507521\pi$
$941$	−1.44949	−0.0472520	−0.0236260	+	0.999721i	$0.507521\pi$
$942$	0	0
$943$	9.79796	0.319065
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.4949	−1.70585	−0.852927	−	0.522029i	$-0.825175\pi$
$947$	−52.4949	−1.70585	−0.852927	+	0.522029i	$0.825175\pi$
$948$	0	0
$949$	−33.7980	−1.09713
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.39388	−0.109938	−0.0549692	−	0.998488i	$-0.517506\pi$
$953$	−3.39388	−0.109938	−0.0549692	+	0.998488i	$0.517506\pi$
$954$	0	0
$955$	5.94439	0.192356
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.79796	0.251809
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25.9444	0.835179
$966$	0	0
$967$	−24.5959	−0.790951	−0.395476	−	0.918476i	$-0.629420\pi$
$967$	−24.5959	−0.790951	−0.395476	+	0.918476i	$0.629420\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.0556128	0.00178470	0.000892350	−	1.00000i	$-0.499716\pi$
$971$	0.0556128	0.00178470	0.000892350		1.00000i	$0.499716\pi$
$972$	0	0
$973$	−4.55051	−0.145883
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.5959	−1.20280	−0.601400	−	0.798948i	$-0.705390\pi$
$977$	−37.5959	−1.20280	−0.601400	+	0.798948i	$0.705390\pi$
$978$	0	0
$979$	33.7980	1.08019
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.1918	1.05866	0.529328	−	0.848418i	$-0.322444\pi$
$983$	33.1918	1.05866	0.529328	+	0.848418i	$0.322444\pi$
$984$	0	0
$985$	−24.2020	−0.771141
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.89898	−0.219375
$990$	0	0
$991$	1.79796	0.0571140	0.0285570	−	0.999592i	$-0.490909\pi$
$991$	1.79796	0.0571140	0.0285570	+	0.999592i	$0.490909\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.20204	−0.133214
$996$	0	0
$997$	52.1464	1.65149	0.825747	−	0.564041i	$-0.190754\pi$
$997$	52.1464	1.65149	0.825747	+	0.564041i	$0.190754\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bk.1.1		2
3.2	odd	2		9072.2.a.bd.1.2		2
4.3	odd	2		1134.2.a.p.1.1		2
9.2	odd	6		3024.2.r.e.1009.1		4
9.4	even	3		1008.2.r.e.673.1		4
9.5	odd	6		3024.2.r.e.2017.1		4
9.7	even	3		1008.2.r.e.337.2		4
12.11	even	2		1134.2.a.i.1.2		2
28.27	even	2		7938.2.a.bn.1.2		2
36.7	odd	6		126.2.f.c.85.1	yes	4
36.11	even	6		378.2.f.d.253.1		4
36.23	even	6		378.2.f.d.127.1		4
36.31	odd	6		126.2.f.c.43.2	✓	4
84.83	odd	2		7938.2.a.bm.1.1		2
252.11	even	6		2646.2.e.l.1549.1		4
252.23	even	6		2646.2.e.l.2125.1		4
252.31	even	6		882.2.h.l.79.1		4
252.47	odd	6		2646.2.h.n.361.1		4
252.59	odd	6		2646.2.h.n.667.1		4
252.67	odd	6		882.2.h.k.79.2		4
252.79	odd	6		882.2.h.k.67.2		4
252.83	odd	6		2646.2.f.k.1765.2		4
252.95	even	6		2646.2.h.m.667.2		4
252.103	even	6		882.2.e.n.655.2		4
252.115	even	6		882.2.e.n.373.2		4
252.131	odd	6		2646.2.e.k.2125.2		4
252.139	even	6		882.2.f.j.295.1		4
252.151	odd	6		882.2.e.m.373.1		4
252.167	odd	6		2646.2.f.k.883.2		4
252.187	even	6		882.2.h.l.67.1		4
252.191	even	6		2646.2.h.m.361.2		4
252.223	even	6		882.2.f.j.589.2		4
252.227	odd	6		2646.2.e.k.1549.2		4
252.247	odd	6		882.2.e.m.655.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.c.43.2	✓	4	36.31	odd	6
126.2.f.c.85.1	yes	4	36.7	odd	6
378.2.f.d.127.1		4	36.23	even	6
378.2.f.d.253.1		4	36.11	even	6
882.2.e.m.373.1		4	252.151	odd	6
882.2.e.m.655.1		4	252.247	odd	6
882.2.e.n.373.2		4	252.115	even	6
882.2.e.n.655.2		4	252.103	even	6
882.2.f.j.295.1		4	252.139	even	6
882.2.f.j.589.2		4	252.223	even	6
882.2.h.k.67.2		4	252.79	odd	6
882.2.h.k.79.2		4	252.67	odd	6
882.2.h.l.67.1		4	252.187	even	6
882.2.h.l.79.1		4	252.31	even	6
1008.2.r.e.337.2		4	9.7	even	3
1008.2.r.e.673.1		4	9.4	even	3
1134.2.a.i.1.2		2	12.11	even	2
1134.2.a.p.1.1		2	4.3	odd	2
2646.2.e.k.1549.2		4	252.227	odd	6
2646.2.e.k.2125.2		4	252.131	odd	6
2646.2.e.l.1549.1		4	252.11	even	6
2646.2.e.l.2125.1		4	252.23	even	6
2646.2.f.k.883.2		4	252.167	odd	6
2646.2.f.k.1765.2		4	252.83	odd	6
2646.2.h.m.361.2		4	252.191	even	6
2646.2.h.m.667.2		4	252.95	even	6
2646.2.h.n.361.1		4	252.47	odd	6
2646.2.h.n.667.1		4	252.59	odd	6
3024.2.r.e.1009.1		4	9.2	odd	6
3024.2.r.e.2017.1		4	9.5	odd	6
7938.2.a.bm.1.1		2	84.83	odd	2
7938.2.a.bn.1.2		2	28.27	even	2
9072.2.a.bd.1.2		2	3.2	odd	2
9072.2.a.bk.1.1		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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44949 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.44949 q^{5} +1.00000 q^{7} -2.00000 q^{11} +4.89898 q^{13} +2.00000 q^{17} -2.55051 q^{19} +1.00000 q^{23} -2.89898 q^{25} -6.89898 q^{29} -6.00000 q^{31} -1.44949 q^{35} +11.7980 q^{37} +9.79796 q^{41} -6.89898 q^{43} -9.79796 q^{47} +1.00000 q^{49} -10.8990 q^{53} +2.89898 q^{55} +2.00000 q^{59} +6.55051 q^{61} -7.10102 q^{65} +12.8990 q^{67} -0.101021 q^{71} -6.89898 q^{73} -2.00000 q^{77} +1.89898 q^{79} -2.00000 q^{83} -2.89898 q^{85} -16.8990 q^{89} +4.89898 q^{91} +3.69694 q^{95} +2.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 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

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