LMFDB - Embedded newform 2268.2.a.g.1.3

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:	$1$
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			1.3
Root			$2.46050$ 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)

$f(q)$	$=$	$q+2.05408 q^{5} +1.00000 q^{7} -5.05408 q^{11} -1.00000 q^{13} +0.273346 q^{17} -5.38151 q^{19} -5.32743 q^{23} -0.780738 q^{25} -8.32743 q^{29} -10.1623 q^{31} +2.05408 q^{35} +8.16225 q^{37} -5.05408 q^{41} +4.60078 q^{43} +1.38151 q^{47} +1.00000 q^{49} +3.43560 q^{53} -10.3815 q^{55} +1.78074 q^{59} +0.780738 q^{61} -2.05408 q^{65} -8.38151 q^{67} -7.78074 q^{71} +9.38151 q^{73} -5.05408 q^{77} +12.9430 q^{79} -5.72665 q^{83} +0.561476 q^{85} -13.8171 q^{89} -1.00000 q^{91} -11.0541 q^{95} -2.21926 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.05408	0.918614	0.459307	−	0.888277i	$-0.348098\pi$
$5$	2.05408	0.918614	0.459307	+	0.888277i	$0.348098\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.05408	−1.52386	−0.761932	−	0.647657i	$-0.775749\pi$
$11$	−5.05408	−1.52386	−0.761932	+	0.647657i	$0.775749\pi$
$12$	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
$17$	0.273346	0.0662962	0.0331481	−	0.999450i	$-0.489447\pi$
$17$	0.273346	0.0662962	0.0331481	+	0.999450i	$0.489447\pi$
$18$	0	0
$19$	−5.38151	−1.23460	−0.617302	−	0.786726i	$-0.711774\pi$
$19$	−5.38151	−1.23460	−0.617302	+	0.786726i	$0.711774\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.32743	−1.11085	−0.555423	−	0.831568i	$-0.687444\pi$
$23$	−5.32743	−1.11085	−0.555423	+	0.831568i	$0.687444\pi$
$24$	0	0
$25$	−0.780738	−0.156148
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.32743	−1.54637	−0.773183	−	0.634184i	$-0.781336\pi$
$29$	−8.32743	−1.54637	−0.773183	+	0.634184i	$0.781336\pi$
$30$	0	0
$31$	−10.1623	−1.82519	−0.912597	−	0.408860i	$-0.865927\pi$
$31$	−10.1623	−1.82519	−0.912597	+	0.408860i	$0.865927\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.05408	0.347204
$36$	0	0
$37$	8.16225	1.34187	0.670933	−	0.741518i	$-0.265894\pi$
$37$	8.16225	1.34187	0.670933	+	0.741518i	$0.265894\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.05408	−0.789315	−0.394658	−	0.918828i	$-0.629137\pi$
$41$	−5.05408	−0.789315	−0.394658	+	0.918828i	$0.629137\pi$
$42$	0	0
$43$	4.60078	0.701612	0.350806	−	0.936448i	$-0.385908\pi$
$43$	4.60078	0.701612	0.350806	+	0.936448i	$0.385908\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.38151	0.201515	0.100757	−	0.994911i	$-0.467873\pi$
$47$	1.38151	0.201515	0.100757	+	0.994911i	$0.467873\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.43560	0.471916	0.235958	−	0.971763i	$-0.424177\pi$
$53$	3.43560	0.471916	0.235958	+	0.971763i	$0.424177\pi$
$54$	0	0
$55$	−10.3815	−1.39984
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.78074	0.231832	0.115916	−	0.993259i	$-0.463020\pi$
$59$	1.78074	0.231832	0.115916	+	0.993259i	$0.463020\pi$
$60$	0	0
$61$	0.780738	0.0999633	0.0499816	−	0.998750i	$-0.484084\pi$
$61$	0.780738	0.0999633	0.0499816	+	0.998750i	$0.484084\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.05408	−0.254778
$66$	0	0
$67$	−8.38151	−1.02396	−0.511982	−	0.858996i	$-0.671089\pi$
$67$	−8.38151	−1.02396	−0.511982	+	0.858996i	$0.671089\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.78074	−0.923404	−0.461702	−	0.887035i	$-0.652761\pi$
$71$	−7.78074	−0.923404	−0.461702	+	0.887035i	$0.652761\pi$
$72$	0	0
$73$	9.38151	1.09802	0.549012	−	0.835815i	$-0.315004\pi$
$73$	9.38151	1.09802	0.549012	+	0.835815i	$0.315004\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.05408	−0.575966
$78$	0	0
$79$	12.9430	1.45620	0.728100	−	0.685471i	$-0.240404\pi$
$79$	12.9430	1.45620	0.728100	+	0.685471i	$0.240404\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.72665	−0.628582	−0.314291	−	0.949327i	$-0.601767\pi$
$83$	−5.72665	−0.628582	−0.314291	+	0.949327i	$0.601767\pi$
$84$	0	0
$85$	0.561476	0.0609006
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.8171	−1.46461	−0.732306	−	0.680976i	$-0.761556\pi$
$89$	−13.8171	−1.46461	−0.732306	+	0.680976i	$0.761556\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.0541	−1.13413
$96$	0	0
$97$	−2.21926	−0.225332	−0.112666	−	0.993633i	$-0.535939\pi$
$97$	−2.21926	−0.225332	−0.112666	+	0.993633i	$0.535939\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.72665	0.271312	0.135656	−	0.990756i	$-0.456686\pi$
$101$	2.72665	0.271312	0.135656	+	0.990756i	$0.456686\pi$
$102$	0	0
$103$	17.9823	1.77185	0.885924	−	0.463831i	$-0.153525\pi$
$103$	17.9823	1.77185	0.885924	+	0.463831i	$0.153525\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.10817	0.107131	0.0535653	−	0.998564i	$-0.482941\pi$
$107$	1.10817	0.107131	0.0535653	+	0.998564i	$0.482941\pi$
$108$	0	0
$109$	3.38151	0.323890	0.161945	−	0.986800i	$-0.448223\pi$
$109$	3.38151	0.323890	0.161945	+	0.986800i	$0.448223\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.8712	1.77525	0.887626	−	0.460564i	$-0.152353\pi$
$113$	18.8712	1.77525	0.887626	+	0.460564i	$0.152353\pi$
$114$	0	0
$115$	−10.9430	−1.02044
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.273346	0.0250576
$120$	0	0
$121$	14.5438	1.32216
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8741	−1.06205
$126$	0	0
$127$	17.1623	1.52290	0.761452	−	0.648221i	$-0.224487\pi$
$127$	17.1623	1.52290	0.761452	+	0.648221i	$0.224487\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.8889	−1.56296	−0.781481	−	0.623930i	$-0.785535\pi$
$131$	−17.8889	−1.56296	−0.781481	+	0.623930i	$0.785535\pi$
$132$	0	0
$133$	−5.38151	−0.466636
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.49261	−0.383829	−0.191915	−	0.981412i	$-0.561470\pi$
$137$	−4.49261	−0.383829	−0.191915	+	0.981412i	$0.561470\pi$
$138$	0	0
$139$	18.1445	1.53900	0.769500	−	0.638647i	$-0.220505\pi$
$139$	18.1445	1.53900	0.769500	+	0.638647i	$0.220505\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.05408	0.422644
$144$	0	0
$145$	−17.1052	−1.42051
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.50739	−0.369260	−0.184630	−	0.982808i	$-0.559109\pi$
$149$	−4.50739	−0.369260	−0.184630	+	0.982808i	$0.559109\pi$
$150$	0	0
$151$	−10.9823	−0.893726	−0.446863	−	0.894602i	$-0.647459\pi$
$151$	−10.9823	−0.893726	−0.446863	+	0.894602i	$0.647459\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−20.8741	−1.67665
$156$	0	0
$157$	4.17996	0.333597	0.166799	−	0.985991i	$-0.446657\pi$
$157$	4.17996	0.333597	0.166799	+	0.985991i	$0.446657\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.32743	−0.419860
$162$	0	0
$163$	−5.61849	−0.440074	−0.220037	−	0.975492i	$-0.570618\pi$
$163$	−5.61849	−0.440074	−0.220037	+	0.975492i	$0.570618\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.8918	−0.842835	−0.421418	−	0.906867i	$-0.638467\pi$
$167$	−10.8918	−0.842835	−0.421418	+	0.906867i	$0.638467\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.6008	1.11008	0.555038	−	0.831825i	$-0.312704\pi$
$173$	14.6008	1.11008	0.555038	+	0.831825i	$0.312704\pi$
$174$	0	0
$175$	−0.780738	−0.0590182
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.4897	−1.75570	−0.877851	−	0.478934i	$-0.841023\pi$
$179$	−23.4897	−1.75570	−0.877851	+	0.478934i	$0.841023\pi$
$180$	0	0
$181$	−1.39922	−0.104003	−0.0520017	−	0.998647i	$-0.516560\pi$
$181$	−1.39922	−0.104003	−0.0520017	+	0.998647i	$0.516560\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.7660	1.23266
$186$	0	0
$187$	−1.38151	−0.101026
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.3638	1.69055	0.845273	−	0.534335i	$-0.179438\pi$
$191$	23.3638	1.69055	0.845273	+	0.534335i	$0.179438\pi$
$192$	0	0
$193$	−14.5438	−1.04688	−0.523442	−	0.852062i	$-0.675352\pi$
$193$	−14.5438	−1.04688	−0.523442	+	0.852062i	$0.675352\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.3422	−1.23558	−0.617791	−	0.786343i	$-0.711972\pi$
$197$	−17.3422	−1.23558	−0.617791	+	0.786343i	$0.711972\pi$
$198$	0	0
$199$	−11.5438	−0.818316	−0.409158	−	0.912464i	$-0.634178\pi$
$199$	−11.5438	−0.818316	−0.409158	+	0.912464i	$0.634178\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.32743	−0.584471
$204$	0	0
$205$	−10.3815	−0.725076
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.1986	1.88137
$210$	0	0
$211$	−24.5261	−1.68844	−0.844222	−	0.535994i	$-0.819937\pi$
$211$	−24.5261	−1.68844	−0.844222	+	0.535994i	$0.819937\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.45038	0.644511
$216$	0	0
$217$	−10.1623	−0.689859
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.273346	−0.0183873
$222$	0	0
$223$	8.56148	0.573319	0.286659	−	0.958033i	$-0.407455\pi$
$223$	8.56148	0.573319	0.286659	+	0.958033i	$0.407455\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.1986	−1.40700	−0.703501	−	0.710694i	$-0.748381\pi$
$227$	−21.1986	−1.40700	−0.703501	+	0.710694i	$0.748381\pi$
$228$	0	0
$229$	−4.56148	−0.301431	−0.150715	−	0.988577i	$-0.548158\pi$
$229$	−4.56148	−0.301431	−0.150715	+	0.988577i	$0.548158\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.5074	0.884899	0.442449	−	0.896794i	$-0.354110\pi$
$233$	13.5074	0.884899	0.442449	+	0.896794i	$0.354110\pi$
$234$	0	0
$235$	2.83775	0.185114
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.6549	−0.883260	−0.441630	−	0.897197i	$-0.645600\pi$
$239$	−13.6549	−0.883260	−0.441630	+	0.897197i	$0.645600\pi$
$240$	0	0
$241$	3.21926	0.207371	0.103685	−	0.994610i	$-0.466936\pi$
$241$	3.21926	0.207371	0.103685	+	0.994610i	$0.466936\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.05408	0.131231
$246$	0	0
$247$	5.38151	0.342418
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.38151	0.276559	0.138279	−	0.990393i	$-0.455843\pi$
$251$	4.38151	0.276559	0.138279	+	0.990393i	$0.455843\pi$
$252$	0	0
$253$	26.9253	1.69278
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.4533	0.714438	0.357219	−	0.934021i	$-0.383725\pi$
$257$	11.4533	0.714438	0.357219	+	0.934021i	$0.383725\pi$
$258$	0	0
$259$	8.16225	0.507178
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.83482	−0.421453	−0.210727	−	0.977545i	$-0.567583\pi$
$263$	−6.83482	−0.421453	−0.210727	+	0.977545i	$0.567583\pi$
$264$	0	0
$265$	7.05701	0.433509
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.67257	0.589747	0.294873	−	0.955536i	$-0.404722\pi$
$269$	9.67257	0.589747	0.294873	+	0.955536i	$0.404722\pi$
$270$	0	0
$271$	−12.8377	−0.779838	−0.389919	−	0.920849i	$-0.627497\pi$
$271$	−12.8377	−0.779838	−0.389919	+	0.920849i	$0.627497\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.94592	0.237948
$276$	0	0
$277$	11.5831	0.695959	0.347980	−	0.937502i	$-0.386868\pi$
$277$	11.5831	0.695959	0.347980	+	0.937502i	$0.386868\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.92821	−0.293992	−0.146996	−	0.989137i	$-0.546960\pi$
$281$	−4.92821	−0.293992	−0.146996	+	0.989137i	$0.546960\pi$
$282$	0	0
$283$	−18.6008	−1.10570	−0.552851	−	0.833280i	$-0.686460\pi$
$283$	−18.6008	−1.10570	−0.552851	+	0.833280i	$0.686460\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.05408	−0.298333
$288$	0	0
$289$	−16.9253	−0.995605
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.7601	−1.44650	−0.723250	−	0.690586i	$-0.757353\pi$
$293$	−24.7601	−1.44650	−0.723250	+	0.690586i	$0.757353\pi$
$294$	0	0
$295$	3.65779	0.212965
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.32743	0.308093
$300$	0	0
$301$	4.60078	0.265184
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.60370	0.0918277
$306$	0	0
$307$	21.9430	1.25235	0.626176	−	0.779681i	$-0.284619\pi$
$307$	21.9430	1.25235	0.626176	+	0.779681i	$0.284619\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.1623	1.54023	0.770115	−	0.637905i	$-0.220199\pi$
$311$	27.1623	1.54023	0.770115	+	0.637905i	$0.220199\pi$
$312$	0	0
$313$	−8.54377	−0.482922	−0.241461	−	0.970410i	$-0.577627\pi$
$313$	−8.54377	−0.482922	−0.241461	+	0.970410i	$0.577627\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.399223	0.0224226	0.0112113	−	0.999937i	$-0.496431\pi$
$317$	0.399223	0.0224226	0.0112113	+	0.999937i	$0.496431\pi$
$318$	0	0
$319$	42.0875	2.35645
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.47102	−0.0818495
$324$	0	0
$325$	0.780738	0.0433076
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.38151	0.0761654
$330$	0	0
$331$	−5.61849	−0.308820	−0.154410	−	0.988007i	$-0.549348\pi$
$331$	−5.61849	−0.308820	−0.154410	+	0.988007i	$0.549348\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.2163	−0.940629
$336$	0	0
$337$	−28.9823	−1.57877	−0.789383	−	0.613901i	$-0.789599\pi$
$337$	−28.9823	−1.57877	−0.789383	+	0.613901i	$0.789599\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.3609	2.78135
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.4690	1.85040	0.925198	−	0.379485i	$-0.123899\pi$
$347$	34.4690	1.85040	0.925198	+	0.379485i	$0.123899\pi$
$348$	0	0
$349$	17.5615	0.940044	0.470022	−	0.882655i	$-0.344246\pi$
$349$	17.5615	0.940044	0.470022	+	0.882655i	$0.344246\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.8889	1.75050	0.875250	−	0.483671i	$-0.160697\pi$
$353$	32.8889	1.75050	0.875250	+	0.483671i	$0.160697\pi$
$354$	0	0
$355$	−15.9823	−0.848252
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.96362	0.156414	0.0782071	−	0.996937i	$-0.475080\pi$
$359$	2.96362	0.156414	0.0782071	+	0.996937i	$0.475080\pi$
$360$	0	0
$361$	9.96070	0.524247
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.2704	1.00866
$366$	0	0
$367$	13.3638	0.697585	0.348792	−	0.937200i	$-0.386592\pi$
$367$	13.3638	0.697585	0.348792	+	0.937200i	$0.386592\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.43560	0.178367
$372$	0	0
$373$	4.60078	0.238219	0.119110	−	0.992881i	$-0.461996\pi$
$373$	4.60078	0.238219	0.119110	+	0.992881i	$0.461996\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.32743	0.428884
$378$	0	0
$379$	−2.21926	−0.113996	−0.0569979	−	0.998374i	$-0.518153\pi$
$379$	−2.21926	−0.113996	−0.0569979	+	0.998374i	$0.518153\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.0335	−1.58574	−0.792868	−	0.609394i	$-0.791413\pi$
$383$	−31.0335	−1.58574	−0.792868	+	0.609394i	$0.791413\pi$
$384$	0	0
$385$	−10.3815	−0.529091
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.8348	−1.25918	−0.629588	−	0.776929i	$-0.716776\pi$
$389$	−24.8348	−1.25918	−0.629588	+	0.776929i	$0.716776\pi$
$390$	0	0
$391$	−1.45623	−0.0736449
$392$	0	0
$393$	0	0
$394$	0	0
$395$	26.5860	1.33769
$396$	0	0
$397$	17.7237	0.889528	0.444764	−	0.895648i	$-0.353287\pi$
$397$	17.7237	0.889528	0.444764	+	0.895648i	$0.353287\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.1770	1.50697	0.753485	−	0.657466i	$-0.228371\pi$
$401$	30.1770	1.50697	0.753485	+	0.657466i	$0.228371\pi$
$402$	0	0
$403$	10.1623	0.506218
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−41.2527	−2.04482
$408$	0	0
$409$	16.7630	0.828878	0.414439	−	0.910077i	$-0.363978\pi$
$409$	16.7630	0.828878	0.414439	+	0.910077i	$0.363978\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.78074	0.0876244
$414$	0	0
$415$	−11.7630	−0.577424
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.88891	0.141132	0.0705662	−	0.997507i	$-0.477519\pi$
$419$	2.88891	0.141132	0.0705662	+	0.997507i	$0.477519\pi$
$420$	0	0
$421$	−0.179961	−0.00877078	−0.00438539	−	0.999990i	$-0.501396\pi$
$421$	−0.179961	−0.00877078	−0.00438539	+	0.999990i	$0.501396\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.213412	−0.0103520
$426$	0	0
$427$	0.780738	0.0377826
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.76595	−0.229568	−0.114784	−	0.993390i	$-0.536618\pi$
$431$	−4.76595	−0.229568	−0.114784	+	0.993390i	$0.536618\pi$
$432$	0	0
$433$	−27.7630	−1.33421	−0.667103	−	0.744965i	$-0.732466\pi$
$433$	−27.7630	−1.33421	−0.667103	+	0.744965i	$0.732466\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.6696	1.37146
$438$	0	0
$439$	−4.65779	−0.222304	−0.111152	−	0.993803i	$-0.535454\pi$
$439$	−4.65779	−0.222304	−0.111152	+	0.993803i	$0.535454\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.76303	−0.131275	−0.0656377	−	0.997844i	$-0.520908\pi$
$443$	−2.76303	−0.131275	−0.0656377	+	0.997844i	$0.520908\pi$
$444$	0	0
$445$	−28.3815	−1.34541
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.9430	0.941168	0.470584	−	0.882355i	$-0.344043\pi$
$449$	19.9430	0.941168	0.470584	+	0.882355i	$0.344043\pi$
$450$	0	0
$451$	25.5438	1.20281
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.05408	−0.0962970
$456$	0	0
$457$	27.3815	1.28085	0.640427	−	0.768019i	$-0.278758\pi$
$457$	27.3815	1.28085	0.640427	+	0.768019i	$0.278758\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.05116	−0.281831	−0.140915	−	0.990022i	$-0.545005\pi$
$461$	−6.05116	−0.281831	−0.140915	+	0.990022i	$0.545005\pi$
$462$	0	0
$463$	−17.5438	−0.815328	−0.407664	−	0.913132i	$-0.633657\pi$
$463$	−17.5438	−0.815328	−0.407664	+	0.913132i	$0.633657\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.6156	−1.09280	−0.546399	−	0.837525i	$-0.684002\pi$
$467$	−23.6156	−1.09280	−0.546399	+	0.837525i	$0.684002\pi$
$468$	0	0
$469$	−8.38151	−0.387022
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−23.2527	−1.06916
$474$	0	0
$475$	4.20155	0.192780
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.2527	1.74781	0.873906	−	0.486096i	$-0.161579\pi$
$479$	38.2527	1.74781	0.873906	+	0.486096i	$0.161579\pi$
$480$	0	0
$481$	−8.16225	−0.372167
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.55855	−0.206993
$486$	0	0
$487$	−6.57918	−0.298131	−0.149066	−	0.988827i	$-0.547627\pi$
$487$	−6.57918	−0.298131	−0.149066	+	0.988827i	$0.547627\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.05408	−0.0926995	−0.0463498	−	0.998925i	$-0.514759\pi$
$491$	−2.05408	−0.0926995	−0.0463498	+	0.998925i	$0.514759\pi$
$492$	0	0
$493$	−2.27627	−0.102518
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.78074	−0.349014
$498$	0	0
$499$	−16.3245	−0.730785	−0.365393	−	0.930854i	$-0.619065\pi$
$499$	−16.3245	−0.730785	−0.365393	+	0.930854i	$0.619065\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.60078	−0.249726	−0.124863	−	0.992174i	$-0.539849\pi$
$503$	−5.60078	−0.249726	−0.124863	+	0.992174i	$0.539849\pi$
$504$	0	0
$505$	5.60078	0.249231
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.672570	0.0298111	0.0149056	−	0.999889i	$-0.495255\pi$
$509$	0.672570	0.0298111	0.0149056	+	0.999889i	$0.495255\pi$
$510$	0	0
$511$	9.38151	0.415014
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.9371	1.62764
$516$	0	0
$517$	−6.98229	−0.307081
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.4533	1.15894	0.579470	−	0.814993i	$-0.303260\pi$
$521$	26.4533	1.15894	0.579470	+	0.814993i	$0.303260\pi$
$522$	0	0
$523$	27.3068	1.19404	0.597021	−	0.802225i	$-0.296351\pi$
$523$	27.3068	1.19404	0.597021	+	0.802225i	$0.296351\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.77781	−0.121003
$528$	0	0
$529$	5.38151	0.233979
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.05408	0.218917
$534$	0	0
$535$	2.27627	0.0984118
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.05408	−0.217695
$540$	0	0
$541$	−19.3245	−0.830825	−0.415413	−	0.909633i	$-0.636363\pi$
$541$	−19.3245	−0.830825	−0.415413	+	0.909633i	$0.636363\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.94592	0.297530
$546$	0	0
$547$	−18.3422	−0.784256	−0.392128	−	0.919911i	$-0.628261\pi$
$547$	−18.3422	−0.784256	−0.392128	+	0.919911i	$0.628261\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.8142	1.90915
$552$	0	0
$553$	12.9430	0.550392
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.19863	0.389758	0.194879	−	0.980827i	$-0.437569\pi$
$557$	9.19863	0.389758	0.194879	+	0.980827i	$0.437569\pi$
$558$	0	0
$559$	−4.60078	−0.194592
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.1623	1.39762	0.698811	−	0.715306i	$-0.253713\pi$
$563$	33.1623	1.39762	0.698811	+	0.715306i	$0.253713\pi$
$564$	0	0
$565$	38.7630	1.63077
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.2016	1.09843	0.549213	−	0.835682i	$-0.314928\pi$
$569$	26.2016	1.09843	0.549213	+	0.835682i	$0.314928\pi$
$570$	0	0
$571$	9.78074	0.409311	0.204656	−	0.978834i	$-0.434393\pi$
$571$	9.78074	0.409311	0.204656	+	0.978834i	$0.434393\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.15933	0.173456
$576$	0	0
$577$	36.3068	1.51147	0.755736	−	0.654877i	$-0.227279\pi$
$577$	36.3068	1.51147	0.755736	+	0.654877i	$0.227279\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.72665	−0.237582
$582$	0	0
$583$	−17.3638	−0.719135
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.1475	−0.996673	−0.498336	−	0.866984i	$-0.666056\pi$
$587$	−24.1475	−0.996673	−0.498336	+	0.866984i	$0.666056\pi$
$588$	0	0
$589$	54.6883	2.25339
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.4897	1.70378	0.851889	−	0.523723i	$-0.175457\pi$
$593$	41.4897	1.70378	0.851889	+	0.523723i	$0.175457\pi$
$594$	0	0
$595$	0.561476	0.0230183
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.6844	−0.926861	−0.463430	−	0.886133i	$-0.653382\pi$
$599$	−22.6844	−0.926861	−0.463430	+	0.886133i	$0.653382\pi$
$600$	0	0
$601$	−40.2498	−1.64182	−0.820912	−	0.571055i	$-0.806534\pi$
$601$	−40.2498	−1.64182	−0.820912	+	0.571055i	$0.806534\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.8741	1.21456
$606$	0	0
$607$	17.3245	0.703180	0.351590	−	0.936154i	$-0.385641\pi$
$607$	17.3245	0.703180	0.351590	+	0.936154i	$0.385641\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.38151	−0.0558901
$612$	0	0
$613$	−32.9646	−1.33143	−0.665713	−	0.746207i	$-0.731873\pi$
$613$	−32.9646	−1.33143	−0.665713	+	0.746207i	$0.731873\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.9401	−1.08457	−0.542283	−	0.840196i	$-0.682440\pi$
$617$	−26.9401	−1.08457	−0.542283	+	0.840196i	$0.682440\pi$
$618$	0	0
$619$	−1.98229	−0.0796750	−0.0398375	−	0.999206i	$-0.512684\pi$
$619$	−1.98229	−0.0796750	−0.0398375	+	0.999206i	$0.512684\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.8171	−0.553571
$624$	0	0
$625$	−20.4868	−0.819470
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.23112	0.0889606
$630$	0	0
$631$	−25.4868	−1.01461	−0.507306	−	0.861766i	$-0.669359\pi$
$631$	−25.4868	−1.01461	−0.507306	+	0.861766i	$0.669359\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	35.2527	1.39896
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.0846	−1.58325	−0.791623	−	0.611009i	$-0.790764\pi$
$641$	−40.0846	−1.58325	−0.791623	+	0.611009i	$0.790764\pi$
$642$	0	0
$643$	7.01771	0.276751	0.138376	−	0.990380i	$-0.455812\pi$
$643$	7.01771	0.276751	0.138376	+	0.990380i	$0.455812\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.8142	−0.464464	−0.232232	−	0.972660i	$-0.574603\pi$
$647$	−11.8142	−0.464464	−0.232232	+	0.972660i	$0.574603\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.273346	0.0106969	0.00534843	−	0.999986i	$-0.498298\pi$
$653$	0.273346	0.0106969	0.00534843	+	0.999986i	$0.498298\pi$
$654$	0	0
$655$	−36.7453	−1.43576
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.7994	0.576503	0.288251	−	0.957555i	$-0.406926\pi$
$659$	14.7994	0.576503	0.288251	+	0.957555i	$0.406926\pi$
$660$	0	0
$661$	−9.01771	−0.350748	−0.175374	−	0.984502i	$-0.556113\pi$
$661$	−9.01771	−0.350748	−0.175374	+	0.984502i	$0.556113\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.0541	−0.428659
$666$	0	0
$667$	44.3638	1.71777
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.94592	−0.152330
$672$	0	0
$673$	23.9607	0.923617	0.461809	−	0.886980i	$-0.347201\pi$
$673$	23.9607	0.923617	0.461809	+	0.886980i	$0.347201\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.65779	−0.255879	−0.127940	−	0.991782i	$-0.540836\pi$
$677$	−6.65779	−0.255879	−0.127940	+	0.991782i	$0.540836\pi$
$678$	0	0
$679$	−2.21926	−0.0851675
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.0728	−1.15070	−0.575351	−	0.817907i	$-0.695135\pi$
$683$	−30.0728	−1.15070	−0.575351	+	0.817907i	$0.695135\pi$
$684$	0	0
$685$	−9.22820	−0.352591
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.43560	−0.130886
$690$	0	0
$691$	−3.27627	−0.124635	−0.0623176	−	0.998056i	$-0.519849\pi$
$691$	−3.27627	−0.124635	−0.0623176	+	0.998056i	$0.519849\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	37.2704	1.41375
$696$	0	0
$697$	−1.38151	−0.0523286
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.2891	−1.21954	−0.609771	−	0.792578i	$-0.708739\pi$
$701$	−32.2891	−1.21954	−0.609771	+	0.792578i	$0.708739\pi$
$702$	0	0
$703$	−43.9253	−1.65667
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.72665	0.102546
$708$	0	0
$709$	−49.0875	−1.84352	−0.921761	−	0.387760i	$-0.873249\pi$
$709$	−49.0875	−1.84352	−0.921761	+	0.387760i	$0.873249\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	54.1387	2.02751
$714$	0	0
$715$	10.3815	0.388247
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.9617	−1.56491	−0.782453	−	0.622710i	$-0.786032\pi$
$719$	−41.9617	−1.56491	−0.782453	+	0.622710i	$0.786032\pi$
$720$	0	0
$721$	17.9823	0.669696
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.50154	0.241461
$726$	0	0
$727$	−28.4868	−1.05652	−0.528258	−	0.849084i	$-0.677154\pi$
$727$	−28.4868	−1.05652	−0.528258	+	0.849084i	$0.677154\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.25760	0.0465142
$732$	0	0
$733$	28.5261	1.05363	0.526817	−	0.849979i	$-0.323385\pi$
$733$	28.5261	1.05363	0.526817	+	0.849979i	$0.323385\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.3609	1.56038
$738$	0	0
$739$	7.85934	0.289110	0.144555	−	0.989497i	$-0.453825\pi$
$739$	7.85934	0.289110	0.144555	+	0.989497i	$0.453825\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.74729	−0.247534	−0.123767	−	0.992311i	$-0.539498\pi$
$743$	−6.74729	−0.247534	−0.123767	+	0.992311i	$0.539498\pi$
$744$	0	0
$745$	−9.25856	−0.339207
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.10817	0.0404916
$750$	0	0
$751$	22.1800	0.809358	0.404679	−	0.914459i	$-0.367383\pi$
$751$	22.1800	0.809358	0.404679	+	0.914459i	$0.367383\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.5586	−0.820990
$756$	0	0
$757$	−20.3815	−0.740779	−0.370389	−	0.928877i	$-0.620776\pi$
$757$	−20.3815	−0.740779	−0.370389	+	0.928877i	$0.620776\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.6549	1.47374	0.736869	−	0.676036i	$-0.236304\pi$
$761$	40.6549	1.47374	0.736869	+	0.676036i	$0.236304\pi$
$762$	0	0
$763$	3.38151	0.122419
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.78074	−0.0642987
$768$	0	0
$769$	−33.9037	−1.22260	−0.611299	−	0.791400i	$-0.709353\pi$
$769$	−33.9037	−1.22260	−0.611299	+	0.791400i	$0.709353\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.74825	−0.314653	−0.157326	−	0.987547i	$-0.550287\pi$
$773$	−8.74825	−0.314653	−0.157326	+	0.987547i	$0.550287\pi$
$774$	0	0
$775$	7.93406	0.285000
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27.1986	0.974492
$780$	0	0
$781$	39.3245	1.40714
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.58599	0.306447
$786$	0	0
$787$	9.28520	0.330982	0.165491	−	0.986211i	$-0.447079\pi$
$787$	9.28520	0.330982	0.165491	+	0.986211i	$0.447079\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.8712	0.670983
$792$	0	0
$793$	−0.780738	−0.0277248
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.4543	0.795371	0.397685	−	0.917522i	$-0.369814\pi$
$797$	22.4543	0.795371	0.397685	+	0.917522i	$0.369814\pi$
$798$	0	0
$799$	0.377632	0.0133597
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−47.4150	−1.67324
$804$	0	0
$805$	−10.9430	−0.385690
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.40215	−0.189929	−0.0949647	−	0.995481i	$-0.530274\pi$
$809$	−5.40215	−0.189929	−0.0949647	+	0.995481i	$0.530274\pi$
$810$	0	0
$811$	−0.0177088	−0.000621841 0	−0.000310920	−	1.00000i	$-0.500099\pi$
$811$	−0.0177088	−0.000621841 0	−0.000310920		1.00000i	$0.500099\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.5408	−0.404258
$816$	0	0
$817$	−24.7591	−0.866213
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.05701	−0.0368899	−0.0184449	−	0.999830i	$-0.505872\pi$
$821$	−1.05701	−0.0368899	−0.0184449	+	0.999830i	$0.505872\pi$
$822$	0	0
$823$	13.5261	0.471489	0.235744	−	0.971815i	$-0.424247\pi$
$823$	13.5261	0.471489	0.235744	+	0.971815i	$0.424247\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−49.3068	−1.71457	−0.857283	−	0.514846i	$-0.827849\pi$
$827$	−49.3068	−1.71457	−0.857283	+	0.514846i	$0.827849\pi$
$828$	0	0
$829$	−53.7060	−1.86529	−0.932644	−	0.360799i	$-0.882504\pi$
$829$	−53.7060	−1.86529	−0.932644	+	0.360799i	$0.882504\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.273346	0.00947088
$834$	0	0
$835$	−22.3727	−0.774241
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.4327	−1.29232	−0.646160	−	0.763202i	$-0.723626\pi$
$839$	−37.4327	−1.29232	−0.646160	+	0.763202i	$0.723626\pi$
$840$	0	0
$841$	40.3461	1.39124
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24.6490	−0.847952
$846$	0	0
$847$	14.5438	0.499730
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43.4838	−1.49061
$852$	0	0
$853$	−20.6185	−0.705963	−0.352982	−	0.935630i	$-0.614832\pi$
$853$	−20.6185	−0.705963	−0.352982	+	0.935630i	$0.614832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.8889	0.713551	0.356776	−	0.934190i	$-0.383876\pi$
$857$	20.8889	0.713551	0.356776	+	0.934190i	$0.383876\pi$
$858$	0	0
$859$	26.8860	0.917338	0.458669	−	0.888607i	$-0.348326\pi$
$859$	26.8860	0.917338	0.458669	+	0.888607i	$0.348326\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.8142	0.912766	0.456383	−	0.889784i	$-0.349145\pi$
$863$	26.8142	0.912766	0.456383	+	0.889784i	$0.349145\pi$
$864$	0	0
$865$	29.9912	1.01973
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−65.4150	−2.21905
$870$	0	0
$871$	8.38151	0.283997
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.8741	−0.401419
$876$	0	0
$877$	−37.9076	−1.28005	−0.640024	−	0.768355i	$-0.721076\pi$
$877$	−37.9076	−1.28005	−0.640024	+	0.768355i	$0.721076\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.47782	0.251934	0.125967	−	0.992034i	$-0.459797\pi$
$881$	7.47782	0.251934	0.125967	+	0.992034i	$0.459797\pi$
$882$	0	0
$883$	5.07472	0.170778	0.0853889	−	0.996348i	$-0.472787\pi$
$883$	5.07472	0.170778	0.0853889	+	0.996348i	$0.472787\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.5231	0.621946	0.310973	−	0.950419i	$-0.399345\pi$
$887$	18.5231	0.621946	0.310973	+	0.950419i	$0.399345\pi$
$888$	0	0
$889$	17.1623	0.575603
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.43464	−0.248791
$894$	0	0
$895$	−48.2498	−1.61281
$896$	0	0
$897$	0	0
$898$	0	0
$899$	84.6255	2.82242
$900$	0	0
$901$	0.939108	0.0312862
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.87412	−0.0955391
$906$	0	0
$907$	−49.6490	−1.64857	−0.824284	−	0.566176i	$-0.808422\pi$
$907$	−49.6490	−1.64857	−0.824284	+	0.566176i	$0.808422\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.9617	0.793885	0.396943	−	0.917843i	$-0.370071\pi$
$911$	23.9617	0.793885	0.396943	+	0.917843i	$0.370071\pi$
$912$	0	0
$913$	28.9430	0.957873
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.8889	−0.590744
$918$	0	0
$919$	−18.8377	−0.621400	−0.310700	−	0.950508i	$-0.600563\pi$
$919$	−18.8377	−0.621400	−0.310700	+	0.950508i	$0.600563\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.78074	0.256106
$924$	0	0
$925$	−6.37258	−0.209529
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.9037	0.488974	0.244487	−	0.969653i	$-0.421380\pi$
$929$	14.9037	0.488974	0.244487	+	0.969653i	$0.421380\pi$
$930$	0	0
$931$	−5.38151	−0.176372
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.83775	−0.0928043
$936$	0	0
$937$	3.94299	0.128812	0.0644059	−	0.997924i	$-0.479485\pi$
$937$	3.94299	0.128812	0.0644059	+	0.997924i	$0.479485\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.8113	−1.39561	−0.697804	−	0.716289i	$-0.745839\pi$
$941$	−42.8113	−1.39561	−0.697804	+	0.716289i	$0.745839\pi$
$942$	0	0
$943$	26.9253	0.876808
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.1838	−0.883356	−0.441678	−	0.897174i	$-0.645617\pi$
$947$	−27.1838	−0.883356	−0.441678	+	0.897174i	$0.645617\pi$
$948$	0	0
$949$	−9.38151	−0.304537
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.12295	0.0363760	0.0181880	−	0.999835i	$-0.494210\pi$
$953$	1.12295	0.0363760	0.0181880	+	0.999835i	$0.494210\pi$
$954$	0	0
$955$	47.9912	1.55296
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.49261	−0.145074
$960$	0	0
$961$	72.2714	2.33133
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−29.8741	−0.961682
$966$	0	0
$967$	−9.50447	−0.305643	−0.152822	−	0.988254i	$-0.548836\pi$
$967$	−9.50447	−0.305643	−0.152822	+	0.988254i	$0.548836\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24.5979	−0.789383	−0.394691	−	0.918814i	$-0.629148\pi$
$971$	−24.5979	−0.789383	−0.394691	+	0.918814i	$0.629148\pi$
$972$	0	0
$973$	18.1445	0.581687
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.4031	−0.812717	−0.406359	−	0.913714i	$-0.633202\pi$
$977$	−25.4031	−0.812717	−0.406359	+	0.913714i	$0.633202\pi$
$978$	0	0
$979$	69.8329	2.23187
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.8535	0.728913	0.364457	−	0.931220i	$-0.381255\pi$
$983$	22.8535	0.728913	0.364457	+	0.931220i	$0.381255\pi$
$984$	0	0
$985$	−35.6224	−1.13502
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.5103	−0.779383
$990$	0	0
$991$	24.4690	0.777285	0.388642	−	0.921389i	$-0.372944\pi$
$991$	24.4690	0.777285	0.388642	+	0.921389i	$0.372944\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−23.7119	−0.751717
$996$	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.g.1.3		3
3.2	odd	2		2268.2.a.j.1.1		3
4.3	odd	2		9072.2.a.bt.1.3		3
9.2	odd	6		756.2.j.a.253.3		6
9.4	even	3		252.2.j.b.169.2	yes	6
9.5	odd	6		756.2.j.a.505.3		6
9.7	even	3		252.2.j.b.85.2	✓	6
12.11	even	2		9072.2.a.bz.1.1		3
36.7	odd	6		1008.2.r.g.337.2		6
36.11	even	6		3024.2.r.i.1009.3		6
36.23	even	6		3024.2.r.i.2017.3		6
36.31	odd	6		1008.2.r.g.673.2		6
63.2	odd	6		5292.2.l.g.361.1		6
63.4	even	3		1764.2.l.d.961.1		6
63.5	even	6		5292.2.i.g.2125.1		6
63.11	odd	6		5292.2.i.d.1549.3		6
63.13	odd	6		1764.2.j.d.1177.2		6
63.16	even	3		1764.2.l.d.949.1		6
63.20	even	6		5292.2.j.e.1765.1		6
63.23	odd	6		5292.2.i.d.2125.3		6
63.25	even	3		1764.2.i.f.373.2		6
63.31	odd	6		1764.2.l.g.961.3		6
63.32	odd	6		5292.2.l.g.3313.1		6
63.34	odd	6		1764.2.j.d.589.2		6
63.38	even	6		5292.2.i.g.1549.1		6
63.40	odd	6		1764.2.i.e.1537.2		6
63.41	even	6		5292.2.j.e.3529.1		6
63.47	even	6		5292.2.l.d.361.3		6
63.52	odd	6		1764.2.i.e.373.2		6
63.58	even	3		1764.2.i.f.1537.2		6
63.59	even	6		5292.2.l.d.3313.3		6
63.61	odd	6		1764.2.l.g.949.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.2	✓	6	9.7	even	3
252.2.j.b.169.2	yes	6	9.4	even	3
756.2.j.a.253.3		6	9.2	odd	6
756.2.j.a.505.3		6	9.5	odd	6
1008.2.r.g.337.2		6	36.7	odd	6
1008.2.r.g.673.2		6	36.31	odd	6
1764.2.i.e.373.2		6	63.52	odd	6
1764.2.i.e.1537.2		6	63.40	odd	6
1764.2.i.f.373.2		6	63.25	even	3
1764.2.i.f.1537.2		6	63.58	even	3
1764.2.j.d.589.2		6	63.34	odd	6
1764.2.j.d.1177.2		6	63.13	odd	6
1764.2.l.d.949.1		6	63.16	even	3
1764.2.l.d.961.1		6	63.4	even	3
1764.2.l.g.949.3		6	63.61	odd	6
1764.2.l.g.961.3		6	63.31	odd	6
2268.2.a.g.1.3		3	1.1	even	1	trivial
2268.2.a.j.1.1		3	3.2	odd	2
3024.2.r.i.1009.3		6	36.11	even	6
3024.2.r.i.2017.3		6	36.23	even	6
5292.2.i.d.1549.3		6	63.11	odd	6
5292.2.i.d.2125.3		6	63.23	odd	6
5292.2.i.g.1549.1		6	63.38	even	6
5292.2.i.g.2125.1		6	63.5	even	6
5292.2.j.e.1765.1		6	63.20	even	6
5292.2.j.e.3529.1		6	63.41	even	6
5292.2.l.d.361.3		6	63.47	even	6
5292.2.l.d.3313.3		6	63.59	even	6
5292.2.l.g.361.1		6	63.2	odd	6
5292.2.l.g.3313.1		6	63.32	odd	6
9072.2.a.bt.1.3		3	4.3	odd	2
9072.2.a.bz.1.1		3	12.11	even	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)

\(f(q)\)	\(=\)	\(q+2.05408 q^{5} +1.00000 q^{7} -5.05408 q^{11} -1.00000 q^{13} +0.273346 q^{17} -5.38151 q^{19} -5.32743 q^{23} -0.780738 q^{25} -8.32743 q^{29} -10.1623 q^{31} +2.05408 q^{35} +8.16225 q^{37} -5.05408 q^{41} +4.60078 q^{43} +1.38151 q^{47} +1.00000 q^{49} +3.43560 q^{53} -10.3815 q^{55} +1.78074 q^{59} +0.780738 q^{61} -2.05408 q^{65} -8.38151 q^{67} -7.78074 q^{71} +9.38151 q^{73} -5.05408 q^{77} +12.9430 q^{79} -5.72665 q^{83} +0.561476 q^{85} -13.8171 q^{89} -1.00000 q^{91} -11.0541 q^{95} -2.21926 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} - 6 q^{11} - 3 q^{13} + 3 q^{19} - 6 q^{23} + 6 q^{25} - 15 q^{29} - 3 q^{31} - 3 q^{35} - 3 q^{37} - 6 q^{41} + 3 q^{43} - 15 q^{47} + 3 q^{49} - 18 q^{53} - 12 q^{55} - 3 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 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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