LMFDB - Embedded newform 2116.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2116,2,Mod(1,2116)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2116.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2116, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2116 = 2^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2116.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.8963450677$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2116.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410 q^{5} -3.46410 q^{7} -2.00000 q^{9} -3.46410 q^{11} -5.00000 q^{13} +3.46410 q^{15} -3.46410 q^{17} +6.92820 q^{19} -3.46410 q^{21} +7.00000 q^{25} -5.00000 q^{27} -3.00000 q^{29} +1.00000 q^{31} -3.46410 q^{33} -12.0000 q^{35} -3.46410 q^{37} -5.00000 q^{39} -9.00000 q^{41} -6.92820 q^{45} -3.00000 q^{47} +5.00000 q^{49} -3.46410 q^{51} +6.92820 q^{53} -12.0000 q^{55} +6.92820 q^{57} -12.0000 q^{59} -13.8564 q^{61} +6.92820 q^{63} -17.3205 q^{65} +3.46410 q^{67} +9.00000 q^{71} -11.0000 q^{73} +7.00000 q^{75} +12.0000 q^{77} +6.92820 q^{79} +1.00000 q^{81} +10.3923 q^{83} -12.0000 q^{85} -3.00000 q^{87} +6.92820 q^{89} +17.3205 q^{91} +1.00000 q^{93} +24.0000 q^{95} -17.3205 q^{97} +6.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{9} - 10 q^{13} + 14 q^{25} - 10 q^{27} - 6 q^{29} + 2 q^{31} - 24 q^{35} - 10 q^{39} - 18 q^{41} - 6 q^{47} + 10 q^{49} - 24 q^{55} - 24 q^{59} + 18 q^{71} - 22 q^{73} + 14 q^{75} + 24 q^{77}+ \cdots + 48 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 - 10 * q^13 + 14 * q^25 - 10 * q^27 - 6 * q^29 + 2 * q^31 - 24 * q^35 - 10 * q^39 - 18 * q^41 - 6 * q^47 + 10 * q^49 - 24 * q^55 - 24 * q^59 + 18 * q^71 - 22 * q^73 + 14 * q^75 + 24 * q^77 + 2 * q^81 - 24 * q^85 - 6 * q^87 + 2 * q^93 + 48 * q^95

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	−3.46410	−0.569495	−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410	−0.569495	−0.284747	+	0.958603i	$0.591910\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−6.92820	−1.03280
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	−3.46410	−0.485071
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	6.92820	0.917663
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−13.8564	−1.77413	−0.887066	−	0.461644i	$-0.847260\pi$
$61$	−13.8564	−1.77413	−0.887066	+	0.461644i	$0.847260\pi$
$62$	0	0
$63$	6.92820	0.872872
$64$	0	0
$65$	−17.3205	−2.14834
$66$	0	0
$67$	3.46410	0.423207	0.211604	−	0.977356i	$-0.432131\pi$
$67$	3.46410	0.423207	0.211604	+	0.977356i	$0.432131\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	17.3205	1.81568
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	24.0000	2.46235
$96$	0	0
$97$	−17.3205	−1.75863	−0.879316	−	0.476240i	$-0.842000\pi$
$97$	−17.3205	−1.75863	−0.879316	+	0.476240i	$0.842000\pi$
$98$	0	0
$99$	6.92820	0.696311
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	3.46410	0.341328	0.170664	−	0.985329i	$-0.445409\pi$
$103$	3.46410	0.341328	0.170664	+	0.985329i	$0.445409\pi$
$104$	0	0
$105$	−12.0000	−1.17108
$106$	0	0
$107$	13.8564	1.33955	0.669775	−	0.742564i	$-0.266391\pi$
$107$	13.8564	1.33955	0.669775	+	0.742564i	$0.266391\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−3.46410	−0.328798
$112$	0	0
$113$	10.3923	0.977626	0.488813	−	0.872389i	$-0.337430\pi$
$113$	10.3923	0.977626	0.488813	+	0.872389i	$0.337430\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.0000	0.924500
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.00000	−0.811503
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	0	0
$135$	−17.3205	−1.49071
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	17.3205	1.44841
$144$	0	0
$145$	−10.3923	−0.863034
$146$	0	0
$147$	5.00000	0.412393
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	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
$153$	6.92820	0.560112
$154$	0	0
$155$	3.46410	0.278243
$156$	0	0
$157$	20.7846	1.65879	0.829396	−	0.558661i	$-0.188685\pi$
$157$	20.7846	1.65879	0.829396	+	0.558661i	$0.188685\pi$
$158$	0	0
$159$	6.92820	0.549442
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	0	0
$165$	−12.0000	−0.934199
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−13.8564	−1.05963
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−24.2487	−1.83303
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−3.46410	−0.257485	−0.128742	−	0.991678i	$-0.541094\pi$
$181$	−3.46410	−0.257485	−0.128742	+	0.991678i	$0.541094\pi$
$182$	0	0
$183$	−13.8564	−1.02430
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	17.3205	1.25988
$190$	0	0
$191$	−17.3205	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$191$	−17.3205	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	−17.3205	−1.24035
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	−17.3205	−1.22782	−0.613909	−	0.789377i	$-0.710404\pi$
$199$	−17.3205	−1.22782	−0.613909	+	0.789377i	$0.710404\pi$
$200$	0	0
$201$	3.46410	0.244339
$202$	0	0
$203$	10.3923	0.729397
$204$	0	0
$205$	−31.1769	−2.17749
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	9.00000	0.616670
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.46410	−0.235159
$218$	0	0
$219$	−11.0000	−0.743311
$220$	0	0
$221$	17.3205	1.16510
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	−14.0000	−0.933333
$226$	0	0
$227$	17.3205	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	−10.3923	−0.677919
$236$	0	0
$237$	6.92820	0.450035
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	−17.3205	−1.11571	−0.557856	−	0.829938i	$-0.688376\pi$
$241$	−17.3205	−1.11571	−0.557856	+	0.829938i	$0.688376\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	17.3205	1.10657
$246$	0	0
$247$	−34.6410	−2.20416
$248$	0	0
$249$	10.3923	0.658586
$250$	0	0
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−6.92820	−0.427211	−0.213606	−	0.976920i	$-0.568521\pi$
$263$	−6.92820	−0.427211	−0.213606	+	0.976920i	$0.568521\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	6.92820	0.423999
$268$	0	0
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	17.3205	1.04828
$274$	0	0
$275$	−24.2487	−1.46225
$276$	0	0
$277$	25.0000	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$277$	25.0000	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	3.46410	0.206651	0.103325	−	0.994648i	$-0.467052\pi$
$281$	3.46410	0.206651	0.103325	+	0.994648i	$0.467052\pi$
$282$	0	0
$283$	10.3923	0.617758	0.308879	−	0.951101i	$-0.400046\pi$
$283$	10.3923	0.617758	0.308879	+	0.951101i	$0.400046\pi$
$284$	0	0
$285$	24.0000	1.42164
$286$	0	0
$287$	31.1769	1.84032
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−17.3205	−1.01535
$292$	0	0
$293$	−6.92820	−0.404750	−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820	−0.404750	−0.202375	+	0.979308i	$0.564866\pi$
$294$	0	0
$295$	−41.5692	−2.42025
$296$	0	0
$297$	17.3205	1.00504
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−48.0000	−2.74847
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	3.46410	0.197066
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	−6.92820	−0.391605	−0.195803	−	0.980643i	$-0.562731\pi$
$313$	−6.92820	−0.391605	−0.195803	+	0.980643i	$0.562731\pi$
$314$	0	0
$315$	24.0000	1.35225
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	10.3923	0.581857
$320$	0	0
$321$	13.8564	0.773389
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−35.0000	−1.94145
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.3923	0.572946
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	0	0
$333$	6.92820	0.379663
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	6.92820	0.377403	0.188702	−	0.982034i	$-0.439572\pi$
$337$	6.92820	0.377403	0.188702	+	0.982034i	$0.439572\pi$
$338$	0	0
$339$	10.3923	0.564433
$340$	0	0
$341$	−3.46410	−0.187592
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	25.0000	1.33440
$352$	0	0
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	31.1769	1.65470
$356$	0	0
$357$	12.0000	0.635107
$358$	0	0
$359$	−6.92820	−0.365657	−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820	−0.365657	−0.182828	+	0.983145i	$0.558525\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−38.1051	−1.99451
$366$	0	0
$367$	3.46410	0.180825	0.0904123	−	0.995904i	$-0.471182\pi$
$367$	3.46410	0.180825	0.0904123	+	0.995904i	$0.471182\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−17.3205	−0.896822	−0.448411	−	0.893828i	$-0.648010\pi$
$373$	−17.3205	−0.896822	−0.448411	+	0.893828i	$0.648010\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	6.92820	0.355878	0.177939	−	0.984042i	$-0.443057\pi$
$379$	6.92820	0.355878	0.177939	+	0.984042i	$0.443057\pi$
$380$	0	0
$381$	−5.00000	−0.256158
$382$	0	0
$383$	−27.7128	−1.41606	−0.708029	−	0.706183i	$-0.750416\pi$
$383$	−27.7128	−1.41606	−0.708029	+	0.706183i	$0.750416\pi$
$384$	0	0
$385$	41.5692	2.11856
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.6410	1.75637	0.878185	−	0.478322i	$-0.158755\pi$
$389$	34.6410	1.75637	0.878185	+	0.478322i	$0.158755\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	24.0000	1.20757
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	−17.3205	−0.864945	−0.432472	−	0.901647i	$-0.642359\pi$
$401$	−17.3205	−0.864945	−0.432472	+	0.901647i	$0.642359\pi$
$402$	0	0
$403$	−5.00000	−0.249068
$404$	0	0
$405$	3.46410	0.172133
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	41.5692	2.04549
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	−13.0000	−0.636613
$418$	0	0
$419$	−34.6410	−1.69232	−0.846162	−	0.532925i	$-0.821093\pi$
$419$	−34.6410	−1.69232	−0.846162	+	0.532925i	$0.821093\pi$
$420$	0	0
$421$	−17.3205	−0.844150	−0.422075	−	0.906561i	$-0.638698\pi$
$421$	−17.3205	−0.844150	−0.422075	+	0.906561i	$0.638698\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	−24.2487	−1.17624
$426$	0	0
$427$	48.0000	2.32288
$428$	0	0
$429$	17.3205	0.836242
$430$	0	0
$431$	34.6410	1.66860	0.834300	−	0.551311i	$-0.185872\pi$
$431$	34.6410	1.66860	0.834300	+	0.551311i	$0.185872\pi$
$432$	0	0
$433$	10.3923	0.499422	0.249711	−	0.968320i	$-0.419664\pi$
$433$	10.3923	0.499422	0.249711	+	0.968320i	$0.419664\pi$
$434$	0	0
$435$	−10.3923	−0.498273
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	0	0
$441$	−10.0000	−0.476190
$442$	0	0
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	31.1769	1.46806
$452$	0	0
$453$	5.00000	0.234920
$454$	0	0
$455$	60.0000	2.81284
$456$	0	0
$457$	−31.1769	−1.45839	−0.729197	−	0.684304i	$-0.760106\pi$
$457$	−31.1769	−1.45839	−0.729197	+	0.684304i	$0.760106\pi$
$458$	0	0
$459$	17.3205	0.808452
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	3.46410	0.160644
$466$	0	0
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	20.7846	0.957704
$472$	0	0
$473$	0	0
$474$	0	0
$475$	48.4974	2.22521
$476$	0	0
$477$	−13.8564	−0.634441
$478$	0	0
$479$	17.3205	0.791394	0.395697	−	0.918381i	$-0.370503\pi$
$479$	17.3205	0.791394	0.395697	+	0.918381i	$0.370503\pi$
$480$	0	0
$481$	17.3205	0.789747
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−60.0000	−2.72446
$486$	0	0
$487$	−25.0000	−1.13286	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000	−1.13286	−0.566429	+	0.824110i	$0.691675\pi$
$488$	0	0
$489$	19.0000	0.859210
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	10.3923	0.468046
$494$	0	0
$495$	24.0000	1.07872
$496$	0	0
$497$	−31.1769	−1.39848
$498$	0	0
$499$	37.0000	1.65635	0.828174	−	0.560471i	$-0.189380\pi$
$499$	37.0000	1.65635	0.828174	+	0.560471i	$0.189380\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.92820	−0.308913	−0.154457	−	0.988000i	$-0.549363\pi$
$503$	−6.92820	−0.308913	−0.154457	+	0.988000i	$0.549363\pi$
$504$	0	0
$505$	−20.7846	−0.924903
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	38.1051	1.68567
$512$	0	0
$513$	−34.6410	−1.52944
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	10.3923	0.457053
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	−17.3205	−0.757373	−0.378686	−	0.925525i	$-0.623624\pi$
$523$	−17.3205	−0.757373	−0.378686	+	0.925525i	$0.623624\pi$
$524$	0	0
$525$	−24.2487	−1.05830
$526$	0	0
$527$	−3.46410	−0.150899
$528$	0	0
$529$	0	0
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	45.0000	1.94917
$534$	0	0
$535$	48.0000	2.07522
$536$	0	0
$537$	15.0000	0.647298
$538$	0	0
$539$	−17.3205	−0.746047
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	−3.46410	−0.148659
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.00000	−0.213785	−0.106892	−	0.994271i	$-0.534090\pi$
$547$	−5.00000	−0.213785	−0.106892	+	0.994271i	$0.534090\pi$
$548$	0	0
$549$	27.7128	1.18275
$550$	0	0
$551$	−20.7846	−0.885454
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	20.7846	0.880672	0.440336	−	0.897833i	$-0.354859\pi$
$557$	20.7846	0.880672	0.440336	+	0.897833i	$0.354859\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−27.7128	−1.16796	−0.583978	−	0.811770i	$-0.698505\pi$
$563$	−27.7128	−1.16796	−0.583978	+	0.811770i	$0.698505\pi$
$564$	0	0
$565$	36.0000	1.51453
$566$	0	0
$567$	−3.46410	−0.145479
$568$	0	0
$569$	−45.0333	−1.88790	−0.943948	−	0.330096i	$-0.892919\pi$
$569$	−45.0333	−1.88790	−0.943948	+	0.330096i	$0.892919\pi$
$570$	0	0
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	−17.3205	−0.723575
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	5.00000	0.207793
$580$	0	0
$581$	−36.0000	−1.49353
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	34.6410	1.43223
$586$	0	0
$587$	−27.0000	−1.11441	−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000	−1.11441	−0.557205	+	0.830375i	$0.688126\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	41.5692	1.70417
$596$	0	0
$597$	−17.3205	−0.708881
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	−6.92820	−0.282138
$604$	0	0
$605$	3.46410	0.140836
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	10.3923	0.421117
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	−17.3205	−0.699569	−0.349784	−	0.936830i	$-0.613745\pi$
$613$	−17.3205	−0.699569	−0.349784	+	0.936830i	$0.613745\pi$
$614$	0	0
$615$	−31.1769	−1.25717
$616$	0	0
$617$	20.7846	0.836757	0.418378	−	0.908273i	$-0.362599\pi$
$617$	20.7846	0.836757	0.418378	+	0.908273i	$0.362599\pi$
$618$	0	0
$619$	−6.92820	−0.278468	−0.139234	−	0.990260i	$-0.544464\pi$
$619$	−6.92820	−0.278468	−0.139234	+	0.990260i	$0.544464\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−24.0000	−0.958468
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−34.6410	−1.37904	−0.689519	−	0.724268i	$-0.742178\pi$
$631$	−34.6410	−1.37904	−0.689519	+	0.724268i	$0.742178\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	−17.3205	−0.687343
$636$	0	0
$637$	−25.0000	−0.990536
$638$	0	0
$639$	−18.0000	−0.712069
$640$	0	0
$641$	20.7846	0.820943	0.410471	−	0.911873i	$-0.365364\pi$
$641$	20.7846	0.820943	0.410471	+	0.911873i	$0.365364\pi$
$642$	0	0
$643$	−10.3923	−0.409832	−0.204916	−	0.978780i	$-0.565692\pi$
$643$	−10.3923	−0.409832	−0.204916	+	0.978780i	$0.565692\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	0	0
$649$	41.5692	1.63173
$650$	0	0
$651$	−3.46410	−0.135769
$652$	0	0
$653$	15.0000	0.586995	0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000	0.586995	0.293498	+	0.955960i	$0.405181\pi$
$654$	0	0
$655$	−51.9615	−2.03030
$656$	0	0
$657$	22.0000	0.858302
$658$	0	0
$659$	−17.3205	−0.674711	−0.337356	−	0.941377i	$-0.609532\pi$
$659$	−17.3205	−0.674711	−0.337356	+	0.941377i	$0.609532\pi$
$660$	0	0
$661$	13.8564	0.538952	0.269476	−	0.963007i	$-0.413150\pi$
$661$	13.8564	0.538952	0.269476	+	0.963007i	$0.413150\pi$
$662$	0	0
$663$	17.3205	0.672673
$664$	0	0
$665$	−83.1384	−3.22397
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.0000	0.773245
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	0	0
$675$	−35.0000	−1.34715
$676$	0	0
$677$	−13.8564	−0.532545	−0.266272	−	0.963898i	$-0.585792\pi$
$677$	−13.8564	−0.532545	−0.266272	+	0.963898i	$0.585792\pi$
$678$	0	0
$679$	60.0000	2.30259
$680$	0	0
$681$	17.3205	0.663723
$682$	0	0
$683$	−21.0000	−0.803543	−0.401771	−	0.915740i	$-0.631605\pi$
$683$	−21.0000	−0.803543	−0.401771	+	0.915740i	$0.631605\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−34.6410	−1.31972
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	−45.0333	−1.70821
$696$	0	0
$697$	31.1769	1.18091
$698$	0	0
$699$	−9.00000	−0.340411
$700$	0	0
$701$	17.3205	0.654187	0.327093	−	0.944992i	$-0.393931\pi$
$701$	17.3205	0.654187	0.327093	+	0.944992i	$0.393931\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	−10.3923	−0.391397
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	−17.3205	−0.650485	−0.325243	−	0.945631i	$-0.605446\pi$
$709$	−17.3205	−0.650485	−0.325243	+	0.945631i	$0.605446\pi$
$710$	0	0
$711$	−13.8564	−0.519656
$712$	0	0
$713$	0	0
$714$	0	0
$715$	60.0000	2.24387
$716$	0	0
$717$	−27.0000	−1.00833
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	−17.3205	−0.644157
$724$	0	0
$725$	−21.0000	−0.779920
$726$	0	0
$727$	31.1769	1.15629	0.578144	−	0.815935i	$-0.303777\pi$
$727$	31.1769	1.15629	0.578144	+	0.815935i	$0.303777\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	10.3923	0.383849	0.191924	−	0.981410i	$-0.438527\pi$
$733$	10.3923	0.383849	0.191924	+	0.981410i	$0.438527\pi$
$734$	0	0
$735$	17.3205	0.638877
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−13.0000	−0.478213	−0.239106	−	0.970993i	$-0.576854\pi$
$739$	−13.0000	−0.478213	−0.239106	+	0.970993i	$0.576854\pi$
$740$	0	0
$741$	−34.6410	−1.27257
$742$	0	0
$743$	−17.3205	−0.635428	−0.317714	−	0.948187i	$-0.602915\pi$
$743$	−17.3205	−0.635428	−0.317714	+	0.948187i	$0.602915\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20.7846	−0.760469
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	34.6410	1.26407	0.632034	−	0.774940i	$-0.282220\pi$
$751$	34.6410	1.26407	0.632034	+	0.774940i	$0.282220\pi$
$752$	0	0
$753$	−17.3205	−0.631194
$754$	0	0
$755$	17.3205	0.630358
$756$	0	0
$757$	38.1051	1.38495	0.692477	−	0.721440i	$-0.256519\pi$
$757$	38.1051	1.38495	0.692477	+	0.721440i	$0.256519\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−51.0000	−1.84875	−0.924374	−	0.381487i	$-0.875412\pi$
$761$	−51.0000	−1.84875	−0.924374	+	0.381487i	$0.875412\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	60.0000	2.16647
$768$	0	0
$769$	−45.0333	−1.62394	−0.811972	−	0.583697i	$-0.801606\pi$
$769$	−45.0333	−1.62394	−0.811972	+	0.583697i	$0.801606\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	−34.6410	−1.24595	−0.622975	−	0.782241i	$-0.714076\pi$
$773$	−34.6410	−1.24595	−0.622975	+	0.782241i	$0.714076\pi$
$774$	0	0
$775$	7.00000	0.251447
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	−62.3538	−2.23406
$780$	0	0
$781$	−31.1769	−1.11560
$782$	0	0
$783$	15.0000	0.536056
$784$	0	0
$785$	72.0000	2.56979
$786$	0	0
$787$	48.4974	1.72875	0.864373	−	0.502851i	$-0.167715\pi$
$787$	48.4974	1.72875	0.864373	+	0.502851i	$0.167715\pi$
$788$	0	0
$789$	−6.92820	−0.246651
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	69.2820	2.46028
$794$	0	0
$795$	24.0000	0.851192
$796$	0	0
$797$	−20.7846	−0.736229	−0.368114	−	0.929781i	$-0.619996\pi$
$797$	−20.7846	−0.736229	−0.368114	+	0.929781i	$0.619996\pi$
$798$	0	0
$799$	10.3923	0.367653
$800$	0	0
$801$	−13.8564	−0.489592
$802$	0	0
$803$	38.1051	1.34470
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.00000	−0.105605
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−11.0000	−0.386262	−0.193131	−	0.981173i	$-0.561864\pi$
$811$	−11.0000	−0.386262	−0.193131	+	0.981173i	$0.561864\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	65.8179	2.30550
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−34.6410	−1.21046
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−11.0000	−0.383436	−0.191718	−	0.981450i	$-0.561406\pi$
$823$	−11.0000	−0.383436	−0.191718	+	0.981450i	$0.561406\pi$
$824$	0	0
$825$	−24.2487	−0.844232
$826$	0	0
$827$	48.4974	1.68642	0.843210	−	0.537584i	$-0.180663\pi$
$827$	48.4974	1.68642	0.843210	+	0.537584i	$0.180663\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	25.0000	0.867240
$832$	0	0
$833$	−17.3205	−0.600120
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.00000	−0.172825
$838$	0	0
$839$	17.3205	0.597970	0.298985	−	0.954258i	$-0.403352\pi$
$839$	17.3205	0.597970	0.298985	+	0.954258i	$0.403352\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	3.46410	0.119310
$844$	0	0
$845$	41.5692	1.43002
$846$	0	0
$847$	−3.46410	−0.119028
$848$	0	0
$849$	10.3923	0.356663
$850$	0	0
$851$	0	0
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−48.0000	−1.64157
$856$	0	0
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	−7.00000	−0.238837	−0.119418	−	0.992844i	$-0.538103\pi$
$859$	−7.00000	−0.238837	−0.119418	+	0.992844i	$0.538103\pi$
$860$	0	0
$861$	31.1769	1.06251
$862$	0	0
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	0	0
$865$	20.7846	0.706698
$866$	0	0
$867$	−5.00000	−0.169809
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−17.3205	−0.586883
$872$	0	0
$873$	34.6410	1.17242
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	−6.92820	−0.233682
$880$	0	0
$881$	3.46410	0.116709	0.0583543	−	0.998296i	$-0.481415\pi$
$881$	3.46410	0.116709	0.0583543	+	0.998296i	$0.481415\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−41.5692	−1.39733
$886$	0	0
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	17.3205	0.580911
$890$	0	0
$891$	−3.46410	−0.116052
$892$	0	0
$893$	−20.7846	−0.695530
$894$	0	0
$895$	51.9615	1.73688
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.00000	−0.100056
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	13.8564	0.460094	0.230047	−	0.973179i	$-0.426112\pi$
$907$	13.8564	0.460094	0.230047	+	0.973179i	$0.426112\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	−48.0000	−1.58683
$916$	0	0
$917$	51.9615	1.71592
$918$	0	0
$919$	51.9615	1.71405	0.857026	−	0.515273i	$-0.172309\pi$
$919$	51.9615	1.71405	0.857026	+	0.515273i	$0.172309\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	−45.0000	−1.48119
$924$	0	0
$925$	−24.2487	−0.797293
$926$	0	0
$927$	−6.92820	−0.227552
$928$	0	0
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	34.6410	1.13531
$932$	0	0
$933$	−15.0000	−0.491078
$934$	0	0
$935$	41.5692	1.35946
$936$	0	0
$937$	17.3205	0.565836	0.282918	−	0.959144i	$-0.408698\pi$
$937$	17.3205	0.565836	0.282918	+	0.959144i	$0.408698\pi$
$938$	0	0
$939$	−6.92820	−0.226093
$940$	0	0
$941$	38.1051	1.24219	0.621096	−	0.783735i	$-0.286688\pi$
$941$	38.1051	1.24219	0.621096	+	0.783735i	$0.286688\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	60.0000	1.95180
$946$	0	0
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	0	0
$949$	55.0000	1.78538
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	34.6410	1.12213	0.561066	−	0.827771i	$-0.310391\pi$
$953$	34.6410	1.12213	0.561066	+	0.827771i	$0.310391\pi$
$954$	0	0
$955$	−60.0000	−1.94155
$956$	0	0
$957$	10.3923	0.335936
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−27.7128	−0.893033
$964$	0	0
$965$	17.3205	0.557567
$966$	0	0
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	13.8564	0.444673	0.222337	−	0.974970i	$-0.428632\pi$
$971$	13.8564	0.444673	0.222337	+	0.974970i	$0.428632\pi$
$972$	0	0
$973$	45.0333	1.44370
$974$	0	0
$975$	−35.0000	−1.12090
$976$	0	0
$977$	−31.1769	−0.997438	−0.498719	−	0.866764i	$-0.666196\pi$
$977$	−31.1769	−0.997438	−0.498719	+	0.866764i	$0.666196\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.7128	0.883901	0.441951	−	0.897039i	$-0.354287\pi$
$983$	27.7128	0.883901	0.441951	+	0.897039i	$0.354287\pi$
$984$	0	0
$985$	−10.3923	−0.331126
$986$	0	0
$987$	10.3923	0.330791
$988$	0	0
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	19.0000	0.602947
$994$	0	0
$995$	−60.0000	−1.90213
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	17.3205	0.547997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2116.2.a.f.1.2	yes	2
4.3	odd	2		8464.2.a.v.1.2		2
23.22	odd	2	inner	2116.2.a.f.1.1	✓	2
92.91	even	2		8464.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2116.2.a.f.1.1	✓	2	23.22	odd	2	inner
2116.2.a.f.1.2	yes	2	1.1	even	1	trivial
8464.2.a.v.1.1		2	92.91	even	2
8464.2.a.v.1.2		2	4.3	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2116 = 2^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2116.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410 q^{5} -3.46410 q^{7} -2.00000 q^{9} -3.46410 q^{11} -5.00000 q^{13} +3.46410 q^{15} -3.46410 q^{17} +6.92820 q^{19} -3.46410 q^{21} +7.00000 q^{25} -5.00000 q^{27} -3.00000 q^{29} +1.00000 q^{31} -3.46410 q^{33} -12.0000 q^{35} -3.46410 q^{37} -5.00000 q^{39} -9.00000 q^{41} -6.92820 q^{45} -3.00000 q^{47} +5.00000 q^{49} -3.46410 q^{51} +6.92820 q^{53} -12.0000 q^{55} +6.92820 q^{57} -12.0000 q^{59} -13.8564 q^{61} +6.92820 q^{63} -17.3205 q^{65} +3.46410 q^{67} +9.00000 q^{71} -11.0000 q^{73} +7.00000 q^{75} +12.0000 q^{77} +6.92820 q^{79} +1.00000 q^{81} +10.3923 q^{83} -12.0000 q^{85} -3.00000 q^{87} +6.92820 q^{89} +17.3205 q^{91} +1.00000 q^{93} +24.0000 q^{95} -17.3205 q^{97} +6.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9} - 10 q^{13} + 14 q^{25} - 10 q^{27} - 6 q^{29} + 2 q^{31} - 24 q^{35} - 10 q^{39} - 18 q^{41} - 6 q^{47} + 10 q^{49} - 24 q^{55} - 24 q^{59} + 18 q^{71} - 22 q^{73} + 14 q^{75} + 24 q^{77}+ \cdots + 48 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 - 10 * q^13 + 14 * q^25 - 10 * q^27 - 6 * q^29 + 2 * q^31 - 24 * q^35 - 10 * q^39 - 18 * q^41 - 6 * q^47 + 10 * q^49 - 24 * q^55 - 24 * q^59 + 18 * q^71 - 22 * q^73 + 14 * q^75 + 24 * q^77 + 2 * q^81 - 24 * q^85 - 6 * q^87 + 2 * q^93 + 48 * q^95

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