LMFDB - Embedded newform 9900.2.a.bv.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9900,2,Mod(1,9900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9900.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,2,0,0,0,2,0,2,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.0518980011$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1980)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	9900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{7} +1.00000 q^{11} -4.19615 q^{13} -4.73205 q^{17} -7.46410 q^{19} +6.92820 q^{23} +6.92820 q^{29} -0.535898 q^{31} +0.535898 q^{37} -6.92820 q^{41} +12.1962 q^{43} -9.46410 q^{47} +0.464102 q^{49} +6.00000 q^{53} +0.928203 q^{59} -7.46410 q^{61} +4.00000 q^{67} -6.00000 q^{71} +9.66025 q^{73} +2.73205 q^{77} -7.46410 q^{79} -4.73205 q^{83} -12.9282 q^{89} -11.4641 q^{91} +4.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} + 2 q^{11} + 2 q^{13} - 6 q^{17} - 8 q^{19} - 8 q^{31} + 8 q^{37} + 14 q^{43} - 12 q^{47} - 6 q^{49} + 12 q^{53} - 12 q^{59} - 8 q^{61} + 8 q^{67} - 12 q^{71} + 2 q^{73} + 2 q^{77} - 8 q^{79}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 2 * q^11 + 2 * q^13 - 6 * q^17 - 8 * q^19 - 8 * q^31 + 8 * q^37 + 14 * q^43 - 12 * q^47 - 6 * q^49 + 12 * q^53 - 12 * q^59 - 8 * q^61 + 8 * q^67 - 12 * q^71 + 2 * q^73 + 2 * q^77 - 8 * q^79 - 6 * q^83 - 12 * q^89 - 16 * q^91 - 4 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.19615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$13$	−4.19615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.73205	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$17$	−4.73205	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$18$	0	0
$19$	−7.46410	−1.71238	−0.856191	−	0.516659i	$-0.827175\pi$
$19$	−7.46410	−1.71238	−0.856191	+	0.516659i	$0.827175\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.92820	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$30$	0	0
$31$	−0.535898	−0.0962502	−0.0481251	−	0.998841i	$-0.515325\pi$
$31$	−0.535898	−0.0962502	−0.0481251	+	0.998841i	$0.515325\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	12.1962	1.85990	0.929948	−	0.367691i	$-0.119852\pi$
$43$	12.1962	1.85990	0.929948	+	0.367691i	$0.119852\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.46410	−1.38048	−0.690241	−	0.723580i	$-0.742495\pi$
$47$	−9.46410	−1.38048	−0.690241	+	0.723580i	$0.742495\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.928203	0.120842	0.0604209	−	0.998173i	$-0.480756\pi$
$59$	0.928203	0.120842	0.0604209	+	0.998173i	$0.480756\pi$
$60$	0	0
$61$	−7.46410	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$61$	−7.46410	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	9.66025	1.13065	0.565324	−	0.824869i	$-0.308751\pi$
$73$	9.66025	1.13065	0.565324	+	0.824869i	$0.308751\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.73205	0.311346
$78$	0	0
$79$	−7.46410	−0.839777	−0.419889	−	0.907576i	$-0.637931\pi$
$79$	−7.46410	−0.839777	−0.419889	+	0.907576i	$0.637931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.73205	−0.519410	−0.259705	−	0.965688i	$-0.583625\pi$
$83$	−4.73205	−0.519410	−0.259705	+	0.965688i	$0.583625\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−11.4641	−1.20176
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.92820	0.500383	0.250192	−	0.968196i	$-0.419506\pi$
$97$	4.92820	0.500383	0.250192	+	0.968196i	$0.419506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.3923	−1.63110	−0.815548	−	0.578690i	$-0.803564\pi$
$101$	−16.3923	−1.63110	−0.815548	+	0.578690i	$0.803564\pi$
$102$	0	0
$103$	13.4641	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$103$	13.4641	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.80385	−0.947774	−0.473887	−	0.880586i	$-0.657149\pi$
$107$	−9.80385	−0.947774	−0.473887	+	0.880586i	$0.657149\pi$
$108$	0	0
$109$	−3.07180	−0.294225	−0.147112	−	0.989120i	$-0.546998\pi$
$109$	−3.07180	−0.294225	−0.147112	+	0.989120i	$0.546998\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.9282	−1.18513
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.80385	0.692479	0.346240	−	0.938146i	$-0.387458\pi$
$127$	7.80385	0.692479	0.346240	+	0.938146i	$0.387458\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.39230	0.383757	0.191879	−	0.981419i	$-0.438542\pi$
$131$	4.39230	0.383757	0.191879	+	0.981419i	$0.438542\pi$
$132$	0	0
$133$	−20.3923	−1.76824
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.8564	−1.69645	−0.848224	−	0.529638i	$-0.822328\pi$
$137$	−19.8564	−1.69645	−0.848224	+	0.529638i	$0.822328\pi$
$138$	0	0
$139$	22.7846	1.93257	0.966283	−	0.257484i	$-0.0828933\pi$
$139$	22.7846	1.93257	0.966283	+	0.257484i	$0.0828933\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.19615	−0.350900
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.46410	−0.775329	−0.387665	−	0.921800i	$-0.626718\pi$
$149$	−9.46410	−0.775329	−0.387665	+	0.921800i	$0.626718\pi$
$150$	0	0
$151$	6.39230	0.520198	0.260099	−	0.965582i	$-0.416245\pi$
$151$	6.39230	0.520198	0.260099	+	0.965582i	$0.416245\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.4641	−0.914935	−0.457467	−	0.889226i	$-0.651243\pi$
$157$	−11.4641	−0.914935	−0.457467	+	0.889226i	$0.651243\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.9282	1.49175
$162$	0	0
$163$	−0.392305	−0.0307277	−0.0153638	−	0.999882i	$-0.504891\pi$
$163$	−0.392305	−0.0307277	−0.0153638	+	0.999882i	$0.504891\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.6603	0.902298	0.451149	−	0.892449i	$-0.351014\pi$
$167$	11.6603	0.902298	0.451149	+	0.892449i	$0.351014\pi$
$168$	0	0
$169$	4.60770	0.354438
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.19615	0.166970	0.0834852	−	0.996509i	$-0.473395\pi$
$173$	2.19615	0.166970	0.0834852	+	0.996509i	$0.473395\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	−20.3923	−1.51575	−0.757874	−	0.652401i	$-0.773762\pi$
$181$	−20.3923	−1.51575	−0.757874	+	0.652401i	$0.773762\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.73205	−0.346042
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	0	0
$193$	−6.73205	−0.484584	−0.242292	−	0.970203i	$-0.577899\pi$
$193$	−6.73205	−0.484584	−0.242292	+	0.970203i	$0.577899\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.7321	−1.19211	−0.596055	−	0.802944i	$-0.703266\pi$
$197$	−16.7321	−1.19211	−0.596055	+	0.802944i	$0.703266\pi$
$198$	0	0
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.9282	1.32850
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.46410	−0.516303
$210$	0	0
$211$	−14.3923	−0.990807	−0.495404	−	0.868663i	$-0.664980\pi$
$211$	−14.3923	−0.990807	−0.495404	+	0.868663i	$0.664980\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.46410	−0.0993897
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.8564	1.33569
$222$	0	0
$223$	−5.46410	−0.365903	−0.182952	−	0.983122i	$-0.558565\pi$
$223$	−5.46410	−0.365903	−0.182952	+	0.983122i	$0.558565\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.7321	−1.11055	−0.555273	−	0.831668i	$-0.687386\pi$
$227$	−16.7321	−1.11055	−0.555273	+	0.831668i	$0.687386\pi$
$228$	0	0
$229$	−18.7846	−1.24132	−0.620661	−	0.784079i	$-0.713136\pi$
$229$	−18.7846	−1.24132	−0.620661	+	0.784079i	$0.713136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.2679	1.26229	0.631143	−	0.775667i	$-0.282586\pi$
$233$	19.2679	1.26229	0.631143	+	0.775667i	$0.282586\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.3205	1.50848	0.754239	−	0.656600i	$-0.228006\pi$
$239$	23.3205	1.50848	0.754239	+	0.656600i	$0.228006\pi$
$240$	0	0
$241$	30.3923	1.95774	0.978870	−	0.204482i	$-0.0655511\pi$
$241$	30.3923	1.95774	0.978870	+	0.204482i	$0.0655511\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	31.3205	1.99288
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8564	1.25333	0.626663	−	0.779291i	$-0.284420\pi$
$251$	19.8564	1.25333	0.626663	+	0.779291i	$0.284420\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.60770	−0.100285	−0.0501426	−	0.998742i	$-0.515968\pi$
$257$	−1.60770	−0.100285	−0.0501426	+	0.998742i	$0.515968\pi$
$258$	0	0
$259$	1.46410	0.0909748
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.12436	−0.562632	−0.281316	−	0.959615i	$-0.590771\pi$
$263$	−9.12436	−0.562632	−0.281316	+	0.959615i	$0.590771\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.0718	−1.04089	−0.520443	−	0.853896i	$-0.674233\pi$
$269$	−17.0718	−1.04089	−0.520443	+	0.853896i	$0.674233\pi$
$270$	0	0
$271$	−2.39230	−0.145322	−0.0726611	−	0.997357i	$-0.523149\pi$
$271$	−2.39230	−0.145322	−0.0726611	+	0.997357i	$0.523149\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.6603	−0.820765	−0.410383	−	0.911913i	$-0.634605\pi$
$277$	−13.6603	−0.820765	−0.410383	+	0.911913i	$0.634605\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	2.73205	0.162404	0.0812018	−	0.996698i	$-0.474124\pi$
$283$	2.73205	0.162404	0.0812018	+	0.996698i	$0.474124\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.9282	−1.11730
$288$	0	0
$289$	5.39230	0.317194
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.0526	0.937801	0.468900	−	0.883251i	$-0.344650\pi$
$293$	16.0526	0.937801	0.468900	+	0.883251i	$0.344650\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.0718	−1.68127
$300$	0	0
$301$	33.3205	1.92056
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.26795	−0.528950	−0.264475	−	0.964393i	$-0.585199\pi$
$307$	−9.26795	−0.528950	−0.264475	+	0.964393i	$0.585199\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.8564	−1.12595	−0.562977	−	0.826473i	$-0.690344\pi$
$311$	−19.8564	−1.12595	−0.562977	+	0.826473i	$0.690344\pi$
$312$	0	0
$313$	−9.60770	−0.543059	−0.271530	−	0.962430i	$-0.587529\pi$
$313$	−9.60770	−0.543059	−0.271530	+	0.962430i	$0.587529\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.3923	0.583690	0.291845	−	0.956466i	$-0.405731\pi$
$317$	10.3923	0.583690	0.291845	+	0.956466i	$0.405731\pi$
$318$	0	0
$319$	6.92820	0.387905
$320$	0	0
$321$	0	0
$322$	0	0
$323$	35.3205	1.96529
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−25.8564	−1.42551
$330$	0	0
$331$	25.3205	1.39174	0.695870	−	0.718167i	$-0.255019\pi$
$331$	25.3205	1.39174	0.695870	+	0.718167i	$0.255019\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.196152	0.0106851	0.00534255	−	0.999986i	$-0.498299\pi$
$337$	0.196152	0.0106851	0.00534255	+	0.999986i	$0.498299\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.535898	−0.0290205
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.58846	0.353687	0.176843	−	0.984239i	$-0.443411\pi$
$347$	6.58846	0.353687	0.176843	+	0.984239i	$0.443411\pi$
$348$	0	0
$349$	23.4641	1.25600	0.628002	−	0.778211i	$-0.283873\pi$
$349$	23.4641	1.25600	0.628002	+	0.778211i	$0.283873\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.39230	0.231817	0.115908	−	0.993260i	$-0.463022\pi$
$359$	4.39230	0.231817	0.115908	+	0.993260i	$0.463022\pi$
$360$	0	0
$361$	36.7128	1.93225
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.3923	−0.646873	−0.323437	−	0.946250i	$-0.604838\pi$
$367$	−12.3923	−0.646873	−0.323437	+	0.946250i	$0.604838\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.3923	0.851046
$372$	0	0
$373$	35.5167	1.83898	0.919491	−	0.393110i	$-0.128601\pi$
$373$	35.5167	1.83898	0.919491	+	0.393110i	$0.128601\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−29.0718	−1.49727
$378$	0	0
$379$	16.7846	0.862167	0.431084	−	0.902312i	$-0.358131\pi$
$379$	16.7846	0.862167	0.431084	+	0.902312i	$0.358131\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.39230	0.224436	0.112218	−	0.993684i	$-0.464204\pi$
$383$	4.39230	0.224436	0.112218	+	0.993684i	$0.464204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−32.7846	−1.65799
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.8564	0.595056	0.297528	−	0.954713i	$-0.403838\pi$
$397$	11.8564	0.595056	0.297528	+	0.954713i	$0.403838\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.0718	−1.15215	−0.576075	−	0.817397i	$-0.695416\pi$
$401$	−23.0718	−1.15215	−0.576075	+	0.817397i	$0.695416\pi$
$402$	0	0
$403$	2.24871	0.112016
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.535898	0.0265635
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.53590	0.124783
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.7846	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$419$	20.7846	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$420$	0	0
$421$	−29.1769	−1.42200	−0.710998	−	0.703194i	$-0.751756\pi$
$421$	−29.1769	−1.42200	−0.710998	+	0.703194i	$0.751756\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−20.3923	−0.986853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.07180	−0.244300	−0.122150	−	0.992512i	$-0.538979\pi$
$431$	−5.07180	−0.244300	−0.122150	+	0.992512i	$0.538979\pi$
$432$	0	0
$433$	−27.8564	−1.33869	−0.669347	−	0.742950i	$-0.733426\pi$
$433$	−27.8564	−1.33869	−0.669347	+	0.742950i	$0.733426\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−51.7128	−2.47376
$438$	0	0
$439$	8.92820	0.426120	0.213060	−	0.977039i	$-0.431657\pi$
$439$	8.92820	0.426120	0.213060	+	0.977039i	$0.431657\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.5359	−0.690621	−0.345311	−	0.938488i	$-0.612226\pi$
$443$	−14.5359	−0.690621	−0.345311	+	0.938488i	$0.612226\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−36.9282	−1.74275	−0.871375	−	0.490618i	$-0.836771\pi$
$449$	−36.9282	−1.74275	−0.871375	+	0.490618i	$0.836771\pi$
$450$	0	0
$451$	−6.92820	−0.326236
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.8038	−1.11350	−0.556749	−	0.830681i	$-0.687951\pi$
$457$	−23.8038	−1.11350	−0.556749	+	0.830681i	$0.687951\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.60770	0.354326	0.177163	−	0.984182i	$-0.443308\pi$
$461$	7.60770	0.354326	0.177163	+	0.984182i	$0.443308\pi$
$462$	0	0
$463$	−0.392305	−0.0182320	−0.00911598	−	0.999958i	$-0.502902\pi$
$463$	−0.392305	−0.0182320	−0.00911598	+	0.999958i	$0.502902\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.7128	1.83769	0.918845	−	0.394619i	$-0.129123\pi$
$467$	39.7128	1.83769	0.918845	+	0.394619i	$0.129123\pi$
$468$	0	0
$469$	10.9282	0.504618
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.1962	0.560780
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.53590	0.115868	0.0579341	−	0.998320i	$-0.481549\pi$
$479$	2.53590	0.115868	0.0579341	+	0.998320i	$0.481549\pi$
$480$	0	0
$481$	−2.24871	−0.102532
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.7128	−1.25066	−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128	−1.25066	−0.625331	+	0.780360i	$0.715036\pi$
$492$	0	0
$493$	−32.7846	−1.47654
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.3923	−0.735295
$498$	0	0
$499$	−17.8564	−0.799363	−0.399681	−	0.916654i	$-0.630879\pi$
$499$	−17.8564	−0.799363	−0.399681	+	0.916654i	$0.630879\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.7321	−1.28110	−0.640549	−	0.767917i	$-0.721293\pi$
$503$	−28.7321	−1.28110	−0.640549	+	0.767917i	$0.721293\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.7846	1.45315	0.726576	−	0.687086i	$-0.241110\pi$
$509$	32.7846	1.45315	0.726576	+	0.687086i	$0.241110\pi$
$510$	0	0
$511$	26.3923	1.16753
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.46410	−0.416231
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.9282	−0.566395	−0.283197	−	0.959062i	$-0.591395\pi$
$521$	−12.9282	−0.566395	−0.283197	+	0.959062i	$0.591395\pi$
$522$	0	0
$523$	−6.73205	−0.294372	−0.147186	−	0.989109i	$-0.547022\pi$
$523$	−6.73205	−0.294372	−0.147186	+	0.989109i	$0.547022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.53590	0.110465
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	29.0718	1.25924
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.464102	0.0199903
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.1962	0.521470	0.260735	−	0.965410i	$-0.416035\pi$
$547$	12.1962	0.521470	0.260735	+	0.965410i	$0.416035\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−51.7128	−2.20304
$552$	0	0
$553$	−20.3923	−0.867169
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.51666	0.0642630	0.0321315	−	0.999484i	$-0.489770\pi$
$557$	1.51666	0.0642630	0.0321315	+	0.999484i	$0.489770\pi$
$558$	0	0
$559$	−51.1769	−2.16455
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.19615	−0.0925568	−0.0462784	−	0.998929i	$-0.514736\pi$
$563$	−2.19615	−0.0925568	−0.0462784	+	0.998929i	$0.514736\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.3205	−0.474580	−0.237290	−	0.971439i	$-0.576259\pi$
$569$	−11.3205	−0.474580	−0.237290	+	0.971439i	$0.576259\pi$
$570$	0	0
$571$	3.85641	0.161386	0.0806928	−	0.996739i	$-0.474287\pi$
$571$	3.85641	0.161386	0.0806928	+	0.996739i	$0.474287\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.3205	−1.05411	−0.527053	−	0.849832i	$-0.676703\pi$
$577$	−25.3205	−1.05411	−0.527053	+	0.849832i	$0.676703\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.9282	−0.536352
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.6410	−1.42979	−0.714894	−	0.699233i	$-0.753525\pi$
$587$	−34.6410	−1.42979	−0.714894	+	0.699233i	$0.753525\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.6603	0.971610	0.485805	−	0.874067i	$-0.338526\pi$
$593$	23.6603	0.971610	0.485805	+	0.874067i	$0.338526\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−42.9282	−1.75400	−0.876999	−	0.480491i	$-0.840458\pi$
$599$	−42.9282	−1.75400	−0.876999	+	0.480491i	$0.840458\pi$
$600$	0	0
$601$	11.4641	0.467630	0.233815	−	0.972281i	$-0.424879\pi$
$601$	11.4641	0.467630	0.233815	+	0.972281i	$0.424879\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.9808	−1.01394	−0.506969	−	0.861964i	$-0.669234\pi$
$607$	−24.9808	−1.01394	−0.506969	+	0.861964i	$0.669234\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.7128	1.60661
$612$	0	0
$613$	17.2679	0.697446	0.348723	−	0.937226i	$-0.386615\pi$
$613$	17.2679	0.697446	0.348723	+	0.937226i	$0.386615\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.6077	−1.03093	−0.515463	−	0.856912i	$-0.672380\pi$
$617$	−25.6077	−1.03093	−0.515463	+	0.856912i	$0.672380\pi$
$618$	0	0
$619$	−26.3923	−1.06080	−0.530398	−	0.847749i	$-0.677958\pi$
$619$	−26.3923	−1.06080	−0.530398	+	0.847749i	$0.677958\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−35.3205	−1.41509
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.53590	−0.101113
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.94744	−0.0771604
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.7128	1.80555	0.902774	−	0.430116i	$-0.141527\pi$
$641$	45.7128	1.80555	0.902774	+	0.430116i	$0.141527\pi$
$642$	0	0
$643$	8.39230	0.330960	0.165480	−	0.986213i	$-0.447083\pi$
$643$	8.39230	0.330960	0.165480	+	0.986213i	$0.447083\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−42.2487	−1.66097	−0.830484	−	0.557042i	$-0.811936\pi$
$647$	−42.2487	−1.66097	−0.830484	+	0.557042i	$0.811936\pi$
$648$	0	0
$649$	0.928203	0.0364352
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.2487	−0.479329	−0.239665	−	0.970856i	$-0.577037\pi$
$653$	−12.2487	−0.479329	−0.239665	+	0.970856i	$0.577037\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.9282	−1.20479	−0.602396	−	0.798197i	$-0.705787\pi$
$659$	−30.9282	−1.20479	−0.602396	+	0.798197i	$0.705787\pi$
$660$	0	0
$661$	−3.07180	−0.119479	−0.0597395	−	0.998214i	$-0.519027\pi$
$661$	−3.07180	−0.119479	−0.0597395	+	0.998214i	$0.519027\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.46410	−0.288148
$672$	0	0
$673$	−2.33975	−0.0901906	−0.0450953	−	0.998983i	$-0.514359\pi$
$673$	−2.33975	−0.0901906	−0.0450953	+	0.998983i	$0.514359\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.7321	−1.10426	−0.552131	−	0.833757i	$-0.686185\pi$
$677$	−28.7321	−1.10426	−0.552131	+	0.833757i	$0.686185\pi$
$678$	0	0
$679$	13.4641	0.516705
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.3205	−0.433167	−0.216584	−	0.976264i	$-0.569491\pi$
$683$	−11.3205	−0.433167	−0.216584	+	0.976264i	$0.569491\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25.1769	−0.959164
$690$	0	0
$691$	−0.784610	−0.0298480	−0.0149240	−	0.999889i	$-0.504751\pi$
$691$	−0.784610	−0.0298480	−0.0149240	+	0.999889i	$0.504751\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	32.7846	1.24181
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.0718	1.09803	0.549013	−	0.835814i	$-0.315004\pi$
$701$	29.0718	1.09803	0.549013	+	0.835814i	$0.315004\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−44.7846	−1.68430
$708$	0	0
$709$	12.3923	0.465403	0.232701	−	0.972548i	$-0.425244\pi$
$709$	12.3923	0.465403	0.232701	+	0.972548i	$0.425244\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.71281	−0.139046
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.7846	−1.22266	−0.611330	−	0.791376i	$-0.709365\pi$
$719$	−32.7846	−1.22266	−0.611330	+	0.791376i	$0.709365\pi$
$720$	0	0
$721$	36.7846	1.36993
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.7846	0.919210	0.459605	−	0.888123i	$-0.347991\pi$
$727$	24.7846	0.919210	0.459605	+	0.888123i	$0.347991\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−57.7128	−2.13459
$732$	0	0
$733$	−4.19615	−0.154988	−0.0774942	−	0.996993i	$-0.524692\pi$
$733$	−4.19615	−0.154988	−0.0774942	+	0.996993i	$0.524692\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	5.21539	0.191851	0.0959256	−	0.995389i	$-0.469419\pi$
$739$	5.21539	0.191851	0.0959256	+	0.995389i	$0.469419\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.3397	−0.452701	−0.226351	−	0.974046i	$-0.572680\pi$
$743$	−12.3397	−0.452701	−0.226351	+	0.974046i	$0.572680\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−26.7846	−0.978688
$750$	0	0
$751$	11.7128	0.427407	0.213703	−	0.976899i	$-0.431447\pi$
$751$	11.7128	0.427407	0.213703	+	0.976899i	$0.431447\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	45.3205	1.64720	0.823601	−	0.567170i	$-0.191962\pi$
$757$	45.3205	1.64720	0.823601	+	0.567170i	$0.191962\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.24871	0.226516	0.113258	−	0.993566i	$-0.463871\pi$
$761$	6.24871	0.226516	0.113258	+	0.993566i	$0.463871\pi$
$762$	0	0
$763$	−8.39230	−0.303822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.89488	−0.140636
$768$	0	0
$769$	−38.3923	−1.38446	−0.692231	−	0.721676i	$-0.743372\pi$
$769$	−38.3923	−1.38446	−0.692231	+	0.721676i	$0.743372\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	50.1051	1.80216	0.901078	−	0.433657i	$-0.142777\pi$
$773$	50.1051	1.80216	0.901078	+	0.433657i	$0.142777\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	51.7128	1.85280
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.5885	−0.733899	−0.366950	−	0.930241i	$-0.619598\pi$
$787$	−20.5885	−0.733899	−0.366950	+	0.930241i	$0.619598\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−47.3205	−1.68252
$792$	0	0
$793$	31.3205	1.11222
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.4641	1.39789	0.698945	−	0.715175i	$-0.253653\pi$
$797$	39.4641	1.39789	0.698945	+	0.715175i	$0.253653\pi$
$798$	0	0
$799$	44.7846	1.58437
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.66025	0.340903
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.7846	1.15265	0.576323	−	0.817222i	$-0.304487\pi$
$809$	32.7846	1.15265	0.576323	+	0.817222i	$0.304487\pi$
$810$	0	0
$811$	−15.0718	−0.529242	−0.264621	−	0.964352i	$-0.585247\pi$
$811$	−15.0718	−0.529242	−0.264621	+	0.964352i	$0.585247\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−91.0333	−3.18485
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.7128	0.967184	0.483592	−	0.875294i	$-0.339332\pi$
$821$	27.7128	0.967184	0.483592	+	0.875294i	$0.339332\pi$
$822$	0	0
$823$	20.3923	0.710831	0.355416	−	0.934708i	$-0.384339\pi$
$823$	20.3923	0.710831	0.355416	+	0.934708i	$0.384339\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.94744	0.276360	0.138180	−	0.990407i	$-0.455875\pi$
$827$	7.94744	0.276360	0.138180	+	0.990407i	$0.455875\pi$
$828$	0	0
$829$	−6.53590	−0.227001	−0.113501	−	0.993538i	$-0.536206\pi$
$829$	−6.53590	−0.227001	−0.113501	+	0.993538i	$0.536206\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.19615	−0.0760922
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.7128	0.749610	0.374805	−	0.927104i	$-0.377710\pi$
$839$	21.7128	0.749610	0.374805	+	0.927104i	$0.377710\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.73205	0.0938744
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.71281	0.127274
$852$	0	0
$853$	7.80385	0.267199	0.133599	−	0.991035i	$-0.457346\pi$
$853$	7.80385	0.267199	0.133599	+	0.991035i	$0.457346\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.1962	−0.894844	−0.447422	−	0.894323i	$-0.647658\pi$
$857$	−26.1962	−0.894844	−0.447422	+	0.894323i	$0.647658\pi$
$858$	0	0
$859$	−36.5359	−1.24659	−0.623294	−	0.781987i	$-0.714206\pi$
$859$	−36.5359	−1.24659	−0.623294	+	0.781987i	$0.714206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.679492	0.0231302	0.0115651	−	0.999933i	$-0.496319\pi$
$863$	0.679492	0.0231302	0.0115651	+	0.999933i	$0.496319\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.46410	−0.253202
$870$	0	0
$871$	−16.7846	−0.568725
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.48334	0.286462	0.143231	−	0.989689i	$-0.454251\pi$
$877$	8.48334	0.286462	0.143231	+	0.989689i	$0.454251\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	46.9282	1.57926	0.789630	−	0.613583i	$-0.210273\pi$
$883$	46.9282	1.57926	0.789630	+	0.613583i	$0.210273\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−58.3013	−1.95757	−0.978783	−	0.204902i	$-0.934312\pi$
$887$	−58.3013	−1.95757	−0.978783	+	0.204902i	$0.934312\pi$
$888$	0	0
$889$	21.3205	0.715067
$890$	0	0
$891$	0	0
$892$	0	0
$893$	70.6410	2.36391
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.71281	−0.123829
$900$	0	0
$901$	−28.3923	−0.945885
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	48.1051	1.59730	0.798652	−	0.601793i	$-0.205547\pi$
$907$	48.1051	1.59730	0.798652	+	0.601793i	$0.205547\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.8564	−0.459083	−0.229542	−	0.973299i	$-0.573723\pi$
$911$	−13.8564	−0.459083	−0.229542	+	0.973299i	$0.573723\pi$
$912$	0	0
$913$	−4.73205	−0.156608
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	17.7128	0.584292	0.292146	−	0.956374i	$-0.405631\pi$
$919$	17.7128	0.584292	0.292146	+	0.956374i	$0.405631\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.1769	0.828708
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	−3.46410	−0.113531
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50.7321	1.65734	0.828672	−	0.559735i	$-0.189097\pi$
$937$	50.7321	1.65734	0.828672	+	0.559735i	$0.189097\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−51.0333	−1.66364	−0.831819	−	0.555046i	$-0.812700\pi$
$941$	−51.0333	−1.66364	−0.831819	+	0.555046i	$0.812700\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.3205	1.53771	0.768855	−	0.639423i	$-0.220827\pi$
$947$	47.3205	1.53771	0.768855	+	0.639423i	$0.220827\pi$
$948$	0	0
$949$	−40.5359	−1.31585
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.01924	0.0330164	0.0165082	−	0.999864i	$-0.494745\pi$
$953$	1.01924	0.0330164	0.0165082	+	0.999864i	$0.494745\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−54.2487	−1.75178
$960$	0	0
$961$	−30.7128	−0.990736
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.73205	0.0878568	0.0439284	−	0.999035i	$-0.486013\pi$
$967$	2.73205	0.0878568	0.0439284	+	0.999035i	$0.486013\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.0718	0.547860	0.273930	−	0.961750i	$-0.411676\pi$
$971$	17.0718	0.547860	0.273930	+	0.961750i	$0.411676\pi$
$972$	0	0
$973$	62.2487	1.99560
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.7846	0.856916	0.428458	−	0.903562i	$-0.359057\pi$
$977$	26.7846	0.856916	0.428458	+	0.903562i	$0.359057\pi$
$978$	0	0
$979$	−12.9282	−0.413187
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.2487	−1.34752	−0.673762	−	0.738948i	$-0.735323\pi$
$983$	−42.2487	−1.34752	−0.673762	+	0.738948i	$0.735323\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	84.4974	2.68686
$990$	0	0
$991$	39.1769	1.24450	0.622248	−	0.782820i	$-0.286220\pi$
$991$	39.1769	1.24450	0.622248	+	0.782820i	$0.286220\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	27.4115	0.868132	0.434066	−	0.900881i	$-0.357078\pi$
$997$	27.4115	0.868132	0.434066	+	0.900881i	$0.357078\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9900.2.a.bv.1.2		2
3.2	odd	2		9900.2.a.bs.1.2		2
5.2	odd	4		9900.2.c.z.5149.4		4
5.3	odd	4		9900.2.c.z.5149.1		4
5.4	even	2		1980.2.a.g.1.1	✓	2
15.2	even	4		9900.2.c.u.5149.4		4
15.8	even	4		9900.2.c.u.5149.1		4
15.14	odd	2		1980.2.a.k.1.1	yes	2
20.19	odd	2		7920.2.a.bv.1.2		2
60.59	even	2		7920.2.a.cd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1980.2.a.g.1.1	✓	2	5.4	even	2
1980.2.a.k.1.1	yes	2	15.14	odd	2
7920.2.a.bv.1.2		2	20.19	odd	2
7920.2.a.cd.1.2		2	60.59	even	2
9900.2.a.bs.1.2		2	3.2	odd	2
9900.2.a.bv.1.2		2	1.1	even	1	trivial
9900.2.c.u.5149.1		4	15.8	even	4
9900.2.c.u.5149.4		4	15.2	even	4
9900.2.c.z.5149.1		4	5.3	odd	4
9900.2.c.z.5149.4		4	5.2	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9900.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{7} +1.00000 q^{11} -4.19615 q^{13} -4.73205 q^{17} -7.46410 q^{19} +6.92820 q^{23} +6.92820 q^{29} -0.535898 q^{31} +0.535898 q^{37} -6.92820 q^{41} +12.1962 q^{43} -9.46410 q^{47} +0.464102 q^{49} +6.00000 q^{53} +0.928203 q^{59} -7.46410 q^{61} +4.00000 q^{67} -6.00000 q^{71} +9.66025 q^{73} +2.73205 q^{77} -7.46410 q^{79} -4.73205 q^{83} -12.9282 q^{89} -11.4641 q^{91} +4.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} + 2 q^{11} + 2 q^{13} - 6 q^{17} - 8 q^{19} - 8 q^{31} + 8 q^{37} + 14 q^{43} - 12 q^{47} - 6 q^{49} + 12 q^{53} - 12 q^{59} - 8 q^{61} + 8 q^{67} - 12 q^{71} + 2 q^{73} + 2 q^{77} - 8 q^{79}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 2 * q^11 + 2 * q^13 - 6 * q^17 - 8 * q^19 - 8 * q^31 + 8 * q^37 + 14 * q^43 - 12 * q^47 - 6 * q^49 + 12 * q^53 - 12 * q^59 - 8 * q^61 + 8 * q^67 - 12 * q^71 + 2 * q^73 + 2 * q^77 - 8 * q^79 - 6 * q^83 - 12 * q^89 - 16 * q^91 - 4 * q^97

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