LMFDB - Embedded newform 3276.2.a.s.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3276,2,Mod(1,3276)] 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("3276.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3276.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,-4,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.31342$ of defining polynomial
Character	$\chi$	$=$	3276.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31342 q^{5} -1.00000 q^{7} +2.62685 q^{11} -1.00000 q^{13} -6.92820 q^{17} -2.27492 q^{19} +1.31342 q^{23} -3.27492 q^{25} +8.24163 q^{29} +6.27492 q^{31} -1.31342 q^{35} +10.5498 q^{37} +6.92820 q^{41} +10.8248 q^{43} -1.31342 q^{47} +1.00000 q^{49} +10.8685 q^{53} +3.45017 q^{55} -9.55505 q^{59} -2.00000 q^{61} -1.31342 q^{65} -0.549834 q^{67} -5.25370 q^{71} -0.274917 q^{73} -2.62685 q^{77} +5.72508 q^{79} +3.94027 q^{83} -9.09967 q^{85} -17.7967 q^{89} +1.00000 q^{91} -2.98793 q^{95} +4.27492 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{13} + 6 q^{19} + 2 q^{25} + 10 q^{31} + 12 q^{37} - 2 q^{43} + 4 q^{49} + 44 q^{55} - 8 q^{61} + 28 q^{67} + 14 q^{73} + 38 q^{79} + 24 q^{85} + 4 q^{91} + 2 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^13 + 6 * q^19 + 2 * q^25 + 10 * q^31 + 12 * q^37 - 2 * q^43 + 4 * q^49 + 44 * q^55 - 8 * q^61 + 28 * q^67 + 14 * q^73 + 38 * q^79 + 24 * q^85 + 4 * q^91 + 2 * 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
$5$	1.31342	0.587381	0.293691	−	0.955901i	$-0.405116\pi$
$5$	1.31342	0.587381	0.293691	+	0.955901i	$0.405116\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.62685	0.792025	0.396012	−	0.918245i	$-0.370394\pi$
$11$	2.62685	0.792025	0.396012	+	0.918245i	$0.370394\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	−2.27492	−0.521902	−0.260951	−	0.965352i	$-0.584036\pi$
$19$	−2.27492	−0.521902	−0.260951	+	0.965352i	$0.584036\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.31342	0.273868	0.136934	−	0.990580i	$-0.456275\pi$
$23$	1.31342	0.273868	0.136934	+	0.990580i	$0.456275\pi$
$24$	0	0
$25$	−3.27492	−0.654983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.24163	1.53043	0.765216	−	0.643774i	$-0.222632\pi$
$29$	8.24163	1.53043	0.765216	+	0.643774i	$0.222632\pi$
$30$	0	0
$31$	6.27492	1.12701	0.563504	−	0.826113i	$-0.309453\pi$
$31$	6.27492	1.12701	0.563504	+	0.826113i	$0.309453\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.31342	−0.222009
$36$	0	0
$37$	10.5498	1.73438	0.867191	−	0.497976i	$-0.165923\pi$
$37$	10.5498	1.73438	0.867191	+	0.497976i	$0.165923\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	10.8248	1.65076	0.825380	−	0.564578i	$-0.190961\pi$
$43$	10.8248	1.65076	0.825380	+	0.564578i	$0.190961\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.31342	−0.191583	−0.0957913	−	0.995401i	$-0.530538\pi$
$47$	−1.31342	−0.191583	−0.0957913	+	0.995401i	$0.530538\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.8685	1.49290	0.746450	−	0.665442i	$-0.231757\pi$
$53$	10.8685	1.49290	0.746450	+	0.665442i	$0.231757\pi$
$54$	0	0
$55$	3.45017	0.465220
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.55505	−1.24396	−0.621981	−	0.783032i	$-0.713672\pi$
$59$	−9.55505	−1.24396	−0.621981	+	0.783032i	$0.713672\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.31342	−0.162910
$66$	0	0
$67$	−0.549834	−0.0671730	−0.0335865	−	0.999436i	$-0.510693\pi$
$67$	−0.549834	−0.0671730	−0.0335865	+	0.999436i	$0.510693\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.25370	−0.623499	−0.311750	−	0.950164i	$-0.600915\pi$
$71$	−5.25370	−0.623499	−0.311750	+	0.950164i	$0.600915\pi$
$72$	0	0
$73$	−0.274917	−0.0321766	−0.0160883	−	0.999871i	$-0.505121\pi$
$73$	−0.274917	−0.0321766	−0.0160883	+	0.999871i	$0.505121\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.62685	−0.299357
$78$	0	0
$79$	5.72508	0.644122	0.322061	−	0.946719i	$-0.395624\pi$
$79$	5.72508	0.644122	0.322061	+	0.946719i	$0.395624\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.94027	0.432501	0.216251	−	0.976338i	$-0.430617\pi$
$83$	3.94027	0.432501	0.216251	+	0.976338i	$0.430617\pi$
$84$	0	0
$85$	−9.09967	−0.986998
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.7967	−1.88644	−0.943222	−	0.332163i	$-0.892222\pi$
$89$	−17.7967	−1.88644	−0.943222	+	0.332163i	$0.892222\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.98793	−0.306555
$96$	0	0
$97$	4.27492	0.434052	0.217026	−	0.976166i	$-0.430364\pi$
$97$	4.27492	0.434052	0.217026	+	0.976166i	$0.430364\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.55505	0.950763	0.475382	−	0.879780i	$-0.342310\pi$
$101$	9.55505	0.950763	0.475382	+	0.879780i	$0.342310\pi$
$102$	0	0
$103$	13.0997	1.29075	0.645374	−	0.763866i	$-0.276701\pi$
$103$	13.0997	1.29075	0.645374	+	0.763866i	$0.276701\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.55505	−0.923722	−0.461861	−	0.886952i	$-0.652818\pi$
$107$	−9.55505	−0.923722	−0.461861	+	0.886952i	$0.652818\pi$
$108$	0	0
$109$	19.0997	1.82942	0.914708	−	0.404115i	$-0.132420\pi$
$109$	19.0997	1.82942	0.914708	+	0.404115i	$0.132420\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.4953	−1.26953	−0.634767	−	0.772704i	$-0.718904\pi$
$113$	−13.4953	−1.26953	−0.634767	+	0.772704i	$0.718904\pi$
$114$	0	0
$115$	1.72508	0.160865
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	−4.09967	−0.372697
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.8685	−0.972106
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.1101	1.66966	0.834829	−	0.550510i	$-0.185567\pi$
$131$	19.1101	1.66966	0.834829	+	0.550510i	$0.185567\pi$
$132$	0	0
$133$	2.27492	0.197260
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	20.5498	1.74301	0.871507	−	0.490383i	$-0.163143\pi$
$139$	20.5498	1.74301	0.871507	+	0.490383i	$0.163143\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.62685	−0.219668
$144$	0	0
$145$	10.8248	0.898947
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.88054	−0.645599	−0.322800	−	0.946467i	$-0.604624\pi$
$149$	−7.88054	−0.645599	−0.322800	+	0.946467i	$0.604624\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.24163	0.661983
$156$	0	0
$157$	−5.45017	−0.434971	−0.217485	−	0.976064i	$-0.569785\pi$
$157$	−5.45017	−0.434971	−0.217485	+	0.976064i	$0.569785\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.31342	−0.103512
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.94027	0.304907	0.152454	−	0.988311i	$-0.451283\pi$
$167$	3.94027	0.304907	0.152454	+	0.988311i	$0.451283\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.67451	−0.127310	−0.0636552	−	0.997972i	$-0.520276\pi$
$173$	−1.67451	−0.127310	−0.0636552	+	0.997972i	$0.520276\pi$
$174$	0	0
$175$	3.27492	0.247560
$176$	0	0
$177$	0	0
$178$	0	0
$179$	25.6772	1.91921	0.959603	−	0.281358i	$-0.0907848\pi$
$179$	25.6772	1.91921	0.959603	+	0.281358i	$0.0907848\pi$
$180$	0	0
$181$	−15.0997	−1.12235	−0.561175	−	0.827697i	$-0.689650\pi$
$181$	−15.0997	−1.12235	−0.561175	+	0.827697i	$0.689650\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	13.8564	1.01874
$186$	0	0
$187$	−18.1993	−1.33087
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.55505	0.691379	0.345690	−	0.938349i	$-0.387645\pi$
$191$	9.55505	0.691379	0.345690	+	0.938349i	$0.387645\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.88054	−0.561466	−0.280733	−	0.959786i	$-0.590577\pi$
$197$	−7.88054	−0.561466	−0.280733	+	0.959786i	$0.590577\pi$
$198$	0	0
$199$	−13.0997	−0.928611	−0.464305	−	0.885675i	$-0.653696\pi$
$199$	−13.0997	−0.928611	−0.464305	+	0.885675i	$0.653696\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.24163	−0.578449
$204$	0	0
$205$	9.09967	0.635548
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.97586	−0.413359
$210$	0	0
$211$	22.2749	1.53347	0.766734	−	0.641965i	$-0.221880\pi$
$211$	22.2749	1.53347	0.766734	+	0.641965i	$0.221880\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.2175	0.969625
$216$	0	0
$217$	−6.27492	−0.425969
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	23.3746	1.56528	0.782639	−	0.622476i	$-0.213873\pi$
$223$	23.3746	1.56528	0.782639	+	0.622476i	$0.213873\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.8087	−0.982891	−0.491446	−	0.870908i	$-0.663531\pi$
$227$	−14.8087	−0.982891	−0.491446	+	0.870908i	$0.663531\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.1222	−1.05620	−0.528099	−	0.849183i	$-0.677095\pi$
$233$	−16.1222	−1.05620	−0.528099	+	0.849183i	$0.677095\pi$
$234$	0	0
$235$	−1.72508	−0.112532
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.8564	−0.896296	−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564	−0.896296	−0.448148	+	0.893959i	$0.647916\pi$
$240$	0	0
$241$	0.824752	0.0531269	0.0265635	−	0.999647i	$-0.491544\pi$
$241$	0.824752	0.0531269	0.0265635	+	0.999647i	$0.491544\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.31342	0.0839116
$246$	0	0
$247$	2.27492	0.144750
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.7370	1.37202	0.686012	−	0.727590i	$-0.259360\pi$
$251$	21.7370	1.37202	0.686012	+	0.727590i	$0.259360\pi$
$252$	0	0
$253$	3.45017	0.216910
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.8087	−0.923744	−0.461872	−	0.886947i	$-0.652822\pi$
$257$	−14.8087	−0.923744	−0.461872	+	0.886947i	$0.652822\pi$
$258$	0	0
$259$	−10.5498	−0.655535
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−31.6531	−1.95181	−0.975906	−	0.218189i	$-0.929985\pi$
$263$	−31.6531	−1.95181	−0.975906	+	0.218189i	$0.929985\pi$
$264$	0	0
$265$	14.2749	0.876901
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.6893	1.38339	0.691695	−	0.722189i	$-0.256864\pi$
$269$	22.6893	1.38339	0.691695	+	0.722189i	$0.256864\pi$
$270$	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
$273$	0	0
$274$	0	0
$275$	−8.60271	−0.518763
$276$	0	0
$277$	−20.2749	−1.21820	−0.609101	−	0.793093i	$-0.708470\pi$
$277$	−20.2749	−1.21820	−0.609101	+	0.793093i	$0.708470\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.9906	−1.61013	−0.805064	−	0.593189i	$-0.797869\pi$
$281$	−26.9906	−1.61013	−0.805064	+	0.593189i	$0.797869\pi$
$282$	0	0
$283$	−17.0997	−1.01647	−0.508235	−	0.861218i	$-0.669702\pi$
$283$	−17.0997	−1.01647	−0.508235	+	0.861218i	$0.669702\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.31342	−0.0767311	−0.0383655	−	0.999264i	$-0.512215\pi$
$293$	−1.31342	−0.0767311	−0.0383655	+	0.999264i	$0.512215\pi$
$294$	0	0
$295$	−12.5498	−0.730680
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.31342	−0.0759573
$300$	0	0
$301$	−10.8248	−0.623928
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.62685	−0.150413
$306$	0	0
$307$	14.8248	0.846093	0.423047	−	0.906108i	$-0.360961\pi$
$307$	14.8248	0.846093	0.423047	+	0.906108i	$0.360961\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.97586	0.338860	0.169430	−	0.985542i	$-0.445807\pi$
$311$	5.97586	0.338860	0.169430	+	0.985542i	$0.445807\pi$
$312$	0	0
$313$	−31.0997	−1.75786	−0.878929	−	0.476953i	$-0.841741\pi$
$313$	−31.0997	−1.75786	−0.878929	+	0.476953i	$0.841741\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.8564	0.778253	0.389127	−	0.921184i	$-0.372777\pi$
$317$	13.8564	0.778253	0.389127	+	0.921184i	$0.372777\pi$
$318$	0	0
$319$	21.6495	1.21214
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15.7611	0.876970
$324$	0	0
$325$	3.27492	0.181660
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.31342	0.0724114
$330$	0	0
$331$	24.5498	1.34938	0.674690	−	0.738101i	$-0.264277\pi$
$331$	24.5498	1.34938	0.674690	+	0.738101i	$0.264277\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.722166	−0.0394561
$336$	0	0
$337$	17.9244	0.976405	0.488203	−	0.872730i	$-0.337653\pi$
$337$	17.9244	0.976405	0.488203	+	0.872730i	$0.337653\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.4833	0.892618
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.8087	−0.794975	−0.397488	−	0.917608i	$-0.630118\pi$
$347$	−14.8087	−0.794975	−0.397488	+	0.917608i	$0.630118\pi$
$348$	0	0
$349$	−25.3746	−1.35827	−0.679135	−	0.734013i	$-0.737645\pi$
$349$	−25.3746	−1.35827	−0.679135	+	0.734013i	$0.737645\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	31.2920	1.66551	0.832753	−	0.553645i	$-0.186764\pi$
$353$	31.2920	1.66551	0.832753	+	0.553645i	$0.186764\pi$
$354$	0	0
$355$	−6.90033	−0.366232
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.7128	1.46263	0.731313	−	0.682042i	$-0.238908\pi$
$359$	27.7128	1.46263	0.731313	+	0.682042i	$0.238908\pi$
$360$	0	0
$361$	−13.8248	−0.727619
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.361083	−0.0188999
$366$	0	0
$367$	0.549834	0.0287011	0.0143506	−	0.999897i	$-0.495432\pi$
$367$	0.549834	0.0287011	0.0143506	+	0.999897i	$0.495432\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.8685	−0.564263
$372$	0	0
$373$	−19.0997	−0.988943	−0.494472	−	0.869194i	$-0.664638\pi$
$373$	−19.0997	−0.988943	−0.494472	+	0.869194i	$0.664638\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.24163	−0.424465
$378$	0	0
$379$	−0.549834	−0.0282431	−0.0141215	−	0.999900i	$-0.504495\pi$
$379$	−0.549834	−0.0282431	−0.0141215	+	0.999900i	$0.504495\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.8087	−0.756692	−0.378346	−	0.925664i	$-0.623507\pi$
$383$	−14.8087	−0.756692	−0.378346	+	0.925664i	$0.623507\pi$
$384$	0	0
$385$	−3.45017	−0.175837
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−27.7128	−1.40510	−0.702548	−	0.711637i	$-0.747954\pi$
$389$	−27.7128	−1.40510	−0.702548	+	0.711637i	$0.747954\pi$
$390$	0	0
$391$	−9.09967	−0.460190
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.51946	0.378345
$396$	0	0
$397$	−19.7251	−0.989974	−0.494987	−	0.868901i	$-0.664827\pi$
$397$	−19.7251	−0.989974	−0.494987	+	0.868901i	$0.664827\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.7128	1.38391	0.691956	−	0.721940i	$-0.256749\pi$
$401$	27.7128	1.38391	0.691956	+	0.721940i	$0.256749\pi$
$402$	0	0
$403$	−6.27492	−0.312576
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.7128	1.37367
$408$	0	0
$409$	21.3746	1.05691	0.528453	−	0.848963i	$-0.322772\pi$
$409$	21.3746	1.05691	0.528453	+	0.848963i	$0.322772\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.55505	0.470173
$414$	0	0
$415$	5.17525	0.254043
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.7370	1.06192	0.530960	−	0.847397i	$-0.321832\pi$
$419$	21.7370	1.06192	0.530960	+	0.847397i	$0.321832\pi$
$420$	0	0
$421$	14.5498	0.709116	0.354558	−	0.935034i	$-0.384631\pi$
$421$	14.5498	0.709116	0.354558	+	0.935034i	$0.384631\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	22.6893	1.10059
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.5934	−1.71447	−0.857236	−	0.514924i	$-0.827820\pi$
$431$	−35.5934	−1.71447	−0.857236	+	0.514924i	$0.827820\pi$
$432$	0	0
$433$	23.6495	1.13652	0.568261	−	0.822848i	$-0.307616\pi$
$433$	23.6495	1.13652	0.568261	+	0.822848i	$0.307616\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.98793	−0.142932
$438$	0	0
$439$	0.549834	0.0262422	0.0131211	−	0.999914i	$-0.495823\pi$
$439$	0.549834	0.0262422	0.0131211	+	0.999914i	$0.495823\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.91613	−0.471130	−0.235565	−	0.971859i	$-0.575694\pi$
$443$	−9.91613	−0.471130	−0.235565	+	0.971859i	$0.575694\pi$
$444$	0	0
$445$	−23.3746	−1.10806
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.3638	1.14980	0.574899	−	0.818224i	$-0.305041\pi$
$449$	24.3638	1.14980	0.574899	+	0.818224i	$0.305041\pi$
$450$	0	0
$451$	18.1993	0.856973
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.31342	0.0615743
$456$	0	0
$457$	−15.6495	−0.732053	−0.366026	−	0.930604i	$-0.619282\pi$
$457$	−15.6495	−0.732053	−0.366026	+	0.930604i	$0.619282\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.6893	1.05675	0.528373	−	0.849012i	$-0.322802\pi$
$461$	22.6893	1.05675	0.528373	+	0.849012i	$0.322802\pi$
$462$	0	0
$463$	−21.6495	−1.00614	−0.503069	−	0.864246i	$-0.667796\pi$
$463$	−21.6495	−1.00614	−0.503069	+	0.864246i	$0.667796\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.2443	−1.49209	−0.746045	−	0.665895i	$-0.768050\pi$
$467$	−32.2443	−1.49209	−0.746045	+	0.665895i	$0.768050\pi$
$468$	0	0
$469$	0.549834	0.0253890
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.4350	1.30744
$474$	0	0
$475$	7.45017	0.341837
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.94027	−0.180036	−0.0900178	−	0.995940i	$-0.528692\pi$
$479$	−3.94027	−0.180036	−0.0900178	+	0.995940i	$0.528692\pi$
$480$	0	0
$481$	−10.5498	−0.481031
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.61478	0.254954
$486$	0	0
$487$	−29.6495	−1.34355	−0.671774	−	0.740757i	$-0.734467\pi$
$487$	−29.6495	−1.34355	−0.671774	+	0.740757i	$0.734467\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.1578	−0.819448	−0.409724	−	0.912209i	$-0.634375\pi$
$491$	−18.1578	−0.819448	−0.409724	+	0.912209i	$0.634375\pi$
$492$	0	0
$493$	−57.0997	−2.57164
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.25370	0.235660
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.62685	0.117125	0.0585627	−	0.998284i	$-0.481348\pi$
$503$	2.62685	0.117125	0.0585627	+	0.998284i	$0.481348\pi$
$504$	0	0
$505$	12.5498	0.558460
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−42.8826	−1.90074	−0.950370	−	0.311122i	$-0.899295\pi$
$509$	−42.8826	−1.90074	−0.950370	+	0.311122i	$0.899295\pi$
$510$	0	0
$511$	0.274917	0.0121616
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.2054	0.758161
$516$	0	0
$517$	−3.45017	−0.151738
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−34.6410	−1.51765	−0.758825	−	0.651294i	$-0.774226\pi$
$521$	−34.6410	−1.51765	−0.758825	+	0.651294i	$0.774226\pi$
$522$	0	0
$523$	37.6495	1.64630	0.823149	−	0.567826i	$-0.192215\pi$
$523$	37.6495	1.64630	0.823149	+	0.567826i	$0.192215\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.4739	−1.89375
$528$	0	0
$529$	−21.2749	−0.924996
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.92820	−0.300094
$534$	0	0
$535$	−12.5498	−0.542577
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.62685	0.113146
$540$	0	0
$541$	15.6495	0.672825	0.336412	−	0.941715i	$-0.390786\pi$
$541$	15.6495	0.672825	0.336412	+	0.941715i	$0.390786\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.0860	1.07456
$546$	0	0
$547$	−13.1752	−0.563333	−0.281666	−	0.959512i	$-0.590887\pi$
$547$	−13.1752	−0.563333	−0.281666	+	0.959512i	$0.590887\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−18.7490	−0.798735
$552$	0	0
$553$	−5.72508	−0.243455
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.6175	1.25493	0.627467	−	0.778643i	$-0.284092\pi$
$557$	29.6175	1.25493	0.627467	+	0.778643i	$0.284092\pi$
$558$	0	0
$559$	−10.8248	−0.457838
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.34901	0.141144	0.0705721	−	0.997507i	$-0.477518\pi$
$563$	3.34901	0.141144	0.0705721	+	0.997507i	$0.477518\pi$
$564$	0	0
$565$	−17.7251	−0.745700
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.1222	−0.675877	−0.337938	−	0.941168i	$-0.609730\pi$
$569$	−16.1222	−0.675877	−0.337938	+	0.941168i	$0.609730\pi$
$570$	0	0
$571$	−5.17525	−0.216577	−0.108289	−	0.994119i	$-0.534537\pi$
$571$	−5.17525	−0.216577	−0.108289	+	0.994119i	$0.534537\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.30136	−0.179379
$576$	0	0
$577$	−24.1993	−1.00743	−0.503716	−	0.863869i	$-0.668034\pi$
$577$	−24.1993	−1.00743	−0.503716	+	0.863869i	$0.668034\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.94027	−0.163470
$582$	0	0
$583$	28.5498	1.18241
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.4477	−0.596319	−0.298159	−	0.954516i	$-0.596373\pi$
$587$	−14.4477	−0.596319	−0.298159	+	0.954516i	$0.596373\pi$
$588$	0	0
$589$	−14.2749	−0.588188
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.28929	0.299335	0.149668	−	0.988736i	$-0.452180\pi$
$593$	7.28929	0.299335	0.149668	+	0.988736i	$0.452180\pi$
$594$	0	0
$595$	9.09967	0.373050
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.3994	1.07865	0.539325	−	0.842098i	$-0.318679\pi$
$599$	26.3994	1.07865	0.539325	+	0.842098i	$0.318679\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.38460	−0.218915
$606$	0	0
$607$	−25.6495	−1.04108	−0.520541	−	0.853837i	$-0.674270\pi$
$607$	−25.6495	−1.04108	−0.520541	+	0.853837i	$0.674270\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.31342	0.0531355
$612$	0	0
$613$	−5.45017	−0.220130	−0.110065	−	0.993924i	$-0.535106\pi$
$613$	−5.45017	−0.220130	−0.110065	+	0.993924i	$0.535106\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.722166	0.0290733	0.0145366	−	0.999894i	$-0.495373\pi$
$617$	0.722166	0.0290733	0.0145366	+	0.999894i	$0.495373\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.7967	0.713009
$624$	0	0
$625$	2.09967	0.0839868
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−73.0914	−2.91434
$630$	0	0
$631$	−12.5498	−0.499601	−0.249801	−	0.968297i	$-0.580365\pi$
$631$	−12.5498	−0.499601	−0.249801	+	0.968297i	$0.580365\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.25370	0.208487
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.8350	−1.73138	−0.865689	−	0.500582i	$-0.833119\pi$
$641$	−43.8350	−1.73138	−0.865689	+	0.500582i	$0.833119\pi$
$642$	0	0
$643$	10.9003	0.429867	0.214934	−	0.976629i	$-0.431047\pi$
$643$	10.9003	0.429867	0.214934	+	0.976629i	$0.431047\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.1342	−0.516360	−0.258180	−	0.966097i	$-0.583123\pi$
$647$	−13.1342	−0.516360	−0.258180	+	0.966097i	$0.583123\pi$
$648$	0	0
$649$	−25.0997	−0.985248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.7128	−1.08449	−0.542243	−	0.840222i	$-0.682425\pi$
$653$	−27.7128	−1.08449	−0.542243	+	0.840222i	$0.682425\pi$
$654$	0	0
$655$	25.0997	0.980725
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.03559	0.0792953	0.0396477	−	0.999214i	$-0.487376\pi$
$659$	2.03559	0.0792953	0.0396477	+	0.999214i	$0.487376\pi$
$660$	0	0
$661$	−24.2749	−0.944185	−0.472092	−	0.881549i	$-0.656501\pi$
$661$	−24.2749	−0.944185	−0.472092	+	0.881549i	$0.656501\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.98793	0.115867
$666$	0	0
$667$	10.8248	0.419136
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.25370	−0.202817
$672$	0	0
$673$	−17.9244	−0.690936	−0.345468	−	0.938431i	$-0.612280\pi$
$673$	−17.9244	−0.690936	−0.345468	+	0.938431i	$0.612280\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.5216	1.63424	0.817118	−	0.576470i	$-0.195570\pi$
$677$	42.5216	1.63424	0.817118	+	0.576470i	$0.195570\pi$
$678$	0	0
$679$	−4.27492	−0.164056
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.4591	0.859374	0.429687	−	0.902978i	$-0.358624\pi$
$683$	22.4591	0.859374	0.429687	+	0.902978i	$0.358624\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.8685	−0.414056
$690$	0	0
$691$	−29.7251	−1.13080	−0.565398	−	0.824818i	$-0.691277\pi$
$691$	−29.7251	−1.13080	−0.565398	+	0.824818i	$0.691277\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26.9906	1.02381
$696$	0	0
$697$	−48.0000	−1.81813
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.7732	0.482436	0.241218	−	0.970471i	$-0.422453\pi$
$701$	12.7732	0.482436	0.241218	+	0.970471i	$0.422453\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.55505	−0.359355
$708$	0	0
$709$	11.0997	0.416857	0.208428	−	0.978038i	$-0.433165\pi$
$709$	11.0997	0.416857	0.208428	+	0.978038i	$0.433165\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.24163	0.308651
$714$	0	0
$715$	−3.45017	−0.129029
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.62685	−0.0979649	−0.0489825	−	0.998800i	$-0.515598\pi$
$719$	−2.62685	−0.0979649	−0.0489825	+	0.998800i	$0.515598\pi$
$720$	0	0
$721$	−13.0997	−0.487857
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−26.9906	−1.00241
$726$	0	0
$727$	−15.4502	−0.573015	−0.286507	−	0.958078i	$-0.592494\pi$
$727$	−15.4502	−0.573015	−0.286507	+	0.958078i	$0.592494\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−74.9961	−2.77383
$732$	0	0
$733$	5.92442	0.218823	0.109412	−	0.993997i	$-0.465103\pi$
$733$	5.92442	0.218823	0.109412	+	0.993997i	$0.465103\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.44433	−0.0532026
$738$	0	0
$739$	7.45017	0.274059	0.137029	−	0.990567i	$-0.456245\pi$
$739$	7.45017	0.274059	0.137029	+	0.990567i	$0.456245\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.2202	1.40216	0.701082	−	0.713081i	$-0.252701\pi$
$743$	38.2202	1.40216	0.701082	+	0.713081i	$0.252701\pi$
$744$	0	0
$745$	−10.3505	−0.379213
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.55505	0.349134
$750$	0	0
$751$	12.6254	0.460708	0.230354	−	0.973107i	$-0.426012\pi$
$751$	12.6254	0.460708	0.230354	+	0.973107i	$0.426012\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	45.9244	1.66915	0.834576	−	0.550893i	$-0.185713\pi$
$757$	45.9244	1.66915	0.834576	+	0.550893i	$0.185713\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.6772	0.930799	0.465399	−	0.885101i	$-0.345911\pi$
$761$	25.6772	0.930799	0.465399	+	0.885101i	$0.345911\pi$
$762$	0	0
$763$	−19.0997	−0.691454
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.55505	0.345013
$768$	0	0
$769$	8.27492	0.298401	0.149201	−	0.988807i	$-0.452330\pi$
$769$	8.27492	0.298401	0.149201	+	0.988807i	$0.452330\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.1819	0.438152	0.219076	−	0.975708i	$-0.429696\pi$
$773$	12.1819	0.438152	0.219076	+	0.975708i	$0.429696\pi$
$774$	0	0
$775$	−20.5498	−0.738172
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15.7611	−0.564700
$780$	0	0
$781$	−13.8007	−0.493827
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.15838	−0.255493
$786$	0	0
$787$	−36.4743	−1.30017	−0.650083	−	0.759863i	$-0.725266\pi$
$787$	−36.4743	−1.30017	−0.650083	+	0.759863i	$0.725266\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.4953	0.479838
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.7605	0.947905	0.473952	−	0.880551i	$-0.342827\pi$
$797$	26.7605	0.947905	0.473952	+	0.880551i	$0.342827\pi$
$798$	0	0
$799$	9.09967	0.321923
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.722166	−0.0254847
$804$	0	0
$805$	−1.72508	−0.0608012
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.7490	0.659180	0.329590	−	0.944124i	$-0.393089\pi$
$809$	18.7490	0.659180	0.329590	+	0.944124i	$0.393089\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.2685	0.920145
$816$	0	0
$817$	−24.6254	−0.861534
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.1342	0.458388	0.229194	−	0.973381i	$-0.426391\pi$
$821$	13.1342	0.458388	0.229194	+	0.973381i	$0.426391\pi$
$822$	0	0
$823$	−21.0997	−0.735488	−0.367744	−	0.929927i	$-0.619870\pi$
$823$	−21.0997	−0.735488	−0.367744	+	0.929927i	$0.619870\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.6175	1.02990	0.514951	−	0.857220i	$-0.327810\pi$
$827$	29.6175	1.02990	0.514951	+	0.857220i	$0.327810\pi$
$828$	0	0
$829$	−23.0997	−0.802285	−0.401142	−	0.916016i	$-0.631387\pi$
$829$	−23.0997	−0.802285	−0.401142	+	0.916016i	$0.631387\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.92820	−0.240048
$834$	0	0
$835$	5.17525	0.179097
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.1725	−1.35239	−0.676193	−	0.736724i	$-0.736372\pi$
$839$	−39.1725	−1.35239	−0.676193	+	0.736724i	$0.736372\pi$
$840$	0	0
$841$	38.9244	1.34222
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.31342	0.0451832
$846$	0	0
$847$	4.09967	0.140866
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13.8564	0.474991
$852$	0	0
$853$	−49.3746	−1.69055	−0.845277	−	0.534329i	$-0.820564\pi$
$853$	−49.3746	−1.69055	−0.845277	+	0.534329i	$0.820564\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.4115	0.799720	0.399860	−	0.916576i	$-0.369059\pi$
$857$	23.4115	0.799720	0.399860	+	0.916576i	$0.369059\pi$
$858$	0	0
$859$	−20.5498	−0.701151	−0.350576	−	0.936534i	$-0.614014\pi$
$859$	−20.5498	−0.701151	−0.350576	+	0.936534i	$0.614014\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.5074	0.357676	0.178838	−	0.983879i	$-0.442766\pi$
$863$	10.5074	0.357676	0.178838	+	0.983879i	$0.442766\pi$
$864$	0	0
$865$	−2.19934	−0.0747797
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.0389	0.510161
$870$	0	0
$871$	0.549834	0.0186304
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.8685	0.367422
$876$	0	0
$877$	25.4502	0.859391	0.429696	−	0.902974i	$-0.358621\pi$
$877$	25.4502	0.859391	0.429696	+	0.902974i	$0.358621\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.952341	−0.0320852	−0.0160426	−	0.999871i	$-0.505107\pi$
$881$	−0.952341	−0.0320852	−0.0160426	+	0.999871i	$0.505107\pi$
$882$	0	0
$883$	−17.0997	−0.575450	−0.287725	−	0.957713i	$-0.592899\pi$
$883$	−17.0997	−0.575450	−0.287725	+	0.957713i	$0.592899\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.3638	0.818056	0.409028	−	0.912522i	$-0.365868\pi$
$887$	24.3638	0.818056	0.409028	+	0.912522i	$0.365868\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.98793	0.0999873
$894$	0	0
$895$	33.7251	1.12731
$896$	0	0
$897$	0	0
$898$	0	0
$899$	51.7155	1.72481
$900$	0	0
$901$	−75.2990	−2.50857
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.8323	−0.659247
$906$	0	0
$907$	−0.625414	−0.0207665	−0.0103833	−	0.999946i	$-0.503305\pi$
$907$	−0.625414	−0.0207665	−0.0103833	+	0.999946i	$0.503305\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.94027	−0.130547	−0.0652735	−	0.997867i	$-0.520792\pi$
$911$	−3.94027	−0.130547	−0.0652735	+	0.997867i	$0.520792\pi$
$912$	0	0
$913$	10.3505	0.342551
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.1101	−0.631071
$918$	0	0
$919$	45.0997	1.48770	0.743850	−	0.668346i	$-0.232997\pi$
$919$	45.0997	1.48770	0.743850	+	0.668346i	$0.232997\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.25370	0.172928
$924$	0	0
$925$	−34.5498	−1.13599
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.4383	1.35955	0.679773	−	0.733422i	$-0.262078\pi$
$929$	41.4383	1.35955	0.679773	+	0.733422i	$0.262078\pi$
$930$	0	0
$931$	−2.27492	−0.0745574
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−23.9034	−0.781726
$936$	0	0
$937$	−25.4502	−0.831421	−0.415710	−	0.909497i	$-0.636467\pi$
$937$	−25.4502	−0.831421	−0.415710	+	0.909497i	$0.636467\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.1457	−0.689330	−0.344665	−	0.938726i	$-0.612007\pi$
$941$	−21.1457	−0.689330	−0.344665	+	0.938726i	$0.612007\pi$
$942$	0	0
$943$	9.09967	0.296326
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.7370	−0.706356	−0.353178	−	0.935556i	$-0.614899\pi$
$947$	−21.7370	−0.706356	−0.353178	+	0.935556i	$0.614899\pi$
$948$	0	0
$949$	0.274917	0.00892419
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.9934	−1.65184	−0.825919	−	0.563789i	$-0.809343\pi$
$953$	−50.9934	−1.65184	−0.825919	+	0.563789i	$0.809343\pi$
$954$	0	0
$955$	12.5498	0.406103
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.37459	0.270148
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.62685	0.0845612
$966$	0	0
$967$	21.6495	0.696201	0.348101	−	0.937457i	$-0.386827\pi$
$967$	21.6495	0.696201	0.348101	+	0.937457i	$0.386827\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−49.4498	−1.58692	−0.793459	−	0.608623i	$-0.791722\pi$
$971$	−49.4498	−1.58692	−0.793459	+	0.608623i	$0.791722\pi$
$972$	0	0
$973$	−20.5498	−0.658797
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.4498	−1.58204	−0.791019	−	0.611792i	$-0.790449\pi$
$977$	−49.4498	−1.58204	−0.791019	+	0.611792i	$0.790449\pi$
$978$	0	0
$979$	−46.7492	−1.49411
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.0410	−1.59606	−0.798030	−	0.602617i	$-0.794125\pi$
$983$	−50.0410	−1.59606	−0.798030	+	0.602617i	$0.794125\pi$
$984$	0	0
$985$	−10.3505	−0.329794
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.2175	0.452090
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.2054	−0.545448
$996$	0	0
$997$	−29.4502	−0.932696	−0.466348	−	0.884601i	$-0.654431\pi$
$997$	−29.4502	−0.932696	−0.466348	+	0.884601i	$0.654431\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3276.2.a.s.1.3	yes	4
3.2	odd	2	inner	3276.2.a.s.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3276.2.a.s.1.2	✓	4	3.2	odd	2	inner
3276.2.a.s.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3276.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31342 q^{5} -1.00000 q^{7} +2.62685 q^{11} -1.00000 q^{13} -6.92820 q^{17} -2.27492 q^{19} +1.31342 q^{23} -3.27492 q^{25} +8.24163 q^{29} +6.27492 q^{31} -1.31342 q^{35} +10.5498 q^{37} +6.92820 q^{41} +10.8248 q^{43} -1.31342 q^{47} +1.00000 q^{49} +10.8685 q^{53} +3.45017 q^{55} -9.55505 q^{59} -2.00000 q^{61} -1.31342 q^{65} -0.549834 q^{67} -5.25370 q^{71} -0.274917 q^{73} -2.62685 q^{77} +5.72508 q^{79} +3.94027 q^{83} -9.09967 q^{85} -17.7967 q^{89} +1.00000 q^{91} -2.98793 q^{95} +4.27492 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{13} + 6 q^{19} + 2 q^{25} + 10 q^{31} + 12 q^{37} - 2 q^{43} + 4 q^{49} + 44 q^{55} - 8 q^{61} + 28 q^{67} + 14 q^{73} + 38 q^{79} + 24 q^{85} + 4 q^{91} + 2 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^13 + 6 * q^19 + 2 * q^25 + 10 * q^31 + 12 * q^37 - 2 * q^43 + 4 * q^49 + 44 * q^55 - 8 * q^61 + 28 * q^67 + 14 * q^73 + 38 * q^79 + 24 * q^85 + 4 * q^91 + 2 * q^97

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