LMFDB - Embedded newform 648.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,2,Mod(1,648)] 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("648.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228 q^{5} +4.37228 q^{7} +1.00000 q^{11} -0.372281 q^{13} -5.37228 q^{17} +0.627719 q^{19} -0.372281 q^{23} +0.627719 q^{25} -4.37228 q^{29} +6.37228 q^{31} +10.3723 q^{35} +8.74456 q^{37} -11.7446 q^{41} -1.74456 q^{43} +4.37228 q^{47} +12.1168 q^{49} +0.744563 q^{53} +2.37228 q^{55} +7.00000 q^{59} +2.37228 q^{61} -0.883156 q^{65} -3.74456 q^{67} -4.00000 q^{71} -12.1168 q^{73} +4.37228 q^{77} -6.37228 q^{79} +9.62772 q^{83} -12.7446 q^{85} +6.00000 q^{89} -1.62772 q^{91} +1.48913 q^{95} +1.74456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} - 5 q^{17} + 7 q^{19} + 5 q^{23} + 7 q^{25} - 3 q^{29} + 7 q^{31} + 15 q^{35} + 6 q^{37} - 12 q^{41} + 8 q^{43} + 3 q^{47} + 7 q^{49} - 10 q^{53} - q^{55} + 14 q^{59}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - q^5 + 3 * q^7 + 2 * q^11 + 5 * q^13 - 5 * q^17 + 7 * q^19 + 5 * q^23 + 7 * q^25 - 3 * q^29 + 7 * q^31 + 15 * q^35 + 6 * q^37 - 12 * q^41 + 8 * q^43 + 3 * q^47 + 7 * q^49 - 10 * q^53 - q^55 + 14 * q^59 - q^61 - 19 * q^65 + 4 * q^67 - 8 * q^71 - 7 * q^73 + 3 * q^77 - 7 * q^79 + 25 * q^83 - 14 * q^85 + 12 * q^89 - 9 * q^91 - 20 * q^95 - 8 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.37228	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$5$	2.37228	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$6$	0	0
$7$	4.37228	1.65257	0.826284	−	0.563254i	$-0.190451\pi$
$7$	4.37228	1.65257	0.826284	+	0.563254i	$0.190451\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−0.372281	−0.103252	−0.0516261	−	0.998666i	$-0.516440\pi$
$13$	−0.372281	−0.103252	−0.0516261	+	0.998666i	$0.516440\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.37228	−1.30297	−0.651485	−	0.758662i	$-0.725854\pi$
$17$	−5.37228	−1.30297	−0.651485	+	0.758662i	$0.725854\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.372281	−0.0776260	−0.0388130	−	0.999246i	$-0.512358\pi$
$23$	−0.372281	−0.0776260	−0.0388130	+	0.999246i	$0.512358\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.37228	−0.811912	−0.405956	−	0.913893i	$-0.633061\pi$
$29$	−4.37228	−0.811912	−0.405956	+	0.913893i	$0.633061\pi$
$30$	0	0
$31$	6.37228	1.14450	0.572248	−	0.820081i	$-0.306072\pi$
$31$	6.37228	1.14450	0.572248	+	0.820081i	$0.306072\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.3723	1.75324
$36$	0	0
$37$	8.74456	1.43760	0.718799	−	0.695218i	$-0.244692\pi$
$37$	8.74456	1.43760	0.718799	+	0.695218i	$0.244692\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.7446	−1.83419	−0.917096	−	0.398666i	$-0.869473\pi$
$41$	−11.7446	−1.83419	−0.917096	+	0.398666i	$0.869473\pi$
$42$	0	0
$43$	−1.74456	−0.266043	−0.133022	−	0.991113i	$-0.542468\pi$
$43$	−1.74456	−0.266043	−0.133022	+	0.991113i	$0.542468\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.37228	0.637763	0.318881	−	0.947795i	$-0.396693\pi$
$47$	4.37228	0.637763	0.318881	+	0.947795i	$0.396693\pi$
$48$	0	0
$49$	12.1168	1.73098
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.744563	0.102274	0.0511368	−	0.998692i	$-0.483716\pi$
$53$	0.744563	0.102274	0.0511368	+	0.998692i	$0.483716\pi$
$54$	0	0
$55$	2.37228	0.319878
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	2.37228	0.303739	0.151870	−	0.988401i	$-0.451471\pi$
$61$	2.37228	0.303739	0.151870	+	0.988401i	$0.451471\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.883156	−0.109542
$66$	0	0
$67$	−3.74456	−0.457471	−0.228736	−	0.973489i	$-0.573459\pi$
$67$	−3.74456	−0.457471	−0.228736	+	0.973489i	$0.573459\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−12.1168	−1.41817	−0.709085	−	0.705123i	$-0.750892\pi$
$73$	−12.1168	−1.41817	−0.709085	+	0.705123i	$0.750892\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.37228	0.498268
$78$	0	0
$79$	−6.37228	−0.716938	−0.358469	−	0.933542i	$-0.616701\pi$
$79$	−6.37228	−0.716938	−0.358469	+	0.933542i	$0.616701\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.62772	1.05678	0.528390	−	0.849002i	$-0.322796\pi$
$83$	9.62772	1.05678	0.528390	+	0.849002i	$0.322796\pi$
$84$	0	0
$85$	−12.7446	−1.38234
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.62772	−0.170631
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.48913	0.152781
$96$	0	0
$97$	1.74456	0.177133	0.0885667	−	0.996070i	$-0.471771\pi$
$97$	1.74456	0.177133	0.0885667	+	0.996070i	$0.471771\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.11684	−0.708152	−0.354076	−	0.935217i	$-0.615205\pi$
$101$	−7.11684	−0.708152	−0.354076	+	0.935217i	$0.615205\pi$
$102$	0	0
$103$	−12.3723	−1.21908	−0.609539	−	0.792756i	$-0.708645\pi$
$103$	−12.3723	−1.21908	−0.609539	+	0.792756i	$0.708645\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8614	−1.24336	−0.621680	−	0.783272i	$-0.713549\pi$
$107$	−12.8614	−1.24336	−0.621680	+	0.783272i	$0.713549\pi$
$108$	0	0
$109$	−4.74456	−0.454447	−0.227223	−	0.973843i	$-0.572965\pi$
$109$	−4.74456	−0.454447	−0.227223	+	0.973843i	$0.572965\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.37228	−0.411310	−0.205655	−	0.978625i	$-0.565932\pi$
$113$	−4.37228	−0.411310	−0.205655	+	0.978625i	$0.565932\pi$
$114$	0	0
$115$	−0.883156	−0.0823547
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−23.4891	−2.15324
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.3723	−0.927725
$126$	0	0
$127$	6.74456	0.598483	0.299242	−	0.954177i	$-0.403266\pi$
$127$	6.74456	0.598483	0.299242	+	0.954177i	$0.403266\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.3723	1.60519	0.802597	−	0.596522i	$-0.203451\pi$
$131$	18.3723	1.60519	0.802597	+	0.596522i	$0.203451\pi$
$132$	0	0
$133$	2.74456	0.237984
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.7446	1.51602	0.758010	−	0.652243i	$-0.226172\pi$
$137$	17.7446	1.51602	0.758010	+	0.652243i	$0.226172\pi$
$138$	0	0
$139$	5.74456	0.487247	0.243624	−	0.969870i	$-0.421664\pi$
$139$	5.74456	0.487247	0.243624	+	0.969870i	$0.421664\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.372281	−0.0311317
$144$	0	0
$145$	−10.3723	−0.861371
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.37228	−0.685884	−0.342942	−	0.939357i	$-0.611423\pi$
$149$	−8.37228	−0.685884	−0.342942	+	0.939357i	$0.611423\pi$
$150$	0	0
$151$	0.372281	0.0302958	0.0151479	−	0.999885i	$-0.495178\pi$
$151$	0.372281	0.0302958	0.0151479	+	0.999885i	$0.495178\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.1168	1.21421
$156$	0	0
$157$	−3.11684	−0.248751	−0.124376	−	0.992235i	$-0.539693\pi$
$157$	−3.11684	−0.248751	−0.124376	+	0.992235i	$0.539693\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.62772	−0.128282
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.1168	−1.47931	−0.739653	−	0.672989i	$-0.765010\pi$
$167$	−19.1168	−1.47931	−0.739653	+	0.672989i	$0.765010\pi$
$168$	0	0
$169$	−12.8614	−0.989339
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.3723	−1.24476	−0.622381	−	0.782715i	$-0.713834\pi$
$173$	−16.3723	−1.24476	−0.622381	+	0.782715i	$0.713834\pi$
$174$	0	0
$175$	2.74456	0.207469
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.9783	1.71748	0.858738	−	0.512416i	$-0.171249\pi$
$179$	22.9783	1.71748	0.858738	+	0.512416i	$0.171249\pi$
$180$	0	0
$181$	−0.510875	−0.0379730	−0.0189865	−	0.999820i	$-0.506044\pi$
$181$	−0.510875	−0.0379730	−0.0189865	+	0.999820i	$0.506044\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.7446	1.52517
$186$	0	0
$187$	−5.37228	−0.392860
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.86141	0.713546	0.356773	−	0.934191i	$-0.383877\pi$
$191$	9.86141	0.713546	0.356773	+	0.934191i	$0.383877\pi$
$192$	0	0
$193$	1.74456	0.125576	0.0627882	−	0.998027i	$-0.480001\pi$
$193$	1.74456	0.125576	0.0627882	+	0.998027i	$0.480001\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.2554	1.08690	0.543452	−	0.839440i	$-0.317117\pi$
$197$	15.2554	1.08690	0.543452	+	0.839440i	$0.317117\pi$
$198$	0	0
$199$	−16.2337	−1.15078	−0.575388	−	0.817881i	$-0.695149\pi$
$199$	−16.2337	−1.15078	−0.575388	+	0.817881i	$0.695149\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−19.1168	−1.34174
$204$	0	0
$205$	−27.8614	−1.94593
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.627719	0.0434202
$210$	0	0
$211$	−9.62772	−0.662799	−0.331400	−	0.943490i	$-0.607521\pi$
$211$	−9.62772	−0.662799	−0.331400	+	0.943490i	$0.607521\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.13859	−0.282250
$216$	0	0
$217$	27.8614	1.89136
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	−14.6060	−0.978088	−0.489044	−	0.872259i	$-0.662654\pi$
$223$	−14.6060	−0.978088	−0.489044	+	0.872259i	$0.662654\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	0	0
$229$	22.6060	1.49384	0.746922	−	0.664911i	$-0.231531\pi$
$229$	22.6060	1.49384	0.746922	+	0.664911i	$0.231531\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.37228	0.351950	0.175975	−	0.984395i	$-0.443692\pi$
$233$	5.37228	0.351950	0.175975	+	0.984395i	$0.443692\pi$
$234$	0	0
$235$	10.3723	0.676613
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.8614	−0.896620	−0.448310	−	0.893878i	$-0.647974\pi$
$239$	−13.8614	−0.896620	−0.448310	+	0.893878i	$0.647974\pi$
$240$	0	0
$241$	5.74456	0.370040	0.185020	−	0.982735i	$-0.440765\pi$
$241$	5.74456	0.370040	0.185020	+	0.982735i	$0.440765\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	28.7446	1.83642
$246$	0	0
$247$	−0.233688	−0.0148692
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.88316	0.623819	0.311910	−	0.950112i	$-0.399031\pi$
$251$	9.88316	0.623819	0.311910	+	0.950112i	$0.399031\pi$
$252$	0	0
$253$	−0.372281	−0.0234051
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.48913	0.280024	0.140012	−	0.990150i	$-0.455286\pi$
$257$	4.48913	0.280024	0.140012	+	0.990150i	$0.455286\pi$
$258$	0	0
$259$	38.2337	2.37573
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.1168	−1.30212	−0.651060	−	0.759026i	$-0.725675\pi$
$263$	−21.1168	−1.30212	−0.651060	+	0.759026i	$0.725675\pi$
$264$	0	0
$265$	1.76631	0.108504
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.7446	−0.777050	−0.388525	−	0.921438i	$-0.627015\pi$
$269$	−12.7446	−0.777050	−0.388525	+	0.921438i	$0.627015\pi$
$270$	0	0
$271$	−21.4891	−1.30537	−0.652686	−	0.757629i	$-0.726358\pi$
$271$	−21.4891	−1.30537	−0.652686	+	0.757629i	$0.726358\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.627719	0.0378529
$276$	0	0
$277$	29.1168	1.74946	0.874731	−	0.484609i	$-0.161038\pi$
$277$	29.1168	1.74946	0.874731	+	0.484609i	$0.161038\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.8614	−0.826902	−0.413451	−	0.910526i	$-0.635677\pi$
$281$	−13.8614	−0.826902	−0.413451	+	0.910526i	$0.635677\pi$
$282$	0	0
$283$	23.8614	1.41841	0.709207	−	0.705001i	$-0.249053\pi$
$283$	23.8614	1.41841	0.709207	+	0.705001i	$0.249053\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−51.3505	−3.03113
$288$	0	0
$289$	11.8614	0.697730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.6277	1.14666	0.573332	−	0.819323i	$-0.305651\pi$
$293$	19.6277	1.14666	0.573332	+	0.819323i	$0.305651\pi$
$294$	0	0
$295$	16.6060	0.966837
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.138593	0.00801506
$300$	0	0
$301$	−7.62772	−0.439654
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.62772	0.322242
$306$	0	0
$307$	31.3723	1.79051	0.895255	−	0.445553i	$-0.146993\pi$
$307$	31.3723	1.79051	0.895255	+	0.445553i	$0.146993\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.8832	−0.617127	−0.308564	−	0.951204i	$-0.599848\pi$
$311$	−10.8832	−0.617127	−0.308564	+	0.951204i	$0.599848\pi$
$312$	0	0
$313$	11.2337	0.634966	0.317483	−	0.948264i	$-0.397162\pi$
$313$	11.2337	0.634966	0.317483	+	0.948264i	$0.397162\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.6060	1.71900	0.859501	−	0.511134i	$-0.170774\pi$
$317$	30.6060	1.71900	0.859501	+	0.511134i	$0.170774\pi$
$318$	0	0
$319$	−4.37228	−0.244801
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.37228	−0.187639
$324$	0	0
$325$	−0.233688	−0.0129627
$326$	0	0
$327$	0	0
$328$	0	0
$329$	19.1168	1.05395
$330$	0	0
$331$	−31.8614	−1.75126	−0.875631	−	0.482981i	$-0.839554\pi$
$331$	−31.8614	−1.75126	−0.875631	+	0.482981i	$0.839554\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.88316	−0.485339
$336$	0	0
$337$	−19.9783	−1.08828	−0.544142	−	0.838993i	$-0.683145\pi$
$337$	−19.9783	−1.08828	−0.544142	+	0.838993i	$0.683145\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.37228	0.345078
$342$	0	0
$343$	22.3723	1.20799
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.7228	1.54192	0.770961	−	0.636883i	$-0.219776\pi$
$347$	28.7228	1.54192	0.770961	+	0.636883i	$0.219776\pi$
$348$	0	0
$349$	16.8832	0.903735	0.451867	−	0.892085i	$-0.350758\pi$
$349$	16.8832	0.903735	0.451867	+	0.892085i	$0.350758\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.9783	−1.27623	−0.638117	−	0.769940i	$-0.720286\pi$
$353$	−23.9783	−1.27623	−0.638117	+	0.769940i	$0.720286\pi$
$354$	0	0
$355$	−9.48913	−0.503630
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.2337	1.06789	0.533947	−	0.845518i	$-0.320708\pi$
$359$	20.2337	1.06789	0.533947	+	0.845518i	$0.320708\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−28.7446	−1.50456
$366$	0	0
$367$	17.6277	0.920159	0.460080	−	0.887878i	$-0.347821\pi$
$367$	17.6277	0.920159	0.460080	+	0.887878i	$0.347821\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.25544	0.169014
$372$	0	0
$373$	21.1168	1.09339	0.546694	−	0.837332i	$-0.315886\pi$
$373$	21.1168	1.09339	0.546694	+	0.837332i	$0.315886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.62772	0.0838318
$378$	0	0
$379$	−5.88316	−0.302197	−0.151099	−	0.988519i	$-0.548281\pi$
$379$	−5.88316	−0.302197	−0.151099	+	0.988519i	$0.548281\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.3505	1.09096	0.545481	−	0.838123i	$-0.316347\pi$
$383$	21.3505	1.09096	0.545481	+	0.838123i	$0.316347\pi$
$384$	0	0
$385$	10.3723	0.528620
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.6060	0.537744	0.268872	−	0.963176i	$-0.413349\pi$
$389$	10.6060	0.537744	0.268872	+	0.963176i	$0.413349\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.1168	−0.760611
$396$	0	0
$397$	−18.2337	−0.915123	−0.457561	−	0.889178i	$-0.651277\pi$
$397$	−18.2337	−0.915123	−0.457561	+	0.889178i	$0.651277\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.2337	−0.860609	−0.430305	−	0.902684i	$-0.641594\pi$
$401$	−17.2337	−0.860609	−0.430305	+	0.902684i	$0.641594\pi$
$402$	0	0
$403$	−2.37228	−0.118172
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.74456	0.433452
$408$	0	0
$409$	5.74456	0.284050	0.142025	−	0.989863i	$-0.454639\pi$
$409$	5.74456	0.284050	0.142025	+	0.989863i	$0.454639\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	30.6060	1.50602
$414$	0	0
$415$	22.8397	1.12115
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.6060	−0.615842	−0.307921	−	0.951412i	$-0.599633\pi$
$419$	−12.6060	−0.615842	−0.307921	+	0.951412i	$0.599633\pi$
$420$	0	0
$421$	−35.1168	−1.71149	−0.855745	−	0.517398i	$-0.826901\pi$
$421$	−35.1168	−1.71149	−0.855745	+	0.517398i	$0.826901\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.37228	−0.163580
$426$	0	0
$427$	10.3723	0.501950
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.25544	−0.0604723	−0.0302361	−	0.999543i	$-0.509626\pi$
$431$	−1.25544	−0.0604723	−0.0302361	+	0.999543i	$0.509626\pi$
$432$	0	0
$433$	32.1168	1.54344	0.771719	−	0.635964i	$-0.219397\pi$
$433$	32.1168	1.54344	0.771719	+	0.635964i	$0.219397\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.233688	−0.0111788
$438$	0	0
$439$	7.11684	0.339668	0.169834	−	0.985473i	$-0.445677\pi$
$439$	7.11684	0.339668	0.169834	+	0.985473i	$0.445677\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.7228	1.45968	0.729842	−	0.683615i	$-0.239593\pi$
$443$	30.7228	1.45968	0.729842	+	0.683615i	$0.239593\pi$
$444$	0	0
$445$	14.2337	0.674742
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.8832	−0.749572	−0.374786	−	0.927111i	$-0.622284\pi$
$449$	−15.8832	−0.749572	−0.374786	+	0.927111i	$0.622284\pi$
$450$	0	0
$451$	−11.7446	−0.553030
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.86141	−0.181026
$456$	0	0
$457$	−3.74456	−0.175163	−0.0875816	−	0.996157i	$-0.527914\pi$
$457$	−3.74456	−0.175163	−0.0875816	+	0.996157i	$0.527914\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.88316	−0.320581	−0.160290	−	0.987070i	$-0.551243\pi$
$461$	−6.88316	−0.320581	−0.160290	+	0.987070i	$0.551243\pi$
$462$	0	0
$463$	41.3505	1.92172	0.960861	−	0.277031i	$-0.0893503\pi$
$463$	41.3505	1.92172	0.960861	+	0.277031i	$0.0893503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.37228	0.341148	0.170574	−	0.985345i	$-0.445438\pi$
$467$	7.37228	0.341148	0.170574	+	0.985345i	$0.445438\pi$
$468$	0	0
$469$	−16.3723	−0.756002
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.74456	−0.0802151
$474$	0	0
$475$	0.394031	0.0180794
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.6060	−0.484599	−0.242300	−	0.970201i	$-0.577902\pi$
$479$	−10.6060	−0.484599	−0.242300	+	0.970201i	$0.577902\pi$
$480$	0	0
$481$	−3.25544	−0.148435
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.13859	0.187924
$486$	0	0
$487$	6.74456	0.305625	0.152813	−	0.988255i	$-0.451167\pi$
$487$	6.74456	0.305625	0.152813	+	0.988255i	$0.451167\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.255437	0.0115277	0.00576386	−	0.999983i	$-0.498165\pi$
$491$	0.255437	0.0115277	0.00576386	+	0.999983i	$0.498165\pi$
$492$	0	0
$493$	23.4891	1.05790
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.4891	−0.784494
$498$	0	0
$499$	−19.9783	−0.894349	−0.447175	−	0.894447i	$-0.647570\pi$
$499$	−19.9783	−0.894349	−0.447175	+	0.894447i	$0.647570\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.51087	0.290306	0.145153	−	0.989409i	$-0.453633\pi$
$503$	6.51087	0.290306	0.145153	+	0.989409i	$0.453633\pi$
$504$	0	0
$505$	−16.8832	−0.751291
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.1168	−1.37923	−0.689615	−	0.724176i	$-0.742220\pi$
$509$	−31.1168	−1.37923	−0.689615	+	0.724176i	$0.742220\pi$
$510$	0	0
$511$	−52.9783	−2.34362
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−29.3505	−1.29334
$516$	0	0
$517$	4.37228	0.192293
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.1168	0.530849	0.265424	−	0.964132i	$-0.414488\pi$
$521$	12.1168	0.530849	0.265424	+	0.964132i	$0.414488\pi$
$522$	0	0
$523$	−13.4891	−0.589838	−0.294919	−	0.955522i	$-0.595293\pi$
$523$	−13.4891	−0.589838	−0.294919	+	0.955522i	$0.595293\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.2337	−1.49124
$528$	0	0
$529$	−22.8614	−0.993974
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.37228	0.189385
$534$	0	0
$535$	−30.5109	−1.31910
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.1168	0.521909
$540$	0	0
$541$	2.23369	0.0960337	0.0480169	−	0.998847i	$-0.484710\pi$
$541$	2.23369	0.0960337	0.0480169	+	0.998847i	$0.484710\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.2554	−0.482130
$546$	0	0
$547$	−16.2554	−0.695032	−0.347516	−	0.937674i	$-0.612975\pi$
$547$	−16.2554	−0.695032	−0.347516	+	0.937674i	$0.612975\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.74456	−0.116922
$552$	0	0
$553$	−27.8614	−1.18479
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	0.649468	0.0274696
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.74456	0.410684	0.205342	−	0.978690i	$-0.434169\pi$
$563$	9.74456	0.410684	0.205342	+	0.978690i	$0.434169\pi$
$564$	0	0
$565$	−10.3723	−0.436365
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.2337	1.14170	0.570848	−	0.821056i	$-0.306615\pi$
$569$	27.2337	1.14170	0.570848	+	0.821056i	$0.306615\pi$
$570$	0	0
$571$	−26.4891	−1.10854	−0.554268	−	0.832338i	$-0.687002\pi$
$571$	−26.4891	−1.10854	−0.554268	+	0.832338i	$0.687002\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.233688	−0.00974546
$576$	0	0
$577$	14.8614	0.618688	0.309344	−	0.950950i	$-0.399890\pi$
$577$	14.8614	0.618688	0.309344	+	0.950950i	$0.399890\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	42.0951	1.74640
$582$	0	0
$583$	0.744563	0.0308366
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.23369	0.381115	0.190558	−	0.981676i	$-0.438970\pi$
$587$	9.23369	0.381115	0.190558	+	0.981676i	$0.438970\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	−55.7228	−2.28441
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.8614	1.62869	0.814346	−	0.580379i	$-0.197096\pi$
$599$	39.8614	1.62869	0.814346	+	0.580379i	$0.197096\pi$
$600$	0	0
$601$	5.97825	0.243858	0.121929	−	0.992539i	$-0.461092\pi$
$601$	5.97825	0.243858	0.121929	+	0.992539i	$0.461092\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23.7228	−0.964470
$606$	0	0
$607$	−11.1168	−0.451219	−0.225609	−	0.974218i	$-0.572437\pi$
$607$	−11.1168	−0.451219	−0.225609	+	0.974218i	$0.572437\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.62772	−0.0658504
$612$	0	0
$613$	−12.7446	−0.514748	−0.257374	−	0.966312i	$-0.582857\pi$
$613$	−12.7446	−0.514748	−0.257374	+	0.966312i	$0.582857\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.9783	1.52895	0.764473	−	0.644655i	$-0.222999\pi$
$617$	37.9783	1.52895	0.764473	+	0.644655i	$0.222999\pi$
$618$	0	0
$619$	21.2337	0.853454	0.426727	−	0.904380i	$-0.359667\pi$
$619$	21.2337	0.853454	0.426727	+	0.904380i	$0.359667\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	26.2337	1.05103
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−46.9783	−1.87315
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	−4.51087	−0.178727
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.23369	0.127723	0.0638615	−	0.997959i	$-0.479658\pi$
$641$	3.23369	0.127723	0.0638615	+	0.997959i	$0.479658\pi$
$642$	0	0
$643$	3.00000	0.118308	0.0591542	−	0.998249i	$-0.481160\pi$
$643$	3.00000	0.118308	0.0591542	+	0.998249i	$0.481160\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.7228	1.32578	0.662890	−	0.748717i	$-0.269330\pi$
$647$	33.7228	1.32578	0.662890	+	0.748717i	$0.269330\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.8832	−0.895487	−0.447744	−	0.894162i	$-0.647772\pi$
$653$	−22.8832	−0.895487	−0.447744	+	0.894162i	$0.647772\pi$
$654$	0	0
$655$	43.5842	1.70298
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19.1168	0.744687	0.372343	−	0.928095i	$-0.378554\pi$
$659$	19.1168	0.744687	0.372343	+	0.928095i	$0.378554\pi$
$660$	0	0
$661$	−46.0951	−1.79289	−0.896446	−	0.443154i	$-0.853860\pi$
$661$	−46.0951	−1.79289	−0.896446	+	0.443154i	$0.853860\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.51087	0.252481
$666$	0	0
$667$	1.62772	0.0630255
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.37228	0.0915809
$672$	0	0
$673$	−0.372281	−0.0143504	−0.00717520	−	0.999974i	$-0.502284\pi$
$673$	−0.372281	−0.0143504	−0.00717520	+	0.999974i	$0.502284\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20.6060	−0.791952	−0.395976	−	0.918261i	$-0.629594\pi$
$677$	−20.6060	−0.791952	−0.395976	+	0.918261i	$0.629594\pi$
$678$	0	0
$679$	7.62772	0.292725
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.3723	−0.588204	−0.294102	−	0.955774i	$-0.595021\pi$
$683$	−15.3723	−0.588204	−0.294102	+	0.955774i	$0.595021\pi$
$684$	0	0
$685$	42.0951	1.60837
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.277187	−0.0105600
$690$	0	0
$691$	15.1168	0.575072	0.287536	−	0.957770i	$-0.407164\pi$
$691$	15.1168	0.575072	0.287536	+	0.957770i	$0.407164\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.6277	0.516929
$696$	0	0
$697$	63.0951	2.38990
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−35.4891	−1.34041	−0.670203	−	0.742178i	$-0.733793\pi$
$701$	−35.4891	−1.34041	−0.670203	+	0.742178i	$0.733793\pi$
$702$	0	0
$703$	5.48913	0.207026
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−31.1168	−1.17027
$708$	0	0
$709$	−28.6060	−1.07432	−0.537160	−	0.843480i	$-0.680503\pi$
$709$	−28.6060	−1.07432	−0.537160	+	0.843480i	$0.680503\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.37228	−0.0888426
$714$	0	0
$715$	−0.883156	−0.0330282
$716$	0	0
$717$	0	0
$718$	0	0
$719$	45.4891	1.69646	0.848229	−	0.529630i	$-0.177669\pi$
$719$	45.4891	1.69646	0.848229	+	0.529630i	$0.177669\pi$
$720$	0	0
$721$	−54.0951	−2.01461
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	10.8832	0.403634	0.201817	−	0.979423i	$-0.435315\pi$
$727$	10.8832	0.403634	0.201817	+	0.979423i	$0.435315\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.37228	0.346646
$732$	0	0
$733$	−37.8614	−1.39844	−0.699221	−	0.714905i	$-0.746470\pi$
$733$	−37.8614	−1.39844	−0.699221	+	0.714905i	$0.746470\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.74456	−0.137933
$738$	0	0
$739$	42.1168	1.54929	0.774647	−	0.632394i	$-0.217928\pi$
$739$	42.1168	1.54929	0.774647	+	0.632394i	$0.217928\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.6277	1.30705	0.653527	−	0.756903i	$-0.273289\pi$
$743$	35.6277	1.30705	0.653527	+	0.756903i	$0.273289\pi$
$744$	0	0
$745$	−19.8614	−0.727666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−56.2337	−2.05473
$750$	0	0
$751$	−37.6277	−1.37305	−0.686527	−	0.727104i	$-0.740866\pi$
$751$	−37.6277	−1.37305	−0.686527	+	0.727104i	$0.740866\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.883156	0.0321413
$756$	0	0
$757$	8.51087	0.309333	0.154667	−	0.987967i	$-0.450570\pi$
$757$	8.51087	0.309333	0.154667	+	0.987967i	$0.450570\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.3505	−0.701456	−0.350728	−	0.936477i	$-0.614066\pi$
$761$	−19.3505	−0.701456	−0.350728	+	0.936477i	$0.614066\pi$
$762$	0	0
$763$	−20.7446	−0.751004
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.60597	−0.0940961
$768$	0	0
$769$	33.1168	1.19422	0.597112	−	0.802158i	$-0.296315\pi$
$769$	33.1168	1.19422	0.597112	+	0.802158i	$0.296315\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.9783	1.04228	0.521138	−	0.853473i	$-0.325508\pi$
$773$	28.9783	1.04228	0.521138	+	0.853473i	$0.325508\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.37228	−0.264139
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.39403	−0.263904
$786$	0	0
$787$	42.8397	1.52707	0.763534	−	0.645767i	$-0.223462\pi$
$787$	42.8397	1.52707	0.763534	+	0.645767i	$0.223462\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19.1168	−0.679717
$792$	0	0
$793$	−0.883156	−0.0313618
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.39403	0.120223	0.0601114	−	0.998192i	$-0.480854\pi$
$797$	3.39403	0.120223	0.0601114	+	0.998192i	$0.480854\pi$
$798$	0	0
$799$	−23.4891	−0.830986
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12.1168	−0.427594
$804$	0	0
$805$	−3.86141	−0.136097
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13.1386	0.461928	0.230964	−	0.972962i	$-0.425812\pi$
$809$	13.1386	0.461928	0.230964	+	0.972962i	$0.425812\pi$
$810$	0	0
$811$	10.3505	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$811$	10.3505	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−28.4674	−0.997169
$816$	0	0
$817$	−1.09509	−0.0383125
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.6277	0.685012	0.342506	−	0.939516i	$-0.388724\pi$
$821$	19.6277	0.685012	0.342506	+	0.939516i	$0.388724\pi$
$822$	0	0
$823$	24.8832	0.867372	0.433686	−	0.901064i	$-0.357213\pi$
$823$	24.8832	0.867372	0.433686	+	0.901064i	$0.357213\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.9783	1.07722	0.538610	−	0.842555i	$-0.318950\pi$
$827$	30.9783	1.07722	0.538610	+	0.842555i	$0.318950\pi$
$828$	0	0
$829$	15.4891	0.537960	0.268980	−	0.963146i	$-0.413314\pi$
$829$	15.4891	0.537960	0.268980	+	0.963146i	$0.413314\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−65.0951	−2.25541
$834$	0	0
$835$	−45.3505	−1.56942
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.0951	1.03900	0.519499	−	0.854471i	$-0.326119\pi$
$839$	30.0951	1.03900	0.519499	+	0.854471i	$0.326119\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−30.5109	−1.04961
$846$	0	0
$847$	−43.7228	−1.50233
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.25544	−0.111595
$852$	0	0
$853$	33.1168	1.13390	0.566950	−	0.823753i	$-0.308123\pi$
$853$	33.1168	1.13390	0.566950	+	0.823753i	$0.308123\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.1168	1.26789	0.633944	−	0.773379i	$-0.281435\pi$
$857$	37.1168	1.26789	0.633944	+	0.773379i	$0.281435\pi$
$858$	0	0
$859$	−2.48913	−0.0849279	−0.0424639	−	0.999098i	$-0.513521\pi$
$859$	−2.48913	−0.0849279	−0.0424639	+	0.999098i	$0.513521\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.9783	−0.646027	−0.323014	−	0.946394i	$-0.604696\pi$
$863$	−18.9783	−0.646027	−0.323014	+	0.946394i	$0.604696\pi$
$864$	0	0
$865$	−38.8397	−1.32059
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.37228	−0.216165
$870$	0	0
$871$	1.39403	0.0472349
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−45.3505	−1.53313
$876$	0	0
$877$	27.8614	0.940813	0.470406	−	0.882450i	$-0.344107\pi$
$877$	27.8614	0.940813	0.470406	+	0.882450i	$0.344107\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24.9783	0.841539	0.420769	−	0.907168i	$-0.361760\pi$
$881$	24.9783	0.841539	0.420769	+	0.907168i	$0.361760\pi$
$882$	0	0
$883$	31.8397	1.07149	0.535745	−	0.844380i	$-0.320031\pi$
$883$	31.8397	1.07149	0.535745	+	0.844380i	$0.320031\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.8614	0.398267	0.199134	−	0.979972i	$-0.436187\pi$
$887$	11.8614	0.398267	0.199134	+	0.979972i	$0.436187\pi$
$888$	0	0
$889$	29.4891	0.989034
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.74456	0.0918433
$894$	0	0
$895$	54.5109	1.82210
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.8614	−0.929230
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.21194	−0.0402862
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.883156	−0.0292603	−0.0146301	−	0.999893i	$-0.504657\pi$
$911$	−0.883156	−0.0292603	−0.0146301	+	0.999893i	$0.504657\pi$
$912$	0	0
$913$	9.62772	0.318631
$914$	0	0
$915$	0	0
$916$	0	0
$917$	80.3288	2.65269
$918$	0	0
$919$	36.2337	1.19524	0.597620	−	0.801780i	$-0.296113\pi$
$919$	36.2337	1.19524	0.597620	+	0.801780i	$0.296113\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.48913	0.0490152
$924$	0	0
$925$	5.48913	0.180481
$926$	0	0
$927$	0	0
$928$	0	0
$929$	37.5842	1.23310	0.616549	−	0.787316i	$-0.288530\pi$
$929$	37.5842	1.23310	0.616549	+	0.787316i	$0.288530\pi$
$930$	0	0
$931$	7.60597	0.249276
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.7446	−0.416792
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.8397	0.874948	0.437474	−	0.899231i	$-0.355873\pi$
$941$	26.8397	0.874948	0.437474	+	0.899231i	$0.355873\pi$
$942$	0	0
$943$	4.37228	0.142381
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.7446	−1.74646	−0.873232	−	0.487305i	$-0.837980\pi$
$947$	−53.7446	−1.74646	−0.873232	+	0.487305i	$0.837980\pi$
$948$	0	0
$949$	4.51087	0.146429
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7.88316	−0.255360	−0.127680	−	0.991815i	$-0.540753\pi$
$953$	−7.88316	−0.255360	−0.127680	+	0.991815i	$0.540753\pi$
$954$	0	0
$955$	23.3940	0.757013
$956$	0	0
$957$	0	0
$958$	0	0
$959$	77.5842	2.50533
$960$	0	0
$961$	9.60597	0.309870
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.13859	0.133226
$966$	0	0
$967$	−8.37228	−0.269234	−0.134617	−	0.990898i	$-0.542980\pi$
$967$	−8.37228	−0.269234	−0.134617	+	0.990898i	$0.542980\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.48913	−0.176154	−0.0880772	−	0.996114i	$-0.528072\pi$
$971$	−5.48913	−0.176154	−0.0880772	+	0.996114i	$0.528072\pi$
$972$	0	0
$973$	25.1168	0.805209
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.255437	0.00817216	0.00408608	−	0.999992i	$-0.498699\pi$
$977$	0.255437	0.00817216	0.00408608	+	0.999992i	$0.498699\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.1168	1.75795	0.878977	−	0.476864i	$-0.158227\pi$
$983$	55.1168	1.75795	0.878977	+	0.476864i	$0.158227\pi$
$984$	0	0
$985$	36.1902	1.15312
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.649468	0.0206519
$990$	0	0
$991$	50.9783	1.61938	0.809689	−	0.586860i	$-0.199636\pi$
$991$	50.9783	1.61938	0.809689	+	0.586860i	$0.199636\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−38.5109	−1.22088
$996$	0	0
$997$	49.3505	1.56295	0.781474	−	0.623938i	$-0.214468\pi$
$997$	49.3505	1.56295	0.781474	+	0.623938i	$0.214468\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.a.f.1.2		2
3.2	odd	2		648.2.a.g.1.1		2
4.3	odd	2		1296.2.a.n.1.2		2
8.3	odd	2		5184.2.a.bs.1.1		2
8.5	even	2		5184.2.a.bt.1.1		2
9.2	odd	6		216.2.i.b.145.2		4
9.4	even	3		72.2.i.b.25.2	✓	4
9.5	odd	6		216.2.i.b.73.2		4
9.7	even	3		72.2.i.b.49.2	yes	4
12.11	even	2		1296.2.a.p.1.1		2
24.5	odd	2		5184.2.a.bp.1.2		2
24.11	even	2		5184.2.a.bo.1.2		2
36.7	odd	6		144.2.i.d.49.1		4
36.11	even	6		432.2.i.d.145.2		4
36.23	even	6		432.2.i.d.289.2		4
36.31	odd	6		144.2.i.d.97.1		4
72.5	odd	6		1728.2.i.i.1153.1		4
72.11	even	6		1728.2.i.j.577.1		4
72.13	even	6		576.2.i.j.385.1		4
72.29	odd	6		1728.2.i.i.577.1		4
72.43	odd	6		576.2.i.l.193.2		4
72.59	even	6		1728.2.i.j.1153.1		4
72.61	even	6		576.2.i.j.193.1		4
72.67	odd	6		576.2.i.l.385.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.b.25.2	✓	4	9.4	even	3
72.2.i.b.49.2	yes	4	9.7	even	3
144.2.i.d.49.1		4	36.7	odd	6
144.2.i.d.97.1		4	36.31	odd	6
216.2.i.b.73.2		4	9.5	odd	6
216.2.i.b.145.2		4	9.2	odd	6
432.2.i.d.145.2		4	36.11	even	6
432.2.i.d.289.2		4	36.23	even	6
576.2.i.j.193.1		4	72.61	even	6
576.2.i.j.385.1		4	72.13	even	6
576.2.i.l.193.2		4	72.43	odd	6
576.2.i.l.385.2		4	72.67	odd	6
648.2.a.f.1.2		2	1.1	even	1	trivial
648.2.a.g.1.1		2	3.2	odd	2
1296.2.a.n.1.2		2	4.3	odd	2
1296.2.a.p.1.1		2	12.11	even	2
1728.2.i.i.577.1		4	72.29	odd	6
1728.2.i.i.1153.1		4	72.5	odd	6
1728.2.i.j.577.1		4	72.11	even	6
1728.2.i.j.1153.1		4	72.59	even	6
5184.2.a.bo.1.2		2	24.11	even	2
5184.2.a.bp.1.2		2	24.5	odd	2
5184.2.a.bs.1.1		2	8.3	odd	2
5184.2.a.bt.1.1		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37228 q^{5} +4.37228 q^{7} +1.00000 q^{11} -0.372281 q^{13} -5.37228 q^{17} +0.627719 q^{19} -0.372281 q^{23} +0.627719 q^{25} -4.37228 q^{29} +6.37228 q^{31} +10.3723 q^{35} +8.74456 q^{37} -11.7446 q^{41} -1.74456 q^{43} +4.37228 q^{47} +12.1168 q^{49} +0.744563 q^{53} +2.37228 q^{55} +7.00000 q^{59} +2.37228 q^{61} -0.883156 q^{65} -3.74456 q^{67} -4.00000 q^{71} -12.1168 q^{73} +4.37228 q^{77} -6.37228 q^{79} +9.62772 q^{83} -12.7446 q^{85} +6.00000 q^{89} -1.62772 q^{91} +1.48913 q^{95} +1.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} - 5 q^{17} + 7 q^{19} + 5 q^{23} + 7 q^{25} - 3 q^{29} + 7 q^{31} + 15 q^{35} + 6 q^{37} - 12 q^{41} + 8 q^{43} + 3 q^{47} + 7 q^{49} - 10 q^{53} - q^{55} + 14 q^{59}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - q^5 + 3 * q^7 + 2 * q^11 + 5 * q^13 - 5 * q^17 + 7 * q^19 + 5 * q^23 + 7 * q^25 - 3 * q^29 + 7 * q^31 + 15 * q^35 + 6 * q^37 - 12 * q^41 + 8 * q^43 + 3 * q^47 + 7 * q^49 - 10 * q^53 - q^55 + 14 * q^59 - q^61 - 19 * q^65 + 4 * q^67 - 8 * q^71 - 7 * q^73 + 3 * q^77 - 7 * q^79 + 25 * q^83 - 14 * q^85 + 12 * q^89 - 9 * q^91 - 20 * q^95 - 8 * q^97

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