LMFDB - Embedded newform 648.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(1,648)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.17430605098$
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			$3.37228$ 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)

$f(q)$	$=$	$q+3.37228 q^{5} -1.37228 q^{7} +O(q^{10})$	$q+3.37228 q^{5} -1.37228 q^{7} -1.00000 q^{11} +5.37228 q^{13} -0.372281 q^{17} +6.37228 q^{19} -5.37228 q^{23} +6.37228 q^{25} -1.37228 q^{29} +0.627719 q^{31} -4.62772 q^{35} -2.74456 q^{37} +0.255437 q^{41} +9.74456 q^{43} +1.37228 q^{47} -5.11684 q^{49} +10.7446 q^{53} -3.37228 q^{55} -7.00000 q^{59} -3.37228 q^{61} +18.1168 q^{65} +7.74456 q^{67} +4.00000 q^{71} +5.11684 q^{73} +1.37228 q^{77} -0.627719 q^{79} -15.3723 q^{83} -1.25544 q^{85} -6.00000 q^{89} -7.37228 q^{91} +21.4891 q^{95} -9.74456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 3 q^{7}+O(q^{10}) $ 2 * q + q^5 + 3 * q^7	$ 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}+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

Display 100 coefficients 10 coefficients

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
$5$	3.37228	1.50813	0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228	1.50813	0.754065	+	0.656800i	$0.228090\pi$
$6$	0	0
$7$	−1.37228	−0.518674	−0.259337	−	0.965787i	$-0.583504\pi$
$7$	−1.37228	−0.518674	−0.259337	+	0.965787i	$0.583504\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	5.37228	1.49000	0.745001	−	0.667063i	$-0.232449\pi$
$13$	5.37228	1.49000	0.745001	+	0.667063i	$0.232449\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.372281	−0.0902915	−0.0451457	−	0.998980i	$-0.514375\pi$
$17$	−0.372281	−0.0902915	−0.0451457	+	0.998980i	$0.514375\pi$
$18$	0	0
$19$	6.37228	1.46190	0.730951	−	0.682430i	$-0.239077\pi$
$19$	6.37228	1.46190	0.730951	+	0.682430i	$0.239077\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.37228	−1.12020	−0.560099	−	0.828426i	$-0.689237\pi$
$23$	−5.37228	−1.12020	−0.560099	+	0.828426i	$0.689237\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	0.627719	0.112742	0.0563708	−	0.998410i	$-0.482047\pi$
$31$	0.627719	0.112742	0.0563708	+	0.998410i	$0.482047\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.62772	−0.782227
$36$	0	0
$37$	−2.74456	−0.451203	−0.225602	−	0.974220i	$-0.572435\pi$
$37$	−2.74456	−0.451203	−0.225602	+	0.974220i	$0.572435\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.255437	0.0398926	0.0199463	−	0.999801i	$-0.493650\pi$
$41$	0.255437	0.0398926	0.0199463	+	0.999801i	$0.493650\pi$
$42$	0	0
$43$	9.74456	1.48603	0.743016	−	0.669274i	$-0.233395\pi$
$43$	9.74456	1.48603	0.743016	+	0.669274i	$0.233395\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.37228	0.200168	0.100084	−	0.994979i	$-0.468089\pi$
$47$	1.37228	0.200168	0.100084	+	0.994979i	$0.468089\pi$
$48$	0	0
$49$	−5.11684	−0.730978
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.7446	1.47588	0.737940	−	0.674867i	$-0.235799\pi$
$53$	10.7446	1.47588	0.737940	+	0.674867i	$0.235799\pi$
$54$	0	0
$55$	−3.37228	−0.454718
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	−3.37228	−0.431776	−0.215888	−	0.976418i	$-0.569265\pi$
$61$	−3.37228	−0.431776	−0.215888	+	0.976418i	$0.569265\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.1168	2.24712
$66$	0	0
$67$	7.74456	0.946149	0.473074	−	0.881022i	$-0.343144\pi$
$67$	7.74456	0.946149	0.473074	+	0.881022i	$0.343144\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	5.11684	0.598881	0.299441	−	0.954115i	$-0.403200\pi$
$73$	5.11684	0.598881	0.299441	+	0.954115i	$0.403200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.37228	0.156386
$78$	0	0
$79$	−0.627719	−0.0706239	−0.0353119	−	0.999376i	$-0.511242\pi$
$79$	−0.627719	−0.0706239	−0.0353119	+	0.999376i	$0.511242\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.3723	−1.68733	−0.843664	−	0.536872i	$-0.819606\pi$
$83$	−15.3723	−1.68733	−0.843664	+	0.536872i	$0.819606\pi$
$84$	0	0
$85$	−1.25544	−0.136171
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−7.37228	−0.772825
$92$	0	0
$93$	0	0
$94$	0	0
$95$	21.4891	2.20474
$96$	0	0
$97$	−9.74456	−0.989410	−0.494705	−	0.869061i	$-0.664724\pi$
$97$	−9.74456	−0.989410	−0.494705	+	0.869061i	$0.664724\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.1168	−1.00666	−0.503332	−	0.864093i	$-0.667893\pi$
$101$	−10.1168	−1.00666	−0.503332	+	0.864093i	$0.667893\pi$
$102$	0	0
$103$	−6.62772	−0.653049	−0.326524	−	0.945189i	$-0.605877\pi$
$103$	−6.62772	−0.653049	−0.326524	+	0.945189i	$0.605877\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.8614	−1.53338	−0.766690	−	0.642017i	$-0.778098\pi$
$107$	−15.8614	−1.53338	−0.766690	+	0.642017i	$0.778098\pi$
$108$	0	0
$109$	6.74456	0.646012	0.323006	−	0.946397i	$-0.395307\pi$
$109$	6.74456	0.646012	0.323006	+	0.946397i	$0.395307\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.37228	−0.129093	−0.0645467	−	0.997915i	$-0.520560\pi$
$113$	−1.37228	−0.129093	−0.0645467	+	0.997915i	$0.520560\pi$
$114$	0	0
$115$	−18.1168	−1.68940
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.510875	0.0468318
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.62772	0.413916
$126$	0	0
$127$	−4.74456	−0.421012	−0.210506	−	0.977593i	$-0.567511\pi$
$127$	−4.74456	−0.421012	−0.210506	+	0.977593i	$0.567511\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.6277	−1.10329	−0.551644	−	0.834079i	$-0.685999\pi$
$131$	−12.6277	−1.10329	−0.551644	+	0.834079i	$0.685999\pi$
$132$	0	0
$133$	−8.74456	−0.758250
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.25544	−0.534438	−0.267219	−	0.963636i	$-0.586105\pi$
$137$	−6.25544	−0.534438	−0.267219	+	0.963636i	$0.586105\pi$
$138$	0	0
$139$	−5.74456	−0.487247	−0.243624	−	0.969870i	$-0.578336\pi$
$139$	−5.74456	−0.487247	−0.243624	+	0.969870i	$0.578336\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.37228	−0.449253
$144$	0	0
$145$	−4.62772	−0.384311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.62772	0.215271	0.107636	−	0.994190i	$-0.465672\pi$
$149$	2.62772	0.215271	0.107636	+	0.994190i	$0.465672\pi$
$150$	0	0
$151$	−5.37228	−0.437190	−0.218595	−	0.975816i	$-0.570147\pi$
$151$	−5.37228	−0.437190	−0.218595	+	0.975816i	$0.570147\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.11684	0.170029
$156$	0	0
$157$	14.1168	1.12665	0.563323	−	0.826236i	$-0.309523\pi$
$157$	14.1168	1.12665	0.563323	+	0.826236i	$0.309523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.37228	0.581017
$162$	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
$167$	1.88316	0.145723	0.0728615	−	0.997342i	$-0.476787\pi$
$167$	1.88316	0.145723	0.0728615	+	0.997342i	$0.476787\pi$
$168$	0	0
$169$	15.8614	1.22011
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.6277	0.808010	0.404005	−	0.914757i	$-0.367618\pi$
$173$	10.6277	0.808010	0.404005	+	0.914757i	$0.367618\pi$
$174$	0	0
$175$	−8.74456	−0.661027
$176$	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
$181$	−23.4891	−1.74593	−0.872966	−	0.487780i	$-0.837807\pi$
$181$	−23.4891	−1.74593	−0.872966	+	0.487780i	$0.837807\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.25544	−0.680473
$186$	0	0
$187$	0.372281	0.0272239
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.8614	1.36476	0.682382	−	0.730996i	$-0.260944\pi$
$191$	18.8614	1.36476	0.682382	+	0.730996i	$0.260944\pi$
$192$	0	0
$193$	−9.74456	−0.701429	−0.350714	−	0.936482i	$-0.614061\pi$
$193$	−9.74456	−0.701429	−0.350714	+	0.936482i	$0.614061\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.7446	−1.90547	−0.952736	−	0.303801i	$-0.901744\pi$
$197$	−26.7446	−1.90547	−0.952736	+	0.303801i	$0.901744\pi$
$198$	0	0
$199$	18.2337	1.29255	0.646276	−	0.763104i	$-0.276326\pi$
$199$	18.2337	1.29255	0.646276	+	0.763104i	$0.276326\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.88316	0.132172
$204$	0	0
$205$	0.861407	0.0601632
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.37228	−0.440780
$210$	0	0
$211$	−15.3723	−1.05827	−0.529136	−	0.848537i	$-0.677484\pi$
$211$	−15.3723	−1.05827	−0.529136	+	0.848537i	$0.677484\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	32.8614	2.24113
$216$	0	0
$217$	−0.861407	−0.0584761
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	25.6060	1.71470	0.857351	−	0.514732i	$-0.172108\pi$
$223$	25.6060	1.71470	0.857351	+	0.514732i	$0.172108\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	−17.6060	−1.16344	−0.581718	−	0.813391i	$-0.697619\pi$
$229$	−17.6060	−1.16344	−0.581718	+	0.813391i	$0.697619\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.372281	0.0243890	0.0121945	−	0.999926i	$-0.496118\pi$
$233$	0.372281	0.0243890	0.0121945	+	0.999926i	$0.496118\pi$
$234$	0	0
$235$	4.62772	0.301879
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.8614	−0.961304	−0.480652	−	0.876911i	$-0.659600\pi$
$239$	−14.8614	−0.961304	−0.480652	+	0.876911i	$0.659600\pi$
$240$	0	0
$241$	−5.74456	−0.370040	−0.185020	−	0.982735i	$-0.559235\pi$
$241$	−5.74456	−0.370040	−0.185020	+	0.982735i	$0.559235\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−17.2554	−1.10241
$246$	0	0
$247$	34.2337	2.17824
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.1168	−1.71160	−0.855800	−	0.517307i	$-0.826935\pi$
$251$	−27.1168	−1.71160	−0.855800	+	0.517307i	$0.826935\pi$
$252$	0	0
$253$	5.37228	0.337752
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.4891	1.15332	0.576660	−	0.816984i	$-0.304356\pi$
$257$	18.4891	1.15332	0.576660	+	0.816984i	$0.304356\pi$
$258$	0	0
$259$	3.76631	0.234027
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.88316	0.239446	0.119723	−	0.992807i	$-0.461799\pi$
$263$	3.88316	0.239446	0.119723	+	0.992807i	$0.461799\pi$
$264$	0	0
$265$	36.2337	2.22582
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.25544	0.0765454	0.0382727	−	0.999267i	$-0.487814\pi$
$269$	1.25544	0.0765454	0.0382727	+	0.999267i	$0.487814\pi$
$270$	0	0
$271$	1.48913	0.0904579	0.0452290	−	0.998977i	$-0.485598\pi$
$271$	1.48913	0.0904579	0.0452290	+	0.998977i	$0.485598\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.37228	−0.384263
$276$	0	0
$277$	11.8832	0.713990	0.356995	−	0.934106i	$-0.383801\pi$
$277$	11.8832	0.713990	0.356995	+	0.934106i	$0.383801\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.8614	−0.886557	−0.443279	−	0.896384i	$-0.646185\pi$
$281$	−14.8614	−0.886557	−0.443279	+	0.896384i	$0.646185\pi$
$282$	0	0
$283$	−4.86141	−0.288981	−0.144490	−	0.989506i	$-0.546154\pi$
$283$	−4.86141	−0.288981	−0.144490	+	0.989506i	$0.546154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.350532	−0.0206912
$288$	0	0
$289$	−16.8614	−0.991847
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.3723	−1.48226	−0.741132	−	0.671359i	$-0.765711\pi$
$293$	−25.3723	−1.48226	−0.741132	+	0.671359i	$0.765711\pi$
$294$	0	0
$295$	−23.6060	−1.37439
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−28.8614	−1.66910
$300$	0	0
$301$	−13.3723	−0.770765
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.3723	−0.651175
$306$	0	0
$307$	25.6277	1.46265	0.731326	−	0.682029i	$-0.238902\pi$
$307$	25.6277	1.46265	0.731326	+	0.682029i	$0.238902\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.1168	1.59436	0.797180	−	0.603742i	$-0.206324\pi$
$311$	28.1168	1.59436	0.797180	+	0.603742i	$0.206324\pi$
$312$	0	0
$313$	−23.2337	−1.31325	−0.656623	−	0.754219i	$-0.728016\pi$
$313$	−23.2337	−1.31325	−0.656623	+	0.754219i	$0.728016\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.60597	0.539525	0.269762	−	0.962927i	$-0.413055\pi$
$317$	9.60597	0.539525	0.269762	+	0.962927i	$0.413055\pi$
$318$	0	0
$319$	1.37228	0.0768330
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.37228	−0.131997
$324$	0	0
$325$	34.2337	1.89894
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.88316	−0.103822
$330$	0	0
$331$	−3.13859	−0.172513	−0.0862563	−	0.996273i	$-0.527490\pi$
$331$	−3.13859	−0.172513	−0.0862563	+	0.996273i	$0.527490\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	26.1168	1.42692
$336$	0	0
$337$	25.9783	1.41513	0.707563	−	0.706651i	$-0.249795\pi$
$337$	25.9783	1.41513	0.707563	+	0.706651i	$0.249795\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.627719	−0.0339929
$342$	0	0
$343$	16.6277	0.897812
$344$	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
$349$	34.1168	1.82623	0.913116	−	0.407699i	$-0.133669\pi$
$349$	34.1168	1.82623	0.913116	+	0.407699i	$0.133669\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.9783	−1.16978	−0.584892	−	0.811111i	$-0.698863\pi$
$353$	−21.9783	−1.16978	−0.584892	+	0.811111i	$0.698863\pi$
$354$	0	0
$355$	13.4891	0.715928
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2337	0.751225	0.375613	−	0.926777i	$-0.377432\pi$
$359$	14.2337	0.751225	0.375613	+	0.926777i	$0.377432\pi$
$360$	0	0
$361$	21.6060	1.13716
$362$	0	0
$363$	0	0
$364$	0	0
$365$	17.2554	0.903191
$366$	0	0
$367$	23.3723	1.22002	0.610012	−	0.792393i	$-0.291165\pi$
$367$	23.3723	1.22002	0.610012	+	0.792393i	$0.291165\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.7446	−0.765500
$372$	0	0
$373$	3.88316	0.201062	0.100531	−	0.994934i	$-0.467946\pi$
$373$	3.88316	0.201062	0.100531	+	0.994934i	$0.467946\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.37228	−0.379692
$378$	0	0
$379$	−23.1168	−1.18743	−0.593716	−	0.804674i	$-0.702340\pi$
$379$	−23.1168	−1.18743	−0.593716	+	0.804674i	$0.702340\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.3505	1.55084	0.775420	−	0.631446i	$-0.217538\pi$
$383$	30.3505	1.55084	0.775420	+	0.631446i	$0.217538\pi$
$384$	0	0
$385$	4.62772	0.235850
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.6060	1.50108	0.750541	−	0.660824i	$-0.229793\pi$
$389$	29.6060	1.50108	0.750541	+	0.660824i	$0.229793\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.11684	−0.106510
$396$	0	0
$397$	16.2337	0.814745	0.407373	−	0.913262i	$-0.366445\pi$
$397$	16.2337	0.814745	0.407373	+	0.913262i	$0.366445\pi$
$398$	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
$403$	3.37228	0.167985
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.74456	0.136043
$408$	0	0
$409$	−5.74456	−0.284050	−0.142025	−	0.989863i	$-0.545361\pi$
$409$	−5.74456	−0.284050	−0.142025	+	0.989863i	$0.545361\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.60597	0.472679
$414$	0	0
$415$	−51.8397	−2.54471
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.6060	−1.34864	−0.674320	−	0.738439i	$-0.735563\pi$
$419$	−27.6060	−1.34864	−0.674320	+	0.738439i	$0.735563\pi$
$420$	0	0
$421$	−17.8832	−0.871572	−0.435786	−	0.900050i	$-0.643529\pi$
$421$	−17.8832	−0.871572	−0.435786	+	0.900050i	$0.643529\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.37228	−0.115073
$426$	0	0
$427$	4.62772	0.223951
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.7446	0.613884	0.306942	−	0.951728i	$-0.400694\pi$
$431$	12.7446	0.613884	0.306942	+	0.951728i	$0.400694\pi$
$432$	0	0
$433$	14.8832	0.715239	0.357619	−	0.933867i	$-0.383589\pi$
$433$	14.8832	0.715239	0.357619	+	0.933867i	$0.383589\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−34.2337	−1.63762
$438$	0	0
$439$	−10.1168	−0.482851	−0.241425	−	0.970419i	$-0.577615\pi$
$439$	−10.1168	−0.482851	−0.241425	+	0.970419i	$0.577615\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.7228	1.26964	0.634820	−	0.772660i	$-0.281074\pi$
$443$	26.7228	1.26964	0.634820	+	0.772660i	$0.281074\pi$
$444$	0	0
$445$	−20.2337	−0.959169
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.1168	1.56288	0.781440	−	0.623980i	$-0.214485\pi$
$449$	33.1168	1.56288	0.781440	+	0.623980i	$0.214485\pi$
$450$	0	0
$451$	−0.255437	−0.0120281
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−24.8614	−1.16552
$456$	0	0
$457$	7.74456	0.362275	0.181138	−	0.983458i	$-0.442022\pi$
$457$	7.74456	0.362275	0.181138	+	0.983458i	$0.442022\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.1168	1.12323	0.561617	−	0.827398i	$-0.310180\pi$
$461$	24.1168	1.12323	0.561617	+	0.827398i	$0.310180\pi$
$462$	0	0
$463$	−10.3505	−0.481030	−0.240515	−	0.970645i	$-0.577316\pi$
$463$	−10.3505	−0.481030	−0.240515	+	0.970645i	$0.577316\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.62772	−0.0753218	−0.0376609	−	0.999291i	$-0.511991\pi$
$467$	−1.62772	−0.0753218	−0.0376609	+	0.999291i	$0.511991\pi$
$468$	0	0
$469$	−10.6277	−0.490742
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.74456	−0.448055
$474$	0	0
$475$	40.6060	1.86313
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.6060	−1.35273	−0.676366	−	0.736566i	$-0.736446\pi$
$479$	−29.6060	−1.35273	−0.676366	+	0.736566i	$0.736446\pi$
$480$	0	0
$481$	−14.7446	−0.672294
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−32.8614	−1.49216
$486$	0	0
$487$	−4.74456	−0.214997	−0.107498	−	0.994205i	$-0.534284\pi$
$487$	−4.74456	−0.214997	−0.107498	+	0.994205i	$0.534284\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.7446	−0.530025	−0.265012	−	0.964245i	$-0.585376\pi$
$491$	−11.7446	−0.530025	−0.265012	+	0.964245i	$0.585376\pi$
$492$	0	0
$493$	0.510875	0.0230086
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.48913	−0.246221
$498$	0	0
$499$	25.9783	1.16295	0.581473	−	0.813566i	$-0.302477\pi$
$499$	25.9783	1.16295	0.581473	+	0.813566i	$0.302477\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.4891	−1.31486	−0.657428	−	0.753518i	$-0.728355\pi$
$503$	−29.4891	−1.31486	−0.657428	+	0.753518i	$0.728355\pi$
$504$	0	0
$505$	−34.1168	−1.51818
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.8832	0.615360	0.307680	−	0.951490i	$-0.400447\pi$
$509$	13.8832	0.615360	0.307680	+	0.951490i	$0.400447\pi$
$510$	0	0
$511$	−7.02175	−0.310624
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22.3505	−0.984882
$516$	0	0
$517$	−1.37228	−0.0603529
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.11684	0.224173	0.112087	−	0.993698i	$-0.464247\pi$
$521$	5.11684	0.224173	0.112087	+	0.993698i	$0.464247\pi$
$522$	0	0
$523$	9.48913	0.414930	0.207465	−	0.978242i	$-0.433479\pi$
$523$	9.48913	0.414930	0.207465	+	0.978242i	$0.433479\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.233688	−0.0101796
$528$	0	0
$529$	5.86141	0.254844
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.37228	0.0594401
$534$	0	0
$535$	−53.4891	−2.31254
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.11684	0.220398
$540$	0	0
$541$	−32.2337	−1.38583	−0.692917	−	0.721017i	$-0.743675\pi$
$541$	−32.2337	−1.38583	−0.692917	+	0.721017i	$0.743675\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.7446	0.974270
$546$	0	0
$547$	−27.7446	−1.18627	−0.593136	−	0.805102i	$-0.702110\pi$
$547$	−27.7446	−1.18627	−0.593136	+	0.805102i	$0.702110\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.74456	−0.372531
$552$	0	0
$553$	0.861407	0.0366307
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	52.3505	2.21419
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.74456	0.0735245	0.0367623	−	0.999324i	$-0.488296\pi$
$563$	1.74456	0.0735245	0.0367623	+	0.999324i	$0.488296\pi$
$564$	0	0
$565$	−4.62772	−0.194690
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.23369	0.303252	0.151626	−	0.988438i	$-0.451549\pi$
$569$	7.23369	0.303252	0.151626	+	0.988438i	$0.451549\pi$
$570$	0	0
$571$	−3.51087	−0.146926	−0.0734628	−	0.997298i	$-0.523405\pi$
$571$	−3.51087	−0.146926	−0.0734628	+	0.997298i	$0.523405\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−34.2337	−1.42764
$576$	0	0
$577$	−13.8614	−0.577058	−0.288529	−	0.957471i	$-0.593166\pi$
$577$	−13.8614	−0.577058	−0.288529	+	0.957471i	$0.593166\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	21.0951	0.875172
$582$	0	0
$583$	−10.7446	−0.444994
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.2337	1.04151	0.520753	−	0.853707i	$-0.325651\pi$
$587$	25.2337	1.04151	0.520753	+	0.853707i	$0.325651\pi$
$588$	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	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	1.72281	0.0706285
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.1386	−0.455111	−0.227555	−	0.973765i	$-0.573073\pi$
$599$	−11.1386	−0.455111	−0.227555	+	0.973765i	$0.573073\pi$
$600$	0	0
$601$	−39.9783	−1.63075	−0.815373	−	0.578935i	$-0.803468\pi$
$601$	−39.9783	−1.63075	−0.815373	+	0.578935i	$0.803468\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−33.7228	−1.37103
$606$	0	0
$607$	6.11684	0.248275	0.124138	−	0.992265i	$-0.460384\pi$
$607$	6.11684	0.248275	0.124138	+	0.992265i	$0.460384\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.37228	0.298251
$612$	0	0
$613$	−1.25544	−0.0507066	−0.0253533	−	0.999679i	$-0.508071\pi$
$613$	−1.25544	−0.0507066	−0.0253533	+	0.999679i	$0.508071\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.97825	0.321192	0.160596	−	0.987020i	$-0.448658\pi$
$617$	7.97825	0.321192	0.160596	+	0.987020i	$0.448658\pi$
$618$	0	0
$619$	−13.2337	−0.531907	−0.265953	−	0.963986i	$-0.585687\pi$
$619$	−13.2337	−0.531907	−0.265953	+	0.963986i	$0.585687\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.23369	0.329876
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.02175	0.0407398
$630$	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
$637$	−27.4891	−1.08916
$638$	0	0
$639$	0	0
$640$	0	0
$641$	31.2337	1.23366	0.616828	−	0.787098i	$-0.288417\pi$
$641$	31.2337	1.23366	0.616828	+	0.787098i	$0.288417\pi$
$642$	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
$647$	23.7228	0.932640	0.466320	−	0.884616i	$-0.345580\pi$
$647$	23.7228	0.932640	0.466320	+	0.884616i	$0.345580\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.1168	1.56989	0.784947	−	0.619563i	$-0.212690\pi$
$653$	40.1168	1.56989	0.784947	+	0.619563i	$0.212690\pi$
$654$	0	0
$655$	−42.5842	−1.66390
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.88316	−0.0733573	−0.0366787	−	0.999327i	$-0.511678\pi$
$659$	−1.88316	−0.0733573	−0.0366787	+	0.999327i	$0.511678\pi$
$660$	0	0
$661$	17.0951	0.664922	0.332461	−	0.943117i	$-0.392121\pi$
$661$	17.0951	0.664922	0.332461	+	0.943117i	$0.392121\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−29.4891	−1.14354
$666$	0	0
$667$	7.37228	0.285456
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.37228	0.130185
$672$	0	0
$673$	5.37228	0.207086	0.103543	−	0.994625i	$-0.466982\pi$
$673$	5.37228	0.207086	0.103543	+	0.994625i	$0.466982\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.6060	−0.753519	−0.376759	−	0.926311i	$-0.622962\pi$
$677$	−19.6060	−0.753519	−0.376759	+	0.926311i	$0.622962\pi$
$678$	0	0
$679$	13.3723	0.513181
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.62772	0.368394	0.184197	−	0.982889i	$-0.441031\pi$
$683$	9.62772	0.368394	0.184197	+	0.982889i	$0.441031\pi$
$684$	0	0
$685$	−21.0951	−0.806002
$686$	0	0
$687$	0	0
$688$	0	0
$689$	57.7228	2.19906
$690$	0	0
$691$	−2.11684	−0.0805285	−0.0402643	−	0.999189i	$-0.512820\pi$
$691$	−2.11684	−0.0805285	−0.0402643	+	0.999189i	$0.512820\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.3723	−0.734833
$696$	0	0
$697$	−0.0950946	−0.00360196
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.5109	0.472529	0.236265	−	0.971689i	$-0.424077\pi$
$701$	12.5109	0.472529	0.236265	+	0.971689i	$0.424077\pi$
$702$	0	0
$703$	−17.4891	−0.659615
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8832	0.522130
$708$	0	0
$709$	11.6060	0.435871	0.217936	−	0.975963i	$-0.430068\pi$
$709$	11.6060	0.435871	0.217936	+	0.975963i	$0.430068\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.37228	−0.126293
$714$	0	0
$715$	−18.1168	−0.677532
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.5109	−0.839514	−0.419757	−	0.907637i	$-0.637885\pi$
$719$	−22.5109	−0.839514	−0.419757	+	0.907637i	$0.637885\pi$
$720$	0	0
$721$	9.09509	0.338719
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.74456	−0.324765
$726$	0	0
$727$	28.1168	1.04280	0.521398	−	0.853314i	$-0.325411\pi$
$727$	28.1168	1.04280	0.521398	+	0.853314i	$0.325411\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.62772	−0.134176
$732$	0	0
$733$	−9.13859	−0.337542	−0.168771	−	0.985655i	$-0.553980\pi$
$733$	−9.13859	−0.337542	−0.168771	+	0.985655i	$0.553980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.74456	−0.285275
$738$	0	0
$739$	24.8832	0.915342	0.457671	−	0.889122i	$-0.348684\pi$
$739$	24.8832	0.915342	0.457671	+	0.889122i	$0.348684\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.3723	−1.51780	−0.758901	−	0.651206i	$-0.774263\pi$
$743$	−41.3723	−1.51780	−0.758901	+	0.651206i	$0.774263\pi$
$744$	0	0
$745$	8.86141	0.324657
$746$	0	0
$747$	0	0
$748$	0	0
$749$	21.7663	0.795324
$750$	0	0
$751$	−43.3723	−1.58268	−0.791339	−	0.611378i	$-0.790615\pi$
$751$	−43.3723	−1.58268	−0.791339	+	0.611378i	$0.790615\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.1168	−0.659339
$756$	0	0
$757$	31.4891	1.14449	0.572246	−	0.820082i	$-0.306072\pi$
$757$	31.4891	1.14449	0.572246	+	0.820082i	$0.306072\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.3505	−1.17271	−0.586353	−	0.810056i	$-0.699437\pi$
$761$	−32.3505	−1.17271	−0.586353	+	0.810056i	$0.699437\pi$
$762$	0	0
$763$	−9.25544	−0.335069
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37.6060	−1.35787
$768$	0	0
$769$	15.8832	0.572761	0.286381	−	0.958116i	$-0.407548\pi$
$769$	15.8832	0.572761	0.286381	+	0.958116i	$0.407548\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.9783	0.610665	0.305333	−	0.952246i	$-0.401232\pi$
$773$	16.9783	0.610665	0.305333	+	0.952246i	$0.401232\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.62772	0.0583191
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	47.6060	1.69913
$786$	0	0
$787$	−31.8397	−1.13496	−0.567481	−	0.823387i	$-0.692082\pi$
$787$	−31.8397	−1.13496	−0.567481	+	0.823387i	$0.692082\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.88316	0.0669573
$792$	0	0
$793$	−18.1168	−0.643348
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−43.6060	−1.54460	−0.772301	−	0.635256i	$-0.780894\pi$
$797$	−43.6060	−1.54460	−0.772301	+	0.635256i	$0.780894\pi$
$798$	0	0
$799$	−0.510875	−0.0180734
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.11684	−0.180570
$804$	0	0
$805$	24.8614	0.876249
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−41.8614	−1.47177	−0.735884	−	0.677107i	$-0.763233\pi$
$809$	−41.8614	−1.47177	−0.735884	+	0.677107i	$0.763233\pi$
$810$	0	0
$811$	−41.3505	−1.45201	−0.726007	−	0.687688i	$-0.758626\pi$
$811$	−41.3505	−1.45201	−0.726007	+	0.687688i	$0.758626\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−40.4674	−1.41751
$816$	0	0
$817$	62.0951	2.17243
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.3723	−0.885499	−0.442749	−	0.896645i	$-0.645997\pi$
$821$	−25.3723	−0.885499	−0.442749	+	0.896645i	$0.645997\pi$
$822$	0	0
$823$	42.1168	1.46810	0.734050	−	0.679095i	$-0.237628\pi$
$823$	42.1168	1.46810	0.734050	+	0.679095i	$0.237628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.9783	0.520845	0.260422	−	0.965495i	$-0.416138\pi$
$827$	14.9783	0.520845	0.260422	+	0.965495i	$0.416138\pi$
$828$	0	0
$829$	−7.48913	−0.260108	−0.130054	−	0.991507i	$-0.541515\pi$
$829$	−7.48913	−0.260108	−0.130054	+	0.991507i	$0.541515\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.90491	0.0660011
$834$	0	0
$835$	6.35053	0.219769
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33.0951	1.14257	0.571285	−	0.820752i	$-0.306445\pi$
$839$	33.0951	1.14257	0.571285	+	0.820752i	$0.306445\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	0	0
$844$	0	0
$845$	53.4891	1.84008
$846$	0	0
$847$	13.7228	0.471521
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.7446	0.505437
$852$	0	0
$853$	15.8832	0.543829	0.271914	−	0.962321i	$-0.412343\pi$
$853$	15.8832	0.543829	0.271914	+	0.962321i	$0.412343\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.8832	−0.679196	−0.339598	−	0.940571i	$-0.610291\pi$
$857$	−19.8832	−0.679196	−0.339598	+	0.940571i	$0.610291\pi$
$858$	0	0
$859$	20.4891	0.699080	0.349540	−	0.936921i	$-0.386338\pi$
$859$	20.4891	0.699080	0.349540	+	0.936921i	$0.386338\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.9783	−0.918350	−0.459175	−	0.888346i	$-0.651855\pi$
$863$	−26.9783	−0.918350	−0.459175	+	0.888346i	$0.651855\pi$
$864$	0	0
$865$	35.8397	1.21858
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.627719	0.0212939
$870$	0	0
$871$	41.6060	1.40976
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.35053	−0.214687
$876$	0	0
$877$	−0.861407	−0.0290876	−0.0145438	−	0.999894i	$-0.504630\pi$
$877$	−0.861407	−0.0290876	−0.0145438	+	0.999894i	$0.504630\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.9783	0.706775	0.353388	−	0.935477i	$-0.385030\pi$
$881$	20.9783	0.706775	0.353388	+	0.935477i	$0.385030\pi$
$882$	0	0
$883$	−42.8397	−1.44167	−0.720835	−	0.693107i	$-0.756241\pi$
$883$	−42.8397	−1.44167	−0.720835	+	0.693107i	$0.756241\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.8614	0.566151	0.283075	−	0.959098i	$-0.408645\pi$
$887$	16.8614	0.566151	0.283075	+	0.959098i	$0.408645\pi$
$888$	0	0
$889$	6.51087	0.218368
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.74456	0.292626
$894$	0	0
$895$	77.4891	2.59018
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.861407	−0.0287295
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−79.2119	−2.63309
$906$	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
$911$	18.1168	0.600238	0.300119	−	0.953902i	$-0.402974\pi$
$911$	18.1168	0.600238	0.300119	+	0.953902i	$0.402974\pi$
$912$	0	0
$913$	15.3723	0.508748
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.3288	0.572247
$918$	0	0
$919$	1.76631	0.0582653	0.0291326	−	0.999576i	$-0.490725\pi$
$919$	1.76631	0.0582653	0.0291326	+	0.999576i	$0.490725\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	21.4891	0.707323
$924$	0	0
$925$	−17.4891	−0.575039
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.5842	1.59400	0.796998	−	0.603982i	$-0.206420\pi$
$929$	48.5842	1.59400	0.796998	+	0.603982i	$0.206420\pi$
$930$	0	0
$931$	−32.6060	−1.06862
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.25544	0.0410572
$936$	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
$941$	47.8397	1.55953	0.779764	−	0.626073i	$-0.215339\pi$
$941$	47.8397	1.55953	0.779764	+	0.626073i	$0.215339\pi$
$942$	0	0
$943$	−1.37228	−0.0446876
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.2554	1.37312	0.686559	−	0.727074i	$-0.259121\pi$
$947$	42.2554	1.37312	0.686559	+	0.727074i	$0.259121\pi$
$948$	0	0
$949$	27.4891	0.892335
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.1168	0.813614	0.406807	−	0.913514i	$-0.366642\pi$
$953$	25.1168	0.813614	0.406807	+	0.913514i	$0.366642\pi$
$954$	0	0
$955$	63.6060	2.05824
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.58422	0.277199
$960$	0	0
$961$	−30.6060	−0.987289
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−32.8614	−1.05785
$966$	0	0
$967$	−2.62772	−0.0845017	−0.0422509	−	0.999107i	$-0.513453\pi$
$967$	−2.62772	−0.0845017	−0.0422509	+	0.999107i	$0.513453\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.4891	−0.561253	−0.280626	−	0.959817i	$-0.590542\pi$
$971$	−17.4891	−0.561253	−0.280626	+	0.959817i	$0.590542\pi$
$972$	0	0
$973$	7.88316	0.252722
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.7446	−0.375742	−0.187871	−	0.982194i	$-0.560159\pi$
$977$	−11.7446	−0.375742	−0.187871	+	0.982194i	$0.560159\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.8832	−1.20829	−0.604143	−	0.796876i	$-0.706484\pi$
$983$	−37.8832	−1.20829	−0.604143	+	0.796876i	$0.706484\pi$
$984$	0	0
$985$	−90.1902	−2.87370
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−52.3505	−1.66465
$990$	0	0
$991$	5.02175	0.159521	0.0797606	−	0.996814i	$-0.474584\pi$
$991$	5.02175	0.159521	0.0797606	+	0.996814i	$0.474584\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	61.4891	1.94934
$996$	0	0
$997$	−2.35053	−0.0744421	−0.0372210	−	0.999307i	$-0.511851\pi$
$997$	−2.35053	−0.0744421	−0.0372210	+	0.999307i	$0.511851\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.a.g.1.2		2
3.2	odd	2		648.2.a.f.1.1		2
4.3	odd	2		1296.2.a.p.1.2		2
8.3	odd	2		5184.2.a.bo.1.1		2
8.5	even	2		5184.2.a.bp.1.1		2
9.2	odd	6		72.2.i.b.49.1	yes	4
9.4	even	3		216.2.i.b.73.1		4
9.5	odd	6		72.2.i.b.25.1	✓	4
9.7	even	3		216.2.i.b.145.1		4
12.11	even	2		1296.2.a.n.1.1		2
24.5	odd	2		5184.2.a.bt.1.2		2
24.11	even	2		5184.2.a.bs.1.2		2
36.7	odd	6		432.2.i.d.145.1		4
36.11	even	6		144.2.i.d.49.2		4
36.23	even	6		144.2.i.d.97.2		4
36.31	odd	6		432.2.i.d.289.1		4
72.5	odd	6		576.2.i.j.385.2		4
72.11	even	6		576.2.i.l.193.1		4
72.13	even	6		1728.2.i.i.1153.2		4
72.29	odd	6		576.2.i.j.193.2		4
72.43	odd	6		1728.2.i.j.577.2		4
72.59	even	6		576.2.i.l.385.1		4
72.61	even	6		1728.2.i.i.577.2		4
72.67	odd	6		1728.2.i.j.1153.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.b.25.1	✓	4	9.5	odd	6
72.2.i.b.49.1	yes	4	9.2	odd	6
144.2.i.d.49.2		4	36.11	even	6
144.2.i.d.97.2		4	36.23	even	6
216.2.i.b.73.1		4	9.4	even	3
216.2.i.b.145.1		4	9.7	even	3
432.2.i.d.145.1		4	36.7	odd	6
432.2.i.d.289.1		4	36.31	odd	6
576.2.i.j.193.2		4	72.29	odd	6
576.2.i.j.385.2		4	72.5	odd	6
576.2.i.l.193.1		4	72.11	even	6
576.2.i.l.385.1		4	72.59	even	6
648.2.a.f.1.1		2	3.2	odd	2
648.2.a.g.1.2		2	1.1	even	1	trivial
1296.2.a.n.1.1		2	12.11	even	2
1296.2.a.p.1.2		2	4.3	odd	2
1728.2.i.i.577.2		4	72.61	even	6
1728.2.i.i.1153.2		4	72.13	even	6
1728.2.i.j.577.2		4	72.43	odd	6
1728.2.i.j.1153.2		4	72.67	odd	6
5184.2.a.bo.1.1		2	8.3	odd	2
5184.2.a.bp.1.1		2	8.5	even	2
5184.2.a.bs.1.2		2	24.11	even	2
5184.2.a.bt.1.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.37228 q^{5} -1.37228 q^{7} +O(q^{10})\)	\(q+3.37228 q^{5} -1.37228 q^{7} -1.00000 q^{11} +5.37228 q^{13} -0.372281 q^{17} +6.37228 q^{19} -5.37228 q^{23} +6.37228 q^{25} -1.37228 q^{29} +0.627719 q^{31} -4.62772 q^{35} -2.74456 q^{37} +0.255437 q^{41} +9.74456 q^{43} +1.37228 q^{47} -5.11684 q^{49} +10.7446 q^{53} -3.37228 q^{55} -7.00000 q^{59} -3.37228 q^{61} +18.1168 q^{65} +7.74456 q^{67} +4.00000 q^{71} +5.11684 q^{73} +1.37228 q^{77} -0.627719 q^{79} -15.3723 q^{83} -1.25544 q^{85} -6.00000 q^{89} -7.37228 q^{91} +21.4891 q^{95} -9.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 3 q^{7}+O(q^{10}) \) 2 * q + q^5 + 3 * q^7	\( 2 q + q^{5} + 3 q^{7} - 2 q^{11} + 5 q^{13} + 5 q^{17} + 7 q^{19} - 5 q^{23} + 7 q^{25} + 3 q^{29} + 7 q^{31} - 15 q^{35} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 3 q^{47} + 7 q^{49} + 10 q^{53} - q^{55} - 14 q^{59} - q^{61} + 19 q^{65} + 4 q^{67} + 8 q^{71} - 7 q^{73} - 3 q^{77} - 7 q^{79} - 25 q^{83} - 14 q^{85} - 12 q^{89} - 9 q^{91} + 20 q^{95} - 8 q^{97}+O(q^{100}) \) 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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