LMFDB - Embedded newform 9072.2.a.bu.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9072,2,Mod(1,9072)] 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("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,-3,0,0,0,6,0,3,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.523976$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20147 q^{5} -1.00000 q^{7} +5.20147 q^{11} +3.15352 q^{13} +3.24943 q^{17} -7.45090 q^{19} +4.40294 q^{23} -0.153520 q^{25} +1.15352 q^{29} -2.00000 q^{31} -2.20147 q^{35} +5.00000 q^{37} +11.4509 q^{41} -9.29738 q^{43} -1.04795 q^{47} +1.00000 q^{49} +0.249425 q^{53} +11.4509 q^{55} -8.09591 q^{59} +8.60442 q^{61} +6.94239 q^{65} +7.60442 q^{67} +9.60442 q^{71} +0.846480 q^{73} -5.20147 q^{77} +7.60442 q^{79} +11.4509 q^{83} +7.15352 q^{85} -9.24943 q^{89} -3.15352 q^{91} -16.4029 q^{95} -3.45090 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{23} + 6 q^{25} - 3 q^{29} - 6 q^{31} + 3 q^{35} + 15 q^{37} + 12 q^{41} - 12 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 3 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 + 6 * q^11 + 3 * q^13 - 3 * q^17 - 6 * q^23 + 6 * q^25 - 3 * q^29 - 6 * q^31 + 3 * q^35 + 15 * q^37 + 12 * q^41 - 12 * q^43 + 3 * q^49 - 12 * q^53 + 12 * q^55 - 18 * q^59 - 3 * q^61 + 21 * q^65 - 6 * q^67 + 9 * q^73 - 6 * q^77 - 6 * q^79 + 12 * q^83 + 15 * q^85 - 15 * q^89 - 3 * q^91 - 30 * q^95 + 12 * 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
$5$	2.20147	0.984528	0.492264	−	0.870446i	$-0.336169\pi$
$5$	2.20147	0.984528	0.492264	+	0.870446i	$0.336169\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.20147	1.56830	0.784151	−	0.620569i	$-0.213099\pi$
$11$	5.20147	1.56830	0.784151	+	0.620569i	$0.213099\pi$
$12$	0	0
$13$	3.15352	0.874629	0.437314	−	0.899309i	$-0.355930\pi$
$13$	3.15352	0.874629	0.437314	+	0.899309i	$0.355930\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.24943	0.788101	0.394051	−	0.919089i	$-0.371073\pi$
$17$	3.24943	0.788101	0.394051	+	0.919089i	$0.371073\pi$
$18$	0	0
$19$	−7.45090	−1.70935	−0.854677	−	0.519161i	$-0.826245\pi$
$19$	−7.45090	−1.70935	−0.854677	+	0.519161i	$0.826245\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.40294	0.918077	0.459039	−	0.888416i	$-0.348194\pi$
$23$	4.40294	0.918077	0.459039	+	0.888416i	$0.348194\pi$
$24$	0	0
$25$	−0.153520	−0.0307039
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.15352	0.214203	0.107102	−	0.994248i	$-0.465843\pi$
$29$	1.15352	0.214203	0.107102	+	0.994248i	$0.465843\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.20147	−0.372117
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.4509	1.78833	0.894165	−	0.447738i	$-0.147770\pi$
$41$	11.4509	1.78833	0.894165	+	0.447738i	$0.147770\pi$
$42$	0	0
$43$	−9.29738	−1.41784	−0.708918	−	0.705290i	$-0.750817\pi$
$43$	−9.29738	−1.41784	−0.708918	+	0.705290i	$0.750817\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.04795	−0.152860	−0.0764298	−	0.997075i	$-0.524352\pi$
$47$	−1.04795	−0.152860	−0.0764298	+	0.997075i	$0.524352\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.249425	0.0342612	0.0171306	−	0.999853i	$-0.494547\pi$
$53$	0.249425	0.0342612	0.0171306	+	0.999853i	$0.494547\pi$
$54$	0	0
$55$	11.4509	1.54404
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.09591	−1.05400	−0.526999	−	0.849866i	$-0.676683\pi$
$59$	−8.09591	−1.05400	−0.526999	+	0.849866i	$0.676683\pi$
$60$	0	0
$61$	8.60442	1.10168	0.550841	−	0.834610i	$-0.314307\pi$
$61$	8.60442	1.10168	0.550841	+	0.834610i	$0.314307\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.94239	0.861097
$66$	0	0
$67$	7.60442	0.929027	0.464514	−	0.885566i	$-0.346229\pi$
$67$	7.60442	0.929027	0.464514	+	0.885566i	$0.346229\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.60442	1.13983	0.569917	−	0.821702i	$-0.306975\pi$
$71$	9.60442	1.13983	0.569917	+	0.821702i	$0.306975\pi$
$72$	0	0
$73$	0.846480	0.0990730	0.0495365	−	0.998772i	$-0.484226\pi$
$73$	0.846480	0.0990730	0.0495365	+	0.998772i	$0.484226\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.20147	−0.592763
$78$	0	0
$79$	7.60442	0.855564	0.427782	−	0.903882i	$-0.359295\pi$
$79$	7.60442	0.855564	0.427782	+	0.903882i	$0.359295\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.4509	1.25690	0.628450	−	0.777850i	$-0.283690\pi$
$83$	11.4509	1.25690	0.628450	+	0.777850i	$0.283690\pi$
$84$	0	0
$85$	7.15352	0.775908
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.24943	−0.980437	−0.490219	−	0.871600i	$-0.663083\pi$
$89$	−9.24943	−0.980437	−0.490219	+	0.871600i	$0.663083\pi$
$90$	0	0
$91$	−3.15352	−0.330579
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.4029	−1.68291
$96$	0	0
$97$	−3.45090	−0.350386	−0.175193	−	0.984534i	$-0.556055\pi$
$97$	−3.45090	−0.350386	−0.175193	+	0.984534i	$0.556055\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.4029	1.63215	0.816077	−	0.577943i	$-0.196144\pi$
$101$	16.4029	1.63215	0.816077	+	0.577943i	$0.196144\pi$
$102$	0	0
$103$	1.14386	0.112708	0.0563539	−	0.998411i	$-0.482053\pi$
$103$	1.14386	0.112708	0.0563539	+	0.998411i	$0.482053\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.10557	0.880268	0.440134	−	0.897932i	$-0.354931\pi$
$107$	9.10557	0.880268	0.440134	+	0.897932i	$0.354931\pi$
$108$	0	0
$109$	−13.7483	−1.31685	−0.658423	−	0.752648i	$-0.728776\pi$
$109$	−13.7483	−1.31685	−0.658423	+	0.752648i	$0.728776\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.9018	−1.87220	−0.936102	−	0.351729i	$-0.885594\pi$
$113$	−19.9018	−1.87220	−0.936102	+	0.351729i	$0.885594\pi$
$114$	0	0
$115$	9.69296	0.903873
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.24943	−0.297874
$120$	0	0
$121$	16.0553	1.45957
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3453	−1.01476
$126$	0	0
$127$	−9.29738	−0.825009	−0.412504	−	0.910956i	$-0.635346\pi$
$127$	−9.29738	−0.825009	−0.412504	+	0.910956i	$0.635346\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.54680	0.659367	0.329684	−	0.944091i	$-0.393058\pi$
$131$	7.54680	0.659367	0.329684	+	0.944091i	$0.393058\pi$
$132$	0	0
$133$	7.45090	0.646075
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.40294	0.119862	0.0599308	−	0.998203i	$-0.480912\pi$
$137$	1.40294	0.119862	0.0599308	+	0.998203i	$0.480912\pi$
$138$	0	0
$139$	15.4509	1.31053	0.655264	−	0.755400i	$-0.272557\pi$
$139$	15.4509	1.31053	0.655264	+	0.755400i	$0.272557\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.4029	1.37168
$144$	0	0
$145$	2.53944	0.210889
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.4029	−1.09801	−0.549006	−	0.835818i	$-0.684994\pi$
$149$	−13.4029	−1.09801	−0.549006	+	0.835818i	$0.684994\pi$
$150$	0	0
$151$	−11.6044	−0.944354	−0.472177	−	0.881504i	$-0.656532\pi$
$151$	−11.6044	−0.944354	−0.472177	+	0.881504i	$0.656532\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.40294	−0.353653
$156$	0	0
$157$	−24.3624	−1.94433	−0.972164	−	0.234302i	$-0.924719\pi$
$157$	−24.3624	−1.94433	−0.972164	+	0.234302i	$0.924719\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.40294	−0.347001
$162$	0	0
$163$	−5.60442	−0.438972	−0.219486	−	0.975616i	$-0.570438\pi$
$163$	−5.60442	−0.438972	−0.219486	+	0.975616i	$0.570438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.4509	−1.81468	−0.907342	−	0.420392i	$-0.861892\pi$
$167$	−23.4509	−1.81468	−0.907342	+	0.420392i	$0.861892\pi$
$168$	0	0
$169$	−3.05531	−0.235024
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.5085	1.71129	0.855645	−	0.517563i	$-0.173161\pi$
$173$	22.5085	1.71129	0.855645	+	0.517563i	$0.173161\pi$
$174$	0	0
$175$	0.153520	0.0116050
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.49885	0.485747	0.242873	−	0.970058i	$-0.421910\pi$
$179$	6.49885	0.485747	0.242873	+	0.970058i	$0.421910\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.0074	0.809277
$186$	0	0
$187$	16.9018	1.23598
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.5085	−0.977442	−0.488721	−	0.872440i	$-0.662536\pi$
$191$	−13.5085	−0.977442	−0.488721	+	0.872440i	$0.662536\pi$
$192$	0	0
$193$	17.7483	1.27755	0.638774	−	0.769394i	$-0.279442\pi$
$193$	17.7483	1.27755	0.638774	+	0.769394i	$0.279442\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.2088	1.15483	0.577416	−	0.816450i	$-0.304061\pi$
$197$	16.2088	1.15483	0.577416	+	0.816450i	$0.304061\pi$
$198$	0	0
$199$	7.14386	0.506415	0.253207	−	0.967412i	$-0.418515\pi$
$199$	7.14386	0.506415	0.253207	+	0.967412i	$0.418515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.15352	−0.0809612
$204$	0	0
$205$	25.2088	1.76066
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−38.7556	−2.68078
$210$	0	0
$211$	−3.29738	−0.227001	−0.113500	−	0.993538i	$-0.536206\pi$
$211$	−3.29738	−0.227001	−0.113500	+	0.993538i	$0.536206\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−20.4679	−1.39590
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.2471	0.689296
$222$	0	0
$223$	23.2088	1.55418	0.777089	−	0.629390i	$-0.216695\pi$
$223$	23.2088	1.55418	0.777089	+	0.629390i	$0.216695\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.6620	0.774036	0.387018	−	0.922072i	$-0.373505\pi$
$227$	11.6620	0.774036	0.387018	+	0.922072i	$0.373505\pi$
$228$	0	0
$229$	1.39558	0.0922227	0.0461114	−	0.998936i	$-0.485317\pi$
$229$	1.39558	0.0922227	0.0461114	+	0.998936i	$0.485317\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.75057	0.573269	0.286635	−	0.958040i	$-0.407463\pi$
$233$	8.75057	0.573269	0.286635	+	0.958040i	$0.407463\pi$
$234$	0	0
$235$	−2.30704	−0.150495
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.70262	−0.304187	−0.152094	−	0.988366i	$-0.548602\pi$
$239$	−4.70262	−0.304187	−0.152094	+	0.988366i	$0.548602\pi$
$240$	0	0
$241$	−4.60442	−0.296597	−0.148298	−	0.988943i	$-0.547380\pi$
$241$	−4.60442	−0.296597	−0.148298	+	0.988943i	$0.547380\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.20147	0.140647
$246$	0	0
$247$	−23.4966	−1.49505
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3.14386	−0.198439	−0.0992193	−	0.995066i	$-0.531635\pi$
$251$	−3.14386	−0.198439	−0.0992193	+	0.995066i	$0.531635\pi$
$252$	0	0
$253$	22.9018	1.43982
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.1512	−1.25700	−0.628499	−	0.777810i	$-0.716330\pi$
$257$	−20.1512	−1.25700	−0.628499	+	0.777810i	$0.716330\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.2015	1.06069	0.530344	−	0.847782i	$-0.322063\pi$
$263$	17.2015	1.06069	0.530344	+	0.847782i	$0.322063\pi$
$264$	0	0
$265$	0.549103	0.0337311
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.75057	−0.167706	−0.0838528	−	0.996478i	$-0.526723\pi$
$269$	−2.75057	−0.167706	−0.0838528	+	0.996478i	$0.526723\pi$
$270$	0	0
$271$	12.8561	0.780955	0.390477	−	0.920612i	$-0.372310\pi$
$271$	12.8561	0.780955	0.390477	+	0.920612i	$0.372310\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.798528	−0.0481530
$276$	0	0
$277$	−16.7483	−1.00631	−0.503153	−	0.864197i	$-0.667827\pi$
$277$	−16.7483	−1.00631	−0.503153	+	0.864197i	$0.667827\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.30704	−0.316591	−0.158296	−	0.987392i	$-0.550600\pi$
$281$	−5.30704	−0.316591	−0.158296	+	0.987392i	$0.550600\pi$
$282$	0	0
$283$	−12.3527	−0.734291	−0.367146	−	0.930163i	$-0.619665\pi$
$283$	−12.3527	−0.734291	−0.367146	+	0.930163i	$0.619665\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.4509	−0.675925
$288$	0	0
$289$	−6.44124	−0.378896
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.2974	−0.601579	−0.300790	−	0.953691i	$-0.597250\pi$
$293$	−10.2974	−0.601579	−0.300790	+	0.953691i	$0.597250\pi$
$294$	0	0
$295$	−17.8229	−1.03769
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.8848	0.802977
$300$	0	0
$301$	9.29738	0.535892
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.9424	1.08464
$306$	0	0
$307$	0.307039	0.0175236	0.00876182	−	0.999962i	$-0.497211\pi$
$307$	0.307039	0.0175236	0.00876182	+	0.999962i	$0.497211\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.04795	0.0594240	0.0297120	−	0.999559i	$-0.490541\pi$
$311$	1.04795	0.0594240	0.0297120	+	0.999559i	$0.490541\pi$
$312$	0	0
$313$	32.0553	1.81187	0.905937	−	0.423413i	$-0.139168\pi$
$313$	32.0553	1.81187	0.905937	+	0.423413i	$0.139168\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.44354	0.361905	0.180953	−	0.983492i	$-0.442082\pi$
$317$	6.44354	0.361905	0.180953	+	0.983492i	$0.442082\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−24.2111	−1.34714
$324$	0	0
$325$	−0.484127	−0.0268545
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.04795	0.0577755
$330$	0	0
$331$	20.6141	1.13305	0.566526	−	0.824044i	$-0.308287\pi$
$331$	20.6141	1.13305	0.566526	+	0.824044i	$0.308287\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.7409	0.914654
$336$	0	0
$337$	26.4606	1.44140	0.720699	−	0.693248i	$-0.243821\pi$
$337$	26.4606	1.44140	0.720699	+	0.693248i	$0.243821\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.4029	−0.563351
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.49149	−0.241116	−0.120558	−	0.992706i	$-0.538468\pi$
$347$	−4.49149	−0.241116	−0.120558	+	0.992706i	$0.538468\pi$
$348$	0	0
$349$	3.75794	0.201158	0.100579	−	0.994929i	$-0.467931\pi$
$349$	3.75794	0.201158	0.100579	+	0.994929i	$0.467931\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.2591	1.02506	0.512529	−	0.858670i	$-0.328709\pi$
$353$	19.2591	1.02506	0.512529	+	0.858670i	$0.328709\pi$
$354$	0	0
$355$	21.1439	1.12220
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.10557	0.480573	0.240287	−	0.970702i	$-0.422759\pi$
$359$	9.10557	0.480573	0.240287	+	0.970702i	$0.422759\pi$
$360$	0	0
$361$	36.5159	1.92189
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.86350	0.0975402
$366$	0	0
$367$	−4.59476	−0.239844	−0.119922	−	0.992783i	$-0.538264\pi$
$367$	−4.59476	−0.239844	−0.119922	+	0.992783i	$0.538264\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.249425	−0.0129495
$372$	0	0
$373$	−21.3624	−1.10610	−0.553050	−	0.833148i	$-0.686536\pi$
$373$	−21.3624	−1.10610	−0.553050	+	0.833148i	$0.686536\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.63765	0.187348
$378$	0	0
$379$	35.1203	1.80401	0.902004	−	0.431728i	$-0.142096\pi$
$379$	35.1203	1.80401	0.902004	+	0.431728i	$0.142096\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.19411	0.163211	0.0816057	−	0.996665i	$-0.473995\pi$
$383$	3.19411	0.163211	0.0816057	+	0.996665i	$0.473995\pi$
$384$	0	0
$385$	−11.4509	−0.583592
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.8059	−0.750688	−0.375344	−	0.926886i	$-0.622475\pi$
$389$	−14.8059	−0.750688	−0.375344	+	0.926886i	$0.622475\pi$
$390$	0	0
$391$	14.3070	0.723538
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.7409	0.842327
$396$	0	0
$397$	26.8921	1.34968	0.674839	−	0.737965i	$-0.264213\pi$
$397$	26.8921	1.34968	0.674839	+	0.737965i	$0.264213\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.4029	−0.968937	−0.484468	−	0.874809i	$-0.660987\pi$
$401$	−19.4029	−0.968937	−0.484468	+	0.874809i	$0.660987\pi$
$402$	0	0
$403$	−6.30704	−0.314176
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.0074	1.28914
$408$	0	0
$409$	−9.39558	−0.464582	−0.232291	−	0.972646i	$-0.574622\pi$
$409$	−9.39558	−0.464582	−0.232291	+	0.972646i	$0.574622\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.09591	0.398373
$414$	0	0
$415$	25.2088	1.23745
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.6597	−1.20471	−0.602353	−	0.798230i	$-0.705770\pi$
$419$	−24.6597	−1.20471	−0.602353	+	0.798230i	$0.705770\pi$
$420$	0	0
$421$	−14.2088	−0.692496	−0.346248	−	0.938143i	$-0.612544\pi$
$421$	−14.2088	−0.692496	−0.346248	+	0.938143i	$0.612544\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.498850	−0.0241978
$426$	0	0
$427$	−8.60442	−0.416397
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.9977	−0.626077	−0.313039	−	0.949740i	$-0.601347\pi$
$431$	−12.9977	−0.626077	−0.313039	+	0.949740i	$0.601347\pi$
$432$	0	0
$433$	−0.911456	−0.0438018	−0.0219009	−	0.999760i	$-0.506972\pi$
$433$	−0.911456	−0.0438018	−0.0219009	+	0.999760i	$0.506972\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−32.8059	−1.56932
$438$	0	0
$439$	10.5491	0.503481	0.251741	−	0.967795i	$-0.418997\pi$
$439$	10.5491	0.503481	0.251741	+	0.967795i	$0.418997\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.4006	−0.826730	−0.413365	−	0.910566i	$-0.635647\pi$
$443$	−17.4006	−0.826730	−0.413365	+	0.910566i	$0.635647\pi$
$444$	0	0
$445$	−20.3624	−0.965268
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.748275	−0.0353133	−0.0176566	−	0.999844i	$-0.505621\pi$
$449$	−0.748275	−0.0353133	−0.0176566	+	0.999844i	$0.505621\pi$
$450$	0	0
$451$	59.5615	2.80464
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.94239	−0.325464
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.0457	1.77196	0.885981	−	0.463721i	$-0.153486\pi$
$461$	38.0457	1.77196	0.885981	+	0.463721i	$0.153486\pi$
$462$	0	0
$463$	25.8921	1.20331	0.601655	−	0.798756i	$-0.294508\pi$
$463$	25.8921	1.20331	0.601655	+	0.798756i	$0.294508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.0627	0.789566	0.394783	−	0.918774i	$-0.370820\pi$
$467$	17.0627	0.789566	0.394783	+	0.918774i	$0.370820\pi$
$468$	0	0
$469$	−7.60442	−0.351139
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−48.3601	−2.22360
$474$	0	0
$475$	1.14386	0.0524838
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.80589	0.128204	0.0641022	−	0.997943i	$-0.479582\pi$
$479$	2.80589	0.128204	0.0641022	+	0.997943i	$0.479582\pi$
$480$	0	0
$481$	15.7676	0.718941
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.59706	−0.344965
$486$	0	0
$487$	6.50621	0.294825	0.147412	−	0.989075i	$-0.452906\pi$
$487$	6.50621	0.294825	0.147412	+	0.989075i	$0.452906\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.61408	0.208230	0.104115	−	0.994565i	$-0.466799\pi$
$491$	4.61408	0.208230	0.104115	+	0.994565i	$0.466799\pi$
$492$	0	0
$493$	3.74828	0.168814
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.60442	−0.430817
$498$	0	0
$499$	35.4966	1.58904	0.794522	−	0.607235i	$-0.207722\pi$
$499$	35.4966	1.58904	0.794522	+	0.607235i	$0.207722\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.9211	−0.843651	−0.421825	−	0.906677i	$-0.638611\pi$
$503$	−18.9211	−0.843651	−0.421825	+	0.906677i	$0.638611\pi$
$504$	0	0
$505$	36.1106	1.60690
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.8059	0.656260	0.328130	−	0.944633i	$-0.393582\pi$
$509$	14.8059	0.656260	0.328130	+	0.944633i	$0.393582\pi$
$510$	0	0
$511$	−0.846480	−0.0374461
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.51817	0.110964
$516$	0	0
$517$	−5.45090	−0.239730
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7100	0.556834	0.278417	−	0.960460i	$-0.410190\pi$
$521$	12.7100	0.556834	0.278417	+	0.960460i	$0.410190\pi$
$522$	0	0
$523$	−0.352692	−0.0154222	−0.00771108	−	0.999970i	$-0.502455\pi$
$523$	−0.352692	−0.0154222	−0.00771108	+	0.999970i	$0.502455\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.49885	−0.283094
$528$	0	0
$529$	−3.61408	−0.157134
$530$	0	0
$531$	0	0
$532$	0	0
$533$	36.1106	1.56412
$534$	0	0
$535$	20.0457	0.866649
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.20147	0.224043
$540$	0	0
$541$	−22.6930	−0.975647	−0.487823	−	0.872942i	$-0.662209\pi$
$541$	−22.6930	−0.975647	−0.487823	+	0.872942i	$0.662209\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−30.2664	−1.29647
$546$	0	0
$547$	1.49379	0.0638698	0.0319349	−	0.999490i	$-0.489833\pi$
$547$	1.49379	0.0638698	0.0319349	+	0.999490i	$0.489833\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.59476	−0.366149
$552$	0	0
$553$	−7.60442	−0.323373
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.50115	0.360205	0.180103	−	0.983648i	$-0.442357\pi$
$557$	8.50115	0.360205	0.180103	+	0.983648i	$0.442357\pi$
$558$	0	0
$559$	−29.3195	−1.24008
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.4006	0.986220	0.493110	−	0.869967i	$-0.335860\pi$
$563$	23.4006	0.986220	0.493110	+	0.869967i	$0.335860\pi$
$564$	0	0
$565$	−43.8133	−1.84324
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.11293	0.340112	0.170056	−	0.985434i	$-0.445605\pi$
$569$	8.11293	0.340112	0.170056	+	0.985434i	$0.445605\pi$
$570$	0	0
$571$	−16.5948	−0.694469	−0.347234	−	0.937778i	$-0.612879\pi$
$571$	−16.5948	−0.694469	−0.347234	+	0.937778i	$0.612879\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.675938	−0.0281886
$576$	0	0
$577$	−10.8921	−0.453445	−0.226723	−	0.973959i	$-0.572801\pi$
$577$	−10.8921	−0.453445	−0.226723	+	0.973959i	$0.572801\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11.4509	−0.475063
$582$	0	0
$583$	1.29738	0.0537319
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.3195	−1.78799	−0.893993	−	0.448081i	$-0.852107\pi$
$587$	−43.3195	−1.78799	−0.893993	+	0.448081i	$0.852107\pi$
$588$	0	0
$589$	14.9018	0.614018
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.4583	−1.16864	−0.584320	−	0.811523i	$-0.698639\pi$
$593$	−28.4583	−1.16864	−0.584320	+	0.811523i	$0.698639\pi$
$594$	0	0
$595$	−7.15352	−0.293266
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.7003	−1.21352	−0.606761	−	0.794884i	$-0.707532\pi$
$599$	−29.7003	−1.21352	−0.606761	+	0.794884i	$0.707532\pi$
$600$	0	0
$601$	37.5062	1.52991	0.764955	−	0.644084i	$-0.222761\pi$
$601$	37.5062	1.52991	0.764955	+	0.644084i	$0.222761\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	35.3453	1.43699
$606$	0	0
$607$	−2.65973	−0.107955	−0.0539776	−	0.998542i	$-0.517190\pi$
$607$	−2.65973	−0.107955	−0.0539776	+	0.998542i	$0.517190\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.30474	−0.133695
$612$	0	0
$613$	−41.6694	−1.68301	−0.841505	−	0.540249i	$-0.818330\pi$
$613$	−41.6694	−1.68301	−0.841505	+	0.540249i	$0.818330\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.5519	1.75333	0.876666	−	0.481099i	$-0.159762\pi$
$617$	43.5519	1.75333	0.876666	+	0.481099i	$0.159762\pi$
$618$	0	0
$619$	14.0650	0.565319	0.282660	−	0.959220i	$-0.408783\pi$
$619$	14.0650	0.565319	0.282660	+	0.959220i	$0.408783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.24943	0.370570
$624$	0	0
$625$	−24.2088	−0.968353
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.2471	0.647815
$630$	0	0
$631$	0.594756	0.0236769	0.0118384	−	0.999930i	$-0.496232\pi$
$631$	0.594756	0.0236769	0.0118384	+	0.999930i	$0.496232\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−20.4679	−0.812245
$636$	0	0
$637$	3.15352	0.124947
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.69066	0.303763	0.151881	−	0.988399i	$-0.451467\pi$
$641$	7.69066	0.303763	0.151881	+	0.988399i	$0.451467\pi$
$642$	0	0
$643$	−49.8229	−1.96482	−0.982412	−	0.186727i	$-0.940212\pi$
$643$	−49.8229	−1.96482	−0.982412	+	0.186727i	$0.940212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.2761	1.11165	0.555824	−	0.831300i	$-0.312403\pi$
$647$	28.2761	1.11165	0.555824	+	0.831300i	$0.312403\pi$
$648$	0	0
$649$	−42.1106	−1.65299
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.55416	0.373883	0.186942	−	0.982371i	$-0.440142\pi$
$653$	9.55416	0.373883	0.186942	+	0.982371i	$0.440142\pi$
$654$	0	0
$655$	16.6141	0.649166
$656$	0	0
$657$	0	0
$658$	0	0
$659$	48.1992	1.87757	0.938787	−	0.344499i	$-0.111951\pi$
$659$	48.1992	1.87757	0.938787	+	0.344499i	$0.111951\pi$
$660$	0	0
$661$	2.05531	0.0799425	0.0399712	−	0.999201i	$-0.487273\pi$
$661$	2.05531	0.0799425	0.0399712	+	0.999201i	$0.487273\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.4029	0.636079
$666$	0	0
$667$	5.07888	0.196655
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.7556	1.72777
$672$	0	0
$673$	3.79117	0.146139	0.0730694	−	0.997327i	$-0.476721\pi$
$673$	3.79117	0.146139	0.0730694	+	0.997327i	$0.476721\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.5136	−0.865267	−0.432633	−	0.901570i	$-0.642416\pi$
$677$	−22.5136	−0.865267	−0.432633	+	0.901570i	$0.642416\pi$
$678$	0	0
$679$	3.45090	0.132433
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.3933	−0.589008	−0.294504	−	0.955650i	$-0.595154\pi$
$683$	−15.3933	−0.589008	−0.294504	+	0.955650i	$0.595154\pi$
$684$	0	0
$685$	3.08854	0.118007
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.786567	0.0299658
$690$	0	0
$691$	23.2088	0.882906	0.441453	−	0.897284i	$-0.354463\pi$
$691$	23.2088	0.882906	0.441453	+	0.897284i	$0.354463\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34.0147	1.29025
$696$	0	0
$697$	37.2088	1.40939
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	−37.2545	−1.40508
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.4029	−0.616896
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.80589	−0.329783
$714$	0	0
$715$	36.1106	1.35046
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.2111	−0.679161	−0.339580	−	0.940577i	$-0.610285\pi$
$719$	−18.2111	−0.679161	−0.339580	+	0.940577i	$0.610285\pi$
$720$	0	0
$721$	−1.14386	−0.0425995
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.177088	−0.00657688
$726$	0	0
$727$	−8.83682	−0.327739	−0.163870	−	0.986482i	$-0.552398\pi$
$727$	−8.83682	−0.327739	−0.163870	+	0.986482i	$0.552398\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−30.2111	−1.11740
$732$	0	0
$733$	−44.6404	−1.64883	−0.824416	−	0.565985i	$-0.808496\pi$
$733$	−44.6404	−1.64883	−0.824416	+	0.565985i	$0.808496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.5542	1.45700
$738$	0	0
$739$	−11.6044	−0.426875	−0.213438	−	0.976957i	$-0.568466\pi$
$739$	−11.6044	−0.426875	−0.213438	+	0.976957i	$0.568466\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.7939	0.982974	0.491487	−	0.870885i	$-0.336454\pi$
$743$	26.7939	0.982974	0.491487	+	0.870885i	$0.336454\pi$
$744$	0	0
$745$	−29.5062	−1.08102
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.10557	−0.332710
$750$	0	0
$751$	18.2185	0.664802	0.332401	−	0.943138i	$-0.392141\pi$
$751$	18.2185	0.664802	0.332401	+	0.943138i	$0.392141\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−25.5468	−0.929743
$756$	0	0
$757$	12.9018	0.468924	0.234462	−	0.972125i	$-0.424667\pi$
$757$	12.9018	0.468924	0.234462	+	0.972125i	$0.424667\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.2162	−0.660337	−0.330168	−	0.943922i	$-0.607106\pi$
$761$	−18.2162	−0.660337	−0.330168	+	0.943922i	$0.607106\pi$
$762$	0	0
$763$	13.7483	0.497721
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.5306	−0.921856
$768$	0	0
$769$	38.6044	1.39211	0.696055	−	0.717988i	$-0.254937\pi$
$769$	38.6044	1.39211	0.696055	+	0.717988i	$0.254937\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.9594	−1.22144	−0.610718	−	0.791849i	$-0.709119\pi$
$773$	−33.9594	−1.22144	−0.610718	+	0.791849i	$0.709119\pi$
$774$	0	0
$775$	0.307039	0.0110292
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−85.3195	−3.05689
$780$	0	0
$781$	49.9571	1.78761
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−53.6330	−1.91425
$786$	0	0
$787$	6.85614	0.244395	0.122198	−	0.992506i	$-0.461006\pi$
$787$	6.85614	0.244395	0.122198	+	0.992506i	$0.461006\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.9018	0.707626
$792$	0	0
$793$	27.1342	0.963564
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.6501	−0.943994	−0.471997	−	0.881600i	$-0.656467\pi$
$797$	−26.6501	−0.943994	−0.471997	+	0.881600i	$0.656467\pi$
$798$	0	0
$799$	−3.40524	−0.120469
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.40294	0.155377
$804$	0	0
$805$	−9.69296	−0.341632
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.7100	1.60708	0.803539	−	0.595252i	$-0.202948\pi$
$809$	45.7100	1.60708	0.803539	+	0.595252i	$0.202948\pi$
$810$	0	0
$811$	−33.2088	−1.16612	−0.583060	−	0.812429i	$-0.698145\pi$
$811$	−33.2088	−1.16612	−0.583060	+	0.812429i	$0.698145\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.3380	−0.432180
$816$	0	0
$817$	69.2738	2.42358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.304740	−0.0106355	−0.00531774	−	0.999986i	$-0.501693\pi$
$821$	−0.304740	−0.0106355	−0.00531774	+	0.999986i	$0.501693\pi$
$822$	0	0
$823$	−46.4177	−1.61802	−0.809009	−	0.587796i	$-0.799996\pi$
$823$	−46.4177	−1.61802	−0.809009	+	0.587796i	$0.799996\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−40.8133	−1.41922	−0.709608	−	0.704597i	$-0.751128\pi$
$827$	−40.8133	−1.41922	−0.709608	+	0.704597i	$0.751128\pi$
$828$	0	0
$829$	−7.40524	−0.257195	−0.128597	−	0.991697i	$-0.541047\pi$
$829$	−7.40524	−0.257195	−0.128597	+	0.991697i	$0.541047\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.24943	0.112586
$834$	0	0
$835$	−51.6265	−1.78661
$836$	0	0
$837$	0	0
$838$	0	0
$839$	18.3720	0.634272	0.317136	−	0.948380i	$-0.397279\pi$
$839$	18.3720	0.634272	0.317136	+	0.948380i	$0.397279\pi$
$840$	0	0
$841$	−27.6694	−0.954117
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.72619	−0.231388
$846$	0	0
$847$	−16.0553	−0.551667
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22.0147	0.754655
$852$	0	0
$853$	37.5615	1.28608	0.643041	−	0.765832i	$-0.277672\pi$
$853$	37.5615	1.28608	0.643041	+	0.765832i	$0.277672\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.15122	0.0734843	0.0367421	−	0.999325i	$-0.488302\pi$
$857$	2.15122	0.0734843	0.0367421	+	0.999325i	$0.488302\pi$
$858$	0	0
$859$	−14.2877	−0.487491	−0.243745	−	0.969839i	$-0.578376\pi$
$859$	−14.2877	−0.487491	−0.243745	+	0.969839i	$0.578376\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.6810	0.942272	0.471136	−	0.882061i	$-0.343844\pi$
$863$	27.6810	0.942272	0.471136	+	0.882061i	$0.343844\pi$
$864$	0	0
$865$	49.5519	1.68481
$866$	0	0
$867$	0	0
$868$	0	0
$869$	39.5542	1.34178
$870$	0	0
$871$	23.9807	0.812554
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.3453	0.383542
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.06498	0.136953	0.0684763	−	0.997653i	$-0.478186\pi$
$881$	4.06498	0.136953	0.0684763	+	0.997653i	$0.478186\pi$
$882$	0	0
$883$	20.5255	0.690739	0.345370	−	0.938467i	$-0.387754\pi$
$883$	20.5255	0.690739	0.345370	+	0.938467i	$0.387754\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.1152	0.339636	0.169818	−	0.985475i	$-0.445682\pi$
$887$	10.1152	0.339636	0.169818	+	0.985475i	$0.445682\pi$
$888$	0	0
$889$	9.29738	0.311824
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.80819	0.261291
$894$	0	0
$895$	14.3070	0.478232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.30704	−0.0769441
$900$	0	0
$901$	0.810488	0.0270013
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.40294	0.146359
$906$	0	0
$907$	−22.3956	−0.743633	−0.371817	−	0.928306i	$-0.621265\pi$
$907$	−22.3956	−0.743633	−0.371817	+	0.928306i	$0.621265\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.8036	1.11996	0.559981	−	0.828505i	$-0.310808\pi$
$911$	33.8036	1.11996	0.559981	+	0.828505i	$0.310808\pi$
$912$	0	0
$913$	59.5615	1.97120
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.54680	−0.249217
$918$	0	0
$919$	−35.6044	−1.17448	−0.587241	−	0.809412i	$-0.699786\pi$
$919$	−35.6044	−1.17448	−0.587241	+	0.809412i	$0.699786\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	30.2877	0.996932
$924$	0	0
$925$	−0.767598	−0.0252385
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.4126	−1.06342	−0.531712	−	0.846926i	$-0.678451\pi$
$929$	−32.4126	−1.06342	−0.531712	+	0.846926i	$0.678451\pi$
$930$	0	0
$931$	−7.45090	−0.244193
$932$	0	0
$933$	0	0
$934$	0	0
$935$	37.2088	1.21686
$936$	0	0
$937$	−4.34303	−0.141881	−0.0709403	−	0.997481i	$-0.522600\pi$
$937$	−4.34303	−0.141881	−0.0709403	+	0.997481i	$0.522600\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.9401	−1.04122	−0.520609	−	0.853796i	$-0.674295\pi$
$941$	−31.9401	−1.04122	−0.520609	+	0.853796i	$0.674295\pi$
$942$	0	0
$943$	50.4177	1.64183
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.5136	1.31651	0.658257	−	0.752793i	$-0.271294\pi$
$947$	40.5136	1.31651	0.658257	+	0.752793i	$0.271294\pi$
$948$	0	0
$949$	2.66939	0.0866522
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.74828	0.315778	0.157889	−	0.987457i	$-0.449531\pi$
$953$	9.74828	0.315778	0.157889	+	0.987457i	$0.449531\pi$
$954$	0	0
$955$	−29.7386	−0.962319
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.40294	−0.0453034
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	39.0723	1.25778
$966$	0	0
$967$	−29.6930	−0.954861	−0.477431	−	0.878669i	$-0.658432\pi$
$967$	−29.6930	−0.954861	−0.477431	+	0.878669i	$0.658432\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.9954	−1.02678	−0.513391	−	0.858155i	$-0.671611\pi$
$971$	−31.9954	−1.02678	−0.513391	+	0.858155i	$0.671611\pi$
$972$	0	0
$973$	−15.4509	−0.495333
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.80819	0.0578491	0.0289245	−	0.999582i	$-0.490792\pi$
$977$	1.80819	0.0578491	0.0289245	+	0.999582i	$0.490792\pi$
$978$	0	0
$979$	−48.1106	−1.53762
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43.5468	−1.38893	−0.694464	−	0.719528i	$-0.744358\pi$
$983$	−43.5468	−1.38893	−0.694464	+	0.719528i	$0.744358\pi$
$984$	0	0
$985$	35.6833	1.13696
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−40.9358	−1.30168
$990$	0	0
$991$	33.1010	1.05149	0.525743	−	0.850643i	$-0.323787\pi$
$991$	33.1010	1.05149	0.525743	+	0.850643i	$0.323787\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.7270	0.498580
$996$	0	0
$997$	−0.539441	−0.0170843	−0.00854214	−	0.999964i	$-0.502719\pi$
$997$	−0.539441	−0.0170843	−0.00854214	+	0.999964i	$0.502719\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bu.1.3		3
3.2	odd	2		9072.2.a.cb.1.1		3
4.3	odd	2		567.2.a.e.1.2	✓	3
12.11	even	2		567.2.a.f.1.2	yes	3
28.27	even	2		3969.2.a.o.1.2		3
36.7	odd	6		567.2.f.m.190.2		6
36.11	even	6		567.2.f.l.190.2		6
36.23	even	6		567.2.f.l.379.2		6
36.31	odd	6		567.2.f.m.379.2		6
84.83	odd	2		3969.2.a.n.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.e.1.2	✓	3	4.3	odd	2
567.2.a.f.1.2	yes	3	12.11	even	2
567.2.f.l.190.2		6	36.11	even	6
567.2.f.l.379.2		6	36.23	even	6
567.2.f.m.190.2		6	36.7	odd	6
567.2.f.m.379.2		6	36.31	odd	6
3969.2.a.n.1.2		3	84.83	odd	2
3969.2.a.o.1.2		3	28.27	even	2
9072.2.a.bu.1.3		3	1.1	even	1	trivial
9072.2.a.cb.1.1		3	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20147 q^{5} -1.00000 q^{7} +5.20147 q^{11} +3.15352 q^{13} +3.24943 q^{17} -7.45090 q^{19} +4.40294 q^{23} -0.153520 q^{25} +1.15352 q^{29} -2.00000 q^{31} -2.20147 q^{35} +5.00000 q^{37} +11.4509 q^{41} -9.29738 q^{43} -1.04795 q^{47} +1.00000 q^{49} +0.249425 q^{53} +11.4509 q^{55} -8.09591 q^{59} +8.60442 q^{61} +6.94239 q^{65} +7.60442 q^{67} +9.60442 q^{71} +0.846480 q^{73} -5.20147 q^{77} +7.60442 q^{79} +11.4509 q^{83} +7.15352 q^{85} -9.24943 q^{89} -3.15352 q^{91} -16.4029 q^{95} -3.45090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{23} + 6 q^{25} - 3 q^{29} - 6 q^{31} + 3 q^{35} + 15 q^{37} + 12 q^{41} - 12 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 3 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 + 6 * q^11 + 3 * q^13 - 3 * q^17 - 6 * q^23 + 6 * q^25 - 3 * q^29 - 6 * q^31 + 3 * q^35 + 15 * q^37 + 12 * q^41 - 12 * q^43 + 3 * q^49 - 12 * q^53 + 12 * q^55 - 18 * q^59 - 3 * q^61 + 21 * q^65 - 6 * q^67 + 9 * q^73 - 6 * q^77 - 6 * q^79 + 12 * q^83 + 15 * q^85 - 15 * q^89 - 3 * q^91 - 30 * q^95 + 12 * q^97

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