LMFDB - Embedded newform 6561.2.a.d.1.43

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6561,2,Mod(1,6561)] 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("6561.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6561, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.a (trivial)

Newform invariants

comment:select newform

sage:traces = [72,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.3898487662$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.43
Character	$\chi$	$=$	6561.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.781548 q^{2} -1.38918 q^{4} +1.48073 q^{5} -3.80126 q^{7} -2.64881 q^{8} +1.15726 q^{10} -5.05444 q^{11} -4.49226 q^{13} -2.97087 q^{14} +0.708195 q^{16} -4.58215 q^{17} -4.46448 q^{19} -2.05700 q^{20} -3.95028 q^{22} +3.45493 q^{23} -2.80745 q^{25} -3.51091 q^{26} +5.28065 q^{28} -0.652939 q^{29} +0.658151 q^{31} +5.85111 q^{32} -3.58117 q^{34} -5.62863 q^{35} +1.00016 q^{37} -3.48921 q^{38} -3.92216 q^{40} +0.682813 q^{41} +8.39308 q^{43} +7.02154 q^{44} +2.70019 q^{46} -7.93477 q^{47} +7.44960 q^{49} -2.19416 q^{50} +6.24057 q^{52} -4.14938 q^{53} -7.48424 q^{55} +10.0688 q^{56} -0.510303 q^{58} -5.26997 q^{59} -6.79097 q^{61} +0.514376 q^{62} +3.15653 q^{64} -6.65180 q^{65} +5.30120 q^{67} +6.36545 q^{68} -4.39904 q^{70} +6.13308 q^{71} -4.40156 q^{73} +0.781671 q^{74} +6.20199 q^{76} +19.2132 q^{77} +11.2425 q^{79} +1.04864 q^{80} +0.533651 q^{82} -1.62164 q^{83} -6.78492 q^{85} +6.55959 q^{86} +13.3882 q^{88} +1.04284 q^{89} +17.0762 q^{91} -4.79953 q^{92} -6.20140 q^{94} -6.61068 q^{95} +4.98814 q^{97} +5.82222 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

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 $

$2$	0.781548	0.552638	0.276319	−	0.961066i	$-0.410885\pi$
$2$	0.781548	0.552638	0.276319	+	0.961066i	$0.410885\pi$
$3$	0	0
$3$	0	0
$4$	−1.38918	−0.694591
$5$	1.48073	0.662201	0.331100	−	0.943596i	$-0.392580\pi$
$5$	1.48073	0.662201	0.331100	+	0.943596i	$0.392580\pi$
$6$	0	0
$7$	−3.80126	−1.43674	−0.718371	−	0.695660i	$-0.755112\pi$
$7$	−3.80126	−1.43674	−0.718371	+	0.695660i	$0.755112\pi$
$8$	−2.64881	−0.936495
$9$	0	0
$10$	1.15726	0.365957
$11$	−5.05444	−1.52397	−0.761985	−	0.647595i	$-0.775775\pi$
$11$	−5.05444	−1.52397	−0.761985	+	0.647595i	$0.775775\pi$
$12$	0	0
$13$	−4.49226	−1.24593	−0.622964	−	0.782251i	$-0.714072\pi$
$13$	−4.49226	−1.24593	−0.622964	+	0.782251i	$0.714072\pi$
$14$	−2.97087	−0.793998
$15$	0	0
$16$	0.708195	0.177049
$17$	−4.58215	−1.11134	−0.555668	−	0.831404i	$-0.687537\pi$
$17$	−4.58215	−1.11134	−0.555668	+	0.831404i	$0.687537\pi$
$18$	0	0
$19$	−4.46448	−1.02422	−0.512112	−	0.858919i	$-0.671137\pi$
$19$	−4.46448	−1.02422	−0.512112	+	0.858919i	$0.671137\pi$
$20$	−2.05700	−0.459959
$21$	0	0
$22$	−3.95028	−0.842203
$23$	3.45493	0.720403	0.360201	−	0.932875i	$-0.382708\pi$
$23$	3.45493	0.720403	0.360201	+	0.932875i	$0.382708\pi$
$24$	0	0
$25$	−2.80745	−0.561490
$26$	−3.51091	−0.688547
$27$	0	0
$28$	5.28065	0.997949
$29$	−0.652939	−0.121248	−0.0606239	−	0.998161i	$-0.519309\pi$
$29$	−0.652939	−0.121248	−0.0606239	+	0.998161i	$0.519309\pi$
$30$	0	0
$31$	0.658151	0.118207	0.0591037	−	0.998252i	$-0.481176\pi$
$31$	0.658151	0.118207	0.0591037	+	0.998252i	$0.481176\pi$
$32$	5.85111	1.03434
$33$	0	0
$34$	−3.58117	−0.614166
$35$	−5.62863	−0.951412
$36$	0	0
$37$	1.00016	0.164425	0.0822124	−	0.996615i	$-0.473801\pi$
$37$	1.00016	0.164425	0.0822124	+	0.996615i	$0.473801\pi$
$38$	−3.48921	−0.566024
$39$	0	0
$40$	−3.92216	−0.620148
$41$	0.682813	0.106637	0.0533187	−	0.998578i	$-0.483020\pi$
$41$	0.682813	0.106637	0.0533187	+	0.998578i	$0.483020\pi$
$42$	0	0
$43$	8.39308	1.27993	0.639966	−	0.768403i	$-0.278948\pi$
$43$	8.39308	1.27993	0.639966	+	0.768403i	$0.278948\pi$
$44$	7.02154	1.05854
$45$	0	0
$46$	2.70019	0.398122
$47$	−7.93477	−1.15741	−0.578703	−	0.815539i	$-0.696441\pi$
$47$	−7.93477	−1.15741	−0.578703	+	0.815539i	$0.696441\pi$
$48$	0	0
$49$	7.44960	1.06423
$50$	−2.19416	−0.310301
$51$	0	0
$52$	6.24057	0.865411
$53$	−4.14938	−0.569962	−0.284981	−	0.958533i	$-0.591987\pi$
$53$	−4.14938	−0.569962	−0.284981	+	0.958533i	$0.591987\pi$
$54$	0	0
$55$	−7.48424	−1.00917
$56$	10.0688	1.34550
$57$	0	0
$58$	−0.510303	−0.0670061
$59$	−5.26997	−0.686092	−0.343046	−	0.939319i	$-0.611459\pi$
$59$	−5.26997	−0.686092	−0.343046	+	0.939319i	$0.611459\pi$
$60$	0	0
$61$	−6.79097	−0.869495	−0.434747	−	0.900553i	$-0.643162\pi$
$61$	−6.79097	−0.869495	−0.434747	+	0.900553i	$0.643162\pi$
$62$	0.514376	0.0653259
$63$	0	0
$64$	3.15653	0.394566
$65$	−6.65180	−0.825055
$66$	0	0
$67$	5.30120	0.647645	0.323822	−	0.946118i	$-0.395032\pi$
$67$	5.30120	0.647645	0.323822	+	0.946118i	$0.395032\pi$
$68$	6.36545	0.771924
$69$	0	0
$70$	−4.39904	−0.525786
$71$	6.13308	0.727863	0.363932	−	0.931426i	$-0.381434\pi$
$71$	6.13308	0.727863	0.363932	+	0.931426i	$0.381434\pi$
$72$	0	0
$73$	−4.40156	−0.515163	−0.257582	−	0.966257i	$-0.582926\pi$
$73$	−4.40156	−0.515163	−0.257582	+	0.966257i	$0.582926\pi$
$74$	0.781671	0.0908674
$75$	0	0
$76$	6.20199	0.711417
$77$	19.2132	2.18955
$78$	0	0
$79$	11.2425	1.26488	0.632442	−	0.774608i	$-0.282053\pi$
$79$	11.2425	1.26488	0.632442	+	0.774608i	$0.282053\pi$
$80$	1.04864	0.117242
$81$	0	0
$82$	0.533651	0.0589319
$83$	−1.62164	−0.177999	−0.0889993	−	0.996032i	$-0.528367\pi$
$83$	−1.62164	−0.177999	−0.0889993	+	0.996032i	$0.528367\pi$
$84$	0	0
$85$	−6.78492	−0.735928
$86$	6.55959	0.707339
$87$	0	0
$88$	13.3882	1.42719
$89$	1.04284	0.110541	0.0552703	−	0.998471i	$-0.482398\pi$
$89$	1.04284	0.110541	0.0552703	+	0.998471i	$0.482398\pi$
$90$	0	0
$91$	17.0762	1.79008
$92$	−4.79953	−0.500386
$93$	0	0
$94$	−6.20140	−0.639626
$95$	−6.61068	−0.678241
$96$	0	0
$97$	4.98814	0.506469	0.253235	−	0.967405i	$-0.418506\pi$
$97$	4.98814	0.506469	0.253235	+	0.967405i	$0.418506\pi$
$98$	5.82222	0.588133
$99$	0	0
$100$	3.90006	0.390006
$101$	9.57048	0.952299	0.476149	−	0.879364i	$-0.342032\pi$
$101$	9.57048	0.952299	0.476149	+	0.879364i	$0.342032\pi$
$102$	0	0
$103$	3.25514	0.320738	0.160369	−	0.987057i	$-0.448732\pi$
$103$	3.25514	0.320738	0.160369	+	0.987057i	$0.448732\pi$
$104$	11.8991	1.16681
$105$	0	0
$106$	−3.24294	−0.314982
$107$	−16.8136	−1.62543	−0.812716	−	0.582660i	$-0.802012\pi$
$107$	−16.8136	−1.62543	−0.812716	+	0.582660i	$0.802012\pi$
$108$	0	0
$109$	7.63544	0.731343	0.365671	−	0.930744i	$-0.380839\pi$
$109$	7.63544	0.731343	0.365671	+	0.930744i	$0.380839\pi$
$110$	−5.84929	−0.557708
$111$	0	0
$112$	−2.69204	−0.254373
$113$	10.0194	0.942543	0.471272	−	0.881988i	$-0.343795\pi$
$113$	10.0194	0.942543	0.471272	+	0.881988i	$0.343795\pi$
$114$	0	0
$115$	5.11581	0.477051
$116$	0.907052	0.0842176
$117$	0	0
$118$	−4.11874	−0.379160
$119$	17.4180	1.59670
$120$	0	0
$121$	14.5473	1.32248
$122$	−5.30747	−0.480516
$123$	0	0
$124$	−0.914292	−0.0821058
$125$	−11.5607	−1.03402
$126$	0	0
$127$	3.88574	0.344803	0.172402	−	0.985027i	$-0.444847\pi$
$127$	3.88574	0.344803	0.172402	+	0.985027i	$0.444847\pi$
$128$	−9.23523	−0.816287
$129$	0	0
$130$	−5.19870	−0.455956
$131$	16.6492	1.45465	0.727324	−	0.686294i	$-0.240764\pi$
$131$	16.6492	1.45465	0.727324	+	0.686294i	$0.240764\pi$
$132$	0	0
$133$	16.9707	1.47154
$134$	4.14314	0.357913
$135$	0	0
$136$	12.1373	1.04076
$137$	−9.27589	−0.792492	−0.396246	−	0.918144i	$-0.629687\pi$
$137$	−9.27589	−0.792492	−0.396246	+	0.918144i	$0.629687\pi$
$138$	0	0
$139$	15.9212	1.35042	0.675208	−	0.737628i	$-0.264054\pi$
$139$	15.9212	1.35042	0.675208	+	0.737628i	$0.264054\pi$
$140$	7.81920	0.660843
$141$	0	0
$142$	4.79330	0.402245
$143$	22.7058	1.89876
$144$	0	0
$145$	−0.966824	−0.0802904
$146$	−3.44003	−0.284699
$147$	0	0
$148$	−1.38940	−0.114208
$149$	−23.3418	−1.91223	−0.956116	−	0.292988i	$-0.905350\pi$
$149$	−23.3418	−1.91223	−0.956116	+	0.292988i	$0.905350\pi$
$150$	0	0
$151$	−2.33209	−0.189783	−0.0948915	−	0.995488i	$-0.530250\pi$
$151$	−2.33209	−0.189783	−0.0948915	+	0.995488i	$0.530250\pi$
$152$	11.8256	0.959180
$153$	0	0
$154$	15.0161	1.21003
$155$	0.974541	0.0782770
$156$	0	0
$157$	−19.6833	−1.57089	−0.785447	−	0.618929i	$-0.787567\pi$
$157$	−19.6833	−1.57089	−0.785447	+	0.618929i	$0.787567\pi$
$158$	8.78658	0.699023
$159$	0	0
$160$	8.66389	0.684940
$161$	−13.1331	−1.03503
$162$	0	0
$163$	−17.6622	−1.38341	−0.691707	−	0.722179i	$-0.743141\pi$
$163$	−17.6622	−1.38341	−0.691707	+	0.722179i	$0.743141\pi$
$164$	−0.948552	−0.0740695
$165$	0	0
$166$	−1.26739	−0.0983688
$167$	−3.39014	−0.262337	−0.131168	−	0.991360i	$-0.541873\pi$
$167$	−3.39014	−0.262337	−0.131168	+	0.991360i	$0.541873\pi$
$168$	0	0
$169$	7.18037	0.552336
$170$	−5.30274	−0.406701
$171$	0	0
$172$	−11.6595	−0.889030
$173$	18.6790	1.42014	0.710069	−	0.704132i	$-0.248664\pi$
$173$	18.6790	1.42014	0.710069	+	0.704132i	$0.248664\pi$
$174$	0	0
$175$	10.6719	0.806716
$176$	−3.57953	−0.269817
$177$	0	0
$178$	0.815027	0.0610889
$179$	−11.8846	−0.888294	−0.444147	−	0.895954i	$-0.646493\pi$
$179$	−11.8846	−0.888294	−0.444147	+	0.895954i	$0.646493\pi$
$180$	0	0
$181$	−15.7848	−1.17328	−0.586638	−	0.809849i	$-0.699549\pi$
$181$	−15.7848	−1.17328	−0.586638	+	0.809849i	$0.699549\pi$
$182$	13.3459	0.989264
$183$	0	0
$184$	−9.15145	−0.674654
$185$	1.48096	0.108882
$186$	0	0
$187$	23.1602	1.69364
$188$	11.0228	0.803924
$189$	0	0
$190$	−5.16656	−0.374822
$191$	−5.76661	−0.417257	−0.208629	−	0.977995i	$-0.566900\pi$
$191$	−5.76661	−0.417257	−0.208629	+	0.977995i	$0.566900\pi$
$192$	0	0
$193$	0.175757	0.0126513	0.00632564	−	0.999980i	$-0.497986\pi$
$193$	0.175757	0.0126513	0.00632564	+	0.999980i	$0.497986\pi$
$194$	3.89847	0.279894
$195$	0	0
$196$	−10.3489	−0.739204
$197$	−11.2155	−0.799069	−0.399535	−	0.916718i	$-0.630828\pi$
$197$	−11.2155	−0.799069	−0.399535	+	0.916718i	$0.630828\pi$
$198$	0	0
$199$	−20.4783	−1.45167	−0.725834	−	0.687870i	$-0.758546\pi$
$199$	−20.4783	−1.45167	−0.725834	+	0.687870i	$0.758546\pi$
$200$	7.43640	0.525833
$201$	0	0
$202$	7.47979	0.526276
$203$	2.48199	0.174202
$204$	0	0
$205$	1.01106	0.0706154
$206$	2.54405	0.177252
$207$	0	0
$208$	−3.18139	−0.220590
$209$	22.5655	1.56089
$210$	0	0
$211$	1.19139	0.0820184	0.0410092	−	0.999159i	$-0.486943\pi$
$211$	1.19139	0.0820184	0.0410092	+	0.999159i	$0.486943\pi$
$212$	5.76425	0.395891
$213$	0	0
$214$	−13.1406	−0.898275
$215$	12.4279	0.847573
$216$	0	0
$217$	−2.50180	−0.169834
$218$	5.96746	0.404168
$219$	0	0
$220$	10.3970	0.700964
$221$	20.5842	1.38464
$222$	0	0
$223$	−14.8999	−0.997770	−0.498885	−	0.866668i	$-0.666257\pi$
$223$	−14.8999	−0.997770	−0.498885	+	0.866668i	$0.666257\pi$
$224$	−22.2416	−1.48608
$225$	0	0
$226$	7.83062	0.520885
$227$	24.5988	1.63268	0.816340	−	0.577572i	$-0.196000\pi$
$227$	24.5988	1.63268	0.816340	+	0.577572i	$0.196000\pi$
$228$	0	0
$229$	1.52698	0.100906	0.0504528	−	0.998726i	$-0.483934\pi$
$229$	1.52698	0.100906	0.0504528	+	0.998726i	$0.483934\pi$
$230$	3.99825	0.263637
$231$	0	0
$232$	1.72951	0.113548
$233$	−14.0599	−0.921094	−0.460547	−	0.887635i	$-0.652347\pi$
$233$	−14.0599	−0.921094	−0.460547	+	0.887635i	$0.652347\pi$
$234$	0	0
$235$	−11.7492	−0.766435
$236$	7.32096	0.476554
$237$	0	0
$238$	13.6130	0.882399
$239$	22.9174	1.48240	0.741201	−	0.671283i	$-0.234257\pi$
$239$	22.9174	1.48240	0.741201	+	0.671283i	$0.234257\pi$
$240$	0	0
$241$	−7.06874	−0.455338	−0.227669	−	0.973739i	$-0.573110\pi$
$241$	−7.06874	−0.455338	−0.227669	+	0.973739i	$0.573110\pi$
$242$	11.3694	0.730855
$243$	0	0
$244$	9.43390	0.603943
$245$	11.0308	0.704733
$246$	0	0
$247$	20.0556	1.27611
$248$	−1.74332	−0.110701
$249$	0	0
$250$	−9.03524	−0.571439
$251$	−0.0664671	−0.00419537	−0.00209768	−	0.999998i	$-0.500668\pi$
$251$	−0.0664671	−0.00419537	−0.00209768	+	0.999998i	$0.500668\pi$
$252$	0	0
$253$	−17.4627	−1.09787
$254$	3.03689	0.190551
$255$	0	0
$256$	−13.5308	−0.845677
$257$	−19.3437	−1.20663	−0.603313	−	0.797504i	$-0.706153\pi$
$257$	−19.3437	−1.20663	−0.603313	+	0.797504i	$0.706153\pi$
$258$	0	0
$259$	−3.80186	−0.236236
$260$	9.24057	0.573076
$261$	0	0
$262$	13.0122	0.803893
$263$	−15.0601	−0.928646	−0.464323	−	0.885666i	$-0.653702\pi$
$263$	−15.0601	−0.928646	−0.464323	+	0.885666i	$0.653702\pi$
$264$	0	0
$265$	−6.14410	−0.377429
$266$	13.2634	0.813231
$267$	0	0
$268$	−7.36434	−0.449848
$269$	11.7365	0.715585	0.357792	−	0.933801i	$-0.383530\pi$
$269$	11.7365	0.715585	0.357792	+	0.933801i	$0.383530\pi$
$270$	0	0
$271$	−2.88026	−0.174963	−0.0874817	−	0.996166i	$-0.527882\pi$
$271$	−2.88026	−0.174963	−0.0874817	+	0.996166i	$0.527882\pi$
$272$	−3.24506	−0.196761
$273$	0	0
$274$	−7.24955	−0.437961
$275$	14.1901	0.855694
$276$	0	0
$277$	−2.30102	−0.138255	−0.0691276	−	0.997608i	$-0.522022\pi$
$277$	−2.30102	−0.138255	−0.0691276	+	0.997608i	$0.522022\pi$
$278$	12.4431	0.746290
$279$	0	0
$280$	14.9092	0.890993
$281$	−6.34551	−0.378541	−0.189271	−	0.981925i	$-0.560612\pi$
$281$	−6.34551	−0.378541	−0.189271	+	0.981925i	$0.560612\pi$
$282$	0	0
$283$	−14.6818	−0.872741	−0.436371	−	0.899767i	$-0.643736\pi$
$283$	−14.6818	−0.872741	−0.436371	+	0.899767i	$0.643736\pi$
$284$	−8.51998	−0.505568
$285$	0	0
$286$	17.7457	1.04932
$287$	−2.59555	−0.153211
$288$	0	0
$289$	3.99614	0.235067
$290$	−0.755619	−0.0443715
$291$	0	0
$292$	6.11457	0.357828
$293$	25.3553	1.48127	0.740635	−	0.671908i	$-0.234525\pi$
$293$	25.3553	1.48127	0.740635	+	0.671908i	$0.234525\pi$
$294$	0	0
$295$	−7.80339	−0.454331
$296$	−2.64923	−0.153983
$297$	0	0
$298$	−18.2427	−1.05677
$299$	−15.5204	−0.897570
$300$	0	0
$301$	−31.9043	−1.83893
$302$	−1.82264	−0.104881
$303$	0	0
$304$	−3.16173	−0.181337
$305$	−10.0556	−0.575780
$306$	0	0
$307$	−12.1833	−0.695339	−0.347670	−	0.937617i	$-0.613027\pi$
$307$	−12.1833	−0.695339	−0.347670	+	0.937617i	$0.613027\pi$
$308$	−26.6907	−1.52084
$309$	0	0
$310$	0.761651	0.0432589
$311$	25.1356	1.42531	0.712656	−	0.701514i	$-0.247492\pi$
$311$	25.1356	1.42531	0.712656	+	0.701514i	$0.247492\pi$
$312$	0	0
$313$	29.5629	1.67099	0.835497	−	0.549495i	$-0.185180\pi$
$313$	29.5629	1.67099	0.835497	+	0.549495i	$0.185180\pi$
$314$	−15.3834	−0.868136
$315$	0	0
$316$	−15.6179	−0.878578
$317$	−4.28813	−0.240845	−0.120423	−	0.992723i	$-0.538425\pi$
$317$	−4.28813	−0.240845	−0.120423	+	0.992723i	$0.538425\pi$
$318$	0	0
$319$	3.30024	0.184778
$320$	4.67396	0.261282
$321$	0	0
$322$	−10.2641	−0.571999
$323$	20.4570	1.13826
$324$	0	0
$325$	12.6118	0.699576
$326$	−13.8039	−0.764526
$327$	0	0
$328$	−1.80864	−0.0998655
$329$	30.1622	1.66289
$330$	0	0
$331$	−27.8313	−1.52975	−0.764874	−	0.644179i	$-0.777199\pi$
$331$	−27.8313	−1.52975	−0.764874	+	0.644179i	$0.777199\pi$
$332$	2.25276	0.123636
$333$	0	0
$334$	−2.64956	−0.144977
$335$	7.84963	0.428871
$336$	0	0
$337$	26.9017	1.46543	0.732715	−	0.680536i	$-0.238253\pi$
$337$	26.9017	1.46543	0.732715	+	0.680536i	$0.238253\pi$
$338$	5.61180	0.305242
$339$	0	0
$340$	9.42549	0.511169
$341$	−3.32658	−0.180144
$342$	0	0
$343$	−1.70905	−0.0922799
$344$	−22.2317	−1.19865
$345$	0	0
$346$	14.5985	0.784822
$347$	30.4987	1.63726	0.818629	−	0.574322i	$-0.194734\pi$
$347$	30.4987	1.63726	0.818629	+	0.574322i	$0.194734\pi$
$348$	0	0
$349$	−7.97533	−0.426910	−0.213455	−	0.976953i	$-0.568472\pi$
$349$	−7.97533	−0.426910	−0.213455	+	0.976953i	$0.568472\pi$
$350$	8.34056	0.445822
$351$	0	0
$352$	−29.5740	−1.57630
$353$	21.8302	1.16190	0.580952	−	0.813938i	$-0.302681\pi$
$353$	21.8302	1.16190	0.580952	+	0.813938i	$0.302681\pi$
$354$	0	0
$355$	9.08142	0.481992
$356$	−1.44869	−0.0767805
$357$	0	0
$358$	−9.28835	−0.490905
$359$	17.3105	0.913611	0.456805	−	0.889567i	$-0.348994\pi$
$359$	17.3105	0.913611	0.456805	+	0.889567i	$0.348994\pi$
$360$	0	0
$361$	0.931623	0.0490328
$362$	−12.3366	−0.648397
$363$	0	0
$364$	−23.7220	−1.24337
$365$	−6.51750	−0.341142
$366$	0	0
$367$	−23.9067	−1.24792	−0.623961	−	0.781456i	$-0.714477\pi$
$367$	−23.9067	−1.24792	−0.623961	+	0.781456i	$0.714477\pi$
$368$	2.44676	0.127546
$369$	0	0
$370$	1.15744	0.0601725
$371$	15.7729	0.818888
$372$	0	0
$373$	−11.5309	−0.597046	−0.298523	−	0.954402i	$-0.596494\pi$
$373$	−11.5309	−0.597046	−0.298523	+	0.954402i	$0.596494\pi$
$374$	18.1008	0.935971
$375$	0	0
$376$	21.0177	1.08390
$377$	2.93317	0.151066
$378$	0	0
$379$	20.7311	1.06489	0.532443	−	0.846466i	$-0.321274\pi$
$379$	20.7311	1.06489	0.532443	+	0.846466i	$0.321274\pi$
$380$	9.18344	0.471101
$381$	0	0
$382$	−4.50688	−0.230592
$383$	15.8552	0.810162	0.405081	−	0.914281i	$-0.367243\pi$
$383$	15.8552	0.810162	0.405081	+	0.914281i	$0.367243\pi$
$384$	0	0
$385$	28.4496	1.44992
$386$	0.137363	0.00699158
$387$	0	0
$388$	−6.92944	−0.351789
$389$	28.5090	1.44547	0.722733	−	0.691128i	$-0.242886\pi$
$389$	28.5090	1.44547	0.722733	+	0.691128i	$0.242886\pi$
$390$	0	0
$391$	−15.8310	−0.800609
$392$	−19.7326	−0.996645
$393$	0	0
$394$	−8.76543	−0.441596
$395$	16.6471	0.837608
$396$	0	0
$397$	31.6207	1.58700	0.793498	−	0.608573i	$-0.208258\pi$
$397$	31.6207	1.58700	0.793498	+	0.608573i	$0.208258\pi$
$398$	−16.0048	−0.802246
$399$	0	0
$400$	−1.98822	−0.0994111
$401$	−17.0478	−0.851325	−0.425662	−	0.904882i	$-0.639959\pi$
$401$	−17.0478	−0.851325	−0.425662	+	0.904882i	$0.639959\pi$
$402$	0	0
$403$	−2.95658	−0.147278
$404$	−13.2952	−0.661458
$405$	0	0
$406$	1.93980	0.0962705
$407$	−5.05523	−0.250579
$408$	0	0
$409$	−23.8370	−1.17866	−0.589331	−	0.807892i	$-0.700608\pi$
$409$	−23.8370	−1.17866	−0.589331	+	0.807892i	$0.700608\pi$
$410$	0.790191	0.0390248
$411$	0	0
$412$	−4.52198	−0.222782
$413$	20.0326	0.985738
$414$	0	0
$415$	−2.40121	−0.117871
$416$	−26.2847	−1.28871
$417$	0	0
$418$	17.6360	0.862604
$419$	15.7362	0.768765	0.384382	−	0.923174i	$-0.374414\pi$
$419$	15.7362	0.768765	0.384382	+	0.923174i	$0.374414\pi$
$420$	0	0
$421$	−23.5578	−1.14814	−0.574070	−	0.818806i	$-0.694636\pi$
$421$	−23.5578	−1.14814	−0.574070	+	0.818806i	$0.694636\pi$
$422$	0.931125	0.0453265
$423$	0	0
$424$	10.9909	0.533767
$425$	12.8642	0.624004
$426$	0	0
$427$	25.8143	1.24924
$428$	23.3572	1.12901
$429$	0	0
$430$	9.71296	0.468401
$431$	1.41170	0.0679990	0.0339995	−	0.999422i	$-0.489176\pi$
$431$	1.41170	0.0679990	0.0339995	+	0.999422i	$0.489176\pi$
$432$	0	0
$433$	13.0105	0.625244	0.312622	−	0.949878i	$-0.398793\pi$
$433$	13.0105	0.625244	0.312622	+	0.949878i	$0.398793\pi$
$434$	−1.95528	−0.0938564
$435$	0	0
$436$	−10.6070	−0.507984
$437$	−15.4245	−0.737853
$438$	0	0
$439$	20.7384	0.989789	0.494895	−	0.868953i	$-0.335207\pi$
$439$	20.7384	0.989789	0.494895	+	0.868953i	$0.335207\pi$
$440$	19.8243	0.945087
$441$	0	0
$442$	16.0875	0.765207
$443$	−33.9675	−1.61384	−0.806921	−	0.590659i	$-0.798868\pi$
$443$	−33.9675	−1.61384	−0.806921	+	0.590659i	$0.798868\pi$
$444$	0	0
$445$	1.54416	0.0732001
$446$	−11.6450	−0.551406
$447$	0	0
$448$	−11.9988	−0.566890
$449$	−1.20059	−0.0566596	−0.0283298	−	0.999599i	$-0.509019\pi$
$449$	−1.20059	−0.0566596	−0.0283298	+	0.999599i	$0.509019\pi$
$450$	0	0
$451$	−3.45123	−0.162512
$452$	−13.9187	−0.654683
$453$	0	0
$454$	19.2251	0.902280
$455$	25.2853	1.18539
$456$	0	0
$457$	−32.0532	−1.49939	−0.749693	−	0.661786i	$-0.769799\pi$
$457$	−32.0532	−1.49939	−0.749693	+	0.661786i	$0.769799\pi$
$458$	1.19341	0.0557643
$459$	0	0
$460$	−7.10679	−0.331356
$461$	−0.336685	−0.0156810	−0.00784049	−	0.999969i	$-0.502496\pi$
$461$	−0.336685	−0.0156810	−0.00784049	+	0.999969i	$0.502496\pi$
$462$	0	0
$463$	−2.95273	−0.137225	−0.0686125	−	0.997643i	$-0.521857\pi$
$463$	−2.95273	−0.137225	−0.0686125	+	0.997643i	$0.521857\pi$
$464$	−0.462408	−0.0214668
$465$	0	0
$466$	−10.9885	−0.509032
$467$	−3.78427	−0.175115	−0.0875576	−	0.996159i	$-0.527906\pi$
$467$	−3.78427	−0.175115	−0.0875576	+	0.996159i	$0.527906\pi$
$468$	0	0
$469$	−20.1513	−0.930498
$470$	−9.18258	−0.423561
$471$	0	0
$472$	13.9592	0.642522
$473$	−42.4223	−1.95058
$474$	0	0
$475$	12.5338	0.575091
$476$	−24.1968	−1.10906
$477$	0	0
$478$	17.9110	0.819231
$479$	3.16676	0.144693	0.0723465	−	0.997380i	$-0.476951\pi$
$479$	3.16676	0.144693	0.0723465	+	0.997380i	$0.476951\pi$
$480$	0	0
$481$	−4.49296	−0.204862
$482$	−5.52456	−0.251637
$483$	0	0
$484$	−20.2089	−0.918586
$485$	7.38607	0.335384
$486$	0	0
$487$	1.22501	0.0555106	0.0277553	−	0.999615i	$-0.491164\pi$
$487$	1.22501	0.0555106	0.0277553	+	0.999615i	$0.491164\pi$
$488$	17.9880	0.814278
$489$	0	0
$490$	8.62111	0.389462
$491$	40.5477	1.82989	0.914946	−	0.403577i	$-0.132233\pi$
$491$	40.5477	1.82989	0.914946	+	0.403577i	$0.132233\pi$
$492$	0	0
$493$	2.99187	0.134747
$494$	15.6744	0.705225
$495$	0	0
$496$	0.466099	0.0209285
$497$	−23.3135	−1.04575
$498$	0	0
$499$	−41.1516	−1.84220	−0.921100	−	0.389325i	$-0.872708\pi$
$499$	−41.1516	−1.84220	−0.921100	+	0.389325i	$0.872708\pi$
$500$	16.0599	0.718222
$501$	0	0
$502$	−0.0519473	−0.00231852
$503$	10.1538	0.452735	0.226368	−	0.974042i	$-0.427315\pi$
$503$	10.1538	0.452735	0.226368	+	0.974042i	$0.427315\pi$
$504$	0	0
$505$	14.1713	0.630613
$506$	−13.6480	−0.606726
$507$	0	0
$508$	−5.39800	−0.239498
$509$	−24.2472	−1.07474	−0.537369	−	0.843347i	$-0.680582\pi$
$509$	−24.2472	−1.07474	−0.537369	+	0.843347i	$0.680582\pi$
$510$	0	0
$511$	16.7315	0.740157
$512$	7.89547	0.348934
$513$	0	0
$514$	−15.1180	−0.666827
$515$	4.81997	0.212393
$516$	0	0
$517$	40.1058	1.76385
$518$	−2.97134	−0.130553
$519$	0	0
$520$	17.6194	0.772660
$521$	6.35050	0.278221	0.139110	−	0.990277i	$-0.455576\pi$
$521$	6.35050	0.278221	0.139110	+	0.990277i	$0.455576\pi$
$522$	0	0
$523$	28.1100	1.22916	0.614582	−	0.788853i	$-0.289325\pi$
$523$	28.1100	1.22916	0.614582	+	0.788853i	$0.289325\pi$
$524$	−23.1288	−1.01039
$525$	0	0
$526$	−11.7702	−0.513205
$527$	−3.01575	−0.131368
$528$	0	0
$529$	−11.0635	−0.481020
$530$	−4.80191	−0.208582
$531$	0	0
$532$	−23.5754	−1.02212
$533$	−3.06737	−0.132863
$534$	0	0
$535$	−24.8963	−1.07636
$536$	−14.0419	−0.606516
$537$	0	0
$538$	9.17261	0.395459
$539$	−37.6535	−1.62185
$540$	0	0
$541$	2.68780	0.115558	0.0577788	−	0.998329i	$-0.481598\pi$
$541$	2.68780	0.115558	0.0577788	+	0.998329i	$0.481598\pi$
$542$	−2.25106	−0.0966914
$543$	0	0
$544$	−26.8107	−1.14950
$545$	11.3060	0.484296
$546$	0	0
$547$	32.8075	1.40275	0.701373	−	0.712794i	$-0.252571\pi$
$547$	32.8075	1.40275	0.701373	+	0.712794i	$0.252571\pi$
$548$	12.8859	0.550458
$549$	0	0
$550$	11.0902	0.472889
$551$	2.91504	0.124185
$552$	0	0
$553$	−42.7358	−1.81731
$554$	−1.79836	−0.0764050
$555$	0	0
$556$	−22.1174	−0.937987
$557$	−9.89968	−0.419463	−0.209731	−	0.977759i	$-0.567259\pi$
$557$	−9.89968	−0.419463	−0.209731	+	0.977759i	$0.567259\pi$
$558$	0	0
$559$	−37.7039	−1.59470
$560$	−3.98617	−0.168446
$561$	0	0
$562$	−4.95932	−0.209196
$563$	7.03550	0.296511	0.148256	−	0.988949i	$-0.452634\pi$
$563$	7.03550	0.296511	0.148256	+	0.988949i	$0.452634\pi$
$564$	0	0
$565$	14.8359	0.624153
$566$	−11.4745	−0.482310
$567$	0	0
$568$	−16.2454	−0.681640
$569$	−3.93812	−0.165095	−0.0825473	−	0.996587i	$-0.526306\pi$
$569$	−3.93812	−0.165095	−0.0825473	+	0.996587i	$0.526306\pi$
$570$	0	0
$571$	25.5675	1.06996	0.534982	−	0.844863i	$-0.320318\pi$
$571$	25.5675	1.06996	0.534982	+	0.844863i	$0.320318\pi$
$572$	−31.5425	−1.31886
$573$	0	0
$574$	−2.02855	−0.0846700
$575$	−9.69954	−0.404499
$576$	0	0
$577$	4.69942	0.195639	0.0978197	−	0.995204i	$-0.468813\pi$
$577$	4.69942	0.195639	0.0978197	+	0.995204i	$0.468813\pi$
$578$	3.12318	0.129907
$579$	0	0
$580$	1.34310	0.0557690
$581$	6.16430	0.255738
$582$	0	0
$583$	20.9728	0.868605
$584$	11.6589	0.482448
$585$	0	0
$586$	19.8163	0.818606
$587$	−27.7731	−1.14632	−0.573159	−	0.819444i	$-0.694282\pi$
$587$	−27.7731	−1.14632	−0.573159	+	0.819444i	$0.694282\pi$
$588$	0	0
$589$	−2.93830	−0.121071
$590$	−6.09872	−0.251080
$591$	0	0
$592$	0.708306	0.0291112
$593$	14.3326	0.588570	0.294285	−	0.955718i	$-0.404918\pi$
$593$	14.3326	0.588570	0.294285	+	0.955718i	$0.404918\pi$
$594$	0	0
$595$	25.7913	1.05734
$596$	32.4260	1.32822
$597$	0	0
$598$	−12.1300	−0.496031
$599$	−27.5031	−1.12375	−0.561873	−	0.827223i	$-0.689919\pi$
$599$	−27.5031	−1.12375	−0.561873	+	0.827223i	$0.689919\pi$
$600$	0	0
$601$	18.7502	0.764836	0.382418	−	0.923989i	$-0.375091\pi$
$601$	18.7502	0.764836	0.382418	+	0.923989i	$0.375091\pi$
$602$	−24.9347	−1.01626
$603$	0	0
$604$	3.23970	0.131822
$605$	21.5406	0.875750
$606$	0	0
$607$	26.4087	1.07190	0.535948	−	0.844251i	$-0.319954\pi$
$607$	26.4087	1.07190	0.535948	+	0.844251i	$0.319954\pi$
$608$	−26.1222	−1.05939
$609$	0	0
$610$	−7.85891	−0.318198
$611$	35.6450	1.44204
$612$	0	0
$613$	−34.9615	−1.41208	−0.706042	−	0.708170i	$-0.749521\pi$
$613$	−34.9615	−1.41208	−0.706042	+	0.708170i	$0.749521\pi$
$614$	−9.52185	−0.384271
$615$	0	0
$616$	−50.8922	−2.05051
$617$	7.15141	0.287905	0.143952	−	0.989585i	$-0.454019\pi$
$617$	7.15141	0.287905	0.143952	+	0.989585i	$0.454019\pi$
$618$	0	0
$619$	−18.8216	−0.756504	−0.378252	−	0.925703i	$-0.623475\pi$
$619$	−18.8216	−0.756504	−0.378252	+	0.925703i	$0.623475\pi$
$620$	−1.35382	−0.0543706
$621$	0	0
$622$	19.6447	0.787681
$623$	−3.96410	−0.158818
$624$	0	0
$625$	−3.08098	−0.123239
$626$	23.1048	0.923454
$627$	0	0
$628$	27.3436	1.09113
$629$	−4.58288	−0.182731
$630$	0	0
$631$	23.4802	0.934732	0.467366	−	0.884064i	$-0.345203\pi$
$631$	23.4802	0.934732	0.467366	+	0.884064i	$0.345203\pi$
$632$	−29.7793	−1.18456
$633$	0	0
$634$	−3.35138	−0.133100
$635$	5.75371	0.228329
$636$	0	0
$637$	−33.4655	−1.32595
$638$	2.57929	0.102115
$639$	0	0
$640$	−13.6749	−0.540546
$641$	4.35901	0.172171	0.0860853	−	0.996288i	$-0.472564\pi$
$641$	4.35901	0.172171	0.0860853	+	0.996288i	$0.472564\pi$
$642$	0	0
$643$	19.0120	0.749760	0.374880	−	0.927073i	$-0.377684\pi$
$643$	19.0120	0.749760	0.374880	+	0.927073i	$0.377684\pi$
$644$	18.2443	0.718925
$645$	0	0
$646$	15.9881	0.629043
$647$	20.6268	0.810924	0.405462	−	0.914112i	$-0.367111\pi$
$647$	20.6268	0.810924	0.405462	+	0.914112i	$0.367111\pi$
$648$	0	0
$649$	26.6367	1.04558
$650$	9.85671	0.386612
$651$	0	0
$652$	24.5361	0.960907
$653$	−5.99076	−0.234437	−0.117218	−	0.993106i	$-0.537398\pi$
$653$	−5.99076	−0.234437	−0.117218	+	0.993106i	$0.537398\pi$
$654$	0	0
$655$	24.6529	0.963269
$656$	0.483565	0.0188800
$657$	0	0
$658$	23.5732	0.918978
$659$	38.8389	1.51295	0.756475	−	0.654022i	$-0.226920\pi$
$659$	38.8389	1.51295	0.756475	+	0.654022i	$0.226920\pi$
$660$	0	0
$661$	30.9957	1.20559	0.602796	−	0.797896i	$-0.294053\pi$
$661$	30.9957	1.20559	0.602796	+	0.797896i	$0.294053\pi$
$662$	−21.7515	−0.845397
$663$	0	0
$664$	4.29543	0.166695
$665$	25.1289	0.974458
$666$	0	0
$667$	−2.25586	−0.0873472
$668$	4.70953	0.182217
$669$	0	0
$670$	6.13486	0.237010
$671$	34.3245	1.32508
$672$	0	0
$673$	−38.5896	−1.48752	−0.743760	−	0.668447i	$-0.766959\pi$
$673$	−38.5896	−1.48752	−0.743760	+	0.668447i	$0.766959\pi$
$674$	21.0250	0.809852
$675$	0	0
$676$	−9.97485	−0.383648
$677$	−3.98869	−0.153298	−0.0766489	−	0.997058i	$-0.524422\pi$
$677$	−3.98869	−0.153298	−0.0766489	+	0.997058i	$0.524422\pi$
$678$	0	0
$679$	−18.9612	−0.727666
$680$	17.9719	0.689193
$681$	0	0
$682$	−2.59988	−0.0995547
$683$	−42.1155	−1.61151	−0.805753	−	0.592252i	$-0.798239\pi$
$683$	−42.1155	−1.61151	−0.805753	+	0.592252i	$0.798239\pi$
$684$	0	0
$685$	−13.7350	−0.524789
$686$	−1.33570	−0.0509974
$687$	0	0
$688$	5.94394	0.226610
$689$	18.6401	0.710131
$690$	0	0
$691$	−8.55061	−0.325281	−0.162640	−	0.986685i	$-0.552001\pi$
$691$	−8.55061	−0.325281	−0.162640	+	0.986685i	$0.552001\pi$
$692$	−25.9485	−0.986415
$693$	0	0
$694$	23.8362	0.904811
$695$	23.5749	0.894246
$696$	0	0
$697$	−3.12875	−0.118510
$698$	−6.23310	−0.235926
$699$	0	0
$700$	−14.8252	−0.560338
$701$	−0.881949	−0.0333108	−0.0166554	−	0.999861i	$-0.505302\pi$
$701$	−0.881949	−0.0333108	−0.0166554	+	0.999861i	$0.505302\pi$
$702$	0	0
$703$	−4.46519	−0.168408
$704$	−15.9545	−0.601307
$705$	0	0
$706$	17.0613	0.642112
$707$	−36.3799	−1.36821
$708$	0	0
$709$	−40.2337	−1.51101	−0.755504	−	0.655144i	$-0.772608\pi$
$709$	−40.2337	−1.51101	−0.755504	+	0.655144i	$0.772608\pi$
$710$	7.09756	0.266367
$711$	0	0
$712$	−2.76228	−0.103521
$713$	2.27387	0.0851569
$714$	0	0
$715$	33.6211	1.25736
$716$	16.5098	0.617001
$717$	0	0
$718$	13.5289	0.504896
$719$	13.7722	0.513615	0.256807	−	0.966463i	$-0.417329\pi$
$719$	13.7722	0.513615	0.256807	+	0.966463i	$0.417329\pi$
$720$	0	0
$721$	−12.3736	−0.460818
$722$	0.728108	0.0270974
$723$	0	0
$724$	21.9280	0.814948
$725$	1.83309	0.0680794
$726$	0	0
$727$	−12.4849	−0.463039	−0.231520	−	0.972830i	$-0.574370\pi$
$727$	−12.4849	−0.463039	−0.231520	+	0.972830i	$0.574370\pi$
$728$	−45.2317	−1.67640
$729$	0	0
$730$	−5.09374	−0.188528
$731$	−38.4584	−1.42243
$732$	0	0
$733$	−26.8730	−0.992575	−0.496288	−	0.868158i	$-0.665304\pi$
$733$	−26.8730	−0.992575	−0.496288	+	0.868158i	$0.665304\pi$
$734$	−18.6843	−0.689649
$735$	0	0
$736$	20.2152	0.745141
$737$	−26.7946	−0.986991
$738$	0	0
$739$	−24.3811	−0.896874	−0.448437	−	0.893815i	$-0.648019\pi$
$739$	−24.3811	−0.896874	−0.448437	+	0.893815i	$0.648019\pi$
$740$	−2.05732	−0.0756287
$741$	0	0
$742$	12.3273	0.452549
$743$	36.0700	1.32328	0.661640	−	0.749822i	$-0.269861\pi$
$743$	36.0700	1.32328	0.661640	+	0.749822i	$0.269861\pi$
$744$	0	0
$745$	−34.5628	−1.26628
$746$	−9.01192	−0.329950
$747$	0	0
$748$	−32.1738	−1.17639
$749$	63.9129	2.33533
$750$	0	0
$751$	−2.33641	−0.0852570	−0.0426285	−	0.999091i	$-0.513573\pi$
$751$	−2.33641	−0.0852570	−0.0426285	+	0.999091i	$0.513573\pi$
$752$	−5.61937	−0.204917
$753$	0	0
$754$	2.29241	0.0834847
$755$	−3.45319	−0.125674
$756$	0	0
$757$	−16.1729	−0.587815	−0.293907	−	0.955834i	$-0.594956\pi$
$757$	−16.1729	−0.587815	−0.293907	+	0.955834i	$0.594956\pi$
$758$	16.2024	0.588496
$759$	0	0
$760$	17.5104	0.635170
$761$	45.3568	1.64418	0.822092	−	0.569355i	$-0.192807\pi$
$761$	45.3568	1.64418	0.822092	+	0.569355i	$0.192807\pi$
$762$	0	0
$763$	−29.0243	−1.05075
$764$	8.01088	0.289823
$765$	0	0
$766$	12.3916	0.447726
$767$	23.6741	0.854821
$768$	0	0
$769$	18.3658	0.662289	0.331144	−	0.943580i	$-0.392565\pi$
$769$	18.3658	0.662289	0.331144	+	0.943580i	$0.392565\pi$
$770$	22.2347	0.801283
$771$	0	0
$772$	−0.244159	−0.00878748
$773$	−29.4789	−1.06028	−0.530141	−	0.847910i	$-0.677861\pi$
$773$	−29.4789	−1.06028	−0.530141	+	0.847910i	$0.677861\pi$
$774$	0	0
$775$	−1.84773	−0.0663722
$776$	−13.2126	−0.474306
$777$	0	0
$778$	22.2812	0.798819
$779$	−3.04841	−0.109221
$780$	0	0
$781$	−30.9993	−1.10924
$782$	−12.3727	−0.442447
$783$	0	0
$784$	5.27577	0.188420
$785$	−29.1455	−1.04025
$786$	0	0
$787$	32.0568	1.14270	0.571351	−	0.820706i	$-0.306419\pi$
$787$	32.0568	1.14270	0.571351	+	0.820706i	$0.306419\pi$
$788$	15.5803	0.555027
$789$	0	0
$790$	13.0105	0.462894
$791$	−38.0863	−1.35419
$792$	0	0
$793$	30.5068	1.08333
$794$	24.7131	0.877034
$795$	0	0
$796$	28.4481	1.00832
$797$	−11.7864	−0.417496	−0.208748	−	0.977969i	$-0.566939\pi$
$797$	−11.7864	−0.417496	−0.208748	+	0.977969i	$0.566939\pi$
$798$	0	0
$799$	36.3584	1.28627
$800$	−16.4267	−0.580771
$801$	0	0
$802$	−13.3236	−0.470474
$803$	22.2474	0.785093
$804$	0	0
$805$	−19.4465	−0.685400
$806$	−2.31071	−0.0813913
$807$	0	0
$808$	−25.3504	−0.891823
$809$	−21.4155	−0.752929	−0.376464	−	0.926431i	$-0.622860\pi$
$809$	−21.4155	−0.752929	−0.376464	+	0.926431i	$0.622860\pi$
$810$	0	0
$811$	1.73790	0.0610258	0.0305129	−	0.999534i	$-0.490286\pi$
$811$	1.73790	0.0610258	0.0305129	+	0.999534i	$0.490286\pi$
$812$	−3.44794	−0.120999
$813$	0	0
$814$	−3.95091	−0.138479
$815$	−26.1529	−0.916097
$816$	0	0
$817$	−37.4708	−1.31094
$818$	−18.6297	−0.651373
$819$	0	0
$820$	−1.40455	−0.0490489
$821$	8.91880	0.311268	0.155634	−	0.987815i	$-0.450258\pi$
$821$	8.91880	0.311268	0.155634	+	0.987815i	$0.450258\pi$
$822$	0	0
$823$	1.85218	0.0645628	0.0322814	−	0.999479i	$-0.489723\pi$
$823$	1.85218	0.0645628	0.0322814	+	0.999479i	$0.489723\pi$
$824$	−8.62224	−0.300370
$825$	0	0
$826$	15.6564	0.544756
$827$	−21.0373	−0.731537	−0.365769	−	0.930706i	$-0.619194\pi$
$827$	−21.0373	−0.731537	−0.365769	+	0.930706i	$0.619194\pi$
$828$	0	0
$829$	5.67239	0.197010	0.0985050	−	0.995137i	$-0.468594\pi$
$829$	5.67239	0.197010	0.0985050	+	0.995137i	$0.468594\pi$
$830$	−1.87666	−0.0651399
$831$	0	0
$832$	−14.1799	−0.491601
$833$	−34.1352	−1.18272
$834$	0	0
$835$	−5.01987	−0.173720
$836$	−31.3475	−1.08418
$837$	0	0
$838$	12.2986	0.424849
$839$	−23.9284	−0.826099	−0.413050	−	0.910709i	$-0.635536\pi$
$839$	−23.9284	−0.826099	−0.413050	+	0.910709i	$0.635536\pi$
$840$	0	0
$841$	−28.5737	−0.985299
$842$	−18.4116	−0.634505
$843$	0	0
$844$	−1.65505	−0.0569693
$845$	10.6322	0.365757
$846$	0	0
$847$	−55.2982	−1.90007
$848$	−2.93857	−0.100911
$849$	0	0
$850$	10.0540	0.344848
$851$	3.45547	0.118452
$852$	0	0
$853$	6.75335	0.231230	0.115615	−	0.993294i	$-0.463116\pi$
$853$	6.75335	0.231230	0.115615	+	0.993294i	$0.463116\pi$
$854$	20.1751	0.690377
$855$	0	0
$856$	44.5360	1.52221
$857$	18.8564	0.644123	0.322062	−	0.946719i	$-0.395624\pi$
$857$	18.8564	0.644123	0.322062	+	0.946719i	$0.395624\pi$
$858$	0	0
$859$	32.4166	1.10604	0.553019	−	0.833168i	$-0.313476\pi$
$859$	32.4166	1.10604	0.553019	+	0.833168i	$0.313476\pi$
$860$	−17.2646	−0.588717
$861$	0	0
$862$	1.10331	0.0375788
$863$	47.2242	1.60753	0.803765	−	0.594947i	$-0.202827\pi$
$863$	47.2242	1.60753	0.803765	+	0.594947i	$0.202827\pi$
$864$	0	0
$865$	27.6585	0.940416
$866$	10.1683	0.345533
$867$	0	0
$868$	3.47546	0.117965
$869$	−56.8247	−1.92765
$870$	0	0
$871$	−23.8143	−0.806918
$872$	−20.2248	−0.684899
$873$	0	0
$874$	−12.0550	−0.407766
$875$	43.9452	1.48562
$876$	0	0
$877$	5.63165	0.190167	0.0950837	−	0.995469i	$-0.469688\pi$
$877$	5.63165	0.190167	0.0950837	+	0.995469i	$0.469688\pi$
$878$	16.2080	0.546995
$879$	0	0
$880$	−5.30030	−0.178673
$881$	5.67058	0.191047	0.0955234	−	0.995427i	$-0.469548\pi$
$881$	5.67058	0.191047	0.0955234	+	0.995427i	$0.469548\pi$
$882$	0	0
$883$	34.3021	1.15436	0.577179	−	0.816618i	$-0.304153\pi$
$883$	34.3021	1.15436	0.577179	+	0.816618i	$0.304153\pi$
$884$	−28.5952	−0.961762
$885$	0	0
$886$	−26.5472	−0.891871
$887$	−26.8838	−0.902670	−0.451335	−	0.892355i	$-0.649052\pi$
$887$	−26.8838	−0.902670	−0.451335	+	0.892355i	$0.649052\pi$
$888$	0	0
$889$	−14.7707	−0.495394
$890$	1.20683	0.0404531
$891$	0	0
$892$	20.6987	0.693043
$893$	35.4247	1.18544
$894$	0	0
$895$	−17.5978	−0.588229
$896$	35.1055	1.17279
$897$	0	0
$898$	−0.938322	−0.0313122
$899$	−0.429732	−0.0143324
$900$	0	0
$901$	19.0131	0.633419
$902$	−2.69731	−0.0898104
$903$	0	0
$904$	−26.5394	−0.882687
$905$	−23.3730	−0.776945
$906$	0	0
$907$	30.4385	1.01070	0.505348	−	0.862916i	$-0.331364\pi$
$907$	30.4385	1.01070	0.505348	+	0.862916i	$0.331364\pi$
$908$	−34.1722	−1.13404
$909$	0	0
$910$	19.7616	0.655092
$911$	−14.0660	−0.466026	−0.233013	−	0.972474i	$-0.574858\pi$
$911$	−14.0660	−0.466026	−0.233013	+	0.972474i	$0.574858\pi$
$912$	0	0
$913$	8.19650	0.271265
$914$	−25.0511	−0.828617
$915$	0	0
$916$	−2.12125	−0.0700882
$917$	−63.2880	−2.08995
$918$	0	0
$919$	29.0015	0.956670	0.478335	−	0.878177i	$-0.341240\pi$
$919$	29.0015	0.956670	0.478335	+	0.878177i	$0.341240\pi$
$920$	−13.5508	−0.446756
$921$	0	0
$922$	−0.263135	−0.00866590
$923$	−27.5514	−0.906865
$924$	0	0
$925$	−2.80789	−0.0923229
$926$	−2.30770	−0.0758357
$927$	0	0
$928$	−3.82041	−0.125411
$929$	44.5589	1.46193	0.730965	−	0.682415i	$-0.239070\pi$
$929$	44.5589	1.46193	0.730965	+	0.682415i	$0.239070\pi$
$930$	0	0
$931$	−33.2586	−1.09001
$932$	19.5318	0.639784
$933$	0	0
$934$	−2.95759	−0.0967753
$935$	34.2939	1.12153
$936$	0	0
$937$	−20.5920	−0.672710	−0.336355	−	0.941735i	$-0.609194\pi$
$937$	−20.5920	−0.672710	−0.336355	+	0.941735i	$0.609194\pi$
$938$	−15.7492	−0.514229
$939$	0	0
$940$	16.3218	0.532359
$941$	−56.7820	−1.85104	−0.925520	−	0.378698i	$-0.876372\pi$
$941$	−56.7820	−1.85104	−0.925520	+	0.378698i	$0.876372\pi$
$942$	0	0
$943$	2.35907	0.0768219
$944$	−3.73217	−0.121472
$945$	0	0
$946$	−33.1550	−1.07796
$947$	15.7756	0.512638	0.256319	−	0.966592i	$-0.417490\pi$
$947$	15.7756	0.512638	0.256319	+	0.966592i	$0.417490\pi$
$948$	0	0
$949$	19.7729	0.641856
$950$	9.79578	0.317817
$951$	0	0
$952$	−46.1369	−1.49530
$953$	50.4349	1.63375	0.816873	−	0.576818i	$-0.195706\pi$
$953$	50.4349	1.63375	0.816873	+	0.576818i	$0.195706\pi$
$954$	0	0
$955$	−8.53878	−0.276308
$956$	−31.8364	−1.02966
$957$	0	0
$958$	2.47497	0.0799628
$959$	35.2601	1.13861
$960$	0	0
$961$	−30.5668	−0.986027
$962$	−3.51147	−0.113214
$963$	0	0
$964$	9.81978	0.316274
$965$	0.260248	0.00837769
$966$	0	0
$967$	21.8942	0.704070	0.352035	−	0.935987i	$-0.385490\pi$
$967$	21.8942	0.704070	0.352035	+	0.935987i	$0.385490\pi$
$968$	−38.5331	−1.23850
$969$	0	0
$970$	5.77257	0.185346
$971$	21.3313	0.684555	0.342277	−	0.939599i	$-0.388802\pi$
$971$	21.3313	0.684555	0.342277	+	0.939599i	$0.388802\pi$
$972$	0	0
$973$	−60.5205	−1.94020
$974$	0.957406	0.0306773
$975$	0	0
$976$	−4.80933	−0.153943
$977$	−7.17129	−0.229430	−0.114715	−	0.993398i	$-0.536595\pi$
$977$	−7.17129	−0.229430	−0.114715	+	0.993398i	$0.536595\pi$
$978$	0	0
$979$	−5.27096	−0.168460
$980$	−15.3238	−0.489502
$981$	0	0
$982$	31.6900	1.01127
$983$	41.4135	1.32089	0.660443	−	0.750876i	$-0.270369\pi$
$983$	41.4135	1.32089	0.660443	+	0.750876i	$0.270369\pi$
$984$	0	0
$985$	−16.6070	−0.529144
$986$	2.33829	0.0744662
$987$	0	0
$988$	−27.8609	−0.886374
$989$	28.9975	0.922067
$990$	0	0
$991$	−54.5793	−1.73377	−0.866884	−	0.498509i	$-0.833881\pi$
$991$	−54.5793	−1.73377	−0.866884	+	0.498509i	$0.833881\pi$
$992$	3.85091	0.122267
$993$	0	0
$994$	−18.2206	−0.577922
$995$	−30.3227	−0.961295
$996$	0	0
$997$	−26.5122	−0.839649	−0.419824	−	0.907605i	$-0.637908\pi$
$997$	−26.5122	−0.839649	−0.419824	+	0.907605i	$0.637908\pi$
$998$	−32.1620	−1.01807
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.43		72
3.2	odd	2		6561.2.a.c.1.30		72
81.5	odd	54		81.2.g.a.25.4	yes	144
81.11	odd	54		729.2.g.d.514.5		144
81.16	even	27		243.2.g.a.10.5		144
81.22	even	27		729.2.g.a.217.4		144
81.32	odd	54		729.2.g.c.703.5		144
81.38	odd	54		729.2.g.c.28.5		144
81.43	even	27		729.2.g.b.28.4		144
81.49	even	27		729.2.g.b.703.4		144
81.59	odd	54		729.2.g.d.217.5		144
81.65	odd	54		81.2.g.a.13.4	✓	144
81.70	even	27		729.2.g.a.514.4		144
81.76	even	27		243.2.g.a.73.5		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.4	✓	144	81.65	odd	54
81.2.g.a.25.4	yes	144	81.5	odd	54
243.2.g.a.10.5		144	81.16	even	27
243.2.g.a.73.5		144	81.76	even	27
729.2.g.a.217.4		144	81.22	even	27
729.2.g.a.514.4		144	81.70	even	27
729.2.g.b.28.4		144	81.43	even	27
729.2.g.b.703.4		144	81.49	even	27
729.2.g.c.28.5		144	81.38	odd	54
729.2.g.c.703.5		144	81.32	odd	54
729.2.g.d.217.5		144	81.59	odd	54
729.2.g.d.514.5		144	81.11	odd	54
6561.2.a.c.1.30		72	3.2	odd	2
6561.2.a.d.1.43		72	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q+0.781548 q^{2} -1.38918 q^{4} +1.48073 q^{5} -3.80126 q^{7} -2.64881 q^{8} +1.15726 q^{10} -5.05444 q^{11} -4.49226 q^{13} -2.97087 q^{14} +0.708195 q^{16} -4.58215 q^{17} -4.46448 q^{19} -2.05700 q^{20} -3.95028 q^{22} +3.45493 q^{23} -2.80745 q^{25} -3.51091 q^{26} +5.28065 q^{28} -0.652939 q^{29} +0.658151 q^{31} +5.85111 q^{32} -3.58117 q^{34} -5.62863 q^{35} +1.00016 q^{37} -3.48921 q^{38} -3.92216 q^{40} +0.682813 q^{41} +8.39308 q^{43} +7.02154 q^{44} +2.70019 q^{46} -7.93477 q^{47} +7.44960 q^{49} -2.19416 q^{50} +6.24057 q^{52} -4.14938 q^{53} -7.48424 q^{55} +10.0688 q^{56} -0.510303 q^{58} -5.26997 q^{59} -6.79097 q^{61} +0.514376 q^{62} +3.15653 q^{64} -6.65180 q^{65} +5.30120 q^{67} +6.36545 q^{68} -4.39904 q^{70} +6.13308 q^{71} -4.40156 q^{73} +0.781671 q^{74} +6.20199 q^{76} +19.2132 q^{77} +11.2425 q^{79} +1.04864 q^{80} +0.533651 q^{82} -1.62164 q^{83} -6.78492 q^{85} +6.55959 q^{86} +13.3882 q^{88} +1.04284 q^{89} +17.0762 q^{91} -4.79953 q^{92} -6.20140 q^{94} -6.61068 q^{95} +4.98814 q^{97} +5.82222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Introduction

L-functions

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Database

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