LMFDB - Embedded newform 6561.2.a.c.1.30

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:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
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	$-0.589115\pi$
$2$	−0.781548	−0.552638	−0.276319	+	0.961066i	$0.589115\pi$
$3$	0	0
$3$	0	0
$4$	−1.38918	−0.694591
$5$	−1.48073	−0.662201	−0.331100	−	0.943596i	$-0.607420\pi$
$5$	−1.48073	−0.662201	−0.331100	+	0.943596i	$0.607420\pi$
$6$	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	$-0.224225\pi$
$11$	5.05444	1.52397	0.761985	+	0.647595i	$0.224225\pi$
$12$	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	$-0.312463\pi$
$17$	4.58215	1.11134	0.555668	+	0.831404i	$0.312463\pi$
$18$	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	$-0.617292\pi$
$23$	−3.45493	−0.720403	−0.360201	+	0.932875i	$0.617292\pi$
$24$	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	$-0.480691\pi$
$29$	0.652939	0.121248	0.0606239	+	0.998161i	$0.480691\pi$
$30$	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	$-0.516980\pi$
$41$	−0.682813	−0.106637	−0.0533187	+	0.998578i	$0.516980\pi$
$42$	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	$-0.303559\pi$
$47$	7.93477	1.15741	0.578703	+	0.815539i	$0.303559\pi$
$48$	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	$-0.408013\pi$
$53$	4.14938	0.569962	0.284981	+	0.958533i	$0.408013\pi$
$54$	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	$-0.388541\pi$
$59$	5.26997	0.686092	0.343046	+	0.939319i	$0.388541\pi$
$60$	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	$-0.618566\pi$
$71$	−6.13308	−0.727863	−0.363932	+	0.931426i	$0.618566\pi$
$72$	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	$-0.471633\pi$
$83$	1.62164	0.177999	0.0889993	+	0.996032i	$0.471633\pi$
$84$	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	$-0.517602\pi$
$89$	−1.04284	−0.110541	−0.0552703	+	0.998471i	$0.517602\pi$
$90$	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	$-0.657968\pi$
$101$	−9.57048	−0.952299	−0.476149	+	0.879364i	$0.657968\pi$
$102$	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	$-0.197988\pi$
$107$	16.8136	1.62543	0.812716	+	0.582660i	$0.197988\pi$
$108$	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	$-0.656205\pi$
$113$	−10.0194	−0.942543	−0.471272	+	0.881988i	$0.656205\pi$
$114$	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	$-0.759236\pi$
$131$	−16.6492	−1.45465	−0.727324	+	0.686294i	$0.759236\pi$
$132$	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	$-0.370313\pi$
$137$	9.27589	0.792492	0.396246	+	0.918144i	$0.370313\pi$
$138$	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	$-0.0946496\pi$
$149$	23.3418	1.91223	0.956116	+	0.292988i	$0.0946496\pi$
$150$	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	$-0.458127\pi$
$167$	3.39014	0.262337	0.131168	+	0.991360i	$0.458127\pi$
$168$	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	$-0.751336\pi$
$173$	−18.6790	−1.42014	−0.710069	+	0.704132i	$0.751336\pi$
$174$	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	$-0.353507\pi$
$179$	11.8846	0.888294	0.444147	+	0.895954i	$0.353507\pi$
$180$	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	$-0.433100\pi$
$191$	5.76661	0.417257	0.208629	+	0.977995i	$0.433100\pi$
$192$	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	$-0.369172\pi$
$197$	11.2155	0.799069	0.399535	+	0.916718i	$0.369172\pi$
$198$	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	$-0.804000\pi$
$227$	−24.5988	−1.63268	−0.816340	+	0.577572i	$0.804000\pi$
$228$	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	$-0.347653\pi$
$233$	14.0599	0.921094	0.460547	+	0.887635i	$0.347653\pi$
$234$	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	$-0.765743\pi$
$239$	−22.9174	−1.48240	−0.741201	+	0.671283i	$0.765743\pi$
$240$	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	$-0.499332\pi$
$251$	0.0664671	0.00419537	0.00209768	+	0.999998i	$0.499332\pi$
$252$	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	$-0.293847\pi$
$257$	19.3437	1.20663	0.603313	+	0.797504i	$0.293847\pi$
$258$	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	$-0.346298\pi$
$263$	15.0601	0.928646	0.464323	+	0.885666i	$0.346298\pi$
$264$	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	$-0.616470\pi$
$269$	−11.7365	−0.715585	−0.357792	+	0.933801i	$0.616470\pi$
$270$	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	$-0.439388\pi$
$281$	6.34551	0.378541	0.189271	+	0.981925i	$0.439388\pi$
$282$	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	$-0.765475\pi$
$293$	−25.3553	−1.48127	−0.740635	+	0.671908i	$0.765475\pi$
$294$	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	$-0.752508\pi$
$311$	−25.1356	−1.42531	−0.712656	+	0.701514i	$0.752508\pi$
$312$	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	$-0.461575\pi$
$317$	4.28813	0.240845	0.120423	+	0.992723i	$0.461575\pi$
$318$	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	$-0.805266\pi$
$347$	−30.4987	−1.63726	−0.818629	+	0.574322i	$0.805266\pi$
$348$	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	$-0.697319\pi$
$353$	−21.8302	−1.16190	−0.580952	+	0.813938i	$0.697319\pi$
$354$	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	$-0.651006\pi$
$359$	−17.3105	−0.913611	−0.456805	+	0.889567i	$0.651006\pi$
$360$	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	$-0.632757\pi$
$383$	−15.8552	−0.810162	−0.405081	+	0.914281i	$0.632757\pi$
$384$	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	$-0.757114\pi$
$389$	−28.5090	−1.44547	−0.722733	+	0.691128i	$0.757114\pi$
$390$	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	$-0.360041\pi$
$401$	17.0478	0.851325	0.425662	+	0.904882i	$0.360041\pi$
$402$	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	$-0.625586\pi$
$419$	−15.7362	−0.768765	−0.384382	+	0.923174i	$0.625586\pi$
$420$	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	$-0.510824\pi$
$431$	−1.41170	−0.0679990	−0.0339995	+	0.999422i	$0.510824\pi$
$432$	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	$-0.201132\pi$
$443$	33.9675	1.61384	0.806921	+	0.590659i	$0.201132\pi$
$444$	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	$-0.490981\pi$
$449$	1.20059	0.0566596	0.0283298	+	0.999599i	$0.490981\pi$
$450$	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	$-0.497504\pi$
$461$	0.336685	0.0156810	0.00784049	+	0.999969i	$0.497504\pi$
$462$	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	$-0.472094\pi$
$467$	3.78427	0.175115	0.0875576	+	0.996159i	$0.472094\pi$
$468$	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	$-0.523049\pi$
$479$	−3.16676	−0.144693	−0.0723465	+	0.997380i	$0.523049\pi$
$480$	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	$-0.867767\pi$
$491$	−40.5477	−1.82989	−0.914946	+	0.403577i	$0.867767\pi$
$492$	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	$-0.572685\pi$
$503$	−10.1538	−0.452735	−0.226368	+	0.974042i	$0.572685\pi$
$504$	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	$-0.319418\pi$
$509$	24.2472	1.07474	0.537369	+	0.843347i	$0.319418\pi$
$510$	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	$-0.544424\pi$
$521$	−6.35050	−0.278221	−0.139110	+	0.990277i	$0.544424\pi$
$522$	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	$-0.432741\pi$
$557$	9.89968	0.419463	0.209731	+	0.977759i	$0.432741\pi$
$558$	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	$-0.547366\pi$
$563$	−7.03550	−0.296511	−0.148256	+	0.988949i	$0.547366\pi$
$564$	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	$-0.473694\pi$
$569$	3.93812	0.165095	0.0825473	+	0.996587i	$0.473694\pi$
$570$	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	$-0.305718\pi$
$587$	27.7731	1.14632	0.573159	+	0.819444i	$0.305718\pi$
$588$	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	$-0.595082\pi$
$593$	−14.3326	−0.588570	−0.294285	+	0.955718i	$0.595082\pi$
$594$	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	$-0.310081\pi$
$599$	27.5031	1.12375	0.561873	+	0.827223i	$0.310081\pi$
$600$	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	$-0.545981\pi$
$617$	−7.15141	−0.287905	−0.143952	+	0.989585i	$0.545981\pi$
$618$	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	$-0.527436\pi$
$641$	−4.35901	−0.172171	−0.0860853	+	0.996288i	$0.527436\pi$
$642$	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	$-0.632889\pi$
$647$	−20.6268	−0.810924	−0.405462	+	0.914112i	$0.632889\pi$
$648$	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	$-0.462602\pi$
$653$	5.99076	0.234437	0.117218	+	0.993106i	$0.462602\pi$
$654$	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	$-0.773080\pi$
$659$	−38.8389	−1.51295	−0.756475	+	0.654022i	$0.773080\pi$
$660$	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	$-0.475578\pi$
$677$	3.98869	0.153298	0.0766489	+	0.997058i	$0.475578\pi$
$678$	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	$-0.201761\pi$
$683$	42.1155	1.61151	0.805753	+	0.592252i	$0.201761\pi$
$684$	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	$-0.494698\pi$
$701$	0.881949	0.0333108	0.0166554	+	0.999861i	$0.494698\pi$
$702$	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	$-0.582671\pi$
$719$	−13.7722	−0.513615	−0.256807	+	0.966463i	$0.582671\pi$
$720$	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	$-0.730139\pi$
$743$	−36.0700	−1.32328	−0.661640	+	0.749822i	$0.730139\pi$
$744$	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	$-0.807193\pi$
$761$	−45.3568	−1.64418	−0.822092	+	0.569355i	$0.807193\pi$
$762$	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	$-0.322139\pi$
$773$	29.4789	1.06028	0.530141	+	0.847910i	$0.322139\pi$
$774$	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	$-0.433061\pi$
$797$	11.7864	0.417496	0.208748	+	0.977969i	$0.433061\pi$
$798$	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	$-0.377140\pi$
$809$	21.4155	0.752929	0.376464	+	0.926431i	$0.377140\pi$
$810$	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	$-0.549742\pi$
$821$	−8.91880	−0.311268	−0.155634	+	0.987815i	$0.549742\pi$
$822$	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	$-0.380806\pi$
$827$	21.0373	0.731537	0.365769	+	0.930706i	$0.380806\pi$
$828$	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	$-0.364464\pi$
$839$	23.9284	0.826099	0.413050	+	0.910709i	$0.364464\pi$
$840$	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	$-0.604376\pi$
$857$	−18.8564	−0.644123	−0.322062	+	0.946719i	$0.604376\pi$
$858$	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	$-0.797173\pi$
$863$	−47.2242	−1.60753	−0.803765	+	0.594947i	$0.797173\pi$
$864$	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	$-0.530452\pi$
$881$	−5.67058	−0.191047	−0.0955234	+	0.995427i	$0.530452\pi$
$882$	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	$-0.350948\pi$
$887$	26.8838	0.902670	0.451335	+	0.892355i	$0.350948\pi$
$888$	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	$-0.425142\pi$
$911$	14.0660	0.466026	0.233013	+	0.972474i	$0.425142\pi$
$912$	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	$-0.760930\pi$
$929$	−44.5589	−1.46193	−0.730965	+	0.682415i	$0.760930\pi$
$930$	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	$-0.123628\pi$
$941$	56.7820	1.85104	0.925520	+	0.378698i	$0.123628\pi$
$942$	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	$-0.582510\pi$
$947$	−15.7756	−0.512638	−0.256319	+	0.966592i	$0.582510\pi$
$948$	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	$-0.804294\pi$
$953$	−50.4349	−1.63375	−0.816873	+	0.576818i	$0.804294\pi$
$954$	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	$-0.611198\pi$
$971$	−21.3313	−0.684555	−0.342277	+	0.939599i	$0.611198\pi$
$972$	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	$-0.463405\pi$
$977$	7.17129	0.229430	0.114715	+	0.993398i	$0.463405\pi$
$978$	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	$-0.729631\pi$
$983$	−41.4135	−1.32089	−0.660443	+	0.750876i	$0.729631\pi$
$984$	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.c.1.30		72
3.2	odd	2		6561.2.a.d.1.43		72
81.5	odd	54		243.2.g.a.73.5		144
81.11	odd	54		729.2.g.a.514.4		144
81.16	even	27		81.2.g.a.13.4	✓	144
81.22	even	27		729.2.g.d.217.5		144
81.32	odd	54		729.2.g.b.703.4		144
81.38	odd	54		729.2.g.b.28.4		144
81.43	even	27		729.2.g.c.28.5		144
81.49	even	27		729.2.g.c.703.5		144
81.59	odd	54		729.2.g.a.217.4		144
81.65	odd	54		243.2.g.a.10.5		144
81.70	even	27		729.2.g.d.514.5		144
81.76	even	27		81.2.g.a.25.4	yes	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.4	✓	144	81.16	even	27
81.2.g.a.25.4	yes	144	81.76	even	27
243.2.g.a.10.5		144	81.65	odd	54
243.2.g.a.73.5		144	81.5	odd	54
729.2.g.a.217.4		144	81.59	odd	54
729.2.g.a.514.4		144	81.11	odd	54
729.2.g.b.28.4		144	81.38	odd	54
729.2.g.b.703.4		144	81.32	odd	54
729.2.g.c.28.5		144	81.43	even	27
729.2.g.c.703.5		144	81.49	even	27
729.2.g.d.217.5		144	81.22	even	27
729.2.g.d.514.5		144	81.70	even	27
6561.2.a.c.1.30		72	1.1	even	1	trivial
6561.2.a.d.1.43		72	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:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
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

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