LMFDB - Embedded newform 6561.2.a.c.1.32

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			1.32
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)

$f(q)$	$=$	\(q-0.567608 q^{2} -1.67782 q^{4} +2.90373 q^{5} +3.12982 q^{7} +2.08756 q^{8} -1.64818 q^{10} -3.43215 q^{11} -3.98780 q^{13} -1.77651 q^{14} +2.17072 q^{16} -0.934819 q^{17} +4.85100 q^{19} -4.87193 q^{20} +1.94811 q^{22} -6.15225 q^{23} +3.43163 q^{25} +2.26351 q^{26} -5.25127 q^{28} +1.79970 q^{29} +0.675872 q^{31} -5.40724 q^{32} +0.530611 q^{34} +9.08813 q^{35} -11.3972 q^{37} -2.75347 q^{38} +6.06171 q^{40} +6.65264 q^{41} -2.23641 q^{43} +5.75853 q^{44} +3.49207 q^{46} -3.16965 q^{47} +2.79575 q^{49} -1.94782 q^{50} +6.69082 q^{52} +8.37927 q^{53} -9.96601 q^{55} +6.53368 q^{56} -1.02153 q^{58} -5.03087 q^{59} -0.570638 q^{61} -0.383631 q^{62} -1.27225 q^{64} -11.5795 q^{65} -13.6689 q^{67} +1.56846 q^{68} -5.15850 q^{70} -13.3479 q^{71} -6.74596 q^{73} +6.46916 q^{74} -8.13910 q^{76} -10.7420 q^{77} +0.745336 q^{79} +6.30319 q^{80} -3.77609 q^{82} -3.63944 q^{83} -2.71446 q^{85} +1.26941 q^{86} -7.16481 q^{88} -13.1174 q^{89} -12.4811 q^{91} +10.3224 q^{92} +1.79912 q^{94} +14.0860 q^{95} +12.5053 q^{97} -1.58689 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.567608	−0.401360	−0.200680	−	0.979657i	$-0.564315\pi$
$2$	−0.567608	−0.401360	−0.200680	+	0.979657i	$0.564315\pi$
$3$	0	0
$3$	0	0
$4$	−1.67782	−0.838910
$5$	2.90373	1.29859	0.649293	−	0.760538i	$-0.275065\pi$
$5$	2.90373	1.29859	0.649293	+	0.760538i	$0.275065\pi$
$6$	0	0
$7$	3.12982	1.18296	0.591480	−	0.806320i	$-0.298544\pi$
$7$	3.12982	1.18296	0.591480	+	0.806320i	$0.298544\pi$
$8$	2.08756	0.738064
$9$	0	0
$10$	−1.64818	−0.521200
$11$	−3.43215	−1.03483	−0.517415	−	0.855734i	$-0.673106\pi$
$11$	−3.43215	−1.03483	−0.517415	+	0.855734i	$0.673106\pi$
$12$	0	0
$13$	−3.98780	−1.10602	−0.553009	−	0.833175i	$-0.686520\pi$
$13$	−3.98780	−1.10602	−0.553009	+	0.833175i	$0.686520\pi$
$14$	−1.77651	−0.474792
$15$	0	0
$16$	2.17072	0.542681
$17$	−0.934819	−0.226727	−0.113363	−	0.993554i	$-0.536162\pi$
$17$	−0.934819	−0.226727	−0.113363	+	0.993554i	$0.536162\pi$
$18$	0	0
$19$	4.85100	1.11290	0.556448	−	0.830883i	$-0.312164\pi$
$19$	4.85100	1.11290	0.556448	+	0.830883i	$0.312164\pi$
$20$	−4.87193	−1.08940
$21$	0	0
$22$	1.94811	0.415339
$23$	−6.15225	−1.28283	−0.641416	−	0.767193i	$-0.721653\pi$
$23$	−6.15225	−1.28283	−0.641416	+	0.767193i	$0.721653\pi$
$24$	0	0
$25$	3.43163	0.686327
$26$	2.26351	0.443911
$27$	0	0
$28$	−5.25127	−0.992397
$29$	1.79970	0.334197	0.167098	−	0.985940i	$-0.446560\pi$
$29$	1.79970	0.334197	0.167098	+	0.985940i	$0.446560\pi$
$30$	0	0
$31$	0.675872	0.121390	0.0606951	−	0.998156i	$-0.480668\pi$
$31$	0.675872	0.121390	0.0606951	+	0.998156i	$0.480668\pi$
$32$	−5.40724	−0.955875
$33$	0	0
$34$	0.530611	0.0909990
$35$	9.08813	1.53617
$36$	0	0
$37$	−11.3972	−1.87369	−0.936847	−	0.349740i	$-0.886270\pi$
$37$	−11.3972	−1.87369	−0.936847	+	0.349740i	$0.886270\pi$
$38$	−2.75347	−0.446671
$39$	0	0
$40$	6.06171	0.958440
$41$	6.65264	1.03897	0.519484	−	0.854480i	$-0.326124\pi$
$41$	6.65264	1.03897	0.519484	+	0.854480i	$0.326124\pi$
$42$	0	0
$43$	−2.23641	−0.341050	−0.170525	−	0.985353i	$-0.554546\pi$
$43$	−2.23641	−0.341050	−0.170525	+	0.985353i	$0.554546\pi$
$44$	5.75853	0.868130
$45$	0	0
$46$	3.49207	0.514877
$47$	−3.16965	−0.462341	−0.231170	−	0.972913i	$-0.574255\pi$
$47$	−3.16965	−0.462341	−0.231170	+	0.972913i	$0.574255\pi$
$48$	0	0
$49$	2.79575	0.399392
$50$	−1.94782	−0.275464
$51$	0	0
$52$	6.69082	0.927850
$53$	8.37927	1.15098	0.575490	−	0.817808i	$-0.304811\pi$
$53$	8.37927	1.15098	0.575490	+	0.817808i	$0.304811\pi$
$54$	0	0
$55$	−9.96601	−1.34382
$56$	6.53368	0.873100
$57$	0	0
$58$	−1.02153	−0.134133
$59$	−5.03087	−0.654963	−0.327482	−	0.944858i	$-0.606200\pi$
$59$	−5.03087	−0.654963	−0.327482	+	0.944858i	$0.606200\pi$
$60$	0	0
$61$	−0.570638	−0.0730627	−0.0365314	−	0.999333i	$-0.511631\pi$
$61$	−0.570638	−0.0730627	−0.0365314	+	0.999333i	$0.511631\pi$
$62$	−0.383631	−0.0487211
$63$	0	0
$64$	−1.27225	−0.159032
$65$	−11.5795	−1.43626
$66$	0	0
$67$	−13.6689	−1.66993	−0.834963	−	0.550306i	$-0.814511\pi$
$67$	−13.6689	−1.66993	−0.834963	+	0.550306i	$0.814511\pi$
$68$	1.56846	0.190204
$69$	0	0
$70$	−5.15850	−0.616558
$71$	−13.3479	−1.58411	−0.792054	−	0.610451i	$-0.790988\pi$
$71$	−13.3479	−1.58411	−0.792054	+	0.610451i	$0.790988\pi$
$72$	0	0
$73$	−6.74596	−0.789555	−0.394777	−	0.918777i	$-0.629178\pi$
$73$	−6.74596	−0.789555	−0.394777	+	0.918777i	$0.629178\pi$
$74$	6.46916	0.752025
$75$	0	0
$76$	−8.13910	−0.933619
$77$	−10.7420	−1.22416
$78$	0	0
$79$	0.745336	0.0838569	0.0419284	−	0.999121i	$-0.486650\pi$
$79$	0.745336	0.0838569	0.0419284	+	0.999121i	$0.486650\pi$
$80$	6.30319	0.704718
$81$	0	0
$82$	−3.77609	−0.417000
$83$	−3.63944	−0.399481	−0.199740	−	0.979849i	$-0.564010\pi$
$83$	−3.63944	−0.399481	−0.199740	+	0.979849i	$0.564010\pi$
$84$	0	0
$85$	−2.71446	−0.294424
$86$	1.26941	0.136884
$87$	0	0
$88$	−7.16481	−0.763772
$89$	−13.1174	−1.39044	−0.695219	−	0.718798i	$-0.744693\pi$
$89$	−13.1174	−1.39044	−0.695219	+	0.718798i	$0.744693\pi$
$90$	0	0
$91$	−12.4811	−1.30837
$92$	10.3224	1.07618
$93$	0	0
$94$	1.79912	0.185565
$95$	14.0860	1.44519
$96$	0	0
$97$	12.5053	1.26972	0.634860	−	0.772627i	$-0.281058\pi$
$97$	12.5053	1.26972	0.634860	+	0.772627i	$0.281058\pi$
$98$	−1.58689	−0.160300
$99$	0	0
$100$	−5.75767	−0.575767
$101$	−5.98380	−0.595410	−0.297705	−	0.954658i	$-0.596221\pi$
$101$	−5.98380	−0.595410	−0.297705	+	0.954658i	$0.596221\pi$
$102$	0	0
$103$	1.76860	0.174265	0.0871326	−	0.996197i	$-0.472230\pi$
$103$	1.76860	0.174265	0.0871326	+	0.996197i	$0.472230\pi$
$104$	−8.32478	−0.816312
$105$	0	0
$106$	−4.75614	−0.461957
$107$	7.15256	0.691464	0.345732	−	0.938333i	$-0.387631\pi$
$107$	7.15256	0.691464	0.345732	+	0.938333i	$0.387631\pi$
$108$	0	0
$109$	11.3723	1.08927	0.544633	−	0.838675i	$-0.316669\pi$
$109$	11.3723	1.08927	0.544633	+	0.838675i	$0.316669\pi$
$110$	5.65679	0.539354
$111$	0	0
$112$	6.79397	0.641970
$113$	−5.82409	−0.547885	−0.273942	−	0.961746i	$-0.588328\pi$
$113$	−5.82409	−0.547885	−0.273942	+	0.961746i	$0.588328\pi$
$114$	0	0
$115$	−17.8645	−1.66587
$116$	−3.01958	−0.280361
$117$	0	0
$118$	2.85556	0.262876
$119$	−2.92581	−0.268209
$120$	0	0
$121$	0.779623	0.0708748
$122$	0.323899	0.0293244
$123$	0	0
$124$	−1.13399	−0.101836
$125$	−4.55411	−0.407332
$126$	0	0
$127$	10.7409	0.953103	0.476551	−	0.879147i	$-0.341887\pi$
$127$	10.7409	0.953103	0.476551	+	0.879147i	$0.341887\pi$
$128$	11.5366	1.01970
$129$	0	0
$130$	6.57261	0.576456
$131$	−7.16262	−0.625801	−0.312900	−	0.949786i	$-0.601301\pi$
$131$	−7.16262	−0.625801	−0.312900	+	0.949786i	$0.601301\pi$
$132$	0	0
$133$	15.1827	1.31651
$134$	7.75860	0.670241
$135$	0	0
$136$	−1.95149	−0.167339
$137$	−8.62912	−0.737235	−0.368618	−	0.929581i	$-0.620169\pi$
$137$	−8.62912	−0.737235	−0.368618	+	0.929581i	$0.620169\pi$
$138$	0	0
$139$	−5.89915	−0.500359	−0.250180	−	0.968199i	$-0.580490\pi$
$139$	−5.89915	−0.500359	−0.250180	+	0.968199i	$0.580490\pi$
$140$	−15.2483	−1.28871
$141$	0	0
$142$	7.57640	0.635797
$143$	13.6867	1.14454
$144$	0	0
$145$	5.22585	0.433983
$146$	3.82906	0.316895
$147$	0	0
$148$	19.1225	1.57186
$149$	−13.7946	−1.13010	−0.565049	−	0.825057i	$-0.691143\pi$
$149$	−13.7946	−1.13010	−0.565049	+	0.825057i	$0.691143\pi$
$150$	0	0
$151$	6.27700	0.510815	0.255408	−	0.966833i	$-0.417790\pi$
$151$	6.27700	0.510815	0.255408	+	0.966833i	$0.417790\pi$
$152$	10.1268	0.821388
$153$	0	0
$154$	6.09724	0.491329
$155$	1.96255	0.157636
$156$	0	0
$157$	−0.166094	−0.0132557	−0.00662787	−	0.999978i	$-0.502110\pi$
$157$	−0.166094	−0.0132557	−0.00662787	+	0.999978i	$0.502110\pi$
$158$	−0.423059	−0.0336568
$159$	0	0
$160$	−15.7012	−1.24129
$161$	−19.2554	−1.51754
$162$	0	0
$163$	16.2014	1.26899	0.634495	−	0.772927i	$-0.281208\pi$
$163$	16.2014	1.26899	0.634495	+	0.772927i	$0.281208\pi$
$164$	−11.1619	−0.871601
$165$	0	0
$166$	2.06578	0.160335
$167$	−18.2760	−1.41424	−0.707122	−	0.707092i	$-0.750007\pi$
$167$	−18.2760	−1.41424	−0.707122	+	0.707092i	$0.750007\pi$
$168$	0	0
$169$	2.90257	0.223274
$170$	1.54075	0.118170
$171$	0	0
$172$	3.75230	0.286110
$173$	18.8538	1.43343	0.716715	−	0.697366i	$-0.245645\pi$
$173$	18.8538	1.43343	0.716715	+	0.697366i	$0.245645\pi$
$174$	0	0
$175$	10.7404	0.811896
$176$	−7.45024	−0.561583
$177$	0	0
$178$	7.44553	0.558066
$179$	−10.2961	−0.769570	−0.384785	−	0.923006i	$-0.625724\pi$
$179$	−10.2961	−0.769570	−0.384785	+	0.923006i	$0.625724\pi$
$180$	0	0
$181$	−0.819628	−0.0609224	−0.0304612	−	0.999536i	$-0.509698\pi$
$181$	−0.819628	−0.0609224	−0.0304612	+	0.999536i	$0.509698\pi$
$182$	7.08437	0.525128
$183$	0	0
$184$	−12.8432	−0.946813
$185$	−33.0945	−2.43315
$186$	0	0
$187$	3.20843	0.234624
$188$	5.31810	0.387862
$189$	0	0
$190$	−7.99531	−0.580041
$191$	−0.765051	−0.0553572	−0.0276786	−	0.999617i	$-0.508811\pi$
$191$	−0.765051	−0.0553572	−0.0276786	+	0.999617i	$0.508811\pi$
$192$	0	0
$193$	9.35469	0.673365	0.336683	−	0.941618i	$-0.390695\pi$
$193$	9.35469	0.673365	0.336683	+	0.941618i	$0.390695\pi$
$194$	−7.09811	−0.509614
$195$	0	0
$196$	−4.69076	−0.335054
$197$	1.11982	0.0797839	0.0398920	−	0.999204i	$-0.487299\pi$
$197$	1.11982	0.0797839	0.0398920	+	0.999204i	$0.487299\pi$
$198$	0	0
$199$	7.68060	0.544463	0.272232	−	0.962232i	$-0.412238\pi$
$199$	7.68060	0.544463	0.272232	+	0.962232i	$0.412238\pi$
$200$	7.16374	0.506553
$201$	0	0
$202$	3.39645	0.238974
$203$	5.63274	0.395341
$204$	0	0
$205$	19.3174	1.34919
$206$	−1.00387	−0.0699430
$207$	0	0
$208$	−8.65642	−0.600215
$209$	−16.6493	−1.15166
$210$	0	0
$211$	−10.2373	−0.704764	−0.352382	−	0.935856i	$-0.614628\pi$
$211$	−10.2373	−0.704764	−0.352382	+	0.935856i	$0.614628\pi$
$212$	−14.0589	−0.965570
$213$	0	0
$214$	−4.05985	−0.277526
$215$	−6.49393	−0.442882
$216$	0	0
$217$	2.11536	0.143600
$218$	−6.45499	−0.437187
$219$	0	0
$220$	16.7212	1.12734
$221$	3.72787	0.250764
$222$	0	0
$223$	−4.65168	−0.311499	−0.155750	−	0.987797i	$-0.549779\pi$
$223$	−4.65168	−0.311499	−0.155750	+	0.987797i	$0.549779\pi$
$224$	−16.9237	−1.13076
$225$	0	0
$226$	3.30580	0.219899
$227$	14.1196	0.937149	0.468575	−	0.883424i	$-0.344768\pi$
$227$	14.1196	0.937149	0.468575	+	0.883424i	$0.344768\pi$
$228$	0	0
$229$	−5.18532	−0.342656	−0.171328	−	0.985214i	$-0.554806\pi$
$229$	−5.18532	−0.342656	−0.171328	+	0.985214i	$0.554806\pi$
$230$	10.1400	0.668613
$231$	0	0
$232$	3.75699	0.246659
$233$	4.45346	0.291756	0.145878	−	0.989303i	$-0.453399\pi$
$233$	4.45346	0.291756	0.145878	+	0.989303i	$0.453399\pi$
$234$	0	0
$235$	−9.20379	−0.600389
$236$	8.44090	0.549456
$237$	0	0
$238$	1.66071	0.107648
$239$	−29.0614	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$239$	−29.0614	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$240$	0	0
$241$	24.9840	1.60936	0.804682	−	0.593706i	$-0.202336\pi$
$241$	24.9840	1.60936	0.804682	+	0.593706i	$0.202336\pi$
$242$	−0.442520	−0.0284463
$243$	0	0
$244$	0.957429	0.0612931
$245$	8.11809	0.518646
$246$	0	0
$247$	−19.3448	−1.23088
$248$	1.41093	0.0895938
$249$	0	0
$250$	2.58495	0.163487
$251$	14.1718	0.894515	0.447258	−	0.894405i	$-0.352401\pi$
$251$	14.1718	0.894515	0.447258	+	0.894405i	$0.352401\pi$
$252$	0	0
$253$	21.1154	1.32751
$254$	−6.09663	−0.382537
$255$	0	0
$256$	−4.00378	−0.250236
$257$	14.6989	0.916895	0.458448	−	0.888721i	$-0.348406\pi$
$257$	14.6989	0.916895	0.458448	+	0.888721i	$0.348406\pi$
$258$	0	0
$259$	−35.6712	−2.21650
$260$	19.4283	1.20489
$261$	0	0
$262$	4.06556	0.251171
$263$	16.3890	1.01059	0.505294	−	0.862947i	$-0.331384\pi$
$263$	16.3890	1.01059	0.505294	+	0.862947i	$0.331384\pi$
$264$	0	0
$265$	24.3311	1.49465
$266$	−8.61784	−0.528394
$267$	0	0
$268$	22.9340	1.40092
$269$	−11.3626	−0.692791	−0.346396	−	0.938088i	$-0.612595\pi$
$269$	−11.3626	−0.692791	−0.346396	+	0.938088i	$0.612595\pi$
$270$	0	0
$271$	2.76531	0.167981	0.0839903	−	0.996467i	$-0.473234\pi$
$271$	2.76531	0.167981	0.0839903	+	0.996467i	$0.473234\pi$
$272$	−2.02923	−0.123040
$273$	0	0
$274$	4.89796	0.295896
$275$	−11.7779	−0.710232
$276$	0	0
$277$	18.3961	1.10531	0.552657	−	0.833409i	$-0.313614\pi$
$277$	18.3961	1.10531	0.552657	+	0.833409i	$0.313614\pi$
$278$	3.34841	0.200824
$279$	0	0
$280$	18.9720	1.13380
$281$	10.7332	0.640287	0.320144	−	0.947369i	$-0.396269\pi$
$281$	10.7332	0.640287	0.320144	+	0.947369i	$0.396269\pi$
$282$	0	0
$283$	−14.2919	−0.849568	−0.424784	−	0.905295i	$-0.639650\pi$
$283$	−14.2919	−0.849568	−0.424784	+	0.905295i	$0.639650\pi$
$284$	22.3954	1.32893
$285$	0	0
$286$	−7.76869	−0.459372
$287$	20.8215	1.22906
$288$	0	0
$289$	−16.1261	−0.948595
$290$	−2.96624	−0.174183
$291$	0	0
$292$	11.3185	0.662366
$293$	−21.8459	−1.27625	−0.638126	−	0.769932i	$-0.720290\pi$
$293$	−21.8459	−1.27625	−0.638126	+	0.769932i	$0.720290\pi$
$294$	0	0
$295$	−14.6083	−0.850527
$296$	−23.7924	−1.38291
$297$	0	0
$298$	7.82994	0.453576
$299$	24.5340	1.41884
$300$	0	0
$301$	−6.99956	−0.403448
$302$	−3.56288	−0.205021
$303$	0	0
$304$	10.5302	0.603947
$305$	−1.65698	−0.0948783
$306$	0	0
$307$	30.4747	1.73928	0.869640	−	0.493686i	$-0.164351\pi$
$307$	30.4747	1.73928	0.869640	+	0.493686i	$0.164351\pi$
$308$	18.0231	1.02696
$309$	0	0
$310$	−1.11396	−0.0632686
$311$	27.1122	1.53739	0.768697	−	0.639613i	$-0.220906\pi$
$311$	27.1122	1.53739	0.768697	+	0.639613i	$0.220906\pi$
$312$	0	0
$313$	−8.10638	−0.458200	−0.229100	−	0.973403i	$-0.573578\pi$
$313$	−8.10638	−0.458200	−0.229100	+	0.973403i	$0.573578\pi$
$314$	0.0942762	0.00532032
$315$	0	0
$316$	−1.25054	−0.0703484
$317$	−19.4191	−1.09069	−0.545343	−	0.838213i	$-0.683600\pi$
$317$	−19.4191	−1.09069	−0.545343	+	0.838213i	$0.683600\pi$
$318$	0	0
$319$	−6.17685	−0.345837
$320$	−3.69428	−0.206516
$321$	0	0
$322$	10.9295	0.609079
$323$	−4.53480	−0.252323
$324$	0	0
$325$	−13.6847	−0.759089
$326$	−9.19603	−0.509321
$327$	0	0
$328$	13.8878	0.766825
$329$	−9.92041	−0.546930
$330$	0	0
$331$	1.98671	0.109200	0.0545998	−	0.998508i	$-0.482612\pi$
$331$	1.98671	0.109200	0.0545998	+	0.998508i	$0.482612\pi$
$332$	6.10633	0.335129
$333$	0	0
$334$	10.3736	0.567620
$335$	−39.6909	−2.16854
$336$	0	0
$337$	−18.3556	−0.999891	−0.499946	−	0.866057i	$-0.666647\pi$
$337$	−18.3556	−0.999891	−0.499946	+	0.866057i	$0.666647\pi$
$338$	−1.64752	−0.0896134
$339$	0	0
$340$	4.55438	0.246996
$341$	−2.31969	−0.125618
$342$	0	0
$343$	−13.1585	−0.710494
$344$	−4.66865	−0.251717
$345$	0	0
$346$	−10.7016	−0.575321
$347$	26.7840	1.43784	0.718920	−	0.695093i	$-0.244637\pi$
$347$	26.7840	1.43784	0.718920	+	0.695093i	$0.244637\pi$
$348$	0	0
$349$	−13.1094	−0.701732	−0.350866	−	0.936426i	$-0.614113\pi$
$349$	−13.1094	−0.701732	−0.350866	+	0.936426i	$0.614113\pi$
$350$	−6.09633	−0.325862
$351$	0	0
$352$	18.5584	0.989169
$353$	−19.4348	−1.03441	−0.517205	−	0.855862i	$-0.673028\pi$
$353$	−19.4348	−1.03441	−0.517205	+	0.855862i	$0.673028\pi$
$354$	0	0
$355$	−38.7588	−2.05710
$356$	22.0086	1.16645
$357$	0	0
$358$	5.84418	0.308874
$359$	−14.6764	−0.774593	−0.387297	−	0.921955i	$-0.626591\pi$
$359$	−14.6764	−0.774593	−0.387297	+	0.921955i	$0.626591\pi$
$360$	0	0
$361$	4.53218	0.238536
$362$	0.465227	0.0244518
$363$	0	0
$364$	20.9410	1.09761
$365$	−19.5884	−1.02531
$366$	0	0
$367$	−10.9940	−0.573884	−0.286942	−	0.957948i	$-0.592639\pi$
$367$	−10.9940	−0.573884	−0.286942	+	0.957948i	$0.592639\pi$
$368$	−13.3548	−0.696169
$369$	0	0
$370$	18.7847	0.976569
$371$	26.2256	1.36156
$372$	0	0
$373$	−20.4182	−1.05721	−0.528607	−	0.848867i	$-0.677286\pi$
$373$	−20.4182	−1.05721	−0.528607	+	0.848867i	$0.677286\pi$
$374$	−1.82113	−0.0941686
$375$	0	0
$376$	−6.61683	−0.341237
$377$	−7.17687	−0.369627
$378$	0	0
$379$	26.0308	1.33711	0.668556	−	0.743662i	$-0.266913\pi$
$379$	26.0308	1.33711	0.668556	+	0.743662i	$0.266913\pi$
$380$	−23.6337	−1.21239
$381$	0	0
$382$	0.434249	0.0222181
$383$	−23.5306	−1.20236	−0.601180	−	0.799114i	$-0.705302\pi$
$383$	−23.5306	−1.20236	−0.601180	+	0.799114i	$0.705302\pi$
$384$	0	0
$385$	−31.1918	−1.58968
$386$	−5.30980	−0.270262
$387$	0	0
$388$	−20.9816	−1.06518
$389$	−33.0703	−1.67673	−0.838366	−	0.545107i	$-0.816489\pi$
$389$	−33.0703	−1.67673	−0.838366	+	0.545107i	$0.816489\pi$
$390$	0	0
$391$	5.75124	0.290853
$392$	5.83629	0.294777
$393$	0	0
$394$	−0.635619	−0.0320220
$395$	2.16425	0.108895
$396$	0	0
$397$	−0.0552892	−0.00277489	−0.00138744	−	0.999999i	$-0.500442\pi$
$397$	−0.0552892	−0.00277489	−0.00138744	+	0.999999i	$0.500442\pi$
$398$	−4.35957	−0.218526
$399$	0	0
$400$	7.44913	0.372457
$401$	−28.6437	−1.43040	−0.715198	−	0.698922i	$-0.753664\pi$
$401$	−28.6437	−1.43040	−0.715198	+	0.698922i	$0.753664\pi$
$402$	0	0
$403$	−2.69525	−0.134260
$404$	10.0397	0.499496
$405$	0	0
$406$	−3.19719	−0.158674
$407$	39.1170	1.93896
$408$	0	0
$409$	31.4369	1.55445	0.777227	−	0.629220i	$-0.216626\pi$
$409$	31.4369	1.55445	0.777227	+	0.629220i	$0.216626\pi$
$410$	−10.9647	−0.541510
$411$	0	0
$412$	−2.96739	−0.146193
$413$	−15.7457	−0.774795
$414$	0	0
$415$	−10.5680	−0.518760
$416$	21.5630	1.05721
$417$	0	0
$418$	9.45030	0.462229
$419$	−3.37426	−0.164843	−0.0824217	−	0.996598i	$-0.526265\pi$
$419$	−3.37426	−0.164843	−0.0824217	+	0.996598i	$0.526265\pi$
$420$	0	0
$421$	25.2636	1.23127	0.615636	−	0.788031i	$-0.288899\pi$
$421$	25.2636	1.23127	0.615636	+	0.788031i	$0.288899\pi$
$422$	5.81077	0.282864
$423$	0	0
$424$	17.4922	0.849498
$425$	−3.20796	−0.155609
$426$	0	0
$427$	−1.78599	−0.0864302
$428$	−12.0007	−0.580076
$429$	0	0
$430$	3.68601	0.177755
$431$	−32.8136	−1.58058	−0.790288	−	0.612736i	$-0.790069\pi$
$431$	−32.8136	−1.58058	−0.790288	+	0.612736i	$0.790069\pi$
$432$	0	0
$433$	−12.9224	−0.621013	−0.310507	−	0.950571i	$-0.600499\pi$
$433$	−12.9224	−0.621013	−0.310507	+	0.950571i	$0.600499\pi$
$434$	−1.20069	−0.0576351
$435$	0	0
$436$	−19.0806	−0.913797
$437$	−29.8446	−1.42766
$438$	0	0
$439$	36.8773	1.76006	0.880030	−	0.474918i	$-0.157522\pi$
$439$	36.8773	1.76006	0.880030	+	0.474918i	$0.157522\pi$
$440$	−20.8047	−0.991824
$441$	0	0
$442$	−2.11597	−0.100647
$443$	−34.3635	−1.63266	−0.816330	−	0.577586i	$-0.803995\pi$
$443$	−34.3635	−1.63266	−0.816330	+	0.577586i	$0.803995\pi$
$444$	0	0
$445$	−38.0893	−1.80560
$446$	2.64033	0.125023
$447$	0	0
$448$	−3.98192	−0.188128
$449$	−17.5603	−0.828723	−0.414361	−	0.910112i	$-0.635995\pi$
$449$	−17.5603	−0.828723	−0.414361	+	0.910112i	$0.635995\pi$
$450$	0	0
$451$	−22.8328	−1.07516
$452$	9.77179	0.459626
$453$	0	0
$454$	−8.01439	−0.376134
$455$	−36.2417	−1.69904
$456$	0	0
$457$	−8.34889	−0.390544	−0.195272	−	0.980749i	$-0.562559\pi$
$457$	−8.34889	−0.390544	−0.195272	+	0.980749i	$0.562559\pi$
$458$	2.94323	0.137528
$459$	0	0
$460$	29.9734	1.39752
$461$	20.3398	0.947320	0.473660	−	0.880708i	$-0.342933\pi$
$461$	20.3398	0.947320	0.473660	+	0.880708i	$0.342933\pi$
$462$	0	0
$463$	−2.58935	−0.120337	−0.0601686	−	0.998188i	$-0.519164\pi$
$463$	−2.58935	−0.120337	−0.0601686	+	0.998188i	$0.519164\pi$
$464$	3.90666	0.181362
$465$	0	0
$466$	−2.52782	−0.117099
$467$	−12.2401	−0.566403	−0.283201	−	0.959060i	$-0.591396\pi$
$467$	−12.2401	−0.566403	−0.283201	+	0.959060i	$0.591396\pi$
$468$	0	0
$469$	−42.7812	−1.97545
$470$	5.22415	0.240972
$471$	0	0
$472$	−10.5023	−0.483405
$473$	7.67569	0.352929
$474$	0	0
$475$	16.6468	0.763809
$476$	4.90899	0.225003
$477$	0	0
$478$	16.4955	0.754486
$479$	−27.4270	−1.25317	−0.626587	−	0.779352i	$-0.715549\pi$
$479$	−27.4270	−1.25317	−0.626587	+	0.779352i	$0.715549\pi$
$480$	0	0
$481$	45.4499	2.07234
$482$	−14.1812	−0.645934
$483$	0	0
$484$	−1.30807	−0.0594576
$485$	36.3120	1.64884
$486$	0	0
$487$	−6.02417	−0.272981	−0.136491	−	0.990641i	$-0.543582\pi$
$487$	−6.02417	−0.272981	−0.136491	+	0.990641i	$0.543582\pi$
$488$	−1.19124	−0.0539250
$489$	0	0
$490$	−4.60789	−0.208163
$491$	−0.487291	−0.0219911	−0.0109956	−	0.999940i	$-0.503500\pi$
$491$	−0.487291	−0.0219911	−0.0109956	+	0.999940i	$0.503500\pi$
$492$	0	0
$493$	−1.68240	−0.0757714
$494$	10.9803	0.494026
$495$	0	0
$496$	1.46713	0.0658762
$497$	−41.7766	−1.87394
$498$	0	0
$499$	5.48473	0.245530	0.122765	−	0.992436i	$-0.460824\pi$
$499$	5.48473	0.245530	0.122765	+	0.992436i	$0.460824\pi$
$500$	7.64098	0.341715
$501$	0	0
$502$	−8.04402	−0.359022
$503$	−10.5682	−0.471215	−0.235607	−	0.971848i	$-0.575708\pi$
$503$	−10.5682	−0.471215	−0.235607	+	0.971848i	$0.575708\pi$
$504$	0	0
$505$	−17.3753	−0.773191
$506$	−11.9853	−0.532811
$507$	0	0
$508$	−18.0213	−0.799568
$509$	−12.4066	−0.549915	−0.274957	−	0.961456i	$-0.588664\pi$
$509$	−12.4066	−0.549915	−0.274957	+	0.961456i	$0.588664\pi$
$510$	0	0
$511$	−21.1136	−0.934011
$512$	−20.8007	−0.919269
$513$	0	0
$514$	−8.34324	−0.368005
$515$	5.13553	0.226298
$516$	0	0
$517$	10.8787	0.478444
$518$	20.2473	0.889615
$519$	0	0
$520$	−24.1729	−1.06005
$521$	−17.7888	−0.779342	−0.389671	−	0.920954i	$-0.627411\pi$
$521$	−17.7888	−0.779342	−0.389671	+	0.920954i	$0.627411\pi$
$522$	0	0
$523$	−13.5765	−0.593658	−0.296829	−	0.954931i	$-0.595929\pi$
$523$	−13.5765	−0.593658	−0.296829	+	0.954931i	$0.595929\pi$
$524$	12.0176	0.524991
$525$	0	0
$526$	−9.30253	−0.405610
$527$	−0.631818	−0.0275224
$528$	0	0
$529$	14.8502	0.645660
$530$	−13.8105	−0.599891
$531$	0	0
$532$	−25.4739	−1.10443
$533$	−26.5294	−1.14912
$534$	0	0
$535$	20.7691	0.897925
$536$	−28.5347	−1.23251
$537$	0	0
$538$	6.44952	0.278059
$539$	−9.59541	−0.413304
$540$	0	0
$541$	−27.3077	−1.17405	−0.587025	−	0.809569i	$-0.699701\pi$
$541$	−27.3077	−1.17405	−0.587025	+	0.809569i	$0.699701\pi$
$542$	−1.56961	−0.0674207
$543$	0	0
$544$	5.05479	0.216723
$545$	33.0220	1.41451
$546$	0	0
$547$	12.6976	0.542910	0.271455	−	0.962451i	$-0.412495\pi$
$547$	12.6976	0.542910	0.271455	+	0.962451i	$0.412495\pi$
$548$	14.4781	0.618474
$549$	0	0
$550$	6.68521	0.285058
$551$	8.73036	0.371926
$552$	0	0
$553$	2.33277	0.0991993
$554$	−10.4418	−0.443628
$555$	0	0
$556$	9.89771	0.419757
$557$	15.1085	0.640166	0.320083	−	0.947389i	$-0.396289\pi$
$557$	15.1085	0.640166	0.320083	+	0.947389i	$0.396289\pi$
$558$	0	0
$559$	8.91837	0.377207
$560$	19.7278	0.833653
$561$	0	0
$562$	−6.09224	−0.256985
$563$	6.13230	0.258445	0.129223	−	0.991616i	$-0.458752\pi$
$563$	6.13230	0.258445	0.129223	+	0.991616i	$0.458752\pi$
$564$	0	0
$565$	−16.9116	−0.711476
$566$	8.11223	0.340982
$567$	0	0
$568$	−27.8646	−1.16917
$569$	1.47937	0.0620185	0.0310092	−	0.999519i	$-0.490128\pi$
$569$	1.47937	0.0620185	0.0310092	+	0.999519i	$0.490128\pi$
$570$	0	0
$571$	8.81608	0.368941	0.184471	−	0.982838i	$-0.440943\pi$
$571$	8.81608	0.368941	0.184471	+	0.982838i	$0.440943\pi$
$572$	−22.9639	−0.960167
$573$	0	0
$574$	−11.8185	−0.493294
$575$	−21.1123	−0.880442
$576$	0	0
$577$	14.3168	0.596015	0.298008	−	0.954563i	$-0.403678\pi$
$577$	14.3168	0.596015	0.298008	+	0.954563i	$0.403678\pi$
$578$	9.15331	0.380728
$579$	0	0
$580$	−8.76804	−0.364073
$581$	−11.3908	−0.472570
$582$	0	0
$583$	−28.7589	−1.19107
$584$	−14.0826	−0.582742
$585$	0	0
$586$	12.3999	0.512236
$587$	−8.53860	−0.352426	−0.176213	−	0.984352i	$-0.556385\pi$
$587$	−8.53860	−0.352426	−0.176213	+	0.984352i	$0.556385\pi$
$588$	0	0
$589$	3.27865	0.135095
$590$	8.29178	0.341367
$591$	0	0
$592$	−24.7403	−1.01682
$593$	39.7479	1.63225	0.816125	−	0.577875i	$-0.196118\pi$
$593$	39.7479	1.63225	0.816125	+	0.577875i	$0.196118\pi$
$594$	0	0
$595$	−8.49576	−0.348292
$596$	23.1449	0.948052
$597$	0	0
$598$	−13.9257	−0.569463
$599$	−20.3776	−0.832607	−0.416304	−	0.909226i	$-0.636675\pi$
$599$	−20.3776	−0.832607	−0.416304	+	0.909226i	$0.636675\pi$
$600$	0	0
$601$	−13.6888	−0.558379	−0.279190	−	0.960236i	$-0.590066\pi$
$601$	−13.6888	−0.558379	−0.279190	+	0.960236i	$0.590066\pi$
$602$	3.97301	0.161928
$603$	0	0
$604$	−10.5317	−0.428528
$605$	2.26381	0.0920370
$606$	0	0
$607$	−4.50009	−0.182653	−0.0913265	−	0.995821i	$-0.529111\pi$
$607$	−4.50009	−0.182653	−0.0913265	+	0.995821i	$0.529111\pi$
$608$	−26.2305	−1.06379
$609$	0	0
$610$	0.940514	0.0380803
$611$	12.6399	0.511357
$612$	0	0
$613$	35.7734	1.44487	0.722436	−	0.691437i	$-0.243022\pi$
$613$	35.7734	1.44487	0.722436	+	0.691437i	$0.243022\pi$
$614$	−17.2977	−0.698077
$615$	0	0
$616$	−22.4245	−0.903511
$617$	27.5829	1.11044	0.555222	−	0.831702i	$-0.312633\pi$
$617$	27.5829	1.11044	0.555222	+	0.831702i	$0.312633\pi$
$618$	0	0
$619$	−33.4281	−1.34359	−0.671794	−	0.740738i	$-0.734476\pi$
$619$	−33.4281	−1.34359	−0.671794	+	0.740738i	$0.734476\pi$
$620$	−3.29281	−0.132242
$621$	0	0
$622$	−15.3891	−0.617048
$623$	−41.0549	−1.64483
$624$	0	0
$625$	−30.3821	−1.21528
$626$	4.60125	0.183903
$627$	0	0
$628$	0.278676	0.0111204
$629$	10.6544	0.424817
$630$	0	0
$631$	−10.9325	−0.435217	−0.217608	−	0.976036i	$-0.569826\pi$
$631$	−10.9325	−0.435217	−0.217608	+	0.976036i	$0.569826\pi$
$632$	1.55594	0.0618918
$633$	0	0
$634$	11.0224	0.437757
$635$	31.1887	1.23769
$636$	0	0
$637$	−11.1489	−0.441735
$638$	3.50603	0.138805
$639$	0	0
$640$	33.4992	1.32417
$641$	−9.31534	−0.367934	−0.183967	−	0.982932i	$-0.558894\pi$
$641$	−9.31534	−0.367934	−0.183967	+	0.982932i	$0.558894\pi$
$642$	0	0
$643$	1.28750	0.0507741	0.0253871	−	0.999678i	$-0.491918\pi$
$643$	1.28750	0.0507741	0.0253871	+	0.999678i	$0.491918\pi$
$644$	32.3071	1.27308
$645$	0	0
$646$	2.57399	0.101272
$647$	−25.7409	−1.01198	−0.505989	−	0.862540i	$-0.668872\pi$
$647$	−25.7409	−1.01198	−0.505989	+	0.862540i	$0.668872\pi$
$648$	0	0
$649$	17.2667	0.677776
$650$	7.76753	0.304668
$651$	0	0
$652$	−27.1830	−1.06457
$653$	−38.9594	−1.52460	−0.762299	−	0.647225i	$-0.775929\pi$
$653$	−38.9594	−1.52460	−0.762299	+	0.647225i	$0.775929\pi$
$654$	0	0
$655$	−20.7983	−0.812657
$656$	14.4410	0.563828
$657$	0	0
$658$	5.63091	0.219516
$659$	−33.5828	−1.30820	−0.654100	−	0.756408i	$-0.726953\pi$
$659$	−33.5828	−1.30820	−0.654100	+	0.756408i	$0.726953\pi$
$660$	0	0
$661$	35.4338	1.37822	0.689108	−	0.724659i	$-0.258003\pi$
$661$	35.4338	1.37822	0.689108	+	0.724659i	$0.258003\pi$
$662$	−1.12768	−0.0438283
$663$	0	0
$664$	−7.59756	−0.294843
$665$	44.0865	1.70960
$666$	0	0
$667$	−11.0722	−0.428719
$668$	30.6639	1.18642
$669$	0	0
$670$	22.5289	0.870366
$671$	1.95851	0.0756076
$672$	0	0
$673$	27.4725	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$673$	27.4725	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$674$	10.4188	0.401316
$675$	0	0
$676$	−4.86999	−0.187307
$677$	−9.55320	−0.367159	−0.183580	−	0.983005i	$-0.558769\pi$
$677$	−9.55320	−0.367159	−0.183580	+	0.983005i	$0.558769\pi$
$678$	0	0
$679$	39.1393	1.50203
$680$	−5.66660	−0.217304
$681$	0	0
$682$	1.31668	0.0504181
$683$	33.2321	1.27159	0.635795	−	0.771858i	$-0.280672\pi$
$683$	33.2321	1.27159	0.635795	+	0.771858i	$0.280672\pi$
$684$	0	0
$685$	−25.0566	−0.957364
$686$	7.46889	0.285164
$687$	0	0
$688$	−4.85464	−0.185081
$689$	−33.4149	−1.27300
$690$	0	0
$691$	−5.42964	−0.206553	−0.103277	−	0.994653i	$-0.532933\pi$
$691$	−5.42964	−0.206553	−0.103277	+	0.994653i	$0.532933\pi$
$692$	−31.6333	−1.20252
$693$	0	0
$694$	−15.2028	−0.577091
$695$	−17.1295	−0.649760
$696$	0	0
$697$	−6.21901	−0.235562
$698$	7.44103	0.281647
$699$	0	0
$700$	−18.0204	−0.681108
$701$	33.7389	1.27430	0.637150	−	0.770740i	$-0.280113\pi$
$701$	33.7389	1.27430	0.637150	+	0.770740i	$0.280113\pi$
$702$	0	0
$703$	−55.2880	−2.08522
$704$	4.36656	0.164571
$705$	0	0
$706$	11.0314	0.415170
$707$	−18.7282	−0.704346
$708$	0	0
$709$	−19.3855	−0.728037	−0.364018	−	0.931392i	$-0.618595\pi$
$709$	−19.3855	−0.728037	−0.364018	+	0.931392i	$0.618595\pi$
$710$	21.9998	0.825638
$711$	0	0
$712$	−27.3833	−1.02623
$713$	−4.15814	−0.155723
$714$	0	0
$715$	39.7425	1.48629
$716$	17.2751	0.645600
$717$	0	0
$718$	8.33047	0.310890
$719$	−4.78753	−0.178545	−0.0892724	−	0.996007i	$-0.528454\pi$
$719$	−4.78753	−0.178545	−0.0892724	+	0.996007i	$0.528454\pi$
$720$	0	0
$721$	5.53539	0.206149
$722$	−2.57250	−0.0957386
$723$	0	0
$724$	1.37519	0.0511085
$725$	6.17593	0.229368
$726$	0	0
$727$	−7.56024	−0.280394	−0.140197	−	0.990124i	$-0.544774\pi$
$727$	−7.56024	−0.280394	−0.140197	+	0.990124i	$0.544774\pi$
$728$	−26.0550	−0.965664
$729$	0	0
$730$	11.1186	0.411516
$731$	2.09064	0.0773251
$732$	0	0
$733$	38.6412	1.42725	0.713623	−	0.700530i	$-0.247053\pi$
$733$	38.6412	1.42725	0.713623	+	0.700530i	$0.247053\pi$
$734$	6.24030	0.230334
$735$	0	0
$736$	33.2667	1.22623
$737$	46.9138	1.72809
$738$	0	0
$739$	−50.6296	−1.86244	−0.931219	−	0.364459i	$-0.881254\pi$
$739$	−50.6296	−1.86244	−0.931219	+	0.364459i	$0.881254\pi$
$740$	55.5266	2.04120
$741$	0	0
$742$	−14.8858	−0.546477
$743$	15.3539	0.563280	0.281640	−	0.959520i	$-0.409122\pi$
$743$	15.3539	0.563280	0.281640	+	0.959520i	$0.409122\pi$
$744$	0	0
$745$	−40.0558	−1.46753
$746$	11.5895	0.424323
$747$	0	0
$748$	−5.38318	−0.196829
$749$	22.3862	0.817973
$750$	0	0
$751$	−46.6556	−1.70249	−0.851244	−	0.524770i	$-0.824151\pi$
$751$	−46.6556	−1.70249	−0.851244	+	0.524770i	$0.824151\pi$
$752$	−6.88043	−0.250904
$753$	0	0
$754$	4.07365	0.148354
$755$	18.2267	0.663338
$756$	0	0
$757$	−11.7045	−0.425407	−0.212703	−	0.977117i	$-0.568227\pi$
$757$	−11.7045	−0.425407	−0.212703	+	0.977117i	$0.568227\pi$
$758$	−14.7753	−0.536662
$759$	0	0
$760$	29.4053	1.06664
$761$	26.1789	0.948983	0.474492	−	0.880260i	$-0.342632\pi$
$761$	26.1789	0.948983	0.474492	+	0.880260i	$0.342632\pi$
$762$	0	0
$763$	35.5931	1.28856
$764$	1.28362	0.0464397
$765$	0	0
$766$	13.3562	0.482578
$767$	20.0621	0.724401
$768$	0	0
$769$	45.5433	1.64233	0.821166	−	0.570689i	$-0.193324\pi$
$769$	45.5433	1.64233	0.821166	+	0.570689i	$0.193324\pi$
$770$	17.7047	0.638034
$771$	0	0
$772$	−15.6955	−0.564893
$773$	13.9880	0.503113	0.251556	−	0.967843i	$-0.419058\pi$
$773$	13.9880	0.503113	0.251556	+	0.967843i	$0.419058\pi$
$774$	0	0
$775$	2.31935	0.0833134
$776$	26.1056	0.937135
$777$	0	0
$778$	18.7710	0.672973
$779$	32.2719	1.15626
$780$	0	0
$781$	45.8121	1.63928
$782$	−3.26445	−0.116737
$783$	0	0
$784$	6.06880	0.216743
$785$	−0.482291	−0.0172137
$786$	0	0
$787$	14.2028	0.506274	0.253137	−	0.967431i	$-0.418538\pi$
$787$	14.2028	0.506274	0.253137	+	0.967431i	$0.418538\pi$
$788$	−1.87886	−0.0669316
$789$	0	0
$790$	−1.22845	−0.0437062
$791$	−18.2283	−0.648125
$792$	0	0
$793$	2.27559	0.0808086
$794$	0.0313826	0.00111373
$795$	0	0
$796$	−12.8867	−0.456756
$797$	−41.3022	−1.46300	−0.731499	−	0.681842i	$-0.761179\pi$
$797$	−41.3022	−1.46300	−0.731499	+	0.681842i	$0.761179\pi$
$798$	0	0
$799$	2.96305	0.104825
$800$	−18.5557	−0.656042
$801$	0	0
$802$	16.2584	0.574103
$803$	23.1531	0.817056
$804$	0	0
$805$	−55.9125	−1.97066
$806$	1.52984	0.0538864
$807$	0	0
$808$	−12.4915	−0.439451
$809$	23.8379	0.838097	0.419048	−	0.907964i	$-0.362364\pi$
$809$	23.8379	0.838097	0.419048	+	0.907964i	$0.362364\pi$
$810$	0	0
$811$	−3.33188	−0.116998	−0.0584991	−	0.998287i	$-0.518631\pi$
$811$	−3.33188	−0.116998	−0.0584991	+	0.998287i	$0.518631\pi$
$812$	−9.45074	−0.331656
$813$	0	0
$814$	−22.2031	−0.778219
$815$	47.0444	1.64789
$816$	0	0
$817$	−10.8488	−0.379553
$818$	−17.8438	−0.623895
$819$	0	0
$820$	−32.4112	−1.13185
$821$	8.97665	0.313287	0.156644	−	0.987655i	$-0.449933\pi$
$821$	8.97665	0.313287	0.156644	+	0.987655i	$0.449933\pi$
$822$	0	0
$823$	−1.29814	−0.0452502	−0.0226251	−	0.999744i	$-0.507202\pi$
$823$	−1.29814	−0.0452502	−0.0226251	+	0.999744i	$0.507202\pi$
$824$	3.69206	0.128619
$825$	0	0
$826$	8.93739	0.310971
$827$	−31.5324	−1.09649	−0.548245	−	0.836318i	$-0.684704\pi$
$827$	−31.5324	−1.09649	−0.548245	+	0.836318i	$0.684704\pi$
$828$	0	0
$829$	26.7574	0.929322	0.464661	−	0.885489i	$-0.346176\pi$
$829$	26.7574	0.929322	0.464661	+	0.885489i	$0.346176\pi$
$830$	5.99846	0.208209
$831$	0	0
$832$	5.07349	0.175892
$833$	−2.61352	−0.0905530
$834$	0	0
$835$	−53.0687	−1.83652
$836$	27.9346	0.966138
$837$	0	0
$838$	1.91526	0.0661615
$839$	3.87938	0.133931	0.0669656	−	0.997755i	$-0.478668\pi$
$839$	3.87938	0.133931	0.0669656	+	0.997755i	$0.478668\pi$
$840$	0	0
$841$	−25.7611	−0.888312
$842$	−14.3398	−0.494183
$843$	0	0
$844$	17.1763	0.591234
$845$	8.42827	0.289941
$846$	0	0
$847$	2.44008	0.0838420
$848$	18.1891	0.624616
$849$	0	0
$850$	1.82086	0.0624551
$851$	70.1186	2.40364
$852$	0	0
$853$	−3.43923	−0.117757	−0.0588784	−	0.998265i	$-0.518752\pi$
$853$	−3.43923	−0.117757	−0.0588784	+	0.998265i	$0.518752\pi$
$854$	1.01374	0.0346896
$855$	0	0
$856$	14.9314	0.510345
$857$	14.6592	0.500748	0.250374	−	0.968149i	$-0.419446\pi$
$857$	14.6592	0.500748	0.250374	+	0.968149i	$0.419446\pi$
$858$	0	0
$859$	9.53506	0.325332	0.162666	−	0.986681i	$-0.447991\pi$
$859$	9.53506	0.325332	0.162666	+	0.986681i	$0.447991\pi$
$860$	10.8957	0.371539
$861$	0	0
$862$	18.6253	0.634379
$863$	16.0175	0.545243	0.272621	−	0.962121i	$-0.412109\pi$
$863$	16.0175	0.545243	0.272621	+	0.962121i	$0.412109\pi$
$864$	0	0
$865$	54.7464	1.86143
$866$	7.33489	0.249250
$867$	0	0
$868$	−3.54919	−0.120467
$869$	−2.55810	−0.0867777
$870$	0	0
$871$	54.5090	1.84697
$872$	23.7403	0.803948
$873$	0	0
$874$	16.9400	0.573004
$875$	−14.2535	−0.481857
$876$	0	0
$877$	−33.1203	−1.11839	−0.559196	−	0.829035i	$-0.688890\pi$
$877$	−33.1203	−1.11839	−0.559196	+	0.829035i	$0.688890\pi$
$878$	−20.9319	−0.706417
$879$	0	0
$880$	−21.6335	−0.729264
$881$	11.5610	0.389500	0.194750	−	0.980853i	$-0.437610\pi$
$881$	11.5610	0.389500	0.194750	+	0.980853i	$0.437610\pi$
$882$	0	0
$883$	−41.7333	−1.40444	−0.702219	−	0.711961i	$-0.747807\pi$
$883$	−41.7333	−1.40444	−0.702219	+	0.711961i	$0.747807\pi$
$884$	−6.25470	−0.210368
$885$	0	0
$886$	19.5050	0.655283
$887$	24.1020	0.809265	0.404632	−	0.914479i	$-0.367399\pi$
$887$	24.1020	0.809265	0.404632	+	0.914479i	$0.367399\pi$
$888$	0	0
$889$	33.6171	1.12748
$890$	21.6198	0.724697
$891$	0	0
$892$	7.80468	0.261320
$893$	−15.3760	−0.514537
$894$	0	0
$895$	−29.8972	−0.999353
$896$	36.1075	1.20627
$897$	0	0
$898$	9.96738	0.332616
$899$	1.21637	0.0405682
$900$	0	0
$901$	−7.83310	−0.260958
$902$	12.9601	0.431524
$903$	0	0
$904$	−12.1582	−0.404374
$905$	−2.37998	−0.0791131
$906$	0	0
$907$	23.0915	0.766740	0.383370	−	0.923595i	$-0.374763\pi$
$907$	23.0915	0.766740	0.383370	+	0.923595i	$0.374763\pi$
$908$	−23.6901	−0.786184
$909$	0	0
$910$	20.5711	0.681924
$911$	−22.8992	−0.758685	−0.379342	−	0.925256i	$-0.623850\pi$
$911$	−22.8992	−0.758685	−0.379342	+	0.925256i	$0.623850\pi$
$912$	0	0
$913$	12.4911	0.413395
$914$	4.73890	0.156749
$915$	0	0
$916$	8.70005	0.287458
$917$	−22.4177	−0.740297
$918$	0	0
$919$	19.6827	0.649273	0.324636	−	0.945839i	$-0.394758\pi$
$919$	19.6827	0.649273	0.324636	+	0.945839i	$0.394758\pi$
$920$	−37.2932	−1.22952
$921$	0	0
$922$	−11.5451	−0.380216
$923$	53.2289	1.75205
$924$	0	0
$925$	−39.1111	−1.28597
$926$	1.46974	0.0482985
$927$	0	0
$928$	−9.73144	−0.319450
$929$	−28.9175	−0.948753	−0.474376	−	0.880322i	$-0.657326\pi$
$929$	−28.9175	−0.948753	−0.474376	+	0.880322i	$0.657326\pi$
$930$	0	0
$931$	13.5622	0.444482
$932$	−7.47212	−0.244757
$933$	0	0
$934$	6.94756	0.227331
$935$	9.31642	0.304680
$936$	0	0
$937$	19.5798	0.639643	0.319822	−	0.947478i	$-0.396377\pi$
$937$	19.5798	0.639643	0.319822	+	0.947478i	$0.396377\pi$
$938$	24.2830	0.792867
$939$	0	0
$940$	15.4423	0.503673
$941$	−10.0961	−0.329124	−0.164562	−	0.986367i	$-0.552621\pi$
$941$	−10.0961	−0.329124	−0.164562	+	0.986367i	$0.552621\pi$
$942$	0	0
$943$	−40.9287	−1.33282
$944$	−10.9206	−0.355436
$945$	0	0
$946$	−4.35679	−0.141651
$947$	21.0071	0.682637	0.341319	−	0.939948i	$-0.389126\pi$
$947$	21.0071	0.682637	0.341319	+	0.939948i	$0.389126\pi$
$948$	0	0
$949$	26.9016	0.873262
$950$	−9.44888	−0.306562
$951$	0	0
$952$	−6.10781	−0.197955
$953$	−6.75297	−0.218750	−0.109375	−	0.994001i	$-0.534885\pi$
$953$	−6.75297	−0.218750	−0.109375	+	0.994001i	$0.534885\pi$
$954$	0	0
$955$	−2.22150	−0.0718860
$956$	48.7598	1.57701
$957$	0	0
$958$	15.5678	0.502973
$959$	−27.0075	−0.872119
$960$	0	0
$961$	−30.5432	−0.985264
$962$	−25.7977	−0.831753
$963$	0	0
$964$	−41.9188	−1.35011
$965$	27.1635	0.874423
$966$	0	0
$967$	11.7933	0.379248	0.189624	−	0.981857i	$-0.439273\pi$
$967$	11.7933	0.379248	0.189624	+	0.981857i	$0.439273\pi$
$968$	1.62751	0.0523102
$969$	0	0
$970$	−20.6110	−0.661778
$971$	24.4687	0.785239	0.392620	−	0.919701i	$-0.371569\pi$
$971$	24.4687	0.785239	0.392620	+	0.919701i	$0.371569\pi$
$972$	0	0
$973$	−18.4632	−0.591905
$974$	3.41937	0.109564
$975$	0	0
$976$	−1.23870	−0.0396498
$977$	38.9169	1.24506	0.622531	−	0.782595i	$-0.286104\pi$
$977$	38.9169	1.24506	0.622531	+	0.782595i	$0.286104\pi$
$978$	0	0
$979$	45.0207	1.43887
$980$	−13.6207	−0.435097
$981$	0	0
$982$	0.276590	0.00882635
$983$	18.7169	0.596976	0.298488	−	0.954413i	$-0.403518\pi$
$983$	18.7169	0.596976	0.298488	+	0.954413i	$0.403518\pi$
$984$	0	0
$985$	3.25165	0.103606
$986$	0.954943	0.0304116
$987$	0	0
$988$	32.4571	1.03260
$989$	13.7590	0.437510
$990$	0	0
$991$	2.29173	0.0727993	0.0363997	−	0.999337i	$-0.488411\pi$
$991$	2.29173	0.0727993	0.0363997	+	0.999337i	$0.488411\pi$
$992$	−3.65461	−0.116034
$993$	0	0
$994$	23.7127	0.752122
$995$	22.3024	0.707033
$996$	0	0
$997$	31.2797	0.990639	0.495319	−	0.868711i	$-0.335051\pi$
$997$	31.2797	0.990639	0.495319	+	0.868711i	$0.335051\pi$
$998$	−3.11318	−0.0985458
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.32		72
3.2	odd	2		6561.2.a.d.1.41		72
81.5	odd	54		243.2.g.a.73.4		144
81.11	odd	54		729.2.g.a.514.5		144
81.16	even	27		81.2.g.a.13.5	✓	144
81.22	even	27		729.2.g.d.217.4		144
81.32	odd	54		729.2.g.b.703.5		144
81.38	odd	54		729.2.g.b.28.5		144
81.43	even	27		729.2.g.c.28.4		144
81.49	even	27		729.2.g.c.703.4		144
81.59	odd	54		729.2.g.a.217.5		144
81.65	odd	54		243.2.g.a.10.4		144
81.70	even	27		729.2.g.d.514.4		144
81.76	even	27		81.2.g.a.25.5	yes	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.5	✓	144	81.16	even	27
81.2.g.a.25.5	yes	144	81.76	even	27
243.2.g.a.10.4		144	81.65	odd	54
243.2.g.a.73.4		144	81.5	odd	54
729.2.g.a.217.5		144	81.59	odd	54
729.2.g.a.514.5		144	81.11	odd	54
729.2.g.b.28.5		144	81.38	odd	54
729.2.g.b.703.5		144	81.32	odd	54
729.2.g.c.28.4		144	81.43	even	27
729.2.g.c.703.4		144	81.49	even	27
729.2.g.d.217.4		144	81.22	even	27
729.2.g.d.514.4		144	81.70	even	27
6561.2.a.c.1.32		72	1.1	even	1	trivial
6561.2.a.d.1.41		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)

\(f(q)\)	\(=\)	\(q-0.567608 q^{2} -1.67782 q^{4} +2.90373 q^{5} +3.12982 q^{7} +2.08756 q^{8} -1.64818 q^{10} -3.43215 q^{11} -3.98780 q^{13} -1.77651 q^{14} +2.17072 q^{16} -0.934819 q^{17} +4.85100 q^{19} -4.87193 q^{20} +1.94811 q^{22} -6.15225 q^{23} +3.43163 q^{25} +2.26351 q^{26} -5.25127 q^{28} +1.79970 q^{29} +0.675872 q^{31} -5.40724 q^{32} +0.530611 q^{34} +9.08813 q^{35} -11.3972 q^{37} -2.75347 q^{38} +6.06171 q^{40} +6.65264 q^{41} -2.23641 q^{43} +5.75853 q^{44} +3.49207 q^{46} -3.16965 q^{47} +2.79575 q^{49} -1.94782 q^{50} +6.69082 q^{52} +8.37927 q^{53} -9.96601 q^{55} +6.53368 q^{56} -1.02153 q^{58} -5.03087 q^{59} -0.570638 q^{61} -0.383631 q^{62} -1.27225 q^{64} -11.5795 q^{65} -13.6689 q^{67} +1.56846 q^{68} -5.15850 q^{70} -13.3479 q^{71} -6.74596 q^{73} +6.46916 q^{74} -8.13910 q^{76} -10.7420 q^{77} +0.745336 q^{79} +6.30319 q^{80} -3.77609 q^{82} -3.63944 q^{83} -2.71446 q^{85} +1.26941 q^{86} -7.16481 q^{88} -13.1174 q^{89} -12.4811 q^{91} +10.3224 q^{92} +1.79912 q^{94} +14.0860 q^{95} +12.5053 q^{97} -1.58689 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 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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