LMFDB - Embedded newform 729.2.a.c.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{36})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 9x^{2} - 3 $ x^6 - 6x^4 + 9x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.28558$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.28558 q^{2} -0.347296 q^{4} +0.446476 q^{5} -3.53209 q^{7} -3.01763 q^{8} +0.573978 q^{10} -2.78006 q^{11} +3.29086 q^{13} -4.54077 q^{14} -3.18479 q^{16} -7.03936 q^{17} -5.18479 q^{19} -0.155059 q^{20} -3.57398 q^{22} +7.27693 q^{23} -4.80066 q^{25} +4.23065 q^{26} +1.22668 q^{28} +3.61916 q^{29} -1.93582 q^{31} +1.94096 q^{32} -9.04963 q^{34} -1.57699 q^{35} -3.22668 q^{37} -6.66544 q^{38} -1.34730 q^{40} -4.86084 q^{41} -5.75877 q^{43} +0.965505 q^{44} +9.35504 q^{46} +3.01763 q^{47} +5.47565 q^{49} -6.17161 q^{50} -1.14290 q^{52} +8.77141 q^{53} -1.24123 q^{55} +10.6585 q^{56} +4.65270 q^{58} +2.96377 q^{59} +7.88713 q^{61} -2.48865 q^{62} +8.86484 q^{64} +1.46929 q^{65} -9.43882 q^{67} +2.44474 q^{68} -2.02734 q^{70} +5.30731 q^{71} -1.55438 q^{73} -4.14814 q^{74} +1.80066 q^{76} +9.81942 q^{77} -11.9017 q^{79} -1.42193 q^{80} -6.24897 q^{82} -16.2573 q^{83} -3.14290 q^{85} -7.40333 q^{86} +8.38919 q^{88} +18.4258 q^{89} -11.6236 q^{91} -2.52725 q^{92} +3.87939 q^{94} -2.31488 q^{95} +10.1138 q^{97} +7.03936 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) $ 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.28558	0.909039	0.454519	−	0.890737i	$-0.349811\pi$
$2$	1.28558	0.909039	0.454519	+	0.890737i	$0.349811\pi$
$3$	0	0
$3$	0	0
$4$	−0.347296	−0.173648
$5$	0.446476	0.199670	0.0998350	−	0.995004i	$-0.468169\pi$
$5$	0.446476	0.199670	0.0998350	+	0.995004i	$0.468169\pi$
$6$	0	0
$7$	−3.53209	−1.33500	−0.667502	−	0.744608i	$-0.732636\pi$
$7$	−3.53209	−1.33500	−0.667502	+	0.744608i	$0.732636\pi$
$8$	−3.01763	−1.06689
$9$	0	0
$10$	0.573978	0.181508
$11$	−2.78006	−0.838220	−0.419110	−	0.907935i	$-0.637658\pi$
$11$	−2.78006	−0.838220	−0.419110	+	0.907935i	$0.637658\pi$
$12$	0	0
$13$	3.29086	0.912720	0.456360	−	0.889795i	$-0.349153\pi$
$13$	3.29086	0.912720	0.456360	+	0.889795i	$0.349153\pi$
$14$	−4.54077	−1.21357
$15$	0	0
$16$	−3.18479	−0.796198
$17$	−7.03936	−1.70730	−0.853648	−	0.520850i	$-0.825615\pi$
$17$	−7.03936	−1.70730	−0.853648	+	0.520850i	$0.825615\pi$
$18$	0	0
$19$	−5.18479	−1.18947	−0.594736	−	0.803921i	$-0.702744\pi$
$19$	−5.18479	−1.18947	−0.594736	+	0.803921i	$0.702744\pi$
$20$	−0.155059	−0.0346723
$21$	0	0
$22$	−3.57398	−0.761975
$23$	7.27693	1.51734	0.758672	−	0.651473i	$-0.225848\pi$
$23$	7.27693	1.51734	0.758672	+	0.651473i	$0.225848\pi$
$24$	0	0
$25$	−4.80066	−0.960132
$26$	4.23065	0.829698
$27$	0	0
$28$	1.22668	0.231821
$29$	3.61916	0.672061	0.336031	−	0.941851i	$-0.390915\pi$
$29$	3.61916	0.672061	0.336031	+	0.941851i	$0.390915\pi$
$30$	0	0
$31$	−1.93582	−0.347684	−0.173842	−	0.984774i	$-0.555618\pi$
$31$	−1.93582	−0.347684	−0.173842	+	0.984774i	$0.555618\pi$
$32$	1.94096	0.343117
$33$	0	0
$34$	−9.04963	−1.55200
$35$	−1.57699	−0.266560
$36$	0	0
$37$	−3.22668	−0.530463	−0.265232	−	0.964185i	$-0.585448\pi$
$37$	−3.22668	−0.530463	−0.265232	+	0.964185i	$0.585448\pi$
$38$	−6.66544	−1.08128
$39$	0	0
$40$	−1.34730	−0.213026
$41$	−4.86084	−0.759135	−0.379568	−	0.925164i	$-0.623927\pi$
$41$	−4.86084	−0.759135	−0.379568	+	0.925164i	$0.623927\pi$
$42$	0	0
$43$	−5.75877	−0.878204	−0.439102	−	0.898437i	$-0.644703\pi$
$43$	−5.75877	−0.878204	−0.439102	+	0.898437i	$0.644703\pi$
$44$	0.965505	0.145555
$45$	0	0
$46$	9.35504	1.37932
$47$	3.01763	0.440166	0.220083	−	0.975481i	$-0.429367\pi$
$47$	3.01763	0.440166	0.220083	+	0.975481i	$0.429367\pi$
$48$	0	0
$49$	5.47565	0.782236
$50$	−6.17161	−0.872797
$51$	0	0
$52$	−1.14290	−0.158492
$53$	8.77141	1.20485	0.602423	−	0.798177i	$-0.294202\pi$
$53$	8.77141	1.20485	0.602423	+	0.798177i	$0.294202\pi$
$54$	0	0
$55$	−1.24123	−0.167367
$56$	10.6585	1.42431
$57$	0	0
$58$	4.65270	0.610930
$59$	2.96377	0.385851	0.192925	−	0.981213i	$-0.438203\pi$
$59$	2.96377	0.385851	0.192925	+	0.981213i	$0.438203\pi$
$60$	0	0
$61$	7.88713	1.00984	0.504922	−	0.863165i	$-0.331521\pi$
$61$	7.88713	1.00984	0.504922	+	0.863165i	$0.331521\pi$
$62$	−2.48865	−0.316058
$63$	0	0
$64$	8.86484	1.10810
$65$	1.46929	0.182243
$66$	0	0
$67$	−9.43882	−1.15313	−0.576567	−	0.817050i	$-0.695608\pi$
$67$	−9.43882	−1.15313	−0.576567	+	0.817050i	$0.695608\pi$
$68$	2.44474	0.296469
$69$	0	0
$70$	−2.02734	−0.242314
$71$	5.30731	0.629862	0.314931	−	0.949115i	$-0.398019\pi$
$71$	5.30731	0.629862	0.314931	+	0.949115i	$0.398019\pi$
$72$	0	0
$73$	−1.55438	−0.181926	−0.0909631	−	0.995854i	$-0.528995\pi$
$73$	−1.55438	−0.181926	−0.0909631	+	0.995854i	$0.528995\pi$
$74$	−4.14814	−0.482212
$75$	0	0
$76$	1.80066	0.206550
$77$	9.81942	1.11903
$78$	0	0
$79$	−11.9017	−1.33904	−0.669521	−	0.742793i	$-0.733501\pi$
$79$	−11.9017	−1.33904	−0.669521	+	0.742793i	$0.733501\pi$
$80$	−1.42193	−0.158977
$81$	0	0
$82$	−6.24897	−0.690083
$83$	−16.2573	−1.78447	−0.892233	−	0.451576i	$-0.850862\pi$
$83$	−16.2573	−1.78447	−0.892233	+	0.451576i	$0.850862\pi$
$84$	0	0
$85$	−3.14290	−0.340896
$86$	−7.40333	−0.798322
$87$	0	0
$88$	8.38919	0.894290
$89$	18.4258	1.95313	0.976567	−	0.215214i	$-0.0690450\pi$
$89$	18.4258	1.95313	0.976567	+	0.215214i	$0.0690450\pi$
$90$	0	0
$91$	−11.6236	−1.21849
$92$	−2.52725	−0.263484
$93$	0	0
$94$	3.87939	0.400128
$95$	−2.31488	−0.237502
$96$	0	0
$97$	10.1138	1.02690	0.513451	−	0.858119i	$-0.328367\pi$
$97$	10.1138	1.02690	0.513451	+	0.858119i	$0.328367\pi$
$98$	7.03936	0.711083
$99$	0	0
$100$	1.66725	0.166725
$101$	−2.08077	−0.207045	−0.103522	−	0.994627i	$-0.533011\pi$
$101$	−2.08077	−0.207045	−0.103522	+	0.994627i	$0.533011\pi$
$102$	0	0
$103$	−1.44831	−0.142706	−0.0713531	−	0.997451i	$-0.522732\pi$
$103$	−1.44831	−0.142706	−0.0713531	+	0.997451i	$0.522732\pi$
$104$	−9.93058	−0.973774
$105$	0	0
$106$	11.2763	1.09525
$107$	−2.23583	−0.216146	−0.108073	−	0.994143i	$-0.534468\pi$
$107$	−2.23583	−0.216146	−0.108073	+	0.994143i	$0.534468\pi$
$108$	0	0
$109$	−11.5030	−1.10179	−0.550893	−	0.834576i	$-0.685713\pi$
$109$	−11.5030	−1.10179	−0.550893	+	0.834576i	$0.685713\pi$
$110$	−1.59569	−0.152143
$111$	0	0
$112$	11.2490	1.06293
$113$	1.68815	0.158808	0.0794039	−	0.996843i	$-0.474698\pi$
$113$	1.68815	0.158808	0.0794039	+	0.996843i	$0.474698\pi$
$114$	0	0
$115$	3.24897	0.302968
$116$	−1.25692	−0.116702
$117$	0	0
$118$	3.81016	0.350753
$119$	24.8637	2.27925
$120$	0	0
$121$	−3.27126	−0.297387
$122$	10.1395	0.917987
$123$	0	0
$124$	0.672304	0.0603747
$125$	−4.37576	−0.391379
$126$	0	0
$127$	−2.67230	−0.237129	−0.118564	−	0.992946i	$-0.537829\pi$
$127$	−2.67230	−0.237129	−0.118564	+	0.992946i	$0.537829\pi$
$128$	7.51449	0.664194
$129$	0	0
$130$	1.88888	0.165666
$131$	−2.91987	−0.255111	−0.127555	−	0.991831i	$-0.540713\pi$
$131$	−2.91987	−0.255111	−0.127555	+	0.991831i	$0.540713\pi$
$132$	0	0
$133$	18.3131	1.58795
$134$	−12.1343	−1.04824
$135$	0	0
$136$	21.2422	1.82150
$137$	−4.65722	−0.397893	−0.198947	−	0.980010i	$-0.563752\pi$
$137$	−4.65722	−0.397893	−0.198947	+	0.980010i	$0.563752\pi$
$138$	0	0
$139$	−8.00269	−0.678779	−0.339390	−	0.940646i	$-0.610220\pi$
$139$	−8.00269	−0.678779	−0.339390	+	0.940646i	$0.610220\pi$
$140$	0.547683	0.0462877
$141$	0	0
$142$	6.82295	0.572569
$143$	−9.14879	−0.765060
$144$	0	0
$145$	1.61587	0.134190
$146$	−1.99827	−0.165378
$147$	0	0
$148$	1.12061	0.0921140
$149$	20.2117	1.65581	0.827905	−	0.560869i	$-0.189533\pi$
$149$	20.2117	1.65581	0.827905	+	0.560869i	$0.189533\pi$
$150$	0	0
$151$	6.70233	0.545428	0.272714	−	0.962095i	$-0.412079\pi$
$151$	6.70233	0.545428	0.272714	+	0.962095i	$0.412079\pi$
$152$	15.6458	1.26904
$153$	0	0
$154$	12.6236	1.01724
$155$	−0.864297	−0.0694220
$156$	0	0
$157$	5.32501	0.424982	0.212491	−	0.977163i	$-0.431842\pi$
$157$	5.32501	0.424982	0.212491	+	0.977163i	$0.431842\pi$
$158$	−15.3005	−1.21724
$159$	0	0
$160$	0.866592	0.0685101
$161$	−25.7028	−2.02566
$162$	0	0
$163$	3.81521	0.298830	0.149415	−	0.988775i	$-0.452261\pi$
$163$	3.81521	0.298830	0.149415	+	0.988775i	$0.452261\pi$
$164$	1.68815	0.131822
$165$	0	0
$166$	−20.8999	−1.62215
$167$	−9.13538	−0.706917	−0.353459	−	0.935450i	$-0.614994\pi$
$167$	−9.13538	−0.706917	−0.353459	+	0.935450i	$0.614994\pi$
$168$	0	0
$169$	−2.17024	−0.166942
$170$	−4.04044	−0.309888
$171$	0	0
$172$	2.00000	0.152499
$173$	−6.07386	−0.461787	−0.230893	−	0.972979i	$-0.574165\pi$
$173$	−6.07386	−0.461787	−0.230893	+	0.972979i	$0.574165\pi$
$174$	0	0
$175$	16.9564	1.28178
$176$	8.85392	0.667389
$177$	0	0
$178$	23.6878	1.77547
$179$	−10.2811	−0.768449	−0.384224	−	0.923240i	$-0.625531\pi$
$179$	−10.2811	−0.768449	−0.384224	+	0.923240i	$0.625531\pi$
$180$	0	0
$181$	−23.1411	−1.72007	−0.860034	−	0.510237i	$-0.829558\pi$
$181$	−23.1411	−1.72007	−0.860034	+	0.510237i	$0.829558\pi$
$182$	−14.9430	−1.10765
$183$	0	0
$184$	−21.9590	−1.61884
$185$	−1.44063	−0.105918
$186$	0	0
$187$	19.5699	1.43109
$188$	−1.04801	−0.0764340
$189$	0	0
$190$	−2.97596	−0.215899
$191$	−14.6703	−1.06151	−0.530753	−	0.847526i	$-0.678091\pi$
$191$	−14.6703	−1.06151	−0.530753	+	0.847526i	$0.678091\pi$
$192$	0	0
$193$	23.4415	1.68736	0.843678	−	0.536849i	$-0.180386\pi$
$193$	23.4415	1.68736	0.843678	+	0.536849i	$0.180386\pi$
$194$	13.0021	0.933494
$195$	0	0
$196$	−1.90167	−0.135834
$197$	9.02768	0.643196	0.321598	−	0.946876i	$-0.395780\pi$
$197$	9.02768	0.643196	0.321598	+	0.946876i	$0.395780\pi$
$198$	0	0
$199$	2.60401	0.184593	0.0922966	−	0.995732i	$-0.470579\pi$
$199$	2.60401	0.184593	0.0922966	+	0.995732i	$0.470579\pi$
$200$	14.4866	1.02436
$201$	0	0
$202$	−2.67499	−0.188212
$203$	−12.7832	−0.897205
$204$	0	0
$205$	−2.17024	−0.151576
$206$	−1.86191	−0.129726
$207$	0	0
$208$	−10.4807	−0.726706
$209$	14.4140	0.997040
$210$	0	0
$211$	−16.3550	−1.12593	−0.562964	−	0.826482i	$-0.690339\pi$
$211$	−16.3550	−1.12593	−0.562964	+	0.826482i	$0.690339\pi$
$212$	−3.04628	−0.209219
$213$	0	0
$214$	−2.87433	−0.196485
$215$	−2.57115	−0.175351
$216$	0	0
$217$	6.83750	0.464159
$218$	−14.7880	−1.00157
$219$	0	0
$220$	0.431074	0.0290630
$221$	−23.1656	−1.55828
$222$	0	0
$223$	3.70233	0.247927	0.123963	−	0.992287i	$-0.460439\pi$
$223$	3.70233	0.247927	0.123963	+	0.992287i	$0.460439\pi$
$224$	−6.85565	−0.458062
$225$	0	0
$226$	2.17024	0.144363
$227$	10.5608	0.700943	0.350472	−	0.936573i	$-0.386021\pi$
$227$	10.5608	0.700943	0.350472	+	0.936573i	$0.386021\pi$
$228$	0	0
$229$	13.5253	0.893776	0.446888	−	0.894590i	$-0.352532\pi$
$229$	13.5253	0.893776	0.446888	+	0.894590i	$0.352532\pi$
$230$	4.17680	0.275410
$231$	0	0
$232$	−10.9213	−0.717017
$233$	−12.7007	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$233$	−12.7007	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$234$	0	0
$235$	1.34730	0.0878879
$236$	−1.02931	−0.0670022
$237$	0	0
$238$	31.9641	2.07192
$239$	5.85154	0.378505	0.189252	−	0.981928i	$-0.439394\pi$
$239$	5.85154	0.378505	0.189252	+	0.981928i	$0.439394\pi$
$240$	0	0
$241$	−8.53983	−0.550099	−0.275049	−	0.961430i	$-0.588694\pi$
$241$	−8.53983	−0.550099	−0.275049	+	0.961430i	$0.588694\pi$
$242$	−4.20545	−0.270337
$243$	0	0
$244$	−2.73917	−0.175357
$245$	2.44474	0.156189
$246$	0	0
$247$	−17.0624	−1.08566
$248$	5.84159	0.370941
$249$	0	0
$250$	−5.62536	−0.355779
$251$	6.75790	0.426555	0.213277	−	0.976992i	$-0.431586\pi$
$251$	6.75790	0.426555	0.213277	+	0.976992i	$0.431586\pi$
$252$	0	0
$253$	−20.2303	−1.27187
$254$	−3.43545	−0.215559
$255$	0	0
$256$	−8.06923	−0.504327
$257$	−3.68296	−0.229737	−0.114868	−	0.993381i	$-0.536645\pi$
$257$	−3.68296	−0.229737	−0.114868	+	0.993381i	$0.536645\pi$
$258$	0	0
$259$	11.3969	0.708171
$260$	−0.510278	−0.0316461
$261$	0	0
$262$	−3.75372	−0.231905
$263$	−3.63786	−0.224320	−0.112160	−	0.993690i	$-0.535777\pi$
$263$	−3.63786	−0.224320	−0.112160	+	0.993690i	$0.535777\pi$
$264$	0	0
$265$	3.91622	0.240572
$266$	23.5429	1.44351
$267$	0	0
$268$	3.27807	0.200240
$269$	7.08672	0.432085	0.216042	−	0.976384i	$-0.430685\pi$
$269$	7.08672	0.432085	0.216042	+	0.976384i	$0.430685\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	22.4189	1.35935
$273$	0	0
$274$	−5.98721	−0.361700
$275$	13.3461	0.804802
$276$	0	0
$277$	−13.8922	−0.834700	−0.417350	−	0.908746i	$-0.637041\pi$
$277$	−13.8922	−0.834700	−0.417350	+	0.908746i	$0.637041\pi$
$278$	−10.2881	−0.617037
$279$	0	0
$280$	4.75877	0.284391
$281$	−22.1275	−1.32002	−0.660008	−	0.751259i	$-0.729447\pi$
$281$	−22.1275	−1.32002	−0.660008	+	0.751259i	$0.729447\pi$
$282$	0	0
$283$	−23.7324	−1.41074	−0.705371	−	0.708838i	$-0.749220\pi$
$283$	−23.7324	−1.41074	−0.705371	+	0.708838i	$0.749220\pi$
$284$	−1.84321	−0.109374
$285$	0	0
$286$	−11.7615	−0.695470
$287$	17.1689	1.01345
$288$	0	0
$289$	32.5526	1.91486
$290$	2.07732	0.121984
$291$	0	0
$292$	0.539830	0.0315911
$293$	−18.6061	−1.08698	−0.543489	−	0.839416i	$-0.682897\pi$
$293$	−18.6061	−1.08698	−0.543489	+	0.839416i	$0.682897\pi$
$294$	0	0
$295$	1.32325	0.0770428
$296$	9.73692	0.565947
$297$	0	0
$298$	25.9837	1.50520
$299$	23.9473	1.38491
$300$	0	0
$301$	20.3405	1.17241
$302$	8.61635	0.495815
$303$	0	0
$304$	16.5125	0.947056
$305$	3.52141	0.201635
$306$	0	0
$307$	20.7469	1.18409	0.592044	−	0.805905i	$-0.298321\pi$
$307$	20.7469	1.18409	0.592044	+	0.805905i	$0.298321\pi$
$308$	−3.41025	−0.194317
$309$	0	0
$310$	−1.11112	−0.0631073
$311$	−20.3855	−1.15596	−0.577978	−	0.816053i	$-0.696158\pi$
$311$	−20.3855	−1.15596	−0.577978	+	0.816053i	$0.696158\pi$
$312$	0	0
$313$	−29.8408	−1.68670	−0.843351	−	0.537364i	$-0.819420\pi$
$313$	−29.8408	−1.68670	−0.843351	+	0.537364i	$0.819420\pi$
$314$	6.84570	0.386325
$315$	0	0
$316$	4.13341	0.232522
$317$	−4.31661	−0.242445	−0.121222	−	0.992625i	$-0.538681\pi$
$317$	−4.31661	−0.242445	−0.121222	+	0.992625i	$0.538681\pi$
$318$	0	0
$319$	−10.0615	−0.563335
$320$	3.95793	0.221255
$321$	0	0
$322$	−33.0428	−1.84140
$323$	36.4976	2.03078
$324$	0	0
$325$	−15.7983	−0.876332
$326$	4.90474	0.271648
$327$	0	0
$328$	14.6682	0.809915
$329$	−10.6585	−0.587623
$330$	0	0
$331$	−2.91891	−0.160438	−0.0802189	−	0.996777i	$-0.525562\pi$
$331$	−2.91891	−0.160438	−0.0802189	+	0.996777i	$0.525562\pi$
$332$	5.64608	0.309869
$333$	0	0
$334$	−11.7442	−0.642615
$335$	−4.21420	−0.230246
$336$	0	0
$337$	14.4415	0.786679	0.393339	−	0.919393i	$-0.371320\pi$
$337$	14.4415	0.786679	0.393339	+	0.919393i	$0.371320\pi$
$338$	−2.79001	−0.151757
$339$	0	0
$340$	1.09152	0.0591959
$341$	5.38170	0.291436
$342$	0	0
$343$	5.38413	0.290716
$344$	17.3778	0.936949
$345$	0	0
$346$	−7.80840	−0.419782
$347$	1.45404	0.0780571	0.0390285	−	0.999238i	$-0.487574\pi$
$347$	1.45404	0.0780571	0.0390285	+	0.999238i	$0.487574\pi$
$348$	0	0
$349$	−27.5817	−1.47642	−0.738208	−	0.674573i	$-0.764328\pi$
$349$	−27.5817	−1.47642	−0.738208	+	0.674573i	$0.764328\pi$
$350$	21.7987	1.16519
$351$	0	0
$352$	−5.39599	−0.287607
$353$	−10.5321	−0.560568	−0.280284	−	0.959917i	$-0.590429\pi$
$353$	−10.5321	−0.560568	−0.280284	+	0.959917i	$0.590429\pi$
$354$	0	0
$355$	2.36959	0.125765
$356$	−6.39922	−0.339158
$357$	0	0
$358$	−13.2172	−0.698550
$359$	−14.7055	−0.776124	−0.388062	−	0.921633i	$-0.626855\pi$
$359$	−14.7055	−0.776124	−0.388062	+	0.921633i	$0.626855\pi$
$360$	0	0
$361$	7.88207	0.414846
$362$	−29.7497	−1.56361
$363$	0	0
$364$	4.03684	0.211588
$365$	−0.693992	−0.0363252
$366$	0	0
$367$	−21.3628	−1.11513	−0.557564	−	0.830134i	$-0.688264\pi$
$367$	−21.3628	−1.11513	−0.557564	+	0.830134i	$0.688264\pi$
$368$	−23.1755	−1.20811
$369$	0	0
$370$	−1.85204	−0.0962832
$371$	−30.9814	−1.60847
$372$	0	0
$373$	22.6245	1.17145	0.585727	−	0.810508i	$-0.300809\pi$
$373$	22.6245	1.17145	0.585727	+	0.810508i	$0.300809\pi$
$374$	25.1585	1.30092
$375$	0	0
$376$	−9.10607	−0.469610
$377$	11.9101	0.613404
$378$	0	0
$379$	−17.0743	−0.877047	−0.438523	−	0.898720i	$-0.644498\pi$
$379$	−17.0743	−0.877047	−0.438523	+	0.898720i	$0.644498\pi$
$380$	0.803951	0.0412418
$381$	0	0
$382$	−18.8598	−0.964951
$383$	38.9916	1.99238	0.996188	−	0.0872303i	$-0.0278016\pi$
$383$	38.9916	1.99238	0.996188	+	0.0872303i	$0.0278016\pi$
$384$	0	0
$385$	4.38413	0.223436
$386$	30.1358	1.53387
$387$	0	0
$388$	−3.51249	−0.178320
$389$	−10.2120	−0.517771	−0.258886	−	0.965908i	$-0.583355\pi$
$389$	−10.2120	−0.517771	−0.258886	+	0.965908i	$0.583355\pi$
$390$	0	0
$391$	−51.2249	−2.59056
$392$	−16.5235	−0.834561
$393$	0	0
$394$	11.6058	0.584690
$395$	−5.31381	−0.267367
$396$	0	0
$397$	−1.14290	−0.0573607	−0.0286803	−	0.999589i	$-0.509130\pi$
$397$	−1.14290	−0.0573607	−0.0286803	+	0.999589i	$0.509130\pi$
$398$	3.34765	0.167802
$399$	0	0
$400$	15.2891	0.764455
$401$	20.5540	1.02642	0.513208	−	0.858264i	$-0.328457\pi$
$401$	20.5540	1.02642	0.513208	+	0.858264i	$0.328457\pi$
$402$	0	0
$403$	−6.37052	−0.317338
$404$	0.722645	0.0359530
$405$	0	0
$406$	−16.4338	−0.815594
$407$	8.97037	0.444645
$408$	0	0
$409$	−8.12836	−0.401921	−0.200961	−	0.979599i	$-0.564406\pi$
$409$	−8.12836	−0.401921	−0.200961	+	0.979599i	$0.564406\pi$
$410$	−2.79001	−0.137789
$411$	0	0
$412$	0.502993	0.0247807
$413$	−10.4683	−0.515112
$414$	0	0
$415$	−7.25847	−0.356304
$416$	6.38743	0.313170
$417$	0	0
$418$	18.5303	0.906348
$419$	−35.7784	−1.74789	−0.873946	−	0.486024i	$-0.838447\pi$
$419$	−35.7784	−1.74789	−0.873946	+	0.486024i	$0.838447\pi$
$420$	0	0
$421$	−13.1189	−0.639374	−0.319687	−	0.947523i	$-0.603578\pi$
$421$	−13.1189	−0.639374	−0.319687	+	0.947523i	$0.603578\pi$
$422$	−21.0256	−1.02351
$423$	0	0
$424$	−26.4688	−1.28544
$425$	33.7936	1.63923
$426$	0	0
$427$	−27.8580	−1.34814
$428$	0.776497	0.0375334
$429$	0	0
$430$	−3.30541	−0.159401
$431$	−9.48411	−0.456833	−0.228417	−	0.973563i	$-0.573355\pi$
$431$	−9.48411	−0.456833	−0.228417	+	0.973563i	$0.573355\pi$
$432$	0	0
$433$	17.6628	0.848820	0.424410	−	0.905470i	$-0.360481\pi$
$433$	17.6628	0.848820	0.424410	+	0.905470i	$0.360481\pi$
$434$	8.79012	0.421939
$435$	0	0
$436$	3.99495	0.191323
$437$	−37.7294	−1.80484
$438$	0	0
$439$	−17.6655	−0.843128	−0.421564	−	0.906799i	$-0.638519\pi$
$439$	−17.6655	−0.843128	−0.421564	+	0.906799i	$0.638519\pi$
$440$	3.74557	0.178563
$441$	0	0
$442$	−29.7811	−1.41654
$443$	12.9921	0.617274	0.308637	−	0.951180i	$-0.400127\pi$
$443$	12.9921	0.617274	0.308637	+	0.951180i	$0.400127\pi$
$444$	0	0
$445$	8.22668	0.389982
$446$	4.75963	0.225375
$447$	0	0
$448$	−31.3114	−1.47932
$449$	4.62857	0.218436	0.109218	−	0.994018i	$-0.465165\pi$
$449$	4.62857	0.218436	0.109218	+	0.994018i	$0.465165\pi$
$450$	0	0
$451$	13.5134	0.636322
$452$	−0.586289	−0.0275767
$453$	0	0
$454$	13.5767	0.637185
$455$	−5.18966	−0.243295
$456$	0	0
$457$	20.5672	0.962092	0.481046	−	0.876695i	$-0.340257\pi$
$457$	20.5672	0.962092	0.481046	+	0.876695i	$0.340257\pi$
$458$	17.3878	0.812477
$459$	0	0
$460$	−1.12836	−0.0526098
$461$	33.5725	1.56363	0.781813	−	0.623513i	$-0.214295\pi$
$461$	33.5725	1.56363	0.781813	+	0.623513i	$0.214295\pi$
$462$	0	0
$463$	13.1310	0.610251	0.305126	−	0.952312i	$-0.401302\pi$
$463$	13.1310	0.610251	0.305126	+	0.952312i	$0.401302\pi$
$464$	−11.5263	−0.535094
$465$	0	0
$466$	−16.3277	−0.756366
$467$	−23.6307	−1.09350	−0.546750	−	0.837296i	$-0.684135\pi$
$467$	−23.6307	−1.09350	−0.546750	+	0.837296i	$0.684135\pi$
$468$	0	0
$469$	33.3387	1.53944
$470$	1.73205	0.0798935
$471$	0	0
$472$	−8.94356	−0.411661
$473$	16.0097	0.736128
$474$	0	0
$475$	24.8904	1.14205
$476$	−8.63506	−0.395787
$477$	0	0
$478$	7.52259	0.344075
$479$	5.88019	0.268673	0.134336	−	0.990936i	$-0.457110\pi$
$479$	5.88019	0.268673	0.134336	+	0.990936i	$0.457110\pi$
$480$	0	0
$481$	−10.6186	−0.484164
$482$	−10.9786	−0.500061
$483$	0	0
$484$	1.13610	0.0516407
$485$	4.51557	0.205041
$486$	0	0
$487$	38.7965	1.75804	0.879020	−	0.476786i	$-0.158198\pi$
$487$	38.7965	1.75804	0.879020	+	0.476786i	$0.158198\pi$
$488$	−23.8004	−1.07739
$489$	0	0
$490$	3.14290	0.141982
$491$	37.5842	1.69615	0.848077	−	0.529873i	$-0.177761\pi$
$491$	37.5842	1.69615	0.848077	+	0.529873i	$0.177761\pi$
$492$	0	0
$493$	−25.4766	−1.14741
$494$	−21.9350	−0.986904
$495$	0	0
$496$	6.16519	0.276825
$497$	−18.7459	−0.840868
$498$	0	0
$499$	33.7520	1.51095	0.755473	−	0.655180i	$-0.227407\pi$
$499$	33.7520	1.51095	0.755473	+	0.655180i	$0.227407\pi$
$500$	1.51968	0.0679623
$501$	0	0
$502$	8.68779	0.387755
$503$	18.7119	0.834324	0.417162	−	0.908832i	$-0.363025\pi$
$503$	18.7119	0.834324	0.417162	+	0.908832i	$0.363025\pi$
$504$	0	0
$505$	−0.929015	−0.0413406
$506$	−26.0076	−1.15618
$507$	0	0
$508$	0.928081	0.0411770
$509$	−21.7682	−0.964858	−0.482429	−	0.875935i	$-0.660245\pi$
$509$	−21.7682	−0.964858	−0.482429	+	0.875935i	$0.660245\pi$
$510$	0	0
$511$	5.49020	0.242872
$512$	−25.4026	−1.12265
$513$	0	0
$514$	−4.73473	−0.208840
$515$	−0.646635	−0.0284942
$516$	0	0
$517$	−8.38919	−0.368956
$518$	14.6516	0.643755
$519$	0	0
$520$	−4.43376	−0.194433
$521$	6.47643	0.283738	0.141869	−	0.989885i	$-0.454689\pi$
$521$	6.47643	0.283738	0.141869	+	0.989885i	$0.454689\pi$
$522$	0	0
$523$	−10.8726	−0.475425	−0.237712	−	0.971336i	$-0.576398\pi$
$523$	−10.8726	−0.475425	−0.237712	+	0.971336i	$0.576398\pi$
$524$	1.01406	0.0442995
$525$	0	0
$526$	−4.67675	−0.203916
$527$	13.6270	0.593599
$528$	0	0
$529$	29.9537	1.30233
$530$	5.03460	0.218689
$531$	0	0
$532$	−6.36009	−0.275745
$533$	−15.9963	−0.692878
$534$	0	0
$535$	−0.998245	−0.0431579
$536$	28.4828	1.23027
$537$	0	0
$538$	9.11051	0.392782
$539$	−15.2226	−0.655686
$540$	0	0
$541$	24.6459	1.05961	0.529805	−	0.848120i	$-0.322265\pi$
$541$	24.6459	1.05961	0.529805	+	0.848120i	$0.322265\pi$
$542$	−24.4259	−1.04918
$543$	0	0
$544$	−13.6631	−0.585802
$545$	−5.13581	−0.219994
$546$	0	0
$547$	30.9273	1.32235	0.661177	−	0.750230i	$-0.270057\pi$
$547$	30.9273	1.32235	0.661177	+	0.750230i	$0.270057\pi$
$548$	1.61744	0.0690934
$549$	0	0
$550$	17.1575	0.731596
$551$	−18.7646	−0.799399
$552$	0	0
$553$	42.0378	1.78763
$554$	−17.8594	−0.758775
$555$	0	0
$556$	2.77930	0.117869
$557$	−23.3627	−0.989908	−0.494954	−	0.868919i	$-0.664815\pi$
$557$	−23.3627	−0.989908	−0.494954	+	0.868919i	$0.664815\pi$
$558$	0	0
$559$	−18.9513	−0.801555
$560$	5.02239	0.212235
$561$	0	0
$562$	−28.4466	−1.19995
$563$	−33.6877	−1.41977	−0.709884	−	0.704319i	$-0.751253\pi$
$563$	−33.6877	−1.41977	−0.709884	+	0.704319i	$0.751253\pi$
$564$	0	0
$565$	0.753718	0.0317092
$566$	−30.5097	−1.28242
$567$	0	0
$568$	−16.0155	−0.671995
$569$	30.7959	1.29103	0.645515	−	0.763748i	$-0.276643\pi$
$569$	30.7959	1.29103	0.645515	+	0.763748i	$0.276643\pi$
$570$	0	0
$571$	−29.8753	−1.25024	−0.625120	−	0.780528i	$-0.714950\pi$
$571$	−29.8753	−1.25024	−0.625120	+	0.780528i	$0.714950\pi$
$572$	3.17734	0.132851
$573$	0	0
$574$	22.0719	0.921264
$575$	−34.9340	−1.45685
$576$	0	0
$577$	4.80747	0.200137	0.100069	−	0.994981i	$-0.468094\pi$
$577$	4.80747	0.200137	0.100069	+	0.994981i	$0.468094\pi$
$578$	41.8488	1.74068
$579$	0	0
$580$	−0.561185	−0.0233019
$581$	57.4221	2.38227
$582$	0	0
$583$	−24.3851	−1.00993
$584$	4.69053	0.194096
$585$	0	0
$586$	−23.9195	−0.988106
$587$	−7.93761	−0.327620	−0.163810	−	0.986492i	$-0.552378\pi$
$587$	−7.93761	−0.327620	−0.163810	+	0.986492i	$0.552378\pi$
$588$	0	0
$589$	10.0368	0.413561
$590$	1.70114	0.0700349
$591$	0	0
$592$	10.2763	0.422354
$593$	−36.2753	−1.48965	−0.744824	−	0.667261i	$-0.767467\pi$
$593$	−36.2753	−1.48965	−0.744824	+	0.667261i	$0.767467\pi$
$594$	0	0
$595$	11.1010	0.455097
$596$	−7.01946	−0.287528
$597$	0	0
$598$	30.7861	1.25894
$599$	−33.6450	−1.37470	−0.687349	−	0.726327i	$-0.741226\pi$
$599$	−33.6450	−1.37470	−0.687349	+	0.726327i	$0.741226\pi$
$600$	0	0
$601$	−2.97771	−0.121463	−0.0607317	−	0.998154i	$-0.519343\pi$
$601$	−2.97771	−0.121463	−0.0607317	+	0.998154i	$0.519343\pi$
$602$	26.1492	1.06576
$603$	0	0
$604$	−2.32770	−0.0947126
$605$	−1.46054	−0.0593793
$606$	0	0
$607$	−15.6486	−0.635156	−0.317578	−	0.948232i	$-0.602870\pi$
$607$	−15.6486	−0.635156	−0.317578	+	0.948232i	$0.602870\pi$
$608$	−10.0635	−0.408128
$609$	0	0
$610$	4.52704	0.183294
$611$	9.93058	0.401748
$612$	0	0
$613$	−1.06687	−0.0430903	−0.0215452	−	0.999768i	$-0.506859\pi$
$613$	−1.06687	−0.0430903	−0.0215452	+	0.999768i	$0.506859\pi$
$614$	26.6717	1.07638
$615$	0	0
$616$	−29.6313	−1.19388
$617$	13.0700	0.526177	0.263088	−	0.964772i	$-0.415259\pi$
$617$	13.0700	0.526177	0.263088	+	0.964772i	$0.415259\pi$
$618$	0	0
$619$	20.5175	0.824670	0.412335	−	0.911032i	$-0.364713\pi$
$619$	20.5175	0.824670	0.412335	+	0.911032i	$0.364713\pi$
$620$	0.300167	0.0120550
$621$	0	0
$622$	−26.2071	−1.05081
$623$	−65.0817	−2.60744
$624$	0	0
$625$	22.0496	0.881985
$626$	−38.3626	−1.53328
$627$	0	0
$628$	−1.84936	−0.0737973
$629$	22.7138	0.905658
$630$	0	0
$631$	−10.3122	−0.410523	−0.205261	−	0.978707i	$-0.565804\pi$
$631$	−10.3122	−0.410523	−0.205261	+	0.978707i	$0.565804\pi$
$632$	35.9148	1.42861
$633$	0	0
$634$	−5.54933	−0.220392
$635$	−1.19312	−0.0473475
$636$	0	0
$637$	18.0196	0.713963
$638$	−12.9348	−0.512094
$639$	0	0
$640$	3.35504	0.132619
$641$	−3.50616	−0.138485	−0.0692426	−	0.997600i	$-0.522058\pi$
$641$	−3.50616	−0.138485	−0.0692426	+	0.997600i	$0.522058\pi$
$642$	0	0
$643$	31.5517	1.24428	0.622139	−	0.782907i	$-0.286264\pi$
$643$	31.5517	1.24428	0.622139	+	0.782907i	$0.286264\pi$
$644$	8.92647	0.351752
$645$	0	0
$646$	46.9205	1.84606
$647$	3.04628	0.119762	0.0598808	−	0.998206i	$-0.480928\pi$
$647$	3.04628	0.119762	0.0598808	+	0.998206i	$0.480928\pi$
$648$	0	0
$649$	−8.23947	−0.323428
$650$	−20.3099	−0.796620
$651$	0	0
$652$	−1.32501	−0.0518913
$653$	30.7374	1.20285	0.601423	−	0.798931i	$-0.294601\pi$
$653$	30.7374	1.20285	0.601423	+	0.798931i	$0.294601\pi$
$654$	0	0
$655$	−1.30365	−0.0509379
$656$	15.4808	0.604422
$657$	0	0
$658$	−13.7023	−0.534173
$659$	33.1839	1.29266	0.646331	−	0.763057i	$-0.276302\pi$
$659$	33.1839	1.29266	0.646331	+	0.763057i	$0.276302\pi$
$660$	0	0
$661$	−14.9077	−0.579841	−0.289920	−	0.957051i	$-0.593629\pi$
$661$	−14.9077	−0.579841	−0.289920	+	0.957051i	$0.593629\pi$
$662$	−3.75248	−0.145844
$663$	0	0
$664$	49.0583	1.90383
$665$	8.17637	0.317066
$666$	0	0
$667$	26.3364	1.01975
$668$	3.17269	0.122755
$669$	0	0
$670$	−5.41767	−0.209303
$671$	−21.9267	−0.846471
$672$	0	0
$673$	6.88888	0.265547	0.132773	−	0.991146i	$-0.457612\pi$
$673$	6.88888	0.265547	0.132773	+	0.991146i	$0.457612\pi$
$674$	18.5656	0.715122
$675$	0	0
$676$	0.753718	0.0289892
$677$	13.1892	0.506903	0.253452	−	0.967348i	$-0.418434\pi$
$677$	13.1892	0.506903	0.253452	+	0.967348i	$0.418434\pi$
$678$	0	0
$679$	−35.7229	−1.37092
$680$	9.48411	0.363699
$681$	0	0
$682$	6.91859	0.264926
$683$	3.37814	0.129261	0.0646305	−	0.997909i	$-0.479413\pi$
$683$	3.37814	0.129261	0.0646305	+	0.997909i	$0.479413\pi$
$684$	0	0
$685$	−2.07934	−0.0794473
$686$	6.92171	0.264272
$687$	0	0
$688$	18.3405	0.699225
$689$	28.8655	1.09969
$690$	0	0
$691$	23.4151	0.890752	0.445376	−	0.895344i	$-0.353070\pi$
$691$	23.4151	0.890752	0.445376	+	0.895344i	$0.353070\pi$
$692$	2.10943	0.0801884
$693$	0	0
$694$	1.86928	0.0709569
$695$	−3.57300	−0.135532
$696$	0	0
$697$	34.2172	1.29607
$698$	−35.4584	−1.34212
$699$	0	0
$700$	−5.88888	−0.222579
$701$	−45.5001	−1.71852	−0.859258	−	0.511543i	$-0.829074\pi$
$701$	−45.5001	−1.71852	−0.859258	+	0.511543i	$0.829074\pi$
$702$	0	0
$703$	16.7297	0.630972
$704$	−24.6448	−0.928836
$705$	0	0
$706$	−13.5398	−0.509578
$707$	7.34948	0.276406
$708$	0	0
$709$	38.6168	1.45028	0.725142	−	0.688599i	$-0.241774\pi$
$709$	38.6168	1.45028	0.725142	+	0.688599i	$0.241774\pi$
$710$	3.04628	0.114325
$711$	0	0
$712$	−55.6023	−2.08378
$713$	−14.0868	−0.527556
$714$	0	0
$715$	−4.08471	−0.152760
$716$	3.57060	0.133440
$717$	0	0
$718$	−18.9050	−0.705527
$719$	49.3182	1.83926	0.919630	−	0.392786i	$-0.128489\pi$
$719$	49.3182	1.83926	0.919630	+	0.392786i	$0.128489\pi$
$720$	0	0
$721$	5.11556	0.190513
$722$	10.1330	0.377111
$723$	0	0
$724$	8.03684	0.298687
$725$	−17.3744	−0.645268
$726$	0	0
$727$	−32.2945	−1.19774	−0.598868	−	0.800848i	$-0.704383\pi$
$727$	−32.2945	−1.19774	−0.598868	+	0.800848i	$0.704383\pi$
$728$	35.0757	1.29999
$729$	0	0
$730$	−0.892178	−0.0330210
$731$	40.5381	1.49935
$732$	0	0
$733$	39.4662	1.45772	0.728858	−	0.684665i	$-0.240051\pi$
$733$	39.4662	1.45772	0.728858	+	0.684665i	$0.240051\pi$
$734$	−27.4635	−1.01369
$735$	0	0
$736$	14.1242	0.520626
$737$	26.2405	0.966581
$738$	0	0
$739$	35.3090	1.29886	0.649432	−	0.760420i	$-0.275007\pi$
$739$	35.3090	1.29886	0.649432	+	0.760420i	$0.275007\pi$
$740$	0.500327	0.0183924
$741$	0	0
$742$	−39.8289	−1.46217
$743$	47.4106	1.73933	0.869663	−	0.493646i	$-0.164336\pi$
$743$	47.4106	1.73933	0.869663	+	0.493646i	$0.164336\pi$
$744$	0	0
$745$	9.02404	0.330615
$746$	29.0855	1.06490
$747$	0	0
$748$	−6.79654	−0.248506
$749$	7.89716	0.288556
$750$	0	0
$751$	−8.92808	−0.325790	−0.162895	−	0.986643i	$-0.552083\pi$
$751$	−8.92808	−0.325790	−0.162895	+	0.986643i	$0.552083\pi$
$752$	−9.61051	−0.350459
$753$	0	0
$754$	15.3114	0.557608
$755$	2.99243	0.108906
$756$	0	0
$757$	−3.63816	−0.132231	−0.0661155	−	0.997812i	$-0.521061\pi$
$757$	−3.63816	−0.132231	−0.0661155	+	0.997812i	$0.521061\pi$
$758$	−21.9503	−0.797270
$759$	0	0
$760$	6.98545	0.253389
$761$	6.68880	0.242469	0.121234	−	0.992624i	$-0.461315\pi$
$761$	6.68880	0.242469	0.121234	+	0.992624i	$0.461315\pi$
$762$	0	0
$763$	40.6296	1.47089
$764$	5.09494	0.184329
$765$	0	0
$766$	50.1266	1.81115
$767$	9.75337	0.352174
$768$	0	0
$769$	−21.4712	−0.774272	−0.387136	−	0.922023i	$-0.626536\pi$
$769$	−21.4712	−0.774272	−0.387136	+	0.922023i	$0.626536\pi$
$770$	5.63613	0.203112
$771$	0	0
$772$	−8.14115	−0.293006
$773$	10.2442	0.368457	0.184228	−	0.982883i	$-0.441021\pi$
$773$	10.2442	0.368457	0.184228	+	0.982883i	$0.441021\pi$
$774$	0	0
$775$	9.29322	0.333822
$776$	−30.5197	−1.09559
$777$	0	0
$778$	−13.1284	−0.470674
$779$	25.2024	0.902971
$780$	0	0
$781$	−14.7547	−0.527963
$782$	−65.8535	−2.35492
$783$	0	0
$784$	−17.4388	−0.622815
$785$	2.37749	0.0848561
$786$	0	0
$787$	0.947682	0.0337812	0.0168906	−	0.999857i	$-0.494623\pi$
$787$	0.947682	0.0337812	0.0168906	+	0.999857i	$0.494623\pi$
$788$	−3.13528	−0.111690
$789$	0	0
$790$	−6.83130	−0.243047
$791$	−5.96270	−0.212009
$792$	0	0
$793$	25.9554	0.921704
$794$	−1.46929	−0.0521431
$795$	0	0
$796$	−0.904362	−0.0320543
$797$	−7.01071	−0.248332	−0.124166	−	0.992261i	$-0.539626\pi$
$797$	−7.01071	−0.248332	−0.124166	+	0.992261i	$0.539626\pi$
$798$	0	0
$799$	−21.2422	−0.751494
$800$	−9.31790	−0.329437
$801$	0	0
$802$	26.4237	0.933052
$803$	4.32127	0.152494
$804$	0	0
$805$	−11.4757	−0.404464
$806$	−8.18978	−0.288473
$807$	0	0
$808$	6.27900	0.220894
$809$	45.1028	1.58573	0.792866	−	0.609396i	$-0.208588\pi$
$809$	45.1028	1.58573	0.792866	+	0.609396i	$0.208588\pi$
$810$	0	0
$811$	8.07285	0.283476	0.141738	−	0.989904i	$-0.454731\pi$
$811$	8.07285	0.283476	0.141738	+	0.989904i	$0.454731\pi$
$812$	4.43956	0.155798
$813$	0	0
$814$	11.5321	0.404200
$815$	1.70340	0.0596674
$816$	0	0
$817$	29.8580	1.04460
$818$	−10.4496	−0.365362
$819$	0	0
$820$	0.753718	0.0263210
$821$	−6.26936	−0.218802	−0.109401	−	0.993998i	$-0.534893\pi$
$821$	−6.26936	−0.218802	−0.109401	+	0.993998i	$0.534893\pi$
$822$	0	0
$823$	−19.7929	−0.689938	−0.344969	−	0.938614i	$-0.612111\pi$
$823$	−19.7929	−0.689938	−0.344969	+	0.938614i	$0.612111\pi$
$824$	4.37046	0.152252
$825$	0	0
$826$	−13.4578	−0.468257
$827$	41.8003	1.45354	0.726769	−	0.686882i	$-0.241021\pi$
$827$	41.8003	1.45354	0.726769	+	0.686882i	$0.241021\pi$
$828$	0	0
$829$	33.7279	1.17142	0.585710	−	0.810521i	$-0.300816\pi$
$829$	33.7279	1.17142	0.585710	+	0.810521i	$0.300816\pi$
$830$	−9.33130	−0.323894
$831$	0	0
$832$	29.1729	1.01139
$833$	−38.5451	−1.33551
$834$	0	0
$835$	−4.07873	−0.141150
$836$	−5.00594	−0.173134
$837$	0	0
$838$	−45.9959	−1.58890
$839$	−29.2868	−1.01109	−0.505546	−	0.862800i	$-0.668709\pi$
$839$	−29.2868	−1.01109	−0.505546	+	0.862800i	$0.668709\pi$
$840$	0	0
$841$	−15.9017	−0.548334
$842$	−16.8653	−0.581216
$843$	0	0
$844$	5.68004	0.195515
$845$	−0.968961	−0.0333333
$846$	0	0
$847$	11.5544	0.397013
$848$	−27.9351	−0.959296
$849$	0	0
$850$	43.4442	1.49012
$851$	−23.4803	−0.804895
$852$	0	0
$853$	37.5981	1.28734	0.643668	−	0.765305i	$-0.277412\pi$
$853$	37.5981	1.28734	0.643668	+	0.765305i	$0.277412\pi$
$854$	−35.8136	−1.22552
$855$	0	0
$856$	6.74691	0.230605
$857$	−29.0100	−0.990961	−0.495481	−	0.868619i	$-0.665008\pi$
$857$	−29.0100	−0.990961	−0.495481	+	0.868619i	$0.665008\pi$
$858$	0	0
$859$	−24.8675	−0.848469	−0.424235	−	0.905552i	$-0.639457\pi$
$859$	−24.8675	−0.848469	−0.424235	+	0.905552i	$0.639457\pi$
$860$	0.892951	0.0304494
$861$	0	0
$862$	−12.1925	−0.415279
$863$	42.4018	1.44337	0.721687	−	0.692219i	$-0.243367\pi$
$863$	42.4018	1.44337	0.721687	+	0.692219i	$0.243367\pi$
$864$	0	0
$865$	−2.71183	−0.0922050
$866$	22.7069	0.771611
$867$	0	0
$868$	−2.37464	−0.0806004
$869$	33.0874	1.12241
$870$	0	0
$871$	−31.0618	−1.05249
$872$	34.7117	1.17549
$873$	0	0
$874$	−48.5039	−1.64067
$875$	15.4556	0.522493
$876$	0	0
$877$	12.0547	0.407058	0.203529	−	0.979069i	$-0.434759\pi$
$877$	12.0547	0.407058	0.203529	+	0.979069i	$0.434759\pi$
$878$	−22.7103	−0.766436
$879$	0	0
$880$	3.95306	0.133258
$881$	−14.7827	−0.498041	−0.249020	−	0.968498i	$-0.580109\pi$
$881$	−14.7827	−0.498041	−0.249020	+	0.968498i	$0.580109\pi$
$882$	0	0
$883$	−25.8462	−0.869793	−0.434896	−	0.900480i	$-0.643215\pi$
$883$	−25.8462	−0.869793	−0.434896	+	0.900480i	$0.643215\pi$
$884$	8.04531	0.270593
$885$	0	0
$886$	16.7023	0.561126
$887$	45.8560	1.53969	0.769846	−	0.638229i	$-0.220333\pi$
$887$	45.8560	1.53969	0.769846	+	0.638229i	$0.220333\pi$
$888$	0	0
$889$	9.43882	0.316568
$890$	10.5760	0.354509
$891$	0	0
$892$	−1.28581	−0.0430520
$893$	−15.6458	−0.523566
$894$	0	0
$895$	−4.59028	−0.153436
$896$	−26.5419	−0.886701
$897$	0	0
$898$	5.95037	0.198566
$899$	−7.00605	−0.233665
$900$	0	0
$901$	−61.7452	−2.05703
$902$	17.3725	0.578442
$903$	0	0
$904$	−5.09421	−0.169431
$905$	−10.3320	−0.343446
$906$	0	0
$907$	−11.7000	−0.388491	−0.194246	−	0.980953i	$-0.562226\pi$
$907$	−11.7000	−0.388491	−0.194246	+	0.980953i	$0.562226\pi$
$908$	−3.66772	−0.121717
$909$	0	0
$910$	−6.67169	−0.221165
$911$	28.6753	0.950054	0.475027	−	0.879971i	$-0.342438\pi$
$911$	28.6753	0.950054	0.475027	+	0.879971i	$0.342438\pi$
$912$	0	0
$913$	45.1962	1.49577
$914$	26.4406	0.874579
$915$	0	0
$916$	−4.69728	−0.155203
$917$	10.3133	0.340574
$918$	0	0
$919$	4.33511	0.143002	0.0715011	−	0.997441i	$-0.477221\pi$
$919$	4.33511	0.143002	0.0715011	+	0.997441i	$0.477221\pi$
$920$	−9.80418	−0.323234
$921$	0	0
$922$	43.1599	1.42140
$923$	17.4656	0.574888
$924$	0	0
$925$	15.4902	0.509315
$926$	16.8809	0.554742
$927$	0	0
$928$	7.02465	0.230596
$929$	−12.8909	−0.422937	−0.211468	−	0.977385i	$-0.567825\pi$
$929$	−12.8909	−0.422937	−0.211468	+	0.977385i	$0.567825\pi$
$930$	0	0
$931$	−28.3901	−0.930449
$932$	4.41090	0.144484
$933$	0	0
$934$	−30.3791	−0.994034
$935$	8.73746	0.285746
$936$	0	0
$937$	−53.2080	−1.73823	−0.869115	−	0.494610i	$-0.835311\pi$
$937$	−53.2080	−1.73823	−0.869115	+	0.494610i	$0.835311\pi$
$938$	42.8595	1.39941
$939$	0	0
$940$	−0.467911	−0.0152616
$941$	33.4075	1.08905	0.544526	−	0.838744i	$-0.316710\pi$
$941$	33.4075	1.08905	0.544526	+	0.838744i	$0.316710\pi$
$942$	0	0
$943$	−35.3719	−1.15187
$944$	−9.43901	−0.307214
$945$	0	0
$946$	20.5817	0.669169
$947$	15.5048	0.503837	0.251918	−	0.967748i	$-0.418939\pi$
$947$	15.5048	0.503837	0.251918	+	0.967748i	$0.418939\pi$
$948$	0	0
$949$	−5.11524	−0.166048
$950$	31.9985	1.03817
$951$	0	0
$952$	−75.0292	−2.43171
$953$	−22.5049	−0.729004	−0.364502	−	0.931203i	$-0.618761\pi$
$953$	−22.5049	−0.729004	−0.364502	+	0.931203i	$0.618761\pi$
$954$	0	0
$955$	−6.54993	−0.211951
$956$	−2.03222	−0.0657266
$957$	0	0
$958$	7.55943	0.244234
$959$	16.4497	0.531189
$960$	0	0
$961$	−27.2526	−0.879116
$962$	−13.6510	−0.440124
$963$	0	0
$964$	2.96585	0.0955237
$965$	10.4661	0.336914
$966$	0	0
$967$	−41.8084	−1.34447	−0.672234	−	0.740339i	$-0.734665\pi$
$967$	−41.8084	−1.34447	−0.672234	+	0.740339i	$0.734665\pi$
$968$	9.87144	0.317280
$969$	0	0
$970$	5.80510	0.186391
$971$	−35.8662	−1.15100	−0.575501	−	0.817801i	$-0.695193\pi$
$971$	−35.8662	−1.15100	−0.575501	+	0.817801i	$0.695193\pi$
$972$	0	0
$973$	28.2662	0.906173
$974$	49.8759	1.59813
$975$	0	0
$976$	−25.1189	−0.804035
$977$	−14.0501	−0.449502	−0.224751	−	0.974416i	$-0.572157\pi$
$977$	−14.0501	−0.449502	−0.224751	+	0.974416i	$0.572157\pi$
$978$	0	0
$979$	−51.2249	−1.63716
$980$	−0.849051	−0.0271219
$981$	0	0
$982$	48.3174	1.54187
$983$	17.6891	0.564196	0.282098	−	0.959386i	$-0.408970\pi$
$983$	17.6891	0.564196	0.282098	+	0.959386i	$0.408970\pi$
$984$	0	0
$985$	4.03064	0.128427
$986$	−32.7521	−1.04304
$987$	0	0
$988$	5.92572	0.188522
$989$	−41.9062	−1.33254
$990$	0	0
$991$	32.8958	1.04497	0.522485	−	0.852649i	$-0.325005\pi$
$991$	32.8958	1.04497	0.522485	+	0.852649i	$0.325005\pi$
$992$	−3.75736	−0.119296
$993$	0	0
$994$	−24.0993	−0.764382
$995$	1.16263	0.0368577
$996$	0	0
$997$	20.0068	0.633622	0.316811	−	0.948489i	$-0.397388\pi$
$997$	20.0068	0.633622	0.316811	+	0.948489i	$0.397388\pi$
$998$	43.3907	1.37351
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.c.1.5	yes	6
3.2	odd	2	inner	729.2.a.c.1.2	✓	6
9.2	odd	6		729.2.c.c.244.5		12
9.4	even	3		729.2.c.c.487.2		12
9.5	odd	6		729.2.c.c.487.5		12
9.7	even	3		729.2.c.c.244.2		12
27.2	odd	18		729.2.e.q.568.2		12
27.4	even	9		729.2.e.m.406.2		12
27.5	odd	18		729.2.e.r.649.1		12
27.7	even	9		729.2.e.m.325.2		12
27.11	odd	18		729.2.e.r.82.1		12
27.13	even	9		729.2.e.q.163.1		12
27.14	odd	18		729.2.e.q.163.2		12
27.16	even	9		729.2.e.r.82.2		12
27.20	odd	18		729.2.e.m.325.1		12
27.22	even	9		729.2.e.r.649.2		12
27.23	odd	18		729.2.e.m.406.1		12
27.25	even	9		729.2.e.q.568.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.c.1.2	✓	6	3.2	odd	2	inner
729.2.a.c.1.5	yes	6	1.1	even	1	trivial
729.2.c.c.244.2		12	9.7	even	3
729.2.c.c.244.5		12	9.2	odd	6
729.2.c.c.487.2		12	9.4	even	3
729.2.c.c.487.5		12	9.5	odd	6
729.2.e.m.325.1		12	27.20	odd	18
729.2.e.m.325.2		12	27.7	even	9
729.2.e.m.406.1		12	27.23	odd	18
729.2.e.m.406.2		12	27.4	even	9
729.2.e.q.163.1		12	27.13	even	9
729.2.e.q.163.2		12	27.14	odd	18
729.2.e.q.568.1		12	27.25	even	9
729.2.e.q.568.2		12	27.2	odd	18
729.2.e.r.82.1		12	27.11	odd	18
729.2.e.r.82.2		12	27.16	even	9
729.2.e.r.649.1		12	27.5	odd	18
729.2.e.r.649.2		12	27.22	even	9

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28558 q^{2} -0.347296 q^{4} +0.446476 q^{5} -3.53209 q^{7} -3.01763 q^{8} +0.573978 q^{10} -2.78006 q^{11} +3.29086 q^{13} -4.54077 q^{14} -3.18479 q^{16} -7.03936 q^{17} -5.18479 q^{19} -0.155059 q^{20} -3.57398 q^{22} +7.27693 q^{23} -4.80066 q^{25} +4.23065 q^{26} +1.22668 q^{28} +3.61916 q^{29} -1.93582 q^{31} +1.94096 q^{32} -9.04963 q^{34} -1.57699 q^{35} -3.22668 q^{37} -6.66544 q^{38} -1.34730 q^{40} -4.86084 q^{41} -5.75877 q^{43} +0.965505 q^{44} +9.35504 q^{46} +3.01763 q^{47} +5.47565 q^{49} -6.17161 q^{50} -1.14290 q^{52} +8.77141 q^{53} -1.24123 q^{55} +10.6585 q^{56} +4.65270 q^{58} +2.96377 q^{59} +7.88713 q^{61} -2.48865 q^{62} +8.86484 q^{64} +1.46929 q^{65} -9.43882 q^{67} +2.44474 q^{68} -2.02734 q^{70} +5.30731 q^{71} -1.55438 q^{73} -4.14814 q^{74} +1.80066 q^{76} +9.81942 q^{77} -11.9017 q^{79} -1.42193 q^{80} -6.24897 q^{82} -16.2573 q^{83} -3.14290 q^{85} -7.40333 q^{86} +8.38919 q^{88} +18.4258 q^{89} -11.6236 q^{91} -2.52725 q^{92} +3.87939 q^{94} -2.31488 q^{95} +10.1138 q^{97} +7.03936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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