LMFDB - Embedded newform 729.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.7459857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 $ x^6 - 3x^5 - 6x^4 + 13x^3 + 12x^2 - 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.12503$ 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)

$f(q)$	$=$	$q+0.777732 q^{2} -1.39513 q^{4} +2.37635 q^{5} -2.50138 q^{7} -2.64050 q^{8} +O(q^{10})$	\(q+0.777732 q^{2} -1.39513 q^{4} +2.37635 q^{5} -2.50138 q^{7} -2.64050 q^{8} +1.84816 q^{10} +3.14137 q^{11} +1.33663 q^{13} -1.94540 q^{14} +0.736659 q^{16} +6.27452 q^{17} +8.06469 q^{19} -3.31532 q^{20} +2.44315 q^{22} +4.05460 q^{23} +0.647028 q^{25} +1.03954 q^{26} +3.48975 q^{28} -9.28723 q^{29} +2.83182 q^{31} +5.85393 q^{32} +4.87990 q^{34} -5.94414 q^{35} +5.53191 q^{37} +6.27217 q^{38} -6.27476 q^{40} -7.10576 q^{41} +2.33690 q^{43} -4.38263 q^{44} +3.15339 q^{46} -4.61421 q^{47} -0.743116 q^{49} +0.503215 q^{50} -1.86478 q^{52} +0.135496 q^{53} +7.46499 q^{55} +6.60490 q^{56} -7.22298 q^{58} -3.99874 q^{59} +0.341798 q^{61} +2.20240 q^{62} +3.07947 q^{64} +3.17630 q^{65} +10.1229 q^{67} -8.75378 q^{68} -4.62295 q^{70} +8.19080 q^{71} -12.3144 q^{73} +4.30235 q^{74} -11.2513 q^{76} -7.85775 q^{77} +4.08070 q^{79} +1.75056 q^{80} -5.52638 q^{82} +0.913228 q^{83} +14.9104 q^{85} +1.81748 q^{86} -8.29480 q^{88} -3.72875 q^{89} -3.34341 q^{91} -5.65670 q^{92} -3.58862 q^{94} +19.1645 q^{95} -5.99630 q^{97} -0.577945 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28} - 21 q^{29} + 15 q^{31} - 30 q^{35} + 3 q^{37} - 15 q^{38} + 3 q^{40} + 12 q^{41} + 6 q^{43} + 33 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 24 q^{50} + 3 q^{52} + 9 q^{53} + 15 q^{55} - 12 q^{56} - 15 q^{58} - 6 q^{59} + 24 q^{61} + 30 q^{62} + 6 q^{64} + 15 q^{65} + 15 q^{67} - 36 q^{68} - 15 q^{70} + 12 q^{73} - 24 q^{74} + 9 q^{76} - 15 q^{77} + 24 q^{79} + 21 q^{80} - 21 q^{82} + 6 q^{83} - 18 q^{85} + 30 q^{86} - 21 q^{88} + 9 q^{89} + 18 q^{91} - 6 q^{92} - 6 q^{94} + 33 q^{95} - 21 q^{97} - 18 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.777732	0.549940	0.274970	−	0.961453i	$-0.411332\pi$
$2$	0.777732	0.549940	0.274970	+	0.961453i	$0.411332\pi$
$3$	0	0
$3$	0	0
$4$	−1.39513	−0.697566
$5$	2.37635	1.06273	0.531367	−	0.847141i	$-0.321678\pi$
$5$	2.37635	1.06273	0.531367	+	0.847141i	$0.321678\pi$
$6$	0	0
$7$	−2.50138	−0.945431	−0.472716	−	0.881215i	$-0.656726\pi$
$7$	−2.50138	−0.945431	−0.472716	+	0.881215i	$0.656726\pi$
$8$	−2.64050	−0.933559
$9$	0	0
$10$	1.84816	0.584440
$11$	3.14137	0.947159	0.473579	−	0.880751i	$-0.342962\pi$
$11$	3.14137	0.947159	0.473579	+	0.880751i	$0.342962\pi$
$12$	0	0
$13$	1.33663	0.370714	0.185357	−	0.982671i	$-0.440656\pi$
$13$	1.33663	0.370714	0.185357	+	0.982671i	$0.440656\pi$
$14$	−1.94540	−0.519930
$15$	0	0
$16$	0.736659	0.184165
$17$	6.27452	1.52179	0.760897	−	0.648873i	$-0.224759\pi$
$17$	6.27452	1.52179	0.760897	+	0.648873i	$0.224759\pi$
$18$	0	0
$19$	8.06469	1.85017	0.925083	−	0.379765i	$-0.123995\pi$
$19$	8.06469	1.85017	0.925083	+	0.379765i	$0.123995\pi$
$20$	−3.31532	−0.741328
$21$	0	0
$22$	2.44315	0.520880
$23$	4.05460	0.845442	0.422721	−	0.906260i	$-0.361075\pi$
$23$	4.05460	0.845442	0.422721	+	0.906260i	$0.361075\pi$
$24$	0	0
$25$	0.647028	0.129406
$26$	1.03954	0.203871
$27$	0	0
$28$	3.48975	0.659501
$29$	−9.28723	−1.72459	−0.862297	−	0.506402i	$-0.830975\pi$
$29$	−9.28723	−1.72459	−0.862297	+	0.506402i	$0.830975\pi$
$30$	0	0
$31$	2.83182	0.508610	0.254305	−	0.967124i	$-0.418153\pi$
$31$	2.83182	0.508610	0.254305	+	0.967124i	$0.418153\pi$
$32$	5.85393	1.03484
$33$	0	0
$34$	4.87990	0.836895
$35$	−5.94414	−1.00474
$36$	0	0
$37$	5.53191	0.909441	0.454720	−	0.890634i	$-0.349739\pi$
$37$	5.53191	0.909441	0.454720	+	0.890634i	$0.349739\pi$
$38$	6.27217	1.01748
$39$	0	0
$40$	−6.27476	−0.992126
$41$	−7.10576	−1.10973	−0.554867	−	0.831939i	$-0.687231\pi$
$41$	−7.10576	−1.10973	−0.554867	+	0.831939i	$0.687231\pi$
$42$	0	0
$43$	2.33690	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$43$	2.33690	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$44$	−4.38263	−0.660706
$45$	0	0
$46$	3.15339	0.464942
$47$	−4.61421	−0.673051	−0.336526	−	0.941674i	$-0.609252\pi$
$47$	−4.61421	−0.673051	−0.336526	+	0.941674i	$0.609252\pi$
$48$	0	0
$49$	−0.743116	−0.106159
$50$	0.503215	0.0711653
$51$	0	0
$52$	−1.86478	−0.258598
$53$	0.135496	0.0186118	0.00930588	−	0.999957i	$-0.497038\pi$
$53$	0.135496	0.0186118	0.00930588	+	0.999957i	$0.497038\pi$
$54$	0	0
$55$	7.46499	1.00658
$56$	6.60490	0.882616
$57$	0	0
$58$	−7.22298	−0.948423
$59$	−3.99874	−0.520591	−0.260296	−	0.965529i	$-0.583820\pi$
$59$	−3.99874	−0.520591	−0.260296	+	0.965529i	$0.583820\pi$
$60$	0	0
$61$	0.341798	0.0437628	0.0218814	−	0.999761i	$-0.493034\pi$
$61$	0.341798	0.0437628	0.0218814	+	0.999761i	$0.493034\pi$
$62$	2.20240	0.279705
$63$	0	0
$64$	3.07947	0.384934
$65$	3.17630	0.393971
$66$	0	0
$67$	10.1229	1.23671	0.618356	−	0.785898i	$-0.287799\pi$
$67$	10.1229	1.23671	0.618356	+	0.785898i	$0.287799\pi$
$68$	−8.75378	−1.06155
$69$	0	0
$70$	−4.62295	−0.552548
$71$	8.19080	0.972069	0.486035	−	0.873940i	$-0.338443\pi$
$71$	8.19080	0.972069	0.486035	+	0.873940i	$0.338443\pi$
$72$	0	0
$73$	−12.3144	−1.44130	−0.720648	−	0.693301i	$-0.756156\pi$
$73$	−12.3144	−1.44130	−0.720648	+	0.693301i	$0.756156\pi$
$74$	4.30235	0.500138
$75$	0	0
$76$	−11.2513	−1.29061
$77$	−7.85775	−0.895474
$78$	0	0
$79$	4.08070	0.459114	0.229557	−	0.973295i	$-0.426272\pi$
$79$	4.08070	0.459114	0.229557	+	0.973295i	$0.426272\pi$
$80$	1.75056	0.195718
$81$	0	0
$82$	−5.52638	−0.610287
$83$	0.913228	0.100240	0.0501199	−	0.998743i	$-0.484040\pi$
$83$	0.913228	0.100240	0.0501199	+	0.998743i	$0.484040\pi$
$84$	0	0
$85$	14.9104	1.61726
$86$	1.81748	0.195984
$87$	0	0
$88$	−8.29480	−0.884229
$89$	−3.72875	−0.395246	−0.197623	−	0.980278i	$-0.563322\pi$
$89$	−3.72875	−0.395246	−0.197623	+	0.980278i	$0.563322\pi$
$90$	0	0
$91$	−3.34341	−0.350485
$92$	−5.65670	−0.589752
$93$	0	0
$94$	−3.58862	−0.370138
$95$	19.1645	1.96624
$96$	0	0
$97$	−5.99630	−0.608832	−0.304416	−	0.952539i	$-0.598461\pi$
$97$	−5.99630	−0.608832	−0.304416	+	0.952539i	$0.598461\pi$
$98$	−0.577945	−0.0583813
$99$	0	0
$100$	−0.902690	−0.0902690
$101$	10.2217	1.01710	0.508549	−	0.861033i	$-0.330182\pi$
$101$	10.2217	1.01710	0.508549	+	0.861033i	$0.330182\pi$
$102$	0	0
$103$	8.53406	0.840886	0.420443	−	0.907319i	$-0.361875\pi$
$103$	8.53406	0.840886	0.420443	+	0.907319i	$0.361875\pi$
$104$	−3.52938	−0.346084
$105$	0	0
$106$	0.105379	0.0102354
$107$	−7.74500	−0.748738	−0.374369	−	0.927280i	$-0.622141\pi$
$107$	−7.74500	−0.748738	−0.374369	+	0.927280i	$0.622141\pi$
$108$	0	0
$109$	1.25438	0.120148	0.0600738	−	0.998194i	$-0.480866\pi$
$109$	1.25438	0.120148	0.0600738	+	0.998194i	$0.480866\pi$
$110$	5.80576	0.553558
$111$	0	0
$112$	−1.84266	−0.174115
$113$	−17.7608	−1.67079	−0.835396	−	0.549648i	$-0.814762\pi$
$113$	−17.7608	−1.67079	−0.835396	+	0.549648i	$0.814762\pi$
$114$	0	0
$115$	9.63514	0.898481
$116$	12.9569	1.20302
$117$	0	0
$118$	−3.10995	−0.286294
$119$	−15.6949	−1.43875
$120$	0	0
$121$	−1.13179	−0.102890
$122$	0.265827	0.0240669
$123$	0	0
$124$	−3.95076	−0.354789
$125$	−10.3442	−0.925211
$126$	0	0
$127$	−3.96558	−0.351888	−0.175944	−	0.984400i	$-0.556298\pi$
$127$	−3.96558	−0.351888	−0.175944	+	0.984400i	$0.556298\pi$
$128$	−9.31286	−0.823148
$129$	0	0
$130$	2.47031	0.216660
$131$	−0.102498	−0.00895533	−0.00447766	−	0.999990i	$-0.501425\pi$
$131$	−0.102498	−0.00895533	−0.00447766	+	0.999990i	$0.501425\pi$
$132$	0	0
$133$	−20.1728	−1.74921
$134$	7.87292	0.680117
$135$	0	0
$136$	−16.5679	−1.42068
$137$	−7.61955	−0.650982	−0.325491	−	0.945545i	$-0.605530\pi$
$137$	−7.61955	−0.650982	−0.325491	+	0.945545i	$0.605530\pi$
$138$	0	0
$139$	−10.4407	−0.885571	−0.442786	−	0.896628i	$-0.646010\pi$
$139$	−10.4407	−0.885571	−0.442786	+	0.896628i	$0.646010\pi$
$140$	8.29286	0.700875
$141$	0	0
$142$	6.37025	0.534580
$143$	4.19885	0.351125
$144$	0	0
$145$	−22.0697	−1.83279
$146$	−9.57734	−0.792626
$147$	0	0
$148$	−7.71775	−0.634395
$149$	9.03258	0.739978	0.369989	−	0.929036i	$-0.379362\pi$
$149$	9.03258	0.739978	0.369989	+	0.929036i	$0.379362\pi$
$150$	0	0
$151$	23.8904	1.94417	0.972086	−	0.234624i	$-0.0753860\pi$
$151$	23.8904	1.94417	0.972086	+	0.234624i	$0.0753860\pi$
$152$	−21.2948	−1.72724
$153$	0	0
$154$	−6.11123	−0.492457
$155$	6.72939	0.540518
$156$	0	0
$157$	2.71199	0.216440	0.108220	−	0.994127i	$-0.465485\pi$
$157$	2.71199	0.216440	0.108220	+	0.994127i	$0.465485\pi$
$158$	3.17369	0.252485
$159$	0	0
$160$	13.9110	1.09976
$161$	−10.1421	−0.799308
$162$	0	0
$163$	−22.0489	−1.72701	−0.863504	−	0.504343i	$-0.831735\pi$
$163$	−22.0489	−1.72701	−0.863504	+	0.504343i	$0.831735\pi$
$164$	9.91348	0.774113
$165$	0	0
$166$	0.710247	0.0551259
$167$	−8.95081	−0.692634	−0.346317	−	0.938118i	$-0.612568\pi$
$167$	−8.95081	−0.692634	−0.346317	+	0.938118i	$0.612568\pi$
$168$	0	0
$169$	−11.2134	−0.862571
$170$	11.5963	0.889398
$171$	0	0
$172$	−3.26029	−0.248595
$173$	−2.63425	−0.200278	−0.100139	−	0.994973i	$-0.531929\pi$
$173$	−2.63425	−0.200278	−0.100139	+	0.994973i	$0.531929\pi$
$174$	0	0
$175$	−1.61846	−0.122344
$176$	2.31412	0.174433
$177$	0	0
$178$	−2.89997	−0.217362
$179$	−3.68453	−0.275395	−0.137697	−	0.990474i	$-0.543970\pi$
$179$	−3.68453	−0.275395	−0.137697	+	0.990474i	$0.543970\pi$
$180$	0	0
$181$	−0.268509	−0.0199581	−0.00997906	−	0.999950i	$-0.503176\pi$
$181$	−0.268509	−0.0199581	−0.00997906	+	0.999950i	$0.503176\pi$
$182$	−2.60028	−0.192746
$183$	0	0
$184$	−10.7062	−0.789270
$185$	13.1457	0.966494
$186$	0	0
$187$	19.7106	1.44138
$188$	6.43743	0.469498
$189$	0	0
$190$	14.9049	1.08131
$191$	2.39799	0.173512	0.0867561	−	0.996230i	$-0.472350\pi$
$191$	2.39799	0.173512	0.0867561	+	0.996230i	$0.472350\pi$
$192$	0	0
$193$	0.496203	0.0357175	0.0178587	−	0.999841i	$-0.494315\pi$
$193$	0.496203	0.0357175	0.0178587	+	0.999841i	$0.494315\pi$
$194$	−4.66351	−0.334821
$195$	0	0
$196$	1.03675	0.0740532
$197$	22.1468	1.57789	0.788946	−	0.614462i	$-0.210627\pi$
$197$	22.1468	1.57789	0.788946	+	0.614462i	$0.210627\pi$
$198$	0	0
$199$	2.13247	0.151167	0.0755834	−	0.997139i	$-0.475918\pi$
$199$	2.13247	0.151167	0.0755834	+	0.997139i	$0.475918\pi$
$200$	−1.70848	−0.120808
$201$	0	0
$202$	7.94976	0.559343
$203$	23.2309	1.63049
$204$	0	0
$205$	−16.8858	−1.17935
$206$	6.63721	0.462437
$207$	0	0
$208$	0.984641	0.0682725
$209$	25.3342	1.75240
$210$	0	0
$211$	20.0086	1.37745	0.688724	−	0.725024i	$-0.258171\pi$
$211$	20.0086	1.37745	0.688724	+	0.725024i	$0.258171\pi$
$212$	−0.189034	−0.0129829
$213$	0	0
$214$	−6.02354	−0.411761
$215$	5.55329	0.378731
$216$	0	0
$217$	−7.08345	−0.480856
$218$	0.975570	0.0660739
$219$	0	0
$220$	−10.4146	−0.702155
$221$	8.38671	0.564151
$222$	0	0
$223$	−13.2785	−0.889192	−0.444596	−	0.895731i	$-0.646653\pi$
$223$	−13.2785	−0.889192	−0.444596	+	0.895731i	$0.646653\pi$
$224$	−14.6429	−0.978369
$225$	0	0
$226$	−13.8131	−0.918835
$227$	12.4709	0.827725	0.413862	−	0.910339i	$-0.364179\pi$
$227$	12.4709	0.827725	0.413862	+	0.910339i	$0.364179\pi$
$228$	0	0
$229$	25.6616	1.69576	0.847882	−	0.530185i	$-0.177878\pi$
$229$	25.6616	1.69576	0.847882	+	0.530185i	$0.177878\pi$
$230$	7.49356	0.494111
$231$	0	0
$232$	24.5230	1.61001
$233$	5.39642	0.353532	0.176766	−	0.984253i	$-0.443436\pi$
$233$	5.39642	0.353532	0.176766	+	0.984253i	$0.443436\pi$
$234$	0	0
$235$	−10.9650	−0.715275
$236$	5.57877	0.363147
$237$	0	0
$238$	−12.2065	−0.791227
$239$	−8.37104	−0.541478	−0.270739	−	0.962653i	$-0.587268\pi$
$239$	−8.37104	−0.541478	−0.270739	+	0.962653i	$0.587268\pi$
$240$	0	0
$241$	−0.442190	−0.0284839	−0.0142420	−	0.999899i	$-0.504534\pi$
$241$	−0.442190	−0.0284839	−0.0142420	+	0.999899i	$0.504534\pi$
$242$	−0.880231	−0.0565834
$243$	0	0
$244$	−0.476854	−0.0305274
$245$	−1.76590	−0.112819
$246$	0	0
$247$	10.7795	0.685883
$248$	−7.47743	−0.474818
$249$	0	0
$250$	−8.04500	−0.508810
$251$	−17.0285	−1.07483	−0.537416	−	0.843317i	$-0.680599\pi$
$251$	−17.0285	−1.07483	−0.537416	+	0.843317i	$0.680599\pi$
$252$	0	0
$253$	12.7370	0.800768
$254$	−3.08416	−0.193517
$255$	0	0
$256$	−13.4019	−0.837616
$257$	−20.8767	−1.30225	−0.651127	−	0.758969i	$-0.725703\pi$
$257$	−20.8767	−1.30225	−0.651127	+	0.758969i	$0.725703\pi$
$258$	0	0
$259$	−13.8374	−0.859814
$260$	−4.43136	−0.274821
$261$	0	0
$262$	−0.0797163	−0.00492489
$263$	−19.3916	−1.19573	−0.597867	−	0.801595i	$-0.703985\pi$
$263$	−19.3916	−1.19573	−0.597867	+	0.801595i	$0.703985\pi$
$264$	0	0
$265$	0.321985	0.0197794
$266$	−15.6891	−0.961958
$267$	0	0
$268$	−14.1228	−0.862688
$269$	−18.6791	−1.13889	−0.569443	−	0.822031i	$-0.692841\pi$
$269$	−18.6791	−1.13889	−0.569443	+	0.822031i	$0.692841\pi$
$270$	0	0
$271$	12.9378	0.785917	0.392958	−	0.919556i	$-0.371452\pi$
$271$	12.9378	0.785917	0.392958	+	0.919556i	$0.371452\pi$
$272$	4.62218	0.280261
$273$	0	0
$274$	−5.92597	−0.358001
$275$	2.03256	0.122568
$276$	0	0
$277$	10.4413	0.627359	0.313679	−	0.949529i	$-0.398438\pi$
$277$	10.4413	0.627359	0.313679	+	0.949529i	$0.398438\pi$
$278$	−8.12009	−0.487011
$279$	0	0
$280$	15.6955	0.937987
$281$	−14.3458	−0.855798	−0.427899	−	0.903826i	$-0.640746\pi$
$281$	−14.3458	−0.855798	−0.427899	+	0.903826i	$0.640746\pi$
$282$	0	0
$283$	−18.6926	−1.11116	−0.555580	−	0.831463i	$-0.687504\pi$
$283$	−18.6926	−1.11116	−0.555580	+	0.831463i	$0.687504\pi$
$284$	−11.4273	−0.678083
$285$	0	0
$286$	3.26558	0.193098
$287$	17.7742	1.04918
$288$	0	0
$289$	22.3696	1.31586
$290$	−17.1643	−1.00792
$291$	0	0
$292$	17.1803	1.00540
$293$	10.7882	0.630252	0.315126	−	0.949050i	$-0.397953\pi$
$293$	10.7882	0.630252	0.315126	+	0.949050i	$0.397953\pi$
$294$	0	0
$295$	−9.50239	−0.553251
$296$	−14.6070	−0.849017
$297$	0	0
$298$	7.02493	0.406943
$299$	5.41950	0.313418
$300$	0	0
$301$	−5.84547	−0.336927
$302$	18.5803	1.06918
$303$	0	0
$304$	5.94092	0.340735
$305$	0.812231	0.0465082
$306$	0	0
$307$	0.0497494	0.00283935	0.00141967	−	0.999999i	$-0.499548\pi$
$307$	0.0497494	0.00283935	0.00141967	+	0.999999i	$0.499548\pi$
$308$	10.9626	0.624652
$309$	0	0
$310$	5.23366	0.297252
$311$	−13.2355	−0.750514	−0.375257	−	0.926921i	$-0.622446\pi$
$311$	−13.2355	−0.750514	−0.375257	+	0.926921i	$0.622446\pi$
$312$	0	0
$313$	−15.0829	−0.852537	−0.426268	−	0.904597i	$-0.640172\pi$
$313$	−15.0829	−0.852537	−0.426268	+	0.904597i	$0.640172\pi$
$314$	2.10920	0.119029
$315$	0	0
$316$	−5.69311	−0.320263
$317$	−8.63082	−0.484755	−0.242378	−	0.970182i	$-0.577927\pi$
$317$	−8.63082	−0.484755	−0.242378	+	0.970182i	$0.577927\pi$
$318$	0	0
$319$	−29.1746	−1.63347
$320$	7.31790	0.409083
$321$	0	0
$322$	−7.88782	−0.439571
$323$	50.6020	2.81557
$324$	0	0
$325$	0.864837	0.0479725
$326$	−17.1482	−0.949750
$327$	0	0
$328$	18.7628	1.03600
$329$	11.5419	0.636324
$330$	0	0
$331$	31.0255	1.70531	0.852657	−	0.522471i	$-0.174990\pi$
$331$	31.0255	1.70531	0.852657	+	0.522471i	$0.174990\pi$
$332$	−1.27407	−0.0699239
$333$	0	0
$334$	−6.96133	−0.380907
$335$	24.0556	1.31430
$336$	0	0
$337$	−23.8673	−1.30014	−0.650068	−	0.759876i	$-0.725260\pi$
$337$	−23.8673	−1.30014	−0.650068	+	0.759876i	$0.725260\pi$
$338$	−8.72104	−0.474362
$339$	0	0
$340$	−20.8020	−1.12815
$341$	8.89580	0.481734
$342$	0	0
$343$	19.3684	1.04580
$344$	−6.17060	−0.332696
$345$	0	0
$346$	−2.04874	−0.110141
$347$	−21.9112	−1.17626	−0.588128	−	0.808768i	$-0.700135\pi$
$347$	−21.9112	−1.17626	−0.588128	+	0.808768i	$0.700135\pi$
$348$	0	0
$349$	15.8469	0.848263	0.424132	−	0.905601i	$-0.360579\pi$
$349$	15.8469	0.848263	0.424132	+	0.905601i	$0.360579\pi$
$350$	−1.25873	−0.0672819
$351$	0	0
$352$	18.3894	0.980157
$353$	−12.9069	−0.686964	−0.343482	−	0.939159i	$-0.611606\pi$
$353$	−12.9069	−0.686964	−0.343482	+	0.939159i	$0.611606\pi$
$354$	0	0
$355$	19.4642	1.03305
$356$	5.20209	0.275710
$357$	0	0
$358$	−2.86558	−0.151451
$359$	−25.8285	−1.36318	−0.681588	−	0.731736i	$-0.738710\pi$
$359$	−25.8285	−1.36318	−0.681588	+	0.731736i	$0.738710\pi$
$360$	0	0
$361$	46.0392	2.42312
$362$	−0.208828	−0.0109758
$363$	0	0
$364$	4.66451	0.244487
$365$	−29.2634	−1.53172
$366$	0	0
$367$	15.9746	0.833866	0.416933	−	0.908937i	$-0.363105\pi$
$367$	15.9746	0.833866	0.416933	+	0.908937i	$0.363105\pi$
$368$	2.98686	0.155701
$369$	0	0
$370$	10.2239	0.531514
$371$	−0.338926	−0.0175961
$372$	0	0
$373$	1.82657	0.0945760	0.0472880	−	0.998881i	$-0.484942\pi$
$373$	1.82657	0.0945760	0.0472880	+	0.998881i	$0.484942\pi$
$374$	15.3296	0.792673
$375$	0	0
$376$	12.1838	0.628333
$377$	−12.4136	−0.639332
$378$	0	0
$379$	1.14694	0.0589141	0.0294571	−	0.999566i	$-0.490622\pi$
$379$	1.14694	0.0589141	0.0294571	+	0.999566i	$0.490622\pi$
$380$	−26.7370	−1.37158
$381$	0	0
$382$	1.86499	0.0954212
$383$	−21.8275	−1.11534	−0.557668	−	0.830064i	$-0.688304\pi$
$383$	−21.8275	−1.11534	−0.557668	+	0.830064i	$0.688304\pi$
$384$	0	0
$385$	−18.6727	−0.951651
$386$	0.385913	0.0196425
$387$	0	0
$388$	8.36563	0.424700
$389$	26.1962	1.32820	0.664099	−	0.747644i	$-0.268815\pi$
$389$	26.1962	1.32820	0.664099	+	0.747644i	$0.268815\pi$
$390$	0	0
$391$	25.4407	1.28659
$392$	1.96220	0.0991061
$393$	0	0
$394$	17.2243	0.867746
$395$	9.69715	0.487917
$396$	0	0
$397$	4.19831	0.210707	0.105353	−	0.994435i	$-0.466403\pi$
$397$	4.19831	0.210707	0.105353	+	0.994435i	$0.466403\pi$
$398$	1.65849	0.0831327
$399$	0	0
$400$	0.476639	0.0238320
$401$	7.82981	0.391002	0.195501	−	0.980703i	$-0.437367\pi$
$401$	7.82981	0.391002	0.195501	+	0.980703i	$0.437367\pi$
$402$	0	0
$403$	3.78510	0.188549
$404$	−14.2606	−0.709493
$405$	0	0
$406$	18.0674	0.896669
$407$	17.3778	0.861385
$408$	0	0
$409$	−17.4290	−0.861808	−0.430904	−	0.902398i	$-0.641805\pi$
$409$	−17.4290	−0.861808	−0.430904	+	0.902398i	$0.641805\pi$
$410$	−13.1326	−0.648573
$411$	0	0
$412$	−11.9061	−0.586574
$413$	10.0024	0.492183
$414$	0	0
$415$	2.17015	0.106528
$416$	7.82454	0.383630
$417$	0	0
$418$	19.7032	0.963715
$419$	−11.4832	−0.560990	−0.280495	−	0.959856i	$-0.590499\pi$
$419$	−11.4832	−0.560990	−0.280495	+	0.959856i	$0.590499\pi$
$420$	0	0
$421$	7.29123	0.355353	0.177676	−	0.984089i	$-0.443142\pi$
$421$	7.29123	0.355353	0.177676	+	0.984089i	$0.443142\pi$
$422$	15.5613	0.757513
$423$	0	0
$424$	−0.357777	−0.0173752
$425$	4.05979	0.196929
$426$	0	0
$427$	−0.854966	−0.0413747
$428$	10.8053	0.522294
$429$	0	0
$430$	4.31897	0.208279
$431$	−0.389084	−0.0187415	−0.00937075	−	0.999956i	$-0.502983\pi$
$431$	−0.389084	−0.0187415	−0.00937075	+	0.999956i	$0.502983\pi$
$432$	0	0
$433$	24.1011	1.15822	0.579112	−	0.815248i	$-0.303400\pi$
$433$	24.1011	1.15822	0.579112	+	0.815248i	$0.303400\pi$
$434$	−5.50903	−0.264442
$435$	0	0
$436$	−1.75002	−0.0838109
$437$	32.6991	1.56421
$438$	0	0
$439$	36.6656	1.74995	0.874977	−	0.484165i	$-0.160877\pi$
$439$	36.6656	1.74995	0.874977	+	0.484165i	$0.160877\pi$
$440$	−19.7113	−0.939701
$441$	0	0
$442$	6.52261	0.310249
$443$	37.7867	1.79530	0.897649	−	0.440711i	$-0.145274\pi$
$443$	37.7867	1.79530	0.897649	+	0.440711i	$0.145274\pi$
$444$	0	0
$445$	−8.86080	−0.420042
$446$	−10.3271	−0.489002
$447$	0	0
$448$	−7.70293	−0.363929
$449$	−11.7858	−0.556205	−0.278103	−	0.960551i	$-0.589706\pi$
$449$	−11.7858	−0.556205	−0.278103	+	0.960551i	$0.589706\pi$
$450$	0	0
$451$	−22.3218	−1.05109
$452$	24.7786	1.16549
$453$	0	0
$454$	9.69905	0.455199
$455$	−7.94512	−0.372473
$456$	0	0
$457$	−19.8559	−0.928820	−0.464410	−	0.885620i	$-0.653734\pi$
$457$	−19.8559	−0.928820	−0.464410	+	0.885620i	$0.653734\pi$
$458$	19.9578	0.932568
$459$	0	0
$460$	−13.4423	−0.626750
$461$	7.64183	0.355915	0.177958	−	0.984038i	$-0.443051\pi$
$461$	7.64183	0.355915	0.177958	+	0.984038i	$0.443051\pi$
$462$	0	0
$463$	−15.9544	−0.741465	−0.370732	−	0.928740i	$-0.620893\pi$
$463$	−15.9544	−0.741465	−0.370732	+	0.928740i	$0.620893\pi$
$464$	−6.84152	−0.317610
$465$	0	0
$466$	4.19697	0.194421
$467$	−26.1406	−1.20964	−0.604822	−	0.796360i	$-0.706756\pi$
$467$	−26.1406	−1.20964	−0.604822	+	0.796360i	$0.706756\pi$
$468$	0	0
$469$	−25.3212	−1.16923
$470$	−8.52780	−0.393358
$471$	0	0
$472$	10.5587	0.486003
$473$	7.34107	0.337543
$474$	0	0
$475$	5.21808	0.239422
$476$	21.8965	1.00362
$477$	0	0
$478$	−6.51043	−0.297780
$479$	−39.3503	−1.79796	−0.898980	−	0.437989i	$-0.855691\pi$
$479$	−39.3503	−1.79796	−0.898980	+	0.437989i	$0.855691\pi$
$480$	0	0
$481$	7.39412	0.337143
$482$	−0.343905	−0.0156644
$483$	0	0
$484$	1.57900	0.0717727
$485$	−14.2493	−0.647027
$486$	0	0
$487$	11.7133	0.530779	0.265389	−	0.964141i	$-0.414500\pi$
$487$	11.7133	0.530779	0.265389	+	0.964141i	$0.414500\pi$
$488$	−0.902519	−0.0408551
$489$	0	0
$490$	−1.37340	−0.0620439
$491$	38.5498	1.73973	0.869865	−	0.493290i	$-0.164206\pi$
$491$	38.5498	1.73973	0.869865	+	0.493290i	$0.164206\pi$
$492$	0	0
$493$	−58.2729	−2.62448
$494$	8.38357	0.377195
$495$	0	0
$496$	2.08609	0.0936680
$497$	−20.4883	−0.919025
$498$	0	0
$499$	−29.5616	−1.32336	−0.661679	−	0.749787i	$-0.730156\pi$
$499$	−29.5616	−1.32336	−0.661679	+	0.749787i	$0.730156\pi$
$500$	14.4315	0.645396
$501$	0	0
$502$	−13.2436	−0.591093
$503$	−35.5775	−1.58632	−0.793161	−	0.609011i	$-0.791566\pi$
$503$	−35.5775	−1.58632	−0.793161	+	0.609011i	$0.791566\pi$
$504$	0	0
$505$	24.2903	1.08091
$506$	9.90597	0.440374
$507$	0	0
$508$	5.53251	0.245465
$509$	28.3823	1.25803	0.629013	−	0.777395i	$-0.283459\pi$
$509$	28.3823	1.25803	0.629013	+	0.777395i	$0.283459\pi$
$510$	0	0
$511$	30.8030	1.36265
$512$	8.20265	0.362510
$513$	0	0
$514$	−16.2365	−0.716161
$515$	20.2799	0.893639
$516$	0	0
$517$	−14.4949	−0.637486
$518$	−10.7618	−0.472846
$519$	0	0
$520$	−8.38703	−0.367795
$521$	25.4351	1.11433	0.557167	−	0.830401i	$-0.311888\pi$
$521$	25.4351	1.11433	0.557167	+	0.830401i	$0.311888\pi$
$522$	0	0
$523$	8.40790	0.367652	0.183826	−	0.982959i	$-0.441152\pi$
$523$	8.40790	0.367652	0.183826	+	0.982959i	$0.441152\pi$
$524$	0.142999	0.00624693
$525$	0	0
$526$	−15.0814	−0.657582
$527$	17.7683	0.774000
$528$	0	0
$529$	−6.56023	−0.285227
$530$	0.250418	0.0108775
$531$	0	0
$532$	28.1438	1.22019
$533$	−9.49777	−0.411394
$534$	0	0
$535$	−18.4048	−0.795710
$536$	−26.7296	−1.15454
$537$	0	0
$538$	−14.5274	−0.626319
$539$	−2.33440	−0.100550
$540$	0	0
$541$	−12.8635	−0.553043	−0.276522	−	0.961008i	$-0.589182\pi$
$541$	−12.8635	−0.553043	−0.276522	+	0.961008i	$0.589182\pi$
$542$	10.0622	0.432207
$543$	0	0
$544$	36.7306	1.57481
$545$	2.98084	0.127685
$546$	0	0
$547$	13.0920	0.559772	0.279886	−	0.960033i	$-0.409703\pi$
$547$	13.0920	0.559772	0.279886	+	0.960033i	$0.409703\pi$
$548$	10.6303	0.454103
$549$	0	0
$550$	1.58078	0.0674049
$551$	−74.8986	−3.19079
$552$	0	0
$553$	−10.2074	−0.434061
$554$	8.12056	0.345010
$555$	0	0
$556$	14.5662	0.617745
$557$	−4.58220	−0.194154	−0.0970769	−	0.995277i	$-0.530949\pi$
$557$	−4.58220	−0.194154	−0.0970769	+	0.995277i	$0.530949\pi$
$558$	0	0
$559$	3.12357	0.132113
$560$	−4.37880	−0.185038
$561$	0	0
$562$	−11.1572	−0.470638
$563$	−12.2839	−0.517705	−0.258853	−	0.965917i	$-0.583344\pi$
$563$	−12.2839	−0.517705	−0.258853	+	0.965917i	$0.583344\pi$
$564$	0	0
$565$	−42.2058	−1.77561
$566$	−14.5378	−0.611071
$567$	0	0
$568$	−21.6278	−0.907484
$569$	−14.4937	−0.607606	−0.303803	−	0.952735i	$-0.598256\pi$
$569$	−14.4937	−0.607606	−0.303803	+	0.952735i	$0.598256\pi$
$570$	0	0
$571$	−22.0454	−0.922569	−0.461285	−	0.887252i	$-0.652611\pi$
$571$	−22.0454	−0.922569	−0.461285	+	0.887252i	$0.652611\pi$
$572$	−5.85795	−0.244933
$573$	0	0
$574$	13.8236	0.576984
$575$	2.62344	0.109405
$576$	0	0
$577$	−31.4835	−1.31068	−0.655338	−	0.755336i	$-0.727474\pi$
$577$	−31.4835	−1.31068	−0.655338	+	0.755336i	$0.727474\pi$
$578$	17.3975	0.723642
$579$	0	0
$580$	30.7901	1.27849
$581$	−2.28433	−0.0947699
$582$	0	0
$583$	0.425642	0.0176283
$584$	32.5163	1.34554
$585$	0	0
$586$	8.39032	0.346601
$587$	14.3694	0.593090	0.296545	−	0.955019i	$-0.404165\pi$
$587$	14.3694	0.593090	0.296545	+	0.955019i	$0.404165\pi$
$588$	0	0
$589$	22.8378	0.941013
$590$	−7.39032	−0.304255
$591$	0	0
$592$	4.07513	0.167487
$593$	41.0988	1.68772	0.843862	−	0.536560i	$-0.180276\pi$
$593$	41.0988	1.68772	0.843862	+	0.536560i	$0.180276\pi$
$594$	0	0
$595$	−37.2966	−1.52901
$596$	−12.6016	−0.516183
$597$	0	0
$598$	4.21492	0.172361
$599$	−8.56216	−0.349840	−0.174920	−	0.984583i	$-0.555967\pi$
$599$	−8.56216	−0.349840	−0.174920	+	0.984583i	$0.555967\pi$
$600$	0	0
$601$	−1.63521	−0.0667016	−0.0333508	−	0.999444i	$-0.510618\pi$
$601$	−1.63521	−0.0667016	−0.0333508	+	0.999444i	$0.510618\pi$
$602$	−4.54621	−0.185290
$603$	0	0
$604$	−33.3303	−1.35619
$605$	−2.68953	−0.109345
$606$	0	0
$607$	6.56943	0.266645	0.133322	−	0.991073i	$-0.457435\pi$
$607$	6.56943	0.266645	0.133322	+	0.991073i	$0.457435\pi$
$608$	47.2101	1.91462
$609$	0	0
$610$	0.631698	0.0255767
$611$	−6.16749	−0.249510
$612$	0	0
$613$	−26.2726	−1.06114	−0.530569	−	0.847642i	$-0.678022\pi$
$613$	−26.2726	−1.06114	−0.530569	+	0.847642i	$0.678022\pi$
$614$	0.0386917	0.00156147
$615$	0	0
$616$	20.7484	0.835978
$617$	6.27303	0.252542	0.126271	−	0.991996i	$-0.459699\pi$
$617$	6.27303	0.252542	0.126271	+	0.991996i	$0.459699\pi$
$618$	0	0
$619$	4.94087	0.198590	0.0992952	−	0.995058i	$-0.468341\pi$
$619$	4.94087	0.198590	0.0992952	+	0.995058i	$0.468341\pi$
$620$	−9.38839	−0.377047
$621$	0	0
$622$	−10.2936	−0.412738
$623$	9.32700	0.373678
$624$	0	0
$625$	−27.8165	−1.11266
$626$	−11.7305	−0.468844
$627$	0	0
$628$	−3.78358	−0.150981
$629$	34.7101	1.38398
$630$	0	0
$631$	−6.92420	−0.275648	−0.137824	−	0.990457i	$-0.544011\pi$
$631$	−6.92420	−0.275648	−0.137824	+	0.990457i	$0.544011\pi$
$632$	−10.7751	−0.428610
$633$	0	0
$634$	−6.71247	−0.266586
$635$	−9.42360	−0.373964
$636$	0	0
$637$	−0.993271	−0.0393548
$638$	−22.6900	−0.898308
$639$	0	0
$640$	−22.1306	−0.874788
$641$	33.9189	1.33972	0.669858	−	0.742490i	$-0.266355\pi$
$641$	33.9189	1.33972	0.669858	+	0.742490i	$0.266355\pi$
$642$	0	0
$643$	7.72322	0.304574	0.152287	−	0.988336i	$-0.451336\pi$
$643$	7.72322	0.304574	0.152287	+	0.988336i	$0.451336\pi$
$644$	14.1495	0.557570
$645$	0	0
$646$	39.3548	1.54840
$647$	−35.1862	−1.38331	−0.691655	−	0.722228i	$-0.743118\pi$
$647$	−35.1862	−1.38331	−0.691655	+	0.722228i	$0.743118\pi$
$648$	0	0
$649$	−12.5615	−0.493083
$650$	0.672612	0.0263820
$651$	0	0
$652$	30.7612	1.20470
$653$	−19.5120	−0.763562	−0.381781	−	0.924253i	$-0.624689\pi$
$653$	−19.5120	−0.763562	−0.381781	+	0.924253i	$0.624689\pi$
$654$	0	0
$655$	−0.243572	−0.00951714
$656$	−5.23452	−0.204374
$657$	0	0
$658$	8.97648	0.349940
$659$	−23.0864	−0.899320	−0.449660	−	0.893200i	$-0.648455\pi$
$659$	−23.0864	−0.899320	−0.449660	+	0.893200i	$0.648455\pi$
$660$	0	0
$661$	−6.87636	−0.267459	−0.133730	−	0.991018i	$-0.542695\pi$
$661$	−6.87636	−0.267459	−0.133730	+	0.991018i	$0.542695\pi$
$662$	24.1295	0.937820
$663$	0	0
$664$	−2.41138	−0.0935798
$665$	−47.9376	−1.85894
$666$	0	0
$667$	−37.6560	−1.45805
$668$	12.4876	0.483158
$669$	0	0
$670$	18.7088	0.722784
$671$	1.07371	0.0414503
$672$	0	0
$673$	−30.6723	−1.18233	−0.591166	−	0.806550i	$-0.701332\pi$
$673$	−30.6723	−1.18233	−0.591166	+	0.806550i	$0.701332\pi$
$674$	−18.5624	−0.714997
$675$	0	0
$676$	15.6442	0.601700
$677$	13.5840	0.522074	0.261037	−	0.965329i	$-0.415935\pi$
$677$	13.5840	0.522074	0.261037	+	0.965329i	$0.415935\pi$
$678$	0	0
$679$	14.9990	0.575608
$680$	−39.3711	−1.50981
$681$	0	0
$682$	6.91855	0.264925
$683$	6.06700	0.232147	0.116074	−	0.993241i	$-0.462969\pi$
$683$	6.06700	0.232147	0.116074	+	0.993241i	$0.462969\pi$
$684$	0	0
$685$	−18.1067	−0.691821
$686$	15.0635	0.575126
$687$	0	0
$688$	1.72150	0.0656316
$689$	0.181108	0.00689965
$690$	0	0
$691$	20.6651	0.786137	0.393068	−	0.919509i	$-0.371414\pi$
$691$	20.6651	0.786137	0.393068	+	0.919509i	$0.371414\pi$
$692$	3.67513	0.139707
$693$	0	0
$694$	−17.0411	−0.646871
$695$	−24.8108	−0.941127
$696$	0	0
$697$	−44.5852	−1.68879
$698$	12.3246	0.466494
$699$	0	0
$700$	2.25797	0.0853431
$701$	−11.0222	−0.416303	−0.208151	−	0.978097i	$-0.566745\pi$
$701$	−11.0222	−0.416303	−0.208151	+	0.978097i	$0.566745\pi$
$702$	0	0
$703$	44.6131	1.68262
$704$	9.67377	0.364594
$705$	0	0
$706$	−10.0381	−0.377789
$707$	−25.5684	−0.961597
$708$	0	0
$709$	−10.9874	−0.412642	−0.206321	−	0.978484i	$-0.566149\pi$
$709$	−10.9874	−0.412642	−0.206321	+	0.978484i	$0.566149\pi$
$710$	15.1379	0.568116
$711$	0	0
$712$	9.84577	0.368986
$713$	11.4819	0.430000
$714$	0	0
$715$	9.97793	0.373153
$716$	5.14041	0.192106
$717$	0	0
$718$	−20.0877	−0.749664
$719$	−32.7057	−1.21972	−0.609859	−	0.792510i	$-0.708774\pi$
$719$	−32.7057	−1.21972	−0.609859	+	0.792510i	$0.708774\pi$
$720$	0	0
$721$	−21.3469	−0.795000
$722$	35.8062	1.33257
$723$	0	0
$724$	0.374606	0.0139221
$725$	−6.00910	−0.223172
$726$	0	0
$727$	38.4606	1.42643	0.713213	−	0.700948i	$-0.247239\pi$
$727$	38.4606	1.42643	0.713213	+	0.700948i	$0.247239\pi$
$728$	8.82830	0.327199
$729$	0	0
$730$	−22.7591	−0.842352
$731$	14.6629	0.542328
$732$	0	0
$733$	14.1077	0.521082	0.260541	−	0.965463i	$-0.416099\pi$
$733$	14.1077	0.521082	0.260541	+	0.965463i	$0.416099\pi$
$734$	12.4239	0.458576
$735$	0	0
$736$	23.7353	0.874896
$737$	31.7998	1.17136
$738$	0	0
$739$	−11.8457	−0.435752	−0.217876	−	0.975977i	$-0.569913\pi$
$739$	−11.8457	−0.435752	−0.217876	+	0.975977i	$0.569913\pi$
$740$	−18.3401	−0.674194
$741$	0	0
$742$	−0.263594	−0.00967682
$743$	21.7676	0.798574	0.399287	−	0.916826i	$-0.369258\pi$
$743$	21.7676	0.798574	0.399287	+	0.916826i	$0.369258\pi$
$744$	0	0
$745$	21.4645	0.786400
$746$	1.42058	0.0520111
$747$	0	0
$748$	−27.4989	−1.00546
$749$	19.3732	0.707880
$750$	0	0
$751$	−42.5490	−1.55264	−0.776318	−	0.630341i	$-0.782915\pi$
$751$	−42.5490	−1.55264	−0.776318	+	0.630341i	$0.782915\pi$
$752$	−3.39910	−0.123952
$753$	0	0
$754$	−9.65445	−0.351594
$755$	56.7719	2.06614
$756$	0	0
$757$	−20.6382	−0.750110	−0.375055	−	0.927003i	$-0.622376\pi$
$757$	−20.6382	−0.750110	−0.375055	+	0.927003i	$0.622376\pi$
$758$	0.892009	0.0323992
$759$	0	0
$760$	−50.6040	−1.83560
$761$	49.4450	1.79238	0.896190	−	0.443670i	$-0.146324\pi$
$761$	49.4450	1.79238	0.896190	+	0.443670i	$0.146324\pi$
$762$	0	0
$763$	−3.13767	−0.113591
$764$	−3.34551	−0.121036
$765$	0	0
$766$	−16.9760	−0.613367
$767$	−5.34483	−0.192991
$768$	0	0
$769$	22.1336	0.798158	0.399079	−	0.916917i	$-0.369330\pi$
$769$	22.1336	0.798158	0.399079	+	0.916917i	$0.369330\pi$
$770$	−14.5224	−0.523351
$771$	0	0
$772$	−0.692269	−0.0249153
$773$	−21.9671	−0.790102	−0.395051	−	0.918659i	$-0.629273\pi$
$773$	−21.9671	−0.790102	−0.395051	+	0.918659i	$0.629273\pi$
$774$	0	0
$775$	1.83227	0.0658170
$776$	15.8332	0.568380
$777$	0	0
$778$	20.3736	0.730430
$779$	−57.3057	−2.05319
$780$	0	0
$781$	25.7303	0.920704
$782$	19.7860	0.707547
$783$	0	0
$784$	−0.547423	−0.0195508
$785$	6.44463	0.230019
$786$	0	0
$787$	0.535531	0.0190896	0.00954480	−	0.999954i	$-0.496962\pi$
$787$	0.535531	0.0190896	0.00954480	+	0.999954i	$0.496962\pi$
$788$	−30.8977	−1.10068
$789$	0	0
$790$	7.54179	0.268325
$791$	44.4264	1.57962
$792$	0	0
$793$	0.456858	0.0162235
$794$	3.26516	0.115876
$795$	0	0
$796$	−2.97508	−0.105449
$797$	−40.0990	−1.42038	−0.710189	−	0.704011i	$-0.751391\pi$
$797$	−40.0990	−1.42038	−0.710189	+	0.704011i	$0.751391\pi$
$798$	0	0
$799$	−28.9519	−1.02425
$800$	3.78766	0.133914
$801$	0	0
$802$	6.08950	0.215028
$803$	−38.6842	−1.36514
$804$	0	0
$805$	−24.1011	−0.849452
$806$	2.94379	0.103691
$807$	0	0
$808$	−26.9905	−0.949522
$809$	−17.1826	−0.604110	−0.302055	−	0.953291i	$-0.597673\pi$
$809$	−17.1826	−0.604110	−0.302055	+	0.953291i	$0.597673\pi$
$810$	0	0
$811$	−19.6169	−0.688842	−0.344421	−	0.938815i	$-0.611925\pi$
$811$	−19.6169	−0.688842	−0.344421	+	0.938815i	$0.611925\pi$
$812$	−32.4101	−1.13737
$813$	0	0
$814$	13.5153	0.473710
$815$	−52.3960	−1.83535
$816$	0	0
$817$	18.8464	0.659351
$818$	−13.5551	−0.473943
$819$	0	0
$820$	23.5579	0.822676
$821$	−35.9059	−1.25313	−0.626563	−	0.779371i	$-0.715539\pi$
$821$	−35.9059	−1.25313	−0.626563	+	0.779371i	$0.715539\pi$
$822$	0	0
$823$	−25.8181	−0.899961	−0.449980	−	0.893038i	$-0.648569\pi$
$823$	−25.8181	−0.899961	−0.449980	+	0.893038i	$0.648569\pi$
$824$	−22.5342	−0.785017
$825$	0	0
$826$	7.77915	0.270671
$827$	24.9586	0.867895	0.433948	−	0.900938i	$-0.357120\pi$
$827$	24.9586	0.867895	0.433948	+	0.900938i	$0.357120\pi$
$828$	0	0
$829$	2.79927	0.0972228	0.0486114	−	0.998818i	$-0.484520\pi$
$829$	2.79927	0.0972228	0.0486114	+	0.998818i	$0.484520\pi$
$830$	1.68779	0.0585842
$831$	0	0
$832$	4.11612	0.142701
$833$	−4.66270	−0.161553
$834$	0	0
$835$	−21.2702	−0.736087
$836$	−35.3445	−1.22242
$837$	0	0
$838$	−8.93083	−0.308511
$839$	44.7079	1.54349	0.771744	−	0.635934i	$-0.219385\pi$
$839$	44.7079	1.54349	0.771744	+	0.635934i	$0.219385\pi$
$840$	0	0
$841$	57.2526	1.97423
$842$	5.67062	0.195423
$843$	0	0
$844$	−27.9146	−0.960861
$845$	−26.6470	−0.916684
$846$	0	0
$847$	2.83104	0.0972756
$848$	0.0998141	0.00342763
$849$	0	0
$850$	3.15743	0.108299
$851$	22.4297	0.768880
$852$	0	0
$853$	−43.5150	−1.48992	−0.744962	−	0.667107i	$-0.767532\pi$
$853$	−43.5150	−1.48992	−0.744962	+	0.667107i	$0.767532\pi$
$854$	−0.664935	−0.0227536
$855$	0	0
$856$	20.4507	0.698991
$857$	−7.31050	−0.249722	−0.124861	−	0.992174i	$-0.539848\pi$
$857$	−7.31050	−0.249722	−0.124861	+	0.992174i	$0.539848\pi$
$858$	0	0
$859$	−9.66310	−0.329701	−0.164850	−	0.986319i	$-0.552714\pi$
$859$	−9.66310	−0.329701	−0.164850	+	0.986319i	$0.552714\pi$
$860$	−7.74758	−0.264190
$861$	0	0
$862$	−0.302603	−0.0103067
$863$	−3.15525	−0.107406	−0.0537030	−	0.998557i	$-0.517102\pi$
$863$	−3.15525	−0.107406	−0.0537030	+	0.998557i	$0.517102\pi$
$864$	0	0
$865$	−6.25989	−0.212843
$866$	18.7442	0.636953
$867$	0	0
$868$	9.88235	0.335429
$869$	12.8190	0.434854
$870$	0	0
$871$	13.5306	0.458467
$872$	−3.31219	−0.112165
$873$	0	0
$874$	25.4311	0.860221
$875$	25.8747	0.874724
$876$	0	0
$877$	−52.9229	−1.78708	−0.893539	−	0.448986i	$-0.851785\pi$
$877$	−52.9229	−1.78708	−0.893539	+	0.448986i	$0.851785\pi$
$878$	28.5160	0.962369
$879$	0	0
$880$	5.49915	0.185376
$881$	36.7014	1.23650	0.618250	−	0.785981i	$-0.287842\pi$
$881$	36.7014	1.23650	0.618250	+	0.785981i	$0.287842\pi$
$882$	0	0
$883$	29.9103	1.00656	0.503280	−	0.864123i	$-0.332126\pi$
$883$	29.9103	1.00656	0.503280	+	0.864123i	$0.332126\pi$
$884$	−11.7006	−0.393533
$885$	0	0
$886$	29.3879	0.987306
$887$	55.7515	1.87195	0.935976	−	0.352064i	$-0.114520\pi$
$887$	55.7515	1.87195	0.935976	+	0.352064i	$0.114520\pi$
$888$	0	0
$889$	9.91941	0.332686
$890$	−6.89133	−0.230998
$891$	0	0
$892$	18.5252	0.620270
$893$	−37.2121	−1.24526
$894$	0	0
$895$	−8.75573	−0.292672
$896$	23.2950	0.778230
$897$	0	0
$898$	−9.16618	−0.305880
$899$	−26.2998	−0.877146
$900$	0	0
$901$	0.850170	0.0283233
$902$	−17.3604	−0.578038
$903$	0	0
$904$	46.8974	1.55978
$905$	−0.638071	−0.0212102
$906$	0	0
$907$	36.0017	1.19542	0.597708	−	0.801714i	$-0.296078\pi$
$907$	36.0017	1.19542	0.597708	+	0.801714i	$0.296078\pi$
$908$	−17.3986	−0.577393
$909$	0	0
$910$	−6.17917	−0.204838
$911$	−16.3768	−0.542589	−0.271294	−	0.962496i	$-0.587452\pi$
$911$	−16.3768	−0.542589	−0.271294	+	0.962496i	$0.587452\pi$
$912$	0	0
$913$	2.86879	0.0949430
$914$	−15.4426	−0.510795
$915$	0	0
$916$	−35.8013	−1.18291
$917$	0.256387	0.00846665
$918$	0	0
$919$	19.5368	0.644461	0.322231	−	0.946661i	$-0.395567\pi$
$919$	19.5368	0.644461	0.322231	+	0.946661i	$0.395567\pi$
$920$	−25.4416	−0.838785
$921$	0	0
$922$	5.94330	0.195732
$923$	10.9481	0.360360
$924$	0	0
$925$	3.57930	0.117687
$926$	−12.4083	−0.407761
$927$	0	0
$928$	−54.3668	−1.78468
$929$	−10.7511	−0.352733	−0.176367	−	0.984325i	$-0.556434\pi$
$929$	−10.7511	−0.352733	−0.176367	+	0.984325i	$0.556434\pi$
$930$	0	0
$931$	−5.99300	−0.196413
$932$	−7.52873	−0.246612
$933$	0	0
$934$	−20.3304	−0.665232
$935$	46.8392	1.53181
$936$	0	0
$937$	−28.4438	−0.929218	−0.464609	−	0.885516i	$-0.653805\pi$
$937$	−28.4438	−0.929218	−0.464609	+	0.885516i	$0.653805\pi$
$938$	−19.6931	−0.643004
$939$	0	0
$940$	15.2976	0.498952
$941$	−27.7605	−0.904968	−0.452484	−	0.891773i	$-0.649462\pi$
$941$	−27.7605	−0.904968	−0.452484	+	0.891773i	$0.649462\pi$
$942$	0	0
$943$	−28.8110	−0.938216
$944$	−2.94571	−0.0958746
$945$	0	0
$946$	5.70939	0.185628
$947$	−42.4755	−1.38027	−0.690135	−	0.723681i	$-0.742449\pi$
$947$	−42.4755	−1.38027	−0.690135	+	0.723681i	$0.742449\pi$
$948$	0	0
$949$	−16.4599	−0.534309
$950$	4.05827	0.131668
$951$	0	0
$952$	41.4425	1.34316
$953$	−49.1516	−1.59218	−0.796088	−	0.605181i	$-0.793101\pi$
$953$	−49.1516	−1.59218	−0.796088	+	0.605181i	$0.793101\pi$
$954$	0	0
$955$	5.69845	0.184397
$956$	11.6787	0.377717
$957$	0	0
$958$	−30.6040	−0.988770
$959$	19.0594	0.615459
$960$	0	0
$961$	−22.9808	−0.741316
$962$	5.75065	0.185408
$963$	0	0
$964$	0.616913	0.0198694
$965$	1.17915	0.0379582
$966$	0	0
$967$	47.9870	1.54316	0.771579	−	0.636134i	$-0.219467\pi$
$967$	47.9870	1.54316	0.771579	+	0.636134i	$0.219467\pi$
$968$	2.98850	0.0960540
$969$	0	0
$970$	−11.0821	−0.355826
$971$	28.9682	0.929633	0.464817	−	0.885407i	$-0.346120\pi$
$971$	28.9682	0.929633	0.464817	+	0.885407i	$0.346120\pi$
$972$	0	0
$973$	26.1162	0.837247
$974$	9.10979	0.291896
$975$	0	0
$976$	0.251789	0.00805956
$977$	15.1972	0.486201	0.243100	−	0.970001i	$-0.421836\pi$
$977$	15.1972	0.486201	0.243100	+	0.970001i	$0.421836\pi$
$978$	0	0
$979$	−11.7134	−0.374361
$980$	2.46367	0.0786990
$981$	0	0
$982$	29.9815	0.956747
$983$	38.5722	1.23026	0.615131	−	0.788425i	$-0.289103\pi$
$983$	38.5722	1.23026	0.615131	+	0.788425i	$0.289103\pi$
$984$	0	0
$985$	52.6284	1.67688
$986$	−45.3207	−1.44331
$987$	0	0
$988$	−15.0388	−0.478449
$989$	9.47520	0.301294
$990$	0	0
$991$	51.0341	1.62115	0.810576	−	0.585633i	$-0.199154\pi$
$991$	51.0341	1.62115	0.810576	+	0.585633i	$0.199154\pi$
$992$	16.5773	0.526329
$993$	0	0
$994$	−15.9344	−0.505408
$995$	5.06749	0.160650
$996$	0	0
$997$	31.7973	1.00703	0.503515	−	0.863987i	$-0.332040\pi$
$997$	31.7973	1.00703	0.503515	+	0.863987i	$0.332040\pi$
$998$	−22.9910	−0.727768
$999$	0	0

Twists

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

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

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.777732 q^{2} -1.39513 q^{4} +2.37635 q^{5} -2.50138 q^{7} -2.64050 q^{8} +O(q^{10})\)	\(q+0.777732 q^{2} -1.39513 q^{4} +2.37635 q^{5} -2.50138 q^{7} -2.64050 q^{8} +1.84816 q^{10} +3.14137 q^{11} +1.33663 q^{13} -1.94540 q^{14} +0.736659 q^{16} +6.27452 q^{17} +8.06469 q^{19} -3.31532 q^{20} +2.44315 q^{22} +4.05460 q^{23} +0.647028 q^{25} +1.03954 q^{26} +3.48975 q^{28} -9.28723 q^{29} +2.83182 q^{31} +5.85393 q^{32} +4.87990 q^{34} -5.94414 q^{35} +5.53191 q^{37} +6.27217 q^{38} -6.27476 q^{40} -7.10576 q^{41} +2.33690 q^{43} -4.38263 q^{44} +3.15339 q^{46} -4.61421 q^{47} -0.743116 q^{49} +0.503215 q^{50} -1.86478 q^{52} +0.135496 q^{53} +7.46499 q^{55} +6.60490 q^{56} -7.22298 q^{58} -3.99874 q^{59} +0.341798 q^{61} +2.20240 q^{62} +3.07947 q^{64} +3.17630 q^{65} +10.1229 q^{67} -8.75378 q^{68} -4.62295 q^{70} +8.19080 q^{71} -12.3144 q^{73} +4.30235 q^{74} -11.2513 q^{76} -7.85775 q^{77} +4.08070 q^{79} +1.75056 q^{80} -5.52638 q^{82} +0.913228 q^{83} +14.9104 q^{85} +1.81748 q^{86} -8.29480 q^{88} -3.72875 q^{89} -3.34341 q^{91} -5.65670 q^{92} -3.58862 q^{94} +19.1645 q^{95} -5.99630 q^{97} -0.577945 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28} - 21 q^{29} + 15 q^{31} - 30 q^{35} + 3 q^{37} - 15 q^{38} + 3 q^{40} + 12 q^{41} + 6 q^{43} + 33 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 24 q^{50} + 3 q^{52} + 9 q^{53} + 15 q^{55} - 12 q^{56} - 15 q^{58} - 6 q^{59} + 24 q^{61} + 30 q^{62} + 6 q^{64} + 15 q^{65} + 15 q^{67} - 36 q^{68} - 15 q^{70} + 12 q^{73} - 24 q^{74} + 9 q^{76} - 15 q^{77} + 24 q^{79} + 21 q^{80} - 21 q^{82} + 6 q^{83} - 18 q^{85} + 30 q^{86} - 21 q^{88} + 9 q^{89} + 18 q^{91} - 6 q^{92} - 6 q^{94} + 33 q^{95} - 21 q^{97} - 18 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * q^98

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