LMFDB - Embedded newform 729.2.a.e.1.2

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			1.2
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)

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

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	$-0.588668\pi$
$2$	−0.777732	−0.549940	−0.274970	+	0.961453i	$0.588668\pi$
$3$	0	0
$3$	0	0
$4$	−1.39513	−0.697566
$5$	−2.37635	−1.06273	−0.531367	−	0.847141i	$-0.678322\pi$
$5$	−2.37635	−1.06273	−0.531367	+	0.847141i	$0.678322\pi$
$6$	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	$-0.657038\pi$
$11$	−3.14137	−0.947159	−0.473579	+	0.880751i	$0.657038\pi$
$12$	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	$-0.775241\pi$
$17$	−6.27452	−1.52179	−0.760897	+	0.648873i	$0.775241\pi$
$18$	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	$-0.638925\pi$
$23$	−4.05460	−0.845442	−0.422721	+	0.906260i	$0.638925\pi$
$24$	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	$-0.169025\pi$
$29$	9.28723	1.72459	0.862297	+	0.506402i	$0.169025\pi$
$30$	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	$-0.312769\pi$
$41$	7.10576	1.10973	0.554867	+	0.831939i	$0.312769\pi$
$42$	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	$-0.390748\pi$
$47$	4.61421	0.673051	0.336526	+	0.941674i	$0.390748\pi$
$48$	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	$-0.502962\pi$
$53$	−0.135496	−0.0186118	−0.00930588	+	0.999957i	$0.502962\pi$
$54$	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	$-0.416180\pi$
$59$	3.99874	0.520591	0.260296	+	0.965529i	$0.416180\pi$
$60$	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	$-0.661557\pi$
$71$	−8.19080	−0.972069	−0.486035	+	0.873940i	$0.661557\pi$
$72$	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	$-0.515960\pi$
$83$	−0.913228	−0.100240	−0.0501199	+	0.998743i	$0.515960\pi$
$84$	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	$-0.436678\pi$
$89$	3.72875	0.395246	0.197623	+	0.980278i	$0.436678\pi$
$90$	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	$-0.669818\pi$
$101$	−10.2217	−1.01710	−0.508549	+	0.861033i	$0.669818\pi$
$102$	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	$-0.377859\pi$
$107$	7.74500	0.748738	0.374369	+	0.927280i	$0.377859\pi$
$108$	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	$-0.185238\pi$
$113$	17.7608	1.67079	0.835396	+	0.549648i	$0.185238\pi$
$114$	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	$-0.498575\pi$
$131$	0.102498	0.00895533	0.00447766	+	0.999990i	$0.498575\pi$
$132$	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	$-0.394470\pi$
$137$	7.61955	0.650982	0.325491	+	0.945545i	$0.394470\pi$
$138$	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	$-0.620638\pi$
$149$	−9.03258	−0.739978	−0.369989	+	0.929036i	$0.620638\pi$
$150$	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	$-0.387432\pi$
$167$	8.95081	0.692634	0.346317	+	0.938118i	$0.387432\pi$
$168$	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	$-0.468071\pi$
$173$	2.63425	0.200278	0.100139	+	0.994973i	$0.468071\pi$
$174$	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	$-0.456030\pi$
$179$	3.68453	0.275395	0.137697	+	0.990474i	$0.456030\pi$
$180$	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	$-0.527650\pi$
$191$	−2.39799	−0.173512	−0.0867561	+	0.996230i	$0.527650\pi$
$192$	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	$-0.789373\pi$
$197$	−22.1468	−1.57789	−0.788946	+	0.614462i	$0.789373\pi$
$198$	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	$-0.635821\pi$
$227$	−12.4709	−0.827725	−0.413862	+	0.910339i	$0.635821\pi$
$228$	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	$-0.556564\pi$
$233$	−5.39642	−0.353532	−0.176766	+	0.984253i	$0.556564\pi$
$234$	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	$-0.412732\pi$
$239$	8.37104	0.541478	0.270739	+	0.962653i	$0.412732\pi$
$240$	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	$-0.319401\pi$
$251$	17.0285	1.07483	0.537416	+	0.843317i	$0.319401\pi$
$252$	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	$-0.274297\pi$
$257$	20.8767	1.30225	0.651127	+	0.758969i	$0.274297\pi$
$258$	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	$-0.296015\pi$
$263$	19.3916	1.19573	0.597867	+	0.801595i	$0.296015\pi$
$264$	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	$-0.307159\pi$
$269$	18.6791	1.13889	0.569443	+	0.822031i	$0.307159\pi$
$270$	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	$-0.359254\pi$
$281$	14.3458	0.855798	0.427899	+	0.903826i	$0.359254\pi$
$282$	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	$-0.602047\pi$
$293$	−10.7882	−0.630252	−0.315126	+	0.949050i	$0.602047\pi$
$294$	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	$-0.377554\pi$
$311$	13.2355	0.750514	0.375257	+	0.926921i	$0.377554\pi$
$312$	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	$-0.422073\pi$
$317$	8.63082	0.484755	0.242378	+	0.970182i	$0.422073\pi$
$318$	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	$-0.299865\pi$
$347$	21.9112	1.17626	0.588128	+	0.808768i	$0.299865\pi$
$348$	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	$-0.388394\pi$
$353$	12.9069	0.686964	0.343482	+	0.939159i	$0.388394\pi$
$354$	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	$-0.261290\pi$
$359$	25.8285	1.36318	0.681588	+	0.731736i	$0.261290\pi$
$360$	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	$-0.311696\pi$
$383$	21.8275	1.11534	0.557668	+	0.830064i	$0.311696\pi$
$384$	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	$-0.731185\pi$
$389$	−26.1962	−1.32820	−0.664099	+	0.747644i	$0.731185\pi$
$390$	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	$-0.562633\pi$
$401$	−7.82981	−0.391002	−0.195501	+	0.980703i	$0.562633\pi$
$402$	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	$-0.409501\pi$
$419$	11.4832	0.560990	0.280495	+	0.959856i	$0.409501\pi$
$420$	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	$-0.497017\pi$
$431$	0.389084	0.0187415	0.00937075	+	0.999956i	$0.497017\pi$
$432$	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	$-0.854726\pi$
$443$	−37.7867	−1.79530	−0.897649	+	0.440711i	$0.854726\pi$
$444$	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	$-0.410294\pi$
$449$	11.7858	0.556205	0.278103	+	0.960551i	$0.410294\pi$
$450$	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	$-0.556949\pi$
$461$	−7.64183	−0.355915	−0.177958	+	0.984038i	$0.556949\pi$
$462$	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	$-0.293244\pi$
$467$	26.1406	1.20964	0.604822	+	0.796360i	$0.293244\pi$
$468$	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	$-0.144309\pi$
$479$	39.3503	1.79796	0.898980	+	0.437989i	$0.144309\pi$
$480$	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	$-0.835794\pi$
$491$	−38.5498	−1.73973	−0.869865	+	0.493290i	$0.835794\pi$
$492$	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	$-0.208434\pi$
$503$	35.5775	1.58632	0.793161	+	0.609011i	$0.208434\pi$
$504$	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	$-0.716541\pi$
$509$	−28.3823	−1.25803	−0.629013	+	0.777395i	$0.716541\pi$
$510$	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	$-0.688112\pi$
$521$	−25.4351	−1.11433	−0.557167	+	0.830401i	$0.688112\pi$
$522$	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	$-0.469051\pi$
$557$	4.58220	0.194154	0.0970769	+	0.995277i	$0.469051\pi$
$558$	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	$-0.416656\pi$
$563$	12.2839	0.517705	0.258853	+	0.965917i	$0.416656\pi$
$564$	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	$-0.401744\pi$
$569$	14.4937	0.607606	0.303803	+	0.952735i	$0.401744\pi$
$570$	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	$-0.595835\pi$
$587$	−14.3694	−0.593090	−0.296545	+	0.955019i	$0.595835\pi$
$588$	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	$-0.819724\pi$
$593$	−41.0988	−1.68772	−0.843862	+	0.536560i	$0.819724\pi$
$594$	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	$-0.444033\pi$
$599$	8.56216	0.349840	0.174920	+	0.984583i	$0.444033\pi$
$600$	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	$-0.540301\pi$
$617$	−6.27303	−0.252542	−0.126271	+	0.991996i	$0.540301\pi$
$618$	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	$-0.733645\pi$
$641$	−33.9189	−1.33972	−0.669858	+	0.742490i	$0.733645\pi$
$642$	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	$-0.256882\pi$
$647$	35.1862	1.38331	0.691655	+	0.722228i	$0.256882\pi$
$648$	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	$-0.375311\pi$
$653$	19.5120	0.763562	0.381781	+	0.924253i	$0.375311\pi$
$654$	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	$-0.351545\pi$
$659$	23.0864	0.899320	0.449660	+	0.893200i	$0.351545\pi$
$660$	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	$-0.584065\pi$
$677$	−13.5840	−0.522074	−0.261037	+	0.965329i	$0.584065\pi$
$678$	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	$-0.537031\pi$
$683$	−6.06700	−0.232147	−0.116074	+	0.993241i	$0.537031\pi$
$684$	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	$-0.433255\pi$
$701$	11.0222	0.416303	0.208151	+	0.978097i	$0.433255\pi$
$702$	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	$-0.291226\pi$
$719$	32.7057	1.21972	0.609859	+	0.792510i	$0.291226\pi$
$720$	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	$-0.630742\pi$
$743$	−21.7676	−0.798574	−0.399287	+	0.916826i	$0.630742\pi$
$744$	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	$-0.853676\pi$
$761$	−49.4450	−1.79238	−0.896190	+	0.443670i	$0.853676\pi$
$762$	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	$-0.370727\pi$
$773$	21.9671	0.790102	0.395051	+	0.918659i	$0.370727\pi$
$774$	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	$-0.248609\pi$
$797$	40.0990	1.42038	0.710189	+	0.704011i	$0.248609\pi$
$798$	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	$-0.402327\pi$
$809$	17.1826	0.604110	0.302055	+	0.953291i	$0.402327\pi$
$810$	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	$-0.284461\pi$
$821$	35.9059	1.25313	0.626563	+	0.779371i	$0.284461\pi$
$822$	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	$-0.642880\pi$
$827$	−24.9586	−0.867895	−0.433948	+	0.900938i	$0.642880\pi$
$828$	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	$-0.780615\pi$
$839$	−44.7079	−1.54349	−0.771744	+	0.635934i	$0.780615\pi$
$840$	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	$-0.460152\pi$
$857$	7.31050	0.249722	0.124861	+	0.992174i	$0.460152\pi$
$858$	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	$-0.482898\pi$
$863$	3.15525	0.107406	0.0537030	+	0.998557i	$0.482898\pi$
$864$	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	$-0.712158\pi$
$881$	−36.7014	−1.23650	−0.618250	+	0.785981i	$0.712158\pi$
$882$	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	$-0.885480\pi$
$887$	−55.7515	−1.87195	−0.935976	+	0.352064i	$0.885480\pi$
$888$	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	$-0.412548\pi$
$911$	16.3768	0.542589	0.271294	+	0.962496i	$0.412548\pi$
$912$	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	$-0.443566\pi$
$929$	10.7511	0.352733	0.176367	+	0.984325i	$0.443566\pi$
$930$	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	$-0.350538\pi$
$941$	27.7605	0.904968	0.452484	+	0.891773i	$0.350538\pi$
$942$	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	$-0.257551\pi$
$947$	42.4755	1.38027	0.690135	+	0.723681i	$0.257551\pi$
$948$	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	$-0.206899\pi$
$953$	49.1516	1.59218	0.796088	+	0.605181i	$0.206899\pi$
$954$	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	$-0.653880\pi$
$971$	−28.9682	−0.929633	−0.464817	+	0.885407i	$0.653880\pi$
$972$	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	$-0.578164\pi$
$977$	−15.1972	−0.486201	−0.243100	+	0.970001i	$0.578164\pi$
$978$	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	$-0.710897\pi$
$983$	−38.5722	−1.23026	−0.615131	+	0.788425i	$0.710897\pi$
$984$	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.e.1.2	yes	6
3.2	odd	2		729.2.a.b.1.5	✓	6
9.2	odd	6		729.2.c.d.244.2		12
9.4	even	3		729.2.c.a.487.5		12
9.5	odd	6		729.2.c.d.487.2		12
9.7	even	3		729.2.c.a.244.5		12
27.2	odd	18		729.2.e.s.568.1		12
27.4	even	9		729.2.e.k.406.1		12
27.5	odd	18		729.2.e.j.649.2		12
27.7	even	9		729.2.e.k.325.1		12
27.11	odd	18		729.2.e.j.82.2		12
27.13	even	9		729.2.e.l.163.2		12
27.14	odd	18		729.2.e.s.163.1		12
27.16	even	9		729.2.e.u.82.1		12
27.20	odd	18		729.2.e.t.325.2		12
27.22	even	9		729.2.e.u.649.1		12
27.23	odd	18		729.2.e.t.406.2		12
27.25	even	9		729.2.e.l.568.2		12

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

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

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

Introduction

L-functions

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