LMFDB - Embedded newform 729.2.a.e.1.4

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 = [6,3,0,9] 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:	$0$
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			1.4
Root			$3.45779$ 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)

$f(q)$	$=$	\(q+1.57840 q^{2} +0.491360 q^{4} -1.67851 q^{5} +2.77928 q^{7} -2.38124 q^{8} -2.64936 q^{10} +4.15122 q^{11} +6.87605 q^{13} +4.38683 q^{14} -4.74129 q^{16} -0.976551 q^{17} +2.68529 q^{19} -0.824751 q^{20} +6.55231 q^{22} -1.61317 q^{23} -2.18261 q^{25} +10.8532 q^{26} +1.36563 q^{28} +8.22788 q^{29} +1.04407 q^{31} -2.72118 q^{32} -1.54139 q^{34} -4.66505 q^{35} -1.30834 q^{37} +4.23847 q^{38} +3.99694 q^{40} +4.84817 q^{41} +9.84023 q^{43} +2.03974 q^{44} -2.54623 q^{46} -12.4977 q^{47} +0.724408 q^{49} -3.44504 q^{50} +3.37861 q^{52} -7.34280 q^{53} -6.96786 q^{55} -6.61815 q^{56} +12.9869 q^{58} -9.05188 q^{59} -1.28574 q^{61} +1.64796 q^{62} +5.18745 q^{64} -11.5415 q^{65} -4.64630 q^{67} -0.479838 q^{68} -7.36333 q^{70} -5.62373 q^{71} -4.56144 q^{73} -2.06510 q^{74} +1.31944 q^{76} +11.5374 q^{77} +4.65680 q^{79} +7.95828 q^{80} +7.65237 q^{82} -5.76439 q^{83} +1.63915 q^{85} +15.5319 q^{86} -9.88507 q^{88} -4.54442 q^{89} +19.1105 q^{91} -0.792647 q^{92} -19.7264 q^{94} -4.50728 q^{95} +8.57544 q^{97} +1.14341 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.57840	1.11610	0.558050	−	0.829807i	$-0.311550\pi$
$2$	1.57840	1.11610	0.558050	+	0.829807i	$0.311550\pi$
$3$	0	0
$3$	0	0
$4$	0.491360	0.245680
$5$	−1.67851	−0.750651	−0.375326	−	0.926893i	$-0.622469\pi$
$5$	−1.67851	−0.750651	−0.375326	+	0.926893i	$0.622469\pi$
$6$	0	0
$7$	2.77928	1.05047	0.525235	−	0.850957i	$-0.323977\pi$
$7$	2.77928	1.05047	0.525235	+	0.850957i	$0.323977\pi$
$8$	−2.38124	−0.841897
$9$	0	0
$10$	−2.64936	−0.837802
$11$	4.15122	1.25164	0.625820	−	0.779967i	$-0.284764\pi$
$11$	4.15122	1.25164	0.625820	+	0.779967i	$0.284764\pi$
$12$	0	0
$13$	6.87605	1.90707	0.953536	−	0.301279i	$-0.0974137\pi$
$13$	6.87605	1.90707	0.953536	+	0.301279i	$0.0974137\pi$
$14$	4.38683	1.17243
$15$	0	0
$16$	−4.74129	−1.18532
$17$	−0.976551	−0.236848	−0.118424	−	0.992963i	$-0.537784\pi$
$17$	−0.976551	−0.236848	−0.118424	+	0.992963i	$0.537784\pi$
$18$	0	0
$19$	2.68529	0.616048	0.308024	−	0.951379i	$-0.400332\pi$
$19$	2.68529	0.616048	0.308024	+	0.951379i	$0.400332\pi$
$20$	−0.824751	−0.184420
$21$	0	0
$22$	6.55231	1.39696
$23$	−1.61317	−0.336369	−0.168185	−	0.985756i	$-0.553790\pi$
$23$	−1.61317	−0.336369	−0.168185	+	0.985756i	$0.553790\pi$
$24$	0	0
$25$	−2.18261	−0.436522
$26$	10.8532	2.12848
$27$	0	0
$28$	1.36563	0.258079
$29$	8.22788	1.52788	0.763939	−	0.645288i	$-0.223263\pi$
$29$	8.22788	1.52788	0.763939	+	0.645288i	$0.223263\pi$
$30$	0	0
$31$	1.04407	0.187520	0.0937602	−	0.995595i	$-0.470111\pi$
$31$	1.04407	0.187520	0.0937602	+	0.995595i	$0.470111\pi$
$32$	−2.72118	−0.481041
$33$	0	0
$34$	−1.54139	−0.264347
$35$	−4.66505	−0.788537
$36$	0	0
$37$	−1.30834	−0.215091	−0.107545	−	0.994200i	$-0.534299\pi$
$37$	−1.30834	−0.215091	−0.107545	+	0.994200i	$0.534299\pi$
$38$	4.23847	0.687571
$39$	0	0
$40$	3.99694	0.631971
$41$	4.84817	0.757157	0.378578	−	0.925569i	$-0.376413\pi$
$41$	4.84817	0.757157	0.378578	+	0.925569i	$0.376413\pi$
$42$	0	0
$43$	9.84023	1.50062	0.750310	−	0.661086i	$-0.229904\pi$
$43$	9.84023	1.50062	0.750310	+	0.661086i	$0.229904\pi$
$44$	2.03974	0.307503
$45$	0	0
$46$	−2.54623	−0.375422
$47$	−12.4977	−1.82298	−0.911488	−	0.411326i	$-0.865066\pi$
$47$	−12.4977	−1.82298	−0.911488	+	0.411326i	$0.865066\pi$
$48$	0	0
$49$	0.724408	0.103487
$50$	−3.44504	−0.487203
$51$	0	0
$52$	3.37861	0.468529
$53$	−7.34280	−1.00861	−0.504305	−	0.863525i	$-0.668251\pi$
$53$	−7.34280	−1.00861	−0.504305	+	0.863525i	$0.668251\pi$
$54$	0	0
$55$	−6.96786	−0.939546
$56$	−6.61815	−0.884387
$57$	0	0
$58$	12.9869	1.70527
$59$	−9.05188	−1.17845	−0.589227	−	0.807968i	$-0.700568\pi$
$59$	−9.05188	−1.17845	−0.589227	+	0.807968i	$0.700568\pi$
$60$	0	0
$61$	−1.28574	−0.164622	−0.0823112	−	0.996607i	$-0.526230\pi$
$61$	−1.28574	−0.164622	−0.0823112	+	0.996607i	$0.526230\pi$
$62$	1.64796	0.209292
$63$	0	0
$64$	5.18745	0.648432
$65$	−11.5415	−1.43155
$66$	0	0
$67$	−4.64630	−0.567636	−0.283818	−	0.958878i	$-0.591601\pi$
$67$	−4.64630	−0.567636	−0.283818	+	0.958878i	$0.591601\pi$
$68$	−0.479838	−0.0581889
$69$	0	0
$70$	−7.36333	−0.880086
$71$	−5.62373	−0.667414	−0.333707	−	0.942677i	$-0.608300\pi$
$71$	−5.62373	−0.667414	−0.333707	+	0.942677i	$0.608300\pi$
$72$	0	0
$73$	−4.56144	−0.533877	−0.266938	−	0.963714i	$-0.586012\pi$
$73$	−4.56144	−0.533877	−0.266938	+	0.963714i	$0.586012\pi$
$74$	−2.06510	−0.240063
$75$	0	0
$76$	1.31944	0.151351
$77$	11.5374	1.31481
$78$	0	0
$79$	4.65680	0.523931	0.261966	−	0.965077i	$-0.415629\pi$
$79$	4.65680	0.523931	0.261966	+	0.965077i	$0.415629\pi$
$80$	7.95828	0.889763
$81$	0	0
$82$	7.65237	0.845063
$83$	−5.76439	−0.632724	−0.316362	−	0.948639i	$-0.602461\pi$
$83$	−5.76439	−0.632724	−0.316362	+	0.948639i	$0.602461\pi$
$84$	0	0
$85$	1.63915	0.177791
$86$	15.5319	1.67484
$87$	0	0
$88$	−9.88507	−1.05375
$89$	−4.54442	−0.481707	−0.240854	−	0.970561i	$-0.577427\pi$
$89$	−4.54442	−0.481707	−0.240854	+	0.970561i	$0.577427\pi$
$90$	0	0
$91$	19.1105	2.00332
$92$	−0.792647	−0.0826391
$93$	0	0
$94$	−19.7264	−2.03463
$95$	−4.50728	−0.462437
$96$	0	0
$97$	8.57544	0.870704	0.435352	−	0.900260i	$-0.356624\pi$
$97$	8.57544	0.870704	0.435352	+	0.900260i	$0.356624\pi$
$98$	1.14341	0.115502
$99$	0	0
$100$	−1.07245	−0.107245
$101$	7.80570	0.776696	0.388348	−	0.921513i	$-0.373046\pi$
$101$	7.80570	0.776696	0.388348	+	0.921513i	$0.373046\pi$
$102$	0	0
$103$	−2.16615	−0.213437	−0.106718	−	0.994289i	$-0.534034\pi$
$103$	−2.16615	−0.213437	−0.106718	+	0.994289i	$0.534034\pi$
$104$	−16.3735	−1.60556
$105$	0	0
$106$	−11.5899	−1.12571
$107$	−12.5849	−1.21663	−0.608317	−	0.793695i	$-0.708155\pi$
$107$	−12.5849	−1.21663	−0.608317	+	0.793695i	$0.708155\pi$
$108$	0	0
$109$	−12.2140	−1.16989	−0.584945	−	0.811073i	$-0.698884\pi$
$109$	−12.2140	−1.16989	−0.584945	+	0.811073i	$0.698884\pi$
$110$	−10.9981	−1.04863
$111$	0	0
$112$	−13.1774	−1.24514
$113$	−0.450833	−0.0424108	−0.0212054	−	0.999775i	$-0.506750\pi$
$113$	−0.450833	−0.0424108	−0.0212054	+	0.999775i	$0.506750\pi$
$114$	0	0
$115$	2.70772	0.252496
$116$	4.04285	0.375369
$117$	0	0
$118$	−14.2875	−1.31527
$119$	−2.71411	−0.248802
$120$	0	0
$121$	6.23265	0.566604
$122$	−2.02942	−0.183735
$123$	0	0
$124$	0.513014	0.0460700
$125$	12.0561	1.07833
$126$	0	0
$127$	−0.531069	−0.0471247	−0.0235624	−	0.999722i	$-0.507501\pi$
$127$	−0.531069	−0.0471247	−0.0235624	+	0.999722i	$0.507501\pi$
$128$	13.6303	1.20476
$129$	0	0
$130$	−18.2171	−1.59775
$131$	11.4160	0.997424	0.498712	−	0.866768i	$-0.333807\pi$
$131$	11.4160	0.997424	0.498712	+	0.866768i	$0.333807\pi$
$132$	0	0
$133$	7.46318	0.647139
$134$	−7.33374	−0.633539
$135$	0	0
$136$	2.32541	0.199402
$137$	4.22924	0.361329	0.180664	−	0.983545i	$-0.442175\pi$
$137$	4.22924	0.361329	0.180664	+	0.983545i	$0.442175\pi$
$138$	0	0
$139$	−11.1555	−0.946200	−0.473100	−	0.881009i	$-0.656865\pi$
$139$	−11.1555	−0.946200	−0.473100	+	0.881009i	$0.656865\pi$
$140$	−2.29222	−0.193728
$141$	0	0
$142$	−8.87653	−0.744902
$143$	28.5440	2.38697
$144$	0	0
$145$	−13.8106	−1.14690
$146$	−7.19980	−0.595860
$147$	0	0
$148$	−0.642868	−0.0528434
$149$	−19.4777	−1.59567	−0.797837	−	0.602873i	$-0.794023\pi$
$149$	−19.4777	−1.59567	−0.797837	+	0.602873i	$0.794023\pi$
$150$	0	0
$151$	−1.24286	−0.101143	−0.0505713	−	0.998720i	$-0.516104\pi$
$151$	−1.24286	−0.101143	−0.0505713	+	0.998720i	$0.516104\pi$
$152$	−6.39433	−0.518649
$153$	0	0
$154$	18.2107	1.46746
$155$	−1.75248	−0.140762
$156$	0	0
$157$	−3.53314	−0.281975	−0.140988	−	0.990011i	$-0.545028\pi$
$157$	−3.53314	−0.281975	−0.140988	+	0.990011i	$0.545028\pi$
$158$	7.35031	0.584760
$159$	0	0
$160$	4.56752	0.361094
$161$	−4.48345	−0.353346
$162$	0	0
$163$	15.9509	1.24937	0.624685	−	0.780877i	$-0.285228\pi$
$163$	15.9509	1.24937	0.624685	+	0.780877i	$0.285228\pi$
$164$	2.38220	0.186018
$165$	0	0
$166$	−9.09854	−0.706184
$167$	14.4846	1.12086	0.560428	−	0.828203i	$-0.310637\pi$
$167$	14.4846	1.12086	0.560428	+	0.828203i	$0.310637\pi$
$168$	0	0
$169$	34.2800	2.63692
$170$	2.58724	0.198432
$171$	0	0
$172$	4.83509	0.368672
$173$	12.6174	0.959283	0.479641	−	0.877465i	$-0.340767\pi$
$173$	12.6174	0.959283	0.479641	+	0.877465i	$0.340767\pi$
$174$	0	0
$175$	−6.06609	−0.458554
$176$	−19.6821	−1.48360
$177$	0	0
$178$	−7.17293	−0.537634
$179$	0.295899	0.0221165	0.0110582	−	0.999939i	$-0.496480\pi$
$179$	0.295899	0.0221165	0.0110582	+	0.999939i	$0.496480\pi$
$180$	0	0
$181$	1.42050	0.105585	0.0527925	−	0.998606i	$-0.483188\pi$
$181$	1.42050	0.105585	0.0527925	+	0.998606i	$0.483188\pi$
$182$	30.1640	2.23591
$183$	0	0
$184$	3.84135	0.283188
$185$	2.19607	0.161458
$186$	0	0
$187$	−4.05388	−0.296449
$188$	−6.14087	−0.447869
$189$	0	0
$190$	−7.11431	−0.516126
$191$	20.6241	1.49231	0.746153	−	0.665774i	$-0.231899\pi$
$191$	20.6241	1.49231	0.746153	+	0.665774i	$0.231899\pi$
$192$	0	0
$193$	−20.9559	−1.50844	−0.754221	−	0.656621i	$-0.771985\pi$
$193$	−20.9559	−1.50844	−0.754221	+	0.656621i	$0.771985\pi$
$194$	13.5355	0.971793
$195$	0	0
$196$	0.355945	0.0254246
$197$	−9.59621	−0.683702	−0.341851	−	0.939754i	$-0.611054\pi$
$197$	−9.59621	−0.683702	−0.341851	+	0.939754i	$0.611054\pi$
$198$	0	0
$199$	−10.6917	−0.757912	−0.378956	−	0.925415i	$-0.623717\pi$
$199$	−10.6917	−0.757912	−0.378956	+	0.925415i	$0.623717\pi$
$200$	5.19733	0.367507
$201$	0	0
$202$	12.3206	0.866871
$203$	22.8676	1.60499
$204$	0	0
$205$	−8.13769	−0.568361
$206$	−3.41906	−0.238217
$207$	0	0
$208$	−32.6013	−2.26049
$209$	11.1472	0.771070
$210$	0	0
$211$	−15.0502	−1.03610	−0.518051	−	0.855350i	$-0.673342\pi$
$211$	−15.0502	−1.03610	−0.518051	+	0.855350i	$0.673342\pi$
$212$	−3.60796	−0.247795
$213$	0	0
$214$	−19.8641	−1.35788
$215$	−16.5169	−1.12644
$216$	0	0
$217$	2.90176	0.196984
$218$	−19.2786	−1.30571
$219$	0	0
$220$	−3.42373	−0.230828
$221$	−6.71481	−0.451687
$222$	0	0
$223$	−12.2881	−0.822873	−0.411436	−	0.911439i	$-0.634973\pi$
$223$	−12.2881	−0.822873	−0.411436	+	0.911439i	$0.634973\pi$
$224$	−7.56292	−0.505319
$225$	0	0
$226$	−0.711597	−0.0473347
$227$	3.75256	0.249066	0.124533	−	0.992215i	$-0.460257\pi$
$227$	3.75256	0.249066	0.124533	+	0.992215i	$0.460257\pi$
$228$	0	0
$229$	−18.6024	−1.22928	−0.614641	−	0.788807i	$-0.710699\pi$
$229$	−18.6024	−1.22928	−0.614641	+	0.788807i	$0.710699\pi$
$230$	4.27387	0.281811
$231$	0	0
$232$	−19.5926	−1.28632
$233$	0.545784	0.0357555	0.0178777	−	0.999840i	$-0.494309\pi$
$233$	0.545784	0.0357555	0.0178777	+	0.999840i	$0.494309\pi$
$234$	0	0
$235$	20.9775	1.36842
$236$	−4.44773	−0.289522
$237$	0	0
$238$	−4.28396	−0.277688
$239$	−20.0947	−1.29982	−0.649911	−	0.760011i	$-0.725194\pi$
$239$	−20.0947	−1.29982	−0.649911	+	0.760011i	$0.725194\pi$
$240$	0	0
$241$	15.2913	0.984999	0.492500	−	0.870313i	$-0.336083\pi$
$241$	15.2913	0.984999	0.492500	+	0.870313i	$0.336083\pi$
$242$	9.83763	0.632387
$243$	0	0
$244$	−0.631762	−0.0404444
$245$	−1.21592	−0.0776825
$246$	0	0
$247$	18.4642	1.17485
$248$	−2.48618	−0.157873
$249$	0	0
$250$	19.0294	1.20352
$251$	12.7563	0.805171	0.402586	−	0.915382i	$-0.368112\pi$
$251$	12.7563	0.805171	0.402586	+	0.915382i	$0.368112\pi$
$252$	0	0
$253$	−6.69663	−0.421013
$254$	−0.838241	−0.0525959
$255$	0	0
$256$	11.1391	0.696196
$257$	−13.1047	−0.817449	−0.408725	−	0.912658i	$-0.634026\pi$
$257$	−13.1047	−0.817449	−0.408725	+	0.912658i	$0.634026\pi$
$258$	0	0
$259$	−3.63626	−0.225946
$260$	−5.67103	−0.351702
$261$	0	0
$262$	18.0191	1.11323
$263$	−9.51418	−0.586669	−0.293335	−	0.956010i	$-0.594765\pi$
$263$	−9.51418	−0.586669	−0.293335	+	0.956010i	$0.594765\pi$
$264$	0	0
$265$	12.3249	0.757115
$266$	11.7799	0.722272
$267$	0	0
$268$	−2.28301	−0.139457
$269$	22.1408	1.34995	0.674973	−	0.737842i	$-0.264155\pi$
$269$	22.1408	1.34995	0.674973	+	0.737842i	$0.264155\pi$
$270$	0	0
$271$	27.9627	1.69861	0.849307	−	0.527899i	$-0.177020\pi$
$271$	27.9627	1.69861	0.849307	+	0.527899i	$0.177020\pi$
$272$	4.63011	0.280742
$273$	0	0
$274$	6.67545	0.403279
$275$	−9.06051	−0.546369
$276$	0	0
$277$	−18.8837	−1.13461	−0.567305	−	0.823508i	$-0.692014\pi$
$277$	−18.8837	−1.13461	−0.567305	+	0.823508i	$0.692014\pi$
$278$	−17.6079	−1.05605
$279$	0	0
$280$	11.1086	0.663867
$281$	−20.0017	−1.19320	−0.596602	−	0.802537i	$-0.703483\pi$
$281$	−20.0017	−1.19320	−0.596602	+	0.802537i	$0.703483\pi$
$282$	0	0
$283$	16.7450	0.995389	0.497695	−	0.867352i	$-0.334180\pi$
$283$	16.7450	0.995389	0.497695	+	0.867352i	$0.334180\pi$
$284$	−2.76328	−0.163970
$285$	0	0
$286$	45.0540	2.66410
$287$	13.4744	0.795371
$288$	0	0
$289$	−16.0463	−0.943903
$290$	−21.7986	−1.28006
$291$	0	0
$292$	−2.24131	−0.131163
$293$	19.5720	1.14341	0.571704	−	0.820460i	$-0.306282\pi$
$293$	19.5720	1.14341	0.571704	+	0.820460i	$0.306282\pi$
$294$	0	0
$295$	15.1936	0.884608
$296$	3.11549	0.181084
$297$	0	0
$298$	−30.7437	−1.78093
$299$	−11.0922	−0.641480
$300$	0	0
$301$	27.3488	1.57636
$302$	−1.96174	−0.112885
$303$	0	0
$304$	−12.7317	−0.730214
$305$	2.15813	0.123574
$306$	0	0
$307$	14.8995	0.850357	0.425179	−	0.905109i	$-0.360211\pi$
$307$	14.8995	0.850357	0.425179	+	0.905109i	$0.360211\pi$
$308$	5.66902	0.323023
$309$	0	0
$310$	−2.76612	−0.157105
$311$	4.59236	0.260409	0.130204	−	0.991487i	$-0.458437\pi$
$311$	4.59236	0.260409	0.130204	+	0.991487i	$0.458437\pi$
$312$	0	0
$313$	11.8687	0.670858	0.335429	−	0.942066i	$-0.391119\pi$
$313$	11.8687	0.670858	0.335429	+	0.942066i	$0.391119\pi$
$314$	−5.57672	−0.314713
$315$	0	0
$316$	2.28817	0.128719
$317$	−14.5173	−0.815375	−0.407688	−	0.913121i	$-0.633665\pi$
$317$	−14.5173	−0.815375	−0.407688	+	0.913121i	$0.633665\pi$
$318$	0	0
$319$	34.1557	1.91235
$320$	−8.70718	−0.486746
$321$	0	0
$322$	−7.07670	−0.394369
$323$	−2.62232	−0.145910
$324$	0	0
$325$	−15.0077	−0.832480
$326$	25.1769	1.39442
$327$	0	0
$328$	−11.5447	−0.637448
$329$	−34.7346	−1.91498
$330$	0	0
$331$	−8.10570	−0.445530	−0.222765	−	0.974872i	$-0.571508\pi$
$331$	−8.10570	−0.445530	−0.222765	+	0.974872i	$0.571508\pi$
$332$	−2.83239	−0.155448
$333$	0	0
$334$	22.8626	1.25099
$335$	7.79885	0.426097
$336$	0	0
$337$	−18.8080	−1.02453	−0.512267	−	0.858826i	$-0.671194\pi$
$337$	−18.8080	−1.02453	−0.512267	+	0.858826i	$0.671194\pi$
$338$	54.1077	2.94307
$339$	0	0
$340$	0.805412	0.0436796
$341$	4.33416	0.234708
$342$	0	0
$343$	−17.4416	−0.941760
$344$	−23.4320	−1.26337
$345$	0	0
$346$	19.9154	1.07066
$347$	−17.6176	−0.945764	−0.472882	−	0.881126i	$-0.656786\pi$
$347$	−17.6176	−0.945764	−0.472882	+	0.881126i	$0.656786\pi$
$348$	0	0
$349$	−16.9893	−0.909416	−0.454708	−	0.890641i	$-0.650256\pi$
$349$	−16.9893	−0.909416	−0.454708	+	0.890641i	$0.650256\pi$
$350$	−9.57475	−0.511792
$351$	0	0
$352$	−11.2962	−0.602090
$353$	−15.1556	−0.806651	−0.403326	−	0.915057i	$-0.632146\pi$
$353$	−15.1556	−0.806651	−0.403326	+	0.915057i	$0.632146\pi$
$354$	0	0
$355$	9.43948	0.500996
$356$	−2.23294	−0.118346
$357$	0	0
$358$	0.467048	0.0246842
$359$	2.45096	0.129357	0.0646783	−	0.997906i	$-0.479398\pi$
$359$	2.45096	0.129357	0.0646783	+	0.997906i	$0.479398\pi$
$360$	0	0
$361$	−11.7892	−0.620485
$362$	2.24213	0.117844
$363$	0	0
$364$	9.39012	0.492176
$365$	7.65642	0.400755
$366$	0	0
$367$	−1.31353	−0.0685659	−0.0342829	−	0.999412i	$-0.510915\pi$
$367$	−1.31353	−0.0685659	−0.0342829	+	0.999412i	$0.510915\pi$
$368$	7.64850	0.398706
$369$	0	0
$370$	3.46628	0.180203
$371$	−20.4077	−1.05952
$372$	0	0
$373$	−9.62617	−0.498424	−0.249212	−	0.968449i	$-0.580172\pi$
$373$	−9.62617	−0.498424	−0.249212	+	0.968449i	$0.580172\pi$
$374$	−6.39866	−0.330867
$375$	0	0
$376$	29.7601	1.53476
$377$	56.5753	2.91377
$378$	0	0
$379$	−8.56311	−0.439857	−0.219929	−	0.975516i	$-0.570582\pi$
$379$	−8.56311	−0.439857	−0.219929	+	0.975516i	$0.570582\pi$
$380$	−2.21470	−0.113611
$381$	0	0
$382$	32.5531	1.66556
$383$	33.6346	1.71865	0.859324	−	0.511431i	$-0.170884\pi$
$383$	33.6346	1.71865	0.859324	+	0.511431i	$0.170884\pi$
$384$	0	0
$385$	−19.3656	−0.986964
$386$	−33.0769	−1.68357
$387$	0	0
$388$	4.21363	0.213915
$389$	−14.9540	−0.758197	−0.379098	−	0.925356i	$-0.623766\pi$
$389$	−14.9540	−0.758197	−0.379098	+	0.925356i	$0.623766\pi$
$390$	0	0
$391$	1.57534	0.0796685
$392$	−1.72499	−0.0871252
$393$	0	0
$394$	−15.1467	−0.763080
$395$	−7.81648	−0.393290
$396$	0	0
$397$	−16.7788	−0.842102	−0.421051	−	0.907037i	$-0.638339\pi$
$397$	−16.7788	−0.842102	−0.421051	+	0.907037i	$0.638339\pi$
$398$	−16.8758	−0.845906
$399$	0	0
$400$	10.3484	0.517419
$401$	13.1263	0.655496	0.327748	−	0.944765i	$-0.393710\pi$
$401$	13.1263	0.655496	0.327748	+	0.944765i	$0.393710\pi$
$402$	0	0
$403$	7.17907	0.357615
$404$	3.83541	0.190819
$405$	0	0
$406$	36.0943	1.79133
$407$	−5.43123	−0.269216
$408$	0	0
$409$	25.4505	1.25845	0.629223	−	0.777225i	$-0.283373\pi$
$409$	25.4505	1.25845	0.629223	+	0.777225i	$0.283373\pi$
$410$	−12.8446	−0.634348
$411$	0	0
$412$	−1.06436	−0.0524371
$413$	−25.1577	−1.23793
$414$	0	0
$415$	9.67558	0.474955
$416$	−18.7109	−0.917379
$417$	0	0
$418$	17.5948	0.860592
$419$	−12.6701	−0.618973	−0.309486	−	0.950904i	$-0.600157\pi$
$419$	−12.6701	−0.618973	−0.309486	+	0.950904i	$0.600157\pi$
$420$	0	0
$421$	−17.6772	−0.861536	−0.430768	−	0.902463i	$-0.641757\pi$
$421$	−17.6772	−0.861536	−0.430768	+	0.902463i	$0.641757\pi$
$422$	−23.7554	−1.15639
$423$	0	0
$424$	17.4850	0.849146
$425$	2.13143	0.103390
$426$	0	0
$427$	−3.57344	−0.172931
$428$	−6.18374	−0.298902
$429$	0	0
$430$	−26.0703	−1.25722
$431$	15.6974	0.756117	0.378059	−	0.925782i	$-0.376592\pi$
$431$	15.6974	0.756117	0.378059	+	0.925782i	$0.376592\pi$
$432$	0	0
$433$	−12.6258	−0.606759	−0.303380	−	0.952870i	$-0.598115\pi$
$433$	−12.6258	−0.606759	−0.303380	+	0.952870i	$0.598115\pi$
$434$	4.58015	0.219854
$435$	0	0
$436$	−6.00147	−0.287418
$437$	−4.33183	−0.207219
$438$	0	0
$439$	26.9369	1.28563	0.642814	−	0.766022i	$-0.277767\pi$
$439$	26.9369	1.28563	0.642814	+	0.766022i	$0.277767\pi$
$440$	16.5922	0.791001
$441$	0	0
$442$	−10.5987	−0.504128
$443$	34.7279	1.64997	0.824986	−	0.565153i	$-0.191183\pi$
$443$	34.7279	1.64997	0.824986	+	0.565153i	$0.191183\pi$
$444$	0	0
$445$	7.62784	0.361594
$446$	−19.3956	−0.918408
$447$	0	0
$448$	14.4174	0.681158
$449$	−20.7461	−0.979070	−0.489535	−	0.871984i	$-0.662834\pi$
$449$	−20.7461	−0.979070	−0.489535	+	0.871984i	$0.662834\pi$
$450$	0	0
$451$	20.1258	0.947688
$452$	−0.221521	−0.0104195
$453$	0	0
$454$	5.92306	0.277983
$455$	−32.0771	−1.50380
$456$	0	0
$457$	−7.20773	−0.337164	−0.168582	−	0.985688i	$-0.553919\pi$
$457$	−7.20773	−0.337164	−0.168582	+	0.985688i	$0.553919\pi$
$458$	−29.3621	−1.37200
$459$	0	0
$460$	1.33046	0.0620332
$461$	23.1268	1.07712	0.538562	−	0.842586i	$-0.318968\pi$
$461$	23.1268	1.07712	0.538562	+	0.842586i	$0.318968\pi$
$462$	0	0
$463$	4.97584	0.231247	0.115623	−	0.993293i	$-0.463113\pi$
$463$	4.97584	0.231247	0.115623	+	0.993293i	$0.463113\pi$
$464$	−39.0107	−1.81103
$465$	0	0
$466$	0.861467	0.0399067
$467$	−12.4814	−0.577569	−0.288784	−	0.957394i	$-0.593251\pi$
$467$	−12.4814	−0.577569	−0.288784	+	0.957394i	$0.593251\pi$
$468$	0	0
$469$	−12.9134	−0.596284
$470$	33.1110	1.52729
$471$	0	0
$472$	21.5547	0.992137
$473$	40.8490	1.87824
$474$	0	0
$475$	−5.86094	−0.268919
$476$	−1.33361	−0.0611257
$477$	0	0
$478$	−31.7176	−1.45073
$479$	28.4713	1.30089	0.650443	−	0.759555i	$-0.274583\pi$
$479$	28.4713	1.30089	0.650443	+	0.759555i	$0.274583\pi$
$480$	0	0
$481$	−8.99624	−0.410193
$482$	24.1359	1.09936
$483$	0	0
$484$	3.06247	0.139203
$485$	−14.3939	−0.653595
$486$	0	0
$487$	29.6841	1.34511	0.672557	−	0.740045i	$-0.265196\pi$
$487$	29.6841	1.34511	0.672557	+	0.740045i	$0.265196\pi$
$488$	3.06166	0.138595
$489$	0	0
$490$	−1.91922	−0.0867015
$491$	12.4050	0.559828	0.279914	−	0.960025i	$-0.409694\pi$
$491$	12.4050	0.559828	0.279914	+	0.960025i	$0.409694\pi$
$492$	0	0
$493$	−8.03494	−0.361876
$494$	29.1439	1.31125
$495$	0	0
$496$	−4.95023	−0.222272
$497$	−15.6299	−0.701099
$498$	0	0
$499$	−2.27982	−0.102059	−0.0510294	−	0.998697i	$-0.516250\pi$
$499$	−2.27982	−0.102059	−0.0510294	+	0.998697i	$0.516250\pi$
$500$	5.92387	0.264923
$501$	0	0
$502$	20.1346	0.898652
$503$	41.2812	1.84064	0.920320	−	0.391167i	$-0.127929\pi$
$503$	41.2812	1.84064	0.920320	+	0.391167i	$0.127929\pi$
$504$	0	0
$505$	−13.1019	−0.583028
$506$	−10.5700	−0.469893
$507$	0	0
$508$	−0.260946	−0.0115776
$509$	16.4242	0.727988	0.363994	−	0.931401i	$-0.381413\pi$
$509$	16.4242	0.727988	0.363994	+	0.931401i	$0.381413\pi$
$510$	0	0
$511$	−12.6775	−0.560821
$512$	−9.67844	−0.427731
$513$	0	0
$514$	−20.6845	−0.912355
$515$	3.63589	0.160217
$516$	0	0
$517$	−51.8807	−2.28171
$518$	−5.73949	−0.252179
$519$	0	0
$520$	27.4831	1.20521
$521$	−9.29672	−0.407297	−0.203648	−	0.979044i	$-0.565280\pi$
$521$	−9.29672	−0.407297	−0.203648	+	0.979044i	$0.565280\pi$
$522$	0	0
$523$	−22.7471	−0.994662	−0.497331	−	0.867561i	$-0.665686\pi$
$523$	−22.7471	−0.994662	−0.497331	+	0.867561i	$0.665686\pi$
$524$	5.60938	0.245047
$525$	0	0
$526$	−15.0172	−0.654782
$527$	−1.01959	−0.0444139
$528$	0	0
$529$	−20.3977	−0.886856
$530$	19.4537	0.845016
$531$	0	0
$532$	3.66710	0.158989
$533$	33.3362	1.44395
$534$	0	0
$535$	21.1239	0.913267
$536$	11.0640	0.477891
$537$	0	0
$538$	34.9471	1.50668
$539$	3.00718	0.129528
$540$	0	0
$541$	2.38959	0.102737	0.0513683	−	0.998680i	$-0.483642\pi$
$541$	2.38959	0.102737	0.0513683	+	0.998680i	$0.483642\pi$
$542$	44.1365	1.89582
$543$	0	0
$544$	2.65737	0.113934
$545$	20.5013	0.878179
$546$	0	0
$547$	29.6668	1.26846	0.634230	−	0.773144i	$-0.281317\pi$
$547$	29.6668	1.26846	0.634230	+	0.773144i	$0.281317\pi$
$548$	2.07808	0.0887712
$549$	0	0
$550$	−14.3011	−0.609803
$551$	22.0942	0.941246
$552$	0	0
$553$	12.9426	0.550374
$554$	−29.8061	−1.26634
$555$	0	0
$556$	−5.48138	−0.232462
$557$	8.41413	0.356518	0.178259	−	0.983984i	$-0.442954\pi$
$557$	8.41413	0.356518	0.178259	+	0.983984i	$0.442954\pi$
$558$	0	0
$559$	67.6619	2.86179
$560$	22.1183	0.934669
$561$	0	0
$562$	−31.5708	−1.33174
$563$	−27.2265	−1.14746	−0.573730	−	0.819044i	$-0.694504\pi$
$563$	−27.2265	−1.14746	−0.573730	+	0.819044i	$0.694504\pi$
$564$	0	0
$565$	0.756727	0.0318357
$566$	26.4304	1.11095
$567$	0	0
$568$	13.3915	0.561894
$569$	−21.9493	−0.920163	−0.460082	−	0.887877i	$-0.652180\pi$
$569$	−21.9493	−0.920163	−0.460082	+	0.887877i	$0.652180\pi$
$570$	0	0
$571$	−44.8535	−1.87706	−0.938530	−	0.345199i	$-0.887811\pi$
$571$	−44.8535	−1.87706	−0.938530	+	0.345199i	$0.887811\pi$
$572$	14.0254	0.586430
$573$	0	0
$574$	21.2681	0.887713
$575$	3.52092	0.146833
$576$	0	0
$577$	12.0191	0.500361	0.250181	−	0.968199i	$-0.419510\pi$
$577$	12.0191	0.500361	0.250181	+	0.968199i	$0.419510\pi$
$578$	−25.3276	−1.05349
$579$	0	0
$580$	−6.78595	−0.281771
$581$	−16.0209	−0.664658
$582$	0	0
$583$	−30.4816	−1.26242
$584$	10.8619	0.449469
$585$	0	0
$586$	30.8925	1.27616
$587$	−17.0299	−0.702898	−0.351449	−	0.936207i	$-0.614311\pi$
$587$	−17.0299	−0.702898	−0.351449	+	0.936207i	$0.614311\pi$
$588$	0	0
$589$	2.80363	0.115521
$590$	23.9817	0.987311
$591$	0	0
$592$	6.20324	0.254951
$593$	−14.9284	−0.613037	−0.306519	−	0.951865i	$-0.599164\pi$
$593$	−14.9284	−0.613037	−0.306519	+	0.951865i	$0.599164\pi$
$594$	0	0
$595$	4.55566	0.186764
$596$	−9.57056	−0.392025
$597$	0	0
$598$	−17.5080	−0.715956
$599$	13.5348	0.553017	0.276508	−	0.961011i	$-0.410823\pi$
$599$	13.5348	0.553017	0.276508	+	0.961011i	$0.410823\pi$
$600$	0	0
$601$	−45.3173	−1.84853	−0.924265	−	0.381752i	$-0.875321\pi$
$601$	−45.3173	−1.84853	−0.924265	+	0.381752i	$0.875321\pi$
$602$	43.1674	1.75937
$603$	0	0
$604$	−0.610691	−0.0248487
$605$	−10.4615	−0.425322
$606$	0	0
$607$	24.4368	0.991859	0.495929	−	0.868363i	$-0.334827\pi$
$607$	24.4368	0.991859	0.495929	+	0.868363i	$0.334827\pi$
$608$	−7.30715	−0.296344
$609$	0	0
$610$	3.40640	0.137921
$611$	−85.9348	−3.47655
$612$	0	0
$613$	−25.1996	−1.01780	−0.508901	−	0.860825i	$-0.669948\pi$
$613$	−25.1996	−1.01780	−0.508901	+	0.860825i	$0.669948\pi$
$614$	23.5174	0.949084
$615$	0	0
$616$	−27.4734	−1.10693
$617$	4.04185	0.162719	0.0813594	−	0.996685i	$-0.474074\pi$
$617$	4.04185	0.162719	0.0813594	+	0.996685i	$0.474074\pi$
$618$	0	0
$619$	−44.7700	−1.79946	−0.899729	−	0.436449i	$-0.856236\pi$
$619$	−44.7700	−1.79946	−0.899729	+	0.436449i	$0.856236\pi$
$620$	−0.861097	−0.0345825
$621$	0	0
$622$	7.24860	0.290642
$623$	−12.6302	−0.506019
$624$	0	0
$625$	−9.32314	−0.372926
$626$	18.7336	0.748745
$627$	0	0
$628$	−1.73604	−0.0692757
$629$	1.27767	0.0509439
$630$	0	0
$631$	31.4116	1.25048	0.625238	−	0.780434i	$-0.285002\pi$
$631$	31.4116	1.25048	0.625238	+	0.780434i	$0.285002\pi$
$632$	−11.0890	−0.441096
$633$	0	0
$634$	−22.9142	−0.910041
$635$	0.891403	0.0353742
$636$	0	0
$637$	4.98106	0.197357
$638$	53.9116	2.13438
$639$	0	0
$640$	−22.8785	−0.904351
$641$	48.6406	1.92119	0.960594	−	0.277954i	$-0.0896564\pi$
$641$	48.6406	1.92119	0.960594	+	0.277954i	$0.0896564\pi$
$642$	0	0
$643$	−28.0324	−1.10549	−0.552744	−	0.833351i	$-0.686419\pi$
$643$	−28.0324	−1.10549	−0.552744	+	0.833351i	$0.686419\pi$
$644$	−2.20299	−0.0868099
$645$	0	0
$646$	−4.13908	−0.162850
$647$	−37.5519	−1.47632	−0.738159	−	0.674627i	$-0.764304\pi$
$647$	−37.5519	−1.47632	−0.738159	+	0.674627i	$0.764304\pi$
$648$	0	0
$649$	−37.5763	−1.47500
$650$	−23.6883	−0.929131
$651$	0	0
$652$	7.83762	0.306945
$653$	−5.11222	−0.200057	−0.100028	−	0.994985i	$-0.531893\pi$
$653$	−5.11222	−0.200057	−0.100028	+	0.994985i	$0.531893\pi$
$654$	0	0
$655$	−19.1619	−0.748718
$656$	−22.9866	−0.897474
$657$	0	0
$658$	−54.8253	−2.13731
$659$	35.0059	1.36364	0.681818	−	0.731522i	$-0.261189\pi$
$659$	35.0059	1.36364	0.681818	+	0.731522i	$0.261189\pi$
$660$	0	0
$661$	1.33346	0.0518656	0.0259328	−	0.999664i	$-0.491744\pi$
$661$	1.33346	0.0518656	0.0259328	+	0.999664i	$0.491744\pi$
$662$	−12.7941	−0.497256
$663$	0	0
$664$	13.7264	0.532689
$665$	−12.5270	−0.485776
$666$	0	0
$667$	−13.2730	−0.513931
$668$	7.11717	0.275372
$669$	0	0
$670$	12.3097	0.475567
$671$	−5.33740	−0.206048
$672$	0	0
$673$	−3.57953	−0.137981	−0.0689904	−	0.997617i	$-0.521978\pi$
$673$	−3.57953	−0.137981	−0.0689904	+	0.997617i	$0.521978\pi$
$674$	−29.6866	−1.14348
$675$	0	0
$676$	16.8438	0.647839
$677$	−36.9389	−1.41968	−0.709839	−	0.704364i	$-0.751232\pi$
$677$	−36.9389	−1.41968	−0.709839	+	0.704364i	$0.751232\pi$
$678$	0	0
$679$	23.8336	0.914648
$680$	−3.90321	−0.149681
$681$	0	0
$682$	6.84106	0.261958
$683$	38.0166	1.45466	0.727332	−	0.686286i	$-0.240760\pi$
$683$	38.0166	1.45466	0.727332	+	0.686286i	$0.240760\pi$
$684$	0	0
$685$	−7.09881	−0.271232
$686$	−27.5300	−1.05110
$687$	0	0
$688$	−46.6553	−1.77872
$689$	−50.4894	−1.92349
$690$	0	0
$691$	0.550464	0.0209406	0.0104703	−	0.999945i	$-0.496667\pi$
$691$	0.550464	0.0209406	0.0104703	+	0.999945i	$0.496667\pi$
$692$	6.19968	0.235677
$693$	0	0
$694$	−27.8077	−1.05557
$695$	18.7246	0.710266
$696$	0	0
$697$	−4.73449	−0.179331
$698$	−26.8160	−1.01500
$699$	0	0
$700$	−2.98064	−0.112657
$701$	19.0242	0.718534	0.359267	−	0.933235i	$-0.383027\pi$
$701$	19.0242	0.718534	0.359267	+	0.933235i	$0.383027\pi$
$702$	0	0
$703$	−3.51328	−0.132506
$704$	21.5343	0.811603
$705$	0	0
$706$	−23.9217	−0.900304
$707$	21.6942	0.815896
$708$	0	0
$709$	12.1568	0.456558	0.228279	−	0.973596i	$-0.426690\pi$
$709$	12.1568	0.456558	0.228279	+	0.973596i	$0.426690\pi$
$710$	14.8993	0.559161
$711$	0	0
$712$	10.8214	0.405548
$713$	−1.68426	−0.0630761
$714$	0	0
$715$	−47.9113	−1.79178
$716$	0.145393	0.00543358
$717$	0	0
$718$	3.86860	0.144375
$719$	9.77667	0.364608	0.182304	−	0.983242i	$-0.441644\pi$
$719$	9.77667	0.364608	0.182304	+	0.983242i	$0.441644\pi$
$720$	0	0
$721$	−6.02033	−0.224209
$722$	−18.6082	−0.692524
$723$	0	0
$724$	0.697978	0.0259401
$725$	−17.9583	−0.666953
$726$	0	0
$727$	4.33493	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$727$	4.33493	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$728$	−45.5067	−1.68659
$729$	0	0
$730$	12.0849	0.447283
$731$	−9.60949	−0.355420
$732$	0	0
$733$	29.1937	1.07829	0.539146	−	0.842212i	$-0.318747\pi$
$733$	29.1937	1.07829	0.539146	+	0.842212i	$0.318747\pi$
$734$	−2.07329	−0.0765264
$735$	0	0
$736$	4.38972	0.161807
$737$	−19.2878	−0.710476
$738$	0	0
$739$	41.5553	1.52864	0.764319	−	0.644838i	$-0.223075\pi$
$739$	41.5553	1.52864	0.764319	+	0.644838i	$0.223075\pi$
$740$	1.07906	0.0396670
$741$	0	0
$742$	−32.2116	−1.18253
$743$	21.3774	0.784259	0.392130	−	0.919910i	$-0.371738\pi$
$743$	21.3774	0.784259	0.392130	+	0.919910i	$0.371738\pi$
$744$	0	0
$745$	32.6935	1.19780
$746$	−15.1940	−0.556291
$747$	0	0
$748$	−1.99191	−0.0728316
$749$	−34.9771	−1.27804
$750$	0	0
$751$	−39.9676	−1.45844	−0.729220	−	0.684280i	$-0.760117\pi$
$751$	−39.9676	−1.45844	−0.729220	+	0.684280i	$0.760117\pi$
$752$	59.2552	2.16081
$753$	0	0
$754$	89.2986	3.25206
$755$	2.08615	0.0759228
$756$	0	0
$757$	6.68348	0.242915	0.121458	−	0.992597i	$-0.461243\pi$
$757$	6.68348	0.242915	0.121458	+	0.992597i	$0.461243\pi$
$758$	−13.5161	−0.490925
$759$	0	0
$760$	10.7329	0.389324
$761$	42.0250	1.52341	0.761703	−	0.647927i	$-0.224364\pi$
$761$	42.0250	1.52341	0.761703	+	0.647927i	$0.224364\pi$
$762$	0	0
$763$	−33.9461	−1.22893
$764$	10.1338	0.366630
$765$	0	0
$766$	53.0890	1.91818
$767$	−62.2411	−2.24740
$768$	0	0
$769$	2.41323	0.0870233	0.0435117	−	0.999053i	$-0.486145\pi$
$769$	2.41323	0.0870233	0.0435117	+	0.999053i	$0.486145\pi$
$770$	−30.5668	−1.10155
$771$	0	0
$772$	−10.2969	−0.370594
$773$	1.39780	0.0502754	0.0251377	−	0.999684i	$-0.491998\pi$
$773$	1.39780	0.0502754	0.0251377	+	0.999684i	$0.491998\pi$
$774$	0	0
$775$	−2.27880	−0.0818569
$776$	−20.4202	−0.733043
$777$	0	0
$778$	−23.6034	−0.846224
$779$	13.0187	0.466445
$780$	0	0
$781$	−23.3454	−0.835363
$782$	2.48653	0.0889181
$783$	0	0
$784$	−3.43462	−0.122665
$785$	5.93040	0.211665
$786$	0	0
$787$	−39.6515	−1.41342	−0.706712	−	0.707501i	$-0.749822\pi$
$787$	−39.6515	−1.41342	−0.706712	+	0.707501i	$0.749822\pi$
$788$	−4.71519	−0.167972
$789$	0	0
$790$	−12.3376	−0.438951
$791$	−1.25299	−0.0445513
$792$	0	0
$793$	−8.84082	−0.313947
$794$	−26.4837	−0.939871
$795$	0	0
$796$	−5.25345	−0.186204
$797$	−3.29387	−0.116675	−0.0583374	−	0.998297i	$-0.518580\pi$
$797$	−3.29387	−0.116675	−0.0583374	+	0.998297i	$0.518580\pi$
$798$	0	0
$799$	12.2046	0.431769
$800$	5.93927	0.209985
$801$	0	0
$802$	20.7186	0.731599
$803$	−18.9356	−0.668222
$804$	0	0
$805$	7.52551	0.265239
$806$	11.3315	0.399134
$807$	0	0
$808$	−18.5873	−0.653898
$809$	−6.54436	−0.230087	−0.115044	−	0.993360i	$-0.536701\pi$
$809$	−6.54436	−0.230087	−0.115044	+	0.993360i	$0.536701\pi$
$810$	0	0
$811$	−44.7516	−1.57144	−0.785721	−	0.618581i	$-0.787708\pi$
$811$	−44.7516	−1.57144	−0.785721	+	0.618581i	$0.787708\pi$
$812$	11.2362	0.394314
$813$	0	0
$814$	−8.57268	−0.300472
$815$	−26.7737	−0.937841
$816$	0	0
$817$	26.4239	0.924454
$818$	40.1712	1.40455
$819$	0	0
$820$	−3.99853	−0.139635
$821$	−49.6840	−1.73398	−0.866991	−	0.498324i	$-0.833949\pi$
$821$	−49.6840	−1.73398	−0.866991	+	0.498324i	$0.833949\pi$
$822$	0	0
$823$	10.6206	0.370211	0.185106	−	0.982719i	$-0.440737\pi$
$823$	10.6206	0.370211	0.185106	+	0.982719i	$0.440737\pi$
$824$	5.15812	0.179692
$825$	0	0
$826$	−39.7090	−1.38165
$827$	−16.4008	−0.570311	−0.285156	−	0.958481i	$-0.592045\pi$
$827$	−16.4008	−0.570311	−0.285156	+	0.958481i	$0.592045\pi$
$828$	0	0
$829$	−2.95645	−0.102682	−0.0513409	−	0.998681i	$-0.516350\pi$
$829$	−2.95645	−0.102682	−0.0513409	+	0.998681i	$0.516350\pi$
$830$	15.2720	0.530098
$831$	0	0
$832$	35.6692	1.23661
$833$	−0.707421	−0.0245107
$834$	0	0
$835$	−24.3126	−0.841372
$836$	5.47730	0.189436
$837$	0	0
$838$	−19.9985	−0.690835
$839$	−32.4464	−1.12018	−0.560088	−	0.828433i	$-0.689233\pi$
$839$	−32.4464	−1.12018	−0.560088	+	0.828433i	$0.689233\pi$
$840$	0	0
$841$	38.6980	1.33441
$842$	−27.9018	−0.961561
$843$	0	0
$844$	−7.39508	−0.254549
$845$	−57.5392	−1.97941
$846$	0	0
$847$	17.3223	0.595201
$848$	34.8143	1.19553
$849$	0	0
$850$	3.36426	0.115393
$851$	2.11058	0.0723498
$852$	0	0
$853$	28.3331	0.970107	0.485053	−	0.874485i	$-0.338800\pi$
$853$	28.3331	0.970107	0.485053	+	0.874485i	$0.338800\pi$
$854$	−5.64033	−0.193008
$855$	0	0
$856$	29.9678	1.02428
$857$	14.0035	0.478349	0.239175	−	0.970977i	$-0.423123\pi$
$857$	14.0035	0.478349	0.239175	+	0.970977i	$0.423123\pi$
$858$	0	0
$859$	−2.15977	−0.0736905	−0.0368452	−	0.999321i	$-0.511731\pi$
$859$	−2.15977	−0.0736905	−0.0368452	+	0.999321i	$0.511731\pi$
$860$	−8.11574	−0.276744
$861$	0	0
$862$	24.7768	0.843903
$863$	−14.9487	−0.508859	−0.254430	−	0.967091i	$-0.581888\pi$
$863$	−14.9487	−0.508859	−0.254430	+	0.967091i	$0.581888\pi$
$864$	0	0
$865$	−21.1784	−0.720087
$866$	−19.9287	−0.677204
$867$	0	0
$868$	1.42581	0.0483951
$869$	19.3314	0.655773
$870$	0	0
$871$	−31.9482	−1.08252
$872$	29.0845	0.984926
$873$	0	0
$874$	−6.83737	−0.231278
$875$	33.5072	1.13275
$876$	0	0
$877$	−1.88708	−0.0637221	−0.0318611	−	0.999492i	$-0.510143\pi$
$877$	−1.88708	−0.0637221	−0.0318611	+	0.999492i	$0.510143\pi$
$878$	42.5173	1.43489
$879$	0	0
$880$	33.0366	1.11366
$881$	46.8258	1.57760	0.788800	−	0.614649i	$-0.210703\pi$
$881$	46.8258	1.57760	0.788800	+	0.614649i	$0.210703\pi$
$882$	0	0
$883$	−30.0635	−1.01172	−0.505858	−	0.862617i	$-0.668824\pi$
$883$	−30.0635	−1.01172	−0.505858	+	0.862617i	$0.668824\pi$
$884$	−3.29939	−0.110970
$885$	0	0
$886$	54.8147	1.84153
$887$	−29.9139	−1.00441	−0.502205	−	0.864749i	$-0.667478\pi$
$887$	−29.9139	−1.00441	−0.502205	+	0.864749i	$0.667478\pi$
$888$	0	0
$889$	−1.47599	−0.0495031
$890$	12.0398	0.403576
$891$	0	0
$892$	−6.03788	−0.202163
$893$	−33.5599	−1.12304
$894$	0	0
$895$	−0.496668	−0.0166018
$896$	37.8823	1.26556
$897$	0	0
$898$	−32.7458	−1.09274
$899$	8.59047	0.286508
$900$	0	0
$901$	7.17062	0.238888
$902$	31.7667	1.05772
$903$	0	0
$904$	1.07354	0.0357055
$905$	−2.38432	−0.0792576
$906$	0	0
$907$	26.7652	0.888725	0.444363	−	0.895847i	$-0.353430\pi$
$907$	26.7652	0.888725	0.444363	+	0.895847i	$0.353430\pi$
$908$	1.84386	0.0611905
$909$	0	0
$910$	−50.6306	−1.67839
$911$	0.441137	0.0146155	0.00730776	−	0.999973i	$-0.497674\pi$
$911$	0.441137	0.0146155	0.00730776	+	0.999973i	$0.497674\pi$
$912$	0	0
$913$	−23.9293	−0.791943
$914$	−11.3767	−0.376308
$915$	0	0
$916$	−9.14048	−0.302010
$917$	31.7284	1.04776
$918$	0	0
$919$	49.0749	1.61883	0.809416	−	0.587236i	$-0.199784\pi$
$919$	49.0749	1.61883	0.809416	+	0.587236i	$0.199784\pi$
$920$	−6.44774	−0.212576
$921$	0	0
$922$	36.5035	1.20218
$923$	−38.6691	−1.27281
$924$	0	0
$925$	2.85561	0.0938919
$926$	7.85389	0.258095
$927$	0	0
$928$	−22.3895	−0.734972
$929$	32.4312	1.06403	0.532017	−	0.846734i	$-0.321434\pi$
$929$	32.4312	1.06403	0.532017	+	0.846734i	$0.321434\pi$
$930$	0	0
$931$	1.94524	0.0637528
$932$	0.268176	0.00878441
$933$	0	0
$934$	−19.7006	−0.644625
$935$	6.80447	0.222530
$936$	0	0
$937$	48.6157	1.58821	0.794103	−	0.607783i	$-0.207941\pi$
$937$	48.6157	1.58821	0.794103	+	0.607783i	$0.207941\pi$
$938$	−20.3825	−0.665513
$939$	0	0
$940$	10.3075	0.336193
$941$	−0.710434	−0.0231595	−0.0115797	−	0.999933i	$-0.503686\pi$
$941$	−0.710434	−0.0231595	−0.0115797	+	0.999933i	$0.503686\pi$
$942$	0	0
$943$	−7.82092	−0.254684
$944$	42.9175	1.39685
$945$	0	0
$946$	64.4762	2.09630
$947$	26.1077	0.848387	0.424194	−	0.905571i	$-0.360558\pi$
$947$	26.1077	0.848387	0.424194	+	0.905571i	$0.360558\pi$
$948$	0	0
$949$	−31.3647	−1.01814
$950$	−9.25094	−0.300140
$951$	0	0
$952$	6.46296	0.209466
$953$	−51.0054	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$953$	−51.0054	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$954$	0	0
$955$	−34.6177	−1.12020
$956$	−9.87375	−0.319340
$957$	0	0
$958$	44.9392	1.45192
$959$	11.7543	0.379565
$960$	0	0
$961$	−29.9099	−0.964836
$962$	−14.1997	−0.457817
$963$	0	0
$964$	7.51353	0.241995
$965$	35.1747	1.13231
$966$	0	0
$967$	10.0048	0.321733	0.160867	−	0.986976i	$-0.448571\pi$
$967$	10.0048	0.321733	0.160867	+	0.986976i	$0.448571\pi$
$968$	−14.8414	−0.477022
$969$	0	0
$970$	−22.7195	−0.729478
$971$	44.6269	1.43215	0.716073	−	0.698025i	$-0.245938\pi$
$971$	44.6269	1.43215	0.716073	+	0.698025i	$0.245938\pi$
$972$	0	0
$973$	−31.0044	−0.993954
$974$	46.8535	1.50128
$975$	0	0
$976$	6.09607	0.195130
$977$	−39.0856	−1.25046	−0.625230	−	0.780440i	$-0.714995\pi$
$977$	−39.0856	−1.25046	−0.625230	+	0.780440i	$0.714995\pi$
$978$	0	0
$979$	−18.8649	−0.602925
$980$	−0.597456	−0.0190850
$981$	0	0
$982$	19.5800	0.624824
$983$	6.79459	0.216714	0.108357	−	0.994112i	$-0.465441\pi$
$983$	6.79459	0.216714	0.108357	+	0.994112i	$0.465441\pi$
$984$	0	0
$985$	16.1073	0.513222
$986$	−12.6824	−0.403890
$987$	0	0
$988$	9.07255	0.288636
$989$	−15.8740	−0.504763
$990$	0	0
$991$	17.2046	0.546522	0.273261	−	0.961940i	$-0.411898\pi$
$991$	17.2046	0.546522	0.273261	+	0.961940i	$0.411898\pi$
$992$	−2.84110	−0.0902049
$993$	0	0
$994$	−24.6704	−0.782497
$995$	17.9460	0.568928
$996$	0	0
$997$	7.33368	0.232260	0.116130	−	0.993234i	$-0.462951\pi$
$997$	7.33368	0.232260	0.116130	+	0.993234i	$0.462951\pi$
$998$	−3.59848	−0.113908
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.e.1.4	yes	6
3.2	odd	2		729.2.a.b.1.3	✓	6
9.2	odd	6		729.2.c.d.244.4		12
9.4	even	3		729.2.c.a.487.3		12
9.5	odd	6		729.2.c.d.487.4		12
9.7	even	3		729.2.c.a.244.3		12
27.2	odd	18		729.2.e.j.568.2		12
27.4	even	9		729.2.e.l.406.2		12
27.5	odd	18		729.2.e.t.649.1		12
27.7	even	9		729.2.e.l.325.2		12
27.11	odd	18		729.2.e.t.82.1		12
27.13	even	9		729.2.e.u.163.1		12
27.14	odd	18		729.2.e.j.163.2		12
27.16	even	9		729.2.e.k.82.2		12
27.20	odd	18		729.2.e.s.325.1		12
27.22	even	9		729.2.e.k.649.2		12
27.23	odd	18		729.2.e.s.406.1		12
27.25	even	9		729.2.e.u.568.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.b.1.3	✓	6	3.2	odd	2
729.2.a.e.1.4	yes	6	1.1	even	1	trivial
729.2.c.a.244.3		12	9.7	even	3
729.2.c.a.487.3		12	9.4	even	3
729.2.c.d.244.4		12	9.2	odd	6
729.2.c.d.487.4		12	9.5	odd	6
729.2.e.j.163.2		12	27.14	odd	18
729.2.e.j.568.2		12	27.2	odd	18
729.2.e.k.82.2		12	27.16	even	9
729.2.e.k.649.2		12	27.22	even	9
729.2.e.l.325.2		12	27.7	even	9
729.2.e.l.406.2		12	27.4	even	9
729.2.e.s.325.1		12	27.20	odd	18
729.2.e.s.406.1		12	27.23	odd	18
729.2.e.t.82.1		12	27.11	odd	18
729.2.e.t.649.1		12	27.5	odd	18
729.2.e.u.163.1		12	27.13	even	9
729.2.e.u.568.1		12	27.25	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)

\(f(q)\)	\(=\)	\(q+1.57840 q^{2} +0.491360 q^{4} -1.67851 q^{5} +2.77928 q^{7} -2.38124 q^{8} -2.64936 q^{10} +4.15122 q^{11} +6.87605 q^{13} +4.38683 q^{14} -4.74129 q^{16} -0.976551 q^{17} +2.68529 q^{19} -0.824751 q^{20} +6.55231 q^{22} -1.61317 q^{23} -2.18261 q^{25} +10.8532 q^{26} +1.36563 q^{28} +8.22788 q^{29} +1.04407 q^{31} -2.72118 q^{32} -1.54139 q^{34} -4.66505 q^{35} -1.30834 q^{37} +4.23847 q^{38} +3.99694 q^{40} +4.84817 q^{41} +9.84023 q^{43} +2.03974 q^{44} -2.54623 q^{46} -12.4977 q^{47} +0.724408 q^{49} -3.44504 q^{50} +3.37861 q^{52} -7.34280 q^{53} -6.96786 q^{55} -6.61815 q^{56} +12.9869 q^{58} -9.05188 q^{59} -1.28574 q^{61} +1.64796 q^{62} +5.18745 q^{64} -11.5415 q^{65} -4.64630 q^{67} -0.479838 q^{68} -7.36333 q^{70} -5.62373 q^{71} -4.56144 q^{73} -2.06510 q^{74} +1.31944 q^{76} +11.5374 q^{77} +4.65680 q^{79} +7.95828 q^{80} +7.65237 q^{82} -5.76439 q^{83} +1.63915 q^{85} +15.5319 q^{86} -9.88507 q^{88} -4.54442 q^{89} +19.1105 q^{91} -0.792647 q^{92} -19.7264 q^{94} -4.50728 q^{95} +8.57544 q^{97} +1.14341 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) 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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