LMFDB - Embedded newform 729.2.a.d.1.4

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:	6.6.1397493.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 $ x^6 - 3x^5 - 3x^4 + 10x^3 + 3x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.68091$ 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)

$f(q)$	$=$	$q+0.801527 q^{2} -1.35755 q^{4} +2.74984 q^{5} +2.37683 q^{7} -2.69117 q^{8} +O(q^{10})$	\(q+0.801527 q^{2} -1.35755 q^{4} +2.74984 q^{5} +2.37683 q^{7} -2.69117 q^{8} +2.20407 q^{10} +0.250159 q^{11} +2.61198 q^{13} +1.90510 q^{14} +0.558064 q^{16} +0.293377 q^{17} -2.78475 q^{19} -3.73306 q^{20} +0.200509 q^{22} +6.68984 q^{23} +2.56163 q^{25} +2.09357 q^{26} -3.22668 q^{28} +0.355057 q^{29} +2.76547 q^{31} +5.82964 q^{32} +0.235149 q^{34} +6.53592 q^{35} -6.99238 q^{37} -2.23205 q^{38} -7.40029 q^{40} +9.71761 q^{41} +0.260706 q^{43} -0.339604 q^{44} +5.36209 q^{46} +11.4256 q^{47} -1.35066 q^{49} +2.05321 q^{50} -3.54591 q^{52} -5.43137 q^{53} +0.687897 q^{55} -6.39646 q^{56} +0.284588 q^{58} -5.97693 q^{59} -11.8468 q^{61} +2.21660 q^{62} +3.55649 q^{64} +7.18254 q^{65} +1.81030 q^{67} -0.398275 q^{68} +5.23871 q^{70} +0.370510 q^{71} +5.02679 q^{73} -5.60458 q^{74} +3.78044 q^{76} +0.594586 q^{77} +0.802822 q^{79} +1.53459 q^{80} +7.78892 q^{82} +2.75565 q^{83} +0.806740 q^{85} +0.208963 q^{86} -0.673220 q^{88} -10.4507 q^{89} +6.20825 q^{91} -9.08183 q^{92} +9.15789 q^{94} -7.65761 q^{95} -14.8346 q^{97} -1.08259 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8}+O(q^{10}) $ 6 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^8	$ 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35} + 3 q^{37} - 3 q^{38} + 6 q^{40} + 15 q^{41} + 3 q^{44} + 3 q^{46} + 21 q^{47} - 12 q^{49} + 3 q^{50} + 12 q^{52} + 9 q^{53} - 6 q^{55} - 6 q^{56} - 12 q^{58} + 24 q^{59} - 9 q^{61} - 12 q^{62} - 12 q^{64} - 6 q^{65} - 9 q^{67} - 9 q^{68} + 15 q^{70} + 27 q^{71} - 6 q^{73} - 12 q^{74} + 6 q^{76} - 12 q^{77} - 21 q^{80} - 6 q^{82} + 12 q^{83} - 21 q^{86} + 12 q^{88} + 9 q^{89} - 6 q^{91} + 6 q^{92} + 6 q^{94} + 12 q^{95} - 45 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^8 + 3 * q^10 + 12 * q^11 + 6 * q^14 - 3 * q^16 + 9 * q^17 + 3 * q^19 + 6 * q^20 + 6 * q^22 + 15 * q^23 - 6 * q^25 + 15 * q^26 - 6 * q^28 + 12 * q^29 + 12 * q^35 + 3 * q^37 - 3 * q^38 + 6 * q^40 + 15 * q^41 + 3 * q^44 + 3 * q^46 + 21 * q^47 - 12 * q^49 + 3 * q^50 + 12 * q^52 + 9 * q^53 - 6 * q^55 - 6 * q^56 - 12 * q^58 + 24 * q^59 - 9 * q^61 - 12 * q^62 - 12 * q^64 - 6 * q^65 - 9 * q^67 - 9 * q^68 + 15 * q^70 + 27 * q^71 - 6 * q^73 - 12 * q^74 + 6 * q^76 - 12 * q^77 - 21 * q^80 - 6 * q^82 + 12 * q^83 - 21 * q^86 + 12 * q^88 + 9 * q^89 - 6 * q^91 + 6 * q^92 + 6 * q^94 + 12 * q^95 - 45 * 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 $

$2$	0.801527	0.566765	0.283383	−	0.959007i	$-0.408543\pi$
$2$	0.801527	0.566765	0.283383	+	0.959007i	$0.408543\pi$
$3$	0	0
$3$	0	0
$4$	−1.35755	−0.678777
$5$	2.74984	1.22977	0.614883	−	0.788618i	$-0.289203\pi$
$5$	2.74984	1.22977	0.614883	+	0.788618i	$0.289203\pi$
$6$	0	0
$7$	2.37683	0.898359	0.449179	−	0.893442i	$-0.351716\pi$
$7$	2.37683	0.898359	0.449179	+	0.893442i	$0.351716\pi$
$8$	−2.69117	−0.951472
$9$	0	0
$10$	2.20407	0.696989
$11$	0.250159	0.0754257	0.0377129	−	0.999289i	$-0.487993\pi$
$11$	0.250159	0.0754257	0.0377129	+	0.999289i	$0.487993\pi$
$12$	0	0
$13$	2.61198	0.724434	0.362217	−	0.932094i	$-0.382020\pi$
$13$	2.61198	0.724434	0.362217	+	0.932094i	$0.382020\pi$
$14$	1.90510	0.509158
$15$	0	0
$16$	0.558064	0.139516
$17$	0.293377	0.0711543	0.0355772	−	0.999367i	$-0.488673\pi$
$17$	0.293377	0.0711543	0.0355772	+	0.999367i	$0.488673\pi$
$18$	0	0
$19$	−2.78475	−0.638864	−0.319432	−	0.947609i	$-0.603492\pi$
$19$	−2.78475	−0.638864	−0.319432	+	0.947609i	$0.603492\pi$
$20$	−3.73306	−0.834737
$21$	0	0
$22$	0.200509	0.0427487
$23$	6.68984	1.39493	0.697464	−	0.716620i	$-0.254312\pi$
$23$	6.68984	1.39493	0.697464	+	0.716620i	$0.254312\pi$
$24$	0	0
$25$	2.56163	0.512325
$26$	2.09357	0.410584
$27$	0	0
$28$	−3.22668	−0.609786
$29$	0.355057	0.0659324	0.0329662	−	0.999456i	$-0.489505\pi$
$29$	0.355057	0.0659324	0.0329662	+	0.999456i	$0.489505\pi$
$30$	0	0
$31$	2.76547	0.496692	0.248346	−	0.968671i	$-0.420113\pi$
$31$	2.76547	0.496692	0.248346	+	0.968671i	$0.420113\pi$
$32$	5.82964	1.03055
$33$	0	0
$34$	0.235149	0.0403278
$35$	6.53592	1.10477
$36$	0	0
$37$	−6.99238	−1.14954	−0.574770	−	0.818315i	$-0.694909\pi$
$37$	−6.99238	−1.14954	−0.574770	+	0.818315i	$0.694909\pi$
$38$	−2.23205	−0.362086
$39$	0	0
$40$	−7.40029	−1.17009
$41$	9.71761	1.51764	0.758818	−	0.651303i	$-0.225777\pi$
$41$	9.71761	1.51764	0.758818	+	0.651303i	$0.225777\pi$
$42$	0	0
$43$	0.260706	0.0397574	0.0198787	−	0.999802i	$-0.493672\pi$
$43$	0.260706	0.0397574	0.0198787	+	0.999802i	$0.493672\pi$
$44$	−0.339604	−0.0511973
$45$	0	0
$46$	5.36209	0.790597
$47$	11.4256	1.66659	0.833295	−	0.552829i	$-0.186452\pi$
$47$	11.4256	1.66659	0.833295	+	0.552829i	$0.186452\pi$
$48$	0	0
$49$	−1.35066	−0.192952
$50$	2.05321	0.290368
$51$	0	0
$52$	−3.54591	−0.491729
$53$	−5.43137	−0.746056	−0.373028	−	0.927820i	$-0.621680\pi$
$53$	−5.43137	−0.746056	−0.373028	+	0.927820i	$0.621680\pi$
$54$	0	0
$55$	0.687897	0.0927560
$56$	−6.39646	−0.854764
$57$	0	0
$58$	0.284588	0.0373682
$59$	−5.97693	−0.778130	−0.389065	−	0.921210i	$-0.627202\pi$
$59$	−5.97693	−0.778130	−0.389065	+	0.921210i	$0.627202\pi$
$60$	0	0
$61$	−11.8468	−1.51682	−0.758411	−	0.651776i	$-0.774024\pi$
$61$	−11.8468	−1.51682	−0.758411	+	0.651776i	$0.774024\pi$
$62$	2.21660	0.281508
$63$	0	0
$64$	3.55649	0.444561
$65$	7.18254	0.890884
$66$	0	0
$67$	1.81030	0.221164	0.110582	−	0.993867i	$-0.464729\pi$
$67$	1.81030	0.221164	0.110582	+	0.993867i	$0.464729\pi$
$68$	−0.398275	−0.0482979
$69$	0	0
$70$	5.23871	0.626146
$71$	0.370510	0.0439714	0.0219857	−	0.999758i	$-0.493001\pi$
$71$	0.370510	0.0439714	0.0219857	+	0.999758i	$0.493001\pi$
$72$	0	0
$73$	5.02679	0.588341	0.294171	−	0.955753i	$-0.404957\pi$
$73$	5.02679	0.588341	0.294171	+	0.955753i	$0.404957\pi$
$74$	−5.60458	−0.651519
$75$	0	0
$76$	3.78044	0.433647
$77$	0.594586	0.0677594
$78$	0	0
$79$	0.802822	0.0903245	0.0451622	−	0.998980i	$-0.485620\pi$
$79$	0.802822	0.0903245	0.0451622	+	0.998980i	$0.485620\pi$
$80$	1.53459	0.171572
$81$	0	0
$82$	7.78892	0.860143
$83$	2.75565	0.302472	0.151236	−	0.988498i	$-0.451675\pi$
$83$	2.75565	0.302472	0.151236	+	0.988498i	$0.451675\pi$
$84$	0	0
$85$	0.806740	0.0875032
$86$	0.208963	0.0225331
$87$	0	0
$88$	−0.673220	−0.0717655
$89$	−10.4507	−1.10777	−0.553884	−	0.832594i	$-0.686855\pi$
$89$	−10.4507	−1.10777	−0.553884	+	0.832594i	$0.686855\pi$
$90$	0	0
$91$	6.20825	0.650801
$92$	−9.08183	−0.946846
$93$	0	0
$94$	9.15789	0.944565
$95$	−7.65761	−0.785654
$96$	0	0
$97$	−14.8346	−1.50623	−0.753113	−	0.657891i	$-0.771449\pi$
$97$	−14.8346	−1.50623	−0.753113	+	0.657891i	$0.771449\pi$
$98$	−1.08259	−0.109358
$99$	0	0
$100$	−3.47755	−0.347755
$101$	−4.00533	−0.398545	−0.199272	−	0.979944i	$-0.563858\pi$
$101$	−4.00533	−0.398545	−0.199272	+	0.979944i	$0.563858\pi$
$102$	0	0
$103$	−5.91936	−0.583252	−0.291626	−	0.956532i	$-0.594196\pi$
$103$	−5.91936	−0.583252	−0.291626	+	0.956532i	$0.594196\pi$
$104$	−7.02929	−0.689279
$105$	0	0
$106$	−4.35339	−0.422838
$107$	0.258978	0.0250364	0.0125182	−	0.999922i	$-0.496015\pi$
$107$	0.258978	0.0250364	0.0125182	+	0.999922i	$0.496015\pi$
$108$	0	0
$109$	−8.55787	−0.819695	−0.409848	−	0.912154i	$-0.634418\pi$
$109$	−8.55787	−0.819695	−0.409848	+	0.912154i	$0.634418\pi$
$110$	0.551368	0.0525709
$111$	0	0
$112$	1.32642	0.125335
$113$	3.11918	0.293428	0.146714	−	0.989179i	$-0.453130\pi$
$113$	3.11918	0.293428	0.146714	+	0.989179i	$0.453130\pi$
$114$	0	0
$115$	18.3960	1.71544
$116$	−0.482009	−0.0447534
$117$	0	0
$118$	−4.79067	−0.441017
$119$	0.697308	0.0639221
$120$	0	0
$121$	−10.9374	−0.994311
$122$	−9.49550	−0.859682
$123$	0	0
$124$	−3.75427	−0.337144
$125$	−6.70514	−0.599726
$126$	0	0
$127$	−18.4545	−1.63757	−0.818787	−	0.574097i	$-0.805353\pi$
$127$	−18.4545	−1.63757	−0.818787	+	0.574097i	$0.805353\pi$
$128$	−8.80867	−0.778583
$129$	0	0
$130$	5.75700	0.504922
$131$	−14.2255	−1.24289	−0.621443	−	0.783460i	$-0.713453\pi$
$131$	−14.2255	−1.24289	−0.621443	+	0.783460i	$0.713453\pi$
$132$	0	0
$133$	−6.61888	−0.573929
$134$	1.45101	0.125348
$135$	0	0
$136$	−0.789527	−0.0677014
$137$	19.6856	1.68185	0.840926	−	0.541149i	$-0.182011\pi$
$137$	19.6856	1.68185	0.840926	+	0.541149i	$0.182011\pi$
$138$	0	0
$139$	−17.9110	−1.51919	−0.759594	−	0.650398i	$-0.774602\pi$
$139$	−17.9110	−1.51919	−0.759594	+	0.650398i	$0.774602\pi$
$140$	−8.87286	−0.749894
$141$	0	0
$142$	0.296974	0.0249215
$143$	0.653411	0.0546410
$144$	0	0
$145$	0.976351	0.0810815
$146$	4.02911	0.333451
$147$	0	0
$148$	9.49254	0.780282
$149$	−16.2895	−1.33448	−0.667242	−	0.744841i	$-0.732525\pi$
$149$	−16.2895	−1.33448	−0.667242	+	0.744841i	$0.732525\pi$
$150$	0	0
$151$	14.2749	1.16167	0.580836	−	0.814021i	$-0.302726\pi$
$151$	14.2749	1.16167	0.580836	+	0.814021i	$0.302726\pi$
$152$	7.49422	0.607862
$153$	0	0
$154$	0.476577	0.0384036
$155$	7.60459	0.610816
$156$	0	0
$157$	0.763354	0.0609223	0.0304612	−	0.999536i	$-0.490302\pi$
$157$	0.763354	0.0609223	0.0304612	+	0.999536i	$0.490302\pi$
$158$	0.643483	0.0511928
$159$	0	0
$160$	16.0306	1.26733
$161$	15.9006	1.25315
$162$	0	0
$163$	5.12834	0.401682	0.200841	−	0.979624i	$-0.435632\pi$
$163$	5.12834	0.401682	0.200841	+	0.979624i	$0.435632\pi$
$164$	−13.1922	−1.03014
$165$	0	0
$166$	2.20873	0.171431
$167$	−8.90112	−0.688790	−0.344395	−	0.938825i	$-0.611916\pi$
$167$	−8.90112	−0.688790	−0.344395	+	0.938825i	$0.611916\pi$
$168$	0	0
$169$	−6.17754	−0.475196
$170$	0.646624	0.0495938
$171$	0	0
$172$	−0.353923	−0.0269864
$173$	6.81124	0.517849	0.258924	−	0.965898i	$-0.416632\pi$
$173$	6.81124	0.517849	0.258924	+	0.965898i	$0.416632\pi$
$174$	0	0
$175$	6.08856	0.460252
$176$	0.139605	0.0105231
$177$	0	0
$178$	−8.37649	−0.627844
$179$	18.3476	1.37137	0.685684	−	0.727900i	$-0.259503\pi$
$179$	18.3476	1.37137	0.685684	+	0.727900i	$0.259503\pi$
$180$	0	0
$181$	11.3256	0.841829	0.420914	−	0.907100i	$-0.361709\pi$
$181$	11.3256	0.841829	0.420914	+	0.907100i	$0.361709\pi$
$182$	4.97608	0.368852
$183$	0	0
$184$	−18.0035	−1.32724
$185$	−19.2279	−1.41367
$186$	0	0
$187$	0.0733908	0.00536687
$188$	−15.5108	−1.13124
$189$	0	0
$190$	−6.13778	−0.445281
$191$	−6.85841	−0.496257	−0.248129	−	0.968727i	$-0.579816\pi$
$191$	−6.85841	−0.496257	−0.248129	+	0.968727i	$0.579816\pi$
$192$	0	0
$193$	20.4128	1.46935	0.734673	−	0.678422i	$-0.237336\pi$
$193$	20.4128	1.46935	0.734673	+	0.678422i	$0.237336\pi$
$194$	−11.8903	−0.853676
$195$	0	0
$196$	1.83360	0.130971
$197$	−3.03573	−0.216287	−0.108143	−	0.994135i	$-0.534491\pi$
$197$	−3.03573	−0.216287	−0.108143	+	0.994135i	$0.534491\pi$
$198$	0	0
$199$	−2.26247	−0.160382	−0.0801912	−	0.996779i	$-0.525553\pi$
$199$	−2.26247	−0.160382	−0.0801912	+	0.996779i	$0.525553\pi$
$200$	−6.89377	−0.487463
$201$	0	0
$202$	−3.21038	−0.225881
$203$	0.843912	0.0592310
$204$	0	0
$205$	26.7219	1.86634
$206$	−4.74453	−0.330567
$207$	0	0
$208$	1.45765	0.101070
$209$	−0.696629	−0.0481868
$210$	0	0
$211$	25.4308	1.75073	0.875364	−	0.483464i	$-0.160621\pi$
$211$	25.4308	1.75073	0.875364	+	0.483464i	$0.160621\pi$
$212$	7.37338	0.506406
$213$	0	0
$214$	0.207578	0.0141898
$215$	0.716901	0.0488923
$216$	0	0
$217$	6.57305	0.446208
$218$	−6.85936	−0.464575
$219$	0	0
$220$	−0.933858	−0.0629607
$221$	0.766295	0.0515466
$222$	0	0
$223$	−3.83134	−0.256565	−0.128283	−	0.991738i	$-0.540946\pi$
$223$	−3.83134	−0.256565	−0.128283	+	0.991738i	$0.540946\pi$
$224$	13.8561	0.925799
$225$	0	0
$226$	2.50011	0.166305
$227$	−2.51599	−0.166992	−0.0834961	−	0.996508i	$-0.526609\pi$
$227$	−2.51599	−0.166992	−0.0834961	+	0.996508i	$0.526609\pi$
$228$	0	0
$229$	−15.9396	−1.05332	−0.526660	−	0.850076i	$-0.676556\pi$
$229$	−15.9396	−1.05332	−0.526660	+	0.850076i	$0.676556\pi$
$230$	14.7449	0.972249
$231$	0	0
$232$	−0.955519	−0.0627329
$233$	−28.1283	−1.84274	−0.921372	−	0.388682i	$-0.872930\pi$
$233$	−28.1283	−1.84274	−0.921372	+	0.388682i	$0.872930\pi$
$234$	0	0
$235$	31.4185	2.04952
$236$	8.11400	0.528177
$237$	0	0
$238$	0.558911	0.0362288
$239$	14.7058	0.951238	0.475619	−	0.879651i	$-0.342224\pi$
$239$	14.7058	0.951238	0.475619	+	0.879651i	$0.342224\pi$
$240$	0	0
$241$	8.44295	0.543858	0.271929	−	0.962317i	$-0.412338\pi$
$241$	8.44295	0.543858	0.271929	+	0.962317i	$0.412338\pi$
$242$	−8.76664	−0.563541
$243$	0	0
$244$	16.0826	1.02958
$245$	−3.71410	−0.237285
$246$	0	0
$247$	−7.27371	−0.462815
$248$	−7.44234	−0.472589
$249$	0	0
$250$	−5.37435	−0.339904
$251$	23.2205	1.46566	0.732832	−	0.680409i	$-0.238198\pi$
$251$	23.2205	1.46566	0.732832	+	0.680409i	$0.238198\pi$
$252$	0	0
$253$	1.67352	0.105213
$254$	−14.7918	−0.928120
$255$	0	0
$256$	−14.1734	−0.885835
$257$	−6.86520	−0.428239	−0.214120	−	0.976807i	$-0.568688\pi$
$257$	−6.86520	−0.428239	−0.214120	+	0.976807i	$0.568688\pi$
$258$	0	0
$259$	−16.6197	−1.03270
$260$	−9.75069	−0.604712
$261$	0	0
$262$	−11.4021	−0.704424
$263$	−3.35294	−0.206751	−0.103376	−	0.994642i	$-0.532964\pi$
$263$	−3.35294	−0.206751	−0.103376	+	0.994642i	$0.532964\pi$
$264$	0	0
$265$	−14.9354	−0.917474
$266$	−5.30521	−0.325283
$267$	0	0
$268$	−2.45758	−0.150121
$269$	−12.7416	−0.776869	−0.388434	−	0.921476i	$-0.626984\pi$
$269$	−12.7416	−0.776869	−0.388434	+	0.921476i	$0.626984\pi$
$270$	0	0
$271$	−23.5566	−1.43096	−0.715481	−	0.698632i	$-0.753792\pi$
$271$	−23.5566	−1.43096	−0.715481	+	0.698632i	$0.753792\pi$
$272$	0.163723	0.00992716
$273$	0	0
$274$	15.7785	0.953216
$275$	0.640814	0.0386425
$276$	0	0
$277$	4.18122	0.251225	0.125613	−	0.992079i	$-0.459910\pi$
$277$	4.18122	0.251225	0.125613	+	0.992079i	$0.459910\pi$
$278$	−14.3561	−0.861023
$279$	0	0
$280$	−17.5893	−1.05116
$281$	21.6360	1.29070	0.645348	−	0.763888i	$-0.276712\pi$
$281$	21.6360	1.29070	0.645348	+	0.763888i	$0.276712\pi$
$282$	0	0
$283$	5.22734	0.310733	0.155366	−	0.987857i	$-0.450344\pi$
$283$	5.22734	0.310733	0.155366	+	0.987857i	$0.450344\pi$
$284$	−0.502987	−0.0298468
$285$	0	0
$286$	0.523726	0.0309686
$287$	23.0971	1.36338
$288$	0	0
$289$	−16.9139	−0.994937
$290$	0.782571	0.0459542
$291$	0	0
$292$	−6.82414	−0.399353
$293$	6.14217	0.358829	0.179415	−	0.983774i	$-0.442580\pi$
$293$	6.14217	0.358829	0.179415	+	0.983774i	$0.442580\pi$
$294$	0	0
$295$	−16.4356	−0.956918
$296$	18.8177	1.09376
$297$	0	0
$298$	−13.0564	−0.756339
$299$	17.4738	1.01053
$300$	0	0
$301$	0.619656	0.0357164
$302$	11.4417	0.658395
$303$	0	0
$304$	−1.55407	−0.0891317
$305$	−32.5767	−1.86534
$306$	0	0
$307$	19.0039	1.08461	0.542304	−	0.840182i	$-0.317552\pi$
$307$	19.0039	1.08461	0.542304	+	0.840182i	$0.317552\pi$
$308$	−0.807183	−0.0459935
$309$	0	0
$310$	6.09529	0.346189
$311$	21.5469	1.22181	0.610907	−	0.791703i	$-0.290805\pi$
$311$	21.5469	1.22181	0.610907	+	0.791703i	$0.290805\pi$
$312$	0	0
$313$	−3.81362	−0.215558	−0.107779	−	0.994175i	$-0.534374\pi$
$313$	−3.81362	−0.215558	−0.107779	+	0.994175i	$0.534374\pi$
$314$	0.611849	0.0345286
$315$	0	0
$316$	−1.08987	−0.0613102
$317$	−4.25449	−0.238956	−0.119478	−	0.992837i	$-0.538122\pi$
$317$	−4.25449	−0.238956	−0.119478	+	0.992837i	$0.538122\pi$
$318$	0	0
$319$	0.0888207	0.00497300
$320$	9.77978	0.546706
$321$	0	0
$322$	12.7448	0.710239
$323$	−0.816980	−0.0454580
$324$	0	0
$325$	6.69092	0.371146
$326$	4.11050	0.227660
$327$	0	0
$328$	−26.1517	−1.44399
$329$	27.1567	1.49719
$330$	0	0
$331$	−14.2938	−0.785658	−0.392829	−	0.919612i	$-0.628504\pi$
$331$	−14.2938	−0.785658	−0.392829	+	0.919612i	$0.628504\pi$
$332$	−3.74095	−0.205311
$333$	0	0
$334$	−7.13449	−0.390382
$335$	4.97804	0.271980
$336$	0	0
$337$	35.7185	1.94571	0.972855	−	0.231414i	$-0.0743352\pi$
$337$	35.7185	1.94571	0.972855	+	0.231414i	$0.0743352\pi$
$338$	−4.95147	−0.269324
$339$	0	0
$340$	−1.09519	−0.0593952
$341$	0.691806	0.0374634
$342$	0	0
$343$	−19.8481	−1.07170
$344$	−0.701606	−0.0378280
$345$	0	0
$346$	5.45939	0.293499
$347$	19.4415	1.04368	0.521838	−	0.853045i	$-0.325247\pi$
$347$	19.4415	1.04368	0.521838	+	0.853045i	$0.325247\pi$
$348$	0	0
$349$	−8.02070	−0.429338	−0.214669	−	0.976687i	$-0.568867\pi$
$349$	−8.02070	−0.429338	−0.214669	+	0.976687i	$0.568867\pi$
$350$	4.88014	0.260855
$351$	0	0
$352$	1.45834	0.0777296
$353$	−8.75381	−0.465919	−0.232959	−	0.972486i	$-0.574841\pi$
$353$	−8.75381	−0.465919	−0.232959	+	0.972486i	$0.574841\pi$
$354$	0	0
$355$	1.01884	0.0540746
$356$	14.1873	0.751928
$357$	0	0
$358$	14.7061	0.777243
$359$	8.27791	0.436892	0.218446	−	0.975849i	$-0.429901\pi$
$359$	8.27791	0.436892	0.218446	+	0.975849i	$0.429901\pi$
$360$	0	0
$361$	−11.2452	−0.591852
$362$	9.07781	0.477119
$363$	0	0
$364$	−8.42804	−0.441749
$365$	13.8229	0.723522
$366$	0	0
$367$	−14.7999	−0.772546	−0.386273	−	0.922384i	$-0.626238\pi$
$367$	−14.7999	−0.772546	−0.386273	+	0.922384i	$0.626238\pi$
$368$	3.73336	0.194615
$369$	0	0
$370$	−15.4117	−0.801216
$371$	−12.9095	−0.670226
$372$	0	0
$373$	25.5334	1.32207	0.661035	−	0.750355i	$-0.270118\pi$
$373$	25.5334	1.32207	0.661035	+	0.750355i	$0.270118\pi$
$374$	0.0588247	0.00304175
$375$	0	0
$376$	−30.7481	−1.58571
$377$	0.927403	0.0477637
$378$	0	0
$379$	20.1244	1.03372	0.516861	−	0.856070i	$-0.327101\pi$
$379$	20.1244	1.03372	0.516861	+	0.856070i	$0.327101\pi$
$380$	10.3956	0.533284
$381$	0	0
$382$	−5.49720	−0.281261
$383$	23.8613	1.21925	0.609627	−	0.792689i	$-0.291319\pi$
$383$	23.8613	1.21925	0.609627	+	0.792689i	$0.291319\pi$
$384$	0	0
$385$	1.63502	0.0833282
$386$	16.3614	0.832774
$387$	0	0
$388$	20.1388	1.02239
$389$	−37.9733	−1.92532	−0.962662	−	0.270708i	$-0.912742\pi$
$389$	−37.9733	−1.92532	−0.962662	+	0.270708i	$0.912742\pi$
$390$	0	0
$391$	1.96264	0.0992552
$392$	3.63486	0.183588
$393$	0	0
$394$	−2.43322	−0.122584
$395$	2.20763	0.111078
$396$	0	0
$397$	20.3493	1.02130	0.510651	−	0.859788i	$-0.329404\pi$
$397$	20.3493	1.02130	0.510651	+	0.859788i	$0.329404\pi$
$398$	−1.81343	−0.0908992
$399$	0	0
$400$	1.42955	0.0714775
$401$	−6.94497	−0.346815	−0.173408	−	0.984850i	$-0.555478\pi$
$401$	−6.94497	−0.346815	−0.173408	+	0.984850i	$0.555478\pi$
$402$	0	0
$403$	7.22335	0.359821
$404$	5.43745	0.270523
$405$	0	0
$406$	0.676418	0.0335701
$407$	−1.74921	−0.0867049
$408$	0	0
$409$	−10.9060	−0.539265	−0.269632	−	0.962963i	$-0.586902\pi$
$409$	−10.9060	−0.539265	−0.269632	+	0.962963i	$0.586902\pi$
$410$	21.4183	1.05777
$411$	0	0
$412$	8.03586	0.395898
$413$	−14.2062	−0.699040
$414$	0	0
$415$	7.57760	0.371970
$416$	15.2269	0.746562
$417$	0	0
$418$	−0.558367	−0.0273106
$419$	10.0692	0.491912	0.245956	−	0.969281i	$-0.420898\pi$
$419$	10.0692	0.491912	0.245956	+	0.969281i	$0.420898\pi$
$420$	0	0
$421$	−3.10756	−0.151453	−0.0757267	−	0.997129i	$-0.524128\pi$
$421$	−3.10756	−0.151453	−0.0757267	+	0.997129i	$0.524128\pi$
$422$	20.3835	0.992252
$423$	0	0
$424$	14.6167	0.709852
$425$	0.751522	0.0364542
$426$	0	0
$427$	−28.1578	−1.36265
$428$	−0.351577	−0.0169941
$429$	0	0
$430$	0.574616	0.0277104
$431$	28.0701	1.35209	0.676044	−	0.736862i	$-0.263693\pi$
$431$	28.0701	1.35209	0.676044	+	0.736862i	$0.263693\pi$
$432$	0	0
$433$	19.5251	0.938317	0.469158	−	0.883114i	$-0.344557\pi$
$433$	19.5251	0.938317	0.469158	+	0.883114i	$0.344557\pi$
$434$	5.26848	0.252895
$435$	0	0
$436$	11.6178	0.556391
$437$	−18.6295	−0.891170
$438$	0	0
$439$	14.6296	0.698232	0.349116	−	0.937080i	$-0.386482\pi$
$439$	14.6296	0.698232	0.349116	+	0.937080i	$0.386482\pi$
$440$	−1.85125	−0.0882548
$441$	0	0
$442$	0.614206	0.0292148
$443$	−18.3559	−0.872117	−0.436059	−	0.899918i	$-0.643626\pi$
$443$	−18.3559	−0.872117	−0.436059	+	0.899918i	$0.643626\pi$
$444$	0	0
$445$	−28.7377	−1.36230
$446$	−3.07092	−0.145412
$447$	0	0
$448$	8.45318	0.399375
$449$	−13.8594	−0.654065	−0.327032	−	0.945013i	$-0.606049\pi$
$449$	−13.8594	−0.654065	−0.327032	+	0.945013i	$0.606049\pi$
$450$	0	0
$451$	2.43095	0.114469
$452$	−4.23445	−0.199172
$453$	0	0
$454$	−2.01663	−0.0946454
$455$	17.0717	0.800334
$456$	0	0
$457$	17.6481	0.825545	0.412772	−	0.910834i	$-0.364561\pi$
$457$	17.6481	0.825545	0.412772	+	0.910834i	$0.364561\pi$
$458$	−12.7760	−0.596985
$459$	0	0
$460$	−24.9736	−1.16440
$461$	−25.6380	−1.19408	−0.597041	−	0.802211i	$-0.703657\pi$
$461$	−25.6380	−1.19408	−0.597041	+	0.802211i	$0.703657\pi$
$462$	0	0
$463$	18.3474	0.852677	0.426339	−	0.904564i	$-0.359803\pi$
$463$	18.3474	0.852677	0.426339	+	0.904564i	$0.359803\pi$
$464$	0.198144	0.00919863
$465$	0	0
$466$	−22.5456	−1.04440
$467$	16.2618	0.752509	0.376254	−	0.926516i	$-0.377212\pi$
$467$	16.2618	0.752509	0.376254	+	0.926516i	$0.377212\pi$
$468$	0	0
$469$	4.30279	0.198684
$470$	25.1828	1.16159
$471$	0	0
$472$	16.0849	0.740369
$473$	0.0652180	0.00299873
$474$	0	0
$475$	−7.13348	−0.327306
$476$	−0.946634	−0.0433889
$477$	0	0
$478$	11.7871	0.539129
$479$	9.47171	0.432773	0.216387	−	0.976308i	$-0.430573\pi$
$479$	9.47171	0.432773	0.216387	+	0.976308i	$0.430573\pi$
$480$	0	0
$481$	−18.2640	−0.832766
$482$	6.76725	0.308240
$483$	0	0
$484$	14.8481	0.674916
$485$	−40.7928	−1.85231
$486$	0	0
$487$	−0.467564	−0.0211874	−0.0105937	−	0.999944i	$-0.503372\pi$
$487$	−0.467564	−0.0211874	−0.0105937	+	0.999944i	$0.503372\pi$
$488$	31.8817	1.44321
$489$	0	0
$490$	−2.97695	−0.134485
$491$	−25.0470	−1.13035	−0.565177	−	0.824969i	$-0.691192\pi$
$491$	−25.0470	−1.13035	−0.565177	+	0.824969i	$0.691192\pi$
$492$	0	0
$493$	0.104166	0.00469138
$494$	−5.83007	−0.262307
$495$	0	0
$496$	1.54331	0.0692965
$497$	0.880640	0.0395021
$498$	0	0
$499$	−14.0342	−0.628255	−0.314128	−	0.949381i	$-0.601712\pi$
$499$	−14.0342	−0.628255	−0.314128	+	0.949381i	$0.601712\pi$
$500$	9.10259	0.407080
$501$	0	0
$502$	18.6119	0.830688
$503$	−28.3116	−1.26235	−0.631176	−	0.775640i	$-0.717427\pi$
$503$	−28.3116	−1.26235	−0.631176	+	0.775640i	$0.717427\pi$
$504$	0	0
$505$	−11.0140	−0.490117
$506$	1.34137	0.0596313
$507$	0	0
$508$	25.0530	1.11155
$509$	28.6875	1.27155	0.635774	−	0.771875i	$-0.280681\pi$
$509$	28.6875	1.27155	0.635774	+	0.771875i	$0.280681\pi$
$510$	0	0
$511$	11.9478	0.528541
$512$	6.25700	0.276523
$513$	0	0
$514$	−5.50264	−0.242711
$515$	−16.2773	−0.717264
$516$	0	0
$517$	2.85820	0.125704
$518$	−13.3212	−0.585298
$519$	0	0
$520$	−19.3294	−0.847652
$521$	24.9096	1.09131	0.545655	−	0.838010i	$-0.316281\pi$
$521$	24.9096	1.09131	0.545655	+	0.838010i	$0.316281\pi$
$522$	0	0
$523$	−25.8648	−1.13099	−0.565494	−	0.824753i	$-0.691314\pi$
$523$	−25.8648	−1.13099	−0.565494	+	0.824753i	$0.691314\pi$
$524$	19.3119	0.843642
$525$	0	0
$526$	−2.68747	−0.117179
$527$	0.811324	0.0353418
$528$	0	0
$529$	21.7540	0.945825
$530$	−11.9711	−0.519992
$531$	0	0
$532$	8.98549	0.389570
$533$	25.3822	1.09943
$534$	0	0
$535$	0.712150	0.0307889
$536$	−4.87183	−0.210431
$537$	0	0
$538$	−10.2127	−0.440302
$539$	−0.337880	−0.0145535
$540$	0	0
$541$	−21.9158	−0.942232	−0.471116	−	0.882071i	$-0.656149\pi$
$541$	−21.9158	−0.942232	−0.471116	+	0.882071i	$0.656149\pi$
$542$	−18.8813	−0.811019
$543$	0	0
$544$	1.71028	0.0733278
$545$	−23.5328	−1.00803
$546$	0	0
$547$	−9.97605	−0.426545	−0.213273	−	0.976993i	$-0.568412\pi$
$547$	−9.97605	−0.426545	−0.213273	+	0.976993i	$0.568412\pi$
$548$	−26.7243	−1.14160
$549$	0	0
$550$	0.513629	0.0219012
$551$	−0.988744	−0.0421219
$552$	0	0
$553$	1.90817	0.0811438
$554$	3.35136	0.142386
$555$	0	0
$556$	24.3151	1.03119
$557$	18.5330	0.785268	0.392634	−	0.919695i	$-0.371564\pi$
$557$	18.5330	0.785268	0.392634	+	0.919695i	$0.371564\pi$
$558$	0	0
$559$	0.680961	0.0288016
$560$	3.64746	0.154133
$561$	0	0
$562$	17.3418	0.731522
$563$	−43.6831	−1.84102	−0.920511	−	0.390716i	$-0.872228\pi$
$563$	−43.6831	−1.84102	−0.920511	+	0.390716i	$0.872228\pi$
$564$	0	0
$565$	8.57724	0.360847
$566$	4.18985	0.176113
$567$	0	0
$568$	−0.997105	−0.0418376
$569$	−13.5667	−0.568745	−0.284373	−	0.958714i	$-0.591785\pi$
$569$	−13.5667	−0.568745	−0.284373	+	0.958714i	$0.591785\pi$
$570$	0	0
$571$	23.7487	0.993853	0.496926	−	0.867793i	$-0.334462\pi$
$571$	23.7487	0.993853	0.496926	+	0.867793i	$0.334462\pi$
$572$	−0.887041	−0.0370890
$573$	0	0
$574$	18.5130	0.772717
$575$	17.1369	0.714657
$576$	0	0
$577$	−8.11902	−0.337999	−0.169000	−	0.985616i	$-0.554054\pi$
$577$	−8.11902	−0.337999	−0.169000	+	0.985616i	$0.554054\pi$
$578$	−13.5570	−0.563896
$579$	0	0
$580$	−1.32545	−0.0550363
$581$	6.54972	0.271728
$582$	0	0
$583$	−1.35871	−0.0562718
$584$	−13.5279	−0.559790
$585$	0	0
$586$	4.92311	0.203372
$587$	3.69199	0.152385	0.0761925	−	0.997093i	$-0.475724\pi$
$587$	3.69199	0.152385	0.0761925	+	0.997093i	$0.475724\pi$
$588$	0	0
$589$	−7.70112	−0.317319
$590$	−13.1736	−0.542348
$591$	0	0
$592$	−3.90219	−0.160379
$593$	−29.4590	−1.20974	−0.604869	−	0.796325i	$-0.706774\pi$
$593$	−29.4590	−1.20974	−0.604869	+	0.796325i	$0.706774\pi$
$594$	0	0
$595$	1.91749	0.0786093
$596$	22.1138	0.905818
$597$	0	0
$598$	14.0057	0.572735
$599$	21.8754	0.893804	0.446902	−	0.894583i	$-0.352527\pi$
$599$	21.8754	0.893804	0.446902	+	0.894583i	$0.352527\pi$
$600$	0	0
$601$	36.5207	1.48971	0.744854	−	0.667227i	$-0.232519\pi$
$601$	36.5207	1.48971	0.744854	+	0.667227i	$0.232519\pi$
$602$	0.496671	0.0202428
$603$	0	0
$604$	−19.3789	−0.788517
$605$	−30.0762	−1.22277
$606$	0	0
$607$	−6.58082	−0.267107	−0.133554	−	0.991042i	$-0.542639\pi$
$607$	−6.58082	−0.267107	−0.133554	+	0.991042i	$0.542639\pi$
$608$	−16.2341	−0.658379
$609$	0	0
$610$	−26.1111	−1.05721
$611$	29.8434	1.20733
$612$	0	0
$613$	−7.14867	−0.288732	−0.144366	−	0.989524i	$-0.546114\pi$
$613$	−7.14867	−0.288732	−0.144366	+	0.989524i	$0.546114\pi$
$614$	15.2321	0.614718
$615$	0	0
$616$	−1.60013	−0.0644712
$617$	−16.5375	−0.665774	−0.332887	−	0.942967i	$-0.608023\pi$
$617$	−16.5375	−0.665774	−0.332887	+	0.942967i	$0.608023\pi$
$618$	0	0
$619$	−1.49935	−0.0602640	−0.0301320	−	0.999546i	$-0.509593\pi$
$619$	−1.49935	−0.0602640	−0.0301320	+	0.999546i	$0.509593\pi$
$620$	−10.3236	−0.414608
$621$	0	0
$622$	17.2704	0.692481
$623$	−24.8395	−0.995173
$624$	0	0
$625$	−31.2462	−1.24985
$626$	−3.05672	−0.122171
$627$	0	0
$628$	−1.03630	−0.0413527
$629$	−2.05140	−0.0817948
$630$	0	0
$631$	−35.8913	−1.42881	−0.714404	−	0.699733i	$-0.753302\pi$
$631$	−35.8913	−1.42881	−0.714404	+	0.699733i	$0.753302\pi$
$632$	−2.16053	−0.0859413
$633$	0	0
$634$	−3.41009	−0.135432
$635$	−50.7470	−2.01383
$636$	0	0
$637$	−3.52790	−0.139781
$638$	0.0711922	0.00281852
$639$	0	0
$640$	−24.2224	−0.957476
$641$	−39.2279	−1.54941	−0.774705	−	0.632322i	$-0.782102\pi$
$641$	−39.2279	−1.54941	−0.774705	+	0.632322i	$0.782102\pi$
$642$	0	0
$643$	−10.4185	−0.410866	−0.205433	−	0.978671i	$-0.565860\pi$
$643$	−10.4185	−0.410866	−0.205433	+	0.978671i	$0.565860\pi$
$644$	−21.5860	−0.850607
$645$	0	0
$646$	−0.654831	−0.0257640
$647$	39.1517	1.53921	0.769606	−	0.638519i	$-0.220453\pi$
$647$	39.1517	1.53921	0.769606	+	0.638519i	$0.220453\pi$
$648$	0	0
$649$	−1.49518	−0.0586910
$650$	5.36296	0.210352
$651$	0	0
$652$	−6.96200	−0.272653
$653$	−32.9099	−1.28786	−0.643932	−	0.765083i	$-0.722698\pi$
$653$	−32.9099	−1.28786	−0.643932	+	0.765083i	$0.722698\pi$
$654$	0	0
$655$	−39.1178	−1.52846
$656$	5.42304	0.211734
$657$	0	0
$658$	21.7668	0.848558
$659$	21.5684	0.840186	0.420093	−	0.907481i	$-0.361997\pi$
$659$	21.5684	0.840186	0.420093	+	0.907481i	$0.361997\pi$
$660$	0	0
$661$	26.2964	1.02281	0.511405	−	0.859340i	$-0.329125\pi$
$661$	26.2964	1.02281	0.511405	+	0.859340i	$0.329125\pi$
$662$	−11.4569	−0.445283
$663$	0	0
$664$	−7.41593	−0.287794
$665$	−18.2009	−0.705799
$666$	0	0
$667$	2.37528	0.0919710
$668$	12.0838	0.467535
$669$	0	0
$670$	3.99004	0.154149
$671$	−2.96357	−0.114407
$672$	0	0
$673$	11.5221	0.444145	0.222073	−	0.975030i	$-0.428718\pi$
$673$	11.5221	0.444145	0.222073	+	0.975030i	$0.428718\pi$
$674$	28.6293	1.10276
$675$	0	0
$676$	8.38635	0.322552
$677$	33.9250	1.30384	0.651922	−	0.758286i	$-0.273963\pi$
$677$	33.9250	1.30384	0.651922	+	0.758286i	$0.273963\pi$
$678$	0	0
$679$	−35.2594	−1.35313
$680$	−2.17107	−0.0832569
$681$	0	0
$682$	0.554501	0.0212329
$683$	−36.7553	−1.40640	−0.703201	−	0.710991i	$-0.748247\pi$
$683$	−36.7553	−1.40640	−0.703201	+	0.710991i	$0.748247\pi$
$684$	0	0
$685$	54.1322	2.06829
$686$	−15.9088	−0.607401
$687$	0	0
$688$	0.145491	0.00554678
$689$	−14.1866	−0.540468
$690$	0	0
$691$	13.3781	0.508928	0.254464	−	0.967082i	$-0.418101\pi$
$691$	13.3781	0.508928	0.254464	+	0.967082i	$0.418101\pi$
$692$	−9.24663	−0.351504
$693$	0	0
$694$	15.5829	0.591519
$695$	−49.2523	−1.86825
$696$	0	0
$697$	2.85092	0.107986
$698$	−6.42881	−0.243334
$699$	0	0
$700$	−8.26555	−0.312409
$701$	−5.00452	−0.189018	−0.0945091	−	0.995524i	$-0.530128\pi$
$701$	−5.00452	−0.189018	−0.0945091	+	0.995524i	$0.530128\pi$
$702$	0	0
$703$	19.4720	0.734400
$704$	0.889687	0.0335314
$705$	0	0
$706$	−7.01642	−0.264066
$707$	−9.52000	−0.358036
$708$	0	0
$709$	17.1439	0.643851	0.321925	−	0.946765i	$-0.395670\pi$
$709$	17.1439	0.643851	0.321925	+	0.946765i	$0.395670\pi$
$710$	0.816630	0.0306476
$711$	0	0
$712$	28.1245	1.05401
$713$	18.5005	0.692850
$714$	0	0
$715$	1.79678	0.0671956
$716$	−24.9079	−0.930853
$717$	0	0
$718$	6.63497	0.247615
$719$	43.3519	1.61675	0.808377	−	0.588665i	$-0.200346\pi$
$719$	43.3519	1.61675	0.808377	+	0.588665i	$0.200346\pi$
$720$	0	0
$721$	−14.0693	−0.523969
$722$	−9.01333	−0.335441
$723$	0	0
$724$	−15.3752	−0.571414
$725$	0.909524	0.0337789
$726$	0	0
$727$	−36.3439	−1.34792	−0.673960	−	0.738768i	$-0.735408\pi$
$727$	−36.3439	−1.34792	−0.673960	+	0.738768i	$0.735408\pi$
$728$	−16.7075	−0.619220
$729$	0	0
$730$	11.0794	0.410067
$731$	0.0764852	0.00282891
$732$	0	0
$733$	3.87561	0.143149	0.0715744	−	0.997435i	$-0.477198\pi$
$733$	3.87561	0.143149	0.0715744	+	0.997435i	$0.477198\pi$
$734$	−11.8625	−0.437852
$735$	0	0
$736$	38.9994	1.43754
$737$	0.452863	0.0166814
$738$	0	0
$739$	26.4482	0.972913	0.486456	−	0.873705i	$-0.338289\pi$
$739$	26.4482	0.972913	0.486456	+	0.873705i	$0.338289\pi$
$740$	26.1030	0.959564
$741$	0	0
$742$	−10.3473	−0.379861
$743$	13.4634	0.493923	0.246961	−	0.969025i	$-0.420568\pi$
$743$	13.4634	0.493923	0.246961	+	0.969025i	$0.420568\pi$
$744$	0	0
$745$	−44.7934	−1.64110
$746$	20.4657	0.749303
$747$	0	0
$748$	−0.0996320	−0.00364291
$749$	0.615549	0.0224917
$750$	0	0
$751$	3.76830	0.137507	0.0687537	−	0.997634i	$-0.478098\pi$
$751$	3.76830	0.137507	0.0687537	+	0.997634i	$0.478098\pi$
$752$	6.37619	0.232516
$753$	0	0
$754$	0.743339	0.0270708
$755$	39.2536	1.42859
$756$	0	0
$757$	−33.7073	−1.22511	−0.612556	−	0.790427i	$-0.709859\pi$
$757$	−33.7073	−1.22511	−0.612556	+	0.790427i	$0.709859\pi$
$758$	16.1303	0.585877
$759$	0	0
$760$	20.6079	0.747528
$761$	9.65543	0.350009	0.175005	−	0.984568i	$-0.444006\pi$
$761$	9.65543	0.350009	0.175005	+	0.984568i	$0.444006\pi$
$762$	0	0
$763$	−20.3406	−0.736381
$764$	9.31067	0.336848
$765$	0	0
$766$	19.1254	0.691030
$767$	−15.6116	−0.563703
$768$	0	0
$769$	38.7110	1.39595	0.697977	−	0.716120i	$-0.254084\pi$
$769$	38.7110	1.39595	0.697977	+	0.716120i	$0.254084\pi$
$770$	1.31051	0.0472275
$771$	0	0
$772$	−27.7115	−0.997358
$773$	−24.3039	−0.874150	−0.437075	−	0.899425i	$-0.643986\pi$
$773$	−24.3039	−0.874150	−0.437075	+	0.899425i	$0.643986\pi$
$774$	0	0
$775$	7.08409	0.254468
$776$	39.9225	1.43313
$777$	0	0
$778$	−30.4366	−1.09121
$779$	−27.0611	−0.969563
$780$	0	0
$781$	0.0926863	0.00331658
$782$	1.57311	0.0562544
$783$	0	0
$784$	−0.753755	−0.0269198
$785$	2.09910	0.0749202
$786$	0	0
$787$	20.9406	0.746452	0.373226	−	0.927740i	$-0.378252\pi$
$787$	20.9406	0.746452	0.373226	+	0.927740i	$0.378252\pi$
$788$	4.12117	0.146810
$789$	0	0
$790$	1.76948	0.0629551
$791$	7.41377	0.263603
$792$	0	0
$793$	−30.9435	−1.09884
$794$	16.3105	0.578839
$795$	0	0
$796$	3.07143	0.108864
$797$	−11.9368	−0.422822	−0.211411	−	0.977397i	$-0.567806\pi$
$797$	−11.9368	−0.422822	−0.211411	+	0.977397i	$0.567806\pi$
$798$	0	0
$799$	3.35199	0.118585
$800$	14.9334	0.527974
$801$	0	0
$802$	−5.56658	−0.196563
$803$	1.25750	0.0443761
$804$	0	0
$805$	43.7242	1.54108
$806$	5.78971	0.203934
$807$	0	0
$808$	10.7790	0.379205
$809$	−8.60808	−0.302644	−0.151322	−	0.988485i	$-0.548353\pi$
$809$	−8.60808	−0.302644	−0.151322	+	0.988485i	$0.548353\pi$
$810$	0	0
$811$	1.53770	0.0539958	0.0269979	−	0.999635i	$-0.491405\pi$
$811$	1.53770	0.0539958	0.0269979	+	0.999635i	$0.491405\pi$
$812$	−1.14566	−0.0402046
$813$	0	0
$814$	−1.40204	−0.0491413
$815$	14.1021	0.493975
$816$	0	0
$817$	−0.726001	−0.0253996
$818$	−8.74141	−0.305636
$819$	0	0
$820$	−36.2764	−1.26683
$821$	−28.9763	−1.01128	−0.505639	−	0.862745i	$-0.668743\pi$
$821$	−28.9763	−1.01128	−0.505639	+	0.862745i	$0.668743\pi$
$822$	0	0
$823$	−11.2568	−0.392386	−0.196193	−	0.980565i	$-0.562858\pi$
$823$	−11.2568	−0.392386	−0.196193	+	0.980565i	$0.562858\pi$
$824$	15.9300	0.554948
$825$	0	0
$826$	−11.3866	−0.396191
$827$	−30.9279	−1.07547	−0.537734	−	0.843114i	$-0.680720\pi$
$827$	−30.9279	−1.07547	−0.537734	+	0.843114i	$0.680720\pi$
$828$	0	0
$829$	−9.83524	−0.341592	−0.170796	−	0.985306i	$-0.554634\pi$
$829$	−9.83524	−0.341592	−0.170796	+	0.985306i	$0.554634\pi$
$830$	6.07365	0.210820
$831$	0	0
$832$	9.28949	0.322055
$833$	−0.396253	−0.0137293
$834$	0	0
$835$	−24.4767	−0.847050
$836$	0.945711	0.0327081
$837$	0	0
$838$	8.07072	0.278798
$839$	13.1432	0.453754	0.226877	−	0.973923i	$-0.427148\pi$
$839$	13.1432	0.453754	0.226877	+	0.973923i	$0.427148\pi$
$840$	0	0
$841$	−28.8739	−0.995653
$842$	−2.49080	−0.0858385
$843$	0	0
$844$	−34.5237	−1.18835
$845$	−16.9873	−0.584380
$846$	0	0
$847$	−25.9964	−0.893248
$848$	−3.03105	−0.104087
$849$	0	0
$850$	0.602365	0.0206609
$851$	−46.7779	−1.60353
$852$	0	0
$853$	15.4423	0.528735	0.264368	−	0.964422i	$-0.414837\pi$
$853$	15.4423	0.528735	0.264368	+	0.964422i	$0.414837\pi$
$854$	−22.5692	−0.772303
$855$	0	0
$856$	−0.696955	−0.0238214
$857$	−21.9827	−0.750916	−0.375458	−	0.926839i	$-0.622515\pi$
$857$	−21.9827	−0.750916	−0.375458	+	0.926839i	$0.622515\pi$
$858$	0	0
$859$	19.5667	0.667606	0.333803	−	0.942643i	$-0.391668\pi$
$859$	19.5667	0.667606	0.333803	+	0.942643i	$0.391668\pi$
$860$	−0.973233	−0.0331870
$861$	0	0
$862$	22.4989	0.766316
$863$	−21.8676	−0.744383	−0.372191	−	0.928156i	$-0.621393\pi$
$863$	−21.8676	−0.744383	−0.372191	+	0.928156i	$0.621393\pi$
$864$	0	0
$865$	18.7298	0.636833
$866$	15.6499	0.531805
$867$	0	0
$868$	−8.92328	−0.302876
$869$	0.200833	0.00681279
$870$	0	0
$871$	4.72848	0.160218
$872$	23.0307	0.779918
$873$	0	0
$874$	−14.9320	−0.505084
$875$	−15.9370	−0.538769
$876$	0	0
$877$	−39.0999	−1.32031	−0.660155	−	0.751130i	$-0.729509\pi$
$877$	−39.0999	−1.32031	−0.660155	+	0.751130i	$0.729509\pi$
$878$	11.7260	0.395733
$879$	0	0
$880$	0.383890	0.0129409
$881$	7.30508	0.246115	0.123057	−	0.992400i	$-0.460730\pi$
$881$	7.30508	0.246115	0.123057	+	0.992400i	$0.460730\pi$
$882$	0	0
$883$	−3.49293	−0.117546	−0.0587732	−	0.998271i	$-0.518719\pi$
$883$	−3.49293	−0.117546	−0.0587732	+	0.998271i	$0.518719\pi$
$884$	−1.04029	−0.0349887
$885$	0	0
$886$	−14.7128	−0.494286
$887$	−28.4213	−0.954293	−0.477147	−	0.878824i	$-0.658329\pi$
$887$	−28.4213	−0.954293	−0.477147	+	0.878824i	$0.658329\pi$
$888$	0	0
$889$	−43.8633	−1.47113
$890$	−23.0340	−0.772102
$891$	0	0
$892$	5.20125	0.174151
$893$	−31.8173	−1.06472
$894$	0	0
$895$	50.4531	1.68646
$896$	−20.9367	−0.699447
$897$	0	0
$898$	−11.1087	−0.370701
$899$	0.981898	0.0327481
$900$	0	0
$901$	−1.59344	−0.0530851
$902$	1.94847	0.0648769
$903$	0	0
$904$	−8.39424	−0.279188
$905$	31.1437	1.03525
$906$	0	0
$907$	53.1167	1.76371	0.881855	−	0.471520i	$-0.156295\pi$
$907$	53.1167	1.76371	0.881855	+	0.471520i	$0.156295\pi$
$908$	3.41560	0.113351
$909$	0	0
$910$	13.6834	0.453601
$911$	8.07589	0.267566	0.133783	−	0.991011i	$-0.457287\pi$
$911$	8.07589	0.267566	0.133783	+	0.991011i	$0.457287\pi$
$912$	0	0
$913$	0.689351	0.0228142
$914$	14.1455	0.467890
$915$	0	0
$916$	21.6389	0.714970
$917$	−33.8116	−1.11656
$918$	0	0
$919$	−47.9961	−1.58325	−0.791623	−	0.611009i	$-0.790764\pi$
$919$	−47.9961	−1.58325	−0.791623	+	0.611009i	$0.790764\pi$
$920$	−49.5068	−1.63219
$921$	0	0
$922$	−20.5496	−0.676764
$923$	0.967766	0.0318544
$924$	0	0
$925$	−17.9119	−0.588938
$926$	14.7060	0.483268
$927$	0	0
$928$	2.06986	0.0679464
$929$	28.9939	0.951260	0.475630	−	0.879645i	$-0.342220\pi$
$929$	28.9939	0.951260	0.475630	+	0.879645i	$0.342220\pi$
$930$	0	0
$931$	3.76125	0.123270
$932$	38.1857	1.25081
$933$	0	0
$934$	13.0343	0.426496
$935$	0.201813	0.00659999
$936$	0	0
$937$	5.02850	0.164274	0.0821369	−	0.996621i	$-0.473826\pi$
$937$	5.02850	0.164274	0.0821369	+	0.996621i	$0.473826\pi$
$938$	3.44880	0.112607
$939$	0	0
$940$	−42.6523	−1.39116
$941$	55.8411	1.82037	0.910183	−	0.414206i	$-0.135941\pi$
$941$	55.8411	1.82037	0.910183	+	0.414206i	$0.135941\pi$
$942$	0	0
$943$	65.0092	2.11699
$944$	−3.33551	−0.108561
$945$	0	0
$946$	0.0522740	0.00169957
$947$	−42.5294	−1.38202	−0.691009	−	0.722846i	$-0.742834\pi$
$947$	−42.5294	−1.38202	−0.691009	+	0.722846i	$0.742834\pi$
$948$	0	0
$949$	13.1299	0.426214
$950$	−5.71767	−0.185506
$951$	0	0
$952$	−1.87657	−0.0608201
$953$	−21.8148	−0.706651	−0.353325	−	0.935501i	$-0.614949\pi$
$953$	−21.8148	−0.706651	−0.353325	+	0.935501i	$0.614949\pi$
$954$	0	0
$955$	−18.8595	−0.610280
$956$	−19.9639	−0.645679
$957$	0	0
$958$	7.59183	0.245281
$959$	46.7894	1.51091
$960$	0	0
$961$	−23.3522	−0.753297
$962$	−14.6391	−0.471983
$963$	0	0
$964$	−11.4618	−0.369158
$965$	56.1319	1.80695
$966$	0	0
$967$	4.62859	0.148845	0.0744227	−	0.997227i	$-0.476289\pi$
$967$	4.62859	0.148845	0.0744227	+	0.997227i	$0.476289\pi$
$968$	29.4345	0.946059
$969$	0	0
$970$	−32.6965	−1.04982
$971$	21.6509	0.694809	0.347405	−	0.937715i	$-0.387063\pi$
$971$	21.6509	0.694809	0.347405	+	0.937715i	$0.387063\pi$
$972$	0	0
$973$	−42.5714	−1.36478
$974$	−0.374766	−0.0120083
$975$	0	0
$976$	−6.61125	−0.211621
$977$	22.0848	0.706556	0.353278	−	0.935518i	$-0.385067\pi$
$977$	22.0848	0.706556	0.353278	+	0.935518i	$0.385067\pi$
$978$	0	0
$979$	−2.61433	−0.0835542
$980$	5.04210	0.161064
$981$	0	0
$982$	−20.0758	−0.640646
$983$	13.8630	0.442163	0.221081	−	0.975255i	$-0.429041\pi$
$983$	13.8630	0.442163	0.221081	+	0.975255i	$0.429041\pi$
$984$	0	0
$985$	−8.34777	−0.265982
$986$	0.0834915	0.00265891
$987$	0	0
$988$	9.87446	0.314148
$989$	1.74408	0.0554587
$990$	0	0
$991$	34.8224	1.10617	0.553084	−	0.833125i	$-0.313451\pi$
$991$	34.8224	1.10617	0.553084	+	0.833125i	$0.313451\pi$
$992$	16.1217	0.511864
$993$	0	0
$994$	0.705857	0.0223884
$995$	−6.22144	−0.197233
$996$	0	0
$997$	−24.7498	−0.783833	−0.391916	−	0.920001i	$-0.628188\pi$
$997$	−24.7498	−0.783833	−0.391916	+	0.920001i	$0.628188\pi$
$998$	−11.2488	−0.356073
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.d.1.4		6
3.2	odd	2		729.2.a.a.1.3		6
9.2	odd	6		729.2.c.e.244.4		12
9.4	even	3		729.2.c.b.487.3		12
9.5	odd	6		729.2.c.e.487.4		12
9.7	even	3		729.2.c.b.244.3		12
27.2	odd	18		27.2.e.a.4.2	✓	12
27.4	even	9		243.2.e.a.136.2		12
27.5	odd	18		243.2.e.c.217.1		12
27.7	even	9		243.2.e.a.109.2		12
27.11	odd	18		243.2.e.c.28.1		12
27.13	even	9		81.2.e.a.19.1		12
27.14	odd	18		27.2.e.a.7.2	yes	12
27.16	even	9		243.2.e.b.28.2		12
27.20	odd	18		243.2.e.d.109.1		12
27.22	even	9		243.2.e.b.217.2		12
27.23	odd	18		243.2.e.d.136.1		12
27.25	even	9		81.2.e.a.64.1		12
108.83	even	18		432.2.u.c.193.2		12
108.95	even	18		432.2.u.c.385.2		12
135.2	even	36		675.2.u.b.274.2		24
135.14	odd	18		675.2.l.c.601.1		12
135.29	odd	18		675.2.l.c.301.1		12
135.68	even	36		675.2.u.b.574.2		24
135.83	even	36		675.2.u.b.274.3		24
135.122	even	36		675.2.u.b.574.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.4.2	✓	12	27.2	odd	18
27.2.e.a.7.2	yes	12	27.14	odd	18
81.2.e.a.19.1		12	27.13	even	9
81.2.e.a.64.1		12	27.25	even	9
243.2.e.a.109.2		12	27.7	even	9
243.2.e.a.136.2		12	27.4	even	9
243.2.e.b.28.2		12	27.16	even	9
243.2.e.b.217.2		12	27.22	even	9
243.2.e.c.28.1		12	27.11	odd	18
243.2.e.c.217.1		12	27.5	odd	18
243.2.e.d.109.1		12	27.20	odd	18
243.2.e.d.136.1		12	27.23	odd	18
432.2.u.c.193.2		12	108.83	even	18
432.2.u.c.385.2		12	108.95	even	18
675.2.l.c.301.1		12	135.29	odd	18
675.2.l.c.601.1		12	135.14	odd	18
675.2.u.b.274.2		24	135.2	even	36
675.2.u.b.274.3		24	135.83	even	36
675.2.u.b.574.2		24	135.68	even	36
675.2.u.b.574.3		24	135.122	even	36
729.2.a.a.1.3		6	3.2	odd	2
729.2.a.d.1.4		6	1.1	even	1	trivial
729.2.c.b.244.3		12	9.7	even	3
729.2.c.b.487.3		12	9.4	even	3
729.2.c.e.244.4		12	9.2	odd	6
729.2.c.e.487.4		12	9.5	odd	6

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)

\(f(q)\)	\(=\)	\(q+0.801527 q^{2} -1.35755 q^{4} +2.74984 q^{5} +2.37683 q^{7} -2.69117 q^{8} +O(q^{10})\)	\(q+0.801527 q^{2} -1.35755 q^{4} +2.74984 q^{5} +2.37683 q^{7} -2.69117 q^{8} +2.20407 q^{10} +0.250159 q^{11} +2.61198 q^{13} +1.90510 q^{14} +0.558064 q^{16} +0.293377 q^{17} -2.78475 q^{19} -3.73306 q^{20} +0.200509 q^{22} +6.68984 q^{23} +2.56163 q^{25} +2.09357 q^{26} -3.22668 q^{28} +0.355057 q^{29} +2.76547 q^{31} +5.82964 q^{32} +0.235149 q^{34} +6.53592 q^{35} -6.99238 q^{37} -2.23205 q^{38} -7.40029 q^{40} +9.71761 q^{41} +0.260706 q^{43} -0.339604 q^{44} +5.36209 q^{46} +11.4256 q^{47} -1.35066 q^{49} +2.05321 q^{50} -3.54591 q^{52} -5.43137 q^{53} +0.687897 q^{55} -6.39646 q^{56} +0.284588 q^{58} -5.97693 q^{59} -11.8468 q^{61} +2.21660 q^{62} +3.55649 q^{64} +7.18254 q^{65} +1.81030 q^{67} -0.398275 q^{68} +5.23871 q^{70} +0.370510 q^{71} +5.02679 q^{73} -5.60458 q^{74} +3.78044 q^{76} +0.594586 q^{77} +0.802822 q^{79} +1.53459 q^{80} +7.78892 q^{82} +2.75565 q^{83} +0.806740 q^{85} +0.208963 q^{86} -0.673220 q^{88} -10.4507 q^{89} +6.20825 q^{91} -9.08183 q^{92} +9.15789 q^{94} -7.65761 q^{95} -14.8346 q^{97} -1.08259 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8}+O(q^{10}) \) 6 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^8	\( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35} + 3 q^{37} - 3 q^{38} + 6 q^{40} + 15 q^{41} + 3 q^{44} + 3 q^{46} + 21 q^{47} - 12 q^{49} + 3 q^{50} + 12 q^{52} + 9 q^{53} - 6 q^{55} - 6 q^{56} - 12 q^{58} + 24 q^{59} - 9 q^{61} - 12 q^{62} - 12 q^{64} - 6 q^{65} - 9 q^{67} - 9 q^{68} + 15 q^{70} + 27 q^{71} - 6 q^{73} - 12 q^{74} + 6 q^{76} - 12 q^{77} - 21 q^{80} - 6 q^{82} + 12 q^{83} - 21 q^{86} + 12 q^{88} + 9 q^{89} - 6 q^{91} + 6 q^{92} + 6 q^{94} + 12 q^{95} - 45 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^8 + 3 * q^10 + 12 * q^11 + 6 * q^14 - 3 * q^16 + 9 * q^17 + 3 * q^19 + 6 * q^20 + 6 * q^22 + 15 * q^23 - 6 * q^25 + 15 * q^26 - 6 * q^28 + 12 * q^29 + 12 * q^35 + 3 * q^37 - 3 * q^38 + 6 * q^40 + 15 * q^41 + 3 * q^44 + 3 * q^46 + 21 * q^47 - 12 * q^49 + 3 * q^50 + 12 * q^52 + 9 * q^53 - 6 * q^55 - 6 * q^56 - 12 * q^58 + 24 * q^59 - 9 * q^61 - 12 * q^62 - 12 * q^64 - 6 * q^65 - 9 * q^67 - 9 * q^68 + 15 * q^70 + 27 * q^71 - 6 * q^73 - 12 * q^74 + 6 * q^76 - 12 * q^77 - 21 * q^80 - 6 * q^82 + 12 * q^83 - 21 * q^86 + 12 * q^88 + 9 * q^89 - 6 * q^91 + 6 * q^92 + 6 * q^94 + 12 * q^95 - 45 * q^98

Introduction

L-functions

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