LMFDB - Embedded newform 1287.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14243 q^{2} +2.59002 q^{4} -0.447591 q^{5} +4.87834 q^{7} -1.26409 q^{8} +O(q^{10})$	\(q-2.14243 q^{2} +2.59002 q^{4} -0.447591 q^{5} +4.87834 q^{7} -1.26409 q^{8} +0.958933 q^{10} -1.00000 q^{11} -1.00000 q^{13} -10.4515 q^{14} -2.47182 q^{16} -6.61426 q^{17} -5.98316 q^{19} -1.15927 q^{20} +2.14243 q^{22} +0.694844 q^{23} -4.79966 q^{25} +2.14243 q^{26} +12.6350 q^{28} -2.14589 q^{29} -2.44759 q^{31} +7.82389 q^{32} +14.1706 q^{34} -2.18350 q^{35} +0.921318 q^{37} +12.8185 q^{38} +0.565795 q^{40} -1.12166 q^{41} +1.14243 q^{43} -2.59002 q^{44} -1.48866 q^{46} -8.67801 q^{47} +16.7982 q^{49} +10.2830 q^{50} -2.59002 q^{52} +5.01887 q^{53} +0.447591 q^{55} -6.16666 q^{56} +4.59742 q^{58} -7.47182 q^{59} -3.85806 q^{61} +5.24380 q^{62} -11.8185 q^{64} +0.447591 q^{65} -4.73246 q^{67} -17.1311 q^{68} +4.67801 q^{70} -8.60278 q^{71} +1.77353 q^{73} -1.97386 q^{74} -15.4965 q^{76} -4.87834 q^{77} -13.9180 q^{79} +1.10636 q^{80} +2.40307 q^{82} +4.57664 q^{83} +2.96048 q^{85} -2.44759 q^{86} +1.26409 q^{88} +12.3056 q^{89} -4.87834 q^{91} +1.79966 q^{92} +18.5921 q^{94} +2.67801 q^{95} -12.2182 q^{97} -35.9891 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8	$ 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8} - 4 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} - 2 q^{19} - 20 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{26} + 10 q^{28} - 4 q^{29} - 14 q^{31} - 6 q^{32} - 10 q^{34} - 6 q^{35} - 2 q^{37} + 8 q^{38} + 18 q^{40} - 18 q^{41} - 2 q^{43} - 8 q^{44} - 14 q^{46} - 2 q^{47} + 24 q^{50} - 8 q^{52} - 4 q^{53} + 6 q^{55} - 24 q^{58} - 16 q^{59} + 22 q^{61} + 4 q^{62} - 4 q^{64} + 6 q^{65} - 10 q^{67} + 4 q^{68} - 14 q^{70} + 2 q^{71} + 2 q^{73} - 22 q^{74} + 14 q^{76} - 6 q^{77} + 2 q^{79} - 6 q^{80} + 8 q^{82} - 4 q^{83} + 6 q^{85} - 14 q^{86} + 12 q^{88} + 16 q^{89} - 6 q^{91} - 12 q^{92} + 26 q^{94} - 22 q^{95} - 2 q^{97} - 34 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8 - 4 * q^11 - 4 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 - 2 * q^19 - 20 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^26 + 10 * q^28 - 4 * q^29 - 14 * q^31 - 6 * q^32 - 10 * q^34 - 6 * q^35 - 2 * q^37 + 8 * q^38 + 18 * q^40 - 18 * q^41 - 2 * q^43 - 8 * q^44 - 14 * q^46 - 2 * q^47 + 24 * q^50 - 8 * q^52 - 4 * q^53 + 6 * q^55 - 24 * q^58 - 16 * q^59 + 22 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 - 10 * q^67 + 4 * q^68 - 14 * q^70 + 2 * q^71 + 2 * q^73 - 22 * q^74 + 14 * q^76 - 6 * q^77 + 2 * q^79 - 6 * q^80 + 8 * q^82 - 4 * q^83 + 6 * q^85 - 14 * q^86 + 12 * q^88 + 16 * q^89 - 6 * q^91 - 12 * q^92 + 26 * q^94 - 22 * q^95 - 2 * q^97 - 34 * 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 $

$2$	−2.14243	−1.51493	−0.757465	−	0.652876i	$-0.773562\pi$
$2$	−2.14243	−1.51493	−0.757465	+	0.652876i	$0.773562\pi$
$3$	0	0
$3$	0	0
$4$	2.59002	1.29501
$5$	−0.447591	−0.200169	−0.100084	−	0.994979i	$-0.531911\pi$
$5$	−0.447591	−0.200169	−0.100084	+	0.994979i	$0.531911\pi$
$6$	0	0
$7$	4.87834	1.84384	0.921921	−	0.387379i	$-0.126620\pi$
$7$	4.87834	1.84384	0.921921	+	0.387379i	$0.126620\pi$
$8$	−1.26409	−0.446923
$9$	0	0
$10$	0.958933	0.303241
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−10.4515	−2.79329
$15$	0	0
$16$	−2.47182	−0.617955
$17$	−6.61426	−1.60419	−0.802096	−	0.597195i	$-0.796282\pi$
$17$	−6.61426	−1.60419	−0.802096	+	0.597195i	$0.796282\pi$
$18$	0	0
$19$	−5.98316	−1.37263	−0.686316	−	0.727304i	$-0.740773\pi$
$19$	−5.98316	−1.37263	−0.686316	+	0.727304i	$0.740773\pi$
$20$	−1.15927	−0.259221
$21$	0	0
$22$	2.14243	0.456769
$23$	0.694844	0.144885	0.0724425	−	0.997373i	$-0.476921\pi$
$23$	0.694844	0.144885	0.0724425	+	0.997373i	$0.476921\pi$
$24$	0	0
$25$	−4.79966	−0.959933
$26$	2.14243	0.420166
$27$	0	0
$28$	12.6350	2.38780
$29$	−2.14589	−0.398481	−0.199240	−	0.979951i	$-0.563847\pi$
$29$	−2.14589	−0.398481	−0.199240	+	0.979951i	$0.563847\pi$
$30$	0	0
$31$	−2.44759	−0.439600	−0.219800	−	0.975545i	$-0.570541\pi$
$31$	−2.44759	−0.439600	−0.219800	+	0.975545i	$0.570541\pi$
$32$	7.82389	1.38308
$33$	0	0
$34$	14.1706	2.43024
$35$	−2.18350	−0.369079
$36$	0	0
$37$	0.921318	0.151464	0.0757319	−	0.997128i	$-0.475871\pi$
$37$	0.921318	0.151464	0.0757319	+	0.997128i	$0.475871\pi$
$38$	12.8185	2.07944
$39$	0	0
$40$	0.565795	0.0894600
$41$	−1.12166	−0.175173	−0.0875866	−	0.996157i	$-0.527915\pi$
$41$	−1.12166	−0.175173	−0.0875866	+	0.996157i	$0.527915\pi$
$42$	0	0
$43$	1.14243	0.174220	0.0871098	−	0.996199i	$-0.472237\pi$
$43$	1.14243	0.174220	0.0871098	+	0.996199i	$0.472237\pi$
$44$	−2.59002	−0.390461
$45$	0	0
$46$	−1.48866	−0.219491
$47$	−8.67801	−1.26582	−0.632909	−	0.774226i	$-0.718139\pi$
$47$	−8.67801	−1.26582	−0.632909	+	0.774226i	$0.718139\pi$
$48$	0	0
$49$	16.7982	2.39975
$50$	10.2830	1.45423
$51$	0	0
$52$	−2.59002	−0.359172
$53$	5.01887	0.689395	0.344698	−	0.938714i	$-0.387981\pi$
$53$	5.01887	0.689395	0.344698	+	0.938714i	$0.387981\pi$
$54$	0	0
$55$	0.447591	0.0603531
$56$	−6.16666	−0.824055
$57$	0	0
$58$	4.59742	0.603671
$59$	−7.47182	−0.972748	−0.486374	−	0.873751i	$-0.661681\pi$
$59$	−7.47182	−0.972748	−0.486374	+	0.873751i	$0.661681\pi$
$60$	0	0
$61$	−3.85806	−0.493974	−0.246987	−	0.969019i	$-0.579440\pi$
$61$	−3.85806	−0.493974	−0.246987	+	0.969019i	$0.579440\pi$
$62$	5.24380	0.665964
$63$	0	0
$64$	−11.8185	−1.47732
$65$	0.447591	0.0555168
$66$	0	0
$67$	−4.73246	−0.578162	−0.289081	−	0.957305i	$-0.593350\pi$
$67$	−4.73246	−0.578162	−0.289081	+	0.957305i	$0.593350\pi$
$68$	−17.1311	−2.07745
$69$	0	0
$70$	4.67801	0.559129
$71$	−8.60278	−1.02096	−0.510481	−	0.859889i	$-0.670533\pi$
$71$	−8.60278	−1.02096	−0.510481	+	0.859889i	$0.670533\pi$
$72$	0	0
$73$	1.77353	0.207576	0.103788	−	0.994599i	$-0.466904\pi$
$73$	1.77353	0.207576	0.103788	+	0.994599i	$0.466904\pi$
$74$	−1.97386	−0.229457
$75$	0	0
$76$	−15.4965	−1.77758
$77$	−4.87834	−0.555939
$78$	0	0
$79$	−13.9180	−1.56590	−0.782948	−	0.622087i	$-0.786285\pi$
$79$	−13.9180	−1.56590	−0.782948	+	0.622087i	$0.786285\pi$
$80$	1.10636	0.123695
$81$	0	0
$82$	2.40307	0.265375
$83$	4.57664	0.502351	0.251176	−	0.967941i	$-0.419183\pi$
$83$	4.57664	0.502351	0.251176	+	0.967941i	$0.419183\pi$
$84$	0	0
$85$	2.96048	0.321109
$86$	−2.44759	−0.263930
$87$	0	0
$88$	1.26409	0.134752
$89$	12.3056	1.30440	0.652198	−	0.758049i	$-0.273847\pi$
$89$	12.3056	1.30440	0.652198	+	0.758049i	$0.273847\pi$
$90$	0	0
$91$	−4.87834	−0.511389
$92$	1.79966	0.187628
$93$	0	0
$94$	18.5921	1.91762
$95$	2.67801	0.274758
$96$	0	0
$97$	−12.2182	−1.24057	−0.620283	−	0.784378i	$-0.712982\pi$
$97$	−12.2182	−1.24057	−0.620283	+	0.784378i	$0.712982\pi$
$98$	−35.9891	−3.63545
$99$	0	0
$100$	−12.4312	−1.24312
$101$	19.0980	1.90033	0.950163	−	0.311753i	$-0.100916\pi$
$101$	19.0980	1.90033	0.950163	+	0.311753i	$0.100916\pi$
$102$	0	0
$103$	−4.81305	−0.474244	−0.237122	−	0.971480i	$-0.576204\pi$
$103$	−4.81305	−0.474244	−0.237122	+	0.971480i	$0.576204\pi$
$104$	1.26409	0.123954
$105$	0	0
$106$	−10.7526	−1.04439
$107$	14.6934	1.42047	0.710234	−	0.703966i	$-0.248589\pi$
$107$	14.6934	1.42047	0.710234	+	0.703966i	$0.248589\pi$
$108$	0	0
$109$	6.34326	0.607574	0.303787	−	0.952740i	$-0.401749\pi$
$109$	6.34326	0.607574	0.303787	+	0.952740i	$0.401749\pi$
$110$	−0.958933	−0.0914307
$111$	0	0
$112$	−12.0584	−1.13941
$113$	−15.0120	−1.41221	−0.706104	−	0.708108i	$-0.749549\pi$
$113$	−15.0120	−1.41221	−0.706104	+	0.708108i	$0.749549\pi$
$114$	0	0
$115$	−0.311006	−0.0290014
$116$	−5.55790	−0.516038
$117$	0	0
$118$	16.0079	1.47365
$119$	−32.2666	−2.95788
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.26563	0.748335
$123$	0	0
$124$	−6.33932	−0.569288
$125$	4.38624	0.392317
$126$	0	0
$127$	14.9253	1.32440	0.662201	−	0.749326i	$-0.269622\pi$
$127$	14.9253	1.32440	0.662201	+	0.749326i	$0.269622\pi$
$128$	9.67265	0.854950
$129$	0	0
$130$	−0.958933	−0.0841040
$131$	0.744719	0.0650664	0.0325332	−	0.999471i	$-0.489643\pi$
$131$	0.744719	0.0650664	0.0325332	+	0.999471i	$0.489643\pi$
$132$	0	0
$133$	−29.1879	−2.53091
$134$	10.1390	0.875875
$135$	0	0
$136$	8.36101	0.716951
$137$	−2.94963	−0.252004	−0.126002	−	0.992030i	$-0.540215\pi$
$137$	−2.94963	−0.252004	−0.126002	+	0.992030i	$0.540215\pi$
$138$	0	0
$139$	7.25070	0.614997	0.307498	−	0.951549i	$-0.400508\pi$
$139$	7.25070	0.614997	0.307498	+	0.951549i	$0.400508\pi$
$140$	−5.65532	−0.477962
$141$	0	0
$142$	18.4309	1.54669
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.960478	0.0797634
$146$	−3.79966	−0.314462
$147$	0	0
$148$	2.38624	0.196147
$149$	−22.3096	−1.82767	−0.913836	−	0.406082i	$-0.866895\pi$
$149$	−22.3096	−1.82767	−0.913836	+	0.406082i	$0.866895\pi$
$150$	0	0
$151$	−21.7330	−1.76860	−0.884301	−	0.466917i	$-0.845365\pi$
$151$	−21.7330	−1.76860	−0.884301	+	0.466917i	$0.845365\pi$
$152$	7.56325	0.613461
$153$	0	0
$154$	10.4515	0.842209
$155$	1.09552	0.0879942
$156$	0	0
$157$	−15.5564	−1.24153	−0.620766	−	0.783996i	$-0.713178\pi$
$157$	−15.5564	−1.24153	−0.620766	+	0.783996i	$0.713178\pi$
$158$	29.8184	2.37222
$159$	0	0
$160$	−3.50190	−0.276850
$161$	3.38969	0.267145
$162$	0	0
$163$	−15.2761	−1.19651	−0.598257	−	0.801305i	$-0.704140\pi$
$163$	−15.2761	−1.19651	−0.598257	+	0.801305i	$0.704140\pi$
$164$	−2.90511	−0.226851
$165$	0	0
$166$	−9.80515	−0.761027
$167$	14.7794	1.14366	0.571831	−	0.820371i	$-0.306233\pi$
$167$	14.7794	1.14366	0.571831	+	0.820371i	$0.306233\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−6.34263	−0.486458
$171$	0	0
$172$	2.95893	0.225617
$173$	4.89912	0.372473	0.186237	−	0.982505i	$-0.440371\pi$
$173$	4.89912	0.372473	0.186237	+	0.982505i	$0.440371\pi$
$174$	0	0
$175$	−23.4144	−1.76996
$176$	2.47182	0.186321
$177$	0	0
$178$	−26.3640	−1.97607
$179$	0.776977	0.0580740	0.0290370	−	0.999578i	$-0.490756\pi$
$179$	0.776977	0.0580740	0.0290370	+	0.999578i	$0.490756\pi$
$180$	0	0
$181$	1.01339	0.0753243	0.0376622	−	0.999291i	$-0.488009\pi$
$181$	1.01339	0.0753243	0.0376622	+	0.999291i	$0.488009\pi$
$182$	10.4515	0.774719
$183$	0	0
$184$	−0.878345	−0.0647524
$185$	−0.412373	−0.0303183
$186$	0	0
$187$	6.61426	0.483682
$188$	−22.4763	−1.63925
$189$	0	0
$190$	−5.73746	−0.416239
$191$	−16.6182	−1.20245	−0.601225	−	0.799080i	$-0.705321\pi$
$191$	−16.6182	−1.20245	−0.601225	+	0.799080i	$0.705321\pi$
$192$	0	0
$193$	−14.6766	−1.05644	−0.528222	−	0.849106i	$-0.677141\pi$
$193$	−14.6766	−1.05644	−0.528222	+	0.849106i	$0.677141\pi$
$194$	26.1766	1.87937
$195$	0	0
$196$	43.5079	3.10771
$197$	−15.5233	−1.10599	−0.552995	−	0.833184i	$-0.686515\pi$
$197$	−15.5233	−1.10599	−0.552995	+	0.833184i	$0.686515\pi$
$198$	0	0
$199$	26.4164	1.87261	0.936306	−	0.351185i	$-0.114221\pi$
$199$	26.4164	1.87261	0.936306	+	0.351185i	$0.114221\pi$
$200$	6.06720	0.429016
$201$	0	0
$202$	−40.9163	−2.87886
$203$	−10.4684	−0.734736
$204$	0	0
$205$	0.502042	0.0350642
$206$	10.3116	0.718446
$207$	0	0
$208$	2.47182	0.171390
$209$	5.98316	0.413864
$210$	0	0
$211$	23.1572	1.59421	0.797104	−	0.603841i	$-0.206364\pi$
$211$	23.1572	1.59421	0.797104	+	0.603841i	$0.206364\pi$
$212$	12.9990	0.892775
$213$	0	0
$214$	−31.4797	−2.15191
$215$	−0.511343	−0.0348733
$216$	0	0
$217$	−11.9402	−0.810553
$218$	−13.5900	−0.920433
$219$	0	0
$220$	1.15927	0.0781580
$221$	6.61426	0.444923
$222$	0	0
$223$	−20.5376	−1.37530	−0.687650	−	0.726042i	$-0.741358\pi$
$223$	−20.5376	−1.37530	−0.687650	+	0.726042i	$0.741358\pi$
$224$	38.1676	2.55018
$225$	0	0
$226$	32.1622	2.13940
$227$	21.1711	1.40518	0.702588	−	0.711597i	$-0.252028\pi$
$227$	21.1711	1.40518	0.702588	+	0.711597i	$0.252028\pi$
$228$	0	0
$229$	7.00345	0.462801	0.231401	−	0.972859i	$-0.425669\pi$
$229$	7.00345	0.462801	0.231401	+	0.972859i	$0.425669\pi$
$230$	0.666309	0.0439351
$231$	0	0
$232$	2.71259	0.178090
$233$	12.9668	0.849485	0.424742	−	0.905314i	$-0.360365\pi$
$233$	12.9668	0.849485	0.424742	+	0.905314i	$0.360365\pi$
$234$	0	0
$235$	3.88419	0.253377
$236$	−19.3522	−1.25972
$237$	0	0
$238$	69.1291	4.48097
$239$	7.33129	0.474222	0.237111	−	0.971483i	$-0.423800\pi$
$239$	7.33129	0.474222	0.237111	+	0.971483i	$0.423800\pi$
$240$	0	0
$241$	−4.51746	−0.290995	−0.145498	−	0.989359i	$-0.546478\pi$
$241$	−4.51746	−0.290995	−0.145498	+	0.989359i	$0.546478\pi$
$242$	−2.14243	−0.137721
$243$	0	0
$244$	−9.99246	−0.639702
$245$	−7.51874	−0.480354
$246$	0	0
$247$	5.98316	0.380700
$248$	3.09397	0.196468
$249$	0	0
$250$	−9.39722	−0.594333
$251$	6.16118	0.388890	0.194445	−	0.980913i	$-0.437709\pi$
$251$	6.16118	0.388890	0.194445	+	0.980913i	$0.437709\pi$
$252$	0	0
$253$	−0.694844	−0.0436845
$254$	−31.9764	−2.00638
$255$	0	0
$256$	2.91405	0.182128
$257$	−24.1760	−1.50806	−0.754028	−	0.656843i	$-0.771892\pi$
$257$	−24.1760	−1.50806	−0.754028	+	0.656843i	$0.771892\pi$
$258$	0	0
$259$	4.49451	0.279275
$260$	1.15927	0.0718949
$261$	0	0
$262$	−1.59551	−0.0985711
$263$	−27.5549	−1.69911	−0.849555	−	0.527500i	$-0.823130\pi$
$263$	−27.5549	−1.69911	−0.849555	+	0.527500i	$0.823130\pi$
$264$	0	0
$265$	−2.24640	−0.137995
$266$	62.5332	3.83416
$267$	0	0
$268$	−12.2572	−0.748727
$269$	24.6251	1.50142	0.750709	−	0.660633i	$-0.229712\pi$
$269$	24.6251	1.50142	0.750709	+	0.660633i	$0.229712\pi$
$270$	0	0
$271$	3.16321	0.192152	0.0960758	−	0.995374i	$-0.469371\pi$
$271$	3.16321	0.192152	0.0960758	+	0.995374i	$0.469371\pi$
$272$	16.3493	0.991319
$273$	0	0
$274$	6.31939	0.381769
$275$	4.79966	0.289431
$276$	0	0
$277$	−10.3333	−0.620870	−0.310435	−	0.950595i	$-0.600475\pi$
$277$	−10.3333	−0.620870	−0.310435	+	0.950595i	$0.600475\pi$
$278$	−15.5342	−0.931677
$279$	0	0
$280$	2.76014	0.164950
$281$	16.1415	0.962922	0.481461	−	0.876468i	$-0.340106\pi$
$281$	16.1415	0.962922	0.481461	+	0.876468i	$0.340106\pi$
$282$	0	0
$283$	4.99359	0.296838	0.148419	−	0.988925i	$-0.452582\pi$
$283$	4.99359	0.296838	0.148419	+	0.988925i	$0.452582\pi$
$284$	−22.2814	−1.32216
$285$	0	0
$286$	−2.14243	−0.126685
$287$	−5.47182	−0.322991
$288$	0	0
$289$	26.7484	1.57343
$290$	−2.05776	−0.120836
$291$	0	0
$292$	4.59348	0.268813
$293$	5.76662	0.336890	0.168445	−	0.985711i	$-0.446125\pi$
$293$	5.76662	0.336890	0.168445	+	0.985711i	$0.446125\pi$
$294$	0	0
$295$	3.34432	0.194714
$296$	−1.16463	−0.0676926
$297$	0	0
$298$	47.7968	2.76880
$299$	−0.694844	−0.0401839
$300$	0	0
$301$	5.57319	0.321233
$302$	46.5614	2.67931
$303$	0	0
$304$	14.7893	0.848225
$305$	1.72683	0.0988780
$306$	0	0
$307$	15.9240	0.908830	0.454415	−	0.890790i	$-0.349848\pi$
$307$	15.9240	0.908830	0.454415	+	0.890790i	$0.349848\pi$
$308$	−12.6350	−0.719948
$309$	0	0
$310$	−2.34708	−0.133305
$311$	−14.5346	−0.824184	−0.412092	−	0.911142i	$-0.635202\pi$
$311$	−14.5346	−0.824184	−0.412092	+	0.911142i	$0.635202\pi$
$312$	0	0
$313$	11.0996	0.627386	0.313693	−	0.949524i	$-0.398434\pi$
$313$	11.0996	0.627386	0.313693	+	0.949524i	$0.398434\pi$
$314$	33.3285	1.88083
$315$	0	0
$316$	−36.0480	−2.02786
$317$	−26.5461	−1.49098	−0.745490	−	0.666517i	$-0.767784\pi$
$317$	−26.5461	−1.49098	−0.745490	+	0.666517i	$0.767784\pi$
$318$	0	0
$319$	2.14589	0.120147
$320$	5.28986	0.295712
$321$	0	0
$322$	−7.26218	−0.404706
$323$	39.5742	2.20197
$324$	0	0
$325$	4.79966	0.266237
$326$	32.7280	1.81263
$327$	0	0
$328$	1.41787	0.0782889
$329$	−42.3343	−2.33397
$330$	0	0
$331$	−20.8716	−1.14721	−0.573603	−	0.819134i	$-0.694455\pi$
$331$	−20.8716	−1.14721	−0.573603	+	0.819134i	$0.694455\pi$
$332$	11.8536	0.650551
$333$	0	0
$334$	−31.6638	−1.73257
$335$	2.11820	0.115730
$336$	0	0
$337$	−8.19892	−0.446624	−0.223312	−	0.974747i	$-0.571687\pi$
$337$	−8.19892	−0.446624	−0.223312	+	0.974747i	$0.571687\pi$
$338$	−2.14243	−0.116533
$339$	0	0
$340$	7.66771	0.415840
$341$	2.44759	0.132544
$342$	0	0
$343$	47.7992	2.58092
$344$	−1.44414	−0.0778628
$345$	0	0
$346$	−10.4961	−0.564271
$347$	−16.7879	−0.901221	−0.450611	−	0.892721i	$-0.648794\pi$
$347$	−16.7879	−0.901221	−0.450611	+	0.892721i	$0.648794\pi$
$348$	0	0
$349$	9.72302	0.520461	0.260231	−	0.965547i	$-0.416201\pi$
$349$	9.72302	0.520461	0.260231	+	0.965547i	$0.416201\pi$
$350$	50.1638	2.68137
$351$	0	0
$352$	−7.82389	−0.417015
$353$	−14.0875	−0.749801	−0.374901	−	0.927065i	$-0.622323\pi$
$353$	−14.0875	−0.749801	−0.374901	+	0.927065i	$0.622323\pi$
$354$	0	0
$355$	3.85052	0.204364
$356$	31.8719	1.68921
$357$	0	0
$358$	−1.66462	−0.0879780
$359$	−6.76156	−0.356861	−0.178431	−	0.983952i	$-0.557102\pi$
$359$	−6.76156	−0.356861	−0.178431	+	0.983952i	$0.557102\pi$
$360$	0	0
$361$	16.7982	0.884118
$362$	−2.17111	−0.114111
$363$	0	0
$364$	−12.6350	−0.662256
$365$	−0.793813	−0.0415501
$366$	0	0
$367$	−24.6705	−1.28779	−0.643894	−	0.765115i	$-0.722682\pi$
$367$	−24.6705	−1.28779	−0.643894	+	0.765115i	$0.722682\pi$
$368$	−1.71753	−0.0895324
$369$	0	0
$370$	0.883482	0.0459301
$371$	24.4838	1.27114
$372$	0	0
$373$	7.09792	0.367516	0.183758	−	0.982971i	$-0.441174\pi$
$373$	7.09792	0.367516	0.183758	+	0.982971i	$0.441174\pi$
$374$	−14.1706	−0.732745
$375$	0	0
$376$	10.9698	0.565723
$377$	2.14589	0.110519
$378$	0	0
$379$	−5.09094	−0.261504	−0.130752	−	0.991415i	$-0.541739\pi$
$379$	−5.09094	−0.261504	−0.130752	+	0.991415i	$0.541739\pi$
$380$	6.93611	0.355815
$381$	0	0
$382$	35.6034	1.82163
$383$	21.7567	1.11171	0.555857	−	0.831278i	$-0.312390\pi$
$383$	21.7567	1.11171	0.555857	+	0.831278i	$0.312390\pi$
$384$	0	0
$385$	2.18350	0.111282
$386$	31.4436	1.60044
$387$	0	0
$388$	−31.6453	−1.60655
$389$	−10.0347	−0.508777	−0.254389	−	0.967102i	$-0.581874\pi$
$389$	−10.0347	−0.508777	−0.254389	+	0.967102i	$0.581874\pi$
$390$	0	0
$391$	−4.59587	−0.232423
$392$	−21.2345	−1.07250
$393$	0	0
$394$	33.2577	1.67550
$395$	6.22956	0.313443
$396$	0	0
$397$	−23.2141	−1.16508	−0.582540	−	0.812802i	$-0.697941\pi$
$397$	−23.2141	−1.16508	−0.582540	+	0.812802i	$0.697941\pi$
$398$	−56.5955	−2.83688
$399$	0	0
$400$	11.8639	0.593195
$401$	−10.3239	−0.515551	−0.257775	−	0.966205i	$-0.582989\pi$
$401$	−10.3239	−0.515551	−0.257775	+	0.966205i	$0.582989\pi$
$402$	0	0
$403$	2.44759	0.121923
$404$	49.4644	2.46095
$405$	0	0
$406$	22.4278	1.11307
$407$	−0.921318	−0.0456680
$408$	0	0
$409$	37.2532	1.84205	0.921027	−	0.389499i	$-0.127352\pi$
$409$	37.2532	1.84205	0.921027	+	0.389499i	$0.127352\pi$
$410$	−1.07559	−0.0531197
$411$	0	0
$412$	−12.4659	−0.614151
$413$	−36.4501	−1.79359
$414$	0	0
$415$	−2.04846	−0.100555
$416$	−7.82389	−0.383598
$417$	0	0
$418$	−12.8185	−0.626975
$419$	36.8264	1.79909	0.899544	−	0.436830i	$-0.143899\pi$
$419$	36.8264	1.79909	0.899544	+	0.436830i	$0.143899\pi$
$420$	0	0
$421$	39.1366	1.90740	0.953702	−	0.300754i	$-0.0972384\pi$
$421$	39.1366	1.90740	0.953702	+	0.300754i	$0.0972384\pi$
$422$	−49.6128	−2.41511
$423$	0	0
$424$	−6.34430	−0.308107
$425$	31.7462	1.53992
$426$	0	0
$427$	−18.8209	−0.910809
$428$	38.0563	1.83952
$429$	0	0
$430$	1.09552	0.0528306
$431$	−31.1987	−1.50279	−0.751393	−	0.659855i	$-0.770618\pi$
$431$	−31.1987	−1.50279	−0.751393	+	0.659855i	$0.770618\pi$
$432$	0	0
$433$	24.4432	1.17467	0.587333	−	0.809345i	$-0.300178\pi$
$433$	24.4432	1.17467	0.587333	+	0.809345i	$0.300178\pi$
$434$	25.5811	1.22793
$435$	0	0
$436$	16.4292	0.786816
$437$	−4.15736	−0.198874
$438$	0	0
$439$	28.0049	1.33660	0.668301	−	0.743891i	$-0.267022\pi$
$439$	28.0049	1.33660	0.668301	+	0.743891i	$0.267022\pi$
$440$	−0.565795	−0.0269732
$441$	0	0
$442$	−14.1706	−0.674027
$443$	0.164267	0.00780454	0.00390227	−	0.999992i	$-0.498758\pi$
$443$	0.164267	0.00780454	0.00390227	+	0.999992i	$0.498758\pi$
$444$	0	0
$445$	−5.50789	−0.261099
$446$	44.0005	2.08348
$447$	0	0
$448$	−57.6549	−2.72394
$449$	−16.0469	−0.757301	−0.378650	−	0.925540i	$-0.623612\pi$
$449$	−16.0469	−0.757301	−0.378650	+	0.925540i	$0.623612\pi$
$450$	0	0
$451$	1.12166	0.0528167
$452$	−38.8814	−1.82883
$453$	0	0
$454$	−45.3577	−2.12874
$455$	2.18350	0.102364
$456$	0	0
$457$	20.4262	0.955496	0.477748	−	0.878497i	$-0.341453\pi$
$457$	20.4262	0.955496	0.477748	+	0.878497i	$0.341453\pi$
$458$	−15.0044	−0.701111
$459$	0	0
$460$	−0.805512	−0.0375572
$461$	4.60136	0.214307	0.107153	−	0.994243i	$-0.465826\pi$
$461$	4.60136	0.214307	0.107153	+	0.994243i	$0.465826\pi$
$462$	0	0
$463$	14.1414	0.657206	0.328603	−	0.944468i	$-0.393422\pi$
$463$	14.1414	0.657206	0.328603	+	0.944468i	$0.393422\pi$
$464$	5.30424	0.246243
$465$	0	0
$466$	−27.7806	−1.28691
$467$	−37.6624	−1.74281	−0.871405	−	0.490565i	$-0.836790\pi$
$467$	−37.6624	−1.74281	−0.871405	+	0.490565i	$0.836790\pi$
$468$	0	0
$469$	−23.0866	−1.06604
$470$	−8.32163	−0.383848
$471$	0	0
$472$	9.44505	0.434744
$473$	−1.14243	−0.0525292
$474$	0	0
$475$	28.7172	1.31763
$476$	−83.5713	−3.83049
$477$	0	0
$478$	−15.7068	−0.718413
$479$	−14.9951	−0.685145	−0.342573	−	0.939491i	$-0.611298\pi$
$479$	−14.9951	−0.685145	−0.342573	+	0.939491i	$0.611298\pi$
$480$	0	0
$481$	−0.921318	−0.0420085
$482$	9.67837	0.440838
$483$	0	0
$484$	2.59002	0.117728
$485$	5.46873	0.248322
$486$	0	0
$487$	−13.6362	−0.617913	−0.308957	−	0.951076i	$-0.599980\pi$
$487$	−13.6362	−0.617913	−0.308957	+	0.951076i	$0.599980\pi$
$488$	4.87693	0.220768
$489$	0	0
$490$	16.1084	0.727703
$491$	27.5257	1.24222	0.621109	−	0.783724i	$-0.286682\pi$
$491$	27.5257	1.24222	0.621109	+	0.783724i	$0.286682\pi$
$492$	0	0
$493$	14.1934	0.639240
$494$	−12.8185	−0.576733
$495$	0	0
$496$	6.05001	0.271653
$497$	−41.9673	−1.88249
$498$	0	0
$499$	−14.5445	−0.651102	−0.325551	−	0.945525i	$-0.605550\pi$
$499$	−14.5445	−0.651102	−0.325551	+	0.945525i	$0.605550\pi$
$500$	11.3605	0.508055
$501$	0	0
$502$	−13.1999	−0.589141
$503$	−14.2512	−0.635429	−0.317715	−	0.948186i	$-0.602915\pi$
$503$	−14.2512	−0.635429	−0.317715	+	0.948186i	$0.602915\pi$
$504$	0	0
$505$	−8.54810	−0.380386
$506$	1.48866	0.0661789
$507$	0	0
$508$	38.6568	1.71512
$509$	−43.6994	−1.93694	−0.968471	−	0.249126i	$-0.919857\pi$
$509$	−43.6994	−1.93694	−0.968471	+	0.249126i	$0.919857\pi$
$510$	0	0
$511$	8.65187	0.382736
$512$	−25.5885	−1.13086
$513$	0	0
$514$	51.7954	2.28460
$515$	2.15427	0.0949287
$516$	0	0
$517$	8.67801	0.381658
$518$	−9.62919	−0.423082
$519$	0	0
$520$	−0.565795	−0.0248117
$521$	18.4570	0.808617	0.404308	−	0.914623i	$-0.367512\pi$
$521$	18.4570	0.808617	0.404308	+	0.914623i	$0.367512\pi$
$522$	0	0
$523$	−0.731044	−0.0319663	−0.0159832	−	0.999872i	$-0.505088\pi$
$523$	−0.731044	−0.0319663	−0.0159832	+	0.999872i	$0.505088\pi$
$524$	1.92884	0.0842618
$525$	0	0
$526$	59.0346	2.57403
$527$	16.1890	0.705203
$528$	0	0
$529$	−22.5172	−0.979008
$530$	4.81276	0.209053
$531$	0	0
$532$	−75.5975	−3.27757
$533$	1.12166	0.0485843
$534$	0	0
$535$	−6.57664	−0.284333
$536$	5.98225	0.258394
$537$	0	0
$538$	−52.7577	−2.27454
$539$	−16.7982	−0.723552
$540$	0	0
$541$	−18.2137	−0.783069	−0.391534	−	0.920164i	$-0.628056\pi$
$541$	−18.2137	−0.783069	−0.391534	+	0.920164i	$0.628056\pi$
$542$	−6.77698	−0.291096
$543$	0	0
$544$	−51.7492	−2.21873
$545$	−2.83918	−0.121617
$546$	0	0
$547$	−26.9668	−1.15302	−0.576509	−	0.817091i	$-0.695585\pi$
$547$	−26.9668	−1.15302	−0.576509	+	0.817091i	$0.695585\pi$
$548$	−7.63962	−0.326348
$549$	0	0
$550$	−10.2830	−0.438467
$551$	12.8392	0.546968
$552$	0	0
$553$	−67.8968	−2.88727
$554$	22.1385	0.940574
$555$	0	0
$556$	18.7795	0.796428
$557$	−45.2869	−1.91887	−0.959434	−	0.281934i	$-0.909024\pi$
$557$	−45.2869	−1.91887	−0.959434	+	0.281934i	$0.909024\pi$
$558$	0	0
$559$	−1.14243	−0.0483198
$560$	5.39722	0.228074
$561$	0	0
$562$	−34.5821	−1.45876
$563$	46.3264	1.95243	0.976213	−	0.216813i	$-0.0695663\pi$
$563$	46.3264	1.95243	0.976213	+	0.216813i	$0.0695663\pi$
$564$	0	0
$565$	6.71922	0.282680
$566$	−10.6984	−0.449689
$567$	0	0
$568$	10.8747	0.456291
$569$	−24.5092	−1.02748	−0.513739	−	0.857947i	$-0.671740\pi$
$569$	−24.5092	−1.02748	−0.513739	+	0.857947i	$0.671740\pi$
$570$	0	0
$571$	−10.3679	−0.433881	−0.216941	−	0.976185i	$-0.569608\pi$
$571$	−10.3679	−0.433881	−0.216941	+	0.976185i	$0.569608\pi$
$572$	2.59002	0.108294
$573$	0	0
$574$	11.7230	0.489309
$575$	−3.33502	−0.139080
$576$	0	0
$577$	−6.23950	−0.259754	−0.129877	−	0.991530i	$-0.541458\pi$
$577$	−6.23950	−0.259754	−0.129877	+	0.991530i	$0.541458\pi$
$578$	−57.3066	−2.38364
$579$	0	0
$580$	2.48766	0.103295
$581$	22.3264	0.926256
$582$	0	0
$583$	−5.01887	−0.207860
$584$	−2.24190	−0.0927703
$585$	0	0
$586$	−12.3546	−0.510364
$587$	20.9676	0.865425	0.432712	−	0.901532i	$-0.357557\pi$
$587$	20.9676	0.865425	0.432712	+	0.901532i	$0.357557\pi$
$588$	0	0
$589$	14.6443	0.603409
$590$	−7.16498	−0.294977
$591$	0	0
$592$	−2.27733	−0.0935978
$593$	−25.6684	−1.05408	−0.527038	−	0.849842i	$-0.676698\pi$
$593$	−25.6684	−1.05408	−0.527038	+	0.849842i	$0.676698\pi$
$594$	0	0
$595$	14.4422	0.592074
$596$	−57.7824	−2.36686
$597$	0	0
$598$	1.48866	0.0608757
$599$	24.4677	0.999725	0.499862	−	0.866105i	$-0.333384\pi$
$599$	24.4677	0.999725	0.499862	+	0.866105i	$0.333384\pi$
$600$	0	0
$601$	−18.7954	−0.766682	−0.383341	−	0.923607i	$-0.625227\pi$
$601$	−18.7954	−0.766682	−0.383341	+	0.923607i	$0.625227\pi$
$602$	−11.9402	−0.486646
$603$	0	0
$604$	−56.2889	−2.29036
$605$	−0.447591	−0.0181971
$606$	0	0
$607$	−26.7166	−1.08439	−0.542197	−	0.840252i	$-0.682407\pi$
$607$	−26.7166	−1.08439	−0.542197	+	0.840252i	$0.682407\pi$
$608$	−46.8116	−1.89846
$609$	0	0
$610$	−3.69962	−0.149793
$611$	8.67801	0.351075
$612$	0	0
$613$	30.2958	1.22364	0.611818	−	0.790999i	$-0.290439\pi$
$613$	30.2958	1.22364	0.611818	+	0.790999i	$0.290439\pi$
$614$	−34.1161	−1.37681
$615$	0	0
$616$	6.16666	0.248462
$617$	−33.4060	−1.34488	−0.672438	−	0.740154i	$-0.734753\pi$
$617$	−33.4060	−1.34488	−0.672438	+	0.740154i	$0.734753\pi$
$618$	0	0
$619$	35.0151	1.40738	0.703688	−	0.710509i	$-0.251535\pi$
$619$	35.0151	1.40738	0.703688	+	0.710509i	$0.251535\pi$
$620$	2.83742	0.113954
$621$	0	0
$622$	31.1395	1.24858
$623$	60.0312	2.40510
$624$	0	0
$625$	22.0351	0.881403
$626$	−23.7801	−0.950446
$627$	0	0
$628$	−40.2913	−1.60780
$629$	−6.09383	−0.242977
$630$	0	0
$631$	−7.69817	−0.306459	−0.153230	−	0.988191i	$-0.548967\pi$
$631$	−7.69817	−0.306459	−0.153230	+	0.988191i	$0.548967\pi$
$632$	17.5936	0.699836
$633$	0	0
$634$	56.8733	2.25873
$635$	−6.68041	−0.265104
$636$	0	0
$637$	−16.7982	−0.665571
$638$	−4.59742	−0.182014
$639$	0	0
$640$	−4.32939	−0.171134
$641$	−41.6034	−1.64324	−0.821618	−	0.570039i	$-0.806928\pi$
$641$	−41.6034	−1.64324	−0.821618	+	0.570039i	$0.806928\pi$
$642$	0	0
$643$	−23.4734	−0.925699	−0.462850	−	0.886437i	$-0.653173\pi$
$643$	−23.4734	−0.925699	−0.462850	+	0.886437i	$0.653173\pi$
$644$	8.77938	0.345956
$645$	0	0
$646$	−84.7851	−3.33582
$647$	13.8360	0.543949	0.271975	−	0.962304i	$-0.412323\pi$
$647$	13.8360	0.543949	0.271975	+	0.962304i	$0.412323\pi$
$648$	0	0
$649$	7.47182	0.293295
$650$	−10.2830	−0.403331
$651$	0	0
$652$	−39.5654	−1.54950
$653$	30.7696	1.20411	0.602055	−	0.798455i	$-0.294349\pi$
$653$	30.7696	1.20411	0.602055	+	0.798455i	$0.294349\pi$
$654$	0	0
$655$	−0.333329	−0.0130243
$656$	2.77253	0.108249
$657$	0	0
$658$	90.6985	3.53580
$659$	20.9965	0.817909	0.408955	−	0.912555i	$-0.365893\pi$
$659$	20.9965	0.817909	0.408955	+	0.912555i	$0.365893\pi$
$660$	0	0
$661$	24.1545	0.939502	0.469751	−	0.882799i	$-0.344344\pi$
$661$	24.1545	0.939502	0.469751	+	0.882799i	$0.344344\pi$
$662$	44.7160	1.73794
$663$	0	0
$664$	−5.78528	−0.224513
$665$	13.0642	0.506610
$666$	0	0
$667$	−1.49106	−0.0577339
$668$	38.2789	1.48106
$669$	0	0
$670$	−4.53811	−0.175323
$671$	3.85806	0.148939
$672$	0	0
$673$	13.4842	0.519776	0.259888	−	0.965639i	$-0.416314\pi$
$673$	13.4842	0.519776	0.259888	+	0.965639i	$0.416314\pi$
$674$	17.5657	0.676604
$675$	0	0
$676$	2.59002	0.0996163
$677$	−6.93216	−0.266425	−0.133212	−	0.991088i	$-0.542529\pi$
$677$	−6.93216	−0.266425	−0.133212	+	0.991088i	$0.542529\pi$
$678$	0	0
$679$	−59.6044	−2.28741
$680$	−3.74231	−0.143511
$681$	0	0
$682$	−5.24380	−0.200796
$683$	21.1095	0.807734	0.403867	−	0.914818i	$-0.367666\pi$
$683$	21.1095	0.807734	0.403867	+	0.914818i	$0.367666\pi$
$684$	0	0
$685$	1.32023	0.0504433
$686$	−102.407	−3.90991
$687$	0	0
$688$	−2.82389	−0.107660
$689$	−5.01887	−0.191204
$690$	0	0
$691$	11.4143	0.434220	0.217110	−	0.976147i	$-0.430337\pi$
$691$	11.4143	0.434220	0.217110	+	0.976147i	$0.430337\pi$
$692$	12.6889	0.482358
$693$	0	0
$694$	35.9670	1.36529
$695$	−3.24535	−0.123103
$696$	0	0
$697$	7.41891	0.281011
$698$	−20.8309	−0.788462
$699$	0	0
$700$	−60.6439	−2.29212
$701$	−17.7270	−0.669538	−0.334769	−	0.942300i	$-0.608658\pi$
$701$	−17.7270	−0.669538	−0.334769	+	0.942300i	$0.608658\pi$
$702$	0	0
$703$	−5.51240	−0.207904
$704$	11.8185	0.445428
$705$	0	0
$706$	30.1815	1.13590
$707$	93.1669	3.50390
$708$	0	0
$709$	−6.65941	−0.250099	−0.125050	−	0.992150i	$-0.539909\pi$
$709$	−6.65941	−0.250099	−0.125050	+	0.992150i	$0.539909\pi$
$710$	−8.24949	−0.309598
$711$	0	0
$712$	−15.5554	−0.582965
$713$	−1.70069	−0.0636915
$714$	0	0
$715$	−0.447591	−0.0167389
$716$	2.01239	0.0752065
$717$	0	0
$718$	14.4862	0.540620
$719$	36.8386	1.37385	0.686923	−	0.726730i	$-0.258961\pi$
$719$	36.8386	1.37385	0.686923	+	0.726730i	$0.258961\pi$
$720$	0	0
$721$	−23.4797	−0.874430
$722$	−35.9891	−1.33938
$723$	0	0
$724$	2.62469	0.0975459
$725$	10.2995	0.382515
$726$	0	0
$727$	21.0444	0.780493	0.390246	−	0.920710i	$-0.372390\pi$
$727$	21.0444	0.780493	0.390246	+	0.920710i	$0.372390\pi$
$728$	6.16666	0.228552
$729$	0	0
$730$	1.70069	0.0629455
$731$	−7.55635	−0.279482
$732$	0	0
$733$	−5.24226	−0.193627	−0.0968136	−	0.995303i	$-0.530865\pi$
$733$	−5.24226	−0.193627	−0.0968136	+	0.995303i	$0.530865\pi$
$734$	52.8549	1.95091
$735$	0	0
$736$	5.43638	0.200388
$737$	4.73246	0.174322
$738$	0	0
$739$	37.1979	1.36835	0.684173	−	0.729320i	$-0.260163\pi$
$739$	37.1979	1.36835	0.684173	+	0.729320i	$0.260163\pi$
$740$	−1.06806	−0.0392625
$741$	0	0
$742$	−52.4549	−1.92568
$743$	25.0268	0.918143	0.459071	−	0.888399i	$-0.348182\pi$
$743$	25.0268	0.918143	0.459071	+	0.888399i	$0.348182\pi$
$744$	0	0
$745$	9.98556	0.365843
$746$	−15.2068	−0.556761
$747$	0	0
$748$	17.1311	0.626375
$749$	71.6796	2.61912
$750$	0	0
$751$	21.7391	0.793270	0.396635	−	0.917976i	$-0.370178\pi$
$751$	21.7391	0.793270	0.396635	+	0.917976i	$0.370178\pi$
$752$	21.4505	0.782219
$753$	0	0
$754$	−4.59742	−0.167428
$755$	9.72746	0.354019
$756$	0	0
$757$	46.2161	1.67975	0.839877	−	0.542778i	$-0.182627\pi$
$757$	46.2161	1.67975	0.839877	+	0.542778i	$0.182627\pi$
$758$	10.9070	0.396160
$759$	0	0
$760$	−3.38524	−0.122796
$761$	8.17900	0.296488	0.148244	−	0.988951i	$-0.452638\pi$
$761$	8.17900	0.296488	0.148244	+	0.988951i	$0.452638\pi$
$762$	0	0
$763$	30.9446	1.12027
$764$	−43.0415	−1.55719
$765$	0	0
$766$	−46.6123	−1.68417
$767$	7.47182	0.269792
$768$	0	0
$769$	−16.9635	−0.611720	−0.305860	−	0.952077i	$-0.598944\pi$
$769$	−16.9635	−0.611720	−0.305860	+	0.952077i	$0.598944\pi$
$770$	−4.67801	−0.168584
$771$	0	0
$772$	−38.0127	−1.36811
$773$	43.7825	1.57475	0.787374	−	0.616476i	$-0.211440\pi$
$773$	43.7825	1.57475	0.787374	+	0.616476i	$0.211440\pi$
$774$	0	0
$775$	11.7476	0.421987
$776$	15.4448	0.554438
$777$	0	0
$778$	21.4986	0.770762
$779$	6.71105	0.240448
$780$	0	0
$781$	8.60278	0.307832
$782$	9.84636	0.352105
$783$	0	0
$784$	−41.5223	−1.48294
$785$	6.96288	0.248516
$786$	0	0
$787$	−31.0713	−1.10757	−0.553787	−	0.832658i	$-0.686818\pi$
$787$	−31.0713	−1.10757	−0.553787	+	0.832658i	$0.686818\pi$
$788$	−40.2058	−1.43227
$789$	0	0
$790$	−13.3464	−0.474845
$791$	−73.2336	−2.60389
$792$	0	0
$793$	3.85806	0.137004
$794$	49.7346	1.76502
$795$	0	0
$796$	68.4193	2.42506
$797$	−31.0227	−1.09888	−0.549440	−	0.835533i	$-0.685159\pi$
$797$	−31.0227	−1.09888	−0.549440	+	0.835533i	$0.685159\pi$
$798$	0	0
$799$	57.3986	2.03061
$800$	−37.5520	−1.32767
$801$	0	0
$802$	22.1183	0.781023
$803$	−1.77353	−0.0625864
$804$	0	0
$805$	−1.51719	−0.0534740
$806$	−5.24380	−0.184705
$807$	0	0
$808$	−24.1416	−0.849300
$809$	−18.2643	−0.642138	−0.321069	−	0.947056i	$-0.604042\pi$
$809$	−18.2643	−0.642138	−0.321069	+	0.947056i	$0.604042\pi$
$810$	0	0
$811$	10.4147	0.365709	0.182854	−	0.983140i	$-0.441466\pi$
$811$	10.4147	0.365709	0.182854	+	0.983140i	$0.441466\pi$
$812$	−27.1133	−0.951492
$813$	0	0
$814$	1.97386	0.0691839
$815$	6.83742	0.239504
$816$	0	0
$817$	−6.83537	−0.239139
$818$	−79.8126	−2.79058
$819$	0	0
$820$	1.30030	0.0454085
$821$	26.2542	0.916279	0.458139	−	0.888880i	$-0.348516\pi$
$821$	26.2542	0.916279	0.458139	+	0.888880i	$0.348516\pi$
$822$	0	0
$823$	−33.4759	−1.16690	−0.583448	−	0.812150i	$-0.698297\pi$
$823$	−33.4759	−1.16690	−0.583448	+	0.812150i	$0.698297\pi$
$824$	6.08412	0.211950
$825$	0	0
$826$	78.0920	2.71717
$827$	−38.7251	−1.34660	−0.673301	−	0.739368i	$-0.735124\pi$
$827$	−38.7251	−1.34660	−0.673301	+	0.739368i	$0.735124\pi$
$828$	0	0
$829$	−3.31755	−0.115223	−0.0576116	−	0.998339i	$-0.518348\pi$
$829$	−3.31755	−0.115223	−0.0576116	+	0.998339i	$0.518348\pi$
$830$	4.38869	0.152334
$831$	0	0
$832$	11.8185	0.409734
$833$	−111.108	−3.84966
$834$	0	0
$835$	−6.61511	−0.228925
$836$	15.4965	0.535959
$837$	0	0
$838$	−78.8982	−2.72549
$839$	37.8367	1.30627	0.653134	−	0.757242i	$-0.273454\pi$
$839$	37.8367	1.30627	0.653134	+	0.757242i	$0.273454\pi$
$840$	0	0
$841$	−24.3952	−0.841213
$842$	−83.8477	−2.88958
$843$	0	0
$844$	59.9778	2.06452
$845$	−0.447591	−0.0153976
$846$	0	0
$847$	4.87834	0.167622
$848$	−12.4058	−0.426015
$849$	0	0
$850$	−68.0141	−2.33287
$851$	0.640172	0.0219448
$852$	0	0
$853$	29.4797	1.00937	0.504683	−	0.863305i	$-0.331609\pi$
$853$	29.4797	1.00937	0.504683	+	0.863305i	$0.331609\pi$
$854$	40.3226	1.37981
$855$	0	0
$856$	−18.5738	−0.634840
$857$	−25.3231	−0.865021	−0.432510	−	0.901629i	$-0.642372\pi$
$857$	−25.3231	−0.865021	−0.432510	+	0.901629i	$0.642372\pi$
$858$	0	0
$859$	35.3185	1.20505	0.602526	−	0.798099i	$-0.294161\pi$
$859$	35.3185	1.20505	0.602526	+	0.798099i	$0.294161\pi$
$860$	−1.32439	−0.0451613
$861$	0	0
$862$	66.8411	2.27662
$863$	−6.49069	−0.220946	−0.110473	−	0.993879i	$-0.535237\pi$
$863$	−6.49069	−0.220946	−0.110473	+	0.993879i	$0.535237\pi$
$864$	0	0
$865$	−2.19280	−0.0745575
$866$	−52.3680	−1.77954
$867$	0	0
$868$	−30.9254	−1.04968
$869$	13.9180	0.472136
$870$	0	0
$871$	4.73246	0.160353
$872$	−8.01845	−0.271539
$873$	0	0
$874$	8.90688	0.301280
$875$	21.3976	0.723370
$876$	0	0
$877$	20.2782	0.684747	0.342374	−	0.939564i	$-0.388769\pi$
$877$	20.2782	0.684747	0.342374	+	0.939564i	$0.388769\pi$
$878$	−59.9987	−2.02486
$879$	0	0
$880$	−1.10636	−0.0372955
$881$	−32.5162	−1.09550	−0.547749	−	0.836643i	$-0.684515\pi$
$881$	−32.5162	−1.09550	−0.547749	+	0.836643i	$0.684515\pi$
$882$	0	0
$883$	36.6982	1.23499	0.617497	−	0.786573i	$-0.288147\pi$
$883$	36.6982	1.23499	0.617497	+	0.786573i	$0.288147\pi$
$884$	17.1311	0.576181
$885$	0	0
$886$	−0.351930	−0.0118233
$887$	6.60561	0.221795	0.110897	−	0.993832i	$-0.464628\pi$
$887$	6.60561	0.221795	0.110897	+	0.993832i	$0.464628\pi$
$888$	0	0
$889$	72.8106	2.44199
$890$	11.8003	0.395547
$891$	0	0
$892$	−53.1929	−1.78103
$893$	51.9219	1.73750
$894$	0	0
$895$	−0.347768	−0.0116246
$896$	47.1865	1.57639
$897$	0	0
$898$	34.3795	1.14726
$899$	5.25225	0.175172
$900$	0	0
$901$	−33.1961	−1.10592
$902$	−2.40307	−0.0800136
$903$	0	0
$904$	18.9765	0.631148
$905$	−0.453582	−0.0150776
$906$	0	0
$907$	−37.0711	−1.23093	−0.615464	−	0.788165i	$-0.711031\pi$
$907$	−37.0711	−1.23093	−0.615464	+	0.788165i	$0.711031\pi$
$908$	54.8337	1.81972
$909$	0	0
$910$	−4.67801	−0.155074
$911$	31.5927	1.04671	0.523356	−	0.852114i	$-0.324680\pi$
$911$	31.5927	1.04671	0.523356	+	0.852114i	$0.324680\pi$
$912$	0	0
$913$	−4.57664	−0.151465
$914$	−43.7617	−1.44751
$915$	0	0
$916$	18.1391	0.599333
$917$	3.63300	0.119972
$918$	0	0
$919$	8.07065	0.266226	0.133113	−	0.991101i	$-0.457503\pi$
$919$	8.07065	0.266226	0.133113	+	0.991101i	$0.457503\pi$
$920$	0.393139	0.0129614
$921$	0	0
$922$	−9.85812	−0.324660
$923$	8.60278	0.283164
$924$	0	0
$925$	−4.42202	−0.145395
$926$	−30.2970	−0.995621
$927$	0	0
$928$	−16.7892	−0.551132
$929$	−49.8874	−1.63675	−0.818376	−	0.574684i	$-0.805125\pi$
$929$	−49.8874	−1.63675	−0.818376	+	0.574684i	$0.805125\pi$
$930$	0	0
$931$	−100.507	−3.29397
$932$	33.5844	1.10009
$933$	0	0
$934$	80.6893	2.64023
$935$	−2.96048	−0.0968180
$936$	0	0
$937$	14.7601	0.482193	0.241096	−	0.970501i	$-0.422493\pi$
$937$	14.7601	0.482193	0.241096	+	0.970501i	$0.422493\pi$
$938$	49.4615	1.61497
$939$	0	0
$940$	10.0602	0.328126
$941$	14.3224	0.466897	0.233449	−	0.972369i	$-0.424999\pi$
$941$	14.3224	0.466897	0.233449	+	0.972369i	$0.424999\pi$
$942$	0	0
$943$	−0.779375	−0.0253799
$944$	18.4690	0.601115
$945$	0	0
$946$	2.44759	0.0795780
$947$	9.66985	0.314228	0.157114	−	0.987580i	$-0.449781\pi$
$947$	9.66985	0.314228	0.157114	+	0.987580i	$0.449781\pi$
$948$	0	0
$949$	−1.77353	−0.0575711
$950$	−61.5246	−1.99612
$951$	0	0
$952$	40.7879	1.32194
$953$	−19.4157	−0.628936	−0.314468	−	0.949268i	$-0.601826\pi$
$953$	−19.4157	−0.628936	−0.314468	+	0.949268i	$0.601826\pi$
$954$	0	0
$955$	7.43815	0.240693
$956$	18.9882	0.614123
$957$	0	0
$958$	32.1261	1.03795
$959$	−14.3893	−0.464656
$960$	0	0
$961$	−25.0093	−0.806752
$962$	1.97386	0.0636399
$963$	0	0
$964$	−11.7003	−0.376843
$965$	6.56910	0.211467
$966$	0	0
$967$	54.5718	1.75491	0.877455	−	0.479659i	$-0.159240\pi$
$967$	54.5718	1.75491	0.877455	+	0.479659i	$0.159240\pi$
$968$	−1.26409	−0.0406294
$969$	0	0
$970$	−11.7164	−0.376191
$971$	11.4893	0.368709	0.184354	−	0.982860i	$-0.440981\pi$
$971$	11.4893	0.368709	0.184354	+	0.982860i	$0.440981\pi$
$972$	0	0
$973$	35.3714	1.13396
$974$	29.2146	0.936095
$975$	0	0
$976$	9.53643	0.305254
$977$	−22.6903	−0.725926	−0.362963	−	0.931804i	$-0.618235\pi$
$977$	−22.6903	−0.725926	−0.362963	+	0.931804i	$0.618235\pi$
$978$	0	0
$979$	−12.3056	−0.393290
$980$	−19.4737	−0.622065
$981$	0	0
$982$	−58.9720	−1.88187
$983$	−0.224426	−0.00715807	−0.00357904	−	0.999994i	$-0.501139\pi$
$983$	−0.224426	−0.00715807	−0.00357904	+	0.999994i	$0.501139\pi$
$984$	0	0
$985$	6.94809	0.221384
$986$	−30.4085	−0.968404
$987$	0	0
$988$	15.4965	0.493011
$989$	0.793813	0.0252418
$990$	0	0
$991$	39.2525	1.24690	0.623448	−	0.781865i	$-0.285731\pi$
$991$	39.2525	1.24690	0.623448	+	0.781865i	$0.285731\pi$
$992$	−19.1497	−0.608003
$993$	0	0
$994$	89.9122	2.85184
$995$	−11.8238	−0.374838
$996$	0	0
$997$	−35.5848	−1.12698	−0.563491	−	0.826122i	$-0.690542\pi$
$997$	−35.5848	−1.12698	−0.563491	+	0.826122i	$0.690542\pi$
$998$	31.1607	0.986374
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.l.1.2	✓	4
3.2	odd	2		1287.2.a.n.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.2.a.l.1.2	✓	4	1.1	even	1	trivial
1287.2.a.n.1.3	yes	4	3.2	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14243 q^{2} +2.59002 q^{4} -0.447591 q^{5} +4.87834 q^{7} -1.26409 q^{8} +O(q^{10})\)	\(q-2.14243 q^{2} +2.59002 q^{4} -0.447591 q^{5} +4.87834 q^{7} -1.26409 q^{8} +0.958933 q^{10} -1.00000 q^{11} -1.00000 q^{13} -10.4515 q^{14} -2.47182 q^{16} -6.61426 q^{17} -5.98316 q^{19} -1.15927 q^{20} +2.14243 q^{22} +0.694844 q^{23} -4.79966 q^{25} +2.14243 q^{26} +12.6350 q^{28} -2.14589 q^{29} -2.44759 q^{31} +7.82389 q^{32} +14.1706 q^{34} -2.18350 q^{35} +0.921318 q^{37} +12.8185 q^{38} +0.565795 q^{40} -1.12166 q^{41} +1.14243 q^{43} -2.59002 q^{44} -1.48866 q^{46} -8.67801 q^{47} +16.7982 q^{49} +10.2830 q^{50} -2.59002 q^{52} +5.01887 q^{53} +0.447591 q^{55} -6.16666 q^{56} +4.59742 q^{58} -7.47182 q^{59} -3.85806 q^{61} +5.24380 q^{62} -11.8185 q^{64} +0.447591 q^{65} -4.73246 q^{67} -17.1311 q^{68} +4.67801 q^{70} -8.60278 q^{71} +1.77353 q^{73} -1.97386 q^{74} -15.4965 q^{76} -4.87834 q^{77} -13.9180 q^{79} +1.10636 q^{80} +2.40307 q^{82} +4.57664 q^{83} +2.96048 q^{85} -2.44759 q^{86} +1.26409 q^{88} +12.3056 q^{89} -4.87834 q^{91} +1.79966 q^{92} +18.5921 q^{94} +2.67801 q^{95} -12.2182 q^{97} -35.9891 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8	\( 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8} - 4 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} - 2 q^{19} - 20 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{26} + 10 q^{28} - 4 q^{29} - 14 q^{31} - 6 q^{32} - 10 q^{34} - 6 q^{35} - 2 q^{37} + 8 q^{38} + 18 q^{40} - 18 q^{41} - 2 q^{43} - 8 q^{44} - 14 q^{46} - 2 q^{47} + 24 q^{50} - 8 q^{52} - 4 q^{53} + 6 q^{55} - 24 q^{58} - 16 q^{59} + 22 q^{61} + 4 q^{62} - 4 q^{64} + 6 q^{65} - 10 q^{67} + 4 q^{68} - 14 q^{70} + 2 q^{71} + 2 q^{73} - 22 q^{74} + 14 q^{76} - 6 q^{77} + 2 q^{79} - 6 q^{80} + 8 q^{82} - 4 q^{83} + 6 q^{85} - 14 q^{86} + 12 q^{88} + 16 q^{89} - 6 q^{91} - 12 q^{92} + 26 q^{94} - 22 q^{95} - 2 q^{97} - 34 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8 - 4 * q^11 - 4 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 - 2 * q^19 - 20 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^26 + 10 * q^28 - 4 * q^29 - 14 * q^31 - 6 * q^32 - 10 * q^34 - 6 * q^35 - 2 * q^37 + 8 * q^38 + 18 * q^40 - 18 * q^41 - 2 * q^43 - 8 * q^44 - 14 * q^46 - 2 * q^47 + 24 * q^50 - 8 * q^52 - 4 * q^53 + 6 * q^55 - 24 * q^58 - 16 * q^59 + 22 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 - 10 * q^67 + 4 * q^68 - 14 * q^70 + 2 * q^71 + 2 * q^73 - 22 * q^74 + 14 * q^76 - 6 * q^77 + 2 * q^79 - 6 * q^80 + 8 * q^82 - 4 * q^83 + 6 * q^85 - 14 * q^86 + 12 * q^88 + 16 * q^89 - 6 * q^91 - 12 * q^92 + 26 * q^94 - 22 * q^95 - 2 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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