LMFDB - Embedded newform 1287.2.a.q.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:	$0$
Dimension:	$6$
Coefficient field:	6.6.194616205.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 2x^{3} + 24x^{2} + 7x - 12 $ x^6 - 10x^4 - 2x^3 + 24x^2 + 7x - 12
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.36536$ 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)

$f(q)$	$=$	$q-1.36536 q^{2} -0.135780 q^{4} +3.38172 q^{5} -1.56538 q^{7} +2.91612 q^{8} +O(q^{10})$	\(q-1.36536 q^{2} -0.135780 q^{4} +3.38172 q^{5} -1.56538 q^{7} +2.91612 q^{8} -4.61728 q^{10} +1.00000 q^{11} +1.00000 q^{13} +2.13732 q^{14} -3.71000 q^{16} +2.73073 q^{17} +4.69036 q^{19} -0.459169 q^{20} -1.36536 q^{22} -4.67555 q^{23} +6.43604 q^{25} -1.36536 q^{26} +0.212548 q^{28} +2.34572 q^{29} -1.16448 q^{31} -0.766728 q^{32} -3.72844 q^{34} -5.29369 q^{35} +5.84089 q^{37} -6.40405 q^{38} +9.86150 q^{40} +5.83694 q^{41} -4.21724 q^{43} -0.135780 q^{44} +6.38382 q^{46} +1.88655 q^{47} -4.54957 q^{49} -8.78754 q^{50} -0.135780 q^{52} +7.54957 q^{53} +3.38172 q^{55} -4.56485 q^{56} -3.20277 q^{58} -10.0533 q^{59} -9.83989 q^{61} +1.58994 q^{62} +8.46687 q^{64} +3.38172 q^{65} +0.889838 q^{67} -0.370778 q^{68} +7.22782 q^{70} -2.62594 q^{71} +8.80381 q^{73} -7.97494 q^{74} -0.636856 q^{76} -1.56538 q^{77} -0.345723 q^{79} -12.5462 q^{80} -7.96956 q^{82} +14.6595 q^{83} +9.23456 q^{85} +5.75807 q^{86} +2.91612 q^{88} +3.54949 q^{89} -1.56538 q^{91} +0.634845 q^{92} -2.57583 q^{94} +15.8615 q^{95} +9.78176 q^{97} +6.21182 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8	$ 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 12 q^{14} + 8 q^{16} - 10 q^{19} - 4 q^{20} - 11 q^{23} + 23 q^{25} + 9 q^{28} - 2 q^{29} - 9 q^{31} + 17 q^{32} - 40 q^{34} + 24 q^{35} + 15 q^{37} + 9 q^{38} + 16 q^{40} + 4 q^{41} - 2 q^{43} + 8 q^{44} - 6 q^{46} - 6 q^{47} + 20 q^{49} + 4 q^{50} + 8 q^{52} - 2 q^{53} - q^{55} + 39 q^{56} + 18 q^{58} - 11 q^{59} + 16 q^{61} - 16 q^{62} + 36 q^{64} - q^{65} + 9 q^{67} - 12 q^{68} + 32 q^{70} + 15 q^{71} + 32 q^{73} - 22 q^{74} - 26 q^{76} + 4 q^{77} + 14 q^{79} - 56 q^{80} - 24 q^{82} + 26 q^{83} - 12 q^{85} + 10 q^{86} + 6 q^{88} + 23 q^{89} + 4 q^{91} - 83 q^{92} + 46 q^{94} + 52 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) $ 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 + 12 * q^14 + 8 * q^16 - 10 * q^19 - 4 * q^20 - 11 * q^23 + 23 * q^25 + 9 * q^28 - 2 * q^29 - 9 * q^31 + 17 * q^32 - 40 * q^34 + 24 * q^35 + 15 * q^37 + 9 * q^38 + 16 * q^40 + 4 * q^41 - 2 * q^43 + 8 * q^44 - 6 * q^46 - 6 * q^47 + 20 * q^49 + 4 * q^50 + 8 * q^52 - 2 * q^53 - q^55 + 39 * q^56 + 18 * q^58 - 11 * q^59 + 16 * q^61 - 16 * q^62 + 36 * q^64 - q^65 + 9 * q^67 - 12 * q^68 + 32 * q^70 + 15 * q^71 + 32 * q^73 - 22 * q^74 - 26 * q^76 + 4 * q^77 + 14 * q^79 - 56 * q^80 - 24 * q^82 + 26 * q^83 - 12 * q^85 + 10 * q^86 + 6 * q^88 + 23 * q^89 + 4 * q^91 - 83 * q^92 + 46 * q^94 + 52 * q^95 + 27 * q^97 + 3 * 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 $

$2$	−1.36536	−0.965458	−0.482729	−	0.875770i	$-0.660354\pi$
$2$	−1.36536	−0.965458	−0.482729	+	0.875770i	$0.660354\pi$
$3$	0	0
$3$	0	0
$4$	−0.135780	−0.0678899
$5$	3.38172	1.51235	0.756176	−	0.654368i	$-0.227065\pi$
$5$	3.38172	1.51235	0.756176	+	0.654368i	$0.227065\pi$
$6$	0	0
$7$	−1.56538	−0.591660	−0.295830	−	0.955241i	$-0.595596\pi$
$7$	−1.56538	−0.591660	−0.295830	+	0.955241i	$0.595596\pi$
$8$	2.91612	1.03100
$9$	0	0
$10$	−4.61728	−1.46011
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.13732	0.571223
$15$	0	0
$16$	−3.71000	−0.927501
$17$	2.73073	0.662299	0.331150	−	0.943578i	$-0.392564\pi$
$17$	2.73073	0.662299	0.331150	+	0.943578i	$0.392564\pi$
$18$	0	0
$19$	4.69036	1.07604	0.538021	−	0.842931i	$-0.319172\pi$
$19$	4.69036	1.07604	0.538021	+	0.842931i	$0.319172\pi$
$20$	−0.459169	−0.102673
$21$	0	0
$22$	−1.36536	−0.291097
$23$	−4.67555	−0.974919	−0.487459	−	0.873146i	$-0.662076\pi$
$23$	−4.67555	−0.974919	−0.487459	+	0.873146i	$0.662076\pi$
$24$	0	0
$25$	6.43604	1.28721
$26$	−1.36536	−0.267770
$27$	0	0
$28$	0.212548	0.0401677
$29$	2.34572	0.435590	0.217795	−	0.975995i	$-0.430114\pi$
$29$	2.34572	0.435590	0.217795	+	0.975995i	$0.430114\pi$
$30$	0	0
$31$	−1.16448	−0.209147	−0.104573	−	0.994517i	$-0.533348\pi$
$31$	−1.16448	−0.209147	−0.104573	+	0.994517i	$0.533348\pi$
$32$	−0.766728	−0.135540
$33$	0	0
$34$	−3.72844	−0.639422
$35$	−5.29369	−0.894798
$36$	0	0
$37$	5.84089	0.960237	0.480118	−	0.877204i	$-0.340594\pi$
$37$	5.84089	0.960237	0.480118	+	0.877204i	$0.340594\pi$
$38$	−6.40405	−1.03887
$39$	0	0
$40$	9.86150	1.55924
$41$	5.83694	0.911578	0.455789	−	0.890088i	$-0.349357\pi$
$41$	5.83694	0.911578	0.455789	+	0.890088i	$0.349357\pi$
$42$	0	0
$43$	−4.21724	−0.643123	−0.321562	−	0.946889i	$-0.604208\pi$
$43$	−4.21724	−0.643123	−0.321562	+	0.946889i	$0.604208\pi$
$44$	−0.135780	−0.0204696
$45$	0	0
$46$	6.38382	0.941244
$47$	1.88655	0.275182	0.137591	−	0.990489i	$-0.456064\pi$
$47$	1.88655	0.275182	0.137591	+	0.990489i	$0.456064\pi$
$48$	0	0
$49$	−4.54957	−0.649939
$50$	−8.78754	−1.24275
$51$	0	0
$52$	−0.135780	−0.0188293
$53$	7.54957	1.03701	0.518507	−	0.855074i	$-0.326488\pi$
$53$	7.54957	1.03701	0.518507	+	0.855074i	$0.326488\pi$
$54$	0	0
$55$	3.38172	0.455991
$56$	−4.56485	−0.610003
$57$	0	0
$58$	−3.20277	−0.420544
$59$	−10.0533	−1.30883	−0.654415	−	0.756135i	$-0.727085\pi$
$59$	−10.0533	−1.30883	−0.654415	+	0.756135i	$0.727085\pi$
$60$	0	0
$61$	−9.83989	−1.25987	−0.629935	−	0.776648i	$-0.716918\pi$
$61$	−9.83989	−1.25987	−0.629935	+	0.776648i	$0.716918\pi$
$62$	1.58994	0.201922
$63$	0	0
$64$	8.46687	1.05836
$65$	3.38172	0.419451
$66$	0	0
$67$	0.889838	0.108711	0.0543555	−	0.998522i	$-0.482690\pi$
$67$	0.889838	0.108711	0.0543555	+	0.998522i	$0.482690\pi$
$68$	−0.370778	−0.0449634
$69$	0	0
$70$	7.22782	0.863890
$71$	−2.62594	−0.311641	−0.155821	−	0.987785i	$-0.549802\pi$
$71$	−2.62594	−0.311641	−0.155821	+	0.987785i	$0.549802\pi$
$72$	0	0
$73$	8.80381	1.03041	0.515204	−	0.857068i	$-0.327716\pi$
$73$	8.80381	1.03041	0.515204	+	0.857068i	$0.327716\pi$
$74$	−7.97494	−0.927068
$75$	0	0
$76$	−0.636856	−0.0730524
$77$	−1.56538	−0.178392
$78$	0	0
$79$	−0.345723	−0.0388968	−0.0194484	−	0.999811i	$-0.506191\pi$
$79$	−0.345723	−0.0388968	−0.0194484	+	0.999811i	$0.506191\pi$
$80$	−12.5462	−1.40271
$81$	0	0
$82$	−7.96956	−0.880090
$83$	14.6595	1.60909	0.804545	−	0.593891i	$-0.202409\pi$
$83$	14.6595	1.60909	0.804545	+	0.593891i	$0.202409\pi$
$84$	0	0
$85$	9.23456	1.00163
$86$	5.75807	0.620909
$87$	0	0
$88$	2.91612	0.310859
$89$	3.54949	0.376245	0.188122	−	0.982146i	$-0.439760\pi$
$89$	3.54949	0.376245	0.188122	+	0.982146i	$0.439760\pi$
$90$	0	0
$91$	−1.56538	−0.164097
$92$	0.634845	0.0661871
$93$	0	0
$94$	−2.57583	−0.265677
$95$	15.8615	1.62736
$96$	0	0
$97$	9.78176	0.993187	0.496594	−	0.867983i	$-0.334584\pi$
$97$	9.78176	0.993187	0.496594	+	0.867983i	$0.334584\pi$
$98$	6.21182	0.627489
$99$	0	0
$100$	−0.873884	−0.0873884
$101$	0.676412	0.0673055	0.0336527	−	0.999434i	$-0.489286\pi$
$101$	0.676412	0.0673055	0.0336527	+	0.999434i	$0.489286\pi$
$102$	0	0
$103$	−2.65013	−0.261125	−0.130562	−	0.991440i	$-0.541678\pi$
$103$	−2.65013	−0.261125	−0.130562	+	0.991440i	$0.541678\pi$
$104$	2.91612	0.285949
$105$	0	0
$106$	−10.3079	−1.00119
$107$	13.8072	1.33479	0.667395	−	0.744704i	$-0.267409\pi$
$107$	13.8072	1.33479	0.667395	+	0.744704i	$0.267409\pi$
$108$	0	0
$109$	14.1845	1.35863	0.679316	−	0.733846i	$-0.262277\pi$
$109$	14.1845	1.35863	0.679316	+	0.733846i	$0.262277\pi$
$110$	−4.61728	−0.440241
$111$	0	0
$112$	5.80758	0.548765
$113$	−15.3414	−1.44319	−0.721597	−	0.692314i	$-0.756591\pi$
$113$	−15.3414	−1.44319	−0.721597	+	0.692314i	$0.756591\pi$
$114$	0	0
$115$	−15.8114	−1.47442
$116$	−0.318502	−0.0295721
$117$	0	0
$118$	13.7264	1.26362
$119$	−4.27464	−0.391856
$120$	0	0
$121$	1.00000	0.0909091
$122$	13.4350	1.21635
$123$	0	0
$124$	0.158113	0.0141989
$125$	4.85628	0.434359
$126$	0	0
$127$	9.80718	0.870247	0.435123	−	0.900371i	$-0.356705\pi$
$127$	9.80718	0.870247	0.435123	+	0.900371i	$0.356705\pi$
$128$	−10.0269	−0.886262
$129$	0	0
$130$	−4.61728	−0.404962
$131$	17.7657	1.55220	0.776100	−	0.630610i	$-0.217195\pi$
$131$	17.7657	1.55220	0.776100	+	0.630610i	$0.217195\pi$
$132$	0	0
$133$	−7.34222	−0.636651
$134$	−1.21495	−0.104956
$135$	0	0
$136$	7.96313	0.682833
$137$	6.65865	0.568887	0.284443	−	0.958693i	$-0.408191\pi$
$137$	6.65865	0.568887	0.284443	+	0.958693i	$0.408191\pi$
$138$	0	0
$139$	18.7120	1.58713	0.793564	−	0.608486i	$-0.208223\pi$
$139$	18.7120	1.58713	0.793564	+	0.608486i	$0.208223\pi$
$140$	0.718777	0.0607477
$141$	0	0
$142$	3.58536	0.300877
$143$	1.00000	0.0836242
$144$	0	0
$145$	7.93258	0.658765
$146$	−12.0204	−0.994816
$147$	0	0
$148$	−0.793075	−0.0651904
$149$	12.9677	1.06236	0.531178	−	0.847260i	$-0.321749\pi$
$149$	12.9677	1.06236	0.531178	+	0.847260i	$0.321749\pi$
$150$	0	0
$151$	−3.08166	−0.250782	−0.125391	−	0.992107i	$-0.540019\pi$
$151$	−3.08166	−0.250782	−0.125391	+	0.992107i	$0.540019\pi$
$152$	13.6776	1.10940
$153$	0	0
$154$	2.13732	0.172230
$155$	−3.93794	−0.316303
$156$	0	0
$157$	−3.27779	−0.261597	−0.130798	−	0.991409i	$-0.541754\pi$
$157$	−3.27779	−0.261597	−0.130798	+	0.991409i	$0.541754\pi$
$158$	0.472037	0.0375533
$159$	0	0
$160$	−2.59286	−0.204984
$161$	7.31903	0.576820
$162$	0	0
$163$	−4.13787	−0.324103	−0.162051	−	0.986782i	$-0.551811\pi$
$163$	−4.13787	−0.324103	−0.162051	+	0.986782i	$0.551811\pi$
$164$	−0.792539	−0.0618869
$165$	0	0
$166$	−20.0156	−1.55351
$167$	−7.56244	−0.585199	−0.292600	−	0.956235i	$-0.594520\pi$
$167$	−7.56244	−0.585199	−0.292600	+	0.956235i	$0.594520\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−12.6085	−0.967031
$171$	0	0
$172$	0.572616	0.0436616
$173$	0.143079	0.0108781	0.00543906	−	0.999985i	$-0.498269\pi$
$173$	0.143079	0.0108781	0.00543906	+	0.999985i	$0.498269\pi$
$174$	0	0
$175$	−10.0749	−0.761589
$176$	−3.71000	−0.279652
$177$	0	0
$178$	−4.84634	−0.363249
$179$	−11.8705	−0.887245	−0.443622	−	0.896214i	$-0.646307\pi$
$179$	−11.8705	−0.887245	−0.443622	+	0.896214i	$0.646307\pi$
$180$	0	0
$181$	20.1923	1.50088	0.750440	−	0.660939i	$-0.229842\pi$
$181$	20.1923	1.50088	0.750440	+	0.660939i	$0.229842\pi$
$182$	2.13732	0.158429
$183$	0	0
$184$	−13.6344	−1.00514
$185$	19.7523	1.45222
$186$	0	0
$187$	2.73073	0.199691
$188$	−0.256156	−0.0186821
$189$	0	0
$190$	−21.6567	−1.57114
$191$	7.70560	0.557558	0.278779	−	0.960355i	$-0.410070\pi$
$191$	7.70560	0.557558	0.278779	+	0.960355i	$0.410070\pi$
$192$	0	0
$193$	−7.62252	−0.548681	−0.274341	−	0.961633i	$-0.588460\pi$
$193$	−7.62252	−0.548681	−0.274341	+	0.961633i	$0.588460\pi$
$194$	−13.3557	−0.958881
$195$	0	0
$196$	0.617740	0.0441243
$197$	−15.9642	−1.13740	−0.568701	−	0.822544i	$-0.692554\pi$
$197$	−15.9642	−1.13740	−0.568701	+	0.822544i	$0.692554\pi$
$198$	0	0
$199$	−11.9792	−0.849186	−0.424593	−	0.905384i	$-0.639583\pi$
$199$	−11.9792	−0.849186	−0.424593	+	0.905384i	$0.639583\pi$
$200$	18.7682	1.32712
$201$	0	0
$202$	−0.923548	−0.0649806
$203$	−3.67196	−0.257721
$204$	0	0
$205$	19.7389	1.37863
$206$	3.61839	0.252105
$207$	0	0
$208$	−3.71000	−0.257243
$209$	4.69036	0.324439
$210$	0	0
$211$	−4.63804	−0.319296	−0.159648	−	0.987174i	$-0.551036\pi$
$211$	−4.63804	−0.319296	−0.159648	+	0.987174i	$0.551036\pi$
$212$	−1.02508	−0.0704027
$213$	0	0
$214$	−18.8518	−1.28868
$215$	−14.2615	−0.972629
$216$	0	0
$217$	1.82286	0.123744
$218$	−19.3671	−1.31170
$219$	0	0
$220$	−0.459169	−0.0309572
$221$	2.73073	0.183689
$222$	0	0
$223$	−14.9893	−1.00376	−0.501880	−	0.864937i	$-0.667358\pi$
$223$	−14.9893	−1.00376	−0.501880	+	0.864937i	$0.667358\pi$
$224$	1.20022	0.0801933
$225$	0	0
$226$	20.9465	1.39334
$227$	7.79150	0.517140	0.258570	−	0.965993i	$-0.416749\pi$
$227$	7.79150	0.517140	0.258570	+	0.965993i	$0.416749\pi$
$228$	0	0
$229$	−24.1321	−1.59469	−0.797346	−	0.603523i	$-0.793763\pi$
$229$	−24.1321	−1.59469	−0.797346	+	0.603523i	$0.793763\pi$
$230$	21.5883	1.42349
$231$	0	0
$232$	6.84040	0.449094
$233$	−23.6056	−1.54646	−0.773228	−	0.634128i	$-0.781359\pi$
$233$	−23.6056	−1.54646	−0.773228	+	0.634128i	$0.781359\pi$
$234$	0	0
$235$	6.37980	0.416172
$236$	1.36504	0.0888564
$237$	0	0
$238$	5.83644	0.378320
$239$	−4.83652	−0.312849	−0.156424	−	0.987690i	$-0.549997\pi$
$239$	−4.83652	−0.312849	−0.156424	+	0.987690i	$0.549997\pi$
$240$	0	0
$241$	−13.6291	−0.877927	−0.438963	−	0.898505i	$-0.644654\pi$
$241$	−13.6291	−0.877927	−0.438963	+	0.898505i	$0.644654\pi$
$242$	−1.36536	−0.0877690
$243$	0	0
$244$	1.33606	0.0855324
$245$	−15.3854	−0.982936
$246$	0	0
$247$	4.69036	0.298441
$248$	−3.39576	−0.215631
$249$	0	0
$250$	−6.63060	−0.419356
$251$	16.7839	1.05939	0.529696	−	0.848188i	$-0.322306\pi$
$251$	16.7839	1.05939	0.529696	+	0.848188i	$0.322306\pi$
$252$	0	0
$253$	−4.67555	−0.293949
$254$	−13.3904	−0.840187
$255$	0	0
$256$	−3.24336	−0.202710
$257$	20.3157	1.26726	0.633630	−	0.773636i	$-0.281564\pi$
$257$	20.3157	1.26726	0.633630	+	0.773636i	$0.281564\pi$
$258$	0	0
$259$	−9.14324	−0.568133
$260$	−0.459169	−0.0284765
$261$	0	0
$262$	−24.2567	−1.49858
$263$	−24.9536	−1.53871	−0.769354	−	0.638823i	$-0.779422\pi$
$263$	−24.9536	−1.53871	−0.769354	+	0.638823i	$0.779422\pi$
$264$	0	0
$265$	25.5305	1.56833
$266$	10.0248	0.614660
$267$	0	0
$268$	−0.120822	−0.00738038
$269$	−14.1033	−0.859893	−0.429947	−	0.902854i	$-0.641468\pi$
$269$	−14.1033	−0.859893	−0.429947	+	0.902854i	$0.641468\pi$
$270$	0	0
$271$	−5.84619	−0.355131	−0.177565	−	0.984109i	$-0.556822\pi$
$271$	−5.84619	−0.355131	−0.177565	+	0.984109i	$0.556822\pi$
$272$	−10.1310	−0.614283
$273$	0	0
$274$	−9.09148	−0.549237
$275$	6.43604	0.388108
$276$	0	0
$277$	29.2099	1.75505	0.877526	−	0.479529i	$-0.159193\pi$
$277$	29.2099	1.75505	0.877526	+	0.479529i	$0.159193\pi$
$278$	−25.5487	−1.53231
$279$	0	0
$280$	−15.4370	−0.922539
$281$	−6.47485	−0.386257	−0.193129	−	0.981173i	$-0.561863\pi$
$281$	−6.47485	−0.386257	−0.193129	+	0.981173i	$0.561863\pi$
$282$	0	0
$283$	−8.25653	−0.490800	−0.245400	−	0.969422i	$-0.578919\pi$
$283$	−8.25653	−0.490800	−0.245400	+	0.969422i	$0.578919\pi$
$284$	0.356549	0.0211573
$285$	0	0
$286$	−1.36536	−0.0807357
$287$	−9.13706	−0.539344
$288$	0	0
$289$	−9.54312	−0.561360
$290$	−10.8309	−0.636010
$291$	0	0
$292$	−1.19538	−0.0699543
$293$	18.6085	1.08712	0.543561	−	0.839369i	$-0.317075\pi$
$293$	18.6085	1.08712	0.543561	+	0.839369i	$0.317075\pi$
$294$	0	0
$295$	−33.9975	−1.97941
$296$	17.0327	0.990007
$297$	0	0
$298$	−17.7057	−1.02566
$299$	−4.67555	−0.270394
$300$	0	0
$301$	6.60161	0.380510
$302$	4.20759	0.242120
$303$	0	0
$304$	−17.4013	−0.998031
$305$	−33.2758	−1.90537
$306$	0	0
$307$	−14.1529	−0.807749	−0.403874	−	0.914814i	$-0.632337\pi$
$307$	−14.1529	−0.807749	−0.403874	+	0.914814i	$0.632337\pi$
$308$	0.212548	0.0121110
$309$	0	0
$310$	5.37673	0.305378
$311$	−1.25532	−0.0711827	−0.0355913	−	0.999366i	$-0.511331\pi$
$311$	−1.25532	−0.0711827	−0.0355913	+	0.999366i	$0.511331\pi$
$312$	0	0
$313$	8.35488	0.472246	0.236123	−	0.971723i	$-0.424123\pi$
$313$	8.35488	0.472246	0.236123	+	0.971723i	$0.424123\pi$
$314$	4.47538	0.252561
$315$	0	0
$316$	0.0469421	0.00264070
$317$	8.94805	0.502573	0.251286	−	0.967913i	$-0.419146\pi$
$317$	8.94805	0.502573	0.251286	+	0.967913i	$0.419146\pi$
$318$	0	0
$319$	2.34572	0.131335
$320$	28.6326	1.60061
$321$	0	0
$322$	−9.99314	−0.556896
$323$	12.8081	0.712662
$324$	0	0
$325$	6.43604	0.357007
$326$	5.64970	0.312908
$327$	0	0
$328$	17.0212	0.939839
$329$	−2.95318	−0.162814
$330$	0	0
$331$	−25.5966	−1.40692	−0.703458	−	0.710737i	$-0.748362\pi$
$331$	−25.5966	−1.40692	−0.703458	+	0.710737i	$0.748362\pi$
$332$	−1.99047	−0.109241
$333$	0	0
$334$	10.3255	0.564985
$335$	3.00919	0.164409
$336$	0	0
$337$	26.4245	1.43943	0.719716	−	0.694268i	$-0.244272\pi$
$337$	26.4245	1.43943	0.719716	+	0.694268i	$0.244272\pi$
$338$	−1.36536	−0.0742660
$339$	0	0
$340$	−1.25387	−0.0680005
$341$	−1.16448	−0.0630601
$342$	0	0
$343$	18.0795	0.976202
$344$	−12.2980	−0.663062
$345$	0	0
$346$	−0.195355	−0.0105024
$347$	−1.42217	−0.0763463	−0.0381732	−	0.999271i	$-0.512154\pi$
$347$	−1.42217	−0.0763463	−0.0381732	+	0.999271i	$0.512154\pi$
$348$	0	0
$349$	0.116794	0.00625186	0.00312593	−	0.999995i	$-0.499005\pi$
$349$	0.116794	0.00625186	0.00312593	+	0.999995i	$0.499005\pi$
$350$	13.7559	0.735283
$351$	0	0
$352$	−0.766728	−0.0408667
$353$	−10.3561	−0.551200	−0.275600	−	0.961272i	$-0.588877\pi$
$353$	−10.3561	−0.551200	−0.275600	+	0.961272i	$0.588877\pi$
$354$	0	0
$355$	−8.88019	−0.471311
$356$	−0.481948	−0.0255432
$357$	0	0
$358$	16.2076	0.856598
$359$	−37.0183	−1.95375	−0.976875	−	0.213811i	$-0.931412\pi$
$359$	−37.0183	−1.95375	−0.976875	+	0.213811i	$0.931412\pi$
$360$	0	0
$361$	2.99950	0.157868
$362$	−27.5698	−1.44904
$363$	0	0
$364$	0.212548	0.0111405
$365$	29.7720	1.55834
$366$	0	0
$367$	−31.3066	−1.63419	−0.817097	−	0.576500i	$-0.804418\pi$
$367$	−31.3066	−1.63419	−0.817097	+	0.576500i	$0.804418\pi$
$368$	17.3463	0.904238
$369$	0	0
$370$	−26.9690	−1.40205
$371$	−11.8180	−0.613559
$372$	0	0
$373$	14.1365	0.731961	0.365980	−	0.930623i	$-0.380734\pi$
$373$	14.1365	0.731961	0.365980	+	0.930623i	$0.380734\pi$
$374$	−3.72844	−0.192793
$375$	0	0
$376$	5.50141	0.283714
$377$	2.34572	0.120811
$378$	0	0
$379$	−7.00329	−0.359735	−0.179867	−	0.983691i	$-0.557567\pi$
$379$	−7.00329	−0.359735	−0.179867	+	0.983691i	$0.557567\pi$
$380$	−2.15367	−0.110481
$381$	0	0
$382$	−10.5210	−0.538299
$383$	−1.58301	−0.0808883	−0.0404441	−	0.999182i	$-0.512877\pi$
$383$	−1.58301	−0.0808883	−0.0404441	+	0.999182i	$0.512877\pi$
$384$	0	0
$385$	−5.29369	−0.269792
$386$	10.4075	0.529729
$387$	0	0
$388$	−1.32817	−0.0674274
$389$	−32.9882	−1.67257	−0.836284	−	0.548297i	$-0.815276\pi$
$389$	−32.9882	−1.67257	−0.836284	+	0.548297i	$0.815276\pi$
$390$	0	0
$391$	−12.7676	−0.645688
$392$	−13.2671	−0.670089
$393$	0	0
$394$	21.7970	1.09812
$395$	−1.16914	−0.0588257
$396$	0	0
$397$	−9.44723	−0.474143	−0.237071	−	0.971492i	$-0.576188\pi$
$397$	−9.44723	−0.474143	−0.237071	+	0.971492i	$0.576188\pi$
$398$	16.3560	0.819854
$399$	0	0
$400$	−23.8777	−1.19389
$401$	−18.7111	−0.934386	−0.467193	−	0.884155i	$-0.654735\pi$
$401$	−18.7111	−0.934386	−0.467193	+	0.884155i	$0.654735\pi$
$402$	0	0
$403$	−1.16448	−0.0580068
$404$	−0.0918430	−0.00456936
$405$	0	0
$406$	5.01356	0.248819
$407$	5.84089	0.289522
$408$	0	0
$409$	−10.6853	−0.528353	−0.264177	−	0.964474i	$-0.585100\pi$
$409$	−10.6853	−0.528353	−0.264177	+	0.964474i	$0.585100\pi$
$410$	−26.9508	−1.33101
$411$	0	0
$412$	0.359834	0.0177277
$413$	15.7373	0.774382
$414$	0	0
$415$	49.5744	2.43351
$416$	−0.766728	−0.0375919
$417$	0	0
$418$	−6.40405	−0.313233
$419$	1.34987	0.0659456	0.0329728	−	0.999456i	$-0.489503\pi$
$419$	1.34987	0.0659456	0.0329728	+	0.999456i	$0.489503\pi$
$420$	0	0
$421$	22.3597	1.08974	0.544872	−	0.838519i	$-0.316578\pi$
$421$	22.3597	1.08974	0.544872	+	0.838519i	$0.316578\pi$
$422$	6.33262	0.308267
$423$	0	0
$424$	22.0154	1.06916
$425$	17.5751	0.852516
$426$	0	0
$427$	15.4032	0.745414
$428$	−1.87474	−0.0906188
$429$	0	0
$430$	19.4722	0.939033
$431$	−20.6788	−0.996065	−0.498032	−	0.867158i	$-0.665944\pi$
$431$	−20.6788	−0.996065	−0.498032	+	0.867158i	$0.665944\pi$
$432$	0	0
$433$	19.9674	0.959571	0.479785	−	0.877386i	$-0.340715\pi$
$433$	19.9674	0.959571	0.479785	+	0.877386i	$0.340715\pi$
$434$	−2.48886	−0.119469
$435$	0	0
$436$	−1.92597	−0.0922374
$437$	−21.9300	−1.04905
$438$	0	0
$439$	−26.8104	−1.27959	−0.639795	−	0.768546i	$-0.720981\pi$
$439$	−26.8104	−1.27959	−0.639795	+	0.768546i	$0.720981\pi$
$440$	9.86150	0.470128
$441$	0	0
$442$	−3.72844	−0.177344
$443$	−2.02074	−0.0960082	−0.0480041	−	0.998847i	$-0.515286\pi$
$443$	−2.02074	−0.0960082	−0.0480041	+	0.998847i	$0.515286\pi$
$444$	0	0
$445$	12.0034	0.569014
$446$	20.4659	0.969089
$447$	0	0
$448$	−13.2539	−0.626188
$449$	−39.5449	−1.86624	−0.933119	−	0.359568i	$-0.882924\pi$
$449$	−39.5449	−1.86624	−0.933119	+	0.359568i	$0.882924\pi$
$450$	0	0
$451$	5.83694	0.274851
$452$	2.08305	0.0979782
$453$	0	0
$454$	−10.6382	−0.499277
$455$	−5.29369	−0.248172
$456$	0	0
$457$	17.3531	0.811743	0.405872	−	0.913930i	$-0.366968\pi$
$457$	17.3531	0.811743	0.405872	+	0.913930i	$0.366968\pi$
$458$	32.9491	1.53961
$459$	0	0
$460$	2.14687	0.100098
$461$	−32.4162	−1.50977	−0.754886	−	0.655856i	$-0.772308\pi$
$461$	−32.4162	−1.50977	−0.754886	+	0.655856i	$0.772308\pi$
$462$	0	0
$463$	23.4772	1.09108	0.545540	−	0.838085i	$-0.316325\pi$
$463$	23.4772	1.09108	0.545540	+	0.838085i	$0.316325\pi$
$464$	−8.70264	−0.404010
$465$	0	0
$466$	32.2303	1.49304
$467$	36.0260	1.66708	0.833542	−	0.552455i	$-0.186309\pi$
$467$	36.0260	1.66708	0.833542	+	0.552455i	$0.186309\pi$
$468$	0	0
$469$	−1.39294	−0.0643200
$470$	−8.71075	−0.401797
$471$	0	0
$472$	−29.3167	−1.34941
$473$	−4.21724	−0.193909
$474$	0	0
$475$	30.1874	1.38509
$476$	0.580410	0.0266030
$477$	0	0
$478$	6.60362	0.302042
$479$	0.174062	0.00795308	0.00397654	−	0.999992i	$-0.498734\pi$
$479$	0.174062	0.00795308	0.00397654	+	0.999992i	$0.498734\pi$
$480$	0	0
$481$	5.84089	0.266322
$482$	18.6087	0.847602
$483$	0	0
$484$	−0.135780	−0.00617181
$485$	33.0792	1.50205
$486$	0	0
$487$	−2.05568	−0.0931519	−0.0465760	−	0.998915i	$-0.514831\pi$
$487$	−2.05568	−0.0931519	−0.0465760	+	0.998915i	$0.514831\pi$
$488$	−28.6943	−1.29893
$489$	0	0
$490$	21.0067	0.948984
$491$	32.6378	1.47292	0.736461	−	0.676480i	$-0.236495\pi$
$491$	32.6378	1.47292	0.736461	+	0.676480i	$0.236495\pi$
$492$	0	0
$493$	6.40553	0.288491
$494$	−6.40405	−0.288132
$495$	0	0
$496$	4.32022	0.193984
$497$	4.11060	0.184386
$498$	0	0
$499$	−6.95908	−0.311531	−0.155766	−	0.987794i	$-0.549784\pi$
$499$	−6.95908	−0.311531	−0.155766	+	0.987794i	$0.549784\pi$
$500$	−0.659385	−0.0294886
$501$	0	0
$502$	−22.9162	−1.02280
$503$	−19.7507	−0.880640	−0.440320	−	0.897841i	$-0.645135\pi$
$503$	−19.7507	−0.880640	−0.440320	+	0.897841i	$0.645135\pi$
$504$	0	0
$505$	2.28744	0.101790
$506$	6.38382	0.283796
$507$	0	0
$508$	−1.33162	−0.0590809
$509$	31.2097	1.38334	0.691672	−	0.722212i	$-0.256874\pi$
$509$	31.2097	1.38334	0.691672	+	0.722212i	$0.256874\pi$
$510$	0	0
$511$	−13.7813	−0.609651
$512$	24.4822	1.08197
$513$	0	0
$514$	−27.7384	−1.22349
$515$	−8.96199	−0.394912
$516$	0	0
$517$	1.88655	0.0829705
$518$	12.4839	0.548509
$519$	0	0
$520$	9.86150	0.432455
$521$	6.21546	0.272304	0.136152	−	0.990688i	$-0.456526\pi$
$521$	6.21546	0.272304	0.136152	+	0.990688i	$0.456526\pi$
$522$	0	0
$523$	20.1836	0.882568	0.441284	−	0.897367i	$-0.354523\pi$
$523$	20.1836	0.882568	0.441284	+	0.897367i	$0.354523\pi$
$524$	−2.41223	−0.105379
$525$	0	0
$526$	34.0708	1.48556
$527$	−3.17988	−0.138518
$528$	0	0
$529$	−1.13927	−0.0495334
$530$	−34.8585	−1.51416
$531$	0	0
$532$	0.996925	0.0432222
$533$	5.83694	0.252826
$534$	0	0
$535$	46.6920	2.01867
$536$	2.59487	0.112081
$537$	0	0
$538$	19.2561	0.830191
$539$	−4.54957	−0.195964
$540$	0	0
$541$	−26.1092	−1.12252	−0.561262	−	0.827638i	$-0.689684\pi$
$541$	−26.1092	−1.12252	−0.561262	+	0.827638i	$0.689684\pi$
$542$	7.98218	0.342864
$543$	0	0
$544$	−2.09373	−0.0897677
$545$	47.9681	2.05473
$546$	0	0
$547$	−10.9298	−0.467326	−0.233663	−	0.972318i	$-0.575071\pi$
$547$	−10.9298	−0.467326	−0.233663	+	0.972318i	$0.575071\pi$
$548$	−0.904110	−0.0386217
$549$	0	0
$550$	−8.78754	−0.374702
$551$	11.0023	0.468713
$552$	0	0
$553$	0.541189	0.0230137
$554$	−39.8821	−1.69443
$555$	0	0
$556$	−2.54071	−0.107750
$557$	33.5088	1.41982	0.709908	−	0.704295i	$-0.248737\pi$
$557$	33.5088	1.41982	0.709908	+	0.704295i	$0.248737\pi$
$558$	0	0
$559$	−4.21724	−0.178370
$560$	19.6396	0.829926
$561$	0	0
$562$	8.84053	0.372915
$563$	−30.0182	−1.26511	−0.632557	−	0.774514i	$-0.717995\pi$
$563$	−30.0182	−1.26511	−0.632557	+	0.774514i	$0.717995\pi$
$564$	0	0
$565$	−51.8802	−2.18262
$566$	11.2732	0.473847
$567$	0	0
$568$	−7.65754	−0.321303
$569$	2.63604	0.110509	0.0552543	−	0.998472i	$-0.482403\pi$
$569$	2.63604	0.110509	0.0552543	+	0.998472i	$0.482403\pi$
$570$	0	0
$571$	−39.0005	−1.63212	−0.816059	−	0.577968i	$-0.803846\pi$
$571$	−39.0005	−1.63212	−0.816059	+	0.577968i	$0.803846\pi$
$572$	−0.135780	−0.00567724
$573$	0	0
$574$	12.4754	0.520714
$575$	−30.0920	−1.25492
$576$	0	0
$577$	−1.36649	−0.0568877	−0.0284439	−	0.999595i	$-0.509055\pi$
$577$	−1.36649	−0.0568877	−0.0284439	+	0.999595i	$0.509055\pi$
$578$	13.0298	0.541970
$579$	0	0
$580$	−1.07708	−0.0447235
$581$	−22.9478	−0.952034
$582$	0	0
$583$	7.54957	0.312671
$584$	25.6729	1.06235
$585$	0	0
$586$	−25.4074	−1.04957
$587$	28.2066	1.16421	0.582105	−	0.813113i	$-0.302229\pi$
$587$	28.2066	1.16421	0.582105	+	0.813113i	$0.302229\pi$
$588$	0	0
$589$	−5.46183	−0.225051
$590$	46.4190	1.91104
$591$	0	0
$592$	−21.6697	−0.890620
$593$	−46.1573	−1.89545	−0.947727	−	0.319081i	$-0.896626\pi$
$593$	−46.1573	−1.89545	−0.947727	+	0.319081i	$0.896626\pi$
$594$	0	0
$595$	−14.4556	−0.592624
$596$	−1.76075	−0.0721233
$597$	0	0
$598$	6.38382	0.261054
$599$	−44.8282	−1.83163	−0.915815	−	0.401601i	$-0.868454\pi$
$599$	−44.8282	−1.83163	−0.915815	+	0.401601i	$0.868454\pi$
$600$	0	0
$601$	17.9228	0.731084	0.365542	−	0.930795i	$-0.380884\pi$
$601$	17.9228	0.731084	0.365542	+	0.930795i	$0.380884\pi$
$602$	−9.01360	−0.367367
$603$	0	0
$604$	0.418427	0.0170256
$605$	3.38172	0.137487
$606$	0	0
$607$	43.0416	1.74701	0.873503	−	0.486819i	$-0.161843\pi$
$607$	43.0416	1.74701	0.873503	+	0.486819i	$0.161843\pi$
$608$	−3.59623	−0.145846
$609$	0	0
$610$	45.4336	1.83955
$611$	1.88655	0.0763218
$612$	0	0
$613$	−9.70883	−0.392136	−0.196068	−	0.980590i	$-0.562817\pi$
$613$	−9.70883	−0.392136	−0.196068	+	0.980590i	$0.562817\pi$
$614$	19.3239	0.779848
$615$	0	0
$616$	−4.56485	−0.183923
$617$	−28.7909	−1.15908	−0.579540	−	0.814944i	$-0.696768\pi$
$617$	−28.7909	−1.15908	−0.579540	+	0.814944i	$0.696768\pi$
$618$	0	0
$619$	−44.0345	−1.76990	−0.884949	−	0.465688i	$-0.845807\pi$
$619$	−44.0345	−1.76990	−0.884949	+	0.465688i	$0.845807\pi$
$620$	0.534693	0.0214738
$621$	0	0
$622$	1.71397	0.0687239
$623$	−5.55631	−0.222609
$624$	0	0
$625$	−15.7576	−0.630304
$626$	−11.4075	−0.455934
$627$	0	0
$628$	0.445058	0.0177598
$629$	15.9499	0.635964
$630$	0	0
$631$	−25.7962	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$631$	−25.7962	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$632$	−1.00817	−0.0401028
$633$	0	0
$634$	−12.2174	−0.485213
$635$	33.1652	1.31612
$636$	0	0
$637$	−4.54957	−0.180261
$638$	−3.20277	−0.126799
$639$	0	0
$640$	−33.9082	−1.34034
$641$	49.0095	1.93576	0.967880	−	0.251412i	$-0.0808949\pi$
$641$	49.0095	1.93576	0.967880	+	0.251412i	$0.0808949\pi$
$642$	0	0
$643$	24.2003	0.954368	0.477184	−	0.878803i	$-0.341658\pi$
$643$	24.2003	0.954368	0.477184	+	0.878803i	$0.341658\pi$
$644$	−0.993776	−0.0391603
$645$	0	0
$646$	−17.4877	−0.688046
$647$	33.6509	1.32295	0.661477	−	0.749965i	$-0.269930\pi$
$647$	33.6509	1.32295	0.661477	+	0.749965i	$0.269930\pi$
$648$	0	0
$649$	−10.0533	−0.394627
$650$	−8.78754	−0.344676
$651$	0	0
$652$	0.561839	0.0220033
$653$	−1.15334	−0.0451338	−0.0225669	−	0.999745i	$-0.507184\pi$
$653$	−1.15334	−0.0451338	−0.0225669	+	0.999745i	$0.507184\pi$
$654$	0	0
$655$	60.0788	2.34747
$656$	−21.6551	−0.845489
$657$	0	0
$658$	4.03217	0.157190
$659$	23.7495	0.925151	0.462575	−	0.886580i	$-0.346925\pi$
$659$	23.7495	0.925151	0.462575	+	0.886580i	$0.346925\pi$
$660$	0	0
$661$	21.2162	0.825213	0.412607	−	0.910909i	$-0.364618\pi$
$661$	21.2162	0.825213	0.412607	+	0.910909i	$0.364618\pi$
$662$	34.9487	1.35832
$663$	0	0
$664$	42.7489	1.65898
$665$	−24.8293	−0.962841
$666$	0	0
$667$	−10.9675	−0.424665
$668$	1.02683	0.0397291
$669$	0	0
$670$	−4.10863	−0.158730
$671$	−9.83989	−0.379865
$672$	0	0
$673$	−31.2618	−1.20505	−0.602527	−	0.798098i	$-0.705839\pi$
$673$	−31.2618	−1.20505	−0.602527	+	0.798098i	$0.705839\pi$
$674$	−36.0790	−1.38971
$675$	0	0
$676$	−0.135780	−0.00522230
$677$	1.55393	0.0597225	0.0298613	−	0.999554i	$-0.490493\pi$
$677$	1.55393	0.0597225	0.0298613	+	0.999554i	$0.490493\pi$
$678$	0	0
$679$	−15.3122	−0.587629
$680$	26.9291	1.03268
$681$	0	0
$682$	1.58994	0.0608819
$683$	13.9286	0.532965	0.266482	−	0.963840i	$-0.414139\pi$
$683$	13.9286	0.532965	0.266482	+	0.963840i	$0.414139\pi$
$684$	0	0
$685$	22.5177	0.860357
$686$	−24.6851	−0.942483
$687$	0	0
$688$	15.6460	0.596498
$689$	7.54957	0.287616
$690$	0	0
$691$	8.99018	0.342003	0.171001	−	0.985271i	$-0.445300\pi$
$691$	8.99018	0.342003	0.171001	+	0.985271i	$0.445300\pi$
$692$	−0.0194273	−0.000738514 0
$693$	0	0
$694$	1.94179	0.0737092
$695$	63.2787	2.40030
$696$	0	0
$697$	15.9391	0.603737
$698$	−0.159467	−0.00603591
$699$	0	0
$700$	1.36796	0.0517042
$701$	19.7640	0.746476	0.373238	−	0.927736i	$-0.378247\pi$
$701$	19.7640	0.746476	0.373238	+	0.927736i	$0.378247\pi$
$702$	0	0
$703$	27.3959	1.03326
$704$	8.46687	0.319107
$705$	0	0
$706$	14.1399	0.532161
$707$	−1.05884	−0.0398219
$708$	0	0
$709$	−10.9372	−0.410753	−0.205377	−	0.978683i	$-0.565842\pi$
$709$	−10.9372	−0.410753	−0.205377	+	0.978683i	$0.565842\pi$
$710$	12.1247	0.455032
$711$	0	0
$712$	10.3507	0.387910
$713$	5.44458	0.203901
$714$	0	0
$715$	3.38172	0.126469
$716$	1.61178	0.0602349
$717$	0	0
$718$	50.5434	1.88626
$719$	40.8149	1.52214	0.761070	−	0.648669i	$-0.224674\pi$
$719$	40.8149	1.52214	0.761070	+	0.648669i	$0.224674\pi$
$720$	0	0
$721$	4.14847	0.154497
$722$	−4.09541	−0.152415
$723$	0	0
$724$	−2.74170	−0.101895
$725$	15.0972	0.560694
$726$	0	0
$727$	−1.61194	−0.0597834	−0.0298917	−	0.999553i	$-0.509516\pi$
$727$	−1.61194	−0.0597834	−0.0298917	+	0.999553i	$0.509516\pi$
$728$	−4.56485	−0.169184
$729$	0	0
$730$	−40.6497	−1.50451
$731$	−11.5161	−0.425940
$732$	0	0
$733$	−1.35925	−0.0502051	−0.0251026	−	0.999685i	$-0.507991\pi$
$733$	−1.35925	−0.0502051	−0.0251026	+	0.999685i	$0.507991\pi$
$734$	42.7450	1.57775
$735$	0	0
$736$	3.58487	0.132140
$737$	0.889838	0.0327776
$738$	0	0
$739$	−18.2169	−0.670119	−0.335059	−	0.942197i	$-0.608756\pi$
$739$	−18.2169	−0.670119	−0.335059	+	0.942197i	$0.608756\pi$
$740$	−2.68196	−0.0985907
$741$	0	0
$742$	16.1359	0.592366
$743$	28.1616	1.03315	0.516575	−	0.856242i	$-0.327207\pi$
$743$	28.1616	1.03315	0.516575	+	0.856242i	$0.327207\pi$
$744$	0	0
$745$	43.8532	1.60666
$746$	−19.3015	−0.706678
$747$	0	0
$748$	−0.370778	−0.0135570
$749$	−21.6135	−0.789742
$750$	0	0
$751$	−22.9509	−0.837490	−0.418745	−	0.908104i	$-0.637530\pi$
$751$	−22.9509	−0.837490	−0.418745	+	0.908104i	$0.637530\pi$
$752$	−6.99912	−0.255232
$753$	0	0
$754$	−3.20277	−0.116638
$755$	−10.4213	−0.379271
$756$	0	0
$757$	25.3032	0.919661	0.459830	−	0.888007i	$-0.347910\pi$
$757$	25.3032	0.919661	0.459830	+	0.888007i	$0.347910\pi$
$758$	9.56204	0.347309
$759$	0	0
$760$	46.2540	1.67781
$761$	10.6708	0.386817	0.193408	−	0.981118i	$-0.438046\pi$
$761$	10.6708	0.386817	0.193408	+	0.981118i	$0.438046\pi$
$762$	0	0
$763$	−22.2042	−0.803848
$764$	−1.04626	−0.0378525
$765$	0	0
$766$	2.16139	0.0780943
$767$	−10.0533	−0.363004
$768$	0	0
$769$	22.1015	0.797002	0.398501	−	0.917168i	$-0.369531\pi$
$769$	22.1015	0.797002	0.398501	+	0.917168i	$0.369531\pi$
$770$	7.22782	0.260473
$771$	0	0
$772$	1.03498	0.0372499
$773$	5.52424	0.198693	0.0993465	−	0.995053i	$-0.468325\pi$
$773$	5.52424	0.198693	0.0993465	+	0.995053i	$0.468325\pi$
$774$	0	0
$775$	−7.49463	−0.269215
$776$	28.5248	1.02398
$777$	0	0
$778$	45.0409	1.61479
$779$	27.3774	0.980897
$780$	0	0
$781$	−2.62594	−0.0939634
$782$	17.4325	0.623385
$783$	0	0
$784$	16.8789	0.602819
$785$	−11.0846	−0.395626
$786$	0	0
$787$	6.13272	0.218608	0.109304	−	0.994008i	$-0.465138\pi$
$787$	6.13272	0.218608	0.109304	+	0.994008i	$0.465138\pi$
$788$	2.16762	0.0772182
$789$	0	0
$790$	1.59630	0.0567938
$791$	24.0151	0.853879
$792$	0	0
$793$	−9.83989	−0.349425
$794$	12.8989	0.457765
$795$	0	0
$796$	1.62654	0.0576512
$797$	−19.5848	−0.693728	−0.346864	−	0.937915i	$-0.612754\pi$
$797$	−19.5848	−0.693728	−0.346864	+	0.937915i	$0.612754\pi$
$798$	0	0
$799$	5.15167	0.182253
$800$	−4.93469	−0.174468
$801$	0	0
$802$	25.5474	0.902110
$803$	8.80381	0.310680
$804$	0	0
$805$	24.7509	0.872355
$806$	1.58994	0.0560032
$807$	0	0
$808$	1.97250	0.0693922
$809$	−7.74183	−0.272188	−0.136094	−	0.990696i	$-0.543455\pi$
$809$	−7.74183	−0.272188	−0.136094	+	0.990696i	$0.543455\pi$
$810$	0	0
$811$	−27.4950	−0.965478	−0.482739	−	0.875764i	$-0.660358\pi$
$811$	−27.4950	−0.965478	−0.482739	+	0.875764i	$0.660358\pi$
$812$	0.498578	0.0174966
$813$	0	0
$814$	−7.97494	−0.279522
$815$	−13.9931	−0.490158
$816$	0	0
$817$	−19.7804	−0.692028
$818$	14.5893	0.510103
$819$	0	0
$820$	−2.68015	−0.0935948
$821$	18.6498	0.650881	0.325441	−	0.945562i	$-0.394487\pi$
$821$	18.6498	0.650881	0.325441	+	0.945562i	$0.394487\pi$
$822$	0	0
$823$	−0.258109	−0.00899712	−0.00449856	−	0.999990i	$-0.501432\pi$
$823$	−0.258109	−0.00899712	−0.00449856	+	0.999990i	$0.501432\pi$
$824$	−7.72808	−0.269220
$825$	0	0
$826$	−21.4872	−0.747634
$827$	−19.8549	−0.690424	−0.345212	−	0.938525i	$-0.612193\pi$
$827$	−19.8549	−0.690424	−0.345212	+	0.938525i	$0.612193\pi$
$828$	0	0
$829$	−10.3895	−0.360843	−0.180422	−	0.983589i	$-0.557746\pi$
$829$	−10.3895	−0.360843	−0.180422	+	0.983589i	$0.557746\pi$
$830$	−67.6871	−2.34945
$831$	0	0
$832$	8.46687	0.293536
$833$	−12.4236	−0.430454
$834$	0	0
$835$	−25.5741	−0.885027
$836$	−0.636856	−0.0220261
$837$	0	0
$838$	−1.84307	−0.0636678
$839$	−25.4028	−0.877003	−0.438502	−	0.898730i	$-0.644491\pi$
$839$	−25.4028	−0.877003	−0.438502	+	0.898730i	$0.644491\pi$
$840$	0	0
$841$	−23.4976	−0.810262
$842$	−30.5291	−1.05210
$843$	0	0
$844$	0.629753	0.0216770
$845$	3.38172	0.116335
$846$	0	0
$847$	−1.56538	−0.0537872
$848$	−28.0089	−0.961831
$849$	0	0
$850$	−23.9964	−0.823069
$851$	−27.3094	−0.936153
$852$	0	0
$853$	−20.3987	−0.698438	−0.349219	−	0.937041i	$-0.613553\pi$
$853$	−20.3987	−0.698438	−0.349219	+	0.937041i	$0.613553\pi$
$854$	−21.0310	−0.719666
$855$	0	0
$856$	40.2634	1.37617
$857$	−46.5680	−1.59073	−0.795366	−	0.606129i	$-0.792721\pi$
$857$	−46.5680	−1.59073	−0.795366	+	0.606129i	$0.792721\pi$
$858$	0	0
$859$	1.66333	0.0567523	0.0283761	−	0.999597i	$-0.490966\pi$
$859$	1.66333	0.0567523	0.0283761	+	0.999597i	$0.490966\pi$
$860$	1.93643	0.0660317
$861$	0	0
$862$	28.2342	0.961659
$863$	−11.3783	−0.387321	−0.193661	−	0.981069i	$-0.562036\pi$
$863$	−11.3783	−0.387321	−0.193661	+	0.981069i	$0.562036\pi$
$864$	0	0
$865$	0.483854	0.0164515
$866$	−27.2627	−0.926426
$867$	0	0
$868$	−0.247507	−0.00840094
$869$	−0.345723	−0.0117278
$870$	0	0
$871$	0.889838	0.0301510
$872$	41.3638	1.40075
$873$	0	0
$874$	29.9425	1.01282
$875$	−7.60195	−0.256993
$876$	0	0
$877$	−26.1045	−0.881487	−0.440744	−	0.897633i	$-0.645285\pi$
$877$	−26.1045	−0.881487	−0.440744	+	0.897633i	$0.645285\pi$
$878$	36.6059	1.23539
$879$	0	0
$880$	−12.5462	−0.422932
$881$	−6.95082	−0.234179	−0.117090	−	0.993121i	$-0.537356\pi$
$881$	−6.95082	−0.234179	−0.117090	+	0.993121i	$0.537356\pi$
$882$	0	0
$883$	17.2427	0.580264	0.290132	−	0.956987i	$-0.406301\pi$
$883$	17.2427	0.580264	0.290132	+	0.956987i	$0.406301\pi$
$884$	−0.370778	−0.0124706
$885$	0	0
$886$	2.75904	0.0926919
$887$	30.9154	1.03804	0.519019	−	0.854763i	$-0.326297\pi$
$887$	30.9154	1.03804	0.519019	+	0.854763i	$0.326297\pi$
$888$	0	0
$889$	−15.3520	−0.514890
$890$	−16.3890	−0.549360
$891$	0	0
$892$	2.03525	0.0681452
$893$	8.84862	0.296108
$894$	0	0
$895$	−40.1428	−1.34183
$896$	15.6960	0.524366
$897$	0	0
$898$	53.9932	1.80177
$899$	−2.73154	−0.0911021
$900$	0	0
$901$	20.6158	0.686813
$902$	−7.96956	−0.265357
$903$	0	0
$904$	−44.7372	−1.48794
$905$	68.2846	2.26986
$906$	0	0
$907$	37.9142	1.25892	0.629460	−	0.777033i	$-0.283276\pi$
$907$	37.9142	1.25892	0.629460	+	0.777033i	$0.283276\pi$
$908$	−1.05793	−0.0351086
$909$	0	0
$910$	7.22782	0.239600
$911$	−37.3558	−1.23765	−0.618826	−	0.785528i	$-0.712391\pi$
$911$	−37.3558	−1.23765	−0.618826	+	0.785528i	$0.712391\pi$
$912$	0	0
$913$	14.6595	0.485159
$914$	−23.6933	−0.783704
$915$	0	0
$916$	3.27665	0.108263
$917$	−27.8102	−0.918374
$918$	0	0
$919$	34.4729	1.13715	0.568577	−	0.822630i	$-0.307494\pi$
$919$	34.4729	1.13715	0.568577	+	0.822630i	$0.307494\pi$
$920$	−46.1079	−1.52013
$921$	0	0
$922$	44.2599	1.45762
$923$	−2.62594	−0.0864338
$924$	0	0
$925$	37.5922	1.23602
$926$	−32.0550	−1.05339
$927$	0	0
$928$	−1.79853	−0.0590397
$929$	−12.5997	−0.413382	−0.206691	−	0.978406i	$-0.566269\pi$
$929$	−12.5997	−0.413382	−0.206691	+	0.978406i	$0.566269\pi$
$930$	0	0
$931$	−21.3391	−0.699362
$932$	3.20517	0.104989
$933$	0	0
$934$	−49.1886	−1.60950
$935$	9.23456	0.302003
$936$	0	0
$937$	−32.9887	−1.07769	−0.538847	−	0.842404i	$-0.681140\pi$
$937$	−32.9887	−1.07769	−0.538847	+	0.842404i	$0.681140\pi$
$938$	1.90187	0.0620982
$939$	0	0
$940$	−0.866247	−0.0282539
$941$	12.4007	0.404251	0.202125	−	0.979360i	$-0.435215\pi$
$941$	12.4007	0.404251	0.202125	+	0.979360i	$0.435215\pi$
$942$	0	0
$943$	−27.2909	−0.888714
$944$	37.2979	1.21394
$945$	0	0
$946$	5.75807	0.187211
$947$	−47.5413	−1.54488	−0.772442	−	0.635085i	$-0.780965\pi$
$947$	−47.5413	−1.54488	−0.772442	+	0.635085i	$0.780965\pi$
$948$	0	0
$949$	8.80381	0.285784
$950$	−41.2167	−1.33725
$951$	0	0
$952$	−12.4654	−0.404005
$953$	−55.9904	−1.81371	−0.906854	−	0.421446i	$-0.861523\pi$
$953$	−55.9904	−1.81371	−0.906854	+	0.421446i	$0.861523\pi$
$954$	0	0
$955$	26.0582	0.843223
$956$	0.656702	0.0212393
$957$	0	0
$958$	−0.237658	−0.00767837
$959$	−10.4233	−0.336587
$960$	0	0
$961$	−29.6440	−0.956258
$962$	−7.97494	−0.257123
$963$	0	0
$964$	1.85056	0.0596024
$965$	−25.7772	−0.829799
$966$	0	0
$967$	−23.8808	−0.767956	−0.383978	−	0.923342i	$-0.625446\pi$
$967$	−23.8808	−0.767956	−0.383978	+	0.923342i	$0.625446\pi$
$968$	2.91612	0.0937276
$969$	0	0
$970$	−45.1652	−1.45017
$971$	−39.4355	−1.26555	−0.632773	−	0.774338i	$-0.718083\pi$
$971$	−39.4355	−1.26555	−0.632773	+	0.774338i	$0.718083\pi$
$972$	0	0
$973$	−29.2914	−0.939040
$974$	2.80676	0.0899343
$975$	0	0
$976$	36.5060	1.16853
$977$	−25.1714	−0.805303	−0.402652	−	0.915353i	$-0.631911\pi$
$977$	−25.1714	−0.805303	−0.402652	+	0.915353i	$0.631911\pi$
$978$	0	0
$979$	3.54949	0.113442
$980$	2.08902	0.0667314
$981$	0	0
$982$	−44.5625	−1.42205
$983$	−32.9259	−1.05017	−0.525087	−	0.851048i	$-0.675967\pi$
$983$	−32.9259	−1.05017	−0.525087	+	0.851048i	$0.675967\pi$
$984$	0	0
$985$	−53.9865	−1.72015
$986$	−8.74589	−0.278526
$987$	0	0
$988$	−0.636856	−0.0202611
$989$	19.7179	0.626993
$990$	0	0
$991$	−28.4513	−0.903784	−0.451892	−	0.892073i	$-0.649251\pi$
$991$	−28.4513	−0.903784	−0.451892	+	0.892073i	$0.649251\pi$
$992$	0.892838	0.0283476
$993$	0	0
$994$	−5.61247	−0.178017
$995$	−40.5105	−1.28427
$996$	0	0
$997$	7.62156	0.241377	0.120689	−	0.992690i	$-0.461490\pi$
$997$	7.62156	0.241377	0.120689	+	0.992690i	$0.461490\pi$
$998$	9.50168	0.300770
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.q.1.2		6
3.2	odd	2		143.2.a.c.1.5	✓	6
12.11	even	2		2288.2.a.z.1.1		6
15.14	odd	2		3575.2.a.p.1.2		6
21.20	even	2		7007.2.a.r.1.5		6
24.5	odd	2		9152.2.a.cm.1.1		6
24.11	even	2		9152.2.a.cs.1.6		6
33.32	even	2		1573.2.a.m.1.2		6
39.38	odd	2		1859.2.a.m.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.a.c.1.5	✓	6	3.2	odd	2
1287.2.a.q.1.2		6	1.1	even	1	trivial
1573.2.a.m.1.2		6	33.32	even	2
1859.2.a.m.1.2		6	39.38	odd	2
2288.2.a.z.1.1		6	12.11	even	2
3575.2.a.p.1.2		6	15.14	odd	2
7007.2.a.r.1.5		6	21.20	even	2
9152.2.a.cm.1.1		6	24.5	odd	2
9152.2.a.cs.1.6		6	24.11	even	2

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

\(f(q)\)	\(=\)	\(q-1.36536 q^{2} -0.135780 q^{4} +3.38172 q^{5} -1.56538 q^{7} +2.91612 q^{8} +O(q^{10})\)	\(q-1.36536 q^{2} -0.135780 q^{4} +3.38172 q^{5} -1.56538 q^{7} +2.91612 q^{8} -4.61728 q^{10} +1.00000 q^{11} +1.00000 q^{13} +2.13732 q^{14} -3.71000 q^{16} +2.73073 q^{17} +4.69036 q^{19} -0.459169 q^{20} -1.36536 q^{22} -4.67555 q^{23} +6.43604 q^{25} -1.36536 q^{26} +0.212548 q^{28} +2.34572 q^{29} -1.16448 q^{31} -0.766728 q^{32} -3.72844 q^{34} -5.29369 q^{35} +5.84089 q^{37} -6.40405 q^{38} +9.86150 q^{40} +5.83694 q^{41} -4.21724 q^{43} -0.135780 q^{44} +6.38382 q^{46} +1.88655 q^{47} -4.54957 q^{49} -8.78754 q^{50} -0.135780 q^{52} +7.54957 q^{53} +3.38172 q^{55} -4.56485 q^{56} -3.20277 q^{58} -10.0533 q^{59} -9.83989 q^{61} +1.58994 q^{62} +8.46687 q^{64} +3.38172 q^{65} +0.889838 q^{67} -0.370778 q^{68} +7.22782 q^{70} -2.62594 q^{71} +8.80381 q^{73} -7.97494 q^{74} -0.636856 q^{76} -1.56538 q^{77} -0.345723 q^{79} -12.5462 q^{80} -7.96956 q^{82} +14.6595 q^{83} +9.23456 q^{85} +5.75807 q^{86} +2.91612 q^{88} +3.54949 q^{89} -1.56538 q^{91} +0.634845 q^{92} -2.57583 q^{94} +15.8615 q^{95} +9.78176 q^{97} +6.21182 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8	\( 6 q + 8 q^{4} - q^{5} + 4 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 12 q^{14} + 8 q^{16} - 10 q^{19} - 4 q^{20} - 11 q^{23} + 23 q^{25} + 9 q^{28} - 2 q^{29} - 9 q^{31} + 17 q^{32} - 40 q^{34} + 24 q^{35} + 15 q^{37} + 9 q^{38} + 16 q^{40} + 4 q^{41} - 2 q^{43} + 8 q^{44} - 6 q^{46} - 6 q^{47} + 20 q^{49} + 4 q^{50} + 8 q^{52} - 2 q^{53} - q^{55} + 39 q^{56} + 18 q^{58} - 11 q^{59} + 16 q^{61} - 16 q^{62} + 36 q^{64} - q^{65} + 9 q^{67} - 12 q^{68} + 32 q^{70} + 15 q^{71} + 32 q^{73} - 22 q^{74} - 26 q^{76} + 4 q^{77} + 14 q^{79} - 56 q^{80} - 24 q^{82} + 26 q^{83} - 12 q^{85} + 10 q^{86} + 6 q^{88} + 23 q^{89} + 4 q^{91} - 83 q^{92} + 46 q^{94} + 52 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) \) 6 * q + 8 * q^4 - q^5 + 4 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 + 12 * q^14 + 8 * q^16 - 10 * q^19 - 4 * q^20 - 11 * q^23 + 23 * q^25 + 9 * q^28 - 2 * q^29 - 9 * q^31 + 17 * q^32 - 40 * q^34 + 24 * q^35 + 15 * q^37 + 9 * q^38 + 16 * q^40 + 4 * q^41 - 2 * q^43 + 8 * q^44 - 6 * q^46 - 6 * q^47 + 20 * q^49 + 4 * q^50 + 8 * q^52 - 2 * q^53 - q^55 + 39 * q^56 + 18 * q^58 - 11 * q^59 + 16 * q^61 - 16 * q^62 + 36 * q^64 - q^65 + 9 * q^67 - 12 * q^68 + 32 * q^70 + 15 * q^71 + 32 * q^73 - 22 * q^74 - 26 * q^76 + 4 * q^77 + 14 * q^79 - 56 * q^80 - 24 * q^82 + 26 * q^83 - 12 * q^85 + 10 * q^86 + 6 * q^88 + 23 * q^89 + 4 * q^91 - 83 * q^92 + 46 * q^94 + 52 * q^95 + 27 * q^97 + 3 * q^98

Introduction

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