LMFDB - Embedded newform 2888.2.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,3,0,2,0,-2,0,9,0,-5,0,10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3022625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 $ x^6 - 2x^5 - 7x^4 + 7x^3 + 17x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.50071$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29838 q^{3} +2.83118 q^{5} -4.51153 q^{7} +2.28257 q^{9} -5.79410 q^{11} +0.740130 q^{13} -6.50714 q^{15} -5.42213 q^{17} +10.3692 q^{21} -3.94732 q^{23} +3.01559 q^{25} +1.64894 q^{27} -5.50802 q^{29} +5.13934 q^{31} +13.3171 q^{33} -12.7730 q^{35} +6.55825 q^{37} -1.70110 q^{39} -6.98726 q^{41} +5.53624 q^{43} +6.46236 q^{45} +0.994182 q^{47} +13.3539 q^{49} +12.4621 q^{51} +1.00231 q^{53} -16.4041 q^{55} +0.364344 q^{59} +1.82500 q^{61} -10.2979 q^{63} +2.09544 q^{65} +12.8940 q^{67} +9.07246 q^{69} -3.46632 q^{71} +1.41070 q^{73} -6.93099 q^{75} +26.1403 q^{77} +6.19548 q^{79} -10.6376 q^{81} -2.62892 q^{83} -15.3510 q^{85} +12.6596 q^{87} -2.10047 q^{89} -3.33912 q^{91} -11.8122 q^{93} -0.874719 q^{97} -13.2254 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 - 2 * q^7 + 9 * q^9 - 5 * q^11 + 10 * q^13 + 4 * q^17 + 23 * q^21 - 15 * q^23 + 12 * q^25 + 18 * q^27 - 7 * q^29 + 5 * q^31 - q^33 - 38 * q^35 + 17 * q^37 + 37 * q^39 + 4 * q^41 - 11 * q^43 + 6 * q^45 + 18 * q^47 + 20 * q^49 + 48 * q^51 - 7 * q^53 - 42 * q^55 + q^59 - 4 * q^61 + 19 * q^63 + 6 * q^65 + 20 * q^67 + 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 - 22 * q^83 + 32 * q^85 + 29 * q^87 + 14 * q^89 + 26 * q^91 - 5 * q^93 + 41 * q^97 - 47 * q^99

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	0
$2$	0	0
$3$	−2.29838	−1.32697	−0.663486	−	0.748189i	$-0.730924\pi$
$3$	−2.29838	−1.32697	−0.663486	+	0.748189i	$0.730924\pi$
$4$	0	0
$5$	2.83118	1.26614	0.633072	−	0.774093i	$-0.281794\pi$
$5$	2.83118	1.26614	0.633072	+	0.774093i	$0.281794\pi$
$6$	0	0
$7$	−4.51153	−1.70520	−0.852599	−	0.522565i	$-0.824975\pi$
$7$	−4.51153	−1.70520	−0.852599	+	0.522565i	$0.824975\pi$
$8$	0	0
$9$	2.28257	0.760855
$10$	0	0
$11$	−5.79410	−1.74699	−0.873493	−	0.486837i	$-0.838151\pi$
$11$	−5.79410	−1.74699	−0.873493	+	0.486837i	$0.838151\pi$
$12$	0	0
$13$	0.740130	0.205275	0.102638	−	0.994719i	$-0.467272\pi$
$13$	0.740130	0.205275	0.102638	+	0.994719i	$0.467272\pi$
$14$	0	0
$15$	−6.50714	−1.68014
$16$	0	0
$17$	−5.42213	−1.31506	−0.657530	−	0.753428i	$-0.728399\pi$
$17$	−5.42213	−1.31506	−0.657530	+	0.753428i	$0.728399\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	10.3692	2.26275
$22$	0	0
$23$	−3.94732	−0.823074	−0.411537	−	0.911393i	$-0.635008\pi$
$23$	−3.94732	−0.823074	−0.411537	+	0.911393i	$0.635008\pi$
$24$	0	0
$25$	3.01559	0.603119
$26$	0	0
$27$	1.64894	0.317338
$28$	0	0
$29$	−5.50802	−1.02281	−0.511407	−	0.859338i	$-0.670876\pi$
$29$	−5.50802	−1.02281	−0.511407	+	0.859338i	$0.670876\pi$
$30$	0	0
$31$	5.13934	0.923053	0.461526	−	0.887126i	$-0.347302\pi$
$31$	5.13934	0.923053	0.461526	+	0.887126i	$0.347302\pi$
$32$	0	0
$33$	13.3171	2.31820
$34$	0	0
$35$	−12.7730	−2.15903
$36$	0	0
$37$	6.55825	1.07817	0.539085	−	0.842251i	$-0.318770\pi$
$37$	6.55825	1.07817	0.539085	+	0.842251i	$0.318770\pi$
$38$	0	0
$39$	−1.70110	−0.272394
$40$	0	0
$41$	−6.98726	−1.09123	−0.545614	−	0.838037i	$-0.683703\pi$
$41$	−6.98726	−1.09123	−0.545614	+	0.838037i	$0.683703\pi$
$42$	0	0
$43$	5.53624	0.844269	0.422135	−	0.906533i	$-0.361281\pi$
$43$	5.53624	0.844269	0.422135	+	0.906533i	$0.361281\pi$
$44$	0	0
$45$	6.46236	0.963352
$46$	0	0
$47$	0.994182	0.145016	0.0725082	−	0.997368i	$-0.476900\pi$
$47$	0.994182	0.145016	0.0725082	+	0.997368i	$0.476900\pi$
$48$	0	0
$49$	13.3539	1.90770
$50$	0	0
$51$	12.4621	1.74505
$52$	0	0
$53$	1.00231	0.137678	0.0688390	−	0.997628i	$-0.478071\pi$
$53$	1.00231	0.137678	0.0688390	+	0.997628i	$0.478071\pi$
$54$	0	0
$55$	−16.4041	−2.21193
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.364344	0.0474335	0.0237168	−	0.999719i	$-0.492450\pi$
$59$	0.364344	0.0474335	0.0237168	+	0.999719i	$0.492450\pi$
$60$	0	0
$61$	1.82500	0.233668	0.116834	−	0.993151i	$-0.462726\pi$
$61$	1.82500	0.233668	0.116834	+	0.993151i	$0.462726\pi$
$62$	0	0
$63$	−10.2979	−1.29741
$64$	0	0
$65$	2.09544	0.259908
$66$	0	0
$67$	12.8940	1.57526	0.787629	−	0.616150i	$-0.211308\pi$
$67$	12.8940	1.57526	0.787629	+	0.616150i	$0.211308\pi$
$68$	0	0
$69$	9.07246	1.09220
$70$	0	0
$71$	−3.46632	−0.411376	−0.205688	−	0.978618i	$-0.565943\pi$
$71$	−3.46632	−0.411376	−0.205688	+	0.978618i	$0.565943\pi$
$72$	0	0
$73$	1.41070	0.165110	0.0825552	−	0.996586i	$-0.473692\pi$
$73$	1.41070	0.165110	0.0825552	+	0.996586i	$0.473692\pi$
$74$	0	0
$75$	−6.93099	−0.800322
$76$	0	0
$77$	26.1403	2.97896
$78$	0	0
$79$	6.19548	0.697046	0.348523	−	0.937300i	$-0.386683\pi$
$79$	6.19548	0.697046	0.348523	+	0.937300i	$0.386683\pi$
$80$	0	0
$81$	−10.6376	−1.18195
$82$	0	0
$83$	−2.62892	−0.288561	−0.144281	−	0.989537i	$-0.546087\pi$
$83$	−2.62892	−0.288561	−0.144281	+	0.989537i	$0.546087\pi$
$84$	0	0
$85$	−15.3510	−1.66505
$86$	0	0
$87$	12.6596	1.35725
$88$	0	0
$89$	−2.10047	−0.222649	−0.111324	−	0.993784i	$-0.535509\pi$
$89$	−2.10047	−0.222649	−0.111324	+	0.993784i	$0.535509\pi$
$90$	0	0
$91$	−3.33912	−0.350035
$92$	0	0
$93$	−11.8122	−1.22487
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.874719	−0.0888143	−0.0444071	−	0.999014i	$-0.514140\pi$
$97$	−0.874719	−0.0888143	−0.0444071	+	0.999014i	$0.514140\pi$
$98$	0	0
$99$	−13.2254	−1.32920
$100$	0	0
$101$	−11.4556	−1.13987	−0.569936	−	0.821689i	$-0.693032\pi$
$101$	−11.4556	−1.13987	−0.569936	+	0.821689i	$0.693032\pi$
$102$	0	0
$103$	8.55696	0.843143	0.421571	−	0.906795i	$-0.361479\pi$
$103$	8.55696	0.843143	0.421571	+	0.906795i	$0.361479\pi$
$104$	0	0
$105$	29.3572	2.86497
$106$	0	0
$107$	−6.01725	−0.581709	−0.290855	−	0.956767i	$-0.593940\pi$
$107$	−6.01725	−0.581709	−0.290855	+	0.956767i	$0.593940\pi$
$108$	0	0
$109$	−19.8749	−1.90367	−0.951833	−	0.306616i	$-0.900803\pi$
$109$	−19.8749	−1.90367	−0.951833	+	0.306616i	$0.900803\pi$
$110$	0	0
$111$	−15.0734	−1.43070
$112$	0	0
$113$	18.1119	1.70382	0.851912	−	0.523685i	$-0.175443\pi$
$113$	18.1119	1.70382	0.851912	+	0.523685i	$0.175443\pi$
$114$	0	0
$115$	−11.1756	−1.04213
$116$	0	0
$117$	1.68939	0.156185
$118$	0	0
$119$	24.4621	2.24244
$120$	0	0
$121$	22.5716	2.05196
$122$	0	0
$123$	16.0594	1.44803
$124$	0	0
$125$	−5.61822	−0.502509
$126$	0	0
$127$	−15.6786	−1.39125	−0.695627	−	0.718403i	$-0.744874\pi$
$127$	−15.6786	−1.39125	−0.695627	+	0.718403i	$0.744874\pi$
$128$	0	0
$129$	−12.7244	−1.12032
$130$	0	0
$131$	10.3981	0.908487	0.454244	−	0.890878i	$-0.349910\pi$
$131$	10.3981	0.908487	0.454244	+	0.890878i	$0.349910\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.66845	0.401796
$136$	0	0
$137$	−16.4792	−1.40791	−0.703955	−	0.710245i	$-0.748584\pi$
$137$	−16.4792	−1.40791	−0.703955	+	0.710245i	$0.748584\pi$
$138$	0	0
$139$	2.12015	0.179829	0.0899145	−	0.995949i	$-0.471341\pi$
$139$	2.12015	0.179829	0.0899145	+	0.995949i	$0.471341\pi$
$140$	0	0
$141$	−2.28501	−0.192433
$142$	0	0
$143$	−4.28838	−0.358613
$144$	0	0
$145$	−15.5942	−1.29503
$146$	0	0
$147$	−30.6924	−2.53147
$148$	0	0
$149$	15.5374	1.27288	0.636439	−	0.771327i	$-0.280407\pi$
$149$	15.5374	1.27288	0.636439	+	0.771327i	$0.280407\pi$
$150$	0	0
$151$	−2.41862	−0.196825	−0.0984124	−	0.995146i	$-0.531376\pi$
$151$	−2.41862	−0.196825	−0.0984124	+	0.995146i	$0.531376\pi$
$152$	0	0
$153$	−12.3764	−1.00057
$154$	0	0
$155$	14.5504	1.16872
$156$	0	0
$157$	4.66351	0.372189	0.186094	−	0.982532i	$-0.440417\pi$
$157$	4.66351	0.372189	0.186094	+	0.982532i	$0.440417\pi$
$158$	0	0
$159$	−2.30369	−0.182695
$160$	0	0
$161$	17.8085	1.40350
$162$	0	0
$163$	17.8733	1.39995	0.699973	−	0.714169i	$-0.253195\pi$
$163$	17.8733	1.39995	0.699973	+	0.714169i	$0.253195\pi$
$164$	0	0
$165$	37.7030	2.93518
$166$	0	0
$167$	11.7221	0.907087	0.453543	−	0.891234i	$-0.350160\pi$
$167$	11.7221	0.907087	0.453543	+	0.891234i	$0.350160\pi$
$168$	0	0
$169$	−12.4522	−0.957862
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.74631	0.588941	0.294471	−	0.955661i	$-0.404857\pi$
$173$	7.74631	0.588941	0.294471	+	0.955661i	$0.404857\pi$
$174$	0	0
$175$	−13.6049	−1.02844
$176$	0	0
$177$	−0.837402	−0.0629430
$178$	0	0
$179$	14.6756	1.09690	0.548452	−	0.836182i	$-0.315217\pi$
$179$	14.6756	1.09690	0.548452	+	0.836182i	$0.315217\pi$
$180$	0	0
$181$	−4.26090	−0.316710	−0.158355	−	0.987382i	$-0.550619\pi$
$181$	−4.26090	−0.316710	−0.158355	+	0.987382i	$0.550619\pi$
$182$	0	0
$183$	−4.19455	−0.310070
$184$	0	0
$185$	18.5676	1.36512
$186$	0	0
$187$	31.4164	2.29739
$188$	0	0
$189$	−7.43924	−0.541125
$190$	0	0
$191$	13.8202	0.999995	0.499998	−	0.866027i	$-0.333334\pi$
$191$	13.8202	0.999995	0.499998	+	0.866027i	$0.333334\pi$
$192$	0	0
$193$	26.5259	1.90938	0.954688	−	0.297608i	$-0.0961889\pi$
$193$	26.5259	1.90938	0.954688	+	0.297608i	$0.0961889\pi$
$194$	0	0
$195$	−4.81613	−0.344890
$196$	0	0
$197$	18.9255	1.34839	0.674194	−	0.738554i	$-0.264491\pi$
$197$	18.9255	1.34839	0.674194	+	0.738554i	$0.264491\pi$
$198$	0	0
$199$	24.8247	1.75978	0.879889	−	0.475180i	$-0.157617\pi$
$199$	24.8247	1.75978	0.879889	+	0.475180i	$0.157617\pi$
$200$	0	0
$201$	−29.6355	−2.09032
$202$	0	0
$203$	24.8496	1.74410
$204$	0	0
$205$	−19.7822	−1.38165
$206$	0	0
$207$	−9.01002	−0.626240
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.48981	−0.446777	−0.223388	−	0.974730i	$-0.571712\pi$
$211$	−6.48981	−0.446777	−0.223388	+	0.974730i	$0.571712\pi$
$212$	0	0
$213$	7.96693	0.545885
$214$	0	0
$215$	15.6741	1.06897
$216$	0	0
$217$	−23.1863	−1.57399
$218$	0	0
$219$	−3.24234	−0.219097
$220$	0	0
$221$	−4.01308	−0.269949
$222$	0	0
$223$	1.44396	0.0966946	0.0483473	−	0.998831i	$-0.484605\pi$
$223$	1.44396	0.0966946	0.0483473	+	0.998831i	$0.484605\pi$
$224$	0	0
$225$	6.88329	0.458886
$226$	0	0
$227$	−6.71093	−0.445420	−0.222710	−	0.974885i	$-0.571490\pi$
$227$	−6.71093	−0.445420	−0.222710	+	0.974885i	$0.571490\pi$
$228$	0	0
$229$	5.82596	0.384990	0.192495	−	0.981298i	$-0.438342\pi$
$229$	5.82596	0.384990	0.192495	+	0.981298i	$0.438342\pi$
$230$	0	0
$231$	−60.0803	−3.95300
$232$	0	0
$233$	−3.44646	−0.225785	−0.112892	−	0.993607i	$-0.536012\pi$
$233$	−3.44646	−0.225785	−0.112892	+	0.993607i	$0.536012\pi$
$234$	0	0
$235$	2.81471	0.183612
$236$	0	0
$237$	−14.2396	−0.924960
$238$	0	0
$239$	−1.24263	−0.0803790	−0.0401895	−	0.999192i	$-0.512796\pi$
$239$	−1.24263	−0.0803790	−0.0401895	+	0.999192i	$0.512796\pi$
$240$	0	0
$241$	14.0302	0.903765	0.451883	−	0.892077i	$-0.350753\pi$
$241$	14.0302	0.903765	0.451883	+	0.892077i	$0.350753\pi$
$242$	0	0
$243$	19.5024	1.25108
$244$	0	0
$245$	37.8074	2.41542
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.04226	0.382913
$250$	0	0
$251$	−25.6175	−1.61696	−0.808481	−	0.588522i	$-0.799710\pi$
$251$	−25.6175	−1.61696	−0.808481	+	0.588522i	$0.799710\pi$
$252$	0	0
$253$	22.8712	1.43790
$254$	0	0
$255$	35.2826	2.20948
$256$	0	0
$257$	3.31950	0.207065	0.103532	−	0.994626i	$-0.466985\pi$
$257$	3.31950	0.207065	0.103532	+	0.994626i	$0.466985\pi$
$258$	0	0
$259$	−29.5878	−1.83849
$260$	0	0
$261$	−12.5724	−0.778214
$262$	0	0
$263$	6.57698	0.405554	0.202777	−	0.979225i	$-0.435003\pi$
$263$	6.57698	0.405554	0.202777	+	0.979225i	$0.435003\pi$
$264$	0	0
$265$	2.83773	0.174320
$266$	0	0
$267$	4.82768	0.295449
$268$	0	0
$269$	−13.0093	−0.793194	−0.396597	−	0.917993i	$-0.629809\pi$
$269$	−13.0093	−0.793194	−0.396597	+	0.917993i	$0.629809\pi$
$270$	0	0
$271$	−22.0499	−1.33944	−0.669718	−	0.742615i	$-0.733585\pi$
$271$	−22.0499	−1.33944	−0.669718	+	0.742615i	$0.733585\pi$
$272$	0	0
$273$	7.67457	0.464486
$274$	0	0
$275$	−17.4726	−1.05364
$276$	0	0
$277$	9.59769	0.576670	0.288335	−	0.957530i	$-0.406898\pi$
$277$	9.59769	0.576670	0.288335	+	0.957530i	$0.406898\pi$
$278$	0	0
$279$	11.7309	0.702310
$280$	0	0
$281$	6.17182	0.368180	0.184090	−	0.982909i	$-0.441066\pi$
$281$	6.17182	0.368180	0.184090	+	0.982909i	$0.441066\pi$
$282$	0	0
$283$	−21.4101	−1.27270	−0.636348	−	0.771402i	$-0.719556\pi$
$283$	−21.4101	−1.27270	−0.636348	+	0.771402i	$0.719556\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.5233	1.86076
$288$	0	0
$289$	12.3995	0.729383
$290$	0	0
$291$	2.01044	0.117854
$292$	0	0
$293$	12.0192	0.702168	0.351084	−	0.936344i	$-0.385813\pi$
$293$	12.0192	0.702168	0.351084	+	0.936344i	$0.385813\pi$
$294$	0	0
$295$	1.03152	0.0600576
$296$	0	0
$297$	−9.55411	−0.554386
$298$	0	0
$299$	−2.92153	−0.168956
$300$	0	0
$301$	−24.9769	−1.43965
$302$	0	0
$303$	26.3293	1.51258
$304$	0	0
$305$	5.16691	0.295857
$306$	0	0
$307$	19.8660	1.13381	0.566906	−	0.823783i	$-0.308140\pi$
$307$	19.8660	1.13381	0.566906	+	0.823783i	$0.308140\pi$
$308$	0	0
$309$	−19.6672	−1.11883
$310$	0	0
$311$	−17.4764	−0.990997	−0.495498	−	0.868609i	$-0.665015\pi$
$311$	−17.4764	−0.990997	−0.495498	+	0.868609i	$0.665015\pi$
$312$	0	0
$313$	10.4090	0.588353	0.294177	−	0.955751i	$-0.404955\pi$
$313$	10.4090	0.588353	0.294177	+	0.955751i	$0.404955\pi$
$314$	0	0
$315$	−29.1551	−1.64271
$316$	0	0
$317$	−29.1520	−1.63734	−0.818670	−	0.574265i	$-0.805288\pi$
$317$	−29.1520	−1.63734	−0.818670	+	0.574265i	$0.805288\pi$
$318$	0	0
$319$	31.9140	1.78684
$320$	0	0
$321$	13.8299	0.771912
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.23193	0.123805
$326$	0	0
$327$	45.6801	2.52611
$328$	0	0
$329$	−4.48529	−0.247282
$330$	0	0
$331$	17.7008	0.972923	0.486462	−	0.873702i	$-0.338287\pi$
$331$	17.7008	0.972923	0.486462	+	0.873702i	$0.338287\pi$
$332$	0	0
$333$	14.9696	0.820332
$334$	0	0
$335$	36.5054	1.99450
$336$	0	0
$337$	29.2129	1.59133	0.795665	−	0.605737i	$-0.207122\pi$
$337$	29.2129	1.59133	0.795665	+	0.605737i	$0.207122\pi$
$338$	0	0
$339$	−41.6281	−2.26093
$340$	0	0
$341$	−29.7778	−1.61256
$342$	0	0
$343$	−28.6659	−1.54781
$344$	0	0
$345$	25.6858	1.38288
$346$	0	0
$347$	−16.4479	−0.882971	−0.441486	−	0.897268i	$-0.645548\pi$
$347$	−16.4479	−0.882971	−0.441486	+	0.897268i	$0.645548\pi$
$348$	0	0
$349$	−20.1094	−1.07643	−0.538217	−	0.842806i	$-0.680902\pi$
$349$	−20.1094	−1.07643	−0.538217	+	0.842806i	$0.680902\pi$
$350$	0	0
$351$	1.22043	0.0651416
$352$	0	0
$353$	−4.68694	−0.249460	−0.124730	−	0.992191i	$-0.539807\pi$
$353$	−4.68694	−0.249460	−0.124730	+	0.992191i	$0.539807\pi$
$354$	0	0
$355$	−9.81378	−0.520861
$356$	0	0
$357$	−56.2233	−2.97565
$358$	0	0
$359$	−5.77416	−0.304749	−0.152374	−	0.988323i	$-0.548692\pi$
$359$	−5.77416	−0.304749	−0.152374	+	0.988323i	$0.548692\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−51.8781	−2.72289
$364$	0	0
$365$	3.99396	0.209053
$366$	0	0
$367$	−21.6204	−1.12858	−0.564289	−	0.825577i	$-0.690850\pi$
$367$	−21.6204	−1.12858	−0.564289	+	0.825577i	$0.690850\pi$
$368$	0	0
$369$	−15.9489	−0.830266
$370$	0	0
$371$	−4.52196	−0.234768
$372$	0	0
$373$	−19.3660	−1.00274	−0.501368	−	0.865234i	$-0.667170\pi$
$373$	−19.3660	−1.00274	−0.501368	+	0.865234i	$0.667170\pi$
$374$	0	0
$375$	12.9128	0.666815
$376$	0	0
$377$	−4.07665	−0.209958
$378$	0	0
$379$	−16.0287	−0.823339	−0.411669	−	0.911333i	$-0.635054\pi$
$379$	−16.0287	−0.823339	−0.411669	+	0.911333i	$0.635054\pi$
$380$	0	0
$381$	36.0355	1.84616
$382$	0	0
$383$	11.2081	0.572708	0.286354	−	0.958124i	$-0.407557\pi$
$383$	11.2081	0.572708	0.286354	+	0.958124i	$0.407557\pi$
$384$	0	0
$385$	74.0078	3.77179
$386$	0	0
$387$	12.6368	0.642367
$388$	0	0
$389$	22.0841	1.11971	0.559854	−	0.828591i	$-0.310857\pi$
$389$	22.0841	1.11971	0.559854	+	0.828591i	$0.310857\pi$
$390$	0	0
$391$	21.4029	1.08239
$392$	0	0
$393$	−23.8988	−1.20554
$394$	0	0
$395$	17.5405	0.882560
$396$	0	0
$397$	14.3925	0.722337	0.361169	−	0.932501i	$-0.382378\pi$
$397$	14.3925	0.722337	0.361169	+	0.932501i	$0.382378\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.5364	0.875728	0.437864	−	0.899041i	$-0.355735\pi$
$401$	17.5364	0.875728	0.437864	+	0.899041i	$0.355735\pi$
$402$	0	0
$403$	3.80378	0.189480
$404$	0	0
$405$	−30.1170	−1.49652
$406$	0	0
$407$	−37.9992	−1.88355
$408$	0	0
$409$	10.7768	0.532879	0.266439	−	0.963852i	$-0.414153\pi$
$409$	10.7768	0.532879	0.266439	+	0.963852i	$0.414153\pi$
$410$	0	0
$411$	37.8754	1.86826
$412$	0	0
$413$	−1.64375	−0.0808836
$414$	0	0
$415$	−7.44294	−0.365360
$416$	0	0
$417$	−4.87292	−0.238628
$418$	0	0
$419$	−9.77713	−0.477644	−0.238822	−	0.971063i	$-0.576761\pi$
$419$	−9.77713	−0.477644	−0.238822	+	0.971063i	$0.576761\pi$
$420$	0	0
$421$	−10.6228	−0.517725	−0.258862	−	0.965914i	$-0.583348\pi$
$421$	−10.6228	−0.517725	−0.258862	+	0.965914i	$0.583348\pi$
$422$	0	0
$423$	2.26929	0.110336
$424$	0	0
$425$	−16.3509	−0.793137
$426$	0	0
$427$	−8.23355	−0.398450
$428$	0	0
$429$	9.85635	0.475869
$430$	0	0
$431$	−12.7250	−0.612941	−0.306470	−	0.951880i	$-0.599148\pi$
$431$	−12.7250	−0.612941	−0.306470	+	0.951880i	$0.599148\pi$
$432$	0	0
$433$	34.8499	1.67478	0.837390	−	0.546606i	$-0.184081\pi$
$433$	34.8499	1.67478	0.837390	+	0.546606i	$0.184081\pi$
$434$	0	0
$435$	35.8415	1.71847
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−27.4710	−1.31112	−0.655559	−	0.755144i	$-0.727567\pi$
$439$	−27.4710	−1.31112	−0.655559	+	0.755144i	$0.727567\pi$
$440$	0	0
$441$	30.4812	1.45149
$442$	0	0
$443$	4.02503	0.191235	0.0956175	−	0.995418i	$-0.469517\pi$
$443$	4.02503	0.191235	0.0956175	+	0.995418i	$0.469517\pi$
$444$	0	0
$445$	−5.94680	−0.281905
$446$	0	0
$447$	−35.7110	−1.68907
$448$	0	0
$449$	25.4138	1.19935	0.599676	−	0.800243i	$-0.295296\pi$
$449$	25.4138	1.19935	0.599676	+	0.800243i	$0.295296\pi$
$450$	0	0
$451$	40.4849	1.90636
$452$	0	0
$453$	5.55893	0.261181
$454$	0	0
$455$	−9.45365	−0.443194
$456$	0	0
$457$	−9.86656	−0.461538	−0.230769	−	0.973009i	$-0.574124\pi$
$457$	−9.86656	−0.461538	−0.230769	+	0.973009i	$0.574124\pi$
$458$	0	0
$459$	−8.94076	−0.417319
$460$	0	0
$461$	0.237626	0.0110673	0.00553367	−	0.999985i	$-0.498239\pi$
$461$	0.237626	0.0110673	0.00553367	+	0.999985i	$0.498239\pi$
$462$	0	0
$463$	9.32971	0.433588	0.216794	−	0.976217i	$-0.430440\pi$
$463$	9.32971	0.433588	0.216794	+	0.976217i	$0.430440\pi$
$464$	0	0
$465$	−33.4424	−1.55086
$466$	0	0
$467$	24.7580	1.14567	0.572833	−	0.819672i	$-0.305844\pi$
$467$	24.7580	1.14567	0.572833	+	0.819672i	$0.305844\pi$
$468$	0	0
$469$	−58.1719	−2.68613
$470$	0	0
$471$	−10.7185	−0.493884
$472$	0	0
$473$	−32.0775	−1.47493
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.28784	0.104753
$478$	0	0
$479$	−16.0531	−0.733487	−0.366743	−	0.930322i	$-0.619527\pi$
$479$	−16.0531	−0.733487	−0.366743	+	0.930322i	$0.619527\pi$
$480$	0	0
$481$	4.85396	0.221321
$482$	0	0
$483$	−40.9307	−1.86241
$484$	0	0
$485$	−2.47649	−0.112452
$486$	0	0
$487$	35.2897	1.59913	0.799565	−	0.600580i	$-0.205064\pi$
$487$	35.2897	1.59913	0.799565	+	0.600580i	$0.205064\pi$
$488$	0	0
$489$	−41.0797	−1.85769
$490$	0	0
$491$	4.55133	0.205399	0.102699	−	0.994712i	$-0.467252\pi$
$491$	4.55133	0.205399	0.102699	+	0.994712i	$0.467252\pi$
$492$	0	0
$493$	29.8652	1.34506
$494$	0	0
$495$	−37.4435	−1.68296
$496$	0	0
$497$	15.6384	0.701478
$498$	0	0
$499$	−22.7080	−1.01655	−0.508276	−	0.861194i	$-0.669717\pi$
$499$	−22.7080	−1.01655	−0.508276	+	0.861194i	$0.669717\pi$
$500$	0	0
$501$	−26.9420	−1.20368
$502$	0	0
$503$	−30.7686	−1.37190	−0.685951	−	0.727648i	$-0.740614\pi$
$503$	−30.7686	−1.37190	−0.685951	+	0.727648i	$0.740614\pi$
$504$	0	0
$505$	−32.4328	−1.44324
$506$	0	0
$507$	28.6199	1.27106
$508$	0	0
$509$	−13.5449	−0.600366	−0.300183	−	0.953882i	$-0.597048\pi$
$509$	−13.5449	−0.600366	−0.300183	+	0.953882i	$0.597048\pi$
$510$	0	0
$511$	−6.36443	−0.281546
$512$	0	0
$513$	0	0
$514$	0	0
$515$	24.2263	1.06754
$516$	0	0
$517$	−5.76039	−0.253342
$518$	0	0
$519$	−17.8040	−0.781508
$520$	0	0
$521$	9.72420	0.426025	0.213013	−	0.977049i	$-0.431672\pi$
$521$	9.72420	0.426025	0.213013	+	0.977049i	$0.431672\pi$
$522$	0	0
$523$	−27.0164	−1.18135	−0.590673	−	0.806911i	$-0.701138\pi$
$523$	−27.0164	−1.18135	−0.590673	+	0.806911i	$0.701138\pi$
$524$	0	0
$525$	31.2694	1.36471
$526$	0	0
$527$	−27.8662	−1.21387
$528$	0	0
$529$	−7.41865	−0.322550
$530$	0	0
$531$	0.831639	0.0360901
$532$	0	0
$533$	−5.17148	−0.224002
$534$	0	0
$535$	−17.0359	−0.736527
$536$	0	0
$537$	−33.7301	−1.45556
$538$	0	0
$539$	−77.3739	−3.33273
$540$	0	0
$541$	−28.9405	−1.24425	−0.622124	−	0.782919i	$-0.713730\pi$
$541$	−28.9405	−1.24425	−0.622124	+	0.782919i	$0.713730\pi$
$542$	0	0
$543$	9.79317	0.420265
$544$	0	0
$545$	−56.2694	−2.41031
$546$	0	0
$547$	−27.9270	−1.19407	−0.597036	−	0.802214i	$-0.703655\pi$
$547$	−27.9270	−1.19407	−0.597036	+	0.802214i	$0.703655\pi$
$548$	0	0
$549$	4.16569	0.177787
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−27.9511	−1.18860
$554$	0	0
$555$	−42.6755	−1.81147
$556$	0	0
$557$	4.24684	0.179945	0.0899723	−	0.995944i	$-0.471322\pi$
$557$	4.24684	0.179945	0.0899723	+	0.995944i	$0.471322\pi$
$558$	0	0
$559$	4.09754	0.173307
$560$	0	0
$561$	−72.2068	−3.04857
$562$	0	0
$563$	−25.7131	−1.08368	−0.541840	−	0.840482i	$-0.682272\pi$
$563$	−25.7131	−1.08368	−0.541840	+	0.840482i	$0.682272\pi$
$564$	0	0
$565$	51.2781	2.15728
$566$	0	0
$567$	47.9918	2.01547
$568$	0	0
$569$	42.0218	1.76165	0.880823	−	0.473446i	$-0.156990\pi$
$569$	42.0218	1.76165	0.880823	+	0.473446i	$0.156990\pi$
$570$	0	0
$571$	29.0050	1.21382	0.606910	−	0.794770i	$-0.292409\pi$
$571$	29.0050	1.21382	0.606910	+	0.794770i	$0.292409\pi$
$572$	0	0
$573$	−31.7641	−1.32697
$574$	0	0
$575$	−11.9035	−0.496411
$576$	0	0
$577$	12.9779	0.540278	0.270139	−	0.962821i	$-0.412930\pi$
$577$	12.9779	0.540278	0.270139	+	0.962821i	$0.412930\pi$
$578$	0	0
$579$	−60.9667	−2.53369
$580$	0	0
$581$	11.8604	0.492054
$582$	0	0
$583$	−5.80749	−0.240522
$584$	0	0
$585$	4.78298	0.197752
$586$	0	0
$587$	2.06328	0.0851605	0.0425802	−	0.999093i	$-0.486442\pi$
$587$	2.06328	0.0851605	0.0425802	+	0.999093i	$0.486442\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−43.4981	−1.78927
$592$	0	0
$593$	35.9271	1.47535	0.737675	−	0.675156i	$-0.235924\pi$
$593$	35.9271	1.47535	0.737675	+	0.675156i	$0.235924\pi$
$594$	0	0
$595$	69.2567	2.83925
$596$	0	0
$597$	−57.0567	−2.33518
$598$	0	0
$599$	−5.04596	−0.206172	−0.103086	−	0.994672i	$-0.532872\pi$
$599$	−5.04596	−0.206172	−0.103086	+	0.994672i	$0.532872\pi$
$600$	0	0
$601$	−46.0189	−1.87715	−0.938575	−	0.345076i	$-0.887853\pi$
$601$	−46.0189	−1.87715	−0.938575	+	0.345076i	$0.887853\pi$
$602$	0	0
$603$	29.4315	1.19854
$604$	0	0
$605$	63.9042	2.59808
$606$	0	0
$607$	40.0288	1.62472	0.812359	−	0.583157i	$-0.198183\pi$
$607$	40.0288	1.62472	0.812359	+	0.583157i	$0.198183\pi$
$608$	0	0
$609$	−57.1140	−2.31437
$610$	0	0
$611$	0.735824	0.0297682
$612$	0	0
$613$	−4.61647	−0.186457	−0.0932286	−	0.995645i	$-0.529719\pi$
$613$	−4.61647	−0.186457	−0.0932286	+	0.995645i	$0.529719\pi$
$614$	0	0
$615$	45.4671	1.83341
$616$	0	0
$617$	−3.01385	−0.121333	−0.0606666	−	0.998158i	$-0.519323\pi$
$617$	−3.01385	−0.121333	−0.0606666	+	0.998158i	$0.519323\pi$
$618$	0	0
$619$	−21.0232	−0.844996	−0.422498	−	0.906364i	$-0.638847\pi$
$619$	−21.0232	−0.844996	−0.422498	+	0.906364i	$0.638847\pi$
$620$	0	0
$621$	−6.50889	−0.261193
$622$	0	0
$623$	9.47632	0.379661
$624$	0	0
$625$	−30.9842	−1.23937
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−35.5597	−1.41786
$630$	0	0
$631$	27.2081	1.08314	0.541568	−	0.840657i	$-0.317831\pi$
$631$	27.2081	1.08314	0.541568	+	0.840657i	$0.317831\pi$
$632$	0	0
$633$	14.9161	0.592860
$634$	0	0
$635$	−44.3891	−1.76153
$636$	0	0
$637$	9.88363	0.391604
$638$	0	0
$639$	−7.91210	−0.312998
$640$	0	0
$641$	−10.5546	−0.416880	−0.208440	−	0.978035i	$-0.566839\pi$
$641$	−10.5546	−0.416880	−0.208440	+	0.978035i	$0.566839\pi$
$642$	0	0
$643$	−42.1222	−1.66114	−0.830569	−	0.556916i	$-0.811984\pi$
$643$	−42.1222	−1.66114	−0.830569	+	0.556916i	$0.811984\pi$
$644$	0	0
$645$	−36.0251	−1.41849
$646$	0	0
$647$	20.1644	0.792743	0.396371	−	0.918090i	$-0.370269\pi$
$647$	20.1644	0.792743	0.396371	+	0.918090i	$0.370269\pi$
$648$	0	0
$649$	−2.11104	−0.0828657
$650$	0	0
$651$	53.2910	2.08864
$652$	0	0
$653$	−18.3945	−0.719834	−0.359917	−	0.932984i	$-0.617195\pi$
$653$	−18.3945	−0.719834	−0.359917	+	0.932984i	$0.617195\pi$
$654$	0	0
$655$	29.4389	1.15027
$656$	0	0
$657$	3.22002	0.125625
$658$	0	0
$659$	10.7294	0.417956	0.208978	−	0.977920i	$-0.432986\pi$
$659$	10.7294	0.417956	0.208978	+	0.977920i	$0.432986\pi$
$660$	0	0
$661$	41.0944	1.59839	0.799194	−	0.601073i	$-0.205260\pi$
$661$	41.0944	1.59839	0.799194	+	0.601073i	$0.205260\pi$
$662$	0	0
$663$	9.22360	0.358215
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.7419	0.841852
$668$	0	0
$669$	−3.31877	−0.128311
$670$	0	0
$671$	−10.5742	−0.408214
$672$	0	0
$673$	11.4764	0.442382	0.221191	−	0.975231i	$-0.429006\pi$
$673$	11.4764	0.442382	0.221191	+	0.975231i	$0.429006\pi$
$674$	0	0
$675$	4.97253	0.191393
$676$	0	0
$677$	9.66705	0.371535	0.185767	−	0.982594i	$-0.440523\pi$
$677$	9.66705	0.371535	0.185767	+	0.982594i	$0.440523\pi$
$678$	0	0
$679$	3.94632	0.151446
$680$	0	0
$681$	15.4243	0.591060
$682$	0	0
$683$	−47.4759	−1.81661	−0.908307	−	0.418304i	$-0.862625\pi$
$683$	−47.4759	−1.81661	−0.908307	+	0.418304i	$0.862625\pi$
$684$	0	0
$685$	−46.6555	−1.78262
$686$	0	0
$687$	−13.3903	−0.510871
$688$	0	0
$689$	0.741840	0.0282619
$690$	0	0
$691$	−36.0080	−1.36981	−0.684905	−	0.728632i	$-0.740156\pi$
$691$	−36.0080	−1.36981	−0.684905	+	0.728632i	$0.740156\pi$
$692$	0	0
$693$	59.6669	2.26656
$694$	0	0
$695$	6.00254	0.227689
$696$	0	0
$697$	37.8859	1.43503
$698$	0	0
$699$	7.92128	0.299610
$700$	0	0
$701$	30.1889	1.14022	0.570109	−	0.821569i	$-0.306901\pi$
$701$	30.1889	1.14022	0.570109	+	0.821569i	$0.306901\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−6.46929	−0.243647
$706$	0	0
$707$	51.6822	1.94371
$708$	0	0
$709$	−4.00549	−0.150429	−0.0752147	−	0.997167i	$-0.523964\pi$
$709$	−4.00549	−0.150429	−0.0752147	+	0.997167i	$0.523964\pi$
$710$	0	0
$711$	14.1416	0.530351
$712$	0	0
$713$	−20.2866	−0.759740
$714$	0	0
$715$	−12.1412	−0.454055
$716$	0	0
$717$	2.85604	0.106661
$718$	0	0
$719$	23.8085	0.887908	0.443954	−	0.896050i	$-0.353575\pi$
$719$	23.8085	0.887908	0.443954	+	0.896050i	$0.353575\pi$
$720$	0	0
$721$	−38.6050	−1.43773
$722$	0	0
$723$	−32.2468	−1.19927
$724$	0	0
$725$	−16.6100	−0.616879
$726$	0	0
$727$	−12.5631	−0.465941	−0.232970	−	0.972484i	$-0.574845\pi$
$727$	−12.5631	−0.465941	−0.232970	+	0.972484i	$0.574845\pi$
$728$	0	0
$729$	−12.9113	−0.478197
$730$	0	0
$731$	−30.0182	−1.11026
$732$	0	0
$733$	−40.4416	−1.49374	−0.746872	−	0.664967i	$-0.768445\pi$
$733$	−40.4416	−1.49374	−0.746872	+	0.664967i	$0.768445\pi$
$734$	0	0
$735$	−86.8958	−3.20520
$736$	0	0
$737$	−74.7093	−2.75195
$738$	0	0
$739$	5.51605	0.202911	0.101456	−	0.994840i	$-0.467650\pi$
$739$	5.51605	0.202911	0.101456	+	0.994840i	$0.467650\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.75575	0.101099	0.0505494	−	0.998722i	$-0.483903\pi$
$743$	2.75575	0.101099	0.0505494	+	0.998722i	$0.483903\pi$
$744$	0	0
$745$	43.9893	1.61164
$746$	0	0
$747$	−6.00068	−0.219553
$748$	0	0
$749$	27.1470	0.991930
$750$	0	0
$751$	−15.9169	−0.580816	−0.290408	−	0.956903i	$-0.593791\pi$
$751$	−15.9169	−0.580816	−0.290408	+	0.956903i	$0.593791\pi$
$752$	0	0
$753$	58.8788	2.14566
$754$	0	0
$755$	−6.84757	−0.249208
$756$	0	0
$757$	−10.1917	−0.370424	−0.185212	−	0.982699i	$-0.559297\pi$
$757$	−10.1917	−0.370424	−0.185212	+	0.982699i	$0.559297\pi$
$758$	0	0
$759$	−52.5667	−1.90805
$760$	0	0
$761$	27.9403	1.01284	0.506418	−	0.862288i	$-0.330969\pi$
$761$	27.9403	1.01284	0.506418	+	0.862288i	$0.330969\pi$
$762$	0	0
$763$	89.6661	3.24613
$764$	0	0
$765$	−35.0398	−1.26687
$766$	0	0
$767$	0.269662	0.00973692
$768$	0	0
$769$	31.5912	1.13921	0.569603	−	0.821920i	$-0.307097\pi$
$769$	31.5912	1.13921	0.569603	+	0.821920i	$0.307097\pi$
$770$	0	0
$771$	−7.62948	−0.274769
$772$	0	0
$773$	28.6451	1.03029	0.515147	−	0.857102i	$-0.327737\pi$
$773$	28.6451	1.03029	0.515147	+	0.857102i	$0.327737\pi$
$774$	0	0
$775$	15.4982	0.556710
$776$	0	0
$777$	68.0040	2.43963
$778$	0	0
$779$	0	0
$780$	0	0
$781$	20.0842	0.718669
$782$	0	0
$783$	−9.08239	−0.324578
$784$	0	0
$785$	13.2032	0.471244
$786$	0	0
$787$	11.7496	0.418829	0.209414	−	0.977827i	$-0.432844\pi$
$787$	11.7496	0.418829	0.209414	+	0.977827i	$0.432844\pi$
$788$	0	0
$789$	−15.1164	−0.538159
$790$	0	0
$791$	−81.7124	−2.90536
$792$	0	0
$793$	1.35074	0.0479661
$794$	0	0
$795$	−6.52218	−0.231318
$796$	0	0
$797$	−2.38658	−0.0845369	−0.0422685	−	0.999106i	$-0.513458\pi$
$797$	−2.38658	−0.0845369	−0.0422685	+	0.999106i	$0.513458\pi$
$798$	0	0
$799$	−5.39059	−0.190705
$800$	0	0
$801$	−4.79445	−0.169404
$802$	0	0
$803$	−8.17375	−0.288445
$804$	0	0
$805$	50.4190	1.77704
$806$	0	0
$807$	29.9005	1.05255
$808$	0	0
$809$	48.9199	1.71993	0.859965	−	0.510352i	$-0.170485\pi$
$809$	48.9199	1.71993	0.859965	+	0.510352i	$0.170485\pi$
$810$	0	0
$811$	−36.9088	−1.29604	−0.648021	−	0.761622i	$-0.724403\pi$
$811$	−36.9088	−1.29604	−0.648021	+	0.761622i	$0.724403\pi$
$812$	0	0
$813$	50.6791	1.77739
$814$	0	0
$815$	50.6026	1.77253
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−7.62176	−0.266326
$820$	0	0
$821$	−52.1507	−1.82007	−0.910036	−	0.414530i	$-0.863946\pi$
$821$	−52.1507	−1.82007	−0.910036	+	0.414530i	$0.863946\pi$
$822$	0	0
$823$	30.2084	1.05300	0.526500	−	0.850175i	$-0.323504\pi$
$823$	30.2084	1.05300	0.526500	+	0.850175i	$0.323504\pi$
$824$	0	0
$825$	40.1588	1.39815
$826$	0	0
$827$	27.1908	0.945517	0.472758	−	0.881192i	$-0.343258\pi$
$827$	27.1908	0.945517	0.472758	+	0.881192i	$0.343258\pi$
$828$	0	0
$829$	9.79460	0.340180	0.170090	−	0.985428i	$-0.445594\pi$
$829$	9.79460	0.340180	0.170090	+	0.985428i	$0.445594\pi$
$830$	0	0
$831$	−22.0592	−0.765224
$832$	0	0
$833$	−72.4067	−2.50874
$834$	0	0
$835$	33.1875	1.14850
$836$	0	0
$837$	8.47446	0.292920
$838$	0	0
$839$	−3.74154	−0.129172	−0.0645861	−	0.997912i	$-0.520573\pi$
$839$	−3.74154	−0.129172	−0.0645861	+	0.997912i	$0.520573\pi$
$840$	0	0
$841$	1.33834	0.0461496
$842$	0	0
$843$	−14.1852	−0.488564
$844$	0	0
$845$	−35.2545	−1.21279
$846$	0	0
$847$	−101.832	−3.49900
$848$	0	0
$849$	49.2085	1.68883
$850$	0	0
$851$	−25.8875	−0.887413
$852$	0	0
$853$	43.6533	1.49466	0.747330	−	0.664453i	$-0.231335\pi$
$853$	43.6533	1.49466	0.747330	+	0.664453i	$0.231335\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37.3889	−1.27718	−0.638591	−	0.769547i	$-0.720482\pi$
$857$	−37.3889	−1.27718	−0.638591	+	0.769547i	$0.720482\pi$
$858$	0	0
$859$	18.2210	0.621691	0.310846	−	0.950460i	$-0.399388\pi$
$859$	18.2210	0.621691	0.310846	+	0.950460i	$0.399388\pi$
$860$	0	0
$861$	−72.4525	−2.46918
$862$	0	0
$863$	23.5925	0.803099	0.401550	−	0.915837i	$-0.368472\pi$
$863$	23.5925	0.803099	0.401550	+	0.915837i	$0.368472\pi$
$864$	0	0
$865$	21.9312	0.745684
$866$	0	0
$867$	−28.4988	−0.967871
$868$	0	0
$869$	−35.8972	−1.21773
$870$	0	0
$871$	9.54326	0.323361
$872$	0	0
$873$	−1.99660	−0.0675748
$874$	0	0
$875$	25.3468	0.856877
$876$	0	0
$877$	15.1131	0.510335	0.255167	−	0.966897i	$-0.417869\pi$
$877$	15.1131	0.510335	0.255167	+	0.966897i	$0.417869\pi$
$878$	0	0
$879$	−27.6247	−0.931757
$880$	0	0
$881$	45.7224	1.54043	0.770213	−	0.637786i	$-0.220150\pi$
$881$	45.7224	1.54043	0.770213	+	0.637786i	$0.220150\pi$
$882$	0	0
$883$	21.4988	0.723493	0.361747	−	0.932276i	$-0.382181\pi$
$883$	21.4988	0.723493	0.361747	+	0.932276i	$0.382181\pi$
$884$	0	0
$885$	−2.37084	−0.0796948
$886$	0	0
$887$	−6.60235	−0.221685	−0.110843	−	0.993838i	$-0.535355\pi$
$887$	−6.60235	−0.221685	−0.110843	+	0.993838i	$0.535355\pi$
$888$	0	0
$889$	70.7347	2.37237
$890$	0	0
$891$	61.6352	2.06486
$892$	0	0
$893$	0	0
$894$	0	0
$895$	41.5493	1.38884
$896$	0	0
$897$	6.71480	0.224201
$898$	0	0
$899$	−28.3076	−0.944112
$900$	0	0
$901$	−5.43466	−0.181055
$902$	0	0
$903$	57.4066	1.91037
$904$	0	0
$905$	−12.0634	−0.401000
$906$	0	0
$907$	28.0852	0.932552	0.466276	−	0.884639i	$-0.345595\pi$
$907$	28.0852	0.932552	0.466276	+	0.884639i	$0.345595\pi$
$908$	0	0
$909$	−26.1481	−0.867278
$910$	0	0
$911$	25.2342	0.836046	0.418023	−	0.908436i	$-0.362723\pi$
$911$	25.2342	0.836046	0.418023	+	0.908436i	$0.362723\pi$
$912$	0	0
$913$	15.2322	0.504112
$914$	0	0
$915$	−11.8755	−0.392594
$916$	0	0
$917$	−46.9114	−1.54915
$918$	0	0
$919$	−0.131215	−0.00432837	−0.00216418	−	0.999998i	$-0.500689\pi$
$919$	−0.131215	−0.00432837	−0.00216418	+	0.999998i	$0.500689\pi$
$920$	0	0
$921$	−45.6597	−1.50454
$922$	0	0
$923$	−2.56553	−0.0844453
$924$	0	0
$925$	19.7770	0.650265
$926$	0	0
$927$	19.5318	0.641510
$928$	0	0
$929$	17.3446	0.569058	0.284529	−	0.958667i	$-0.408163\pi$
$929$	17.3446	0.569058	0.284529	+	0.958667i	$0.408163\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	40.1675	1.31503
$934$	0	0
$935$	88.9454	2.90883
$936$	0	0
$937$	37.0190	1.20936	0.604680	−	0.796469i	$-0.293301\pi$
$937$	37.0190	1.20936	0.604680	+	0.796469i	$0.293301\pi$
$938$	0	0
$939$	−23.9239	−0.780729
$940$	0	0
$941$	−4.89144	−0.159456	−0.0797282	−	0.996817i	$-0.525405\pi$
$941$	−4.89144	−0.159456	−0.0797282	+	0.996817i	$0.525405\pi$
$942$	0	0
$943$	27.5810	0.898160
$944$	0	0
$945$	−21.0618	−0.685142
$946$	0	0
$947$	−4.12717	−0.134115	−0.0670575	−	0.997749i	$-0.521361\pi$
$947$	−4.12717	−0.134115	−0.0670575	+	0.997749i	$0.521361\pi$
$948$	0	0
$949$	1.04410	0.0338930
$950$	0	0
$951$	67.0025	2.17270
$952$	0	0
$953$	−40.6091	−1.31546	−0.657729	−	0.753255i	$-0.728483\pi$
$953$	−40.6091	−1.31546	−0.657729	+	0.753255i	$0.728483\pi$
$954$	0	0
$955$	39.1275	1.26614
$956$	0	0
$957$	−73.3507	−2.37109
$958$	0	0
$959$	74.3462	2.40077
$960$	0	0
$961$	−4.58717	−0.147973
$962$	0	0
$963$	−13.7348	−0.442596
$964$	0	0
$965$	75.0997	2.41754
$966$	0	0
$967$	−6.60049	−0.212257	−0.106129	−	0.994352i	$-0.533846\pi$
$967$	−6.60049	−0.212257	−0.106129	+	0.994352i	$0.533846\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−49.9885	−1.60421	−0.802103	−	0.597186i	$-0.796286\pi$
$971$	−49.9885	−1.60421	−0.802103	+	0.597186i	$0.796286\pi$
$972$	0	0
$973$	−9.56514	−0.306644
$974$	0	0
$975$	−5.12983	−0.164286
$976$	0	0
$977$	−18.0781	−0.578370	−0.289185	−	0.957273i	$-0.593384\pi$
$977$	−18.0781	−0.578370	−0.289185	+	0.957273i	$0.593384\pi$
$978$	0	0
$979$	12.1703	0.388965
$980$	0	0
$981$	−45.3657	−1.44841
$982$	0	0
$983$	−27.0749	−0.863556	−0.431778	−	0.901980i	$-0.642114\pi$
$983$	−27.0749	−0.863556	−0.431778	+	0.901980i	$0.642114\pi$
$984$	0	0
$985$	53.5817	1.70725
$986$	0	0
$987$	10.3089	0.328136
$988$	0	0
$989$	−21.8533	−0.694895
$990$	0	0
$991$	26.5909	0.844688	0.422344	−	0.906436i	$-0.361207\pi$
$991$	26.5909	0.844688	0.422344	+	0.906436i	$0.361207\pi$
$992$	0	0
$993$	−40.6832	−1.29104
$994$	0	0
$995$	70.2833	2.22813
$996$	0	0
$997$	8.26982	0.261908	0.130954	−	0.991388i	$-0.458196\pi$
$997$	8.26982	0.261908	0.130954	+	0.991388i	$0.458196\pi$
$998$	0	0
$999$	10.8142	0.342145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.u.1.1	yes	6
4.3	odd	2		5776.2.a.bx.1.6		6
19.18	odd	2		2888.2.a.t.1.6	✓	6
76.75	even	2		5776.2.a.bz.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.t.1.6	✓	6	19.18	odd	2
2888.2.a.u.1.1	yes	6	1.1	even	1	trivial
5776.2.a.bx.1.6		6	4.3	odd	2
5776.2.a.bz.1.1		6	76.75	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q-2.29838 q^{3} +2.83118 q^{5} -4.51153 q^{7} +2.28257 q^{9} -5.79410 q^{11} +0.740130 q^{13} -6.50714 q^{15} -5.42213 q^{17} +10.3692 q^{21} -3.94732 q^{23} +3.01559 q^{25} +1.64894 q^{27} -5.50802 q^{29} +5.13934 q^{31} +13.3171 q^{33} -12.7730 q^{35} +6.55825 q^{37} -1.70110 q^{39} -6.98726 q^{41} +5.53624 q^{43} +6.46236 q^{45} +0.994182 q^{47} +13.3539 q^{49} +12.4621 q^{51} +1.00231 q^{53} -16.4041 q^{55} +0.364344 q^{59} +1.82500 q^{61} -10.2979 q^{63} +2.09544 q^{65} +12.8940 q^{67} +9.07246 q^{69} -3.46632 q^{71} +1.41070 q^{73} -6.93099 q^{75} +26.1403 q^{77} +6.19548 q^{79} -10.6376 q^{81} -2.62892 q^{83} -15.3510 q^{85} +12.6596 q^{87} -2.10047 q^{89} -3.33912 q^{91} -11.8122 q^{93} -0.874719 q^{97} -13.2254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 - 2 * q^7 + 9 * q^9 - 5 * q^11 + 10 * q^13 + 4 * q^17 + 23 * q^21 - 15 * q^23 + 12 * q^25 + 18 * q^27 - 7 * q^29 + 5 * q^31 - q^33 - 38 * q^35 + 17 * q^37 + 37 * q^39 + 4 * q^41 - 11 * q^43 + 6 * q^45 + 18 * q^47 + 20 * q^49 + 48 * q^51 - 7 * q^53 - 42 * q^55 + q^59 - 4 * q^61 + 19 * q^63 + 6 * q^65 + 20 * q^67 + 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 - 22 * q^83 + 32 * q^85 + 29 * q^87 + 14 * q^89 + 26 * q^91 - 5 * q^93 + 41 * q^97 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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