LMFDB - Embedded newform 1984.2.a.z.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1984,2,Mod(1,1984)] 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("1984.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1984 = 2^{6} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1984.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,-2,0,-4,0,4,0,10,0,0,0,-2,0,2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.8423197610$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 992)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.27743$ of defining polynomial
Character	$\chi$	$=$	1984.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{3} +2.27743 q^{5} +0.610235 q^{7} +4.46410 q^{9} -0.399243 q^{11} -4.06644 q^{13} +6.22205 q^{15} +4.88766 q^{17} +5.94462 q^{19} +1.66719 q^{21} -1.42356 q^{23} +0.186674 q^{25} +4.00000 q^{27} -3.82280 q^{29} -1.00000 q^{31} -1.09075 q^{33} +1.38977 q^{35} +3.30849 q^{37} -11.1097 q^{39} +0.943042 q^{41} +9.82280 q^{43} +10.1667 q^{45} -3.70773 q^{47} -6.62761 q^{49} +13.3533 q^{51} +12.6618 q^{53} -0.909247 q^{55} +16.2410 q^{57} -7.96358 q^{59} -4.06644 q^{61} +2.72415 q^{63} -9.26101 q^{65} +9.22047 q^{67} -3.88924 q^{69} +11.1651 q^{71} -4.92820 q^{73} +0.510004 q^{75} -0.243632 q^{77} -10.7769 q^{79} -2.46410 q^{81} -11.8665 q^{83} +11.1313 q^{85} -10.4441 q^{87} -8.64403 q^{89} -2.48148 q^{91} -2.73205 q^{93} +13.5384 q^{95} -12.1667 q^{97} -1.78226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 10 q^{11} - 2 q^{15} + 2 q^{17} + 8 q^{19} + 2 q^{21} - 2 q^{23} - 2 q^{25} + 16 q^{27} - 4 q^{31} + 4 q^{33} + 12 q^{35} + 10 q^{37} + 2 q^{41} + 24 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 10 * q^11 - 2 * q^15 + 2 * q^17 + 8 * q^19 + 2 * q^21 - 2 * q^23 - 2 * q^25 + 16 * q^27 - 4 * q^31 + 4 * q^33 + 12 * q^35 + 10 * q^37 + 2 * q^41 + 24 * q^43 - 2 * q^45 + 2 * q^49 + 8 * q^51 + 2 * q^53 - 12 * q^55 + 2 * q^57 + 20 * q^59 + 8 * q^63 - 14 * q^65 + 24 * q^67 + 16 * q^69 + 16 * q^71 + 8 * q^73 + 22 * q^75 + 6 * q^79 + 4 * q^81 + 6 * q^83 + 26 * q^85 + 12 * q^87 - 18 * q^89 + 26 * q^91 - 4 * q^93 + 20 * q^95 - 6 * q^97 - 2 * 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
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	2.27743	1.01850	0.509248	−	0.860620i	$-0.329924\pi$
$5$	2.27743	1.01850	0.509248	+	0.860620i	$0.329924\pi$
$6$	0	0
$7$	0.610235	0.230647	0.115324	−	0.993328i	$-0.463210\pi$
$7$	0.610235	0.230647	0.115324	+	0.993328i	$0.463210\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−0.399243	−0.120376	−0.0601882	−	0.998187i	$-0.519170\pi$
$11$	−0.399243	−0.120376	−0.0601882	+	0.998187i	$0.519170\pi$
$12$	0	0
$13$	−4.06644	−1.12783	−0.563913	−	0.825834i	$-0.690705\pi$
$13$	−4.06644	−1.12783	−0.563913	+	0.825834i	$0.690705\pi$
$14$	0	0
$15$	6.22205	1.60653
$16$	0	0
$17$	4.88766	1.18543	0.592716	−	0.805411i	$-0.298056\pi$
$17$	4.88766	1.18543	0.592716	+	0.805411i	$0.298056\pi$
$18$	0	0
$19$	5.94462	1.36379	0.681895	−	0.731450i	$-0.261156\pi$
$19$	5.94462	1.36379	0.681895	+	0.731450i	$0.261156\pi$
$20$	0	0
$21$	1.66719	0.363811
$22$	0	0
$23$	−1.42356	−0.296833	−0.148416	−	0.988925i	$-0.547418\pi$
$23$	−1.42356	−0.296833	−0.148416	+	0.988925i	$0.547418\pi$
$24$	0	0
$25$	0.186674	0.0373349
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−3.82280	−0.709877	−0.354938	−	0.934890i	$-0.615498\pi$
$29$	−3.82280	−0.709877	−0.354938	+	0.934890i	$0.615498\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.09075	−0.189876
$34$	0	0
$35$	1.38977	0.234913
$36$	0	0
$37$	3.30849	0.543912	0.271956	−	0.962310i	$-0.412329\pi$
$37$	3.30849	0.543912	0.271956	+	0.962310i	$0.412329\pi$
$38$	0	0
$39$	−11.1097	−1.77898
$40$	0	0
$41$	0.943042	0.147278	0.0736392	−	0.997285i	$-0.476539\pi$
$41$	0.943042	0.147278	0.0736392	+	0.997285i	$0.476539\pi$
$42$	0	0
$43$	9.82280	1.49796	0.748982	−	0.662591i	$-0.230543\pi$
$43$	9.82280	1.49796	0.748982	+	0.662591i	$0.230543\pi$
$44$	0	0
$45$	10.1667	1.51556
$46$	0	0
$47$	−3.70773	−0.540829	−0.270414	−	0.962744i	$-0.587161\pi$
$47$	−3.70773	−0.540829	−0.270414	+	0.962744i	$0.587161\pi$
$48$	0	0
$49$	−6.62761	−0.946802
$50$	0	0
$51$	13.3533	1.86984
$52$	0	0
$53$	12.6618	1.73924	0.869618	−	0.493725i	$-0.164365\pi$
$53$	12.6618	1.73924	0.869618	+	0.493725i	$0.164365\pi$
$54$	0	0
$55$	−0.909247	−0.122603
$56$	0	0
$57$	16.2410	2.15117
$58$	0	0
$59$	−7.96358	−1.03677	−0.518385	−	0.855148i	$-0.673466\pi$
$59$	−7.96358	−1.03677	−0.518385	+	0.855148i	$0.673466\pi$
$60$	0	0
$61$	−4.06644	−0.520654	−0.260327	−	0.965521i	$-0.583830\pi$
$61$	−4.06644	−0.520654	−0.260327	+	0.965521i	$0.583830\pi$
$62$	0	0
$63$	2.72415	0.343211
$64$	0	0
$65$	−9.26101	−1.14869
$66$	0	0
$67$	9.22047	1.12646	0.563230	−	0.826300i	$-0.309559\pi$
$67$	9.22047	1.12646	0.563230	+	0.826300i	$0.309559\pi$
$68$	0	0
$69$	−3.88924	−0.468209
$70$	0	0
$71$	11.1651	1.32505	0.662526	−	0.749039i	$-0.269484\pi$
$71$	11.1651	1.32505	0.662526	+	0.749039i	$0.269484\pi$
$72$	0	0
$73$	−4.92820	−0.576803	−0.288401	−	0.957510i	$-0.593124\pi$
$73$	−4.92820	−0.576803	−0.288401	+	0.957510i	$0.593124\pi$
$74$	0	0
$75$	0.510004	0.0588902
$76$	0	0
$77$	−0.243632	−0.0277645
$78$	0	0
$79$	−10.7769	−1.21250	−0.606248	−	0.795276i	$-0.707326\pi$
$79$	−10.7769	−1.21250	−0.606248	+	0.795276i	$0.707326\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−11.8665	−1.30252	−0.651259	−	0.758856i	$-0.725759\pi$
$83$	−11.8665	−1.30252	−0.651259	+	0.758856i	$0.725759\pi$
$84$	0	0
$85$	11.1313	1.20736
$86$	0	0
$87$	−10.4441	−1.11972
$88$	0	0
$89$	−8.64403	−0.916265	−0.458133	−	0.888884i	$-0.651482\pi$
$89$	−8.64403	−0.916265	−0.458133	+	0.888884i	$0.651482\pi$
$90$	0	0
$91$	−2.48148	−0.260130
$92$	0	0
$93$	−2.73205	−0.283300
$94$	0	0
$95$	13.5384	1.38901
$96$	0	0
$97$	−12.1667	−1.23534	−0.617669	−	0.786438i	$-0.711923\pi$
$97$	−12.1667	−1.23534	−0.617669	+	0.786438i	$0.711923\pi$
$98$	0	0
$99$	−1.78226	−0.179124
$100$	0	0
$101$	−6.76469	−0.673112	−0.336556	−	0.941663i	$-0.609262\pi$
$101$	−6.76469	−0.673112	−0.336556	+	0.941663i	$0.609262\pi$
$102$	0	0
$103$	9.42768	0.928937	0.464468	−	0.885590i	$-0.346245\pi$
$103$	9.42768	0.928937	0.464468	+	0.885590i	$0.346245\pi$
$104$	0	0
$105$	3.79691	0.370540
$106$	0	0
$107$	7.34113	0.709694	0.354847	−	0.934924i	$-0.384533\pi$
$107$	7.34113	0.709694	0.354847	+	0.934924i	$0.384533\pi$
$108$	0	0
$109$	4.70256	0.450424	0.225212	−	0.974310i	$-0.427693\pi$
$109$	4.70256	0.450424	0.225212	+	0.974310i	$0.427693\pi$
$110$	0	0
$111$	9.03896	0.857941
$112$	0	0
$113$	−15.6929	−1.47626	−0.738132	−	0.674657i	$-0.764292\pi$
$113$	−15.6929	−1.47626	−0.738132	+	0.674657i	$0.764292\pi$
$114$	0	0
$115$	−3.24205	−0.302323
$116$	0	0
$117$	−18.1530	−1.67824
$118$	0	0
$119$	2.98262	0.273416
$120$	0	0
$121$	−10.8406	−0.985510
$122$	0	0
$123$	2.57644	0.232310
$124$	0	0
$125$	−10.9620	−0.980471
$126$	0	0
$127$	9.75952	0.866018	0.433009	−	0.901390i	$-0.357452\pi$
$127$	9.75952	0.866018	0.433009	+	0.901390i	$0.357452\pi$
$128$	0	0
$129$	26.8364	2.36281
$130$	0	0
$131$	−0.259434	−0.0226668	−0.0113334	−	0.999936i	$-0.503608\pi$
$131$	−0.259434	−0.0226668	−0.0113334	+	0.999936i	$0.503608\pi$
$132$	0	0
$133$	3.62761	0.314554
$134$	0	0
$135$	9.10971	0.784039
$136$	0	0
$137$	−0.628228	−0.0536732	−0.0268366	−	0.999640i	$-0.508543\pi$
$137$	−0.628228	−0.0536732	−0.0268366	+	0.999640i	$0.508543\pi$
$138$	0	0
$139$	17.7684	1.50709	0.753547	−	0.657394i	$-0.228341\pi$
$139$	17.7684	1.50709	0.753547	+	0.657394i	$0.228341\pi$
$140$	0	0
$141$	−10.1297	−0.853076
$142$	0	0
$143$	1.62350	0.135764
$144$	0	0
$145$	−8.70616	−0.723007
$146$	0	0
$147$	−18.1070	−1.49344
$148$	0	0
$149$	−7.35334	−0.602409	−0.301205	−	0.953560i	$-0.597389\pi$
$149$	−7.35334	−0.602409	−0.301205	+	0.953560i	$0.597389\pi$
$150$	0	0
$151$	−1.84554	−0.150188	−0.0750941	−	0.997176i	$-0.523926\pi$
$151$	−1.84554	−0.150188	−0.0750941	+	0.997176i	$0.523926\pi$
$152$	0	0
$153$	21.8190	1.76396
$154$	0	0
$155$	−2.27743	−0.182927
$156$	0	0
$157$	−19.8902	−1.58741	−0.793705	−	0.608302i	$-0.791851\pi$
$157$	−19.8902	−1.58741	−0.793705	+	0.608302i	$0.791851\pi$
$158$	0	0
$159$	34.5928	2.74338
$160$	0	0
$161$	−0.868706	−0.0684636
$162$	0	0
$163$	15.0025	1.17509	0.587545	−	0.809192i	$-0.300095\pi$
$163$	15.0025	1.17509	0.587545	+	0.809192i	$0.300095\pi$
$164$	0	0
$165$	−2.48411	−0.193388
$166$	0	0
$167$	−22.9661	−1.77717	−0.888586	−	0.458711i	$-0.848311\pi$
$167$	−22.9661	−1.77717	−0.888586	+	0.458711i	$0.848311\pi$
$168$	0	0
$169$	3.53590	0.271992
$170$	0	0
$171$	26.5374	2.02936
$172$	0	0
$173$	5.48306	0.416869	0.208435	−	0.978036i	$-0.433163\pi$
$173$	5.48306	0.416869	0.208435	+	0.978036i	$0.433163\pi$
$174$	0	0
$175$	0.113915	0.00861118
$176$	0	0
$177$	−21.7569	−1.63535
$178$	0	0
$179$	3.84176	0.287147	0.143573	−	0.989640i	$-0.454141\pi$
$179$	3.84176	0.287147	0.143573	+	0.989640i	$0.454141\pi$
$180$	0	0
$181$	−20.0526	−1.49049	−0.745247	−	0.666788i	$-0.767669\pi$
$181$	−20.0526	−1.49049	−0.745247	+	0.666788i	$0.767669\pi$
$182$	0	0
$183$	−11.1097	−0.821253
$184$	0	0
$185$	7.53485	0.553973
$186$	0	0
$187$	−1.95137	−0.142698
$188$	0	0
$189$	2.44094	0.177552
$190$	0	0
$191$	10.7242	0.775972	0.387986	−	0.921665i	$-0.373171\pi$
$191$	10.7242	0.775972	0.387986	+	0.921665i	$0.373171\pi$
$192$	0	0
$193$	3.63077	0.261348	0.130674	−	0.991425i	$-0.458286\pi$
$193$	3.63077	0.261348	0.130674	+	0.991425i	$0.458286\pi$
$194$	0	0
$195$	−25.3016	−1.81188
$196$	0	0
$197$	23.4140	1.66818	0.834089	−	0.551630i	$-0.185994\pi$
$197$	23.4140	1.66818	0.834089	+	0.551630i	$0.185994\pi$
$198$	0	0
$199$	0.733205	0.0519756	0.0259878	−	0.999662i	$-0.491727\pi$
$199$	0.733205	0.0519756	0.0259878	+	0.999662i	$0.491727\pi$
$200$	0	0
$201$	25.1908	1.77682
$202$	0	0
$203$	−2.33281	−0.163731
$204$	0	0
$205$	2.14771	0.150003
$206$	0	0
$207$	−6.35492	−0.441697
$208$	0	0
$209$	−2.37335	−0.164168
$210$	0	0
$211$	−19.4763	−1.34080	−0.670402	−	0.741998i	$-0.733878\pi$
$211$	−19.4763	−1.34080	−0.670402	+	0.741998i	$0.733878\pi$
$212$	0	0
$213$	30.5036	2.09007
$214$	0	0
$215$	22.3707	1.52567
$216$	0	0
$217$	−0.610235	−0.0414254
$218$	0	0
$219$	−13.4641	−0.909820
$220$	0	0
$221$	−19.8754	−1.33696
$222$	0	0
$223$	−24.2140	−1.62149	−0.810743	−	0.585402i	$-0.800937\pi$
$223$	−24.2140	−1.62149	−0.810743	+	0.585402i	$0.800937\pi$
$224$	0	0
$225$	0.833334	0.0555556
$226$	0	0
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	0	0
$229$	20.4182	1.34927	0.674636	−	0.738150i	$-0.264300\pi$
$229$	20.4182	1.34927	0.674636	+	0.738150i	$0.264300\pi$
$230$	0	0
$231$	−0.665615	−0.0437943
$232$	0	0
$233$	−20.9810	−1.37451	−0.687254	−	0.726417i	$-0.741184\pi$
$233$	−20.9810	−1.37451	−0.687254	+	0.726417i	$0.741184\pi$
$234$	0	0
$235$	−8.44409	−0.550832
$236$	0	0
$237$	−29.4430	−1.91253
$238$	0	0
$239$	14.8201	0.958631	0.479315	−	0.877643i	$-0.340885\pi$
$239$	14.8201	0.958631	0.479315	+	0.877643i	$0.340885\pi$
$240$	0	0
$241$	−3.05792	−0.196978	−0.0984890	−	0.995138i	$-0.531401\pi$
$241$	−3.05792	−0.196978	−0.0984890	+	0.995138i	$0.531401\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	−15.0939	−0.964314
$246$	0	0
$247$	−24.1734	−1.53812
$248$	0	0
$249$	−32.4199	−2.05453
$250$	0	0
$251$	24.7078	1.55954	0.779772	−	0.626063i	$-0.215335\pi$
$251$	24.7078	1.55954	0.779772	+	0.626063i	$0.215335\pi$
$252$	0	0
$253$	0.568347	0.0357317
$254$	0	0
$255$	30.4113	1.90443
$256$	0	0
$257$	22.7373	1.41832	0.709158	−	0.705050i	$-0.249075\pi$
$257$	22.7373	1.41832	0.709158	+	0.705050i	$0.249075\pi$
$258$	0	0
$259$	2.01896	0.125452
$260$	0	0
$261$	−17.0654	−1.05632
$262$	0	0
$263$	10.3031	0.635318	0.317659	−	0.948205i	$-0.397103\pi$
$263$	10.3031	0.635318	0.317659	+	0.948205i	$0.397103\pi$
$264$	0	0
$265$	28.8364	1.77141
$266$	0	0
$267$	−23.6159	−1.44527
$268$	0	0
$269$	−1.50580	−0.0918101	−0.0459050	−	0.998946i	$-0.514617\pi$
$269$	−1.50580	−0.0918101	−0.0459050	+	0.998946i	$0.514617\pi$
$270$	0	0
$271$	25.7862	1.56640	0.783200	−	0.621770i	$-0.213586\pi$
$271$	25.7862	1.56640	0.783200	+	0.621770i	$0.213586\pi$
$272$	0	0
$273$	−6.77953	−0.410316
$274$	0	0
$275$	−0.0745285	−0.00449424
$276$	0	0
$277$	3.64445	0.218974	0.109487	−	0.993988i	$-0.465079\pi$
$277$	3.64445	0.218974	0.109487	+	0.993988i	$0.465079\pi$
$278$	0	0
$279$	−4.46410	−0.267259
$280$	0	0
$281$	−14.7373	−0.879155	−0.439577	−	0.898205i	$-0.644872\pi$
$281$	−14.7373	−0.879155	−0.439577	+	0.898205i	$0.644872\pi$
$282$	0	0
$283$	17.6488	1.04911	0.524555	−	0.851377i	$-0.324232\pi$
$283$	17.6488	1.04911	0.524555	+	0.851377i	$0.324232\pi$
$284$	0	0
$285$	36.9877	2.19096
$286$	0	0
$287$	0.575477	0.0339693
$288$	0	0
$289$	6.88924	0.405249
$290$	0	0
$291$	−33.2400	−1.94856
$292$	0	0
$293$	16.8174	0.982485	0.491243	−	0.871023i	$-0.336543\pi$
$293$	16.8174	0.982485	0.491243	+	0.871023i	$0.336543\pi$
$294$	0	0
$295$	−18.1365	−1.05595
$296$	0	0
$297$	−1.59697	−0.0926658
$298$	0	0
$299$	5.78882	0.334776
$300$	0	0
$301$	5.99422	0.345501
$302$	0	0
$303$	−18.4815	−1.06173
$304$	0	0
$305$	−9.26101	−0.530284
$306$	0	0
$307$	−30.9594	−1.76694	−0.883472	−	0.468483i	$-0.844801\pi$
$307$	−30.9594	−1.76694	−0.883472	+	0.468483i	$0.844801\pi$
$308$	0	0
$309$	25.7569	1.46526
$310$	0	0
$311$	−7.42452	−0.421006	−0.210503	−	0.977593i	$-0.567510\pi$
$311$	−7.42452	−0.421006	−0.210503	+	0.977593i	$0.567510\pi$
$312$	0	0
$313$	−24.0437	−1.35903	−0.679515	−	0.733662i	$-0.737810\pi$
$313$	−24.0437	−1.35903	−0.679515	+	0.733662i	$0.737810\pi$
$314$	0	0
$315$	6.20405	0.349559
$316$	0	0
$317$	29.5011	1.65694	0.828472	−	0.560030i	$-0.189211\pi$
$317$	29.5011	1.65694	0.828472	+	0.560030i	$0.189211\pi$
$318$	0	0
$319$	1.52623	0.0854524
$320$	0	0
$321$	20.0563	1.11944
$322$	0	0
$323$	29.0553	1.61668
$324$	0	0
$325$	−0.759100	−0.0421073
$326$	0	0
$327$	12.8476	0.710476
$328$	0	0
$329$	−2.26259	−0.124741
$330$	0	0
$331$	17.7684	0.976639	0.488319	−	0.872665i	$-0.337610\pi$
$331$	17.7684	0.976639	0.488319	+	0.872665i	$0.337610\pi$
$332$	0	0
$333$	14.7694	0.809360
$334$	0	0
$335$	20.9989	1.14730
$336$	0	0
$337$	8.01633	0.436677	0.218339	−	0.975873i	$-0.429936\pi$
$337$	8.01633	0.436677	0.218339	+	0.975873i	$0.429936\pi$
$338$	0	0
$339$	−42.8738	−2.32858
$340$	0	0
$341$	0.399243	0.0216202
$342$	0	0
$343$	−8.31604	−0.449024
$344$	0	0
$345$	−8.85746	−0.476870
$346$	0	0
$347$	33.6171	1.80466	0.902330	−	0.431046i	$-0.141855\pi$
$347$	33.6171	1.80466	0.902330	+	0.431046i	$0.141855\pi$
$348$	0	0
$349$	−8.68457	−0.464874	−0.232437	−	0.972611i	$-0.574670\pi$
$349$	−8.68457	−0.464874	−0.232437	+	0.972611i	$0.574670\pi$
$350$	0	0
$351$	−16.2657	−0.868201
$352$	0	0
$353$	0.860613	0.0458058	0.0229029	−	0.999738i	$-0.492709\pi$
$353$	0.860613	0.0458058	0.0229029	+	0.999738i	$0.492709\pi$
$354$	0	0
$355$	25.4277	1.34956
$356$	0	0
$357$	8.14867	0.431273
$358$	0	0
$359$	27.1323	1.43199	0.715993	−	0.698108i	$-0.245974\pi$
$359$	27.1323	1.43199	0.715993	+	0.698108i	$0.245974\pi$
$360$	0	0
$361$	16.3385	0.859921
$362$	0	0
$363$	−29.6171	−1.55449
$364$	0	0
$365$	−11.2236	−0.587471
$366$	0	0
$367$	−35.4399	−1.84995	−0.924974	−	0.380032i	$-0.875913\pi$
$367$	−35.4399	−1.84995	−0.924974	+	0.380032i	$0.875913\pi$
$368$	0	0
$369$	4.20984	0.219155
$370$	0	0
$371$	7.72669	0.401150
$372$	0	0
$373$	11.9199	0.617188	0.308594	−	0.951194i	$-0.400142\pi$
$373$	11.9199	0.617188	0.308594	+	0.951194i	$0.400142\pi$
$374$	0	0
$375$	−29.9487	−1.54655
$376$	0	0
$377$	15.5452	0.800618
$378$	0	0
$379$	−29.9461	−1.53823	−0.769114	−	0.639112i	$-0.779302\pi$
$379$	−29.9461	−1.53823	−0.769114	+	0.639112i	$0.779302\pi$
$380$	0	0
$381$	26.6635	1.36601
$382$	0	0
$383$	−20.4283	−1.04384	−0.521918	−	0.852995i	$-0.674783\pi$
$383$	−20.4283	−1.04384	−0.521918	+	0.852995i	$0.674783\pi$
$384$	0	0
$385$	−0.554854	−0.0282780
$386$	0	0
$387$	43.8500	2.22902
$388$	0	0
$389$	−19.1166	−0.969252	−0.484626	−	0.874721i	$-0.661044\pi$
$389$	−19.1166	−0.969252	−0.484626	+	0.874721i	$0.661044\pi$
$390$	0	0
$391$	−6.95788	−0.351875
$392$	0	0
$393$	−0.708786	−0.0357535
$394$	0	0
$395$	−24.5436	−1.23492
$396$	0	0
$397$	17.7605	0.891373	0.445687	−	0.895189i	$-0.352960\pi$
$397$	17.7605	0.891373	0.445687	+	0.895189i	$0.352960\pi$
$398$	0	0
$399$	9.91082	0.496162
$400$	0	0
$401$	10.5143	0.525060	0.262530	−	0.964924i	$-0.415443\pi$
$401$	10.5143	0.525060	0.262530	+	0.964924i	$0.415443\pi$
$402$	0	0
$403$	4.06644	0.202564
$404$	0	0
$405$	−5.61181	−0.278853
$406$	0	0
$407$	−1.32089	−0.0654742
$408$	0	0
$409$	21.4128	1.05880	0.529398	−	0.848373i	$-0.322418\pi$
$409$	21.4128	1.05880	0.529398	+	0.848373i	$0.322418\pi$
$410$	0	0
$411$	−1.71635	−0.0846614
$412$	0	0
$413$	−4.85965	−0.239128
$414$	0	0
$415$	−27.0251	−1.32661
$416$	0	0
$417$	48.5441	2.37722
$418$	0	0
$419$	−16.2590	−0.794304	−0.397152	−	0.917753i	$-0.630001\pi$
$419$	−16.2590	−0.794304	−0.397152	+	0.917753i	$0.630001\pi$
$420$	0	0
$421$	−29.3840	−1.43209	−0.716044	−	0.698055i	$-0.754049\pi$
$421$	−29.3840	−1.43209	−0.716044	+	0.698055i	$0.754049\pi$
$422$	0	0
$423$	−16.5517	−0.804771
$424$	0	0
$425$	0.912402	0.0442580
$426$	0	0
$427$	−2.48148	−0.120087
$428$	0	0
$429$	4.43548	0.214147
$430$	0	0
$431$	−19.1908	−0.924388	−0.462194	−	0.886779i	$-0.652938\pi$
$431$	−19.1908	−0.924388	−0.462194	+	0.886779i	$0.652938\pi$
$432$	0	0
$433$	−29.2210	−1.40427	−0.702136	−	0.712043i	$-0.747770\pi$
$433$	−29.2210	−1.40427	−0.702136	+	0.712043i	$0.747770\pi$
$434$	0	0
$435$	−23.7857	−1.14044
$436$	0	0
$437$	−8.46252	−0.404817
$438$	0	0
$439$	−27.1682	−1.29667	−0.648335	−	0.761355i	$-0.724534\pi$
$439$	−27.1682	−1.29667	−0.648335	+	0.761355i	$0.724534\pi$
$440$	0	0
$441$	−29.5863	−1.40887
$442$	0	0
$443$	−17.7682	−0.844192	−0.422096	−	0.906551i	$-0.638706\pi$
$443$	−17.7682	−0.844192	−0.422096	+	0.906551i	$0.638706\pi$
$444$	0	0
$445$	−19.6861	−0.933213
$446$	0	0
$447$	−20.0897	−0.950210
$448$	0	0
$449$	11.6425	0.549441	0.274721	−	0.961524i	$-0.411415\pi$
$449$	11.6425	0.549441	0.274721	+	0.961524i	$0.411415\pi$
$450$	0	0
$451$	−0.376503	−0.0177288
$452$	0	0
$453$	−5.04212	−0.236899
$454$	0	0
$455$	−5.65139	−0.264941
$456$	0	0
$457$	−7.93787	−0.371318	−0.185659	−	0.982614i	$-0.559442\pi$
$457$	−7.93787	−0.371318	−0.185659	+	0.982614i	$0.559442\pi$
$458$	0	0
$459$	19.5506	0.912546
$460$	0	0
$461$	4.74330	0.220917	0.110459	−	0.993881i	$-0.464768\pi$
$461$	4.74330	0.220917	0.110459	+	0.993881i	$0.464768\pi$
$462$	0	0
$463$	−6.55170	−0.304483	−0.152242	−	0.988343i	$-0.548649\pi$
$463$	−6.55170	−0.304483	−0.152242	+	0.988343i	$0.548649\pi$
$464$	0	0
$465$	−6.22205	−0.288541
$466$	0	0
$467$	−18.6755	−0.864200	−0.432100	−	0.901826i	$-0.642227\pi$
$467$	−18.6755	−0.864200	−0.432100	+	0.901826i	$0.642227\pi$
$468$	0	0
$469$	5.62665	0.259815
$470$	0	0
$471$	−54.3410	−2.50390
$472$	0	0
$473$	−3.92169	−0.180319
$474$	0	0
$475$	1.10971	0.0509169
$476$	0	0
$477$	56.5237	2.58804
$478$	0	0
$479$	3.50368	0.160087	0.0800436	−	0.996791i	$-0.474494\pi$
$479$	3.50368	0.160087	0.0800436	+	0.996791i	$0.474494\pi$
$480$	0	0
$481$	−13.4538	−0.613439
$482$	0	0
$483$	−2.37335	−0.107991
$484$	0	0
$485$	−27.7087	−1.25819
$486$	0	0
$487$	19.4507	0.881398	0.440699	−	0.897655i	$-0.354731\pi$
$487$	19.4507	0.881398	0.440699	+	0.897655i	$0.354731\pi$
$488$	0	0
$489$	40.9877	1.85353
$490$	0	0
$491$	−8.27461	−0.373428	−0.186714	−	0.982414i	$-0.559784\pi$
$491$	−8.27461	−0.373428	−0.186714	+	0.982414i	$0.559784\pi$
$492$	0	0
$493$	−18.6846	−0.841511
$494$	0	0
$495$	−4.05897	−0.182437
$496$	0	0
$497$	6.81333	0.305619
$498$	0	0
$499$	5.45716	0.244296	0.122148	−	0.992512i	$-0.461022\pi$
$499$	5.45716	0.244296	0.122148	+	0.992512i	$0.461022\pi$
$500$	0	0
$501$	−62.7446	−2.80322
$502$	0	0
$503$	13.9446	0.621760	0.310880	−	0.950449i	$-0.399376\pi$
$503$	13.9446	0.621760	0.310880	+	0.950449i	$0.399376\pi$
$504$	0	0
$505$	−15.4061	−0.685562
$506$	0	0
$507$	9.66025	0.429027
$508$	0	0
$509$	11.8205	0.523934	0.261967	−	0.965077i	$-0.415629\pi$
$509$	11.8205	0.523934	0.261967	+	0.965077i	$0.415629\pi$
$510$	0	0
$511$	−3.00736	−0.133038
$512$	0	0
$513$	23.7785	1.04985
$514$	0	0
$515$	21.4708	0.946119
$516$	0	0
$517$	1.48029	0.0651030
$518$	0	0
$519$	14.9800	0.657549
$520$	0	0
$521$	−14.5169	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$521$	−14.5169	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$522$	0	0
$523$	24.3290	1.06383	0.531917	−	0.846797i	$-0.321472\pi$
$523$	24.3290	1.06383	0.531917	+	0.846797i	$0.321472\pi$
$524$	0	0
$525$	0.311222	0.0135829
$526$	0	0
$527$	−4.88766	−0.212910
$528$	0	0
$529$	−20.9735	−0.911890
$530$	0	0
$531$	−35.5502	−1.54275
$532$	0	0
$533$	−3.83482	−0.166105
$534$	0	0
$535$	16.7189	0.722821
$536$	0	0
$537$	10.4959	0.452931
$538$	0	0
$539$	2.64603	0.113973
$540$	0	0
$541$	−14.8976	−0.640496	−0.320248	−	0.947334i	$-0.603766\pi$
$541$	−14.8976	−0.640496	−0.320248	+	0.947334i	$0.603766\pi$
$542$	0	0
$543$	−54.7846	−2.35103
$544$	0	0
$545$	10.7097	0.458755
$546$	0	0
$547$	−15.7421	−0.673082	−0.336541	−	0.941669i	$-0.609257\pi$
$547$	−15.7421	−0.673082	−0.336541	+	0.941669i	$0.609257\pi$
$548$	0	0
$549$	−18.1530	−0.774750
$550$	0	0
$551$	−22.7251	−0.968122
$552$	0	0
$553$	−6.57644	−0.279659
$554$	0	0
$555$	20.5856	0.873809
$556$	0	0
$557$	−27.7252	−1.17476	−0.587378	−	0.809313i	$-0.699840\pi$
$557$	−27.7252	−1.17476	−0.587378	+	0.809313i	$0.699840\pi$
$558$	0	0
$559$	−39.9438	−1.68944
$560$	0	0
$561$	−5.33123	−0.225085
$562$	0	0
$563$	26.6616	1.12365	0.561827	−	0.827255i	$-0.310099\pi$
$563$	26.6616	1.12365	0.561827	+	0.827255i	$0.310099\pi$
$564$	0	0
$565$	−35.7394	−1.50357
$566$	0	0
$567$	−1.50368	−0.0631486
$568$	0	0
$569$	11.3154	0.474367	0.237184	−	0.971465i	$-0.423776\pi$
$569$	11.3154	0.474367	0.237184	+	0.971465i	$0.423776\pi$
$570$	0	0
$571$	45.8742	1.91978	0.959888	−	0.280384i	$-0.0904618\pi$
$571$	45.8742	1.91978	0.959888	+	0.280384i	$0.0904618\pi$
$572$	0	0
$573$	29.2989	1.22398
$574$	0	0
$575$	−0.265742	−0.0110822
$576$	0	0
$577$	46.1739	1.92225	0.961123	−	0.276120i	$-0.0890488\pi$
$577$	46.1739	1.92225	0.961123	+	0.276120i	$0.0890488\pi$
$578$	0	0
$579$	9.91944	0.412238
$580$	0	0
$581$	−7.24135	−0.300422
$582$	0	0
$583$	−5.05515	−0.209363
$584$	0	0
$585$	−41.3421	−1.70929
$586$	0	0
$587$	−16.5479	−0.683006	−0.341503	−	0.939881i	$-0.610936\pi$
$587$	−16.5479	−0.683006	−0.341503	+	0.939881i	$0.610936\pi$
$588$	0	0
$589$	−5.94462	−0.244944
$590$	0	0
$591$	63.9682	2.63130
$592$	0	0
$593$	−29.6466	−1.21744	−0.608719	−	0.793386i	$-0.708316\pi$
$593$	−29.6466	−1.21744	−0.608719	+	0.793386i	$0.708316\pi$
$594$	0	0
$595$	6.79270	0.278474
$596$	0	0
$597$	2.00315	0.0819836
$598$	0	0
$599$	34.2451	1.39922	0.699609	−	0.714526i	$-0.253358\pi$
$599$	34.2451	1.39922	0.699609	+	0.714526i	$0.253358\pi$
$600$	0	0
$601$	−11.8811	−0.484642	−0.242321	−	0.970196i	$-0.577909\pi$
$601$	−11.8811	−0.484642	−0.242321	+	0.970196i	$0.577909\pi$
$602$	0	0
$603$	41.1611	1.67621
$604$	0	0
$605$	−24.6887	−1.00374
$606$	0	0
$607$	38.8257	1.57589	0.787943	−	0.615748i	$-0.211146\pi$
$607$	38.8257	1.57589	0.787943	+	0.615748i	$0.211146\pi$
$608$	0	0
$609$	−6.37335	−0.258261
$610$	0	0
$611$	15.0773	0.609961
$612$	0	0
$613$	−14.4961	−0.585493	−0.292747	−	0.956190i	$-0.594569\pi$
$613$	−14.4961	−0.585493	−0.292747	+	0.956190i	$0.594569\pi$
$614$	0	0
$615$	5.86765	0.236607
$616$	0	0
$617$	12.1950	0.490952	0.245476	−	0.969403i	$-0.421056\pi$
$617$	12.1950	0.490952	0.245476	+	0.969403i	$0.421056\pi$
$618$	0	0
$619$	−5.82049	−0.233945	−0.116973	−	0.993135i	$-0.537319\pi$
$619$	−5.82049	−0.233945	−0.116973	+	0.993135i	$0.537319\pi$
$620$	0	0
$621$	−5.69424	−0.228502
$622$	0	0
$623$	−5.27489	−0.211334
$624$	0	0
$625$	−25.8985	−1.03594
$626$	0	0
$627$	−6.48411	−0.258950
$628$	0	0
$629$	16.1708	0.644771
$630$	0	0
$631$	−12.4599	−0.496021	−0.248010	−	0.968757i	$-0.579777\pi$
$631$	−12.4599	−0.496021	−0.248010	+	0.968757i	$0.579777\pi$
$632$	0	0
$633$	−53.2103	−2.11492
$634$	0	0
$635$	22.2266	0.882036
$636$	0	0
$637$	26.9508	1.06783
$638$	0	0
$639$	49.8421	1.97172
$640$	0	0
$641$	12.6909	0.501260	0.250630	−	0.968083i	$-0.419362\pi$
$641$	12.6909	0.501260	0.250630	+	0.968083i	$0.419362\pi$
$642$	0	0
$643$	−28.5205	−1.12474	−0.562370	−	0.826886i	$-0.690110\pi$
$643$	−28.5205	−1.12474	−0.562370	+	0.826886i	$0.690110\pi$
$644$	0	0
$645$	61.1179	2.40652
$646$	0	0
$647$	21.6425	0.850853	0.425426	−	0.904993i	$-0.360124\pi$
$647$	21.6425	0.850853	0.425426	+	0.904993i	$0.360124\pi$
$648$	0	0
$649$	3.17940	0.124802
$650$	0	0
$651$	−1.66719	−0.0653424
$652$	0	0
$653$	10.6814	0.417996	0.208998	−	0.977916i	$-0.432980\pi$
$653$	10.6814	0.417996	0.208998	+	0.977916i	$0.432980\pi$
$654$	0	0
$655$	−0.590841	−0.0230861
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	36.1344	1.40759	0.703797	−	0.710401i	$-0.251486\pi$
$659$	36.1344	1.40759	0.703797	+	0.710401i	$0.251486\pi$
$660$	0	0
$661$	−13.8851	−0.540069	−0.270034	−	0.962851i	$-0.587035\pi$
$661$	−13.8851	−0.540069	−0.270034	+	0.962851i	$0.587035\pi$
$662$	0	0
$663$	−54.3005	−2.10886
$664$	0	0
$665$	8.26163	0.320372
$666$	0	0
$667$	5.44199	0.210715
$668$	0	0
$669$	−66.1538	−2.55765
$670$	0	0
$671$	1.62350	0.0626744
$672$	0	0
$673$	38.8990	1.49945	0.749724	−	0.661751i	$-0.230186\pi$
$673$	38.8990	1.49945	0.749724	+	0.661751i	$0.230186\pi$
$674$	0	0
$675$	0.746698	0.0287404
$676$	0	0
$677$	42.3643	1.62819	0.814096	−	0.580730i	$-0.197233\pi$
$677$	42.3643	1.62819	0.814096	+	0.580730i	$0.197233\pi$
$678$	0	0
$679$	−7.42452	−0.284927
$680$	0	0
$681$	−44.7846	−1.71615
$682$	0	0
$683$	−47.9583	−1.83507	−0.917537	−	0.397650i	$-0.869826\pi$
$683$	−47.9583	−1.83507	−0.917537	+	0.397650i	$0.869826\pi$
$684$	0	0
$685$	−1.43074	−0.0546659
$686$	0	0
$687$	55.7836	2.12828
$688$	0	0
$689$	−51.4885	−1.96156
$690$	0	0
$691$	−26.3211	−1.00130	−0.500651	−	0.865649i	$-0.666906\pi$
$691$	−26.3211	−1.00130	−0.500651	+	0.865649i	$0.666906\pi$
$692$	0	0
$693$	−1.08760	−0.0413144
$694$	0	0
$695$	40.4662	1.53497
$696$	0	0
$697$	4.60927	0.174589
$698$	0	0
$699$	−57.3210	−2.16808
$700$	0	0
$701$	−2.65624	−0.100325	−0.0501624	−	0.998741i	$-0.515974\pi$
$701$	−2.65624	−0.100325	−0.0501624	+	0.998741i	$0.515974\pi$
$702$	0	0
$703$	19.6677	0.741782
$704$	0	0
$705$	−23.0697	−0.868855
$706$	0	0
$707$	−4.12805	−0.155251
$708$	0	0
$709$	36.4858	1.37025	0.685126	−	0.728424i	$-0.259747\pi$
$709$	36.4858	1.37025	0.685126	+	0.728424i	$0.259747\pi$
$710$	0	0
$711$	−48.1092	−1.80424
$712$	0	0
$713$	1.42356	0.0533128
$714$	0	0
$715$	3.69740	0.138275
$716$	0	0
$717$	40.4892	1.51210
$718$	0	0
$719$	19.3160	0.720364	0.360182	−	0.932882i	$-0.382715\pi$
$719$	19.3160	0.720364	0.360182	+	0.932882i	$0.382715\pi$
$720$	0	0
$721$	5.75310	0.214257
$722$	0	0
$723$	−8.35439	−0.310703
$724$	0	0
$725$	−0.713620	−0.0265032
$726$	0	0
$727$	−8.38556	−0.311003	−0.155502	−	0.987836i	$-0.549699\pi$
$727$	−8.38556	−0.311003	−0.155502	+	0.987836i	$0.549699\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	48.0105	1.77573
$732$	0	0
$733$	7.43031	0.274445	0.137222	−	0.990540i	$-0.456183\pi$
$733$	7.43031	0.274445	0.137222	+	0.990540i	$0.456183\pi$
$734$	0	0
$735$	−41.2373	−1.52106
$736$	0	0
$737$	−3.68121	−0.135599
$738$	0	0
$739$	−2.69606	−0.0991763	−0.0495882	−	0.998770i	$-0.515791\pi$
$739$	−2.69606	−0.0991763	−0.0495882	+	0.998770i	$0.515791\pi$
$740$	0	0
$741$	−66.0430	−2.42615
$742$	0	0
$743$	−29.5647	−1.08462	−0.542311	−	0.840178i	$-0.682451\pi$
$743$	−29.5647	−1.08462	−0.542311	+	0.840178i	$0.682451\pi$
$744$	0	0
$745$	−16.7467	−0.613552
$746$	0	0
$747$	−52.9733	−1.93819
$748$	0	0
$749$	4.47981	0.163689
$750$	0	0
$751$	−27.3845	−0.999275	−0.499637	−	0.866235i	$-0.666533\pi$
$751$	−27.3845	−0.999275	−0.499637	+	0.866235i	$0.666533\pi$
$752$	0	0
$753$	67.5031	2.45995
$754$	0	0
$755$	−4.20309	−0.152966
$756$	0	0
$757$	−0.849120	−0.0308618	−0.0154309	−	0.999881i	$-0.504912\pi$
$757$	−0.849120	−0.0308618	−0.0154309	+	0.999881i	$0.504912\pi$
$758$	0	0
$759$	1.55275	0.0563613
$760$	0	0
$761$	29.8619	1.08249	0.541246	−	0.840864i	$-0.317953\pi$
$761$	29.8619	1.08249	0.541246	+	0.840864i	$0.317953\pi$
$762$	0	0
$763$	2.86967	0.103889
$764$	0	0
$765$	49.6912	1.79659
$766$	0	0
$767$	32.3834	1.16930
$768$	0	0
$769$	−13.8365	−0.498957	−0.249478	−	0.968380i	$-0.580259\pi$
$769$	−13.8365	−0.498957	−0.249478	+	0.968380i	$0.580259\pi$
$770$	0	0
$771$	62.1195	2.23718
$772$	0	0
$773$	−36.7921	−1.32332	−0.661659	−	0.749804i	$-0.730148\pi$
$773$	−36.7921	−1.32332	−0.661659	+	0.749804i	$0.730148\pi$
$774$	0	0
$775$	−0.186674	−0.00670554
$776$	0	0
$777$	5.51589	0.197881
$778$	0	0
$779$	5.60603	0.200857
$780$	0	0
$781$	−4.45759	−0.159505
$782$	0	0
$783$	−15.2912	−0.546463
$784$	0	0
$785$	−45.2985	−1.61677
$786$	0	0
$787$	1.77963	0.0634371	0.0317185	−	0.999497i	$-0.489902\pi$
$787$	1.77963	0.0634371	0.0317185	+	0.999497i	$0.489902\pi$
$788$	0	0
$789$	28.1487	1.00212
$790$	0	0
$791$	−9.57635	−0.340496
$792$	0	0
$793$	16.5359	0.587207
$794$	0	0
$795$	78.7825	2.79413
$796$	0	0
$797$	−42.2904	−1.49800	−0.749002	−	0.662568i	$-0.769466\pi$
$797$	−42.2904	−1.49800	−0.749002	+	0.662568i	$0.769466\pi$
$798$	0	0
$799$	−18.1221	−0.641116
$800$	0	0
$801$	−38.5878	−1.36343
$802$	0	0
$803$	1.96755	0.0694334
$804$	0	0
$805$	−1.97841	−0.0697300
$806$	0	0
$807$	−4.11392	−0.144817
$808$	0	0
$809$	47.9543	1.68598	0.842992	−	0.537925i	$-0.180792\pi$
$809$	47.9543	1.68598	0.842992	+	0.537925i	$0.180792\pi$
$810$	0	0
$811$	11.9545	0.419780	0.209890	−	0.977725i	$-0.432689\pi$
$811$	11.9545	0.419780	0.209890	+	0.977725i	$0.432689\pi$
$812$	0	0
$813$	70.4492	2.47076
$814$	0	0
$815$	34.1672	1.19682
$816$	0	0
$817$	58.3928	2.04291
$818$	0	0
$819$	−11.0776	−0.387082
$820$	0	0
$821$	14.8028	0.516621	0.258311	−	0.966062i	$-0.416834\pi$
$821$	14.8028	0.516621	0.258311	+	0.966062i	$0.416834\pi$
$822$	0	0
$823$	−53.6259	−1.86928	−0.934641	−	0.355591i	$-0.884279\pi$
$823$	−53.6259	−1.86928	−0.934641	+	0.355591i	$0.884279\pi$
$824$	0	0
$825$	−0.203616	−0.00708899
$826$	0	0
$827$	9.83630	0.342042	0.171021	−	0.985267i	$-0.445293\pi$
$827$	9.83630	0.342042	0.171021	+	0.985267i	$0.445293\pi$
$828$	0	0
$829$	5.67361	0.197052	0.0985262	−	0.995134i	$-0.468587\pi$
$829$	5.67361	0.197052	0.0985262	+	0.995134i	$0.468587\pi$
$830$	0	0
$831$	9.95683	0.345399
$832$	0	0
$833$	−32.3935	−1.12237
$834$	0	0
$835$	−52.3037	−1.81004
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	8.91997	0.307952	0.153976	−	0.988075i	$-0.450792\pi$
$839$	8.91997	0.307952	0.153976	+	0.988075i	$0.450792\pi$
$840$	0	0
$841$	−14.3862	−0.496075
$842$	0	0
$843$	−40.2631	−1.38674
$844$	0	0
$845$	8.05275	0.277023
$846$	0	0
$847$	−6.61531	−0.227305
$848$	0	0
$849$	48.2173	1.65481
$850$	0	0
$851$	−4.70984	−0.161451
$852$	0	0
$853$	−10.7850	−0.369271	−0.184636	−	0.982807i	$-0.559110\pi$
$853$	−10.7850	−0.369271	−0.184636	+	0.982807i	$0.559110\pi$
$854$	0	0
$855$	60.4370	2.06690
$856$	0	0
$857$	29.3858	1.00380	0.501900	−	0.864926i	$-0.332635\pi$
$857$	29.3858	1.00380	0.501900	+	0.864926i	$0.332635\pi$
$858$	0	0
$859$	−41.7928	−1.42595	−0.712976	−	0.701188i	$-0.752653\pi$
$859$	−41.7928	−1.42595	−0.712976	+	0.701188i	$0.752653\pi$
$860$	0	0
$861$	1.57223	0.0535816
$862$	0	0
$863$	14.8148	0.504302	0.252151	−	0.967688i	$-0.418862\pi$
$863$	14.8148	0.504302	0.252151	+	0.967688i	$0.418862\pi$
$864$	0	0
$865$	12.4873	0.424580
$866$	0	0
$867$	18.8218	0.639220
$868$	0	0
$869$	4.30260	0.145956
$870$	0	0
$871$	−37.4944	−1.27045
$872$	0	0
$873$	−54.3132	−1.83822
$874$	0	0
$875$	−6.68939	−0.226143
$876$	0	0
$877$	−27.2269	−0.919387	−0.459693	−	0.888078i	$-0.652041\pi$
$877$	−27.2269	−0.919387	−0.459693	+	0.888078i	$0.652041\pi$
$878$	0	0
$879$	45.9461	1.54972
$880$	0	0
$881$	41.7551	1.40677	0.703383	−	0.710811i	$-0.251672\pi$
$881$	41.7551	1.40677	0.703383	+	0.710811i	$0.251672\pi$
$882$	0	0
$883$	−34.8191	−1.17176	−0.585878	−	0.810399i	$-0.699250\pi$
$883$	−34.8191	−1.17176	−0.585878	+	0.810399i	$0.699250\pi$
$884$	0	0
$885$	−49.5497	−1.66560
$886$	0	0
$887$	−22.9783	−0.771537	−0.385768	−	0.922596i	$-0.626064\pi$
$887$	−22.9783	−0.771537	−0.385768	+	0.922596i	$0.626064\pi$
$888$	0	0
$889$	5.95560	0.199744
$890$	0	0
$891$	0.983776	0.0329577
$892$	0	0
$893$	−22.0411	−0.737576
$894$	0	0
$895$	8.74933	0.292458
$896$	0	0
$897$	15.8153	0.528059
$898$	0	0
$899$	3.82280	0.127498
$900$	0	0
$901$	61.8868	2.06175
$902$	0	0
$903$	16.3765	0.544976
$904$	0	0
$905$	−45.6682	−1.51806
$906$	0	0
$907$	52.5247	1.74406	0.872028	−	0.489457i	$-0.162805\pi$
$907$	52.5247	1.74406	0.872028	+	0.489457i	$0.162805\pi$
$908$	0	0
$909$	−30.1983	−1.00161
$910$	0	0
$911$	10.3846	0.344057	0.172029	−	0.985092i	$-0.444968\pi$
$911$	10.3846	0.344057	0.172029	+	0.985092i	$0.444968\pi$
$912$	0	0
$913$	4.73762	0.156792
$914$	0	0
$915$	−25.3016	−0.836444
$916$	0	0
$917$	−0.158315	−0.00522804
$918$	0	0
$919$	36.9630	1.21930	0.609648	−	0.792672i	$-0.291311\pi$
$919$	36.9630	1.21930	0.609648	+	0.792672i	$0.291311\pi$
$920$	0	0
$921$	−84.5826	−2.78709
$922$	0	0
$923$	−45.4021	−1.49443
$924$	0	0
$925$	0.617611	0.0203069
$926$	0	0
$927$	42.0861	1.38229
$928$	0	0
$929$	23.1634	0.759967	0.379984	−	0.924993i	$-0.375930\pi$
$929$	23.1634	0.759967	0.379984	+	0.924993i	$0.375930\pi$
$930$	0	0
$931$	−39.3986	−1.29124
$932$	0	0
$933$	−20.2842	−0.664074
$934$	0	0
$935$	−4.44409	−0.145337
$936$	0	0
$937$	17.6452	0.576444	0.288222	−	0.957564i	$-0.406936\pi$
$937$	17.6452	0.576444	0.288222	+	0.957564i	$0.406936\pi$
$938$	0	0
$939$	−65.6886	−2.14367
$940$	0	0
$941$	−8.31462	−0.271049	−0.135524	−	0.990774i	$-0.543272\pi$
$941$	−8.31462	−0.271049	−0.135524	+	0.990774i	$0.543272\pi$
$942$	0	0
$943$	−1.34248	−0.0437171
$944$	0	0
$945$	5.55906	0.180836
$946$	0	0
$947$	51.9107	1.68687	0.843436	−	0.537230i	$-0.180529\pi$
$947$	51.9107	1.68687	0.843436	+	0.537230i	$0.180529\pi$
$948$	0	0
$949$	20.0402	0.650533
$950$	0	0
$951$	80.5984	2.61358
$952$	0	0
$953$	47.8601	1.55034	0.775170	−	0.631753i	$-0.217664\pi$
$953$	47.8601	1.55034	0.775170	+	0.631753i	$0.217664\pi$
$954$	0	0
$955$	24.4235	0.790325
$956$	0	0
$957$	4.16973	0.134788
$958$	0	0
$959$	−0.383367	−0.0123796
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	32.7716	1.05605
$964$	0	0
$965$	8.26881	0.266182
$966$	0	0
$967$	26.4990	0.852152	0.426076	−	0.904687i	$-0.359896\pi$
$967$	26.4990	0.852152	0.426076	+	0.904687i	$0.359896\pi$
$968$	0	0
$969$	79.3805	2.55007
$970$	0	0
$971$	−28.8882	−0.927066	−0.463533	−	0.886080i	$-0.653418\pi$
$971$	−28.8882	−0.927066	−0.463533	+	0.886080i	$0.653418\pi$
$972$	0	0
$973$	10.8429	0.347607
$974$	0	0
$975$	−2.07390	−0.0664179
$976$	0	0
$977$	27.7363	0.887362	0.443681	−	0.896185i	$-0.353672\pi$
$977$	27.7363	0.887362	0.443681	+	0.896185i	$0.353672\pi$
$978$	0	0
$979$	3.45107	0.110297
$980$	0	0
$981$	20.9927	0.670246
$982$	0	0
$983$	55.7890	1.77939	0.889697	−	0.456552i	$-0.150916\pi$
$983$	55.7890	1.77939	0.889697	+	0.456552i	$0.150916\pi$
$984$	0	0
$985$	53.3237	1.69903
$986$	0	0
$987$	−6.18151	−0.196759
$988$	0	0
$989$	−13.9834	−0.444645
$990$	0	0
$991$	2.80427	0.0890806	0.0445403	−	0.999008i	$-0.485818\pi$
$991$	2.80427	0.0890806	0.0445403	+	0.999008i	$0.485818\pi$
$992$	0	0
$993$	48.5441	1.54050
$994$	0	0
$995$	1.66982	0.0529369
$996$	0	0
$997$	−60.3343	−1.91081	−0.955403	−	0.295305i	$-0.904579\pi$
$997$	−60.3343	−1.91081	−0.955403	+	0.295305i	$0.904579\pi$
$998$	0	0
$999$	13.2340	0.418704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.a.z.1.4		4
4.3	odd	2		1984.2.a.y.1.2		4
8.3	odd	2		992.2.a.f.1.3	yes	4
8.5	even	2		992.2.a.e.1.1	✓	4
24.5	odd	2		8928.2.a.bm.1.4		4
24.11	even	2		8928.2.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
992.2.a.e.1.1	✓	4	8.5	even	2
992.2.a.f.1.3	yes	4	8.3	odd	2
1984.2.a.y.1.2		4	4.3	odd	2
1984.2.a.z.1.4		4	1.1	even	1	trivial
8928.2.a.bm.1.4		4	24.5	odd	2
8928.2.a.bn.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1984 = 2^{6} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1984.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} +2.27743 q^{5} +0.610235 q^{7} +4.46410 q^{9} -0.399243 q^{11} -4.06644 q^{13} +6.22205 q^{15} +4.88766 q^{17} +5.94462 q^{19} +1.66719 q^{21} -1.42356 q^{23} +0.186674 q^{25} +4.00000 q^{27} -3.82280 q^{29} -1.00000 q^{31} -1.09075 q^{33} +1.38977 q^{35} +3.30849 q^{37} -11.1097 q^{39} +0.943042 q^{41} +9.82280 q^{43} +10.1667 q^{45} -3.70773 q^{47} -6.62761 q^{49} +13.3533 q^{51} +12.6618 q^{53} -0.909247 q^{55} +16.2410 q^{57} -7.96358 q^{59} -4.06644 q^{61} +2.72415 q^{63} -9.26101 q^{65} +9.22047 q^{67} -3.88924 q^{69} +11.1651 q^{71} -4.92820 q^{73} +0.510004 q^{75} -0.243632 q^{77} -10.7769 q^{79} -2.46410 q^{81} -11.8665 q^{83} +11.1313 q^{85} -10.4441 q^{87} -8.64403 q^{89} -2.48148 q^{91} -2.73205 q^{93} +13.5384 q^{95} -12.1667 q^{97} -1.78226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 10 q^{11} - 2 q^{15} + 2 q^{17} + 8 q^{19} + 2 q^{21} - 2 q^{23} - 2 q^{25} + 16 q^{27} - 4 q^{31} + 4 q^{33} + 12 q^{35} + 10 q^{37} + 2 q^{41} + 24 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^5 - 4 * q^7 + 4 * q^9 + 10 * q^11 - 2 * q^15 + 2 * q^17 + 8 * q^19 + 2 * q^21 - 2 * q^23 - 2 * q^25 + 16 * q^27 - 4 * q^31 + 4 * q^33 + 12 * q^35 + 10 * q^37 + 2 * q^41 + 24 * q^43 - 2 * q^45 + 2 * q^49 + 8 * q^51 + 2 * q^53 - 12 * q^55 + 2 * q^57 + 20 * q^59 + 8 * q^63 - 14 * q^65 + 24 * q^67 + 16 * q^69 + 16 * q^71 + 8 * q^73 + 22 * q^75 + 6 * q^79 + 4 * q^81 + 6 * q^83 + 26 * q^85 + 12 * q^87 - 18 * q^89 + 26 * q^91 - 4 * q^93 + 20 * q^95 - 6 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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