LMFDB - Embedded newform 1984.2.a.z.1.3

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			1.3
Root			$-2.27743$ 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)

$f(q)$	$=$	$q+2.73205 q^{3} -3.27743 q^{5} -0.878184 q^{7} +4.46410 q^{9} +3.66719 q^{11} +4.06644 q^{13} -8.95410 q^{15} -2.15561 q^{17} -3.67667 q^{19} -2.39924 q^{21} +5.61971 q^{23} +5.74153 q^{25} +4.00000 q^{27} +7.28691 q^{29} -1.00000 q^{31} +10.0190 q^{33} +2.87818 q^{35} +10.3518 q^{37} +11.1097 q^{39} +3.52106 q^{41} -1.28691 q^{43} -14.6308 q^{45} -6.68457 q^{47} -6.22879 q^{49} -5.88924 q^{51} +0.462524 q^{53} -12.0190 q^{55} -10.0449 q^{57} +12.7674 q^{59} +4.06644 q^{61} -3.92030 q^{63} -13.3274 q^{65} +6.24363 q^{67} +15.3533 q^{69} -1.43304 q^{71} -4.92820 q^{73} +15.6861 q^{75} -3.22047 q^{77} +15.5090 q^{79} -2.46410 q^{81} +16.5985 q^{83} +7.06486 q^{85} +19.9082 q^{87} +1.37608 q^{89} -3.57108 q^{91} -2.73205 q^{93} +12.0500 q^{95} +12.6308 q^{97} +16.3707 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.27743	−1.46571	−0.732855	−	0.680385i	$-0.761813\pi$
$5$	−3.27743	−1.46571	−0.732855	+	0.680385i	$0.761813\pi$
$6$	0	0
$7$	−0.878184	−0.331922	−0.165961	−	0.986132i	$-0.553073\pi$
$7$	−0.878184	−0.331922	−0.165961	+	0.986132i	$0.553073\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	3.66719	1.10570	0.552850	−	0.833281i	$-0.313540\pi$
$11$	3.66719	1.10570	0.552850	+	0.833281i	$0.313540\pi$
$12$	0	0
$13$	4.06644	1.12783	0.563913	−	0.825834i	$-0.309295\pi$
$13$	4.06644	1.12783	0.563913	+	0.825834i	$0.309295\pi$
$14$	0	0
$15$	−8.95410	−2.31194
$16$	0	0
$17$	−2.15561	−0.522812	−0.261406	−	0.965229i	$-0.584186\pi$
$17$	−2.15561	−0.522812	−0.261406	+	0.965229i	$0.584186\pi$
$18$	0	0
$19$	−3.67667	−0.843486	−0.421743	−	0.906715i	$-0.638582\pi$
$19$	−3.67667	−0.843486	−0.421743	+	0.906715i	$0.638582\pi$
$20$	0	0
$21$	−2.39924	−0.523558
$22$	0	0
$23$	5.61971	1.17179	0.585896	−	0.810387i	$-0.300743\pi$
$23$	5.61971	1.17179	0.585896	+	0.810387i	$0.300743\pi$
$24$	0	0
$25$	5.74153	1.14831
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	7.28691	1.35314	0.676572	−	0.736376i	$-0.263465\pi$
$29$	7.28691	1.35314	0.676572	+	0.736376i	$0.263465\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	10.0190	1.74408
$34$	0	0
$35$	2.87818	0.486502
$36$	0	0
$37$	10.3518	1.70182	0.850910	−	0.525312i	$-0.176051\pi$
$37$	10.3518	1.70182	0.850910	+	0.525312i	$0.176051\pi$
$38$	0	0
$39$	11.1097	1.77898
$40$	0	0
$41$	3.52106	0.549897	0.274949	−	0.961459i	$-0.411339\pi$
$41$	3.52106	0.549897	0.274949	+	0.961459i	$0.411339\pi$
$42$	0	0
$43$	−1.28691	−0.196251	−0.0981256	−	0.995174i	$-0.531285\pi$
$43$	−1.28691	−0.196251	−0.0981256	+	0.995174i	$0.531285\pi$
$44$	0	0
$45$	−14.6308	−2.18103
$46$	0	0
$47$	−6.68457	−0.975045	−0.487522	−	0.873110i	$-0.662099\pi$
$47$	−6.68457	−0.975045	−0.487522	+	0.873110i	$0.662099\pi$
$48$	0	0
$49$	−6.22879	−0.889828
$50$	0	0
$51$	−5.88924	−0.824658
$52$	0	0
$53$	0.462524	0.0635326	0.0317663	−	0.999495i	$-0.489887\pi$
$53$	0.462524	0.0635326	0.0317663	+	0.999495i	$0.489887\pi$
$54$	0	0
$55$	−12.0190	−1.62064
$56$	0	0
$57$	−10.0449	−1.33047
$58$	0	0
$59$	12.7674	1.66218	0.831088	−	0.556140i	$-0.187718\pi$
$59$	12.7674	1.66218	0.831088	+	0.556140i	$0.187718\pi$
$60$	0	0
$61$	4.06644	0.520654	0.260327	−	0.965521i	$-0.416170\pi$
$61$	4.06644	0.520654	0.260327	+	0.965521i	$0.416170\pi$
$62$	0	0
$63$	−3.92030	−0.493912
$64$	0	0
$65$	−13.3274	−1.65307
$66$	0	0
$67$	6.24363	0.762781	0.381391	−	0.924414i	$-0.375445\pi$
$67$	6.24363	0.762781	0.381391	+	0.924414i	$0.375445\pi$
$68$	0	0
$69$	15.3533	1.84833
$70$	0	0
$71$	−1.43304	−0.170070	−0.0850352	−	0.996378i	$-0.527100\pi$
$71$	−1.43304	−0.170070	−0.0850352	+	0.996378i	$0.527100\pi$
$72$	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
$75$	15.6861	1.81128
$76$	0	0
$77$	−3.22047	−0.367007
$78$	0	0
$79$	15.5090	1.74489	0.872447	−	0.488709i	$-0.162532\pi$
$79$	15.5090	1.74489	0.872447	+	0.488709i	$0.162532\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	16.5985	1.82193	0.910964	−	0.412486i	$-0.135340\pi$
$83$	16.5985	1.82193	0.910964	+	0.412486i	$0.135340\pi$
$84$	0	0
$85$	7.06486	0.766291
$86$	0	0
$87$	19.9082	2.13438
$88$	0	0
$89$	1.37608	0.145864	0.0729321	−	0.997337i	$-0.476764\pi$
$89$	1.37608	0.145864	0.0729321	+	0.997337i	$0.476764\pi$
$90$	0	0
$91$	−3.57108	−0.374351
$92$	0	0
$93$	−2.73205	−0.283300
$94$	0	0
$95$	12.0500	1.23631
$96$	0	0
$97$	12.6308	1.28246	0.641230	−	0.767349i	$-0.278424\pi$
$97$	12.6308	1.28246	0.641230	+	0.767349i	$0.278424\pi$
$98$	0	0
$99$	16.3707	1.64532
$100$	0	0
$101$	−7.16351	−0.712796	−0.356398	−	0.934334i	$-0.615995\pi$
$101$	−7.16351	−0.712796	−0.356398	+	0.934334i	$0.615995\pi$
$102$	0	0
$103$	−11.3033	−1.11375	−0.556875	−	0.830597i	$-0.688000\pi$
$103$	−11.3033	−1.11375	−0.556875	+	0.830597i	$0.688000\pi$
$104$	0	0
$105$	7.86334	0.767384
$106$	0	0
$107$	14.7832	1.42915	0.714574	−	0.699560i	$-0.246621\pi$
$107$	14.7832	1.42915	0.714574	+	0.699560i	$0.246621\pi$
$108$	0	0
$109$	−20.0949	−1.92474	−0.962370	−	0.271744i	$-0.912400\pi$
$109$	−20.0949	−1.92474	−0.962370	+	0.271744i	$0.912400\pi$
$110$	0	0
$111$	28.2815	2.68437
$112$	0	0
$113$	−16.0917	−1.51378	−0.756891	−	0.653542i	$-0.773282\pi$
$113$	−16.0917	−1.51378	−0.756891	+	0.653542i	$0.773282\pi$
$114$	0	0
$115$	−18.4182	−1.71751
$116$	0	0
$117$	18.1530	1.67824
$118$	0	0
$119$	1.89302	0.173533
$120$	0	0
$121$	2.44830	0.222573
$122$	0	0
$123$	9.61971	0.867380
$124$	0	0
$125$	−2.43031	−0.217373
$126$	0	0
$127$	−17.6159	−1.56316	−0.781580	−	0.623804i	$-0.785586\pi$
$127$	−17.6159	−1.56316	−0.781580	+	0.623804i	$0.785586\pi$
$128$	0	0
$129$	−3.51589	−0.309557
$130$	0	0
$131$	−16.5252	−1.44381	−0.721906	−	0.691991i	$-0.756734\pi$
$131$	−16.5252	−1.44381	−0.721906	+	0.691991i	$0.756734\pi$
$132$	0	0
$133$	3.22879	0.279972
$134$	0	0
$135$	−13.1097	−1.12830
$136$	0	0
$137$	22.6808	1.93775	0.968875	−	0.247550i	$-0.0796253\pi$
$137$	22.6808	1.93775	0.968875	+	0.247550i	$0.0796253\pi$
$138$	0	0
$139$	7.74828	0.657200	0.328600	−	0.944469i	$-0.393423\pi$
$139$	7.74828	0.657200	0.328600	+	0.944469i	$0.393423\pi$
$140$	0	0
$141$	−18.2626	−1.53799
$142$	0	0
$143$	14.9124	1.24704
$144$	0	0
$145$	−23.8823	−1.98332
$146$	0	0
$147$	−17.0174	−1.40357
$148$	0	0
$149$	11.8892	0.974004	0.487002	−	0.873401i	$-0.338090\pi$
$149$	11.8892	0.974004	0.487002	+	0.873401i	$0.338090\pi$
$150$	0	0
$151$	0.0416959	0.00339316	0.00169658	−	0.999999i	$-0.499460\pi$
$151$	0.0416959	0.00339316	0.00169658	+	0.999999i	$0.499460\pi$
$152$	0	0
$153$	−9.62287	−0.777963
$154$	0	0
$155$	3.27743	0.263249
$156$	0	0
$157$	−11.3585	−0.906508	−0.453254	−	0.891381i	$-0.649737\pi$
$157$	−11.3585	−0.906508	−0.453254	+	0.891381i	$0.649737\pi$
$158$	0	0
$159$	1.26364	0.100213
$160$	0	0
$161$	−4.93514	−0.388944
$162$	0	0
$163$	13.5141	1.05851	0.529254	−	0.848464i	$-0.322472\pi$
$163$	13.5141	1.05851	0.529254	+	0.848464i	$0.322472\pi$
$164$	0	0
$165$	−32.8364	−2.55631
$166$	0	0
$167$	−0.746698	−0.0577812	−0.0288906	−	0.999583i	$-0.509197\pi$
$167$	−0.746698	−0.0577812	−0.0288906	+	0.999583i	$0.509197\pi$
$168$	0	0
$169$	3.53590	0.271992
$170$	0	0
$171$	−16.4130	−1.25514
$172$	0	0
$173$	−5.62665	−0.427786	−0.213893	−	0.976857i	$-0.568614\pi$
$173$	−5.62665	−0.427786	−0.213893	+	0.976857i	$0.568614\pi$
$174$	0	0
$175$	−5.04212	−0.381148
$176$	0	0
$177$	34.8812	2.62184
$178$	0	0
$179$	−18.3777	−1.37361	−0.686805	−	0.726841i	$-0.740988\pi$
$179$	−18.3777	−1.37361	−0.686805	+	0.726841i	$0.740988\pi$
$180$	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
$185$	−33.9272	−2.49437
$186$	0	0
$187$	−7.90504	−0.578074
$188$	0	0
$189$	−3.51274	−0.255514
$190$	0	0
$191$	4.07970	0.295197	0.147598	−	0.989047i	$-0.452846\pi$
$191$	4.07970	0.295197	0.147598	+	0.989047i	$0.452846\pi$
$192$	0	0
$193$	−21.1667	−1.52361	−0.761805	−	0.647806i	$-0.775687\pi$
$193$	−21.1667	−1.52361	−0.761805	+	0.647806i	$0.775687\pi$
$194$	0	0
$195$	−36.4113	−2.60746
$196$	0	0
$197$	−8.82554	−0.628793	−0.314397	−	0.949292i	$-0.601802\pi$
$197$	−8.82554	−0.628793	−0.314397	+	0.949292i	$0.601802\pi$
$198$	0	0
$199$	−8.19731	−0.581092	−0.290546	−	0.956861i	$-0.593837\pi$
$199$	−8.19731	−0.581092	−0.290546	+	0.956861i	$0.593837\pi$
$200$	0	0
$201$	17.0579	1.20317
$202$	0	0
$203$	−6.39924	−0.449139
$204$	0	0
$205$	−11.5400	−0.805990
$206$	0	0
$207$	25.0870	1.74366
$208$	0	0
$209$	−13.4831	−0.932643
$210$	0	0
$211$	7.20836	0.496244	0.248122	−	0.968729i	$-0.420187\pi$
$211$	7.20836	0.496244	0.248122	+	0.968729i	$0.420187\pi$
$212$	0	0
$213$	−3.91513	−0.268261
$214$	0	0
$215$	4.21774	0.287647
$216$	0	0
$217$	0.878184	0.0596150
$218$	0	0
$219$	−13.4641	−0.909820
$220$	0	0
$221$	−8.76565	−0.589642
$222$	0	0
$223$	−22.0348	−1.47556	−0.737778	−	0.675043i	$-0.764125\pi$
$223$	−22.0348	−1.47556	−0.737778	+	0.675043i	$0.764125\pi$
$224$	0	0
$225$	25.6308	1.70872
$226$	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
$229$	5.24205	0.346405	0.173202	−	0.984886i	$-0.444589\pi$
$229$	5.24205	0.346405	0.173202	+	0.984886i	$0.444589\pi$
$230$	0	0
$231$	−8.79849	−0.578898
$232$	0	0
$233$	−1.33955	−0.0877571	−0.0438785	−	0.999037i	$-0.513971\pi$
$233$	−1.33955	−0.0877571	−0.0438785	+	0.999037i	$0.513971\pi$
$234$	0	0
$235$	21.9082	1.42913
$236$	0	0
$237$	42.3712	2.75231
$238$	0	0
$239$	24.8402	1.60678	0.803389	−	0.595455i	$-0.203028\pi$
$239$	24.8402	1.60678	0.803389	+	0.595455i	$0.203028\pi$
$240$	0	0
$241$	−11.1908	−0.720862	−0.360431	−	0.932786i	$-0.617370\pi$
$241$	−11.1908	−0.720862	−0.360431	+	0.932786i	$0.617370\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	20.4144	1.30423
$246$	0	0
$247$	−14.9509	−0.951306
$248$	0	0
$249$	45.3481	2.87382
$250$	0	0
$251$	−22.7078	−1.43331	−0.716653	−	0.697430i	$-0.754327\pi$
$251$	−22.7078	−1.43331	−0.716653	+	0.697430i	$0.754327\pi$
$252$	0	0
$253$	20.6086	1.29565
$254$	0	0
$255$	19.3016	1.20871
$256$	0	0
$257$	0.119084	0.00742827	0.00371414	−	0.999993i	$-0.498818\pi$
$257$	0.119084	0.00742827	0.00371414	+	0.999993i	$0.498818\pi$
$258$	0	0
$259$	−9.09075	−0.564872
$260$	0	0
$261$	32.5295	2.01352
$262$	0	0
$263$	9.21353	0.568131	0.284065	−	0.958805i	$-0.408317\pi$
$263$	9.21353	0.568131	0.284065	+	0.958805i	$0.408317\pi$
$264$	0	0
$265$	−1.51589	−0.0931204
$266$	0	0
$267$	3.75952	0.230079
$268$	0	0
$269$	0.381442	0.0232569	0.0116285	−	0.999932i	$-0.496298\pi$
$269$	0.381442	0.0232569	0.0116285	+	0.999932i	$0.496298\pi$
$270$	0	0
$271$	13.5869	0.825344	0.412672	−	0.910880i	$-0.364596\pi$
$271$	13.5869	0.825344	0.412672	+	0.910880i	$0.364596\pi$
$272$	0	0
$273$	−9.75637	−0.590482
$274$	0	0
$275$	21.0553	1.26968
$276$	0	0
$277$	−9.64445	−0.579479	−0.289739	−	0.957106i	$-0.593569\pi$
$277$	−9.64445	−0.579479	−0.289739	+	0.957106i	$0.593569\pi$
$278$	0	0
$279$	−4.46410	−0.267259
$280$	0	0
$281$	7.88092	0.470136	0.235068	−	0.971979i	$-0.424469\pi$
$281$	7.88092	0.470136	0.235068	+	0.971979i	$0.424469\pi$
$282$	0	0
$283$	−28.9693	−1.72204	−0.861022	−	0.508567i	$-0.830175\pi$
$283$	−28.9693	−1.72204	−0.861022	+	0.508567i	$0.830175\pi$
$284$	0	0
$285$	32.9213	1.95009
$286$	0	0
$287$	−3.09214	−0.182523
$288$	0	0
$289$	−12.3533	−0.726667
$290$	0	0
$291$	34.5079	2.02289
$292$	0	0
$293$	−2.42514	−0.141678	−0.0708390	−	0.997488i	$-0.522568\pi$
$293$	−2.42514	−0.141678	−0.0708390	+	0.997488i	$0.522568\pi$
$294$	0	0
$295$	−41.8443	−2.43627
$296$	0	0
$297$	14.6688	0.851168
$298$	0	0
$299$	22.8522	1.32158
$300$	0	0
$301$	1.13014	0.0651402
$302$	0	0
$303$	−19.5711	−1.12433
$304$	0	0
$305$	−13.3274	−0.763127
$306$	0	0
$307$	6.83501	0.390095	0.195047	−	0.980794i	$-0.437514\pi$
$307$	6.83501	0.390095	0.195047	+	0.980794i	$0.437514\pi$
$308$	0	0
$309$	−30.8812	−1.75677
$310$	0	0
$311$	−11.0921	−0.628977	−0.314489	−	0.949261i	$-0.601833\pi$
$311$	−11.0921	−0.628977	−0.314489	+	0.949261i	$0.601833\pi$
$312$	0	0
$313$	−6.68836	−0.378048	−0.189024	−	0.981972i	$-0.560532\pi$
$313$	−6.68836	−0.378048	−0.189024	+	0.981972i	$0.560532\pi$
$314$	0	0
$315$	12.8485	0.723931
$316$	0	0
$317$	−3.42925	−0.192606	−0.0963031	−	0.995352i	$-0.530702\pi$
$317$	−3.42925	−0.192606	−0.0963031	+	0.995352i	$0.530702\pi$
$318$	0	0
$319$	26.7225	1.49617
$320$	0	0
$321$	40.3885	2.25427
$322$	0	0
$323$	7.92547	0.440985
$324$	0	0
$325$	23.3476	1.29509
$326$	0	0
$327$	−54.9002	−3.03599
$328$	0	0
$329$	5.87028	0.323639
$330$	0	0
$331$	7.74828	0.425884	0.212942	−	0.977065i	$-0.431696\pi$
$331$	7.74828	0.425884	0.212942	+	0.977065i	$0.431696\pi$
$332$	0	0
$333$	46.2113	2.53236
$334$	0	0
$335$	−20.4630	−1.11802
$336$	0	0
$337$	−32.3561	−1.76255	−0.881274	−	0.472606i	$-0.843313\pi$
$337$	−32.3561	−1.76255	−0.881274	+	0.472606i	$0.843313\pi$
$338$	0	0
$339$	−43.9634	−2.38776
$340$	0	0
$341$	−3.66719	−0.198590
$342$	0	0
$343$	11.6173	0.627276
$344$	0	0
$345$	−50.3195	−2.70911
$346$	0	0
$347$	−2.68888	−0.144347	−0.0721733	−	0.997392i	$-0.522993\pi$
$347$	−2.68888	−0.144347	−0.0721733	+	0.997392i	$0.522993\pi$
$348$	0	0
$349$	−5.70773	−0.305528	−0.152764	−	0.988263i	$-0.548817\pi$
$349$	−5.70773	−0.305528	−0.152764	+	0.988263i	$0.548817\pi$
$350$	0	0
$351$	16.2657	0.868201
$352$	0	0
$353$	17.9240	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$353$	17.9240	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$354$	0	0
$355$	4.69668	0.249274
$356$	0	0
$357$	5.17184	0.273723
$358$	0	0
$359$	33.7767	1.78267	0.891333	−	0.453349i	$-0.149771\pi$
$359$	33.7767	1.78267	0.891333	+	0.453349i	$0.149771\pi$
$360$	0	0
$361$	−5.48210	−0.288531
$362$	0	0
$363$	6.68888	0.351075
$364$	0	0
$365$	16.1518	0.845425
$366$	0	0
$367$	11.9758	0.625131	0.312565	−	0.949896i	$-0.398812\pi$
$367$	11.9758	0.625131	0.312565	+	0.949896i	$0.398812\pi$
$368$	0	0
$369$	15.7184	0.818265
$370$	0	0
$371$	−0.406182	−0.0210879
$372$	0	0
$373$	8.54422	0.442403	0.221202	−	0.975228i	$-0.429002\pi$
$373$	8.54422	0.442403	0.221202	+	0.975228i	$0.429002\pi$
$374$	0	0
$375$	−6.63972	−0.342874
$376$	0	0
$377$	29.6317	1.52611
$378$	0	0
$379$	22.6256	1.16220	0.581099	−	0.813833i	$-0.302623\pi$
$379$	22.6256	1.16220	0.581099	+	0.813833i	$0.302623\pi$
$380$	0	0
$381$	−48.1276	−2.46565
$382$	0	0
$383$	23.2129	1.18612	0.593062	−	0.805157i	$-0.297919\pi$
$383$	23.2129	1.18612	0.593062	+	0.805157i	$0.297919\pi$
$384$	0	0
$385$	10.5549	0.537925
$386$	0	0
$387$	−5.74488	−0.292028
$388$	0	0
$389$	7.16921	0.363493	0.181747	−	0.983345i	$-0.441825\pi$
$389$	7.16921	0.363493	0.181747	+	0.983345i	$0.441825\pi$
$390$	0	0
$391$	−12.1139	−0.612627
$392$	0	0
$393$	−45.1476	−2.27740
$394$	0	0
$395$	−50.8295	−2.55751
$396$	0	0
$397$	1.09592	0.0550027	0.0275014	−	0.999622i	$-0.491245\pi$
$397$	1.09592	0.0550027	0.0275014	+	0.999622i	$0.491245\pi$
$398$	0	0
$399$	8.82123	0.441614
$400$	0	0
$401$	−7.63867	−0.381457	−0.190728	−	0.981643i	$-0.561085\pi$
$401$	−7.63867	−0.381457	−0.190728	+	0.981643i	$0.561085\pi$
$402$	0	0
$403$	−4.06644	−0.202564
$404$	0	0
$405$	8.07591	0.401295
$406$	0	0
$407$	37.9619	1.88170
$408$	0	0
$409$	−1.89618	−0.0937599	−0.0468800	−	0.998901i	$-0.514928\pi$
$409$	−1.89618	−0.0937599	−0.0468800	+	0.998901i	$0.514928\pi$
$410$	0	0
$411$	61.9651	3.05651
$412$	0	0
$413$	−11.2121	−0.551714
$414$	0	0
$415$	−54.4005	−2.67042
$416$	0	0
$417$	21.1687	1.03663
$418$	0	0
$419$	31.8475	1.55585	0.777925	−	0.628357i	$-0.216272\pi$
$419$	31.8475	1.55585	0.777925	+	0.628357i	$0.216272\pi$
$420$	0	0
$421$	−26.0083	−1.26757	−0.633784	−	0.773510i	$-0.718499\pi$
$421$	−26.0083	−1.26757	−0.633784	+	0.773510i	$0.718499\pi$
$422$	0	0
$423$	−29.8406	−1.45090
$424$	0	0
$425$	−12.3765	−0.600349
$426$	0	0
$427$	−3.57108	−0.172817
$428$	0	0
$429$	40.7414	1.96702
$430$	0	0
$431$	−11.0579	−0.532641	−0.266321	−	0.963884i	$-0.585808\pi$
$431$	−11.0579	−0.532641	−0.266321	+	0.963884i	$0.585808\pi$
$432$	0	0
$433$	27.4171	1.31758	0.658792	−	0.752325i	$-0.271068\pi$
$433$	27.4171	1.31758	0.658792	+	0.752325i	$0.271068\pi$
$434$	0	0
$435$	−65.2477	−3.12839
$436$	0	0
$437$	−20.6618	−0.988389
$438$	0	0
$439$	9.82850	0.469089	0.234544	−	0.972105i	$-0.424640\pi$
$439$	9.82850	0.469089	0.234544	+	0.972105i	$0.424640\pi$
$440$	0	0
$441$	−27.8060	−1.32409
$442$	0	0
$443$	−27.3895	−1.30131	−0.650657	−	0.759372i	$-0.725506\pi$
$443$	−27.3895	−1.30131	−0.650657	+	0.759372i	$0.725506\pi$
$444$	0	0
$445$	−4.51000	−0.213795
$446$	0	0
$447$	32.4820	1.53635
$448$	0	0
$449$	13.8216	0.652284	0.326142	−	0.945321i	$-0.394251\pi$
$449$	13.8216	0.652284	0.326142	+	0.945321i	$0.394251\pi$
$450$	0	0
$451$	12.9124	0.608021
$452$	0	0
$453$	0.113915	0.00535221
$454$	0	0
$455$	11.7039	0.548690
$456$	0	0
$457$	17.2584	0.807313	0.403657	−	0.914911i	$-0.367739\pi$
$457$	17.2584	0.807313	0.403657	+	0.914911i	$0.367739\pi$
$458$	0	0
$459$	−8.62244	−0.402461
$460$	0	0
$461$	−32.6523	−1.52077	−0.760384	−	0.649474i	$-0.774989\pi$
$461$	−32.6523	−1.52077	−0.760384	+	0.649474i	$0.774989\pi$
$462$	0	0
$463$	−19.8406	−0.922071	−0.461036	−	0.887382i	$-0.652522\pi$
$463$	−19.8406	−0.922071	−0.461036	+	0.887382i	$0.652522\pi$
$464$	0	0
$465$	8.95410	0.415236
$466$	0	0
$467$	−17.9847	−0.832234	−0.416117	−	0.909311i	$-0.636609\pi$
$467$	−17.9847	−0.832234	−0.416117	+	0.909311i	$0.636609\pi$
$468$	0	0
$469$	−5.48306	−0.253184
$470$	0	0
$471$	−31.0320	−1.42988
$472$	0	0
$473$	−4.71933	−0.216995
$474$	0	0
$475$	−21.1097	−0.968580
$476$	0	0
$477$	2.06476	0.0945387
$478$	0	0
$479$	−0.163935	−0.00749036	−0.00374518	−	0.999993i	$-0.501192\pi$
$479$	−0.163935	−0.00749036	−0.00374518	+	0.999993i	$0.501192\pi$
$480$	0	0
$481$	42.0948	1.91936
$482$	0	0
$483$	−13.4831	−0.613500
$484$	0	0
$485$	−41.3964	−1.87971
$486$	0	0
$487$	−26.0777	−1.18169	−0.590846	−	0.806784i	$-0.701206\pi$
$487$	−26.0777	−1.18169	−0.590846	+	0.806784i	$0.701206\pi$
$488$	0	0
$489$	36.9213	1.66964
$490$	0	0
$491$	6.90154	0.311462	0.155731	−	0.987799i	$-0.450227\pi$
$491$	6.90154	0.311462	0.155731	+	0.987799i	$0.450227\pi$
$492$	0	0
$493$	−15.7077	−0.707441
$494$	0	0
$495$	−53.6538	−2.41156
$496$	0	0
$497$	1.25847	0.0564501
$498$	0	0
$499$	9.52360	0.426335	0.213167	−	0.977016i	$-0.431622\pi$
$499$	9.52360	0.426335	0.213167	+	0.977016i	$0.431622\pi$
$500$	0	0
$501$	−2.04002	−0.0911412
$502$	0	0
$503$	4.32333	0.192768	0.0963839	−	0.995344i	$-0.469272\pi$
$503$	4.32333	0.192768	0.0963839	+	0.995344i	$0.469272\pi$
$504$	0	0
$505$	23.4779	1.04475
$506$	0	0
$507$	9.66025	0.429027
$508$	0	0
$509$	18.5718	0.823181	0.411591	−	0.911369i	$-0.364973\pi$
$509$	18.5718	0.823181	0.411591	+	0.911369i	$0.364973\pi$
$510$	0	0
$511$	4.32787	0.191454
$512$	0	0
$513$	−14.7067	−0.649316
$514$	0	0
$515$	37.0458	1.63243
$516$	0	0
$517$	−24.5136	−1.07811
$518$	0	0
$519$	−15.3723	−0.674769
$520$	0	0
$521$	−25.6267	−1.12272	−0.561362	−	0.827570i	$-0.689722\pi$
$521$	−25.6267	−1.12272	−0.561362	+	0.827570i	$0.689722\pi$
$522$	0	0
$523$	8.06328	0.352583	0.176291	−	0.984338i	$-0.443590\pi$
$523$	8.06328	0.352583	0.176291	+	0.984338i	$0.443590\pi$
$524$	0	0
$525$	−13.7753	−0.601204
$526$	0	0
$527$	2.15561	0.0938999
$528$	0	0
$529$	8.58117	0.373094
$530$	0	0
$531$	56.9951	2.47338
$532$	0	0
$533$	14.3182	0.620188
$534$	0	0
$535$	−48.4509	−2.09472
$536$	0	0
$537$	−50.2087	−2.16667
$538$	0	0
$539$	−22.8422	−0.983882
$540$	0	0
$541$	0.969359	0.0416760	0.0208380	−	0.999783i	$-0.493367\pi$
$541$	0.969359	0.0416760	0.0208380	+	0.999783i	$0.493367\pi$
$542$	0	0
$543$	−54.7846	−2.35103
$544$	0	0
$545$	65.8595	2.82111
$546$	0	0
$547$	43.4741	1.85882	0.929409	−	0.369051i	$-0.120317\pi$
$547$	43.4741	1.85882	0.929409	+	0.369051i	$0.120317\pi$
$548$	0	0
$549$	18.1530	0.774750
$550$	0	0
$551$	−26.7915	−1.14136
$552$	0	0
$553$	−13.6197	−0.579169
$554$	0	0
$555$	−92.6907	−3.93450
$556$	0	0
$557$	18.6009	0.788144	0.394072	−	0.919080i	$-0.371066\pi$
$557$	18.6009	0.788144	0.394072	+	0.919080i	$0.371066\pi$
$558$	0	0
$559$	−5.23312	−0.221337
$560$	0	0
$561$	−21.5970	−0.911825
$562$	0	0
$563$	34.1037	1.43730	0.718650	−	0.695371i	$-0.244760\pi$
$563$	34.1037	1.43730	0.718650	+	0.695371i	$0.244760\pi$
$564$	0	0
$565$	52.7394	2.21876
$566$	0	0
$567$	2.16393	0.0908767
$568$	0	0
$569$	14.2923	0.599163	0.299581	−	0.954071i	$-0.403153\pi$
$569$	14.2923	0.599163	0.299581	+	0.954071i	$0.403153\pi$
$570$	0	0
$571$	43.6950	1.82858	0.914290	−	0.405061i	$-0.132750\pi$
$571$	43.6950	1.82858	0.914290	+	0.405061i	$0.132750\pi$
$572$	0	0
$573$	11.1459	0.465628
$574$	0	0
$575$	32.2657	1.34557
$576$	0	0
$577$	−16.7098	−0.695640	−0.347820	−	0.937561i	$-0.613078\pi$
$577$	−16.7098	−0.695640	−0.347820	+	0.937561i	$0.613078\pi$
$578$	0	0
$579$	−57.8284	−2.40327
$580$	0	0
$581$	−14.5766	−0.604738
$582$	0	0
$583$	1.69617	0.0702480
$584$	0	0
$585$	−59.4951	−2.45982
$586$	0	0
$587$	−9.50464	−0.392299	−0.196149	−	0.980574i	$-0.562844\pi$
$587$	−9.50464	−0.392299	−0.196149	+	0.980574i	$0.562844\pi$
$588$	0	0
$589$	3.67667	0.151495
$590$	0	0
$591$	−24.1118	−0.991827
$592$	0	0
$593$	−18.1380	−0.744840	−0.372420	−	0.928064i	$-0.621472\pi$
$593$	−18.1380	−0.744840	−0.372420	+	0.928064i	$0.621472\pi$
$594$	0	0
$595$	−6.20425	−0.254349
$596$	0	0
$597$	−22.3955	−0.916585
$598$	0	0
$599$	−5.72846	−0.234058	−0.117029	−	0.993128i	$-0.537337\pi$
$599$	−5.72846	−0.234058	−0.117029	+	0.993128i	$0.537337\pi$
$600$	0	0
$601$	−5.63551	−0.229877	−0.114939	−	0.993373i	$-0.536667\pi$
$601$	−5.63551	−0.229877	−0.114939	+	0.993373i	$0.536667\pi$
$602$	0	0
$603$	27.8722	1.13504
$604$	0	0
$605$	−8.02412	−0.326227
$606$	0	0
$607$	−7.79235	−0.316282	−0.158141	−	0.987417i	$-0.550550\pi$
$607$	−7.79235	−0.316282	−0.158141	+	0.987417i	$0.550550\pi$
$608$	0	0
$609$	−17.4831	−0.708449
$610$	0	0
$611$	−27.1824	−1.09968
$612$	0	0
$613$	−37.8051	−1.52694	−0.763468	−	0.645846i	$-0.776505\pi$
$613$	−37.8051	−1.52694	−0.763468	+	0.645846i	$0.776505\pi$
$614$	0	0
$615$	−31.5279	−1.27133
$616$	0	0
$617$	21.1255	0.850481	0.425241	−	0.905080i	$-0.360190\pi$
$617$	21.1255	0.850481	0.425241	+	0.905080i	$0.360190\pi$
$618$	0	0
$619$	−12.5718	−0.505304	−0.252652	−	0.967557i	$-0.581303\pi$
$619$	−12.5718	−0.505304	−0.252652	+	0.967557i	$0.581303\pi$
$620$	0	0
$621$	22.4789	0.902045
$622$	0	0
$623$	−1.20845	−0.0484156
$624$	0	0
$625$	−20.7425	−0.829700
$626$	0	0
$627$	−36.8364	−1.47110
$628$	0	0
$629$	−22.3144	−0.889732
$630$	0	0
$631$	4.60349	0.183262	0.0916310	−	0.995793i	$-0.470792\pi$
$631$	4.60349	0.183262	0.0916310	+	0.995793i	$0.470792\pi$
$632$	0	0
$633$	19.6936	0.782751
$634$	0	0
$635$	57.7349	2.29114
$636$	0	0
$637$	−25.3290	−1.00357
$638$	0	0
$639$	−6.39723	−0.253070
$640$	0	0
$641$	−39.0832	−1.54369	−0.771846	−	0.635809i	$-0.780667\pi$
$641$	−39.0832	−1.54369	−0.771846	+	0.635809i	$0.780667\pi$
$642$	0	0
$643$	1.53978	0.0607232	0.0303616	−	0.999539i	$-0.490334\pi$
$643$	1.53978	0.0607232	0.0303616	+	0.999539i	$0.490334\pi$
$644$	0	0
$645$	11.5231	0.453721
$646$	0	0
$647$	23.8216	0.936526	0.468263	−	0.883589i	$-0.344880\pi$
$647$	23.8216	0.936526	0.468263	+	0.883589i	$0.344880\pi$
$648$	0	0
$649$	46.8206	1.83787
$650$	0	0
$651$	2.39924	0.0940338
$652$	0	0
$653$	32.1032	1.25630	0.628148	−	0.778094i	$-0.283813\pi$
$653$	32.1032	1.25630	0.628148	+	0.778094i	$0.283813\pi$
$654$	0	0
$655$	54.1601	2.11621
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−23.0818	−0.899139	−0.449570	−	0.893245i	$-0.648423\pi$
$659$	−23.0818	−0.899139	−0.449570	+	0.893245i	$0.648423\pi$
$660$	0	0
$661$	−8.33027	−0.324010	−0.162005	−	0.986790i	$-0.551796\pi$
$661$	−8.33027	−0.324010	−0.162005	+	0.986790i	$0.551796\pi$
$662$	0	0
$663$	−23.9482	−0.930071
$664$	0	0
$665$	−10.5821	−0.410358
$666$	0	0
$667$	40.9503	1.58560
$668$	0	0
$669$	−60.2001	−2.32747
$670$	0	0
$671$	14.9124	0.575687
$672$	0	0
$673$	−19.9183	−0.767794	−0.383897	−	0.923376i	$-0.625418\pi$
$673$	−19.9183	−0.767794	−0.383897	+	0.923376i	$0.625418\pi$
$674$	0	0
$675$	22.9661	0.883966
$676$	0	0
$677$	−25.3835	−0.975569	−0.487784	−	0.872964i	$-0.662195\pi$
$677$	−25.3835	−0.975569	−0.487784	+	0.872964i	$0.662195\pi$
$678$	0	0
$679$	−11.0921	−0.425677
$680$	0	0
$681$	−44.7846	−1.71615
$682$	0	0
$683$	31.2981	1.19759	0.598794	−	0.800903i	$-0.295647\pi$
$683$	31.2981	1.19759	0.598794	+	0.800903i	$0.295647\pi$
$684$	0	0
$685$	−74.3346	−2.84018
$686$	0	0
$687$	14.3216	0.546402
$688$	0	0
$689$	1.88083	0.0716538
$690$	0	0
$691$	−3.41093	−0.129758	−0.0648789	−	0.997893i	$-0.520666\pi$
$691$	−3.41093	−0.129758	−0.0648789	+	0.997893i	$0.520666\pi$
$692$	0	0
$693$	−14.3765	−0.546118
$694$	0	0
$695$	−25.3944	−0.963265
$696$	0	0
$697$	−7.59003	−0.287493
$698$	0	0
$699$	−3.65973	−0.138424
$700$	0	0
$701$	34.0485	1.28600	0.642998	−	0.765868i	$-0.277690\pi$
$701$	34.0485	1.28600	0.642998	+	0.765868i	$0.277690\pi$
$702$	0	0
$703$	−38.0600	−1.43546
$704$	0	0
$705$	59.8543	2.25424
$706$	0	0
$707$	6.29088	0.236593
$708$	0	0
$709$	4.24626	0.159472	0.0797358	−	0.996816i	$-0.474592\pi$
$709$	4.24626	0.159472	0.0797358	+	0.996816i	$0.474592\pi$
$710$	0	0
$711$	69.2335	2.59646
$712$	0	0
$713$	−5.61971	−0.210460
$714$	0	0
$715$	−48.8743	−1.82780
$716$	0	0
$717$	67.8646	2.53445
$718$	0	0
$719$	−31.3685	−1.16985	−0.584924	−	0.811088i	$-0.698876\pi$
$719$	−31.3685	−1.16985	−0.584924	+	0.811088i	$0.698876\pi$
$720$	0	0
$721$	9.92640	0.369678
$722$	0	0
$723$	−30.5738	−1.13705
$724$	0	0
$725$	41.8380	1.55382
$726$	0	0
$727$	7.18941	0.266640	0.133320	−	0.991073i	$-0.457436\pi$
$727$	7.18941	0.266640	0.133320	+	0.991073i	$0.457436\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	2.77407	0.102603
$732$	0	0
$733$	15.9620	0.589570	0.294785	−	0.955564i	$-0.404752\pi$
$733$	15.9620	0.589570	0.294785	+	0.955564i	$0.404752\pi$
$734$	0	0
$735$	55.7732	2.05723
$736$	0	0
$737$	22.8966	0.843407
$738$	0	0
$739$	−46.3373	−1.70454	−0.852272	−	0.523099i	$-0.824776\pi$
$739$	−46.3373	−1.70454	−0.852272	+	0.523099i	$0.824776\pi$
$740$	0	0
$741$	−40.8467	−1.50054
$742$	0	0
$743$	21.1198	0.774810	0.387405	−	0.921910i	$-0.373371\pi$
$743$	21.1198	0.774810	0.387405	+	0.921910i	$0.373371\pi$
$744$	0	0
$745$	−38.9661	−1.42761
$746$	0	0
$747$	74.0976	2.71109
$748$	0	0
$749$	−12.9824	−0.474366
$750$	0	0
$751$	29.6525	1.08203	0.541017	−	0.841012i	$-0.318040\pi$
$751$	29.6525	1.08203	0.541017	+	0.841012i	$0.318040\pi$
$752$	0	0
$753$	−62.0390	−2.26083
$754$	0	0
$755$	−0.136655	−0.00497339
$756$	0	0
$757$	28.7055	1.04332	0.521660	−	0.853153i	$-0.325313\pi$
$757$	28.7055	1.04332	0.521660	+	0.853153i	$0.325313\pi$
$758$	0	0
$759$	56.3037	2.04369
$760$	0	0
$761$	−12.3978	−0.449419	−0.224709	−	0.974426i	$-0.572143\pi$
$761$	−12.3978	−0.449419	−0.224709	+	0.974426i	$0.572143\pi$
$762$	0	0
$763$	17.6470	0.638864
$764$	0	0
$765$	31.5382	1.14027
$766$	0	0
$767$	51.9179	1.87465
$768$	0	0
$769$	−14.2353	−0.513338	−0.256669	−	0.966499i	$-0.582625\pi$
$769$	−14.2353	−0.513338	−0.256669	+	0.966499i	$0.582625\pi$
$770$	0	0
$771$	0.325344	0.0117170
$772$	0	0
$773$	20.9357	0.753004	0.376502	−	0.926416i	$-0.377127\pi$
$773$	20.9357	0.753004	0.376502	+	0.926416i	$0.377127\pi$
$774$	0	0
$775$	−5.74153	−0.206242
$776$	0	0
$777$	−24.8364	−0.891001
$778$	0	0
$779$	−12.9458	−0.463830
$780$	0	0
$781$	−5.25523	−0.188047
$782$	0	0
$783$	29.1476	1.04165
$784$	0	0
$785$	37.2267	1.32868
$786$	0	0
$787$	−45.6360	−1.62675	−0.813375	−	0.581740i	$-0.802372\pi$
$787$	−45.6360	−1.62675	−0.813375	+	0.581740i	$0.802372\pi$
$788$	0	0
$789$	25.1718	0.896141
$790$	0	0
$791$	14.1315	0.502458
$792$	0	0
$793$	16.5359	0.587207
$794$	0	0
$795$	−4.14149	−0.146883
$796$	0	0
$797$	−40.4032	−1.43115	−0.715577	−	0.698534i	$-0.753836\pi$
$797$	−40.4032	−1.43115	−0.715577	+	0.698534i	$0.753836\pi$
$798$	0	0
$799$	14.4093	0.509766
$800$	0	0
$801$	6.14296	0.217051
$802$	0	0
$803$	−18.0727	−0.637771
$804$	0	0
$805$	16.1746	0.570079
$806$	0	0
$807$	1.04212	0.0366843
$808$	0	0
$809$	−31.9928	−1.12481	−0.562404	−	0.826863i	$-0.690123\pi$
$809$	−31.9928	−1.12481	−0.562404	+	0.826863i	$0.690123\pi$
$810$	0	0
$811$	−6.49042	−0.227909	−0.113955	−	0.993486i	$-0.536352\pi$
$811$	−6.49042	−0.227909	−0.113955	+	0.993486i	$0.536352\pi$
$812$	0	0
$813$	37.1200	1.30186
$814$	0	0
$815$	−44.2915	−1.55147
$816$	0	0
$817$	4.73153	0.165535
$818$	0	0
$819$	−15.9417	−0.557047
$820$	0	0
$821$	−26.6592	−0.930413	−0.465206	−	0.885202i	$-0.654020\pi$
$821$	−26.6592	−0.930413	−0.465206	+	0.885202i	$0.654020\pi$
$822$	0	0
$823$	−34.6753	−1.20871	−0.604353	−	0.796717i	$-0.706568\pi$
$823$	−34.6753	−1.20871	−0.604353	+	0.796717i	$0.706568\pi$
$824$	0	0
$825$	57.5241	2.00273
$826$	0	0
$827$	29.8765	1.03891	0.519454	−	0.854498i	$-0.326135\pi$
$827$	29.8765	1.03891	0.519454	+	0.854498i	$0.326135\pi$
$828$	0	0
$829$	51.2020	1.77832	0.889160	−	0.457596i	$-0.151289\pi$
$829$	51.2020	1.77832	0.889160	+	0.457596i	$0.151289\pi$
$830$	0	0
$831$	−26.3491	−0.914041
$832$	0	0
$833$	13.4269	0.465213
$834$	0	0
$835$	2.44725	0.0846905
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	36.2954	1.25306	0.626529	−	0.779398i	$-0.284475\pi$
$839$	36.2954	1.25306	0.626529	+	0.779398i	$0.284475\pi$
$840$	0	0
$841$	24.0990	0.831000
$842$	0	0
$843$	21.5311	0.741569
$844$	0	0
$845$	−11.5886	−0.398662
$846$	0	0
$847$	−2.15006	−0.0738769
$848$	0	0
$849$	−79.1455	−2.71627
$850$	0	0
$851$	58.1739	1.99418
$852$	0	0
$853$	28.4978	0.975746	0.487873	−	0.872915i	$-0.337773\pi$
$853$	28.4978	0.975746	0.487873	+	0.872915i	$0.337773\pi$
$854$	0	0
$855$	53.7925	1.83967
$856$	0	0
$857$	30.1834	1.03105	0.515523	−	0.856876i	$-0.327598\pi$
$857$	30.1834	1.03105	0.515523	+	0.856876i	$0.327598\pi$
$858$	0	0
$859$	21.5967	0.736868	0.368434	−	0.929654i	$-0.379894\pi$
$859$	21.5967	0.736868	0.368434	+	0.929654i	$0.379894\pi$
$860$	0	0
$861$	−8.44788	−0.287903
$862$	0	0
$863$	−33.6905	−1.14684	−0.573418	−	0.819263i	$-0.694383\pi$
$863$	−33.6905	−1.14684	−0.573418	+	0.819263i	$0.694383\pi$
$864$	0	0
$865$	18.4409	0.627011
$866$	0	0
$867$	−33.7500	−1.14621
$868$	0	0
$869$	56.8743	1.92933
$870$	0	0
$871$	25.3893	0.860285
$872$	0	0
$873$	56.3850	1.90834
$874$	0	0
$875$	2.13426	0.0721510
$876$	0	0
$877$	7.29869	0.246459	0.123230	−	0.992378i	$-0.460675\pi$
$877$	7.29869	0.246459	0.123230	+	0.992378i	$0.460675\pi$
$878$	0	0
$879$	−6.62560	−0.223476
$880$	0	0
$881$	−50.6833	−1.70756	−0.853782	−	0.520631i	$-0.825697\pi$
$881$	−50.6833	−1.70756	−0.853782	+	0.520631i	$0.825697\pi$
$882$	0	0
$883$	47.0153	1.58219	0.791095	−	0.611693i	$-0.209511\pi$
$883$	47.0153	1.58219	0.791095	+	0.611693i	$0.209511\pi$
$884$	0	0
$885$	−114.321	−3.84285
$886$	0	0
$887$	25.9258	0.870502	0.435251	−	0.900309i	$-0.356660\pi$
$887$	25.9258	0.870502	0.435251	+	0.900309i	$0.356660\pi$
$888$	0	0
$889$	15.4700	0.518848
$890$	0	0
$891$	−9.03633	−0.302729
$892$	0	0
$893$	24.5770	0.822437
$894$	0	0
$895$	60.2314	2.01331
$896$	0	0
$897$	62.4334	2.08459
$898$	0	0
$899$	−7.28691	−0.243032
$900$	0	0
$901$	−0.997023	−0.0332156
$902$	0	0
$903$	3.08760	0.102749
$904$	0	0
$905$	65.7208	2.18463
$906$	0	0
$907$	−28.1132	−0.933483	−0.466742	−	0.884394i	$-0.654572\pi$
$907$	−28.1132	−0.933483	−0.466742	+	0.884394i	$0.654572\pi$
$908$	0	0
$909$	−31.9786	−1.06066
$910$	0	0
$911$	−15.9013	−0.526832	−0.263416	−	0.964682i	$-0.584849\pi$
$911$	−15.9013	−0.526832	−0.263416	+	0.964682i	$0.584849\pi$
$912$	0	0
$913$	60.8701	2.01451
$914$	0	0
$915$	−36.4113	−1.20372
$916$	0	0
$917$	14.5121	0.479233
$918$	0	0
$919$	39.1422	1.29118	0.645590	−	0.763684i	$-0.276611\pi$
$919$	39.1422	1.29118	0.645590	+	0.763684i	$0.276611\pi$
$920$	0	0
$921$	18.6736	0.615316
$922$	0	0
$923$	−5.82736	−0.191810
$924$	0	0
$925$	59.4349	1.95421
$926$	0	0
$927$	−50.4592	−1.65730
$928$	0	0
$929$	−7.98651	−0.262029	−0.131014	−	0.991380i	$-0.541823\pi$
$929$	−7.98651	−0.262029	−0.131014	+	0.991380i	$0.541823\pi$
$930$	0	0
$931$	22.9012	0.750557
$932$	0	0
$933$	−30.3043	−0.992118
$934$	0	0
$935$	25.9082	0.847289
$936$	0	0
$937$	34.7086	1.13388	0.566940	−	0.823759i	$-0.308127\pi$
$937$	34.7086	1.13388	0.566940	+	0.823759i	$0.308127\pi$
$938$	0	0
$939$	−18.2729	−0.596314
$940$	0	0
$941$	−53.8431	−1.75523	−0.877617	−	0.479363i	$-0.840868\pi$
$941$	−53.8431	−1.75523	−0.877617	+	0.479363i	$0.840868\pi$
$942$	0	0
$943$	19.7873	0.644364
$944$	0	0
$945$	11.5127	0.374509
$946$	0	0
$947$	−47.5710	−1.54585	−0.772924	−	0.634498i	$-0.781207\pi$
$947$	−47.5710	−1.54585	−0.772924	+	0.634498i	$0.781207\pi$
$948$	0	0
$949$	−20.0402	−0.650533
$950$	0	0
$951$	−9.36890	−0.303807
$952$	0	0
$953$	−30.1998	−0.978269	−0.489134	−	0.872209i	$-0.662687\pi$
$953$	−30.1998	−0.978269	−0.489134	+	0.872209i	$0.662687\pi$
$954$	0	0
$955$	−13.3709	−0.432673
$956$	0	0
$957$	73.0072	2.35999
$958$	0	0
$959$	−19.9179	−0.643183
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	65.9938	2.12662
$964$	0	0
$965$	69.3722	2.23317
$966$	0	0
$967$	−58.6042	−1.88458	−0.942291	−	0.334794i	$-0.891333\pi$
$967$	−58.6042	−1.88458	−0.942291	+	0.334794i	$0.891333\pi$
$968$	0	0
$969$	21.6528	0.695588
$970$	0	0
$971$	31.8164	1.02104	0.510518	−	0.859867i	$-0.329454\pi$
$971$	31.8164	1.02104	0.510518	+	0.859867i	$0.329454\pi$
$972$	0	0
$973$	−6.80441	−0.218139
$974$	0	0
$975$	63.7867	2.04281
$976$	0	0
$977$	−36.3440	−1.16275	−0.581373	−	0.813637i	$-0.697484\pi$
$977$	−36.3440	−1.16275	−0.581373	+	0.813637i	$0.697484\pi$
$978$	0	0
$979$	5.04635	0.161282
$980$	0	0
$981$	−89.7055	−2.86408
$982$	0	0
$983$	−27.9326	−0.890912	−0.445456	−	0.895304i	$-0.646958\pi$
$983$	−27.9326	−0.890912	−0.445456	+	0.895304i	$0.646958\pi$
$984$	0	0
$985$	28.9250	0.921629
$986$	0	0
$987$	16.0379	0.510492
$988$	0	0
$989$	−7.23204	−0.229965
$990$	0	0
$991$	−0.464524	−0.0147561	−0.00737805	−	0.999973i	$-0.502349\pi$
$991$	−0.464524	−0.0147561	−0.00737805	+	0.999973i	$0.502349\pi$
$992$	0	0
$993$	21.1687	0.671768
$994$	0	0
$995$	26.8661	0.851712
$996$	0	0
$997$	−21.4503	−0.679338	−0.339669	−	0.940545i	$-0.610315\pi$
$997$	−21.4503	−0.679338	−0.339669	+	0.940545i	$0.610315\pi$
$998$	0	0
$999$	41.4071	1.31006

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} -3.27743 q^{5} -0.878184 q^{7} +4.46410 q^{9} +3.66719 q^{11} +4.06644 q^{13} -8.95410 q^{15} -2.15561 q^{17} -3.67667 q^{19} -2.39924 q^{21} +5.61971 q^{23} +5.74153 q^{25} +4.00000 q^{27} +7.28691 q^{29} -1.00000 q^{31} +10.0190 q^{33} +2.87818 q^{35} +10.3518 q^{37} +11.1097 q^{39} +3.52106 q^{41} -1.28691 q^{43} -14.6308 q^{45} -6.68457 q^{47} -6.22879 q^{49} -5.88924 q^{51} +0.462524 q^{53} -12.0190 q^{55} -10.0449 q^{57} +12.7674 q^{59} +4.06644 q^{61} -3.92030 q^{63} -13.3274 q^{65} +6.24363 q^{67} +15.3533 q^{69} -1.43304 q^{71} -4.92820 q^{73} +15.6861 q^{75} -3.22047 q^{77} +15.5090 q^{79} -2.46410 q^{81} +16.5985 q^{83} +7.06486 q^{85} +19.9082 q^{87} +1.37608 q^{89} -3.57108 q^{91} -2.73205 q^{93} +12.0500 q^{95} +12.6308 q^{97} +16.3707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{5} - 4 q^{7} + 4 q^{9} + 10 q^{11} - 2 q^{15} + 2 q^{17} + 8 q^{19} + 2 q^{21} - 2 q^{23} - 2 q^{25} + 16 q^{27} - 4 q^{31} + 4 q^{33} + 12 q^{35} + 10 q^{37} + 2 q^{41} + 24 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 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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Representations

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Database

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