LMFDB - Embedded newform 1984.2.a.bb.1.6

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 = [6,0,2,0,-2,0,2,0,10,0,10,0,-10,0,8,0,4,0,2] 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:	$6$
Coefficient field:	6.6.66862976.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 20x^{2} - 16x - 15 $ x^6 - 2x^5 - 8x^4 + 12x^3 + 20x^2 - 16*x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 992)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.71608$ 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)

$f(q)$	$=$	$q+3.07118 q^{3} -1.24104 q^{5} +3.86022 q^{7} +6.43216 q^{9} +2.58173 q^{11} +1.07118 q^{13} -3.81147 q^{15} +5.81147 q^{17} -2.07362 q^{19} +11.8554 q^{21} -8.53343 q^{23} -3.45981 q^{25} +10.5408 q^{27} -2.25084 q^{29} +1.00000 q^{31} +7.92897 q^{33} -4.79070 q^{35} -5.65814 q^{37} +3.28980 q^{39} +3.02765 q^{41} -12.0941 q^{43} -7.98259 q^{45} +3.19229 q^{47} +7.90132 q^{49} +17.8481 q^{51} -0.183308 q^{53} -3.20404 q^{55} -6.36845 q^{57} +2.64834 q^{59} +1.39471 q^{61} +24.8296 q^{63} -1.32938 q^{65} -8.38223 q^{67} -26.2077 q^{69} +11.0525 q^{71} -15.5084 q^{73} -10.6257 q^{75} +9.96606 q^{77} +16.1937 q^{79} +13.0762 q^{81} -11.7666 q^{83} -7.21229 q^{85} -6.91274 q^{87} -1.84881 q^{89} +4.13500 q^{91} +3.07118 q^{93} +2.57345 q^{95} +19.5571 q^{97} +16.6061 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 10 q^{9} + 10 q^{11} - 10 q^{13} + 8 q^{15} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} + 20 q^{25} + 8 q^{27} - 10 q^{29} + 6 q^{31} + 8 q^{33} + 14 q^{35} - 26 q^{37}+ \cdots + 46 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 10 * q^9 + 10 * q^11 - 10 * q^13 + 8 * q^15 + 4 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 + 20 * q^25 + 8 * q^27 - 10 * q^29 + 6 * q^31 + 8 * q^33 + 14 * q^35 - 26 * q^37 + 24 * q^39 + 6 * q^41 + 10 * q^43 - 22 * q^45 + 4 * q^47 + 20 * q^49 + 32 * q^51 - 18 * q^53 + 16 * q^55 + 24 * q^57 + 6 * q^59 - 10 * q^61 - 2 * q^63 + 12 * q^65 - 4 * q^67 + 4 * q^69 + 30 * q^71 + 4 * q^73 - 14 * q^75 + 20 * q^79 + 10 * q^81 + 30 * q^83 + 16 * q^87 - 20 * q^89 - 8 * q^91 + 2 * q^93 + 26 * q^95 + 30 * q^97 + 46 * 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
$3$	3.07118	1.77315	0.886574	−	0.462587i	$-0.153079\pi$
$3$	3.07118	1.77315	0.886574	+	0.462587i	$0.153079\pi$
$4$	0	0
$5$	−1.24104	−0.555012	−0.277506	−	0.960724i	$-0.589508\pi$
$5$	−1.24104	−0.555012	−0.277506	+	0.960724i	$0.589508\pi$
$6$	0	0
$7$	3.86022	1.45903	0.729514	−	0.683966i	$-0.239747\pi$
$7$	3.86022	1.45903	0.729514	+	0.683966i	$0.239747\pi$
$8$	0	0
$9$	6.43216	2.14405
$10$	0	0
$11$	2.58173	0.778422	0.389211	−	0.921149i	$-0.372748\pi$
$11$	2.58173	0.778422	0.389211	+	0.921149i	$0.372748\pi$
$12$	0	0
$13$	1.07118	0.297092	0.148546	−	0.988905i	$-0.452541\pi$
$13$	1.07118	0.297092	0.148546	+	0.988905i	$0.452541\pi$
$14$	0	0
$15$	−3.81147	−0.984118
$16$	0	0
$17$	5.81147	1.40949	0.704744	−	0.709461i	$-0.251062\pi$
$17$	5.81147	1.40949	0.704744	+	0.709461i	$0.251062\pi$
$18$	0	0
$19$	−2.07362	−0.475720	−0.237860	−	0.971299i	$-0.576446\pi$
$19$	−2.07362	−0.475720	−0.237860	+	0.971299i	$0.576446\pi$
$20$	0	0
$21$	11.8554	2.58707
$22$	0	0
$23$	−8.53343	−1.77934	−0.889671	−	0.456602i	$-0.849066\pi$
$23$	−8.53343	−1.77934	−0.889671	+	0.456602i	$0.849066\pi$
$24$	0	0
$25$	−3.45981	−0.691962
$26$	0	0
$27$	10.5408	2.02858
$28$	0	0
$29$	−2.25084	−0.417970	−0.208985	−	0.977919i	$-0.567016\pi$
$29$	−2.25084	−0.417970	−0.208985	+	0.977919i	$0.567016\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	7.92897	1.38026
$34$	0	0
$35$	−4.79070	−0.809777
$36$	0	0
$37$	−5.65814	−0.930192	−0.465096	−	0.885260i	$-0.653980\pi$
$37$	−5.65814	−0.930192	−0.465096	+	0.885260i	$0.653980\pi$
$38$	0	0
$39$	3.28980	0.526789
$40$	0	0
$41$	3.02765	0.472840	0.236420	−	0.971651i	$-0.424026\pi$
$41$	3.02765	0.472840	0.236420	+	0.971651i	$0.424026\pi$
$42$	0	0
$43$	−12.0941	−1.84433	−0.922163	−	0.386801i	$-0.873580\pi$
$43$	−12.0941	−1.84433	−0.922163	+	0.386801i	$0.873580\pi$
$44$	0	0
$45$	−7.98259	−1.18997
$46$	0	0
$47$	3.19229	0.465644	0.232822	−	0.972519i	$-0.425204\pi$
$47$	3.19229	0.465644	0.232822	+	0.972519i	$0.425204\pi$
$48$	0	0
$49$	7.90132	1.12876
$50$	0	0
$51$	17.8481	2.49923
$52$	0	0
$53$	−0.183308	−0.0251793	−0.0125897	−	0.999921i	$-0.504008\pi$
$53$	−0.183308	−0.0251793	−0.0125897	+	0.999921i	$0.504008\pi$
$54$	0	0
$55$	−3.20404	−0.432033
$56$	0	0
$57$	−6.36845	−0.843522
$58$	0	0
$59$	2.64834	0.344785	0.172392	−	0.985028i	$-0.444850\pi$
$59$	2.64834	0.344785	0.172392	+	0.985028i	$0.444850\pi$
$60$	0	0
$61$	1.39471	0.178575	0.0892874	−	0.996006i	$-0.471541\pi$
$61$	1.39471	0.178575	0.0892874	+	0.996006i	$0.471541\pi$
$62$	0	0
$63$	24.8296	3.12823
$64$	0	0
$65$	−1.32938	−0.164890
$66$	0	0
$67$	−8.38223	−1.02405	−0.512026	−	0.858970i	$-0.671105\pi$
$67$	−8.38223	−1.02405	−0.512026	+	0.858970i	$0.671105\pi$
$68$	0	0
$69$	−26.2077	−3.15504
$70$	0	0
$71$	11.0525	1.31169	0.655846	−	0.754895i	$-0.272312\pi$
$71$	11.0525	1.31169	0.655846	+	0.754895i	$0.272312\pi$
$72$	0	0
$73$	−15.5084	−1.81512	−0.907558	−	0.419926i	$-0.862056\pi$
$73$	−15.5084	−1.81512	−0.907558	+	0.419926i	$0.862056\pi$
$74$	0	0
$75$	−10.6257	−1.22695
$76$	0	0
$77$	9.96606	1.13574
$78$	0	0
$79$	16.1937	1.82193	0.910967	−	0.412479i	$-0.135337\pi$
$79$	16.1937	1.82193	0.910967	+	0.412479i	$0.135337\pi$
$80$	0	0
$81$	13.0762	1.45291
$82$	0	0
$83$	−11.7666	−1.29155	−0.645774	−	0.763529i	$-0.723465\pi$
$83$	−11.7666	−1.29155	−0.645774	+	0.763529i	$0.723465\pi$
$84$	0	0
$85$	−7.21229	−0.782282
$86$	0	0
$87$	−6.91274	−0.741123
$88$	0	0
$89$	−1.84881	−0.195973	−0.0979866	−	0.995188i	$-0.531240\pi$
$89$	−1.84881	−0.195973	−0.0979866	+	0.995188i	$0.531240\pi$
$90$	0	0
$91$	4.13500	0.433466
$92$	0	0
$93$	3.07118	0.318467
$94$	0	0
$95$	2.57345	0.264030
$96$	0	0
$97$	19.5571	1.98572	0.992862	−	0.119268i	$-0.0380548\pi$
$97$	19.5571	1.98572	0.992862	+	0.119268i	$0.0380548\pi$
$98$	0	0
$99$	16.6061	1.66898
$100$	0	0
$101$	10.6997	1.06466	0.532329	−	0.846538i	$-0.321317\pi$
$101$	10.6997	1.06466	0.532329	+	0.846538i	$0.321317\pi$
$102$	0	0
$103$	11.7143	1.15424	0.577122	−	0.816658i	$-0.304176\pi$
$103$	11.7143	1.15424	0.577122	+	0.816658i	$0.304176\pi$
$104$	0	0
$105$	−14.7131	−1.43585
$106$	0	0
$107$	6.14851	0.594399	0.297199	−	0.954815i	$-0.403947\pi$
$107$	6.14851	0.594399	0.297199	+	0.954815i	$0.403947\pi$
$108$	0	0
$109$	−2.81912	−0.270023	−0.135012	−	0.990844i	$-0.543107\pi$
$109$	−2.81912	−0.270023	−0.135012	+	0.990844i	$0.543107\pi$
$110$	0	0
$111$	−17.3772	−1.64937
$112$	0	0
$113$	7.17488	0.674956	0.337478	−	0.941333i	$-0.390426\pi$
$113$	7.17488	0.674956	0.337478	+	0.941333i	$0.390426\pi$
$114$	0	0
$115$	10.5904	0.987556
$116$	0	0
$117$	6.89002	0.636982
$118$	0	0
$119$	22.4336	2.05648
$120$	0	0
$121$	−4.33466	−0.394060
$122$	0	0
$123$	9.29847	0.838415
$124$	0	0
$125$	10.4990	0.939059
$126$	0	0
$127$	−17.1042	−1.51775	−0.758876	−	0.651236i	$-0.774251\pi$
$127$	−17.1042	−1.51775	−0.758876	+	0.651236i	$0.774251\pi$
$128$	0	0
$129$	−37.1431	−3.27026
$130$	0	0
$131$	19.1814	1.67589	0.837945	−	0.545755i	$-0.183757\pi$
$131$	19.1814	1.67589	0.837945	+	0.545755i	$0.183757\pi$
$132$	0	0
$133$	−8.00462	−0.694088
$134$	0	0
$135$	−13.0816	−1.12588
$136$	0	0
$137$	−11.8789	−1.01489	−0.507443	−	0.861685i	$-0.669409\pi$
$137$	−11.8789	−1.01489	−0.507443	+	0.861685i	$0.669409\pi$
$138$	0	0
$139$	−6.47848	−0.549497	−0.274749	−	0.961516i	$-0.588595\pi$
$139$	−6.47848	−0.549497	−0.274749	+	0.961516i	$0.588595\pi$
$140$	0	0
$141$	9.80411	0.825655
$142$	0	0
$143$	2.76551	0.231263
$144$	0	0
$145$	2.79339	0.231978
$146$	0	0
$147$	24.2664	2.00146
$148$	0	0
$149$	−19.3889	−1.58840	−0.794201	−	0.607655i	$-0.792110\pi$
$149$	−19.3889	−1.58840	−0.794201	+	0.607655i	$0.792110\pi$
$150$	0	0
$151$	3.46747	0.282178	0.141089	−	0.989997i	$-0.454940\pi$
$151$	3.46747	0.282178	0.141089	+	0.989997i	$0.454940\pi$
$152$	0	0
$153$	37.3803	3.02202
$154$	0	0
$155$	−1.24104	−0.0996830
$156$	0	0
$157$	−14.0275	−1.11951	−0.559756	−	0.828657i	$-0.689105\pi$
$157$	−14.0275	−1.11951	−0.559756	+	0.828657i	$0.689105\pi$
$158$	0	0
$159$	−0.562973	−0.0446467
$160$	0	0
$161$	−32.9409	−2.59611
$162$	0	0
$163$	−6.97762	−0.546529	−0.273265	−	0.961939i	$-0.588104\pi$
$163$	−6.97762	−0.546529	−0.273265	+	0.961939i	$0.588104\pi$
$164$	0	0
$165$	−9.84020	−0.766058
$166$	0	0
$167$	11.6082	0.898271	0.449135	−	0.893464i	$-0.351732\pi$
$167$	11.6082	0.898271	0.449135	+	0.893464i	$0.351732\pi$
$168$	0	0
$169$	−11.8526	−0.911736
$170$	0	0
$171$	−13.3378	−1.01997
$172$	0	0
$173$	−7.78792	−0.592104	−0.296052	−	0.955172i	$-0.595670\pi$
$173$	−7.78792	−0.592104	−0.296052	+	0.955172i	$0.595670\pi$
$174$	0	0
$175$	−13.3556	−1.00959
$176$	0	0
$177$	8.13354	0.611354
$178$	0	0
$179$	20.3201	1.51879	0.759397	−	0.650627i	$-0.225494\pi$
$179$	20.3201	1.51879	0.759397	+	0.650627i	$0.225494\pi$
$180$	0	0
$181$	4.70457	0.349688	0.174844	−	0.984596i	$-0.444058\pi$
$181$	4.70457	0.349688	0.174844	+	0.984596i	$0.444058\pi$
$182$	0	0
$183$	4.28342	0.316639
$184$	0	0
$185$	7.02199	0.516267
$186$	0	0
$187$	15.0037	1.09718
$188$	0	0
$189$	40.6898	2.95975
$190$	0	0
$191$	−13.1091	−0.948543	−0.474272	−	0.880379i	$-0.657288\pi$
$191$	−13.1091	−0.948543	−0.474272	+	0.880379i	$0.657288\pi$
$192$	0	0
$193$	5.75745	0.414430	0.207215	−	0.978295i	$-0.433560\pi$
$193$	5.75745	0.414430	0.207215	+	0.978295i	$0.433560\pi$
$194$	0	0
$195$	−4.08278	−0.292374
$196$	0	0
$197$	−23.8658	−1.70037	−0.850185	−	0.526483i	$-0.823510\pi$
$197$	−23.8658	−1.70037	−0.850185	+	0.526483i	$0.823510\pi$
$198$	0	0
$199$	−5.56764	−0.394680	−0.197340	−	0.980335i	$-0.563230\pi$
$199$	−5.56764	−0.394680	−0.197340	+	0.980335i	$0.563230\pi$
$200$	0	0
$201$	−25.7434	−1.81580
$202$	0	0
$203$	−8.68874	−0.609830
$204$	0	0
$205$	−3.75745	−0.262431
$206$	0	0
$207$	−54.8884	−3.81501
$208$	0	0
$209$	−5.35352	−0.370311
$210$	0	0
$211$	−18.2617	−1.25719	−0.628595	−	0.777733i	$-0.716370\pi$
$211$	−18.2617	−1.25719	−0.628595	+	0.777733i	$0.716370\pi$
$212$	0	0
$213$	33.9443	2.32582
$214$	0	0
$215$	15.0093	1.02362
$216$	0	0
$217$	3.86022	0.262049
$218$	0	0
$219$	−47.6290	−3.21847
$220$	0	0
$221$	6.22514	0.418748
$222$	0	0
$223$	−16.0602	−1.07547	−0.537734	−	0.843114i	$-0.680720\pi$
$223$	−16.0602	−1.07547	−0.537734	+	0.843114i	$0.680720\pi$
$224$	0	0
$225$	−22.2541	−1.48360
$226$	0	0
$227$	−15.9034	−1.05554	−0.527772	−	0.849386i	$-0.676973\pi$
$227$	−15.9034	−1.05554	−0.527772	+	0.849386i	$0.676973\pi$
$228$	0	0
$229$	6.62659	0.437898	0.218949	−	0.975736i	$-0.429737\pi$
$229$	6.62659	0.437898	0.218949	+	0.975736i	$0.429737\pi$
$230$	0	0
$231$	30.6076	2.01383
$232$	0	0
$233$	−17.6970	−1.15937	−0.579683	−	0.814842i	$-0.696824\pi$
$233$	−17.6970	−1.15937	−0.579683	+	0.814842i	$0.696824\pi$
$234$	0	0
$235$	−3.96177	−0.258438
$236$	0	0
$237$	49.7338	3.23056
$238$	0	0
$239$	6.72034	0.434703	0.217351	−	0.976093i	$-0.430258\pi$
$239$	6.72034	0.434703	0.217351	+	0.976093i	$0.430258\pi$
$240$	0	0
$241$	−8.12764	−0.523547	−0.261774	−	0.965129i	$-0.584307\pi$
$241$	−8.12764	−0.523547	−0.261774	+	0.965129i	$0.584307\pi$
$242$	0	0
$243$	8.53703	0.547651
$244$	0	0
$245$	−9.80588	−0.626475
$246$	0	0
$247$	−2.22122	−0.141333
$248$	0	0
$249$	−36.1373	−2.29011
$250$	0	0
$251$	17.2575	1.08929	0.544643	−	0.838668i	$-0.316665\pi$
$251$	17.2575	1.08929	0.544643	+	0.838668i	$0.316665\pi$
$252$	0	0
$253$	−22.0310	−1.38508
$254$	0	0
$255$	−22.1503	−1.38710
$256$	0	0
$257$	16.4120	1.02375	0.511877	−	0.859059i	$-0.328950\pi$
$257$	16.4120	1.02375	0.511877	+	0.859059i	$0.328950\pi$
$258$	0	0
$259$	−21.8417	−1.35718
$260$	0	0
$261$	−14.4778	−0.896151
$262$	0	0
$263$	8.23826	0.507993	0.253996	−	0.967205i	$-0.418255\pi$
$263$	8.23826	0.507993	0.253996	+	0.967205i	$0.418255\pi$
$264$	0	0
$265$	0.227494	0.0139748
$266$	0	0
$267$	−5.67802	−0.347489
$268$	0	0
$269$	23.5850	1.43800	0.719002	−	0.695008i	$-0.244599\pi$
$269$	23.5850	1.43800	0.719002	+	0.695008i	$0.244599\pi$
$270$	0	0
$271$	11.0580	0.671727	0.335864	−	0.941911i	$-0.390972\pi$
$271$	11.0580	0.671727	0.335864	+	0.941911i	$0.390972\pi$
$272$	0	0
$273$	12.6993	0.768599
$274$	0	0
$275$	−8.93231	−0.538638
$276$	0	0
$277$	−27.1690	−1.63243	−0.816213	−	0.577751i	$-0.803931\pi$
$277$	−27.1690	−1.63243	−0.816213	+	0.577751i	$0.803931\pi$
$278$	0	0
$279$	6.43216	0.385083
$280$	0	0
$281$	18.8512	1.12457	0.562284	−	0.826944i	$-0.309923\pi$
$281$	18.8512	1.12457	0.562284	+	0.826944i	$0.309923\pi$
$282$	0	0
$283$	−1.78703	−0.106228	−0.0531138	−	0.998588i	$-0.516915\pi$
$283$	−1.78703	−0.106228	−0.0531138	+	0.998588i	$0.516915\pi$
$284$	0	0
$285$	7.90352	0.468164
$286$	0	0
$287$	11.6874	0.689886
$288$	0	0
$289$	16.7732	0.986658
$290$	0	0
$291$	60.0635	3.52098
$292$	0	0
$293$	20.0579	1.17180	0.585898	−	0.810385i	$-0.300742\pi$
$293$	20.0579	1.17180	0.585898	+	0.810385i	$0.300742\pi$
$294$	0	0
$295$	−3.28671	−0.191359
$296$	0	0
$297$	27.2135	1.57909
$298$	0	0
$299$	−9.14085	−0.528629
$300$	0	0
$301$	−46.6858	−2.69092
$302$	0	0
$303$	32.8607	1.88780
$304$	0	0
$305$	−1.73090	−0.0991111
$306$	0	0
$307$	15.8389	0.903975	0.451987	−	0.892024i	$-0.350715\pi$
$307$	15.8389	0.903975	0.451987	+	0.892024i	$0.350715\pi$
$308$	0	0
$309$	35.9768	2.04665
$310$	0	0
$311$	−6.87046	−0.389588	−0.194794	−	0.980844i	$-0.562404\pi$
$311$	−6.87046	−0.389588	−0.194794	+	0.980844i	$0.562404\pi$
$312$	0	0
$313$	−4.93525	−0.278957	−0.139478	−	0.990225i	$-0.544543\pi$
$313$	−4.93525	−0.278957	−0.139478	+	0.990225i	$0.544543\pi$
$314$	0	0
$315$	−30.8146	−1.73620
$316$	0	0
$317$	−0.841152	−0.0472438	−0.0236219	−	0.999721i	$-0.507520\pi$
$317$	−0.841152	−0.0472438	−0.0236219	+	0.999721i	$0.507520\pi$
$318$	0	0
$319$	−5.81107	−0.325357
$320$	0	0
$321$	18.8832	1.05396
$322$	0	0
$323$	−12.0508	−0.670522
$324$	0	0
$325$	−3.70609	−0.205577
$326$	0	0
$327$	−8.65805	−0.478791
$328$	0	0
$329$	12.3230	0.679387
$330$	0	0
$331$	−2.51766	−0.138383	−0.0691916	−	0.997603i	$-0.522042\pi$
$331$	−2.51766	−0.138383	−0.0691916	+	0.997603i	$0.522042\pi$
$332$	0	0
$333$	−36.3940	−1.99438
$334$	0	0
$335$	10.4027	0.568361
$336$	0	0
$337$	6.56825	0.357795	0.178898	−	0.983868i	$-0.442747\pi$
$337$	6.56825	0.357795	0.178898	+	0.983868i	$0.442747\pi$
$338$	0	0
$339$	22.0354	1.19680
$340$	0	0
$341$	2.58173	0.139809
$342$	0	0
$343$	3.47930	0.187865
$344$	0	0
$345$	32.5249	1.75108
$346$	0	0
$347$	3.25323	0.174643	0.0873213	−	0.996180i	$-0.472169\pi$
$347$	3.25323	0.174643	0.0873213	+	0.996180i	$0.472169\pi$
$348$	0	0
$349$	32.1973	1.72348	0.861740	−	0.507350i	$-0.169375\pi$
$349$	32.1973	1.72348	0.861740	+	0.507350i	$0.169375\pi$
$350$	0	0
$351$	11.2911	0.602675
$352$	0	0
$353$	−24.8313	−1.32163	−0.660817	−	0.750547i	$-0.729790\pi$
$353$	−24.8313	−1.32163	−0.660817	+	0.750547i	$0.729790\pi$
$354$	0	0
$355$	−13.7167	−0.728004
$356$	0	0
$357$	68.8976	3.64645
$358$	0	0
$359$	−5.90760	−0.311791	−0.155896	−	0.987774i	$-0.549826\pi$
$359$	−5.90760	−0.311791	−0.155896	+	0.987774i	$0.549826\pi$
$360$	0	0
$361$	−14.7001	−0.773690
$362$	0	0
$363$	−13.3125	−0.698726
$364$	0	0
$365$	19.2466	1.00741
$366$	0	0
$367$	−7.39653	−0.386096	−0.193048	−	0.981189i	$-0.561837\pi$
$367$	−7.39653	−0.386096	−0.193048	+	0.981189i	$0.561837\pi$
$368$	0	0
$369$	19.4743	1.01379
$370$	0	0
$371$	−0.707611	−0.0367373
$372$	0	0
$373$	3.33886	0.172879	0.0864397	−	0.996257i	$-0.472451\pi$
$373$	3.33886	0.172879	0.0864397	+	0.996257i	$0.472451\pi$
$374$	0	0
$375$	32.2443	1.66509
$376$	0	0
$377$	−2.41106	−0.124176
$378$	0	0
$379$	17.0888	0.877792	0.438896	−	0.898538i	$-0.355370\pi$
$379$	17.0888	0.877792	0.438896	+	0.898538i	$0.355370\pi$
$380$	0	0
$381$	−52.5301	−2.69120
$382$	0	0
$383$	2.40719	0.123002	0.0615009	−	0.998107i	$-0.480411\pi$
$383$	2.40719	0.123002	0.0615009	+	0.998107i	$0.480411\pi$
$384$	0	0
$385$	−12.3683	−0.630348
$386$	0	0
$387$	−77.7909	−3.95433
$388$	0	0
$389$	0.301128	0.0152678	0.00763391	−	0.999971i	$-0.497570\pi$
$389$	0.301128	0.0152678	0.00763391	+	0.999971i	$0.497570\pi$
$390$	0	0
$391$	−49.5918	−2.50796
$392$	0	0
$393$	58.9097	2.97160
$394$	0	0
$395$	−20.0971	−1.01119
$396$	0	0
$397$	4.12009	0.206781	0.103391	−	0.994641i	$-0.467031\pi$
$397$	4.12009	0.206781	0.103391	+	0.994641i	$0.467031\pi$
$398$	0	0
$399$	−24.5836	−1.23072
$400$	0	0
$401$	−19.5132	−0.974444	−0.487222	−	0.873278i	$-0.661990\pi$
$401$	−19.5132	−0.974444	−0.487222	+	0.873278i	$0.661990\pi$
$402$	0	0
$403$	1.07118	0.0533594
$404$	0	0
$405$	−16.2281	−0.806383
$406$	0	0
$407$	−14.6078	−0.724082
$408$	0	0
$409$	9.50502	0.469993	0.234997	−	0.971996i	$-0.424492\pi$
$409$	9.50502	0.469993	0.234997	+	0.971996i	$0.424492\pi$
$410$	0	0
$411$	−36.4824	−1.79954
$412$	0	0
$413$	10.2232	0.503050
$414$	0	0
$415$	14.6028	0.716824
$416$	0	0
$417$	−19.8966	−0.974340
$418$	0	0
$419$	27.1448	1.32611	0.663054	−	0.748572i	$-0.269260\pi$
$419$	27.1448	1.32611	0.663054	+	0.748572i	$0.269260\pi$
$420$	0	0
$421$	−6.07271	−0.295966	−0.147983	−	0.988990i	$-0.547278\pi$
$421$	−6.07271	−0.295966	−0.147983	+	0.988990i	$0.547278\pi$
$422$	0	0
$423$	20.5333	0.998365
$424$	0	0
$425$	−20.1066	−0.975313
$426$	0	0
$427$	5.38390	0.260545
$428$	0	0
$429$	8.49337	0.410064
$430$	0	0
$431$	16.9549	0.816691	0.408345	−	0.912828i	$-0.366106\pi$
$431$	16.9549	0.816691	0.408345	+	0.912828i	$0.366106\pi$
$432$	0	0
$433$	8.45034	0.406097	0.203049	−	0.979169i	$-0.434915\pi$
$433$	8.45034	0.406097	0.203049	+	0.979169i	$0.434915\pi$
$434$	0	0
$435$	8.57901	0.411332
$436$	0	0
$437$	17.6950	0.846469
$438$	0	0
$439$	2.77165	0.132284	0.0661418	−	0.997810i	$-0.478931\pi$
$439$	2.77165	0.132284	0.0661418	+	0.997810i	$0.478931\pi$
$440$	0	0
$441$	50.8226	2.42012
$442$	0	0
$443$	3.11764	0.148123	0.0740617	−	0.997254i	$-0.476404\pi$
$443$	3.11764	0.148123	0.0740617	+	0.997254i	$0.476404\pi$
$444$	0	0
$445$	2.29445	0.108767
$446$	0	0
$447$	−59.5469	−2.81647
$448$	0	0
$449$	−30.7717	−1.45221	−0.726103	−	0.687586i	$-0.758670\pi$
$449$	−30.7717	−1.45221	−0.726103	+	0.687586i	$0.758670\pi$
$450$	0	0
$451$	7.81659	0.368069
$452$	0	0
$453$	10.6492	0.500344
$454$	0	0
$455$	−5.13172	−0.240579
$456$	0	0
$457$	35.4681	1.65913	0.829564	−	0.558412i	$-0.188589\pi$
$457$	35.4681	1.65913	0.829564	+	0.558412i	$0.188589\pi$
$458$	0	0
$459$	61.2575	2.85925
$460$	0	0
$461$	−32.3513	−1.50675	−0.753376	−	0.657590i	$-0.771576\pi$
$461$	−32.3513	−1.50675	−0.753376	+	0.657590i	$0.771576\pi$
$462$	0	0
$463$	11.9407	0.554932	0.277466	−	0.960735i	$-0.410505\pi$
$463$	11.9407	0.554932	0.277466	+	0.960735i	$0.410505\pi$
$464$	0	0
$465$	−3.81147	−0.176753
$466$	0	0
$467$	25.5918	1.18425	0.592124	−	0.805847i	$-0.298289\pi$
$467$	25.5918	1.18425	0.592124	+	0.805847i	$0.298289\pi$
$468$	0	0
$469$	−32.3573	−1.49412
$470$	0	0
$471$	−43.0809	−1.98506
$472$	0	0
$473$	−31.2236	−1.43566
$474$	0	0
$475$	7.17432	0.329180
$476$	0	0
$477$	−1.17907	−0.0539858
$478$	0	0
$479$	−20.5861	−0.940605	−0.470303	−	0.882505i	$-0.655855\pi$
$479$	−20.5861	−0.940605	−0.470303	+	0.882505i	$0.655855\pi$
$480$	0	0
$481$	−6.06089	−0.276353
$482$	0	0
$483$	−101.168	−4.60328
$484$	0	0
$485$	−24.2712	−1.10210
$486$	0	0
$487$	14.7317	0.667556	0.333778	−	0.942652i	$-0.391676\pi$
$487$	14.7317	0.667556	0.333778	+	0.942652i	$0.391676\pi$
$488$	0	0
$489$	−21.4295	−0.969078
$490$	0	0
$491$	37.3846	1.68715	0.843573	−	0.537015i	$-0.180448\pi$
$491$	37.3846	1.68715	0.843573	+	0.537015i	$0.180448\pi$
$492$	0	0
$493$	−13.0807	−0.589124
$494$	0	0
$495$	−20.6089	−0.926302
$496$	0	0
$497$	42.6652	1.91379
$498$	0	0
$499$	6.25178	0.279868	0.139934	−	0.990161i	$-0.455311\pi$
$499$	6.25178	0.279868	0.139934	+	0.990161i	$0.455311\pi$
$500$	0	0
$501$	35.6509	1.59277
$502$	0	0
$503$	−27.3844	−1.22101	−0.610504	−	0.792013i	$-0.709033\pi$
$503$	−27.3844	−1.22101	−0.610504	+	0.792013i	$0.709033\pi$
$504$	0	0
$505$	−13.2788	−0.590897
$506$	0	0
$507$	−36.4014	−1.61664
$508$	0	0
$509$	15.8700	0.703424	0.351712	−	0.936108i	$-0.385600\pi$
$509$	15.8700	0.703424	0.351712	+	0.936108i	$0.385600\pi$
$510$	0	0
$511$	−59.8657	−2.64830
$512$	0	0
$513$	−21.8575	−0.965034
$514$	0	0
$515$	−14.5380	−0.640619
$516$	0	0
$517$	8.24164	0.362467
$518$	0	0
$519$	−23.9181	−1.04989
$520$	0	0
$521$	−18.6914	−0.818886	−0.409443	−	0.912336i	$-0.634277\pi$
$521$	−18.6914	−0.818886	−0.409443	+	0.912336i	$0.634277\pi$
$522$	0	0
$523$	−2.95749	−0.129322	−0.0646609	−	0.997907i	$-0.520597\pi$
$523$	−2.95749	−0.129322	−0.0646609	+	0.997907i	$0.520597\pi$
$524$	0	0
$525$	−41.0176	−1.79016
$526$	0	0
$527$	5.81147	0.253152
$528$	0	0
$529$	49.8194	2.16606
$530$	0	0
$531$	17.0345	0.739236
$532$	0	0
$533$	3.24317	0.140477
$534$	0	0
$535$	−7.63057	−0.329898
$536$	0	0
$537$	62.4067	2.69305
$538$	0	0
$539$	20.3991	0.878651
$540$	0	0
$541$	−7.39296	−0.317848	−0.158924	−	0.987291i	$-0.550802\pi$
$541$	−7.39296	−0.317848	−0.158924	+	0.987291i	$0.550802\pi$
$542$	0	0
$543$	14.4486	0.620048
$544$	0	0
$545$	3.49866	0.149866
$546$	0	0
$547$	−7.65413	−0.327267	−0.163634	−	0.986521i	$-0.552321\pi$
$547$	−7.65413	−0.327267	−0.163634	+	0.986521i	$0.552321\pi$
$548$	0	0
$549$	8.97102	0.382874
$550$	0	0
$551$	4.66737	0.198837
$552$	0	0
$553$	62.5113	2.65825
$554$	0	0
$555$	21.5658	0.915418
$556$	0	0
$557$	−8.17941	−0.346573	−0.173286	−	0.984871i	$-0.555439\pi$
$557$	−8.17941	−0.346573	−0.173286	+	0.984871i	$0.555439\pi$
$558$	0	0
$559$	−12.9549	−0.547936
$560$	0	0
$561$	46.0790	1.94546
$562$	0	0
$563$	21.3577	0.900121	0.450060	−	0.892998i	$-0.351402\pi$
$563$	21.3577	0.900121	0.450060	+	0.892998i	$0.351402\pi$
$564$	0	0
$565$	−8.90434	−0.374608
$566$	0	0
$567$	50.4771	2.11984
$568$	0	0
$569$	−16.2546	−0.681428	−0.340714	−	0.940167i	$-0.610669\pi$
$569$	−16.2546	−0.681428	−0.340714	+	0.940167i	$0.610669\pi$
$570$	0	0
$571$	−39.6822	−1.66065	−0.830324	−	0.557281i	$-0.811845\pi$
$571$	−39.6822	−1.66065	−0.830324	+	0.557281i	$0.811845\pi$
$572$	0	0
$573$	−40.2605	−1.68191
$574$	0	0
$575$	29.5240	1.23124
$576$	0	0
$577$	−46.7231	−1.94511	−0.972553	−	0.232680i	$-0.925251\pi$
$577$	−46.7231	−1.94511	−0.972553	+	0.232680i	$0.925251\pi$
$578$	0	0
$579$	17.6822	0.734846
$580$	0	0
$581$	−45.4216	−1.88440
$582$	0	0
$583$	−0.473253	−0.0196001
$584$	0	0
$585$	−8.55081	−0.353532
$586$	0	0
$587$	18.0103	0.743366	0.371683	−	0.928360i	$-0.378781\pi$
$587$	18.0103	0.743366	0.371683	+	0.928360i	$0.378781\pi$
$588$	0	0
$589$	−2.07362	−0.0854418
$590$	0	0
$591$	−73.2964	−3.01501
$592$	0	0
$593$	−4.51074	−0.185234	−0.0926170	−	0.995702i	$-0.529523\pi$
$593$	−4.51074	−0.185234	−0.0926170	+	0.995702i	$0.529523\pi$
$594$	0	0
$595$	−27.8410	−1.14137
$596$	0	0
$597$	−17.0992	−0.699825
$598$	0	0
$599$	−37.1770	−1.51901	−0.759505	−	0.650501i	$-0.774559\pi$
$599$	−37.1770	−1.51901	−0.759505	+	0.650501i	$0.774559\pi$
$600$	0	0
$601$	−41.4176	−1.68946	−0.844731	−	0.535192i	$-0.820239\pi$
$601$	−41.4176	−1.68946	−0.844731	+	0.535192i	$0.820239\pi$
$602$	0	0
$603$	−53.9159	−2.19562
$604$	0	0
$605$	5.37950	0.218708
$606$	0	0
$607$	32.2230	1.30789	0.653946	−	0.756541i	$-0.273112\pi$
$607$	32.2230	1.30789	0.653946	+	0.756541i	$0.273112\pi$
$608$	0	0
$609$	−26.6847	−1.08132
$610$	0	0
$611$	3.41953	0.138339
$612$	0	0
$613$	−46.1486	−1.86393	−0.931963	−	0.362553i	$-0.881905\pi$
$613$	−46.1486	−1.86393	−0.931963	+	0.362553i	$0.881905\pi$
$614$	0	0
$615$	−11.5398	−0.465330
$616$	0	0
$617$	−5.70572	−0.229704	−0.114852	−	0.993383i	$-0.536639\pi$
$617$	−5.70572	−0.229704	−0.114852	+	0.993383i	$0.536639\pi$
$618$	0	0
$619$	−11.2110	−0.450610	−0.225305	−	0.974288i	$-0.572338\pi$
$619$	−11.2110	−0.450610	−0.225305	+	0.974288i	$0.572338\pi$
$620$	0	0
$621$	−89.9490	−3.60953
$622$	0	0
$623$	−7.13681	−0.285930
$624$	0	0
$625$	4.26935	0.170774
$626$	0	0
$627$	−16.4416	−0.656616
$628$	0	0
$629$	−32.8821	−1.31109
$630$	0	0
$631$	39.2823	1.56380	0.781901	−	0.623402i	$-0.214250\pi$
$631$	39.2823	1.56380	0.781901	+	0.623402i	$0.214250\pi$
$632$	0	0
$633$	−56.0851	−2.22918
$634$	0	0
$635$	21.2270	0.842369
$636$	0	0
$637$	8.46375	0.335346
$638$	0	0
$639$	71.0915	2.81234
$640$	0	0
$641$	−30.7592	−1.21492	−0.607458	−	0.794352i	$-0.707811\pi$
$641$	−30.7592	−1.21492	−0.607458	+	0.794352i	$0.707811\pi$
$642$	0	0
$643$	38.4960	1.51814	0.759068	−	0.651011i	$-0.225655\pi$
$643$	38.4960	1.51814	0.759068	+	0.651011i	$0.225655\pi$
$644$	0	0
$645$	46.0961	1.81503
$646$	0	0
$647$	−33.7906	−1.32844	−0.664222	−	0.747535i	$-0.731237\pi$
$647$	−33.7906	−1.32844	−0.664222	+	0.747535i	$0.731237\pi$
$648$	0	0
$649$	6.83731	0.268388
$650$	0	0
$651$	11.8554	0.464652
$652$	0	0
$653$	−9.15608	−0.358305	−0.179153	−	0.983821i	$-0.557336\pi$
$653$	−9.15608	−0.358305	−0.179153	+	0.983821i	$0.557336\pi$
$654$	0	0
$655$	−23.8050	−0.930138
$656$	0	0
$657$	−99.7523	−3.89171
$658$	0	0
$659$	8.36148	0.325717	0.162859	−	0.986649i	$-0.447929\pi$
$659$	8.36148	0.325717	0.162859	+	0.986649i	$0.447929\pi$
$660$	0	0
$661$	6.57195	0.255619	0.127810	−	0.991799i	$-0.459205\pi$
$661$	6.57195	0.255619	0.127810	+	0.991799i	$0.459205\pi$
$662$	0	0
$663$	19.1186	0.742503
$664$	0	0
$665$	9.93408	0.385227
$666$	0	0
$667$	19.2074	0.743712
$668$	0	0
$669$	−49.3237	−1.90696
$670$	0	0
$671$	3.60078	0.139006
$672$	0	0
$673$	17.1798	0.662233	0.331116	−	0.943590i	$-0.392575\pi$
$673$	17.1798	0.662233	0.331116	+	0.943590i	$0.392575\pi$
$674$	0	0
$675$	−36.4691	−1.40370
$676$	0	0
$677$	23.2374	0.893085	0.446542	−	0.894763i	$-0.352655\pi$
$677$	23.2374	0.893085	0.446542	+	0.894763i	$0.352655\pi$
$678$	0	0
$679$	75.4948	2.89723
$680$	0	0
$681$	−48.8422	−1.87164
$682$	0	0
$683$	26.4209	1.01097	0.505485	−	0.862836i	$-0.331314\pi$
$683$	26.4209	1.01097	0.505485	+	0.862836i	$0.331314\pi$
$684$	0	0
$685$	14.7423	0.563274
$686$	0	0
$687$	20.3515	0.776457
$688$	0	0
$689$	−0.196357	−0.00748059
$690$	0	0
$691$	26.6015	1.01197	0.505984	−	0.862543i	$-0.331130\pi$
$691$	26.6015	1.01197	0.505984	+	0.862543i	$0.331130\pi$
$692$	0	0
$693$	64.1033	2.43508
$694$	0	0
$695$	8.04007	0.304977
$696$	0	0
$697$	17.5951	0.666462
$698$	0	0
$699$	−54.3506	−2.05573
$700$	0	0
$701$	−17.0983	−0.645793	−0.322897	−	0.946434i	$-0.604657\pi$
$701$	−17.0983	−0.645793	−0.322897	+	0.946434i	$0.604657\pi$
$702$	0	0
$703$	11.7328	0.442511
$704$	0	0
$705$	−12.1673	−0.458248
$706$	0	0
$707$	41.3031	1.55336
$708$	0	0
$709$	33.6632	1.26425	0.632123	−	0.774868i	$-0.282184\pi$
$709$	33.6632	1.26425	0.632123	+	0.774868i	$0.282184\pi$
$710$	0	0
$711$	104.160	3.90632
$712$	0	0
$713$	−8.53343	−0.319579
$714$	0	0
$715$	−3.43211	−0.128354
$716$	0	0
$717$	20.6394	0.770792
$718$	0	0
$719$	37.2607	1.38959	0.694794	−	0.719209i	$-0.255496\pi$
$719$	37.2607	1.38959	0.694794	+	0.719209i	$0.255496\pi$
$720$	0	0
$721$	45.2198	1.68407
$722$	0	0
$723$	−24.9615	−0.928327
$724$	0	0
$725$	7.78748	0.289220
$726$	0	0
$727$	−27.2456	−1.01048	−0.505241	−	0.862978i	$-0.668597\pi$
$727$	−27.2456	−1.01048	−0.505241	+	0.862978i	$0.668597\pi$
$728$	0	0
$729$	−13.0098	−0.481845
$730$	0	0
$731$	−70.2843	−2.59956
$732$	0	0
$733$	25.7964	0.952813	0.476407	−	0.879225i	$-0.341939\pi$
$733$	25.7964	0.952813	0.476407	+	0.879225i	$0.341939\pi$
$734$	0	0
$735$	−30.1157	−1.11083
$736$	0	0
$737$	−21.6407	−0.797145
$738$	0	0
$739$	26.4631	0.973462	0.486731	−	0.873552i	$-0.338189\pi$
$739$	26.4631	0.973462	0.486731	+	0.873552i	$0.338189\pi$
$740$	0	0
$741$	−6.82177	−0.250604
$742$	0	0
$743$	16.4932	0.605075	0.302538	−	0.953137i	$-0.402166\pi$
$743$	16.4932	0.605075	0.302538	+	0.953137i	$0.402166\pi$
$744$	0	0
$745$	24.0625	0.881581
$746$	0	0
$747$	−75.6844	−2.76915
$748$	0	0
$749$	23.7346	0.867244
$750$	0	0
$751$	−2.59903	−0.0948400	−0.0474200	−	0.998875i	$-0.515100\pi$
$751$	−2.59903	−0.0948400	−0.0474200	+	0.998875i	$0.515100\pi$
$752$	0	0
$753$	53.0010	1.93146
$754$	0	0
$755$	−4.30328	−0.156612
$756$	0	0
$757$	−31.4161	−1.14184	−0.570919	−	0.821007i	$-0.693413\pi$
$757$	−31.4161	−1.14184	−0.570919	+	0.821007i	$0.693413\pi$
$758$	0	0
$759$	−67.6613	−2.45595
$760$	0	0
$761$	45.4939	1.64915	0.824577	−	0.565750i	$-0.191413\pi$
$761$	45.4939	1.64915	0.824577	+	0.565750i	$0.191413\pi$
$762$	0	0
$763$	−10.8825	−0.393971
$764$	0	0
$765$	−46.3906	−1.67726
$766$	0	0
$767$	2.83685	0.102433
$768$	0	0
$769$	9.91887	0.357683	0.178842	−	0.983878i	$-0.442765\pi$
$769$	9.91887	0.357683	0.178842	+	0.983878i	$0.442765\pi$
$770$	0	0
$771$	50.4043	1.81527
$772$	0	0
$773$	−14.1257	−0.508068	−0.254034	−	0.967195i	$-0.581757\pi$
$773$	−14.1257	−0.508068	−0.254034	+	0.967195i	$0.581757\pi$
$774$	0	0
$775$	−3.45981	−0.124280
$776$	0	0
$777$	−67.0797	−2.40647
$778$	0	0
$779$	−6.27818	−0.224939
$780$	0	0
$781$	28.5346	1.02105
$782$	0	0
$783$	−23.7256	−0.847885
$784$	0	0
$785$	17.4087	0.621342
$786$	0	0
$787$	34.9135	1.24453	0.622266	−	0.782806i	$-0.286212\pi$
$787$	34.9135	1.24453	0.622266	+	0.782806i	$0.286212\pi$
$788$	0	0
$789$	25.3012	0.900746
$790$	0	0
$791$	27.6966	0.984779
$792$	0	0
$793$	1.49399	0.0530532
$794$	0	0
$795$	0.698674	0.0247794
$796$	0	0
$797$	−8.39550	−0.297384	−0.148692	−	0.988884i	$-0.547506\pi$
$797$	−8.39550	−0.297384	−0.148692	+	0.988884i	$0.547506\pi$
$798$	0	0
$799$	18.5519	0.656319
$800$	0	0
$801$	−11.8918	−0.420177
$802$	0	0
$803$	−40.0384	−1.41293
$804$	0	0
$805$	40.8811	1.44087
$806$	0	0
$807$	72.4339	2.54979
$808$	0	0
$809$	−3.64857	−0.128277	−0.0641385	−	0.997941i	$-0.520430\pi$
$809$	−3.64857	−0.128277	−0.0641385	+	0.997941i	$0.520430\pi$
$810$	0	0
$811$	−37.2315	−1.30737	−0.653687	−	0.756765i	$-0.726779\pi$
$811$	−37.2315	−1.30737	−0.653687	+	0.756765i	$0.726779\pi$
$812$	0	0
$813$	33.9612	1.19107
$814$	0	0
$815$	8.65953	0.303330
$816$	0	0
$817$	25.0784	0.877383
$818$	0	0
$819$	26.5970	0.929374
$820$	0	0
$821$	−23.0624	−0.804884	−0.402442	−	0.915446i	$-0.631838\pi$
$821$	−23.0624	−0.804884	−0.402442	+	0.915446i	$0.631838\pi$
$822$	0	0
$823$	−36.9878	−1.28931	−0.644657	−	0.764472i	$-0.723000\pi$
$823$	−36.9878	−1.28931	−0.644657	+	0.764472i	$0.723000\pi$
$824$	0	0
$825$	−27.4327	−0.955086
$826$	0	0
$827$	−19.1291	−0.665184	−0.332592	−	0.943071i	$-0.607923\pi$
$827$	−19.1291	−0.665184	−0.332592	+	0.943071i	$0.607923\pi$
$828$	0	0
$829$	−39.9822	−1.38864	−0.694320	−	0.719667i	$-0.744295\pi$
$829$	−39.9822	−1.38864	−0.694320	+	0.719667i	$0.744295\pi$
$830$	0	0
$831$	−83.4409	−2.89453
$832$	0	0
$833$	45.9183	1.59097
$834$	0	0
$835$	−14.4063	−0.498551
$836$	0	0
$837$	10.5408	0.364343
$838$	0	0
$839$	−7.02485	−0.242525	−0.121262	−	0.992620i	$-0.538694\pi$
$839$	−7.02485	−0.242525	−0.121262	+	0.992620i	$0.538694\pi$
$840$	0	0
$841$	−23.9337	−0.825301
$842$	0	0
$843$	57.8954	1.99403
$844$	0	0
$845$	14.7096	0.506024
$846$	0	0
$847$	−16.7327	−0.574944
$848$	0	0
$849$	−5.48828	−0.188357
$850$	0	0
$851$	48.2833	1.65513
$852$	0	0
$853$	25.3929	0.869436	0.434718	−	0.900567i	$-0.356848\pi$
$853$	25.3929	0.869436	0.434718	+	0.900567i	$0.356848\pi$
$854$	0	0
$855$	16.5528	0.566095
$856$	0	0
$857$	17.5775	0.600435	0.300217	−	0.953871i	$-0.402941\pi$
$857$	17.5775	0.600435	0.300217	+	0.953871i	$0.402941\pi$
$858$	0	0
$859$	1.88009	0.0641479	0.0320740	−	0.999485i	$-0.489789\pi$
$859$	1.88009	0.0641479	0.0320740	+	0.999485i	$0.489789\pi$
$860$	0	0
$861$	35.8942	1.22327
$862$	0	0
$863$	28.4117	0.967145	0.483573	−	0.875304i	$-0.339339\pi$
$863$	28.4117	0.967145	0.483573	+	0.875304i	$0.339339\pi$
$864$	0	0
$865$	9.66514	0.328625
$866$	0	0
$867$	51.5135	1.74949
$868$	0	0
$869$	41.8078	1.41823
$870$	0	0
$871$	−8.97890	−0.304238
$872$	0	0
$873$	125.794	4.25750
$874$	0	0
$875$	40.5285	1.37011
$876$	0	0
$877$	49.0719	1.65704	0.828520	−	0.559959i	$-0.189183\pi$
$877$	49.0719	1.65704	0.828520	+	0.559959i	$0.189183\pi$
$878$	0	0
$879$	61.6015	2.07777
$880$	0	0
$881$	−28.1295	−0.947706	−0.473853	−	0.880604i	$-0.657137\pi$
$881$	−28.1295	−0.947706	−0.473853	+	0.880604i	$0.657137\pi$
$882$	0	0
$883$	−32.5579	−1.09566	−0.547831	−	0.836589i	$-0.684546\pi$
$883$	−32.5579	−1.09566	−0.547831	+	0.836589i	$0.684546\pi$
$884$	0	0
$885$	−10.0941	−0.339308
$886$	0	0
$887$	27.5919	0.926445	0.463222	−	0.886242i	$-0.346693\pi$
$887$	27.5919	0.926445	0.463222	+	0.886242i	$0.346693\pi$
$888$	0	0
$889$	−66.0260	−2.21444
$890$	0	0
$891$	33.7593	1.13098
$892$	0	0
$893$	−6.61958	−0.221516
$894$	0	0
$895$	−25.2181	−0.842948
$896$	0	0
$897$	−28.0732	−0.937338
$898$	0	0
$899$	−2.25084	−0.0750697
$900$	0	0
$901$	−1.06529	−0.0354900
$902$	0	0
$903$	−143.380	−4.77140
$904$	0	0
$905$	−5.83857	−0.194081
$906$	0	0
$907$	−0.0313108	−0.00103966	−0.000519829	−	1.00000i	$-0.500165\pi$
$907$	−0.0313108	−0.00103966	−0.000519829		1.00000i	$0.500165\pi$
$908$	0	0
$909$	68.8221	2.28268
$910$	0	0
$911$	44.7801	1.48363	0.741816	−	0.670604i	$-0.233965\pi$
$911$	44.7801	1.48363	0.741816	+	0.670604i	$0.233965\pi$
$912$	0	0
$913$	−30.3781	−1.00537
$914$	0	0
$915$	−5.31591	−0.175739
$916$	0	0
$917$	74.0446	2.44517
$918$	0	0
$919$	−33.6755	−1.11085	−0.555426	−	0.831566i	$-0.687445\pi$
$919$	−33.6755	−1.11085	−0.555426	+	0.831566i	$0.687445\pi$
$920$	0	0
$921$	48.6442	1.60288
$922$	0	0
$923$	11.8393	0.389694
$924$	0	0
$925$	19.5761	0.643658
$926$	0	0
$927$	75.3483	2.47476
$928$	0	0
$929$	−5.13585	−0.168502	−0.0842508	−	0.996445i	$-0.526850\pi$
$929$	−5.13585	−0.168502	−0.0842508	+	0.996445i	$0.526850\pi$
$930$	0	0
$931$	−16.3843	−0.536974
$932$	0	0
$933$	−21.1004	−0.690798
$934$	0	0
$935$	−18.6202	−0.608946
$936$	0	0
$937$	11.1367	0.363819	0.181910	−	0.983315i	$-0.441772\pi$
$937$	11.1367	0.363819	0.181910	+	0.983315i	$0.441772\pi$
$938$	0	0
$939$	−15.1571	−0.494632
$940$	0	0
$941$	−43.6608	−1.42330	−0.711651	−	0.702533i	$-0.752052\pi$
$941$	−43.6608	−1.42330	−0.711651	+	0.702533i	$0.752052\pi$
$942$	0	0
$943$	−25.8362	−0.841344
$944$	0	0
$945$	−50.4978	−1.64269
$946$	0	0
$947$	37.8252	1.22915	0.614576	−	0.788857i	$-0.289327\pi$
$947$	37.8252	1.22915	0.614576	+	0.788857i	$0.289327\pi$
$948$	0	0
$949$	−16.6123	−0.539257
$950$	0	0
$951$	−2.58333	−0.0837703
$952$	0	0
$953$	31.4722	1.01949	0.509743	−	0.860327i	$-0.329741\pi$
$953$	31.4722	1.01949	0.509743	+	0.860327i	$0.329741\pi$
$954$	0	0
$955$	16.2690	0.526452
$956$	0	0
$957$	−17.8468	−0.576906
$958$	0	0
$959$	−45.8554	−1.48075
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	39.5482	1.27442
$964$	0	0
$965$	−7.14524	−0.230013
$966$	0	0
$967$	34.6404	1.11396	0.556980	−	0.830526i	$-0.311960\pi$
$967$	34.6404	1.11396	0.556980	+	0.830526i	$0.311960\pi$
$968$	0	0
$969$	−37.0101	−1.18893
$970$	0	0
$971$	51.9007	1.66557	0.832786	−	0.553595i	$-0.186744\pi$
$971$	51.9007	1.66557	0.832786	+	0.553595i	$0.186744\pi$
$972$	0	0
$973$	−25.0084	−0.801732
$974$	0	0
$975$	−11.3821	−0.364518
$976$	0	0
$977$	5.90194	0.188820	0.0944099	−	0.995533i	$-0.469904\pi$
$977$	5.90194	0.188820	0.0944099	+	0.995533i	$0.469904\pi$
$978$	0	0
$979$	−4.77312	−0.152550
$980$	0	0
$981$	−18.1331	−0.578944
$982$	0	0
$983$	37.6032	1.19935	0.599677	−	0.800242i	$-0.295296\pi$
$983$	37.6032	1.19935	0.599677	+	0.800242i	$0.295296\pi$
$984$	0	0
$985$	29.6186	0.943725
$986$	0	0
$987$	37.8460	1.20465
$988$	0	0
$989$	103.204	3.28169
$990$	0	0
$991$	−11.3304	−0.359921	−0.179960	−	0.983674i	$-0.557597\pi$
$991$	−11.3304	−0.359921	−0.179960	+	0.983674i	$0.557597\pi$
$992$	0	0
$993$	−7.73220	−0.245374
$994$	0	0
$995$	6.90968	0.219052
$996$	0	0
$997$	9.18624	0.290931	0.145466	−	0.989363i	$-0.453532\pi$
$997$	9.18624	0.290931	0.145466	+	0.989363i	$0.453532\pi$
$998$	0	0
$999$	−59.6412	−1.88696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1984.2.a.bb.1.6		6
4.3	odd	2		1984.2.a.ba.1.1		6
8.3	odd	2		992.2.a.h.1.6	yes	6
8.5	even	2		992.2.a.g.1.1	✓	6
24.5	odd	2		8928.2.a.br.1.3		6
24.11	even	2		8928.2.a.bq.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
992.2.a.g.1.1	✓	6	8.5	even	2
992.2.a.h.1.6	yes	6	8.3	odd	2
1984.2.a.ba.1.1		6	4.3	odd	2
1984.2.a.bb.1.6		6	1.1	even	1	trivial
8928.2.a.bq.1.3		6	24.11	even	2
8928.2.a.br.1.3		6	24.5	odd	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)

\(f(q)\)	\(=\)	\(q+3.07118 q^{3} -1.24104 q^{5} +3.86022 q^{7} +6.43216 q^{9} +2.58173 q^{11} +1.07118 q^{13} -3.81147 q^{15} +5.81147 q^{17} -2.07362 q^{19} +11.8554 q^{21} -8.53343 q^{23} -3.45981 q^{25} +10.5408 q^{27} -2.25084 q^{29} +1.00000 q^{31} +7.92897 q^{33} -4.79070 q^{35} -5.65814 q^{37} +3.28980 q^{39} +3.02765 q^{41} -12.0941 q^{43} -7.98259 q^{45} +3.19229 q^{47} +7.90132 q^{49} +17.8481 q^{51} -0.183308 q^{53} -3.20404 q^{55} -6.36845 q^{57} +2.64834 q^{59} +1.39471 q^{61} +24.8296 q^{63} -1.32938 q^{65} -8.38223 q^{67} -26.2077 q^{69} +11.0525 q^{71} -15.5084 q^{73} -10.6257 q^{75} +9.96606 q^{77} +16.1937 q^{79} +13.0762 q^{81} -11.7666 q^{83} -7.21229 q^{85} -6.91274 q^{87} -1.84881 q^{89} +4.13500 q^{91} +3.07118 q^{93} +2.57345 q^{95} +19.5571 q^{97} +16.6061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 10 q^{9} + 10 q^{11} - 10 q^{13} + 8 q^{15} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} + 20 q^{25} + 8 q^{27} - 10 q^{29} + 6 q^{31} + 8 q^{33} + 14 q^{35} - 26 q^{37}+ \cdots + 46 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 10 * q^9 + 10 * q^11 - 10 * q^13 + 8 * q^15 + 4 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 + 20 * q^25 + 8 * q^27 - 10 * q^29 + 6 * q^31 + 8 * q^33 + 14 * q^35 - 26 * q^37 + 24 * q^39 + 6 * q^41 + 10 * q^43 - 22 * q^45 + 4 * q^47 + 20 * q^49 + 32 * q^51 - 18 * q^53 + 16 * q^55 + 24 * q^57 + 6 * q^59 - 10 * q^61 - 2 * q^63 + 12 * q^65 - 4 * q^67 + 4 * q^69 + 30 * q^71 + 4 * q^73 - 14 * q^75 + 20 * q^79 + 10 * q^81 + 30 * q^83 + 16 * q^87 - 20 * q^89 - 8 * q^91 + 2 * q^93 + 26 * q^95 + 30 * q^97 + 46 * q^99

Introduction

L-functions

Modular forms

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Database

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