LMFDB - Embedded newform 1984.2.a.y.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:	$1$
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			$-0.386509$ 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)

$f(q)$	$=$	$q+0.732051 q^{3} +0.386509 q^{5} -1.44247 q^{7} -2.46410 q^{9} -5.78801 q^{11} +4.84398 q^{13} +0.282944 q^{15} +3.82898 q^{17} -1.33055 q^{19} -1.05596 q^{21} +7.29308 q^{23} -4.85061 q^{25} -4.00000 q^{27} -3.50507 q^{29} +1.00000 q^{31} -4.23712 q^{33} -0.557528 q^{35} -6.02513 q^{37} +3.54604 q^{39} +4.49843 q^{41} -9.50507 q^{43} -0.952398 q^{45} -11.8131 q^{47} -4.91928 q^{49} +2.80301 q^{51} -12.8281 q^{53} -2.23712 q^{55} -0.974028 q^{57} -7.36054 q^{59} +4.84398 q^{61} +3.55440 q^{63} +1.87224 q^{65} -10.8849 q^{67} +5.33891 q^{69} -8.21549 q^{71} +8.92820 q^{73} -3.55089 q^{75} +8.34904 q^{77} +0.490074 q^{79} +4.46410 q^{81} -17.5879 q^{83} +1.47994 q^{85} -2.56589 q^{87} -16.1780 q^{89} -6.98730 q^{91} +0.732051 q^{93} -0.514268 q^{95} -1.04760 q^{97} +14.2623 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	0.386509	0.172852	0.0864261	−	0.996258i	$-0.472455\pi$
$5$	0.386509	0.172852	0.0864261	+	0.996258i	$0.472455\pi$
$6$	0	0
$7$	−1.44247	−0.545203	−0.272602	−	0.962127i	$-0.587884\pi$
$7$	−1.44247	−0.545203	−0.272602	+	0.962127i	$0.587884\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−5.78801	−1.74515	−0.872576	−	0.488479i	$-0.837552\pi$
$11$	−5.78801	−1.74515	−0.872576	+	0.488479i	$0.837552\pi$
$12$	0	0
$13$	4.84398	1.34348	0.671739	−	0.740788i	$-0.265548\pi$
$13$	4.84398	1.34348	0.671739	+	0.740788i	$0.265548\pi$
$14$	0	0
$15$	0.282944	0.0730559
$16$	0	0
$17$	3.82898	0.928664	0.464332	−	0.885661i	$-0.346294\pi$
$17$	3.82898	0.928664	0.464332	+	0.885661i	$0.346294\pi$
$18$	0	0
$19$	−1.33055	−0.305248	−0.152624	−	0.988284i	$-0.548772\pi$
$19$	−1.33055	−0.305248	−0.152624	+	0.988284i	$0.548772\pi$
$20$	0	0
$21$	−1.05596	−0.230430
$22$	0	0
$23$	7.29308	1.52071	0.760356	−	0.649506i	$-0.225024\pi$
$23$	7.29308	1.52071	0.760356	+	0.649506i	$0.225024\pi$
$24$	0	0
$25$	−4.85061	−0.970122
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.50507	−0.650875	−0.325437	−	0.945564i	$-0.605512\pi$
$29$	−3.50507	−0.650875	−0.325437	+	0.945564i	$0.605512\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−4.23712	−0.737588
$34$	0	0
$35$	−0.557528	−0.0942395
$36$	0	0
$37$	−6.02513	−0.990526	−0.495263	−	0.868743i	$-0.664928\pi$
$37$	−6.02513	−0.990526	−0.495263	+	0.868743i	$0.664928\pi$
$38$	0	0
$39$	3.54604	0.567820
$40$	0	0
$41$	4.49843	0.702537	0.351269	−	0.936275i	$-0.385750\pi$
$41$	4.49843	0.702537	0.351269	+	0.936275i	$0.385750\pi$
$42$	0	0
$43$	−9.50507	−1.44951	−0.724755	−	0.689007i	$-0.758047\pi$
$43$	−9.50507	−1.44951	−0.724755	+	0.689007i	$0.758047\pi$
$44$	0	0
$45$	−0.952398	−0.141975
$46$	0	0
$47$	−11.8131	−1.72312	−0.861562	−	0.507652i	$-0.830514\pi$
$47$	−11.8131	−1.72312	−0.861562	+	0.507652i	$0.830514\pi$
$48$	0	0
$49$	−4.91928	−0.702754
$50$	0	0
$51$	2.80301	0.392500
$52$	0	0
$53$	−12.8281	−1.76208	−0.881040	−	0.473041i	$-0.843156\pi$
$53$	−12.8281	−1.76208	−0.881040	+	0.473041i	$0.843156\pi$
$54$	0	0
$55$	−2.23712	−0.301653
$56$	0	0
$57$	−0.974028	−0.129013
$58$	0	0
$59$	−7.36054	−0.958260	−0.479130	−	0.877744i	$-0.659048\pi$
$59$	−7.36054	−0.958260	−0.479130	+	0.877744i	$0.659048\pi$
$60$	0	0
$61$	4.84398	0.620208	0.310104	−	0.950703i	$-0.399636\pi$
$61$	4.84398	0.620208	0.310104	+	0.950703i	$0.399636\pi$
$62$	0	0
$63$	3.55440	0.447812
$64$	0	0
$65$	1.87224	0.232223
$66$	0	0
$67$	−10.8849	−1.32981	−0.664904	−	0.746929i	$-0.731527\pi$
$67$	−10.8849	−1.32981	−0.664904	+	0.746929i	$0.731527\pi$
$68$	0	0
$69$	5.33891	0.642729
$70$	0	0
$71$	−8.21549	−0.974999	−0.487500	−	0.873123i	$-0.662091\pi$
$71$	−8.21549	−0.974999	−0.487500	+	0.873123i	$0.662091\pi$
$72$	0	0
$73$	8.92820	1.04497	0.522484	−	0.852649i	$-0.325006\pi$
$73$	8.92820	1.04497	0.522484	+	0.852649i	$0.325006\pi$
$74$	0	0
$75$	−3.55089	−0.410022
$76$	0	0
$77$	8.34904	0.951462
$78$	0	0
$79$	0.490074	0.0551376	0.0275688	−	0.999620i	$-0.491223\pi$
$79$	0.490074	0.0551376	0.0275688	+	0.999620i	$0.491223\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−17.5879	−1.93052	−0.965261	−	0.261288i	$-0.915853\pi$
$83$	−17.5879	−1.93052	−0.965261	+	0.261288i	$0.915853\pi$
$84$	0	0
$85$	1.47994	0.160522
$86$	0	0
$87$	−2.56589	−0.275092
$88$	0	0
$89$	−16.1780	−1.71487	−0.857434	−	0.514595i	$-0.827942\pi$
$89$	−16.1780	−1.71487	−0.857434	+	0.514595i	$0.827942\pi$
$90$	0	0
$91$	−6.98730	−0.732468
$92$	0	0
$93$	0.732051	0.0759101
$94$	0	0
$95$	−0.514268	−0.0527628
$96$	0	0
$97$	−1.04760	−0.106368	−0.0531839	−	0.998585i	$-0.516937\pi$
$97$	−1.04760	−0.106368	−0.0531839	+	0.998585i	$0.516937\pi$
$98$	0	0
$99$	14.2623	1.43341
$100$	0	0
$101$	12.3116	1.22505	0.612524	−	0.790452i	$-0.290154\pi$
$101$	12.3116	1.22505	0.612524	+	0.790452i	$0.290154\pi$
$102$	0	0
$103$	12.8246	1.26365	0.631825	−	0.775111i	$-0.282306\pi$
$103$	12.8246	1.26365	0.631825	+	0.775111i	$0.282306\pi$
$104$	0	0
$105$	−0.408139	−0.0398303
$106$	0	0
$107$	17.6047	1.70191	0.850953	−	0.525241i	$-0.176025\pi$
$107$	17.6047	1.70191	0.850953	+	0.525241i	$0.176025\pi$
$108$	0	0
$109$	0.511704	0.0490123	0.0245062	−	0.999700i	$-0.492199\pi$
$109$	0.511704	0.0490123	0.0245062	+	0.999700i	$0.492199\pi$
$110$	0	0
$111$	−4.41070	−0.418645
$112$	0	0
$113$	17.2398	1.62178	0.810891	−	0.585197i	$-0.198983\pi$
$113$	17.2398	1.62178	0.810891	+	0.585197i	$0.198983\pi$
$114$	0	0
$115$	2.81884	0.262858
$116$	0	0
$117$	−11.9360	−1.10349
$118$	0	0
$119$	−5.52320	−0.506311
$120$	0	0
$121$	22.5011	2.04555
$122$	0	0
$123$	3.29308	0.296927
$124$	0	0
$125$	−3.80735	−0.340540
$126$	0	0
$127$	−2.01327	−0.178649	−0.0893244	−	0.996003i	$-0.528471\pi$
$127$	−2.01327	−0.178649	−0.0893244	+	0.996003i	$0.528471\pi$
$128$	0	0
$129$	−6.95819	−0.612635
$130$	0	0
$131$	−2.70435	−0.236280	−0.118140	−	0.992997i	$-0.537693\pi$
$131$	−2.70435	−0.236280	−0.118140	+	0.992997i	$0.537693\pi$
$132$	0	0
$133$	1.91928	0.166422
$134$	0	0
$135$	−1.54604	−0.133062
$136$	0	0
$137$	−2.53333	−0.216437	−0.108219	−	0.994127i	$-0.534515\pi$
$137$	−2.53333	−0.216437	−0.108219	+	0.994127i	$0.534515\pi$
$138$	0	0
$139$	−1.05367	−0.0893711	−0.0446856	−	0.999001i	$-0.514229\pi$
$139$	−1.05367	−0.0893711	−0.0446856	+	0.999001i	$0.514229\pi$
$140$	0	0
$141$	−8.64782	−0.728278
$142$	0	0
$143$	−28.0370	−2.34457
$144$	0	0
$145$	−1.35474	−0.112505
$146$	0	0
$147$	−3.60116	−0.297019
$148$	0	0
$149$	8.80301	0.721171	0.360585	−	0.932726i	$-0.382577\pi$
$149$	8.80301	0.721171	0.360585	+	0.932726i	$0.382577\pi$
$150$	0	0
$151$	21.7541	1.77032	0.885160	−	0.465287i	$-0.154049\pi$
$151$	21.7541	1.77032	0.885160	+	0.465287i	$0.154049\pi$
$152$	0	0
$153$	−9.43500	−0.762774
$154$	0	0
$155$	0.386509	0.0310452
$156$	0	0
$157$	1.12085	0.0894538	0.0447269	−	0.998999i	$-0.485758\pi$
$157$	1.12085	0.0894538	0.0447269	+	0.998999i	$0.485758\pi$
$158$	0	0
$159$	−9.39085	−0.744743
$160$	0	0
$161$	−10.5201	−0.829097
$162$	0	0
$163$	4.94983	0.387701	0.193850	−	0.981031i	$-0.437902\pi$
$163$	4.94983	0.387701	0.193850	+	0.981031i	$0.437902\pi$
$164$	0	0
$165$	−1.63769	−0.127494
$166$	0	0
$167$	−12.3104	−0.952605	−0.476303	−	0.879281i	$-0.658023\pi$
$167$	−12.3104	−0.952605	−0.476303	+	0.879281i	$0.658023\pi$
$168$	0	0
$169$	10.4641	0.804931
$170$	0	0
$171$	3.27860	0.250721
$172$	0	0
$173$	−12.1552	−0.924142	−0.462071	−	0.886843i	$-0.652893\pi$
$173$	−12.1552	−0.924142	−0.462071	+	0.886843i	$0.652893\pi$
$174$	0	0
$175$	6.99687	0.528914
$176$	0	0
$177$	−5.38829	−0.405008
$178$	0	0
$179$	7.18601	0.537108	0.268554	−	0.963265i	$-0.413454\pi$
$179$	7.18601	0.537108	0.268554	+	0.963265i	$0.413454\pi$
$180$	0	0
$181$	18.0526	1.34184	0.670918	−	0.741531i	$-0.265900\pi$
$181$	18.0526	1.34184	0.670918	+	0.741531i	$0.265900\pi$
$182$	0	0
$183$	3.54604	0.262131
$184$	0	0
$185$	−2.32877	−0.171214
$186$	0	0
$187$	−22.1622	−1.62066
$188$	0	0
$189$	5.76989	0.419697
$190$	0	0
$191$	−4.44560	−0.321673	−0.160836	−	0.986981i	$-0.551419\pi$
$191$	−4.44560	−0.321673	−0.160836	+	0.986981i	$0.551419\pi$
$192$	0	0
$193$	−14.4165	−1.03772	−0.518861	−	0.854859i	$-0.673644\pi$
$193$	−14.4165	−1.03772	−0.518861	+	0.854859i	$0.673644\pi$
$194$	0	0
$195$	1.37058	0.0981489
$196$	0	0
$197$	6.06381	0.432028	0.216014	−	0.976390i	$-0.430694\pi$
$197$	6.06381	0.432028	0.216014	+	0.976390i	$0.430694\pi$
$198$	0	0
$199$	−19.5830	−1.38820	−0.694102	−	0.719876i	$-0.744198\pi$
$199$	−19.5830	−1.38820	−0.694102	+	0.719876i	$0.744198\pi$
$200$	0	0
$201$	−7.96833	−0.562043
$202$	0	0
$203$	5.05596	0.354859
$204$	0	0
$205$	1.73869	0.121435
$206$	0	0
$207$	−17.9709	−1.24906
$208$	0	0
$209$	7.70122	0.532705
$210$	0	0
$211$	21.3376	1.46894	0.734469	−	0.678642i	$-0.237431\pi$
$211$	21.3376	1.46894	0.734469	+	0.678642i	$0.237431\pi$
$212$	0	0
$213$	−6.01416	−0.412083
$214$	0	0
$215$	−3.67380	−0.250551
$216$	0	0
$217$	−1.44247	−0.0979214
$218$	0	0
$219$	6.53590	0.441655
$220$	0	0
$221$	18.5475	1.24764
$222$	0	0
$223$	−19.2023	−1.28588	−0.642941	−	0.765916i	$-0.722286\pi$
$223$	−19.2023	−1.28588	−0.642941	+	0.765916i	$0.722286\pi$
$224$	0	0
$225$	11.9524	0.796827
$226$	0	0
$227$	−4.39230	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$227$	−4.39230	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$228$	0	0
$229$	3.52090	0.232668	0.116334	−	0.993210i	$-0.462886\pi$
$229$	3.52090	0.232668	0.116334	+	0.993210i	$0.462886\pi$
$230$	0	0
$231$	6.11192	0.402135
$232$	0	0
$233$	−3.11627	−0.204153	−0.102077	−	0.994777i	$-0.532549\pi$
$233$	−3.11627	−0.204153	−0.102077	+	0.994777i	$0.532549\pi$
$234$	0	0
$235$	−4.56589	−0.297846
$236$	0	0
$237$	0.358759	0.0233039
$238$	0	0
$239$	−0.357873	−0.0231489	−0.0115744	−	0.999933i	$-0.503684\pi$
$239$	−0.357873	−0.0231489	−0.0115744	+	0.999933i	$0.503684\pi$
$240$	0	0
$241$	12.2804	0.791049	0.395524	−	0.918455i	$-0.370563\pi$
$241$	12.2804	0.791049	0.395524	+	0.918455i	$0.370563\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	−1.90134	−0.121472
$246$	0	0
$247$	−6.44514	−0.410094
$248$	0	0
$249$	−12.8752	−0.815934
$250$	0	0
$251$	−14.7091	−0.928428	−0.464214	−	0.885723i	$-0.653663\pi$
$251$	−14.7091	−0.928428	−0.464214	+	0.885723i	$0.653663\pi$
$252$	0	0
$253$	−42.2125	−2.65387
$254$	0	0
$255$	1.08339	0.0678444
$256$	0	0
$257$	13.4653	0.839943	0.419972	−	0.907537i	$-0.362040\pi$
$257$	13.4653	0.839943	0.419972	+	0.907537i	$0.362040\pi$
$258$	0	0
$259$	8.69108	0.540038
$260$	0	0
$261$	8.63685	0.534607
$262$	0	0
$263$	21.7973	1.34408	0.672040	−	0.740515i	$-0.265418\pi$
$263$	21.7973	1.34408	0.672040	+	0.740515i	$0.265418\pi$
$264$	0	0
$265$	−4.95819	−0.304579
$266$	0	0
$267$	−11.8431	−0.724788
$268$	0	0
$269$	−4.09380	−0.249603	−0.124802	−	0.992182i	$-0.539829\pi$
$269$	−4.09380	−0.249603	−0.124802	+	0.992182i	$0.539829\pi$
$270$	0	0
$271$	23.9525	1.45501	0.727505	−	0.686102i	$-0.240680\pi$
$271$	23.9525	1.45501	0.727505	+	0.686102i	$0.240680\pi$
$272$	0	0
$273$	−5.11506	−0.309577
$274$	0	0
$275$	28.0754	1.69301
$276$	0	0
$277$	−19.3049	−1.15992	−0.579961	−	0.814644i	$-0.696932\pi$
$277$	−19.3049	−1.15992	−0.579961	+	0.814644i	$0.696932\pi$
$278$	0	0
$279$	−2.46410	−0.147522
$280$	0	0
$281$	−5.46531	−0.326033	−0.163017	−	0.986623i	$-0.552122\pi$
$281$	−5.46531	−0.326033	−0.163017	+	0.986623i	$0.552122\pi$
$282$	0	0
$283$	−0.674363	−0.0400867	−0.0200434	−	0.999799i	$-0.506380\pi$
$283$	−0.674363	−0.0400867	−0.0200434	+	0.999799i	$0.506380\pi$
$284$	0	0
$285$	−0.376471	−0.0223002
$286$	0	0
$287$	−6.48886	−0.383025
$288$	0	0
$289$	−2.33891	−0.137583
$290$	0	0
$291$	−0.766898	−0.0449564
$292$	0	0
$293$	−6.26711	−0.366128	−0.183064	−	0.983101i	$-0.558602\pi$
$293$	−6.26711	−0.366128	−0.183064	+	0.983101i	$0.558602\pi$
$294$	0	0
$295$	−2.84491	−0.165637
$296$	0	0
$297$	23.1521	1.34342
$298$	0	0
$299$	35.3275	2.04304
$300$	0	0
$301$	13.7108	0.790277
$302$	0	0
$303$	9.01270	0.517766
$304$	0	0
$305$	1.87224	0.107204
$306$	0	0
$307$	15.1824	0.866503	0.433252	−	0.901273i	$-0.357366\pi$
$307$	15.1824	0.866503	0.433252	+	0.901273i	$0.357366\pi$
$308$	0	0
$309$	9.38829	0.534081
$310$	0	0
$311$	1.51114	0.0856887	0.0428443	−	0.999082i	$-0.486358\pi$
$311$	1.51114	0.0856887	0.0428443	+	0.999082i	$0.486358\pi$
$312$	0	0
$313$	5.09296	0.287871	0.143936	−	0.989587i	$-0.454024\pi$
$313$	5.09296	0.287871	0.143936	+	0.989587i	$0.454024\pi$
$314$	0	0
$315$	1.37381	0.0774052
$316$	0	0
$317$	12.9357	0.726540	0.363270	−	0.931684i	$-0.381660\pi$
$317$	12.9357	0.726540	0.363270	+	0.931684i	$0.381660\pi$
$318$	0	0
$319$	20.2874	1.13588
$320$	0	0
$321$	12.8875	0.719311
$322$	0	0
$323$	−5.09464	−0.283473
$324$	0	0
$325$	−23.4962	−1.30334
$326$	0	0
$327$	0.374593	0.0207151
$328$	0	0
$329$	17.0401	0.939453
$330$	0	0
$331$	−1.05367	−0.0579149	−0.0289575	−	0.999581i	$-0.509219\pi$
$331$	−1.05367	−0.0579149	−0.0289575	+	0.999581i	$0.509219\pi$
$332$	0	0
$333$	14.8465	0.813585
$334$	0	0
$335$	−4.20713	−0.229860
$336$	0	0
$337$	−11.3161	−0.616425	−0.308213	−	0.951317i	$-0.599731\pi$
$337$	−11.3161	−0.616425	−0.308213	+	0.951317i	$0.599731\pi$
$338$	0	0
$339$	12.6204	0.685446
$340$	0	0
$341$	−5.78801	−0.313438
$342$	0	0
$343$	17.1932	0.928346
$344$	0	0
$345$	2.06354	0.111097
$346$	0	0
$347$	−20.4719	−1.09899	−0.549496	−	0.835496i	$-0.685180\pi$
$347$	−20.4719	−1.09899	−0.549496	+	0.835496i	$0.685180\pi$
$348$	0	0
$349$	−3.42084	−0.183113	−0.0915567	−	0.995800i	$-0.529184\pi$
$349$	−3.42084	−0.183113	−0.0915567	+	0.995800i	$0.529184\pi$
$350$	0	0
$351$	−19.3759	−1.03421
$352$	0	0
$353$	−26.3993	−1.40509	−0.702547	−	0.711638i	$-0.747954\pi$
$353$	−26.3993	−1.40509	−0.702547	+	0.711638i	$0.747954\pi$
$354$	0	0
$355$	−3.17536	−0.168531
$356$	0	0
$357$	−4.04326	−0.213992
$358$	0	0
$359$	−5.69799	−0.300728	−0.150364	−	0.988631i	$-0.548045\pi$
$359$	−5.69799	−0.300728	−0.150364	+	0.988631i	$0.548045\pi$
$360$	0	0
$361$	−17.2296	−0.906823
$362$	0	0
$363$	16.4719	0.864553
$364$	0	0
$365$	3.45083	0.180625
$366$	0	0
$367$	21.9770	1.14719	0.573595	−	0.819139i	$-0.305548\pi$
$367$	21.9770	1.14719	0.573595	+	0.819139i	$0.305548\pi$
$368$	0	0
$369$	−11.0846	−0.577041
$370$	0	0
$371$	18.5042	0.960692
$372$	0	0
$373$	25.7324	1.33237	0.666187	−	0.745785i	$-0.267925\pi$
$373$	25.7324	1.33237	0.666187	+	0.745785i	$0.267925\pi$
$374$	0	0
$375$	−2.78717	−0.143929
$376$	0	0
$377$	−16.9785	−0.874436
$378$	0	0
$379$	−11.4122	−0.586203	−0.293102	−	0.956081i	$-0.594687\pi$
$379$	−11.4122	−0.586203	−0.293102	+	0.956081i	$0.594687\pi$
$380$	0	0
$381$	−1.47382	−0.0755059
$382$	0	0
$383$	1.78942	0.0914351	0.0457175	−	0.998954i	$-0.485443\pi$
$383$	1.78942	0.0914351	0.0457175	+	0.998954i	$0.485443\pi$
$384$	0	0
$385$	3.22698	0.164462
$386$	0	0
$387$	23.4215	1.19058
$388$	0	0
$389$	−26.1503	−1.32587	−0.662937	−	0.748675i	$-0.730690\pi$
$389$	−26.1503	−1.32587	−0.662937	+	0.748675i	$0.730690\pi$
$390$	0	0
$391$	27.9251	1.41223
$392$	0	0
$393$	−1.97972	−0.0998639
$394$	0	0
$395$	0.189418	0.00953066
$396$	0	0
$397$	−1.76868	−0.0887673	−0.0443837	−	0.999015i	$-0.514132\pi$
$397$	−1.76868	−0.0887673	−0.0443837	+	0.999015i	$0.514132\pi$
$398$	0	0
$399$	1.40501	0.0703383
$400$	0	0
$401$	19.5302	0.975292	0.487646	−	0.873042i	$-0.337856\pi$
$401$	19.5302	0.975292	0.487646	+	0.873042i	$0.337856\pi$
$402$	0	0
$403$	4.84398	0.241296
$404$	0	0
$405$	1.72542	0.0857366
$406$	0	0
$407$	34.8736	1.72862
$408$	0	0
$409$	−18.2513	−0.902468	−0.451234	−	0.892406i	$-0.649016\pi$
$409$	−18.2513	−0.902468	−0.451234	+	0.892406i	$0.649016\pi$
$410$	0	0
$411$	−1.85453	−0.0914772
$412$	0	0
$413$	10.6174	0.522446
$414$	0	0
$415$	−6.79788	−0.333695
$416$	0	0
$417$	−0.771340	−0.0377727
$418$	0	0
$419$	0.116835	0.00570774	0.00285387	−	0.999996i	$-0.499092\pi$
$419$	0.116835	0.00570774	0.00285387	+	0.999996i	$0.499092\pi$
$420$	0	0
$421$	−36.2683	−1.76761	−0.883805	−	0.467856i	$-0.845027\pi$
$421$	−36.2683	−1.76761	−0.883805	+	0.467856i	$0.845027\pi$
$422$	0	0
$423$	29.1088	1.41532
$424$	0	0
$425$	−18.5729	−0.900918
$426$	0	0
$427$	−6.98730	−0.338139
$428$	0	0
$429$	−20.5245	−0.990932
$430$	0	0
$431$	−13.9683	−0.672831	−0.336415	−	0.941714i	$-0.609215\pi$
$431$	−13.9683	−0.672831	−0.336415	+	0.941714i	$0.609215\pi$
$432$	0	0
$433$	−5.92419	−0.284698	−0.142349	−	0.989817i	$-0.545466\pi$
$433$	−5.92419	−0.284698	−0.142349	+	0.989817i	$0.545466\pi$
$434$	0	0
$435$	−0.991739	−0.0475503
$436$	0	0
$437$	−9.70379	−0.464195
$438$	0	0
$439$	7.87971	0.376078	0.188039	−	0.982162i	$-0.439787\pi$
$439$	7.87971	0.376078	0.188039	+	0.982162i	$0.439787\pi$
$440$	0	0
$441$	12.1216	0.577219
$442$	0	0
$443$	−33.0434	−1.56994	−0.784969	−	0.619535i	$-0.787321\pi$
$443$	−33.0434	−1.56994	−0.784969	+	0.619535i	$0.787321\pi$
$444$	0	0
$445$	−6.25295	−0.296418
$446$	0	0
$447$	6.44425	0.304803
$448$	0	0
$449$	27.3459	1.29053	0.645267	−	0.763957i	$-0.276746\pi$
$449$	27.3459	1.29053	0.645267	+	0.763957i	$0.276746\pi$
$450$	0	0
$451$	−26.0370	−1.22603
$452$	0	0
$453$	15.9251	0.748225
$454$	0	0
$455$	−2.70065	−0.126609
$456$	0	0
$457$	−22.8233	−1.06763	−0.533814	−	0.845602i	$-0.679242\pi$
$457$	−22.8233	−1.06763	−0.533814	+	0.845602i	$0.679242\pi$
$458$	0	0
$459$	−15.3159	−0.714886
$460$	0	0
$461$	21.8515	1.01773	0.508864	−	0.860847i	$-0.330066\pi$
$461$	21.8515	1.01773	0.508864	+	0.860847i	$0.330066\pi$
$462$	0	0
$463$	19.1088	0.888061	0.444030	−	0.896012i	$-0.353548\pi$
$463$	19.1088	0.888061	0.444030	+	0.896012i	$0.353548\pi$
$464$	0	0
$465$	0.282944	0.0131212
$466$	0	0
$467$	−11.7166	−0.542179	−0.271090	−	0.962554i	$-0.587384\pi$
$467$	−11.7166	−0.542179	−0.271090	+	0.962554i	$0.587384\pi$
$468$	0	0
$469$	15.7012	0.725015
$470$	0	0
$471$	0.820521	0.0378076
$472$	0	0
$473$	55.0155	2.52961
$474$	0	0
$475$	6.45396	0.296128
$476$	0	0
$477$	31.6098	1.44732
$478$	0	0
$479$	4.43934	0.202839	0.101419	−	0.994844i	$-0.467662\pi$
$479$	4.43934	0.202839	0.101419	+	0.994844i	$0.467662\pi$
$480$	0	0
$481$	−29.1856	−1.33075
$482$	0	0
$483$	−7.70122	−0.350418
$484$	0	0
$485$	−0.404908	−0.0183859
$486$	0	0
$487$	41.6334	1.88659	0.943296	−	0.331954i	$-0.107708\pi$
$487$	41.6334	1.88659	0.943296	+	0.331954i	$0.107708\pi$
$488$	0	0
$489$	3.62353	0.163862
$490$	0	0
$491$	−36.3355	−1.63980	−0.819899	−	0.572508i	$-0.805971\pi$
$491$	−36.3355	−1.63980	−0.819899	+	0.572508i	$0.805971\pi$
$492$	0	0
$493$	−13.4208	−0.604444
$494$	0	0
$495$	5.51249	0.247768
$496$	0	0
$497$	11.8506	0.531573
$498$	0	0
$499$	16.0684	0.719320	0.359660	−	0.933083i	$-0.382893\pi$
$499$	16.0684	0.719320	0.359660	+	0.933083i	$0.382893\pi$
$500$	0	0
$501$	−9.01182	−0.402618
$502$	0	0
$503$	−9.33055	−0.416029	−0.208014	−	0.978126i	$-0.566700\pi$
$503$	−9.33055	−0.416029	−0.208014	+	0.978126i	$0.566700\pi$
$504$	0	0
$505$	4.75854	0.211752
$506$	0	0
$507$	7.66025	0.340204
$508$	0	0
$509$	−33.1251	−1.46824	−0.734122	−	0.679018i	$-0.762406\pi$
$509$	−33.1251	−1.46824	−0.734122	+	0.679018i	$0.762406\pi$
$510$	0	0
$511$	−12.8787	−0.569719
$512$	0	0
$513$	5.32219	0.234980
$514$	0	0
$515$	4.95684	0.218424
$516$	0	0
$517$	68.3746	3.00711
$518$	0	0
$519$	−8.89821	−0.390588
$520$	0	0
$521$	−32.1552	−1.40874	−0.704372	−	0.709831i	$-0.748771\pi$
$521$	−32.1552	−1.40874	−0.704372	+	0.709831i	$0.748771\pi$
$522$	0	0
$523$	3.88410	0.169840	0.0849200	−	0.996388i	$-0.472937\pi$
$523$	3.88410	0.169840	0.0849200	+	0.996388i	$0.472937\pi$
$524$	0	0
$525$	5.12206	0.223545
$526$	0	0
$527$	3.82898	0.166793
$528$	0	0
$529$	30.1890	1.31257
$530$	0	0
$531$	18.1371	0.787084
$532$	0	0
$533$	21.7903	0.943843
$534$	0	0
$535$	6.80436	0.294178
$536$	0	0
$537$	5.26053	0.227009
$538$	0	0
$539$	28.4728	1.22641
$540$	0	0
$541$	21.9995	0.945834	0.472917	−	0.881107i	$-0.343201\pi$
$541$	21.9995	0.945834	0.472917	+	0.881107i	$0.343201\pi$
$542$	0	0
$543$	13.2154	0.567127
$544$	0	0
$545$	0.197778	0.00847189
$546$	0	0
$547$	−18.0384	−0.771264	−0.385632	−	0.922653i	$-0.626017\pi$
$547$	−18.0384	−0.771264	−0.385632	+	0.922653i	$0.626017\pi$
$548$	0	0
$549$	−11.9360	−0.509418
$550$	0	0
$551$	4.66366	0.198678
$552$	0	0
$553$	−0.706918	−0.0300612
$554$	0	0
$555$	−1.70478	−0.0723638
$556$	0	0
$557$	−15.1859	−0.643446	−0.321723	−	0.946834i	$-0.604262\pi$
$557$	−15.1859	−0.643446	−0.321723	+	0.946834i	$0.604262\pi$
$558$	0	0
$559$	−46.0423	−1.94738
$560$	0	0
$561$	−16.2238	−0.684971
$562$	0	0
$563$	32.9252	1.38763	0.693815	−	0.720153i	$-0.255928\pi$
$563$	32.9252	1.38763	0.693815	+	0.720153i	$0.255928\pi$
$564$	0	0
$565$	6.66333	0.280329
$566$	0	0
$567$	−6.43934	−0.270427
$568$	0	0
$569$	16.5792	0.695034	0.347517	−	0.937674i	$-0.387025\pi$
$569$	16.5792	0.695034	0.347517	+	0.937674i	$0.387025\pi$
$570$	0	0
$571$	14.8626	0.621979	0.310990	−	0.950413i	$-0.399340\pi$
$571$	14.8626	0.621979	0.310990	+	0.950413i	$0.399340\pi$
$572$	0	0
$573$	−3.25441	−0.135955
$574$	0	0
$575$	−35.3759	−1.47528
$576$	0	0
$577$	−9.40589	−0.391572	−0.195786	−	0.980647i	$-0.562726\pi$
$577$	−9.40589	−0.391572	−0.195786	+	0.980647i	$0.562726\pi$
$578$	0	0
$579$	−10.5536	−0.438593
$580$	0	0
$581$	25.3700	1.05253
$582$	0	0
$583$	74.2495	3.07510
$584$	0	0
$585$	−4.61339	−0.190740
$586$	0	0
$587$	−1.83127	−0.0755847	−0.0377924	−	0.999286i	$-0.512033\pi$
$587$	−1.83127	−0.0755847	−0.0377924	+	0.999286i	$0.512033\pi$
$588$	0	0
$589$	−1.33055	−0.0548242
$590$	0	0
$591$	4.43902	0.182597
$592$	0	0
$593$	−17.2282	−0.707477	−0.353739	−	0.935344i	$-0.615090\pi$
$593$	−17.2282	−0.707477	−0.353739	+	0.935344i	$0.615090\pi$
$594$	0	0
$595$	−2.13477	−0.0875169
$596$	0	0
$597$	−14.3358	−0.586724
$598$	0	0
$599$	11.0917	0.453197	0.226598	−	0.973988i	$-0.427240\pi$
$599$	11.0917	0.453197	0.226598	+	0.973988i	$0.427240\pi$
$600$	0	0
$601$	34.2583	1.39742	0.698712	−	0.715403i	$-0.253757\pi$
$601$	34.2583	1.39742	0.698712	+	0.715403i	$0.253757\pi$
$602$	0	0
$603$	26.8216	1.09226
$604$	0	0
$605$	8.69688	0.353578
$606$	0	0
$607$	40.5026	1.64395	0.821974	−	0.569525i	$-0.192873\pi$
$607$	40.5026	1.64395	0.821974	+	0.569525i	$0.192873\pi$
$608$	0	0
$609$	3.70122	0.149981
$610$	0	0
$611$	−57.2226	−2.31498
$612$	0	0
$613$	11.6577	0.470850	0.235425	−	0.971893i	$-0.424352\pi$
$613$	11.6577	0.470850	0.235425	+	0.971893i	$0.424352\pi$
$614$	0	0
$615$	1.27281	0.0513245
$616$	0	0
$617$	−20.5112	−0.825751	−0.412876	−	0.910787i	$-0.635476\pi$
$617$	−20.5112	−0.825751	−0.412876	+	0.910787i	$0.635476\pi$
$618$	0	0
$619$	−39.1251	−1.57257	−0.786285	−	0.617864i	$-0.787998\pi$
$619$	−39.1251	−1.57257	−0.786285	+	0.617864i	$0.787998\pi$
$620$	0	0
$621$	−29.1723	−1.17065
$622$	0	0
$623$	23.3363	0.934951
$624$	0	0
$625$	22.7815	0.911259
$626$	0	0
$627$	5.63769	0.225147
$628$	0	0
$629$	−23.0701	−0.919866
$630$	0	0
$631$	5.07880	0.202184	0.101092	−	0.994877i	$-0.467766\pi$
$631$	5.07880	0.202184	0.101092	+	0.994877i	$0.467766\pi$
$632$	0	0
$633$	15.6202	0.620846
$634$	0	0
$635$	−0.778147	−0.0308798
$636$	0	0
$637$	−23.8289	−0.944134
$638$	0	0
$639$	20.2438	0.800832
$640$	0	0
$641$	−25.2507	−0.997343	−0.498672	−	0.866791i	$-0.666179\pi$
$641$	−25.2507	−0.997343	−0.498672	+	0.866791i	$0.666179\pi$
$642$	0	0
$643$	19.9456	0.786578	0.393289	−	0.919415i	$-0.371337\pi$
$643$	19.9456	0.786578	0.393289	+	0.919415i	$0.371337\pi$
$644$	0	0
$645$	−2.68941	−0.105895
$646$	0	0
$647$	−37.3459	−1.46822	−0.734110	−	0.679031i	$-0.762400\pi$
$647$	−37.3459	−1.46822	−0.734110	+	0.679031i	$0.762400\pi$
$648$	0	0
$649$	42.6029	1.67231
$650$	0	0
$651$	−1.05596	−0.0413864
$652$	0	0
$653$	21.7566	0.851402	0.425701	−	0.904864i	$-0.360028\pi$
$653$	21.7566	0.851402	0.425701	+	0.904864i	$0.360028\pi$
$654$	0	0
$655$	−1.04526	−0.0408416
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	18.4307	0.717956	0.358978	−	0.933346i	$-0.383125\pi$
$659$	18.4307	0.717956	0.358978	+	0.933346i	$0.383125\pi$
$660$	0	0
$661$	−32.7788	−1.27495	−0.637474	−	0.770472i	$-0.720021\pi$
$661$	−32.7788	−1.27495	−0.637474	+	0.770472i	$0.720021\pi$
$662$	0	0
$663$	13.5777	0.527314
$664$	0	0
$665$	0.741818	0.0287664
$666$	0	0
$667$	−25.5628	−0.989794
$668$	0	0
$669$	−14.0571	−0.543478
$670$	0	0
$671$	−28.0370	−1.08236
$672$	0	0
$673$	−34.7422	−1.33921	−0.669607	−	0.742715i	$-0.733538\pi$
$673$	−34.7422	−1.33921	−0.669607	+	0.742715i	$0.733538\pi$
$674$	0	0
$675$	19.4024	0.746800
$676$	0	0
$677$	−15.8913	−0.610750	−0.305375	−	0.952232i	$-0.598782\pi$
$677$	−15.8913	−0.610750	−0.305375	+	0.952232i	$0.598782\pi$
$678$	0	0
$679$	1.51114	0.0579921
$680$	0	0
$681$	−3.21539	−0.123214
$682$	0	0
$683$	15.3895	0.588863	0.294431	−	0.955673i	$-0.404870\pi$
$683$	15.3895	0.588863	0.294431	+	0.955673i	$0.404870\pi$
$684$	0	0
$685$	−0.979157	−0.0374117
$686$	0	0
$687$	2.57748	0.0983370
$688$	0	0
$689$	−62.1392	−2.36732
$690$	0	0
$691$	−4.70645	−0.179042	−0.0895209	−	0.995985i	$-0.528534\pi$
$691$	−4.70645	−0.179042	−0.0895209	+	0.995985i	$0.528534\pi$
$692$	0	0
$693$	−20.5729	−0.781500
$694$	0	0
$695$	−0.407253	−0.0154480
$696$	0	0
$697$	17.2244	0.652421
$698$	0	0
$699$	−2.28127	−0.0862854
$700$	0	0
$701$	−18.9797	−0.716853	−0.358426	−	0.933558i	$-0.616687\pi$
$701$	−18.9797	−0.716853	−0.358426	+	0.933558i	$0.616687\pi$
$702$	0	0
$703$	8.01672	0.302356
$704$	0	0
$705$	−3.34246	−0.125884
$706$	0	0
$707$	−17.7591	−0.667900
$708$	0	0
$709$	32.9920	1.23904	0.619520	−	0.784980i	$-0.287327\pi$
$709$	32.9920	1.23904	0.619520	+	0.784980i	$0.287327\pi$
$710$	0	0
$711$	−1.20759	−0.0452882
$712$	0	0
$713$	7.29308	0.273128
$714$	0	0
$715$	−10.8366	−0.405264
$716$	0	0
$717$	−0.261981	−0.00978386
$718$	0	0
$719$	0.381599	0.0142313	0.00711563	−	0.999975i	$-0.497735\pi$
$719$	0.381599	0.0142313	0.00711563	+	0.999975i	$0.497735\pi$
$720$	0	0
$721$	−18.4992	−0.688945
$722$	0	0
$723$	8.98986	0.334337
$724$	0	0
$725$	17.0017	0.631428
$726$	0	0
$727$	7.10043	0.263340	0.131670	−	0.991294i	$-0.457966\pi$
$727$	7.10043	0.263340	0.131670	+	0.991294i	$0.457966\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−36.3947	−1.34611
$732$	0	0
$733$	−6.19966	−0.228990	−0.114495	−	0.993424i	$-0.536525\pi$
$733$	−6.19966	−0.228990	−0.114495	+	0.993424i	$0.536525\pi$
$734$	0	0
$735$	−1.39188	−0.0513403
$736$	0	0
$737$	63.0022	2.32072
$738$	0	0
$739$	−2.91378	−0.107185	−0.0535925	−	0.998563i	$-0.517067\pi$
$739$	−2.91378	−0.107185	−0.0535925	+	0.998563i	$0.517067\pi$
$740$	0	0
$741$	−4.71817	−0.173326
$742$	0	0
$743$	−38.6303	−1.41721	−0.708604	−	0.705606i	$-0.750675\pi$
$743$	−38.6303	−1.41721	−0.708604	+	0.705606i	$0.750675\pi$
$744$	0	0
$745$	3.40244	0.124656
$746$	0	0
$747$	43.3383	1.58567
$748$	0	0
$749$	−25.3942	−0.927885
$750$	0	0
$751$	9.30756	0.339638	0.169819	−	0.985475i	$-0.445682\pi$
$751$	9.30756	0.339638	0.169819	+	0.985475i	$0.445682\pi$
$752$	0	0
$753$	−10.7678	−0.392400
$754$	0	0
$755$	8.40814	0.306004
$756$	0	0
$757$	26.0647	0.947337	0.473669	−	0.880703i	$-0.342929\pi$
$757$	26.0647	0.947337	0.473669	+	0.880703i	$0.342929\pi$
$758$	0	0
$759$	−30.9017	−1.12166
$760$	0	0
$761$	30.4380	1.10338	0.551688	−	0.834051i	$-0.313984\pi$
$761$	30.4380	1.10338	0.551688	+	0.834051i	$0.313984\pi$
$762$	0	0
$763$	−0.738118	−0.0267217
$764$	0	0
$765$	−3.64671	−0.131847
$766$	0	0
$767$	−35.6543	−1.28740
$768$	0	0
$769$	−8.61662	−0.310723	−0.155362	−	0.987858i	$-0.549654\pi$
$769$	−8.61662	−0.310723	−0.155362	+	0.987858i	$0.549654\pi$
$770$	0	0
$771$	9.85729	0.355002
$772$	0	0
$773$	15.1411	0.544587	0.272293	−	0.962214i	$-0.412218\pi$
$773$	15.1411	0.544587	0.272293	+	0.962214i	$0.412218\pi$
$774$	0	0
$775$	−4.85061	−0.174239
$776$	0	0
$777$	6.36231	0.228247
$778$	0	0
$779$	−5.98538	−0.214448
$780$	0	0
$781$	47.5514	1.70152
$782$	0	0
$783$	14.0203	0.501044
$784$	0	0
$785$	0.433220	0.0154623
$786$	0	0
$787$	−5.63727	−0.200947	−0.100473	−	0.994940i	$-0.532036\pi$
$787$	−5.63727	−0.200947	−0.100473	+	0.994940i	$0.532036\pi$
$788$	0	0
$789$	15.9567	0.568075
$790$	0	0
$791$	−24.8679	−0.884201
$792$	0	0
$793$	23.4641	0.833235
$794$	0	0
$795$	−3.62965	−0.128730
$796$	0	0
$797$	−3.30919	−0.117217	−0.0586087	−	0.998281i	$-0.518666\pi$
$797$	−3.30919	−0.117217	−0.0586087	+	0.998281i	$0.518666\pi$
$798$	0	0
$799$	−45.2323	−1.60020
$800$	0	0
$801$	39.8643	1.40854
$802$	0	0
$803$	−51.6766	−1.82363
$804$	0	0
$805$	−4.06610	−0.143311
$806$	0	0
$807$	−2.99687	−0.105495
$808$	0	0
$809$	−49.6476	−1.74552	−0.872758	−	0.488153i	$-0.837671\pi$
$809$	−49.6476	−1.74552	−0.872758	+	0.488153i	$0.837671\pi$
$810$	0	0
$811$	28.4980	1.00070	0.500349	−	0.865824i	$-0.333205\pi$
$811$	28.4980	1.00070	0.500349	+	0.865824i	$0.333205\pi$
$812$	0	0
$813$	17.5344	0.614960
$814$	0	0
$815$	1.91316	0.0670149
$816$	0	0
$817$	12.6469	0.442460
$818$	0	0
$819$	17.2174	0.601625
$820$	0	0
$821$	8.40328	0.293277	0.146638	−	0.989190i	$-0.453155\pi$
$821$	8.40328	0.293277	0.146638	+	0.989190i	$0.453155\pi$
$822$	0	0
$823$	31.5123	1.09845	0.549226	−	0.835674i	$-0.314923\pi$
$823$	31.5123	1.09845	0.549226	+	0.835674i	$0.314923\pi$
$824$	0	0
$825$	20.5526	0.715550
$826$	0	0
$827$	29.4804	1.02513	0.512567	−	0.858647i	$-0.328695\pi$
$827$	29.4804	1.02513	0.512567	+	0.858647i	$0.328695\pi$
$828$	0	0
$829$	42.5091	1.47640	0.738201	−	0.674581i	$-0.235676\pi$
$829$	42.5091	1.47640	0.738201	+	0.674581i	$0.235676\pi$
$830$	0	0
$831$	−14.1322	−0.490241
$832$	0	0
$833$	−18.8358	−0.652622
$834$	0	0
$835$	−4.75807	−0.164660
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	−51.3072	−1.77132	−0.885661	−	0.464332i	$-0.846294\pi$
$839$	−51.3072	−1.77132	−0.885661	+	0.464332i	$0.846294\pi$
$840$	0	0
$841$	−16.7145	−0.576362
$842$	0	0
$843$	−4.00089	−0.137798
$844$	0	0
$845$	4.04447	0.139134
$846$	0	0
$847$	−32.4572	−1.11524
$848$	0	0
$849$	−0.493668	−0.0169426
$850$	0	0
$851$	−43.9418	−1.50631
$852$	0	0
$853$	−37.4094	−1.28088	−0.640438	−	0.768010i	$-0.721247\pi$
$853$	−37.4094	−1.28088	−0.640438	+	0.768010i	$0.721247\pi$
$854$	0	0
$855$	1.26721	0.0433377
$856$	0	0
$857$	−36.4796	−1.24612	−0.623059	−	0.782175i	$-0.714110\pi$
$857$	−36.4796	−1.24612	−0.623059	+	0.782175i	$0.714110\pi$
$858$	0	0
$859$	42.6570	1.45544	0.727719	−	0.685876i	$-0.240581\pi$
$859$	42.6570	1.45544	0.727719	+	0.685876i	$0.240581\pi$
$860$	0	0
$861$	−4.75018	−0.161886
$862$	0	0
$863$	16.8921	0.575014	0.287507	−	0.957779i	$-0.407174\pi$
$863$	16.8921	0.575014	0.287507	+	0.957779i	$0.407174\pi$
$864$	0	0
$865$	−4.69809	−0.159740
$866$	0	0
$867$	−1.71220	−0.0581493
$868$	0	0
$869$	−2.83655	−0.0962235
$870$	0	0
$871$	−52.7264	−1.78657
$872$	0	0
$873$	2.58140	0.0873671
$874$	0	0
$875$	5.49200	0.185663
$876$	0	0
$877$	−45.3974	−1.53296	−0.766480	−	0.642268i	$-0.777994\pi$
$877$	−45.3974	−1.53296	−0.766480	+	0.642268i	$0.777994\pi$
$878$	0	0
$879$	−4.58784	−0.154744
$880$	0	0
$881$	−44.2026	−1.48922	−0.744612	−	0.667497i	$-0.767366\pi$
$881$	−44.2026	−1.48922	−0.744612	+	0.667497i	$0.767366\pi$
$882$	0	0
$883$	9.08722	0.305809	0.152905	−	0.988241i	$-0.451137\pi$
$883$	9.08722	0.305809	0.152905	+	0.988241i	$0.451137\pi$
$884$	0	0
$885$	−2.08262	−0.0700066
$886$	0	0
$887$	−3.50872	−0.117811	−0.0589056	−	0.998264i	$-0.518761\pi$
$887$	−3.50872	−0.117811	−0.0589056	+	0.998264i	$0.518761\pi$
$888$	0	0
$889$	2.90408	0.0973999
$890$	0	0
$891$	−25.8383	−0.865615
$892$	0	0
$893$	15.7179	0.525981
$894$	0	0
$895$	2.77746	0.0928402
$896$	0	0
$897$	25.8615	0.863492
$898$	0	0
$899$	−3.50507	−0.116901
$900$	0	0
$901$	−49.1187	−1.63638
$902$	0	0
$903$	10.0370	0.334010
$904$	0	0
$905$	6.97748	0.231939
$906$	0	0
$907$	−0.740931	−0.0246022	−0.0123011	−	0.999924i	$-0.503916\pi$
$907$	−0.740931	−0.0246022	−0.0123011	+	0.999924i	$0.503916\pi$
$908$	0	0
$909$	−30.3370	−1.00621
$910$	0	0
$911$	−20.8824	−0.691864	−0.345932	−	0.938260i	$-0.612437\pi$
$911$	−20.8824	−0.691864	−0.345932	+	0.938260i	$0.612437\pi$
$912$	0	0
$913$	101.799	3.36905
$914$	0	0
$915$	1.37058	0.0453098
$916$	0	0
$917$	3.90095	0.128821
$918$	0	0
$919$	−18.0254	−0.594603	−0.297302	−	0.954784i	$-0.596087\pi$
$919$	−18.0254	−0.594603	−0.297302	+	0.954784i	$0.596087\pi$
$920$	0	0
$921$	11.1143	0.366227
$922$	0	0
$923$	−39.7956	−1.30989
$924$	0	0
$925$	29.2256	0.960931
$926$	0	0
$927$	−31.6012	−1.03792
$928$	0	0
$929$	−0.191439	−0.00628090	−0.00314045	−	0.999995i	$-0.501000\pi$
$929$	−0.191439	−0.00628090	−0.00314045	+	0.999995i	$0.501000\pi$
$930$	0	0
$931$	6.54533	0.214514
$932$	0	0
$933$	1.10623	0.0362163
$934$	0	0
$935$	−8.56589	−0.280135
$936$	0	0
$937$	−51.1839	−1.67211	−0.836053	−	0.548649i	$-0.815142\pi$
$937$	−51.1839	−1.67211	−0.836053	+	0.548649i	$0.815142\pi$
$938$	0	0
$939$	3.72830	0.121669
$940$	0	0
$941$	24.1319	0.786678	0.393339	−	0.919393i	$-0.371320\pi$
$941$	24.1319	0.786678	0.393339	+	0.919393i	$0.371320\pi$
$942$	0	0
$943$	32.8075	1.06836
$944$	0	0
$945$	2.23011	0.0725456
$946$	0	0
$947$	31.6416	1.02821	0.514107	−	0.857726i	$-0.328123\pi$
$947$	31.6416	1.02821	0.514107	+	0.857726i	$0.328123\pi$
$948$	0	0
$949$	43.2480	1.40389
$950$	0	0
$951$	9.46957	0.307072
$952$	0	0
$953$	−21.1529	−0.685211	−0.342605	−	0.939479i	$-0.611309\pi$
$953$	−21.1529	−0.685211	−0.342605	+	0.939479i	$0.611309\pi$
$954$	0	0
$955$	−1.71827	−0.0556018
$956$	0	0
$957$	14.8514	0.480077
$958$	0	0
$959$	3.65426	0.118002
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−43.3797	−1.39789
$964$	0	0
$965$	−5.57211	−0.179373
$966$	0	0
$967$	−4.92475	−0.158369	−0.0791847	−	0.996860i	$-0.525232\pi$
$967$	−4.92475	−0.158369	−0.0791847	+	0.996860i	$0.525232\pi$
$968$	0	0
$969$	−3.72953	−0.119810
$970$	0	0
$971$	2.86822	0.0920457	0.0460228	−	0.998940i	$-0.485345\pi$
$971$	2.86822	0.0920457	0.0460228	+	0.998940i	$0.485345\pi$
$972$	0	0
$973$	1.51989	0.0487254
$974$	0	0
$975$	−17.2004	−0.550855
$976$	0	0
$977$	1.67244	0.0535061	0.0267531	−	0.999642i	$-0.491483\pi$
$977$	1.67244	0.0535061	0.0267531	+	0.999642i	$0.491483\pi$
$978$	0	0
$979$	93.6386	2.99270
$980$	0	0
$981$	−1.26089	−0.0402571
$982$	0	0
$983$	−25.7169	−0.820242	−0.410121	−	0.912031i	$-0.634514\pi$
$983$	−25.7169	−0.820242	−0.410121	+	0.912031i	$0.634514\pi$
$984$	0	0
$985$	2.34372	0.0746770
$986$	0	0
$987$	12.4742	0.397059
$988$	0	0
$989$	−69.3213	−2.20429
$990$	0	0
$991$	17.2868	0.549134	0.274567	−	0.961568i	$-0.411466\pi$
$991$	17.2868	0.549134	0.274567	+	0.961568i	$0.411466\pi$
$992$	0	0
$993$	−0.771340	−0.0244777
$994$	0	0
$995$	−7.56902	−0.239954
$996$	0	0
$997$	−26.3133	−0.833349	−0.416675	−	0.909056i	$-0.636805\pi$
$997$	−26.3133	−0.833349	−0.416675	+	0.909056i	$0.636805\pi$
$998$	0	0
$999$	24.1005	0.762507

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +0.386509 q^{5} -1.44247 q^{7} -2.46410 q^{9} -5.78801 q^{11} +4.84398 q^{13} +0.282944 q^{15} +3.82898 q^{17} -1.33055 q^{19} -1.05596 q^{21} +7.29308 q^{23} -4.85061 q^{25} -4.00000 q^{27} -3.50507 q^{29} +1.00000 q^{31} -4.23712 q^{33} -0.557528 q^{35} -6.02513 q^{37} +3.54604 q^{39} +4.49843 q^{41} -9.50507 q^{43} -0.952398 q^{45} -11.8131 q^{47} -4.91928 q^{49} +2.80301 q^{51} -12.8281 q^{53} -2.23712 q^{55} -0.974028 q^{57} -7.36054 q^{59} +4.84398 q^{61} +3.55440 q^{63} +1.87224 q^{65} -10.8849 q^{67} +5.33891 q^{69} -8.21549 q^{71} +8.92820 q^{73} -3.55089 q^{75} +8.34904 q^{77} +0.490074 q^{79} +4.46410 q^{81} -17.5879 q^{83} +1.47994 q^{85} -2.56589 q^{87} -16.1780 q^{89} -6.98730 q^{91} +0.732051 q^{93} -0.514268 q^{95} -1.04760 q^{97} +14.2623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} - 10 q^{11} + 2 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 16 q^{27} + 4 q^{31} + 4 q^{33} - 12 q^{35} + 10 q^{37} + 2 q^{41} - 24 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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