LMFDB - Embedded newform 8092.2.a.z.1.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8092,2,Mod(1,8092)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8092, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8092.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8092 = 2^{2} \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8092.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0,8,0,8,0,20,0,28,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.6149453156$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 12 x^{18} + 240 x^{17} - 224 x^{16} - 2776 x^{15} + 5324 x^{14} + 15280 x^{13} + \cdots + 544 $ x^20 - 8x^19 - 12x^18 + 240x^17 - 224x^16 - 2776x^15 + 5324x^14 + 15280x^13 - 42463x^12 - 36576x^11 + 166532x^10 - 1048x^9 - 328476x^8 + 166744x^7 + 277764x^6 - 258464x^5 - 36367x^4 + 111904x^3 - 40080x^2 + 2688*x + 544
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.76359$ of defining polynomial
Character	$\chi$	$=$	8092.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76359 q^{3} -1.73772 q^{5} +1.00000 q^{7} +4.63745 q^{9} +5.10276 q^{11} +0.561096 q^{13} -4.80236 q^{15} +7.01515 q^{19} +2.76359 q^{21} +6.87639 q^{23} -1.98033 q^{25} +4.52525 q^{27} -7.63470 q^{29} +4.09027 q^{31} +14.1019 q^{33} -1.73772 q^{35} +5.85298 q^{37} +1.55064 q^{39} -12.4480 q^{41} -6.65715 q^{43} -8.05860 q^{45} +5.76036 q^{47} +1.00000 q^{49} +11.7885 q^{53} -8.86716 q^{55} +19.3870 q^{57} +2.65437 q^{59} +5.20183 q^{61} +4.63745 q^{63} -0.975028 q^{65} -11.9894 q^{67} +19.0035 q^{69} +4.46918 q^{71} -5.37549 q^{73} -5.47282 q^{75} +5.10276 q^{77} -9.17100 q^{79} -1.40640 q^{81} -0.667347 q^{83} -21.0992 q^{87} -1.94005 q^{89} +0.561096 q^{91} +11.3038 q^{93} -12.1904 q^{95} +16.3329 q^{97} +23.6638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 20 * q + 8 * q^3 + 8 * q^5 + 20 * q^7 + 28 * q^9 + 16 * q^11 + 8 * q^13 + 8 * q^15 + 8 * q^19 + 8 * q^21 + 40 * q^23 + 36 * q^25 + 32 * q^27 + 16 * q^29 + 16 * q^31 + 8 * q^33 + 8 * q^35 + 24 * q^37 + 16 * q^39 - 16 * q^41 + 16 * q^43 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 8 * q^55 + 24 * q^57 + 8 * q^61 + 28 * q^63 + 8 * q^65 - 8 * q^67 + 8 * q^69 + 72 * q^71 + 40 * q^75 + 16 * q^77 + 60 * q^81 + 8 * q^83 - 24 * q^89 + 8 * q^91 + 32 * q^93 + 48 * q^95 - 16 * q^97 + 72 * 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
$3$	2.76359	1.59556	0.797781	−	0.602948i	$-0.206007\pi$
$3$	2.76359	1.59556	0.797781	+	0.602948i	$0.206007\pi$
$4$	0	0
$5$	−1.73772	−0.777132	−0.388566	−	0.921421i	$-0.627030\pi$
$5$	−1.73772	−0.777132	−0.388566	+	0.921421i	$0.627030\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.63745	1.54582
$10$	0	0
$11$	5.10276	1.53854	0.769269	−	0.638925i	$-0.220620\pi$
$11$	5.10276	1.53854	0.769269	+	0.638925i	$0.220620\pi$
$12$	0	0
$13$	0.561096	0.155620	0.0778100	−	0.996968i	$-0.475207\pi$
$13$	0.561096	0.155620	0.0778100	+	0.996968i	$0.475207\pi$
$14$	0	0
$15$	−4.80236	−1.23996
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	7.01515	1.60939	0.804693	−	0.593691i	$-0.202330\pi$
$19$	7.01515	1.60939	0.804693	+	0.593691i	$0.202330\pi$
$20$	0	0
$21$	2.76359	0.603066
$22$	0	0
$23$	6.87639	1.43383	0.716913	−	0.697163i	$-0.245554\pi$
$23$	6.87639	1.43383	0.716913	+	0.697163i	$0.245554\pi$
$24$	0	0
$25$	−1.98033	−0.396065
$26$	0	0
$27$	4.52525	0.870884
$28$	0	0
$29$	−7.63470	−1.41773	−0.708864	−	0.705345i	$-0.750792\pi$
$29$	−7.63470	−1.41773	−0.708864	+	0.705345i	$0.750792\pi$
$30$	0	0
$31$	4.09027	0.734634	0.367317	−	0.930096i	$-0.380276\pi$
$31$	4.09027	0.734634	0.367317	+	0.930096i	$0.380276\pi$
$32$	0	0
$33$	14.1019	2.45483
$34$	0	0
$35$	−1.73772	−0.293728
$36$	0	0
$37$	5.85298	0.962225	0.481112	−	0.876659i	$-0.340233\pi$
$37$	5.85298	0.962225	0.481112	+	0.876659i	$0.340233\pi$
$38$	0	0
$39$	1.55064	0.248301
$40$	0	0
$41$	−12.4480	−1.94405	−0.972027	−	0.234869i	$-0.924534\pi$
$41$	−12.4480	−1.94405	−0.972027	+	0.234869i	$0.924534\pi$
$42$	0	0
$43$	−6.65715	−1.01521	−0.507603	−	0.861591i	$-0.669468\pi$
$43$	−6.65715	−1.01521	−0.507603	+	0.861591i	$0.669468\pi$
$44$	0	0
$45$	−8.05860	−1.20130
$46$	0	0
$47$	5.76036	0.840235	0.420118	−	0.907470i	$-0.361989\pi$
$47$	5.76036	0.840235	0.420118	+	0.907470i	$0.361989\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.7885	1.61927	0.809636	−	0.586932i	$-0.199664\pi$
$53$	11.7885	1.61927	0.809636	+	0.586932i	$0.199664\pi$
$54$	0	0
$55$	−8.86716	−1.19565
$56$	0	0
$57$	19.3870	2.56788
$58$	0	0
$59$	2.65437	0.345569	0.172785	−	0.984960i	$-0.444724\pi$
$59$	2.65437	0.345569	0.172785	+	0.984960i	$0.444724\pi$
$60$	0	0
$61$	5.20183	0.666027	0.333013	−	0.942922i	$-0.391935\pi$
$61$	5.20183	0.666027	0.333013	+	0.942922i	$0.391935\pi$
$62$	0	0
$63$	4.63745	0.584264
$64$	0	0
$65$	−0.975028	−0.120937
$66$	0	0
$67$	−11.9894	−1.46474	−0.732369	−	0.680908i	$-0.761585\pi$
$67$	−11.9894	−1.46474	−0.732369	+	0.680908i	$0.761585\pi$
$68$	0	0
$69$	19.0035	2.28776
$70$	0	0
$71$	4.46918	0.530394	0.265197	−	0.964194i	$-0.414563\pi$
$71$	4.46918	0.530394	0.265197	+	0.964194i	$0.414563\pi$
$72$	0	0
$73$	−5.37549	−0.629153	−0.314577	−	0.949232i	$-0.601863\pi$
$73$	−5.37549	−0.629153	−0.314577	+	0.949232i	$0.601863\pi$
$74$	0	0
$75$	−5.47282	−0.631946
$76$	0	0
$77$	5.10276	0.581513
$78$	0	0
$79$	−9.17100	−1.03182	−0.515909	−	0.856643i	$-0.672546\pi$
$79$	−9.17100	−1.03182	−0.515909	+	0.856643i	$0.672546\pi$
$80$	0	0
$81$	−1.40640	−0.156267
$82$	0	0
$83$	−0.667347	−0.0732509	−0.0366254	−	0.999329i	$-0.511661\pi$
$83$	−0.667347	−0.0732509	−0.0366254	+	0.999329i	$0.511661\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−21.0992	−2.26207
$88$	0	0
$89$	−1.94005	−0.205645	−0.102823	−	0.994700i	$-0.532787\pi$
$89$	−1.94005	−0.205645	−0.102823	+	0.994700i	$0.532787\pi$
$90$	0	0
$91$	0.561096	0.0588189
$92$	0	0
$93$	11.3038	1.17215
$94$	0	0
$95$	−12.1904	−1.25071
$96$	0	0
$97$	16.3329	1.65835	0.829175	−	0.558989i	$-0.188811\pi$
$97$	16.3329	1.65835	0.829175	+	0.558989i	$0.188811\pi$
$98$	0	0
$99$	23.6638	2.37830
$100$	0	0
$101$	−1.03096	−0.102584	−0.0512921	−	0.998684i	$-0.516334\pi$
$101$	−1.03096	−0.102584	−0.0512921	+	0.998684i	$0.516334\pi$
$102$	0	0
$103$	2.82191	0.278051	0.139026	−	0.990289i	$-0.455603\pi$
$103$	2.82191	0.278051	0.139026	+	0.990289i	$0.455603\pi$
$104$	0	0
$105$	−4.80236	−0.468662
$106$	0	0
$107$	−6.75748	−0.653270	−0.326635	−	0.945151i	$-0.605915\pi$
$107$	−6.75748	−0.653270	−0.326635	+	0.945151i	$0.605915\pi$
$108$	0	0
$109$	−4.63805	−0.444245	−0.222122	−	0.975019i	$-0.571298\pi$
$109$	−4.63805	−0.444245	−0.222122	+	0.975019i	$0.571298\pi$
$110$	0	0
$111$	16.1753	1.53529
$112$	0	0
$113$	7.30142	0.686860	0.343430	−	0.939178i	$-0.388411\pi$
$113$	7.30142	0.686860	0.343430	+	0.939178i	$0.388411\pi$
$114$	0	0
$115$	−11.9492	−1.11427
$116$	0	0
$117$	2.60206	0.240560
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.0381	1.36710
$122$	0	0
$123$	−34.4013	−3.10186
$124$	0	0
$125$	12.1299	1.08493
$126$	0	0
$127$	12.2749	1.08922	0.544611	−	0.838689i	$-0.316677\pi$
$127$	12.2749	1.08922	0.544611	+	0.838689i	$0.316677\pi$
$128$	0	0
$129$	−18.3977	−1.61982
$130$	0	0
$131$	3.35001	0.292692	0.146346	−	0.989233i	$-0.453249\pi$
$131$	3.35001	0.292692	0.146346	+	0.989233i	$0.453249\pi$
$132$	0	0
$133$	7.01515	0.608291
$134$	0	0
$135$	−7.86362	−0.676793
$136$	0	0
$137$	−1.81252	−0.154854	−0.0774270	−	0.996998i	$-0.524670\pi$
$137$	−1.81252	−0.154854	−0.0774270	+	0.996998i	$0.524670\pi$
$138$	0	0
$139$	−7.45961	−0.632716	−0.316358	−	0.948640i	$-0.602460\pi$
$139$	−7.45961	−0.632716	−0.316358	+	0.948640i	$0.602460\pi$
$140$	0	0
$141$	15.9193	1.34065
$142$	0	0
$143$	2.86314	0.239427
$144$	0	0
$145$	13.2670	1.10176
$146$	0	0
$147$	2.76359	0.227937
$148$	0	0
$149$	−8.70129	−0.712837	−0.356419	−	0.934326i	$-0.616002\pi$
$149$	−8.70129	−0.712837	−0.356419	+	0.934326i	$0.616002\pi$
$150$	0	0
$151$	−1.67737	−0.136502	−0.0682512	−	0.997668i	$-0.521742\pi$
$151$	−1.67737	−0.136502	−0.0682512	+	0.997668i	$0.521742\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.10775	−0.570908
$156$	0	0
$157$	−7.45089	−0.594646	−0.297323	−	0.954777i	$-0.596094\pi$
$157$	−7.45089	−0.594646	−0.297323	+	0.954777i	$0.596094\pi$
$158$	0	0
$159$	32.5786	2.58365
$160$	0	0
$161$	6.87639	0.541935
$162$	0	0
$163$	−0.0371483	−0.00290968	−0.00145484	−	0.999999i	$-0.500463\pi$
$163$	−0.0371483	−0.00290968	−0.00145484	+	0.999999i	$0.500463\pi$
$164$	0	0
$165$	−24.5052	−1.90773
$166$	0	0
$167$	−7.87688	−0.609531	−0.304766	−	0.952427i	$-0.598578\pi$
$167$	−7.87688	−0.609531	−0.304766	+	0.952427i	$0.598578\pi$
$168$	0	0
$169$	−12.6852	−0.975782
$170$	0	0
$171$	32.5324	2.48782
$172$	0	0
$173$	4.33545	0.329618	0.164809	−	0.986325i	$-0.447299\pi$
$173$	4.33545	0.329618	0.164809	+	0.986325i	$0.447299\pi$
$174$	0	0
$175$	−1.98033	−0.149699
$176$	0	0
$177$	7.33559	0.551377
$178$	0	0
$179$	22.0983	1.65170	0.825852	−	0.563887i	$-0.190695\pi$
$179$	22.0983	1.65170	0.825852	+	0.563887i	$0.190695\pi$
$180$	0	0
$181$	−15.3679	−1.14228	−0.571142	−	0.820851i	$-0.693499\pi$
$181$	−15.3679	−1.14228	−0.571142	+	0.820851i	$0.693499\pi$
$182$	0	0
$183$	14.3758	1.06269
$184$	0	0
$185$	−10.1709	−0.747776
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.52525	0.329163
$190$	0	0
$191$	17.8407	1.29090	0.645452	−	0.763801i	$-0.276669\pi$
$191$	17.8407	1.29090	0.645452	+	0.763801i	$0.276669\pi$
$192$	0	0
$193$	−22.1174	−1.59204	−0.796022	−	0.605268i	$-0.793066\pi$
$193$	−22.1174	−1.59204	−0.796022	+	0.605268i	$0.793066\pi$
$194$	0	0
$195$	−2.69458	−0.192963
$196$	0	0
$197$	19.0706	1.35872	0.679362	−	0.733803i	$-0.262257\pi$
$197$	19.0706	1.35872	0.679362	+	0.733803i	$0.262257\pi$
$198$	0	0
$199$	6.84328	0.485108	0.242554	−	0.970138i	$-0.422015\pi$
$199$	6.84328	0.485108	0.242554	+	0.970138i	$0.422015\pi$
$200$	0	0
$201$	−33.1338	−2.33708
$202$	0	0
$203$	−7.63470	−0.535851
$204$	0	0
$205$	21.6312	1.51079
$206$	0	0
$207$	31.8889	2.21643
$208$	0	0
$209$	35.7966	2.47610
$210$	0	0
$211$	−0.270884	−0.0186484	−0.00932421	−	0.999957i	$-0.502968\pi$
$211$	−0.270884	−0.0186484	−0.00932421	+	0.999957i	$0.502968\pi$
$212$	0	0
$213$	12.3510	0.846276
$214$	0	0
$215$	11.5683	0.788950
$216$	0	0
$217$	4.09027	0.277666
$218$	0	0
$219$	−14.8557	−1.00385
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.4758	−0.969373	−0.484686	−	0.874688i	$-0.661066\pi$
$223$	−14.4758	−0.969373	−0.484686	+	0.874688i	$0.661066\pi$
$224$	0	0
$225$	−9.18366	−0.612244
$226$	0	0
$227$	10.0316	0.665820	0.332910	−	0.942959i	$-0.391969\pi$
$227$	10.0316	0.665820	0.332910	+	0.942959i	$0.391969\pi$
$228$	0	0
$229$	19.6530	1.29870	0.649352	−	0.760488i	$-0.275040\pi$
$229$	19.6530	1.29870	0.649352	+	0.760488i	$0.275040\pi$
$230$	0	0
$231$	14.1019	0.927840
$232$	0	0
$233$	−12.9993	−0.851613	−0.425806	−	0.904814i	$-0.640009\pi$
$233$	−12.9993	−0.851613	−0.425806	+	0.904814i	$0.640009\pi$
$234$	0	0
$235$	−10.0099	−0.652974
$236$	0	0
$237$	−25.3449	−1.64633
$238$	0	0
$239$	−14.5987	−0.944309	−0.472154	−	0.881516i	$-0.656524\pi$
$239$	−14.5987	−0.944309	−0.472154	+	0.881516i	$0.656524\pi$
$240$	0	0
$241$	14.2479	0.917788	0.458894	−	0.888491i	$-0.348246\pi$
$241$	14.2479	0.917788	0.458894	+	0.888491i	$0.348246\pi$
$242$	0	0
$243$	−17.4625	−1.12022
$244$	0	0
$245$	−1.73772	−0.111019
$246$	0	0
$247$	3.93618	0.250453
$248$	0	0
$249$	−1.84428	−0.116876
$250$	0	0
$251$	2.88099	0.181846	0.0909231	−	0.995858i	$-0.471018\pi$
$251$	2.88099	0.181846	0.0909231	+	0.995858i	$0.471018\pi$
$252$	0	0
$253$	35.0885	2.20600
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.4037	0.711343	0.355671	−	0.934611i	$-0.384252\pi$
$257$	11.4037	0.711343	0.355671	+	0.934611i	$0.384252\pi$
$258$	0	0
$259$	5.85298	0.363687
$260$	0	0
$261$	−35.4055	−2.19155
$262$	0	0
$263$	−15.1008	−0.931154	−0.465577	−	0.885007i	$-0.654153\pi$
$263$	−15.1008	−0.931154	−0.465577	+	0.885007i	$0.654153\pi$
$264$	0	0
$265$	−20.4851	−1.25839
$266$	0	0
$267$	−5.36152	−0.328120
$268$	0	0
$269$	21.4708	1.30910	0.654548	−	0.756020i	$-0.272859\pi$
$269$	21.4708	1.30910	0.654548	+	0.756020i	$0.272859\pi$
$270$	0	0
$271$	−4.10050	−0.249088	−0.124544	−	0.992214i	$-0.539747\pi$
$271$	−4.10050	−0.249088	−0.124544	+	0.992214i	$0.539747\pi$
$272$	0	0
$273$	1.55064	0.0938491
$274$	0	0
$275$	−10.1051	−0.609361
$276$	0	0
$277$	26.3814	1.58510	0.792551	−	0.609805i	$-0.208752\pi$
$277$	26.3814	1.58510	0.792551	+	0.609805i	$0.208752\pi$
$278$	0	0
$279$	18.9684	1.13561
$280$	0	0
$281$	−3.53361	−0.210797	−0.105399	−	0.994430i	$-0.533612\pi$
$281$	−3.53361	−0.210797	−0.105399	+	0.994430i	$0.533612\pi$
$282$	0	0
$283$	−28.7263	−1.70760	−0.853801	−	0.520600i	$-0.825708\pi$
$283$	−28.7263	−1.70760	−0.853801	+	0.520600i	$0.825708\pi$
$284$	0	0
$285$	−33.6893	−1.99558
$286$	0	0
$287$	−12.4480	−0.734783
$288$	0	0
$289$	0	0
$290$	0	0
$291$	45.1374	2.64600
$292$	0	0
$293$	−28.6986	−1.67659	−0.838294	−	0.545219i	$-0.816447\pi$
$293$	−28.6986	−1.67659	−0.838294	+	0.545219i	$0.816447\pi$
$294$	0	0
$295$	−4.61255	−0.268553
$296$	0	0
$297$	23.0912	1.33989
$298$	0	0
$299$	3.85831	0.223132
$300$	0	0
$301$	−6.65715	−0.383712
$302$	0	0
$303$	−2.84915	−0.163679
$304$	0	0
$305$	−9.03934	−0.517591
$306$	0	0
$307$	−30.3698	−1.73330	−0.866648	−	0.498920i	$-0.833730\pi$
$307$	−30.3698	−1.73330	−0.866648	+	0.498920i	$0.833730\pi$
$308$	0	0
$309$	7.79862	0.443648
$310$	0	0
$311$	20.9900	1.19023	0.595116	−	0.803640i	$-0.297106\pi$
$311$	20.9900	1.19023	0.595116	+	0.803640i	$0.297106\pi$
$312$	0	0
$313$	−2.65641	−0.150149	−0.0750747	−	0.997178i	$-0.523920\pi$
$313$	−2.65641	−0.150149	−0.0750747	+	0.997178i	$0.523920\pi$
$314$	0	0
$315$	−8.05860	−0.454050
$316$	0	0
$317$	5.33352	0.299561	0.149780	−	0.988719i	$-0.452143\pi$
$317$	5.33352	0.299561	0.149780	+	0.988719i	$0.452143\pi$
$318$	0	0
$319$	−38.9580	−2.18123
$320$	0	0
$321$	−18.6749	−1.04233
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.11115	−0.0616357
$326$	0	0
$327$	−12.8177	−0.708820
$328$	0	0
$329$	5.76036	0.317579
$330$	0	0
$331$	5.15506	0.283348	0.141674	−	0.989913i	$-0.454752\pi$
$331$	5.15506	0.283348	0.141674	+	0.989913i	$0.454752\pi$
$332$	0	0
$333$	27.1429	1.48742
$334$	0	0
$335$	20.8342	1.13829
$336$	0	0
$337$	22.4126	1.22089	0.610445	−	0.792058i	$-0.290991\pi$
$337$	22.4126	1.22089	0.610445	+	0.792058i	$0.290991\pi$
$338$	0	0
$339$	20.1782	1.09593
$340$	0	0
$341$	20.8716	1.13026
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−33.0229	−1.77789
$346$	0	0
$347$	−4.61077	−0.247519	−0.123760	−	0.992312i	$-0.539495\pi$
$347$	−4.61077	−0.247519	−0.123760	+	0.992312i	$0.539495\pi$
$348$	0	0
$349$	−15.0190	−0.803947	−0.401973	−	0.915651i	$-0.631676\pi$
$349$	−15.0190	−0.803947	−0.401973	+	0.915651i	$0.631676\pi$
$350$	0	0
$351$	2.53910	0.135527
$352$	0	0
$353$	−18.5455	−0.987079	−0.493540	−	0.869723i	$-0.664297\pi$
$353$	−18.5455	−0.987079	−0.493540	+	0.869723i	$0.664297\pi$
$354$	0	0
$355$	−7.76618	−0.412186
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.5890	0.558867	0.279433	−	0.960165i	$-0.409853\pi$
$359$	10.5890	0.558867	0.279433	+	0.960165i	$0.409853\pi$
$360$	0	0
$361$	30.2124	1.59013
$362$	0	0
$363$	41.5592	2.18129
$364$	0	0
$365$	9.34110	0.488935
$366$	0	0
$367$	27.3075	1.42544	0.712719	−	0.701450i	$-0.247464\pi$
$367$	27.3075	1.42544	0.712719	+	0.701450i	$0.247464\pi$
$368$	0	0
$369$	−57.7271	−3.00515
$370$	0	0
$371$	11.7885	0.612028
$372$	0	0
$373$	31.4213	1.62693	0.813467	−	0.581612i	$-0.197578\pi$
$373$	31.4213	1.62693	0.813467	+	0.581612i	$0.197578\pi$
$374$	0	0
$375$	33.5220	1.73107
$376$	0	0
$377$	−4.28380	−0.220627
$378$	0	0
$379$	−0.806313	−0.0414175	−0.0207088	−	0.999786i	$-0.506592\pi$
$379$	−0.806313	−0.0414175	−0.0207088	+	0.999786i	$0.506592\pi$
$380$	0	0
$381$	33.9229	1.73792
$382$	0	0
$383$	−7.69358	−0.393124	−0.196562	−	0.980491i	$-0.562978\pi$
$383$	−7.69358	−0.393124	−0.196562	+	0.980491i	$0.562978\pi$
$384$	0	0
$385$	−8.86716	−0.451913
$386$	0	0
$387$	−30.8722	−1.56932
$388$	0	0
$389$	27.7323	1.40608	0.703042	−	0.711148i	$-0.251825\pi$
$389$	27.7323	1.40608	0.703042	+	0.711148i	$0.251825\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	9.25807	0.467008
$394$	0	0
$395$	15.9366	0.801860
$396$	0	0
$397$	−23.2174	−1.16525	−0.582625	−	0.812741i	$-0.697974\pi$
$397$	−23.2174	−1.16525	−0.582625	+	0.812741i	$0.697974\pi$
$398$	0	0
$399$	19.3870	0.970566
$400$	0	0
$401$	30.2217	1.50920	0.754600	−	0.656185i	$-0.227831\pi$
$401$	30.2217	1.50920	0.754600	+	0.656185i	$0.227831\pi$
$402$	0	0
$403$	2.29503	0.114324
$404$	0	0
$405$	2.44394	0.121440
$406$	0	0
$407$	29.8663	1.48042
$408$	0	0
$409$	0.590806	0.0292135	0.0146067	−	0.999893i	$-0.495350\pi$
$409$	0.590806	0.0292135	0.0146067	+	0.999893i	$0.495350\pi$
$410$	0	0
$411$	−5.00907	−0.247079
$412$	0	0
$413$	2.65437	0.130613
$414$	0	0
$415$	1.15966	0.0569256
$416$	0	0
$417$	−20.6153	−1.00954
$418$	0	0
$419$	20.3379	0.993571	0.496785	−	0.867873i	$-0.334514\pi$
$419$	20.3379	0.993571	0.496785	+	0.867873i	$0.334514\pi$
$420$	0	0
$421$	25.7759	1.25624	0.628120	−	0.778117i	$-0.283825\pi$
$421$	25.7759	1.25624	0.628120	+	0.778117i	$0.283825\pi$
$422$	0	0
$423$	26.7134	1.29885
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.20183	0.251734
$428$	0	0
$429$	7.91254	0.382021
$430$	0	0
$431$	10.1811	0.490408	0.245204	−	0.969472i	$-0.421145\pi$
$431$	10.1811	0.490408	0.245204	+	0.969472i	$0.421145\pi$
$432$	0	0
$433$	41.3181	1.98562	0.992810	−	0.119701i	$-0.0381936\pi$
$433$	41.3181	1.98562	0.992810	+	0.119701i	$0.0381936\pi$
$434$	0	0
$435$	36.6645	1.75793
$436$	0	0
$437$	48.2389	2.30758
$438$	0	0
$439$	−13.7822	−0.657787	−0.328893	−	0.944367i	$-0.606676\pi$
$439$	−13.7822	−0.657787	−0.328893	+	0.944367i	$0.606676\pi$
$440$	0	0
$441$	4.63745	0.220831
$442$	0	0
$443$	−25.6596	−1.21912	−0.609562	−	0.792738i	$-0.708655\pi$
$443$	−25.6596	−1.21912	−0.609562	+	0.792738i	$0.708655\pi$
$444$	0	0
$445$	3.37127	0.159814
$446$	0	0
$447$	−24.0468	−1.13738
$448$	0	0
$449$	15.0919	0.712230	0.356115	−	0.934442i	$-0.384101\pi$
$449$	15.0919	0.712230	0.356115	+	0.934442i	$0.384101\pi$
$450$	0	0
$451$	−63.5192	−2.99100
$452$	0	0
$453$	−4.63557	−0.217798
$454$	0	0
$455$	−0.975028	−0.0457100
$456$	0	0
$457$	3.86976	0.181020	0.0905099	−	0.995896i	$-0.471150\pi$
$457$	3.86976	0.181020	0.0905099	+	0.995896i	$0.471150\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.9729	−1.53570	−0.767851	−	0.640629i	$-0.778674\pi$
$461$	−32.9729	−1.53570	−0.767851	+	0.640629i	$0.778674\pi$
$462$	0	0
$463$	−1.29237	−0.0600613	−0.0300306	−	0.999549i	$-0.509560\pi$
$463$	−1.29237	−0.0600613	−0.0300306	+	0.999549i	$0.509560\pi$
$464$	0	0
$465$	−19.6429	−0.910919
$466$	0	0
$467$	−16.9911	−0.786255	−0.393127	−	0.919484i	$-0.628607\pi$
$467$	−16.9911	−0.786255	−0.393127	+	0.919484i	$0.628607\pi$
$468$	0	0
$469$	−11.9894	−0.553619
$470$	0	0
$471$	−20.5912	−0.948794
$472$	0	0
$473$	−33.9698	−1.56193
$474$	0	0
$475$	−13.8923	−0.637422
$476$	0	0
$477$	54.6685	2.50310
$478$	0	0
$479$	16.9471	0.774333	0.387167	−	0.922010i	$-0.373454\pi$
$479$	16.9471	0.774333	0.387167	+	0.922010i	$0.373454\pi$
$480$	0	0
$481$	3.28409	0.149741
$482$	0	0
$483$	19.0035	0.864691
$484$	0	0
$485$	−28.3819	−1.28876
$486$	0	0
$487$	16.4316	0.744586	0.372293	−	0.928115i	$-0.378572\pi$
$487$	16.4316	0.744586	0.372293	+	0.928115i	$0.378572\pi$
$488$	0	0
$489$	−0.102663	−0.00464257
$490$	0	0
$491$	−32.3194	−1.45855	−0.729276	−	0.684220i	$-0.760143\pi$
$491$	−32.3194	−1.45855	−0.729276	+	0.684220i	$0.760143\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−41.1210	−1.84825
$496$	0	0
$497$	4.46918	0.200470
$498$	0	0
$499$	42.0042	1.88036	0.940182	−	0.340672i	$-0.110655\pi$
$499$	42.0042	1.88036	0.940182	+	0.340672i	$0.110655\pi$
$500$	0	0
$501$	−21.7685	−0.972545
$502$	0	0
$503$	18.4516	0.822716	0.411358	−	0.911474i	$-0.365055\pi$
$503$	18.4516	0.822716	0.411358	+	0.911474i	$0.365055\pi$
$504$	0	0
$505$	1.79152	0.0797215
$506$	0	0
$507$	−35.0567	−1.55692
$508$	0	0
$509$	−6.99787	−0.310175	−0.155088	−	0.987901i	$-0.549566\pi$
$509$	−6.99787	−0.310175	−0.155088	+	0.987901i	$0.549566\pi$
$510$	0	0
$511$	−5.37549	−0.237798
$512$	0	0
$513$	31.7453	1.40159
$514$	0	0
$515$	−4.90370	−0.216083
$516$	0	0
$517$	29.3937	1.29273
$518$	0	0
$519$	11.9814	0.525927
$520$	0	0
$521$	−27.9243	−1.22339	−0.611693	−	0.791096i	$-0.709511\pi$
$521$	−27.9243	−1.22339	−0.611693	+	0.791096i	$0.709511\pi$
$522$	0	0
$523$	−12.4125	−0.542761	−0.271380	−	0.962472i	$-0.587480\pi$
$523$	−12.4125	−0.542761	−0.271380	+	0.962472i	$0.587480\pi$
$524$	0	0
$525$	−5.47282	−0.238853
$526$	0	0
$527$	0	0
$528$	0	0
$529$	24.2847	1.05586
$530$	0	0
$531$	12.3095	0.534187
$532$	0	0
$533$	−6.98453	−0.302534
$534$	0	0
$535$	11.7426	0.507678
$536$	0	0
$537$	61.0707	2.63540
$538$	0	0
$539$	5.10276	0.219791
$540$	0	0
$541$	24.6130	1.05820	0.529098	−	0.848561i	$-0.322531\pi$
$541$	24.6130	1.05820	0.529098	+	0.848561i	$0.322531\pi$
$542$	0	0
$543$	−42.4705	−1.82258
$544$	0	0
$545$	8.05964	0.345237
$546$	0	0
$547$	−35.3223	−1.51027	−0.755136	−	0.655569i	$-0.772429\pi$
$547$	−35.3223	−1.51027	−0.755136	+	0.655569i	$0.772429\pi$
$548$	0	0
$549$	24.1232	1.02956
$550$	0	0
$551$	−53.5586	−2.28167
$552$	0	0
$553$	−9.17100	−0.389991
$554$	0	0
$555$	−28.1081	−1.19312
$556$	0	0
$557$	−37.4499	−1.58680	−0.793402	−	0.608699i	$-0.791692\pi$
$557$	−37.4499	−1.58680	−0.793402	+	0.608699i	$0.791692\pi$
$558$	0	0
$559$	−3.73530	−0.157986
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.7093	−0.577779	−0.288889	−	0.957363i	$-0.593286\pi$
$563$	−13.7093	−0.577779	−0.288889	+	0.957363i	$0.593286\pi$
$564$	0	0
$565$	−12.6878	−0.533781
$566$	0	0
$567$	−1.40640	−0.0590634
$568$	0	0
$569$	8.06785	0.338222	0.169111	−	0.985597i	$-0.445910\pi$
$569$	8.06785	0.338222	0.169111	+	0.985597i	$0.445910\pi$
$570$	0	0
$571$	31.9855	1.33855	0.669275	−	0.743014i	$-0.266605\pi$
$571$	31.9855	1.33855	0.669275	+	0.743014i	$0.266605\pi$
$572$	0	0
$573$	49.3043	2.05972
$574$	0	0
$575$	−13.6175	−0.567888
$576$	0	0
$577$	−12.7661	−0.531461	−0.265730	−	0.964047i	$-0.585613\pi$
$577$	−12.7661	−0.531461	−0.265730	+	0.964047i	$0.585613\pi$
$578$	0	0
$579$	−61.1235	−2.54020
$580$	0	0
$581$	−0.667347	−0.0276862
$582$	0	0
$583$	60.1537	2.49131
$584$	0	0
$585$	−4.52165	−0.186947
$586$	0	0
$587$	−4.83699	−0.199644	−0.0998219	−	0.995005i	$-0.531827\pi$
$587$	−4.83699	−0.199644	−0.0998219	+	0.995005i	$0.531827\pi$
$588$	0	0
$589$	28.6939	1.18231
$590$	0	0
$591$	52.7034	2.16793
$592$	0	0
$593$	−37.8310	−1.55353	−0.776766	−	0.629789i	$-0.783141\pi$
$593$	−37.8310	−1.55353	−0.776766	+	0.629789i	$0.783141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.9121	0.774019
$598$	0	0
$599$	−3.91799	−0.160085	−0.0800424	−	0.996791i	$-0.525506\pi$
$599$	−3.91799	−0.160085	−0.0800424	+	0.996791i	$0.525506\pi$
$600$	0	0
$601$	−1.37398	−0.0560458	−0.0280229	−	0.999607i	$-0.508921\pi$
$601$	−1.37398	−0.0560458	−0.0280229	+	0.999607i	$0.508921\pi$
$602$	0	0
$603$	−55.6002	−2.26422
$604$	0	0
$605$	−26.1320	−1.06242
$606$	0	0
$607$	−37.1829	−1.50921	−0.754605	−	0.656180i	$-0.772171\pi$
$607$	−37.1829	−1.50921	−0.754605	+	0.656180i	$0.772171\pi$
$608$	0	0
$609$	−21.0992	−0.854983
$610$	0	0
$611$	3.23212	0.130757
$612$	0	0
$613$	−11.6669	−0.471221	−0.235610	−	0.971848i	$-0.575709\pi$
$613$	−11.6669	−0.471221	−0.235610	+	0.971848i	$0.575709\pi$
$614$	0	0
$615$	59.7798	2.41055
$616$	0	0
$617$	20.0304	0.806392	0.403196	−	0.915114i	$-0.367899\pi$
$617$	20.0304	0.806392	0.403196	+	0.915114i	$0.367899\pi$
$618$	0	0
$619$	26.0880	1.04857	0.524283	−	0.851544i	$-0.324333\pi$
$619$	26.0880	1.04857	0.524283	+	0.851544i	$0.324333\pi$
$620$	0	0
$621$	31.1174	1.24870
$622$	0	0
$623$	−1.94005	−0.0777266
$624$	0	0
$625$	−11.1767	−0.447067
$626$	0	0
$627$	98.9273	3.95078
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−33.7273	−1.34266	−0.671330	−	0.741158i	$-0.734277\pi$
$631$	−33.7273	−1.34266	−0.671330	+	0.741158i	$0.734277\pi$
$632$	0	0
$633$	−0.748613	−0.0297547
$634$	0	0
$635$	−21.3304	−0.846470
$636$	0	0
$637$	0.561096	0.0222314
$638$	0	0
$639$	20.7256	0.819891
$640$	0	0
$641$	−26.8691	−1.06126	−0.530632	−	0.847602i	$-0.678045\pi$
$641$	−26.8691	−1.06126	−0.530632	+	0.847602i	$0.678045\pi$
$642$	0	0
$643$	−32.9547	−1.29961	−0.649803	−	0.760102i	$-0.725149\pi$
$643$	−32.9547	−1.29961	−0.649803	+	0.760102i	$0.725149\pi$
$644$	0	0
$645$	31.9700	1.25882
$646$	0	0
$647$	−6.21592	−0.244373	−0.122186	−	0.992507i	$-0.538991\pi$
$647$	−6.21592	−0.244373	−0.122186	+	0.992507i	$0.538991\pi$
$648$	0	0
$649$	13.5446	0.531671
$650$	0	0
$651$	11.3038	0.443032
$652$	0	0
$653$	7.58890	0.296977	0.148488	−	0.988914i	$-0.452559\pi$
$653$	7.58890	0.296977	0.148488	+	0.988914i	$0.452559\pi$
$654$	0	0
$655$	−5.82139	−0.227460
$656$	0	0
$657$	−24.9286	−0.972556
$658$	0	0
$659$	−34.9451	−1.36127	−0.680634	−	0.732623i	$-0.738296\pi$
$659$	−34.9451	−1.36127	−0.680634	+	0.732623i	$0.738296\pi$
$660$	0	0
$661$	−36.7390	−1.42898	−0.714492	−	0.699644i	$-0.753342\pi$
$661$	−36.7390	−1.42898	−0.714492	+	0.699644i	$0.753342\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.1904	−0.472723
$666$	0	0
$667$	−52.4991	−2.03277
$668$	0	0
$669$	−40.0053	−1.54669
$670$	0	0
$671$	26.5437	1.02471
$672$	0	0
$673$	−28.3186	−1.09160	−0.545802	−	0.837914i	$-0.683775\pi$
$673$	−28.3186	−1.09160	−0.545802	+	0.837914i	$0.683775\pi$
$674$	0	0
$675$	−8.96146	−0.344927
$676$	0	0
$677$	26.0492	1.00115	0.500577	−	0.865692i	$-0.333121\pi$
$677$	26.0492	1.00115	0.500577	+	0.865692i	$0.333121\pi$
$678$	0	0
$679$	16.3329	0.626797
$680$	0	0
$681$	27.7232	1.06236
$682$	0	0
$683$	−3.34382	−0.127948	−0.0639738	−	0.997952i	$-0.520377\pi$
$683$	−3.34382	−0.127948	−0.0639738	+	0.997952i	$0.520377\pi$
$684$	0	0
$685$	3.14965	0.120342
$686$	0	0
$687$	54.3128	2.07216
$688$	0	0
$689$	6.61447	0.251991
$690$	0	0
$691$	15.5473	0.591448	0.295724	−	0.955273i	$-0.404439\pi$
$691$	15.5473	0.591448	0.295724	+	0.955273i	$0.404439\pi$
$692$	0	0
$693$	23.6638	0.898912
$694$	0	0
$695$	12.9627	0.491704
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−35.9248	−1.35880
$700$	0	0
$701$	8.46852	0.319852	0.159926	−	0.987129i	$-0.448875\pi$
$701$	8.46852	0.319852	0.159926	+	0.987129i	$0.448875\pi$
$702$	0	0
$703$	41.0596	1.54859
$704$	0	0
$705$	−27.6633	−1.04186
$706$	0	0
$707$	−1.03096	−0.0387732
$708$	0	0
$709$	−14.2678	−0.535837	−0.267918	−	0.963442i	$-0.586336\pi$
$709$	−14.2678	−0.535837	−0.267918	+	0.963442i	$0.586336\pi$
$710$	0	0
$711$	−42.5301	−1.59500
$712$	0	0
$713$	28.1263	1.05334
$714$	0	0
$715$	−4.97533	−0.186067
$716$	0	0
$717$	−40.3448	−1.50670
$718$	0	0
$719$	−49.0421	−1.82896	−0.914480	−	0.404630i	$-0.867400\pi$
$719$	−49.0421	−1.82896	−0.914480	+	0.404630i	$0.867400\pi$
$720$	0	0
$721$	2.82191	0.105094
$722$	0	0
$723$	39.3754	1.46439
$724$	0	0
$725$	15.1192	0.561513
$726$	0	0
$727$	−1.83148	−0.0679259	−0.0339630	−	0.999423i	$-0.510813\pi$
$727$	−1.83148	−0.0679259	−0.0339630	+	0.999423i	$0.510813\pi$
$728$	0	0
$729$	−44.0400	−1.63111
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−12.8655	−0.475198	−0.237599	−	0.971363i	$-0.576360\pi$
$733$	−12.8655	−0.475198	−0.237599	+	0.971363i	$0.576360\pi$
$734$	0	0
$735$	−4.80236	−0.177138
$736$	0	0
$737$	−61.1789	−2.25355
$738$	0	0
$739$	−39.3486	−1.44746	−0.723731	−	0.690082i	$-0.757574\pi$
$739$	−39.3486	−1.44746	−0.723731	+	0.690082i	$0.757574\pi$
$740$	0	0
$741$	10.8780	0.399613
$742$	0	0
$743$	38.9850	1.43022	0.715110	−	0.699012i	$-0.246376\pi$
$743$	38.9850	1.43022	0.715110	+	0.699012i	$0.246376\pi$
$744$	0	0
$745$	15.1204	0.553969
$746$	0	0
$747$	−3.09479	−0.113232
$748$	0	0
$749$	−6.75748	−0.246913
$750$	0	0
$751$	−14.4585	−0.527600	−0.263800	−	0.964577i	$-0.584976\pi$
$751$	−14.4585	−0.527600	−0.263800	+	0.964577i	$0.584976\pi$
$752$	0	0
$753$	7.96188	0.290147
$754$	0	0
$755$	2.91480	0.106080
$756$	0	0
$757$	−38.8314	−1.41135	−0.705676	−	0.708535i	$-0.749357\pi$
$757$	−38.8314	−1.41135	−0.705676	+	0.708535i	$0.749357\pi$
$758$	0	0
$759$	96.9704	3.51980
$760$	0	0
$761$	−23.0643	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$761$	−23.0643	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$762$	0	0
$763$	−4.63805	−0.167909
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.48935	0.0537775
$768$	0	0
$769$	−38.7458	−1.39721	−0.698605	−	0.715507i	$-0.746196\pi$
$769$	−38.7458	−1.39721	−0.698605	+	0.715507i	$0.746196\pi$
$770$	0	0
$771$	31.5152	1.13499
$772$	0	0
$773$	−17.9149	−0.644355	−0.322177	−	0.946679i	$-0.604415\pi$
$773$	−17.9149	−0.644355	−0.322177	+	0.946679i	$0.604415\pi$
$774$	0	0
$775$	−8.10006	−0.290963
$776$	0	0
$777$	16.1753	0.580285
$778$	0	0
$779$	−87.3248	−3.12873
$780$	0	0
$781$	22.8051	0.816031
$782$	0	0
$783$	−34.5489	−1.23468
$784$	0	0
$785$	12.9476	0.462119
$786$	0	0
$787$	13.0117	0.463816	0.231908	−	0.972738i	$-0.425503\pi$
$787$	13.0117	0.463816	0.231908	+	0.972738i	$0.425503\pi$
$788$	0	0
$789$	−41.7324	−1.48571
$790$	0	0
$791$	7.30142	0.259609
$792$	0	0
$793$	2.91873	0.103647
$794$	0	0
$795$	−56.6125	−2.00784
$796$	0	0
$797$	−7.69443	−0.272551	−0.136275	−	0.990671i	$-0.543513\pi$
$797$	−7.69443	−0.272551	−0.136275	+	0.990671i	$0.543513\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−8.99691	−0.317890
$802$	0	0
$803$	−27.4298	−0.967977
$804$	0	0
$805$	−11.9492	−0.421155
$806$	0	0
$807$	59.3365	2.08874
$808$	0	0
$809$	−1.23286	−0.0433451	−0.0216726	−	0.999765i	$-0.506899\pi$
$809$	−1.23286	−0.0433451	−0.0216726	+	0.999765i	$0.506899\pi$
$810$	0	0
$811$	−32.4554	−1.13966	−0.569831	−	0.821762i	$-0.692991\pi$
$811$	−32.4554	−1.13966	−0.569831	+	0.821762i	$0.692991\pi$
$812$	0	0
$813$	−11.3321	−0.397435
$814$	0	0
$815$	0.0645534	0.00226121
$816$	0	0
$817$	−46.7010	−1.63386
$818$	0	0
$819$	2.60206	0.0909232
$820$	0	0
$821$	52.9470	1.84786	0.923932	−	0.382556i	$-0.124956\pi$
$821$	52.9470	1.84786	0.923932	+	0.382556i	$0.124956\pi$
$822$	0	0
$823$	−45.8098	−1.59683	−0.798415	−	0.602108i	$-0.794328\pi$
$823$	−45.8098	−1.59683	−0.798415	+	0.602108i	$0.794328\pi$
$824$	0	0
$825$	−27.9264	−0.972274
$826$	0	0
$827$	35.1923	1.22376	0.611878	−	0.790952i	$-0.290414\pi$
$827$	35.1923	1.22376	0.611878	+	0.790952i	$0.290414\pi$
$828$	0	0
$829$	−13.1718	−0.457476	−0.228738	−	0.973488i	$-0.573460\pi$
$829$	−13.1718	−0.457476	−0.228738	+	0.973488i	$0.573460\pi$
$830$	0	0
$831$	72.9074	2.52913
$832$	0	0
$833$	0	0
$834$	0	0
$835$	13.6878	0.473687
$836$	0	0
$837$	18.5095	0.639781
$838$	0	0
$839$	−25.9218	−0.894920	−0.447460	−	0.894304i	$-0.647671\pi$
$839$	−25.9218	−0.894920	−0.447460	+	0.894304i	$0.647671\pi$
$840$	0	0
$841$	29.2886	1.00995
$842$	0	0
$843$	−9.76546	−0.336340
$844$	0	0
$845$	22.0433	0.758312
$846$	0	0
$847$	15.0381	0.516716
$848$	0	0
$849$	−79.3878	−2.72458
$850$	0	0
$851$	40.2474	1.37966
$852$	0	0
$853$	−20.7962	−0.712050	−0.356025	−	0.934476i	$-0.615868\pi$
$853$	−20.7962	−0.712050	−0.356025	+	0.934476i	$0.615868\pi$
$854$	0	0
$855$	−56.5323	−1.93336
$856$	0	0
$857$	23.6143	0.806649	0.403324	−	0.915057i	$-0.367855\pi$
$857$	23.6143	0.806649	0.403324	+	0.915057i	$0.367855\pi$
$858$	0	0
$859$	−17.1481	−0.585085	−0.292543	−	0.956252i	$-0.594501\pi$
$859$	−17.1481	−0.585085	−0.292543	+	0.956252i	$0.594501\pi$
$860$	0	0
$861$	−34.4013	−1.17239
$862$	0	0
$863$	−15.9665	−0.543506	−0.271753	−	0.962367i	$-0.587603\pi$
$863$	−15.9665	−0.543506	−0.271753	+	0.962367i	$0.587603\pi$
$864$	0	0
$865$	−7.53381	−0.256157
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−46.7974	−1.58749
$870$	0	0
$871$	−6.72720	−0.227942
$872$	0	0
$873$	75.7428	2.56351
$874$	0	0
$875$	12.1299	0.410064
$876$	0	0
$877$	−8.26638	−0.279136	−0.139568	−	0.990213i	$-0.544571\pi$
$877$	−8.26638	−0.279136	−0.139568	+	0.990213i	$0.544571\pi$
$878$	0	0
$879$	−79.3112	−2.67510
$880$	0	0
$881$	21.3991	0.720955	0.360478	−	0.932768i	$-0.382614\pi$
$881$	21.3991	0.720955	0.360478	+	0.932768i	$0.382614\pi$
$882$	0	0
$883$	−42.2441	−1.42163	−0.710813	−	0.703381i	$-0.751673\pi$
$883$	−42.2441	−1.42163	−0.710813	+	0.703381i	$0.751673\pi$
$884$	0	0
$885$	−12.7472	−0.428493
$886$	0	0
$887$	17.3563	0.582768	0.291384	−	0.956606i	$-0.405884\pi$
$887$	17.3563	0.582768	0.291384	+	0.956606i	$0.405884\pi$
$888$	0	0
$889$	12.2749	0.411687
$890$	0	0
$891$	−7.17654	−0.240423
$892$	0	0
$893$	40.4098	1.35226
$894$	0	0
$895$	−38.4007	−1.28359
$896$	0	0
$897$	10.6628	0.356021
$898$	0	0
$899$	−31.2280	−1.04151
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−18.3977	−0.612236
$904$	0	0
$905$	26.7051	0.887706
$906$	0	0
$907$	−24.2818	−0.806263	−0.403132	−	0.915142i	$-0.632078\pi$
$907$	−24.2818	−0.806263	−0.403132	+	0.915142i	$0.632078\pi$
$908$	0	0
$909$	−4.78102	−0.158576
$910$	0	0
$911$	−50.4775	−1.67239	−0.836197	−	0.548430i	$-0.815226\pi$
$911$	−50.4775	−1.67239	−0.836197	+	0.548430i	$0.815226\pi$
$912$	0	0
$913$	−3.40531	−0.112699
$914$	0	0
$915$	−24.9811	−0.825848
$916$	0	0
$917$	3.35001	0.110627
$918$	0	0
$919$	2.03508	0.0671310	0.0335655	−	0.999437i	$-0.489314\pi$
$919$	2.03508	0.0671310	0.0335655	+	0.999437i	$0.489314\pi$
$920$	0	0
$921$	−83.9298	−2.76558
$922$	0	0
$923$	2.50764	0.0825399
$924$	0	0
$925$	−11.5908	−0.381104
$926$	0	0
$927$	13.0865	0.429816
$928$	0	0
$929$	−7.61578	−0.249866	−0.124933	−	0.992165i	$-0.539872\pi$
$929$	−7.61578	−0.249866	−0.124933	+	0.992165i	$0.539872\pi$
$930$	0	0
$931$	7.01515	0.229912
$932$	0	0
$933$	58.0078	1.89909
$934$	0	0
$935$	0	0
$936$	0	0
$937$	17.1317	0.559667	0.279833	−	0.960049i	$-0.409721\pi$
$937$	17.1317	0.559667	0.279833	+	0.960049i	$0.409721\pi$
$938$	0	0
$939$	−7.34125	−0.239573
$940$	0	0
$941$	−23.1803	−0.755657	−0.377829	−	0.925876i	$-0.623329\pi$
$941$	−23.1803	−0.755657	−0.377829	+	0.925876i	$0.623329\pi$
$942$	0	0
$943$	−85.5974	−2.78743
$944$	0	0
$945$	−7.86362	−0.255804
$946$	0	0
$947$	−4.50836	−0.146502	−0.0732510	−	0.997314i	$-0.523337\pi$
$947$	−4.50836	−0.146502	−0.0732510	+	0.997314i	$0.523337\pi$
$948$	0	0
$949$	−3.01617	−0.0979089
$950$	0	0
$951$	14.7397	0.477967
$952$	0	0
$953$	21.1276	0.684391	0.342196	−	0.939629i	$-0.388829\pi$
$953$	21.1276	0.684391	0.342196	+	0.939629i	$0.388829\pi$
$954$	0	0
$955$	−31.0021	−1.00320
$956$	0	0
$957$	−107.664	−3.48029
$958$	0	0
$959$	−1.81252	−0.0585293
$960$	0	0
$961$	−14.2697	−0.460313
$962$	0	0
$963$	−31.3375	−1.00984
$964$	0	0
$965$	38.4338	1.23723
$966$	0	0
$967$	6.46713	0.207969	0.103984	−	0.994579i	$-0.466841\pi$
$967$	6.46713	0.207969	0.103984	+	0.994579i	$0.466841\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.9855	0.705550	0.352775	−	0.935708i	$-0.385238\pi$
$971$	21.9855	0.705550	0.352775	+	0.935708i	$0.385238\pi$
$972$	0	0
$973$	−7.45961	−0.239144
$974$	0	0
$975$	−3.07078	−0.0983435
$976$	0	0
$977$	9.05212	0.289603	0.144802	−	0.989461i	$-0.453746\pi$
$977$	9.05212	0.289603	0.144802	+	0.989461i	$0.453746\pi$
$978$	0	0
$979$	−9.89962	−0.316393
$980$	0	0
$981$	−21.5087	−0.686721
$982$	0	0
$983$	38.1934	1.21818	0.609090	−	0.793101i	$-0.291535\pi$
$983$	38.1934	1.21818	0.609090	+	0.793101i	$0.291535\pi$
$984$	0	0
$985$	−33.1394	−1.05591
$986$	0	0
$987$	15.9193	0.506717
$988$	0	0
$989$	−45.7772	−1.45563
$990$	0	0
$991$	−48.9693	−1.55556	−0.777780	−	0.628536i	$-0.783654\pi$
$991$	−48.9693	−1.55556	−0.777780	+	0.628536i	$0.783654\pi$
$992$	0	0
$993$	14.2465	0.452099
$994$	0	0
$995$	−11.8917	−0.376993
$996$	0	0
$997$	3.83570	0.121478	0.0607389	−	0.998154i	$-0.480654\pi$
$997$	3.83570	0.121478	0.0607389	+	0.998154i	$0.480654\pi$
$998$	0	0
$999$	26.4862	0.837986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.z.1.16		20
17.5	odd	16		476.2.u.a.365.1	yes	40
17.7	odd	16		476.2.u.a.253.1	✓	40
17.16	even	2		8092.2.a.y.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.u.a.253.1	✓	40	17.7	odd	16
476.2.u.a.365.1	yes	40	17.5	odd	16
8092.2.a.y.1.5		20	17.16	even	2
8092.2.a.z.1.16		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8092.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76359 q^{3} -1.73772 q^{5} +1.00000 q^{7} +4.63745 q^{9} +5.10276 q^{11} +0.561096 q^{13} -4.80236 q^{15} +7.01515 q^{19} +2.76359 q^{21} +6.87639 q^{23} -1.98033 q^{25} +4.52525 q^{27} -7.63470 q^{29} +4.09027 q^{31} +14.1019 q^{33} -1.73772 q^{35} +5.85298 q^{37} +1.55064 q^{39} -12.4480 q^{41} -6.65715 q^{43} -8.05860 q^{45} +5.76036 q^{47} +1.00000 q^{49} +11.7885 q^{53} -8.86716 q^{55} +19.3870 q^{57} +2.65437 q^{59} +5.20183 q^{61} +4.63745 q^{63} -0.975028 q^{65} -11.9894 q^{67} +19.0035 q^{69} +4.46918 q^{71} -5.37549 q^{73} -5.47282 q^{75} +5.10276 q^{77} -9.17100 q^{79} -1.40640 q^{81} -0.667347 q^{83} -21.0992 q^{87} -1.94005 q^{89} +0.561096 q^{91} +11.3038 q^{93} -12.1904 q^{95} +16.3329 q^{97} +23.6638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 20 * q + 8 * q^3 + 8 * q^5 + 20 * q^7 + 28 * q^9 + 16 * q^11 + 8 * q^13 + 8 * q^15 + 8 * q^19 + 8 * q^21 + 40 * q^23 + 36 * q^25 + 32 * q^27 + 16 * q^29 + 16 * q^31 + 8 * q^33 + 8 * q^35 + 24 * q^37 + 16 * q^39 - 16 * q^41 + 16 * q^43 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 8 * q^55 + 24 * q^57 + 8 * q^61 + 28 * q^63 + 8 * q^65 - 8 * q^67 + 8 * q^69 + 72 * q^71 + 40 * q^75 + 16 * q^77 + 60 * q^81 + 8 * q^83 - 24 * q^89 + 8 * q^91 + 32 * q^93 + 48 * q^95 - 16 * q^97 + 72 * q^99

Introduction

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Database

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