LMFDB - Embedded newform 512.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [512,2,Mod(1,512)] 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("512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.08834058349$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} +3.46410 q^{5} -2.82843 q^{7} +3.00000 q^{9} +2.44949 q^{11} -3.46410 q^{13} +8.48528 q^{15} +4.00000 q^{17} -2.44949 q^{19} -6.92820 q^{21} -2.82843 q^{23} +7.00000 q^{25} +3.46410 q^{29} -5.65685 q^{31} +6.00000 q^{33} -9.79796 q^{35} -3.46410 q^{37} -8.48528 q^{39} -2.00000 q^{41} -12.2474 q^{43} +10.3923 q^{45} +11.3137 q^{47} +1.00000 q^{49} +9.79796 q^{51} +10.3923 q^{53} +8.48528 q^{55} -6.00000 q^{57} -2.44949 q^{59} -3.46410 q^{61} -8.48528 q^{63} -12.0000 q^{65} +7.34847 q^{67} -6.92820 q^{69} +2.82843 q^{71} +8.00000 q^{73} +17.1464 q^{75} -6.92820 q^{77} -5.65685 q^{79} -9.00000 q^{81} -7.34847 q^{83} +13.8564 q^{85} +8.48528 q^{87} -8.00000 q^{89} +9.79796 q^{91} -13.8564 q^{93} -8.48528 q^{95} +12.0000 q^{97} +7.34847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9} + 16 q^{17} + 28 q^{25} + 24 q^{33} - 8 q^{41} + 4 q^{49} - 24 q^{57} - 48 q^{65} + 32 q^{73} - 36 q^{81} - 32 q^{89} + 48 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 + 16 * q^17 + 28 * q^25 + 24 * q^33 - 8 * q^41 + 4 * q^49 - 24 * q^57 - 48 * q^65 + 32 * q^73 - 36 * q^81 - 32 * q^89 + 48 * q^97

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
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	8.48528	2.19089
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−2.44949	−0.561951	−0.280976	−	0.959715i	$-0.590658\pi$
$19$	−2.44949	−0.561951	−0.280976	+	0.959715i	$0.590658\pi$
$20$	0	0
$21$	−6.92820	−1.51186
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−9.79796	−1.65616
$36$	0	0
$37$	−3.46410	−0.569495	−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410	−0.569495	−0.284747	+	0.958603i	$0.591910\pi$
$38$	0	0
$39$	−8.48528	−1.35873
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−12.2474	−1.86772	−0.933859	−	0.357641i	$-0.883581\pi$
$43$	−12.2474	−1.86772	−0.933859	+	0.357641i	$0.883581\pi$
$44$	0	0
$45$	10.3923	1.54919
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.79796	1.37199
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	8.48528	1.14416
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−2.44949	−0.318896	−0.159448	−	0.987206i	$-0.550971\pi$
$59$	−2.44949	−0.318896	−0.159448	+	0.987206i	$0.550971\pi$
$60$	0	0
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	0	0
$63$	−8.48528	−1.06904
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	7.34847	0.897758	0.448879	−	0.893592i	$-0.351823\pi$
$67$	7.34847	0.897758	0.448879	+	0.893592i	$0.351823\pi$
$68$	0	0
$69$	−6.92820	−0.834058
$70$	0	0
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	17.1464	1.97990
$76$	0	0
$77$	−6.92820	−0.789542
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−7.34847	−0.806599	−0.403300	−	0.915068i	$-0.632137\pi$
$83$	−7.34847	−0.806599	−0.403300	+	0.915068i	$0.632137\pi$
$84$	0	0
$85$	13.8564	1.50294
$86$	0	0
$87$	8.48528	0.909718
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	9.79796	1.02711
$92$	0	0
$93$	−13.8564	−1.43684
$94$	0	0
$95$	−8.48528	−0.870572
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	7.34847	0.738549
$100$	0	0
$101$	−3.46410	−0.344691	−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410	−0.344691	−0.172345	+	0.985037i	$0.555135\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	−24.0000	−2.34216
$106$	0	0
$107$	2.44949	0.236801	0.118401	−	0.992966i	$-0.462223\pi$
$107$	2.44949	0.236801	0.118401	+	0.992966i	$0.462223\pi$
$108$	0	0
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	0	0
$111$	−8.48528	−0.805387
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−9.79796	−0.913664
$116$	0	0
$117$	−10.3923	−0.960769
$118$	0	0
$119$	−11.3137	−1.03713
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	−4.89898	−0.441726
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	0	0
$129$	−30.0000	−2.64135
$130$	0	0
$131$	7.34847	0.642039	0.321019	−	0.947073i	$-0.395975\pi$
$131$	7.34847	0.642039	0.321019	+	0.947073i	$0.395975\pi$
$132$	0	0
$133$	6.92820	0.600751
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−12.2474	−1.03882	−0.519408	−	0.854527i	$-0.673847\pi$
$139$	−12.2474	−1.03882	−0.519408	+	0.854527i	$0.673847\pi$
$140$	0	0
$141$	27.7128	2.33384
$142$	0	0
$143$	−8.48528	−0.709575
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	2.44949	0.202031
$148$	0	0
$149$	−10.3923	−0.851371	−0.425685	−	0.904871i	$-0.639967\pi$
$149$	−10.3923	−0.851371	−0.425685	+	0.904871i	$0.639967\pi$
$150$	0	0
$151$	19.7990	1.61122	0.805609	−	0.592447i	$-0.201838\pi$
$151$	19.7990	1.61122	0.805609	+	0.592447i	$0.201838\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	−19.5959	−1.57398
$156$	0	0
$157$	24.2487	1.93526	0.967629	−	0.252377i	$-0.0812124\pi$
$157$	24.2487	1.93526	0.967629	+	0.252377i	$0.0812124\pi$
$158$	0	0
$159$	25.4558	2.01878
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	−17.1464	−1.34301	−0.671506	−	0.740999i	$-0.734352\pi$
$163$	−17.1464	−1.34301	−0.671506	+	0.740999i	$0.734352\pi$
$164$	0	0
$165$	20.7846	1.61808
$166$	0	0
$167$	8.48528	0.656611	0.328305	−	0.944572i	$-0.393522\pi$
$167$	8.48528	0.656611	0.328305	+	0.944572i	$0.393522\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−7.34847	−0.561951
$172$	0	0
$173$	−17.3205	−1.31685	−0.658427	−	0.752645i	$-0.728778\pi$
$173$	−17.3205	−1.31685	−0.658427	+	0.752645i	$0.728778\pi$
$174$	0	0
$175$	−19.7990	−1.49666
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	12.2474	0.915417	0.457709	−	0.889102i	$-0.348670\pi$
$179$	12.2474	0.915417	0.457709	+	0.889102i	$0.348670\pi$
$180$	0	0
$181$	−24.2487	−1.80239	−0.901196	−	0.433411i	$-0.857310\pi$
$181$	−24.2487	−1.80239	−0.901196	+	0.433411i	$0.857310\pi$
$182$	0	0
$183$	−8.48528	−0.627250
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	9.79796	0.716498
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	−29.3939	−2.10494
$196$	0	0
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	−2.82843	−0.200502	−0.100251	−	0.994962i	$-0.531965\pi$
$199$	−2.82843	−0.200502	−0.100251	+	0.994962i	$0.531965\pi$
$200$	0	0
$201$	18.0000	1.26962
$202$	0	0
$203$	−9.79796	−0.687682
$204$	0	0
$205$	−6.92820	−0.483887
$206$	0	0
$207$	−8.48528	−0.589768
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	12.2474	0.843149	0.421575	−	0.906794i	$-0.361478\pi$
$211$	12.2474	0.843149	0.421575	+	0.906794i	$0.361478\pi$
$212$	0	0
$213$	6.92820	0.474713
$214$	0	0
$215$	−42.4264	−2.89346
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	19.5959	1.32417
$220$	0	0
$221$	−13.8564	−0.932083
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	21.0000	1.40000
$226$	0	0
$227$	22.0454	1.46321	0.731603	−	0.681731i	$-0.238773\pi$
$227$	22.0454	1.46321	0.731603	+	0.681731i	$0.238773\pi$
$228$	0	0
$229$	3.46410	0.228914	0.114457	−	0.993428i	$-0.463487\pi$
$229$	3.46410	0.228914	0.114457	+	0.993428i	$0.463487\pi$
$230$	0	0
$231$	−16.9706	−1.11658
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	39.1918	2.55659
$236$	0	0
$237$	−13.8564	−0.900070
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	3.46410	0.221313
$246$	0	0
$247$	8.48528	0.539906
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	17.1464	1.08227	0.541136	−	0.840935i	$-0.317994\pi$
$251$	17.1464	1.08227	0.541136	+	0.840935i	$0.317994\pi$
$252$	0	0
$253$	−6.92820	−0.435572
$254$	0	0
$255$	33.9411	2.12548
$256$	0	0
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	9.79796	0.608816
$260$	0	0
$261$	10.3923	0.643268
$262$	0	0
$263$	25.4558	1.56967	0.784837	−	0.619702i	$-0.212746\pi$
$263$	25.4558	1.56967	0.784837	+	0.619702i	$0.212746\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	−19.5959	−1.19925
$268$	0	0
$269$	3.46410	0.211210	0.105605	−	0.994408i	$-0.466322\pi$
$269$	3.46410	0.211210	0.105605	+	0.994408i	$0.466322\pi$
$270$	0	0
$271$	−11.3137	−0.687259	−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137	−0.687259	−0.343629	+	0.939105i	$0.611656\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	17.1464	1.03397
$276$	0	0
$277$	24.2487	1.45696	0.728482	−	0.685065i	$-0.240226\pi$
$277$	24.2487	1.45696	0.728482	+	0.685065i	$0.240226\pi$
$278$	0	0
$279$	−16.9706	−1.01600
$280$	0	0
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	0	0
$283$	−26.9444	−1.60168	−0.800839	−	0.598880i	$-0.795613\pi$
$283$	−26.9444	−1.60168	−0.800839	+	0.598880i	$0.795613\pi$
$284$	0	0
$285$	−20.7846	−1.23117
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	29.3939	1.72310
$292$	0	0
$293$	−10.3923	−0.607125	−0.303562	−	0.952812i	$-0.598176\pi$
$293$	−10.3923	−0.607125	−0.303562	+	0.952812i	$0.598176\pi$
$294$	0	0
$295$	−8.48528	−0.494032
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.79796	0.566631
$300$	0	0
$301$	34.6410	1.99667
$302$	0	0
$303$	−8.48528	−0.487467
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−2.44949	−0.139800	−0.0698999	−	0.997554i	$-0.522268\pi$
$307$	−2.44949	−0.139800	−0.0698999	+	0.997554i	$0.522268\pi$
$308$	0	0
$309$	34.6410	1.97066
$310$	0	0
$311$	−8.48528	−0.481156	−0.240578	−	0.970630i	$-0.577337\pi$
$311$	−8.48528	−0.481156	−0.240578	+	0.970630i	$0.577337\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	−29.3939	−1.65616
$316$	0	0
$317$	17.3205	0.972817	0.486408	−	0.873732i	$-0.338307\pi$
$317$	17.3205	0.972817	0.486408	+	0.873732i	$0.338307\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	−9.79796	−0.545173
$324$	0	0
$325$	−24.2487	−1.34508
$326$	0	0
$327$	−25.4558	−1.40771
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	2.44949	0.134636	0.0673181	−	0.997732i	$-0.478556\pi$
$331$	2.44949	0.134636	0.0673181	+	0.997732i	$0.478556\pi$
$332$	0	0
$333$	−10.3923	−0.569495
$334$	0	0
$335$	25.4558	1.39080
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−24.4949	−1.33038
$340$	0	0
$341$	−13.8564	−0.750366
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	17.1464	0.920468	0.460234	−	0.887798i	$-0.347765\pi$
$347$	17.1464	0.920468	0.460234	+	0.887798i	$0.347765\pi$
$348$	0	0
$349$	17.3205	0.927146	0.463573	−	0.886059i	$-0.346567\pi$
$349$	17.3205	0.927146	0.463573	+	0.886059i	$0.346567\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	9.79796	0.520022
$356$	0	0
$357$	−27.7128	−1.46672
$358$	0	0
$359$	2.82843	0.149279	0.0746393	−	0.997211i	$-0.476219\pi$
$359$	2.82843	0.149279	0.0746393	+	0.997211i	$0.476219\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	−12.2474	−0.642824
$364$	0	0
$365$	27.7128	1.45055
$366$	0	0
$367$	−22.6274	−1.18114	−0.590571	−	0.806986i	$-0.701097\pi$
$367$	−22.6274	−1.18114	−0.590571	+	0.806986i	$0.701097\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−29.3939	−1.52605
$372$	0	0
$373$	−3.46410	−0.179364	−0.0896822	−	0.995970i	$-0.528585\pi$
$373$	−3.46410	−0.179364	−0.0896822	+	0.995970i	$0.528585\pi$
$374$	0	0
$375$	16.9706	0.876356
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−2.44949	−0.125822	−0.0629109	−	0.998019i	$-0.520038\pi$
$379$	−2.44949	−0.125822	−0.0629109	+	0.998019i	$0.520038\pi$
$380$	0	0
$381$	−13.8564	−0.709885
$382$	0	0
$383$	−33.9411	−1.73431	−0.867155	−	0.498038i	$-0.834054\pi$
$383$	−33.9411	−1.73431	−0.867155	+	0.498038i	$0.834054\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	−36.7423	−1.86772
$388$	0	0
$389$	−3.46410	−0.175637	−0.0878185	−	0.996136i	$-0.527990\pi$
$389$	−3.46410	−0.175637	−0.0878185	+	0.996136i	$0.527990\pi$
$390$	0	0
$391$	−11.3137	−0.572159
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	−19.5959	−0.985978
$396$	0	0
$397$	−24.2487	−1.21701	−0.608504	−	0.793551i	$-0.708230\pi$
$397$	−24.2487	−1.21701	−0.608504	+	0.793551i	$0.708230\pi$
$398$	0	0
$399$	16.9706	0.849591
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	0	0
$403$	19.5959	0.976142
$404$	0	0
$405$	−31.1769	−1.54919
$406$	0	0
$407$	−8.48528	−0.420600
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	4.89898	0.241649
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	−25.4558	−1.24958
$416$	0	0
$417$	−30.0000	−1.46911
$418$	0	0
$419$	−31.8434	−1.55565	−0.777825	−	0.628481i	$-0.783677\pi$
$419$	−31.8434	−1.55565	−0.777825	+	0.628481i	$0.783677\pi$
$420$	0	0
$421$	−10.3923	−0.506490	−0.253245	−	0.967402i	$-0.581498\pi$
$421$	−10.3923	−0.506490	−0.253245	+	0.967402i	$0.581498\pi$
$422$	0	0
$423$	33.9411	1.65027
$424$	0	0
$425$	28.0000	1.35820
$426$	0	0
$427$	9.79796	0.474156
$428$	0	0
$429$	−20.7846	−1.00349
$430$	0	0
$431$	5.65685	0.272481	0.136241	−	0.990676i	$-0.456498\pi$
$431$	5.65685	0.272481	0.136241	+	0.990676i	$0.456498\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	29.3939	1.40933
$436$	0	0
$437$	6.92820	0.331421
$438$	0	0
$439$	2.82843	0.134993	0.0674967	−	0.997719i	$-0.478499\pi$
$439$	2.82843	0.134993	0.0674967	+	0.997719i	$0.478499\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	31.8434	1.51292	0.756462	−	0.654038i	$-0.226926\pi$
$443$	31.8434	1.51292	0.756462	+	0.654038i	$0.226926\pi$
$444$	0	0
$445$	−27.7128	−1.31371
$446$	0	0
$447$	−25.4558	−1.20402
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	−4.89898	−0.230684
$452$	0	0
$453$	48.4974	2.27861
$454$	0	0
$455$	33.9411	1.59118
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.3205	0.806696	0.403348	−	0.915047i	$-0.367846\pi$
$461$	17.3205	0.806696	0.403348	+	0.915047i	$0.367846\pi$
$462$	0	0
$463$	5.65685	0.262896	0.131448	−	0.991323i	$-0.458037\pi$
$463$	5.65685	0.262896	0.131448	+	0.991323i	$0.458037\pi$
$464$	0	0
$465$	−48.0000	−2.22595
$466$	0	0
$467$	17.1464	0.793442	0.396721	−	0.917939i	$-0.370148\pi$
$467$	17.1464	0.793442	0.396721	+	0.917939i	$0.370148\pi$
$468$	0	0
$469$	−20.7846	−0.959744
$470$	0	0
$471$	59.3970	2.73687
$472$	0	0
$473$	−30.0000	−1.37940
$474$	0	0
$475$	−17.1464	−0.786732
$476$	0	0
$477$	31.1769	1.42749
$478$	0	0
$479$	−5.65685	−0.258468	−0.129234	−	0.991614i	$-0.541252\pi$
$479$	−5.65685	−0.258468	−0.129234	+	0.991614i	$0.541252\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	19.5959	0.891645
$484$	0	0
$485$	41.5692	1.88756
$486$	0	0
$487$	−14.1421	−0.640841	−0.320421	−	0.947275i	$-0.603824\pi$
$487$	−14.1421	−0.640841	−0.320421	+	0.947275i	$0.603824\pi$
$488$	0	0
$489$	−42.0000	−1.89931
$490$	0	0
$491$	7.34847	0.331632	0.165816	−	0.986157i	$-0.446974\pi$
$491$	7.34847	0.331632	0.165816	+	0.986157i	$0.446974\pi$
$492$	0	0
$493$	13.8564	0.624061
$494$	0	0
$495$	25.4558	1.14416
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	31.8434	1.42550	0.712752	−	0.701416i	$-0.247448\pi$
$499$	31.8434	1.42550	0.712752	+	0.701416i	$0.247448\pi$
$500$	0	0
$501$	20.7846	0.928588
$502$	0	0
$503$	−14.1421	−0.630567	−0.315283	−	0.948998i	$-0.602100\pi$
$503$	−14.1421	−0.630567	−0.315283	+	0.948998i	$0.602100\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−2.44949	−0.108786
$508$	0	0
$509$	10.3923	0.460631	0.230315	−	0.973116i	$-0.426024\pi$
$509$	10.3923	0.460631	0.230315	+	0.973116i	$0.426024\pi$
$510$	0	0
$511$	−22.6274	−1.00098
$512$	0	0
$513$	0	0
$514$	0	0
$515$	48.9898	2.15875
$516$	0	0
$517$	27.7128	1.21881
$518$	0	0
$519$	−42.4264	−1.86231
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	22.0454	0.963978	0.481989	−	0.876177i	$-0.339914\pi$
$523$	22.0454	0.963978	0.481989	+	0.876177i	$0.339914\pi$
$524$	0	0
$525$	−48.4974	−2.11660
$526$	0	0
$527$	−22.6274	−0.985666
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−7.34847	−0.318896
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	8.48528	0.366851
$536$	0	0
$537$	30.0000	1.29460
$538$	0	0
$539$	2.44949	0.105507
$540$	0	0
$541$	−31.1769	−1.34040	−0.670200	−	0.742180i	$-0.733792\pi$
$541$	−31.1769	−1.34040	−0.670200	+	0.742180i	$0.733792\pi$
$542$	0	0
$543$	−59.3970	−2.54897
$544$	0	0
$545$	−36.0000	−1.54207
$546$	0	0
$547$	−12.2474	−0.523663	−0.261832	−	0.965114i	$-0.584326\pi$
$547$	−12.2474	−0.523663	−0.261832	+	0.965114i	$0.584326\pi$
$548$	0	0
$549$	−10.3923	−0.443533
$550$	0	0
$551$	−8.48528	−0.361485
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−29.3939	−1.24770
$556$	0	0
$557$	−17.3205	−0.733893	−0.366947	−	0.930242i	$-0.619597\pi$
$557$	−17.3205	−0.733893	−0.366947	+	0.930242i	$0.619597\pi$
$558$	0	0
$559$	42.4264	1.79445
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	−22.0454	−0.929103	−0.464552	−	0.885546i	$-0.653784\pi$
$563$	−22.0454	−0.929103	−0.464552	+	0.885546i	$0.653784\pi$
$564$	0	0
$565$	−34.6410	−1.45736
$566$	0	0
$567$	25.4558	1.06904
$568$	0	0
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	−7.34847	−0.307524	−0.153762	−	0.988108i	$-0.549139\pi$
$571$	−7.34847	−0.307524	−0.153762	+	0.988108i	$0.549139\pi$
$572$	0	0
$573$	41.5692	1.73658
$574$	0	0
$575$	−19.7990	−0.825675
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	29.3939	1.22157
$580$	0	0
$581$	20.7846	0.862291
$582$	0	0
$583$	25.4558	1.05427
$584$	0	0
$585$	−36.0000	−1.48842
$586$	0	0
$587$	22.0454	0.909911	0.454956	−	0.890514i	$-0.349655\pi$
$587$	22.0454	0.909911	0.454956	+	0.890514i	$0.349655\pi$
$588$	0	0
$589$	13.8564	0.570943
$590$	0	0
$591$	−25.4558	−1.04711
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	−39.1918	−1.60671
$596$	0	0
$597$	−6.92820	−0.283552
$598$	0	0
$599$	2.82843	0.115566	0.0577832	−	0.998329i	$-0.481597\pi$
$599$	2.82843	0.115566	0.0577832	+	0.998329i	$0.481597\pi$
$600$	0	0
$601$	−24.0000	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$602$	0	0
$603$	22.0454	0.897758
$604$	0	0
$605$	−17.3205	−0.704179
$606$	0	0
$607$	45.2548	1.83684	0.918419	−	0.395610i	$-0.129467\pi$
$607$	45.2548	1.83684	0.918419	+	0.395610i	$0.129467\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−39.1918	−1.58553
$612$	0	0
$613$	−10.3923	−0.419741	−0.209871	−	0.977729i	$-0.567304\pi$
$613$	−10.3923	−0.419741	−0.209871	+	0.977729i	$0.567304\pi$
$614$	0	0
$615$	−16.9706	−0.684319
$616$	0	0
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	0	0
$619$	−17.1464	−0.689173	−0.344587	−	0.938755i	$-0.611981\pi$
$619$	−17.1464	−0.689173	−0.344587	+	0.938755i	$0.611981\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	22.6274	0.906548
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−14.6969	−0.586939
$628$	0	0
$629$	−13.8564	−0.552491
$630$	0	0
$631$	19.7990	0.788185	0.394093	−	0.919071i	$-0.371059\pi$
$631$	19.7990	0.788185	0.394093	+	0.919071i	$0.371059\pi$
$632$	0	0
$633$	30.0000	1.19239
$634$	0	0
$635$	−19.5959	−0.777640
$636$	0	0
$637$	−3.46410	−0.137253
$638$	0	0
$639$	8.48528	0.335673
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	7.34847	0.289795	0.144898	−	0.989447i	$-0.453715\pi$
$643$	7.34847	0.289795	0.144898	+	0.989447i	$0.453715\pi$
$644$	0	0
$645$	−103.923	−4.09197
$646$	0	0
$647$	14.1421	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	39.1918	1.53605
$652$	0	0
$653$	31.1769	1.22005	0.610023	−	0.792383i	$-0.291160\pi$
$653$	31.1769	1.22005	0.610023	+	0.792383i	$0.291160\pi$
$654$	0	0
$655$	25.4558	0.994642
$656$	0	0
$657$	24.0000	0.936329
$658$	0	0
$659$	−46.5403	−1.81295	−0.906476	−	0.422256i	$-0.861238\pi$
$659$	−46.5403	−1.81295	−0.906476	+	0.422256i	$0.861238\pi$
$660$	0	0
$661$	31.1769	1.21264	0.606321	−	0.795220i	$-0.292645\pi$
$661$	31.1769	1.21264	0.606321	+	0.795220i	$0.292645\pi$
$662$	0	0
$663$	−33.9411	−1.31816
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	−9.79796	−0.379378
$668$	0	0
$669$	13.8564	0.535720
$670$	0	0
$671$	−8.48528	−0.327571
$672$	0	0
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.46410	−0.133136	−0.0665681	−	0.997782i	$-0.521205\pi$
$677$	−3.46410	−0.133136	−0.0665681	+	0.997782i	$0.521205\pi$
$678$	0	0
$679$	−33.9411	−1.30254
$680$	0	0
$681$	54.0000	2.06928
$682$	0	0
$683$	−12.2474	−0.468636	−0.234318	−	0.972160i	$-0.575286\pi$
$683$	−12.2474	−0.468636	−0.234318	+	0.972160i	$0.575286\pi$
$684$	0	0
$685$	6.92820	0.264713
$686$	0	0
$687$	8.48528	0.323734
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−7.34847	−0.279549	−0.139774	−	0.990183i	$-0.544638\pi$
$691$	−7.34847	−0.279549	−0.139774	+	0.990183i	$0.544638\pi$
$692$	0	0
$693$	−20.7846	−0.789542
$694$	0	0
$695$	−42.4264	−1.60933
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	−19.5959	−0.741186
$700$	0	0
$701$	−45.0333	−1.70089	−0.850443	−	0.526068i	$-0.823666\pi$
$701$	−45.0333	−1.70089	−0.850443	+	0.526068i	$0.823666\pi$
$702$	0	0
$703$	8.48528	0.320028
$704$	0	0
$705$	96.0000	3.61557
$706$	0	0
$707$	9.79796	0.368490
$708$	0	0
$709$	10.3923	0.390291	0.195146	−	0.980774i	$-0.437482\pi$
$709$	10.3923	0.390291	0.195146	+	0.980774i	$0.437482\pi$
$710$	0	0
$711$	−16.9706	−0.636446
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	−29.3939	−1.09927
$716$	0	0
$717$	41.5692	1.55243
$718$	0	0
$719$	39.5980	1.47676	0.738378	−	0.674387i	$-0.235592\pi$
$719$	39.5980	1.47676	0.738378	+	0.674387i	$0.235592\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	48.9898	1.82195
$724$	0	0
$725$	24.2487	0.900575
$726$	0	0
$727$	−19.7990	−0.734304	−0.367152	−	0.930161i	$-0.619667\pi$
$727$	−19.7990	−0.734304	−0.367152	+	0.930161i	$0.619667\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−48.9898	−1.81195
$732$	0	0
$733$	17.3205	0.639748	0.319874	−	0.947460i	$-0.396359\pi$
$733$	17.3205	0.639748	0.319874	+	0.947460i	$0.396359\pi$
$734$	0	0
$735$	8.48528	0.312984
$736$	0	0
$737$	18.0000	0.663039
$738$	0	0
$739$	−36.7423	−1.35159	−0.675795	−	0.737090i	$-0.736199\pi$
$739$	−36.7423	−1.35159	−0.675795	+	0.737090i	$0.736199\pi$
$740$	0	0
$741$	20.7846	0.763542
$742$	0	0
$743$	42.4264	1.55647	0.778237	−	0.627971i	$-0.216114\pi$
$743$	42.4264	1.55647	0.778237	+	0.627971i	$0.216114\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	−22.0454	−0.806599
$748$	0	0
$749$	−6.92820	−0.253151
$750$	0	0
$751$	−28.2843	−1.03211	−0.516054	−	0.856556i	$-0.672600\pi$
$751$	−28.2843	−1.03211	−0.516054	+	0.856556i	$0.672600\pi$
$752$	0	0
$753$	42.0000	1.53057
$754$	0	0
$755$	68.5857	2.49609
$756$	0	0
$757$	45.0333	1.63676	0.818382	−	0.574675i	$-0.194871\pi$
$757$	45.0333	1.63676	0.818382	+	0.574675i	$0.194871\pi$
$758$	0	0
$759$	−16.9706	−0.615992
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	29.3939	1.06413
$764$	0	0
$765$	41.5692	1.50294
$766$	0	0
$767$	8.48528	0.306386
$768$	0	0
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	−24.4949	−0.882162
$772$	0	0
$773$	−45.0333	−1.61974	−0.809868	−	0.586612i	$-0.800461\pi$
$773$	−45.0333	−1.61974	−0.809868	+	0.586612i	$0.800461\pi$
$774$	0	0
$775$	−39.5980	−1.42240
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	4.89898	0.175524
$780$	0	0
$781$	6.92820	0.247911
$782$	0	0
$783$	0	0
$784$	0	0
$785$	84.0000	2.99809
$786$	0	0
$787$	−7.34847	−0.261945	−0.130972	−	0.991386i	$-0.541810\pi$
$787$	−7.34847	−0.261945	−0.130972	+	0.991386i	$0.541810\pi$
$788$	0	0
$789$	62.3538	2.21986
$790$	0	0
$791$	28.2843	1.00567
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	88.1816	3.12748
$796$	0	0
$797$	−45.0333	−1.59516	−0.797581	−	0.603212i	$-0.793887\pi$
$797$	−45.0333	−1.59516	−0.797581	+	0.603212i	$0.793887\pi$
$798$	0	0
$799$	45.2548	1.60100
$800$	0	0
$801$	−24.0000	−0.847998
$802$	0	0
$803$	19.5959	0.691525
$804$	0	0
$805$	27.7128	0.976748
$806$	0	0
$807$	8.48528	0.298696
$808$	0	0
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	−36.7423	−1.29020	−0.645099	−	0.764099i	$-0.723184\pi$
$811$	−36.7423	−1.29020	−0.645099	+	0.764099i	$0.723184\pi$
$812$	0	0
$813$	−27.7128	−0.971931
$814$	0	0
$815$	−59.3970	−2.08059
$816$	0	0
$817$	30.0000	1.04957
$818$	0	0
$819$	29.3939	1.02711
$820$	0	0
$821$	−3.46410	−0.120898	−0.0604490	−	0.998171i	$-0.519253\pi$
$821$	−3.46410	−0.120898	−0.0604490	+	0.998171i	$0.519253\pi$
$822$	0	0
$823$	−19.7990	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$823$	−19.7990	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$824$	0	0
$825$	42.0000	1.46225
$826$	0	0
$827$	−7.34847	−0.255531	−0.127766	−	0.991804i	$-0.540781\pi$
$827$	−7.34847	−0.255531	−0.127766	+	0.991804i	$0.540781\pi$
$828$	0	0
$829$	−17.3205	−0.601566	−0.300783	−	0.953693i	$-0.597248\pi$
$829$	−17.3205	−0.601566	−0.300783	+	0.953693i	$0.597248\pi$
$830$	0	0
$831$	59.3970	2.06046
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	29.3939	1.01722
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.7696	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$839$	−36.7696	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	78.3837	2.69968
$844$	0	0
$845$	−3.46410	−0.119169
$846$	0	0
$847$	14.1421	0.485930
$848$	0	0
$849$	−66.0000	−2.26511
$850$	0	0
$851$	9.79796	0.335870
$852$	0	0
$853$	−17.3205	−0.593043	−0.296521	−	0.955026i	$-0.595827\pi$
$853$	−17.3205	−0.593043	−0.296521	+	0.955026i	$0.595827\pi$
$854$	0	0
$855$	−25.4558	−0.870572
$856$	0	0
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	17.1464	0.585029	0.292514	−	0.956261i	$-0.405508\pi$
$859$	17.1464	0.585029	0.292514	+	0.956261i	$0.405508\pi$
$860$	0	0
$861$	13.8564	0.472225
$862$	0	0
$863$	−16.9706	−0.577685	−0.288842	−	0.957377i	$-0.593270\pi$
$863$	−16.9706	−0.577685	−0.288842	+	0.957377i	$0.593270\pi$
$864$	0	0
$865$	−60.0000	−2.04006
$866$	0	0
$867$	−2.44949	−0.0831890
$868$	0	0
$869$	−13.8564	−0.470046
$870$	0	0
$871$	−25.4558	−0.862538
$872$	0	0
$873$	36.0000	1.21842
$874$	0	0
$875$	−19.5959	−0.662463
$876$	0	0
$877$	−38.1051	−1.28672	−0.643359	−	0.765564i	$-0.722460\pi$
$877$	−38.1051	−1.28672	−0.643359	+	0.765564i	$0.722460\pi$
$878$	0	0
$879$	−25.4558	−0.858604
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−2.44949	−0.0824319	−0.0412159	−	0.999150i	$-0.513123\pi$
$883$	−2.44949	−0.0824319	−0.0412159	+	0.999150i	$0.513123\pi$
$884$	0	0
$885$	−20.7846	−0.698667
$886$	0	0
$887$	25.4558	0.854724	0.427362	−	0.904081i	$-0.359443\pi$
$887$	25.4558	0.854724	0.427362	+	0.904081i	$0.359443\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−22.0454	−0.738549
$892$	0	0
$893$	−27.7128	−0.927374
$894$	0	0
$895$	42.4264	1.41816
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	−19.5959	−0.653560
$900$	0	0
$901$	41.5692	1.38487
$902$	0	0
$903$	84.8528	2.82372
$904$	0	0
$905$	−84.0000	−2.79225
$906$	0	0
$907$	7.34847	0.244002	0.122001	−	0.992530i	$-0.461069\pi$
$907$	7.34847	0.244002	0.122001	+	0.992530i	$0.461069\pi$
$908$	0	0
$909$	−10.3923	−0.344691
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	−29.3939	−0.971732
$916$	0	0
$917$	−20.7846	−0.686368
$918$	0	0
$919$	−36.7696	−1.21292	−0.606458	−	0.795116i	$-0.707410\pi$
$919$	−36.7696	−1.21292	−0.606458	+	0.795116i	$0.707410\pi$
$920$	0	0
$921$	−6.00000	−0.197707
$922$	0	0
$923$	−9.79796	−0.322504
$924$	0	0
$925$	−24.2487	−0.797293
$926$	0	0
$927$	42.4264	1.39347
$928$	0	0
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	−2.44949	−0.0802788
$932$	0	0
$933$	−20.7846	−0.680458
$934$	0	0
$935$	33.9411	1.10999
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	0	0
$939$	73.4847	2.39808
$940$	0	0
$941$	−38.1051	−1.24219	−0.621096	−	0.783735i	$-0.713312\pi$
$941$	−38.1051	−1.24219	−0.621096	+	0.783735i	$0.713312\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.6413	−1.35316	−0.676581	−	0.736369i	$-0.736539\pi$
$947$	−41.6413	−1.35316	−0.676581	+	0.736369i	$0.736539\pi$
$948$	0	0
$949$	−27.7128	−0.899596
$950$	0	0
$951$	42.4264	1.37577
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	58.7878	1.90233
$956$	0	0
$957$	20.7846	0.671871
$958$	0	0
$959$	−5.65685	−0.182669
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	7.34847	0.236801
$964$	0	0
$965$	41.5692	1.33816
$966$	0	0
$967$	−14.1421	−0.454780	−0.227390	−	0.973804i	$-0.573019\pi$
$967$	−14.1421	−0.454780	−0.227390	+	0.973804i	$0.573019\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−51.4393	−1.65077	−0.825383	−	0.564574i	$-0.809041\pi$
$971$	−51.4393	−1.65077	−0.825383	+	0.564574i	$0.809041\pi$
$972$	0	0
$973$	34.6410	1.11054
$974$	0	0
$975$	−59.3970	−1.90223
$976$	0	0
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	−19.5959	−0.626288
$980$	0	0
$981$	−31.1769	−0.995402
$982$	0	0
$983$	−42.4264	−1.35319	−0.676596	−	0.736354i	$-0.736546\pi$
$983$	−42.4264	−1.35319	−0.676596	+	0.736354i	$0.736546\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	−78.3837	−2.49498
$988$	0	0
$989$	34.6410	1.10152
$990$	0	0
$991$	−45.2548	−1.43757	−0.718784	−	0.695234i	$-0.755301\pi$
$991$	−45.2548	−1.43757	−0.718784	+	0.695234i	$0.755301\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	−9.79796	−0.310616
$996$	0	0
$997$	10.3923	0.329128	0.164564	−	0.986366i	$-0.447378\pi$
$997$	10.3923	0.329128	0.164564	+	0.986366i	$0.447378\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	512.2.a.g.1.4	yes	4
3.2	odd	2		4608.2.a.w.1.1		4
4.3	odd	2	inner	512.2.a.g.1.2	yes	4
8.3	odd	2	inner	512.2.a.g.1.3	yes	4
8.5	even	2	inner	512.2.a.g.1.1	✓	4
12.11	even	2		4608.2.a.w.1.2		4
16.3	odd	4		512.2.b.d.257.1		4
16.5	even	4		512.2.b.d.257.2		4
16.11	odd	4		512.2.b.d.257.4		4
16.13	even	4		512.2.b.d.257.3		4
24.5	odd	2		4608.2.a.w.1.3		4
24.11	even	2		4608.2.a.w.1.4		4
32.3	odd	8		1024.2.e.p.257.1		8
32.5	even	8		1024.2.e.p.769.2		8
32.11	odd	8		1024.2.e.p.769.1		8
32.13	even	8		1024.2.e.p.257.2		8
32.19	odd	8		1024.2.e.p.257.4		8
32.21	even	8		1024.2.e.p.769.3		8
32.27	odd	8		1024.2.e.p.769.4		8
32.29	even	8		1024.2.e.p.257.3		8
48.5	odd	4		4608.2.d.d.2305.2		4
48.11	even	4		4608.2.d.d.2305.1		4
48.29	odd	4		4608.2.d.d.2305.4		4
48.35	even	4		4608.2.d.d.2305.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.g.1.1	✓	4	8.5	even	2	inner
512.2.a.g.1.2	yes	4	4.3	odd	2	inner
512.2.a.g.1.3	yes	4	8.3	odd	2	inner
512.2.a.g.1.4	yes	4	1.1	even	1	trivial
512.2.b.d.257.1		4	16.3	odd	4
512.2.b.d.257.2		4	16.5	even	4
512.2.b.d.257.3		4	16.13	even	4
512.2.b.d.257.4		4	16.11	odd	4
1024.2.e.p.257.1		8	32.3	odd	8
1024.2.e.p.257.2		8	32.13	even	8
1024.2.e.p.257.3		8	32.29	even	8
1024.2.e.p.257.4		8	32.19	odd	8
1024.2.e.p.769.1		8	32.11	odd	8
1024.2.e.p.769.2		8	32.5	even	8
1024.2.e.p.769.3		8	32.21	even	8
1024.2.e.p.769.4		8	32.27	odd	8
4608.2.a.w.1.1		4	3.2	odd	2
4608.2.a.w.1.2		4	12.11	even	2
4608.2.a.w.1.3		4	24.5	odd	2
4608.2.a.w.1.4		4	24.11	even	2
4608.2.d.d.2305.1		4	48.11	even	4
4608.2.d.d.2305.2		4	48.5	odd	4
4608.2.d.d.2305.3		4	48.35	even	4
4608.2.d.d.2305.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} +3.46410 q^{5} -2.82843 q^{7} +3.00000 q^{9} +2.44949 q^{11} -3.46410 q^{13} +8.48528 q^{15} +4.00000 q^{17} -2.44949 q^{19} -6.92820 q^{21} -2.82843 q^{23} +7.00000 q^{25} +3.46410 q^{29} -5.65685 q^{31} +6.00000 q^{33} -9.79796 q^{35} -3.46410 q^{37} -8.48528 q^{39} -2.00000 q^{41} -12.2474 q^{43} +10.3923 q^{45} +11.3137 q^{47} +1.00000 q^{49} +9.79796 q^{51} +10.3923 q^{53} +8.48528 q^{55} -6.00000 q^{57} -2.44949 q^{59} -3.46410 q^{61} -8.48528 q^{63} -12.0000 q^{65} +7.34847 q^{67} -6.92820 q^{69} +2.82843 q^{71} +8.00000 q^{73} +17.1464 q^{75} -6.92820 q^{77} -5.65685 q^{79} -9.00000 q^{81} -7.34847 q^{83} +13.8564 q^{85} +8.48528 q^{87} -8.00000 q^{89} +9.79796 q^{91} -13.8564 q^{93} -8.48528 q^{95} +12.0000 q^{97} +7.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9} + 16 q^{17} + 28 q^{25} + 24 q^{33} - 8 q^{41} + 4 q^{49} - 24 q^{57} - 48 q^{65} + 32 q^{73} - 36 q^{81} - 32 q^{89} + 48 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 + 16 * q^17 + 28 * q^25 + 24 * q^33 - 8 * q^41 + 4 * q^49 - 24 * q^57 - 48 * q^65 + 32 * q^73 - 36 * q^81 - 32 * q^89 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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