LMFDB - Embedded newform 1000.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1000,2,Mod(1,1000)] 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("1000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1000 = 2^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,1,0,0,0,9] 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:	$7.98504020213$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.10025.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 10x + 20 $ x^4 - x^3 - 11x^2 + 10x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.65792$ of defining polynomial
Character	$\chi$	$=$	1000.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.65792 q^{3} +4.68257 q^{7} +4.06454 q^{9} -0.0398855 q^{11} -1.02465 q^{13} +4.95852 q^{17} -8.21924 q^{19} +12.4459 q^{21} -4.15471 q^{23} +2.82945 q^{27} -0.878753 q^{29} +5.45690 q^{31} -0.106013 q^{33} -9.51202 q^{37} -2.72344 q^{39} -5.28378 q^{41} -8.95852 q^{43} +9.09017 q^{47} +14.9265 q^{49} +13.1794 q^{51} +1.18095 q^{53} -21.8461 q^{57} +9.34049 q^{59} +2.57815 q^{61} +19.0325 q^{63} -8.46015 q^{67} -11.0429 q^{69} +15.0383 q^{71} +3.58597 q^{73} -0.186767 q^{77} -10.2439 q^{79} -4.67315 q^{81} -9.65890 q^{83} -2.33565 q^{87} +0.759096 q^{89} -4.79800 q^{91} +14.5040 q^{93} -6.67217 q^{97} -0.162116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 9 q^{7} + 11 q^{9} + 5 q^{11} - 4 q^{13} - 4 q^{17} + 3 q^{21} + 11 q^{23} - 2 q^{27} + 10 q^{29} + 9 q^{31} - 19 q^{33} - 15 q^{37} + 21 q^{39} + 17 q^{41} - 12 q^{43} + 14 q^{47} + 17 q^{49}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + q^3 + 9 * q^7 + 11 * q^9 + 5 * q^11 - 4 * q^13 - 4 * q^17 + 3 * q^21 + 11 * q^23 - 2 * q^27 + 10 * q^29 + 9 * q^31 - 19 * q^33 - 15 * q^37 + 21 * q^39 + 17 * q^41 - 12 * q^43 + 14 * q^47 + 17 * q^49 + 25 * q^51 + 6 * q^53 - 9 * q^57 + 18 * q^59 + 11 * q^61 + 52 * q^63 + q^67 - 8 * q^69 + 26 * q^71 - 9 * q^73 + 8 * q^77 - 8 * q^79 - 4 * q^81 - 12 * q^83 + 17 * q^87 + 5 * q^89 - 33 * q^91 - 30 * q^93 - 29 * q^97 + 8 * 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
$3$	2.65792	1.53455	0.767275	−	0.641318i	$-0.221612\pi$
$3$	2.65792	1.53455	0.767275	+	0.641318i	$0.221612\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.68257	1.76985	0.884923	−	0.465738i	$-0.154211\pi$
$7$	4.68257	1.76985	0.884923	+	0.465738i	$0.154211\pi$
$8$	0	0
$9$	4.06454	1.35485
$10$	0	0
$11$	−0.0398855	−0.0120259	−0.00601297	−	0.999982i	$-0.501914\pi$
$11$	−0.0398855	−0.0120259	−0.00601297	+	0.999982i	$0.501914\pi$
$12$	0	0
$13$	−1.02465	−0.284187	−0.142093	−	0.989853i	$-0.545383\pi$
$13$	−1.02465	−0.284187	−0.142093	+	0.989853i	$0.545383\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.95852	1.20262	0.601309	−	0.799016i	$-0.294646\pi$
$17$	4.95852	1.20262	0.601309	+	0.799016i	$0.294646\pi$
$18$	0	0
$19$	−8.21924	−1.88562	−0.942812	−	0.333326i	$-0.891829\pi$
$19$	−8.21924	−1.88562	−0.942812	+	0.333326i	$0.891829\pi$
$20$	0	0
$21$	12.4459	2.71592
$22$	0	0
$23$	−4.15471	−0.866316	−0.433158	−	0.901318i	$-0.642601\pi$
$23$	−4.15471	−0.866316	−0.433158	+	0.901318i	$0.642601\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.82945	0.544528
$28$	0	0
$29$	−0.878753	−0.163180	−0.0815901	−	0.996666i	$-0.526000\pi$
$29$	−0.878753	−0.163180	−0.0815901	+	0.996666i	$0.526000\pi$
$30$	0	0
$31$	5.45690	0.980088	0.490044	−	0.871698i	$-0.336981\pi$
$31$	5.45690	0.980088	0.490044	+	0.871698i	$0.336981\pi$
$32$	0	0
$33$	−0.106013	−0.0184544
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.51202	−1.56377	−0.781883	−	0.623425i	$-0.785741\pi$
$37$	−9.51202	−1.56377	−0.781883	+	0.623425i	$0.785741\pi$
$38$	0	0
$39$	−2.72344	−0.436099
$40$	0	0
$41$	−5.28378	−0.825188	−0.412594	−	0.910915i	$-0.635377\pi$
$41$	−5.28378	−0.825188	−0.412594	+	0.910915i	$0.635377\pi$
$42$	0	0
$43$	−8.95852	−1.36616	−0.683081	−	0.730343i	$-0.739360\pi$
$43$	−8.95852	−1.36616	−0.683081	+	0.730343i	$0.739360\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.09017	1.32594	0.662969	−	0.748647i	$-0.269296\pi$
$47$	9.09017	1.32594	0.662969	+	0.748647i	$0.269296\pi$
$48$	0	0
$49$	14.9265	2.13235
$50$	0	0
$51$	13.1794	1.84548
$52$	0	0
$53$	1.18095	0.162216	0.0811078	−	0.996705i	$-0.474154\pi$
$53$	1.18095	0.162216	0.0811078	+	0.996705i	$0.474154\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−21.8461	−2.89358
$58$	0	0
$59$	9.34049	1.21603	0.608014	−	0.793926i	$-0.291966\pi$
$59$	9.34049	1.21603	0.608014	+	0.793926i	$0.291966\pi$
$60$	0	0
$61$	2.57815	0.330098	0.165049	−	0.986285i	$-0.447222\pi$
$61$	2.57815	0.330098	0.165049	+	0.986285i	$0.447222\pi$
$62$	0	0
$63$	19.0325	2.39787
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.46015	−1.03357	−0.516786	−	0.856115i	$-0.672872\pi$
$67$	−8.46015	−1.03357	−0.516786	+	0.856115i	$0.672872\pi$
$68$	0	0
$69$	−11.0429	−1.32941
$70$	0	0
$71$	15.0383	1.78472	0.892359	−	0.451327i	$-0.149049\pi$
$71$	15.0383	1.78472	0.892359	+	0.451327i	$0.149049\pi$
$72$	0	0
$73$	3.58597	0.419706	0.209853	−	0.977733i	$-0.432701\pi$
$73$	3.58597	0.419706	0.209853	+	0.977733i	$0.432701\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.186767	−0.0212840
$78$	0	0
$79$	−10.2439	−1.15253	−0.576264	−	0.817264i	$-0.695490\pi$
$79$	−10.2439	−1.15253	−0.576264	+	0.817264i	$0.695490\pi$
$80$	0	0
$81$	−4.67315	−0.519239
$82$	0	0
$83$	−9.65890	−1.06020	−0.530101	−	0.847934i	$-0.677846\pi$
$83$	−9.65890	−1.06020	−0.530101	+	0.847934i	$0.677846\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.33565	−0.250408
$88$	0	0
$89$	0.759096	0.0804640	0.0402320	−	0.999190i	$-0.487190\pi$
$89$	0.759096	0.0804640	0.0402320	+	0.999190i	$0.487190\pi$
$90$	0	0
$91$	−4.79800	−0.502967
$92$	0	0
$93$	14.5040	1.50400
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.67217	−0.677456	−0.338728	−	0.940884i	$-0.609997\pi$
$97$	−6.67217	−0.677456	−0.338728	+	0.940884i	$0.609997\pi$
$98$	0	0
$99$	−0.162116	−0.0162933
$100$	0	0
$101$	11.5597	1.15024	0.575118	−	0.818070i	$-0.304956\pi$
$101$	11.5597	1.15024	0.575118	+	0.818070i	$0.304956\pi$
$102$	0	0
$103$	10.3829	1.02306	0.511531	−	0.859265i	$-0.329078\pi$
$103$	10.3829	1.02306	0.511531	+	0.859265i	$0.329078\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.94329	0.574559	0.287280	−	0.957847i	$-0.407249\pi$
$107$	5.94329	0.574559	0.287280	+	0.957847i	$0.407249\pi$
$108$	0	0
$109$	−9.17936	−0.879223	−0.439611	−	0.898188i	$-0.644884\pi$
$109$	−9.17936	−0.879223	−0.439611	+	0.898188i	$0.644884\pi$
$110$	0	0
$111$	−25.2822	−2.39968
$112$	0	0
$113$	−14.9249	−1.40401	−0.702007	−	0.712170i	$-0.747712\pi$
$113$	−14.9249	−1.40401	−0.702007	+	0.712170i	$0.747712\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.16473	−0.385029
$118$	0	0
$119$	23.2186	2.12845
$120$	0	0
$121$	−10.9984	−0.999855
$122$	0	0
$123$	−14.0439	−1.26629
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.68416	0.593123	0.296562	−	0.955014i	$-0.404160\pi$
$127$	6.68416	0.593123	0.296562	+	0.955014i	$0.404160\pi$
$128$	0	0
$129$	−23.8110	−2.09644
$130$	0	0
$131$	0.552515	0.0482734	0.0241367	−	0.999709i	$-0.492316\pi$
$131$	0.552515	0.0482734	0.0241367	+	0.999709i	$0.492316\pi$
$132$	0	0
$133$	−38.4872	−3.33726
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.95168	−0.593922	−0.296961	−	0.954890i	$-0.595973\pi$
$137$	−6.95168	−0.593922	−0.296961	+	0.954890i	$0.595973\pi$
$138$	0	0
$139$	7.19300	0.610102	0.305051	−	0.952336i	$-0.401326\pi$
$139$	7.19300	0.610102	0.305051	+	0.952336i	$0.401326\pi$
$140$	0	0
$141$	24.1609	2.03472
$142$	0	0
$143$	0.0408687	0.00341761
$144$	0	0
$145$	0	0
$146$	0	0
$147$	39.6733	3.27220
$148$	0	0
$149$	9.22182	0.755481	0.377740	−	0.925912i	$-0.376701\pi$
$149$	9.22182	0.755481	0.377740	+	0.925912i	$0.376701\pi$
$150$	0	0
$151$	3.09501	0.251868	0.125934	−	0.992039i	$-0.459807\pi$
$151$	3.09501	0.251868	0.125934	+	0.992039i	$0.459807\pi$
$152$	0	0
$153$	20.1541	1.62936
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.39463	0.191112	0.0955560	−	0.995424i	$-0.469537\pi$
$157$	2.39463	0.191112	0.0955560	+	0.995424i	$0.469537\pi$
$158$	0	0
$159$	3.13886	0.248928
$160$	0	0
$161$	−19.4547	−1.53325
$162$	0	0
$163$	6.59240	0.516357	0.258178	−	0.966097i	$-0.416878\pi$
$163$	6.59240	0.516357	0.258178	+	0.966097i	$0.416878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.24548	0.173761	0.0868804	−	0.996219i	$-0.472310\pi$
$167$	2.24548	0.173761	0.0868804	+	0.996219i	$0.472310\pi$
$168$	0	0
$169$	−11.9501	−0.919238
$170$	0	0
$171$	−33.4074	−2.55473
$172$	0	0
$173$	10.0782	0.766230	0.383115	−	0.923701i	$-0.374851\pi$
$173$	10.0782	0.766230	0.383115	+	0.923701i	$0.374851\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.8263	1.86606
$178$	0	0
$179$	11.0256	0.824095	0.412047	−	0.911162i	$-0.364814\pi$
$179$	11.0256	0.824095	0.412047	+	0.911162i	$0.364814\pi$
$180$	0	0
$181$	−12.8950	−0.958476	−0.479238	−	0.877685i	$-0.659087\pi$
$181$	−12.8950	−0.958476	−0.479238	+	0.877685i	$0.659087\pi$
$182$	0	0
$183$	6.85251	0.506552
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.197773	−0.0144626
$188$	0	0
$189$	13.2491	0.963731
$190$	0	0
$191$	−16.6032	−1.20136	−0.600682	−	0.799488i	$-0.705104\pi$
$191$	−16.6032	−1.20136	−0.600682	+	0.799488i	$0.705104\pi$
$192$	0	0
$193$	−4.98477	−0.358811	−0.179406	−	0.983775i	$-0.557417\pi$
$193$	−4.98477	−0.358811	−0.179406	+	0.983775i	$0.557417\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.95528	0.566790	0.283395	−	0.959003i	$-0.408539\pi$
$197$	7.95528	0.566790	0.283395	+	0.959003i	$0.408539\pi$
$198$	0	0
$199$	−4.26329	−0.302217	−0.151108	−	0.988517i	$-0.548284\pi$
$199$	−4.26329	−0.302217	−0.151108	+	0.988517i	$0.548284\pi$
$200$	0	0
$201$	−22.4864	−1.58607
$202$	0	0
$203$	−4.11482	−0.288804
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−16.8870	−1.17372
$208$	0	0
$209$	0.327829	0.0226764
$210$	0	0
$211$	20.6315	1.42033	0.710165	−	0.704035i	$-0.248620\pi$
$211$	20.6315	1.42033	0.710165	+	0.704035i	$0.248620\pi$
$212$	0	0
$213$	39.9706	2.73874
$214$	0	0
$215$	0	0
$216$	0	0
$217$	25.5523	1.73460
$218$	0	0
$219$	9.53123	0.644061
$220$	0	0
$221$	−5.08075	−0.341769
$222$	0	0
$223$	−5.79482	−0.388050	−0.194025	−	0.980997i	$-0.562154\pi$
$223$	−5.79482	−0.388050	−0.194025	+	0.980997i	$0.562154\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.95754	−0.660905	−0.330453	−	0.943823i	$-0.607201\pi$
$227$	−9.95754	−0.660905	−0.330453	+	0.943823i	$0.607201\pi$
$228$	0	0
$229$	−1.91142	−0.126310	−0.0631551	−	0.998004i	$-0.520116\pi$
$229$	−1.91142	−0.126310	−0.0631551	+	0.998004i	$0.520116\pi$
$230$	0	0
$231$	−0.496411	−0.0326614
$232$	0	0
$233$	2.74650	0.179929	0.0899646	−	0.995945i	$-0.471325\pi$
$233$	2.74650	0.179929	0.0899646	+	0.995945i	$0.471325\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−27.2274	−1.76861
$238$	0	0
$239$	−16.8189	−1.08792	−0.543961	−	0.839111i	$-0.683076\pi$
$239$	−16.8189	−1.08792	−0.543961	+	0.839111i	$0.683076\pi$
$240$	0	0
$241$	1.69879	0.109429	0.0547143	−	0.998502i	$-0.482575\pi$
$241$	1.69879	0.109429	0.0547143	+	0.998502i	$0.482575\pi$
$242$	0	0
$243$	−20.9092	−1.34133
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.42185	0.535870
$248$	0	0
$249$	−25.6726	−1.62693
$250$	0	0
$251$	−8.25270	−0.520906	−0.260453	−	0.965487i	$-0.583872\pi$
$251$	−8.25270	−0.520906	−0.260453	+	0.965487i	$0.583872\pi$
$252$	0	0
$253$	0.165713	0.0104183
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.6824	−1.53964	−0.769822	−	0.638258i	$-0.779655\pi$
$257$	−24.6824	−1.53964	−0.769822	+	0.638258i	$0.779655\pi$
$258$	0	0
$259$	−44.5407	−2.76762
$260$	0	0
$261$	−3.57172	−0.221084
$262$	0	0
$263$	12.0759	0.744633	0.372317	−	0.928106i	$-0.378564\pi$
$263$	12.0759	0.744633	0.372317	+	0.928106i	$0.378564\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.01762	0.123476
$268$	0	0
$269$	17.3577	1.05832	0.529160	−	0.848522i	$-0.322507\pi$
$269$	17.3577	1.05832	0.529160	+	0.848522i	$0.322507\pi$
$270$	0	0
$271$	−7.40302	−0.449701	−0.224851	−	0.974393i	$-0.572189\pi$
$271$	−7.40302	−0.449701	−0.224851	+	0.974393i	$0.572189\pi$
$272$	0	0
$273$	−12.7527	−0.771828
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.3208	1.10079	0.550395	−	0.834904i	$-0.314477\pi$
$277$	18.3208	1.10079	0.550395	+	0.834904i	$0.314477\pi$
$278$	0	0
$279$	22.1798	1.32787
$280$	0	0
$281$	−20.3539	−1.21421	−0.607105	−	0.794621i	$-0.707669\pi$
$281$	−20.3539	−1.21421	−0.607105	+	0.794621i	$0.707669\pi$
$282$	0	0
$283$	3.33149	0.198036	0.0990182	−	0.995086i	$-0.468430\pi$
$283$	3.33149	0.198036	0.0990182	+	0.995086i	$0.468430\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.7417	−1.46045
$288$	0	0
$289$	7.58696	0.446292
$290$	0	0
$291$	−17.7341	−1.03959
$292$	0	0
$293$	−25.3525	−1.48111	−0.740556	−	0.671995i	$-0.765438\pi$
$293$	−25.3525	−1.48111	−0.740556	+	0.671995i	$0.765438\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.112854	−0.00654846
$298$	0	0
$299$	4.25712	0.246196
$300$	0	0
$301$	−41.9489	−2.41790
$302$	0	0
$303$	30.7248	1.76510
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.2611	−1.09929	−0.549644	−	0.835399i	$-0.685237\pi$
$307$	−19.2611	−1.09929	−0.549644	+	0.835399i	$0.685237\pi$
$308$	0	0
$309$	27.5970	1.56994
$310$	0	0
$311$	29.6626	1.68201	0.841005	−	0.541028i	$-0.181965\pi$
$311$	29.6626	1.68201	0.841005	+	0.541028i	$0.181965\pi$
$312$	0	0
$313$	−15.5759	−0.880405	−0.440202	−	0.897899i	$-0.645093\pi$
$313$	−15.5759	−0.880405	−0.440202	+	0.897899i	$0.645093\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.2994	−1.81412	−0.907058	−	0.421005i	$-0.861677\pi$
$317$	−32.2994	−1.81412	−0.907058	+	0.421005i	$0.861677\pi$
$318$	0	0
$319$	0.0350495	0.00196240
$320$	0	0
$321$	15.7968	0.881690
$322$	0	0
$323$	−40.7553	−2.26769
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−24.3980	−1.34921
$328$	0	0
$329$	42.5654	2.34670
$330$	0	0
$331$	20.3804	1.12021	0.560103	−	0.828423i	$-0.310761\pi$
$331$	20.3804	1.12021	0.560103	+	0.828423i	$0.310761\pi$
$332$	0	0
$333$	−38.6620	−2.11866
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.85795	−0.264630	−0.132315	−	0.991208i	$-0.542241\pi$
$337$	−4.85795	−0.264630	−0.132315	+	0.991208i	$0.542241\pi$
$338$	0	0
$339$	−39.6691	−2.15453
$340$	0	0
$341$	−0.217651	−0.0117865
$342$	0	0
$343$	37.1162	2.00409
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.5630	−1.42598	−0.712988	−	0.701177i	$-0.752658\pi$
$347$	−26.5630	−1.42598	−0.712988	+	0.701177i	$0.752658\pi$
$348$	0	0
$349$	4.01622	0.214983	0.107492	−	0.994206i	$-0.465718\pi$
$349$	4.01622	0.214983	0.107492	+	0.994206i	$0.465718\pi$
$350$	0	0
$351$	−2.89920	−0.154748
$352$	0	0
$353$	23.1038	1.22969	0.614846	−	0.788647i	$-0.289218\pi$
$353$	23.1038	1.22969	0.614846	+	0.788647i	$0.289218\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	61.7133	3.26621
$358$	0	0
$359$	23.2611	1.22767	0.613837	−	0.789433i	$-0.289625\pi$
$359$	23.2611	1.22767	0.613837	+	0.789433i	$0.289625\pi$
$360$	0	0
$361$	48.5559	2.55558
$362$	0	0
$363$	−29.2329	−1.53433
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.305184	−0.0159305	−0.00796523	−	0.999968i	$-0.502535\pi$
$367$	−0.305184	−0.0159305	−0.00796523	+	0.999968i	$0.502535\pi$
$368$	0	0
$369$	−21.4761	−1.11800
$370$	0	0
$371$	5.52987	0.287097
$372$	0	0
$373$	6.97859	0.361338	0.180669	−	0.983544i	$-0.442174\pi$
$373$	6.97859	0.361338	0.180669	+	0.983544i	$0.442174\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.900414	0.0463737
$378$	0	0
$379$	27.8509	1.43061	0.715303	−	0.698815i	$-0.246289\pi$
$379$	27.8509	1.43061	0.715303	+	0.698815i	$0.246289\pi$
$380$	0	0
$381$	17.7660	0.910178
$382$	0	0
$383$	7.54707	0.385637	0.192819	−	0.981234i	$-0.438237\pi$
$383$	7.54707	0.385637	0.192819	+	0.981234i	$0.438237\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−36.4122	−1.85094
$388$	0	0
$389$	11.4831	0.582218	0.291109	−	0.956690i	$-0.405976\pi$
$389$	11.4831	0.582218	0.291109	+	0.956690i	$0.405976\pi$
$390$	0	0
$391$	−20.6012	−1.04185
$392$	0	0
$393$	1.46854	0.0740780
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.39077	0.220367	0.110183	−	0.993911i	$-0.464856\pi$
$397$	4.39077	0.220367	0.110183	+	0.993911i	$0.464856\pi$
$398$	0	0
$399$	−102.296	−5.12120
$400$	0	0
$401$	−9.31883	−0.465360	−0.232680	−	0.972553i	$-0.574749\pi$
$401$	−9.31883	−0.465360	−0.232680	+	0.972553i	$0.574749\pi$
$402$	0	0
$403$	−5.59142	−0.278528
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.379392	0.0188058
$408$	0	0
$409$	−22.7657	−1.12569	−0.562846	−	0.826562i	$-0.690294\pi$
$409$	−22.7657	−1.12569	−0.562846	+	0.826562i	$0.690294\pi$
$410$	0	0
$411$	−18.4770	−0.911404
$412$	0	0
$413$	43.7375	2.15218
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.1184	0.936233
$418$	0	0
$419$	18.1849	0.888391	0.444196	−	0.895930i	$-0.353489\pi$
$419$	18.1849	0.888391	0.444196	+	0.895930i	$0.353489\pi$
$420$	0	0
$421$	−14.6177	−0.712424	−0.356212	−	0.934405i	$-0.615932\pi$
$421$	−14.6177	−0.712424	−0.356212	+	0.934405i	$0.615932\pi$
$422$	0	0
$423$	36.9473	1.79644
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.0724	0.584222
$428$	0	0
$429$	0.108626	0.00524450
$430$	0	0
$431$	9.73641	0.468986	0.234493	−	0.972118i	$-0.424657\pi$
$431$	9.73641	0.468986	0.234493	+	0.972118i	$0.424657\pi$
$432$	0	0
$433$	20.1544	0.968558	0.484279	−	0.874914i	$-0.339082\pi$
$433$	20.1544	0.968558	0.484279	+	0.874914i	$0.339082\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.1485	1.63355
$438$	0	0
$439$	8.93773	0.426574	0.213287	−	0.976990i	$-0.431583\pi$
$439$	8.93773	0.426574	0.213287	+	0.976990i	$0.431583\pi$
$440$	0	0
$441$	60.6691	2.88901
$442$	0	0
$443$	4.43390	0.210661	0.105331	−	0.994437i	$-0.466410\pi$
$443$	4.43390	0.210661	0.105331	+	0.994437i	$0.466410\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	24.5108	1.15932
$448$	0	0
$449$	38.3645	1.81053	0.905267	−	0.424843i	$-0.139671\pi$
$449$	38.3645	1.81053	0.905267	+	0.424843i	$0.139671\pi$
$450$	0	0
$451$	0.210746	0.00992365
$452$	0	0
$453$	8.22628	0.386504
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.8529	−1.49002	−0.745009	−	0.667055i	$-0.767555\pi$
$457$	−31.8529	−1.49002	−0.745009	+	0.667055i	$0.767555\pi$
$458$	0	0
$459$	14.0299	0.654860
$460$	0	0
$461$	35.4680	1.65191	0.825954	−	0.563737i	$-0.190637\pi$
$461$	35.4680	1.65191	0.825954	+	0.563737i	$0.190637\pi$
$462$	0	0
$463$	−0.845101	−0.0392752	−0.0196376	−	0.999807i	$-0.506251\pi$
$463$	−0.845101	−0.0392752	−0.0196376	+	0.999807i	$0.506251\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.7848	0.637884	0.318942	−	0.947774i	$-0.396672\pi$
$467$	13.7848	0.637884	0.318942	+	0.947774i	$0.396672\pi$
$468$	0	0
$469$	−39.6152	−1.82926
$470$	0	0
$471$	6.36473	0.293271
$472$	0	0
$473$	0.357315	0.0164294
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.80000	0.219777
$478$	0	0
$479$	7.08075	0.323528	0.161764	−	0.986829i	$-0.448282\pi$
$479$	7.08075	0.323528	0.161764	+	0.986829i	$0.448282\pi$
$480$	0	0
$481$	9.74650	0.444402
$482$	0	0
$483$	−51.7090	−2.35284
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.3059	1.14672	0.573359	−	0.819304i	$-0.305640\pi$
$487$	25.3059	1.14672	0.573359	+	0.819304i	$0.305640\pi$
$488$	0	0
$489$	17.5221	0.792375
$490$	0	0
$491$	24.4890	1.10517	0.552587	−	0.833455i	$-0.313641\pi$
$491$	24.4890	1.10517	0.552587	+	0.833455i	$0.313641\pi$
$492$	0	0
$493$	−4.35732	−0.196244
$494$	0	0
$495$	0	0
$496$	0	0
$497$	70.4179	3.15867
$498$	0	0
$499$	3.14230	0.140669	0.0703344	−	0.997523i	$-0.477593\pi$
$499$	3.14230	0.140669	0.0703344	+	0.997523i	$0.477593\pi$
$500$	0	0
$501$	5.96831	0.266645
$502$	0	0
$503$	−20.7790	−0.926489	−0.463244	−	0.886231i	$-0.653315\pi$
$503$	−20.7790	−0.926489	−0.463244	+	0.886231i	$0.653315\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.7624	−1.41062
$508$	0	0
$509$	−27.9139	−1.23726	−0.618630	−	0.785682i	$-0.712312\pi$
$509$	−27.9139	−1.23726	−0.618630	+	0.785682i	$0.712312\pi$
$510$	0	0
$511$	16.7916	0.742815
$512$	0	0
$513$	−23.2559	−1.02678
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.362566	−0.0159456
$518$	0	0
$519$	26.7870	1.17582
$520$	0	0
$521$	24.0675	1.05442	0.527209	−	0.849736i	$-0.323239\pi$
$521$	24.0675	1.05442	0.527209	+	0.849736i	$0.323239\pi$
$522$	0	0
$523$	40.1150	1.75411	0.877053	−	0.480393i	$-0.159506\pi$
$523$	40.1150	1.75411	0.877053	+	0.480393i	$0.159506\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.0582	1.17867
$528$	0	0
$529$	−5.73842	−0.249496
$530$	0	0
$531$	37.9648	1.64753
$532$	0	0
$533$	5.41403	0.234508
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.3052	1.26461
$538$	0	0
$539$	−0.595350	−0.0256435
$540$	0	0
$541$	−28.9359	−1.24405	−0.622027	−	0.782996i	$-0.713690\pi$
$541$	−28.9359	−1.24405	−0.622027	+	0.782996i	$0.713690\pi$
$542$	0	0
$543$	−34.2738	−1.47083
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−25.9290	−1.10865	−0.554323	−	0.832302i	$-0.687023\pi$
$547$	−25.9290	−1.10865	−0.554323	+	0.832302i	$0.687023\pi$
$548$	0	0
$549$	10.4790	0.447232
$550$	0	0
$551$	7.22268	0.307697
$552$	0	0
$553$	−47.9677	−2.03980
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.7002	0.580496	0.290248	−	0.956952i	$-0.406262\pi$
$557$	13.7002	0.580496	0.290248	+	0.956952i	$0.406262\pi$
$558$	0	0
$559$	9.17936	0.388245
$560$	0	0
$561$	−0.525666	−0.0221936
$562$	0	0
$563$	24.8866	1.04885	0.524423	−	0.851458i	$-0.324281\pi$
$563$	24.8866	1.04885	0.524423	+	0.851458i	$0.324281\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.8824	−0.918973
$568$	0	0
$569$	28.2649	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$569$	28.2649	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$570$	0	0
$571$	−7.04643	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$571$	−7.04643	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$572$	0	0
$573$	−44.1299	−1.84355
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.86295	−0.0775556	−0.0387778	−	0.999248i	$-0.512346\pi$
$577$	−1.86295	−0.0775556	−0.0387778	+	0.999248i	$0.512346\pi$
$578$	0	0
$579$	−13.2491	−0.550614
$580$	0	0
$581$	−45.2285	−1.87639
$582$	0	0
$583$	−0.0471027	−0.00195079
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.9871	0.742409	0.371204	−	0.928551i	$-0.378945\pi$
$587$	17.9871	0.742409	0.371204	+	0.928551i	$0.378945\pi$
$588$	0	0
$589$	−44.8516	−1.84808
$590$	0	0
$591$	21.1445	0.869768
$592$	0	0
$593$	29.1161	1.19565	0.597827	−	0.801625i	$-0.296031\pi$
$593$	29.1161	1.19565	0.597827	+	0.801625i	$0.296031\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.3315	−0.463767
$598$	0	0
$599$	29.0913	1.18864	0.594318	−	0.804230i	$-0.297422\pi$
$599$	29.0913	1.18864	0.594318	+	0.804230i	$0.297422\pi$
$600$	0	0
$601$	32.3954	1.32144	0.660718	−	0.750634i	$-0.270252\pi$
$601$	32.3954	1.32144	0.660718	+	0.750634i	$0.270252\pi$
$602$	0	0
$603$	−34.3866	−1.40033
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.0239	−0.772154	−0.386077	−	0.922467i	$-0.626170\pi$
$607$	−19.0239	−0.772154	−0.386077	+	0.922467i	$0.626170\pi$
$608$	0	0
$609$	−10.9369	−0.443184
$610$	0	0
$611$	−9.31425	−0.376814
$612$	0	0
$613$	2.03474	0.0821823	0.0410911	−	0.999155i	$-0.486917\pi$
$613$	2.03474	0.0821823	0.0410911	+	0.999155i	$0.486917\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.73728	−0.392008	−0.196004	−	0.980603i	$-0.562797\pi$
$617$	−9.73728	−0.392008	−0.196004	+	0.980603i	$0.562797\pi$
$618$	0	0
$619$	31.1593	1.25240	0.626200	−	0.779662i	$-0.284609\pi$
$619$	31.1593	1.25240	0.626200	+	0.779662i	$0.284609\pi$
$620$	0	0
$621$	−11.7555	−0.471734
$622$	0	0
$623$	3.55452	0.142409
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.871343	0.0347981
$628$	0	0
$629$	−47.1656	−1.88061
$630$	0	0
$631$	28.6829	1.14185	0.570925	−	0.821002i	$-0.306585\pi$
$631$	28.6829	1.14185	0.570925	+	0.821002i	$0.306585\pi$
$632$	0	0
$633$	54.8368	2.17957
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15.2944	−0.605987
$638$	0	0
$639$	61.1237	2.41802
$640$	0	0
$641$	22.3152	0.881399	0.440699	−	0.897655i	$-0.354731\pi$
$641$	22.3152	0.881399	0.440699	+	0.897655i	$0.354731\pi$
$642$	0	0
$643$	36.8046	1.45143	0.725716	−	0.687994i	$-0.241509\pi$
$643$	36.8046	1.45143	0.725716	+	0.687994i	$0.241509\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−49.8385	−1.95935	−0.979676	−	0.200584i	$-0.935716\pi$
$647$	−49.8385	−1.95935	−0.979676	+	0.200584i	$0.935716\pi$
$648$	0	0
$649$	−0.372550	−0.0146239
$650$	0	0
$651$	67.9160	2.66184
$652$	0	0
$653$	15.8293	0.619447	0.309723	−	0.950827i	$-0.399764\pi$
$653$	15.8293	0.619447	0.309723	+	0.950827i	$0.399764\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.5753	0.568637
$658$	0	0
$659$	−10.1311	−0.394651	−0.197325	−	0.980338i	$-0.563226\pi$
$659$	−10.1311	−0.394651	−0.197325	+	0.980338i	$0.563226\pi$
$660$	0	0
$661$	−28.7912	−1.11985	−0.559924	−	0.828544i	$-0.689170\pi$
$661$	−28.7912	−1.11985	−0.559924	+	0.828544i	$0.689170\pi$
$662$	0	0
$663$	−13.5042	−0.524461
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.65096	0.141366
$668$	0	0
$669$	−15.4022	−0.595482
$670$	0	0
$671$	−0.102831	−0.00396974
$672$	0	0
$673$	−20.6035	−0.794205	−0.397103	−	0.917774i	$-0.629984\pi$
$673$	−20.6035	−0.794205	−0.397103	+	0.917774i	$0.629984\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.5265	−1.13479	−0.567397	−	0.823444i	$-0.692049\pi$
$677$	−29.5265	−1.13479	−0.567397	+	0.823444i	$0.692049\pi$
$678$	0	0
$679$	−31.2429	−1.19899
$680$	0	0
$681$	−26.4663	−1.01419
$682$	0	0
$683$	−3.62067	−0.138541	−0.0692706	−	0.997598i	$-0.522067\pi$
$683$	−3.62067	−0.138541	−0.0692706	+	0.997598i	$0.522067\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.08040	−0.193829
$688$	0	0
$689$	−1.21006	−0.0460996
$690$	0	0
$691$	4.34306	0.165218	0.0826090	−	0.996582i	$-0.473675\pi$
$691$	4.34306	0.165218	0.0826090	+	0.996582i	$0.473675\pi$
$692$	0	0
$693$	−0.759120	−0.0288366
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−26.1997	−0.992386
$698$	0	0
$699$	7.29997	0.276110
$700$	0	0
$701$	−15.6469	−0.590976	−0.295488	−	0.955346i	$-0.595482\pi$
$701$	−15.6469	−0.590976	−0.295488	+	0.955346i	$0.595482\pi$
$702$	0	0
$703$	78.1816	2.94868
$704$	0	0
$705$	0	0
$706$	0	0
$707$	54.1293	2.03574
$708$	0	0
$709$	−10.6054	−0.398296	−0.199148	−	0.979969i	$-0.563817\pi$
$709$	−10.6054	−0.398296	−0.199148	+	0.979969i	$0.563817\pi$
$710$	0	0
$711$	−41.6367	−1.56150
$712$	0	0
$713$	−22.6718	−0.849066
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−44.7032	−1.66947
$718$	0	0
$719$	−28.6068	−1.06685	−0.533427	−	0.845846i	$-0.679096\pi$
$719$	−28.6068	−1.06685	−0.533427	+	0.845846i	$0.679096\pi$
$720$	0	0
$721$	48.6189	1.81066
$722$	0	0
$723$	4.51524	0.167924
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.410174	0.0152125	0.00760625	−	0.999971i	$-0.497579\pi$
$727$	0.410174	0.0152125	0.00760625	+	0.999971i	$0.497579\pi$
$728$	0	0
$729$	−41.5556	−1.53910
$730$	0	0
$731$	−44.4211	−1.64297
$732$	0	0
$733$	7.18559	0.265406	0.132703	−	0.991156i	$-0.457634\pi$
$733$	7.18559	0.265406	0.132703	+	0.991156i	$0.457634\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.337437	0.0124297
$738$	0	0
$739$	−45.5555	−1.67579	−0.837894	−	0.545834i	$-0.816213\pi$
$739$	−45.5555	−1.67579	−0.837894	+	0.545834i	$0.816213\pi$
$740$	0	0
$741$	22.3846	0.822319
$742$	0	0
$743$	11.8535	0.434864	0.217432	−	0.976075i	$-0.430232\pi$
$743$	11.8535	0.434864	0.217432	+	0.976075i	$0.430232\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−39.2590	−1.43641
$748$	0	0
$749$	27.8299	1.01688
$750$	0	0
$751$	0.674433	0.0246104	0.0123052	−	0.999924i	$-0.496083\pi$
$751$	0.674433	0.0246104	0.0123052	+	0.999924i	$0.496083\pi$
$752$	0	0
$753$	−21.9350	−0.799356
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.41666	−0.124181	−0.0620904	−	0.998071i	$-0.519777\pi$
$757$	−3.41666	−0.124181	−0.0620904	+	0.998071i	$0.519777\pi$
$758$	0	0
$759$	0.440451	0.0159874
$760$	0	0
$761$	−26.1930	−0.949496	−0.474748	−	0.880122i	$-0.657461\pi$
$761$	−26.1930	−0.949496	−0.474748	+	0.880122i	$0.657461\pi$
$762$	0	0
$763$	−42.9830	−1.55609
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.57074	−0.345579
$768$	0	0
$769$	15.1742	0.547195	0.273597	−	0.961844i	$-0.411786\pi$
$769$	15.1742	0.547195	0.273597	+	0.961844i	$0.411786\pi$
$770$	0	0
$771$	−65.6038	−2.36266
$772$	0	0
$773$	29.0032	1.04317	0.521587	−	0.853198i	$-0.325340\pi$
$773$	29.0032	1.04317	0.521587	+	0.853198i	$0.325340\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−118.386	−4.24706
$778$	0	0
$779$	43.4287	1.55599
$780$	0	0
$781$	−0.599810	−0.0214629
$782$	0	0
$783$	−2.48639	−0.0888562
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−19.2660	−0.686758	−0.343379	−	0.939197i	$-0.611571\pi$
$787$	−19.2660	−0.686758	−0.343379	+	0.939197i	$0.611571\pi$
$788$	0	0
$789$	32.0968	1.14268
$790$	0	0
$791$	−69.8868	−2.48489
$792$	0	0
$793$	−2.64170	−0.0938096
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.54279	0.0546483	0.0273242	−	0.999627i	$-0.491301\pi$
$797$	1.54279	0.0546483	0.0273242	+	0.999627i	$0.491301\pi$
$798$	0	0
$799$	45.0738	1.59460
$800$	0	0
$801$	3.08537	0.109016
$802$	0	0
$803$	−0.143028	−0.00504736
$804$	0	0
$805$	0	0
$806$	0	0
$807$	46.1354	1.62404
$808$	0	0
$809$	36.7800	1.29312	0.646558	−	0.762865i	$-0.276208\pi$
$809$	36.7800	1.29312	0.646558	+	0.762865i	$0.276208\pi$
$810$	0	0
$811$	−17.4212	−0.611743	−0.305871	−	0.952073i	$-0.598948\pi$
$811$	−17.4212	−0.611743	−0.305871	+	0.952073i	$0.598948\pi$
$812$	0	0
$813$	−19.6766	−0.690090
$814$	0	0
$815$	0	0
$816$	0	0
$817$	73.6323	2.57607
$818$	0	0
$819$	−19.5016	−0.681442
$820$	0	0
$821$	18.6670	0.651484	0.325742	−	0.945459i	$-0.394386\pi$
$821$	18.6670	0.651484	0.325742	+	0.945459i	$0.394386\pi$
$822$	0	0
$823$	39.4544	1.37530	0.687648	−	0.726044i	$-0.258643\pi$
$823$	39.4544	1.37530	0.687648	+	0.726044i	$0.258643\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.1204	0.908297	0.454148	−	0.890926i	$-0.349944\pi$
$827$	26.1204	0.908297	0.454148	+	0.890926i	$0.349944\pi$
$828$	0	0
$829$	−9.81880	−0.341021	−0.170510	−	0.985356i	$-0.554542\pi$
$829$	−9.81880	−0.341021	−0.170510	+	0.985356i	$0.554542\pi$
$830$	0	0
$831$	48.6952	1.68922
$832$	0	0
$833$	74.0132	2.56441
$834$	0	0
$835$	0	0
$836$	0	0
$837$	15.4400	0.533686
$838$	0	0
$839$	13.9336	0.481042	0.240521	−	0.970644i	$-0.422682\pi$
$839$	13.9336	0.481042	0.240521	+	0.970644i	$0.422682\pi$
$840$	0	0
$841$	−28.2278	−0.973372
$842$	0	0
$843$	−54.0990	−1.86327
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−51.5008	−1.76959
$848$	0	0
$849$	8.85483	0.303897
$850$	0	0
$851$	39.5197	1.35472
$852$	0	0
$853$	−18.1138	−0.620205	−0.310102	−	0.950703i	$-0.600363\pi$
$853$	−18.1138	−0.620205	−0.310102	+	0.950703i	$0.600363\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.72704	−0.332269	−0.166135	−	0.986103i	$-0.553129\pi$
$857$	−9.72704	−0.332269	−0.166135	+	0.986103i	$0.553129\pi$
$858$	0	0
$859$	16.7910	0.572902	0.286451	−	0.958095i	$-0.407524\pi$
$859$	16.7910	0.572902	0.286451	+	0.958095i	$0.407524\pi$
$860$	0	0
$861$	−65.7613	−2.24114
$862$	0	0
$863$	−3.06190	−0.104228	−0.0521141	−	0.998641i	$-0.516596\pi$
$863$	−3.06190	−0.104228	−0.0521141	+	0.998641i	$0.516596\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20.1655	0.684857
$868$	0	0
$869$	0.408583	0.0138602
$870$	0	0
$871$	8.66869	0.293727
$872$	0	0
$873$	−27.1193	−0.917849
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.7836	0.735580	0.367790	−	0.929909i	$-0.380114\pi$
$877$	21.7836	0.735580	0.367790	+	0.929909i	$0.380114\pi$
$878$	0	0
$879$	−67.3850	−2.27284
$880$	0	0
$881$	47.4065	1.59717	0.798583	−	0.601885i	$-0.205584\pi$
$881$	47.4065	1.59717	0.798583	+	0.601885i	$0.205584\pi$
$882$	0	0
$883$	−16.8303	−0.566384	−0.283192	−	0.959063i	$-0.591393\pi$
$883$	−16.8303	−0.566384	−0.283192	+	0.959063i	$0.591393\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.25584	0.0421671	0.0210835	−	0.999778i	$-0.493288\pi$
$887$	1.25584	0.0421671	0.0210835	+	0.999778i	$0.493288\pi$
$888$	0	0
$889$	31.2991	1.04974
$890$	0	0
$891$	0.186391	0.00624434
$892$	0	0
$893$	−74.7143	−2.50022
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.3151	0.377800
$898$	0	0
$899$	−4.79527	−0.159931
$900$	0	0
$901$	5.85576	0.195084
$902$	0	0
$903$	−111.497	−3.71038
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.0199	0.930385	0.465192	−	0.885210i	$-0.345985\pi$
$907$	28.0199	0.930385	0.465192	+	0.885210i	$0.345985\pi$
$908$	0	0
$909$	46.9849	1.55839
$910$	0	0
$911$	−20.7419	−0.687210	−0.343605	−	0.939114i	$-0.611648\pi$
$911$	−20.7419	−0.687210	−0.343605	+	0.939114i	$0.611648\pi$
$912$	0	0
$913$	0.385250	0.0127499
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.58719	0.0854365
$918$	0	0
$919$	−0.224493	−0.00740534	−0.00370267	−	0.999993i	$-0.501179\pi$
$919$	−0.224493	−0.00740534	−0.00370267	+	0.999993i	$0.501179\pi$
$920$	0	0
$921$	−51.1944	−1.68691
$922$	0	0
$923$	−15.4090	−0.507193
$924$	0	0
$925$	0	0
$926$	0	0
$927$	42.2019	1.38609
$928$	0	0
$929$	−33.2747	−1.09171	−0.545854	−	0.837880i	$-0.683795\pi$
$929$	−33.2747	−1.09171	−0.545854	+	0.837880i	$0.683795\pi$
$930$	0	0
$931$	−122.684	−4.02081
$932$	0	0
$933$	78.8407	2.58113
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.4776	−0.603636	−0.301818	−	0.953366i	$-0.597594\pi$
$937$	−18.4776	−0.603636	−0.301818	+	0.953366i	$0.597594\pi$
$938$	0	0
$939$	−41.3996	−1.35103
$940$	0	0
$941$	−13.3752	−0.436018	−0.218009	−	0.975947i	$-0.569956\pi$
$941$	−13.3752	−0.436018	−0.218009	+	0.975947i	$0.569956\pi$
$942$	0	0
$943$	21.9525	0.714873
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.0656	1.20447	0.602235	−	0.798319i	$-0.294277\pi$
$947$	37.0656	1.20447	0.602235	+	0.798319i	$0.294277\pi$
$948$	0	0
$949$	−3.67437	−0.119275
$950$	0	0
$951$	−85.8493	−2.78385
$952$	0	0
$953$	−27.8429	−0.901920	−0.450960	−	0.892544i	$-0.648918\pi$
$953$	−27.8429	−0.901920	−0.450960	+	0.892544i	$0.648918\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.0931588	0.00301140
$958$	0	0
$959$	−32.5517	−1.05115
$960$	0	0
$961$	−1.22223	−0.0394268
$962$	0	0
$963$	24.1567	0.778439
$964$	0	0
$965$	0	0
$966$	0	0
$967$	54.0629	1.73855	0.869274	−	0.494331i	$-0.164587\pi$
$967$	54.0629	1.73855	0.869274	+	0.494331i	$0.164587\pi$
$968$	0	0
$969$	−108.324	−3.47988
$970$	0	0
$971$	53.4131	1.71411	0.857054	−	0.515227i	$-0.172292\pi$
$971$	53.4131	1.71411	0.857054	+	0.515227i	$0.172292\pi$
$972$	0	0
$973$	33.6817	1.07979
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.7391	−1.30336	−0.651679	−	0.758494i	$-0.725935\pi$
$977$	−40.7391	−1.30336	−0.651679	+	0.758494i	$0.725935\pi$
$978$	0	0
$979$	−0.0302769	−0.000967655 0
$980$	0	0
$981$	−37.3098	−1.19121
$982$	0	0
$983$	6.22306	0.198485	0.0992423	−	0.995063i	$-0.468358\pi$
$983$	6.22306	0.198485	0.0992423	+	0.995063i	$0.468358\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	113.135	3.60114
$988$	0	0
$989$	37.2200	1.18353
$990$	0	0
$991$	−27.7894	−0.882759	−0.441379	−	0.897321i	$-0.645511\pi$
$991$	−27.7894	−0.882759	−0.441379	+	0.897321i	$0.645511\pi$
$992$	0	0
$993$	54.1694	1.71901
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.56549	−0.271272	−0.135636	−	0.990759i	$-0.543308\pi$
$997$	−8.56549	−0.271272	−0.135636	+	0.990759i	$0.543308\pi$
$998$	0	0
$999$	−26.9138	−0.851515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.2.a.g.1.4	yes	4
3.2	odd	2		9000.2.a.bb.1.4		4
4.3	odd	2		2000.2.a.n.1.1		4
5.2	odd	4		1000.2.c.c.249.2		8
5.3	odd	4		1000.2.c.c.249.7		8
5.4	even	2		1000.2.a.f.1.1	✓	4
8.3	odd	2		8000.2.a.bo.1.4		4
8.5	even	2		8000.2.a.bd.1.1		4
15.14	odd	2		9000.2.a.q.1.1		4
20.3	even	4		2000.2.c.i.1249.2		8
20.7	even	4		2000.2.c.i.1249.7		8
20.19	odd	2		2000.2.a.q.1.4		4
40.19	odd	2		8000.2.a.be.1.1		4
40.29	even	2		8000.2.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.f.1.1	✓	4	5.4	even	2
1000.2.a.g.1.4	yes	4	1.1	even	1	trivial
1000.2.c.c.249.2		8	5.2	odd	4
1000.2.c.c.249.7		8	5.3	odd	4
2000.2.a.n.1.1		4	4.3	odd	2
2000.2.a.q.1.4		4	20.19	odd	2
2000.2.c.i.1249.2		8	20.3	even	4
2000.2.c.i.1249.7		8	20.7	even	4
8000.2.a.bd.1.1		4	8.5	even	2
8000.2.a.be.1.1		4	40.19	odd	2
8000.2.a.bn.1.4		4	40.29	even	2
8000.2.a.bo.1.4		4	8.3	odd	2
9000.2.a.q.1.1		4	15.14	odd	2
9000.2.a.bb.1.4		4	3.2	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:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.65792 q^{3} +4.68257 q^{7} +4.06454 q^{9} -0.0398855 q^{11} -1.02465 q^{13} +4.95852 q^{17} -8.21924 q^{19} +12.4459 q^{21} -4.15471 q^{23} +2.82945 q^{27} -0.878753 q^{29} +5.45690 q^{31} -0.106013 q^{33} -9.51202 q^{37} -2.72344 q^{39} -5.28378 q^{41} -8.95852 q^{43} +9.09017 q^{47} +14.9265 q^{49} +13.1794 q^{51} +1.18095 q^{53} -21.8461 q^{57} +9.34049 q^{59} +2.57815 q^{61} +19.0325 q^{63} -8.46015 q^{67} -11.0429 q^{69} +15.0383 q^{71} +3.58597 q^{73} -0.186767 q^{77} -10.2439 q^{79} -4.67315 q^{81} -9.65890 q^{83} -2.33565 q^{87} +0.759096 q^{89} -4.79800 q^{91} +14.5040 q^{93} -6.67217 q^{97} -0.162116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 9 q^{7} + 11 q^{9} + 5 q^{11} - 4 q^{13} - 4 q^{17} + 3 q^{21} + 11 q^{23} - 2 q^{27} + 10 q^{29} + 9 q^{31} - 19 q^{33} - 15 q^{37} + 21 q^{39} + 17 q^{41} - 12 q^{43} + 14 q^{47} + 17 q^{49}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + q^3 + 9 * q^7 + 11 * q^9 + 5 * q^11 - 4 * q^13 - 4 * q^17 + 3 * q^21 + 11 * q^23 - 2 * q^27 + 10 * q^29 + 9 * q^31 - 19 * q^33 - 15 * q^37 + 21 * q^39 + 17 * q^41 - 12 * q^43 + 14 * q^47 + 17 * q^49 + 25 * q^51 + 6 * q^53 - 9 * q^57 + 18 * q^59 + 11 * q^61 + 52 * q^63 + q^67 - 8 * q^69 + 26 * q^71 - 9 * q^73 + 8 * q^77 - 8 * q^79 - 4 * q^81 - 12 * q^83 + 17 * q^87 + 5 * q^89 - 33 * q^91 - 30 * q^93 - 29 * q^97 + 8 * q^99

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