LMFDB - Embedded newform 800.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(1,800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$47.2015280046$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.21110 q^{3} +7.21110 q^{7} +25.0000 q^{9} -43.2666 q^{11} -34.0000 q^{13} -114.000 q^{17} +52.0000 q^{21} -209.122 q^{23} -14.4222 q^{27} -26.0000 q^{29} -100.955 q^{31} -312.000 q^{33} +150.000 q^{37} -245.177 q^{39} +342.000 q^{41} -454.299 q^{43} +584.099 q^{47} -291.000 q^{49} -822.066 q^{51} +262.000 q^{53} +490.355 q^{59} -262.000 q^{61} +180.278 q^{63} -497.566 q^{67} -1508.00 q^{69} +1052.82 q^{71} -682.000 q^{73} -312.000 q^{77} +201.911 q^{79} -779.000 q^{81} +151.433 q^{83} -187.489 q^{87} -630.000 q^{89} -245.177 q^{91} -728.000 q^{93} +966.000 q^{97} -1081.67 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 50 q^{9} - 68 q^{13} - 228 q^{17} + 104 q^{21} - 52 q^{29} - 624 q^{33} + 300 q^{37} + 684 q^{41} - 582 q^{49} + 524 q^{53} - 524 q^{61} - 3016 q^{69} - 1364 q^{73} - 624 q^{77} - 1558 q^{81} - 1260 q^{89}+ \cdots + 1932 q^{97}+O(q^{100}) $ 2 * q + 50 * q^9 - 68 * q^13 - 228 * q^17 + 104 * q^21 - 52 * q^29 - 624 * q^33 + 300 * q^37 + 684 * q^41 - 582 * q^49 + 524 * q^53 - 524 * q^61 - 3016 * q^69 - 1364 * q^73 - 624 * q^77 - 1558 * q^81 - 1260 * q^89 - 1456 * q^93 + 1932 * 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
$3$	7.21110	1.38778	0.693889	−	0.720082i	$-0.255896\pi$
$3$	7.21110	1.38778	0.693889	+	0.720082i	$0.255896\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.21110	0.389363	0.194681	−	0.980867i	$-0.437633\pi$
$7$	7.21110	0.389363	0.194681	+	0.980867i	$0.437633\pi$
$8$	0	0
$9$	25.0000	0.925926
$10$	0	0
$11$	−43.2666	−1.18594	−0.592972	−	0.805223i	$-0.702045\pi$
$11$	−43.2666	−1.18594	−0.592972	+	0.805223i	$0.702045\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	52.0000	0.540349
$22$	0	0
$23$	−209.122	−1.89587	−0.947934	−	0.318468i	$-0.896832\pi$
$23$	−209.122	−1.89587	−0.947934	+	0.318468i	$0.896832\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−14.4222	−0.102798
$28$	0	0
$29$	−26.0000	−0.166485	−0.0832427	−	0.996529i	$-0.526528\pi$
$29$	−26.0000	−0.166485	−0.0832427	+	0.996529i	$0.526528\pi$
$30$	0	0
$31$	−100.955	−0.584907	−0.292454	−	0.956280i	$-0.594472\pi$
$31$	−100.955	−0.584907	−0.292454	+	0.956280i	$0.594472\pi$
$32$	0	0
$33$	−312.000	−1.64583
$34$	0	0
$35$	0	0
$36$	0	0
$37$	150.000	0.666482	0.333241	−	0.942842i	$-0.391858\pi$
$37$	150.000	0.666482	0.333241	+	0.942842i	$0.391858\pi$
$38$	0	0
$39$	−245.177	−1.00666
$40$	0	0
$41$	342.000	1.30272	0.651359	−	0.758770i	$-0.274199\pi$
$41$	342.000	1.30272	0.651359	+	0.758770i	$0.274199\pi$
$42$	0	0
$43$	−454.299	−1.61116	−0.805582	−	0.592485i	$-0.798147\pi$
$43$	−454.299	−1.61116	−0.805582	+	0.592485i	$0.798147\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	584.099	1.81276	0.906379	−	0.422465i	$-0.138835\pi$
$47$	584.099	1.81276	0.906379	+	0.422465i	$0.138835\pi$
$48$	0	0
$49$	−291.000	−0.848397
$50$	0	0
$51$	−822.066	−2.25710
$52$	0	0
$53$	262.000	0.679028	0.339514	−	0.940601i	$-0.389737\pi$
$53$	262.000	0.679028	0.339514	+	0.940601i	$0.389737\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	490.355	1.08201	0.541007	−	0.841018i	$-0.318043\pi$
$59$	490.355	1.08201	0.541007	+	0.841018i	$0.318043\pi$
$60$	0	0
$61$	−262.000	−0.549929	−0.274964	−	0.961454i	$-0.588666\pi$
$61$	−262.000	−0.549929	−0.274964	+	0.961454i	$0.588666\pi$
$62$	0	0
$63$	180.278	0.360521
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−497.566	−0.907274	−0.453637	−	0.891187i	$-0.649874\pi$
$67$	−497.566	−0.907274	−0.453637	+	0.891187i	$0.649874\pi$
$68$	0	0
$69$	−1508.00	−2.63104
$70$	0	0
$71$	1052.82	1.75981	0.879907	−	0.475145i	$-0.157604\pi$
$71$	1052.82	1.75981	0.879907	+	0.475145i	$0.157604\pi$
$72$	0	0
$73$	−682.000	−1.09345	−0.546726	−	0.837311i	$-0.684126\pi$
$73$	−682.000	−1.09345	−0.546726	+	0.837311i	$0.684126\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−312.000	−0.461762
$78$	0	0
$79$	201.911	0.287554	0.143777	−	0.989610i	$-0.454075\pi$
$79$	201.911	0.287554	0.143777	+	0.989610i	$0.454075\pi$
$80$	0	0
$81$	−779.000	−1.06859
$82$	0	0
$83$	151.433	0.200264	0.100132	−	0.994974i	$-0.468073\pi$
$83$	151.433	0.200264	0.100132	+	0.994974i	$0.468073\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−187.489	−0.231045
$88$	0	0
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	0	0
$91$	−245.177	−0.282435
$92$	0	0
$93$	−728.000	−0.811721
$94$	0	0
$95$	0	0
$96$	0	0
$97$	966.000	1.01116	0.505580	−	0.862780i	$-0.331279\pi$
$97$	966.000	1.01116	0.505580	+	0.862780i	$0.331279\pi$
$98$	0	0
$99$	−1081.67	−1.09810
$100$	0	0
$101$	1638.00	1.61373	0.806867	−	0.590733i	$-0.201162\pi$
$101$	1638.00	1.61373	0.806867	+	0.590733i	$0.201162\pi$
$102$	0	0
$103$	685.055	0.655344	0.327672	−	0.944792i	$-0.393736\pi$
$103$	685.055	0.655344	0.327672	+	0.944792i	$0.393736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−612.944	−0.553790	−0.276895	−	0.960900i	$-0.589305\pi$
$107$	−612.944	−0.553790	−0.276895	+	0.960900i	$0.589305\pi$
$108$	0	0
$109$	−342.000	−0.300529	−0.150264	−	0.988646i	$-0.548013\pi$
$109$	−342.000	−0.300529	−0.150264	+	0.988646i	$0.548013\pi$
$110$	0	0
$111$	1081.67	0.924929
$112$	0	0
$113$	−2106.00	−1.75324	−0.876619	−	0.481186i	$-0.840206\pi$
$113$	−2106.00	−1.75324	−0.876619	+	0.481186i	$0.840206\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−850.000	−0.671646
$118$	0	0
$119$	−822.066	−0.633266
$120$	0	0
$121$	541.000	0.406461
$122$	0	0
$123$	2466.20	1.80788
$124$	0	0
$125$	0	0
$126$	0	0
$127$	439.877	0.307345	0.153672	−	0.988122i	$-0.450890\pi$
$127$	439.877	0.307345	0.153672	+	0.988122i	$0.450890\pi$
$128$	0	0
$129$	−3276.00	−2.23594
$130$	0	0
$131$	1773.93	1.18312	0.591561	−	0.806260i	$-0.298512\pi$
$131$	1773.93	1.18312	0.591561	+	0.806260i	$0.298512\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1618.00	−1.00902	−0.504508	−	0.863407i	$-0.668326\pi$
$137$	−1618.00	−1.00902	−0.504508	+	0.863407i	$0.668326\pi$
$138$	0	0
$139$	2509.46	1.53129	0.765647	−	0.643261i	$-0.222419\pi$
$139$	2509.46	1.53129	0.765647	+	0.643261i	$0.222419\pi$
$140$	0	0
$141$	4212.00	2.51570
$142$	0	0
$143$	1471.06	0.860256
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2098.43	−1.17739
$148$	0	0
$149$	1010.00	0.555318	0.277659	−	0.960680i	$-0.410441\pi$
$149$	1010.00	0.555318	0.277659	+	0.960680i	$0.410441\pi$
$150$	0	0
$151$	14.4222	0.00777260	0.00388630	−	0.999992i	$-0.498763\pi$
$151$	14.4222	0.00777260	0.00388630	+	0.999992i	$0.498763\pi$
$152$	0	0
$153$	−2850.00	−1.50594
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1794.00	−0.911954	−0.455977	−	0.889992i	$-0.650710\pi$
$157$	−1794.00	−0.911954	−0.455977	+	0.889992i	$0.650710\pi$
$158$	0	0
$159$	1889.31	0.942339
$160$	0	0
$161$	−1508.00	−0.738180
$162$	0	0
$163$	−1983.05	−0.952912	−0.476456	−	0.879198i	$-0.658079\pi$
$163$	−1983.05	−0.952912	−0.476456	+	0.879198i	$0.658079\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	757.166	0.350846	0.175423	−	0.984493i	$-0.443871\pi$
$167$	757.166	0.350846	0.175423	+	0.984493i	$0.443871\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2834.00	−1.24546	−0.622731	−	0.782436i	$-0.713977\pi$
$173$	−2834.00	−1.24546	−0.622731	+	0.782436i	$0.713977\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3536.00	1.50159
$178$	0	0
$179$	−951.866	−0.397462	−0.198731	−	0.980054i	$-0.563682\pi$
$179$	−951.866	−0.397462	−0.198731	+	0.980054i	$0.563682\pi$
$180$	0	0
$181$	−1466.00	−0.602027	−0.301014	−	0.953620i	$-0.597325\pi$
$181$	−1466.00	−0.602027	−0.301014	+	0.953620i	$0.597325\pi$
$182$	0	0
$183$	−1889.31	−0.763179
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4932.39	1.92884
$188$	0	0
$189$	−104.000	−0.0400259
$190$	0	0
$191$	−3475.75	−1.31674	−0.658368	−	0.752696i	$-0.728753\pi$
$191$	−3475.75	−1.31674	−0.658368	+	0.752696i	$0.728753\pi$
$192$	0	0
$193$	46.0000	0.0171562	0.00857812	−	0.999963i	$-0.497269\pi$
$193$	46.0000	0.0171562	0.00857812	+	0.999963i	$0.497269\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1122.00	−0.405783	−0.202891	−	0.979201i	$-0.565034\pi$
$197$	−1122.00	−0.405783	−0.202891	+	0.979201i	$0.565034\pi$
$198$	0	0
$199$	−2999.82	−1.06860	−0.534300	−	0.845295i	$-0.679425\pi$
$199$	−2999.82	−1.06860	−0.534300	+	0.845295i	$0.679425\pi$
$200$	0	0
$201$	−3588.00	−1.25909
$202$	0	0
$203$	−187.489	−0.0648233
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−5228.05	−1.75543
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−418.244	−0.136460	−0.0682301	−	0.997670i	$-0.521735\pi$
$211$	−418.244	−0.136460	−0.0682301	+	0.997670i	$0.521735\pi$
$212$	0	0
$213$	7592.00	2.44223
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−728.000	−0.227741
$218$	0	0
$219$	−4917.97	−1.51747
$220$	0	0
$221$	3876.00	1.17976
$222$	0	0
$223$	−2545.52	−0.764397	−0.382199	−	0.924080i	$-0.624833\pi$
$223$	−2545.52	−0.764397	−0.382199	+	0.924080i	$0.624833\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5069.41	1.48224	0.741119	−	0.671373i	$-0.234295\pi$
$227$	5069.41	1.48224	0.741119	+	0.671373i	$0.234295\pi$
$228$	0	0
$229$	−6194.00	−1.78738	−0.893692	−	0.448681i	$-0.851894\pi$
$229$	−6194.00	−1.78738	−0.893692	+	0.448681i	$0.851894\pi$
$230$	0	0
$231$	−2249.86	−0.640823
$232$	0	0
$233$	−4290.00	−1.20621	−0.603106	−	0.797661i	$-0.706070\pi$
$233$	−4290.00	−1.20621	−0.603106	+	0.797661i	$0.706070\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1456.00	0.399061
$238$	0	0
$239$	5278.53	1.42862	0.714309	−	0.699831i	$-0.246741\pi$
$239$	5278.53	1.42862	0.714309	+	0.699831i	$0.246741\pi$
$240$	0	0
$241$	−3074.00	−0.821634	−0.410817	−	0.911718i	$-0.634756\pi$
$241$	−3074.00	−0.821634	−0.410817	+	0.911718i	$0.634756\pi$
$242$	0	0
$243$	−5228.05	−1.38016
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1092.00	0.277922
$250$	0	0
$251$	−2062.38	−0.518629	−0.259315	−	0.965793i	$-0.583497\pi$
$251$	−2062.38	−0.518629	−0.259315	+	0.965793i	$0.583497\pi$
$252$	0	0
$253$	9048.00	2.24839
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3718.00	0.902422	0.451211	−	0.892417i	$-0.350992\pi$
$257$	3718.00	0.902422	0.451211	+	0.892417i	$0.350992\pi$
$258$	0	0
$259$	1081.67	0.259504
$260$	0	0
$261$	−650.000	−0.154153
$262$	0	0
$263$	−7045.25	−1.65182	−0.825910	−	0.563802i	$-0.809338\pi$
$263$	−7045.25	−1.65182	−0.825910	+	0.563802i	$0.809338\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4542.99	−1.04130
$268$	0	0
$269$	6058.00	1.37310	0.686548	−	0.727085i	$-0.259125\pi$
$269$	6058.00	1.37310	0.686548	+	0.727085i	$0.259125\pi$
$270$	0	0
$271$	−5206.42	−1.16704	−0.583519	−	0.812100i	$-0.698325\pi$
$271$	−5206.42	−1.16704	−0.583519	+	0.812100i	$0.698325\pi$
$272$	0	0
$273$	−1768.00	−0.391957
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2990.00	0.648562	0.324281	−	0.945961i	$-0.394878\pi$
$277$	2990.00	0.648562	0.324281	+	0.945961i	$0.394878\pi$
$278$	0	0
$279$	−2523.89	−0.541581
$280$	0	0
$281$	2710.00	0.575320	0.287660	−	0.957733i	$-0.407123\pi$
$281$	2710.00	0.575320	0.287660	+	0.957733i	$0.407123\pi$
$282$	0	0
$283$	4593.47	0.964854	0.482427	−	0.875936i	$-0.339755\pi$
$283$	4593.47	0.964854	0.482427	+	0.875936i	$0.339755\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2466.20	0.507230
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	6965.93	1.40326
$292$	0	0
$293$	3750.00	0.747704	0.373852	−	0.927488i	$-0.378037\pi$
$293$	3750.00	0.747704	0.373852	+	0.927488i	$0.378037\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	624.000	0.121913
$298$	0	0
$299$	7110.15	1.37522
$300$	0	0
$301$	−3276.00	−0.627327
$302$	0	0
$303$	11811.8	2.23950
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4405.98	0.819097	0.409548	−	0.912288i	$-0.365686\pi$
$307$	4405.98	0.819097	0.409548	+	0.912288i	$0.365686\pi$
$308$	0	0
$309$	4940.00	0.909472
$310$	0	0
$311$	−2956.55	−0.539070	−0.269535	−	0.962991i	$-0.586870\pi$
$311$	−2956.55	−0.539070	−0.269535	+	0.962991i	$0.586870\pi$
$312$	0	0
$313$	−5642.00	−1.01886	−0.509432	−	0.860511i	$-0.670145\pi$
$313$	−5642.00	−1.01886	−0.509432	+	0.860511i	$0.670145\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4650.00	−0.823880	−0.411940	−	0.911211i	$-0.635149\pi$
$317$	−4650.00	−0.823880	−0.411940	+	0.911211i	$0.635149\pi$
$318$	0	0
$319$	1124.93	0.197442
$320$	0	0
$321$	−4420.00	−0.768537
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2466.20	−0.417067
$328$	0	0
$329$	4212.00	0.705821
$330$	0	0
$331$	5523.70	0.917252	0.458626	−	0.888630i	$-0.348342\pi$
$331$	5523.70	0.917252	0.458626	+	0.888630i	$0.348342\pi$
$332$	0	0
$333$	3750.00	0.617113
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9266.00	−1.49778	−0.748889	−	0.662695i	$-0.769412\pi$
$337$	−9266.00	−1.49778	−0.748889	+	0.662695i	$0.769412\pi$
$338$	0	0
$339$	−15186.6	−2.43310
$340$	0	0
$341$	4368.00	0.693667
$342$	0	0
$343$	−4571.84	−0.719697
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−814.855	−0.126062	−0.0630312	−	0.998012i	$-0.520077\pi$
$347$	−814.855	−0.126062	−0.0630312	+	0.998012i	$0.520077\pi$
$348$	0	0
$349$	7494.00	1.14941	0.574706	−	0.818360i	$-0.305117\pi$
$349$	7494.00	1.14941	0.574706	+	0.818360i	$0.305117\pi$
$350$	0	0
$351$	490.355	0.0745676
$352$	0	0
$353$	6270.00	0.945378	0.472689	−	0.881229i	$-0.343283\pi$
$353$	6270.00	0.945378	0.472689	+	0.881229i	$0.343283\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5928.00	−0.878832
$358$	0	0
$359$	692.266	0.101773	0.0508863	−	0.998704i	$-0.483795\pi$
$359$	692.266	0.101773	0.0508863	+	0.998704i	$0.483795\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	3901.21	0.564078
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2141.70	0.304620	0.152310	−	0.988333i	$-0.451329\pi$
$367$	2141.70	0.304620	0.152310	+	0.988333i	$0.451329\pi$
$368$	0	0
$369$	8550.00	1.20622
$370$	0	0
$371$	1889.31	0.264388
$372$	0	0
$373$	2574.00	0.357310	0.178655	−	0.983912i	$-0.442825\pi$
$373$	2574.00	0.357310	0.178655	+	0.983912i	$0.442825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	884.000	0.120765
$378$	0	0
$379$	13729.9	1.86084	0.930422	−	0.366491i	$-0.119441\pi$
$379$	13729.9	1.86084	0.930422	+	0.366491i	$0.119441\pi$
$380$	0	0
$381$	3172.00	0.426526
$382$	0	0
$383$	4204.07	0.560883	0.280441	−	0.959871i	$-0.409519\pi$
$383$	4204.07	0.560883	0.280441	+	0.959871i	$0.409519\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11357.5	−1.49182
$388$	0	0
$389$	5314.00	0.692623	0.346312	−	0.938120i	$-0.387434\pi$
$389$	5314.00	0.692623	0.346312	+	0.938120i	$0.387434\pi$
$390$	0	0
$391$	23839.9	3.08347
$392$	0	0
$393$	12792.0	1.64191
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8638.00	1.09201	0.546006	−	0.837781i	$-0.316148\pi$
$397$	8638.00	1.09201	0.546006	+	0.837781i	$0.316148\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2802.00	0.348941	0.174470	−	0.984662i	$-0.444179\pi$
$401$	2802.00	0.348941	0.174470	+	0.984662i	$0.444179\pi$
$402$	0	0
$403$	3432.48	0.424279
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6489.99	−0.790410
$408$	0	0
$409$	−82.0000	−0.00991354	−0.00495677	−	0.999988i	$-0.501578\pi$
$409$	−82.0000	−0.00991354	−0.00495677	+	0.999988i	$0.501578\pi$
$410$	0	0
$411$	−11667.6	−1.40029
$412$	0	0
$413$	3536.00	0.421296
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18096.0	2.12510
$418$	0	0
$419$	−1067.24	−0.124435	−0.0622175	−	0.998063i	$-0.519817\pi$
$419$	−1067.24	−0.124435	−0.0622175	+	0.998063i	$0.519817\pi$
$420$	0	0
$421$	−5742.00	−0.664722	−0.332361	−	0.943152i	$-0.607845\pi$
$421$	−5742.00	−0.664722	−0.332361	+	0.943152i	$0.607845\pi$
$422$	0	0
$423$	14602.5	1.67848
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1889.31	−0.214122
$428$	0	0
$429$	10608.0	1.19384
$430$	0	0
$431$	14234.7	1.59086	0.795432	−	0.606043i	$-0.207244\pi$
$431$	14234.7	1.59086	0.795432	+	0.606043i	$0.207244\pi$
$432$	0	0
$433$	−7098.00	−0.787779	−0.393889	−	0.919158i	$-0.628871\pi$
$433$	−7098.00	−0.787779	−0.393889	+	0.919158i	$0.628871\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−461.511	−0.0501747	−0.0250874	−	0.999685i	$-0.507986\pi$
$439$	−461.511	−0.0501747	−0.0250874	+	0.999685i	$0.507986\pi$
$440$	0	0
$441$	−7275.00	−0.785552
$442$	0	0
$443$	6064.54	0.650417	0.325209	−	0.945642i	$-0.394565\pi$
$443$	6064.54	0.650417	0.325209	+	0.945642i	$0.394565\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7283.21	0.770658
$448$	0	0
$449$	−11706.0	−1.23038	−0.615190	−	0.788379i	$-0.710921\pi$
$449$	−11706.0	−1.23038	−0.615190	+	0.788379i	$0.710921\pi$
$450$	0	0
$451$	−14797.2	−1.54495
$452$	0	0
$453$	104.000	0.0107866
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5066.00	−0.518550	−0.259275	−	0.965803i	$-0.583484\pi$
$457$	−5066.00	−0.518550	−0.259275	+	0.965803i	$0.583484\pi$
$458$	0	0
$459$	1644.13	0.167193
$460$	0	0
$461$	−594.000	−0.0600116	−0.0300058	−	0.999550i	$-0.509553\pi$
$461$	−594.000	−0.0600116	−0.0300058	+	0.999550i	$0.509553\pi$
$462$	0	0
$463$	483.144	0.0484959	0.0242479	−	0.999706i	$-0.492281\pi$
$463$	483.144	0.0484959	0.0242479	+	0.999706i	$0.492281\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2271.50	0.225080	0.112540	−	0.993647i	$-0.464101\pi$
$467$	2271.50	0.225080	0.112540	+	0.993647i	$0.464101\pi$
$468$	0	0
$469$	−3588.00	−0.353259
$470$	0	0
$471$	−12936.7	−1.26559
$472$	0	0
$473$	19656.0	1.91075
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6550.00	0.628729
$478$	0	0
$479$	−4067.06	−0.387952	−0.193976	−	0.981006i	$-0.562138\pi$
$479$	−4067.06	−0.387952	−0.193976	+	0.981006i	$0.562138\pi$
$480$	0	0
$481$	−5100.00	−0.483451
$482$	0	0
$483$	−10874.3	−1.02443
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11170.0	1.03934	0.519672	−	0.854366i	$-0.326054\pi$
$487$	11170.0	1.03934	0.519672	+	0.854366i	$0.326054\pi$
$488$	0	0
$489$	−14300.0	−1.32243
$490$	0	0
$491$	−4600.68	−0.422863	−0.211432	−	0.977393i	$-0.567813\pi$
$491$	−4600.68	−0.422863	−0.211432	+	0.977393i	$0.567813\pi$
$492$	0	0
$493$	2964.00	0.270775
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7592.00	0.685207
$498$	0	0
$499$	−288.444	−0.0258768	−0.0129384	−	0.999916i	$-0.504119\pi$
$499$	−288.444	−0.0258768	−0.0129384	+	0.999916i	$0.504119\pi$
$500$	0	0
$501$	5460.00	0.486896
$502$	0	0
$503$	−18409.9	−1.63193	−0.815963	−	0.578104i	$-0.803793\pi$
$503$	−18409.9	−1.63193	−0.815963	+	0.578104i	$0.803793\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7506.76	−0.657568
$508$	0	0
$509$	−8714.00	−0.758824	−0.379412	−	0.925228i	$-0.623874\pi$
$509$	−8714.00	−0.758824	−0.379412	+	0.925228i	$0.623874\pi$
$510$	0	0
$511$	−4917.97	−0.425750
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−25272.0	−2.14983
$518$	0	0
$519$	−20436.3	−1.72842
$520$	0	0
$521$	−11830.0	−0.994783	−0.497391	−	0.867526i	$-0.665709\pi$
$521$	−11830.0	−0.994783	−0.497391	+	0.867526i	$0.665709\pi$
$522$	0	0
$523$	−8963.40	−0.749411	−0.374706	−	0.927144i	$-0.622256\pi$
$523$	−8963.40	−0.749411	−0.374706	+	0.927144i	$0.622256\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11508.9	0.951302
$528$	0	0
$529$	31565.0	2.59431
$530$	0	0
$531$	12258.9	1.00186
$532$	0	0
$533$	−11628.0	−0.944962
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6864.00	−0.551589
$538$	0	0
$539$	12590.6	1.00615
$540$	0	0
$541$	−15490.0	−1.23099	−0.615496	−	0.788140i	$-0.711044\pi$
$541$	−15490.0	−1.23099	−0.615496	+	0.788140i	$0.711044\pi$
$542$	0	0
$543$	−10571.5	−0.835480
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11429.6	−0.893408	−0.446704	−	0.894682i	$-0.647402\pi$
$547$	−11429.6	−0.893408	−0.446704	+	0.894682i	$0.647402\pi$
$548$	0	0
$549$	−6550.00	−0.509193
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1456.00	0.111963
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23862.0	1.81520	0.907599	−	0.419838i	$-0.137913\pi$
$557$	23862.0	1.81520	0.907599	+	0.419838i	$0.137913\pi$
$558$	0	0
$559$	15446.2	1.16870
$560$	0	0
$561$	35568.0	2.67680
$562$	0	0
$563$	−7261.58	−0.543586	−0.271793	−	0.962356i	$-0.587617\pi$
$563$	−7261.58	−0.543586	−0.271793	+	0.962356i	$0.587617\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5617.45	−0.416068
$568$	0	0
$569$	−9074.00	−0.668545	−0.334272	−	0.942477i	$-0.608491\pi$
$569$	−9074.00	−0.668545	−0.334272	+	0.942477i	$0.608491\pi$
$570$	0	0
$571$	−7860.10	−0.576068	−0.288034	−	0.957620i	$-0.593002\pi$
$571$	−7860.10	−0.576068	−0.288034	+	0.957620i	$0.593002\pi$
$572$	0	0
$573$	−25064.0	−1.82734
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4762.00	−0.343578	−0.171789	−	0.985134i	$-0.554955\pi$
$577$	−4762.00	−0.343578	−0.171789	+	0.985134i	$0.554955\pi$
$578$	0	0
$579$	331.711	0.0238090
$580$	0	0
$581$	1092.00	0.0779755
$582$	0	0
$583$	−11335.9	−0.805288
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21481.9	1.51048	0.755240	−	0.655448i	$-0.227520\pi$
$587$	21481.9	1.51048	0.755240	+	0.655448i	$0.227520\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−8090.86	−0.563136
$592$	0	0
$593$	−11954.0	−0.827811	−0.413906	−	0.910320i	$-0.635836\pi$
$593$	−11954.0	−0.827811	−0.413906	+	0.910320i	$0.635836\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−21632.0	−1.48298
$598$	0	0
$599$	−10759.0	−0.733889	−0.366944	−	0.930243i	$-0.619596\pi$
$599$	−10759.0	−0.733889	−0.366944	+	0.930243i	$0.619596\pi$
$600$	0	0
$601$	17862.0	1.21232	0.606162	−	0.795342i	$-0.292708\pi$
$601$	17862.0	1.21232	0.606162	+	0.795342i	$0.292708\pi$
$602$	0	0
$603$	−12439.2	−0.840069
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7506.76	0.501960	0.250980	−	0.967992i	$-0.419247\pi$
$607$	7506.76	0.501960	0.250980	+	0.967992i	$0.419247\pi$
$608$	0	0
$609$	−1352.00	−0.0899603
$610$	0	0
$611$	−19859.4	−1.31493
$612$	0	0
$613$	−11522.0	−0.759167	−0.379583	−	0.925158i	$-0.623933\pi$
$613$	−11522.0	−0.759167	−0.379583	+	0.925158i	$0.623933\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8290.00	−0.540912	−0.270456	−	0.962732i	$-0.587174\pi$
$617$	−8290.00	−0.540912	−0.270456	+	0.962732i	$0.587174\pi$
$618$	0	0
$619$	−24171.6	−1.56953	−0.784765	−	0.619793i	$-0.787216\pi$
$619$	−24171.6	−1.56953	−0.784765	+	0.619793i	$0.787216\pi$
$620$	0	0
$621$	3016.00	0.194892
$622$	0	0
$623$	−4542.99	−0.292153
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17100.0	−1.08398
$630$	0	0
$631$	−12388.7	−0.781593	−0.390797	−	0.920477i	$-0.627800\pi$
$631$	−12388.7	−0.781593	−0.390797	+	0.920477i	$0.627800\pi$
$632$	0	0
$633$	−3016.00	−0.189376
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9894.00	0.615407
$638$	0	0
$639$	26320.5	1.62946
$640$	0	0
$641$	6750.00	0.415927	0.207963	−	0.978137i	$-0.433317\pi$
$641$	6750.00	0.415927	0.207963	+	0.978137i	$0.433317\pi$
$642$	0	0
$643$	23428.9	1.43693	0.718464	−	0.695564i	$-0.244846\pi$
$643$	23428.9	1.43693	0.718464	+	0.695564i	$0.244846\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8689.38	0.527998	0.263999	−	0.964523i	$-0.414958\pi$
$647$	8689.38	0.527998	0.263999	+	0.964523i	$0.414958\pi$
$648$	0	0
$649$	−21216.0	−1.28321
$650$	0	0
$651$	−5249.68	−0.316054
$652$	0	0
$653$	−22282.0	−1.33532	−0.667659	−	0.744468i	$-0.732703\pi$
$653$	−22282.0	−1.33532	−0.667659	+	0.744468i	$0.732703\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−17050.0	−1.01246
$658$	0	0
$659$	15835.6	0.936065	0.468032	−	0.883711i	$-0.344963\pi$
$659$	15835.6	0.936065	0.468032	+	0.883711i	$0.344963\pi$
$660$	0	0
$661$	−11758.0	−0.691881	−0.345940	−	0.938256i	$-0.612440\pi$
$661$	−11758.0	−0.691881	−0.345940	+	0.938256i	$0.612440\pi$
$662$	0	0
$663$	27950.2	1.63725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5437.17	0.315634
$668$	0	0
$669$	−18356.0	−1.06081
$670$	0	0
$671$	11335.9	0.652184
$672$	0	0
$673$	−11866.0	−0.679644	−0.339822	−	0.940490i	$-0.610367\pi$
$673$	−11866.0	−0.679644	−0.339822	+	0.940490i	$0.610367\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26574.0	1.50860	0.754300	−	0.656530i	$-0.227977\pi$
$677$	26574.0	1.50860	0.754300	+	0.656530i	$0.227977\pi$
$678$	0	0
$679$	6965.93	0.393708
$680$	0	0
$681$	36556.0	2.05702
$682$	0	0
$683$	13737.2	0.769601	0.384800	−	0.923000i	$-0.374270\pi$
$683$	13737.2	0.769601	0.384800	+	0.923000i	$0.374270\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−44665.6	−2.48049
$688$	0	0
$689$	−8908.00	−0.492551
$690$	0	0
$691$	22570.8	1.24259	0.621297	−	0.783576i	$-0.286606\pi$
$691$	22570.8	1.24259	0.621297	+	0.783576i	$0.286606\pi$
$692$	0	0
$693$	−7800.00	−0.427558
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−38988.0	−2.11876
$698$	0	0
$699$	−30935.6	−1.67395
$700$	0	0
$701$	−7062.00	−0.380497	−0.190248	−	0.981736i	$-0.560929\pi$
$701$	−7062.00	−0.380497	−0.190248	+	0.981736i	$0.560929\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11811.8	0.628328
$708$	0	0
$709$	−1554.00	−0.0823155	−0.0411578	−	0.999153i	$-0.513105\pi$
$709$	−1554.00	−0.0823155	−0.0411578	+	0.999153i	$0.513105\pi$
$710$	0	0
$711$	5047.77	0.266253
$712$	0	0
$713$	21112.0	1.10891
$714$	0	0
$715$	0	0
$716$	0	0
$717$	38064.0	1.98260
$718$	0	0
$719$	−31065.4	−1.61133	−0.805664	−	0.592373i	$-0.798191\pi$
$719$	−31065.4	−1.61133	−0.805664	+	0.592373i	$0.798191\pi$
$720$	0	0
$721$	4940.00	0.255167
$722$	0	0
$723$	−22166.9	−1.14024
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−24452.8	−1.24746	−0.623732	−	0.781638i	$-0.714384\pi$
$727$	−24452.8	−1.24746	−0.623732	+	0.781638i	$0.714384\pi$
$728$	0	0
$729$	−16667.0	−0.846771
$730$	0	0
$731$	51790.1	2.62042
$732$	0	0
$733$	−15058.0	−0.758772	−0.379386	−	0.925238i	$-0.623865\pi$
$733$	−15058.0	−0.758772	−0.379386	+	0.925238i	$0.623865\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21528.0	1.07598
$738$	0	0
$739$	11912.7	0.592987	0.296493	−	0.955035i	$-0.404183\pi$
$739$	11912.7	0.592987	0.296493	+	0.955035i	$0.404183\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17270.6	0.852754	0.426377	−	0.904545i	$-0.359790\pi$
$743$	17270.6	0.852754	0.426377	+	0.904545i	$0.359790\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3785.83	0.185430
$748$	0	0
$749$	−4420.00	−0.215625
$750$	0	0
$751$	−17869.1	−0.868247	−0.434123	−	0.900853i	$-0.642942\pi$
$751$	−17869.1	−0.868247	−0.434123	+	0.900853i	$0.642942\pi$
$752$	0	0
$753$	−14872.0	−0.719742
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−35410.0	−1.70013	−0.850065	−	0.526678i	$-0.823437\pi$
$757$	−35410.0	−1.70013	−0.850065	+	0.526678i	$0.823437\pi$
$758$	0	0
$759$	65246.1	3.12027
$760$	0	0
$761$	3386.00	0.161291	0.0806455	−	0.996743i	$-0.474302\pi$
$761$	3386.00	0.161291	0.0806455	+	0.996743i	$0.474302\pi$
$762$	0	0
$763$	−2466.20	−0.117015
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16672.1	−0.784868
$768$	0	0
$769$	11522.0	0.540304	0.270152	−	0.962818i	$-0.412926\pi$
$769$	11522.0	0.540304	0.270152	+	0.962818i	$0.412926\pi$
$770$	0	0
$771$	26810.9	1.25236
$772$	0	0
$773$	14382.0	0.669191	0.334595	−	0.942362i	$-0.391400\pi$
$773$	14382.0	0.669191	0.334595	+	0.942362i	$0.391400\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7800.00	0.360133
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−45552.0	−2.08704
$782$	0	0
$783$	374.977	0.0171144
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−814.855	−0.0369078	−0.0184539	−	0.999830i	$-0.505874\pi$
$787$	−814.855	−0.0369078	−0.0184539	+	0.999830i	$0.505874\pi$
$788$	0	0
$789$	−50804.0	−2.29236
$790$	0	0
$791$	−15186.6	−0.682646
$792$	0	0
$793$	8908.00	0.398906
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14550.0	0.646659	0.323330	−	0.946286i	$-0.395198\pi$
$797$	14550.0	0.646659	0.323330	+	0.946286i	$0.395198\pi$
$798$	0	0
$799$	−66587.3	−2.94830
$800$	0	0
$801$	−15750.0	−0.694755
$802$	0	0
$803$	29507.8	1.29677
$804$	0	0
$805$	0	0
$806$	0	0
$807$	43684.9	1.90555
$808$	0	0
$809$	−37622.0	−1.63501	−0.817503	−	0.575925i	$-0.804642\pi$
$809$	−37622.0	−1.63501	−0.817503	+	0.575925i	$0.804642\pi$
$810$	0	0
$811$	−331.711	−0.0143624	−0.00718122	−	0.999974i	$-0.502286\pi$
$811$	−331.711	−0.0143624	−0.00718122	+	0.999974i	$0.502286\pi$
$812$	0	0
$813$	−37544.0	−1.61959
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6129.44	−0.261514
$820$	0	0
$821$	18690.0	0.794501	0.397251	−	0.917710i	$-0.369964\pi$
$821$	18690.0	0.794501	0.397251	+	0.917710i	$0.369964\pi$
$822$	0	0
$823$	16866.8	0.714385	0.357192	−	0.934031i	$-0.383734\pi$
$823$	16866.8	0.714385	0.357192	+	0.934031i	$0.383734\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39235.6	−1.64977	−0.824883	−	0.565304i	$-0.808759\pi$
$827$	−39235.6	−1.64977	−0.824883	+	0.565304i	$0.808759\pi$
$828$	0	0
$829$	−3718.00	−0.155768	−0.0778839	−	0.996962i	$-0.524816\pi$
$829$	−3718.00	−0.155768	−0.0778839	+	0.996962i	$0.524816\pi$
$830$	0	0
$831$	21561.2	0.900060
$832$	0	0
$833$	33174.0	1.37985
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1456.00	0.0601275
$838$	0	0
$839$	14335.7	0.589896	0.294948	−	0.955513i	$-0.404698\pi$
$839$	14335.7	0.589896	0.294948	+	0.955513i	$0.404698\pi$
$840$	0	0
$841$	−23713.0	−0.972283
$842$	0	0
$843$	19542.1	0.798417
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3901.21	0.158261
$848$	0	0
$849$	33124.0	1.33900
$850$	0	0
$851$	−31368.3	−1.26356
$852$	0	0
$853$	−26786.0	−1.07519	−0.537594	−	0.843204i	$-0.680667\pi$
$853$	−26786.0	−1.07519	−0.537594	+	0.843204i	$0.680667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19682.0	−0.784509	−0.392255	−	0.919857i	$-0.628305\pi$
$857$	−19682.0	−0.784509	−0.392255	+	0.919857i	$0.628305\pi$
$858$	0	0
$859$	33199.9	1.31870	0.659352	−	0.751834i	$-0.270831\pi$
$859$	33199.9	1.31870	0.659352	+	0.751834i	$0.270831\pi$
$860$	0	0
$861$	17784.0	0.703922
$862$	0	0
$863$	10203.7	0.402478	0.201239	−	0.979542i	$-0.435503\pi$
$863$	10203.7	0.402478	0.201239	+	0.979542i	$0.435503\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	58287.3	2.28321
$868$	0	0
$869$	−8736.00	−0.341022
$870$	0	0
$871$	16917.2	0.658116
$872$	0	0
$873$	24150.0	0.936258
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41986.0	−1.61661	−0.808305	−	0.588764i	$-0.799615\pi$
$877$	−41986.0	−1.61661	−0.808305	+	0.588764i	$0.799615\pi$
$878$	0	0
$879$	27041.6	1.03765
$880$	0	0
$881$	38142.0	1.45861	0.729306	−	0.684188i	$-0.239843\pi$
$881$	38142.0	1.45861	0.729306	+	0.684188i	$0.239843\pi$
$882$	0	0
$883$	39235.6	1.49534	0.747669	−	0.664072i	$-0.231173\pi$
$883$	39235.6	1.49534	0.747669	+	0.664072i	$0.231173\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41528.7	−1.57204	−0.786020	−	0.618202i	$-0.787861\pi$
$887$	−41528.7	−1.57204	−0.786020	+	0.618202i	$0.787861\pi$
$888$	0	0
$889$	3172.00	0.119669
$890$	0	0
$891$	33704.7	1.26728
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	51272.0	1.90850
$898$	0	0
$899$	2624.84	0.0973786
$900$	0	0
$901$	−29868.0	−1.10438
$902$	0	0
$903$	−23623.6	−0.870591
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4319.45	0.158131	0.0790656	−	0.996869i	$-0.474806\pi$
$907$	4319.45	0.158131	0.0790656	+	0.996869i	$0.474806\pi$
$908$	0	0
$909$	40950.0	1.49420
$910$	0	0
$911$	−49713.3	−1.80799	−0.903994	−	0.427546i	$-0.859378\pi$
$911$	−49713.3	−1.80799	−0.903994	+	0.427546i	$0.859378\pi$
$912$	0	0
$913$	−6552.00	−0.237502
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12792.0	0.460664
$918$	0	0
$919$	5220.84	0.187399	0.0936994	−	0.995601i	$-0.470131\pi$
$919$	5220.84	0.187399	0.0936994	+	0.995601i	$0.470131\pi$
$920$	0	0
$921$	31772.0	1.13672
$922$	0	0
$923$	−35795.9	−1.27653
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17126.4	0.606800
$928$	0	0
$929$	−17546.0	−0.619662	−0.309831	−	0.950792i	$-0.600272\pi$
$929$	−17546.0	−0.619662	−0.309831	+	0.950792i	$0.600272\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−21320.0	−0.748109
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11390.0	0.397113	0.198557	−	0.980089i	$-0.436375\pi$
$937$	11390.0	0.397113	0.198557	+	0.980089i	$0.436375\pi$
$938$	0	0
$939$	−40685.0	−1.41396
$940$	0	0
$941$	41838.0	1.44939	0.724697	−	0.689068i	$-0.241980\pi$
$941$	41838.0	1.44939	0.724697	+	0.689068i	$0.241980\pi$
$942$	0	0
$943$	−71519.7	−2.46978
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35226.2	−1.20876	−0.604382	−	0.796695i	$-0.706580\pi$
$947$	−35226.2	−1.20876	−0.604382	+	0.796695i	$0.706580\pi$
$948$	0	0
$949$	23188.0	0.793166
$950$	0	0
$951$	−33531.6	−1.14336
$952$	0	0
$953$	−28522.0	−0.969484	−0.484742	−	0.874657i	$-0.661086\pi$
$953$	−28522.0	−0.969484	−0.484742	+	0.874657i	$0.661086\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8112.00	0.274006
$958$	0	0
$959$	−11667.6	−0.392873
$960$	0	0
$961$	−19599.0	−0.657883
$962$	0	0
$963$	−15323.6	−0.512768
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4579.05	−0.152277	−0.0761387	−	0.997097i	$-0.524259\pi$
$967$	−4579.05	−0.152277	−0.0761387	+	0.997097i	$0.524259\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22282.3	−0.736430	−0.368215	−	0.929741i	$-0.620031\pi$
$971$	−22282.3	−0.736430	−0.368215	+	0.929741i	$0.620031\pi$
$972$	0	0
$973$	18096.0	0.596229
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2214.00	0.0724996	0.0362498	−	0.999343i	$-0.488459\pi$
$977$	2214.00	0.0724996	0.0362498	+	0.999343i	$0.488459\pi$
$978$	0	0
$979$	27258.0	0.889855
$980$	0	0
$981$	−8550.00	−0.278268
$982$	0	0
$983$	−24871.1	−0.806983	−0.403492	−	0.914983i	$-0.632204\pi$
$983$	−24871.1	−0.806983	−0.403492	+	0.914983i	$0.632204\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30373.2	0.979522
$988$	0	0
$989$	95004.0	3.05455
$990$	0	0
$991$	46453.9	1.48906	0.744530	−	0.667590i	$-0.232674\pi$
$991$	46453.9	1.48906	0.744530	+	0.667590i	$0.232674\pi$
$992$	0	0
$993$	39832.0	1.27294
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−57930.0	−1.84018	−0.920091	−	0.391705i	$-0.871886\pi$
$997$	−57930.0	−1.84018	−0.920091	+	0.391705i	$0.871886\pi$
$998$	0	0
$999$	−2163.33	−0.0685133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.a.q.1.2		2
4.3	odd	2	inner	800.4.a.q.1.1		2
5.2	odd	4		800.4.c.h.449.1		4
5.3	odd	4		800.4.c.h.449.3		4
5.4	even	2		160.4.a.e.1.1	✓	2
8.3	odd	2		1600.4.a.ci.1.2		2
8.5	even	2		1600.4.a.ci.1.1		2
15.14	odd	2		1440.4.a.bd.1.1		2
20.3	even	4		800.4.c.h.449.2		4
20.7	even	4		800.4.c.h.449.4		4
20.19	odd	2		160.4.a.e.1.2	yes	2
40.19	odd	2		320.4.a.r.1.1		2
40.29	even	2		320.4.a.r.1.2		2
60.59	even	2		1440.4.a.bd.1.2		2
80.19	odd	4		1280.4.d.t.641.4		4
80.29	even	4		1280.4.d.t.641.2		4
80.59	odd	4		1280.4.d.t.641.1		4
80.69	even	4		1280.4.d.t.641.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.e.1.1	✓	2	5.4	even	2
160.4.a.e.1.2	yes	2	20.19	odd	2
320.4.a.r.1.1		2	40.19	odd	2
320.4.a.r.1.2		2	40.29	even	2
800.4.a.q.1.1		2	4.3	odd	2	inner
800.4.a.q.1.2		2	1.1	even	1	trivial
800.4.c.h.449.1		4	5.2	odd	4
800.4.c.h.449.2		4	20.3	even	4
800.4.c.h.449.3		4	5.3	odd	4
800.4.c.h.449.4		4	20.7	even	4
1280.4.d.t.641.1		4	80.59	odd	4
1280.4.d.t.641.2		4	80.29	even	4
1280.4.d.t.641.3		4	80.69	even	4
1280.4.d.t.641.4		4	80.19	odd	4
1440.4.a.bd.1.1		2	15.14	odd	2
1440.4.a.bd.1.2		2	60.59	even	2
1600.4.a.ci.1.1		2	8.5	even	2
1600.4.a.ci.1.2		2	8.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q+7.21110 q^{3} +7.21110 q^{7} +25.0000 q^{9} -43.2666 q^{11} -34.0000 q^{13} -114.000 q^{17} +52.0000 q^{21} -209.122 q^{23} -14.4222 q^{27} -26.0000 q^{29} -100.955 q^{31} -312.000 q^{33} +150.000 q^{37} -245.177 q^{39} +342.000 q^{41} -454.299 q^{43} +584.099 q^{47} -291.000 q^{49} -822.066 q^{51} +262.000 q^{53} +490.355 q^{59} -262.000 q^{61} +180.278 q^{63} -497.566 q^{67} -1508.00 q^{69} +1052.82 q^{71} -682.000 q^{73} -312.000 q^{77} +201.911 q^{79} -779.000 q^{81} +151.433 q^{83} -187.489 q^{87} -630.000 q^{89} -245.177 q^{91} -728.000 q^{93} +966.000 q^{97} -1081.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{9} - 68 q^{13} - 228 q^{17} + 104 q^{21} - 52 q^{29} - 624 q^{33} + 300 q^{37} + 684 q^{41} - 582 q^{49} + 524 q^{53} - 524 q^{61} - 3016 q^{69} - 1364 q^{73} - 624 q^{77} - 1558 q^{81} - 1260 q^{89}+ \cdots + 1932 q^{97}+O(q^{100}) \) 2 * q + 50 * q^9 - 68 * q^13 - 228 * q^17 + 104 * q^21 - 52 * q^29 - 624 * q^33 + 300 * q^37 + 684 * q^41 - 582 * q^49 + 524 * q^53 - 524 * q^61 - 3016 * q^69 - 1364 * q^73 - 624 * q^77 - 1558 * q^81 - 1260 * q^89 - 1456 * q^93 + 1932 * q^97

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