LMFDB - Embedded newform 800.4.c.h.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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, 1]))

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.h.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.21110i q^{3} -7.21110i q^{7} -25.0000 q^{9} +O(q^{10})$	$q+7.21110i q^{3} -7.21110i q^{7} -25.0000 q^{9} -43.2666 q^{11} -34.0000i q^{13} +114.000i q^{17} +52.0000 q^{21} -209.122i q^{23} +14.4222i q^{27} +26.0000 q^{29} -100.955 q^{31} -312.000i q^{33} -150.000i q^{37} +245.177 q^{39} +342.000 q^{41} -454.299i q^{43} -584.099i q^{47} +291.000 q^{49} -822.066 q^{51} +262.000i q^{53} -490.355 q^{59} -262.000 q^{61} +180.278i q^{63} +497.566i q^{67} +1508.00 q^{69} +1052.82 q^{71} -682.000i q^{73} +312.000i q^{77} -201.911 q^{79} -779.000 q^{81} +151.433i q^{83} +187.489i q^{87} +630.000 q^{89} -245.177 q^{91} -728.000i q^{93} -966.000i q^{97} +1081.67 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 100 q^{9}+O(q^{10}) $ 4 * q - 100 * q^9	$ 4 q - 100 q^{9} + 208 q^{21} + 104 q^{29} + 1368 q^{41} + 1164 q^{49} - 1048 q^{61} + 6032 q^{69} - 3116 q^{81} + 2520 q^{89}+O(q^{100}) $ 4 * q - 100 * q^9 + 208 * q^21 + 104 * q^29 + 1368 * q^41 + 1164 * q^49 - 1048 * q^61 + 6032 * q^69 - 3116 * q^81 + 2520 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	7.21110i	1.38778i	0.720082	+	0.693889i	$0.244104\pi$
$3$	7.21110i	1.38778i	−0.720082	+	0.693889i	$0.755896\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.21110i	− 0.389363i	−0.980867	−	0.194681i	$-0.937633\pi$
$7$	− 7.21110i	− 0.389363i	0.980867	−	0.194681i	$-0.0623673\pi$
$8$	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
$13$	− 34.0000i	− 0.725377i	−0.931910	−	0.362689i	$-0.881859\pi$
$13$	− 34.0000i	− 0.725377i	0.931910	−	0.362689i	$-0.118141\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	114.000i	1.62642i	0.581974	+	0.813208i	$0.302281\pi$
$17$	114.000i	1.62642i	−0.581974	+	0.813208i	$0.697719\pi$
$18$	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
$23$	− 209.122i	− 1.89587i	−0.318468	−	0.947934i	$-0.603168\pi$
$23$	− 209.122i	− 1.89587i	0.318468	−	0.947934i	$-0.396832\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.4222i	0.102798i
$28$	0	0
$29$	26.0000	0.166485	0.0832427	−	0.996529i	$-0.473472\pi$
$29$	26.0000	0.166485	0.0832427	+	0.996529i	$0.473472\pi$
$30$	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
$33$	− 312.000i	− 1.64583i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 150.000i	− 0.666482i	−0.942842	−	0.333241i	$-0.891858\pi$
$37$	− 150.000i	− 0.666482i	0.942842	−	0.333241i	$-0.108142\pi$
$38$	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
$43$	− 454.299i	− 1.61116i	−0.592485	−	0.805582i	$-0.701853\pi$
$43$	− 454.299i	− 1.61116i	0.592485	−	0.805582i	$-0.298147\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 584.099i	− 1.81276i	−0.422465	−	0.906379i	$-0.638835\pi$
$47$	− 584.099i	− 1.81276i	0.422465	−	0.906379i	$-0.361165\pi$
$48$	0	0
$49$	291.000	0.848397
$50$	0	0
$51$	−822.066	−2.25710
$52$	0	0
$53$	262.000i	0.679028i	0.940601	+	0.339514i	$0.110263\pi$
$53$	262.000i	0.679028i	−0.940601	+	0.339514i	$0.889737\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−490.355	−1.08201	−0.541007	−	0.841018i	$-0.681957\pi$
$59$	−490.355	−1.08201	−0.541007	+	0.841018i	$0.681957\pi$
$60$	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
$63$	180.278i	0.360521i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	497.566i	0.907274i	0.891187	+	0.453637i	$0.149874\pi$
$67$	497.566i	0.907274i	−0.891187	+	0.453637i	$0.850126\pi$
$68$	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
$73$	− 682.000i	− 1.09345i	−0.837311	−	0.546726i	$-0.815874\pi$
$73$	− 682.000i	− 1.09345i	0.837311	−	0.546726i	$-0.184126\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	312.000i	0.461762i
$78$	0	0
$79$	−201.911	−0.287554	−0.143777	−	0.989610i	$-0.545925\pi$
$79$	−201.911	−0.287554	−0.143777	+	0.989610i	$0.545925\pi$
$80$	0	0
$81$	−779.000	−1.06859
$82$	0	0
$83$	151.433i	0.200264i	0.994974	+	0.100132i	$0.0319266\pi$
$83$	151.433i	0.200264i	−0.994974	+	0.100132i	$0.968073\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	187.489i	0.231045i
$88$	0	0
$89$	630.000	0.750336	0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000	0.750336	0.375168	+	0.926957i	$0.377585\pi$
$90$	0	0
$91$	−245.177	−0.282435
$92$	0	0
$93$	− 728.000i	− 0.811721i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 966.000i	− 1.01116i	−0.862780	−	0.505580i	$-0.831279\pi$
$97$	− 966.000i	− 1.01116i	0.862780	−	0.505580i	$-0.168721\pi$
$98$	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
$103$	685.055i	0.655344i	0.944792	+	0.327672i	$0.106264\pi$
$103$	685.055i	0.655344i	−0.944792	+	0.327672i	$0.893736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	612.944i	0.553790i	0.960900	+	0.276895i	$0.0893054\pi$
$107$	612.944i	0.553790i	−0.960900	+	0.276895i	$0.910695\pi$
$108$	0	0
$109$	342.000	0.300529	0.150264	−	0.988646i	$-0.451987\pi$
$109$	342.000	0.300529	0.150264	+	0.988646i	$0.451987\pi$
$110$	0	0
$111$	1081.67	0.924929
$112$	0	0
$113$	− 2106.00i	− 1.75324i	−0.481186	−	0.876619i	$-0.659794\pi$
$113$	− 2106.00i	− 1.75324i	0.481186	−	0.876619i	$-0.340206\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	850.000i	0.671646i
$118$	0	0
$119$	822.066	0.633266
$120$	0	0
$121$	541.000	0.406461
$122$	0	0
$123$	2466.20i	1.80788i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 439.877i	− 0.307345i	−0.988122	−	0.153672i	$-0.950890\pi$
$127$	− 439.877i	− 0.307345i	0.988122	−	0.153672i	$-0.0491101\pi$
$128$	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
$137$	1618.00i	1.00902i	0.863407	+	0.504508i	$0.168326\pi$
$137$	1618.00i	1.00902i	−0.863407	+	0.504508i	$0.831674\pi$
$138$	0	0
$139$	−2509.46	−1.53129	−0.765647	−	0.643261i	$-0.777581\pi$
$139$	−2509.46	−1.53129	−0.765647	+	0.643261i	$0.777581\pi$
$140$	0	0
$141$	4212.00	2.51570
$142$	0	0
$143$	1471.06i	0.860256i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2098.43i	1.17739i
$148$	0	0
$149$	−1010.00	−0.555318	−0.277659	−	0.960680i	$-0.589559\pi$
$149$	−1010.00	−0.555318	−0.277659	+	0.960680i	$0.589559\pi$
$150$	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
$153$	− 2850.00i	− 1.50594i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1794.00i	0.911954i	0.889992	+	0.455977i	$0.150710\pi$
$157$	1794.00i	0.911954i	−0.889992	+	0.455977i	$0.849290\pi$
$158$	0	0
$159$	−1889.31	−0.942339
$160$	0	0
$161$	−1508.00	−0.738180
$162$	0	0
$163$	− 1983.05i	− 0.952912i	−0.879198	−	0.476456i	$-0.841921\pi$
$163$	− 1983.05i	− 0.952912i	0.879198	−	0.476456i	$-0.158079\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 757.166i	− 0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$167$	− 757.166i	− 0.350846i	0.984493	−	0.175423i	$-0.0561293\pi$
$168$	0	0
$169$	1041.00	0.473828
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2834.00i	− 1.24546i	−0.782436	−	0.622731i	$-0.786023\pi$
$173$	− 2834.00i	− 1.24546i	0.782436	−	0.622731i	$-0.213977\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3536.00i	− 1.50159i
$178$	0	0
$179$	951.866	0.397462	0.198731	−	0.980054i	$-0.436318\pi$
$179$	951.866	0.397462	0.198731	+	0.980054i	$0.436318\pi$
$180$	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
$183$	− 1889.31i	− 0.763179i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4932.39i	− 1.92884i
$188$	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
$193$	46.0000i	0.0171562i	0.999963	+	0.00857812i	$0.00273053\pi$
$193$	46.0000i	0.0171562i	−0.999963	+	0.00857812i	$0.997269\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1122.00i	0.405783i	0.979201	+	0.202891i	$0.0650338\pi$
$197$	1122.00i	0.405783i	−0.979201	+	0.202891i	$0.934966\pi$
$198$	0	0
$199$	2999.82	1.06860	0.534300	−	0.845295i	$-0.320575\pi$
$199$	2999.82	1.06860	0.534300	+	0.845295i	$0.320575\pi$
$200$	0	0
$201$	−3588.00	−1.25909
$202$	0	0
$203$	− 187.489i	− 0.0648233i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5228.05i	1.75543i
$208$	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
$213$	7592.00i	2.44223i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	728.000i	0.227741i
$218$	0	0
$219$	4917.97	1.51747
$220$	0	0
$221$	3876.00	1.17976
$222$	0	0
$223$	− 2545.52i	− 0.764397i	−0.924080	−	0.382199i	$-0.875167\pi$
$223$	− 2545.52i	− 0.764397i	0.924080	−	0.382199i	$-0.124833\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5069.41i	− 1.48224i	−0.671373	−	0.741119i	$-0.734295\pi$
$227$	− 5069.41i	− 1.48224i	0.671373	−	0.741119i	$-0.265705\pi$
$228$	0	0
$229$	6194.00	1.78738	0.893692	−	0.448681i	$-0.148106\pi$
$229$	6194.00	1.78738	0.893692	+	0.448681i	$0.148106\pi$
$230$	0	0
$231$	−2249.86	−0.640823
$232$	0	0
$233$	− 4290.00i	− 1.20621i	−0.797661	−	0.603106i	$-0.793930\pi$
$233$	− 4290.00i	− 1.20621i	0.797661	−	0.603106i	$-0.206070\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1456.00i	− 0.399061i
$238$	0	0
$239$	−5278.53	−1.42862	−0.714309	−	0.699831i	$-0.753259\pi$
$239$	−5278.53	−1.42862	−0.714309	+	0.699831i	$0.753259\pi$
$240$	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
$243$	− 5228.05i	− 1.38016i
$244$	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
$253$	9048.00i	2.24839i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3718.00i	− 0.902422i	−0.892417	−	0.451211i	$-0.850992\pi$
$257$	− 3718.00i	− 0.902422i	0.892417	−	0.451211i	$-0.149008\pi$
$258$	0	0
$259$	−1081.67	−0.259504
$260$	0	0
$261$	−650.000	−0.154153
$262$	0	0
$263$	− 7045.25i	− 1.65182i	−0.563802	−	0.825910i	$-0.690662\pi$
$263$	− 7045.25i	− 1.65182i	0.563802	−	0.825910i	$-0.309338\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4542.99i	1.04130i
$268$	0	0
$269$	−6058.00	−1.37310	−0.686548	−	0.727085i	$-0.740875\pi$
$269$	−6058.00	−1.37310	−0.686548	+	0.727085i	$0.740875\pi$
$270$	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
$273$	− 1768.00i	− 0.391957i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2990.00i	− 0.648562i	−0.945961	−	0.324281i	$-0.894878\pi$
$277$	− 2990.00i	− 0.648562i	0.945961	−	0.324281i	$-0.105122\pi$
$278$	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
$283$	4593.47i	0.964854i	0.875936	+	0.482427i	$0.160245\pi$
$283$	4593.47i	0.964854i	−0.875936	+	0.482427i	$0.839755\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2466.20i	− 0.507230i
$288$	0	0
$289$	−8083.00	−1.64523
$290$	0	0
$291$	6965.93	1.40326
$292$	0	0
$293$	3750.00i	0.747704i	0.927488	+	0.373852i	$0.121963\pi$
$293$	3750.00i	0.747704i	−0.927488	+	0.373852i	$0.878037\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 624.000i	− 0.121913i
$298$	0	0
$299$	−7110.15	−1.37522
$300$	0	0
$301$	−3276.00	−0.627327
$302$	0	0
$303$	11811.8i	2.23950i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4405.98i	− 0.819097i	−0.912288	−	0.409548i	$-0.865686\pi$
$307$	− 4405.98i	− 0.819097i	0.912288	−	0.409548i	$-0.134314\pi$
$308$	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
$313$	− 5642.00i	− 1.01886i	−0.860511	−	0.509432i	$-0.829855\pi$
$313$	− 5642.00i	− 1.01886i	0.860511	−	0.509432i	$-0.170145\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4650.00i	0.823880i	0.911211	+	0.411940i	$0.135149\pi$
$317$	4650.00i	0.823880i	−0.911211	+	0.411940i	$0.864851\pi$
$318$	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
$327$	2466.20i	0.417067i
$328$	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
$333$	3750.00i	0.617113i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9266.00i	1.49778i	0.662695	+	0.748889i	$0.269412\pi$
$337$	9266.00i	1.49778i	−0.662695	+	0.748889i	$0.730588\pi$
$338$	0	0
$339$	15186.6	2.43310
$340$	0	0
$341$	4368.00	0.693667
$342$	0	0
$343$	− 4571.84i	− 0.719697i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	814.855i	0.126062i	0.998012	+	0.0630312i	$0.0200768\pi$
$347$	814.855i	0.126062i	−0.998012	+	0.0630312i	$0.979923\pi$
$348$	0	0
$349$	−7494.00	−1.14941	−0.574706	−	0.818360i	$-0.694883\pi$
$349$	−7494.00	−1.14941	−0.574706	+	0.818360i	$0.694883\pi$
$350$	0	0
$351$	490.355	0.0745676
$352$	0	0
$353$	6270.00i	0.945378i	0.881229	+	0.472689i	$0.156717\pi$
$353$	6270.00i	0.945378i	−0.881229	+	0.472689i	$0.843283\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5928.00i	0.878832i
$358$	0	0
$359$	−692.266	−0.101773	−0.0508863	−	0.998704i	$-0.516205\pi$
$359$	−692.266	−0.101773	−0.0508863	+	0.998704i	$0.516205\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	3901.21i	0.564078i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 2141.70i	− 0.304620i	−0.988333	−	0.152310i	$-0.951329\pi$
$367$	− 2141.70i	− 0.304620i	0.988333	−	0.152310i	$-0.0486713\pi$
$368$	0	0
$369$	−8550.00	−1.20622
$370$	0	0
$371$	1889.31	0.264388
$372$	0	0
$373$	2574.00i	0.357310i	0.983912	+	0.178655i	$0.0571746\pi$
$373$	2574.00i	0.357310i	−0.983912	+	0.178655i	$0.942825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 884.000i	− 0.120765i
$378$	0	0
$379$	−13729.9	−1.86084	−0.930422	−	0.366491i	$-0.880559\pi$
$379$	−13729.9	−1.86084	−0.930422	+	0.366491i	$0.880559\pi$
$380$	0	0
$381$	3172.00	0.426526
$382$	0	0
$383$	4204.07i	0.560883i	0.959871	+	0.280441i	$0.0904808\pi$
$383$	4204.07i	0.560883i	−0.959871	+	0.280441i	$0.909519\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11357.5i	1.49182i
$388$	0	0
$389$	−5314.00	−0.692623	−0.346312	−	0.938120i	$-0.612566\pi$
$389$	−5314.00	−0.692623	−0.346312	+	0.938120i	$0.612566\pi$
$390$	0	0
$391$	23839.9	3.08347
$392$	0	0
$393$	12792.0i	1.64191i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8638.00i	− 1.09201i	−0.837781	−	0.546006i	$-0.816148\pi$
$397$	− 8638.00i	− 1.09201i	0.837781	−	0.546006i	$-0.183852\pi$
$398$	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
$403$	3432.48i	0.424279i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6489.99i	0.790410i
$408$	0	0
$409$	82.0000	0.00991354	0.00495677	−	0.999988i	$-0.498422\pi$
$409$	82.0000	0.00991354	0.00495677	+	0.999988i	$0.498422\pi$
$410$	0	0
$411$	−11667.6	−1.40029
$412$	0	0
$413$	3536.00i	0.421296i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 18096.0i	− 2.12510i
$418$	0	0
$419$	1067.24	0.124435	0.0622175	−	0.998063i	$-0.480183\pi$
$419$	1067.24	0.124435	0.0622175	+	0.998063i	$0.480183\pi$
$420$	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
$423$	14602.5i	1.67848i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1889.31i	0.214122i
$428$	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
$433$	− 7098.00i	− 0.787779i	−0.919158	−	0.393889i	$-0.871129\pi$
$433$	− 7098.00i	− 0.787779i	0.919158	−	0.393889i	$-0.128871\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	461.511	0.0501747	0.0250874	−	0.999685i	$-0.492014\pi$
$439$	461.511	0.0501747	0.0250874	+	0.999685i	$0.492014\pi$
$440$	0	0
$441$	−7275.00	−0.785552
$442$	0	0
$443$	6064.54i	0.650417i	0.945642	+	0.325209i	$0.105435\pi$
$443$	6064.54i	0.650417i	−0.945642	+	0.325209i	$0.894565\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 7283.21i	− 0.770658i
$448$	0	0
$449$	11706.0	1.23038	0.615190	−	0.788379i	$-0.289079\pi$
$449$	11706.0	1.23038	0.615190	+	0.788379i	$0.289079\pi$
$450$	0	0
$451$	−14797.2	−1.54495
$452$	0	0
$453$	104.000i	0.0107866i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5066.00i	0.518550i	0.965803	+	0.259275i	$0.0834836\pi$
$457$	5066.00i	0.518550i	−0.965803	+	0.259275i	$0.916516\pi$
$458$	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
$463$	483.144i	0.0484959i	0.999706	+	0.0242479i	$0.00771912\pi$
$463$	483.144i	0.0484959i	−0.999706	+	0.0242479i	$0.992281\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2271.50i	− 0.225080i	−0.993647	−	0.112540i	$-0.964101\pi$
$467$	− 2271.50i	− 0.225080i	0.993647	−	0.112540i	$-0.0358986\pi$
$468$	0	0
$469$	3588.00	0.353259
$470$	0	0
$471$	−12936.7	−1.26559
$472$	0	0
$473$	19656.0i	1.91075i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6550.00i	− 0.628729i
$478$	0	0
$479$	4067.06	0.387952	0.193976	−	0.981006i	$-0.437862\pi$
$479$	4067.06	0.387952	0.193976	+	0.981006i	$0.437862\pi$
$480$	0	0
$481$	−5100.00	−0.483451
$482$	0	0
$483$	− 10874.3i	− 1.02443i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11170.0i	− 1.03934i	−0.854366	−	0.519672i	$-0.826054\pi$
$487$	− 11170.0i	− 1.03934i	0.854366	−	0.519672i	$-0.173946\pi$
$488$	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
$493$	2964.00i	0.270775i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7592.00i	− 0.685207i
$498$	0	0
$499$	288.444	0.0258768	0.0129384	−	0.999916i	$-0.495881\pi$
$499$	288.444	0.0258768	0.0129384	+	0.999916i	$0.495881\pi$
$500$	0	0
$501$	5460.00	0.486896
$502$	0	0
$503$	− 18409.9i	− 1.63193i	−0.578104	−	0.815963i	$-0.696207\pi$
$503$	− 18409.9i	− 1.63193i	0.578104	−	0.815963i	$-0.303793\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7506.76i	0.657568i
$508$	0	0
$509$	8714.00	0.758824	0.379412	−	0.925228i	$-0.376126\pi$
$509$	8714.00	0.758824	0.379412	+	0.925228i	$0.376126\pi$
$510$	0	0
$511$	−4917.97	−0.425750
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	25272.0i	2.14983i
$518$	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
$523$	− 8963.40i	− 0.749411i	−0.927144	−	0.374706i	$-0.877744\pi$
$523$	− 8963.40i	− 0.749411i	0.927144	−	0.374706i	$-0.122256\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11508.9i	− 0.951302i
$528$	0	0
$529$	−31565.0	−2.59431
$530$	0	0
$531$	12258.9	1.00186
$532$	0	0
$533$	− 11628.0i	− 0.944962i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6864.00i	0.551589i
$538$	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
$543$	− 10571.5i	− 0.835480i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11429.6i	0.893408i	0.894682	+	0.446704i	$0.147402\pi$
$547$	11429.6i	0.893408i	−0.894682	+	0.446704i	$0.852598\pi$
$548$	0	0
$549$	6550.00	0.509193
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1456.00i	0.111963i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23862.0i	− 1.81520i	−0.419838	−	0.907599i	$-0.637913\pi$
$557$	− 23862.0i	− 1.81520i	0.419838	−	0.907599i	$-0.362087\pi$
$558$	0	0
$559$	−15446.2	−1.16870
$560$	0	0
$561$	35568.0	2.67680
$562$	0	0
$563$	− 7261.58i	− 0.543586i	−0.962356	−	0.271793i	$-0.912383\pi$
$563$	− 7261.58i	− 0.543586i	0.962356	−	0.271793i	$-0.0876167\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5617.45i	0.416068i
$568$	0	0
$569$	9074.00	0.668545	0.334272	−	0.942477i	$-0.391509\pi$
$569$	9074.00	0.668545	0.334272	+	0.942477i	$0.391509\pi$
$570$	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
$573$	− 25064.0i	− 1.82734i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4762.00i	0.343578i	0.985134	+	0.171789i	$0.0549548\pi$
$577$	4762.00i	0.343578i	−0.985134	+	0.171789i	$0.945045\pi$
$578$	0	0
$579$	−331.711	−0.0238090
$580$	0	0
$581$	1092.00	0.0779755
$582$	0	0
$583$	− 11335.9i	− 0.805288i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21481.9i	− 1.51048i	−0.655448	−	0.755240i	$-0.727520\pi$
$587$	− 21481.9i	− 1.51048i	0.655448	−	0.755240i	$-0.272480\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−8090.86	−0.563136
$592$	0	0
$593$	− 11954.0i	− 0.827811i	−0.910320	−	0.413906i	$-0.864164\pi$
$593$	− 11954.0i	− 0.827811i	0.910320	−	0.413906i	$-0.135836\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21632.0i	1.48298i
$598$	0	0
$599$	10759.0	0.733889	0.366944	−	0.930243i	$-0.380404\pi$
$599$	10759.0	0.733889	0.366944	+	0.930243i	$0.380404\pi$
$600$	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
$603$	− 12439.2i	− 0.840069i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7506.76i	− 0.501960i	−0.967992	−	0.250980i	$-0.919247\pi$
$607$	− 7506.76i	− 0.501960i	0.967992	−	0.250980i	$-0.0807529\pi$
$608$	0	0
$609$	1352.00	0.0899603
$610$	0	0
$611$	−19859.4	−1.31493
$612$	0	0
$613$	− 11522.0i	− 0.759167i	−0.925158	−	0.379583i	$-0.876067\pi$
$613$	− 11522.0i	− 0.759167i	0.925158	−	0.379583i	$-0.123933\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8290.00i	0.540912i	0.962732	+	0.270456i	$0.0871745\pi$
$617$	8290.00i	0.540912i	−0.962732	+	0.270456i	$0.912826\pi$
$618$	0	0
$619$	24171.6	1.56953	0.784765	−	0.619793i	$-0.212784\pi$
$619$	24171.6	1.56953	0.784765	+	0.619793i	$0.212784\pi$
$620$	0	0
$621$	3016.00	0.194892
$622$	0	0
$623$	− 4542.99i	− 0.292153i
$624$	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
$633$	− 3016.00i	− 0.189376i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9894.00i	− 0.615407i
$638$	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
$643$	23428.9i	1.43693i	0.695564	+	0.718464i	$0.255154\pi$
$643$	23428.9i	1.43693i	−0.695564	+	0.718464i	$0.744846\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8689.38i	− 0.527998i	−0.964523	−	0.263999i	$-0.914958\pi$
$647$	− 8689.38i	− 0.527998i	0.964523	−	0.263999i	$-0.0850416\pi$
$648$	0	0
$649$	21216.0	1.28321
$650$	0	0
$651$	−5249.68	−0.316054
$652$	0	0
$653$	− 22282.0i	− 1.33532i	−0.744468	−	0.667659i	$-0.767297\pi$
$653$	− 22282.0i	− 1.33532i	0.744468	−	0.667659i	$-0.232703\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	17050.0i	1.01246i
$658$	0	0
$659$	−15835.6	−0.936065	−0.468032	−	0.883711i	$-0.655037\pi$
$659$	−15835.6	−0.936065	−0.468032	+	0.883711i	$0.655037\pi$
$660$	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
$663$	27950.2i	1.63725i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5437.17i	− 0.315634i
$668$	0	0
$669$	18356.0	1.06081
$670$	0	0
$671$	11335.9	0.652184
$672$	0	0
$673$	− 11866.0i	− 0.679644i	−0.940490	−	0.339822i	$-0.889633\pi$
$673$	− 11866.0i	− 0.679644i	0.940490	−	0.339822i	$-0.110367\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 26574.0i	− 1.50860i	−0.656530	−	0.754300i	$-0.727977\pi$
$677$	− 26574.0i	− 1.50860i	0.656530	−	0.754300i	$-0.272023\pi$
$678$	0	0
$679$	−6965.93	−0.393708
$680$	0	0
$681$	36556.0	2.05702
$682$	0	0
$683$	13737.2i	0.769601i	0.923000	+	0.384800i	$0.125730\pi$
$683$	13737.2i	0.769601i	−0.923000	+	0.384800i	$0.874270\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	44665.6i	2.48049i
$688$	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
$693$	− 7800.00i	− 0.427558i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	38988.0i	2.11876i
$698$	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
$707$	− 11811.8i	− 0.628328i
$708$	0	0
$709$	1554.00	0.0823155	0.0411578	−	0.999153i	$-0.486895\pi$
$709$	1554.00	0.0823155	0.0411578	+	0.999153i	$0.486895\pi$
$710$	0	0
$711$	5047.77	0.266253
$712$	0	0
$713$	21112.0i	1.10891i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 38064.0i	− 1.98260i
$718$	0	0
$719$	31065.4	1.61133	0.805664	−	0.592373i	$-0.201809\pi$
$719$	31065.4	1.61133	0.805664	+	0.592373i	$0.201809\pi$
$720$	0	0
$721$	4940.00	0.255167
$722$	0	0
$723$	− 22166.9i	− 1.14024i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24452.8i	1.24746i	0.781638	+	0.623732i	$0.214384\pi$
$727$	24452.8i	1.24746i	−0.781638	+	0.623732i	$0.785616\pi$
$728$	0	0
$729$	16667.0	0.846771
$730$	0	0
$731$	51790.1	2.62042
$732$	0	0
$733$	− 15058.0i	− 0.758772i	−0.925238	−	0.379386i	$-0.876135\pi$
$733$	− 15058.0i	− 0.758772i	0.925238	−	0.379386i	$-0.123865\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 21528.0i	− 1.07598i
$738$	0	0
$739$	−11912.7	−0.592987	−0.296493	−	0.955035i	$-0.595817\pi$
$739$	−11912.7	−0.592987	−0.296493	+	0.955035i	$0.595817\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17270.6i	0.852754i	0.904545	+	0.426377i	$0.140210\pi$
$743$	17270.6i	0.852754i	−0.904545	+	0.426377i	$0.859790\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3785.83i	− 0.185430i
$748$	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
$753$	− 14872.0i	− 0.719742i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35410.0i	1.70013i	0.526678	+	0.850065i	$0.323437\pi$
$757$	35410.0i	1.70013i	−0.526678	+	0.850065i	$0.676563\pi$
$758$	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
$763$	− 2466.20i	− 0.117015i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16672.1i	0.784868i
$768$	0	0
$769$	−11522.0	−0.540304	−0.270152	−	0.962818i	$-0.587074\pi$
$769$	−11522.0	−0.540304	−0.270152	+	0.962818i	$0.587074\pi$
$770$	0	0
$771$	26810.9	1.25236
$772$	0	0
$773$	14382.0i	0.669191i	0.942362	+	0.334595i	$0.108600\pi$
$773$	14382.0i	0.669191i	−0.942362	+	0.334595i	$0.891400\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7800.00i	− 0.360133i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−45552.0	−2.08704
$782$	0	0
$783$	374.977i	0.0171144i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	814.855i	0.0369078i	0.999830	+	0.0184539i	$0.00587439\pi$
$787$	814.855i	0.0369078i	−0.999830	+	0.0184539i	$0.994126\pi$
$788$	0	0
$789$	50804.0	2.29236
$790$	0	0
$791$	−15186.6	−0.682646
$792$	0	0
$793$	8908.00i	0.398906i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14550.0i	− 0.646659i	−0.946286	−	0.323330i	$-0.895198\pi$
$797$	− 14550.0i	− 0.646659i	0.946286	−	0.323330i	$-0.104802\pi$
$798$	0	0
$799$	66587.3	2.94830
$800$	0	0
$801$	−15750.0	−0.694755
$802$	0	0
$803$	29507.8i	1.29677i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 43684.9i	− 1.90555i
$808$	0	0
$809$	37622.0	1.63501	0.817503	−	0.575925i	$-0.195358\pi$
$809$	37622.0	1.63501	0.817503	+	0.575925i	$0.195358\pi$
$810$	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
$813$	− 37544.0i	− 1.61959i
$814$	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
$823$	16866.8i	0.714385i	0.934031	+	0.357192i	$0.116266\pi$
$823$	16866.8i	0.714385i	−0.934031	+	0.357192i	$0.883734\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39235.6i	1.64977i	0.565304	+	0.824883i	$0.308759\pi$
$827$	39235.6i	1.64977i	−0.565304	+	0.824883i	$0.691241\pi$
$828$	0	0
$829$	3718.00	0.155768	0.0778839	−	0.996962i	$-0.475184\pi$
$829$	3718.00	0.155768	0.0778839	+	0.996962i	$0.475184\pi$
$830$	0	0
$831$	21561.2	0.900060
$832$	0	0
$833$	33174.0i	1.37985i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1456.00i	− 0.0601275i
$838$	0	0
$839$	−14335.7	−0.589896	−0.294948	−	0.955513i	$-0.595302\pi$
$839$	−14335.7	−0.589896	−0.294948	+	0.955513i	$0.595302\pi$
$840$	0	0
$841$	−23713.0	−0.972283
$842$	0	0
$843$	19542.1i	0.798417i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3901.21i	− 0.158261i
$848$	0	0
$849$	−33124.0	−1.33900
$850$	0	0
$851$	−31368.3	−1.26356
$852$	0	0
$853$	− 26786.0i	− 1.07519i	−0.843204	−	0.537594i	$-0.819333\pi$
$853$	− 26786.0i	− 1.07519i	0.843204	−	0.537594i	$-0.180667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19682.0i	0.784509i	0.919857	+	0.392255i	$0.128305\pi$
$857$	19682.0i	0.784509i	−0.919857	+	0.392255i	$0.871695\pi$
$858$	0	0
$859$	−33199.9	−1.31870	−0.659352	−	0.751834i	$-0.729169\pi$
$859$	−33199.9	−1.31870	−0.659352	+	0.751834i	$0.729169\pi$
$860$	0	0
$861$	17784.0	0.703922
$862$	0	0
$863$	10203.7i	0.402478i	0.979542	+	0.201239i	$0.0644968\pi$
$863$	10203.7i	0.402478i	−0.979542	+	0.201239i	$0.935503\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 58287.3i	− 2.28321i
$868$	0	0
$869$	8736.00	0.341022
$870$	0	0
$871$	16917.2	0.658116
$872$	0	0
$873$	24150.0i	0.936258i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41986.0i	1.61661i	0.588764	+	0.808305i	$0.299615\pi$
$877$	41986.0i	1.61661i	−0.588764	+	0.808305i	$0.700385\pi$
$878$	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
$883$	39235.6i	1.49534i	0.664072	+	0.747669i	$0.268827\pi$
$883$	39235.6i	1.49534i	−0.664072	+	0.747669i	$0.731173\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41528.7i	1.57204i	0.618202	+	0.786020i	$0.287861\pi$
$887$	41528.7i	1.57204i	−0.618202	+	0.786020i	$0.712139\pi$
$888$	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
$897$	− 51272.0i	− 1.90850i
$898$	0	0
$899$	−2624.84	−0.0973786
$900$	0	0
$901$	−29868.0	−1.10438
$902$	0	0
$903$	− 23623.6i	− 0.870591i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4319.45i	− 0.158131i	−0.996869	−	0.0790656i	$-0.974806\pi$
$907$	− 4319.45i	− 0.158131i	0.996869	−	0.0790656i	$-0.0251937\pi$
$908$	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
$913$	− 6552.00i	− 0.237502i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12792.0i	− 0.460664i
$918$	0	0
$919$	−5220.84	−0.187399	−0.0936994	−	0.995601i	$-0.529869\pi$
$919$	−5220.84	−0.187399	−0.0936994	+	0.995601i	$0.529869\pi$
$920$	0	0
$921$	31772.0	1.13672
$922$	0	0
$923$	− 35795.9i	− 1.27653i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 17126.4i	− 0.606800i
$928$	0	0
$929$	17546.0	0.619662	0.309831	−	0.950792i	$-0.399728\pi$
$929$	17546.0	0.619662	0.309831	+	0.950792i	$0.399728\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 21320.0i	− 0.748109i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 11390.0i	− 0.397113i	−0.980089	−	0.198557i	$-0.936375\pi$
$937$	− 11390.0i	− 0.397113i	0.980089	−	0.198557i	$-0.0636254\pi$
$938$	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
$943$	− 71519.7i	− 2.46978i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35226.2i	1.20876i	0.796695	+	0.604382i	$0.206580\pi$
$947$	35226.2i	1.20876i	−0.796695	+	0.604382i	$0.793420\pi$
$948$	0	0
$949$	−23188.0	−0.793166
$950$	0	0
$951$	−33531.6	−1.14336
$952$	0	0
$953$	− 28522.0i	− 0.969484i	−0.874657	−	0.484742i	$-0.838914\pi$
$953$	− 28522.0i	− 0.969484i	0.874657	−	0.484742i	$-0.161086\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8112.00i	− 0.274006i
$958$	0	0
$959$	11667.6	0.392873
$960$	0	0
$961$	−19599.0	−0.657883
$962$	0	0
$963$	− 15323.6i	− 0.512768i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4579.05i	0.152277i	0.997097	+	0.0761387i	$0.0242592\pi$
$967$	4579.05i	0.152277i	−0.997097	+	0.0761387i	$0.975741\pi$
$968$	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
$973$	18096.0i	0.596229i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2214.00i	− 0.0724996i	−0.999343	−	0.0362498i	$-0.988459\pi$
$977$	− 2214.00i	− 0.0724996i	0.999343	−	0.0362498i	$-0.0115412\pi$
$978$	0	0
$979$	−27258.0	−0.889855
$980$	0	0
$981$	−8550.00	−0.278268
$982$	0	0
$983$	− 24871.1i	− 0.806983i	−0.914983	−	0.403492i	$-0.867796\pi$
$983$	− 24871.1i	− 0.806983i	0.914983	−	0.403492i	$-0.132204\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 30373.2i	− 0.979522i
$988$	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
$993$	39832.0i	1.27294i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	57930.0i	1.84018i	0.391705	+	0.920091i	$0.371886\pi$
$997$	57930.0i	1.84018i	−0.391705	+	0.920091i	$0.628114\pi$
$998$	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.c.h.449.3		4
4.3	odd	2	inner	800.4.c.h.449.2		4
5.2	odd	4		800.4.a.q.1.2		2
5.3	odd	4		160.4.a.e.1.1	✓	2
5.4	even	2	inner	800.4.c.h.449.1		4
15.8	even	4		1440.4.a.bd.1.1		2
20.3	even	4		160.4.a.e.1.2	yes	2
20.7	even	4		800.4.a.q.1.1		2
20.19	odd	2	inner	800.4.c.h.449.4		4
40.3	even	4		320.4.a.r.1.1		2
40.13	odd	4		320.4.a.r.1.2		2
40.27	even	4		1600.4.a.ci.1.2		2
40.37	odd	4		1600.4.a.ci.1.1		2
60.23	odd	4		1440.4.a.bd.1.2		2
80.3	even	4		1280.4.d.t.641.4		4
80.13	odd	4		1280.4.d.t.641.2		4
80.43	even	4		1280.4.d.t.641.1		4
80.53	odd	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	5.3	odd	4
160.4.a.e.1.2	yes	2	20.3	even	4
320.4.a.r.1.1		2	40.3	even	4
320.4.a.r.1.2		2	40.13	odd	4
800.4.a.q.1.1		2	20.7	even	4
800.4.a.q.1.2		2	5.2	odd	4
800.4.c.h.449.1		4	5.4	even	2	inner
800.4.c.h.449.2		4	4.3	odd	2	inner
800.4.c.h.449.3		4	1.1	even	1	trivial
800.4.c.h.449.4		4	20.19	odd	2	inner
1280.4.d.t.641.1		4	80.43	even	4
1280.4.d.t.641.2		4	80.13	odd	4
1280.4.d.t.641.3		4	80.53	odd	4
1280.4.d.t.641.4		4	80.3	even	4
1440.4.a.bd.1.1		2	15.8	even	4
1440.4.a.bd.1.2		2	60.23	odd	4
1600.4.a.ci.1.1		2	40.37	odd	4
1600.4.a.ci.1.2		2	40.27	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.21110i q^{3} -7.21110i q^{7} -25.0000 q^{9} +O(q^{10})\)	\(q+7.21110i q^{3} -7.21110i q^{7} -25.0000 q^{9} -43.2666 q^{11} -34.0000i q^{13} +114.000i q^{17} +52.0000 q^{21} -209.122i q^{23} +14.4222i q^{27} +26.0000 q^{29} -100.955 q^{31} -312.000i q^{33} -150.000i q^{37} +245.177 q^{39} +342.000 q^{41} -454.299i q^{43} -584.099i q^{47} +291.000 q^{49} -822.066 q^{51} +262.000i q^{53} -490.355 q^{59} -262.000 q^{61} +180.278i q^{63} +497.566i q^{67} +1508.00 q^{69} +1052.82 q^{71} -682.000i q^{73} +312.000i q^{77} -201.911 q^{79} -779.000 q^{81} +151.433i q^{83} +187.489i q^{87} +630.000 q^{89} -245.177 q^{91} -728.000i q^{93} -966.000i q^{97} +1081.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 100 q^{9}+O(q^{10}) \) 4 * q - 100 * q^9	\( 4 q - 100 q^{9} + 208 q^{21} + 104 q^{29} + 1368 q^{41} + 1164 q^{49} - 1048 q^{61} + 6032 q^{69} - 3116 q^{81} + 2520 q^{89}+O(q^{100}) \) 4 * q - 100 * q^9 + 208 * q^21 + 104 * q^29 + 1368 * q^41 + 1164 * q^49 - 1048 * q^61 + 6032 * q^69 - 3116 * q^81 + 2520 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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