LMFDB - Embedded newform 6897.2.a.bc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6897,2,Mod(1,6897)] 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("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6897 = 3 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6897.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-1,-8,9,-2,1,-6,-6,8,-3,0,-9,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.0728222741$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 9x^{5} + 40x^{4} - 22x^{3} - 28x^{2} + 12x - 1 $ x^8 - x^7 - 12x^6 + 9x^5 + 40x^4 - 22x^3 - 28x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.267052$ of defining polynomial
Character	$\chi$	$=$	6897.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.267052 q^{2} -1.00000 q^{3} -1.92868 q^{4} +3.50077 q^{5} +0.267052 q^{6} -4.33946 q^{7} +1.04916 q^{8} +1.00000 q^{9} -0.934889 q^{10} +1.92868 q^{12} -0.267052 q^{13} +1.15886 q^{14} -3.50077 q^{15} +3.57718 q^{16} -5.99455 q^{17} -0.267052 q^{18} +1.00000 q^{19} -6.75188 q^{20} +4.33946 q^{21} +6.72739 q^{23} -1.04916 q^{24} +7.25539 q^{25} +0.0713170 q^{26} -1.00000 q^{27} +8.36944 q^{28} -1.85653 q^{29} +0.934889 q^{30} -5.19887 q^{31} -3.05362 q^{32} +1.60086 q^{34} -15.1914 q^{35} -1.92868 q^{36} +8.06741 q^{37} -0.267052 q^{38} +0.267052 q^{39} +3.67288 q^{40} +10.8939 q^{41} -1.15886 q^{42} -9.05466 q^{43} +3.50077 q^{45} -1.79657 q^{46} -5.43596 q^{47} -3.57718 q^{48} +11.8309 q^{49} -1.93757 q^{50} +5.99455 q^{51} +0.515059 q^{52} -5.54849 q^{53} +0.267052 q^{54} -4.55280 q^{56} -1.00000 q^{57} +0.495790 q^{58} +9.45890 q^{59} +6.75188 q^{60} -1.15697 q^{61} +1.38837 q^{62} -4.33946 q^{63} -6.33889 q^{64} -0.934889 q^{65} +2.53651 q^{67} +11.5616 q^{68} -6.72739 q^{69} +4.05691 q^{70} +8.03442 q^{71} +1.04916 q^{72} -5.93529 q^{73} -2.15442 q^{74} -7.25539 q^{75} -1.92868 q^{76} -0.0713170 q^{78} +8.04694 q^{79} +12.5229 q^{80} +1.00000 q^{81} -2.90923 q^{82} -5.90959 q^{83} -8.36944 q^{84} -20.9855 q^{85} +2.41807 q^{86} +1.85653 q^{87} +7.54793 q^{89} -0.934889 q^{90} +1.15886 q^{91} -12.9750 q^{92} +5.19887 q^{93} +1.45169 q^{94} +3.50077 q^{95} +3.05362 q^{96} -18.5499 q^{97} -3.15947 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) $ 8 * q - q^2 - 8 * q^3 + 9 * q^4 - 2 * q^5 + q^6 - 6 * q^7 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 9 * q^12 - q^13 + 7 * q^14 + 2 * q^15 + 23 * q^16 - 15 * q^17 - q^18 + 8 * q^19 - 4 * q^20 + 6 * q^21 - 5 * q^23 + 6 * q^24 + 4 * q^25 + 25 * q^26 - 8 * q^27 - 8 * q^28 - 17 * q^29 + 3 * q^30 - 4 * q^31 - 38 * q^32 - 18 * q^34 - 2 * q^35 + 9 * q^36 - 4 * q^37 - q^38 + q^39 - 14 * q^40 - 2 * q^41 - 7 * q^42 - 10 * q^43 - 2 * q^45 + 10 * q^46 + 5 * q^47 - 23 * q^48 + 28 * q^49 - 7 * q^50 + 15 * q^51 - 8 * q^52 - 19 * q^53 + q^54 - 49 * q^56 - 8 * q^57 + 35 * q^58 + 12 * q^59 + 4 * q^60 + 5 * q^61 - 22 * q^62 - 6 * q^63 + 24 * q^64 - 3 * q^65 - 12 * q^67 + 3 * q^68 + 5 * q^69 - 22 * q^70 + 40 * q^71 - 6 * q^72 + 12 * q^73 + 7 * q^74 - 4 * q^75 + 9 * q^76 - 25 * q^78 - 14 * q^79 + 37 * q^80 + 8 * q^81 + 31 * q^82 - 17 * q^83 + 8 * q^84 + 35 * q^85 - q^86 + 17 * q^87 - 9 * q^89 - 3 * q^90 + 7 * q^91 + 12 * q^92 + 4 * q^93 - 51 * q^94 - 2 * q^95 + 38 * q^96 + 8 * q^97 - 55 * q^98

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 $

$2$	−0.267052	−0.188835	−0.0944173	−	0.995533i	$-0.530099\pi$
$2$	−0.267052	−0.188835	−0.0944173	+	0.995533i	$0.530099\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.92868	−0.964341
$5$	3.50077	1.56559	0.782796	−	0.622278i	$-0.213793\pi$
$5$	3.50077	1.56559	0.782796	+	0.622278i	$0.213793\pi$
$6$	0.267052	0.109024
$7$	−4.33946	−1.64016	−0.820080	−	0.572249i	$-0.806071\pi$
$7$	−4.33946	−1.64016	−0.820080	+	0.572249i	$0.806071\pi$
$8$	1.04916	0.370936
$9$	1.00000	0.333333
$10$	−0.934889	−0.295638
$11$	0	0
$11$	0	0
$12$	1.92868	0.556763
$13$	−0.267052	−0.0740670	−0.0370335	−	0.999314i	$-0.511791\pi$
$13$	−0.267052	−0.0740670	−0.0370335	+	0.999314i	$0.511791\pi$
$14$	1.15886	0.309719
$15$	−3.50077	−0.903895
$16$	3.57718	0.894296
$17$	−5.99455	−1.45389	−0.726946	−	0.686695i	$-0.759061\pi$
$17$	−5.99455	−1.45389	−0.726946	+	0.686695i	$0.759061\pi$
$18$	−0.267052	−0.0629449
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−6.75188	−1.50977
$21$	4.33946	0.946947
$22$	0	0
$23$	6.72739	1.40276	0.701379	−	0.712788i	$-0.252568\pi$
$23$	6.72739	1.40276	0.701379	+	0.712788i	$0.252568\pi$
$24$	−1.04916	−0.214160
$25$	7.25539	1.45108
$26$	0.0713170	0.0139864
$27$	−1.00000	−0.192450
$28$	8.36944	1.58167
$29$	−1.85653	−0.344749	−0.172374	−	0.985032i	$-0.555144\pi$
$29$	−1.85653	−0.344749	−0.172374	+	0.985032i	$0.555144\pi$
$30$	0.934889	0.170687
$31$	−5.19887	−0.933745	−0.466872	−	0.884325i	$-0.654619\pi$
$31$	−5.19887	−0.933745	−0.466872	+	0.884325i	$0.654619\pi$
$32$	−3.05362	−0.539810
$33$	0	0
$34$	1.60086	0.274545
$35$	−15.1914	−2.56782
$36$	−1.92868	−0.321447
$37$	8.06741	1.32627	0.663137	−	0.748498i	$-0.269225\pi$
$37$	8.06741	1.32627	0.663137	+	0.748498i	$0.269225\pi$
$38$	−0.267052	−0.0433216
$39$	0.267052	0.0427626
$40$	3.67288	0.580734
$41$	10.8939	1.70133	0.850667	−	0.525705i	$-0.176198\pi$
$41$	10.8939	1.70133	0.850667	+	0.525705i	$0.176198\pi$
$42$	−1.15886	−0.178816
$43$	−9.05466	−1.38082	−0.690412	−	0.723417i	$-0.742571\pi$
$43$	−9.05466	−1.38082	−0.690412	+	0.723417i	$0.742571\pi$
$44$	0	0
$45$	3.50077	0.521864
$46$	−1.79657	−0.264889
$47$	−5.43596	−0.792917	−0.396458	−	0.918053i	$-0.629761\pi$
$47$	−5.43596	−0.792917	−0.396458	+	0.918053i	$0.629761\pi$
$48$	−3.57718	−0.516322
$49$	11.8309	1.69013
$50$	−1.93757	−0.274014
$51$	5.99455	0.839405
$52$	0.515059	0.0714259
$53$	−5.54849	−0.762144	−0.381072	−	0.924545i	$-0.624445\pi$
$53$	−5.54849	−0.762144	−0.381072	+	0.924545i	$0.624445\pi$
$54$	0.267052	0.0363412
$55$	0	0
$56$	−4.55280	−0.608394
$57$	−1.00000	−0.132453
$58$	0.495790	0.0651005
$59$	9.45890	1.23144	0.615722	−	0.787964i	$-0.288865\pi$
$59$	9.45890	1.23144	0.615722	+	0.787964i	$0.288865\pi$
$60$	6.75188	0.871664
$61$	−1.15697	−0.148135	−0.0740675	−	0.997253i	$-0.523598\pi$
$61$	−1.15697	−0.148135	−0.0740675	+	0.997253i	$0.523598\pi$
$62$	1.38837	0.176323
$63$	−4.33946	−0.546720
$64$	−6.33889	−0.792361
$65$	−0.934889	−0.115959
$66$	0	0
$67$	2.53651	0.309883	0.154942	−	0.987924i	$-0.450481\pi$
$67$	2.53651	0.309883	0.154942	+	0.987924i	$0.450481\pi$
$68$	11.5616	1.40205
$69$	−6.72739	−0.809883
$70$	4.05691	0.484894
$71$	8.03442	0.953510	0.476755	−	0.879036i	$-0.341813\pi$
$71$	8.03442	0.953510	0.476755	+	0.879036i	$0.341813\pi$
$72$	1.04916	0.123645
$73$	−5.93529	−0.694673	−0.347337	−	0.937741i	$-0.612914\pi$
$73$	−5.93529	−0.694673	−0.347337	+	0.937741i	$0.612914\pi$
$74$	−2.15442	−0.250446
$75$	−7.25539	−0.837781
$76$	−1.92868	−0.221235
$77$	0	0
$78$	−0.0713170	−0.00807506
$79$	8.04694	0.905351	0.452676	−	0.891675i	$-0.350470\pi$
$79$	8.04694	0.905351	0.452676	+	0.891675i	$0.350470\pi$
$80$	12.5229	1.40010
$81$	1.00000	0.111111
$82$	−2.90923	−0.321271
$83$	−5.90959	−0.648661	−0.324331	−	0.945944i	$-0.605139\pi$
$83$	−5.90959	−0.648661	−0.324331	+	0.945944i	$0.605139\pi$
$84$	−8.36944	−0.913180
$85$	−20.9855	−2.27620
$86$	2.41807	0.260747
$87$	1.85653	0.199041
$88$	0	0
$89$	7.54793	0.800079	0.400040	−	0.916498i	$-0.368996\pi$
$89$	7.54793	0.800079	0.400040	+	0.916498i	$0.368996\pi$
$90$	−0.934889	−0.0985460
$91$	1.15886	0.121482
$92$	−12.9750	−1.35274
$93$	5.19887	0.539098
$94$	1.45169	0.149730
$95$	3.50077	0.359171
$96$	3.05362	0.311659
$97$	−18.5499	−1.88346	−0.941730	−	0.336369i	$-0.890801\pi$
$97$	−18.5499	−1.88346	−0.941730	+	0.336369i	$0.890801\pi$
$98$	−3.15947	−0.319154
$99$	0	0
$100$	−13.9934	−1.39934
$101$	11.8791	1.18201	0.591005	−	0.806668i	$-0.298731\pi$
$101$	11.8791	1.18201	0.591005	+	0.806668i	$0.298731\pi$
$102$	−1.60086	−0.158509
$103$	−17.4360	−1.71802	−0.859011	−	0.511958i	$-0.828920\pi$
$103$	−17.4360	−1.71802	−0.859011	+	0.511958i	$0.828920\pi$
$104$	−0.280182	−0.0274741
$105$	15.1914	1.48253
$106$	1.48174	0.143919
$107$	−2.12558	−0.205487	−0.102744	−	0.994708i	$-0.532762\pi$
$107$	−2.12558	−0.205487	−0.102744	+	0.994708i	$0.532762\pi$
$108$	1.92868	0.185588
$109$	10.4849	1.00427	0.502135	−	0.864789i	$-0.332548\pi$
$109$	10.4849	1.00427	0.502135	+	0.864789i	$0.332548\pi$
$110$	0	0
$111$	−8.06741	−0.765724
$112$	−15.5230	−1.46679
$113$	8.34370	0.784909	0.392454	−	0.919771i	$-0.371626\pi$
$113$	8.34370	0.784909	0.392454	+	0.919771i	$0.371626\pi$
$114$	0.267052	0.0250118
$115$	23.5511	2.19615
$116$	3.58065	0.332455
$117$	−0.267052	−0.0246890
$118$	−2.52602	−0.232539
$119$	26.0131	2.38462
$120$	−3.67288	−0.335287
$121$	0	0
$122$	0.308972	0.0279730
$123$	−10.8939	−0.982266
$124$	10.0270	0.900449
$125$	7.89562	0.706206
$126$	1.15886	0.103240
$127$	−16.1705	−1.43490	−0.717451	−	0.696609i	$-0.754691\pi$
$127$	−16.1705	−1.43490	−0.717451	+	0.696609i	$0.754691\pi$
$128$	7.80007	0.689435
$129$	9.05466	0.797219
$130$	0.249664	0.0218970
$131$	16.2061	1.41593	0.707967	−	0.706245i	$-0.249613\pi$
$131$	16.2061	1.41593	0.707967	+	0.706245i	$0.249613\pi$
$132$	0	0
$133$	−4.33946	−0.376279
$134$	−0.677380	−0.0585167
$135$	−3.50077	−0.301298
$136$	−6.28927	−0.539300
$137$	−13.3427	−1.13994	−0.569970	−	0.821665i	$-0.693045\pi$
$137$	−13.3427	−1.13994	−0.569970	+	0.821665i	$0.693045\pi$
$138$	1.79657	0.152934
$139$	−10.9931	−0.932420	−0.466210	−	0.884674i	$-0.654381\pi$
$139$	−10.9931	−0.932420	−0.466210	+	0.884674i	$0.654381\pi$
$140$	29.2995	2.47626
$141$	5.43596	0.457791
$142$	−2.14561	−0.180056
$143$	0	0
$144$	3.57718	0.298099
$145$	−6.49928	−0.539736
$146$	1.58503	0.131178
$147$	−11.8309	−0.975795
$148$	−15.5595	−1.27898
$149$	−2.31790	−0.189890	−0.0949448	−	0.995483i	$-0.530267\pi$
$149$	−2.31790	−0.189890	−0.0949448	+	0.995483i	$0.530267\pi$
$150$	1.93757	0.158202
$151$	−5.58536	−0.454530	−0.227265	−	0.973833i	$-0.572978\pi$
$151$	−5.58536	−0.454530	−0.227265	+	0.973833i	$0.572978\pi$
$152$	1.04916	0.0850985
$153$	−5.99455	−0.484631
$154$	0	0
$155$	−18.2001	−1.46186
$156$	−0.515059	−0.0412378
$157$	4.46648	0.356464	0.178232	−	0.983989i	$-0.442962\pi$
$157$	4.46648	0.356464	0.178232	+	0.983989i	$0.442962\pi$
$158$	−2.14895	−0.170962
$159$	5.54849	0.440024
$160$	−10.6900	−0.845122
$161$	−29.1932	−2.30075
$162$	−0.267052	−0.0209816
$163$	−14.7741	−1.15720	−0.578598	−	0.815613i	$-0.696400\pi$
$163$	−14.7741	−1.15720	−0.578598	+	0.815613i	$0.696400\pi$
$164$	−21.0108	−1.64067
$165$	0	0
$166$	1.57817	0.122490
$167$	8.42779	0.652162	0.326081	−	0.945342i	$-0.394272\pi$
$167$	8.42779	0.652162	0.326081	+	0.945342i	$0.394272\pi$
$168$	4.55280	0.351256
$169$	−12.9287	−0.994514
$170$	5.60424	0.429826
$171$	1.00000	0.0764719
$172$	17.4636	1.33159
$173$	1.00425	0.0763515	0.0381758	−	0.999271i	$-0.487845\pi$
$173$	1.00425	0.0763515	0.0381758	+	0.999271i	$0.487845\pi$
$174$	−0.495790	−0.0375858
$175$	−31.4845	−2.38000
$176$	0	0
$177$	−9.45890	−0.710974
$178$	−2.01569	−0.151083
$179$	−12.4522	−0.930724	−0.465362	−	0.885120i	$-0.654076\pi$
$179$	−12.4522	−0.930724	−0.465362	+	0.885120i	$0.654076\pi$
$180$	−6.75188	−0.503255
$181$	−13.8256	−1.02765	−0.513824	−	0.857896i	$-0.671772\pi$
$181$	−13.8256	−1.02765	−0.513824	+	0.857896i	$0.671772\pi$
$182$	−0.309477	−0.0229400
$183$	1.15697	0.0855258
$184$	7.05814	0.520333
$185$	28.2421	2.07640
$186$	−1.38837	−0.101800
$187$	0	0
$188$	10.4842	0.764642
$189$	4.33946	0.315649
$190$	−0.934889	−0.0678240
$191$	−17.8558	−1.29200	−0.646002	−	0.763336i	$-0.723560\pi$
$191$	−17.8558	−1.29200	−0.646002	+	0.763336i	$0.723560\pi$
$192$	6.33889	0.457470
$193$	25.8076	1.85767	0.928835	−	0.370493i	$-0.120811\pi$
$193$	25.8076	1.85767	0.928835	+	0.370493i	$0.120811\pi$
$194$	4.95380	0.355662
$195$	0.934889	0.0669488
$196$	−22.8180	−1.62986
$197$	6.93992	0.494449	0.247224	−	0.968958i	$-0.420482\pi$
$197$	6.93992	0.494449	0.247224	+	0.968958i	$0.420482\pi$
$198$	0	0
$199$	17.3605	1.23065	0.615325	−	0.788274i	$-0.289025\pi$
$199$	17.3605	1.23065	0.615325	+	0.788274i	$0.289025\pi$
$200$	7.61210	0.538257
$201$	−2.53651	−0.178911
$202$	−3.17233	−0.223205
$203$	8.05632	0.565443
$204$	−11.5616	−0.809473
$205$	38.1369	2.66360
$206$	4.65633	0.324422
$207$	6.72739	0.467586
$208$	−0.955296	−0.0662378
$209$	0	0
$210$	−4.05691	−0.279954
$211$	−14.4749	−0.996494	−0.498247	−	0.867035i	$-0.666023\pi$
$211$	−14.4749	−0.996494	−0.498247	+	0.867035i	$0.666023\pi$
$212$	10.7013	0.734967
$213$	−8.03442	−0.550510
$214$	0.567640	0.0388031
$215$	−31.6983	−2.16181
$216$	−1.04916	−0.0713866
$217$	22.5603	1.53149
$218$	−2.80001	−0.189641
$219$	5.93529	0.401070
$220$	0	0
$221$	1.60086	0.107685
$222$	2.15442	0.144595
$223$	−4.53976	−0.304005	−0.152002	−	0.988380i	$-0.548572\pi$
$223$	−4.53976	−0.304005	−0.152002	+	0.988380i	$0.548572\pi$
$224$	13.2511	0.885374
$225$	7.25539	0.483693
$226$	−2.22820	−0.148218
$227$	−4.69587	−0.311676	−0.155838	−	0.987783i	$-0.549808\pi$
$227$	−4.69587	−0.311676	−0.155838	+	0.987783i	$0.549808\pi$
$228$	1.92868	0.127730
$229$	16.3432	1.07999	0.539995	−	0.841668i	$-0.318426\pi$
$229$	16.3432	1.07999	0.539995	+	0.841668i	$0.318426\pi$
$230$	−6.28937	−0.414709
$231$	0	0
$232$	−1.94780	−0.127880
$233$	29.6495	1.94240	0.971201	−	0.238261i	$-0.0765772\pi$
$233$	29.6495	1.94240	0.971201	+	0.238261i	$0.0765772\pi$
$234$	0.0713170	0.00466214
$235$	−19.0301	−1.24138
$236$	−18.2432	−1.18753
$237$	−8.04694	−0.522705
$238$	−6.94686	−0.450298
$239$	5.32010	0.344129	0.172064	−	0.985086i	$-0.444956\pi$
$239$	5.32010	0.344129	0.172064	+	0.985086i	$0.444956\pi$
$240$	−12.5229	−0.808350
$241$	−20.1099	−1.29539	−0.647695	−	0.761900i	$-0.724267\pi$
$241$	−20.1099	−1.29539	−0.647695	+	0.761900i	$0.724267\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	2.23143	0.142853
$245$	41.4172	2.64605
$246$	2.90923	0.185486
$247$	−0.267052	−0.0169921
$248$	−5.45447	−0.346359
$249$	5.90959	0.374505
$250$	−2.10854	−0.133356
$251$	19.7230	1.24490	0.622452	−	0.782658i	$-0.286137\pi$
$251$	19.7230	1.24490	0.622452	+	0.782658i	$0.286137\pi$
$252$	8.36944	0.527225
$253$	0	0
$254$	4.31838	0.270959
$255$	20.9855	1.31417
$256$	10.5948	0.662172
$257$	0.715435	0.0446276	0.0223138	−	0.999751i	$-0.492897\pi$
$257$	0.715435	0.0446276	0.0223138	+	0.999751i	$0.492897\pi$
$258$	−2.41807	−0.150542
$259$	−35.0082	−2.17530
$260$	1.80311	0.111824
$261$	−1.85653	−0.114916
$262$	−4.32788	−0.267377
$263$	−4.21464	−0.259886	−0.129943	−	0.991521i	$-0.541479\pi$
$263$	−4.21464	−0.259886	−0.129943	+	0.991521i	$0.541479\pi$
$264$	0	0
$265$	−19.4240	−1.19321
$266$	1.15886	0.0710544
$267$	−7.54793	−0.461926
$268$	−4.89212	−0.298834
$269$	13.5913	0.828674	0.414337	−	0.910123i	$-0.364013\pi$
$269$	13.5913	0.828674	0.414337	+	0.910123i	$0.364013\pi$
$270$	0.934889	0.0568956
$271$	−8.12563	−0.493597	−0.246798	−	0.969067i	$-0.579379\pi$
$271$	−8.12563	−0.493597	−0.246798	+	0.969067i	$0.579379\pi$
$272$	−21.4436	−1.30021
$273$	−1.15886	−0.0701376
$274$	3.56319	0.215260
$275$	0	0
$276$	12.9750	0.781004
$277$	−14.4741	−0.869667	−0.434833	−	0.900511i	$-0.643193\pi$
$277$	−14.4741	−0.869667	−0.434833	+	0.900511i	$0.643193\pi$
$278$	2.93573	0.176073
$279$	−5.19887	−0.311248
$280$	−15.9383	−0.952497
$281$	−20.2636	−1.20882	−0.604412	−	0.796672i	$-0.706592\pi$
$281$	−20.2636	−1.20882	−0.604412	+	0.796672i	$0.706592\pi$
$282$	−1.45169	−0.0864467
$283$	−22.9360	−1.36341	−0.681703	−	0.731629i	$-0.738760\pi$
$283$	−22.9360	−1.36341	−0.681703	+	0.731629i	$0.738760\pi$
$284$	−15.4959	−0.919510
$285$	−3.50077	−0.207368
$286$	0	0
$287$	−47.2734	−2.79046
$288$	−3.05362	−0.179937
$289$	18.9346	1.11380
$290$	1.73565	0.101921
$291$	18.5499	1.08742
$292$	11.4473	0.669902
$293$	−2.65708	−0.155229	−0.0776143	−	0.996983i	$-0.524730\pi$
$293$	−2.65708	−0.155229	−0.0776143	+	0.996983i	$0.524730\pi$
$294$	3.15947	0.184264
$295$	33.1134	1.92794
$296$	8.46404	0.491962
$297$	0	0
$298$	0.619000	0.0358577
$299$	−1.79657	−0.103898
$300$	13.9934	0.807907
$301$	39.2923	2.26477
$302$	1.49158	0.0858311
$303$	−11.8791	−0.682434
$304$	3.57718	0.205166
$305$	−4.05029	−0.231919
$306$	1.60086	0.0915150
$307$	−2.72869	−0.155735	−0.0778673	−	0.996964i	$-0.524811\pi$
$307$	−2.72869	−0.155735	−0.0778673	+	0.996964i	$0.524811\pi$
$308$	0	0
$309$	17.4360	0.991900
$310$	4.86037	0.276050
$311$	−34.2024	−1.93944	−0.969721	−	0.244214i	$-0.921470\pi$
$311$	−34.2024	−1.93944	−0.969721	+	0.244214i	$0.921470\pi$
$312$	0.280182	0.0158622
$313$	0.308011	0.0174098	0.00870490	−	0.999962i	$-0.497229\pi$
$313$	0.308011	0.0174098	0.00870490	+	0.999962i	$0.497229\pi$
$314$	−1.19278	−0.0673127
$315$	−15.1914	−0.855941
$316$	−15.5200	−0.873068
$317$	6.89031	0.386998	0.193499	−	0.981100i	$-0.438016\pi$
$317$	6.89031	0.386998	0.193499	+	0.981100i	$0.438016\pi$
$318$	−1.48174	−0.0830918
$319$	0	0
$320$	−22.1910	−1.24051
$321$	2.12558	0.118638
$322$	7.79612	0.434461
$323$	−5.99455	−0.333546
$324$	−1.92868	−0.107149
$325$	−1.93757	−0.107477
$326$	3.94545	0.218518
$327$	−10.4849	−0.579816
$328$	11.4294	0.631085
$329$	23.5891	1.30051
$330$	0	0
$331$	−12.4542	−0.684547	−0.342274	−	0.939600i	$-0.611197\pi$
$331$	−12.4542	−0.684547	−0.342274	+	0.939600i	$0.611197\pi$
$332$	11.3977	0.625531
$333$	8.06741	0.442091
$334$	−2.25066	−0.123151
$335$	8.87972	0.485151
$336$	15.5230	0.846851
$337$	8.79181	0.478920	0.239460	−	0.970906i	$-0.423030\pi$
$337$	8.79181	0.478920	0.239460	+	0.970906i	$0.423030\pi$
$338$	3.45264	0.187799
$339$	−8.34370	−0.453167
$340$	40.4745	2.19504
$341$	0	0
$342$	−0.267052	−0.0144405
$343$	−20.9634	−1.13192
$344$	−9.49983	−0.512197
$345$	−23.5511	−1.26795
$346$	−0.268187	−0.0144178
$347$	−7.07490	−0.379801	−0.189900	−	0.981803i	$-0.560816\pi$
$347$	−7.07490	−0.379801	−0.189900	+	0.981803i	$0.560816\pi$
$348$	−3.58065	−0.191943
$349$	0.331115	0.0177242	0.00886208	−	0.999961i	$-0.497179\pi$
$349$	0.331115	0.0177242	0.00886208	+	0.999961i	$0.497179\pi$
$350$	8.40800	0.449427
$351$	0.267052	0.0142542
$352$	0	0
$353$	−27.1334	−1.44417	−0.722083	−	0.691806i	$-0.756815\pi$
$353$	−27.1334	−1.44417	−0.722083	+	0.691806i	$0.756815\pi$
$354$	2.52602	0.134257
$355$	28.1267	1.49281
$356$	−14.5576	−0.771550
$357$	−26.0131	−1.37676
$358$	3.32540	0.175753
$359$	−26.7863	−1.41373	−0.706864	−	0.707350i	$-0.749891\pi$
$359$	−26.7863	−1.41373	−0.706864	+	0.707350i	$0.749891\pi$
$360$	3.67288	0.193578
$361$	1.00000	0.0526316
$362$	3.69216	0.194055
$363$	0	0
$364$	−2.23508	−0.117150
$365$	−20.7781	−1.08757
$366$	−0.308972	−0.0161502
$367$	1.26596	0.0660825	0.0330412	−	0.999454i	$-0.489481\pi$
$367$	1.26596	0.0660825	0.0330412	+	0.999454i	$0.489481\pi$
$368$	24.0651	1.25448
$369$	10.8939	0.567111
$370$	−7.54213	−0.392097
$371$	24.0774	1.25004
$372$	−10.0270	−0.519874
$373$	−19.4350	−1.00630	−0.503152	−	0.864198i	$-0.667827\pi$
$373$	−19.4350	−1.00630	−0.503152	+	0.864198i	$0.667827\pi$
$374$	0	0
$375$	−7.89562	−0.407728
$376$	−5.70322	−0.294121
$377$	0.495790	0.0255345
$378$	−1.15886	−0.0596055
$379$	−29.2353	−1.50172	−0.750859	−	0.660462i	$-0.770360\pi$
$379$	−29.2353	−1.50172	−0.750859	+	0.660462i	$0.770360\pi$
$380$	−6.75188	−0.346364
$381$	16.1705	0.828441
$382$	4.76845	0.243975
$383$	13.2862	0.678893	0.339446	−	0.940625i	$-0.389760\pi$
$383$	13.2862	0.678893	0.339446	+	0.940625i	$0.389760\pi$
$384$	−7.80007	−0.398045
$385$	0	0
$386$	−6.89198	−0.350792
$387$	−9.05466	−0.460274
$388$	35.7769	1.81630
$389$	−20.9644	−1.06294	−0.531468	−	0.847078i	$-0.678360\pi$
$389$	−20.9644	−1.06294	−0.531468	+	0.847078i	$0.678360\pi$
$390$	−0.249664	−0.0126423
$391$	−40.3277	−2.03946
$392$	12.4125	0.626928
$393$	−16.2061	−0.817490
$394$	−1.85332	−0.0933690
$395$	28.1705	1.41741
$396$	0	0
$397$	−34.4498	−1.72898	−0.864492	−	0.502647i	$-0.832360\pi$
$397$	−34.4498	−1.72898	−0.864492	+	0.502647i	$0.832360\pi$
$398$	−4.63615	−0.232389
$399$	4.33946	0.217245
$400$	25.9539	1.29769
$401$	−18.8837	−0.943009	−0.471505	−	0.881864i	$-0.656289\pi$
$401$	−18.8837	−0.943009	−0.471505	+	0.881864i	$0.656289\pi$
$402$	0.677380	0.0337846
$403$	1.38837	0.0691597
$404$	−22.9109	−1.13986
$405$	3.50077	0.173955
$406$	−2.15146	−0.106775
$407$	0	0
$408$	6.28927	0.311365
$409$	−29.6809	−1.46763	−0.733813	−	0.679352i	$-0.762261\pi$
$409$	−29.6809	−1.46763	−0.733813	+	0.679352i	$0.762261\pi$
$410$	−10.1845	−0.502979
$411$	13.3427	0.658145
$412$	33.6285	1.65676
$413$	−41.0465	−2.01977
$414$	−1.79657	−0.0882964
$415$	−20.6881	−1.01554
$416$	0.815478	0.0399821
$417$	10.9931	0.538333
$418$	0	0
$419$	13.0696	0.638492	0.319246	−	0.947672i	$-0.396570\pi$
$419$	13.0696	0.638492	0.319246	+	0.947672i	$0.396570\pi$
$420$	−29.2995	−1.42967
$421$	−26.5423	−1.29359	−0.646795	−	0.762663i	$-0.723891\pi$
$421$	−26.5423	−1.29359	−0.646795	+	0.762663i	$0.723891\pi$
$422$	3.86556	0.188173
$423$	−5.43596	−0.264306
$424$	−5.82128	−0.282706
$425$	−43.4928	−2.10971
$426$	2.14561	0.103955
$427$	5.02063	0.242965
$428$	4.09956	0.198160
$429$	0	0
$430$	8.46511	0.408224
$431$	−14.6053	−0.703512	−0.351756	−	0.936092i	$-0.614415\pi$
$431$	−14.6053	−0.703512	−0.351756	+	0.936092i	$0.614415\pi$
$432$	−3.57718	−0.172107
$433$	−8.58401	−0.412521	−0.206261	−	0.978497i	$-0.566129\pi$
$433$	−8.58401	−0.412521	−0.206261	+	0.978497i	$0.566129\pi$
$434$	−6.02478	−0.289199
$435$	6.49928	0.311617
$436$	−20.2220	−0.968459
$437$	6.72739	0.321815
$438$	−1.58503	−0.0757358
$439$	−11.5667	−0.552048	−0.276024	−	0.961151i	$-0.589017\pi$
$439$	−11.5667	−0.552048	−0.276024	+	0.961151i	$0.589017\pi$
$440$	0	0
$441$	11.8309	0.563376
$442$	−0.427513	−0.0203347
$443$	37.5962	1.78625	0.893125	−	0.449809i	$-0.148508\pi$
$443$	37.5962	1.78625	0.893125	+	0.449809i	$0.148508\pi$
$444$	15.5595	0.738420
$445$	26.4236	1.25260
$446$	1.21235	0.0574066
$447$	2.31790	0.109633
$448$	27.5073	1.29960
$449$	27.5753	1.30136	0.650680	−	0.759352i	$-0.274484\pi$
$449$	27.5753	1.30136	0.650680	+	0.759352i	$0.274484\pi$
$450$	−1.93757	−0.0913380
$451$	0	0
$452$	−16.0923	−0.756920
$453$	5.58536	0.262423
$454$	1.25404	0.0588551
$455$	4.05691	0.190191
$456$	−1.04916	−0.0491316
$457$	12.1594	0.568795	0.284397	−	0.958706i	$-0.408206\pi$
$457$	12.1594	0.568795	0.284397	+	0.958706i	$0.408206\pi$
$458$	−4.36450	−0.203940
$459$	5.99455	0.279802
$460$	−45.4225	−2.11784
$461$	−34.1168	−1.58898	−0.794488	−	0.607279i	$-0.792261\pi$
$461$	−34.1168	−1.58898	−0.794488	+	0.607279i	$0.792261\pi$
$462$	0	0
$463$	38.7312	1.79999	0.899996	−	0.435898i	$-0.143569\pi$
$463$	38.7312	1.79999	0.899996	+	0.435898i	$0.143569\pi$
$464$	−6.64114	−0.308307
$465$	18.2001	0.844007
$466$	−7.91797	−0.366793
$467$	−20.1371	−0.931832	−0.465916	−	0.884829i	$-0.654275\pi$
$467$	−20.1371	−0.931832	−0.465916	+	0.884829i	$0.654275\pi$
$468$	0.515059	0.0238086
$469$	−11.0071	−0.508259
$470$	5.08202	0.234416
$471$	−4.46648	−0.205805
$472$	9.92394	0.456786
$473$	0	0
$474$	2.14895	0.0987047
$475$	7.25539	0.332900
$476$	−50.1710	−2.29958
$477$	−5.54849	−0.254048
$478$	−1.42075	−0.0649835
$479$	−14.7177	−0.672467	−0.336234	−	0.941779i	$-0.609153\pi$
$479$	−14.7177	−0.672467	−0.336234	+	0.941779i	$0.609153\pi$
$480$	10.6900	0.487931
$481$	−2.15442	−0.0982331
$482$	5.37039	0.244614
$483$	29.1932	1.32834
$484$	0	0
$485$	−64.9391	−2.94873
$486$	0.267052	0.0121137
$487$	−33.3368	−1.51064	−0.755318	−	0.655359i	$-0.772518\pi$
$487$	−33.3368	−1.51064	−0.755318	+	0.655359i	$0.772518\pi$
$488$	−1.21385	−0.0549486
$489$	14.7741	0.668107
$490$	−11.0606	−0.499666
$491$	−3.67946	−0.166052	−0.0830259	−	0.996547i	$-0.526458\pi$
$491$	−3.67946	−0.166052	−0.0830259	+	0.996547i	$0.526458\pi$
$492$	21.0108	0.947240
$493$	11.1290	0.501227
$494$	0.0713170	0.00320870
$495$	0	0
$496$	−18.5973	−0.835044
$497$	−34.8650	−1.56391
$498$	−1.57817	−0.0707195
$499$	−20.5391	−0.919454	−0.459727	−	0.888060i	$-0.652053\pi$
$499$	−20.5391	−0.919454	−0.459727	+	0.888060i	$0.652053\pi$
$500$	−15.2281	−0.681023
$501$	−8.42779	−0.376526
$502$	−5.26707	−0.235081
$503$	−6.16314	−0.274801	−0.137400	−	0.990516i	$-0.543875\pi$
$503$	−6.16314	−0.274801	−0.137400	+	0.990516i	$0.543875\pi$
$504$	−4.55280	−0.202798
$505$	41.5859	1.85055
$506$	0	0
$507$	12.9287	0.574183
$508$	31.1878	1.38373
$509$	22.3506	0.990671	0.495336	−	0.868702i	$-0.335045\pi$
$509$	22.3506	0.990671	0.495336	+	0.868702i	$0.335045\pi$
$510$	−5.60424	−0.248160
$511$	25.7559	1.13938
$512$	−18.4295	−0.814476
$513$	−1.00000	−0.0441511
$514$	−0.191059	−0.00842724
$515$	−61.0395	−2.68972
$516$	−17.4636	−0.768791
$517$	0	0
$518$	9.34902	0.410772
$519$	−1.00425	−0.0440816
$520$	−0.980853	−0.0430132
$521$	15.0243	0.658225	0.329112	−	0.944291i	$-0.393251\pi$
$521$	15.0243	0.658225	0.329112	+	0.944291i	$0.393251\pi$
$522$	0.495790	0.0217002
$523$	−33.1908	−1.45133	−0.725665	−	0.688048i	$-0.758468\pi$
$523$	−33.1908	−1.45133	−0.725665	+	0.688048i	$0.758468\pi$
$524$	−31.2564	−1.36544
$525$	31.4845	1.37410
$526$	1.12553	0.0490754
$527$	31.1649	1.35756
$528$	0	0
$529$	22.2578	0.967732
$530$	5.18723	0.225319
$531$	9.45890	0.410481
$532$	8.36944	0.362861
$533$	−2.90923	−0.126013
$534$	2.01569	0.0872276
$535$	−7.44115	−0.321709
$536$	2.66121	0.114947
$537$	12.4522	0.537354
$538$	−3.62958	−0.156482
$539$	0	0
$540$	6.75188	0.290555
$541$	44.0971	1.89588	0.947942	−	0.318443i	$-0.103160\pi$
$541$	44.0971	1.89588	0.947942	+	0.318443i	$0.103160\pi$
$542$	2.16997	0.0932081
$543$	13.8256	0.593313
$544$	18.3051	0.784825
$545$	36.7052	1.57228
$546$	0.309477	0.0132444
$547$	16.5107	0.705946	0.352973	−	0.935634i	$-0.385171\pi$
$547$	16.5107	0.705946	0.352973	+	0.935634i	$0.385171\pi$
$548$	25.7338	1.09929
$549$	−1.15697	−0.0493784
$550$	0	0
$551$	−1.85653	−0.0790908
$552$	−7.05814	−0.300414
$553$	−34.9193	−1.48492
$554$	3.86535	0.164223
$555$	−28.2421	−1.19881
$556$	21.2022	0.899171
$557$	−6.37629	−0.270172	−0.135086	−	0.990834i	$-0.543131\pi$
$557$	−6.37629	−0.270172	−0.135086	+	0.990834i	$0.543131\pi$
$558$	1.38837	0.0587744
$559$	2.41807	0.102273
$560$	−54.3426	−2.29639
$561$	0	0
$562$	5.41144	0.228268
$563$	10.8744	0.458300	0.229150	−	0.973391i	$-0.426405\pi$
$563$	10.8744	0.458300	0.229150	+	0.973391i	$0.426405\pi$
$564$	−10.4842	−0.441466
$565$	29.2094	1.22885
$566$	6.12513	0.257458
$567$	−4.33946	−0.182240
$568$	8.42943	0.353691
$569$	−6.78459	−0.284425	−0.142212	−	0.989836i	$-0.545422\pi$
$569$	−6.78459	−0.284425	−0.142212	+	0.989836i	$0.545422\pi$
$570$	0.934889	0.0391582
$571$	−32.5219	−1.36100	−0.680500	−	0.732749i	$-0.738237\pi$
$571$	−32.5219	−1.36100	−0.680500	+	0.732749i	$0.738237\pi$
$572$	0	0
$573$	17.8558	0.745938
$574$	12.6245	0.526936
$575$	48.8099	2.03551
$576$	−6.33889	−0.264120
$577$	9.80306	0.408107	0.204053	−	0.978960i	$-0.434588\pi$
$577$	9.80306	0.408107	0.204053	+	0.978960i	$0.434588\pi$
$578$	−5.05653	−0.210324
$579$	−25.8076	−1.07253
$580$	12.5350	0.520490
$581$	25.6444	1.06391
$582$	−4.95380	−0.205342
$583$	0	0
$584$	−6.22709	−0.257679
$585$	−0.934889	−0.0386529
$586$	0.709581	0.0293125
$587$	0.207231	0.00855334	0.00427667	−	0.999991i	$-0.498639\pi$
$587$	0.207231	0.00855334	0.00427667	+	0.999991i	$0.498639\pi$
$588$	22.8180	0.941000
$589$	−5.19887	−0.214216
$590$	−8.84302	−0.364061
$591$	−6.93992	−0.285470
$592$	28.8586	1.18608
$593$	−14.2272	−0.584240	−0.292120	−	0.956382i	$-0.594361\pi$
$593$	−14.2272	−0.584240	−0.292120	+	0.956382i	$0.594361\pi$
$594$	0	0
$595$	91.0658	3.73334
$596$	4.47049	0.183118
$597$	−17.3605	−0.710516
$598$	0.479778	0.0196196
$599$	−4.27608	−0.174716	−0.0873579	−	0.996177i	$-0.527842\pi$
$599$	−4.27608	−0.174716	−0.0873579	+	0.996177i	$0.527842\pi$
$600$	−7.61210	−0.310763
$601$	−38.2289	−1.55939	−0.779696	−	0.626159i	$-0.784626\pi$
$601$	−38.2289	−1.55939	−0.779696	+	0.626159i	$0.784626\pi$
$602$	−10.4931	−0.427667
$603$	2.53651	0.103294
$604$	10.7724	0.438323
$605$	0	0
$606$	3.17233	0.128867
$607$	−45.0362	−1.82796	−0.913982	−	0.405754i	$-0.867009\pi$
$607$	−45.0362	−1.82796	−0.913982	+	0.405754i	$0.867009\pi$
$608$	−3.05362	−0.123841
$609$	−8.05632	−0.326459
$610$	1.08164	0.0437944
$611$	1.45169	0.0587290
$612$	11.5616	0.467349
$613$	14.8215	0.598633	0.299316	−	0.954154i	$-0.403241\pi$
$613$	14.8215	0.598633	0.299316	+	0.954154i	$0.403241\pi$
$614$	0.728703	0.0294081
$615$	−38.1369	−1.53783
$616$	0	0
$617$	6.22743	0.250707	0.125353	−	0.992112i	$-0.459994\pi$
$617$	6.22743	0.250707	0.125353	+	0.992112i	$0.459994\pi$
$618$	−4.65633	−0.187305
$619$	10.2682	0.412712	0.206356	−	0.978477i	$-0.433839\pi$
$619$	10.2682	0.412712	0.206356	+	0.978477i	$0.433839\pi$
$620$	35.1021	1.40974
$621$	−6.72739	−0.269961
$622$	9.13385	0.366234
$623$	−32.7539	−1.31226
$624$	0.955296	0.0382424
$625$	−8.63622	−0.345449
$626$	−0.0822551	−0.00328757
$627$	0	0
$628$	−8.61442	−0.343753
$629$	−48.3605	−1.92826
$630$	4.05691	0.161631
$631$	−7.49717	−0.298458	−0.149229	−	0.988803i	$-0.547679\pi$
$631$	−7.49717	−0.298458	−0.149229	+	0.988803i	$0.547679\pi$
$632$	8.44256	0.335827
$633$	14.4749	0.575326
$634$	−1.84007	−0.0730786
$635$	−56.6093	−2.24647
$636$	−10.7013	−0.424333
$637$	−3.15947	−0.125183
$638$	0	0
$639$	8.03442	0.317837
$640$	27.3062	1.07937
$641$	−22.7799	−0.899753	−0.449877	−	0.893091i	$-0.648532\pi$
$641$	−22.7799	−0.899753	−0.449877	+	0.893091i	$0.648532\pi$
$642$	−0.567640	−0.0224030
$643$	12.8646	0.507331	0.253665	−	0.967292i	$-0.418364\pi$
$643$	12.8646	0.507331	0.253665	+	0.967292i	$0.418364\pi$
$644$	56.3045	2.21871
$645$	31.6983	1.24812
$646$	1.60086	0.0629849
$647$	−20.9538	−0.823777	−0.411889	−	0.911234i	$-0.635131\pi$
$647$	−20.9538	−0.823777	−0.411889	+	0.911234i	$0.635131\pi$
$648$	1.04916	0.0412151
$649$	0	0
$650$	0.517433	0.0202954
$651$	−22.5603	−0.884207
$652$	28.4945	1.11593
$653$	6.09467	0.238503	0.119251	−	0.992864i	$-0.461951\pi$
$653$	6.09467	0.238503	0.119251	+	0.992864i	$0.461951\pi$
$654$	2.80001	0.109489
$655$	56.7339	2.21678
$656$	38.9693	1.52150
$657$	−5.93529	−0.231558
$658$	−6.29953	−0.245581
$659$	31.3776	1.22230	0.611149	−	0.791515i	$-0.290707\pi$
$659$	31.3776	1.22230	0.611149	+	0.791515i	$0.290707\pi$
$660$	0	0
$661$	3.92352	0.152607	0.0763036	−	0.997085i	$-0.475688\pi$
$661$	3.92352	0.152607	0.0763036	+	0.997085i	$0.475688\pi$
$662$	3.32594	0.129266
$663$	−1.60086	−0.0621722
$664$	−6.20013	−0.240612
$665$	−15.1914	−0.589099
$666$	−2.15442	−0.0834821
$667$	−12.4896	−0.483599
$668$	−16.2545	−0.628907
$669$	4.53976	0.175517
$670$	−2.37135	−0.0916133
$671$	0	0
$672$	−13.2511	−0.511171
$673$	10.1222	0.390182	0.195091	−	0.980785i	$-0.437500\pi$
$673$	10.1222	0.390182	0.195091	+	0.980785i	$0.437500\pi$
$674$	−2.34787	−0.0904367
$675$	−7.25539	−0.279260
$676$	24.9353	0.959051
$677$	13.5448	0.520568	0.260284	−	0.965532i	$-0.416184\pi$
$677$	13.5448	0.520568	0.260284	+	0.965532i	$0.416184\pi$
$678$	2.22820	0.0855737
$679$	80.4966	3.08918
$680$	−22.0173	−0.844324
$681$	4.69587	0.179946
$682$	0	0
$683$	30.4131	1.16373	0.581863	−	0.813287i	$-0.302324\pi$
$683$	30.4131	1.16373	0.581863	+	0.813287i	$0.302324\pi$
$684$	−1.92868	−0.0737450
$685$	−46.7096	−1.78468
$686$	5.59833	0.213745
$687$	−16.3432	−0.623533
$688$	−32.3902	−1.23486
$689$	1.48174	0.0564497
$690$	6.28937	0.239432
$691$	−44.3096	−1.68562	−0.842809	−	0.538213i	$-0.819100\pi$
$691$	−44.3096	−1.68562	−0.842809	+	0.538213i	$0.819100\pi$
$692$	−1.93688	−0.0736290
$693$	0	0
$694$	1.88937	0.0717195
$695$	−38.4842	−1.45979
$696$	1.94780	0.0738313
$697$	−65.3037	−2.47356
$698$	−0.0884250	−0.00334693
$699$	−29.6495	−1.12145
$700$	60.7236	2.29514
$701$	−40.8688	−1.54359	−0.771797	−	0.635870i	$-0.780642\pi$
$701$	−40.8688	−1.54359	−0.771797	+	0.635870i	$0.780642\pi$
$702$	−0.0713170	−0.00269169
$703$	8.06741	0.304268
$704$	0	0
$705$	19.0301	0.716713
$706$	7.24605	0.272709
$707$	−51.5487	−1.93869
$708$	18.2432	0.685622
$709$	7.33348	0.275415	0.137707	−	0.990473i	$-0.456027\pi$
$709$	7.33348	0.275415	0.137707	+	0.990473i	$0.456027\pi$
$710$	−7.51129	−0.281894
$711$	8.04694	0.301784
$712$	7.91902	0.296778
$713$	−34.9749	−1.30982
$714$	6.94686	0.259980
$715$	0	0
$716$	24.0164	0.897536
$717$	−5.32010	−0.198683
$718$	7.15335	0.266961
$719$	−1.12757	−0.0420512	−0.0210256	−	0.999779i	$-0.506693\pi$
$719$	−1.12757	−0.0420512	−0.0210256	+	0.999779i	$0.506693\pi$
$720$	12.5229	0.466701
$721$	75.6628	2.81783
$722$	−0.267052	−0.00993866
$723$	20.1099	0.747894
$724$	26.6652	0.991004
$725$	−13.4698	−0.500257
$726$	0	0
$727$	−16.7692	−0.621934	−0.310967	−	0.950421i	$-0.600653\pi$
$727$	−16.7692	−0.621934	−0.310967	+	0.950421i	$0.600653\pi$
$728$	1.21584	0.0450619
$729$	1.00000	0.0370370
$730$	5.54884	0.205372
$731$	54.2786	2.00757
$732$	−2.23143	−0.0824761
$733$	39.0705	1.44310	0.721551	−	0.692362i	$-0.243430\pi$
$733$	39.0705	1.44310	0.721551	+	0.692362i	$0.243430\pi$
$734$	−0.338077	−0.0124787
$735$	−41.4172	−1.52770
$736$	−20.5429	−0.757223
$737$	0	0
$738$	−2.90923	−0.107090
$739$	30.6693	1.12819	0.564094	−	0.825711i	$-0.309226\pi$
$739$	30.6693	1.12819	0.564094	+	0.825711i	$0.309226\pi$
$740$	−54.4701	−2.00236
$741$	0.267052	0.00981042
$742$	−6.42994	−0.236051
$743$	−45.1549	−1.65657	−0.828286	−	0.560306i	$-0.810684\pi$
$743$	−45.1549	−1.65657	−0.828286	+	0.560306i	$0.810684\pi$
$744$	5.45447	0.199971
$745$	−8.11443	−0.297290
$746$	5.19016	0.190025
$747$	−5.90959	−0.216220
$748$	0	0
$749$	9.22384	0.337032
$750$	2.10854	0.0769932
$751$	17.3991	0.634902	0.317451	−	0.948275i	$-0.397173\pi$
$751$	17.3991	0.634902	0.317451	+	0.948275i	$0.397173\pi$
$752$	−19.4454	−0.709102
$753$	−19.7230	−0.718745
$754$	−0.132402	−0.00482180
$755$	−19.5531	−0.711609
$756$	−8.36944	−0.304393
$757$	15.9807	0.580827	0.290414	−	0.956901i	$-0.406207\pi$
$757$	15.9807	0.580827	0.290414	+	0.956901i	$0.406207\pi$
$758$	7.80737	0.283576
$759$	0	0
$760$	3.67288	0.133229
$761$	−40.9778	−1.48544	−0.742721	−	0.669601i	$-0.766465\pi$
$761$	−40.9778	−1.48544	−0.742721	+	0.669601i	$0.766465\pi$
$762$	−4.31838	−0.156438
$763$	−45.4987	−1.64716
$764$	34.4383	1.24593
$765$	−20.9855	−0.758734
$766$	−3.54811	−0.128198
$767$	−2.52602	−0.0912094
$768$	−10.5948	−0.382305
$769$	−22.3231	−0.804991	−0.402496	−	0.915422i	$-0.631857\pi$
$769$	−22.3231	−0.804991	−0.402496	+	0.915422i	$0.631857\pi$
$770$	0	0
$771$	−0.715435	−0.0257658
$772$	−49.7747	−1.79143
$773$	−47.2234	−1.69851	−0.849253	−	0.527986i	$-0.822947\pi$
$773$	−47.2234	−1.69851	−0.849253	+	0.527986i	$0.822947\pi$
$774$	2.41807	0.0869157
$775$	−37.7199	−1.35494
$776$	−19.4619	−0.698643
$777$	35.0082	1.25591
$778$	5.59859	0.200719
$779$	10.8939	0.390313
$780$	−1.80311	−0.0645615
$781$	0	0
$782$	10.7696	0.385120
$783$	1.85653	0.0663469
$784$	42.3213	1.51147
$785$	15.6361	0.558077
$786$	4.32788	0.154370
$787$	39.1052	1.39395	0.696975	−	0.717095i	$-0.254529\pi$
$787$	39.1052	1.39395	0.696975	+	0.717095i	$0.254529\pi$
$788$	−13.3849	−0.476817
$789$	4.21464	0.150045
$790$	−7.52299	−0.267656
$791$	−36.2071	−1.28738
$792$	0	0
$793$	0.308972	0.0109719
$794$	9.19989	0.326492
$795$	19.4240	0.688898
$796$	−33.4828	−1.18677
$797$	−21.3798	−0.757310	−0.378655	−	0.925538i	$-0.623613\pi$
$797$	−21.3798	−0.757310	−0.378655	+	0.925538i	$0.623613\pi$
$798$	−1.15886	−0.0410233
$799$	32.5861	1.15281
$800$	−22.1553	−0.783306
$801$	7.54793	0.266693
$802$	5.04295	0.178073
$803$	0	0
$804$	4.89212	0.172532
$805$	−102.199	−3.60204
$806$	−0.370768	−0.0130597
$807$	−13.5913	−0.478435
$808$	12.4631	0.438450
$809$	55.5057	1.95148	0.975738	−	0.218940i	$-0.0702600\pi$
$809$	55.5057	1.95148	0.975738	+	0.218940i	$0.0702600\pi$
$810$	−0.934889	−0.0328487
$811$	−50.0379	−1.75707	−0.878534	−	0.477681i	$-0.841478\pi$
$811$	−50.0379	−1.75707	−0.878534	+	0.477681i	$0.841478\pi$
$812$	−15.5381	−0.545280
$813$	8.12563	0.284978
$814$	0	0
$815$	−51.7207	−1.81170
$816$	21.4436	0.750676
$817$	−9.05466	−0.316783
$818$	7.92635	0.277138
$819$	1.15886	0.0404939
$820$	−73.5540	−2.56862
$821$	−7.66181	−0.267399	−0.133699	−	0.991022i	$-0.542686\pi$
$821$	−7.66181	−0.267399	−0.133699	+	0.991022i	$0.542686\pi$
$822$	−3.56319	−0.124280
$823$	32.3144	1.12641	0.563205	−	0.826317i	$-0.309568\pi$
$823$	32.3144	1.12641	0.563205	+	0.826317i	$0.309568\pi$
$824$	−18.2932	−0.637275
$825$	0	0
$826$	10.9616	0.381402
$827$	12.5102	0.435021	0.217510	−	0.976058i	$-0.430206\pi$
$827$	12.5102	0.435021	0.217510	+	0.976058i	$0.430206\pi$
$828$	−12.9750	−0.450913
$829$	−20.0992	−0.698073	−0.349036	−	0.937109i	$-0.613491\pi$
$829$	−20.0992	−0.698073	−0.349036	+	0.937109i	$0.613491\pi$
$830$	5.52481	0.191769
$831$	14.4741	0.502102
$832$	1.69282	0.0586878
$833$	−70.9208	−2.45726
$834$	−2.93573	−0.101656
$835$	29.5038	1.02102
$836$	0	0
$837$	5.19887	0.179699
$838$	−3.49027	−0.120569
$839$	54.7318	1.88955	0.944775	−	0.327719i	$-0.106280\pi$
$839$	54.7318	1.88955	0.944775	+	0.327719i	$0.106280\pi$
$840$	15.9383	0.549924
$841$	−25.5533	−0.881148
$842$	7.08818	0.244275
$843$	20.2636	0.697915
$844$	27.9175	0.960960
$845$	−45.2604	−1.55700
$846$	1.45169	0.0499100
$847$	0	0
$848$	−19.8480	−0.681582
$849$	22.9360	0.787163
$850$	11.6149	0.398386
$851$	54.2726	1.86044
$852$	15.4959	0.530879
$853$	0.512388	0.0175438	0.00877192	−	0.999962i	$-0.497208\pi$
$853$	0.512388	0.0175438	0.00877192	+	0.999962i	$0.497208\pi$
$854$	−1.34077	−0.0458803
$855$	3.50077	0.119724
$856$	−2.23008	−0.0762225
$857$	−3.46441	−0.118342	−0.0591710	−	0.998248i	$-0.518846\pi$
$857$	−3.46441	−0.118342	−0.0591710	+	0.998248i	$0.518846\pi$
$858$	0	0
$859$	−26.9736	−0.920329	−0.460164	−	0.887834i	$-0.652210\pi$
$859$	−26.9736	−0.920329	−0.460164	+	0.887834i	$0.652210\pi$
$860$	61.1360	2.08472
$861$	47.2734	1.61107
$862$	3.90038	0.132847
$863$	2.01950	0.0687446	0.0343723	−	0.999409i	$-0.489057\pi$
$863$	2.01950	0.0687446	0.0343723	+	0.999409i	$0.489057\pi$
$864$	3.05362	0.103886
$865$	3.51564	0.119535
$866$	2.29238	0.0778983
$867$	−18.9346	−0.643053
$868$	−43.5116	−1.47688
$869$	0	0
$870$	−1.73565	−0.0588440
$871$	−0.677380	−0.0229521
$872$	11.0004	0.372520
$873$	−18.5499	−0.627820
$874$	−1.79657	−0.0607698
$875$	−34.2627	−1.15829
$876$	−11.4473	−0.386768
$877$	−15.7654	−0.532359	−0.266179	−	0.963923i	$-0.585761\pi$
$877$	−15.7654	−0.532359	−0.266179	+	0.963923i	$0.585761\pi$
$878$	3.08892	0.104246
$879$	2.65708	0.0896213
$880$	0	0
$881$	17.3866	0.585771	0.292885	−	0.956148i	$-0.405385\pi$
$881$	17.3866	0.585771	0.292885	+	0.956148i	$0.405385\pi$
$882$	−3.15947	−0.106385
$883$	−10.7650	−0.362273	−0.181136	−	0.983458i	$-0.557978\pi$
$883$	−10.7650	−0.362273	−0.181136	+	0.983458i	$0.557978\pi$
$884$	−3.08755	−0.103846
$885$	−33.1134	−1.11310
$886$	−10.0402	−0.337306
$887$	−46.8270	−1.57230	−0.786149	−	0.618037i	$-0.787928\pi$
$887$	−46.8270	−1.57230	−0.786149	+	0.618037i	$0.787928\pi$
$888$	−8.46404	−0.284034
$889$	70.1713	2.35347
$890$	−7.05648	−0.236534
$891$	0	0
$892$	8.75575	0.293164
$893$	−5.43596	−0.181908
$894$	−0.619000	−0.0207025
$895$	−43.5925	−1.45713
$896$	−33.8480	−1.13078
$897$	1.79657	0.0599856
$898$	−7.36406	−0.245742
$899$	9.65185	0.321907
$900$	−13.9934	−0.466445
$901$	33.2607	1.10807
$902$	0	0
$903$	−39.2923	−1.30757
$904$	8.75391	0.291151
$905$	−48.4002	−1.60888
$906$	−1.49158	−0.0495546
$907$	−57.0332	−1.89376	−0.946878	−	0.321593i	$-0.895782\pi$
$907$	−57.0332	−1.89376	−0.946878	+	0.321593i	$0.895782\pi$
$908$	9.05684	0.300562
$909$	11.8791	0.394004
$910$	−1.08341	−0.0359146
$911$	31.9365	1.05810	0.529052	−	0.848589i	$-0.322548\pi$
$911$	31.9365	1.05810	0.529052	+	0.848589i	$0.322548\pi$
$912$	−3.57718	−0.118452
$913$	0	0
$914$	−3.24721	−0.107408
$915$	4.05029	0.133899
$916$	−31.5209	−1.04148
$917$	−70.3257	−2.32236
$918$	−1.60086	−0.0528362
$919$	9.03651	0.298087	0.149043	−	0.988831i	$-0.452381\pi$
$919$	9.03651	0.298087	0.149043	+	0.988831i	$0.452381\pi$
$920$	24.7089	0.814630
$921$	2.72869	0.0899134
$922$	9.11097	0.300054
$923$	−2.14561	−0.0706237
$924$	0	0
$925$	58.5322	1.92453
$926$	−10.3433	−0.339901
$927$	−17.4360	−0.572674
$928$	5.66914	0.186099
$929$	−40.4462	−1.32700	−0.663498	−	0.748178i	$-0.730929\pi$
$929$	−40.4462	−1.32700	−0.663498	+	0.748178i	$0.730929\pi$
$930$	−4.86037	−0.159378
$931$	11.8309	0.387742
$932$	−57.1845	−1.87314
$933$	34.2024	1.11974
$934$	5.37765	0.175962
$935$	0	0
$936$	−0.280182	−0.00915803
$937$	29.7273	0.971147	0.485574	−	0.874196i	$-0.338611\pi$
$937$	29.7273	0.971147	0.485574	+	0.874196i	$0.338611\pi$
$938$	2.93946	0.0959768
$939$	−0.308011	−0.0100516
$940$	36.7029	1.19712
$941$	55.2813	1.80212	0.901059	−	0.433697i	$-0.142791\pi$
$941$	55.2813	1.80212	0.901059	+	0.433697i	$0.142791\pi$
$942$	1.19278	0.0388630
$943$	73.2872	2.38656
$944$	33.8362	1.10128
$945$	15.1914	0.494178
$946$	0	0
$947$	−30.1999	−0.981365	−0.490683	−	0.871338i	$-0.663253\pi$
$947$	−30.1999	−0.981365	−0.490683	+	0.871338i	$0.663253\pi$
$948$	15.5200	0.504066
$949$	1.58503	0.0514524
$950$	−1.93757	−0.0628631
$951$	−6.89031	−0.223433
$952$	27.2920	0.884539
$953$	−28.5729	−0.925566	−0.462783	−	0.886472i	$-0.653149\pi$
$953$	−28.5729	−0.925566	−0.462783	+	0.886472i	$0.653149\pi$
$954$	1.48174	0.0479731
$955$	−62.5092	−2.02275
$956$	−10.2608	−0.331858
$957$	0	0
$958$	3.93039	0.126985
$959$	57.8999	1.86968
$960$	22.1910	0.716211
$961$	−3.97174	−0.128121
$962$	0.575343	0.0185498
$963$	−2.12558	−0.0684957
$964$	38.7855	1.24920
$965$	90.3464	2.90835
$966$	−7.79612	−0.250836
$967$	42.3751	1.36269	0.681345	−	0.731963i	$-0.261395\pi$
$967$	42.3751	1.36269	0.681345	+	0.731963i	$0.261395\pi$
$968$	0	0
$969$	5.99455	0.192573
$970$	17.3421	0.556822
$971$	−23.0528	−0.739799	−0.369899	−	0.929072i	$-0.620608\pi$
$971$	−23.0528	−0.739799	−0.369899	+	0.929072i	$0.620608\pi$
$972$	1.92868	0.0618625
$973$	47.7040	1.52932
$974$	8.90268	0.285260
$975$	1.93757	0.0620519
$976$	−4.13870	−0.132477
$977$	4.93602	0.157917	0.0789586	−	0.996878i	$-0.474840\pi$
$977$	4.93602	0.157917	0.0789586	+	0.996878i	$0.474840\pi$
$978$	−3.94545	−0.126162
$979$	0	0
$980$	−79.8807	−2.55169
$981$	10.4849	0.334757
$982$	0.982609	0.0313563
$983$	48.1403	1.53544	0.767718	−	0.640787i	$-0.221392\pi$
$983$	48.1403	1.53544	0.767718	+	0.640787i	$0.221392\pi$
$984$	−11.4294	−0.364357
$985$	24.2951	0.774105
$986$	−2.97204	−0.0946490
$987$	−23.5891	−0.750850
$988$	0.515059	0.0163862
$989$	−60.9143	−1.93696
$990$	0	0
$991$	36.5026	1.15954	0.579771	−	0.814780i	$-0.303142\pi$
$991$	36.5026	1.15954	0.579771	+	0.814780i	$0.303142\pi$
$992$	15.8754	0.504044
$993$	12.4542	0.395224
$994$	9.31079	0.295320
$995$	60.7750	1.92670
$996$	−11.3977	−0.361151
$997$	−5.85519	−0.185436	−0.0927179	−	0.995692i	$-0.529555\pi$
$997$	−5.85519	−0.185436	−0.0927179	+	0.995692i	$0.529555\pi$
$998$	5.48501	0.173625
$999$	−8.06741	−0.255241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.bc.1.4	✓	8
11.10	odd	2		6897.2.a.bd.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6897.2.a.bc.1.4	✓	8	1.1	even	1	trivial
6897.2.a.bd.1.5	yes	8	11.10	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:	\( N \)	\(=\)	\( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6897.a (trivial)

\(f(q)\)	\(=\)	\(q-0.267052 q^{2} -1.00000 q^{3} -1.92868 q^{4} +3.50077 q^{5} +0.267052 q^{6} -4.33946 q^{7} +1.04916 q^{8} +1.00000 q^{9} -0.934889 q^{10} +1.92868 q^{12} -0.267052 q^{13} +1.15886 q^{14} -3.50077 q^{15} +3.57718 q^{16} -5.99455 q^{17} -0.267052 q^{18} +1.00000 q^{19} -6.75188 q^{20} +4.33946 q^{21} +6.72739 q^{23} -1.04916 q^{24} +7.25539 q^{25} +0.0713170 q^{26} -1.00000 q^{27} +8.36944 q^{28} -1.85653 q^{29} +0.934889 q^{30} -5.19887 q^{31} -3.05362 q^{32} +1.60086 q^{34} -15.1914 q^{35} -1.92868 q^{36} +8.06741 q^{37} -0.267052 q^{38} +0.267052 q^{39} +3.67288 q^{40} +10.8939 q^{41} -1.15886 q^{42} -9.05466 q^{43} +3.50077 q^{45} -1.79657 q^{46} -5.43596 q^{47} -3.57718 q^{48} +11.8309 q^{49} -1.93757 q^{50} +5.99455 q^{51} +0.515059 q^{52} -5.54849 q^{53} +0.267052 q^{54} -4.55280 q^{56} -1.00000 q^{57} +0.495790 q^{58} +9.45890 q^{59} +6.75188 q^{60} -1.15697 q^{61} +1.38837 q^{62} -4.33946 q^{63} -6.33889 q^{64} -0.934889 q^{65} +2.53651 q^{67} +11.5616 q^{68} -6.72739 q^{69} +4.05691 q^{70} +8.03442 q^{71} +1.04916 q^{72} -5.93529 q^{73} -2.15442 q^{74} -7.25539 q^{75} -1.92868 q^{76} -0.0713170 q^{78} +8.04694 q^{79} +12.5229 q^{80} +1.00000 q^{81} -2.90923 q^{82} -5.90959 q^{83} -8.36944 q^{84} -20.9855 q^{85} +2.41807 q^{86} +1.85653 q^{87} +7.54793 q^{89} -0.934889 q^{90} +1.15886 q^{91} -12.9750 q^{92} +5.19887 q^{93} +1.45169 q^{94} +3.50077 q^{95} +3.05362 q^{96} -18.5499 q^{97} -3.15947 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) \) 8 * q - q^2 - 8 * q^3 + 9 * q^4 - 2 * q^5 + q^6 - 6 * q^7 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 9 * q^12 - q^13 + 7 * q^14 + 2 * q^15 + 23 * q^16 - 15 * q^17 - q^18 + 8 * q^19 - 4 * q^20 + 6 * q^21 - 5 * q^23 + 6 * q^24 + 4 * q^25 + 25 * q^26 - 8 * q^27 - 8 * q^28 - 17 * q^29 + 3 * q^30 - 4 * q^31 - 38 * q^32 - 18 * q^34 - 2 * q^35 + 9 * q^36 - 4 * q^37 - q^38 + q^39 - 14 * q^40 - 2 * q^41 - 7 * q^42 - 10 * q^43 - 2 * q^45 + 10 * q^46 + 5 * q^47 - 23 * q^48 + 28 * q^49 - 7 * q^50 + 15 * q^51 - 8 * q^52 - 19 * q^53 + q^54 - 49 * q^56 - 8 * q^57 + 35 * q^58 + 12 * q^59 + 4 * q^60 + 5 * q^61 - 22 * q^62 - 6 * q^63 + 24 * q^64 - 3 * q^65 - 12 * q^67 + 3 * q^68 + 5 * q^69 - 22 * q^70 + 40 * q^71 - 6 * q^72 + 12 * q^73 + 7 * q^74 - 4 * q^75 + 9 * q^76 - 25 * q^78 - 14 * q^79 + 37 * q^80 + 8 * q^81 + 31 * q^82 - 17 * q^83 + 8 * q^84 + 35 * q^85 - q^86 + 17 * q^87 - 9 * q^89 - 3 * q^90 + 7 * q^91 + 12 * q^92 + 4 * q^93 - 51 * q^94 - 2 * q^95 + 38 * q^96 + 8 * q^97 - 55 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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