LMFDB - Embedded newform 285.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	285.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:	$2.27573645761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	285.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -2.73205 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} +4.73205 q^{11} +1.00000 q^{12} +0.732051 q^{13} +4.73205 q^{14} +1.00000 q^{15} -5.00000 q^{16} -1.73205 q^{18} +1.00000 q^{19} +1.00000 q^{20} -2.73205 q^{21} -8.19615 q^{22} +3.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.26795 q^{26} +1.00000 q^{27} -2.73205 q^{28} +8.19615 q^{29} -1.73205 q^{30} +8.92820 q^{31} +5.19615 q^{32} +4.73205 q^{33} -2.73205 q^{35} +1.00000 q^{36} -6.19615 q^{37} -1.73205 q^{38} +0.732051 q^{39} +1.73205 q^{40} +1.26795 q^{41} +4.73205 q^{42} +4.19615 q^{43} +4.73205 q^{44} +1.00000 q^{45} -6.00000 q^{46} -3.46410 q^{47} -5.00000 q^{48} +0.464102 q^{49} -1.73205 q^{50} +0.732051 q^{52} -9.46410 q^{53} -1.73205 q^{54} +4.73205 q^{55} -4.73205 q^{56} +1.00000 q^{57} -14.1962 q^{58} +2.53590 q^{59} +1.00000 q^{60} -6.53590 q^{61} -15.4641 q^{62} -2.73205 q^{63} +1.00000 q^{64} +0.732051 q^{65} -8.19615 q^{66} +8.00000 q^{67} +3.46410 q^{69} +4.73205 q^{70} -4.39230 q^{71} +1.73205 q^{72} -16.9282 q^{73} +10.7321 q^{74} +1.00000 q^{75} +1.00000 q^{76} -12.9282 q^{77} -1.26795 q^{78} -10.9282 q^{79} -5.00000 q^{80} +1.00000 q^{81} -2.19615 q^{82} -12.9282 q^{83} -2.73205 q^{84} -7.26795 q^{86} +8.19615 q^{87} +8.19615 q^{88} +10.7321 q^{89} -1.73205 q^{90} -2.00000 q^{91} +3.46410 q^{92} +8.92820 q^{93} +6.00000 q^{94} +1.00000 q^{95} +5.19615 q^{96} -6.19615 q^{97} -0.803848 q^{98} +4.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} + 2 q^{19} + 2 q^{20} - 2 q^{21} - 6 q^{22} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 2 q^{28}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 2 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 + 2 * q^19 + 2 * q^20 - 2 * q^21 - 6 * q^22 + 2 * q^25 - 6 * q^26 + 2 * q^27 - 2 * q^28 + 6 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^39 + 6 * q^41 + 6 * q^42 - 2 * q^43 + 6 * q^44 + 2 * q^45 - 12 * q^46 - 10 * q^48 - 6 * q^49 - 2 * q^52 - 12 * q^53 + 6 * q^55 - 6 * q^56 + 2 * q^57 - 18 * q^58 + 12 * q^59 + 2 * q^60 - 20 * q^61 - 24 * q^62 - 2 * q^63 + 2 * q^64 - 2 * q^65 - 6 * q^66 + 16 * q^67 + 6 * q^70 + 12 * q^71 - 20 * q^73 + 18 * q^74 + 2 * q^75 + 2 * q^76 - 12 * q^77 - 6 * q^78 - 8 * q^79 - 10 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 - 2 * q^84 - 18 * q^86 + 6 * q^87 + 6 * q^88 + 18 * q^89 - 4 * q^91 + 4 * q^93 + 12 * q^94 + 2 * q^95 - 2 * q^97 - 12 * q^98 + 6 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.73205	−0.707107
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	1.73205	0.612372
$9$	1.00000	0.333333
$10$	−1.73205	−0.547723
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	1.00000	0.288675
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	4.73205	1.26469
$15$	1.00000	0.258199
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.73205	−0.408248
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.00000	0.223607
$21$	−2.73205	−0.596182
$22$	−8.19615	−1.74743
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	−1.26795	−0.248665
$27$	1.00000	0.192450
$28$	−2.73205	−0.516309
$29$	8.19615	1.52199	0.760994	−	0.648759i	$-0.224712\pi$
$29$	8.19615	1.52199	0.760994	+	0.648759i	$0.224712\pi$
$30$	−1.73205	−0.316228
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	5.19615	0.918559
$33$	4.73205	0.823744
$34$	0	0
$35$	−2.73205	−0.461801
$36$	1.00000	0.166667
$37$	−6.19615	−1.01864	−0.509321	−	0.860577i	$-0.670103\pi$
$37$	−6.19615	−1.01864	−0.509321	+	0.860577i	$0.670103\pi$
$38$	−1.73205	−0.280976
$39$	0.732051	0.117222
$40$	1.73205	0.273861
$41$	1.26795	0.198020	0.0990102	−	0.995086i	$-0.468432\pi$
$41$	1.26795	0.198020	0.0990102	+	0.995086i	$0.468432\pi$
$42$	4.73205	0.730171
$43$	4.19615	0.639907	0.319954	−	0.947433i	$-0.396333\pi$
$43$	4.19615	0.639907	0.319954	+	0.947433i	$0.396333\pi$
$44$	4.73205	0.713384
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	−5.00000	−0.721688
$49$	0.464102	0.0663002
$50$	−1.73205	−0.244949
$51$	0	0
$52$	0.732051	0.101517
$53$	−9.46410	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$53$	−9.46410	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$54$	−1.73205	−0.235702
$55$	4.73205	0.638070
$56$	−4.73205	−0.632347
$57$	1.00000	0.132453
$58$	−14.1962	−1.86405
$59$	2.53590	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$60$	1.00000	0.129099
$61$	−6.53590	−0.836836	−0.418418	−	0.908255i	$-0.637415\pi$
$61$	−6.53590	−0.836836	−0.418418	+	0.908255i	$0.637415\pi$
$62$	−15.4641	−1.96394
$63$	−2.73205	−0.344206
$64$	1.00000	0.125000
$65$	0.732051	0.0907997
$66$	−8.19615	−1.00888
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	4.73205	0.565588
$71$	−4.39230	−0.521271	−0.260635	−	0.965437i	$-0.583932\pi$
$71$	−4.39230	−0.521271	−0.260635	+	0.965437i	$0.583932\pi$
$72$	1.73205	0.204124
$73$	−16.9282	−1.98130	−0.990648	−	0.136441i	$-0.956434\pi$
$73$	−16.9282	−1.98130	−0.990648	+	0.136441i	$0.956434\pi$
$74$	10.7321	1.24758
$75$	1.00000	0.115470
$76$	1.00000	0.114708
$77$	−12.9282	−1.47331
$78$	−1.26795	−0.143567
$79$	−10.9282	−1.22952	−0.614759	−	0.788715i	$-0.710747\pi$
$79$	−10.9282	−1.22952	−0.614759	+	0.788715i	$0.710747\pi$
$80$	−5.00000	−0.559017
$81$	1.00000	0.111111
$82$	−2.19615	−0.242524
$83$	−12.9282	−1.41905	−0.709527	−	0.704678i	$-0.751092\pi$
$83$	−12.9282	−1.41905	−0.709527	+	0.704678i	$0.751092\pi$
$84$	−2.73205	−0.298091
$85$	0	0
$86$	−7.26795	−0.783723
$87$	8.19615	0.878720
$88$	8.19615	0.873713
$89$	10.7321	1.13760	0.568798	−	0.822478i	$-0.307409\pi$
$89$	10.7321	1.13760	0.568798	+	0.822478i	$0.307409\pi$
$90$	−1.73205	−0.182574
$91$	−2.00000	−0.209657
$92$	3.46410	0.361158
$93$	8.92820	0.925812
$94$	6.00000	0.618853
$95$	1.00000	0.102598
$96$	5.19615	0.530330
$97$	−6.19615	−0.629124	−0.314562	−	0.949237i	$-0.601858\pi$
$97$	−6.19615	−0.629124	−0.314562	+	0.949237i	$0.601858\pi$
$98$	−0.803848	−0.0812009
$99$	4.73205	0.475589
$100$	1.00000	0.100000
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	9.85641	0.971181	0.485590	−	0.874187i	$-0.338605\pi$
$103$	9.85641	0.971181	0.485590	+	0.874187i	$0.338605\pi$
$104$	1.26795	0.124333
$105$	−2.73205	−0.266621
$106$	16.3923	1.59216
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	−14.3923	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$110$	−8.19615	−0.781472
$111$	−6.19615	−0.588113
$112$	13.6603	1.29077
$113$	18.9282	1.78062	0.890308	−	0.455359i	$-0.150489\pi$
$113$	18.9282	1.78062	0.890308	+	0.455359i	$0.150489\pi$
$114$	−1.73205	−0.162221
$115$	3.46410	0.323029
$116$	8.19615	0.760994
$117$	0.732051	0.0676781
$118$	−4.39230	−0.404344
$119$	0	0
$120$	1.73205	0.158114
$121$	11.3923	1.03566
$122$	11.3205	1.02491
$123$	1.26795	0.114327
$124$	8.92820	0.801776
$125$	1.00000	0.0894427
$126$	4.73205	0.421565
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−12.1244	−1.07165
$129$	4.19615	0.369451
$130$	−1.26795	−0.111207
$131$	−9.12436	−0.797199	−0.398599	−	0.917125i	$-0.630504\pi$
$131$	−9.12436	−0.797199	−0.398599	+	0.917125i	$0.630504\pi$
$132$	4.73205	0.411872
$133$	−2.73205	−0.236899
$134$	−13.8564	−1.19701
$135$	1.00000	0.0860663
$136$	0	0
$137$	19.8564	1.69645	0.848224	−	0.529638i	$-0.177672\pi$
$137$	19.8564	1.69645	0.848224	+	0.529638i	$0.177672\pi$
$138$	−6.00000	−0.510754
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	−2.73205	−0.230900
$141$	−3.46410	−0.291730
$142$	7.60770	0.638424
$143$	3.46410	0.289683
$144$	−5.00000	−0.416667
$145$	8.19615	0.680653
$146$	29.3205	2.42658
$147$	0.464102	0.0382785
$148$	−6.19615	−0.509321
$149$	−19.8564	−1.62670	−0.813350	−	0.581775i	$-0.802359\pi$
$149$	−19.8564	−1.62670	−0.813350	+	0.581775i	$0.802359\pi$
$150$	−1.73205	−0.141421
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	22.3923	1.80442
$155$	8.92820	0.717131
$156$	0.732051	0.0586110
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	18.9282	1.50585
$159$	−9.46410	−0.750552
$160$	5.19615	0.410792
$161$	−9.46410	−0.745876
$162$	−1.73205	−0.136083
$163$	9.26795	0.725922	0.362961	−	0.931804i	$-0.381766\pi$
$163$	9.26795	0.725922	0.362961	+	0.931804i	$0.381766\pi$
$164$	1.26795	0.0990102
$165$	4.73205	0.368390
$166$	22.3923	1.73798
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	−4.73205	−0.365086
$169$	−12.4641	−0.958777
$170$	0	0
$171$	1.00000	0.0764719
$172$	4.19615	0.319954
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	−14.1962	−1.07621
$175$	−2.73205	−0.206524
$176$	−23.6603	−1.78346
$177$	2.53590	0.190610
$178$	−18.5885	−1.39326
$179$	23.3205	1.74306	0.871528	−	0.490345i	$-0.163129\pi$
$179$	23.3205	1.74306	0.871528	+	0.490345i	$0.163129\pi$
$180$	1.00000	0.0745356
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	3.46410	0.256776
$183$	−6.53590	−0.483148
$184$	6.00000	0.442326
$185$	−6.19615	−0.455550
$186$	−15.4641	−1.13388
$187$	0	0
$188$	−3.46410	−0.252646
$189$	−2.73205	−0.198727
$190$	−1.73205	−0.125656
$191$	0.339746	0.0245832	0.0122916	−	0.999924i	$-0.496087\pi$
$191$	0.339746	0.0245832	0.0122916	+	0.999924i	$0.496087\pi$
$192$	1.00000	0.0721688
$193$	17.1244	1.23264	0.616319	−	0.787497i	$-0.288623\pi$
$193$	17.1244	1.23264	0.616319	+	0.787497i	$0.288623\pi$
$194$	10.7321	0.770516
$195$	0.732051	0.0524232
$196$	0.464102	0.0331501
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	−8.19615	−0.582475
$199$	−15.3205	−1.08604	−0.543021	−	0.839719i	$-0.682720\pi$
$199$	−15.3205	−1.08604	−0.543021	+	0.839719i	$0.682720\pi$
$200$	1.73205	0.122474
$201$	8.00000	0.564276
$202$	18.0000	1.26648
$203$	−22.3923	−1.57163
$204$	0	0
$205$	1.26795	0.0885574
$206$	−17.0718	−1.18945
$207$	3.46410	0.240772
$208$	−3.66025	−0.253793
$209$	4.73205	0.327323
$210$	4.73205	0.326543
$211$	1.07180	0.0737855	0.0368928	−	0.999319i	$-0.488254\pi$
$211$	1.07180	0.0737855	0.0368928	+	0.999319i	$0.488254\pi$
$212$	−9.46410	−0.649997
$213$	−4.39230	−0.300956
$214$	0	0
$215$	4.19615	0.286175
$216$	1.73205	0.117851
$217$	−24.3923	−1.65586
$218$	24.9282	1.68835
$219$	−16.9282	−1.14390
$220$	4.73205	0.319035
$221$	0	0
$222$	10.7321	0.720288
$223$	−17.8564	−1.19575	−0.597877	−	0.801588i	$-0.703989\pi$
$223$	−17.8564	−1.19575	−0.597877	+	0.801588i	$0.703989\pi$
$224$	−14.1962	−0.948520
$225$	1.00000	0.0666667
$226$	−32.7846	−2.18080
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	1.00000	0.0662266
$229$	−18.5359	−1.22489	−0.612443	−	0.790515i	$-0.709813\pi$
$229$	−18.5359	−1.22489	−0.612443	+	0.790515i	$0.709813\pi$
$230$	−6.00000	−0.395628
$231$	−12.9282	−0.850613
$232$	14.1962	0.932023
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	−1.26795	−0.0828884
$235$	−3.46410	−0.225973
$236$	2.53590	0.165073
$237$	−10.9282	−0.709863
$238$	0	0
$239$	−9.80385	−0.634158	−0.317079	−	0.948399i	$-0.602702\pi$
$239$	−9.80385	−0.634158	−0.317079	+	0.948399i	$0.602702\pi$
$240$	−5.00000	−0.322749
$241$	−3.07180	−0.197872	−0.0989359	−	0.995094i	$-0.531544\pi$
$241$	−3.07180	−0.197872	−0.0989359	+	0.995094i	$0.531544\pi$
$242$	−19.7321	−1.26842
$243$	1.00000	0.0641500
$244$	−6.53590	−0.418418
$245$	0.464102	0.0296504
$246$	−2.19615	−0.140022
$247$	0.732051	0.0465793
$248$	15.4641	0.981971
$249$	−12.9282	−0.819292
$250$	−1.73205	−0.109545
$251$	28.0526	1.77066	0.885331	−	0.464961i	$-0.153932\pi$
$251$	28.0526	1.77066	0.885331	+	0.464961i	$0.153932\pi$
$252$	−2.73205	−0.172103
$253$	16.3923	1.03058
$254$	6.92820	0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	−7.26795	−0.452483
$259$	16.9282	1.05187
$260$	0.732051	0.0453999
$261$	8.19615	0.507329
$262$	15.8038	0.976365
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	8.19615	0.504438
$265$	−9.46410	−0.581375
$266$	4.73205	0.290141
$267$	10.7321	0.656791
$268$	8.00000	0.488678
$269$	0.588457	0.0358789	0.0179394	−	0.999839i	$-0.494289\pi$
$269$	0.588457	0.0358789	0.0179394	+	0.999839i	$0.494289\pi$
$270$	−1.73205	−0.105409
$271$	0.392305	0.0238308	0.0119154	−	0.999929i	$-0.496207\pi$
$271$	0.392305	0.0238308	0.0119154	+	0.999929i	$0.496207\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	−34.3923	−2.07772
$275$	4.73205	0.285353
$276$	3.46410	0.208514
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	14.5359	0.871805
$279$	8.92820	0.534518
$280$	−4.73205	−0.282794
$281$	1.26795	0.0756395	0.0378198	−	0.999285i	$-0.487959\pi$
$281$	1.26795	0.0756395	0.0378198	+	0.999285i	$0.487959\pi$
$282$	6.00000	0.357295
$283$	24.9808	1.48495	0.742476	−	0.669873i	$-0.233651\pi$
$283$	24.9808	1.48495	0.742476	+	0.669873i	$0.233651\pi$
$284$	−4.39230	−0.260635
$285$	1.00000	0.0592349
$286$	−6.00000	−0.354787
$287$	−3.46410	−0.204479
$288$	5.19615	0.306186
$289$	−17.0000	−1.00000
$290$	−14.1962	−0.833627
$291$	−6.19615	−0.363225
$292$	−16.9282	−0.990648
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	−0.803848	−0.0468813
$295$	2.53590	0.147646
$296$	−10.7321	−0.623788
$297$	4.73205	0.274581
$298$	34.3923	1.99229
$299$	2.53590	0.146655
$300$	1.00000	0.0577350
$301$	−11.4641	−0.660780
$302$	−24.2487	−1.39536
$303$	−10.3923	−0.597022
$304$	−5.00000	−0.286770
$305$	−6.53590	−0.374244
$306$	0	0
$307$	−32.3923	−1.84873	−0.924363	−	0.381514i	$-0.875403\pi$
$307$	−32.3923	−1.84873	−0.924363	+	0.381514i	$0.875403\pi$
$308$	−12.9282	−0.736653
$309$	9.85641	0.560711
$310$	−15.4641	−0.878302
$311$	32.4449	1.83978	0.919890	−	0.392177i	$-0.128278\pi$
$311$	32.4449	1.83978	0.919890	+	0.392177i	$0.128278\pi$
$312$	1.26795	0.0717835
$313$	6.39230	0.361314	0.180657	−	0.983546i	$-0.442178\pi$
$313$	6.39230	0.361314	0.180657	+	0.983546i	$0.442178\pi$
$314$	−11.0718	−0.624818
$315$	−2.73205	−0.153934
$316$	−10.9282	−0.614759
$317$	−11.3205	−0.635823	−0.317912	−	0.948120i	$-0.602982\pi$
$317$	−11.3205	−0.635823	−0.317912	+	0.948120i	$0.602982\pi$
$318$	16.3923	0.919235
$319$	38.7846	2.17152
$320$	1.00000	0.0559017
$321$	0	0
$322$	16.3923	0.913507
$323$	0	0
$324$	1.00000	0.0555556
$325$	0.732051	0.0406069
$326$	−16.0526	−0.889069
$327$	−14.3923	−0.795896
$328$	2.19615	0.121262
$329$	9.46410	0.521773
$330$	−8.19615	−0.451183
$331$	−25.7128	−1.41330	−0.706652	−	0.707561i	$-0.749795\pi$
$331$	−25.7128	−1.41330	−0.706652	+	0.707561i	$0.749795\pi$
$332$	−12.9282	−0.709527
$333$	−6.19615	−0.339547
$334$	−6.00000	−0.328305
$335$	8.00000	0.437087
$336$	13.6603	0.745228
$337$	5.12436	0.279141	0.139571	−	0.990212i	$-0.455428\pi$
$337$	5.12436	0.279141	0.139571	+	0.990212i	$0.455428\pi$
$338$	21.5885	1.17426
$339$	18.9282	1.02804
$340$	0	0
$341$	42.2487	2.28790
$342$	−1.73205	−0.0936586
$343$	17.8564	0.964155
$344$	7.26795	0.391862
$345$	3.46410	0.186501
$346$	−12.0000	−0.645124
$347$	−0.928203	−0.0498286	−0.0249143	−	0.999690i	$-0.507931\pi$
$347$	−0.928203	−0.0498286	−0.0249143	+	0.999690i	$0.507931\pi$
$348$	8.19615	0.439360
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	4.73205	0.252939
$351$	0.732051	0.0390740
$352$	24.5885	1.31057
$353$	−14.7846	−0.786905	−0.393453	−	0.919345i	$-0.628719\pi$
$353$	−14.7846	−0.786905	−0.393453	+	0.919345i	$0.628719\pi$
$354$	−4.39230	−0.233448
$355$	−4.39230	−0.233119
$356$	10.7321	0.568798
$357$	0	0
$358$	−40.3923	−2.13480
$359$	0.339746	0.0179311	0.00896555	−	0.999960i	$-0.497146\pi$
$359$	0.339746	0.0179311	0.00896555	+	0.999960i	$0.497146\pi$
$360$	1.73205	0.0912871
$361$	1.00000	0.0526316
$362$	4.14359	0.217782
$363$	11.3923	0.597941
$364$	−2.00000	−0.104828
$365$	−16.9282	−0.886063
$366$	11.3205	0.591732
$367$	16.1962	0.845432	0.422716	−	0.906262i	$-0.361077\pi$
$367$	16.1962	0.845432	0.422716	+	0.906262i	$0.361077\pi$
$368$	−17.3205	−0.902894
$369$	1.26795	0.0660068
$370$	10.7321	0.557933
$371$	25.8564	1.34240
$372$	8.92820	0.462906
$373$	−6.19615	−0.320825	−0.160412	−	0.987050i	$-0.551282\pi$
$373$	−6.19615	−0.320825	−0.160412	+	0.987050i	$0.551282\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−6.00000	−0.309426
$377$	6.00000	0.309016
$378$	4.73205	0.243390
$379$	20.9282	1.07501	0.537505	−	0.843261i	$-0.319367\pi$
$379$	20.9282	1.07501	0.537505	+	0.843261i	$0.319367\pi$
$380$	1.00000	0.0512989
$381$	−4.00000	−0.204926
$382$	−0.588457	−0.0301081
$383$	−17.0718	−0.872328	−0.436164	−	0.899867i	$-0.643663\pi$
$383$	−17.0718	−0.872328	−0.436164	+	0.899867i	$0.643663\pi$
$384$	−12.1244	−0.618718
$385$	−12.9282	−0.658882
$386$	−29.6603	−1.50967
$387$	4.19615	0.213302
$388$	−6.19615	−0.314562
$389$	7.85641	0.398336	0.199168	−	0.979965i	$-0.436176\pi$
$389$	7.85641	0.398336	0.199168	+	0.979965i	$0.436176\pi$
$390$	−1.26795	−0.0642051
$391$	0	0
$392$	0.803848	0.0406004
$393$	−9.12436	−0.460263
$394$	41.5692	2.09423
$395$	−10.9282	−0.549858
$396$	4.73205	0.237795
$397$	8.92820	0.448094	0.224047	−	0.974578i	$-0.428073\pi$
$397$	8.92820	0.448094	0.224047	+	0.974578i	$0.428073\pi$
$398$	26.5359	1.33012
$399$	−2.73205	−0.136774
$400$	−5.00000	−0.250000
$401$	−34.0526	−1.70050	−0.850252	−	0.526376i	$-0.823550\pi$
$401$	−34.0526	−1.70050	−0.850252	+	0.526376i	$0.823550\pi$
$402$	−13.8564	−0.691095
$403$	6.53590	0.325576
$404$	−10.3923	−0.517036
$405$	1.00000	0.0496904
$406$	38.7846	1.92485
$407$	−29.3205	−1.45336
$408$	0	0
$409$	−26.3923	−1.30502	−0.652508	−	0.757782i	$-0.726283\pi$
$409$	−26.3923	−1.30502	−0.652508	+	0.757782i	$0.726283\pi$
$410$	−2.19615	−0.108460
$411$	19.8564	0.979444
$412$	9.85641	0.485590
$413$	−6.92820	−0.340915
$414$	−6.00000	−0.294884
$415$	−12.9282	−0.634621
$416$	3.80385	0.186499
$417$	−8.39230	−0.410973
$418$	−8.19615	−0.400887
$419$	−28.0526	−1.37046	−0.685229	−	0.728328i	$-0.740298\pi$
$419$	−28.0526	−1.37046	−0.685229	+	0.728328i	$0.740298\pi$
$420$	−2.73205	−0.133310
$421$	−18.7846	−0.915506	−0.457753	−	0.889079i	$-0.651346\pi$
$421$	−18.7846	−0.915506	−0.457753	+	0.889079i	$0.651346\pi$
$422$	−1.85641	−0.0903685
$423$	−3.46410	−0.168430
$424$	−16.3923	−0.796081
$425$	0	0
$426$	7.60770	0.368594
$427$	17.8564	0.864132
$428$	0	0
$429$	3.46410	0.167248
$430$	−7.26795	−0.350492
$431$	−11.3205	−0.545290	−0.272645	−	0.962115i	$-0.587898\pi$
$431$	−11.3205	−0.545290	−0.272645	+	0.962115i	$0.587898\pi$
$432$	−5.00000	−0.240563
$433$	−10.5885	−0.508849	−0.254424	−	0.967093i	$-0.581886\pi$
$433$	−10.5885	−0.508849	−0.254424	+	0.967093i	$0.581886\pi$
$434$	42.2487	2.02800
$435$	8.19615	0.392975
$436$	−14.3923	−0.689266
$437$	3.46410	0.165710
$438$	29.3205	1.40099
$439$	26.9282	1.28521	0.642607	−	0.766196i	$-0.277853\pi$
$439$	26.9282	1.28521	0.642607	+	0.766196i	$0.277853\pi$
$440$	8.19615	0.390736
$441$	0.464102	0.0221001
$442$	0	0
$443$	−5.32051	−0.252785	−0.126392	−	0.991980i	$-0.540340\pi$
$443$	−5.32051	−0.252785	−0.126392	+	0.991980i	$0.540340\pi$
$444$	−6.19615	−0.294056
$445$	10.7321	0.508748
$446$	30.9282	1.46449
$447$	−19.8564	−0.939176
$448$	−2.73205	−0.129077
$449$	−5.66025	−0.267124	−0.133562	−	0.991040i	$-0.542642\pi$
$449$	−5.66025	−0.267124	−0.133562	+	0.991040i	$0.542642\pi$
$450$	−1.73205	−0.0816497
$451$	6.00000	0.282529
$452$	18.9282	0.890308
$453$	14.0000	0.657777
$454$	−18.0000	−0.844782
$455$	−2.00000	−0.0937614
$456$	1.73205	0.0811107
$457$	4.53590	0.212180	0.106090	−	0.994357i	$-0.466167\pi$
$457$	4.53590	0.212180	0.106090	+	0.994357i	$0.466167\pi$
$458$	32.1051	1.50017
$459$	0	0
$460$	3.46410	0.161515
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	22.3923	1.04178
$463$	−35.5167	−1.65060	−0.825300	−	0.564695i	$-0.808994\pi$
$463$	−35.5167	−1.65060	−0.825300	+	0.564695i	$0.808994\pi$
$464$	−40.9808	−1.90248
$465$	8.92820	0.414036
$466$	13.6077	0.630364
$467$	20.5359	0.950288	0.475144	−	0.879908i	$-0.342396\pi$
$467$	20.5359	0.950288	0.475144	+	0.879908i	$0.342396\pi$
$468$	0.732051	0.0338391
$469$	−21.8564	−1.00924
$470$	6.00000	0.276759
$471$	6.39230	0.294542
$472$	4.39230	0.202172
$473$	19.8564	0.912999
$474$	18.9282	0.869401
$475$	1.00000	0.0458831
$476$	0	0
$477$	−9.46410	−0.433331
$478$	16.9808	0.776682
$479$	25.5167	1.16589	0.582943	−	0.812513i	$-0.301901\pi$
$479$	25.5167	1.16589	0.582943	+	0.812513i	$0.301901\pi$
$480$	5.19615	0.237171
$481$	−4.53590	−0.206819
$482$	5.32051	0.242343
$483$	−9.46410	−0.430632
$484$	11.3923	0.517832
$485$	−6.19615	−0.281353
$486$	−1.73205	−0.0785674
$487$	−32.3923	−1.46784	−0.733918	−	0.679238i	$-0.762310\pi$
$487$	−32.3923	−1.46784	−0.733918	+	0.679238i	$0.762310\pi$
$488$	−11.3205	−0.512455
$489$	9.26795	0.419111
$490$	−0.803848	−0.0363141
$491$	16.0526	0.724442	0.362221	−	0.932092i	$-0.382019\pi$
$491$	16.0526	0.724442	0.362221	+	0.932092i	$0.382019\pi$
$492$	1.26795	0.0571636
$493$	0	0
$494$	−1.26795	−0.0570477
$495$	4.73205	0.212690
$496$	−44.6410	−2.00444
$497$	12.0000	0.538274
$498$	22.3923	1.00342
$499$	10.5359	0.471652	0.235826	−	0.971795i	$-0.424221\pi$
$499$	10.5359	0.471652	0.235826	+	0.971795i	$0.424221\pi$
$500$	1.00000	0.0447214
$501$	3.46410	0.154765
$502$	−48.5885	−2.16861
$503$	−23.0718	−1.02872	−0.514360	−	0.857574i	$-0.671971\pi$
$503$	−23.0718	−1.02872	−0.514360	+	0.857574i	$0.671971\pi$
$504$	−4.73205	−0.210782
$505$	−10.3923	−0.462451
$506$	−28.3923	−1.26219
$507$	−12.4641	−0.553550
$508$	−4.00000	−0.177471
$509$	−10.0526	−0.445572	−0.222786	−	0.974867i	$-0.571515\pi$
$509$	−10.0526	−0.445572	−0.222786	+	0.974867i	$0.571515\pi$
$510$	0	0
$511$	46.2487	2.04592
$512$	−8.66025	−0.382733
$513$	1.00000	0.0441511
$514$	41.5692	1.83354
$515$	9.85641	0.434325
$516$	4.19615	0.184725
$517$	−16.3923	−0.720933
$518$	−29.3205	−1.28827
$519$	6.92820	0.304114
$520$	1.26795	0.0556033
$521$	37.2679	1.63274	0.816369	−	0.577530i	$-0.195983\pi$
$521$	37.2679	1.63274	0.816369	+	0.577530i	$0.195983\pi$
$522$	−14.1962	−0.621349
$523$	8.67949	0.379528	0.189764	−	0.981830i	$-0.439228\pi$
$523$	8.67949	0.379528	0.189764	+	0.981830i	$0.439228\pi$
$524$	−9.12436	−0.398599
$525$	−2.73205	−0.119236
$526$	−10.3923	−0.453126
$527$	0	0
$528$	−23.6603	−1.02968
$529$	−11.0000	−0.478261
$530$	16.3923	0.712036
$531$	2.53590	0.110049
$532$	−2.73205	−0.118449
$533$	0.928203	0.0402049
$534$	−18.5885	−0.804401
$535$	0	0
$536$	13.8564	0.598506
$537$	23.3205	1.00635
$538$	−1.01924	−0.0439425
$539$	2.19615	0.0945950
$540$	1.00000	0.0430331
$541$	41.7128	1.79337	0.896687	−	0.442665i	$-0.145967\pi$
$541$	41.7128	1.79337	0.896687	+	0.442665i	$0.145967\pi$
$542$	−0.679492	−0.0291867
$543$	−2.39230	−0.102664
$544$	0	0
$545$	−14.3923	−0.616499
$546$	3.46410	0.148250
$547$	43.3205	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$547$	43.3205	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$548$	19.8564	0.848224
$549$	−6.53590	−0.278945
$550$	−8.19615	−0.349485
$551$	8.19615	0.349168
$552$	6.00000	0.255377
$553$	29.8564	1.26962
$554$	−3.46410	−0.147176
$555$	−6.19615	−0.263012
$556$	−8.39230	−0.355913
$557$	0.928203	0.0393292	0.0196646	−	0.999807i	$-0.493740\pi$
$557$	0.928203	0.0393292	0.0196646	+	0.999807i	$0.493740\pi$
$558$	−15.4641	−0.654648
$559$	3.07180	0.129923
$560$	13.6603	0.577251
$561$	0	0
$562$	−2.19615	−0.0926391
$563$	−27.4641	−1.15747	−0.578737	−	0.815514i	$-0.696454\pi$
$563$	−27.4641	−1.15747	−0.578737	+	0.815514i	$0.696454\pi$
$564$	−3.46410	−0.145865
$565$	18.9282	0.796315
$566$	−43.2679	−1.81869
$567$	−2.73205	−0.114735
$568$	−7.60770	−0.319212
$569$	22.0526	0.924491	0.462246	−	0.886752i	$-0.347044\pi$
$569$	22.0526	0.924491	0.462246	+	0.886752i	$0.347044\pi$
$570$	−1.73205	−0.0725476
$571$	−34.2487	−1.43326	−0.716632	−	0.697452i	$-0.754317\pi$
$571$	−34.2487	−1.43326	−0.716632	+	0.697452i	$0.754317\pi$
$572$	3.46410	0.144841
$573$	0.339746	0.0141931
$574$	6.00000	0.250435
$575$	3.46410	0.144463
$576$	1.00000	0.0416667
$577$	15.1769	0.631823	0.315912	−	0.948789i	$-0.397690\pi$
$577$	15.1769	0.631823	0.315912	+	0.948789i	$0.397690\pi$
$578$	29.4449	1.22474
$579$	17.1244	0.711664
$580$	8.19615	0.340327
$581$	35.3205	1.46534
$582$	10.7321	0.444858
$583$	−44.7846	−1.85479
$584$	−29.3205	−1.21329
$585$	0.732051	0.0302666
$586$	−48.0000	−1.98286
$587$	−3.46410	−0.142979	−0.0714894	−	0.997441i	$-0.522775\pi$
$587$	−3.46410	−0.142979	−0.0714894	+	0.997441i	$0.522775\pi$
$588$	0.464102	0.0191392
$589$	8.92820	0.367880
$590$	−4.39230	−0.180828
$591$	−24.0000	−0.987228
$592$	30.9808	1.27330
$593$	−38.7846	−1.59269	−0.796347	−	0.604841i	$-0.793237\pi$
$593$	−38.7846	−1.59269	−0.796347	+	0.604841i	$0.793237\pi$
$594$	−8.19615	−0.336292
$595$	0	0
$596$	−19.8564	−0.813350
$597$	−15.3205	−0.627027
$598$	−4.39230	−0.179615
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	1.73205	0.0707107
$601$	−47.1769	−1.92439	−0.962193	−	0.272368i	$-0.912193\pi$
$601$	−47.1769	−1.92439	−0.962193	+	0.272368i	$0.912193\pi$
$602$	19.8564	0.809287
$603$	8.00000	0.325785
$604$	14.0000	0.569652
$605$	11.3923	0.463163
$606$	18.0000	0.731200
$607$	−11.6077	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$607$	−11.6077	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$608$	5.19615	0.210732
$609$	−22.3923	−0.907382
$610$	11.3205	0.458354
$611$	−2.53590	−0.102591
$612$	0	0
$613$	42.3923	1.71221	0.856105	−	0.516803i	$-0.172878\pi$
$613$	42.3923	1.71221	0.856105	+	0.516803i	$0.172878\pi$
$614$	56.1051	2.26422
$615$	1.26795	0.0511286
$616$	−22.3923	−0.902212
$617$	27.7128	1.11568	0.557838	−	0.829950i	$-0.311631\pi$
$617$	27.7128	1.11568	0.557838	+	0.829950i	$0.311631\pi$
$618$	−17.0718	−0.686728
$619$	−15.3205	−0.615783	−0.307892	−	0.951421i	$-0.599623\pi$
$619$	−15.3205	−0.615783	−0.307892	+	0.951421i	$0.599623\pi$
$620$	8.92820	0.358565
$621$	3.46410	0.139010
$622$	−56.1962	−2.25326
$623$	−29.3205	−1.17470
$624$	−3.66025	−0.146527
$625$	1.00000	0.0400000
$626$	−11.0718	−0.442518
$627$	4.73205	0.188980
$628$	6.39230	0.255081
$629$	0	0
$630$	4.73205	0.188529
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	−18.9282	−0.752923
$633$	1.07180	0.0426001
$634$	19.6077	0.778721
$635$	−4.00000	−0.158735
$636$	−9.46410	−0.375276
$637$	0.339746	0.0134612
$638$	−67.1769	−2.65956
$639$	−4.39230	−0.173757
$640$	−12.1244	−0.479257
$641$	48.5885	1.91913	0.959564	−	0.281489i	$-0.0908284\pi$
$641$	48.5885	1.91913	0.959564	+	0.281489i	$0.0908284\pi$
$642$	0	0
$643$	−12.1962	−0.480969	−0.240485	−	0.970653i	$-0.577306\pi$
$643$	−12.1962	−0.480969	−0.240485	+	0.970653i	$0.577306\pi$
$644$	−9.46410	−0.372938
$645$	4.19615	0.165223
$646$	0	0
$647$	−4.14359	−0.162901	−0.0814507	−	0.996677i	$-0.525955\pi$
$647$	−4.14359	−0.162901	−0.0814507	+	0.996677i	$0.525955\pi$
$648$	1.73205	0.0680414
$649$	12.0000	0.471041
$650$	−1.26795	−0.0497331
$651$	−24.3923	−0.956010
$652$	9.26795	0.362961
$653$	17.0718	0.668071	0.334036	−	0.942560i	$-0.391589\pi$
$653$	17.0718	0.668071	0.334036	+	0.942560i	$0.391589\pi$
$654$	24.9282	0.974770
$655$	−9.12436	−0.356518
$656$	−6.33975	−0.247525
$657$	−16.9282	−0.660432
$658$	−16.3923	−0.639039
$659$	5.07180	0.197569	0.0987846	−	0.995109i	$-0.468505\pi$
$659$	5.07180	0.197569	0.0987846	+	0.995109i	$0.468505\pi$
$660$	4.73205	0.184195
$661$	39.1769	1.52381	0.761903	−	0.647692i	$-0.224265\pi$
$661$	39.1769	1.52381	0.761903	+	0.647692i	$0.224265\pi$
$662$	44.5359	1.73094
$663$	0	0
$664$	−22.3923	−0.868990
$665$	−2.73205	−0.105944
$666$	10.7321	0.415859
$667$	28.3923	1.09935
$668$	3.46410	0.134030
$669$	−17.8564	−0.690369
$670$	−13.8564	−0.535320
$671$	−30.9282	−1.19397
$672$	−14.1962	−0.547628
$673$	17.1244	0.660095	0.330048	−	0.943964i	$-0.392935\pi$
$673$	17.1244	0.660095	0.330048	+	0.943964i	$0.392935\pi$
$674$	−8.87564	−0.341877
$675$	1.00000	0.0384900
$676$	−12.4641	−0.479389
$677$	0.679492	0.0261150	0.0130575	−	0.999915i	$-0.495844\pi$
$677$	0.679492	0.0261150	0.0130575	+	0.999915i	$0.495844\pi$
$678$	−32.7846	−1.25909
$679$	16.9282	0.649645
$680$	0	0
$681$	10.3923	0.398234
$682$	−73.1769	−2.80209
$683$	−5.07180	−0.194067	−0.0970335	−	0.995281i	$-0.530935\pi$
$683$	−5.07180	−0.194067	−0.0970335	+	0.995281i	$0.530935\pi$
$684$	1.00000	0.0382360
$685$	19.8564	0.758674
$686$	−30.9282	−1.18084
$687$	−18.5359	−0.707189
$688$	−20.9808	−0.799884
$689$	−6.92820	−0.263944
$690$	−6.00000	−0.228416
$691$	12.3923	0.471425	0.235713	−	0.971823i	$-0.424258\pi$
$691$	12.3923	0.471425	0.235713	+	0.971823i	$0.424258\pi$
$692$	6.92820	0.263371
$693$	−12.9282	−0.491102
$694$	1.60770	0.0610273
$695$	−8.39230	−0.318338
$696$	14.1962	0.538104
$697$	0	0
$698$	38.1051	1.44230
$699$	−7.85641	−0.297157
$700$	−2.73205	−0.103262
$701$	−33.7128	−1.27332	−0.636658	−	0.771147i	$-0.719684\pi$
$701$	−33.7128	−1.27332	−0.636658	+	0.771147i	$0.719684\pi$
$702$	−1.26795	−0.0478557
$703$	−6.19615	−0.233692
$704$	4.73205	0.178346
$705$	−3.46410	−0.130466
$706$	25.6077	0.963758
$707$	28.3923	1.06780
$708$	2.53590	0.0953049
$709$	−29.1769	−1.09576	−0.547881	−	0.836556i	$-0.684565\pi$
$709$	−29.1769	−1.09576	−0.547881	+	0.836556i	$0.684565\pi$
$710$	7.60770	0.285512
$711$	−10.9282	−0.409840
$712$	18.5885	0.696632
$713$	30.9282	1.15827
$714$	0	0
$715$	3.46410	0.129550
$716$	23.3205	0.871528
$717$	−9.80385	−0.366131
$718$	−0.588457	−0.0219610
$719$	−11.6603	−0.434854	−0.217427	−	0.976077i	$-0.569766\pi$
$719$	−11.6603	−0.434854	−0.217427	+	0.976077i	$0.569766\pi$
$720$	−5.00000	−0.186339
$721$	−26.9282	−1.00286
$722$	−1.73205	−0.0644603
$723$	−3.07180	−0.114241
$724$	−2.39230	−0.0889093
$725$	8.19615	0.304397
$726$	−19.7321	−0.732325
$727$	25.6603	0.951686	0.475843	−	0.879530i	$-0.342143\pi$
$727$	25.6603	0.951686	0.475843	+	0.879530i	$0.342143\pi$
$728$	−3.46410	−0.128388
$729$	1.00000	0.0370370
$730$	29.3205	1.08520
$731$	0	0
$732$	−6.53590	−0.241574
$733$	−18.7846	−0.693825	−0.346913	−	0.937897i	$-0.612770\pi$
$733$	−18.7846	−0.693825	−0.346913	+	0.937897i	$0.612770\pi$
$734$	−28.0526	−1.03544
$735$	0.464102	0.0171186
$736$	18.0000	0.663489
$737$	37.8564	1.39446
$738$	−2.19615	−0.0808415
$739$	6.14359	0.225996	0.112998	−	0.993595i	$-0.463955\pi$
$739$	6.14359	0.225996	0.112998	+	0.993595i	$0.463955\pi$
$740$	−6.19615	−0.227775
$741$	0.732051	0.0268926
$742$	−44.7846	−1.64409
$743$	−3.21539	−0.117961	−0.0589806	−	0.998259i	$-0.518785\pi$
$743$	−3.21539	−0.117961	−0.0589806	+	0.998259i	$0.518785\pi$
$744$	15.4641	0.566941
$745$	−19.8564	−0.727482
$746$	10.7321	0.392928
$747$	−12.9282	−0.473018
$748$	0	0
$749$	0	0
$750$	−1.73205	−0.0632456
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	17.3205	0.631614
$753$	28.0526	1.02229
$754$	−10.3923	−0.378465
$755$	14.0000	0.509512
$756$	−2.73205	−0.0993637
$757$	32.2487	1.17210	0.586050	−	0.810275i	$-0.300682\pi$
$757$	32.2487	1.17210	0.586050	+	0.810275i	$0.300682\pi$
$758$	−36.2487	−1.31661
$759$	16.3923	0.595003
$760$	1.73205	0.0628281
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	6.92820	0.250982
$763$	39.3205	1.42350
$764$	0.339746	0.0122916
$765$	0	0
$766$	29.5692	1.06838
$767$	1.85641	0.0670310
$768$	19.0000	0.685603
$769$	−20.6410	−0.744334	−0.372167	−	0.928166i	$-0.621385\pi$
$769$	−20.6410	−0.744334	−0.372167	+	0.928166i	$0.621385\pi$
$770$	22.3923	0.806963
$771$	−24.0000	−0.864339
$772$	17.1244	0.616319
$773$	−25.1769	−0.905551	−0.452775	−	0.891625i	$-0.649566\pi$
$773$	−25.1769	−0.905551	−0.452775	+	0.891625i	$0.649566\pi$
$774$	−7.26795	−0.261241
$775$	8.92820	0.320711
$776$	−10.7321	−0.385258
$777$	16.9282	0.607296
$778$	−13.6077	−0.487860
$779$	1.26795	0.0454290
$780$	0.732051	0.0262116
$781$	−20.7846	−0.743732
$782$	0	0
$783$	8.19615	0.292907
$784$	−2.32051	−0.0828753
$785$	6.39230	0.228151
$786$	15.8038	0.563705
$787$	8.67949	0.309390	0.154695	−	0.987962i	$-0.450560\pi$
$787$	8.67949	0.309390	0.154695	+	0.987962i	$0.450560\pi$
$788$	−24.0000	−0.854965
$789$	6.00000	0.213606
$790$	18.9282	0.673435
$791$	−51.7128	−1.83870
$792$	8.19615	0.291238
$793$	−4.78461	−0.169906
$794$	−15.4641	−0.548800
$795$	−9.46410	−0.335657
$796$	−15.3205	−0.543021
$797$	44.7846	1.58635	0.793176	−	0.608992i	$-0.208426\pi$
$797$	44.7846	1.58635	0.793176	+	0.608992i	$0.208426\pi$
$798$	4.73205	0.167513
$799$	0	0
$800$	5.19615	0.183712
$801$	10.7321	0.379198
$802$	58.9808	2.08268
$803$	−80.1051	−2.82685
$804$	8.00000	0.282138
$805$	−9.46410	−0.333566
$806$	−11.3205	−0.398748
$807$	0.588457	0.0207147
$808$	−18.0000	−0.633238
$809$	14.7846	0.519799	0.259900	−	0.965636i	$-0.416311\pi$
$809$	14.7846	0.519799	0.259900	+	0.965636i	$0.416311\pi$
$810$	−1.73205	−0.0608581
$811$	37.5692	1.31923	0.659617	−	0.751602i	$-0.270719\pi$
$811$	37.5692	1.31923	0.659617	+	0.751602i	$0.270719\pi$
$812$	−22.3923	−0.785816
$813$	0.392305	0.0137587
$814$	50.7846	1.78000
$815$	9.26795	0.324642
$816$	0	0
$817$	4.19615	0.146805
$818$	45.7128	1.59831
$819$	−2.00000	−0.0698857
$820$	1.26795	0.0442787
$821$	−32.5359	−1.13551	−0.567755	−	0.823197i	$-0.692188\pi$
$821$	−32.5359	−1.13551	−0.567755	+	0.823197i	$0.692188\pi$
$822$	−34.3923	−1.19957
$823$	12.9808	0.452481	0.226240	−	0.974071i	$-0.427356\pi$
$823$	12.9808	0.452481	0.226240	+	0.974071i	$0.427356\pi$
$824$	17.0718	0.594724
$825$	4.73205	0.164749
$826$	12.0000	0.417533
$827$	−5.32051	−0.185012	−0.0925061	−	0.995712i	$-0.529488\pi$
$827$	−5.32051	−0.185012	−0.0925061	+	0.995712i	$0.529488\pi$
$828$	3.46410	0.120386
$829$	34.1051	1.18452	0.592260	−	0.805747i	$-0.298236\pi$
$829$	34.1051	1.18452	0.592260	+	0.805747i	$0.298236\pi$
$830$	22.3923	0.777248
$831$	2.00000	0.0693792
$832$	0.732051	0.0253793
$833$	0	0
$834$	14.5359	0.503337
$835$	3.46410	0.119880
$836$	4.73205	0.163661
$837$	8.92820	0.308604
$838$	48.5885	1.67846
$839$	−19.6077	−0.676933	−0.338466	−	0.940978i	$-0.609908\pi$
$839$	−19.6077	−0.676933	−0.338466	+	0.940978i	$0.609908\pi$
$840$	−4.73205	−0.163271
$841$	38.1769	1.31645
$842$	32.5359	1.12126
$843$	1.26795	0.0436705
$844$	1.07180	0.0368928
$845$	−12.4641	−0.428778
$846$	6.00000	0.206284
$847$	−31.1244	−1.06945
$848$	47.3205	1.62499
$849$	24.9808	0.857338
$850$	0	0
$851$	−21.4641	−0.735780
$852$	−4.39230	−0.150478
$853$	27.1769	0.930520	0.465260	−	0.885174i	$-0.345961\pi$
$853$	27.1769	0.930520	0.465260	+	0.885174i	$0.345961\pi$
$854$	−30.9282	−1.05834
$855$	1.00000	0.0341993
$856$	0	0
$857$	−42.2487	−1.44319	−0.721594	−	0.692316i	$-0.756590\pi$
$857$	−42.2487	−1.44319	−0.721594	+	0.692316i	$0.756590\pi$
$858$	−6.00000	−0.204837
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	4.19615	0.143088
$861$	−3.46410	−0.118056
$862$	19.6077	0.667841
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	5.19615	0.176777
$865$	6.92820	0.235566
$866$	18.3397	0.623210
$867$	−17.0000	−0.577350
$868$	−24.3923	−0.827929
$869$	−51.7128	−1.75424
$870$	−14.1962	−0.481295
$871$	5.85641	0.198437
$872$	−24.9282	−0.844175
$873$	−6.19615	−0.209708
$874$	−6.00000	−0.202953
$875$	−2.73205	−0.0923602
$876$	−16.9282	−0.571951
$877$	53.1244	1.79388	0.896941	−	0.442150i	$-0.145784\pi$
$877$	53.1244	1.79388	0.896941	+	0.442150i	$0.145784\pi$
$878$	−46.6410	−1.57406
$879$	27.7128	0.934730
$880$	−23.6603	−0.797587
$881$	8.53590	0.287582	0.143791	−	0.989608i	$-0.454071\pi$
$881$	8.53590	0.287582	0.143791	+	0.989608i	$0.454071\pi$
$882$	−0.803848	−0.0270670
$883$	36.9808	1.24450	0.622251	−	0.782818i	$-0.286218\pi$
$883$	36.9808	1.24450	0.622251	+	0.782818i	$0.286218\pi$
$884$	0	0
$885$	2.53590	0.0852433
$886$	9.21539	0.309597
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−10.7321	−0.360144
$889$	10.9282	0.366520
$890$	−18.5885	−0.623087
$891$	4.73205	0.158530
$892$	−17.8564	−0.597877
$893$	−3.46410	−0.115922
$894$	34.3923	1.15025
$895$	23.3205	0.779519
$896$	33.1244	1.10661
$897$	2.53590	0.0846712
$898$	9.80385	0.327159
$899$	73.1769	2.44059
$900$	1.00000	0.0333333
$901$	0	0
$902$	−10.3923	−0.346026
$903$	−11.4641	−0.381501
$904$	32.7846	1.09040
$905$	−2.39230	−0.0795229
$906$	−24.2487	−0.805609
$907$	−32.3923	−1.07557	−0.537784	−	0.843082i	$-0.680739\pi$
$907$	−32.3923	−1.07557	−0.537784	+	0.843082i	$0.680739\pi$
$908$	10.3923	0.344881
$909$	−10.3923	−0.344691
$910$	3.46410	0.114834
$911$	−54.9282	−1.81985	−0.909926	−	0.414770i	$-0.863862\pi$
$911$	−54.9282	−1.81985	−0.909926	+	0.414770i	$0.863862\pi$
$912$	−5.00000	−0.165567
$913$	−61.1769	−2.02466
$914$	−7.85641	−0.259867
$915$	−6.53590	−0.216070
$916$	−18.5359	−0.612443
$917$	24.9282	0.823202
$918$	0	0
$919$	51.4256	1.69637	0.848187	−	0.529696i	$-0.177694\pi$
$919$	51.4256	1.69637	0.848187	+	0.529696i	$0.177694\pi$
$920$	6.00000	0.197814
$921$	−32.3923	−1.06736
$922$	−10.3923	−0.342252
$923$	−3.21539	−0.105836
$924$	−12.9282	−0.425307
$925$	−6.19615	−0.203728
$926$	61.5167	2.02156
$927$	9.85641	0.323727
$928$	42.5885	1.39803
$929$	−1.60770	−0.0527468	−0.0263734	−	0.999652i	$-0.508396\pi$
$929$	−1.60770	−0.0527468	−0.0263734	+	0.999652i	$0.508396\pi$
$930$	−15.4641	−0.507088
$931$	0.464102	0.0152103
$932$	−7.85641	−0.257345
$933$	32.4449	1.06220
$934$	−35.5692	−1.16386
$935$	0	0
$936$	1.26795	0.0414442
$937$	−16.2487	−0.530822	−0.265411	−	0.964135i	$-0.585508\pi$
$937$	−16.2487	−0.530822	−0.265411	+	0.964135i	$0.585508\pi$
$938$	37.8564	1.23606
$939$	6.39230	0.208605
$940$	−3.46410	−0.112987
$941$	0.588457	0.0191832	0.00959158	−	0.999954i	$-0.496947\pi$
$941$	0.588457	0.0191832	0.00959158	+	0.999954i	$0.496947\pi$
$942$	−11.0718	−0.360739
$943$	4.39230	0.143033
$944$	−12.6795	−0.412682
$945$	−2.73205	−0.0888736
$946$	−34.3923	−1.11819
$947$	28.1436	0.914544	0.457272	−	0.889327i	$-0.348827\pi$
$947$	28.1436	0.914544	0.457272	+	0.889327i	$0.348827\pi$
$948$	−10.9282	−0.354932
$949$	−12.3923	−0.402271
$950$	−1.73205	−0.0561951
$951$	−11.3205	−0.367093
$952$	0	0
$953$	37.8564	1.22629	0.613145	−	0.789971i	$-0.289904\pi$
$953$	37.8564	1.22629	0.613145	+	0.789971i	$0.289904\pi$
$954$	16.3923	0.530720
$955$	0.339746	0.0109939
$956$	−9.80385	−0.317079
$957$	38.7846	1.25373
$958$	−44.1962	−1.42791
$959$	−54.2487	−1.75178
$960$	1.00000	0.0322749
$961$	48.7128	1.57138
$962$	7.85641	0.253301
$963$	0	0
$964$	−3.07180	−0.0989359
$965$	17.1244	0.551253
$966$	16.3923	0.527414
$967$	4.87564	0.156790	0.0783951	−	0.996922i	$-0.475020\pi$
$967$	4.87564	0.156790	0.0783951	+	0.996922i	$0.475020\pi$
$968$	19.7321	0.634212
$969$	0	0
$970$	10.7321	0.344585
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	1.00000	0.0320750
$973$	22.9282	0.735044
$974$	56.1051	1.79772
$975$	0.732051	0.0234444
$976$	32.6795	1.04605
$977$	−39.0333	−1.24879	−0.624393	−	0.781110i	$-0.714654\pi$
$977$	−39.0333	−1.24879	−0.624393	+	0.781110i	$0.714654\pi$
$978$	−16.0526	−0.513304
$979$	50.7846	1.62308
$980$	0.464102	0.0148252
$981$	−14.3923	−0.459511
$982$	−27.8038	−0.887256
$983$	−41.3205	−1.31792	−0.658960	−	0.752178i	$-0.729003\pi$
$983$	−41.3205	−1.31792	−0.658960	+	0.752178i	$0.729003\pi$
$984$	2.19615	0.0700108
$985$	−24.0000	−0.764704
$986$	0	0
$987$	9.46410	0.301246
$988$	0.732051	0.0232896
$989$	14.5359	0.462215
$990$	−8.19615	−0.260491
$991$	13.0718	0.415239	0.207620	−	0.978210i	$-0.433428\pi$
$991$	13.0718	0.415239	0.207620	+	0.978210i	$0.433428\pi$
$992$	46.3923	1.47296
$993$	−25.7128	−0.815971
$994$	−20.7846	−0.659248
$995$	−15.3205	−0.485693
$996$	−12.9282	−0.409646
$997$	−17.6077	−0.557641	−0.278821	−	0.960343i	$-0.589944\pi$
$997$	−17.6077	−0.557641	−0.278821	+	0.960343i	$0.589944\pi$
$998$	−18.2487	−0.577653
$999$	−6.19615	−0.196038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	285.2.a.e.1.1	✓	2
3.2	odd	2		855.2.a.f.1.2		2
4.3	odd	2		4560.2.a.bh.1.2		2
5.2	odd	4		1425.2.c.k.799.1		4
5.3	odd	4		1425.2.c.k.799.4		4
5.4	even	2		1425.2.a.o.1.2		2
15.14	odd	2		4275.2.a.t.1.1		2
19.18	odd	2		5415.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.1	✓	2	1.1	even	1	trivial
855.2.a.f.1.2		2	3.2	odd	2
1425.2.a.o.1.2		2	5.4	even	2
1425.2.c.k.799.1		4	5.2	odd	4
1425.2.c.k.799.4		4	5.3	odd	4
4275.2.a.t.1.1		2	15.14	odd	2
4560.2.a.bh.1.2		2	4.3	odd	2
5415.2.a.r.1.2		2	19.18	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:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -2.73205 q^{7} +1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} +4.73205 q^{11} +1.00000 q^{12} +0.732051 q^{13} +4.73205 q^{14} +1.00000 q^{15} -5.00000 q^{16} -1.73205 q^{18} +1.00000 q^{19} +1.00000 q^{20} -2.73205 q^{21} -8.19615 q^{22} +3.46410 q^{23} +1.73205 q^{24} +1.00000 q^{25} -1.26795 q^{26} +1.00000 q^{27} -2.73205 q^{28} +8.19615 q^{29} -1.73205 q^{30} +8.92820 q^{31} +5.19615 q^{32} +4.73205 q^{33} -2.73205 q^{35} +1.00000 q^{36} -6.19615 q^{37} -1.73205 q^{38} +0.732051 q^{39} +1.73205 q^{40} +1.26795 q^{41} +4.73205 q^{42} +4.19615 q^{43} +4.73205 q^{44} +1.00000 q^{45} -6.00000 q^{46} -3.46410 q^{47} -5.00000 q^{48} +0.464102 q^{49} -1.73205 q^{50} +0.732051 q^{52} -9.46410 q^{53} -1.73205 q^{54} +4.73205 q^{55} -4.73205 q^{56} +1.00000 q^{57} -14.1962 q^{58} +2.53590 q^{59} +1.00000 q^{60} -6.53590 q^{61} -15.4641 q^{62} -2.73205 q^{63} +1.00000 q^{64} +0.732051 q^{65} -8.19615 q^{66} +8.00000 q^{67} +3.46410 q^{69} +4.73205 q^{70} -4.39230 q^{71} +1.73205 q^{72} -16.9282 q^{73} +10.7321 q^{74} +1.00000 q^{75} +1.00000 q^{76} -12.9282 q^{77} -1.26795 q^{78} -10.9282 q^{79} -5.00000 q^{80} +1.00000 q^{81} -2.19615 q^{82} -12.9282 q^{83} -2.73205 q^{84} -7.26795 q^{86} +8.19615 q^{87} +8.19615 q^{88} +10.7321 q^{89} -1.73205 q^{90} -2.00000 q^{91} +3.46410 q^{92} +8.92820 q^{93} +6.00000 q^{94} +1.00000 q^{95} +5.19615 q^{96} -6.19615 q^{97} -0.803848 q^{98} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} + 2 q^{19} + 2 q^{20} - 2 q^{21} - 6 q^{22} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 2 q^{28}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 2 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 + 2 * q^19 + 2 * q^20 - 2 * q^21 - 6 * q^22 + 2 * q^25 - 6 * q^26 + 2 * q^27 - 2 * q^28 + 6 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^39 + 6 * q^41 + 6 * q^42 - 2 * q^43 + 6 * q^44 + 2 * q^45 - 12 * q^46 - 10 * q^48 - 6 * q^49 - 2 * q^52 - 12 * q^53 + 6 * q^55 - 6 * q^56 + 2 * q^57 - 18 * q^58 + 12 * q^59 + 2 * q^60 - 20 * q^61 - 24 * q^62 - 2 * q^63 + 2 * q^64 - 2 * q^65 - 6 * q^66 + 16 * q^67 + 6 * q^70 + 12 * q^71 - 20 * q^73 + 18 * q^74 + 2 * q^75 + 2 * q^76 - 12 * q^77 - 6 * q^78 - 8 * q^79 - 10 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 - 2 * q^84 - 18 * q^86 + 6 * q^87 + 6 * q^88 + 18 * q^89 - 4 * q^91 + 4 * q^93 + 12 * q^94 + 2 * q^95 - 2 * q^97 - 12 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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