LMFDB - Embedded newform 1617.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.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:	$12.9118100068$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
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			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.11491 q^{2} -1.00000 q^{3} +2.47283 q^{4} -1.64207 q^{5} -2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.11491 q^{2} -1.00000 q^{3} +2.47283 q^{4} -1.64207 q^{5} -2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.47283 q^{10} +1.00000 q^{11} -2.47283 q^{12} -4.58774 q^{13} +1.64207 q^{15} -2.83076 q^{16} -0.715853 q^{17} +2.11491 q^{18} -1.64207 q^{19} -4.06058 q^{20} +2.11491 q^{22} -1.00000 q^{24} -2.30359 q^{25} -9.70265 q^{26} -1.00000 q^{27} -5.53341 q^{29} +3.47283 q^{30} +5.17548 q^{31} -7.98680 q^{32} -1.00000 q^{33} -1.51396 q^{34} +2.47283 q^{36} -6.58774 q^{37} -3.47283 q^{38} +4.58774 q^{39} -1.64207 q^{40} -5.17548 q^{41} +5.17548 q^{43} +2.47283 q^{44} -1.64207 q^{45} -7.87189 q^{47} +2.83076 q^{48} -4.87189 q^{50} +0.715853 q^{51} -11.3447 q^{52} +2.71585 q^{53} -2.11491 q^{54} -1.64207 q^{55} +1.64207 q^{57} -11.7026 q^{58} +3.15604 q^{59} +4.06058 q^{60} -13.1755 q^{61} +10.9457 q^{62} -11.2298 q^{64} +7.53341 q^{65} -2.11491 q^{66} +4.21037 q^{67} -1.77018 q^{68} -1.85244 q^{71} +1.00000 q^{72} +10.4791 q^{73} -13.9325 q^{74} +2.30359 q^{75} -4.06058 q^{76} +9.70265 q^{78} -12.4596 q^{79} +4.64832 q^{80} +1.00000 q^{81} -10.9457 q^{82} +9.17548 q^{83} +1.17548 q^{85} +10.9457 q^{86} +5.53341 q^{87} +1.00000 q^{88} -4.71585 q^{89} -3.47283 q^{90} -5.17548 q^{93} -16.6483 q^{94} +2.69641 q^{95} +7.98680 q^{96} +13.6351 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{4} - 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^4 - 4 * q^5 + 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{4} - 4 q^{5} + 3 q^{8} + 3 q^{9} - 5 q^{10} + 3 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{15} - 4 q^{16} - 4 q^{17} - 4 q^{19} + 5 q^{20} - 3 q^{24} + 3 q^{25} - 11 q^{26} - 3 q^{27} + 6 q^{29} + 5 q^{30} - 8 q^{31} - 4 q^{32} - 3 q^{33} + 10 q^{34} + 2 q^{36} - 8 q^{37} - 5 q^{38} + 2 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 2 q^{44} - 4 q^{45} - 10 q^{47} + 4 q^{48} - q^{50} + 4 q^{51} - 15 q^{52} + 10 q^{53} - 4 q^{55} + 4 q^{57} - 17 q^{58} - 6 q^{59} - 5 q^{60} - 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 3 q^{72} - 2 q^{73} - 11 q^{74} - 3 q^{75} + 5 q^{76} + 11 q^{78} - 12 q^{79} - 15 q^{80} + 3 q^{81} - 22 q^{82} + 4 q^{83} - 20 q^{85} + 22 q^{86} - 6 q^{87} + 3 q^{88} - 16 q^{89} - 5 q^{90} + 8 q^{93} - 21 q^{94} + 18 q^{95} + 4 q^{96} - 8 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^4 - 4 * q^5 + 3 * q^8 + 3 * q^9 - 5 * q^10 + 3 * q^11 - 2 * q^12 - 2 * q^13 + 4 * q^15 - 4 * q^16 - 4 * q^17 - 4 * q^19 + 5 * q^20 - 3 * q^24 + 3 * q^25 - 11 * q^26 - 3 * q^27 + 6 * q^29 + 5 * q^30 - 8 * q^31 - 4 * q^32 - 3 * q^33 + 10 * q^34 + 2 * q^36 - 8 * q^37 - 5 * q^38 + 2 * q^39 - 4 * q^40 + 8 * q^41 - 8 * q^43 + 2 * q^44 - 4 * q^45 - 10 * q^47 + 4 * q^48 - q^50 + 4 * q^51 - 15 * q^52 + 10 * q^53 - 4 * q^55 + 4 * q^57 - 17 * q^58 - 6 * q^59 - 5 * q^60 - 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 3 * q^72 - 2 * q^73 - 11 * q^74 - 3 * q^75 + 5 * q^76 + 11 * q^78 - 12 * q^79 - 15 * q^80 + 3 * q^81 - 22 * q^82 + 4 * q^83 - 20 * q^85 + 22 * q^86 - 6 * q^87 + 3 * q^88 - 16 * q^89 - 5 * q^90 + 8 * q^93 - 21 * q^94 + 18 * q^95 + 4 * q^96 - 8 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.11491	1.49547	0.747733	−	0.664000i	$-0.231142\pi$
$2$	2.11491	1.49547	0.747733	+	0.664000i	$0.231142\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.47283	1.23642
$5$	−1.64207	−0.734358	−0.367179	−	0.930150i	$-0.619676\pi$
$5$	−1.64207	−0.734358	−0.367179	+	0.930150i	$0.619676\pi$
$6$	−2.11491	−0.863407
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.47283	−1.09821
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.47283	−0.713846
$13$	−4.58774	−1.27241	−0.636205	−	0.771520i	$-0.719497\pi$
$13$	−4.58774	−1.27241	−0.636205	+	0.771520i	$0.719497\pi$
$14$	0	0
$15$	1.64207	0.423982
$16$	−2.83076	−0.707690
$17$	−0.715853	−0.173620	−0.0868099	−	0.996225i	$-0.527667\pi$
$17$	−0.715853	−0.173620	−0.0868099	+	0.996225i	$0.527667\pi$
$18$	2.11491	0.498488
$19$	−1.64207	−0.376718	−0.188359	−	0.982100i	$-0.560317\pi$
$19$	−1.64207	−0.376718	−0.188359	+	0.982100i	$0.560317\pi$
$20$	−4.06058	−0.907972
$21$	0	0
$22$	2.11491	0.450900
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−2.30359	−0.460719
$26$	−9.70265	−1.90285
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.53341	−1.02753	−0.513764	−	0.857931i	$-0.671749\pi$
$29$	−5.53341	−1.02753	−0.513764	+	0.857931i	$0.671749\pi$
$30$	3.47283	0.634050
$31$	5.17548	0.929544	0.464772	−	0.885430i	$-0.346136\pi$
$31$	5.17548	0.929544	0.464772	+	0.885430i	$0.346136\pi$
$32$	−7.98680	−1.41188
$33$	−1.00000	−0.174078
$34$	−1.51396	−0.259642
$35$	0	0
$36$	2.47283	0.412139
$37$	−6.58774	−1.08302	−0.541509	−	0.840695i	$-0.682147\pi$
$37$	−6.58774	−1.08302	−0.541509	+	0.840695i	$0.682147\pi$
$38$	−3.47283	−0.563368
$39$	4.58774	0.734627
$40$	−1.64207	−0.259635
$41$	−5.17548	−0.808275	−0.404137	−	0.914698i	$-0.632428\pi$
$41$	−5.17548	−0.808275	−0.404137	+	0.914698i	$0.632428\pi$
$42$	0	0
$43$	5.17548	0.789254	0.394627	−	0.918841i	$-0.370874\pi$
$43$	5.17548	0.789254	0.394627	+	0.918841i	$0.370874\pi$
$44$	2.47283	0.372794
$45$	−1.64207	−0.244786
$46$	0	0
$47$	−7.87189	−1.14823	−0.574116	−	0.818774i	$-0.694654\pi$
$47$	−7.87189	−1.14823	−0.574116	+	0.818774i	$0.694654\pi$
$48$	2.83076	0.408585
$49$	0	0
$50$	−4.87189	−0.688989
$51$	0.715853	0.100239
$52$	−11.3447	−1.57323
$53$	2.71585	0.373051	0.186526	−	0.982450i	$-0.440277\pi$
$53$	2.71585	0.373051	0.186526	+	0.982450i	$0.440277\pi$
$54$	−2.11491	−0.287802
$55$	−1.64207	−0.221417
$56$	0	0
$57$	1.64207	0.217498
$58$	−11.7026	−1.53663
$59$	3.15604	0.410881	0.205440	−	0.978670i	$-0.434137\pi$
$59$	3.15604	0.410881	0.205440	+	0.978670i	$0.434137\pi$
$60$	4.06058	0.524218
$61$	−13.1755	−1.68695	−0.843474	−	0.537170i	$-0.819493\pi$
$61$	−13.1755	−1.68695	−0.843474	+	0.537170i	$0.819493\pi$
$62$	10.9457	1.39010
$63$	0	0
$64$	−11.2298	−1.40373
$65$	7.53341	0.934404
$66$	−2.11491	−0.260327
$67$	4.21037	0.514378	0.257189	−	0.966361i	$-0.417204\pi$
$67$	4.21037	0.514378	0.257189	+	0.966361i	$0.417204\pi$
$68$	−1.77018	−0.214666
$69$	0	0
$70$	0	0
$71$	−1.85244	−0.219844	−0.109922	−	0.993940i	$-0.535060\pi$
$71$	−1.85244	−0.219844	−0.109922	+	0.993940i	$0.535060\pi$
$72$	1.00000	0.117851
$73$	10.4791	1.22648	0.613242	−	0.789895i	$-0.289865\pi$
$73$	10.4791	1.22648	0.613242	+	0.789895i	$0.289865\pi$
$74$	−13.9325	−1.61962
$75$	2.30359	0.265996
$76$	−4.06058	−0.465780
$77$	0	0
$78$	9.70265	1.09861
$79$	−12.4596	−1.40182	−0.700909	−	0.713251i	$-0.747222\pi$
$79$	−12.4596	−1.40182	−0.700909	+	0.713251i	$0.747222\pi$
$80$	4.64832	0.519698
$81$	1.00000	0.111111
$82$	−10.9457	−1.20875
$83$	9.17548	1.00714	0.503570	−	0.863954i	$-0.332020\pi$
$83$	9.17548	1.00714	0.503570	+	0.863954i	$0.332020\pi$
$84$	0	0
$85$	1.17548	0.127499
$86$	10.9457	1.18030
$87$	5.53341	0.593244
$88$	1.00000	0.106600
$89$	−4.71585	−0.499879	−0.249940	−	0.968261i	$-0.580411\pi$
$89$	−4.71585	−0.499879	−0.249940	+	0.968261i	$0.580411\pi$
$90$	−3.47283	−0.366069
$91$	0	0
$92$	0	0
$93$	−5.17548	−0.536673
$94$	−16.6483	−1.71714
$95$	2.69641	0.276645
$96$	7.98680	0.815149
$97$	13.6351	1.38444	0.692218	−	0.721688i	$-0.256634\pi$
$97$	13.6351	1.38444	0.692218	+	0.721688i	$0.256634\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−5.69641	−0.569641
$101$	14.6072	1.45347	0.726735	−	0.686918i	$-0.241037\pi$
$101$	14.6072	1.45347	0.726735	+	0.686918i	$0.241037\pi$
$102$	1.51396	0.149905
$103$	−17.8913	−1.76289	−0.881443	−	0.472291i	$-0.843427\pi$
$103$	−17.8913	−1.76289	−0.881443	+	0.472291i	$0.843427\pi$
$104$	−4.58774	−0.449865
$105$	0	0
$106$	5.74378	0.557885
$107$	11.1560	1.07849	0.539247	−	0.842147i	$-0.318709\pi$
$107$	11.1560	1.07849	0.539247	+	0.842147i	$0.318709\pi$
$108$	−2.47283	−0.237949
$109$	−0.108664	−0.0104082	−0.00520408	−	0.999986i	$-0.501657\pi$
$109$	−0.108664	−0.0104082	−0.00520408	+	0.999986i	$0.501657\pi$
$110$	−3.47283	−0.331122
$111$	6.58774	0.625281
$112$	0	0
$113$	−13.0668	−1.22922	−0.614611	−	0.788830i	$-0.710687\pi$
$113$	−13.0668	−1.22922	−0.614611	+	0.788830i	$0.710687\pi$
$114$	3.47283	0.325261
$115$	0	0
$116$	−13.6832	−1.27045
$117$	−4.58774	−0.424137
$118$	6.67472	0.614458
$119$	0	0
$120$	1.64207	0.149900
$121$	1.00000	0.0909091
$122$	−27.8649	−2.52277
$123$	5.17548	0.466658
$124$	12.7981	1.14930
$125$	11.9930	1.07269
$126$	0	0
$127$	−20.4596	−1.81550	−0.907749	−	0.419513i	$-0.862201\pi$
$127$	−20.4596	−1.81550	−0.907749	+	0.419513i	$0.862201\pi$
$128$	−7.77643	−0.687346
$129$	−5.17548	−0.455676
$130$	15.9325	1.39737
$131$	15.0668	1.31639	0.658197	−	0.752846i	$-0.271319\pi$
$131$	15.0668	1.31639	0.658197	+	0.752846i	$0.271319\pi$
$132$	−2.47283	−0.215233
$133$	0	0
$134$	8.90454	0.769235
$135$	1.64207	0.141327
$136$	−0.715853	−0.0613839
$137$	−14.4596	−1.23537	−0.617685	−	0.786426i	$-0.711929\pi$
$137$	−14.4596	−1.23537	−0.617685	+	0.786426i	$0.711929\pi$
$138$	0	0
$139$	5.13659	0.435680	0.217840	−	0.975985i	$-0.430099\pi$
$139$	5.13659	0.435680	0.217840	+	0.975985i	$0.430099\pi$
$140$	0	0
$141$	7.87189	0.662933
$142$	−3.91774	−0.328770
$143$	−4.58774	−0.383646
$144$	−2.83076	−0.235897
$145$	9.08627	0.754573
$146$	22.1623	1.83416
$147$	0	0
$148$	−16.2904	−1.33906
$149$	16.8176	1.37775	0.688874	−	0.724881i	$-0.258105\pi$
$149$	16.8176	1.37775	0.688874	+	0.724881i	$0.258105\pi$
$150$	4.87189	0.397788
$151$	4.71585	0.383771	0.191885	−	0.981417i	$-0.438540\pi$
$151$	4.71585	0.383771	0.191885	+	0.981417i	$0.438540\pi$
$152$	−1.64207	−0.133190
$153$	−0.715853	−0.0578733
$154$	0	0
$155$	−8.49852	−0.682618
$156$	11.3447	0.908305
$157$	11.0279	0.880124	0.440062	−	0.897967i	$-0.354956\pi$
$157$	11.0279	0.880124	0.440062	+	0.897967i	$0.354956\pi$
$158$	−26.3510	−2.09637
$159$	−2.71585	−0.215381
$160$	13.1149	1.03682
$161$	0	0
$162$	2.11491	0.166163
$163$	5.64207	0.441921	0.220961	−	0.975283i	$-0.429081\pi$
$163$	5.64207	0.441921	0.220961	+	0.975283i	$0.429081\pi$
$164$	−12.7981	−0.999364
$165$	1.64207	0.127835
$166$	19.4053	1.50614
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	8.04737	0.619029
$170$	2.48604	0.190670
$171$	−1.64207	−0.125573
$172$	12.7981	0.975847
$173$	15.0279	1.14255	0.571276	−	0.820758i	$-0.306449\pi$
$173$	15.0279	1.14255	0.571276	+	0.820758i	$0.306449\pi$
$174$	11.7026	0.887176
$175$	0	0
$176$	−2.83076	−0.213377
$177$	−3.15604	−0.237222
$178$	−9.97359	−0.747552
$179$	9.89134	0.739313	0.369657	−	0.929168i	$-0.379475\pi$
$179$	9.89134	0.739313	0.369657	+	0.929168i	$0.379475\pi$
$180$	−4.06058	−0.302657
$181$	−6.82452	−0.507262	−0.253631	−	0.967301i	$-0.581625\pi$
$181$	−6.82452	−0.507262	−0.253631	+	0.967301i	$0.581625\pi$
$182$	0	0
$183$	13.1755	0.973960
$184$	0	0
$185$	10.8176	0.795323
$186$	−10.9457	−0.802575
$187$	−0.715853	−0.0523483
$188$	−19.4659	−1.41969
$189$	0	0
$190$	5.70265	0.413714
$191$	11.0279	0.797953	0.398976	−	0.916961i	$-0.369366\pi$
$191$	11.0279	0.797953	0.398976	+	0.916961i	$0.369366\pi$
$192$	11.2298	0.810442
$193$	−7.85244	−0.565231	−0.282616	−	0.959233i	$-0.591202\pi$
$193$	−7.85244	−0.565231	−0.282616	+	0.959233i	$0.591202\pi$
$194$	28.8370	2.07038
$195$	−7.53341	−0.539479
$196$	0	0
$197$	19.8913	1.41720	0.708599	−	0.705611i	$-0.249327\pi$
$197$	19.8913	1.41720	0.708599	+	0.705611i	$0.249327\pi$
$198$	2.11491	0.150300
$199$	−23.5264	−1.66775	−0.833873	−	0.551957i	$-0.813881\pi$
$199$	−23.5264	−1.66775	−0.833873	+	0.551957i	$0.813881\pi$
$200$	−2.30359	−0.162889
$201$	−4.21037	−0.296976
$202$	30.8929	2.17361
$203$	0	0
$204$	1.77018	0.123938
$205$	8.49852	0.593563
$206$	−37.8385	−2.63633
$207$	0	0
$208$	12.9868	0.900472
$209$	−1.64207	−0.113585
$210$	0	0
$211$	−4.25622	−0.293010	−0.146505	−	0.989210i	$-0.546803\pi$
$211$	−4.25622	−0.293010	−0.146505	+	0.989210i	$0.546803\pi$
$212$	6.71585	0.461247
$213$	1.85244	0.126927
$214$	23.5940	1.61285
$215$	−8.49852	−0.579595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−0.229815	−0.0155650
$219$	−10.4791	−0.708110
$220$	−4.06058	−0.273764
$221$	3.28415	0.220916
$222$	13.9325	0.935086
$223$	3.74378	0.250702	0.125351	−	0.992112i	$-0.459994\pi$
$223$	3.74378	0.250702	0.125351	+	0.992112i	$0.459994\pi$
$224$	0	0
$225$	−2.30359	−0.153573
$226$	−27.6351	−1.83826
$227$	15.4876	1.02795	0.513973	−	0.857807i	$-0.328173\pi$
$227$	15.4876	1.02795	0.513973	+	0.857807i	$0.328173\pi$
$228$	4.06058	0.268918
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	0	0
$232$	−5.53341	−0.363286
$233$	−5.54037	−0.362962	−0.181481	−	0.983394i	$-0.558089\pi$
$233$	−5.54037	−0.362962	−0.181481	+	0.983394i	$0.558089\pi$
$234$	−9.70265	−0.634282
$235$	12.9262	0.843214
$236$	7.80435	0.508020
$237$	12.4596	0.809340
$238$	0	0
$239$	−1.26470	−0.0818067	−0.0409033	−	0.999163i	$-0.513024\pi$
$239$	−1.26470	−0.0818067	−0.0409033	+	0.999163i	$0.513024\pi$
$240$	−4.64832	−0.300048
$241$	23.8719	1.53772	0.768862	−	0.639415i	$-0.220823\pi$
$241$	23.8719	1.53772	0.768862	+	0.639415i	$0.220823\pi$
$242$	2.11491	0.135951
$243$	−1.00000	−0.0641500
$244$	−32.5808	−2.08577
$245$	0	0
$246$	10.9457	0.697870
$247$	7.53341	0.479339
$248$	5.17548	0.328643
$249$	−9.17548	−0.581473
$250$	25.3642	1.60417
$251$	−22.0194	−1.38986	−0.694928	−	0.719080i	$-0.744564\pi$
$251$	−22.0194	−1.38986	−0.694928	+	0.719080i	$0.744564\pi$
$252$	0	0
$253$	0	0
$254$	−43.2702	−2.71502
$255$	−1.17548	−0.0736116
$256$	6.01320	0.375825
$257$	−27.3161	−1.70393	−0.851965	−	0.523599i	$-0.824589\pi$
$257$	−27.3161	−1.70393	−0.851965	+	0.523599i	$0.824589\pi$
$258$	−10.9457	−0.681448
$259$	0	0
$260$	18.6289	1.15531
$261$	−5.53341	−0.342509
$262$	31.8649	1.96862
$263$	−11.8719	−0.732052	−0.366026	−	0.930605i	$-0.619282\pi$
$263$	−11.8719	−0.732052	−0.366026	+	0.930605i	$0.619282\pi$
$264$	−1.00000	−0.0615457
$265$	−4.45963	−0.273953
$266$	0	0
$267$	4.71585	0.288606
$268$	10.4115	0.635986
$269$	−5.63511	−0.343579	−0.171789	−	0.985134i	$-0.554955\pi$
$269$	−5.63511	−0.343579	−0.171789	+	0.985134i	$0.554955\pi$
$270$	3.47283	0.211350
$271$	5.68097	0.345094	0.172547	−	0.985001i	$-0.444800\pi$
$271$	5.68097	0.345094	0.172547	+	0.985001i	$0.444800\pi$
$272$	2.02641	0.122869
$273$	0	0
$274$	−30.5808	−1.84745
$275$	−2.30359	−0.138912
$276$	0	0
$277$	−25.4876	−1.53140	−0.765699	−	0.643199i	$-0.777607\pi$
$277$	−25.4876	−1.53140	−0.765699	+	0.643199i	$0.777607\pi$
$278$	10.8634	0.651544
$279$	5.17548	0.309848
$280$	0	0
$281$	17.5723	1.04828	0.524138	−	0.851633i	$-0.324388\pi$
$281$	17.5723	1.04828	0.524138	+	0.851633i	$0.324388\pi$
$282$	16.6483	0.991393
$283$	−25.1296	−1.49380	−0.746901	−	0.664936i	$-0.768459\pi$
$283$	−25.1296	−1.49380	−0.746901	+	0.664936i	$0.768459\pi$
$284$	−4.58078	−0.271819
$285$	−2.69641	−0.159721
$286$	−9.70265	−0.573730
$287$	0	0
$288$	−7.98680	−0.470626
$289$	−16.4876	−0.969856
$290$	19.2166	1.12844
$291$	−13.6351	−0.799304
$292$	25.9130	1.51644
$293$	14.3510	0.838392	0.419196	−	0.907896i	$-0.362312\pi$
$293$	14.3510	0.838392	0.419196	+	0.907896i	$0.362312\pi$
$294$	0	0
$295$	−5.18244	−0.301734
$296$	−6.58774	−0.382905
$297$	−1.00000	−0.0580259
$298$	35.5676	2.06037
$299$	0	0
$300$	5.69641	0.328882
$301$	0	0
$302$	9.97359	0.573916
$303$	−14.6072	−0.839161
$304$	4.64832	0.266599
$305$	21.6351	1.23882
$306$	−1.51396	−0.0865475
$307$	−13.1366	−0.749745	−0.374872	−	0.927076i	$-0.622313\pi$
$307$	−13.1366	−0.749745	−0.374872	+	0.927076i	$0.622313\pi$
$308$	0	0
$309$	17.8913	1.01780
$310$	−17.9736	−1.02083
$311$	14.3510	0.813769	0.406884	−	0.913480i	$-0.366615\pi$
$311$	14.3510	0.813769	0.406884	+	0.913480i	$0.366615\pi$
$312$	4.58774	0.259730
$313$	−23.7438	−1.34208	−0.671039	−	0.741422i	$-0.734152\pi$
$313$	−23.7438	−1.34208	−0.671039	+	0.741422i	$0.734152\pi$
$314$	23.3230	1.31620
$315$	0	0
$316$	−30.8106	−1.73323
$317$	−6.45963	−0.362809	−0.181404	−	0.983409i	$-0.558064\pi$
$317$	−6.45963	−0.362809	−0.181404	+	0.983409i	$0.558064\pi$
$318$	−5.74378	−0.322095
$319$	−5.53341	−0.309811
$320$	18.4402	1.03084
$321$	−11.1560	−0.622669
$322$	0	0
$323$	1.17548	0.0654056
$324$	2.47283	0.137380
$325$	10.5683	0.586224
$326$	11.9325	0.660878
$327$	0.108664	0.00600915
$328$	−5.17548	−0.285768
$329$	0	0
$330$	3.47283	0.191173
$331$	31.2702	1.71877	0.859384	−	0.511332i	$-0.170848\pi$
$331$	31.2702	1.71877	0.859384	+	0.511332i	$0.170848\pi$
$332$	22.6894	1.24525
$333$	−6.58774	−0.361006
$334$	−33.8385	−1.85156
$335$	−6.91373	−0.377738
$336$	0	0
$337$	−16.3121	−0.888575	−0.444288	−	0.895884i	$-0.646543\pi$
$337$	−16.3121	−0.888575	−0.444288	+	0.895884i	$0.646543\pi$
$338$	17.0194	0.925736
$339$	13.0668	0.709692
$340$	2.90677	0.157642
$341$	5.17548	0.280268
$342$	−3.47283	−0.187789
$343$	0	0
$344$	5.17548	0.279043
$345$	0	0
$346$	31.7827	1.70865
$347$	−6.86341	−0.368447	−0.184224	−	0.982884i	$-0.558977\pi$
$347$	−6.86341	−0.368447	−0.184224	+	0.982884i	$0.558977\pi$
$348$	13.6832	0.733497
$349$	−23.8580	−1.27709	−0.638544	−	0.769585i	$-0.720463\pi$
$349$	−23.8580	−1.27709	−0.638544	+	0.769585i	$0.720463\pi$
$350$	0	0
$351$	4.58774	0.244876
$352$	−7.98680	−0.425698
$353$	−11.7368	−0.624688	−0.312344	−	0.949969i	$-0.601114\pi$
$353$	−11.7368	−0.624688	−0.312344	+	0.949969i	$0.601114\pi$
$354$	−6.67472	−0.354758
$355$	3.04185	0.161444
$356$	−11.6615	−0.618059
$357$	0	0
$358$	20.9193	1.10562
$359$	−27.7827	−1.46631	−0.733157	−	0.680060i	$-0.761954\pi$
$359$	−27.7827	−1.46631	−0.733157	+	0.680060i	$0.761954\pi$
$360$	−1.64207	−0.0865449
$361$	−16.3036	−0.858084
$362$	−14.4332	−0.758593
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−17.2074	−0.900677
$366$	27.8649	1.45652
$367$	26.5544	1.38613	0.693064	−	0.720877i	$-0.256261\pi$
$367$	26.5544	1.38613	0.693064	+	0.720877i	$0.256261\pi$
$368$	0	0
$369$	−5.17548	−0.269425
$370$	22.8781	1.18938
$371$	0	0
$372$	−12.7981	−0.663551
$373$	5.06682	0.262350	0.131175	−	0.991359i	$-0.458125\pi$
$373$	5.06682	0.262350	0.131175	+	0.991359i	$0.458125\pi$
$374$	−1.51396	−0.0782851
$375$	−11.9930	−0.619318
$376$	−7.87189	−0.405962
$377$	25.3859	1.30744
$378$	0	0
$379$	−30.5613	−1.56983	−0.784915	−	0.619603i	$-0.787294\pi$
$379$	−30.5613	−1.56983	−0.784915	+	0.619603i	$0.787294\pi$
$380$	6.66776	0.342049
$381$	20.4596	1.04818
$382$	23.3230	1.19331
$383$	−9.13659	−0.466858	−0.233429	−	0.972374i	$-0.574995\pi$
$383$	−9.13659	−0.466858	−0.233429	+	0.972374i	$0.574995\pi$
$384$	7.77643	0.396839
$385$	0	0
$386$	−16.6072	−0.845284
$387$	5.17548	0.263085
$388$	33.7174	1.71174
$389$	21.2702	1.07844	0.539222	−	0.842164i	$-0.318719\pi$
$389$	21.2702	1.07844	0.539222	+	0.842164i	$0.318719\pi$
$390$	−15.9325	−0.806772
$391$	0	0
$392$	0	0
$393$	−15.0668	−0.760020
$394$	42.0683	2.11937
$395$	20.4596	1.02944
$396$	2.47283	0.124265
$397$	−28.9582	−1.45337	−0.726684	−	0.686972i	$-0.758940\pi$
$397$	−28.9582	−1.45337	−0.726684	+	0.686972i	$0.758940\pi$
$398$	−49.7563	−2.49406
$399$	0	0
$400$	6.52092	0.326046
$401$	32.3510	1.61553	0.807765	−	0.589505i	$-0.200677\pi$
$401$	32.3510	1.61553	0.807765	+	0.589505i	$0.200677\pi$
$402$	−8.90454	−0.444118
$403$	−23.7438	−1.18276
$404$	36.1212	1.79709
$405$	−1.64207	−0.0815953
$406$	0	0
$407$	−6.58774	−0.326542
$408$	0.715853	0.0354400
$409$	−2.31207	−0.114325	−0.0571623	−	0.998365i	$-0.518205\pi$
$409$	−2.31207	−0.114325	−0.0571623	+	0.998365i	$0.518205\pi$
$410$	17.9736	0.887652
$411$	14.4596	0.713241
$412$	−44.2423	−2.17966
$413$	0	0
$414$	0	0
$415$	−15.0668	−0.739601
$416$	36.6414	1.79649
$417$	−5.13659	−0.251540
$418$	−3.47283	−0.169862
$419$	−38.4263	−1.87725	−0.938623	−	0.344945i	$-0.887898\pi$
$419$	−38.4263	−1.87725	−0.938623	+	0.344945i	$0.887898\pi$
$420$	0	0
$421$	15.5070	0.755765	0.377883	−	0.925854i	$-0.376652\pi$
$421$	15.5070	0.755765	0.377883	+	0.925854i	$0.376652\pi$
$422$	−9.00152	−0.438187
$423$	−7.87189	−0.382744
$424$	2.71585	0.131893
$425$	1.64903	0.0799899
$426$	3.91774	0.189815
$427$	0	0
$428$	27.5870	1.33347
$429$	4.58774	0.221498
$430$	−17.9736	−0.866764
$431$	37.9666	1.82879	0.914394	−	0.404825i	$-0.132668\pi$
$431$	37.9666	1.82879	0.914394	+	0.404825i	$0.132668\pi$
$432$	2.83076	0.136195
$433$	−10.1087	−0.485791	−0.242896	−	0.970052i	$-0.578097\pi$
$433$	−10.1087	−0.485791	−0.242896	+	0.970052i	$0.578097\pi$
$434$	0	0
$435$	−9.08627	−0.435653
$436$	−0.268709	−0.0128688
$437$	0	0
$438$	−22.1623	−1.05895
$439$	22.5224	1.07494	0.537469	−	0.843284i	$-0.319381\pi$
$439$	22.5224	1.07494	0.537469	+	0.843284i	$0.319381\pi$
$440$	−1.64207	−0.0782828
$441$	0	0
$442$	6.94567	0.330372
$443$	20.6630	0.981731	0.490865	−	0.871235i	$-0.336681\pi$
$443$	20.6630	0.981731	0.490865	+	0.871235i	$0.336681\pi$
$444$	16.2904	0.773108
$445$	7.74378	0.367090
$446$	7.91774	0.374916
$447$	−16.8176	−0.795443
$448$	0	0
$449$	40.8106	1.92597	0.962986	−	0.269553i	$-0.0868759\pi$
$449$	40.8106	1.92597	0.962986	+	0.269553i	$0.0868759\pi$
$450$	−4.87189	−0.229663
$451$	−5.17548	−0.243704
$452$	−32.3121	−1.51983
$453$	−4.71585	−0.221570
$454$	32.7547	1.53726
$455$	0	0
$456$	1.64207	0.0768971
$457$	−28.3899	−1.32802	−0.664011	−	0.747723i	$-0.731147\pi$
$457$	−28.3899	−1.32802	−0.664011	+	0.747723i	$0.731147\pi$
$458$	16.9193	0.790585
$459$	0.715853	0.0334131
$460$	0	0
$461$	−21.4178	−0.997526	−0.498763	−	0.866739i	$-0.666212\pi$
$461$	−21.4178	−0.997526	−0.498763	+	0.866739i	$0.666212\pi$
$462$	0	0
$463$	20.7478	0.964231	0.482116	−	0.876108i	$-0.339868\pi$
$463$	20.7478	0.964231	0.482116	+	0.876108i	$0.339868\pi$
$464$	15.6638	0.727172
$465$	8.49852	0.394110
$466$	−11.7174	−0.542797
$467$	−26.8998	−1.24477	−0.622387	−	0.782709i	$-0.713837\pi$
$467$	−26.8998	−1.24477	−0.622387	+	0.782709i	$0.713837\pi$
$468$	−11.3447	−0.524410
$469$	0	0
$470$	27.3378	1.26100
$471$	−11.0279	−0.508140
$472$	3.15604	0.145268
$473$	5.17548	0.237969
$474$	26.3510	1.21034
$475$	3.78267	0.173561
$476$	0	0
$477$	2.71585	0.124350
$478$	−2.67472	−0.122339
$479$	−30.8106	−1.40777	−0.703886	−	0.710313i	$-0.748553\pi$
$479$	−30.8106	−1.40777	−0.703886	+	0.710313i	$0.748553\pi$
$480$	−13.1149	−0.598611
$481$	30.2229	1.37804
$482$	50.4868	2.29961
$483$	0	0
$484$	2.47283	0.112402
$485$	−22.3899	−1.01667
$486$	−2.11491	−0.0959342
$487$	−23.4876	−1.06432	−0.532161	−	0.846643i	$-0.678620\pi$
$487$	−23.4876	−1.06432	−0.532161	+	0.846643i	$0.678620\pi$
$488$	−13.1755	−0.596426
$489$	−5.64207	−0.255143
$490$	0	0
$491$	−16.6266	−0.750350	−0.375175	−	0.926954i	$-0.622417\pi$
$491$	−16.6266	−0.750350	−0.375175	+	0.926954i	$0.622417\pi$
$492$	12.7981	0.576983
$493$	3.96111	0.178399
$494$	15.9325	0.716835
$495$	−1.64207	−0.0738057
$496$	−14.6506	−0.657829
$497$	0	0
$498$	−19.4053	−0.869572
$499$	−7.99304	−0.357818	−0.178909	−	0.983866i	$-0.557257\pi$
$499$	−7.99304	−0.357818	−0.178909	+	0.983866i	$0.557257\pi$
$500$	29.6568	1.32629
$501$	16.0000	0.714827
$502$	−46.5691	−2.07848
$503$	−24.6630	−1.09967	−0.549835	−	0.835273i	$-0.685310\pi$
$503$	−24.6630	−1.09967	−0.549835	+	0.835273i	$0.685310\pi$
$504$	0	0
$505$	−23.9861	−1.06737
$506$	0	0
$507$	−8.04737	−0.357396
$508$	−50.5933	−2.24471
$509$	−15.0668	−0.667825	−0.333912	−	0.942604i	$-0.608369\pi$
$509$	−15.0668	−0.667825	−0.333912	+	0.942604i	$0.608369\pi$
$510$	−2.48604	−0.110084
$511$	0	0
$512$	28.2702	1.24938
$513$	1.64207	0.0724993
$514$	−57.7710	−2.54817
$515$	29.3789	1.29459
$516$	−12.7981	−0.563405
$517$	−7.87189	−0.346205
$518$	0	0
$519$	−15.0279	−0.659653
$520$	7.53341	0.330362
$521$	42.3051	1.85342	0.926710	−	0.375777i	$-0.122624\pi$
$521$	42.3051	1.85342	0.926710	+	0.375777i	$0.122624\pi$
$522$	−11.7026	−0.512211
$523$	−5.84548	−0.255605	−0.127803	−	0.991800i	$-0.540792\pi$
$523$	−5.84548	−0.255605	−0.127803	+	0.991800i	$0.540792\pi$
$524$	37.2577	1.62761
$525$	0	0
$526$	−25.1079	−1.09476
$527$	−3.70488	−0.161387
$528$	2.83076	0.123193
$529$	−23.0000	−1.00000
$530$	−9.43171	−0.409687
$531$	3.15604	0.136960
$532$	0	0
$533$	23.7438	1.02846
$534$	9.97359	0.431600
$535$	−18.3190	−0.792001
$536$	4.21037	0.181860
$537$	−9.89134	−0.426843
$538$	−11.9177	−0.513810
$539$	0	0
$540$	4.06058	0.174739
$541$	−40.2982	−1.73255	−0.866276	−	0.499565i	$-0.833493\pi$
$541$	−40.2982	−1.73255	−0.866276	+	0.499565i	$0.833493\pi$
$542$	12.0147	0.516076
$543$	6.82452	0.292868
$544$	5.71737	0.245130
$545$	0.178435	0.00764331
$546$	0	0
$547$	2.98903	0.127802	0.0639009	−	0.997956i	$-0.479646\pi$
$547$	2.98903	0.127802	0.0639009	+	0.997956i	$0.479646\pi$
$548$	−35.7563	−1.52743
$549$	−13.1755	−0.562316
$550$	−4.87189	−0.207738
$551$	9.08627	0.387088
$552$	0	0
$553$	0	0
$554$	−53.9038	−2.29015
$555$	−10.8176	−0.459180
$556$	12.7019	0.538682
$557$	21.1127	0.894573	0.447286	−	0.894391i	$-0.352390\pi$
$557$	21.1127	0.894573	0.447286	+	0.894391i	$0.352390\pi$
$558$	10.9457	0.463367
$559$	−23.7438	−1.00425
$560$	0	0
$561$	0.715853	0.0302233
$562$	37.1638	1.56766
$563$	−29.8774	−1.25918	−0.629591	−	0.776926i	$-0.716778\pi$
$563$	−29.8774	−1.25918	−0.629591	+	0.776926i	$0.716778\pi$
$564$	19.4659	0.819661
$565$	21.4567	0.902689
$566$	−53.1468	−2.23393
$567$	0	0
$568$	−1.85244	−0.0777267
$569$	0.186452	0.00781648	0.00390824	−	0.999992i	$-0.498756\pi$
$569$	0.186452	0.00781648	0.00390824	+	0.999992i	$0.498756\pi$
$570$	−5.70265	−0.238858
$571$	3.06682	0.128342	0.0641712	−	0.997939i	$-0.479560\pi$
$571$	3.06682	0.128342	0.0641712	+	0.997939i	$0.479560\pi$
$572$	−11.3447	−0.474347
$573$	−11.0279	−0.460698
$574$	0	0
$575$	0	0
$576$	−11.2298	−0.467909
$577$	−18.8495	−0.784715	−0.392357	−	0.919813i	$-0.628340\pi$
$577$	−18.8495	−0.784715	−0.392357	+	0.919813i	$0.628340\pi$
$578$	−34.8697	−1.45039
$579$	7.85244	0.326336
$580$	22.4688	0.932967
$581$	0	0
$582$	−28.8370	−1.19533
$583$	2.71585	0.112479
$584$	10.4791	0.433627
$585$	7.53341	0.311468
$586$	30.3510	1.25379
$587$	8.11419	0.334908	0.167454	−	0.985880i	$-0.446445\pi$
$587$	8.11419	0.334908	0.167454	+	0.985880i	$0.446445\pi$
$588$	0	0
$589$	−8.49852	−0.350176
$590$	−10.9604	−0.451232
$591$	−19.8913	−0.818220
$592$	18.6483	0.766441
$593$	−6.36489	−0.261375	−0.130687	−	0.991424i	$-0.541718\pi$
$593$	−6.36489	−0.261375	−0.130687	+	0.991424i	$0.541718\pi$
$594$	−2.11491	−0.0867757
$595$	0	0
$596$	41.5870	1.70347
$597$	23.5264	0.962873
$598$	0	0
$599$	0.676959	0.0276598	0.0138299	−	0.999904i	$-0.495598\pi$
$599$	0.676959	0.0276598	0.0138299	+	0.999904i	$0.495598\pi$
$600$	2.30359	0.0940438
$601$	22.6964	0.925806	0.462903	−	0.886409i	$-0.346808\pi$
$601$	22.6964	0.925806	0.462903	+	0.886409i	$0.346808\pi$
$602$	0	0
$603$	4.21037	0.171459
$604$	11.6615	0.474501
$605$	−1.64207	−0.0667598
$606$	−30.8929	−1.25494
$607$	−14.6141	−0.593170	−0.296585	−	0.955006i	$-0.595848\pi$
$607$	−14.6141	−0.593170	−0.296585	+	0.955006i	$0.595848\pi$
$608$	13.1149	0.531880
$609$	0	0
$610$	45.7563	1.85262
$611$	36.1142	1.46102
$612$	−1.77018	−0.0715555
$613$	−28.3899	−1.14666	−0.573328	−	0.819326i	$-0.694348\pi$
$613$	−28.3899	−1.14666	−0.573328	+	0.819326i	$0.694348\pi$
$614$	−27.7827	−1.12122
$615$	−8.49852	−0.342694
$616$	0	0
$617$	−21.5404	−0.867183	−0.433591	−	0.901110i	$-0.642754\pi$
$617$	−21.5404	−0.867183	−0.433591	+	0.901110i	$0.642754\pi$
$618$	37.8385	1.52209
$619$	−23.7827	−0.955906	−0.477953	−	0.878385i	$-0.658621\pi$
$619$	−23.7827	−0.955906	−0.477953	+	0.878385i	$0.658621\pi$
$620$	−21.0154	−0.844000
$621$	0	0
$622$	30.3510	1.21696
$623$	0	0
$624$	−12.9868	−0.519888
$625$	−8.17548	−0.327019
$626$	−50.2159	−2.00703
$627$	1.64207	0.0655781
$628$	27.2702	1.08820
$629$	4.71585	0.188033
$630$	0	0
$631$	−40.4068	−1.60857	−0.804285	−	0.594244i	$-0.797451\pi$
$631$	−40.4068	−1.60857	−0.804285	+	0.594244i	$0.797451\pi$
$632$	−12.4596	−0.495617
$633$	4.25622	0.169170
$634$	−13.6615	−0.542568
$635$	33.5962	1.33323
$636$	−6.71585	−0.266301
$637$	0	0
$638$	−11.7026	−0.463312
$639$	−1.85244	−0.0732815
$640$	12.7695	0.504758
$641$	13.4876	0.532726	0.266363	−	0.963873i	$-0.414178\pi$
$641$	13.4876	0.532726	0.266363	+	0.963873i	$0.414178\pi$
$642$	−23.5940	−0.931180
$643$	−26.8245	−1.05786	−0.528928	−	0.848667i	$-0.677406\pi$
$643$	−26.8245	−1.05786	−0.528928	+	0.848667i	$0.677406\pi$
$644$	0	0
$645$	8.49852	0.334629
$646$	2.48604	0.0978118
$647$	−42.4791	−1.67002	−0.835012	−	0.550231i	$-0.814540\pi$
$647$	−42.4791	−1.67002	−0.835012	+	0.550231i	$0.814540\pi$
$648$	1.00000	0.0392837
$649$	3.15604	0.123885
$650$	22.3510	0.876677
$651$	0	0
$652$	13.9519	0.546399
$653$	7.39281	0.289303	0.144652	−	0.989483i	$-0.453794\pi$
$653$	7.39281	0.289303	0.144652	+	0.989483i	$0.453794\pi$
$654$	0.229815	0.00898648
$655$	−24.7408	−0.966704
$656$	14.6506	0.572008
$657$	10.4791	0.408828
$658$	0	0
$659$	18.2368	0.710404	0.355202	−	0.934790i	$-0.384412\pi$
$659$	18.2368	0.710404	0.355202	+	0.934790i	$0.384412\pi$
$660$	4.06058	0.158058
$661$	−8.16451	−0.317563	−0.158781	−	0.987314i	$-0.550757\pi$
$661$	−8.16451	−0.317563	−0.158781	+	0.987314i	$0.550757\pi$
$662$	66.1336	2.57036
$663$	−3.28415	−0.127546
$664$	9.17548	0.356078
$665$	0	0
$666$	−13.9325	−0.539872
$667$	0	0
$668$	−39.5653	−1.53083
$669$	−3.74378	−0.144743
$670$	−14.6219	−0.564894
$671$	−13.1755	−0.508634
$672$	0	0
$673$	43.6212	1.68147	0.840737	−	0.541444i	$-0.182122\pi$
$673$	43.6212	1.68147	0.840737	+	0.541444i	$0.182122\pi$
$674$	−34.4985	−1.32883
$675$	2.30359	0.0886654
$676$	19.8998	0.765377
$677$	23.1057	0.888025	0.444012	−	0.896021i	$-0.353555\pi$
$677$	23.1057	0.888025	0.444012	+	0.896021i	$0.353555\pi$
$678$	27.6351	1.06132
$679$	0	0
$680$	1.17548	0.0450777
$681$	−15.4876	−0.593484
$682$	10.9457	0.419131
$683$	27.5653	1.05476	0.527379	−	0.849630i	$-0.323175\pi$
$683$	27.5653	1.05476	0.527379	+	0.849630i	$0.323175\pi$
$684$	−4.06058	−0.155260
$685$	23.7438	0.907203
$686$	0	0
$687$	−8.00000	−0.305219
$688$	−14.6506	−0.558547
$689$	−12.4596	−0.474674
$690$	0	0
$691$	5.85244	0.222637	0.111319	−	0.993785i	$-0.464493\pi$
$691$	5.85244	0.222637	0.111319	+	0.993785i	$0.464493\pi$
$692$	37.1616	1.41267
$693$	0	0
$694$	−14.5155	−0.551000
$695$	−8.43466	−0.319945
$696$	5.53341	0.209743
$697$	3.70488	0.140332
$698$	−50.4574	−1.90984
$699$	5.54037	0.209556
$700$	0	0
$701$	9.32304	0.352126	0.176063	−	0.984379i	$-0.443664\pi$
$701$	9.32304	0.352126	0.176063	+	0.984379i	$0.443664\pi$
$702$	9.70265	0.366203
$703$	10.8176	0.407992
$704$	−11.2298	−0.423240
$705$	−12.9262	−0.486830
$706$	−24.8223	−0.934199
$707$	0	0
$708$	−7.80435	−0.293306
$709$	41.5629	1.56093	0.780463	−	0.625202i	$-0.214983\pi$
$709$	41.5629	1.56093	0.780463	+	0.625202i	$0.214983\pi$
$710$	6.43322	0.241435
$711$	−12.4596	−0.467273
$712$	−4.71585	−0.176734
$713$	0	0
$714$	0	0
$715$	7.53341	0.281734
$716$	24.4596	0.914099
$717$	1.26470	0.0472311
$718$	−58.7578	−2.19282
$719$	31.4372	1.17241	0.586205	−	0.810162i	$-0.300621\pi$
$719$	31.4372	1.17241	0.586205	+	0.810162i	$0.300621\pi$
$720$	4.64832	0.173233
$721$	0	0
$722$	−34.4806	−1.28323
$723$	−23.8719	−0.887805
$724$	−16.8759	−0.627188
$725$	12.7467	0.473402
$726$	−2.11491	−0.0784916
$727$	14.2732	0.529363	0.264681	−	0.964336i	$-0.414733\pi$
$727$	14.2732	0.529363	0.264681	+	0.964336i	$0.414733\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−36.3921	−1.34693
$731$	−3.70488	−0.137030
$732$	32.5808	1.20422
$733$	2.31207	0.0853983	0.0426992	−	0.999088i	$-0.486404\pi$
$733$	2.31207	0.0853983	0.0426992	+	0.999088i	$0.486404\pi$
$734$	56.1600	2.07291
$735$	0	0
$736$	0	0
$737$	4.21037	0.155091
$738$	−10.9457	−0.402916
$739$	−26.0558	−0.958480	−0.479240	−	0.877684i	$-0.659088\pi$
$739$	−26.0558	−0.958480	−0.479240	+	0.877684i	$0.659088\pi$
$740$	26.7500	0.983350
$741$	−7.53341	−0.276747
$742$	0	0
$743$	5.04737	0.185170	0.0925851	−	0.995705i	$-0.470487\pi$
$743$	5.04737	0.185170	0.0925851	+	0.995705i	$0.470487\pi$
$744$	−5.17548	−0.189742
$745$	−27.6157	−1.01176
$746$	10.7159	0.392335
$747$	9.17548	0.335713
$748$	−1.77018	−0.0647244
$749$	0	0
$750$	−25.3642	−0.926169
$751$	−40.4387	−1.47563	−0.737815	−	0.675002i	$-0.764143\pi$
$751$	−40.4387	−1.47563	−0.737815	+	0.675002i	$0.764143\pi$
$752$	22.2834	0.812593
$753$	22.0194	0.802433
$754$	53.6887	1.95523
$755$	−7.74378	−0.281825
$756$	0	0
$757$	−24.4263	−0.887788	−0.443894	−	0.896079i	$-0.646403\pi$
$757$	−24.4263	−0.887788	−0.443894	+	0.896079i	$0.646403\pi$
$758$	−64.6344	−2.34763
$759$	0	0
$760$	2.69641	0.0978089
$761$	−5.18940	−0.188116	−0.0940579	−	0.995567i	$-0.529984\pi$
$761$	−5.18940	−0.188116	−0.0940579	+	0.995567i	$0.529984\pi$
$762$	43.2702	1.56751
$763$	0	0
$764$	27.2702	0.986602
$765$	1.17548	0.0424997
$766$	−19.3230	−0.698170
$767$	−14.4791	−0.522809
$768$	−6.01320	−0.216983
$769$	28.1670	1.01573	0.507864	−	0.861437i	$-0.330435\pi$
$769$	28.1670	1.01573	0.507864	+	0.861437i	$0.330435\pi$
$770$	0	0
$771$	27.3161	0.983765
$772$	−19.4178	−0.698861
$773$	−38.2662	−1.37634	−0.688170	−	0.725549i	$-0.741586\pi$
$773$	−38.2662	−1.37634	−0.688170	+	0.725549i	$0.741586\pi$
$774$	10.9457	0.393434
$775$	−11.9222	−0.428259
$776$	13.6351	0.489472
$777$	0	0
$778$	44.9846	1.61277
$779$	8.49852	0.304491
$780$	−18.6289	−0.667021
$781$	−1.85244	−0.0662856
$782$	0	0
$783$	5.53341	0.197748
$784$	0	0
$785$	−18.1087	−0.646326
$786$	−31.8649	−1.13658
$787$	13.4247	0.478540	0.239270	−	0.970953i	$-0.423092\pi$
$787$	13.4247	0.478540	0.239270	+	0.970953i	$0.423092\pi$
$788$	49.1880	1.75225
$789$	11.8719	0.422650
$790$	43.2702	1.53949
$791$	0	0
$792$	1.00000	0.0355335
$793$	60.4457	2.14649
$794$	−61.2438	−2.17346
$795$	4.45963	0.158167
$796$	−58.1770	−2.06203
$797$	22.3440	0.791465	0.395733	−	0.918366i	$-0.370491\pi$
$797$	22.3440	0.791465	0.395733	+	0.918366i	$0.370491\pi$
$798$	0	0
$799$	5.63511	0.199356
$800$	18.3983	0.650479
$801$	−4.71585	−0.166626
$802$	68.4193	2.41597
$803$	10.4791	0.369799
$804$	−10.4115	−0.367187
$805$	0	0
$806$	−50.2159	−1.76878
$807$	5.63511	0.198365
$808$	14.6072	0.513879
$809$	31.8455	1.11963	0.559814	−	0.828618i	$-0.310873\pi$
$809$	31.8455	1.11963	0.559814	+	0.828618i	$0.310873\pi$
$810$	−3.47283	−0.122023
$811$	−8.70889	−0.305811	−0.152905	−	0.988241i	$-0.548863\pi$
$811$	−8.70889	−0.305811	−0.152905	+	0.988241i	$0.548863\pi$
$812$	0	0
$813$	−5.68097	−0.199240
$814$	−13.9325	−0.488333
$815$	−9.26470	−0.324528
$816$	−2.02641	−0.0709385
$817$	−8.49852	−0.297326
$818$	−4.88982	−0.170968
$819$	0	0
$820$	21.0154	0.733891
$821$	6.45267	0.225200	0.112600	−	0.993640i	$-0.464082\pi$
$821$	6.45267	0.225200	0.112600	+	0.993640i	$0.464082\pi$
$822$	30.5808	1.06663
$823$	25.3719	0.884410	0.442205	−	0.896914i	$-0.354196\pi$
$823$	25.3719	0.884410	0.442205	+	0.896914i	$0.354196\pi$
$824$	−17.8913	−0.623274
$825$	2.30359	0.0802009
$826$	0	0
$827$	14.0194	0.487504	0.243752	−	0.969838i	$-0.421622\pi$
$827$	14.0194	0.487504	0.243752	+	0.969838i	$0.421622\pi$
$828$	0	0
$829$	48.9970	1.70174	0.850869	−	0.525378i	$-0.176076\pi$
$829$	48.9970	1.70174	0.850869	+	0.525378i	$0.176076\pi$
$830$	−31.8649	−1.10605
$831$	25.4876	0.884153
$832$	51.5195	1.78612
$833$	0	0
$834$	−10.8634	−0.376169
$835$	26.2732	0.909221
$836$	−4.06058	−0.140438
$837$	−5.17548	−0.178891
$838$	−81.2680	−2.80736
$839$	13.2647	0.457948	0.228974	−	0.973433i	$-0.426463\pi$
$839$	13.2647	0.457948	0.228974	+	0.973433i	$0.426463\pi$
$840$	0	0
$841$	1.61862	0.0558144
$842$	32.7959	1.13022
$843$	−17.5723	−0.605222
$844$	−10.5249	−0.362283
$845$	−13.2144	−0.454588
$846$	−16.6483	−0.572381
$847$	0	0
$848$	−7.68793	−0.264005
$849$	25.1296	0.862447
$850$	3.48755	0.119622
$851$	0	0
$852$	4.58078	0.156935
$853$	4.25622	0.145730	0.0728651	−	0.997342i	$-0.476786\pi$
$853$	4.25622	0.145730	0.0728651	+	0.997342i	$0.476786\pi$
$854$	0	0
$855$	2.69641	0.0922151
$856$	11.1560	0.381305
$857$	23.5264	0.803648	0.401824	−	0.915717i	$-0.368376\pi$
$857$	23.5264	0.803648	0.401824	+	0.915717i	$0.368376\pi$
$858$	9.70265	0.331243
$859$	31.0279	1.05866	0.529330	−	0.848416i	$-0.322444\pi$
$859$	31.0279	1.05866	0.529330	+	0.848416i	$0.322444\pi$
$860$	−21.0154	−0.716620
$861$	0	0
$862$	80.2959	2.73489
$863$	−46.3121	−1.57648	−0.788241	−	0.615367i	$-0.789008\pi$
$863$	−46.3121	−1.57648	−0.788241	+	0.615367i	$0.789008\pi$
$864$	7.98680	0.271716
$865$	−24.6770	−0.839042
$866$	−21.3789	−0.726484
$867$	16.4876	0.559947
$868$	0	0
$869$	−12.4596	−0.422664
$870$	−19.2166	−0.651504
$871$	−19.3161	−0.654500
$872$	−0.108664	−0.00367984
$873$	13.6351	0.461479
$874$	0	0
$875$	0	0
$876$	−25.9130	−0.875520
$877$	−12.8245	−0.433053	−0.216527	−	0.976277i	$-0.569473\pi$
$877$	−12.8245	−0.433053	−0.216527	+	0.976277i	$0.569473\pi$
$878$	47.6329	1.60753
$879$	−14.3510	−0.484046
$880$	4.64832	0.156695
$881$	56.9512	1.91873	0.959367	−	0.282160i	$-0.0910510\pi$
$881$	56.9512	1.91873	0.959367	+	0.282160i	$0.0910510\pi$
$882$	0	0
$883$	8.00696	0.269456	0.134728	−	0.990883i	$-0.456984\pi$
$883$	8.00696	0.269456	0.134728	+	0.990883i	$0.456984\pi$
$884$	8.12115	0.273144
$885$	5.18244	0.174206
$886$	43.7004	1.46814
$887$	−35.7827	−1.20146	−0.600732	−	0.799450i	$-0.705124\pi$
$887$	−35.7827	−1.20146	−0.600732	+	0.799450i	$0.705124\pi$
$888$	6.58774	0.221070
$889$	0	0
$890$	16.3774	0.548971
$891$	1.00000	0.0335013
$892$	9.25774	0.309972
$893$	12.9262	0.432559
$894$	−35.5676	−1.18956
$895$	−16.2423	−0.542920
$896$	0	0
$897$	0	0
$898$	86.3106	2.88022
$899$	−28.6381	−0.955133
$900$	−5.69641	−0.189880
$901$	−1.94415	−0.0647690
$902$	−10.9457	−0.364451
$903$	0	0
$904$	−13.0668	−0.434596
$905$	11.2064	0.372512
$906$	−9.97359	−0.331350
$907$	5.43171	0.180357	0.0901784	−	0.995926i	$-0.471256\pi$
$907$	5.43171	0.180357	0.0901784	+	0.995926i	$0.471256\pi$
$908$	38.2982	1.27097
$909$	14.6072	0.484490
$910$	0	0
$911$	16.6770	0.552532	0.276266	−	0.961081i	$-0.410903\pi$
$911$	16.6770	0.552532	0.276266	+	0.961081i	$0.410903\pi$
$912$	−4.64832	−0.153921
$913$	9.17548	0.303664
$914$	−60.0419	−1.98601
$915$	−21.6351	−0.715235
$916$	19.7827	0.653638
$917$	0	0
$918$	1.51396	0.0499682
$919$	14.4038	0.475137	0.237568	−	0.971371i	$-0.423650\pi$
$919$	14.4038	0.475137	0.237568	+	0.971371i	$0.423650\pi$
$920$	0	0
$921$	13.1366	0.432865
$922$	−45.2966	−1.49177
$923$	8.49852	0.279732
$924$	0	0
$925$	15.1755	0.498967
$926$	43.8796	1.44197
$927$	−17.8913	−0.587629
$928$	44.1942	1.45075
$929$	−1.89830	−0.0622811	−0.0311405	−	0.999515i	$-0.509914\pi$
$929$	−1.89830	−0.0622811	−0.0311405	+	0.999515i	$0.509914\pi$
$930$	17.9736	0.589377
$931$	0	0
$932$	−13.7004	−0.448772
$933$	−14.3510	−0.469830
$934$	−56.8906	−1.86152
$935$	1.17548	0.0384424
$936$	−4.58774	−0.149955
$937$	13.1755	0.430424	0.215212	−	0.976567i	$-0.430956\pi$
$937$	13.1755	0.430424	0.215212	+	0.976567i	$0.430956\pi$
$938$	0	0
$939$	23.7438	0.774849
$940$	31.9644	1.04256
$941$	−0.381842	−0.0124477	−0.00622385	−	0.999981i	$-0.501981\pi$
$941$	−0.381842	−0.0124477	−0.00622385	+	0.999981i	$0.501981\pi$
$942$	−23.3230	−0.759906
$943$	0	0
$944$	−8.93398	−0.290776
$945$	0	0
$946$	10.9457	0.355874
$947$	−52.2284	−1.69719	−0.848597	−	0.529040i	$-0.822552\pi$
$947$	−52.2284	−1.69719	−0.848597	+	0.529040i	$0.822552\pi$
$948$	30.8106	1.00068
$949$	−48.0753	−1.56059
$950$	8.00000	0.259554
$951$	6.45963	0.209468
$952$	0	0
$953$	−45.7757	−1.48282	−0.741410	−	0.671052i	$-0.765843\pi$
$953$	−45.7757	−1.48282	−0.741410	+	0.671052i	$0.765843\pi$
$954$	5.74378	0.185962
$955$	−18.1087	−0.585983
$956$	−3.12739	−0.101147
$957$	5.53341	0.178870
$958$	−65.1616	−2.10527
$959$	0	0
$960$	−18.4402	−0.595154
$961$	−4.21438	−0.135948
$962$	63.9185	2.06082
$963$	11.1560	0.359498
$964$	59.0312	1.90127
$965$	12.8943	0.415082
$966$	0	0
$967$	−15.3230	−0.492756	−0.246378	−	0.969174i	$-0.579240\pi$
$967$	−15.3230	−0.492756	−0.246378	+	0.969174i	$0.579240\pi$
$968$	1.00000	0.0321412
$969$	−1.17548	−0.0377620
$970$	−47.3525	−1.52040
$971$	−12.0753	−0.387515	−0.193757	−	0.981049i	$-0.562067\pi$
$971$	−12.0753	−0.387515	−0.193757	+	0.981049i	$0.562067\pi$
$972$	−2.47283	−0.0793162
$973$	0	0
$974$	−49.6740	−1.59166
$975$	−10.5683	−0.338456
$976$	37.2966	1.19384
$977$	−39.3650	−1.25940	−0.629698	−	0.776840i	$-0.716822\pi$
$977$	−39.3650	−1.25940	−0.629698	+	0.776840i	$0.716822\pi$
$978$	−11.9325	−0.381558
$979$	−4.71585	−0.150719
$980$	0	0
$981$	−0.108664	−0.00346939
$982$	−35.1638	−1.12212
$983$	−10.5683	−0.337076	−0.168538	−	0.985695i	$-0.553905\pi$
$983$	−10.5683	−0.337076	−0.168538	+	0.985695i	$0.553905\pi$
$984$	5.17548	0.164988
$985$	−32.6630	−1.04073
$986$	8.37737	0.266790
$987$	0	0
$988$	18.6289	0.592663
$989$	0	0
$990$	−3.47283	−0.110374
$991$	42.8036	1.35970	0.679851	−	0.733350i	$-0.262044\pi$
$991$	42.8036	1.35970	0.679851	+	0.733350i	$0.262044\pi$
$992$	−41.3355	−1.31240
$993$	−31.2702	−0.992331
$994$	0	0
$995$	38.6322	1.22472
$996$	−22.6894	−0.718943
$997$	10.3121	0.326587	0.163293	−	0.986578i	$-0.447788\pi$
$997$	10.3121	0.326587	0.163293	+	0.986578i	$0.447788\pi$
$998$	−16.9045	−0.535104
$999$	6.58774	0.208427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.a.q.1.3	✓	3
3.2	odd	2		4851.2.a.bm.1.1		3
7.6	odd	2		1617.2.a.r.1.3	yes	3
21.20	even	2		4851.2.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.a.q.1.3	✓	3	1.1	even	1	trivial
1617.2.a.r.1.3	yes	3	7.6	odd	2
4851.2.a.bl.1.1		3	21.20	even	2
4851.2.a.bm.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11491 q^{2} -1.00000 q^{3} +2.47283 q^{4} -1.64207 q^{5} -2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.11491 q^{2} -1.00000 q^{3} +2.47283 q^{4} -1.64207 q^{5} -2.11491 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.47283 q^{10} +1.00000 q^{11} -2.47283 q^{12} -4.58774 q^{13} +1.64207 q^{15} -2.83076 q^{16} -0.715853 q^{17} +2.11491 q^{18} -1.64207 q^{19} -4.06058 q^{20} +2.11491 q^{22} -1.00000 q^{24} -2.30359 q^{25} -9.70265 q^{26} -1.00000 q^{27} -5.53341 q^{29} +3.47283 q^{30} +5.17548 q^{31} -7.98680 q^{32} -1.00000 q^{33} -1.51396 q^{34} +2.47283 q^{36} -6.58774 q^{37} -3.47283 q^{38} +4.58774 q^{39} -1.64207 q^{40} -5.17548 q^{41} +5.17548 q^{43} +2.47283 q^{44} -1.64207 q^{45} -7.87189 q^{47} +2.83076 q^{48} -4.87189 q^{50} +0.715853 q^{51} -11.3447 q^{52} +2.71585 q^{53} -2.11491 q^{54} -1.64207 q^{55} +1.64207 q^{57} -11.7026 q^{58} +3.15604 q^{59} +4.06058 q^{60} -13.1755 q^{61} +10.9457 q^{62} -11.2298 q^{64} +7.53341 q^{65} -2.11491 q^{66} +4.21037 q^{67} -1.77018 q^{68} -1.85244 q^{71} +1.00000 q^{72} +10.4791 q^{73} -13.9325 q^{74} +2.30359 q^{75} -4.06058 q^{76} +9.70265 q^{78} -12.4596 q^{79} +4.64832 q^{80} +1.00000 q^{81} -10.9457 q^{82} +9.17548 q^{83} +1.17548 q^{85} +10.9457 q^{86} +5.53341 q^{87} +1.00000 q^{88} -4.71585 q^{89} -3.47283 q^{90} -5.17548 q^{93} -16.6483 q^{94} +2.69641 q^{95} +7.98680 q^{96} +13.6351 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{4} - 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^4 - 4 * q^5 + 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{4} - 4 q^{5} + 3 q^{8} + 3 q^{9} - 5 q^{10} + 3 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{15} - 4 q^{16} - 4 q^{17} - 4 q^{19} + 5 q^{20} - 3 q^{24} + 3 q^{25} - 11 q^{26} - 3 q^{27} + 6 q^{29} + 5 q^{30} - 8 q^{31} - 4 q^{32} - 3 q^{33} + 10 q^{34} + 2 q^{36} - 8 q^{37} - 5 q^{38} + 2 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 2 q^{44} - 4 q^{45} - 10 q^{47} + 4 q^{48} - q^{50} + 4 q^{51} - 15 q^{52} + 10 q^{53} - 4 q^{55} + 4 q^{57} - 17 q^{58} - 6 q^{59} - 5 q^{60} - 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 3 q^{72} - 2 q^{73} - 11 q^{74} - 3 q^{75} + 5 q^{76} + 11 q^{78} - 12 q^{79} - 15 q^{80} + 3 q^{81} - 22 q^{82} + 4 q^{83} - 20 q^{85} + 22 q^{86} - 6 q^{87} + 3 q^{88} - 16 q^{89} - 5 q^{90} + 8 q^{93} - 21 q^{94} + 18 q^{95} + 4 q^{96} - 8 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^4 - 4 * q^5 + 3 * q^8 + 3 * q^9 - 5 * q^10 + 3 * q^11 - 2 * q^12 - 2 * q^13 + 4 * q^15 - 4 * q^16 - 4 * q^17 - 4 * q^19 + 5 * q^20 - 3 * q^24 + 3 * q^25 - 11 * q^26 - 3 * q^27 + 6 * q^29 + 5 * q^30 - 8 * q^31 - 4 * q^32 - 3 * q^33 + 10 * q^34 + 2 * q^36 - 8 * q^37 - 5 * q^38 + 2 * q^39 - 4 * q^40 + 8 * q^41 - 8 * q^43 + 2 * q^44 - 4 * q^45 - 10 * q^47 + 4 * q^48 - q^50 + 4 * q^51 - 15 * q^52 + 10 * q^53 - 4 * q^55 + 4 * q^57 - 17 * q^58 - 6 * q^59 - 5 * q^60 - 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 3 * q^72 - 2 * q^73 - 11 * q^74 - 3 * q^75 + 5 * q^76 + 11 * q^78 - 12 * q^79 - 15 * q^80 + 3 * q^81 - 22 * q^82 + 4 * q^83 - 20 * q^85 + 22 * q^86 - 6 * q^87 + 3 * q^88 - 16 * q^89 - 5 * q^90 + 8 * q^93 - 21 * q^94 + 18 * q^95 + 4 * q^96 - 8 * q^97 + 3 * q^99

Introduction

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