LMFDB - Embedded newform 867.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [867,2,Mod(1,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("867.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.857616$ of defining polynomial
Character	$\chi$	$=$	867.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.907065 q^{2} -1.00000 q^{3} -1.17723 q^{4} -3.19333 q^{5} -0.907065 q^{6} -3.56234 q^{7} -2.88196 q^{8} +1.00000 q^{9} -2.89656 q^{10} +3.27613 q^{11} +1.17723 q^{12} +5.58411 q^{13} -3.23128 q^{14} +3.19333 q^{15} -0.259660 q^{16} +0.907065 q^{18} -4.23574 q^{19} +3.75929 q^{20} +3.56234 q^{21} +2.97167 q^{22} +4.60128 q^{23} +2.88196 q^{24} +5.19735 q^{25} +5.06515 q^{26} -1.00000 q^{27} +4.19370 q^{28} -2.08430 q^{29} +2.89656 q^{30} -0.448868 q^{31} +5.52839 q^{32} -3.27613 q^{33} +11.3757 q^{35} -1.17723 q^{36} +0.742912 q^{37} -3.84209 q^{38} -5.58411 q^{39} +9.20304 q^{40} +4.49315 q^{41} +3.23128 q^{42} +6.10953 q^{43} -3.85677 q^{44} -3.19333 q^{45} +4.17367 q^{46} -2.26801 q^{47} +0.259660 q^{48} +5.69028 q^{49} +4.71434 q^{50} -6.57379 q^{52} +7.55680 q^{53} -0.907065 q^{54} -10.4618 q^{55} +10.2665 q^{56} +4.23574 q^{57} -1.89059 q^{58} -2.83196 q^{59} -3.75929 q^{60} +3.91386 q^{61} -0.407152 q^{62} -3.56234 q^{63} +5.53393 q^{64} -17.8319 q^{65} -2.97167 q^{66} -14.5019 q^{67} -4.60128 q^{69} +10.3185 q^{70} +3.64422 q^{71} -2.88196 q^{72} -11.6532 q^{73} +0.673870 q^{74} -5.19735 q^{75} +4.98645 q^{76} -11.6707 q^{77} -5.06515 q^{78} +5.63303 q^{79} +0.829180 q^{80} +1.00000 q^{81} +4.07558 q^{82} -3.92274 q^{83} -4.19370 q^{84} +5.54175 q^{86} +2.08430 q^{87} -9.44167 q^{88} +14.6181 q^{89} -2.89656 q^{90} -19.8925 q^{91} -5.41678 q^{92} +0.448868 q^{93} -2.05724 q^{94} +13.5261 q^{95} -5.52839 q^{96} +0.828387 q^{97} +5.16145 q^{98} +3.27613 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 6 q^{3} + 9 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{7} + 12 q^{8} + 6 q^{9} - 12 q^{10} + 9 q^{11} - 9 q^{12} + 9 q^{13} + 6 q^{14} - 3 q^{15} + 15 q^{16} + 3 q^{18} + 9 q^{19} - 6 q^{20} + 3 q^{21}+ \cdots + 9 q^{99}+O(q^{100}) $ 6 * q + 3 * q^2 - 6 * q^3 + 9 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^7 + 12 * q^8 + 6 * q^9 - 12 * q^10 + 9 * q^11 - 9 * q^12 + 9 * q^13 + 6 * q^14 - 3 * q^15 + 15 * q^16 + 3 * q^18 + 9 * q^19 - 6 * q^20 + 3 * q^21 + 18 * q^22 + 9 * q^23 - 12 * q^24 + 15 * q^25 - 12 * q^26 - 6 * q^27 + 15 * q^28 + 6 * q^29 + 12 * q^30 - 24 * q^31 + 42 * q^32 - 9 * q^33 + 9 * q^36 + 3 * q^37 - 6 * q^38 - 9 * q^39 + 3 * q^40 + 18 * q^41 - 6 * q^42 + 3 * q^44 + 3 * q^45 - 15 * q^46 + 24 * q^47 - 15 * q^48 + 21 * q^49 + 12 * q^50 - 18 * q^52 + 24 * q^53 - 3 * q^54 - 24 * q^55 + 54 * q^56 - 9 * q^57 - 3 * q^58 - 9 * q^59 + 6 * q^60 - 21 * q^61 + 30 * q^62 - 3 * q^63 + 24 * q^64 - 9 * q^65 - 18 * q^66 - 6 * q^67 - 9 * q^69 - 3 * q^70 + 27 * q^71 + 12 * q^72 - 18 * q^73 + 36 * q^74 - 15 * q^75 - 3 * q^76 + 33 * q^77 + 12 * q^78 - 24 * q^79 + 3 * q^80 + 6 * q^81 + 15 * q^82 + 6 * q^83 - 15 * q^84 + 6 * q^86 - 6 * q^87 + 24 * q^88 - 12 * q^90 - 39 * q^91 + 24 * q^93 - 15 * q^94 + 42 * q^95 - 42 * q^96 + 33 * q^97 - 57 * q^98 + 9 * 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 $

$2$	0.907065	0.641392	0.320696	−	0.947182i	$-0.396083\pi$
$2$	0.907065	0.641392	0.320696	+	0.947182i	$0.396083\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.17723	−0.588616
$5$	−3.19333	−1.42810	−0.714050	−	0.700095i	$-0.753141\pi$
$5$	−3.19333	−1.42810	−0.714050	+	0.700095i	$0.753141\pi$
$6$	−0.907065	−0.370308
$7$	−3.56234	−1.34644	−0.673219	−	0.739443i	$-0.735089\pi$
$7$	−3.56234	−1.34644	−0.673219	+	0.739443i	$0.735089\pi$
$8$	−2.88196	−1.01893
$9$	1.00000	0.333333
$10$	−2.89656	−0.915972
$11$	3.27613	0.987791	0.493895	−	0.869521i	$-0.335573\pi$
$11$	3.27613	0.987791	0.493895	+	0.869521i	$0.335573\pi$
$12$	1.17723	0.339838
$13$	5.58411	1.54875	0.774377	−	0.632725i	$-0.218064\pi$
$13$	5.58411	1.54875	0.774377	+	0.632725i	$0.218064\pi$
$14$	−3.23128	−0.863595
$15$	3.19333	0.824514
$16$	−0.259660	−0.0649150
$17$	0	0
$17$	0	0
$18$	0.907065	0.213797
$19$	−4.23574	−0.971745	−0.485872	−	0.874030i	$-0.661498\pi$
$19$	−4.23574	−0.971745	−0.485872	+	0.874030i	$0.661498\pi$
$20$	3.75929	0.840603
$21$	3.56234	0.777367
$22$	2.97167	0.633561
$23$	4.60128	0.959434	0.479717	−	0.877423i	$-0.340739\pi$
$23$	4.60128	0.959434	0.479717	+	0.877423i	$0.340739\pi$
$24$	2.88196	0.588277
$25$	5.19735	1.03947
$26$	5.06515	0.993358
$27$	−1.00000	−0.192450
$28$	4.19370	0.792535
$29$	−2.08430	−0.387044	−0.193522	−	0.981096i	$-0.561991\pi$
$29$	−2.08430	−0.387044	−0.193522	+	0.981096i	$0.561991\pi$
$30$	2.89656	0.528837
$31$	−0.448868	−0.0806190	−0.0403095	−	0.999187i	$-0.512834\pi$
$31$	−0.448868	−0.0806190	−0.0403095	+	0.999187i	$0.512834\pi$
$32$	5.52839	0.977290
$33$	−3.27613	−0.570301
$34$	0	0
$35$	11.3757	1.92285
$36$	−1.17723	−0.196205
$37$	0.742912	0.122134	0.0610670	−	0.998134i	$-0.480550\pi$
$37$	0.742912	0.122134	0.0610670	+	0.998134i	$0.480550\pi$
$38$	−3.84209	−0.623270
$39$	−5.58411	−0.894173
$40$	9.20304	1.45513
$41$	4.49315	0.701712	0.350856	−	0.936430i	$-0.385891\pi$
$41$	4.49315	0.701712	0.350856	+	0.936430i	$0.385891\pi$
$42$	3.23128	0.498597
$43$	6.10953	0.931695	0.465848	−	0.884865i	$-0.345750\pi$
$43$	6.10953	0.931695	0.465848	+	0.884865i	$0.345750\pi$
$44$	−3.85677	−0.581429
$45$	−3.19333	−0.476033
$46$	4.17367	0.615374
$47$	−2.26801	−0.330824	−0.165412	−	0.986225i	$-0.552895\pi$
$47$	−2.26801	−0.330824	−0.165412	+	0.986225i	$0.552895\pi$
$48$	0.259660	0.0374787
$49$	5.69028	0.812897
$50$	4.71434	0.666708
$51$	0	0
$52$	−6.57379	−0.911621
$53$	7.55680	1.03801	0.519003	−	0.854772i	$-0.326303\pi$
$53$	7.55680	1.03801	0.519003	+	0.854772i	$0.326303\pi$
$54$	−0.907065	−0.123436
$55$	−10.4618	−1.41066
$56$	10.2665	1.37192
$57$	4.23574	0.561037
$58$	−1.89059	−0.248247
$59$	−2.83196	−0.368690	−0.184345	−	0.982862i	$-0.559016\pi$
$59$	−2.83196	−0.368690	−0.184345	+	0.982862i	$0.559016\pi$
$60$	−3.75929	−0.485322
$61$	3.91386	0.501118	0.250559	−	0.968101i	$-0.419386\pi$
$61$	3.91386	0.501118	0.250559	+	0.968101i	$0.419386\pi$
$62$	−0.407152	−0.0517084
$63$	−3.56234	−0.448813
$64$	5.53393	0.691741
$65$	−17.8319	−2.21178
$66$	−2.97167	−0.365787
$67$	−14.5019	−1.77168	−0.885842	−	0.463986i	$-0.846419\pi$
$67$	−14.5019	−1.77168	−0.885842	+	0.463986i	$0.846419\pi$
$68$	0	0
$69$	−4.60128	−0.553930
$70$	10.3185	1.23330
$71$	3.64422	0.432489	0.216245	−	0.976339i	$-0.430619\pi$
$71$	3.64422	0.432489	0.216245	+	0.976339i	$0.430619\pi$
$72$	−2.88196	−0.339642
$73$	−11.6532	−1.36390	−0.681951	−	0.731398i	$-0.738868\pi$
$73$	−11.6532	−1.36390	−0.681951	+	0.731398i	$0.738868\pi$
$74$	0.673870	0.0783358
$75$	−5.19735	−0.600139
$76$	4.98645	0.571985
$77$	−11.6707	−1.33000
$78$	−5.06515	−0.573516
$79$	5.63303	0.633766	0.316883	−	0.948465i	$-0.397364\pi$
$79$	5.63303	0.633766	0.316883	+	0.948465i	$0.397364\pi$
$80$	0.829180	0.0927051
$81$	1.00000	0.111111
$82$	4.07558	0.450072
$83$	−3.92274	−0.430577	−0.215288	−	0.976551i	$-0.569069\pi$
$83$	−3.92274	−0.430577	−0.215288	+	0.976551i	$0.569069\pi$
$84$	−4.19370	−0.457571
$85$	0	0
$86$	5.54175	0.597582
$87$	2.08430	0.223460
$88$	−9.44167	−1.00649
$89$	14.6181	1.54951	0.774757	−	0.632259i	$-0.217872\pi$
$89$	14.6181	1.54951	0.774757	+	0.632259i	$0.217872\pi$
$90$	−2.89656	−0.305324
$91$	−19.8925	−2.08530
$92$	−5.41678	−0.564738
$93$	0.448868	0.0465454
$94$	−2.05724	−0.212188
$95$	13.5261	1.38775
$96$	−5.52839	−0.564239
$97$	0.828387	0.0841099	0.0420550	−	0.999115i	$-0.486610\pi$
$97$	0.828387	0.0841099	0.0420550	+	0.999115i	$0.486610\pi$
$98$	5.16145	0.521386
$99$	3.27613	0.329264
$100$	−6.11849	−0.611849
$101$	4.92944	0.490497	0.245249	−	0.969460i	$-0.421130\pi$
$101$	4.92944	0.490497	0.245249	+	0.969460i	$0.421130\pi$
$102$	0	0
$103$	11.4347	1.12669	0.563347	−	0.826221i	$-0.309514\pi$
$103$	11.4347	1.12669	0.563347	+	0.826221i	$0.309514\pi$
$104$	−16.0932	−1.57807
$105$	−11.3757	−1.11016
$106$	6.85452	0.665769
$107$	17.5925	1.70073	0.850366	−	0.526192i	$-0.176381\pi$
$107$	17.5925	1.70073	0.850366	+	0.526192i	$0.176381\pi$
$108$	1.17723	0.113279
$109$	14.1515	1.35546	0.677732	−	0.735309i	$-0.262963\pi$
$109$	14.1515	1.35546	0.677732	+	0.735309i	$0.262963\pi$
$110$	−9.48951	−0.904789
$111$	−0.742912	−0.0705141
$112$	0.924998	0.0874041
$113$	20.4015	1.91921	0.959607	−	0.281346i	$-0.0907807\pi$
$113$	20.4015	1.91921	0.959607	+	0.281346i	$0.0907807\pi$
$114$	3.84209	0.359845
$115$	−14.6934	−1.37017
$116$	2.45370	0.227821
$117$	5.58411	0.516251
$118$	−2.56877	−0.236475
$119$	0	0
$120$	−9.20304	−0.840119
$121$	−0.266966	−0.0242696
$122$	3.55013	0.321413
$123$	−4.49315	−0.405133
$124$	0.528421	0.0474536
$125$	−0.630210	−0.0563677
$126$	−3.23128	−0.287865
$127$	2.93340	0.260297	0.130148	−	0.991495i	$-0.458455\pi$
$127$	2.93340	0.260297	0.130148	+	0.991495i	$0.458455\pi$
$128$	−6.03714	−0.533613
$129$	−6.10953	−0.537914
$130$	−16.1747	−1.41862
$131$	−6.61671	−0.578104	−0.289052	−	0.957313i	$-0.593340\pi$
$131$	−6.61671	−0.578104	−0.289052	+	0.957313i	$0.593340\pi$
$132$	3.85677	0.335688
$133$	15.0891	1.30839
$134$	−13.1541	−1.13634
$135$	3.19333	0.274838
$136$	0	0
$137$	−10.4851	−0.895804	−0.447902	−	0.894083i	$-0.647829\pi$
$137$	−10.4851	−0.895804	−0.447902	+	0.894083i	$0.647829\pi$
$138$	−4.17367	−0.355286
$139$	−5.28597	−0.448350	−0.224175	−	0.974549i	$-0.571969\pi$
$139$	−5.28597	−0.448350	−0.224175	+	0.974549i	$0.571969\pi$
$140$	−13.3919	−1.13182
$141$	2.26801	0.191001
$142$	3.30555	0.277395
$143$	18.2943	1.52984
$144$	−0.259660	−0.0216383
$145$	6.65585	0.552738
$146$	−10.5702	−0.874795
$147$	−5.69028	−0.469326
$148$	−0.874580	−0.0718901
$149$	−22.4592	−1.83993	−0.919965	−	0.392000i	$-0.871783\pi$
$149$	−22.4592	−1.83993	−0.919965	+	0.392000i	$0.871783\pi$
$150$	−4.71434	−0.384924
$151$	−9.52717	−0.775310	−0.387655	−	0.921804i	$-0.626715\pi$
$151$	−9.52717	−0.775310	−0.387655	+	0.921804i	$0.626715\pi$
$152$	12.2072	0.990136
$153$	0	0
$154$	−10.5861	−0.853051
$155$	1.43338	0.115132
$156$	6.57379	0.526325
$157$	14.1185	1.12678	0.563390	−	0.826191i	$-0.309497\pi$
$157$	14.1185	1.12678	0.563390	+	0.826191i	$0.309497\pi$
$158$	5.10953	0.406492
$159$	−7.55680	−0.599294
$160$	−17.6540	−1.39567
$161$	−16.3913	−1.29182
$162$	0.907065	0.0712658
$163$	−3.53391	−0.276797	−0.138398	−	0.990377i	$-0.544195\pi$
$163$	−3.53391	−0.276797	−0.138398	+	0.990377i	$0.544195\pi$
$164$	−5.28948	−0.413039
$165$	10.4618	0.814447
$166$	−3.55818	−0.276169
$167$	16.1872	1.25260	0.626300	−	0.779582i	$-0.284568\pi$
$167$	16.1872	1.25260	0.626300	+	0.779582i	$0.284568\pi$
$168$	−10.2665	−0.792079
$169$	18.1823	1.39864
$170$	0	0
$171$	−4.23574	−0.323915
$172$	−7.19234	−0.548411
$173$	18.3180	1.39269	0.696346	−	0.717706i	$-0.254808\pi$
$173$	18.3180	1.39269	0.696346	+	0.717706i	$0.254808\pi$
$174$	1.89059	0.143326
$175$	−18.5147	−1.39958
$176$	−0.850680	−0.0641224
$177$	2.83196	0.212863
$178$	13.2596	0.993846
$179$	20.7057	1.54762	0.773809	−	0.633419i	$-0.218349\pi$
$179$	20.7057	1.54762	0.773809	+	0.633419i	$0.218349\pi$
$180$	3.75929	0.280201
$181$	−18.6500	−1.38624	−0.693122	−	0.720820i	$-0.743765\pi$
$181$	−18.6500	−1.38624	−0.693122	+	0.720820i	$0.743765\pi$
$182$	−18.0438	−1.33750
$183$	−3.91386	−0.289321
$184$	−13.2607	−0.977592
$185$	−2.37236	−0.174420
$186$	0.407152	0.0298539
$187$	0	0
$188$	2.66998	0.194728
$189$	3.56234	0.259122
$190$	12.2691	0.890091
$191$	2.19374	0.158734	0.0793669	−	0.996845i	$-0.474710\pi$
$191$	2.19374	0.158734	0.0793669	+	0.996845i	$0.474710\pi$
$192$	−5.53393	−0.399377
$193$	−3.31205	−0.238407	−0.119203	−	0.992870i	$-0.538034\pi$
$193$	−3.31205	−0.238407	−0.119203	+	0.992870i	$0.538034\pi$
$194$	0.751401	0.0539474
$195$	17.8319	1.27697
$196$	−6.69878	−0.478484
$197$	18.0182	1.28374	0.641872	−	0.766811i	$-0.278158\pi$
$197$	18.0182	1.28374	0.641872	+	0.766811i	$0.278158\pi$
$198$	2.97167	0.211187
$199$	3.98156	0.282245	0.141123	−	0.989992i	$-0.454929\pi$
$199$	3.98156	0.282245	0.141123	+	0.989992i	$0.454929\pi$
$200$	−14.9785	−1.05914
$201$	14.5019	1.02288
$202$	4.47132	0.314601
$203$	7.42498	0.521131
$204$	0	0
$205$	−14.3481	−1.00211
$206$	10.3720	0.722652
$207$	4.60128	0.319811
$208$	−1.44997	−0.100537
$209$	−13.8768	−0.959881
$210$	−10.3185	−0.712046
$211$	0.200896	0.0138303	0.00691514	−	0.999976i	$-0.497799\pi$
$211$	0.200896	0.0138303	0.00691514	+	0.999976i	$0.497799\pi$
$212$	−8.89611	−0.610988
$213$	−3.64422	−0.249698
$214$	15.9576	1.09084
$215$	−19.5097	−1.33055
$216$	2.88196	0.196092
$217$	1.59902	0.108549
$218$	12.8363	0.869384
$219$	11.6532	0.787449
$220$	12.3159	0.830340
$221$	0	0
$222$	−0.673870	−0.0452272
$223$	−11.5997	−0.776770	−0.388385	−	0.921497i	$-0.626967\pi$
$223$	−11.5997	−0.776770	−0.388385	+	0.921497i	$0.626967\pi$
$224$	−19.6940	−1.31586
$225$	5.19735	0.346490
$226$	18.5055	1.23097
$227$	−9.51394	−0.631462	−0.315731	−	0.948849i	$-0.602250\pi$
$227$	−9.51394	−0.631462	−0.315731	+	0.948849i	$0.602250\pi$
$228$	−4.98645	−0.330236
$229$	−16.7700	−1.10819	−0.554097	−	0.832452i	$-0.686936\pi$
$229$	−16.7700	−1.10819	−0.554097	+	0.832452i	$0.686936\pi$
$230$	−13.3279	−0.878815
$231$	11.6707	0.767876
$232$	6.00686	0.394370
$233$	9.44867	0.619003	0.309501	−	0.950899i	$-0.399838\pi$
$233$	9.44867	0.619003	0.309501	+	0.950899i	$0.399838\pi$
$234$	5.06515	0.331119
$235$	7.24251	0.472450
$236$	3.33387	0.217017
$237$	−5.63303	−0.365905
$238$	0	0
$239$	6.42079	0.415326	0.207663	−	0.978200i	$-0.433414\pi$
$239$	6.42079	0.415326	0.207663	+	0.978200i	$0.433414\pi$
$240$	−0.829180	−0.0535233
$241$	−15.4831	−0.997355	−0.498678	−	0.866788i	$-0.666181\pi$
$241$	−15.4831	−0.997355	−0.498678	+	0.866788i	$0.666181\pi$
$242$	−0.242155	−0.0155663
$243$	−1.00000	−0.0641500
$244$	−4.60752	−0.294966
$245$	−18.1709	−1.16090
$246$	−4.07558	−0.259849
$247$	−23.6528	−1.50499
$248$	1.29362	0.0821448
$249$	3.92274	0.248594
$250$	−0.571642	−0.0361538
$251$	25.7024	1.62232	0.811162	−	0.584821i	$-0.198835\pi$
$251$	25.7024	1.62232	0.811162	+	0.584821i	$0.198835\pi$
$252$	4.19370	0.264178
$253$	15.0744	0.947720
$254$	2.66078	0.166952
$255$	0	0
$256$	−16.5439	−1.03400
$257$	2.50466	0.156236	0.0781182	−	0.996944i	$-0.475109\pi$
$257$	2.50466	0.156236	0.0781182	+	0.996944i	$0.475109\pi$
$258$	−5.54175	−0.345014
$259$	−2.64651	−0.164446
$260$	20.9923	1.30189
$261$	−2.08430	−0.129015
$262$	−6.00179	−0.370792
$263$	6.09867	0.376060	0.188030	−	0.982163i	$-0.439790\pi$
$263$	6.09867	0.376060	0.188030	+	0.982163i	$0.439790\pi$
$264$	9.44167	0.581095
$265$	−24.1314	−1.48238
$266$	13.6868	0.839194
$267$	−14.6181	−0.894612
$268$	17.0721	1.04284
$269$	4.11451	0.250866	0.125433	−	0.992102i	$-0.459968\pi$
$269$	4.11451	0.250866	0.125433	+	0.992102i	$0.459968\pi$
$270$	2.89656	0.176279
$271$	−11.8548	−0.720128	−0.360064	−	0.932928i	$-0.617245\pi$
$271$	−11.8548	−0.720128	−0.360064	+	0.932928i	$0.617245\pi$
$272$	0	0
$273$	19.8925	1.20395
$274$	−9.51068	−0.574561
$275$	17.0272	1.02678
$276$	5.41678	0.326052
$277$	16.2756	0.977908	0.488954	−	0.872309i	$-0.337379\pi$
$277$	16.2756	0.977908	0.488954	+	0.872309i	$0.337379\pi$
$278$	−4.79472	−0.287568
$279$	−0.448868	−0.0268730
$280$	−32.7844	−1.95924
$281$	15.2506	0.909774	0.454887	−	0.890549i	$-0.349680\pi$
$281$	15.2506	0.909774	0.454887	+	0.890549i	$0.349680\pi$
$282$	2.05724	0.122507
$283$	−20.8911	−1.24185	−0.620924	−	0.783871i	$-0.713242\pi$
$283$	−20.8911	−1.24185	−0.620924	+	0.783871i	$0.713242\pi$
$284$	−4.29009	−0.254570
$285$	−13.5261	−0.801217
$286$	16.5941	0.981230
$287$	−16.0061	−0.944811
$288$	5.52839	0.325763
$289$	0	0
$290$	6.03729	0.354522
$291$	−0.828387	−0.0485609
$292$	13.7185	0.802814
$293$	22.4544	1.31180	0.655900	−	0.754848i	$-0.272289\pi$
$293$	22.4544	1.31180	0.655900	+	0.754848i	$0.272289\pi$
$294$	−5.16145	−0.301022
$295$	9.04338	0.526526
$296$	−2.14104	−0.124446
$297$	−3.27613	−0.190100
$298$	−20.3720	−1.18012
$299$	25.6941	1.48593
$300$	6.11849	0.353251
$301$	−21.7642	−1.25447
$302$	−8.64177	−0.497278
$303$	−4.92944	−0.283189
$304$	1.09985	0.0630808
$305$	−12.4982	−0.715647
$306$	0	0
$307$	−0.733467	−0.0418611	−0.0209306	−	0.999781i	$-0.506663\pi$
$307$	−0.733467	−0.0418611	−0.0209306	+	0.999781i	$0.506663\pi$
$308$	13.7391	0.782859
$309$	−11.4347	−0.650497
$310$	1.30017	0.0738448
$311$	−24.2195	−1.37336	−0.686681	−	0.726959i	$-0.740933\pi$
$311$	−24.2195	−1.37336	−0.686681	+	0.726959i	$0.740933\pi$
$312$	16.0932	0.911096
$313$	3.82843	0.216396	0.108198	−	0.994129i	$-0.465492\pi$
$313$	3.82843	0.216396	0.108198	+	0.994129i	$0.465492\pi$
$314$	12.8064	0.722708
$315$	11.3757	0.640950
$316$	−6.63139	−0.373045
$317$	3.49283	0.196177	0.0980885	−	0.995178i	$-0.468727\pi$
$317$	3.49283	0.196177	0.0980885	+	0.995178i	$0.468727\pi$
$318$	−6.85452	−0.384382
$319$	−6.82843	−0.382319
$320$	−17.6717	−0.987876
$321$	−17.5925	−0.981918
$322$	−14.8680	−0.828563
$323$	0	0
$324$	−1.17723	−0.0654018
$325$	29.0226	1.60988
$326$	−3.20548	−0.177535
$327$	−14.1515	−0.782578
$328$	−12.9491	−0.714992
$329$	8.07944	0.445434
$330$	9.48951	0.522380
$331$	5.00357	0.275021	0.137511	−	0.990500i	$-0.456090\pi$
$331$	5.00357	0.275021	0.137511	+	0.990500i	$0.456090\pi$
$332$	4.61798	0.253444
$333$	0.742912	0.0407113
$334$	14.6828	0.803408
$335$	46.3092	2.53014
$336$	−0.924998	−0.0504628
$337$	−27.0434	−1.47315	−0.736573	−	0.676358i	$-0.763557\pi$
$337$	−27.0434	−1.47315	−0.736573	+	0.676358i	$0.763557\pi$
$338$	16.4925	0.897075
$339$	−20.4015	−1.10806
$340$	0	0
$341$	−1.47055	−0.0796347
$342$	−3.84209	−0.207757
$343$	4.66568	0.251923
$344$	−17.6074	−0.949328
$345$	14.6934	0.791067
$346$	16.6156	0.893262
$347$	−4.31305	−0.231537	−0.115768	−	0.993276i	$-0.536933\pi$
$347$	−4.31305	−0.231537	−0.115768	+	0.993276i	$0.536933\pi$
$348$	−2.45370	−0.131532
$349$	29.1251	1.55903	0.779516	−	0.626382i	$-0.215465\pi$
$349$	29.1251	1.55903	0.779516	+	0.626382i	$0.215465\pi$
$350$	−16.7941	−0.897682
$351$	−5.58411	−0.298058
$352$	18.1117	0.965358
$353$	−11.5730	−0.615971	−0.307985	−	0.951391i	$-0.599655\pi$
$353$	−11.5730	−0.615971	−0.307985	+	0.951391i	$0.599655\pi$
$354$	2.56877	0.136529
$355$	−11.6372	−0.617638
$356$	−17.2089	−0.912069
$357$	0	0
$358$	18.7814	0.992630
$359$	−26.1470	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$359$	−26.1470	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$360$	9.20304	0.485043
$361$	−1.05852	−0.0557118
$362$	−16.9168	−0.889126
$363$	0.266966	0.0140121
$364$	23.4181	1.22744
$365$	37.2124	1.94779
$366$	−3.55013	−0.185568
$367$	−21.8427	−1.14018	−0.570089	−	0.821583i	$-0.693091\pi$
$367$	−21.8427	−1.14018	−0.570089	+	0.821583i	$0.693091\pi$
$368$	−1.19477	−0.0622817
$369$	4.49315	0.233904
$370$	−2.15189	−0.111871
$371$	−26.9199	−1.39761
$372$	−0.528421	−0.0273974
$373$	31.0065	1.60545	0.802727	−	0.596347i	$-0.203382\pi$
$373$	31.0065	1.60545	0.802727	+	0.596347i	$0.203382\pi$
$374$	0	0
$375$	0.630210	0.0325439
$376$	6.53632	0.337085
$377$	−11.6389	−0.599436
$378$	3.23128	0.166199
$379$	11.7686	0.604513	0.302256	−	0.953227i	$-0.402260\pi$
$379$	11.7686	0.604513	0.302256	+	0.953227i	$0.402260\pi$
$380$	−15.9234	−0.816852
$381$	−2.93340	−0.150282
$382$	1.98987	0.101811
$383$	−20.0262	−1.02329	−0.511645	−	0.859197i	$-0.670964\pi$
$383$	−20.0262	−1.02329	−0.511645	+	0.859197i	$0.670964\pi$
$384$	6.03714	0.308081
$385$	37.2684	1.89937
$386$	−3.00425	−0.152912
$387$	6.10953	0.310565
$388$	−0.975203	−0.0495085
$389$	12.7561	0.646758	0.323379	−	0.946270i	$-0.395181\pi$
$389$	12.7561	0.646758	0.323379	+	0.946270i	$0.395181\pi$
$390$	16.1747	0.819038
$391$	0	0
$392$	−16.3991	−0.828282
$393$	6.61671	0.333769
$394$	16.3437	0.823384
$395$	−17.9881	−0.905081
$396$	−3.85677	−0.193810
$397$	27.3141	1.37086	0.685428	−	0.728140i	$-0.259615\pi$
$397$	27.3141	1.37086	0.685428	+	0.728140i	$0.259615\pi$
$398$	3.61153	0.181030
$399$	−15.0891	−0.755402
$400$	−1.34954	−0.0674772
$401$	−31.3594	−1.56601	−0.783007	−	0.622013i	$-0.786315\pi$
$401$	−31.3594	−1.56601	−0.783007	+	0.622013i	$0.786315\pi$
$402$	13.1541	0.656069
$403$	−2.50653	−0.124859
$404$	−5.80309	−0.288715
$405$	−3.19333	−0.158678
$406$	6.73494	0.334250
$407$	2.43388	0.120643
$408$	0	0
$409$	17.1030	0.845688	0.422844	−	0.906202i	$-0.361032\pi$
$409$	17.1030	0.845688	0.422844	+	0.906202i	$0.361032\pi$
$410$	−13.0147	−0.642748
$411$	10.4851	0.517192
$412$	−13.4613	−0.663190
$413$	10.0884	0.496418
$414$	4.17367	0.205125
$415$	12.5266	0.614907
$416$	30.8711	1.51358
$417$	5.28597	0.258855
$418$	−12.5872	−0.615660
$419$	−15.2428	−0.744660	−0.372330	−	0.928100i	$-0.621441\pi$
$419$	−15.2428	−0.744660	−0.372330	+	0.928100i	$0.621441\pi$
$420$	13.3919	0.653457
$421$	6.96252	0.339332	0.169666	−	0.985502i	$-0.445731\pi$
$421$	6.96252	0.339332	0.169666	+	0.985502i	$0.445731\pi$
$422$	0.182226	0.00887063
$423$	−2.26801	−0.110275
$424$	−21.7784	−1.05765
$425$	0	0
$426$	−3.30555	−0.160154
$427$	−13.9425	−0.674725
$428$	−20.7105	−1.00108
$429$	−18.2943	−0.883256
$430$	−17.6966	−0.853407
$431$	21.4684	1.03410	0.517049	−	0.855956i	$-0.327031\pi$
$431$	21.4684	1.03410	0.517049	+	0.855956i	$0.327031\pi$
$432$	0.259660	0.0124929
$433$	33.1954	1.59527	0.797635	−	0.603140i	$-0.206084\pi$
$433$	33.1954	1.59527	0.797635	+	0.603140i	$0.206084\pi$
$434$	1.45042	0.0696222
$435$	−6.65585	−0.319124
$436$	−16.6596	−0.797848
$437$	−19.4898	−0.932325
$438$	10.5702	0.505063
$439$	−35.7203	−1.70484	−0.852418	−	0.522861i	$-0.824865\pi$
$439$	−35.7203	−1.70484	−0.852418	+	0.522861i	$0.824865\pi$
$440$	30.1504	1.43736
$441$	5.69028	0.270966
$442$	0	0
$443$	13.8366	0.657398	0.328699	−	0.944435i	$-0.393390\pi$
$443$	13.8366	0.657398	0.328699	+	0.944435i	$0.393390\pi$
$444$	0.874580	0.0415057
$445$	−46.6803	−2.21286
$446$	−10.5216	−0.498214
$447$	22.4592	1.06228
$448$	−19.7137	−0.931387
$449$	−4.67471	−0.220613	−0.110307	−	0.993898i	$-0.535183\pi$
$449$	−4.67471	−0.220613	−0.110307	+	0.993898i	$0.535183\pi$
$450$	4.71434	0.222236
$451$	14.7201	0.693144
$452$	−24.0173	−1.12968
$453$	9.52717	0.447626
$454$	−8.62977	−0.405015
$455$	63.5233	2.97802
$456$	−12.2072	−0.571655
$457$	−5.17186	−0.241930	−0.120965	−	0.992657i	$-0.538599\pi$
$457$	−5.17186	−0.241930	−0.120965	+	0.992657i	$0.538599\pi$
$458$	−15.2115	−0.710787
$459$	0	0
$460$	17.2976	0.806503
$461$	22.6450	1.05468	0.527341	−	0.849654i	$-0.323189\pi$
$461$	22.6450	1.05468	0.527341	+	0.849654i	$0.323189\pi$
$462$	10.5861	0.492509
$463$	38.8961	1.80766	0.903828	−	0.427897i	$-0.140745\pi$
$463$	38.8961	1.80766	0.903828	+	0.427897i	$0.140745\pi$
$464$	0.541209	0.0251250
$465$	−1.43338	−0.0664715
$466$	8.57056	0.397023
$467$	−16.7116	−0.773322	−0.386661	−	0.922222i	$-0.626372\pi$
$467$	−16.7116	−0.773322	−0.386661	+	0.922222i	$0.626372\pi$
$468$	−6.57379	−0.303874
$469$	51.6606	2.38546
$470$	6.56944	0.303025
$471$	−14.1185	−0.650547
$472$	8.16159	0.375667
$473$	20.0156	0.920320
$474$	−5.10953	−0.234688
$475$	−22.0146	−1.01010
$476$	0	0
$477$	7.55680	0.346002
$478$	5.82408	0.266387
$479$	−10.5031	−0.479899	−0.239949	−	0.970785i	$-0.577131\pi$
$479$	−10.5031	−0.479899	−0.239949	+	0.970785i	$0.577131\pi$
$480$	17.6540	0.805789
$481$	4.14851	0.189156
$482$	−14.0442	−0.639696
$483$	16.3913	0.745832
$484$	0.314280	0.0142855
$485$	−2.64531	−0.120117
$486$	−0.907065	−0.0411453
$487$	24.2456	1.09867	0.549336	−	0.835601i	$-0.314881\pi$
$487$	24.2456	1.09867	0.549336	+	0.835601i	$0.314881\pi$
$488$	−11.2796	−0.510603
$489$	3.53391	0.159809
$490$	−16.4822	−0.744591
$491$	−14.7261	−0.664580	−0.332290	−	0.943177i	$-0.607821\pi$
$491$	−14.7261	−0.664580	−0.332290	+	0.943177i	$0.607821\pi$
$492$	5.28948	0.238468
$493$	0	0
$494$	−21.4547	−0.965291
$495$	−10.4618	−0.470221
$496$	0.116553	0.00523338
$497$	−12.9820	−0.582320
$498$	3.55818	0.159446
$499$	−14.3585	−0.642775	−0.321388	−	0.946948i	$-0.604149\pi$
$499$	−14.3585	−0.642775	−0.321388	+	0.946948i	$0.604149\pi$
$500$	0.741904	0.0331789
$501$	−16.1872	−0.723189
$502$	23.3138	1.04055
$503$	−16.5970	−0.740022	−0.370011	−	0.929027i	$-0.620646\pi$
$503$	−16.5970	−0.740022	−0.370011	+	0.929027i	$0.620646\pi$
$504$	10.2665	0.457307
$505$	−15.7413	−0.700479
$506$	13.6735	0.607860
$507$	−18.1823	−0.807504
$508$	−3.45329	−0.153215
$509$	−33.1048	−1.46734	−0.733672	−	0.679503i	$-0.762195\pi$
$509$	−33.1048	−1.46734	−0.733672	+	0.679503i	$0.762195\pi$
$510$	0	0
$511$	41.5126	1.83641
$512$	−2.93216	−0.129584
$513$	4.23574	0.187012
$514$	2.27189	0.100209
$515$	−36.5147	−1.60903
$516$	7.19234	0.316625
$517$	−7.43031	−0.326785
$518$	−2.40056	−0.105474
$519$	−18.3180	−0.804072
$520$	51.3908	2.25364
$521$	24.2541	1.06259	0.531295	−	0.847187i	$-0.321706\pi$
$521$	24.2541	1.06259	0.531295	+	0.847187i	$0.321706\pi$
$522$	−1.89059	−0.0827491
$523$	−2.14935	−0.0939843	−0.0469922	−	0.998895i	$-0.514964\pi$
$523$	−2.14935	−0.0939843	−0.0469922	+	0.998895i	$0.514964\pi$
$524$	7.78940	0.340282
$525$	18.5147	0.808050
$526$	5.53189	0.241202
$527$	0	0
$528$	0.850680	0.0370211
$529$	−1.82818	−0.0794862
$530$	−21.8887	−0.950786
$531$	−2.83196	−0.122897
$532$	−17.7634	−0.770142
$533$	25.0902	1.08678
$534$	−13.2596	−0.573797
$535$	−56.1787	−2.42882
$536$	41.7938	1.80522
$537$	−20.7057	−0.893517
$538$	3.73213	0.160904
$539$	18.6421	0.802972
$540$	−3.75929	−0.161774
$541$	4.04781	0.174029	0.0870145	−	0.996207i	$-0.472267\pi$
$541$	4.04781	0.174029	0.0870145	+	0.996207i	$0.472267\pi$
$542$	−10.7531	−0.461884
$543$	18.6500	0.800348
$544$	0	0
$545$	−45.1903	−1.93574
$546$	18.0438	0.772204
$547$	38.1034	1.62918	0.814591	−	0.580036i	$-0.196961\pi$
$547$	38.1034	1.62918	0.814591	+	0.580036i	$0.196961\pi$
$548$	12.3434	0.527284
$549$	3.91386	0.167039
$550$	15.4448	0.658568
$551$	8.82854	0.376108
$552$	13.2607	0.564413
$553$	−20.0668	−0.853327
$554$	14.7631	0.627223
$555$	2.37236	0.100701
$556$	6.22282	0.263906
$557$	47.0274	1.99262	0.996308	−	0.0858461i	$-0.0273593\pi$
$557$	47.0274	1.99262	0.996308	+	0.0858461i	$0.0273593\pi$
$558$	−0.407152	−0.0172361
$559$	34.1163	1.44297
$560$	−2.95382	−0.124822
$561$	0	0
$562$	13.8333	0.583522
$563$	28.7543	1.21185	0.605925	−	0.795521i	$-0.292803\pi$
$563$	28.7543	1.21185	0.605925	+	0.795521i	$0.292803\pi$
$564$	−2.66998	−0.112426
$565$	−65.1488	−2.74083
$566$	−18.9496	−0.796512
$567$	−3.56234	−0.149604
$568$	−10.5025	−0.440675
$569$	−1.04600	−0.0438505	−0.0219252	−	0.999760i	$-0.506980\pi$
$569$	−1.04600	−0.0438505	−0.0219252	+	0.999760i	$0.506980\pi$
$570$	−12.2691	−0.513895
$571$	−26.2531	−1.09866	−0.549330	−	0.835606i	$-0.685117\pi$
$571$	−26.2531	−1.09866	−0.549330	+	0.835606i	$0.685117\pi$
$572$	−21.5366	−0.900491
$573$	−2.19374	−0.0916450
$574$	−14.5186	−0.605995
$575$	23.9145	0.997303
$576$	5.53393	0.230580
$577$	38.7607	1.61363	0.806814	−	0.590806i	$-0.201190\pi$
$577$	38.7607	1.61363	0.806814	+	0.590806i	$0.201190\pi$
$578$	0	0
$579$	3.31205	0.137644
$580$	−7.83548	−0.325351
$581$	13.9741	0.579745
$582$	−0.751401	−0.0311466
$583$	24.7571	1.02533
$584$	33.5840	1.38971
$585$	−17.8319	−0.737259
$586$	20.3676	0.841378
$587$	−41.5716	−1.71584	−0.857921	−	0.513781i	$-0.828244\pi$
$587$	−41.5716	−1.71584	−0.857921	+	0.513781i	$0.828244\pi$
$588$	6.69878	0.276253
$589$	1.90129	0.0783411
$590$	8.20294	0.337710
$591$	−18.0182	−0.741170
$592$	−0.192905	−0.00792833
$593$	−13.7886	−0.566230	−0.283115	−	0.959086i	$-0.591368\pi$
$593$	−13.7886	−0.566230	−0.283115	+	0.959086i	$0.591368\pi$
$594$	−2.97167	−0.121929
$595$	0	0
$596$	26.4397	1.08301
$597$	−3.98156	−0.162954
$598$	23.3062	0.953062
$599$	0.372690	0.0152277	0.00761386	−	0.999971i	$-0.497576\pi$
$599$	0.372690	0.0152277	0.00761386	+	0.999971i	$0.497576\pi$
$600$	14.9785	0.611497
$601$	−23.0325	−0.939515	−0.469757	−	0.882796i	$-0.655659\pi$
$601$	−23.0325	−0.939515	−0.469757	+	0.882796i	$0.655659\pi$
$602$	−19.7416	−0.804607
$603$	−14.5019	−0.590562
$604$	11.2157	0.456360
$605$	0.852509	0.0346594
$606$	−4.47132	−0.181635
$607$	16.1050	0.653681	0.326840	−	0.945080i	$-0.394016\pi$
$607$	16.1050	0.653681	0.326840	+	0.945080i	$0.394016\pi$
$608$	−23.4168	−0.949677
$609$	−7.42498	−0.300875
$610$	−11.3367	−0.459011
$611$	−12.6648	−0.512365
$612$	0	0
$613$	−14.9606	−0.604251	−0.302126	−	0.953268i	$-0.597696\pi$
$613$	−14.9606	−0.604251	−0.302126	+	0.953268i	$0.597696\pi$
$614$	−0.665302	−0.0268494
$615$	14.3481	0.578571
$616$	33.6345	1.35517
$617$	4.75542	0.191446	0.0957230	−	0.995408i	$-0.469484\pi$
$617$	4.75542	0.191446	0.0957230	+	0.995408i	$0.469484\pi$
$618$	−10.3720	−0.417223
$619$	−43.0995	−1.73232	−0.866158	−	0.499770i	$-0.833418\pi$
$619$	−43.0995	−1.73232	−0.866158	+	0.499770i	$0.833418\pi$
$620$	−1.68742	−0.0677686
$621$	−4.60128	−0.184643
$622$	−21.9687	−0.880864
$623$	−52.0746	−2.08632
$624$	1.44997	0.0580453
$625$	−23.9743	−0.958972
$626$	3.47264	0.138795
$627$	13.8768	0.554187
$628$	−16.6208	−0.663241
$629$	0	0
$630$	10.3185	0.411100
$631$	11.3212	0.450690	0.225345	−	0.974279i	$-0.427649\pi$
$631$	11.3212	0.450690	0.225345	+	0.974279i	$0.427649\pi$
$632$	−16.2342	−0.645760
$633$	−0.200896	−0.00798491
$634$	3.16823	0.125826
$635$	−9.36730	−0.371730
$636$	8.89611	0.352754
$637$	31.7751	1.25898
$638$	−6.19384	−0.245216
$639$	3.64422	0.144163
$640$	19.2786	0.762052
$641$	15.8764	0.627081	0.313541	−	0.949575i	$-0.398485\pi$
$641$	15.8764	0.627081	0.313541	+	0.949575i	$0.398485\pi$
$642$	−15.9576	−0.629795
$643$	25.7352	1.01490	0.507448	−	0.861682i	$-0.330589\pi$
$643$	25.7352	1.01490	0.507448	+	0.861682i	$0.330589\pi$
$644$	19.2964	0.760386
$645$	19.5097	0.768196
$646$	0	0
$647$	−26.9015	−1.05761	−0.528804	−	0.848744i	$-0.677359\pi$
$647$	−26.9015	−1.05761	−0.528804	+	0.848744i	$0.677359\pi$
$648$	−2.88196	−0.113214
$649$	−9.27787	−0.364188
$650$	26.3254	1.03257
$651$	−1.59902	−0.0626705
$652$	4.16023	0.162927
$653$	−20.6599	−0.808484	−0.404242	−	0.914652i	$-0.632465\pi$
$653$	−20.6599	−0.808484	−0.404242	+	0.914652i	$0.632465\pi$
$654$	−12.8363	−0.501939
$655$	21.1293	0.825591
$656$	−1.16669	−0.0455516
$657$	−11.6532	−0.454634
$658$	7.32858	0.285698
$659$	−25.5651	−0.995875	−0.497937	−	0.867213i	$-0.665909\pi$
$659$	−25.5651	−0.995875	−0.497937	+	0.867213i	$0.665909\pi$
$660$	−12.3159	−0.479397
$661$	27.9387	1.08669	0.543345	−	0.839510i	$-0.317158\pi$
$661$	27.9387	1.08669	0.543345	+	0.839510i	$0.317158\pi$
$662$	4.53857	0.176396
$663$	0	0
$664$	11.3052	0.438726
$665$	−48.1846	−1.86852
$666$	0.673870	0.0261119
$667$	−9.59045	−0.371344
$668$	−19.0561	−0.737301
$669$	11.5997	0.448468
$670$	42.0055	1.62281
$671$	12.8223	0.495000
$672$	19.6940	0.759713
$673$	−2.58625	−0.0996927	−0.0498464	−	0.998757i	$-0.515873\pi$
$673$	−2.58625	−0.0996927	−0.0498464	+	0.998757i	$0.515873\pi$
$674$	−24.5301	−0.944865
$675$	−5.19735	−0.200046
$676$	−21.4048	−0.823261
$677$	−0.886837	−0.0340839	−0.0170420	−	0.999855i	$-0.505425\pi$
$677$	−0.886837	−0.0340839	−0.0170420	+	0.999855i	$0.505425\pi$
$678$	−18.5055	−0.710700
$679$	−2.95100	−0.113249
$680$	0	0
$681$	9.51394	0.364575
$682$	−1.33388	−0.0510771
$683$	26.9328	1.03055	0.515277	−	0.857024i	$-0.327689\pi$
$683$	26.9328	1.03055	0.515277	+	0.857024i	$0.327689\pi$
$684$	4.98645	0.190662
$685$	33.4824	1.27930
$686$	4.23208	0.161581
$687$	16.7700	0.639816
$688$	−1.58640	−0.0604810
$689$	42.1980	1.60762
$690$	13.3279	0.507384
$691$	10.3832	0.394994	0.197497	−	0.980303i	$-0.436719\pi$
$691$	10.3832	0.394994	0.197497	+	0.980303i	$0.436719\pi$
$692$	−21.5646	−0.819761
$693$	−11.6707	−0.443333
$694$	−3.91222	−0.148506
$695$	16.8798	0.640289
$696$	−6.00686	−0.227689
$697$	0	0
$698$	26.4184	0.999951
$699$	−9.44867	−0.357381
$700$	21.7962	0.823817
$701$	49.5351	1.87091	0.935457	−	0.353440i	$-0.114988\pi$
$701$	49.5351	1.87091	0.935457	+	0.353440i	$0.114988\pi$
$702$	−5.06515	−0.191172
$703$	−3.14678	−0.118683
$704$	18.1299	0.683295
$705$	−7.24251	−0.272769
$706$	−10.4975	−0.395079
$707$	−17.5603	−0.660424
$708$	−3.33387	−0.125295
$709$	36.3527	1.36526	0.682628	−	0.730766i	$-0.260837\pi$
$709$	36.3527	1.36526	0.682628	+	0.730766i	$0.260837\pi$
$710$	−10.5557	−0.396148
$711$	5.63303	0.211255
$712$	−42.1287	−1.57884
$713$	−2.06537	−0.0773486
$714$	0	0
$715$	−58.4197	−2.18477
$716$	−24.3754	−0.910953
$717$	−6.42079	−0.239789
$718$	−23.7170	−0.885112
$719$	−18.0506	−0.673175	−0.336588	−	0.941652i	$-0.609273\pi$
$719$	−18.0506	−0.673175	−0.336588	+	0.941652i	$0.609273\pi$
$720$	0.829180	0.0309017
$721$	−40.7343	−1.51702
$722$	−0.960151	−0.0357331
$723$	15.4831	0.575823
$724$	21.9554	0.815966
$725$	−10.8328	−0.402321
$726$	0.242155	0.00898723
$727$	25.6114	0.949873	0.474936	−	0.880020i	$-0.342471\pi$
$727$	25.6114	0.949873	0.474936	+	0.880020i	$0.342471\pi$
$728$	57.3294	2.12477
$729$	1.00000	0.0370370
$730$	33.7541	1.24930
$731$	0	0
$732$	4.60752	0.170299
$733$	31.6733	1.16988	0.584941	−	0.811076i	$-0.301118\pi$
$733$	31.6733	1.16988	0.584941	+	0.811076i	$0.301118\pi$
$734$	−19.8127	−0.731301
$735$	18.1709	0.670245
$736$	25.4377	0.937645
$737$	−47.5100	−1.75005
$738$	4.07558	0.150024
$739$	−31.3462	−1.15309	−0.576543	−	0.817066i	$-0.695599\pi$
$739$	−31.3462	−1.15309	−0.576543	+	0.817066i	$0.695599\pi$
$740$	2.79282	0.102666
$741$	23.6528	0.868908
$742$	−24.4181	−0.896418
$743$	−3.20001	−0.117397	−0.0586984	−	0.998276i	$-0.518695\pi$
$743$	−3.20001	−0.117397	−0.0586984	+	0.998276i	$0.518695\pi$
$744$	−1.29362	−0.0474263
$745$	71.7196	2.62760
$746$	28.1249	1.02973
$747$	−3.92274	−0.143526
$748$	0	0
$749$	−62.6705	−2.28993
$750$	0.571642	0.0208734
$751$	27.5933	1.00689	0.503446	−	0.864026i	$-0.332065\pi$
$751$	27.5933	1.00689	0.503446	+	0.864026i	$0.332065\pi$
$752$	0.588913	0.0214754
$753$	−25.7024	−0.936649
$754$	−10.5573	−0.384474
$755$	30.4234	1.10722
$756$	−4.19370	−0.152524
$757$	−10.1597	−0.369259	−0.184629	−	0.982808i	$-0.559108\pi$
$757$	−10.1597	−0.369259	−0.184629	+	0.982808i	$0.559108\pi$
$758$	10.6749	0.387730
$759$	−15.0744	−0.547166
$760$	−38.9817	−1.41401
$761$	8.12911	0.294680	0.147340	−	0.989086i	$-0.452929\pi$
$761$	8.12911	0.294680	0.147340	+	0.989086i	$0.452929\pi$
$762$	−2.66078	−0.0963900
$763$	−50.4124	−1.82505
$764$	−2.58255	−0.0934333
$765$	0	0
$766$	−18.1650	−0.656330
$767$	−15.8140	−0.571009
$768$	16.5439	0.596978
$769$	16.6656	0.600978	0.300489	−	0.953785i	$-0.402850\pi$
$769$	16.6656	0.600978	0.300489	+	0.953785i	$0.402850\pi$
$770$	33.8049	1.21824
$771$	−2.50466	−0.0902031
$772$	3.89906	0.140330
$773$	42.9077	1.54328	0.771641	−	0.636058i	$-0.219436\pi$
$773$	42.9077	1.54328	0.771641	+	0.636058i	$0.219436\pi$
$774$	5.54175	0.199194
$775$	−2.33292	−0.0838011
$776$	−2.38738	−0.0857018
$777$	2.64651	0.0949429
$778$	11.5706	0.414826
$779$	−19.0318	−0.681885
$780$	−20.9923	−0.751645
$781$	11.9389	0.427209
$782$	0	0
$783$	2.08430	0.0744867
$784$	−1.47754	−0.0527692
$785$	−45.0851	−1.60916
$786$	6.00179	0.214077
$787$	10.6447	0.379444	0.189722	−	0.981838i	$-0.439241\pi$
$787$	10.6447	0.379444	0.189722	+	0.981838i	$0.439241\pi$
$788$	−21.2116	−0.755633
$789$	−6.09867	−0.217118
$790$	−16.3164	−0.580512
$791$	−72.6772	−2.58410
$792$	−9.44167	−0.335495
$793$	21.8554	0.776109
$794$	24.7757	0.879257
$795$	24.1314	0.855851
$796$	−4.68722	−0.166134
$797$	−29.6872	−1.05157	−0.525787	−	0.850617i	$-0.676229\pi$
$797$	−29.6872	−1.05157	−0.525787	+	0.850617i	$0.676229\pi$
$798$	−13.6868	−0.484509
$799$	0	0
$800$	28.7330	1.01586
$801$	14.6181	0.516505
$802$	−28.4450	−1.00443
$803$	−38.1773	−1.34725
$804$	−17.0721	−0.602085
$805$	52.3430	1.84485
$806$	−2.27358	−0.0800836
$807$	−4.11451	−0.144838
$808$	−14.2064	−0.499780
$809$	22.8310	0.802694	0.401347	−	0.915926i	$-0.368542\pi$
$809$	22.8310	0.802694	0.401347	+	0.915926i	$0.368542\pi$
$810$	−2.89656	−0.101775
$811$	−37.2506	−1.30805	−0.654023	−	0.756475i	$-0.726920\pi$
$811$	−37.2506	−1.30805	−0.654023	+	0.756475i	$0.726920\pi$
$812$	−8.74093	−0.306746
$813$	11.8548	0.415766
$814$	2.20769	0.0773794
$815$	11.2849	0.395294
$816$	0	0
$817$	−25.8784	−0.905370
$818$	15.5135	0.542418
$819$	−19.8925	−0.695101
$820$	16.8910	0.589861
$821$	3.29267	0.114915	0.0574574	−	0.998348i	$-0.481701\pi$
$821$	3.29267	0.114915	0.0574574	+	0.998348i	$0.481701\pi$
$822$	9.51068	0.331723
$823$	12.7887	0.445786	0.222893	−	0.974843i	$-0.428450\pi$
$823$	12.7887	0.445786	0.222893	+	0.974843i	$0.428450\pi$
$824$	−32.9543	−1.14802
$825$	−17.0272	−0.592811
$826$	9.15084	0.318399
$827$	36.4922	1.26896	0.634480	−	0.772939i	$-0.281214\pi$
$827$	36.4922	1.26896	0.634480	+	0.772939i	$0.281214\pi$
$828$	−5.41678	−0.188246
$829$	−0.425058	−0.0147629	−0.00738144	−	0.999973i	$-0.502350\pi$
$829$	−0.425058	−0.0147629	−0.00738144	+	0.999973i	$0.502350\pi$
$830$	11.3624	0.394396
$831$	−16.2756	−0.564596
$832$	30.9021	1.07134
$833$	0	0
$834$	4.79472	0.166028
$835$	−51.6909	−1.78884
$836$	16.3363	0.565001
$837$	0.448868	0.0155151
$838$	−13.8262	−0.477619
$839$	−31.6812	−1.09376	−0.546879	−	0.837212i	$-0.684184\pi$
$839$	−31.6812	−1.09376	−0.546879	+	0.837212i	$0.684184\pi$
$840$	32.7844	1.13117
$841$	−24.6557	−0.850197
$842$	6.31546	0.217645
$843$	−15.2506	−0.525258
$844$	−0.236502	−0.00814072
$845$	−58.0620	−1.99740
$846$	−2.05724	−0.0707293
$847$	0.951023	0.0326775
$848$	−1.96220	−0.0673822
$849$	20.8911	0.716981
$850$	0	0
$851$	3.41835	0.117180
$852$	4.29009	0.146976
$853$	15.3947	0.527104	0.263552	−	0.964645i	$-0.415106\pi$
$853$	15.3947	0.527104	0.263552	+	0.964645i	$0.415106\pi$
$854$	−12.6468	−0.432763
$855$	13.5261	0.462583
$856$	−50.7009	−1.73292
$857$	−40.7518	−1.39205	−0.696027	−	0.718016i	$-0.745051\pi$
$857$	−40.7518	−1.39205	−0.696027	+	0.718016i	$0.745051\pi$
$858$	−16.5941	−0.566514
$859$	−46.4431	−1.58462	−0.792310	−	0.610119i	$-0.791122\pi$
$859$	−46.4431	−1.58462	−0.792310	+	0.610119i	$0.791122\pi$
$860$	22.9675	0.783185
$861$	16.0061	0.545487
$862$	19.4733	0.663262
$863$	−47.2341	−1.60787	−0.803934	−	0.594719i	$-0.797263\pi$
$863$	−47.2341	−1.60787	−0.803934	+	0.594719i	$0.797263\pi$
$864$	−5.52839	−0.188080
$865$	−58.4954	−1.98890
$866$	30.1104	1.02319
$867$	0	0
$868$	−1.88242	−0.0638934
$869$	18.4546	0.626028
$870$	−6.03729	−0.204683
$871$	−80.9800	−2.74390
$872$	−40.7839	−1.38112
$873$	0.828387	0.0280366
$874$	−17.6786	−0.597986
$875$	2.24502	0.0758957
$876$	−13.7185	−0.463505
$877$	0.316438	0.0106854	0.00534268	−	0.999986i	$-0.498299\pi$
$877$	0.316438	0.0106854	0.00534268	+	0.999986i	$0.498299\pi$
$878$	−32.4006	−1.09347
$879$	−22.4544	−0.757368
$880$	2.71650	0.0915733
$881$	−36.1357	−1.21744	−0.608721	−	0.793385i	$-0.708317\pi$
$881$	−36.1357	−1.21744	−0.608721	+	0.793385i	$0.708317\pi$
$882$	5.16145	0.173795
$883$	0.156899	0.00528008	0.00264004	−	0.999997i	$-0.499160\pi$
$883$	0.156899	0.00528008	0.00264004	+	0.999997i	$0.499160\pi$
$884$	0	0
$885$	−9.04338	−0.303990
$886$	12.5507	0.421650
$887$	−30.1071	−1.01090	−0.505448	−	0.862857i	$-0.668673\pi$
$887$	−30.1071	−1.01090	−0.505448	+	0.862857i	$0.668673\pi$
$888$	2.14104	0.0718487
$889$	−10.4498	−0.350474
$890$	−42.3421	−1.41931
$891$	3.27613	0.109755
$892$	13.6555	0.457219
$893$	9.60671	0.321476
$894$	20.3720	0.681341
$895$	−66.1202	−2.21015
$896$	21.5063	0.718477
$897$	−25.6941	−0.857900
$898$	−4.24027	−0.141500
$899$	0.935574	0.0312031
$900$	−6.11849	−0.203950
$901$	0	0
$902$	13.3521	0.444577
$903$	21.7642	0.724269
$904$	−58.7963	−1.95554
$905$	59.5556	1.97970
$906$	8.64177	0.287104
$907$	−38.4765	−1.27759	−0.638795	−	0.769377i	$-0.720567\pi$
$907$	−38.4765	−1.27759	−0.638795	+	0.769377i	$0.720567\pi$
$908$	11.2001	0.371689
$909$	4.92944	0.163499
$910$	57.6198	1.91008
$911$	1.87182	0.0620162	0.0310081	−	0.999519i	$-0.490128\pi$
$911$	1.87182	0.0620162	0.0310081	+	0.999519i	$0.490128\pi$
$912$	−1.09985	−0.0364197
$913$	−12.8514	−0.425320
$914$	−4.69122	−0.155172
$915$	12.4982	0.413179
$916$	19.7422	0.652301
$917$	23.5710	0.778382
$918$	0	0
$919$	6.41003	0.211447	0.105724	−	0.994396i	$-0.466284\pi$
$919$	6.41003	0.211447	0.105724	+	0.994396i	$0.466284\pi$
$920$	42.3458	1.39610
$921$	0.733467	0.0241685
$922$	20.5405	0.676464
$923$	20.3497	0.669819
$924$	−13.7391	−0.451984
$925$	3.86118	0.126955
$926$	35.2813	1.15942
$927$	11.4347	0.375564
$928$	−11.5228	−0.378255
$929$	−5.74229	−0.188398	−0.0941992	−	0.995553i	$-0.530029\pi$
$929$	−5.74229	−0.188398	−0.0941992	+	0.995553i	$0.530029\pi$
$930$	−1.30017	−0.0426343
$931$	−24.1025	−0.789928
$932$	−11.1233	−0.364355
$933$	24.2195	0.792911
$934$	−15.1585	−0.496003
$935$	0	0
$936$	−16.0932	−0.526022
$937$	−35.7264	−1.16713	−0.583566	−	0.812066i	$-0.698343\pi$
$937$	−35.7264	−1.16713	−0.583566	+	0.812066i	$0.698343\pi$
$938$	46.8595	1.53002
$939$	−3.82843	−0.124936
$940$	−8.52612	−0.278091
$941$	10.9669	0.357511	0.178756	−	0.983893i	$-0.442793\pi$
$941$	10.9669	0.357511	0.178756	+	0.983893i	$0.442793\pi$
$942$	−12.8064	−0.417256
$943$	20.6742	0.673246
$944$	0.735347	0.0239335
$945$	−11.3757	−0.370053
$946$	18.1555	0.590286
$947$	39.7162	1.29060	0.645301	−	0.763928i	$-0.276732\pi$
$947$	39.7162	1.29060	0.645301	+	0.763928i	$0.276732\pi$
$948$	6.63139	0.215377
$949$	−65.0726	−2.11235
$950$	−19.9687	−0.647870
$951$	−3.49283	−0.113263
$952$	0	0
$953$	−5.33601	−0.172850	−0.0864252	−	0.996258i	$-0.527544\pi$
$953$	−5.33601	−0.172850	−0.0864252	+	0.996258i	$0.527544\pi$
$954$	6.85452	0.221923
$955$	−7.00535	−0.226688
$956$	−7.55876	−0.244468
$957$	6.82843	0.220732
$958$	−9.52700	−0.307803
$959$	37.3516	1.20614
$960$	17.6717	0.570350
$961$	−30.7985	−0.993501
$962$	3.76297	0.121323
$963$	17.5925	0.566911
$964$	18.2272	0.587059
$965$	10.5765	0.340469
$966$	14.8680	0.478371
$967$	49.3477	1.58692	0.793458	−	0.608625i	$-0.208279\pi$
$967$	49.3477	1.58692	0.793458	+	0.608625i	$0.208279\pi$
$968$	0.769384	0.0247289
$969$	0	0
$970$	−2.39947	−0.0770424
$971$	−15.7566	−0.505654	−0.252827	−	0.967512i	$-0.581360\pi$
$971$	−15.7566	−0.505654	−0.252827	+	0.967512i	$0.581360\pi$
$972$	1.17723	0.0377597
$973$	18.8304	0.603676
$974$	21.9923	0.704680
$975$	−29.0226	−0.929467
$976$	−1.01627	−0.0325301
$977$	−37.9092	−1.21282	−0.606411	−	0.795151i	$-0.707391\pi$
$977$	−37.9092	−1.21282	−0.606411	+	0.795151i	$0.707391\pi$
$978$	3.20548	0.102500
$979$	47.8908	1.53060
$980$	21.3914	0.683323
$981$	14.1515	0.451822
$982$	−13.3575	−0.426257
$983$	1.08178	0.0345035	0.0172518	−	0.999851i	$-0.494508\pi$
$983$	1.08178	0.0345035	0.0172518	+	0.999851i	$0.494508\pi$
$984$	12.9491	0.412801
$985$	−57.5381	−1.83332
$986$	0	0
$987$	−8.07944	−0.257171
$988$	27.8449	0.885863
$989$	28.1117	0.893900
$990$	−9.48951	−0.301596
$991$	9.64197	0.306287	0.153144	−	0.988204i	$-0.451060\pi$
$991$	9.64197	0.306287	0.153144	+	0.988204i	$0.451060\pi$
$992$	−2.48151	−0.0787882
$993$	−5.00357	−0.158784
$994$	−11.7755	−0.373496
$995$	−12.7144	−0.403074
$996$	−4.61798	−0.146326
$997$	−6.84545	−0.216798	−0.108399	−	0.994107i	$-0.534572\pi$
$997$	−6.84545	−0.216798	−0.108399	+	0.994107i	$0.534572\pi$
$998$	−13.0241	−0.412271
$999$	−0.742912	−0.0235047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.a.o.1.4	✓	6
3.2	odd	2		2601.2.a.bh.1.3		6
17.2	even	8		867.2.e.k.616.7		24
17.3	odd	16		867.2.h.m.757.7		48
17.4	even	4		867.2.d.g.577.6		12
17.5	odd	16		867.2.h.m.688.6		48
17.6	odd	16		867.2.h.m.733.7		48
17.7	odd	16		867.2.h.m.712.6		48
17.8	even	8		867.2.e.k.829.6		24
17.9	even	8		867.2.e.k.829.5		24
17.10	odd	16		867.2.h.m.712.5		48
17.11	odd	16		867.2.h.m.733.8		48
17.12	odd	16		867.2.h.m.688.5		48
17.13	even	4		867.2.d.g.577.5		12
17.14	odd	16		867.2.h.m.757.8		48
17.15	even	8		867.2.e.k.616.8		24
17.16	even	2		867.2.a.p.1.4	yes	6
51.50	odd	2		2601.2.a.bi.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
867.2.a.o.1.4	✓	6	1.1	even	1	trivial
867.2.a.p.1.4	yes	6	17.16	even	2
867.2.d.g.577.5		12	17.13	even	4
867.2.d.g.577.6		12	17.4	even	4
867.2.e.k.616.7		24	17.2	even	8
867.2.e.k.616.8		24	17.15	even	8
867.2.e.k.829.5		24	17.9	even	8
867.2.e.k.829.6		24	17.8	even	8
867.2.h.m.688.5		48	17.12	odd	16
867.2.h.m.688.6		48	17.5	odd	16
867.2.h.m.712.5		48	17.10	odd	16
867.2.h.m.712.6		48	17.7	odd	16
867.2.h.m.733.7		48	17.6	odd	16
867.2.h.m.733.8		48	17.11	odd	16
867.2.h.m.757.7		48	17.3	odd	16
867.2.h.m.757.8		48	17.14	odd	16
2601.2.a.bh.1.3		6	3.2	odd	2
2601.2.a.bi.1.3		6	51.50	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q+0.907065 q^{2} -1.00000 q^{3} -1.17723 q^{4} -3.19333 q^{5} -0.907065 q^{6} -3.56234 q^{7} -2.88196 q^{8} +1.00000 q^{9} -2.89656 q^{10} +3.27613 q^{11} +1.17723 q^{12} +5.58411 q^{13} -3.23128 q^{14} +3.19333 q^{15} -0.259660 q^{16} +0.907065 q^{18} -4.23574 q^{19} +3.75929 q^{20} +3.56234 q^{21} +2.97167 q^{22} +4.60128 q^{23} +2.88196 q^{24} +5.19735 q^{25} +5.06515 q^{26} -1.00000 q^{27} +4.19370 q^{28} -2.08430 q^{29} +2.89656 q^{30} -0.448868 q^{31} +5.52839 q^{32} -3.27613 q^{33} +11.3757 q^{35} -1.17723 q^{36} +0.742912 q^{37} -3.84209 q^{38} -5.58411 q^{39} +9.20304 q^{40} +4.49315 q^{41} +3.23128 q^{42} +6.10953 q^{43} -3.85677 q^{44} -3.19333 q^{45} +4.17367 q^{46} -2.26801 q^{47} +0.259660 q^{48} +5.69028 q^{49} +4.71434 q^{50} -6.57379 q^{52} +7.55680 q^{53} -0.907065 q^{54} -10.4618 q^{55} +10.2665 q^{56} +4.23574 q^{57} -1.89059 q^{58} -2.83196 q^{59} -3.75929 q^{60} +3.91386 q^{61} -0.407152 q^{62} -3.56234 q^{63} +5.53393 q^{64} -17.8319 q^{65} -2.97167 q^{66} -14.5019 q^{67} -4.60128 q^{69} +10.3185 q^{70} +3.64422 q^{71} -2.88196 q^{72} -11.6532 q^{73} +0.673870 q^{74} -5.19735 q^{75} +4.98645 q^{76} -11.6707 q^{77} -5.06515 q^{78} +5.63303 q^{79} +0.829180 q^{80} +1.00000 q^{81} +4.07558 q^{82} -3.92274 q^{83} -4.19370 q^{84} +5.54175 q^{86} +2.08430 q^{87} -9.44167 q^{88} +14.6181 q^{89} -2.89656 q^{90} -19.8925 q^{91} -5.41678 q^{92} +0.448868 q^{93} -2.05724 q^{94} +13.5261 q^{95} -5.52839 q^{96} +0.828387 q^{97} +5.16145 q^{98} +3.27613 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 6 q^{3} + 9 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{7} + 12 q^{8} + 6 q^{9} - 12 q^{10} + 9 q^{11} - 9 q^{12} + 9 q^{13} + 6 q^{14} - 3 q^{15} + 15 q^{16} + 3 q^{18} + 9 q^{19} - 6 q^{20} + 3 q^{21}+ \cdots + 9 q^{99}+O(q^{100}) \) 6 * q + 3 * q^2 - 6 * q^3 + 9 * q^4 + 3 * q^5 - 3 * q^6 - 3 * q^7 + 12 * q^8 + 6 * q^9 - 12 * q^10 + 9 * q^11 - 9 * q^12 + 9 * q^13 + 6 * q^14 - 3 * q^15 + 15 * q^16 + 3 * q^18 + 9 * q^19 - 6 * q^20 + 3 * q^21 + 18 * q^22 + 9 * q^23 - 12 * q^24 + 15 * q^25 - 12 * q^26 - 6 * q^27 + 15 * q^28 + 6 * q^29 + 12 * q^30 - 24 * q^31 + 42 * q^32 - 9 * q^33 + 9 * q^36 + 3 * q^37 - 6 * q^38 - 9 * q^39 + 3 * q^40 + 18 * q^41 - 6 * q^42 + 3 * q^44 + 3 * q^45 - 15 * q^46 + 24 * q^47 - 15 * q^48 + 21 * q^49 + 12 * q^50 - 18 * q^52 + 24 * q^53 - 3 * q^54 - 24 * q^55 + 54 * q^56 - 9 * q^57 - 3 * q^58 - 9 * q^59 + 6 * q^60 - 21 * q^61 + 30 * q^62 - 3 * q^63 + 24 * q^64 - 9 * q^65 - 18 * q^66 - 6 * q^67 - 9 * q^69 - 3 * q^70 + 27 * q^71 + 12 * q^72 - 18 * q^73 + 36 * q^74 - 15 * q^75 - 3 * q^76 + 33 * q^77 + 12 * q^78 - 24 * q^79 + 3 * q^80 + 6 * q^81 + 15 * q^82 + 6 * q^83 - 15 * q^84 + 6 * q^86 - 6 * q^87 + 24 * q^88 - 12 * q^90 - 39 * q^91 + 24 * q^93 - 15 * q^94 + 42 * q^95 - 42 * q^96 + 33 * q^97 - 57 * q^98 + 9 * q^99

Introduction

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