LMFDB - Embedded newform 8670.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -3.41421 q^{13} +4.82843 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} -4.82843 q^{21} +2.00000 q^{22} +2.24264 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.41421 q^{26} +1.00000 q^{27} -4.82843 q^{28} +1.17157 q^{29} -1.00000 q^{30} +6.24264 q^{31} -1.00000 q^{32} -2.00000 q^{33} -4.82843 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.82843 q^{38} -3.41421 q^{39} -1.00000 q^{40} +6.82843 q^{41} +4.82843 q^{42} +8.24264 q^{43} -2.00000 q^{44} +1.00000 q^{45} -2.24264 q^{46} +5.65685 q^{47} +1.00000 q^{48} +16.3137 q^{49} -1.00000 q^{50} -3.41421 q^{52} -7.65685 q^{53} -1.00000 q^{54} -2.00000 q^{55} +4.82843 q^{56} +2.82843 q^{57} -1.17157 q^{58} -12.2426 q^{59} +1.00000 q^{60} -6.00000 q^{61} -6.24264 q^{62} -4.82843 q^{63} +1.00000 q^{64} -3.41421 q^{65} +2.00000 q^{66} -15.0711 q^{67} +2.24264 q^{69} +4.82843 q^{70} +10.0000 q^{71} -1.00000 q^{72} +15.3137 q^{73} +2.00000 q^{74} +1.00000 q^{75} +2.82843 q^{76} +9.65685 q^{77} +3.41421 q^{78} +10.2426 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.82843 q^{82} +2.34315 q^{83} -4.82843 q^{84} -8.24264 q^{86} +1.17157 q^{87} +2.00000 q^{88} -10.4853 q^{89} -1.00000 q^{90} +16.4853 q^{91} +2.24264 q^{92} +6.24264 q^{93} -5.65685 q^{94} +2.82843 q^{95} -1.00000 q^{96} -13.6569 q^{97} -16.3137 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 4 q^{21} + 4 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 4 q^{28} + 8 q^{29} - 2 q^{30} + 4 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{39} - 2 q^{40} + 8 q^{41} + 4 q^{42} + 8 q^{43} - 4 q^{44} + 2 q^{45} + 4 q^{46} + 2 q^{48} + 10 q^{49} - 2 q^{50} - 4 q^{52} - 4 q^{53} - 2 q^{54} - 4 q^{55} + 4 q^{56} - 8 q^{58} - 16 q^{59} + 2 q^{60} - 12 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{65} + 4 q^{66} - 16 q^{67} - 4 q^{69} + 4 q^{70} + 20 q^{71} - 2 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} + 8 q^{77} + 4 q^{78} + 12 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 16 q^{83} - 4 q^{84} - 8 q^{86} + 8 q^{87} + 4 q^{88} - 4 q^{89} - 2 q^{90} + 16 q^{91} - 4 q^{92} + 4 q^{93} - 2 q^{96} - 16 q^{97} - 10 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^18 + 2 * q^20 - 4 * q^21 + 4 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 4 * q^28 + 8 * q^29 - 2 * q^30 + 4 * q^31 - 2 * q^32 - 4 * q^33 - 4 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^39 - 2 * q^40 + 8 * q^41 + 4 * q^42 + 8 * q^43 - 4 * q^44 + 2 * q^45 + 4 * q^46 + 2 * q^48 + 10 * q^49 - 2 * q^50 - 4 * q^52 - 4 * q^53 - 2 * q^54 - 4 * q^55 + 4 * q^56 - 8 * q^58 - 16 * q^59 + 2 * q^60 - 12 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 - 4 * q^65 + 4 * q^66 - 16 * q^67 - 4 * q^69 + 4 * q^70 + 20 * q^71 - 2 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 + 8 * q^77 + 4 * q^78 + 12 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^82 + 16 * q^83 - 4 * q^84 - 8 * q^86 + 8 * q^87 + 4 * q^88 - 4 * q^89 - 2 * q^90 + 16 * q^91 - 4 * q^92 + 4 * q^93 - 2 * q^96 - 16 * q^97 - 10 * q^98 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	4.82843	1.29045
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	1.00000	0.223607
$21$	−4.82843	−1.05365
$22$	2.00000	0.426401
$23$	2.24264	0.467623	0.233811	−	0.972282i	$-0.424880\pi$
$23$	2.24264	0.467623	0.233811	+	0.972282i	$0.424880\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	3.41421	0.669582
$27$	1.00000	0.192450
$28$	−4.82843	−0.912487
$29$	1.17157	0.217556	0.108778	−	0.994066i	$-0.465306\pi$
$29$	1.17157	0.217556	0.108778	+	0.994066i	$0.465306\pi$
$30$	−1.00000	−0.182574
$31$	6.24264	1.12121	0.560606	−	0.828083i	$-0.310568\pi$
$31$	6.24264	1.12121	0.560606	+	0.828083i	$0.310568\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−4.82843	−0.816153
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.82843	−0.458831
$39$	−3.41421	−0.546712
$40$	−1.00000	−0.158114
$41$	6.82843	1.06642	0.533211	−	0.845983i	$-0.320985\pi$
$41$	6.82843	1.06642	0.533211	+	0.845983i	$0.320985\pi$
$42$	4.82843	0.745042
$43$	8.24264	1.25699	0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264	1.25699	0.628495	+	0.777813i	$0.283671\pi$
$44$	−2.00000	−0.301511
$45$	1.00000	0.149071
$46$	−2.24264	−0.330659
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	1.00000	0.144338
$49$	16.3137	2.33053
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.41421	−0.473466
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	−1.00000	−0.136083
$55$	−2.00000	−0.269680
$56$	4.82843	0.645226
$57$	2.82843	0.374634
$58$	−1.17157	−0.153835
$59$	−12.2426	−1.59386	−0.796928	−	0.604074i	$-0.793543\pi$
$59$	−12.2426	−1.59386	−0.796928	+	0.604074i	$0.793543\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−6.24264	−0.792816
$63$	−4.82843	−0.608325
$64$	1.00000	0.125000
$65$	−3.41421	−0.423481
$66$	2.00000	0.246183
$67$	−15.0711	−1.84122	−0.920612	−	0.390479i	$-0.872310\pi$
$67$	−15.0711	−1.84122	−0.920612	+	0.390479i	$0.872310\pi$
$68$	0	0
$69$	2.24264	0.269982
$70$	4.82843	0.577107
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	−1.00000	−0.117851
$73$	15.3137	1.79233	0.896167	−	0.443717i	$-0.146340\pi$
$73$	15.3137	1.79233	0.896167	+	0.443717i	$0.146340\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	2.82843	0.324443
$77$	9.65685	1.10050
$78$	3.41421	0.386584
$79$	10.2426	1.15239	0.576194	−	0.817313i	$-0.304537\pi$
$79$	10.2426	1.15239	0.576194	+	0.817313i	$0.304537\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.82843	−0.754074
$83$	2.34315	0.257194	0.128597	−	0.991697i	$-0.458953\pi$
$83$	2.34315	0.257194	0.128597	+	0.991697i	$0.458953\pi$
$84$	−4.82843	−0.526825
$85$	0	0
$86$	−8.24264	−0.888827
$87$	1.17157	0.125606
$88$	2.00000	0.213201
$89$	−10.4853	−1.11144	−0.555719	−	0.831370i	$-0.687557\pi$
$89$	−10.4853	−1.11144	−0.555719	+	0.831370i	$0.687557\pi$
$90$	−1.00000	−0.105409
$91$	16.4853	1.72813
$92$	2.24264	0.233811
$93$	6.24264	0.647332
$94$	−5.65685	−0.583460
$95$	2.82843	0.290191
$96$	−1.00000	−0.102062
$97$	−13.6569	−1.38664	−0.693322	−	0.720628i	$-0.743854\pi$
$97$	−13.6569	−1.38664	−0.693322	+	0.720628i	$0.743854\pi$
$98$	−16.3137	−1.64793
$99$	−2.00000	−0.201008
$100$	1.00000	0.100000
$101$	−15.4142	−1.53377	−0.766886	−	0.641784i	$-0.778195\pi$
$101$	−15.4142	−1.53377	−0.766886	+	0.641784i	$0.778195\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	3.41421	0.334791
$105$	−4.82843	−0.471206
$106$	7.65685	0.743699
$107$	−6.82843	−0.660129	−0.330064	−	0.943958i	$-0.607070\pi$
$107$	−6.82843	−0.660129	−0.330064	+	0.943958i	$0.607070\pi$
$108$	1.00000	0.0962250
$109$	9.31371	0.892091	0.446046	−	0.895010i	$-0.352832\pi$
$109$	9.31371	0.892091	0.446046	+	0.895010i	$0.352832\pi$
$110$	2.00000	0.190693
$111$	−2.00000	−0.189832
$112$	−4.82843	−0.456243
$113$	−7.75736	−0.729751	−0.364875	−	0.931056i	$-0.618888\pi$
$113$	−7.75736	−0.729751	−0.364875	+	0.931056i	$0.618888\pi$
$114$	−2.82843	−0.264906
$115$	2.24264	0.209127
$116$	1.17157	0.108778
$117$	−3.41421	−0.315644
$118$	12.2426	1.12703
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	6.00000	0.543214
$123$	6.82843	0.615699
$124$	6.24264	0.560606
$125$	1.00000	0.0894427
$126$	4.82843	0.430150
$127$	−18.8284	−1.67075	−0.835376	−	0.549678i	$-0.814750\pi$
$127$	−18.8284	−1.67075	−0.835376	+	0.549678i	$0.814750\pi$
$128$	−1.00000	−0.0883883
$129$	8.24264	0.725724
$130$	3.41421	0.299446
$131$	2.34315	0.204722	0.102361	−	0.994747i	$-0.467360\pi$
$131$	2.34315	0.204722	0.102361	+	0.994747i	$0.467360\pi$
$132$	−2.00000	−0.174078
$133$	−13.6569	−1.18420
$134$	15.0711	1.30194
$135$	1.00000	0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−2.24264	−0.190906
$139$	−10.8284	−0.918455	−0.459228	−	0.888319i	$-0.651874\pi$
$139$	−10.8284	−0.918455	−0.459228	+	0.888319i	$0.651874\pi$
$140$	−4.82843	−0.408077
$141$	5.65685	0.476393
$142$	−10.0000	−0.839181
$143$	6.82843	0.571022
$144$	1.00000	0.0833333
$145$	1.17157	0.0972938
$146$	−15.3137	−1.26737
$147$	16.3137	1.34553
$148$	−2.00000	−0.164399
$149$	23.2132	1.90170	0.950850	−	0.309652i	$-0.100213\pi$
$149$	23.2132	1.90170	0.950850	+	0.309652i	$0.100213\pi$
$150$	−1.00000	−0.0816497
$151$	−2.48528	−0.202249	−0.101125	−	0.994874i	$-0.532244\pi$
$151$	−2.48528	−0.202249	−0.101125	+	0.994874i	$0.532244\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	−9.65685	−0.778171
$155$	6.24264	0.501421
$156$	−3.41421	−0.273356
$157$	−1.07107	−0.0854805	−0.0427403	−	0.999086i	$-0.513609\pi$
$157$	−1.07107	−0.0854805	−0.0427403	+	0.999086i	$0.513609\pi$
$158$	−10.2426	−0.814861
$159$	−7.65685	−0.607228
$160$	−1.00000	−0.0790569
$161$	−10.8284	−0.853400
$162$	−1.00000	−0.0785674
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	6.82843	0.533211
$165$	−2.00000	−0.155700
$166$	−2.34315	−0.181863
$167$	−7.41421	−0.573729	−0.286865	−	0.957971i	$-0.592613\pi$
$167$	−7.41421	−0.573729	−0.286865	+	0.957971i	$0.592613\pi$
$168$	4.82843	0.372521
$169$	−1.34315	−0.103319
$170$	0	0
$171$	2.82843	0.216295
$172$	8.24264	0.628495
$173$	10.4853	0.797181	0.398591	−	0.917129i	$-0.369499\pi$
$173$	10.4853	0.797181	0.398591	+	0.917129i	$0.369499\pi$
$174$	−1.17157	−0.0888167
$175$	−4.82843	−0.364995
$176$	−2.00000	−0.150756
$177$	−12.2426	−0.920213
$178$	10.4853	0.785905
$179$	−5.41421	−0.404677	−0.202339	−	0.979316i	$-0.564854\pi$
$179$	−5.41421	−0.404677	−0.202339	+	0.979316i	$0.564854\pi$
$180$	1.00000	0.0745356
$181$	−18.4853	−1.37400	−0.687000	−	0.726657i	$-0.741073\pi$
$181$	−18.4853	−1.37400	−0.687000	+	0.726657i	$0.741073\pi$
$182$	−16.4853	−1.22197
$183$	−6.00000	−0.443533
$184$	−2.24264	−0.165330
$185$	−2.00000	−0.147043
$186$	−6.24264	−0.457733
$187$	0	0
$188$	5.65685	0.412568
$189$	−4.82843	−0.351216
$190$	−2.82843	−0.205196
$191$	−10.1421	−0.733859	−0.366930	−	0.930249i	$-0.619591\pi$
$191$	−10.1421	−0.733859	−0.366930	+	0.930249i	$0.619591\pi$
$192$	1.00000	0.0721688
$193$	−21.6569	−1.55889	−0.779447	−	0.626468i	$-0.784500\pi$
$193$	−21.6569	−1.55889	−0.779447	+	0.626468i	$0.784500\pi$
$194$	13.6569	0.980505
$195$	−3.41421	−0.244497
$196$	16.3137	1.16526
$197$	6.97056	0.496632	0.248316	−	0.968679i	$-0.420123\pi$
$197$	6.97056	0.496632	0.248316	+	0.968679i	$0.420123\pi$
$198$	2.00000	0.142134
$199$	−17.5563	−1.24454	−0.622268	−	0.782804i	$-0.713789\pi$
$199$	−17.5563	−1.24454	−0.622268	+	0.782804i	$0.713789\pi$
$200$	−1.00000	−0.0707107
$201$	−15.0711	−1.06303
$202$	15.4142	1.08454
$203$	−5.65685	−0.397033
$204$	0	0
$205$	6.82843	0.476918
$206$	−2.82843	−0.197066
$207$	2.24264	0.155874
$208$	−3.41421	−0.236733
$209$	−5.65685	−0.391293
$210$	4.82843	0.333193
$211$	6.14214	0.422842	0.211421	−	0.977395i	$-0.432191\pi$
$211$	6.14214	0.422842	0.211421	+	0.977395i	$0.432191\pi$
$212$	−7.65685	−0.525875
$213$	10.0000	0.685189
$214$	6.82843	0.466782
$215$	8.24264	0.562143
$216$	−1.00000	−0.0680414
$217$	−30.1421	−2.04618
$218$	−9.31371	−0.630804
$219$	15.3137	1.03480
$220$	−2.00000	−0.134840
$221$	0	0
$222$	2.00000	0.134231
$223$	−22.1421	−1.48275	−0.741374	−	0.671093i	$-0.765825\pi$
$223$	−22.1421	−1.48275	−0.741374	+	0.671093i	$0.765825\pi$
$224$	4.82843	0.322613
$225$	1.00000	0.0666667
$226$	7.75736	0.516012
$227$	−4.97056	−0.329908	−0.164954	−	0.986301i	$-0.552748\pi$
$227$	−4.97056	−0.329908	−0.164954	+	0.986301i	$0.552748\pi$
$228$	2.82843	0.187317
$229$	25.3137	1.67278	0.836388	−	0.548137i	$-0.184663\pi$
$229$	25.3137	1.67278	0.836388	+	0.548137i	$0.184663\pi$
$230$	−2.24264	−0.147875
$231$	9.65685	0.635374
$232$	−1.17157	−0.0769175
$233$	−6.38478	−0.418281	−0.209140	−	0.977886i	$-0.567067\pi$
$233$	−6.38478	−0.418281	−0.209140	+	0.977886i	$0.567067\pi$
$234$	3.41421	0.223194
$235$	5.65685	0.369012
$236$	−12.2426	−0.796928
$237$	10.2426	0.665331
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	1.00000	0.0645497
$241$	−2.58579	−0.166565	−0.0832826	−	0.996526i	$-0.526540\pi$
$241$	−2.58579	−0.166565	−0.0832826	+	0.996526i	$0.526540\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	16.3137	1.04224
$246$	−6.82843	−0.435365
$247$	−9.65685	−0.614451
$248$	−6.24264	−0.396408
$249$	2.34315	0.148491
$250$	−1.00000	−0.0632456
$251$	−27.0711	−1.70871	−0.854355	−	0.519689i	$-0.826048\pi$
$251$	−27.0711	−1.70871	−0.854355	+	0.519689i	$0.826048\pi$
$252$	−4.82843	−0.304162
$253$	−4.48528	−0.281987
$254$	18.8284	1.18140
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.65685	0.103352	0.0516759	−	0.998664i	$-0.483544\pi$
$257$	1.65685	0.103352	0.0516759	+	0.998664i	$0.483544\pi$
$258$	−8.24264	−0.513164
$259$	9.65685	0.600048
$260$	−3.41421	−0.211741
$261$	1.17157	0.0725185
$262$	−2.34315	−0.144760
$263$	−18.4853	−1.13985	−0.569926	−	0.821696i	$-0.693028\pi$
$263$	−18.4853	−1.13985	−0.569926	+	0.821696i	$0.693028\pi$
$264$	2.00000	0.123091
$265$	−7.65685	−0.470357
$266$	13.6569	0.837355
$267$	−10.4853	−0.641689
$268$	−15.0711	−0.920612
$269$	−11.6569	−0.710731	−0.355365	−	0.934727i	$-0.615644\pi$
$269$	−11.6569	−0.710731	−0.355365	+	0.934727i	$0.615644\pi$
$270$	−1.00000	−0.0608581
$271$	−1.51472	−0.0920126	−0.0460063	−	0.998941i	$-0.514649\pi$
$271$	−1.51472	−0.0920126	−0.0460063	+	0.998941i	$0.514649\pi$
$272$	0	0
$273$	16.4853	0.997735
$274$	12.0000	0.724947
$275$	−2.00000	−0.120605
$276$	2.24264	0.134991
$277$	−11.5147	−0.691852	−0.345926	−	0.938262i	$-0.612435\pi$
$277$	−11.5147	−0.691852	−0.345926	+	0.938262i	$0.612435\pi$
$278$	10.8284	0.649446
$279$	6.24264	0.373737
$280$	4.82843	0.288554
$281$	−11.1716	−0.666440	−0.333220	−	0.942849i	$-0.608135\pi$
$281$	−11.1716	−0.666440	−0.333220	+	0.942849i	$0.608135\pi$
$282$	−5.65685	−0.336861
$283$	−15.6569	−0.930703	−0.465352	−	0.885126i	$-0.654072\pi$
$283$	−15.6569	−0.930703	−0.465352	+	0.885126i	$0.654072\pi$
$284$	10.0000	0.593391
$285$	2.82843	0.167542
$286$	−6.82843	−0.403773
$287$	−32.9706	−1.94619
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−1.17157	−0.0687971
$291$	−13.6569	−0.800579
$292$	15.3137	0.896167
$293$	20.6274	1.20507	0.602533	−	0.798094i	$-0.294158\pi$
$293$	20.6274	1.20507	0.602533	+	0.798094i	$0.294158\pi$
$294$	−16.3137	−0.951435
$295$	−12.2426	−0.712794
$296$	2.00000	0.116248
$297$	−2.00000	−0.116052
$298$	−23.2132	−1.34470
$299$	−7.65685	−0.442807
$300$	1.00000	0.0577350
$301$	−39.7990	−2.29398
$302$	2.48528	0.143012
$303$	−15.4142	−0.885523
$304$	2.82843	0.162221
$305$	−6.00000	−0.343559
$306$	0	0
$307$	7.75736	0.442736	0.221368	−	0.975190i	$-0.428948\pi$
$307$	7.75736	0.442736	0.221368	+	0.975190i	$0.428948\pi$
$308$	9.65685	0.550250
$309$	2.82843	0.160904
$310$	−6.24264	−0.354558
$311$	23.6569	1.34146	0.670729	−	0.741703i	$-0.265982\pi$
$311$	23.6569	1.34146	0.670729	+	0.741703i	$0.265982\pi$
$312$	3.41421	0.193292
$313$	1.65685	0.0936509	0.0468255	−	0.998903i	$-0.485090\pi$
$313$	1.65685	0.0936509	0.0468255	+	0.998903i	$0.485090\pi$
$314$	1.07107	0.0604439
$315$	−4.82843	−0.272051
$316$	10.2426	0.576194
$317$	−13.3137	−0.747772	−0.373886	−	0.927475i	$-0.621975\pi$
$317$	−13.3137	−0.747772	−0.373886	+	0.927475i	$0.621975\pi$
$318$	7.65685	0.429375
$319$	−2.34315	−0.131191
$320$	1.00000	0.0559017
$321$	−6.82843	−0.381126
$322$	10.8284	0.603445
$323$	0	0
$324$	1.00000	0.0555556
$325$	−3.41421	−0.189386
$326$	2.00000	0.110770
$327$	9.31371	0.515049
$328$	−6.82843	−0.377037
$329$	−27.3137	−1.50585
$330$	2.00000	0.110096
$331$	−13.1716	−0.723975	−0.361988	−	0.932183i	$-0.617902\pi$
$331$	−13.1716	−0.723975	−0.361988	+	0.932183i	$0.617902\pi$
$332$	2.34315	0.128597
$333$	−2.00000	−0.109599
$334$	7.41421	0.405688
$335$	−15.0711	−0.823420
$336$	−4.82843	−0.263412
$337$	−3.31371	−0.180509	−0.0902546	−	0.995919i	$-0.528768\pi$
$337$	−3.31371	−0.180509	−0.0902546	+	0.995919i	$0.528768\pi$
$338$	1.34315	0.0730575
$339$	−7.75736	−0.421322
$340$	0	0
$341$	−12.4853	−0.676116
$342$	−2.82843	−0.152944
$343$	−44.9706	−2.42818
$344$	−8.24264	−0.444413
$345$	2.24264	0.120740
$346$	−10.4853	−0.563692
$347$	19.3137	1.03681	0.518407	−	0.855134i	$-0.326525\pi$
$347$	19.3137	1.03681	0.518407	+	0.855134i	$0.326525\pi$
$348$	1.17157	0.0628029
$349$	−7.65685	−0.409862	−0.204931	−	0.978776i	$-0.565697\pi$
$349$	−7.65685	−0.409862	−0.204931	+	0.978776i	$0.565697\pi$
$350$	4.82843	0.258090
$351$	−3.41421	−0.182237
$352$	2.00000	0.106600
$353$	−30.9706	−1.64840	−0.824198	−	0.566301i	$-0.808374\pi$
$353$	−30.9706	−1.64840	−0.824198	+	0.566301i	$0.808374\pi$
$354$	12.2426	0.650689
$355$	10.0000	0.530745
$356$	−10.4853	−0.555719
$357$	0	0
$358$	5.41421	0.286150
$359$	21.1716	1.11739	0.558696	−	0.829372i	$-0.311302\pi$
$359$	21.1716	1.11739	0.558696	+	0.829372i	$0.311302\pi$
$360$	−1.00000	−0.0527046
$361$	−11.0000	−0.578947
$362$	18.4853	0.971565
$363$	−7.00000	−0.367405
$364$	16.4853	0.864064
$365$	15.3137	0.801556
$366$	6.00000	0.313625
$367$	−4.82843	−0.252042	−0.126021	−	0.992028i	$-0.540221\pi$
$367$	−4.82843	−0.252042	−0.126021	+	0.992028i	$0.540221\pi$
$368$	2.24264	0.116906
$369$	6.82843	0.355474
$370$	2.00000	0.103975
$371$	36.9706	1.91942
$372$	6.24264	0.323666
$373$	9.55635	0.494809	0.247405	−	0.968912i	$-0.420422\pi$
$373$	9.55635	0.494809	0.247405	+	0.968912i	$0.420422\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−5.65685	−0.291730
$377$	−4.00000	−0.206010
$378$	4.82843	0.248347
$379$	−14.3431	−0.736758	−0.368379	−	0.929676i	$-0.620087\pi$
$379$	−14.3431	−0.736758	−0.368379	+	0.929676i	$0.620087\pi$
$380$	2.82843	0.145095
$381$	−18.8284	−0.964610
$382$	10.1421	0.518917
$383$	24.1421	1.23361	0.616803	−	0.787118i	$-0.288428\pi$
$383$	24.1421	1.23361	0.616803	+	0.787118i	$0.288428\pi$
$384$	−1.00000	−0.0510310
$385$	9.65685	0.492159
$386$	21.6569	1.10230
$387$	8.24264	0.418997
$388$	−13.6569	−0.693322
$389$	26.7279	1.35516	0.677580	−	0.735449i	$-0.263029\pi$
$389$	26.7279	1.35516	0.677580	+	0.735449i	$0.263029\pi$
$390$	3.41421	0.172885
$391$	0	0
$392$	−16.3137	−0.823967
$393$	2.34315	0.118196
$394$	−6.97056	−0.351172
$395$	10.2426	0.515363
$396$	−2.00000	−0.100504
$397$	−7.79899	−0.391420	−0.195710	−	0.980662i	$-0.562701\pi$
$397$	−7.79899	−0.391420	−0.195710	+	0.980662i	$0.562701\pi$
$398$	17.5563	0.880020
$399$	−13.6569	−0.683698
$400$	1.00000	0.0500000
$401$	−4.97056	−0.248218	−0.124109	−	0.992269i	$-0.539607\pi$
$401$	−4.97056	−0.248218	−0.124109	+	0.992269i	$0.539607\pi$
$402$	15.0711	0.751677
$403$	−21.3137	−1.06171
$404$	−15.4142	−0.766886
$405$	1.00000	0.0496904
$406$	5.65685	0.280745
$407$	4.00000	0.198273
$408$	0	0
$409$	−24.6274	−1.21775	−0.608874	−	0.793267i	$-0.708378\pi$
$409$	−24.6274	−1.21775	−0.608874	+	0.793267i	$0.708378\pi$
$410$	−6.82843	−0.337232
$411$	−12.0000	−0.591916
$412$	2.82843	0.139347
$413$	59.1127	2.90875
$414$	−2.24264	−0.110220
$415$	2.34315	0.115021
$416$	3.41421	0.167396
$417$	−10.8284	−0.530270
$418$	5.65685	0.276686
$419$	22.9706	1.12219	0.561093	−	0.827753i	$-0.310381\pi$
$419$	22.9706	1.12219	0.561093	+	0.827753i	$0.310381\pi$
$420$	−4.82843	−0.235603
$421$	−12.1421	−0.591771	−0.295886	−	0.955223i	$-0.595615\pi$
$421$	−12.1421	−0.591771	−0.295886	+	0.955223i	$0.595615\pi$
$422$	−6.14214	−0.298994
$423$	5.65685	0.275046
$424$	7.65685	0.371850
$425$	0	0
$426$	−10.0000	−0.484502
$427$	28.9706	1.40198
$428$	−6.82843	−0.330064
$429$	6.82843	0.329680
$430$	−8.24264	−0.397495
$431$	20.3431	0.979895	0.489947	−	0.871752i	$-0.337016\pi$
$431$	20.3431	0.979895	0.489947	+	0.871752i	$0.337016\pi$
$432$	1.00000	0.0481125
$433$	−20.3431	−0.977629	−0.488815	−	0.872388i	$-0.662571\pi$
$433$	−20.3431	−0.977629	−0.488815	+	0.872388i	$0.662571\pi$
$434$	30.1421	1.44687
$435$	1.17157	0.0561726
$436$	9.31371	0.446046
$437$	6.34315	0.303434
$438$	−15.3137	−0.731717
$439$	38.7279	1.84838	0.924191	−	0.381930i	$-0.124740\pi$
$439$	38.7279	1.84838	0.924191	+	0.381930i	$0.124740\pi$
$440$	2.00000	0.0953463
$441$	16.3137	0.776843
$442$	0	0
$443$	33.4558	1.58954	0.794768	−	0.606914i	$-0.207593\pi$
$443$	33.4558	1.58954	0.794768	+	0.606914i	$0.207593\pi$
$444$	−2.00000	−0.0949158
$445$	−10.4853	−0.497050
$446$	22.1421	1.04846
$447$	23.2132	1.09795
$448$	−4.82843	−0.228122
$449$	−6.62742	−0.312767	−0.156384	−	0.987696i	$-0.549984\pi$
$449$	−6.62742	−0.312767	−0.156384	+	0.987696i	$0.549984\pi$
$450$	−1.00000	−0.0471405
$451$	−13.6569	−0.643076
$452$	−7.75736	−0.364875
$453$	−2.48528	−0.116769
$454$	4.97056	0.233280
$455$	16.4853	0.772842
$456$	−2.82843	−0.132453
$457$	29.3137	1.37124	0.685619	−	0.727961i	$-0.259532\pi$
$457$	29.3137	1.37124	0.685619	+	0.727961i	$0.259532\pi$
$458$	−25.3137	−1.18283
$459$	0	0
$460$	2.24264	0.104564
$461$	10.0416	0.467685	0.233843	−	0.972274i	$-0.424870\pi$
$461$	10.0416	0.467685	0.233843	+	0.972274i	$0.424870\pi$
$462$	−9.65685	−0.449278
$463$	23.7990	1.10603	0.553016	−	0.833170i	$-0.313477\pi$
$463$	23.7990	1.10603	0.553016	+	0.833170i	$0.313477\pi$
$464$	1.17157	0.0543889
$465$	6.24264	0.289496
$466$	6.38478	0.295769
$467$	−24.9706	−1.15550	−0.577750	−	0.816214i	$-0.696069\pi$
$467$	−24.9706	−1.15550	−0.577750	+	0.816214i	$0.696069\pi$
$468$	−3.41421	−0.157822
$469$	72.7696	3.36019
$470$	−5.65685	−0.260931
$471$	−1.07107	−0.0493522
$472$	12.2426	0.563513
$473$	−16.4853	−0.757994
$474$	−10.2426	−0.470460
$475$	2.82843	0.129777
$476$	0	0
$477$	−7.65685	−0.350583
$478$	4.00000	0.182956
$479$	38.0000	1.73626	0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000	1.73626	0.868132	+	0.496333i	$0.165321\pi$
$480$	−1.00000	−0.0456435
$481$	6.82843	0.311349
$482$	2.58579	0.117779
$483$	−10.8284	−0.492710
$484$	−7.00000	−0.318182
$485$	−13.6569	−0.620126
$486$	−1.00000	−0.0453609
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	6.00000	0.271607
$489$	−2.00000	−0.0904431
$490$	−16.3137	−0.736978
$491$	32.2426	1.45509	0.727545	−	0.686060i	$-0.240661\pi$
$491$	32.2426	1.45509	0.727545	+	0.686060i	$0.240661\pi$
$492$	6.82843	0.307849
$493$	0	0
$494$	9.65685	0.434482
$495$	−2.00000	−0.0898933
$496$	6.24264	0.280303
$497$	−48.2843	−2.16585
$498$	−2.34315	−0.104999
$499$	17.1716	0.768705	0.384353	−	0.923186i	$-0.374425\pi$
$499$	17.1716	0.768705	0.384353	+	0.923186i	$0.374425\pi$
$500$	1.00000	0.0447214
$501$	−7.41421	−0.331243
$502$	27.0711	1.20824
$503$	4.38478	0.195508	0.0977538	−	0.995211i	$-0.468834\pi$
$503$	4.38478	0.195508	0.0977538	+	0.995211i	$0.468834\pi$
$504$	4.82843	0.215075
$505$	−15.4142	−0.685924
$506$	4.48528	0.199395
$507$	−1.34315	−0.0596512
$508$	−18.8284	−0.835376
$509$	−21.0711	−0.933959	−0.466979	−	0.884268i	$-0.654658\pi$
$509$	−21.0711	−0.933959	−0.466979	+	0.884268i	$0.654658\pi$
$510$	0	0
$511$	−73.9411	−3.27096
$512$	−1.00000	−0.0441942
$513$	2.82843	0.124878
$514$	−1.65685	−0.0730807
$515$	2.82843	0.124635
$516$	8.24264	0.362862
$517$	−11.3137	−0.497576
$518$	−9.65685	−0.424298
$519$	10.4853	0.460253
$520$	3.41421	0.149723
$521$	−21.6569	−0.948804	−0.474402	−	0.880308i	$-0.657336\pi$
$521$	−21.6569	−0.948804	−0.474402	+	0.880308i	$0.657336\pi$
$522$	−1.17157	−0.0512784
$523$	−24.5269	−1.07249	−0.536243	−	0.844063i	$-0.680157\pi$
$523$	−24.5269	−1.07249	−0.536243	+	0.844063i	$0.680157\pi$
$524$	2.34315	0.102361
$525$	−4.82843	−0.210730
$526$	18.4853	0.805997
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	−17.9706	−0.781329
$530$	7.65685	0.332592
$531$	−12.2426	−0.531285
$532$	−13.6569	−0.592100
$533$	−23.3137	−1.00983
$534$	10.4853	0.453743
$535$	−6.82843	−0.295219
$536$	15.0711	0.650971
$537$	−5.41421	−0.233641
$538$	11.6569	0.502563
$539$	−32.6274	−1.40536
$540$	1.00000	0.0430331
$541$	25.3137	1.08832	0.544161	−	0.838981i	$-0.316848\pi$
$541$	25.3137	1.08832	0.544161	+	0.838981i	$0.316848\pi$
$542$	1.51472	0.0650627
$543$	−18.4853	−0.793279
$544$	0	0
$545$	9.31371	0.398955
$546$	−16.4853	−0.705505
$547$	36.6274	1.56608	0.783038	−	0.621974i	$-0.213669\pi$
$547$	36.6274	1.56608	0.783038	+	0.621974i	$0.213669\pi$
$548$	−12.0000	−0.512615
$549$	−6.00000	−0.256074
$550$	2.00000	0.0852803
$551$	3.31371	0.141169
$552$	−2.24264	−0.0954531
$553$	−49.4558	−2.10308
$554$	11.5147	0.489214
$555$	−2.00000	−0.0848953
$556$	−10.8284	−0.459228
$557$	19.6569	0.832888	0.416444	−	0.909161i	$-0.363276\pi$
$557$	19.6569	0.832888	0.416444	+	0.909161i	$0.363276\pi$
$558$	−6.24264	−0.264272
$559$	−28.1421	−1.19029
$560$	−4.82843	−0.204038
$561$	0	0
$562$	11.1716	0.471244
$563$	30.6274	1.29079	0.645396	−	0.763848i	$-0.276692\pi$
$563$	30.6274	1.29079	0.645396	+	0.763848i	$0.276692\pi$
$564$	5.65685	0.238197
$565$	−7.75736	−0.326355
$566$	15.6569	0.658107
$567$	−4.82843	−0.202775
$568$	−10.0000	−0.419591
$569$	−25.5147	−1.06963	−0.534816	−	0.844968i	$-0.679619\pi$
$569$	−25.5147	−1.06963	−0.534816	+	0.844968i	$0.679619\pi$
$570$	−2.82843	−0.118470
$571$	5.65685	0.236732	0.118366	−	0.992970i	$-0.462234\pi$
$571$	5.65685	0.236732	0.118366	+	0.992970i	$0.462234\pi$
$572$	6.82843	0.285511
$573$	−10.1421	−0.423694
$574$	32.9706	1.37616
$575$	2.24264	0.0935246
$576$	1.00000	0.0416667
$577$	−30.9706	−1.28932	−0.644661	−	0.764469i	$-0.723001\pi$
$577$	−30.9706	−1.28932	−0.644661	+	0.764469i	$0.723001\pi$
$578$	0	0
$579$	−21.6569	−0.900028
$580$	1.17157	0.0486469
$581$	−11.3137	−0.469372
$582$	13.6569	0.566095
$583$	15.3137	0.634229
$584$	−15.3137	−0.633686
$585$	−3.41421	−0.141160
$586$	−20.6274	−0.852111
$587$	3.79899	0.156801	0.0784005	−	0.996922i	$-0.475019\pi$
$587$	3.79899	0.156801	0.0784005	+	0.996922i	$0.475019\pi$
$588$	16.3137	0.672766
$589$	17.6569	0.727538
$590$	12.2426	0.504022
$591$	6.97056	0.286731
$592$	−2.00000	−0.0821995
$593$	−32.6274	−1.33985	−0.669924	−	0.742430i	$-0.733673\pi$
$593$	−32.6274	−1.33985	−0.669924	+	0.742430i	$0.733673\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	23.2132	0.950850
$597$	−17.5563	−0.718534
$598$	7.65685	0.313112
$599$	−42.4264	−1.73350	−0.866748	−	0.498746i	$-0.833794\pi$
$599$	−42.4264	−1.73350	−0.866748	+	0.498746i	$0.833794\pi$
$600$	−1.00000	−0.0408248
$601$	1.41421	0.0576870	0.0288435	−	0.999584i	$-0.490818\pi$
$601$	1.41421	0.0576870	0.0288435	+	0.999584i	$0.490818\pi$
$602$	39.7990	1.62209
$603$	−15.0711	−0.613741
$604$	−2.48528	−0.101125
$605$	−7.00000	−0.284590
$606$	15.4142	0.626160
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−2.82843	−0.114708
$609$	−5.65685	−0.229227
$610$	6.00000	0.242933
$611$	−19.3137	−0.781349
$612$	0	0
$613$	−34.7279	−1.40265	−0.701324	−	0.712843i	$-0.747407\pi$
$613$	−34.7279	−1.40265	−0.701324	+	0.712843i	$0.747407\pi$
$614$	−7.75736	−0.313062
$615$	6.82843	0.275349
$616$	−9.65685	−0.389086
$617$	32.2426	1.29804	0.649020	−	0.760771i	$-0.275179\pi$
$617$	32.2426	1.29804	0.649020	+	0.760771i	$0.275179\pi$
$618$	−2.82843	−0.113776
$619$	10.3431	0.415726	0.207863	−	0.978158i	$-0.433349\pi$
$619$	10.3431	0.415726	0.207863	+	0.978158i	$0.433349\pi$
$620$	6.24264	0.250710
$621$	2.24264	0.0899941
$622$	−23.6569	−0.948553
$623$	50.6274	2.02834
$624$	−3.41421	−0.136678
$625$	1.00000	0.0400000
$626$	−1.65685	−0.0662212
$627$	−5.65685	−0.225913
$628$	−1.07107	−0.0427403
$629$	0	0
$630$	4.82843	0.192369
$631$	41.1127	1.63667	0.818335	−	0.574741i	$-0.194897\pi$
$631$	41.1127	1.63667	0.818335	+	0.574741i	$0.194897\pi$
$632$	−10.2426	−0.407430
$633$	6.14214	0.244128
$634$	13.3137	0.528755
$635$	−18.8284	−0.747183
$636$	−7.65685	−0.303614
$637$	−55.6985	−2.20685
$638$	2.34315	0.0927660
$639$	10.0000	0.395594
$640$	−1.00000	−0.0395285
$641$	25.6569	1.01338	0.506692	−	0.862127i	$-0.330868\pi$
$641$	25.6569	1.01338	0.506692	+	0.862127i	$0.330868\pi$
$642$	6.82843	0.269497
$643$	16.6274	0.655721	0.327861	−	0.944726i	$-0.393672\pi$
$643$	16.6274	0.655721	0.327861	+	0.944726i	$0.393672\pi$
$644$	−10.8284	−0.426700
$645$	8.24264	0.324554
$646$	0	0
$647$	29.1127	1.14454	0.572269	−	0.820066i	$-0.306063\pi$
$647$	29.1127	1.14454	0.572269	+	0.820066i	$0.306063\pi$
$648$	−1.00000	−0.0392837
$649$	24.4853	0.961131
$650$	3.41421	0.133916
$651$	−30.1421	−1.18136
$652$	−2.00000	−0.0783260
$653$	−22.2843	−0.872051	−0.436025	−	0.899934i	$-0.643614\pi$
$653$	−22.2843	−0.872051	−0.436025	+	0.899934i	$0.643614\pi$
$654$	−9.31371	−0.364195
$655$	2.34315	0.0915543
$656$	6.82843	0.266605
$657$	15.3137	0.597445
$658$	27.3137	1.06480
$659$	25.4142	0.989997	0.494999	−	0.868894i	$-0.335169\pi$
$659$	25.4142	0.989997	0.494999	+	0.868894i	$0.335169\pi$
$660$	−2.00000	−0.0778499
$661$	26.2843	1.02234	0.511170	−	0.859480i	$-0.329212\pi$
$661$	26.2843	1.02234	0.511170	+	0.859480i	$0.329212\pi$
$662$	13.1716	0.511928
$663$	0	0
$664$	−2.34315	−0.0909317
$665$	−13.6569	−0.529590
$666$	2.00000	0.0774984
$667$	2.62742	0.101734
$668$	−7.41421	−0.286865
$669$	−22.1421	−0.856064
$670$	15.0711	0.582246
$671$	12.0000	0.463255
$672$	4.82843	0.186261
$673$	−38.6274	−1.48898	−0.744489	−	0.667635i	$-0.767307\pi$
$673$	−38.6274	−1.48898	−0.744489	+	0.667635i	$0.767307\pi$
$674$	3.31371	0.127639
$675$	1.00000	0.0384900
$676$	−1.34315	−0.0516595
$677$	42.9706	1.65149	0.825746	−	0.564041i	$-0.190754\pi$
$677$	42.9706	1.65149	0.825746	+	0.564041i	$0.190754\pi$
$678$	7.75736	0.297920
$679$	65.9411	2.53059
$680$	0	0
$681$	−4.97056	−0.190472
$682$	12.4853	0.478086
$683$	−6.34315	−0.242714	−0.121357	−	0.992609i	$-0.538725\pi$
$683$	−6.34315	−0.242714	−0.121357	+	0.992609i	$0.538725\pi$
$684$	2.82843	0.108148
$685$	−12.0000	−0.458496
$686$	44.9706	1.71698
$687$	25.3137	0.965778
$688$	8.24264	0.314248
$689$	26.1421	0.995936
$690$	−2.24264	−0.0853759
$691$	14.1421	0.537992	0.268996	−	0.963141i	$-0.413308\pi$
$691$	14.1421	0.537992	0.268996	+	0.963141i	$0.413308\pi$
$692$	10.4853	0.398591
$693$	9.65685	0.366834
$694$	−19.3137	−0.733138
$695$	−10.8284	−0.410746
$696$	−1.17157	−0.0444084
$697$	0	0
$698$	7.65685	0.289816
$699$	−6.38478	−0.241494
$700$	−4.82843	−0.182497
$701$	−42.0416	−1.58789	−0.793945	−	0.607989i	$-0.791976\pi$
$701$	−42.0416	−1.58789	−0.793945	+	0.607989i	$0.791976\pi$
$702$	3.41421	0.128861
$703$	−5.65685	−0.213352
$704$	−2.00000	−0.0753778
$705$	5.65685	0.213049
$706$	30.9706	1.16559
$707$	74.4264	2.79909
$708$	−12.2426	−0.460107
$709$	−35.4558	−1.33157	−0.665786	−	0.746143i	$-0.731904\pi$
$709$	−35.4558	−1.33157	−0.665786	+	0.746143i	$0.731904\pi$
$710$	−10.0000	−0.375293
$711$	10.2426	0.384129
$712$	10.4853	0.392953
$713$	14.0000	0.524304
$714$	0	0
$715$	6.82843	0.255369
$716$	−5.41421	−0.202339
$717$	−4.00000	−0.149383
$718$	−21.1716	−0.790116
$719$	14.9706	0.558308	0.279154	−	0.960246i	$-0.409946\pi$
$719$	14.9706	0.558308	0.279154	+	0.960246i	$0.409946\pi$
$720$	1.00000	0.0372678
$721$	−13.6569	−0.508608
$722$	11.0000	0.409378
$723$	−2.58579	−0.0961664
$724$	−18.4853	−0.687000
$725$	1.17157	0.0435111
$726$	7.00000	0.259794
$727$	44.0833	1.63496	0.817479	−	0.575959i	$-0.195371\pi$
$727$	44.0833	1.63496	0.817479	+	0.575959i	$0.195371\pi$
$728$	−16.4853	−0.610985
$729$	1.00000	0.0370370
$730$	−15.3137	−0.566786
$731$	0	0
$732$	−6.00000	−0.221766
$733$	7.21320	0.266426	0.133213	−	0.991087i	$-0.457471\pi$
$733$	7.21320	0.266426	0.133213	+	0.991087i	$0.457471\pi$
$734$	4.82843	0.178220
$735$	16.3137	0.601740
$736$	−2.24264	−0.0826648
$737$	30.1421	1.11030
$738$	−6.82843	−0.251358
$739$	14.3431	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$739$	14.3431	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$740$	−2.00000	−0.0735215
$741$	−9.65685	−0.354753
$742$	−36.9706	−1.35723
$743$	−10.7279	−0.393569	−0.196785	−	0.980447i	$-0.563050\pi$
$743$	−10.7279	−0.393569	−0.196785	+	0.980447i	$0.563050\pi$
$744$	−6.24264	−0.228866
$745$	23.2132	0.850466
$746$	−9.55635	−0.349883
$747$	2.34315	0.0857312
$748$	0	0
$749$	32.9706	1.20472
$750$	−1.00000	−0.0365148
$751$	−38.5269	−1.40587	−0.702933	−	0.711256i	$-0.748127\pi$
$751$	−38.5269	−1.40587	−0.702933	+	0.711256i	$0.748127\pi$
$752$	5.65685	0.206284
$753$	−27.0711	−0.986525
$754$	4.00000	0.145671
$755$	−2.48528	−0.0904487
$756$	−4.82843	−0.175608
$757$	−5.75736	−0.209255	−0.104627	−	0.994511i	$-0.533365\pi$
$757$	−5.75736	−0.209255	−0.104627	+	0.994511i	$0.533365\pi$
$758$	14.3431	0.520967
$759$	−4.48528	−0.162805
$760$	−2.82843	−0.102598
$761$	−28.8284	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$761$	−28.8284	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$762$	18.8284	0.682082
$763$	−44.9706	−1.62804
$764$	−10.1421	−0.366930
$765$	0	0
$766$	−24.1421	−0.872291
$767$	41.7990	1.50927
$768$	1.00000	0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	−9.65685	−0.348009
$771$	1.65685	0.0596701
$772$	−21.6569	−0.779447
$773$	41.1127	1.47872	0.739360	−	0.673310i	$-0.235128\pi$
$773$	41.1127	1.47872	0.739360	+	0.673310i	$0.235128\pi$
$774$	−8.24264	−0.296276
$775$	6.24264	0.224242
$776$	13.6569	0.490252
$777$	9.65685	0.346438
$778$	−26.7279	−0.958242
$779$	19.3137	0.691985
$780$	−3.41421	−0.122248
$781$	−20.0000	−0.715656
$782$	0	0
$783$	1.17157	0.0418686
$784$	16.3137	0.582632
$785$	−1.07107	−0.0382280
$786$	−2.34315	−0.0835772
$787$	−16.9706	−0.604935	−0.302468	−	0.953160i	$-0.597810\pi$
$787$	−16.9706	−0.604935	−0.302468	+	0.953160i	$0.597810\pi$
$788$	6.97056	0.248316
$789$	−18.4853	−0.658093
$790$	−10.2426	−0.364417
$791$	37.4558	1.33178
$792$	2.00000	0.0710669
$793$	20.4853	0.727454
$794$	7.79899	0.276776
$795$	−7.65685	−0.271561
$796$	−17.5563	−0.622268
$797$	−12.8284	−0.454406	−0.227203	−	0.973847i	$-0.572958\pi$
$797$	−12.8284	−0.454406	−0.227203	+	0.973847i	$0.572958\pi$
$798$	13.6569	0.483447
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−10.4853	−0.370479
$802$	4.97056	0.175517
$803$	−30.6274	−1.08082
$804$	−15.0711	−0.531516
$805$	−10.8284	−0.381652
$806$	21.3137	0.750743
$807$	−11.6569	−0.410341
$808$	15.4142	0.542270
$809$	−27.1127	−0.953232	−0.476616	−	0.879112i	$-0.658137\pi$
$809$	−27.1127	−0.953232	−0.476616	+	0.879112i	$0.658137\pi$
$810$	−1.00000	−0.0351364
$811$	24.9706	0.876835	0.438418	−	0.898771i	$-0.355539\pi$
$811$	24.9706	0.876835	0.438418	+	0.898771i	$0.355539\pi$
$812$	−5.65685	−0.198517
$813$	−1.51472	−0.0531235
$814$	−4.00000	−0.140200
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	23.3137	0.815643
$818$	24.6274	0.861077
$819$	16.4853	0.576042
$820$	6.82843	0.238459
$821$	2.82843	0.0987128	0.0493564	−	0.998781i	$-0.484283\pi$
$821$	2.82843	0.0987128	0.0493564	+	0.998781i	$0.484283\pi$
$822$	12.0000	0.418548
$823$	−5.51472	−0.192231	−0.0961155	−	0.995370i	$-0.530642\pi$
$823$	−5.51472	−0.192231	−0.0961155	+	0.995370i	$0.530642\pi$
$824$	−2.82843	−0.0985329
$825$	−2.00000	−0.0696311
$826$	−59.1127	−2.05679
$827$	−18.8284	−0.654729	−0.327364	−	0.944898i	$-0.606160\pi$
$827$	−18.8284	−0.654729	−0.327364	+	0.944898i	$0.606160\pi$
$828$	2.24264	0.0779372
$829$	−30.4853	−1.05880	−0.529399	−	0.848373i	$-0.677582\pi$
$829$	−30.4853	−1.05880	−0.529399	+	0.848373i	$0.677582\pi$
$830$	−2.34315	−0.0813318
$831$	−11.5147	−0.399441
$832$	−3.41421	−0.118367
$833$	0	0
$834$	10.8284	0.374958
$835$	−7.41421	−0.256579
$836$	−5.65685	−0.195646
$837$	6.24264	0.215777
$838$	−22.9706	−0.793505
$839$	23.8579	0.823665	0.411832	−	0.911260i	$-0.364889\pi$
$839$	23.8579	0.823665	0.411832	+	0.911260i	$0.364889\pi$
$840$	4.82843	0.166597
$841$	−27.6274	−0.952670
$842$	12.1421	0.418446
$843$	−11.1716	−0.384769
$844$	6.14214	0.211421
$845$	−1.34315	−0.0462056
$846$	−5.65685	−0.194487
$847$	33.7990	1.16135
$848$	−7.65685	−0.262937
$849$	−15.6569	−0.537342
$850$	0	0
$851$	−4.48528	−0.153753
$852$	10.0000	0.342594
$853$	−4.20101	−0.143840	−0.0719199	−	0.997410i	$-0.522913\pi$
$853$	−4.20101	−0.143840	−0.0719199	+	0.997410i	$0.522913\pi$
$854$	−28.9706	−0.991352
$855$	2.82843	0.0967302
$856$	6.82843	0.233391
$857$	12.2426	0.418201	0.209100	−	0.977894i	$-0.432946\pi$
$857$	12.2426	0.418201	0.209100	+	0.977894i	$0.432946\pi$
$858$	−6.82843	−0.233119
$859$	−34.4264	−1.17461	−0.587307	−	0.809364i	$-0.699812\pi$
$859$	−34.4264	−1.17461	−0.587307	+	0.809364i	$0.699812\pi$
$860$	8.24264	0.281072
$861$	−32.9706	−1.12363
$862$	−20.3431	−0.692890
$863$	−28.4264	−0.967646	−0.483823	−	0.875166i	$-0.660752\pi$
$863$	−28.4264	−0.967646	−0.483823	+	0.875166i	$0.660752\pi$
$864$	−1.00000	−0.0340207
$865$	10.4853	0.356510
$866$	20.3431	0.691288
$867$	0	0
$868$	−30.1421	−1.02309
$869$	−20.4853	−0.694916
$870$	−1.17157	−0.0397200
$871$	51.4558	1.74351
$872$	−9.31371	−0.315402
$873$	−13.6569	−0.462214
$874$	−6.34315	−0.214560
$875$	−4.82843	−0.163231
$876$	15.3137	0.517402
$877$	−32.6274	−1.10175	−0.550875	−	0.834588i	$-0.685706\pi$
$877$	−32.6274	−1.10175	−0.550875	+	0.834588i	$0.685706\pi$
$878$	−38.7279	−1.30700
$879$	20.6274	0.695746
$880$	−2.00000	−0.0674200
$881$	20.4853	0.690167	0.345083	−	0.938572i	$-0.387851\pi$
$881$	20.4853	0.690167	0.345083	+	0.938572i	$0.387851\pi$
$882$	−16.3137	−0.549311
$883$	−27.7574	−0.934110	−0.467055	−	0.884228i	$-0.654685\pi$
$883$	−27.7574	−0.934110	−0.467055	+	0.884228i	$0.654685\pi$
$884$	0	0
$885$	−12.2426	−0.411532
$886$	−33.4558	−1.12397
$887$	−45.3553	−1.52288	−0.761442	−	0.648233i	$-0.775508\pi$
$887$	−45.3553	−1.52288	−0.761442	+	0.648233i	$0.775508\pi$
$888$	2.00000	0.0671156
$889$	90.9117	3.04908
$890$	10.4853	0.351467
$891$	−2.00000	−0.0670025
$892$	−22.1421	−0.741374
$893$	16.0000	0.535420
$894$	−23.2132	−0.776366
$895$	−5.41421	−0.180977
$896$	4.82843	0.161306
$897$	−7.65685	−0.255655
$898$	6.62742	0.221160
$899$	7.31371	0.243926
$900$	1.00000	0.0333333
$901$	0	0
$902$	13.6569	0.454724
$903$	−39.7990	−1.32443
$904$	7.75736	0.258006
$905$	−18.4853	−0.614472
$906$	2.48528	0.0825679
$907$	−45.2548	−1.50266	−0.751331	−	0.659925i	$-0.770588\pi$
$907$	−45.2548	−1.50266	−0.751331	+	0.659925i	$0.770588\pi$
$908$	−4.97056	−0.164954
$909$	−15.4142	−0.511257
$910$	−16.4853	−0.546482
$911$	−9.02944	−0.299159	−0.149579	−	0.988750i	$-0.547792\pi$
$911$	−9.02944	−0.299159	−0.149579	+	0.988750i	$0.547792\pi$
$912$	2.82843	0.0936586
$913$	−4.68629	−0.155094
$914$	−29.3137	−0.969611
$915$	−6.00000	−0.198354
$916$	25.3137	0.836388
$917$	−11.3137	−0.373612
$918$	0	0
$919$	1.51472	0.0499660	0.0249830	−	0.999688i	$-0.492047\pi$
$919$	1.51472	0.0499660	0.0249830	+	0.999688i	$0.492047\pi$
$920$	−2.24264	−0.0739377
$921$	7.75736	0.255614
$922$	−10.0416	−0.330703
$923$	−34.1421	−1.12380
$924$	9.65685	0.317687
$925$	−2.00000	−0.0657596
$926$	−23.7990	−0.782083
$927$	2.82843	0.0928977
$928$	−1.17157	−0.0384588
$929$	−43.7990	−1.43700	−0.718499	−	0.695528i	$-0.755171\pi$
$929$	−43.7990	−1.43700	−0.718499	+	0.695528i	$0.755171\pi$
$930$	−6.24264	−0.204704
$931$	46.1421	1.51225
$932$	−6.38478	−0.209140
$933$	23.6569	0.774491
$934$	24.9706	0.817062
$935$	0	0
$936$	3.41421	0.111597
$937$	−48.1421	−1.57274	−0.786368	−	0.617759i	$-0.788041\pi$
$937$	−48.1421	−1.57274	−0.786368	+	0.617759i	$0.788041\pi$
$938$	−72.7696	−2.37601
$939$	1.65685	0.0540694
$940$	5.65685	0.184506
$941$	−18.8284	−0.613789	−0.306895	−	0.951744i	$-0.599290\pi$
$941$	−18.8284	−0.613789	−0.306895	+	0.951744i	$0.599290\pi$
$942$	1.07107	0.0348973
$943$	15.3137	0.498683
$944$	−12.2426	−0.398464
$945$	−4.82843	−0.157069
$946$	16.4853	0.535983
$947$	45.4558	1.47712	0.738558	−	0.674190i	$-0.235507\pi$
$947$	45.4558	1.47712	0.738558	+	0.674190i	$0.235507\pi$
$948$	10.2426	0.332666
$949$	−52.2843	−1.69722
$950$	−2.82843	−0.0917663
$951$	−13.3137	−0.431727
$952$	0	0
$953$	47.2548	1.53073	0.765367	−	0.643594i	$-0.222557\pi$
$953$	47.2548	1.53073	0.765367	+	0.643594i	$0.222557\pi$
$954$	7.65685	0.247900
$955$	−10.1421	−0.328192
$956$	−4.00000	−0.129369
$957$	−2.34315	−0.0757431
$958$	−38.0000	−1.22772
$959$	57.9411	1.87102
$960$	1.00000	0.0322749
$961$	7.97056	0.257115
$962$	−6.82843	−0.220157
$963$	−6.82843	−0.220043
$964$	−2.58579	−0.0832826
$965$	−21.6569	−0.697159
$966$	10.8284	0.348399
$967$	−12.6863	−0.407964	−0.203982	−	0.978975i	$-0.565388\pi$
$967$	−12.6863	−0.407964	−0.203982	+	0.978975i	$0.565388\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	13.6569	0.438495
$971$	−13.6152	−0.436933	−0.218467	−	0.975844i	$-0.570105\pi$
$971$	−13.6152	−0.436933	−0.218467	+	0.975844i	$0.570105\pi$
$972$	1.00000	0.0320750
$973$	52.2843	1.67616
$974$	26.0000	0.833094
$975$	−3.41421	−0.109342
$976$	−6.00000	−0.192055
$977$	35.9411	1.14986	0.574929	−	0.818203i	$-0.305030\pi$
$977$	35.9411	1.14986	0.574929	+	0.818203i	$0.305030\pi$
$978$	2.00000	0.0639529
$979$	20.9706	0.670222
$980$	16.3137	0.521122
$981$	9.31371	0.297364
$982$	−32.2426	−1.02890
$983$	30.0416	0.958179	0.479090	−	0.877766i	$-0.340967\pi$
$983$	30.0416	0.958179	0.479090	+	0.877766i	$0.340967\pi$
$984$	−6.82843	−0.217682
$985$	6.97056	0.222101
$986$	0	0
$987$	−27.3137	−0.869405
$988$	−9.65685	−0.307225
$989$	18.4853	0.587798
$990$	2.00000	0.0635642
$991$	−49.3553	−1.56782	−0.783912	−	0.620872i	$-0.786779\pi$
$991$	−49.3553	−1.56782	−0.783912	+	0.620872i	$0.786779\pi$
$992$	−6.24264	−0.198204
$993$	−13.1716	−0.417987
$994$	48.2843	1.53148
$995$	−17.5563	−0.556574
$996$	2.34315	0.0742454
$997$	40.3431	1.27768	0.638840	−	0.769340i	$-0.279415\pi$
$997$	40.3431	1.27768	0.638840	+	0.769340i	$0.279415\pi$
$998$	−17.1716	−0.543557
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bg.1.1		2
17.2	even	8		510.2.p.b.361.2	✓	4
17.9	even	8		510.2.p.b.421.2	yes	4
17.16	even	2		8670.2.a.bc.1.2		2
51.2	odd	8		1530.2.q.e.361.2		4
51.26	odd	8		1530.2.q.e.1441.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.p.b.361.2	✓	4	17.2	even	8
510.2.p.b.421.2	yes	4	17.9	even	8
1530.2.q.e.361.2		4	51.2	odd	8
1530.2.q.e.1441.2		4	51.26	odd	8
8670.2.a.bc.1.2		2	17.16	even	2
8670.2.a.bg.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -3.41421 q^{13} +4.82843 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} -4.82843 q^{21} +2.00000 q^{22} +2.24264 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.41421 q^{26} +1.00000 q^{27} -4.82843 q^{28} +1.17157 q^{29} -1.00000 q^{30} +6.24264 q^{31} -1.00000 q^{32} -2.00000 q^{33} -4.82843 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.82843 q^{38} -3.41421 q^{39} -1.00000 q^{40} +6.82843 q^{41} +4.82843 q^{42} +8.24264 q^{43} -2.00000 q^{44} +1.00000 q^{45} -2.24264 q^{46} +5.65685 q^{47} +1.00000 q^{48} +16.3137 q^{49} -1.00000 q^{50} -3.41421 q^{52} -7.65685 q^{53} -1.00000 q^{54} -2.00000 q^{55} +4.82843 q^{56} +2.82843 q^{57} -1.17157 q^{58} -12.2426 q^{59} +1.00000 q^{60} -6.00000 q^{61} -6.24264 q^{62} -4.82843 q^{63} +1.00000 q^{64} -3.41421 q^{65} +2.00000 q^{66} -15.0711 q^{67} +2.24264 q^{69} +4.82843 q^{70} +10.0000 q^{71} -1.00000 q^{72} +15.3137 q^{73} +2.00000 q^{74} +1.00000 q^{75} +2.82843 q^{76} +9.65685 q^{77} +3.41421 q^{78} +10.2426 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.82843 q^{82} +2.34315 q^{83} -4.82843 q^{84} -8.24264 q^{86} +1.17157 q^{87} +2.00000 q^{88} -10.4853 q^{89} -1.00000 q^{90} +16.4853 q^{91} +2.24264 q^{92} +6.24264 q^{93} -5.65685 q^{94} +2.82843 q^{95} -1.00000 q^{96} -13.6569 q^{97} -16.3137 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 4 q^{21} + 4 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 4 q^{28} + 8 q^{29} - 2 q^{30} + 4 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{39} - 2 q^{40} + 8 q^{41} + 4 q^{42} + 8 q^{43} - 4 q^{44} + 2 q^{45} + 4 q^{46} + 2 q^{48} + 10 q^{49} - 2 q^{50} - 4 q^{52} - 4 q^{53} - 2 q^{54} - 4 q^{55} + 4 q^{56} - 8 q^{58} - 16 q^{59} + 2 q^{60} - 12 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{65} + 4 q^{66} - 16 q^{67} - 4 q^{69} + 4 q^{70} + 20 q^{71} - 2 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} + 8 q^{77} + 4 q^{78} + 12 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 16 q^{83} - 4 q^{84} - 8 q^{86} + 8 q^{87} + 4 q^{88} - 4 q^{89} - 2 q^{90} + 16 q^{91} - 4 q^{92} + 4 q^{93} - 2 q^{96} - 16 q^{97} - 10 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 4 * q^13 + 4 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^18 + 2 * q^20 - 4 * q^21 + 4 * q^22 - 4 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 4 * q^28 + 8 * q^29 - 2 * q^30 + 4 * q^31 - 2 * q^32 - 4 * q^33 - 4 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^39 - 2 * q^40 + 8 * q^41 + 4 * q^42 + 8 * q^43 - 4 * q^44 + 2 * q^45 + 4 * q^46 + 2 * q^48 + 10 * q^49 - 2 * q^50 - 4 * q^52 - 4 * q^53 - 2 * q^54 - 4 * q^55 + 4 * q^56 - 8 * q^58 - 16 * q^59 + 2 * q^60 - 12 * q^61 - 4 * q^62 - 4 * q^63 + 2 * q^64 - 4 * q^65 + 4 * q^66 - 16 * q^67 - 4 * q^69 + 4 * q^70 + 20 * q^71 - 2 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 + 8 * q^77 + 4 * q^78 + 12 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^82 + 16 * q^83 - 4 * q^84 - 8 * q^86 + 8 * q^87 + 4 * q^88 - 4 * q^89 - 2 * q^90 + 16 * q^91 - 4 * q^92 + 4 * q^93 - 2 * q^96 - 16 * q^97 - 10 * q^98 - 4 * q^99

Introduction

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