LMFDB - Embedded newform 8670.2.a.cj.1.2

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:	$0$
Dimension:	$8$
Coefficient field:	8.8.20417871872.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 38x^{4} + 40x^{3} - 20x^{2} - 24x + 2 $ x^8 - 12x^6 - 8x^5 + 38x^4 + 40x^3 - 20x^2 - 24x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.792510$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.80701 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.80701 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.65672 q^{11} -1.00000 q^{12} +4.55356 q^{13} +4.80701 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +6.04449 q^{19} -1.00000 q^{20} +4.80701 q^{21} -2.65672 q^{22} +4.32010 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.55356 q^{26} -1.00000 q^{27} -4.80701 q^{28} -4.06491 q^{29} -1.00000 q^{30} +3.93892 q^{31} -1.00000 q^{32} -2.65672 q^{33} +4.80701 q^{35} +1.00000 q^{36} -5.53073 q^{37} -6.04449 q^{38} -4.55356 q^{39} +1.00000 q^{40} +11.3393 q^{41} -4.80701 q^{42} -7.80625 q^{43} +2.65672 q^{44} -1.00000 q^{45} -4.32010 q^{46} +1.83866 q^{47} -1.00000 q^{48} +16.1074 q^{49} -1.00000 q^{50} +4.55356 q^{52} +5.88254 q^{53} +1.00000 q^{54} -2.65672 q^{55} +4.80701 q^{56} -6.04449 q^{57} +4.06491 q^{58} +3.50381 q^{59} +1.00000 q^{60} -7.66447 q^{61} -3.93892 q^{62} -4.80701 q^{63} +1.00000 q^{64} -4.55356 q^{65} +2.65672 q^{66} +13.8487 q^{67} -4.32010 q^{69} -4.80701 q^{70} -4.86663 q^{71} -1.00000 q^{72} -8.43486 q^{73} +5.53073 q^{74} -1.00000 q^{75} +6.04449 q^{76} -12.7709 q^{77} +4.55356 q^{78} +13.1363 q^{79} -1.00000 q^{80} +1.00000 q^{81} -11.3393 q^{82} -17.6812 q^{83} +4.80701 q^{84} +7.80625 q^{86} +4.06491 q^{87} -2.65672 q^{88} +15.5491 q^{89} +1.00000 q^{90} -21.8890 q^{91} +4.32010 q^{92} -3.93892 q^{93} -1.83866 q^{94} -6.04449 q^{95} +1.00000 q^{96} +17.1612 q^{97} -16.1074 q^{98} +2.65672 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	$ 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{14} + 8 q^{15} + 8 q^{16} - 8 q^{18} - 8 q^{20} + 8 q^{21} - 8 q^{22} + 8 q^{24} + 8 q^{25} - 8 q^{27} - 8 q^{28} + 8 q^{29} - 8 q^{30} - 8 q^{31} - 8 q^{32} - 8 q^{33} + 8 q^{35} + 8 q^{36} - 32 q^{37} + 8 q^{40} + 8 q^{41} - 8 q^{42} - 8 q^{43} + 8 q^{44} - 8 q^{45} - 8 q^{48} + 40 q^{49} - 8 q^{50} + 16 q^{53} + 8 q^{54} - 8 q^{55} + 8 q^{56} - 8 q^{58} + 32 q^{59} + 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 8 q^{64} + 8 q^{66} - 8 q^{67} - 8 q^{70} + 24 q^{71} - 8 q^{72} - 40 q^{73} + 32 q^{74} - 8 q^{75} + 16 q^{77} - 24 q^{79} - 8 q^{80} + 8 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{84} + 8 q^{86} - 8 q^{87} - 8 q^{88} + 48 q^{89} + 8 q^{90} - 24 q^{91} + 8 q^{93} + 8 q^{96} - 24 q^{97} - 40 q^{98} + 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + 8 * q^11 - 8 * q^12 + 8 * q^14 + 8 * q^15 + 8 * q^16 - 8 * q^18 - 8 * q^20 + 8 * q^21 - 8 * q^22 + 8 * q^24 + 8 * q^25 - 8 * q^27 - 8 * q^28 + 8 * q^29 - 8 * q^30 - 8 * q^31 - 8 * q^32 - 8 * q^33 + 8 * q^35 + 8 * q^36 - 32 * q^37 + 8 * q^40 + 8 * q^41 - 8 * q^42 - 8 * q^43 + 8 * q^44 - 8 * q^45 - 8 * q^48 + 40 * q^49 - 8 * q^50 + 16 * q^53 + 8 * q^54 - 8 * q^55 + 8 * q^56 - 8 * q^58 + 32 * q^59 + 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 8 * q^64 + 8 * q^66 - 8 * q^67 - 8 * q^70 + 24 * q^71 - 8 * q^72 - 40 * q^73 + 32 * q^74 - 8 * q^75 + 16 * q^77 - 24 * q^79 - 8 * q^80 + 8 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^84 + 8 * q^86 - 8 * q^87 - 8 * q^88 + 48 * q^89 + 8 * q^90 - 24 * q^91 + 8 * q^93 + 8 * q^96 - 24 * q^97 - 40 * q^98 + 8 * 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
$7$	−4.80701	−1.81688	−0.908440	−	0.418015i	$-0.862726\pi$
$7$	−4.80701	−1.81688	−0.908440	+	0.418015i	$0.862726\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	2.65672	0.801030	0.400515	−	0.916290i	$-0.368831\pi$
$11$	2.65672	0.801030	0.400515	+	0.916290i	$0.368831\pi$
$12$	−1.00000	−0.288675
$13$	4.55356	1.26293	0.631465	−	0.775405i	$-0.282454\pi$
$13$	4.55356	1.26293	0.631465	+	0.775405i	$0.282454\pi$
$14$	4.80701	1.28473
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	6.04449	1.38670	0.693350	−	0.720601i	$-0.256134\pi$
$19$	6.04449	1.38670	0.693350	+	0.720601i	$0.256134\pi$
$20$	−1.00000	−0.223607
$21$	4.80701	1.04898
$22$	−2.65672	−0.566414
$23$	4.32010	0.900804	0.450402	−	0.892826i	$-0.351281\pi$
$23$	4.32010	0.900804	0.450402	+	0.892826i	$0.351281\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−4.55356	−0.893026
$27$	−1.00000	−0.192450
$28$	−4.80701	−0.908440
$29$	−4.06491	−0.754834	−0.377417	−	0.926043i	$-0.623188\pi$
$29$	−4.06491	−0.754834	−0.377417	+	0.926043i	$0.623188\pi$
$30$	−1.00000	−0.182574
$31$	3.93892	0.707451	0.353725	−	0.935349i	$-0.384915\pi$
$31$	3.93892	0.707451	0.353725	+	0.935349i	$0.384915\pi$
$32$	−1.00000	−0.176777
$33$	−2.65672	−0.462475
$34$	0	0
$35$	4.80701	0.812533
$36$	1.00000	0.166667
$37$	−5.53073	−0.909247	−0.454624	−	0.890684i	$-0.650226\pi$
$37$	−5.53073	−0.909247	−0.454624	+	0.890684i	$0.650226\pi$
$38$	−6.04449	−0.980546
$39$	−4.55356	−0.729153
$40$	1.00000	0.158114
$41$	11.3393	1.77091	0.885453	−	0.464728i	$-0.153848\pi$
$41$	11.3393	1.77091	0.885453	+	0.464728i	$0.153848\pi$
$42$	−4.80701	−0.741738
$43$	−7.80625	−1.19044	−0.595221	−	0.803562i	$-0.702935\pi$
$43$	−7.80625	−1.19044	−0.595221	+	0.803562i	$0.702935\pi$
$44$	2.65672	0.400515
$45$	−1.00000	−0.149071
$46$	−4.32010	−0.636964
$47$	1.83866	0.268196	0.134098	−	0.990968i	$-0.457186\pi$
$47$	1.83866	0.268196	0.134098	+	0.990968i	$0.457186\pi$
$48$	−1.00000	−0.144338
$49$	16.1074	2.30105
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.55356	0.631465
$53$	5.88254	0.808029	0.404014	−	0.914753i	$-0.367615\pi$
$53$	5.88254	0.808029	0.404014	+	0.914753i	$0.367615\pi$
$54$	1.00000	0.136083
$55$	−2.65672	−0.358231
$56$	4.80701	0.642364
$57$	−6.04449	−0.800612
$58$	4.06491	0.533748
$59$	3.50381	0.456157	0.228078	−	0.973643i	$-0.426756\pi$
$59$	3.50381	0.456157	0.228078	+	0.973643i	$0.426756\pi$
$60$	1.00000	0.129099
$61$	−7.66447	−0.981334	−0.490667	−	0.871347i	$-0.663247\pi$
$61$	−7.66447	−0.981334	−0.490667	+	0.871347i	$0.663247\pi$
$62$	−3.93892	−0.500243
$63$	−4.80701	−0.605627
$64$	1.00000	0.125000
$65$	−4.55356	−0.564799
$66$	2.65672	0.327019
$67$	13.8487	1.69189	0.845946	−	0.533268i	$-0.179036\pi$
$67$	13.8487	1.69189	0.845946	+	0.533268i	$0.179036\pi$
$68$	0	0
$69$	−4.32010	−0.520079
$70$	−4.80701	−0.574548
$71$	−4.86663	−0.577563	−0.288781	−	0.957395i	$-0.593250\pi$
$71$	−4.86663	−0.577563	−0.288781	+	0.957395i	$0.593250\pi$
$72$	−1.00000	−0.117851
$73$	−8.43486	−0.987226	−0.493613	−	0.869682i	$-0.664324\pi$
$73$	−8.43486	−0.987226	−0.493613	+	0.869682i	$0.664324\pi$
$74$	5.53073	0.642935
$75$	−1.00000	−0.115470
$76$	6.04449	0.693350
$77$	−12.7709	−1.45537
$78$	4.55356	0.515589
$79$	13.1363	1.47795	0.738975	−	0.673733i	$-0.235310\pi$
$79$	13.1363	1.47795	0.738975	+	0.673733i	$0.235310\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−11.3393	−1.25222
$83$	−17.6812	−1.94077	−0.970385	−	0.241565i	$-0.922339\pi$
$83$	−17.6812	−1.94077	−0.970385	+	0.241565i	$0.922339\pi$
$84$	4.80701	0.524488
$85$	0	0
$86$	7.80625	0.841769
$87$	4.06491	0.435804
$88$	−2.65672	−0.283207
$89$	15.5491	1.64820	0.824101	−	0.566443i	$-0.191681\pi$
$89$	15.5491	1.64820	0.824101	+	0.566443i	$0.191681\pi$
$90$	1.00000	0.105409
$91$	−21.8890	−2.29459
$92$	4.32010	0.450402
$93$	−3.93892	−0.408447
$94$	−1.83866	−0.189643
$95$	−6.04449	−0.620152
$96$	1.00000	0.102062
$97$	17.1612	1.74246	0.871229	−	0.490876i	$-0.163323\pi$
$97$	17.1612	1.74246	0.871229	+	0.490876i	$0.163323\pi$
$98$	−16.1074	−1.62709
$99$	2.65672	0.267010
$100$	1.00000	0.100000
$101$	−6.08472	−0.605452	−0.302726	−	0.953078i	$-0.597897\pi$
$101$	−6.08472	−0.605452	−0.302726	+	0.953078i	$0.597897\pi$
$102$	0	0
$103$	−1.35517	−0.133529	−0.0667645	−	0.997769i	$-0.521268\pi$
$103$	−1.35517	−0.133529	−0.0667645	+	0.997769i	$0.521268\pi$
$104$	−4.55356	−0.446513
$105$	−4.80701	−0.469116
$106$	−5.88254	−0.571362
$107$	−3.10006	−0.299694	−0.149847	−	0.988709i	$-0.547878\pi$
$107$	−3.10006	−0.299694	−0.149847	+	0.988709i	$0.547878\pi$
$108$	−1.00000	−0.0962250
$109$	−15.4449	−1.47935	−0.739676	−	0.672964i	$-0.765021\pi$
$109$	−15.4449	−1.47935	−0.739676	+	0.672964i	$0.765021\pi$
$110$	2.65672	0.253308
$111$	5.53073	0.524954
$112$	−4.80701	−0.454220
$113$	4.15242	0.390627	0.195313	−	0.980741i	$-0.437428\pi$
$113$	4.15242	0.390627	0.195313	+	0.980741i	$0.437428\pi$
$114$	6.04449	0.566118
$115$	−4.32010	−0.402852
$116$	−4.06491	−0.377417
$117$	4.55356	0.420977
$118$	−3.50381	−0.322551
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−3.94187	−0.358351
$122$	7.66447	0.693908
$123$	−11.3393	−1.02243
$124$	3.93892	0.353725
$125$	−1.00000	−0.0894427
$126$	4.80701	0.428243
$127$	−13.0929	−1.16181	−0.580903	−	0.813973i	$-0.697301\pi$
$127$	−13.0929	−1.16181	−0.580903	+	0.813973i	$0.697301\pi$
$128$	−1.00000	−0.0883883
$129$	7.80625	0.687302
$130$	4.55356	0.399373
$131$	−5.41965	−0.473517	−0.236758	−	0.971569i	$-0.576085\pi$
$131$	−5.41965	−0.473517	−0.236758	+	0.971569i	$0.576085\pi$
$132$	−2.65672	−0.231237
$133$	−29.0559	−2.51947
$134$	−13.8487	−1.19635
$135$	1.00000	0.0860663
$136$	0	0
$137$	7.13674	0.609733	0.304866	−	0.952395i	$-0.401388\pi$
$137$	7.13674	0.609733	0.304866	+	0.952395i	$0.401388\pi$
$138$	4.32010	0.367751
$139$	−7.28029	−0.617506	−0.308753	−	0.951142i	$-0.599912\pi$
$139$	−7.28029	−0.617506	−0.308753	+	0.951142i	$0.599912\pi$
$140$	4.80701	0.406267
$141$	−1.83866	−0.154843
$142$	4.86663	0.408398
$143$	12.0975	1.01164
$144$	1.00000	0.0833333
$145$	4.06491	0.337572
$146$	8.43486	0.698074
$147$	−16.1074	−1.32851
$148$	−5.53073	−0.454624
$149$	0.266600	0.0218407	0.0109204	−	0.999940i	$-0.496524\pi$
$149$	0.266600	0.0218407	0.0109204	+	0.999940i	$0.496524\pi$
$150$	1.00000	0.0816497
$151$	−13.1064	−1.06659	−0.533293	−	0.845931i	$-0.679046\pi$
$151$	−13.1064	−1.06659	−0.533293	+	0.845931i	$0.679046\pi$
$152$	−6.04449	−0.490273
$153$	0	0
$154$	12.7709	1.02911
$155$	−3.93892	−0.316382
$156$	−4.55356	−0.364576
$157$	2.93377	0.234140	0.117070	−	0.993124i	$-0.462650\pi$
$157$	2.93377	0.234140	0.117070	+	0.993124i	$0.462650\pi$
$158$	−13.1363	−1.04507
$159$	−5.88254	−0.466516
$160$	1.00000	0.0790569
$161$	−20.7668	−1.63665
$162$	−1.00000	−0.0785674
$163$	9.13905	0.715825	0.357913	−	0.933755i	$-0.383489\pi$
$163$	9.13905	0.715825	0.357913	+	0.933755i	$0.383489\pi$
$164$	11.3393	0.885453
$165$	2.65672	0.206825
$166$	17.6812	1.37233
$167$	−19.0664	−1.47540	−0.737700	−	0.675128i	$-0.764088\pi$
$167$	−19.0664	−1.47540	−0.737700	+	0.675128i	$0.764088\pi$
$168$	−4.80701	−0.370869
$169$	7.73489	0.594991
$170$	0	0
$171$	6.04449	0.462234
$172$	−7.80625	−0.595221
$173$	−5.19673	−0.395100	−0.197550	−	0.980293i	$-0.563299\pi$
$173$	−5.19673	−0.395100	−0.197550	+	0.980293i	$0.563299\pi$
$174$	−4.06491	−0.308160
$175$	−4.80701	−0.363376
$176$	2.65672	0.200257
$177$	−3.50381	−0.263362
$178$	−15.5491	−1.16545
$179$	0.209059	0.0156258	0.00781292	−	0.999969i	$-0.497513\pi$
$179$	0.209059	0.0156258	0.00781292	+	0.999969i	$0.497513\pi$
$180$	−1.00000	−0.0745356
$181$	−14.4208	−1.07189	−0.535945	−	0.844253i	$-0.680044\pi$
$181$	−14.4208	−1.07189	−0.535945	+	0.844253i	$0.680044\pi$
$182$	21.8890	1.62252
$183$	7.66447	0.566574
$184$	−4.32010	−0.318482
$185$	5.53073	0.406628
$186$	3.93892	0.288816
$187$	0	0
$188$	1.83866	0.134098
$189$	4.80701	0.349659
$190$	6.04449	0.438513
$191$	12.3810	0.895857	0.447928	−	0.894069i	$-0.352162\pi$
$191$	12.3810	0.895857	0.447928	+	0.894069i	$0.352162\pi$
$192$	−1.00000	−0.0721688
$193$	−3.59378	−0.258686	−0.129343	−	0.991600i	$-0.541287\pi$
$193$	−3.59378	−0.258686	−0.129343	+	0.991600i	$0.541287\pi$
$194$	−17.1612	−1.23210
$195$	4.55356	0.326087
$196$	16.1074	1.15053
$197$	4.80943	0.342658	0.171329	−	0.985214i	$-0.445194\pi$
$197$	4.80943	0.342658	0.171329	+	0.985214i	$0.445194\pi$
$198$	−2.65672	−0.188805
$199$	−4.44449	−0.315062	−0.157531	−	0.987514i	$-0.550353\pi$
$199$	−4.44449	−0.315062	−0.157531	+	0.987514i	$0.550353\pi$
$200$	−1.00000	−0.0707107
$201$	−13.8487	−0.976814
$202$	6.08472	0.428119
$203$	19.5401	1.37144
$204$	0	0
$205$	−11.3393	−0.791974
$206$	1.35517	0.0944193
$207$	4.32010	0.300268
$208$	4.55356	0.315732
$209$	16.0585	1.11079
$210$	4.80701	0.331715
$211$	15.5928	1.07346	0.536728	−	0.843755i	$-0.319660\pi$
$211$	15.5928	1.07346	0.536728	+	0.843755i	$0.319660\pi$
$212$	5.88254	0.404014
$213$	4.86663	0.333456
$214$	3.10006	0.211916
$215$	7.80625	0.532382
$216$	1.00000	0.0680414
$217$	−18.9344	−1.28535
$218$	15.4449	1.04606
$219$	8.43486	0.569975
$220$	−2.65672	−0.179116
$221$	0	0
$222$	−5.53073	−0.371199
$223$	−0.246992	−0.0165398	−0.00826992	−	0.999966i	$-0.502632\pi$
$223$	−0.246992	−0.0165398	−0.00826992	+	0.999966i	$0.502632\pi$
$224$	4.80701	0.321182
$225$	1.00000	0.0666667
$226$	−4.15242	−0.276215
$227$	−0.206352	−0.0136961	−0.00684804	−	0.999977i	$-0.502180\pi$
$227$	−0.206352	−0.0136961	−0.00684804	+	0.999977i	$0.502180\pi$
$228$	−6.04449	−0.400306
$229$	12.8458	0.848875	0.424438	−	0.905457i	$-0.360472\pi$
$229$	12.8458	0.848875	0.424438	+	0.905457i	$0.360472\pi$
$230$	4.32010	0.284859
$231$	12.7709	0.840261
$232$	4.06491	0.266874
$233$	0.165078	0.0108146	0.00540732	−	0.999985i	$-0.498279\pi$
$233$	0.165078	0.0108146	0.00540732	+	0.999985i	$0.498279\pi$
$234$	−4.55356	−0.297675
$235$	−1.83866	−0.119941
$236$	3.50381	0.228078
$237$	−13.1363	−0.853294
$238$	0	0
$239$	19.2098	1.24258	0.621288	−	0.783582i	$-0.286610\pi$
$239$	19.2098	1.24258	0.621288	+	0.783582i	$0.286610\pi$
$240$	1.00000	0.0645497
$241$	−4.55105	−0.293159	−0.146579	−	0.989199i	$-0.546826\pi$
$241$	−4.55105	−0.293159	−0.146579	+	0.989199i	$0.546826\pi$
$242$	3.94187	0.253393
$243$	−1.00000	−0.0641500
$244$	−7.66447	−0.490667
$245$	−16.1074	−1.02906
$246$	11.3393	0.722970
$247$	27.5239	1.75131
$248$	−3.93892	−0.250122
$249$	17.6812	1.12050
$250$	1.00000	0.0632456
$251$	−12.6713	−0.799805	−0.399903	−	0.916558i	$-0.630956\pi$
$251$	−12.6713	−0.799805	−0.399903	+	0.916558i	$0.630956\pi$
$252$	−4.80701	−0.302813
$253$	11.4773	0.721570
$254$	13.0929	0.821521
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.7579	0.671061	0.335531	−	0.942029i	$-0.391084\pi$
$257$	10.7579	0.671061	0.335531	+	0.942029i	$0.391084\pi$
$258$	−7.80625	−0.485996
$259$	26.5863	1.65199
$260$	−4.55356	−0.282400
$261$	−4.06491	−0.251611
$262$	5.41965	0.334827
$263$	−8.77699	−0.541212	−0.270606	−	0.962690i	$-0.587224\pi$
$263$	−8.77699	−0.541212	−0.270606	+	0.962690i	$0.587224\pi$
$264$	2.65672	0.163510
$265$	−5.88254	−0.361361
$266$	29.0559	1.78153
$267$	−15.5491	−0.951590
$268$	13.8487	0.845946
$269$	17.3898	1.06027	0.530136	−	0.847913i	$-0.322141\pi$
$269$	17.3898	1.06027	0.530136	+	0.847913i	$0.322141\pi$
$270$	−1.00000	−0.0608581
$271$	−17.9138	−1.08819	−0.544093	−	0.839025i	$-0.683126\pi$
$271$	−17.9138	−1.08819	−0.544093	+	0.839025i	$0.683126\pi$
$272$	0	0
$273$	21.8890	1.32478
$274$	−7.13674	−0.431146
$275$	2.65672	0.160206
$276$	−4.32010	−0.260040
$277$	26.2658	1.57816	0.789080	−	0.614290i	$-0.210558\pi$
$277$	26.2658	1.57816	0.789080	+	0.614290i	$0.210558\pi$
$278$	7.28029	0.436642
$279$	3.93892	0.235817
$280$	−4.80701	−0.287274
$281$	28.5009	1.70022	0.850111	−	0.526603i	$-0.176535\pi$
$281$	28.5009	1.70022	0.850111	+	0.526603i	$0.176535\pi$
$282$	1.83866	0.109490
$283$	3.66103	0.217625	0.108813	−	0.994062i	$-0.465295\pi$
$283$	3.66103	0.217625	0.108813	+	0.994062i	$0.465295\pi$
$284$	−4.86663	−0.288781
$285$	6.04449	0.358045
$286$	−12.0975	−0.715340
$287$	−54.5083	−3.21752
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−4.06491	−0.238700
$291$	−17.1612	−1.00601
$292$	−8.43486	−0.493613
$293$	15.6314	0.913197	0.456598	−	0.889673i	$-0.349068\pi$
$293$	15.6314	0.913197	0.456598	+	0.889673i	$0.349068\pi$
$294$	16.1074	0.939401
$295$	−3.50381	−0.203999
$296$	5.53073	0.321467
$297$	−2.65672	−0.154158
$298$	−0.266600	−0.0154437
$299$	19.6718	1.13765
$300$	−1.00000	−0.0577350
$301$	37.5247	2.16289
$302$	13.1064	0.754190
$303$	6.08472	0.349558
$304$	6.04449	0.346675
$305$	7.66447	0.438866
$306$	0	0
$307$	−1.39589	−0.0796676	−0.0398338	−	0.999206i	$-0.512683\pi$
$307$	−1.39589	−0.0796676	−0.0398338	+	0.999206i	$0.512683\pi$
$308$	−12.7709	−0.727687
$309$	1.35517	0.0770930
$310$	3.93892	0.223716
$311$	3.49665	0.198277	0.0991385	−	0.995074i	$-0.468391\pi$
$311$	3.49665	0.198277	0.0991385	+	0.995074i	$0.468391\pi$
$312$	4.55356	0.257794
$313$	23.2668	1.31512	0.657558	−	0.753404i	$-0.271589\pi$
$313$	23.2668	1.31512	0.657558	+	0.753404i	$0.271589\pi$
$314$	−2.93377	−0.165562
$315$	4.80701	0.270844
$316$	13.1363	0.738975
$317$	15.1546	0.851168	0.425584	−	0.904919i	$-0.360069\pi$
$317$	15.1546	0.851168	0.425584	+	0.904919i	$0.360069\pi$
$318$	5.88254	0.329876
$319$	−10.7993	−0.604645
$320$	−1.00000	−0.0559017
$321$	3.10006	0.173028
$322$	20.7668	1.15729
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.55356	0.252586
$326$	−9.13905	−0.506165
$327$	15.4449	0.854104
$328$	−11.3393	−0.626110
$329$	−8.83844	−0.487279
$330$	−2.65672	−0.146247
$331$	−24.0083	−1.31961	−0.659807	−	0.751435i	$-0.729362\pi$
$331$	−24.0083	−1.31961	−0.659807	+	0.751435i	$0.729362\pi$
$332$	−17.6812	−0.970385
$333$	−5.53073	−0.303082
$334$	19.0664	1.04327
$335$	−13.8487	−0.756637
$336$	4.80701	0.262244
$337$	26.8726	1.46384	0.731922	−	0.681388i	$-0.238623\pi$
$337$	26.8726	1.46384	0.731922	+	0.681388i	$0.238623\pi$
$338$	−7.73489	−0.420722
$339$	−4.15242	−0.225529
$340$	0	0
$341$	10.4646	0.566689
$342$	−6.04449	−0.326849
$343$	−43.7792	−2.36386
$344$	7.80625	0.420885
$345$	4.32010	0.232586
$346$	5.19673	0.279378
$347$	−7.98128	−0.428457	−0.214229	−	0.976784i	$-0.568724\pi$
$347$	−7.98128	−0.428457	−0.214229	+	0.976784i	$0.568724\pi$
$348$	4.06491	0.217902
$349$	−13.3176	−0.712874	−0.356437	−	0.934319i	$-0.616009\pi$
$349$	−13.3176	−0.712874	−0.356437	+	0.934319i	$0.616009\pi$
$350$	4.80701	0.256946
$351$	−4.55356	−0.243051
$352$	−2.65672	−0.141603
$353$	−16.9293	−0.901054	−0.450527	−	0.892763i	$-0.648764\pi$
$353$	−16.9293	−0.901054	−0.450527	+	0.892763i	$0.648764\pi$
$354$	3.50381	0.186225
$355$	4.86663	0.258294
$356$	15.5491	0.824101
$357$	0	0
$358$	−0.209059	−0.0110491
$359$	20.8739	1.10168	0.550842	−	0.834609i	$-0.314307\pi$
$359$	20.8739	1.10168	0.550842	+	0.834609i	$0.314307\pi$
$360$	1.00000	0.0527046
$361$	17.5358	0.922939
$362$	14.4208	0.757940
$363$	3.94187	0.206894
$364$	−21.8890	−1.14730
$365$	8.43486	0.441501
$366$	−7.66447	−0.400628
$367$	9.48837	0.495289	0.247645	−	0.968851i	$-0.420344\pi$
$367$	9.48837	0.495289	0.247645	+	0.968851i	$0.420344\pi$
$368$	4.32010	0.225201
$369$	11.3393	0.590302
$370$	−5.53073	−0.287529
$371$	−28.2774	−1.46809
$372$	−3.93892	−0.204223
$373$	−8.88776	−0.460191	−0.230095	−	0.973168i	$-0.573904\pi$
$373$	−8.88776	−0.460191	−0.230095	+	0.973168i	$0.573904\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−1.83866	−0.0948214
$377$	−18.5098	−0.953302
$378$	−4.80701	−0.247246
$379$	20.5944	1.05786	0.528931	−	0.848665i	$-0.322593\pi$
$379$	20.5944	1.05786	0.528931	+	0.848665i	$0.322593\pi$
$380$	−6.04449	−0.310076
$381$	13.0929	0.670769
$382$	−12.3810	−0.633467
$383$	−32.6635	−1.66903	−0.834513	−	0.550989i	$-0.814251\pi$
$383$	−32.6635	−1.66903	−0.834513	+	0.550989i	$0.814251\pi$
$384$	1.00000	0.0510310
$385$	12.7709	0.650863
$386$	3.59378	0.182919
$387$	−7.80625	−0.396814
$388$	17.1612	0.871229
$389$	26.9553	1.36669	0.683344	−	0.730097i	$-0.260525\pi$
$389$	26.9553	1.36669	0.683344	+	0.730097i	$0.260525\pi$
$390$	−4.55356	−0.230578
$391$	0	0
$392$	−16.1074	−0.813545
$393$	5.41965	0.273385
$394$	−4.80943	−0.242296
$395$	−13.1363	−0.660959
$396$	2.65672	0.133505
$397$	−30.6356	−1.53755	−0.768777	−	0.639517i	$-0.779134\pi$
$397$	−30.6356	−1.53755	−0.768777	+	0.639517i	$0.779134\pi$
$398$	4.44449	0.222782
$399$	29.0559	1.45462
$400$	1.00000	0.0500000
$401$	28.4374	1.42010	0.710048	−	0.704153i	$-0.248673\pi$
$401$	28.4374	1.42010	0.710048	+	0.704153i	$0.248673\pi$
$402$	13.8487	0.690712
$403$	17.9361	0.893461
$404$	−6.08472	−0.302726
$405$	−1.00000	−0.0496904
$406$	−19.5401	−0.969757
$407$	−14.6936	−0.728334
$408$	0	0
$409$	−30.3744	−1.50192	−0.750960	−	0.660348i	$-0.770409\pi$
$409$	−30.3744	−1.50192	−0.750960	+	0.660348i	$0.770409\pi$
$410$	11.3393	0.560010
$411$	−7.13674	−0.352029
$412$	−1.35517	−0.0667645
$413$	−16.8428	−0.828782
$414$	−4.32010	−0.212321
$415$	17.6812	0.867938
$416$	−4.55356	−0.223257
$417$	7.28029	0.356517
$418$	−16.0585	−0.785446
$419$	22.0797	1.07866	0.539332	−	0.842093i	$-0.318677\pi$
$419$	22.0797	1.07866	0.539332	+	0.842093i	$0.318677\pi$
$420$	−4.80701	−0.234558
$421$	−2.61967	−0.127675	−0.0638374	−	0.997960i	$-0.520334\pi$
$421$	−2.61967	−0.127675	−0.0638374	+	0.997960i	$0.520334\pi$
$422$	−15.5928	−0.759048
$423$	1.83866	0.0893985
$424$	−5.88254	−0.285681
$425$	0	0
$426$	−4.86663	−0.235789
$427$	36.8432	1.78297
$428$	−3.10006	−0.149847
$429$	−12.0975	−0.584073
$430$	−7.80625	−0.376451
$431$	−13.3377	−0.642456	−0.321228	−	0.947002i	$-0.604096\pi$
$431$	−13.3377	−0.642456	−0.321228	+	0.947002i	$0.604096\pi$
$432$	−1.00000	−0.0481125
$433$	−11.1422	−0.535461	−0.267730	−	0.963494i	$-0.586274\pi$
$433$	−11.1422	−0.535461	−0.267730	+	0.963494i	$0.586274\pi$
$434$	18.9344	0.908882
$435$	−4.06491	−0.194897
$436$	−15.4449	−0.739676
$437$	26.1128	1.24915
$438$	−8.43486	−0.403033
$439$	−30.1107	−1.43711	−0.718554	−	0.695471i	$-0.755196\pi$
$439$	−30.1107	−1.43711	−0.718554	+	0.695471i	$0.755196\pi$
$440$	2.65672	0.126654
$441$	16.1074	0.767017
$442$	0	0
$443$	28.4432	1.35138	0.675689	−	0.737187i	$-0.263846\pi$
$443$	28.4432	1.35138	0.675689	+	0.737187i	$0.263846\pi$
$444$	5.53073	0.262477
$445$	−15.5491	−0.737098
$446$	0.246992	0.0116954
$447$	−0.266600	−0.0126097
$448$	−4.80701	−0.227110
$449$	32.6308	1.53994	0.769971	−	0.638079i	$-0.220271\pi$
$449$	32.6308	1.53994	0.769971	+	0.638079i	$0.220271\pi$
$450$	−1.00000	−0.0471405
$451$	30.1254	1.41855
$452$	4.15242	0.195313
$453$	13.1064	0.615794
$454$	0.206352	0.00968458
$455$	21.8890	1.02617
$456$	6.04449	0.283059
$457$	0.770239	0.0360302	0.0180151	−	0.999838i	$-0.494265\pi$
$457$	0.770239	0.0360302	0.0180151	+	0.999838i	$0.494265\pi$
$458$	−12.8458	−0.600246
$459$	0	0
$460$	−4.32010	−0.201426
$461$	21.3672	0.995169	0.497584	−	0.867416i	$-0.334220\pi$
$461$	21.3672	0.995169	0.497584	+	0.867416i	$0.334220\pi$
$462$	−12.7709	−0.594154
$463$	−5.59618	−0.260076	−0.130038	−	0.991509i	$-0.541510\pi$
$463$	−5.59618	−0.260076	−0.130038	+	0.991509i	$0.541510\pi$
$464$	−4.06491	−0.188709
$465$	3.93892	0.182663
$466$	−0.165078	−0.00764711
$467$	17.4394	0.806998	0.403499	−	0.914980i	$-0.367794\pi$
$467$	17.4394	0.806998	0.403499	+	0.914980i	$0.367794\pi$
$468$	4.55356	0.210488
$469$	−66.5710	−3.07397
$470$	1.83866	0.0848109
$471$	−2.93377	−0.135181
$472$	−3.50381	−0.161276
$473$	−20.7390	−0.953579
$474$	13.1363	0.603370
$475$	6.04449	0.277340
$476$	0	0
$477$	5.88254	0.269343
$478$	−19.2098	−0.878635
$479$	26.6130	1.21598	0.607990	−	0.793945i	$-0.291976\pi$
$479$	26.6130	1.21598	0.607990	+	0.793945i	$0.291976\pi$
$480$	−1.00000	−0.0456435
$481$	−25.1845	−1.14832
$482$	4.55105	0.207294
$483$	20.7668	0.944921
$484$	−3.94187	−0.179176
$485$	−17.1612	−0.779251
$486$	1.00000	0.0453609
$487$	9.44787	0.428124	0.214062	−	0.976820i	$-0.431331\pi$
$487$	9.44787	0.428124	0.214062	+	0.976820i	$0.431331\pi$
$488$	7.66447	0.346954
$489$	−9.13905	−0.413282
$490$	16.1074	0.727657
$491$	11.4423	0.516383	0.258192	−	0.966094i	$-0.416873\pi$
$491$	11.4423	0.516383	0.258192	+	0.966094i	$0.416873\pi$
$492$	−11.3393	−0.511217
$493$	0	0
$494$	−27.5239	−1.23836
$495$	−2.65672	−0.119410
$496$	3.93892	0.176863
$497$	23.3939	1.04936
$498$	−17.6812	−0.792316
$499$	13.8201	0.618672	0.309336	−	0.950953i	$-0.399893\pi$
$499$	13.8201	0.618672	0.309336	+	0.950953i	$0.399893\pi$
$500$	−1.00000	−0.0447214
$501$	19.0664	0.851823
$502$	12.6713	0.565548
$503$	−3.80954	−0.169859	−0.0849295	−	0.996387i	$-0.527067\pi$
$503$	−3.80954	−0.169859	−0.0849295	+	0.996387i	$0.527067\pi$
$504$	4.80701	0.214121
$505$	6.08472	0.270766
$506$	−11.4773	−0.510227
$507$	−7.73489	−0.343518
$508$	−13.0929	−0.580903
$509$	27.4822	1.21813	0.609064	−	0.793121i	$-0.291545\pi$
$509$	27.4822	1.21813	0.609064	+	0.793121i	$0.291545\pi$
$510$	0	0
$511$	40.5465	1.79367
$512$	−1.00000	−0.0441942
$513$	−6.04449	−0.266871
$514$	−10.7579	−0.474512
$515$	1.35517	0.0597160
$516$	7.80625	0.343651
$517$	4.88478	0.214833
$518$	−26.5863	−1.16814
$519$	5.19673	0.228111
$520$	4.55356	0.199687
$521$	−9.33363	−0.408914	−0.204457	−	0.978876i	$-0.565543\pi$
$521$	−9.33363	−0.408914	−0.204457	+	0.978876i	$0.565543\pi$
$522$	4.06491	0.177916
$523$	−29.1806	−1.27598	−0.637990	−	0.770045i	$-0.720234\pi$
$523$	−29.1806	−1.27598	−0.637990	+	0.770045i	$0.720234\pi$
$524$	−5.41965	−0.236758
$525$	4.80701	0.209795
$526$	8.77699	0.382695
$527$	0	0
$528$	−2.65672	−0.115619
$529$	−4.33672	−0.188553
$530$	5.88254	0.255521
$531$	3.50381	0.152052
$532$	−29.0559	−1.25973
$533$	51.6343	2.23653
$534$	15.5491	0.672876
$535$	3.10006	0.134027
$536$	−13.8487	−0.598174
$537$	−0.209059	−0.00902158
$538$	−17.3898	−0.749725
$539$	42.7927	1.84321
$540$	1.00000	0.0430331
$541$	26.1951	1.12621	0.563107	−	0.826384i	$-0.309606\pi$
$541$	26.1951	1.12621	0.563107	+	0.826384i	$0.309606\pi$
$542$	17.9138	0.769464
$543$	14.4208	0.618856
$544$	0	0
$545$	15.4449	0.661586
$546$	−21.8890	−0.936763
$547$	−27.0508	−1.15661	−0.578304	−	0.815821i	$-0.696285\pi$
$547$	−27.0508	−1.15661	−0.578304	+	0.815821i	$0.696285\pi$
$548$	7.13674	0.304866
$549$	−7.66447	−0.327111
$550$	−2.65672	−0.113283
$551$	−24.5703	−1.04673
$552$	4.32010	0.183876
$553$	−63.1464	−2.68526
$554$	−26.2658	−1.11593
$555$	−5.53073	−0.234767
$556$	−7.28029	−0.308753
$557$	−6.91753	−0.293105	−0.146552	−	0.989203i	$-0.546818\pi$
$557$	−6.91753	−0.293105	−0.146552	+	0.989203i	$0.546818\pi$
$558$	−3.93892	−0.166748
$559$	−35.5462	−1.50344
$560$	4.80701	0.203133
$561$	0	0
$562$	−28.5009	−1.20224
$563$	34.1322	1.43850	0.719251	−	0.694750i	$-0.244485\pi$
$563$	34.1322	1.43850	0.719251	+	0.694750i	$0.244485\pi$
$564$	−1.83866	−0.0774214
$565$	−4.15242	−0.174694
$566$	−3.66103	−0.153884
$567$	−4.80701	−0.201876
$568$	4.86663	0.204199
$569$	−27.7102	−1.16167	−0.580836	−	0.814021i	$-0.697274\pi$
$569$	−27.7102	−1.16167	−0.580836	+	0.814021i	$0.697274\pi$
$570$	−6.04449	−0.253176
$571$	−23.2801	−0.974243	−0.487122	−	0.873334i	$-0.661953\pi$
$571$	−23.2801	−0.974243	−0.487122	+	0.873334i	$0.661953\pi$
$572$	12.0975	0.505822
$573$	−12.3810	−0.517223
$574$	54.5083	2.27513
$575$	4.32010	0.180161
$576$	1.00000	0.0416667
$577$	−5.51048	−0.229404	−0.114702	−	0.993400i	$-0.536591\pi$
$577$	−5.51048	−0.229404	−0.114702	+	0.993400i	$0.536591\pi$
$578$	0	0
$579$	3.59378	0.149353
$580$	4.06491	0.168786
$581$	84.9940	3.52614
$582$	17.1612	0.711356
$583$	15.6282	0.647255
$584$	8.43486	0.349037
$585$	−4.55356	−0.188266
$586$	−15.6314	−0.645728
$587$	1.39173	0.0574427	0.0287214	−	0.999587i	$-0.490856\pi$
$587$	1.39173	0.0574427	0.0287214	+	0.999587i	$0.490856\pi$
$588$	−16.1074	−0.664257
$589$	23.8088	0.981023
$590$	3.50381	0.144249
$591$	−4.80943	−0.197834
$592$	−5.53073	−0.227312
$593$	46.3361	1.90280	0.951399	−	0.307962i	$-0.0996470\pi$
$593$	46.3361	1.90280	0.951399	+	0.307962i	$0.0996470\pi$
$594$	2.65672	0.109006
$595$	0	0
$596$	0.266600	0.0109204
$597$	4.44449	0.181901
$598$	−19.6718	−0.804441
$599$	−41.5318	−1.69694	−0.848472	−	0.529240i	$-0.822477\pi$
$599$	−41.5318	−1.69694	−0.848472	+	0.529240i	$0.822477\pi$
$600$	1.00000	0.0408248
$601$	21.6941	0.884921	0.442461	−	0.896788i	$-0.354106\pi$
$601$	21.6941	0.884921	0.442461	+	0.896788i	$0.354106\pi$
$602$	−37.5247	−1.52939
$603$	13.8487	0.563964
$604$	−13.1064	−0.533293
$605$	3.94187	0.160260
$606$	−6.08472	−0.247175
$607$	−2.93865	−0.119276	−0.0596381	−	0.998220i	$-0.518995\pi$
$607$	−2.93865	−0.119276	−0.0596381	+	0.998220i	$0.518995\pi$
$608$	−6.04449	−0.245136
$609$	−19.5401	−0.791803
$610$	−7.66447	−0.310325
$611$	8.37243	0.338712
$612$	0	0
$613$	25.6455	1.03581	0.517906	−	0.855438i	$-0.326712\pi$
$613$	25.6455	1.03581	0.517906	+	0.855438i	$0.326712\pi$
$614$	1.39589	0.0563335
$615$	11.3393	0.457246
$616$	12.7709	0.514553
$617$	−2.65698	−0.106966	−0.0534829	−	0.998569i	$-0.517032\pi$
$617$	−2.65698	−0.106966	−0.0534829	+	0.998569i	$0.517032\pi$
$618$	−1.35517	−0.0545130
$619$	−8.27530	−0.332612	−0.166306	−	0.986074i	$-0.553184\pi$
$619$	−8.27530	−0.332612	−0.166306	+	0.986074i	$0.553184\pi$
$620$	−3.93892	−0.158191
$621$	−4.32010	−0.173360
$622$	−3.49665	−0.140203
$623$	−74.7447	−2.99458
$624$	−4.55356	−0.182288
$625$	1.00000	0.0400000
$626$	−23.2668	−0.929928
$627$	−16.0585	−0.641314
$628$	2.93377	0.117070
$629$	0	0
$630$	−4.80701	−0.191516
$631$	15.7145	0.625583	0.312791	−	0.949822i	$-0.398736\pi$
$631$	15.7145	0.625583	0.312791	+	0.949822i	$0.398736\pi$
$632$	−13.1363	−0.522534
$633$	−15.5928	−0.619760
$634$	−15.1546	−0.601867
$635$	13.0929	0.519576
$636$	−5.88254	−0.233258
$637$	73.3458	2.90607
$638$	10.7993	0.427548
$639$	−4.86663	−0.192521
$640$	1.00000	0.0395285
$641$	21.1603	0.835780	0.417890	−	0.908498i	$-0.362770\pi$
$641$	21.1603	0.835780	0.417890	+	0.908498i	$0.362770\pi$
$642$	−3.10006	−0.122349
$643$	−22.0202	−0.868392	−0.434196	−	0.900818i	$-0.642968\pi$
$643$	−22.0202	−0.868392	−0.434196	+	0.900818i	$0.642968\pi$
$644$	−20.7668	−0.818326
$645$	−7.80625	−0.307371
$646$	0	0
$647$	38.7558	1.52365	0.761824	−	0.647784i	$-0.224304\pi$
$647$	38.7558	1.52365	0.761824	+	0.647784i	$0.224304\pi$
$648$	−1.00000	−0.0392837
$649$	9.30861	0.365395
$650$	−4.55356	−0.178605
$651$	18.9344	0.742099
$652$	9.13905	0.357913
$653$	11.7703	0.460608	0.230304	−	0.973119i	$-0.426028\pi$
$653$	11.7703	0.460608	0.230304	+	0.973119i	$0.426028\pi$
$654$	−15.4449	−0.603943
$655$	5.41965	0.211763
$656$	11.3393	0.442727
$657$	−8.43486	−0.329075
$658$	8.83844	0.344558
$659$	−39.9393	−1.55581	−0.777906	−	0.628380i	$-0.783718\pi$
$659$	−39.9393	−1.55581	−0.777906	+	0.628380i	$0.783718\pi$
$660$	2.65672	0.103412
$661$	4.91728	0.191260	0.0956301	−	0.995417i	$-0.469513\pi$
$661$	4.91728	0.191260	0.0956301	+	0.995417i	$0.469513\pi$
$662$	24.0083	0.933108
$663$	0	0
$664$	17.6812	0.686166
$665$	29.0559	1.12674
$666$	5.53073	0.214312
$667$	−17.5608	−0.679957
$668$	−19.0664	−0.737700
$669$	0.246992	0.00954928
$670$	13.8487	0.535023
$671$	−20.3623	−0.786078
$672$	−4.80701	−0.185435
$673$	−37.1460	−1.43187	−0.715936	−	0.698166i	$-0.754000\pi$
$673$	−37.1460	−1.43187	−0.715936	+	0.698166i	$0.754000\pi$
$674$	−26.8726	−1.03509
$675$	−1.00000	−0.0384900
$676$	7.73489	0.297496
$677$	−4.15436	−0.159665	−0.0798325	−	0.996808i	$-0.525439\pi$
$677$	−4.15436	−0.159665	−0.0798325	+	0.996808i	$0.525439\pi$
$678$	4.15242	0.159473
$679$	−82.4942	−3.16584
$680$	0	0
$681$	0.206352	0.00790743
$682$	−10.4646	−0.400710
$683$	−36.1292	−1.38245	−0.691223	−	0.722642i	$-0.742928\pi$
$683$	−36.1292	−1.38245	−0.691223	+	0.722642i	$0.742928\pi$
$684$	6.04449	0.231117
$685$	−7.13674	−0.272681
$686$	43.7792	1.67150
$687$	−12.8458	−0.490098
$688$	−7.80625	−0.297610
$689$	26.7865	1.02048
$690$	−4.32010	−0.164463
$691$	20.2855	0.771697	0.385849	−	0.922562i	$-0.373909\pi$
$691$	20.2855	0.771697	0.385849	+	0.922562i	$0.373909\pi$
$692$	−5.19673	−0.197550
$693$	−12.7709	−0.485125
$694$	7.98128	0.302965
$695$	7.28029	0.276157
$696$	−4.06491	−0.154080
$697$	0	0
$698$	13.3176	0.504078
$699$	−0.165078	−0.00624384
$700$	−4.80701	−0.181688
$701$	15.9377	0.601958	0.300979	−	0.953631i	$-0.402687\pi$
$701$	15.9377	0.601958	0.300979	+	0.953631i	$0.402687\pi$
$702$	4.55356	0.171863
$703$	−33.4305	−1.26085
$704$	2.65672	0.100129
$705$	1.83866	0.0692478
$706$	16.9293	0.637141
$707$	29.2493	1.10003
$708$	−3.50381	−0.131681
$709$	−43.9963	−1.65231	−0.826157	−	0.563439i	$-0.809478\pi$
$709$	−43.9963	−1.65231	−0.826157	+	0.563439i	$0.809478\pi$
$710$	−4.86663	−0.182641
$711$	13.1363	0.492650
$712$	−15.5491	−0.582727
$713$	17.0165	0.637274
$714$	0	0
$715$	−12.0975	−0.452421
$716$	0.209059	0.00781292
$717$	−19.2098	−0.717402
$718$	−20.8739	−0.779009
$719$	21.3486	0.796169	0.398084	−	0.917349i	$-0.369675\pi$
$719$	21.3486	0.796169	0.398084	+	0.917349i	$0.369675\pi$
$720$	−1.00000	−0.0372678
$721$	6.51433	0.242606
$722$	−17.5358	−0.652617
$723$	4.55105	0.169255
$724$	−14.4208	−0.535945
$725$	−4.06491	−0.150967
$726$	−3.94187	−0.146296
$727$	−3.91209	−0.145091	−0.0725457	−	0.997365i	$-0.523112\pi$
$727$	−3.91209	−0.145091	−0.0725457	+	0.997365i	$0.523112\pi$
$728$	21.8890	0.811261
$729$	1.00000	0.0370370
$730$	−8.43486	−0.312188
$731$	0	0
$732$	7.66447	0.283287
$733$	6.92135	0.255646	0.127823	−	0.991797i	$-0.459201\pi$
$733$	6.92135	0.255646	0.127823	+	0.991797i	$0.459201\pi$
$734$	−9.48837	−0.350222
$735$	16.1074	0.594129
$736$	−4.32010	−0.159241
$737$	36.7921	1.35526
$738$	−11.3393	−0.417407
$739$	−2.94473	−0.108324	−0.0541619	−	0.998532i	$-0.517249\pi$
$739$	−2.94473	−0.108324	−0.0541619	+	0.998532i	$0.517249\pi$
$740$	5.53073	0.203314
$741$	−27.5239	−1.01112
$742$	28.2774	1.03810
$743$	39.4041	1.44560	0.722798	−	0.691059i	$-0.242856\pi$
$743$	39.4041	1.44560	0.722798	+	0.691059i	$0.242856\pi$
$744$	3.93892	0.144408
$745$	−0.266600	−0.00976746
$746$	8.88776	0.325404
$747$	−17.6812	−0.646923
$748$	0	0
$749$	14.9020	0.544508
$750$	−1.00000	−0.0365148
$751$	−33.2917	−1.21483	−0.607416	−	0.794384i	$-0.707794\pi$
$751$	−33.2917	−1.21483	−0.607416	+	0.794384i	$0.707794\pi$
$752$	1.83866	0.0670489
$753$	12.6713	0.461768
$754$	18.5098	0.674087
$755$	13.1064	0.476992
$756$	4.80701	0.174829
$757$	46.6744	1.69641	0.848206	−	0.529667i	$-0.177683\pi$
$757$	46.6744	1.69641	0.848206	+	0.529667i	$0.177683\pi$
$758$	−20.5944	−0.748022
$759$	−11.4773	−0.416599
$760$	6.04449	0.219257
$761$	−7.34541	−0.266271	−0.133135	−	0.991098i	$-0.542505\pi$
$761$	−7.34541	−0.266271	−0.133135	+	0.991098i	$0.542505\pi$
$762$	−13.0929	−0.474305
$763$	74.2437	2.68780
$764$	12.3810	0.447928
$765$	0	0
$766$	32.6635	1.18018
$767$	15.9548	0.576094
$768$	−1.00000	−0.0360844
$769$	39.8949	1.43865	0.719324	−	0.694675i	$-0.244452\pi$
$769$	39.8949	1.43865	0.719324	+	0.694675i	$0.244452\pi$
$770$	−12.7709	−0.460230
$771$	−10.7579	−0.387437
$772$	−3.59378	−0.129343
$773$	37.0561	1.33281	0.666407	−	0.745588i	$-0.267831\pi$
$773$	37.0561	1.33281	0.666407	+	0.745588i	$0.267831\pi$
$774$	7.80625	0.280590
$775$	3.93892	0.141490
$776$	−17.1612	−0.616052
$777$	−26.5863	−0.953778
$778$	−26.9553	−0.966394
$779$	68.5405	2.45572
$780$	4.55356	0.163044
$781$	−12.9292	−0.462645
$782$	0	0
$783$	4.06491	0.145268
$784$	16.1074	0.575263
$785$	−2.93377	−0.104711
$786$	−5.41965	−0.193312
$787$	−38.9395	−1.38804	−0.694021	−	0.719954i	$-0.744163\pi$
$787$	−38.9395	−1.38804	−0.694021	+	0.719954i	$0.744163\pi$
$788$	4.80943	0.171329
$789$	8.77699	0.312469
$790$	13.1363	0.467369
$791$	−19.9607	−0.709722
$792$	−2.65672	−0.0944023
$793$	−34.9006	−1.23936
$794$	30.6356	1.08722
$795$	5.88254	0.208632
$796$	−4.44449	−0.157531
$797$	19.9096	0.705235	0.352617	−	0.935768i	$-0.385292\pi$
$797$	19.9096	0.705235	0.352617	+	0.935768i	$0.385292\pi$
$798$	−29.0559	−1.02857
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	15.5491	0.549401
$802$	−28.4374	−1.00416
$803$	−22.4090	−0.790797
$804$	−13.8487	−0.488407
$805$	20.7668	0.731933
$806$	−17.9361	−0.631772
$807$	−17.3898	−0.612148
$808$	6.08472	0.214060
$809$	47.0466	1.65407	0.827036	−	0.562149i	$-0.190025\pi$
$809$	47.0466	1.65407	0.827036	+	0.562149i	$0.190025\pi$
$810$	1.00000	0.0351364
$811$	28.1042	0.986873	0.493436	−	0.869782i	$-0.335741\pi$
$811$	28.1042	0.986873	0.493436	+	0.869782i	$0.335741\pi$
$812$	19.5401	0.685722
$813$	17.9138	0.628265
$814$	14.6936	0.515010
$815$	−9.13905	−0.320127
$816$	0	0
$817$	−47.1848	−1.65079
$818$	30.3744	1.06202
$819$	−21.8890	−0.764864
$820$	−11.3393	−0.395987
$821$	−10.3014	−0.359521	−0.179760	−	0.983710i	$-0.557532\pi$
$821$	−10.3014	−0.359521	−0.179760	+	0.983710i	$0.557532\pi$
$822$	7.13674	0.248922
$823$	−38.9160	−1.35653	−0.678263	−	0.734819i	$-0.737267\pi$
$823$	−38.9160	−1.35653	−0.678263	+	0.734819i	$0.737267\pi$
$824$	1.35517	0.0472097
$825$	−2.65672	−0.0924949
$826$	16.8428	0.586037
$827$	−37.3497	−1.29878	−0.649388	−	0.760457i	$-0.724975\pi$
$827$	−37.3497	−1.29878	−0.649388	+	0.760457i	$0.724975\pi$
$828$	4.32010	0.150134
$829$	39.3644	1.36718	0.683590	−	0.729866i	$-0.260418\pi$
$829$	39.3644	1.36718	0.683590	+	0.729866i	$0.260418\pi$
$830$	−17.6812	−0.613725
$831$	−26.2658	−0.911151
$832$	4.55356	0.157866
$833$	0	0
$834$	−7.28029	−0.252096
$835$	19.0664	0.659819
$836$	16.0585	0.555394
$837$	−3.93892	−0.136149
$838$	−22.0797	−0.762731
$839$	−42.8275	−1.47857	−0.739285	−	0.673392i	$-0.764837\pi$
$839$	−42.8275	−1.47857	−0.739285	+	0.673392i	$0.764837\pi$
$840$	4.80701	0.165858
$841$	−12.4765	−0.430225
$842$	2.61967	0.0902797
$843$	−28.5009	−0.981624
$844$	15.5928	0.536728
$845$	−7.73489	−0.266088
$846$	−1.83866	−0.0632143
$847$	18.9486	0.651081
$848$	5.88254	0.202007
$849$	−3.66103	−0.125646
$850$	0	0
$851$	−23.8933	−0.819053
$852$	4.86663	0.166728
$853$	11.3847	0.389804	0.194902	−	0.980823i	$-0.437561\pi$
$853$	11.3847	0.389804	0.194902	+	0.980823i	$0.437561\pi$
$854$	−36.8432	−1.26075
$855$	−6.04449	−0.206717
$856$	3.10006	0.105958
$857$	−14.5298	−0.496327	−0.248164	−	0.968718i	$-0.579827\pi$
$857$	−14.5298	−0.496327	−0.248164	+	0.968718i	$0.579827\pi$
$858$	12.0975	0.413002
$859$	−18.8024	−0.641530	−0.320765	−	0.947159i	$-0.603940\pi$
$859$	−18.8024	−0.641530	−0.320765	+	0.947159i	$0.603940\pi$
$860$	7.80625	0.266191
$861$	54.5083	1.85764
$862$	13.3377	0.454285
$863$	−3.91801	−0.133371	−0.0666853	−	0.997774i	$-0.521242\pi$
$863$	−3.91801	−0.133371	−0.0666853	+	0.997774i	$0.521242\pi$
$864$	1.00000	0.0340207
$865$	5.19673	0.176694
$866$	11.1422	0.378628
$867$	0	0
$868$	−18.9344	−0.642677
$869$	34.8994	1.18388
$870$	4.06491	0.137813
$871$	63.0610	2.13674
$872$	15.4449	0.523030
$873$	17.1612	0.580820
$874$	−26.1128	−0.883279
$875$	4.80701	0.162507
$876$	8.43486	0.284988
$877$	−10.7928	−0.364449	−0.182224	−	0.983257i	$-0.558330\pi$
$877$	−10.7928	−0.364449	−0.182224	+	0.983257i	$0.558330\pi$
$878$	30.1107	1.01619
$879$	−15.6314	−0.527235
$880$	−2.65672	−0.0895578
$881$	30.3602	1.02286	0.511430	−	0.859325i	$-0.329116\pi$
$881$	30.3602	1.02286	0.511430	+	0.859325i	$0.329116\pi$
$882$	−16.1074	−0.542363
$883$	38.7408	1.30373	0.651866	−	0.758334i	$-0.273986\pi$
$883$	38.7408	1.30373	0.651866	+	0.758334i	$0.273986\pi$
$884$	0	0
$885$	3.50381	0.117779
$886$	−28.4432	−0.955569
$887$	55.1041	1.85021	0.925107	−	0.379706i	$-0.123975\pi$
$887$	55.1041	1.85021	0.925107	+	0.379706i	$0.123975\pi$
$888$	−5.53073	−0.185599
$889$	62.9377	2.11086
$890$	15.5491	0.521207
$891$	2.65672	0.0890033
$892$	−0.246992	−0.00826992
$893$	11.1137	0.371907
$894$	0.266600	0.00891643
$895$	−0.209059	−0.00698809
$896$	4.80701	0.160591
$897$	−19.6718	−0.656823
$898$	−32.6308	−1.08890
$899$	−16.0113	−0.534008
$900$	1.00000	0.0333333
$901$	0	0
$902$	−30.1254	−1.00307
$903$	−37.5247	−1.24874
$904$	−4.15242	−0.138107
$905$	14.4208	0.479364
$906$	−13.1064	−0.435432
$907$	29.6139	0.983315	0.491657	−	0.870789i	$-0.336391\pi$
$907$	29.6139	0.983315	0.491657	+	0.870789i	$0.336391\pi$
$908$	−0.206352	−0.00684804
$909$	−6.08472	−0.201817
$910$	−21.8890	−0.725614
$911$	−2.96899	−0.0983669	−0.0491835	−	0.998790i	$-0.515662\pi$
$911$	−2.96899	−0.0983669	−0.0491835	+	0.998790i	$0.515662\pi$
$912$	−6.04449	−0.200153
$913$	−46.9740	−1.55461
$914$	−0.770239	−0.0254772
$915$	−7.66447	−0.253379
$916$	12.8458	0.424438
$917$	26.0523	0.860323
$918$	0	0
$919$	−1.58747	−0.0523659	−0.0261830	−	0.999657i	$-0.508335\pi$
$919$	−1.58747	−0.0523659	−0.0261830	+	0.999657i	$0.508335\pi$
$920$	4.32010	0.142430
$921$	1.39589	0.0459961
$922$	−21.3672	−0.703691
$923$	−22.1605	−0.729421
$924$	12.7709	0.420131
$925$	−5.53073	−0.181849
$926$	5.59618	0.183902
$927$	−1.35517	−0.0445097
$928$	4.06491	0.133437
$929$	27.4847	0.901744	0.450872	−	0.892589i	$-0.351113\pi$
$929$	27.4847	0.901744	0.450872	+	0.892589i	$0.351113\pi$
$930$	−3.93892	−0.129162
$931$	97.3608	3.19087
$932$	0.165078	0.00540732
$933$	−3.49665	−0.114475
$934$	−17.4394	−0.570634
$935$	0	0
$936$	−4.55356	−0.148838
$937$	−43.8053	−1.43106	−0.715529	−	0.698583i	$-0.753814\pi$
$937$	−43.8053	−1.43106	−0.715529	+	0.698583i	$0.753814\pi$
$938$	66.5710	2.17362
$939$	−23.2668	−0.759283
$940$	−1.83866	−0.0599703
$941$	−31.1296	−1.01479	−0.507397	−	0.861712i	$-0.669392\pi$
$941$	−31.1296	−1.01479	−0.507397	+	0.861712i	$0.669392\pi$
$942$	2.93377	0.0955872
$943$	48.9871	1.59524
$944$	3.50381	0.114039
$945$	−4.80701	−0.156372
$946$	20.7390	0.674282
$947$	5.39288	0.175245	0.0876225	−	0.996154i	$-0.472073\pi$
$947$	5.39288	0.175245	0.0876225	+	0.996154i	$0.472073\pi$
$948$	−13.1363	−0.426647
$949$	−38.4086	−1.24680
$950$	−6.04449	−0.196109
$951$	−15.1546	−0.491422
$952$	0	0
$953$	−30.3834	−0.984215	−0.492107	−	0.870535i	$-0.663773\pi$
$953$	−30.3834	−0.984215	−0.492107	+	0.870535i	$0.663773\pi$
$954$	−5.88254	−0.190454
$955$	−12.3810	−0.400639
$956$	19.2098	0.621288
$957$	10.7993	0.349092
$958$	−26.6130	−0.859827
$959$	−34.3064	−1.10781
$960$	1.00000	0.0322749
$961$	−15.4849	−0.499513
$962$	25.1845	0.811981
$963$	−3.10006	−0.0998979
$964$	−4.55105	−0.146579
$965$	3.59378	0.115688
$966$	−20.7668	−0.668160
$967$	18.8257	0.605395	0.302698	−	0.953087i	$-0.402113\pi$
$967$	18.8257	0.605395	0.302698	+	0.953087i	$0.402113\pi$
$968$	3.94187	0.126696
$969$	0	0
$970$	17.1612	0.551014
$971$	−34.5809	−1.10975	−0.554876	−	0.831933i	$-0.687234\pi$
$971$	−34.5809	−1.10975	−0.554876	+	0.831933i	$0.687234\pi$
$972$	−1.00000	−0.0320750
$973$	34.9964	1.12193
$974$	−9.44787	−0.302729
$975$	−4.55356	−0.145831
$976$	−7.66447	−0.245334
$977$	8.43629	0.269901	0.134950	−	0.990852i	$-0.456912\pi$
$977$	8.43629	0.269901	0.134950	+	0.990852i	$0.456912\pi$
$978$	9.13905	0.292234
$979$	41.3095	1.32026
$980$	−16.1074	−0.514531
$981$	−15.4449	−0.493117
$982$	−11.4423	−0.365138
$983$	26.8259	0.855614	0.427807	−	0.903870i	$-0.359286\pi$
$983$	26.8259	0.855614	0.427807	+	0.903870i	$0.359286\pi$
$984$	11.3393	0.361485
$985$	−4.80943	−0.153241
$986$	0	0
$987$	8.83844	0.281331
$988$	27.5239	0.875653
$989$	−33.7238	−1.07235
$990$	2.65672	0.0844359
$991$	8.83209	0.280561	0.140280	−	0.990112i	$-0.455200\pi$
$991$	8.83209	0.280561	0.140280	+	0.990112i	$0.455200\pi$
$992$	−3.93892	−0.125061
$993$	24.0083	0.761879
$994$	−23.3939	−0.742011
$995$	4.44449	0.140900
$996$	17.6812	0.560252
$997$	42.7597	1.35421	0.677107	−	0.735885i	$-0.263233\pi$
$997$	42.7597	1.35421	0.677107	+	0.735885i	$0.263233\pi$
$998$	−13.8201	−0.437467
$999$	5.53073	0.174985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.cj.1.2		8
17.5	odd	16		510.2.u.c.331.4	yes	16
17.7	odd	16		510.2.u.c.151.4	✓	16
17.16	even	2		8670.2.a.ck.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.u.c.151.4	✓	16	17.7	odd	16
510.2.u.c.331.4	yes	16	17.5	odd	16
8670.2.a.cj.1.2		8	1.1	even	1	trivial
8670.2.a.ck.1.7		8	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.80701 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.80701 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.65672 q^{11} -1.00000 q^{12} +4.55356 q^{13} +4.80701 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +6.04449 q^{19} -1.00000 q^{20} +4.80701 q^{21} -2.65672 q^{22} +4.32010 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.55356 q^{26} -1.00000 q^{27} -4.80701 q^{28} -4.06491 q^{29} -1.00000 q^{30} +3.93892 q^{31} -1.00000 q^{32} -2.65672 q^{33} +4.80701 q^{35} +1.00000 q^{36} -5.53073 q^{37} -6.04449 q^{38} -4.55356 q^{39} +1.00000 q^{40} +11.3393 q^{41} -4.80701 q^{42} -7.80625 q^{43} +2.65672 q^{44} -1.00000 q^{45} -4.32010 q^{46} +1.83866 q^{47} -1.00000 q^{48} +16.1074 q^{49} -1.00000 q^{50} +4.55356 q^{52} +5.88254 q^{53} +1.00000 q^{54} -2.65672 q^{55} +4.80701 q^{56} -6.04449 q^{57} +4.06491 q^{58} +3.50381 q^{59} +1.00000 q^{60} -7.66447 q^{61} -3.93892 q^{62} -4.80701 q^{63} +1.00000 q^{64} -4.55356 q^{65} +2.65672 q^{66} +13.8487 q^{67} -4.32010 q^{69} -4.80701 q^{70} -4.86663 q^{71} -1.00000 q^{72} -8.43486 q^{73} +5.53073 q^{74} -1.00000 q^{75} +6.04449 q^{76} -12.7709 q^{77} +4.55356 q^{78} +13.1363 q^{79} -1.00000 q^{80} +1.00000 q^{81} -11.3393 q^{82} -17.6812 q^{83} +4.80701 q^{84} +7.80625 q^{86} +4.06491 q^{87} -2.65672 q^{88} +15.5491 q^{89} +1.00000 q^{90} -21.8890 q^{91} +4.32010 q^{92} -3.93892 q^{93} -1.83866 q^{94} -6.04449 q^{95} +1.00000 q^{96} +17.1612 q^{97} -16.1074 q^{98} +2.65672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{14} + 8 q^{15} + 8 q^{16} - 8 q^{18} - 8 q^{20} + 8 q^{21} - 8 q^{22} + 8 q^{24} + 8 q^{25} - 8 q^{27} - 8 q^{28} + 8 q^{29} - 8 q^{30} - 8 q^{31} - 8 q^{32} - 8 q^{33} + 8 q^{35} + 8 q^{36} - 32 q^{37} + 8 q^{40} + 8 q^{41} - 8 q^{42} - 8 q^{43} + 8 q^{44} - 8 q^{45} - 8 q^{48} + 40 q^{49} - 8 q^{50} + 16 q^{53} + 8 q^{54} - 8 q^{55} + 8 q^{56} - 8 q^{58} + 32 q^{59} + 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 8 q^{64} + 8 q^{66} - 8 q^{67} - 8 q^{70} + 24 q^{71} - 8 q^{72} - 40 q^{73} + 32 q^{74} - 8 q^{75} + 16 q^{77} - 24 q^{79} - 8 q^{80} + 8 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{84} + 8 q^{86} - 8 q^{87} - 8 q^{88} + 48 q^{89} + 8 q^{90} - 24 q^{91} + 8 q^{93} + 8 q^{96} - 24 q^{97} - 40 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + 8 * q^11 - 8 * q^12 + 8 * q^14 + 8 * q^15 + 8 * q^16 - 8 * q^18 - 8 * q^20 + 8 * q^21 - 8 * q^22 + 8 * q^24 + 8 * q^25 - 8 * q^27 - 8 * q^28 + 8 * q^29 - 8 * q^30 - 8 * q^31 - 8 * q^32 - 8 * q^33 + 8 * q^35 + 8 * q^36 - 32 * q^37 + 8 * q^40 + 8 * q^41 - 8 * q^42 - 8 * q^43 + 8 * q^44 - 8 * q^45 - 8 * q^48 + 40 * q^49 - 8 * q^50 + 16 * q^53 + 8 * q^54 - 8 * q^55 + 8 * q^56 - 8 * q^58 + 32 * q^59 + 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 8 * q^64 + 8 * q^66 - 8 * q^67 - 8 * q^70 + 24 * q^71 - 8 * q^72 - 40 * q^73 + 32 * q^74 - 8 * q^75 + 16 * q^77 - 24 * q^79 - 8 * q^80 + 8 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^84 + 8 * q^86 - 8 * q^87 - 8 * q^88 + 48 * q^89 + 8 * q^90 - 24 * q^91 + 8 * q^93 + 8 * q^96 - 24 * q^97 - 40 * q^98 + 8 * q^99

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