LMFDB - Embedded newform 8670.2.a.bu.1.4

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:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*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			1.4
Root			$1.84776$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.41421 q^{11} -1.00000 q^{12} -1.23463 q^{13} -2.61313 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -2.82843 q^{19} +1.00000 q^{20} -2.61313 q^{21} +1.41421 q^{22} +0.0630603 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.23463 q^{26} -1.00000 q^{27} +2.61313 q^{28} -2.75057 q^{29} +1.00000 q^{30} -1.75858 q^{31} -1.00000 q^{32} +1.41421 q^{33} +2.61313 q^{35} +1.00000 q^{36} -3.06147 q^{37} +2.82843 q^{38} +1.23463 q^{39} -1.00000 q^{40} +5.24718 q^{41} +2.61313 q^{42} +6.58701 q^{43} -1.41421 q^{44} +1.00000 q^{45} -0.0630603 q^{46} -1.06147 q^{47} -1.00000 q^{48} -0.171573 q^{49} -1.00000 q^{50} -1.23463 q^{52} +4.52395 q^{53} +1.00000 q^{54} -1.41421 q^{55} -2.61313 q^{56} +2.82843 q^{57} +2.75057 q^{58} -2.05121 q^{59} -1.00000 q^{60} +0.304482 q^{61} +1.75858 q^{62} +2.61313 q^{63} +1.00000 q^{64} -1.23463 q^{65} -1.41421 q^{66} +9.94110 q^{67} -0.0630603 q^{69} -2.61313 q^{70} -7.83938 q^{71} -1.00000 q^{72} -11.7711 q^{73} +3.06147 q^{74} -1.00000 q^{75} -2.82843 q^{76} -3.69552 q^{77} -1.23463 q^{78} +7.27157 q^{79} +1.00000 q^{80} +1.00000 q^{81} -5.24718 q^{82} -15.7502 q^{83} -2.61313 q^{84} -6.58701 q^{86} +2.75057 q^{87} +1.41421 q^{88} -15.0775 q^{89} -1.00000 q^{90} -3.22625 q^{91} +0.0630603 q^{92} +1.75858 q^{93} +1.06147 q^{94} -2.82843 q^{95} +1.00000 q^{96} +2.21530 q^{97} +0.171573 q^{98} -1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{12} - 8 q^{13} - 4 q^{15} + 4 q^{16} - 4 q^{18} + 4 q^{20} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 8 q^{26} - 4 q^{27} - 8 q^{29} + 4 q^{30} + 16 q^{31} - 4 q^{32} + 4 q^{36} + 8 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{46} + 8 q^{47} - 4 q^{48} - 12 q^{49} - 4 q^{50} - 8 q^{52} - 8 q^{53} + 4 q^{54} + 8 q^{58} - 4 q^{60} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 8 q^{65} + 8 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{73} - 4 q^{75} - 8 q^{78} + 4 q^{80} + 4 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{86} + 8 q^{87} - 8 q^{89} - 4 q^{90} + 8 q^{91} - 8 q^{92} - 16 q^{93} - 8 q^{94} + 4 q^{96} + 8 q^{97} + 12 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^10 - 4 * q^12 - 8 * q^13 - 4 * q^15 + 4 * q^16 - 4 * q^18 + 4 * q^20 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 8 * q^26 - 4 * q^27 - 8 * q^29 + 4 * q^30 + 16 * q^31 - 4 * q^32 + 4 * q^36 + 8 * q^39 - 4 * q^40 + 8 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^46 + 8 * q^47 - 4 * q^48 - 12 * q^49 - 4 * q^50 - 8 * q^52 - 8 * q^53 + 4 * q^54 + 8 * q^58 - 4 * q^60 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 8 * q^65 + 8 * q^67 + 8 * q^69 - 4 * q^72 - 8 * q^73 - 4 * q^75 - 8 * q^78 + 4 * q^80 + 4 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^86 + 8 * q^87 - 8 * q^89 - 4 * q^90 + 8 * q^91 - 8 * q^92 - 16 * q^93 - 8 * q^94 + 4 * q^96 + 8 * q^97 + 12 * q^98

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
$7$	2.61313	0.987669	0.493834	−	0.869556i	$-0.335595\pi$
$7$	2.61313	0.987669	0.493834	+	0.869556i	$0.335595\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	−1.00000	−0.288675
$13$	−1.23463	−0.342426	−0.171213	−	0.985234i	$-0.554769\pi$
$13$	−1.23463	−0.342426	−0.171213	+	0.985234i	$0.554769\pi$
$14$	−2.61313	−0.698387
$15$	−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	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.00000	0.223607
$21$	−2.61313	−0.570231
$22$	1.41421	0.301511
$23$	0.0630603	0.0131490	0.00657449	−	0.999978i	$-0.497907\pi$
$23$	0.0630603	0.0131490	0.00657449	+	0.999978i	$0.497907\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	1.23463	0.242131
$27$	−1.00000	−0.192450
$28$	2.61313	0.493834
$29$	−2.75057	−0.510768	−0.255384	−	0.966840i	$-0.582202\pi$
$29$	−2.75057	−0.510768	−0.255384	+	0.966840i	$0.582202\pi$
$30$	1.00000	0.182574
$31$	−1.75858	−0.315850	−0.157925	−	0.987451i	$-0.550480\pi$
$31$	−1.75858	−0.315850	−0.157925	+	0.987451i	$0.550480\pi$
$32$	−1.00000	−0.176777
$33$	1.41421	0.246183
$34$	0	0
$35$	2.61313	0.441699
$36$	1.00000	0.166667
$37$	−3.06147	−0.503302	−0.251651	−	0.967818i	$-0.580974\pi$
$37$	−3.06147	−0.503302	−0.251651	+	0.967818i	$0.580974\pi$
$38$	2.82843	0.458831
$39$	1.23463	0.197700
$40$	−1.00000	−0.158114
$41$	5.24718	0.819471	0.409736	−	0.912204i	$-0.365621\pi$
$41$	5.24718	0.819471	0.409736	+	0.912204i	$0.365621\pi$
$42$	2.61313	0.403214
$43$	6.58701	1.00451	0.502254	−	0.864720i	$-0.332504\pi$
$43$	6.58701	1.00451	0.502254	+	0.864720i	$0.332504\pi$
$44$	−1.41421	−0.213201
$45$	1.00000	0.149071
$46$	−0.0630603	−0.00929773
$47$	−1.06147	−0.154831	−0.0774155	−	0.996999i	$-0.524667\pi$
$47$	−1.06147	−0.154831	−0.0774155	+	0.996999i	$0.524667\pi$
$48$	−1.00000	−0.144338
$49$	−0.171573	−0.0245104
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.23463	−0.171213
$53$	4.52395	0.621412	0.310706	−	0.950506i	$-0.399435\pi$
$53$	4.52395	0.621412	0.310706	+	0.950506i	$0.399435\pi$
$54$	1.00000	0.136083
$55$	−1.41421	−0.190693
$56$	−2.61313	−0.349194
$57$	2.82843	0.374634
$58$	2.75057	0.361168
$59$	−2.05121	−0.267044	−0.133522	−	0.991046i	$-0.542629\pi$
$59$	−2.05121	−0.267044	−0.133522	+	0.991046i	$0.542629\pi$
$60$	−1.00000	−0.129099
$61$	0.304482	0.0389849	0.0194925	−	0.999810i	$-0.493795\pi$
$61$	0.304482	0.0389849	0.0194925	+	0.999810i	$0.493795\pi$
$62$	1.75858	0.223340
$63$	2.61313	0.329223
$64$	1.00000	0.125000
$65$	−1.23463	−0.153137
$66$	−1.41421	−0.174078
$67$	9.94110	1.21450	0.607249	−	0.794511i	$-0.292273\pi$
$67$	9.94110	1.21450	0.607249	+	0.794511i	$0.292273\pi$
$68$	0	0
$69$	−0.0630603	−0.00759156
$70$	−2.61313	−0.312328
$71$	−7.83938	−0.930363	−0.465181	−	0.885215i	$-0.654011\pi$
$71$	−7.83938	−0.930363	−0.465181	+	0.885215i	$0.654011\pi$
$72$	−1.00000	−0.117851
$73$	−11.7711	−1.37771	−0.688853	−	0.724901i	$-0.741885\pi$
$73$	−11.7711	−1.37771	−0.688853	+	0.724901i	$0.741885\pi$
$74$	3.06147	0.355888
$75$	−1.00000	−0.115470
$76$	−2.82843	−0.324443
$77$	−3.69552	−0.421143
$78$	−1.23463	−0.139795
$79$	7.27157	0.818116	0.409058	−	0.912508i	$-0.365857\pi$
$79$	7.27157	0.818116	0.409058	+	0.912508i	$0.365857\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−5.24718	−0.579454
$83$	−15.7502	−1.72881	−0.864404	−	0.502797i	$-0.832304\pi$
$83$	−15.7502	−1.72881	−0.864404	+	0.502797i	$0.832304\pi$
$84$	−2.61313	−0.285115
$85$	0	0
$86$	−6.58701	−0.710295
$87$	2.75057	0.294892
$88$	1.41421	0.150756
$89$	−15.0775	−1.59821	−0.799105	−	0.601192i	$-0.794693\pi$
$89$	−15.0775	−1.59821	−0.799105	+	0.601192i	$0.794693\pi$
$90$	−1.00000	−0.105409
$91$	−3.22625	−0.338203
$92$	0.0630603	0.00657449
$93$	1.75858	0.182356
$94$	1.06147	0.109482
$95$	−2.82843	−0.290191
$96$	1.00000	0.102062
$97$	2.21530	0.224930	0.112465	−	0.993656i	$-0.464125\pi$
$97$	2.21530	0.224930	0.112465	+	0.993656i	$0.464125\pi$
$98$	0.171573	0.0173315
$99$	−1.41421	−0.142134
$100$	1.00000	0.100000
$101$	1.27330	0.126698	0.0633489	−	0.997991i	$-0.479822\pi$
$101$	1.27330	0.126698	0.0633489	+	0.997991i	$0.479822\pi$
$102$	0	0
$103$	−12.6887	−1.25026	−0.625129	−	0.780522i	$-0.714954\pi$
$103$	−12.6887	−1.25026	−0.625129	+	0.780522i	$0.714954\pi$
$104$	1.23463	0.121066
$105$	−2.61313	−0.255015
$106$	−4.52395	−0.439404
$107$	−8.72739	−0.843709	−0.421854	−	0.906664i	$-0.638621\pi$
$107$	−8.72739	−0.843709	−0.421854	+	0.906664i	$0.638621\pi$
$108$	−1.00000	−0.0962250
$109$	10.6173	1.01695	0.508476	−	0.861076i	$-0.330209\pi$
$109$	10.6173	1.01695	0.508476	+	0.861076i	$0.330209\pi$
$110$	1.41421	0.134840
$111$	3.06147	0.290582
$112$	2.61313	0.246917
$113$	−7.28515	−0.685329	−0.342665	−	0.939458i	$-0.611329\pi$
$113$	−7.28515	−0.685329	−0.342665	+	0.939458i	$0.611329\pi$
$114$	−2.82843	−0.264906
$115$	0.0630603	0.00588040
$116$	−2.75057	−0.255384
$117$	−1.23463	−0.114142
$118$	2.05121	0.188829
$119$	0	0
$120$	1.00000	0.0912871
$121$	−9.00000	−0.818182
$122$	−0.304482	−0.0275665
$123$	−5.24718	−0.473122
$124$	−1.75858	−0.157925
$125$	1.00000	0.0894427
$126$	−2.61313	−0.232796
$127$	−2.74097	−0.243222	−0.121611	−	0.992578i	$-0.538806\pi$
$127$	−2.74097	−0.243222	−0.121611	+	0.992578i	$0.538806\pi$
$128$	−1.00000	−0.0883883
$129$	−6.58701	−0.579953
$130$	1.23463	0.108284
$131$	14.8149	1.29438	0.647190	−	0.762329i	$-0.275944\pi$
$131$	14.8149	1.29438	0.647190	+	0.762329i	$0.275944\pi$
$132$	1.41421	0.123091
$133$	−7.39104	−0.640884
$134$	−9.94110	−0.858780
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	9.63049	0.822788	0.411394	−	0.911458i	$-0.365042\pi$
$137$	9.63049	0.822788	0.411394	+	0.911458i	$0.365042\pi$
$138$	0.0630603	0.00536805
$139$	4.54071	0.385138	0.192569	−	0.981283i	$-0.438318\pi$
$139$	4.54071	0.385138	0.192569	+	0.981283i	$0.438318\pi$
$140$	2.61313	0.220849
$141$	1.06147	0.0893917
$142$	7.83938	0.657866
$143$	1.74603	0.146011
$144$	1.00000	0.0833333
$145$	−2.75057	−0.228422
$146$	11.7711	0.974185
$147$	0.171573	0.0141511
$148$	−3.06147	−0.251651
$149$	0.113577	0.00930459	0.00465230	−	0.999989i	$-0.498519\pi$
$149$	0.113577	0.00930459	0.00465230	+	0.999989i	$0.498519\pi$
$150$	1.00000	0.0816497
$151$	15.5236	1.26329	0.631645	−	0.775258i	$-0.282380\pi$
$151$	15.5236	1.26329	0.631645	+	0.775258i	$0.282380\pi$
$152$	2.82843	0.229416
$153$	0	0
$154$	3.69552	0.297793
$155$	−1.75858	−0.141252
$156$	1.23463	0.0988498
$157$	−19.6990	−1.57215	−0.786075	−	0.618131i	$-0.787890\pi$
$157$	−19.6990	−1.57215	−0.786075	+	0.618131i	$0.787890\pi$
$158$	−7.27157	−0.578495
$159$	−4.52395	−0.358772
$160$	−1.00000	−0.0790569
$161$	0.164784	0.0129868
$162$	−1.00000	−0.0785674
$163$	18.3907	1.44047	0.720234	−	0.693731i	$-0.244034\pi$
$163$	18.3907	1.44047	0.720234	+	0.693731i	$0.244034\pi$
$164$	5.24718	0.409736
$165$	1.41421	0.110096
$166$	15.7502	1.22245
$167$	−22.4264	−1.73541	−0.867703	−	0.497083i	$-0.834404\pi$
$167$	−22.4264	−1.73541	−0.867703	+	0.497083i	$0.834404\pi$
$168$	2.61313	0.201607
$169$	−11.4757	−0.882745
$170$	0	0
$171$	−2.82843	−0.216295
$172$	6.58701	0.502254
$173$	−24.4434	−1.85840	−0.929200	−	0.369578i	$-0.879502\pi$
$173$	−24.4434	−1.85840	−0.929200	+	0.369578i	$0.879502\pi$
$174$	−2.75057	−0.208520
$175$	2.61313	0.197534
$176$	−1.41421	−0.106600
$177$	2.05121	0.154178
$178$	15.0775	1.13011
$179$	−11.8268	−0.883979	−0.441990	−	0.897020i	$-0.645727\pi$
$179$	−11.8268	−0.883979	−0.441990	+	0.897020i	$0.645727\pi$
$180$	1.00000	0.0745356
$181$	14.4912	1.07712	0.538560	−	0.842587i	$-0.318969\pi$
$181$	14.4912	1.07712	0.538560	+	0.842587i	$0.318969\pi$
$182$	3.22625	0.239146
$183$	−0.304482	−0.0225079
$184$	−0.0630603	−0.00464886
$185$	−3.06147	−0.225084
$186$	−1.75858	−0.128945
$187$	0	0
$188$	−1.06147	−0.0774155
$189$	−2.61313	−0.190077
$190$	2.82843	0.205196
$191$	−7.93174	−0.573921	−0.286960	−	0.957942i	$-0.592645\pi$
$191$	−7.93174	−0.573921	−0.286960	+	0.957942i	$0.592645\pi$
$192$	−1.00000	−0.0721688
$193$	7.18343	0.517074	0.258537	−	0.966001i	$-0.416760\pi$
$193$	7.18343	0.517074	0.258537	+	0.966001i	$0.416760\pi$
$194$	−2.21530	−0.159049
$195$	1.23463	0.0884139
$196$	−0.171573	−0.0122552
$197$	−9.29769	−0.662433	−0.331217	−	0.943555i	$-0.607459\pi$
$197$	−9.29769	−0.662433	−0.331217	+	0.943555i	$0.607459\pi$
$198$	1.41421	0.100504
$199$	−0.454999	−0.0322540	−0.0161270	−	0.999870i	$-0.505134\pi$
$199$	−0.454999	−0.0322540	−0.0161270	+	0.999870i	$0.505134\pi$
$200$	−1.00000	−0.0707107
$201$	−9.94110	−0.701191
$202$	−1.27330	−0.0895889
$203$	−7.18759	−0.504470
$204$	0	0
$205$	5.24718	0.366479
$206$	12.6887	0.884066
$207$	0.0630603	0.00438299
$208$	−1.23463	−0.0856064
$209$	4.00000	0.276686
$210$	2.61313	0.180323
$211$	9.80562	0.675047	0.337523	−	0.941317i	$-0.390411\pi$
$211$	9.80562	0.675047	0.337523	+	0.941317i	$0.390411\pi$
$212$	4.52395	0.310706
$213$	7.83938	0.537145
$214$	8.72739	0.596592
$215$	6.58701	0.449230
$216$	1.00000	0.0680414
$217$	−4.59539	−0.311955
$218$	−10.6173	−0.719093
$219$	11.7711	0.795419
$220$	−1.41421	−0.0953463
$221$	0	0
$222$	−3.06147	−0.205472
$223$	−4.68873	−0.313981	−0.156990	−	0.987600i	$-0.550179\pi$
$223$	−4.68873	−0.313981	−0.156990	+	0.987600i	$0.550179\pi$
$224$	−2.61313	−0.174597
$225$	1.00000	0.0666667
$226$	7.28515	0.484601
$227$	−26.6424	−1.76832	−0.884158	−	0.467187i	$-0.845267\pi$
$227$	−26.6424	−1.76832	−0.884158	+	0.467187i	$0.845267\pi$
$228$	2.82843	0.187317
$229$	−7.73418	−0.511089	−0.255545	−	0.966797i	$-0.582255\pi$
$229$	−7.73418	−0.511089	−0.255545	+	0.966797i	$0.582255\pi$
$230$	−0.0630603	−0.00415807
$231$	3.69552	0.243147
$232$	2.75057	0.180584
$233$	14.3180	0.938003	0.469002	−	0.883197i	$-0.344614\pi$
$233$	14.3180	0.938003	0.469002	+	0.883197i	$0.344614\pi$
$234$	1.23463	0.0807105
$235$	−1.06147	−0.0692425
$236$	−2.05121	−0.133522
$237$	−7.27157	−0.472339
$238$	0	0
$239$	0.368233	0.0238190	0.0119095	−	0.999929i	$-0.496209\pi$
$239$	0.368233	0.0238190	0.0119095	+	0.999929i	$0.496209\pi$
$240$	−1.00000	−0.0645497
$241$	9.64431	0.621245	0.310622	−	0.950533i	$-0.399463\pi$
$241$	9.64431	0.621245	0.310622	+	0.950533i	$0.399463\pi$
$242$	9.00000	0.578542
$243$	−1.00000	−0.0641500
$244$	0.304482	0.0194925
$245$	−0.171573	−0.0109614
$246$	5.24718	0.334548
$247$	3.49207	0.222195
$248$	1.75858	0.111670
$249$	15.7502	0.998128
$250$	−1.00000	−0.0632456
$251$	1.38168	0.0872108	0.0436054	−	0.999049i	$-0.486116\pi$
$251$	1.38168	0.0872108	0.0436054	+	0.999049i	$0.486116\pi$
$252$	2.61313	0.164611
$253$	−0.0891807	−0.00560674
$254$	2.74097	0.171984
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.2130	−0.761829	−0.380915	−	0.924610i	$-0.624391\pi$
$257$	−12.2130	−0.761829	−0.380915	+	0.924610i	$0.624391\pi$
$258$	6.58701	0.410089
$259$	−8.00000	−0.497096
$260$	−1.23463	−0.0765687
$261$	−2.75057	−0.170256
$262$	−14.8149	−0.915265
$263$	11.4985	0.709027	0.354513	−	0.935051i	$-0.384647\pi$
$263$	11.4985	0.709027	0.354513	+	0.935051i	$0.384647\pi$
$264$	−1.41421	−0.0870388
$265$	4.52395	0.277904
$266$	7.39104	0.453174
$267$	15.0775	0.922727
$268$	9.94110	0.607249
$269$	−5.22175	−0.318376	−0.159188	−	0.987248i	$-0.550888\pi$
$269$	−5.22175	−0.318376	−0.159188	+	0.987248i	$0.550888\pi$
$270$	1.00000	0.0608581
$271$	−5.24492	−0.318606	−0.159303	−	0.987230i	$-0.550925\pi$
$271$	−5.24492	−0.318606	−0.159303	+	0.987230i	$0.550925\pi$
$272$	0	0
$273$	3.22625	0.195262
$274$	−9.63049	−0.581799
$275$	−1.41421	−0.0852803
$276$	−0.0630603	−0.00379578
$277$	6.67913	0.401310	0.200655	−	0.979662i	$-0.435693\pi$
$277$	6.67913	0.401310	0.200655	+	0.979662i	$0.435693\pi$
$278$	−4.54071	−0.272334
$279$	−1.75858	−0.105283
$280$	−2.61313	−0.156164
$281$	1.56351	0.0932713	0.0466356	−	0.998912i	$-0.485150\pi$
$281$	1.56351	0.0932713	0.0466356	+	0.998912i	$0.485150\pi$
$282$	−1.06147	−0.0632095
$283$	−8.29143	−0.492875	−0.246437	−	0.969159i	$-0.579260\pi$
$283$	−8.29143	−0.492875	−0.246437	+	0.969159i	$0.579260\pi$
$284$	−7.83938	−0.465181
$285$	2.82843	0.167542
$286$	−1.74603	−0.103245
$287$	13.7115	0.809366
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	2.75057	0.161519
$291$	−2.21530	−0.129863
$292$	−11.7711	−0.688853
$293$	1.44572	0.0844596	0.0422298	−	0.999108i	$-0.486554\pi$
$293$	1.44572	0.0844596	0.0422298	+	0.999108i	$0.486554\pi$
$294$	−0.171573	−0.0100063
$295$	−2.05121	−0.119426
$296$	3.06147	0.177944
$297$	1.41421	0.0820610
$298$	−0.113577	−0.00657934
$299$	−0.0778563	−0.00450255
$300$	−1.00000	−0.0577350
$301$	17.2127	0.992122
$302$	−15.5236	−0.893281
$303$	−1.27330	−0.0731490
$304$	−2.82843	−0.162221
$305$	0.304482	0.0174346
$306$	0	0
$307$	−30.0993	−1.71786	−0.858928	−	0.512097i	$-0.828869\pi$
$307$	−30.0993	−1.71786	−0.858928	+	0.512097i	$0.828869\pi$
$308$	−3.69552	−0.210572
$309$	12.6887	0.721837
$310$	1.75858	0.0998805
$311$	20.2887	1.15047	0.575233	−	0.817989i	$-0.304911\pi$
$311$	20.2887	1.15047	0.575233	+	0.817989i	$0.304911\pi$
$312$	−1.23463	−0.0698973
$313$	23.7485	1.34234	0.671172	−	0.741302i	$-0.265791\pi$
$313$	23.7485	1.34234	0.671172	+	0.741302i	$0.265791\pi$
$314$	19.6990	1.11168
$315$	2.61313	0.147233
$316$	7.27157	0.409058
$317$	11.1959	0.628825	0.314413	−	0.949286i	$-0.398192\pi$
$317$	11.1959	0.628825	0.314413	+	0.949286i	$0.398192\pi$
$318$	4.52395	0.253690
$319$	3.88989	0.217792
$320$	1.00000	0.0559017
$321$	8.72739	0.487116
$322$	−0.164784	−0.00918308
$323$	0	0
$324$	1.00000	0.0555556
$325$	−1.23463	−0.0684851
$326$	−18.3907	−1.01856
$327$	−10.6173	−0.587137
$328$	−5.24718	−0.289727
$329$	−2.77375	−0.152922
$330$	−1.41421	−0.0778499
$331$	13.8854	0.763210	0.381605	−	0.924325i	$-0.375371\pi$
$331$	13.8854	0.763210	0.381605	+	0.924325i	$0.375371\pi$
$332$	−15.7502	−0.864404
$333$	−3.06147	−0.167767
$334$	22.4264	1.22712
$335$	9.94110	0.543140
$336$	−2.61313	−0.142558
$337$	−24.0066	−1.30772	−0.653861	−	0.756614i	$-0.726852\pi$
$337$	−24.0066	−1.30772	−0.653861	+	0.756614i	$0.726852\pi$
$338$	11.4757	0.624195
$339$	7.28515	0.395675
$340$	0	0
$341$	2.48701	0.134679
$342$	2.82843	0.152944
$343$	−18.7402	−1.01188
$344$	−6.58701	−0.355148
$345$	−0.0630603	−0.00339505
$346$	24.4434	1.31409
$347$	31.6356	1.69829	0.849144	−	0.528162i	$-0.177119\pi$
$347$	31.6356	1.69829	0.849144	+	0.528162i	$0.177119\pi$
$348$	2.75057	0.147446
$349$	−18.7730	−1.00490	−0.502448	−	0.864608i	$-0.667567\pi$
$349$	−18.7730	−1.00490	−0.502448	+	0.864608i	$0.667567\pi$
$350$	−2.61313	−0.139677
$351$	1.23463	0.0658998
$352$	1.41421	0.0753778
$353$	6.19681	0.329823	0.164912	−	0.986308i	$-0.447266\pi$
$353$	6.19681	0.329823	0.164912	+	0.986308i	$0.447266\pi$
$354$	−2.05121	−0.109020
$355$	−7.83938	−0.416071
$356$	−15.0775	−0.799105
$357$	0	0
$358$	11.8268	0.625068
$359$	−5.14649	−0.271621	−0.135811	−	0.990735i	$-0.543364\pi$
$359$	−5.14649	−0.271621	−0.135811	+	0.990735i	$0.543364\pi$
$360$	−1.00000	−0.0527046
$361$	−11.0000	−0.578947
$362$	−14.4912	−0.761639
$363$	9.00000	0.472377
$364$	−3.22625	−0.169102
$365$	−11.7711	−0.616129
$366$	0.304482	0.0159155
$367$	27.9655	1.45979	0.729894	−	0.683561i	$-0.239570\pi$
$367$	27.9655	1.45979	0.729894	+	0.683561i	$0.239570\pi$
$368$	0.0630603	0.00328724
$369$	5.24718	0.273157
$370$	3.06147	0.159158
$371$	11.8216	0.613749
$372$	1.75858	0.0911780
$373$	−22.3131	−1.15533	−0.577664	−	0.816275i	$-0.696036\pi$
$373$	−22.3131	−1.15533	−0.577664	+	0.816275i	$0.696036\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	1.06147	0.0547410
$377$	3.39595	0.174900
$378$	2.61313	0.134405
$379$	27.8968	1.43296	0.716482	−	0.697605i	$-0.245751\pi$
$379$	27.8968	1.43296	0.716482	+	0.697605i	$0.245751\pi$
$380$	−2.82843	−0.145095
$381$	2.74097	0.140424
$382$	7.93174	0.405823
$383$	−1.65494	−0.0845637	−0.0422819	−	0.999106i	$-0.513463\pi$
$383$	−1.65494	−0.0845637	−0.0422819	+	0.999106i	$0.513463\pi$
$384$	1.00000	0.0510310
$385$	−3.69552	−0.188341
$386$	−7.18343	−0.365627
$387$	6.58701	0.334836
$388$	2.21530	0.112465
$389$	−32.9771	−1.67201	−0.836004	−	0.548724i	$-0.815114\pi$
$389$	−32.9771	−1.67201	−0.836004	+	0.548724i	$0.815114\pi$
$390$	−1.23463	−0.0625181
$391$	0	0
$392$	0.171573	0.00866574
$393$	−14.8149	−0.747310
$394$	9.29769	0.468411
$395$	7.27157	0.365873
$396$	−1.41421	−0.0710669
$397$	24.4872	1.22898	0.614488	−	0.788926i	$-0.289362\pi$
$397$	24.4872	1.22898	0.614488	+	0.788926i	$0.289362\pi$
$398$	0.454999	0.0228070
$399$	7.39104	0.370015
$400$	1.00000	0.0500000
$401$	−22.5281	−1.12500	−0.562500	−	0.826797i	$-0.690160\pi$
$401$	−22.5281	−1.12500	−0.562500	+	0.826797i	$0.690160\pi$
$402$	9.94110	0.495817
$403$	2.17120	0.108155
$404$	1.27330	0.0633489
$405$	1.00000	0.0496904
$406$	7.18759	0.356714
$407$	4.32957	0.214609
$408$	0	0
$409$	−33.7334	−1.66801	−0.834005	−	0.551756i	$-0.813958\pi$
$409$	−33.7334	−1.66801	−0.834005	+	0.551756i	$0.813958\pi$
$410$	−5.24718	−0.259140
$411$	−9.63049	−0.475037
$412$	−12.6887	−0.625129
$413$	−5.36006	−0.263751
$414$	−0.0630603	−0.00309924
$415$	−15.7502	−0.773147
$416$	1.23463	0.0605329
$417$	−4.54071	−0.222359
$418$	−4.00000	−0.195646
$419$	28.1235	1.37392	0.686961	−	0.726695i	$-0.258945\pi$
$419$	28.1235	1.37392	0.686961	+	0.726695i	$0.258945\pi$
$420$	−2.61313	−0.127507
$421$	34.5173	1.68227	0.841135	−	0.540825i	$-0.181888\pi$
$421$	34.5173	1.68227	0.841135	+	0.540825i	$0.181888\pi$
$422$	−9.80562	−0.477330
$423$	−1.06147	−0.0516103
$424$	−4.52395	−0.219702
$425$	0	0
$426$	−7.83938	−0.379819
$427$	0.795649	0.0385042
$428$	−8.72739	−0.421854
$429$	−1.74603	−0.0842994
$430$	−6.58701	−0.317654
$431$	−23.8882	−1.15065	−0.575326	−	0.817924i	$-0.695125\pi$
$431$	−23.8882	−1.15065	−0.575326	+	0.817924i	$0.695125\pi$
$432$	−1.00000	−0.0481125
$433$	−25.5436	−1.22755	−0.613773	−	0.789483i	$-0.710349\pi$
$433$	−25.5436	−1.22755	−0.613773	+	0.789483i	$0.710349\pi$
$434$	4.59539	0.220586
$435$	2.75057	0.131880
$436$	10.6173	0.508476
$437$	−0.178361	−0.00853218
$438$	−11.7711	−0.562446
$439$	35.2548	1.68262	0.841310	−	0.540553i	$-0.181785\pi$
$439$	35.2548	1.68262	0.841310	+	0.540553i	$0.181785\pi$
$440$	1.41421	0.0674200
$441$	−0.171573	−0.00817014
$442$	0	0
$443$	−33.1991	−1.57734	−0.788668	−	0.614819i	$-0.789229\pi$
$443$	−33.1991	−1.57734	−0.788668	+	0.614819i	$0.789229\pi$
$444$	3.06147	0.145291
$445$	−15.0775	−0.714741
$446$	4.68873	0.222018
$447$	−0.113577	−0.00537201
$448$	2.61313	0.123459
$449$	−17.7428	−0.837337	−0.418668	−	0.908139i	$-0.637503\pi$
$449$	−17.7428	−0.837337	−0.418668	+	0.908139i	$0.637503\pi$
$450$	−1.00000	−0.0471405
$451$	−7.42063	−0.349424
$452$	−7.28515	−0.342665
$453$	−15.5236	−0.729361
$454$	26.6424	1.25039
$455$	−3.22625	−0.151249
$456$	−2.82843	−0.132453
$457$	18.7653	0.877804	0.438902	−	0.898535i	$-0.355367\pi$
$457$	18.7653	0.877804	0.438902	+	0.898535i	$0.355367\pi$
$458$	7.73418	0.361395
$459$	0	0
$460$	0.0630603	0.00294020
$461$	10.2724	0.478433	0.239217	−	0.970966i	$-0.423109\pi$
$461$	10.2724	0.478433	0.239217	+	0.970966i	$0.423109\pi$
$462$	−3.69552	−0.171931
$463$	21.6164	1.00460	0.502299	−	0.864694i	$-0.332488\pi$
$463$	21.6164	1.00460	0.502299	+	0.864694i	$0.332488\pi$
$464$	−2.75057	−0.127692
$465$	1.75858	0.0815521
$466$	−14.3180	−0.663269
$467$	−12.6695	−0.586276	−0.293138	−	0.956070i	$-0.594700\pi$
$467$	−12.6695	−0.586276	−0.293138	+	0.956070i	$0.594700\pi$
$468$	−1.23463	−0.0570709
$469$	25.9774	1.19952
$470$	1.06147	0.0489618
$471$	19.6990	0.907682
$472$	2.05121	0.0944145
$473$	−9.31543	−0.428324
$474$	7.27157	0.333994
$475$	−2.82843	−0.129777
$476$	0	0
$477$	4.52395	0.207137
$478$	−0.368233	−0.0168426
$479$	−34.8068	−1.59036	−0.795181	−	0.606373i	$-0.792624\pi$
$479$	−34.8068	−1.59036	−0.795181	+	0.606373i	$0.792624\pi$
$480$	1.00000	0.0456435
$481$	3.77979	0.172344
$482$	−9.64431	−0.439286
$483$	−0.164784	−0.00749795
$484$	−9.00000	−0.409091
$485$	2.21530	0.100592
$486$	1.00000	0.0453609
$487$	−22.5441	−1.02157	−0.510786	−	0.859708i	$-0.670645\pi$
$487$	−22.5441	−1.02157	−0.510786	+	0.859708i	$0.670645\pi$
$488$	−0.304482	−0.0137832
$489$	−18.3907	−0.831655
$490$	0.171573	0.00775087
$491$	32.6545	1.47368	0.736839	−	0.676068i	$-0.236318\pi$
$491$	32.6545	1.47368	0.736839	+	0.676068i	$0.236318\pi$
$492$	−5.24718	−0.236561
$493$	0	0
$494$	−3.49207	−0.157116
$495$	−1.41421	−0.0635642
$496$	−1.75858	−0.0789625
$497$	−20.4853	−0.918890
$498$	−15.7502	−0.705783
$499$	−8.29016	−0.371118	−0.185559	−	0.982633i	$-0.559410\pi$
$499$	−8.29016	−0.371118	−0.185559	+	0.982633i	$0.559410\pi$
$500$	1.00000	0.0447214
$501$	22.4264	1.00194
$502$	−1.38168	−0.0616673
$503$	−31.7537	−1.41583	−0.707913	−	0.706300i	$-0.750363\pi$
$503$	−31.7537	−1.41583	−0.707913	+	0.706300i	$0.750363\pi$
$504$	−2.61313	−0.116398
$505$	1.27330	0.0566610
$506$	0.0891807	0.00396456
$507$	11.4757	0.509653
$508$	−2.74097	−0.121611
$509$	42.5158	1.88448	0.942240	−	0.334939i	$-0.108716\pi$
$509$	42.5158	1.88448	0.942240	+	0.334939i	$0.108716\pi$
$510$	0	0
$511$	−30.7594	−1.36072
$512$	−1.00000	−0.0441942
$513$	2.82843	0.124878
$514$	12.2130	0.538694
$515$	−12.6887	−0.559132
$516$	−6.58701	−0.289977
$517$	1.50114	0.0660201
$518$	8.00000	0.351500
$519$	24.4434	1.07295
$520$	1.23463	0.0541422
$521$	9.12800	0.399905	0.199952	−	0.979806i	$-0.435921\pi$
$521$	9.12800	0.399905	0.199952	+	0.979806i	$0.435921\pi$
$522$	2.75057	0.120389
$523$	20.0627	0.877279	0.438639	−	0.898663i	$-0.355461\pi$
$523$	20.0627	0.877279	0.438639	+	0.898663i	$0.355461\pi$
$524$	14.8149	0.647190
$525$	−2.61313	−0.114046
$526$	−11.4985	−0.501358
$527$	0	0
$528$	1.41421	0.0615457
$529$	−22.9960	−0.999827
$530$	−4.52395	−0.196508
$531$	−2.05121	−0.0890148
$532$	−7.39104	−0.320442
$533$	−6.47834	−0.280608
$534$	−15.0775	−0.652466
$535$	−8.72739	−0.377318
$536$	−9.94110	−0.429390
$537$	11.8268	0.510366
$538$	5.22175	0.225125
$539$	0.242641	0.0104513
$540$	−1.00000	−0.0430331
$541$	−1.90666	−0.0819736	−0.0409868	−	0.999160i	$-0.513050\pi$
$541$	−1.90666	−0.0819736	−0.0409868	+	0.999160i	$0.513050\pi$
$542$	5.24492	0.225289
$543$	−14.4912	−0.621876
$544$	0	0
$545$	10.6173	0.454795
$546$	−3.22625	−0.138071
$547$	5.45018	0.233033	0.116516	−	0.993189i	$-0.462827\pi$
$547$	5.45018	0.233033	0.116516	+	0.993189i	$0.462827\pi$
$548$	9.63049	0.411394
$549$	0.304482	0.0129950
$550$	1.41421	0.0603023
$551$	7.77979	0.331430
$552$	0.0630603	0.00268402
$553$	19.0015	0.808027
$554$	−6.67913	−0.283769
$555$	3.06147	0.129952
$556$	4.54071	0.192569
$557$	37.2866	1.57988	0.789941	−	0.613183i	$-0.210111\pi$
$557$	37.2866	1.57988	0.789941	+	0.613183i	$0.210111\pi$
$558$	1.75858	0.0744466
$559$	−8.13254	−0.343970
$560$	2.61313	0.110425
$561$	0	0
$562$	−1.56351	−0.0659527
$563$	−16.3533	−0.689208	−0.344604	−	0.938748i	$-0.611987\pi$
$563$	−16.3533	−0.689208	−0.344604	+	0.938748i	$0.611987\pi$
$564$	1.06147	0.0446958
$565$	−7.28515	−0.306489
$566$	8.29143	0.348515
$567$	2.61313	0.109741
$568$	7.83938	0.328933
$569$	−45.3967	−1.90313	−0.951563	−	0.307452i	$-0.900524\pi$
$569$	−45.3967	−1.90313	−0.951563	+	0.307452i	$0.900524\pi$
$570$	−2.82843	−0.118470
$571$	−9.99607	−0.418322	−0.209161	−	0.977881i	$-0.567073\pi$
$571$	−9.99607	−0.418322	−0.209161	+	0.977881i	$0.567073\pi$
$572$	1.74603	0.0730054
$573$	7.93174	0.331353
$574$	−13.7115	−0.572308
$575$	0.0630603	0.00262979
$576$	1.00000	0.0416667
$577$	−16.4867	−0.686349	−0.343174	−	0.939272i	$-0.611502\pi$
$577$	−16.4867	−0.686349	−0.343174	+	0.939272i	$0.611502\pi$
$578$	0	0
$579$	−7.18343	−0.298533
$580$	−2.75057	−0.114211
$581$	−41.1572	−1.70749
$582$	2.21530	0.0918272
$583$	−6.39782	−0.264971
$584$	11.7711	0.487092
$585$	−1.23463	−0.0510458
$586$	−1.44572	−0.0597219
$587$	−20.0083	−0.825832	−0.412916	−	0.910769i	$-0.635490\pi$
$587$	−20.0083	−0.825832	−0.412916	+	0.910769i	$0.635490\pi$
$588$	0.171573	0.00707555
$589$	4.97401	0.204951
$590$	2.05121	0.0844469
$591$	9.29769	0.382456
$592$	−3.06147	−0.125826
$593$	−41.2482	−1.69386	−0.846929	−	0.531705i	$-0.821551\pi$
$593$	−41.2482	−1.69386	−0.846929	+	0.531705i	$0.821551\pi$
$594$	−1.41421	−0.0580259
$595$	0	0
$596$	0.113577	0.00465230
$597$	0.454999	0.0186219
$598$	0.0778563	0.00318378
$599$	3.06391	0.125188	0.0625939	−	0.998039i	$-0.480063\pi$
$599$	3.06391	0.125188	0.0625939	+	0.998039i	$0.480063\pi$
$600$	1.00000	0.0408248
$601$	−33.0319	−1.34740	−0.673700	−	0.739005i	$-0.735296\pi$
$601$	−33.0319	−1.34740	−0.673700	+	0.739005i	$0.735296\pi$
$602$	−17.2127	−0.701536
$603$	9.94110	0.404833
$604$	15.5236	0.631645
$605$	−9.00000	−0.365902
$606$	1.27330	0.0517242
$607$	−24.1167	−0.978867	−0.489433	−	0.872041i	$-0.662796\pi$
$607$	−24.1167	−0.978867	−0.489433	+	0.872041i	$0.662796\pi$
$608$	2.82843	0.114708
$609$	7.18759	0.291256
$610$	−0.304482	−0.0123281
$611$	1.31052	0.0530181
$612$	0	0
$613$	−27.4502	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$613$	−27.4502	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$614$	30.0993	1.21471
$615$	−5.24718	−0.211587
$616$	3.69552	0.148897
$617$	−8.57688	−0.345292	−0.172646	−	0.984984i	$-0.555232\pi$
$617$	−8.57688	−0.345292	−0.172646	+	0.984984i	$0.555232\pi$
$618$	−12.6887	−0.510416
$619$	4.16749	0.167505	0.0837527	−	0.996487i	$-0.473309\pi$
$619$	4.16749	0.167505	0.0837527	+	0.996487i	$0.473309\pi$
$620$	−1.75858	−0.0706262
$621$	−0.0630603	−0.00253052
$622$	−20.2887	−0.813503
$623$	−39.3994	−1.57850
$624$	1.23463	0.0494249
$625$	1.00000	0.0400000
$626$	−23.7485	−0.949180
$627$	−4.00000	−0.159745
$628$	−19.6990	−0.786075
$629$	0	0
$630$	−2.61313	−0.104109
$631$	−9.17210	−0.365136	−0.182568	−	0.983193i	$-0.558441\pi$
$631$	−9.17210	−0.365136	−0.182568	+	0.983193i	$0.558441\pi$
$632$	−7.27157	−0.289248
$633$	−9.80562	−0.389739
$634$	−11.1959	−0.444646
$635$	−2.74097	−0.108772
$636$	−4.52395	−0.179386
$637$	0.211830	0.00839299
$638$	−3.88989	−0.154002
$639$	−7.83938	−0.310121
$640$	−1.00000	−0.0395285
$641$	−3.70812	−0.146462	−0.0732309	−	0.997315i	$-0.523331\pi$
$641$	−3.70812	−0.146462	−0.0732309	+	0.997315i	$0.523331\pi$
$642$	−8.72739	−0.344443
$643$	−9.32994	−0.367937	−0.183969	−	0.982932i	$-0.558894\pi$
$643$	−9.32994	−0.367937	−0.183969	+	0.982932i	$0.558894\pi$
$644$	0.164784	0.00649342
$645$	−6.58701	−0.259363
$646$	0	0
$647$	−22.1712	−0.871640	−0.435820	−	0.900034i	$-0.643542\pi$
$647$	−22.1712	−0.871640	−0.435820	+	0.900034i	$0.643542\pi$
$648$	−1.00000	−0.0392837
$649$	2.90085	0.113868
$650$	1.23463	0.0484263
$651$	4.59539	0.180107
$652$	18.3907	0.720234
$653$	15.3548	0.600880	0.300440	−	0.953801i	$-0.402866\pi$
$653$	15.3548	0.600880	0.300440	+	0.953801i	$0.402866\pi$
$654$	10.6173	0.415169
$655$	14.8149	0.578864
$656$	5.24718	0.204868
$657$	−11.7711	−0.459235
$658$	2.77375	0.108132
$659$	−10.6006	−0.412940	−0.206470	−	0.978453i	$-0.566198\pi$
$659$	−10.6006	−0.412940	−0.206470	+	0.978453i	$0.566198\pi$
$660$	1.41421	0.0550482
$661$	15.5982	0.606701	0.303351	−	0.952879i	$-0.401895\pi$
$661$	15.5982	0.606701	0.303351	+	0.952879i	$0.401895\pi$
$662$	−13.8854	−0.539671
$663$	0	0
$664$	15.7502	0.611226
$665$	−7.39104	−0.286612
$666$	3.06147	0.118629
$667$	−0.173452	−0.00671608
$668$	−22.4264	−0.867703
$669$	4.68873	0.181277
$670$	−9.94110	−0.384058
$671$	−0.430602	−0.0166232
$672$	2.61313	0.100804
$673$	−49.4635	−1.90668	−0.953338	−	0.301905i	$-0.902377\pi$
$673$	−49.4635	−1.90668	−0.953338	+	0.301905i	$0.902377\pi$
$674$	24.0066	0.924700
$675$	−1.00000	−0.0384900
$676$	−11.4757	−0.441372
$677$	20.1744	0.775366	0.387683	−	0.921793i	$-0.373275\pi$
$677$	20.1744	0.775366	0.387683	+	0.921793i	$0.373275\pi$
$678$	−7.28515	−0.279784
$679$	5.78886	0.222156
$680$	0	0
$681$	26.6424	1.02094
$682$	−2.48701	−0.0952324
$683$	−28.0965	−1.07508	−0.537542	−	0.843237i	$-0.680647\pi$
$683$	−28.0965	−1.07508	−0.537542	+	0.843237i	$0.680647\pi$
$684$	−2.82843	−0.108148
$685$	9.63049	0.367962
$686$	18.7402	0.715505
$687$	7.73418	0.295077
$688$	6.58701	0.251127
$689$	−5.58541	−0.212787
$690$	0.0630603	0.00240066
$691$	2.05600	0.0782139	0.0391069	−	0.999235i	$-0.487549\pi$
$691$	2.05600	0.0782139	0.0391069	+	0.999235i	$0.487549\pi$
$692$	−24.4434	−0.929200
$693$	−3.69552	−0.140381
$694$	−31.6356	−1.20087
$695$	4.54071	0.172239
$696$	−2.75057	−0.104260
$697$	0	0
$698$	18.7730	0.710569
$699$	−14.3180	−0.541557
$700$	2.61313	0.0987669
$701$	−25.9654	−0.980701	−0.490351	−	0.871525i	$-0.663131\pi$
$701$	−25.9654	−0.980701	−0.490351	+	0.871525i	$0.663131\pi$
$702$	−1.23463	−0.0465982
$703$	8.65914	0.326586
$704$	−1.41421	−0.0533002
$705$	1.06147	0.0399772
$706$	−6.19681	−0.233220
$707$	3.32729	0.125135
$708$	2.05121	0.0770891
$709$	15.1935	0.570603	0.285301	−	0.958438i	$-0.407906\pi$
$709$	15.1935	0.570603	0.285301	+	0.958438i	$0.407906\pi$
$710$	7.83938	0.294207
$711$	7.27157	0.272705
$712$	15.0775	0.565053
$713$	−0.110896	−0.00415310
$714$	0	0
$715$	1.74603	0.0652980
$716$	−11.8268	−0.441990
$717$	−0.368233	−0.0137519
$718$	5.14649	0.192065
$719$	23.1403	0.862986	0.431493	−	0.902116i	$-0.357987\pi$
$719$	23.1403	0.862986	0.431493	+	0.902116i	$0.357987\pi$
$720$	1.00000	0.0372678
$721$	−33.1572	−1.23484
$722$	11.0000	0.409378
$723$	−9.64431	−0.358676
$724$	14.4912	0.538560
$725$	−2.75057	−0.102154
$726$	−9.00000	−0.334021
$727$	−27.7179	−1.02800	−0.514000	−	0.857790i	$-0.671837\pi$
$727$	−27.7179	−1.02800	−0.514000	+	0.857790i	$0.671837\pi$
$728$	3.22625	0.119573
$729$	1.00000	0.0370370
$730$	11.7711	0.435669
$731$	0	0
$732$	−0.304482	−0.0112540
$733$	−33.8889	−1.25171	−0.625857	−	0.779938i	$-0.715251\pi$
$733$	−33.8889	−1.25171	−0.625857	+	0.779938i	$0.715251\pi$
$734$	−27.9655	−1.03223
$735$	0.171573	0.00632856
$736$	−0.0630603	−0.00232443
$737$	−14.0588	−0.517864
$738$	−5.24718	−0.193151
$739$	39.0053	1.43483	0.717416	−	0.696645i	$-0.245325\pi$
$739$	39.0053	1.43483	0.717416	+	0.696645i	$0.245325\pi$
$740$	−3.06147	−0.112542
$741$	−3.49207	−0.128284
$742$	−11.8216	−0.433986
$743$	−45.9958	−1.68742	−0.843711	−	0.536798i	$-0.819634\pi$
$743$	−45.9958	−1.68742	−0.843711	+	0.536798i	$0.819634\pi$
$744$	−1.75858	−0.0644726
$745$	0.113577	0.00416114
$746$	22.3131	0.816940
$747$	−15.7502	−0.576270
$748$	0	0
$749$	−22.8058	−0.833305
$750$	1.00000	0.0365148
$751$	28.1579	1.02750	0.513748	−	0.857941i	$-0.328257\pi$
$751$	28.1579	1.02750	0.513748	+	0.857941i	$0.328257\pi$
$752$	−1.06147	−0.0387077
$753$	−1.38168	−0.0503512
$754$	−3.39595	−0.123673
$755$	15.5236	0.564961
$756$	−2.61313	−0.0950385
$757$	−5.95149	−0.216311	−0.108155	−	0.994134i	$-0.534494\pi$
$757$	−5.95149	−0.216311	−0.108155	+	0.994134i	$0.534494\pi$
$758$	−27.8968	−1.01326
$759$	0.0891807	0.00323705
$760$	2.82843	0.102598
$761$	−5.03319	−0.182453	−0.0912266	−	0.995830i	$-0.529079\pi$
$761$	−5.03319	−0.182453	−0.0912266	+	0.995830i	$0.529079\pi$
$762$	−2.74097	−0.0992949
$763$	27.7443	1.00441
$764$	−7.93174	−0.286960
$765$	0	0
$766$	1.65494	0.0597956
$767$	2.53249	0.0914429
$768$	−1.00000	−0.0360844
$769$	50.9562	1.83753	0.918764	−	0.394806i	$-0.129188\pi$
$769$	50.9562	1.83753	0.918764	+	0.394806i	$0.129188\pi$
$770$	3.69552	0.133177
$771$	12.2130	0.439842
$772$	7.18343	0.258537
$773$	−22.6128	−0.813325	−0.406663	−	0.913578i	$-0.633308\pi$
$773$	−22.6128	−0.813325	−0.406663	+	0.913578i	$0.633308\pi$
$774$	−6.58701	−0.236765
$775$	−1.75858	−0.0631700
$776$	−2.21530	−0.0795247
$777$	8.00000	0.286998
$778$	32.9771	1.18229
$779$	−14.8413	−0.531743
$780$	1.23463	0.0442070
$781$	11.0866	0.396708
$782$	0	0
$783$	2.75057	0.0982974
$784$	−0.171573	−0.00612760
$785$	−19.6990	−0.703087
$786$	14.8149	0.528428
$787$	−2.13651	−0.0761584	−0.0380792	−	0.999275i	$-0.512124\pi$
$787$	−2.13651	−0.0761584	−0.0380792	+	0.999275i	$0.512124\pi$
$788$	−9.29769	−0.331217
$789$	−11.4985	−0.409357
$790$	−7.27157	−0.258711
$791$	−19.0370	−0.676878
$792$	1.41421	0.0502519
$793$	−0.375923	−0.0133494
$794$	−24.4872	−0.869018
$795$	−4.52395	−0.160448
$796$	−0.454999	−0.0161270
$797$	16.1121	0.570718	0.285359	−	0.958421i	$-0.407887\pi$
$797$	16.1121	0.570718	0.285359	+	0.958421i	$0.407887\pi$
$798$	−7.39104	−0.261640
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−15.0775	−0.532737
$802$	22.5281	0.795495
$803$	16.6469	0.587456
$804$	−9.94110	−0.350596
$805$	0.164784	0.00580789
$806$	−2.17120	−0.0764772
$807$	5.22175	0.183814
$808$	−1.27330	−0.0447944
$809$	−9.38613	−0.329999	−0.164999	−	0.986294i	$-0.552762\pi$
$809$	−9.38613	−0.329999	−0.164999	+	0.986294i	$0.552762\pi$
$810$	−1.00000	−0.0351364
$811$	−5.02355	−0.176401	−0.0882004	−	0.996103i	$-0.528112\pi$
$811$	−5.02355	−0.176401	−0.0882004	+	0.996103i	$0.528112\pi$
$812$	−7.18759	−0.252235
$813$	5.24492	0.183948
$814$	−4.32957	−0.151751
$815$	18.3907	0.644197
$816$	0	0
$817$	−18.6309	−0.651811
$818$	33.7334	1.17946
$819$	−3.22625	−0.112734
$820$	5.24718	0.183239
$821$	16.1517	0.563699	0.281850	−	0.959459i	$-0.409052\pi$
$821$	16.1517	0.563699	0.281850	+	0.959459i	$0.409052\pi$
$822$	9.63049	0.335902
$823$	10.8318	0.377574	0.188787	−	0.982018i	$-0.439544\pi$
$823$	10.8318	0.377574	0.188787	+	0.982018i	$0.439544\pi$
$824$	12.6887	0.442033
$825$	1.41421	0.0492366
$826$	5.36006	0.186500
$827$	43.1559	1.50068	0.750339	−	0.661053i	$-0.229890\pi$
$827$	43.1559	1.50068	0.750339	+	0.661053i	$0.229890\pi$
$828$	0.0630603	0.00219150
$829$	−24.5717	−0.853410	−0.426705	−	0.904391i	$-0.640326\pi$
$829$	−24.5717	−0.853410	−0.426705	+	0.904391i	$0.640326\pi$
$830$	15.7502	0.546697
$831$	−6.67913	−0.231696
$832$	−1.23463	−0.0428032
$833$	0	0
$834$	4.54071	0.157232
$835$	−22.4264	−0.776097
$836$	4.00000	0.138343
$837$	1.75858	0.0607854
$838$	−28.1235	−0.971509
$839$	23.4235	0.808668	0.404334	−	0.914611i	$-0.367503\pi$
$839$	23.4235	0.808668	0.404334	+	0.914611i	$0.367503\pi$
$840$	2.61313	0.0901614
$841$	−21.4344	−0.739116
$842$	−34.5173	−1.18955
$843$	−1.56351	−0.0538502
$844$	9.80562	0.337523
$845$	−11.4757	−0.394775
$846$	1.06147	0.0364940
$847$	−23.5181	−0.808093
$848$	4.52395	0.155353
$849$	8.29143	0.284561
$850$	0	0
$851$	−0.193057	−0.00661791
$852$	7.83938	0.268573
$853$	19.1964	0.657274	0.328637	−	0.944456i	$-0.393411\pi$
$853$	19.1964	0.657274	0.328637	+	0.944456i	$0.393411\pi$
$854$	−0.795649	−0.0272266
$855$	−2.82843	−0.0967302
$856$	8.72739	0.298296
$857$	−5.98295	−0.204374	−0.102187	−	0.994765i	$-0.532584\pi$
$857$	−5.98295	−0.204374	−0.102187	+	0.994765i	$0.532584\pi$
$858$	1.74603	0.0596086
$859$	−42.6250	−1.45435	−0.727173	−	0.686455i	$-0.759166\pi$
$859$	−42.6250	−1.45435	−0.727173	+	0.686455i	$0.759166\pi$
$860$	6.58701	0.224615
$861$	−13.7115	−0.467288
$862$	23.8882	0.813634
$863$	−35.7281	−1.21620	−0.608099	−	0.793861i	$-0.708068\pi$
$863$	−35.7281	−1.21620	−0.608099	+	0.793861i	$0.708068\pi$
$864$	1.00000	0.0340207
$865$	−24.4434	−0.831102
$866$	25.5436	0.868006
$867$	0	0
$868$	−4.59539	−0.155978
$869$	−10.2836	−0.348846
$870$	−2.75057	−0.0932531
$871$	−12.2736	−0.415876
$872$	−10.6173	−0.359547
$873$	2.21530	0.0749766
$874$	0.178361	0.00603316
$875$	2.61313	0.0883398
$876$	11.7711	0.397709
$877$	23.3785	0.789436	0.394718	−	0.918802i	$-0.370842\pi$
$877$	23.3785	0.789436	0.394718	+	0.918802i	$0.370842\pi$
$878$	−35.2548	−1.18979
$879$	−1.44572	−0.0487628
$880$	−1.41421	−0.0476731
$881$	−8.23582	−0.277472	−0.138736	−	0.990329i	$-0.544304\pi$
$881$	−8.23582	−0.277472	−0.138736	+	0.990329i	$0.544304\pi$
$882$	0.171573	0.00577716
$883$	−17.0331	−0.573211	−0.286605	−	0.958049i	$-0.592527\pi$
$883$	−17.0331	−0.573211	−0.286605	+	0.958049i	$0.592527\pi$
$884$	0	0
$885$	2.05121	0.0689506
$886$	33.1991	1.11535
$887$	6.59780	0.221532	0.110766	−	0.993846i	$-0.464670\pi$
$887$	6.59780	0.221532	0.110766	+	0.993846i	$0.464670\pi$
$888$	−3.06147	−0.102736
$889$	−7.16250	−0.240223
$890$	15.0775	0.505398
$891$	−1.41421	−0.0473779
$892$	−4.68873	−0.156990
$893$	3.00228	0.100468
$894$	0.113577	0.00379858
$895$	−11.8268	−0.395327
$896$	−2.61313	−0.0872984
$897$	0.0778563	0.00259955
$898$	17.7428	0.592087
$899$	4.83709	0.161326
$900$	1.00000	0.0333333
$901$	0	0
$902$	7.42063	0.247080
$903$	−17.2127	−0.572802
$904$	7.28515	0.242300
$905$	14.4912	0.481703
$906$	15.5236	0.515736
$907$	−15.5241	−0.515469	−0.257735	−	0.966216i	$-0.582976\pi$
$907$	−15.5241	−0.515469	−0.257735	+	0.966216i	$0.582976\pi$
$908$	−26.6424	−0.884158
$909$	1.27330	0.0422326
$910$	3.22625	0.106949
$911$	0.838476	0.0277799	0.0138900	−	0.999904i	$-0.495579\pi$
$911$	0.838476	0.0277799	0.0138900	+	0.999904i	$0.495579\pi$
$912$	2.82843	0.0936586
$913$	22.2741	0.737167
$914$	−18.7653	−0.620701
$915$	−0.304482	−0.0100659
$916$	−7.73418	−0.255545
$917$	38.7131	1.27842
$918$	0	0
$919$	−5.75736	−0.189918	−0.0949589	−	0.995481i	$-0.530272\pi$
$919$	−5.75736	−0.189918	−0.0949589	+	0.995481i	$0.530272\pi$
$920$	−0.0630603	−0.00207904
$921$	30.0993	0.991804
$922$	−10.2724	−0.338303
$923$	9.67876	0.318580
$924$	3.69552	0.121574
$925$	−3.06147	−0.100660
$926$	−21.6164	−0.710359
$927$	−12.6887	−0.416753
$928$	2.75057	0.0902919
$929$	26.4532	0.867902	0.433951	−	0.900936i	$-0.357119\pi$
$929$	26.4532	0.867902	0.433951	+	0.900936i	$0.357119\pi$
$930$	−1.75858	−0.0576661
$931$	0.485281	0.0159045
$932$	14.3180	0.469002
$933$	−20.2887	−0.664222
$934$	12.6695	0.414560
$935$	0	0
$936$	1.23463	0.0403552
$937$	−32.9392	−1.07608	−0.538038	−	0.842921i	$-0.680834\pi$
$937$	−32.9392	−1.07608	−0.538038	+	0.842921i	$0.680834\pi$
$938$	−25.9774	−0.848191
$939$	−23.7485	−0.775002
$940$	−1.06147	−0.0346213
$941$	−26.5530	−0.865603	−0.432802	−	0.901489i	$-0.642475\pi$
$941$	−26.5530	−0.865603	−0.432802	+	0.901489i	$0.642475\pi$
$942$	−19.6990	−0.641828
$943$	0.330888	0.0107752
$944$	−2.05121	−0.0667611
$945$	−2.61313	−0.0850050
$946$	9.31543	0.302871
$947$	−10.6828	−0.347146	−0.173573	−	0.984821i	$-0.555531\pi$
$947$	−10.6828	−0.347146	−0.173573	+	0.984821i	$0.555531\pi$
$948$	−7.27157	−0.236170
$949$	14.5330	0.471762
$950$	2.82843	0.0917663
$951$	−11.1959	−0.363052
$952$	0	0
$953$	−15.6440	−0.506760	−0.253380	−	0.967367i	$-0.581542\pi$
$953$	−15.6440	−0.506760	−0.253380	+	0.967367i	$0.581542\pi$
$954$	−4.52395	−0.146468
$955$	−7.93174	−0.256665
$956$	0.368233	0.0119095
$957$	−3.88989	−0.125742
$958$	34.8068	1.12456
$959$	25.1657	0.812642
$960$	−1.00000	−0.0322749
$961$	−27.9074	−0.900239
$962$	−3.77979	−0.121865
$963$	−8.72739	−0.281236
$964$	9.64431	0.310622
$965$	7.18343	0.231243
$966$	0.164784	0.00530185
$967$	12.3888	0.398395	0.199198	−	0.979959i	$-0.436166\pi$
$967$	12.3888	0.398395	0.199198	+	0.979959i	$0.436166\pi$
$968$	9.00000	0.289271
$969$	0	0
$970$	−2.21530	−0.0711290
$971$	37.4412	1.20154	0.600772	−	0.799420i	$-0.294860\pi$
$971$	37.4412	1.20154	0.600772	+	0.799420i	$0.294860\pi$
$972$	−1.00000	−0.0320750
$973$	11.8654	0.380389
$974$	22.5441	0.722360
$975$	1.23463	0.0395399
$976$	0.304482	0.00974623
$977$	−32.0857	−1.02651	−0.513255	−	0.858236i	$-0.671561\pi$
$977$	−32.0857	−1.02651	−0.513255	+	0.858236i	$0.671561\pi$
$978$	18.3907	0.588069
$979$	21.3228	0.681479
$980$	−0.171573	−0.00548069
$981$	10.6173	0.338984
$982$	−32.6545	−1.04205
$983$	−4.66017	−0.148636	−0.0743182	−	0.997235i	$-0.523678\pi$
$983$	−4.66017	−0.148636	−0.0743182	+	0.997235i	$0.523678\pi$
$984$	5.24718	0.167274
$985$	−9.29769	−0.296249
$986$	0	0
$987$	2.77375	0.0882894
$988$	3.49207	0.111098
$989$	0.415378	0.0132083
$990$	1.41421	0.0449467
$991$	37.2058	1.18188	0.590940	−	0.806716i	$-0.298757\pi$
$991$	37.2058	1.18188	0.590940	+	0.806716i	$0.298757\pi$
$992$	1.75858	0.0558349
$993$	−13.8854	−0.440640
$994$	20.4853	0.649754
$995$	−0.454999	−0.0144244
$996$	15.7502	0.499064
$997$	−34.2051	−1.08329	−0.541644	−	0.840608i	$-0.682198\pi$
$997$	−34.2051	−1.08329	−0.541644	+	0.840608i	$0.682198\pi$
$998$	8.29016	0.262420
$999$	3.06147	0.0968605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bu.1.4		4
17.10	odd	16		510.2.u.b.151.1	✓	8
17.12	odd	16		510.2.u.b.331.1	yes	8
17.16	even	2		8670.2.a.bv.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.u.b.151.1	✓	8	17.10	odd	16
510.2.u.b.331.1	yes	8	17.12	odd	16
8670.2.a.bu.1.4		4	1.1	even	1	trivial
8670.2.a.bv.1.1		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.41421 q^{11} -1.00000 q^{12} -1.23463 q^{13} -2.61313 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -2.82843 q^{19} +1.00000 q^{20} -2.61313 q^{21} +1.41421 q^{22} +0.0630603 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.23463 q^{26} -1.00000 q^{27} +2.61313 q^{28} -2.75057 q^{29} +1.00000 q^{30} -1.75858 q^{31} -1.00000 q^{32} +1.41421 q^{33} +2.61313 q^{35} +1.00000 q^{36} -3.06147 q^{37} +2.82843 q^{38} +1.23463 q^{39} -1.00000 q^{40} +5.24718 q^{41} +2.61313 q^{42} +6.58701 q^{43} -1.41421 q^{44} +1.00000 q^{45} -0.0630603 q^{46} -1.06147 q^{47} -1.00000 q^{48} -0.171573 q^{49} -1.00000 q^{50} -1.23463 q^{52} +4.52395 q^{53} +1.00000 q^{54} -1.41421 q^{55} -2.61313 q^{56} +2.82843 q^{57} +2.75057 q^{58} -2.05121 q^{59} -1.00000 q^{60} +0.304482 q^{61} +1.75858 q^{62} +2.61313 q^{63} +1.00000 q^{64} -1.23463 q^{65} -1.41421 q^{66} +9.94110 q^{67} -0.0630603 q^{69} -2.61313 q^{70} -7.83938 q^{71} -1.00000 q^{72} -11.7711 q^{73} +3.06147 q^{74} -1.00000 q^{75} -2.82843 q^{76} -3.69552 q^{77} -1.23463 q^{78} +7.27157 q^{79} +1.00000 q^{80} +1.00000 q^{81} -5.24718 q^{82} -15.7502 q^{83} -2.61313 q^{84} -6.58701 q^{86} +2.75057 q^{87} +1.41421 q^{88} -15.0775 q^{89} -1.00000 q^{90} -3.22625 q^{91} +0.0630603 q^{92} +1.75858 q^{93} +1.06147 q^{94} -2.82843 q^{95} +1.00000 q^{96} +2.21530 q^{97} +0.171573 q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{12} - 8 q^{13} - 4 q^{15} + 4 q^{16} - 4 q^{18} + 4 q^{20} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 8 q^{26} - 4 q^{27} - 8 q^{29} + 4 q^{30} + 16 q^{31} - 4 q^{32} + 4 q^{36} + 8 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{46} + 8 q^{47} - 4 q^{48} - 12 q^{49} - 4 q^{50} - 8 q^{52} - 8 q^{53} + 4 q^{54} + 8 q^{58} - 4 q^{60} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 8 q^{65} + 8 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{73} - 4 q^{75} - 8 q^{78} + 4 q^{80} + 4 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{86} + 8 q^{87} - 8 q^{89} - 4 q^{90} + 8 q^{91} - 8 q^{92} - 16 q^{93} - 8 q^{94} + 4 q^{96} + 8 q^{97} + 12 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^10 - 4 * q^12 - 8 * q^13 - 4 * q^15 + 4 * q^16 - 4 * q^18 + 4 * q^20 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 8 * q^26 - 4 * q^27 - 8 * q^29 + 4 * q^30 + 16 * q^31 - 4 * q^32 + 4 * q^36 + 8 * q^39 - 4 * q^40 + 8 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^46 + 8 * q^47 - 4 * q^48 - 12 * q^49 - 4 * q^50 - 8 * q^52 - 8 * q^53 + 4 * q^54 + 8 * q^58 - 4 * q^60 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 8 * q^65 + 8 * q^67 + 8 * q^69 - 4 * q^72 - 8 * q^73 - 4 * q^75 - 8 * q^78 + 4 * q^80 + 4 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^86 + 8 * q^87 - 8 * q^89 - 4 * q^90 + 8 * q^91 - 8 * q^92 - 16 * q^93 - 8 * q^94 + 4 * q^96 + 8 * q^97 + 12 * q^98

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