LMFDB - Embedded newform 8670.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$69.2302985525$
Analytic rank:	$0$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.84776$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.69552 q^{11} +1.00000 q^{12} -5.55807 q^{13} +2.61313 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.99321 q^{19} -1.00000 q^{20} -2.61313 q^{21} -5.69552 q^{22} -6.47568 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.55807 q^{26} +1.00000 q^{27} -2.61313 q^{28} +4.12612 q^{29} +1.00000 q^{30} -8.33598 q^{31} -1.00000 q^{32} +5.69552 q^{33} +2.61313 q^{35} +1.00000 q^{36} +9.15800 q^{37} +4.99321 q^{38} -5.55807 q^{39} +1.00000 q^{40} +3.67459 q^{41} +2.61313 q^{42} -10.5899 q^{43} +5.69552 q^{44} -1.00000 q^{45} +6.47568 q^{46} -0.993212 q^{47} +1.00000 q^{48} -0.171573 q^{49} -1.00000 q^{50} -5.55807 q^{52} -1.06147 q^{53} -1.00000 q^{54} -5.69552 q^{55} +2.61313 q^{56} -4.99321 q^{57} -4.12612 q^{58} +6.23079 q^{59} -1.00000 q^{60} +0.0386639 q^{61} +8.33598 q^{62} -2.61313 q^{63} +1.00000 q^{64} +5.55807 q^{65} -5.69552 q^{66} +8.37049 q^{67} -6.47568 q^{69} -2.61313 q^{70} -1.71644 q^{71} -1.00000 q^{72} +11.0656 q^{73} -9.15800 q^{74} +1.00000 q^{75} -4.99321 q^{76} -14.8831 q^{77} +5.55807 q^{78} -4.76659 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.67459 q^{82} +3.39104 q^{83} -2.61313 q^{84} +10.5899 q^{86} +4.12612 q^{87} -5.69552 q^{88} +15.7115 q^{89} +1.00000 q^{90} +14.5239 q^{91} -6.47568 q^{92} -8.33598 q^{93} +0.993212 q^{94} +4.99321 q^{95} -1.00000 q^{96} +6.70058 q^{97} +0.171573 q^{98} +5.69552 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{11} + 4 q^{12} - 4 q^{15} + 4 q^{16} - 4 q^{18} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27} + 4 q^{30} + 8 q^{31} - 4 q^{32} + 8 q^{33} + 4 q^{36} + 8 q^{37} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 8 q^{44} - 4 q^{45} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 12 q^{49} - 4 q^{50} + 8 q^{53} - 4 q^{54} - 8 q^{55} + 8 q^{59} - 4 q^{60} + 8 q^{61} - 8 q^{62} + 4 q^{64} - 8 q^{66} + 40 q^{67} - 8 q^{69} - 4 q^{72} - 8 q^{73} - 8 q^{74} + 4 q^{75} - 16 q^{77} + 24 q^{79} - 4 q^{80} + 4 q^{81} + 8 q^{82} - 16 q^{83} + 8 q^{86} - 8 q^{88} + 8 q^{89} + 4 q^{90} + 32 q^{91} - 8 q^{92} + 8 q^{93} - 16 q^{94} - 4 q^{96} - 8 q^{97} + 12 q^{98} + 8 q^{99}+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 + 8 * q^11 + 4 * q^12 - 4 * q^15 + 4 * q^16 - 4 * q^18 - 4 * q^20 - 8 * q^22 - 8 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 + 4 * q^30 + 8 * q^31 - 4 * q^32 + 8 * q^33 + 4 * q^36 + 8 * q^37 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 8 * q^44 - 4 * q^45 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^49 - 4 * q^50 + 8 * q^53 - 4 * q^54 - 8 * q^55 + 8 * q^59 - 4 * q^60 + 8 * q^61 - 8 * q^62 + 4 * q^64 - 8 * q^66 + 40 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^73 - 8 * q^74 + 4 * q^75 - 16 * q^77 + 24 * q^79 - 4 * q^80 + 4 * q^81 + 8 * q^82 - 16 * q^83 + 8 * q^86 - 8 * q^88 + 8 * q^89 + 4 * q^90 + 32 * q^91 - 8 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 - 8 * q^97 + 12 * 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
$7$	−2.61313	−0.987669	−0.493834	−	0.869556i	$-0.664405\pi$
$7$	−2.61313	−0.987669	−0.493834	+	0.869556i	$0.664405\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	5.69552	1.71726	0.858632	−	0.512593i	$-0.171315\pi$
$11$	5.69552	1.71726	0.858632	+	0.512593i	$0.171315\pi$
$12$	1.00000	0.288675
$13$	−5.55807	−1.54153	−0.770766	−	0.637118i	$-0.780126\pi$
$13$	−5.55807	−1.54153	−0.770766	+	0.637118i	$0.780126\pi$
$14$	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
$19$	−4.99321	−1.14552	−0.572761	−	0.819723i	$-0.694128\pi$
$19$	−4.99321	−1.14552	−0.572761	+	0.819723i	$0.694128\pi$
$20$	−1.00000	−0.223607
$21$	−2.61313	−0.570231
$22$	−5.69552	−1.21429
$23$	−6.47568	−1.35027	−0.675136	−	0.737693i	$-0.735915\pi$
$23$	−6.47568	−1.35027	−0.675136	+	0.737693i	$0.735915\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	5.55807	1.09003
$27$	1.00000	0.192450
$28$	−2.61313	−0.493834
$29$	4.12612	0.766201	0.383101	−	0.923707i	$-0.374856\pi$
$29$	4.12612	0.766201	0.383101	+	0.923707i	$0.374856\pi$
$30$	1.00000	0.182574
$31$	−8.33598	−1.49719	−0.748593	−	0.663029i	$-0.769270\pi$
$31$	−8.33598	−1.49719	−0.748593	+	0.663029i	$0.769270\pi$
$32$	−1.00000	−0.176777
$33$	5.69552	0.991462
$34$	0	0
$35$	2.61313	0.441699
$36$	1.00000	0.166667
$37$	9.15800	1.50557	0.752783	−	0.658269i	$-0.228711\pi$
$37$	9.15800	1.50557	0.752783	+	0.658269i	$0.228711\pi$
$38$	4.99321	0.810006
$39$	−5.55807	−0.890004
$40$	1.00000	0.158114
$41$	3.67459	0.573875	0.286938	−	0.957949i	$-0.407363\pi$
$41$	3.67459	0.573875	0.286938	+	0.957949i	$0.407363\pi$
$42$	2.61313	0.403214
$43$	−10.5899	−1.61495	−0.807476	−	0.589900i	$-0.799167\pi$
$43$	−10.5899	−1.61495	−0.807476	+	0.589900i	$0.799167\pi$
$44$	5.69552	0.858632
$45$	−1.00000	−0.149071
$46$	6.47568	0.954787
$47$	−0.993212	−0.144875	−0.0724374	−	0.997373i	$-0.523078\pi$
$47$	−0.993212	−0.144875	−0.0724374	+	0.997373i	$0.523078\pi$
$48$	1.00000	0.144338
$49$	−0.171573	−0.0245104
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−5.55807	−0.770766
$53$	−1.06147	−0.145804	−0.0729019	−	0.997339i	$-0.523226\pi$
$53$	−1.06147	−0.145804	−0.0729019	+	0.997339i	$0.523226\pi$
$54$	−1.00000	−0.136083
$55$	−5.69552	−0.767984
$56$	2.61313	0.349194
$57$	−4.99321	−0.661367
$58$	−4.12612	−0.541786
$59$	6.23079	0.811179	0.405590	−	0.914055i	$-0.367066\pi$
$59$	6.23079	0.811179	0.405590	+	0.914055i	$0.367066\pi$
$60$	−1.00000	−0.129099
$61$	0.0386639	0.00495040	0.00247520	−	0.999997i	$-0.499212\pi$
$61$	0.0386639	0.00495040	0.00247520	+	0.999997i	$0.499212\pi$
$62$	8.33598	1.05867
$63$	−2.61313	−0.329223
$64$	1.00000	0.125000
$65$	5.55807	0.689394
$66$	−5.69552	−0.701070
$67$	8.37049	1.02262	0.511309	−	0.859397i	$-0.329161\pi$
$67$	8.37049	1.02262	0.511309	+	0.859397i	$0.329161\pi$
$68$	0	0
$69$	−6.47568	−0.779580
$70$	−2.61313	−0.312328
$71$	−1.71644	−0.203704	−0.101852	−	0.994800i	$-0.532477\pi$
$71$	−1.71644	−0.203704	−0.101852	+	0.994800i	$0.532477\pi$
$72$	−1.00000	−0.117851
$73$	11.0656	1.29513	0.647567	−	0.762009i	$-0.275787\pi$
$73$	11.0656	1.29513	0.647567	+	0.762009i	$0.275787\pi$
$74$	−9.15800	−1.06460
$75$	1.00000	0.115470
$76$	−4.99321	−0.572761
$77$	−14.8831	−1.69609
$78$	5.55807	0.629328
$79$	−4.76659	−0.536283	−0.268141	−	0.963380i	$-0.586409\pi$
$79$	−4.76659	−0.536283	−0.268141	+	0.963380i	$0.586409\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−3.67459	−0.405791
$83$	3.39104	0.372215	0.186107	−	0.982529i	$-0.440413\pi$
$83$	3.39104	0.372215	0.186107	+	0.982529i	$0.440413\pi$
$84$	−2.61313	−0.285115
$85$	0	0
$86$	10.5899	1.14194
$87$	4.12612	0.442367
$88$	−5.69552	−0.607144
$89$	15.7115	1.66542	0.832710	−	0.553710i	$-0.186788\pi$
$89$	15.7115	1.66542	0.832710	+	0.553710i	$0.186788\pi$
$90$	1.00000	0.105409
$91$	14.5239	1.52252
$92$	−6.47568	−0.675136
$93$	−8.33598	−0.864401
$94$	0.993212	0.102442
$95$	4.99321	0.512293
$96$	−1.00000	−0.102062
$97$	6.70058	0.680341	0.340171	−	0.940364i	$-0.389515\pi$
$97$	6.70058	0.680341	0.340171	+	0.940364i	$0.389515\pi$
$98$	0.171573	0.0173315
$99$	5.69552	0.572421
$100$	1.00000	0.100000
$101$	0.636303	0.0633145	0.0316573	−	0.999499i	$-0.489921\pi$
$101$	0.636303	0.0633145	0.0316573	+	0.999499i	$0.489921\pi$
$102$	0	0
$103$	19.5467	1.92600	0.962999	−	0.269504i	$-0.0868598\pi$
$103$	19.5467	1.92600	0.962999	+	0.269504i	$0.0868598\pi$
$104$	5.55807	0.545014
$105$	2.61313	0.255015
$106$	1.06147	0.103099
$107$	−15.8512	−1.53240	−0.766198	−	0.642604i	$-0.777854\pi$
$107$	−15.8512	−1.53240	−0.766198	+	0.642604i	$0.777854\pi$
$108$	1.00000	0.0962250
$109$	2.50793	0.240216	0.120108	−	0.992761i	$-0.461676\pi$
$109$	2.50793	0.240216	0.120108	+	0.992761i	$0.461676\pi$
$110$	5.69552	0.543046
$111$	9.15800	0.869239
$112$	−2.61313	−0.246917
$113$	−6.84391	−0.643821	−0.321911	−	0.946770i	$-0.604325\pi$
$113$	−6.84391	−0.643821	−0.321911	+	0.946770i	$0.604325\pi$
$114$	4.99321	0.467657
$115$	6.47568	0.603860
$116$	4.12612	0.383101
$117$	−5.55807	−0.513844
$118$	−6.23079	−0.573591
$119$	0	0
$120$	1.00000	0.0912871
$121$	21.4389	1.94899
$122$	−0.0386639	−0.00350046
$123$	3.67459	0.331327
$124$	−8.33598	−0.748593
$125$	−1.00000	−0.0894427
$126$	2.61313	0.232796
$127$	−14.5786	−1.29364	−0.646822	−	0.762641i	$-0.723902\pi$
$127$	−14.5786	−1.29364	−0.646822	+	0.762641i	$0.723902\pi$
$128$	−1.00000	−0.0883883
$129$	−10.5899	−0.932393
$130$	−5.55807	−0.487475
$131$	15.3684	1.34274	0.671371	−	0.741121i	$-0.265706\pi$
$131$	15.3684	1.34274	0.671371	+	0.741121i	$0.265706\pi$
$132$	5.69552	0.495731
$133$	13.0479	1.13140
$134$	−8.37049	−0.723100
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	4.66364	0.398442	0.199221	−	0.979955i	$-0.436159\pi$
$137$	4.66364	0.398442	0.199221	+	0.979955i	$0.436159\pi$
$138$	6.47568	0.551247
$139$	−2.85351	−0.242032	−0.121016	−	0.992651i	$-0.538615\pi$
$139$	−2.85351	−0.242032	−0.121016	+	0.992651i	$0.538615\pi$
$140$	2.61313	0.220849
$141$	−0.993212	−0.0836435
$142$	1.71644	0.144041
$143$	−31.6561	−2.64722
$144$	1.00000	0.0833333
$145$	−4.12612	−0.342656
$146$	−11.0656	−0.915798
$147$	−0.171573	−0.0141511
$148$	9.15800	0.752783
$149$	21.2536	1.74116	0.870581	−	0.492025i	$-0.163743\pi$
$149$	21.2536	1.74116	0.870581	+	0.492025i	$0.163743\pi$
$150$	−1.00000	−0.0816497
$151$	−10.3296	−0.840608	−0.420304	−	0.907383i	$-0.638077\pi$
$151$	−10.3296	−0.840608	−0.420304	+	0.907383i	$0.638077\pi$
$152$	4.99321	0.405003
$153$	0	0
$154$	14.8831	1.19931
$155$	8.33598	0.669562
$156$	−5.55807	−0.445002
$157$	21.0456	1.67962	0.839812	−	0.542877i	$-0.182665\pi$
$157$	21.0456	1.67962	0.839812	+	0.542877i	$0.182665\pi$
$158$	4.76659	0.379209
$159$	−1.06147	−0.0841798
$160$	1.00000	0.0790569
$161$	16.9218	1.33362
$162$	−1.00000	−0.0785674
$163$	−11.9150	−0.933253	−0.466627	−	0.884454i	$-0.654531\pi$
$163$	−11.9150	−0.933253	−0.466627	+	0.884454i	$0.654531\pi$
$164$	3.67459	0.286938
$165$	−5.69552	−0.443395
$166$	−3.39104	−0.263195
$167$	−8.97003	−0.694122	−0.347061	−	0.937843i	$-0.612820\pi$
$167$	−8.97003	−0.694122	−0.347061	+	0.937843i	$0.612820\pi$
$168$	2.61313	0.201607
$169$	17.8922	1.37632
$170$	0	0
$171$	−4.99321	−0.381840
$172$	−10.5899	−0.807476
$173$	−21.4206	−1.62858	−0.814290	−	0.580458i	$-0.802873\pi$
$173$	−21.4206	−1.62858	−0.814290	+	0.580458i	$0.802873\pi$
$174$	−4.12612	−0.312800
$175$	−2.61313	−0.197534
$176$	5.69552	0.429316
$177$	6.23079	0.468335
$178$	−15.7115	−1.17763
$179$	−1.86574	−0.139452	−0.0697260	−	0.997566i	$-0.522212\pi$
$179$	−1.86574	−0.139452	−0.0697260	+	0.997566i	$0.522212\pi$
$180$	−1.00000	−0.0745356
$181$	−11.7889	−0.876259	−0.438130	−	0.898912i	$-0.644359\pi$
$181$	−11.7889	−0.876259	−0.438130	+	0.898912i	$0.644359\pi$
$182$	−14.5239	−1.07659
$183$	0.0386639	0.00285812
$184$	6.47568	0.477394
$185$	−9.15800	−0.673309
$186$	8.33598	0.611224
$187$	0	0
$188$	−0.993212	−0.0724374
$189$	−2.61313	−0.190077
$190$	−4.99321	−0.362246
$191$	−12.4912	−0.903829	−0.451915	−	0.892061i	$-0.649259\pi$
$191$	−12.4912	−0.903829	−0.451915	+	0.892061i	$0.649259\pi$
$192$	1.00000	0.0721688
$193$	−4.65498	−0.335072	−0.167536	−	0.985866i	$-0.553581\pi$
$193$	−4.65498	−0.335072	−0.167536	+	0.985866i	$0.553581\pi$
$194$	−6.70058	−0.481074
$195$	5.55807	0.398022
$196$	−0.171573	−0.0122552
$197$	−5.40780	−0.385290	−0.192645	−	0.981269i	$-0.561707\pi$
$197$	−5.40780	−0.385290	−0.192645	+	0.981269i	$0.561707\pi$
$198$	−5.69552	−0.404763
$199$	21.1972	1.50263	0.751314	−	0.659944i	$-0.229420\pi$
$199$	21.1972	1.50263	0.751314	+	0.659944i	$0.229420\pi$
$200$	−1.00000	−0.0707107
$201$	8.37049	0.590409
$202$	−0.636303	−0.0447701
$203$	−10.7821	−0.756753
$204$	0	0
$205$	−3.67459	−0.256645
$206$	−19.5467	−1.36189
$207$	−6.47568	−0.450091
$208$	−5.55807	−0.385383
$209$	−28.4389	−1.96716
$210$	−2.61313	−0.180323
$211$	13.2809	0.914297	0.457148	−	0.889390i	$-0.348871\pi$
$211$	13.2809	0.914297	0.457148	+	0.889390i	$0.348871\pi$
$212$	−1.06147	−0.0729019
$213$	−1.71644	−0.117609
$214$	15.8512	1.08357
$215$	10.5899	0.722229
$216$	−1.00000	−0.0680414
$217$	21.7830	1.47872
$218$	−2.50793	−0.169858
$219$	11.0656	0.747746
$220$	−5.69552	−0.383992
$221$	0	0
$222$	−9.15800	−0.614644
$223$	−25.6105	−1.71501	−0.857503	−	0.514479i	$-0.827985\pi$
$223$	−25.6105	−1.71501	−0.857503	+	0.514479i	$0.827985\pi$
$224$	2.61313	0.174597
$225$	1.00000	0.0666667
$226$	6.84391	0.455250
$227$	14.1480	0.939037	0.469519	−	0.882923i	$-0.344427\pi$
$227$	14.1480	0.939037	0.469519	+	0.882923i	$0.344427\pi$
$228$	−4.99321	−0.330684
$229$	17.0615	1.12745	0.563727	−	0.825961i	$-0.309367\pi$
$229$	17.0615	1.12745	0.563727	+	0.825961i	$0.309367\pi$
$230$	−6.47568	−0.426994
$231$	−14.8831	−0.979236
$232$	−4.12612	−0.270893
$233$	−9.94723	−0.651665	−0.325832	−	0.945428i	$-0.605645\pi$
$233$	−9.94723	−0.651665	−0.325832	+	0.945428i	$0.605645\pi$
$234$	5.55807	0.363343
$235$	0.993212	0.0647900
$236$	6.23079	0.405590
$237$	−4.76659	−0.309623
$238$	0	0
$239$	−1.29769	−0.0839408	−0.0419704	−	0.999119i	$-0.513364\pi$
$239$	−1.29769	−0.0839408	−0.0419704	+	0.999119i	$0.513364\pi$
$240$	−1.00000	−0.0645497
$241$	7.47796	0.481698	0.240849	−	0.970563i	$-0.422574\pi$
$241$	7.47796	0.481698	0.240849	+	0.970563i	$0.422574\pi$
$242$	−21.4389	−1.37815
$243$	1.00000	0.0641500
$244$	0.0386639	0.00247520
$245$	0.171573	0.0109614
$246$	−3.67459	−0.234284
$247$	27.7526	1.76586
$248$	8.33598	0.529335
$249$	3.39104	0.214898
$250$	1.00000	0.0632456
$251$	−8.45476	−0.533659	−0.266830	−	0.963744i	$-0.585976\pi$
$251$	−8.45476	−0.533659	−0.266830	+	0.963744i	$0.585976\pi$
$252$	−2.61313	−0.164611
$253$	−36.8824	−2.31877
$254$	14.5786	0.914744
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.6150	1.34831	0.674153	−	0.738591i	$-0.264509\pi$
$257$	21.6150	1.34831	0.674153	+	0.738591i	$0.264509\pi$
$258$	10.5899	0.659301
$259$	−23.9310	−1.48700
$260$	5.55807	0.344697
$261$	4.12612	0.255400
$262$	−15.3684	−0.949462
$263$	26.8740	1.65712	0.828562	−	0.559897i	$-0.189159\pi$
$263$	26.8740	1.65712	0.828562	+	0.559897i	$0.189159\pi$
$264$	−5.69552	−0.350535
$265$	1.06147	0.0652054
$266$	−13.0479	−0.800017
$267$	15.7115	0.961530
$268$	8.37049	0.511309
$269$	14.0502	0.856654	0.428327	−	0.903624i	$-0.359103\pi$
$269$	14.0502	0.856654	0.428327	+	0.903624i	$0.359103\pi$
$270$	1.00000	0.0608581
$271$	−3.65685	−0.222138	−0.111069	−	0.993813i	$-0.535427\pi$
$271$	−3.65685	−0.222138	−0.111069	+	0.993813i	$0.535427\pi$
$272$	0	0
$273$	14.5239	0.879029
$274$	−4.66364	−0.281741
$275$	5.69552	0.343453
$276$	−6.47568	−0.389790
$277$	−5.48438	−0.329524	−0.164762	−	0.986333i	$-0.552686\pi$
$277$	−5.48438	−0.329524	−0.164762	+	0.986333i	$0.552686\pi$
$278$	2.85351	0.171142
$279$	−8.33598	−0.499062
$280$	−2.61313	−0.156164
$281$	−13.1161	−0.782444	−0.391222	−	0.920296i	$-0.627947\pi$
$281$	−13.1161	−0.782444	−0.391222	+	0.920296i	$0.627947\pi$
$282$	0.993212	0.0591449
$283$	21.2186	1.26131	0.630656	−	0.776062i	$-0.282786\pi$
$283$	21.2186	1.26131	0.630656	+	0.776062i	$0.282786\pi$
$284$	−1.71644	−0.101852
$285$	4.99321	0.295772
$286$	31.6561	1.87186
$287$	−9.60218	−0.566798
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	4.12612	0.242294
$291$	6.70058	0.392795
$292$	11.0656	0.647567
$293$	7.57953	0.442801	0.221400	−	0.975183i	$-0.428937\pi$
$293$	7.57953	0.442801	0.221400	+	0.975183i	$0.428937\pi$
$294$	0.171573	0.0100063
$295$	−6.23079	−0.362770
$296$	−9.15800	−0.532298
$297$	5.69552	0.330487
$298$	−21.2536	−1.23119
$299$	35.9923	2.08149
$300$	1.00000	0.0577350
$301$	27.6729	1.59504
$302$	10.3296	0.594400
$303$	0.636303	0.0365547
$304$	−4.99321	−0.286380
$305$	−0.0386639	−0.00221389
$306$	0	0
$307$	12.8558	0.733717	0.366859	−	0.930277i	$-0.380433\pi$
$307$	12.8558	0.733717	0.366859	+	0.930277i	$0.380433\pi$
$308$	−14.8831	−0.848044
$309$	19.5467	1.11198
$310$	−8.33598	−0.473452
$311$	15.4088	0.873751	0.436876	−	0.899522i	$-0.356085\pi$
$311$	15.4088	0.873751	0.436876	+	0.899522i	$0.356085\pi$
$312$	5.55807	0.314664
$313$	−0.00416219	−0.000235261 0	−0.000117630	−	1.00000i	$-0.500037\pi$
$313$	−0.00416219	−0.000235261 0	−0.000117630		1.00000i	$0.500037\pi$
$314$	−21.0456	−1.18767
$315$	2.61313	0.147233
$316$	−4.76659	−0.268141
$317$	−11.6401	−0.653773	−0.326886	−	0.945064i	$-0.605999\pi$
$317$	−11.6401	−0.653773	−0.326886	+	0.945064i	$0.605999\pi$
$318$	1.06147	0.0595241
$319$	23.5004	1.31577
$320$	−1.00000	−0.0559017
$321$	−15.8512	−0.884729
$322$	−16.9218	−0.943013
$323$	0	0
$324$	1.00000	0.0555556
$325$	−5.55807	−0.308306
$326$	11.9150	0.659910
$327$	2.50793	0.138689
$328$	−3.67459	−0.202895
$329$	2.59539	0.143088
$330$	5.69552	0.313528
$331$	16.0965	0.884745	0.442373	−	0.896831i	$-0.354137\pi$
$331$	16.0965	0.884745	0.442373	+	0.896831i	$0.354137\pi$
$332$	3.39104	0.186107
$333$	9.15800	0.501855
$334$	8.97003	0.490818
$335$	−8.37049	−0.457328
$336$	−2.61313	−0.142558
$337$	11.9623	0.651629	0.325814	−	0.945434i	$-0.394362\pi$
$337$	11.9623	0.651629	0.325814	+	0.945434i	$0.394362\pi$
$338$	−17.8922	−0.973206
$339$	−6.84391	−0.371710
$340$	0	0
$341$	−47.4777	−2.57106
$342$	4.99321	0.270002
$343$	18.7402	1.01188
$344$	10.5899	0.570972
$345$	6.47568	0.348639
$346$	21.4206	1.15158
$347$	−19.5590	−1.04998	−0.524991	−	0.851108i	$-0.675931\pi$
$347$	−19.5590	−1.04998	−0.524991	+	0.851108i	$0.675931\pi$
$348$	4.12612	0.221183
$349$	12.1648	0.651166	0.325583	−	0.945513i	$-0.394439\pi$
$349$	12.1648	0.651166	0.325583	+	0.945513i	$0.394439\pi$
$350$	2.61313	0.139677
$351$	−5.55807	−0.296668
$352$	−5.69552	−0.303572
$353$	−20.0185	−1.06547	−0.532737	−	0.846281i	$-0.678837\pi$
$353$	−20.0185	−1.06547	−0.532737	+	0.846281i	$0.678837\pi$
$354$	−6.23079	−0.331163
$355$	1.71644	0.0910993
$356$	15.7115	0.832710
$357$	0	0
$358$	1.86574	0.0986075
$359$	2.47515	0.130634	0.0653168	−	0.997865i	$-0.479194\pi$
$359$	2.47515	0.130634	0.0653168	+	0.997865i	$0.479194\pi$
$360$	1.00000	0.0527046
$361$	5.93216	0.312219
$362$	11.7889	0.619609
$363$	21.4389	1.12525
$364$	14.5239	0.761262
$365$	−11.0656	−0.579202
$366$	−0.0386639	−0.00202099
$367$	−7.61084	−0.397283	−0.198641	−	0.980072i	$-0.563653\pi$
$367$	−7.61084	−0.397283	−0.198641	+	0.980072i	$0.563653\pi$
$368$	−6.47568	−0.337568
$369$	3.67459	0.191292
$370$	9.15800	0.476102
$371$	2.77375	0.144006
$372$	−8.33598	−0.432201
$373$	12.0332	0.623057	0.311528	−	0.950237i	$-0.399159\pi$
$373$	12.0332	0.623057	0.311528	+	0.950237i	$0.399159\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0.993212	0.0512210
$377$	−22.9333	−1.18112
$378$	2.61313	0.134405
$379$	30.9437	1.58947	0.794735	−	0.606957i	$-0.207610\pi$
$379$	30.9437	1.58947	0.794735	+	0.606957i	$0.207610\pi$
$380$	4.99321	0.256146
$381$	−14.5786	−0.746886
$382$	12.4912	0.639104
$383$	23.6878	1.21039	0.605196	−	0.796077i	$-0.293095\pi$
$383$	23.6878	1.21039	0.605196	+	0.796077i	$0.293095\pi$
$384$	−1.00000	−0.0510310
$385$	14.8831	0.758513
$386$	4.65498	0.236932
$387$	−10.5899	−0.538317
$388$	6.70058	0.340171
$389$	23.4376	1.18833	0.594166	−	0.804342i	$-0.297482\pi$
$389$	23.4376	1.18833	0.594166	+	0.804342i	$0.297482\pi$
$390$	−5.55807	−0.281444
$391$	0	0
$392$	0.171573	0.00866574
$393$	15.3684	0.775233
$394$	5.40780	0.272441
$395$	4.76659	0.239833
$396$	5.69552	0.286211
$397$	−8.03792	−0.403411	−0.201706	−	0.979446i	$-0.564648\pi$
$397$	−8.03792	−0.403411	−0.201706	+	0.979446i	$0.564648\pi$
$398$	−21.1972	−1.06252
$399$	13.0479	0.653212
$400$	1.00000	0.0500000
$401$	−18.4675	−0.922225	−0.461113	−	0.887342i	$-0.652550\pi$
$401$	−18.4675	−0.922225	−0.461113	+	0.887342i	$0.652550\pi$
$402$	−8.37049	−0.417482
$403$	46.3320	2.30796
$404$	0.636303	0.0316573
$405$	−1.00000	−0.0496904
$406$	10.7821	0.535105
$407$	52.1595	2.58545
$408$	0	0
$409$	4.64174	0.229519	0.114760	−	0.993393i	$-0.463390\pi$
$409$	4.64174	0.229519	0.114760	+	0.993393i	$0.463390\pi$
$410$	3.67459	0.181475
$411$	4.66364	0.230041
$412$	19.5467	0.962999
$413$	−16.2818	−0.801177
$414$	6.47568	0.318262
$415$	−3.39104	−0.166459
$416$	5.55807	0.272507
$417$	−2.85351	−0.139737
$418$	28.4389	1.39099
$419$	13.7374	0.671114	0.335557	−	0.942020i	$-0.391075\pi$
$419$	13.7374	0.671114	0.335557	+	0.942020i	$0.391075\pi$
$420$	2.61313	0.127507
$421$	6.33407	0.308704	0.154352	−	0.988016i	$-0.450671\pi$
$421$	6.33407	0.308704	0.154352	+	0.988016i	$0.450671\pi$
$422$	−13.2809	−0.646506
$423$	−0.993212	−0.0482916
$424$	1.06147	0.0515494
$425$	0	0
$426$	1.71644	0.0831619
$427$	−0.101034	−0.00488936
$428$	−15.8512	−0.766198
$429$	−31.6561	−1.52837
$430$	−10.5899	−0.510693
$431$	13.5199	0.651232	0.325616	−	0.945502i	$-0.394428\pi$
$431$	13.5199	0.651232	0.325616	+	0.945502i	$0.394428\pi$
$432$	1.00000	0.0481125
$433$	41.1058	1.97542	0.987708	−	0.156309i	$-0.0499594\pi$
$433$	41.1058	1.97542	0.987708	+	0.156309i	$0.0499594\pi$
$434$	−21.7830	−1.04562
$435$	−4.12612	−0.197832
$436$	2.50793	0.120108
$437$	32.3344	1.54677
$438$	−11.0656	−0.528736
$439$	−22.9860	−1.09706	−0.548532	−	0.836129i	$-0.684813\pi$
$439$	−22.9860	−1.09706	−0.548532	+	0.836129i	$0.684813\pi$
$440$	5.69552	0.271523
$441$	−0.171573	−0.00817014
$442$	0	0
$443$	20.8922	0.992617	0.496309	−	0.868146i	$-0.334688\pi$
$443$	20.8922	0.992617	0.496309	+	0.868146i	$0.334688\pi$
$444$	9.15800	0.434619
$445$	−15.7115	−0.744798
$446$	25.6105	1.21269
$447$	21.2536	1.00526
$448$	−2.61313	−0.123459
$449$	12.9974	0.613384	0.306692	−	0.951809i	$-0.400778\pi$
$449$	12.9974	0.613384	0.306692	+	0.951809i	$0.400778\pi$
$450$	−1.00000	−0.0471405
$451$	20.9287	0.985495
$452$	−6.84391	−0.321911
$453$	−10.3296	−0.485325
$454$	−14.1480	−0.664000
$455$	−14.5239	−0.680893
$456$	4.99321	0.233829
$457$	8.38106	0.392050	0.196025	−	0.980599i	$-0.437197\pi$
$457$	8.38106	0.392050	0.196025	+	0.980599i	$0.437197\pi$
$458$	−17.0615	−0.797230
$459$	0	0
$460$	6.47568	0.301930
$461$	−37.8709	−1.76382	−0.881911	−	0.471415i	$-0.843743\pi$
$461$	−37.8709	−1.76382	−0.881911	+	0.471415i	$0.843743\pi$
$462$	14.8831	0.692425
$463$	19.1580	0.890348	0.445174	−	0.895444i	$-0.353142\pi$
$463$	19.1580	0.890348	0.445174	+	0.895444i	$0.353142\pi$
$464$	4.12612	0.191550
$465$	8.33598	0.386572
$466$	9.94723	0.460797
$467$	11.2398	0.520117	0.260059	−	0.965593i	$-0.416258\pi$
$467$	11.2398	0.520117	0.260059	+	0.965593i	$0.416258\pi$
$468$	−5.55807	−0.256922
$469$	−21.8731	−1.01001
$470$	−0.993212	−0.0458134
$471$	21.0456	0.969732
$472$	−6.23079	−0.286795
$473$	−60.3152	−2.77330
$474$	4.76659	0.218937
$475$	−4.99321	−0.229104
$476$	0	0
$477$	−1.06147	−0.0486013
$478$	1.29769	0.0593551
$479$	−3.54529	−0.161988	−0.0809942	−	0.996715i	$-0.525810\pi$
$479$	−3.54529	−0.161988	−0.0809942	+	0.996715i	$0.525810\pi$
$480$	1.00000	0.0456435
$481$	−50.9008	−2.32088
$482$	−7.47796	−0.340612
$483$	16.9218	0.769967
$484$	21.4389	0.974497
$485$	−6.70058	−0.304258
$486$	−1.00000	−0.0453609
$487$	−13.5928	−0.615947	−0.307973	−	0.951395i	$-0.599651\pi$
$487$	−13.5928	−0.615947	−0.307973	+	0.951395i	$0.599651\pi$
$488$	−0.0386639	−0.00175023
$489$	−11.9150	−0.538814
$490$	−0.171573	−0.00775087
$491$	−3.20480	−0.144631	−0.0723153	−	0.997382i	$-0.523039\pi$
$491$	−3.20480	−0.144631	−0.0723153	+	0.997382i	$0.523039\pi$
$492$	3.67459	0.165663
$493$	0	0
$494$	−27.7526	−1.24865
$495$	−5.69552	−0.255995
$496$	−8.33598	−0.374297
$497$	4.48528	0.201192
$498$	−3.39104	−0.151956
$499$	33.1263	1.48294	0.741468	−	0.670988i	$-0.234130\pi$
$499$	33.1263	1.48294	0.741468	+	0.670988i	$0.234130\pi$
$500$	−1.00000	−0.0447214
$501$	−8.97003	−0.400752
$502$	8.45476	0.377354
$503$	−24.1635	−1.07740	−0.538699	−	0.842498i	$-0.681084\pi$
$503$	−24.1635	−1.07740	−0.538699	+	0.842498i	$0.681084\pi$
$504$	2.61313	0.116398
$505$	−0.636303	−0.0283151
$506$	36.8824	1.63962
$507$	17.8922	0.794620
$508$	−14.5786	−0.646822
$509$	17.4557	0.773708	0.386854	−	0.922141i	$-0.373562\pi$
$509$	17.4557	0.773708	0.386854	+	0.922141i	$0.373562\pi$
$510$	0	0
$511$	−28.9159	−1.27916
$512$	−1.00000	−0.0441942
$513$	−4.99321	−0.220456
$514$	−21.6150	−0.953397
$515$	−19.5467	−0.861333
$516$	−10.5899	−0.466196
$517$	−5.65685	−0.248788
$518$	23.9310	1.05147
$519$	−21.4206	−0.940261
$520$	−5.55807	−0.243738
$521$	7.29867	0.319761	0.159880	−	0.987136i	$-0.448889\pi$
$521$	7.29867	0.319761	0.159880	+	0.987136i	$0.448889\pi$
$522$	−4.12612	−0.180595
$523$	−3.60687	−0.157717	−0.0788586	−	0.996886i	$-0.525128\pi$
$523$	−3.60687	−0.157717	−0.0788586	+	0.996886i	$0.525128\pi$
$524$	15.3684	0.671371
$525$	−2.61313	−0.114046
$526$	−26.8740	−1.17176
$527$	0	0
$528$	5.69552	0.247866
$529$	18.9344	0.823237
$530$	−1.06147	−0.0461072
$531$	6.23079	0.270393
$532$	13.0479	0.565698
$533$	−20.4237	−0.884647
$534$	−15.7115	−0.679905
$535$	15.8512	0.685308
$536$	−8.37049	−0.361550
$537$	−1.86574	−0.0805127
$538$	−14.0502	−0.605746
$539$	−0.977196	−0.0420908
$540$	−1.00000	−0.0430331
$541$	−10.4912	−0.451051	−0.225525	−	0.974237i	$-0.572410\pi$
$541$	−10.4912	−0.451051	−0.225525	+	0.974237i	$0.572410\pi$
$542$	3.65685	0.157075
$543$	−11.7889	−0.505908
$544$	0	0
$545$	−2.50793	−0.107428
$546$	−14.5239	−0.621567
$547$	1.95226	0.0834728	0.0417364	−	0.999129i	$-0.486711\pi$
$547$	1.95226	0.0834728	0.0417364	+	0.999129i	$0.486711\pi$
$548$	4.66364	0.199221
$549$	0.0386639	0.00165013
$550$	−5.69552	−0.242858
$551$	−20.6026	−0.877700
$552$	6.47568	0.275623
$553$	12.4557	0.529670
$554$	5.48438	0.233009
$555$	−9.15800	−0.388735
$556$	−2.85351	−0.121016
$557$	18.5400	0.785563	0.392782	−	0.919632i	$-0.371513\pi$
$557$	18.5400	0.785563	0.392782	+	0.919632i	$0.371513\pi$
$558$	8.33598	0.352890
$559$	58.8597	2.48950
$560$	2.61313	0.110425
$561$	0	0
$562$	13.1161	0.553271
$563$	−25.4450	−1.07238	−0.536189	−	0.844098i	$-0.680136\pi$
$563$	−25.4450	−1.07238	−0.536189	+	0.844098i	$0.680136\pi$
$564$	−0.993212	−0.0418217
$565$	6.84391	0.287926
$566$	−21.2186	−0.891883
$567$	−2.61313	−0.109741
$568$	1.71644	0.0720203
$569$	−22.4995	−0.943228	−0.471614	−	0.881805i	$-0.656328\pi$
$569$	−22.4995	−0.943228	−0.471614	+	0.881805i	$0.656328\pi$
$570$	−4.99321	−0.209143
$571$	8.95725	0.374849	0.187425	−	0.982279i	$-0.439986\pi$
$571$	8.95725	0.374849	0.187425	+	0.982279i	$0.439986\pi$
$572$	−31.6561	−1.32361
$573$	−12.4912	−0.521826
$574$	9.60218	0.400787
$575$	−6.47568	−0.270055
$576$	1.00000	0.0416667
$577$	−35.6676	−1.48486	−0.742431	−	0.669922i	$-0.766327\pi$
$577$	−35.6676	−1.48486	−0.742431	+	0.669922i	$0.766327\pi$
$578$	0	0
$579$	−4.65498	−0.193454
$580$	−4.12612	−0.171328
$581$	−8.86120	−0.367625
$582$	−6.70058	−0.277748
$583$	−6.04561	−0.250383
$584$	−11.0656	−0.457899
$585$	5.55807	0.229798
$586$	−7.57953	−0.313107
$587$	−22.7357	−0.938404	−0.469202	−	0.883091i	$-0.655458\pi$
$587$	−22.7357	−0.938404	−0.469202	+	0.883091i	$0.655458\pi$
$588$	−0.171573	−0.00707555
$589$	41.6233	1.71506
$590$	6.23079	0.256517
$591$	−5.40780	−0.222447
$592$	9.15800	0.376391
$593$	15.1844	0.623549	0.311774	−	0.950156i	$-0.399077\pi$
$593$	15.1844	0.623549	0.311774	+	0.950156i	$0.399077\pi$
$594$	−5.69552	−0.233690
$595$	0	0
$596$	21.2536	0.870581
$597$	21.1972	0.867543
$598$	−35.9923	−1.47183
$599$	26.3205	1.07543	0.537713	−	0.843128i	$-0.319288\pi$
$599$	26.3205	1.07543	0.537713	+	0.843128i	$0.319288\pi$
$600$	−1.00000	−0.0408248
$601$	37.6156	1.53437	0.767186	−	0.641425i	$-0.221656\pi$
$601$	37.6156	1.53437	0.767186	+	0.641425i	$0.221656\pi$
$602$	−27.6729	−1.12786
$603$	8.37049	0.340873
$604$	−10.3296	−0.420304
$605$	−21.4389	−0.871616
$606$	−0.636303	−0.0258481
$607$	23.3360	0.947177	0.473589	−	0.880746i	$-0.342958\pi$
$607$	23.3360	0.947177	0.473589	+	0.880746i	$0.342958\pi$
$608$	4.99321	0.202501
$609$	−10.7821	−0.436912
$610$	0.0386639	0.00156545
$611$	5.52034	0.223329
$612$	0	0
$613$	23.2006	0.937064	0.468532	−	0.883447i	$-0.344783\pi$
$613$	23.2006	0.937064	0.468532	+	0.883447i	$0.344783\pi$
$614$	−12.8558	−0.518817
$615$	−3.67459	−0.148174
$616$	14.8831	0.599657
$617$	−36.5245	−1.47042	−0.735210	−	0.677839i	$-0.762916\pi$
$617$	−36.5245	−1.47042	−0.735210	+	0.677839i	$0.762916\pi$
$618$	−19.5467	−0.786286
$619$	5.26173	0.211487	0.105743	−	0.994393i	$-0.466278\pi$
$619$	5.26173	0.211487	0.105743	+	0.994393i	$0.466278\pi$
$620$	8.33598	0.334781
$621$	−6.47568	−0.259860
$622$	−15.4088	−0.617836
$623$	−41.0562	−1.64488
$624$	−5.55807	−0.222501
$625$	1.00000	0.0400000
$626$	0.00416219	0.000166355 0
$627$	−28.4389	−1.13574
$628$	21.0456	0.839812
$629$	0	0
$630$	−2.61313	−0.104109
$631$	−34.4344	−1.37081	−0.685406	−	0.728161i	$-0.740375\pi$
$631$	−34.4344	−1.37081	−0.685406	+	0.728161i	$0.740375\pi$
$632$	4.76659	0.189605
$633$	13.2809	0.527870
$634$	11.6401	0.462287
$635$	14.5786	0.578535
$636$	−1.06147	−0.0420899
$637$	0.953615	0.0377836
$638$	−23.5004	−0.930390
$639$	−1.71644	−0.0679014
$640$	1.00000	0.0395285
$641$	−25.0984	−0.991327	−0.495664	−	0.868515i	$-0.665075\pi$
$641$	−25.0984	−0.991327	−0.495664	+	0.868515i	$0.665075\pi$
$642$	15.8512	0.625598
$643$	−13.0556	−0.514862	−0.257431	−	0.966297i	$-0.582876\pi$
$643$	−13.0556	−0.514862	−0.257431	+	0.966297i	$0.582876\pi$
$644$	16.9218	0.666811
$645$	10.5899	0.416979
$646$	0	0
$647$	5.31265	0.208862	0.104431	−	0.994532i	$-0.466698\pi$
$647$	5.31265	0.208862	0.104431	+	0.994532i	$0.466698\pi$
$648$	−1.00000	−0.0392837
$649$	35.4876	1.39301
$650$	5.55807	0.218006
$651$	21.7830	0.853742
$652$	−11.9150	−0.466627
$653$	12.8141	0.501455	0.250727	−	0.968058i	$-0.419330\pi$
$653$	12.8141	0.501455	0.250727	+	0.968058i	$0.419330\pi$
$654$	−2.50793	−0.0980678
$655$	−15.3684	−0.600493
$656$	3.67459	0.143469
$657$	11.0656	0.431711
$658$	−2.59539	−0.101179
$659$	4.03053	0.157007	0.0785035	−	0.996914i	$-0.474986\pi$
$659$	4.03053	0.157007	0.0785035	+	0.996914i	$0.474986\pi$
$660$	−5.69552	−0.221698
$661$	−5.64328	−0.219498	−0.109749	−	0.993959i	$-0.535005\pi$
$661$	−5.64328	−0.219498	−0.109749	+	0.993959i	$0.535005\pi$
$662$	−16.0965	−0.625609
$663$	0	0
$664$	−3.39104	−0.131598
$665$	−13.0479	−0.505975
$666$	−9.15800	−0.354865
$667$	−26.7194	−1.03458
$668$	−8.97003	−0.347061
$669$	−25.6105	−0.990159
$670$	8.37049	0.323380
$671$	0.220211	0.00850114
$672$	2.61313	0.100804
$673$	−12.7560	−0.491708	−0.245854	−	0.969307i	$-0.579068\pi$
$673$	−12.7560	−0.491708	−0.245854	+	0.969307i	$0.579068\pi$
$674$	−11.9623	−0.460771
$675$	1.00000	0.0384900
$676$	17.8922	0.688161
$677$	36.4086	1.39930	0.699648	−	0.714488i	$-0.253340\pi$
$677$	36.4086	1.39930	0.699648	+	0.714488i	$0.253340\pi$
$678$	6.84391	0.262839
$679$	−17.5095	−0.671952
$680$	0	0
$681$	14.1480	0.542153
$682$	47.4777	1.81802
$683$	22.2051	0.849656	0.424828	−	0.905274i	$-0.360335\pi$
$683$	22.2051	0.849656	0.424828	+	0.905274i	$0.360335\pi$
$684$	−4.99321	−0.190920
$685$	−4.66364	−0.178189
$686$	−18.7402	−0.715505
$687$	17.0615	0.650936
$688$	−10.5899	−0.403738
$689$	5.89971	0.224761
$690$	−6.47568	−0.246525
$691$	−39.1580	−1.48964	−0.744820	−	0.667265i	$-0.767465\pi$
$691$	−39.1580	−1.48964	−0.744820	+	0.667265i	$0.767465\pi$
$692$	−21.4206	−0.814290
$693$	−14.8831	−0.565362
$694$	19.5590	0.742450
$695$	2.85351	0.108240
$696$	−4.12612	−0.156400
$697$	0	0
$698$	−12.1648	−0.460444
$699$	−9.94723	−0.376239
$700$	−2.61313	−0.0987669
$701$	1.18608	0.0447977	0.0223989	−	0.999749i	$-0.492870\pi$
$701$	1.18608	0.0447977	0.0223989	+	0.999749i	$0.492870\pi$
$702$	5.55807	0.209776
$703$	−45.7278	−1.72466
$704$	5.69552	0.214658
$705$	0.993212	0.0374065
$706$	20.0185	0.753404
$707$	−1.66274	−0.0625338
$708$	6.23079	0.234167
$709$	6.72421	0.252533	0.126266	−	0.991996i	$-0.459701\pi$
$709$	6.72421	0.252533	0.126266	+	0.991996i	$0.459701\pi$
$710$	−1.71644	−0.0644170
$711$	−4.76659	−0.178761
$712$	−15.7115	−0.588815
$713$	53.9812	2.02161
$714$	0	0
$715$	31.6561	1.18387
$716$	−1.86574	−0.0697260
$717$	−1.29769	−0.0484632
$718$	−2.47515	−0.0923719
$719$	−29.6237	−1.10478	−0.552389	−	0.833586i	$-0.686284\pi$
$719$	−29.6237	−1.10478	−0.552389	+	0.833586i	$0.686284\pi$
$720$	−1.00000	−0.0372678
$721$	−51.0781	−1.90225
$722$	−5.93216	−0.220772
$723$	7.47796	0.278108
$724$	−11.7889	−0.438130
$725$	4.12612	0.153240
$726$	−21.4389	−0.795673
$727$	4.70793	0.174607	0.0873037	−	0.996182i	$-0.472175\pi$
$727$	4.70793	0.174607	0.0873037	+	0.996182i	$0.472175\pi$
$728$	−14.5239	−0.538293
$729$	1.00000	0.0370370
$730$	11.0656	0.409557
$731$	0	0
$732$	0.0386639	0.00142906
$733$	45.7403	1.68945	0.844727	−	0.535197i	$-0.179763\pi$
$733$	45.7403	1.68945	0.844727	+	0.535197i	$0.179763\pi$
$734$	7.61084	0.280921
$735$	0.171573	0.00632856
$736$	6.47568	0.238697
$737$	47.6743	1.75610
$738$	−3.67459	−0.135264
$739$	17.9627	0.660769	0.330385	−	0.943846i	$-0.392822\pi$
$739$	17.9627	0.660769	0.330385	+	0.943846i	$0.392822\pi$
$740$	−9.15800	−0.336655
$741$	27.7526	1.01952
$742$	−2.77375	−0.101827
$743$	−1.85839	−0.0681778	−0.0340889	−	0.999419i	$-0.510853\pi$
$743$	−1.85839	−0.0681778	−0.0340889	+	0.999419i	$0.510853\pi$
$744$	8.33598	0.305612
$745$	−21.2536	−0.778671
$746$	−12.0332	−0.440568
$747$	3.39104	0.124072
$748$	0	0
$749$	41.4213	1.51350
$750$	1.00000	0.0365148
$751$	−0.856743	−0.0312630	−0.0156315	−	0.999878i	$-0.504976\pi$
$751$	−0.856743	−0.0312630	−0.0156315	+	0.999878i	$0.504976\pi$
$752$	−0.993212	−0.0362187
$753$	−8.45476	−0.308108
$754$	22.9333	0.835181
$755$	10.3296	0.375931
$756$	−2.61313	−0.0950385
$757$	21.9068	0.796218	0.398109	−	0.917338i	$-0.369667\pi$
$757$	21.9068	0.796218	0.398109	+	0.917338i	$0.369667\pi$
$758$	−30.9437	−1.12392
$759$	−36.8824	−1.33874
$760$	−4.99321	−0.181123
$761$	−6.39013	−0.231642	−0.115821	−	0.993270i	$-0.536950\pi$
$761$	−6.39013	−0.231642	−0.115821	+	0.993270i	$0.536950\pi$
$762$	14.5786	0.528128
$763$	−6.55354	−0.237254
$764$	−12.4912	−0.451915
$765$	0	0
$766$	−23.6878	−0.855876
$767$	−34.6312	−1.25046
$768$	1.00000	0.0360844
$769$	5.77873	0.208386	0.104193	−	0.994557i	$-0.466774\pi$
$769$	5.77873	0.208386	0.104193	+	0.994557i	$0.466774\pi$
$770$	−14.8831	−0.536350
$771$	21.6150	0.778445
$772$	−4.65498	−0.167536
$773$	40.2696	1.44840	0.724199	−	0.689591i	$-0.242210\pi$
$773$	40.2696	1.44840	0.724199	+	0.689591i	$0.242210\pi$
$774$	10.5899	0.380648
$775$	−8.33598	−0.299437
$776$	−6.70058	−0.240537
$777$	−23.9310	−0.858520
$778$	−23.4376	−0.840278
$779$	−18.3480	−0.657386
$780$	5.55807	0.199011
$781$	−9.77603	−0.349814
$782$	0	0
$783$	4.12612	0.147456
$784$	−0.171573	−0.00612760
$785$	−21.0456	−0.751151
$786$	−15.3684	−0.548172
$787$	6.22397	0.221861	0.110930	−	0.993828i	$-0.464617\pi$
$787$	6.22397	0.221861	0.110930	+	0.993828i	$0.464617\pi$
$788$	−5.40780	−0.192645
$789$	26.8740	0.956741
$790$	−4.76659	−0.169588
$791$	17.8840	0.635882
$792$	−5.69552	−0.202381
$793$	−0.214897	−0.00763120
$794$	8.03792	0.285255
$795$	1.06147	0.0376464
$796$	21.1972	0.751314
$797$	−10.5709	−0.374442	−0.187221	−	0.982318i	$-0.559948\pi$
$797$	−10.5709	−0.374442	−0.187221	+	0.982318i	$0.559948\pi$
$798$	−13.0479	−0.461890
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	15.7115	0.555140
$802$	18.4675	0.652112
$803$	63.0245	2.22409
$804$	8.37049	0.295204
$805$	−16.9218	−0.596414
$806$	−46.3320	−1.63198
$807$	14.0502	0.494590
$808$	−0.636303	−0.0223851
$809$	−6.37934	−0.224286	−0.112143	−	0.993692i	$-0.535771\pi$
$809$	−6.37934	−0.224286	−0.112143	+	0.993692i	$0.535771\pi$
$810$	1.00000	0.0351364
$811$	−52.4305	−1.84108	−0.920542	−	0.390644i	$-0.872252\pi$
$811$	−52.4305	−1.84108	−0.920542	+	0.390644i	$0.872252\pi$
$812$	−10.7821	−0.378377
$813$	−3.65685	−0.128251
$814$	−52.1595	−1.82819
$815$	11.9150	0.417364
$816$	0	0
$817$	52.8779	1.84996
$818$	−4.64174	−0.162295
$819$	14.5239	0.507508
$820$	−3.67459	−0.128322
$821$	−22.3501	−0.780023	−0.390012	−	0.920810i	$-0.627529\pi$
$821$	−22.3501	−0.780023	−0.390012	+	0.920810i	$0.627529\pi$
$822$	−4.66364	−0.162663
$823$	27.2979	0.951546	0.475773	−	0.879568i	$-0.342168\pi$
$823$	27.2979	0.951546	0.475773	+	0.879568i	$0.342168\pi$
$824$	−19.5467	−0.680943
$825$	5.69552	0.198292
$826$	16.2818	0.566517
$827$	15.4812	0.538334	0.269167	−	0.963094i	$-0.413252\pi$
$827$	15.4812	0.538334	0.269167	+	0.963094i	$0.413252\pi$
$828$	−6.47568	−0.225045
$829$	31.5652	1.09630	0.548152	−	0.836378i	$-0.315331\pi$
$829$	31.5652	1.09630	0.548152	+	0.836378i	$0.315331\pi$
$830$	3.39104	0.117705
$831$	−5.48438	−0.190251
$832$	−5.55807	−0.192692
$833$	0	0
$834$	2.85351	0.0988091
$835$	8.97003	0.310421
$836$	−28.4389	−0.983581
$837$	−8.33598	−0.288134
$838$	−13.7374	−0.474549
$839$	9.84845	0.340006	0.170003	−	0.985444i	$-0.445622\pi$
$839$	9.84845	0.340006	0.170003	+	0.985444i	$0.445622\pi$
$840$	−2.61313	−0.0901614
$841$	−11.9751	−0.412935
$842$	−6.33407	−0.218287
$843$	−13.1161	−0.451744
$844$	13.2809	0.457148
$845$	−17.8922	−0.615510
$846$	0.993212	0.0341473
$847$	−56.0226	−1.92496
$848$	−1.06147	−0.0364509
$849$	21.2186	0.728219
$850$	0	0
$851$	−59.3043	−2.03292
$852$	−1.71644	−0.0588044
$853$	−32.6232	−1.11700	−0.558498	−	0.829506i	$-0.688622\pi$
$853$	−32.6232	−1.11700	−0.558498	+	0.829506i	$0.688622\pi$
$854$	0.101034	0.00345730
$855$	4.99321	0.170764
$856$	15.8512	0.541784
$857$	30.3217	1.03577	0.517884	−	0.855451i	$-0.326720\pi$
$857$	30.3217	1.03577	0.517884	+	0.855451i	$0.326720\pi$
$858$	31.6561	1.08072
$859$	−34.6347	−1.18172	−0.590860	−	0.806774i	$-0.701211\pi$
$859$	−34.6347	−1.18172	−0.590860	+	0.806774i	$0.701211\pi$
$860$	10.5899	0.361114
$861$	−9.60218	−0.327241
$862$	−13.5199	−0.460491
$863$	0.158413	0.00539245	0.00269622	−	0.999996i	$-0.499142\pi$
$863$	0.158413	0.00539245	0.00269622	+	0.999996i	$0.499142\pi$
$864$	−1.00000	−0.0340207
$865$	21.4206	0.728323
$866$	−41.1058	−1.39683
$867$	0	0
$868$	21.7830	0.739362
$869$	−27.1482	−0.920939
$870$	4.12612	0.139889
$871$	−46.5238	−1.57640
$872$	−2.50793	−0.0849292
$873$	6.70058	0.226780
$874$	−32.3344	−1.09373
$875$	2.61313	0.0883398
$876$	11.0656	0.373873
$877$	2.80015	0.0945545	0.0472773	−	0.998882i	$-0.484946\pi$
$877$	2.80015	0.0945545	0.0472773	+	0.998882i	$0.484946\pi$
$878$	22.9860	0.775742
$879$	7.57953	0.255651
$880$	−5.69552	−0.191996
$881$	29.1029	0.980502	0.490251	−	0.871581i	$-0.336905\pi$
$881$	29.1029	0.980502	0.490251	+	0.871581i	$0.336905\pi$
$882$	0.171573	0.00577716
$883$	12.5572	0.422582	0.211291	−	0.977423i	$-0.432233\pi$
$883$	12.5572	0.422582	0.211291	+	0.977423i	$0.432233\pi$
$884$	0	0
$885$	−6.23079	−0.209446
$886$	−20.8922	−0.701886
$887$	52.5753	1.76531	0.882653	−	0.470026i	$-0.155755\pi$
$887$	52.5753	1.76531	0.882653	+	0.470026i	$0.155755\pi$
$888$	−9.15800	−0.307322
$889$	38.0958	1.27769
$890$	15.7115	0.526652
$891$	5.69552	0.190807
$892$	−25.6105	−0.857503
$893$	4.95932	0.165957
$894$	−21.2536	−0.710826
$895$	1.86574	0.0623648
$896$	2.61313	0.0872984
$897$	35.9923	1.20175
$898$	−12.9974	−0.433728
$899$	−34.3953	−1.14715
$900$	1.00000	0.0333333
$901$	0	0
$902$	−20.9287	−0.696850
$903$	27.6729	0.920895
$904$	6.84391	0.227625
$905$	11.7889	0.391875
$906$	10.3296	0.343177
$907$	20.3469	0.675608	0.337804	−	0.941217i	$-0.390316\pi$
$907$	20.3469	0.675608	0.337804	+	0.941217i	$0.390316\pi$
$908$	14.1480	0.469519
$909$	0.636303	0.0211048
$910$	14.5239	0.481464
$911$	−23.1112	−0.765709	−0.382855	−	0.923809i	$-0.625059\pi$
$911$	−23.1112	−0.765709	−0.382855	+	0.923809i	$0.625059\pi$
$912$	−4.99321	−0.165342
$913$	19.3137	0.639190
$914$	−8.38106	−0.277221
$915$	−0.0386639	−0.00127819
$916$	17.0615	0.563727
$917$	−40.1595	−1.32618
$918$	0	0
$919$	0.628912	0.0207459	0.0103729	−	0.999946i	$-0.496698\pi$
$919$	0.628912	0.0207459	0.0103729	+	0.999946i	$0.496698\pi$
$920$	−6.47568	−0.213497
$921$	12.8558	0.423612
$922$	37.8709	1.24721
$923$	9.54012	0.314017
$924$	−14.8831	−0.489618
$925$	9.15800	0.301113
$926$	−19.1580	−0.629571
$927$	19.5467	0.641999
$928$	−4.12612	−0.135447
$929$	18.8371	0.618025	0.309013	−	0.951058i	$-0.400002\pi$
$929$	18.8371	0.618025	0.309013	+	0.951058i	$0.400002\pi$
$930$	−8.33598	−0.273348
$931$	0.856700	0.0280772
$932$	−9.94723	−0.325832
$933$	15.4088	0.504461
$934$	−11.2398	−0.367778
$935$	0	0
$936$	5.55807	0.181671
$937$	−8.50718	−0.277918	−0.138959	−	0.990298i	$-0.544376\pi$
$937$	−8.50718	−0.277918	−0.138959	+	0.990298i	$0.544376\pi$
$938$	21.8731	0.714183
$939$	−0.00416219	−0.000135828 0
$940$	0.993212	0.0323950
$941$	9.55444	0.311466	0.155733	−	0.987799i	$-0.450226\pi$
$941$	9.55444	0.311466	0.155733	+	0.987799i	$0.450226\pi$
$942$	−21.0456	−0.685704
$943$	−23.7955	−0.774888
$944$	6.23079	0.202795
$945$	2.61313	0.0850050
$946$	60.3152	1.96102
$947$	23.8512	0.775061	0.387530	−	0.921857i	$-0.373328\pi$
$947$	23.8512	0.775061	0.387530	+	0.921857i	$0.373328\pi$
$948$	−4.76659	−0.154812
$949$	−61.5036	−1.99649
$950$	4.99321	0.162001
$951$	−11.6401	−0.377456
$952$	0	0
$953$	−39.2655	−1.27194	−0.635968	−	0.771715i	$-0.719399\pi$
$953$	−39.2655	−1.27194	−0.635968	+	0.771715i	$0.719399\pi$
$954$	1.06147	0.0343663
$955$	12.4912	0.404205
$956$	−1.29769	−0.0419704
$957$	23.5004	0.759660
$958$	3.54529	0.114543
$959$	−12.1867	−0.393529
$960$	−1.00000	−0.0322749
$961$	38.4886	1.24157
$962$	50.9008	1.64111
$963$	−15.8512	−0.510799
$964$	7.47796	0.240849
$965$	4.65498	0.149849
$966$	−16.9218	−0.544449
$967$	17.9356	0.576769	0.288384	−	0.957515i	$-0.406882\pi$
$967$	17.9356	0.576769	0.288384	+	0.957515i	$0.406882\pi$
$968$	−21.4389	−0.689073
$969$	0	0
$970$	6.70058	0.215143
$971$	41.4336	1.32967	0.664834	−	0.746991i	$-0.268502\pi$
$971$	41.4336	1.32967	0.664834	+	0.746991i	$0.268502\pi$
$972$	1.00000	0.0320750
$973$	7.45659	0.239047
$974$	13.5928	0.435540
$975$	−5.55807	−0.178001
$976$	0.0386639	0.00123760
$977$	−13.9547	−0.446450	−0.223225	−	0.974767i	$-0.571659\pi$
$977$	−13.9547	−0.446450	−0.223225	+	0.974767i	$0.571659\pi$
$978$	11.9150	0.380999
$979$	89.4853	2.85996
$980$	0.171573	0.00548069
$981$	2.50793	0.0800720
$982$	3.20480	0.102269
$983$	−43.0620	−1.37346	−0.686732	−	0.726911i	$-0.740955\pi$
$983$	−43.0620	−1.37346	−0.686732	+	0.726911i	$0.740955\pi$
$984$	−3.67459	−0.117142
$985$	5.40780	0.172307
$986$	0	0
$987$	2.59539	0.0826121
$988$	27.7526	0.882929
$989$	68.5771	2.18063
$990$	5.69552	0.181015
$991$	10.9777	0.348719	0.174359	−	0.984682i	$-0.444215\pi$
$991$	10.9777	0.348719	0.174359	+	0.984682i	$0.444215\pi$
$992$	8.33598	0.264668
$993$	16.0965	0.510808
$994$	−4.48528	−0.142264
$995$	−21.1972	−0.671996
$996$	3.39104	0.107449
$997$	−33.1181	−1.04886	−0.524430	−	0.851453i	$-0.675722\pi$
$997$	−33.1181	−1.04886	−0.524430	+	0.851453i	$0.675722\pi$
$998$	−33.1263	−1.04859
$999$	9.15800	0.289746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bw.1.1		4
17.2	even	8		510.2.p.d.361.4	✓	8
17.9	even	8		510.2.p.d.421.4	yes	8
17.16	even	2		8670.2.a.bt.1.4		4
51.2	odd	8		1530.2.q.i.361.2		8
51.26	odd	8		1530.2.q.i.1441.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.p.d.361.4	✓	8	17.2	even	8
510.2.p.d.421.4	yes	8	17.9	even	8
1530.2.q.i.361.2		8	51.2	odd	8
1530.2.q.i.1441.2		8	51.26	odd	8
8670.2.a.bt.1.4		4	17.16	even	2
8670.2.a.bw.1.1		4	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.61313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.69552 q^{11} +1.00000 q^{12} -5.55807 q^{13} +2.61313 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.99321 q^{19} -1.00000 q^{20} -2.61313 q^{21} -5.69552 q^{22} -6.47568 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.55807 q^{26} +1.00000 q^{27} -2.61313 q^{28} +4.12612 q^{29} +1.00000 q^{30} -8.33598 q^{31} -1.00000 q^{32} +5.69552 q^{33} +2.61313 q^{35} +1.00000 q^{36} +9.15800 q^{37} +4.99321 q^{38} -5.55807 q^{39} +1.00000 q^{40} +3.67459 q^{41} +2.61313 q^{42} -10.5899 q^{43} +5.69552 q^{44} -1.00000 q^{45} +6.47568 q^{46} -0.993212 q^{47} +1.00000 q^{48} -0.171573 q^{49} -1.00000 q^{50} -5.55807 q^{52} -1.06147 q^{53} -1.00000 q^{54} -5.69552 q^{55} +2.61313 q^{56} -4.99321 q^{57} -4.12612 q^{58} +6.23079 q^{59} -1.00000 q^{60} +0.0386639 q^{61} +8.33598 q^{62} -2.61313 q^{63} +1.00000 q^{64} +5.55807 q^{65} -5.69552 q^{66} +8.37049 q^{67} -6.47568 q^{69} -2.61313 q^{70} -1.71644 q^{71} -1.00000 q^{72} +11.0656 q^{73} -9.15800 q^{74} +1.00000 q^{75} -4.99321 q^{76} -14.8831 q^{77} +5.55807 q^{78} -4.76659 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.67459 q^{82} +3.39104 q^{83} -2.61313 q^{84} +10.5899 q^{86} +4.12612 q^{87} -5.69552 q^{88} +15.7115 q^{89} +1.00000 q^{90} +14.5239 q^{91} -6.47568 q^{92} -8.33598 q^{93} +0.993212 q^{94} +4.99321 q^{95} -1.00000 q^{96} +6.70058 q^{97} +0.171573 q^{98} +5.69552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{10} + 8 q^{11} + 4 q^{12} - 4 q^{15} + 4 q^{16} - 4 q^{18} - 4 q^{20} - 8 q^{22} - 8 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27} + 4 q^{30} + 8 q^{31} - 4 q^{32} + 8 q^{33} + 4 q^{36} + 8 q^{37} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 8 q^{44} - 4 q^{45} + 8 q^{46} + 16 q^{47} + 4 q^{48} - 12 q^{49} - 4 q^{50} + 8 q^{53} - 4 q^{54} - 8 q^{55} + 8 q^{59} - 4 q^{60} + 8 q^{61} - 8 q^{62} + 4 q^{64} - 8 q^{66} + 40 q^{67} - 8 q^{69} - 4 q^{72} - 8 q^{73} - 8 q^{74} + 4 q^{75} - 16 q^{77} + 24 q^{79} - 4 q^{80} + 4 q^{81} + 8 q^{82} - 16 q^{83} + 8 q^{86} - 8 q^{88} + 8 q^{89} + 4 q^{90} + 32 q^{91} - 8 q^{92} + 8 q^{93} - 16 q^{94} - 4 q^{96} - 8 q^{97} + 12 q^{98} + 8 q^{99}+O(q^{100}) \) 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 + 8 * q^11 + 4 * q^12 - 4 * q^15 + 4 * q^16 - 4 * q^18 - 4 * q^20 - 8 * q^22 - 8 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 + 4 * q^30 + 8 * q^31 - 4 * q^32 + 8 * q^33 + 4 * q^36 + 8 * q^37 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 8 * q^44 - 4 * q^45 + 8 * q^46 + 16 * q^47 + 4 * q^48 - 12 * q^49 - 4 * q^50 + 8 * q^53 - 4 * q^54 - 8 * q^55 + 8 * q^59 - 4 * q^60 + 8 * q^61 - 8 * q^62 + 4 * q^64 - 8 * q^66 + 40 * q^67 - 8 * q^69 - 4 * q^72 - 8 * q^73 - 8 * q^74 + 4 * q^75 - 16 * q^77 + 24 * q^79 - 4 * q^80 + 4 * q^81 + 8 * q^82 - 16 * q^83 + 8 * q^86 - 8 * q^88 + 8 * q^89 + 4 * q^90 + 32 * q^91 - 8 * q^92 + 8 * q^93 - 16 * q^94 - 4 * q^96 - 8 * q^97 + 12 * q^98 + 8 * q^99

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