LMFDB - Embedded newform 8670.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$69.2302985525$
Analytic rank:	$0$
Dimension:	$4$
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			1.2
Root			$0.765367$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.469266 q^{11} +1.00000 q^{12} -3.36370 q^{13} +1.08239 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +8.05468 q^{19} -1.00000 q^{20} -1.08239 q^{21} -0.469266 q^{22} -7.97682 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.36370 q^{26} +1.00000 q^{27} -1.08239 q^{28} -9.35237 q^{29} +1.00000 q^{30} +2.78016 q^{31} -1.00000 q^{32} +0.469266 q^{33} +1.08239 q^{35} +1.00000 q^{36} -11.2809 q^{37} -8.05468 q^{38} -3.36370 q^{39} +1.00000 q^{40} +6.47343 q^{41} +1.08239 q^{42} -1.43514 q^{43} +0.469266 q^{44} -1.00000 q^{45} +7.97682 q^{46} +12.0547 q^{47} +1.00000 q^{48} -5.82843 q^{49} -1.00000 q^{50} -3.36370 q^{52} -5.39104 q^{53} -1.00000 q^{54} -0.469266 q^{55} +1.08239 q^{56} +8.05468 q^{57} +9.35237 q^{58} +0.568047 q^{59} -1.00000 q^{60} +6.12612 q^{61} -2.78016 q^{62} -1.08239 q^{63} +1.00000 q^{64} +3.36370 q^{65} -0.469266 q^{66} +15.3250 q^{67} -7.97682 q^{69} -1.08239 q^{70} +11.5349 q^{71} -1.00000 q^{72} +3.41196 q^{73} +11.2809 q^{74} +1.00000 q^{75} +8.05468 q^{76} -0.507930 q^{77} +3.36370 q^{78} +14.6018 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.47343 q^{82} -7.06147 q^{83} -1.08239 q^{84} +1.43514 q^{86} -9.35237 q^{87} -0.469266 q^{88} -4.32050 q^{89} +1.00000 q^{90} +3.64084 q^{91} -7.97682 q^{92} +2.78016 q^{93} -12.0547 q^{94} -8.05468 q^{95} -1.00000 q^{96} -14.3961 q^{97} +5.82843 q^{98} +0.469266 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−1.08239	−0.409106	−0.204553	−	0.978856i	$-0.565574\pi$
$7$	−1.08239	−0.409106	−0.204553	+	0.978856i	$0.565574\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0.469266	0.141489	0.0707446	−	0.997494i	$-0.477462\pi$
$11$	0.469266	0.141489	0.0707446	+	0.997494i	$0.477462\pi$
$12$	1.00000	0.288675
$13$	−3.36370	−0.932922	−0.466461	−	0.884542i	$-0.654471\pi$
$13$	−3.36370	−0.932922	−0.466461	+	0.884542i	$0.654471\pi$
$14$	1.08239	0.289281
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	8.05468	1.84787	0.923935	−	0.382549i	$-0.124954\pi$
$19$	8.05468	1.84787	0.923935	+	0.382549i	$0.124954\pi$
$20$	−1.00000	−0.223607
$21$	−1.08239	−0.236197
$22$	−0.469266	−0.100048
$23$	−7.97682	−1.66328	−0.831641	−	0.555313i	$-0.812598\pi$
$23$	−7.97682	−1.66328	−0.831641	+	0.555313i	$0.812598\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	3.36370	0.659675
$27$	1.00000	0.192450
$28$	−1.08239	−0.204553
$29$	−9.35237	−1.73669	−0.868346	−	0.495959i	$-0.834817\pi$
$29$	−9.35237	−1.73669	−0.868346	+	0.495959i	$0.834817\pi$
$30$	1.00000	0.182574
$31$	2.78016	0.499332	0.249666	−	0.968332i	$-0.419679\pi$
$31$	2.78016	0.499332	0.249666	+	0.968332i	$0.419679\pi$
$32$	−1.00000	−0.176777
$33$	0.469266	0.0816888
$34$	0	0
$35$	1.08239	0.182958
$36$	1.00000	0.166667
$37$	−11.2809	−1.85457	−0.927287	−	0.374352i	$-0.877865\pi$
$37$	−11.2809	−1.85457	−0.927287	+	0.374352i	$0.877865\pi$
$38$	−8.05468	−1.30664
$39$	−3.36370	−0.538623
$40$	1.00000	0.158114
$41$	6.47343	1.01098	0.505490	−	0.862833i	$-0.331312\pi$
$41$	6.47343	1.01098	0.505490	+	0.862833i	$0.331312\pi$
$42$	1.08239	0.167017
$43$	−1.43514	−0.218857	−0.109428	−	0.993995i	$-0.534902\pi$
$43$	−1.43514	−0.218857	−0.109428	+	0.993995i	$0.534902\pi$
$44$	0.469266	0.0707446
$45$	−1.00000	−0.149071
$46$	7.97682	1.17612
$47$	12.0547	1.75836	0.879178	−	0.476494i	$-0.158093\pi$
$47$	12.0547	1.75836	0.879178	+	0.476494i	$0.158093\pi$
$48$	1.00000	0.144338
$49$	−5.82843	−0.832632
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.36370	−0.466461
$53$	−5.39104	−0.740516	−0.370258	−	0.928929i	$-0.620731\pi$
$53$	−5.39104	−0.740516	−0.370258	+	0.928929i	$0.620731\pi$
$54$	−1.00000	−0.136083
$55$	−0.469266	−0.0632759
$56$	1.08239	0.144641
$57$	8.05468	1.06687
$58$	9.35237	1.22803
$59$	0.568047	0.0739535	0.0369767	−	0.999316i	$-0.488227\pi$
$59$	0.568047	0.0739535	0.0369767	+	0.999316i	$0.488227\pi$
$60$	−1.00000	−0.129099
$61$	6.12612	0.784369	0.392185	−	0.919887i	$-0.371719\pi$
$61$	6.12612	0.784369	0.392185	+	0.919887i	$0.371719\pi$
$62$	−2.78016	−0.353081
$63$	−1.08239	−0.136369
$64$	1.00000	0.125000
$65$	3.36370	0.417215
$66$	−0.469266	−0.0577627
$67$	15.3250	1.87225	0.936125	−	0.351666i	$-0.114385\pi$
$67$	15.3250	1.87225	0.936125	+	0.351666i	$0.114385\pi$
$68$	0	0
$69$	−7.97682	−0.960297
$70$	−1.08239	−0.129371
$71$	11.5349	1.36894	0.684470	−	0.729041i	$-0.260034\pi$
$71$	11.5349	1.36894	0.684470	+	0.729041i	$0.260034\pi$
$72$	−1.00000	−0.117851
$73$	3.41196	0.399340	0.199670	−	0.979863i	$-0.436013\pi$
$73$	3.41196	0.399340	0.199670	+	0.979863i	$0.436013\pi$
$74$	11.2809	1.31138
$75$	1.00000	0.115470
$76$	8.05468	0.923935
$77$	−0.507930	−0.0578840
$78$	3.36370	0.380864
$79$	14.6018	1.64283	0.821416	−	0.570330i	$-0.193185\pi$
$79$	14.6018	1.64283	0.821416	+	0.570330i	$0.193185\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.47343	−0.714871
$83$	−7.06147	−0.775097	−0.387548	−	0.921849i	$-0.626678\pi$
$83$	−7.06147	−0.775097	−0.387548	+	0.921849i	$0.626678\pi$
$84$	−1.08239	−0.118099
$85$	0	0
$86$	1.43514	0.154755
$87$	−9.35237	−1.00268
$88$	−0.469266	−0.0500240
$89$	−4.32050	−0.457972	−0.228986	−	0.973430i	$-0.573541\pi$
$89$	−4.32050	−0.457972	−0.228986	+	0.973430i	$0.573541\pi$
$90$	1.00000	0.105409
$91$	3.64084	0.381664
$92$	−7.97682	−0.831641
$93$	2.78016	0.288289
$94$	−12.0547	−1.24335
$95$	−8.05468	−0.826393
$96$	−1.00000	−0.102062
$97$	−14.3961	−1.46170	−0.730851	−	0.682537i	$-0.760877\pi$
$97$	−14.3961	−1.46170	−0.730851	+	0.682537i	$0.760877\pi$
$98$	5.82843	0.588760
$99$	0.469266	0.0471630
$100$	1.00000	0.100000
$101$	6.72965	0.669625	0.334812	−	0.942285i	$-0.391327\pi$
$101$	6.72965	0.669625	0.334812	+	0.942285i	$0.391327\pi$
$102$	0	0
$103$	6.90575	0.680444	0.340222	−	0.940345i	$-0.389498\pi$
$103$	6.90575	0.680444	0.340222	+	0.940345i	$0.389498\pi$
$104$	3.36370	0.329838
$105$	1.08239	0.105631
$106$	5.39104	0.523624
$107$	−8.43649	−0.815586	−0.407793	−	0.913074i	$-0.633702\pi$
$107$	−8.43649	−0.815586	−0.407793	+	0.913074i	$0.633702\pi$
$108$	1.00000	0.0962250
$109$	6.43060	0.615940	0.307970	−	0.951396i	$-0.400350\pi$
$109$	6.43060	0.615940	0.307970	+	0.951396i	$0.400350\pi$
$110$	0.469266	0.0447428
$111$	−11.2809	−1.07074
$112$	−1.08239	−0.102276
$113$	0.349561	0.0328839	0.0164419	−	0.999865i	$-0.494766\pi$
$113$	0.349561	0.0328839	0.0164419	+	0.999865i	$0.494766\pi$
$114$	−8.05468	−0.754390
$115$	7.97682	0.743843
$116$	−9.35237	−0.868346
$117$	−3.36370	−0.310974
$118$	−0.568047	−0.0522930
$119$	0	0
$120$	1.00000	0.0912871
$121$	−10.7798	−0.979981
$122$	−6.12612	−0.554633
$123$	6.47343	0.583689
$124$	2.78016	0.249666
$125$	−1.00000	−0.0894427
$126$	1.08239	0.0964272
$127$	5.02280	0.445702	0.222851	−	0.974853i	$-0.428464\pi$
$127$	5.02280	0.445702	0.222851	+	0.974853i	$0.428464\pi$
$128$	−1.00000	−0.0883883
$129$	−1.43514	−0.126357
$130$	−3.36370	−0.295016
$131$	−15.9774	−1.39595	−0.697974	−	0.716123i	$-0.745915\pi$
$131$	−15.9774	−1.39595	−0.697974	+	0.716123i	$0.745915\pi$
$132$	0.469266	0.0408444
$133$	−8.71832	−0.755974
$134$	−15.3250	−1.32388
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	6.39782	0.546603	0.273302	−	0.961928i	$-0.411884\pi$
$137$	6.39782	0.546603	0.273302	+	0.961928i	$0.411884\pi$
$138$	7.97682	0.679032
$139$	22.8117	1.93486	0.967430	−	0.253138i	$-0.0814628\pi$
$139$	22.8117	1.93486	0.967430	+	0.253138i	$0.0814628\pi$
$140$	1.08239	0.0914788
$141$	12.0547	1.01519
$142$	−11.5349	−0.967987
$143$	−1.57847	−0.131998
$144$	1.00000	0.0833333
$145$	9.35237	0.776672
$146$	−3.41196	−0.282376
$147$	−5.82843	−0.480721
$148$	−11.2809	−0.927287
$149$	13.8330	1.13324	0.566620	−	0.823979i	$-0.308251\pi$
$149$	13.8330	1.13324	0.566620	+	0.823979i	$0.308251\pi$
$150$	−1.00000	−0.0816497
$151$	4.45250	0.362340	0.181170	−	0.983452i	$-0.442012\pi$
$151$	4.45250	0.362340	0.181170	+	0.983452i	$0.442012\pi$
$152$	−8.05468	−0.653321
$153$	0	0
$154$	0.507930	0.0409302
$155$	−2.78016	−0.223308
$156$	−3.36370	−0.269311
$157$	−16.3697	−1.30645	−0.653224	−	0.757165i	$-0.726584\pi$
$157$	−16.3697	−1.30645	−0.653224	+	0.757165i	$0.726584\pi$
$158$	−14.6018	−1.16166
$159$	−5.39104	−0.427537
$160$	1.00000	0.0790569
$161$	8.63405	0.680459
$162$	−1.00000	−0.0785674
$163$	9.42063	0.737881	0.368940	−	0.929453i	$-0.379721\pi$
$163$	9.42063	0.737881	0.368940	+	0.929453i	$0.379721\pi$
$164$	6.47343	0.505490
$165$	−0.469266	−0.0365323
$166$	7.06147	0.548076
$167$	11.7019	0.905523	0.452761	−	0.891632i	$-0.350439\pi$
$167$	11.7019	0.905523	0.452761	+	0.891632i	$0.350439\pi$
$168$	1.08239	0.0835084
$169$	−1.68554	−0.129657
$170$	0	0
$171$	8.05468	0.615957
$172$	−1.43514	−0.109428
$173$	−22.2581	−1.69225	−0.846127	−	0.532981i	$-0.821072\pi$
$173$	−22.2581	−1.69225	−0.846127	+	0.532981i	$0.821072\pi$
$174$	9.35237	0.709002
$175$	−1.08239	−0.0818212
$176$	0.469266	0.0353723
$177$	0.568047	0.0426970
$178$	4.32050	0.323835
$179$	17.2400	1.28858	0.644290	−	0.764782i	$-0.277153\pi$
$179$	17.2400	1.28858	0.644290	+	0.764782i	$0.277153\pi$
$180$	−1.00000	−0.0745356
$181$	−3.93174	−0.292244	−0.146122	−	0.989267i	$-0.546679\pi$
$181$	−3.93174	−0.292244	−0.146122	+	0.989267i	$0.546679\pi$
$182$	−3.64084	−0.269877
$183$	6.12612	0.452856
$184$	7.97682	0.588059
$185$	11.2809	0.829391
$186$	−2.78016	−0.203851
$187$	0	0
$188$	12.0547	0.879178
$189$	−1.08239	−0.0787324
$190$	8.05468	0.584348
$191$	−12.4557	−0.901262	−0.450631	−	0.892710i	$-0.648801\pi$
$191$	−12.4557	−0.901262	−0.450631	+	0.892710i	$0.648801\pi$
$192$	1.00000	0.0721688
$193$	12.9259	0.930429	0.465214	−	0.885198i	$-0.345977\pi$
$193$	12.9259	0.930429	0.465214	+	0.885198i	$0.345977\pi$
$194$	14.3961	1.03358
$195$	3.36370	0.240879
$196$	−5.82843	−0.416316
$197$	1.08655	0.0774138	0.0387069	−	0.999251i	$-0.487676\pi$
$197$	1.08655	0.0774138	0.0387069	+	0.999251i	$0.487676\pi$
$198$	−0.469266	−0.0333493
$199$	−6.42344	−0.455346	−0.227673	−	0.973738i	$-0.573112\pi$
$199$	−6.42344	−0.455346	−0.227673	+	0.973738i	$0.573112\pi$
$200$	−1.00000	−0.0707107
$201$	15.3250	1.08094
$202$	−6.72965	−0.473496
$203$	10.1229	0.710491
$204$	0	0
$205$	−6.47343	−0.452124
$206$	−6.90575	−0.481147
$207$	−7.97682	−0.554428
$208$	−3.36370	−0.233230
$209$	3.77979	0.261453
$210$	−1.08239	−0.0746922
$211$	1.50114	0.103343	0.0516714	−	0.998664i	$-0.483545\pi$
$211$	1.50114	0.103343	0.0516714	+	0.998664i	$0.483545\pi$
$212$	−5.39104	−0.370258
$213$	11.5349	0.790358
$214$	8.43649	0.576706
$215$	1.43514	0.0978756
$216$	−1.00000	−0.0680414
$217$	−3.00923	−0.204280
$218$	−6.43060	−0.435535
$219$	3.41196	0.230559
$220$	−0.469266	−0.0316379
$221$	0	0
$222$	11.2809	0.757127
$223$	0.951362	0.0637079	0.0318540	−	0.999493i	$-0.489859\pi$
$223$	0.951362	0.0637079	0.0318540	+	0.999493i	$0.489859\pi$
$224$	1.08239	0.0723204
$225$	1.00000	0.0666667
$226$	−0.349561	−0.0232524
$227$	2.79884	0.185765	0.0928826	−	0.995677i	$-0.470392\pi$
$227$	2.79884	0.185765	0.0928826	+	0.995677i	$0.470392\pi$
$228$	8.05468	0.533434
$229$	21.3910	1.41356	0.706780	−	0.707434i	$-0.250147\pi$
$229$	21.3910	1.41356	0.706780	+	0.707434i	$0.250147\pi$
$230$	−7.97682	−0.525976
$231$	−0.507930	−0.0334194
$232$	9.35237	0.614013
$233$	8.96685	0.587438	0.293719	−	0.955892i	$-0.405107\pi$
$233$	8.96685	0.587438	0.293719	+	0.955892i	$0.405107\pi$
$234$	3.36370	0.219892
$235$	−12.0547	−0.786361
$236$	0.568047	0.0369767
$237$	14.6018	0.948489
$238$	0	0
$239$	6.52395	0.421999	0.210999	−	0.977486i	$-0.432328\pi$
$239$	6.52395	0.421999	0.210999	+	0.977486i	$0.432328\pi$
$240$	−1.00000	−0.0645497
$241$	−9.27133	−0.597219	−0.298609	−	0.954375i	$-0.596523\pi$
$241$	−9.27133	−0.597219	−0.298609	+	0.954375i	$0.596523\pi$
$242$	10.7798	0.692951
$243$	1.00000	0.0641500
$244$	6.12612	0.392185
$245$	5.82843	0.372365
$246$	−6.47343	−0.412731
$247$	−27.0935	−1.72392
$248$	−2.78016	−0.176541
$249$	−7.06147	−0.447502
$250$	1.00000	0.0632456
$251$	−17.9810	−1.13495	−0.567475	−	0.823391i	$-0.692080\pi$
$251$	−17.9810	−1.13495	−0.567475	+	0.823391i	$0.692080\pi$
$252$	−1.08239	−0.0681843
$253$	−3.74325	−0.235336
$254$	−5.02280	−0.315159
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.3515	1.64376	0.821880	−	0.569661i	$-0.192925\pi$
$257$	26.3515	1.64376	0.821880	+	0.569661i	$0.192925\pi$
$258$	1.43514	0.0893478
$259$	12.2104	0.758717
$260$	3.36370	0.208608
$261$	−9.35237	−0.578897
$262$	15.9774	0.987084
$263$	17.7014	1.09152	0.545758	−	0.837943i	$-0.316242\pi$
$263$	17.7014	1.09152	0.545758	+	0.837943i	$0.316242\pi$
$264$	−0.469266	−0.0288813
$265$	5.39104	0.331169
$266$	8.71832	0.534555
$267$	−4.32050	−0.264410
$268$	15.3250	0.936125
$269$	−25.9665	−1.58320	−0.791602	−	0.611037i	$-0.790753\pi$
$269$	−25.9665	−1.58320	−0.791602	+	0.611037i	$0.790753\pi$
$270$	1.00000	0.0608581
$271$	7.65685	0.465121	0.232560	−	0.972582i	$-0.425290\pi$
$271$	7.65685	0.465121	0.232560	+	0.972582i	$0.425290\pi$
$272$	0	0
$273$	3.64084	0.220354
$274$	−6.39782	−0.386507
$275$	0.469266	0.0282978
$276$	−7.97682	−0.480148
$277$	7.59899	0.456579	0.228290	−	0.973593i	$-0.426687\pi$
$277$	7.59899	0.456579	0.228290	+	0.973593i	$0.426687\pi$
$278$	−22.8117	−1.36815
$279$	2.78016	0.166444
$280$	−1.08239	−0.0646853
$281$	−8.72739	−0.520633	−0.260316	−	0.965523i	$-0.583827\pi$
$281$	−8.72739	−0.520633	−0.260316	+	0.965523i	$0.583827\pi$
$282$	−12.0547	−0.717846
$283$	8.99640	0.534780	0.267390	−	0.963588i	$-0.413839\pi$
$283$	8.99640	0.534780	0.267390	+	0.963588i	$0.413839\pi$
$284$	11.5349	0.684470
$285$	−8.05468	−0.477118
$286$	1.57847	0.0933369
$287$	−7.00679	−0.413598
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−9.35237	−0.549190
$291$	−14.3961	−0.843914
$292$	3.41196	0.199670
$293$	−15.9091	−0.929419	−0.464710	−	0.885463i	$-0.653841\pi$
$293$	−15.9091	−0.929419	−0.464710	+	0.885463i	$0.653841\pi$
$294$	5.82843	0.339921
$295$	−0.568047	−0.0330730
$296$	11.2809	0.655691
$297$	0.469266	0.0272296
$298$	−13.8330	−0.801322
$299$	26.8316	1.55171
$300$	1.00000	0.0577350
$301$	1.55338	0.0895355
$302$	−4.45250	−0.256213
$303$	6.72965	0.386608
$304$	8.05468	0.461968
$305$	−6.12612	−0.350781
$306$	0	0
$307$	2.83975	0.162073	0.0810366	−	0.996711i	$-0.474177\pi$
$307$	2.83975	0.162073	0.0810366	+	0.996711i	$0.474177\pi$
$308$	−0.507930	−0.0289420
$309$	6.90575	0.392855
$310$	2.78016	0.157903
$311$	19.0688	1.08129	0.540647	−	0.841250i	$-0.318180\pi$
$311$	19.0688	1.08129	0.540647	+	0.841250i	$0.318180\pi$
$312$	3.36370	0.190432
$313$	11.9791	0.677097	0.338549	−	0.940949i	$-0.390064\pi$
$313$	11.9791	0.677097	0.338549	+	0.940949i	$0.390064\pi$
$314$	16.3697	0.923798
$315$	1.08239	0.0609859
$316$	14.6018	0.821416
$317$	3.63177	0.203980	0.101990	−	0.994785i	$-0.467479\pi$
$317$	3.63177	0.203980	0.101990	+	0.994785i	$0.467479\pi$
$318$	5.39104	0.302314
$319$	−4.38875	−0.245723
$320$	−1.00000	−0.0559017
$321$	−8.43649	−0.470879
$322$	−8.63405	−0.481157
$323$	0	0
$324$	1.00000	0.0555556
$325$	−3.36370	−0.186584
$326$	−9.42063	−0.521760
$327$	6.43060	0.355613
$328$	−6.47343	−0.357435
$329$	−13.0479	−0.719353
$330$	0.469266	0.0258323
$331$	−8.67197	−0.476654	−0.238327	−	0.971185i	$-0.576599\pi$
$331$	−8.67197	−0.476654	−0.238327	+	0.971185i	$0.576599\pi$
$332$	−7.06147	−0.387548
$333$	−11.2809	−0.618191
$334$	−11.7019	−0.640301
$335$	−15.3250	−0.837296
$336$	−1.08239	−0.0590493
$337$	16.0292	0.873169	0.436584	−	0.899663i	$-0.356188\pi$
$337$	16.0292	0.873169	0.436584	+	0.899663i	$0.356188\pi$
$338$	1.68554	0.0916815
$339$	0.349561	0.0189855
$340$	0	0
$341$	1.30464	0.0706500
$342$	−8.05468	−0.435547
$343$	13.8854	0.749741
$344$	1.43514	0.0773775
$345$	7.97682	0.429458
$346$	22.2581	1.19660
$347$	20.4222	1.09632	0.548160	−	0.836374i	$-0.315329\pi$
$347$	20.4222	1.09632	0.548160	+	0.836374i	$0.315329\pi$
$348$	−9.35237	−0.501340
$349$	4.77375	0.255533	0.127766	−	0.991804i	$-0.459219\pi$
$349$	4.77375	0.255533	0.127766	+	0.991804i	$0.459219\pi$
$350$	1.08239	0.0578563
$351$	−3.36370	−0.179541
$352$	−0.469266	−0.0250120
$353$	35.6889	1.89953	0.949764	−	0.312968i	$-0.101323\pi$
$353$	35.6889	1.89953	0.949764	+	0.312968i	$0.101323\pi$
$354$	−0.568047	−0.0301914
$355$	−11.5349	−0.612209
$356$	−4.32050	−0.228986
$357$	0	0
$358$	−17.2400	−0.911163
$359$	17.2455	0.910180	0.455090	−	0.890445i	$-0.349607\pi$
$359$	17.2455	0.910180	0.455090	+	0.890445i	$0.349607\pi$
$360$	1.00000	0.0527046
$361$	45.8779	2.41462
$362$	3.93174	0.206648
$363$	−10.7798	−0.565792
$364$	3.64084	0.190832
$365$	−3.41196	−0.178590
$366$	−6.12612	−0.320217
$367$	−24.3305	−1.27004	−0.635022	−	0.772494i	$-0.719009\pi$
$367$	−24.3305	−1.27004	−0.635022	+	0.772494i	$0.719009\pi$
$368$	−7.97682	−0.415821
$369$	6.47343	0.336993
$370$	−11.2809	−0.586468
$371$	5.83522	0.302949
$372$	2.78016	0.144145
$373$	24.6091	1.27421	0.637107	−	0.770776i	$-0.280131\pi$
$373$	24.6091	1.27421	0.637107	+	0.770776i	$0.280131\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−12.0547	−0.621673
$377$	31.4585	1.62020
$378$	1.08239	0.0556722
$379$	24.7853	1.27313	0.636567	−	0.771222i	$-0.280354\pi$
$379$	24.7853	1.27313	0.636567	+	0.771222i	$0.280354\pi$
$380$	−8.05468	−0.413196
$381$	5.02280	0.257326
$382$	12.4557	0.637289
$383$	9.30088	0.475253	0.237626	−	0.971357i	$-0.423631\pi$
$383$	9.30088	0.475253	0.237626	+	0.971357i	$0.423631\pi$
$384$	−1.00000	−0.0510310
$385$	0.507930	0.0258865
$386$	−12.9259	−0.657913
$387$	−1.43514	−0.0729522
$388$	−14.3961	−0.730851
$389$	−28.3175	−1.43575	−0.717877	−	0.696170i	$-0.754886\pi$
$389$	−28.3175	−1.43575	−0.717877	+	0.696170i	$0.754886\pi$
$390$	−3.36370	−0.170327
$391$	0	0
$392$	5.82843	0.294380
$393$	−15.9774	−0.805951
$394$	−1.08655	−0.0547398
$395$	−14.6018	−0.734696
$396$	0.469266	0.0235815
$397$	4.63856	0.232802	0.116401	−	0.993202i	$-0.462864\pi$
$397$	4.63856	0.232802	0.116401	+	0.993202i	$0.462864\pi$
$398$	6.42344	0.321978
$399$	−8.71832	−0.436462
$400$	1.00000	0.0500000
$401$	12.6156	0.629991	0.314996	−	0.949093i	$-0.397997\pi$
$401$	12.6156	0.629991	0.314996	+	0.949093i	$0.397997\pi$
$402$	−15.3250	−0.764343
$403$	−9.35162	−0.465838
$404$	6.72965	0.334812
$405$	−1.00000	−0.0496904
$406$	−10.1229	−0.502393
$407$	−5.29376	−0.262402
$408$	0	0
$409$	4.24662	0.209982	0.104991	−	0.994473i	$-0.466519\pi$
$409$	4.24662	0.209982	0.104991	+	0.994473i	$0.466519\pi$
$410$	6.47343	0.319700
$411$	6.39782	0.315581
$412$	6.90575	0.340222
$413$	−0.614850	−0.0302548
$414$	7.97682	0.392039
$415$	7.06147	0.346634
$416$	3.36370	0.164919
$417$	22.8117	1.11709
$418$	−3.77979	−0.184876
$419$	−7.53906	−0.368307	−0.184154	−	0.982897i	$-0.558954\pi$
$419$	−7.53906	−0.368307	−0.184154	+	0.982897i	$0.558954\pi$
$420$	1.08239	0.0528153
$421$	22.8503	1.11366	0.556828	−	0.830628i	$-0.312018\pi$
$421$	22.8503	1.11366	0.556828	+	0.830628i	$0.312018\pi$
$422$	−1.50114	−0.0730744
$423$	12.0547	0.586119
$424$	5.39104	0.261812
$425$	0	0
$426$	−11.5349	−0.558868
$427$	−6.63087	−0.320890
$428$	−8.43649	−0.407793
$429$	−1.57847	−0.0762092
$430$	−1.43514	−0.0692085
$431$	−8.03105	−0.386842	−0.193421	−	0.981116i	$-0.561958\pi$
$431$	−8.03105	−0.386842	−0.193421	+	0.981116i	$0.561958\pi$
$432$	1.00000	0.0481125
$433$	−11.5164	−0.553443	−0.276722	−	0.960950i	$-0.589248\pi$
$433$	−11.5164	−0.553443	−0.276722	+	0.960950i	$0.589248\pi$
$434$	3.00923	0.144448
$435$	9.35237	0.448412
$436$	6.43060	0.307970
$437$	−64.2507	−3.07353
$438$	−3.41196	−0.163030
$439$	12.4917	0.596196	0.298098	−	0.954535i	$-0.403648\pi$
$439$	12.4917	0.596196	0.298098	+	0.954535i	$0.403648\pi$
$440$	0.469266	0.0223714
$441$	−5.82843	−0.277544
$442$	0	0
$443$	1.31446	0.0624517	0.0312258	−	0.999512i	$-0.490059\pi$
$443$	1.31446	0.0624517	0.0312258	+	0.999512i	$0.490059\pi$
$444$	−11.2809	−0.535369
$445$	4.32050	0.204811
$446$	−0.951362	−0.0450483
$447$	13.8330	0.654277
$448$	−1.08239	−0.0511382
$449$	−12.0338	−0.567908	−0.283954	−	0.958838i	$-0.591646\pi$
$449$	−12.0338	−0.567908	−0.283954	+	0.958838i	$0.591646\pi$
$450$	−1.00000	−0.0471405
$451$	3.03776	0.143043
$452$	0.349561	0.0164419
$453$	4.45250	0.209197
$454$	−2.79884	−0.131356
$455$	−3.64084	−0.170685
$456$	−8.05468	−0.377195
$457$	7.01830	0.328302	0.164151	−	0.986435i	$-0.447512\pi$
$457$	7.01830	0.328302	0.164151	+	0.986435i	$0.447512\pi$
$458$	−21.3910	−0.999537
$459$	0	0
$460$	7.97682	0.371921
$461$	−16.9363	−0.788801	−0.394401	−	0.918939i	$-0.629048\pi$
$461$	−16.9363	−0.788801	−0.394401	+	0.918939i	$0.629048\pi$
$462$	0.507930	0.0236310
$463$	−1.28093	−0.0595299	−0.0297650	−	0.999557i	$-0.509476\pi$
$463$	−1.28093	−0.0595299	−0.0297650	+	0.999557i	$0.509476\pi$
$464$	−9.35237	−0.434173
$465$	−2.78016	−0.128927
$466$	−8.96685	−0.415381
$467$	34.2741	1.58602	0.793009	−	0.609210i	$-0.208513\pi$
$467$	34.2741	1.58602	0.793009	+	0.609210i	$0.208513\pi$
$468$	−3.36370	−0.155487
$469$	−16.5877	−0.765949
$470$	12.0547	0.556041
$471$	−16.3697	−0.754278
$472$	−0.568047	−0.0261465
$473$	−0.673462	−0.0309658
$474$	−14.6018	−0.670683
$475$	8.05468	0.369574
$476$	0	0
$477$	−5.39104	−0.246839
$478$	−6.52395	−0.298398
$479$	−41.9602	−1.91721	−0.958606	−	0.284735i	$-0.908094\pi$
$479$	−41.9602	−1.91721	−0.958606	+	0.284735i	$0.908094\pi$
$480$	1.00000	0.0456435
$481$	37.9456	1.73017
$482$	9.27133	0.422298
$483$	8.63405	0.392863
$484$	−10.7798	−0.489990
$485$	14.3961	0.653693
$486$	−1.00000	−0.0453609
$487$	27.0816	1.22719	0.613593	−	0.789622i	$-0.289723\pi$
$487$	27.0816	1.22719	0.613593	+	0.789622i	$0.289723\pi$
$488$	−6.12612	−0.277316
$489$	9.42063	0.426016
$490$	−5.82843	−0.263301
$491$	−21.4376	−0.967464	−0.483732	−	0.875216i	$-0.660719\pi$
$491$	−21.4376	−0.967464	−0.483732	+	0.875216i	$0.660719\pi$
$492$	6.47343	0.291845
$493$	0	0
$494$	27.0935	1.21899
$495$	−0.469266	−0.0210920
$496$	2.78016	0.124833
$497$	−12.4853	−0.560041
$498$	7.06147	0.316432
$499$	−3.00334	−0.134448	−0.0672240	−	0.997738i	$-0.521414\pi$
$499$	−3.00334	−0.134448	−0.0672240	+	0.997738i	$0.521414\pi$
$500$	−1.00000	−0.0447214
$501$	11.7019	0.522804
$502$	17.9810	0.802531
$503$	−11.2777	−0.502848	−0.251424	−	0.967877i	$-0.580899\pi$
$503$	−11.2777	−0.502848	−0.251424	+	0.967877i	$0.580899\pi$
$504$	1.08239	0.0482136
$505$	−6.72965	−0.299465
$506$	3.74325	0.166408
$507$	−1.68554	−0.0748576
$508$	5.02280	0.222851
$509$	23.0947	1.02365	0.511827	−	0.859088i	$-0.328969\pi$
$509$	23.0947	1.02365	0.511827	+	0.859088i	$0.328969\pi$
$510$	0	0
$511$	−3.69308	−0.163372
$512$	−1.00000	−0.0441942
$513$	8.05468	0.355623
$514$	−26.3515	−1.16231
$515$	−6.90575	−0.304304
$516$	−1.43514	−0.0631784
$517$	5.65685	0.248788
$518$	−12.2104	−0.536494
$519$	−22.2581	−0.977023
$520$	−3.36370	−0.147508
$521$	9.63142	0.421960	0.210980	−	0.977490i	$-0.432334\pi$
$521$	9.63142	0.421960	0.210980	+	0.977490i	$0.432334\pi$
$522$	9.35237	0.409342
$523$	24.2409	1.05998	0.529991	−	0.848003i	$-0.322195\pi$
$523$	24.2409	1.05998	0.529991	+	0.848003i	$0.322195\pi$
$524$	−15.9774	−0.697974
$525$	−1.08239	−0.0472395
$526$	−17.7014	−0.771818
$527$	0	0
$528$	0.469266	0.0204222
$529$	40.6297	1.76651
$530$	−5.39104	−0.234172
$531$	0.568047	0.0246512
$532$	−8.71832	−0.377987
$533$	−21.7747	−0.943165
$534$	4.32050	0.186966
$535$	8.43649	0.364741
$536$	−15.3250	−0.661941
$537$	17.2400	0.743962
$538$	25.9665	1.11949
$539$	−2.73508	−0.117808
$540$	−1.00000	−0.0430331
$541$	−10.4557	−0.449525	−0.224763	−	0.974414i	$-0.572161\pi$
$541$	−10.4557	−0.449525	−0.224763	+	0.974414i	$0.572161\pi$
$542$	−7.65685	−0.328890
$543$	−3.93174	−0.168727
$544$	0	0
$545$	−6.43060	−0.275457
$546$	−3.64084	−0.155814
$547$	1.06735	0.0456367	0.0228184	−	0.999740i	$-0.492736\pi$
$547$	1.06735	0.0456367	0.0228184	+	0.999740i	$0.492736\pi$
$548$	6.39782	0.273302
$549$	6.12612	0.261456
$550$	−0.469266	−0.0200096
$551$	−75.3304	−3.20918
$552$	7.97682	0.339516
$553$	−15.8049	−0.672092
$554$	−7.59899	−0.322850
$555$	11.2809	0.478849
$556$	22.8117	0.967430
$557$	−7.14892	−0.302910	−0.151455	−	0.988464i	$-0.548396\pi$
$557$	−7.14892	−0.302910	−0.151455	+	0.988464i	$0.548396\pi$
$558$	−2.78016	−0.117694
$559$	4.82737	0.204176
$560$	1.08239	0.0457394
$561$	0	0
$562$	8.72739	0.368143
$563$	12.4898	0.526382	0.263191	−	0.964744i	$-0.415225\pi$
$563$	12.4898	0.526382	0.263191	+	0.964744i	$0.415225\pi$
$564$	12.0547	0.507594
$565$	−0.349561	−0.0147061
$566$	−8.99640	−0.378147
$567$	−1.08239	−0.0454562
$568$	−11.5349	−0.483993
$569$	1.50246	0.0629864	0.0314932	−	0.999504i	$-0.489974\pi$
$569$	1.50246	0.0629864	0.0314932	+	0.999504i	$0.489974\pi$
$570$	8.05468	0.337373
$571$	28.8946	1.20920	0.604601	−	0.796528i	$-0.293333\pi$
$571$	28.8946	1.20920	0.604601	+	0.796528i	$0.293333\pi$
$572$	−1.57847	−0.0659991
$573$	−12.4557	−0.520344
$574$	7.00679	0.292458
$575$	−7.97682	−0.332657
$576$	1.00000	0.0416667
$577$	40.5141	1.68663	0.843313	−	0.537423i	$-0.180602\pi$
$577$	40.5141	1.68663	0.843313	+	0.537423i	$0.180602\pi$
$578$	0	0
$579$	12.9259	0.537183
$580$	9.35237	0.388336
$581$	7.64328	0.317097
$582$	14.3961	0.596738
$583$	−2.52983	−0.104775
$584$	−3.41196	−0.141188
$585$	3.36370	0.139072
$586$	15.9091	0.657199
$587$	13.4174	0.553797	0.276899	−	0.960899i	$-0.410693\pi$
$587$	13.4174	0.553797	0.276899	+	0.960899i	$0.410693\pi$
$588$	−5.82843	−0.240360
$589$	22.3933	0.922701
$590$	0.568047	0.0233861
$591$	1.08655	0.0446949
$592$	−11.2809	−0.463643
$593$	28.1731	1.15693	0.578465	−	0.815707i	$-0.303652\pi$
$593$	28.1731	1.15693	0.578465	+	0.815707i	$0.303652\pi$
$594$	−0.469266	−0.0192542
$595$	0	0
$596$	13.8330	0.566620
$597$	−6.42344	−0.262894
$598$	−26.8316	−1.09723
$599$	16.7410	0.684018	0.342009	−	0.939697i	$-0.388893\pi$
$599$	16.7410	0.684018	0.342009	+	0.939697i	$0.388893\pi$
$600$	−1.00000	−0.0408248
$601$	−38.7163	−1.57927	−0.789635	−	0.613576i	$-0.789730\pi$
$601$	−38.7163	−1.57927	−0.789635	+	0.613576i	$0.789730\pi$
$602$	−1.55338	−0.0633111
$603$	15.3250	0.624084
$604$	4.45250	0.181170
$605$	10.7798	0.438261
$606$	−6.72965	−0.273373
$607$	46.1194	1.87193	0.935964	−	0.352094i	$-0.114530\pi$
$607$	46.1194	1.87193	0.935964	+	0.352094i	$0.114530\pi$
$608$	−8.05468	−0.326660
$609$	10.1229	0.410202
$610$	6.12612	0.248039
$611$	−40.5483	−1.64041
$612$	0	0
$613$	−35.1672	−1.42039	−0.710195	−	0.704005i	$-0.751393\pi$
$613$	−35.1672	−1.42039	−0.710195	+	0.704005i	$0.751393\pi$
$614$	−2.83975	−0.114603
$615$	−6.47343	−0.261034
$616$	0.507930	0.0204651
$617$	−12.3722	−0.498086	−0.249043	−	0.968492i	$-0.580116\pi$
$617$	−12.3722	−0.498086	−0.249043	+	0.968492i	$0.580116\pi$
$618$	−6.90575	−0.277790
$619$	30.4253	1.22290	0.611449	−	0.791284i	$-0.290587\pi$
$619$	30.4253	1.22290	0.611449	+	0.791284i	$0.290587\pi$
$620$	−2.78016	−0.111654
$621$	−7.97682	−0.320099
$622$	−19.0688	−0.764590
$623$	4.67647	0.187359
$624$	−3.36370	−0.134656
$625$	1.00000	0.0400000
$626$	−11.9791	−0.478780
$627$	3.77979	0.150950
$628$	−16.3697	−0.653224
$629$	0	0
$630$	−1.08239	−0.0431235
$631$	29.0826	1.15776	0.578880	−	0.815412i	$-0.303490\pi$
$631$	29.0826	1.15776	0.578880	+	0.815412i	$0.303490\pi$
$632$	−14.6018	−0.580828
$633$	1.50114	0.0596650
$634$	−3.63177	−0.144236
$635$	−5.02280	−0.199324
$636$	−5.39104	−0.213768
$637$	19.6051	0.776781
$638$	4.38875	0.173752
$639$	11.5349	0.456313
$640$	1.00000	0.0395285
$641$	−6.59711	−0.260570	−0.130285	−	0.991477i	$-0.541589\pi$
$641$	−6.59711	−0.260570	−0.130285	+	0.991477i	$0.541589\pi$
$642$	8.43649	0.332962
$643$	−0.450066	−0.0177489	−0.00887443	−	0.999961i	$-0.502825\pi$
$643$	−0.450066	−0.0177489	−0.00887443	+	0.999961i	$0.502825\pi$
$644$	8.63405	0.340229
$645$	1.43514	0.0565085
$646$	0	0
$647$	9.22355	0.362615	0.181308	−	0.983426i	$-0.441967\pi$
$647$	9.22355	0.362615	0.181308	+	0.983426i	$0.441967\pi$
$648$	−1.00000	−0.0392837
$649$	0.266565	0.0104636
$650$	3.36370	0.131935
$651$	−3.00923	−0.117941
$652$	9.42063	0.368940
$653$	−37.7025	−1.47541	−0.737706	−	0.675122i	$-0.764091\pi$
$653$	−37.7025	−1.47541	−0.737706	+	0.675122i	$0.764091\pi$
$654$	−6.43060	−0.251456
$655$	15.9774	0.624287
$656$	6.47343	0.252745
$657$	3.41196	0.133113
$658$	13.0479	0.508660
$659$	−22.4663	−0.875161	−0.437581	−	0.899179i	$-0.644165\pi$
$659$	−22.4663	−0.875161	−0.437581	+	0.899179i	$0.644165\pi$
$660$	−0.469266	−0.0182662
$661$	31.7662	1.23556	0.617781	−	0.786350i	$-0.288032\pi$
$661$	31.7662	1.23556	0.617781	+	0.786350i	$0.288032\pi$
$662$	8.67197	0.337046
$663$	0	0
$664$	7.06147	0.274038
$665$	8.71832	0.338082
$666$	11.2809	0.437127
$667$	74.6022	2.88861
$668$	11.7019	0.452761
$669$	0.951362	0.0367818
$670$	15.3250	0.592058
$671$	2.87478	0.110980
$672$	1.08239	0.0417542
$673$	−1.70493	−0.0657203	−0.0328602	−	0.999460i	$-0.510462\pi$
$673$	−1.70493	−0.0657203	−0.0328602	+	0.999460i	$0.510462\pi$
$674$	−16.0292	−0.617424
$675$	1.00000	0.0384900
$676$	−1.68554	−0.0648286
$677$	−25.8641	−0.994037	−0.497018	−	0.867740i	$-0.665572\pi$
$677$	−25.8641	−0.994037	−0.497018	+	0.867740i	$0.665572\pi$
$678$	−0.349561	−0.0134248
$679$	15.5822	0.597991
$680$	0	0
$681$	2.79884	0.107252
$682$	−1.30464	−0.0499571
$683$	−38.7639	−1.48326	−0.741630	−	0.670809i	$-0.765947\pi$
$683$	−38.7639	−1.48326	−0.741630	+	0.670809i	$0.765947\pi$
$684$	8.05468	0.307978
$685$	−6.39782	−0.244448
$686$	−13.8854	−0.530147
$687$	21.3910	0.816119
$688$	−1.43514	−0.0547141
$689$	18.1338	0.690843
$690$	−7.97682	−0.303672
$691$	−18.7191	−0.712107	−0.356053	−	0.934466i	$-0.615878\pi$
$691$	−18.7191	−0.712107	−0.356053	+	0.934466i	$0.615878\pi$
$692$	−22.2581	−0.846127
$693$	−0.507930	−0.0192947
$694$	−20.4222	−0.775215
$695$	−22.8117	−0.865296
$696$	9.35237	0.354501
$697$	0	0
$698$	−4.77375	−0.180689
$699$	8.96685	0.339157
$700$	−1.08239	−0.0409106
$701$	−4.84808	−0.183109	−0.0915546	−	0.995800i	$-0.529184\pi$
$701$	−4.84808	−0.183109	−0.0915546	+	0.995800i	$0.529184\pi$
$702$	3.36370	0.126955
$703$	−90.8643	−3.42701
$704$	0.469266	0.0176861
$705$	−12.0547	−0.454005
$706$	−35.6889	−1.34317
$707$	−7.28412	−0.273947
$708$	0.568047	0.0213485
$709$	16.6752	0.626249	0.313124	−	0.949712i	$-0.398624\pi$
$709$	16.6752	0.626249	0.313124	+	0.949712i	$0.398624\pi$
$710$	11.5349	0.432897
$711$	14.6018	0.547610
$712$	4.32050	0.161917
$713$	−22.1769	−0.830530
$714$	0	0
$715$	1.57847	0.0590314
$716$	17.2400	0.644290
$717$	6.52395	0.243641
$718$	−17.2455	−0.643595
$719$	14.1239	0.526733	0.263367	−	0.964696i	$-0.415167\pi$
$719$	14.1239	0.526733	0.263367	+	0.964696i	$0.415167\pi$
$720$	−1.00000	−0.0372678
$721$	−7.47474	−0.278374
$722$	−45.8779	−1.70740
$723$	−9.27133	−0.344804
$724$	−3.93174	−0.146122
$725$	−9.35237	−0.347338
$726$	10.7798	0.400075
$727$	−50.5096	−1.87330	−0.936649	−	0.350269i	$-0.886090\pi$
$727$	−50.5096	−1.87330	−0.936649	+	0.350269i	$0.886090\pi$
$728$	−3.64084	−0.134938
$729$	1.00000	0.0370370
$730$	3.41196	0.126282
$731$	0	0
$732$	6.12612	0.226428
$733$	6.98582	0.258027	0.129014	−	0.991643i	$-0.458819\pi$
$733$	6.98582	0.258027	0.129014	+	0.991643i	$0.458819\pi$
$734$	24.3305	0.898057
$735$	5.82843	0.214985
$736$	7.97682	0.294030
$737$	7.19152	0.264903
$738$	−6.47343	−0.238290
$739$	−2.48798	−0.0915219	−0.0457609	−	0.998952i	$-0.514571\pi$
$739$	−2.48798	−0.0915219	−0.0457609	+	0.998952i	$0.514571\pi$
$740$	11.2809	0.414695
$741$	−27.0935	−0.995305
$742$	−5.83522	−0.214218
$743$	−16.8735	−0.619029	−0.309514	−	0.950895i	$-0.600167\pi$
$743$	−16.8735	−0.619029	−0.309514	+	0.950895i	$0.600167\pi$
$744$	−2.78016	−0.101926
$745$	−13.8330	−0.506801
$746$	−24.6091	−0.901005
$747$	−7.06147	−0.258366
$748$	0	0
$749$	9.13159	0.333661
$750$	1.00000	0.0365148
$751$	−0.995121	−0.0363125	−0.0181562	−	0.999835i	$-0.505780\pi$
$751$	−0.995121	−0.0363125	−0.0181562	+	0.999835i	$0.505780\pi$
$752$	12.0547	0.439589
$753$	−17.9810	−0.655264
$754$	−31.4585	−1.14565
$755$	−4.45250	−0.162043
$756$	−1.08239	−0.0393662
$757$	−32.0130	−1.16353	−0.581766	−	0.813356i	$-0.697638\pi$
$757$	−32.0130	−1.16353	−0.581766	+	0.813356i	$0.697638\pi$
$758$	−24.7853	−0.900241
$759$	−3.74325	−0.135872
$760$	8.05468	0.292174
$761$	0.175176	0.00635012	0.00317506	−	0.999995i	$-0.498989\pi$
$761$	0.175176	0.00635012	0.00317506	+	0.999995i	$0.498989\pi$
$762$	−5.02280	−0.181957
$763$	−6.96043	−0.251985
$764$	−12.4557	−0.450631
$765$	0	0
$766$	−9.30088	−0.336054
$767$	−1.91074	−0.0689928
$768$	1.00000	0.0360844
$769$	29.6625	1.06966	0.534828	−	0.844961i	$-0.320376\pi$
$769$	29.6625	1.06966	0.534828	+	0.844961i	$0.320376\pi$
$770$	−0.507930	−0.0183045
$771$	26.3515	0.949025
$772$	12.9259	0.465214
$773$	−15.8564	−0.570314	−0.285157	−	0.958481i	$-0.592046\pi$
$773$	−15.8564	−0.570314	−0.285157	+	0.958481i	$0.592046\pi$
$774$	1.43514	0.0515850
$775$	2.78016	0.0998664
$776$	14.3961	0.516790
$777$	12.2104	0.438045
$778$	28.3175	1.01523
$779$	52.1414	1.86816
$780$	3.36370	0.120440
$781$	5.41294	0.193690
$782$	0	0
$783$	−9.35237	−0.334227
$784$	−5.82843	−0.208158
$785$	16.3697	0.584261
$786$	15.9774	0.569893
$787$	21.4129	0.763289	0.381644	−	0.924309i	$-0.375358\pi$
$787$	21.4129	0.763289	0.381644	+	0.924309i	$0.375358\pi$
$788$	1.08655	0.0387069
$789$	17.7014	0.630187
$790$	14.6018	0.519509
$791$	−0.378362	−0.0134530
$792$	−0.469266	−0.0166747
$793$	−20.6064	−0.731755
$794$	−4.63856	−0.164616
$795$	5.39104	0.191200
$796$	−6.42344	−0.227673
$797$	18.1912	0.644365	0.322183	−	0.946678i	$-0.395584\pi$
$797$	18.1912	0.644365	0.322183	+	0.946678i	$0.395584\pi$
$798$	8.71832	0.308625
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−4.32050	−0.152657
$802$	−12.6156	−0.445471
$803$	1.60112	0.0565022
$804$	15.3250	0.540472
$805$	−8.63405	−0.304310
$806$	9.35162	0.329397
$807$	−25.9665	−0.914063
$808$	−6.72965	−0.236748
$809$	23.9017	0.840341	0.420170	−	0.907445i	$-0.361970\pi$
$809$	23.9017	0.840341	0.420170	+	0.907445i	$0.361970\pi$
$810$	1.00000	0.0351364
$811$	7.71285	0.270835	0.135417	−	0.990789i	$-0.456762\pi$
$811$	7.71285	0.270835	0.135417	+	0.990789i	$0.456762\pi$
$812$	10.1229	0.355245
$813$	7.65685	0.268538
$814$	5.29376	0.185546
$815$	−9.42063	−0.329990
$816$	0	0
$817$	−11.5596	−0.404418
$818$	−4.24662	−0.148479
$819$	3.64084	0.127221
$820$	−6.47343	−0.226062
$821$	−24.0606	−0.839720	−0.419860	−	0.907589i	$-0.637921\pi$
$821$	−24.0606	−0.839720	−0.419860	+	0.907589i	$0.637921\pi$
$822$	−6.39782	−0.223150
$823$	10.8667	0.378791	0.189396	−	0.981901i	$-0.439347\pi$
$823$	10.8667	0.378791	0.189396	+	0.981901i	$0.439347\pi$
$824$	−6.90575	−0.240573
$825$	0.469266	0.0163378
$826$	0.614850	0.0213934
$827$	24.5355	0.853181	0.426591	−	0.904445i	$-0.359714\pi$
$827$	24.5355	0.853181	0.426591	+	0.904445i	$0.359714\pi$
$828$	−7.97682	−0.277214
$829$	−36.7831	−1.27753	−0.638765	−	0.769402i	$-0.720555\pi$
$829$	−36.7831	−1.27753	−0.638765	+	0.769402i	$0.720555\pi$
$830$	−7.06147	−0.245107
$831$	7.59899	0.263606
$832$	−3.36370	−0.116615
$833$	0	0
$834$	−22.8117	−0.789903
$835$	−11.7019	−0.404962
$836$	3.77979	0.130727
$837$	2.78016	0.0960965
$838$	7.53906	0.260432
$839$	0.0537024	0.00185401	0.000927006	−	1.00000i	$-0.499705\pi$
$839$	0.0537024	0.00185401	0.000927006		1.00000i	$0.499705\pi$
$840$	−1.08239	−0.0373461
$841$	58.4669	2.01610
$842$	−22.8503	−0.787474
$843$	−8.72739	−0.300587
$844$	1.50114	0.0516714
$845$	1.68554	0.0579845
$846$	−12.0547	−0.414448
$847$	11.6680	0.400916
$848$	−5.39104	−0.185129
$849$	8.99640	0.308756
$850$	0	0
$851$	89.9860	3.08468
$852$	11.5349	0.395179
$853$	−36.0443	−1.23413	−0.617066	−	0.786911i	$-0.711679\pi$
$853$	−36.0443	−1.23413	−0.617066	+	0.786911i	$0.711679\pi$
$854$	6.63087	0.226904
$855$	−8.05468	−0.275464
$856$	8.43649	0.288353
$857$	−25.6542	−0.876331	−0.438165	−	0.898894i	$-0.644372\pi$
$857$	−25.6542	−0.876331	−0.438165	+	0.898894i	$0.644372\pi$
$858$	1.57847	0.0538881
$859$	8.04831	0.274605	0.137302	−	0.990529i	$-0.456157\pi$
$859$	8.04831	0.274605	0.137302	+	0.990529i	$0.456157\pi$
$860$	1.43514	0.0489378
$861$	−7.00679	−0.238791
$862$	8.03105	0.273539
$863$	37.0426	1.26095	0.630473	−	0.776211i	$-0.282861\pi$
$863$	37.0426	1.26095	0.630473	+	0.776211i	$0.282861\pi$
$864$	−1.00000	−0.0340207
$865$	22.2581	0.756799
$866$	11.5164	0.391344
$867$	0	0
$868$	−3.00923	−0.102140
$869$	6.85213	0.232443
$870$	−9.35237	−0.317075
$871$	−51.5488	−1.74666
$872$	−6.43060	−0.217768
$873$	−14.3961	−0.487234
$874$	64.2507	2.17331
$875$	1.08239	0.0365915
$876$	3.41196	0.115279
$877$	39.2893	1.32670	0.663352	−	0.748308i	$-0.269133\pi$
$877$	39.2893	1.32670	0.663352	+	0.748308i	$0.269133\pi$
$878$	−12.4917	−0.421574
$879$	−15.9091	−0.536600
$880$	−0.469266	−0.0158190
$881$	41.8999	1.41164	0.705822	−	0.708389i	$-0.250578\pi$
$881$	41.8999	1.41164	0.705822	+	0.708389i	$0.250578\pi$
$882$	5.82843	0.196253
$883$	14.2500	0.479550	0.239775	−	0.970828i	$-0.422926\pi$
$883$	14.2500	0.479550	0.239775	+	0.970828i	$0.422926\pi$
$884$	0	0
$885$	−0.568047	−0.0190947
$886$	−1.31446	−0.0441600
$887$	−44.7208	−1.50158	−0.750789	−	0.660542i	$-0.770326\pi$
$887$	−44.7208	−1.50158	−0.750789	+	0.660542i	$0.770326\pi$
$888$	11.2809	0.378563
$889$	−5.43664	−0.182339
$890$	−4.32050	−0.144823
$891$	0.469266	0.0157210
$892$	0.951362	0.0318540
$893$	97.0966	3.24921
$894$	−13.8330	−0.462643
$895$	−17.2400	−0.576270
$896$	1.08239	0.0361602
$897$	26.8316	0.895881
$898$	12.0338	0.401572
$899$	−26.0011	−0.867186
$900$	1.00000	0.0333333
$901$	0	0
$902$	−3.03776	−0.101146
$903$	1.55338	0.0516933
$904$	−0.349561	−0.0116262
$905$	3.93174	0.130696
$906$	−4.45250	−0.147925
$907$	44.1950	1.46747	0.733736	−	0.679435i	$-0.237775\pi$
$907$	44.1950	1.46747	0.733736	+	0.679435i	$0.237775\pi$
$908$	2.79884	0.0928826
$909$	6.72965	0.223208
$910$	3.64084	0.120693
$911$	−11.9418	−0.395649	−0.197825	−	0.980237i	$-0.563388\pi$
$911$	−11.9418	−0.395649	−0.197825	+	0.980237i	$0.563388\pi$
$912$	8.05468	0.266717
$913$	−3.31371	−0.109668
$914$	−7.01830	−0.232145
$915$	−6.12612	−0.202523
$916$	21.3910	0.706780
$917$	17.2938	0.571090
$918$	0	0
$919$	−7.09806	−0.234144	−0.117072	−	0.993123i	$-0.537351\pi$
$919$	−7.09806	−0.234144	−0.117072	+	0.993123i	$0.537351\pi$
$920$	−7.97682	−0.262988
$921$	2.83975	0.0935730
$922$	16.9363	0.557767
$923$	−38.7999	−1.27711
$924$	−0.507930	−0.0167097
$925$	−11.2809	−0.370915
$926$	1.28093	0.0420940
$927$	6.90575	0.226815
$928$	9.35237	0.307007
$929$	32.4953	1.06614	0.533069	−	0.846072i	$-0.321039\pi$
$929$	32.4953	1.06614	0.533069	+	0.846072i	$0.321039\pi$
$930$	2.78016	0.0911651
$931$	−46.9461	−1.53860
$932$	8.96685	0.293719
$933$	19.0688	0.624285
$934$	−34.2741	−1.12148
$935$	0	0
$936$	3.36370	0.109946
$937$	6.33407	0.206925	0.103463	−	0.994633i	$-0.467008\pi$
$937$	6.33407	0.206925	0.103463	+	0.994633i	$0.467008\pi$
$938$	16.5877	0.541607
$939$	11.9791	0.390922
$940$	−12.0547	−0.393180
$941$	6.07414	0.198011	0.0990057	−	0.995087i	$-0.468434\pi$
$941$	6.07414	0.198011	0.0990057	+	0.995087i	$0.468434\pi$
$942$	16.3697	0.533355
$943$	−51.6374	−1.68154
$944$	0.568047	0.0184884
$945$	1.08239	0.0352102
$946$	0.673462	0.0218961
$947$	16.4365	0.534114	0.267057	−	0.963681i	$-0.413949\pi$
$947$	16.4365	0.534114	0.267057	+	0.963681i	$0.413949\pi$
$948$	14.6018	0.474244
$949$	−11.4768	−0.372553
$950$	−8.05468	−0.261328
$951$	3.63177	0.117768
$952$	0	0
$953$	−9.16437	−0.296863	−0.148431	−	0.988923i	$-0.547422\pi$
$953$	−9.16437	−0.296863	−0.148431	+	0.988923i	$0.547422\pi$
$954$	5.39104	0.174541
$955$	12.4557	0.403057
$956$	6.52395	0.210999
$957$	−4.38875	−0.141868
$958$	41.9602	1.35567
$959$	−6.92496	−0.223618
$960$	−1.00000	−0.0322749
$961$	−23.2707	−0.750668
$962$	−37.9456	−1.22342
$963$	−8.43649	−0.271862
$964$	−9.27133	−0.298609
$965$	−12.9259	−0.416100
$966$	−8.63405	−0.277796
$967$	−54.7067	−1.75925	−0.879624	−	0.475669i	$-0.842206\pi$
$967$	−54.7067	−1.75925	−0.879624	+	0.475669i	$0.842206\pi$
$968$	10.7798	0.346476
$969$	0	0
$970$	−14.3961	−0.462231
$971$	−6.94773	−0.222963	−0.111482	−	0.993766i	$-0.535560\pi$
$971$	−6.94773	−0.222963	−0.111482	+	0.993766i	$0.535560\pi$
$972$	1.00000	0.0320750
$973$	−24.6912	−0.791562
$974$	−27.0816	−0.867752
$975$	−3.36370	−0.107725
$976$	6.12612	0.196092
$977$	27.8318	0.890417	0.445209	−	0.895427i	$-0.353130\pi$
$977$	27.8318	0.890417	0.445209	+	0.895427i	$0.353130\pi$
$978$	−9.42063	−0.301238
$979$	−2.02746	−0.0647980
$980$	5.82843	0.186182
$981$	6.43060	0.205313
$982$	21.4376	0.684101
$983$	−34.1224	−1.08834	−0.544168	−	0.838976i	$-0.683155\pi$
$983$	−34.1224	−1.08834	−0.544168	+	0.838976i	$0.683155\pi$
$984$	−6.47343	−0.206365
$985$	−1.08655	−0.0346205
$986$	0	0
$987$	−13.0479	−0.415319
$988$	−27.0935	−0.861959
$989$	11.4478	0.364020
$990$	0.469266	0.0149143
$991$	−0.533546	−0.0169486	−0.00847432	−	0.999964i	$-0.502697\pi$
$991$	−0.533546	−0.0169486	−0.00847432	+	0.999964i	$0.502697\pi$
$992$	−2.78016	−0.0882703
$993$	−8.67197	−0.275197
$994$	12.4853	0.396009
$995$	6.42344	0.203637
$996$	−7.06147	−0.223751
$997$	−49.0381	−1.55305	−0.776527	−	0.630084i	$-0.783020\pi$
$997$	−49.0381	−1.55305	−0.776527	+	0.630084i	$0.783020\pi$
$998$	3.00334	0.0950691
$999$	−11.2809	−0.356913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bw.1.2		4
17.8	even	8		510.2.p.d.421.1	yes	8
17.15	even	8		510.2.p.d.361.1	✓	8
17.16	even	2		8670.2.a.bt.1.3		4
51.8	odd	8		1530.2.q.i.1441.3		8
51.32	odd	8		1530.2.q.i.361.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.p.d.361.1	✓	8	17.15	even	8
510.2.p.d.421.1	yes	8	17.8	even	8
1530.2.q.i.361.3		8	51.32	odd	8
1530.2.q.i.1441.3		8	51.8	odd	8
8670.2.a.bt.1.3		4	17.16	even	2
8670.2.a.bw.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.469266 q^{11} +1.00000 q^{12} -3.36370 q^{13} +1.08239 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +8.05468 q^{19} -1.00000 q^{20} -1.08239 q^{21} -0.469266 q^{22} -7.97682 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.36370 q^{26} +1.00000 q^{27} -1.08239 q^{28} -9.35237 q^{29} +1.00000 q^{30} +2.78016 q^{31} -1.00000 q^{32} +0.469266 q^{33} +1.08239 q^{35} +1.00000 q^{36} -11.2809 q^{37} -8.05468 q^{38} -3.36370 q^{39} +1.00000 q^{40} +6.47343 q^{41} +1.08239 q^{42} -1.43514 q^{43} +0.469266 q^{44} -1.00000 q^{45} +7.97682 q^{46} +12.0547 q^{47} +1.00000 q^{48} -5.82843 q^{49} -1.00000 q^{50} -3.36370 q^{52} -5.39104 q^{53} -1.00000 q^{54} -0.469266 q^{55} +1.08239 q^{56} +8.05468 q^{57} +9.35237 q^{58} +0.568047 q^{59} -1.00000 q^{60} +6.12612 q^{61} -2.78016 q^{62} -1.08239 q^{63} +1.00000 q^{64} +3.36370 q^{65} -0.469266 q^{66} +15.3250 q^{67} -7.97682 q^{69} -1.08239 q^{70} +11.5349 q^{71} -1.00000 q^{72} +3.41196 q^{73} +11.2809 q^{74} +1.00000 q^{75} +8.05468 q^{76} -0.507930 q^{77} +3.36370 q^{78} +14.6018 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.47343 q^{82} -7.06147 q^{83} -1.08239 q^{84} +1.43514 q^{86} -9.35237 q^{87} -0.469266 q^{88} -4.32050 q^{89} +1.00000 q^{90} +3.64084 q^{91} -7.97682 q^{92} +2.78016 q^{93} -12.0547 q^{94} -8.05468 q^{95} -1.00000 q^{96} -14.3961 q^{97} +5.82843 q^{98} +0.469266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 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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