LMFDB - Embedded newform 5070.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.35690 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.35690 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.69202 q^{11} +1.00000 q^{12} +3.35690 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.939001 q^{17} -1.00000 q^{18} +4.85086 q^{19} +1.00000 q^{20} -3.35690 q^{21} +1.69202 q^{22} +4.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -3.35690 q^{28} -8.12498 q^{29} -1.00000 q^{30} -4.08815 q^{31} -1.00000 q^{32} -1.69202 q^{33} -0.939001 q^{34} -3.35690 q^{35} +1.00000 q^{36} -11.6310 q^{37} -4.85086 q^{38} -1.00000 q^{40} -3.86294 q^{41} +3.35690 q^{42} +4.02177 q^{43} -1.69202 q^{44} +1.00000 q^{45} -4.04892 q^{46} +1.27413 q^{47} +1.00000 q^{48} +4.26875 q^{49} -1.00000 q^{50} +0.939001 q^{51} +5.74094 q^{53} -1.00000 q^{54} -1.69202 q^{55} +3.35690 q^{56} +4.85086 q^{57} +8.12498 q^{58} -0.417895 q^{59} +1.00000 q^{60} +0.198062 q^{61} +4.08815 q^{62} -3.35690 q^{63} +1.00000 q^{64} +1.69202 q^{66} -8.93900 q^{67} +0.939001 q^{68} +4.04892 q^{69} +3.35690 q^{70} +5.15883 q^{71} -1.00000 q^{72} +11.5308 q^{73} +11.6310 q^{74} +1.00000 q^{75} +4.85086 q^{76} +5.67994 q^{77} -4.94869 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.86294 q^{82} -13.1739 q^{83} -3.35690 q^{84} +0.939001 q^{85} -4.02177 q^{86} -8.12498 q^{87} +1.69202 q^{88} +9.47650 q^{89} -1.00000 q^{90} +4.04892 q^{92} -4.08815 q^{93} -1.27413 q^{94} +4.85086 q^{95} -1.00000 q^{96} -15.4547 q^{97} -4.26875 q^{98} -1.69202 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} + 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + q^{19} + 3 q^{20} - 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} - 6 q^{28} - 3 q^{30} - 16 q^{31} - 3 q^{32} + 7 q^{34} - 6 q^{35} + 3 q^{36} - 20 q^{37} - q^{38} - 3 q^{40} - 17 q^{41} + 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} - 7 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} + 6 q^{56} + q^{57} - 7 q^{59} + 3 q^{60} + 5 q^{61} + 16 q^{62} - 6 q^{63} + 3 q^{64} - 17 q^{67} - 7 q^{68} + 3 q^{69} + 6 q^{70} + 7 q^{71} - 3 q^{72} - 3 q^{73} + 20 q^{74} + 3 q^{75} + q^{76} - 7 q^{77} + 17 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} - 6 q^{83} - 6 q^{84} - 7 q^{85} - 9 q^{86} + 3 q^{89} - 3 q^{90} + 3 q^{92} - 16 q^{93} + 7 q^{94} + q^{95} - 3 q^{96} - 24 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 + 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + q^19 + 3 * q^20 - 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 - 6 * q^28 - 3 * q^30 - 16 * q^31 - 3 * q^32 + 7 * q^34 - 6 * q^35 + 3 * q^36 - 20 * q^37 - q^38 - 3 * q^40 - 17 * q^41 + 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 - 7 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 + 6 * q^56 + q^57 - 7 * q^59 + 3 * q^60 + 5 * q^61 + 16 * q^62 - 6 * q^63 + 3 * q^64 - 17 * q^67 - 7 * q^68 + 3 * q^69 + 6 * q^70 + 7 * q^71 - 3 * q^72 - 3 * q^73 + 20 * q^74 + 3 * q^75 + q^76 - 7 * q^77 + 17 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 - 6 * q^83 - 6 * q^84 - 7 * q^85 - 9 * q^86 + 3 * q^89 - 3 * q^90 + 3 * q^92 - 16 * q^93 + 7 * q^94 + q^95 - 3 * q^96 - 24 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−3.35690	−1.26879	−0.634394	−	0.773010i	$-0.718750\pi$
$7$	−3.35690	−1.26879	−0.634394	+	0.773010i	$0.718750\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.69202	−0.510164	−0.255082	−	0.966919i	$-0.582102\pi$
$11$	−1.69202	−0.510164	−0.255082	+	0.966919i	$0.582102\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	3.35690	0.897168
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0.939001	0.227741	0.113871	−	0.993496i	$-0.463675\pi$
$17$	0.939001	0.227741	0.113871	+	0.993496i	$0.463675\pi$
$18$	−1.00000	−0.235702
$19$	4.85086	1.11286	0.556431	−	0.830894i	$-0.312170\pi$
$19$	4.85086	1.11286	0.556431	+	0.830894i	$0.312170\pi$
$20$	1.00000	0.223607
$21$	−3.35690	−0.732535
$22$	1.69202	0.360740
$23$	4.04892	0.844258	0.422129	−	0.906536i	$-0.361283\pi$
$23$	4.04892	0.844258	0.422129	+	0.906536i	$0.361283\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−3.35690	−0.634394
$29$	−8.12498	−1.50877	−0.754386	−	0.656432i	$-0.772065\pi$
$29$	−8.12498	−1.50877	−0.754386	+	0.656432i	$0.772065\pi$
$30$	−1.00000	−0.182574
$31$	−4.08815	−0.734253	−0.367126	−	0.930171i	$-0.619658\pi$
$31$	−4.08815	−0.734253	−0.367126	+	0.930171i	$0.619658\pi$
$32$	−1.00000	−0.176777
$33$	−1.69202	−0.294543
$34$	−0.939001	−0.161037
$35$	−3.35690	−0.567419
$36$	1.00000	0.166667
$37$	−11.6310	−1.91213	−0.956064	−	0.293157i	$-0.905294\pi$
$37$	−11.6310	−1.91213	−0.956064	+	0.293157i	$0.905294\pi$
$38$	−4.85086	−0.786913
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.86294	−0.603289	−0.301645	−	0.953420i	$-0.597536\pi$
$41$	−3.86294	−0.603289	−0.301645	+	0.953420i	$0.597536\pi$
$42$	3.35690	0.517980
$43$	4.02177	0.613314	0.306657	−	0.951820i	$-0.400790\pi$
$43$	4.02177	0.613314	0.306657	+	0.951820i	$0.400790\pi$
$44$	−1.69202	−0.255082
$45$	1.00000	0.149071
$46$	−4.04892	−0.596980
$47$	1.27413	0.185850	0.0929252	−	0.995673i	$-0.470378\pi$
$47$	1.27413	0.185850	0.0929252	+	0.995673i	$0.470378\pi$
$48$	1.00000	0.144338
$49$	4.26875	0.609821
$50$	−1.00000	−0.141421
$51$	0.939001	0.131486
$52$	0	0
$53$	5.74094	0.788579	0.394289	−	0.918986i	$-0.370991\pi$
$53$	5.74094	0.788579	0.394289	+	0.918986i	$0.370991\pi$
$54$	−1.00000	−0.136083
$55$	−1.69202	−0.228152
$56$	3.35690	0.448584
$57$	4.85086	0.642511
$58$	8.12498	1.06686
$59$	−0.417895	−0.0544053	−0.0272026	−	0.999630i	$-0.508660\pi$
$59$	−0.417895	−0.0544053	−0.0272026	+	0.999630i	$0.508660\pi$
$60$	1.00000	0.129099
$61$	0.198062	0.0253593	0.0126796	−	0.999920i	$-0.495964\pi$
$61$	0.198062	0.0253593	0.0126796	+	0.999920i	$0.495964\pi$
$62$	4.08815	0.519195
$63$	−3.35690	−0.422929
$64$	1.00000	0.125000
$65$	0	0
$66$	1.69202	0.208273
$67$	−8.93900	−1.09207	−0.546036	−	0.837761i	$-0.683864\pi$
$67$	−8.93900	−1.09207	−0.546036	+	0.837761i	$0.683864\pi$
$68$	0.939001	0.113871
$69$	4.04892	0.487432
$70$	3.35690	0.401226
$71$	5.15883	0.612241	0.306120	−	0.951993i	$-0.400969\pi$
$71$	5.15883	0.612241	0.306120	+	0.951993i	$0.400969\pi$
$72$	−1.00000	−0.117851
$73$	11.5308	1.34958	0.674789	−	0.738011i	$-0.264235\pi$
$73$	11.5308	1.34958	0.674789	+	0.738011i	$0.264235\pi$
$74$	11.6310	1.35208
$75$	1.00000	0.115470
$76$	4.85086	0.556431
$77$	5.67994	0.647289
$78$	0	0
$79$	−4.94869	−0.556771	−0.278386	−	0.960469i	$-0.589799\pi$
$79$	−4.94869	−0.556771	−0.278386	+	0.960469i	$0.589799\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	3.86294	0.426590
$83$	−13.1739	−1.44602	−0.723012	−	0.690836i	$-0.757243\pi$
$83$	−13.1739	−1.44602	−0.723012	+	0.690836i	$0.757243\pi$
$84$	−3.35690	−0.366267
$85$	0.939001	0.101849
$86$	−4.02177	−0.433679
$87$	−8.12498	−0.871089
$88$	1.69202	0.180370
$89$	9.47650	1.00451	0.502254	−	0.864720i	$-0.332504\pi$
$89$	9.47650	1.00451	0.502254	+	0.864720i	$0.332504\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	4.04892	0.422129
$93$	−4.08815	−0.423921
$94$	−1.27413	−0.131416
$95$	4.85086	0.497687
$96$	−1.00000	−0.102062
$97$	−15.4547	−1.56919	−0.784595	−	0.620009i	$-0.787129\pi$
$97$	−15.4547	−1.56919	−0.784595	+	0.620009i	$0.787129\pi$
$98$	−4.26875	−0.431209
$99$	−1.69202	−0.170055
$100$	1.00000	0.100000
$101$	5.38404	0.535732	0.267866	−	0.963456i	$-0.413682\pi$
$101$	5.38404	0.535732	0.267866	+	0.963456i	$0.413682\pi$
$102$	−0.939001	−0.0929750
$103$	4.08575	0.402581	0.201291	−	0.979532i	$-0.435486\pi$
$103$	4.08575	0.402581	0.201291	+	0.979532i	$0.435486\pi$
$104$	0	0
$105$	−3.35690	−0.327599
$106$	−5.74094	−0.557609
$107$	−16.1196	−1.55834	−0.779171	−	0.626812i	$-0.784359\pi$
$107$	−16.1196	−1.55834	−0.779171	+	0.626812i	$0.784359\pi$
$108$	1.00000	0.0962250
$109$	14.2325	1.36323	0.681613	−	0.731713i	$-0.261279\pi$
$109$	14.2325	1.36323	0.681613	+	0.731713i	$0.261279\pi$
$110$	1.69202	0.161328
$111$	−11.6310	−1.10397
$112$	−3.35690	−0.317197
$113$	−13.3274	−1.25373	−0.626866	−	0.779127i	$-0.715663\pi$
$113$	−13.3274	−1.25373	−0.626866	+	0.779127i	$0.715663\pi$
$114$	−4.85086	−0.454324
$115$	4.04892	0.377563
$116$	−8.12498	−0.754386
$117$	0	0
$118$	0.417895	0.0384703
$119$	−3.15213	−0.288955
$120$	−1.00000	−0.0912871
$121$	−8.13706	−0.739733
$122$	−0.198062	−0.0179317
$123$	−3.86294	−0.348309
$124$	−4.08815	−0.367126
$125$	1.00000	0.0894427
$126$	3.35690	0.299056
$127$	−12.8509	−1.14033	−0.570164	−	0.821531i	$-0.693121\pi$
$127$	−12.8509	−1.14033	−0.570164	+	0.821531i	$0.693121\pi$
$128$	−1.00000	−0.0883883
$129$	4.02177	0.354097
$130$	0	0
$131$	−0.570024	−0.0498032	−0.0249016	−	0.999690i	$-0.507927\pi$
$131$	−0.570024	−0.0498032	−0.0249016	+	0.999690i	$0.507927\pi$
$132$	−1.69202	−0.147272
$133$	−16.2838	−1.41199
$134$	8.93900	0.772212
$135$	1.00000	0.0860663
$136$	−0.939001	−0.0805187
$137$	−21.5405	−1.84033	−0.920164	−	0.391533i	$-0.871945\pi$
$137$	−21.5405	−1.84033	−0.920164	+	0.391533i	$0.871945\pi$
$138$	−4.04892	−0.344667
$139$	7.09783	0.602030	0.301015	−	0.953619i	$-0.402674\pi$
$139$	7.09783	0.602030	0.301015	+	0.953619i	$0.402674\pi$
$140$	−3.35690	−0.283709
$141$	1.27413	0.107301
$142$	−5.15883	−0.432920
$143$	0	0
$144$	1.00000	0.0833333
$145$	−8.12498	−0.674743
$146$	−11.5308	−0.954295
$147$	4.26875	0.352081
$148$	−11.6310	−0.956064
$149$	1.14914	0.0941416	0.0470708	−	0.998892i	$-0.485011\pi$
$149$	1.14914	0.0941416	0.0470708	+	0.998892i	$0.485011\pi$
$150$	−1.00000	−0.0816497
$151$	−0.0217703	−0.00177164	−0.000885819	−	1.00000i	$-0.500282\pi$
$151$	−0.0217703	−0.00177164	−0.000885819		1.00000i	$0.500282\pi$
$152$	−4.85086	−0.393456
$153$	0.939001	0.0759137
$154$	−5.67994	−0.457703
$155$	−4.08815	−0.328368
$156$	0	0
$157$	−10.1588	−0.810763	−0.405382	−	0.914148i	$-0.632861\pi$
$157$	−10.1588	−0.810763	−0.405382	+	0.914148i	$0.632861\pi$
$158$	4.94869	0.393697
$159$	5.74094	0.455286
$160$	−1.00000	−0.0790569
$161$	−13.5918	−1.07118
$162$	−1.00000	−0.0785674
$163$	−19.1511	−1.50003	−0.750014	−	0.661422i	$-0.769953\pi$
$163$	−19.1511	−1.50003	−0.750014	+	0.661422i	$0.769953\pi$
$164$	−3.86294	−0.301645
$165$	−1.69202	−0.131724
$166$	13.1739	1.02249
$167$	15.2174	1.17756	0.588780	−	0.808293i	$-0.299608\pi$
$167$	15.2174	1.17756	0.588780	+	0.808293i	$0.299608\pi$
$168$	3.35690	0.258990
$169$	0	0
$170$	−0.939001	−0.0720181
$171$	4.85086	0.370954
$172$	4.02177	0.306657
$173$	21.8213	1.65904	0.829522	−	0.558474i	$-0.188613\pi$
$173$	21.8213	1.65904	0.829522	+	0.558474i	$0.188613\pi$
$174$	8.12498	0.615953
$175$	−3.35690	−0.253757
$176$	−1.69202	−0.127541
$177$	−0.417895	−0.0314109
$178$	−9.47650	−0.710294
$179$	−20.2513	−1.51365	−0.756826	−	0.653616i	$-0.773251\pi$
$179$	−20.2513	−1.51365	−0.756826	+	0.653616i	$0.773251\pi$
$180$	1.00000	0.0745356
$181$	14.4969	1.07755	0.538775	−	0.842450i	$-0.318887\pi$
$181$	14.4969	1.07755	0.538775	+	0.842450i	$0.318887\pi$
$182$	0	0
$183$	0.198062	0.0146412
$184$	−4.04892	−0.298490
$185$	−11.6310	−0.855130
$186$	4.08815	0.299757
$187$	−1.58881	−0.116185
$188$	1.27413	0.0929252
$189$	−3.35690	−0.244178
$190$	−4.85086	−0.351918
$191$	−10.1806	−0.736643	−0.368321	−	0.929699i	$-0.620067\pi$
$191$	−10.1806	−0.736643	−0.368321	+	0.929699i	$0.620067\pi$
$192$	1.00000	0.0721688
$193$	−27.6407	−1.98962	−0.994811	−	0.101741i	$-0.967559\pi$
$193$	−27.6407	−1.98962	−0.994811	+	0.101741i	$0.967559\pi$
$194$	15.4547	1.10958
$195$	0	0
$196$	4.26875	0.304911
$197$	13.0261	0.928070	0.464035	−	0.885817i	$-0.346401\pi$
$197$	13.0261	0.928070	0.464035	+	0.885817i	$0.346401\pi$
$198$	1.69202	0.120247
$199$	6.67563	0.473223	0.236611	−	0.971604i	$-0.423963\pi$
$199$	6.67563	0.473223	0.236611	+	0.971604i	$0.423963\pi$
$200$	−1.00000	−0.0707107
$201$	−8.93900	−0.630509
$202$	−5.38404	−0.378820
$203$	27.2747	1.91431
$204$	0.939001	0.0657432
$205$	−3.86294	−0.269799
$206$	−4.08575	−0.284668
$207$	4.04892	0.281419
$208$	0	0
$209$	−8.20775	−0.567742
$210$	3.35690	0.231648
$211$	−11.3056	−0.778309	−0.389154	−	0.921173i	$-0.627233\pi$
$211$	−11.3056	−0.778309	−0.389154	+	0.921173i	$0.627233\pi$
$212$	5.74094	0.394289
$213$	5.15883	0.353477
$214$	16.1196	1.10191
$215$	4.02177	0.274282
$216$	−1.00000	−0.0680414
$217$	13.7235	0.931611
$218$	−14.2325	−0.963947
$219$	11.5308	0.779179
$220$	−1.69202	−0.114076
$221$	0	0
$222$	11.6310	0.780623
$223$	−18.0911	−1.21147	−0.605736	−	0.795666i	$-0.707121\pi$
$223$	−18.0911	−1.21147	−0.605736	+	0.795666i	$0.707121\pi$
$224$	3.35690	0.224292
$225$	1.00000	0.0666667
$226$	13.3274	0.886523
$227$	9.37196	0.622039	0.311019	−	0.950404i	$-0.399330\pi$
$227$	9.37196	0.622039	0.311019	+	0.950404i	$0.399330\pi$
$228$	4.85086	0.321256
$229$	−21.2664	−1.40532	−0.702660	−	0.711526i	$-0.748005\pi$
$229$	−21.2664	−1.40532	−0.702660	+	0.711526i	$0.748005\pi$
$230$	−4.04892	−0.266978
$231$	5.67994	0.373713
$232$	8.12498	0.533431
$233$	2.03146	0.133085	0.0665427	−	0.997784i	$-0.478803\pi$
$233$	2.03146	0.133085	0.0665427	+	0.997784i	$0.478803\pi$
$234$	0	0
$235$	1.27413	0.0831149
$236$	−0.417895	−0.0272026
$237$	−4.94869	−0.321452
$238$	3.15213	0.204322
$239$	11.9903	0.775589	0.387794	−	0.921746i	$-0.373237\pi$
$239$	11.9903	0.775589	0.387794	+	0.921746i	$0.373237\pi$
$240$	1.00000	0.0645497
$241$	−21.7627	−1.40186	−0.700930	−	0.713230i	$-0.747231\pi$
$241$	−21.7627	−1.40186	−0.700930	+	0.713230i	$0.747231\pi$
$242$	8.13706	0.523070
$243$	1.00000	0.0641500
$244$	0.198062	0.0126796
$245$	4.26875	0.272720
$246$	3.86294	0.246292
$247$	0	0
$248$	4.08815	0.259598
$249$	−13.1739	−0.834862
$250$	−1.00000	−0.0632456
$251$	13.7506	0.867932	0.433966	−	0.900929i	$-0.357114\pi$
$251$	13.7506	0.867932	0.433966	+	0.900929i	$0.357114\pi$
$252$	−3.35690	−0.211465
$253$	−6.85086	−0.430710
$254$	12.8509	0.806334
$255$	0.939001	0.0588025
$256$	1.00000	0.0625000
$257$	−24.1782	−1.50820	−0.754098	−	0.656762i	$-0.771925\pi$
$257$	−24.1782	−1.50820	−0.754098	+	0.656762i	$0.771925\pi$
$258$	−4.02177	−0.250384
$259$	39.0441	2.42608
$260$	0	0
$261$	−8.12498	−0.502924
$262$	0.570024	0.0352162
$263$	0.960771	0.0592437	0.0296218	−	0.999561i	$-0.490570\pi$
$263$	0.960771	0.0592437	0.0296218	+	0.999561i	$0.490570\pi$
$264$	1.69202	0.104137
$265$	5.74094	0.352663
$266$	16.2838	0.998425
$267$	9.47650	0.579952
$268$	−8.93900	−0.546036
$269$	10.0761	0.614348	0.307174	−	0.951653i	$-0.400617\pi$
$269$	10.0761	0.614348	0.307174	+	0.951653i	$0.400617\pi$
$270$	−1.00000	−0.0608581
$271$	14.9584	0.908657	0.454328	−	0.890834i	$-0.349879\pi$
$271$	14.9584	0.908657	0.454328	+	0.890834i	$0.349879\pi$
$272$	0.939001	0.0569353
$273$	0	0
$274$	21.5405	1.30131
$275$	−1.69202	−0.102033
$276$	4.04892	0.243716
$277$	−13.0465	−0.783890	−0.391945	−	0.919989i	$-0.628198\pi$
$277$	−13.0465	−0.783890	−0.391945	+	0.919989i	$0.628198\pi$
$278$	−7.09783	−0.425700
$279$	−4.08815	−0.244751
$280$	3.35690	0.200613
$281$	−16.2881	−0.971668	−0.485834	−	0.874051i	$-0.661484\pi$
$281$	−16.2881	−0.971668	−0.485834	+	0.874051i	$0.661484\pi$
$282$	−1.27413	−0.0758731
$283$	−12.3773	−0.735756	−0.367878	−	0.929874i	$-0.619916\pi$
$283$	−12.3773	−0.735756	−0.367878	+	0.929874i	$0.619916\pi$
$284$	5.15883	0.306120
$285$	4.85086	0.287340
$286$	0	0
$287$	12.9675	0.765446
$288$	−1.00000	−0.0589256
$289$	−16.1183	−0.948134
$290$	8.12498	0.477115
$291$	−15.4547	−0.905972
$292$	11.5308	0.674789
$293$	1.73663	0.101455	0.0507274	−	0.998713i	$-0.483846\pi$
$293$	1.73663	0.101455	0.0507274	+	0.998713i	$0.483846\pi$
$294$	−4.26875	−0.248959
$295$	−0.417895	−0.0243308
$296$	11.6310	0.676039
$297$	−1.69202	−0.0981810
$298$	−1.14914	−0.0665682
$299$	0	0
$300$	1.00000	0.0577350
$301$	−13.5007	−0.778165
$302$	0.0217703	0.00125274
$303$	5.38404	0.309305
$304$	4.85086	0.278216
$305$	0.198062	0.0113410
$306$	−0.939001	−0.0536791
$307$	−25.4403	−1.45195	−0.725976	−	0.687720i	$-0.758612\pi$
$307$	−25.4403	−1.45195	−0.725976	+	0.687720i	$0.758612\pi$
$308$	5.67994	0.323645
$309$	4.08575	0.232430
$310$	4.08815	0.232191
$311$	−10.0325	−0.568892	−0.284446	−	0.958692i	$-0.591810\pi$
$311$	−10.0325	−0.568892	−0.284446	+	0.958692i	$0.591810\pi$
$312$	0	0
$313$	−22.3274	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$313$	−22.3274	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$314$	10.1588	0.573296
$315$	−3.35690	−0.189140
$316$	−4.94869	−0.278386
$317$	5.33811	0.299818	0.149909	−	0.988700i	$-0.452102\pi$
$317$	5.33811	0.299818	0.149909	+	0.988700i	$0.452102\pi$
$318$	−5.74094	−0.321936
$319$	13.7476	0.769720
$320$	1.00000	0.0559017
$321$	−16.1196	−0.899709
$322$	13.5918	0.757441
$323$	4.55496	0.253445
$324$	1.00000	0.0555556
$325$	0	0
$326$	19.1511	1.06068
$327$	14.2325	0.787059
$328$	3.86294	0.213295
$329$	−4.27711	−0.235805
$330$	1.69202	0.0931427
$331$	4.49827	0.247247	0.123624	−	0.992329i	$-0.460548\pi$
$331$	4.49827	0.247247	0.123624	+	0.992329i	$0.460548\pi$
$332$	−13.1739	−0.723012
$333$	−11.6310	−0.637376
$334$	−15.2174	−0.832661
$335$	−8.93900	−0.488390
$336$	−3.35690	−0.183134
$337$	−11.5676	−0.630129	−0.315064	−	0.949070i	$-0.602026\pi$
$337$	−11.5676	−0.630129	−0.315064	+	0.949070i	$0.602026\pi$
$338$	0	0
$339$	−13.3274	−0.723843
$340$	0.939001	0.0509245
$341$	6.91723	0.374589
$342$	−4.85086	−0.262304
$343$	9.16852	0.495054
$344$	−4.02177	−0.216839
$345$	4.04892	0.217986
$346$	−21.8213	−1.17312
$347$	27.1226	1.45602	0.728008	−	0.685568i	$-0.240446\pi$
$347$	27.1226	1.45602	0.728008	+	0.685568i	$0.240446\pi$
$348$	−8.12498	−0.435545
$349$	−12.8562	−0.688178	−0.344089	−	0.938937i	$-0.611812\pi$
$349$	−12.8562	−0.688178	−0.344089	+	0.938937i	$0.611812\pi$
$350$	3.35690	0.179434
$351$	0	0
$352$	1.69202	0.0901850
$353$	−24.6262	−1.31072	−0.655361	−	0.755316i	$-0.727484\pi$
$353$	−24.6262	−1.31072	−0.655361	+	0.755316i	$0.727484\pi$
$354$	0.417895	0.0222109
$355$	5.15883	0.273802
$356$	9.47650	0.502254
$357$	−3.15213	−0.166828
$358$	20.2513	1.07031
$359$	23.8629	1.25944	0.629719	−	0.776823i	$-0.283170\pi$
$359$	23.8629	1.25944	0.629719	+	0.776823i	$0.283170\pi$
$360$	−1.00000	−0.0527046
$361$	4.53079	0.238463
$362$	−14.4969	−0.761942
$363$	−8.13706	−0.427085
$364$	0	0
$365$	11.5308	0.603549
$366$	−0.198062	−0.0103529
$367$	33.1933	1.73267	0.866337	−	0.499459i	$-0.166468\pi$
$367$	33.1933	1.73267	0.866337	+	0.499459i	$0.166468\pi$
$368$	4.04892	0.211064
$369$	−3.86294	−0.201096
$370$	11.6310	0.604668
$371$	−19.2717	−1.00054
$372$	−4.08815	−0.211960
$373$	−10.8465	−0.561613	−0.280806	−	0.959764i	$-0.590602\pi$
$373$	−10.8465	−0.561613	−0.280806	+	0.959764i	$0.590602\pi$
$374$	1.58881	0.0821554
$375$	1.00000	0.0516398
$376$	−1.27413	−0.0657081
$377$	0	0
$378$	3.35690	0.172660
$379$	6.09485	0.313071	0.156536	−	0.987672i	$-0.449967\pi$
$379$	6.09485	0.313071	0.156536	+	0.987672i	$0.449967\pi$
$380$	4.85086	0.248844
$381$	−12.8509	−0.658369
$382$	10.1806	0.520885
$383$	7.95407	0.406434	0.203217	−	0.979134i	$-0.434860\pi$
$383$	7.95407	0.406434	0.203217	+	0.979134i	$0.434860\pi$
$384$	−1.00000	−0.0510310
$385$	5.67994	0.289477
$386$	27.6407	1.40688
$387$	4.02177	0.204438
$388$	−15.4547	−0.784595
$389$	1.50365	0.0762380	0.0381190	−	0.999273i	$-0.487863\pi$
$389$	1.50365	0.0762380	0.0381190	+	0.999273i	$0.487863\pi$
$390$	0	0
$391$	3.80194	0.192272
$392$	−4.26875	−0.215604
$393$	−0.570024	−0.0287539
$394$	−13.0261	−0.656245
$395$	−4.94869	−0.248996
$396$	−1.69202	−0.0850273
$397$	−34.9627	−1.75473	−0.877364	−	0.479826i	$-0.840700\pi$
$397$	−34.9627	−1.75473	−0.877364	+	0.479826i	$0.840700\pi$
$398$	−6.67563	−0.334619
$399$	−16.2838	−0.815210
$400$	1.00000	0.0500000
$401$	31.1148	1.55380	0.776900	−	0.629624i	$-0.216791\pi$
$401$	31.1148	1.55380	0.776900	+	0.629624i	$0.216791\pi$
$402$	8.93900	0.445837
$403$	0	0
$404$	5.38404	0.267866
$405$	1.00000	0.0496904
$406$	−27.2747	−1.35362
$407$	19.6799	0.975498
$408$	−0.939001	−0.0464875
$409$	32.7318	1.61849	0.809243	−	0.587474i	$-0.199878\pi$
$409$	32.7318	1.61849	0.809243	+	0.587474i	$0.199878\pi$
$410$	3.86294	0.190777
$411$	−21.5405	−1.06251
$412$	4.08575	0.201291
$413$	1.40283	0.0690287
$414$	−4.04892	−0.198993
$415$	−13.1739	−0.646681
$416$	0	0
$417$	7.09783	0.347582
$418$	8.20775	0.401454
$419$	13.0694	0.638480	0.319240	−	0.947674i	$-0.396572\pi$
$419$	13.0694	0.638480	0.319240	+	0.947674i	$0.396572\pi$
$420$	−3.35690	−0.163800
$421$	−16.4789	−0.803132	−0.401566	−	0.915830i	$-0.631534\pi$
$421$	−16.4789	−0.803132	−0.401566	+	0.915830i	$0.631534\pi$
$422$	11.3056	0.550347
$423$	1.27413	0.0619502
$424$	−5.74094	−0.278805
$425$	0.939001	0.0455482
$426$	−5.15883	−0.249946
$427$	−0.664874	−0.0321755
$428$	−16.1196	−0.779171
$429$	0	0
$430$	−4.02177	−0.193947
$431$	41.0640	1.97798	0.988991	−	0.147974i	$-0.0472752\pi$
$431$	41.0640	1.97798	0.988991	+	0.147974i	$0.0472752\pi$
$432$	1.00000	0.0481125
$433$	−6.04593	−0.290549	−0.145275	−	0.989391i	$-0.546407\pi$
$433$	−6.04593	−0.290549	−0.145275	+	0.989391i	$0.546407\pi$
$434$	−13.7235	−0.658748
$435$	−8.12498	−0.389563
$436$	14.2325	0.681613
$437$	19.6407	0.939543
$438$	−11.5308	−0.550963
$439$	−26.2543	−1.25305	−0.626524	−	0.779402i	$-0.715523\pi$
$439$	−26.2543	−1.25305	−0.626524	+	0.779402i	$0.715523\pi$
$440$	1.69202	0.0806640
$441$	4.26875	0.203274
$442$	0	0
$443$	−39.3846	−1.87122	−0.935610	−	0.353035i	$-0.885150\pi$
$443$	−39.3846	−1.87122	−0.935610	+	0.353035i	$0.885150\pi$
$444$	−11.6310	−0.551984
$445$	9.47650	0.449229
$446$	18.0911	0.856640
$447$	1.14914	0.0543527
$448$	−3.35690	−0.158598
$449$	9.39075	0.443177	0.221588	−	0.975140i	$-0.428876\pi$
$449$	9.39075	0.443177	0.221588	+	0.975140i	$0.428876\pi$
$450$	−1.00000	−0.0471405
$451$	6.53617	0.307776
$452$	−13.3274	−0.626866
$453$	−0.0217703	−0.00102286
$454$	−9.37196	−0.439848
$455$	0	0
$456$	−4.85086	−0.227162
$457$	2.66487	0.124658	0.0623288	−	0.998056i	$-0.480147\pi$
$457$	2.66487	0.124658	0.0623288	+	0.998056i	$0.480147\pi$
$458$	21.2664	0.993712
$459$	0.939001	0.0438288
$460$	4.04892	0.188782
$461$	24.5284	1.14240	0.571201	−	0.820810i	$-0.306478\pi$
$461$	24.5284	1.14240	0.571201	+	0.820810i	$0.306478\pi$
$462$	−5.67994	−0.264255
$463$	4.50498	0.209364	0.104682	−	0.994506i	$-0.466618\pi$
$463$	4.50498	0.209364	0.104682	+	0.994506i	$0.466618\pi$
$464$	−8.12498	−0.377193
$465$	−4.08815	−0.189583
$466$	−2.03146	−0.0941055
$467$	21.3317	0.987112	0.493556	−	0.869714i	$-0.335697\pi$
$467$	21.3317	0.987112	0.493556	+	0.869714i	$0.335697\pi$
$468$	0	0
$469$	30.0073	1.38561
$470$	−1.27413	−0.0587711
$471$	−10.1588	−0.468094
$472$	0.417895	0.0192352
$473$	−6.80492	−0.312891
$474$	4.94869	0.227301
$475$	4.85086	0.222572
$476$	−3.15213	−0.144478
$477$	5.74094	0.262860
$478$	−11.9903	−0.548424
$479$	32.9517	1.50560	0.752800	−	0.658249i	$-0.228703\pi$
$479$	32.9517	1.50560	0.752800	+	0.658249i	$0.228703\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	21.7627	0.991264
$483$	−13.5918	−0.618448
$484$	−8.13706	−0.369867
$485$	−15.4547	−0.701763
$486$	−1.00000	−0.0453609
$487$	9.33034	0.422798	0.211399	−	0.977400i	$-0.432198\pi$
$487$	9.33034	0.422798	0.211399	+	0.977400i	$0.432198\pi$
$488$	−0.198062	−0.00896586
$489$	−19.1511	−0.866041
$490$	−4.26875	−0.192842
$491$	−20.4349	−0.922213	−0.461107	−	0.887345i	$-0.652547\pi$
$491$	−20.4349	−0.922213	−0.461107	+	0.887345i	$0.652547\pi$
$492$	−3.86294	−0.174155
$493$	−7.62937	−0.343609
$494$	0	0
$495$	−1.69202	−0.0760507
$496$	−4.08815	−0.183563
$497$	−17.3177	−0.776804
$498$	13.1739	0.590337
$499$	−3.54958	−0.158901	−0.0794505	−	0.996839i	$-0.525317\pi$
$499$	−3.54958	−0.158901	−0.0794505	+	0.996839i	$0.525317\pi$
$500$	1.00000	0.0447214
$501$	15.2174	0.679865
$502$	−13.7506	−0.613721
$503$	−19.8146	−0.883490	−0.441745	−	0.897141i	$-0.645640\pi$
$503$	−19.8146	−0.883490	−0.441745	+	0.897141i	$0.645640\pi$
$504$	3.35690	0.149528
$505$	5.38404	0.239587
$506$	6.85086	0.304558
$507$	0	0
$508$	−12.8509	−0.570164
$509$	−32.4325	−1.43754	−0.718772	−	0.695246i	$-0.755296\pi$
$509$	−32.4325	−1.43754	−0.718772	+	0.695246i	$0.755296\pi$
$510$	−0.939001	−0.0415797
$511$	−38.7077	−1.71233
$512$	−1.00000	−0.0441942
$513$	4.85086	0.214170
$514$	24.1782	1.06646
$515$	4.08575	0.180040
$516$	4.02177	0.177049
$517$	−2.15585	−0.0948142
$518$	−39.0441	−1.71550
$519$	21.8213	0.957849
$520$	0	0
$521$	7.87694	0.345095	0.172547	−	0.985001i	$-0.444800\pi$
$521$	7.87694	0.345095	0.172547	+	0.985001i	$0.444800\pi$
$522$	8.12498	0.355621
$523$	−23.4276	−1.02442	−0.512208	−	0.858861i	$-0.671172\pi$
$523$	−23.4276	−1.02442	−0.512208	+	0.858861i	$0.671172\pi$
$524$	−0.570024	−0.0249016
$525$	−3.35690	−0.146507
$526$	−0.960771	−0.0418916
$527$	−3.83877	−0.167220
$528$	−1.69202	−0.0736358
$529$	−6.60627	−0.287229
$530$	−5.74094	−0.249370
$531$	−0.417895	−0.0181351
$532$	−16.2838	−0.705993
$533$	0	0
$534$	−9.47650	−0.410088
$535$	−16.1196	−0.696911
$536$	8.93900	0.386106
$537$	−20.2513	−0.873908
$538$	−10.0761	−0.434410
$539$	−7.22282	−0.311109
$540$	1.00000	0.0430331
$541$	1.21552	0.0522593	0.0261297	−	0.999659i	$-0.491682\pi$
$541$	1.21552	0.0522593	0.0261297	+	0.999659i	$0.491682\pi$
$542$	−14.9584	−0.642517
$543$	14.4969	0.622123
$544$	−0.939001	−0.0402593
$545$	14.2325	0.609654
$546$	0	0
$547$	−15.1371	−0.647214	−0.323607	−	0.946192i	$-0.604896\pi$
$547$	−15.1371	−0.647214	−0.323607	+	0.946192i	$0.604896\pi$
$548$	−21.5405	−0.920164
$549$	0.198062	0.00845309
$550$	1.69202	0.0721480
$551$	−39.4131	−1.67905
$552$	−4.04892	−0.172333
$553$	16.6122	0.706424
$554$	13.0465	0.554294
$555$	−11.6310	−0.493709
$556$	7.09783	0.301015
$557$	−0.147817	−0.00626321	−0.00313160	−	0.999995i	$-0.500997\pi$
$557$	−0.147817	−0.00626321	−0.00313160	+	0.999995i	$0.500997\pi$
$558$	4.08815	0.173065
$559$	0	0
$560$	−3.35690	−0.141855
$561$	−1.58881	−0.0670796
$562$	16.2881	0.687073
$563$	40.6945	1.71507	0.857535	−	0.514426i	$-0.171995\pi$
$563$	40.6945	1.71507	0.857535	+	0.514426i	$0.171995\pi$
$564$	1.27413	0.0536504
$565$	−13.3274	−0.560686
$566$	12.3773	0.520258
$567$	−3.35690	−0.140976
$568$	−5.15883	−0.216460
$569$	−26.2433	−1.10017	−0.550087	−	0.835107i	$-0.685406\pi$
$569$	−26.2433	−1.10017	−0.550087	+	0.835107i	$0.685406\pi$
$570$	−4.85086	−0.203180
$571$	2.31575	0.0969110	0.0484555	−	0.998825i	$-0.484570\pi$
$571$	2.31575	0.0969110	0.0484555	+	0.998825i	$0.484570\pi$
$572$	0	0
$573$	−10.1806	−0.425301
$574$	−12.9675	−0.541252
$575$	4.04892	0.168852
$576$	1.00000	0.0416667
$577$	14.5241	0.604646	0.302323	−	0.953206i	$-0.402238\pi$
$577$	14.5241	0.604646	0.302323	+	0.953206i	$0.402238\pi$
$578$	16.1183	0.670432
$579$	−27.6407	−1.14871
$580$	−8.12498	−0.337372
$581$	44.2234	1.83470
$582$	15.4547	0.640619
$583$	−9.71379	−0.402304
$584$	−11.5308	−0.477148
$585$	0	0
$586$	−1.73663	−0.0717394
$587$	32.9028	1.35804	0.679021	−	0.734119i	$-0.262404\pi$
$587$	32.9028	1.35804	0.679021	+	0.734119i	$0.262404\pi$
$588$	4.26875	0.176040
$589$	−19.8310	−0.817122
$590$	0.417895	0.0172045
$591$	13.0261	0.535821
$592$	−11.6310	−0.478032
$593$	36.0146	1.47894	0.739471	−	0.673188i	$-0.235076\pi$
$593$	36.0146	1.47894	0.739471	+	0.673188i	$0.235076\pi$
$594$	1.69202	0.0694245
$595$	−3.15213	−0.129225
$596$	1.14914	0.0470708
$597$	6.67563	0.273215
$598$	0	0
$599$	−29.4644	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$599$	−29.4644	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$600$	−1.00000	−0.0408248
$601$	−28.4004	−1.15848	−0.579239	−	0.815158i	$-0.696650\pi$
$601$	−28.4004	−1.15848	−0.579239	+	0.815158i	$0.696650\pi$
$602$	13.5007	0.550246
$603$	−8.93900	−0.364024
$604$	−0.0217703	−0.000885819 0
$605$	−8.13706	−0.330819
$606$	−5.38404	−0.218712
$607$	15.6450	0.635012	0.317506	−	0.948256i	$-0.397155\pi$
$607$	15.6450	0.635012	0.317506	+	0.948256i	$0.397155\pi$
$608$	−4.85086	−0.196728
$609$	27.2747	1.10523
$610$	−0.198062	−0.00801931
$611$	0	0
$612$	0.939001	0.0379569
$613$	17.6329	0.712188	0.356094	−	0.934450i	$-0.384108\pi$
$613$	17.6329	0.712188	0.356094	+	0.934450i	$0.384108\pi$
$614$	25.4403	1.02669
$615$	−3.86294	−0.155769
$616$	−5.67994	−0.228851
$617$	−10.8582	−0.437133	−0.218566	−	0.975822i	$-0.570138\pi$
$617$	−10.8582	−0.437133	−0.218566	+	0.975822i	$0.570138\pi$
$618$	−4.08575	−0.164353
$619$	34.2083	1.37495	0.687475	−	0.726208i	$-0.258719\pi$
$619$	34.2083	1.37495	0.687475	+	0.726208i	$0.258719\pi$
$620$	−4.08815	−0.164184
$621$	4.04892	0.162477
$622$	10.0325	0.402268
$623$	−31.8116	−1.27451
$624$	0	0
$625$	1.00000	0.0400000
$626$	22.3274	0.892381
$627$	−8.20775	−0.327786
$628$	−10.1588	−0.405382
$629$	−10.9215	−0.435470
$630$	3.35690	0.133742
$631$	−31.9571	−1.27219	−0.636095	−	0.771611i	$-0.719451\pi$
$631$	−31.9571	−1.27219	−0.636095	+	0.771611i	$0.719451\pi$
$632$	4.94869	0.196848
$633$	−11.3056	−0.449357
$634$	−5.33811	−0.212003
$635$	−12.8509	−0.509971
$636$	5.74094	0.227643
$637$	0	0
$638$	−13.7476	−0.544274
$639$	5.15883	0.204080
$640$	−1.00000	−0.0395285
$641$	−45.8297	−1.81016	−0.905082	−	0.425238i	$-0.860190\pi$
$641$	−45.8297	−1.81016	−0.905082	+	0.425238i	$0.860190\pi$
$642$	16.1196	0.636190
$643$	−20.6418	−0.814032	−0.407016	−	0.913421i	$-0.633431\pi$
$643$	−20.6418	−0.814032	−0.407016	+	0.913421i	$0.633431\pi$
$644$	−13.5918	−0.535592
$645$	4.02177	0.158357
$646$	−4.55496	−0.179212
$647$	1.40389	0.0551928	0.0275964	−	0.999619i	$-0.491215\pi$
$647$	1.40389	0.0551928	0.0275964	+	0.999619i	$0.491215\pi$
$648$	−1.00000	−0.0392837
$649$	0.707087	0.0277556
$650$	0	0
$651$	13.7235	0.537866
$652$	−19.1511	−0.750014
$653$	−4.54229	−0.177753	−0.0888767	−	0.996043i	$-0.528328\pi$
$653$	−4.54229	−0.177753	−0.0888767	+	0.996043i	$0.528328\pi$
$654$	−14.2325	−0.556535
$655$	−0.570024	−0.0222727
$656$	−3.86294	−0.150822
$657$	11.5308	0.449859
$658$	4.27711	0.166739
$659$	17.4161	0.678435	0.339217	−	0.940708i	$-0.389838\pi$
$659$	17.4161	0.678435	0.339217	+	0.940708i	$0.389838\pi$
$660$	−1.69202	−0.0658618
$661$	−43.3682	−1.68683	−0.843415	−	0.537263i	$-0.819458\pi$
$661$	−43.3682	−1.68683	−0.843415	+	0.537263i	$0.819458\pi$
$662$	−4.49827	−0.174830
$663$	0	0
$664$	13.1739	0.511246
$665$	−16.2838	−0.631459
$666$	11.6310	0.450693
$667$	−32.8974	−1.27379
$668$	15.2174	0.588780
$669$	−18.0911	−0.699443
$670$	8.93900	0.345344
$671$	−0.335126	−0.0129374
$672$	3.35690	0.129495
$673$	−3.49875	−0.134867	−0.0674334	−	0.997724i	$-0.521481\pi$
$673$	−3.49875	−0.134867	−0.0674334	+	0.997724i	$0.521481\pi$
$674$	11.5676	0.445568
$675$	1.00000	0.0384900
$676$	0	0
$677$	−13.3284	−0.512253	−0.256126	−	0.966643i	$-0.582446\pi$
$677$	−13.3284	−0.512253	−0.256126	+	0.966643i	$0.582446\pi$
$678$	13.3274	0.511834
$679$	51.8799	1.99097
$680$	−0.939001	−0.0360090
$681$	9.37196	0.359134
$682$	−6.91723	−0.264874
$683$	24.1739	0.924989	0.462494	−	0.886622i	$-0.346955\pi$
$683$	24.1739	0.924989	0.462494	+	0.886622i	$0.346955\pi$
$684$	4.85086	0.185477
$685$	−21.5405	−0.823020
$686$	−9.16852	−0.350056
$687$	−21.2664	−0.811362
$688$	4.02177	0.153329
$689$	0	0
$690$	−4.04892	−0.154140
$691$	37.8810	1.44106	0.720530	−	0.693423i	$-0.243898\pi$
$691$	37.8810	1.44106	0.720530	+	0.693423i	$0.243898\pi$
$692$	21.8213	0.829522
$693$	5.67994	0.215763
$694$	−27.1226	−1.02956
$695$	7.09783	0.269236
$696$	8.12498	0.307977
$697$	−3.62730	−0.137394
$698$	12.8562	0.486616
$699$	2.03146	0.0768368
$700$	−3.35690	−0.126879
$701$	−21.5967	−0.815696	−0.407848	−	0.913050i	$-0.633721\pi$
$701$	−21.5967	−0.815696	−0.407848	+	0.913050i	$0.633721\pi$
$702$	0	0
$703$	−56.4204	−2.12794
$704$	−1.69202	−0.0637705
$705$	1.27413	0.0479864
$706$	24.6262	0.926821
$707$	−18.0737	−0.679730
$708$	−0.417895	−0.0157054
$709$	−41.3062	−1.55129	−0.775643	−	0.631172i	$-0.782574\pi$
$709$	−41.3062	−1.55129	−0.775643	+	0.631172i	$0.782574\pi$
$710$	−5.15883	−0.193608
$711$	−4.94869	−0.185590
$712$	−9.47650	−0.355147
$713$	−16.5526	−0.619898
$714$	3.15213	0.117965
$715$	0	0
$716$	−20.2513	−0.756826
$717$	11.9903	0.447786
$718$	−23.8629	−0.890557
$719$	−35.5338	−1.32519	−0.662593	−	0.748980i	$-0.730544\pi$
$719$	−35.5338	−1.32519	−0.662593	+	0.748980i	$0.730544\pi$
$720$	1.00000	0.0372678
$721$	−13.7154	−0.510790
$722$	−4.53079	−0.168619
$723$	−21.7627	−0.809364
$724$	14.4969	0.538775
$725$	−8.12498	−0.301754
$726$	8.13706	0.301995
$727$	19.6635	0.729281	0.364640	−	0.931148i	$-0.381192\pi$
$727$	19.6635	0.729281	0.364640	+	0.931148i	$0.381192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−11.5308	−0.426774
$731$	3.77645	0.139677
$732$	0.198062	0.00732059
$733$	46.8702	1.73119	0.865596	−	0.500743i	$-0.166940\pi$
$733$	46.8702	1.73119	0.865596	+	0.500743i	$0.166940\pi$
$734$	−33.1933	−1.22519
$735$	4.26875	0.157455
$736$	−4.04892	−0.149245
$737$	15.1250	0.557136
$738$	3.86294	0.142197
$739$	8.57779	0.315539	0.157770	−	0.987476i	$-0.449570\pi$
$739$	8.57779	0.315539	0.157770	+	0.987476i	$0.449570\pi$
$740$	−11.6310	−0.427565
$741$	0	0
$742$	19.2717	0.707488
$743$	44.9506	1.64908	0.824539	−	0.565805i	$-0.191435\pi$
$743$	44.9506	1.64908	0.824539	+	0.565805i	$0.191435\pi$
$744$	4.08815	0.149879
$745$	1.14914	0.0421014
$746$	10.8465	0.397120
$747$	−13.1739	−0.482008
$748$	−1.58881	−0.0580926
$749$	54.1118	1.97720
$750$	−1.00000	−0.0365148
$751$	44.0374	1.60695	0.803474	−	0.595339i	$-0.202982\pi$
$751$	44.0374	1.60695	0.803474	+	0.595339i	$0.202982\pi$
$752$	1.27413	0.0464626
$753$	13.7506	0.501101
$754$	0	0
$755$	−0.0217703	−0.000792301 0
$756$	−3.35690	−0.122089
$757$	−13.2798	−0.482661	−0.241331	−	0.970443i	$-0.577584\pi$
$757$	−13.2798	−0.482661	−0.241331	+	0.970443i	$0.577584\pi$
$758$	−6.09485	−0.221375
$759$	−6.85086	−0.248670
$760$	−4.85086	−0.175959
$761$	−9.11231	−0.330321	−0.165160	−	0.986267i	$-0.552814\pi$
$761$	−9.11231	−0.330321	−0.165160	+	0.986267i	$0.552814\pi$
$762$	12.8509	0.465537
$763$	−47.7770	−1.72964
$764$	−10.1806	−0.368321
$765$	0.939001	0.0339497
$766$	−7.95407	−0.287392
$767$	0	0
$768$	1.00000	0.0360844
$769$	31.7784	1.14596	0.572979	−	0.819570i	$-0.305788\pi$
$769$	31.7784	1.14596	0.572979	+	0.819570i	$0.305788\pi$
$770$	−5.67994	−0.204691
$771$	−24.1782	−0.870757
$772$	−27.6407	−0.994811
$773$	−7.11960	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$773$	−7.11960	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$774$	−4.02177	−0.144560
$775$	−4.08815	−0.146851
$776$	15.4547	0.554792
$777$	39.0441	1.40070
$778$	−1.50365	−0.0539084
$779$	−18.7385	−0.671378
$780$	0	0
$781$	−8.72886	−0.312343
$782$	−3.80194	−0.135957
$783$	−8.12498	−0.290363
$784$	4.26875	0.152455
$785$	−10.1588	−0.362584
$786$	0.570024	0.0203321
$787$	44.2121	1.57599	0.787995	−	0.615682i	$-0.211119\pi$
$787$	44.2121	1.57599	0.787995	+	0.615682i	$0.211119\pi$
$788$	13.0261	0.464035
$789$	0.960771	0.0342044
$790$	4.94869	0.176066
$791$	44.7385	1.59072
$792$	1.69202	0.0601234
$793$	0	0
$794$	34.9627	1.24078
$795$	5.74094	0.203610
$796$	6.67563	0.236611
$797$	19.2241	0.680954	0.340477	−	0.940253i	$-0.389411\pi$
$797$	19.2241	0.680954	0.340477	+	0.940253i	$0.389411\pi$
$798$	16.2838	0.576441
$799$	1.19641	0.0423258
$800$	−1.00000	−0.0353553
$801$	9.47650	0.334836
$802$	−31.1148	−1.09870
$803$	−19.5104	−0.688505
$804$	−8.93900	−0.315254
$805$	−13.5918	−0.479048
$806$	0	0
$807$	10.0761	0.354694
$808$	−5.38404	−0.189410
$809$	0.0988996	0.00347713	0.00173856	−	0.999998i	$-0.499447\pi$
$809$	0.0988996	0.00347713	0.00173856	+	0.999998i	$0.499447\pi$
$810$	−1.00000	−0.0351364
$811$	−6.70304	−0.235376	−0.117688	−	0.993051i	$-0.537548\pi$
$811$	−6.70304	−0.235376	−0.117688	+	0.993051i	$0.537548\pi$
$812$	27.2747	0.957155
$813$	14.9584	0.524613
$814$	−19.6799	−0.689782
$815$	−19.1511	−0.670833
$816$	0.939001	0.0328716
$817$	19.5090	0.682534
$818$	−32.7318	−1.14444
$819$	0	0
$820$	−3.86294	−0.134900
$821$	23.7450	0.828706	0.414353	−	0.910116i	$-0.364008\pi$
$821$	23.7450	0.828706	0.414353	+	0.910116i	$0.364008\pi$
$822$	21.5405	0.751311
$823$	18.0049	0.627611	0.313806	−	0.949487i	$-0.398396\pi$
$823$	18.0049	0.627611	0.313806	+	0.949487i	$0.398396\pi$
$824$	−4.08575	−0.142334
$825$	−1.69202	−0.0589086
$826$	−1.40283	−0.0488107
$827$	−40.4566	−1.40682	−0.703408	−	0.710787i	$-0.748339\pi$
$827$	−40.4566	−1.40682	−0.703408	+	0.710787i	$0.748339\pi$
$828$	4.04892	0.140710
$829$	8.56704	0.297546	0.148773	−	0.988871i	$-0.452468\pi$
$829$	8.56704	0.297546	0.148773	+	0.988871i	$0.452468\pi$
$830$	13.1739	0.457273
$831$	−13.0465	−0.452579
$832$	0	0
$833$	4.00836	0.138881
$834$	−7.09783	−0.245778
$835$	15.2174	0.526621
$836$	−8.20775	−0.283871
$837$	−4.08815	−0.141307
$838$	−13.0694	−0.451474
$839$	6.75600	0.233243	0.116622	−	0.993176i	$-0.462794\pi$
$839$	6.75600	0.233243	0.116622	+	0.993176i	$0.462794\pi$
$840$	3.35690	0.115824
$841$	37.0153	1.27639
$842$	16.4789	0.567900
$843$	−16.2881	−0.560993
$844$	−11.3056	−0.389154
$845$	0	0
$846$	−1.27413	−0.0438054
$847$	27.3153	0.938564
$848$	5.74094	0.197145
$849$	−12.3773	−0.424789
$850$	−0.939001	−0.0322075
$851$	−47.0930	−1.61433
$852$	5.15883	0.176739
$853$	37.1269	1.27120	0.635600	−	0.772018i	$-0.280753\pi$
$853$	37.1269	1.27120	0.635600	+	0.772018i	$0.280753\pi$
$854$	0.664874	0.0227515
$855$	4.85086	0.165896
$856$	16.1196	0.550957
$857$	−24.3937	−0.833274	−0.416637	−	0.909073i	$-0.636791\pi$
$857$	−24.3937	−0.833274	−0.416637	+	0.909073i	$0.636791\pi$
$858$	0	0
$859$	43.9366	1.49910	0.749549	−	0.661949i	$-0.230270\pi$
$859$	43.9366	1.49910	0.749549	+	0.661949i	$0.230270\pi$
$860$	4.02177	0.137141
$861$	12.9675	0.441930
$862$	−41.0640	−1.39864
$863$	−16.3846	−0.557739	−0.278870	−	0.960329i	$-0.589960\pi$
$863$	−16.3846	−0.557739	−0.278870	+	0.960329i	$0.589960\pi$
$864$	−1.00000	−0.0340207
$865$	21.8213	0.741947
$866$	6.04593	0.205449
$867$	−16.1183	−0.547405
$868$	13.7235	0.465805
$869$	8.37329	0.284044
$870$	8.12498	0.275463
$871$	0	0
$872$	−14.2325	−0.481973
$873$	−15.4547	−0.523063
$874$	−19.6407	−0.664357
$875$	−3.35690	−0.113484
$876$	11.5308	0.389589
$877$	−28.1347	−0.950040	−0.475020	−	0.879975i	$-0.657559\pi$
$877$	−28.1347	−0.950040	−0.475020	+	0.879975i	$0.657559\pi$
$878$	26.2543	0.886039
$879$	1.73663	0.0585750
$880$	−1.69202	−0.0570380
$881$	39.1454	1.31884	0.659421	−	0.751773i	$-0.270801\pi$
$881$	39.1454	1.31884	0.659421	+	0.751773i	$0.270801\pi$
$882$	−4.26875	−0.143736
$883$	46.2833	1.55756	0.778779	−	0.627298i	$-0.215839\pi$
$883$	46.2833	1.55756	0.778779	+	0.627298i	$0.215839\pi$
$884$	0	0
$885$	−0.417895	−0.0140474
$886$	39.3846	1.32315
$887$	−3.08097	−0.103449	−0.0517244	−	0.998661i	$-0.516472\pi$
$887$	−3.08097	−0.103449	−0.0517244	+	0.998661i	$0.516472\pi$
$888$	11.6310	0.390312
$889$	43.1390	1.44684
$890$	−9.47650	−0.317653
$891$	−1.69202	−0.0566849
$892$	−18.0911	−0.605736
$893$	6.18060	0.206826
$894$	−1.14914	−0.0384332
$895$	−20.2513	−0.676926
$896$	3.35690	0.112146
$897$	0	0
$898$	−9.39075	−0.313373
$899$	33.2161	1.10782
$900$	1.00000	0.0333333
$901$	5.39075	0.179592
$902$	−6.53617	−0.217631
$903$	−13.5007	−0.449274
$904$	13.3274	0.443261
$905$	14.4969	0.481895
$906$	0.0217703	0.000723269 0
$907$	−2.61224	−0.0867379	−0.0433689	−	0.999059i	$-0.513809\pi$
$907$	−2.61224	−0.0867379	−0.0433689	+	0.999059i	$0.513809\pi$
$908$	9.37196	0.311019
$909$	5.38404	0.178577
$910$	0	0
$911$	24.9302	0.825973	0.412987	−	0.910737i	$-0.364486\pi$
$911$	24.9302	0.825973	0.412987	+	0.910737i	$0.364486\pi$
$912$	4.85086	0.160628
$913$	22.2905	0.737709
$914$	−2.66487	−0.0881462
$915$	0.198062	0.00654774
$916$	−21.2664	−0.702660
$917$	1.91351	0.0631897
$918$	−0.939001	−0.0309917
$919$	−54.4462	−1.79602	−0.898008	−	0.439980i	$-0.854986\pi$
$919$	−54.4462	−1.79602	−0.898008	+	0.439980i	$0.854986\pi$
$920$	−4.04892	−0.133489
$921$	−25.4403	−0.838285
$922$	−24.5284	−0.807800
$923$	0	0
$924$	5.67994	0.186856
$925$	−11.6310	−0.382426
$926$	−4.50498	−0.148043
$927$	4.08575	0.134194
$928$	8.12498	0.266716
$929$	40.8267	1.33948	0.669740	−	0.742596i	$-0.266405\pi$
$929$	40.8267	1.33948	0.669740	+	0.742596i	$0.266405\pi$
$930$	4.08815	0.134056
$931$	20.7071	0.678647
$932$	2.03146	0.0665427
$933$	−10.0325	−0.328450
$934$	−21.3317	−0.697993
$935$	−1.58881	−0.0519596
$936$	0	0
$937$	−46.9571	−1.53402	−0.767010	−	0.641635i	$-0.778256\pi$
$937$	−46.9571	−1.53402	−0.767010	+	0.641635i	$0.778256\pi$
$938$	−30.0073	−0.979773
$939$	−22.3274	−0.728626
$940$	1.27413	0.0415574
$941$	−35.8525	−1.16876	−0.584379	−	0.811481i	$-0.698662\pi$
$941$	−35.8525	−1.16876	−0.584379	+	0.811481i	$0.698662\pi$
$942$	10.1588	0.330993
$943$	−15.6407	−0.509332
$944$	−0.417895	−0.0136013
$945$	−3.35690	−0.109200
$946$	6.80492	0.221247
$947$	8.73019	0.283693	0.141846	−	0.989889i	$-0.454696\pi$
$947$	8.73019	0.283693	0.141846	+	0.989889i	$0.454696\pi$
$948$	−4.94869	−0.160726
$949$	0	0
$950$	−4.85086	−0.157383
$951$	5.33811	0.173100
$952$	3.15213	0.102161
$953$	10.2107	0.330758	0.165379	−	0.986230i	$-0.447115\pi$
$953$	10.2107	0.330758	0.165379	+	0.986230i	$0.447115\pi$
$954$	−5.74094	−0.185870
$955$	−10.1806	−0.329437
$956$	11.9903	0.387794
$957$	13.7476	0.444398
$958$	−32.9517	−1.06462
$959$	72.3092	2.33498
$960$	1.00000	0.0322749
$961$	−14.2871	−0.460873
$962$	0	0
$963$	−16.1196	−0.519447
$964$	−21.7627	−0.700930
$965$	−27.6407	−0.889786
$966$	13.5918	0.437309
$967$	34.6907	1.11558	0.557789	−	0.829983i	$-0.311650\pi$
$967$	34.6907	1.11558	0.557789	+	0.829983i	$0.311650\pi$
$968$	8.13706	0.261535
$969$	4.55496	0.146326
$970$	15.4547	0.496221
$971$	34.8039	1.11691	0.558454	−	0.829535i	$-0.311395\pi$
$971$	34.8039	1.11691	0.558454	+	0.829535i	$0.311395\pi$
$972$	1.00000	0.0320750
$973$	−23.8267	−0.763849
$974$	−9.33034	−0.298963
$975$	0	0
$976$	0.198062	0.00633982
$977$	−23.3086	−0.745707	−0.372854	−	0.927890i	$-0.621621\pi$
$977$	−23.3086	−0.745707	−0.372854	+	0.927890i	$0.621621\pi$
$978$	19.1511	0.612383
$979$	−16.0344	−0.512463
$980$	4.26875	0.136360
$981$	14.2325	0.454409
$982$	20.4349	0.652103
$983$	−41.3639	−1.31930	−0.659652	−	0.751571i	$-0.729296\pi$
$983$	−41.3639	−1.31930	−0.659652	+	0.751571i	$0.729296\pi$
$984$	3.86294	0.123146
$985$	13.0261	0.415045
$986$	7.62937	0.242969
$987$	−4.27711	−0.136142
$988$	0	0
$989$	16.2838	0.517795
$990$	1.69202	0.0537760
$991$	49.4209	1.56991	0.784953	−	0.619555i	$-0.212687\pi$
$991$	49.4209	1.56991	0.784953	+	0.619555i	$0.212687\pi$
$992$	4.08815	0.129799
$993$	4.49827	0.142748
$994$	17.3177	0.549283
$995$	6.67563	0.211632
$996$	−13.1739	−0.417431
$997$	44.1879	1.39944	0.699722	−	0.714415i	$-0.253307\pi$
$997$	44.1879	1.39944	0.699722	+	0.714415i	$0.253307\pi$
$998$	3.54958	0.112360
$999$	−11.6310	−0.367989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bp.1.2	✓	3
13.5	odd	4		5070.2.b.z.1351.5		6
13.8	odd	4		5070.2.b.z.1351.2		6
13.12	even	2		5070.2.a.bw.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bp.1.2	✓	3	1.1	even	1	trivial
5070.2.a.bw.1.2	yes	3	13.12	even	2
5070.2.b.z.1351.2		6	13.8	odd	4
5070.2.b.z.1351.5		6	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.35690 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.35690 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.69202 q^{11} +1.00000 q^{12} +3.35690 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.939001 q^{17} -1.00000 q^{18} +4.85086 q^{19} +1.00000 q^{20} -3.35690 q^{21} +1.69202 q^{22} +4.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -3.35690 q^{28} -8.12498 q^{29} -1.00000 q^{30} -4.08815 q^{31} -1.00000 q^{32} -1.69202 q^{33} -0.939001 q^{34} -3.35690 q^{35} +1.00000 q^{36} -11.6310 q^{37} -4.85086 q^{38} -1.00000 q^{40} -3.86294 q^{41} +3.35690 q^{42} +4.02177 q^{43} -1.69202 q^{44} +1.00000 q^{45} -4.04892 q^{46} +1.27413 q^{47} +1.00000 q^{48} +4.26875 q^{49} -1.00000 q^{50} +0.939001 q^{51} +5.74094 q^{53} -1.00000 q^{54} -1.69202 q^{55} +3.35690 q^{56} +4.85086 q^{57} +8.12498 q^{58} -0.417895 q^{59} +1.00000 q^{60} +0.198062 q^{61} +4.08815 q^{62} -3.35690 q^{63} +1.00000 q^{64} +1.69202 q^{66} -8.93900 q^{67} +0.939001 q^{68} +4.04892 q^{69} +3.35690 q^{70} +5.15883 q^{71} -1.00000 q^{72} +11.5308 q^{73} +11.6310 q^{74} +1.00000 q^{75} +4.85086 q^{76} +5.67994 q^{77} -4.94869 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.86294 q^{82} -13.1739 q^{83} -3.35690 q^{84} +0.939001 q^{85} -4.02177 q^{86} -8.12498 q^{87} +1.69202 q^{88} +9.47650 q^{89} -1.00000 q^{90} +4.04892 q^{92} -4.08815 q^{93} -1.27413 q^{94} +4.85086 q^{95} -1.00000 q^{96} -15.4547 q^{97} -4.26875 q^{98} -1.69202 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} + 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + q^{19} + 3 q^{20} - 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} - 6 q^{28} - 3 q^{30} - 16 q^{31} - 3 q^{32} + 7 q^{34} - 6 q^{35} + 3 q^{36} - 20 q^{37} - q^{38} - 3 q^{40} - 17 q^{41} + 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} - 7 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} + 6 q^{56} + q^{57} - 7 q^{59} + 3 q^{60} + 5 q^{61} + 16 q^{62} - 6 q^{63} + 3 q^{64} - 17 q^{67} - 7 q^{68} + 3 q^{69} + 6 q^{70} + 7 q^{71} - 3 q^{72} - 3 q^{73} + 20 q^{74} + 3 q^{75} + q^{76} - 7 q^{77} + 17 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} - 6 q^{83} - 6 q^{84} - 7 q^{85} - 9 q^{86} + 3 q^{89} - 3 q^{90} + 3 q^{92} - 16 q^{93} + 7 q^{94} + q^{95} - 3 q^{96} - 24 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 - 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 + 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + q^19 + 3 * q^20 - 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 - 6 * q^28 - 3 * q^30 - 16 * q^31 - 3 * q^32 + 7 * q^34 - 6 * q^35 + 3 * q^36 - 20 * q^37 - q^38 - 3 * q^40 - 17 * q^41 + 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 - 7 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 + 6 * q^56 + q^57 - 7 * q^59 + 3 * q^60 + 5 * q^61 + 16 * q^62 - 6 * q^63 + 3 * q^64 - 17 * q^67 - 7 * q^68 + 3 * q^69 + 6 * q^70 + 7 * q^71 - 3 * q^72 - 3 * q^73 + 20 * q^74 + 3 * q^75 + q^76 - 7 * q^77 + 17 * q^79 + 3 * q^80 + 3 * q^81 + 17 * q^82 - 6 * q^83 - 6 * q^84 - 7 * q^85 - 9 * q^86 + 3 * q^89 - 3 * q^90 + 3 * q^92 - 16 * q^93 + 7 * q^94 + q^95 - 3 * q^96 - 24 * q^97 - 5 * q^98

Introduction

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