LMFDB - Embedded newform 6069.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6069,2,Mod(1,6069)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6069, 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("6069.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.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:	$48.4612089867$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	6069.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.554958 q^{2} +1.00000 q^{3} -1.69202 q^{4} -3.85086 q^{5} -0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.554958 q^{2} +1.00000 q^{3} -1.69202 q^{4} -3.85086 q^{5} -0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} +2.13706 q^{10} +0.951083 q^{11} -1.69202 q^{12} +7.15883 q^{13} -0.554958 q^{14} -3.85086 q^{15} +2.24698 q^{16} -0.554958 q^{18} -4.51573 q^{19} +6.51573 q^{20} +1.00000 q^{21} -0.527811 q^{22} -2.02715 q^{23} +2.04892 q^{24} +9.82908 q^{25} -3.97285 q^{26} +1.00000 q^{27} -1.69202 q^{28} +9.76809 q^{29} +2.13706 q^{30} +6.87800 q^{31} -5.34481 q^{32} +0.951083 q^{33} -3.85086 q^{35} -1.69202 q^{36} +3.62565 q^{37} +2.50604 q^{38} +7.15883 q^{39} -7.89008 q^{40} +1.96077 q^{41} -0.554958 q^{42} -9.28382 q^{43} -1.60925 q^{44} -3.85086 q^{45} +1.12498 q^{46} -8.24698 q^{47} +2.24698 q^{48} +1.00000 q^{49} -5.45473 q^{50} -12.1129 q^{52} -8.13706 q^{53} -0.554958 q^{54} -3.66248 q^{55} +2.04892 q^{56} -4.51573 q^{57} -5.42088 q^{58} -1.55496 q^{59} +6.51573 q^{60} -6.14675 q^{61} -3.81700 q^{62} +1.00000 q^{63} -1.52781 q^{64} -27.5676 q^{65} -0.527811 q^{66} +2.32975 q^{67} -2.02715 q^{69} +2.13706 q^{70} -4.38404 q^{71} +2.04892 q^{72} -2.16421 q^{73} -2.01208 q^{74} +9.82908 q^{75} +7.64071 q^{76} +0.951083 q^{77} -3.97285 q^{78} +1.98792 q^{79} -8.65279 q^{80} +1.00000 q^{81} -1.08815 q^{82} -5.04892 q^{83} -1.69202 q^{84} +5.15213 q^{86} +9.76809 q^{87} +1.94869 q^{88} +3.31767 q^{89} +2.13706 q^{90} +7.15883 q^{91} +3.42998 q^{92} +6.87800 q^{93} +4.57673 q^{94} +17.3894 q^{95} -5.34481 q^{96} +14.7899 q^{97} -0.554958 q^{98} +0.951083 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 3 q^{3} + 2 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 3 * q^3 + 2 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 2 q^{2} + 3 q^{3} + 2 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + q^{10} + 12 q^{11} + 13 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} - q^{19} + 7 q^{20} + 3 q^{21} - 8 q^{22} - 3 q^{24} + 19 q^{25} - 18 q^{26} + 3 q^{27} + 9 q^{29} + q^{30} + q^{31} + 7 q^{32} + 12 q^{33} + 2 q^{35} - q^{37} + 17 q^{38} + 13 q^{39} - 23 q^{40} - 7 q^{41} - 2 q^{42} + 5 q^{43} + 7 q^{44} + 2 q^{45} - 21 q^{46} - 20 q^{47} + 2 q^{48} + 3 q^{49} + 6 q^{50} + 7 q^{52} - 19 q^{53} - 2 q^{54} + 29 q^{55} - 3 q^{56} - q^{57} + 22 q^{58} - 5 q^{59} + 7 q^{60} + 9 q^{61} + 18 q^{62} + 3 q^{63} - 11 q^{64} - 17 q^{65} - 8 q^{66} + 9 q^{67} + q^{70} - 3 q^{71} - 3 q^{72} + 5 q^{73} - 25 q^{74} + 19 q^{75} - 14 q^{76} + 12 q^{77} - 18 q^{78} - 13 q^{79} - 8 q^{80} + 3 q^{81} - 7 q^{82} - 6 q^{83} - 15 q^{86} + 9 q^{87} - 26 q^{88} - 7 q^{89} + q^{90} + 13 q^{91} + 35 q^{92} + q^{93} + 11 q^{94} + 39 q^{95} + 7 q^{96} + 21 q^{97} - 2 q^{98} + 12 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 3 * q^3 + 2 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + q^10 + 12 * q^11 + 13 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^18 - q^19 + 7 * q^20 + 3 * q^21 - 8 * q^22 - 3 * q^24 + 19 * q^25 - 18 * q^26 + 3 * q^27 + 9 * q^29 + q^30 + q^31 + 7 * q^32 + 12 * q^33 + 2 * q^35 - q^37 + 17 * q^38 + 13 * q^39 - 23 * q^40 - 7 * q^41 - 2 * q^42 + 5 * q^43 + 7 * q^44 + 2 * q^45 - 21 * q^46 - 20 * q^47 + 2 * q^48 + 3 * q^49 + 6 * q^50 + 7 * q^52 - 19 * q^53 - 2 * q^54 + 29 * q^55 - 3 * q^56 - q^57 + 22 * q^58 - 5 * q^59 + 7 * q^60 + 9 * q^61 + 18 * q^62 + 3 * q^63 - 11 * q^64 - 17 * q^65 - 8 * q^66 + 9 * q^67 + q^70 - 3 * q^71 - 3 * q^72 + 5 * q^73 - 25 * q^74 + 19 * q^75 - 14 * q^76 + 12 * q^77 - 18 * q^78 - 13 * q^79 - 8 * q^80 + 3 * q^81 - 7 * q^82 - 6 * q^83 - 15 * q^86 + 9 * q^87 - 26 * q^88 - 7 * q^89 + q^90 + 13 * q^91 + 35 * q^92 + q^93 + 11 * q^94 + 39 * q^95 + 7 * q^96 + 21 * q^97 - 2 * q^98 + 12 * 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 $

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.69202	−0.846011
$5$	−3.85086	−1.72215	−0.861077	−	0.508474i	$-0.830210\pi$
$5$	−3.85086	−1.72215	−0.861077	+	0.508474i	$0.830210\pi$
$6$	−0.554958	−0.226561
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.04892	0.724402
$9$	1.00000	0.333333
$10$	2.13706	0.675799
$11$	0.951083	0.286762	0.143381	−	0.989668i	$-0.454203\pi$
$11$	0.951083	0.286762	0.143381	+	0.989668i	$0.454203\pi$
$12$	−1.69202	−0.488445
$13$	7.15883	1.98550	0.992752	−	0.120184i	$-0.0383485\pi$
$13$	7.15883	1.98550	0.992752	+	0.120184i	$0.0383485\pi$
$14$	−0.554958	−0.148319
$15$	−3.85086	−0.994287
$16$	2.24698	0.561745
$17$	0	0
$17$	0	0
$18$	−0.554958	−0.130805
$19$	−4.51573	−1.03598	−0.517990	−	0.855387i	$-0.673319\pi$
$19$	−4.51573	−1.03598	−0.517990	+	0.855387i	$0.673319\pi$
$20$	6.51573	1.45696
$21$	1.00000	0.218218
$22$	−0.527811	−0.112530
$23$	−2.02715	−0.422689	−0.211345	−	0.977412i	$-0.567784\pi$
$23$	−2.02715	−0.422689	−0.211345	+	0.977412i	$0.567784\pi$
$24$	2.04892	0.418234
$25$	9.82908	1.96582
$26$	−3.97285	−0.779141
$27$	1.00000	0.192450
$28$	−1.69202	−0.319762
$29$	9.76809	1.81389	0.906944	−	0.421251i	$-0.138409\pi$
$29$	9.76809	1.81389	0.906944	+	0.421251i	$0.138409\pi$
$30$	2.13706	0.390173
$31$	6.87800	1.23533	0.617663	−	0.786443i	$-0.288080\pi$
$31$	6.87800	1.23533	0.617663	+	0.786443i	$0.288080\pi$
$32$	−5.34481	−0.944839
$33$	0.951083	0.165562
$34$	0	0
$35$	−3.85086	−0.650913
$36$	−1.69202	−0.282004
$37$	3.62565	0.596052	0.298026	−	0.954558i	$-0.403672\pi$
$37$	3.62565	0.596052	0.298026	+	0.954558i	$0.403672\pi$
$38$	2.50604	0.406533
$39$	7.15883	1.14633
$40$	−7.89008	−1.24753
$41$	1.96077	0.306221	0.153111	−	0.988209i	$-0.451071\pi$
$41$	1.96077	0.306221	0.153111	+	0.988209i	$0.451071\pi$
$42$	−0.554958	−0.0856319
$43$	−9.28382	−1.41577	−0.707884	−	0.706328i	$-0.750350\pi$
$43$	−9.28382	−1.41577	−0.707884	+	0.706328i	$0.750350\pi$
$44$	−1.60925	−0.242604
$45$	−3.85086	−0.574052
$46$	1.12498	0.165870
$47$	−8.24698	−1.20295	−0.601473	−	0.798893i	$-0.705419\pi$
$47$	−8.24698	−1.20295	−0.601473	+	0.798893i	$0.705419\pi$
$48$	2.24698	0.324324
$49$	1.00000	0.142857
$50$	−5.45473	−0.771415
$51$	0	0
$52$	−12.1129	−1.67976
$53$	−8.13706	−1.11771	−0.558856	−	0.829265i	$-0.688759\pi$
$53$	−8.13706	−1.11771	−0.558856	+	0.829265i	$0.688759\pi$
$54$	−0.554958	−0.0755202
$55$	−3.66248	−0.493849
$56$	2.04892	0.273798
$57$	−4.51573	−0.598123
$58$	−5.42088	−0.711796
$59$	−1.55496	−0.202438	−0.101219	−	0.994864i	$-0.532274\pi$
$59$	−1.55496	−0.202438	−0.101219	+	0.994864i	$0.532274\pi$
$60$	6.51573	0.841177
$61$	−6.14675	−0.787011	−0.393505	−	0.919322i	$-0.628738\pi$
$61$	−6.14675	−0.787011	−0.393505	+	0.919322i	$0.628738\pi$
$62$	−3.81700	−0.484760
$63$	1.00000	0.125988
$64$	−1.52781	−0.190976
$65$	−27.5676	−3.41934
$66$	−0.527811	−0.0649690
$67$	2.32975	0.284624	0.142312	−	0.989822i	$-0.454546\pi$
$67$	2.32975	0.284624	0.142312	+	0.989822i	$0.454546\pi$
$68$	0	0
$69$	−2.02715	−0.244040
$70$	2.13706	0.255428
$71$	−4.38404	−0.520290	−0.260145	−	0.965570i	$-0.583770\pi$
$71$	−4.38404	−0.520290	−0.260145	+	0.965570i	$0.583770\pi$
$72$	2.04892	0.241467
$73$	−2.16421	−0.253302	−0.126651	−	0.991947i	$-0.540423\pi$
$73$	−2.16421	−0.253302	−0.126651	+	0.991947i	$0.540423\pi$
$74$	−2.01208	−0.233900
$75$	9.82908	1.13496
$76$	7.64071	0.876450
$77$	0.951083	0.108386
$78$	−3.97285	−0.449837
$79$	1.98792	0.223658	0.111829	−	0.993727i	$-0.464329\pi$
$79$	1.98792	0.223658	0.111829	+	0.993727i	$0.464329\pi$
$80$	−8.65279	−0.967412
$81$	1.00000	0.111111
$82$	−1.08815	−0.120166
$83$	−5.04892	−0.554191	−0.277095	−	0.960842i	$-0.589372\pi$
$83$	−5.04892	−0.554191	−0.277095	+	0.960842i	$0.589372\pi$
$84$	−1.69202	−0.184615
$85$	0	0
$86$	5.15213	0.555568
$87$	9.76809	1.04725
$88$	1.94869	0.207731
$89$	3.31767	0.351672	0.175836	−	0.984419i	$-0.443737\pi$
$89$	3.31767	0.351672	0.175836	+	0.984419i	$0.443737\pi$
$90$	2.13706	0.225266
$91$	7.15883	0.750450
$92$	3.42998	0.357600
$93$	6.87800	0.713216
$94$	4.57673	0.472053
$95$	17.3894	1.78412
$96$	−5.34481	−0.545503
$97$	14.7899	1.50168	0.750841	−	0.660483i	$-0.229648\pi$
$97$	14.7899	1.50168	0.750841	+	0.660483i	$0.229648\pi$
$98$	−0.554958	−0.0560592
$99$	0.951083	0.0955874
$100$	−16.6310	−1.66310
$101$	9.22521	0.917943	0.458971	−	0.888451i	$-0.348218\pi$
$101$	9.22521	0.917943	0.458971	+	0.888451i	$0.348218\pi$
$102$	0	0
$103$	4.86294	0.479159	0.239580	−	0.970877i	$-0.422990\pi$
$103$	4.86294	0.479159	0.239580	+	0.970877i	$0.422990\pi$
$104$	14.6679	1.43830
$105$	−3.85086	−0.375805
$106$	4.51573	0.438606
$107$	1.34481	0.130008	0.0650041	−	0.997885i	$-0.479294\pi$
$107$	1.34481	0.130008	0.0650041	+	0.997885i	$0.479294\pi$
$108$	−1.69202	−0.162815
$109$	18.2228	1.74543	0.872715	−	0.488231i	$-0.162358\pi$
$109$	18.2228	1.74543	0.872715	+	0.488231i	$0.162358\pi$
$110$	2.03252	0.193794
$111$	3.62565	0.344131
$112$	2.24698	0.212320
$113$	−7.47650	−0.703330	−0.351665	−	0.936126i	$-0.614384\pi$
$113$	−7.47650	−0.703330	−0.351665	+	0.936126i	$0.614384\pi$
$114$	2.50604	0.234712
$115$	7.80625	0.727937
$116$	−16.5278	−1.53457
$117$	7.15883	0.661834
$118$	0.862937	0.0794398
$119$	0	0
$120$	−7.89008	−0.720263
$121$	−10.0954	−0.917767
$122$	3.41119	0.308835
$123$	1.96077	0.176797
$124$	−11.6377	−1.04510
$125$	−18.5961	−1.66329
$126$	−0.554958	−0.0494396
$127$	−14.7289	−1.30697	−0.653487	−	0.756937i	$-0.726695\pi$
$127$	−14.7289	−1.30697	−0.653487	+	0.756937i	$0.726695\pi$
$128$	11.5375	1.01978
$129$	−9.28382	−0.817394
$130$	15.2989	1.34180
$131$	10.4601	0.913904	0.456952	−	0.889491i	$-0.348941\pi$
$131$	10.4601	0.913904	0.456952	+	0.889491i	$0.348941\pi$
$132$	−1.60925	−0.140067
$133$	−4.51573	−0.391563
$134$	−1.29291	−0.111691
$135$	−3.85086	−0.331429
$136$	0	0
$137$	−9.35690	−0.799414	−0.399707	−	0.916643i	$-0.630888\pi$
$137$	−9.35690	−0.799414	−0.399707	+	0.916643i	$0.630888\pi$
$138$	1.12498	0.0957648
$139$	17.5864	1.49166	0.745830	−	0.666136i	$-0.232053\pi$
$139$	17.5864	1.49166	0.745830	+	0.666136i	$0.232053\pi$
$140$	6.51573	0.550680
$141$	−8.24698	−0.694521
$142$	2.43296	0.204169
$143$	6.80864	0.569367
$144$	2.24698	0.187248
$145$	−37.6155	−3.12380
$146$	1.20105	0.0993993
$147$	1.00000	0.0824786
$148$	−6.13467	−0.504267
$149$	−10.1564	−0.832048	−0.416024	−	0.909354i	$-0.636577\pi$
$149$	−10.1564	−0.832048	−0.416024	+	0.909354i	$0.636577\pi$
$150$	−5.45473	−0.445377
$151$	14.5187	1.18152	0.590758	−	0.806849i	$-0.298829\pi$
$151$	14.5187	1.18152	0.590758	+	0.806849i	$0.298829\pi$
$152$	−9.25236	−0.750465
$153$	0	0
$154$	−0.527811	−0.0425322
$155$	−26.4862	−2.12742
$156$	−12.1129	−0.969808
$157$	6.53319	0.521405	0.260703	−	0.965419i	$-0.416046\pi$
$157$	6.53319	0.521405	0.260703	+	0.965419i	$0.416046\pi$
$158$	−1.10321	−0.0877668
$159$	−8.13706	−0.645311
$160$	20.5821	1.62716
$161$	−2.02715	−0.159762
$162$	−0.554958	−0.0436016
$163$	7.49396	0.586972	0.293486	−	0.955963i	$-0.405185\pi$
$163$	7.49396	0.586972	0.293486	+	0.955963i	$0.405185\pi$
$164$	−3.31767	−0.259066
$165$	−3.66248	−0.285124
$166$	2.80194	0.217473
$167$	15.1903	1.17546	0.587730	−	0.809057i	$-0.300022\pi$
$167$	15.1903	1.17546	0.587730	+	0.809057i	$0.300022\pi$
$168$	2.04892	0.158077
$169$	38.2489	2.94222
$170$	0	0
$171$	−4.51573	−0.345326
$172$	15.7084	1.19776
$173$	−19.5308	−1.48490	−0.742449	−	0.669902i	$-0.766336\pi$
$173$	−19.5308	−1.48490	−0.742449	+	0.669902i	$0.766336\pi$
$174$	−5.42088	−0.410956
$175$	9.82908	0.743009
$176$	2.13706	0.161087
$177$	−1.55496	−0.116878
$178$	−1.84117	−0.138001
$179$	24.9191	1.86254	0.931272	−	0.364324i	$-0.118700\pi$
$179$	24.9191	1.86254	0.931272	+	0.364324i	$0.118700\pi$
$180$	6.51573	0.485654
$181$	5.59179	0.415635	0.207817	−	0.978168i	$-0.433364\pi$
$181$	5.59179	0.415635	0.207817	+	0.978168i	$0.433364\pi$
$182$	−3.97285	−0.294487
$183$	−6.14675	−0.454381
$184$	−4.15346	−0.306197
$185$	−13.9618	−1.02649
$186$	−3.81700	−0.279876
$187$	0	0
$188$	13.9541	1.01770
$189$	1.00000	0.0727393
$190$	−9.65040	−0.700114
$191$	22.4426	1.62389	0.811947	−	0.583732i	$-0.198408\pi$
$191$	22.4426	1.62389	0.811947	+	0.583732i	$0.198408\pi$
$192$	−1.52781	−0.110260
$193$	−10.6920	−0.769629	−0.384814	−	0.922994i	$-0.625735\pi$
$193$	−10.6920	−0.769629	−0.384814	+	0.922994i	$0.625735\pi$
$194$	−8.20775	−0.589282
$195$	−27.5676	−1.97416
$196$	−1.69202	−0.120859
$197$	3.73125	0.265841	0.132920	−	0.991127i	$-0.457565\pi$
$197$	3.73125	0.265841	0.132920	+	0.991127i	$0.457565\pi$
$198$	−0.527811	−0.0375099
$199$	−1.10023	−0.0779931	−0.0389965	−	0.999239i	$-0.512416\pi$
$199$	−1.10023	−0.0779931	−0.0389965	+	0.999239i	$0.512416\pi$
$200$	20.1390	1.42404
$201$	2.32975	0.164328
$202$	−5.11960	−0.360214
$203$	9.76809	0.685585
$204$	0	0
$205$	−7.55065	−0.527360
$206$	−2.69873	−0.188029
$207$	−2.02715	−0.140896
$208$	16.0858	1.11535
$209$	−4.29483	−0.297080
$210$	2.13706	0.147471
$211$	−8.22282	−0.566082	−0.283041	−	0.959108i	$-0.591343\pi$
$211$	−8.22282	−0.566082	−0.283041	+	0.959108i	$0.591343\pi$
$212$	13.7681	0.945596
$213$	−4.38404	−0.300390
$214$	−0.746316	−0.0510171
$215$	35.7506	2.43817
$216$	2.04892	0.139411
$217$	6.87800	0.466909
$218$	−10.1129	−0.684932
$219$	−2.16421	−0.146244
$220$	6.19700	0.417801
$221$	0	0
$222$	−2.01208	−0.135042
$223$	−3.46250	−0.231866	−0.115933	−	0.993257i	$-0.536986\pi$
$223$	−3.46250	−0.231866	−0.115933	+	0.993257i	$0.536986\pi$
$224$	−5.34481	−0.357115
$225$	9.82908	0.655272
$226$	4.14914	0.275997
$227$	−13.4058	−0.889775	−0.444888	−	0.895586i	$-0.646756\pi$
$227$	−13.4058	−0.889775	−0.444888	+	0.895586i	$0.646756\pi$
$228$	7.64071	0.506018
$229$	−17.1618	−1.13408	−0.567042	−	0.823689i	$-0.691912\pi$
$229$	−17.1618	−1.13408	−0.567042	+	0.823689i	$0.691912\pi$
$230$	−4.33214	−0.285653
$231$	0.951083	0.0625766
$232$	20.0140	1.31398
$233$	−19.7453	−1.29355	−0.646777	−	0.762679i	$-0.723884\pi$
$233$	−19.7453	−1.29355	−0.646777	+	0.762679i	$0.723884\pi$
$234$	−3.97285	−0.259714
$235$	31.7579	2.07166
$236$	2.63102	0.171265
$237$	1.98792	0.129129
$238$	0	0
$239$	23.2228	1.50216	0.751080	−	0.660212i	$-0.229533\pi$
$239$	23.2228	1.50216	0.751080	+	0.660212i	$0.229533\pi$
$240$	−8.65279	−0.558535
$241$	−14.0489	−0.904970	−0.452485	−	0.891772i	$-0.649462\pi$
$241$	−14.0489	−0.904970	−0.452485	+	0.891772i	$0.649462\pi$
$242$	5.60255	0.360145
$243$	1.00000	0.0641500
$244$	10.4004	0.665820
$245$	−3.85086	−0.246022
$246$	−1.08815	−0.0693777
$247$	−32.3274	−2.05694
$248$	14.0925	0.894872
$249$	−5.04892	−0.319962
$250$	10.3201	0.652698
$251$	−26.3274	−1.66177	−0.830884	−	0.556446i	$-0.812165\pi$
$251$	−26.3274	−1.66177	−0.830884	+	0.556446i	$0.812165\pi$
$252$	−1.69202	−0.106587
$253$	−1.92798	−0.121211
$254$	8.17390	0.512876
$255$	0	0
$256$	−3.34721	−0.209200
$257$	14.1317	0.881510	0.440755	−	0.897627i	$-0.354711\pi$
$257$	14.1317	0.881510	0.440755	+	0.897627i	$0.354711\pi$
$258$	5.15213	0.320758
$259$	3.62565	0.225287
$260$	46.6450	2.89280
$261$	9.76809	0.604629
$262$	−5.80492	−0.358629
$263$	−13.2121	−0.814691	−0.407345	−	0.913274i	$-0.633545\pi$
$263$	−13.2121	−0.814691	−0.407345	+	0.913274i	$0.633545\pi$
$264$	1.94869	0.119934
$265$	31.3347	1.92487
$266$	2.50604	0.153655
$267$	3.31767	0.203038
$268$	−3.94198	−0.240795
$269$	30.7211	1.87310	0.936549	−	0.350537i	$-0.114001\pi$
$269$	30.7211	1.87310	0.936549	+	0.350537i	$0.114001\pi$
$270$	2.13706	0.130058
$271$	31.7071	1.92607	0.963034	−	0.269379i	$-0.0868185\pi$
$271$	31.7071	1.92607	0.963034	+	0.269379i	$0.0868185\pi$
$272$	0	0
$273$	7.15883	0.433272
$274$	5.19269	0.313702
$275$	9.34827	0.563722
$276$	3.42998	0.206460
$277$	−24.4131	−1.46684	−0.733421	−	0.679775i	$-0.762077\pi$
$277$	−24.4131	−1.46684	−0.733421	+	0.679775i	$0.762077\pi$
$278$	−9.75973	−0.585349
$279$	6.87800	0.411775
$280$	−7.89008	−0.471523
$281$	−16.5036	−0.984525	−0.492263	−	0.870447i	$-0.663830\pi$
$281$	−16.5036	−0.984525	−0.492263	+	0.870447i	$0.663830\pi$
$282$	4.57673	0.272540
$283$	17.3840	1.03337	0.516687	−	0.856174i	$-0.327165\pi$
$283$	17.3840	1.03337	0.516687	+	0.856174i	$0.327165\pi$
$284$	7.41789	0.440171
$285$	17.3894	1.03006
$286$	−3.77851	−0.223428
$287$	1.96077	0.115741
$288$	−5.34481	−0.314946
$289$	0	0
$290$	20.8750	1.22582
$291$	14.7899	0.866997
$292$	3.66189	0.214296
$293$	−33.2301	−1.94132	−0.970662	−	0.240448i	$-0.922706\pi$
$293$	−33.2301	−1.94132	−0.970662	+	0.240448i	$0.922706\pi$
$294$	−0.554958	−0.0323658
$295$	5.98792	0.348630
$296$	7.42865	0.431781
$297$	0.951083	0.0551874
$298$	5.63640	0.326508
$299$	−14.5120	−0.839251
$300$	−16.6310	−0.960193
$301$	−9.28382	−0.535110
$302$	−8.05728	−0.463644
$303$	9.22521	0.529974
$304$	−10.1468	−0.581956
$305$	23.6703	1.35535
$306$	0	0
$307$	−14.9825	−0.855099	−0.427549	−	0.903992i	$-0.640623\pi$
$307$	−14.9825	−0.855099	−0.427549	+	0.903992i	$0.640623\pi$
$308$	−1.60925	−0.0916957
$309$	4.86294	0.276643
$310$	14.6987	0.834832
$311$	8.61356	0.488430	0.244215	−	0.969721i	$-0.421470\pi$
$311$	8.61356	0.488430	0.244215	+	0.969721i	$0.421470\pi$
$312$	14.6679	0.830404
$313$	12.3207	0.696404	0.348202	−	0.937419i	$-0.386792\pi$
$313$	12.3207	0.696404	0.348202	+	0.937419i	$0.386792\pi$
$314$	−3.62565	−0.204607
$315$	−3.85086	−0.216971
$316$	−3.36360	−0.189217
$317$	16.7536	0.940977	0.470488	−	0.882406i	$-0.344078\pi$
$317$	16.7536	0.940977	0.470488	+	0.882406i	$0.344078\pi$
$318$	4.51573	0.253230
$319$	9.29026	0.520155
$320$	5.88338	0.328891
$321$	1.34481	0.0750602
$322$	1.12498	0.0626928
$323$	0	0
$324$	−1.69202	−0.0940012
$325$	70.3648	3.90314
$326$	−4.15883	−0.230336
$327$	18.2228	1.00772
$328$	4.01746	0.221827
$329$	−8.24698	−0.454671
$330$	2.03252	0.111887
$331$	14.4547	0.794504	0.397252	−	0.917710i	$-0.369964\pi$
$331$	14.4547	0.794504	0.397252	+	0.917710i	$0.369964\pi$
$332$	8.54288	0.468851
$333$	3.62565	0.198684
$334$	−8.42998	−0.461268
$335$	−8.97152	−0.490167
$336$	2.24698	0.122583
$337$	−2.32304	−0.126544	−0.0632721	−	0.997996i	$-0.520154\pi$
$337$	−2.32304	−0.126544	−0.0632721	+	0.997996i	$0.520154\pi$
$338$	−21.2265	−1.15457
$339$	−7.47650	−0.406068
$340$	0	0
$341$	6.54155	0.354245
$342$	2.50604	0.135511
$343$	1.00000	0.0539949
$344$	−19.0218	−1.02559
$345$	7.80625	0.420274
$346$	10.8388	0.582696
$347$	26.2161	1.40735	0.703677	−	0.710520i	$-0.251540\pi$
$347$	26.2161	1.40735	0.703677	+	0.710520i	$0.251540\pi$
$348$	−16.5278	−0.885984
$349$	29.6601	1.58767	0.793834	−	0.608134i	$-0.208082\pi$
$349$	29.6601	1.58767	0.793834	+	0.608134i	$0.208082\pi$
$350$	−5.45473	−0.291568
$351$	7.15883	0.382110
$352$	−5.08336	−0.270944
$353$	−25.0301	−1.33222	−0.666110	−	0.745854i	$-0.732042\pi$
$353$	−25.0301	−1.33222	−0.666110	+	0.745854i	$0.732042\pi$
$354$	0.862937	0.0458646
$355$	16.8823	0.896020
$356$	−5.61356	−0.297518
$357$	0	0
$358$	−13.8291	−0.730890
$359$	0.987918	0.0521403	0.0260702	−	0.999660i	$-0.491701\pi$
$359$	0.987918	0.0521403	0.0260702	+	0.999660i	$0.491701\pi$
$360$	−7.89008	−0.415844
$361$	1.39181	0.0732533
$362$	−3.10321	−0.163101
$363$	−10.0954	−0.529873
$364$	−12.1129	−0.634888
$365$	8.33406	0.436225
$366$	3.41119	0.178306
$367$	21.3086	1.11230	0.556149	−	0.831083i	$-0.312278\pi$
$367$	21.3086	1.11230	0.556149	+	0.831083i	$0.312278\pi$
$368$	−4.55496	−0.237444
$369$	1.96077	0.102074
$370$	7.74823	0.402812
$371$	−8.13706	−0.422455
$372$	−11.6377	−0.603388
$373$	−8.88040	−0.459809	−0.229905	−	0.973213i	$-0.573841\pi$
$373$	−8.88040	−0.459809	−0.229905	+	0.973213i	$0.573841\pi$
$374$	0	0
$375$	−18.5961	−0.960299
$376$	−16.8974	−0.871416
$377$	69.9281	3.60148
$378$	−0.554958	−0.0285440
$379$	28.9976	1.48951	0.744754	−	0.667340i	$-0.232567\pi$
$379$	28.9976	1.48951	0.744754	+	0.667340i	$0.232567\pi$
$380$	−29.4233	−1.50938
$381$	−14.7289	−0.754582
$382$	−12.4547	−0.637239
$383$	27.5163	1.40602	0.703009	−	0.711181i	$-0.251839\pi$
$383$	27.5163	1.40602	0.703009	+	0.711181i	$0.251839\pi$
$384$	11.5375	0.588771
$385$	−3.66248	−0.186657
$386$	5.93362	0.302014
$387$	−9.28382	−0.471923
$388$	−25.0248	−1.27044
$389$	−15.3220	−0.776855	−0.388428	−	0.921479i	$-0.626982\pi$
$389$	−15.3220	−0.776855	−0.388428	+	0.921479i	$0.626982\pi$
$390$	15.2989	0.774689
$391$	0	0
$392$	2.04892	0.103486
$393$	10.4601	0.527643
$394$	−2.07069	−0.104320
$395$	−7.65519	−0.385174
$396$	−1.60925	−0.0808680
$397$	9.64742	0.484190	0.242095	−	0.970253i	$-0.422165\pi$
$397$	9.64742	0.484190	0.242095	+	0.970253i	$0.422165\pi$
$398$	0.610580	0.0306056
$399$	−4.51573	−0.226069
$400$	22.0858	1.10429
$401$	−19.1903	−0.958317	−0.479159	−	0.877728i	$-0.659058\pi$
$401$	−19.1903	−0.958317	−0.479159	+	0.877728i	$0.659058\pi$
$402$	−1.29291	−0.0644846
$403$	49.2385	2.45274
$404$	−15.6093	−0.776589
$405$	−3.85086	−0.191351
$406$	−5.42088	−0.269034
$407$	3.44829	0.170925
$408$	0	0
$409$	36.1812	1.78904	0.894522	−	0.447023i	$-0.147516\pi$
$409$	36.1812	1.78904	0.894522	+	0.447023i	$0.147516\pi$
$410$	4.19029	0.206944
$411$	−9.35690	−0.461542
$412$	−8.22819	−0.405374
$413$	−1.55496	−0.0765145
$414$	1.12498	0.0552898
$415$	19.4426	0.954402
$416$	−38.2626	−1.87598
$417$	17.5864	0.861211
$418$	2.38345	0.116578
$419$	35.8726	1.75249	0.876246	−	0.481864i	$-0.160040\pi$
$419$	35.8726	1.75249	0.876246	+	0.481864i	$0.160040\pi$
$420$	6.51573	0.317935
$421$	−15.4015	−0.750623	−0.375312	−	0.926899i	$-0.622464\pi$
$421$	−15.4015	−0.750623	−0.375312	+	0.926899i	$0.622464\pi$
$422$	4.56332	0.222139
$423$	−8.24698	−0.400982
$424$	−16.6722	−0.809672
$425$	0	0
$426$	2.43296	0.117877
$427$	−6.14675	−0.297462
$428$	−2.27545	−0.109988
$429$	6.80864	0.328724
$430$	−19.8401	−0.956775
$431$	−11.8877	−0.572610	−0.286305	−	0.958139i	$-0.592427\pi$
$431$	−11.8877	−0.572610	−0.286305	+	0.958139i	$0.592427\pi$
$432$	2.24698	0.108108
$433$	−6.15213	−0.295652	−0.147826	−	0.989013i	$-0.547228\pi$
$433$	−6.15213	−0.295652	−0.147826	+	0.989013i	$0.547228\pi$
$434$	−3.81700	−0.183222
$435$	−37.6155	−1.80352
$436$	−30.8334	−1.47665
$437$	9.15405	0.437897
$438$	1.20105	0.0573882
$439$	−8.52542	−0.406896	−0.203448	−	0.979086i	$-0.565215\pi$
$439$	−8.52542	−0.406896	−0.203448	+	0.979086i	$0.565215\pi$
$440$	−7.50412	−0.357745
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.4155	0.874947	0.437473	−	0.899231i	$-0.355873\pi$
$443$	18.4155	0.874947	0.437473	+	0.899231i	$0.355873\pi$
$444$	−6.13467	−0.291139
$445$	−12.7759	−0.605634
$446$	1.92154	0.0909877
$447$	−10.1564	−0.480383
$448$	−1.52781	−0.0721823
$449$	28.3153	1.33628	0.668140	−	0.744035i	$-0.267091\pi$
$449$	28.3153	1.33628	0.668140	+	0.744035i	$0.267091\pi$
$450$	−5.45473	−0.257138
$451$	1.86486	0.0878126
$452$	12.6504	0.595025
$453$	14.5187	0.682149
$454$	7.43967	0.349161
$455$	−27.5676	−1.29239
$456$	−9.25236	−0.433281
$457$	0.652793	0.0305364	0.0152682	−	0.999883i	$-0.495140\pi$
$457$	0.652793	0.0305364	0.0152682	+	0.999883i	$0.495140\pi$
$458$	9.52409	0.445032
$459$	0	0
$460$	−13.2083	−0.615842
$461$	−0.292913	−0.0136423	−0.00682116	−	0.999977i	$-0.502171\pi$
$461$	−0.292913	−0.0136423	−0.00682116	+	0.999977i	$0.502171\pi$
$462$	−0.527811	−0.0245560
$463$	16.4969	0.766678	0.383339	−	0.923608i	$-0.374774\pi$
$463$	16.4969	0.766678	0.383339	+	0.923608i	$0.374774\pi$
$464$	21.9487	1.01894
$465$	−26.4862	−1.22827
$466$	10.9578	0.507610
$467$	17.9801	0.832022	0.416011	−	0.909360i	$-0.363428\pi$
$467$	17.9801	0.832022	0.416011	+	0.909360i	$0.363428\pi$
$468$	−12.1129	−0.559919
$469$	2.32975	0.107578
$470$	−17.6243	−0.812949
$471$	6.53319	0.301033
$472$	−3.18598	−0.146647
$473$	−8.82968	−0.405989
$474$	−1.10321	−0.0506722
$475$	−44.3855	−2.03655
$476$	0	0
$477$	−8.13706	−0.372571
$478$	−12.8877	−0.589469
$479$	−4.23596	−0.193546	−0.0967730	−	0.995306i	$-0.530852\pi$
$479$	−4.23596	−0.193546	−0.0967730	+	0.995306i	$0.530852\pi$
$480$	20.5821	0.939440
$481$	25.9554	1.18346
$482$	7.79656	0.355124
$483$	−2.02715	−0.0922384
$484$	17.0817	0.776441
$485$	−56.9536	−2.58613
$486$	−0.554958	−0.0251734
$487$	−3.47112	−0.157292	−0.0786458	−	0.996903i	$-0.525060\pi$
$487$	−3.47112	−0.157292	−0.0786458	+	0.996903i	$0.525060\pi$
$488$	−12.5942	−0.570112
$489$	7.49396	0.338889
$490$	2.13706	0.0965427
$491$	11.0814	0.500098	0.250049	−	0.968233i	$-0.419553\pi$
$491$	11.0814	0.500098	0.250049	+	0.968233i	$0.419553\pi$
$492$	−3.31767	−0.149572
$493$	0	0
$494$	17.9403	0.807174
$495$	−3.66248	−0.164616
$496$	15.4547	0.693938
$497$	−4.38404	−0.196651
$498$	2.80194	0.125558
$499$	6.10262	0.273191	0.136595	−	0.990627i	$-0.456384\pi$
$499$	6.10262	0.273191	0.136595	+	0.990627i	$0.456384\pi$
$500$	31.4650	1.40716
$501$	15.1903	0.678652
$502$	14.6106	0.652102
$503$	42.0140	1.87331	0.936656	−	0.350251i	$-0.113904\pi$
$503$	42.0140	1.87331	0.936656	+	0.350251i	$0.113904\pi$
$504$	2.04892	0.0912660
$505$	−35.5249	−1.58084
$506$	1.06995	0.0475651
$507$	38.2489	1.69869
$508$	24.9215	1.10571
$509$	−4.48858	−0.198953	−0.0994765	−	0.995040i	$-0.531717\pi$
$509$	−4.48858	−0.198953	−0.0994765	+	0.995040i	$0.531717\pi$
$510$	0	0
$511$	−2.16421	−0.0957390
$512$	−21.2174	−0.937687
$513$	−4.51573	−0.199374
$514$	−7.84249	−0.345918
$515$	−18.7265	−0.825187
$516$	15.7084	0.691524
$517$	−7.84356	−0.344959
$518$	−2.01208	−0.0884058
$519$	−19.5308	−0.857307
$520$	−56.4838	−2.47698
$521$	−4.80492	−0.210507	−0.105254	−	0.994445i	$-0.533565\pi$
$521$	−4.80492	−0.210507	−0.105254	+	0.994445i	$0.533565\pi$
$522$	−5.42088	−0.237265
$523$	4.75600	0.207966	0.103983	−	0.994579i	$-0.466841\pi$
$523$	4.75600	0.207966	0.103983	+	0.994579i	$0.466841\pi$
$524$	−17.6987	−0.773172
$525$	9.82908	0.428976
$526$	7.33214	0.319697
$527$	0	0
$528$	2.13706	0.0930037
$529$	−18.8907	−0.821334
$530$	−17.3894	−0.755348
$531$	−1.55496	−0.0674794
$532$	7.64071	0.331267
$533$	14.0368	0.608003
$534$	−1.84117	−0.0796751
$535$	−5.17868	−0.223894
$536$	4.77346	0.206182
$537$	24.9191	1.07534
$538$	−17.0489	−0.735031
$539$	0.951083	0.0409660
$540$	6.51573	0.280392
$541$	−26.4620	−1.13769	−0.568846	−	0.822444i	$-0.692610\pi$
$541$	−26.4620	−1.13769	−0.568846	+	0.822444i	$0.692610\pi$
$542$	−17.5961	−0.755817
$543$	5.59179	0.239967
$544$	0	0
$545$	−70.1734	−3.00590
$546$	−3.97285	−0.170022
$547$	16.3134	0.697509	0.348754	−	0.937214i	$-0.386605\pi$
$547$	16.3134	0.697509	0.348754	+	0.937214i	$0.386605\pi$
$548$	15.8321	0.676312
$549$	−6.14675	−0.262337
$550$	−5.18790	−0.221213
$551$	−44.1100	−1.87915
$552$	−4.15346	−0.176783
$553$	1.98792	0.0845349
$554$	13.5483	0.575610
$555$	−13.9618	−0.592647
$556$	−29.7566	−1.26196
$557$	1.43057	0.0606151	0.0303075	−	0.999541i	$-0.490351\pi$
$557$	1.43057	0.0606151	0.0303075	+	0.999541i	$0.490351\pi$
$558$	−3.81700	−0.161587
$559$	−66.4613	−2.81101
$560$	−8.65279	−0.365647
$561$	0	0
$562$	9.15883	0.386342
$563$	−26.4969	−1.11671	−0.558357	−	0.829601i	$-0.688568\pi$
$563$	−26.4969	−1.11671	−0.558357	+	0.829601i	$0.688568\pi$
$564$	13.9541	0.587572
$565$	28.7909	1.21124
$566$	−9.64742	−0.405511
$567$	1.00000	0.0419961
$568$	−8.98254	−0.376899
$569$	18.8267	0.789256	0.394628	−	0.918841i	$-0.370873\pi$
$569$	18.8267	0.789256	0.394628	+	0.918841i	$0.370873\pi$
$570$	−9.65040	−0.404211
$571$	−29.4403	−1.23204	−0.616018	−	0.787732i	$-0.711255\pi$
$571$	−29.4403	−1.23204	−0.616018	+	0.787732i	$0.711255\pi$
$572$	−11.5204	−0.481691
$573$	22.4426	0.937555
$574$	−1.08815	−0.0454183
$575$	−19.9250	−0.830930
$576$	−1.52781	−0.0636588
$577$	15.8436	0.659576	0.329788	−	0.944055i	$-0.393023\pi$
$577$	15.8436	0.659576	0.329788	+	0.944055i	$0.393023\pi$
$578$	0	0
$579$	−10.6920	−0.444345
$580$	63.6462	2.64276
$581$	−5.04892	−0.209464
$582$	−8.20775	−0.340222
$583$	−7.73902	−0.320517
$584$	−4.43429	−0.183492
$585$	−27.5676	−1.13978
$586$	18.4413	0.761804
$587$	−18.0151	−0.743561	−0.371781	−	0.928321i	$-0.621253\pi$
$587$	−18.0151	−0.743561	−0.371781	+	0.928321i	$0.621253\pi$
$588$	−1.69202	−0.0697778
$589$	−31.0592	−1.27977
$590$	−3.32304	−0.136808
$591$	3.73125	0.153483
$592$	8.14675	0.334829
$593$	−12.8086	−0.525988	−0.262994	−	0.964797i	$-0.584710\pi$
$593$	−12.8086	−0.525988	−0.262994	+	0.964797i	$0.584710\pi$
$594$	−0.527811	−0.0216563
$595$	0	0
$596$	17.1849	0.703922
$597$	−1.10023	−0.0450293
$598$	8.05356	0.329334
$599$	−16.6160	−0.678910	−0.339455	−	0.940622i	$-0.610243\pi$
$599$	−16.6160	−0.678910	−0.339455	+	0.940622i	$0.610243\pi$
$600$	20.1390	0.822171
$601$	35.1094	1.43214	0.716072	−	0.698026i	$-0.245938\pi$
$601$	35.1094	1.43214	0.716072	+	0.698026i	$0.245938\pi$
$602$	5.15213	0.209985
$603$	2.32975	0.0948747
$604$	−24.5660	−0.999575
$605$	38.8761	1.58054
$606$	−5.11960	−0.207970
$607$	14.8702	0.603564	0.301782	−	0.953377i	$-0.402419\pi$
$607$	14.8702	0.603564	0.301782	+	0.953377i	$0.402419\pi$
$608$	24.1357	0.978833
$609$	9.76809	0.395823
$610$	−13.1360	−0.531861
$611$	−59.0388	−2.38845
$612$	0	0
$613$	22.6698	0.915624	0.457812	−	0.889049i	$-0.348633\pi$
$613$	22.6698	0.915624	0.457812	+	0.889049i	$0.348633\pi$
$614$	8.31468	0.335553
$615$	−7.55065	−0.304471
$616$	1.94869	0.0785149
$617$	14.6571	0.590073	0.295036	−	0.955486i	$-0.404668\pi$
$617$	14.6571	0.590073	0.295036	+	0.955486i	$0.404668\pi$
$618$	−2.69873	−0.108559
$619$	−17.3599	−0.697752	−0.348876	−	0.937169i	$-0.613437\pi$
$619$	−17.3599	−0.697752	−0.348876	+	0.937169i	$0.613437\pi$
$620$	44.8152	1.79982
$621$	−2.02715	−0.0813466
$622$	−4.78017	−0.191667
$623$	3.31767	0.132920
$624$	16.0858	0.643945
$625$	22.4655	0.898619
$626$	−6.83745	−0.273279
$627$	−4.29483	−0.171519
$628$	−11.0543	−0.441114
$629$	0	0
$630$	2.13706	0.0851426
$631$	13.4789	0.536586	0.268293	−	0.963337i	$-0.413540\pi$
$631$	13.4789	0.536586	0.268293	+	0.963337i	$0.413540\pi$
$632$	4.07308	0.162018
$633$	−8.22282	−0.326828
$634$	−9.29755	−0.369253
$635$	56.7187	2.25081
$636$	13.7681	0.545940
$637$	7.15883	0.283643
$638$	−5.15570	−0.204116
$639$	−4.38404	−0.173430
$640$	−44.4292	−1.75622
$641$	2.20775	0.0872009	0.0436005	−	0.999049i	$-0.486117\pi$
$641$	2.20775	0.0872009	0.0436005	+	0.999049i	$0.486117\pi$
$642$	−0.746316	−0.0294547
$643$	10.7597	0.424322	0.212161	−	0.977235i	$-0.431950\pi$
$643$	10.7597	0.424322	0.212161	+	0.977235i	$0.431950\pi$
$644$	3.42998	0.135160
$645$	35.7506	1.40768
$646$	0	0
$647$	17.0067	0.668603	0.334301	−	0.942466i	$-0.391500\pi$
$647$	17.0067	0.668603	0.334301	+	0.942466i	$0.391500\pi$
$648$	2.04892	0.0804891
$649$	−1.47889	−0.0580517
$650$	−39.0495	−1.53165
$651$	6.87800	0.269570
$652$	−12.6799	−0.496585
$653$	−7.49934	−0.293472	−0.146736	−	0.989176i	$-0.546877\pi$
$653$	−7.49934	−0.293472	−0.146736	+	0.989176i	$0.546877\pi$
$654$	−10.1129	−0.395446
$655$	−40.2804	−1.57388
$656$	4.40581	0.172018
$657$	−2.16421	−0.0844339
$658$	4.57673	0.178419
$659$	39.5013	1.53875	0.769375	−	0.638797i	$-0.220568\pi$
$659$	39.5013	1.53875	0.769375	+	0.638797i	$0.220568\pi$
$660$	6.19700	0.241218
$661$	−0.587745	−0.0228606	−0.0114303	−	0.999935i	$-0.503638\pi$
$661$	−0.587745	−0.0228606	−0.0114303	+	0.999935i	$0.503638\pi$
$662$	−8.02177	−0.311775
$663$	0	0
$664$	−10.3448	−0.401457
$665$	17.3894	0.674333
$666$	−2.01208	−0.0779666
$667$	−19.8013	−0.766711
$668$	−25.7023	−0.994452
$669$	−3.46250	−0.133868
$670$	4.97882	0.192349
$671$	−5.84607	−0.225685
$672$	−5.34481	−0.206181
$673$	−32.2664	−1.24378	−0.621888	−	0.783106i	$-0.713634\pi$
$673$	−32.2664	−1.24378	−0.621888	+	0.783106i	$0.713634\pi$
$674$	1.28919	0.0496578
$675$	9.82908	0.378322
$676$	−64.7180	−2.48915
$677$	−35.4959	−1.36422	−0.682109	−	0.731251i	$-0.738937\pi$
$677$	−35.4959	−1.36422	−0.682109	+	0.731251i	$0.738937\pi$
$678$	4.14914	0.159347
$679$	14.7899	0.567583
$680$	0	0
$681$	−13.4058	−0.513712
$682$	−3.63029	−0.139011
$683$	22.8605	0.874734	0.437367	−	0.899283i	$-0.355911\pi$
$683$	22.8605	0.874734	0.437367	+	0.899283i	$0.355911\pi$
$684$	7.64071	0.292150
$685$	36.0320	1.37671
$686$	−0.554958	−0.0211884
$687$	−17.1618	−0.654764
$688$	−20.8605	−0.795301
$689$	−58.2519	−2.21922
$690$	−4.33214	−0.164922
$691$	12.7269	0.484156	0.242078	−	0.970257i	$-0.422171\pi$
$691$	12.7269	0.484156	0.242078	+	0.970257i	$0.422171\pi$
$692$	33.0465	1.25624
$693$	0.951083	0.0361286
$694$	−14.5488	−0.552267
$695$	−67.7227	−2.56887
$696$	20.0140	0.758629
$697$	0	0
$698$	−16.4601	−0.623024
$699$	−19.7453	−0.746834
$700$	−16.6310	−0.628594
$701$	−30.1914	−1.14031	−0.570156	−	0.821537i	$-0.693117\pi$
$701$	−30.1914	−1.14031	−0.570156	+	0.821537i	$0.693117\pi$
$702$	−3.97285	−0.149946
$703$	−16.3724	−0.617498
$704$	−1.45307	−0.0547648
$705$	31.7579	1.19607
$706$	13.8907	0.522782
$707$	9.22521	0.346950
$708$	2.63102	0.0988799
$709$	39.6034	1.48734	0.743668	−	0.668549i	$-0.233084\pi$
$709$	39.6034	1.48734	0.743668	+	0.668549i	$0.233084\pi$
$710$	−9.36898	−0.351611
$711$	1.98792	0.0745528
$712$	6.79763	0.254752
$713$	−13.9427	−0.522159
$714$	0	0
$715$	−26.2191	−0.980539
$716$	−42.1637	−1.57573
$717$	23.2228	0.867272
$718$	−0.548253	−0.0204606
$719$	−11.1787	−0.416895	−0.208447	−	0.978034i	$-0.566841\pi$
$719$	−11.1787	−0.416895	−0.208447	+	0.978034i	$0.566841\pi$
$720$	−8.65279	−0.322471
$721$	4.86294	0.181105
$722$	−0.772398	−0.0287457
$723$	−14.0489	−0.522485
$724$	−9.46144	−0.351631
$725$	96.0113	3.56577
$726$	5.60255	0.207930
$727$	16.3690	0.607092	0.303546	−	0.952817i	$-0.401829\pi$
$727$	16.3690	0.607092	0.303546	+	0.952817i	$0.401829\pi$
$728$	14.6679	0.543627
$729$	1.00000	0.0370370
$730$	−4.62505	−0.171181
$731$	0	0
$732$	10.4004	0.384411
$733$	4.65950	0.172102	0.0860512	−	0.996291i	$-0.472575\pi$
$733$	4.65950	0.172102	0.0860512	+	0.996291i	$0.472575\pi$
$734$	−11.8254	−0.436482
$735$	−3.85086	−0.142041
$736$	10.8347	0.399373
$737$	2.21578	0.0816194
$738$	−1.08815	−0.0400552
$739$	−28.9903	−1.06643	−0.533213	−	0.845981i	$-0.679016\pi$
$739$	−28.9903	−1.06643	−0.533213	+	0.845981i	$0.679016\pi$
$740$	23.6237	0.868425
$741$	−32.3274	−1.18758
$742$	4.51573	0.165778
$743$	−12.0965	−0.443778	−0.221889	−	0.975072i	$-0.571222\pi$
$743$	−12.0965	−0.443778	−0.221889	+	0.975072i	$0.571222\pi$
$744$	14.0925	0.516655
$745$	39.1110	1.43292
$746$	4.92825	0.180436
$747$	−5.04892	−0.184730
$748$	0	0
$749$	1.34481	0.0491384
$750$	10.3201	0.376835
$751$	−17.4499	−0.636758	−0.318379	−	0.947964i	$-0.603138\pi$
$751$	−17.4499	−0.636758	−0.318379	+	0.947964i	$0.603138\pi$
$752$	−18.5308	−0.675749
$753$	−26.3274	−0.959422
$754$	−38.8072	−1.41327
$755$	−55.9095	−2.03475
$756$	−1.69202	−0.0615382
$757$	18.1148	0.658394	0.329197	−	0.944261i	$-0.393222\pi$
$757$	18.1148	0.658394	0.329197	+	0.944261i	$0.393222\pi$
$758$	−16.0925	−0.584504
$759$	−1.92798	−0.0699814
$760$	35.6295	1.29242
$761$	−19.0291	−0.689803	−0.344902	−	0.938639i	$-0.612088\pi$
$761$	−19.0291	−0.689803	−0.344902	+	0.938639i	$0.612088\pi$
$762$	8.17390	0.296109
$763$	18.2228	0.659710
$764$	−37.9734	−1.37383
$765$	0	0
$766$	−15.2704	−0.551742
$767$	−11.1317	−0.401942
$768$	−3.34721	−0.120782
$769$	7.64204	0.275579	0.137789	−	0.990462i	$-0.456000\pi$
$769$	7.64204	0.275579	0.137789	+	0.990462i	$0.456000\pi$
$770$	2.03252	0.0732471
$771$	14.1317	0.508940
$772$	18.0911	0.651114
$773$	28.3142	1.01839	0.509196	−	0.860651i	$-0.329943\pi$
$773$	28.3142	1.01839	0.509196	+	0.860651i	$0.329943\pi$
$774$	5.15213	0.185189
$775$	67.6045	2.42842
$776$	30.3032	1.08782
$777$	3.62565	0.130069
$778$	8.50306	0.304849
$779$	−8.85431	−0.317239
$780$	46.6450	1.67016
$781$	−4.16959	−0.149200
$782$	0	0
$783$	9.76809	0.349083
$784$	2.24698	0.0802493
$785$	−25.1584	−0.897940
$786$	−5.80492	−0.207055
$787$	26.9982	0.962382	0.481191	−	0.876616i	$-0.340204\pi$
$787$	26.9982	0.962382	0.481191	+	0.876616i	$0.340204\pi$
$788$	−6.31336	−0.224904
$789$	−13.2121	−0.470362
$790$	4.24831	0.151148
$791$	−7.47650	−0.265834
$792$	1.94869	0.0692437
$793$	−44.0036	−1.56261
$794$	−5.35391	−0.190003
$795$	31.3347	1.11133
$796$	1.86161	0.0659830
$797$	4.55065	0.161192	0.0805961	−	0.996747i	$-0.474318\pi$
$797$	4.55065	0.161192	0.0805961	+	0.996747i	$0.474318\pi$
$798$	2.50604	0.0887129
$799$	0	0
$800$	−52.5346	−1.85738
$801$	3.31767	0.117224
$802$	10.6498	0.376058
$803$	−2.05834	−0.0726373
$804$	−3.94198	−0.139023
$805$	7.80625	0.275134
$806$	−27.3253	−0.962492
$807$	30.7211	1.08143
$808$	18.9017	0.664959
$809$	29.4010	1.03369	0.516843	−	0.856080i	$-0.327107\pi$
$809$	29.4010	1.03369	0.516843	+	0.856080i	$0.327107\pi$
$810$	2.13706	0.0750888
$811$	−4.31229	−0.151425	−0.0757125	−	0.997130i	$-0.524123\pi$
$811$	−4.31229	−0.151425	−0.0757125	+	0.997130i	$0.524123\pi$
$812$	−16.5278	−0.580012
$813$	31.7071	1.11202
$814$	−1.91366	−0.0670736
$815$	−28.8582	−1.01086
$816$	0	0
$817$	41.9232	1.46671
$818$	−20.0790	−0.702047
$819$	7.15883	0.250150
$820$	12.7759	0.446152
$821$	0.165275	0.00576815	0.00288407	−	0.999996i	$-0.499082\pi$
$821$	0.165275	0.00576815	0.00288407	+	0.999996i	$0.499082\pi$
$822$	5.19269	0.181116
$823$	14.7248	0.513275	0.256637	−	0.966508i	$-0.417385\pi$
$823$	14.7248	0.513275	0.256637	+	0.966508i	$0.417385\pi$
$824$	9.96376	0.347104
$825$	9.34827	0.325465
$826$	0.862937	0.0300254
$827$	16.7764	0.583374	0.291687	−	0.956514i	$-0.405783\pi$
$827$	16.7764	0.583374	0.291687	+	0.956514i	$0.405783\pi$
$828$	3.42998	0.119200
$829$	35.1672	1.22141	0.610704	−	0.791859i	$-0.290887\pi$
$829$	35.1672	1.22141	0.610704	+	0.791859i	$0.290887\pi$
$830$	−10.7899	−0.374521
$831$	−24.4131	−0.846881
$832$	−10.9373	−0.379184
$833$	0	0
$834$	−9.75973	−0.337952
$835$	−58.4956	−2.02432
$836$	7.26695	0.251333
$837$	6.87800	0.237739
$838$	−19.9078	−0.687704
$839$	−22.6233	−0.781041	−0.390521	−	0.920594i	$-0.627705\pi$
$839$	−22.6233	−0.781041	−0.390521	+	0.920594i	$0.627705\pi$
$840$	−7.89008	−0.272234
$841$	66.4155	2.29019
$842$	8.54719	0.294556
$843$	−16.5036	−0.568416
$844$	13.9132	0.478911
$845$	−147.291	−5.06696
$846$	4.57673	0.157351
$847$	−10.0954	−0.346883
$848$	−18.2838	−0.627869
$849$	17.3840	0.596619
$850$	0	0
$851$	−7.34972	−0.251945
$852$	7.41789	0.254133
$853$	−16.0941	−0.551052	−0.275526	−	0.961294i	$-0.588852\pi$
$853$	−16.0941	−0.551052	−0.275526	+	0.961294i	$0.588852\pi$
$854$	3.41119	0.116729
$855$	17.3894	0.594706
$856$	2.75541	0.0941781
$857$	−6.55389	−0.223877	−0.111938	−	0.993715i	$-0.535706\pi$
$857$	−6.55389	−0.223877	−0.111938	+	0.993715i	$0.535706\pi$
$858$	−3.77851	−0.128996
$859$	25.2704	0.862215	0.431108	−	0.902301i	$-0.358123\pi$
$859$	25.2704	0.862215	0.431108	+	0.902301i	$0.358123\pi$
$860$	−60.4908	−2.06272
$861$	1.96077	0.0668229
$862$	6.59717	0.224701
$863$	10.0379	0.341694	0.170847	−	0.985298i	$-0.445350\pi$
$863$	10.0379	0.341694	0.170847	+	0.985298i	$0.445350\pi$
$864$	−5.34481	−0.181834
$865$	75.2103	2.55723
$866$	3.41417	0.116018
$867$	0	0
$868$	−11.6377	−0.395010
$869$	1.89067	0.0641367
$870$	20.8750	0.707729
$871$	16.6783	0.565122
$872$	37.3370	1.26439
$873$	14.7899	0.500561
$874$	−5.08011	−0.171837
$875$	−18.5961	−0.628663
$876$	3.66189	0.123724
$877$	−15.2457	−0.514809	−0.257405	−	0.966304i	$-0.582867\pi$
$877$	−15.2457	−0.514809	−0.257405	+	0.966304i	$0.582867\pi$
$878$	4.73125	0.159672
$879$	−33.2301	−1.12082
$880$	−8.22952	−0.277417
$881$	−3.21014	−0.108152	−0.0540762	−	0.998537i	$-0.517221\pi$
$881$	−3.21014	−0.108152	−0.0540762	+	0.998537i	$0.517221\pi$
$882$	−0.554958	−0.0186864
$883$	23.9124	0.804718	0.402359	−	0.915482i	$-0.368190\pi$
$883$	23.9124	0.804718	0.402359	+	0.915482i	$0.368190\pi$
$884$	0	0
$885$	5.98792	0.201282
$886$	−10.2198	−0.343342
$887$	−2.39612	−0.0804540	−0.0402270	−	0.999191i	$-0.512808\pi$
$887$	−2.39612	−0.0804540	−0.0402270	+	0.999191i	$0.512808\pi$
$888$	7.42865	0.249289
$889$	−14.7289	−0.493990
$890$	7.09006	0.237660
$891$	0.951083	0.0318625
$892$	5.85862	0.196161
$893$	37.2411	1.24623
$894$	5.63640	0.188509
$895$	−95.9600	−3.20759
$896$	11.5375	0.385441
$897$	−14.5120	−0.484542
$898$	−15.7138	−0.524376
$899$	67.1849	2.24074
$900$	−16.6310	−0.554367
$901$	0	0
$902$	−1.03492	−0.0344590
$903$	−9.28382	−0.308946
$904$	−15.3187	−0.509493
$905$	−21.5332	−0.715787
$906$	−8.05728	−0.267685
$907$	−56.7464	−1.88423	−0.942117	−	0.335284i	$-0.891168\pi$
$907$	−56.7464	−1.88423	−0.942117	+	0.335284i	$0.891168\pi$
$908$	22.6829	0.752759
$909$	9.22521	0.305981
$910$	15.2989	0.507153
$911$	−31.2825	−1.03644	−0.518218	−	0.855249i	$-0.673404\pi$
$911$	−31.2825	−1.03644	−0.518218	+	0.855249i	$0.673404\pi$
$912$	−10.1468	−0.335993
$913$	−4.80194	−0.158921
$914$	−0.362273	−0.0119829
$915$	23.6703	0.782514
$916$	29.0382	0.959448
$917$	10.4601	0.345423
$918$	0	0
$919$	−36.4838	−1.20349	−0.601745	−	0.798688i	$-0.705528\pi$
$919$	−36.4838	−1.20349	−0.601745	+	0.798688i	$0.705528\pi$
$920$	15.9944	0.527318
$921$	−14.9825	−0.493692
$922$	0.162554	0.00535345
$923$	−31.3846	−1.03304
$924$	−1.60925	−0.0529405
$925$	35.6368	1.17173
$926$	−9.15511	−0.300856
$927$	4.86294	0.159720
$928$	−52.2086	−1.71383
$929$	−1.46921	−0.0482031	−0.0241015	−	0.999710i	$-0.507672\pi$
$929$	−1.46921	−0.0482031	−0.0241015	+	0.999710i	$0.507672\pi$
$930$	14.6987	0.481990
$931$	−4.51573	−0.147997
$932$	33.4094	1.09436
$933$	8.61356	0.281995
$934$	−9.97823	−0.326498
$935$	0	0
$936$	14.6679	0.479434
$937$	42.1347	1.37648	0.688240	−	0.725483i	$-0.258384\pi$
$937$	42.1347	1.37648	0.688240	+	0.725483i	$0.258384\pi$
$938$	−1.29291	−0.0422151
$939$	12.3207	0.402069
$940$	−53.7351	−1.75265
$941$	10.4480	0.340596	0.170298	−	0.985393i	$-0.445527\pi$
$941$	10.4480	0.340596	0.170298	+	0.985393i	$0.445527\pi$
$942$	−3.62565	−0.118130
$943$	−3.97477	−0.129436
$944$	−3.49396	−0.113719
$945$	−3.85086	−0.125268
$946$	4.90010	0.159316
$947$	42.1444	1.36951	0.684754	−	0.728774i	$-0.259910\pi$
$947$	42.1444	1.36951	0.684754	+	0.728774i	$0.259910\pi$
$948$	−3.36360	−0.109245
$949$	−15.4932	−0.502931
$950$	24.6321	0.799170
$951$	16.7536	0.543273
$952$	0	0
$953$	−24.7571	−0.801960	−0.400980	−	0.916087i	$-0.631330\pi$
$953$	−24.7571	−0.801960	−0.400980	+	0.916087i	$0.631330\pi$
$954$	4.51573	0.146202
$955$	−86.4234	−2.79660
$956$	−39.2935	−1.27084
$957$	9.29026	0.300311
$958$	2.35078	0.0759503
$959$	−9.35690	−0.302150
$960$	5.88338	0.189885
$961$	16.3069	0.526029
$962$	−14.4042	−0.464409
$963$	1.34481	0.0433360
$964$	23.7711	0.765615
$965$	41.1734	1.32542
$966$	1.12498	0.0361957
$967$	0.0687686	0.00221145	0.00110573	−	0.999999i	$-0.499648\pi$
$967$	0.0687686	0.00221145	0.00110573	+	0.999999i	$0.499648\pi$
$968$	−20.6847	−0.664832
$969$	0	0
$970$	31.6069	1.01484
$971$	−41.5211	−1.33248	−0.666238	−	0.745739i	$-0.732096\pi$
$971$	−41.5211	−1.33248	−0.666238	+	0.745739i	$0.732096\pi$
$972$	−1.69202	−0.0542716
$973$	17.5864	0.563795
$974$	1.92633	0.0617235
$975$	70.3648	2.25348
$976$	−13.8116	−0.442099
$977$	10.7832	0.344984	0.172492	−	0.985011i	$-0.444818\pi$
$977$	10.7832	0.344984	0.172492	+	0.985011i	$0.444818\pi$
$978$	−4.15883	−0.132985
$979$	3.15538	0.100846
$980$	6.51573	0.208137
$981$	18.2228	0.581810
$982$	−6.14974	−0.196246
$983$	−1.42327	−0.0453953	−0.0226977	−	0.999742i	$-0.507226\pi$
$983$	−1.42327	−0.0453953	−0.0226977	+	0.999742i	$0.507226\pi$
$984$	4.01746	0.128072
$985$	−14.3685	−0.457819
$986$	0	0
$987$	−8.24698	−0.262504
$988$	54.6986	1.74019
$989$	18.8197	0.598430
$990$	2.03252	0.0645978
$991$	15.5418	0.493702	0.246851	−	0.969053i	$-0.420604\pi$
$991$	15.5418	0.493702	0.246851	+	0.969053i	$0.420604\pi$
$992$	−36.7616	−1.16718
$993$	14.4547	0.458707
$994$	2.43296	0.0771688
$995$	4.23682	0.134316
$996$	8.54288	0.270691
$997$	−1.02416	−0.0324356	−0.0162178	−	0.999868i	$-0.505163\pi$
$997$	−1.02416	−0.0324356	−0.0162178	+	0.999868i	$0.505163\pi$
$998$	−3.38670	−0.107204
$999$	3.62565	0.114710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.l.1.2	yes	3
17.16	even	2		6069.2.a.j.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6069.2.a.j.1.2	✓	3	17.16	even	2
6069.2.a.l.1.2	yes	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

\(f(q)\)	\(=\)	\(q-0.554958 q^{2} +1.00000 q^{3} -1.69202 q^{4} -3.85086 q^{5} -0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.554958 q^{2} +1.00000 q^{3} -1.69202 q^{4} -3.85086 q^{5} -0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} +2.13706 q^{10} +0.951083 q^{11} -1.69202 q^{12} +7.15883 q^{13} -0.554958 q^{14} -3.85086 q^{15} +2.24698 q^{16} -0.554958 q^{18} -4.51573 q^{19} +6.51573 q^{20} +1.00000 q^{21} -0.527811 q^{22} -2.02715 q^{23} +2.04892 q^{24} +9.82908 q^{25} -3.97285 q^{26} +1.00000 q^{27} -1.69202 q^{28} +9.76809 q^{29} +2.13706 q^{30} +6.87800 q^{31} -5.34481 q^{32} +0.951083 q^{33} -3.85086 q^{35} -1.69202 q^{36} +3.62565 q^{37} +2.50604 q^{38} +7.15883 q^{39} -7.89008 q^{40} +1.96077 q^{41} -0.554958 q^{42} -9.28382 q^{43} -1.60925 q^{44} -3.85086 q^{45} +1.12498 q^{46} -8.24698 q^{47} +2.24698 q^{48} +1.00000 q^{49} -5.45473 q^{50} -12.1129 q^{52} -8.13706 q^{53} -0.554958 q^{54} -3.66248 q^{55} +2.04892 q^{56} -4.51573 q^{57} -5.42088 q^{58} -1.55496 q^{59} +6.51573 q^{60} -6.14675 q^{61} -3.81700 q^{62} +1.00000 q^{63} -1.52781 q^{64} -27.5676 q^{65} -0.527811 q^{66} +2.32975 q^{67} -2.02715 q^{69} +2.13706 q^{70} -4.38404 q^{71} +2.04892 q^{72} -2.16421 q^{73} -2.01208 q^{74} +9.82908 q^{75} +7.64071 q^{76} +0.951083 q^{77} -3.97285 q^{78} +1.98792 q^{79} -8.65279 q^{80} +1.00000 q^{81} -1.08815 q^{82} -5.04892 q^{83} -1.69202 q^{84} +5.15213 q^{86} +9.76809 q^{87} +1.94869 q^{88} +3.31767 q^{89} +2.13706 q^{90} +7.15883 q^{91} +3.42998 q^{92} +6.87800 q^{93} +4.57673 q^{94} +17.3894 q^{95} -5.34481 q^{96} +14.7899 q^{97} -0.554958 q^{98} +0.951083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 3 q^{3} + 2 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 3 * q^3 + 2 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 2 q^{2} + 3 q^{3} + 2 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + q^{10} + 12 q^{11} + 13 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} - q^{19} + 7 q^{20} + 3 q^{21} - 8 q^{22} - 3 q^{24} + 19 q^{25} - 18 q^{26} + 3 q^{27} + 9 q^{29} + q^{30} + q^{31} + 7 q^{32} + 12 q^{33} + 2 q^{35} - q^{37} + 17 q^{38} + 13 q^{39} - 23 q^{40} - 7 q^{41} - 2 q^{42} + 5 q^{43} + 7 q^{44} + 2 q^{45} - 21 q^{46} - 20 q^{47} + 2 q^{48} + 3 q^{49} + 6 q^{50} + 7 q^{52} - 19 q^{53} - 2 q^{54} + 29 q^{55} - 3 q^{56} - q^{57} + 22 q^{58} - 5 q^{59} + 7 q^{60} + 9 q^{61} + 18 q^{62} + 3 q^{63} - 11 q^{64} - 17 q^{65} - 8 q^{66} + 9 q^{67} + q^{70} - 3 q^{71} - 3 q^{72} + 5 q^{73} - 25 q^{74} + 19 q^{75} - 14 q^{76} + 12 q^{77} - 18 q^{78} - 13 q^{79} - 8 q^{80} + 3 q^{81} - 7 q^{82} - 6 q^{83} - 15 q^{86} + 9 q^{87} - 26 q^{88} - 7 q^{89} + q^{90} + 13 q^{91} + 35 q^{92} + q^{93} + 11 q^{94} + 39 q^{95} + 7 q^{96} + 21 q^{97} - 2 q^{98} + 12 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 3 * q^3 + 2 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 + q^10 + 12 * q^11 + 13 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^18 - q^19 + 7 * q^20 + 3 * q^21 - 8 * q^22 - 3 * q^24 + 19 * q^25 - 18 * q^26 + 3 * q^27 + 9 * q^29 + q^30 + q^31 + 7 * q^32 + 12 * q^33 + 2 * q^35 - q^37 + 17 * q^38 + 13 * q^39 - 23 * q^40 - 7 * q^41 - 2 * q^42 + 5 * q^43 + 7 * q^44 + 2 * q^45 - 21 * q^46 - 20 * q^47 + 2 * q^48 + 3 * q^49 + 6 * q^50 + 7 * q^52 - 19 * q^53 - 2 * q^54 + 29 * q^55 - 3 * q^56 - q^57 + 22 * q^58 - 5 * q^59 + 7 * q^60 + 9 * q^61 + 18 * q^62 + 3 * q^63 - 11 * q^64 - 17 * q^65 - 8 * q^66 + 9 * q^67 + q^70 - 3 * q^71 - 3 * q^72 + 5 * q^73 - 25 * q^74 + 19 * q^75 - 14 * q^76 + 12 * q^77 - 18 * q^78 - 13 * q^79 - 8 * q^80 + 3 * q^81 - 7 * q^82 - 6 * q^83 - 15 * q^86 + 9 * q^87 - 26 * q^88 - 7 * q^89 + q^90 + 13 * q^91 + 35 * q^92 + q^93 + 11 * q^94 + 39 * q^95 + 7 * q^96 + 21 * q^97 - 2 * q^98 + 12 * q^99

Introduction

L-functions

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