LMFDB - Embedded newform 539.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("539.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.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:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +1.23607 q^{3} +3.00000 q^{4} +2.00000 q^{5} +2.76393 q^{6} +2.23607 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q+2.23607 q^{2} +1.23607 q^{3} +3.00000 q^{4} +2.00000 q^{5} +2.76393 q^{6} +2.23607 q^{8} -1.47214 q^{9} +4.47214 q^{10} -1.00000 q^{11} +3.70820 q^{12} -3.23607 q^{13} +2.47214 q^{15} -1.00000 q^{16} +3.23607 q^{17} -3.29180 q^{18} -6.47214 q^{19} +6.00000 q^{20} -2.23607 q^{22} +2.47214 q^{23} +2.76393 q^{24} -1.00000 q^{25} -7.23607 q^{26} -5.52786 q^{27} +8.47214 q^{29} +5.52786 q^{30} +2.76393 q^{31} -6.70820 q^{32} -1.23607 q^{33} +7.23607 q^{34} -4.41641 q^{36} -8.47214 q^{37} -14.4721 q^{38} -4.00000 q^{39} +4.47214 q^{40} +11.2361 q^{41} +8.00000 q^{43} -3.00000 q^{44} -2.94427 q^{45} +5.52786 q^{46} -2.76393 q^{47} -1.23607 q^{48} -2.23607 q^{50} +4.00000 q^{51} -9.70820 q^{52} -0.472136 q^{53} -12.3607 q^{54} -2.00000 q^{55} -8.00000 q^{57} +18.9443 q^{58} +1.23607 q^{59} +7.41641 q^{60} +7.23607 q^{61} +6.18034 q^{62} -13.0000 q^{64} -6.47214 q^{65} -2.76393 q^{66} +14.4721 q^{67} +9.70820 q^{68} +3.05573 q^{69} -10.4721 q^{71} -3.29180 q^{72} +0.763932 q^{73} -18.9443 q^{74} -1.23607 q^{75} -19.4164 q^{76} -8.94427 q^{78} -8.94427 q^{79} -2.00000 q^{80} -2.41641 q^{81} +25.1246 q^{82} +11.4164 q^{83} +6.47214 q^{85} +17.8885 q^{86} +10.4721 q^{87} -2.23607 q^{88} -2.00000 q^{89} -6.58359 q^{90} +7.41641 q^{92} +3.41641 q^{93} -6.18034 q^{94} -12.9443 q^{95} -8.29180 q^{96} -17.4164 q^{97} +1.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} + 10 q^{6} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^4 + 4 * q^5 + 10 * q^6 + 6 * q^9	$ 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} + 10 q^{6} + 6 q^{9} - 2 q^{11} - 6 q^{12} - 2 q^{13} - 4 q^{15} - 2 q^{16} + 2 q^{17} - 20 q^{18} - 4 q^{19} + 12 q^{20} - 4 q^{23} + 10 q^{24} - 2 q^{25} - 10 q^{26} - 20 q^{27} + 8 q^{29} + 20 q^{30} + 10 q^{31} + 2 q^{33} + 10 q^{34} + 18 q^{36} - 8 q^{37} - 20 q^{38} - 8 q^{39} + 18 q^{41} + 16 q^{43} - 6 q^{44} + 12 q^{45} + 20 q^{46} - 10 q^{47} + 2 q^{48} + 8 q^{51} - 6 q^{52} + 8 q^{53} + 20 q^{54} - 4 q^{55} - 16 q^{57} + 20 q^{58} - 2 q^{59} - 12 q^{60} + 10 q^{61} - 10 q^{62} - 26 q^{64} - 4 q^{65} - 10 q^{66} + 20 q^{67} + 6 q^{68} + 24 q^{69} - 12 q^{71} - 20 q^{72} + 6 q^{73} - 20 q^{74} + 2 q^{75} - 12 q^{76} - 4 q^{80} + 22 q^{81} + 10 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{87} - 4 q^{89} - 40 q^{90} - 12 q^{92} - 20 q^{93} + 10 q^{94} - 8 q^{95} - 30 q^{96} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^4 + 4 * q^5 + 10 * q^6 + 6 * q^9 - 2 * q^11 - 6 * q^12 - 2 * q^13 - 4 * q^15 - 2 * q^16 + 2 * q^17 - 20 * q^18 - 4 * q^19 + 12 * q^20 - 4 * q^23 + 10 * q^24 - 2 * q^25 - 10 * q^26 - 20 * q^27 + 8 * q^29 + 20 * q^30 + 10 * q^31 + 2 * q^33 + 10 * q^34 + 18 * q^36 - 8 * q^37 - 20 * q^38 - 8 * q^39 + 18 * q^41 + 16 * q^43 - 6 * q^44 + 12 * q^45 + 20 * q^46 - 10 * q^47 + 2 * q^48 + 8 * q^51 - 6 * q^52 + 8 * q^53 + 20 * q^54 - 4 * q^55 - 16 * q^57 + 20 * q^58 - 2 * q^59 - 12 * q^60 + 10 * q^61 - 10 * q^62 - 26 * q^64 - 4 * q^65 - 10 * q^66 + 20 * q^67 + 6 * q^68 + 24 * q^69 - 12 * q^71 - 20 * q^72 + 6 * q^73 - 20 * q^74 + 2 * q^75 - 12 * q^76 - 4 * q^80 + 22 * q^81 + 10 * q^82 - 4 * q^83 + 4 * q^85 + 12 * q^87 - 4 * q^89 - 40 * q^90 - 12 * q^92 - 20 * q^93 + 10 * q^94 - 8 * q^95 - 30 * q^96 - 8 * q^97 - 6 * 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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	3.00000	1.50000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	2.76393	1.12837
$7$	0	0
$7$	0	0
$8$	2.23607	0.790569
$9$	−1.47214	−0.490712
$10$	4.47214	1.41421
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.70820	1.07047
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	2.47214	0.638303
$16$	−1.00000	−0.250000
$17$	3.23607	0.784862	0.392431	−	0.919781i	$-0.371634\pi$
$17$	3.23607	0.784862	0.392431	+	0.919781i	$0.371634\pi$
$18$	−3.29180	−0.775884
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	6.00000	1.34164
$21$	0	0
$22$	−2.23607	−0.476731
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	2.76393	0.564185
$25$	−1.00000	−0.200000
$26$	−7.23607	−1.41911
$27$	−5.52786	−1.06384
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	5.52786	1.00925
$31$	2.76393	0.496417	0.248208	−	0.968707i	$-0.420158\pi$
$31$	2.76393	0.496417	0.248208	+	0.968707i	$0.420158\pi$
$32$	−6.70820	−1.18585
$33$	−1.23607	−0.215172
$34$	7.23607	1.24098
$35$	0	0
$36$	−4.41641	−0.736068
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	−14.4721	−2.34769
$39$	−4.00000	−0.640513
$40$	4.47214	0.707107
$41$	11.2361	1.75478	0.877390	−	0.479779i	$-0.159283\pi$
$41$	11.2361	1.75478	0.877390	+	0.479779i	$0.159283\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−3.00000	−0.452267
$45$	−2.94427	−0.438906
$46$	5.52786	0.815039
$47$	−2.76393	−0.403161	−0.201580	−	0.979472i	$-0.564608\pi$
$47$	−2.76393	−0.403161	−0.201580	+	0.979472i	$0.564608\pi$
$48$	−1.23607	−0.178411
$49$	0	0
$50$	−2.23607	−0.316228
$51$	4.00000	0.560112
$52$	−9.70820	−1.34629
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	−12.3607	−1.68208
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−8.00000	−1.05963
$58$	18.9443	2.48750
$59$	1.23607	0.160922	0.0804612	−	0.996758i	$-0.474361\pi$
$59$	1.23607	0.160922	0.0804612	+	0.996758i	$0.474361\pi$
$60$	7.41641	0.957454
$61$	7.23607	0.926484	0.463242	−	0.886232i	$-0.346686\pi$
$61$	7.23607	0.926484	0.463242	+	0.886232i	$0.346686\pi$
$62$	6.18034	0.784904
$63$	0	0
$64$	−13.0000	−1.62500
$65$	−6.47214	−0.802770
$66$	−2.76393	−0.340217
$67$	14.4721	1.76805	0.884026	−	0.467437i	$-0.154823\pi$
$67$	14.4721	1.76805	0.884026	+	0.467437i	$0.154823\pi$
$68$	9.70820	1.17729
$69$	3.05573	0.367866
$70$	0	0
$71$	−10.4721	−1.24281	−0.621407	−	0.783488i	$-0.713439\pi$
$71$	−10.4721	−1.24281	−0.621407	+	0.783488i	$0.713439\pi$
$72$	−3.29180	−0.387942
$73$	0.763932	0.0894115	0.0447057	−	0.999000i	$-0.485765\pi$
$73$	0.763932	0.0894115	0.0447057	+	0.999000i	$0.485765\pi$
$74$	−18.9443	−2.20223
$75$	−1.23607	−0.142729
$76$	−19.4164	−2.22721
$77$	0	0
$78$	−8.94427	−1.01274
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	−2.00000	−0.223607
$81$	−2.41641	−0.268490
$82$	25.1246	2.77455
$83$	11.4164	1.25311	0.626557	−	0.779376i	$-0.284464\pi$
$83$	11.4164	1.25311	0.626557	+	0.779376i	$0.284464\pi$
$84$	0	0
$85$	6.47214	0.702002
$86$	17.8885	1.92897
$87$	10.4721	1.12273
$88$	−2.23607	−0.238366
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−6.58359	−0.693972
$91$	0	0
$92$	7.41641	0.773214
$93$	3.41641	0.354265
$94$	−6.18034	−0.637453
$95$	−12.9443	−1.32805
$96$	−8.29180	−0.846278
$97$	−17.4164	−1.76837	−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164	−1.76837	−0.884184	+	0.467139i	$0.845285\pi$
$98$	0	0
$99$	1.47214	0.147955
$100$	−3.00000	−0.300000
$101$	4.76393	0.474029	0.237014	−	0.971506i	$-0.423831\pi$
$101$	4.76393	0.474029	0.237014	+	0.971506i	$0.423831\pi$
$102$	8.94427	0.885615
$103$	−7.70820	−0.759512	−0.379756	−	0.925087i	$-0.623992\pi$
$103$	−7.70820	−0.759512	−0.379756	+	0.925087i	$0.623992\pi$
$104$	−7.23607	−0.709555
$105$	0	0
$106$	−1.05573	−0.102541
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−16.5836	−1.59576
$109$	−4.47214	−0.428353	−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214	−0.428353	−0.214176	+	0.976795i	$0.568707\pi$
$110$	−4.47214	−0.426401
$111$	−10.4721	−0.993971
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−17.8885	−1.67542
$115$	4.94427	0.461056
$116$	25.4164	2.35985
$117$	4.76393	0.440426
$118$	2.76393	0.254441
$119$	0	0
$120$	5.52786	0.504623
$121$	1.00000	0.0909091
$122$	16.1803	1.46490
$123$	13.8885	1.25229
$124$	8.29180	0.744625
$125$	−12.0000	−1.07331
$126$	0	0
$127$	3.05573	0.271152	0.135576	−	0.990767i	$-0.456712\pi$
$127$	3.05573	0.271152	0.135576	+	0.990767i	$0.456712\pi$
$128$	−15.6525	−1.38350
$129$	9.88854	0.870638
$130$	−14.4721	−1.26929
$131$	−21.8885	−1.91241	−0.956205	−	0.292696i	$-0.905448\pi$
$131$	−21.8885	−1.91241	−0.956205	+	0.292696i	$0.905448\pi$
$132$	−3.70820	−0.322758
$133$	0	0
$134$	32.3607	2.79554
$135$	−11.0557	−0.951526
$136$	7.23607	0.620488
$137$	16.4721	1.40731	0.703655	−	0.710542i	$-0.251550\pi$
$137$	16.4721	1.40731	0.703655	+	0.710542i	$0.251550\pi$
$138$	6.83282	0.581648
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	0	0
$141$	−3.41641	−0.287713
$142$	−23.4164	−1.96506
$143$	3.23607	0.270614
$144$	1.47214	0.122678
$145$	16.9443	1.40715
$146$	1.70820	0.141372
$147$	0	0
$148$	−25.4164	−2.08922
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	−2.76393	−0.225674
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	−14.4721	−1.17385
$153$	−4.76393	−0.385141
$154$	0	0
$155$	5.52786	0.444009
$156$	−12.0000	−0.960769
$157$	−10.9443	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$158$	−20.0000	−1.59111
$159$	−0.583592	−0.0462819
$160$	−13.4164	−1.06066
$161$	0	0
$162$	−5.40325	−0.424520
$163$	−3.41641	−0.267594	−0.133797	−	0.991009i	$-0.542717\pi$
$163$	−3.41641	−0.267594	−0.133797	+	0.991009i	$0.542717\pi$
$164$	33.7082	2.63217
$165$	−2.47214	−0.192456
$166$	25.5279	1.98135
$167$	−4.94427	−0.382599	−0.191300	−	0.981532i	$-0.561270\pi$
$167$	−4.94427	−0.382599	−0.191300	+	0.981532i	$0.561270\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	14.4721	1.10996
$171$	9.52786	0.728614
$172$	24.0000	1.82998
$173$	12.7639	0.970424	0.485212	−	0.874397i	$-0.338742\pi$
$173$	12.7639	0.970424	0.485212	+	0.874397i	$0.338742\pi$
$174$	23.4164	1.77519
$175$	0	0
$176$	1.00000	0.0753778
$177$	1.52786	0.114841
$178$	−4.47214	−0.335201
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	−8.83282	−0.658359
$181$	25.4164	1.88919	0.944593	−	0.328243i	$-0.106456\pi$
$181$	25.4164	1.88919	0.944593	+	0.328243i	$0.106456\pi$
$182$	0	0
$183$	8.94427	0.661180
$184$	5.52786	0.407520
$185$	−16.9443	−1.24577
$186$	7.63932	0.560142
$187$	−3.23607	−0.236645
$188$	−8.29180	−0.604741
$189$	0	0
$190$	−28.9443	−2.09984
$191$	−3.05573	−0.221105	−0.110552	−	0.993870i	$-0.535262\pi$
$191$	−3.05573	−0.221105	−0.110552	+	0.993870i	$0.535262\pi$
$192$	−16.0689	−1.15967
$193$	11.8885	0.855756	0.427878	−	0.903836i	$-0.359261\pi$
$193$	11.8885	0.855756	0.427878	+	0.903836i	$0.359261\pi$
$194$	−38.9443	−2.79604
$195$	−8.00000	−0.572892
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	3.29180	0.233938
$199$	2.18034	0.154560	0.0772801	−	0.997009i	$-0.475376\pi$
$199$	2.18034	0.154560	0.0772801	+	0.997009i	$0.475376\pi$
$200$	−2.23607	−0.158114
$201$	17.8885	1.26176
$202$	10.6525	0.749506
$203$	0	0
$204$	12.0000	0.840168
$205$	22.4721	1.56952
$206$	−17.2361	−1.20089
$207$	−3.63932	−0.252950
$208$	3.23607	0.224381
$209$	6.47214	0.447687
$210$	0	0
$211$	−13.8885	−0.956127	−0.478063	−	0.878325i	$-0.658661\pi$
$211$	−13.8885	−0.956127	−0.478063	+	0.878325i	$0.658661\pi$
$212$	−1.41641	−0.0972793
$213$	−12.9443	−0.886927
$214$	−8.94427	−0.611418
$215$	16.0000	1.09119
$216$	−12.3607	−0.841038
$217$	0	0
$218$	−10.0000	−0.677285
$219$	0.944272	0.0638080
$220$	−6.00000	−0.404520
$221$	−10.4721	−0.704432
$222$	−23.4164	−1.57161
$223$	10.1803	0.681726	0.340863	−	0.940113i	$-0.389281\pi$
$223$	10.1803	0.681726	0.340863	+	0.940113i	$0.389281\pi$
$224$	0	0
$225$	1.47214	0.0981424
$226$	4.47214	0.297482
$227$	−5.88854	−0.390836	−0.195418	−	0.980720i	$-0.562606\pi$
$227$	−5.88854	−0.390836	−0.195418	+	0.980720i	$0.562606\pi$
$228$	−24.0000	−1.58944
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	11.0557	0.728993
$231$	0	0
$232$	18.9443	1.24375
$233$	9.41641	0.616889	0.308445	−	0.951242i	$-0.400192\pi$
$233$	9.41641	0.616889	0.308445	+	0.951242i	$0.400192\pi$
$234$	10.6525	0.696374
$235$	−5.52786	−0.360598
$236$	3.70820	0.241384
$237$	−11.0557	−0.718147
$238$	0	0
$239$	−9.88854	−0.639637	−0.319818	−	0.947479i	$-0.603622\pi$
$239$	−9.88854	−0.639637	−0.319818	+	0.947479i	$0.603622\pi$
$240$	−2.47214	−0.159576
$241$	13.1246	0.845431	0.422715	−	0.906263i	$-0.361077\pi$
$241$	13.1246	0.845431	0.422715	+	0.906263i	$0.361077\pi$
$242$	2.23607	0.143740
$243$	13.5967	0.872232
$244$	21.7082	1.38973
$245$	0	0
$246$	31.0557	1.98004
$247$	20.9443	1.33265
$248$	6.18034	0.392452
$249$	14.1115	0.894277
$250$	−26.8328	−1.69706
$251$	−4.29180	−0.270896	−0.135448	−	0.990784i	$-0.543247\pi$
$251$	−4.29180	−0.270896	−0.135448	+	0.990784i	$0.543247\pi$
$252$	0	0
$253$	−2.47214	−0.155422
$254$	6.83282	0.428729
$255$	8.00000	0.500979
$256$	−9.00000	−0.562500
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	22.1115	1.37660
$259$	0	0
$260$	−19.4164	−1.20415
$261$	−12.4721	−0.772006
$262$	−48.9443	−3.02379
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−2.76393	−0.170108
$265$	−0.944272	−0.0580062
$266$	0	0
$267$	−2.47214	−0.151292
$268$	43.4164	2.65208
$269$	−13.4164	−0.818013	−0.409006	−	0.912532i	$-0.634125\pi$
$269$	−13.4164	−0.818013	−0.409006	+	0.912532i	$0.634125\pi$
$270$	−24.7214	−1.50449
$271$	10.4721	0.636137	0.318068	−	0.948068i	$-0.396966\pi$
$271$	10.4721	0.636137	0.318068	+	0.948068i	$0.396966\pi$
$272$	−3.23607	−0.196215
$273$	0	0
$274$	36.8328	2.22515
$275$	1.00000	0.0603023
$276$	9.16718	0.551800
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	3.41641	0.204903
$279$	−4.06888	−0.243598
$280$	0	0
$281$	−3.52786	−0.210455	−0.105227	−	0.994448i	$-0.533557\pi$
$281$	−3.52786	−0.210455	−0.105227	+	0.994448i	$0.533557\pi$
$282$	−7.63932	−0.454915
$283$	−29.8885	−1.77669	−0.888345	−	0.459177i	$-0.848144\pi$
$283$	−29.8885	−1.77669	−0.888345	+	0.459177i	$0.848144\pi$
$284$	−31.4164	−1.86422
$285$	−16.0000	−0.947758
$286$	7.23607	0.427878
$287$	0	0
$288$	9.87539	0.581913
$289$	−6.52786	−0.383992
$290$	37.8885	2.22489
$291$	−21.5279	−1.26199
$292$	2.29180	0.134117
$293$	25.1246	1.46780	0.733898	−	0.679260i	$-0.237699\pi$
$293$	25.1246	1.46780	0.733898	+	0.679260i	$0.237699\pi$
$294$	0	0
$295$	2.47214	0.143933
$296$	−18.9443	−1.10111
$297$	5.52786	0.320759
$298$	31.3050	1.81345
$299$	−8.00000	−0.462652
$300$	−3.70820	−0.214093
$301$	0	0
$302$	20.0000	1.15087
$303$	5.88854	0.338288
$304$	6.47214	0.371202
$305$	14.4721	0.828672
$306$	−10.6525	−0.608962
$307$	8.94427	0.510477	0.255238	−	0.966878i	$-0.417846\pi$
$307$	8.94427	0.510477	0.255238	+	0.966878i	$0.417846\pi$
$308$	0	0
$309$	−9.52786	−0.542021
$310$	12.3607	0.702039
$311$	8.29180	0.470185	0.235092	−	0.971973i	$-0.424461\pi$
$311$	8.29180	0.470185	0.235092	+	0.971973i	$0.424461\pi$
$312$	−8.94427	−0.506370
$313$	−14.9443	−0.844700	−0.422350	−	0.906433i	$-0.638795\pi$
$313$	−14.9443	−0.844700	−0.422350	+	0.906433i	$0.638795\pi$
$314$	−24.4721	−1.38104
$315$	0	0
$316$	−26.8328	−1.50946
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	−1.30495	−0.0731781
$319$	−8.47214	−0.474349
$320$	−26.0000	−1.45344
$321$	−4.94427	−0.275962
$322$	0	0
$323$	−20.9443	−1.16537
$324$	−7.24922	−0.402735
$325$	3.23607	0.179505
$326$	−7.63932	−0.423103
$327$	−5.52786	−0.305692
$328$	25.1246	1.38727
$329$	0	0
$330$	−5.52786	−0.304299
$331$	−13.8885	−0.763383	−0.381692	−	0.924290i	$-0.624658\pi$
$331$	−13.8885	−0.763383	−0.381692	+	0.924290i	$0.624658\pi$
$332$	34.2492	1.87967
$333$	12.4721	0.683469
$334$	−11.0557	−0.604943
$335$	28.9443	1.58139
$336$	0	0
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	−5.65248	−0.307454
$339$	2.47214	0.134268
$340$	19.4164	1.05300
$341$	−2.76393	−0.149675
$342$	21.3050	1.15204
$343$	0	0
$344$	17.8885	0.964486
$345$	6.11146	0.329030
$346$	28.5410	1.53437
$347$	20.9443	1.12435	0.562174	−	0.827019i	$-0.309965\pi$
$347$	20.9443	1.12435	0.562174	+	0.827019i	$0.309965\pi$
$348$	31.4164	1.68410
$349$	7.23607	0.387338	0.193669	−	0.981067i	$-0.437961\pi$
$349$	7.23607	0.387338	0.193669	+	0.981067i	$0.437961\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	6.70820	0.357548
$353$	−19.8885	−1.05856	−0.529280	−	0.848447i	$-0.677538\pi$
$353$	−19.8885	−1.05856	−0.529280	+	0.848447i	$0.677538\pi$
$354$	3.41641	0.181580
$355$	−20.9443	−1.11161
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−24.9443	−1.31651	−0.658254	−	0.752796i	$-0.728705\pi$
$359$	−24.9443	−1.31651	−0.658254	+	0.752796i	$0.728705\pi$
$360$	−6.58359	−0.346986
$361$	22.8885	1.20466
$362$	56.8328	2.98707
$363$	1.23607	0.0648767
$364$	0	0
$365$	1.52786	0.0799721
$366$	20.0000	1.04542
$367$	23.1246	1.20709	0.603547	−	0.797327i	$-0.293753\pi$
$367$	23.1246	1.20709	0.603547	+	0.797327i	$0.293753\pi$
$368$	−2.47214	−0.128869
$369$	−16.5410	−0.861091
$370$	−37.8885	−1.96973
$371$	0	0
$372$	10.2492	0.531397
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−7.23607	−0.374168
$375$	−14.8328	−0.765963
$376$	−6.18034	−0.318727
$377$	−27.4164	−1.41202
$378$	0	0
$379$	37.3050	1.91623	0.958113	−	0.286389i	$-0.0924551\pi$
$379$	37.3050	1.91623	0.958113	+	0.286389i	$0.0924551\pi$
$380$	−38.8328	−1.99208
$381$	3.77709	0.193506
$382$	−6.83282	−0.349597
$383$	4.65248	0.237730	0.118865	−	0.992910i	$-0.462074\pi$
$383$	4.65248	0.237730	0.118865	+	0.992910i	$0.462074\pi$
$384$	−19.3475	−0.987324
$385$	0	0
$386$	26.5836	1.35307
$387$	−11.7771	−0.598663
$388$	−52.2492	−2.65255
$389$	15.8885	0.805581	0.402791	−	0.915292i	$-0.368040\pi$
$389$	15.8885	0.805581	0.402791	+	0.915292i	$0.368040\pi$
$390$	−17.8885	−0.905822
$391$	8.00000	0.404577
$392$	0	0
$393$	−27.0557	−1.36478
$394$	−4.47214	−0.225303
$395$	−17.8885	−0.900070
$396$	4.41641	0.221933
$397$	35.8885	1.80119	0.900597	−	0.434655i	$-0.143130\pi$
$397$	35.8885	1.80119	0.900597	+	0.434655i	$0.143130\pi$
$398$	4.87539	0.244381
$399$	0	0
$400$	1.00000	0.0500000
$401$	22.9443	1.14578	0.572891	−	0.819631i	$-0.305822\pi$
$401$	22.9443	1.14578	0.572891	+	0.819631i	$0.305822\pi$
$402$	40.0000	1.99502
$403$	−8.94427	−0.445546
$404$	14.2918	0.711043
$405$	−4.83282	−0.240145
$406$	0	0
$407$	8.47214	0.419948
$408$	8.94427	0.442807
$409$	−9.12461	−0.451183	−0.225592	−	0.974222i	$-0.572431\pi$
$409$	−9.12461	−0.451183	−0.225592	+	0.974222i	$0.572431\pi$
$410$	50.2492	2.48163
$411$	20.3607	1.00432
$412$	−23.1246	−1.13927
$413$	0	0
$414$	−8.13777	−0.399949
$415$	22.8328	1.12082
$416$	21.7082	1.06433
$417$	1.88854	0.0924824
$418$	14.4721	0.707855
$419$	−24.6525	−1.20435	−0.602176	−	0.798363i	$-0.705700\pi$
$419$	−24.6525	−1.20435	−0.602176	+	0.798363i	$0.705700\pi$
$420$	0	0
$421$	22.3607	1.08979	0.544896	−	0.838503i	$-0.316569\pi$
$421$	22.3607	1.08979	0.544896	+	0.838503i	$0.316569\pi$
$422$	−31.0557	−1.51177
$423$	4.06888	0.197836
$424$	−1.05573	−0.0512707
$425$	−3.23607	−0.156972
$426$	−28.9443	−1.40235
$427$	0	0
$428$	−12.0000	−0.580042
$429$	4.00000	0.193122
$430$	35.7771	1.72532
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	5.52786	0.265959
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	0	0
$435$	20.9443	1.00420
$436$	−13.4164	−0.642529
$437$	−16.0000	−0.765384
$438$	2.11146	0.100889
$439$	−10.4721	−0.499808	−0.249904	−	0.968271i	$-0.580399\pi$
$439$	−10.4721	−0.499808	−0.249904	+	0.968271i	$0.580399\pi$
$440$	−4.47214	−0.213201
$441$	0	0
$442$	−23.4164	−1.11380
$443$	−24.9443	−1.18514	−0.592569	−	0.805520i	$-0.701886\pi$
$443$	−24.9443	−1.18514	−0.592569	+	0.805520i	$0.701886\pi$
$444$	−31.4164	−1.49096
$445$	−4.00000	−0.189618
$446$	22.7639	1.07790
$447$	17.3050	0.818496
$448$	0	0
$449$	−28.4721	−1.34368	−0.671842	−	0.740695i	$-0.734496\pi$
$449$	−28.4721	−1.34368	−0.671842	+	0.740695i	$0.734496\pi$
$450$	3.29180	0.155177
$451$	−11.2361	−0.529086
$452$	6.00000	0.282216
$453$	11.0557	0.519443
$454$	−13.1672	−0.617967
$455$	0	0
$456$	−17.8885	−0.837708
$457$	−28.8328	−1.34874	−0.674371	−	0.738393i	$-0.735585\pi$
$457$	−28.8328	−1.34874	−0.674371	+	0.738393i	$0.735585\pi$
$458$	10.0000	0.467269
$459$	−17.8885	−0.834966
$460$	14.8328	0.691584
$461$	12.1803	0.567295	0.283647	−	0.958929i	$-0.408455\pi$
$461$	12.1803	0.567295	0.283647	+	0.958929i	$0.408455\pi$
$462$	0	0
$463$	−5.52786	−0.256902	−0.128451	−	0.991716i	$-0.541000\pi$
$463$	−5.52786	−0.256902	−0.128451	+	0.991716i	$0.541000\pi$
$464$	−8.47214	−0.393309
$465$	6.83282	0.316864
$466$	21.0557	0.975388
$467$	−24.0689	−1.11378	−0.556888	−	0.830588i	$-0.688005\pi$
$467$	−24.0689	−1.11378	−0.556888	+	0.830588i	$0.688005\pi$
$468$	14.2918	0.660639
$469$	0	0
$470$	−12.3607	−0.570156
$471$	−13.5279	−0.623331
$472$	2.76393	0.127220
$473$	−8.00000	−0.367840
$474$	−24.7214	−1.13549
$475$	6.47214	0.296962
$476$	0	0
$477$	0.695048	0.0318241
$478$	−22.1115	−1.01135
$479$	−13.5279	−0.618104	−0.309052	−	0.951045i	$-0.600012\pi$
$479$	−13.5279	−0.618104	−0.309052	+	0.951045i	$0.600012\pi$
$480$	−16.5836	−0.756934
$481$	27.4164	1.25008
$482$	29.3475	1.33674
$483$	0	0
$484$	3.00000	0.136364
$485$	−34.8328	−1.58168
$486$	30.4033	1.37912
$487$	−36.3607	−1.64766	−0.823830	−	0.566837i	$-0.808167\pi$
$487$	−36.3607	−1.64766	−0.823830	+	0.566837i	$0.808167\pi$
$488$	16.1803	0.732450
$489$	−4.22291	−0.190967
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	41.6656	1.87843
$493$	27.4164	1.23477
$494$	46.8328	2.10711
$495$	2.94427	0.132335
$496$	−2.76393	−0.124104
$497$	0	0
$498$	31.5542	1.41398
$499$	1.52786	0.0683966	0.0341983	−	0.999415i	$-0.489112\pi$
$499$	1.52786	0.0683966	0.0341983	+	0.999415i	$0.489112\pi$
$500$	−36.0000	−1.60997
$501$	−6.11146	−0.273040
$502$	−9.59675	−0.428324
$503$	−23.4164	−1.04409	−0.522043	−	0.852919i	$-0.674830\pi$
$503$	−23.4164	−1.04409	−0.522043	+	0.852919i	$0.674830\pi$
$504$	0	0
$505$	9.52786	0.423984
$506$	−5.52786	−0.245744
$507$	−3.12461	−0.138769
$508$	9.16718	0.406728
$509$	−40.4721	−1.79390	−0.896948	−	0.442136i	$-0.854221\pi$
$509$	−40.4721	−1.79390	−0.896948	+	0.442136i	$0.854221\pi$
$510$	17.8885	0.792118
$511$	0	0
$512$	11.1803	0.494106
$513$	35.7771	1.57960
$514$	13.4164	0.591772
$515$	−15.4164	−0.679328
$516$	29.6656	1.30596
$517$	2.76393	0.121558
$518$	0	0
$519$	15.7771	0.692537
$520$	−14.4721	−0.634645
$521$	−30.3607	−1.33013	−0.665063	−	0.746787i	$-0.731595\pi$
$521$	−30.3607	−1.33013	−0.665063	+	0.746787i	$0.731595\pi$
$522$	−27.8885	−1.22065
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	−65.6656	−2.86862
$525$	0	0
$526$	0	0
$527$	8.94427	0.389619
$528$	1.23607	0.0537930
$529$	−16.8885	−0.734285
$530$	−2.11146	−0.0917158
$531$	−1.81966	−0.0789665
$532$	0	0
$533$	−36.3607	−1.57496
$534$	−5.52786	−0.239214
$535$	−8.00000	−0.345870
$536$	32.3607	1.39777
$537$	−11.0557	−0.477090
$538$	−30.0000	−1.29339
$539$	0	0
$540$	−33.1672	−1.42729
$541$	20.8328	0.895673	0.447836	−	0.894116i	$-0.352195\pi$
$541$	20.8328	0.895673	0.447836	+	0.894116i	$0.352195\pi$
$542$	23.4164	1.00582
$543$	31.4164	1.34821
$544$	−21.7082	−0.930732
$545$	−8.94427	−0.383131
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	49.4164	2.11096
$549$	−10.6525	−0.454637
$550$	2.23607	0.0953463
$551$	−54.8328	−2.33596
$552$	6.83282	0.290824
$553$	0	0
$554$	−44.4721	−1.88944
$555$	−20.9443	−0.889035
$556$	4.58359	0.194388
$557$	−38.9443	−1.65012	−0.825061	−	0.565044i	$-0.808859\pi$
$557$	−38.9443	−1.65012	−0.825061	+	0.565044i	$0.808859\pi$
$558$	−9.09830	−0.385162
$559$	−25.8885	−1.09497
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−7.88854	−0.332758
$563$	−12.5836	−0.530335	−0.265168	−	0.964202i	$-0.585427\pi$
$563$	−12.5836	−0.530335	−0.265168	+	0.964202i	$0.585427\pi$
$564$	−10.2492	−0.431570
$565$	4.00000	0.168281
$566$	−66.8328	−2.80919
$567$	0	0
$568$	−23.4164	−0.982531
$569$	7.52786	0.315584	0.157792	−	0.987472i	$-0.449562\pi$
$569$	7.52786	0.315584	0.157792	+	0.987472i	$0.449562\pi$
$570$	−35.7771	−1.49854
$571$	−15.0557	−0.630063	−0.315031	−	0.949081i	$-0.602015\pi$
$571$	−15.0557	−0.630063	−0.315031	+	0.949081i	$0.602015\pi$
$572$	9.70820	0.405920
$573$	−3.77709	−0.157790
$574$	0	0
$575$	−2.47214	−0.103095
$576$	19.1378	0.797407
$577$	19.5279	0.812956	0.406478	−	0.913661i	$-0.366757\pi$
$577$	19.5279	0.812956	0.406478	+	0.913661i	$0.366757\pi$
$578$	−14.5967	−0.607145
$579$	14.6950	0.610705
$580$	50.8328	2.11072
$581$	0	0
$582$	−48.1378	−1.99537
$583$	0.472136	0.0195539
$584$	1.70820	0.0706860
$585$	9.52786	0.393929
$586$	56.1803	2.32079
$587$	−27.1246	−1.11955	−0.559776	−	0.828644i	$-0.689113\pi$
$587$	−27.1246	−1.11955	−0.559776	+	0.828644i	$0.689113\pi$
$588$	0	0
$589$	−17.8885	−0.737085
$590$	5.52786	0.227579
$591$	−2.47214	−0.101690
$592$	8.47214	0.348203
$593$	45.7082	1.87701	0.938505	−	0.345264i	$-0.112211\pi$
$593$	45.7082	1.87701	0.938505	+	0.345264i	$0.112211\pi$
$594$	12.3607	0.507165
$595$	0	0
$596$	42.0000	1.72039
$597$	2.69505	0.110301
$598$	−17.8885	−0.731517
$599$	−23.4164	−0.956768	−0.478384	−	0.878151i	$-0.658777\pi$
$599$	−23.4164	−0.956768	−0.478384	+	0.878151i	$0.658777\pi$
$600$	−2.76393	−0.112837
$601$	37.1246	1.51434	0.757172	−	0.653215i	$-0.226580\pi$
$601$	37.1246	1.51434	0.757172	+	0.653215i	$0.226580\pi$
$602$	0	0
$603$	−21.3050	−0.867605
$604$	26.8328	1.09181
$605$	2.00000	0.0813116
$606$	13.1672	0.534880
$607$	−12.9443	−0.525392	−0.262696	−	0.964879i	$-0.584612\pi$
$607$	−12.9443	−0.525392	−0.262696	+	0.964879i	$0.584612\pi$
$608$	43.4164	1.76077
$609$	0	0
$610$	32.3607	1.31025
$611$	8.94427	0.361847
$612$	−14.2918	−0.577712
$613$	15.3050	0.618161	0.309081	−	0.951036i	$-0.399979\pi$
$613$	15.3050	0.618161	0.309081	+	0.951036i	$0.399979\pi$
$614$	20.0000	0.807134
$615$	27.7771	1.12008
$616$	0	0
$617$	6.58359	0.265045	0.132523	−	0.991180i	$-0.457692\pi$
$617$	6.58359	0.265045	0.132523	+	0.991180i	$0.457692\pi$
$618$	−21.3050	−0.857011
$619$	−11.1246	−0.447136	−0.223568	−	0.974688i	$-0.571770\pi$
$619$	−11.1246	−0.447136	−0.223568	+	0.974688i	$0.571770\pi$
$620$	16.5836	0.666013
$621$	−13.6656	−0.548383
$622$	18.5410	0.743427
$623$	0	0
$624$	4.00000	0.160128
$625$	−19.0000	−0.760000
$626$	−33.4164	−1.33559
$627$	8.00000	0.319489
$628$	−32.8328	−1.31017
$629$	−27.4164	−1.09316
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	−20.0000	−0.795557
$633$	−17.1672	−0.682334
$634$	31.3050	1.24328
$635$	6.11146	0.242526
$636$	−1.75078	−0.0694228
$637$	0	0
$638$	−18.9443	−0.750011
$639$	15.4164	0.609864
$640$	−31.3050	−1.23744
$641$	−15.5279	−0.613314	−0.306657	−	0.951820i	$-0.599210\pi$
$641$	−15.5279	−0.613314	−0.306657	+	0.951820i	$0.599210\pi$
$642$	−11.0557	−0.436335
$643$	−11.1246	−0.438712	−0.219356	−	0.975645i	$-0.570396\pi$
$643$	−11.1246	−0.438712	−0.219356	+	0.975645i	$0.570396\pi$
$644$	0	0
$645$	19.7771	0.778722
$646$	−46.8328	−1.84261
$647$	−36.0689	−1.41801	−0.709007	−	0.705201i	$-0.750857\pi$
$647$	−36.0689	−1.41801	−0.709007	+	0.705201i	$0.750857\pi$
$648$	−5.40325	−0.212260
$649$	−1.23607	−0.0485199
$650$	7.23607	0.283822
$651$	0	0
$652$	−10.2492	−0.401391
$653$	25.0557	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$653$	25.0557	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$654$	−12.3607	−0.483341
$655$	−43.7771	−1.71051
$656$	−11.2361	−0.438695
$657$	−1.12461	−0.0438753
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	−7.41641	−0.288683
$661$	40.8328	1.58821	0.794106	−	0.607779i	$-0.207939\pi$
$661$	40.8328	1.58821	0.794106	+	0.607779i	$0.207939\pi$
$662$	−31.0557	−1.20702
$663$	−12.9443	−0.502714
$664$	25.5279	0.990673
$665$	0	0
$666$	27.8885	1.08066
$667$	20.9443	0.810965
$668$	−14.8328	−0.573899
$669$	12.5836	0.486510
$670$	64.7214	2.50040
$671$	−7.23607	−0.279345
$672$	0	0
$673$	−21.4164	−0.825542	−0.412771	−	0.910835i	$-0.635439\pi$
$673$	−21.4164	−0.825542	−0.412771	+	0.910835i	$0.635439\pi$
$674$	−25.7771	−0.992896
$675$	5.52786	0.212768
$676$	−7.58359	−0.291677
$677$	9.70820	0.373117	0.186558	−	0.982444i	$-0.440267\pi$
$677$	9.70820	0.373117	0.186558	+	0.982444i	$0.440267\pi$
$678$	5.52786	0.212296
$679$	0	0
$680$	14.4721	0.554981
$681$	−7.27864	−0.278918
$682$	−6.18034	−0.236657
$683$	−5.88854	−0.225319	−0.112659	−	0.993634i	$-0.535937\pi$
$683$	−5.88854	−0.225319	−0.112659	+	0.993634i	$0.535937\pi$
$684$	28.5836	1.09292
$685$	32.9443	1.25874
$686$	0	0
$687$	5.52786	0.210901
$688$	−8.00000	−0.304997
$689$	1.52786	0.0582070
$690$	13.6656	0.520242
$691$	−18.5410	−0.705334	−0.352667	−	0.935749i	$-0.614725\pi$
$691$	−18.5410	−0.705334	−0.352667	+	0.935749i	$0.614725\pi$
$692$	38.2918	1.45564
$693$	0	0
$694$	46.8328	1.77775
$695$	3.05573	0.115910
$696$	23.4164	0.887597
$697$	36.3607	1.37726
$698$	16.1803	0.612435
$699$	11.6393	0.440240
$700$	0	0
$701$	−15.5279	−0.586479	−0.293240	−	0.956039i	$-0.594733\pi$
$701$	−15.5279	−0.586479	−0.293240	+	0.956039i	$0.594733\pi$
$702$	40.0000	1.50970
$703$	54.8328	2.06806
$704$	13.0000	0.489956
$705$	−6.83282	−0.257339
$706$	−44.4721	−1.67373
$707$	0	0
$708$	4.58359	0.172262
$709$	−14.9443	−0.561244	−0.280622	−	0.959818i	$-0.590541\pi$
$709$	−14.9443	−0.561244	−0.280622	+	0.959818i	$0.590541\pi$
$710$	−46.8328	−1.75760
$711$	13.1672	0.493808
$712$	−4.47214	−0.167600
$713$	6.83282	0.255891
$714$	0	0
$715$	6.47214	0.242044
$716$	−26.8328	−1.00279
$717$	−12.2229	−0.456473
$718$	−55.7771	−2.08158
$719$	51.4853	1.92008	0.960039	−	0.279867i	$-0.0902905\pi$
$719$	51.4853	1.92008	0.960039	+	0.279867i	$0.0902905\pi$
$720$	2.94427	0.109727
$721$	0	0
$722$	51.1803	1.90474
$723$	16.2229	0.603337
$724$	76.2492	2.83378
$725$	−8.47214	−0.314647
$726$	2.76393	0.102579
$727$	−25.0132	−0.927687	−0.463843	−	0.885917i	$-0.653530\pi$
$727$	−25.0132	−0.927687	−0.463843	+	0.885917i	$0.653530\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	3.41641	0.126447
$731$	25.8885	0.957522
$732$	26.8328	0.991769
$733$	−8.76393	−0.323703	−0.161852	−	0.986815i	$-0.551747\pi$
$733$	−8.76393	−0.323703	−0.161852	+	0.986815i	$0.551747\pi$
$734$	51.7082	1.90858
$735$	0	0
$736$	−16.5836	−0.611279
$737$	−14.4721	−0.533088
$738$	−36.9868	−1.36150
$739$	24.9443	0.917590	0.458795	−	0.888542i	$-0.348281\pi$
$739$	24.9443	0.917590	0.458795	+	0.888542i	$0.348281\pi$
$740$	−50.8328	−1.86865
$741$	25.8885	0.951039
$742$	0	0
$743$	1.88854	0.0692840	0.0346420	−	0.999400i	$-0.488971\pi$
$743$	1.88854	0.0692840	0.0346420	+	0.999400i	$0.488971\pi$
$744$	7.63932	0.280071
$745$	28.0000	1.02584
$746$	13.4164	0.491210
$747$	−16.8065	−0.614918
$748$	−9.70820	−0.354967
$749$	0	0
$750$	−33.1672	−1.21109
$751$	29.5279	1.07749	0.538744	−	0.842470i	$-0.318899\pi$
$751$	29.5279	1.07749	0.538744	+	0.842470i	$0.318899\pi$
$752$	2.76393	0.100790
$753$	−5.30495	−0.193323
$754$	−61.3050	−2.23259
$755$	17.8885	0.651031
$756$	0	0
$757$	15.8885	0.577479	0.288739	−	0.957408i	$-0.406764\pi$
$757$	15.8885	0.577479	0.288739	+	0.957408i	$0.406764\pi$
$758$	83.4164	3.02982
$759$	−3.05573	−0.110916
$760$	−28.9443	−1.04992
$761$	31.5967	1.14538	0.572691	−	0.819772i	$-0.305900\pi$
$761$	31.5967	1.14538	0.572691	+	0.819772i	$0.305900\pi$
$762$	8.44582	0.305960
$763$	0	0
$764$	−9.16718	−0.331657
$765$	−9.52786	−0.344481
$766$	10.4033	0.375885
$767$	−4.00000	−0.144432
$768$	−11.1246	−0.401425
$769$	−18.2918	−0.659619	−0.329810	−	0.944047i	$-0.606985\pi$
$769$	−18.2918	−0.659619	−0.329810	+	0.944047i	$0.606985\pi$
$770$	0	0
$771$	7.41641	0.267095
$772$	35.6656	1.28363
$773$	38.3607	1.37974	0.689869	−	0.723934i	$-0.257668\pi$
$773$	38.3607	1.37974	0.689869	+	0.723934i	$0.257668\pi$
$774$	−26.3344	−0.946569
$775$	−2.76393	−0.0992834
$776$	−38.9443	−1.39802
$777$	0	0
$778$	35.5279	1.27374
$779$	−72.7214	−2.60551
$780$	−24.0000	−0.859338
$781$	10.4721	0.374722
$782$	17.8885	0.639693
$783$	−46.8328	−1.67367
$784$	0	0
$785$	−21.8885	−0.781236
$786$	−60.4984	−2.15791
$787$	43.4164	1.54763	0.773814	−	0.633413i	$-0.218347\pi$
$787$	43.4164	1.54763	0.773814	+	0.633413i	$0.218347\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−40.0000	−1.42314
$791$	0	0
$792$	3.29180	0.116969
$793$	−23.4164	−0.831541
$794$	80.2492	2.84794
$795$	−1.16718	−0.0413958
$796$	6.54102	0.231840
$797$	14.9443	0.529353	0.264677	−	0.964337i	$-0.414735\pi$
$797$	14.9443	0.529353	0.264677	+	0.964337i	$0.414735\pi$
$798$	0	0
$799$	−8.94427	−0.316426
$800$	6.70820	0.237171
$801$	2.94427	0.104031
$802$	51.3050	1.81164
$803$	−0.763932	−0.0269586
$804$	53.6656	1.89264
$805$	0	0
$806$	−20.0000	−0.704470
$807$	−16.5836	−0.583770
$808$	10.6525	0.374753
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	−10.8065	−0.379702
$811$	34.8328	1.22315	0.611573	−	0.791188i	$-0.290537\pi$
$811$	34.8328	1.22315	0.611573	+	0.791188i	$0.290537\pi$
$812$	0	0
$813$	12.9443	0.453975
$814$	18.9443	0.663996
$815$	−6.83282	−0.239343
$816$	−4.00000	−0.140028
$817$	−51.7771	−1.81145
$818$	−20.4033	−0.713383
$819$	0	0
$820$	67.4164	2.35428
$821$	44.8328	1.56468	0.782338	−	0.622854i	$-0.214027\pi$
$821$	44.8328	1.56468	0.782338	+	0.622854i	$0.214027\pi$
$822$	45.5279	1.58797
$823$	−14.1115	−0.491894	−0.245947	−	0.969283i	$-0.579099\pi$
$823$	−14.1115	−0.491894	−0.245947	+	0.969283i	$0.579099\pi$
$824$	−17.2361	−0.600447
$825$	1.23607	0.0430344
$826$	0	0
$827$	−12.9443	−0.450116	−0.225058	−	0.974345i	$-0.572257\pi$
$827$	−12.9443	−0.450116	−0.225058	+	0.974345i	$0.572257\pi$
$828$	−10.9180	−0.379425
$829$	−36.8328	−1.27926	−0.639628	−	0.768684i	$-0.720912\pi$
$829$	−36.8328	−1.27926	−0.639628	+	0.768684i	$0.720912\pi$
$830$	51.0557	1.77217
$831$	−24.5836	−0.852795
$832$	42.0689	1.45848
$833$	0	0
$834$	4.22291	0.146227
$835$	−9.88854	−0.342207
$836$	19.4164	0.671531
$837$	−15.2786	−0.528107
$838$	−55.1246	−1.90425
$839$	−44.0689	−1.52143	−0.760713	−	0.649088i	$-0.775151\pi$
$839$	−44.0689	−1.52143	−0.760713	+	0.649088i	$0.775151\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	50.0000	1.72311
$843$	−4.36068	−0.150190
$844$	−41.6656	−1.43419
$845$	−5.05573	−0.173922
$846$	9.09830	0.312806
$847$	0	0
$848$	0.472136	0.0162132
$849$	−36.9443	−1.26792
$850$	−7.23607	−0.248195
$851$	−20.9443	−0.717960
$852$	−38.8328	−1.33039
$853$	30.6525	1.04952	0.524760	−	0.851250i	$-0.324155\pi$
$853$	30.6525	1.04952	0.524760	+	0.851250i	$0.324155\pi$
$854$	0	0
$855$	19.0557	0.651692
$856$	−8.94427	−0.305709
$857$	−15.2361	−0.520454	−0.260227	−	0.965547i	$-0.583797\pi$
$857$	−15.2361	−0.520454	−0.260227	+	0.965547i	$0.583797\pi$
$858$	8.94427	0.305352
$859$	26.5410	0.905568	0.452784	−	0.891620i	$-0.350431\pi$
$859$	26.5410	0.905568	0.452784	+	0.891620i	$0.350431\pi$
$860$	48.0000	1.63679
$861$	0	0
$862$	26.8328	0.913929
$863$	−3.05573	−0.104018	−0.0520091	−	0.998647i	$-0.516562\pi$
$863$	−3.05573	−0.104018	−0.0520091	+	0.998647i	$0.516562\pi$
$864$	37.0820	1.26156
$865$	25.5279	0.867973
$866$	18.9443	0.643753
$867$	−8.06888	−0.274034
$868$	0	0
$869$	8.94427	0.303414
$870$	46.8328	1.58778
$871$	−46.8328	−1.58687
$872$	−10.0000	−0.338643
$873$	25.6393	0.867760
$874$	−35.7771	−1.21018
$875$	0	0
$876$	2.83282	0.0957120
$877$	14.5836	0.492453	0.246226	−	0.969212i	$-0.420809\pi$
$877$	14.5836	0.492453	0.246226	+	0.969212i	$0.420809\pi$
$878$	−23.4164	−0.790265
$879$	31.0557	1.04748
$880$	2.00000	0.0674200
$881$	−2.58359	−0.0870434	−0.0435217	−	0.999052i	$-0.513858\pi$
$881$	−2.58359	−0.0870434	−0.0435217	+	0.999052i	$0.513858\pi$
$882$	0	0
$883$	−8.94427	−0.300999	−0.150499	−	0.988610i	$-0.548088\pi$
$883$	−8.94427	−0.300999	−0.150499	+	0.988610i	$0.548088\pi$
$884$	−31.4164	−1.05665
$885$	3.05573	0.102717
$886$	−55.7771	−1.87387
$887$	4.36068	0.146417	0.0732086	−	0.997317i	$-0.476676\pi$
$887$	4.36068	0.146417	0.0732086	+	0.997317i	$0.476676\pi$
$888$	−23.4164	−0.785803
$889$	0	0
$890$	−8.94427	−0.299813
$891$	2.41641	0.0809527
$892$	30.5410	1.02259
$893$	17.8885	0.598617
$894$	38.6950	1.29416
$895$	−17.8885	−0.597948
$896$	0	0
$897$	−9.88854	−0.330169
$898$	−63.6656	−2.12455
$899$	23.4164	0.780981
$900$	4.41641	0.147214
$901$	−1.52786	−0.0509005
$902$	−25.1246	−0.836558
$903$	0	0
$904$	4.47214	0.148741
$905$	50.8328	1.68974
$906$	24.7214	0.821312
$907$	22.4721	0.746175	0.373088	−	0.927796i	$-0.378299\pi$
$907$	22.4721	0.746175	0.373088	+	0.927796i	$0.378299\pi$
$908$	−17.6656	−0.586255
$909$	−7.01316	−0.232612
$910$	0	0
$911$	42.4721	1.40716	0.703582	−	0.710614i	$-0.251583\pi$
$911$	42.4721	1.40716	0.703582	+	0.710614i	$0.251583\pi$
$912$	8.00000	0.264906
$913$	−11.4164	−0.377828
$914$	−64.4721	−2.13255
$915$	17.8885	0.591377
$916$	13.4164	0.443291
$917$	0	0
$918$	−40.0000	−1.32020
$919$	41.8885	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$919$	41.8885	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$920$	11.0557	0.364497
$921$	11.0557	0.364299
$922$	27.2361	0.896972
$923$	33.8885	1.11546
$924$	0	0
$925$	8.47214	0.278562
$926$	−12.3607	−0.406197
$927$	11.3475	0.372702
$928$	−56.8328	−1.86563
$929$	52.2492	1.71424	0.857121	−	0.515116i	$-0.172251\pi$
$929$	52.2492	1.71424	0.857121	+	0.515116i	$0.172251\pi$
$930$	15.2786	0.501006
$931$	0	0
$932$	28.2492	0.925334
$933$	10.2492	0.335545
$934$	−53.8197	−1.76103
$935$	−6.47214	−0.211661
$936$	10.6525	0.348187
$937$	10.6525	0.348001	0.174001	−	0.984746i	$-0.444331\pi$
$937$	10.6525	0.348001	0.174001	+	0.984746i	$0.444331\pi$
$938$	0	0
$939$	−18.4721	−0.602815
$940$	−16.5836	−0.540897
$941$	−7.59675	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$941$	−7.59675	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$942$	−30.2492	−0.985573
$943$	27.7771	0.904546
$944$	−1.23607	−0.0402306
$945$	0	0
$946$	−17.8885	−0.581607
$947$	−5.16718	−0.167911	−0.0839555	−	0.996470i	$-0.526755\pi$
$947$	−5.16718	−0.167911	−0.0839555	+	0.996470i	$0.526755\pi$
$948$	−33.1672	−1.07722
$949$	−2.47214	−0.0802489
$950$	14.4721	0.469538
$951$	17.3050	0.561152
$952$	0	0
$953$	22.9443	0.743238	0.371619	−	0.928385i	$-0.378803\pi$
$953$	22.9443	0.743238	0.371619	+	0.928385i	$0.378803\pi$
$954$	1.55418	0.0503183
$955$	−6.11146	−0.197762
$956$	−29.6656	−0.959455
$957$	−10.4721	−0.338516
$958$	−30.2492	−0.977308
$959$	0	0
$960$	−32.1378	−1.03724
$961$	−23.3607	−0.753570
$962$	61.3050	1.97655
$963$	5.88854	0.189756
$964$	39.3738	1.26815
$965$	23.7771	0.765412
$966$	0	0
$967$	−13.8885	−0.446625	−0.223313	−	0.974747i	$-0.571687\pi$
$967$	−13.8885	−0.446625	−0.223313	+	0.974747i	$0.571687\pi$
$968$	2.23607	0.0718699
$969$	−25.8885	−0.831660
$970$	−77.8885	−2.50085
$971$	−11.1246	−0.357006	−0.178503	−	0.983939i	$-0.557125\pi$
$971$	−11.1246	−0.357006	−0.178503	+	0.983939i	$0.557125\pi$
$972$	40.7902	1.30835
$973$	0	0
$974$	−81.3050	−2.60518
$975$	4.00000	0.128103
$976$	−7.23607	−0.231621
$977$	22.9443	0.734052	0.367026	−	0.930211i	$-0.380376\pi$
$977$	22.9443	0.734052	0.367026	+	0.930211i	$0.380376\pi$
$978$	−9.44272	−0.301945
$979$	2.00000	0.0639203
$980$	0	0
$981$	6.58359	0.210198
$982$	0	0
$983$	−21.8197	−0.695939	−0.347970	−	0.937506i	$-0.613129\pi$
$983$	−21.8197	−0.695939	−0.347970	+	0.937506i	$0.613129\pi$
$984$	31.0557	0.990020
$985$	−4.00000	−0.127451
$986$	61.3050	1.95235
$987$	0	0
$988$	62.8328	1.99898
$989$	19.7771	0.628875
$990$	6.58359	0.209240
$991$	54.2492	1.72328	0.861642	−	0.507517i	$-0.169437\pi$
$991$	54.2492	1.72328	0.861642	+	0.507517i	$0.169437\pi$
$992$	−18.5410	−0.588678
$993$	−17.1672	−0.544784
$994$	0	0
$995$	4.36068	0.138243
$996$	42.3344	1.34142
$997$	−1.34752	−0.0426765	−0.0213383	−	0.999772i	$-0.506793\pi$
$997$	−1.34752	−0.0426765	−0.0213383	+	0.999772i	$0.506793\pi$
$998$	3.41641	0.108145
$999$	46.8328	1.48172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.f.1.2		2
3.2	odd	2		4851.2.a.y.1.1		2
4.3	odd	2		8624.2.a.ce.1.1		2
7.2	even	3		539.2.e.j.67.1		4
7.3	odd	6		539.2.e.i.177.1		4
7.4	even	3		539.2.e.j.177.1		4
7.5	odd	6		539.2.e.i.67.1		4
7.6	odd	2		77.2.a.d.1.2	✓	2
11.10	odd	2		5929.2.a.m.1.1		2
21.20	even	2		693.2.a.h.1.1		2
28.27	even	2		1232.2.a.m.1.2		2
35.13	even	4		1925.2.b.h.1849.2		4
35.27	even	4		1925.2.b.h.1849.3		4
35.34	odd	2		1925.2.a.r.1.1		2
56.13	odd	2		4928.2.a.bm.1.2		2
56.27	even	2		4928.2.a.bv.1.1		2
77.6	even	10		847.2.f.m.729.1		4
77.13	even	10		847.2.f.m.323.1		4
77.20	odd	10		847.2.f.a.323.1		4
77.27	odd	10		847.2.f.a.729.1		4
77.41	even	10		847.2.f.b.372.1		4
77.48	odd	10		847.2.f.n.148.1		4
77.62	even	10		847.2.f.b.148.1		4
77.69	odd	10		847.2.f.n.372.1		4
77.76	even	2		847.2.a.f.1.1		2
231.230	odd	2		7623.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.d.1.2	✓	2	7.6	odd	2
539.2.a.f.1.2		2	1.1	even	1	trivial
539.2.e.i.67.1		4	7.5	odd	6
539.2.e.i.177.1		4	7.3	odd	6
539.2.e.j.67.1		4	7.2	even	3
539.2.e.j.177.1		4	7.4	even	3
693.2.a.h.1.1		2	21.20	even	2
847.2.a.f.1.1		2	77.76	even	2
847.2.f.a.323.1		4	77.20	odd	10
847.2.f.a.729.1		4	77.27	odd	10
847.2.f.b.148.1		4	77.62	even	10
847.2.f.b.372.1		4	77.41	even	10
847.2.f.m.323.1		4	77.13	even	10
847.2.f.m.729.1		4	77.6	even	10
847.2.f.n.148.1		4	77.48	odd	10
847.2.f.n.372.1		4	77.69	odd	10
1232.2.a.m.1.2		2	28.27	even	2
1925.2.a.r.1.1		2	35.34	odd	2
1925.2.b.h.1849.2		4	35.13	even	4
1925.2.b.h.1849.3		4	35.27	even	4
4851.2.a.y.1.1		2	3.2	odd	2
4928.2.a.bm.1.2		2	56.13	odd	2
4928.2.a.bv.1.1		2	56.27	even	2
5929.2.a.m.1.1		2	11.10	odd	2
7623.2.a.bl.1.2		2	231.230	odd	2
8624.2.a.ce.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +1.23607 q^{3} +3.00000 q^{4} +2.00000 q^{5} +2.76393 q^{6} +2.23607 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} +1.23607 q^{3} +3.00000 q^{4} +2.00000 q^{5} +2.76393 q^{6} +2.23607 q^{8} -1.47214 q^{9} +4.47214 q^{10} -1.00000 q^{11} +3.70820 q^{12} -3.23607 q^{13} +2.47214 q^{15} -1.00000 q^{16} +3.23607 q^{17} -3.29180 q^{18} -6.47214 q^{19} +6.00000 q^{20} -2.23607 q^{22} +2.47214 q^{23} +2.76393 q^{24} -1.00000 q^{25} -7.23607 q^{26} -5.52786 q^{27} +8.47214 q^{29} +5.52786 q^{30} +2.76393 q^{31} -6.70820 q^{32} -1.23607 q^{33} +7.23607 q^{34} -4.41641 q^{36} -8.47214 q^{37} -14.4721 q^{38} -4.00000 q^{39} +4.47214 q^{40} +11.2361 q^{41} +8.00000 q^{43} -3.00000 q^{44} -2.94427 q^{45} +5.52786 q^{46} -2.76393 q^{47} -1.23607 q^{48} -2.23607 q^{50} +4.00000 q^{51} -9.70820 q^{52} -0.472136 q^{53} -12.3607 q^{54} -2.00000 q^{55} -8.00000 q^{57} +18.9443 q^{58} +1.23607 q^{59} +7.41641 q^{60} +7.23607 q^{61} +6.18034 q^{62} -13.0000 q^{64} -6.47214 q^{65} -2.76393 q^{66} +14.4721 q^{67} +9.70820 q^{68} +3.05573 q^{69} -10.4721 q^{71} -3.29180 q^{72} +0.763932 q^{73} -18.9443 q^{74} -1.23607 q^{75} -19.4164 q^{76} -8.94427 q^{78} -8.94427 q^{79} -2.00000 q^{80} -2.41641 q^{81} +25.1246 q^{82} +11.4164 q^{83} +6.47214 q^{85} +17.8885 q^{86} +10.4721 q^{87} -2.23607 q^{88} -2.00000 q^{89} -6.58359 q^{90} +7.41641 q^{92} +3.41641 q^{93} -6.18034 q^{94} -12.9443 q^{95} -8.29180 q^{96} -17.4164 q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^4 + 4 * q^5 + 10 * q^6 + 6 * q^9	\( 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} + 10 q^{6} + 6 q^{9} - 2 q^{11} - 6 q^{12} - 2 q^{13} - 4 q^{15} - 2 q^{16} + 2 q^{17} - 20 q^{18} - 4 q^{19} + 12 q^{20} - 4 q^{23} + 10 q^{24} - 2 q^{25} - 10 q^{26} - 20 q^{27} + 8 q^{29} + 20 q^{30} + 10 q^{31} + 2 q^{33} + 10 q^{34} + 18 q^{36} - 8 q^{37} - 20 q^{38} - 8 q^{39} + 18 q^{41} + 16 q^{43} - 6 q^{44} + 12 q^{45} + 20 q^{46} - 10 q^{47} + 2 q^{48} + 8 q^{51} - 6 q^{52} + 8 q^{53} + 20 q^{54} - 4 q^{55} - 16 q^{57} + 20 q^{58} - 2 q^{59} - 12 q^{60} + 10 q^{61} - 10 q^{62} - 26 q^{64} - 4 q^{65} - 10 q^{66} + 20 q^{67} + 6 q^{68} + 24 q^{69} - 12 q^{71} - 20 q^{72} + 6 q^{73} - 20 q^{74} + 2 q^{75} - 12 q^{76} - 4 q^{80} + 22 q^{81} + 10 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{87} - 4 q^{89} - 40 q^{90} - 12 q^{92} - 20 q^{93} + 10 q^{94} - 8 q^{95} - 30 q^{96} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^4 + 4 * q^5 + 10 * q^6 + 6 * q^9 - 2 * q^11 - 6 * q^12 - 2 * q^13 - 4 * q^15 - 2 * q^16 + 2 * q^17 - 20 * q^18 - 4 * q^19 + 12 * q^20 - 4 * q^23 + 10 * q^24 - 2 * q^25 - 10 * q^26 - 20 * q^27 + 8 * q^29 + 20 * q^30 + 10 * q^31 + 2 * q^33 + 10 * q^34 + 18 * q^36 - 8 * q^37 - 20 * q^38 - 8 * q^39 + 18 * q^41 + 16 * q^43 - 6 * q^44 + 12 * q^45 + 20 * q^46 - 10 * q^47 + 2 * q^48 + 8 * q^51 - 6 * q^52 + 8 * q^53 + 20 * q^54 - 4 * q^55 - 16 * q^57 + 20 * q^58 - 2 * q^59 - 12 * q^60 + 10 * q^61 - 10 * q^62 - 26 * q^64 - 4 * q^65 - 10 * q^66 + 20 * q^67 + 6 * q^68 + 24 * q^69 - 12 * q^71 - 20 * q^72 + 6 * q^73 - 20 * q^74 + 2 * q^75 - 12 * q^76 - 4 * q^80 + 22 * q^81 + 10 * q^82 - 4 * q^83 + 4 * q^85 + 12 * q^87 - 4 * q^89 - 40 * q^90 - 12 * q^92 - 20 * q^93 + 10 * q^94 - 8 * q^95 - 30 * q^96 - 8 * q^97 - 6 * q^99

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