LMFDB - Embedded newform 1859.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,2,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, 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("1859.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1859.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:	$14.8441897358$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.756.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 2 $ x^3 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.339877$ of defining polynomial
Character	$\chi$	$=$	1859.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.339877 q^{3} -1.00000 q^{4} +2.88448 q^{5} +0.339877 q^{6} +3.54461 q^{7} +3.00000 q^{8} -2.88448 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.339877 q^{3} -1.00000 q^{4} +2.88448 q^{5} +0.339877 q^{6} +3.54461 q^{7} +3.00000 q^{8} -2.88448 q^{9} -2.88448 q^{10} -1.00000 q^{11} +0.339877 q^{12} -3.54461 q^{14} -0.980369 q^{15} -1.00000 q^{16} -5.22436 q^{17} +2.88448 q^{18} -6.22436 q^{19} -2.88448 q^{20} -1.20473 q^{21} +1.00000 q^{22} +0.679754 q^{23} -1.01963 q^{24} +3.32025 q^{25} +2.00000 q^{27} -3.54461 q^{28} -8.42909 q^{29} +0.980369 q^{30} -8.40946 q^{31} -5.00000 q^{32} +0.339877 q^{33} +5.22436 q^{34} +10.2244 q^{35} +2.88448 q^{36} -3.52498 q^{37} +6.22436 q^{38} +8.65345 q^{40} +5.74934 q^{41} +1.20473 q^{42} -1.32025 q^{43} +1.00000 q^{44} -8.32025 q^{45} -0.679754 q^{46} -6.10884 q^{47} +0.339877 q^{48} +5.56424 q^{49} -3.32025 q^{50} +1.77564 q^{51} -6.08921 q^{53} -2.00000 q^{54} -2.88448 q^{55} +10.6338 q^{56} +2.11552 q^{57} +8.42909 q^{58} +14.1088 q^{59} +0.980369 q^{60} +4.01963 q^{61} +8.40946 q^{62} -10.2244 q^{63} +7.00000 q^{64} -0.339877 q^{66} -12.1088 q^{67} +5.22436 q^{68} -0.231033 q^{69} -10.2244 q^{70} -6.00000 q^{71} -8.65345 q^{72} +16.0825 q^{73} +3.52498 q^{74} -1.12847 q^{75} +6.22436 q^{76} -3.54461 q^{77} +0.904114 q^{79} -2.88448 q^{80} +7.97370 q^{81} -5.74934 q^{82} -4.90411 q^{83} +1.20473 q^{84} -15.0696 q^{85} +1.32025 q^{86} +2.86485 q^{87} -3.00000 q^{88} +5.65345 q^{89} +8.32025 q^{90} -0.679754 q^{92} +2.85818 q^{93} +6.10884 q^{94} -17.9541 q^{95} +1.69938 q^{96} -2.79527 q^{97} -5.56424 q^{98} +2.88448 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{4} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^4 - 3 * q^5 + 9 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} - 3 q^{4} - 3 q^{5} + 9 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} - 6 q^{15} - 3 q^{16} - 3 q^{17} - 3 q^{18} - 6 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{22} + 12 q^{25} + 6 q^{27} - 3 q^{29} + 6 q^{30} - 6 q^{31} - 15 q^{32} + 3 q^{34} + 18 q^{35} - 3 q^{36} - 3 q^{37} + 6 q^{38} - 9 q^{40} - 3 q^{41} - 6 q^{42} - 6 q^{43} + 3 q^{44} - 27 q^{45} + 6 q^{47} + 3 q^{49} - 12 q^{50} + 18 q^{51} + 3 q^{53} - 6 q^{54} + 3 q^{55} + 18 q^{57} + 3 q^{58} + 18 q^{59} + 6 q^{60} + 9 q^{61} + 6 q^{62} - 18 q^{63} + 21 q^{64} - 12 q^{67} + 3 q^{68} - 24 q^{69} - 18 q^{70} - 18 q^{71} + 9 q^{72} - 9 q^{73} + 3 q^{74} + 24 q^{75} + 6 q^{76} - 12 q^{79} + 3 q^{80} - 9 q^{81} + 3 q^{82} - 6 q^{84} - 27 q^{85} + 6 q^{86} - 9 q^{88} - 18 q^{89} + 27 q^{90} - 36 q^{93} - 6 q^{94} - 24 q^{95} - 18 q^{97} - 3 q^{98} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^4 - 3 * q^5 + 9 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 - 6 * q^15 - 3 * q^16 - 3 * q^17 - 3 * q^18 - 6 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^22 + 12 * q^25 + 6 * q^27 - 3 * q^29 + 6 * q^30 - 6 * q^31 - 15 * q^32 + 3 * q^34 + 18 * q^35 - 3 * q^36 - 3 * q^37 + 6 * q^38 - 9 * q^40 - 3 * q^41 - 6 * q^42 - 6 * q^43 + 3 * q^44 - 27 * q^45 + 6 * q^47 + 3 * q^49 - 12 * q^50 + 18 * q^51 + 3 * q^53 - 6 * q^54 + 3 * q^55 + 18 * q^57 + 3 * q^58 + 18 * q^59 + 6 * q^60 + 9 * q^61 + 6 * q^62 - 18 * q^63 + 21 * q^64 - 12 * q^67 + 3 * q^68 - 24 * q^69 - 18 * q^70 - 18 * q^71 + 9 * q^72 - 9 * q^73 + 3 * q^74 + 24 * q^75 + 6 * q^76 - 12 * q^79 + 3 * q^80 - 9 * q^81 + 3 * q^82 - 6 * q^84 - 27 * q^85 + 6 * q^86 - 9 * q^88 - 18 * q^89 + 27 * q^90 - 36 * q^93 - 6 * q^94 - 24 * q^95 - 18 * q^97 - 3 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−0.339877	−0.196228	−0.0981140	−	0.995175i	$-0.531281\pi$
$3$	−0.339877	−0.196228	−0.0981140	+	0.995175i	$0.531281\pi$
$4$	−1.00000	−0.500000
$5$	2.88448	1.28998	0.644990	−	0.764191i	$-0.276861\pi$
$5$	2.88448	1.28998	0.644990	+	0.764191i	$0.276861\pi$
$6$	0.339877	0.138754
$7$	3.54461	1.33974	0.669868	−	0.742480i	$-0.266351\pi$
$7$	3.54461	1.33974	0.669868	+	0.742480i	$0.266351\pi$
$8$	3.00000	1.06066
$9$	−2.88448	−0.961495
$10$	−2.88448	−0.912154
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.339877	0.0981140
$13$	0	0
$13$	0	0
$14$	−3.54461	−0.947336
$15$	−0.980369	−0.253130
$16$	−1.00000	−0.250000
$17$	−5.22436	−1.26709	−0.633547	−	0.773704i	$-0.718402\pi$
$17$	−5.22436	−1.26709	−0.633547	+	0.773704i	$0.718402\pi$
$18$	2.88448	0.679879
$19$	−6.22436	−1.42797	−0.713983	−	0.700163i	$-0.753111\pi$
$19$	−6.22436	−1.42797	−0.713983	+	0.700163i	$0.753111\pi$
$20$	−2.88448	−0.644990
$21$	−1.20473	−0.262894
$22$	1.00000	0.213201
$23$	0.679754	0.141738	0.0708692	−	0.997486i	$-0.477423\pi$
$23$	0.679754	0.141738	0.0708692	+	0.997486i	$0.477423\pi$
$24$	−1.01963	−0.208131
$25$	3.32025	0.664049
$26$	0	0
$27$	2.00000	0.384900
$28$	−3.54461	−0.669868
$29$	−8.42909	−1.56524	−0.782621	−	0.622498i	$-0.786118\pi$
$29$	−8.42909	−1.56524	−0.782621	+	0.622498i	$0.786118\pi$
$30$	0.980369	0.178990
$31$	−8.40946	−1.51038	−0.755192	−	0.655504i	$-0.772456\pi$
$31$	−8.40946	−1.51038	−0.755192	+	0.655504i	$0.772456\pi$
$32$	−5.00000	−0.883883
$33$	0.339877	0.0591650
$34$	5.22436	0.895970
$35$	10.2244	1.72823
$36$	2.88448	0.480747
$37$	−3.52498	−0.579503	−0.289751	−	0.957102i	$-0.593573\pi$
$37$	−3.52498	−0.579503	−0.289751	+	0.957102i	$0.593573\pi$
$38$	6.22436	1.00972
$39$	0	0
$40$	8.65345	1.36823
$41$	5.74934	0.897896	0.448948	−	0.893558i	$-0.351799\pi$
$41$	5.74934	0.897896	0.448948	+	0.893558i	$0.351799\pi$
$42$	1.20473	0.185894
$43$	−1.32025	−0.201336	−0.100668	−	0.994920i	$-0.532098\pi$
$43$	−1.32025	−0.201336	−0.100668	+	0.994920i	$0.532098\pi$
$44$	1.00000	0.150756
$45$	−8.32025	−1.24031
$46$	−0.679754	−0.100224
$47$	−6.10884	−0.891067	−0.445533	−	0.895265i	$-0.646986\pi$
$47$	−6.10884	−0.891067	−0.445533	+	0.895265i	$0.646986\pi$
$48$	0.339877	0.0490570
$49$	5.56424	0.794891
$50$	−3.32025	−0.469554
$51$	1.77564	0.248639
$52$	0	0
$53$	−6.08921	−0.836418	−0.418209	−	0.908351i	$-0.637342\pi$
$53$	−6.08921	−0.836418	−0.418209	+	0.908351i	$0.637342\pi$
$54$	−2.00000	−0.272166
$55$	−2.88448	−0.388944
$56$	10.6338	1.42100
$57$	2.11552	0.280207
$58$	8.42909	1.10679
$59$	14.1088	1.83682	0.918408	−	0.395636i	$-0.129476\pi$
$59$	14.1088	1.83682	0.918408	+	0.395636i	$0.129476\pi$
$60$	0.980369	0.126565
$61$	4.01963	0.514661	0.257330	−	0.966323i	$-0.417157\pi$
$61$	4.01963	0.514661	0.257330	+	0.966323i	$0.417157\pi$
$62$	8.40946	1.06800
$63$	−10.2244	−1.28815
$64$	7.00000	0.875000
$65$	0	0
$66$	−0.339877	−0.0418360
$67$	−12.1088	−1.47933	−0.739665	−	0.672975i	$-0.765016\pi$
$67$	−12.1088	−1.47933	−0.739665	+	0.672975i	$0.765016\pi$
$68$	5.22436	0.633547
$69$	−0.231033	−0.0278131
$70$	−10.2244	−1.22204
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−8.65345	−1.01982
$73$	16.0825	1.88232	0.941160	−	0.337963i	$-0.109738\pi$
$73$	16.0825	1.88232	0.941160	+	0.337963i	$0.109738\pi$
$74$	3.52498	0.409770
$75$	−1.12847	−0.130305
$76$	6.22436	0.713983
$77$	−3.54461	−0.403945
$78$	0	0
$79$	0.904114	0.101721	0.0508604	−	0.998706i	$-0.483804\pi$
$79$	0.904114	0.101721	0.0508604	+	0.998706i	$0.483804\pi$
$80$	−2.88448	−0.322495
$81$	7.97370	0.885966
$82$	−5.74934	−0.634908
$83$	−4.90411	−0.538296	−0.269148	−	0.963099i	$-0.586742\pi$
$83$	−4.90411	−0.538296	−0.269148	+	0.963099i	$0.586742\pi$
$84$	1.20473	0.131447
$85$	−15.0696	−1.63453
$86$	1.32025	0.142366
$87$	2.86485	0.307144
$88$	−3.00000	−0.319801
$89$	5.65345	0.599265	0.299632	−	0.954055i	$-0.403136\pi$
$89$	5.65345	0.599265	0.299632	+	0.954055i	$0.403136\pi$
$90$	8.32025	0.877031
$91$	0	0
$92$	−0.679754	−0.0708692
$93$	2.85818	0.296380
$94$	6.10884	0.630079
$95$	−17.9541	−1.84205
$96$	1.69938	0.173443
$97$	−2.79527	−0.283817	−0.141908	−	0.989880i	$-0.545324\pi$
$97$	−2.79527	−0.283817	−0.141908	+	0.989880i	$0.545324\pi$
$98$	−5.56424	−0.562073
$99$	2.88448	0.289902
$100$	−3.32025	−0.332025
$101$	−9.86485	−0.981590	−0.490795	−	0.871275i	$-0.663294\pi$
$101$	−9.86485	−0.981590	−0.490795	+	0.871275i	$0.663294\pi$
$102$	−1.77564	−0.175815
$103$	5.35951	0.528088	0.264044	−	0.964511i	$-0.414944\pi$
$103$	5.35951	0.528088	0.264044	+	0.964511i	$0.414944\pi$
$104$	0	0
$105$	−3.47502	−0.339128
$106$	6.08921	0.591437
$107$	3.31357	0.320335	0.160168	−	0.987090i	$-0.448797\pi$
$107$	3.31357	0.320335	0.160168	+	0.987090i	$0.448797\pi$
$108$	−2.00000	−0.192450
$109$	−2.15478	−0.206390	−0.103195	−	0.994661i	$-0.532907\pi$
$109$	−2.15478	−0.206390	−0.103195	+	0.994661i	$0.532907\pi$
$110$	2.88448	0.275025
$111$	1.19806	0.113715
$112$	−3.54461	−0.334934
$113$	3.79527	0.357029	0.178514	−	0.983937i	$-0.442871\pi$
$113$	3.79527	0.357029	0.178514	+	0.983937i	$0.442871\pi$
$114$	−2.11552	−0.198136
$115$	1.96074	0.182840
$116$	8.42909	0.782621
$117$	0	0
$118$	−14.1088	−1.29882
$119$	−18.5183	−1.69757
$120$	−2.94111	−0.268485
$121$	1.00000	0.0909091
$122$	−4.01963	−0.363920
$123$	−1.95407	−0.176192
$124$	8.40946	0.755192
$125$	−4.84522	−0.433370
$126$	10.2244	0.910858
$127$	−16.1784	−1.43560	−0.717802	−	0.696248i	$-0.754852\pi$
$127$	−16.1784	−1.43560	−0.717802	+	0.696248i	$0.754852\pi$
$128$	3.00000	0.265165
$129$	0.448721	0.0395077
$130$	0	0
$131$	−3.31357	−0.289508	−0.144754	−	0.989468i	$-0.546239\pi$
$131$	−3.31357	−0.289508	−0.144754	+	0.989468i	$0.546239\pi$
$132$	−0.339877	−0.0295825
$133$	−22.0629	−1.91310
$134$	12.1088	1.04604
$135$	5.76897	0.496514
$136$	−15.6731	−1.34396
$137$	−13.7427	−1.17412	−0.587058	−	0.809545i	$-0.699714\pi$
$137$	−13.7427	−1.17412	−0.587058	+	0.809545i	$0.699714\pi$
$138$	0.231033	0.0196668
$139$	2.40946	0.204368	0.102184	−	0.994766i	$-0.467417\pi$
$139$	2.40946	0.204368	0.102184	+	0.994766i	$0.467417\pi$
$140$	−10.2244	−0.864116
$141$	2.07625	0.174852
$142$	6.00000	0.503509
$143$	0	0
$144$	2.88448	0.240374
$145$	−24.3136	−2.01913
$146$	−16.0825	−1.33100
$147$	−1.89116	−0.155980
$148$	3.52498	0.289751
$149$	2.54461	0.208462	0.104231	−	0.994553i	$-0.466762\pi$
$149$	2.54461	0.208462	0.104231	+	0.994553i	$0.466762\pi$
$150$	1.12847	0.0921396
$151$	−8.44872	−0.687547	−0.343774	−	0.939053i	$-0.611705\pi$
$151$	−8.44872	−0.687547	−0.343774	+	0.939053i	$0.611705\pi$
$152$	−18.6731	−1.51459
$153$	15.0696	1.21830
$154$	3.54461	0.285633
$155$	−24.2569	−1.94837
$156$	0	0
$157$	−16.3595	−1.30563	−0.652815	−	0.757517i	$-0.726412\pi$
$157$	−16.3595	−1.30563	−0.652815	+	0.757517i	$0.726412\pi$
$158$	−0.904114	−0.0719275
$159$	2.06958	0.164129
$160$	−14.4224	−1.14019
$161$	2.40946	0.189892
$162$	−7.97370	−0.626473
$163$	10.7886	0.845028	0.422514	−	0.906356i	$-0.361148\pi$
$163$	10.7886	0.845028	0.422514	+	0.906356i	$0.361148\pi$
$164$	−5.74934	−0.448948
$165$	0.980369	0.0763216
$166$	4.90411	0.380633
$167$	−7.04995	−0.545542	−0.272771	−	0.962079i	$-0.587940\pi$
$167$	−7.04995	−0.545542	−0.272771	+	0.962079i	$0.587940\pi$
$168$	−3.61419	−0.278841
$169$	0	0
$170$	15.0696	1.15578
$171$	17.9541	1.37298
$172$	1.32025	0.100668
$173$	−1.84522	−0.140290	−0.0701448	−	0.997537i	$-0.522346\pi$
$173$	−1.84522	−0.140290	−0.0701448	+	0.997537i	$0.522346\pi$
$174$	−2.86485	−0.217184
$175$	11.7690	0.889650
$176$	1.00000	0.0753778
$177$	−4.79527	−0.360435
$178$	−5.65345	−0.423744
$179$	5.42909	0.405789	0.202895	−	0.979201i	$-0.434965\pi$
$179$	5.42909	0.405789	0.202895	+	0.979201i	$0.434965\pi$
$180$	8.32025	0.620155
$181$	−4.47502	−0.332626	−0.166313	−	0.986073i	$-0.553186\pi$
$181$	−4.47502	−0.332626	−0.166313	+	0.986073i	$0.553186\pi$
$182$	0	0
$183$	−1.36618	−0.100991
$184$	2.03926	0.150336
$185$	−10.1677	−0.747547
$186$	−2.85818	−0.209572
$187$	5.22436	0.382043
$188$	6.10884	0.445533
$189$	7.08921	0.515664
$190$	17.9541	1.30252
$191$	9.89116	0.715699	0.357849	−	0.933779i	$-0.383510\pi$
$191$	9.89116	0.715699	0.357849	+	0.933779i	$0.383510\pi$
$192$	−2.37914	−0.171700
$193$	−5.10884	−0.367743	−0.183871	−	0.982950i	$-0.558863\pi$
$193$	−5.10884	−0.367743	−0.183871	+	0.982950i	$0.558863\pi$
$194$	2.79527	0.200689
$195$	0	0
$196$	−5.56424	−0.397446
$197$	−2.23103	−0.158954	−0.0794772	−	0.996837i	$-0.525325\pi$
$197$	−2.23103	−0.158954	−0.0794772	+	0.996837i	$0.525325\pi$
$198$	−2.88448	−0.204991
$199$	17.8778	1.26732	0.633662	−	0.773610i	$-0.281551\pi$
$199$	17.8778	1.26732	0.633662	+	0.773610i	$0.281551\pi$
$200$	9.96074	0.704331
$201$	4.11552	0.290286
$202$	9.86485	0.694089
$203$	−29.8778	−2.09701
$204$	−1.77564	−0.124320
$205$	16.5839	1.15827
$206$	−5.35951	−0.373415
$207$	−1.96074	−0.136281
$208$	0	0
$209$	6.22436	0.430548
$210$	3.47502	0.239799
$211$	10.9041	0.750670	0.375335	−	0.926889i	$-0.377528\pi$
$211$	10.9041	0.750670	0.375335	+	0.926889i	$0.377528\pi$
$212$	6.08921	0.418209
$213$	2.03926	0.139728
$214$	−3.31357	−0.226511
$215$	−3.80823	−0.259719
$216$	6.00000	0.408248
$217$	−29.8082	−2.02351
$218$	2.15478	0.145940
$219$	−5.46608	−0.369364
$220$	2.88448	0.194472
$221$	0	0
$222$	−1.19806	−0.0804084
$223$	19.4291	1.30107	0.650534	−	0.759477i	$-0.274545\pi$
$223$	19.4291	1.30107	0.650534	+	0.759477i	$0.274545\pi$
$224$	−17.7230	−1.18417
$225$	−9.57720	−0.638480
$226$	−3.79527	−0.252458
$227$	7.32025	0.485862	0.242931	−	0.970044i	$-0.421891\pi$
$227$	7.32025	0.485862	0.242931	+	0.970044i	$0.421891\pi$
$228$	−2.11552	−0.140103
$229$	−14.5642	−0.962432	−0.481216	−	0.876602i	$-0.659805\pi$
$229$	−14.5642	−0.962432	−0.481216	+	0.876602i	$0.659805\pi$
$230$	−1.96074	−0.129287
$231$	1.20473	0.0792654
$232$	−25.2873	−1.66019
$233$	6.44872	0.422470	0.211235	−	0.977435i	$-0.432252\pi$
$233$	6.44872	0.422470	0.211235	+	0.977435i	$0.432252\pi$
$234$	0	0
$235$	−17.6209	−1.14946
$236$	−14.1088	−0.918408
$237$	−0.307288	−0.0199605
$238$	18.5183	1.20036
$239$	0.270294	0.0174839	0.00874193	−	0.999962i	$-0.497217\pi$
$239$	0.270294	0.0174839	0.00874193	+	0.999962i	$0.497217\pi$
$240$	0.980369	0.0632826
$241$	−30.5313	−1.96669	−0.983346	−	0.181745i	$-0.941826\pi$
$241$	−30.5313	−1.96669	−0.983346	+	0.181745i	$0.941826\pi$
$242$	−1.00000	−0.0642824
$243$	−8.71008	−0.558752
$244$	−4.01963	−0.257330
$245$	16.0500	1.02539
$246$	1.95407	0.124587
$247$	0	0
$248$	−25.2284	−1.60200
$249$	1.66680	0.105629
$250$	4.84522	0.306439
$251$	10.4790	0.661431	0.330716	−	0.943730i	$-0.392710\pi$
$251$	10.4790	0.661431	0.330716	+	0.943730i	$0.392710\pi$
$252$	10.2244	0.644074
$253$	−0.679754	−0.0427358
$254$	16.1784	1.01512
$255$	5.12180	0.320740
$256$	−17.0000	−1.06250
$257$	5.75601	0.359050	0.179525	−	0.983753i	$-0.442544\pi$
$257$	5.75601	0.359050	0.179525	+	0.983753i	$0.442544\pi$
$258$	−0.448721	−0.0279362
$259$	−12.4947	−0.776380
$260$	0	0
$261$	24.3136	1.50497
$262$	3.31357	0.204713
$263$	8.44872	0.520970	0.260485	−	0.965478i	$-0.416117\pi$
$263$	8.44872	0.520970	0.260485	+	0.965478i	$0.416117\pi$
$264$	1.01963	0.0627539
$265$	−17.5642	−1.07896
$266$	22.0629	1.35276
$267$	−1.92148	−0.117593
$268$	12.1088	0.739665
$269$	25.0892	1.52972	0.764858	−	0.644199i	$-0.222809\pi$
$269$	25.0892	1.52972	0.764858	+	0.644199i	$0.222809\pi$
$270$	−5.76897	−0.351088
$271$	10.4487	0.634715	0.317357	−	0.948306i	$-0.397205\pi$
$271$	10.4487	0.634715	0.317357	+	0.948306i	$0.397205\pi$
$272$	5.22436	0.316773
$273$	0	0
$274$	13.7427	0.830225
$275$	−3.32025	−0.200218
$276$	0.231033	0.0139065
$277$	8.62086	0.517977	0.258989	−	0.965880i	$-0.416611\pi$
$277$	8.62086	0.517977	0.258989	+	0.965880i	$0.416611\pi$
$278$	−2.40946	−0.144510
$279$	24.2569	1.45223
$280$	30.6731	1.83307
$281$	1.98037	0.118139	0.0590695	−	0.998254i	$-0.481187\pi$
$281$	1.98037	0.118139	0.0590695	+	0.998254i	$0.481187\pi$
$282$	−2.07625	−0.123639
$283$	2.18510	0.129891	0.0649453	−	0.997889i	$-0.479313\pi$
$283$	2.18510	0.129891	0.0649453	+	0.997889i	$0.479313\pi$
$284$	6.00000	0.356034
$285$	6.10217	0.361461
$286$	0	0
$287$	20.3791	1.20294
$288$	14.4224	0.849849
$289$	10.2939	0.605526
$290$	24.3136	1.42774
$291$	0.950048	0.0556928
$292$	−16.0825	−0.941160
$293$	−6.13515	−0.358419	−0.179210	−	0.983811i	$-0.557354\pi$
$293$	−6.13515	−0.358419	−0.179210	+	0.983811i	$0.557354\pi$
$294$	1.89116	0.110294
$295$	40.6967	2.36946
$296$	−10.5749	−0.614655
$297$	−2.00000	−0.116052
$298$	−2.54461	−0.147405
$299$	0	0
$300$	1.12847	0.0651525
$301$	−4.67975	−0.269737
$302$	8.44872	0.486169
$303$	3.35284	0.192615
$304$	6.22436	0.356992
$305$	11.5946	0.663903
$306$	−15.0696	−0.861471
$307$	−6.18510	−0.353002	−0.176501	−	0.984300i	$-0.556478\pi$
$307$	−6.18510	−0.353002	−0.176501	+	0.984300i	$0.556478\pi$
$308$	3.54461	0.201973
$309$	−1.82157	−0.103626
$310$	24.2569	1.37770
$311$	−1.28992	−0.0731449	−0.0365725	−	0.999331i	$-0.511644\pi$
$311$	−1.28992	−0.0731449	−0.0365725	+	0.999331i	$0.511644\pi$
$312$	0	0
$313$	−0.293944	−0.0166147	−0.00830734	−	0.999965i	$-0.502644\pi$
$313$	−0.293944	−0.0166147	−0.00830734	+	0.999965i	$0.502644\pi$
$314$	16.3595	0.923220
$315$	−29.4920	−1.66169
$316$	−0.904114	−0.0508604
$317$	22.0522	1.23858	0.619288	−	0.785164i	$-0.287421\pi$
$317$	22.0522	1.23858	0.619288	+	0.785164i	$0.287421\pi$
$318$	−2.06958	−0.116056
$319$	8.42909	0.471938
$320$	20.1914	1.12873
$321$	−1.12621	−0.0628588
$322$	−2.40946	−0.134274
$323$	32.5183	1.80937
$324$	−7.97370	−0.442983
$325$	0	0
$326$	−10.7886	−0.597525
$327$	0.732359	0.0404996
$328$	17.2480	0.952362
$329$	−21.6535	−1.19379
$330$	−0.980369	−0.0539676
$331$	14.4487	0.794174	0.397087	−	0.917781i	$-0.370021\pi$
$331$	14.4487	0.794174	0.397087	+	0.917781i	$0.370021\pi$
$332$	4.90411	0.269148
$333$	10.1677	0.557189
$334$	7.04995	0.385756
$335$	−34.9278	−1.90831
$336$	1.20473	0.0657234
$337$	−18.5053	−1.00805	−0.504025	−	0.863689i	$-0.668148\pi$
$337$	−18.5053	−1.00805	−0.504025	+	0.863689i	$0.668148\pi$
$338$	0	0
$339$	−1.28992	−0.0700591
$340$	15.0696	0.817263
$341$	8.40946	0.455398
$342$	−17.9541	−0.970845
$343$	−5.08921	−0.274792
$344$	−3.96074	−0.213549
$345$	−0.666410	−0.0358783
$346$	1.84522	0.0991998
$347$	−15.5446	−0.834478	−0.417239	−	0.908797i	$-0.637002\pi$
$347$	−15.5446	−0.834478	−0.417239	+	0.908797i	$0.637002\pi$
$348$	−2.86485	−0.153572
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−11.7690	−0.629078
$351$	0	0
$352$	5.00000	0.266501
$353$	4.32025	0.229944	0.114972	−	0.993369i	$-0.463322\pi$
$353$	4.32025	0.229944	0.114972	+	0.993369i	$0.463322\pi$
$354$	4.79527	0.254866
$355$	−17.3069	−0.918555
$356$	−5.65345	−0.299632
$357$	6.29394	0.333111
$358$	−5.42909	−0.286936
$359$	0.231033	0.0121934	0.00609672	−	0.999981i	$-0.498059\pi$
$359$	0.231033	0.0121934	0.00609672	+	0.999981i	$0.498059\pi$
$360$	−24.9607	−1.31555
$361$	19.7427	1.03909
$362$	4.47502	0.235202
$363$	−0.339877	−0.0178389
$364$	0	0
$365$	46.3898	2.42815
$366$	1.36618	0.0714113
$367$	19.8082	1.03398	0.516991	−	0.855991i	$-0.327052\pi$
$367$	19.8082	1.03398	0.516991	+	0.855991i	$0.327052\pi$
$368$	−0.679754	−0.0354346
$369$	−16.5839	−0.863322
$370$	10.1677	0.528595
$371$	−21.5839	−1.12058
$372$	−2.85818	−0.148190
$373$	−14.9148	−0.772259	−0.386130	−	0.922445i	$-0.626188\pi$
$373$	−14.9148	−0.772259	−0.386130	+	0.922445i	$0.626188\pi$
$374$	−5.22436	−0.270145
$375$	1.64678	0.0850393
$376$	−18.3265	−0.945119
$377$	0	0
$378$	−7.08921	−0.364630
$379$	−6.64049	−0.341099	−0.170550	−	0.985349i	$-0.554554\pi$
$379$	−6.64049	−0.341099	−0.170550	+	0.985349i	$0.554554\pi$
$380$	17.9541	0.921024
$381$	5.49867	0.281706
$382$	−9.89116	−0.506076
$383$	5.14182	0.262735	0.131367	−	0.991334i	$-0.458063\pi$
$383$	5.14182	0.262735	0.131367	+	0.991334i	$0.458063\pi$
$384$	−1.01963	−0.0520328
$385$	−10.2244	−0.521082
$386$	5.10884	0.260033
$387$	3.80823	0.193583
$388$	2.79527	0.141908
$389$	−33.2806	−1.68739	−0.843697	−	0.536820i	$-0.819625\pi$
$389$	−33.2806	−1.68739	−0.843697	+	0.536820i	$0.819625\pi$
$390$	0	0
$391$	−3.55128	−0.179596
$392$	16.6927	0.843109
$393$	1.12621	0.0568096
$394$	2.23103	0.112398
$395$	2.60790	0.131218
$396$	−2.88448	−0.144951
$397$	−8.60350	−0.431797	−0.215899	−	0.976416i	$-0.569268\pi$
$397$	−8.60350	−0.431797	−0.215899	+	0.976416i	$0.569268\pi$
$398$	−17.8778	−0.896134
$399$	7.49867	0.375403
$400$	−3.32025	−0.166012
$401$	5.67975	0.283633	0.141817	−	0.989893i	$-0.454706\pi$
$401$	5.67975	0.283633	0.141817	+	0.989893i	$0.454706\pi$
$402$	−4.11552	−0.205263
$403$	0	0
$404$	9.86485	0.490795
$405$	23.0000	1.14288
$406$	29.8778	1.48281
$407$	3.52498	0.174727
$408$	5.32692	0.263722
$409$	−22.9933	−1.13695	−0.568473	−	0.822702i	$-0.692466\pi$
$409$	−22.9933	−1.13695	−0.568473	+	0.822702i	$0.692466\pi$
$410$	−16.5839	−0.819019
$411$	4.67081	0.230394
$412$	−5.35951	−0.264044
$413$	50.0103	2.46085
$414$	1.96074	0.0963650
$415$	−14.1458	−0.694392
$416$	0	0
$417$	−0.818920	−0.0401027
$418$	−6.22436	−0.304443
$419$	30.8582	1.50752	0.753760	−	0.657149i	$-0.228238\pi$
$419$	30.8582	1.50752	0.753760	+	0.657149i	$0.228238\pi$
$420$	3.47502	0.169564
$421$	−23.6035	−1.15036	−0.575182	−	0.818025i	$-0.695069\pi$
$421$	−23.6035	−1.15036	−0.575182	+	0.818025i	$0.695069\pi$
$422$	−10.9041	−0.530804
$423$	17.6209	0.856756
$424$	−18.2676	−0.887155
$425$	−17.3462	−0.841413
$426$	−2.03926	−0.0988025
$427$	14.2480	0.689510
$428$	−3.31357	−0.160168
$429$	0	0
$430$	3.80823	0.183649
$431$	−28.6338	−1.37924	−0.689621	−	0.724170i	$-0.742223\pi$
$431$	−28.6338	−1.37924	−0.689621	+	0.724170i	$0.742223\pi$
$432$	−2.00000	−0.0962250
$433$	−28.6535	−1.37700	−0.688498	−	0.725238i	$-0.741730\pi$
$433$	−28.6535	−1.37700	−0.688498	+	0.725238i	$0.741730\pi$
$434$	29.8082	1.43084
$435$	8.26362	0.396210
$436$	2.15478	0.103195
$437$	−4.23103	−0.202398
$438$	5.46608	0.261180
$439$	−9.09589	−0.434123	−0.217061	−	0.976158i	$-0.569647\pi$
$439$	−9.09589	−0.434123	−0.217061	+	0.976158i	$0.569647\pi$
$440$	−8.65345	−0.412537
$441$	−16.0500	−0.764283
$442$	0	0
$443$	0.518304	0.0246254	0.0123127	−	0.999924i	$-0.496081\pi$
$443$	0.518304	0.0246254	0.0123127	+	0.999924i	$0.496081\pi$
$444$	−1.19806	−0.0568573
$445$	16.3073	0.773040
$446$	−19.4291	−0.919994
$447$	−0.864853	−0.0409061
$448$	24.8122	1.17227
$449$	33.0522	1.55983	0.779915	−	0.625885i	$-0.215262\pi$
$449$	33.0522	1.55983	0.779915	+	0.625885i	$0.215262\pi$
$450$	9.57720	0.451473
$451$	−5.74934	−0.270726
$452$	−3.79527	−0.178514
$453$	2.87153	0.134916
$454$	−7.32025	−0.343556
$455$	0	0
$456$	6.34655	0.297204
$457$	27.3265	1.27828	0.639141	−	0.769090i	$-0.279290\pi$
$457$	27.3265	1.27828	0.639141	+	0.769090i	$0.279290\pi$
$458$	14.5642	0.680542
$459$	−10.4487	−0.487705
$460$	−1.96074	−0.0914199
$461$	−13.2873	−0.618850	−0.309425	−	0.950924i	$-0.600137\pi$
$461$	−13.2873	−0.618850	−0.309425	+	0.950924i	$0.600137\pi$
$462$	−1.20473	−0.0560491
$463$	−18.2480	−0.848057	−0.424028	−	0.905649i	$-0.639384\pi$
$463$	−18.2480	−0.848057	−0.424028	+	0.905649i	$0.639384\pi$
$464$	8.42909	0.391311
$465$	8.24438	0.382324
$466$	−6.44872	−0.298731
$467$	28.2873	1.30898	0.654489	−	0.756071i	$-0.272884\pi$
$467$	28.2873	1.30898	0.654489	+	0.756071i	$0.272884\pi$
$468$	0	0
$469$	−42.9211	−1.98191
$470$	17.6209	0.812790
$471$	5.56022	0.256201
$472$	42.3265	1.94824
$473$	1.32025	0.0607050
$474$	0.307288	0.0141142
$475$	−20.6664	−0.948240
$476$	18.5183	0.848785
$477$	17.5642	0.804211
$478$	−0.270294	−0.0123630
$479$	−15.5446	−0.710251	−0.355126	−	0.934819i	$-0.615562\pi$
$479$	−15.5446	−0.710251	−0.355126	+	0.934819i	$0.615562\pi$
$480$	4.90185	0.223738
$481$	0	0
$482$	30.5313	1.39066
$483$	−0.818920	−0.0372621
$484$	−1.00000	−0.0454545
$485$	−8.06291	−0.366118
$486$	8.71008	0.395097
$487$	−32.3265	−1.46486	−0.732428	−	0.680845i	$-0.761613\pi$
$487$	−32.3265	−1.46486	−0.732428	+	0.680845i	$0.761613\pi$
$488$	12.0589	0.545880
$489$	−3.66680	−0.165818
$490$	−16.0500	−0.725063
$491$	11.5446	0.521001	0.260500	−	0.965474i	$-0.416112\pi$
$491$	11.5446	0.521001	0.260500	+	0.965474i	$0.416112\pi$
$492$	1.95407	0.0880961
$493$	44.0366	1.98331
$494$	0	0
$495$	8.32025	0.373967
$496$	8.40946	0.377596
$497$	−21.2676	−0.953984
$498$	−1.66680	−0.0746909
$499$	27.7859	1.24387	0.621935	−	0.783069i	$-0.286347\pi$
$499$	27.7859	1.24387	0.621935	+	0.783069i	$0.286347\pi$
$500$	4.84522	0.216685
$501$	2.39612	0.107051
$502$	−10.4790	−0.467703
$503$	−5.40544	−0.241017	−0.120508	−	0.992712i	$-0.538452\pi$
$503$	−5.40544	−0.241017	−0.120508	+	0.992712i	$0.538452\pi$
$504$	−30.6731	−1.36629
$505$	−28.4550	−1.26623
$506$	0.679754	0.0302187
$507$	0	0
$508$	16.1784	0.717802
$509$	25.4095	1.12625	0.563127	−	0.826370i	$-0.309598\pi$
$509$	25.4095	1.12625	0.563127	+	0.826370i	$0.309598\pi$
$510$	−5.12180	−0.226797
$511$	57.0063	2.52181
$512$	11.0000	0.486136
$513$	−12.4487	−0.549624
$514$	−5.75601	−0.253887
$515$	15.4594	0.681223
$516$	−0.448721	−0.0197538
$517$	6.10884	0.268667
$518$	12.4947	0.548984
$519$	0.627148	0.0275288
$520$	0	0
$521$	33.1784	1.45357	0.726787	−	0.686863i	$-0.241013\pi$
$521$	33.1784	1.45357	0.726787	+	0.686863i	$0.241013\pi$
$522$	−24.3136	−1.06418
$523$	−18.7730	−0.820885	−0.410443	−	0.911886i	$-0.634626\pi$
$523$	−18.7730	−0.820885	−0.410443	+	0.911886i	$0.634626\pi$
$524$	3.31357	0.144754
$525$	−4.00000	−0.174574
$526$	−8.44872	−0.368382
$527$	43.9341	1.91380
$528$	−0.339877	−0.0147912
$529$	−22.5379	−0.979910
$530$	17.5642	0.762942
$531$	−40.6967	−1.76609
$532$	22.0629	0.956549
$533$	0	0
$534$	1.92148	0.0831505
$535$	9.55795	0.413226
$536$	−36.3265	−1.56907
$537$	−1.84522	−0.0796272
$538$	−25.0892	−1.08167
$539$	−5.56424	−0.239669
$540$	−5.76897	−0.248257
$541$	−42.8016	−1.84018	−0.920091	−	0.391704i	$-0.871886\pi$
$541$	−42.8016	−1.84018	−0.920091	+	0.391704i	$0.871886\pi$
$542$	−10.4487	−0.448811
$543$	1.52096	0.0652705
$544$	26.1218	1.11996
$545$	−6.21542	−0.266239
$546$	0	0
$547$	−33.7556	−1.44329	−0.721643	−	0.692265i	$-0.756613\pi$
$547$	−33.7556	−1.44329	−0.721643	+	0.692265i	$0.756613\pi$
$548$	13.7427	0.587058
$549$	−11.5946	−0.494844
$550$	3.32025	0.141576
$551$	52.4657	2.23511
$552$	−0.693098	−0.0295002
$553$	3.20473	0.136279
$554$	−8.62086	−0.366265
$555$	3.45578	0.146690
$556$	−2.40946	−0.102184
$557$	1.78860	0.0757853	0.0378927	−	0.999282i	$-0.487936\pi$
$557$	1.78860	0.0757853	0.0378927	+	0.999282i	$0.487936\pi$
$558$	−24.2569	−1.02688
$559$	0	0
$560$	−10.2244	−0.432058
$561$	−1.77564	−0.0749676
$562$	−1.98037	−0.0835368
$563$	−7.68377	−0.323832	−0.161916	−	0.986805i	$-0.551767\pi$
$563$	−7.68377	−0.323832	−0.161916	+	0.986805i	$0.551767\pi$
$564$	−2.07625	−0.0874261
$565$	10.9474	0.460560
$566$	−2.18510	−0.0918466
$567$	28.2636	1.18696
$568$	−18.0000	−0.755263
$569$	−26.1258	−1.09525	−0.547626	−	0.836723i	$-0.684468\pi$
$569$	−26.1258	−1.09525	−0.547626	+	0.836723i	$0.684468\pi$
$570$	−6.10217	−0.255592
$571$	−11.6624	−0.488056	−0.244028	−	0.969768i	$-0.578469\pi$
$571$	−11.6624	−0.488056	−0.244028	+	0.969768i	$0.578469\pi$
$572$	0	0
$573$	−3.36178	−0.140440
$574$	−20.3791	−0.850609
$575$	2.25695	0.0941213
$576$	−20.1914	−0.841308
$577$	−28.6008	−1.19067	−0.595334	−	0.803478i	$-0.702980\pi$
$577$	−28.6008	−1.19067	−0.595334	+	0.803478i	$0.702980\pi$
$578$	−10.2939	−0.428172
$579$	1.73638	0.0721614
$580$	24.3136	1.00957
$581$	−17.3832	−0.721175
$582$	−0.950048	−0.0393807
$583$	6.08921	0.252189
$584$	48.2476	1.99650
$585$	0	0
$586$	6.13515	0.253441
$587$	35.1455	1.45061	0.725304	−	0.688429i	$-0.241699\pi$
$587$	35.1455	1.45061	0.725304	+	0.688429i	$0.241699\pi$
$588$	1.89116	0.0779899
$589$	52.3435	2.15678
$590$	−40.6967	−1.67546
$591$	0.758276	0.0311913
$592$	3.52498	0.144876
$593$	25.2087	1.03520	0.517600	−	0.855623i	$-0.326826\pi$
$593$	25.2087	1.03520	0.517600	+	0.855623i	$0.326826\pi$
$594$	2.00000	0.0820610
$595$	−53.4157	−2.18983
$596$	−2.54461	−0.104231
$597$	−6.07625	−0.248685
$598$	0	0
$599$	26.8582	1.09740	0.548698	−	0.836021i	$-0.315124\pi$
$599$	26.8582	1.09740	0.548698	+	0.836021i	$0.315124\pi$
$600$	−3.38542	−0.138209
$601$	32.5683	1.32849	0.664243	−	0.747516i	$-0.268754\pi$
$601$	32.5683	1.32849	0.664243	+	0.747516i	$0.268754\pi$
$602$	4.67975	0.190733
$603$	34.9278	1.42237
$604$	8.44872	0.343774
$605$	2.88448	0.117271
$606$	−3.35284	−0.136200
$607$	40.7730	1.65492	0.827462	−	0.561521i	$-0.189784\pi$
$607$	40.7730	1.65492	0.827462	+	0.561521i	$0.189784\pi$
$608$	31.1218	1.26216
$609$	10.1548	0.411492
$610$	−11.5946	−0.469450
$611$	0	0
$612$	−15.0696	−0.609152
$613$	23.3635	0.943644	0.471822	−	0.881694i	$-0.343597\pi$
$613$	23.3635	0.943644	0.471822	+	0.881694i	$0.343597\pi$
$614$	6.18510	0.249610
$615$	−5.63647	−0.227285
$616$	−10.6338	−0.428449
$617$	19.4095	0.781395	0.390698	−	0.920519i	$-0.372234\pi$
$617$	19.4095	0.781395	0.390698	+	0.920519i	$0.372234\pi$
$618$	1.82157	0.0732744
$619$	0.941108	0.0378263	0.0189132	−	0.999821i	$-0.493979\pi$
$619$	0.941108	0.0378263	0.0189132	+	0.999821i	$0.493979\pi$
$620$	24.2569	0.974183
$621$	1.35951	0.0545552
$622$	1.28992	0.0517213
$623$	20.0393	0.802856
$624$	0	0
$625$	−30.5772	−1.22309
$626$	0.293944	0.0117483
$627$	−2.11552	−0.0844856
$628$	16.3595	0.652815
$629$	18.4157	0.734284
$630$	29.4920	1.17499
$631$	21.4684	0.854642	0.427321	−	0.904100i	$-0.359457\pi$
$631$	21.4684	0.854642	0.427321	+	0.904100i	$0.359457\pi$
$632$	2.71234	0.107891
$633$	−3.70606	−0.147303
$634$	−22.0522	−0.875806
$635$	−46.6664	−1.85190
$636$	−2.06958	−0.0820643
$637$	0	0
$638$	−8.42909	−0.333711
$639$	17.3069	0.684650
$640$	8.65345	0.342058
$641$	−24.5379	−0.969190	−0.484595	−	0.874739i	$-0.661033\pi$
$641$	−24.5379	−0.969190	−0.484595	+	0.874739i	$0.661033\pi$
$642$	1.12621	0.0444479
$643$	16.7493	0.660529	0.330265	−	0.943888i	$-0.392862\pi$
$643$	16.7493	0.660529	0.330265	+	0.943888i	$0.392862\pi$
$644$	−2.40946	−0.0949460
$645$	1.29433	0.0509642
$646$	−32.5183	−1.27942
$647$	35.5986	1.39952	0.699762	−	0.714376i	$-0.253289\pi$
$647$	35.5986	1.39952	0.699762	+	0.714376i	$0.253289\pi$
$648$	23.9211	0.939709
$649$	−14.1088	−0.553821
$650$	0	0
$651$	10.1311	0.397070
$652$	−10.7886	−0.422514
$653$	−7.02630	−0.274960	−0.137480	−	0.990505i	$-0.543900\pi$
$653$	−7.02630	−0.274960	−0.137480	+	0.990505i	$0.543900\pi$
$654$	−0.732359	−0.0286375
$655$	−9.55795	−0.373460
$656$	−5.74934	−0.224474
$657$	−46.3898	−1.80984
$658$	21.6535	0.844139
$659$	42.8908	1.67079	0.835394	−	0.549652i	$-0.185240\pi$
$659$	42.8908	1.67079	0.835394	+	0.549652i	$0.185240\pi$
$660$	−0.980369	−0.0381608
$661$	−36.6535	−1.42565	−0.712827	−	0.701340i	$-0.752586\pi$
$661$	−36.6535	−1.42565	−0.712827	+	0.701340i	$0.752586\pi$
$662$	−14.4487	−0.561565
$663$	0	0
$664$	−14.7123	−0.570950
$665$	−63.6401	−2.46786
$666$	−10.1677	−0.393992
$667$	−5.72971	−0.221855
$668$	7.04995	0.272771
$669$	−6.60350	−0.255306
$670$	34.9278	1.34938
$671$	−4.01963	−0.155176
$672$	6.02365	0.232367
$673$	−2.73865	−0.105567	−0.0527835	−	0.998606i	$-0.516809\pi$
$673$	−2.73865	−0.105567	−0.0527835	+	0.998606i	$0.516809\pi$
$674$	18.5053	0.712799
$675$	6.64049	0.255593
$676$	0	0
$677$	17.3069	0.665158	0.332579	−	0.943075i	$-0.392081\pi$
$677$	17.3069	0.665158	0.332579	+	0.943075i	$0.392081\pi$
$678$	1.28992	0.0495393
$679$	−9.90813	−0.380239
$680$	−45.2087	−1.73368
$681$	−2.48798	−0.0953397
$682$	−8.40946	−0.322015
$683$	−46.7271	−1.78796	−0.893980	−	0.448106i	$-0.852099\pi$
$683$	−46.7271	−1.78796	−0.893980	+	0.448106i	$0.852099\pi$
$684$	−17.9541	−0.686491
$685$	−39.6405	−1.51459
$686$	5.08921	0.194307
$687$	4.95005	0.188856
$688$	1.32025	0.0503339
$689$	0	0
$690$	0.666410	0.0253698
$691$	11.9474	0.454500	0.227250	−	0.973836i	$-0.427026\pi$
$691$	11.9474	0.454500	0.227250	+	0.973836i	$0.427026\pi$
$692$	1.84522	0.0701448
$693$	10.2244	0.388391
$694$	15.5446	0.590065
$695$	6.95005	0.263630
$696$	8.59456	0.325776
$697$	−30.0366	−1.13772
$698$	30.0000	1.13552
$699$	−2.19177	−0.0829004
$700$	−11.7690	−0.444825
$701$	−19.2833	−0.728318	−0.364159	−	0.931337i	$-0.618644\pi$
$701$	−19.2833	−0.728318	−0.364159	+	0.931337i	$0.618644\pi$
$702$	0	0
$703$	21.9407	0.827510
$704$	−7.00000	−0.263822
$705$	5.98892	0.225556
$706$	−4.32025	−0.162595
$707$	−34.9670	−1.31507
$708$	4.79527	0.180217
$709$	41.7057	1.56629	0.783145	−	0.621840i	$-0.213614\pi$
$709$	41.7057	1.56629	0.783145	+	0.621840i	$0.213614\pi$
$710$	17.3069	0.649516
$711$	−2.60790	−0.0978040
$712$	16.9604	0.635616
$713$	−5.71636	−0.214079
$714$	−6.29394	−0.235545
$715$	0	0
$716$	−5.42909	−0.202895
$717$	−0.0918667	−0.00343082
$718$	−0.231033	−0.00862206
$719$	−24.6271	−0.918438	−0.459219	−	0.888323i	$-0.651871\pi$
$719$	−24.6271	−0.918438	−0.459219	+	0.888323i	$0.651871\pi$
$720$	8.32025	0.310077
$721$	18.9973	0.707498
$722$	−19.7427	−0.734746
$723$	10.3769	0.385920
$724$	4.47502	0.166313
$725$	−27.9867	−1.03940
$726$	0.339877	0.0126140
$727$	5.82157	0.215910	0.107955	−	0.994156i	$-0.465570\pi$
$727$	5.82157	0.215910	0.107955	+	0.994156i	$0.465570\pi$
$728$	0	0
$729$	−20.9607	−0.776324
$730$	−46.3898	−1.71696
$731$	6.89744	0.255111
$732$	1.36618	0.0504954
$733$	−32.1588	−1.18781	−0.593906	−	0.804534i	$-0.702415\pi$
$733$	−32.1588	−1.18781	−0.593906	+	0.804534i	$0.702415\pi$
$734$	−19.8082	−0.731135
$735$	−5.45501	−0.201211
$736$	−3.39877	−0.125280
$737$	12.1088	0.446035
$738$	16.5839	0.610461
$739$	0.370199	0.0136180	0.00680899	−	0.999977i	$-0.497833\pi$
$739$	0.370199	0.0136180	0.00680899	+	0.999977i	$0.497833\pi$
$740$	10.1677	0.373773
$741$	0	0
$742$	21.5839	0.792369
$743$	−32.0932	−1.17739	−0.588693	−	0.808357i	$-0.700357\pi$
$743$	−32.0932	−1.17739	−0.588693	+	0.808357i	$0.700357\pi$
$744$	8.57454	0.314358
$745$	7.33988	0.268912
$746$	14.9148	0.546070
$747$	14.1458	0.517569
$748$	−5.22436	−0.191022
$749$	11.7453	0.429165
$750$	−1.64678	−0.0601319
$751$	10.7797	0.393355	0.196678	−	0.980468i	$-0.436985\pi$
$751$	10.7797	0.393355	0.196678	+	0.980468i	$0.436985\pi$
$752$	6.10884	0.222767
$753$	−3.56158	−0.129791
$754$	0	0
$755$	−24.3702	−0.886922
$756$	−7.08921	−0.257832
$757$	−51.6794	−1.87832	−0.939159	−	0.343482i	$-0.888394\pi$
$757$	−51.6794	−1.87832	−0.939159	+	0.343482i	$0.888394\pi$
$758$	6.64049	0.241194
$759$	0.231033	0.00838595
$760$	−53.8622	−1.95379
$761$	38.2783	1.38759	0.693794	−	0.720173i	$-0.255938\pi$
$761$	38.2783	1.38759	0.693794	+	0.720173i	$0.255938\pi$
$762$	−5.49867	−0.199196
$763$	−7.63784	−0.276508
$764$	−9.89116	−0.357849
$765$	43.4680	1.57159
$766$	−5.14182	−0.185781
$767$	0	0
$768$	5.77791	0.208492
$769$	−46.5745	−1.67952	−0.839760	−	0.542957i	$-0.817305\pi$
$769$	−46.5745	−1.67952	−0.839760	+	0.542957i	$0.817305\pi$
$770$	10.2244	0.368460
$771$	−1.95633	−0.0704557
$772$	5.10884	0.183871
$773$	−22.7819	−0.819409	−0.409704	−	0.912218i	$-0.634368\pi$
$773$	−22.7819	−0.819409	−0.409704	+	0.912218i	$0.634368\pi$
$774$	−3.80823	−0.136884
$775$	−27.9215	−1.00297
$776$	−8.38581	−0.301033
$777$	4.24664	0.152348
$778$	33.2806	1.19317
$779$	−35.7859	−1.28216
$780$	0	0
$781$	6.00000	0.214697
$782$	3.55128	0.126993
$783$	−16.8582	−0.602462
$784$	−5.56424	−0.198723
$785$	−47.1887	−1.68424
$786$	−1.12621	−0.0401705
$787$	−4.46874	−0.159293	−0.0796466	−	0.996823i	$-0.525379\pi$
$787$	−4.46874	−0.159293	−0.0796466	+	0.996823i	$0.525379\pi$
$788$	2.23103	0.0794772
$789$	−2.87153	−0.102229
$790$	−2.60790	−0.0927850
$791$	13.4527	0.478324
$792$	8.65345	0.307487
$793$	0	0
$794$	8.60350	0.305327
$795$	5.96968	0.211723
$796$	−17.8778	−0.633662
$797$	31.2306	1.10625	0.553123	−	0.833100i	$-0.313436\pi$
$797$	31.2306	1.10625	0.553123	+	0.833100i	$0.313436\pi$
$798$	−7.49867	−0.265450
$799$	31.9148	1.12906
$800$	−16.6012	−0.586942
$801$	−16.3073	−0.576190
$802$	−5.67975	−0.200559
$803$	−16.0825	−0.567541
$804$	−4.11552	−0.145143
$805$	6.95005	0.244957
$806$	0	0
$807$	−8.52724	−0.300173
$808$	−29.5946	−1.04113
$809$	−11.4161	−0.401370	−0.200685	−	0.979656i	$-0.564317\pi$
$809$	−11.4161	−0.401370	−0.200685	+	0.979656i	$0.564317\pi$
$810$	−23.0000	−0.808138
$811$	−2.13917	−0.0751163	−0.0375581	−	0.999294i	$-0.511958\pi$
$811$	−2.13917	−0.0751163	−0.0375581	+	0.999294i	$0.511958\pi$
$812$	29.8778	1.04851
$813$	−3.55128	−0.124549
$814$	−3.52498	−0.123550
$815$	31.1195	1.09007
$816$	−1.77564	−0.0621598
$817$	8.21769	0.287501
$818$	22.9933	0.803943
$819$	0	0
$820$	−16.5839	−0.579134
$821$	−25.9867	−0.906941	−0.453470	−	0.891271i	$-0.649814\pi$
$821$	−25.9867	−0.906941	−0.453470	+	0.891271i	$0.649814\pi$
$822$	−4.67081	−0.162913
$823$	−25.0759	−0.874090	−0.437045	−	0.899440i	$-0.643975\pi$
$823$	−25.0759	−0.874090	−0.437045	+	0.899440i	$0.643975\pi$
$824$	16.0785	0.560122
$825$	1.12847	0.0392885
$826$	−50.0103	−1.74008
$827$	43.0692	1.49766	0.748831	−	0.662761i	$-0.230615\pi$
$827$	43.0692	1.49766	0.748831	+	0.662761i	$0.230615\pi$
$828$	1.96074	0.0681404
$829$	−2.43576	−0.0845975	−0.0422988	−	0.999105i	$-0.513468\pi$
$829$	−2.43576	−0.0845975	−0.0422988	+	0.999105i	$0.513468\pi$
$830$	14.1458	0.491009
$831$	−2.93003	−0.101642
$832$	0	0
$833$	−29.0696	−1.00720
$834$	0.818920	0.0283569
$835$	−20.3355	−0.703738
$836$	−6.22436	−0.215274
$837$	−16.8189	−0.581347
$838$	−30.8582	−1.06598
$839$	−46.2347	−1.59620	−0.798099	−	0.602526i	$-0.794161\pi$
$839$	−46.2347	−1.59620	−0.798099	+	0.602526i	$0.794161\pi$
$840$	−10.4251	−0.359699
$841$	42.0496	1.44999
$842$	23.6035	0.813430
$843$	−0.673082	−0.0231822
$844$	−10.9041	−0.375335
$845$	0	0
$846$	−17.6209	−0.605818
$847$	3.54461	0.121794
$848$	6.08921	0.209104
$849$	−0.742665	−0.0254882
$850$	17.3462	0.594969
$851$	−2.39612	−0.0821378
$852$	−2.03926	−0.0698639
$853$	19.0562	0.652473	0.326237	−	0.945288i	$-0.394219\pi$
$853$	19.0562	0.652473	0.326237	+	0.945288i	$0.394219\pi$
$854$	−14.2480	−0.487557
$855$	51.7882	1.77112
$856$	9.94072	0.339767
$857$	−7.86712	−0.268736	−0.134368	−	0.990932i	$-0.542900\pi$
$857$	−7.86712	−0.268736	−0.134368	+	0.990932i	$0.542900\pi$
$858$	0	0
$859$	−0.993713	−0.0339051	−0.0169525	−	0.999856i	$-0.505396\pi$
$859$	−0.993713	−0.0339051	−0.0169525	+	0.999856i	$0.505396\pi$
$860$	3.80823	0.129860
$861$	−6.92640	−0.236051
$862$	28.6338	0.975272
$863$	29.3069	0.997619	0.498809	−	0.866712i	$-0.333771\pi$
$863$	29.3069	0.997619	0.498809	+	0.866712i	$0.333771\pi$
$864$	−10.0000	−0.340207
$865$	−5.32251	−0.180971
$866$	28.6535	0.973684
$867$	−3.49867	−0.118821
$868$	29.8082	1.01176
$869$	−0.904114	−0.0306700
$870$	−8.26362	−0.280163
$871$	0	0
$872$	−6.46433	−0.218910
$873$	8.06291	0.272888
$874$	4.23103	0.143117
$875$	−17.1744	−0.580601
$876$	5.46608	0.184682
$877$	−50.4135	−1.70234	−0.851171	−	0.524888i	$-0.824107\pi$
$877$	−50.4135	−1.70234	−0.851171	+	0.524888i	$0.824107\pi$
$878$	9.09589	0.306971
$879$	2.08519	0.0703319
$880$	2.88448	0.0972359
$881$	−17.5879	−0.592551	−0.296275	−	0.955103i	$-0.595745\pi$
$881$	−17.5879	−0.592551	−0.296275	+	0.955103i	$0.595745\pi$
$882$	16.0500	0.540430
$883$	42.3702	1.42587	0.712935	−	0.701230i	$-0.247365\pi$
$883$	42.3702	1.42587	0.712935	+	0.701230i	$0.247365\pi$
$884$	0	0
$885$	−13.8319	−0.464954
$886$	−0.518304	−0.0174128
$887$	2.35685	0.0791354	0.0395677	−	0.999217i	$-0.487402\pi$
$887$	2.35685	0.0791354	0.0395677	+	0.999217i	$0.487402\pi$
$888$	3.59417	0.120613
$889$	−57.3462	−1.92333
$890$	−16.3073	−0.546622
$891$	−7.97370	−0.267129
$892$	−19.4291	−0.650534
$893$	38.0236	1.27241
$894$	0.864853	0.0289250
$895$	15.6601	0.523460
$896$	10.6338	0.355251
$897$	0	0
$898$	−33.0522	−1.10297
$899$	70.8841	2.36412
$900$	9.57720	0.319240
$901$	31.8122	1.05982
$902$	5.74934	0.191432
$903$	1.59054	0.0529299
$904$	11.3858	0.378686
$905$	−12.9081	−0.429081
$906$	−2.87153	−0.0954000
$907$	−17.5210	−0.581774	−0.290887	−	0.956757i	$-0.593950\pi$
$907$	−17.5210	−0.581774	−0.290887	+	0.956757i	$0.593950\pi$
$908$	−7.32025	−0.242931
$909$	28.4550	0.943793
$910$	0	0
$911$	10.2873	0.340833	0.170416	−	0.985372i	$-0.445489\pi$
$911$	10.2873	0.340833	0.170416	+	0.985372i	$0.445489\pi$
$912$	−2.11552	−0.0700517
$913$	4.90411	0.162302
$914$	−27.3265	−0.903881
$915$	−3.94072	−0.130276
$916$	14.5642	0.481216
$917$	−11.7453	−0.387865
$918$	10.4487	0.344859
$919$	−31.1544	−1.02769	−0.513844	−	0.857883i	$-0.671779\pi$
$919$	−31.1544	−1.02769	−0.513844	+	0.857883i	$0.671779\pi$
$920$	5.88222	0.193931
$921$	2.10217	0.0692689
$922$	13.2873	0.437593
$923$	0	0
$924$	−1.20473	−0.0396327
$925$	−11.7038	−0.384818
$926$	18.2480	0.599667
$927$	−15.4594	−0.507754
$928$	42.1455	1.38349
$929$	−38.4594	−1.26181	−0.630906	−	0.775859i	$-0.717317\pi$
$929$	−38.4594	−1.26181	−0.630906	+	0.775859i	$0.717317\pi$
$930$	−8.24438	−0.270344
$931$	−34.6338	−1.13508
$932$	−6.44872	−0.211235
$933$	0.438416	0.0143531
$934$	−28.2873	−0.925588
$935$	15.0696	0.492828
$936$	0	0
$937$	21.6182	0.706236	0.353118	−	0.935579i	$-0.385121\pi$
$937$	21.6182	0.706236	0.353118	+	0.935579i	$0.385121\pi$
$938$	42.9211	1.40142
$939$	0.0999046	0.00326026
$940$	17.6209	0.574729
$941$	6.44872	0.210222	0.105111	−	0.994460i	$-0.466480\pi$
$941$	6.44872	0.210222	0.105111	+	0.994460i	$0.466480\pi$
$942$	−5.56022	−0.181162
$943$	3.90813	0.127266
$944$	−14.1088	−0.459204
$945$	20.4487	0.665197
$946$	−1.32025	−0.0429249
$947$	−15.5379	−0.504915	−0.252458	−	0.967608i	$-0.581239\pi$
$947$	−15.5379	−0.504915	−0.252458	+	0.967608i	$0.581239\pi$
$948$	0.307288	0.00998024
$949$	0	0
$950$	20.6664	0.670507
$951$	−7.49504	−0.243043
$952$	−55.5549	−1.80055
$953$	−15.8868	−0.514622	−0.257311	−	0.966329i	$-0.582837\pi$
$953$	−15.8868	−0.514622	−0.257311	+	0.966329i	$0.582837\pi$
$954$	−17.5642	−0.568663
$955$	28.5309	0.923238
$956$	−0.270294	−0.00874193
$957$	−2.86485	−0.0926075
$958$	15.5446	0.502223
$959$	−48.7123	−1.57300
$960$	−6.86259	−0.221489
$961$	39.7190	1.28126
$962$	0	0
$963$	−9.55795	−0.308001
$964$	30.5313	0.983346
$965$	−14.7364	−0.474381
$966$	0.818920	0.0263483
$967$	−1.28766	−0.0414083	−0.0207041	−	0.999786i	$-0.506591\pi$
$967$	−1.28766	−0.0414083	−0.0207041	+	0.999786i	$0.506591\pi$
$968$	3.00000	0.0964237
$969$	−11.0522	−0.355048
$970$	8.06291	0.258884
$971$	−19.1195	−0.613575	−0.306788	−	0.951778i	$-0.599254\pi$
$971$	−19.1195	−0.613575	−0.306788	+	0.951778i	$0.599254\pi$
$972$	8.71008	0.279376
$973$	8.54059	0.273799
$974$	32.3265	1.03581
$975$	0	0
$976$	−4.01963	−0.128665
$977$	−27.0103	−0.864136	−0.432068	−	0.901841i	$-0.642216\pi$
$977$	−27.0103	−0.864136	−0.432068	+	0.901841i	$0.642216\pi$
$978$	3.66680	0.117251
$979$	−5.65345	−0.180685
$980$	−16.0500	−0.512697
$981$	6.21542	0.198443
$982$	−11.5446	−0.368403
$983$	9.35951	0.298522	0.149261	−	0.988798i	$-0.452311\pi$
$983$	9.35951	0.298522	0.149261	+	0.988798i	$0.452311\pi$
$984$	−5.86220	−0.186880
$985$	−6.43538	−0.205048
$986$	−44.0366	−1.40241
$987$	7.35951	0.234256
$988$	0	0
$989$	−0.897442	−0.0285370
$990$	−8.32025	−0.264435
$991$	14.8885	0.472949	0.236474	−	0.971638i	$-0.424008\pi$
$991$	14.8885	0.472949	0.236474	+	0.971638i	$0.424008\pi$
$992$	42.0473	1.33500
$993$	−4.91079	−0.155839
$994$	21.2676	0.674569
$995$	51.5683	1.63482
$996$	−1.66680	−0.0528144
$997$	−20.2137	−0.640173	−0.320087	−	0.947388i	$-0.603712\pi$
$997$	−20.2137	−0.640173	−0.320087	+	0.947388i	$0.603712\pi$
$998$	−27.7859	−0.879549
$999$	−7.04995	−0.223051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.2.a.e.1.2		3
13.4	even	6		143.2.e.a.133.2	yes	6
13.10	even	6		143.2.e.a.100.2	✓	6
13.12	even	2		1859.2.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.e.a.100.2	✓	6	13.10	even	6
143.2.e.a.133.2	yes	6	13.4	even	6
1859.2.a.e.1.2		3	1.1	even	1	trivial
1859.2.a.h.1.2		3	13.12	even	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:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.339877 q^{3} -1.00000 q^{4} +2.88448 q^{5} +0.339877 q^{6} +3.54461 q^{7} +3.00000 q^{8} -2.88448 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.339877 q^{3} -1.00000 q^{4} +2.88448 q^{5} +0.339877 q^{6} +3.54461 q^{7} +3.00000 q^{8} -2.88448 q^{9} -2.88448 q^{10} -1.00000 q^{11} +0.339877 q^{12} -3.54461 q^{14} -0.980369 q^{15} -1.00000 q^{16} -5.22436 q^{17} +2.88448 q^{18} -6.22436 q^{19} -2.88448 q^{20} -1.20473 q^{21} +1.00000 q^{22} +0.679754 q^{23} -1.01963 q^{24} +3.32025 q^{25} +2.00000 q^{27} -3.54461 q^{28} -8.42909 q^{29} +0.980369 q^{30} -8.40946 q^{31} -5.00000 q^{32} +0.339877 q^{33} +5.22436 q^{34} +10.2244 q^{35} +2.88448 q^{36} -3.52498 q^{37} +6.22436 q^{38} +8.65345 q^{40} +5.74934 q^{41} +1.20473 q^{42} -1.32025 q^{43} +1.00000 q^{44} -8.32025 q^{45} -0.679754 q^{46} -6.10884 q^{47} +0.339877 q^{48} +5.56424 q^{49} -3.32025 q^{50} +1.77564 q^{51} -6.08921 q^{53} -2.00000 q^{54} -2.88448 q^{55} +10.6338 q^{56} +2.11552 q^{57} +8.42909 q^{58} +14.1088 q^{59} +0.980369 q^{60} +4.01963 q^{61} +8.40946 q^{62} -10.2244 q^{63} +7.00000 q^{64} -0.339877 q^{66} -12.1088 q^{67} +5.22436 q^{68} -0.231033 q^{69} -10.2244 q^{70} -6.00000 q^{71} -8.65345 q^{72} +16.0825 q^{73} +3.52498 q^{74} -1.12847 q^{75} +6.22436 q^{76} -3.54461 q^{77} +0.904114 q^{79} -2.88448 q^{80} +7.97370 q^{81} -5.74934 q^{82} -4.90411 q^{83} +1.20473 q^{84} -15.0696 q^{85} +1.32025 q^{86} +2.86485 q^{87} -3.00000 q^{88} +5.65345 q^{89} +8.32025 q^{90} -0.679754 q^{92} +2.85818 q^{93} +6.10884 q^{94} -17.9541 q^{95} +1.69938 q^{96} -2.79527 q^{97} -5.56424 q^{98} +2.88448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{4} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^4 - 3 * q^5 + 9 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{4} - 3 q^{5} + 9 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} - 6 q^{15} - 3 q^{16} - 3 q^{17} - 3 q^{18} - 6 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{22} + 12 q^{25} + 6 q^{27} - 3 q^{29} + 6 q^{30} - 6 q^{31} - 15 q^{32} + 3 q^{34} + 18 q^{35} - 3 q^{36} - 3 q^{37} + 6 q^{38} - 9 q^{40} - 3 q^{41} - 6 q^{42} - 6 q^{43} + 3 q^{44} - 27 q^{45} + 6 q^{47} + 3 q^{49} - 12 q^{50} + 18 q^{51} + 3 q^{53} - 6 q^{54} + 3 q^{55} + 18 q^{57} + 3 q^{58} + 18 q^{59} + 6 q^{60} + 9 q^{61} + 6 q^{62} - 18 q^{63} + 21 q^{64} - 12 q^{67} + 3 q^{68} - 24 q^{69} - 18 q^{70} - 18 q^{71} + 9 q^{72} - 9 q^{73} + 3 q^{74} + 24 q^{75} + 6 q^{76} - 12 q^{79} + 3 q^{80} - 9 q^{81} + 3 q^{82} - 6 q^{84} - 27 q^{85} + 6 q^{86} - 9 q^{88} - 18 q^{89} + 27 q^{90} - 36 q^{93} - 6 q^{94} - 24 q^{95} - 18 q^{97} - 3 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^4 - 3 * q^5 + 9 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 - 6 * q^15 - 3 * q^16 - 3 * q^17 - 3 * q^18 - 6 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^22 + 12 * q^25 + 6 * q^27 - 3 * q^29 + 6 * q^30 - 6 * q^31 - 15 * q^32 + 3 * q^34 + 18 * q^35 - 3 * q^36 - 3 * q^37 + 6 * q^38 - 9 * q^40 - 3 * q^41 - 6 * q^42 - 6 * q^43 + 3 * q^44 - 27 * q^45 + 6 * q^47 + 3 * q^49 - 12 * q^50 + 18 * q^51 + 3 * q^53 - 6 * q^54 + 3 * q^55 + 18 * q^57 + 3 * q^58 + 18 * q^59 + 6 * q^60 + 9 * q^61 + 6 * q^62 - 18 * q^63 + 21 * q^64 - 12 * q^67 + 3 * q^68 - 24 * q^69 - 18 * q^70 - 18 * q^71 + 9 * q^72 - 9 * q^73 + 3 * q^74 + 24 * q^75 + 6 * q^76 - 12 * q^79 + 3 * q^80 - 9 * q^81 + 3 * q^82 - 6 * q^84 - 27 * q^85 + 6 * q^86 - 9 * q^88 - 18 * q^89 + 27 * q^90 - 36 * q^93 - 6 * q^94 - 24 * q^95 - 18 * q^97 - 3 * q^98 - 3 * q^99

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