LMFDB - Embedded newform 8085.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.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:	$64.5590500342$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} -1.00000 q^{5} -1.90321 q^{6} -0.719004 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} -1.00000 q^{5} -1.90321 q^{6} -0.719004 q^{8} +1.00000 q^{9} -1.90321 q^{10} +1.00000 q^{11} -1.62222 q^{12} +0.622216 q^{13} +1.00000 q^{15} -4.61285 q^{16} +5.18421 q^{17} +1.90321 q^{18} -7.05086 q^{19} -1.62222 q^{20} +1.90321 q^{22} +8.85728 q^{23} +0.719004 q^{24} +1.00000 q^{25} +1.18421 q^{26} -1.00000 q^{27} -7.80642 q^{29} +1.90321 q^{30} -2.75557 q^{31} -7.34122 q^{32} -1.00000 q^{33} +9.86665 q^{34} +1.62222 q^{36} -2.00000 q^{37} -13.4193 q^{38} -0.622216 q^{39} +0.719004 q^{40} +0.193576 q^{41} +5.67307 q^{43} +1.62222 q^{44} -1.00000 q^{45} +16.8573 q^{46} +2.75557 q^{47} +4.61285 q^{48} +1.90321 q^{50} -5.18421 q^{51} +1.00937 q^{52} -10.8573 q^{53} -1.90321 q^{54} -1.00000 q^{55} +7.05086 q^{57} -14.8573 q^{58} +4.85728 q^{59} +1.62222 q^{60} -6.85728 q^{61} -5.24443 q^{62} -4.74620 q^{64} -0.622216 q^{65} -1.90321 q^{66} -1.24443 q^{67} +8.40990 q^{68} -8.85728 q^{69} +2.75557 q^{71} -0.719004 q^{72} -4.23506 q^{73} -3.80642 q^{74} -1.00000 q^{75} -11.4380 q^{76} -1.18421 q^{78} +8.56199 q^{79} +4.61285 q^{80} +1.00000 q^{81} +0.368416 q^{82} -0.133353 q^{83} -5.18421 q^{85} +10.7971 q^{86} +7.80642 q^{87} -0.719004 q^{88} -5.61285 q^{89} -1.90321 q^{90} +14.3684 q^{92} +2.75557 q^{93} +5.24443 q^{94} +7.05086 q^{95} +7.34122 q^{96} -7.24443 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9	$ 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9 + q^10 + 3 * q^11 - 5 * q^12 + 2 * q^13 + 3 * q^15 + 13 * q^16 + 2 * q^17 - q^18 - 8 * q^19 - 5 * q^20 - q^22 + 9 * q^24 + 3 * q^25 - 10 * q^26 - 3 * q^27 - 10 * q^29 - q^30 - 8 * q^31 - 29 * q^32 - 3 * q^33 + 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 + 9 * q^40 + 14 * q^41 + 4 * q^43 + 5 * q^44 - 3 * q^45 + 24 * q^46 + 8 * q^47 - 13 * q^48 - q^50 - 2 * q^51 + 30 * q^52 - 6 * q^53 + q^54 - 3 * q^55 + 8 * q^57 - 18 * q^58 - 12 * q^59 + 5 * q^60 + 6 * q^61 - 16 * q^62 + 13 * q^64 - 2 * q^65 + q^66 - 4 * q^67 - 42 * q^68 + 8 * q^71 - 9 * q^72 + 14 * q^73 + 2 * q^74 - 3 * q^75 - 48 * q^76 + 10 * q^78 + 12 * q^79 - 13 * q^80 + 3 * q^81 - 26 * q^82 - 2 * q^85 - 8 * q^86 + 10 * q^87 - 9 * q^88 + 10 * q^89 + q^90 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + 29 * q^96 - 22 * q^97 + 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 $

$2$	1.90321	1.34577	0.672887	−	0.739745i	$-0.265054\pi$
$2$	1.90321	1.34577	0.672887	+	0.739745i	$0.265054\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.62222	0.811108
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.90321	−0.776983
$7$	0	0
$7$	0	0
$8$	−0.719004	−0.254206
$9$	1.00000	0.333333
$10$	−1.90321	−0.601848
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.62222	−0.468293
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	−4.61285	−1.15321
$17$	5.18421	1.25736	0.628678	−	0.777666i	$-0.283597\pi$
$17$	5.18421	1.25736	0.628678	+	0.777666i	$0.283597\pi$
$18$	1.90321	0.448591
$19$	−7.05086	−1.61758	−0.808789	−	0.588100i	$-0.799876\pi$
$19$	−7.05086	−1.61758	−0.808789	+	0.588100i	$0.799876\pi$
$20$	−1.62222	−0.362738
$21$	0	0
$22$	1.90321	0.405766
$23$	8.85728	1.84687	0.923435	−	0.383754i	$-0.125369\pi$
$23$	8.85728	1.84687	0.923435	+	0.383754i	$0.125369\pi$
$24$	0.719004	0.146766
$25$	1.00000	0.200000
$26$	1.18421	0.232242
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.80642	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$29$	−7.80642	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$30$	1.90321	0.347477
$31$	−2.75557	−0.494915	−0.247457	−	0.968899i	$-0.579595\pi$
$31$	−2.75557	−0.494915	−0.247457	+	0.968899i	$0.579595\pi$
$32$	−7.34122	−1.29776
$33$	−1.00000	−0.174078
$34$	9.86665	1.69212
$35$	0	0
$36$	1.62222	0.270369
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−13.4193	−2.17689
$39$	−0.622216	−0.0996342
$40$	0.719004	0.113684
$41$	0.193576	0.0302315	0.0151158	−	0.999886i	$-0.495188\pi$
$41$	0.193576	0.0302315	0.0151158	+	0.999886i	$0.495188\pi$
$42$	0	0
$43$	5.67307	0.865135	0.432568	−	0.901602i	$-0.357608\pi$
$43$	5.67307	0.865135	0.432568	+	0.901602i	$0.357608\pi$
$44$	1.62222	0.244558
$45$	−1.00000	−0.149071
$46$	16.8573	2.48547
$47$	2.75557	0.401941	0.200971	−	0.979597i	$-0.435590\pi$
$47$	2.75557	0.401941	0.200971	+	0.979597i	$0.435590\pi$
$48$	4.61285	0.665807
$49$	0	0
$50$	1.90321	0.269155
$51$	−5.18421	−0.725934
$52$	1.00937	0.139974
$53$	−10.8573	−1.49136	−0.745681	−	0.666303i	$-0.767876\pi$
$53$	−10.8573	−1.49136	−0.745681	+	0.666303i	$0.767876\pi$
$54$	−1.90321	−0.258994
$55$	−1.00000	−0.134840
$56$	0	0
$57$	7.05086	0.933909
$58$	−14.8573	−1.95086
$59$	4.85728	0.632364	0.316182	−	0.948699i	$-0.397599\pi$
$59$	4.85728	0.632364	0.316182	+	0.948699i	$0.397599\pi$
$60$	1.62222	0.209427
$61$	−6.85728	−0.877985	−0.438992	−	0.898491i	$-0.644664\pi$
$61$	−6.85728	−0.877985	−0.438992	+	0.898491i	$0.644664\pi$
$62$	−5.24443	−0.666043
$63$	0	0
$64$	−4.74620	−0.593275
$65$	−0.622216	−0.0771764
$66$	−1.90321	−0.234269
$67$	−1.24443	−0.152031	−0.0760157	−	0.997107i	$-0.524220\pi$
$67$	−1.24443	−0.152031	−0.0760157	+	0.997107i	$0.524220\pi$
$68$	8.40990	1.01985
$69$	−8.85728	−1.06629
$70$	0	0
$71$	2.75557	0.327026	0.163513	−	0.986541i	$-0.447717\pi$
$71$	2.75557	0.327026	0.163513	+	0.986541i	$0.447717\pi$
$72$	−0.719004	−0.0847354
$73$	−4.23506	−0.495677	−0.247838	−	0.968801i	$-0.579720\pi$
$73$	−4.23506	−0.495677	−0.247838	+	0.968801i	$0.579720\pi$
$74$	−3.80642	−0.442488
$75$	−1.00000	−0.115470
$76$	−11.4380	−1.31203
$77$	0	0
$78$	−1.18421	−0.134085
$79$	8.56199	0.963299	0.481650	−	0.876364i	$-0.340038\pi$
$79$	8.56199	0.963299	0.481650	+	0.876364i	$0.340038\pi$
$80$	4.61285	0.515732
$81$	1.00000	0.111111
$82$	0.368416	0.0406848
$83$	−0.133353	−0.0146374	−0.00731870	−	0.999973i	$-0.502330\pi$
$83$	−0.133353	−0.0146374	−0.00731870	+	0.999973i	$0.502330\pi$
$84$	0	0
$85$	−5.18421	−0.562306
$86$	10.7971	1.16428
$87$	7.80642	0.836936
$88$	−0.719004	−0.0766461
$89$	−5.61285	−0.594961	−0.297480	−	0.954728i	$-0.596146\pi$
$89$	−5.61285	−0.594961	−0.297480	+	0.954728i	$0.596146\pi$
$90$	−1.90321	−0.200616
$91$	0	0
$92$	14.3684	1.49801
$93$	2.75557	0.285739
$94$	5.24443	0.540922
$95$	7.05086	0.723402
$96$	7.34122	0.749260
$97$	−7.24443	−0.735561	−0.367780	−	0.929913i	$-0.619882\pi$
$97$	−7.24443	−0.735561	−0.367780	+	0.929913i	$0.619882\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	1.62222	0.162222
$101$	−4.66370	−0.464056	−0.232028	−	0.972709i	$-0.574536\pi$
$101$	−4.66370	−0.464056	−0.232028	+	0.972709i	$0.574536\pi$
$102$	−9.86665	−0.976944
$103$	11.6128	1.14425	0.572124	−	0.820167i	$-0.306120\pi$
$103$	11.6128	1.14425	0.572124	+	0.820167i	$0.306120\pi$
$104$	−0.447375	−0.0438688
$105$	0	0
$106$	−20.6637	−2.00704
$107$	2.62222	0.253499	0.126750	−	0.991935i	$-0.459546\pi$
$107$	2.62222	0.253499	0.126750	+	0.991935i	$0.459546\pi$
$108$	−1.62222	−0.156098
$109$	−19.7146	−1.88831	−0.944156	−	0.329499i	$-0.893120\pi$
$109$	−19.7146	−1.88831	−0.944156	+	0.329499i	$0.893120\pi$
$110$	−1.90321	−0.181464
$111$	2.00000	0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	13.4193	1.25683
$115$	−8.85728	−0.825946
$116$	−12.6637	−1.17580
$117$	0.622216	0.0575239
$118$	9.24443	0.851019
$119$	0	0
$120$	−0.719004	−0.0656358
$121$	1.00000	0.0909091
$122$	−13.0509	−1.18157
$123$	−0.193576	−0.0174542
$124$	−4.47013	−0.401429
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.1842	−1.34738	−0.673690	−	0.739014i	$-0.735292\pi$
$127$	−15.1842	−1.34738	−0.673690	+	0.739014i	$0.735292\pi$
$128$	5.64941	0.499342
$129$	−5.67307	−0.499486
$130$	−1.18421	−0.103862
$131$	−1.24443	−0.108726	−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443	−0.108726	−0.0543632	+	0.998521i	$0.517313\pi$
$132$	−1.62222	−0.141196
$133$	0	0
$134$	−2.36842	−0.204600
$135$	1.00000	0.0860663
$136$	−3.72746	−0.319627
$137$	−0.488863	−0.0417663	−0.0208832	−	0.999782i	$-0.506648\pi$
$137$	−0.488863	−0.0417663	−0.0208832	+	0.999782i	$0.506648\pi$
$138$	−16.8573	−1.43499
$139$	−17.8064	−1.51032	−0.755161	−	0.655540i	$-0.772441\pi$
$139$	−17.8064	−1.51032	−0.755161	+	0.655540i	$0.772441\pi$
$140$	0	0
$141$	−2.75557	−0.232061
$142$	5.24443	0.440103
$143$	0.622216	0.0520323
$144$	−4.61285	−0.384404
$145$	7.80642	0.648288
$146$	−8.06022	−0.667069
$147$	0	0
$148$	−3.24443	−0.266691
$149$	−1.43801	−0.117806	−0.0589031	−	0.998264i	$-0.518760\pi$
$149$	−1.43801	−0.117806	−0.0589031	+	0.998264i	$0.518760\pi$
$150$	−1.90321	−0.155397
$151$	−12.1748	−0.990774	−0.495387	−	0.868672i	$-0.664974\pi$
$151$	−12.1748	−0.990774	−0.495387	+	0.868672i	$0.664974\pi$
$152$	5.06959	0.411198
$153$	5.18421	0.419118
$154$	0	0
$155$	2.75557	0.221333
$156$	−1.00937	−0.0808141
$157$	−18.4701	−1.47408	−0.737038	−	0.675851i	$-0.763776\pi$
$157$	−18.4701	−1.47408	−0.737038	+	0.675851i	$0.763776\pi$
$158$	16.2953	1.29638
$159$	10.8573	0.861038
$160$	7.34122	0.580374
$161$	0	0
$162$	1.90321	0.149530
$163$	−10.1017	−0.791227	−0.395614	−	0.918417i	$-0.629468\pi$
$163$	−10.1017	−0.791227	−0.395614	+	0.918417i	$0.629468\pi$
$164$	0.314022	0.0245210
$165$	1.00000	0.0778499
$166$	−0.253799	−0.0196986
$167$	16.3368	1.26418	0.632089	−	0.774896i	$-0.282198\pi$
$167$	16.3368	1.26418	0.632089	+	0.774896i	$0.282198\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	−9.86665	−0.756737
$171$	−7.05086	−0.539192
$172$	9.20294	0.701718
$173$	9.18421	0.698262	0.349131	−	0.937074i	$-0.386477\pi$
$173$	9.18421	0.698262	0.349131	+	0.937074i	$0.386477\pi$
$174$	14.8573	1.12633
$175$	0	0
$176$	−4.61285	−0.347706
$177$	−4.85728	−0.365095
$178$	−10.6824	−0.800683
$179$	−25.3274	−1.89306	−0.946530	−	0.322617i	$-0.895437\pi$
$179$	−25.3274	−1.89306	−0.946530	+	0.322617i	$0.895437\pi$
$180$	−1.62222	−0.120913
$181$	13.6128	1.01184	0.505918	−	0.862582i	$-0.331154\pi$
$181$	13.6128	1.01184	0.505918	+	0.862582i	$0.331154\pi$
$182$	0	0
$183$	6.85728	0.506905
$184$	−6.36842	−0.469486
$185$	2.00000	0.147043
$186$	5.24443	0.384540
$187$	5.18421	0.379107
$188$	4.47013	0.326017
$189$	0	0
$190$	13.4193	0.973536
$191$	6.10171	0.441504	0.220752	−	0.975330i	$-0.429149\pi$
$191$	6.10171	0.441504	0.220752	+	0.975330i	$0.429149\pi$
$192$	4.74620	0.342528
$193$	18.3368	1.31991	0.659955	−	0.751305i	$-0.270575\pi$
$193$	18.3368	1.31991	0.659955	+	0.751305i	$0.270575\pi$
$194$	−13.7877	−0.989898
$195$	0.622216	0.0445578
$196$	0	0
$197$	−6.69535	−0.477024	−0.238512	−	0.971140i	$-0.576660\pi$
$197$	−6.69535	−0.477024	−0.238512	+	0.971140i	$0.576660\pi$
$198$	1.90321	0.135255
$199$	−14.1017	−0.999644	−0.499822	−	0.866128i	$-0.666601\pi$
$199$	−14.1017	−0.999644	−0.499822	+	0.866128i	$0.666601\pi$
$200$	−0.719004	−0.0508412
$201$	1.24443	0.0877754
$202$	−8.87601	−0.624514
$203$	0	0
$204$	−8.40990	−0.588811
$205$	−0.193576	−0.0135199
$206$	22.1017	1.53990
$207$	8.85728	0.615623
$208$	−2.87019	−0.199012
$209$	−7.05086	−0.487718
$210$	0	0
$211$	−10.6637	−0.734120	−0.367060	−	0.930197i	$-0.619636\pi$
$211$	−10.6637	−0.734120	−0.367060	+	0.930197i	$0.619636\pi$
$212$	−17.6128	−1.20966
$213$	−2.75557	−0.188808
$214$	4.99063	0.341153
$215$	−5.67307	−0.386900
$216$	0.719004	0.0489220
$217$	0	0
$218$	−37.5210	−2.54124
$219$	4.23506	0.286179
$220$	−1.62222	−0.109370
$221$	3.22570	0.216984
$222$	3.80642	0.255470
$223$	−8.85728	−0.593127	−0.296564	−	0.955013i	$-0.595841\pi$
$223$	−8.85728	−0.593127	−0.296564	+	0.955013i	$0.595841\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−11.4193	−0.759599
$227$	−13.3778	−0.887915	−0.443957	−	0.896048i	$-0.646426\pi$
$227$	−13.3778	−0.887915	−0.443957	+	0.896048i	$0.646426\pi$
$228$	11.4380	0.757501
$229$	−11.5111	−0.760677	−0.380339	−	0.924847i	$-0.624193\pi$
$229$	−11.5111	−0.760677	−0.380339	+	0.924847i	$0.624193\pi$
$230$	−16.8573	−1.11154
$231$	0	0
$232$	5.61285	0.368502
$233$	−4.32693	−0.283467	−0.141733	−	0.989905i	$-0.545268\pi$
$233$	−4.32693	−0.283467	−0.141733	+	0.989905i	$0.545268\pi$
$234$	1.18421	0.0774141
$235$	−2.75557	−0.179753
$236$	7.87955	0.512915
$237$	−8.56199	−0.556161
$238$	0	0
$239$	−3.34614	−0.216444	−0.108222	−	0.994127i	$-0.534516\pi$
$239$	−3.34614	−0.216444	−0.108222	+	0.994127i	$0.534516\pi$
$240$	−4.61285	−0.297758
$241$	1.34614	0.0867126	0.0433563	−	0.999060i	$-0.486195\pi$
$241$	1.34614	0.0867126	0.0433563	+	0.999060i	$0.486195\pi$
$242$	1.90321	0.122343
$243$	−1.00000	−0.0641500
$244$	−11.1240	−0.712140
$245$	0	0
$246$	−0.368416	−0.0234894
$247$	−4.38715	−0.279148
$248$	1.98126	0.125810
$249$	0.133353	0.00845091
$250$	−1.90321	−0.120370
$251$	−22.7556	−1.43632	−0.718159	−	0.695879i	$-0.755015\pi$
$251$	−22.7556	−1.43632	−0.718159	+	0.695879i	$0.755015\pi$
$252$	0	0
$253$	8.85728	0.556852
$254$	−28.8988	−1.81327
$255$	5.18421	0.324648
$256$	20.2444	1.26528
$257$	6.85728	0.427745	0.213873	−	0.976862i	$-0.431392\pi$
$257$	6.85728	0.427745	0.213873	+	0.976862i	$0.431392\pi$
$258$	−10.7971	−0.672195
$259$	0	0
$260$	−1.00937	−0.0625983
$261$	−7.80642	−0.483206
$262$	−2.36842	−0.146321
$263$	−29.5812	−1.82406	−0.912028	−	0.410129i	$-0.865484\pi$
$263$	−29.5812	−1.82406	−0.912028	+	0.410129i	$0.865484\pi$
$264$	0.719004	0.0442516
$265$	10.8573	0.666957
$266$	0	0
$267$	5.61285	0.343501
$268$	−2.01874	−0.123314
$269$	−8.48886	−0.517575	−0.258788	−	0.965934i	$-0.583323\pi$
$269$	−8.48886	−0.517575	−0.258788	+	0.965934i	$0.583323\pi$
$270$	1.90321	0.115826
$271$	−14.6637	−0.890757	−0.445378	−	0.895343i	$-0.646931\pi$
$271$	−14.6637	−0.890757	−0.445378	+	0.895343i	$0.646931\pi$
$272$	−23.9140	−1.45000
$273$	0	0
$274$	−0.930409	−0.0562081
$275$	1.00000	0.0603023
$276$	−14.3684	−0.864877
$277$	14.6035	0.877438	0.438719	−	0.898624i	$-0.355432\pi$
$277$	14.6035	0.877438	0.438719	+	0.898624i	$0.355432\pi$
$278$	−33.8894	−2.03255
$279$	−2.75557	−0.164972
$280$	0	0
$281$	−0.193576	−0.0115478	−0.00577389	−	0.999983i	$-0.501838\pi$
$281$	−0.193576	−0.0115478	−0.00577389	+	0.999983i	$0.501838\pi$
$282$	−5.24443	−0.312301
$283$	−27.1842	−1.61593	−0.807967	−	0.589228i	$-0.799432\pi$
$283$	−27.1842	−1.61593	−0.807967	+	0.589228i	$0.799432\pi$
$284$	4.47013	0.265253
$285$	−7.05086	−0.417657
$286$	1.18421	0.0700237
$287$	0	0
$288$	−7.34122	−0.432585
$289$	9.87601	0.580942
$290$	14.8573	0.872449
$291$	7.24443	0.424676
$292$	−6.87019	−0.402047
$293$	2.81579	0.164500	0.0822502	−	0.996612i	$-0.473789\pi$
$293$	2.81579	0.164500	0.0822502	+	0.996612i	$0.473789\pi$
$294$	0	0
$295$	−4.85728	−0.282802
$296$	1.43801	0.0835825
$297$	−1.00000	−0.0580259
$298$	−2.73683	−0.158540
$299$	5.51114	0.318717
$300$	−1.62222	−0.0936587
$301$	0	0
$302$	−23.1713	−1.33336
$303$	4.66370	0.267923
$304$	32.5245	1.86541
$305$	6.85728	0.392647
$306$	9.86665	0.564039
$307$	24.4286	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$307$	24.4286	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$308$	0	0
$309$	−11.6128	−0.660632
$310$	5.24443	0.297864
$311$	−19.8796	−1.12727	−0.563633	−	0.826025i	$-0.690597\pi$
$311$	−19.8796	−1.12727	−0.563633	+	0.826025i	$0.690597\pi$
$312$	0.447375	0.0253276
$313$	15.7146	0.888239	0.444120	−	0.895967i	$-0.353517\pi$
$313$	15.7146	0.888239	0.444120	+	0.895967i	$0.353517\pi$
$314$	−35.1526	−1.98377
$315$	0	0
$316$	13.8894	0.781340
$317$	16.4889	0.926107	0.463053	−	0.886330i	$-0.346754\pi$
$317$	16.4889	0.926107	0.463053	+	0.886330i	$0.346754\pi$
$318$	20.6637	1.15876
$319$	−7.80642	−0.437076
$320$	4.74620	0.265321
$321$	−2.62222	−0.146358
$322$	0	0
$323$	−36.5531	−2.03387
$324$	1.62222	0.0901231
$325$	0.622216	0.0345143
$326$	−19.2257	−1.06481
$327$	19.7146	1.09022
$328$	−0.139182	−0.00768504
$329$	0	0
$330$	1.90321	0.104768
$331$	15.3461	0.843500	0.421750	−	0.906712i	$-0.361416\pi$
$331$	15.3461	0.843500	0.421750	+	0.906712i	$0.361416\pi$
$332$	−0.216327	−0.0118725
$333$	−2.00000	−0.109599
$334$	31.0923	1.70130
$335$	1.24443	0.0679905
$336$	0	0
$337$	28.2351	1.53806	0.769031	−	0.639212i	$-0.220739\pi$
$337$	28.2351	1.53806	0.769031	+	0.639212i	$0.220739\pi$
$338$	−24.0049	−1.30570
$339$	6.00000	0.325875
$340$	−8.40990	−0.456091
$341$	−2.75557	−0.149222
$342$	−13.4193	−0.725631
$343$	0	0
$344$	−4.07896	−0.219923
$345$	8.85728	0.476860
$346$	17.4795	0.939703
$347$	2.62222	0.140768	0.0703840	−	0.997520i	$-0.477578\pi$
$347$	2.62222	0.140768	0.0703840	+	0.997520i	$0.477578\pi$
$348$	12.6637	0.678846
$349$	−5.14272	−0.275284	−0.137642	−	0.990482i	$-0.543952\pi$
$349$	−5.14272	−0.275284	−0.137642	+	0.990482i	$0.543952\pi$
$350$	0	0
$351$	−0.622216	−0.0332114
$352$	−7.34122	−0.391288
$353$	9.34614	0.497445	0.248722	−	0.968575i	$-0.419989\pi$
$353$	9.34614	0.497445	0.248722	+	0.968575i	$0.419989\pi$
$354$	−9.24443	−0.491336
$355$	−2.75557	−0.146250
$356$	−9.10525	−0.482577
$357$	0	0
$358$	−48.2034	−2.54763
$359$	10.7556	0.567657	0.283829	−	0.958875i	$-0.408395\pi$
$359$	10.7556	0.567657	0.283829	+	0.958875i	$0.408395\pi$
$360$	0.719004	0.0378948
$361$	30.7146	1.61656
$362$	25.9081	1.36170
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	4.23506	0.221673
$366$	13.0509	0.682179
$367$	−33.7975	−1.76422	−0.882108	−	0.471046i	$-0.843876\pi$
$367$	−33.7975	−1.76422	−0.882108	+	0.471046i	$0.843876\pi$
$368$	−40.8573	−2.12983
$369$	0.193576	0.0100772
$370$	3.80642	0.197887
$371$	0	0
$372$	4.47013	0.231765
$373$	33.9496	1.75784	0.878922	−	0.476965i	$-0.158263\pi$
$373$	33.9496	1.75784	0.878922	+	0.476965i	$0.158263\pi$
$374$	9.86665	0.510192
$375$	1.00000	0.0516398
$376$	−1.98126	−0.102176
$377$	−4.85728	−0.250163
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	11.4380	0.586757
$381$	15.1842	0.777911
$382$	11.6128	0.594165
$383$	14.6351	0.747820	0.373910	−	0.927465i	$-0.378017\pi$
$383$	14.6351	0.747820	0.373910	+	0.927465i	$0.378017\pi$
$384$	−5.64941	−0.288295
$385$	0	0
$386$	34.8988	1.77630
$387$	5.67307	0.288378
$388$	−11.7520	−0.596619
$389$	−5.61285	−0.284583	−0.142291	−	0.989825i	$-0.545447\pi$
$389$	−5.61285	−0.284583	−0.142291	+	0.989825i	$0.545447\pi$
$390$	1.18421	0.0599647
$391$	45.9180	2.32217
$392$	0	0
$393$	1.24443	0.0627733
$394$	−12.7427	−0.641966
$395$	−8.56199	−0.430801
$396$	1.62222	0.0815194
$397$	12.7556	0.640184	0.320092	−	0.947387i	$-0.396286\pi$
$397$	12.7556	0.640184	0.320092	+	0.947387i	$0.396286\pi$
$398$	−26.8385	−1.34529
$399$	0	0
$400$	−4.61285	−0.230642
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	2.36842	0.118126
$403$	−1.71456	−0.0854082
$404$	−7.56553	−0.376399
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	3.72746	0.184537
$409$	7.12399	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$409$	7.12399	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$410$	−0.368416	−0.0181948
$411$	0.488863	0.0241138
$412$	18.8385	0.928108
$413$	0	0
$414$	16.8573	0.828490
$415$	0.133353	0.00654605
$416$	−4.56782	−0.223956
$417$	17.8064	0.871984
$418$	−13.4193	−0.656358
$419$	−15.6128	−0.762738	−0.381369	−	0.924423i	$-0.624547\pi$
$419$	−15.6128	−0.762738	−0.381369	+	0.924423i	$0.624547\pi$
$420$	0	0
$421$	7.89829	0.384939	0.192470	−	0.981303i	$-0.438350\pi$
$421$	7.89829	0.384939	0.192470	+	0.981303i	$0.438350\pi$
$422$	−20.2953	−0.987959
$423$	2.75557	0.133980
$424$	7.80642	0.379113
$425$	5.18421	0.251471
$426$	−5.24443	−0.254094
$427$	0	0
$428$	4.25380	0.205615
$429$	−0.622216	−0.0300409
$430$	−10.7971	−0.520680
$431$	34.3051	1.65242	0.826210	−	0.563362i	$-0.190492\pi$
$431$	34.3051	1.65242	0.826210	+	0.563362i	$0.190492\pi$
$432$	4.61285	0.221936
$433$	−14.4701	−0.695390	−0.347695	−	0.937608i	$-0.613036\pi$
$433$	−14.4701	−0.695390	−0.347695	+	0.937608i	$0.613036\pi$
$434$	0	0
$435$	−7.80642	−0.374289
$436$	−31.9813	−1.53162
$437$	−62.4514	−2.98746
$438$	8.06022	0.385132
$439$	19.3176	0.921977	0.460988	−	0.887406i	$-0.347495\pi$
$439$	19.3176	0.921977	0.460988	+	0.887406i	$0.347495\pi$
$440$	0.719004	0.0342772
$441$	0	0
$442$	6.13918	0.292011
$443$	−13.1240	−0.623539	−0.311770	−	0.950158i	$-0.600922\pi$
$443$	−13.1240	−0.623539	−0.311770	+	0.950158i	$0.600922\pi$
$444$	3.24443	0.153974
$445$	5.61285	0.266074
$446$	−16.8573	−0.798215
$447$	1.43801	0.0680154
$448$	0	0
$449$	−32.3051	−1.52457	−0.762287	−	0.647240i	$-0.775923\pi$
$449$	−32.3051	−1.52457	−0.762287	+	0.647240i	$0.775923\pi$
$450$	1.90321	0.0897183
$451$	0.193576	0.00911514
$452$	−9.73329	−0.457816
$453$	12.1748	0.572024
$454$	−25.4608	−1.19493
$455$	0	0
$456$	−5.06959	−0.237405
$457$	−23.4608	−1.09745	−0.548724	−	0.836004i	$-0.684886\pi$
$457$	−23.4608	−1.09745	−0.548724	+	0.836004i	$0.684886\pi$
$458$	−21.9081	−1.02370
$459$	−5.18421	−0.241978
$460$	−14.3684	−0.669931
$461$	28.8671	1.34448	0.672238	−	0.740335i	$-0.265333\pi$
$461$	28.8671	1.34448	0.672238	+	0.740335i	$0.265333\pi$
$462$	0	0
$463$	−19.3461	−0.899091	−0.449546	−	0.893257i	$-0.648414\pi$
$463$	−19.3461	−0.899091	−0.449546	+	0.893257i	$0.648414\pi$
$464$	36.0098	1.67172
$465$	−2.75557	−0.127786
$466$	−8.23506	−0.381482
$467$	−3.14272	−0.145428	−0.0727139	−	0.997353i	$-0.523166\pi$
$467$	−3.14272	−0.145428	−0.0727139	+	0.997353i	$0.523166\pi$
$468$	1.00937	0.0466580
$469$	0	0
$470$	−5.24443	−0.241908
$471$	18.4701	0.851059
$472$	−3.49240	−0.160751
$473$	5.67307	0.260848
$474$	−16.2953	−0.748467
$475$	−7.05086	−0.323515
$476$	0	0
$477$	−10.8573	−0.497121
$478$	−6.36842	−0.291285
$479$	24.8573	1.13576	0.567879	−	0.823112i	$-0.307764\pi$
$479$	24.8573	1.13576	0.567879	+	0.823112i	$0.307764\pi$
$480$	−7.34122	−0.335079
$481$	−1.24443	−0.0567412
$482$	2.56199	0.116696
$483$	0	0
$484$	1.62222	0.0737371
$485$	7.24443	0.328953
$486$	−1.90321	−0.0863314
$487$	−11.5299	−0.522468	−0.261234	−	0.965275i	$-0.584129\pi$
$487$	−11.5299	−0.522468	−0.261234	+	0.965275i	$0.584129\pi$
$488$	4.93041	0.223189
$489$	10.1017	0.456815
$490$	0	0
$491$	16.3872	0.739542	0.369771	−	0.929123i	$-0.379436\pi$
$491$	16.3872	0.739542	0.369771	+	0.929123i	$0.379436\pi$
$492$	−0.314022	−0.0141572
$493$	−40.4701	−1.82268
$494$	−8.34968	−0.375670
$495$	−1.00000	−0.0449467
$496$	12.7110	0.570742
$497$	0	0
$498$	0.253799	0.0113730
$499$	−25.3274	−1.13381	−0.566905	−	0.823783i	$-0.691859\pi$
$499$	−25.3274	−1.13381	−0.566905	+	0.823783i	$0.691859\pi$
$500$	−1.62222	−0.0725477
$501$	−16.3368	−0.729873
$502$	−43.3087	−1.93296
$503$	−19.0923	−0.851285	−0.425643	−	0.904891i	$-0.639952\pi$
$503$	−19.0923	−0.851285	−0.425643	+	0.904891i	$0.639952\pi$
$504$	0	0
$505$	4.66370	0.207532
$506$	16.8573	0.749397
$507$	12.6128	0.560156
$508$	−24.6321	−1.09287
$509$	32.4514	1.43838	0.719191	−	0.694812i	$-0.244512\pi$
$509$	32.4514	1.43838	0.719191	+	0.694812i	$0.244512\pi$
$510$	9.86665	0.436902
$511$	0	0
$512$	27.2306	1.20343
$513$	7.05086	0.311303
$514$	13.0509	0.575649
$515$	−11.6128	−0.511723
$516$	−9.20294	−0.405137
$517$	2.75557	0.121190
$518$	0	0
$519$	−9.18421	−0.403142
$520$	0.447375	0.0196187
$521$	29.2257	1.28040	0.640200	−	0.768208i	$-0.278851\pi$
$521$	29.2257	1.28040	0.640200	+	0.768208i	$0.278851\pi$
$522$	−14.8573	−0.650285
$523$	−6.71408	−0.293586	−0.146793	−	0.989167i	$-0.546895\pi$
$523$	−6.71408	−0.293586	−0.146793	+	0.989167i	$0.546895\pi$
$524$	−2.01874	−0.0881889
$525$	0	0
$526$	−56.2993	−2.45477
$527$	−14.2854	−0.622284
$528$	4.61285	0.200748
$529$	55.4514	2.41093
$530$	20.6637	0.897574
$531$	4.85728	0.210788
$532$	0	0
$533$	0.120446	0.00521710
$534$	10.6824	0.462274
$535$	−2.62222	−0.113368
$536$	0.894751	0.0386473
$537$	25.3274	1.09296
$538$	−16.1561	−0.696539
$539$	0	0
$540$	1.62222	0.0698090
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−27.9081	−1.19876
$543$	−13.6128	−0.584183
$544$	−38.0584	−1.63174
$545$	19.7146	0.844479
$546$	0	0
$547$	41.3689	1.76881	0.884403	−	0.466724i	$-0.154566\pi$
$547$	41.3689	1.76881	0.884403	+	0.466724i	$0.154566\pi$
$548$	−0.793040	−0.0338770
$549$	−6.85728	−0.292662
$550$	1.90321	0.0811532
$551$	55.0420	2.34487
$552$	6.36842	0.271058
$553$	0	0
$554$	27.7935	1.18083
$555$	−2.00000	−0.0848953
$556$	−28.8859	−1.22503
$557$	−20.7971	−0.881200	−0.440600	−	0.897704i	$-0.645234\pi$
$557$	−20.7971	−0.881200	−0.440600	+	0.897704i	$0.645234\pi$
$558$	−5.24443	−0.222014
$559$	3.52987	0.149298
$560$	0	0
$561$	−5.18421	−0.218877
$562$	−0.368416	−0.0155407
$563$	−37.7275	−1.59002	−0.795012	−	0.606594i	$-0.792535\pi$
$563$	−37.7275	−1.59002	−0.795012	+	0.606594i	$0.792535\pi$
$564$	−4.47013	−0.188226
$565$	6.00000	0.252422
$566$	−51.7373	−2.17468
$567$	0	0
$568$	−1.98126	−0.0831320
$569$	−7.33630	−0.307554	−0.153777	−	0.988106i	$-0.549144\pi$
$569$	−7.33630	−0.307554	−0.153777	+	0.988106i	$0.549144\pi$
$570$	−13.4193	−0.562071
$571$	36.6450	1.53354	0.766772	−	0.641919i	$-0.221862\pi$
$571$	36.6450	1.53354	0.766772	+	0.641919i	$0.221862\pi$
$572$	1.00937	0.0422038
$573$	−6.10171	−0.254903
$574$	0	0
$575$	8.85728	0.369374
$576$	−4.74620	−0.197758
$577$	−4.22216	−0.175771	−0.0878853	−	0.996131i	$-0.528011\pi$
$577$	−4.22216	−0.175771	−0.0878853	+	0.996131i	$0.528011\pi$
$578$	18.7961	0.781817
$579$	−18.3368	−0.762050
$580$	12.6637	0.525832
$581$	0	0
$582$	13.7877	0.571518
$583$	−10.8573	−0.449663
$584$	3.04503	0.126004
$585$	−0.622216	−0.0257255
$586$	5.35905	0.221380
$587$	−34.3684	−1.41854	−0.709268	−	0.704939i	$-0.750974\pi$
$587$	−34.3684	−1.41854	−0.709268	+	0.704939i	$0.750974\pi$
$588$	0	0
$589$	19.4291	0.800563
$590$	−9.24443	−0.380587
$591$	6.69535	0.275410
$592$	9.22570	0.379174
$593$	−27.9398	−1.14735	−0.573675	−	0.819083i	$-0.694483\pi$
$593$	−27.9398	−1.14735	−0.573675	+	0.819083i	$0.694483\pi$
$594$	−1.90321	−0.0780897
$595$	0	0
$596$	−2.33276	−0.0955535
$597$	14.1017	0.577145
$598$	10.4889	0.428921
$599$	−31.2257	−1.27585	−0.637924	−	0.770100i	$-0.720206\pi$
$599$	−31.2257	−1.27585	−0.637924	+	0.770100i	$0.720206\pi$
$600$	0.719004	0.0293532
$601$	8.75557	0.357147	0.178574	−	0.983927i	$-0.442852\pi$
$601$	8.75557	0.357147	0.178574	+	0.983927i	$0.442852\pi$
$602$	0	0
$603$	−1.24443	−0.0506772
$604$	−19.7502	−0.803625
$605$	−1.00000	−0.0406558
$606$	8.87601	0.360563
$607$	15.1842	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$607$	15.1842	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$608$	51.7619	2.09922
$609$	0	0
$610$	13.0509	0.528414
$611$	1.71456	0.0693636
$612$	8.40990	0.339950
$613$	42.7239	1.72560	0.862802	−	0.505543i	$-0.168708\pi$
$613$	42.7239	1.72560	0.862802	+	0.505543i	$0.168708\pi$
$614$	46.4929	1.87630
$615$	0.193576	0.00780574
$616$	0	0
$617$	−3.51114	−0.141353	−0.0706765	−	0.997499i	$-0.522516\pi$
$617$	−3.51114	−0.141353	−0.0706765	+	0.997499i	$0.522516\pi$
$618$	−22.1017	−0.889061
$619$	−17.5941	−0.707167	−0.353584	−	0.935403i	$-0.615037\pi$
$619$	−17.5941	−0.707167	−0.353584	+	0.935403i	$0.615037\pi$
$620$	4.47013	0.179525
$621$	−8.85728	−0.355430
$622$	−37.8350	−1.51705
$623$	0	0
$624$	2.87019	0.114899
$625$	1.00000	0.0400000
$626$	29.9081	1.19537
$627$	7.05086	0.281584
$628$	−29.9625	−1.19564
$629$	−10.3684	−0.413416
$630$	0	0
$631$	15.8163	0.629636	0.314818	−	0.949152i	$-0.398057\pi$
$631$	15.8163	0.629636	0.314818	+	0.949152i	$0.398057\pi$
$632$	−6.15610	−0.244877
$633$	10.6637	0.423844
$634$	31.3818	1.24633
$635$	15.1842	0.602567
$636$	17.6128	0.698395
$637$	0	0
$638$	−14.8573	−0.588205
$639$	2.75557	0.109009
$640$	−5.64941	−0.223313
$641$	25.8163	1.01968	0.509841	−	0.860269i	$-0.329704\pi$
$641$	25.8163	1.01968	0.509841	+	0.860269i	$0.329704\pi$
$642$	−4.99063	−0.196965
$643$	−18.1017	−0.713862	−0.356931	−	0.934131i	$-0.616177\pi$
$643$	−18.1017	−0.713862	−0.356931	+	0.934131i	$0.616177\pi$
$644$	0	0
$645$	5.67307	0.223377
$646$	−69.5683	−2.73713
$647$	47.0420	1.84941	0.924705	−	0.380684i	$-0.124311\pi$
$647$	47.0420	1.84941	0.924705	+	0.380684i	$0.124311\pi$
$648$	−0.719004	−0.0282451
$649$	4.85728	0.190665
$650$	1.18421	0.0464485
$651$	0	0
$652$	−16.3872	−0.641770
$653$	30.0830	1.17724	0.588619	−	0.808411i	$-0.299672\pi$
$653$	30.0830	1.17724	0.588619	+	0.808411i	$0.299672\pi$
$654$	37.5210	1.46719
$655$	1.24443	0.0486240
$656$	−0.892937	−0.0348633
$657$	−4.23506	−0.165226
$658$	0	0
$659$	−10.2854	−0.400664	−0.200332	−	0.979728i	$-0.564202\pi$
$659$	−10.2854	−0.400664	−0.200332	+	0.979728i	$0.564202\pi$
$660$	1.62222	0.0631447
$661$	27.7146	1.07797	0.538986	−	0.842315i	$-0.318808\pi$
$661$	27.7146	1.07797	0.538986	+	0.842315i	$0.318808\pi$
$662$	29.2070	1.13516
$663$	−3.22570	−0.125276
$664$	0.0958814	0.00372092
$665$	0	0
$666$	−3.80642	−0.147496
$667$	−69.1437	−2.67725
$668$	26.5018	1.02538
$669$	8.85728	0.342442
$670$	2.36842	0.0914999
$671$	−6.85728	−0.264722
$672$	0	0
$673$	−9.86665	−0.380331	−0.190166	−	0.981752i	$-0.560903\pi$
$673$	−9.86665	−0.380331	−0.190166	+	0.981752i	$0.560903\pi$
$674$	53.7373	2.06988
$675$	−1.00000	−0.0384900
$676$	−20.4608	−0.786952
$677$	5.65433	0.217314	0.108657	−	0.994079i	$-0.465345\pi$
$677$	5.65433	0.217314	0.108657	+	0.994079i	$0.465345\pi$
$678$	11.4193	0.438554
$679$	0	0
$680$	3.72746	0.142942
$681$	13.3778	0.512638
$682$	−5.24443	−0.200820
$683$	34.1847	1.30804	0.654020	−	0.756477i	$-0.273081\pi$
$683$	34.1847	1.30804	0.654020	+	0.756477i	$0.273081\pi$
$684$	−11.4380	−0.437343
$685$	0.488863	0.0186785
$686$	0	0
$687$	11.5111	0.439177
$688$	−26.1690	−0.997684
$689$	−6.75557	−0.257367
$690$	16.8573	0.641746
$691$	19.2257	0.731380	0.365690	−	0.930737i	$-0.380833\pi$
$691$	19.2257	0.731380	0.365690	+	0.930737i	$0.380833\pi$
$692$	14.8988	0.566366
$693$	0	0
$694$	4.99063	0.189442
$695$	17.8064	0.675436
$696$	−5.61285	−0.212754
$697$	1.00354	0.0380118
$698$	−9.78769	−0.370469
$699$	4.32693	0.163659
$700$	0	0
$701$	−29.9081	−1.12961	−0.564807	−	0.825223i	$-0.691050\pi$
$701$	−29.9081	−1.12961	−0.564807	+	0.825223i	$0.691050\pi$
$702$	−1.18421	−0.0446951
$703$	14.1017	0.531856
$704$	−4.74620	−0.178879
$705$	2.75557	0.103781
$706$	17.7877	0.669448
$707$	0	0
$708$	−7.87955	−0.296132
$709$	−15.3274	−0.575633	−0.287816	−	0.957686i	$-0.592929\pi$
$709$	−15.3274	−0.575633	−0.287816	+	0.957686i	$0.592929\pi$
$710$	−5.24443	−0.196820
$711$	8.56199	0.321100
$712$	4.03566	0.151243
$713$	−24.4068	−0.914043
$714$	0	0
$715$	−0.622216	−0.0232695
$716$	−41.0865	−1.53548
$717$	3.34614	0.124964
$718$	20.4701	0.763938
$719$	−23.8163	−0.888197	−0.444098	−	0.895978i	$-0.646476\pi$
$719$	−23.8163	−0.888197	−0.444098	+	0.895978i	$0.646476\pi$
$720$	4.61285	0.171911
$721$	0	0
$722$	58.4563	2.17552
$723$	−1.34614	−0.0500635
$724$	22.0830	0.820707
$725$	−7.80642	−0.289923
$726$	−1.90321	−0.0706348
$727$	32.9403	1.22169	0.610843	−	0.791752i	$-0.290831\pi$
$727$	32.9403	1.22169	0.610843	+	0.791752i	$0.290831\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.06022	0.298322
$731$	29.4104	1.08778
$732$	11.1240	0.411154
$733$	29.8666	1.10315	0.551575	−	0.834125i	$-0.314027\pi$
$733$	29.8666	1.10315	0.551575	+	0.834125i	$0.314027\pi$
$734$	−64.3239	−2.37424
$735$	0	0
$736$	−65.0232	−2.39679
$737$	−1.24443	−0.0458392
$738$	0.368416	0.0135616
$739$	−5.06959	−0.186488	−0.0932440	−	0.995643i	$-0.529724\pi$
$739$	−5.06959	−0.186488	−0.0932440	+	0.995643i	$0.529724\pi$
$740$	3.24443	0.119268
$741$	4.38715	0.161166
$742$	0	0
$743$	22.4385	0.823188	0.411594	−	0.911367i	$-0.364972\pi$
$743$	22.4385	0.823188	0.411594	+	0.911367i	$0.364972\pi$
$744$	−1.98126	−0.0726367
$745$	1.43801	0.0526845
$746$	64.6133	2.36566
$747$	−0.133353	−0.00487913
$748$	8.40990	0.307497
$749$	0	0
$750$	1.90321	0.0694955
$751$	−6.63512	−0.242119	−0.121060	−	0.992645i	$-0.538629\pi$
$751$	−6.63512	−0.242119	−0.121060	+	0.992645i	$0.538629\pi$
$752$	−12.7110	−0.463523
$753$	22.7556	0.829259
$754$	−9.24443	−0.336662
$755$	12.1748	0.443088
$756$	0	0
$757$	8.75557	0.318227	0.159113	−	0.987260i	$-0.449136\pi$
$757$	8.75557	0.318227	0.159113	+	0.987260i	$0.449136\pi$
$758$	−38.0642	−1.38256
$759$	−8.85728	−0.321499
$760$	−5.06959	−0.183893
$761$	−3.15257	−0.114280	−0.0571402	−	0.998366i	$-0.518198\pi$
$761$	−3.15257	−0.114280	−0.0571402	+	0.998366i	$0.518198\pi$
$762$	28.8988	1.04689
$763$	0	0
$764$	9.89829	0.358108
$765$	−5.18421	−0.187435
$766$	27.8537	1.00640
$767$	3.02227	0.109128
$768$	−20.2444	−0.730508
$769$	28.9590	1.04429	0.522144	−	0.852857i	$-0.325132\pi$
$769$	28.9590	1.04429	0.522144	+	0.852857i	$0.325132\pi$
$770$	0	0
$771$	−6.85728	−0.246959
$772$	29.7462	1.07059
$773$	−29.1427	−1.04819	−0.524095	−	0.851660i	$-0.675596\pi$
$773$	−29.1427	−1.04819	−0.524095	+	0.851660i	$0.675596\pi$
$774$	10.7971	0.388092
$775$	−2.75557	−0.0989830
$776$	5.20877	0.186984
$777$	0	0
$778$	−10.6824	−0.382984
$779$	−1.36488	−0.0489018
$780$	1.00937	0.0361412
$781$	2.75557	0.0986020
$782$	87.3916	3.12512
$783$	7.80642	0.278979
$784$	0	0
$785$	18.4701	0.659227
$786$	2.36842	0.0844786
$787$	11.2672	0.401632	0.200816	−	0.979629i	$-0.435641\pi$
$787$	11.2672	0.401632	0.200816	+	0.979629i	$0.435641\pi$
$788$	−10.8613	−0.386918
$789$	29.5812	1.05312
$790$	−16.2953	−0.579760
$791$	0	0
$792$	−0.719004	−0.0255487
$793$	−4.26671	−0.151515
$794$	24.2766	0.861543
$795$	−10.8573	−0.385068
$796$	−22.8760	−0.810819
$797$	−41.9625	−1.48639	−0.743195	−	0.669075i	$-0.766690\pi$
$797$	−41.9625	−1.48639	−0.743195	+	0.669075i	$0.766690\pi$
$798$	0	0
$799$	14.2854	0.505383
$800$	−7.34122	−0.259551
$801$	−5.61285	−0.198320
$802$	3.80642	0.134409
$803$	−4.23506	−0.149452
$804$	2.01874	0.0711953
$805$	0	0
$806$	−3.26317	−0.114940
$807$	8.48886	0.298822
$808$	3.35322	0.117966
$809$	−27.8064	−0.977622	−0.488811	−	0.872390i	$-0.662569\pi$
$809$	−27.8064	−0.977622	−0.488811	+	0.872390i	$0.662569\pi$
$810$	−1.90321	−0.0668721
$811$	−6.78415	−0.238224	−0.119112	−	0.992881i	$-0.538005\pi$
$811$	−6.78415	−0.238224	−0.119112	+	0.992881i	$0.538005\pi$
$812$	0	0
$813$	14.6637	0.514279
$814$	−3.80642	−0.133415
$815$	10.1017	0.353847
$816$	23.9140	0.837156
$817$	−40.0000	−1.39942
$818$	13.5585	0.474060
$819$	0	0
$820$	−0.314022	−0.0109661
$821$	3.62269	0.126433	0.0632164	−	0.998000i	$-0.479864\pi$
$821$	3.62269	0.126433	0.0632164	+	0.998000i	$0.479864\pi$
$822$	0.930409	0.0324517
$823$	42.0642	1.46627	0.733134	−	0.680085i	$-0.238057\pi$
$823$	42.0642	1.46627	0.733134	+	0.680085i	$0.238057\pi$
$824$	−8.34968	−0.290875
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	30.8256	1.07191	0.535956	−	0.844246i	$-0.319951\pi$
$827$	30.8256	1.07191	0.535956	+	0.844246i	$0.319951\pi$
$828$	14.3684	0.499337
$829$	−7.12399	−0.247426	−0.123713	−	0.992318i	$-0.539480\pi$
$829$	−7.12399	−0.247426	−0.123713	+	0.992318i	$0.539480\pi$
$830$	0.253799	0.00880950
$831$	−14.6035	−0.506589
$832$	−2.95316	−0.102382
$833$	0	0
$834$	33.8894	1.17349
$835$	−16.3368	−0.565357
$836$	−11.4380	−0.395592
$837$	2.75557	0.0952464
$838$	−29.7146	−1.02647
$839$	−3.34614	−0.115522	−0.0577608	−	0.998330i	$-0.518396\pi$
$839$	−3.34614	−0.115522	−0.0577608	+	0.998330i	$0.518396\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	15.0321	0.518041
$843$	0.193576	0.00666712
$844$	−17.2988	−0.595450
$845$	12.6128	0.433895
$846$	5.24443	0.180307
$847$	0	0
$848$	50.0830	1.71986
$849$	27.1842	0.932960
$850$	9.86665	0.338423
$851$	−17.7146	−0.607247
$852$	−4.47013	−0.153144
$853$	26.4197	0.904595	0.452297	−	0.891867i	$-0.350605\pi$
$853$	26.4197	0.904595	0.452297	+	0.891867i	$0.350605\pi$
$854$	0	0
$855$	7.05086	0.241134
$856$	−1.88538	−0.0644411
$857$	−38.7783	−1.32464	−0.662321	−	0.749220i	$-0.730429\pi$
$857$	−38.7783	−1.32464	−0.662321	+	0.749220i	$0.730429\pi$
$858$	−1.18421	−0.0404282
$859$	27.3087	0.931760	0.465880	−	0.884848i	$-0.345738\pi$
$859$	27.3087	0.931760	0.465880	+	0.884848i	$0.345738\pi$
$860$	−9.20294	−0.313818
$861$	0	0
$862$	65.2899	2.22378
$863$	−49.5308	−1.68605	−0.843024	−	0.537875i	$-0.819227\pi$
$863$	−49.5308	−1.68605	−0.843024	+	0.537875i	$0.819227\pi$
$864$	7.34122	0.249753
$865$	−9.18421	−0.312272
$866$	−27.5397	−0.935838
$867$	−9.87601	−0.335407
$868$	0	0
$869$	8.56199	0.290446
$870$	−14.8573	−0.503709
$871$	−0.774305	−0.0262363
$872$	14.1748	0.480021
$873$	−7.24443	−0.245187
$874$	−118.858	−4.02044
$875$	0	0
$876$	6.87019	0.232122
$877$	−4.50177	−0.152014	−0.0760070	−	0.997107i	$-0.524217\pi$
$877$	−4.50177	−0.152014	−0.0760070	+	0.997107i	$0.524217\pi$
$878$	36.7654	1.24077
$879$	−2.81579	−0.0949743
$880$	4.61285	0.155499
$881$	15.1240	0.509540	0.254770	−	0.967002i	$-0.418000\pi$
$881$	15.1240	0.509540	0.254770	+	0.967002i	$0.418000\pi$
$882$	0	0
$883$	−30.2480	−1.01793	−0.508963	−	0.860789i	$-0.669971\pi$
$883$	−30.2480	−1.01793	−0.508963	+	0.860789i	$0.669971\pi$
$884$	5.23277	0.175997
$885$	4.85728	0.163276
$886$	−24.9777	−0.839143
$887$	−57.1941	−1.92039	−0.960194	−	0.279333i	$-0.909887\pi$
$887$	−57.1941	−1.92039	−0.960194	+	0.279333i	$0.909887\pi$
$888$	−1.43801	−0.0482564
$889$	0	0
$890$	10.6824	0.358076
$891$	1.00000	0.0335013
$892$	−14.3684	−0.481090
$893$	−19.4291	−0.650171
$894$	2.73683	0.0915334
$895$	25.3274	0.846602
$896$	0	0
$897$	−5.51114	−0.184012
$898$	−61.4835	−2.05173
$899$	21.5111	0.717437
$900$	1.62222	0.0540739
$901$	−56.2864	−1.87517
$902$	0.368416	0.0122669
$903$	0	0
$904$	4.31402	0.143482
$905$	−13.6128	−0.452506
$906$	23.1713	0.769815
$907$	−53.2641	−1.76861	−0.884303	−	0.466913i	$-0.845366\pi$
$907$	−53.2641	−1.76861	−0.884303	+	0.466913i	$0.845366\pi$
$908$	−21.7017	−0.720195
$909$	−4.66370	−0.154685
$910$	0	0
$911$	−0.590573	−0.0195665	−0.00978327	−	0.999952i	$-0.503114\pi$
$911$	−0.590573	−0.0195665	−0.00978327	+	0.999952i	$0.503114\pi$
$912$	−32.5245	−1.07699
$913$	−0.133353	−0.00441334
$914$	−44.6508	−1.47692
$915$	−6.85728	−0.226695
$916$	−18.6735	−0.616991
$917$	0	0
$918$	−9.86665	−0.325648
$919$	55.8707	1.84300	0.921502	−	0.388375i	$-0.126963\pi$
$919$	55.8707	1.84300	0.921502	+	0.388375i	$0.126963\pi$
$920$	6.36842	0.209960
$921$	−24.4286	−0.804951
$922$	54.9403	1.80936
$923$	1.71456	0.0564354
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−36.8198	−1.20997
$927$	11.6128	0.381416
$928$	57.3087	1.88125
$929$	−15.3274	−0.502876	−0.251438	−	0.967873i	$-0.580903\pi$
$929$	−15.3274	−0.502876	−0.251438	+	0.967873i	$0.580903\pi$
$930$	−5.24443	−0.171972
$931$	0	0
$932$	−7.01921	−0.229922
$933$	19.8796	0.650827
$934$	−5.98126	−0.195713
$935$	−5.18421	−0.169542
$936$	−0.447375	−0.0146229
$937$	27.8479	0.909752	0.454876	−	0.890555i	$-0.349684\pi$
$937$	27.8479	0.909752	0.454876	+	0.890555i	$0.349684\pi$
$938$	0	0
$939$	−15.7146	−0.512825
$940$	−4.47013	−0.145799
$941$	−10.4157	−0.339543	−0.169772	−	0.985483i	$-0.554303\pi$
$941$	−10.4157	−0.339543	−0.169772	+	0.985483i	$0.554303\pi$
$942$	35.1526	1.14533
$943$	1.71456	0.0558337
$944$	−22.4059	−0.729250
$945$	0	0
$946$	10.7971	0.351043
$947$	8.47013	0.275242	0.137621	−	0.990485i	$-0.456054\pi$
$947$	8.47013	0.275242	0.137621	+	0.990485i	$0.456054\pi$
$948$	−13.8894	−0.451107
$949$	−2.63512	−0.0855397
$950$	−13.4193	−0.435379
$951$	−16.4889	−0.534688
$952$	0	0
$953$	−8.71408	−0.282277	−0.141138	−	0.989990i	$-0.545076\pi$
$953$	−8.71408	−0.282277	−0.141138	+	0.989990i	$0.545076\pi$
$954$	−20.6637	−0.669012
$955$	−6.10171	−0.197447
$956$	−5.42816	−0.175559
$957$	7.80642	0.252346
$958$	47.3087	1.52847
$959$	0	0
$960$	−4.74620	−0.153183
$961$	−23.4068	−0.755059
$962$	−2.36842	−0.0763608
$963$	2.62222	0.0844997
$964$	2.18373	0.0703333
$965$	−18.3368	−0.590282
$966$	0	0
$967$	44.2449	1.42282	0.711410	−	0.702777i	$-0.248057\pi$
$967$	44.2449	1.42282	0.711410	+	0.702777i	$0.248057\pi$
$968$	−0.719004	−0.0231097
$969$	36.5531	1.17425
$970$	13.7877	0.442696
$971$	57.1437	1.83383	0.916914	−	0.399085i	$-0.130672\pi$
$971$	57.1437	1.83383	0.916914	+	0.399085i	$0.130672\pi$
$972$	−1.62222	−0.0520326
$973$	0	0
$974$	−21.9438	−0.703124
$975$	−0.622216	−0.0199268
$976$	31.6316	1.01250
$977$	−16.2480	−0.519819	−0.259909	−	0.965633i	$-0.583693\pi$
$977$	−16.2480	−0.519819	−0.259909	+	0.965633i	$0.583693\pi$
$978$	19.2257	0.614770
$979$	−5.61285	−0.179387
$980$	0	0
$981$	−19.7146	−0.629437
$982$	31.1882	0.995256
$983$	−1.12399	−0.0358496	−0.0179248	−	0.999839i	$-0.505706\pi$
$983$	−1.12399	−0.0358496	−0.0179248	+	0.999839i	$0.505706\pi$
$984$	0.139182	0.00443696
$985$	6.69535	0.213331
$986$	−77.0232	−2.45292
$987$	0	0
$988$	−7.11691	−0.226419
$989$	50.2480	1.59779
$990$	−1.90321	−0.0604880
$991$	−53.6513	−1.70429	−0.852144	−	0.523307i	$-0.824698\pi$
$991$	−53.6513	−1.70429	−0.852144	+	0.523307i	$0.824698\pi$
$992$	20.2292	0.642279
$993$	−15.3461	−0.486995
$994$	0	0
$995$	14.1017	0.447054
$996$	0.216327	0.00685460
$997$	35.7275	1.13150	0.565750	−	0.824577i	$-0.308587\pi$
$997$	35.7275	1.13150	0.565750	+	0.824577i	$0.308587\pi$
$998$	−48.2034	−1.52585
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.bk.1.3		3
7.6	odd	2		165.2.a.c.1.3	✓	3
21.20	even	2		495.2.a.e.1.1		3
28.27	even	2		2640.2.a.be.1.3		3
35.13	even	4		825.2.c.g.199.2		6
35.27	even	4		825.2.c.g.199.5		6
35.34	odd	2		825.2.a.k.1.1		3
77.76	even	2		1815.2.a.m.1.1		3
84.83	odd	2		7920.2.a.cj.1.3		3
105.62	odd	4		2475.2.c.r.199.2		6
105.83	odd	4		2475.2.c.r.199.5		6
105.104	even	2		2475.2.a.bb.1.3		3
231.230	odd	2		5445.2.a.z.1.3		3
385.384	even	2		9075.2.a.cf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.3	✓	3	7.6	odd	2
495.2.a.e.1.1		3	21.20	even	2
825.2.a.k.1.1		3	35.34	odd	2
825.2.c.g.199.2		6	35.13	even	4
825.2.c.g.199.5		6	35.27	even	4
1815.2.a.m.1.1		3	77.76	even	2
2475.2.a.bb.1.3		3	105.104	even	2
2475.2.c.r.199.2		6	105.62	odd	4
2475.2.c.r.199.5		6	105.83	odd	4
2640.2.a.be.1.3		3	28.27	even	2
5445.2.a.z.1.3		3	231.230	odd	2
7920.2.a.cj.1.3		3	84.83	odd	2
8085.2.a.bk.1.3		3	1.1	even	1	trivial
9075.2.a.cf.1.3		3	385.384	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:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} -1.00000 q^{5} -1.90321 q^{6} -0.719004 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} -1.00000 q^{5} -1.90321 q^{6} -0.719004 q^{8} +1.00000 q^{9} -1.90321 q^{10} +1.00000 q^{11} -1.62222 q^{12} +0.622216 q^{13} +1.00000 q^{15} -4.61285 q^{16} +5.18421 q^{17} +1.90321 q^{18} -7.05086 q^{19} -1.62222 q^{20} +1.90321 q^{22} +8.85728 q^{23} +0.719004 q^{24} +1.00000 q^{25} +1.18421 q^{26} -1.00000 q^{27} -7.80642 q^{29} +1.90321 q^{30} -2.75557 q^{31} -7.34122 q^{32} -1.00000 q^{33} +9.86665 q^{34} +1.62222 q^{36} -2.00000 q^{37} -13.4193 q^{38} -0.622216 q^{39} +0.719004 q^{40} +0.193576 q^{41} +5.67307 q^{43} +1.62222 q^{44} -1.00000 q^{45} +16.8573 q^{46} +2.75557 q^{47} +4.61285 q^{48} +1.90321 q^{50} -5.18421 q^{51} +1.00937 q^{52} -10.8573 q^{53} -1.90321 q^{54} -1.00000 q^{55} +7.05086 q^{57} -14.8573 q^{58} +4.85728 q^{59} +1.62222 q^{60} -6.85728 q^{61} -5.24443 q^{62} -4.74620 q^{64} -0.622216 q^{65} -1.90321 q^{66} -1.24443 q^{67} +8.40990 q^{68} -8.85728 q^{69} +2.75557 q^{71} -0.719004 q^{72} -4.23506 q^{73} -3.80642 q^{74} -1.00000 q^{75} -11.4380 q^{76} -1.18421 q^{78} +8.56199 q^{79} +4.61285 q^{80} +1.00000 q^{81} +0.368416 q^{82} -0.133353 q^{83} -5.18421 q^{85} +10.7971 q^{86} +7.80642 q^{87} -0.719004 q^{88} -5.61285 q^{89} -1.90321 q^{90} +14.3684 q^{92} +2.75557 q^{93} +5.24443 q^{94} +7.05086 q^{95} +7.34122 q^{96} -7.24443 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9	\( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + 5 * q^4 - 3 * q^5 + q^6 - 9 * q^8 + 3 * q^9 + q^10 + 3 * q^11 - 5 * q^12 + 2 * q^13 + 3 * q^15 + 13 * q^16 + 2 * q^17 - q^18 - 8 * q^19 - 5 * q^20 - q^22 + 9 * q^24 + 3 * q^25 - 10 * q^26 - 3 * q^27 - 10 * q^29 - q^30 - 8 * q^31 - 29 * q^32 - 3 * q^33 + 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 + 9 * q^40 + 14 * q^41 + 4 * q^43 + 5 * q^44 - 3 * q^45 + 24 * q^46 + 8 * q^47 - 13 * q^48 - q^50 - 2 * q^51 + 30 * q^52 - 6 * q^53 + q^54 - 3 * q^55 + 8 * q^57 - 18 * q^58 - 12 * q^59 + 5 * q^60 + 6 * q^61 - 16 * q^62 + 13 * q^64 - 2 * q^65 + q^66 - 4 * q^67 - 42 * q^68 + 8 * q^71 - 9 * q^72 + 14 * q^73 + 2 * q^74 - 3 * q^75 - 48 * q^76 + 10 * q^78 + 12 * q^79 - 13 * q^80 + 3 * q^81 - 26 * q^82 - 2 * q^85 - 8 * q^86 + 10 * q^87 - 9 * q^88 + 10 * q^89 + q^90 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 8 * q^95 + 29 * q^96 - 22 * q^97 + 3 * q^99

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