LMFDB - Embedded newform 8670.2.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.53209 q^{11} -1.00000 q^{12} +4.10607 q^{13} +3.87939 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -3.87939 q^{21} -1.53209 q^{22} +9.41147 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.10607 q^{26} -1.00000 q^{27} +3.87939 q^{28} +5.75877 q^{29} -1.00000 q^{30} +6.45336 q^{31} +1.00000 q^{32} +1.53209 q^{33} +3.87939 q^{35} +1.00000 q^{36} +0.305407 q^{37} +1.00000 q^{38} -4.10607 q^{39} +1.00000 q^{40} -11.5817 q^{41} -3.87939 q^{42} -12.5817 q^{43} -1.53209 q^{44} +1.00000 q^{45} +9.41147 q^{46} +5.63816 q^{47} -1.00000 q^{48} +8.04963 q^{49} +1.00000 q^{50} +4.10607 q^{52} +4.22668 q^{53} -1.00000 q^{54} -1.53209 q^{55} +3.87939 q^{56} -1.00000 q^{57} +5.75877 q^{58} -13.0496 q^{59} -1.00000 q^{60} +4.61081 q^{61} +6.45336 q^{62} +3.87939 q^{63} +1.00000 q^{64} +4.10607 q^{65} +1.53209 q^{66} -4.08378 q^{67} -9.41147 q^{69} +3.87939 q^{70} +14.2567 q^{71} +1.00000 q^{72} -15.4047 q^{73} +0.305407 q^{74} -1.00000 q^{75} +1.00000 q^{76} -5.94356 q^{77} -4.10607 q^{78} -7.19253 q^{79} +1.00000 q^{80} +1.00000 q^{81} -11.5817 q^{82} -3.87939 q^{84} -12.5817 q^{86} -5.75877 q^{87} -1.53209 q^{88} +6.24897 q^{89} +1.00000 q^{90} +15.9290 q^{91} +9.41147 q^{92} -6.45336 q^{93} +5.63816 q^{94} +1.00000 q^{95} -1.00000 q^{96} +2.24123 q^{97} +8.04963 q^{98} -1.53209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{12} + 6 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{18} + 3 q^{19} + 3 q^{20} - 6 q^{21} + 18 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} + 6 q^{29} - 3 q^{30} + 6 q^{31} + 3 q^{32} + 6 q^{35} + 3 q^{36} + 3 q^{37} + 3 q^{38} + 3 q^{40} - 3 q^{41} - 6 q^{42} - 6 q^{43} + 3 q^{45} + 18 q^{46} - 3 q^{48} - 3 q^{49} + 3 q^{50} + 6 q^{53} - 3 q^{54} + 6 q^{56} - 3 q^{57} + 6 q^{58} - 12 q^{59} - 3 q^{60} + 18 q^{61} + 6 q^{62} + 6 q^{63} + 3 q^{64} - 6 q^{67} - 18 q^{69} + 6 q^{70} + 6 q^{71} + 3 q^{72} + 6 q^{73} + 3 q^{74} - 3 q^{75} + 3 q^{76} - 3 q^{77} + 6 q^{79} + 3 q^{80} + 3 q^{81} - 3 q^{82} - 6 q^{84} - 6 q^{86} - 6 q^{87} + 6 q^{89} + 3 q^{90} + 15 q^{91} + 18 q^{92} - 6 q^{93} + 3 q^{95} - 3 q^{96} + 18 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^12 + 6 * q^14 - 3 * q^15 + 3 * q^16 + 3 * q^18 + 3 * q^19 + 3 * q^20 - 6 * q^21 + 18 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 + 6 * q^29 - 3 * q^30 + 6 * q^31 + 3 * q^32 + 6 * q^35 + 3 * q^36 + 3 * q^37 + 3 * q^38 + 3 * q^40 - 3 * q^41 - 6 * q^42 - 6 * q^43 + 3 * q^45 + 18 * q^46 - 3 * q^48 - 3 * q^49 + 3 * q^50 + 6 * q^53 - 3 * q^54 + 6 * q^56 - 3 * q^57 + 6 * q^58 - 12 * q^59 - 3 * q^60 + 18 * q^61 + 6 * q^62 + 6 * q^63 + 3 * q^64 - 6 * q^67 - 18 * q^69 + 6 * q^70 + 6 * q^71 + 3 * q^72 + 6 * q^73 + 3 * q^74 - 3 * q^75 + 3 * q^76 - 3 * q^77 + 6 * q^79 + 3 * q^80 + 3 * q^81 - 3 * q^82 - 6 * q^84 - 6 * q^86 - 6 * q^87 + 6 * q^89 + 3 * q^90 + 15 * q^91 + 18 * q^92 - 6 * q^93 + 3 * q^95 - 3 * q^96 + 18 * q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	3.87939	1.46627	0.733135	−	0.680083i	$-0.238056\pi$
$7$	3.87939	1.46627	0.733135	+	0.680083i	$0.238056\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.53209	−0.461942	−0.230971	−	0.972961i	$-0.574190\pi$
$11$	−1.53209	−0.461942	−0.230971	+	0.972961i	$0.574190\pi$
$12$	−1.00000	−0.288675
$13$	4.10607	1.13882	0.569409	−	0.822054i	$-0.307172\pi$
$13$	4.10607	1.13882	0.569409	+	0.822054i	$0.307172\pi$
$14$	3.87939	1.03681
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	1.00000	0.223607
$21$	−3.87939	−0.846551
$22$	−1.53209	−0.326642
$23$	9.41147	1.96243	0.981214	−	0.192922i	$-0.0617966\pi$
$23$	9.41147	1.96243	0.981214	+	0.192922i	$0.0617966\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.10607	0.805266
$27$	−1.00000	−0.192450
$28$	3.87939	0.733135
$29$	5.75877	1.06938	0.534688	−	0.845049i	$-0.320429\pi$
$29$	5.75877	1.06938	0.534688	+	0.845049i	$0.320429\pi$
$30$	−1.00000	−0.182574
$31$	6.45336	1.15906	0.579529	−	0.814952i	$-0.303236\pi$
$31$	6.45336	1.15906	0.579529	+	0.814952i	$0.303236\pi$
$32$	1.00000	0.176777
$33$	1.53209	0.266702
$34$	0	0
$35$	3.87939	0.655736
$36$	1.00000	0.166667
$37$	0.305407	0.0502086	0.0251043	−	0.999685i	$-0.492008\pi$
$37$	0.305407	0.0502086	0.0251043	+	0.999685i	$0.492008\pi$
$38$	1.00000	0.162221
$39$	−4.10607	−0.657497
$40$	1.00000	0.158114
$41$	−11.5817	−1.80876	−0.904380	−	0.426727i	$-0.859666\pi$
$41$	−11.5817	−1.80876	−0.904380	+	0.426727i	$0.859666\pi$
$42$	−3.87939	−0.598602
$43$	−12.5817	−1.91869	−0.959347	−	0.282229i	$-0.908926\pi$
$43$	−12.5817	−1.91869	−0.959347	+	0.282229i	$0.908926\pi$
$44$	−1.53209	−0.230971
$45$	1.00000	0.149071
$46$	9.41147	1.38765
$47$	5.63816	0.822410	0.411205	−	0.911543i	$-0.365108\pi$
$47$	5.63816	0.822410	0.411205	+	0.911543i	$0.365108\pi$
$48$	−1.00000	−0.144338
$49$	8.04963	1.14995
$50$	1.00000	0.141421
$51$	0	0
$52$	4.10607	0.569409
$53$	4.22668	0.580579	0.290290	−	0.956939i	$-0.406248\pi$
$53$	4.22668	0.580579	0.290290	+	0.956939i	$0.406248\pi$
$54$	−1.00000	−0.136083
$55$	−1.53209	−0.206587
$56$	3.87939	0.518405
$57$	−1.00000	−0.132453
$58$	5.75877	0.756164
$59$	−13.0496	−1.69892	−0.849459	−	0.527655i	$-0.823071\pi$
$59$	−13.0496	−1.69892	−0.849459	+	0.527655i	$0.823071\pi$
$60$	−1.00000	−0.129099
$61$	4.61081	0.590354	0.295177	−	0.955443i	$-0.404621\pi$
$61$	4.61081	0.590354	0.295177	+	0.955443i	$0.404621\pi$
$62$	6.45336	0.819578
$63$	3.87939	0.488757
$64$	1.00000	0.125000
$65$	4.10607	0.509295
$66$	1.53209	0.188587
$67$	−4.08378	−0.498913	−0.249456	−	0.968386i	$-0.580252\pi$
$67$	−4.08378	−0.498913	−0.249456	+	0.968386i	$0.580252\pi$
$68$	0	0
$69$	−9.41147	−1.13301
$70$	3.87939	0.463675
$71$	14.2567	1.69196	0.845980	−	0.533214i	$-0.179016\pi$
$71$	14.2567	1.69196	0.845980	+	0.533214i	$0.179016\pi$
$72$	1.00000	0.117851
$73$	−15.4047	−1.80298	−0.901490	−	0.432800i	$-0.857526\pi$
$73$	−15.4047	−1.80298	−0.901490	+	0.432800i	$0.857526\pi$
$74$	0.305407	0.0355029
$75$	−1.00000	−0.115470
$76$	1.00000	0.114708
$77$	−5.94356	−0.677332
$78$	−4.10607	−0.464921
$79$	−7.19253	−0.809223	−0.404612	−	0.914489i	$-0.632593\pi$
$79$	−7.19253	−0.809223	−0.404612	+	0.914489i	$0.632593\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−11.5817	−1.27899
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−3.87939	−0.423276
$85$	0	0
$86$	−12.5817	−1.35672
$87$	−5.75877	−0.617405
$88$	−1.53209	−0.163321
$89$	6.24897	0.662390	0.331195	−	0.943562i	$-0.392548\pi$
$89$	6.24897	0.662390	0.331195	+	0.943562i	$0.392548\pi$
$90$	1.00000	0.105409
$91$	15.9290	1.66981
$92$	9.41147	0.981214
$93$	−6.45336	−0.669183
$94$	5.63816	0.581531
$95$	1.00000	0.102598
$96$	−1.00000	−0.102062
$97$	2.24123	0.227562	0.113781	−	0.993506i	$-0.463704\pi$
$97$	2.24123	0.227562	0.113781	+	0.993506i	$0.463704\pi$
$98$	8.04963	0.813135
$99$	−1.53209	−0.153981
$100$	1.00000	0.100000
$101$	11.5175	1.14604	0.573019	−	0.819542i	$-0.305772\pi$
$101$	11.5175	1.14604	0.573019	+	0.819542i	$0.305772\pi$
$102$	0	0
$103$	−7.69459	−0.758171	−0.379085	−	0.925362i	$-0.623761\pi$
$103$	−7.69459	−0.758171	−0.379085	+	0.925362i	$0.623761\pi$
$104$	4.10607	0.402633
$105$	−3.87939	−0.378589
$106$	4.22668	0.410532
$107$	0.694593	0.0671488	0.0335744	−	0.999436i	$-0.489311\pi$
$107$	0.694593	0.0671488	0.0335744	+	0.999436i	$0.489311\pi$
$108$	−1.00000	−0.0962250
$109$	12.1284	1.16169	0.580843	−	0.814016i	$-0.302723\pi$
$109$	12.1284	1.16169	0.580843	+	0.814016i	$0.302723\pi$
$110$	−1.53209	−0.146079
$111$	−0.305407	−0.0289880
$112$	3.87939	0.366567
$113$	−3.43376	−0.323021	−0.161511	−	0.986871i	$-0.551637\pi$
$113$	−3.43376	−0.323021	−0.161511	+	0.986871i	$0.551637\pi$
$114$	−1.00000	−0.0936586
$115$	9.41147	0.877624
$116$	5.75877	0.534688
$117$	4.10607	0.379606
$118$	−13.0496	−1.20132
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−8.65270	−0.786609
$122$	4.61081	0.417444
$123$	11.5817	1.04429
$124$	6.45336	0.579529
$125$	1.00000	0.0894427
$126$	3.87939	0.345603
$127$	−7.34049	−0.651363	−0.325682	−	0.945480i	$-0.605594\pi$
$127$	−7.34049	−0.651363	−0.325682	+	0.945480i	$0.605594\pi$
$128$	1.00000	0.0883883
$129$	12.5817	1.10776
$130$	4.10607	0.360126
$131$	12.4757	1.09000	0.545001	−	0.838435i	$-0.316529\pi$
$131$	12.4757	1.09000	0.545001	+	0.838435i	$0.316529\pi$
$132$	1.53209	0.133351
$133$	3.87939	0.336385
$134$	−4.08378	−0.352785
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.45336	−0.551348	−0.275674	−	0.961251i	$-0.588901\pi$
$137$	−6.45336	−0.551348	−0.275674	+	0.961251i	$0.588901\pi$
$138$	−9.41147	−0.801158
$139$	11.6108	0.984816	0.492408	−	0.870364i	$-0.336117\pi$
$139$	11.6108	0.984816	0.492408	+	0.870364i	$0.336117\pi$
$140$	3.87939	0.327868
$141$	−5.63816	−0.474818
$142$	14.2567	1.19640
$143$	−6.29086	−0.526068
$144$	1.00000	0.0833333
$145$	5.75877	0.478240
$146$	−15.4047	−1.27490
$147$	−8.04963	−0.663922
$148$	0.305407	0.0251043
$149$	−19.9317	−1.63287	−0.816434	−	0.577438i	$-0.804052\pi$
$149$	−19.9317	−1.63287	−0.816434	+	0.577438i	$0.804052\pi$
$150$	−1.00000	−0.0816497
$151$	−22.8229	−1.85731	−0.928653	−	0.370951i	$-0.879032\pi$
$151$	−22.8229	−1.85731	−0.928653	+	0.370951i	$0.879032\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−5.94356	−0.478946
$155$	6.45336	0.518347
$156$	−4.10607	−0.328748
$157$	−0.573978	−0.0458084	−0.0229042	−	0.999738i	$-0.507291\pi$
$157$	−0.573978	−0.0458084	−0.0229042	+	0.999738i	$0.507291\pi$
$158$	−7.19253	−0.572207
$159$	−4.22668	−0.335198
$160$	1.00000	0.0790569
$161$	36.5107	2.87745
$162$	1.00000	0.0785674
$163$	−13.4338	−1.05221	−0.526107	−	0.850419i	$-0.676349\pi$
$163$	−13.4338	−1.05221	−0.526107	+	0.850419i	$0.676349\pi$
$164$	−11.5817	−0.904380
$165$	1.53209	0.119273
$166$	0	0
$167$	8.01455	0.620184	0.310092	−	0.950706i	$-0.399640\pi$
$167$	8.01455	0.620184	0.310092	+	0.950706i	$0.399640\pi$
$168$	−3.87939	−0.299301
$169$	3.85978	0.296907
$170$	0	0
$171$	1.00000	0.0764719
$172$	−12.5817	−0.959347
$173$	−4.43107	−0.336888	−0.168444	−	0.985711i	$-0.553874\pi$
$173$	−4.43107	−0.336888	−0.168444	+	0.985711i	$0.553874\pi$
$174$	−5.75877	−0.436571
$175$	3.87939	0.293254
$176$	−1.53209	−0.115486
$177$	13.0496	0.980870
$178$	6.24897	0.468380
$179$	4.95811	0.370587	0.185293	−	0.982683i	$-0.440676\pi$
$179$	4.95811	0.370587	0.185293	+	0.982683i	$0.440676\pi$
$180$	1.00000	0.0745356
$181$	4.28581	0.318562	0.159281	−	0.987233i	$-0.449082\pi$
$181$	4.28581	0.318562	0.159281	+	0.987233i	$0.449082\pi$
$182$	15.9290	1.18074
$183$	−4.61081	−0.340841
$184$	9.41147	0.693823
$185$	0.305407	0.0224540
$186$	−6.45336	−0.473184
$187$	0	0
$188$	5.63816	0.411205
$189$	−3.87939	−0.282184
$190$	1.00000	0.0725476
$191$	10.0155	0.724695	0.362347	−	0.932043i	$-0.381975\pi$
$191$	10.0155	0.724695	0.362347	+	0.932043i	$0.381975\pi$
$192$	−1.00000	−0.0721688
$193$	−12.6108	−0.907746	−0.453873	−	0.891066i	$-0.649958\pi$
$193$	−12.6108	−0.907746	−0.453873	+	0.891066i	$0.649958\pi$
$194$	2.24123	0.160911
$195$	−4.10607	−0.294042
$196$	8.04963	0.574974
$197$	2.55169	0.181800	0.0909002	−	0.995860i	$-0.471026\pi$
$197$	2.55169	0.181800	0.0909002	+	0.995860i	$0.471026\pi$
$198$	−1.53209	−0.108881
$199$	−16.0993	−1.14125	−0.570623	−	0.821212i	$-0.693298\pi$
$199$	−16.0993	−1.14125	−0.570623	+	0.821212i	$0.693298\pi$
$200$	1.00000	0.0707107
$201$	4.08378	0.288048
$202$	11.5175	0.810371
$203$	22.3405	1.56799
$204$	0	0
$205$	−11.5817	−0.808902
$206$	−7.69459	−0.536108
$207$	9.41147	0.654143
$208$	4.10607	0.284705
$209$	−1.53209	−0.105977
$210$	−3.87939	−0.267703
$211$	20.2199	1.39199	0.695997	−	0.718045i	$-0.254963\pi$
$211$	20.2199	1.39199	0.695997	+	0.718045i	$0.254963\pi$
$212$	4.22668	0.290290
$213$	−14.2567	−0.976854
$214$	0.694593	0.0474814
$215$	−12.5817	−0.858066
$216$	−1.00000	−0.0680414
$217$	25.0351	1.69949
$218$	12.1284	0.821436
$219$	15.4047	1.04095
$220$	−1.53209	−0.103293
$221$	0	0
$222$	−0.305407	−0.0204976
$223$	2.47565	0.165782	0.0828910	−	0.996559i	$-0.473585\pi$
$223$	2.47565	0.165782	0.0828910	+	0.996559i	$0.473585\pi$
$224$	3.87939	0.259202
$225$	1.00000	0.0666667
$226$	−3.43376	−0.228411
$227$	10.1284	0.672243	0.336121	−	0.941819i	$-0.390885\pi$
$227$	10.1284	0.672243	0.336121	+	0.941819i	$0.390885\pi$
$228$	−1.00000	−0.0662266
$229$	−11.5621	−0.764046	−0.382023	−	0.924153i	$-0.624773\pi$
$229$	−11.5621	−0.764046	−0.382023	+	0.924153i	$0.624773\pi$
$230$	9.41147	0.620574
$231$	5.94356	0.391058
$232$	5.75877	0.378082
$233$	4.65539	0.304985	0.152492	−	0.988305i	$-0.451270\pi$
$233$	4.65539	0.304985	0.152492	+	0.988305i	$0.451270\pi$
$234$	4.10607	0.268422
$235$	5.63816	0.367793
$236$	−13.0496	−0.849459
$237$	7.19253	0.467205
$238$	0	0
$239$	−18.4534	−1.19365	−0.596824	−	0.802372i	$-0.703571\pi$
$239$	−18.4534	−1.19365	−0.596824	+	0.802372i	$0.703571\pi$
$240$	−1.00000	−0.0645497
$241$	−27.0428	−1.74198	−0.870991	−	0.491300i	$-0.836522\pi$
$241$	−27.0428	−1.74198	−0.870991	+	0.491300i	$0.836522\pi$
$242$	−8.65270	−0.556217
$243$	−1.00000	−0.0641500
$244$	4.61081	0.295177
$245$	8.04963	0.514272
$246$	11.5817	0.738423
$247$	4.10607	0.261263
$248$	6.45336	0.409789
$249$	0	0
$250$	1.00000	0.0632456
$251$	27.7989	1.75465	0.877326	−	0.479895i	$-0.159325\pi$
$251$	27.7989	1.75465	0.877326	+	0.479895i	$0.159325\pi$
$252$	3.87939	0.244378
$253$	−14.4192	−0.906528
$254$	−7.34049	−0.460583
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.99050	−0.560812	−0.280406	−	0.959881i	$-0.590469\pi$
$257$	−8.99050	−0.560812	−0.280406	+	0.959881i	$0.590469\pi$
$258$	12.5817	0.783304
$259$	1.18479	0.0736194
$260$	4.10607	0.254647
$261$	5.75877	0.356459
$262$	12.4757	0.770748
$263$	2.37639	0.146535	0.0732673	−	0.997312i	$-0.476657\pi$
$263$	2.37639	0.146535	0.0732673	+	0.997312i	$0.476657\pi$
$264$	1.53209	0.0942936
$265$	4.22668	0.259643
$266$	3.87939	0.237860
$267$	−6.24897	−0.382431
$268$	−4.08378	−0.249456
$269$	24.7493	1.50899	0.754495	−	0.656306i	$-0.227882\pi$
$269$	24.7493	1.50899	0.754495	+	0.656306i	$0.227882\pi$
$270$	−1.00000	−0.0608581
$271$	−12.6209	−0.766666	−0.383333	−	0.923610i	$-0.625224\pi$
$271$	−12.6209	−0.766666	−0.383333	+	0.923610i	$0.625224\pi$
$272$	0	0
$273$	−15.9290	−0.964068
$274$	−6.45336	−0.389862
$275$	−1.53209	−0.0923884
$276$	−9.41147	−0.566504
$277$	10.1898	0.612248	0.306124	−	0.951992i	$-0.400968\pi$
$277$	10.1898	0.612248	0.306124	+	0.951992i	$0.400968\pi$
$278$	11.6108	0.696370
$279$	6.45336	0.386353
$280$	3.87939	0.231838
$281$	−27.2226	−1.62396	−0.811981	−	0.583684i	$-0.801611\pi$
$281$	−27.2226	−1.62396	−0.811981	+	0.583684i	$0.801611\pi$
$282$	−5.63816	−0.335747
$283$	−10.6554	−0.633397	−0.316699	−	0.948526i	$-0.602574\pi$
$283$	−10.6554	−0.633397	−0.316699	+	0.948526i	$0.602574\pi$
$284$	14.2567	0.845980
$285$	−1.00000	−0.0592349
$286$	−6.29086	−0.371986
$287$	−44.9299	−2.65213
$288$	1.00000	0.0589256
$289$	0	0
$290$	5.75877	0.338167
$291$	−2.24123	−0.131383
$292$	−15.4047	−0.901490
$293$	3.12330	0.182465	0.0912327	−	0.995830i	$-0.470919\pi$
$293$	3.12330	0.182465	0.0912327	+	0.995830i	$0.470919\pi$
$294$	−8.04963	−0.469464
$295$	−13.0496	−0.759779
$296$	0.305407	0.0177514
$297$	1.53209	0.0889008
$298$	−19.9317	−1.15461
$299$	38.6441	2.23485
$300$	−1.00000	−0.0577350
$301$	−48.8093	−2.81332
$302$	−22.8229	−1.31331
$303$	−11.5175	−0.661665
$304$	1.00000	0.0573539
$305$	4.61081	0.264014
$306$	0	0
$307$	−27.9864	−1.59727	−0.798634	−	0.601817i	$-0.794443\pi$
$307$	−27.9864	−1.59727	−0.798634	+	0.601817i	$0.794443\pi$
$308$	−5.94356	−0.338666
$309$	7.69459	0.437730
$310$	6.45336	0.366526
$311$	6.98040	0.395822	0.197911	−	0.980220i	$-0.436584\pi$
$311$	6.98040	0.395822	0.197911	+	0.980220i	$0.436584\pi$
$312$	−4.10607	−0.232460
$313$	27.8188	1.57241	0.786207	−	0.617964i	$-0.212042\pi$
$313$	27.8188	1.57241	0.786207	+	0.617964i	$0.212042\pi$
$314$	−0.573978	−0.0323914
$315$	3.87939	0.218579
$316$	−7.19253	−0.404612
$317$	−2.27631	−0.127850	−0.0639252	−	0.997955i	$-0.520362\pi$
$317$	−2.27631	−0.127850	−0.0639252	+	0.997955i	$0.520362\pi$
$318$	−4.22668	−0.237021
$319$	−8.82295	−0.493990
$320$	1.00000	0.0559017
$321$	−0.694593	−0.0387684
$322$	36.5107	2.03466
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.10607	0.227764
$326$	−13.4338	−0.744027
$327$	−12.1284	−0.670700
$328$	−11.5817	−0.639493
$329$	21.8726	1.20587
$330$	1.53209	0.0843387
$331$	−9.20027	−0.505693	−0.252846	−	0.967506i	$-0.581367\pi$
$331$	−9.20027	−0.505693	−0.252846	+	0.967506i	$0.581367\pi$
$332$	0	0
$333$	0.305407	0.0167362
$334$	8.01455	0.438537
$335$	−4.08378	−0.223121
$336$	−3.87939	−0.211638
$337$	−16.9513	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$337$	−16.9513	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$338$	3.85978	0.209945
$339$	3.43376	0.186496
$340$	0	0
$341$	−9.88713	−0.535418
$342$	1.00000	0.0540738
$343$	4.07192	0.219863
$344$	−12.5817	−0.678361
$345$	−9.41147	−0.506697
$346$	−4.43107	−0.238216
$347$	24.9067	1.33706	0.668532	−	0.743684i	$-0.266923\pi$
$347$	24.9067	1.33706	0.668532	+	0.743684i	$0.266923\pi$
$348$	−5.75877	−0.308703
$349$	9.77425	0.523204	0.261602	−	0.965176i	$-0.415749\pi$
$349$	9.77425	0.523204	0.261602	+	0.965176i	$0.415749\pi$
$350$	3.87939	0.207362
$351$	−4.10607	−0.219166
$352$	−1.53209	−0.0816606
$353$	11.3601	0.604637	0.302318	−	0.953207i	$-0.402239\pi$
$353$	11.3601	0.604637	0.302318	+	0.953207i	$0.402239\pi$
$354$	13.0496	0.693580
$355$	14.2567	0.756668
$356$	6.24897	0.331195
$357$	0	0
$358$	4.95811	0.262044
$359$	11.3892	0.601098	0.300549	−	0.953766i	$-0.402830\pi$
$359$	11.3892	0.601098	0.300549	+	0.953766i	$0.402830\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	4.28581	0.225257
$363$	8.65270	0.454149
$364$	15.9290	0.834907
$365$	−15.4047	−0.806317
$366$	−4.61081	−0.241011
$367$	−23.0651	−1.20399	−0.601995	−	0.798500i	$-0.705627\pi$
$367$	−23.0651	−1.20399	−0.601995	+	0.798500i	$0.705627\pi$
$368$	9.41147	0.490607
$369$	−11.5817	−0.602920
$370$	0.305407	0.0158774
$371$	16.3969	0.851286
$372$	−6.45336	−0.334591
$373$	13.0719	0.676838	0.338419	−	0.940995i	$-0.390108\pi$
$373$	13.0719	0.676838	0.338419	+	0.940995i	$0.390108\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	5.63816	0.290766
$377$	23.6459	1.21783
$378$	−3.87939	−0.199534
$379$	14.1284	0.725725	0.362862	−	0.931843i	$-0.381799\pi$
$379$	14.1284	0.725725	0.362862	+	0.931843i	$0.381799\pi$
$380$	1.00000	0.0512989
$381$	7.34049	0.376065
$382$	10.0155	0.512437
$383$	1.72369	0.0880764	0.0440382	−	0.999030i	$-0.485978\pi$
$383$	1.72369	0.0880764	0.0440382	+	0.999030i	$0.485978\pi$
$384$	−1.00000	−0.0510310
$385$	−5.94356	−0.302912
$386$	−12.6108	−0.641874
$387$	−12.5817	−0.639565
$388$	2.24123	0.113781
$389$	−7.06418	−0.358168	−0.179084	−	0.983834i	$-0.557313\pi$
$389$	−7.06418	−0.358168	−0.179084	+	0.983834i	$0.557313\pi$
$390$	−4.10607	−0.207919
$391$	0	0
$392$	8.04963	0.406568
$393$	−12.4757	−0.629313
$394$	2.55169	0.128552
$395$	−7.19253	−0.361896
$396$	−1.53209	−0.0769904
$397$	30.7033	1.54095	0.770476	−	0.637469i	$-0.220018\pi$
$397$	30.7033	1.54095	0.770476	+	0.637469i	$0.220018\pi$
$398$	−16.0993	−0.806983
$399$	−3.87939	−0.194212
$400$	1.00000	0.0500000
$401$	8.97596	0.448238	0.224119	−	0.974562i	$-0.428050\pi$
$401$	8.97596	0.448238	0.224119	+	0.974562i	$0.428050\pi$
$402$	4.08378	0.203680
$403$	26.4979	1.31996
$404$	11.5175	0.573019
$405$	1.00000	0.0496904
$406$	22.3405	1.10874
$407$	−0.467911	−0.0231935
$408$	0	0
$409$	10.4311	0.515783	0.257892	−	0.966174i	$-0.416972\pi$
$409$	10.4311	0.515783	0.257892	+	0.966174i	$0.416972\pi$
$410$	−11.5817	−0.571980
$411$	6.45336	0.318321
$412$	−7.69459	−0.379085
$413$	−50.6245	−2.49107
$414$	9.41147	0.462549
$415$	0	0
$416$	4.10607	0.201316
$417$	−11.6108	−0.568584
$418$	−1.53209	−0.0749369
$419$	−1.86484	−0.0911033	−0.0455516	−	0.998962i	$-0.514505\pi$
$419$	−1.86484	−0.0911033	−0.0455516	+	0.998962i	$0.514505\pi$
$420$	−3.87939	−0.189295
$421$	3.84255	0.187274	0.0936372	−	0.995606i	$-0.470151\pi$
$421$	3.84255	0.187274	0.0936372	+	0.995606i	$0.470151\pi$
$422$	20.2199	0.984288
$423$	5.63816	0.274137
$424$	4.22668	0.205266
$425$	0	0
$426$	−14.2567	−0.690740
$427$	17.8871	0.865619
$428$	0.694593	0.0335744
$429$	6.29086	0.303726
$430$	−12.5817	−0.606744
$431$	−17.9608	−0.865141	−0.432571	−	0.901600i	$-0.642393\pi$
$431$	−17.9608	−0.865141	−0.432571	+	0.901600i	$0.642393\pi$
$432$	−1.00000	−0.0481125
$433$	0.723689	0.0347783	0.0173891	−	0.999849i	$-0.494465\pi$
$433$	0.723689	0.0347783	0.0173891	+	0.999849i	$0.494465\pi$
$434$	25.0351	1.20172
$435$	−5.75877	−0.276112
$436$	12.1284	0.580843
$437$	9.41147	0.450212
$438$	15.4047	0.736063
$439$	29.5722	1.41141	0.705703	−	0.708508i	$-0.250631\pi$
$439$	29.5722	1.41141	0.705703	+	0.708508i	$0.250631\pi$
$440$	−1.53209	−0.0730395
$441$	8.04963	0.383316
$442$	0	0
$443$	24.3851	1.15857	0.579285	−	0.815125i	$-0.303332\pi$
$443$	24.3851	1.15857	0.579285	+	0.815125i	$0.303332\pi$
$444$	−0.305407	−0.0144940
$445$	6.24897	0.296230
$446$	2.47565	0.117226
$447$	19.9317	0.942737
$448$	3.87939	0.183284
$449$	12.3851	0.584487	0.292244	−	0.956344i	$-0.405598\pi$
$449$	12.3851	0.584487	0.292244	+	0.956344i	$0.405598\pi$
$450$	1.00000	0.0471405
$451$	17.7442	0.835543
$452$	−3.43376	−0.161511
$453$	22.8229	1.07232
$454$	10.1284	0.475347
$455$	15.9290	0.746764
$456$	−1.00000	−0.0468293
$457$	18.3797	0.859766	0.429883	−	0.902885i	$-0.358555\pi$
$457$	18.3797	0.859766	0.429883	+	0.902885i	$0.358555\pi$
$458$	−11.5621	−0.540262
$459$	0	0
$460$	9.41147	0.438812
$461$	−11.9263	−0.555464	−0.277732	−	0.960659i	$-0.589583\pi$
$461$	−11.9263	−0.555464	−0.277732	+	0.960659i	$0.589583\pi$
$462$	5.94356	0.276520
$463$	30.5996	1.42208	0.711041	−	0.703150i	$-0.248224\pi$
$463$	30.5996	1.42208	0.711041	+	0.703150i	$0.248224\pi$
$464$	5.75877	0.267344
$465$	−6.45336	−0.299268
$466$	4.65539	0.215657
$467$	14.8075	0.685208	0.342604	−	0.939480i	$-0.388691\pi$
$467$	14.8075	0.685208	0.342604	+	0.939480i	$0.388691\pi$
$468$	4.10607	0.189803
$469$	−15.8425	−0.731541
$470$	5.63816	0.260069
$471$	0.573978	0.0264475
$472$	−13.0496	−0.600658
$473$	19.2763	0.886326
$474$	7.19253	0.330364
$475$	1.00000	0.0458831
$476$	0	0
$477$	4.22668	0.193526
$478$	−18.4534	−0.844037
$479$	−28.8093	−1.31633	−0.658166	−	0.752873i	$-0.728668\pi$
$479$	−28.8093	−1.31633	−0.658166	+	0.752873i	$0.728668\pi$
$480$	−1.00000	−0.0456435
$481$	1.25402	0.0571785
$482$	−27.0428	−1.23177
$483$	−36.5107	−1.66130
$484$	−8.65270	−0.393305
$485$	2.24123	0.101769
$486$	−1.00000	−0.0453609
$487$	17.8229	0.807635	0.403817	−	0.914840i	$-0.367683\pi$
$487$	17.8229	0.807635	0.403817	+	0.914840i	$0.367683\pi$
$488$	4.61081	0.208722
$489$	13.4338	0.607496
$490$	8.04963	0.363645
$491$	−10.8699	−0.490551	−0.245276	−	0.969453i	$-0.578878\pi$
$491$	−10.8699	−0.490551	−0.245276	+	0.969453i	$0.578878\pi$
$492$	11.5817	0.522144
$493$	0	0
$494$	4.10607	0.184741
$495$	−1.53209	−0.0688623
$496$	6.45336	0.289765
$497$	55.3073	2.48087
$498$	0	0
$499$	23.0719	1.03284	0.516420	−	0.856335i	$-0.327264\pi$
$499$	23.0719	1.03284	0.516420	+	0.856335i	$0.327264\pi$
$500$	1.00000	0.0447214
$501$	−8.01455	−0.358064
$502$	27.7989	1.24073
$503$	−27.1465	−1.21040	−0.605202	−	0.796072i	$-0.706908\pi$
$503$	−27.1465	−1.21040	−0.605202	+	0.796072i	$0.706908\pi$
$504$	3.87939	0.172802
$505$	11.5175	0.512524
$506$	−14.4192	−0.641012
$507$	−3.85978	−0.171419
$508$	−7.34049	−0.325682
$509$	20.6946	0.917272	0.458636	−	0.888624i	$-0.348338\pi$
$509$	20.6946	0.917272	0.458636	+	0.888624i	$0.348338\pi$
$510$	0	0
$511$	−59.7606	−2.64365
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−8.99050	−0.396554
$515$	−7.69459	−0.339064
$516$	12.5817	0.553879
$517$	−8.63816	−0.379906
$518$	1.18479	0.0520568
$519$	4.43107	0.194503
$520$	4.10607	0.180063
$521$	−1.49256	−0.0653904	−0.0326952	−	0.999465i	$-0.510409\pi$
$521$	−1.49256	−0.0653904	−0.0326952	+	0.999465i	$0.510409\pi$
$522$	5.75877	0.252055
$523$	36.5134	1.59662	0.798310	−	0.602246i	$-0.205728\pi$
$523$	36.5134	1.59662	0.798310	+	0.602246i	$0.205728\pi$
$524$	12.4757	0.545001
$525$	−3.87939	−0.169310
$526$	2.37639	0.103616
$527$	0	0
$528$	1.53209	0.0666756
$529$	65.5758	2.85112
$530$	4.22668	0.183595
$531$	−13.0496	−0.566306
$532$	3.87939	0.168193
$533$	−47.5553	−2.05985
$534$	−6.24897	−0.270419
$535$	0.694593	0.0300299
$536$	−4.08378	−0.176392
$537$	−4.95811	−0.213958
$538$	24.7493	1.06702
$539$	−12.3327	−0.531209
$540$	−1.00000	−0.0430331
$541$	−5.23173	−0.224930	−0.112465	−	0.993656i	$-0.535875\pi$
$541$	−5.23173	−0.224930	−0.112465	+	0.993656i	$0.535875\pi$
$542$	−12.6209	−0.542115
$543$	−4.28581	−0.183922
$544$	0	0
$545$	12.1284	0.519522
$546$	−15.9290	−0.681699
$547$	9.63579	0.411997	0.205998	−	0.978552i	$-0.433956\pi$
$547$	9.63579	0.411997	0.205998	+	0.978552i	$0.433956\pi$
$548$	−6.45336	−0.275674
$549$	4.61081	0.196785
$550$	−1.53209	−0.0653285
$551$	5.75877	0.245332
$552$	−9.41147	−0.400579
$553$	−27.9026	−1.18654
$554$	10.1898	0.432925
$555$	−0.305407	−0.0129638
$556$	11.6108	0.492408
$557$	−26.5895	−1.12663	−0.563316	−	0.826242i	$-0.690475\pi$
$557$	−26.5895	−1.12663	−0.563316	+	0.826242i	$0.690475\pi$
$558$	6.45336	0.273193
$559$	−51.6614	−2.18504
$560$	3.87939	0.163934
$561$	0	0
$562$	−27.2226	−1.14831
$563$	27.9263	1.17695	0.588477	−	0.808514i	$-0.299728\pi$
$563$	27.9263	1.17695	0.588477	+	0.808514i	$0.299728\pi$
$564$	−5.63816	−0.237409
$565$	−3.43376	−0.144459
$566$	−10.6554	−0.447880
$567$	3.87939	0.162919
$568$	14.2567	0.598198
$569$	17.6209	0.738707	0.369354	−	0.929289i	$-0.379579\pi$
$569$	17.6209	0.738707	0.369354	+	0.929289i	$0.379579\pi$
$570$	−1.00000	−0.0418854
$571$	5.97771	0.250159	0.125080	−	0.992147i	$-0.460081\pi$
$571$	5.97771	0.250159	0.125080	+	0.992147i	$0.460081\pi$
$572$	−6.29086	−0.263034
$573$	−10.0155	−0.418403
$574$	−44.9299	−1.87534
$575$	9.41147	0.392486
$576$	1.00000	0.0416667
$577$	−7.75877	−0.323002	−0.161501	−	0.986873i	$-0.551633\pi$
$577$	−7.75877	−0.323002	−0.161501	+	0.986873i	$0.551633\pi$
$578$	0	0
$579$	12.6108	0.524088
$580$	5.75877	0.239120
$581$	0	0
$582$	−2.24123	−0.0929020
$583$	−6.47565	−0.268194
$584$	−15.4047	−0.637450
$585$	4.10607	0.169765
$586$	3.12330	0.129022
$587$	29.6013	1.22178	0.610889	−	0.791717i	$-0.290812\pi$
$587$	29.6013	1.22178	0.610889	+	0.791717i	$0.290812\pi$
$588$	−8.04963	−0.331961
$589$	6.45336	0.265906
$590$	−13.0496	−0.537245
$591$	−2.55169	−0.104962
$592$	0.305407	0.0125522
$593$	−43.2336	−1.77539	−0.887696	−	0.460431i	$-0.847695\pi$
$593$	−43.2336	−1.77539	−0.887696	+	0.460431i	$0.847695\pi$
$594$	1.53209	0.0628624
$595$	0	0
$596$	−19.9317	−0.816434
$597$	16.0993	0.658899
$598$	38.6441	1.58028
$599$	27.3209	1.11630	0.558151	−	0.829740i	$-0.311511\pi$
$599$	27.3209	1.11630	0.558151	+	0.829740i	$0.311511\pi$
$600$	−1.00000	−0.0408248
$601$	−31.6604	−1.29146	−0.645728	−	0.763567i	$-0.723446\pi$
$601$	−31.6604	−1.29146	−0.645728	+	0.763567i	$0.723446\pi$
$602$	−48.8093	−1.98932
$603$	−4.08378	−0.166304
$604$	−22.8229	−0.928653
$605$	−8.65270	−0.351782
$606$	−11.5175	−0.467868
$607$	10.9135	0.442967	0.221483	−	0.975164i	$-0.428910\pi$
$607$	10.9135	0.442967	0.221483	+	0.975164i	$0.428910\pi$
$608$	1.00000	0.0405554
$609$	−22.3405	−0.905282
$610$	4.61081	0.186686
$611$	23.1506	0.936575
$612$	0	0
$613$	−0.243594	−0.00983865	−0.00491933	−	0.999988i	$-0.501566\pi$
$613$	−0.243594	−0.00983865	−0.00491933	+	0.999988i	$0.501566\pi$
$614$	−27.9864	−1.12944
$615$	11.5817	0.467020
$616$	−5.94356	−0.239473
$617$	−24.1830	−0.973572	−0.486786	−	0.873521i	$-0.661831\pi$
$617$	−24.1830	−0.973572	−0.486786	+	0.873521i	$0.661831\pi$
$618$	7.69459	0.309522
$619$	31.3286	1.25920	0.629602	−	0.776918i	$-0.283218\pi$
$619$	31.3286	1.25920	0.629602	+	0.776918i	$0.283218\pi$
$620$	6.45336	0.259173
$621$	−9.41147	−0.377669
$622$	6.98040	0.279889
$623$	24.2422	0.971242
$624$	−4.10607	−0.164374
$625$	1.00000	0.0400000
$626$	27.8188	1.11186
$627$	1.53209	0.0611857
$628$	−0.573978	−0.0229042
$629$	0	0
$630$	3.87939	0.154558
$631$	−22.1967	−0.883635	−0.441817	−	0.897105i	$-0.645666\pi$
$631$	−22.1967	−0.883635	−0.441817	+	0.897105i	$0.645666\pi$
$632$	−7.19253	−0.286104
$633$	−20.2199	−0.803668
$634$	−2.27631	−0.0904039
$635$	−7.34049	−0.291298
$636$	−4.22668	−0.167599
$637$	33.0523	1.30958
$638$	−8.82295	−0.349304
$639$	14.2567	0.563987
$640$	1.00000	0.0395285
$641$	−39.1557	−1.54656	−0.773279	−	0.634067i	$-0.781385\pi$
$641$	−39.1557	−1.54656	−0.773279	+	0.634067i	$0.781385\pi$
$642$	−0.694593	−0.0274134
$643$	−17.3054	−0.682459	−0.341229	−	0.939980i	$-0.610843\pi$
$643$	−17.3054	−0.682459	−0.341229	+	0.939980i	$0.610843\pi$
$644$	36.5107	1.43872
$645$	12.5817	0.495405
$646$	0	0
$647$	31.1388	1.22419	0.612096	−	0.790783i	$-0.290327\pi$
$647$	31.1388	1.22419	0.612096	+	0.790783i	$0.290327\pi$
$648$	1.00000	0.0392837
$649$	19.9932	0.784801
$650$	4.10607	0.161053
$651$	−25.0351	−0.981202
$652$	−13.4338	−0.526107
$653$	1.27395	0.0498534	0.0249267	−	0.999689i	$-0.492065\pi$
$653$	1.27395	0.0498534	0.0249267	+	0.999689i	$0.492065\pi$
$654$	−12.1284	−0.474256
$655$	12.4757	0.487464
$656$	−11.5817	−0.452190
$657$	−15.4047	−0.600993
$658$	21.8726	0.852682
$659$	26.5466	1.03411	0.517055	−	0.855952i	$-0.327028\pi$
$659$	26.5466	1.03411	0.517055	+	0.855952i	$0.327028\pi$
$660$	1.53209	0.0596365
$661$	−14.2412	−0.553920	−0.276960	−	0.960882i	$-0.589327\pi$
$661$	−14.2412	−0.553920	−0.276960	+	0.960882i	$0.589327\pi$
$662$	−9.20027	−0.357579
$663$	0	0
$664$	0	0
$665$	3.87939	0.150436
$666$	0.305407	0.0118343
$667$	54.1985	2.09858
$668$	8.01455	0.310092
$669$	−2.47565	−0.0957142
$670$	−4.08378	−0.157770
$671$	−7.06418	−0.272710
$672$	−3.87939	−0.149651
$673$	−50.5134	−1.94715	−0.973575	−	0.228369i	$-0.926661\pi$
$673$	−50.5134	−1.94715	−0.973575	+	0.228369i	$0.926661\pi$
$674$	−16.9513	−0.652940
$675$	−1.00000	−0.0384900
$676$	3.85978	0.148453
$677$	−19.4962	−0.749299	−0.374650	−	0.927166i	$-0.622237\pi$
$677$	−19.4962	−0.749299	−0.374650	+	0.927166i	$0.622237\pi$
$678$	3.43376	0.131873
$679$	8.69459	0.333668
$680$	0	0
$681$	−10.1284	−0.388119
$682$	−9.88713	−0.378598
$683$	−42.9905	−1.64499	−0.822493	−	0.568775i	$-0.807417\pi$
$683$	−42.9905	−1.64499	−0.822493	+	0.568775i	$0.807417\pi$
$684$	1.00000	0.0382360
$685$	−6.45336	−0.246570
$686$	4.07192	0.155467
$687$	11.5621	0.441122
$688$	−12.5817	−0.479674
$689$	17.3550	0.661174
$690$	−9.41147	−0.358289
$691$	19.9385	0.758497	0.379248	−	0.925295i	$-0.376183\pi$
$691$	19.9385	0.758497	0.379248	+	0.925295i	$0.376183\pi$
$692$	−4.43107	−0.168444
$693$	−5.94356	−0.225777
$694$	24.9067	0.945446
$695$	11.6108	0.440423
$696$	−5.75877	−0.218286
$697$	0	0
$698$	9.77425	0.369961
$699$	−4.65539	−0.176083
$700$	3.87939	0.146627
$701$	4.30129	0.162457	0.0812287	−	0.996695i	$-0.474116\pi$
$701$	4.30129	0.162457	0.0812287	+	0.996695i	$0.474116\pi$
$702$	−4.10607	−0.154974
$703$	0.305407	0.0115187
$704$	−1.53209	−0.0577428
$705$	−5.63816	−0.212345
$706$	11.3601	0.427543
$707$	44.6810	1.68040
$708$	13.0496	0.490435
$709$	−41.6032	−1.56244	−0.781220	−	0.624256i	$-0.785402\pi$
$709$	−41.6032	−1.56244	−0.781220	+	0.624256i	$0.785402\pi$
$710$	14.2567	0.535045
$711$	−7.19253	−0.269741
$712$	6.24897	0.234190
$713$	60.7357	2.27457
$714$	0	0
$715$	−6.29086	−0.235265
$716$	4.95811	0.185293
$717$	18.4534	0.689153
$718$	11.3892	0.425041
$719$	−45.0060	−1.67844	−0.839220	−	0.543792i	$-0.816988\pi$
$719$	−45.0060	−1.67844	−0.839220	+	0.543792i	$0.816988\pi$
$720$	1.00000	0.0372678
$721$	−29.8503	−1.11168
$722$	−18.0000	−0.669891
$723$	27.0428	1.00573
$724$	4.28581	0.159281
$725$	5.75877	0.213875
$726$	8.65270	0.321132
$727$	49.5553	1.83790	0.918952	−	0.394369i	$-0.129037\pi$
$727$	49.5553	1.83790	0.918952	+	0.394369i	$0.129037\pi$
$728$	15.9290	0.590369
$729$	1.00000	0.0370370
$730$	−15.4047	−0.570152
$731$	0	0
$732$	−4.61081	−0.170421
$733$	39.7648	1.46874	0.734372	−	0.678747i	$-0.237477\pi$
$733$	39.7648	1.46874	0.734372	+	0.678747i	$0.237477\pi$
$734$	−23.0651	−0.851349
$735$	−8.04963	−0.296915
$736$	9.41147	0.346912
$737$	6.25671	0.230469
$738$	−11.5817	−0.426329
$739$	−23.9617	−0.881447	−0.440723	−	0.897643i	$-0.645278\pi$
$739$	−23.9617	−0.881447	−0.440723	+	0.897643i	$0.645278\pi$
$740$	0.305407	0.0112270
$741$	−4.10607	−0.150840
$742$	16.3969	0.601950
$743$	54.0556	1.98311	0.991554	−	0.129694i	$-0.0413994\pi$
$743$	54.0556	1.98311	0.991554	+	0.129694i	$0.0413994\pi$
$744$	−6.45336	−0.236592
$745$	−19.9317	−0.730241
$746$	13.0719	0.478597
$747$	0	0
$748$	0	0
$749$	2.69459	0.0984583
$750$	−1.00000	−0.0365148
$751$	49.0351	1.78932	0.894658	−	0.446752i	$-0.147419\pi$
$751$	49.0351	1.78932	0.894658	+	0.446752i	$0.147419\pi$
$752$	5.63816	0.205602
$753$	−27.7989	−1.01305
$754$	23.6459	0.861133
$755$	−22.8229	−0.830612
$756$	−3.87939	−0.141092
$757$	−1.52198	−0.0553174	−0.0276587	−	0.999617i	$-0.508805\pi$
$757$	−1.52198	−0.0553174	−0.0276587	+	0.999617i	$0.508805\pi$
$758$	14.1284	0.513165
$759$	14.4192	0.523384
$760$	1.00000	0.0362738
$761$	−20.6186	−0.747422	−0.373711	−	0.927545i	$-0.621915\pi$
$761$	−20.6186	−0.747422	−0.373711	+	0.927545i	$0.621915\pi$
$762$	7.34049	0.265918
$763$	47.0506	1.70334
$764$	10.0155	0.362347
$765$	0	0
$766$	1.72369	0.0622794
$767$	−53.5827	−1.93476
$768$	−1.00000	−0.0360844
$769$	−43.0428	−1.55216	−0.776082	−	0.630632i	$-0.782796\pi$
$769$	−43.0428	−1.55216	−0.776082	+	0.630632i	$0.782796\pi$
$770$	−5.94356	−0.214191
$771$	8.99050	0.323785
$772$	−12.6108	−0.453873
$773$	−19.9222	−0.716552	−0.358276	−	0.933616i	$-0.616635\pi$
$773$	−19.9222	−0.716552	−0.358276	+	0.933616i	$0.616635\pi$
$774$	−12.5817	−0.452241
$775$	6.45336	0.231812
$776$	2.24123	0.0804555
$777$	−1.18479	−0.0425042
$778$	−7.06418	−0.253263
$779$	−11.5817	−0.414958
$780$	−4.10607	−0.147021
$781$	−21.8425	−0.781588
$782$	0	0
$783$	−5.75877	−0.205802
$784$	8.04963	0.287487
$785$	−0.573978	−0.0204862
$786$	−12.4757	−0.444992
$787$	−55.0215	−1.96130	−0.980652	−	0.195760i	$-0.937283\pi$
$787$	−55.0215	−1.96130	−0.980652	+	0.195760i	$0.937283\pi$
$788$	2.55169	0.0909002
$789$	−2.37639	−0.0846018
$790$	−7.19253	−0.255899
$791$	−13.3209	−0.473636
$792$	−1.53209	−0.0544404
$793$	18.9323	0.672306
$794$	30.7033	1.08962
$795$	−4.22668	−0.149905
$796$	−16.0993	−0.570623
$797$	−9.75641	−0.345590	−0.172795	−	0.984958i	$-0.555280\pi$
$797$	−9.75641	−0.345590	−0.172795	+	0.984958i	$0.555280\pi$
$798$	−3.87939	−0.137329
$799$	0	0
$800$	1.00000	0.0353553
$801$	6.24897	0.220797
$802$	8.97596	0.316952
$803$	23.6013	0.832872
$804$	4.08378	0.144024
$805$	36.5107	1.28683
$806$	26.4979	0.933350
$807$	−24.7493	−0.871216
$808$	11.5175	0.405186
$809$	−4.12237	−0.144935	−0.0724674	−	0.997371i	$-0.523087\pi$
$809$	−4.12237	−0.144935	−0.0724674	+	0.997371i	$0.523087\pi$
$810$	1.00000	0.0351364
$811$	−50.9353	−1.78858	−0.894291	−	0.447487i	$-0.852319\pi$
$811$	−50.9353	−1.78858	−0.894291	+	0.447487i	$0.852319\pi$
$812$	22.3405	0.783997
$813$	12.6209	0.442635
$814$	−0.467911	−0.0164003
$815$	−13.4338	−0.470564
$816$	0	0
$817$	−12.5817	−0.440179
$818$	10.4311	0.364714
$819$	15.9290	0.556605
$820$	−11.5817	−0.404451
$821$	10.2276	0.356946	0.178473	−	0.983945i	$-0.442884\pi$
$821$	10.2276	0.356946	0.178473	+	0.983945i	$0.442884\pi$
$822$	6.45336	0.225087
$823$	−3.26083	−0.113665	−0.0568327	−	0.998384i	$-0.518100\pi$
$823$	−3.26083	−0.113665	−0.0568327	+	0.998384i	$0.518100\pi$
$824$	−7.69459	−0.268054
$825$	1.53209	0.0533405
$826$	−50.6245	−1.76145
$827$	−45.0351	−1.56602	−0.783012	−	0.622007i	$-0.786317\pi$
$827$	−45.0351	−1.56602	−0.783012	+	0.622007i	$0.786317\pi$
$828$	9.41147	0.327071
$829$	−12.5763	−0.436794	−0.218397	−	0.975860i	$-0.570083\pi$
$829$	−12.5763	−0.436794	−0.218397	+	0.975860i	$0.570083\pi$
$830$	0	0
$831$	−10.1898	−0.353482
$832$	4.10607	0.142352
$833$	0	0
$834$	−11.6108	−0.402050
$835$	8.01455	0.277355
$836$	−1.53209	−0.0529884
$837$	−6.45336	−0.223061
$838$	−1.86484	−0.0644197
$839$	−5.45748	−0.188413	−0.0942066	−	0.995553i	$-0.530031\pi$
$839$	−5.45748	−0.188413	−0.0942066	+	0.995553i	$0.530031\pi$
$840$	−3.87939	−0.133852
$841$	4.16344	0.143567
$842$	3.84255	0.132423
$843$	27.2226	0.937595
$844$	20.2199	0.695997
$845$	3.85978	0.132781
$846$	5.63816	0.193844
$847$	−33.5672	−1.15338
$848$	4.22668	0.145145
$849$	10.6554	0.365692
$850$	0	0
$851$	2.87433	0.0985309
$852$	−14.2567	−0.488427
$853$	−17.1179	−0.586107	−0.293053	−	0.956096i	$-0.594671\pi$
$853$	−17.1179	−0.586107	−0.293053	+	0.956096i	$0.594671\pi$
$854$	17.8871	0.612085
$855$	1.00000	0.0341993
$856$	0.694593	0.0237407
$857$	52.1931	1.78288	0.891442	−	0.453135i	$-0.149694\pi$
$857$	52.1931	1.78288	0.891442	+	0.453135i	$0.149694\pi$
$858$	6.29086	0.214766
$859$	49.7701	1.69814	0.849068	−	0.528284i	$-0.177164\pi$
$859$	49.7701	1.69814	0.849068	+	0.528284i	$0.177164\pi$
$860$	−12.5817	−0.429033
$861$	44.9299	1.53121
$862$	−17.9608	−0.611747
$863$	12.8922	0.438855	0.219427	−	0.975629i	$-0.429581\pi$
$863$	12.8922	0.438855	0.219427	+	0.975629i	$0.429581\pi$
$864$	−1.00000	−0.0340207
$865$	−4.43107	−0.150661
$866$	0.723689	0.0245919
$867$	0	0
$868$	25.0351	0.849746
$869$	11.0196	0.373814
$870$	−5.75877	−0.195241
$871$	−16.7683	−0.568171
$872$	12.1284	0.410718
$873$	2.24123	0.0758541
$874$	9.41147	0.318348
$875$	3.87939	0.131147
$876$	15.4047	0.520475
$877$	52.4529	1.77121	0.885604	−	0.464442i	$-0.153745\pi$
$877$	52.4529	1.77121	0.885604	+	0.464442i	$0.153745\pi$
$878$	29.5722	0.998014
$879$	−3.12330	−0.105346
$880$	−1.53209	−0.0516467
$881$	−35.4816	−1.19541	−0.597703	−	0.801717i	$-0.703920\pi$
$881$	−35.4816	−1.19541	−0.597703	+	0.801717i	$0.703920\pi$
$882$	8.04963	0.271045
$883$	−50.6709	−1.70521	−0.852605	−	0.522555i	$-0.824979\pi$
$883$	−50.6709	−1.70521	−0.852605	+	0.522555i	$0.824979\pi$
$884$	0	0
$885$	13.0496	0.438659
$886$	24.3851	0.819232
$887$	−12.7897	−0.429437	−0.214719	−	0.976676i	$-0.568883\pi$
$887$	−12.7897	−0.429437	−0.214719	+	0.976676i	$0.568883\pi$
$888$	−0.305407	−0.0102488
$889$	−28.4766	−0.955074
$890$	6.24897	0.209466
$891$	−1.53209	−0.0513269
$892$	2.47565	0.0828910
$893$	5.63816	0.188674
$894$	19.9317	0.666616
$895$	4.95811	0.165731
$896$	3.87939	0.129601
$897$	−38.6441	−1.29029
$898$	12.3851	0.413295
$899$	37.1634	1.23947
$900$	1.00000	0.0333333
$901$	0	0
$902$	17.7442	0.590818
$903$	48.8093	1.62427
$904$	−3.43376	−0.114205
$905$	4.28581	0.142465
$906$	22.8229	0.758242
$907$	12.8830	0.427773	0.213887	−	0.976858i	$-0.431388\pi$
$907$	12.8830	0.427773	0.213887	+	0.976858i	$0.431388\pi$
$908$	10.1284	0.336121
$909$	11.5175	0.382013
$910$	15.9290	0.528042
$911$	19.2026	0.636212	0.318106	−	0.948055i	$-0.396953\pi$
$911$	19.2026	0.636212	0.318106	+	0.948055i	$0.396953\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	18.3797	0.607946
$915$	−4.61081	−0.152429
$916$	−11.5621	−0.382023
$917$	48.3979	1.59824
$918$	0	0
$919$	56.6709	1.86940	0.934700	−	0.355438i	$-0.115668\pi$
$919$	56.6709	1.86940	0.934700	+	0.355438i	$0.115668\pi$
$920$	9.41147	0.310287
$921$	27.9864	0.922183
$922$	−11.9263	−0.392773
$923$	58.5390	1.92683
$924$	5.94356	0.195529
$925$	0.305407	0.0100417
$926$	30.5996	1.00556
$927$	−7.69459	−0.252724
$928$	5.75877	0.189041
$929$	−31.8716	−1.04567	−0.522837	−	0.852432i	$-0.675126\pi$
$929$	−31.8716	−1.04567	−0.522837	+	0.852432i	$0.675126\pi$
$930$	−6.45336	−0.211614
$931$	8.04963	0.263816
$932$	4.65539	0.152492
$933$	−6.98040	−0.228528
$934$	14.8075	0.484515
$935$	0	0
$936$	4.10607	0.134211
$937$	−44.8093	−1.46386	−0.731929	−	0.681381i	$-0.761380\pi$
$937$	−44.8093	−1.46386	−0.731929	+	0.681381i	$0.761380\pi$
$938$	−15.8425	−0.517278
$939$	−27.8188	−0.907833
$940$	5.63816	0.183896
$941$	24.4688	0.797662	0.398831	−	0.917025i	$-0.369416\pi$
$941$	24.4688	0.797662	0.398831	+	0.917025i	$0.369416\pi$
$942$	0.573978	0.0187012
$943$	−109.001	−3.54956
$944$	−13.0496	−0.424729
$945$	−3.87939	−0.126196
$946$	19.2763	0.626727
$947$	−38.7837	−1.26030	−0.630151	−	0.776472i	$-0.717007\pi$
$947$	−38.7837	−1.26030	−0.630151	+	0.776472i	$0.717007\pi$
$948$	7.19253	0.233603
$949$	−63.2526	−2.05327
$950$	1.00000	0.0324443
$951$	2.27631	0.0738145
$952$	0	0
$953$	−23.1581	−0.750163	−0.375082	−	0.926992i	$-0.622385\pi$
$953$	−23.1581	−0.750163	−0.375082	+	0.926992i	$0.622385\pi$
$954$	4.22668	0.136844
$955$	10.0155	0.324093
$956$	−18.4534	−0.596824
$957$	8.82295	0.285205
$958$	−28.8093	−0.930787
$959$	−25.0351	−0.808425
$960$	−1.00000	−0.0322749
$961$	10.6459	0.343416
$962$	1.25402	0.0404313
$963$	0.694593	0.0223829
$964$	−27.0428	−0.870991
$965$	−12.6108	−0.405956
$966$	−36.5107	−1.17471
$967$	−15.3987	−0.495188	−0.247594	−	0.968864i	$-0.579640\pi$
$967$	−15.3987	−0.495188	−0.247594	+	0.968864i	$0.579640\pi$
$968$	−8.65270	−0.278108
$969$	0	0
$970$	2.24123	0.0719615
$971$	−21.2276	−0.681227	−0.340613	−	0.940203i	$-0.610635\pi$
$971$	−21.2276	−0.681227	−0.340613	+	0.940203i	$0.610635\pi$
$972$	−1.00000	−0.0320750
$973$	45.0428	1.44401
$974$	17.8229	0.571084
$975$	−4.10607	−0.131499
$976$	4.61081	0.147589
$977$	29.5431	0.945168	0.472584	−	0.881286i	$-0.343321\pi$
$977$	29.5431	0.945168	0.472584	+	0.881286i	$0.343321\pi$
$978$	13.4338	0.429564
$979$	−9.57398	−0.305986
$980$	8.04963	0.257136
$981$	12.1284	0.387229
$982$	−10.8699	−0.346872
$983$	−6.03920	−0.192621	−0.0963103	−	0.995351i	$-0.530704\pi$
$983$	−6.03920	−0.192621	−0.0963103	+	0.995351i	$0.530704\pi$
$984$	11.5817	0.369212
$985$	2.55169	0.0813036
$986$	0	0
$987$	−21.8726	−0.696212
$988$	4.10607	0.130631
$989$	−118.413	−3.76530
$990$	−1.53209	−0.0486930
$991$	52.0464	1.65331	0.826655	−	0.562709i	$-0.190241\pi$
$991$	52.0464	1.65331	0.826655	+	0.562709i	$0.190241\pi$
$992$	6.45336	0.204894
$993$	9.20027	0.291962
$994$	55.3073	1.75424
$995$	−16.0993	−0.510381
$996$	0	0
$997$	−35.3645	−1.12001	−0.560003	−	0.828491i	$-0.689200\pi$
$997$	−35.3645	−1.12001	−0.560003	+	0.828491i	$0.689200\pi$
$998$	23.0719	0.730329
$999$	−0.305407	−0.00966266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bq.1.3	✓	3
17.16	even	2		8670.2.a.br.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.bq.1.3	✓	3	1.1	even	1	trivial
8670.2.a.br.1.1	yes	3	17.16	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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.53209 q^{11} -1.00000 q^{12} +4.10607 q^{13} +3.87939 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -3.87939 q^{21} -1.53209 q^{22} +9.41147 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.10607 q^{26} -1.00000 q^{27} +3.87939 q^{28} +5.75877 q^{29} -1.00000 q^{30} +6.45336 q^{31} +1.00000 q^{32} +1.53209 q^{33} +3.87939 q^{35} +1.00000 q^{36} +0.305407 q^{37} +1.00000 q^{38} -4.10607 q^{39} +1.00000 q^{40} -11.5817 q^{41} -3.87939 q^{42} -12.5817 q^{43} -1.53209 q^{44} +1.00000 q^{45} +9.41147 q^{46} +5.63816 q^{47} -1.00000 q^{48} +8.04963 q^{49} +1.00000 q^{50} +4.10607 q^{52} +4.22668 q^{53} -1.00000 q^{54} -1.53209 q^{55} +3.87939 q^{56} -1.00000 q^{57} +5.75877 q^{58} -13.0496 q^{59} -1.00000 q^{60} +4.61081 q^{61} +6.45336 q^{62} +3.87939 q^{63} +1.00000 q^{64} +4.10607 q^{65} +1.53209 q^{66} -4.08378 q^{67} -9.41147 q^{69} +3.87939 q^{70} +14.2567 q^{71} +1.00000 q^{72} -15.4047 q^{73} +0.305407 q^{74} -1.00000 q^{75} +1.00000 q^{76} -5.94356 q^{77} -4.10607 q^{78} -7.19253 q^{79} +1.00000 q^{80} +1.00000 q^{81} -11.5817 q^{82} -3.87939 q^{84} -12.5817 q^{86} -5.75877 q^{87} -1.53209 q^{88} +6.24897 q^{89} +1.00000 q^{90} +15.9290 q^{91} +9.41147 q^{92} -6.45336 q^{93} +5.63816 q^{94} +1.00000 q^{95} -1.00000 q^{96} +2.24123 q^{97} +8.04963 q^{98} -1.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{12} + 6 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{18} + 3 q^{19} + 3 q^{20} - 6 q^{21} + 18 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} + 6 q^{29} - 3 q^{30} + 6 q^{31} + 3 q^{32} + 6 q^{35} + 3 q^{36} + 3 q^{37} + 3 q^{38} + 3 q^{40} - 3 q^{41} - 6 q^{42} - 6 q^{43} + 3 q^{45} + 18 q^{46} - 3 q^{48} - 3 q^{49} + 3 q^{50} + 6 q^{53} - 3 q^{54} + 6 q^{56} - 3 q^{57} + 6 q^{58} - 12 q^{59} - 3 q^{60} + 18 q^{61} + 6 q^{62} + 6 q^{63} + 3 q^{64} - 6 q^{67} - 18 q^{69} + 6 q^{70} + 6 q^{71} + 3 q^{72} + 6 q^{73} + 3 q^{74} - 3 q^{75} + 3 q^{76} - 3 q^{77} + 6 q^{79} + 3 q^{80} + 3 q^{81} - 3 q^{82} - 6 q^{84} - 6 q^{86} - 6 q^{87} + 6 q^{89} + 3 q^{90} + 15 q^{91} + 18 q^{92} - 6 q^{93} + 3 q^{95} - 3 q^{96} + 18 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^12 + 6 * q^14 - 3 * q^15 + 3 * q^16 + 3 * q^18 + 3 * q^19 + 3 * q^20 - 6 * q^21 + 18 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 + 6 * q^29 - 3 * q^30 + 6 * q^31 + 3 * q^32 + 6 * q^35 + 3 * q^36 + 3 * q^37 + 3 * q^38 + 3 * q^40 - 3 * q^41 - 6 * q^42 - 6 * q^43 + 3 * q^45 + 18 * q^46 - 3 * q^48 - 3 * q^49 + 3 * q^50 + 6 * q^53 - 3 * q^54 + 6 * q^56 - 3 * q^57 + 6 * q^58 - 12 * q^59 - 3 * q^60 + 18 * q^61 + 6 * q^62 + 6 * q^63 + 3 * q^64 - 6 * q^67 - 18 * q^69 + 6 * q^70 + 6 * q^71 + 3 * q^72 + 6 * q^73 + 3 * q^74 - 3 * q^75 + 3 * q^76 - 3 * q^77 + 6 * q^79 + 3 * q^80 + 3 * q^81 - 3 * q^82 - 6 * q^84 - 6 * q^86 - 6 * q^87 + 6 * q^89 + 3 * q^90 + 15 * q^91 + 18 * q^92 - 6 * q^93 + 3 * q^95 - 3 * q^96 + 18 * q^97 - 3 * q^98

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