LMFDB - Embedded newform 6762.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.41421 q^{10} +2.00000 q^{11} -1.00000 q^{12} -3.82843 q^{13} -2.41421 q^{15} +1.00000 q^{16} -5.24264 q^{17} -1.00000 q^{18} +7.65685 q^{19} +2.41421 q^{20} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +0.828427 q^{25} +3.82843 q^{26} -1.00000 q^{27} -6.82843 q^{29} +2.41421 q^{30} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +5.24264 q^{34} +1.00000 q^{36} -8.82843 q^{37} -7.65685 q^{38} +3.82843 q^{39} -2.41421 q^{40} +5.17157 q^{41} -0.343146 q^{43} +2.00000 q^{44} +2.41421 q^{45} -1.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} -0.828427 q^{50} +5.24264 q^{51} -3.82843 q^{52} +4.07107 q^{53} +1.00000 q^{54} +4.82843 q^{55} -7.65685 q^{57} +6.82843 q^{58} +9.65685 q^{59} -2.41421 q^{60} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -9.24264 q^{65} +2.00000 q^{66} -4.89949 q^{67} -5.24264 q^{68} -1.00000 q^{69} -4.17157 q^{71} -1.00000 q^{72} -0.656854 q^{73} +8.82843 q^{74} -0.828427 q^{75} +7.65685 q^{76} -3.82843 q^{78} -10.0000 q^{79} +2.41421 q^{80} +1.00000 q^{81} -5.17157 q^{82} -7.17157 q^{83} -12.6569 q^{85} +0.343146 q^{86} +6.82843 q^{87} -2.00000 q^{88} -6.82843 q^{89} -2.41421 q^{90} +1.00000 q^{92} -2.00000 q^{93} +9.00000 q^{94} +18.4853 q^{95} +1.00000 q^{96} -1.65685 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 2 q^{23} + 2 q^{24} - 4 q^{25} + 2 q^{26} - 2 q^{27} - 8 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} - 4 q^{33} + 2 q^{34} + 2 q^{36} - 12 q^{37} - 4 q^{38} + 2 q^{39} - 2 q^{40} + 16 q^{41} - 12 q^{43} + 4 q^{44} + 2 q^{45} - 2 q^{46} - 18 q^{47} - 2 q^{48} + 4 q^{50} + 2 q^{51} - 2 q^{52} - 6 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} + 8 q^{59} - 2 q^{60} + 4 q^{61} - 4 q^{62} + 2 q^{64} - 10 q^{65} + 4 q^{66} + 10 q^{67} - 2 q^{68} - 2 q^{69} - 14 q^{71} - 2 q^{72} + 10 q^{73} + 12 q^{74} + 4 q^{75} + 4 q^{76} - 2 q^{78} - 20 q^{79} + 2 q^{80} + 2 q^{81} - 16 q^{82} - 20 q^{83} - 14 q^{85} + 12 q^{86} + 8 q^{87} - 4 q^{88} - 8 q^{89} - 2 q^{90} + 2 q^{92} - 4 q^{93} + 18 q^{94} + 20 q^{95} + 2 q^{96} + 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 - 2 * q^12 - 2 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 2 * q^23 + 2 * q^24 - 4 * q^25 + 2 * q^26 - 2 * q^27 - 8 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 - 4 * q^33 + 2 * q^34 + 2 * q^36 - 12 * q^37 - 4 * q^38 + 2 * q^39 - 2 * q^40 + 16 * q^41 - 12 * q^43 + 4 * q^44 + 2 * q^45 - 2 * q^46 - 18 * q^47 - 2 * q^48 + 4 * q^50 + 2 * q^51 - 2 * q^52 - 6 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 + 8 * q^59 - 2 * q^60 + 4 * q^61 - 4 * q^62 + 2 * q^64 - 10 * q^65 + 4 * q^66 + 10 * q^67 - 2 * q^68 - 2 * q^69 - 14 * q^71 - 2 * q^72 + 10 * q^73 + 12 * q^74 + 4 * q^75 + 4 * q^76 - 2 * q^78 - 20 * q^79 + 2 * q^80 + 2 * q^81 - 16 * q^82 - 20 * q^83 - 14 * q^85 + 12 * q^86 + 8 * q^87 - 4 * q^88 - 8 * q^89 - 2 * q^90 + 2 * q^92 - 4 * q^93 + 18 * q^94 + 20 * q^95 + 2 * q^96 + 8 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.41421	1.07967	0.539835	−	0.841771i	$-0.318487\pi$
$5$	2.41421	1.07967	0.539835	+	0.841771i	$0.318487\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.41421	−0.763441
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.00000	−0.288675
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0	0
$15$	−2.41421	−0.623347
$16$	1.00000	0.250000
$17$	−5.24264	−1.27153	−0.635764	−	0.771884i	$-0.719315\pi$
$17$	−5.24264	−1.27153	−0.635764	+	0.771884i	$0.719315\pi$
$18$	−1.00000	−0.235702
$19$	7.65685	1.75660	0.878301	−	0.478107i	$-0.158677\pi$
$19$	7.65685	1.75660	0.878301	+	0.478107i	$0.158677\pi$
$20$	2.41421	0.539835
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	0.828427	0.165685
$26$	3.82843	0.750816
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.82843	−1.26801	−0.634004	−	0.773330i	$-0.718590\pi$
$29$	−6.82843	−1.26801	−0.634004	+	0.773330i	$0.718590\pi$
$30$	2.41421	0.440773
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	5.24264	0.899105
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.82843	−1.45138	−0.725692	−	0.688019i	$-0.758480\pi$
$37$	−8.82843	−1.45138	−0.725692	+	0.688019i	$0.758480\pi$
$38$	−7.65685	−1.24211
$39$	3.82843	0.613039
$40$	−2.41421	−0.381721
$41$	5.17157	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$41$	5.17157	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$42$	0	0
$43$	−0.343146	−0.0523292	−0.0261646	−	0.999658i	$-0.508329\pi$
$43$	−0.343146	−0.0523292	−0.0261646	+	0.999658i	$0.508329\pi$
$44$	2.00000	0.301511
$45$	2.41421	0.359890
$46$	−1.00000	−0.147442
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−0.828427	−0.117157
$51$	5.24264	0.734117
$52$	−3.82843	−0.530907
$53$	4.07107	0.559204	0.279602	−	0.960116i	$-0.409797\pi$
$53$	4.07107	0.559204	0.279602	+	0.960116i	$0.409797\pi$
$54$	1.00000	0.136083
$55$	4.82843	0.651065
$56$	0	0
$57$	−7.65685	−1.01418
$58$	6.82843	0.896616
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	−2.41421	−0.311674
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−9.24264	−1.14641
$66$	2.00000	0.246183
$67$	−4.89949	−0.598569	−0.299284	−	0.954164i	$-0.596748\pi$
$67$	−4.89949	−0.598569	−0.299284	+	0.954164i	$0.596748\pi$
$68$	−5.24264	−0.635764
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−4.17157	−0.495075	−0.247537	−	0.968878i	$-0.579621\pi$
$71$	−4.17157	−0.495075	−0.247537	+	0.968878i	$0.579621\pi$
$72$	−1.00000	−0.117851
$73$	−0.656854	−0.0768790	−0.0384395	−	0.999261i	$-0.512239\pi$
$73$	−0.656854	−0.0768790	−0.0384395	+	0.999261i	$0.512239\pi$
$74$	8.82843	1.02628
$75$	−0.828427	−0.0956585
$76$	7.65685	0.878301
$77$	0	0
$78$	−3.82843	−0.433484
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	2.41421	0.269917
$81$	1.00000	0.111111
$82$	−5.17157	−0.571105
$83$	−7.17157	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$83$	−7.17157	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$84$	0	0
$85$	−12.6569	−1.37283
$86$	0.343146	0.0370024
$87$	6.82843	0.732084
$88$	−2.00000	−0.213201
$89$	−6.82843	−0.723812	−0.361906	−	0.932215i	$-0.617874\pi$
$89$	−6.82843	−0.723812	−0.361906	+	0.932215i	$0.617874\pi$
$90$	−2.41421	−0.254480
$91$	0	0
$92$	1.00000	0.104257
$93$	−2.00000	−0.207390
$94$	9.00000	0.928279
$95$	18.4853	1.89655
$96$	1.00000	0.102062
$97$	−1.65685	−0.168228	−0.0841140	−	0.996456i	$-0.526806\pi$
$97$	−1.65685	−0.168228	−0.0841140	+	0.996456i	$0.526806\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0.828427	0.0828427
$101$	−8.82843	−0.878461	−0.439231	−	0.898374i	$-0.644749\pi$
$101$	−8.82843	−0.878461	−0.439231	+	0.898374i	$0.644749\pi$
$102$	−5.24264	−0.519099
$103$	−10.8995	−1.07396	−0.536980	−	0.843595i	$-0.680435\pi$
$103$	−10.8995	−1.07396	−0.536980	+	0.843595i	$0.680435\pi$
$104$	3.82843	0.375408
$105$	0	0
$106$	−4.07107	−0.395417
$107$	−5.65685	−0.546869	−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685	−0.546869	−0.273434	+	0.961891i	$0.588160\pi$
$108$	−1.00000	−0.0962250
$109$	9.17157	0.878477	0.439239	−	0.898370i	$-0.355248\pi$
$109$	9.17157	0.878477	0.439239	+	0.898370i	$0.355248\pi$
$110$	−4.82843	−0.460372
$111$	8.82843	0.837957
$112$	0	0
$113$	11.7279	1.10327	0.551635	−	0.834086i	$-0.314004\pi$
$113$	11.7279	1.10327	0.551635	+	0.834086i	$0.314004\pi$
$114$	7.65685	0.717130
$115$	2.41421	0.225127
$116$	−6.82843	−0.634004
$117$	−3.82843	−0.353938
$118$	−9.65685	−0.888985
$119$	0	0
$120$	2.41421	0.220387
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	−5.17157	−0.466305
$124$	2.00000	0.179605
$125$	−10.0711	−0.900784
$126$	0	0
$127$	−0.828427	−0.0735110	−0.0367555	−	0.999324i	$-0.511702\pi$
$127$	−0.828427	−0.0735110	−0.0367555	+	0.999324i	$0.511702\pi$
$128$	−1.00000	−0.0883883
$129$	0.343146	0.0302123
$130$	9.24264	0.810633
$131$	9.14214	0.798752	0.399376	−	0.916787i	$-0.369227\pi$
$131$	9.14214	0.798752	0.399376	+	0.916787i	$0.369227\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	4.89949	0.423252
$135$	−2.41421	−0.207782
$136$	5.24264	0.449553
$137$	−7.58579	−0.648097	−0.324049	−	0.946040i	$-0.605044\pi$
$137$	−7.58579	−0.648097	−0.324049	+	0.946040i	$0.605044\pi$
$138$	1.00000	0.0851257
$139$	−9.31371	−0.789978	−0.394989	−	0.918686i	$-0.629252\pi$
$139$	−9.31371	−0.789978	−0.394989	+	0.918686i	$0.629252\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	4.17157	0.350071
$143$	−7.65685	−0.640298
$144$	1.00000	0.0833333
$145$	−16.4853	−1.36903
$146$	0.656854	0.0543616
$147$	0	0
$148$	−8.82843	−0.725692
$149$	9.24264	0.757187	0.378593	−	0.925563i	$-0.376408\pi$
$149$	9.24264	0.757187	0.378593	+	0.925563i	$0.376408\pi$
$150$	0.828427	0.0676408
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	−7.65685	−0.621053
$153$	−5.24264	−0.423842
$154$	0	0
$155$	4.82843	0.387829
$156$	3.82843	0.306519
$157$	15.6569	1.24955	0.624777	−	0.780804i	$-0.285190\pi$
$157$	15.6569	1.24955	0.624777	+	0.780804i	$0.285190\pi$
$158$	10.0000	0.795557
$159$	−4.07107	−0.322857
$160$	−2.41421	−0.190860
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	1.65685	0.129775	0.0648874	−	0.997893i	$-0.479331\pi$
$163$	1.65685	0.129775	0.0648874	+	0.997893i	$0.479331\pi$
$164$	5.17157	0.403832
$165$	−4.82843	−0.375893
$166$	7.17157	0.556622
$167$	−24.7990	−1.91900	−0.959502	−	0.281703i	$-0.909101\pi$
$167$	−24.7990	−1.91900	−0.959502	+	0.281703i	$0.909101\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	12.6569	0.970736
$171$	7.65685	0.585534
$172$	−0.343146	−0.0261646
$173$	7.31371	0.556051	0.278025	−	0.960574i	$-0.410320\pi$
$173$	7.31371	0.556051	0.278025	+	0.960574i	$0.410320\pi$
$174$	−6.82843	−0.517662
$175$	0	0
$176$	2.00000	0.150756
$177$	−9.65685	−0.725854
$178$	6.82843	0.511812
$179$	−21.9706	−1.64216	−0.821078	−	0.570815i	$-0.806627\pi$
$179$	−21.9706	−1.64216	−0.821078	+	0.570815i	$0.806627\pi$
$180$	2.41421	0.179945
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	−1.00000	−0.0737210
$185$	−21.3137	−1.56702
$186$	2.00000	0.146647
$187$	−10.4853	−0.766760
$188$	−9.00000	−0.656392
$189$	0	0
$190$	−18.4853	−1.34106
$191$	−5.17157	−0.374202	−0.187101	−	0.982341i	$-0.559909\pi$
$191$	−5.17157	−0.374202	−0.187101	+	0.982341i	$0.559909\pi$
$192$	−1.00000	−0.0721688
$193$	−5.48528	−0.394839	−0.197420	−	0.980319i	$-0.563256\pi$
$193$	−5.48528	−0.394839	−0.197420	+	0.980319i	$0.563256\pi$
$194$	1.65685	0.118955
$195$	9.24264	0.661879
$196$	0	0
$197$	−12.8284	−0.913988	−0.456994	−	0.889470i	$-0.651074\pi$
$197$	−12.8284	−0.913988	−0.456994	+	0.889470i	$0.651074\pi$
$198$	−2.00000	−0.142134
$199$	27.6569	1.96054	0.980271	−	0.197657i	$-0.0633333\pi$
$199$	27.6569	1.96054	0.980271	+	0.197657i	$0.0633333\pi$
$200$	−0.828427	−0.0585786
$201$	4.89949	0.345584
$202$	8.82843	0.621166
$203$	0	0
$204$	5.24264	0.367058
$205$	12.4853	0.872010
$206$	10.8995	0.759404
$207$	1.00000	0.0695048
$208$	−3.82843	−0.265454
$209$	15.3137	1.05927
$210$	0	0
$211$	−11.1716	−0.769083	−0.384541	−	0.923108i	$-0.625640\pi$
$211$	−11.1716	−0.769083	−0.384541	+	0.923108i	$0.625640\pi$
$212$	4.07107	0.279602
$213$	4.17157	0.285831
$214$	5.65685	0.386695
$215$	−0.828427	−0.0564983
$216$	1.00000	0.0680414
$217$	0	0
$218$	−9.17157	−0.621177
$219$	0.656854	0.0443861
$220$	4.82843	0.325532
$221$	20.0711	1.35013
$222$	−8.82843	−0.592525
$223$	12.4853	0.836076	0.418038	−	0.908429i	$-0.362718\pi$
$223$	12.4853	0.836076	0.418038	+	0.908429i	$0.362718\pi$
$224$	0	0
$225$	0.828427	0.0552285
$226$	−11.7279	−0.780130
$227$	10.9706	0.728142	0.364071	−	0.931371i	$-0.381387\pi$
$227$	10.9706	0.728142	0.364071	+	0.931371i	$0.381387\pi$
$228$	−7.65685	−0.507088
$229$	−2.14214	−0.141556	−0.0707782	−	0.997492i	$-0.522548\pi$
$229$	−2.14214	−0.141556	−0.0707782	+	0.997492i	$0.522548\pi$
$230$	−2.41421	−0.159189
$231$	0	0
$232$	6.82843	0.448308
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	3.82843	0.250272
$235$	−21.7279	−1.41737
$236$	9.65685	0.628608
$237$	10.0000	0.649570
$238$	0	0
$239$	10.3431	0.669042	0.334521	−	0.942388i	$-0.391425\pi$
$239$	10.3431	0.669042	0.334521	+	0.942388i	$0.391425\pi$
$240$	−2.41421	−0.155837
$241$	12.8284	0.826352	0.413176	−	0.910651i	$-0.364419\pi$
$241$	12.8284	0.826352	0.413176	+	0.910651i	$0.364419\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	5.17157	0.329727
$247$	−29.3137	−1.86519
$248$	−2.00000	−0.127000
$249$	7.17157	0.454480
$250$	10.0711	0.636950
$251$	−17.3137	−1.09283	−0.546416	−	0.837514i	$-0.684008\pi$
$251$	−17.3137	−1.09283	−0.546416	+	0.837514i	$0.684008\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0.828427	0.0519801
$255$	12.6569	0.792603
$256$	1.00000	0.0625000
$257$	−5.51472	−0.343999	−0.171999	−	0.985097i	$-0.555023\pi$
$257$	−5.51472	−0.343999	−0.171999	+	0.985097i	$0.555023\pi$
$258$	−0.343146	−0.0213633
$259$	0	0
$260$	−9.24264	−0.573204
$261$	−6.82843	−0.422669
$262$	−9.14214	−0.564803
$263$	19.3137	1.19093	0.595467	−	0.803380i	$-0.296967\pi$
$263$	19.3137	1.19093	0.595467	+	0.803380i	$0.296967\pi$
$264$	2.00000	0.123091
$265$	9.82843	0.603755
$266$	0	0
$267$	6.82843	0.417893
$268$	−4.89949	−0.299284
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	2.41421	0.146924
$271$	30.2843	1.83964	0.919819	−	0.392342i	$-0.128335\pi$
$271$	30.2843	1.83964	0.919819	+	0.392342i	$0.128335\pi$
$272$	−5.24264	−0.317882
$273$	0	0
$274$	7.58579	0.458274
$275$	1.65685	0.0999121
$276$	−1.00000	−0.0601929
$277$	25.2843	1.51918	0.759592	−	0.650400i	$-0.225398\pi$
$277$	25.2843	1.51918	0.759592	+	0.650400i	$0.225398\pi$
$278$	9.31371	0.558599
$279$	2.00000	0.119737
$280$	0	0
$281$	−13.5858	−0.810460	−0.405230	−	0.914215i	$-0.632809\pi$
$281$	−13.5858	−0.810460	−0.405230	+	0.914215i	$0.632809\pi$
$282$	−9.00000	−0.535942
$283$	29.3848	1.74674	0.873372	−	0.487054i	$-0.161929\pi$
$283$	29.3848	1.74674	0.873372	+	0.487054i	$0.161929\pi$
$284$	−4.17157	−0.247537
$285$	−18.4853	−1.09497
$286$	7.65685	0.452759
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	10.4853	0.616781
$290$	16.4853	0.968049
$291$	1.65685	0.0971265
$292$	−0.656854	−0.0384395
$293$	3.58579	0.209484	0.104742	−	0.994499i	$-0.466598\pi$
$293$	3.58579	0.209484	0.104742	+	0.994499i	$0.466598\pi$
$294$	0	0
$295$	23.3137	1.35738
$296$	8.82843	0.513142
$297$	−2.00000	−0.116052
$298$	−9.24264	−0.535412
$299$	−3.82843	−0.221404
$300$	−0.828427	−0.0478293
$301$	0	0
$302$	14.1421	0.813788
$303$	8.82843	0.507180
$304$	7.65685	0.439151
$305$	4.82843	0.276475
$306$	5.24264	0.299702
$307$	14.1421	0.807134	0.403567	−	0.914950i	$-0.367770\pi$
$307$	14.1421	0.807134	0.403567	+	0.914950i	$0.367770\pi$
$308$	0	0
$309$	10.8995	0.620051
$310$	−4.82843	−0.274236
$311$	16.3137	0.925066	0.462533	−	0.886602i	$-0.346941\pi$
$311$	16.3137	0.925066	0.462533	+	0.886602i	$0.346941\pi$
$312$	−3.82843	−0.216742
$313$	−32.9706	−1.86361	−0.931803	−	0.362964i	$-0.881765\pi$
$313$	−32.9706	−1.86361	−0.931803	+	0.362964i	$0.881765\pi$
$314$	−15.6569	−0.883567
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−32.6274	−1.83254	−0.916269	−	0.400563i	$-0.868815\pi$
$317$	−32.6274	−1.83254	−0.916269	+	0.400563i	$0.868815\pi$
$318$	4.07107	0.228294
$319$	−13.6569	−0.764637
$320$	2.41421	0.134959
$321$	5.65685	0.315735
$322$	0	0
$323$	−40.1421	−2.23357
$324$	1.00000	0.0555556
$325$	−3.17157	−0.175927
$326$	−1.65685	−0.0917647
$327$	−9.17157	−0.507189
$328$	−5.17157	−0.285552
$329$	0	0
$330$	4.82843	0.265796
$331$	−34.4853	−1.89548	−0.947741	−	0.319040i	$-0.896640\pi$
$331$	−34.4853	−1.89548	−0.947741	+	0.319040i	$0.896640\pi$
$332$	−7.17157	−0.393591
$333$	−8.82843	−0.483795
$334$	24.7990	1.35694
$335$	−11.8284	−0.646256
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−1.65685	−0.0901210
$339$	−11.7279	−0.636973
$340$	−12.6569	−0.686414
$341$	4.00000	0.216612
$342$	−7.65685	−0.414035
$343$	0	0
$344$	0.343146	0.0185012
$345$	−2.41421	−0.129977
$346$	−7.31371	−0.393187
$347$	−32.6569	−1.75311	−0.876556	−	0.481300i	$-0.840165\pi$
$347$	−32.6569	−1.75311	−0.876556	+	0.481300i	$0.840165\pi$
$348$	6.82843	0.366042
$349$	25.1421	1.34583	0.672914	−	0.739721i	$-0.265042\pi$
$349$	25.1421	1.34583	0.672914	+	0.739721i	$0.265042\pi$
$350$	0	0
$351$	3.82843	0.204346
$352$	−2.00000	−0.106600
$353$	−2.14214	−0.114014	−0.0570072	−	0.998374i	$-0.518156\pi$
$353$	−2.14214	−0.114014	−0.0570072	+	0.998374i	$0.518156\pi$
$354$	9.65685	0.513256
$355$	−10.0711	−0.534517
$356$	−6.82843	−0.361906
$357$	0	0
$358$	21.9706	1.16118
$359$	26.9706	1.42345	0.711726	−	0.702457i	$-0.247914\pi$
$359$	26.9706	1.42345	0.711726	+	0.702457i	$0.247914\pi$
$360$	−2.41421	−0.127240
$361$	39.6274	2.08565
$362$	−9.31371	−0.489518
$363$	7.00000	0.367405
$364$	0	0
$365$	−1.58579	−0.0830039
$366$	2.00000	0.104542
$367$	−26.6985	−1.39365	−0.696825	−	0.717241i	$-0.745404\pi$
$367$	−26.6985	−1.39365	−0.696825	+	0.717241i	$0.745404\pi$
$368$	1.00000	0.0521286
$369$	5.17157	0.269221
$370$	21.3137	1.10805
$371$	0	0
$372$	−2.00000	−0.103695
$373$	−0.343146	−0.0177674	−0.00888371	−	0.999961i	$-0.502828\pi$
$373$	−0.343146	−0.0177674	−0.00888371	+	0.999961i	$0.502828\pi$
$374$	10.4853	0.542181
$375$	10.0711	0.520068
$376$	9.00000	0.464140
$377$	26.1421	1.34639
$378$	0	0
$379$	−18.8995	−0.970802	−0.485401	−	0.874292i	$-0.661326\pi$
$379$	−18.8995	−0.970802	−0.485401	+	0.874292i	$0.661326\pi$
$380$	18.4853	0.948275
$381$	0.828427	0.0424416
$382$	5.17157	0.264601
$383$	−3.65685	−0.186857	−0.0934283	−	0.995626i	$-0.529783\pi$
$383$	−3.65685	−0.186857	−0.0934283	+	0.995626i	$0.529783\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	5.48528	0.279193
$387$	−0.343146	−0.0174431
$388$	−1.65685	−0.0841140
$389$	−37.4558	−1.89909	−0.949543	−	0.313636i	$-0.898453\pi$
$389$	−37.4558	−1.89909	−0.949543	+	0.313636i	$0.898453\pi$
$390$	−9.24264	−0.468019
$391$	−5.24264	−0.265132
$392$	0	0
$393$	−9.14214	−0.461160
$394$	12.8284	0.646287
$395$	−24.1421	−1.21472
$396$	2.00000	0.100504
$397$	8.65685	0.434475	0.217238	−	0.976119i	$-0.430295\pi$
$397$	8.65685	0.434475	0.217238	+	0.976119i	$0.430295\pi$
$398$	−27.6569	−1.38631
$399$	0	0
$400$	0.828427	0.0414214
$401$	−32.6985	−1.63288	−0.816442	−	0.577427i	$-0.804057\pi$
$401$	−32.6985	−1.63288	−0.816442	+	0.577427i	$0.804057\pi$
$402$	−4.89949	−0.244365
$403$	−7.65685	−0.381415
$404$	−8.82843	−0.439231
$405$	2.41421	0.119963
$406$	0	0
$407$	−17.6569	−0.875218
$408$	−5.24264	−0.259549
$409$	26.7990	1.32512	0.662562	−	0.749007i	$-0.269469\pi$
$409$	26.7990	1.32512	0.662562	+	0.749007i	$0.269469\pi$
$410$	−12.4853	−0.616604
$411$	7.58579	0.374179
$412$	−10.8995	−0.536980
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−17.3137	−0.849897
$416$	3.82843	0.187704
$417$	9.31371	0.456094
$418$	−15.3137	−0.749018
$419$	−7.51472	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$419$	−7.51472	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$420$	0	0
$421$	−7.31371	−0.356448	−0.178224	−	0.983990i	$-0.557035\pi$
$421$	−7.31371	−0.356448	−0.178224	+	0.983990i	$0.557035\pi$
$422$	11.1716	0.543824
$423$	−9.00000	−0.437595
$424$	−4.07107	−0.197709
$425$	−4.34315	−0.210674
$426$	−4.17157	−0.202113
$427$	0	0
$428$	−5.65685	−0.273434
$429$	7.65685	0.369676
$430$	0.828427	0.0399503
$431$	−27.3137	−1.31566	−0.657828	−	0.753169i	$-0.728524\pi$
$431$	−27.3137	−1.31566	−0.657828	+	0.753169i	$0.728524\pi$
$432$	−1.00000	−0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	16.4853	0.790409
$436$	9.17157	0.439239
$437$	7.65685	0.366277
$438$	−0.656854	−0.0313857
$439$	−30.9706	−1.47814	−0.739072	−	0.673626i	$-0.764736\pi$
$439$	−30.9706	−1.47814	−0.739072	+	0.673626i	$0.764736\pi$
$440$	−4.82843	−0.230186
$441$	0	0
$442$	−20.0711	−0.954683
$443$	11.1421	0.529379	0.264689	−	0.964334i	$-0.414731\pi$
$443$	11.1421	0.529379	0.264689	+	0.964334i	$0.414731\pi$
$444$	8.82843	0.418979
$445$	−16.4853	−0.781477
$446$	−12.4853	−0.591195
$447$	−9.24264	−0.437162
$448$	0	0
$449$	9.51472	0.449027	0.224514	−	0.974471i	$-0.427921\pi$
$449$	9.51472	0.449027	0.224514	+	0.974471i	$0.427921\pi$
$450$	−0.828427	−0.0390524
$451$	10.3431	0.487040
$452$	11.7279	0.551635
$453$	14.1421	0.664455
$454$	−10.9706	−0.514874
$455$	0	0
$456$	7.65685	0.358565
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	2.14214	0.100095
$459$	5.24264	0.244706
$460$	2.41421	0.112563
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−34.6274	−1.60927	−0.804636	−	0.593768i	$-0.797640\pi$
$463$	−34.6274	−1.60927	−0.804636	+	0.593768i	$0.797640\pi$
$464$	−6.82843	−0.317002
$465$	−4.82843	−0.223913
$466$	8.00000	0.370593
$467$	−21.6569	−1.00216	−0.501080	−	0.865401i	$-0.667064\pi$
$467$	−21.6569	−1.00216	−0.501080	+	0.865401i	$0.667064\pi$
$468$	−3.82843	−0.176969
$469$	0	0
$470$	21.7279	1.00223
$471$	−15.6569	−0.721430
$472$	−9.65685	−0.444493
$473$	−0.686292	−0.0315557
$474$	−10.0000	−0.459315
$475$	6.34315	0.291043
$476$	0	0
$477$	4.07107	0.186401
$478$	−10.3431	−0.473084
$479$	−5.85786	−0.267653	−0.133826	−	0.991005i	$-0.542726\pi$
$479$	−5.85786	−0.267653	−0.133826	+	0.991005i	$0.542726\pi$
$480$	2.41421	0.110193
$481$	33.7990	1.54110
$482$	−12.8284	−0.584319
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−4.00000	−0.181631
$486$	1.00000	0.0453609
$487$	40.7696	1.84744	0.923722	−	0.383063i	$-0.125131\pi$
$487$	40.7696	1.84744	0.923722	+	0.383063i	$0.125131\pi$
$488$	−2.00000	−0.0905357
$489$	−1.65685	−0.0749255
$490$	0	0
$491$	−22.6569	−1.02249	−0.511245	−	0.859435i	$-0.670815\pi$
$491$	−22.6569	−1.02249	−0.511245	+	0.859435i	$0.670815\pi$
$492$	−5.17157	−0.233153
$493$	35.7990	1.61231
$494$	29.3137	1.31889
$495$	4.82843	0.217022
$496$	2.00000	0.0898027
$497$	0	0
$498$	−7.17157	−0.321366
$499$	6.82843	0.305682	0.152841	−	0.988251i	$-0.451158\pi$
$499$	6.82843	0.305682	0.152841	+	0.988251i	$0.451158\pi$
$500$	−10.0711	−0.450392
$501$	24.7990	1.10794
$502$	17.3137	0.772749
$503$	−32.3431	−1.44211	−0.721055	−	0.692878i	$-0.756342\pi$
$503$	−32.3431	−1.44211	−0.721055	+	0.692878i	$0.756342\pi$
$504$	0	0
$505$	−21.3137	−0.948448
$506$	−2.00000	−0.0889108
$507$	−1.65685	−0.0735835
$508$	−0.828427	−0.0367555
$509$	−44.1421	−1.95657	−0.978283	−	0.207274i	$-0.933541\pi$
$509$	−44.1421	−1.95657	−0.978283	+	0.207274i	$0.933541\pi$
$510$	−12.6569	−0.560455
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−7.65685	−0.338058
$514$	5.51472	0.243244
$515$	−26.3137	−1.15952
$516$	0.343146	0.0151061
$517$	−18.0000	−0.791639
$518$	0	0
$519$	−7.31371	−0.321036
$520$	9.24264	0.405317
$521$	−28.4142	−1.24485	−0.622425	−	0.782680i	$-0.713852\pi$
$521$	−28.4142	−1.24485	−0.622425	+	0.782680i	$0.713852\pi$
$522$	6.82843	0.298872
$523$	−0.0710678	−0.00310758	−0.00155379	−	0.999999i	$-0.500495\pi$
$523$	−0.0710678	−0.00310758	−0.00155379	+	0.999999i	$0.500495\pi$
$524$	9.14214	0.399376
$525$	0	0
$526$	−19.3137	−0.842118
$527$	−10.4853	−0.456746
$528$	−2.00000	−0.0870388
$529$	1.00000	0.0434783
$530$	−9.82843	−0.426920
$531$	9.65685	0.419072
$532$	0	0
$533$	−19.7990	−0.857589
$534$	−6.82843	−0.295495
$535$	−13.6569	−0.590437
$536$	4.89949	0.211626
$537$	21.9706	0.948100
$538$	−12.0000	−0.517357
$539$	0	0
$540$	−2.41421	−0.103891
$541$	41.1421	1.76884	0.884419	−	0.466693i	$-0.154555\pi$
$541$	41.1421	1.76884	0.884419	+	0.466693i	$0.154555\pi$
$542$	−30.2843	−1.30082
$543$	−9.31371	−0.399689
$544$	5.24264	0.224776
$545$	22.1421	0.948465
$546$	0	0
$547$	22.2843	0.952807	0.476403	−	0.879227i	$-0.341940\pi$
$547$	22.2843	0.952807	0.476403	+	0.879227i	$0.341940\pi$
$548$	−7.58579	−0.324049
$549$	2.00000	0.0853579
$550$	−1.65685	−0.0706485
$551$	−52.2843	−2.22738
$552$	1.00000	0.0425628
$553$	0	0
$554$	−25.2843	−1.07423
$555$	21.3137	0.904717
$556$	−9.31371	−0.394989
$557$	15.7990	0.669425	0.334712	−	0.942320i	$-0.391361\pi$
$557$	15.7990	0.669425	0.334712	+	0.942320i	$0.391361\pi$
$558$	−2.00000	−0.0846668
$559$	1.31371	0.0555639
$560$	0	0
$561$	10.4853	0.442689
$562$	13.5858	0.573082
$563$	−45.4558	−1.91574	−0.957868	−	0.287210i	$-0.907272\pi$
$563$	−45.4558	−1.91574	−0.957868	+	0.287210i	$0.907272\pi$
$564$	9.00000	0.378968
$565$	28.3137	1.19117
$566$	−29.3848	−1.23513
$567$	0	0
$568$	4.17157	0.175035
$569$	7.58579	0.318013	0.159006	−	0.987278i	$-0.449171\pi$
$569$	7.58579	0.318013	0.159006	+	0.987278i	$0.449171\pi$
$570$	18.4853	0.774263
$571$	−1.10051	−0.0460547	−0.0230274	−	0.999735i	$-0.507330\pi$
$571$	−1.10051	−0.0460547	−0.0230274	+	0.999735i	$0.507330\pi$
$572$	−7.65685	−0.320149
$573$	5.17157	0.216046
$574$	0	0
$575$	0.828427	0.0345478
$576$	1.00000	0.0416667
$577$	−35.9411	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$577$	−35.9411	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$578$	−10.4853	−0.436130
$579$	5.48528	0.227961
$580$	−16.4853	−0.684514
$581$	0	0
$582$	−1.65685	−0.0686788
$583$	8.14214	0.337213
$584$	0.656854	0.0271808
$585$	−9.24264	−0.382136
$586$	−3.58579	−0.148127
$587$	27.1421	1.12028	0.560138	−	0.828399i	$-0.310748\pi$
$587$	27.1421	1.12028	0.560138	+	0.828399i	$0.310748\pi$
$588$	0	0
$589$	15.3137	0.630990
$590$	−23.3137	−0.959810
$591$	12.8284	0.527691
$592$	−8.82843	−0.362846
$593$	32.1421	1.31992	0.659960	−	0.751301i	$-0.270573\pi$
$593$	32.1421	1.31992	0.659960	+	0.751301i	$0.270573\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	9.24264	0.378593
$597$	−27.6569	−1.13192
$598$	3.82843	0.156556
$599$	−11.8284	−0.483296	−0.241648	−	0.970364i	$-0.577688\pi$
$599$	−11.8284	−0.483296	−0.241648	+	0.970364i	$0.577688\pi$
$600$	0.828427	0.0338204
$601$	−42.2843	−1.72481	−0.862406	−	0.506218i	$-0.831043\pi$
$601$	−42.2843	−1.72481	−0.862406	+	0.506218i	$0.831043\pi$
$602$	0	0
$603$	−4.89949	−0.199523
$604$	−14.1421	−0.575435
$605$	−16.8995	−0.687062
$606$	−8.82843	−0.358630
$607$	−7.85786	−0.318941	−0.159470	−	0.987203i	$-0.550979\pi$
$607$	−7.85786	−0.318941	−0.159470	+	0.987203i	$0.550979\pi$
$608$	−7.65685	−0.310526
$609$	0	0
$610$	−4.82843	−0.195497
$611$	34.4558	1.39393
$612$	−5.24264	−0.211921
$613$	21.7990	0.880453	0.440226	−	0.897887i	$-0.354898\pi$
$613$	21.7990	0.880453	0.440226	+	0.897887i	$0.354898\pi$
$614$	−14.1421	−0.570730
$615$	−12.4853	−0.503455
$616$	0	0
$617$	−21.9289	−0.882826	−0.441413	−	0.897304i	$-0.645523\pi$
$617$	−21.9289	−0.882826	−0.441413	+	0.897304i	$0.645523\pi$
$618$	−10.8995	−0.438442
$619$	−44.0711	−1.77137	−0.885683	−	0.464291i	$-0.846309\pi$
$619$	−44.0711	−1.77137	−0.885683	+	0.464291i	$0.846309\pi$
$620$	4.82843	0.193914
$621$	−1.00000	−0.0401286
$622$	−16.3137	−0.654120
$623$	0	0
$624$	3.82843	0.153260
$625$	−28.4558	−1.13823
$626$	32.9706	1.31777
$627$	−15.3137	−0.611571
$628$	15.6569	0.624777
$629$	46.2843	1.84547
$630$	0	0
$631$	−15.4437	−0.614802	−0.307401	−	0.951580i	$-0.599459\pi$
$631$	−15.4437	−0.614802	−0.307401	+	0.951580i	$0.599459\pi$
$632$	10.0000	0.397779
$633$	11.1716	0.444030
$634$	32.6274	1.29580
$635$	−2.00000	−0.0793676
$636$	−4.07107	−0.161428
$637$	0	0
$638$	13.6569	0.540680
$639$	−4.17157	−0.165025
$640$	−2.41421	−0.0954302
$641$	−47.3848	−1.87159	−0.935793	−	0.352550i	$-0.885315\pi$
$641$	−47.3848	−1.87159	−0.935793	+	0.352550i	$0.885315\pi$
$642$	−5.65685	−0.223258
$643$	−29.3137	−1.15602	−0.578010	−	0.816030i	$-0.696171\pi$
$643$	−29.3137	−1.15602	−0.578010	+	0.816030i	$0.696171\pi$
$644$	0	0
$645$	0.828427	0.0326193
$646$	40.1421	1.57937
$647$	10.3431	0.406631	0.203316	−	0.979113i	$-0.434828\pi$
$647$	10.3431	0.406631	0.203316	+	0.979113i	$0.434828\pi$
$648$	−1.00000	−0.0392837
$649$	19.3137	0.758129
$650$	3.17157	0.124399
$651$	0	0
$652$	1.65685	0.0648874
$653$	35.9411	1.40648	0.703242	−	0.710950i	$-0.251735\pi$
$653$	35.9411	1.40648	0.703242	+	0.710950i	$0.251735\pi$
$654$	9.17157	0.358637
$655$	22.0711	0.862388
$656$	5.17157	0.201916
$657$	−0.656854	−0.0256263
$658$	0	0
$659$	−23.4558	−0.913710	−0.456855	−	0.889541i	$-0.651024\pi$
$659$	−23.4558	−0.913710	−0.456855	+	0.889541i	$0.651024\pi$
$660$	−4.82843	−0.187946
$661$	33.4558	1.30128	0.650641	−	0.759386i	$-0.274500\pi$
$661$	33.4558	1.30128	0.650641	+	0.759386i	$0.274500\pi$
$662$	34.4853	1.34031
$663$	−20.0711	−0.779496
$664$	7.17157	0.278311
$665$	0	0
$666$	8.82843	0.342095
$667$	−6.82843	−0.264398
$668$	−24.7990	−0.959502
$669$	−12.4853	−0.482709
$670$	11.8284	0.456972
$671$	4.00000	0.154418
$672$	0	0
$673$	−6.68629	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$673$	−6.68629	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$674$	0	0
$675$	−0.828427	−0.0318862
$676$	1.65685	0.0637252
$677$	45.7279	1.75747	0.878733	−	0.477313i	$-0.158389\pi$
$677$	45.7279	1.75747	0.878733	+	0.477313i	$0.158389\pi$
$678$	11.7279	0.450408
$679$	0	0
$680$	12.6569	0.485368
$681$	−10.9706	−0.420393
$682$	−4.00000	−0.153168
$683$	−16.8579	−0.645048	−0.322524	−	0.946561i	$-0.604531\pi$
$683$	−16.8579	−0.645048	−0.322524	+	0.946561i	$0.604531\pi$
$684$	7.65685	0.292767
$685$	−18.3137	−0.699731
$686$	0	0
$687$	2.14214	0.0817276
$688$	−0.343146	−0.0130823
$689$	−15.5858	−0.593771
$690$	2.41421	0.0919075
$691$	12.1421	0.461909	0.230954	−	0.972965i	$-0.425815\pi$
$691$	12.1421	0.461909	0.230954	+	0.972965i	$0.425815\pi$
$692$	7.31371	0.278025
$693$	0	0
$694$	32.6569	1.23964
$695$	−22.4853	−0.852915
$696$	−6.82843	−0.258831
$697$	−27.1127	−1.02697
$698$	−25.1421	−0.951644
$699$	8.00000	0.302588
$700$	0	0
$701$	8.61522	0.325393	0.162696	−	0.986676i	$-0.447981\pi$
$701$	8.61522	0.325393	0.162696	+	0.986676i	$0.447981\pi$
$702$	−3.82843	−0.144495
$703$	−67.5980	−2.54951
$704$	2.00000	0.0753778
$705$	21.7279	0.818321
$706$	2.14214	0.0806203
$707$	0	0
$708$	−9.65685	−0.362927
$709$	−9.85786	−0.370220	−0.185110	−	0.982718i	$-0.559264\pi$
$709$	−9.85786	−0.370220	−0.185110	+	0.982718i	$0.559264\pi$
$710$	10.0711	0.377960
$711$	−10.0000	−0.375029
$712$	6.82843	0.255906
$713$	2.00000	0.0749006
$714$	0	0
$715$	−18.4853	−0.691310
$716$	−21.9706	−0.821078
$717$	−10.3431	−0.386272
$718$	−26.9706	−1.00653
$719$	−14.6569	−0.546608	−0.273304	−	0.961928i	$-0.588117\pi$
$719$	−14.6569	−0.546608	−0.273304	+	0.961928i	$0.588117\pi$
$720$	2.41421	0.0899724
$721$	0	0
$722$	−39.6274	−1.47478
$723$	−12.8284	−0.477094
$724$	9.31371	0.346141
$725$	−5.65685	−0.210090
$726$	−7.00000	−0.259794
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.58579	0.0586926
$731$	1.79899	0.0665380
$732$	−2.00000	−0.0739221
$733$	8.48528	0.313411	0.156706	−	0.987645i	$-0.449913\pi$
$733$	8.48528	0.313411	0.156706	+	0.987645i	$0.449913\pi$
$734$	26.6985	0.985459
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−9.79899	−0.360950
$738$	−5.17157	−0.190368
$739$	−45.2548	−1.66473	−0.832363	−	0.554231i	$-0.813012\pi$
$739$	−45.2548	−1.66473	−0.832363	+	0.554231i	$0.813012\pi$
$740$	−21.3137	−0.783508
$741$	29.3137	1.07687
$742$	0	0
$743$	15.4558	0.567020	0.283510	−	0.958969i	$-0.408501\pi$
$743$	15.4558	0.567020	0.283510	+	0.958969i	$0.408501\pi$
$744$	2.00000	0.0733236
$745$	22.3137	0.817511
$746$	0.343146	0.0125635
$747$	−7.17157	−0.262394
$748$	−10.4853	−0.383380
$749$	0	0
$750$	−10.0711	−0.367743
$751$	37.5980	1.37197	0.685985	−	0.727616i	$-0.259372\pi$
$751$	37.5980	1.37197	0.685985	+	0.727616i	$0.259372\pi$
$752$	−9.00000	−0.328196
$753$	17.3137	0.630947
$754$	−26.1421	−0.952040
$755$	−34.1421	−1.24256
$756$	0	0
$757$	28.9706	1.05295	0.526477	−	0.850190i	$-0.323513\pi$
$757$	28.9706	1.05295	0.526477	+	0.850190i	$0.323513\pi$
$758$	18.8995	0.686461
$759$	−2.00000	−0.0725954
$760$	−18.4853	−0.670532
$761$	21.6569	0.785060	0.392530	−	0.919739i	$-0.371600\pi$
$761$	21.6569	0.785060	0.392530	+	0.919739i	$0.371600\pi$
$762$	−0.828427	−0.0300107
$763$	0	0
$764$	−5.17157	−0.187101
$765$	−12.6569	−0.457610
$766$	3.65685	0.132128
$767$	−36.9706	−1.33493
$768$	−1.00000	−0.0360844
$769$	−33.3137	−1.20132	−0.600662	−	0.799503i	$-0.705096\pi$
$769$	−33.3137	−1.20132	−0.600662	+	0.799503i	$0.705096\pi$
$770$	0	0
$771$	5.51472	0.198608
$772$	−5.48528	−0.197420
$773$	0.757359	0.0272403	0.0136202	−	0.999907i	$-0.495664\pi$
$773$	0.757359	0.0272403	0.0136202	+	0.999907i	$0.495664\pi$
$774$	0.343146	0.0123341
$775$	1.65685	0.0595160
$776$	1.65685	0.0594776
$777$	0	0
$778$	37.4558	1.34286
$779$	39.5980	1.41874
$780$	9.24264	0.330940
$781$	−8.34315	−0.298541
$782$	5.24264	0.187476
$783$	6.82843	0.244028
$784$	0	0
$785$	37.7990	1.34910
$786$	9.14214	0.326089
$787$	37.0416	1.32039	0.660196	−	0.751094i	$-0.270473\pi$
$787$	37.0416	1.32039	0.660196	+	0.751094i	$0.270473\pi$
$788$	−12.8284	−0.456994
$789$	−19.3137	−0.687586
$790$	24.1421	0.858939
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	−7.65685	−0.271903
$794$	−8.65685	−0.307220
$795$	−9.82843	−0.348578
$796$	27.6569	0.980271
$797$	39.8701	1.41227	0.706135	−	0.708077i	$-0.250437\pi$
$797$	39.8701	1.41227	0.706135	+	0.708077i	$0.250437\pi$
$798$	0	0
$799$	47.1838	1.66924
$800$	−0.828427	−0.0292893
$801$	−6.82843	−0.241271
$802$	32.6985	1.15462
$803$	−1.31371	−0.0463598
$804$	4.89949	0.172792
$805$	0	0
$806$	7.65685	0.269701
$807$	−12.0000	−0.422420
$808$	8.82843	0.310583
$809$	−24.1421	−0.848792	−0.424396	−	0.905477i	$-0.639514\pi$
$809$	−24.1421	−0.848792	−0.424396	+	0.905477i	$0.639514\pi$
$810$	−2.41421	−0.0848268
$811$	−21.1127	−0.741367	−0.370684	−	0.928759i	$-0.620877\pi$
$811$	−21.1127	−0.741367	−0.370684	+	0.928759i	$0.620877\pi$
$812$	0	0
$813$	−30.2843	−1.06212
$814$	17.6569	0.618872
$815$	4.00000	0.140114
$816$	5.24264	0.183529
$817$	−2.62742	−0.0919217
$818$	−26.7990	−0.937005
$819$	0	0
$820$	12.4853	0.436005
$821$	−2.28427	−0.0797216	−0.0398608	−	0.999205i	$-0.512691\pi$
$821$	−2.28427	−0.0797216	−0.0398608	+	0.999205i	$0.512691\pi$
$822$	−7.58579	−0.264585
$823$	−13.7990	−0.481003	−0.240501	−	0.970649i	$-0.577312\pi$
$823$	−13.7990	−0.481003	−0.240501	+	0.970649i	$0.577312\pi$
$824$	10.8995	0.379702
$825$	−1.65685	−0.0576843
$826$	0	0
$827$	−42.6274	−1.48230	−0.741150	−	0.671339i	$-0.765719\pi$
$827$	−42.6274	−1.48230	−0.741150	+	0.671339i	$0.765719\pi$
$828$	1.00000	0.0347524
$829$	−16.1716	−0.561662	−0.280831	−	0.959757i	$-0.590610\pi$
$829$	−16.1716	−0.561662	−0.280831	+	0.959757i	$0.590610\pi$
$830$	17.3137	0.600968
$831$	−25.2843	−0.877102
$832$	−3.82843	−0.132727
$833$	0	0
$834$	−9.31371	−0.322507
$835$	−59.8701	−2.07189
$836$	15.3137	0.529636
$837$	−2.00000	−0.0691301
$838$	7.51472	0.259592
$839$	−24.9706	−0.862080	−0.431040	−	0.902333i	$-0.641853\pi$
$839$	−24.9706	−0.862080	−0.431040	+	0.902333i	$0.641853\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	7.31371	0.252047
$843$	13.5858	0.467919
$844$	−11.1716	−0.384541
$845$	4.00000	0.137604
$846$	9.00000	0.309426
$847$	0	0
$848$	4.07107	0.139801
$849$	−29.3848	−1.00848
$850$	4.34315	0.148969
$851$	−8.82843	−0.302635
$852$	4.17157	0.142916
$853$	52.9117	1.81166	0.905831	−	0.423640i	$-0.139248\pi$
$853$	52.9117	1.81166	0.905831	+	0.423640i	$0.139248\pi$
$854$	0	0
$855$	18.4853	0.632183
$856$	5.65685	0.193347
$857$	0.627417	0.0214322	0.0107161	−	0.999943i	$-0.496589\pi$
$857$	0.627417	0.0214322	0.0107161	+	0.999943i	$0.496589\pi$
$858$	−7.65685	−0.261401
$859$	35.3137	1.20489	0.602444	−	0.798161i	$-0.294194\pi$
$859$	35.3137	1.20489	0.602444	+	0.798161i	$0.294194\pi$
$860$	−0.828427	−0.0282491
$861$	0	0
$862$	27.3137	0.930309
$863$	42.6569	1.45206	0.726028	−	0.687665i	$-0.241364\pi$
$863$	42.6569	1.45206	0.726028	+	0.687665i	$0.241364\pi$
$864$	1.00000	0.0340207
$865$	17.6569	0.600351
$866$	−14.0000	−0.475739
$867$	−10.4853	−0.356099
$868$	0	0
$869$	−20.0000	−0.678454
$870$	−16.4853	−0.558903
$871$	18.7574	0.635569
$872$	−9.17157	−0.310589
$873$	−1.65685	−0.0560760
$874$	−7.65685	−0.258997
$875$	0	0
$876$	0.656854	0.0221930
$877$	−17.6863	−0.597224	−0.298612	−	0.954375i	$-0.596524\pi$
$877$	−17.6863	−0.597224	−0.298612	+	0.954375i	$0.596524\pi$
$878$	30.9706	1.04521
$879$	−3.58579	−0.120946
$880$	4.82843	0.162766
$881$	6.27208	0.211312	0.105656	−	0.994403i	$-0.466306\pi$
$881$	6.27208	0.211312	0.105656	+	0.994403i	$0.466306\pi$
$882$	0	0
$883$	−44.2843	−1.49028	−0.745142	−	0.666906i	$-0.767618\pi$
$883$	−44.2843	−1.49028	−0.745142	+	0.666906i	$0.767618\pi$
$884$	20.0711	0.675063
$885$	−23.3137	−0.783682
$886$	−11.1421	−0.374327
$887$	−40.9706	−1.37566	−0.687828	−	0.725873i	$-0.741436\pi$
$887$	−40.9706	−1.37566	−0.687828	+	0.725873i	$0.741436\pi$
$888$	−8.82843	−0.296263
$889$	0	0
$890$	16.4853	0.552588
$891$	2.00000	0.0670025
$892$	12.4853	0.418038
$893$	−68.9117	−2.30604
$894$	9.24264	0.309120
$895$	−53.0416	−1.77299
$896$	0	0
$897$	3.82843	0.127827
$898$	−9.51472	−0.317510
$899$	−13.6569	−0.455482
$900$	0.828427	0.0276142
$901$	−21.3431	−0.711043
$902$	−10.3431	−0.344389
$903$	0	0
$904$	−11.7279	−0.390065
$905$	22.4853	0.747436
$906$	−14.1421	−0.469841
$907$	42.8995	1.42445	0.712227	−	0.701949i	$-0.247687\pi$
$907$	42.8995	1.42445	0.712227	+	0.701949i	$0.247687\pi$
$908$	10.9706	0.364071
$909$	−8.82843	−0.292820
$910$	0	0
$911$	8.48528	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$911$	8.48528	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$912$	−7.65685	−0.253544
$913$	−14.3431	−0.474689
$914$	0	0
$915$	−4.82843	−0.159623
$916$	−2.14214	−0.0707782
$917$	0	0
$918$	−5.24264	−0.173033
$919$	−23.7279	−0.782712	−0.391356	−	0.920239i	$-0.627994\pi$
$919$	−23.7279	−0.782712	−0.391356	+	0.920239i	$0.627994\pi$
$920$	−2.41421	−0.0795943
$921$	−14.1421	−0.465999
$922$	14.0000	0.461065
$923$	15.9706	0.525677
$924$	0	0
$925$	−7.31371	−0.240473
$926$	34.6274	1.13793
$927$	−10.8995	−0.357986
$928$	6.82843	0.224154
$929$	5.37258	0.176269	0.0881344	−	0.996109i	$-0.471910\pi$
$929$	5.37258	0.176269	0.0881344	+	0.996109i	$0.471910\pi$
$930$	4.82843	0.158330
$931$	0	0
$932$	−8.00000	−0.262049
$933$	−16.3137	−0.534087
$934$	21.6569	0.708634
$935$	−25.3137	−0.827847
$936$	3.82843	0.125136
$937$	−4.48528	−0.146528	−0.0732639	−	0.997313i	$-0.523342\pi$
$937$	−4.48528	−0.146528	−0.0732639	+	0.997313i	$0.523342\pi$
$938$	0	0
$939$	32.9706	1.07595
$940$	−21.7279	−0.708687
$941$	−31.5147	−1.02735	−0.513675	−	0.857985i	$-0.671716\pi$
$941$	−31.5147	−1.02735	−0.513675	+	0.857985i	$0.671716\pi$
$942$	15.6569	0.510128
$943$	5.17157	0.168410
$944$	9.65685	0.314304
$945$	0	0
$946$	0.686292	0.0223133
$947$	−11.6863	−0.379753	−0.189877	−	0.981808i	$-0.560809\pi$
$947$	−11.6863	−0.379753	−0.189877	+	0.981808i	$0.560809\pi$
$948$	10.0000	0.324785
$949$	2.51472	0.0816312
$950$	−6.34315	−0.205799
$951$	32.6274	1.05802
$952$	0	0
$953$	11.7990	0.382207	0.191103	−	0.981570i	$-0.438793\pi$
$953$	11.7990	0.382207	0.191103	+	0.981570i	$0.438793\pi$
$954$	−4.07107	−0.131806
$955$	−12.4853	−0.404014
$956$	10.3431	0.334521
$957$	13.6569	0.441463
$958$	5.85786	0.189259
$959$	0	0
$960$	−2.41421	−0.0779184
$961$	−27.0000	−0.870968
$962$	−33.7990	−1.08972
$963$	−5.65685	−0.182290
$964$	12.8284	0.413176
$965$	−13.2426	−0.426296
$966$	0	0
$967$	−10.8284	−0.348219	−0.174109	−	0.984726i	$-0.555705\pi$
$967$	−10.8284	−0.348219	−0.174109	+	0.984726i	$0.555705\pi$
$968$	7.00000	0.224989
$969$	40.1421	1.28955
$970$	4.00000	0.128432
$971$	−13.6569	−0.438269	−0.219135	−	0.975695i	$-0.570323\pi$
$971$	−13.6569	−0.438269	−0.219135	+	0.975695i	$0.570323\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−40.7696	−1.30634
$975$	3.17157	0.101572
$976$	2.00000	0.0640184
$977$	24.3553	0.779196	0.389598	−	0.920985i	$-0.372614\pi$
$977$	24.3553	0.779196	0.389598	+	0.920985i	$0.372614\pi$
$978$	1.65685	0.0529804
$979$	−13.6569	−0.436475
$980$	0	0
$981$	9.17157	0.292826
$982$	22.6569	0.723009
$983$	−0.544156	−0.0173559	−0.00867794	−	0.999962i	$-0.502762\pi$
$983$	−0.544156	−0.0173559	−0.00867794	+	0.999962i	$0.502762\pi$
$984$	5.17157	0.164864
$985$	−30.9706	−0.986804
$986$	−35.7990	−1.14007
$987$	0	0
$988$	−29.3137	−0.932593
$989$	−0.343146	−0.0109114
$990$	−4.82843	−0.153457
$991$	19.9411	0.633451	0.316725	−	0.948517i	$-0.397417\pi$
$991$	19.9411	0.633451	0.316725	+	0.948517i	$0.397417\pi$
$992$	−2.00000	−0.0635001
$993$	34.4853	1.09436
$994$	0	0
$995$	66.7696	2.11674
$996$	7.17157	0.227240
$997$	24.6274	0.779958	0.389979	−	0.920824i	$-0.372482\pi$
$997$	24.6274	0.779958	0.389979	+	0.920824i	$0.372482\pi$
$998$	−6.82843	−0.216150
$999$	8.82843	0.279319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bt.1.2		2
7.3	odd	6		966.2.i.j.415.2	yes	4
7.5	odd	6		966.2.i.j.277.2	✓	4
7.6	odd	2		6762.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.i.j.277.2	✓	4	7.5	odd	6
966.2.i.j.415.2	yes	4	7.3	odd	6
6762.2.a.bt.1.2		2	1.1	even	1	trivial
6762.2.a.bv.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41421 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.41421 q^{10} +2.00000 q^{11} -1.00000 q^{12} -3.82843 q^{13} -2.41421 q^{15} +1.00000 q^{16} -5.24264 q^{17} -1.00000 q^{18} +7.65685 q^{19} +2.41421 q^{20} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +0.828427 q^{25} +3.82843 q^{26} -1.00000 q^{27} -6.82843 q^{29} +2.41421 q^{30} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +5.24264 q^{34} +1.00000 q^{36} -8.82843 q^{37} -7.65685 q^{38} +3.82843 q^{39} -2.41421 q^{40} +5.17157 q^{41} -0.343146 q^{43} +2.00000 q^{44} +2.41421 q^{45} -1.00000 q^{46} -9.00000 q^{47} -1.00000 q^{48} -0.828427 q^{50} +5.24264 q^{51} -3.82843 q^{52} +4.07107 q^{53} +1.00000 q^{54} +4.82843 q^{55} -7.65685 q^{57} +6.82843 q^{58} +9.65685 q^{59} -2.41421 q^{60} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -9.24264 q^{65} +2.00000 q^{66} -4.89949 q^{67} -5.24264 q^{68} -1.00000 q^{69} -4.17157 q^{71} -1.00000 q^{72} -0.656854 q^{73} +8.82843 q^{74} -0.828427 q^{75} +7.65685 q^{76} -3.82843 q^{78} -10.0000 q^{79} +2.41421 q^{80} +1.00000 q^{81} -5.17157 q^{82} -7.17157 q^{83} -12.6569 q^{85} +0.343146 q^{86} +6.82843 q^{87} -2.00000 q^{88} -6.82843 q^{89} -2.41421 q^{90} +1.00000 q^{92} -2.00000 q^{93} +9.00000 q^{94} +18.4853 q^{95} +1.00000 q^{96} -1.65685 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 2 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 2 q^{23} + 2 q^{24} - 4 q^{25} + 2 q^{26} - 2 q^{27} - 8 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} - 4 q^{33} + 2 q^{34} + 2 q^{36} - 12 q^{37} - 4 q^{38} + 2 q^{39} - 2 q^{40} + 16 q^{41} - 12 q^{43} + 4 q^{44} + 2 q^{45} - 2 q^{46} - 18 q^{47} - 2 q^{48} + 4 q^{50} + 2 q^{51} - 2 q^{52} - 6 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} + 8 q^{59} - 2 q^{60} + 4 q^{61} - 4 q^{62} + 2 q^{64} - 10 q^{65} + 4 q^{66} + 10 q^{67} - 2 q^{68} - 2 q^{69} - 14 q^{71} - 2 q^{72} + 10 q^{73} + 12 q^{74} + 4 q^{75} + 4 q^{76} - 2 q^{78} - 20 q^{79} + 2 q^{80} + 2 q^{81} - 16 q^{82} - 20 q^{83} - 14 q^{85} + 12 q^{86} + 8 q^{87} - 4 q^{88} - 8 q^{89} - 2 q^{90} + 2 q^{92} - 4 q^{93} + 18 q^{94} + 20 q^{95} + 2 q^{96} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 - 2 * q^12 - 2 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 2 * q^23 + 2 * q^24 - 4 * q^25 + 2 * q^26 - 2 * q^27 - 8 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 - 4 * q^33 + 2 * q^34 + 2 * q^36 - 12 * q^37 - 4 * q^38 + 2 * q^39 - 2 * q^40 + 16 * q^41 - 12 * q^43 + 4 * q^44 + 2 * q^45 - 2 * q^46 - 18 * q^47 - 2 * q^48 + 4 * q^50 + 2 * q^51 - 2 * q^52 - 6 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 + 8 * q^59 - 2 * q^60 + 4 * q^61 - 4 * q^62 + 2 * q^64 - 10 * q^65 + 4 * q^66 + 10 * q^67 - 2 * q^68 - 2 * q^69 - 14 * q^71 - 2 * q^72 + 10 * q^73 + 12 * q^74 + 4 * q^75 + 4 * q^76 - 2 * q^78 - 20 * q^79 + 2 * q^80 + 2 * q^81 - 16 * q^82 - 20 * q^83 - 14 * q^85 + 12 * q^86 + 8 * q^87 - 4 * q^88 - 8 * q^89 - 2 * q^90 + 2 * q^92 - 4 * q^93 + 18 * q^94 + 20 * q^95 + 2 * q^96 + 8 * q^97 + 4 * q^99

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