LMFDB - Embedded newform 8002.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.19543 q^{3} +1.00000 q^{4} -0.753590 q^{5} +3.19543 q^{6} -0.614331 q^{7} -1.00000 q^{8} +7.21079 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.19543 q^{3} +1.00000 q^{4} -0.753590 q^{5} +3.19543 q^{6} -0.614331 q^{7} -1.00000 q^{8} +7.21079 q^{9} +0.753590 q^{10} +2.80168 q^{11} -3.19543 q^{12} -0.521877 q^{13} +0.614331 q^{14} +2.40805 q^{15} +1.00000 q^{16} +4.60958 q^{17} -7.21079 q^{18} -4.91550 q^{19} -0.753590 q^{20} +1.96305 q^{21} -2.80168 q^{22} +2.21094 q^{23} +3.19543 q^{24} -4.43210 q^{25} +0.521877 q^{26} -13.4553 q^{27} -0.614331 q^{28} +8.80780 q^{29} -2.40805 q^{30} +3.63131 q^{31} -1.00000 q^{32} -8.95257 q^{33} -4.60958 q^{34} +0.462954 q^{35} +7.21079 q^{36} -4.71059 q^{37} +4.91550 q^{38} +1.66762 q^{39} +0.753590 q^{40} -3.30996 q^{41} -1.96305 q^{42} -2.17937 q^{43} +2.80168 q^{44} -5.43398 q^{45} -2.21094 q^{46} -2.38550 q^{47} -3.19543 q^{48} -6.62260 q^{49} +4.43210 q^{50} -14.7296 q^{51} -0.521877 q^{52} +9.77192 q^{53} +13.4553 q^{54} -2.11132 q^{55} +0.614331 q^{56} +15.7071 q^{57} -8.80780 q^{58} -2.95303 q^{59} +2.40805 q^{60} +2.85484 q^{61} -3.63131 q^{62} -4.42981 q^{63} +1.00000 q^{64} +0.393282 q^{65} +8.95257 q^{66} -7.75229 q^{67} +4.60958 q^{68} -7.06489 q^{69} -0.462954 q^{70} -5.02121 q^{71} -7.21079 q^{72} -11.0441 q^{73} +4.71059 q^{74} +14.1625 q^{75} -4.91550 q^{76} -1.72116 q^{77} -1.66762 q^{78} +8.27007 q^{79} -0.753590 q^{80} +21.3631 q^{81} +3.30996 q^{82} -13.0291 q^{83} +1.96305 q^{84} -3.47373 q^{85} +2.17937 q^{86} -28.1447 q^{87} -2.80168 q^{88} -16.5895 q^{89} +5.43398 q^{90} +0.320605 q^{91} +2.21094 q^{92} -11.6036 q^{93} +2.38550 q^{94} +3.70427 q^{95} +3.19543 q^{96} +10.5924 q^{97} +6.62260 q^{98} +20.2023 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−3.19543	−1.84488	−0.922442	−	0.386136i	$-0.873809\pi$
$3$	−3.19543	−1.84488	−0.922442	+	0.386136i	$0.873809\pi$
$4$	1.00000	0.500000
$5$	−0.753590	−0.337016	−0.168508	−	0.985700i	$-0.553895\pi$
$5$	−0.753590	−0.337016	−0.168508	+	0.985700i	$0.553895\pi$
$6$	3.19543	1.30453
$7$	−0.614331	−0.232195	−0.116098	−	0.993238i	$-0.537039\pi$
$7$	−0.614331	−0.232195	−0.116098	+	0.993238i	$0.537039\pi$
$8$	−1.00000	−0.353553
$9$	7.21079	2.40360
$10$	0.753590	0.238306
$11$	2.80168	0.844738	0.422369	−	0.906424i	$-0.361199\pi$
$11$	2.80168	0.844738	0.422369	+	0.906424i	$0.361199\pi$
$12$	−3.19543	−0.922442
$13$	−0.521877	−0.144743	−0.0723713	−	0.997378i	$-0.523057\pi$
$13$	−0.521877	−0.144743	−0.0723713	+	0.997378i	$0.523057\pi$
$14$	0.614331	0.164187
$15$	2.40805	0.621755
$16$	1.00000	0.250000
$17$	4.60958	1.11799	0.558993	−	0.829172i	$-0.311188\pi$
$17$	4.60958	1.11799	0.558993	+	0.829172i	$0.311188\pi$
$18$	−7.21079	−1.69960
$19$	−4.91550	−1.12769	−0.563846	−	0.825880i	$-0.690679\pi$
$19$	−4.91550	−1.12769	−0.563846	+	0.825880i	$0.690679\pi$
$20$	−0.753590	−0.168508
$21$	1.96305	0.428373
$22$	−2.80168	−0.597320
$23$	2.21094	0.461012	0.230506	−	0.973071i	$-0.425962\pi$
$23$	2.21094	0.461012	0.230506	+	0.973071i	$0.425962\pi$
$24$	3.19543	0.652265
$25$	−4.43210	−0.886420
$26$	0.521877	0.102349
$27$	−13.4553	−2.58947
$28$	−0.614331	−0.116098
$29$	8.80780	1.63557	0.817784	−	0.575525i	$-0.195202\pi$
$29$	8.80780	1.63557	0.817784	+	0.575525i	$0.195202\pi$
$30$	−2.40805	−0.439647
$31$	3.63131	0.652202	0.326101	−	0.945335i	$-0.394265\pi$
$31$	3.63131	0.652202	0.326101	+	0.945335i	$0.394265\pi$
$32$	−1.00000	−0.176777
$33$	−8.95257	−1.55844
$34$	−4.60958	−0.790536
$35$	0.462954	0.0782535
$36$	7.21079	1.20180
$37$	−4.71059	−0.774416	−0.387208	−	0.921992i	$-0.626560\pi$
$37$	−4.71059	−0.774416	−0.387208	+	0.921992i	$0.626560\pi$
$38$	4.91550	0.797399
$39$	1.66762	0.267033
$40$	0.753590	0.119153
$41$	−3.30996	−0.516930	−0.258465	−	0.966021i	$-0.583217\pi$
$41$	−3.30996	−0.516930	−0.258465	+	0.966021i	$0.583217\pi$
$42$	−1.96305	−0.302906
$43$	−2.17937	−0.332351	−0.166175	−	0.986096i	$-0.553142\pi$
$43$	−2.17937	−0.332351	−0.166175	+	0.986096i	$0.553142\pi$
$44$	2.80168	0.422369
$45$	−5.43398	−0.810050
$46$	−2.21094	−0.325985
$47$	−2.38550	−0.347961	−0.173980	−	0.984749i	$-0.555663\pi$
$47$	−2.38550	−0.347961	−0.173980	+	0.984749i	$0.555663\pi$
$48$	−3.19543	−0.461221
$49$	−6.62260	−0.946085
$50$	4.43210	0.626794
$51$	−14.7296	−2.06256
$52$	−0.521877	−0.0723713
$53$	9.77192	1.34228	0.671138	−	0.741332i	$-0.265806\pi$
$53$	9.77192	1.34228	0.671138	+	0.741332i	$0.265806\pi$
$54$	13.4553	1.83103
$55$	−2.11132	−0.284690
$56$	0.614331	0.0820934
$57$	15.7071	2.08046
$58$	−8.80780	−1.15652
$59$	−2.95303	−0.384452	−0.192226	−	0.981351i	$-0.561571\pi$
$59$	−2.95303	−0.384452	−0.192226	+	0.981351i	$0.561571\pi$
$60$	2.40805	0.310877
$61$	2.85484	0.365524	0.182762	−	0.983157i	$-0.441496\pi$
$61$	2.85484	0.365524	0.182762	+	0.983157i	$0.441496\pi$
$62$	−3.63131	−0.461177
$63$	−4.42981	−0.558103
$64$	1.00000	0.125000
$65$	0.393282	0.0487806
$66$	8.95257	1.10199
$67$	−7.75229	−0.947092	−0.473546	−	0.880769i	$-0.657026\pi$
$67$	−7.75229	−0.947092	−0.473546	+	0.880769i	$0.657026\pi$
$68$	4.60958	0.558993
$69$	−7.06489	−0.850513
$70$	−0.462954	−0.0553336
$71$	−5.02121	−0.595907	−0.297954	−	0.954580i	$-0.596304\pi$
$71$	−5.02121	−0.595907	−0.297954	+	0.954580i	$0.596304\pi$
$72$	−7.21079	−0.849799
$73$	−11.0441	−1.29262	−0.646308	−	0.763076i	$-0.723688\pi$
$73$	−11.0441	−1.29262	−0.646308	+	0.763076i	$0.723688\pi$
$74$	4.71059	0.547595
$75$	14.1625	1.63534
$76$	−4.91550	−0.563846
$77$	−1.72116	−0.196144
$78$	−1.66762	−0.188821
$79$	8.27007	0.930455	0.465228	−	0.885191i	$-0.345972\pi$
$79$	8.27007	0.930455	0.465228	+	0.885191i	$0.345972\pi$
$80$	−0.753590	−0.0842540
$81$	21.3631	2.37367
$82$	3.30996	0.365524
$83$	−13.0291	−1.43013	−0.715066	−	0.699056i	$-0.753604\pi$
$83$	−13.0291	−1.43013	−0.715066	+	0.699056i	$0.753604\pi$
$84$	1.96305	0.214187
$85$	−3.47373	−0.376779
$86$	2.17937	0.235007
$87$	−28.1447	−3.01743
$88$	−2.80168	−0.298660
$89$	−16.5895	−1.75848	−0.879239	−	0.476380i	$-0.841949\pi$
$89$	−16.5895	−1.75848	−0.879239	+	0.476380i	$0.841949\pi$
$90$	5.43398	0.572792
$91$	0.320605	0.0336086
$92$	2.21094	0.230506
$93$	−11.6036	−1.20324
$94$	2.38550	0.246045
$95$	3.70427	0.380050
$96$	3.19543	0.326132
$97$	10.5924	1.07549	0.537747	−	0.843106i	$-0.319276\pi$
$97$	10.5924	1.07549	0.537747	+	0.843106i	$0.319276\pi$
$98$	6.62260	0.668983
$99$	20.2023	2.03041
$100$	−4.43210	−0.443210
$101$	12.5997	1.25372	0.626860	−	0.779132i	$-0.284340\pi$
$101$	12.5997	1.25372	0.626860	+	0.779132i	$0.284340\pi$
$102$	14.7296	1.45845
$103$	−7.14668	−0.704183	−0.352092	−	0.935966i	$-0.614529\pi$
$103$	−7.14668	−0.704183	−0.352092	+	0.935966i	$0.614529\pi$
$104$	0.521877	0.0511743
$105$	−1.47934	−0.144369
$106$	−9.77192	−0.949133
$107$	−9.20363	−0.889748	−0.444874	−	0.895593i	$-0.646752\pi$
$107$	−9.20363	−0.889748	−0.444874	+	0.895593i	$0.646752\pi$
$108$	−13.4553	−1.29473
$109$	14.6212	1.40046	0.700230	−	0.713917i	$-0.253081\pi$
$109$	14.6212	1.40046	0.700230	+	0.713917i	$0.253081\pi$
$110$	2.11132	0.201306
$111$	15.0524	1.42871
$112$	−0.614331	−0.0580488
$113$	8.80333	0.828147	0.414074	−	0.910243i	$-0.364106\pi$
$113$	8.80333	0.828147	0.414074	+	0.910243i	$0.364106\pi$
$114$	−15.7071	−1.47111
$115$	−1.66614	−0.155368
$116$	8.80780	0.817784
$117$	−3.76314	−0.347903
$118$	2.95303	0.271849
$119$	−2.83181	−0.259591
$120$	−2.40805	−0.219824
$121$	−3.15060	−0.286418
$122$	−2.85484	−0.258465
$123$	10.5768	0.953675
$124$	3.63131	0.326101
$125$	7.10794	0.635754
$126$	4.42981	0.394639
$127$	−7.08961	−0.629101	−0.314550	−	0.949241i	$-0.601854\pi$
$127$	−7.08961	−0.629101	−0.314550	+	0.949241i	$0.601854\pi$
$128$	−1.00000	−0.0883883
$129$	6.96403	0.613148
$130$	−0.393282	−0.0344931
$131$	4.88867	0.427125	0.213562	−	0.976929i	$-0.431493\pi$
$131$	4.88867	0.427125	0.213562	+	0.976929i	$0.431493\pi$
$132$	−8.95257	−0.779221
$133$	3.01974	0.261845
$134$	7.75229	0.669696
$135$	10.1398	0.872692
$136$	−4.60958	−0.395268
$137$	−14.5708	−1.24487	−0.622434	−	0.782672i	$-0.713856\pi$
$137$	−14.5708	−1.24487	−0.622434	+	0.782672i	$0.713856\pi$
$138$	7.06489	0.601404
$139$	23.2706	1.97378	0.986891	−	0.161386i	$-0.0515964\pi$
$139$	23.2706	1.97378	0.986891	+	0.161386i	$0.0515964\pi$
$140$	0.462954	0.0391267
$141$	7.62270	0.641947
$142$	5.02121	0.421370
$143$	−1.46213	−0.122270
$144$	7.21079	0.600899
$145$	−6.63748	−0.551212
$146$	11.0441	0.914018
$147$	21.1621	1.74542
$148$	−4.71059	−0.387208
$149$	15.5970	1.27776	0.638879	−	0.769307i	$-0.279398\pi$
$149$	15.5970	1.27776	0.638879	+	0.769307i	$0.279398\pi$
$150$	−14.1625	−1.15636
$151$	4.66480	0.379616	0.189808	−	0.981821i	$-0.439213\pi$
$151$	4.66480	0.379616	0.189808	+	0.981821i	$0.439213\pi$
$152$	4.91550	0.398700
$153$	33.2387	2.68719
$154$	1.72116	0.138695
$155$	−2.73652	−0.219802
$156$	1.66762	0.133517
$157$	−13.2129	−1.05451	−0.527253	−	0.849709i	$-0.676778\pi$
$157$	−13.2129	−1.05451	−0.527253	+	0.849709i	$0.676778\pi$
$158$	−8.27007	−0.657931
$159$	−31.2255	−2.47634
$160$	0.753590	0.0595765
$161$	−1.35825	−0.107045
$162$	−21.3631	−1.67844
$163$	−3.28559	−0.257347	−0.128674	−	0.991687i	$-0.541072\pi$
$163$	−3.28559	−0.257347	−0.128674	+	0.991687i	$0.541072\pi$
$164$	−3.30996	−0.258465
$165$	6.74657	0.525220
$166$	13.0291	1.01126
$167$	12.4208	0.961152	0.480576	−	0.876953i	$-0.340428\pi$
$167$	12.4208	0.961152	0.480576	+	0.876953i	$0.340428\pi$
$168$	−1.96305	−0.151453
$169$	−12.7276	−0.979050
$170$	3.47373	0.266423
$171$	−35.4446	−2.71052
$172$	−2.17937	−0.166175
$173$	10.1804	0.774000	0.387000	−	0.922080i	$-0.373511\pi$
$173$	10.1804	0.774000	0.387000	+	0.922080i	$0.373511\pi$
$174$	28.1447	2.13365
$175$	2.72278	0.205823
$176$	2.80168	0.211184
$177$	9.43622	0.709269
$178$	16.5895	1.24343
$179$	13.8440	1.03475	0.517376	−	0.855758i	$-0.326909\pi$
$179$	13.8440	1.03475	0.517376	+	0.855758i	$0.326909\pi$
$180$	−5.43398	−0.405025
$181$	7.95875	0.591569	0.295785	−	0.955255i	$-0.404419\pi$
$181$	7.95875	0.591569	0.295785	+	0.955255i	$0.404419\pi$
$182$	−0.320605	−0.0237648
$183$	−9.12244	−0.674350
$184$	−2.21094	−0.162992
$185$	3.54985	0.260990
$186$	11.6036	0.850817
$187$	12.9146	0.944406
$188$	−2.38550	−0.173980
$189$	8.26599	0.601263
$190$	−3.70427	−0.268736
$191$	−4.12532	−0.298498	−0.149249	−	0.988800i	$-0.547686\pi$
$191$	−4.12532	−0.298498	−0.149249	+	0.988800i	$0.547686\pi$
$192$	−3.19543	−0.230610
$193$	15.7369	1.13277	0.566383	−	0.824142i	$-0.308342\pi$
$193$	15.7369	1.13277	0.566383	+	0.824142i	$0.308342\pi$
$194$	−10.5924	−0.760489
$195$	−1.25670	−0.0899945
$196$	−6.62260	−0.473043
$197$	17.5196	1.24822	0.624108	−	0.781338i	$-0.285462\pi$
$197$	17.5196	1.24822	0.624108	+	0.781338i	$0.285462\pi$
$198$	−20.2023	−1.43572
$199$	−14.2033	−1.00685	−0.503424	−	0.864039i	$-0.667927\pi$
$199$	−14.2033	−1.00685	−0.503424	+	0.864039i	$0.667927\pi$
$200$	4.43210	0.313397
$201$	24.7719	1.74728
$202$	−12.5997	−0.886514
$203$	−5.41091	−0.379771
$204$	−14.7296	−1.03128
$205$	2.49436	0.174213
$206$	7.14668	0.497933
$207$	15.9426	1.10809
$208$	−0.521877	−0.0361857
$209$	−13.7716	−0.952605
$210$	1.47934	0.102084
$211$	−14.1662	−0.975244	−0.487622	−	0.873055i	$-0.662136\pi$
$211$	−14.1662	−0.975244	−0.487622	+	0.873055i	$0.662136\pi$
$212$	9.77192	0.671138
$213$	16.0449	1.09938
$214$	9.20363	0.629147
$215$	1.64235	0.112007
$216$	13.4553	0.915516
$217$	−2.23083	−0.151438
$218$	−14.6212	−0.990275
$219$	35.2907	2.38473
$220$	−2.11132	−0.142345
$221$	−2.40563	−0.161820
$222$	−15.0524	−1.01025
$223$	−14.2587	−0.954832	−0.477416	−	0.878677i	$-0.658427\pi$
$223$	−14.2587	−0.954832	−0.477416	+	0.878677i	$0.658427\pi$
$224$	0.614331	0.0410467
$225$	−31.9589	−2.13060
$226$	−8.80333	−0.585589
$227$	14.7756	0.980691	0.490346	−	0.871528i	$-0.336871\pi$
$227$	14.7756	0.980691	0.490346	+	0.871528i	$0.336871\pi$
$228$	15.7071	1.04023
$229$	−8.99913	−0.594679	−0.297340	−	0.954772i	$-0.596099\pi$
$229$	−8.99913	−0.594679	−0.297340	+	0.954772i	$0.596099\pi$
$230$	1.66614	0.109862
$231$	5.49984	0.361863
$232$	−8.80780	−0.578261
$233$	2.53368	0.165987	0.0829934	−	0.996550i	$-0.473552\pi$
$233$	2.53368	0.165987	0.0829934	+	0.996550i	$0.473552\pi$
$234$	3.76314	0.246004
$235$	1.79769	0.117268
$236$	−2.95303	−0.192226
$237$	−26.4264	−1.71658
$238$	2.83181	0.183559
$239$	1.92244	0.124352	0.0621762	−	0.998065i	$-0.480196\pi$
$239$	1.92244	0.124352	0.0621762	+	0.998065i	$0.480196\pi$
$240$	2.40805	0.155439
$241$	8.83135	0.568877	0.284439	−	0.958694i	$-0.408193\pi$
$241$	8.83135	0.568877	0.284439	+	0.958694i	$0.408193\pi$
$242$	3.15060	0.202528
$243$	−27.8984	−1.78968
$244$	2.85484	0.182762
$245$	4.99073	0.318846
$246$	−10.5768	−0.674350
$247$	2.56529	0.163225
$248$	−3.63131	−0.230588
$249$	41.6337	2.63843
$250$	−7.10794	−0.449546
$251$	−22.9669	−1.44966	−0.724828	−	0.688929i	$-0.758081\pi$
$251$	−22.9669	−1.44966	−0.724828	+	0.688929i	$0.758081\pi$
$252$	−4.42981	−0.279052
$253$	6.19433	0.389434
$254$	7.08961	0.444842
$255$	11.1001	0.695114
$256$	1.00000	0.0625000
$257$	−22.0029	−1.37250	−0.686250	−	0.727365i	$-0.740745\pi$
$257$	−22.0029	−1.37250	−0.686250	+	0.727365i	$0.740745\pi$
$258$	−6.96403	−0.433561
$259$	2.89386	0.179816
$260$	0.393282	0.0243903
$261$	63.5112	3.93124
$262$	−4.88867	−0.302023
$263$	−3.79362	−0.233925	−0.116962	−	0.993136i	$-0.537316\pi$
$263$	−3.79362	−0.233925	−0.116962	+	0.993136i	$0.537316\pi$
$264$	8.95257	0.550993
$265$	−7.36403	−0.452369
$266$	−3.01974	−0.185152
$267$	53.0105	3.24419
$268$	−7.75229	−0.473546
$269$	−4.51591	−0.275340	−0.137670	−	0.990478i	$-0.543961\pi$
$269$	−4.51591	−0.275340	−0.137670	+	0.990478i	$0.543961\pi$
$270$	−10.1398	−0.617087
$271$	22.3526	1.35782	0.678911	−	0.734220i	$-0.262452\pi$
$271$	22.3526	1.35782	0.678911	+	0.734220i	$0.262452\pi$
$272$	4.60958	0.279497
$273$	−1.02447	−0.0620039
$274$	14.5708	0.880254
$275$	−12.4173	−0.748793
$276$	−7.06489	−0.425257
$277$	28.3828	1.70536	0.852680	−	0.522433i	$-0.174976\pi$
$277$	28.3828	1.70536	0.852680	+	0.522433i	$0.174976\pi$
$278$	−23.2706	−1.39568
$279$	26.1846	1.56763
$280$	−0.462954	−0.0276668
$281$	−18.2330	−1.08769	−0.543846	−	0.839185i	$-0.683032\pi$
$281$	−18.2330	−1.08769	−0.543846	+	0.839185i	$0.683032\pi$
$282$	−7.62270	−0.453925
$283$	9.58264	0.569629	0.284814	−	0.958583i	$-0.408068\pi$
$283$	9.58264	0.569629	0.284814	+	0.958583i	$0.408068\pi$
$284$	−5.02121	−0.297954
$285$	−11.8367	−0.701149
$286$	1.46213	0.0864577
$287$	2.03341	0.120029
$288$	−7.21079	−0.424900
$289$	4.24820	0.249894
$290$	6.63748	0.389766
$291$	−33.8473	−1.98416
$292$	−11.0441	−0.646308
$293$	6.47750	0.378419	0.189210	−	0.981937i	$-0.439407\pi$
$293$	6.47750	0.378419	0.189210	+	0.981937i	$0.439407\pi$
$294$	−21.1621	−1.23420
$295$	2.22538	0.129566
$296$	4.71059	0.273797
$297$	−37.6974	−2.18742
$298$	−15.5970	−0.903511
$299$	−1.15384	−0.0667281
$300$	14.1625	0.817671
$301$	1.33885	0.0771703
$302$	−4.66480	−0.268429
$303$	−40.2616	−2.31297
$304$	−4.91550	−0.281923
$305$	−2.15138	−0.123188
$306$	−33.2387	−1.90013
$307$	−6.24834	−0.356612	−0.178306	−	0.983975i	$-0.557062\pi$
$307$	−6.24834	−0.356612	−0.178306	+	0.983975i	$0.557062\pi$
$308$	−1.72116	−0.0980721
$309$	22.8367	1.29914
$310$	2.73652	0.155424
$311$	30.4229	1.72512	0.862562	−	0.505951i	$-0.168858\pi$
$311$	30.4229	1.72512	0.862562	+	0.505951i	$0.168858\pi$
$312$	−1.66762	−0.0944106
$313$	−20.1095	−1.13665	−0.568327	−	0.822803i	$-0.692409\pi$
$313$	−20.1095	−1.13665	−0.568327	+	0.822803i	$0.692409\pi$
$314$	13.2129	0.745648
$315$	3.33826	0.188090
$316$	8.27007	0.465228
$317$	−22.5061	−1.26407	−0.632035	−	0.774940i	$-0.717780\pi$
$317$	−22.5061	−1.26407	−0.632035	+	0.774940i	$0.717780\pi$
$318$	31.2255	1.75104
$319$	24.6766	1.38163
$320$	−0.753590	−0.0421270
$321$	29.4096	1.64148
$322$	1.35825	0.0756921
$323$	−22.6584	−1.26075
$324$	21.3631	1.18684
$325$	2.31301	0.128303
$326$	3.28559	0.181972
$327$	−46.7211	−2.58369
$328$	3.30996	0.182762
$329$	1.46549	0.0807949
$330$	−6.74657	−0.371387
$331$	9.28691	0.510455	0.255227	−	0.966881i	$-0.417850\pi$
$331$	9.28691	0.510455	0.255227	+	0.966881i	$0.417850\pi$
$332$	−13.0291	−0.715066
$333$	−33.9670	−1.86138
$334$	−12.4208	−0.679637
$335$	5.84205	0.319185
$336$	1.96305	0.107093
$337$	21.4415	1.16799	0.583996	−	0.811756i	$-0.301488\pi$
$337$	21.4415	1.16799	0.583996	+	0.811756i	$0.301488\pi$
$338$	12.7276	0.692293
$339$	−28.1304	−1.52784
$340$	−3.47373	−0.188390
$341$	10.1738	0.550940
$342$	35.4446	1.91662
$343$	8.36878	0.451872
$344$	2.17937	0.117504
$345$	5.32404	0.286636
$346$	−10.1804	−0.547300
$347$	−24.2857	−1.30373	−0.651864	−	0.758336i	$-0.726013\pi$
$347$	−24.2857	−1.30373	−0.651864	+	0.758336i	$0.726013\pi$
$348$	−28.1447	−1.50872
$349$	15.5532	0.832541	0.416271	−	0.909241i	$-0.363337\pi$
$349$	15.5532	0.832541	0.416271	+	0.909241i	$0.363337\pi$
$350$	−2.72278	−0.145539
$351$	7.02200	0.374807
$352$	−2.80168	−0.149330
$353$	−7.67418	−0.408455	−0.204228	−	0.978923i	$-0.565468\pi$
$353$	−7.67418	−0.408455	−0.204228	+	0.978923i	$0.565468\pi$
$354$	−9.43622	−0.501529
$355$	3.78393	0.200830
$356$	−16.5895	−0.879239
$357$	9.04884	0.478916
$358$	−13.8440	−0.731680
$359$	−18.6913	−0.986491	−0.493245	−	0.869890i	$-0.664190\pi$
$359$	−18.6913	−0.986491	−0.493245	+	0.869890i	$0.664190\pi$
$360$	5.43398	0.286396
$361$	5.16213	0.271691
$362$	−7.95875	−0.418303
$363$	10.0675	0.528408
$364$	0.320605	0.0168043
$365$	8.32274	0.435632
$366$	9.12244	0.476837
$367$	−1.79254	−0.0935696	−0.0467848	−	0.998905i	$-0.514898\pi$
$367$	−1.79254	−0.0935696	−0.0467848	+	0.998905i	$0.514898\pi$
$368$	2.21094	0.115253
$369$	−23.8674	−1.24249
$370$	−3.54985	−0.184548
$371$	−6.00319	−0.311670
$372$	−11.6036	−0.601619
$373$	10.9179	0.565305	0.282653	−	0.959222i	$-0.408786\pi$
$373$	10.9179	0.565305	0.282653	+	0.959222i	$0.408786\pi$
$374$	−12.9146	−0.667796
$375$	−22.7129	−1.17289
$376$	2.38550	0.123023
$377$	−4.59659	−0.236737
$378$	−8.26599	−0.425157
$379$	−2.96150	−0.152122	−0.0760610	−	0.997103i	$-0.524234\pi$
$379$	−2.96150	−0.152122	−0.0760610	+	0.997103i	$0.524234\pi$
$380$	3.70427	0.190025
$381$	22.6544	1.16062
$382$	4.12532	0.211070
$383$	−19.0884	−0.975371	−0.487685	−	0.873019i	$-0.662159\pi$
$383$	−19.0884	−0.975371	−0.487685	+	0.873019i	$0.662159\pi$
$384$	3.19543	0.163066
$385$	1.29705	0.0661037
$386$	−15.7369	−0.800986
$387$	−15.7150	−0.798837
$388$	10.5924	0.537747
$389$	11.0370	0.559598	0.279799	−	0.960059i	$-0.409732\pi$
$389$	11.0370	0.559598	0.279799	+	0.960059i	$0.409732\pi$
$390$	1.25670	0.0636357
$391$	10.1915	0.515405
$392$	6.62260	0.334492
$393$	−15.6214	−0.787996
$394$	−17.5196	−0.882622
$395$	−6.23224	−0.313578
$396$	20.2023	1.01520
$397$	12.3819	0.621428	0.310714	−	0.950504i	$-0.399432\pi$
$397$	12.3819	0.621428	0.310714	+	0.950504i	$0.399432\pi$
$398$	14.2033	0.711949
$399$	−9.64938	−0.483073
$400$	−4.43210	−0.221605
$401$	−11.4045	−0.569516	−0.284758	−	0.958599i	$-0.591913\pi$
$401$	−11.4045	−0.569516	−0.284758	+	0.958599i	$0.591913\pi$
$402$	−24.7719	−1.23551
$403$	−1.89510	−0.0944015
$404$	12.5997	0.626860
$405$	−16.0990	−0.799966
$406$	5.41091	0.268539
$407$	−13.1976	−0.654178
$408$	14.7296	0.729223
$409$	−25.3280	−1.25239	−0.626195	−	0.779667i	$-0.715389\pi$
$409$	−25.3280	−1.25239	−0.626195	+	0.779667i	$0.715389\pi$
$410$	−2.49436	−0.123187
$411$	46.5600	2.29664
$412$	−7.14668	−0.352092
$413$	1.81414	0.0892680
$414$	−15.9426	−0.783535
$415$	9.81863	0.481977
$416$	0.521877	0.0255871
$417$	−74.3595	−3.64140
$418$	13.7716	0.673593
$419$	−37.6788	−1.84073	−0.920365	−	0.391060i	$-0.872109\pi$
$419$	−37.6788	−1.84073	−0.920365	+	0.391060i	$0.872109\pi$
$420$	−1.47934	−0.0721843
$421$	10.4859	0.511051	0.255526	−	0.966802i	$-0.417751\pi$
$421$	10.4859	0.511051	0.255526	+	0.966802i	$0.417751\pi$
$422$	14.1662	0.689602
$423$	−17.2013	−0.836357
$424$	−9.77192	−0.474567
$425$	−20.4301	−0.991006
$426$	−16.0449	−0.777379
$427$	−1.75381	−0.0848730
$428$	−9.20363	−0.444874
$429$	4.67214	0.225573
$430$	−1.64235	−0.0792012
$431$	−24.4266	−1.17659	−0.588295	−	0.808647i	$-0.700201\pi$
$431$	−24.4266	−1.17659	−0.588295	+	0.808647i	$0.700201\pi$
$432$	−13.4553	−0.647367
$433$	−2.58590	−0.124271	−0.0621353	−	0.998068i	$-0.519791\pi$
$433$	−2.58590	−0.124271	−0.0621353	+	0.998068i	$0.519791\pi$
$434$	2.23083	0.107083
$435$	21.2096	1.01692
$436$	14.6212	0.700230
$437$	−10.8679	−0.519880
$438$	−35.2907	−1.68626
$439$	−12.2174	−0.583104	−0.291552	−	0.956555i	$-0.594172\pi$
$439$	−12.2174	−0.583104	−0.291552	+	0.956555i	$0.594172\pi$
$440$	2.11132	0.100653
$441$	−47.7541	−2.27401
$442$	2.40563	0.114424
$443$	5.10147	0.242378	0.121189	−	0.992629i	$-0.461329\pi$
$443$	5.10147	0.242378	0.121189	+	0.992629i	$0.461329\pi$
$444$	15.0524	0.714354
$445$	12.5017	0.592635
$446$	14.2587	0.675169
$447$	−49.8392	−2.35731
$448$	−0.614331	−0.0290244
$449$	9.29676	0.438741	0.219371	−	0.975642i	$-0.429600\pi$
$449$	9.29676	0.438741	0.219371	+	0.975642i	$0.429600\pi$
$450$	31.9589	1.50656
$451$	−9.27345	−0.436670
$452$	8.80333	0.414074
$453$	−14.9061	−0.700348
$454$	−14.7756	−0.693453
$455$	−0.241605	−0.0113266
$456$	−15.7071	−0.735554
$457$	21.6529	1.01288	0.506439	−	0.862276i	$-0.330961\pi$
$457$	21.6529	1.01288	0.506439	+	0.862276i	$0.330961\pi$
$458$	8.99913	0.420502
$459$	−62.0231	−2.89499
$460$	−1.66614	−0.0776842
$461$	−13.3440	−0.621491	−0.310745	−	0.950493i	$-0.600579\pi$
$461$	−13.3440	−0.621491	−0.310745	+	0.950493i	$0.600579\pi$
$462$	−5.49984	−0.255876
$463$	1.33623	0.0621000	0.0310500	−	0.999518i	$-0.490115\pi$
$463$	1.33623	0.0621000	0.0310500	+	0.999518i	$0.490115\pi$
$464$	8.80780	0.408892
$465$	8.74436	0.405510
$466$	−2.53368	−0.117370
$467$	−30.1692	−1.39606	−0.698032	−	0.716067i	$-0.745941\pi$
$467$	−30.1692	−1.39606	−0.698032	+	0.716067i	$0.745941\pi$
$468$	−3.76314	−0.173951
$469$	4.76247	0.219910
$470$	−1.79769	−0.0829212
$471$	42.2210	1.94544
$472$	2.95303	0.135924
$473$	−6.10589	−0.280749
$474$	26.4264	1.21381
$475$	21.7860	0.999610
$476$	−2.83181	−0.129796
$477$	70.4632	3.22629
$478$	−1.92244	−0.0879304
$479$	0.0798274	0.00364741	0.00182370	−	0.999998i	$-0.499419\pi$
$479$	0.0798274	0.00364741	0.00182370	+	0.999998i	$0.499419\pi$
$480$	−2.40805	−0.109912
$481$	2.45835	0.112091
$482$	−8.83135	−0.402257
$483$	4.34018	0.197485
$484$	−3.15060	−0.143209
$485$	−7.98232	−0.362459
$486$	27.8984	1.26550
$487$	−3.63507	−0.164721	−0.0823603	−	0.996603i	$-0.526246\pi$
$487$	−3.63507	−0.164721	−0.0823603	+	0.996603i	$0.526246\pi$
$488$	−2.85484	−0.129232
$489$	10.4989	0.474776
$490$	−4.99073	−0.225458
$491$	−1.76052	−0.0794511	−0.0397255	−	0.999211i	$-0.512648\pi$
$491$	−1.76052	−0.0794511	−0.0397255	+	0.999211i	$0.512648\pi$
$492$	10.5768	0.476837
$493$	40.6003	1.82854
$494$	−2.56529	−0.115418
$495$	−15.2243	−0.684280
$496$	3.63131	0.163051
$497$	3.08468	0.138367
$498$	−41.6337	−1.86565
$499$	26.5470	1.18841	0.594204	−	0.804314i	$-0.297467\pi$
$499$	26.5470	1.18841	0.594204	+	0.804314i	$0.297467\pi$
$500$	7.10794	0.317877
$501$	−39.6899	−1.77321
$502$	22.9669	1.02506
$503$	−13.8846	−0.619085	−0.309542	−	0.950886i	$-0.600176\pi$
$503$	−13.8846	−0.619085	−0.309542	+	0.950886i	$0.600176\pi$
$504$	4.42981	0.197319
$505$	−9.49504	−0.422524
$506$	−6.19433	−0.275372
$507$	40.6703	1.80623
$508$	−7.08961	−0.314550
$509$	35.4642	1.57192	0.785962	−	0.618275i	$-0.212168\pi$
$509$	35.4642	1.57192	0.785962	+	0.618275i	$0.212168\pi$
$510$	−11.1001	−0.491520
$511$	6.78475	0.300140
$512$	−1.00000	−0.0441942
$513$	66.1394	2.92013
$514$	22.0029	0.970505
$515$	5.38567	0.237321
$516$	6.96403	0.306574
$517$	−6.68340	−0.293936
$518$	−2.89386	−0.127149
$519$	−32.5307	−1.42794
$520$	−0.393282	−0.0172465
$521$	20.7449	0.908848	0.454424	−	0.890785i	$-0.349845\pi$
$521$	20.7449	0.908848	0.454424	+	0.890785i	$0.349845\pi$
$522$	−63.5112	−2.77981
$523$	−43.3451	−1.89535	−0.947674	−	0.319239i	$-0.896573\pi$
$523$	−43.3451	−1.89535	−0.947674	+	0.319239i	$0.896573\pi$
$524$	4.88867	0.213562
$525$	−8.70045	−0.379719
$526$	3.79362	0.165410
$527$	16.7388	0.729153
$528$	−8.95257	−0.389611
$529$	−18.1118	−0.787468
$530$	7.36403	0.319873
$531$	−21.2937	−0.924067
$532$	3.01974	0.130922
$533$	1.72739	0.0748218
$534$	−53.0105	−2.29399
$535$	6.93576	0.299859
$536$	7.75229	0.334848
$537$	−44.2377	−1.90900
$538$	4.51591	0.194695
$539$	−18.5544	−0.799194
$540$	10.1398	0.436346
$541$	1.76485	0.0758769	0.0379385	−	0.999280i	$-0.487921\pi$
$541$	1.76485	0.0758769	0.0379385	+	0.999280i	$0.487921\pi$
$542$	−22.3526	−0.960125
$543$	−25.4316	−1.09138
$544$	−4.60958	−0.197634
$545$	−11.0184	−0.471977
$546$	1.02447	0.0438434
$547$	24.5254	1.04863	0.524315	−	0.851524i	$-0.324321\pi$
$547$	24.5254	1.04863	0.524315	+	0.851524i	$0.324321\pi$
$548$	−14.5708	−0.622434
$549$	20.5856	0.878573
$550$	12.4173	0.529476
$551$	−43.2948	−1.84442
$552$	7.06489	0.300702
$553$	−5.08056	−0.216047
$554$	−28.3828	−1.20587
$555$	−11.3433	−0.481497
$556$	23.2706	0.986891
$557$	26.3698	1.11732	0.558662	−	0.829395i	$-0.311315\pi$
$557$	26.3698	1.11732	0.558662	+	0.829395i	$0.311315\pi$
$558$	−26.1846	−1.10848
$559$	1.13736	0.0481053
$560$	0.462954	0.0195634
$561$	−41.2676	−1.74232
$562$	18.2330	0.769114
$563$	−32.7490	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$563$	−32.7490	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$564$	7.62270	0.320974
$565$	−6.63410	−0.279099
$566$	−9.58264	−0.402788
$567$	−13.1240	−0.551156
$568$	5.02121	0.210685
$569$	32.3619	1.35668	0.678340	−	0.734748i	$-0.262700\pi$
$569$	32.3619	1.35668	0.678340	+	0.734748i	$0.262700\pi$
$570$	11.8367	0.495787
$571$	−2.73505	−0.114458	−0.0572292	−	0.998361i	$-0.518227\pi$
$571$	−2.73505	−0.114458	−0.0572292	+	0.998361i	$0.518227\pi$
$572$	−1.46213	−0.0611348
$573$	13.1822	0.550694
$574$	−2.03341	−0.0848730
$575$	−9.79909	−0.408650
$576$	7.21079	0.300449
$577$	−24.8640	−1.03510	−0.517550	−	0.855653i	$-0.673156\pi$
$577$	−24.8640	−1.03510	−0.517550	+	0.855653i	$0.673156\pi$
$578$	−4.24820	−0.176702
$579$	−50.2861	−2.08982
$580$	−6.63748	−0.275606
$581$	8.00420	0.332070
$582$	33.8473	1.40301
$583$	27.3778	1.13387
$584$	11.0441	0.457009
$585$	2.83587	0.117249
$586$	−6.47750	−0.267583
$587$	−8.88240	−0.366616	−0.183308	−	0.983056i	$-0.558681\pi$
$587$	−8.88240	−0.366616	−0.183308	+	0.983056i	$0.558681\pi$
$588$	21.1621	0.872709
$589$	−17.8497	−0.735484
$590$	−2.22538	−0.0916173
$591$	−55.9825	−2.30281
$592$	−4.71059	−0.193604
$593$	19.0034	0.780375	0.390188	−	0.920735i	$-0.372410\pi$
$593$	19.0034	0.780375	0.390188	+	0.920735i	$0.372410\pi$
$594$	37.6974	1.54674
$595$	2.13402	0.0874863
$596$	15.5970	0.638879
$597$	45.3858	1.85752
$598$	1.15384	0.0471839
$599$	1.81902	0.0743229	0.0371615	−	0.999309i	$-0.488168\pi$
$599$	1.81902	0.0743229	0.0371615	+	0.999309i	$0.488168\pi$
$600$	−14.1625	−0.578181
$601$	−20.2039	−0.824136	−0.412068	−	0.911153i	$-0.635193\pi$
$601$	−20.2039	−0.824136	−0.412068	+	0.911153i	$0.635193\pi$
$602$	−1.33885	−0.0545676
$603$	−55.9001	−2.27643
$604$	4.66480	0.189808
$605$	2.37426	0.0965274
$606$	40.2616	1.63552
$607$	−41.5972	−1.68838	−0.844188	−	0.536047i	$-0.819917\pi$
$607$	−41.5972	−1.68838	−0.844188	+	0.536047i	$0.819917\pi$
$608$	4.91550	0.199350
$609$	17.2902	0.700634
$610$	2.15138	0.0871067
$611$	1.24494	0.0503648
$612$	33.2387	1.34359
$613$	−16.2208	−0.655152	−0.327576	−	0.944825i	$-0.606232\pi$
$613$	−16.2208	−0.655152	−0.327576	+	0.944825i	$0.606232\pi$
$614$	6.24834	0.252163
$615$	−7.97055	−0.321403
$616$	1.72116	0.0693474
$617$	−43.2911	−1.74284	−0.871418	−	0.490542i	$-0.836799\pi$
$617$	−43.2911	−1.74284	−0.871418	+	0.490542i	$0.836799\pi$
$618$	−22.8367	−0.918628
$619$	−15.5992	−0.626985	−0.313493	−	0.949591i	$-0.601499\pi$
$619$	−15.5992	−0.626985	−0.313493	+	0.949591i	$0.601499\pi$
$620$	−2.73652	−0.109901
$621$	−29.7488	−1.19378
$622$	−30.4229	−1.21985
$623$	10.1914	0.408310
$624$	1.66762	0.0667583
$625$	16.8040	0.672161
$626$	20.1095	0.803736
$627$	44.0064	1.75744
$628$	−13.2129	−0.527253
$629$	−21.7138	−0.865787
$630$	−3.33826	−0.132999
$631$	−24.6115	−0.979770	−0.489885	−	0.871787i	$-0.662961\pi$
$631$	−24.6115	−0.979770	−0.489885	+	0.871787i	$0.662961\pi$
$632$	−8.27007	−0.328966
$633$	45.2673	1.79921
$634$	22.5061	0.893832
$635$	5.34266	0.212017
$636$	−31.2255	−1.23817
$637$	3.45618	0.136939
$638$	−24.6766	−0.976957
$639$	−36.2068	−1.43232
$640$	0.753590	0.0297883
$641$	−12.9516	−0.511557	−0.255778	−	0.966735i	$-0.582332\pi$
$641$	−12.9516	−0.511557	−0.255778	+	0.966735i	$0.582332\pi$
$642$	−29.4096	−1.16070
$643$	8.29637	0.327177	0.163588	−	0.986529i	$-0.447693\pi$
$643$	8.29637	0.327177	0.163588	+	0.986529i	$0.447693\pi$
$644$	−1.35825	−0.0535224
$645$	−5.24802	−0.206641
$646$	22.6584	0.891482
$647$	−14.8927	−0.585494	−0.292747	−	0.956190i	$-0.594569\pi$
$647$	−14.8927	−0.585494	−0.292747	+	0.956190i	$0.594569\pi$
$648$	−21.3631	−0.839221
$649$	−8.27345	−0.324761
$650$	−2.31301	−0.0907238
$651$	7.12845	0.279386
$652$	−3.28559	−0.128674
$653$	−13.4530	−0.526456	−0.263228	−	0.964734i	$-0.584787\pi$
$653$	−13.4530	−0.526456	−0.263228	+	0.964734i	$0.584787\pi$
$654$	46.7211	1.82694
$655$	−3.68405	−0.143948
$656$	−3.30996	−0.129232
$657$	−79.6368	−3.10693
$658$	−1.46549	−0.0571306
$659$	4.12333	0.160622	0.0803111	−	0.996770i	$-0.474409\pi$
$659$	4.12333	0.160622	0.0803111	+	0.996770i	$0.474409\pi$
$660$	6.74657	0.262610
$661$	−18.5605	−0.721920	−0.360960	−	0.932581i	$-0.617551\pi$
$661$	−18.5605	−0.721920	−0.360960	+	0.932581i	$0.617551\pi$
$662$	−9.28691	−0.360946
$663$	7.68704	0.298540
$664$	13.0291	0.505628
$665$	−2.27565	−0.0882459
$666$	33.9670	1.31620
$667$	19.4735	0.754016
$668$	12.4208	0.480576
$669$	45.5627	1.76155
$670$	−5.84205	−0.225698
$671$	7.99833	0.308772
$672$	−1.96305	−0.0757264
$673$	−7.71875	−0.297536	−0.148768	−	0.988872i	$-0.547531\pi$
$673$	−7.71875	−0.297536	−0.148768	+	0.988872i	$0.547531\pi$
$674$	−21.4415	−0.825895
$675$	59.6352	2.29536
$676$	−12.7276	−0.489525
$677$	14.0302	0.539226	0.269613	−	0.962969i	$-0.413104\pi$
$677$	14.0302	0.539226	0.269613	+	0.962969i	$0.413104\pi$
$678$	28.1304	1.08034
$679$	−6.50723	−0.249725
$680$	3.47373	0.133212
$681$	−47.2144	−1.80926
$682$	−10.1738	−0.389573
$683$	−46.9418	−1.79618	−0.898089	−	0.439813i	$-0.855045\pi$
$683$	−46.9418	−1.79618	−0.898089	+	0.439813i	$0.855045\pi$
$684$	−35.4446	−1.35526
$685$	10.9804	0.419540
$686$	−8.36878	−0.319522
$687$	28.7561	1.09711
$688$	−2.17937	−0.0830877
$689$	−5.09974	−0.194285
$690$	−5.32404	−0.202683
$691$	−15.5584	−0.591870	−0.295935	−	0.955208i	$-0.595631\pi$
$691$	−15.5584	−0.591870	−0.295935	+	0.955208i	$0.595631\pi$
$692$	10.1804	0.387000
$693$	−12.4109	−0.471451
$694$	24.2857	0.921874
$695$	−17.5365	−0.665196
$696$	28.1447	1.06682
$697$	−15.2575	−0.577920
$698$	−15.5532	−0.588696
$699$	−8.09620	−0.306226
$700$	2.72278	0.102911
$701$	−23.7680	−0.897703	−0.448852	−	0.893606i	$-0.648167\pi$
$701$	−23.7680	−0.897703	−0.448852	+	0.893606i	$0.648167\pi$
$702$	−7.02200	−0.265028
$703$	23.1549	0.873303
$704$	2.80168	0.105592
$705$	−5.74439	−0.216346
$706$	7.67418	0.288822
$707$	−7.74041	−0.291108
$708$	9.43622	0.354635
$709$	−20.6904	−0.777045	−0.388522	−	0.921439i	$-0.627014\pi$
$709$	−20.6904	−0.777045	−0.388522	+	0.921439i	$0.627014\pi$
$710$	−3.78393	−0.142008
$711$	59.6337	2.23644
$712$	16.5895	0.621716
$713$	8.02859	0.300673
$714$	−9.04884	−0.338644
$715$	1.10185	0.0412068
$716$	13.8440	0.517376
$717$	−6.14303	−0.229416
$718$	18.6913	0.697554
$719$	29.2483	1.09078	0.545389	−	0.838183i	$-0.316382\pi$
$719$	29.2483	1.09078	0.545389	+	0.838183i	$0.316382\pi$
$720$	−5.43398	−0.202512
$721$	4.39042	0.163508
$722$	−5.16213	−0.192115
$723$	−28.2200	−1.04951
$724$	7.95875	0.295785
$725$	−39.0371	−1.44980
$726$	−10.0675	−0.373641
$727$	15.9690	0.592257	0.296128	−	0.955148i	$-0.404304\pi$
$727$	15.9690	0.592257	0.296128	+	0.955148i	$0.404304\pi$
$728$	−0.320605	−0.0118824
$729$	25.0583	0.928083
$730$	−8.32274	−0.308039
$731$	−10.0460	−0.371564
$732$	−9.12244	−0.337175
$733$	−8.22236	−0.303700	−0.151850	−	0.988404i	$-0.548523\pi$
$733$	−8.22236	−0.303700	−0.151850	+	0.988404i	$0.548523\pi$
$734$	1.79254	0.0661637
$735$	−15.9475	−0.588233
$736$	−2.21094	−0.0814962
$737$	−21.7194	−0.800045
$738$	23.8674	0.878573
$739$	36.3606	1.33755	0.668774	−	0.743466i	$-0.266819\pi$
$739$	36.3606	1.33755	0.668774	+	0.743466i	$0.266819\pi$
$740$	3.54985	0.130495
$741$	−8.19720	−0.301132
$742$	6.00319	0.220384
$743$	11.4107	0.418619	0.209310	−	0.977849i	$-0.432878\pi$
$743$	11.4107	0.418619	0.209310	+	0.977849i	$0.432878\pi$
$744$	11.6036	0.425409
$745$	−11.7538	−0.430625
$746$	−10.9179	−0.399731
$747$	−93.9503	−3.43746
$748$	12.9146	0.472203
$749$	5.65407	0.206595
$750$	22.7129	0.829359
$751$	34.9227	1.27435	0.637174	−	0.770720i	$-0.280103\pi$
$751$	34.9227	1.27435	0.637174	+	0.770720i	$0.280103\pi$
$752$	−2.38550	−0.0869902
$753$	73.3891	2.67445
$754$	4.59659	0.167398
$755$	−3.51535	−0.127937
$756$	8.26599	0.300631
$757$	7.87616	0.286264	0.143132	−	0.989704i	$-0.454283\pi$
$757$	7.87616	0.286264	0.143132	+	0.989704i	$0.454283\pi$
$758$	2.96150	0.107567
$759$	−19.7936	−0.718461
$760$	−3.70427	−0.134368
$761$	−29.3448	−1.06375	−0.531873	−	0.846824i	$-0.678512\pi$
$761$	−29.3448	−1.06375	−0.531873	+	0.846824i	$0.678512\pi$
$762$	−22.6544	−0.820681
$763$	−8.98227	−0.325180
$764$	−4.12532	−0.149249
$765$	−25.0483	−0.905625
$766$	19.0884	0.689691
$767$	1.54112	0.0556466
$768$	−3.19543	−0.115305
$769$	16.5791	0.597857	0.298929	−	0.954275i	$-0.403371\pi$
$769$	16.5791	0.597857	0.298929	+	0.954275i	$0.403371\pi$
$770$	−1.29705	−0.0467424
$771$	70.3086	2.53210
$772$	15.7369	0.566383
$773$	−50.5429	−1.81790	−0.908950	−	0.416905i	$-0.863115\pi$
$773$	−50.5429	−1.81790	−0.908950	+	0.416905i	$0.863115\pi$
$774$	15.7150	0.564863
$775$	−16.0943	−0.578125
$776$	−10.5924	−0.380245
$777$	−9.24713	−0.331739
$778$	−11.0370	−0.395695
$779$	16.2701	0.582938
$780$	−1.25670	−0.0449972
$781$	−14.0678	−0.503386
$782$	−10.1915	−0.364446
$783$	−118.511	−4.23525
$784$	−6.62260	−0.236521
$785$	9.95712	0.355385
$786$	15.6214	0.557197
$787$	−7.47450	−0.266437	−0.133219	−	0.991087i	$-0.542531\pi$
$787$	−7.47450	−0.266437	−0.133219	+	0.991087i	$0.542531\pi$
$788$	17.5196	0.624108
$789$	12.1222	0.431563
$790$	6.23224	0.221733
$791$	−5.40816	−0.192292
$792$	−20.2023	−0.717858
$793$	−1.48987	−0.0529070
$794$	−12.3819	−0.439416
$795$	23.5312	0.834567
$796$	−14.2033	−0.503424
$797$	25.7040	0.910481	0.455240	−	0.890369i	$-0.349553\pi$
$797$	25.7040	0.910481	0.455240	+	0.890369i	$0.349553\pi$
$798$	9.64938	0.341584
$799$	−10.9961	−0.389016
$800$	4.43210	0.156698
$801$	−119.623	−4.22667
$802$	11.4045	0.402709
$803$	−30.9421	−1.09192
$804$	24.7719	0.873638
$805$	1.02356	0.0360758
$806$	1.89510	0.0667519
$807$	14.4303	0.507970
$808$	−12.5997	−0.443257
$809$	−3.15696	−0.110993	−0.0554963	−	0.998459i	$-0.517674\pi$
$809$	−3.15696	−0.110993	−0.0554963	+	0.998459i	$0.517674\pi$
$810$	16.0990	0.565661
$811$	−41.0492	−1.44143	−0.720717	−	0.693230i	$-0.756187\pi$
$811$	−41.0492	−1.44143	−0.720717	+	0.693230i	$0.756187\pi$
$812$	−5.41091	−0.189886
$813$	−71.4261	−2.50502
$814$	13.1976	0.462574
$815$	2.47599	0.0867301
$816$	−14.7296	−0.515639
$817$	10.7127	0.374790
$818$	25.3280	0.885573
$819$	2.31182	0.0807814
$820$	2.49436	0.0871067
$821$	52.0810	1.81764	0.908820	−	0.417188i	$-0.136984\pi$
$821$	52.0810	1.81764	0.908820	+	0.417188i	$0.136984\pi$
$822$	−46.5600	−1.62397
$823$	−44.6713	−1.55714	−0.778571	−	0.627556i	$-0.784055\pi$
$823$	−44.6713	−1.55714	−0.778571	+	0.627556i	$0.784055\pi$
$824$	7.14668	0.248966
$825$	39.6787	1.38144
$826$	−1.81414	−0.0631220
$827$	23.0927	0.803012	0.401506	−	0.915856i	$-0.368487\pi$
$827$	23.0927	0.803012	0.401506	+	0.915856i	$0.368487\pi$
$828$	15.9426	0.554043
$829$	12.9158	0.448583	0.224292	−	0.974522i	$-0.427993\pi$
$829$	12.9158	0.448583	0.224292	+	0.974522i	$0.427993\pi$
$830$	−9.81863	−0.340810
$831$	−90.6955	−3.14619
$832$	−0.521877	−0.0180928
$833$	−30.5274	−1.05771
$834$	74.3595	2.57486
$835$	−9.36021	−0.323924
$836$	−13.7716	−0.476302
$837$	−48.8603	−1.68886
$838$	37.6788	1.30159
$839$	−24.8984	−0.859589	−0.429795	−	0.902927i	$-0.641414\pi$
$839$	−24.8984	−0.859589	−0.429795	+	0.902927i	$0.641414\pi$
$840$	1.47934	0.0510420
$841$	48.5774	1.67508
$842$	−10.4859	−0.361368
$843$	58.2624	2.00666
$844$	−14.1662	−0.487622
$845$	9.59143	0.329955
$846$	17.2013	0.591394
$847$	1.93551	0.0665049
$848$	9.77192	0.335569
$849$	−30.6207	−1.05090
$850$	20.4301	0.700747
$851$	−10.4148	−0.357015
$852$	16.0449	0.549690
$853$	−7.57955	−0.259519	−0.129759	−	0.991546i	$-0.541421\pi$
$853$	−7.57955	−0.259519	−0.129759	+	0.991546i	$0.541421\pi$
$854$	1.75381	0.0600143
$855$	26.7107	0.913487
$856$	9.20363	0.314574
$857$	14.1193	0.482307	0.241154	−	0.970487i	$-0.422474\pi$
$857$	14.1193	0.482307	0.241154	+	0.970487i	$0.422474\pi$
$858$	−4.67214	−0.159504
$859$	14.8858	0.507898	0.253949	−	0.967218i	$-0.418270\pi$
$859$	14.8858	0.507898	0.253949	+	0.967218i	$0.418270\pi$
$860$	1.64235	0.0560037
$861$	−6.49763	−0.221439
$862$	24.4266	0.831975
$863$	0.972315	0.0330980	0.0165490	−	0.999863i	$-0.494732\pi$
$863$	0.972315	0.0330980	0.0165490	+	0.999863i	$0.494732\pi$
$864$	13.4553	0.457758
$865$	−7.67183	−0.260850
$866$	2.58590	0.0878726
$867$	−13.5748	−0.461026
$868$	−2.23083	−0.0757191
$869$	23.1701	0.785991
$870$	−21.2096	−0.719073
$871$	4.04574	0.137085
$872$	−14.6212	−0.495137
$873$	76.3794	2.58505
$874$	10.8679	0.367611
$875$	−4.36663	−0.147619
$876$	35.2907	1.19236
$877$	−1.81225	−0.0611953	−0.0305976	−	0.999532i	$-0.509741\pi$
$877$	−1.81225	−0.0611953	−0.0305976	+	0.999532i	$0.509741\pi$
$878$	12.2174	0.412317
$879$	−20.6984	−0.698140
$880$	−2.11132	−0.0711725
$881$	−14.8664	−0.500861	−0.250430	−	0.968135i	$-0.580572\pi$
$881$	−14.8664	−0.500861	−0.250430	+	0.968135i	$0.580572\pi$
$882$	47.7541	1.60797
$883$	24.7405	0.832584	0.416292	−	0.909231i	$-0.363329\pi$
$883$	24.7405	0.832584	0.416292	+	0.909231i	$0.363329\pi$
$884$	−2.40563	−0.0809102
$885$	−7.11104	−0.239035
$886$	−5.10147	−0.171387
$887$	28.1823	0.946271	0.473135	−	0.880990i	$-0.343122\pi$
$887$	28.1823	0.946271	0.473135	+	0.880990i	$0.343122\pi$
$888$	−15.0524	−0.505124
$889$	4.35536	0.146074
$890$	−12.5017	−0.419056
$891$	59.8525	2.00513
$892$	−14.2587	−0.477416
$893$	11.7259	0.392393
$894$	49.8392	1.66687
$895$	−10.4327	−0.348728
$896$	0.614331	0.0205234
$897$	3.68701	0.123106
$898$	−9.29676	−0.310237
$899$	31.9839	1.06672
$900$	−31.9589	−1.06530
$901$	45.0444	1.50065
$902$	9.27345	0.308772
$903$	−4.27822	−0.142370
$904$	−8.80333	−0.292794
$905$	−5.99764	−0.199368
$906$	14.9061	0.495221
$907$	−19.9878	−0.663685	−0.331842	−	0.943335i	$-0.607670\pi$
$907$	−19.9878	−0.663685	−0.331842	+	0.943335i	$0.607670\pi$
$908$	14.7756	0.490346
$909$	90.8540	3.01344
$910$	0.241605	0.00800913
$911$	55.8879	1.85165	0.925824	−	0.377955i	$-0.123373\pi$
$911$	55.8879	1.85165	0.925824	+	0.377955i	$0.123373\pi$
$912$	15.7071	0.520115
$913$	−36.5034	−1.20809
$914$	−21.6529	−0.716214
$915$	6.87458	0.227267
$916$	−8.99913	−0.297340
$917$	−3.00326	−0.0991764
$918$	62.0231	2.04707
$919$	−49.2573	−1.62485	−0.812424	−	0.583067i	$-0.801852\pi$
$919$	−49.2573	−1.62485	−0.812424	+	0.583067i	$0.801852\pi$
$920$	1.66614	0.0549310
$921$	19.9662	0.657907
$922$	13.3440	0.439460
$923$	2.62045	0.0862532
$924$	5.49984	0.180932
$925$	20.8778	0.686458
$926$	−1.33623	−0.0439114
$927$	−51.5332	−1.69257
$928$	−8.80780	−0.289130
$929$	−2.94830	−0.0967305	−0.0483653	−	0.998830i	$-0.515401\pi$
$929$	−2.94830	−0.0967305	−0.0483653	+	0.998830i	$0.515401\pi$
$930$	−8.74436	−0.286739
$931$	32.5534	1.06689
$932$	2.53368	0.0829934
$933$	−97.2143	−3.18265
$934$	30.1692	0.987166
$935$	−9.73228	−0.318280
$936$	3.76314	0.123002
$937$	52.2080	1.70556	0.852781	−	0.522268i	$-0.174914\pi$
$937$	52.2080	1.70556	0.852781	+	0.522268i	$0.174914\pi$
$938$	−4.76247	−0.155500
$939$	64.2585	2.09700
$940$	1.79769	0.0586342
$941$	10.1434	0.330665	0.165333	−	0.986238i	$-0.447130\pi$
$941$	10.1434	0.330665	0.165333	+	0.986238i	$0.447130\pi$
$942$	−42.2210	−1.37563
$943$	−7.31812	−0.238311
$944$	−2.95303	−0.0961130
$945$	−6.22917	−0.202635
$946$	6.10589	0.198520
$947$	−38.3042	−1.24472	−0.622359	−	0.782732i	$-0.713826\pi$
$947$	−38.3042	−1.24472	−0.622359	+	0.782732i	$0.713826\pi$
$948$	−26.4264	−0.858291
$949$	5.76368	0.187097
$950$	−21.7860	−0.706831
$951$	71.9168	2.33206
$952$	2.83181	0.0917793
$953$	6.95857	0.225410	0.112705	−	0.993628i	$-0.464048\pi$
$953$	6.95857	0.225410	0.112705	+	0.993628i	$0.464048\pi$
$954$	−70.4632	−2.28133
$955$	3.10880	0.100598
$956$	1.92244	0.0621762
$957$	−78.8525	−2.54894
$958$	−0.0798274	−0.00257911
$959$	8.95130	0.289052
$960$	2.40805	0.0777194
$961$	−17.8136	−0.574632
$962$	−2.45835	−0.0792603
$963$	−66.3654	−2.13859
$964$	8.83135	0.284439
$965$	−11.8592	−0.381760
$966$	−4.34018	−0.139643
$967$	22.6721	0.729084	0.364542	−	0.931187i	$-0.381225\pi$
$967$	22.6721	0.729084	0.364542	+	0.931187i	$0.381225\pi$
$968$	3.15060	0.101264
$969$	72.4033	2.32593
$970$	7.98232	0.256297
$971$	−3.36485	−0.107983	−0.0539916	−	0.998541i	$-0.517194\pi$
$971$	−3.36485	−0.107983	−0.0539916	+	0.998541i	$0.517194\pi$
$972$	−27.8984	−0.894842
$973$	−14.2958	−0.458303
$974$	3.63507	0.116475
$975$	−7.39107	−0.236704
$976$	2.85484	0.0913811
$977$	20.7244	0.663032	0.331516	−	0.943450i	$-0.392440\pi$
$977$	20.7244	0.663032	0.331516	+	0.943450i	$0.392440\pi$
$978$	−10.4989	−0.335717
$979$	−46.4783	−1.48545
$980$	4.99073	0.159423
$981$	105.431	3.36614
$982$	1.76052	0.0561804
$983$	−45.8770	−1.46325	−0.731624	−	0.681709i	$-0.761237\pi$
$983$	−45.8770	−1.46325	−0.731624	+	0.681709i	$0.761237\pi$
$984$	−10.5768	−0.337175
$985$	−13.2026	−0.420669
$986$	−40.6003	−1.29298
$987$	−4.68286	−0.149057
$988$	2.56529	0.0816126
$989$	−4.81845	−0.153218
$990$	15.2243	0.483859
$991$	53.6991	1.70581	0.852903	−	0.522069i	$-0.174840\pi$
$991$	53.6991	1.70581	0.852903	+	0.522069i	$0.174840\pi$
$992$	−3.63131	−0.115294
$993$	−29.6757	−0.941729
$994$	−3.08468	−0.0978402
$995$	10.7035	0.339324
$996$	41.6337	1.31921
$997$	3.60231	0.114086	0.0570431	−	0.998372i	$-0.481833\pi$
$997$	3.60231	0.114086	0.0570431	+	0.998372i	$0.481833\pi$
$998$	−26.5470	−0.840332
$999$	63.3823	2.00533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.6	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.6	✓	89	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.19543 q^{3} +1.00000 q^{4} -0.753590 q^{5} +3.19543 q^{6} -0.614331 q^{7} -1.00000 q^{8} +7.21079 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.19543 q^{3} +1.00000 q^{4} -0.753590 q^{5} +3.19543 q^{6} -0.614331 q^{7} -1.00000 q^{8} +7.21079 q^{9} +0.753590 q^{10} +2.80168 q^{11} -3.19543 q^{12} -0.521877 q^{13} +0.614331 q^{14} +2.40805 q^{15} +1.00000 q^{16} +4.60958 q^{17} -7.21079 q^{18} -4.91550 q^{19} -0.753590 q^{20} +1.96305 q^{21} -2.80168 q^{22} +2.21094 q^{23} +3.19543 q^{24} -4.43210 q^{25} +0.521877 q^{26} -13.4553 q^{27} -0.614331 q^{28} +8.80780 q^{29} -2.40805 q^{30} +3.63131 q^{31} -1.00000 q^{32} -8.95257 q^{33} -4.60958 q^{34} +0.462954 q^{35} +7.21079 q^{36} -4.71059 q^{37} +4.91550 q^{38} +1.66762 q^{39} +0.753590 q^{40} -3.30996 q^{41} -1.96305 q^{42} -2.17937 q^{43} +2.80168 q^{44} -5.43398 q^{45} -2.21094 q^{46} -2.38550 q^{47} -3.19543 q^{48} -6.62260 q^{49} +4.43210 q^{50} -14.7296 q^{51} -0.521877 q^{52} +9.77192 q^{53} +13.4553 q^{54} -2.11132 q^{55} +0.614331 q^{56} +15.7071 q^{57} -8.80780 q^{58} -2.95303 q^{59} +2.40805 q^{60} +2.85484 q^{61} -3.63131 q^{62} -4.42981 q^{63} +1.00000 q^{64} +0.393282 q^{65} +8.95257 q^{66} -7.75229 q^{67} +4.60958 q^{68} -7.06489 q^{69} -0.462954 q^{70} -5.02121 q^{71} -7.21079 q^{72} -11.0441 q^{73} +4.71059 q^{74} +14.1625 q^{75} -4.91550 q^{76} -1.72116 q^{77} -1.66762 q^{78} +8.27007 q^{79} -0.753590 q^{80} +21.3631 q^{81} +3.30996 q^{82} -13.0291 q^{83} +1.96305 q^{84} -3.47373 q^{85} +2.17937 q^{86} -28.1447 q^{87} -2.80168 q^{88} -16.5895 q^{89} +5.43398 q^{90} +0.320605 q^{91} +2.21094 q^{92} -11.6036 q^{93} +2.38550 q^{94} +3.70427 q^{95} +3.19543 q^{96} +10.5924 q^{97} +6.62260 q^{98} +20.2023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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