LMFDB - Embedded newform 8002.2.a.f.1.3

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			1.3
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.29759 q^{3} +1.00000 q^{4} -2.40937 q^{5} +3.29759 q^{6} +1.21018 q^{7} -1.00000 q^{8} +7.87411 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.29759 q^{3} +1.00000 q^{4} -2.40937 q^{5} +3.29759 q^{6} +1.21018 q^{7} -1.00000 q^{8} +7.87411 q^{9} +2.40937 q^{10} -4.49235 q^{11} -3.29759 q^{12} +1.87412 q^{13} -1.21018 q^{14} +7.94512 q^{15} +1.00000 q^{16} +0.135500 q^{17} -7.87411 q^{18} +6.90709 q^{19} -2.40937 q^{20} -3.99067 q^{21} +4.49235 q^{22} +2.56525 q^{23} +3.29759 q^{24} +0.805068 q^{25} -1.87412 q^{26} -16.0728 q^{27} +1.21018 q^{28} -6.72694 q^{29} -7.94512 q^{30} -5.71019 q^{31} -1.00000 q^{32} +14.8139 q^{33} -0.135500 q^{34} -2.91577 q^{35} +7.87411 q^{36} +10.2543 q^{37} -6.90709 q^{38} -6.18008 q^{39} +2.40937 q^{40} -10.6956 q^{41} +3.99067 q^{42} +5.12576 q^{43} -4.49235 q^{44} -18.9717 q^{45} -2.56525 q^{46} -5.98727 q^{47} -3.29759 q^{48} -5.53547 q^{49} -0.805068 q^{50} -0.446825 q^{51} +1.87412 q^{52} +2.09275 q^{53} +16.0728 q^{54} +10.8237 q^{55} -1.21018 q^{56} -22.7768 q^{57} +6.72694 q^{58} +11.4393 q^{59} +7.94512 q^{60} +12.8461 q^{61} +5.71019 q^{62} +9.52907 q^{63} +1.00000 q^{64} -4.51545 q^{65} -14.8139 q^{66} -10.5365 q^{67} +0.135500 q^{68} -8.45914 q^{69} +2.91577 q^{70} -14.8715 q^{71} -7.87411 q^{72} -12.8913 q^{73} -10.2543 q^{74} -2.65479 q^{75} +6.90709 q^{76} -5.43654 q^{77} +6.18008 q^{78} +5.65417 q^{79} -2.40937 q^{80} +29.3793 q^{81} +10.6956 q^{82} -3.04871 q^{83} -3.99067 q^{84} -0.326470 q^{85} -5.12576 q^{86} +22.1827 q^{87} +4.49235 q^{88} +17.2371 q^{89} +18.9717 q^{90} +2.26802 q^{91} +2.56525 q^{92} +18.8299 q^{93} +5.98727 q^{94} -16.6417 q^{95} +3.29759 q^{96} -9.19641 q^{97} +5.53547 q^{98} -35.3733 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.29759	−1.90387	−0.951933	−	0.306307i	$-0.900907\pi$
$3$	−3.29759	−1.90387	−0.951933	+	0.306307i	$0.900907\pi$
$4$	1.00000	0.500000
$5$	−2.40937	−1.07750	−0.538752	−	0.842465i	$-0.681104\pi$
$5$	−2.40937	−1.07750	−0.538752	+	0.842465i	$0.681104\pi$
$6$	3.29759	1.34624
$7$	1.21018	0.457404	0.228702	−	0.973496i	$-0.426552\pi$
$7$	1.21018	0.457404	0.228702	+	0.973496i	$0.426552\pi$
$8$	−1.00000	−0.353553
$9$	7.87411	2.62470
$10$	2.40937	0.761910
$11$	−4.49235	−1.35450	−0.677248	−	0.735755i	$-0.736827\pi$
$11$	−4.49235	−1.35450	−0.677248	+	0.735755i	$0.736827\pi$
$12$	−3.29759	−0.951933
$13$	1.87412	0.519787	0.259894	−	0.965637i	$-0.416312\pi$
$13$	1.87412	0.519787	0.259894	+	0.965637i	$0.416312\pi$
$14$	−1.21018	−0.323434
$15$	7.94512	2.05142
$16$	1.00000	0.250000
$17$	0.135500	0.0328636	0.0164318	−	0.999865i	$-0.494769\pi$
$17$	0.135500	0.0328636	0.0164318	+	0.999865i	$0.494769\pi$
$18$	−7.87411	−1.85595
$19$	6.90709	1.58460	0.792298	−	0.610135i	$-0.208885\pi$
$19$	6.90709	1.58460	0.792298	+	0.610135i	$0.208885\pi$
$20$	−2.40937	−0.538752
$21$	−3.99067	−0.870836
$22$	4.49235	0.957773
$23$	2.56525	0.534891	0.267446	−	0.963573i	$-0.413820\pi$
$23$	2.56525	0.534891	0.267446	+	0.963573i	$0.413820\pi$
$24$	3.29759	0.673118
$25$	0.805068	0.161014
$26$	−1.87412	−0.367545
$27$	−16.0728	−3.09322
$28$	1.21018	0.228702
$29$	−6.72694	−1.24916	−0.624581	−	0.780960i	$-0.714730\pi$
$29$	−6.72694	−1.24916	−0.624581	+	0.780960i	$0.714730\pi$
$30$	−7.94512	−1.45057
$31$	−5.71019	−1.02558	−0.512790	−	0.858514i	$-0.671388\pi$
$31$	−5.71019	−1.02558	−0.512790	+	0.858514i	$0.671388\pi$
$32$	−1.00000	−0.176777
$33$	14.8139	2.57878
$34$	−0.135500	−0.0232381
$35$	−2.91577	−0.492854
$36$	7.87411	1.31235
$37$	10.2543	1.68580	0.842902	−	0.538067i	$-0.180845\pi$
$37$	10.2543	1.68580	0.842902	+	0.538067i	$0.180845\pi$
$38$	−6.90709	−1.12048
$39$	−6.18008	−0.989606
$40$	2.40937	0.380955
$41$	−10.6956	−1.67037	−0.835185	−	0.549970i	$-0.814639\pi$
$41$	−10.6956	−1.67037	−0.835185	+	0.549970i	$0.814639\pi$
$42$	3.99067	0.615774
$43$	5.12576	0.781672	0.390836	−	0.920460i	$-0.372186\pi$
$43$	5.12576	0.781672	0.390836	+	0.920460i	$0.372186\pi$
$44$	−4.49235	−0.677248
$45$	−18.9717	−2.82813
$46$	−2.56525	−0.378225
$47$	−5.98727	−0.873334	−0.436667	−	0.899623i	$-0.643841\pi$
$47$	−5.98727	−0.873334	−0.436667	+	0.899623i	$0.643841\pi$
$48$	−3.29759	−0.475966
$49$	−5.53547	−0.790782
$50$	−0.805068	−0.113854
$51$	−0.446825	−0.0625680
$52$	1.87412	0.259894
$53$	2.09275	0.287462	0.143731	−	0.989617i	$-0.454090\pi$
$53$	2.09275	0.287462	0.143731	+	0.989617i	$0.454090\pi$
$54$	16.0728	2.18724
$55$	10.8237	1.45947
$56$	−1.21018	−0.161717
$57$	−22.7768	−3.01686
$58$	6.72694	0.883291
$59$	11.4393	1.48927	0.744637	−	0.667470i	$-0.232623\pi$
$59$	11.4393	1.48927	0.744637	+	0.667470i	$0.232623\pi$
$60$	7.94512	1.02571
$61$	12.8461	1.64477	0.822385	−	0.568931i	$-0.192643\pi$
$61$	12.8461	1.64477	0.822385	+	0.568931i	$0.192643\pi$
$62$	5.71019	0.725195
$63$	9.52907	1.20055
$64$	1.00000	0.125000
$65$	−4.51545	−0.560073
$66$	−14.8139	−1.82347
$67$	−10.5365	−1.28724	−0.643618	−	0.765347i	$-0.722567\pi$
$67$	−10.5365	−1.28724	−0.643618	+	0.765347i	$0.722567\pi$
$68$	0.135500	0.0164318
$69$	−8.45914	−1.01836
$70$	2.91577	0.348501
$71$	−14.8715	−1.76493	−0.882464	−	0.470381i	$-0.844117\pi$
$71$	−14.8715	−1.76493	−0.882464	+	0.470381i	$0.844117\pi$
$72$	−7.87411	−0.927973
$73$	−12.8913	−1.50881	−0.754407	−	0.656407i	$-0.772076\pi$
$73$	−12.8913	−1.50881	−0.754407	+	0.656407i	$0.772076\pi$
$74$	−10.2543	−1.19204
$75$	−2.65479	−0.306548
$76$	6.90709	0.792298
$77$	−5.43654	−0.619552
$78$	6.18008	0.699757
$79$	5.65417	0.636144	0.318072	−	0.948067i	$-0.396965\pi$
$79$	5.65417	0.636144	0.318072	+	0.948067i	$0.396965\pi$
$80$	−2.40937	−0.269376
$81$	29.3793	3.26437
$82$	10.6956	1.18113
$83$	−3.04871	−0.334639	−0.167320	−	0.985903i	$-0.553511\pi$
$83$	−3.04871	−0.334639	−0.167320	+	0.985903i	$0.553511\pi$
$84$	−3.99067	−0.435418
$85$	−0.326470	−0.0354107
$86$	−5.12576	−0.552725
$87$	22.1827	2.37824
$88$	4.49235	0.478886
$89$	17.2371	1.82713	0.913567	−	0.406689i	$-0.133317\pi$
$89$	17.2371	1.82713	0.913567	+	0.406689i	$0.133317\pi$
$90$	18.9717	1.99979
$91$	2.26802	0.237753
$92$	2.56525	0.267446
$93$	18.8299	1.95257
$94$	5.98727	0.617540
$95$	−16.6417	−1.70741
$96$	3.29759	0.336559
$97$	−9.19641	−0.933754	−0.466877	−	0.884322i	$-0.654621\pi$
$97$	−9.19641	−0.933754	−0.466877	+	0.884322i	$0.654621\pi$
$98$	5.53547	0.559167
$99$	−35.3733	−3.55515
$100$	0.805068	0.0805068
$101$	1.09450	0.108907	0.0544533	−	0.998516i	$-0.482658\pi$
$101$	1.09450	0.108907	0.0544533	+	0.998516i	$0.482658\pi$
$102$	0.446825	0.0442422
$103$	−0.250501	−0.0246826	−0.0123413	−	0.999924i	$-0.503928\pi$
$103$	−0.250501	−0.0246826	−0.0123413	+	0.999924i	$0.503928\pi$
$104$	−1.87412	−0.183773
$105$	9.61501	0.938329
$106$	−2.09275	−0.203266
$107$	−7.16532	−0.692698	−0.346349	−	0.938106i	$-0.612579\pi$
$107$	−7.16532	−0.692698	−0.346349	+	0.938106i	$0.612579\pi$
$108$	−16.0728	−1.54661
$109$	7.82466	0.749467	0.374733	−	0.927133i	$-0.377734\pi$
$109$	7.82466	0.749467	0.374733	+	0.927133i	$0.377734\pi$
$110$	−10.8237	−1.03200
$111$	−33.8147	−3.20955
$112$	1.21018	0.114351
$113$	10.5537	0.992811	0.496405	−	0.868091i	$-0.334653\pi$
$113$	10.5537	0.992811	0.496405	+	0.868091i	$0.334653\pi$
$114$	22.7768	2.13324
$115$	−6.18064	−0.576347
$116$	−6.72694	−0.624581
$117$	14.7570	1.36429
$118$	−11.4393	−1.05308
$119$	0.163979	0.0150320
$120$	−7.94512	−0.725287
$121$	9.18124	0.834658
$122$	−12.8461	−1.16303
$123$	35.2697	3.18016
$124$	−5.71019	−0.512790
$125$	10.1071	0.904011
$126$	−9.52907	−0.848917
$127$	12.7483	1.13123	0.565615	−	0.824670i	$-0.308639\pi$
$127$	12.7483	1.13123	0.565615	+	0.824670i	$0.308639\pi$
$128$	−1.00000	−0.0883883
$129$	−16.9027	−1.48820
$130$	4.51545	0.396031
$131$	10.9824	0.959537	0.479769	−	0.877395i	$-0.340721\pi$
$131$	10.9824	0.959537	0.479769	+	0.877395i	$0.340721\pi$
$132$	14.8139	1.28939
$133$	8.35880	0.724800
$134$	10.5365	0.910213
$135$	38.7254	3.33295
$136$	−0.135500	−0.0116191
$137$	−8.34612	−0.713057	−0.356528	−	0.934284i	$-0.616040\pi$
$137$	−8.34612	−0.713057	−0.356528	+	0.934284i	$0.616040\pi$
$138$	8.45914	0.720090
$139$	13.2173	1.12108	0.560540	−	0.828128i	$-0.310594\pi$
$139$	13.2173	1.12108	0.560540	+	0.828128i	$0.310594\pi$
$140$	−2.91577	−0.246427
$141$	19.7436	1.66271
$142$	14.8715	1.24799
$143$	−8.41921	−0.704050
$144$	7.87411	0.656176
$145$	16.2077	1.34598
$146$	12.8913	1.06689
$147$	18.2537	1.50554
$148$	10.2543	0.842902
$149$	−3.13614	−0.256923	−0.128461	−	0.991714i	$-0.541004\pi$
$149$	−3.13614	−0.256923	−0.128461	+	0.991714i	$0.541004\pi$
$150$	2.65479	0.216762
$151$	−15.9882	−1.30110	−0.650549	−	0.759464i	$-0.725461\pi$
$151$	−15.9882	−1.30110	−0.650549	+	0.759464i	$0.725461\pi$
$152$	−6.90709	−0.560239
$153$	1.06694	0.0862573
$154$	5.43654	0.438089
$155$	13.7580	1.10507
$156$	−6.18008	−0.494803
$157$	−4.84567	−0.386727	−0.193363	−	0.981127i	$-0.561940\pi$
$157$	−4.84567	−0.386727	−0.193363	+	0.981127i	$0.561940\pi$
$158$	−5.65417	−0.449821
$159$	−6.90105	−0.547288
$160$	2.40937	0.190477
$161$	3.10441	0.244661
$162$	−29.3793	−2.30826
$163$	−4.54335	−0.355862	−0.177931	−	0.984043i	$-0.556940\pi$
$163$	−4.54335	−0.355862	−0.177931	+	0.984043i	$0.556940\pi$
$164$	−10.6956	−0.835185
$165$	−35.6923	−2.77864
$166$	3.04871	0.236626
$167$	−10.1532	−0.785679	−0.392840	−	0.919607i	$-0.628507\pi$
$167$	−10.1532	−0.785679	−0.392840	+	0.919607i	$0.628507\pi$
$168$	3.99067	0.307887
$169$	−9.48767	−0.729821
$170$	0.326470	0.0250391
$171$	54.3872	4.15909
$172$	5.12576	0.390836
$173$	4.98779	0.379215	0.189607	−	0.981860i	$-0.439278\pi$
$173$	4.98779	0.379215	0.189607	+	0.981860i	$0.439278\pi$
$174$	−22.1827	−1.68167
$175$	0.974276	0.0736483
$176$	−4.49235	−0.338624
$177$	−37.7222	−2.83538
$178$	−17.2371	−1.29198
$179$	−9.73183	−0.727391	−0.363695	−	0.931518i	$-0.618485\pi$
$179$	−9.73183	−0.727391	−0.363695	+	0.931518i	$0.618485\pi$
$180$	−18.9717	−1.41406
$181$	16.9143	1.25723	0.628615	−	0.777717i	$-0.283622\pi$
$181$	16.9143	1.25723	0.628615	+	0.777717i	$0.283622\pi$
$182$	−2.26802	−0.168117
$183$	−42.3611	−3.13142
$184$	−2.56525	−0.189113
$185$	−24.7065	−1.81646
$186$	−18.8299	−1.38067
$187$	−0.608715	−0.0445137
$188$	−5.98727	−0.436667
$189$	−19.4510	−1.41485
$190$	16.6417	1.20732
$191$	25.7352	1.86213	0.931065	−	0.364852i	$-0.118881\pi$
$191$	25.7352	1.86213	0.931065	+	0.364852i	$0.118881\pi$
$192$	−3.29759	−0.237983
$193$	25.7510	1.85360	0.926798	−	0.375561i	$-0.122550\pi$
$193$	25.7510	1.85360	0.926798	+	0.375561i	$0.122550\pi$
$194$	9.19641	0.660264
$195$	14.8901	1.06630
$196$	−5.53547	−0.395391
$197$	8.21809	0.585514	0.292757	−	0.956187i	$-0.405427\pi$
$197$	8.21809	0.585514	0.292757	+	0.956187i	$0.405427\pi$
$198$	35.3733	2.51387
$199$	−1.39332	−0.0987699	−0.0493850	−	0.998780i	$-0.515726\pi$
$199$	−1.39332	−0.0987699	−0.0493850	+	0.998780i	$0.515726\pi$
$200$	−0.805068	−0.0569269
$201$	34.7450	2.45072
$202$	−1.09450	−0.0770086
$203$	−8.14080	−0.571372
$204$	−0.446825	−0.0312840
$205$	25.7696	1.79983
$206$	0.250501	0.0174532
$207$	20.1991	1.40393
$208$	1.87412	0.129947
$209$	−31.0291	−2.14633
$210$	−9.61501	−0.663499
$211$	9.67767	0.666239	0.333119	−	0.942885i	$-0.391899\pi$
$211$	9.67767	0.666239	0.333119	+	0.942885i	$0.391899\pi$
$212$	2.09275	0.143731
$213$	49.0403	3.36018
$214$	7.16532	0.489811
$215$	−12.3499	−0.842254
$216$	16.0728	1.09362
$217$	−6.91034	−0.469104
$218$	−7.82466	−0.529953
$219$	42.5103	2.87258
$220$	10.8237	0.729737
$221$	0.253944	0.0170821
$222$	33.8147	2.26949
$223$	0.370955	0.0248410	0.0124205	−	0.999923i	$-0.496046\pi$
$223$	0.370955	0.0248410	0.0124205	+	0.999923i	$0.496046\pi$
$224$	−1.21018	−0.0808584
$225$	6.33920	0.422613
$226$	−10.5537	−0.702023
$227$	−23.2490	−1.54309	−0.771546	−	0.636173i	$-0.780516\pi$
$227$	−23.2490	−1.54309	−0.771546	+	0.636173i	$0.780516\pi$
$228$	−22.7768	−1.50843
$229$	8.79006	0.580863	0.290432	−	0.956896i	$-0.406201\pi$
$229$	8.79006	0.580863	0.290432	+	0.956896i	$0.406201\pi$
$230$	6.18064	0.407539
$231$	17.9275	1.17954
$232$	6.72694	0.441646
$233$	7.51585	0.492380	0.246190	−	0.969222i	$-0.420821\pi$
$233$	7.51585	0.492380	0.246190	+	0.969222i	$0.420821\pi$
$234$	−14.7570	−0.964698
$235$	14.4256	0.941020
$236$	11.4393	0.744637
$237$	−18.6451	−1.21113
$238$	−0.163979	−0.0106292
$239$	20.5193	1.32728	0.663641	−	0.748051i	$-0.269010\pi$
$239$	20.5193	1.32728	0.663641	+	0.748051i	$0.269010\pi$
$240$	7.94512	0.512855
$241$	12.7356	0.820372	0.410186	−	0.912002i	$-0.365464\pi$
$241$	12.7356	0.820372	0.410186	+	0.912002i	$0.365464\pi$
$242$	−9.18124	−0.590192
$243$	−48.6625	−3.12170
$244$	12.8461	0.822385
$245$	13.3370	0.852070
$246$	−35.2697	−2.24871
$247$	12.9447	0.823653
$248$	5.71019	0.362597
$249$	10.0534	0.637109
$250$	−10.1071	−0.639232
$251$	2.35404	0.148586	0.0742928	−	0.997236i	$-0.476330\pi$
$251$	2.35404	0.148586	0.0742928	+	0.997236i	$0.476330\pi$
$252$	9.52907	0.600275
$253$	−11.5240	−0.724508
$254$	−12.7483	−0.799900
$255$	1.07657	0.0674172
$256$	1.00000	0.0625000
$257$	5.08059	0.316919	0.158459	−	0.987366i	$-0.449347\pi$
$257$	5.08059	0.316919	0.158459	+	0.987366i	$0.449347\pi$
$258$	16.9027	1.05231
$259$	12.4096	0.771094
$260$	−4.51545	−0.280036
$261$	−52.9687	−3.27868
$262$	−10.9824	−0.678495
$263$	8.37496	0.516422	0.258211	−	0.966089i	$-0.416867\pi$
$263$	8.37496	0.516422	0.258211	+	0.966089i	$0.416867\pi$
$264$	−14.8139	−0.911735
$265$	−5.04222	−0.309741
$266$	−8.35880	−0.512511
$267$	−56.8411	−3.47862
$268$	−10.5365	−0.643618
$269$	16.4949	1.00571	0.502855	−	0.864371i	$-0.332283\pi$
$269$	16.4949	1.00571	0.502855	+	0.864371i	$0.332283\pi$
$270$	−38.7254	−2.35675
$271$	−18.7866	−1.14121	−0.570603	−	0.821226i	$-0.693290\pi$
$271$	−18.7866	−1.14121	−0.570603	+	0.821226i	$0.693290\pi$
$272$	0.135500	0.00821591
$273$	−7.47900	−0.452650
$274$	8.34612	0.504207
$275$	−3.61665	−0.218092
$276$	−8.45914	−0.509181
$277$	13.8140	0.830006	0.415003	−	0.909820i	$-0.363781\pi$
$277$	13.8140	0.830006	0.415003	+	0.909820i	$0.363781\pi$
$278$	−13.2173	−0.792723
$279$	−44.9627	−2.69184
$280$	2.91577	0.174250
$281$	8.31218	0.495863	0.247932	−	0.968778i	$-0.420249\pi$
$281$	8.31218	0.495863	0.247932	+	0.968778i	$0.420249\pi$
$282$	−19.7436	−1.17571
$283$	−19.4846	−1.15824	−0.579119	−	0.815243i	$-0.696603\pi$
$283$	−19.4846	−1.15824	−0.579119	+	0.815243i	$0.696603\pi$
$284$	−14.8715	−0.882464
$285$	54.8777	3.25067
$286$	8.41921	0.497838
$287$	−12.9435	−0.764034
$288$	−7.87411	−0.463987
$289$	−16.9816	−0.998920
$290$	−16.2077	−0.951749
$291$	30.3260	1.77774
$292$	−12.8913	−0.754407
$293$	26.1648	1.52856	0.764282	−	0.644882i	$-0.223093\pi$
$293$	26.1648	1.52856	0.764282	+	0.644882i	$0.223093\pi$
$294$	−18.2537	−1.06458
$295$	−27.5616	−1.60470
$296$	−10.2543	−0.596022
$297$	72.2049	4.18975
$298$	3.13614	0.181672
$299$	4.80758	0.278030
$300$	−2.65479	−0.153274
$301$	6.20308	0.357540
$302$	15.9882	0.920015
$303$	−3.60921	−0.207343
$304$	6.90709	0.396149
$305$	−30.9510	−1.77225
$306$	−1.06694	−0.0609932
$307$	−20.0891	−1.14655	−0.573273	−	0.819365i	$-0.694326\pi$
$307$	−20.0891	−1.14655	−0.573273	+	0.819365i	$0.694326\pi$
$308$	−5.43654	−0.309776
$309$	0.826051	0.0469924
$310$	−13.7580	−0.781400
$311$	−31.6388	−1.79407	−0.897037	−	0.441956i	$-0.854285\pi$
$311$	−31.6388	−1.79407	−0.897037	+	0.441956i	$0.854285\pi$
$312$	6.18008	0.349878
$313$	−16.0607	−0.907802	−0.453901	−	0.891052i	$-0.649968\pi$
$313$	−16.0607	−0.907802	−0.453901	+	0.891052i	$0.649968\pi$
$314$	4.84567	0.273457
$315$	−22.9591	−1.29360
$316$	5.65417	0.318072
$317$	−15.0302	−0.844178	−0.422089	−	0.906554i	$-0.638703\pi$
$317$	−15.0302	−0.844178	−0.422089	+	0.906554i	$0.638703\pi$
$318$	6.90105	0.386991
$319$	30.2198	1.69198
$320$	−2.40937	−0.134688
$321$	23.6283	1.31880
$322$	−3.10441	−0.173002
$323$	0.935913	0.0520756
$324$	29.3793	1.63218
$325$	1.50880	0.0836929
$326$	4.54335	0.251633
$327$	−25.8026	−1.42688
$328$	10.6956	0.590565
$329$	−7.24566	−0.399466
$330$	35.6923	1.96480
$331$	−8.26141	−0.454088	−0.227044	−	0.973885i	$-0.572906\pi$
$331$	−8.26141	−0.454088	−0.227044	+	0.973885i	$0.572906\pi$
$332$	−3.04871	−0.167320
$333$	80.7439	4.42474
$334$	10.1532	0.555559
$335$	25.3863	1.38700
$336$	−3.99067	−0.217709
$337$	−1.57327	−0.0857016	−0.0428508	−	0.999081i	$-0.513644\pi$
$337$	−1.57327	−0.0857016	−0.0428508	+	0.999081i	$0.513644\pi$
$338$	9.48767	0.516061
$339$	−34.8019	−1.89018
$340$	−0.326470	−0.0177053
$341$	25.6522	1.38914
$342$	−54.3872	−2.94092
$343$	−15.1701	−0.819111
$344$	−5.12576	−0.276363
$345$	20.3812	1.09729
$346$	−4.98779	−0.268145
$347$	2.56754	0.137833	0.0689163	−	0.997622i	$-0.478046\pi$
$347$	2.56754	0.137833	0.0689163	+	0.997622i	$0.478046\pi$
$348$	22.1827	1.18912
$349$	15.3449	0.821394	0.410697	−	0.911772i	$-0.365285\pi$
$349$	15.3449	0.821394	0.410697	+	0.911772i	$0.365285\pi$
$350$	−0.974276	−0.0520772
$351$	−30.1224	−1.60782
$352$	4.49235	0.239443
$353$	26.9640	1.43515	0.717575	−	0.696482i	$-0.245252\pi$
$353$	26.9640	1.43515	0.717575	+	0.696482i	$0.245252\pi$
$354$	37.7222	2.00491
$355$	35.8311	1.90172
$356$	17.2371	0.913567
$357$	−0.540737	−0.0286188
$358$	9.73183	0.514343
$359$	24.9909	1.31897	0.659485	−	0.751718i	$-0.270774\pi$
$359$	24.9909	1.31897	0.659485	+	0.751718i	$0.270774\pi$
$360$	18.9717	0.999894
$361$	28.7079	1.51094
$362$	−16.9143	−0.888996
$363$	−30.2760	−1.58908
$364$	2.26802	0.118876
$365$	31.0600	1.62575
$366$	42.3611	2.21425
$367$	−19.3936	−1.01234	−0.506169	−	0.862434i	$-0.668939\pi$
$367$	−19.3936	−1.01234	−0.506169	+	0.862434i	$0.668939\pi$
$368$	2.56525	0.133723
$369$	−84.2182	−4.38423
$370$	24.7065	1.28443
$371$	2.53260	0.131486
$372$	18.8299	0.976283
$373$	23.7024	1.22726	0.613632	−	0.789592i	$-0.289708\pi$
$373$	23.7024	1.22726	0.613632	+	0.789592i	$0.289708\pi$
$374$	0.608715	0.0314759
$375$	−33.3292	−1.72111
$376$	5.98727	0.308770
$377$	−12.6071	−0.649299
$378$	19.4510	1.00045
$379$	−24.8475	−1.27633	−0.638164	−	0.769900i	$-0.720306\pi$
$379$	−24.8475	−1.27633	−0.638164	+	0.769900i	$0.720306\pi$
$380$	−16.6417	−0.853703
$381$	−42.0387	−2.15371
$382$	−25.7352	−1.31673
$383$	26.7830	1.36855	0.684273	−	0.729226i	$-0.260120\pi$
$383$	26.7830	1.36855	0.684273	+	0.729226i	$0.260120\pi$
$384$	3.29759	0.168280
$385$	13.0987	0.667569
$386$	−25.7510	−1.31069
$387$	40.3609	2.05166
$388$	−9.19641	−0.466877
$389$	−21.3732	−1.08366	−0.541832	−	0.840487i	$-0.682269\pi$
$389$	−21.3732	−1.08366	−0.541832	+	0.840487i	$0.682269\pi$
$390$	−14.8901	−0.753990
$391$	0.347592	0.0175785
$392$	5.53547	0.279583
$393$	−36.2155	−1.82683
$394$	−8.21809	−0.414021
$395$	−13.6230	−0.685447
$396$	−35.3733	−1.77758
$397$	−14.3845	−0.721939	−0.360970	−	0.932578i	$-0.617554\pi$
$397$	−14.3845	−0.721939	−0.360970	+	0.932578i	$0.617554\pi$
$398$	1.39332	0.0698409
$399$	−27.5639	−1.37992
$400$	0.805068	0.0402534
$401$	35.2862	1.76211	0.881056	−	0.473013i	$-0.156834\pi$
$401$	35.2862	1.76211	0.881056	+	0.473013i	$0.156834\pi$
$402$	−34.7450	−1.73292
$403$	−10.7016	−0.533084
$404$	1.09450	0.0544533
$405$	−70.7857	−3.51737
$406$	8.14080	0.404021
$407$	−46.0662	−2.28341
$408$	0.446825	0.0221211
$409$	−36.5764	−1.80859	−0.904293	−	0.426913i	$-0.859601\pi$
$409$	−36.5764	−1.80859	−0.904293	+	0.426913i	$0.859601\pi$
$410$	−25.7696	−1.27267
$411$	27.5221	1.35756
$412$	−0.250501	−0.0123413
$413$	13.8436	0.681200
$414$	−20.1991	−0.992730
$415$	7.34548	0.360575
$416$	−1.87412	−0.0918863
$417$	−43.5854	−2.13438
$418$	31.0291	1.51768
$419$	35.8410	1.75095	0.875474	−	0.483265i	$-0.160549\pi$
$419$	35.8410	1.75095	0.875474	+	0.483265i	$0.160549\pi$
$420$	9.61501	0.469164
$421$	12.6682	0.617410	0.308705	−	0.951158i	$-0.400104\pi$
$421$	12.6682	0.617410	0.308705	+	0.951158i	$0.400104\pi$
$422$	−9.67767	−0.471102
$423$	−47.1445	−2.29224
$424$	−2.09275	−0.101633
$425$	0.109087	0.00529150
$426$	−49.0403	−2.37601
$427$	15.5460	0.752325
$428$	−7.16532	−0.346349
$429$	27.7631	1.34042
$430$	12.3499	0.595564
$431$	12.8887	0.620828	0.310414	−	0.950601i	$-0.399532\pi$
$431$	12.8887	0.620828	0.310414	+	0.950601i	$0.399532\pi$
$432$	−16.0728	−0.773305
$433$	0.303876	0.0146033	0.00730167	−	0.999973i	$-0.497676\pi$
$433$	0.303876	0.0146033	0.00730167	+	0.999973i	$0.497676\pi$
$434$	6.91034	0.331707
$435$	−53.4464	−2.56256
$436$	7.82466	0.374733
$437$	17.7184	0.847586
$438$	−42.5103	−2.03122
$439$	−5.49296	−0.262165	−0.131082	−	0.991371i	$-0.541845\pi$
$439$	−5.49296	−0.262165	−0.131082	+	0.991371i	$0.541845\pi$
$440$	−10.8237	−0.516002
$441$	−43.5869	−2.07557
$442$	−0.253944	−0.0120789
$443$	−25.3068	−1.20236	−0.601182	−	0.799112i	$-0.705303\pi$
$443$	−25.3068	−1.20236	−0.601182	+	0.799112i	$0.705303\pi$
$444$	−33.8147	−1.60477
$445$	−41.5307	−1.96874
$446$	−0.370955	−0.0175652
$447$	10.3417	0.489147
$448$	1.21018	0.0571755
$449$	30.5642	1.44241	0.721207	−	0.692720i	$-0.243588\pi$
$449$	30.5642	1.44241	0.721207	+	0.692720i	$0.243588\pi$
$450$	−6.33920	−0.298833
$451$	48.0483	2.26251
$452$	10.5537	0.496405
$453$	52.7224	2.47712
$454$	23.2490	1.09113
$455$	−5.46450	−0.256180
$456$	22.7768	1.06662
$457$	8.63653	0.404000	0.202000	−	0.979386i	$-0.435256\pi$
$457$	8.63653	0.404000	0.202000	+	0.979386i	$0.435256\pi$
$458$	−8.79006	−0.410732
$459$	−2.17787	−0.101654
$460$	−6.18064	−0.288174
$461$	−35.9245	−1.67317	−0.836584	−	0.547838i	$-0.815451\pi$
$461$	−35.9245	−1.67317	−0.836584	+	0.547838i	$0.815451\pi$
$462$	−17.9275	−0.834063
$463$	−18.5262	−0.860984	−0.430492	−	0.902594i	$-0.641660\pi$
$463$	−18.5262	−0.860984	−0.430492	+	0.902594i	$0.641660\pi$
$464$	−6.72694	−0.312291
$465$	−45.3681	−2.10390
$466$	−7.51585	−0.348165
$467$	−19.7948	−0.915996	−0.457998	−	0.888953i	$-0.651433\pi$
$467$	−19.7948	−0.915996	−0.457998	+	0.888953i	$0.651433\pi$
$468$	14.7570	0.682144
$469$	−12.7510	−0.588787
$470$	−14.4256	−0.665402
$471$	15.9791	0.736276
$472$	−11.4393	−0.526538
$473$	−23.0267	−1.05877
$474$	18.6451	0.856400
$475$	5.56068	0.255142
$476$	0.163979	0.00751598
$477$	16.4786	0.754502
$478$	−20.5193	−0.938530
$479$	−25.9786	−1.18699	−0.593495	−	0.804837i	$-0.702252\pi$
$479$	−25.9786	−1.18699	−0.593495	+	0.804837i	$0.702252\pi$
$480$	−7.94512	−0.362644
$481$	19.2179	0.876260
$482$	−12.7356	−0.580091
$483$	−10.2371	−0.465803
$484$	9.18124	0.417329
$485$	22.1576	1.00612
$486$	48.6625	2.20738
$487$	−14.6046	−0.661796	−0.330898	−	0.943666i	$-0.607352\pi$
$487$	−14.6046	−0.661796	−0.330898	+	0.943666i	$0.607352\pi$
$488$	−12.8461	−0.581514
$489$	14.9821	0.677514
$490$	−13.3370	−0.602504
$491$	30.9627	1.39733	0.698664	−	0.715450i	$-0.253778\pi$
$491$	30.9627	1.39733	0.698664	+	0.715450i	$0.253778\pi$
$492$	35.2697	1.59008
$493$	−0.911503	−0.0410520
$494$	−12.9447	−0.582410
$495$	85.2274	3.83069
$496$	−5.71019	−0.256395
$497$	−17.9972	−0.807285
$498$	−10.0534	−0.450504
$499$	11.2315	0.502790	0.251395	−	0.967885i	$-0.419111\pi$
$499$	11.2315	0.502790	0.251395	+	0.967885i	$0.419111\pi$
$500$	10.1071	0.452005
$501$	33.4812	1.49583
$502$	−2.35404	−0.105066
$503$	−43.5496	−1.94178	−0.970889	−	0.239528i	$-0.923007\pi$
$503$	−43.5496	−1.94178	−0.970889	+	0.239528i	$0.923007\pi$
$504$	−9.52907	−0.424459
$505$	−2.63705	−0.117347
$506$	11.5240	0.512304
$507$	31.2865	1.38948
$508$	12.7483	0.565615
$509$	−18.1542	−0.804671	−0.402335	−	0.915492i	$-0.631801\pi$
$509$	−18.1542	−0.804671	−0.402335	+	0.915492i	$0.631801\pi$
$510$	−1.07657	−0.0476712
$511$	−15.6008	−0.690138
$512$	−1.00000	−0.0441942
$513$	−111.017	−4.90150
$514$	−5.08059	−0.224095
$515$	0.603550	0.0265956
$516$	−16.9027	−0.744099
$517$	26.8969	1.18293
$518$	−12.4096	−0.545246
$519$	−16.4477	−0.721974
$520$	4.51545	0.198016
$521$	−35.2985	−1.54645	−0.773227	−	0.634130i	$-0.781359\pi$
$521$	−35.2985	−1.54645	−0.773227	+	0.634130i	$0.781359\pi$
$522$	52.9687	2.31838
$523$	−20.6293	−0.902056	−0.451028	−	0.892510i	$-0.648943\pi$
$523$	−20.6293	−0.902056	−0.451028	+	0.892510i	$0.648943\pi$
$524$	10.9824	0.479769
$525$	−3.21276	−0.140217
$526$	−8.37496	−0.365166
$527$	−0.773732	−0.0337043
$528$	14.8139	0.644694
$529$	−16.4195	−0.713891
$530$	5.04222	0.219020
$531$	90.0746	3.90890
$532$	8.35880	0.362400
$533$	−20.0448	−0.868237
$534$	56.8411	2.45975
$535$	17.2639	0.746384
$536$	10.5365	0.455107
$537$	32.0916	1.38485
$538$	−16.4949	−0.711145
$539$	24.8673	1.07111
$540$	38.7254	1.66648
$541$	−28.7501	−1.23606	−0.618032	−	0.786153i	$-0.712070\pi$
$541$	−28.7501	−1.23606	−0.618032	+	0.786153i	$0.712070\pi$
$542$	18.7866	0.806954
$543$	−55.7765	−2.39360
$544$	−0.135500	−0.00580953
$545$	−18.8525	−0.807553
$546$	7.47900	0.320072
$547$	−23.5169	−1.00551	−0.502756	−	0.864428i	$-0.667681\pi$
$547$	−23.5169	−1.00551	−0.502756	+	0.864428i	$0.667681\pi$
$548$	−8.34612	−0.356528
$549$	101.151	4.31704
$550$	3.61665	0.154215
$551$	−46.4636	−1.97942
$552$	8.45914	0.360045
$553$	6.84255	0.290975
$554$	−13.8140	−0.586903
$555$	81.4720	3.45830
$556$	13.2173	0.560540
$557$	−25.4660	−1.07903	−0.539514	−	0.841977i	$-0.681392\pi$
$557$	−25.4660	−1.07903	−0.539514	+	0.841977i	$0.681392\pi$
$558$	44.9627	1.90342
$559$	9.60630	0.406303
$560$	−2.91577	−0.123214
$561$	2.00729	0.0847480
$562$	−8.31218	−0.350628
$563$	−24.0715	−1.01449	−0.507247	−	0.861801i	$-0.669337\pi$
$563$	−24.0715	−1.01449	−0.507247	+	0.861801i	$0.669337\pi$
$564$	19.7436	0.831355
$565$	−25.4278	−1.06976
$566$	19.4846	0.818998
$567$	35.5542	1.49314
$568$	14.8715	0.623996
$569$	28.3393	1.18804	0.594022	−	0.804449i	$-0.297539\pi$
$569$	28.3393	1.18804	0.594022	+	0.804449i	$0.297539\pi$
$570$	−54.8777	−2.29857
$571$	39.7571	1.66378	0.831892	−	0.554937i	$-0.187258\pi$
$571$	39.7571	1.66378	0.831892	+	0.554937i	$0.187258\pi$
$572$	−8.41921	−0.352025
$573$	−84.8641	−3.54525
$574$	12.9435	0.540253
$575$	2.06520	0.0861248
$576$	7.87411	0.328088
$577$	−27.3702	−1.13944	−0.569718	−	0.821840i	$-0.692948\pi$
$577$	−27.3702	−1.13944	−0.569718	+	0.821840i	$0.692948\pi$
$578$	16.9816	0.706343
$579$	−84.9162	−3.52900
$580$	16.2077	0.672988
$581$	−3.68948	−0.153065
$582$	−30.3260	−1.25705
$583$	−9.40138	−0.389366
$584$	12.8913	0.533447
$585$	−35.5552	−1.47003
$586$	−26.1648	−1.08086
$587$	−22.5668	−0.931433	−0.465716	−	0.884934i	$-0.654203\pi$
$587$	−22.5668	−0.931433	−0.465716	+	0.884934i	$0.654203\pi$
$588$	18.2537	0.752771
$589$	−39.4408	−1.62513
$590$	27.5616	1.13469
$591$	−27.0999	−1.11474
$592$	10.2543	0.421451
$593$	−9.97834	−0.409761	−0.204881	−	0.978787i	$-0.565681\pi$
$593$	−9.97834	−0.409761	−0.204881	+	0.978787i	$0.565681\pi$
$594$	−72.2049	−2.96260
$595$	−0.395087	−0.0161970
$596$	−3.13614	−0.128461
$597$	4.59460	0.188045
$598$	−4.80758	−0.196597
$599$	22.5219	0.920222	0.460111	−	0.887861i	$-0.347810\pi$
$599$	22.5219	0.920222	0.460111	+	0.887861i	$0.347810\pi$
$600$	2.65479	0.108381
$601$	36.9999	1.50926	0.754628	−	0.656153i	$-0.227817\pi$
$601$	36.9999	1.50926	0.754628	+	0.656153i	$0.227817\pi$
$602$	−6.20308	−0.252819
$603$	−82.9654	−3.37861
$604$	−15.9882	−0.650549
$605$	−22.1210	−0.899347
$606$	3.60921	0.146614
$607$	23.9249	0.971081	0.485541	−	0.874214i	$-0.338623\pi$
$607$	23.9249	0.971081	0.485541	+	0.874214i	$0.338623\pi$
$608$	−6.90709	−0.280120
$609$	26.8450	1.08782
$610$	30.9510	1.25317
$611$	−11.2209	−0.453948
$612$	1.06694	0.0431287
$613$	3.19407	0.129007	0.0645036	−	0.997917i	$-0.479454\pi$
$613$	3.19407	0.129007	0.0645036	+	0.997917i	$0.479454\pi$
$614$	20.0891	0.810730
$615$	−84.9777	−3.42663
$616$	5.43654	0.219045
$617$	−37.3149	−1.50224	−0.751120	−	0.660166i	$-0.770486\pi$
$617$	−37.3149	−1.50224	−0.751120	+	0.660166i	$0.770486\pi$
$618$	−0.826051	−0.0332286
$619$	11.3861	0.457645	0.228822	−	0.973468i	$-0.426513\pi$
$619$	11.3861	0.457645	0.228822	+	0.973468i	$0.426513\pi$
$620$	13.7580	0.552533
$621$	−41.2308	−1.65454
$622$	31.6388	1.26860
$623$	20.8600	0.835738
$624$	−6.18008	−0.247401
$625$	−28.3772	−1.13509
$626$	16.0607	0.641913
$627$	102.321	4.08632
$628$	−4.84567	−0.193363
$629$	1.38947	0.0554017
$630$	22.9591	0.914711
$631$	16.7859	0.668237	0.334118	−	0.942531i	$-0.391561\pi$
$631$	16.7859	0.668237	0.334118	+	0.942531i	$0.391561\pi$
$632$	−5.65417	−0.224911
$633$	−31.9130	−1.26843
$634$	15.0302	0.596924
$635$	−30.7154	−1.21890
$636$	−6.90105	−0.273644
$637$	−10.3741	−0.411038
$638$	−30.2198	−1.19641
$639$	−117.100	−4.63241
$640$	2.40937	0.0952387
$641$	−33.4951	−1.32298	−0.661488	−	0.749956i	$-0.730075\pi$
$641$	−33.4951	−1.32298	−0.661488	+	0.749956i	$0.730075\pi$
$642$	−23.6283	−0.932535
$643$	−28.0076	−1.10451	−0.552257	−	0.833674i	$-0.686233\pi$
$643$	−28.0076	−1.10451	−0.552257	+	0.833674i	$0.686233\pi$
$644$	3.10441	0.122331
$645$	40.7248	1.60354
$646$	−0.935913	−0.0368230
$647$	5.18859	0.203985	0.101992	−	0.994785i	$-0.467478\pi$
$647$	5.18859	0.203985	0.101992	+	0.994785i	$0.467478\pi$
$648$	−29.3793	−1.15413
$649$	−51.3895	−2.01721
$650$	−1.50880	−0.0591798
$651$	22.7875	0.893112
$652$	−4.54335	−0.177931
$653$	25.7319	1.00697	0.503484	−	0.864005i	$-0.332051\pi$
$653$	25.7319	1.00697	0.503484	+	0.864005i	$0.332051\pi$
$654$	25.8026	1.00896
$655$	−26.4607	−1.03390
$656$	−10.6956	−0.417592
$657$	−101.508	−3.96019
$658$	7.24566	0.282465
$659$	−22.6070	−0.880643	−0.440321	−	0.897840i	$-0.645136\pi$
$659$	−22.6070	−0.880643	−0.440321	+	0.897840i	$0.645136\pi$
$660$	−35.6923	−1.38932
$661$	18.8418	0.732861	0.366430	−	0.930446i	$-0.380580\pi$
$661$	18.8418	0.732861	0.366430	+	0.930446i	$0.380580\pi$
$662$	8.26141	0.321089
$663$	−0.837403	−0.0325220
$664$	3.04871	0.118313
$665$	−20.1395	−0.780975
$666$	−80.7439	−3.12876
$667$	−17.2563	−0.668166
$668$	−10.1532	−0.392840
$669$	−1.22326	−0.0472939
$670$	−25.3863	−0.980758
$671$	−57.7091	−2.22783
$672$	3.99067	0.153943
$673$	−7.62860	−0.294061	−0.147031	−	0.989132i	$-0.546972\pi$
$673$	−7.62860	−0.294061	−0.147031	+	0.989132i	$0.546972\pi$
$674$	1.57327	0.0606002
$675$	−12.9397	−0.498051
$676$	−9.48767	−0.364910
$677$	−46.6679	−1.79359	−0.896796	−	0.442445i	$-0.854111\pi$
$677$	−46.6679	−1.79359	−0.896796	+	0.442445i	$0.854111\pi$
$678$	34.8019	1.33656
$679$	−11.1293	−0.427103
$680$	0.326470	0.0125196
$681$	76.6658	2.93784
$682$	−25.6522	−0.982273
$683$	7.76309	0.297046	0.148523	−	0.988909i	$-0.452548\pi$
$683$	7.76309	0.297046	0.148523	+	0.988909i	$0.452548\pi$
$684$	54.3872	2.07955
$685$	20.1089	0.768321
$686$	15.1701	0.579199
$687$	−28.9860	−1.10589
$688$	5.12576	0.195418
$689$	3.92207	0.149419
$690$	−20.3812	−0.775900
$691$	−17.2473	−0.656118	−0.328059	−	0.944657i	$-0.606395\pi$
$691$	−17.2473	−0.656118	−0.328059	+	0.944657i	$0.606395\pi$
$692$	4.98779	0.189607
$693$	−42.8080	−1.62614
$694$	−2.56754	−0.0974624
$695$	−31.8455	−1.20797
$696$	−22.1827	−0.840834
$697$	−1.44925	−0.0548944
$698$	−15.3449	−0.580813
$699$	−24.7842	−0.937425
$700$	0.974276	0.0368242
$701$	44.7282	1.68936	0.844681	−	0.535270i	$-0.179790\pi$
$701$	44.7282	1.68936	0.844681	+	0.535270i	$0.179790\pi$
$702$	30.1224	1.13690
$703$	70.8277	2.67132
$704$	−4.49235	−0.169312
$705$	−47.5696	−1.79158
$706$	−26.9640	−1.01480
$707$	1.32454	0.0498143
$708$	−37.7222	−1.41769
$709$	−35.0212	−1.31525	−0.657625	−	0.753346i	$-0.728439\pi$
$709$	−35.0212	−1.31525	−0.657625	+	0.753346i	$0.728439\pi$
$710$	−35.8311	−1.34472
$711$	44.5216	1.66969
$712$	−17.2371	−0.645989
$713$	−14.6480	−0.548574
$714$	0.540737	0.0202366
$715$	20.2850	0.758616
$716$	−9.73183	−0.363695
$717$	−67.6642	−2.52697
$718$	−24.9909	−0.932653
$719$	28.8095	1.07441	0.537206	−	0.843451i	$-0.319480\pi$
$719$	28.8095	1.07441	0.537206	+	0.843451i	$0.319480\pi$
$720$	−18.9717	−0.707032
$721$	−0.303151	−0.0112899
$722$	−28.7079	−1.06840
$723$	−41.9968	−1.56188
$724$	16.9143	0.628615
$725$	−5.41565	−0.201132
$726$	30.2760	1.12365
$727$	7.88482	0.292432	0.146216	−	0.989253i	$-0.453291\pi$
$727$	7.88482	0.292432	0.146216	+	0.989253i	$0.453291\pi$
$728$	−2.26802	−0.0840583
$729$	72.3311	2.67893
$730$	−31.0600	−1.14958
$731$	0.694542	0.0256886
$732$	−42.3611	−1.56571
$733$	−22.6409	−0.836261	−0.418131	−	0.908387i	$-0.637315\pi$
$733$	−22.6409	−0.836261	−0.418131	+	0.908387i	$0.637315\pi$
$734$	19.3936	0.715831
$735$	−43.9800	−1.62223
$736$	−2.56525	−0.0945563
$737$	47.3336	1.74355
$738$	84.2182	3.10012
$739$	−40.3096	−1.48281	−0.741406	−	0.671057i	$-0.765841\pi$
$739$	−40.3096	−1.48281	−0.741406	+	0.671057i	$0.765841\pi$
$740$	−24.7065	−0.908230
$741$	−42.6864	−1.56812
$742$	−2.53260	−0.0929747
$743$	−42.4358	−1.55682	−0.778410	−	0.627756i	$-0.783973\pi$
$743$	−42.4358	−1.55682	−0.778410	+	0.627756i	$0.783973\pi$
$744$	−18.8299	−0.690336
$745$	7.55614	0.276835
$746$	−23.7024	−0.867807
$747$	−24.0059	−0.878330
$748$	−0.608715	−0.0222568
$749$	−8.67131	−0.316843
$750$	33.3292	1.21701
$751$	−25.0311	−0.913400	−0.456700	−	0.889621i	$-0.650969\pi$
$751$	−25.0311	−0.913400	−0.456700	+	0.889621i	$0.650969\pi$
$752$	−5.98727	−0.218333
$753$	−7.76266	−0.282887
$754$	12.6071	0.459124
$755$	38.5214	1.40194
$756$	−19.4510	−0.707425
$757$	−12.3901	−0.450324	−0.225162	−	0.974321i	$-0.572291\pi$
$757$	−12.3901	−0.450324	−0.225162	+	0.974321i	$0.572291\pi$
$758$	24.8475	0.902500
$759$	38.0015	1.37937
$760$	16.6417	0.603659
$761$	3.63385	0.131727	0.0658635	−	0.997829i	$-0.479020\pi$
$761$	3.63385	0.131727	0.0658635	+	0.997829i	$0.479020\pi$
$762$	42.0387	1.52290
$763$	9.46923	0.342809
$764$	25.7352	0.931065
$765$	−2.57066	−0.0929426
$766$	−26.7830	−0.967708
$767$	21.4387	0.774106
$768$	−3.29759	−0.118992
$769$	−2.31063	−0.0833233	−0.0416616	−	0.999132i	$-0.513265\pi$
$769$	−2.31063	−0.0833233	−0.0416616	+	0.999132i	$0.513265\pi$
$770$	−13.0987	−0.472043
$771$	−16.7537	−0.603371
$772$	25.7510	0.926798
$773$	18.9276	0.680778	0.340389	−	0.940285i	$-0.389441\pi$
$773$	18.9276	0.680778	0.340389	+	0.940285i	$0.389441\pi$
$774$	−40.3609	−1.45074
$775$	−4.59709	−0.165132
$776$	9.19641	0.330132
$777$	−40.9217	−1.46806
$778$	21.3732	0.766265
$779$	−73.8753	−2.64686
$780$	14.8901	0.533152
$781$	66.8082	2.39059
$782$	−0.347592	−0.0124299
$783$	108.121	3.86393
$784$	−5.53547	−0.197695
$785$	11.6750	0.416699
$786$	36.2155	1.29176
$787$	−1.93381	−0.0689329	−0.0344664	−	0.999406i	$-0.510973\pi$
$787$	−1.93381	−0.0689329	−0.0344664	+	0.999406i	$0.510973\pi$
$788$	8.21809	0.292757
$789$	−27.6172	−0.983199
$790$	13.6230	0.484684
$791$	12.7719	0.454116
$792$	35.3733	1.25694
$793$	24.0751	0.854931
$794$	14.3845	0.510488
$795$	16.6272	0.589705
$796$	−1.39332	−0.0493850
$797$	−12.5428	−0.444289	−0.222144	−	0.975014i	$-0.571306\pi$
$797$	−12.5428	−0.444289	−0.222144	+	0.975014i	$0.571306\pi$
$798$	27.5639	0.975752
$799$	−0.811277	−0.0287009
$800$	−0.805068	−0.0284635
$801$	135.727	4.79568
$802$	−35.2862	−1.24600
$803$	57.9124	2.04368
$804$	34.7450	1.22536
$805$	−7.47967	−0.263624
$806$	10.7016	0.376947
$807$	−54.3934	−1.91474
$808$	−1.09450	−0.0385043
$809$	21.1945	0.745157	0.372579	−	0.928001i	$-0.378474\pi$
$809$	21.1945	0.745157	0.372579	+	0.928001i	$0.378474\pi$
$810$	70.7857	2.48716
$811$	11.5067	0.404054	0.202027	−	0.979380i	$-0.435247\pi$
$811$	11.5067	0.404054	0.202027	+	0.979380i	$0.435247\pi$
$812$	−8.14080	−0.285686
$813$	61.9506	2.17270
$814$	46.0662	1.61462
$815$	10.9466	0.383443
$816$	−0.446825	−0.0156420
$817$	35.4041	1.23863
$818$	36.5764	1.27886
$819$	17.8586	0.624031
$820$	25.7696	0.899914
$821$	−56.1084	−1.95820	−0.979098	−	0.203389i	$-0.934804\pi$
$821$	−56.1084	−1.95820	−0.979098	+	0.203389i	$0.934804\pi$
$822$	−27.5221	−0.959943
$823$	34.7118	1.20998	0.604989	−	0.796234i	$-0.293178\pi$
$823$	34.7118	1.20998	0.604989	+	0.796234i	$0.293178\pi$
$824$	0.250501	0.00872662
$825$	11.9262	0.415218
$826$	−13.8436	−0.481681
$827$	18.1653	0.631668	0.315834	−	0.948815i	$-0.397716\pi$
$827$	18.1653	0.631668	0.315834	+	0.948815i	$0.397716\pi$
$828$	20.1991	0.701966
$829$	10.1306	0.351851	0.175925	−	0.984403i	$-0.443708\pi$
$829$	10.1306	0.351851	0.175925	+	0.984403i	$0.443708\pi$
$830$	−7.34548	−0.254965
$831$	−45.5531	−1.58022
$832$	1.87412	0.0649734
$833$	−0.750058	−0.0259880
$834$	43.5854	1.50924
$835$	24.4629	0.846572
$836$	−31.0291	−1.07316
$837$	91.7789	3.17234
$838$	−35.8410	−1.23811
$839$	3.32816	0.114901	0.0574503	−	0.998348i	$-0.481703\pi$
$839$	3.32816	0.114901	0.0574503	+	0.998348i	$0.481703\pi$
$840$	−9.61501	−0.331749
$841$	16.2518	0.560406
$842$	−12.6682	−0.436575
$843$	−27.4102	−0.944057
$844$	9.67767	0.333119
$845$	22.8593	0.786385
$846$	47.1445	1.62086
$847$	11.1109	0.381776
$848$	2.09275	0.0718654
$849$	64.2522	2.20513
$850$	−0.109087	−0.00374165
$851$	26.3049	0.901722
$852$	49.0403	1.68009
$853$	−29.9103	−1.02411	−0.512055	−	0.858953i	$-0.671116\pi$
$853$	−29.9103	−1.02411	−0.512055	+	0.858953i	$0.671116\pi$
$854$	−15.5460	−0.531974
$855$	−131.039	−4.48144
$856$	7.16532	0.244906
$857$	25.8677	0.883625	0.441813	−	0.897107i	$-0.354336\pi$
$857$	25.8677	0.883625	0.441813	+	0.897107i	$0.354336\pi$
$858$	−27.7631	−0.947817
$859$	28.7288	0.980215	0.490108	−	0.871662i	$-0.336957\pi$
$859$	28.7288	0.980215	0.490108	+	0.871662i	$0.336957\pi$
$860$	−12.3499	−0.421127
$861$	42.6825	1.45462
$862$	−12.8887	−0.438992
$863$	42.1206	1.43380	0.716901	−	0.697175i	$-0.245560\pi$
$863$	42.1206	1.43380	0.716901	+	0.697175i	$0.245560\pi$
$864$	16.0728	0.546809
$865$	−12.0174	−0.408605
$866$	−0.303876	−0.0103261
$867$	55.9985	1.90181
$868$	−6.91034	−0.234552
$869$	−25.4005	−0.861654
$870$	53.4464	1.81200
$871$	−19.7466	−0.669089
$872$	−7.82466	−0.264977
$873$	−72.4136	−2.45083
$874$	−17.7184	−0.599334
$875$	12.2314	0.413498
$876$	42.5103	1.43629
$877$	33.5154	1.13173	0.565867	−	0.824496i	$-0.308541\pi$
$877$	33.5154	1.13173	0.565867	+	0.824496i	$0.308541\pi$
$878$	5.49296	0.185379
$879$	−86.2809	−2.91018
$880$	10.8237	0.364868
$881$	−5.07207	−0.170882	−0.0854412	−	0.996343i	$-0.527230\pi$
$881$	−5.07207	−0.170882	−0.0854412	+	0.996343i	$0.527230\pi$
$882$	43.5869	1.46765
$883$	−20.7833	−0.699414	−0.349707	−	0.936859i	$-0.613719\pi$
$883$	−20.7833	−0.699414	−0.349707	+	0.936859i	$0.613719\pi$
$884$	0.253944	0.00854105
$885$	90.8869	3.05513
$886$	25.3068	0.850200
$887$	−29.1754	−0.979615	−0.489807	−	0.871831i	$-0.662933\pi$
$887$	−29.1754	−0.979615	−0.489807	+	0.871831i	$0.662933\pi$
$888$	33.8147	1.13475
$889$	15.4277	0.517429
$890$	41.5307	1.39211
$891$	−131.982	−4.42157
$892$	0.370955	0.0124205
$893$	−41.3546	−1.38388
$894$	−10.3417	−0.345879
$895$	23.4476	0.783766
$896$	−1.21018	−0.0404292
$897$	−15.8535	−0.529331
$898$	−30.5642	−1.01994
$899$	38.4121	1.28112
$900$	6.33920	0.211307
$901$	0.283569	0.00944704
$902$	−48.0483	−1.59983
$903$	−20.4552	−0.680708
$904$	−10.5537	−0.351012
$905$	−40.7528	−1.35467
$906$	−52.7224	−1.75159
$907$	10.2108	0.339043	0.169521	−	0.985527i	$-0.445778\pi$
$907$	10.2108	0.339043	0.169521	+	0.985527i	$0.445778\pi$
$908$	−23.2490	−0.771546
$909$	8.61820	0.285848
$910$	5.46450	0.181146
$911$	−20.9344	−0.693587	−0.346794	−	0.937941i	$-0.612730\pi$
$911$	−20.9344	−0.693587	−0.346794	+	0.937941i	$0.612730\pi$
$912$	−22.7768	−0.754214
$913$	13.6959	0.453268
$914$	−8.63653	−0.285671
$915$	102.064	3.37412
$916$	8.79006	0.290432
$917$	13.2907	0.438896
$918$	2.17787	0.0718805
$919$	−13.1339	−0.433247	−0.216623	−	0.976255i	$-0.569504\pi$
$919$	−13.1339	−0.433247	−0.216623	+	0.976255i	$0.569504\pi$
$920$	6.18064	0.203770
$921$	66.2456	2.18287
$922$	35.9245	1.18311
$923$	−27.8711	−0.917387
$924$	17.9275	0.589772
$925$	8.25545	0.271438
$926$	18.5262	0.608808
$927$	−1.97247	−0.0647846
$928$	6.72694	0.220823
$929$	1.31500	0.0431439	0.0215719	−	0.999767i	$-0.493133\pi$
$929$	1.31500	0.0431439	0.0215719	+	0.999767i	$0.493133\pi$
$930$	45.3681	1.48768
$931$	−38.2340	−1.25307
$932$	7.51585	0.246190
$933$	104.332	3.41567
$934$	19.7948	0.647707
$935$	1.46662	0.0479636
$936$	−14.7570	−0.482349
$937$	27.7737	0.907326	0.453663	−	0.891173i	$-0.350117\pi$
$937$	27.7737	0.907326	0.453663	+	0.891173i	$0.350117\pi$
$938$	12.7510	0.416335
$939$	52.9615	1.72833
$940$	14.4256	0.470510
$941$	20.1913	0.658217	0.329108	−	0.944292i	$-0.393252\pi$
$941$	20.1913	0.658217	0.329108	+	0.944292i	$0.393252\pi$
$942$	−15.9791	−0.520626
$943$	−27.4368	−0.893466
$944$	11.4393	0.372319
$945$	46.8646	1.52451
$946$	23.0267	0.748664
$947$	37.2731	1.21121	0.605607	−	0.795764i	$-0.292930\pi$
$947$	37.2731	1.21121	0.605607	+	0.795764i	$0.292930\pi$
$948$	−18.6451	−0.605566
$949$	−24.1599	−0.784263
$950$	−5.56068	−0.180412
$951$	49.5633	1.60720
$952$	−0.163979	−0.00531460
$953$	47.4189	1.53605	0.768024	−	0.640421i	$-0.221240\pi$
$953$	47.4189	1.53605	0.768024	+	0.640421i	$0.221240\pi$
$954$	−16.4786	−0.533514
$955$	−62.0055	−2.00645
$956$	20.5193	0.663641
$957$	−99.6526	−3.22131
$958$	25.9786	0.839329
$959$	−10.1003	−0.326155
$960$	7.94512	0.256428
$961$	1.60624	0.0518142
$962$	−19.2179	−0.619609
$963$	−56.4206	−1.81813
$964$	12.7356	0.410186
$965$	−62.0436	−1.99726
$966$	10.2371	0.329372
$967$	−29.7180	−0.955665	−0.477833	−	0.878451i	$-0.658577\pi$
$967$	−29.7180	−0.955665	−0.477833	+	0.878451i	$0.658577\pi$
$968$	−9.18124	−0.295096
$969$	−3.08626	−0.0991449
$970$	−22.1576	−0.711437
$971$	−44.5894	−1.43094	−0.715472	−	0.698642i	$-0.753788\pi$
$971$	−44.5894	−1.43094	−0.715472	+	0.698642i	$0.753788\pi$
$972$	−48.6625	−1.56085
$973$	15.9953	0.512786
$974$	14.6046	0.467961
$975$	−4.97539	−0.159340
$976$	12.8461	0.411193
$977$	−9.72486	−0.311126	−0.155563	−	0.987826i	$-0.549719\pi$
$977$	−9.72486	−0.311126	−0.155563	+	0.987826i	$0.549719\pi$
$978$	−14.9821	−0.479075
$979$	−77.4353	−2.47484
$980$	13.3370	0.426035
$981$	61.6123	1.96713
$982$	−30.9627	−0.988059
$983$	−58.6104	−1.86938	−0.934691	−	0.355461i	$-0.884324\pi$
$983$	−58.6104	−1.86938	−0.934691	+	0.355461i	$0.884324\pi$
$984$	−35.2697	−1.12436
$985$	−19.8004	−0.630894
$986$	0.911503	0.0290282
$987$	23.8932	0.760530
$988$	12.9447	0.411826
$989$	13.1489	0.418109
$990$	−85.2274	−2.70870
$991$	−40.1997	−1.27699	−0.638493	−	0.769628i	$-0.720442\pi$
$991$	−40.1997	−1.27699	−0.638493	+	0.769628i	$0.720442\pi$
$992$	5.71019	0.181299
$993$	27.2427	0.864522
$994$	17.9972	0.570837
$995$	3.35703	0.106425
$996$	10.0534	0.318554
$997$	7.36519	0.233258	0.116629	−	0.993176i	$-0.462791\pi$
$997$	7.36519	0.233258	0.116629	+	0.993176i	$0.462791\pi$
$998$	−11.2315	−0.355526
$999$	−164.816	−5.21456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.3	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.29759 q^{3} +1.00000 q^{4} -2.40937 q^{5} +3.29759 q^{6} +1.21018 q^{7} -1.00000 q^{8} +7.87411 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.29759 q^{3} +1.00000 q^{4} -2.40937 q^{5} +3.29759 q^{6} +1.21018 q^{7} -1.00000 q^{8} +7.87411 q^{9} +2.40937 q^{10} -4.49235 q^{11} -3.29759 q^{12} +1.87412 q^{13} -1.21018 q^{14} +7.94512 q^{15} +1.00000 q^{16} +0.135500 q^{17} -7.87411 q^{18} +6.90709 q^{19} -2.40937 q^{20} -3.99067 q^{21} +4.49235 q^{22} +2.56525 q^{23} +3.29759 q^{24} +0.805068 q^{25} -1.87412 q^{26} -16.0728 q^{27} +1.21018 q^{28} -6.72694 q^{29} -7.94512 q^{30} -5.71019 q^{31} -1.00000 q^{32} +14.8139 q^{33} -0.135500 q^{34} -2.91577 q^{35} +7.87411 q^{36} +10.2543 q^{37} -6.90709 q^{38} -6.18008 q^{39} +2.40937 q^{40} -10.6956 q^{41} +3.99067 q^{42} +5.12576 q^{43} -4.49235 q^{44} -18.9717 q^{45} -2.56525 q^{46} -5.98727 q^{47} -3.29759 q^{48} -5.53547 q^{49} -0.805068 q^{50} -0.446825 q^{51} +1.87412 q^{52} +2.09275 q^{53} +16.0728 q^{54} +10.8237 q^{55} -1.21018 q^{56} -22.7768 q^{57} +6.72694 q^{58} +11.4393 q^{59} +7.94512 q^{60} +12.8461 q^{61} +5.71019 q^{62} +9.52907 q^{63} +1.00000 q^{64} -4.51545 q^{65} -14.8139 q^{66} -10.5365 q^{67} +0.135500 q^{68} -8.45914 q^{69} +2.91577 q^{70} -14.8715 q^{71} -7.87411 q^{72} -12.8913 q^{73} -10.2543 q^{74} -2.65479 q^{75} +6.90709 q^{76} -5.43654 q^{77} +6.18008 q^{78} +5.65417 q^{79} -2.40937 q^{80} +29.3793 q^{81} +10.6956 q^{82} -3.04871 q^{83} -3.99067 q^{84} -0.326470 q^{85} -5.12576 q^{86} +22.1827 q^{87} +4.49235 q^{88} +17.2371 q^{89} +18.9717 q^{90} +2.26802 q^{91} +2.56525 q^{92} +18.8299 q^{93} +5.98727 q^{94} -16.6417 q^{95} +3.29759 q^{96} -9.19641 q^{97} +5.53547 q^{98} -35.3733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 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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