LMFDB - Embedded newform 8002.2.a.f.1.12

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			1.12
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.76801 q^{3} +1.00000 q^{4} -2.71168 q^{5} +2.76801 q^{6} -4.48343 q^{7} -1.00000 q^{8} +4.66186 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.76801 q^{3} +1.00000 q^{4} -2.71168 q^{5} +2.76801 q^{6} -4.48343 q^{7} -1.00000 q^{8} +4.66186 q^{9} +2.71168 q^{10} -5.20634 q^{11} -2.76801 q^{12} -6.04655 q^{13} +4.48343 q^{14} +7.50595 q^{15} +1.00000 q^{16} -6.59491 q^{17} -4.66186 q^{18} -7.84954 q^{19} -2.71168 q^{20} +12.4102 q^{21} +5.20634 q^{22} +6.28504 q^{23} +2.76801 q^{24} +2.35321 q^{25} +6.04655 q^{26} -4.60003 q^{27} -4.48343 q^{28} +6.85860 q^{29} -7.50595 q^{30} -8.89300 q^{31} -1.00000 q^{32} +14.4112 q^{33} +6.59491 q^{34} +12.1576 q^{35} +4.66186 q^{36} +0.0170320 q^{37} +7.84954 q^{38} +16.7369 q^{39} +2.71168 q^{40} -7.90688 q^{41} -12.4102 q^{42} -10.3610 q^{43} -5.20634 q^{44} -12.6415 q^{45} -6.28504 q^{46} -9.47316 q^{47} -2.76801 q^{48} +13.1012 q^{49} -2.35321 q^{50} +18.2548 q^{51} -6.04655 q^{52} +2.89615 q^{53} +4.60003 q^{54} +14.1179 q^{55} +4.48343 q^{56} +21.7276 q^{57} -6.85860 q^{58} +6.04705 q^{59} +7.50595 q^{60} +12.3377 q^{61} +8.89300 q^{62} -20.9011 q^{63} +1.00000 q^{64} +16.3963 q^{65} -14.4112 q^{66} +0.170647 q^{67} -6.59491 q^{68} -17.3970 q^{69} -12.1576 q^{70} +7.13938 q^{71} -4.66186 q^{72} +6.20456 q^{73} -0.0170320 q^{74} -6.51371 q^{75} -7.84954 q^{76} +23.3423 q^{77} -16.7369 q^{78} -12.4184 q^{79} -2.71168 q^{80} -1.25266 q^{81} +7.90688 q^{82} +4.17524 q^{83} +12.4102 q^{84} +17.8833 q^{85} +10.3610 q^{86} -18.9846 q^{87} +5.20634 q^{88} -5.91623 q^{89} +12.6415 q^{90} +27.1093 q^{91} +6.28504 q^{92} +24.6159 q^{93} +9.47316 q^{94} +21.2855 q^{95} +2.76801 q^{96} -13.3374 q^{97} -13.1012 q^{98} -24.2712 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.76801	−1.59811	−0.799054	−	0.601259i	$-0.794666\pi$
$3$	−2.76801	−1.59811	−0.799054	+	0.601259i	$0.794666\pi$
$4$	1.00000	0.500000
$5$	−2.71168	−1.21270	−0.606350	−	0.795198i	$-0.707367\pi$
$5$	−2.71168	−1.21270	−0.606350	+	0.795198i	$0.707367\pi$
$6$	2.76801	1.13003
$7$	−4.48343	−1.69458	−0.847289	−	0.531132i	$-0.821767\pi$
$7$	−4.48343	−1.69458	−0.847289	+	0.531132i	$0.821767\pi$
$8$	−1.00000	−0.353553
$9$	4.66186	1.55395
$10$	2.71168	0.857509
$11$	−5.20634	−1.56977	−0.784885	−	0.619641i	$-0.787278\pi$
$11$	−5.20634	−1.56977	−0.784885	+	0.619641i	$0.787278\pi$
$12$	−2.76801	−0.799054
$13$	−6.04655	−1.67701	−0.838506	−	0.544892i	$-0.816571\pi$
$13$	−6.04655	−1.67701	−0.838506	+	0.544892i	$0.816571\pi$
$14$	4.48343	1.19825
$15$	7.50595	1.93803
$16$	1.00000	0.250000
$17$	−6.59491	−1.59950	−0.799750	−	0.600333i	$-0.795035\pi$
$17$	−6.59491	−1.59950	−0.799750	+	0.600333i	$0.795035\pi$
$18$	−4.66186	−1.09881
$19$	−7.84954	−1.80081	−0.900404	−	0.435054i	$-0.856729\pi$
$19$	−7.84954	−1.80081	−0.900404	+	0.435054i	$0.856729\pi$
$20$	−2.71168	−0.606350
$21$	12.4102	2.70812
$22$	5.20634	1.11000
$23$	6.28504	1.31052	0.655261	−	0.755403i	$-0.272559\pi$
$23$	6.28504	1.31052	0.655261	+	0.755403i	$0.272559\pi$
$24$	2.76801	0.565017
$25$	2.35321	0.470643
$26$	6.04655	1.18583
$27$	−4.60003	−0.885276
$28$	−4.48343	−0.847289
$29$	6.85860	1.27361	0.636805	−	0.771025i	$-0.280256\pi$
$29$	6.85860	1.27361	0.636805	+	0.771025i	$0.280256\pi$
$30$	−7.50595	−1.37039
$31$	−8.89300	−1.59723	−0.798615	−	0.601842i	$-0.794434\pi$
$31$	−8.89300	−1.59723	−0.798615	+	0.601842i	$0.794434\pi$
$32$	−1.00000	−0.176777
$33$	14.4112	2.50866
$34$	6.59491	1.13102
$35$	12.1576	2.05502
$36$	4.66186	0.776976
$37$	0.0170320	0.00280005	0.00140002	−	0.999999i	$-0.499554\pi$
$37$	0.0170320	0.00280005	0.00140002	+	0.999999i	$0.499554\pi$
$38$	7.84954	1.27336
$39$	16.7369	2.68005
$40$	2.71168	0.428754
$41$	−7.90688	−1.23485	−0.617424	−	0.786631i	$-0.711824\pi$
$41$	−7.90688	−1.23485	−0.617424	+	0.786631i	$0.711824\pi$
$42$	−12.4102	−1.91493
$43$	−10.3610	−1.58004	−0.790022	−	0.613078i	$-0.789931\pi$
$43$	−10.3610	−1.58004	−0.790022	+	0.613078i	$0.789931\pi$
$44$	−5.20634	−0.784885
$45$	−12.6415	−1.88448
$46$	−6.28504	−0.926678
$47$	−9.47316	−1.38180	−0.690901	−	0.722949i	$-0.742786\pi$
$47$	−9.47316	−1.38180	−0.690901	+	0.722949i	$0.742786\pi$
$48$	−2.76801	−0.399527
$49$	13.1012	1.87160
$50$	−2.35321	−0.332795
$51$	18.2548	2.55618
$52$	−6.04655	−0.838506
$53$	2.89615	0.397816	0.198908	−	0.980018i	$-0.436260\pi$
$53$	2.89615	0.397816	0.198908	+	0.980018i	$0.436260\pi$
$54$	4.60003	0.625985
$55$	14.1179	1.90366
$56$	4.48343	0.599124
$57$	21.7276	2.87789
$58$	−6.85860	−0.900578
$59$	6.04705	0.787258	0.393629	−	0.919269i	$-0.371219\pi$
$59$	6.04705	0.787258	0.393629	+	0.919269i	$0.371219\pi$
$60$	7.50595	0.969014
$61$	12.3377	1.57969	0.789844	−	0.613308i	$-0.210162\pi$
$61$	12.3377	1.57969	0.789844	+	0.613308i	$0.210162\pi$
$62$	8.89300	1.12941
$63$	−20.9011	−2.63329
$64$	1.00000	0.125000
$65$	16.3963	2.03371
$66$	−14.4112	−1.77389
$67$	0.170647	0.0208478	0.0104239	−	0.999946i	$-0.496682\pi$
$67$	0.170647	0.0208478	0.0104239	+	0.999946i	$0.496682\pi$
$68$	−6.59491	−0.799750
$69$	−17.3970	−2.09436
$70$	−12.1576	−1.45312
$71$	7.13938	0.847288	0.423644	−	0.905829i	$-0.360751\pi$
$71$	7.13938	0.847288	0.423644	+	0.905829i	$0.360751\pi$
$72$	−4.66186	−0.549405
$73$	6.20456	0.726188	0.363094	−	0.931752i	$-0.381720\pi$
$73$	6.20456	0.726188	0.363094	+	0.931752i	$0.381720\pi$
$74$	−0.0170320	−0.00197993
$75$	−6.51371	−0.752138
$76$	−7.84954	−0.900404
$77$	23.3423	2.66010
$78$	−16.7369	−1.89508
$79$	−12.4184	−1.39718	−0.698589	−	0.715523i	$-0.746188\pi$
$79$	−12.4184	−1.39718	−0.698589	+	0.715523i	$0.746188\pi$
$80$	−2.71168	−0.303175
$81$	−1.25266	−0.139184
$82$	7.90688	0.873169
$83$	4.17524	0.458292	0.229146	−	0.973392i	$-0.426407\pi$
$83$	4.17524	0.458292	0.229146	+	0.973392i	$0.426407\pi$
$84$	12.4102	1.35406
$85$	17.8833	1.93972
$86$	10.3610	1.11726
$87$	−18.9846	−2.03537
$88$	5.20634	0.554998
$89$	−5.91623	−0.627119	−0.313559	−	0.949569i	$-0.601521\pi$
$89$	−5.91623	−0.627119	−0.313559	+	0.949569i	$0.601521\pi$
$90$	12.6415	1.33253
$91$	27.1093	2.84183
$92$	6.28504	0.655261
$93$	24.6159	2.55255
$94$	9.47316	0.977082
$95$	21.2855	2.18384
$96$	2.76801	0.282508
$97$	−13.3374	−1.35421	−0.677105	−	0.735886i	$-0.736766\pi$
$97$	−13.3374	−1.35421	−0.677105	+	0.735886i	$0.736766\pi$
$98$	−13.1012	−1.32342
$99$	−24.2712	−2.43935
$100$	2.35321	0.235321
$101$	−6.23542	−0.620448	−0.310224	−	0.950664i	$-0.600404\pi$
$101$	−6.23542	−0.620448	−0.310224	+	0.950664i	$0.600404\pi$
$102$	−18.2548	−1.80749
$103$	−0.0244290	−0.00240706	−0.00120353	−	0.999999i	$-0.500383\pi$
$103$	−0.0244290	−0.00240706	−0.00120353	+	0.999999i	$0.500383\pi$
$104$	6.04655	0.592913
$105$	−33.6524	−3.28414
$106$	−2.89615	−0.281299
$107$	−11.2118	−1.08389	−0.541944	−	0.840415i	$-0.682312\pi$
$107$	−11.2118	−1.08389	−0.541944	+	0.840415i	$0.682312\pi$
$108$	−4.60003	−0.442638
$109$	14.1039	1.35091	0.675454	−	0.737402i	$-0.263948\pi$
$109$	14.1039	1.35091	0.675454	+	0.737402i	$0.263948\pi$
$110$	−14.1179	−1.34609
$111$	−0.0471448	−0.00447478
$112$	−4.48343	−0.423645
$113$	−9.96505	−0.937433	−0.468716	−	0.883349i	$-0.655283\pi$
$113$	−9.96505	−0.937433	−0.468716	+	0.883349i	$0.655283\pi$
$114$	−21.7276	−2.03497
$115$	−17.0430	−1.58927
$116$	6.85860	0.636805
$117$	−28.1882	−2.60600
$118$	−6.04705	−0.556676
$119$	29.5678	2.71048
$120$	−7.50595	−0.685196
$121$	16.1060	1.46418
$122$	−12.3377	−1.11701
$123$	21.8863	1.97342
$124$	−8.89300	−0.798615
$125$	7.17724	0.641952
$126$	20.9011	1.86202
$127$	−7.81793	−0.693729	−0.346865	−	0.937915i	$-0.612754\pi$
$127$	−7.81793	−0.693729	−0.346865	+	0.937915i	$0.612754\pi$
$128$	−1.00000	−0.0883883
$129$	28.6794	2.52508
$130$	−16.3963	−1.43805
$131$	−2.94059	−0.256920	−0.128460	−	0.991715i	$-0.541003\pi$
$131$	−2.94059	−0.256920	−0.128460	+	0.991715i	$0.541003\pi$
$132$	14.4112	1.25433
$133$	35.1929	3.05161
$134$	−0.170647	−0.0147416
$135$	12.4738	1.07357
$136$	6.59491	0.565509
$137$	−9.04575	−0.772830	−0.386415	−	0.922325i	$-0.626287\pi$
$137$	−9.04575	−0.772830	−0.386415	+	0.922325i	$0.626287\pi$
$138$	17.3970	1.48093
$139$	−5.07454	−0.430416	−0.215208	−	0.976568i	$-0.569043\pi$
$139$	−5.07454	−0.430416	−0.215208	+	0.976568i	$0.569043\pi$
$140$	12.1576	1.02751
$141$	26.2218	2.20827
$142$	−7.13938	−0.599123
$143$	31.4804	2.63252
$144$	4.66186	0.388488
$145$	−18.5983	−1.54451
$146$	−6.20456	−0.513493
$147$	−36.2641	−2.99102
$148$	0.0170320	0.00140002
$149$	−1.65472	−0.135560	−0.0677800	−	0.997700i	$-0.521592\pi$
$149$	−1.65472	−0.135560	−0.0677800	+	0.997700i	$0.521592\pi$
$150$	6.51371	0.531842
$151$	8.94798	0.728177	0.364088	−	0.931364i	$-0.381381\pi$
$151$	8.94798	0.728177	0.364088	+	0.931364i	$0.381381\pi$
$152$	7.84954	0.636682
$153$	−30.7445	−2.48555
$154$	−23.3423	−1.88097
$155$	24.1150	1.93696
$156$	16.7369	1.34002
$157$	2.40312	0.191789	0.0958947	−	0.995391i	$-0.469429\pi$
$157$	2.40312	0.191789	0.0958947	+	0.995391i	$0.469429\pi$
$158$	12.4184	0.987954
$159$	−8.01655	−0.635754
$160$	2.71168	0.214377
$161$	−28.1786	−2.22078
$162$	1.25266	0.0984183
$163$	0.0725103	0.00567944	0.00283972	−	0.999996i	$-0.499096\pi$
$163$	0.0725103	0.00567944	0.00283972	+	0.999996i	$0.499096\pi$
$164$	−7.90688	−0.617424
$165$	−39.0785	−3.04226
$166$	−4.17524	−0.324061
$167$	−23.8554	−1.84599	−0.922993	−	0.384818i	$-0.874264\pi$
$167$	−23.8554	−1.84599	−0.922993	+	0.384818i	$0.874264\pi$
$168$	−12.4102	−0.957465
$169$	23.5608	1.81237
$170$	−17.8833	−1.37159
$171$	−36.5934	−2.79837
$172$	−10.3610	−0.790022
$173$	12.5832	0.956684	0.478342	−	0.878174i	$-0.341238\pi$
$173$	12.5832	0.956684	0.478342	+	0.878174i	$0.341238\pi$
$174$	18.9846	1.43922
$175$	−10.5505	−0.797541
$176$	−5.20634	−0.392443
$177$	−16.7383	−1.25812
$178$	5.91623	0.443440
$179$	9.21041	0.688419	0.344209	−	0.938893i	$-0.388147\pi$
$179$	9.21041	0.688419	0.344209	+	0.938893i	$0.388147\pi$
$180$	−12.6415	−0.942239
$181$	1.50118	0.111582	0.0557909	−	0.998442i	$-0.482232\pi$
$181$	1.50118	0.111582	0.0557909	+	0.998442i	$0.482232\pi$
$182$	−27.1093	−2.00948
$183$	−34.1510	−2.52451
$184$	−6.28504	−0.463339
$185$	−0.0461854	−0.00339562
$186$	−24.6159	−1.80492
$187$	34.3354	2.51085
$188$	−9.47316	−0.690901
$189$	20.6239	1.50017
$190$	−21.2855	−1.54421
$191$	−2.07411	−0.150078	−0.0750388	−	0.997181i	$-0.523908\pi$
$191$	−2.07411	−0.150078	−0.0750388	+	0.997181i	$0.523908\pi$
$192$	−2.76801	−0.199764
$193$	−8.13858	−0.585828	−0.292914	−	0.956139i	$-0.594625\pi$
$193$	−8.13858	−0.585828	−0.292914	+	0.956139i	$0.594625\pi$
$194$	13.3374	0.957571
$195$	−45.3851	−3.25010
$196$	13.1012	0.935798
$197$	−24.3148	−1.73236	−0.866180	−	0.499732i	$-0.833432\pi$
$197$	−24.3148	−1.73236	−0.866180	+	0.499732i	$0.833432\pi$
$198$	24.2712	1.72488
$199$	0.452270	0.0320606	0.0160303	−	0.999872i	$-0.494897\pi$
$199$	0.452270	0.0320606	0.0160303	+	0.999872i	$0.494897\pi$
$200$	−2.35321	−0.166397
$201$	−0.472351	−0.0333171
$202$	6.23542	0.438723
$203$	−30.7501	−2.15823
$204$	18.2548	1.27809
$205$	21.4409	1.49750
$206$	0.0244290	0.00170205
$207$	29.3000	2.03649
$208$	−6.04655	−0.419253
$209$	40.8674	2.82686
$210$	33.6524	2.32224
$211$	21.3662	1.47091	0.735455	−	0.677573i	$-0.236968\pi$
$211$	21.3662	1.47091	0.735455	+	0.677573i	$0.236968\pi$
$212$	2.89615	0.198908
$213$	−19.7618	−1.35406
$214$	11.2118	0.766425
$215$	28.0958	1.91612
$216$	4.60003	0.312992
$217$	39.8712	2.70663
$218$	−14.1039	−0.955236
$219$	−17.1742	−1.16053
$220$	14.1179	0.951831
$221$	39.8765	2.68238
$222$	0.0471448	0.00316415
$223$	9.17276	0.614253	0.307127	−	0.951669i	$-0.400632\pi$
$223$	9.17276	0.614253	0.307127	+	0.951669i	$0.400632\pi$
$224$	4.48343	0.299562
$225$	10.9703	0.731356
$226$	9.96505	0.662865
$227$	27.8952	1.85147	0.925734	−	0.378175i	$-0.123448\pi$
$227$	27.8952	1.85147	0.925734	+	0.378175i	$0.123448\pi$
$228$	21.7276	1.43894
$229$	7.84217	0.518225	0.259112	−	0.965847i	$-0.416570\pi$
$229$	7.84217	0.518225	0.259112	+	0.965847i	$0.416570\pi$
$230$	17.0430	1.12378
$231$	−64.6116	−4.25113
$232$	−6.85860	−0.450289
$233$	−16.7757	−1.09901	−0.549505	−	0.835490i	$-0.685184\pi$
$233$	−16.7757	−1.09901	−0.549505	+	0.835490i	$0.685184\pi$
$234$	28.1882	1.84272
$235$	25.6882	1.67571
$236$	6.04705	0.393629
$237$	34.3742	2.23284
$238$	−29.5678	−1.91660
$239$	−5.82364	−0.376700	−0.188350	−	0.982102i	$-0.560314\pi$
$239$	−5.82364	−0.376700	−0.188350	+	0.982102i	$0.560314\pi$
$240$	7.50595	0.484507
$241$	−11.4606	−0.738240	−0.369120	−	0.929382i	$-0.620341\pi$
$241$	−11.4606	−0.738240	−0.369120	+	0.929382i	$0.620341\pi$
$242$	−16.1060	−1.03533
$243$	17.2675	1.10771
$244$	12.3377	0.789844
$245$	−35.5262	−2.26969
$246$	−21.8863	−1.39542
$247$	47.4627	3.01998
$248$	8.89300	0.564706
$249$	−11.5571	−0.732400
$250$	−7.17724	−0.453929
$251$	−1.22945	−0.0776021	−0.0388011	−	0.999247i	$-0.512354\pi$
$251$	−1.22945	−0.0776021	−0.0388011	+	0.999247i	$0.512354\pi$
$252$	−20.9011	−1.31665
$253$	−32.7220	−2.05722
$254$	7.81793	0.490541
$255$	−49.5011	−3.09988
$256$	1.00000	0.0625000
$257$	0.814736	0.0508218	0.0254109	−	0.999677i	$-0.491911\pi$
$257$	0.814736	0.0508218	0.0254109	+	0.999677i	$0.491911\pi$
$258$	−28.6794	−1.78550
$259$	−0.0763620	−0.00474490
$260$	16.3963	1.01686
$261$	31.9738	1.97913
$262$	2.94059	0.181670
$263$	24.1566	1.48956	0.744780	−	0.667310i	$-0.232554\pi$
$263$	24.1566	1.48956	0.744780	+	0.667310i	$0.232554\pi$
$264$	−14.4112	−0.886947
$265$	−7.85342	−0.482432
$266$	−35.1929	−2.15782
$267$	16.3761	1.00220
$268$	0.170647	0.0104239
$269$	16.3324	0.995803	0.497901	−	0.867234i	$-0.334104\pi$
$269$	16.3324	0.995803	0.497901	+	0.867234i	$0.334104\pi$
$270$	−12.4738	−0.759132
$271$	−8.84844	−0.537505	−0.268752	−	0.963209i	$-0.586611\pi$
$271$	−8.84844	−0.537505	−0.268752	+	0.963209i	$0.586611\pi$
$272$	−6.59491	−0.399875
$273$	−75.0388	−4.54155
$274$	9.04575	0.546474
$275$	−12.2516	−0.738801
$276$	−17.3970	−1.04718
$277$	−2.95873	−0.177773	−0.0888865	−	0.996042i	$-0.528331\pi$
$277$	−2.95873	−0.177773	−0.0888865	+	0.996042i	$0.528331\pi$
$278$	5.07454	0.304350
$279$	−41.4579	−2.48202
$280$	−12.1576	−0.726558
$281$	−15.9776	−0.953146	−0.476573	−	0.879135i	$-0.658121\pi$
$281$	−15.9776	−0.953146	−0.476573	+	0.879135i	$0.658121\pi$
$282$	−26.2218	−1.56148
$283$	15.9438	0.947763	0.473881	−	0.880589i	$-0.342853\pi$
$283$	15.9438	0.947763	0.473881	+	0.880589i	$0.342853\pi$
$284$	7.13938	0.423644
$285$	−58.9183	−3.49002
$286$	−31.4804	−1.86148
$287$	35.4500	2.09255
$288$	−4.66186	−0.274703
$289$	26.4929	1.55840
$290$	18.5983	1.09213
$291$	36.9181	2.16418
$292$	6.20456	0.363094
$293$	−0.537550	−0.0314040	−0.0157020	−	0.999877i	$-0.504998\pi$
$293$	−0.537550	−0.0314040	−0.0157020	+	0.999877i	$0.504998\pi$
$294$	36.2641	2.11497
$295$	−16.3977	−0.954709
$296$	−0.0170320	−0.000989967 0
$297$	23.9493	1.38968
$298$	1.65472	0.0958554
$299$	−38.0028	−2.19776
$300$	−6.51371	−0.376069
$301$	46.4531	2.67751
$302$	−8.94798	−0.514899
$303$	17.2597	0.991543
$304$	−7.84954	−0.450202
$305$	−33.4560	−1.91569
$306$	30.7445	1.75755
$307$	−12.6772	−0.723524	−0.361762	−	0.932270i	$-0.617825\pi$
$307$	−12.6772	−0.723524	−0.361762	+	0.932270i	$0.617825\pi$
$308$	23.3423	1.33005
$309$	0.0676195	0.00384674
$310$	−24.1150	−1.36964
$311$	−19.2429	−1.09116	−0.545581	−	0.838058i	$-0.683691\pi$
$311$	−19.2429	−1.09116	−0.545581	+	0.838058i	$0.683691\pi$
$312$	−16.7369	−0.947540
$313$	12.2206	0.690749	0.345375	−	0.938465i	$-0.387752\pi$
$313$	12.2206	0.690749	0.345375	+	0.938465i	$0.387752\pi$
$314$	−2.40312	−0.135616
$315$	56.6772	3.19340
$316$	−12.4184	−0.698589
$317$	−20.4292	−1.14742	−0.573710	−	0.819059i	$-0.694496\pi$
$317$	−20.4292	−1.14742	−0.573710	+	0.819059i	$0.694496\pi$
$318$	8.01655	0.449546
$319$	−35.7082	−1.99927
$320$	−2.71168	−0.151588
$321$	31.0344	1.73217
$322$	28.1786	1.57033
$323$	51.7670	2.88040
$324$	−1.25266	−0.0695922
$325$	−14.2288	−0.789273
$326$	−0.0725103	−0.00401597
$327$	−39.0397	−2.15890
$328$	7.90688	0.436585
$329$	42.4723	2.34157
$330$	39.0785	2.15120
$331$	−9.63064	−0.529348	−0.264674	−	0.964338i	$-0.585264\pi$
$331$	−9.63064	−0.529348	−0.264674	+	0.964338i	$0.585264\pi$
$332$	4.17524	0.229146
$333$	0.0794009	0.00435114
$334$	23.8554	1.30531
$335$	−0.462739	−0.0252821
$336$	12.4102	0.677030
$337$	21.3884	1.16510	0.582550	−	0.812795i	$-0.302055\pi$
$337$	21.3884	1.16510	0.582550	+	0.812795i	$0.302055\pi$
$338$	−23.5608	−1.28154
$339$	27.5833	1.49812
$340$	17.8833	0.969858
$341$	46.3000	2.50728
$342$	36.5934	1.97875
$343$	−27.3542	−1.47699
$344$	10.3610	0.558630
$345$	47.1752	2.53983
$346$	−12.5832	−0.676478
$347$	−19.8717	−1.06677	−0.533383	−	0.845874i	$-0.679080\pi$
$347$	−19.8717	−1.06677	−0.533383	+	0.845874i	$0.679080\pi$
$348$	−18.9846	−1.01768
$349$	−16.6532	−0.891425	−0.445713	−	0.895176i	$-0.647050\pi$
$349$	−16.6532	−0.891425	−0.445713	+	0.895176i	$0.647050\pi$
$350$	10.5505	0.563947
$351$	27.8143	1.48462
$352$	5.20634	0.277499
$353$	−30.9781	−1.64880	−0.824398	−	0.566011i	$-0.808486\pi$
$353$	−30.9781	−1.64880	−0.824398	+	0.566011i	$0.808486\pi$
$354$	16.7383	0.889629
$355$	−19.3597	−1.02751
$356$	−5.91623	−0.313559
$357$	−81.8440	−4.33164
$358$	−9.21041	−0.486786
$359$	2.06464	0.108968	0.0544838	−	0.998515i	$-0.482649\pi$
$359$	2.06464	0.108968	0.0544838	+	0.998515i	$0.482649\pi$
$360$	12.6415	0.666264
$361$	42.6153	2.24291
$362$	−1.50118	−0.0789002
$363$	−44.5814	−2.33992
$364$	27.1093	1.42091
$365$	−16.8248	−0.880649
$366$	34.1510	1.78510
$367$	10.9507	0.571622	0.285811	−	0.958286i	$-0.407737\pi$
$367$	10.9507	0.571622	0.285811	+	0.958286i	$0.407737\pi$
$368$	6.28504	0.327630
$369$	−36.8608	−1.91889
$370$	0.0461854	0.00240107
$371$	−12.9847	−0.674131
$372$	24.6159	1.27627
$373$	8.84596	0.458026	0.229013	−	0.973423i	$-0.426450\pi$
$373$	8.84596	0.458026	0.229013	+	0.973423i	$0.426450\pi$
$374$	−34.3354	−1.77544
$375$	−19.8666	−1.02591
$376$	9.47316	0.488541
$377$	−41.4709	−2.13586
$378$	−20.6239	−1.06078
$379$	−15.8900	−0.816216	−0.408108	−	0.912934i	$-0.633811\pi$
$379$	−15.8900	−0.816216	−0.408108	+	0.912934i	$0.633811\pi$
$380$	21.2855	1.09192
$381$	21.6401	1.10865
$382$	2.07411	0.106121
$383$	8.99735	0.459743	0.229871	−	0.973221i	$-0.426169\pi$
$383$	8.99735	0.459743	0.229871	+	0.973221i	$0.426169\pi$
$384$	2.76801	0.141254
$385$	−63.2968	−3.22590
$386$	8.13858	0.414243
$387$	−48.3017	−2.45531
$388$	−13.3374	−0.677105
$389$	12.7191	0.644883	0.322442	−	0.946589i	$-0.395496\pi$
$389$	12.7191	0.644883	0.322442	+	0.946589i	$0.395496\pi$
$390$	45.3851	2.29816
$391$	−41.4493	−2.09618
$392$	−13.1012	−0.661709
$393$	8.13957	0.410587
$394$	24.3148	1.22496
$395$	33.6747	1.69436
$396$	−24.2712	−1.21967
$397$	11.2057	0.562400	0.281200	−	0.959649i	$-0.409268\pi$
$397$	11.2057	0.562400	0.281200	+	0.959649i	$0.409268\pi$
$398$	−0.452270	−0.0226703
$399$	−97.4142	−4.87681
$400$	2.35321	0.117661
$401$	9.54544	0.476677	0.238338	−	0.971182i	$-0.423397\pi$
$401$	9.54544	0.476677	0.238338	+	0.971182i	$0.423397\pi$
$402$	0.472351	0.0235587
$403$	53.7720	2.67857
$404$	−6.23542	−0.310224
$405$	3.39681	0.168789
$406$	30.7501	1.52610
$407$	−0.0886745	−0.00439543
$408$	−18.2548	−0.903745
$409$	5.21468	0.257849	0.128925	−	0.991654i	$-0.458847\pi$
$409$	5.21468	0.257849	0.128925	+	0.991654i	$0.458847\pi$
$410$	−21.4409	−1.05889
$411$	25.0387	1.23507
$412$	−0.0244290	−0.00120353
$413$	−27.1115	−1.33407
$414$	−29.3000	−1.44001
$415$	−11.3219	−0.555771
$416$	6.04655	0.296457
$417$	14.0463	0.687852
$418$	−40.8674	−1.99889
$419$	−39.1646	−1.91331	−0.956657	−	0.291217i	$-0.905940\pi$
$419$	−39.1646	−1.91331	−0.956657	+	0.291217i	$0.905940\pi$
$420$	−33.6524	−1.64207
$421$	−0.199839	−0.00973956	−0.00486978	−	0.999988i	$-0.501550\pi$
$421$	−0.199839	−0.00973956	−0.00486978	+	0.999988i	$0.501550\pi$
$422$	−21.3662	−1.04009
$423$	−44.1625	−2.14725
$424$	−2.89615	−0.140649
$425$	−15.5192	−0.752793
$426$	19.7618	0.957464
$427$	−55.3155	−2.67690
$428$	−11.2118	−0.541944
$429$	−87.1380	−4.20706
$430$	−28.0958	−1.35490
$431$	3.32693	0.160253	0.0801264	−	0.996785i	$-0.474468\pi$
$431$	3.32693	0.160253	0.0801264	+	0.996785i	$0.474468\pi$
$432$	−4.60003	−0.221319
$433$	26.8500	1.29033	0.645165	−	0.764043i	$-0.276788\pi$
$433$	26.8500	1.29033	0.645165	+	0.764043i	$0.276788\pi$
$434$	−39.8712	−1.91388
$435$	51.4803	2.46829
$436$	14.1039	0.675454
$437$	−49.3347	−2.36000
$438$	17.1742	0.820617
$439$	−22.2487	−1.06187	−0.530936	−	0.847412i	$-0.678160\pi$
$439$	−22.2487	−1.06187	−0.530936	+	0.847412i	$0.678160\pi$
$440$	−14.1179	−0.673046
$441$	61.0758	2.90837
$442$	−39.8765	−1.89673
$443$	0.767745	0.0364767	0.0182383	−	0.999834i	$-0.494194\pi$
$443$	0.767745	0.0364767	0.0182383	+	0.999834i	$0.494194\pi$
$444$	−0.0471448	−0.00223739
$445$	16.0429	0.760507
$446$	−9.17276	−0.434343
$447$	4.58028	0.216640
$448$	−4.48343	−0.211822
$449$	−22.9421	−1.08270	−0.541351	−	0.840797i	$-0.682087\pi$
$449$	−22.9421	−1.08270	−0.541351	+	0.840797i	$0.682087\pi$
$450$	−10.9703	−0.517147
$451$	41.1659	1.93843
$452$	−9.96505	−0.468716
$453$	−24.7681	−1.16371
$454$	−27.8952	−1.30919
$455$	−73.5118	−3.44629
$456$	−21.7276	−1.01749
$457$	19.3281	0.904129	0.452065	−	0.891985i	$-0.350688\pi$
$457$	19.3281	0.904129	0.452065	+	0.891985i	$0.350688\pi$
$458$	−7.84217	−0.366440
$459$	30.3368	1.41600
$460$	−17.0430	−0.794635
$461$	32.6236	1.51943	0.759717	−	0.650254i	$-0.225338\pi$
$461$	32.6236	1.51943	0.759717	+	0.650254i	$0.225338\pi$
$462$	64.6116	3.00600
$463$	−35.9761	−1.67195	−0.835975	−	0.548767i	$-0.815097\pi$
$463$	−35.9761	−1.67195	−0.835975	+	0.548767i	$0.815097\pi$
$464$	6.85860	0.318402
$465$	−66.7504	−3.09548
$466$	16.7757	0.777117
$467$	−24.1814	−1.11898	−0.559490	−	0.828837i	$-0.689003\pi$
$467$	−24.1814	−1.11898	−0.559490	+	0.828837i	$0.689003\pi$
$468$	−28.1882	−1.30300
$469$	−0.765083	−0.0353282
$470$	−25.6882	−1.18491
$471$	−6.65184	−0.306500
$472$	−6.04705	−0.278338
$473$	53.9431	2.48031
$474$	−34.3742	−1.57886
$475$	−18.4716	−0.847537
$476$	29.5678	1.35524
$477$	13.5014	0.618187
$478$	5.82364	0.266367
$479$	−23.9134	−1.09263	−0.546315	−	0.837580i	$-0.683970\pi$
$479$	−23.9134	−1.09263	−0.546315	+	0.837580i	$0.683970\pi$
$480$	−7.50595	−0.342598
$481$	−0.102985	−0.00469572
$482$	11.4606	0.522014
$483$	77.9984	3.54905
$484$	16.1060	0.732090
$485$	36.1668	1.64225
$486$	−17.2675	−0.783268
$487$	1.56352	0.0708499	0.0354249	−	0.999372i	$-0.488722\pi$
$487$	1.56352	0.0708499	0.0354249	+	0.999372i	$0.488722\pi$
$488$	−12.3377	−0.558504
$489$	−0.200709	−0.00907637
$490$	35.5262	1.60491
$491$	−35.6676	−1.60966	−0.804828	−	0.593508i	$-0.797743\pi$
$491$	−35.6676	−1.60966	−0.804828	+	0.593508i	$0.797743\pi$
$492$	21.8863	0.986711
$493$	−45.2318	−2.03714
$494$	−47.4627	−2.13545
$495$	65.8158	2.95820
$496$	−8.89300	−0.399307
$497$	−32.0089	−1.43580
$498$	11.5571	0.517885
$499$	7.43746	0.332947	0.166473	−	0.986046i	$-0.446762\pi$
$499$	7.43746	0.332947	0.166473	+	0.986046i	$0.446762\pi$
$500$	7.17724	0.320976
$501$	66.0318	2.95009
$502$	1.22945	0.0548730
$503$	−35.9626	−1.60349	−0.801747	−	0.597663i	$-0.796096\pi$
$503$	−35.9626	−1.60349	−0.801747	+	0.597663i	$0.796096\pi$
$504$	20.9011	0.931010
$505$	16.9085	0.752417
$506$	32.7220	1.45467
$507$	−65.2165	−2.89636
$508$	−7.81793	−0.346865
$509$	−20.0229	−0.887500	−0.443750	−	0.896151i	$-0.646352\pi$
$509$	−20.0229	−0.887500	−0.443750	+	0.896151i	$0.646352\pi$
$510$	49.5011	2.19194
$511$	−27.8177	−1.23058
$512$	−1.00000	−0.0441942
$513$	36.1081	1.59421
$514$	−0.814736	−0.0359365
$515$	0.0662436	0.00291904
$516$	28.6794	1.26254
$517$	49.3205	2.16911
$518$	0.0763620	0.00335515
$519$	−34.8304	−1.52889
$520$	−16.3963	−0.719026
$521$	13.1563	0.576388	0.288194	−	0.957572i	$-0.406945\pi$
$521$	13.1563	0.576388	0.288194	+	0.957572i	$0.406945\pi$
$522$	−31.9738	−1.39945
$523$	25.8429	1.13003	0.565015	−	0.825081i	$-0.308870\pi$
$523$	25.8429	1.13003	0.565015	+	0.825081i	$0.308870\pi$
$524$	−2.94059	−0.128460
$525$	29.2038	1.27456
$526$	−24.1566	−1.05328
$527$	58.6485	2.55477
$528$	14.4112	0.627166
$529$	16.5017	0.717466
$530$	7.85342	0.341131
$531$	28.1905	1.22336
$532$	35.1929	1.52581
$533$	47.8094	2.07085
$534$	−16.3761	−0.708665
$535$	30.4029	1.31443
$536$	−0.170647	−0.00737081
$537$	−25.4945	−1.10017
$538$	−16.3324	−0.704139
$539$	−68.2092	−2.93798
$540$	12.4738	0.536787
$541$	8.85348	0.380641	0.190320	−	0.981722i	$-0.439047\pi$
$541$	8.85348	0.380641	0.190320	+	0.981722i	$0.439047\pi$
$542$	8.84844	0.380073
$543$	−4.15527	−0.178320
$544$	6.59491	0.282754
$545$	−38.2453	−1.63825
$546$	75.0388	3.21136
$547$	−29.8075	−1.27448	−0.637238	−	0.770667i	$-0.719923\pi$
$547$	−29.8075	−1.27448	−0.637238	+	0.770667i	$0.719923\pi$
$548$	−9.04575	−0.386415
$549$	57.5168	2.45476
$550$	12.2516	0.522411
$551$	−53.8368	−2.29353
$552$	17.3970	0.740467
$553$	55.6770	2.36763
$554$	2.95873	0.125704
$555$	0.127842	0.00542657
$556$	−5.07454	−0.215208
$557$	10.1061	0.428211	0.214105	−	0.976811i	$-0.431316\pi$
$557$	10.1061	0.428211	0.214105	+	0.976811i	$0.431316\pi$
$558$	41.4579	1.75505
$559$	62.6486	2.64975
$560$	12.1576	0.513754
$561$	−95.0405	−4.01261
$562$	15.9776	0.673976
$563$	8.51339	0.358796	0.179398	−	0.983777i	$-0.442585\pi$
$563$	8.51339	0.358796	0.179398	+	0.983777i	$0.442585\pi$
$564$	26.2218	1.10414
$565$	27.0220	1.13683
$566$	−15.9438	−0.670169
$567$	5.61622	0.235859
$568$	−7.13938	−0.299562
$569$	5.25910	0.220473	0.110236	−	0.993905i	$-0.464839\pi$
$569$	5.25910	0.220473	0.110236	+	0.993905i	$0.464839\pi$
$570$	58.9183	2.46781
$571$	−2.59509	−0.108601	−0.0543005	−	0.998525i	$-0.517293\pi$
$571$	−2.59509	−0.108601	−0.0543005	+	0.998525i	$0.517293\pi$
$572$	31.4804	1.31626
$573$	5.74116	0.239840
$574$	−35.4500	−1.47965
$575$	14.7900	0.616787
$576$	4.66186	0.194244
$577$	−30.7251	−1.27910	−0.639551	−	0.768749i	$-0.720880\pi$
$577$	−30.7251	−1.27910	−0.639551	+	0.768749i	$0.720880\pi$
$578$	−26.4929	−1.10196
$579$	22.5276	0.936217
$580$	−18.5983	−0.772253
$581$	−18.7194	−0.776611
$582$	−36.9181	−1.53030
$583$	−15.0783	−0.624480
$584$	−6.20456	−0.256746
$585$	76.4373	3.16029
$586$	0.537550	0.0222060
$587$	−28.5561	−1.17864	−0.589318	−	0.807901i	$-0.700604\pi$
$587$	−28.5561	−1.17864	−0.589318	+	0.807901i	$0.700604\pi$
$588$	−36.2641	−1.49551
$589$	69.8060	2.87631
$590$	16.3977	0.675081
$591$	67.3036	2.76850
$592$	0.0170320	0.000700012 0
$593$	−33.3027	−1.36758	−0.683789	−	0.729680i	$-0.739669\pi$
$593$	−33.3027	−1.36758	−0.683789	+	0.729680i	$0.739669\pi$
$594$	−23.9493	−0.982653
$595$	−80.1786	−3.28700
$596$	−1.65472	−0.0677800
$597$	−1.25189	−0.0512363
$598$	38.0028	1.55405
$599$	0.307711	0.0125727	0.00628637	−	0.999980i	$-0.497999\pi$
$599$	0.307711	0.0125727	0.00628637	+	0.999980i	$0.497999\pi$
$600$	6.51371	0.265921
$601$	−33.4286	−1.36358	−0.681790	−	0.731548i	$-0.738798\pi$
$601$	−33.4286	−1.36358	−0.681790	+	0.731548i	$0.738798\pi$
$602$	−46.4531	−1.89329
$603$	0.795530	0.0323965
$604$	8.94798	0.364088
$605$	−43.6743	−1.77561
$606$	−17.2597	−0.701127
$607$	−25.9757	−1.05432	−0.527160	−	0.849766i	$-0.676743\pi$
$607$	−25.9757	−1.05432	−0.527160	+	0.849766i	$0.676743\pi$
$608$	7.84954	0.318341
$609$	85.1163	3.44909
$610$	33.4560	1.35460
$611$	57.2800	2.31730
$612$	−30.7445	−1.24277
$613$	−6.01811	−0.243069	−0.121535	−	0.992587i	$-0.538782\pi$
$613$	−6.01811	−0.243069	−0.121535	+	0.992587i	$0.538782\pi$
$614$	12.6772	0.511609
$615$	−59.3487	−2.39317
$616$	−23.3423	−0.940487
$617$	−38.3608	−1.54435	−0.772173	−	0.635412i	$-0.780830\pi$
$617$	−38.3608	−1.54435	−0.772173	+	0.635412i	$0.780830\pi$
$618$	−0.0676195	−0.00272006
$619$	21.9377	0.881750	0.440875	−	0.897569i	$-0.354668\pi$
$619$	21.9377	0.881750	0.440875	+	0.897569i	$0.354668\pi$
$620$	24.1150	0.968481
$621$	−28.9114	−1.16017
$622$	19.2429	0.771568
$623$	26.5250	1.06270
$624$	16.7369	0.670012
$625$	−31.2285	−1.24914
$626$	−12.2206	−0.488433
$627$	−113.121	−4.51763
$628$	2.40312	0.0958947
$629$	−0.112325	−0.00447868
$630$	−56.6772	−2.25807
$631$	35.7767	1.42425	0.712124	−	0.702054i	$-0.247734\pi$
$631$	35.7767	1.42425	0.712124	+	0.702054i	$0.247734\pi$
$632$	12.4184	0.493977
$633$	−59.1418	−2.35067
$634$	20.4292	0.811348
$635$	21.1997	0.841286
$636$	−8.01655	−0.317877
$637$	−79.2170	−3.13869
$638$	35.7082	1.41370
$639$	33.2828	1.31665
$640$	2.71168	0.107189
$641$	−33.1385	−1.30889	−0.654446	−	0.756109i	$-0.727098\pi$
$641$	−33.1385	−1.30889	−0.654446	+	0.756109i	$0.727098\pi$
$642$	−31.0344	−1.22483
$643$	9.08129	0.358131	0.179066	−	0.983837i	$-0.442693\pi$
$643$	9.08129	0.358131	0.179066	+	0.983837i	$0.442693\pi$
$644$	−28.1786	−1.11039
$645$	−77.7695	−3.06217
$646$	−51.7670	−2.03675
$647$	−4.54994	−0.178877	−0.0894384	−	0.995992i	$-0.528507\pi$
$647$	−4.54994	−0.178877	−0.0894384	+	0.995992i	$0.528507\pi$
$648$	1.25266	0.0492091
$649$	−31.4830	−1.23582
$650$	14.2288	0.558101
$651$	−110.364	−4.32549
$652$	0.0725103	0.00283972
$653$	36.8424	1.44176	0.720878	−	0.693062i	$-0.243739\pi$
$653$	36.8424	1.44176	0.720878	+	0.693062i	$0.243739\pi$
$654$	39.0397	1.52657
$655$	7.97394	0.311568
$656$	−7.90688	−0.308712
$657$	28.9247	1.12846
$658$	−42.4723	−1.65574
$659$	−5.65853	−0.220425	−0.110212	−	0.993908i	$-0.535153\pi$
$659$	−5.65853	−0.220425	−0.110212	+	0.993908i	$0.535153\pi$
$660$	−39.0785	−1.52113
$661$	2.42544	0.0943388	0.0471694	−	0.998887i	$-0.484980\pi$
$661$	2.42544	0.0943388	0.0471694	+	0.998887i	$0.484980\pi$
$662$	9.63064	0.374305
$663$	−110.378	−4.28674
$664$	−4.17524	−0.162031
$665$	−95.4319	−3.70069
$666$	−0.0794009	−0.00307672
$667$	43.1065	1.66909
$668$	−23.8554	−0.922993
$669$	−25.3903	−0.981644
$670$	0.462739	0.0178772
$671$	−64.2345	−2.47975
$672$	−12.4102	−0.478733
$673$	−19.3685	−0.746600	−0.373300	−	0.927711i	$-0.621774\pi$
$673$	−19.3685	−0.746600	−0.373300	+	0.927711i	$0.621774\pi$
$674$	−21.3884	−0.823850
$675$	−10.8249	−0.416649
$676$	23.5608	0.906185
$677$	−1.95936	−0.0753044	−0.0376522	−	0.999291i	$-0.511988\pi$
$677$	−1.95936	−0.0753044	−0.0376522	+	0.999291i	$0.511988\pi$
$678$	−27.5833	−1.05933
$679$	59.7974	2.29482
$680$	−17.8833	−0.685793
$681$	−77.2140	−2.95885
$682$	−46.3000	−1.77292
$683$	23.7282	0.907933	0.453967	−	0.891019i	$-0.350008\pi$
$683$	23.7282	0.907933	0.453967	+	0.891019i	$0.350008\pi$
$684$	−36.5934	−1.39919
$685$	24.5292	0.937212
$686$	27.3542	1.04439
$687$	−21.7072	−0.828180
$688$	−10.3610	−0.395011
$689$	−17.5117	−0.667143
$690$	−47.1752	−1.79593
$691$	−27.3534	−1.04057	−0.520287	−	0.853991i	$-0.674175\pi$
$691$	−27.3534	−1.04057	−0.520287	+	0.853991i	$0.674175\pi$
$692$	12.5832	0.478342
$693$	108.818	4.13367
$694$	19.8717	0.754318
$695$	13.7605	0.521966
$696$	18.9846	0.719611
$697$	52.1452	1.97514
$698$	16.6532	0.630333
$699$	46.4351	1.75634
$700$	−10.5505	−0.398770
$701$	32.5218	1.22833	0.614166	−	0.789177i	$-0.289493\pi$
$701$	32.5218	1.22833	0.614166	+	0.789177i	$0.289493\pi$
$702$	−27.8143	−1.04978
$703$	−0.133694	−0.00504235
$704$	−5.20634	−0.196221
$705$	−71.1050	−2.67797
$706$	30.9781	1.16587
$707$	27.9561	1.05140
$708$	−16.7383	−0.629062
$709$	13.9673	0.524555	0.262277	−	0.964993i	$-0.415526\pi$
$709$	13.9673	0.524555	0.262277	+	0.964993i	$0.415526\pi$
$710$	19.3597	0.726557
$711$	−57.8927	−2.17115
$712$	5.91623	0.221720
$713$	−55.8928	−2.09320
$714$	81.8440	3.06293
$715$	−85.3648	−3.19246
$716$	9.21041	0.344209
$717$	16.1199	0.602008
$718$	−2.06464	−0.0770517
$719$	−28.3325	−1.05663	−0.528313	−	0.849050i	$-0.677175\pi$
$719$	−28.3325	−1.05663	−0.528313	+	0.849050i	$0.677175\pi$
$720$	−12.6415	−0.471120
$721$	0.109526	0.00407895
$722$	−42.6153	−1.58598
$723$	31.7229	1.17979
$724$	1.50118	0.0557909
$725$	16.1397	0.599415
$726$	44.5814	1.65457
$727$	5.98444	0.221951	0.110975	−	0.993823i	$-0.464603\pi$
$727$	5.98444	0.221951	0.110975	+	0.993823i	$0.464603\pi$
$728$	−27.1093	−1.00474
$729$	−44.0385	−1.63105
$730$	16.8248	0.622713
$731$	68.3302	2.52728
$732$	−34.1510	−1.26226
$733$	14.7078	0.543245	0.271623	−	0.962404i	$-0.412440\pi$
$733$	14.7078	0.543245	0.271623	+	0.962404i	$0.412440\pi$
$734$	−10.9507	−0.404198
$735$	98.3367	3.62721
$736$	−6.28504	−0.231670
$737$	−0.888444	−0.0327263
$738$	36.8608	1.35686
$739$	6.74895	0.248264	0.124132	−	0.992266i	$-0.460385\pi$
$739$	6.74895	0.248264	0.124132	+	0.992266i	$0.460385\pi$
$740$	−0.0461854	−0.00169781
$741$	−131.377	−4.82625
$742$	12.9847	0.476682
$743$	−17.3070	−0.634933	−0.317467	−	0.948269i	$-0.602832\pi$
$743$	−17.3070	−0.634933	−0.317467	+	0.948269i	$0.602832\pi$
$744$	−24.6159	−0.902462
$745$	4.48708	0.164394
$746$	−8.84596	−0.323874
$747$	19.4644	0.712163
$748$	34.3354	1.25542
$749$	50.2675	1.83673
$750$	19.8666	0.725427
$751$	8.50329	0.310289	0.155145	−	0.987892i	$-0.450416\pi$
$751$	8.50329	0.310289	0.155145	+	0.987892i	$0.450416\pi$
$752$	−9.47316	−0.345451
$753$	3.40312	0.124017
$754$	41.4709	1.51028
$755$	−24.2641	−0.883060
$756$	20.6239	0.750085
$757$	−3.01404	−0.109547	−0.0547736	−	0.998499i	$-0.517444\pi$
$757$	−3.01404	−0.109547	−0.0547736	+	0.998499i	$0.517444\pi$
$758$	15.8900	0.577152
$759$	90.5748	3.28766
$760$	−21.2855	−0.772105
$761$	50.5940	1.83403	0.917016	−	0.398850i	$-0.130591\pi$
$761$	50.5940	1.83403	0.917016	+	0.398850i	$0.130591\pi$
$762$	−21.6401	−0.783937
$763$	−63.2339	−2.28922
$764$	−2.07411	−0.0750388
$765$	83.3694	3.01423
$766$	−8.99735	−0.325087
$767$	−36.5638	−1.32024
$768$	−2.76801	−0.0998818
$769$	48.2530	1.74005	0.870024	−	0.493009i	$-0.164103\pi$
$769$	48.2530	1.74005	0.870024	+	0.493009i	$0.164103\pi$
$770$	63.2968	2.28106
$771$	−2.25519	−0.0812188
$772$	−8.13858	−0.292914
$773$	40.7788	1.46671	0.733356	−	0.679845i	$-0.237953\pi$
$773$	40.7788	1.46671	0.733356	+	0.679845i	$0.237953\pi$
$774$	48.3017	1.73617
$775$	−20.9271	−0.751724
$776$	13.3374	0.478786
$777$	0.211370	0.00758287
$778$	−12.7191	−0.456001
$779$	62.0654	2.22372
$780$	−45.3851	−1.62505
$781$	−37.1700	−1.33005
$782$	41.4493	1.48222
$783$	−31.5497	−1.12750
$784$	13.1012	0.467899
$785$	−6.51648	−0.232583
$786$	−8.13957	−0.290329
$787$	−36.4462	−1.29917	−0.649584	−	0.760290i	$-0.725057\pi$
$787$	−36.4462	−1.29917	−0.649584	+	0.760290i	$0.725057\pi$
$788$	−24.3148	−0.866180
$789$	−66.8656	−2.38048
$790$	−33.6747	−1.19809
$791$	44.6776	1.58855
$792$	24.2712	0.862440
$793$	−74.6009	−2.64915
$794$	−11.2057	−0.397677
$795$	21.7383	0.770979
$796$	0.452270	0.0160303
$797$	−14.2694	−0.505447	−0.252723	−	0.967539i	$-0.581326\pi$
$797$	−14.2694	−0.505447	−0.252723	+	0.967539i	$0.581326\pi$
$798$	97.4142	3.44842
$799$	62.4746	2.21019
$800$	−2.35321	−0.0831987
$801$	−27.5806	−0.974512
$802$	−9.54544	−0.337061
$803$	−32.3030	−1.13995
$804$	−0.472351	−0.0166585
$805$	76.4112	2.69314
$806$	−53.7720	−1.89404
$807$	−45.2081	−1.59140
$808$	6.23542	0.219361
$809$	23.4963	0.826086	0.413043	−	0.910712i	$-0.364466\pi$
$809$	23.4963	0.826086	0.413043	+	0.910712i	$0.364466\pi$
$810$	−3.39681	−0.119352
$811$	41.5943	1.46057	0.730287	−	0.683140i	$-0.239386\pi$
$811$	41.5943	1.46057	0.730287	+	0.683140i	$0.239386\pi$
$812$	−30.7501	−1.07912
$813$	24.4925	0.858991
$814$	0.0886745	0.00310804
$815$	−0.196625	−0.00688746
$816$	18.2548	0.639044
$817$	81.3295	2.84536
$818$	−5.21468	−0.182327
$819$	126.380	4.41607
$820$	21.4409	0.748750
$821$	4.39841	0.153506	0.0767528	−	0.997050i	$-0.475545\pi$
$821$	4.39841	0.153506	0.0767528	+	0.997050i	$0.475545\pi$
$822$	−25.0387	−0.873324
$823$	−8.21581	−0.286385	−0.143193	−	0.989695i	$-0.545737\pi$
$823$	−8.21581	−0.286385	−0.143193	+	0.989695i	$0.545737\pi$
$824$	0.0244290	0.000851024 0
$825$	33.9126	1.18068
$826$	27.1115	0.943331
$827$	3.97953	0.138382	0.0691908	−	0.997603i	$-0.477958\pi$
$827$	3.97953	0.138382	0.0691908	+	0.997603i	$0.477958\pi$
$828$	29.3000	1.01824
$829$	−15.6726	−0.544332	−0.272166	−	0.962250i	$-0.587740\pi$
$829$	−15.6726	−0.544332	−0.272166	+	0.962250i	$0.587740\pi$
$830$	11.3219	0.392989
$831$	8.18979	0.284101
$832$	−6.04655	−0.209627
$833$	−86.4011	−2.99362
$834$	−14.0463	−0.486385
$835$	64.6882	2.23863
$836$	40.8674	1.41343
$837$	40.9081	1.41399
$838$	39.1646	1.35292
$839$	26.5458	0.916462	0.458231	−	0.888833i	$-0.348483\pi$
$839$	26.5458	0.916462	0.458231	+	0.888833i	$0.348483\pi$
$840$	33.6524	1.16112
$841$	18.0403	0.622080
$842$	0.199839	0.00688691
$843$	44.2262	1.52323
$844$	21.3662	0.735455
$845$	−63.8894	−2.19786
$846$	44.1625	1.51834
$847$	−72.2101	−2.48117
$848$	2.89615	0.0994541
$849$	−44.1326	−1.51463
$850$	15.5192	0.532305
$851$	0.107047	0.00366952
$852$	−19.7618	−0.677029
$853$	−28.9316	−0.990599	−0.495299	−	0.868722i	$-0.664942\pi$
$853$	−28.9316	−0.990599	−0.495299	+	0.868722i	$0.664942\pi$
$854$	55.3155	1.89286
$855$	99.2298	3.39359
$856$	11.2118	0.383212
$857$	−33.9224	−1.15877	−0.579383	−	0.815055i	$-0.696707\pi$
$857$	−33.9224	−1.15877	−0.579383	+	0.815055i	$0.696707\pi$
$858$	87.1380	2.97484
$859$	−49.2241	−1.67951	−0.839753	−	0.542969i	$-0.817300\pi$
$859$	−49.2241	−1.67951	−0.839753	+	0.542969i	$0.817300\pi$
$860$	28.0958	0.958060
$861$	−98.1258	−3.34412
$862$	−3.32693	−0.113316
$863$	−16.6474	−0.566684	−0.283342	−	0.959019i	$-0.591443\pi$
$863$	−16.6474	−0.566684	−0.283342	+	0.959019i	$0.591443\pi$
$864$	4.60003	0.156496
$865$	−34.1217	−1.16017
$866$	−26.8500	−0.912402
$867$	−73.3324	−2.49050
$868$	39.8712	1.35332
$869$	64.6543	2.19325
$870$	−51.4803	−1.74534
$871$	−1.03182	−0.0349620
$872$	−14.1039	−0.477618
$873$	−62.1772	−2.10438
$874$	49.3347	1.66877
$875$	−32.1787	−1.08784
$876$	−17.1742	−0.580264
$877$	8.64229	0.291829	0.145915	−	0.989297i	$-0.453388\pi$
$877$	8.64229	0.291829	0.145915	+	0.989297i	$0.453388\pi$
$878$	22.2487	0.750858
$879$	1.48794	0.0501870
$880$	14.1179	0.475915
$881$	39.7553	1.33939	0.669695	−	0.742637i	$-0.266425\pi$
$881$	39.7553	1.33939	0.669695	+	0.742637i	$0.266425\pi$
$882$	−61.0758	−2.05653
$883$	34.7200	1.16842	0.584210	−	0.811602i	$-0.301404\pi$
$883$	34.7200	1.16842	0.584210	+	0.811602i	$0.301404\pi$
$884$	39.8765	1.34119
$885$	45.3888	1.52573
$886$	−0.767745	−0.0257929
$887$	−30.0366	−1.00853	−0.504265	−	0.863549i	$-0.668237\pi$
$887$	−30.0366	−1.00853	−0.504265	+	0.863549i	$0.668237\pi$
$888$	0.0471448	0.00158207
$889$	35.0512	1.17558
$890$	−16.0429	−0.537760
$891$	6.52177	0.218488
$892$	9.17276	0.307127
$893$	74.3600	2.48836
$894$	−4.58028	−0.153187
$895$	−24.9757	−0.834846
$896$	4.48343	0.149781
$897$	105.192	3.51226
$898$	22.9421	0.765586
$899$	−60.9935	−2.03425
$900$	10.9703	0.365678
$901$	−19.0998	−0.636307
$902$	−41.1659	−1.37068
$903$	−128.582	−4.27895
$904$	9.96505	0.331433
$905$	−4.07072	−0.135315
$906$	24.7681	0.822864
$907$	−51.5106	−1.71038	−0.855190	−	0.518314i	$-0.826560\pi$
$907$	−51.5106	−1.71038	−0.855190	+	0.518314i	$0.826560\pi$
$908$	27.8952	0.925734
$909$	−29.0687	−0.964146
$910$	73.5118	2.43689
$911$	−23.2323	−0.769721	−0.384861	−	0.922975i	$-0.625751\pi$
$911$	−23.2323	−0.769721	−0.384861	+	0.922975i	$0.625751\pi$
$912$	21.7276	0.719472
$913$	−21.7377	−0.719413
$914$	−19.3281	−0.639316
$915$	92.6065	3.06148
$916$	7.84217	0.259112
$917$	13.1839	0.435372
$918$	−30.3368	−1.00126
$919$	43.8591	1.44678	0.723388	−	0.690441i	$-0.242584\pi$
$919$	43.8591	1.44678	0.723388	+	0.690441i	$0.242584\pi$
$920$	17.0430	0.561892
$921$	35.0905	1.15627
$922$	−32.6236	−1.07440
$923$	−43.1686	−1.42091
$924$	−64.6116	−2.12556
$925$	0.0400800	0.00131782
$926$	35.9761	1.18225
$927$	−0.113884	−0.00374045
$928$	−6.85860	−0.225144
$929$	5.20055	0.170624	0.0853122	−	0.996354i	$-0.472811\pi$
$929$	5.20055	0.170624	0.0853122	+	0.996354i	$0.472811\pi$
$930$	66.7504	2.18883
$931$	−102.838	−3.37039
$932$	−16.7757	−0.549505
$933$	53.2643	1.74380
$934$	24.1814	0.791238
$935$	−93.1065	−3.04491
$936$	28.1882	0.921359
$937$	1.56794	0.0512225	0.0256112	−	0.999672i	$-0.491847\pi$
$937$	1.56794	0.0512225	0.0256112	+	0.999672i	$0.491847\pi$
$938$	0.765083	0.0249808
$939$	−33.8267	−1.10389
$940$	25.6882	0.837856
$941$	37.6213	1.22642	0.613210	−	0.789920i	$-0.289878\pi$
$941$	37.6213	1.22642	0.613210	+	0.789920i	$0.289878\pi$
$942$	6.65184	0.216729
$943$	−49.6951	−1.61829
$944$	6.04705	0.196815
$945$	−55.9255	−1.81926
$946$	−53.9431	−1.75384
$947$	3.74693	0.121759	0.0608794	−	0.998145i	$-0.480609\pi$
$947$	3.74693	0.121759	0.0608794	+	0.998145i	$0.480609\pi$
$948$	34.3742	1.11642
$949$	−37.5162	−1.21783
$950$	18.4716	0.599299
$951$	56.5482	1.83370
$952$	−29.5678	−0.958299
$953$	35.9401	1.16421	0.582107	−	0.813112i	$-0.302229\pi$
$953$	35.9401	1.16421	0.582107	+	0.813112i	$0.302229\pi$
$954$	−13.5014	−0.437125
$955$	5.62434	0.181999
$956$	−5.82364	−0.188350
$957$	98.8405	3.19506
$958$	23.9134	0.772606
$959$	40.5560	1.30962
$960$	7.50595	0.242253
$961$	48.0854	1.55114
$962$	0.102985	0.00332037
$963$	−52.2679	−1.68431
$964$	−11.4606	−0.369120
$965$	22.0692	0.710434
$966$	−77.9984	−2.50956
$967$	18.1868	0.584849	0.292425	−	0.956289i	$-0.405538\pi$
$967$	18.1868	0.584849	0.292425	+	0.956289i	$0.405538\pi$
$968$	−16.1060	−0.517666
$969$	−143.291	−4.60319
$970$	−36.1668	−1.16125
$971$	10.0521	0.322587	0.161294	−	0.986906i	$-0.448433\pi$
$971$	10.0521	0.322587	0.161294	+	0.986906i	$0.448433\pi$
$972$	17.2675	0.553854
$973$	22.7513	0.729374
$974$	−1.56352	−0.0500984
$975$	39.3855	1.26134
$976$	12.3377	0.394922
$977$	−3.93141	−0.125777	−0.0628885	−	0.998021i	$-0.520031\pi$
$977$	−3.93141	−0.125777	−0.0628885	+	0.998021i	$0.520031\pi$
$978$	0.200709	0.00641796
$979$	30.8019	0.984432
$980$	−35.5262	−1.13484
$981$	65.7503	2.09925
$982$	35.6676	1.13820
$983$	42.5206	1.35620	0.678098	−	0.734971i	$-0.262804\pi$
$983$	42.5206	1.35620	0.678098	+	0.734971i	$0.262804\pi$
$984$	−21.8863	−0.697710
$985$	65.9341	2.10083
$986$	45.2318	1.44047
$987$	−117.564	−3.74209
$988$	47.4627	1.50999
$989$	−65.1196	−2.07068
$990$	−65.8158	−2.09176
$991$	−25.2704	−0.802740	−0.401370	−	0.915916i	$-0.631466\pi$
$991$	−25.2704	−0.802740	−0.401370	+	0.915916i	$0.631466\pi$
$992$	8.89300	0.282353
$993$	26.6577	0.845956
$994$	32.0089	1.01526
$995$	−1.22641	−0.0388799
$996$	−11.5571	−0.366200
$997$	39.4553	1.24956	0.624781	−	0.780800i	$-0.285188\pi$
$997$	39.4553	1.24956	0.624781	+	0.780800i	$0.285188\pi$
$998$	−7.43746	−0.235429
$999$	−0.0783478	−0.00247882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.12	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.76801 q^{3} +1.00000 q^{4} -2.71168 q^{5} +2.76801 q^{6} -4.48343 q^{7} -1.00000 q^{8} +4.66186 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.76801 q^{3} +1.00000 q^{4} -2.71168 q^{5} +2.76801 q^{6} -4.48343 q^{7} -1.00000 q^{8} +4.66186 q^{9} +2.71168 q^{10} -5.20634 q^{11} -2.76801 q^{12} -6.04655 q^{13} +4.48343 q^{14} +7.50595 q^{15} +1.00000 q^{16} -6.59491 q^{17} -4.66186 q^{18} -7.84954 q^{19} -2.71168 q^{20} +12.4102 q^{21} +5.20634 q^{22} +6.28504 q^{23} +2.76801 q^{24} +2.35321 q^{25} +6.04655 q^{26} -4.60003 q^{27} -4.48343 q^{28} +6.85860 q^{29} -7.50595 q^{30} -8.89300 q^{31} -1.00000 q^{32} +14.4112 q^{33} +6.59491 q^{34} +12.1576 q^{35} +4.66186 q^{36} +0.0170320 q^{37} +7.84954 q^{38} +16.7369 q^{39} +2.71168 q^{40} -7.90688 q^{41} -12.4102 q^{42} -10.3610 q^{43} -5.20634 q^{44} -12.6415 q^{45} -6.28504 q^{46} -9.47316 q^{47} -2.76801 q^{48} +13.1012 q^{49} -2.35321 q^{50} +18.2548 q^{51} -6.04655 q^{52} +2.89615 q^{53} +4.60003 q^{54} +14.1179 q^{55} +4.48343 q^{56} +21.7276 q^{57} -6.85860 q^{58} +6.04705 q^{59} +7.50595 q^{60} +12.3377 q^{61} +8.89300 q^{62} -20.9011 q^{63} +1.00000 q^{64} +16.3963 q^{65} -14.4112 q^{66} +0.170647 q^{67} -6.59491 q^{68} -17.3970 q^{69} -12.1576 q^{70} +7.13938 q^{71} -4.66186 q^{72} +6.20456 q^{73} -0.0170320 q^{74} -6.51371 q^{75} -7.84954 q^{76} +23.3423 q^{77} -16.7369 q^{78} -12.4184 q^{79} -2.71168 q^{80} -1.25266 q^{81} +7.90688 q^{82} +4.17524 q^{83} +12.4102 q^{84} +17.8833 q^{85} +10.3610 q^{86} -18.9846 q^{87} +5.20634 q^{88} -5.91623 q^{89} +12.6415 q^{90} +27.1093 q^{91} +6.28504 q^{92} +24.6159 q^{93} +9.47316 q^{94} +21.2855 q^{95} +2.76801 q^{96} -13.3374 q^{97} -13.1012 q^{98} -24.2712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 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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