LMFDB - Embedded newform 8038.2.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.88709 q^{3} +1.00000 q^{4} -0.769273 q^{5} +1.88709 q^{6} -1.40180 q^{7} -1.00000 q^{8} +0.561125 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.88709 q^{3} +1.00000 q^{4} -0.769273 q^{5} +1.88709 q^{6} -1.40180 q^{7} -1.00000 q^{8} +0.561125 q^{9} +0.769273 q^{10} -4.14341 q^{11} -1.88709 q^{12} -3.47439 q^{13} +1.40180 q^{14} +1.45169 q^{15} +1.00000 q^{16} -2.85618 q^{17} -0.561125 q^{18} +3.29153 q^{19} -0.769273 q^{20} +2.64533 q^{21} +4.14341 q^{22} +4.55928 q^{23} +1.88709 q^{24} -4.40822 q^{25} +3.47439 q^{26} +4.60239 q^{27} -1.40180 q^{28} +4.29529 q^{29} -1.45169 q^{30} -6.09531 q^{31} -1.00000 q^{32} +7.81901 q^{33} +2.85618 q^{34} +1.07837 q^{35} +0.561125 q^{36} +6.85771 q^{37} -3.29153 q^{38} +6.55651 q^{39} +0.769273 q^{40} -2.80459 q^{41} -2.64533 q^{42} -9.87958 q^{43} -4.14341 q^{44} -0.431658 q^{45} -4.55928 q^{46} -4.87416 q^{47} -1.88709 q^{48} -5.03495 q^{49} +4.40822 q^{50} +5.38989 q^{51} -3.47439 q^{52} -14.1023 q^{53} -4.60239 q^{54} +3.18742 q^{55} +1.40180 q^{56} -6.21142 q^{57} -4.29529 q^{58} +1.62988 q^{59} +1.45169 q^{60} -12.5278 q^{61} +6.09531 q^{62} -0.786585 q^{63} +1.00000 q^{64} +2.67276 q^{65} -7.81901 q^{66} -9.12061 q^{67} -2.85618 q^{68} -8.60379 q^{69} -1.07837 q^{70} -10.8181 q^{71} -0.561125 q^{72} +5.64152 q^{73} -6.85771 q^{74} +8.31872 q^{75} +3.29153 q^{76} +5.80824 q^{77} -6.55651 q^{78} -4.78407 q^{79} -0.769273 q^{80} -10.3685 q^{81} +2.80459 q^{82} -14.4061 q^{83} +2.64533 q^{84} +2.19719 q^{85} +9.87958 q^{86} -8.10562 q^{87} +4.14341 q^{88} -17.5670 q^{89} +0.431658 q^{90} +4.87041 q^{91} +4.55928 q^{92} +11.5024 q^{93} +4.87416 q^{94} -2.53208 q^{95} +1.88709 q^{96} +1.21158 q^{97} +5.03495 q^{98} -2.32497 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.88709	−1.08951	−0.544757	−	0.838594i	$-0.683378\pi$
$3$	−1.88709	−1.08951	−0.544757	+	0.838594i	$0.683378\pi$
$4$	1.00000	0.500000
$5$	−0.769273	−0.344029	−0.172015	−	0.985094i	$-0.555028\pi$
$5$	−0.769273	−0.344029	−0.172015	+	0.985094i	$0.555028\pi$
$6$	1.88709	0.770403
$7$	−1.40180	−0.529831	−0.264916	−	0.964272i	$-0.585344\pi$
$7$	−1.40180	−0.529831	−0.264916	+	0.964272i	$0.585344\pi$
$8$	−1.00000	−0.353553
$9$	0.561125	0.187042
$10$	0.769273	0.243266
$11$	−4.14341	−1.24929	−0.624643	−	0.780910i	$-0.714755\pi$
$11$	−4.14341	−1.24929	−0.624643	+	0.780910i	$0.714755\pi$
$12$	−1.88709	−0.544757
$13$	−3.47439	−0.963623	−0.481812	−	0.876275i	$-0.660021\pi$
$13$	−3.47439	−0.963623	−0.481812	+	0.876275i	$0.660021\pi$
$14$	1.40180	0.374647
$15$	1.45169	0.374825
$16$	1.00000	0.250000
$17$	−2.85618	−0.692726	−0.346363	−	0.938101i	$-0.612583\pi$
$17$	−2.85618	−0.692726	−0.346363	+	0.938101i	$0.612583\pi$
$18$	−0.561125	−0.132258
$19$	3.29153	0.755128	0.377564	−	0.925983i	$-0.376762\pi$
$19$	3.29153	0.755128	0.377564	+	0.925983i	$0.376762\pi$
$20$	−0.769273	−0.172015
$21$	2.64533	0.577258
$22$	4.14341	0.883379
$23$	4.55928	0.950675	0.475338	−	0.879803i	$-0.342326\pi$
$23$	4.55928	0.950675	0.475338	+	0.879803i	$0.342326\pi$
$24$	1.88709	0.385201
$25$	−4.40822	−0.881644
$26$	3.47439	0.681384
$27$	4.60239	0.885730
$28$	−1.40180	−0.264916
$29$	4.29529	0.797615	0.398808	−	0.917035i	$-0.369424\pi$
$29$	4.29529	0.797615	0.398808	+	0.917035i	$0.369424\pi$
$30$	−1.45169	−0.265041
$31$	−6.09531	−1.09475	−0.547375	−	0.836887i	$-0.684373\pi$
$31$	−6.09531	−1.09475	−0.547375	+	0.836887i	$0.684373\pi$
$32$	−1.00000	−0.176777
$33$	7.81901	1.36111
$34$	2.85618	0.489831
$35$	1.07837	0.182277
$36$	0.561125	0.0935208
$37$	6.85771	1.12740	0.563700	−	0.825979i	$-0.309377\pi$
$37$	6.85771	1.12740	0.563700	+	0.825979i	$0.309377\pi$
$38$	−3.29153	−0.533956
$39$	6.55651	1.04988
$40$	0.769273	0.121633
$41$	−2.80459	−0.438003	−0.219002	−	0.975724i	$-0.570280\pi$
$41$	−2.80459	−0.438003	−0.219002	+	0.975724i	$0.570280\pi$
$42$	−2.64533	−0.408183
$43$	−9.87958	−1.50662	−0.753311	−	0.657664i	$-0.771544\pi$
$43$	−9.87958	−1.50662	−0.753311	+	0.657664i	$0.771544\pi$
$44$	−4.14341	−0.624643
$45$	−0.431658	−0.0643478
$46$	−4.55928	−0.672229
$47$	−4.87416	−0.710969	−0.355484	−	0.934682i	$-0.615684\pi$
$47$	−4.87416	−0.710969	−0.355484	+	0.934682i	$0.615684\pi$
$48$	−1.88709	−0.272379
$49$	−5.03495	−0.719279
$50$	4.40822	0.623416
$51$	5.38989	0.754735
$52$	−3.47439	−0.481812
$53$	−14.1023	−1.93710	−0.968548	−	0.248827i	$-0.919955\pi$
$53$	−14.1023	−1.93710	−0.968548	+	0.248827i	$0.919955\pi$
$54$	−4.60239	−0.626306
$55$	3.18742	0.429791
$56$	1.40180	0.187324
$57$	−6.21142	−0.822723
$58$	−4.29529	−0.563999
$59$	1.62988	0.212193	0.106096	−	0.994356i	$-0.466165\pi$
$59$	1.62988	0.212193	0.106096	+	0.994356i	$0.466165\pi$
$60$	1.45169	0.187413
$61$	−12.5278	−1.60402	−0.802010	−	0.597311i	$-0.796236\pi$
$61$	−12.5278	−1.60402	−0.802010	+	0.597311i	$0.796236\pi$
$62$	6.09531	0.774105
$63$	−0.786585	−0.0991004
$64$	1.00000	0.125000
$65$	2.67276	0.331515
$66$	−7.81901	−0.962454
$67$	−9.12061	−1.11426	−0.557130	−	0.830425i	$-0.688098\pi$
$67$	−9.12061	−1.11426	−0.557130	+	0.830425i	$0.688098\pi$
$68$	−2.85618	−0.346363
$69$	−8.60379	−1.03577
$70$	−1.07837	−0.128890
$71$	−10.8181	−1.28387	−0.641933	−	0.766761i	$-0.721867\pi$
$71$	−10.8181	−1.28387	−0.641933	+	0.766761i	$0.721867\pi$
$72$	−0.561125	−0.0661292
$73$	5.64152	0.660291	0.330145	−	0.943930i	$-0.392902\pi$
$73$	5.64152	0.660291	0.330145	+	0.943930i	$0.392902\pi$
$74$	−6.85771	−0.797193
$75$	8.31872	0.960563
$76$	3.29153	0.377564
$77$	5.80824	0.661910
$78$	−6.55651	−0.742378
$79$	−4.78407	−0.538250	−0.269125	−	0.963105i	$-0.586734\pi$
$79$	−4.78407	−0.538250	−0.269125	+	0.963105i	$0.586734\pi$
$80$	−0.769273	−0.0860074
$81$	−10.3685	−1.15206
$82$	2.80459	0.309715
$83$	−14.4061	−1.58128	−0.790638	−	0.612284i	$-0.790251\pi$
$83$	−14.4061	−1.58128	−0.790638	+	0.612284i	$0.790251\pi$
$84$	2.64533	0.288629
$85$	2.19719	0.238318
$86$	9.87958	1.06534
$87$	−8.10562	−0.869013
$88$	4.14341	0.441689
$89$	−17.5670	−1.86210	−0.931049	−	0.364894i	$-0.881105\pi$
$89$	−17.5670	−1.86210	−0.931049	+	0.364894i	$0.881105\pi$
$90$	0.431658	0.0455008
$91$	4.87041	0.510557
$92$	4.55928	0.475338
$93$	11.5024	1.19275
$94$	4.87416	0.502731
$95$	−2.53208	−0.259786
$96$	1.88709	0.192601
$97$	1.21158	0.123017	0.0615085	−	0.998107i	$-0.480409\pi$
$97$	1.21158	0.123017	0.0615085	+	0.998107i	$0.480409\pi$
$98$	5.03495	0.508607
$99$	−2.32497	−0.233668
$100$	−4.40822	−0.440822
$101$	11.1373	1.10820	0.554101	−	0.832449i	$-0.313062\pi$
$101$	11.1373	1.10820	0.554101	+	0.832449i	$0.313062\pi$
$102$	−5.38989	−0.533678
$103$	16.2781	1.60393	0.801963	−	0.597373i	$-0.203789\pi$
$103$	16.2781	1.60393	0.801963	+	0.597373i	$0.203789\pi$
$104$	3.47439	0.340692
$105$	−2.03498	−0.198594
$106$	14.1023	1.36973
$107$	3.70336	0.358018	0.179009	−	0.983847i	$-0.442711\pi$
$107$	3.70336	0.358018	0.179009	+	0.983847i	$0.442711\pi$
$108$	4.60239	0.442865
$109$	−12.6011	−1.20697	−0.603485	−	0.797374i	$-0.706222\pi$
$109$	−12.6011	−1.20697	−0.603485	+	0.797374i	$0.706222\pi$
$110$	−3.18742	−0.303908
$111$	−12.9411	−1.22832
$112$	−1.40180	−0.132458
$113$	8.83720	0.831334	0.415667	−	0.909517i	$-0.363548\pi$
$113$	8.83720	0.831334	0.415667	+	0.909517i	$0.363548\pi$
$114$	6.21142	0.581753
$115$	−3.50733	−0.327060
$116$	4.29529	0.398808
$117$	−1.94957	−0.180238
$118$	−1.62988	−0.150043
$119$	4.00380	0.367028
$120$	−1.45169	−0.132521
$121$	6.16787	0.560715
$122$	12.5278	1.13421
$123$	5.29253	0.477211
$124$	−6.09531	−0.547375
$125$	7.23749	0.647341
$126$	0.786585	0.0700746
$127$	−18.7263	−1.66169	−0.830844	−	0.556506i	$-0.812142\pi$
$127$	−18.7263	−1.66169	−0.830844	+	0.556506i	$0.812142\pi$
$128$	−1.00000	−0.0883883
$129$	18.6437	1.64149
$130$	−2.67276	−0.234416
$131$	−6.92432	−0.604981	−0.302490	−	0.953152i	$-0.597818\pi$
$131$	−6.92432	−0.604981	−0.302490	+	0.953152i	$0.597818\pi$
$132$	7.81901	0.680557
$133$	−4.61407	−0.400090
$134$	9.12061	0.787901
$135$	−3.54049	−0.304717
$136$	2.85618	0.244916
$137$	11.7741	1.00593	0.502963	−	0.864308i	$-0.332243\pi$
$137$	11.7741	1.00593	0.502963	+	0.864308i	$0.332243\pi$
$138$	8.60379	0.732403
$139$	−15.1254	−1.28292	−0.641460	−	0.767157i	$-0.721671\pi$
$139$	−15.1254	−1.28292	−0.641460	+	0.767157i	$0.721671\pi$
$140$	1.07837	0.0911387
$141$	9.19800	0.774611
$142$	10.8181	0.907831
$143$	14.3958	1.20384
$144$	0.561125	0.0467604
$145$	−3.30425	−0.274403
$146$	−5.64152	−0.466896
$147$	9.50143	0.783665
$148$	6.85771	0.563700
$149$	−15.2303	−1.24772	−0.623859	−	0.781537i	$-0.714436\pi$
$149$	−15.2303	−1.24772	−0.623859	+	0.781537i	$0.714436\pi$
$150$	−8.31872	−0.679221
$151$	0.207109	0.0168543	0.00842714	−	0.999964i	$-0.497318\pi$
$151$	0.207109	0.0168543	0.00842714	+	0.999964i	$0.497318\pi$
$152$	−3.29153	−0.266978
$153$	−1.60267	−0.129569
$154$	−5.80824	−0.468041
$155$	4.68896	0.376626
$156$	6.55651	0.524941
$157$	3.89308	0.310702	0.155351	−	0.987859i	$-0.450349\pi$
$157$	3.89308	0.310702	0.155351	+	0.987859i	$0.450349\pi$
$158$	4.78407	0.380600
$159$	26.6123	2.11049
$160$	0.769273	0.0608164
$161$	−6.39120	−0.503697
$162$	10.3685	0.814627
$163$	10.8582	0.850477	0.425238	−	0.905081i	$-0.360190\pi$
$163$	10.8582	0.850477	0.425238	+	0.905081i	$0.360190\pi$
$164$	−2.80459	−0.219002
$165$	−6.01496	−0.468264
$166$	14.4061	1.11813
$167$	−15.9362	−1.23318	−0.616592	−	0.787283i	$-0.711487\pi$
$167$	−15.9362	−1.23318	−0.616592	+	0.787283i	$0.711487\pi$
$168$	−2.64533	−0.204092
$169$	−0.928597	−0.0714306
$170$	−2.19719	−0.168516
$171$	1.84696	0.141240
$172$	−9.87958	−0.753311
$173$	−22.6631	−1.72305	−0.861523	−	0.507719i	$-0.830489\pi$
$173$	−22.6631	−1.72305	−0.861523	+	0.507719i	$0.830489\pi$
$174$	8.10562	0.614485
$175$	6.17945	0.467122
$176$	−4.14341	−0.312321
$177$	−3.07575	−0.231187
$178$	17.5670	1.31670
$179$	6.85785	0.512580	0.256290	−	0.966600i	$-0.417500\pi$
$179$	6.85785	0.512580	0.256290	+	0.966600i	$0.417500\pi$
$180$	−0.431658	−0.0321739
$181$	9.19811	0.683690	0.341845	−	0.939756i	$-0.388948\pi$
$181$	9.19811	0.683690	0.341845	+	0.939756i	$0.388948\pi$
$182$	−4.87041	−0.361019
$183$	23.6411	1.74760
$184$	−4.55928	−0.336114
$185$	−5.27545	−0.387859
$186$	−11.5024	−0.843399
$187$	11.8343	0.865413
$188$	−4.87416	−0.355484
$189$	−6.45163	−0.469287
$190$	2.53208	0.183697
$191$	21.0811	1.52538	0.762688	−	0.646767i	$-0.223879\pi$
$191$	21.0811	1.52538	0.762688	+	0.646767i	$0.223879\pi$
$192$	−1.88709	−0.136189
$193$	−21.2491	−1.52954	−0.764772	−	0.644301i	$-0.777149\pi$
$193$	−21.2491	−1.52954	−0.764772	+	0.644301i	$0.777149\pi$
$194$	−1.21158	−0.0869862
$195$	−5.04374	−0.361190
$196$	−5.03495	−0.359640
$197$	11.8225	0.842318	0.421159	−	0.906987i	$-0.361623\pi$
$197$	11.8225	0.842318	0.421159	+	0.906987i	$0.361623\pi$
$198$	2.32497	0.165228
$199$	−25.3283	−1.79547	−0.897737	−	0.440531i	$-0.854790\pi$
$199$	−25.3283	−1.79547	−0.897737	+	0.440531i	$0.854790\pi$
$200$	4.40822	0.311708
$201$	17.2115	1.21400
$202$	−11.1373	−0.783618
$203$	−6.02114	−0.422601
$204$	5.38989	0.377368
$205$	2.15750	0.150686
$206$	−16.2781	−1.13415
$207$	2.55832	0.177816
$208$	−3.47439	−0.240906
$209$	−13.6382	−0.943371
$210$	2.03498	0.140427
$211$	−2.69895	−0.185803	−0.0929017	−	0.995675i	$-0.529614\pi$
$211$	−2.69895	−0.185803	−0.0929017	+	0.995675i	$0.529614\pi$
$212$	−14.1023	−0.968548
$213$	20.4147	1.39879
$214$	−3.70336	−0.253157
$215$	7.60010	0.518322
$216$	−4.60239	−0.313153
$217$	8.54441	0.580033
$218$	12.6011	0.853457
$219$	−10.6461	−0.719396
$220$	3.18742	0.214896
$221$	9.92350	0.667527
$222$	12.9411	0.868553
$223$	2.27359	0.152251	0.0761255	−	0.997098i	$-0.475745\pi$
$223$	2.27359	0.152251	0.0761255	+	0.997098i	$0.475745\pi$
$224$	1.40180	0.0936618
$225$	−2.47356	−0.164904
$226$	−8.83720	−0.587842
$227$	−12.7390	−0.845517	−0.422759	−	0.906242i	$-0.638938\pi$
$227$	−12.7390	−0.845517	−0.422759	+	0.906242i	$0.638938\pi$
$228$	−6.21142	−0.411362
$229$	9.63415	0.636642	0.318321	−	0.947983i	$-0.396881\pi$
$229$	9.63415	0.636642	0.318321	+	0.947983i	$0.396881\pi$
$230$	3.50733	0.231267
$231$	−10.9607	−0.721161
$232$	−4.29529	−0.282000
$233$	−20.4570	−1.34018	−0.670090	−	0.742280i	$-0.733744\pi$
$233$	−20.4570	−1.34018	−0.670090	+	0.742280i	$0.733744\pi$
$234$	1.94957	0.127447
$235$	3.74956	0.244594
$236$	1.62988	0.106096
$237$	9.02799	0.586431
$238$	−4.00380	−0.259528
$239$	8.07931	0.522607	0.261304	−	0.965257i	$-0.415848\pi$
$239$	8.07931	0.522607	0.261304	+	0.965257i	$0.415848\pi$
$240$	1.45169	0.0937063
$241$	17.0549	1.09861	0.549303	−	0.835623i	$-0.314893\pi$
$241$	17.0549	1.09861	0.549303	+	0.835623i	$0.314893\pi$
$242$	−6.16787	−0.396486
$243$	5.75920	0.369453
$244$	−12.5278	−0.802010
$245$	3.87326	0.247453
$246$	−5.29253	−0.337439
$247$	−11.4361	−0.727659
$248$	6.09531	0.387053
$249$	27.1857	1.72282
$250$	−7.23749	−0.457739
$251$	9.71528	0.613223	0.306612	−	0.951835i	$-0.400805\pi$
$251$	9.71528	0.613223	0.306612	+	0.951835i	$0.400805\pi$
$252$	−0.786585	−0.0495502
$253$	−18.8910	−1.18767
$254$	18.7263	1.17499
$255$	−4.14630	−0.259651
$256$	1.00000	0.0625000
$257$	26.6599	1.66300	0.831498	−	0.555527i	$-0.187484\pi$
$257$	26.6599	1.66300	0.831498	+	0.555527i	$0.187484\pi$
$258$	−18.6437	−1.16071
$259$	−9.61315	−0.597332
$260$	2.67276	0.165757
$261$	2.41019	0.149187
$262$	6.92432	0.427786
$263$	−5.50686	−0.339568	−0.169784	−	0.985481i	$-0.554307\pi$
$263$	−5.50686	−0.339568	−0.169784	+	0.985481i	$0.554307\pi$
$264$	−7.81901	−0.481227
$265$	10.8485	0.666418
$266$	4.61407	0.282907
$267$	33.1506	2.02878
$268$	−9.12061	−0.557130
$269$	−19.5044	−1.18921	−0.594604	−	0.804019i	$-0.702691\pi$
$269$	−19.5044	−1.18921	−0.594604	+	0.804019i	$0.702691\pi$
$270$	3.54049	0.215468
$271$	−23.7375	−1.44195	−0.720976	−	0.692960i	$-0.756306\pi$
$271$	−23.7375	−1.44195	−0.720976	+	0.692960i	$0.756306\pi$
$272$	−2.85618	−0.173182
$273$	−9.19092	−0.556260
$274$	−11.7741	−0.711298
$275$	18.2651	1.10143
$276$	−8.60379	−0.517887
$277$	−17.9574	−1.07896	−0.539478	−	0.842000i	$-0.681378\pi$
$277$	−17.9574	−1.07896	−0.539478	+	0.842000i	$0.681378\pi$
$278$	15.1254	0.907161
$279$	−3.42023	−0.204764
$280$	−1.07837	−0.0644448
$281$	20.2309	1.20688	0.603438	−	0.797410i	$-0.293797\pi$
$281$	20.2309	1.20688	0.603438	+	0.797410i	$0.293797\pi$
$282$	−9.19800	−0.547733
$283$	−25.7768	−1.53227	−0.766136	−	0.642678i	$-0.777823\pi$
$283$	−25.7768	−1.53227	−0.766136	+	0.642678i	$0.777823\pi$
$284$	−10.8181	−0.641933
$285$	4.77828	0.283041
$286$	−14.3958	−0.851244
$287$	3.93148	0.232068
$288$	−0.561125	−0.0330646
$289$	−8.84222	−0.520131
$290$	3.30425	0.194032
$291$	−2.28636	−0.134029
$292$	5.64152	0.330145
$293$	18.9229	1.10549	0.552743	−	0.833352i	$-0.313581\pi$
$293$	18.9229	1.10549	0.552743	+	0.833352i	$0.313581\pi$
$294$	−9.50143	−0.554135
$295$	−1.25383	−0.0730006
$296$	−6.85771	−0.398596
$297$	−19.0696	−1.10653
$298$	15.2303	0.882270
$299$	−15.8407	−0.916093
$300$	8.31872	0.480282
$301$	13.8492	0.798255
$302$	−0.207109	−0.0119178
$303$	−21.0171	−1.20740
$304$	3.29153	0.188782
$305$	9.63729	0.551830
$306$	1.60267	0.0916188
$307$	2.49091	0.142164	0.0710819	−	0.997470i	$-0.477355\pi$
$307$	2.49091	0.142164	0.0710819	+	0.997470i	$0.477355\pi$
$308$	5.80824	0.330955
$309$	−30.7183	−1.74750
$310$	−4.68896	−0.266315
$311$	−9.79834	−0.555613	−0.277807	−	0.960637i	$-0.589607\pi$
$311$	−9.79834	−0.555613	−0.277807	+	0.960637i	$0.589607\pi$
$312$	−6.55651	−0.371189
$313$	21.4807	1.21416	0.607080	−	0.794641i	$-0.292341\pi$
$313$	21.4807	1.21416	0.607080	+	0.794641i	$0.292341\pi$
$314$	−3.89308	−0.219699
$315$	0.605099	0.0340935
$316$	−4.78407	−0.269125
$317$	22.7436	1.27741	0.638703	−	0.769453i	$-0.279471\pi$
$317$	22.7436	1.27741	0.638703	+	0.769453i	$0.279471\pi$
$318$	−26.6123	−1.49234
$319$	−17.7972	−0.996449
$320$	−0.769273	−0.0430037
$321$	−6.98860	−0.390065
$322$	6.39120	0.356168
$323$	−9.40121	−0.523097
$324$	−10.3685	−0.576028
$325$	15.3159	0.849572
$326$	−10.8582	−0.601378
$327$	23.7795	1.31501
$328$	2.80459	0.154858
$329$	6.83260	0.376693
$330$	6.01496	0.331112
$331$	1.86500	0.102509	0.0512547	−	0.998686i	$-0.483678\pi$
$331$	1.86500	0.102509	0.0512547	+	0.998686i	$0.483678\pi$
$332$	−14.4061	−0.790638
$333$	3.84803	0.210871
$334$	15.9362	0.871993
$335$	7.01624	0.383338
$336$	2.64533	0.144315
$337$	17.3900	0.947292	0.473646	−	0.880715i	$-0.342938\pi$
$337$	17.3900	0.947292	0.473646	+	0.880715i	$0.342938\pi$
$338$	0.928597	0.0505090
$339$	−16.6766	−0.905751
$340$	2.19719	0.119159
$341$	25.2554	1.36766
$342$	−1.84696	−0.0998720
$343$	16.8706	0.910927
$344$	9.87958	0.532671
$345$	6.61866	0.356337
$346$	22.6631	1.21838
$347$	−9.47935	−0.508878	−0.254439	−	0.967089i	$-0.581891\pi$
$347$	−9.47935	−0.508878	−0.254439	+	0.967089i	$0.581891\pi$
$348$	−8.10562	−0.434507
$349$	−7.97497	−0.426891	−0.213445	−	0.976955i	$-0.568469\pi$
$349$	−7.97497	−0.426891	−0.213445	+	0.976955i	$0.568469\pi$
$350$	−6.17945	−0.330305
$351$	−15.9905	−0.853510
$352$	4.14341	0.220845
$353$	1.07605	0.0572722	0.0286361	−	0.999590i	$-0.490884\pi$
$353$	1.07605	0.0572722	0.0286361	+	0.999590i	$0.490884\pi$
$354$	3.07575	0.163474
$355$	8.32204	0.441688
$356$	−17.5670	−0.931049
$357$	−7.55555	−0.399882
$358$	−6.85785	−0.362449
$359$	13.2079	0.697088	0.348544	−	0.937292i	$-0.386676\pi$
$359$	13.2079	0.697088	0.348544	+	0.937292i	$0.386676\pi$
$360$	0.431658	0.0227504
$361$	−8.16585	−0.429781
$362$	−9.19811	−0.483442
$363$	−11.6393	−0.610907
$364$	4.87041	0.255279
$365$	−4.33987	−0.227159
$366$	−23.6411	−1.23574
$367$	28.7119	1.49875	0.749375	−	0.662146i	$-0.230354\pi$
$367$	28.7119	1.49875	0.749375	+	0.662146i	$0.230354\pi$
$368$	4.55928	0.237669
$369$	−1.57372	−0.0819248
$370$	5.27545	0.274258
$371$	19.7686	1.02633
$372$	11.5024	0.596373
$373$	−16.8455	−0.872225	−0.436112	−	0.899892i	$-0.643645\pi$
$373$	−16.8455	−0.872225	−0.436112	+	0.899892i	$0.643645\pi$
$374$	−11.8343	−0.611939
$375$	−13.6578	−0.705287
$376$	4.87416	0.251365
$377$	−14.9235	−0.768600
$378$	6.45163	0.331836
$379$	−29.9590	−1.53889	−0.769445	−	0.638713i	$-0.779467\pi$
$379$	−29.9590	−1.53889	−0.769445	+	0.638713i	$0.779467\pi$
$380$	−2.53208	−0.129893
$381$	35.3382	1.81043
$382$	−21.0811	−1.07860
$383$	6.77946	0.346414	0.173207	−	0.984885i	$-0.444587\pi$
$383$	6.77946	0.346414	0.173207	+	0.984885i	$0.444587\pi$
$384$	1.88709	0.0963004
$385$	−4.46812	−0.227717
$386$	21.2491	1.08155
$387$	−5.54368	−0.281801
$388$	1.21158	0.0615085
$389$	−3.16241	−0.160340	−0.0801702	−	0.996781i	$-0.525546\pi$
$389$	−3.16241	−0.160340	−0.0801702	+	0.996781i	$0.525546\pi$
$390$	5.04374	0.255400
$391$	−13.0221	−0.658558
$392$	5.03495	0.254304
$393$	13.0668	0.659135
$394$	−11.8225	−0.595609
$395$	3.68026	0.185174
$396$	−2.32497	−0.116834
$397$	24.7903	1.24419	0.622094	−	0.782942i	$-0.286282\pi$
$397$	24.7903	1.24419	0.622094	+	0.782942i	$0.286282\pi$
$398$	25.3283	1.26959
$399$	8.70718	0.435904
$400$	−4.40822	−0.220411
$401$	23.0616	1.15164	0.575822	−	0.817575i	$-0.304682\pi$
$401$	23.0616	1.15164	0.575822	+	0.817575i	$0.304682\pi$
$402$	−17.2115	−0.858429
$403$	21.1775	1.05493
$404$	11.1373	0.554101
$405$	7.97622	0.396342
$406$	6.02114	0.298824
$407$	−28.4143	−1.40845
$408$	−5.38989	−0.266839
$409$	−22.3217	−1.10374	−0.551869	−	0.833931i	$-0.686085\pi$
$409$	−22.3217	−1.10374	−0.551869	+	0.833931i	$0.686085\pi$
$410$	−2.15750	−0.106551
$411$	−22.2188	−1.09597
$412$	16.2781	0.801963
$413$	−2.28477	−0.112426
$414$	−2.55832	−0.125735
$415$	11.0822	0.544005
$416$	3.47439	0.170346
$417$	28.5430	1.39776
$418$	13.6382	0.667064
$419$	−4.74192	−0.231658	−0.115829	−	0.993269i	$-0.536952\pi$
$419$	−4.74192	−0.231658	−0.115829	+	0.993269i	$0.536952\pi$
$420$	−2.03498	−0.0992970
$421$	−4.62189	−0.225257	−0.112629	−	0.993637i	$-0.535927\pi$
$421$	−4.62189	−0.225257	−0.112629	+	0.993637i	$0.535927\pi$
$422$	2.69895	0.131383
$423$	−2.73501	−0.132981
$424$	14.1023	0.684867
$425$	12.5907	0.610738
$426$	−20.4147	−0.989094
$427$	17.5615	0.849859
$428$	3.70336	0.179009
$429$	−27.1663	−1.31160
$430$	−7.60010	−0.366509
$431$	−23.8811	−1.15031	−0.575156	−	0.818044i	$-0.695059\pi$
$431$	−23.8811	−1.15031	−0.575156	+	0.818044i	$0.695059\pi$
$432$	4.60239	0.221432
$433$	0.0562602	0.00270369	0.00135185	−	0.999999i	$-0.499570\pi$
$433$	0.0562602	0.00270369	0.00135185	+	0.999999i	$0.499570\pi$
$434$	−8.54441	−0.410145
$435$	6.23543	0.298966
$436$	−12.6011	−0.603485
$437$	15.0070	0.717882
$438$	10.6461	0.508690
$439$	20.8598	0.995582	0.497791	−	0.867297i	$-0.334145\pi$
$439$	20.8598	0.995582	0.497791	+	0.867297i	$0.334145\pi$
$440$	−3.18742	−0.151954
$441$	−2.82524	−0.134535
$442$	−9.92350	−0.472013
$443$	29.4220	1.39788	0.698940	−	0.715181i	$-0.253656\pi$
$443$	29.4220	1.39788	0.698940	+	0.715181i	$0.253656\pi$
$444$	−12.9411	−0.614160
$445$	13.5138	0.640617
$446$	−2.27359	−0.107658
$447$	28.7411	1.35941
$448$	−1.40180	−0.0662289
$449$	−20.4153	−0.963460	−0.481730	−	0.876320i	$-0.659991\pi$
$449$	−20.4153	−0.963460	−0.481730	+	0.876320i	$0.659991\pi$
$450$	2.47356	0.116605
$451$	11.6206	0.547192
$452$	8.83720	0.415667
$453$	−0.390834	−0.0183630
$454$	12.7390	0.597871
$455$	−3.74667	−0.175647
$456$	6.21142	0.290877
$457$	12.7398	0.595943	0.297972	−	0.954575i	$-0.403690\pi$
$457$	12.7398	0.595943	0.297972	+	0.954575i	$0.403690\pi$
$458$	−9.63415	−0.450174
$459$	−13.1453	−0.613568
$460$	−3.50733	−0.163530
$461$	0.852919	0.0397244	0.0198622	−	0.999803i	$-0.493677\pi$
$461$	0.852919	0.0397244	0.0198622	+	0.999803i	$0.493677\pi$
$462$	10.9607	0.509938
$463$	−18.5501	−0.862098	−0.431049	−	0.902329i	$-0.641856\pi$
$463$	−18.5501	−0.862098	−0.431049	+	0.902329i	$0.641856\pi$
$464$	4.29529	0.199404
$465$	−8.84851	−0.410340
$466$	20.4570	0.947650
$467$	−4.85397	−0.224615	−0.112308	−	0.993673i	$-0.535824\pi$
$467$	−4.85397	−0.224615	−0.112308	+	0.993673i	$0.535824\pi$
$468$	−1.94957	−0.0901188
$469$	12.7853	0.590370
$470$	−3.74956	−0.172954
$471$	−7.34661	−0.338514
$472$	−1.62988	−0.0750215
$473$	40.9352	1.88220
$474$	−9.02799	−0.414669
$475$	−14.5098	−0.665754
$476$	4.00380	0.183514
$477$	−7.91313	−0.362317
$478$	−8.07931	−0.369539
$479$	−21.6277	−0.988197	−0.494098	−	0.869406i	$-0.664502\pi$
$479$	−21.6277	−0.988197	−0.494098	+	0.869406i	$0.664502\pi$
$480$	−1.45169	−0.0662603
$481$	−23.8264	−1.08639
$482$	−17.0549	−0.776831
$483$	12.0608	0.548785
$484$	6.16787	0.280358
$485$	−0.932034	−0.0423215
$486$	−5.75920	−0.261242
$487$	−12.1615	−0.551090	−0.275545	−	0.961288i	$-0.588858\pi$
$487$	−12.1615	−0.551090	−0.275545	+	0.961288i	$0.588858\pi$
$488$	12.5278	0.567106
$489$	−20.4904	−0.926607
$490$	−3.87326	−0.174976
$491$	40.1455	1.81174	0.905871	−	0.423553i	$-0.139217\pi$
$491$	40.1455	1.81174	0.905871	+	0.423553i	$0.139217\pi$
$492$	5.29253	0.238606
$493$	−12.2681	−0.552529
$494$	11.4361	0.514533
$495$	1.78854	0.0803888
$496$	−6.09531	−0.273688
$497$	15.1648	0.680232
$498$	−27.1857	−1.21822
$499$	−9.63752	−0.431435	−0.215717	−	0.976456i	$-0.569209\pi$
$499$	−9.63752	−0.431435	−0.215717	+	0.976456i	$0.569209\pi$
$500$	7.23749	0.323670
$501$	30.0732	1.34357
$502$	−9.71528	−0.433614
$503$	1.24263	0.0554061	0.0277031	−	0.999616i	$-0.491181\pi$
$503$	1.24263	0.0554061	0.0277031	+	0.999616i	$0.491181\pi$
$504$	0.786585	0.0350373
$505$	−8.56763	−0.381254
$506$	18.8910	0.839806
$507$	1.75235	0.0778246
$508$	−18.7263	−0.830844
$509$	0.638107	0.0282836	0.0141418	−	0.999900i	$-0.495498\pi$
$509$	0.638107	0.0282836	0.0141418	+	0.999900i	$0.495498\pi$
$510$	4.14630	0.183601
$511$	−7.90830	−0.349842
$512$	−1.00000	−0.0441942
$513$	15.1489	0.668840
$514$	−26.6599	−1.17592
$515$	−12.5223	−0.551798
$516$	18.6437	0.820743
$517$	20.1956	0.888204
$518$	9.61315	0.422377
$519$	42.7675	1.87728
$520$	−2.67276	−0.117208
$521$	−22.7011	−0.994552	−0.497276	−	0.867592i	$-0.665666\pi$
$521$	−22.7011	−0.994552	−0.497276	+	0.867592i	$0.665666\pi$
$522$	−2.41019	−0.105491
$523$	−21.0386	−0.919954	−0.459977	−	0.887931i	$-0.652142\pi$
$523$	−21.0386	−0.919954	−0.459977	+	0.887931i	$0.652142\pi$
$524$	−6.92432	−0.302490
$525$	−11.6612	−0.508936
$526$	5.50686	0.240111
$527$	17.4093	0.758362
$528$	7.81901	0.340279
$529$	−2.21299	−0.0962167
$530$	−10.8485	−0.471229
$531$	0.914568	0.0396889
$532$	−4.61407	−0.200045
$533$	9.74425	0.422070
$534$	−33.1506	−1.43457
$535$	−2.84890	−0.123169
$536$	9.12061	0.393951
$537$	−12.9414	−0.558463
$538$	19.5044	0.840896
$539$	20.8619	0.898585
$540$	−3.54049	−0.152359
$541$	−15.2905	−0.657388	−0.328694	−	0.944436i	$-0.606609\pi$
$541$	−15.2905	−0.657388	−0.328694	+	0.944436i	$0.606609\pi$
$542$	23.7375	1.01961
$543$	−17.3577	−0.744890
$544$	2.85618	0.122458
$545$	9.69372	0.415233
$546$	9.19092	0.393335
$547$	20.9245	0.894668	0.447334	−	0.894367i	$-0.352373\pi$
$547$	20.9245	0.894668	0.447334	+	0.894367i	$0.352373\pi$
$548$	11.7741	0.502963
$549$	−7.02965	−0.300018
$550$	−18.2651	−0.778825
$551$	14.1381	0.602302
$552$	8.60379	0.366201
$553$	6.70632	0.285182
$554$	17.9574	0.762937
$555$	9.95528	0.422578
$556$	−15.1254	−0.641460
$557$	−4.98790	−0.211344	−0.105672	−	0.994401i	$-0.533699\pi$
$557$	−4.98790	−0.211344	−0.105672	+	0.994401i	$0.533699\pi$
$558$	3.42023	0.144790
$559$	34.3255	1.45182
$560$	1.07837	0.0455694
$561$	−22.3325	−0.942880
$562$	−20.2309	−0.853390
$563$	45.4850	1.91697	0.958483	−	0.285149i	$-0.0920432\pi$
$563$	45.4850	1.91697	0.958483	+	0.285149i	$0.0920432\pi$
$564$	9.19800	0.387305
$565$	−6.79822	−0.286003
$566$	25.7768	1.08348
$567$	14.5346	0.610396
$568$	10.8181	0.453915
$569$	30.5721	1.28165	0.640825	−	0.767687i	$-0.278592\pi$
$569$	30.5721	1.28165	0.640825	+	0.767687i	$0.278592\pi$
$570$	−4.77828	−0.200140
$571$	−27.0066	−1.13019	−0.565096	−	0.825025i	$-0.691161\pi$
$571$	−27.0066	−1.13019	−0.565096	+	0.825025i	$0.691161\pi$
$572$	14.3958	0.601920
$573$	−39.7821	−1.66192
$574$	−3.93148	−0.164097
$575$	−20.0983	−0.838157
$576$	0.561125	0.0233802
$577$	−36.1915	−1.50667	−0.753336	−	0.657636i	$-0.771557\pi$
$577$	−36.1915	−1.50667	−0.753336	+	0.657636i	$0.771557\pi$
$578$	8.84222	0.367788
$579$	40.0991	1.66646
$580$	−3.30425	−0.137202
$581$	20.1945	0.837809
$582$	2.28636	0.0947727
$583$	58.4315	2.41999
$584$	−5.64152	−0.233448
$585$	1.49975	0.0620070
$586$	−18.9229	−0.781696
$587$	−8.00648	−0.330463	−0.165231	−	0.986255i	$-0.552837\pi$
$587$	−8.00648	−0.330463	−0.165231	+	0.986255i	$0.552837\pi$
$588$	9.50143	0.391832
$589$	−20.0629	−0.826677
$590$	1.25383	0.0516192
$591$	−22.3102	−0.917717
$592$	6.85771	0.281850
$593$	29.8016	1.22380	0.611902	−	0.790934i	$-0.290405\pi$
$593$	29.8016	1.22380	0.611902	+	0.790934i	$0.290405\pi$
$594$	19.0696	0.782435
$595$	−3.08002	−0.126268
$596$	−15.2303	−0.623859
$597$	47.7969	1.95620
$598$	15.8407	0.647775
$599$	−7.16340	−0.292689	−0.146344	−	0.989234i	$-0.546751\pi$
$599$	−7.16340	−0.292689	−0.146344	+	0.989234i	$0.546751\pi$
$600$	−8.31872	−0.339610
$601$	−2.14259	−0.0873982	−0.0436991	−	0.999045i	$-0.513914\pi$
$601$	−2.14259	−0.0873982	−0.0436991	+	0.999045i	$0.513914\pi$
$602$	−13.8492	−0.564452
$603$	−5.11780	−0.208413
$604$	0.207109	0.00842714
$605$	−4.74478	−0.192903
$606$	21.0171	0.853763
$607$	−39.6482	−1.60927	−0.804636	−	0.593768i	$-0.797640\pi$
$607$	−39.6482	−1.60927	−0.804636	+	0.593768i	$0.797640\pi$
$608$	−3.29153	−0.133489
$609$	11.3625	0.460430
$610$	−9.63729	−0.390203
$611$	16.9347	0.685106
$612$	−1.60267	−0.0647843
$613$	4.27390	0.172621	0.0863107	−	0.996268i	$-0.472492\pi$
$613$	4.27390	0.172621	0.0863107	+	0.996268i	$0.472492\pi$
$614$	−2.49091	−0.100525
$615$	−4.07140	−0.164175
$616$	−5.80824	−0.234021
$617$	3.76038	0.151387	0.0756937	−	0.997131i	$-0.475883\pi$
$617$	3.76038	0.151387	0.0756937	+	0.997131i	$0.475883\pi$
$618$	30.7183	1.23567
$619$	2.02215	0.0812772	0.0406386	−	0.999174i	$-0.487061\pi$
$619$	2.02215	0.0812772	0.0406386	+	0.999174i	$0.487061\pi$
$620$	4.68896	0.188313
$621$	20.9836	0.842041
$622$	9.79834	0.392878
$623$	24.6254	0.986597
$624$	6.55651	0.262470
$625$	16.4735	0.658939
$626$	−21.4807	−0.858541
$627$	25.7365	1.02782
$628$	3.89308	0.155351
$629$	−19.5869	−0.780980
$630$	−0.605099	−0.0241077
$631$	−19.8956	−0.792033	−0.396017	−	0.918243i	$-0.629608\pi$
$631$	−19.8956	−0.792033	−0.396017	+	0.918243i	$0.629608\pi$
$632$	4.78407	0.190300
$633$	5.09317	0.202435
$634$	−22.7436	−0.903263
$635$	14.4056	0.571669
$636$	26.6123	1.05525
$637$	17.4934	0.693114
$638$	17.7972	0.704596
$639$	−6.07027	−0.240136
$640$	0.769273	0.0304082
$641$	38.1130	1.50537	0.752686	−	0.658380i	$-0.228758\pi$
$641$	38.1130	1.50537	0.752686	+	0.658380i	$0.228758\pi$
$642$	6.98860	0.275818
$643$	36.1666	1.42627	0.713135	−	0.701027i	$-0.247275\pi$
$643$	36.1666	1.42627	0.713135	+	0.701027i	$0.247275\pi$
$644$	−6.39120	−0.251849
$645$	−14.3421	−0.564720
$646$	9.40121	0.369885
$647$	2.04275	0.0803089	0.0401544	−	0.999193i	$-0.487215\pi$
$647$	2.04275	0.0803089	0.0401544	+	0.999193i	$0.487215\pi$
$648$	10.3685	0.407314
$649$	−6.75329	−0.265090
$650$	−15.3159	−0.600738
$651$	−16.1241	−0.631954
$652$	10.8582	0.425238
$653$	23.7979	0.931285	0.465642	−	0.884973i	$-0.345823\pi$
$653$	23.7979	0.931285	0.465642	+	0.884973i	$0.345823\pi$
$654$	−23.7795	−0.929853
$655$	5.32669	0.208131
$656$	−2.80459	−0.109501
$657$	3.16560	0.123502
$658$	−6.83260	−0.266362
$659$	−3.03883	−0.118376	−0.0591881	−	0.998247i	$-0.518851\pi$
$659$	−3.03883	−0.118376	−0.0591881	+	0.998247i	$0.518851\pi$
$660$	−6.01496	−0.234132
$661$	29.3397	1.14118	0.570591	−	0.821234i	$-0.306714\pi$
$661$	29.3397	1.14118	0.570591	+	0.821234i	$0.306714\pi$
$662$	−1.86500	−0.0724851
$663$	−18.7266	−0.727280
$664$	14.4061	0.559065
$665$	3.54948	0.137643
$666$	−3.84803	−0.149108
$667$	19.5834	0.758273
$668$	−15.9362	−0.616592
$669$	−4.29048	−0.165880
$670$	−7.01624	−0.271061
$671$	51.9078	2.00388
$672$	−2.64533	−0.102046
$673$	19.5086	0.751999	0.376000	−	0.926620i	$-0.377299\pi$
$673$	19.5086	0.751999	0.376000	+	0.926620i	$0.377299\pi$
$674$	−17.3900	−0.669837
$675$	−20.2883	−0.780898
$676$	−0.928597	−0.0357153
$677$	43.6910	1.67918	0.839591	−	0.543219i	$-0.182795\pi$
$677$	43.6910	1.67918	0.839591	+	0.543219i	$0.182795\pi$
$678$	16.6766	0.640462
$679$	−1.69839	−0.0651783
$680$	−2.19719	−0.0842582
$681$	24.0397	0.921203
$682$	−25.2554	−0.967079
$683$	−10.9309	−0.418260	−0.209130	−	0.977888i	$-0.567063\pi$
$683$	−10.9309	−0.418260	−0.209130	+	0.977888i	$0.567063\pi$
$684$	1.84696	0.0706202
$685$	−9.05748	−0.346068
$686$	−16.8706	−0.644123
$687$	−18.1805	−0.693631
$688$	−9.87958	−0.376656
$689$	48.9968	1.86663
$690$	−6.61866	−0.251968
$691$	−18.8905	−0.718627	−0.359313	−	0.933217i	$-0.616989\pi$
$691$	−18.8905	−0.718627	−0.359313	+	0.933217i	$0.616989\pi$
$692$	−22.6631	−0.861523
$693$	3.25915	0.123805
$694$	9.47935	0.359831
$695$	11.6356	0.441362
$696$	8.10562	0.307243
$697$	8.01042	0.303416
$698$	7.97497	0.301857
$699$	38.6042	1.46014
$700$	6.17945	0.233561
$701$	−10.1539	−0.383508	−0.191754	−	0.981443i	$-0.561418\pi$
$701$	−10.1539	−0.383508	−0.191754	+	0.981443i	$0.561418\pi$
$702$	15.9905	0.603523
$703$	22.5723	0.851332
$704$	−4.14341	−0.156161
$705$	−7.07577	−0.266489
$706$	−1.07605	−0.0404976
$707$	−15.6123	−0.587160
$708$	−3.07575	−0.115594
$709$	32.8867	1.23509	0.617543	−	0.786537i	$-0.288128\pi$
$709$	32.8867	1.23509	0.617543	+	0.786537i	$0.288128\pi$
$710$	−8.32204	−0.312320
$711$	−2.68446	−0.100675
$712$	17.5670	0.658351
$713$	−27.7902	−1.04075
$714$	7.55555	0.282759
$715$	−11.0743	−0.414157
$716$	6.85785	0.256290
$717$	−15.2464	−0.569388
$718$	−13.2079	−0.492916
$719$	−24.2802	−0.905500	−0.452750	−	0.891637i	$-0.649557\pi$
$719$	−24.2802	−0.905500	−0.452750	+	0.891637i	$0.649557\pi$
$720$	−0.431658	−0.0160869
$721$	−22.8186	−0.849810
$722$	8.16585	0.303901
$723$	−32.1843	−1.19695
$724$	9.19811	0.341845
$725$	−18.9346	−0.703212
$726$	11.6393	0.431977
$727$	−36.7415	−1.36267	−0.681334	−	0.731973i	$-0.738600\pi$
$727$	−36.7415	−1.36267	−0.681334	+	0.731973i	$0.738600\pi$
$728$	−4.87041	−0.180509
$729$	20.2374	0.749533
$730$	4.33987	0.160626
$731$	28.2179	1.04368
$732$	23.6411	0.873801
$733$	27.2123	1.00511	0.502554	−	0.864546i	$-0.332394\pi$
$733$	27.2123	1.00511	0.502554	+	0.864546i	$0.332394\pi$
$734$	−28.7119	−1.05978
$735$	−7.30920	−0.269604
$736$	−4.55928	−0.168057
$737$	37.7905	1.39203
$738$	1.57372	0.0579296
$739$	−2.00201	−0.0736450	−0.0368225	−	0.999322i	$-0.511724\pi$
$739$	−2.00201	−0.0736450	−0.0368225	+	0.999322i	$0.511724\pi$
$740$	−5.27545	−0.193930
$741$	21.5809	0.792795
$742$	−19.7686	−0.725727
$743$	−37.0867	−1.36058	−0.680290	−	0.732943i	$-0.738146\pi$
$743$	−37.0867	−1.36058	−0.680290	+	0.732943i	$0.738146\pi$
$744$	−11.5024	−0.421699
$745$	11.7163	0.429252
$746$	16.8455	0.616756
$747$	−8.08362	−0.295764
$748$	11.8343	0.432707
$749$	−5.19138	−0.189689
$750$	13.6578	0.498713
$751$	−42.1714	−1.53886	−0.769428	−	0.638734i	$-0.779459\pi$
$751$	−42.1714	−1.53886	−0.769428	+	0.638734i	$0.779459\pi$
$752$	−4.87416	−0.177742
$753$	−18.3337	−0.668116
$754$	14.9235	0.543483
$755$	−0.159323	−0.00579837
$756$	−6.45163	−0.234644
$757$	27.7829	1.00979	0.504894	−	0.863181i	$-0.331532\pi$
$757$	27.7829	1.00979	0.504894	+	0.863181i	$0.331532\pi$
$758$	29.9590	1.08816
$759$	35.6490	1.29398
$760$	2.53208	0.0918484
$761$	−14.9680	−0.542590	−0.271295	−	0.962496i	$-0.587452\pi$
$761$	−14.9680	−0.542590	−0.271295	+	0.962496i	$0.587452\pi$
$762$	−35.3382	−1.28017
$763$	17.6643	0.639490
$764$	21.0811	0.762688
$765$	1.23289	0.0445754
$766$	−6.77946	−0.244952
$767$	−5.66286	−0.204474
$768$	−1.88709	−0.0680946
$769$	−19.5581	−0.705284	−0.352642	−	0.935758i	$-0.614717\pi$
$769$	−19.5581	−0.705284	−0.352642	+	0.935758i	$0.614717\pi$
$770$	4.46812	0.161020
$771$	−50.3097	−1.81186
$772$	−21.2491	−0.764772
$773$	9.31828	0.335155	0.167578	−	0.985859i	$-0.446406\pi$
$773$	9.31828	0.335155	0.167578	+	0.985859i	$0.446406\pi$
$774$	5.54368	0.199263
$775$	26.8695	0.965180
$776$	−1.21158	−0.0434931
$777$	18.1409	0.650802
$778$	3.16241	0.113378
$779$	−9.23139	−0.330749
$780$	−5.04374	−0.180595
$781$	44.8236	1.60392
$782$	13.0221	0.465670
$783$	19.7686	0.706472
$784$	−5.03495	−0.179820
$785$	−2.99484	−0.106891
$786$	−13.0668	−0.466079
$787$	8.06599	0.287522	0.143761	−	0.989612i	$-0.454080\pi$
$787$	8.06599	0.287522	0.143761	+	0.989612i	$0.454080\pi$
$788$	11.8225	0.421159
$789$	10.3920	0.369964
$790$	−3.68026	−0.130938
$791$	−12.3880	−0.440467
$792$	2.32497	0.0826142
$793$	43.5265	1.54567
$794$	−24.7903	−0.879774
$795$	−20.4721	−0.726072
$796$	−25.3283	−0.897737
$797$	32.6889	1.15790	0.578950	−	0.815363i	$-0.303463\pi$
$797$	32.6889	1.15790	0.578950	+	0.815363i	$0.303463\pi$
$798$	−8.70718	−0.308231
$799$	13.9215	0.492507
$800$	4.40822	0.155854
$801$	−9.85727	−0.348290
$802$	−23.0616	−0.814335
$803$	−23.3752	−0.824892
$804$	17.2115	0.607001
$805$	4.91658	0.173287
$806$	−21.1775	−0.745946
$807$	36.8067	1.29566
$808$	−11.1373	−0.391809
$809$	27.1756	0.955445	0.477722	−	0.878511i	$-0.341462\pi$
$809$	27.1756	0.955445	0.477722	+	0.878511i	$0.341462\pi$
$810$	−7.97622	−0.280256
$811$	2.59964	0.0912858	0.0456429	−	0.998958i	$-0.485466\pi$
$811$	2.59964	0.0912858	0.0456429	+	0.998958i	$0.485466\pi$
$812$	−6.02114	−0.211301
$813$	44.7950	1.57103
$814$	28.4143	0.995922
$815$	−8.35289	−0.292589
$816$	5.38989	0.188684
$817$	−32.5189	−1.13769
$818$	22.3217	0.780460
$819$	2.73290	0.0954954
$820$	2.15750	0.0753431
$821$	−46.1280	−1.60988	−0.804939	−	0.593357i	$-0.797802\pi$
$821$	−46.1280	−1.60988	−0.804939	+	0.593357i	$0.797802\pi$
$822$	22.2188	0.774969
$823$	−47.3668	−1.65110	−0.825551	−	0.564327i	$-0.809136\pi$
$823$	−47.3668	−1.65110	−0.825551	+	0.564327i	$0.809136\pi$
$824$	−16.2781	−0.567074
$825$	−34.4679	−1.20002
$826$	2.28477	0.0794975
$827$	−37.5699	−1.30644	−0.653218	−	0.757170i	$-0.726581\pi$
$827$	−37.5699	−1.30644	−0.653218	+	0.757170i	$0.726581\pi$
$828$	2.55832	0.0889079
$829$	28.9472	1.00538	0.502689	−	0.864467i	$-0.332344\pi$
$829$	28.9472	1.00538	0.502689	+	0.864467i	$0.332344\pi$
$830$	−11.0822	−0.384670
$831$	33.8873	1.17554
$832$	−3.47439	−0.120453
$833$	14.3807	0.498263
$834$	−28.5430	−0.988365
$835$	12.2593	0.424252
$836$	−13.6382	−0.471686
$837$	−28.0530	−0.969653
$838$	4.74192	0.163807
$839$	−21.6577	−0.747706	−0.373853	−	0.927488i	$-0.621964\pi$
$839$	−21.6577	−0.747706	−0.373853	+	0.927488i	$0.621964\pi$
$840$	2.03498	0.0702136
$841$	−10.5505	−0.363810
$842$	4.62189	0.159281
$843$	−38.1777	−1.31491
$844$	−2.69895	−0.0929017
$845$	0.714345	0.0245742
$846$	2.73501	0.0940316
$847$	−8.64613	−0.297084
$848$	−14.1023	−0.484274
$849$	48.6433	1.66943
$850$	−12.5907	−0.431857
$851$	31.2662	1.07179
$852$	20.4147	0.699395
$853$	1.32932	0.0455152	0.0227576	−	0.999741i	$-0.492755\pi$
$853$	1.32932	0.0455152	0.0227576	+	0.999741i	$0.492755\pi$
$854$	−17.5615	−0.600941
$855$	−1.42081	−0.0485908
$856$	−3.70336	−0.126578
$857$	−2.86279	−0.0977909	−0.0488955	−	0.998804i	$-0.515570\pi$
$857$	−2.86279	−0.0977909	−0.0488955	+	0.998804i	$0.515570\pi$
$858$	27.1663	0.927443
$859$	5.02624	0.171493	0.0857465	−	0.996317i	$-0.472672\pi$
$859$	5.02624	0.171493	0.0857465	+	0.996317i	$0.472672\pi$
$860$	7.60010	0.259161
$861$	−7.41907	−0.252841
$862$	23.8811	0.813394
$863$	−38.0610	−1.29561	−0.647807	−	0.761805i	$-0.724314\pi$
$863$	−38.0610	−1.29561	−0.647807	+	0.761805i	$0.724314\pi$
$864$	−4.60239	−0.156576
$865$	17.4341	0.592779
$866$	−0.0562602	−0.00191180
$867$	16.6861	0.566690
$868$	8.54441	0.290016
$869$	19.8224	0.672428
$870$	−6.23543	−0.211401
$871$	31.6886	1.07373
$872$	12.6011	0.426728
$873$	0.679846	0.0230093
$874$	−15.0070	−0.507619
$875$	−10.1455	−0.342981
$876$	−10.6461	−0.359698
$877$	−53.7754	−1.81587	−0.907933	−	0.419116i	$-0.862340\pi$
$877$	−53.7754	−1.81587	−0.907933	+	0.419116i	$0.862340\pi$
$878$	−20.8598	−0.703983
$879$	−35.7092	−1.20444
$880$	3.18742	0.107448
$881$	−26.4482	−0.891063	−0.445531	−	0.895266i	$-0.646985\pi$
$881$	−26.4482	−0.891063	−0.445531	+	0.895266i	$0.646985\pi$
$882$	2.82524	0.0951306
$883$	−20.0777	−0.675668	−0.337834	−	0.941206i	$-0.609694\pi$
$883$	−20.0777	−0.675668	−0.337834	+	0.941206i	$0.609694\pi$
$884$	9.92350	0.333763
$885$	2.36609	0.0795352
$886$	−29.4220	−0.988450
$887$	21.1278	0.709401	0.354701	−	0.934980i	$-0.384583\pi$
$887$	21.1278	0.709401	0.354701	+	0.934980i	$0.384583\pi$
$888$	12.9411	0.434276
$889$	26.2505	0.880413
$890$	−13.5138	−0.452984
$891$	42.9610	1.43925
$892$	2.27359	0.0761255
$893$	−16.0434	−0.536873
$894$	−28.7411	−0.961246
$895$	−5.27556	−0.176342
$896$	1.40180	0.0468309
$897$	29.8929	0.998096
$898$	20.4153	0.681269
$899$	−26.1811	−0.873189
$900$	−2.47356	−0.0824520
$901$	40.2787	1.34188
$902$	−11.6206	−0.386923
$903$	−26.1348	−0.869710
$904$	−8.83720	−0.293921
$905$	−7.07586	−0.235210
$906$	0.390834	0.0129846
$907$	−9.44916	−0.313754	−0.156877	−	0.987618i	$-0.550143\pi$
$907$	−9.44916	−0.313754	−0.156877	+	0.987618i	$0.550143\pi$
$908$	−12.7390	−0.422759
$909$	6.24941	0.207280
$910$	3.74667	0.124201
$911$	−56.4215	−1.86933	−0.934663	−	0.355534i	$-0.884299\pi$
$911$	−56.4215	−1.86933	−0.934663	+	0.355534i	$0.884299\pi$
$912$	−6.21142	−0.205681
$913$	59.6904	1.97547
$914$	−12.7398	−0.421396
$915$	−18.1865	−0.601227
$916$	9.63415	0.318321
$917$	9.70652	0.320537
$918$	13.1453	0.433858
$919$	26.1977	0.864184	0.432092	−	0.901830i	$-0.357776\pi$
$919$	26.1977	0.864184	0.432092	+	0.901830i	$0.357776\pi$
$920$	3.50733	0.115633
$921$	−4.70059	−0.154890
$922$	−0.852919	−0.0280894
$923$	37.5862	1.23716
$924$	−10.9607	−0.360580
$925$	−30.2303	−0.993966
$926$	18.5501	0.609595
$927$	9.13403	0.300001
$928$	−4.29529	−0.141000
$929$	0.193858	0.00636028	0.00318014	−	0.999995i	$-0.498988\pi$
$929$	0.193858	0.00636028	0.00318014	+	0.999995i	$0.498988\pi$
$930$	8.84851	0.290154
$931$	−16.5727	−0.543148
$932$	−20.4570	−0.670090
$933$	18.4904	0.605348
$934$	4.85397	0.158827
$935$	−9.10385	−0.297728
$936$	1.94957	0.0637236
$937$	−40.1656	−1.31215	−0.656076	−	0.754695i	$-0.727785\pi$
$937$	−40.1656	−1.31215	−0.656076	+	0.754695i	$0.727785\pi$
$938$	−12.7853	−0.417454
$939$	−40.5361	−1.32284
$940$	3.74956	0.122297
$941$	−17.4025	−0.567307	−0.283653	−	0.958927i	$-0.591546\pi$
$941$	−17.4025	−0.567307	−0.283653	+	0.958927i	$0.591546\pi$
$942$	7.34661	0.239366
$943$	−12.7869	−0.416399
$944$	1.62988	0.0530482
$945$	4.96307	0.161449
$946$	−40.9352	−1.33092
$947$	36.2659	1.17848	0.589242	−	0.807957i	$-0.299427\pi$
$947$	36.2659	1.17848	0.589242	+	0.807957i	$0.299427\pi$
$948$	9.02799	0.293216
$949$	−19.6009	−0.636271
$950$	14.5098	0.470759
$951$	−42.9193	−1.39175
$952$	−4.00380	−0.129764
$953$	20.9190	0.677634	0.338817	−	0.940852i	$-0.389973\pi$
$953$	20.9190	0.677634	0.338817	+	0.940852i	$0.389973\pi$
$954$	7.91313	0.256197
$955$	−16.2171	−0.524774
$956$	8.07931	0.261304
$957$	33.5849	1.08565
$958$	21.6277	0.698761
$959$	−16.5049	−0.532971
$960$	1.45169	0.0468531
$961$	6.15281	0.198478
$962$	23.8264	0.768193
$963$	2.07805	0.0669642
$964$	17.0549	0.549303
$965$	16.3464	0.526208
$966$	−12.0608	−0.388050
$967$	−34.5872	−1.11225	−0.556124	−	0.831099i	$-0.687712\pi$
$967$	−34.5872	−1.11225	−0.556124	+	0.831099i	$0.687712\pi$
$968$	−6.16787	−0.198243
$969$	17.7410	0.569922
$970$	0.932034	0.0299258
$971$	7.79431	0.250131	0.125066	−	0.992148i	$-0.460086\pi$
$971$	7.79431	0.250131	0.125066	+	0.992148i	$0.460086\pi$
$972$	5.75920	0.184726
$973$	21.2028	0.679730
$974$	12.1615	0.389679
$975$	−28.9025	−0.925621
$976$	−12.5278	−0.401005
$977$	−5.37016	−0.171807	−0.0859033	−	0.996303i	$-0.527378\pi$
$977$	−5.37016	−0.171807	−0.0859033	+	0.996303i	$0.527378\pi$
$978$	20.4904	0.655210
$979$	72.7873	2.32629
$980$	3.87326	0.123727
$981$	−7.07081	−0.225753
$982$	−40.1455	−1.28110
$983$	−53.2043	−1.69695	−0.848476	−	0.529233i	$-0.822480\pi$
$983$	−53.2043	−1.69695	−0.848476	+	0.529233i	$0.822480\pi$
$984$	−5.29253	−0.168720
$985$	−9.09473	−0.289782
$986$	12.2681	0.390697
$987$	−12.8938	−0.410413
$988$	−11.4361	−0.363829
$989$	−45.0438	−1.43231
$990$	−1.78854	−0.0568435
$991$	−16.3473	−0.519288	−0.259644	−	0.965704i	$-0.583605\pi$
$991$	−16.3473	−0.519288	−0.259644	+	0.965704i	$0.583605\pi$
$992$	6.09531	0.193526
$993$	−3.51942	−0.111686
$994$	−15.1648	−0.480997
$995$	19.4844	0.617696
$996$	27.1857	0.861411
$997$	40.4321	1.28050	0.640249	−	0.768168i	$-0.278831\pi$
$997$	40.4321	1.28050	0.640249	+	0.768168i	$0.278831\pi$
$998$	9.63752	0.305071
$999$	31.5618	0.998573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.18	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.18	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.88709 q^{3} +1.00000 q^{4} -0.769273 q^{5} +1.88709 q^{6} -1.40180 q^{7} -1.00000 q^{8} +0.561125 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.88709 q^{3} +1.00000 q^{4} -0.769273 q^{5} +1.88709 q^{6} -1.40180 q^{7} -1.00000 q^{8} +0.561125 q^{9} +0.769273 q^{10} -4.14341 q^{11} -1.88709 q^{12} -3.47439 q^{13} +1.40180 q^{14} +1.45169 q^{15} +1.00000 q^{16} -2.85618 q^{17} -0.561125 q^{18} +3.29153 q^{19} -0.769273 q^{20} +2.64533 q^{21} +4.14341 q^{22} +4.55928 q^{23} +1.88709 q^{24} -4.40822 q^{25} +3.47439 q^{26} +4.60239 q^{27} -1.40180 q^{28} +4.29529 q^{29} -1.45169 q^{30} -6.09531 q^{31} -1.00000 q^{32} +7.81901 q^{33} +2.85618 q^{34} +1.07837 q^{35} +0.561125 q^{36} +6.85771 q^{37} -3.29153 q^{38} +6.55651 q^{39} +0.769273 q^{40} -2.80459 q^{41} -2.64533 q^{42} -9.87958 q^{43} -4.14341 q^{44} -0.431658 q^{45} -4.55928 q^{46} -4.87416 q^{47} -1.88709 q^{48} -5.03495 q^{49} +4.40822 q^{50} +5.38989 q^{51} -3.47439 q^{52} -14.1023 q^{53} -4.60239 q^{54} +3.18742 q^{55} +1.40180 q^{56} -6.21142 q^{57} -4.29529 q^{58} +1.62988 q^{59} +1.45169 q^{60} -12.5278 q^{61} +6.09531 q^{62} -0.786585 q^{63} +1.00000 q^{64} +2.67276 q^{65} -7.81901 q^{66} -9.12061 q^{67} -2.85618 q^{68} -8.60379 q^{69} -1.07837 q^{70} -10.8181 q^{71} -0.561125 q^{72} +5.64152 q^{73} -6.85771 q^{74} +8.31872 q^{75} +3.29153 q^{76} +5.80824 q^{77} -6.55651 q^{78} -4.78407 q^{79} -0.769273 q^{80} -10.3685 q^{81} +2.80459 q^{82} -14.4061 q^{83} +2.64533 q^{84} +2.19719 q^{85} +9.87958 q^{86} -8.10562 q^{87} +4.14341 q^{88} -17.5670 q^{89} +0.431658 q^{90} +4.87041 q^{91} +4.55928 q^{92} +11.5024 q^{93} +4.87416 q^{94} -2.53208 q^{95} +1.88709 q^{96} +1.21158 q^{97} +5.03495 q^{98} -2.32497 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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