LMFDB - Embedded newform 8038.2.a.d.1.1

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:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.24576 q^{3} +1.00000 q^{4} -1.41185 q^{5} -3.24576 q^{6} -1.89864 q^{7} +1.00000 q^{8} +7.53496 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.24576 q^{3} +1.00000 q^{4} -1.41185 q^{5} -3.24576 q^{6} -1.89864 q^{7} +1.00000 q^{8} +7.53496 q^{9} -1.41185 q^{10} -1.03451 q^{11} -3.24576 q^{12} -2.23657 q^{13} -1.89864 q^{14} +4.58252 q^{15} +1.00000 q^{16} +5.19837 q^{17} +7.53496 q^{18} +1.28227 q^{19} -1.41185 q^{20} +6.16254 q^{21} -1.03451 q^{22} +2.39435 q^{23} -3.24576 q^{24} -3.00668 q^{25} -2.23657 q^{26} -14.7194 q^{27} -1.89864 q^{28} -6.02635 q^{29} +4.58252 q^{30} +8.29120 q^{31} +1.00000 q^{32} +3.35778 q^{33} +5.19837 q^{34} +2.68060 q^{35} +7.53496 q^{36} +1.99941 q^{37} +1.28227 q^{38} +7.25938 q^{39} -1.41185 q^{40} -10.0706 q^{41} +6.16254 q^{42} -2.80092 q^{43} -1.03451 q^{44} -10.6382 q^{45} +2.39435 q^{46} +3.62461 q^{47} -3.24576 q^{48} -3.39516 q^{49} -3.00668 q^{50} -16.8727 q^{51} -2.23657 q^{52} -2.66057 q^{53} -14.7194 q^{54} +1.46058 q^{55} -1.89864 q^{56} -4.16195 q^{57} -6.02635 q^{58} +4.87034 q^{59} +4.58252 q^{60} +6.89395 q^{61} +8.29120 q^{62} -14.3062 q^{63} +1.00000 q^{64} +3.15771 q^{65} +3.35778 q^{66} -4.72057 q^{67} +5.19837 q^{68} -7.77150 q^{69} +2.68060 q^{70} +7.37877 q^{71} +7.53496 q^{72} -6.51693 q^{73} +1.99941 q^{74} +9.75896 q^{75} +1.28227 q^{76} +1.96417 q^{77} +7.25938 q^{78} -10.5735 q^{79} -1.41185 q^{80} +25.1707 q^{81} -10.0706 q^{82} -5.77316 q^{83} +6.16254 q^{84} -7.33932 q^{85} -2.80092 q^{86} +19.5601 q^{87} -1.03451 q^{88} -13.9370 q^{89} -10.6382 q^{90} +4.24645 q^{91} +2.39435 q^{92} -26.9112 q^{93} +3.62461 q^{94} -1.81038 q^{95} -3.24576 q^{96} +13.7129 q^{97} -3.39516 q^{98} -7.79500 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.24576	−1.87394	−0.936970	−	0.349409i	$-0.886382\pi$
$3$	−3.24576	−1.87394	−0.936970	+	0.349409i	$0.886382\pi$
$4$	1.00000	0.500000
$5$	−1.41185	−0.631398	−0.315699	−	0.948859i	$-0.602239\pi$
$5$	−1.41185	−0.631398	−0.315699	+	0.948859i	$0.602239\pi$
$6$	−3.24576	−1.32508
$7$	−1.89864	−0.717619	−0.358810	−	0.933411i	$-0.616817\pi$
$7$	−1.89864	−0.717619	−0.358810	+	0.933411i	$0.616817\pi$
$8$	1.00000	0.353553
$9$	7.53496	2.51165
$10$	−1.41185	−0.446466
$11$	−1.03451	−0.311917	−0.155959	−	0.987764i	$-0.549847\pi$
$11$	−1.03451	−0.311917	−0.155959	+	0.987764i	$0.549847\pi$
$12$	−3.24576	−0.936970
$13$	−2.23657	−0.620314	−0.310157	−	0.950685i	$-0.600382\pi$
$13$	−2.23657	−0.620314	−0.310157	+	0.950685i	$0.600382\pi$
$14$	−1.89864	−0.507433
$15$	4.58252	1.18320
$16$	1.00000	0.250000
$17$	5.19837	1.26079	0.630395	−	0.776274i	$-0.282893\pi$
$17$	5.19837	1.26079	0.630395	+	0.776274i	$0.282893\pi$
$18$	7.53496	1.77601
$19$	1.28227	0.294173	0.147087	−	0.989124i	$-0.453010\pi$
$19$	1.28227	0.294173	0.147087	+	0.989124i	$0.453010\pi$
$20$	−1.41185	−0.315699
$21$	6.16254	1.34478
$22$	−1.03451	−0.220559
$23$	2.39435	0.499257	0.249629	−	0.968342i	$-0.419691\pi$
$23$	2.39435	0.499257	0.249629	+	0.968342i	$0.419691\pi$
$24$	−3.24576	−0.662538
$25$	−3.00668	−0.601336
$26$	−2.23657	−0.438628
$27$	−14.7194	−2.83275
$28$	−1.89864	−0.358810
$29$	−6.02635	−1.11907	−0.559533	−	0.828808i	$-0.689019\pi$
$29$	−6.02635	−1.11907	−0.559533	+	0.828808i	$0.689019\pi$
$30$	4.58252	0.836651
$31$	8.29120	1.48914	0.744571	−	0.667543i	$-0.232654\pi$
$31$	8.29120	1.48914	0.744571	+	0.667543i	$0.232654\pi$
$32$	1.00000	0.176777
$33$	3.35778	0.584514
$34$	5.19837	0.891513
$35$	2.68060	0.453104
$36$	7.53496	1.25583
$37$	1.99941	0.328702	0.164351	−	0.986402i	$-0.447447\pi$
$37$	1.99941	0.328702	0.164351	+	0.986402i	$0.447447\pi$
$38$	1.28227	0.208012
$39$	7.25938	1.16243
$40$	−1.41185	−0.223233
$41$	−10.0706	−1.57277	−0.786383	−	0.617739i	$-0.788049\pi$
$41$	−10.0706	−1.57277	−0.786383	+	0.617739i	$0.788049\pi$
$42$	6.16254	0.950900
$43$	−2.80092	−0.427137	−0.213568	−	0.976928i	$-0.568509\pi$
$43$	−2.80092	−0.427137	−0.213568	+	0.976928i	$0.568509\pi$
$44$	−1.03451	−0.155959
$45$	−10.6382	−1.58585
$46$	2.39435	0.353028
$47$	3.62461	0.528704	0.264352	−	0.964426i	$-0.414842\pi$
$47$	3.62461	0.528704	0.264352	+	0.964426i	$0.414842\pi$
$48$	−3.24576	−0.468485
$49$	−3.39516	−0.485023
$50$	−3.00668	−0.425209
$51$	−16.8727	−2.36265
$52$	−2.23657	−0.310157
$53$	−2.66057	−0.365457	−0.182729	−	0.983163i	$-0.558493\pi$
$53$	−2.66057	−0.365457	−0.182729	+	0.983163i	$0.558493\pi$
$54$	−14.7194	−2.00305
$55$	1.46058	0.196944
$56$	−1.89864	−0.253717
$57$	−4.16195	−0.551263
$58$	−6.02635	−0.791299
$59$	4.87034	0.634064	0.317032	−	0.948415i	$-0.397314\pi$
$59$	4.87034	0.634064	0.317032	+	0.948415i	$0.397314\pi$
$60$	4.58252	0.591601
$61$	6.89395	0.882679	0.441340	−	0.897340i	$-0.354503\pi$
$61$	6.89395	0.882679	0.441340	+	0.897340i	$0.354503\pi$
$62$	8.29120	1.05298
$63$	−14.3062	−1.80241
$64$	1.00000	0.125000
$65$	3.15771	0.391665
$66$	3.35778	0.413314
$67$	−4.72057	−0.576709	−0.288354	−	0.957524i	$-0.593108\pi$
$67$	−4.72057	−0.576709	−0.288354	+	0.957524i	$0.593108\pi$
$68$	5.19837	0.630395
$69$	−7.77150	−0.935579
$70$	2.68060	0.320393
$71$	7.37877	0.875699	0.437850	−	0.899048i	$-0.355740\pi$
$71$	7.37877	0.875699	0.437850	+	0.899048i	$0.355740\pi$
$72$	7.53496	0.888003
$73$	−6.51693	−0.762750	−0.381375	−	0.924421i	$-0.624549\pi$
$73$	−6.51693	−0.762750	−0.381375	+	0.924421i	$0.624549\pi$
$74$	1.99941	0.232427
$75$	9.75896	1.12687
$76$	1.28227	0.147087
$77$	1.96417	0.223838
$78$	7.25938	0.821963
$79$	−10.5735	−1.18961	−0.594807	−	0.803868i	$-0.702772\pi$
$79$	−10.5735	−1.18961	−0.594807	+	0.803868i	$0.702772\pi$
$80$	−1.41185	−0.157850
$81$	25.1707	2.79675
$82$	−10.0706	−1.11211
$83$	−5.77316	−0.633687	−0.316844	−	0.948478i	$-0.602623\pi$
$83$	−5.77316	−0.633687	−0.316844	+	0.948478i	$0.602623\pi$
$84$	6.16254	0.672388
$85$	−7.33932	−0.796061
$86$	−2.80092	−0.302031
$87$	19.5601	2.09706
$88$	−1.03451	−0.110279
$89$	−13.9370	−1.47732	−0.738660	−	0.674078i	$-0.764541\pi$
$89$	−13.9370	−1.47732	−0.738660	+	0.674078i	$0.764541\pi$
$90$	−10.6382	−1.12137
$91$	4.24645	0.445149
$92$	2.39435	0.249629
$93$	−26.9112	−2.79057
$94$	3.62461	0.373850
$95$	−1.81038	−0.185741
$96$	−3.24576	−0.331269
$97$	13.7129	1.39233	0.696167	−	0.717880i	$-0.254887\pi$
$97$	13.7129	1.39233	0.696167	+	0.717880i	$0.254887\pi$
$98$	−3.39516	−0.342963
$99$	−7.79500	−0.783427
$100$	−3.00668	−0.300668
$101$	−1.57023	−0.156243	−0.0781216	−	0.996944i	$-0.524892\pi$
$101$	−1.57023	−0.156243	−0.0781216	+	0.996944i	$0.524892\pi$
$102$	−16.8727	−1.67064
$103$	3.41774	0.336760	0.168380	−	0.985722i	$-0.446146\pi$
$103$	3.41774	0.336760	0.168380	+	0.985722i	$0.446146\pi$
$104$	−2.23657	−0.219314
$105$	−8.70057	−0.849089
$106$	−2.66057	−0.258417
$107$	−5.29099	−0.511500	−0.255750	−	0.966743i	$-0.582322\pi$
$107$	−5.29099	−0.511500	−0.255750	+	0.966743i	$0.582322\pi$
$108$	−14.7194	−1.41637
$109$	−9.86287	−0.944692	−0.472346	−	0.881413i	$-0.656593\pi$
$109$	−9.86287	−0.944692	−0.472346	+	0.881413i	$0.656593\pi$
$110$	1.46058	0.139260
$111$	−6.48962	−0.615967
$112$	−1.89864	−0.179405
$113$	7.43457	0.699386	0.349693	−	0.936864i	$-0.386286\pi$
$113$	7.43457	0.699386	0.349693	+	0.936864i	$0.386286\pi$
$114$	−4.16195	−0.389802
$115$	−3.38047	−0.315230
$116$	−6.02635	−0.559533
$117$	−16.8525	−1.55801
$118$	4.87034	0.448351
$119$	−9.86984	−0.904767
$120$	4.58252	0.418325
$121$	−9.92978	−0.902708
$122$	6.89395	0.624148
$123$	32.6868	2.94727
$124$	8.29120	0.744571
$125$	11.3042	1.01108
$126$	−14.3062	−1.27450
$127$	−7.11279	−0.631158	−0.315579	−	0.948899i	$-0.602199\pi$
$127$	−7.11279	−0.631158	−0.315579	+	0.948899i	$0.602199\pi$
$128$	1.00000	0.0883883
$129$	9.09112	0.800429
$130$	3.15771	0.276949
$131$	−19.0311	−1.66275	−0.831376	−	0.555710i	$-0.812446\pi$
$131$	−19.0311	−1.66275	−0.831376	+	0.555710i	$0.812446\pi$
$132$	3.35778	0.292257
$133$	−2.43458	−0.211104
$134$	−4.72057	−0.407795
$135$	20.7816	1.78859
$136$	5.19837	0.445757
$137$	0.0766558	0.00654915	0.00327457	−	0.999995i	$-0.498958\pi$
$137$	0.0766558	0.00654915	0.00327457	+	0.999995i	$0.498958\pi$
$138$	−7.77150	−0.661554
$139$	11.2461	0.953880	0.476940	−	0.878936i	$-0.341746\pi$
$139$	11.2461	0.953880	0.476940	+	0.878936i	$0.341746\pi$
$140$	2.68060	0.226552
$141$	−11.7646	−0.990760
$142$	7.37877	0.619213
$143$	2.31376	0.193487
$144$	7.53496	0.627913
$145$	8.50831	0.706576
$146$	−6.51693	−0.539345
$147$	11.0199	0.908904
$148$	1.99941	0.164351
$149$	−1.87745	−0.153807	−0.0769035	−	0.997039i	$-0.524503\pi$
$149$	−1.87745	−0.153807	−0.0769035	+	0.997039i	$0.524503\pi$
$150$	9.75896	0.796816
$151$	9.80607	0.798007	0.399003	−	0.916949i	$-0.369356\pi$
$151$	9.80607	0.798007	0.399003	+	0.916949i	$0.369356\pi$
$152$	1.28227	0.104006
$153$	39.1695	3.16667
$154$	1.96417	0.158277
$155$	−11.7059	−0.940242
$156$	7.25938	0.581216
$157$	−3.26950	−0.260934	−0.130467	−	0.991453i	$-0.541648\pi$
$157$	−3.26950	−0.260934	−0.130467	+	0.991453i	$0.541648\pi$
$158$	−10.5735	−0.841185
$159$	8.63556	0.684845
$160$	−1.41185	−0.111617
$161$	−4.54602	−0.358277
$162$	25.1707	1.97760
$163$	19.4225	1.52129	0.760643	−	0.649170i	$-0.224884\pi$
$163$	19.4225	1.52129	0.760643	+	0.649170i	$0.224884\pi$
$164$	−10.0706	−0.786383
$165$	−4.74068	−0.369061
$166$	−5.77316	−0.448084
$167$	15.3098	1.18471	0.592353	−	0.805679i	$-0.298199\pi$
$167$	15.3098	1.18471	0.592353	+	0.805679i	$0.298199\pi$
$168$	6.16254	0.475450
$169$	−7.99774	−0.615211
$170$	−7.33932	−0.562900
$171$	9.66187	0.738861
$172$	−2.80092	−0.213568
$173$	21.8620	1.66213	0.831067	−	0.556172i	$-0.187730\pi$
$173$	21.8620	1.66213	0.831067	+	0.556172i	$0.187730\pi$
$174$	19.5601	1.48285
$175$	5.70861	0.431530
$176$	−1.03451	−0.0779793
$177$	−15.8080	−1.18820
$178$	−13.9370	−1.04462
$179$	10.8154	0.808384	0.404192	−	0.914674i	$-0.367553\pi$
$179$	10.8154	0.808384	0.404192	+	0.914674i	$0.367553\pi$
$180$	−10.6382	−0.792927
$181$	−7.86366	−0.584501	−0.292251	−	0.956342i	$-0.594404\pi$
$181$	−7.86366	−0.584501	−0.292251	+	0.956342i	$0.594404\pi$
$182$	4.24645	0.314768
$183$	−22.3761	−1.65409
$184$	2.39435	0.176514
$185$	−2.82287	−0.207542
$186$	−26.9112	−1.97323
$187$	−5.37778	−0.393262
$188$	3.62461	0.264352
$189$	27.9468	2.03283
$190$	−1.81038	−0.131338
$191$	4.30563	0.311545	0.155772	−	0.987793i	$-0.450213\pi$
$191$	4.30563	0.311545	0.155772	+	0.987793i	$0.450213\pi$
$192$	−3.24576	−0.234243
$193$	3.25746	0.234477	0.117239	−	0.993104i	$-0.462596\pi$
$193$	3.25746	0.234477	0.117239	+	0.993104i	$0.462596\pi$
$194$	13.7129	0.984528
$195$	−10.2492	−0.733957
$196$	−3.39516	−0.242511
$197$	−4.49809	−0.320476	−0.160238	−	0.987078i	$-0.551226\pi$
$197$	−4.49809	−0.320476	−0.160238	+	0.987078i	$0.551226\pi$
$198$	−7.79500	−0.553967
$199$	0.213906	0.0151634	0.00758170	−	0.999971i	$-0.497587\pi$
$199$	0.213906	0.0151634	0.00758170	+	0.999971i	$0.497587\pi$
$200$	−3.00668	−0.212604
$201$	15.3218	1.08072
$202$	−1.57023	−0.110481
$203$	11.4419	0.803063
$204$	−16.8727	−1.18132
$205$	14.2182	0.993042
$206$	3.41774	0.238125
$207$	18.0414	1.25396
$208$	−2.23657	−0.155078
$209$	−1.32653	−0.0917577
$210$	−8.70057	−0.600397
$211$	0.926507	0.0637834	0.0318917	−	0.999491i	$-0.489847\pi$
$211$	0.926507	0.0637834	0.0318917	+	0.999491i	$0.489847\pi$
$212$	−2.66057	−0.182729
$213$	−23.9497	−1.64101
$214$	−5.29099	−0.361685
$215$	3.95448	0.269693
$216$	−14.7194	−1.00153
$217$	−15.7420	−1.06864
$218$	−9.86287	−0.667998
$219$	21.1524	1.42935
$220$	1.46058	0.0984720
$221$	−11.6265	−0.782086
$222$	−6.48962	−0.435555
$223$	0.415483	0.0278228	0.0139114	−	0.999903i	$-0.495572\pi$
$223$	0.415483	0.0278228	0.0139114	+	0.999903i	$0.495572\pi$
$224$	−1.89864	−0.126858
$225$	−22.6552	−1.51035
$226$	7.43457	0.494540
$227$	17.1043	1.13525	0.567625	−	0.823287i	$-0.307862\pi$
$227$	17.1043	1.13525	0.567625	+	0.823287i	$0.307862\pi$
$228$	−4.16195	−0.275632
$229$	−6.46452	−0.427187	−0.213594	−	0.976923i	$-0.568517\pi$
$229$	−6.46452	−0.427187	−0.213594	+	0.976923i	$0.568517\pi$
$230$	−3.38047	−0.222901
$231$	−6.37522	−0.419458
$232$	−6.02635	−0.395650
$233$	−2.88549	−0.189035	−0.0945173	−	0.995523i	$-0.530131\pi$
$233$	−2.88549	−0.189035	−0.0945173	+	0.995523i	$0.530131\pi$
$234$	−16.8525	−1.10168
$235$	−5.11741	−0.333823
$236$	4.87034	0.317032
$237$	34.3191	2.22927
$238$	−9.86984	−0.639767
$239$	2.83778	0.183561	0.0917804	−	0.995779i	$-0.470744\pi$
$239$	2.83778	0.183561	0.0917804	+	0.995779i	$0.470744\pi$
$240$	4.58252	0.295801
$241$	5.23672	0.337327	0.168663	−	0.985674i	$-0.446055\pi$
$241$	5.23672	0.337327	0.168663	+	0.985674i	$0.446055\pi$
$242$	−9.92978	−0.638311
$243$	−37.5399	−2.40819
$244$	6.89395	0.441340
$245$	4.79346	0.306243
$246$	32.6868	2.08404
$247$	−2.86790	−0.182480
$248$	8.29120	0.526492
$249$	18.7383	1.18749
$250$	11.3042	0.714942
$251$	−9.57568	−0.604411	−0.302206	−	0.953243i	$-0.597723\pi$
$251$	−9.57568	−0.604411	−0.302206	+	0.953243i	$0.597723\pi$
$252$	−14.3062	−0.901205
$253$	−2.47699	−0.155727
$254$	−7.11279	−0.446296
$255$	23.8217	1.49177
$256$	1.00000	0.0625000
$257$	−2.23721	−0.139553	−0.0697766	−	0.997563i	$-0.522229\pi$
$257$	−2.23721	−0.139553	−0.0697766	+	0.997563i	$0.522229\pi$
$258$	9.09112	0.565988
$259$	−3.79617	−0.235883
$260$	3.15771	0.195833
$261$	−45.4083	−2.81070
$262$	−19.0311	−1.17574
$263$	27.4302	1.69142	0.845710	−	0.533642i	$-0.179177\pi$
$263$	27.4302	1.69142	0.845710	+	0.533642i	$0.179177\pi$
$264$	3.35778	0.206657
$265$	3.75632	0.230749
$266$	−2.43458	−0.149273
$267$	45.2362	2.76841
$268$	−4.72057	−0.288354
$269$	15.7403	0.959706	0.479853	−	0.877349i	$-0.340690\pi$
$269$	15.7403	0.959706	0.479853	+	0.877349i	$0.340690\pi$
$270$	20.7816	1.26473
$271$	14.0189	0.851586	0.425793	−	0.904821i	$-0.359995\pi$
$271$	14.0189	0.851586	0.425793	+	0.904821i	$0.359995\pi$
$272$	5.19837	0.315198
$273$	−13.7830	−0.834183
$274$	0.0766558	0.00463095
$275$	3.11045	0.187567
$276$	−7.77150	−0.467789
$277$	−8.37428	−0.503162	−0.251581	−	0.967836i	$-0.580950\pi$
$277$	−8.37428	−0.503162	−0.251581	+	0.967836i	$0.580950\pi$
$278$	11.2461	0.674495
$279$	62.4738	3.74021
$280$	2.68060	0.160196
$281$	13.8304	0.825053	0.412526	−	0.910946i	$-0.364646\pi$
$281$	13.8304	0.825053	0.412526	+	0.910946i	$0.364646\pi$
$282$	−11.7646	−0.700573
$283$	14.5876	0.867144	0.433572	−	0.901119i	$-0.357253\pi$
$283$	14.5876	0.867144	0.433572	+	0.901119i	$0.357253\pi$
$284$	7.37877	0.437850
$285$	5.87604	0.348067
$286$	2.31376	0.136816
$287$	19.1205	1.12865
$288$	7.53496	0.444002
$289$	10.0231	0.589592
$290$	8.50831	0.499625
$291$	−44.5088	−2.60915
$292$	−6.51693	−0.381375
$293$	−16.9759	−0.991745	−0.495873	−	0.868395i	$-0.665152\pi$
$293$	−16.9759	−0.991745	−0.495873	+	0.868395i	$0.665152\pi$
$294$	11.0199	0.642692
$295$	−6.87619	−0.400347
$296$	1.99941	0.116214
$297$	15.2274	0.883582
$298$	−1.87745	−0.108758
$299$	−5.35515	−0.309696
$300$	9.75896	0.563434
$301$	5.31795	0.306521
$302$	9.80607	0.564276
$303$	5.09657	0.292791
$304$	1.28227	0.0735434
$305$	−9.73321	−0.557322
$306$	39.1695	2.23917
$307$	25.8754	1.47679	0.738393	−	0.674371i	$-0.235585\pi$
$307$	25.8754	1.47679	0.738393	+	0.674371i	$0.235585\pi$
$308$	1.96417	0.111919
$309$	−11.0932	−0.631069
$310$	−11.7059	−0.664852
$311$	24.4932	1.38888	0.694442	−	0.719549i	$-0.255651\pi$
$311$	24.4932	1.38888	0.694442	+	0.719549i	$0.255651\pi$
$312$	7.25938	0.410982
$313$	13.1672	0.744254	0.372127	−	0.928182i	$-0.378629\pi$
$313$	13.1672	0.744254	0.372127	+	0.928182i	$0.378629\pi$
$314$	−3.26950	−0.184509
$315$	20.1982	1.13804
$316$	−10.5735	−0.594807
$317$	−0.189914	−0.0106667	−0.00533333	−	0.999986i	$-0.501698\pi$
$317$	−0.189914	−0.0106667	−0.00533333	+	0.999986i	$0.501698\pi$
$318$	8.63556	0.484258
$319$	6.23434	0.349056
$320$	−1.41185	−0.0789248
$321$	17.1733	0.958520
$322$	−4.54602	−0.253340
$323$	6.66573	0.370891
$324$	25.1707	1.39837
$325$	6.72466	0.373017
$326$	19.4225	1.07571
$327$	32.0125	1.77030
$328$	−10.0706	−0.556057
$329$	−6.88184	−0.379408
$330$	−4.74068	−0.260966
$331$	−19.0774	−1.04859	−0.524295	−	0.851536i	$-0.675671\pi$
$331$	−19.0774	−1.04859	−0.524295	+	0.851536i	$0.675671\pi$
$332$	−5.77316	−0.316844
$333$	15.0655	0.825584
$334$	15.3098	0.837714
$335$	6.66473	0.364133
$336$	6.16254	0.336194
$337$	22.2556	1.21234	0.606171	−	0.795335i	$-0.292705\pi$
$337$	22.2556	1.21234	0.606171	+	0.795335i	$0.292705\pi$
$338$	−7.99774	−0.435020
$339$	−24.1308	−1.31061
$340$	−7.33932	−0.398030
$341$	−8.57734	−0.464489
$342$	9.66187	0.522454
$343$	19.7367	1.06568
$344$	−2.80092	−0.151016
$345$	10.9722	0.590723
$346$	21.8620	1.17531
$347$	−24.3315	−1.30618	−0.653092	−	0.757278i	$-0.726529\pi$
$347$	−24.3315	−1.30618	−0.653092	+	0.757278i	$0.726529\pi$
$348$	19.5601	1.04853
$349$	8.94988	0.479076	0.239538	−	0.970887i	$-0.423004\pi$
$349$	8.94988	0.479076	0.239538	+	0.970887i	$0.423004\pi$
$350$	5.70861	0.305138
$351$	32.9210	1.75719
$352$	−1.03451	−0.0551397
$353$	30.2617	1.61067	0.805333	−	0.592822i	$-0.201986\pi$
$353$	30.2617	1.61067	0.805333	+	0.592822i	$0.201986\pi$
$354$	−15.8080	−0.840183
$355$	−10.4177	−0.552915
$356$	−13.9370	−0.738660
$357$	32.0351	1.69548
$358$	10.8154	0.571614
$359$	−12.6456	−0.667408	−0.333704	−	0.942678i	$-0.608299\pi$
$359$	−12.6456	−0.667408	−0.333704	+	0.942678i	$0.608299\pi$
$360$	−10.6382	−0.560684
$361$	−17.3558	−0.913462
$362$	−7.86366	−0.413305
$363$	32.2297	1.69162
$364$	4.24645	0.222575
$365$	9.20093	0.481599
$366$	−22.3761	−1.16962
$367$	15.5176	0.810012	0.405006	−	0.914314i	$-0.367269\pi$
$367$	15.5176	0.810012	0.405006	+	0.914314i	$0.367269\pi$
$368$	2.39435	0.124814
$369$	−75.8817	−3.95024
$370$	−2.82287	−0.146754
$371$	5.05146	0.262259
$372$	−26.9112	−1.39528
$373$	−17.1549	−0.888246	−0.444123	−	0.895966i	$-0.646485\pi$
$373$	−17.1549	−0.888246	−0.444123	+	0.895966i	$0.646485\pi$
$374$	−5.37778	−0.278078
$375$	−36.6908	−1.89471
$376$	3.62461	0.186925
$377$	13.4784	0.694172
$378$	27.9468	1.43743
$379$	−13.3227	−0.684340	−0.342170	−	0.939638i	$-0.611162\pi$
$379$	−13.3227	−0.684340	−0.342170	+	0.939638i	$0.611162\pi$
$380$	−1.81038	−0.0928703
$381$	23.0864	1.18275
$382$	4.30563	0.220295
$383$	3.67169	0.187614	0.0938072	−	0.995590i	$-0.470096\pi$
$383$	3.67169	0.187614	0.0938072	+	0.995590i	$0.470096\pi$
$384$	−3.24576	−0.165634
$385$	−2.77311	−0.141331
$386$	3.25746	0.165800
$387$	−21.1048	−1.07282
$388$	13.7129	0.696167
$389$	−30.0064	−1.52138	−0.760691	−	0.649114i	$-0.775140\pi$
$389$	−30.0064	−1.52138	−0.760691	+	0.649114i	$0.775140\pi$
$390$	−10.2492	−0.518986
$391$	12.4467	0.629459
$392$	−3.39516	−0.171481
$393$	61.7703	3.11590
$394$	−4.49809	−0.226611
$395$	14.9282	0.751121
$396$	−7.79500	−0.391714
$397$	8.38078	0.420619	0.210310	−	0.977635i	$-0.432553\pi$
$397$	8.38078	0.420619	0.210310	+	0.977635i	$0.432553\pi$
$398$	0.213906	0.0107221
$399$	7.90205	0.395597
$400$	−3.00668	−0.150334
$401$	−9.52957	−0.475884	−0.237942	−	0.971279i	$-0.576473\pi$
$401$	−9.52957	−0.475884	−0.237942	+	0.971279i	$0.576473\pi$
$402$	15.3218	0.764183
$403$	−18.5439	−0.923736
$404$	−1.57023	−0.0781216
$405$	−35.5373	−1.76586
$406$	11.4419	0.567851
$407$	−2.06842	−0.102528
$408$	−16.8727	−0.835321
$409$	20.5932	1.01827	0.509133	−	0.860688i	$-0.329966\pi$
$409$	20.5932	1.01827	0.509133	+	0.860688i	$0.329966\pi$
$410$	14.2182	0.702187
$411$	−0.248806	−0.0122727
$412$	3.41774	0.168380
$413$	−9.24703	−0.455017
$414$	18.0414	0.886684
$415$	8.15084	0.400109
$416$	−2.23657	−0.109657
$417$	−36.5021	−1.78751
$418$	−1.32653	−0.0648825
$419$	−4.84239	−0.236566	−0.118283	−	0.992980i	$-0.537739\pi$
$419$	−4.84239	−0.236566	−0.118283	+	0.992980i	$0.537739\pi$
$420$	−8.70057	−0.424544
$421$	−6.02342	−0.293563	−0.146782	−	0.989169i	$-0.546891\pi$
$421$	−6.02342	−0.293563	−0.146782	+	0.989169i	$0.546891\pi$
$422$	0.926507	0.0451017
$423$	27.3113	1.32792
$424$	−2.66057	−0.129209
$425$	−15.6298	−0.758159
$426$	−23.9497	−1.16037
$427$	−13.0891	−0.633427
$428$	−5.29099	−0.255750
$429$	−7.50992	−0.362582
$430$	3.95448	0.190702
$431$	39.4209	1.89884	0.949420	−	0.314010i	$-0.101673\pi$
$431$	39.4209	1.89884	0.949420	+	0.314010i	$0.101673\pi$
$432$	−14.7194	−0.708187
$433$	21.8747	1.05123	0.525616	−	0.850722i	$-0.323835\pi$
$433$	21.8747	1.05123	0.525616	+	0.850722i	$0.323835\pi$
$434$	−15.7420	−0.755641
$435$	−27.6159	−1.32408
$436$	−9.86287	−0.472346
$437$	3.07021	0.146868
$438$	21.1524	1.01070
$439$	9.09124	0.433901	0.216951	−	0.976183i	$-0.430389\pi$
$439$	9.09124	0.433901	0.216951	+	0.976183i	$0.430389\pi$
$440$	1.46058	0.0696302
$441$	−25.5824	−1.21821
$442$	−11.6265	−0.553018
$443$	−20.4808	−0.973072	−0.486536	−	0.873661i	$-0.661740\pi$
$443$	−20.4808	−0.973072	−0.486536	+	0.873661i	$0.661740\pi$
$444$	−6.48962	−0.307984
$445$	19.6770	0.932778
$446$	0.415483	0.0196737
$447$	6.09376	0.288225
$448$	−1.89864	−0.0897024
$449$	−6.67598	−0.315059	−0.157530	−	0.987514i	$-0.550353\pi$
$449$	−6.67598	−0.315059	−0.157530	+	0.987514i	$0.550353\pi$
$450$	−22.6552	−1.06798
$451$	10.4182	0.490573
$452$	7.43457	0.349693
$453$	−31.8282	−1.49542
$454$	17.1043	0.802743
$455$	−5.99535	−0.281066
$456$	−4.16195	−0.194901
$457$	−8.06570	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$457$	−8.06570	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$458$	−6.46452	−0.302067
$459$	−76.5168	−3.57150
$460$	−3.38047	−0.157615
$461$	−1.28037	−0.0596330	−0.0298165	−	0.999555i	$-0.509492\pi$
$461$	−1.28037	−0.0596330	−0.0298165	+	0.999555i	$0.509492\pi$
$462$	−6.37522	−0.296602
$463$	20.7724	0.965373	0.482687	−	0.875793i	$-0.339661\pi$
$463$	20.7724	0.965373	0.482687	+	0.875793i	$0.339661\pi$
$464$	−6.02635	−0.279766
$465$	37.9946	1.76196
$466$	−2.88549	−0.133668
$467$	−17.7093	−0.819490	−0.409745	−	0.912200i	$-0.634382\pi$
$467$	−17.7093	−0.819490	−0.409745	+	0.912200i	$0.634382\pi$
$468$	−16.8525	−0.779007
$469$	8.96266	0.413857
$470$	−5.11741	−0.236048
$471$	10.6120	0.488976
$472$	4.87034	0.224176
$473$	2.89759	0.133231
$474$	34.3191	1.57633
$475$	−3.85538	−0.176897
$476$	−9.86984	−0.452384
$477$	−20.0473	−0.917901
$478$	2.83778	0.129797
$479$	19.7025	0.900229	0.450115	−	0.892971i	$-0.351383\pi$
$479$	19.7025	0.900229	0.450115	+	0.892971i	$0.351383\pi$
$480$	4.58252	0.209163
$481$	−4.47184	−0.203898
$482$	5.23672	0.238526
$483$	14.7553	0.671389
$484$	−9.92978	−0.451354
$485$	−19.3605	−0.879117
$486$	−37.5399	−1.70285
$487$	40.2656	1.82461	0.912304	−	0.409514i	$-0.134302\pi$
$487$	40.2656	1.82461	0.912304	+	0.409514i	$0.134302\pi$
$488$	6.89395	0.312074
$489$	−63.0407	−2.85080
$490$	4.79346	0.216546
$491$	22.9582	1.03609	0.518045	−	0.855353i	$-0.326660\pi$
$491$	22.9582	1.03609	0.518045	+	0.855353i	$0.326660\pi$
$492$	32.6868	1.47364
$493$	−31.3272	−1.41091
$494$	−2.86790	−0.129033
$495$	11.0054	0.494655
$496$	8.29120	0.372286
$497$	−14.0096	−0.628418
$498$	18.7383	0.839684
$499$	11.7759	0.527162	0.263581	−	0.964637i	$-0.415096\pi$
$499$	11.7759	0.527162	0.263581	+	0.964637i	$0.415096\pi$
$500$	11.3042	0.505540
$501$	−49.6919	−2.22007
$502$	−9.57568	−0.427383
$503$	7.18593	0.320405	0.160202	−	0.987084i	$-0.448785\pi$
$503$	7.18593	0.320405	0.160202	+	0.987084i	$0.448785\pi$
$504$	−14.3062	−0.637248
$505$	2.21692	0.0986517
$506$	−2.47699	−0.110116
$507$	25.9587	1.15287
$508$	−7.11279	−0.315579
$509$	−26.2665	−1.16424	−0.582121	−	0.813103i	$-0.697777\pi$
$509$	−26.2665	−1.16424	−0.582121	+	0.813103i	$0.697777\pi$
$510$	23.8217	1.05484
$511$	12.3733	0.547364
$512$	1.00000	0.0441942
$513$	−18.8743	−0.833319
$514$	−2.23721	−0.0986791
$515$	−4.82534	−0.212630
$516$	9.09112	0.400214
$517$	−3.74971	−0.164912
$518$	−3.79617	−0.166794
$519$	−70.9587	−3.11474
$520$	3.15771	0.138475
$521$	−11.1226	−0.487288	−0.243644	−	0.969865i	$-0.578343\pi$
$521$	−11.1226	−0.487288	−0.243644	+	0.969865i	$0.578343\pi$
$522$	−45.4083	−1.98747
$523$	15.9908	0.699227	0.349613	−	0.936894i	$-0.386313\pi$
$523$	15.9908	0.699227	0.349613	+	0.936894i	$0.386313\pi$
$524$	−19.0311	−0.831376
$525$	−18.5288	−0.808662
$526$	27.4302	1.19602
$527$	43.1007	1.87750
$528$	3.35778	0.146129
$529$	−17.2671	−0.750742
$530$	3.75632	0.163164
$531$	36.6978	1.59255
$532$	−2.43458	−0.105552
$533$	22.5237	0.975609
$534$	45.2362	1.95756
$535$	7.47009	0.322960
$536$	−4.72057	−0.203897
$537$	−35.1043	−1.51486
$538$	15.7403	0.678615
$539$	3.51233	0.151287
$540$	20.7816	0.894296
$541$	31.3622	1.34837	0.674183	−	0.738565i	$-0.264496\pi$
$541$	31.3622	1.34837	0.674183	+	0.738565i	$0.264496\pi$
$542$	14.0189	0.602162
$543$	25.5235	1.09532
$544$	5.19837	0.222878
$545$	13.9249	0.596477
$546$	−13.7830	−0.589856
$547$	19.0462	0.814355	0.407178	−	0.913349i	$-0.366513\pi$
$547$	19.0462	0.814355	0.407178	+	0.913349i	$0.366513\pi$
$548$	0.0766558	0.00327457
$549$	51.9456	2.21698
$550$	3.11045	0.132630
$551$	−7.72743	−0.329199
$552$	−7.77150	−0.330777
$553$	20.0753	0.853690
$554$	−8.37428	−0.355789
$555$	9.16237	0.388921
$556$	11.2461	0.476940
$557$	5.55380	0.235322	0.117661	−	0.993054i	$-0.462460\pi$
$557$	5.55380	0.235322	0.117661	+	0.993054i	$0.462460\pi$
$558$	62.4738	2.64473
$559$	6.26447	0.264959
$560$	2.68060	0.113276
$561$	17.4550	0.736950
$562$	13.8304	0.583400
$563$	−25.3248	−1.06731	−0.533656	−	0.845701i	$-0.679182\pi$
$563$	−25.3248	−1.06731	−0.533656	+	0.845701i	$0.679182\pi$
$564$	−11.7646	−0.495380
$565$	−10.4965	−0.441591
$566$	14.5876	0.613163
$567$	−47.7902	−2.00700
$568$	7.37877	0.309606
$569$	−3.84152	−0.161045	−0.0805225	−	0.996753i	$-0.525659\pi$
$569$	−3.84152	−0.161045	−0.0805225	+	0.996753i	$0.525659\pi$
$570$	5.87604	0.246120
$571$	3.06340	0.128199	0.0640997	−	0.997944i	$-0.479582\pi$
$571$	3.06340	0.128199	0.0640997	+	0.997944i	$0.479582\pi$
$572$	2.31376	0.0967433
$573$	−13.9751	−0.583816
$574$	19.1205	0.798074
$575$	−7.19906	−0.300221
$576$	7.53496	0.313957
$577$	−20.0090	−0.832987	−0.416494	−	0.909139i	$-0.636741\pi$
$577$	−20.0090	−0.832987	−0.416494	+	0.909139i	$0.636741\pi$
$578$	10.0231	0.416904
$579$	−10.5729	−0.439396
$580$	8.50831	0.353288
$581$	10.9612	0.454746
$582$	−44.5088	−1.84495
$583$	2.75239	0.113992
$584$	−6.51693	−0.269673
$585$	23.7932	0.983727
$586$	−16.9759	−0.701270
$587$	27.6241	1.14017	0.570084	−	0.821586i	$-0.306911\pi$
$587$	27.6241	1.14017	0.570084	+	0.821586i	$0.306911\pi$
$588$	11.0199	0.454452
$589$	10.6316	0.438066
$590$	−6.87619	−0.283088
$591$	14.5997	0.600552
$592$	1.99941	0.0821754
$593$	29.9904	1.23156	0.615779	−	0.787919i	$-0.288842\pi$
$593$	29.9904	1.23156	0.615779	+	0.787919i	$0.288842\pi$
$594$	15.2274	0.624787
$595$	13.9347	0.571268
$596$	−1.87745	−0.0769035
$597$	−0.694288	−0.0284153
$598$	−5.35515	−0.218988
$599$	−30.4264	−1.24319	−0.621595	−	0.783339i	$-0.713515\pi$
$599$	−30.4264	−1.24319	−0.621595	+	0.783339i	$0.713515\pi$
$600$	9.75896	0.398408
$601$	12.7298	0.519259	0.259629	−	0.965708i	$-0.416400\pi$
$601$	12.7298	0.519259	0.259629	+	0.965708i	$0.416400\pi$
$602$	5.31795	0.216743
$603$	−35.5693	−1.44849
$604$	9.80607	0.399003
$605$	14.0194	0.569968
$606$	5.09657	0.207034
$607$	−32.7741	−1.33026	−0.665130	−	0.746728i	$-0.731624\pi$
$607$	−32.7741	−1.33026	−0.665130	+	0.746728i	$0.731624\pi$
$608$	1.28227	0.0520030
$609$	−37.1376	−1.50489
$610$	−9.73321	−0.394086
$611$	−8.10671	−0.327963
$612$	39.1695	1.58333
$613$	16.8553	0.680780	0.340390	−	0.940284i	$-0.389441\pi$
$613$	16.8553	0.680780	0.340390	+	0.940284i	$0.389441\pi$
$614$	25.8754	1.04424
$615$	−46.1489	−1.86090
$616$	1.96417	0.0791386
$617$	1.80968	0.0728550	0.0364275	−	0.999336i	$-0.488402\pi$
$617$	1.80968	0.0728550	0.0364275	+	0.999336i	$0.488402\pi$
$618$	−11.0932	−0.446233
$619$	19.5225	0.784676	0.392338	−	0.919821i	$-0.371666\pi$
$619$	19.5225	0.784676	0.392338	+	0.919821i	$0.371666\pi$
$620$	−11.7059	−0.470121
$621$	−35.2434	−1.41427
$622$	24.4932	0.982089
$623$	26.4614	1.06015
$624$	7.25938	0.290608
$625$	−0.926468	−0.0370587
$626$	13.1672	0.526267
$627$	4.30558	0.171948
$628$	−3.26950	−0.130467
$629$	10.3937	0.414424
$630$	20.1982	0.804715
$631$	28.6244	1.13952	0.569759	−	0.821812i	$-0.307036\pi$
$631$	28.6244	1.13952	0.569759	+	0.821812i	$0.307036\pi$
$632$	−10.5735	−0.420592
$633$	−3.00722	−0.119526
$634$	−0.189914	−0.00754246
$635$	10.0422	0.398512
$636$	8.63556	0.342422
$637$	7.59352	0.300866
$638$	6.23434	0.246820
$639$	55.5987	2.19945
$640$	−1.41185	−0.0558083
$641$	39.7891	1.57158	0.785788	−	0.618496i	$-0.212258\pi$
$641$	39.7891	1.57158	0.785788	+	0.618496i	$0.212258\pi$
$642$	17.1733	0.677776
$643$	0.722342	0.0284864	0.0142432	−	0.999899i	$-0.495466\pi$
$643$	0.722342	0.0284864	0.0142432	+	0.999899i	$0.495466\pi$
$644$	−4.54602	−0.179138
$645$	−12.8353	−0.505389
$646$	6.66573	0.262259
$647$	−11.6058	−0.456271	−0.228135	−	0.973629i	$-0.573263\pi$
$647$	−11.6058	−0.456271	−0.228135	+	0.973629i	$0.573263\pi$
$648$	25.1707	0.988799
$649$	−5.03843	−0.197775
$650$	6.72466	0.263763
$651$	51.0948	2.00256
$652$	19.4225	0.760643
$653$	21.6345	0.846624	0.423312	−	0.905984i	$-0.360867\pi$
$653$	21.6345	0.846624	0.423312	+	0.905984i	$0.360867\pi$
$654$	32.0125	1.25179
$655$	26.8690	1.04986
$656$	−10.0706	−0.393192
$657$	−49.1048	−1.91576
$658$	−6.88184	−0.268282
$659$	−9.00912	−0.350946	−0.175473	−	0.984484i	$-0.556145\pi$
$659$	−9.00912	−0.350946	−0.175473	+	0.984484i	$0.556145\pi$
$660$	−4.74068	−0.184531
$661$	1.07051	0.0416380	0.0208190	−	0.999783i	$-0.493373\pi$
$661$	1.07051	0.0416380	0.0208190	+	0.999783i	$0.493373\pi$
$662$	−19.0774	−0.741466
$663$	37.7370	1.46558
$664$	−5.77316	−0.224042
$665$	3.43725	0.133291
$666$	15.0655	0.583776
$667$	−14.4292	−0.558702
$668$	15.3098	0.592353
$669$	−1.34856	−0.0521383
$670$	6.66473	0.257481
$671$	−7.13187	−0.275323
$672$	6.16254	0.237725
$673$	42.7559	1.64812	0.824060	−	0.566503i	$-0.191704\pi$
$673$	42.7559	1.64812	0.824060	+	0.566503i	$0.191704\pi$
$674$	22.2556	0.857255
$675$	44.2565	1.70343
$676$	−7.99774	−0.307605
$677$	22.6461	0.870361	0.435181	−	0.900343i	$-0.356685\pi$
$677$	22.6461	0.870361	0.435181	+	0.900343i	$0.356685\pi$
$678$	−24.1308	−0.926739
$679$	−26.0359	−0.999165
$680$	−7.33932	−0.281450
$681$	−55.5163	−2.12739
$682$	−8.57734	−0.328443
$683$	−0.260187	−0.00995577	−0.00497789	−	0.999988i	$-0.501585\pi$
$683$	−0.260187	−0.00995577	−0.00497789	+	0.999988i	$0.501585\pi$
$684$	9.66187	0.369431
$685$	−0.108226	−0.00413512
$686$	19.7367	0.753550
$687$	20.9823	0.800524
$688$	−2.80092	−0.106784
$689$	5.95055	0.226698
$690$	10.9722	0.417704
$691$	38.0302	1.44674	0.723368	−	0.690463i	$-0.242593\pi$
$691$	38.0302	1.44674	0.723368	+	0.690463i	$0.242593\pi$
$692$	21.8620	0.831067
$693$	14.7999	0.562202
$694$	−24.3315	−0.923612
$695$	−15.8778	−0.602278
$696$	19.5601	0.741424
$697$	−52.3508	−1.98293
$698$	8.94988	0.338758
$699$	9.36560	0.354239
$700$	5.70861	0.215765
$701$	21.2285	0.801791	0.400895	−	0.916124i	$-0.368699\pi$
$701$	21.2285	0.801791	0.400895	+	0.916124i	$0.368699\pi$
$702$	32.9210	1.24252
$703$	2.56379	0.0966953
$704$	−1.03451	−0.0389896
$705$	16.6099	0.625564
$706$	30.2617	1.13891
$707$	2.98130	0.112123
$708$	−15.8080	−0.594099
$709$	27.4177	1.02969	0.514846	−	0.857283i	$-0.327849\pi$
$709$	27.4177	1.02969	0.514846	+	0.857283i	$0.327849\pi$
$710$	−10.4177	−0.390970
$711$	−79.6711	−2.98790
$712$	−13.9370	−0.522312
$713$	19.8521	0.743466
$714$	32.0351	1.19889
$715$	−3.26668	−0.122167
$716$	10.8154	0.404192
$717$	−9.21076	−0.343982
$718$	−12.6456	−0.471929
$719$	−22.5308	−0.840258	−0.420129	−	0.907464i	$-0.638015\pi$
$719$	−22.5308	−0.840258	−0.420129	+	0.907464i	$0.638015\pi$
$720$	−10.6382	−0.396463
$721$	−6.48907	−0.241666
$722$	−17.3558	−0.645915
$723$	−16.9971	−0.632130
$724$	−7.86366	−0.292251
$725$	18.1193	0.672935
$726$	32.2297	1.19616
$727$	−34.6238	−1.28412	−0.642062	−	0.766653i	$-0.721921\pi$
$727$	−34.6238	−1.28412	−0.642062	+	0.766653i	$0.721921\pi$
$728$	4.24645	0.157384
$729$	46.3335	1.71606
$730$	9.20093	0.340542
$731$	−14.5602	−0.538530
$732$	−22.3761	−0.827044
$733$	16.0947	0.594471	0.297235	−	0.954804i	$-0.403935\pi$
$733$	16.0947	0.594471	0.297235	+	0.954804i	$0.403935\pi$
$734$	15.5176	0.572765
$735$	−15.5584	−0.573880
$736$	2.39435	0.0882571
$737$	4.88348	0.179885
$738$	−75.8817	−2.79324
$739$	−4.06216	−0.149429	−0.0747145	−	0.997205i	$-0.523805\pi$
$739$	−4.06216	−0.149429	−0.0747145	+	0.997205i	$0.523805\pi$
$740$	−2.82287	−0.103771
$741$	9.30850	0.341956
$742$	5.05146	0.185445
$743$	32.6430	1.19755	0.598777	−	0.800916i	$-0.295654\pi$
$743$	32.6430	1.19755	0.598777	+	0.800916i	$0.295654\pi$
$744$	−26.9112	−0.986614
$745$	2.65068	0.0971135
$746$	−17.1549	−0.628085
$747$	−43.5005	−1.59160
$748$	−5.37778	−0.196631
$749$	10.0457	0.367062
$750$	−36.6908	−1.33976
$751$	2.62225	0.0956871	0.0478436	−	0.998855i	$-0.484765\pi$
$751$	2.62225	0.0956871	0.0478436	+	0.998855i	$0.484765\pi$
$752$	3.62461	0.132176
$753$	31.0803	1.13263
$754$	13.4784	0.490854
$755$	−13.8447	−0.503860
$756$	27.9468	1.01642
$757$	−21.9644	−0.798308	−0.399154	−	0.916884i	$-0.630696\pi$
$757$	−21.9644	−0.798308	−0.399154	+	0.916884i	$0.630696\pi$
$758$	−13.3227	−0.483901
$759$	8.03971	0.291823
$760$	−1.81038	−0.0656692
$761$	9.36280	0.339401	0.169701	−	0.985496i	$-0.445720\pi$
$761$	9.36280	0.339401	0.169701	+	0.985496i	$0.445720\pi$
$762$	23.0864	0.836332
$763$	18.7261	0.677929
$764$	4.30563	0.155772
$765$	−55.3014	−1.99943
$766$	3.67169	0.132663
$767$	−10.8929	−0.393319
$768$	−3.24576	−0.117121
$769$	16.6292	0.599664	0.299832	−	0.953992i	$-0.403069\pi$
$769$	16.6292	0.599664	0.299832	+	0.953992i	$0.403069\pi$
$770$	−2.77311	−0.0999359
$771$	7.26145	0.261515
$772$	3.25746	0.117239
$773$	35.1641	1.26476	0.632382	−	0.774657i	$-0.282077\pi$
$773$	35.1641	1.26476	0.632382	+	0.774657i	$0.282077\pi$
$774$	−21.1048	−0.758597
$775$	−24.9290	−0.895475
$776$	13.7129	0.492264
$777$	12.3215	0.442030
$778$	−30.0064	−1.07578
$779$	−12.9133	−0.462666
$780$	−10.2492	−0.366979
$781$	−7.63343	−0.273146
$782$	12.4467	0.445095
$783$	88.7042	3.17003
$784$	−3.39516	−0.121256
$785$	4.61604	0.164754
$786$	61.7703	2.20327
$787$	−25.3029	−0.901952	−0.450976	−	0.892536i	$-0.648924\pi$
$787$	−25.3029	−0.901952	−0.450976	+	0.892536i	$0.648924\pi$
$788$	−4.49809	−0.160238
$789$	−89.0320	−3.16962
$790$	14.9282	0.531123
$791$	−14.1156	−0.501893
$792$	−7.79500	−0.276983
$793$	−15.4188	−0.547538
$794$	8.38078	0.297423
$795$	−12.1921	−0.432410
$796$	0.213906	0.00758170
$797$	−42.3672	−1.50072	−0.750361	−	0.661028i	$-0.770121\pi$
$797$	−42.3672	−1.50072	−0.750361	+	0.661028i	$0.770121\pi$
$798$	7.90205	0.279729
$799$	18.8421	0.666585
$800$	−3.00668	−0.106302
$801$	−105.015	−3.71052
$802$	−9.52957	−0.336501
$803$	6.74185	0.237915
$804$	15.3218	0.540359
$805$	6.41830	0.226215
$806$	−18.5439	−0.653180
$807$	−51.0894	−1.79843
$808$	−1.57023	−0.0552403
$809$	−26.4199	−0.928875	−0.464437	−	0.885606i	$-0.653743\pi$
$809$	−26.4199	−0.928875	−0.464437	+	0.885606i	$0.653743\pi$
$810$	−35.5373	−1.24865
$811$	−44.4027	−1.55919	−0.779595	−	0.626284i	$-0.784575\pi$
$811$	−44.4027	−1.55919	−0.779595	+	0.626284i	$0.784575\pi$
$812$	11.4419	0.401532
$813$	−45.5019	−1.59582
$814$	−2.06842	−0.0724980
$815$	−27.4216	−0.960538
$816$	−16.8727	−0.590661
$817$	−3.59154	−0.125652
$818$	20.5932	0.720023
$819$	31.9968	1.11806
$820$	14.2182	0.496521
$821$	−5.98317	−0.208814	−0.104407	−	0.994535i	$-0.533294\pi$
$821$	−5.98317	−0.208814	−0.104407	+	0.994535i	$0.533294\pi$
$822$	−0.248806	−0.00867812
$823$	−31.1621	−1.08624	−0.543121	−	0.839655i	$-0.682757\pi$
$823$	−31.1621	−1.08624	−0.543121	+	0.839655i	$0.682757\pi$
$824$	3.41774	0.119063
$825$	−10.0958	−0.351489
$826$	−9.24703	−0.321745
$827$	−6.70487	−0.233151	−0.116576	−	0.993182i	$-0.537192\pi$
$827$	−6.70487	−0.233151	−0.116576	+	0.993182i	$0.537192\pi$
$828$	18.0414	0.626980
$829$	3.57668	0.124223	0.0621116	−	0.998069i	$-0.480217\pi$
$829$	3.57668	0.124223	0.0621116	+	0.998069i	$0.480217\pi$
$830$	8.15084	0.282920
$831$	27.1809	0.942895
$832$	−2.23657	−0.0775392
$833$	−17.6493	−0.611512
$834$	−36.5021	−1.26396
$835$	−21.6151	−0.748021
$836$	−1.32653	−0.0458789
$837$	−122.041	−4.21836
$838$	−4.84239	−0.167278
$839$	−13.0681	−0.451161	−0.225580	−	0.974225i	$-0.572428\pi$
$839$	−13.0681	−0.451161	−0.225580	+	0.974225i	$0.572428\pi$
$840$	−8.70057	−0.300198
$841$	7.31695	0.252309
$842$	−6.02342	−0.207581
$843$	−44.8902	−1.54610
$844$	0.926507	0.0318917
$845$	11.2916	0.388443
$846$	27.3113	0.938982
$847$	18.8531	0.647800
$848$	−2.66057	−0.0913643
$849$	−47.3479	−1.62498
$850$	−15.6298	−0.536099
$851$	4.78731	0.164107
$852$	−23.9497	−0.820504
$853$	6.46125	0.221229	0.110615	−	0.993863i	$-0.464718\pi$
$853$	6.46125	0.221229	0.110615	+	0.993863i	$0.464718\pi$
$854$	−13.0891	−0.447901
$855$	−13.6411	−0.466516
$856$	−5.29099	−0.180842
$857$	44.0161	1.50356	0.751781	−	0.659413i	$-0.229195\pi$
$857$	44.0161	1.50356	0.751781	+	0.659413i	$0.229195\pi$
$858$	−7.50992	−0.256384
$859$	14.0046	0.477830	0.238915	−	0.971041i	$-0.423208\pi$
$859$	14.0046	0.477830	0.238915	+	0.971041i	$0.423208\pi$
$860$	3.95448	0.134847
$861$	−62.0606	−2.11502
$862$	39.4209	1.34268
$863$	36.1384	1.23016	0.615082	−	0.788463i	$-0.289123\pi$
$863$	36.1384	1.23016	0.615082	+	0.788463i	$0.289123\pi$
$864$	−14.7194	−0.500764
$865$	−30.8658	−1.04947
$866$	21.8747	0.743333
$867$	−32.5324	−1.10486
$868$	−15.7420	−0.534319
$869$	10.9384	0.371061
$870$	−27.6159	−0.936267
$871$	10.5579	0.357741
$872$	−9.86287	−0.333999
$873$	103.326	3.49706
$874$	3.07021	0.103852
$875$	−21.4627	−0.725571
$876$	21.1524	0.714674
$877$	−44.3977	−1.49920	−0.749601	−	0.661890i	$-0.769755\pi$
$877$	−44.3977	−1.49920	−0.749601	+	0.661890i	$0.769755\pi$
$878$	9.09124	0.306815
$879$	55.0998	1.85847
$880$	1.46058	0.0492360
$881$	11.4476	0.385680	0.192840	−	0.981230i	$-0.438230\pi$
$881$	11.4476	0.385680	0.192840	+	0.981230i	$0.438230\pi$
$882$	−25.5824	−0.861404
$883$	−41.3408	−1.39123	−0.695614	−	0.718415i	$-0.744868\pi$
$883$	−41.3408	−1.39123	−0.695614	+	0.718415i	$0.744868\pi$
$884$	−11.6265	−0.391043
$885$	22.3185	0.750227
$886$	−20.4808	−0.688066
$887$	−11.3492	−0.381070	−0.190535	−	0.981680i	$-0.561022\pi$
$887$	−11.3492	−0.381070	−0.190535	+	0.981680i	$0.561022\pi$
$888$	−6.48962	−0.217777
$889$	13.5046	0.452931
$890$	19.6770	0.659574
$891$	−26.0394	−0.872353
$892$	0.415483	0.0139114
$893$	4.64774	0.155531
$894$	6.09376	0.203806
$895$	−15.2698	−0.510413
$896$	−1.89864	−0.0634292
$897$	17.3815	0.580352
$898$	−6.67598	−0.222780
$899$	−49.9657	−1.66645
$900$	−22.6552	−0.755174
$901$	−13.8306	−0.460765
$902$	10.4182	0.346887
$903$	−17.2608	−0.574403
$904$	7.43457	0.247270
$905$	11.1023	0.369053
$906$	−31.8282	−1.05742
$907$	29.4489	0.977835	0.488918	−	0.872330i	$-0.337392\pi$
$907$	29.4489	0.977835	0.488918	+	0.872330i	$0.337392\pi$
$908$	17.1043	0.567625
$909$	−11.8316	−0.392429
$910$	−5.99535	−0.198744
$911$	−7.22517	−0.239380	−0.119690	−	0.992811i	$-0.538190\pi$
$911$	−7.22517	−0.239380	−0.119690	+	0.992811i	$0.538190\pi$
$912$	−4.16195	−0.137816
$913$	5.97241	0.197658
$914$	−8.06570	−0.266790
$915$	31.5917	1.04439
$916$	−6.46452	−0.213594
$917$	36.1332	1.19322
$918$	−76.5168	−2.52543
$919$	−54.5197	−1.79844	−0.899220	−	0.437497i	$-0.855865\pi$
$919$	−54.5197	−1.79844	−0.899220	+	0.437497i	$0.855865\pi$
$920$	−3.38047	−0.111451
$921$	−83.9852	−2.76741
$922$	−1.28037	−0.0421669
$923$	−16.5032	−0.543208
$924$	−6.37522	−0.209729
$925$	−6.01160	−0.197660
$926$	20.7724	0.682622
$927$	25.7526	0.845825
$928$	−6.02635	−0.197825
$929$	17.2637	0.566404	0.283202	−	0.959060i	$-0.408603\pi$
$929$	17.2637	0.566404	0.283202	+	0.959060i	$0.408603\pi$
$930$	37.9946	1.24589
$931$	−4.35352	−0.142681
$932$	−2.88549	−0.0945173
$933$	−79.4991	−2.60269
$934$	−17.7093	−0.579467
$935$	7.59261	0.248305
$936$	−16.8525	−0.550841
$937$	4.52106	0.147697	0.0738483	−	0.997269i	$-0.476472\pi$
$937$	4.52106	0.147697	0.0738483	+	0.997269i	$0.476472\pi$
$938$	8.96266	0.292641
$939$	−42.7376	−1.39469
$940$	−5.11741	−0.166911
$941$	26.8762	0.876139	0.438069	−	0.898941i	$-0.355662\pi$
$941$	26.8762	0.876139	0.438069	+	0.898941i	$0.355662\pi$
$942$	10.6120	0.345758
$943$	−24.1126	−0.785215
$944$	4.87034	0.158516
$945$	−39.4567	−1.28353
$946$	2.89759	0.0942087
$947$	−8.37801	−0.272249	−0.136124	−	0.990692i	$-0.543465\pi$
$947$	−8.37801	−0.272249	−0.136124	+	0.990692i	$0.543465\pi$
$948$	34.3191	1.11463
$949$	14.5756	0.473144
$950$	−3.85538	−0.125085
$951$	0.616416	0.0199887
$952$	−9.86984	−0.319883
$953$	12.9668	0.420036	0.210018	−	0.977698i	$-0.432648\pi$
$953$	12.9668	0.420036	0.210018	+	0.977698i	$0.432648\pi$
$954$	−20.0473	−0.649054
$955$	−6.07891	−0.196709
$956$	2.83778	0.0917804
$957$	−20.2352	−0.654110
$958$	19.7025	0.636558
$959$	−0.145542	−0.00469979
$960$	4.58252	0.147900
$961$	37.7440	1.21755
$962$	−4.47184	−0.144178
$963$	−39.8674	−1.28471
$964$	5.23672	0.168663
$965$	−4.59904	−0.148048
$966$	14.7553	0.474744
$967$	−30.9615	−0.995655	−0.497828	−	0.867276i	$-0.665869\pi$
$967$	−30.9615	−0.995655	−0.497828	+	0.867276i	$0.665869\pi$
$968$	−9.92978	−0.319155
$969$	−21.6353	−0.695028
$970$	−19.3605	−0.621630
$971$	55.3010	1.77469	0.887347	−	0.461102i	$-0.152546\pi$
$971$	55.3010	1.77469	0.887347	+	0.461102i	$0.152546\pi$
$972$	−37.5399	−1.20409
$973$	−21.3523	−0.684523
$974$	40.2656	1.29019
$975$	−21.8266	−0.699012
$976$	6.89395	0.220670
$977$	−29.2597	−0.936100	−0.468050	−	0.883702i	$-0.655043\pi$
$977$	−29.2597	−0.936100	−0.468050	+	0.883702i	$0.655043\pi$
$978$	−63.0407	−2.01582
$979$	14.4180	0.460802
$980$	4.79346	0.153121
$981$	−74.3163	−2.37274
$982$	22.9582	0.732627
$983$	−21.0873	−0.672581	−0.336290	−	0.941758i	$-0.609172\pi$
$983$	−21.0873	−0.672581	−0.336290	+	0.941758i	$0.609172\pi$
$984$	32.6868	1.04202
$985$	6.35063	0.202348
$986$	−31.3272	−0.997662
$987$	22.3368	0.710988
$988$	−2.86790	−0.0912399
$989$	−6.70640	−0.213251
$990$	11.0054	0.349774
$991$	−10.6864	−0.339464	−0.169732	−	0.985490i	$-0.554290\pi$
$991$	−10.6864	−0.339464	−0.169732	+	0.985490i	$0.554290\pi$
$992$	8.29120	0.263246
$993$	61.9208	1.96500
$994$	−14.0096	−0.444359
$995$	−0.302003	−0.00957415
$996$	18.7383	0.593746
$997$	−42.2894	−1.33932	−0.669659	−	0.742669i	$-0.733560\pi$
$997$	−42.2894	−1.33932	−0.669659	+	0.742669i	$0.733560\pi$
$998$	11.7759	0.372760
$999$	−29.4301	−0.931129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.1	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.1	✓	92	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.24576 q^{3} +1.00000 q^{4} -1.41185 q^{5} -3.24576 q^{6} -1.89864 q^{7} +1.00000 q^{8} +7.53496 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.24576 q^{3} +1.00000 q^{4} -1.41185 q^{5} -3.24576 q^{6} -1.89864 q^{7} +1.00000 q^{8} +7.53496 q^{9} -1.41185 q^{10} -1.03451 q^{11} -3.24576 q^{12} -2.23657 q^{13} -1.89864 q^{14} +4.58252 q^{15} +1.00000 q^{16} +5.19837 q^{17} +7.53496 q^{18} +1.28227 q^{19} -1.41185 q^{20} +6.16254 q^{21} -1.03451 q^{22} +2.39435 q^{23} -3.24576 q^{24} -3.00668 q^{25} -2.23657 q^{26} -14.7194 q^{27} -1.89864 q^{28} -6.02635 q^{29} +4.58252 q^{30} +8.29120 q^{31} +1.00000 q^{32} +3.35778 q^{33} +5.19837 q^{34} +2.68060 q^{35} +7.53496 q^{36} +1.99941 q^{37} +1.28227 q^{38} +7.25938 q^{39} -1.41185 q^{40} -10.0706 q^{41} +6.16254 q^{42} -2.80092 q^{43} -1.03451 q^{44} -10.6382 q^{45} +2.39435 q^{46} +3.62461 q^{47} -3.24576 q^{48} -3.39516 q^{49} -3.00668 q^{50} -16.8727 q^{51} -2.23657 q^{52} -2.66057 q^{53} -14.7194 q^{54} +1.46058 q^{55} -1.89864 q^{56} -4.16195 q^{57} -6.02635 q^{58} +4.87034 q^{59} +4.58252 q^{60} +6.89395 q^{61} +8.29120 q^{62} -14.3062 q^{63} +1.00000 q^{64} +3.15771 q^{65} +3.35778 q^{66} -4.72057 q^{67} +5.19837 q^{68} -7.77150 q^{69} +2.68060 q^{70} +7.37877 q^{71} +7.53496 q^{72} -6.51693 q^{73} +1.99941 q^{74} +9.75896 q^{75} +1.28227 q^{76} +1.96417 q^{77} +7.25938 q^{78} -10.5735 q^{79} -1.41185 q^{80} +25.1707 q^{81} -10.0706 q^{82} -5.77316 q^{83} +6.16254 q^{84} -7.33932 q^{85} -2.80092 q^{86} +19.5601 q^{87} -1.03451 q^{88} -13.9370 q^{89} -10.6382 q^{90} +4.24645 q^{91} +2.39435 q^{92} -26.9112 q^{93} +3.62461 q^{94} -1.81038 q^{95} -3.24576 q^{96} +13.7129 q^{97} -3.39516 q^{98} -7.79500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

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