LMFDB - Embedded newform 8038.2.a.c.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:	$1$
Dimension:	$84$
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.25732 q^{3} +1.00000 q^{4} -3.39963 q^{5} +2.25732 q^{6} +3.01257 q^{7} -1.00000 q^{8} +2.09550 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.25732 q^{3} +1.00000 q^{4} -3.39963 q^{5} +2.25732 q^{6} +3.01257 q^{7} -1.00000 q^{8} +2.09550 q^{9} +3.39963 q^{10} +4.16619 q^{11} -2.25732 q^{12} +5.10966 q^{13} -3.01257 q^{14} +7.67405 q^{15} +1.00000 q^{16} -7.62489 q^{17} -2.09550 q^{18} +1.60426 q^{19} -3.39963 q^{20} -6.80033 q^{21} -4.16619 q^{22} +1.35396 q^{23} +2.25732 q^{24} +6.55745 q^{25} -5.10966 q^{26} +2.04174 q^{27} +3.01257 q^{28} -1.65335 q^{29} -7.67405 q^{30} -2.96797 q^{31} -1.00000 q^{32} -9.40442 q^{33} +7.62489 q^{34} -10.2416 q^{35} +2.09550 q^{36} +2.53390 q^{37} -1.60426 q^{38} -11.5341 q^{39} +3.39963 q^{40} -2.39033 q^{41} +6.80033 q^{42} -0.735148 q^{43} +4.16619 q^{44} -7.12392 q^{45} -1.35396 q^{46} -2.78981 q^{47} -2.25732 q^{48} +2.07556 q^{49} -6.55745 q^{50} +17.2118 q^{51} +5.10966 q^{52} +11.2449 q^{53} -2.04174 q^{54} -14.1635 q^{55} -3.01257 q^{56} -3.62132 q^{57} +1.65335 q^{58} -11.9716 q^{59} +7.67405 q^{60} -3.24789 q^{61} +2.96797 q^{62} +6.31284 q^{63} +1.00000 q^{64} -17.3709 q^{65} +9.40442 q^{66} +0.353132 q^{67} -7.62489 q^{68} -3.05632 q^{69} +10.2416 q^{70} -3.85526 q^{71} -2.09550 q^{72} -13.5364 q^{73} -2.53390 q^{74} -14.8023 q^{75} +1.60426 q^{76} +12.5509 q^{77} +11.5341 q^{78} -4.89001 q^{79} -3.39963 q^{80} -10.8954 q^{81} +2.39033 q^{82} -7.98975 q^{83} -6.80033 q^{84} +25.9218 q^{85} +0.735148 q^{86} +3.73214 q^{87} -4.16619 q^{88} +16.0276 q^{89} +7.12392 q^{90} +15.3932 q^{91} +1.35396 q^{92} +6.69967 q^{93} +2.78981 q^{94} -5.45387 q^{95} +2.25732 q^{96} -0.0540933 q^{97} -2.07556 q^{98} +8.73025 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.25732	−1.30327	−0.651633	−	0.758535i	$-0.725916\pi$
$3$	−2.25732	−1.30327	−0.651633	+	0.758535i	$0.725916\pi$
$4$	1.00000	0.500000
$5$	−3.39963	−1.52036	−0.760179	−	0.649713i	$-0.774889\pi$
$5$	−3.39963	−1.52036	−0.760179	+	0.649713i	$0.774889\pi$
$6$	2.25732	0.921548
$7$	3.01257	1.13864	0.569322	−	0.822115i	$-0.307206\pi$
$7$	3.01257	1.13864	0.569322	+	0.822115i	$0.307206\pi$
$8$	−1.00000	−0.353553
$9$	2.09550	0.698501
$10$	3.39963	1.07506
$11$	4.16619	1.25615	0.628076	−	0.778152i	$-0.283843\pi$
$11$	4.16619	1.25615	0.628076	+	0.778152i	$0.283843\pi$
$12$	−2.25732	−0.651633
$13$	5.10966	1.41716	0.708582	−	0.705628i	$-0.249335\pi$
$13$	5.10966	1.41716	0.708582	+	0.705628i	$0.249335\pi$
$14$	−3.01257	−0.805142
$15$	7.67405	1.98143
$16$	1.00000	0.250000
$17$	−7.62489	−1.84931	−0.924654	−	0.380808i	$-0.875646\pi$
$17$	−7.62489	−1.84931	−0.924654	+	0.380808i	$0.875646\pi$
$18$	−2.09550	−0.493915
$19$	1.60426	0.368042	0.184021	−	0.982922i	$-0.441089\pi$
$19$	1.60426	0.368042	0.184021	+	0.982922i	$0.441089\pi$
$20$	−3.39963	−0.760179
$21$	−6.80033	−1.48395
$22$	−4.16619	−0.888234
$23$	1.35396	0.282320	0.141160	−	0.989987i	$-0.454917\pi$
$23$	1.35396	0.282320	0.141160	+	0.989987i	$0.454917\pi$
$24$	2.25732	0.460774
$25$	6.55745	1.31149
$26$	−5.10966	−1.00209
$27$	2.04174	0.392934
$28$	3.01257	0.569322
$29$	−1.65335	−0.307019	−0.153510	−	0.988147i	$-0.549058\pi$
$29$	−1.65335	−0.307019	−0.153510	+	0.988147i	$0.549058\pi$
$30$	−7.67405	−1.40108
$31$	−2.96797	−0.533064	−0.266532	−	0.963826i	$-0.585878\pi$
$31$	−2.96797	−0.533064	−0.266532	+	0.963826i	$0.585878\pi$
$32$	−1.00000	−0.176777
$33$	−9.40442	−1.63710
$34$	7.62489	1.30766
$35$	−10.2416	−1.73115
$36$	2.09550	0.349250
$37$	2.53390	0.416570	0.208285	−	0.978068i	$-0.433212\pi$
$37$	2.53390	0.416570	0.208285	+	0.978068i	$0.433212\pi$
$38$	−1.60426	−0.260245
$39$	−11.5341	−1.84694
$40$	3.39963	0.537528
$41$	−2.39033	−0.373308	−0.186654	−	0.982426i	$-0.559764\pi$
$41$	−2.39033	−0.373308	−0.186654	+	0.982426i	$0.559764\pi$
$42$	6.80033	1.04931
$43$	−0.735148	−0.112109	−0.0560545	−	0.998428i	$-0.517852\pi$
$43$	−0.735148	−0.112109	−0.0560545	+	0.998428i	$0.517852\pi$
$44$	4.16619	0.628076
$45$	−7.12392	−1.06197
$46$	−1.35396	−0.199630
$47$	−2.78981	−0.406935	−0.203468	−	0.979082i	$-0.565221\pi$
$47$	−2.78981	−0.406935	−0.203468	+	0.979082i	$0.565221\pi$
$48$	−2.25732	−0.325816
$49$	2.07556	0.296508
$50$	−6.55745	−0.927364
$51$	17.2118	2.41014
$52$	5.10966	0.708582
$53$	11.2449	1.54460	0.772302	−	0.635255i	$-0.219105\pi$
$53$	11.2449	1.54460	0.772302	+	0.635255i	$0.219105\pi$
$54$	−2.04174	−0.277846
$55$	−14.1635	−1.90980
$56$	−3.01257	−0.402571
$57$	−3.62132	−0.479656
$58$	1.65335	0.217095
$59$	−11.9716	−1.55857	−0.779285	−	0.626669i	$-0.784418\pi$
$59$	−11.9716	−1.55857	−0.779285	+	0.626669i	$0.784418\pi$
$60$	7.67405	0.990715
$61$	−3.24789	−0.415850	−0.207925	−	0.978145i	$-0.566671\pi$
$61$	−3.24789	−0.415850	−0.207925	+	0.978145i	$0.566671\pi$
$62$	2.96797	0.376933
$63$	6.31284	0.795343
$64$	1.00000	0.125000
$65$	−17.3709	−2.15460
$66$	9.40442	1.15760
$67$	0.353132	0.0431419	0.0215710	−	0.999767i	$-0.493133\pi$
$67$	0.353132	0.0431419	0.0215710	+	0.999767i	$0.493133\pi$
$68$	−7.62489	−0.924654
$69$	−3.05632	−0.367938
$70$	10.2416	1.22411
$71$	−3.85526	−0.457535	−0.228767	−	0.973481i	$-0.573470\pi$
$71$	−3.85526	−0.457535	−0.228767	+	0.973481i	$0.573470\pi$
$72$	−2.09550	−0.246957
$73$	−13.5364	−1.58431	−0.792157	−	0.610318i	$-0.791042\pi$
$73$	−13.5364	−1.58431	−0.792157	+	0.610318i	$0.791042\pi$
$74$	−2.53390	−0.294560
$75$	−14.8023	−1.70922
$76$	1.60426	0.184021
$77$	12.5509	1.43031
$78$	11.5341	1.30598
$79$	−4.89001	−0.550169	−0.275085	−	0.961420i	$-0.588706\pi$
$79$	−4.89001	−0.550169	−0.275085	+	0.961420i	$0.588706\pi$
$80$	−3.39963	−0.380090
$81$	−10.8954	−1.21060
$82$	2.39033	0.263968
$83$	−7.98975	−0.876989	−0.438495	−	0.898734i	$-0.644488\pi$
$83$	−7.98975	−0.876989	−0.438495	+	0.898734i	$0.644488\pi$
$84$	−6.80033	−0.741977
$85$	25.9218	2.81161
$86$	0.735148	0.0792730
$87$	3.73214	0.400127
$88$	−4.16619	−0.444117
$89$	16.0276	1.69892	0.849462	−	0.527650i	$-0.176927\pi$
$89$	16.0276	1.69892	0.849462	+	0.527650i	$0.176927\pi$
$90$	7.12392	0.750927
$91$	15.3932	1.61364
$92$	1.35396	0.141160
$93$	6.69967	0.694723
$94$	2.78981	0.287747
$95$	−5.45387	−0.559555
$96$	2.25732	0.230387
$97$	−0.0540933	−0.00549235	−0.00274617	−	0.999996i	$-0.500874\pi$
$97$	−0.0540933	−0.00549235	−0.00274617	+	0.999996i	$0.500874\pi$
$98$	−2.07556	−0.209663
$99$	8.73025	0.877423
$100$	6.55745	0.655745
$101$	−0.875370	−0.0871026	−0.0435513	−	0.999051i	$-0.513867\pi$
$101$	−0.875370	−0.0871026	−0.0435513	+	0.999051i	$0.513867\pi$
$102$	−17.2118	−1.70423
$103$	2.30046	0.226671	0.113336	−	0.993557i	$-0.463846\pi$
$103$	2.30046	0.226671	0.113336	+	0.993557i	$0.463846\pi$
$104$	−5.10966	−0.501043
$105$	23.1186	2.25614
$106$	−11.2449	−1.09220
$107$	−14.1693	−1.36980	−0.684899	−	0.728638i	$-0.740154\pi$
$107$	−14.1693	−1.36980	−0.684899	+	0.728638i	$0.740154\pi$
$108$	2.04174	0.196467
$109$	−2.76873	−0.265196	−0.132598	−	0.991170i	$-0.542332\pi$
$109$	−2.76873	−0.265196	−0.132598	+	0.991170i	$0.542332\pi$
$110$	14.1635	1.35043
$111$	−5.71982	−0.542901
$112$	3.01257	0.284661
$113$	−5.95901	−0.560576	−0.280288	−	0.959916i	$-0.590430\pi$
$113$	−5.95901	−0.560576	−0.280288	+	0.959916i	$0.590430\pi$
$114$	3.62132	0.339168
$115$	−4.60296	−0.429228
$116$	−1.65335	−0.153510
$117$	10.7073	0.989890
$118$	11.9716	1.10208
$119$	−22.9705	−2.10570
$120$	−7.67405	−0.700542
$121$	6.35711	0.577919
$122$	3.24789	0.294051
$123$	5.39576	0.486519
$124$	−2.96797	−0.266532
$125$	−5.29475	−0.473577
$126$	−6.31284	−0.562393
$127$	9.14594	0.811571	0.405786	−	0.913968i	$-0.366998\pi$
$127$	9.14594	0.811571	0.405786	+	0.913968i	$0.366998\pi$
$128$	−1.00000	−0.0883883
$129$	1.65947	0.146108
$130$	17.3709	1.52353
$131$	6.25909	0.546860	0.273430	−	0.961892i	$-0.411842\pi$
$131$	6.25909	0.546860	0.273430	+	0.961892i	$0.411842\pi$
$132$	−9.40442	−0.818550
$133$	4.83293	0.419068
$134$	−0.353132	−0.0305060
$135$	−6.94116	−0.597400
$136$	7.62489	0.653829
$137$	12.6017	1.07664	0.538319	−	0.842741i	$-0.319060\pi$
$137$	12.6017	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$138$	3.05632	0.260171
$139$	−11.2300	−0.952516	−0.476258	−	0.879306i	$-0.658007\pi$
$139$	−11.2300	−0.952516	−0.476258	+	0.879306i	$0.658007\pi$
$140$	−10.2416	−0.865573
$141$	6.29750	0.530345
$142$	3.85526	0.323526
$143$	21.2878	1.78017
$144$	2.09550	0.174625
$145$	5.62077	0.466779
$146$	13.5364	1.12028
$147$	−4.68521	−0.386429
$148$	2.53390	0.208285
$149$	8.13479	0.666428	0.333214	−	0.942851i	$-0.391867\pi$
$149$	8.13479	0.666428	0.333214	+	0.942851i	$0.391867\pi$
$150$	14.8023	1.20860
$151$	23.7976	1.93662	0.968311	−	0.249747i	$-0.0803475\pi$
$151$	23.7976	1.93662	0.968311	+	0.249747i	$0.0803475\pi$
$152$	−1.60426	−0.130122
$153$	−15.9780	−1.29174
$154$	−12.5509	−1.01138
$155$	10.0900	0.810448
$156$	−11.5341	−0.923471
$157$	12.7425	1.01696	0.508481	−	0.861073i	$-0.330207\pi$
$157$	12.7425	1.01696	0.508481	+	0.861073i	$0.330207\pi$
$158$	4.89001	0.389028
$159$	−25.3833	−2.01303
$160$	3.39963	0.268764
$161$	4.07889	0.321462
$162$	10.8954	0.856022
$163$	−2.51320	−0.196849	−0.0984244	−	0.995145i	$-0.531380\pi$
$163$	−2.51320	−0.196849	−0.0984244	+	0.995145i	$0.531380\pi$
$164$	−2.39033	−0.186654
$165$	31.9715	2.48898
$166$	7.98975	0.620125
$167$	−17.7397	−1.37274	−0.686371	−	0.727252i	$-0.740797\pi$
$167$	−17.7397	−1.37274	−0.686371	+	0.727252i	$0.740797\pi$
$168$	6.80033	0.524657
$169$	13.1086	1.00835
$170$	−25.9218	−1.98811
$171$	3.36172	0.257077
$172$	−0.735148	−0.0560545
$173$	−3.33923	−0.253877	−0.126938	−	0.991911i	$-0.540515\pi$
$173$	−3.33923	−0.253877	−0.126938	+	0.991911i	$0.540515\pi$
$174$	−3.73214	−0.282933
$175$	19.7548	1.49332
$176$	4.16619	0.314038
$177$	27.0238	2.03123
$178$	−16.0276	−1.20132
$179$	−2.85724	−0.213560	−0.106780	−	0.994283i	$-0.534054\pi$
$179$	−2.85724	−0.213560	−0.106780	+	0.994283i	$0.534054\pi$
$180$	−7.12392	−0.530986
$181$	21.5030	1.59831	0.799154	−	0.601126i	$-0.205281\pi$
$181$	21.5030	1.59831	0.799154	+	0.601126i	$0.205281\pi$
$182$	−15.3932	−1.14102
$183$	7.33154	0.541963
$184$	−1.35396	−0.0998152
$185$	−8.61430	−0.633336
$186$	−6.69967	−0.491244
$187$	−31.7667	−2.32301
$188$	−2.78981	−0.203468
$189$	6.15089	0.447411
$190$	5.45387	0.395665
$191$	18.5293	1.34073	0.670365	−	0.742031i	$-0.266138\pi$
$191$	18.5293	1.34073	0.670365	+	0.742031i	$0.266138\pi$
$192$	−2.25732	−0.162908
$193$	8.37556	0.602886	0.301443	−	0.953484i	$-0.402532\pi$
$193$	8.37556	0.602886	0.301443	+	0.953484i	$0.402532\pi$
$194$	0.0540933	0.00388367
$195$	39.2118	2.80801
$196$	2.07556	0.148254
$197$	−0.431387	−0.0307351	−0.0153675	−	0.999882i	$-0.504892\pi$
$197$	−0.431387	−0.0307351	−0.0153675	+	0.999882i	$0.504892\pi$
$198$	−8.73025	−0.620432
$199$	19.1587	1.35812	0.679062	−	0.734081i	$-0.262387\pi$
$199$	19.1587	1.35812	0.679062	+	0.734081i	$0.262387\pi$
$200$	−6.55745	−0.463682
$201$	−0.797133	−0.0562254
$202$	0.875370	0.0615908
$203$	−4.98082	−0.349585
$204$	17.2118	1.20507
$205$	8.12624	0.567561
$206$	−2.30046	−0.160281
$207$	2.83723	0.197201
$208$	5.10966	0.354291
$209$	6.68363	0.462316
$210$	−23.1186	−1.59533
$211$	−14.1464	−0.973879	−0.486939	−	0.873436i	$-0.661887\pi$
$211$	−14.1464	−0.973879	−0.486939	+	0.873436i	$0.661887\pi$
$212$	11.2449	0.772302
$213$	8.70255	0.596289
$214$	14.1693	0.968593
$215$	2.49923	0.170446
$216$	−2.04174	−0.138923
$217$	−8.94121	−0.606969
$218$	2.76873	0.187522
$219$	30.5560	2.06478
$220$	−14.1635	−0.954901
$221$	−38.9606	−2.62077
$222$	5.71982	0.383889
$223$	13.8229	0.925651	0.462825	−	0.886450i	$-0.346836\pi$
$223$	13.8229	0.925651	0.462825	+	0.886450i	$0.346836\pi$
$224$	−3.01257	−0.201286
$225$	13.7412	0.916077
$226$	5.95901	0.396387
$227$	−17.0945	−1.13460	−0.567300	−	0.823511i	$-0.692012\pi$
$227$	−17.0945	−1.13460	−0.567300	+	0.823511i	$0.692012\pi$
$228$	−3.62132	−0.239828
$229$	0.0325964	0.00215403	0.00107701	−	0.999999i	$-0.499657\pi$
$229$	0.0325964	0.00215403	0.00107701	+	0.999999i	$0.499657\pi$
$230$	4.60296	0.303510
$231$	−28.3315	−1.86407
$232$	1.65335	0.108548
$233$	−6.25048	−0.409483	−0.204741	−	0.978816i	$-0.565635\pi$
$233$	−6.25048	−0.409483	−0.204741	+	0.978816i	$0.565635\pi$
$234$	−10.7073	−0.699958
$235$	9.48430	0.618688
$236$	−11.9716	−0.779285
$237$	11.0383	0.717016
$238$	22.9705	1.48896
$239$	8.39388	0.542955	0.271478	−	0.962445i	$-0.412488\pi$
$239$	8.39388	0.542955	0.271478	+	0.962445i	$0.412488\pi$
$240$	7.67405	0.495358
$241$	−12.6023	−0.811788	−0.405894	−	0.913920i	$-0.633040\pi$
$241$	−12.6023	−0.811788	−0.405894	+	0.913920i	$0.633040\pi$
$242$	−6.35711	−0.408651
$243$	18.4691	1.18480
$244$	−3.24789	−0.207925
$245$	−7.05612	−0.450799
$246$	−5.39576	−0.344021
$247$	8.19720	0.521575
$248$	2.96797	0.188466
$249$	18.0354	1.14295
$250$	5.29475	0.334869
$251$	−9.27420	−0.585382	−0.292691	−	0.956207i	$-0.594551\pi$
$251$	−9.27420	−0.585382	−0.292691	+	0.956207i	$0.594551\pi$
$252$	6.31284	0.397672
$253$	5.64085	0.354637
$254$	−9.14594	−0.573868
$255$	−58.5138	−3.66428
$256$	1.00000	0.0625000
$257$	4.78182	0.298282	0.149141	−	0.988816i	$-0.452349\pi$
$257$	4.78182	0.298282	0.149141	+	0.988816i	$0.452349\pi$
$258$	−1.65947	−0.103314
$259$	7.63353	0.474325
$260$	−17.3709	−1.07730
$261$	−3.46460	−0.214453
$262$	−6.25909	−0.386688
$263$	−2.62380	−0.161790	−0.0808951	−	0.996723i	$-0.525778\pi$
$263$	−2.62380	−0.161790	−0.0808951	+	0.996723i	$0.525778\pi$
$264$	9.40442	0.578802
$265$	−38.2284	−2.34835
$266$	−4.83293	−0.296326
$267$	−36.1795	−2.21415
$268$	0.353132	0.0215710
$269$	−22.6169	−1.37898	−0.689489	−	0.724296i	$-0.742165\pi$
$269$	−22.6169	−1.37898	−0.689489	+	0.724296i	$0.742165\pi$
$270$	6.94116	0.422426
$271$	4.89524	0.297365	0.148682	−	0.988885i	$-0.452497\pi$
$271$	4.89524	0.297365	0.148682	+	0.988885i	$0.452497\pi$
$272$	−7.62489	−0.462327
$273$	−34.7474	−2.10301
$274$	−12.6017	−0.761298
$275$	27.3196	1.64743
$276$	−3.05632	−0.183969
$277$	−24.9917	−1.50160	−0.750802	−	0.660527i	$-0.770333\pi$
$277$	−24.9917	−1.50160	−0.750802	+	0.660527i	$0.770333\pi$
$278$	11.2300	0.673530
$279$	−6.21939	−0.372345
$280$	10.2416	0.612053
$281$	19.0330	1.13541	0.567706	−	0.823232i	$-0.307831\pi$
$281$	19.0330	1.13541	0.567706	+	0.823232i	$0.307831\pi$
$282$	−6.29750	−0.375010
$283$	−12.4021	−0.737226	−0.368613	−	0.929583i	$-0.620167\pi$
$283$	−12.4021	−0.737226	−0.368613	+	0.929583i	$0.620167\pi$
$284$	−3.85526	−0.228767
$285$	12.3111	0.729249
$286$	−21.2878	−1.25877
$287$	−7.20104	−0.425064
$288$	−2.09550	−0.123479
$289$	41.1390	2.41994
$290$	−5.62077	−0.330063
$291$	0.122106	0.00715798
$292$	−13.5364	−0.792157
$293$	11.8090	0.689892	0.344946	−	0.938623i	$-0.387897\pi$
$293$	11.8090	0.689892	0.344946	+	0.938623i	$0.387897\pi$
$294$	4.68521	0.273247
$295$	40.6990	2.36959
$296$	−2.53390	−0.147280
$297$	8.50628	0.493585
$298$	−8.13479	−0.471236
$299$	6.91827	0.400094
$300$	−14.8023	−0.854610
$301$	−2.21468	−0.127652
$302$	−23.7976	−1.36940
$303$	1.97599	0.113518
$304$	1.60426	0.0920104
$305$	11.0416	0.632242
$306$	15.9780	0.913400
$307$	−3.01736	−0.172210	−0.0861049	−	0.996286i	$-0.527442\pi$
$307$	−3.01736	−0.172210	−0.0861049	+	0.996286i	$0.527442\pi$
$308$	12.5509	0.715155
$309$	−5.19288	−0.295413
$310$	−10.0900	−0.573073
$311$	−16.0797	−0.911797	−0.455899	−	0.890032i	$-0.650682\pi$
$311$	−16.0797	−0.911797	−0.455899	+	0.890032i	$0.650682\pi$
$312$	11.5341	0.652992
$313$	−14.4361	−0.815978	−0.407989	−	0.912987i	$-0.633770\pi$
$313$	−14.4361	−0.815978	−0.407989	+	0.912987i	$0.633770\pi$
$314$	−12.7425	−0.719101
$315$	−21.4613	−1.20921
$316$	−4.89001	−0.275085
$317$	−13.5693	−0.762128	−0.381064	−	0.924549i	$-0.624442\pi$
$317$	−13.5693	−0.762128	−0.381064	+	0.924549i	$0.624442\pi$
$318$	25.3833	1.42343
$319$	−6.88816	−0.385663
$320$	−3.39963	−0.190045
$321$	31.9847	1.78521
$322$	−4.07889	−0.227308
$323$	−12.2323	−0.680622
$324$	−10.8954	−0.605299
$325$	33.5063	1.85860
$326$	2.51320	0.139193
$327$	6.24990	0.345621
$328$	2.39033	0.131984
$329$	−8.40449	−0.463354
$330$	−31.9715	−1.75997
$331$	27.1225	1.49079	0.745394	−	0.666624i	$-0.232261\pi$
$331$	27.1225	1.49079	0.745394	+	0.666624i	$0.232261\pi$
$332$	−7.98975	−0.438495
$333$	5.30979	0.290975
$334$	17.7397	0.970675
$335$	−1.20052	−0.0655912
$336$	−6.80033	−0.370989
$337$	−14.0098	−0.763164	−0.381582	−	0.924335i	$-0.624621\pi$
$337$	−14.0098	−0.763164	−0.381582	+	0.924335i	$0.624621\pi$
$338$	−13.1086	−0.713014
$339$	13.4514	0.730579
$340$	25.9218	1.40581
$341$	−12.3651	−0.669609
$342$	−3.36172	−0.181781
$343$	−14.8352	−0.801026
$344$	0.735148	0.0396365
$345$	10.3904	0.559398
$346$	3.33923	0.179518
$347$	14.7285	0.790669	0.395335	−	0.918537i	$-0.370629\pi$
$347$	14.7285	0.790669	0.395335	+	0.918537i	$0.370629\pi$
$348$	3.73214	0.200064
$349$	11.3677	0.608501	0.304250	−	0.952592i	$-0.401594\pi$
$349$	11.3677	0.608501	0.304250	+	0.952592i	$0.401594\pi$
$350$	−19.7548	−1.05594
$351$	10.4326	0.556851
$352$	−4.16619	−0.222058
$353$	−29.0859	−1.54808	−0.774042	−	0.633134i	$-0.781768\pi$
$353$	−29.0859	−1.54808	−0.774042	+	0.633134i	$0.781768\pi$
$354$	−27.0238	−1.43630
$355$	13.1064	0.695617
$356$	16.0276	0.849462
$357$	51.8518	2.74429
$358$	2.85724	0.151010
$359$	−24.0181	−1.26763	−0.633813	−	0.773486i	$-0.718511\pi$
$359$	−24.0181	−1.26763	−0.633813	+	0.773486i	$0.718511\pi$
$360$	7.12392	0.375464
$361$	−16.4264	−0.864545
$362$	−21.5030	−1.13017
$363$	−14.3500	−0.753182
$364$	15.3932	0.806822
$365$	46.0186	2.40872
$366$	−7.33154	−0.383226
$367$	−4.31979	−0.225491	−0.112746	−	0.993624i	$-0.535964\pi$
$367$	−4.31979	−0.225491	−0.112746	+	0.993624i	$0.535964\pi$
$368$	1.35396	0.0705800
$369$	−5.00895	−0.260756
$370$	8.61430	0.447836
$371$	33.8760	1.75875
$372$	6.69967	0.347362
$373$	−24.2731	−1.25681	−0.628406	−	0.777886i	$-0.716292\pi$
$373$	−24.2731	−1.25681	−0.628406	+	0.777886i	$0.716292\pi$
$374$	31.7667	1.64262
$375$	11.9519	0.617196
$376$	2.78981	0.143873
$377$	−8.44805	−0.435097
$378$	−6.15089	−0.316367
$379$	−36.5410	−1.87698	−0.938492	−	0.345301i	$-0.887777\pi$
$379$	−36.5410	−1.87698	−0.938492	+	0.345301i	$0.887777\pi$
$380$	−5.45387	−0.279778
$381$	−20.6453	−1.05769
$382$	−18.5293	−0.948039
$383$	−30.8397	−1.57583	−0.787917	−	0.615781i	$-0.788840\pi$
$383$	−30.8397	−1.57583	−0.787917	+	0.615781i	$0.788840\pi$
$384$	2.25732	0.115193
$385$	−42.6684	−2.17458
$386$	−8.37556	−0.426305
$387$	−1.54050	−0.0783082
$388$	−0.0540933	−0.00274617
$389$	−23.5188	−1.19245	−0.596225	−	0.802817i	$-0.703334\pi$
$389$	−23.5188	−1.19245	−0.596225	+	0.802817i	$0.703334\pi$
$390$	−39.2118	−1.98556
$391$	−10.3238	−0.522097
$392$	−2.07556	−0.104832
$393$	−14.1288	−0.712703
$394$	0.431387	0.0217330
$395$	16.6242	0.836454
$396$	8.73025	0.438712
$397$	1.12932	0.0566787	0.0283394	−	0.999598i	$-0.490978\pi$
$397$	1.12932	0.0566787	0.0283394	+	0.999598i	$0.490978\pi$
$398$	−19.1587	−0.960339
$399$	−10.9095	−0.546157
$400$	6.55745	0.327873
$401$	−2.99685	−0.149656	−0.0748278	−	0.997196i	$-0.523841\pi$
$401$	−2.99685	−0.149656	−0.0748278	+	0.997196i	$0.523841\pi$
$402$	0.797133	0.0397574
$403$	−15.1653	−0.755439
$404$	−0.875370	−0.0435513
$405$	37.0402	1.84054
$406$	4.98082	0.247194
$407$	10.5567	0.523276
$408$	−17.2118	−0.852113
$409$	−0.671672	−0.0332121	−0.0166060	−	0.999862i	$-0.505286\pi$
$409$	−0.671672	−0.0332121	−0.0166060	+	0.999862i	$0.505286\pi$
$410$	−8.12624	−0.401327
$411$	−28.4461	−1.40314
$412$	2.30046	0.113336
$413$	−36.0653	−1.77466
$414$	−2.83723	−0.139442
$415$	27.1622	1.33334
$416$	−5.10966	−0.250522
$417$	25.3497	1.24138
$418$	−6.68363	−0.326907
$419$	8.91735	0.435641	0.217821	−	0.975989i	$-0.430105\pi$
$419$	8.91735	0.435641	0.217821	+	0.975989i	$0.430105\pi$
$420$	23.1186	1.12807
$421$	−8.63228	−0.420712	−0.210356	−	0.977625i	$-0.567462\pi$
$421$	−8.63228	−0.420712	−0.210356	+	0.977625i	$0.567462\pi$
$422$	14.1464	0.688636
$423$	−5.84605	−0.284245
$424$	−11.2449	−0.546100
$425$	−49.9999	−2.42535
$426$	−8.70255	−0.421640
$427$	−9.78450	−0.473505
$428$	−14.1693	−0.684899
$429$	−48.0534	−2.32004
$430$	−2.49923	−0.120523
$431$	2.97642	0.143369	0.0716846	−	0.997427i	$-0.477162\pi$
$431$	2.97642	0.143369	0.0716846	+	0.997427i	$0.477162\pi$
$432$	2.04174	0.0982334
$433$	−11.4056	−0.548119	−0.274060	−	0.961713i	$-0.588367\pi$
$433$	−11.4056	−0.548119	−0.274060	+	0.961713i	$0.588367\pi$
$434$	8.94121	0.429192
$435$	−12.6879	−0.608337
$436$	−2.76873	−0.132598
$437$	2.17210	0.103906
$438$	−30.5560	−1.46002
$439$	28.9601	1.38219	0.691096	−	0.722763i	$-0.257128\pi$
$439$	28.9601	1.38219	0.691096	+	0.722763i	$0.257128\pi$
$440$	14.1635	0.675217
$441$	4.34934	0.207111
$442$	38.9606	1.85317
$443$	−20.7667	−0.986654	−0.493327	−	0.869844i	$-0.664219\pi$
$443$	−20.7667	−0.986654	−0.493327	+	0.869844i	$0.664219\pi$
$444$	−5.71982	−0.271451
$445$	−54.4879	−2.58297
$446$	−13.8229	−0.654534
$447$	−18.3628	−0.868532
$448$	3.01257	0.142330
$449$	29.6950	1.40140	0.700698	−	0.713458i	$-0.252872\pi$
$449$	29.6950	1.40140	0.700698	+	0.713458i	$0.252872\pi$
$450$	−13.7412	−0.647764
$451$	−9.95858	−0.468931
$452$	−5.95901	−0.280288
$453$	−53.7189	−2.52393
$454$	17.0945	0.802283
$455$	−52.3311	−2.45332
$456$	3.62132	0.169584
$457$	6.14617	0.287506	0.143753	−	0.989614i	$-0.454083\pi$
$457$	6.14617	0.287506	0.143753	+	0.989614i	$0.454083\pi$
$458$	−0.0325964	−0.00152313
$459$	−15.5681	−0.726655
$460$	−4.60296	−0.214614
$461$	34.2649	1.59588	0.797939	−	0.602739i	$-0.205924\pi$
$461$	34.2649	1.59588	0.797939	+	0.602739i	$0.205924\pi$
$462$	28.3315	1.31810
$463$	2.17470	0.101067	0.0505335	−	0.998722i	$-0.483908\pi$
$463$	2.17470	0.101067	0.0505335	+	0.998722i	$0.483908\pi$
$464$	−1.65335	−0.0767548
$465$	−22.7764	−1.05623
$466$	6.25048	0.289548
$467$	−32.3688	−1.49785	−0.748924	−	0.662656i	$-0.769429\pi$
$467$	−32.3688	−1.49785	−0.748924	+	0.662656i	$0.769429\pi$
$468$	10.7073	0.494945
$469$	1.06383	0.0491233
$470$	−9.48430	−0.437478
$471$	−28.7639	−1.32537
$472$	11.9716	0.551038
$473$	−3.06276	−0.140826
$474$	−11.0383	−0.507007
$475$	10.5198	0.482683
$476$	−22.9705	−1.05285
$477$	23.5637	1.07891
$478$	−8.39388	−0.383927
$479$	−13.8014	−0.630600	−0.315300	−	0.948992i	$-0.602105\pi$
$479$	−13.8014	−0.630600	−0.315300	+	0.948992i	$0.602105\pi$
$480$	−7.67405	−0.350271
$481$	12.9473	0.590348
$482$	12.6023	0.574021
$483$	−9.20738	−0.418950
$484$	6.35711	0.288960
$485$	0.183897	0.00835033
$486$	−18.4691	−0.837777
$487$	14.6551	0.664087	0.332044	−	0.943264i	$-0.392262\pi$
$487$	14.6551	0.664087	0.332044	+	0.943264i	$0.392262\pi$
$488$	3.24789	0.147025
$489$	5.67309	0.256546
$490$	7.05612	0.318763
$491$	−6.19311	−0.279491	−0.139746	−	0.990187i	$-0.544628\pi$
$491$	−6.19311	−0.279491	−0.139746	+	0.990187i	$0.544628\pi$
$492$	5.39576	0.243259
$493$	12.6066	0.567773
$494$	−8.19720	−0.368809
$495$	−29.6796	−1.33400
$496$	−2.96797	−0.133266
$497$	−11.6142	−0.520969
$498$	−18.0354	−0.808187
$499$	−13.6261	−0.609989	−0.304995	−	0.952354i	$-0.598655\pi$
$499$	−13.6261	−0.609989	−0.304995	+	0.952354i	$0.598655\pi$
$500$	−5.29475	−0.236788
$501$	40.0443	1.78905
$502$	9.27420	0.413928
$503$	19.9720	0.890507	0.445253	−	0.895405i	$-0.353114\pi$
$503$	19.9720	0.890507	0.445253	+	0.895405i	$0.353114\pi$
$504$	−6.31284	−0.281196
$505$	2.97593	0.132427
$506$	−5.64085	−0.250766
$507$	−29.5903	−1.31415
$508$	9.14594	0.405786
$509$	−16.2908	−0.722076	−0.361038	−	0.932551i	$-0.617578\pi$
$509$	−16.2908	−0.722076	−0.361038	+	0.932551i	$0.617578\pi$
$510$	58.5138	2.59103
$511$	−40.7792	−1.80397
$512$	−1.00000	−0.0441942
$513$	3.27548	0.144616
$514$	−4.78182	−0.210917
$515$	−7.82070	−0.344621
$516$	1.65947	0.0730539
$517$	−11.6229	−0.511173
$518$	−7.63353	−0.335398
$519$	7.53771	0.330869
$520$	17.3709	0.761765
$521$	3.60840	0.158087	0.0790435	−	0.996871i	$-0.474813\pi$
$521$	3.60840	0.158087	0.0790435	+	0.996871i	$0.474813\pi$
$522$	3.46460	0.151641
$523$	−37.5837	−1.64342	−0.821711	−	0.569904i	$-0.806980\pi$
$523$	−37.5837	−1.64342	−0.821711	+	0.569904i	$0.806980\pi$
$524$	6.25909	0.273430
$525$	−44.5928	−1.94619
$526$	2.62380	0.114403
$527$	22.6305	0.985799
$528$	−9.40442	−0.409275
$529$	−21.1668	−0.920295
$530$	38.2284	1.66054
$531$	−25.0865	−1.08866
$532$	4.83293	0.209534
$533$	−12.2138	−0.529038
$534$	36.1795	1.56564
$535$	48.1703	2.08258
$536$	−0.353132	−0.0152530
$537$	6.44972	0.278326
$538$	22.6169	0.975085
$539$	8.64717	0.372460
$540$	−6.94116	−0.298700
$541$	−11.4288	−0.491363	−0.245681	−	0.969351i	$-0.579012\pi$
$541$	−11.4288	−0.491363	−0.245681	+	0.969351i	$0.579012\pi$
$542$	−4.89524	−0.210269
$543$	−48.5393	−2.08302
$544$	7.62489	0.326915
$545$	9.41263	0.403193
$546$	34.7474	1.48705
$547$	5.12653	0.219195	0.109597	−	0.993976i	$-0.465044\pi$
$547$	5.12653	0.219195	0.109597	+	0.993976i	$0.465044\pi$
$548$	12.6017	0.538319
$549$	−6.80597	−0.290472
$550$	−27.3196	−1.16491
$551$	−2.65239	−0.112996
$552$	3.05632	0.130086
$553$	−14.7315	−0.626446
$554$	24.9917	1.06179
$555$	19.4452	0.825405
$556$	−11.2300	−0.476258
$557$	−21.7495	−0.921557	−0.460779	−	0.887515i	$-0.652430\pi$
$557$	−21.7495	−0.921557	−0.460779	+	0.887515i	$0.652430\pi$
$558$	6.21939	0.263288
$559$	−3.75635	−0.158877
$560$	−10.2416	−0.432786
$561$	71.7077	3.02750
$562$	−19.0330	−0.802857
$563$	34.2164	1.44205	0.721025	−	0.692909i	$-0.243671\pi$
$563$	34.2164	1.44205	0.721025	+	0.692909i	$0.243671\pi$
$564$	6.29750	0.265172
$565$	20.2584	0.852277
$566$	12.4021	0.521298
$567$	−32.8231	−1.37844
$568$	3.85526	0.161763
$569$	29.2502	1.22623	0.613116	−	0.789993i	$-0.289916\pi$
$569$	29.2502	1.22623	0.613116	+	0.789993i	$0.289916\pi$
$570$	−12.3111	−0.515657
$571$	−23.4463	−0.981197	−0.490599	−	0.871386i	$-0.663222\pi$
$571$	−23.4463	−0.981197	−0.490599	+	0.871386i	$0.663222\pi$
$572$	21.2878	0.890087
$573$	−41.8265	−1.74733
$574$	7.20104	0.300566
$575$	8.87852	0.370260
$576$	2.09550	0.0873126
$577$	34.7166	1.44527	0.722635	−	0.691230i	$-0.242931\pi$
$577$	34.7166	1.44527	0.722635	+	0.691230i	$0.242931\pi$
$578$	−41.1390	−1.71116
$579$	−18.9063	−0.785720
$580$	5.62077	0.233390
$581$	−24.0697	−0.998578
$582$	−0.122106	−0.00506146
$583$	46.8483	1.94026
$584$	13.5364	0.560139
$585$	−36.4008	−1.50499
$586$	−11.8090	−0.487827
$587$	2.13075	0.0879453	0.0439727	−	0.999033i	$-0.485999\pi$
$587$	2.13075	0.0879453	0.0439727	+	0.999033i	$0.485999\pi$
$588$	−4.68521	−0.193215
$589$	−4.76139	−0.196190
$590$	−40.6990	−1.67555
$591$	0.973780	0.0400560
$592$	2.53390	0.104143
$593$	−31.6878	−1.30126	−0.650632	−	0.759393i	$-0.725496\pi$
$593$	−31.6878	−1.30126	−0.650632	+	0.759393i	$0.725496\pi$
$594$	−8.50628	−0.349017
$595$	78.0911	3.20142
$596$	8.13479	0.333214
$597$	−43.2473	−1.77000
$598$	−6.91827	−0.282909
$599$	43.3852	1.77267	0.886335	−	0.463045i	$-0.153243\pi$
$599$	43.3852	1.77267	0.886335	+	0.463045i	$0.153243\pi$
$600$	14.8023	0.604300
$601$	−30.0543	−1.22594	−0.612971	−	0.790106i	$-0.710026\pi$
$601$	−30.0543	−1.22594	−0.612971	+	0.790106i	$0.710026\pi$
$602$	2.21468	0.0902637
$603$	0.739989	0.0301347
$604$	23.7976	0.968311
$605$	−21.6118	−0.878644
$606$	−1.97599	−0.0802692
$607$	−15.5735	−0.632108	−0.316054	−	0.948741i	$-0.602358\pi$
$607$	−15.5735	−0.632108	−0.316054	+	0.948741i	$0.602358\pi$
$608$	−1.60426	−0.0650612
$609$	11.2433	0.455602
$610$	−11.0416	−0.447062
$611$	−14.2550	−0.576694
$612$	−15.9780	−0.645871
$613$	−13.2736	−0.536116	−0.268058	−	0.963403i	$-0.586382\pi$
$613$	−13.2736	−0.536116	−0.268058	+	0.963403i	$0.586382\pi$
$614$	3.01736	0.121771
$615$	−18.3435	−0.739683
$616$	−12.5509	−0.505691
$617$	−16.0415	−0.645805	−0.322902	−	0.946432i	$-0.604659\pi$
$617$	−16.0415	−0.645805	−0.322902	+	0.946432i	$0.604659\pi$
$618$	5.19288	0.208888
$619$	0.873608	0.0351133	0.0175566	−	0.999846i	$-0.494411\pi$
$619$	0.873608	0.0351133	0.0175566	+	0.999846i	$0.494411\pi$
$620$	10.0900	0.405224
$621$	2.76444	0.110933
$622$	16.0797	0.644738
$623$	48.2842	1.93447
$624$	−11.5341	−0.461735
$625$	−14.7871	−0.591484
$626$	14.4361	0.576984
$627$	−15.0871	−0.602521
$628$	12.7425	0.508481
$629$	−19.3207	−0.770366
$630$	21.4613	0.855038
$631$	10.4944	0.417777	0.208889	−	0.977939i	$-0.433015\pi$
$631$	10.4944	0.417777	0.208889	+	0.977939i	$0.433015\pi$
$632$	4.89001	0.194514
$633$	31.9330	1.26922
$634$	13.5693	0.538906
$635$	−31.0928	−1.23388
$636$	−25.3833	−1.00651
$637$	10.6054	0.420201
$638$	6.88816	0.272705
$639$	−8.07870	−0.319588
$640$	3.39963	0.134382
$641$	30.0346	1.18630	0.593148	−	0.805094i	$-0.297885\pi$
$641$	30.0346	1.18630	0.593148	+	0.805094i	$0.297885\pi$
$642$	−31.9847	−1.26233
$643$	44.9893	1.77421	0.887103	−	0.461571i	$-0.152714\pi$
$643$	44.9893	1.77421	0.887103	+	0.461571i	$0.152714\pi$
$644$	4.07889	0.160731
$645$	−5.64156	−0.222136
$646$	12.2323	0.481273
$647$	2.57234	0.101129	0.0505645	−	0.998721i	$-0.483898\pi$
$647$	2.57234	0.101129	0.0505645	+	0.998721i	$0.483898\pi$
$648$	10.8954	0.428011
$649$	−49.8760	−1.95780
$650$	−33.5063	−1.31423
$651$	20.1832	0.791042
$652$	−2.51320	−0.0984244
$653$	−14.7389	−0.576777	−0.288389	−	0.957513i	$-0.593120\pi$
$653$	−14.7389	−0.576777	−0.288389	+	0.957513i	$0.593120\pi$
$654$	−6.24990	−0.244391
$655$	−21.2786	−0.831423
$656$	−2.39033	−0.0933269
$657$	−28.3655	−1.10664
$658$	8.40449	0.327641
$659$	42.9284	1.67225	0.836127	−	0.548536i	$-0.184815\pi$
$659$	42.9284	1.67225	0.836127	+	0.548536i	$0.184815\pi$
$660$	31.9715	1.24449
$661$	−7.95555	−0.309435	−0.154718	−	0.987959i	$-0.549447\pi$
$661$	−7.95555	−0.309435	−0.154718	+	0.987959i	$0.549447\pi$
$662$	−27.1225	−1.05415
$663$	87.9466	3.41556
$664$	7.98975	0.310062
$665$	−16.4301	−0.637134
$666$	−5.30979	−0.205750
$667$	−2.23857	−0.0866777
$668$	−17.7397	−0.686371
$669$	−31.2028	−1.20637
$670$	1.20052	0.0463800
$671$	−13.5313	−0.522371
$672$	6.80033	0.262329
$673$	8.89014	0.342690	0.171345	−	0.985211i	$-0.445189\pi$
$673$	8.89014	0.342690	0.171345	+	0.985211i	$0.445189\pi$
$674$	14.0098	0.539638
$675$	13.3886	0.515328
$676$	13.1086	0.504177
$677$	4.75592	0.182785	0.0913923	−	0.995815i	$-0.470868\pi$
$677$	4.75592	0.182785	0.0913923	+	0.995815i	$0.470868\pi$
$678$	−13.4514	−0.516598
$679$	−0.162960	−0.00625382
$680$	−25.9218	−0.994055
$681$	38.5877	1.47868
$682$	12.3651	0.473485
$683$	−17.8334	−0.682376	−0.341188	−	0.939995i	$-0.610829\pi$
$683$	−17.8334	−0.682376	−0.341188	+	0.939995i	$0.610829\pi$
$684$	3.36172	0.128539
$685$	−42.8411	−1.63688
$686$	14.8352	0.566411
$687$	−0.0735805	−0.00280727
$688$	−0.735148	−0.0280272
$689$	57.4575	2.18896
$690$	−10.3904	−0.395554
$691$	26.0131	0.989584	0.494792	−	0.869011i	$-0.335244\pi$
$691$	26.0131	0.989584	0.494792	+	0.869011i	$0.335244\pi$
$692$	−3.33923	−0.126938
$693$	26.3005	0.999072
$694$	−14.7285	−0.559088
$695$	38.1778	1.44817
$696$	−3.73214	−0.141466
$697$	18.2260	0.690361
$698$	−11.3677	−0.430275
$699$	14.1094	0.533665
$700$	19.7548	0.746660
$701$	31.0610	1.17316	0.586579	−	0.809892i	$-0.300474\pi$
$701$	31.0610	1.17316	0.586579	+	0.809892i	$0.300474\pi$
$702$	−10.4326	−0.393753
$703$	4.06502	0.153315
$704$	4.16619	0.157019
$705$	−21.4091	−0.806314
$706$	29.0859	1.09466
$707$	−2.63711	−0.0991788
$708$	27.0238	1.01562
$709$	26.9172	1.01090	0.505448	−	0.862857i	$-0.331327\pi$
$709$	26.9172	1.01090	0.505448	+	0.862857i	$0.331327\pi$
$710$	−13.1064	−0.491875
$711$	−10.2470	−0.384293
$712$	−16.0276	−0.600660
$713$	−4.01851	−0.150495
$714$	−51.8518	−1.94051
$715$	−72.3705	−2.70650
$716$	−2.85724	−0.106780
$717$	−18.9477	−0.707615
$718$	24.0181	0.896347
$719$	−13.9336	−0.519634	−0.259817	−	0.965658i	$-0.583662\pi$
$719$	−13.9336	−0.519634	−0.259817	+	0.965658i	$0.583662\pi$
$720$	−7.12392	−0.265493
$721$	6.93029	0.258098
$722$	16.4264	0.611326
$723$	28.4475	1.05798
$724$	21.5030	0.799154
$725$	−10.8418	−0.402653
$726$	14.3500	0.532580
$727$	−35.9964	−1.33503	−0.667516	−	0.744595i	$-0.732642\pi$
$727$	−35.9964	−1.33503	−0.667516	+	0.744595i	$0.732642\pi$
$728$	−15.3932	−0.570509
$729$	−9.00467	−0.333506
$730$	−46.0186	−1.70323
$731$	5.60542	0.207324
$732$	7.33154	0.270982
$733$	14.6852	0.542410	0.271205	−	0.962522i	$-0.412578\pi$
$733$	14.6852	0.542410	0.271205	+	0.962522i	$0.412578\pi$
$734$	4.31979	0.159446
$735$	15.9279	0.587511
$736$	−1.35396	−0.0499076
$737$	1.47121	0.0541929
$738$	5.00895	0.184382
$739$	−31.4975	−1.15865	−0.579327	−	0.815095i	$-0.696685\pi$
$739$	−31.4975	−1.15865	−0.579327	+	0.815095i	$0.696685\pi$
$740$	−8.61430	−0.316668
$741$	−18.5037	−0.679751
$742$	−33.8760	−1.24363
$743$	50.6992	1.85997	0.929986	−	0.367595i	$-0.119819\pi$
$743$	50.6992	1.85997	0.929986	+	0.367595i	$0.119819\pi$
$744$	−6.69967	−0.245622
$745$	−27.6552	−1.01321
$746$	24.2731	0.888700
$747$	−16.7425	−0.612578
$748$	−31.7667	−1.16151
$749$	−42.6860	−1.55971
$750$	−11.9519	−0.436423
$751$	−9.26026	−0.337912	−0.168956	−	0.985624i	$-0.554040\pi$
$751$	−9.26026	−0.337912	−0.168956	+	0.985624i	$0.554040\pi$
$752$	−2.78981	−0.101734
$753$	20.9349	0.762908
$754$	8.44805	0.307660
$755$	−80.9030	−2.94436
$756$	6.15089	0.223706
$757$	20.0724	0.729542	0.364771	−	0.931097i	$-0.381147\pi$
$757$	20.0724	0.729542	0.364771	+	0.931097i	$0.381147\pi$
$758$	36.5410	1.32723
$759$	−12.7332	−0.462186
$760$	5.45387	0.197833
$761$	30.8830	1.11951	0.559754	−	0.828659i	$-0.310896\pi$
$761$	30.8830	1.11951	0.559754	+	0.828659i	$0.310896\pi$
$762$	20.6453	0.747902
$763$	−8.34097	−0.301963
$764$	18.5293	0.670365
$765$	54.3191	1.96391
$766$	30.8397	1.11428
$767$	−61.1708	−2.20875
$768$	−2.25732	−0.0814541
$769$	−18.5920	−0.670446	−0.335223	−	0.942139i	$-0.608812\pi$
$769$	−18.5920	−0.670446	−0.335223	+	0.942139i	$0.608812\pi$
$770$	42.6684	1.53766
$771$	−10.7941	−0.388740
$772$	8.37556	0.301443
$773$	−30.5935	−1.10037	−0.550186	−	0.835042i	$-0.685443\pi$
$773$	−30.5935	−1.10037	−0.550186	+	0.835042i	$0.685443\pi$
$774$	1.54050	0.0553723
$775$	−19.4623	−0.699108
$776$	0.0540933	0.00194184
$777$	−17.2313	−0.618171
$778$	23.5188	0.843190
$779$	−3.83471	−0.137393
$780$	39.2118	1.40401
$781$	−16.0617	−0.574733
$782$	10.3238	0.369178
$783$	−3.37571	−0.120638
$784$	2.07556	0.0741271
$785$	−43.3197	−1.54615
$786$	14.1288	0.503957
$787$	−3.75188	−0.133740	−0.0668700	−	0.997762i	$-0.521301\pi$
$787$	−3.75188	−0.133740	−0.0668700	+	0.997762i	$0.521301\pi$
$788$	−0.431387	−0.0153675
$789$	5.92275	0.210855
$790$	−16.6242	−0.591462
$791$	−17.9519	−0.638296
$792$	−8.73025	−0.310216
$793$	−16.5956	−0.589328
$794$	−1.12932	−0.0400779
$795$	86.2938	3.06053
$796$	19.1587	0.679062
$797$	40.4021	1.43112	0.715559	−	0.698553i	$-0.246172\pi$
$797$	40.4021	1.43112	0.715559	+	0.698553i	$0.246172\pi$
$798$	10.9095	0.386191
$799$	21.2720	0.752549
$800$	−6.55745	−0.231841
$801$	33.5859	1.18670
$802$	2.99685	0.105823
$803$	−56.3951	−1.99014
$804$	−0.797133	−0.0281127
$805$	−13.8667	−0.488737
$806$	15.1653	0.534176
$807$	51.0537	1.79717
$808$	0.875370	0.0307954
$809$	37.8275	1.32995	0.664973	−	0.746868i	$-0.268443\pi$
$809$	37.8275	1.32995	0.664973	+	0.746868i	$0.268443\pi$
$810$	−37.0402	−1.30146
$811$	1.46398	0.0514075	0.0257037	−	0.999670i	$-0.491817\pi$
$811$	1.46398	0.0514075	0.0257037	+	0.999670i	$0.491817\pi$
$812$	−4.98082	−0.174793
$813$	−11.0501	−0.387545
$814$	−10.5567	−0.370012
$815$	8.54393	0.299281
$816$	17.2118	0.602535
$817$	−1.17937	−0.0412608
$818$	0.671672	0.0234845
$819$	32.2565	1.12713
$820$	8.12624	0.283781
$821$	−11.0306	−0.384971	−0.192485	−	0.981300i	$-0.561655\pi$
$821$	−11.0306	−0.384971	−0.192485	+	0.981300i	$0.561655\pi$
$822$	28.4461	0.992173
$823$	40.9250	1.42656	0.713279	−	0.700881i	$-0.247209\pi$
$823$	40.9250	1.42656	0.713279	+	0.700881i	$0.247209\pi$
$824$	−2.30046	−0.0801403
$825$	−61.6690	−2.14704
$826$	36.0653	1.25487
$827$	−17.6682	−0.614382	−0.307191	−	0.951648i	$-0.599389\pi$
$827$	−17.6682	−0.614382	−0.307191	+	0.951648i	$0.599389\pi$
$828$	2.83723	0.0986004
$829$	−18.0459	−0.626760	−0.313380	−	0.949628i	$-0.601461\pi$
$829$	−18.0459	−0.626760	−0.313380	+	0.949628i	$0.601461\pi$
$830$	−27.1622	−0.942812
$831$	56.4142	1.95699
$832$	5.10966	0.177146
$833$	−15.8259	−0.548335
$834$	−25.3497	−0.877789
$835$	60.3084	2.08706
$836$	6.68363	0.231158
$837$	−6.05984	−0.209459
$838$	−8.91735	−0.308045
$839$	−9.76621	−0.337167	−0.168584	−	0.985687i	$-0.553919\pi$
$839$	−9.76621	−0.337167	−0.168584	+	0.985687i	$0.553919\pi$
$840$	−23.1186	−0.797667
$841$	−26.2664	−0.905739
$842$	8.63228	0.297488
$843$	−42.9635	−1.47974
$844$	−14.1464	−0.486939
$845$	−44.5644	−1.53306
$846$	5.84605	0.200991
$847$	19.1512	0.658044
$848$	11.2449	0.386151
$849$	27.9955	0.960802
$850$	49.9999	1.71498
$851$	3.43079	0.117606
$852$	8.70255	0.298145
$853$	−55.8110	−1.91093	−0.955466	−	0.295102i	$-0.904646\pi$
$853$	−55.8110	−1.91093	−0.955466	+	0.295102i	$0.904646\pi$
$854$	9.78450	0.334819
$855$	−11.4286	−0.390850
$856$	14.1693	0.484297
$857$	29.1816	0.996825	0.498413	−	0.866940i	$-0.333916\pi$
$857$	29.1816	0.996825	0.498413	+	0.866940i	$0.333916\pi$
$858$	48.0534	1.64052
$859$	−37.3961	−1.27594	−0.637970	−	0.770061i	$-0.720226\pi$
$859$	−37.3961	−1.27594	−0.637970	+	0.770061i	$0.720226\pi$
$860$	2.49923	0.0852229
$861$	16.2551	0.553971
$862$	−2.97642	−0.101377
$863$	−9.99316	−0.340171	−0.170086	−	0.985429i	$-0.554404\pi$
$863$	−9.99316	−0.340171	−0.170086	+	0.985429i	$0.554404\pi$
$864$	−2.04174	−0.0694615
$865$	11.3521	0.385984
$866$	11.4056	0.387579
$867$	−92.8639	−3.15382
$868$	−8.94121	−0.303485
$869$	−20.3727	−0.691096
$870$	12.6879	0.430159
$871$	1.80438	0.0611392
$872$	2.76873	0.0937609
$873$	−0.113353	−0.00383641
$874$	−2.17210	−0.0734723
$875$	−15.9508	−0.539235
$876$	30.5560	1.03239
$877$	−52.5262	−1.77368	−0.886841	−	0.462074i	$-0.847105\pi$
$877$	−52.5262	−1.77368	−0.886841	+	0.462074i	$0.847105\pi$
$878$	−28.9601	−0.977357
$879$	−26.6568	−0.899112
$880$	−14.1635	−0.477451
$881$	−0.541869	−0.0182560	−0.00912802	−	0.999958i	$-0.502906\pi$
$881$	−0.541869	−0.0182560	−0.00912802	+	0.999958i	$0.502906\pi$
$882$	−4.34934	−0.146450
$883$	−55.4424	−1.86579	−0.932893	−	0.360154i	$-0.882724\pi$
$883$	−55.4424	−1.86579	−0.932893	+	0.360154i	$0.882724\pi$
$884$	−38.9606	−1.31039
$885$	−91.8707	−3.08820
$886$	20.7667	0.697670
$887$	−55.9674	−1.87920	−0.939600	−	0.342274i	$-0.888803\pi$
$887$	−55.9674	−1.87920	−0.939600	+	0.342274i	$0.888803\pi$
$888$	5.71982	0.191945
$889$	27.5528	0.924090
$890$	54.4879	1.82644
$891$	−45.3922	−1.52070
$892$	13.8229	0.462825
$893$	−4.47557	−0.149769
$894$	18.3628	0.614145
$895$	9.71356	0.324688
$896$	−3.01257	−0.100643
$897$	−15.6168	−0.521429
$898$	−29.6950	−0.990937
$899$	4.90709	0.163661
$900$	13.7412	0.458038
$901$	−85.7411	−2.85645
$902$	9.95858	0.331584
$903$	4.99925	0.166365
$904$	5.95901	0.198194
$905$	−73.1023	−2.43000
$906$	53.7189	1.78469
$907$	−23.2863	−0.773209	−0.386604	−	0.922246i	$-0.626352\pi$
$907$	−23.2863	−0.773209	−0.386604	+	0.922246i	$0.626352\pi$
$908$	−17.0945	−0.567300
$909$	−1.83434	−0.0608412
$910$	52.3311	1.73476
$911$	22.9887	0.761649	0.380824	−	0.924647i	$-0.375640\pi$
$911$	22.9887	0.761649	0.380824	+	0.924647i	$0.375640\pi$
$912$	−3.62132	−0.119914
$913$	−33.2868	−1.10163
$914$	−6.14617	−0.203297
$915$	−24.9245	−0.823979
$916$	0.0325964	0.00107701
$917$	18.8559	0.622678
$918$	15.5681	0.513823
$919$	−11.3333	−0.373851	−0.186926	−	0.982374i	$-0.559852\pi$
$919$	−11.3333	−0.373851	−0.186926	+	0.982374i	$0.559852\pi$
$920$	4.60296	0.151755
$921$	6.81115	0.224435
$922$	−34.2649	−1.12846
$923$	−19.6990	−0.648402
$924$	−28.3315	−0.932037
$925$	16.6159	0.546328
$926$	−2.17470	−0.0714652
$927$	4.82062	0.158330
$928$	1.65335	0.0542738
$929$	−38.5964	−1.26631	−0.633154	−	0.774026i	$-0.718240\pi$
$929$	−38.5964	−1.26631	−0.633154	+	0.774026i	$0.718240\pi$
$930$	22.7764	0.746866
$931$	3.32973	0.109127
$932$	−6.25048	−0.204741
$933$	36.2971	1.18831
$934$	32.3688	1.05914
$935$	107.995	3.53181
$936$	−10.7073	−0.349979
$937$	50.3828	1.64593	0.822967	−	0.568089i	$-0.192317\pi$
$937$	50.3828	1.64593	0.822967	+	0.568089i	$0.192317\pi$
$938$	−1.06383	−0.0347354
$939$	32.5870	1.06344
$940$	9.48430	0.309344
$941$	−37.2863	−1.21550	−0.607749	−	0.794129i	$-0.707927\pi$
$941$	−37.2863	−1.21550	−0.607749	+	0.794129i	$0.707927\pi$
$942$	28.7639	0.937180
$943$	−3.23642	−0.105392
$944$	−11.9716	−0.389643
$945$	−20.9107	−0.680225
$946$	3.06276	0.0995790
$947$	27.2548	0.885663	0.442832	−	0.896605i	$-0.353974\pi$
$947$	27.2548	0.885663	0.442832	+	0.896605i	$0.353974\pi$
$948$	11.0383	0.358508
$949$	−69.1663	−2.24523
$950$	−10.5198	−0.341308
$951$	30.6303	0.993255
$952$	22.9705	0.744478
$953$	−12.7213	−0.412085	−0.206042	−	0.978543i	$-0.566058\pi$
$953$	−12.7213	−0.412085	−0.206042	+	0.978543i	$0.566058\pi$
$954$	−23.5637	−0.762903
$955$	−62.9925	−2.03839
$956$	8.39388	0.271478
$957$	15.5488	0.502621
$958$	13.8014	0.445901
$959$	37.9635	1.22591
$960$	7.67405	0.247679
$961$	−22.1911	−0.715843
$962$	−12.9473	−0.417439
$963$	−29.6918	−0.956805
$964$	−12.6023	−0.405894
$965$	−28.4738	−0.916603
$966$	9.20738	0.296243
$967$	60.4129	1.94275	0.971374	−	0.237556i	$-0.0763464\pi$
$967$	60.4129	1.94275	0.971374	+	0.237556i	$0.0763464\pi$
$968$	−6.35711	−0.204325
$969$	27.6122	0.887031
$970$	−0.183897	−0.00590458
$971$	−3.84176	−0.123288	−0.0616440	−	0.998098i	$-0.519634\pi$
$971$	−3.84176	−0.123288	−0.0616440	+	0.998098i	$0.519634\pi$
$972$	18.4691	0.592398
$973$	−33.8311	−1.08458
$974$	−14.6551	−0.469581
$975$	−75.6346	−2.42224
$976$	−3.24789	−0.103963
$977$	−51.3169	−1.64177	−0.820887	−	0.571091i	$-0.806520\pi$
$977$	−51.3169	−1.64177	−0.820887	+	0.571091i	$0.806520\pi$
$978$	−5.67309	−0.181406
$979$	66.7740	2.13411
$980$	−7.05612	−0.225400
$981$	−5.80187	−0.185239
$982$	6.19311	0.197630
$983$	7.46426	0.238073	0.119036	−	0.992890i	$-0.462019\pi$
$983$	7.46426	0.238073	0.119036	+	0.992890i	$0.462019\pi$
$984$	−5.39576	−0.172010
$985$	1.46656	0.0467283
$986$	−12.6066	−0.401476
$987$	18.9716	0.603874
$988$	8.19720	0.260788
$989$	−0.995360	−0.0316506
$990$	29.6796	0.943279
$991$	4.64476	0.147546	0.0737728	−	0.997275i	$-0.476496\pi$
$991$	4.64476	0.147546	0.0737728	+	0.997275i	$0.476496\pi$
$992$	2.96797	0.0942332
$993$	−61.2243	−1.94289
$994$	11.6142	0.368381
$995$	−65.1324	−2.06484
$996$	18.0354	0.571475
$997$	−8.41481	−0.266500	−0.133250	−	0.991082i	$-0.542541\pi$
$997$	−8.41481	−0.266500	−0.133250	+	0.991082i	$0.542541\pi$
$998$	13.6261	0.431328
$999$	5.17357	0.163684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.18	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.18	✓	84	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.25732 q^{3} +1.00000 q^{4} -3.39963 q^{5} +2.25732 q^{6} +3.01257 q^{7} -1.00000 q^{8} +2.09550 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.25732 q^{3} +1.00000 q^{4} -3.39963 q^{5} +2.25732 q^{6} +3.01257 q^{7} -1.00000 q^{8} +2.09550 q^{9} +3.39963 q^{10} +4.16619 q^{11} -2.25732 q^{12} +5.10966 q^{13} -3.01257 q^{14} +7.67405 q^{15} +1.00000 q^{16} -7.62489 q^{17} -2.09550 q^{18} +1.60426 q^{19} -3.39963 q^{20} -6.80033 q^{21} -4.16619 q^{22} +1.35396 q^{23} +2.25732 q^{24} +6.55745 q^{25} -5.10966 q^{26} +2.04174 q^{27} +3.01257 q^{28} -1.65335 q^{29} -7.67405 q^{30} -2.96797 q^{31} -1.00000 q^{32} -9.40442 q^{33} +7.62489 q^{34} -10.2416 q^{35} +2.09550 q^{36} +2.53390 q^{37} -1.60426 q^{38} -11.5341 q^{39} +3.39963 q^{40} -2.39033 q^{41} +6.80033 q^{42} -0.735148 q^{43} +4.16619 q^{44} -7.12392 q^{45} -1.35396 q^{46} -2.78981 q^{47} -2.25732 q^{48} +2.07556 q^{49} -6.55745 q^{50} +17.2118 q^{51} +5.10966 q^{52} +11.2449 q^{53} -2.04174 q^{54} -14.1635 q^{55} -3.01257 q^{56} -3.62132 q^{57} +1.65335 q^{58} -11.9716 q^{59} +7.67405 q^{60} -3.24789 q^{61} +2.96797 q^{62} +6.31284 q^{63} +1.00000 q^{64} -17.3709 q^{65} +9.40442 q^{66} +0.353132 q^{67} -7.62489 q^{68} -3.05632 q^{69} +10.2416 q^{70} -3.85526 q^{71} -2.09550 q^{72} -13.5364 q^{73} -2.53390 q^{74} -14.8023 q^{75} +1.60426 q^{76} +12.5509 q^{77} +11.5341 q^{78} -4.89001 q^{79} -3.39963 q^{80} -10.8954 q^{81} +2.39033 q^{82} -7.98975 q^{83} -6.80033 q^{84} +25.9218 q^{85} +0.735148 q^{86} +3.73214 q^{87} -4.16619 q^{88} +16.0276 q^{89} +7.12392 q^{90} +15.3932 q^{91} +1.35396 q^{92} +6.69967 q^{93} +2.78981 q^{94} -5.45387 q^{95} +2.25732 q^{96} -0.0540933 q^{97} -2.07556 q^{98} +8.73025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

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