LMFDB - Embedded newform 8006.2.a.a.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, 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("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9282318582$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.72077 q^{3} +1.00000 q^{4} +0.316882 q^{5} -2.72077 q^{6} -4.75223 q^{7} +1.00000 q^{8} +4.40260 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.72077 q^{3} +1.00000 q^{4} +0.316882 q^{5} -2.72077 q^{6} -4.75223 q^{7} +1.00000 q^{8} +4.40260 q^{9} +0.316882 q^{10} +0.196056 q^{11} -2.72077 q^{12} -3.60545 q^{13} -4.75223 q^{14} -0.862164 q^{15} +1.00000 q^{16} +5.11303 q^{17} +4.40260 q^{18} -0.722898 q^{19} +0.316882 q^{20} +12.9297 q^{21} +0.196056 q^{22} +5.37168 q^{23} -2.72077 q^{24} -4.89959 q^{25} -3.60545 q^{26} -3.81616 q^{27} -4.75223 q^{28} +1.19084 q^{29} -0.862164 q^{30} -6.65272 q^{31} +1.00000 q^{32} -0.533424 q^{33} +5.11303 q^{34} -1.50590 q^{35} +4.40260 q^{36} +6.68793 q^{37} -0.722898 q^{38} +9.80960 q^{39} +0.316882 q^{40} -3.40050 q^{41} +12.9297 q^{42} +2.22717 q^{43} +0.196056 q^{44} +1.39511 q^{45} +5.37168 q^{46} +5.80777 q^{47} -2.72077 q^{48} +15.5837 q^{49} -4.89959 q^{50} -13.9114 q^{51} -3.60545 q^{52} -1.99628 q^{53} -3.81616 q^{54} +0.0621266 q^{55} -4.75223 q^{56} +1.96684 q^{57} +1.19084 q^{58} +9.67428 q^{59} -0.862164 q^{60} -0.661462 q^{61} -6.65272 q^{62} -20.9222 q^{63} +1.00000 q^{64} -1.14250 q^{65} -0.533424 q^{66} -5.97378 q^{67} +5.11303 q^{68} -14.6151 q^{69} -1.50590 q^{70} -5.86978 q^{71} +4.40260 q^{72} +4.76979 q^{73} +6.68793 q^{74} +13.3307 q^{75} -0.722898 q^{76} -0.931704 q^{77} +9.80960 q^{78} +7.73505 q^{79} +0.316882 q^{80} -2.82490 q^{81} -3.40050 q^{82} -6.99848 q^{83} +12.9297 q^{84} +1.62023 q^{85} +2.22717 q^{86} -3.24000 q^{87} +0.196056 q^{88} +12.5037 q^{89} +1.39511 q^{90} +17.1339 q^{91} +5.37168 q^{92} +18.1005 q^{93} +5.80777 q^{94} -0.229073 q^{95} -2.72077 q^{96} +11.0442 q^{97} +15.5837 q^{98} +0.863156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * 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
$3$	−2.72077	−1.57084	−0.785419	−	0.618964i	$-0.787553\pi$
$3$	−2.72077	−1.57084	−0.785419	+	0.618964i	$0.787553\pi$
$4$	1.00000	0.500000
$5$	0.316882	0.141714	0.0708570	−	0.997486i	$-0.477427\pi$
$5$	0.316882	0.141714	0.0708570	+	0.997486i	$0.477427\pi$
$6$	−2.72077	−1.11075
$7$	−4.75223	−1.79618	−0.898088	−	0.439816i	$-0.855044\pi$
$7$	−4.75223	−1.79618	−0.898088	+	0.439816i	$0.855044\pi$
$8$	1.00000	0.353553
$9$	4.40260	1.46753
$10$	0.316882	0.100207
$11$	0.196056	0.0591131	0.0295565	−	0.999563i	$-0.490590\pi$
$11$	0.196056	0.0591131	0.0295565	+	0.999563i	$0.490590\pi$
$12$	−2.72077	−0.785419
$13$	−3.60545	−0.999971	−0.499985	−	0.866034i	$-0.666661\pi$
$13$	−3.60545	−0.999971	−0.499985	+	0.866034i	$0.666661\pi$
$14$	−4.75223	−1.27009
$15$	−0.862164	−0.222610
$16$	1.00000	0.250000
$17$	5.11303	1.24009	0.620046	−	0.784565i	$-0.287114\pi$
$17$	5.11303	1.24009	0.620046	+	0.784565i	$0.287114\pi$
$18$	4.40260	1.03770
$19$	−0.722898	−0.165844	−0.0829221	−	0.996556i	$-0.526425\pi$
$19$	−0.722898	−0.165844	−0.0829221	+	0.996556i	$0.526425\pi$
$20$	0.316882	0.0708570
$21$	12.9297	2.82150
$22$	0.196056	0.0417993
$23$	5.37168	1.12007	0.560037	−	0.828468i	$-0.310787\pi$
$23$	5.37168	1.12007	0.560037	+	0.828468i	$0.310787\pi$
$24$	−2.72077	−0.555375
$25$	−4.89959	−0.979917
$26$	−3.60545	−0.707086
$27$	−3.81616	−0.734420
$28$	−4.75223	−0.898088
$29$	1.19084	0.221133	0.110566	−	0.993869i	$-0.464734\pi$
$29$	1.19084	0.221133	0.110566	+	0.993869i	$0.464734\pi$
$30$	−0.862164	−0.157409
$31$	−6.65272	−1.19486	−0.597432	−	0.801919i	$-0.703812\pi$
$31$	−6.65272	−1.19486	−0.597432	+	0.801919i	$0.703812\pi$
$32$	1.00000	0.176777
$33$	−0.533424	−0.0928571
$34$	5.11303	0.876878
$35$	−1.50590	−0.254543
$36$	4.40260	0.733767
$37$	6.68793	1.09949	0.549745	−	0.835333i	$-0.314725\pi$
$37$	6.68793	1.09949	0.549745	+	0.835333i	$0.314725\pi$
$38$	−0.722898	−0.117270
$39$	9.80960	1.57079
$40$	0.316882	0.0501034
$41$	−3.40050	−0.531068	−0.265534	−	0.964101i	$-0.585548\pi$
$41$	−3.40050	−0.531068	−0.265534	+	0.964101i	$0.585548\pi$
$42$	12.9297	1.99510
$43$	2.22717	0.339640	0.169820	−	0.985475i	$-0.445681\pi$
$43$	2.22717	0.339640	0.169820	+	0.985475i	$0.445681\pi$
$44$	0.196056	0.0295565
$45$	1.39511	0.207970
$46$	5.37168	0.792012
$47$	5.80777	0.847150	0.423575	−	0.905861i	$-0.360775\pi$
$47$	5.80777	0.847150	0.423575	+	0.905861i	$0.360775\pi$
$48$	−2.72077	−0.392710
$49$	15.5837	2.22625
$50$	−4.89959	−0.692906
$51$	−13.9114	−1.94799
$52$	−3.60545	−0.499985
$53$	−1.99628	−0.274211	−0.137105	−	0.990556i	$-0.543780\pi$
$53$	−1.99628	−0.274211	−0.137105	+	0.990556i	$0.543780\pi$
$54$	−3.81616	−0.519314
$55$	0.0621266	0.00837715
$56$	−4.75223	−0.635044
$57$	1.96684	0.260514
$58$	1.19084	0.156365
$59$	9.67428	1.25948	0.629742	−	0.776804i	$-0.283161\pi$
$59$	9.67428	1.25948	0.629742	+	0.776804i	$0.283161\pi$
$60$	−0.862164	−0.111305
$61$	−0.661462	−0.0846916	−0.0423458	−	0.999103i	$-0.513483\pi$
$61$	−0.661462	−0.0846916	−0.0423458	+	0.999103i	$0.513483\pi$
$62$	−6.65272	−0.844897
$63$	−20.9222	−2.63595
$64$	1.00000	0.125000
$65$	−1.14250	−0.141710
$66$	−0.533424	−0.0656599
$67$	−5.97378	−0.729813	−0.364906	−	0.931044i	$-0.618899\pi$
$67$	−5.97378	−0.729813	−0.364906	+	0.931044i	$0.618899\pi$
$68$	5.11303	0.620046
$69$	−14.6151	−1.75945
$70$	−1.50590	−0.179989
$71$	−5.86978	−0.696615	−0.348307	−	0.937380i	$-0.613243\pi$
$71$	−5.86978	−0.696615	−0.348307	+	0.937380i	$0.613243\pi$
$72$	4.40260	0.518852
$73$	4.76979	0.558261	0.279131	−	0.960253i	$-0.409954\pi$
$73$	4.76979	0.558261	0.279131	+	0.960253i	$0.409954\pi$
$74$	6.68793	0.777457
$75$	13.3307	1.53929
$76$	−0.722898	−0.0829221
$77$	−0.931704	−0.106178
$78$	9.80960	1.11072
$79$	7.73505	0.870261	0.435130	−	0.900367i	$-0.356702\pi$
$79$	7.73505	0.870261	0.435130	+	0.900367i	$0.356702\pi$
$80$	0.316882	0.0354285
$81$	−2.82490	−0.313878
$82$	−3.40050	−0.375522
$83$	−6.99848	−0.768183	−0.384091	−	0.923295i	$-0.625485\pi$
$83$	−6.99848	−0.768183	−0.384091	+	0.923295i	$0.625485\pi$
$84$	12.9297	1.41075
$85$	1.62023	0.175738
$86$	2.22717	0.240162
$87$	−3.24000	−0.347364
$88$	0.196056	0.0208996
$89$	12.5037	1.32539	0.662694	−	0.748890i	$-0.269413\pi$
$89$	12.5037	1.32539	0.662694	+	0.748890i	$0.269413\pi$
$90$	1.39511	0.147057
$91$	17.1339	1.79612
$92$	5.37168	0.560037
$93$	18.1005	1.87694
$94$	5.80777	0.599025
$95$	−0.229073	−0.0235024
$96$	−2.72077	−0.277688
$97$	11.0442	1.12136	0.560682	−	0.828031i	$-0.310539\pi$
$97$	11.0442	1.12136	0.560682	+	0.828031i	$0.310539\pi$
$98$	15.5837	1.57419
$99$	0.863156	0.0867505
$100$	−4.89959	−0.489959
$101$	3.78907	0.377026	0.188513	−	0.982071i	$-0.439633\pi$
$101$	3.78907	0.377026	0.188513	+	0.982071i	$0.439633\pi$
$102$	−13.9114	−1.37743
$103$	2.68251	0.264316	0.132158	−	0.991229i	$-0.457809\pi$
$103$	2.68251	0.264316	0.132158	+	0.991229i	$0.457809\pi$
$104$	−3.60545	−0.353543
$105$	4.09720	0.399846
$106$	−1.99628	−0.193896
$107$	−8.30648	−0.803018	−0.401509	−	0.915855i	$-0.631514\pi$
$107$	−8.30648	−0.803018	−0.401509	+	0.915855i	$0.631514\pi$
$108$	−3.81616	−0.367210
$109$	−11.2209	−1.07477	−0.537383	−	0.843339i	$-0.680587\pi$
$109$	−11.2209	−1.07477	−0.537383	+	0.843339i	$0.680587\pi$
$110$	0.0621266	0.00592354
$111$	−18.1963	−1.72712
$112$	−4.75223	−0.449044
$113$	−5.79338	−0.544995	−0.272498	−	0.962156i	$-0.587850\pi$
$113$	−5.79338	−0.544995	−0.272498	+	0.962156i	$0.587850\pi$
$114$	1.96684	0.184212
$115$	1.70219	0.158730
$116$	1.19084	0.110566
$117$	−15.8733	−1.46749
$118$	9.67428	0.890590
$119$	−24.2983	−2.22742
$120$	−0.862164	−0.0787044
$121$	−10.9616	−0.996506
$122$	−0.661462	−0.0598860
$123$	9.25198	0.834223
$124$	−6.65272	−0.597432
$125$	−3.13700	−0.280582
$126$	−20.9222	−1.86390
$127$	−5.03969	−0.447200	−0.223600	−	0.974681i	$-0.571781\pi$
$127$	−5.03969	−0.447200	−0.223600	+	0.974681i	$0.571781\pi$
$128$	1.00000	0.0883883
$129$	−6.05962	−0.533520
$130$	−1.14250	−0.100204
$131$	−4.87520	−0.425948	−0.212974	−	0.977058i	$-0.568315\pi$
$131$	−4.87520	−0.425948	−0.212974	+	0.977058i	$0.568315\pi$
$132$	−0.533424	−0.0464286
$133$	3.43538	0.297885
$134$	−5.97378	−0.516056
$135$	−1.20927	−0.104078
$136$	5.11303	0.438439
$137$	−2.52143	−0.215420	−0.107710	−	0.994182i	$-0.534352\pi$
$137$	−2.52143	−0.215420	−0.107710	+	0.994182i	$0.534352\pi$
$138$	−14.6151	−1.24412
$139$	−0.245745	−0.0208438	−0.0104219	−	0.999946i	$-0.503317\pi$
$139$	−0.245745	−0.0208438	−0.0104219	+	0.999946i	$0.503317\pi$
$140$	−1.50590	−0.127272
$141$	−15.8016	−1.33074
$142$	−5.86978	−0.492581
$143$	−0.706869	−0.0591114
$144$	4.40260	0.366883
$145$	0.377355	0.0313376
$146$	4.76979	0.394750
$147$	−42.3998	−3.49708
$148$	6.68793	0.549745
$149$	19.2607	1.57790	0.788948	−	0.614460i	$-0.210626\pi$
$149$	19.2607	1.57790	0.788948	+	0.614460i	$0.210626\pi$
$150$	13.3307	1.08844
$151$	−15.5494	−1.26539	−0.632696	−	0.774400i	$-0.718052\pi$
$151$	−15.5494	−1.26539	−0.632696	+	0.774400i	$0.718052\pi$
$152$	−0.722898	−0.0586348
$153$	22.5107	1.81988
$154$	−0.931704	−0.0750788
$155$	−2.10813	−0.169329
$156$	9.80960	0.785396
$157$	20.8275	1.66221	0.831107	−	0.556112i	$-0.187707\pi$
$157$	20.8275	1.66221	0.831107	+	0.556112i	$0.187707\pi$
$158$	7.73505	0.615367
$159$	5.43143	0.430741
$160$	0.316882	0.0250517
$161$	−25.5275	−2.01185
$162$	−2.82490	−0.221945
$163$	−1.95549	−0.153166	−0.0765829	−	0.997063i	$-0.524401\pi$
$163$	−1.95549	−0.153166	−0.0765829	+	0.997063i	$0.524401\pi$
$164$	−3.40050	−0.265534
$165$	−0.169032	−0.0131592
$166$	−6.99848	−0.543187
$167$	−11.4700	−0.887578	−0.443789	−	0.896131i	$-0.646366\pi$
$167$	−11.4700	−0.887578	−0.443789	+	0.896131i	$0.646366\pi$
$168$	12.9297	0.997552
$169$	−0.000758399 0	−5.83384e−5 0
$170$	1.62023	0.124266
$171$	−3.18263	−0.243382
$172$	2.22717	0.169820
$173$	24.8576	1.88989	0.944944	−	0.327232i	$-0.106116\pi$
$173$	24.8576	1.88989	0.944944	+	0.327232i	$0.106116\pi$
$174$	−3.24000	−0.245623
$175$	23.2840	1.76010
$176$	0.196056	0.0147783
$177$	−26.3215	−1.97845
$178$	12.5037	0.937191
$179$	−25.2519	−1.88742	−0.943708	−	0.330781i	$-0.892688\pi$
$179$	−25.2519	−1.88742	−0.943708	+	0.330781i	$0.892688\pi$
$180$	1.39511	0.103985
$181$	14.1502	1.05178	0.525888	−	0.850554i	$-0.323733\pi$
$181$	14.1502	1.05178	0.525888	+	0.850554i	$0.323733\pi$
$182$	17.1339	1.27005
$183$	1.79969	0.133037
$184$	5.37168	0.396006
$185$	2.11929	0.155813
$186$	18.1005	1.32720
$187$	1.00244	0.0733057
$188$	5.80777	0.423575
$189$	18.1353	1.31915
$190$	−0.229073	−0.0166187
$191$	−11.3739	−0.822987	−0.411494	−	0.911413i	$-0.634993\pi$
$191$	−11.3739	−0.822987	−0.411494	+	0.911413i	$0.634993\pi$
$192$	−2.72077	−0.196355
$193$	−13.7856	−0.992306	−0.496153	−	0.868235i	$-0.665255\pi$
$193$	−13.7856	−0.992306	−0.496153	+	0.868235i	$0.665255\pi$
$194$	11.0442	0.792924
$195$	3.10849	0.222603
$196$	15.5837	1.11312
$197$	−2.39387	−0.170557	−0.0852783	−	0.996357i	$-0.527178\pi$
$197$	−2.39387	−0.170557	−0.0852783	+	0.996357i	$0.527178\pi$
$198$	0.863156	0.0613418
$199$	4.11065	0.291396	0.145698	−	0.989329i	$-0.453457\pi$
$199$	4.11065	0.291396	0.145698	+	0.989329i	$0.453457\pi$
$200$	−4.89959	−0.346453
$201$	16.2533	1.14642
$202$	3.78907	0.266598
$203$	−5.65914	−0.397193
$204$	−13.9114	−0.973993
$205$	−1.07756	−0.0752598
$206$	2.68251	0.186899
$207$	23.6494	1.64375
$208$	−3.60545	−0.249993
$209$	−0.141728	−0.00980356
$210$	4.09720	0.282734
$211$	−16.4507	−1.13252	−0.566258	−	0.824228i	$-0.691609\pi$
$211$	−16.4507	−1.13252	−0.566258	+	0.824228i	$0.691609\pi$
$212$	−1.99628	−0.137105
$213$	15.9703	1.09427
$214$	−8.30648	−0.567820
$215$	0.705750	0.0481318
$216$	−3.81616	−0.259657
$217$	31.6153	2.14619
$218$	−11.2209	−0.759974
$219$	−12.9775	−0.876939
$220$	0.0621266	0.00418857
$221$	−18.4348	−1.24006
$222$	−18.1963	−1.22126
$223$	−11.5707	−0.774832	−0.387416	−	0.921905i	$-0.626632\pi$
$223$	−11.5707	−0.774832	−0.387416	+	0.921905i	$0.626632\pi$
$224$	−4.75223	−0.317522
$225$	−21.5709	−1.43806
$226$	−5.79338	−0.385370
$227$	−4.07634	−0.270556	−0.135278	−	0.990808i	$-0.543193\pi$
$227$	−4.07634	−0.270556	−0.135278	+	0.990808i	$0.543193\pi$
$228$	1.96684	0.130257
$229$	−23.2026	−1.53327	−0.766635	−	0.642083i	$-0.778070\pi$
$229$	−23.2026	−1.53327	−0.766635	+	0.642083i	$0.778070\pi$
$230$	1.70219	0.112239
$231$	2.53495	0.166788
$232$	1.19084	0.0781823
$233$	−21.9760	−1.43970	−0.719848	−	0.694132i	$-0.755788\pi$
$233$	−21.9760	−1.43970	−0.719848	+	0.694132i	$0.755788\pi$
$234$	−15.8733	−1.03767
$235$	1.84038	0.120053
$236$	9.67428	0.629742
$237$	−21.0453	−1.36704
$238$	−24.2983	−1.57503
$239$	7.62410	0.493162	0.246581	−	0.969122i	$-0.420693\pi$
$239$	7.62410	0.493162	0.246581	+	0.969122i	$0.420693\pi$
$240$	−0.862164	−0.0556524
$241$	7.98770	0.514533	0.257267	−	0.966340i	$-0.417178\pi$
$241$	7.98770	0.514533	0.257267	+	0.966340i	$0.417178\pi$
$242$	−10.9616	−0.704636
$243$	19.1344	1.22747
$244$	−0.661462	−0.0423458
$245$	4.93820	0.315490
$246$	9.25198	0.589885
$247$	2.60637	0.165839
$248$	−6.65272	−0.422448
$249$	19.0413	1.20669
$250$	−3.13700	−0.198401
$251$	−21.7588	−1.37341	−0.686703	−	0.726938i	$-0.740943\pi$
$251$	−21.7588	−1.37341	−0.686703	+	0.726938i	$0.740943\pi$
$252$	−20.9222	−1.31797
$253$	1.05315	0.0662110
$254$	−5.03969	−0.316218
$255$	−4.40827	−0.276057
$256$	1.00000	0.0625000
$257$	−8.49082	−0.529643	−0.264821	−	0.964298i	$-0.585313\pi$
$257$	−8.49082	−0.529643	−0.264821	+	0.964298i	$0.585313\pi$
$258$	−6.05962	−0.377256
$259$	−31.7826	−1.97488
$260$	−1.14250	−0.0708549
$261$	5.24278	0.324520
$262$	−4.87520	−0.301191
$263$	−25.7922	−1.59042	−0.795208	−	0.606337i	$-0.792638\pi$
$263$	−25.7922	−1.59042	−0.795208	+	0.606337i	$0.792638\pi$
$264$	−0.533424	−0.0328300
$265$	−0.632586	−0.0388595
$266$	3.43538	0.210637
$267$	−34.0197	−2.08197
$268$	−5.97378	−0.364906
$269$	−8.17103	−0.498196	−0.249098	−	0.968478i	$-0.580134\pi$
$269$	−8.17103	−0.498196	−0.249098	+	0.968478i	$0.580134\pi$
$270$	−1.20927	−0.0735940
$271$	8.44208	0.512820	0.256410	−	0.966568i	$-0.417460\pi$
$271$	8.44208	0.512820	0.256410	+	0.966568i	$0.417460\pi$
$272$	5.11303	0.310023
$273$	−46.6175	−2.82142
$274$	−2.52143	−0.152325
$275$	−0.960593	−0.0579259
$276$	−14.6151	−0.879727
$277$	−0.713567	−0.0428741	−0.0214370	−	0.999770i	$-0.506824\pi$
$277$	−0.713567	−0.0428741	−0.0214370	+	0.999770i	$0.506824\pi$
$278$	−0.245745	−0.0147388
$279$	−29.2893	−1.75350
$280$	−1.50590	−0.0899946
$281$	16.8918	1.00768	0.503841	−	0.863796i	$-0.331920\pi$
$281$	16.8918	1.00768	0.503841	+	0.863796i	$0.331920\pi$
$282$	−15.8016	−0.940972
$283$	12.1434	0.721850	0.360925	−	0.932595i	$-0.382461\pi$
$283$	12.1434	0.721850	0.360925	+	0.932595i	$0.382461\pi$
$284$	−5.86978	−0.348307
$285$	0.623256	0.0369185
$286$	−0.706869	−0.0417980
$287$	16.1600	0.953892
$288$	4.40260	0.259426
$289$	9.14312	0.537830
$290$	0.377355	0.0221590
$291$	−30.0486	−1.76148
$292$	4.76979	0.279131
$293$	−4.32195	−0.252491	−0.126246	−	0.991999i	$-0.540293\pi$
$293$	−4.32195	−0.252491	−0.126246	+	0.991999i	$0.540293\pi$
$294$	−42.3998	−2.47281
$295$	3.06561	0.178486
$296$	6.68793	0.388728
$297$	−0.748181	−0.0434139
$298$	19.2607	1.11574
$299$	−19.3673	−1.12004
$300$	13.3307	0.769646
$301$	−10.5840	−0.610053
$302$	−15.5494	−0.894767
$303$	−10.3092	−0.592247
$304$	−0.722898	−0.0414610
$305$	−0.209606	−0.0120020
$306$	22.5107	1.28685
$307$	−17.7127	−1.01092	−0.505458	−	0.862851i	$-0.668676\pi$
$307$	−17.7127	−1.01092	−0.505458	+	0.862851i	$0.668676\pi$
$308$	−0.931704	−0.0530888
$309$	−7.29850	−0.415197
$310$	−2.10813	−0.119734
$311$	−31.6678	−1.79572	−0.897858	−	0.440286i	$-0.854877\pi$
$311$	−31.6678	−1.79572	−0.897858	+	0.440286i	$0.854877\pi$
$312$	9.80960	0.555359
$313$	27.5333	1.55627	0.778136	−	0.628096i	$-0.216166\pi$
$313$	27.5333	1.55627	0.778136	+	0.628096i	$0.216166\pi$
$314$	20.8275	1.17536
$315$	−6.62987	−0.373551
$316$	7.73505	0.435130
$317$	3.98188	0.223644	0.111822	−	0.993728i	$-0.464331\pi$
$317$	3.98188	0.223644	0.111822	+	0.993728i	$0.464331\pi$
$318$	5.43143	0.304580
$319$	0.233471	0.0130718
$320$	0.316882	0.0177142
$321$	22.6000	1.26141
$322$	−25.5275	−1.42259
$323$	−3.69620	−0.205662
$324$	−2.82490	−0.156939
$325$	17.6652	0.979889
$326$	−1.95549	−0.108305
$327$	30.5295	1.68828
$328$	−3.40050	−0.187761
$329$	−27.5999	−1.52163
$330$	−0.169032	−0.00930492
$331$	3.96189	0.217765	0.108883	−	0.994055i	$-0.465273\pi$
$331$	3.96189	0.217765	0.108883	+	0.994055i	$0.465273\pi$
$332$	−6.99848	−0.384091
$333$	29.4443	1.61354
$334$	−11.4700	−0.627612
$335$	−1.89298	−0.103425
$336$	12.9297	0.705376
$337$	21.0857	1.14861	0.574305	−	0.818641i	$-0.305272\pi$
$337$	21.0857	1.14861	0.574305	+	0.818641i	$0.305272\pi$
$338$	−0.000758399 0	−4.12515e−5 0
$339$	15.7625	0.856100
$340$	1.62023	0.0878692
$341$	−1.30431	−0.0706321
$342$	−3.18263	−0.172097
$343$	−40.7919	−2.20256
$344$	2.22717	0.120081
$345$	−4.63127	−0.249339
$346$	24.8576	1.33635
$347$	−0.701496	−0.0376583	−0.0188291	−	0.999823i	$-0.505994\pi$
$347$	−0.701496	−0.0376583	−0.0188291	+	0.999823i	$0.505994\pi$
$348$	−3.24000	−0.173682
$349$	−7.74778	−0.414729	−0.207365	−	0.978264i	$-0.566489\pi$
$349$	−7.74778	−0.414729	−0.207365	+	0.978264i	$0.566489\pi$
$350$	23.2840	1.24458
$351$	13.7590	0.734399
$352$	0.196056	0.0104498
$353$	−7.28930	−0.387970	−0.193985	−	0.981004i	$-0.562141\pi$
$353$	−7.28930	−0.387970	−0.193985	+	0.981004i	$0.562141\pi$
$354$	−26.3215	−1.39897
$355$	−1.86003	−0.0987200
$356$	12.5037	0.662694
$357$	66.1102	3.49892
$358$	−25.2519	−1.33460
$359$	−18.7220	−0.988107	−0.494054	−	0.869431i	$-0.664485\pi$
$359$	−18.7220	−0.988107	−0.494054	+	0.869431i	$0.664485\pi$
$360$	1.39511	0.0735285
$361$	−18.4774	−0.972496
$362$	14.1502	0.743718
$363$	29.8239	1.56535
$364$	17.1339	0.898062
$365$	1.51146	0.0791134
$366$	1.79969	0.0940712
$367$	27.4019	1.43037	0.715185	−	0.698935i	$-0.246342\pi$
$367$	27.4019	1.43037	0.715185	+	0.698935i	$0.246342\pi$
$368$	5.37168	0.280018
$369$	−14.9710	−0.779361
$370$	2.11929	0.110176
$371$	9.48681	0.492530
$372$	18.1005	0.938470
$373$	31.9987	1.65683	0.828414	−	0.560116i	$-0.189244\pi$
$373$	31.9987	1.65683	0.828414	+	0.560116i	$0.189244\pi$
$374$	1.00244	0.0518350
$375$	8.53507	0.440749
$376$	5.80777	0.299513
$377$	−4.29350	−0.221126
$378$	18.1353	0.932778
$379$	−27.6133	−1.41840	−0.709201	−	0.705007i	$-0.750944\pi$
$379$	−27.6133	−1.41840	−0.709201	+	0.705007i	$0.750944\pi$
$380$	−0.229073	−0.0117512
$381$	13.7119	0.702480
$382$	−11.3739	−0.581940
$383$	−19.1482	−0.978426	−0.489213	−	0.872164i	$-0.662716\pi$
$383$	−19.1482	−0.978426	−0.489213	+	0.872164i	$0.662716\pi$
$384$	−2.72077	−0.138844
$385$	−0.295240	−0.0150468
$386$	−13.7856	−0.701667
$387$	9.80534	0.498433
$388$	11.0442	0.560682
$389$	−30.9589	−1.56968	−0.784840	−	0.619699i	$-0.787255\pi$
$389$	−30.9589	−1.56968	−0.784840	+	0.619699i	$0.787255\pi$
$390$	3.10849	0.157404
$391$	27.4656	1.38900
$392$	15.5837	0.787097
$393$	13.2643	0.669095
$394$	−2.39387	−0.120602
$395$	2.45110	0.123328
$396$	0.863156	0.0433752
$397$	−6.05248	−0.303765	−0.151883	−	0.988399i	$-0.548534\pi$
$397$	−6.05248	−0.303765	−0.151883	+	0.988399i	$0.548534\pi$
$398$	4.11065	0.206048
$399$	−9.34689	−0.467930
$400$	−4.89959	−0.244979
$401$	−3.56302	−0.177929	−0.0889645	−	0.996035i	$-0.528356\pi$
$401$	−3.56302	−0.177929	−0.0889645	+	0.996035i	$0.528356\pi$
$402$	16.2533	0.810640
$403$	23.9860	1.19483
$404$	3.78907	0.188513
$405$	−0.895161	−0.0444809
$406$	−5.65914	−0.280858
$407$	1.31121	0.0649942
$408$	−13.9114	−0.688717
$409$	12.2249	0.604484	0.302242	−	0.953231i	$-0.402265\pi$
$409$	12.2249	0.604484	0.302242	+	0.953231i	$0.402265\pi$
$410$	−1.07756	−0.0532167
$411$	6.86023	0.338390
$412$	2.68251	0.132158
$413$	−45.9744	−2.26225
$414$	23.6494	1.16230
$415$	−2.21769	−0.108862
$416$	−3.60545	−0.176772
$417$	0.668617	0.0327423
$418$	−0.141728	−0.00693216
$419$	7.37793	0.360435	0.180218	−	0.983627i	$-0.442320\pi$
$419$	7.37793	0.360435	0.180218	+	0.983627i	$0.442320\pi$
$420$	4.09720	0.199923
$421$	−18.3985	−0.896686	−0.448343	−	0.893862i	$-0.647986\pi$
$421$	−18.3985	−0.896686	−0.448343	+	0.893862i	$0.647986\pi$
$422$	−16.4507	−0.800810
$423$	25.5693	1.24322
$424$	−1.99628	−0.0969481
$425$	−25.0517	−1.21519
$426$	15.9703	0.773765
$427$	3.14342	0.152121
$428$	−8.30648	−0.401509
$429$	1.92323	0.0928544
$430$	0.705750	0.0340343
$431$	17.8345	0.859059	0.429529	−	0.903053i	$-0.358679\pi$
$431$	17.8345	0.859059	0.429529	+	0.903053i	$0.358679\pi$
$432$	−3.81616	−0.183605
$433$	−21.8467	−1.04988	−0.524942	−	0.851138i	$-0.675913\pi$
$433$	−21.8467	−1.04988	−0.524942	+	0.851138i	$0.675913\pi$
$434$	31.6153	1.51758
$435$	−1.02670	−0.0492263
$436$	−11.2209	−0.537383
$437$	−3.88318	−0.185758
$438$	−12.9775	−0.620089
$439$	2.16133	0.103155	0.0515774	−	0.998669i	$-0.483575\pi$
$439$	2.16133	0.103155	0.0515774	+	0.998669i	$0.483575\pi$
$440$	0.0621266	0.00296177
$441$	68.6090	3.26709
$442$	−18.4348	−0.876852
$443$	7.90190	0.375431	0.187715	−	0.982223i	$-0.439892\pi$
$443$	7.90190	0.375431	0.187715	+	0.982223i	$0.439892\pi$
$444$	−18.1963	−0.863560
$445$	3.96219	0.187826
$446$	−11.5707	−0.547889
$447$	−52.4039	−2.47862
$448$	−4.75223	−0.224522
$449$	−6.97746	−0.329287	−0.164643	−	0.986353i	$-0.552647\pi$
$449$	−6.97746	−0.329287	−0.164643	+	0.986353i	$0.552647\pi$
$450$	−21.5709	−1.01686
$451$	−0.666688	−0.0313931
$452$	−5.79338	−0.272498
$453$	42.3064	1.98773
$454$	−4.07634	−0.191312
$455$	5.42943	0.254536
$456$	1.96684	0.0921058
$457$	17.7812	0.831771	0.415886	−	0.909417i	$-0.363472\pi$
$457$	17.7812	0.831771	0.415886	+	0.909417i	$0.363472\pi$
$458$	−23.2026	−1.08419
$459$	−19.5122	−0.910749
$460$	1.70219	0.0793650
$461$	−24.5253	−1.14226	−0.571129	−	0.820860i	$-0.693494\pi$
$461$	−24.5253	−1.14226	−0.571129	+	0.820860i	$0.693494\pi$
$462$	2.53495	0.117937
$463$	−3.67389	−0.170740	−0.0853700	−	0.996349i	$-0.527207\pi$
$463$	−3.67389	−0.170740	−0.0853700	+	0.996349i	$0.527207\pi$
$464$	1.19084	0.0552832
$465$	5.73574	0.265988
$466$	−21.9760	−1.01802
$467$	0.474797	0.0219710	0.0109855	−	0.999940i	$-0.496503\pi$
$467$	0.474797	0.0219710	0.0109855	+	0.999940i	$0.496503\pi$
$468$	−15.8733	−0.733746
$469$	28.3888	1.31087
$470$	1.84038	0.0848902
$471$	−56.6669	−2.61107
$472$	9.67428	0.445295
$473$	0.436650	0.0200772
$474$	−21.0453	−0.966643
$475$	3.54190	0.162514
$476$	−24.2983	−1.11371
$477$	−8.78884	−0.402413
$478$	7.62410	0.348718
$479$	7.33172	0.334995	0.167497	−	0.985873i	$-0.446431\pi$
$479$	7.33172	0.334995	0.167497	+	0.985873i	$0.446431\pi$
$480$	−0.862164	−0.0393522
$481$	−24.1130	−1.09946
$482$	7.98770	0.363830
$483$	69.4545	3.16029
$484$	−10.9616	−0.498253
$485$	3.49970	0.158913
$486$	19.1344	0.867954
$487$	−26.5025	−1.20094	−0.600471	−	0.799647i	$-0.705020\pi$
$487$	−26.5025	−1.20094	−0.600471	+	0.799647i	$0.705020\pi$
$488$	−0.661462	−0.0299430
$489$	5.32044	0.240599
$490$	4.93820	0.223085
$491$	18.9903	0.857022	0.428511	−	0.903537i	$-0.359038\pi$
$491$	18.9903	0.857022	0.428511	+	0.903537i	$0.359038\pi$
$492$	9.25198	0.417111
$493$	6.08879	0.274225
$494$	2.60637	0.117266
$495$	0.273519	0.0122938
$496$	−6.65272	−0.298716
$497$	27.8946	1.25124
$498$	19.0413	0.853260
$499$	23.9385	1.07164	0.535818	−	0.844334i	$-0.320003\pi$
$499$	23.9385	1.07164	0.535818	+	0.844334i	$0.320003\pi$
$500$	−3.13700	−0.140291
$501$	31.2073	1.39424
$502$	−21.7588	−0.971145
$503$	26.4485	1.17928	0.589641	−	0.807666i	$-0.299269\pi$
$503$	26.4485	1.17928	0.589641	+	0.807666i	$0.299269\pi$
$504$	−20.9222	−0.931949
$505$	1.20069	0.0534299
$506$	1.05315	0.0468182
$507$	0.00206343	9.16402e−5 0
$508$	−5.03969	−0.223600
$509$	6.19767	0.274707	0.137353	−	0.990522i	$-0.456140\pi$
$509$	6.19767	0.274707	0.137353	+	0.990522i	$0.456140\pi$
$510$	−4.40827	−0.195202
$511$	−22.6672	−1.00274
$512$	1.00000	0.0441942
$513$	2.75869	0.121799
$514$	−8.49082	−0.374514
$515$	0.850039	0.0374572
$516$	−6.05962	−0.266760
$517$	1.13865	0.0500776
$518$	−31.7826	−1.39645
$519$	−67.6319	−2.96871
$520$	−1.14250	−0.0501020
$521$	−2.06771	−0.0905881	−0.0452941	−	0.998974i	$-0.514422\pi$
$521$	−2.06771	−0.0905881	−0.0452941	+	0.998974i	$0.514422\pi$
$522$	5.24278	0.229470
$523$	37.1465	1.62430	0.812152	−	0.583446i	$-0.198296\pi$
$523$	37.1465	1.62430	0.812152	+	0.583446i	$0.198296\pi$
$524$	−4.87520	−0.212974
$525$	−63.3504	−2.76484
$526$	−25.7922	−1.12459
$527$	−34.0156	−1.48174
$528$	−0.533424	−0.0232143
$529$	5.85498	0.254565
$530$	−0.632586	−0.0274778
$531$	42.5920	1.84834
$532$	3.43538	0.148943
$533$	12.2603	0.531053
$534$	−34.0197	−1.47218
$535$	−2.63218	−0.113799
$536$	−5.97378	−0.258028
$537$	68.7047	2.96482
$538$	−8.17103	−0.352278
$539$	3.05528	0.131600
$540$	−1.20927	−0.0520388
$541$	−20.3017	−0.872837	−0.436418	−	0.899744i	$-0.643753\pi$
$541$	−20.3017	−0.872837	−0.436418	+	0.899744i	$0.643753\pi$
$542$	8.44208	0.362618
$543$	−38.4995	−1.65217
$544$	5.11303	0.219220
$545$	−3.55569	−0.152309
$546$	−46.6175	−1.99505
$547$	11.9084	0.509166	0.254583	−	0.967051i	$-0.418062\pi$
$547$	11.9084	0.509166	0.254583	+	0.967051i	$0.418062\pi$
$548$	−2.52143	−0.107710
$549$	−2.91216	−0.124288
$550$	−0.960593	−0.0409598
$551$	−0.860853	−0.0366736
$552$	−14.6151	−0.622061
$553$	−36.7588	−1.56314
$554$	−0.713567	−0.0303166
$555$	−5.76610	−0.244757
$556$	−0.245745	−0.0104219
$557$	−19.9796	−0.846564	−0.423282	−	0.905998i	$-0.639122\pi$
$557$	−19.9796	−0.846564	−0.423282	+	0.905998i	$0.639122\pi$
$558$	−29.2893	−1.23991
$559$	−8.02994	−0.339630
$560$	−1.50590	−0.0636358
$561$	−2.72741	−0.115151
$562$	16.8918	0.712539
$563$	−23.8225	−1.00400	−0.502000	−	0.864868i	$-0.667402\pi$
$563$	−23.8225	−1.00400	−0.502000	+	0.864868i	$0.667402\pi$
$564$	−15.8016	−0.665368
$565$	−1.83582	−0.0772334
$566$	12.1434	0.510425
$567$	13.4246	0.563780
$568$	−5.86978	−0.246291
$569$	37.9515	1.59101	0.795505	−	0.605947i	$-0.207206\pi$
$569$	37.9515	1.59101	0.795505	+	0.605947i	$0.207206\pi$
$570$	0.623256	0.0261053
$571$	−41.6750	−1.74404	−0.872022	−	0.489468i	$-0.837191\pi$
$571$	−41.6750	−1.74404	−0.872022	+	0.489468i	$0.837191\pi$
$572$	−0.706869	−0.0295557
$573$	30.9458	1.29278
$574$	16.1600	0.674504
$575$	−26.3190	−1.09758
$576$	4.40260	0.183442
$577$	−18.8643	−0.785332	−0.392666	−	0.919681i	$-0.628447\pi$
$577$	−18.8643	−0.785332	−0.392666	+	0.919681i	$0.628447\pi$
$578$	9.14312	0.380303
$579$	37.5074	1.55875
$580$	0.377355	0.0156688
$581$	33.2584	1.37979
$582$	−30.0486	−1.24556
$583$	−0.391383	−0.0162094
$584$	4.76979	0.197375
$585$	−5.02998	−0.207964
$586$	−4.32195	−0.178538
$587$	−34.0378	−1.40489	−0.702445	−	0.711738i	$-0.747908\pi$
$587$	−34.0378	−1.40489	−0.702445	+	0.711738i	$0.747908\pi$
$588$	−42.3998	−1.74854
$589$	4.80924	0.198161
$590$	3.06561	0.126209
$591$	6.51319	0.267917
$592$	6.68793	0.274872
$593$	−35.6082	−1.46225	−0.731126	−	0.682242i	$-0.761005\pi$
$593$	−35.6082	−1.46225	−0.731126	+	0.682242i	$0.761005\pi$
$594$	−0.748181	−0.0306982
$595$	−7.69971	−0.315657
$596$	19.2607	0.788948
$597$	−11.1841	−0.457736
$598$	−19.3673	−0.791988
$599$	−37.3683	−1.52683	−0.763415	−	0.645909i	$-0.776479\pi$
$599$	−37.3683	−1.52683	−0.763415	+	0.645909i	$0.776479\pi$
$600$	13.3307	0.544222
$601$	−22.3170	−0.910331	−0.455166	−	0.890407i	$-0.650420\pi$
$601$	−22.3170	−0.910331	−0.455166	+	0.890407i	$0.650420\pi$
$602$	−10.5840	−0.431373
$603$	−26.3002	−1.07103
$604$	−15.5494	−0.632696
$605$	−3.47352	−0.141219
$606$	−10.3092	−0.418782
$607$	−36.7308	−1.49086	−0.745428	−	0.666586i	$-0.767755\pi$
$607$	−36.7308	−1.49086	−0.745428	+	0.666586i	$0.767755\pi$
$608$	−0.722898	−0.0293174
$609$	15.3972	0.623927
$610$	−0.209606	−0.00848668
$611$	−20.9396	−0.847125
$612$	22.5107	0.909939
$613$	16.5205	0.667256	0.333628	−	0.942705i	$-0.391727\pi$
$613$	16.5205	0.667256	0.333628	+	0.942705i	$0.391727\pi$
$614$	−17.7127	−0.714826
$615$	2.93179	0.118221
$616$	−0.931704	−0.0375394
$617$	−9.07919	−0.365514	−0.182757	−	0.983158i	$-0.558502\pi$
$617$	−9.07919	−0.365514	−0.182757	+	0.983158i	$0.558502\pi$
$618$	−7.29850	−0.293589
$619$	29.2530	1.17578	0.587889	−	0.808942i	$-0.299959\pi$
$619$	29.2530	1.17578	0.587889	+	0.808942i	$0.299959\pi$
$620$	−2.10813	−0.0846645
$621$	−20.4992	−0.822605
$622$	−31.6678	−1.26976
$623$	−59.4205	−2.38063
$624$	9.80960	0.392698
$625$	23.5039	0.940155
$626$	27.5333	1.10045
$627$	0.385611	0.0153998
$628$	20.8275	0.831107
$629$	34.1956	1.36347
$630$	−6.62987	−0.264140
$631$	−13.8803	−0.552568	−0.276284	−	0.961076i	$-0.589103\pi$
$631$	−13.8803	−0.552568	−0.276284	+	0.961076i	$0.589103\pi$
$632$	7.73505	0.307684
$633$	44.7587	1.77900
$634$	3.98188	0.158140
$635$	−1.59699	−0.0633745
$636$	5.43143	0.215370
$637$	−56.1863	−2.22618
$638$	0.233471	0.00924319
$639$	−25.8423	−1.02231
$640$	0.316882	0.0125259
$641$	9.82304	0.387987	0.193993	−	0.981003i	$-0.437856\pi$
$641$	9.82304	0.387987	0.193993	+	0.981003i	$0.437856\pi$
$642$	22.6000	0.891953
$643$	−6.95705	−0.274359	−0.137180	−	0.990546i	$-0.543804\pi$
$643$	−6.95705	−0.274359	−0.137180	+	0.990546i	$0.543804\pi$
$644$	−25.5275	−1.00592
$645$	−1.92019	−0.0756072
$646$	−3.69620	−0.145425
$647$	33.3108	1.30958	0.654791	−	0.755810i	$-0.272757\pi$
$647$	33.3108	1.30958	0.654791	+	0.755810i	$0.272757\pi$
$648$	−2.82490	−0.110973
$649$	1.89670	0.0744520
$650$	17.6652	0.692886
$651$	−86.0180	−3.37131
$652$	−1.95549	−0.0765829
$653$	8.95320	0.350366	0.175183	−	0.984536i	$-0.443948\pi$
$653$	8.95320	0.350366	0.175183	+	0.984536i	$0.443948\pi$
$654$	30.5295	1.19380
$655$	−1.54486	−0.0603627
$656$	−3.40050	−0.132767
$657$	20.9995	0.819268
$658$	−27.5999	−1.07595
$659$	−28.5962	−1.11395	−0.556975	−	0.830529i	$-0.688038\pi$
$659$	−28.5962	−1.11395	−0.556975	+	0.830529i	$0.688038\pi$
$660$	−0.169032	−0.00657958
$661$	−12.6731	−0.492926	−0.246463	−	0.969152i	$-0.579268\pi$
$661$	−12.6731	−0.492926	−0.246463	+	0.969152i	$0.579268\pi$
$662$	3.96189	0.153983
$663$	50.1568	1.94793
$664$	−6.99848	−0.271594
$665$	1.08861	0.0422145
$666$	29.4443	1.14094
$667$	6.39680	0.247685
$668$	−11.4700	−0.443789
$669$	31.4813	1.21714
$670$	−1.89298	−0.0731323
$671$	−0.129684	−0.00500638
$672$	12.9297	0.498776
$673$	−30.7144	−1.18395	−0.591977	−	0.805955i	$-0.701652\pi$
$673$	−30.7144	−1.18395	−0.591977	+	0.805955i	$0.701652\pi$
$674$	21.0857	0.812190
$675$	18.6976	0.719671
$676$	−0.000758399 0	−2.91692e−5 0
$677$	39.8704	1.53234	0.766172	−	0.642636i	$-0.222159\pi$
$677$	39.8704	1.53234	0.766172	+	0.642636i	$0.222159\pi$
$678$	15.7625	0.605354
$679$	−52.4844	−2.01417
$680$	1.62023	0.0621329
$681$	11.0908	0.425000
$682$	−1.30431	−0.0499445
$683$	−20.6126	−0.788719	−0.394359	−	0.918956i	$-0.629033\pi$
$683$	−20.6126	−0.788719	−0.394359	+	0.918956i	$0.629033\pi$
$684$	−3.18263	−0.121691
$685$	−0.798995	−0.0305280
$686$	−40.7919	−1.55744
$687$	63.1289	2.40852
$688$	2.22717	0.0849100
$689$	7.19749	0.274203
$690$	−4.63127	−0.176309
$691$	−37.2368	−1.41655	−0.708277	−	0.705934i	$-0.750527\pi$
$691$	−37.2368	−1.41655	−0.708277	+	0.705934i	$0.750527\pi$
$692$	24.8576	0.944944
$693$	−4.10192	−0.155819
$694$	−0.701496	−0.0266284
$695$	−0.0778723	−0.00295386
$696$	−3.24000	−0.122812
$697$	−17.3869	−0.658574
$698$	−7.74778	−0.293258
$699$	59.7917	2.26153
$700$	23.2840	0.880052
$701$	−20.7743	−0.784636	−0.392318	−	0.919830i	$-0.628327\pi$
$701$	−20.7743	−0.784636	−0.392318	+	0.919830i	$0.628327\pi$
$702$	13.7590	0.519298
$703$	−4.83469	−0.182344
$704$	0.196056	0.00738914
$705$	−5.00725	−0.188584
$706$	−7.28930	−0.274336
$707$	−18.0065	−0.677205
$708$	−26.3215	−0.989223
$709$	−9.57055	−0.359430	−0.179715	−	0.983719i	$-0.557517\pi$
$709$	−9.57055	−0.359430	−0.179715	+	0.983719i	$0.557517\pi$
$710$	−1.86003	−0.0698056
$711$	34.0543	1.27714
$712$	12.5037	0.468595
$713$	−35.7363	−1.33834
$714$	66.1102	2.47411
$715$	−0.223994	−0.00837691
$716$	−25.2519	−0.943708
$717$	−20.7434	−0.774678
$718$	−18.7220	−0.698697
$719$	−14.5103	−0.541141	−0.270571	−	0.962700i	$-0.587212\pi$
$719$	−14.5103	−0.541141	−0.270571	+	0.962700i	$0.587212\pi$
$720$	1.39511	0.0519925
$721$	−12.7479	−0.474757
$722$	−18.4774	−0.687658
$723$	−21.7327	−0.808249
$724$	14.1502	0.525888
$725$	−5.83461	−0.216692
$726$	29.8239	1.10687
$727$	43.1380	1.59990	0.799949	−	0.600068i	$-0.204860\pi$
$727$	43.1380	1.59990	0.799949	+	0.600068i	$0.204860\pi$
$728$	17.1339	0.635026
$729$	−43.5856	−1.61428
$730$	1.51146	0.0559417
$731$	11.3876	0.421185
$732$	1.79969	0.0665184
$733$	−34.6015	−1.27804	−0.639018	−	0.769191i	$-0.720659\pi$
$733$	−34.6015	−1.27804	−0.639018	+	0.769191i	$0.720659\pi$
$734$	27.4019	1.01142
$735$	−13.4357	−0.495584
$736$	5.37168	0.198003
$737$	−1.17119	−0.0431415
$738$	−14.9710	−0.551091
$739$	35.8238	1.31780	0.658901	−	0.752230i	$-0.271022\pi$
$739$	35.8238	1.31780	0.658901	+	0.752230i	$0.271022\pi$
$740$	2.11929	0.0779065
$741$	−7.09134	−0.260507
$742$	9.48681	0.348272
$743$	−5.36419	−0.196793	−0.0983966	−	0.995147i	$-0.531371\pi$
$743$	−5.36419	−0.196793	−0.0983966	+	0.995147i	$0.531371\pi$
$744$	18.1005	0.663598
$745$	6.10336	0.223610
$746$	31.9987	1.17155
$747$	−30.8115	−1.12733
$748$	1.00244	0.0366529
$749$	39.4744	1.44236
$750$	8.53507	0.311657
$751$	−13.3333	−0.486537	−0.243269	−	0.969959i	$-0.578220\pi$
$751$	−13.3333	−0.486537	−0.243269	+	0.969959i	$0.578220\pi$
$752$	5.80777	0.211787
$753$	59.2009	2.15740
$754$	−4.29350	−0.156360
$755$	−4.92732	−0.179324
$756$	18.1353	0.659574
$757$	−12.8395	−0.466659	−0.233329	−	0.972398i	$-0.574962\pi$
$757$	−12.8395	−0.466659	−0.233329	+	0.972398i	$0.574962\pi$
$758$	−27.6133	−1.00296
$759$	−2.86538	−0.104007
$760$	−0.229073	−0.00830936
$761$	11.6518	0.422376	0.211188	−	0.977445i	$-0.432267\pi$
$761$	11.6518	0.422376	0.211188	+	0.977445i	$0.432267\pi$
$762$	13.7119	0.496728
$763$	53.3242	1.93047
$764$	−11.3739	−0.411494
$765$	7.13322	0.257902
$766$	−19.1482	−0.691852
$767$	−34.8801	−1.25945
$768$	−2.72077	−0.0981774
$769$	−2.60297	−0.0938654	−0.0469327	−	0.998898i	$-0.514945\pi$
$769$	−2.60297	−0.0938654	−0.0469327	+	0.998898i	$0.514945\pi$
$770$	−0.295240	−0.0106397
$771$	23.1016	0.831983
$772$	−13.7856	−0.496153
$773$	11.0418	0.397147	0.198574	−	0.980086i	$-0.436369\pi$
$773$	11.0418	0.397147	0.198574	+	0.980086i	$0.436369\pi$
$774$	9.80534	0.352446
$775$	32.5956	1.17087
$776$	11.0442	0.396462
$777$	86.4733	3.10221
$778$	−30.9589	−1.10993
$779$	2.45821	0.0880746
$780$	3.10849	0.111302
$781$	−1.15081	−0.0411791
$782$	27.4656	0.982168
$783$	−4.54442	−0.162404
$784$	15.5837	0.556562
$785$	6.59986	0.235559
$786$	13.2643	0.473122
$787$	−4.95967	−0.176793	−0.0883965	−	0.996085i	$-0.528174\pi$
$787$	−4.95967	−0.176793	−0.0883965	+	0.996085i	$0.528174\pi$
$788$	−2.39387	−0.0852783
$789$	70.1747	2.49829
$790$	2.45110	0.0872061
$791$	27.5315	0.978907
$792$	0.863156	0.0306709
$793$	2.38487	0.0846891
$794$	−6.05248	−0.214795
$795$	1.72112	0.0610420
$796$	4.11065	0.145698
$797$	38.6671	1.36966	0.684829	−	0.728704i	$-0.259877\pi$
$797$	38.6671	1.36966	0.684829	+	0.728704i	$0.259877\pi$
$798$	−9.34689	−0.330876
$799$	29.6953	1.05054
$800$	−4.89959	−0.173227
$801$	55.0488	1.94505
$802$	−3.56302	−0.125815
$803$	0.935145	0.0330006
$804$	16.2533	0.573209
$805$	−8.08921	−0.285107
$806$	23.9860	0.844872
$807$	22.2315	0.782586
$808$	3.78907	0.133299
$809$	−5.23503	−0.184054	−0.0920269	−	0.995757i	$-0.529335\pi$
$809$	−5.23503	−0.184054	−0.0920269	+	0.995757i	$0.529335\pi$
$810$	−0.895161	−0.0314528
$811$	−4.03316	−0.141623	−0.0708117	−	0.997490i	$-0.522559\pi$
$811$	−4.03316	−0.141623	−0.0708117	+	0.997490i	$0.522559\pi$
$812$	−5.65914	−0.198597
$813$	−22.9690	−0.805557
$814$	1.31121	0.0459579
$815$	−0.619659	−0.0217057
$816$	−13.9114	−0.486996
$817$	−1.61002	−0.0563273
$818$	12.2249	0.427435
$819$	75.4338	2.63587
$820$	−1.07756	−0.0376299
$821$	−12.1519	−0.424106	−0.212053	−	0.977258i	$-0.568015\pi$
$821$	−12.1519	−0.424106	−0.212053	+	0.977258i	$0.568015\pi$
$822$	6.86023	0.239278
$823$	7.39702	0.257844	0.128922	−	0.991655i	$-0.458848\pi$
$823$	7.39702	0.257844	0.128922	+	0.991655i	$0.458848\pi$
$824$	2.68251	0.0934497
$825$	2.61355	0.0909923
$826$	−45.9744	−1.59966
$827$	13.3443	0.464028	0.232014	−	0.972712i	$-0.425468\pi$
$827$	13.3443	0.464028	0.232014	+	0.972712i	$0.425468\pi$
$828$	23.6494	0.821873
$829$	−24.9653	−0.867081	−0.433540	−	0.901134i	$-0.642736\pi$
$829$	−24.9653	−0.867081	−0.433540	+	0.901134i	$0.642736\pi$
$830$	−2.21769	−0.0769772
$831$	1.94145	0.0673483
$832$	−3.60545	−0.124996
$833$	79.6801	2.76075
$834$	0.668617	0.0231523
$835$	−3.63465	−0.125782
$836$	−0.141728	−0.00490178
$837$	25.3879	0.877533
$838$	7.37793	0.254866
$839$	−8.15039	−0.281383	−0.140691	−	0.990053i	$-0.544933\pi$
$839$	−8.15039	−0.281383	−0.140691	+	0.990053i	$0.544933\pi$
$840$	4.09720	0.141367
$841$	−27.5819	−0.951100
$842$	−18.3985	−0.634053
$843$	−45.9588	−1.58291
$844$	−16.4507	−0.566258
$845$	−0.000240323 0	−8.26737e−6 0
$846$	25.5693	0.879090
$847$	52.0919	1.78990
$848$	−1.99628	−0.0685527
$849$	−33.0394	−1.13391
$850$	−25.0517	−0.859268
$851$	35.9255	1.23151
$852$	15.9703	0.547135
$853$	6.24558	0.213845	0.106922	−	0.994267i	$-0.465900\pi$
$853$	6.24558	0.213845	0.106922	+	0.994267i	$0.465900\pi$
$854$	3.14342	0.107566
$855$	−1.00852	−0.0344906
$856$	−8.30648	−0.283910
$857$	−43.4899	−1.48559	−0.742793	−	0.669522i	$-0.766499\pi$
$857$	−43.4899	−1.48559	−0.742793	+	0.669522i	$0.766499\pi$
$858$	1.92323	0.0656580
$859$	−10.0674	−0.343494	−0.171747	−	0.985141i	$-0.554941\pi$
$859$	−10.0674	−0.343494	−0.171747	+	0.985141i	$0.554941\pi$
$860$	0.705750	0.0240659
$861$	−43.9676	−1.49841
$862$	17.8345	0.607446
$863$	8.15238	0.277510	0.138755	−	0.990327i	$-0.455690\pi$
$863$	8.15238	0.277510	0.138755	+	0.990327i	$0.455690\pi$
$864$	−3.81616	−0.129828
$865$	7.87693	0.267824
$866$	−21.8467	−0.742380
$867$	−24.8763	−0.844845
$868$	31.6153	1.07309
$869$	1.51650	0.0514438
$870$	−1.02670	−0.0348083
$871$	21.5381	0.729792
$872$	−11.2209	−0.379987
$873$	48.6230	1.64564
$874$	−3.88318	−0.131350
$875$	14.9078	0.503974
$876$	−12.9775	−0.438469
$877$	8.22893	0.277871	0.138936	−	0.990301i	$-0.455632\pi$
$877$	8.22893	0.277871	0.138936	+	0.990301i	$0.455632\pi$
$878$	2.16133	0.0729414
$879$	11.7590	0.396623
$880$	0.0621266	0.00209429
$881$	−14.5860	−0.491413	−0.245707	−	0.969344i	$-0.579020\pi$
$881$	−14.5860	−0.491413	−0.245707	+	0.969344i	$0.579020\pi$
$882$	68.6090	2.31018
$883$	38.8232	1.30651	0.653253	−	0.757140i	$-0.273404\pi$
$883$	38.8232	1.30651	0.653253	+	0.757140i	$0.273404\pi$
$884$	−18.4348	−0.620028
$885$	−8.34081	−0.280373
$886$	7.90190	0.265470
$887$	−7.01234	−0.235451	−0.117726	−	0.993046i	$-0.537560\pi$
$887$	−7.01234	−0.235451	−0.117726	+	0.993046i	$0.537560\pi$
$888$	−18.1963	−0.610629
$889$	23.9498	0.803250
$890$	3.96219	0.132813
$891$	−0.553839	−0.0185543
$892$	−11.5707	−0.387416
$893$	−4.19842	−0.140495
$894$	−52.4039	−1.75265
$895$	−8.00187	−0.267473
$896$	−4.75223	−0.158761
$897$	52.6941	1.75940
$898$	−6.97746	−0.232841
$899$	−7.92231	−0.264224
$900$	−21.5709	−0.719031
$901$	−10.2071	−0.340047
$902$	−0.666688	−0.0221983
$903$	28.7967	0.958295
$904$	−5.79338	−0.192685
$905$	4.48394	0.149051
$906$	42.3064	1.40553
$907$	5.60052	0.185962	0.0929810	−	0.995668i	$-0.470360\pi$
$907$	5.60052	0.185962	0.0929810	+	0.995668i	$0.470360\pi$
$908$	−4.07634	−0.135278
$909$	16.6817	0.553299
$910$	5.42943	0.179984
$911$	49.9448	1.65475	0.827373	−	0.561653i	$-0.189834\pi$
$911$	49.9448	1.65475	0.827373	+	0.561653i	$0.189834\pi$
$912$	1.96684	0.0651286
$913$	−1.37209	−0.0454097
$914$	17.7812	0.588151
$915$	0.570289	0.0188532
$916$	−23.2026	−0.766635
$917$	23.1681	0.765077
$918$	−19.5122	−0.643997
$919$	30.5323	1.00717	0.503584	−	0.863946i	$-0.332015\pi$
$919$	30.5323	1.00717	0.503584	+	0.863946i	$0.332015\pi$
$920$	1.70219	0.0561195
$921$	48.1922	1.58799
$922$	−24.5253	−0.807698
$923$	21.1632	0.696594
$924$	2.53495	0.0833939
$925$	−32.7681	−1.07741
$926$	−3.67389	−0.120731
$927$	11.8100	0.387892
$928$	1.19084	0.0390911
$929$	2.65144	0.0869911	0.0434955	−	0.999054i	$-0.486151\pi$
$929$	2.65144	0.0869911	0.0434955	+	0.999054i	$0.486151\pi$
$930$	5.73574	0.188082
$931$	−11.2654	−0.369210
$932$	−21.9760	−0.719848
$933$	86.1608	2.82078
$934$	0.474797	0.0155358
$935$	0.317655	0.0103884
$936$	−15.8733	−0.518836
$937$	−26.3778	−0.861725	−0.430863	−	0.902418i	$-0.641791\pi$
$937$	−26.3778	−0.861725	−0.430863	+	0.902418i	$0.641791\pi$
$938$	28.3888	0.926927
$939$	−74.9117	−2.44465
$940$	1.84038	0.0600265
$941$	44.2565	1.44272	0.721361	−	0.692559i	$-0.243517\pi$
$941$	44.2565	1.44272	0.721361	+	0.692559i	$0.243517\pi$
$942$	−56.6669	−1.84631
$943$	−18.2664	−0.594836
$944$	9.67428	0.314871
$945$	5.74675	0.186942
$946$	0.436650	0.0141967
$947$	7.70512	0.250383	0.125191	−	0.992133i	$-0.460045\pi$
$947$	7.70512	0.250383	0.125191	+	0.992133i	$0.460045\pi$
$948$	−21.0453	−0.683520
$949$	−17.1972	−0.558245
$950$	3.54190	0.114914
$951$	−10.8338	−0.351309
$952$	−24.2983	−0.787514
$953$	−58.1282	−1.88296	−0.941479	−	0.337072i	$-0.890563\pi$
$953$	−58.1282	−1.88296	−0.941479	+	0.337072i	$0.890563\pi$
$954$	−8.78884	−0.284549
$955$	−3.60419	−0.116629
$956$	7.62410	0.246581
$957$	−0.635220	−0.0205338
$958$	7.33172	0.236877
$959$	11.9824	0.386932
$960$	−0.862164	−0.0278262
$961$	13.2587	0.427701
$962$	−24.1130	−0.777434
$963$	−36.5701	−1.17846
$964$	7.98770	0.257267
$965$	−4.36840	−0.140624
$966$	69.4545	2.23466
$967$	55.0355	1.76982	0.884911	−	0.465761i	$-0.154219\pi$
$967$	55.0355	1.76982	0.884911	+	0.465761i	$0.154219\pi$
$968$	−10.9616	−0.352318
$969$	10.0565	0.323062
$970$	3.49970	0.112368
$971$	44.5662	1.43020	0.715099	−	0.699024i	$-0.246382\pi$
$971$	44.5662	1.43020	0.715099	+	0.699024i	$0.246382\pi$
$972$	19.1344	0.613736
$973$	1.16784	0.0374392
$974$	−26.5025	−0.849194
$975$	−48.0630	−1.53925
$976$	−0.661462	−0.0211729
$977$	2.75140	0.0880252	0.0440126	−	0.999031i	$-0.485986\pi$
$977$	2.75140	0.0880252	0.0440126	+	0.999031i	$0.485986\pi$
$978$	5.32044	0.170129
$979$	2.45142	0.0783478
$980$	4.93820	0.157745
$981$	−49.4011	−1.57725
$982$	18.9903	0.606006
$983$	16.2185	0.517291	0.258645	−	0.965972i	$-0.416724\pi$
$983$	16.2185	0.517291	0.258645	+	0.965972i	$0.416724\pi$
$984$	9.25198	0.294942
$985$	−0.758576	−0.0241702
$986$	6.08879	0.193907
$987$	75.0929	2.39023
$988$	2.60637	0.0829197
$989$	11.9636	0.380422
$990$	0.273519	0.00869299
$991$	−8.77957	−0.278892	−0.139446	−	0.990230i	$-0.544532\pi$
$991$	−8.77957	−0.278892	−0.139446	+	0.990230i	$0.544532\pi$
$992$	−6.65272	−0.211224
$993$	−10.7794	−0.342074
$994$	27.8946	0.884762
$995$	1.30259	0.0412949
$996$	19.0413	0.603346
$997$	9.44107	0.299002	0.149501	−	0.988762i	$-0.452233\pi$
$997$	9.44107	0.299002	0.149501	+	0.988762i	$0.452233\pi$
$998$	23.9385	0.757761
$999$	−25.5222	−0.807487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.a.1.9	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.a.1.9	✓	69	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:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.72077 q^{3} +1.00000 q^{4} +0.316882 q^{5} -2.72077 q^{6} -4.75223 q^{7} +1.00000 q^{8} +4.40260 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.72077 q^{3} +1.00000 q^{4} +0.316882 q^{5} -2.72077 q^{6} -4.75223 q^{7} +1.00000 q^{8} +4.40260 q^{9} +0.316882 q^{10} +0.196056 q^{11} -2.72077 q^{12} -3.60545 q^{13} -4.75223 q^{14} -0.862164 q^{15} +1.00000 q^{16} +5.11303 q^{17} +4.40260 q^{18} -0.722898 q^{19} +0.316882 q^{20} +12.9297 q^{21} +0.196056 q^{22} +5.37168 q^{23} -2.72077 q^{24} -4.89959 q^{25} -3.60545 q^{26} -3.81616 q^{27} -4.75223 q^{28} +1.19084 q^{29} -0.862164 q^{30} -6.65272 q^{31} +1.00000 q^{32} -0.533424 q^{33} +5.11303 q^{34} -1.50590 q^{35} +4.40260 q^{36} +6.68793 q^{37} -0.722898 q^{38} +9.80960 q^{39} +0.316882 q^{40} -3.40050 q^{41} +12.9297 q^{42} +2.22717 q^{43} +0.196056 q^{44} +1.39511 q^{45} +5.37168 q^{46} +5.80777 q^{47} -2.72077 q^{48} +15.5837 q^{49} -4.89959 q^{50} -13.9114 q^{51} -3.60545 q^{52} -1.99628 q^{53} -3.81616 q^{54} +0.0621266 q^{55} -4.75223 q^{56} +1.96684 q^{57} +1.19084 q^{58} +9.67428 q^{59} -0.862164 q^{60} -0.661462 q^{61} -6.65272 q^{62} -20.9222 q^{63} +1.00000 q^{64} -1.14250 q^{65} -0.533424 q^{66} -5.97378 q^{67} +5.11303 q^{68} -14.6151 q^{69} -1.50590 q^{70} -5.86978 q^{71} +4.40260 q^{72} +4.76979 q^{73} +6.68793 q^{74} +13.3307 q^{75} -0.722898 q^{76} -0.931704 q^{77} +9.80960 q^{78} +7.73505 q^{79} +0.316882 q^{80} -2.82490 q^{81} -3.40050 q^{82} -6.99848 q^{83} +12.9297 q^{84} +1.62023 q^{85} +2.22717 q^{86} -3.24000 q^{87} +0.196056 q^{88} +12.5037 q^{89} +1.39511 q^{90} +17.1339 q^{91} +5.37168 q^{92} +18.1005 q^{93} +5.80777 q^{94} -0.229073 q^{95} -2.72077 q^{96} +11.0442 q^{97} +15.5837 q^{98} +0.863156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

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