LMFDB - Embedded newform 4009.2.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93894 q^{2} -1.61421 q^{3} +1.75950 q^{4} -0.710068 q^{5} +3.12986 q^{6} -1.51648 q^{7} +0.466323 q^{8} -0.394323 q^{9} +O(q^{10})$	\(q-1.93894 q^{2} -1.61421 q^{3} +1.75950 q^{4} -0.710068 q^{5} +3.12986 q^{6} -1.51648 q^{7} +0.466323 q^{8} -0.394323 q^{9} +1.37678 q^{10} +4.83786 q^{11} -2.84020 q^{12} -3.42790 q^{13} +2.94037 q^{14} +1.14620 q^{15} -4.42317 q^{16} +2.25296 q^{17} +0.764570 q^{18} +1.00000 q^{19} -1.24936 q^{20} +2.44792 q^{21} -9.38033 q^{22} +5.02944 q^{23} -0.752744 q^{24} -4.49580 q^{25} +6.64650 q^{26} +5.47915 q^{27} -2.66824 q^{28} -9.09232 q^{29} -2.22241 q^{30} -7.95846 q^{31} +7.64362 q^{32} -7.80933 q^{33} -4.36836 q^{34} +1.07680 q^{35} -0.693810 q^{36} +0.118446 q^{37} -1.93894 q^{38} +5.53335 q^{39} -0.331121 q^{40} +4.81219 q^{41} -4.74637 q^{42} +7.87596 q^{43} +8.51220 q^{44} +0.279996 q^{45} -9.75180 q^{46} -4.37650 q^{47} +7.13992 q^{48} -4.70029 q^{49} +8.71710 q^{50} -3.63675 q^{51} -6.03137 q^{52} -2.80310 q^{53} -10.6238 q^{54} -3.43521 q^{55} -0.707169 q^{56} -1.61421 q^{57} +17.6295 q^{58} +13.7278 q^{59} +2.01673 q^{60} +5.59544 q^{61} +15.4310 q^{62} +0.597983 q^{63} -5.97420 q^{64} +2.43404 q^{65} +15.1418 q^{66} -4.65901 q^{67} +3.96408 q^{68} -8.11858 q^{69} -2.08786 q^{70} +4.73205 q^{71} -0.183882 q^{72} +8.30981 q^{73} -0.229659 q^{74} +7.25717 q^{75} +1.75950 q^{76} -7.33651 q^{77} -10.7288 q^{78} -1.73416 q^{79} +3.14075 q^{80} -7.66154 q^{81} -9.33056 q^{82} -10.6561 q^{83} +4.30710 q^{84} -1.59976 q^{85} -15.2710 q^{86} +14.6769 q^{87} +2.25601 q^{88} -0.175995 q^{89} -0.542897 q^{90} +5.19834 q^{91} +8.84929 q^{92} +12.8466 q^{93} +8.48579 q^{94} -0.710068 q^{95} -12.3384 q^{96} -6.23723 q^{97} +9.11359 q^{98} -1.90768 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−1.93894	−1.37104	−0.685520	−	0.728054i	$-0.740425\pi$
$2$	−1.93894	−1.37104	−0.685520	+	0.728054i	$0.740425\pi$
$3$	−1.61421	−0.931965	−0.465983	−	0.884794i	$-0.654299\pi$
$3$	−1.61421	−0.931965	−0.465983	+	0.884794i	$0.654299\pi$
$4$	1.75950	0.879748
$5$	−0.710068	−0.317552	−0.158776	−	0.987315i	$-0.550755\pi$
$5$	−0.710068	−0.317552	−0.158776	+	0.987315i	$0.550755\pi$
$6$	3.12986	1.27776
$7$	−1.51648	−0.573175	−0.286588	−	0.958054i	$-0.592521\pi$
$7$	−1.51648	−0.573175	−0.286588	+	0.958054i	$0.592521\pi$
$8$	0.466323	0.164870
$9$	−0.394323	−0.131441
$10$	1.37678	0.435376
$11$	4.83786	1.45867	0.729335	−	0.684157i	$-0.239830\pi$
$11$	4.83786	1.45867	0.729335	+	0.684157i	$0.239830\pi$
$12$	−2.84020	−0.819894
$13$	−3.42790	−0.950728	−0.475364	−	0.879789i	$-0.657684\pi$
$13$	−3.42790	−0.950728	−0.475364	+	0.879789i	$0.657684\pi$
$14$	2.94037	0.785846
$15$	1.14620	0.295948
$16$	−4.42317	−1.10579
$17$	2.25296	0.546423	0.273212	−	0.961954i	$-0.411914\pi$
$17$	2.25296	0.546423	0.273212	+	0.961954i	$0.411914\pi$
$18$	0.764570	0.180211
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.24936	−0.279366
$21$	2.44792	0.534179
$22$	−9.38033	−1.99989
$23$	5.02944	1.04871	0.524356	−	0.851499i	$-0.324306\pi$
$23$	5.02944	1.04871	0.524356	+	0.851499i	$0.324306\pi$
$24$	−0.752744	−0.153653
$25$	−4.49580	−0.899161
$26$	6.64650	1.30349
$27$	5.47915	1.05446
$28$	−2.66824	−0.504250
$29$	−9.09232	−1.68840	−0.844200	−	0.536028i	$-0.819924\pi$
$29$	−9.09232	−1.68840	−0.844200	+	0.536028i	$0.819924\pi$
$30$	−2.22241	−0.405756
$31$	−7.95846	−1.42938	−0.714691	−	0.699440i	$-0.753433\pi$
$31$	−7.95846	−1.42938	−0.714691	+	0.699440i	$0.753433\pi$
$32$	7.64362	1.35121
$33$	−7.80933	−1.35943
$34$	−4.36836	−0.749168
$35$	1.07680	0.182013
$36$	−0.693810	−0.115635
$37$	0.118446	0.0194724	0.00973618	−	0.999953i	$-0.496901\pi$
$37$	0.118446	0.0194724	0.00973618	+	0.999953i	$0.496901\pi$
$38$	−1.93894	−0.314538
$39$	5.53335	0.886045
$40$	−0.331121	−0.0523549
$41$	4.81219	0.751538	0.375769	−	0.926713i	$-0.377379\pi$
$41$	4.81219	0.751538	0.375769	+	0.926713i	$0.377379\pi$
$42$	−4.74637	−0.732381
$43$	7.87596	1.20107	0.600537	−	0.799597i	$-0.294954\pi$
$43$	7.87596	1.20107	0.600537	+	0.799597i	$0.294954\pi$
$44$	8.51220	1.28326
$45$	0.279996	0.0417394
$46$	−9.75180	−1.43782
$47$	−4.37650	−0.638379	−0.319189	−	0.947691i	$-0.603411\pi$
$47$	−4.37650	−0.638379	−0.319189	+	0.947691i	$0.603411\pi$
$48$	7.13992	1.03056
$49$	−4.70029	−0.671470
$50$	8.71710	1.23278
$51$	−3.63675	−0.509248
$52$	−6.03137	−0.836401
$53$	−2.80310	−0.385036	−0.192518	−	0.981293i	$-0.561665\pi$
$53$	−2.80310	−0.385036	−0.192518	+	0.981293i	$0.561665\pi$
$54$	−10.6238	−1.44571
$55$	−3.43521	−0.463204
$56$	−0.707169	−0.0944995
$57$	−1.61421	−0.213807
$58$	17.6295	2.31486
$59$	13.7278	1.78721	0.893607	−	0.448851i	$-0.148167\pi$
$59$	13.7278	1.78721	0.893607	+	0.448851i	$0.148167\pi$
$60$	2.01673	0.260359
$61$	5.59544	0.716423	0.358211	−	0.933640i	$-0.383387\pi$
$61$	5.59544	0.716423	0.358211	+	0.933640i	$0.383387\pi$
$62$	15.4310	1.95974
$63$	0.597983	0.0753388
$64$	−5.97420	−0.746774
$65$	2.43404	0.301906
$66$	15.1418	1.86383
$67$	−4.65901	−0.569189	−0.284594	−	0.958648i	$-0.591859\pi$
$67$	−4.65901	−0.569189	−0.284594	+	0.958648i	$0.591859\pi$
$68$	3.96408	0.480715
$69$	−8.11858	−0.977362
$70$	−2.08786	−0.249547
$71$	4.73205	0.561591	0.280795	−	0.959768i	$-0.409402\pi$
$71$	4.73205	0.561591	0.280795	+	0.959768i	$0.409402\pi$
$72$	−0.183882	−0.0216707
$73$	8.30981	0.972589	0.486295	−	0.873795i	$-0.338348\pi$
$73$	8.30981	0.972589	0.486295	+	0.873795i	$0.338348\pi$
$74$	−0.229659	−0.0266974
$75$	7.25717	0.837986
$76$	1.75950	0.201828
$77$	−7.33651	−0.836073
$78$	−10.7288	−1.21480
$79$	−1.73416	−0.195108	−0.0975539	−	0.995230i	$-0.531102\pi$
$79$	−1.73416	−0.195108	−0.0975539	+	0.995230i	$0.531102\pi$
$80$	3.14075	0.351146
$81$	−7.66154	−0.851282
$82$	−9.33056	−1.03039
$83$	−10.6561	−1.16966	−0.584831	−	0.811155i	$-0.698839\pi$
$83$	−10.6561	−1.16966	−0.584831	+	0.811155i	$0.698839\pi$
$84$	4.30710	0.469943
$85$	−1.59976	−0.173518
$86$	−15.2710	−1.64672
$87$	14.6769	1.57353
$88$	2.25601	0.240491
$89$	−0.175995	−0.0186555	−0.00932773	−	0.999956i	$-0.502969\pi$
$89$	−0.175995	−0.0186555	−0.00932773	+	0.999956i	$0.502969\pi$
$90$	−0.542897	−0.0572263
$91$	5.19834	0.544934
$92$	8.84929	0.922602
$93$	12.8466	1.33213
$94$	8.48579	0.875242
$95$	−0.710068	−0.0728515
$96$	−12.3384	−1.25928
$97$	−6.23723	−0.633295	−0.316648	−	0.948543i	$-0.602557\pi$
$97$	−6.23723	−0.633295	−0.316648	+	0.948543i	$0.602557\pi$
$98$	9.11359	0.920612
$99$	−1.90768	−0.191729
$100$	−7.91035	−0.791035
$101$	10.6438	1.05910	0.529551	−	0.848278i	$-0.322360\pi$
$101$	10.6438	1.05910	0.529551	+	0.848278i	$0.322360\pi$
$102$	7.05146	0.698198
$103$	10.3097	1.01585	0.507923	−	0.861402i	$-0.330413\pi$
$103$	10.3097	1.01585	0.507923	+	0.861402i	$0.330413\pi$
$104$	−1.59851	−0.156747
$105$	−1.73819	−0.169630
$106$	5.43505	0.527899
$107$	−11.2971	−1.09213	−0.546065	−	0.837743i	$-0.683875\pi$
$107$	−11.2971	−1.09213	−0.546065	+	0.837743i	$0.683875\pi$
$108$	9.64055	0.927662
$109$	−8.43282	−0.807717	−0.403859	−	0.914821i	$-0.632331\pi$
$109$	−8.43282	−0.807717	−0.403859	+	0.914821i	$0.632331\pi$
$110$	6.66067	0.635070
$111$	−0.191196	−0.0181476
$112$	6.70764	0.633812
$113$	1.82578	0.171755	0.0858774	−	0.996306i	$-0.472631\pi$
$113$	1.82578	0.171755	0.0858774	+	0.996306i	$0.472631\pi$
$114$	3.12986	0.293138
$115$	−3.57125	−0.333021
$116$	−15.9979	−1.48537
$117$	1.35170	0.124965
$118$	−26.6175	−2.45034
$119$	−3.41657	−0.313196
$120$	0.534499	0.0487929
$121$	12.4049	1.12772
$122$	−10.8492	−0.982244
$123$	−7.76789	−0.700407
$124$	−14.0029	−1.25750
$125$	6.74267	0.603083
$126$	−1.15945	−0.103292
$127$	18.6381	1.65387	0.826933	−	0.562300i	$-0.190083\pi$
$127$	18.6381	1.65387	0.826933	+	0.562300i	$0.190083\pi$
$128$	−3.70361	−0.327356
$129$	−12.7135	−1.11936
$130$	−4.71947	−0.413925
$131$	13.0963	1.14423	0.572113	−	0.820174i	$-0.306124\pi$
$131$	13.0963	1.14423	0.572113	+	0.820174i	$0.306124\pi$
$132$	−13.7405	−1.19596
$133$	−1.51648	−0.131495
$134$	9.03356	0.780380
$135$	−3.89057	−0.334847
$136$	1.05061	0.0900889
$137$	12.1805	1.04065	0.520325	−	0.853968i	$-0.325811\pi$
$137$	12.1805	1.04065	0.520325	+	0.853968i	$0.325811\pi$
$138$	15.7415	1.34000
$139$	6.69201	0.567609	0.283804	−	0.958882i	$-0.408403\pi$
$139$	6.69201	0.567609	0.283804	+	0.958882i	$0.408403\pi$
$140$	1.89463	0.160126
$141$	7.06460	0.594947
$142$	−9.17517	−0.769963
$143$	−16.5837	−1.38680
$144$	1.74416	0.145346
$145$	6.45616	0.536155
$146$	−16.1122	−1.33346
$147$	7.58726	0.625787
$148$	0.208405	0.0171308
$149$	15.0153	1.23010	0.615050	−	0.788488i	$-0.289136\pi$
$149$	15.0153	1.23010	0.615050	+	0.788488i	$0.289136\pi$
$150$	−14.0712	−1.14891
$151$	−1.59359	−0.129685	−0.0648423	−	0.997896i	$-0.520654\pi$
$151$	−1.59359	−0.129685	−0.0648423	+	0.997896i	$0.520654\pi$
$152$	0.466323	0.0378238
$153$	−0.888395	−0.0718225
$154$	14.2251	1.14629
$155$	5.65105	0.453903
$156$	9.73591	0.779497
$157$	−18.4721	−1.47423	−0.737115	−	0.675767i	$-0.763812\pi$
$157$	−18.4721	−1.47423	−0.737115	+	0.675767i	$0.763812\pi$
$158$	3.36243	0.267500
$159$	4.52480	0.358840
$160$	−5.42749	−0.429081
$161$	−7.62705	−0.601095
$162$	14.8553	1.16714
$163$	19.3453	1.51524	0.757621	−	0.652695i	$-0.226362\pi$
$163$	19.3453	1.51524	0.757621	+	0.652695i	$0.226362\pi$
$164$	8.46703	0.661164
$165$	5.54515	0.431690
$166$	20.6616	1.60365
$167$	22.4293	1.73563	0.867817	−	0.496884i	$-0.165523\pi$
$167$	22.4293	1.73563	0.867817	+	0.496884i	$0.165523\pi$
$168$	1.14152	0.0880702
$169$	−1.24951	−0.0961161
$170$	3.10183	0.237900
$171$	−0.394323	−0.0301546
$172$	13.8577	1.05664
$173$	−6.57092	−0.499578	−0.249789	−	0.968300i	$-0.580361\pi$
$173$	−6.57092	−0.499578	−0.249789	+	0.968300i	$0.580361\pi$
$174$	−28.4577	−2.15737
$175$	6.81779	0.515377
$176$	−21.3987	−1.61298
$177$	−22.1596	−1.66562
$178$	0.341245	0.0255774
$179$	−19.4022	−1.45019	−0.725096	−	0.688648i	$-0.758205\pi$
$179$	−19.4022	−1.45019	−0.725096	+	0.688648i	$0.758205\pi$
$180$	0.492652	0.0367201
$181$	−22.0532	−1.63920	−0.819600	−	0.572936i	$-0.805804\pi$
$181$	−22.0532	−1.63920	−0.819600	+	0.572936i	$0.805804\pi$
$182$	−10.0793	−0.747126
$183$	−9.03222	−0.667681
$184$	2.34535	0.172901
$185$	−0.0841046	−0.00618349
$186$	−24.9089	−1.82641
$187$	10.8995	0.797051
$188$	−7.70044	−0.561613
$189$	−8.30902	−0.604392
$190$	1.37678	0.0998822
$191$	5.74002	0.415334	0.207667	−	0.978200i	$-0.433413\pi$
$191$	5.74002	0.415334	0.207667	+	0.978200i	$0.433413\pi$
$192$	9.64361	0.695968
$193$	12.5726	0.904995	0.452497	−	0.891766i	$-0.350533\pi$
$193$	12.5726	0.904995	0.452497	+	0.891766i	$0.350533\pi$
$194$	12.0936	0.868272
$195$	−3.92906	−0.281366
$196$	−8.27014	−0.590724
$197$	−22.4687	−1.60083	−0.800414	−	0.599448i	$-0.795387\pi$
$197$	−22.4687	−1.60083	−0.800414	+	0.599448i	$0.795387\pi$
$198$	3.69888	0.262868
$199$	−14.4119	−1.02163	−0.510816	−	0.859690i	$-0.670657\pi$
$199$	−14.4119	−1.02163	−0.510816	+	0.859690i	$0.670657\pi$
$200$	−2.09650	−0.148245
$201$	7.52063	0.530464
$202$	−20.6378	−1.45207
$203$	13.7883	0.967750
$204$	−6.39886	−0.448010
$205$	−3.41699	−0.238653
$206$	−19.9899	−1.39277
$207$	−1.98323	−0.137844
$208$	15.1622	1.05131
$209$	4.83786	0.334642
$210$	3.37025	0.232569
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−4.93205	−0.338734
$213$	−7.63852	−0.523383
$214$	21.9044	1.49735
$215$	−5.59247	−0.381403
$216$	2.55506	0.173850
$217$	12.0688	0.819287
$218$	16.3507	1.10741
$219$	−13.4138	−0.906419
$220$	−6.04424	−0.407503
$221$	−7.72292	−0.519500
$222$	0.370719	0.0248810
$223$	22.4017	1.50013	0.750065	−	0.661364i	$-0.230022\pi$
$223$	22.4017	1.50013	0.750065	+	0.661364i	$0.230022\pi$
$224$	−11.5914	−0.774482
$225$	1.77280	0.118187
$226$	−3.54008	−0.235482
$227$	4.24493	0.281746	0.140873	−	0.990028i	$-0.455009\pi$
$227$	4.24493	0.281746	0.140873	+	0.990028i	$0.455009\pi$
$228$	−2.84020	−0.188097
$229$	24.9751	1.65040	0.825199	−	0.564842i	$-0.191063\pi$
$229$	24.9751	1.65040	0.825199	+	0.564842i	$0.191063\pi$
$230$	6.92444	0.456584
$231$	11.8427	0.779191
$232$	−4.23996	−0.278367
$233$	−9.42191	−0.617250	−0.308625	−	0.951184i	$-0.599869\pi$
$233$	−9.42191	−0.617250	−0.308625	+	0.951184i	$0.599869\pi$
$234$	−2.62087	−0.171332
$235$	3.10762	0.202719
$236$	24.1541	1.57230
$237$	2.79929	0.181834
$238$	6.62453	0.429405
$239$	−16.2998	−1.05435	−0.527173	−	0.849758i	$-0.676748\pi$
$239$	−16.2998	−1.05435	−0.527173	+	0.849758i	$0.676748\pi$
$240$	−5.06983	−0.327256
$241$	−12.7871	−0.823690	−0.411845	−	0.911254i	$-0.635115\pi$
$241$	−12.7871	−0.823690	−0.411845	+	0.911254i	$0.635115\pi$
$242$	−24.0524	−1.54614
$243$	−4.07012	−0.261098
$244$	9.84516	0.630272
$245$	3.33753	0.213227
$246$	15.0615	0.960286
$247$	−3.42790	−0.218112
$248$	−3.71122	−0.235662
$249$	17.2012	1.09008
$250$	−13.0736	−0.826850
$251$	−23.6809	−1.49472	−0.747362	−	0.664417i	$-0.768680\pi$
$251$	−23.6809	−1.49472	−0.747362	+	0.664417i	$0.768680\pi$
$252$	1.05215	0.0662791
$253$	24.3317	1.52972
$254$	−36.1383	−2.26752
$255$	2.58234	0.161713
$256$	19.1295	1.19559
$257$	−3.38495	−0.211147	−0.105574	−	0.994411i	$-0.533668\pi$
$257$	−3.38495	−0.211147	−0.105574	+	0.994411i	$0.533668\pi$
$258$	24.6507	1.53468
$259$	−0.179621	−0.0111611
$260$	4.28269	0.265601
$261$	3.58531	0.221925
$262$	−25.3929	−1.56878
$263$	27.4938	1.69534	0.847670	−	0.530524i	$-0.178005\pi$
$263$	27.4938	1.69534	0.847670	+	0.530524i	$0.178005\pi$
$264$	−3.64167	−0.224129
$265$	1.99039	0.122269
$266$	2.94037	0.180285
$267$	0.284093	0.0173862
$268$	−8.19751	−0.500743
$269$	−26.1034	−1.59155	−0.795775	−	0.605592i	$-0.792936\pi$
$269$	−26.1034	−1.59155	−0.795775	+	0.605592i	$0.792936\pi$
$270$	7.54359	0.459089
$271$	−19.7467	−1.19953	−0.599763	−	0.800177i	$-0.704739\pi$
$271$	−19.7467	−1.19953	−0.599763	+	0.800177i	$0.704739\pi$
$272$	−9.96522	−0.604230
$273$	−8.39121	−0.507859
$274$	−23.6173	−1.42677
$275$	−21.7501	−1.31158
$276$	−14.2846	−0.859833
$277$	22.9531	1.37912	0.689558	−	0.724230i	$-0.257805\pi$
$277$	22.9531	1.37912	0.689558	+	0.724230i	$0.257805\pi$
$278$	−12.9754	−0.778214
$279$	3.13821	0.187880
$280$	0.502138	0.0300085
$281$	−14.0236	−0.836576	−0.418288	−	0.908314i	$-0.637370\pi$
$281$	−14.0236	−0.836576	−0.418288	+	0.908314i	$0.637370\pi$
$282$	−13.6979	−0.815695
$283$	−14.4795	−0.860716	−0.430358	−	0.902658i	$-0.641613\pi$
$283$	−14.4795	−0.860716	−0.430358	+	0.902658i	$0.641613\pi$
$284$	8.32602	0.494058
$285$	1.14620	0.0678950
$286$	32.1548	1.90135
$287$	−7.29759	−0.430763
$288$	−3.01405	−0.177605
$289$	−11.9242	−0.701421
$290$	−12.5181	−0.735090
$291$	10.0682	0.590209
$292$	14.6211	0.855634
$293$	−26.4389	−1.54458	−0.772288	−	0.635273i	$-0.780888\pi$
$293$	−26.4389	−1.54458	−0.772288	+	0.635273i	$0.780888\pi$
$294$	−14.7113	−0.857978
$295$	−9.74771	−0.567533
$296$	0.0552340	0.00321041
$297$	26.5074	1.53811
$298$	−29.1137	−1.68651
$299$	−17.2404	−0.997039
$300$	12.7690	0.737217
$301$	−11.9437	−0.688426
$302$	3.08988	0.177803
$303$	−17.1814	−0.987046
$304$	−4.42317	−0.253686
$305$	−3.97315	−0.227502
$306$	1.72255	0.0984714
$307$	−2.14032	−0.122155	−0.0610773	−	0.998133i	$-0.519454\pi$
$307$	−2.14032	−0.122155	−0.0610773	+	0.998133i	$0.519454\pi$
$308$	−12.9086	−0.735534
$309$	−16.6421	−0.946734
$310$	−10.9571	−0.622319
$311$	−16.8102	−0.953217	−0.476609	−	0.879116i	$-0.658134\pi$
$311$	−16.8102	−0.953217	−0.476609	+	0.879116i	$0.658134\pi$
$312$	2.58033	0.146082
$313$	19.2019	1.08536	0.542678	−	0.839941i	$-0.317410\pi$
$313$	19.2019	1.08536	0.542678	+	0.839941i	$0.317410\pi$
$314$	35.8162	2.02123
$315$	−0.424609	−0.0239240
$316$	−3.05124	−0.171646
$317$	−7.18411	−0.403500	−0.201750	−	0.979437i	$-0.564663\pi$
$317$	−7.18411	−0.403500	−0.201750	+	0.979437i	$0.564663\pi$
$318$	−8.77332	−0.491983
$319$	−43.9874	−2.46282
$320$	4.24209	0.237140
$321$	18.2359	1.01783
$322$	14.7884	0.824125
$323$	2.25296	0.125358
$324$	−13.4804	−0.748914
$325$	15.4112	0.854857
$326$	−37.5094	−2.07746
$327$	13.6123	0.752764
$328$	2.24404	0.123906
$329$	6.63688	0.365903
$330$	−10.7517	−0.591863
$331$	−22.1654	−1.21832	−0.609162	−	0.793046i	$-0.708494\pi$
$331$	−22.1654	−1.21832	−0.609162	+	0.793046i	$0.708494\pi$
$332$	−18.7494	−1.02901
$333$	−0.0467059	−0.00255947
$334$	−43.4892	−2.37962
$335$	3.30822	0.180747
$336$	−10.8275	−0.590691
$337$	21.0069	1.14432	0.572159	−	0.820143i	$-0.306106\pi$
$337$	21.0069	1.14432	0.572159	+	0.820143i	$0.306106\pi$
$338$	2.42273	0.131779
$339$	−2.94719	−0.160069
$340$	−2.81476	−0.152652
$341$	−38.5019	−2.08500
$342$	0.764570	0.0413432
$343$	17.7432	0.958045
$344$	3.67274	0.198021
$345$	5.76475	0.310364
$346$	12.7406	0.684941
$347$	9.85372	0.528975	0.264488	−	0.964389i	$-0.414797\pi$
$347$	9.85372	0.528975	0.264488	+	0.964389i	$0.414797\pi$
$348$	25.8240	1.38431
$349$	−14.4455	−0.773250	−0.386625	−	0.922237i	$-0.626359\pi$
$349$	−14.4455	−0.773250	−0.386625	+	0.922237i	$0.626359\pi$
$350$	−13.2193	−0.706602
$351$	−18.7820	−1.00251
$352$	36.9787	1.97097
$353$	3.85238	0.205041	0.102521	−	0.994731i	$-0.467309\pi$
$353$	3.85238	0.205041	0.102521	+	0.994731i	$0.467309\pi$
$354$	42.9663	2.28363
$355$	−3.36008	−0.178334
$356$	−0.309663	−0.0164121
$357$	5.51506	0.291888
$358$	37.6198	1.98827
$359$	−15.9317	−0.840840	−0.420420	−	0.907329i	$-0.638117\pi$
$359$	−15.9317	−0.840840	−0.420420	+	0.907329i	$0.638117\pi$
$360$	0.130569	0.00688158
$361$	1.00000	0.0526316
$362$	42.7598	2.24741
$363$	−20.0241	−1.05099
$364$	9.14646	0.479405
$365$	−5.90053	−0.308848
$366$	17.5130	0.915417
$367$	−25.5007	−1.33112	−0.665562	−	0.746343i	$-0.731808\pi$
$367$	−25.5007	−1.33112	−0.665562	+	0.746343i	$0.731808\pi$
$368$	−22.2461	−1.15966
$369$	−1.89756	−0.0987830
$370$	0.163074	0.00847781
$371$	4.25085	0.220693
$372$	22.6036	1.17194
$373$	28.4844	1.47486	0.737432	−	0.675421i	$-0.236038\pi$
$373$	28.4844	1.47486	0.737432	+	0.675421i	$0.236038\pi$
$374$	−21.1335	−1.09279
$375$	−10.8841	−0.562052
$376$	−2.04087	−0.105250
$377$	31.1675	1.60521
$378$	16.1107	0.828646
$379$	16.3569	0.840200	0.420100	−	0.907478i	$-0.361995\pi$
$379$	16.3569	0.840200	0.420100	+	0.907478i	$0.361995\pi$
$380$	−1.24936	−0.0640909
$381$	−30.0859	−1.54135
$382$	−11.1296	−0.569439
$383$	−3.96299	−0.202499	−0.101250	−	0.994861i	$-0.532284\pi$
$383$	−3.96299	−0.202499	−0.101250	+	0.994861i	$0.532284\pi$
$384$	5.97841	0.305084
$385$	5.20943	0.265497
$386$	−24.3775	−1.24078
$387$	−3.10568	−0.157870
$388$	−10.9744	−0.557140
$389$	2.67002	0.135375	0.0676877	−	0.997707i	$-0.478438\pi$
$389$	2.67002	0.135375	0.0676877	+	0.997707i	$0.478438\pi$
$390$	7.61821	0.385763
$391$	11.3311	0.573040
$392$	−2.19185	−0.110705
$393$	−21.1402	−1.06638
$394$	43.5655	2.19480
$395$	1.23137	0.0619569
$396$	−3.35656	−0.168673
$397$	−1.51398	−0.0759846	−0.0379923	−	0.999278i	$-0.512096\pi$
$397$	−1.51398	−0.0759846	−0.0379923	+	0.999278i	$0.512096\pi$
$398$	27.9438	1.40070
$399$	2.44792	0.122549
$400$	19.8857	0.994284
$401$	34.0918	1.70246	0.851232	−	0.524790i	$-0.175856\pi$
$401$	34.0918	1.70246	0.851232	+	0.524790i	$0.175856\pi$
$402$	−14.5821	−0.727287
$403$	27.2808	1.35895
$404$	18.7278	0.931743
$405$	5.44022	0.270326
$406$	−26.7347	−1.32682
$407$	0.573024	0.0284037
$408$	−1.69590	−0.0839597
$409$	−30.4921	−1.50774	−0.753869	−	0.657024i	$-0.771815\pi$
$409$	−30.4921	−1.50774	−0.753869	+	0.657024i	$0.771815\pi$
$410$	6.62534	0.327202
$411$	−19.6619	−0.969849
$412$	18.1399	0.893689
$413$	−20.8180	−1.02439
$414$	3.84536	0.188989
$415$	7.56658	0.371429
$416$	−26.2015	−1.28464
$417$	−10.8023	−0.528991
$418$	−9.38033	−0.458807
$419$	24.0107	1.17300	0.586499	−	0.809950i	$-0.300506\pi$
$419$	24.0107	1.17300	0.586499	+	0.809950i	$0.300506\pi$
$420$	−3.05834	−0.149231
$421$	−23.6431	−1.15230	−0.576148	−	0.817346i	$-0.695445\pi$
$421$	−23.6431	−1.15230	−0.576148	+	0.817346i	$0.695445\pi$
$422$	−1.93894	−0.0943862
$423$	1.72576	0.0839092
$424$	−1.30715	−0.0634809
$425$	−10.1289	−0.491322
$426$	14.8107	0.717579
$427$	−8.48537	−0.410636
$428$	−19.8772	−0.960799
$429$	26.7696	1.29245
$430$	10.8435	0.522919
$431$	−35.5519	−1.71247	−0.856237	−	0.516583i	$-0.827204\pi$
$431$	−35.5519	−1.71247	−0.856237	+	0.516583i	$0.827204\pi$
$432$	−24.2352	−1.16602
$433$	−22.7300	−1.09233	−0.546166	−	0.837677i	$-0.683913\pi$
$433$	−22.7300	−1.09233	−0.546166	+	0.837677i	$0.683913\pi$
$434$	−23.4008	−1.12327
$435$	−10.4216	−0.499678
$436$	−14.8375	−0.710588
$437$	5.02944	0.240591
$438$	26.0085	1.24274
$439$	−34.1119	−1.62807	−0.814036	−	0.580814i	$-0.802734\pi$
$439$	−34.1119	−1.62807	−0.814036	+	0.580814i	$0.802734\pi$
$440$	−1.60192	−0.0763684
$441$	1.85343	0.0882587
$442$	14.9743	0.712255
$443$	−27.3154	−1.29779	−0.648897	−	0.760876i	$-0.724769\pi$
$443$	−27.3154	−1.29779	−0.648897	+	0.760876i	$0.724769\pi$
$444$	−0.336409	−0.0159653
$445$	0.124969	0.00592408
$446$	−43.4356	−2.05674
$447$	−24.2378	−1.14641
$448$	9.05974	0.428033
$449$	−26.5782	−1.25430	−0.627151	−	0.778897i	$-0.715779\pi$
$449$	−26.5782	−1.25430	−0.627151	+	0.778897i	$0.715779\pi$
$450$	−3.43736	−0.162038
$451$	23.2807	1.09625
$452$	3.21245	0.151101
$453$	2.57239	0.120862
$454$	−8.23067	−0.386284
$455$	−3.69117	−0.173045
$456$	−0.752744	−0.0352505
$457$	−11.0655	−0.517624	−0.258812	−	0.965928i	$-0.583331\pi$
$457$	−11.0655	−0.517624	−0.258812	+	0.965928i	$0.583331\pi$
$458$	−48.4252	−2.26276
$459$	12.3443	0.576184
$460$	−6.28360	−0.292974
$461$	20.9443	0.975473	0.487737	−	0.872991i	$-0.337823\pi$
$461$	20.9443	0.975473	0.487737	+	0.872991i	$0.337823\pi$
$462$	−22.9623	−1.06830
$463$	−30.7131	−1.42736	−0.713680	−	0.700472i	$-0.752973\pi$
$463$	−30.7131	−1.42736	−0.713680	+	0.700472i	$0.752973\pi$
$464$	40.2168	1.86702
$465$	−9.12199	−0.423022
$466$	18.2685	0.846273
$467$	−15.3672	−0.711111	−0.355555	−	0.934655i	$-0.615708\pi$
$467$	−15.3672	−0.711111	−0.355555	+	0.934655i	$0.615708\pi$
$468$	2.37831	0.109937
$469$	7.06530	0.326245
$470$	−6.02549	−0.277935
$471$	29.8178	1.37393
$472$	6.40161	0.294658
$473$	38.1028	1.75197
$474$	−5.42767	−0.249301
$475$	−4.49580	−0.206282
$476$	−6.01144	−0.275534
$477$	1.10533	0.0506095
$478$	31.6043	1.44555
$479$	−21.2338	−0.970196	−0.485098	−	0.874460i	$-0.661216\pi$
$479$	−21.2338	−0.970196	−0.485098	+	0.874460i	$0.661216\pi$
$480$	8.76111	0.399888
$481$	−0.406020	−0.0185129
$482$	24.7935	1.12931
$483$	12.3117	0.560200
$484$	21.8264	0.992107
$485$	4.42886	0.201104
$486$	7.89173	0.357976
$487$	−16.8974	−0.765696	−0.382848	−	0.923811i	$-0.625057\pi$
$487$	−16.8974	−0.765696	−0.382848	+	0.923811i	$0.625057\pi$
$488$	2.60928	0.118117
$489$	−31.2274	−1.41215
$490$	−6.47127	−0.292342
$491$	−22.5677	−1.01847	−0.509234	−	0.860628i	$-0.670071\pi$
$491$	−22.5677	−1.01847	−0.509234	+	0.860628i	$0.670071\pi$
$492$	−13.6676	−0.616182
$493$	−20.4846	−0.922582
$494$	6.64650	0.299040
$495$	1.35458	0.0608840
$496$	35.2016	1.58060
$497$	−7.17605	−0.321890
$498$	−33.3522	−1.49455
$499$	−13.4511	−0.602156	−0.301078	−	0.953600i	$-0.597346\pi$
$499$	−13.4511	−0.602156	−0.301078	+	0.953600i	$0.597346\pi$
$500$	11.8637	0.530561
$501$	−36.2057	−1.61755
$502$	45.9159	2.04933
$503$	−4.75979	−0.212228	−0.106114	−	0.994354i	$-0.533841\pi$
$503$	−4.75979	−0.212228	−0.106114	+	0.994354i	$0.533841\pi$
$504$	0.278853	0.0124211
$505$	−7.55786	−0.336320
$506$	−47.1778	−2.09731
$507$	2.01697	0.0895768
$508$	32.7937	1.45499
$509$	−1.27382	−0.0564612	−0.0282306	−	0.999601i	$-0.508987\pi$
$509$	−1.27382	−0.0564612	−0.0282306	+	0.999601i	$0.508987\pi$
$510$	−5.00702	−0.221714
$511$	−12.6017	−0.557464
$512$	−29.6837	−1.31185
$513$	5.47915	0.241911
$514$	6.56322	0.289491
$515$	−7.32060	−0.322584
$516$	−22.3693	−0.984753
$517$	−21.1729	−0.931184
$518$	0.348274	0.0153023
$519$	10.6068	0.465589
$520$	1.13505	0.0497752
$521$	−12.1115	−0.530616	−0.265308	−	0.964164i	$-0.585474\pi$
$521$	−12.1115	−0.530616	−0.265308	+	0.964164i	$0.585474\pi$
$522$	−6.95171	−0.304268
$523$	35.7349	1.56258	0.781288	−	0.624170i	$-0.214563\pi$
$523$	35.7349	1.56258	0.781288	+	0.624170i	$0.214563\pi$
$524$	23.0428	1.00663
$525$	−11.0054	−0.480313
$526$	−53.3089	−2.32438
$527$	−17.9301	−0.781048
$528$	34.5419	1.50325
$529$	2.29530	0.0997956
$530$	−3.85926	−0.167635
$531$	−5.41321	−0.234913
$532$	−2.66824	−0.115683
$533$	−16.4957	−0.714509
$534$	−0.550841	−0.0238372
$535$	8.02170	0.346808
$536$	−2.17261	−0.0938423
$537$	31.3193	1.35153
$538$	50.6129	2.18208
$539$	−22.7393	−0.979453
$540$	−6.84545	−0.294581
$541$	1.29855	0.0558291	0.0279145	−	0.999610i	$-0.491113\pi$
$541$	1.29855	0.0558291	0.0279145	+	0.999610i	$0.491113\pi$
$542$	38.2877	1.64460
$543$	35.5985	1.52768
$544$	17.2208	0.738334
$545$	5.98788	0.256492
$546$	16.2701	0.696295
$547$	−25.5778	−1.09363	−0.546814	−	0.837254i	$-0.684160\pi$
$547$	−25.5778	−1.09363	−0.546814	+	0.837254i	$0.684160\pi$
$548$	21.4315	0.915509
$549$	−2.20641	−0.0941674
$550$	42.1721	1.79823
$551$	−9.09232	−0.387346
$552$	−3.78588	−0.161138
$553$	2.62981	0.111831
$554$	−44.5047	−1.89082
$555$	0.135762	0.00576280
$556$	11.7746	0.499353
$557$	−39.8206	−1.68725	−0.843626	−	0.536931i	$-0.819583\pi$
$557$	−39.8206	−1.68725	−0.843626	+	0.536931i	$0.819583\pi$
$558$	−6.08480	−0.257590
$559$	−26.9980	−1.14189
$560$	−4.76288	−0.201268
$561$	−17.5941	−0.742824
$562$	27.1909	1.14698
$563$	11.4528	0.482680	0.241340	−	0.970441i	$-0.422413\pi$
$563$	11.4528	0.482680	0.241340	+	0.970441i	$0.422413\pi$
$564$	12.4301	0.523403
$565$	−1.29643	−0.0545411
$566$	28.0749	1.18008
$567$	11.6186	0.487934
$568$	2.20666	0.0925895
$569$	−2.46885	−0.103500	−0.0517498	−	0.998660i	$-0.516480\pi$
$569$	−2.46885	−0.103500	−0.0517498	+	0.998660i	$0.516480\pi$
$570$	−2.22241	−0.0930867
$571$	−28.1346	−1.17739	−0.588697	−	0.808354i	$-0.700359\pi$
$571$	−28.1346	−1.17739	−0.588697	+	0.808354i	$0.700359\pi$
$572$	−29.1789	−1.22003
$573$	−9.26561	−0.387076
$574$	14.1496	0.590593
$575$	−22.6114	−0.942960
$576$	2.35576	0.0981568
$577$	20.2880	0.844603	0.422301	−	0.906456i	$-0.361222\pi$
$577$	20.2880	0.844603	0.422301	+	0.906456i	$0.361222\pi$
$578$	23.1203	0.961676
$579$	−20.2948	−0.843423
$580$	11.3596	0.471682
$581$	16.1598	0.670422
$582$	−19.5217	−0.809199
$583$	−13.5610	−0.561640
$584$	3.87506	0.160351
$585$	−0.959799	−0.0396828
$586$	51.2634	2.11767
$587$	15.9598	0.658730	0.329365	−	0.944203i	$-0.393165\pi$
$587$	15.9598	0.658730	0.329365	+	0.944203i	$0.393165\pi$
$588$	13.3498	0.550535
$589$	−7.95846	−0.327923
$590$	18.9002	0.778111
$591$	36.2692	1.49192
$592$	−0.523905	−0.0215324
$593$	15.8008	0.648860	0.324430	−	0.945910i	$-0.394827\pi$
$593$	15.8008	0.648860	0.324430	+	0.945910i	$0.394827\pi$
$594$	−51.3963	−2.10881
$595$	2.42600	0.0994562
$596$	26.4193	1.08218
$597$	23.2638	0.952125
$598$	33.4282	1.36698
$599$	−4.63147	−0.189237	−0.0946184	−	0.995514i	$-0.530163\pi$
$599$	−4.63147	−0.189237	−0.0946184	+	0.995514i	$0.530163\pi$
$600$	3.38419	0.138159
$601$	−12.3540	−0.503929	−0.251964	−	0.967737i	$-0.581077\pi$
$601$	−12.3540	−0.503929	−0.251964	+	0.967737i	$0.581077\pi$
$602$	23.1582	0.943858
$603$	1.83716	0.0748148
$604$	−2.80392	−0.114090
$605$	−8.80832	−0.358109
$606$	33.3138	1.35328
$607$	−11.3892	−0.462274	−0.231137	−	0.972921i	$-0.574245\pi$
$607$	−11.3892	−0.462274	−0.231137	+	0.972921i	$0.574245\pi$
$608$	7.64362	0.309990
$609$	−22.2572	−0.901909
$610$	7.70370	0.311914
$611$	15.0022	0.606925
$612$	−1.56313	−0.0631857
$613$	10.3331	0.417350	0.208675	−	0.977985i	$-0.433085\pi$
$613$	10.3331	0.417350	0.208675	+	0.977985i	$0.433085\pi$
$614$	4.14996	0.167479
$615$	5.51573	0.222416
$616$	−3.42119	−0.137844
$617$	−5.53466	−0.222817	−0.111408	−	0.993775i	$-0.535536\pi$
$617$	−5.53466	−0.222817	−0.111408	+	0.993775i	$0.535536\pi$
$618$	32.2680	1.29801
$619$	14.9342	0.600256	0.300128	−	0.953899i	$-0.402971\pi$
$619$	14.9342	0.600256	0.300128	+	0.953899i	$0.402971\pi$
$620$	9.94300	0.399321
$621$	27.5571	1.10583
$622$	32.5939	1.30690
$623$	0.266893	0.0106928
$624$	−24.4749	−0.979781
$625$	17.6913	0.707650
$626$	−37.2314	−1.48807
$627$	−7.80933	−0.311874
$628$	−32.5015	−1.29695
$629$	0.266854	0.0106402
$630$	0.823292	0.0328007
$631$	−22.5759	−0.898732	−0.449366	−	0.893348i	$-0.648350\pi$
$631$	−22.5759	−0.898732	−0.449366	+	0.893348i	$0.648350\pi$
$632$	−0.808677	−0.0321674
$633$	−1.61421	−0.0641591
$634$	13.9296	0.553214
$635$	−13.2343	−0.525189
$636$	7.96136	0.315689
$637$	16.1121	0.638385
$638$	85.2889	3.37662
$639$	−1.86596	−0.0738161
$640$	2.62982	0.103953
$641$	7.33130	0.289569	0.144784	−	0.989463i	$-0.453751\pi$
$641$	7.33130	0.289569	0.144784	+	0.989463i	$0.453751\pi$
$642$	−35.3583	−1.39548
$643$	31.4635	1.24080	0.620399	−	0.784286i	$-0.286971\pi$
$643$	31.4635	1.24080	0.620399	+	0.784286i	$0.286971\pi$
$644$	−13.4198	−0.528813
$645$	9.02743	0.355455
$646$	−4.36836	−0.171871
$647$	−32.1732	−1.26486	−0.632429	−	0.774618i	$-0.717942\pi$
$647$	−32.1732	−1.26486	−0.632429	+	0.774618i	$0.717942\pi$
$648$	−3.57275	−0.140351
$649$	66.4134	2.60695
$650$	−29.8813	−1.17204
$651$	−19.4817	−0.763546
$652$	34.0380	1.33303
$653$	−40.2209	−1.57396	−0.786982	−	0.616976i	$-0.788358\pi$
$653$	−40.2209	−1.57396	−0.786982	+	0.616976i	$0.788358\pi$
$654$	−26.3936	−1.03207
$655$	−9.29925	−0.363352
$656$	−21.2851	−0.831045
$657$	−3.27675	−0.127838
$658$	−12.8685	−0.501667
$659$	−41.6504	−1.62247	−0.811235	−	0.584720i	$-0.801204\pi$
$659$	−41.6504	−1.62247	−0.811235	+	0.584720i	$0.801204\pi$
$660$	9.75668	0.379778
$661$	−15.5677	−0.605513	−0.302757	−	0.953068i	$-0.597907\pi$
$661$	−15.5677	−0.605513	−0.302757	+	0.953068i	$0.597907\pi$
$662$	42.9775	1.67037
$663$	12.4664	0.484156
$664$	−4.96920	−0.192842
$665$	1.07680	0.0417567
$666$	0.0905600	0.00350913
$667$	−45.7293	−1.77064
$668$	39.4643	1.52692
$669$	−36.1611	−1.39807
$670$	−6.41444	−0.247811
$671$	27.0700	1.04502
$672$	18.7109	0.721790
$673$	−21.7671	−0.839059	−0.419530	−	0.907742i	$-0.637805\pi$
$673$	−21.7671	−0.839059	−0.419530	+	0.907742i	$0.637805\pi$
$674$	−40.7311	−1.56890
$675$	−24.6332	−0.948132
$676$	−2.19851	−0.0845579
$677$	14.2408	0.547319	0.273659	−	0.961827i	$-0.411766\pi$
$677$	14.2408	0.547319	0.273659	+	0.961827i	$0.411766\pi$
$678$	5.71443	0.219461
$679$	9.45863	0.362989
$680$	−0.746003	−0.0286079
$681$	−6.85221	−0.262577
$682$	74.6530	2.85861
$683$	−27.6097	−1.05646	−0.528228	−	0.849102i	$-0.677144\pi$
$683$	−27.6097	−1.05646	−0.528228	+	0.849102i	$0.677144\pi$
$684$	−0.693810	−0.0265285
$685$	−8.64898	−0.330460
$686$	−34.4031	−1.31352
$687$	−40.3150	−1.53811
$688$	−34.8367	−1.32814
$689$	9.60875	0.366064
$690$	−11.1775	−0.425521
$691$	30.2222	1.14971	0.574854	−	0.818256i	$-0.305059\pi$
$691$	30.2222	1.14971	0.574854	+	0.818256i	$0.305059\pi$
$692$	−11.5615	−0.439503
$693$	2.89296	0.109894
$694$	−19.1058	−0.725246
$695$	−4.75178	−0.180245
$696$	6.84418	0.259428
$697$	10.8417	0.410658
$698$	28.0090	1.06016
$699$	15.2089	0.575255
$700$	11.9959	0.453402
$701$	49.8089	1.88126	0.940628	−	0.339438i	$-0.110237\pi$
$701$	49.8089	1.88126	0.940628	+	0.339438i	$0.110237\pi$
$702$	36.4172	1.37448
$703$	0.118446	0.00446727
$704$	−28.9023	−1.08930
$705$	−5.01635	−0.188927
$706$	−7.46954	−0.281120
$707$	−16.1412	−0.607051
$708$	−38.9898	−1.46533
$709$	33.8181	1.27007	0.635033	−	0.772485i	$-0.280987\pi$
$709$	33.8181	1.27007	0.635033	+	0.772485i	$0.280987\pi$
$710$	6.51499	0.244503
$711$	0.683818	0.0256452
$712$	−0.0820706	−0.00307573
$713$	−40.0266	−1.49901
$714$	−10.6934	−0.400190
$715$	11.7756	0.440381
$716$	−34.1382	−1.27580
$717$	26.3113	0.982613
$718$	30.8906	1.15283
$719$	−18.0136	−0.671794	−0.335897	−	0.941899i	$-0.609039\pi$
$719$	−18.0136	−0.671794	−0.335897	+	0.941899i	$0.609039\pi$
$720$	−1.23847	−0.0461551
$721$	−15.6345	−0.582258
$722$	−1.93894	−0.0721599
$723$	20.6411	0.767650
$724$	−38.8025	−1.44208
$725$	40.8773	1.51814
$726$	38.8256	1.44095
$727$	27.6422	1.02519	0.512595	−	0.858630i	$-0.328684\pi$
$727$	27.6422	1.02519	0.512595	+	0.858630i	$0.328684\pi$
$728$	2.42411	0.0898433
$729$	29.5547	1.09462
$730$	11.4408	0.423442
$731$	17.7442	0.656295
$732$	−15.8922	−0.587391
$733$	32.4675	1.19922	0.599608	−	0.800294i	$-0.295323\pi$
$733$	32.4675	1.19922	0.599608	+	0.800294i	$0.295323\pi$
$734$	49.4443	1.82502
$735$	−5.38747	−0.198720
$736$	38.4431	1.41703
$737$	−22.5397	−0.830259
$738$	3.67926	0.135435
$739$	8.02656	0.295262	0.147631	−	0.989043i	$-0.452835\pi$
$739$	8.02656	0.295262	0.147631	+	0.989043i	$0.452835\pi$
$740$	−0.147982	−0.00543991
$741$	5.53335	0.203273
$742$	−8.24214	−0.302579
$743$	45.1744	1.65729	0.828644	−	0.559776i	$-0.189113\pi$
$743$	45.1744	1.65729	0.828644	+	0.559776i	$0.189113\pi$
$744$	5.99068	0.219629
$745$	−10.6619	−0.390621
$746$	−55.2295	−2.02210
$747$	4.20196	0.153742
$748$	19.1776	0.701204
$749$	17.1318	0.625982
$750$	21.1036	0.770595
$751$	29.4782	1.07568	0.537838	−	0.843048i	$-0.319241\pi$
$751$	29.4782	1.07568	0.537838	+	0.843048i	$0.319241\pi$
$752$	19.3580	0.705914
$753$	38.2260	1.39303
$754$	−60.4321	−2.20081
$755$	1.13156	0.0411816
$756$	−14.6197	−0.531713
$757$	−22.6655	−0.823793	−0.411896	−	0.911231i	$-0.635133\pi$
$757$	−22.6655	−0.823793	−0.411896	+	0.911231i	$0.635133\pi$
$758$	−31.7152	−1.15195
$759$	−39.2766	−1.42565
$760$	−0.331121	−0.0120110
$761$	23.1047	0.837543	0.418772	−	0.908092i	$-0.362461\pi$
$761$	23.1047	0.837543	0.418772	+	0.908092i	$0.362461\pi$
$762$	58.3348	2.11325
$763$	12.7882	0.462964
$764$	10.0996	0.365389
$765$	0.630821	0.0228074
$766$	7.68401	0.277635
$767$	−47.0577	−1.69915
$768$	−30.8790	−1.11425
$769$	20.8502	0.751877	0.375939	−	0.926645i	$-0.377320\pi$
$769$	20.8502	0.751877	0.375939	+	0.926645i	$0.377320\pi$
$770$	−10.1008	−0.364007
$771$	5.46402	0.196782
$772$	22.1214	0.796167
$773$	34.7891	1.25128	0.625639	−	0.780113i	$-0.284838\pi$
$773$	34.7891	1.25128	0.625639	+	0.780113i	$0.284838\pi$
$774$	6.02172	0.216446
$775$	35.7797	1.28524
$776$	−2.90857	−0.104411
$777$	0.289945	0.0104017
$778$	−5.17701	−0.185605
$779$	4.81219	0.172415
$780$	−6.91316	−0.247531
$781$	22.8930	0.819175
$782$	−21.9704	−0.785661
$783$	−49.8182	−1.78036
$784$	20.7902	0.742506
$785$	13.1164	0.468145
$786$	40.9895	1.46205
$787$	−9.81657	−0.349923	−0.174961	−	0.984575i	$-0.555980\pi$
$787$	−9.81657	−0.349923	−0.174961	+	0.984575i	$0.555980\pi$
$788$	−39.5336	−1.40832
$789$	−44.3808	−1.58000
$790$	−2.38755	−0.0849453
$791$	−2.76875	−0.0984456
$792$	−0.889596	−0.0316104
$793$	−19.1806	−0.681123
$794$	2.93552	0.104178
$795$	−3.21291	−0.113950
$796$	−25.3577	−0.898778
$797$	29.2697	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$797$	29.2697	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$798$	−4.74637	−0.168020
$799$	−9.86010	−0.348825
$800$	−34.3642	−1.21496
$801$	0.0693990	0.00245209
$802$	−66.1020	−2.33414
$803$	40.2017	1.41869
$804$	13.2325	0.466675
$805$	5.41572	0.190879
$806$	−52.8959	−1.86318
$807$	42.1364	1.48327
$808$	4.96347	0.174614
$809$	44.9580	1.58064	0.790320	−	0.612695i	$-0.209915\pi$
$809$	44.9580	1.58064	0.790320	+	0.612695i	$0.209915\pi$
$810$	−10.5483	−0.370628
$811$	12.0436	0.422908	0.211454	−	0.977388i	$-0.432180\pi$
$811$	12.0436	0.422908	0.211454	+	0.977388i	$0.432180\pi$
$812$	24.2605	0.851376
$813$	31.8753	1.11792
$814$	−1.11106	−0.0389426
$815$	−13.7365	−0.481168
$816$	16.0860	0.563122
$817$	7.87596	0.275545
$818$	59.1225	2.06717
$819$	−2.04983	−0.0716267
$820$	−6.01217	−0.209954
$821$	9.76338	0.340744	0.170372	−	0.985380i	$-0.445503\pi$
$821$	9.76338	0.340744	0.170372	+	0.985380i	$0.445503\pi$
$822$	38.1232	1.32970
$823$	6.09808	0.212566	0.106283	−	0.994336i	$-0.466105\pi$
$823$	6.09808	0.212566	0.106283	+	0.994336i	$0.466105\pi$
$824$	4.80766	0.167483
$825$	35.1092	1.22235
$826$	40.3649	1.40447
$827$	7.68780	0.267331	0.133666	−	0.991026i	$-0.457325\pi$
$827$	7.68780	0.267331	0.133666	+	0.991026i	$0.457325\pi$
$828$	−3.48948	−0.121268
$829$	−41.3837	−1.43731	−0.718657	−	0.695365i	$-0.755243\pi$
$829$	−41.3837	−1.43731	−0.718657	+	0.695365i	$0.755243\pi$
$830$	−14.6712	−0.509243
$831$	−37.0511	−1.28529
$832$	20.4789	0.709979
$833$	−10.5896	−0.366907
$834$	20.9451	0.725268
$835$	−15.9264	−0.551154
$836$	8.51220	0.294400
$837$	−43.6056	−1.50723
$838$	−46.5553	−1.60823
$839$	−26.4281	−0.912399	−0.456199	−	0.889878i	$-0.650790\pi$
$839$	−26.4281	−0.912399	−0.456199	+	0.889878i	$0.650790\pi$
$840$	−0.810557	−0.0279669
$841$	53.6702	1.85070
$842$	45.8426	1.57984
$843$	22.6370	0.779660
$844$	1.75950	0.0605644
$845$	0.887237	0.0305219
$846$	−3.34614	−0.115043
$847$	−18.8118	−0.646380
$848$	12.3986	0.425769
$849$	23.3729	0.802157
$850$	19.6393	0.673622
$851$	0.595716	0.0204209
$852$	−13.4400	−0.460445
$853$	44.8321	1.53502	0.767510	−	0.641037i	$-0.221495\pi$
$853$	44.8321	1.53502	0.767510	+	0.641037i	$0.221495\pi$
$854$	16.4526	0.562998
$855$	0.279996	0.00957567
$856$	−5.26809	−0.180060
$857$	36.7482	1.25529	0.627647	−	0.778498i	$-0.284018\pi$
$857$	36.7482	1.25529	0.627647	+	0.778498i	$0.284018\pi$
$858$	−51.9047	−1.77200
$859$	−43.4808	−1.48355	−0.741774	−	0.670650i	$-0.766015\pi$
$859$	−43.4808	−1.48355	−0.741774	+	0.670650i	$0.766015\pi$
$860$	−9.83993	−0.335539
$861$	11.7799	0.401456
$862$	68.9330	2.34787
$863$	9.85100	0.335332	0.167666	−	0.985844i	$-0.446377\pi$
$863$	9.85100	0.335332	0.167666	+	0.985844i	$0.446377\pi$
$864$	41.8805	1.42480
$865$	4.66580	0.158642
$866$	44.0721	1.49763
$867$	19.2481	0.653700
$868$	21.2351	0.720766
$869$	−8.38960	−0.284598
$870$	20.2069	0.685078
$871$	15.9706	0.541144
$872$	−3.93242	−0.133168
$873$	2.45949	0.0832410
$874$	−9.75180	−0.329860
$875$	−10.2251	−0.345672
$876$	−23.6015	−0.797421
$877$	−46.1145	−1.55718	−0.778589	−	0.627534i	$-0.784064\pi$
$877$	−46.1145	−1.55718	−0.778589	+	0.627534i	$0.784064\pi$
$878$	66.1410	2.23215
$879$	42.6779	1.43949
$880$	15.1945	0.512207
$881$	18.8370	0.634634	0.317317	−	0.948319i	$-0.397218\pi$
$881$	18.8370	0.634634	0.317317	+	0.948319i	$0.397218\pi$
$882$	−3.59370	−0.121006
$883$	15.8320	0.532789	0.266394	−	0.963864i	$-0.414168\pi$
$883$	15.8320	0.532789	0.266394	+	0.963864i	$0.414168\pi$
$884$	−13.5885	−0.457029
$885$	15.7349	0.528921
$886$	52.9630	1.77933
$887$	−38.9792	−1.30879	−0.654397	−	0.756151i	$-0.727077\pi$
$887$	−38.9792	−1.30879	−0.654397	+	0.756151i	$0.727077\pi$
$888$	−0.0891593	−0.00299199
$889$	−28.2643	−0.947956
$890$	−0.242307	−0.00812215
$891$	−37.0655	−1.24174
$892$	39.4157	1.31974
$893$	−4.37650	−0.146454
$894$	46.9957	1.57177
$895$	13.7769	0.460512
$896$	5.61645	0.187632
$897$	27.8297	0.929206
$898$	51.5336	1.71970
$899$	72.3609	2.41337
$900$	3.11923	0.103974
$901$	−6.31528	−0.210392
$902$	−45.1400	−1.50300
$903$	19.2797	0.641589
$904$	0.851402	0.0283172
$905$	15.6593	0.520532
$906$	−4.98772	−0.165706
$907$	16.2101	0.538246	0.269123	−	0.963106i	$-0.413266\pi$
$907$	16.2101	0.538246	0.269123	+	0.963106i	$0.413266\pi$
$908$	7.46893	0.247865
$909$	−4.19712	−0.139210
$910$	7.15697	0.237251
$911$	−1.93959	−0.0642616	−0.0321308	−	0.999484i	$-0.510229\pi$
$911$	−1.93959	−0.0642616	−0.0321308	+	0.999484i	$0.510229\pi$
$912$	7.13992	0.236426
$913$	−51.5529	−1.70615
$914$	21.4554	0.709683
$915$	6.41349	0.212024
$916$	43.9435	1.45193
$917$	−19.8602	−0.655843
$918$	−23.9349	−0.789970
$919$	−14.6607	−0.483613	−0.241807	−	0.970324i	$-0.577740\pi$
$919$	−14.6607	−0.483613	−0.241807	+	0.970324i	$0.577740\pi$
$920$	−1.66536	−0.0549051
$921$	3.45493	0.113844
$922$	−40.6098	−1.33741
$923$	−16.2210	−0.533920
$924$	20.8372	0.685492
$925$	−0.532509	−0.0175088
$926$	59.5509	1.95696
$927$	−4.06536	−0.133524
$928$	−69.4982	−2.28139
$929$	25.6528	0.841642	0.420821	−	0.907144i	$-0.361742\pi$
$929$	25.6528	0.841642	0.420821	+	0.907144i	$0.361742\pi$
$930$	17.6870	0.579980
$931$	−4.70029	−0.154046
$932$	−16.5778	−0.543024
$933$	27.1352	0.888365
$934$	29.7962	0.974961
$935$	−7.73940	−0.253105
$936$	0.630329	0.0206029
$937$	−24.2289	−0.791525	−0.395763	−	0.918353i	$-0.629520\pi$
$937$	−24.2289	−0.791525	−0.395763	+	0.918353i	$0.629520\pi$
$938$	−13.6992	−0.447295
$939$	−30.9959	−1.01151
$940$	5.46784	0.178341
$941$	−54.4971	−1.77655	−0.888277	−	0.459308i	$-0.848097\pi$
$941$	−54.4971	−1.77655	−0.888277	+	0.459308i	$0.848097\pi$
$942$	−57.8150	−1.88371
$943$	24.2027	0.788147
$944$	−60.7205	−1.97629
$945$	5.89997	0.191926
$946$	−73.8791	−2.40202
$947$	6.51946	0.211854	0.105927	−	0.994374i	$-0.466219\pi$
$947$	6.51946	0.211854	0.105927	+	0.994374i	$0.466219\pi$
$948$	4.92535	0.159968
$949$	−28.4852	−0.924668
$950$	8.71710	0.282820
$951$	11.5967	0.376048
$952$	−1.59323	−0.0516367
$953$	−36.3798	−1.17846	−0.589229	−	0.807966i	$-0.700569\pi$
$953$	−36.3798	−1.17846	−0.589229	+	0.807966i	$0.700569\pi$
$954$	−2.14317	−0.0693876
$955$	−4.07581	−0.131890
$956$	−28.6794	−0.927558
$957$	71.0049	2.29526
$958$	41.1711	1.33018
$959$	−18.4715	−0.596475
$960$	−6.84762	−0.221006
$961$	32.3371	1.04313
$962$	0.787249	0.0253819
$963$	4.45470	0.143551
$964$	−22.4989	−0.724640
$965$	−8.92739	−0.287383
$966$	−23.8716	−0.768056
$967$	18.2294	0.586219	0.293110	−	0.956079i	$-0.405310\pi$
$967$	18.2294	0.586219	0.293110	+	0.956079i	$0.405310\pi$
$968$	5.78469	0.185927
$969$	−3.63675	−0.116829
$970$	−8.58730	−0.275722
$971$	−41.7816	−1.34084	−0.670418	−	0.741984i	$-0.733885\pi$
$971$	−41.7816	−1.34084	−0.670418	+	0.741984i	$0.733885\pi$
$972$	−7.16136	−0.229701
$973$	−10.1483	−0.325339
$974$	32.7631	1.04980
$975$	−24.8769	−0.796697
$976$	−24.7496	−0.792214
$977$	19.6861	0.629813	0.314906	−	0.949123i	$-0.398027\pi$
$977$	19.6861	0.629813	0.314906	+	0.949123i	$0.398027\pi$
$978$	60.5481	1.93612
$979$	−0.851440	−0.0272122
$980$	5.87237	0.187586
$981$	3.32526	0.106167
$982$	43.7576	1.39636
$983$	−49.8680	−1.59054	−0.795271	−	0.606254i	$-0.792671\pi$
$983$	−49.8680	−1.59054	−0.795271	+	0.606254i	$0.792671\pi$
$984$	−3.62235	−0.115476
$985$	15.9543	0.508346
$986$	39.7185	1.26490
$987$	−10.7133	−0.341009
$988$	−6.03137	−0.191884
$989$	39.6117	1.25958
$990$	−2.62646	−0.0834743
$991$	−4.85857	−0.154337	−0.0771687	−	0.997018i	$-0.524588\pi$
$991$	−4.85857	−0.154337	−0.0771687	+	0.997018i	$0.524588\pi$
$992$	−60.8314	−1.93140
$993$	35.7797	1.13543
$994$	13.9140	0.441324
$995$	10.2334	0.324421
$996$	30.2655	0.959000
$997$	−7.93651	−0.251352	−0.125676	−	0.992071i	$-0.540110\pi$
$997$	−7.93651	−0.251352	−0.125676	+	0.992071i	$0.540110\pi$
$998$	26.0810	0.825579
$999$	0.648982	0.0205329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.16	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.16	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93894 q^{2} -1.61421 q^{3} +1.75950 q^{4} -0.710068 q^{5} +3.12986 q^{6} -1.51648 q^{7} +0.466323 q^{8} -0.394323 q^{9} +O(q^{10})\)	\(q-1.93894 q^{2} -1.61421 q^{3} +1.75950 q^{4} -0.710068 q^{5} +3.12986 q^{6} -1.51648 q^{7} +0.466323 q^{8} -0.394323 q^{9} +1.37678 q^{10} +4.83786 q^{11} -2.84020 q^{12} -3.42790 q^{13} +2.94037 q^{14} +1.14620 q^{15} -4.42317 q^{16} +2.25296 q^{17} +0.764570 q^{18} +1.00000 q^{19} -1.24936 q^{20} +2.44792 q^{21} -9.38033 q^{22} +5.02944 q^{23} -0.752744 q^{24} -4.49580 q^{25} +6.64650 q^{26} +5.47915 q^{27} -2.66824 q^{28} -9.09232 q^{29} -2.22241 q^{30} -7.95846 q^{31} +7.64362 q^{32} -7.80933 q^{33} -4.36836 q^{34} +1.07680 q^{35} -0.693810 q^{36} +0.118446 q^{37} -1.93894 q^{38} +5.53335 q^{39} -0.331121 q^{40} +4.81219 q^{41} -4.74637 q^{42} +7.87596 q^{43} +8.51220 q^{44} +0.279996 q^{45} -9.75180 q^{46} -4.37650 q^{47} +7.13992 q^{48} -4.70029 q^{49} +8.71710 q^{50} -3.63675 q^{51} -6.03137 q^{52} -2.80310 q^{53} -10.6238 q^{54} -3.43521 q^{55} -0.707169 q^{56} -1.61421 q^{57} +17.6295 q^{58} +13.7278 q^{59} +2.01673 q^{60} +5.59544 q^{61} +15.4310 q^{62} +0.597983 q^{63} -5.97420 q^{64} +2.43404 q^{65} +15.1418 q^{66} -4.65901 q^{67} +3.96408 q^{68} -8.11858 q^{69} -2.08786 q^{70} +4.73205 q^{71} -0.183882 q^{72} +8.30981 q^{73} -0.229659 q^{74} +7.25717 q^{75} +1.75950 q^{76} -7.33651 q^{77} -10.7288 q^{78} -1.73416 q^{79} +3.14075 q^{80} -7.66154 q^{81} -9.33056 q^{82} -10.6561 q^{83} +4.30710 q^{84} -1.59976 q^{85} -15.2710 q^{86} +14.6769 q^{87} +2.25601 q^{88} -0.175995 q^{89} -0.542897 q^{90} +5.19834 q^{91} +8.84929 q^{92} +12.8466 q^{93} +8.48579 q^{94} -0.710068 q^{95} -12.3384 q^{96} -6.23723 q^{97} +9.11359 q^{98} -1.90768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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