LMFDB - Embedded newform 4009.2.a.f.1.12

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:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.15681 q^{2} -0.442776 q^{3} +2.65181 q^{4} -2.84036 q^{5} +0.954981 q^{6} +0.461477 q^{7} -1.40583 q^{8} -2.80395 q^{9} +O(q^{10})$	\(q-2.15681 q^{2} -0.442776 q^{3} +2.65181 q^{4} -2.84036 q^{5} +0.954981 q^{6} +0.461477 q^{7} -1.40583 q^{8} -2.80395 q^{9} +6.12610 q^{10} +1.89005 q^{11} -1.17416 q^{12} +5.61800 q^{13} -0.995317 q^{14} +1.25764 q^{15} -2.27152 q^{16} +7.29760 q^{17} +6.04757 q^{18} -1.00000 q^{19} -7.53209 q^{20} -0.204331 q^{21} -4.07648 q^{22} +4.13767 q^{23} +0.622466 q^{24} +3.06764 q^{25} -12.1169 q^{26} +2.56985 q^{27} +1.22375 q^{28} +5.17784 q^{29} -2.71249 q^{30} -6.84962 q^{31} +7.71089 q^{32} -0.836870 q^{33} -15.7395 q^{34} -1.31076 q^{35} -7.43554 q^{36} +4.05305 q^{37} +2.15681 q^{38} -2.48751 q^{39} +3.99306 q^{40} +9.69644 q^{41} +0.440702 q^{42} +10.0122 q^{43} +5.01206 q^{44} +7.96423 q^{45} -8.92414 q^{46} +3.40075 q^{47} +1.00577 q^{48} -6.78704 q^{49} -6.61631 q^{50} -3.23120 q^{51} +14.8979 q^{52} -14.1167 q^{53} -5.54266 q^{54} -5.36843 q^{55} -0.648758 q^{56} +0.442776 q^{57} -11.1676 q^{58} -8.79793 q^{59} +3.33503 q^{60} -3.34815 q^{61} +14.7733 q^{62} -1.29396 q^{63} -12.0878 q^{64} -15.9571 q^{65} +1.80497 q^{66} +6.31079 q^{67} +19.3518 q^{68} -1.83206 q^{69} +2.82706 q^{70} +8.58718 q^{71} +3.94187 q^{72} -8.41218 q^{73} -8.74163 q^{74} -1.35828 q^{75} -2.65181 q^{76} +0.872217 q^{77} +5.36508 q^{78} +1.24212 q^{79} +6.45194 q^{80} +7.27398 q^{81} -20.9133 q^{82} -4.62979 q^{83} -0.541847 q^{84} -20.7278 q^{85} -21.5944 q^{86} -2.29262 q^{87} -2.65709 q^{88} -7.62626 q^{89} -17.1773 q^{90} +2.59258 q^{91} +10.9723 q^{92} +3.03284 q^{93} -7.33476 q^{94} +2.84036 q^{95} -3.41419 q^{96} -0.493240 q^{97} +14.6383 q^{98} -5.29962 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) $ 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * 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 $

$2$	−2.15681	−1.52509	−0.762546	−	0.646934i	$-0.776051\pi$
$2$	−2.15681	−1.52509	−0.762546	+	0.646934i	$0.776051\pi$
$3$	−0.442776	−0.255637	−0.127818	−	0.991798i	$-0.540797\pi$
$3$	−0.442776	−0.255637	−0.127818	+	0.991798i	$0.540797\pi$
$4$	2.65181	1.32591
$5$	−2.84036	−1.27025	−0.635124	−	0.772410i	$-0.719051\pi$
$5$	−2.84036	−1.27025	−0.635124	+	0.772410i	$0.719051\pi$
$6$	0.954981	0.389869
$7$	0.461477	0.174422	0.0872110	−	0.996190i	$-0.472205\pi$
$7$	0.461477	0.174422	0.0872110	+	0.996190i	$0.472205\pi$
$8$	−1.40583	−0.497035
$9$	−2.80395	−0.934650
$10$	6.12610	1.93724
$11$	1.89005	0.569873	0.284936	−	0.958546i	$-0.408028\pi$
$11$	1.89005	0.569873	0.284936	+	0.958546i	$0.408028\pi$
$12$	−1.17416	−0.338950
$13$	5.61800	1.55815	0.779076	−	0.626930i	$-0.215689\pi$
$13$	5.61800	1.55815	0.779076	+	0.626930i	$0.215689\pi$
$14$	−0.995317	−0.266010
$15$	1.25764	0.324722
$16$	−2.27152	−0.567881
$17$	7.29760	1.76993	0.884964	−	0.465660i	$-0.154183\pi$
$17$	7.29760	1.76993	0.884964	+	0.465660i	$0.154183\pi$
$18$	6.04757	1.42543
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−7.53209	−1.68423
$21$	−0.204331	−0.0445887
$22$	−4.07648	−0.869108
$23$	4.13767	0.862763	0.431382	−	0.902170i	$-0.358026\pi$
$23$	4.13767	0.862763	0.431382	+	0.902170i	$0.358026\pi$
$24$	0.622466	0.127060
$25$	3.06764	0.613528
$26$	−12.1169	−2.37632
$27$	2.56985	0.494567
$28$	1.22375	0.231267
$29$	5.17784	0.961501	0.480750	−	0.876857i	$-0.340364\pi$
$29$	5.17784	0.961501	0.480750	+	0.876857i	$0.340364\pi$
$30$	−2.71249	−0.495230
$31$	−6.84962	−1.23023	−0.615114	−	0.788438i	$-0.710890\pi$
$31$	−6.84962	−1.23023	−0.615114	+	0.788438i	$0.710890\pi$
$32$	7.71089	1.36311
$33$	−0.836870	−0.145680
$34$	−15.7395	−2.69930
$35$	−1.31076	−0.221559
$36$	−7.43554	−1.23926
$37$	4.05305	0.666317	0.333158	−	0.942871i	$-0.391886\pi$
$37$	4.05305	0.666317	0.333158	+	0.942871i	$0.391886\pi$
$38$	2.15681	0.349880
$39$	−2.48751	−0.398321
$40$	3.99306	0.631358
$41$	9.69644	1.51433	0.757165	−	0.653224i	$-0.226584\pi$
$41$	9.69644	1.51433	0.757165	+	0.653224i	$0.226584\pi$
$42$	0.440702	0.0680018
$43$	10.0122	1.52685	0.763424	−	0.645897i	$-0.223517\pi$
$43$	10.0122	1.52685	0.763424	+	0.645897i	$0.223517\pi$
$44$	5.01206	0.755597
$45$	7.96423	1.18724
$46$	−8.92414	−1.31579
$47$	3.40075	0.496050	0.248025	−	0.968754i	$-0.420218\pi$
$47$	3.40075	0.496050	0.248025	+	0.968754i	$0.420218\pi$
$48$	1.00577	0.145171
$49$	−6.78704	−0.969577
$50$	−6.61631	−0.935687
$51$	−3.23120	−0.452458
$52$	14.8979	2.06596
$53$	−14.1167	−1.93907	−0.969536	−	0.244947i	$-0.921229\pi$
$53$	−14.1167	−1.93907	−0.969536	+	0.244947i	$0.921229\pi$
$54$	−5.54266	−0.754261
$55$	−5.36843	−0.723879
$56$	−0.648758	−0.0866939
$57$	0.442776	0.0586471
$58$	−11.1676	−1.46638
$59$	−8.79793	−1.14539	−0.572696	−	0.819768i	$-0.694103\pi$
$59$	−8.79793	−1.14539	−0.572696	+	0.819768i	$0.694103\pi$
$60$	3.33503	0.430550
$61$	−3.34815	−0.428686	−0.214343	−	0.976758i	$-0.568761\pi$
$61$	−3.34815	−0.428686	−0.214343	+	0.976758i	$0.568761\pi$
$62$	14.7733	1.87621
$63$	−1.29396	−0.163024
$64$	−12.0878	−1.51098
$65$	−15.9571	−1.97924
$66$	1.80497	0.222176
$67$	6.31079	0.770986	0.385493	−	0.922711i	$-0.374031\pi$
$67$	6.31079	0.770986	0.385493	+	0.922711i	$0.374031\pi$
$68$	19.3518	2.34676
$69$	−1.83206	−0.220554
$70$	2.82706	0.337898
$71$	8.58718	1.01911	0.509556	−	0.860438i	$-0.329810\pi$
$71$	8.58718	1.01911	0.509556	+	0.860438i	$0.329810\pi$
$72$	3.94187	0.464554
$73$	−8.41218	−0.984572	−0.492286	−	0.870434i	$-0.663839\pi$
$73$	−8.41218	−0.984572	−0.492286	+	0.870434i	$0.663839\pi$
$74$	−8.74163	−1.01619
$75$	−1.35828	−0.156840
$76$	−2.65181	−0.304184
$77$	0.872217	0.0993984
$78$	5.36508	0.607475
$79$	1.24212	0.139749	0.0698746	−	0.997556i	$-0.477740\pi$
$79$	1.24212	0.139749	0.0698746	+	0.997556i	$0.477740\pi$
$80$	6.45194	0.721349
$81$	7.27398	0.808220
$82$	−20.9133	−2.30949
$83$	−4.62979	−0.508185	−0.254093	−	0.967180i	$-0.581777\pi$
$83$	−4.62979	−0.508185	−0.254093	+	0.967180i	$0.581777\pi$
$84$	−0.541847	−0.0591203
$85$	−20.7278	−2.24825
$86$	−21.5944	−2.32858
$87$	−2.29262	−0.245795
$88$	−2.65709	−0.283247
$89$	−7.62626	−0.808382	−0.404191	−	0.914675i	$-0.632447\pi$
$89$	−7.62626	−0.808382	−0.404191	+	0.914675i	$0.632447\pi$
$90$	−17.1773	−1.81064
$91$	2.59258	0.271776
$92$	10.9723	1.14394
$93$	3.03284	0.314491
$94$	−7.33476	−0.756523
$95$	2.84036	0.291415
$96$	−3.41419	−0.348460
$97$	−0.493240	−0.0500809	−0.0250404	−	0.999686i	$-0.507971\pi$
$97$	−0.493240	−0.0500809	−0.0250404	+	0.999686i	$0.507971\pi$
$98$	14.6383	1.47869
$99$	−5.29962	−0.532631
$100$	8.13480	0.813480
$101$	−5.20073	−0.517492	−0.258746	−	0.965945i	$-0.583309\pi$
$101$	−5.20073	−0.517492	−0.258746	+	0.965945i	$0.583309\pi$
$102$	6.96907	0.690040
$103$	−2.89192	−0.284950	−0.142475	−	0.989798i	$-0.545506\pi$
$103$	−2.89192	−0.284950	−0.142475	+	0.989798i	$0.545506\pi$
$104$	−7.89794	−0.774456
$105$	0.580373	0.0566386
$106$	30.4469	2.95726
$107$	13.8810	1.34193	0.670963	−	0.741491i	$-0.265881\pi$
$107$	13.8810	1.34193	0.670963	+	0.741491i	$0.265881\pi$
$108$	6.81475	0.655749
$109$	16.4140	1.57217	0.786086	−	0.618117i	$-0.212104\pi$
$109$	16.4140	1.57217	0.786086	+	0.618117i	$0.212104\pi$
$110$	11.5787	1.10398
$111$	−1.79459	−0.170335
$112$	−1.04826	−0.0990509
$113$	5.59133	0.525988	0.262994	−	0.964797i	$-0.415290\pi$
$113$	5.59133	0.525988	0.262994	+	0.964797i	$0.415290\pi$
$114$	−0.954981	−0.0894421
$115$	−11.7525	−1.09592
$116$	13.7307	1.27486
$117$	−15.7526	−1.45633
$118$	18.9754	1.74683
$119$	3.36768	0.308714
$120$	−1.76803	−0.161398
$121$	−7.42770	−0.675245
$122$	7.22130	0.653786
$123$	−4.29335	−0.387118
$124$	−18.1639	−1.63117
$125$	5.48859	0.490915
$126$	2.79082	0.248626
$127$	2.69962	0.239552	0.119776	−	0.992801i	$-0.461782\pi$
$127$	2.69962	0.239552	0.119776	+	0.992801i	$0.461782\pi$
$128$	10.6493	0.941278
$129$	−4.43316	−0.390318
$130$	34.4164	3.01852
$131$	−5.90532	−0.515950	−0.257975	−	0.966152i	$-0.583055\pi$
$131$	−5.90532	−0.515950	−0.257975	+	0.966152i	$0.583055\pi$
$132$	−2.21922	−0.193158
$133$	−0.461477	−0.0400152
$134$	−13.6112	−1.17582
$135$	−7.29929	−0.628223
$136$	−10.2592	−0.879716
$137$	0.165245	0.0141178	0.00705891	−	0.999975i	$-0.497753\pi$
$137$	0.165245	0.0141178	0.00705891	+	0.999975i	$0.497753\pi$
$138$	3.95139	0.336365
$139$	−9.79018	−0.830392	−0.415196	−	0.909732i	$-0.636287\pi$
$139$	−9.79018	−0.830392	−0.415196	+	0.909732i	$0.636287\pi$
$140$	−3.47589	−0.293766
$141$	−1.50577	−0.126809
$142$	−18.5209	−1.55424
$143$	10.6183	0.887948
$144$	6.36924	0.530770
$145$	−14.7069	−1.22134
$146$	18.1434	1.50156
$147$	3.00513	0.247859
$148$	10.7479	0.883473
$149$	7.62241	0.624452	0.312226	−	0.950008i	$-0.398925\pi$
$149$	7.62241	0.624452	0.312226	+	0.950008i	$0.398925\pi$
$150$	2.92954	0.239196
$151$	−14.2255	−1.15765	−0.578827	−	0.815450i	$-0.696489\pi$
$151$	−14.2255	−1.15765	−0.578827	+	0.815450i	$0.696489\pi$
$152$	1.40583	0.114028
$153$	−20.4621	−1.65426
$154$	−1.88120	−0.151592
$155$	19.4554	1.56269
$156$	−6.59641	−0.528135
$157$	10.7779	0.860174	0.430087	−	0.902787i	$-0.358483\pi$
$157$	10.7779	0.860174	0.430087	+	0.902787i	$0.358483\pi$
$158$	−2.67901	−0.213130
$159$	6.25051	0.495698
$160$	−21.9017	−1.73148
$161$	1.90944	0.150485
$162$	−15.6886	−1.23261
$163$	−8.40898	−0.658642	−0.329321	−	0.944218i	$-0.606820\pi$
$163$	−8.40898	−0.658642	−0.329321	+	0.944218i	$0.606820\pi$
$164$	25.7131	2.00786
$165$	2.37701	0.185050
$166$	9.98555	0.775029
$167$	17.2881	1.33780	0.668898	−	0.743354i	$-0.266766\pi$
$167$	17.2881	1.33780	0.668898	+	0.743354i	$0.266766\pi$
$168$	0.287254	0.0221621
$169$	18.5619	1.42784
$170$	44.7058	3.42878
$171$	2.80395	0.214423
$172$	26.5505	2.02446
$173$	5.60511	0.426149	0.213074	−	0.977036i	$-0.431652\pi$
$173$	5.60511	0.426149	0.213074	+	0.977036i	$0.431652\pi$
$174$	4.94474	0.374860
$175$	1.41565	0.107013
$176$	−4.29330	−0.323620
$177$	3.89551	0.292804
$178$	16.4484	1.23286
$179$	21.0428	1.57281	0.786406	−	0.617710i	$-0.211940\pi$
$179$	21.0428	1.57281	0.786406	+	0.617710i	$0.211940\pi$
$180$	21.1196	1.57416
$181$	−22.8753	−1.70031	−0.850153	−	0.526536i	$-0.823491\pi$
$181$	−22.8753	−1.70031	−0.850153	+	0.526536i	$0.823491\pi$
$182$	−5.59169	−0.414483
$183$	1.48248	0.109588
$184$	−5.81685	−0.428824
$185$	−11.5121	−0.846387
$186$	−6.54126	−0.479628
$187$	13.7929	1.00863
$188$	9.01815	0.657716
$189$	1.18593	0.0862634
$190$	−6.12610	−0.444434
$191$	−8.82303	−0.638412	−0.319206	−	0.947685i	$-0.603416\pi$
$191$	−8.82303	−0.638412	−0.319206	+	0.947685i	$0.603416\pi$
$192$	5.35220	0.386262
$193$	−12.1674	−0.875829	−0.437915	−	0.899017i	$-0.644283\pi$
$193$	−12.1674	−0.875829	−0.437915	+	0.899017i	$0.644283\pi$
$194$	1.06382	0.0763780
$195$	7.06543	0.505966
$196$	−17.9979	−1.28557
$197$	−7.49599	−0.534067	−0.267034	−	0.963687i	$-0.586043\pi$
$197$	−7.49599	−0.534067	−0.267034	+	0.963687i	$0.586043\pi$
$198$	11.4302	0.812312
$199$	−24.7277	−1.75290	−0.876451	−	0.481491i	$-0.840096\pi$
$199$	−24.7277	−1.75290	−0.876451	+	0.481491i	$0.840096\pi$
$200$	−4.31258	−0.304945
$201$	−2.79426	−0.197092
$202$	11.2170	0.789222
$203$	2.38946	0.167707
$204$	−8.56852	−0.599917
$205$	−27.5414	−1.92357
$206$	6.23732	0.434574
$207$	−11.6018	−0.806382
$208$	−12.7614	−0.884844
$209$	−1.89005	−0.130738
$210$	−1.25175	−0.0863791
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−37.4347	−2.57103
$213$	−3.80220	−0.260522
$214$	−29.9386	−2.04656
$215$	−28.4383	−1.93948
$216$	−3.61276	−0.245817
$217$	−3.16095	−0.214579
$218$	−35.4017	−2.39771
$219$	3.72471	0.251693
$220$	−14.2361	−0.959795
$221$	40.9979	2.75782
$222$	3.87058	0.259776
$223$	20.6137	1.38039	0.690197	−	0.723621i	$-0.257524\pi$
$223$	20.6137	1.38039	0.690197	+	0.723621i	$0.257524\pi$
$224$	3.55840	0.237756
$225$	−8.60151	−0.573434
$226$	−12.0594	−0.802180
$227$	−7.28015	−0.483201	−0.241600	−	0.970376i	$-0.577672\pi$
$227$	−7.28015	−0.483201	−0.241600	+	0.970376i	$0.577672\pi$
$228$	1.17416	0.0777604
$229$	7.45011	0.492317	0.246159	−	0.969230i	$-0.420832\pi$
$229$	7.45011	0.492317	0.246159	+	0.969230i	$0.420832\pi$
$230$	25.3478	1.67138
$231$	−0.386196	−0.0254099
$232$	−7.27915	−0.477900
$233$	28.2363	1.84982	0.924911	−	0.380183i	$-0.124139\pi$
$233$	28.2363	1.84982	0.924911	+	0.380183i	$0.124139\pi$
$234$	33.9752	2.22103
$235$	−9.65935	−0.630107
$236$	−23.3304	−1.51868
$237$	−0.549980	−0.0357250
$238$	−7.26342	−0.470818
$239$	23.8224	1.54094	0.770471	−	0.637475i	$-0.220021\pi$
$239$	23.8224	1.54094	0.770471	+	0.637475i	$0.220021\pi$
$240$	−2.85676	−0.184403
$241$	−12.4450	−0.801650	−0.400825	−	0.916155i	$-0.631276\pi$
$241$	−12.4450	−0.801650	−0.400825	+	0.916155i	$0.631276\pi$
$242$	16.0201	1.02981
$243$	−10.9303	−0.701178
$244$	−8.87865	−0.568397
$245$	19.2776	1.23160
$246$	9.25991	0.590390
$247$	−5.61800	−0.357465
$248$	9.62939	0.611467
$249$	2.04996	0.129911
$250$	−11.8378	−0.748690
$251$	12.3160	0.777380	0.388690	−	0.921369i	$-0.372928\pi$
$251$	12.3160	0.777380	0.388690	+	0.921369i	$0.372928\pi$
$252$	−3.43134	−0.216154
$253$	7.82041	0.491665
$254$	−5.82255	−0.365339
$255$	9.17776	0.574734
$256$	1.20711	0.0754443
$257$	4.76392	0.297165	0.148582	−	0.988900i	$-0.452529\pi$
$257$	4.76392	0.297165	0.148582	+	0.988900i	$0.452529\pi$
$258$	9.56147	0.595271
$259$	1.87039	0.116220
$260$	−42.3153	−2.62428
$261$	−14.5184	−0.898667
$262$	12.7366	0.786871
$263$	17.1968	1.06040	0.530201	−	0.847872i	$-0.322116\pi$
$263$	17.1968	1.06040	0.530201	+	0.847872i	$0.322116\pi$
$264$	1.17649	0.0724083
$265$	40.0964	2.46310
$266$	0.995317	0.0610268
$267$	3.37672	0.206652
$268$	16.7350	1.02225
$269$	−4.89843	−0.298663	−0.149331	−	0.988787i	$-0.547712\pi$
$269$	−4.89843	−0.298663	−0.149331	+	0.988787i	$0.547712\pi$
$270$	15.7431	0.958097
$271$	−30.1700	−1.83270	−0.916349	−	0.400380i	$-0.868878\pi$
$271$	−30.1700	−1.83270	−0.916349	+	0.400380i	$0.868878\pi$
$272$	−16.5767	−1.00511
$273$	−1.14793	−0.0694759
$274$	−0.356401	−0.0215310
$275$	5.79801	0.349633
$276$	−4.85827	−0.292434
$277$	−21.9010	−1.31591	−0.657953	−	0.753059i	$-0.728577\pi$
$277$	−21.9010	−1.31591	−0.657953	+	0.753059i	$0.728577\pi$
$278$	21.1155	1.26642
$279$	19.2060	1.14983
$280$	1.84271	0.110123
$281$	−11.9215	−0.711177	−0.355588	−	0.934643i	$-0.615719\pi$
$281$	−11.9215	−0.711177	−0.355588	+	0.934643i	$0.615719\pi$
$282$	3.24765	0.193395
$283$	16.0748	0.955548	0.477774	−	0.878483i	$-0.341444\pi$
$283$	16.0748	0.955548	0.477774	+	0.878483i	$0.341444\pi$
$284$	22.7716	1.35125
$285$	−1.25764	−0.0744963
$286$	−22.9016	−1.35420
$287$	4.47469	0.264132
$288$	−21.6209	−1.27403
$289$	36.2549	2.13264
$290$	31.7200	1.86266
$291$	0.218394	0.0128025
$292$	−22.3075	−1.30545
$293$	25.7841	1.50632	0.753162	−	0.657835i	$-0.228527\pi$
$293$	25.7841	1.50632	0.753162	+	0.657835i	$0.228527\pi$
$294$	−6.48149	−0.378008
$295$	24.9893	1.45493
$296$	−5.69789	−0.331183
$297$	4.85715	0.281840
$298$	−16.4401	−0.952347
$299$	23.2454	1.34432
$300$	−3.60189	−0.207955
$301$	4.62041	0.266316
$302$	30.6816	1.76553
$303$	2.30275	0.132290
$304$	2.27152	0.130281
$305$	9.50994	0.544537
$306$	44.1328	2.52290
$307$	−8.84612	−0.504874	−0.252437	−	0.967613i	$-0.581232\pi$
$307$	−8.84612	−0.504874	−0.252437	+	0.967613i	$0.581232\pi$
$308$	2.31295	0.131793
$309$	1.28047	0.0728435
$310$	−41.9615	−2.38325
$311$	−19.2563	−1.09192	−0.545962	−	0.837810i	$-0.683836\pi$
$311$	−19.2563	−1.09192	−0.545962	+	0.837810i	$0.683836\pi$
$312$	3.49701	0.197979
$313$	20.8155	1.17656	0.588280	−	0.808658i	$-0.299805\pi$
$313$	20.8155	1.17656	0.588280	+	0.808658i	$0.299805\pi$
$314$	−23.2459	−1.31184
$315$	3.67531	0.207080
$316$	3.29386	0.185294
$317$	−30.9468	−1.73815	−0.869073	−	0.494684i	$-0.835284\pi$
$317$	−30.9468	−1.73815	−0.869073	+	0.494684i	$0.835284\pi$
$318$	−13.4811	−0.755985
$319$	9.78640	0.547933
$320$	34.3338	1.91932
$321$	−6.14616	−0.343045
$322$	−4.11829	−0.229503
$323$	−7.29760	−0.406049
$324$	19.2892	1.07162
$325$	17.2340	0.955970
$326$	18.1365	1.00449
$327$	−7.26770	−0.401905
$328$	−13.6315	−0.752675
$329$	1.56937	0.0865222
$330$	−5.12675	−0.282218
$331$	14.5832	0.801565	0.400782	−	0.916173i	$-0.368738\pi$
$331$	14.5832	0.801565	0.400782	+	0.916173i	$0.368738\pi$
$332$	−12.2773	−0.673805
$333$	−11.3645	−0.622773
$334$	−37.2871	−2.04026
$335$	−17.9249	−0.979343
$336$	0.464142	0.0253210
$337$	−20.9618	−1.14186	−0.570930	−	0.820999i	$-0.693417\pi$
$337$	−20.9618	−1.14186	−0.570930	+	0.820999i	$0.693417\pi$
$338$	−40.0344	−2.17758
$339$	−2.47570	−0.134462
$340$	−54.9662	−2.98096
$341$	−12.9462	−0.701073
$342$	−6.04757	−0.327015
$343$	−6.36241	−0.343538
$344$	−14.0755	−0.758898
$345$	5.20370	0.280158
$346$	−12.0891	−0.649916
$347$	−23.6043	−1.26715	−0.633573	−	0.773683i	$-0.718413\pi$
$347$	−23.6043	−1.26715	−0.633573	+	0.773683i	$0.718413\pi$
$348$	−6.07960	−0.325901
$349$	34.6309	1.85375	0.926875	−	0.375371i	$-0.122485\pi$
$349$	34.6309	1.85375	0.926875	+	0.375371i	$0.122485\pi$
$350$	−3.05328	−0.163204
$351$	14.4374	0.770611
$352$	14.5740	0.776797
$353$	12.7354	0.677834	0.338917	−	0.940816i	$-0.389939\pi$
$353$	12.7354	0.677834	0.338917	+	0.940816i	$0.389939\pi$
$354$	−8.40185	−0.446553
$355$	−24.3907	−1.29452
$356$	−20.2234	−1.07184
$357$	−1.49112	−0.0789187
$358$	−45.3852	−2.39868
$359$	31.6873	1.67239	0.836195	−	0.548432i	$-0.184775\pi$
$359$	31.6873	1.67239	0.836195	+	0.548432i	$0.184775\pi$
$360$	−11.1963	−0.590099
$361$	1.00000	0.0526316
$362$	49.3375	2.59312
$363$	3.28880	0.172617
$364$	6.87503	0.360349
$365$	23.8936	1.25065
$366$	−3.19742	−0.167132
$367$	28.6749	1.49682	0.748410	−	0.663237i	$-0.230818\pi$
$367$	28.6749	1.49682	0.748410	+	0.663237i	$0.230818\pi$
$368$	−9.39881	−0.489947
$369$	−27.1883	−1.41537
$370$	24.8294	1.29082
$371$	−6.51452	−0.338217
$372$	8.04253	0.416986
$373$	−22.5517	−1.16768	−0.583842	−	0.811868i	$-0.698451\pi$
$373$	−22.5517	−1.16768	−0.583842	+	0.811868i	$0.698451\pi$
$374$	−29.7485	−1.53826
$375$	−2.43021	−0.125496
$376$	−4.78087	−0.246555
$377$	29.0891	1.49816
$378$	−2.55781	−0.131560
$379$	4.68450	0.240626	0.120313	−	0.992736i	$-0.461610\pi$
$379$	4.68450	0.240626	0.120313	+	0.992736i	$0.461610\pi$
$380$	7.53209	0.386388
$381$	−1.19532	−0.0612383
$382$	19.0296	0.973637
$383$	−9.81152	−0.501345	−0.250673	−	0.968072i	$-0.580652\pi$
$383$	−9.81152	−0.501345	−0.250673	+	0.968072i	$0.580652\pi$
$384$	−4.71527	−0.240625
$385$	−2.47741	−0.126261
$386$	26.2427	1.33572
$387$	−28.0738	−1.42707
$388$	−1.30798	−0.0664025
$389$	15.5862	0.790253	0.395126	−	0.918627i	$-0.370701\pi$
$389$	15.5862	0.790253	0.395126	+	0.918627i	$0.370701\pi$
$390$	−15.2387	−0.771644
$391$	30.1950	1.52703
$392$	9.54141	0.481914
$393$	2.61473	0.131896
$394$	16.1674	0.814501
$395$	−3.52806	−0.177516
$396$	−14.0536	−0.706219
$397$	28.5738	1.43408	0.717039	−	0.697033i	$-0.245497\pi$
$397$	28.5738	1.43408	0.717039	+	0.697033i	$0.245497\pi$
$398$	53.3329	2.67334
$399$	0.204331	0.0102293
$400$	−6.96822	−0.348411
$401$	−22.4834	−1.12277	−0.561384	−	0.827555i	$-0.689731\pi$
$401$	−22.4834	−1.12277	−0.561384	+	0.827555i	$0.689731\pi$
$402$	6.02669	0.300584
$403$	−38.4811	−1.91688
$404$	−13.7913	−0.686145
$405$	−20.6607	−1.02664
$406$	−5.15359	−0.255769
$407$	7.66047	0.379716
$408$	4.54251	0.224888
$409$	30.3135	1.49891	0.749454	−	0.662057i	$-0.230316\pi$
$409$	30.3135	1.49891	0.749454	+	0.662057i	$0.230316\pi$
$410$	59.4014	2.93362
$411$	−0.0731664	−0.00360903
$412$	−7.66883	−0.377816
$413$	−4.06004	−0.199782
$414$	25.0229	1.22981
$415$	13.1503	0.645521
$416$	43.3197	2.12393
$417$	4.33485	0.212279
$418$	4.07648	0.199387
$419$	−21.8390	−1.06690	−0.533452	−	0.845830i	$-0.679106\pi$
$419$	−21.8390	−1.06690	−0.533452	+	0.845830i	$0.679106\pi$
$420$	1.53904	0.0750974
$421$	8.24437	0.401806	0.200903	−	0.979611i	$-0.435612\pi$
$421$	8.24437	0.401806	0.200903	+	0.979611i	$0.435612\pi$
$422$	−2.15681	−0.104992
$423$	−9.53553	−0.463634
$424$	19.8456	0.963788
$425$	22.3864	1.08590
$426$	8.20060	0.397320
$427$	−1.54509	−0.0747723
$428$	36.8097	1.77927
$429$	−4.70153	−0.226992
$430$	61.3359	2.95788
$431$	12.0987	0.582776	0.291388	−	0.956605i	$-0.405883\pi$
$431$	12.0987	0.582776	0.291388	+	0.956605i	$0.405883\pi$
$432$	−5.83747	−0.280855
$433$	−12.8001	−0.615134	−0.307567	−	0.951526i	$-0.599515\pi$
$433$	−12.8001	−0.615134	−0.307567	+	0.951526i	$0.599515\pi$
$434$	6.81754	0.327253
$435$	6.51187	0.312220
$436$	43.5267	2.08455
$437$	−4.13767	−0.197931
$438$	−8.03347	−0.383854
$439$	14.4795	0.691069	0.345535	−	0.938406i	$-0.387698\pi$
$439$	14.4795	0.691069	0.345535	+	0.938406i	$0.387698\pi$
$440$	7.54709	0.359794
$441$	19.0305	0.906215
$442$	−88.4244	−4.20592
$443$	−5.99393	−0.284780	−0.142390	−	0.989811i	$-0.545479\pi$
$443$	−5.99393	−0.284780	−0.142390	+	0.989811i	$0.545479\pi$
$444$	−4.75891	−0.225848
$445$	21.6613	1.02684
$446$	−44.4597	−2.10523
$447$	−3.37502	−0.159633
$448$	−5.57827	−0.263548
$449$	30.7604	1.45167	0.725837	−	0.687867i	$-0.241453\pi$
$449$	30.7604	1.45167	0.725837	+	0.687867i	$0.241453\pi$
$450$	18.5518	0.874540
$451$	18.3268	0.862975
$452$	14.8271	0.697410
$453$	6.29870	0.295939
$454$	15.7019	0.736926
$455$	−7.36385	−0.345223
$456$	−0.622466	−0.0291497
$457$	12.5895	0.588911	0.294455	−	0.955665i	$-0.404862\pi$
$457$	12.5895	0.588911	0.294455	+	0.955665i	$0.404862\pi$
$458$	−16.0684	−0.750829
$459$	18.7537	0.875348
$460$	−31.1653	−1.45309
$461$	−13.6588	−0.636155	−0.318077	−	0.948065i	$-0.603037\pi$
$461$	−13.6588	−0.636155	−0.318077	+	0.948065i	$0.603037\pi$
$462$	0.832951	0.0387524
$463$	−33.4905	−1.55644	−0.778218	−	0.627994i	$-0.783876\pi$
$463$	−33.4905	−1.55644	−0.778218	+	0.627994i	$0.783876\pi$
$464$	−11.7616	−0.546018
$465$	−8.61437	−0.399482
$466$	−60.9002	−2.82115
$467$	13.8757	0.642090	0.321045	−	0.947064i	$-0.395966\pi$
$467$	13.8757	0.642090	0.321045	+	0.947064i	$0.395966\pi$
$468$	−41.7728	−1.93095
$469$	2.91229	0.134477
$470$	20.8334	0.960971
$471$	−4.77221	−0.219892
$472$	12.3684	0.569300
$473$	18.9236	0.870109
$474$	1.18620	0.0544839
$475$	−3.06764	−0.140753
$476$	8.93044	0.409326
$477$	39.5824	1.81235
$478$	−51.3803	−2.35008
$479$	41.9182	1.91529	0.957646	−	0.287947i	$-0.0929726\pi$
$479$	41.9182	1.91529	0.957646	+	0.287947i	$0.0929726\pi$
$480$	9.69754	0.442630
$481$	22.7700	1.03822
$482$	26.8413	1.22259
$483$	−0.845453	−0.0384695
$484$	−19.6968	−0.895311
$485$	1.40098	0.0636151
$486$	23.5745	1.06936
$487$	−5.52381	−0.250308	−0.125154	−	0.992137i	$-0.539942\pi$
$487$	−5.52381	−0.250308	−0.125154	+	0.992137i	$0.539942\pi$
$488$	4.70692	0.213072
$489$	3.72329	0.168373
$490$	−41.5781	−1.87831
$491$	7.19090	0.324521	0.162260	−	0.986748i	$-0.448122\pi$
$491$	7.19090	0.324521	0.162260	+	0.986748i	$0.448122\pi$
$492$	−11.3851	−0.513282
$493$	37.7858	1.70179
$494$	12.1169	0.545166
$495$	15.0528	0.676574
$496$	15.5591	0.698623
$497$	3.96279	0.177756
$498$	−4.42136	−0.198126
$499$	34.5621	1.54721	0.773606	−	0.633667i	$-0.218451\pi$
$499$	34.5621	1.54721	0.773606	+	0.633667i	$0.218451\pi$
$500$	14.5547	0.650906
$501$	−7.65476	−0.341989
$502$	−26.5633	−1.18558
$503$	12.2914	0.548046	0.274023	−	0.961723i	$-0.411646\pi$
$503$	12.2914	0.548046	0.274023	+	0.961723i	$0.411646\pi$
$504$	1.81908	0.0810285
$505$	14.7719	0.657342
$506$	−16.8671	−0.749835
$507$	−8.21874	−0.365007
$508$	7.15887	0.317623
$509$	−6.33893	−0.280968	−0.140484	−	0.990083i	$-0.544866\pi$
$509$	−6.33893	−0.280968	−0.140484	+	0.990083i	$0.544866\pi$
$510$	−19.7947	−0.876522
$511$	−3.88203	−0.171731
$512$	−23.9022	−1.05634
$513$	−2.56985	−0.113462
$514$	−10.2748	−0.453204
$515$	8.21410	0.361956
$516$	−11.7559	−0.517525
$517$	6.42760	0.282686
$518$	−4.03407	−0.177247
$519$	−2.48181	−0.108939
$520$	22.4330	0.983751
$521$	21.8445	0.957024	0.478512	−	0.878081i	$-0.341176\pi$
$521$	21.8445	0.957024	0.478512	+	0.878081i	$0.341176\pi$
$522$	31.3134	1.37055
$523$	−14.8473	−0.649227	−0.324613	−	0.945847i	$-0.605234\pi$
$523$	−14.8473	−0.649227	−0.324613	+	0.945847i	$0.605234\pi$
$524$	−15.6598	−0.684101
$525$	−0.626814	−0.0273564
$526$	−37.0902	−1.61721
$527$	−49.9858	−2.17741
$528$	1.90097	0.0827290
$529$	−5.87971	−0.255639
$530$	−86.4801	−3.75646
$531$	24.6689	1.07054
$532$	−1.22375	−0.0530563
$533$	54.4745	2.35955
$534$	−7.28293	−0.315163
$535$	−39.4270	−1.70458
$536$	−8.87189	−0.383207
$537$	−9.31723	−0.402068
$538$	10.5650	0.455488
$539$	−12.8279	−0.552535
$540$	−19.3563	−0.832964
$541$	39.0826	1.68029	0.840146	−	0.542360i	$-0.182469\pi$
$541$	39.0826	1.68029	0.840146	+	0.542360i	$0.182469\pi$
$542$	65.0709	2.79503
$543$	10.1286	0.434660
$544$	56.2710	2.41260
$545$	−46.6216	−1.99705
$546$	2.47586	0.105957
$547$	−21.0819	−0.901398	−0.450699	−	0.892676i	$-0.648825\pi$
$547$	−21.0819	−0.901398	−0.450699	+	0.892676i	$0.648825\pi$
$548$	0.438198	0.0187189
$549$	9.38803	0.400672
$550$	−12.5052	−0.533222
$551$	−5.17784	−0.220583
$552$	2.57556	0.109623
$553$	0.573209	0.0243753
$554$	47.2363	2.00688
$555$	5.09728	0.216367
$556$	−25.9617	−1.10102
$557$	32.2178	1.36511	0.682556	−	0.730834i	$-0.260868\pi$
$557$	32.2178	1.36511	0.682556	+	0.730834i	$0.260868\pi$
$558$	−41.4236	−1.75360
$559$	56.2486	2.37906
$560$	2.97742	0.125819
$561$	−6.10714	−0.257844
$562$	25.7123	1.08461
$563$	16.2297	0.684001	0.342001	−	0.939700i	$-0.388896\pi$
$563$	16.2297	0.684001	0.342001	+	0.939700i	$0.388896\pi$
$564$	−3.99301	−0.168136
$565$	−15.8814	−0.668135
$566$	−34.6702	−1.45730
$567$	3.35678	0.140971
$568$	−12.0721	−0.506534
$569$	−38.7382	−1.62399	−0.811995	−	0.583664i	$-0.801618\pi$
$569$	−38.7382	−1.62399	−0.811995	+	0.583664i	$0.801618\pi$
$570$	2.71249	0.113614
$571$	−8.43855	−0.353142	−0.176571	−	0.984288i	$-0.556501\pi$
$571$	−8.43855	−0.353142	−0.176571	+	0.984288i	$0.556501\pi$
$572$	28.1578	1.17733
$573$	3.90662	0.163201
$574$	−9.65103	−0.402826
$575$	12.6929	0.529330
$576$	33.8937	1.41224
$577$	45.9002	1.91085	0.955425	−	0.295234i	$-0.0953975\pi$
$577$	45.9002	1.91085	0.955425	+	0.295234i	$0.0953975\pi$
$578$	−78.1949	−3.25248
$579$	5.38743	0.223894
$580$	−39.0000	−1.61939
$581$	−2.13654	−0.0886387
$582$	−0.471034	−0.0195250
$583$	−26.6813	−1.10502
$584$	11.8261	0.489367
$585$	44.7430	1.84989
$586$	−55.6113	−2.29728
$587$	−8.09459	−0.334099	−0.167050	−	0.985948i	$-0.553424\pi$
$587$	−8.09459	−0.334099	−0.167050	+	0.985948i	$0.553424\pi$
$588$	7.96905	0.328638
$589$	6.84962	0.282234
$590$	−53.8970	−2.21890
$591$	3.31904	0.136527
$592$	−9.20659	−0.378388
$593$	−5.82106	−0.239042	−0.119521	−	0.992832i	$-0.538136\pi$
$593$	−5.82106	−0.239042	−0.119521	+	0.992832i	$0.538136\pi$
$594$	−10.4759	−0.429832
$595$	−9.56541	−0.392144
$596$	20.2132	0.827964
$597$	10.9488	0.448106
$598$	−50.1358	−2.05021
$599$	24.5597	1.00348	0.501740	−	0.865018i	$-0.332693\pi$
$599$	24.5597	1.00348	0.501740	+	0.865018i	$0.332693\pi$
$600$	1.90950	0.0779552
$601$	10.8283	0.441695	0.220848	−	0.975308i	$-0.429118\pi$
$601$	10.8283	0.441695	0.220848	+	0.975308i	$0.429118\pi$
$602$	−9.96533	−0.406156
$603$	−17.6951	−0.720602
$604$	−37.7233	−1.53494
$605$	21.0973	0.857728
$606$	−4.96659	−0.201754
$607$	−28.1348	−1.14195	−0.570977	−	0.820966i	$-0.693436\pi$
$607$	−28.1348	−1.14195	−0.570977	+	0.820966i	$0.693436\pi$
$608$	−7.71089	−0.312718
$609$	−1.05799	−0.0428720
$610$	−20.5111	−0.830470
$611$	19.1054	0.772922
$612$	−54.2616	−2.19340
$613$	18.7209	0.756129	0.378065	−	0.925779i	$-0.376590\pi$
$613$	18.7209	0.756129	0.378065	+	0.925779i	$0.376590\pi$
$614$	19.0794	0.769980
$615$	12.1946	0.491735
$616$	−1.22619	−0.0494045
$617$	2.20398	0.0887290	0.0443645	−	0.999015i	$-0.485874\pi$
$617$	2.20398	0.0887290	0.0443645	+	0.999015i	$0.485874\pi$
$618$	−2.76173	−0.111093
$619$	24.7552	0.994996	0.497498	−	0.867465i	$-0.334252\pi$
$619$	24.7552	0.994996	0.497498	+	0.867465i	$0.334252\pi$
$620$	51.5920	2.07198
$621$	10.6332	0.426695
$622$	41.5321	1.66528
$623$	−3.51935	−0.141000
$624$	5.65044	0.226199
$625$	−30.9278	−1.23711
$626$	−44.8949	−1.79436
$627$	0.836870	0.0334214
$628$	28.5811	1.14051
$629$	29.5775	1.17933
$630$	−7.92693	−0.315816
$631$	15.0785	0.600265	0.300132	−	0.953898i	$-0.402969\pi$
$631$	15.0785	0.600265	0.300132	+	0.953898i	$0.402969\pi$
$632$	−1.74620	−0.0694603
$633$	−0.442776	−0.0175987
$634$	66.7462	2.65083
$635$	−7.66788	−0.304290
$636$	16.5752	0.657248
$637$	−38.1296	−1.51075
$638$	−21.1074	−0.835648
$639$	−24.0780	−0.952512
$640$	−30.2480	−1.19566
$641$	7.15893	0.282761	0.141380	−	0.989955i	$-0.454846\pi$
$641$	7.15893	0.282761	0.141380	+	0.989955i	$0.454846\pi$
$642$	13.2561	0.523175
$643$	11.6976	0.461310	0.230655	−	0.973036i	$-0.425913\pi$
$643$	11.6976	0.461310	0.230655	+	0.973036i	$0.425913\pi$
$644$	5.06347	0.199529
$645$	12.5918	0.495801
$646$	15.7395	0.619262
$647$	−41.8623	−1.64578	−0.822889	−	0.568202i	$-0.807639\pi$
$647$	−41.8623	−1.64578	−0.822889	+	0.568202i	$0.807639\pi$
$648$	−10.2260	−0.401714
$649$	−16.6286	−0.652728
$650$	−37.1704	−1.45794
$651$	1.39959	0.0548542
$652$	−22.2990	−0.873297
$653$	−9.94405	−0.389141	−0.194570	−	0.980889i	$-0.562331\pi$
$653$	−9.94405	−0.389141	−0.194570	+	0.980889i	$0.562331\pi$
$654$	15.6750	0.612942
$655$	16.7732	0.655384
$656$	−22.0257	−0.859958
$657$	23.5873	0.920230
$658$	−3.38483	−0.131954
$659$	−13.8264	−0.538601	−0.269301	−	0.963056i	$-0.586793\pi$
$659$	−13.8264	−0.538601	−0.269301	+	0.963056i	$0.586793\pi$
$660$	6.30338	0.245359
$661$	5.83971	0.227138	0.113569	−	0.993530i	$-0.463772\pi$
$661$	5.83971	0.227138	0.113569	+	0.993530i	$0.463772\pi$
$662$	−31.4531	−1.22246
$663$	−18.1529	−0.704999
$664$	6.50869	0.252586
$665$	1.31076	0.0508292
$666$	24.5111	0.949786
$667$	21.4242	0.829548
$668$	45.8448	1.77379
$669$	−9.12724	−0.352879
$670$	38.6606	1.49359
$671$	−6.32818	−0.244297
$672$	−1.57557	−0.0607790
$673$	5.21753	0.201121	0.100561	−	0.994931i	$-0.467936\pi$
$673$	5.21753	0.201121	0.100561	+	0.994931i	$0.467936\pi$
$674$	45.2105	1.74144
$675$	7.88337	0.303431
$676$	49.2226	1.89318
$677$	1.97959	0.0760820	0.0380410	−	0.999276i	$-0.487888\pi$
$677$	1.97959	0.0760820	0.0380410	+	0.999276i	$0.487888\pi$
$678$	5.33961	0.205067
$679$	−0.227619	−0.00873521
$680$	29.1397	1.11746
$681$	3.22347	0.123524
$682$	27.9223	1.06920
$683$	−36.5033	−1.39676	−0.698380	−	0.715728i	$-0.746095\pi$
$683$	−36.5033	−1.39676	−0.698380	+	0.715728i	$0.746095\pi$
$684$	7.43554	0.284305
$685$	−0.469355	−0.0179331
$686$	13.7225	0.523927
$687$	−3.29873	−0.125854
$688$	−22.7430	−0.867068
$689$	−79.3073	−3.02137
$690$	−11.2234	−0.427267
$691$	17.3289	0.659224	0.329612	−	0.944117i	$-0.393082\pi$
$691$	17.3289	0.659224	0.329612	+	0.944117i	$0.393082\pi$
$692$	14.8637	0.565033
$693$	−2.44565	−0.0929027
$694$	50.9099	1.93252
$695$	27.8076	1.05480
$696$	3.22303	0.122169
$697$	70.7607	2.68025
$698$	−74.6921	−2.82714
$699$	−12.5023	−0.472882
$700$	3.75403	0.141889
$701$	−4.14044	−0.156382	−0.0781911	−	0.996938i	$-0.524914\pi$
$701$	−4.14044	−0.156382	−0.0781911	+	0.996938i	$0.524914\pi$
$702$	−31.1386	−1.17525
$703$	−4.05305	−0.152864
$704$	−22.8467	−0.861066
$705$	4.27693	0.161078
$706$	−27.4677	−1.03376
$707$	−2.40002	−0.0902620
$708$	10.3301	0.388231
$709$	−37.3966	−1.40446	−0.702229	−	0.711951i	$-0.747812\pi$
$709$	−37.3966	−1.40446	−0.702229	+	0.711951i	$0.747812\pi$
$710$	52.6060	1.97427
$711$	−3.48284	−0.130617
$712$	10.7212	0.401794
$713$	−28.3415	−1.06140
$714$	3.21607	0.120358
$715$	−30.1598	−1.12791
$716$	55.8015	2.08540
$717$	−10.5480	−0.393921
$718$	−68.3433	−2.55055
$719$	40.8335	1.52283	0.761416	−	0.648264i	$-0.224505\pi$
$719$	40.8335	1.52283	0.761416	+	0.648264i	$0.224505\pi$
$720$	−18.0909	−0.674209
$721$	−1.33456	−0.0497015
$722$	−2.15681	−0.0802680
$723$	5.51032	0.204931
$724$	−60.6609	−2.25444
$725$	15.8838	0.589908
$726$	−7.09331	−0.263257
$727$	4.57529	0.169688	0.0848440	−	0.996394i	$-0.472961\pi$
$727$	4.57529	0.169688	0.0848440	+	0.996394i	$0.472961\pi$
$728$	−3.64472	−0.135082
$729$	−16.9823	−0.628974
$730$	−51.5339	−1.90736
$731$	73.0651	2.70241
$732$	3.93125	0.145303
$733$	−1.49974	−0.0553942	−0.0276971	−	0.999616i	$-0.508817\pi$
$733$	−1.49974	−0.0553942	−0.0276971	+	0.999616i	$0.508817\pi$
$734$	−61.8462	−2.28279
$735$	−8.53566	−0.314843
$736$	31.9051	1.17604
$737$	11.9277	0.439364
$738$	58.6399	2.15857
$739$	40.8650	1.50324	0.751622	−	0.659594i	$-0.229272\pi$
$739$	40.8650	1.50324	0.751622	+	0.659594i	$0.229272\pi$
$740$	−30.5279	−1.12223
$741$	2.48751	0.0913810
$742$	14.0506	0.515812
$743$	−0.622955	−0.0228540	−0.0114270	−	0.999935i	$-0.503637\pi$
$743$	−0.622955	−0.0228540	−0.0114270	+	0.999935i	$0.503637\pi$
$744$	−4.26366	−0.156313
$745$	−21.6504	−0.793208
$746$	48.6396	1.78082
$747$	12.9817	0.474975
$748$	36.5760	1.33735
$749$	6.40576	0.234061
$750$	5.24150	0.191393
$751$	−6.92446	−0.252677	−0.126339	−	0.991987i	$-0.540323\pi$
$751$	−6.92446	−0.252677	−0.126339	+	0.991987i	$0.540323\pi$
$752$	−7.72488	−0.281697
$753$	−5.45323	−0.198727
$754$	−62.7395	−2.28484
$755$	40.4055	1.47051
$756$	3.14485	0.114377
$757$	34.5670	1.25636	0.628180	−	0.778068i	$-0.283800\pi$
$757$	34.5670	1.25636	0.628180	+	0.778068i	$0.283800\pi$
$758$	−10.1036	−0.366977
$759$	−3.46269	−0.125688
$760$	−3.99306	−0.144843
$761$	−49.8928	−1.80861	−0.904306	−	0.426886i	$-0.859611\pi$
$761$	−49.8928	−1.80861	−0.904306	+	0.426886i	$0.859611\pi$
$762$	2.57808	0.0933940
$763$	7.57467	0.274222
$764$	−23.3970	−0.846474
$765$	58.1197	2.10132
$766$	21.1615	0.764598
$767$	−49.4267	−1.78470
$768$	−0.534479	−0.0192863
$769$	−38.7276	−1.39655	−0.698276	−	0.715829i	$-0.746049\pi$
$769$	−38.7276	−1.39655	−0.698276	+	0.715829i	$0.746049\pi$
$770$	5.34329	0.192559
$771$	−2.10935	−0.0759662
$772$	−32.2657	−1.16127
$773$	25.0985	0.902730	0.451365	−	0.892339i	$-0.350937\pi$
$773$	25.0985	0.902730	0.451365	+	0.892339i	$0.350937\pi$
$774$	60.5496	2.17641
$775$	−21.0122	−0.754780
$776$	0.693410	0.0248920
$777$	−0.828163	−0.0297102
$778$	−33.6164	−1.20521
$779$	−9.69644	−0.347411
$780$	18.7362	0.670862
$781$	16.2302	0.580764
$782$	−65.1248	−2.32886
$783$	13.3063	0.475527
$784$	15.4169	0.550604
$785$	−30.6132	−1.09263
$786$	−5.63946	−0.201153
$787$	−43.5299	−1.55167	−0.775836	−	0.630934i	$-0.782672\pi$
$787$	−43.5299	−1.55167	−0.775836	+	0.630934i	$0.782672\pi$
$788$	−19.8779	−0.708122
$789$	−7.61434	−0.271078
$790$	7.60934	0.270728
$791$	2.58027	0.0917439
$792$	7.45035	0.264737
$793$	−18.8099	−0.667958
$794$	−61.6281	−2.18710
$795$	−17.7537	−0.629659
$796$	−65.5733	−2.32418
$797$	−42.6456	−1.51059	−0.755293	−	0.655387i	$-0.772506\pi$
$797$	−42.6456	−1.51059	−0.755293	+	0.655387i	$0.772506\pi$
$798$	−0.440702	−0.0156007
$799$	24.8173	0.877973
$800$	23.6542	0.836304
$801$	21.3836	0.755554
$802$	48.4924	1.71233
$803$	−15.8995	−0.561080
$804$	−7.40986	−0.261326
$805$	−5.42350	−0.191153
$806$	82.9963	2.92342
$807$	2.16891	0.0763491
$808$	7.31133	0.257212
$809$	5.91490	0.207957	0.103978	−	0.994580i	$-0.466843\pi$
$809$	5.91490	0.207957	0.103978	+	0.994580i	$0.466843\pi$
$810$	44.5612	1.56572
$811$	−9.08685	−0.319082	−0.159541	−	0.987191i	$-0.551001\pi$
$811$	−9.08685	−0.319082	−0.159541	+	0.987191i	$0.551001\pi$
$812$	6.33638	0.222364
$813$	13.3585	0.468505
$814$	−16.5222	−0.579101
$815$	23.8845	0.836638
$816$	7.33974	0.256942
$817$	−10.0122	−0.350283
$818$	−65.3804	−2.28597
$819$	−7.26946	−0.254015
$820$	−73.0345	−2.55047
$821$	−26.6831	−0.931248	−0.465624	−	0.884983i	$-0.654170\pi$
$821$	−26.6831	−0.931248	−0.465624	+	0.884983i	$0.654170\pi$
$822$	0.157806	0.00550410
$823$	41.3210	1.44036	0.720180	−	0.693787i	$-0.244059\pi$
$823$	41.3210	1.44036	0.720180	+	0.693787i	$0.244059\pi$
$824$	4.06555	0.141630
$825$	−2.56722	−0.0893790
$826$	8.75673	0.304685
$827$	−50.7781	−1.76573	−0.882863	−	0.469630i	$-0.844387\pi$
$827$	−50.7781	−1.76573	−0.882863	+	0.469630i	$0.844387\pi$
$828$	−30.7658	−1.06919
$829$	−25.7003	−0.892608	−0.446304	−	0.894881i	$-0.647260\pi$
$829$	−25.7003	−0.892608	−0.446304	+	0.894881i	$0.647260\pi$
$830$	−28.3626	−0.984479
$831$	9.69724	0.336394
$832$	−67.9095	−2.35434
$833$	−49.5291	−1.71608
$834$	−9.34944	−0.323744
$835$	−49.1045	−1.69933
$836$	−5.01206	−0.173346
$837$	−17.6025	−0.608431
$838$	47.1025	1.62713
$839$	−21.7069	−0.749404	−0.374702	−	0.927145i	$-0.622255\pi$
$839$	−21.7069	−0.749404	−0.374702	+	0.927145i	$0.622255\pi$
$840$	−0.815905	−0.0281514
$841$	−2.18997	−0.0755162
$842$	−17.7815	−0.612791
$843$	5.27855	0.181803
$844$	2.65181	0.0912791
$845$	−52.7224	−1.81371
$846$	20.5663	0.707084
$847$	−3.42771	−0.117778
$848$	32.0663	1.10116
$849$	−7.11753	−0.244273
$850$	−48.2831	−1.65610
$851$	16.7702	0.574874
$852$	−10.0827	−0.345428
$853$	−33.5891	−1.15007	−0.575034	−	0.818129i	$-0.695011\pi$
$853$	−33.5891	−1.15007	−0.575034	+	0.818129i	$0.695011\pi$
$854$	3.33247	0.114035
$855$	−7.96423	−0.272371
$856$	−19.5143	−0.666984
$857$	−0.143612	−0.00490570	−0.00245285	−	0.999997i	$-0.500781\pi$
$857$	−0.143612	−0.00490570	−0.00245285	+	0.999997i	$0.500781\pi$
$858$	10.1403	0.346184
$859$	27.4273	0.935806	0.467903	−	0.883780i	$-0.345010\pi$
$859$	27.4273	0.935806	0.467903	+	0.883780i	$0.345010\pi$
$860$	−75.4130	−2.57156
$861$	−1.98128	−0.0675219
$862$	−26.0946	−0.888787
$863$	12.7810	0.435071	0.217535	−	0.976052i	$-0.430198\pi$
$863$	12.7810	0.435071	0.217535	+	0.976052i	$0.430198\pi$
$864$	19.8158	0.674147
$865$	−15.9205	−0.541314
$866$	27.6074	0.938136
$867$	−16.0528	−0.545182
$868$	−8.38223	−0.284511
$869$	2.34767	0.0796393
$870$	−14.0448	−0.476164
$871$	35.4540	1.20131
$872$	−23.0752	−0.781425
$873$	1.38302	0.0468081
$874$	8.92414	0.301864
$875$	2.53286	0.0856264
$876$	9.87722	0.333720
$877$	−37.1170	−1.25335	−0.626676	−	0.779280i	$-0.715585\pi$
$877$	−37.1170	−1.25335	−0.626676	+	0.779280i	$0.715585\pi$
$878$	−31.2295	−1.05394
$879$	−11.4166	−0.385072
$880$	12.1945	0.411077
$881$	32.8950	1.10826	0.554131	−	0.832429i	$-0.313051\pi$
$881$	32.8950	1.10826	0.554131	+	0.832429i	$0.313051\pi$
$882$	−41.0451	−1.38206
$883$	23.1751	0.779904	0.389952	−	0.920835i	$-0.372492\pi$
$883$	23.1751	0.779904	0.389952	+	0.920835i	$0.372492\pi$
$884$	108.719	3.65660
$885$	−11.0646	−0.371934
$886$	12.9277	0.434316
$887$	27.8587	0.935405	0.467702	−	0.883886i	$-0.345082\pi$
$887$	27.8587	0.935405	0.467702	+	0.883886i	$0.345082\pi$
$888$	2.52288	0.0846625
$889$	1.24581	0.0417832
$890$	−46.7192	−1.56603
$891$	13.7482	0.460583
$892$	54.6636	1.83027
$893$	−3.40075	−0.113802
$894$	7.27925	0.243455
$895$	−59.7691	−1.99786
$896$	4.91443	0.164180
$897$	−10.2925	−0.343656
$898$	−66.3442	−2.21394
$899$	−35.4662	−1.18287
$900$	−22.8096	−0.760319
$901$	−103.018	−3.43202
$902$	−39.5273	−1.31612
$903$	−2.04581	−0.0680801
$904$	−7.86045	−0.261435
$905$	64.9740	2.15981
$906$	−13.5851	−0.451334
$907$	26.0243	0.864121	0.432061	−	0.901845i	$-0.357787\pi$
$907$	26.0243	0.864121	0.432061	+	0.901845i	$0.357787\pi$
$908$	−19.3056	−0.640679
$909$	14.5826	0.483674
$910$	15.8824	0.526496
$911$	−14.0475	−0.465414	−0.232707	−	0.972547i	$-0.574758\pi$
$911$	−14.0475	−0.465414	−0.232707	+	0.972547i	$0.574758\pi$
$912$	−1.00577	−0.0333045
$913$	−8.75055	−0.289601
$914$	−27.1530	−0.898143
$915$	−4.21077	−0.139204
$916$	19.7563	0.652766
$917$	−2.72517	−0.0899930
$918$	−40.4481	−1.33499
$919$	−37.5968	−1.24021	−0.620103	−	0.784521i	$-0.712909\pi$
$919$	−37.5968	−1.24021	−0.620103	+	0.784521i	$0.712909\pi$
$920$	16.5219	0.544712
$921$	3.91684	0.129064
$922$	29.4594	0.970195
$923$	48.2428	1.58793
$924$	−1.02412	−0.0336911
$925$	12.4333	0.408804
$926$	72.2326	2.37371
$927$	8.10881	0.266328
$928$	39.9258	1.31063
$929$	9.96854	0.327057	0.163529	−	0.986539i	$-0.447712\pi$
$929$	9.96854	0.327057	0.163529	+	0.986539i	$0.447712\pi$
$930$	18.5795	0.609246
$931$	6.78704	0.222436
$932$	74.8774	2.45269
$933$	8.52621	0.279136
$934$	−29.9271	−0.979246
$935$	−39.1767	−1.28121
$936$	22.1454	0.723846
$937$	−57.8200	−1.88890	−0.944448	−	0.328661i	$-0.893403\pi$
$937$	−57.8200	−1.88890	−0.944448	+	0.328661i	$0.893403\pi$
$938$	−6.28124	−0.205090
$939$	−9.21657	−0.300772
$940$	−25.6148	−0.835462
$941$	−24.3716	−0.794492	−0.397246	−	0.917712i	$-0.630034\pi$
$941$	−24.3716	−0.794492	−0.397246	+	0.917712i	$0.630034\pi$
$942$	10.2927	0.335355
$943$	40.1206	1.30651
$944$	19.9847	0.650446
$945$	−3.36846	−0.109576
$946$	−40.8146	−1.32700
$947$	−0.505234	−0.0164179	−0.00820894	−	0.999966i	$-0.502613\pi$
$947$	−0.505234	−0.0164179	−0.00820894	+	0.999966i	$0.502613\pi$
$948$	−1.45844	−0.0473680
$949$	−47.2596	−1.53411
$950$	6.61631	0.214661
$951$	13.7025	0.444334
$952$	−4.73437	−0.153442
$953$	52.5262	1.70149	0.850746	−	0.525577i	$-0.176151\pi$
$953$	52.5262	1.70149	0.850746	+	0.525577i	$0.176151\pi$
$954$	−85.3716	−2.76401
$955$	25.0606	0.810941
$956$	63.1725	2.04314
$957$	−4.33318	−0.140072
$958$	−90.4095	−2.92100
$959$	0.0762568	0.00246246
$960$	−15.2022	−0.490648
$961$	15.9173	0.513461
$962$	−49.1105	−1.58338
$963$	−38.9216	−1.25423
$964$	−33.0016	−1.06291
$965$	34.5598	1.11252
$966$	1.82348	0.0586695
$967$	27.1712	0.873768	0.436884	−	0.899518i	$-0.356082\pi$
$967$	27.1712	0.873768	0.436884	+	0.899518i	$0.356082\pi$
$968$	10.4421	0.335621
$969$	3.23120	0.103801
$970$	−3.02164	−0.0970189
$971$	−25.9178	−0.831743	−0.415871	−	0.909423i	$-0.636523\pi$
$971$	−25.9178	−0.831743	−0.415871	+	0.909423i	$0.636523\pi$
$972$	−28.9850	−0.929696
$973$	−4.51795	−0.144839
$974$	11.9138	0.381742
$975$	−7.63079	−0.244381
$976$	7.60539	0.243443
$977$	16.0938	0.514888	0.257444	−	0.966293i	$-0.417120\pi$
$977$	16.0938	0.514888	0.257444	+	0.966293i	$0.417120\pi$
$978$	−8.03041	−0.256784
$979$	−14.4140	−0.460675
$980$	51.1206	1.63299
$981$	−46.0239	−1.46943
$982$	−15.5094	−0.494924
$983$	−0.582049	−0.0185645	−0.00928225	−	0.999957i	$-0.502955\pi$
$983$	−0.582049	−0.0185645	−0.00928225	+	0.999957i	$0.502955\pi$
$984$	6.03571	0.192411
$985$	21.2913	0.678397
$986$	−81.4966	−2.59538
$987$	−0.694879	−0.0221182
$988$	−14.8979	−0.473964
$989$	41.4272	1.31731
$990$	−32.4660	−1.03184
$991$	0.394625	0.0125357	0.00626783	−	0.999980i	$-0.498005\pi$
$991$	0.394625	0.0125357	0.00626783	+	0.999980i	$0.498005\pi$
$992$	−52.8167	−1.67693
$993$	−6.45708	−0.204909
$994$	−8.54697	−0.271093
$995$	70.2356	2.22662
$996$	5.43610	0.172249
$997$	11.0139	0.348815	0.174408	−	0.984674i	$-0.444199\pi$
$997$	11.0139	0.348815	0.174408	+	0.984674i	$0.444199\pi$
$998$	−74.5437	−2.35964
$999$	10.4157	0.329538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.12	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.12	✓	83	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15681 q^{2} -0.442776 q^{3} +2.65181 q^{4} -2.84036 q^{5} +0.954981 q^{6} +0.461477 q^{7} -1.40583 q^{8} -2.80395 q^{9} +O(q^{10})\)	\(q-2.15681 q^{2} -0.442776 q^{3} +2.65181 q^{4} -2.84036 q^{5} +0.954981 q^{6} +0.461477 q^{7} -1.40583 q^{8} -2.80395 q^{9} +6.12610 q^{10} +1.89005 q^{11} -1.17416 q^{12} +5.61800 q^{13} -0.995317 q^{14} +1.25764 q^{15} -2.27152 q^{16} +7.29760 q^{17} +6.04757 q^{18} -1.00000 q^{19} -7.53209 q^{20} -0.204331 q^{21} -4.07648 q^{22} +4.13767 q^{23} +0.622466 q^{24} +3.06764 q^{25} -12.1169 q^{26} +2.56985 q^{27} +1.22375 q^{28} +5.17784 q^{29} -2.71249 q^{30} -6.84962 q^{31} +7.71089 q^{32} -0.836870 q^{33} -15.7395 q^{34} -1.31076 q^{35} -7.43554 q^{36} +4.05305 q^{37} +2.15681 q^{38} -2.48751 q^{39} +3.99306 q^{40} +9.69644 q^{41} +0.440702 q^{42} +10.0122 q^{43} +5.01206 q^{44} +7.96423 q^{45} -8.92414 q^{46} +3.40075 q^{47} +1.00577 q^{48} -6.78704 q^{49} -6.61631 q^{50} -3.23120 q^{51} +14.8979 q^{52} -14.1167 q^{53} -5.54266 q^{54} -5.36843 q^{55} -0.648758 q^{56} +0.442776 q^{57} -11.1676 q^{58} -8.79793 q^{59} +3.33503 q^{60} -3.34815 q^{61} +14.7733 q^{62} -1.29396 q^{63} -12.0878 q^{64} -15.9571 q^{65} +1.80497 q^{66} +6.31079 q^{67} +19.3518 q^{68} -1.83206 q^{69} +2.82706 q^{70} +8.58718 q^{71} +3.94187 q^{72} -8.41218 q^{73} -8.74163 q^{74} -1.35828 q^{75} -2.65181 q^{76} +0.872217 q^{77} +5.36508 q^{78} +1.24212 q^{79} +6.45194 q^{80} +7.27398 q^{81} -20.9133 q^{82} -4.62979 q^{83} -0.541847 q^{84} -20.7278 q^{85} -21.5944 q^{86} -2.29262 q^{87} -2.65709 q^{88} -7.62626 q^{89} -17.1773 q^{90} +2.59258 q^{91} +10.9723 q^{92} +3.03284 q^{93} -7.33476 q^{94} +2.84036 q^{95} -3.41419 q^{96} -0.493240 q^{97} +14.6383 q^{98} -5.29962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * q^99

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