LMFDB - Embedded newform 4009.2.a.f.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.39927 q^{2} -1.08640 q^{3} +3.75651 q^{4} +4.25844 q^{5} +2.60656 q^{6} +3.20578 q^{7} -4.21434 q^{8} -1.81974 q^{9} +O(q^{10})$	\(q-2.39927 q^{2} -1.08640 q^{3} +3.75651 q^{4} +4.25844 q^{5} +2.60656 q^{6} +3.20578 q^{7} -4.21434 q^{8} -1.81974 q^{9} -10.2171 q^{10} +4.71479 q^{11} -4.08106 q^{12} +5.09146 q^{13} -7.69155 q^{14} -4.62635 q^{15} +2.59834 q^{16} -0.313021 q^{17} +4.36606 q^{18} -1.00000 q^{19} +15.9969 q^{20} -3.48275 q^{21} -11.3121 q^{22} -0.770596 q^{23} +4.57844 q^{24} +13.1343 q^{25} -12.2158 q^{26} +5.23615 q^{27} +12.0426 q^{28} -5.05855 q^{29} +11.0999 q^{30} -10.3620 q^{31} +2.19457 q^{32} -5.12213 q^{33} +0.751023 q^{34} +13.6516 q^{35} -6.83588 q^{36} -0.698955 q^{37} +2.39927 q^{38} -5.53134 q^{39} -17.9465 q^{40} -1.82492 q^{41} +8.35607 q^{42} -2.31219 q^{43} +17.7111 q^{44} -7.74926 q^{45} +1.84887 q^{46} +1.70108 q^{47} -2.82282 q^{48} +3.27705 q^{49} -31.5127 q^{50} +0.340065 q^{51} +19.1261 q^{52} -5.26474 q^{53} -12.5630 q^{54} +20.0776 q^{55} -13.5103 q^{56} +1.08640 q^{57} +12.1369 q^{58} +11.2654 q^{59} -17.3789 q^{60} +3.95087 q^{61} +24.8613 q^{62} -5.83371 q^{63} -10.4620 q^{64} +21.6817 q^{65} +12.2894 q^{66} -0.288949 q^{67} -1.17587 q^{68} +0.837173 q^{69} -32.7540 q^{70} +8.91954 q^{71} +7.66902 q^{72} +13.5273 q^{73} +1.67698 q^{74} -14.2690 q^{75} -3.75651 q^{76} +15.1146 q^{77} +13.2712 q^{78} +12.9045 q^{79} +11.0649 q^{80} -0.229301 q^{81} +4.37848 q^{82} +10.0861 q^{83} -13.0830 q^{84} -1.33298 q^{85} +5.54758 q^{86} +5.49559 q^{87} -19.8697 q^{88} +9.27838 q^{89} +18.5926 q^{90} +16.3221 q^{91} -2.89475 q^{92} +11.2572 q^{93} -4.08136 q^{94} -4.25844 q^{95} -2.38417 q^{96} -6.43312 q^{97} -7.86254 q^{98} -8.57971 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.39927	−1.69654	−0.848271	−	0.529563i	$-0.822356\pi$
$2$	−2.39927	−1.69654	−0.848271	+	0.529563i	$0.822356\pi$
$3$	−1.08640	−0.627231	−0.313615	−	0.949550i	$-0.601540\pi$
$3$	−1.08640	−0.627231	−0.313615	+	0.949550i	$0.601540\pi$
$4$	3.75651	1.87825
$5$	4.25844	1.90443	0.952215	−	0.305427i	$-0.0987993\pi$
$5$	4.25844	1.90443	0.952215	+	0.305427i	$0.0987993\pi$
$6$	2.60656	1.06412
$7$	3.20578	1.21167	0.605836	−	0.795589i	$-0.292839\pi$
$7$	3.20578	1.21167	0.605836	+	0.795589i	$0.292839\pi$
$8$	−4.21434	−1.48999
$9$	−1.81974	−0.606581
$10$	−10.2171	−3.23095
$11$	4.71479	1.42156	0.710781	−	0.703413i	$-0.248342\pi$
$11$	4.71479	1.42156	0.710781	+	0.703413i	$0.248342\pi$
$12$	−4.08106	−1.17810
$13$	5.09146	1.41212	0.706059	−	0.708153i	$-0.250471\pi$
$13$	5.09146	1.41212	0.706059	+	0.708153i	$0.250471\pi$
$14$	−7.69155	−2.05565
$15$	−4.62635	−1.19452
$16$	2.59834	0.649584
$17$	−0.313021	−0.0759188	−0.0379594	−	0.999279i	$-0.512086\pi$
$17$	−0.313021	−0.0759188	−0.0379594	+	0.999279i	$0.512086\pi$
$18$	4.36606	1.02909
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	15.9969	3.57700
$21$	−3.48275	−0.759999
$22$	−11.3121	−2.41174
$23$	−0.770596	−0.160680	−0.0803402	−	0.996768i	$-0.525601\pi$
$23$	−0.770596	−0.160680	−0.0803402	+	0.996768i	$0.525601\pi$
$24$	4.57844	0.934571
$25$	13.1343	2.62686
$26$	−12.2158	−2.39572
$27$	5.23615	1.00770
$28$	12.0426	2.27583
$29$	−5.05855	−0.939350	−0.469675	−	0.882839i	$-0.655629\pi$
$29$	−5.05855	−0.939350	−0.469675	+	0.882839i	$0.655629\pi$
$30$	11.0999	2.02655
$31$	−10.3620	−1.86107	−0.930536	−	0.366200i	$-0.880659\pi$
$31$	−10.3620	−1.86107	−0.930536	+	0.366200i	$0.880659\pi$
$32$	2.19457	0.387948
$33$	−5.12213	−0.891648
$34$	0.751023	0.128799
$35$	13.6516	2.30755
$36$	−6.83588	−1.13931
$37$	−0.698955	−0.114908	−0.0574538	−	0.998348i	$-0.518298\pi$
$37$	−0.698955	−0.114908	−0.0574538	+	0.998348i	$0.518298\pi$
$38$	2.39927	0.389213
$39$	−5.53134	−0.885724
$40$	−17.9465	−2.83759
$41$	−1.82492	−0.285005	−0.142502	−	0.989794i	$-0.545515\pi$
$41$	−1.82492	−0.285005	−0.142502	+	0.989794i	$0.545515\pi$
$42$	8.35607	1.28937
$43$	−2.31219	−0.352606	−0.176303	−	0.984336i	$-0.556414\pi$
$43$	−2.31219	−0.352606	−0.176303	+	0.984336i	$0.556414\pi$
$44$	17.7111	2.67005
$45$	−7.74926	−1.15519
$46$	1.84887	0.272601
$47$	1.70108	0.248128	0.124064	−	0.992274i	$-0.460407\pi$
$47$	1.70108	0.248128	0.124064	+	0.992274i	$0.460407\pi$
$48$	−2.82282	−0.407439
$49$	3.27705	0.468150
$50$	−31.5127	−4.45657
$51$	0.340065	0.0476186
$52$	19.1261	2.65231
$53$	−5.26474	−0.723168	−0.361584	−	0.932339i	$-0.617764\pi$
$53$	−5.26474	−0.723168	−0.361584	+	0.932339i	$0.617764\pi$
$54$	−12.5630	−1.70960
$55$	20.0776	2.70727
$56$	−13.5103	−1.80539
$57$	1.08640	0.143897
$58$	12.1369	1.59365
$59$	11.2654	1.46663	0.733313	−	0.679891i	$-0.237973\pi$
$59$	11.2654	1.46663	0.733313	+	0.679891i	$0.237973\pi$
$60$	−17.3789	−2.24361
$61$	3.95087	0.505857	0.252929	−	0.967485i	$-0.418606\pi$
$61$	3.95087	0.505857	0.252929	+	0.967485i	$0.418606\pi$
$62$	24.8613	3.15739
$63$	−5.83371	−0.734978
$64$	−10.4620	−1.30775
$65$	21.6817	2.68928
$66$	12.2894	1.51272
$67$	−0.288949	−0.0353007	−0.0176504	−	0.999844i	$-0.505619\pi$
$67$	−0.288949	−0.0353007	−0.0176504	+	0.999844i	$0.505619\pi$
$68$	−1.17587	−0.142595
$69$	0.837173	0.100784
$70$	−32.7540	−3.91485
$71$	8.91954	1.05855	0.529277	−	0.848449i	$-0.322463\pi$
$71$	8.91954	1.05855	0.529277	+	0.848449i	$0.322463\pi$
$72$	7.66902	0.903803
$73$	13.5273	1.58325	0.791625	−	0.611008i	$-0.209236\pi$
$73$	13.5273	1.58325	0.791625	+	0.611008i	$0.209236\pi$
$74$	1.67698	0.194945
$75$	−14.2690	−1.64765
$76$	−3.75651	−0.430901
$77$	15.1146	1.72247
$78$	13.2712	1.50267
$79$	12.9045	1.45186	0.725932	−	0.687766i	$-0.241409\pi$
$79$	12.9045	1.45186	0.725932	+	0.687766i	$0.241409\pi$
$80$	11.0649	1.23709
$81$	−0.229301	−0.0254779
$82$	4.37848	0.483523
$83$	10.0861	1.10709	0.553544	−	0.832820i	$-0.313275\pi$
$83$	10.0861	1.10709	0.553544	+	0.832820i	$0.313275\pi$
$84$	−13.0830	−1.42747
$85$	−1.33298	−0.144582
$86$	5.54758	0.598211
$87$	5.49559	0.589189
$88$	−19.8697	−2.11812
$89$	9.27838	0.983507	0.491753	−	0.870735i	$-0.336356\pi$
$89$	9.27838	0.983507	0.491753	+	0.870735i	$0.336356\pi$
$90$	18.5926	1.95983
$91$	16.3221	1.71102
$92$	−2.89475	−0.301799
$93$	11.2572	1.16732
$94$	−4.08136	−0.420960
$95$	−4.25844	−0.436906
$96$	−2.38417	−0.243333
$97$	−6.43312	−0.653184	−0.326592	−	0.945165i	$-0.605900\pi$
$97$	−6.43312	−0.653184	−0.326592	+	0.945165i	$0.605900\pi$
$98$	−7.86254	−0.794237
$99$	−8.57971	−0.862293
$100$	49.3390	4.93390
$101$	−17.0310	−1.69465	−0.847324	−	0.531076i	$-0.821788\pi$
$101$	−17.0310	−1.69465	−0.847324	+	0.531076i	$0.821788\pi$
$102$	−0.815909	−0.0807870
$103$	5.80604	0.572086	0.286043	−	0.958217i	$-0.407660\pi$
$103$	5.80604	0.572086	0.286043	+	0.958217i	$0.407660\pi$
$104$	−21.4572	−2.10405
$105$	−14.8311	−1.44736
$106$	12.6316	1.22689
$107$	−11.3930	−1.10140	−0.550701	−	0.834703i	$-0.685639\pi$
$107$	−11.3930	−1.10140	−0.550701	+	0.834703i	$0.685639\pi$
$108$	19.6696	1.89271
$109$	−10.5083	−1.00651	−0.503255	−	0.864138i	$-0.667864\pi$
$109$	−10.5083	−1.00651	−0.503255	+	0.864138i	$0.667864\pi$
$110$	−48.1717	−4.59299
$111$	0.759342	0.0720736
$112$	8.32971	0.787083
$113$	3.44879	0.324435	0.162218	−	0.986755i	$-0.448135\pi$
$113$	3.44879	0.324435	0.162218	+	0.986755i	$0.448135\pi$
$114$	−2.60656	−0.244127
$115$	−3.28154	−0.306005
$116$	−19.0025	−1.76434
$117$	−9.26515	−0.856564
$118$	−27.0287	−2.48819
$119$	−1.00348	−0.0919887
$120$	19.4970	1.77983
$121$	11.2292	1.02084
$122$	−9.47921	−0.858208
$123$	1.98259	0.178764
$124$	−38.9250	−3.49557
$125$	34.6393	3.09824
$126$	13.9966	1.24692
$127$	−18.6765	−1.65727	−0.828636	−	0.559788i	$-0.810883\pi$
$127$	−18.6765	−1.65727	−0.828636	+	0.559788i	$0.810883\pi$
$128$	20.7121	1.83071
$129$	2.51196	0.221166
$130$	−52.0202	−4.56248
$131$	9.23273	0.806667	0.403334	−	0.915053i	$-0.367851\pi$
$131$	9.23273	0.806667	0.403334	+	0.915053i	$0.367851\pi$
$132$	−19.2413	−1.67474
$133$	−3.20578	−0.277977
$134$	0.693267	0.0598891
$135$	22.2978	1.91909
$136$	1.31918	0.113119
$137$	6.03340	0.515468	0.257734	−	0.966216i	$-0.417024\pi$
$137$	6.03340	0.515468	0.257734	+	0.966216i	$0.417024\pi$
$138$	−2.00861	−0.170984
$139$	−0.0415311	−0.00352262	−0.00176131	−	0.999998i	$-0.500561\pi$
$139$	−0.0415311	−0.00352262	−0.00176131	+	0.999998i	$0.500561\pi$
$140$	51.2825	4.33416
$141$	−1.84805	−0.155634
$142$	−21.4004	−1.79588
$143$	24.0052	2.00741
$144$	−4.72831	−0.394026
$145$	−21.5415	−1.78893
$146$	−32.4556	−2.68605
$147$	−3.56018	−0.293638
$148$	−2.62563	−0.215826
$149$	9.87166	0.808718	0.404359	−	0.914600i	$-0.367495\pi$
$149$	9.87166	0.808718	0.404359	+	0.914600i	$0.367495\pi$
$150$	34.2353	2.79530
$151$	10.0103	0.814626	0.407313	−	0.913289i	$-0.366466\pi$
$151$	10.0103	0.814626	0.407313	+	0.913289i	$0.366466\pi$
$152$	4.21434	0.341828
$153$	0.569619	0.0460509
$154$	−36.2640	−2.92224
$155$	−44.1260	−3.54428
$156$	−20.7785	−1.66361
$157$	11.7289	0.936070	0.468035	−	0.883710i	$-0.344962\pi$
$157$	11.7289	0.936070	0.468035	+	0.883710i	$0.344962\pi$
$158$	−30.9613	−2.46315
$159$	5.71960	0.453594
$160$	9.34542	0.738820
$161$	−2.47037	−0.194692
$162$	0.550155	0.0432243
$163$	−18.8247	−1.47447	−0.737233	−	0.675639i	$-0.763868\pi$
$163$	−18.8247	−1.47447	−0.737233	+	0.675639i	$0.763868\pi$
$164$	−6.85533	−0.535312
$165$	−21.8123	−1.69808
$166$	−24.1992	−1.87822
$167$	0.0415605	0.00321605	0.00160802	−	0.999999i	$-0.499488\pi$
$167$	0.0415605	0.00321605	0.00160802	+	0.999999i	$0.499488\pi$
$168$	14.6775	1.13239
$169$	12.9230	0.994075
$170$	3.19819	0.245290
$171$	1.81974	0.139159
$172$	−8.68578	−0.662284
$173$	−24.9165	−1.89436	−0.947182	−	0.320697i	$-0.896083\pi$
$173$	−24.9165	−1.89436	−0.947182	+	0.320697i	$0.896083\pi$
$174$	−13.1854	−0.999585
$175$	42.1057	3.18289
$176$	12.2506	0.923424
$177$	−12.2386	−0.919913
$178$	−22.2614	−1.66856
$179$	25.3535	1.89501	0.947506	−	0.319739i	$-0.103595\pi$
$179$	25.3535	1.89501	0.947506	+	0.319739i	$0.103595\pi$
$180$	−29.1102	−2.16974
$181$	17.3338	1.28841	0.644206	−	0.764852i	$-0.277188\pi$
$181$	17.3338	1.28841	0.644206	+	0.764852i	$0.277188\pi$
$182$	−39.1612	−2.90282
$183$	−4.29221	−0.317289
$184$	3.24756	0.239413
$185$	−2.97646	−0.218833
$186$	−27.0092	−1.98041
$187$	−1.47583	−0.107923
$188$	6.39013	0.466048
$189$	16.7860	1.22100
$190$	10.2171	0.741230
$191$	−15.3284	−1.10912	−0.554562	−	0.832142i	$-0.687114\pi$
$191$	−15.3284	−1.10912	−0.554562	+	0.832142i	$0.687114\pi$
$192$	11.3659	0.820264
$193$	−9.71278	−0.699141	−0.349571	−	0.936910i	$-0.613673\pi$
$193$	−9.71278	−0.699141	−0.349571	+	0.936910i	$0.613673\pi$
$194$	15.4348	1.10815
$195$	−23.5549	−1.68680
$196$	12.3103	0.879305
$197$	−2.17581	−0.155020	−0.0775099	−	0.996992i	$-0.524697\pi$
$197$	−2.17581	−0.155020	−0.0775099	+	0.996992i	$0.524697\pi$
$198$	20.5851	1.46292
$199$	−13.0302	−0.923689	−0.461845	−	0.886961i	$-0.652812\pi$
$199$	−13.0302	−0.923689	−0.461845	+	0.886961i	$0.652812\pi$
$200$	−55.3524	−3.91400
$201$	0.313913	0.0221417
$202$	40.8620	2.87504
$203$	−16.2166	−1.13818
$204$	1.27746	0.0894399
$205$	−7.77131	−0.542772
$206$	−13.9303	−0.970568
$207$	1.40229	0.0974657
$208$	13.2293	0.917289
$209$	−4.71479	−0.326129
$210$	35.5838	2.45551
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−19.7771	−1.35829
$213$	−9.69015	−0.663958
$214$	27.3349	1.86857
$215$	−9.84633	−0.671514
$216$	−22.0669	−1.50146
$217$	−33.2184	−2.25501
$218$	25.2122	1.70759
$219$	−14.6960	−0.993063
$220$	75.4218	5.08493
$221$	−1.59374	−0.107206
$222$	−1.82187	−0.122276
$223$	4.06963	0.272523	0.136261	−	0.990673i	$-0.456491\pi$
$223$	4.06963	0.272523	0.136261	+	0.990673i	$0.456491\pi$
$224$	7.03530	0.470066
$225$	−23.9010	−1.59340
$226$	−8.27459	−0.550418
$227$	7.63604	0.506822	0.253411	−	0.967359i	$-0.418448\pi$
$227$	7.63604	0.506822	0.253411	+	0.967359i	$0.418448\pi$
$228$	4.08106	0.270274
$229$	2.86130	0.189080	0.0945399	−	0.995521i	$-0.469862\pi$
$229$	2.86130	0.189080	0.0945399	+	0.995521i	$0.469862\pi$
$230$	7.87330	0.519150
$231$	−16.4204	−1.08039
$232$	21.3185	1.39963
$233$	−4.27774	−0.280244	−0.140122	−	0.990134i	$-0.544749\pi$
$233$	−4.27774	−0.280244	−0.140122	+	0.990134i	$0.544749\pi$
$234$	22.2296	1.45320
$235$	7.24395	0.472543
$236$	42.3184	2.75470
$237$	−14.0193	−0.910654
$238$	2.40762	0.156063
$239$	16.7774	1.08524	0.542619	−	0.839979i	$-0.317433\pi$
$239$	16.7774	1.08524	0.542619	+	0.839979i	$0.317433\pi$
$240$	−12.0208	−0.775940
$241$	−20.7945	−1.33949	−0.669747	−	0.742590i	$-0.733597\pi$
$241$	−20.7945	−1.33949	−0.669747	+	0.742590i	$0.733597\pi$
$242$	−26.9420	−1.73190
$243$	−15.4593	−0.991717
$244$	14.8415	0.950128
$245$	13.9551	0.891560
$246$	−4.75677	−0.303280
$247$	−5.09146	−0.323962
$248$	43.6691	2.77299
$249$	−10.9575	−0.694400
$250$	−83.1092	−5.25629
$251$	−15.9677	−1.00787	−0.503935	−	0.863742i	$-0.668115\pi$
$251$	−15.9677	−1.00787	−0.503935	+	0.863742i	$0.668115\pi$
$252$	−21.9144	−1.38048
$253$	−3.63320	−0.228417
$254$	44.8100	2.81163
$255$	1.44815	0.0906864
$256$	−28.7700	−1.79812
$257$	4.22195	0.263358	0.131679	−	0.991292i	$-0.457963\pi$
$257$	4.22195	0.263358	0.131679	+	0.991292i	$0.457963\pi$
$258$	−6.02687	−0.375217
$259$	−2.24070	−0.139230
$260$	81.4473	5.05115
$261$	9.20527	0.569792
$262$	−22.1518	−1.36854
$263$	−14.6328	−0.902297	−0.451149	−	0.892449i	$-0.648986\pi$
$263$	−14.6328	−0.902297	−0.451149	+	0.892449i	$0.648986\pi$
$264$	21.5864	1.32855
$265$	−22.4196	−1.37722
$266$	7.69155	0.471599
$267$	−10.0800	−0.616886
$268$	−1.08544	−0.0663037
$269$	19.3813	1.18170	0.590848	−	0.806783i	$-0.298793\pi$
$269$	19.3813	1.18170	0.590848	+	0.806783i	$0.298793\pi$
$270$	−53.4985	−3.25582
$271$	−4.69035	−0.284918	−0.142459	−	0.989801i	$-0.545501\pi$
$271$	−4.69035	−0.284918	−0.142459	+	0.989801i	$0.545501\pi$
$272$	−0.813335	−0.0493156
$273$	−17.7323	−1.07321
$274$	−14.4758	−0.874513
$275$	61.9254	3.73424
$276$	3.14485	0.189297
$277$	1.07602	0.0646520	0.0323260	−	0.999477i	$-0.489709\pi$
$277$	1.07602	0.0646520	0.0323260	+	0.999477i	$0.489709\pi$
$278$	0.0996445	0.00597628
$279$	18.8562	1.12889
$280$	−57.5326	−3.43823
$281$	13.6521	0.814419	0.407209	−	0.913335i	$-0.366502\pi$
$281$	13.6521	0.814419	0.407209	+	0.913335i	$0.366502\pi$
$282$	4.43397	0.264039
$283$	12.9578	0.770261	0.385130	−	0.922862i	$-0.374157\pi$
$283$	12.9578	0.770261	0.385130	+	0.922862i	$0.374157\pi$
$284$	33.5063	1.98823
$285$	4.62635	0.274041
$286$	−57.5949	−3.40566
$287$	−5.85031	−0.345333
$288$	−3.99355	−0.235322
$289$	−16.9020	−0.994236
$290$	51.6840	3.03499
$291$	6.98891	0.409697
$292$	50.8154	2.97374
$293$	−4.13612	−0.241634	−0.120817	−	0.992675i	$-0.538551\pi$
$293$	−4.13612	−0.241634	−0.120817	+	0.992675i	$0.538551\pi$
$294$	8.54183	0.498170
$295$	47.9728	2.79309
$296$	2.94564	0.171212
$297$	24.6873	1.43250
$298$	−23.6848	−1.37202
$299$	−3.92346	−0.226900
$300$	−53.6017	−3.09470
$301$	−7.41240	−0.427243
$302$	−24.0174	−1.38205
$303$	18.5024	1.06294
$304$	−2.59834	−0.149025
$305$	16.8245	0.963370
$306$	−1.36667	−0.0781273
$307$	20.7721	1.18553	0.592763	−	0.805377i	$-0.298037\pi$
$307$	20.7721	1.18553	0.592763	+	0.805377i	$0.298037\pi$
$308$	56.7781	3.23523
$309$	−6.30766	−0.358830
$310$	105.870	6.01303
$311$	21.4002	1.21349	0.606747	−	0.794895i	$-0.292474\pi$
$311$	21.4002	1.21349	0.606747	+	0.794895i	$0.292474\pi$
$312$	23.3110	1.31972
$313$	−32.7548	−1.85141	−0.925707	−	0.378242i	$-0.876529\pi$
$313$	−32.7548	−1.85141	−0.925707	+	0.378242i	$0.876529\pi$
$314$	−28.1409	−1.58808
$315$	−24.8425	−1.39971
$316$	48.4757	2.72697
$317$	−2.42747	−0.136340	−0.0681700	−	0.997674i	$-0.521716\pi$
$317$	−2.42747	−0.136340	−0.0681700	+	0.997674i	$0.521716\pi$
$318$	−13.7229	−0.769540
$319$	−23.8500	−1.33534
$320$	−44.5519	−2.49053
$321$	12.3773	0.690833
$322$	5.92708	0.330303
$323$	0.313021	0.0174170
$324$	−0.861371	−0.0478539
$325$	66.8727	3.70943
$326$	45.1656	2.50149
$327$	11.4161	0.631314
$328$	7.69084	0.424656
$329$	5.45330	0.300650
$330$	52.3335	2.88087
$331$	−22.8209	−1.25435	−0.627176	−	0.778878i	$-0.715789\pi$
$331$	−22.8209	−1.25435	−0.627176	+	0.778878i	$0.715789\pi$
$332$	37.8884	2.07939
$333$	1.27192	0.0697008
$334$	−0.0997149	−0.00545616
$335$	−1.23047	−0.0672278
$336$	−9.04936	−0.493683
$337$	−25.3703	−1.38201	−0.691004	−	0.722851i	$-0.742831\pi$
$337$	−25.3703	−1.38201	−0.691004	+	0.722851i	$0.742831\pi$
$338$	−31.0057	−1.68649
$339$	−3.74675	−0.203496
$340$	−5.00736	−0.271562
$341$	−48.8547	−2.64563
$342$	−4.36606	−0.236090
$343$	−11.9350	−0.644428
$344$	9.74438	0.525382
$345$	3.56505	0.191936
$346$	59.7814	3.21387
$347$	21.6986	1.16484	0.582421	−	0.812887i	$-0.302106\pi$
$347$	21.6986	1.16484	0.582421	+	0.812887i	$0.302106\pi$
$348$	20.6442	1.10665
$349$	25.0936	1.34323	0.671615	−	0.740901i	$-0.265601\pi$
$349$	25.0936	1.34323	0.671615	+	0.740901i	$0.265601\pi$
$350$	−101.023	−5.39991
$351$	26.6597	1.42299
$352$	10.3469	0.551492
$353$	−15.2110	−0.809601	−0.404801	−	0.914405i	$-0.632659\pi$
$353$	−15.2110	−0.809601	−0.404801	+	0.914405i	$0.632659\pi$
$354$	29.3638	1.56067
$355$	37.9833	2.01594
$356$	34.8543	1.84728
$357$	1.09018	0.0576982
$358$	−60.8300	−3.21497
$359$	3.65211	0.192751	0.0963755	−	0.995345i	$-0.469275\pi$
$359$	3.65211	0.192751	0.0963755	+	0.995345i	$0.469275\pi$
$360$	32.6580	1.72123
$361$	1.00000	0.0526316
$362$	−41.5885	−2.18584
$363$	−12.1994	−0.640302
$364$	61.3142	3.21374
$365$	57.6051	3.01519
$366$	10.2982	0.538294
$367$	−22.5543	−1.17733	−0.588663	−	0.808379i	$-0.700345\pi$
$367$	−22.5543	−1.17733	−0.588663	+	0.808379i	$0.700345\pi$
$368$	−2.00227	−0.104375
$369$	3.32089	0.172879
$370$	7.14133	0.371260
$371$	−16.8776	−0.876243
$372$	42.2879	2.19253
$373$	−29.4795	−1.52639	−0.763194	−	0.646169i	$-0.776370\pi$
$373$	−29.4795	−1.52639	−0.763194	+	0.646169i	$0.776370\pi$
$374$	3.54092	0.183096
$375$	−37.6320	−1.94331
$376$	−7.16894	−0.369710
$377$	−25.7554	−1.32647
$378$	−40.2741	−2.07148
$379$	18.8950	0.970572	0.485286	−	0.874355i	$-0.338715\pi$
$379$	18.8950	0.970572	0.485286	+	0.874355i	$0.338715\pi$
$380$	−15.9969	−0.820621
$381$	20.2901	1.03949
$382$	36.7770	1.88168
$383$	30.5263	1.55982	0.779910	−	0.625892i	$-0.215265\pi$
$383$	30.5263	1.55982	0.779910	+	0.625892i	$0.215265\pi$
$384$	−22.5016	−1.14828
$385$	64.3645	3.28032
$386$	23.3036	1.18612
$387$	4.20760	0.213884
$388$	−24.1661	−1.22685
$389$	−8.67507	−0.439844	−0.219922	−	0.975518i	$-0.570580\pi$
$389$	−8.67507	−0.439844	−0.219922	+	0.975518i	$0.570580\pi$
$390$	56.5146	2.86173
$391$	0.241213	0.0121987
$392$	−13.8106	−0.697542
$393$	−10.0304	−0.505967
$394$	5.22035	0.262997
$395$	54.9528	2.76497
$396$	−32.2297	−1.61961
$397$	33.7169	1.69220	0.846100	−	0.533024i	$-0.178944\pi$
$397$	33.7169	1.69220	0.846100	+	0.533024i	$0.178944\pi$
$398$	31.2631	1.56708
$399$	3.48275	0.174356
$400$	34.1273	1.70636
$401$	2.53300	0.126492	0.0632459	−	0.997998i	$-0.479855\pi$
$401$	2.53300	0.126492	0.0632459	+	0.997998i	$0.479855\pi$
$402$	−0.753162	−0.0375643
$403$	−52.7578	−2.62805
$404$	−63.9771	−3.18298
$405$	−0.976463	−0.0485209
$406$	38.9081	1.93098
$407$	−3.29543	−0.163348
$408$	−1.43315	−0.0709515
$409$	−16.4163	−0.811732	−0.405866	−	0.913933i	$-0.633030\pi$
$409$	−16.4163	−0.811732	−0.405866	+	0.913933i	$0.633030\pi$
$410$	18.6455	0.920836
$411$	−6.55466	−0.323317
$412$	21.8104	1.07452
$413$	36.1143	1.77707
$414$	−3.36447	−0.165355
$415$	42.9508	2.10837
$416$	11.1735	0.547828
$417$	0.0451192	0.00220950
$418$	11.3121	0.553291
$419$	38.3414	1.87310	0.936550	−	0.350533i	$-0.114000\pi$
$419$	38.3414	1.87310	0.936550	+	0.350533i	$0.114000\pi$
$420$	−55.7131	−2.71852
$421$	13.4500	0.655513	0.327756	−	0.944762i	$-0.393707\pi$
$421$	13.4500	0.655513	0.327756	+	0.944762i	$0.393707\pi$
$422$	−2.39927	−0.116795
$423$	−3.09553	−0.150510
$424$	22.1874	1.07752
$425$	−4.11131	−0.199428
$426$	23.2493	1.12643
$427$	12.6656	0.612933
$428$	−42.7978	−2.06871
$429$	−26.0791	−1.25911
$430$	23.6240	1.13925
$431$	13.8413	0.666711	0.333355	−	0.942801i	$-0.391819\pi$
$431$	13.8413	0.666711	0.333355	+	0.942801i	$0.391819\pi$
$432$	13.6053	0.654584
$433$	−21.8016	−1.04772	−0.523860	−	0.851804i	$-0.675508\pi$
$433$	−21.8016	−1.04772	−0.523860	+	0.851804i	$0.675508\pi$
$434$	79.6999	3.82572
$435$	23.4026	1.12207
$436$	−39.4744	−1.89048
$437$	0.770596	0.0368626
$438$	35.2597	1.68477
$439$	−14.3529	−0.685029	−0.342514	−	0.939513i	$-0.611279\pi$
$439$	−14.3529	−0.685029	−0.342514	+	0.939513i	$0.611279\pi$
$440$	−84.6140	−4.03381
$441$	−5.96340	−0.283971
$442$	3.82381	0.181880
$443$	−16.7598	−0.796280	−0.398140	−	0.917325i	$-0.630344\pi$
$443$	−16.7598	−0.796280	−0.398140	+	0.917325i	$0.630344\pi$
$444$	2.85248	0.135372
$445$	39.5114	1.87302
$446$	−9.76415	−0.462346
$447$	−10.7245	−0.507253
$448$	−33.5390	−1.58457
$449$	−37.9774	−1.79227	−0.896133	−	0.443785i	$-0.853635\pi$
$449$	−37.9774	−1.79227	−0.896133	+	0.443785i	$0.853635\pi$
$450$	57.3451	2.70327
$451$	−8.60412	−0.405152
$452$	12.9554	0.609371
$453$	−10.8751	−0.510959
$454$	−18.3209	−0.859844
$455$	69.5067	3.25853
$456$	−4.57844	−0.214405
$457$	4.57113	0.213829	0.106914	−	0.994268i	$-0.465903\pi$
$457$	4.57113	0.213829	0.106914	+	0.994268i	$0.465903\pi$
$458$	−6.86503	−0.320782
$459$	−1.63903	−0.0765032
$460$	−12.3271	−0.574755
$461$	12.0530	0.561362	0.280681	−	0.959801i	$-0.409440\pi$
$461$	12.0530	0.561362	0.280681	+	0.959801i	$0.409440\pi$
$462$	39.3971	1.83292
$463$	1.14113	0.0530328	0.0265164	−	0.999648i	$-0.491559\pi$
$463$	1.14113	0.0530328	0.0265164	+	0.999648i	$0.491559\pi$
$464$	−13.1438	−0.610187
$465$	47.9383	2.22308
$466$	10.2635	0.475445
$467$	38.3044	1.77252	0.886258	−	0.463192i	$-0.153296\pi$
$467$	38.3044	1.77252	0.886258	+	0.463192i	$0.153296\pi$
$468$	−34.8046	−1.60884
$469$	−0.926307	−0.0427729
$470$	−17.3802	−0.801689
$471$	−12.7423	−0.587132
$472$	−47.4761	−2.18526
$473$	−10.9015	−0.501252
$474$	33.6362	1.54496
$475$	−13.1343	−0.602642
$476$	−3.76958	−0.172778
$477$	9.58049	0.438660
$478$	−40.2535	−1.84115
$479$	−9.58931	−0.438147	−0.219073	−	0.975708i	$-0.570303\pi$
$479$	−9.58931	−0.438147	−0.219073	+	0.975708i	$0.570303\pi$
$480$	−10.1528	−0.463411
$481$	−3.55870	−0.162263
$482$	49.8917	2.27251
$483$	2.68379	0.122117
$484$	42.1827	1.91739
$485$	−27.3950	−1.24394
$486$	37.0912	1.68249
$487$	15.2605	0.691521	0.345761	−	0.938323i	$-0.387621\pi$
$487$	15.2605	0.691521	0.345761	+	0.938323i	$0.387621\pi$
$488$	−16.6503	−0.753724
$489$	20.4511	0.924831
$490$	−33.4821	−1.51257
$491$	28.0582	1.26625	0.633124	−	0.774050i	$-0.281772\pi$
$491$	28.0582	1.26625	0.633124	+	0.774050i	$0.281772\pi$
$492$	7.44761	0.335764
$493$	1.58344	0.0713143
$494$	12.2158	0.549615
$495$	−36.5361	−1.64218
$496$	−26.9240	−1.20892
$497$	28.5941	1.28262
$498$	26.2899	1.17808
$499$	−38.2244	−1.71116	−0.855579	−	0.517672i	$-0.826799\pi$
$499$	−38.2244	−1.71116	−0.855579	+	0.517672i	$0.826799\pi$
$500$	130.123	5.81927
$501$	−0.0451511	−0.00201720
$502$	38.3108	1.70989
$503$	−34.1037	−1.52061	−0.760303	−	0.649568i	$-0.774950\pi$
$503$	−34.1037	−1.52061	−0.760303	+	0.649568i	$0.774950\pi$
$504$	24.5852	1.09511
$505$	−72.5254	−3.22734
$506$	8.71703	0.387519
$507$	−14.0395	−0.623515
$508$	−70.1584	−3.11278
$509$	16.4078	0.727264	0.363632	−	0.931543i	$-0.381537\pi$
$509$	16.4078	0.727264	0.363632	+	0.931543i	$0.381537\pi$
$510$	−3.47450	−0.153853
$511$	43.3656	1.91838
$512$	27.6028	1.21988
$513$	−5.23615	−0.231182
$514$	−10.1296	−0.446798
$515$	24.7246	1.08950
$516$	9.43619	0.415405
$517$	8.02024	0.352730
$518$	5.37605	0.236210
$519$	27.0691	1.18820
$520$	−91.3739	−4.00701
$521$	−34.3683	−1.50570	−0.752851	−	0.658191i	$-0.771322\pi$
$521$	−34.3683	−1.50570	−0.752851	+	0.658191i	$0.771322\pi$
$522$	−22.0860	−0.966676
$523$	35.4957	1.55212	0.776059	−	0.630660i	$-0.217216\pi$
$523$	35.4957	1.55212	0.776059	+	0.630660i	$0.217216\pi$
$524$	34.6828	1.51513
$525$	−45.7434	−1.99641
$526$	35.1081	1.53079
$527$	3.24353	0.141290
$528$	−13.3090	−0.579200
$529$	−22.4062	−0.974182
$530$	53.7907	2.33652
$531$	−20.5001	−0.889628
$532$	−12.0426	−0.522111
$533$	−9.29152	−0.402460
$534$	24.1847	1.04657
$535$	−48.5163	−2.09754
$536$	1.21773	0.0525979
$537$	−27.5440	−1.18861
$538$	−46.5009	−2.00480
$539$	15.4506	0.665505
$540$	83.7619	3.60454
$541$	−23.6354	−1.01617	−0.508083	−	0.861308i	$-0.669646\pi$
$541$	−23.6354	−1.01617	−0.508083	+	0.861308i	$0.669646\pi$
$542$	11.2534	0.483376
$543$	−18.8314	−0.808132
$544$	−0.686946	−0.0294526
$545$	−44.7488	−1.91683
$546$	42.5446	1.82074
$547$	−8.03511	−0.343557	−0.171778	−	0.985136i	$-0.554951\pi$
$547$	−8.03511	−0.343557	−0.171778	+	0.985136i	$0.554951\pi$
$548$	22.6645	0.968180
$549$	−7.18957	−0.306843
$550$	−148.576	−6.33529
$551$	5.05855	0.215502
$552$	−3.52813	−0.150167
$553$	41.3689	1.75918
$554$	−2.58168	−0.109685
$555$	3.23361	0.137259
$556$	−0.156012	−0.00661638
$557$	10.6208	0.450020	0.225010	−	0.974356i	$-0.427759\pi$
$557$	10.6208	0.450020	0.225010	+	0.974356i	$0.427759\pi$
$558$	−45.2412	−1.91521
$559$	−11.7724	−0.497922
$560$	35.4715	1.49895
$561$	1.60333	0.0676928
$562$	−32.7552	−1.38170
$563$	39.9422	1.68336	0.841682	−	0.539973i	$-0.181565\pi$
$563$	39.9422	1.68336	0.841682	+	0.539973i	$0.181565\pi$
$564$	−6.94221	−0.292320
$565$	14.6865	0.617864
$566$	−31.0893	−1.30678
$567$	−0.735089	−0.0308708
$568$	−37.5900	−1.57724
$569$	16.2479	0.681148	0.340574	−	0.940218i	$-0.389379\pi$
$569$	16.2479	0.681148	0.340574	+	0.940218i	$0.389379\pi$
$570$	−11.0999	−0.464922
$571$	−15.2785	−0.639387	−0.319694	−	0.947521i	$-0.603580\pi$
$571$	−15.2785	−0.639387	−0.319694	+	0.947521i	$0.603580\pi$
$572$	90.1756	3.77043
$573$	16.6527	0.695677
$574$	14.0365	0.585871
$575$	−10.1212	−0.422084
$576$	19.0382	0.793259
$577$	−12.5119	−0.520879	−0.260440	−	0.965490i	$-0.583867\pi$
$577$	−12.5119	−0.520879	−0.260440	+	0.965490i	$0.583867\pi$
$578$	40.5525	1.68676
$579$	10.5519	0.438523
$580$	−80.9210	−3.36006
$581$	32.3337	1.34143
$582$	−16.7683	−0.695069
$583$	−24.8222	−1.02803
$584$	−57.0086	−2.35903
$585$	−39.4551	−1.63127
$586$	9.92367	0.409943
$587$	−27.7152	−1.14393	−0.571965	−	0.820278i	$-0.693819\pi$
$587$	−27.7152	−1.14393	−0.571965	+	0.820278i	$0.693819\pi$
$588$	−13.3738	−0.551528
$589$	10.3620	0.426959
$590$	−115.100	−4.73859
$591$	2.36379	0.0972332
$592$	−1.81612	−0.0746421
$593$	−10.9711	−0.450531	−0.225265	−	0.974297i	$-0.572325\pi$
$593$	−10.9711	−0.450531	−0.225265	+	0.974297i	$0.572325\pi$
$594$	−59.2316	−2.43030
$595$	−4.27325	−0.175186
$596$	37.0830	1.51898
$597$	14.1560	0.579367
$598$	9.41345	0.384945
$599$	41.8862	1.71142	0.855711	−	0.517453i	$-0.173120\pi$
$599$	41.8862	1.71142	0.855711	+	0.517453i	$0.173120\pi$
$600$	60.1346	2.45498
$601$	−12.2204	−0.498482	−0.249241	−	0.968441i	$-0.580181\pi$
$601$	−12.2204	−0.498482	−0.249241	+	0.968441i	$0.580181\pi$
$602$	17.7844	0.724836
$603$	0.525813	0.0214127
$604$	37.6038	1.53008
$605$	47.8189	1.94412
$606$	−44.3923	−1.80331
$607$	16.9182	0.686687	0.343343	−	0.939210i	$-0.388441\pi$
$607$	16.9182	0.686687	0.343343	+	0.939210i	$0.388441\pi$
$608$	−2.19457	−0.0890014
$609$	17.6177	0.713905
$610$	−40.3666	−1.63440
$611$	8.66099	0.350386
$612$	2.13978	0.0864953
$613$	6.28080	0.253679	0.126839	−	0.991923i	$-0.459517\pi$
$613$	6.28080	0.253679	0.126839	+	0.991923i	$0.459517\pi$
$614$	−49.8379	−2.01129
$615$	8.44272	0.340444
$616$	−63.6980	−2.56647
$617$	10.0863	0.406059	0.203029	−	0.979173i	$-0.434921\pi$
$617$	10.0863	0.406059	0.203029	+	0.979173i	$0.434921\pi$
$618$	15.1338	0.608770
$619$	7.78575	0.312936	0.156468	−	0.987683i	$-0.449989\pi$
$619$	7.78575	0.312936	0.156468	+	0.987683i	$0.449989\pi$
$620$	−165.760	−6.65707
$621$	−4.03496	−0.161917
$622$	−51.3449	−2.05874
$623$	29.7445	1.19169
$624$	−14.3723	−0.575352
$625$	81.8380	3.27352
$626$	78.5878	3.14100
$627$	5.12213	0.204558
$628$	44.0598	1.75818
$629$	0.218788	0.00872364
$630$	59.6038	2.37467
$631$	−35.5388	−1.41478	−0.707389	−	0.706824i	$-0.750127\pi$
$631$	−35.5388	−1.41478	−0.707389	+	0.706824i	$0.750127\pi$
$632$	−54.3838	−2.16327
$633$	−1.08640	−0.0431804
$634$	5.82415	0.231307
$635$	−79.5327	−3.15616
$636$	21.4857	0.851964
$637$	16.6850	0.661083
$638$	57.2227	2.26547
$639$	−16.2313	−0.642099
$640$	88.2013	3.48646
$641$	5.44677	0.215135	0.107567	−	0.994198i	$-0.465694\pi$
$641$	5.44677	0.215135	0.107567	+	0.994198i	$0.465694\pi$
$642$	−29.6965	−1.17203
$643$	−23.4449	−0.924578	−0.462289	−	0.886729i	$-0.652972\pi$
$643$	−23.4449	−0.924578	−0.462289	+	0.886729i	$0.652972\pi$
$644$	−9.27995	−0.365681
$645$	10.6970	0.421195
$646$	−0.751023	−0.0295486
$647$	−13.6258	−0.535687	−0.267843	−	0.963462i	$-0.586311\pi$
$647$	−13.6258	−0.535687	−0.267843	+	0.963462i	$0.586311\pi$
$648$	0.966352	0.0379619
$649$	53.1138	2.08490
$650$	−160.446	−6.29320
$651$	36.0883	1.41441
$652$	−70.7152	−2.76942
$653$	−25.0963	−0.982093	−0.491047	−	0.871133i	$-0.663386\pi$
$653$	−25.0963	−0.982093	−0.491047	+	0.871133i	$0.663386\pi$
$654$	−27.3904	−1.07105
$655$	39.3170	1.53624
$656$	−4.74176	−0.185135
$657$	−24.6162	−0.960369
$658$	−13.0840	−0.510066
$659$	29.9565	1.16694	0.583470	−	0.812135i	$-0.301695\pi$
$659$	29.9565	1.16694	0.583470	+	0.812135i	$0.301695\pi$
$660$	−81.9379	−3.18943
$661$	18.7562	0.729532	0.364766	−	0.931099i	$-0.381149\pi$
$661$	18.7562	0.729532	0.364766	+	0.931099i	$0.381149\pi$
$662$	54.7536	2.12806
$663$	1.73143	0.0672431
$664$	−42.5061	−1.64956
$665$	−13.6516	−0.529387
$666$	−3.05168	−0.118250
$667$	3.89810	0.150935
$668$	0.156122	0.00604055
$669$	−4.42123	−0.170935
$670$	2.95223	0.114055
$671$	18.6275	0.719107
$672$	−7.64313	−0.294840
$673$	9.19819	0.354564	0.177282	−	0.984160i	$-0.443269\pi$
$673$	9.19819	0.354564	0.177282	+	0.984160i	$0.443269\pi$
$674$	60.8703	2.34464
$675$	68.7731	2.64708
$676$	48.5453	1.86713
$677$	21.0363	0.808491	0.404245	−	0.914651i	$-0.367534\pi$
$677$	21.0363	0.808491	0.404245	+	0.914651i	$0.367534\pi$
$678$	8.98948	0.345239
$679$	−20.6232	−0.791445
$680$	5.61764	0.215427
$681$	−8.29576	−0.317894
$682$	117.216	4.48842
$683$	4.71849	0.180548	0.0902740	−	0.995917i	$-0.471226\pi$
$683$	4.71849	0.180548	0.0902740	+	0.995917i	$0.471226\pi$
$684$	6.83588	0.261376
$685$	25.6928	0.981673
$686$	28.6352	1.09330
$687$	−3.10850	−0.118597
$688$	−6.00786	−0.229047
$689$	−26.8052	−1.02120
$690$	−8.55352	−0.325627
$691$	−48.0366	−1.82740	−0.913698	−	0.406393i	$-0.866786\pi$
$691$	−48.0366	−1.82740	−0.913698	+	0.406393i	$0.866786\pi$
$692$	−93.5989	−3.55810
$693$	−27.5047	−1.04482
$694$	−52.0609	−1.97620
$695$	−0.176858	−0.00670859
$696$	−23.1603	−0.877889
$697$	0.571239	0.0216372
$698$	−60.2064	−2.27884
$699$	4.64732	0.175778
$700$	158.170	5.97828
$701$	−34.3427	−1.29711	−0.648553	−	0.761169i	$-0.724626\pi$
$701$	−34.3427	−1.29711	−0.648553	+	0.761169i	$0.724626\pi$
$702$	−63.9638	−2.41416
$703$	0.698955	0.0263616
$704$	−49.3263	−1.85905
$705$	−7.86980	−0.296394
$706$	36.4954	1.37352
$707$	−54.5977	−2.05336
$708$	−45.9746	−1.72783
$709$	23.9834	0.900717	0.450358	−	0.892848i	$-0.351296\pi$
$709$	23.9834	0.900717	0.450358	+	0.892848i	$0.351296\pi$
$710$	−91.1323	−3.42013
$711$	−23.4828	−0.880674
$712$	−39.1023	−1.46542
$713$	7.98493	0.299038
$714$	−2.61563	−0.0978874
$715$	102.224	3.82298
$716$	95.2407	3.55931
$717$	−18.2269	−0.680695
$718$	−8.76241	−0.327010
$719$	−2.81257	−0.104891	−0.0524457	−	0.998624i	$-0.516702\pi$
$719$	−2.81257	−0.104891	−0.0524457	+	0.998624i	$0.516702\pi$
$720$	−20.1352	−0.750394
$721$	18.6129	0.693181
$722$	−2.39927	−0.0892917
$723$	22.5911	0.840172
$724$	65.1146	2.41996
$725$	−66.4405	−2.46754
$726$	29.2696	1.08630
$727$	−7.75996	−0.287801	−0.143900	−	0.989592i	$-0.545965\pi$
$727$	−7.75996	−0.287801	−0.143900	+	0.989592i	$0.545965\pi$
$728$	−68.7870	−2.54942
$729$	17.4829	0.647514
$730$	−138.210	−5.11539
$731$	0.723766	0.0267695
$732$	−16.1237	−0.595950
$733$	−48.6463	−1.79679	−0.898397	−	0.439184i	$-0.855267\pi$
$733$	−48.6463	−1.79679	−0.898397	+	0.439184i	$0.855267\pi$
$734$	54.1140	1.99738
$735$	−15.1608	−0.559214
$736$	−1.69112	−0.0623357
$737$	−1.36233	−0.0501821
$738$	−7.96772	−0.293296
$739$	0.252299	0.00928096	0.00464048	−	0.999989i	$-0.498523\pi$
$739$	0.252299	0.00928096	0.00464048	+	0.999989i	$0.498523\pi$
$740$	−11.1811	−0.411025
$741$	5.53134	0.203199
$742$	40.4940	1.48658
$743$	−46.0228	−1.68841	−0.844206	−	0.536019i	$-0.819928\pi$
$743$	−46.0228	−1.68841	−0.844206	+	0.536019i	$0.819928\pi$
$744$	−47.4419	−1.73930
$745$	42.0379	1.54015
$746$	70.7292	2.58958
$747$	−18.3540	−0.671539
$748$	−5.54396	−0.202707
$749$	−36.5235	−1.33454
$750$	90.2895	3.29691
$751$	−13.8166	−0.504174	−0.252087	−	0.967705i	$-0.581117\pi$
$751$	−13.8166	−0.504174	−0.252087	+	0.967705i	$0.581117\pi$
$752$	4.41998	0.161180
$753$	17.3472	0.632167
$754$	61.7943	2.25042
$755$	42.6282	1.55140
$756$	63.0566	2.29335
$757$	42.2652	1.53615	0.768077	−	0.640358i	$-0.221214\pi$
$757$	42.2652	1.53615	0.768077	+	0.640358i	$0.221214\pi$
$758$	−45.3343	−1.64662
$759$	3.94709	0.143270
$760$	17.9465	0.650988
$761$	−10.0854	−0.365595	−0.182797	−	0.983151i	$-0.558515\pi$
$761$	−10.0854	−0.365595	−0.182797	+	0.983151i	$0.558515\pi$
$762$	−48.6814	−1.76354
$763$	−33.6872	−1.21956
$764$	−57.5813	−2.08322
$765$	2.42568	0.0877008
$766$	−73.2409	−2.64630
$767$	57.3572	2.07105
$768$	31.2556	1.12784
$769$	−21.1450	−0.762508	−0.381254	−	0.924470i	$-0.624508\pi$
$769$	−21.1450	−0.762508	−0.381254	+	0.924470i	$0.624508\pi$
$770$	−154.428	−5.56520
$771$	−4.58671	−0.165186
$772$	−36.4861	−1.31317
$773$	−37.3448	−1.34320	−0.671599	−	0.740915i	$-0.734392\pi$
$773$	−37.3448	−1.34320	−0.671599	+	0.740915i	$0.734392\pi$
$774$	−10.0952	−0.362864
$775$	−136.098	−4.88877
$776$	27.1113	0.973241
$777$	2.43429	0.0873296
$778$	20.8139	0.746213
$779$	1.82492	0.0653846
$780$	−88.4841	−3.16824
$781$	42.0537	1.50480
$782$	−0.578736	−0.0206955
$783$	−26.4874	−0.946581
$784$	8.51488	0.304103
$785$	49.9469	1.78268
$786$	24.0657	0.858393
$787$	20.7529	0.739762	0.369881	−	0.929079i	$-0.379398\pi$
$787$	20.7529	0.739762	0.369881	+	0.929079i	$0.379398\pi$
$788$	−8.17343	−0.291166
$789$	15.8970	0.565949
$790$	−131.847	−4.69090
$791$	11.0561	0.393109
$792$	36.1578	1.28481
$793$	20.1157	0.714329
$794$	−80.8959	−2.87089
$795$	24.3565	0.863838
$796$	−48.9482	−1.73492
$797$	−53.6696	−1.90107	−0.950537	−	0.310610i	$-0.899467\pi$
$797$	−53.6696	−1.90107	−0.950537	+	0.310610i	$0.899467\pi$
$798$	−8.35607	−0.295802
$799$	−0.532475	−0.0188376
$800$	28.8241	1.01908
$801$	−16.8843	−0.596577
$802$	−6.07735	−0.214599
$803$	63.7783	2.25069
$804$	1.17922	0.0415877
$805$	−10.5199	−0.370778
$806$	126.580	4.45860
$807$	−21.0557	−0.741197
$808$	71.7744	2.52502
$809$	16.9619	0.596350	0.298175	−	0.954511i	$-0.403622\pi$
$809$	16.9619	0.596350	0.298175	+	0.954511i	$0.403622\pi$
$810$	2.34280	0.0823177
$811$	−44.6463	−1.56774	−0.783872	−	0.620923i	$-0.786758\pi$
$811$	−44.6463	−1.56774	−0.783872	+	0.620923i	$0.786758\pi$
$812$	−60.9179	−2.13780
$813$	5.09558	0.178710
$814$	7.90662	0.277127
$815$	−80.1639	−2.80802
$816$	0.883603	0.0309323
$817$	2.31219	0.0808934
$818$	39.3871	1.37714
$819$	−29.7021	−1.03787
$820$	−29.1930	−1.01946
$821$	13.2026	0.460772	0.230386	−	0.973099i	$-0.426001\pi$
$821$	13.2026	0.460772	0.230386	+	0.973099i	$0.426001\pi$
$822$	15.7264	0.548522
$823$	36.5295	1.27334	0.636669	−	0.771138i	$-0.280312\pi$
$823$	36.5295	1.27334	0.636669	+	0.771138i	$0.280312\pi$
$824$	−24.4686	−0.852405
$825$	−67.2755	−2.34223
$826$	−86.6481	−3.01487
$827$	−13.8399	−0.481260	−0.240630	−	0.970617i	$-0.577354\pi$
$827$	−13.8399	−0.481260	−0.240630	+	0.970617i	$0.577354\pi$
$828$	5.26770	0.183065
$829$	−14.8449	−0.515584	−0.257792	−	0.966200i	$-0.582995\pi$
$829$	−14.8449	−0.515584	−0.257792	+	0.966200i	$0.582995\pi$
$830$	−103.051	−3.57694
$831$	−1.16899	−0.0405518
$832$	−53.2670	−1.84670
$833$	−1.02579	−0.0355414
$834$	−0.108253	−0.00374851
$835$	0.176983	0.00612474
$836$	−17.7111	−0.612553
$837$	−54.2571	−1.87540
$838$	−91.9915	−3.17779
$839$	−8.92073	−0.307978	−0.153989	−	0.988073i	$-0.549212\pi$
$839$	−8.92073	−0.307978	−0.153989	+	0.988073i	$0.549212\pi$
$840$	62.5032	2.15657
$841$	−3.41102	−0.117621
$842$	−32.2702	−1.11211
$843$	−14.8316	−0.510829
$844$	3.75651	0.129304
$845$	55.0317	1.89315
$846$	7.42703	0.255346
$847$	35.9985	1.23692
$848$	−13.6796	−0.469759
$849$	−14.0773	−0.483131
$850$	9.86415	0.338338
$851$	0.538612	0.0184634
$852$	−36.4011	−1.24708
$853$	27.5726	0.944069	0.472034	−	0.881580i	$-0.343520\pi$
$853$	27.5726	0.944069	0.472034	+	0.881580i	$0.343520\pi$
$854$	−30.3883	−1.03987
$855$	7.74926	0.265019
$856$	48.0139	1.64108
$857$	43.6906	1.49244	0.746220	−	0.665699i	$-0.231866\pi$
$857$	43.6906	1.49244	0.746220	+	0.665699i	$0.231866\pi$
$858$	62.5709	2.13613
$859$	−27.7800	−0.947840	−0.473920	−	0.880568i	$-0.657161\pi$
$859$	−27.7800	−0.947840	−0.473920	+	0.880568i	$0.657161\pi$
$860$	−36.9878	−1.26127
$861$	6.35575	0.216603
$862$	−33.2090	−1.13110
$863$	−4.78957	−0.163039	−0.0815194	−	0.996672i	$-0.525977\pi$
$863$	−4.78957	−0.163039	−0.0815194	+	0.996672i	$0.525977\pi$
$864$	11.4911	0.390934
$865$	−106.105	−3.60768
$866$	52.3081	1.77750
$867$	18.3623	0.623616
$868$	−124.785	−4.23548
$869$	60.8417	2.06391
$870$	−56.1493	−1.90364
$871$	−1.47117	−0.0498487
$872$	44.2854	1.49969
$873$	11.7066	0.396209
$874$	−1.84887	−0.0625390
$875$	111.046	3.75405
$876$	−55.2056	−1.86522
$877$	−31.8983	−1.07713	−0.538564	−	0.842584i	$-0.681033\pi$
$877$	−31.8983	−1.07713	−0.538564	+	0.842584i	$0.681033\pi$
$878$	34.4366	1.16218
$879$	4.49346	0.151561
$880$	52.1684	1.75860
$881$	−37.4699	−1.26239	−0.631196	−	0.775623i	$-0.717436\pi$
$881$	−37.4699	−1.26239	−0.631196	+	0.775623i	$0.717436\pi$
$882$	14.3078	0.481769
$883$	−24.8683	−0.836886	−0.418443	−	0.908243i	$-0.637424\pi$
$883$	−24.8683	−0.836886	−0.418443	+	0.908243i	$0.637424\pi$
$884$	−5.98688	−0.201361
$885$	−52.1175	−1.75191
$886$	40.2112	1.35092
$887$	1.30978	0.0439780	0.0219890	−	0.999758i	$-0.493000\pi$
$887$	1.30978	0.0439780	0.0219890	+	0.999758i	$0.493000\pi$
$888$	−3.20013	−0.107389
$889$	−59.8729	−2.00807
$890$	−94.7986	−3.17766
$891$	−1.08111	−0.0362184
$892$	15.2876	0.511867
$893$	−1.70108	−0.0569245
$894$	25.7311	0.860576
$895$	107.966	3.60892
$896$	66.3986	2.21822
$897$	4.26243	0.142318
$898$	91.1182	3.04066
$899$	52.4168	1.74820
$900$	−89.7844	−2.99281
$901$	1.64798	0.0549021
$902$	20.6436	0.687358
$903$	8.05280	0.267980
$904$	−14.5344	−0.483406
$905$	73.8149	2.45369
$906$	26.0924	0.866863
$907$	12.6054	0.418556	0.209278	−	0.977856i	$-0.432889\pi$
$907$	12.6054	0.418556	0.209278	+	0.977856i	$0.432889\pi$
$908$	28.6848	0.951940
$909$	30.9921	1.02794
$910$	−166.766	−5.52823
$911$	58.7982	1.94807	0.974036	−	0.226394i	$-0.0726937\pi$
$911$	58.7982	1.94807	0.974036	+	0.226394i	$0.0726937\pi$
$912$	2.82282	0.0934730
$913$	47.5536	1.57380
$914$	−10.9674	−0.362769
$915$	−18.2781	−0.604255
$916$	10.7485	0.355140
$917$	29.5981	0.977416
$918$	3.93247	0.129791
$919$	57.7779	1.90592	0.952959	−	0.303099i	$-0.0980213\pi$
$919$	57.7779	1.90592	0.952959	+	0.303099i	$0.0980213\pi$
$920$	13.8295	0.455945
$921$	−22.5667	−0.743598
$922$	−28.9183	−0.952375
$923$	45.4135	1.49480
$924$	−61.6835	−2.02924
$925$	−9.18028	−0.301846
$926$	−2.73788	−0.0899723
$927$	−10.5655	−0.347017
$928$	−11.1013	−0.364419
$929$	21.2092	0.695851	0.347926	−	0.937522i	$-0.386886\pi$
$929$	21.2092	0.695851	0.347926	+	0.937522i	$0.386886\pi$
$930$	−115.017	−3.77156
$931$	−3.27705	−0.107401
$932$	−16.0694	−0.526369
$933$	−23.2491	−0.761140
$934$	−91.9027	−3.00715
$935$	−6.28472	−0.205532
$936$	39.0465	1.27628
$937$	45.7361	1.49413	0.747067	−	0.664748i	$-0.231461\pi$
$937$	45.7361	1.49413	0.747067	+	0.664748i	$0.231461\pi$
$938$	2.22246	0.0725660
$939$	35.5847	1.16126
$940$	27.2119	0.887556
$941$	24.6510	0.803599	0.401799	−	0.915728i	$-0.368385\pi$
$941$	24.6510	0.803599	0.401799	+	0.915728i	$0.368385\pi$
$942$	30.5722	0.996094
$943$	1.40628	0.0457947
$944$	29.2712	0.952697
$945$	71.4820	2.32531
$946$	26.1557	0.850395
$947$	−9.65725	−0.313819	−0.156909	−	0.987613i	$-0.550153\pi$
$947$	−9.65725	−0.313819	−0.156909	+	0.987613i	$0.550153\pi$
$948$	−52.6638	−1.71044
$949$	68.8737	2.23573
$950$	31.5127	1.02241
$951$	2.63719	0.0855167
$952$	4.22900	0.137063
$953$	−11.9029	−0.385571	−0.192786	−	0.981241i	$-0.561752\pi$
$953$	−11.9029	−0.385571	−0.192786	+	0.981241i	$0.561752\pi$
$954$	−22.9862	−0.744206
$955$	−65.2750	−2.11225
$956$	63.0244	2.03835
$957$	25.9106	0.837569
$958$	23.0074	0.743335
$959$	19.3418	0.624578
$960$	48.4010	1.56214
$961$	76.3713	2.46359
$962$	8.53830	0.275286
$963$	20.7323	0.668090
$964$	−78.1148	−2.51591
$965$	−41.3613	−1.33147
$966$	−6.43915	−0.207176
$967$	−44.3226	−1.42532	−0.712660	−	0.701510i	$-0.752510\pi$
$967$	−44.3226	−1.42532	−0.712660	+	0.701510i	$0.752510\pi$
$968$	−47.3238	−1.52104
$969$	−0.340065	−0.0109245
$970$	65.7281	2.11040
$971$	12.3155	0.395222	0.197611	−	0.980281i	$-0.436682\pi$
$971$	12.3155	0.395222	0.197611	+	0.980281i	$0.436682\pi$
$972$	−58.0731	−1.86270
$973$	−0.133140	−0.00426827
$974$	−36.6142	−1.17320
$975$	−72.6502	−2.32667
$976$	10.2657	0.328597
$977$	−50.0887	−1.60248	−0.801240	−	0.598343i	$-0.795826\pi$
$977$	−50.0887	−1.60248	−0.801240	+	0.598343i	$0.795826\pi$
$978$	−49.0678	−1.56901
$979$	43.7456	1.39812
$980$	52.4225	1.67458
$981$	19.1224	0.610530
$982$	−67.3193	−2.14824
$983$	13.1355	0.418958	0.209479	−	0.977813i	$-0.432823\pi$
$983$	13.1355	0.418958	0.209479	+	0.977813i	$0.432823\pi$
$984$	−8.35530	−0.266357
$985$	−9.26553	−0.295224
$986$	−3.79909	−0.120988
$987$	−5.92444	−0.188577
$988$	−19.1261	−0.608483
$989$	1.78177	0.0566569
$990$	87.6601	2.78602
$991$	−55.3678	−1.75882	−0.879408	−	0.476069i	$-0.842061\pi$
$991$	−55.3678	−1.75882	−0.879408	+	0.476069i	$0.842061\pi$
$992$	−22.7401	−0.722000
$993$	24.7926	0.786768
$994$	−68.6051	−2.17602
$995$	−55.4885	−1.75910
$996$	−41.1618	−1.30426
$997$	−49.3589	−1.56321	−0.781605	−	0.623773i	$-0.785599\pi$
$997$	−49.3589	−1.56321	−0.781605	+	0.623773i	$0.785599\pi$
$998$	91.7107	2.90305
$999$	−3.65984	−0.115792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.7	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-2.39927 q^{2} -1.08640 q^{3} +3.75651 q^{4} +4.25844 q^{5} +2.60656 q^{6} +3.20578 q^{7} -4.21434 q^{8} -1.81974 q^{9} +O(q^{10})\)	\(q-2.39927 q^{2} -1.08640 q^{3} +3.75651 q^{4} +4.25844 q^{5} +2.60656 q^{6} +3.20578 q^{7} -4.21434 q^{8} -1.81974 q^{9} -10.2171 q^{10} +4.71479 q^{11} -4.08106 q^{12} +5.09146 q^{13} -7.69155 q^{14} -4.62635 q^{15} +2.59834 q^{16} -0.313021 q^{17} +4.36606 q^{18} -1.00000 q^{19} +15.9969 q^{20} -3.48275 q^{21} -11.3121 q^{22} -0.770596 q^{23} +4.57844 q^{24} +13.1343 q^{25} -12.2158 q^{26} +5.23615 q^{27} +12.0426 q^{28} -5.05855 q^{29} +11.0999 q^{30} -10.3620 q^{31} +2.19457 q^{32} -5.12213 q^{33} +0.751023 q^{34} +13.6516 q^{35} -6.83588 q^{36} -0.698955 q^{37} +2.39927 q^{38} -5.53134 q^{39} -17.9465 q^{40} -1.82492 q^{41} +8.35607 q^{42} -2.31219 q^{43} +17.7111 q^{44} -7.74926 q^{45} +1.84887 q^{46} +1.70108 q^{47} -2.82282 q^{48} +3.27705 q^{49} -31.5127 q^{50} +0.340065 q^{51} +19.1261 q^{52} -5.26474 q^{53} -12.5630 q^{54} +20.0776 q^{55} -13.5103 q^{56} +1.08640 q^{57} +12.1369 q^{58} +11.2654 q^{59} -17.3789 q^{60} +3.95087 q^{61} +24.8613 q^{62} -5.83371 q^{63} -10.4620 q^{64} +21.6817 q^{65} +12.2894 q^{66} -0.288949 q^{67} -1.17587 q^{68} +0.837173 q^{69} -32.7540 q^{70} +8.91954 q^{71} +7.66902 q^{72} +13.5273 q^{73} +1.67698 q^{74} -14.2690 q^{75} -3.75651 q^{76} +15.1146 q^{77} +13.2712 q^{78} +12.9045 q^{79} +11.0649 q^{80} -0.229301 q^{81} +4.37848 q^{82} +10.0861 q^{83} -13.0830 q^{84} -1.33298 q^{85} +5.54758 q^{86} +5.49559 q^{87} -19.8697 q^{88} +9.27838 q^{89} +18.5926 q^{90} +16.3221 q^{91} -2.89475 q^{92} +11.2572 q^{93} -4.08136 q^{94} -4.25844 q^{95} -2.38417 q^{96} -6.43312 q^{97} -7.86254 q^{98} -8.57971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 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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