LMFDB - Embedded newform 4009.2.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.61964 q^{2} -2.82043 q^{3} +4.86252 q^{4} -3.49446 q^{5} +7.38851 q^{6} +2.54132 q^{7} -7.49878 q^{8} +4.95481 q^{9} +O(q^{10})$	\(q-2.61964 q^{2} -2.82043 q^{3} +4.86252 q^{4} -3.49446 q^{5} +7.38851 q^{6} +2.54132 q^{7} -7.49878 q^{8} +4.95481 q^{9} +9.15424 q^{10} +3.23368 q^{11} -13.7144 q^{12} +5.10597 q^{13} -6.65735 q^{14} +9.85587 q^{15} +9.91908 q^{16} +3.26225 q^{17} -12.9798 q^{18} +1.00000 q^{19} -16.9919 q^{20} -7.16761 q^{21} -8.47108 q^{22} +0.140416 q^{23} +21.1498 q^{24} +7.21126 q^{25} -13.3758 q^{26} -5.51340 q^{27} +12.3572 q^{28} -7.46266 q^{29} -25.8189 q^{30} -7.63648 q^{31} -10.9869 q^{32} -9.12035 q^{33} -8.54593 q^{34} -8.88055 q^{35} +24.0929 q^{36} -1.59139 q^{37} -2.61964 q^{38} -14.4010 q^{39} +26.2042 q^{40} +0.548737 q^{41} +18.7766 q^{42} -1.07983 q^{43} +15.7238 q^{44} -17.3144 q^{45} -0.367839 q^{46} -2.92282 q^{47} -27.9760 q^{48} -0.541686 q^{49} -18.8909 q^{50} -9.20094 q^{51} +24.8279 q^{52} +14.3235 q^{53} +14.4431 q^{54} -11.3000 q^{55} -19.0568 q^{56} -2.82043 q^{57} +19.5495 q^{58} -7.74329 q^{59} +47.9244 q^{60} +10.3831 q^{61} +20.0048 q^{62} +12.5918 q^{63} +8.94349 q^{64} -17.8426 q^{65} +23.8921 q^{66} -0.497786 q^{67} +15.8628 q^{68} -0.396033 q^{69} +23.2639 q^{70} -14.8364 q^{71} -37.1550 q^{72} -4.71769 q^{73} +4.16886 q^{74} -20.3388 q^{75} +4.86252 q^{76} +8.21781 q^{77} +37.7255 q^{78} +0.388482 q^{79} -34.6618 q^{80} +0.685703 q^{81} -1.43750 q^{82} -2.18298 q^{83} -34.8527 q^{84} -11.3998 q^{85} +2.82878 q^{86} +21.0479 q^{87} -24.2486 q^{88} -4.22218 q^{89} +45.3575 q^{90} +12.9759 q^{91} +0.682776 q^{92} +21.5381 q^{93} +7.65673 q^{94} -3.49446 q^{95} +30.9877 q^{96} +4.05461 q^{97} +1.41902 q^{98} +16.0223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.61964	−1.85237	−0.926183	−	0.377074i	$-0.876930\pi$
$2$	−2.61964	−1.85237	−0.926183	+	0.377074i	$0.876930\pi$
$3$	−2.82043	−1.62837	−0.814187	−	0.580603i	$-0.802817\pi$
$3$	−2.82043	−1.62837	−0.814187	+	0.580603i	$0.802817\pi$
$4$	4.86252	2.43126
$5$	−3.49446	−1.56277	−0.781385	−	0.624049i	$-0.785487\pi$
$5$	−3.49446	−1.56277	−0.781385	+	0.624049i	$0.785487\pi$
$6$	7.38851	3.01635
$7$	2.54132	0.960529	0.480265	−	0.877124i	$-0.340541\pi$
$7$	2.54132	0.960529	0.480265	+	0.877124i	$0.340541\pi$
$8$	−7.49878	−2.65122
$9$	4.95481	1.65160
$10$	9.15424	2.89482
$11$	3.23368	0.974990	0.487495	−	0.873126i	$-0.337911\pi$
$11$	3.23368	0.974990	0.487495	+	0.873126i	$0.337911\pi$
$12$	−13.7144	−3.95900
$13$	5.10597	1.41614	0.708070	−	0.706142i	$-0.249566\pi$
$13$	5.10597	1.41614	0.708070	+	0.706142i	$0.249566\pi$
$14$	−6.65735	−1.77925
$15$	9.85587	2.54478
$16$	9.91908	2.47977
$17$	3.26225	0.791212	0.395606	−	0.918420i	$-0.370535\pi$
$17$	3.26225	0.791212	0.395606	+	0.918420i	$0.370535\pi$
$18$	−12.9798	−3.05937
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−16.9919	−3.79950
$21$	−7.16761	−1.56410
$22$	−8.47108	−1.80604
$23$	0.140416	0.0292787	0.0146394	−	0.999893i	$-0.495340\pi$
$23$	0.140416	0.0292787	0.0146394	+	0.999893i	$0.495340\pi$
$24$	21.1498	4.31718
$25$	7.21126	1.44225
$26$	−13.3758	−2.62321
$27$	−5.51340	−1.06105
$28$	12.3572	2.33530
$29$	−7.46266	−1.38578	−0.692891	−	0.721043i	$-0.743663\pi$
$29$	−7.46266	−1.38578	−0.692891	+	0.721043i	$0.743663\pi$
$30$	−25.8189	−4.71386
$31$	−7.63648	−1.37155	−0.685776	−	0.727813i	$-0.740537\pi$
$31$	−7.63648	−1.37155	−0.685776	+	0.727813i	$0.740537\pi$
$32$	−10.9869	−1.94222
$33$	−9.12035	−1.58765
$34$	−8.54593	−1.46561
$35$	−8.88055	−1.50109
$36$	24.0929	4.01548
$37$	−1.59139	−0.261622	−0.130811	−	0.991407i	$-0.541758\pi$
$37$	−1.59139	−0.261622	−0.130811	+	0.991407i	$0.541758\pi$
$38$	−2.61964	−0.424962
$39$	−14.4010	−2.30601
$40$	26.2042	4.14325
$41$	0.548737	0.0856984	0.0428492	−	0.999082i	$-0.486357\pi$
$41$	0.548737	0.0856984	0.0428492	+	0.999082i	$0.486357\pi$
$42$	18.7766	2.89729
$43$	−1.07983	−0.164673	−0.0823366	−	0.996605i	$-0.526238\pi$
$43$	−1.07983	−0.164673	−0.0823366	+	0.996605i	$0.526238\pi$
$44$	15.7238	2.37046
$45$	−17.3144	−2.58108
$46$	−0.367839	−0.0542350
$47$	−2.92282	−0.426336	−0.213168	−	0.977016i	$-0.568378\pi$
$47$	−2.92282	−0.426336	−0.213168	+	0.977016i	$0.568378\pi$
$48$	−27.9760	−4.03799
$49$	−0.541686	−0.0773837
$50$	−18.8909	−2.67158
$51$	−9.20094	−1.28839
$52$	24.8279	3.44301
$53$	14.3235	1.96748	0.983742	−	0.179588i	$-0.0574763\pi$
$53$	14.3235	1.96748	0.983742	+	0.179588i	$0.0574763\pi$
$54$	14.4431	1.96546
$55$	−11.3000	−1.52369
$56$	−19.0568	−2.54657
$57$	−2.82043	−0.373575
$58$	19.5495	2.56697
$59$	−7.74329	−1.00809	−0.504045	−	0.863677i	$-0.668156\pi$
$59$	−7.74329	−1.00809	−0.504045	+	0.863677i	$0.668156\pi$
$60$	47.9244	6.18701
$61$	10.3831	1.32942	0.664712	−	0.747099i	$-0.268554\pi$
$61$	10.3831	1.32942	0.664712	+	0.747099i	$0.268554\pi$
$62$	20.0048	2.54062
$63$	12.5918	1.58641
$64$	8.94349	1.11794
$65$	−17.8426	−2.21310
$66$	23.8921	2.94091
$67$	−0.497786	−0.0608142	−0.0304071	−	0.999538i	$-0.509680\pi$
$67$	−0.497786	−0.0608142	−0.0304071	+	0.999538i	$0.509680\pi$
$68$	15.8628	1.92364
$69$	−0.396033	−0.0476768
$70$	23.2639	2.78056
$71$	−14.8364	−1.76076	−0.880380	−	0.474269i	$-0.842712\pi$
$71$	−14.8364	−1.76076	−0.880380	+	0.474269i	$0.842712\pi$
$72$	−37.1550	−4.37876
$73$	−4.71769	−0.552164	−0.276082	−	0.961134i	$-0.589036\pi$
$73$	−4.71769	−0.552164	−0.276082	+	0.961134i	$0.589036\pi$
$74$	4.16886	0.484620
$75$	−20.3388	−2.34853
$76$	4.86252	0.557770
$77$	8.21781	0.936507
$78$	37.7255	4.27157
$79$	0.388482	0.0437076	0.0218538	−	0.999761i	$-0.493043\pi$
$79$	0.388482	0.0437076	0.0218538	+	0.999761i	$0.493043\pi$
$80$	−34.6618	−3.87531
$81$	0.685703	0.0761893
$82$	−1.43750	−0.158745
$83$	−2.18298	−0.239613	−0.119806	−	0.992797i	$-0.538227\pi$
$83$	−2.18298	−0.239613	−0.119806	+	0.992797i	$0.538227\pi$
$84$	−34.8527	−3.80274
$85$	−11.3998	−1.23648
$86$	2.82878	0.305035
$87$	21.0479	2.25657
$88$	−24.2486	−2.58491
$89$	−4.22218	−0.447550	−0.223775	−	0.974641i	$-0.571838\pi$
$89$	−4.22218	−0.447550	−0.223775	+	0.974641i	$0.571838\pi$
$90$	45.3575	4.78110
$91$	12.9759	1.36024
$92$	0.682776	0.0711843
$93$	21.5381	2.23340
$94$	7.65673	0.789731
$95$	−3.49446	−0.358524
$96$	30.9877	3.16266
$97$	4.05461	0.411683	0.205842	−	0.978585i	$-0.434007\pi$
$97$	4.05461	0.411683	0.205842	+	0.978585i	$0.434007\pi$
$98$	1.41902	0.143343
$99$	16.0223	1.61030
$100$	35.0649	3.50649
$101$	−14.3717	−1.43003	−0.715017	−	0.699107i	$-0.753581\pi$
$101$	−14.3717	−1.43003	−0.715017	+	0.699107i	$0.753581\pi$
$102$	24.1032	2.38657
$103$	10.9220	1.07618	0.538090	−	0.842887i	$-0.319146\pi$
$103$	10.9220	1.07618	0.538090	+	0.842887i	$0.319146\pi$
$104$	−38.2885	−3.75450
$105$	25.0469	2.44433
$106$	−37.5224	−3.64450
$107$	5.10147	0.493178	0.246589	−	0.969120i	$-0.420690\pi$
$107$	5.10147	0.493178	0.246589	+	0.969120i	$0.420690\pi$
$108$	−26.8090	−2.57970
$109$	−17.6997	−1.69533	−0.847664	−	0.530533i	$-0.821992\pi$
$109$	−17.6997	−1.69533	−0.847664	+	0.530533i	$0.821992\pi$
$110$	29.6018	2.82243
$111$	4.48839	0.426019
$112$	25.2076	2.38189
$113$	−11.2553	−1.05881	−0.529404	−	0.848370i	$-0.677584\pi$
$113$	−11.2553	−1.05881	−0.529404	+	0.848370i	$0.677584\pi$
$114$	7.38851	0.691997
$115$	−0.490678	−0.0457560
$116$	−36.2873	−3.36920
$117$	25.2991	2.33890
$118$	20.2846	1.86735
$119$	8.29043	0.759982
$120$	−73.9070	−6.74676
$121$	−0.543330	−0.0493936
$122$	−27.2001	−2.46258
$123$	−1.54767	−0.139549
$124$	−37.1325	−3.33460
$125$	−7.72715	−0.691137
$126$	−32.9859	−2.93862
$127$	4.67121	0.414503	0.207251	−	0.978288i	$-0.433548\pi$
$127$	4.67121	0.414503	0.207251	+	0.978288i	$0.433548\pi$
$128$	−1.45502	−0.128607
$129$	3.04559	0.268150
$130$	46.7412	4.09948
$131$	18.3170	1.60036	0.800180	−	0.599759i	$-0.204737\pi$
$131$	18.3170	1.60036	0.800180	+	0.599759i	$0.204737\pi$
$132$	−44.3479	−3.85999
$133$	2.54132	0.220361
$134$	1.30402	0.112650
$135$	19.2663	1.65818
$136$	−24.4629	−2.09768
$137$	−18.4773	−1.57863	−0.789313	−	0.613991i	$-0.789563\pi$
$137$	−18.4773	−1.57863	−0.789313	+	0.613991i	$0.789563\pi$
$138$	1.03746	0.0883148
$139$	3.22495	0.273537	0.136768	−	0.990603i	$-0.456328\pi$
$139$	3.22495	0.273537	0.136768	+	0.990603i	$0.456328\pi$
$140$	−43.1819	−3.64953
$141$	8.24359	0.694235
$142$	38.8661	3.26157
$143$	16.5110	1.38072
$144$	49.1471	4.09559
$145$	26.0780	2.16566
$146$	12.3587	1.02281
$147$	1.52779	0.126010
$148$	−7.73815	−0.636072
$149$	10.5860	0.867242	0.433621	−	0.901095i	$-0.357236\pi$
$149$	10.5860	0.867242	0.433621	+	0.901095i	$0.357236\pi$
$150$	53.2804	4.35033
$151$	9.09239	0.739928	0.369964	−	0.929046i	$-0.379370\pi$
$151$	9.09239	0.739928	0.369964	+	0.929046i	$0.379370\pi$
$152$	−7.49878	−0.608232
$153$	16.1638	1.30677
$154$	−21.5277	−1.73475
$155$	26.6854	2.14342
$156$	−70.0252	−5.60650
$157$	21.9591	1.75252	0.876262	−	0.481835i	$-0.160030\pi$
$157$	21.9591	1.75252	0.876262	+	0.481835i	$0.160030\pi$
$158$	−1.01768	−0.0809625
$159$	−40.3984	−3.20380
$160$	38.3932	3.03525
$161$	0.356842	0.0281231
$162$	−1.79630	−0.141130
$163$	−8.51150	−0.666672	−0.333336	−	0.942808i	$-0.608174\pi$
$163$	−8.51150	−0.666672	−0.333336	+	0.942808i	$0.608174\pi$
$164$	2.66825	0.208355
$165$	31.8707	2.48113
$166$	5.71861	0.443851
$167$	−22.2400	−1.72098	−0.860490	−	0.509468i	$-0.829842\pi$
$167$	−22.2400	−1.72098	−0.860490	+	0.509468i	$0.829842\pi$
$168$	53.7484	4.14678
$169$	13.0709	1.00545
$170$	29.8634	2.29042
$171$	4.95481	0.378904
$172$	−5.25072	−0.400364
$173$	12.0437	0.915663	0.457831	−	0.889039i	$-0.348626\pi$
$173$	12.0437	0.915663	0.457831	+	0.889039i	$0.348626\pi$
$174$	−55.1379	−4.17999
$175$	18.3261	1.38532
$176$	32.0751	2.41775
$177$	21.8394	1.64155
$178$	11.0606	0.829027
$179$	−8.94236	−0.668383	−0.334192	−	0.942505i	$-0.608463\pi$
$179$	−8.94236	−0.668383	−0.334192	+	0.942505i	$0.608463\pi$
$180$	−84.1916	−6.27527
$181$	−18.5439	−1.37835	−0.689177	−	0.724593i	$-0.742028\pi$
$181$	−18.5439	−1.37835	−0.689177	+	0.724593i	$0.742028\pi$
$182$	−33.9922	−2.51967
$183$	−29.2849	−2.16480
$184$	−1.05295	−0.0776244
$185$	5.56103	0.408855
$186$	−56.4222	−4.13707
$187$	10.5491	0.771424
$188$	−14.2123	−1.03654
$189$	−14.0113	−1.01917
$190$	9.15424	0.664118
$191$	−4.36184	−0.315612	−0.157806	−	0.987470i	$-0.550442\pi$
$191$	−4.36184	−0.315612	−0.157806	+	0.987470i	$0.550442\pi$
$192$	−25.2245	−1.82042
$193$	−21.7113	−1.56281	−0.781406	−	0.624023i	$-0.785497\pi$
$193$	−21.7113	−1.56281	−0.781406	+	0.624023i	$0.785497\pi$
$194$	−10.6216	−0.762589
$195$	50.3237	3.60376
$196$	−2.63396	−0.188140
$197$	−0.956526	−0.0681496	−0.0340748	−	0.999419i	$-0.510848\pi$
$197$	−0.956526	−0.0681496	−0.0340748	+	0.999419i	$0.510848\pi$
$198$	−41.9726	−2.98286
$199$	4.57302	0.324173	0.162086	−	0.986777i	$-0.448178\pi$
$199$	4.57302	0.324173	0.162086	+	0.986777i	$0.448178\pi$
$200$	−54.0756	−3.82373
$201$	1.40397	0.0990283
$202$	37.6486	2.64895
$203$	−18.9650	−1.33108
$204$	−44.7398	−3.13241
$205$	−1.91754	−0.133927
$206$	−28.6118	−1.99348
$207$	0.695734	0.0483569
$208$	50.6465	3.51170
$209$	3.23368	0.223678
$210$	−65.6140	−4.52780
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	69.6483	4.78347
$213$	41.8451	2.86718
$214$	−13.3640	−0.913546
$215$	3.77344	0.257346
$216$	41.3438	2.81309
$217$	−19.4067	−1.31742
$218$	46.3670	3.14037
$219$	13.3059	0.899130
$220$	−54.9463	−3.70448
$221$	16.6569	1.12047
$222$	−11.7580	−0.789143
$223$	−15.1464	−1.01428	−0.507138	−	0.861865i	$-0.669296\pi$
$223$	−15.1464	−1.01428	−0.507138	+	0.861865i	$0.669296\pi$
$224$	−27.9212	−1.86556
$225$	35.7304	2.38203
$226$	29.4848	1.96130
$227$	−2.48470	−0.164915	−0.0824577	−	0.996595i	$-0.526277\pi$
$227$	−2.48470	−0.164915	−0.0824577	+	0.996595i	$0.526277\pi$
$228$	−13.7144	−0.908258
$229$	12.6784	0.837812	0.418906	−	0.908030i	$-0.362414\pi$
$229$	12.6784	0.837812	0.418906	+	0.908030i	$0.362414\pi$
$230$	1.28540	0.0847568
$231$	−23.1777	−1.52498
$232$	55.9609	3.67401
$233$	−17.2086	−1.12737	−0.563687	−	0.825988i	$-0.690618\pi$
$233$	−17.2086	−1.12737	−0.563687	+	0.825988i	$0.690618\pi$
$234$	−66.2745	−4.33250
$235$	10.2137	0.666266
$236$	−37.6519	−2.45093
$237$	−1.09568	−0.0711723
$238$	−21.7180	−1.40777
$239$	4.23894	0.274194	0.137097	−	0.990558i	$-0.456223\pi$
$239$	4.23894	0.274194	0.137097	+	0.990558i	$0.456223\pi$
$240$	97.7612	6.31046
$241$	−10.8967	−0.701918	−0.350959	−	0.936391i	$-0.614144\pi$
$241$	−10.8967	−0.701918	−0.350959	+	0.936391i	$0.614144\pi$
$242$	1.42333	0.0914950
$243$	14.6062	0.936989
$244$	50.4882	3.23218
$245$	1.89290	0.120933
$246$	4.05435	0.258496
$247$	5.10597	0.324885
$248$	57.2643	3.63629
$249$	6.15692	0.390179
$250$	20.2424	1.28024
$251$	19.3873	1.22372	0.611859	−	0.790967i	$-0.290422\pi$
$251$	19.3873	1.22372	0.611859	+	0.790967i	$0.290422\pi$
$252$	61.2277	3.85698
$253$	0.454060	0.0285465
$254$	−12.2369	−0.767811
$255$	32.1523	2.01346
$256$	−14.0754	−0.879710
$257$	−18.3122	−1.14228	−0.571141	−	0.820852i	$-0.693499\pi$
$257$	−18.3122	−1.14228	−0.571141	+	0.820852i	$0.693499\pi$
$258$	−7.97837	−0.496711
$259$	−4.04422	−0.251296
$260$	−86.7600	−5.38063
$261$	−36.9760	−2.28876
$262$	−47.9839	−2.96445
$263$	4.34806	0.268113	0.134057	−	0.990974i	$-0.457200\pi$
$263$	4.34806	0.268113	0.134057	+	0.990974i	$0.457200\pi$
$264$	68.3915	4.20921
$265$	−50.0529	−3.07473
$266$	−6.65735	−0.408188
$267$	11.9084	0.728780
$268$	−2.42049	−0.147855
$269$	−4.64805	−0.283397	−0.141698	−	0.989910i	$-0.545256\pi$
$269$	−4.64805	−0.283397	−0.141698	+	0.989910i	$0.545256\pi$
$270$	−50.4709	−3.07156
$271$	21.5938	1.31173	0.655865	−	0.754878i	$-0.272304\pi$
$271$	21.5938	1.31173	0.655865	+	0.754878i	$0.272304\pi$
$272$	32.3585	1.96202
$273$	−36.5976	−2.21499
$274$	48.4040	2.92419
$275$	23.3189	1.40618
$276$	−1.92572	−0.115915
$277$	−13.9013	−0.835250	−0.417625	−	0.908619i	$-0.637137\pi$
$277$	−13.9013	−0.835250	−0.417625	+	0.908619i	$0.637137\pi$
$278$	−8.44822	−0.506690
$279$	−37.8373	−2.26526
$280$	66.5933	3.97971
$281$	20.0441	1.19573	0.597865	−	0.801597i	$-0.296016\pi$
$281$	20.0441	1.19573	0.597865	+	0.801597i	$0.296016\pi$
$282$	−21.5952	−1.28598
$283$	−1.86018	−0.110576	−0.0552882	−	0.998470i	$-0.517608\pi$
$283$	−1.86018	−0.110576	−0.0552882	+	0.998470i	$0.517608\pi$
$284$	−72.1424	−4.28087
$285$	9.85587	0.583811
$286$	−43.2530	−2.55761
$287$	1.39452	0.0823158
$288$	−54.4378	−3.20778
$289$	−6.35771	−0.373983
$290$	−68.3149	−4.01159
$291$	−11.4357	−0.670375
$292$	−22.9399	−1.34245
$293$	32.2980	1.88687	0.943435	−	0.331558i	$-0.107574\pi$
$293$	32.2980	1.88687	0.943435	+	0.331558i	$0.107574\pi$
$294$	−4.00225	−0.233416
$295$	27.0586	1.57541
$296$	11.9335	0.693618
$297$	−17.8285	−1.03452
$298$	−27.7316	−1.60645
$299$	0.716959	0.0414628
$300$	−98.8980	−5.70988
$301$	−2.74421	−0.158173
$302$	−23.8188	−1.37062
$303$	40.5342	2.32863
$304$	9.91908	0.568898
$305$	−36.2835	−2.07759
$306$	−42.3434	−2.42061
$307$	−16.3508	−0.933191	−0.466595	−	0.884471i	$-0.654520\pi$
$307$	−16.3508	−0.933191	−0.466595	+	0.884471i	$0.654520\pi$
$308$	39.9593	2.27689
$309$	−30.8048	−1.75242
$310$	−69.9061	−3.97040
$311$	−26.0022	−1.47445	−0.737223	−	0.675649i	$-0.763863\pi$
$311$	−26.0022	−1.47445	−0.737223	+	0.675649i	$0.763863\pi$
$312$	107.990	6.11373
$313$	5.37721	0.303938	0.151969	−	0.988385i	$-0.451439\pi$
$313$	5.37721	0.303938	0.151969	+	0.988385i	$0.451439\pi$
$314$	−57.5249	−3.24632
$315$	−44.0014	−2.47920
$316$	1.88900	0.106265
$317$	31.0435	1.74358	0.871788	−	0.489883i	$-0.162961\pi$
$317$	31.0435	1.74358	0.871788	+	0.489883i	$0.162961\pi$
$318$	105.829	5.93461
$319$	−24.1318	−1.35112
$320$	−31.2527	−1.74708
$321$	−14.3883	−0.803078
$322$	−0.934798	−0.0520943
$323$	3.26225	0.181517
$324$	3.33425	0.185236
$325$	36.8204	2.04243
$326$	22.2971	1.23492
$327$	49.9208	2.76063
$328$	−4.11486	−0.227205
$329$	−7.42781	−0.409509
$330$	−83.4898	−4.59596
$331$	−23.8215	−1.30935	−0.654674	−	0.755912i	$-0.727194\pi$
$331$	−23.8215	−1.30935	−0.654674	+	0.755912i	$0.727194\pi$
$332$	−10.6148	−0.582561
$333$	−7.88501	−0.432096
$334$	58.2607	3.18788
$335$	1.73949	0.0950387
$336$	−71.0961	−3.87861
$337$	28.6715	1.56184	0.780919	−	0.624633i	$-0.214751\pi$
$337$	28.6715	1.56184	0.780919	+	0.624633i	$0.214751\pi$
$338$	−34.2410	−1.86247
$339$	31.7447	1.72414
$340$	−55.4318	−3.00621
$341$	−24.6939	−1.33725
$342$	−12.9798	−0.701868
$343$	−19.1658	−1.03486
$344$	8.09744	0.436585
$345$	1.38392	0.0745078
$346$	−31.5501	−1.69614
$347$	−15.4470	−0.829238	−0.414619	−	0.909995i	$-0.636085\pi$
$347$	−15.4470	−0.829238	−0.414619	+	0.909995i	$0.636085\pi$
$348$	102.346	5.48631
$349$	−28.4809	−1.52455	−0.762273	−	0.647256i	$-0.775917\pi$
$349$	−28.4809	−1.52455	−0.762273	+	0.647256i	$0.775917\pi$
$350$	−48.0079	−2.56613
$351$	−28.1512	−1.50260
$352$	−35.5280	−1.89365
$353$	26.1593	1.39232	0.696160	−	0.717886i	$-0.254890\pi$
$353$	26.1593	1.39232	0.696160	+	0.717886i	$0.254890\pi$
$354$	−57.2114	−3.04075
$355$	51.8453	2.75166
$356$	−20.5305	−1.08811
$357$	−23.3826	−1.23754
$358$	23.4258	1.23809
$359$	−10.4251	−0.550216	−0.275108	−	0.961413i	$-0.588714\pi$
$359$	−10.4251	−0.550216	−0.275108	+	0.961413i	$0.588714\pi$
$360$	129.837	6.84300
$361$	1.00000	0.0526316
$362$	48.5783	2.55322
$363$	1.53242	0.0804313
$364$	63.0956	3.30711
$365$	16.4858	0.862905
$366$	76.7159	4.01000
$367$	−31.0491	−1.62075	−0.810376	−	0.585910i	$-0.800737\pi$
$367$	−31.0491	−1.62075	−0.810376	+	0.585910i	$0.800737\pi$
$368$	1.39280	0.0726045
$369$	2.71889	0.141540
$370$	−14.5679	−0.757350
$371$	36.4006	1.88983
$372$	104.730	5.42998
$373$	19.4727	1.00826	0.504130	−	0.863628i	$-0.331813\pi$
$373$	19.4727	1.00826	0.504130	+	0.863628i	$0.331813\pi$
$374$	−27.6348	−1.42896
$375$	21.7939	1.12543
$376$	21.9176	1.13031
$377$	−38.1041	−1.96246
$378$	36.7046	1.88788
$379$	6.99458	0.359288	0.179644	−	0.983732i	$-0.442505\pi$
$379$	6.99458	0.359288	0.179644	+	0.983732i	$0.442505\pi$
$380$	−16.9919	−0.871666
$381$	−13.1748	−0.674965
$382$	11.4265	0.584628
$383$	−9.33025	−0.476754	−0.238377	−	0.971173i	$-0.576615\pi$
$383$	−9.33025	−0.476754	−0.238377	+	0.971173i	$0.576615\pi$
$384$	4.10377	0.209420
$385$	−28.7168	−1.46355
$386$	56.8758	2.89490
$387$	−5.35037	−0.271975
$388$	19.7156	1.00091
$389$	17.3548	0.879921	0.439961	−	0.898017i	$-0.354992\pi$
$389$	17.3548	0.879921	0.439961	+	0.898017i	$0.354992\pi$
$390$	−131.830	−6.67548
$391$	0.458072	0.0231657
$392$	4.06199	0.205161
$393$	−51.6617	−2.60599
$394$	2.50575	0.126238
$395$	−1.35753	−0.0683050
$396$	77.9086	3.91505
$397$	−33.2702	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$397$	−33.2702	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$398$	−11.9797	−0.600486
$399$	−7.16761	−0.358829
$400$	71.5290	3.57645
$401$	−12.0296	−0.600729	−0.300365	−	0.953824i	$-0.597108\pi$
$401$	−12.0296	−0.600729	−0.300365	+	0.953824i	$0.597108\pi$
$402$	−3.67789	−0.183437
$403$	−38.9916	−1.94231
$404$	−69.8826	−3.47679
$405$	−2.39616	−0.119066
$406$	49.6815	2.46565
$407$	−5.14603	−0.255079
$408$	68.9959	3.41580
$409$	22.2804	1.10169	0.550847	−	0.834607i	$-0.314305\pi$
$409$	22.2804	1.10169	0.550847	+	0.834607i	$0.314305\pi$
$410$	5.02327	0.248082
$411$	52.1140	2.57059
$412$	53.1086	2.61647
$413$	−19.6782	−0.968300
$414$	−1.82257	−0.0895746
$415$	7.62832	0.374460
$416$	−56.0986	−2.75046
$417$	−9.09574	−0.445420
$418$	−8.47108	−0.414334
$419$	−20.8185	−1.01705	−0.508525	−	0.861048i	$-0.669809\pi$
$419$	−20.8185	−1.01705	−0.508525	+	0.861048i	$0.669809\pi$
$420$	121.791	5.94281
$421$	−31.1740	−1.51933	−0.759664	−	0.650316i	$-0.774636\pi$
$421$	−31.1740	−1.51933	−0.759664	+	0.650316i	$0.774636\pi$
$422$	−2.61964	−0.127522
$423$	−14.4820	−0.704138
$424$	−107.409	−5.21623
$425$	23.5249	1.14113
$426$	−109.619	−5.31106
$427$	26.3869	1.27695
$428$	24.8060	1.19904
$429$	−46.5682	−2.24833
$430$	−9.88506	−0.476700
$431$	−14.7514	−0.710548	−0.355274	−	0.934762i	$-0.615612\pi$
$431$	−14.7514	−0.710548	−0.355274	+	0.934762i	$0.615612\pi$
$432$	−54.6878	−2.63117
$433$	−3.88447	−0.186676	−0.0933379	−	0.995634i	$-0.529754\pi$
$433$	−3.88447	−0.186676	−0.0933379	+	0.995634i	$0.529754\pi$
$434$	50.8387	2.44034
$435$	−73.5510	−3.52650
$436$	−86.0654	−4.12179
$437$	0.140416	0.00671700
$438$	−34.8567	−1.66552
$439$	5.96184	0.284543	0.142271	−	0.989828i	$-0.454559\pi$
$439$	5.96184	0.284543	0.142271	+	0.989828i	$0.454559\pi$
$440$	84.7359	4.03963
$441$	−2.68395	−0.127807
$442$	−43.6352	−2.07552
$443$	−10.9439	−0.519959	−0.259980	−	0.965614i	$-0.583716\pi$
$443$	−10.9439	−0.519959	−0.259980	+	0.965614i	$0.583716\pi$
$444$	21.8249	1.03576
$445$	14.7543	0.699419
$446$	39.6780	1.87881
$447$	−29.8571	−1.41219
$448$	22.7283	1.07381
$449$	20.0939	0.948292	0.474146	−	0.880446i	$-0.342757\pi$
$449$	20.0939	0.948292	0.474146	+	0.880446i	$0.342757\pi$
$450$	−93.6008	−4.41239
$451$	1.77444	0.0835551
$452$	−54.7291	−2.57424
$453$	−25.6444	−1.20488
$454$	6.50903	0.305484
$455$	−45.3438	−2.12575
$456$	21.1498	0.990429
$457$	−18.7847	−0.878713	−0.439356	−	0.898313i	$-0.644793\pi$
$457$	−18.7847	−0.878713	−0.439356	+	0.898313i	$0.644793\pi$
$458$	−33.2128	−1.55193
$459$	−17.9861	−0.839518
$460$	−2.38593	−0.111245
$461$	5.34488	0.248936	0.124468	−	0.992224i	$-0.460278\pi$
$461$	5.34488	0.248936	0.124468	+	0.992224i	$0.460278\pi$
$462$	60.7174	2.82483
$463$	4.73731	0.220161	0.110081	−	0.993923i	$-0.464889\pi$
$463$	4.73731	0.220161	0.110081	+	0.993923i	$0.464889\pi$
$464$	−74.0227	−3.43642
$465$	−75.2641	−3.49029
$466$	45.0804	2.08831
$467$	33.3733	1.54433	0.772165	−	0.635422i	$-0.219174\pi$
$467$	33.3733	1.54433	0.772165	+	0.635422i	$0.219174\pi$
$468$	123.017	5.68648
$469$	−1.26503	−0.0584138
$470$	−26.7561	−1.23417
$471$	−61.9339	−2.85376
$472$	58.0653	2.67267
$473$	−3.49184	−0.160555
$474$	2.87030	0.131837
$475$	7.21126	0.330875
$476$	40.3124	1.84772
$477$	70.9702	3.24950
$478$	−11.1045	−0.507908
$479$	27.7874	1.26964	0.634818	−	0.772661i	$-0.281075\pi$
$479$	27.7874	1.26964	0.634818	+	0.772661i	$0.281075\pi$
$480$	−108.285	−4.94252
$481$	−8.12556	−0.370494
$482$	28.5455	1.30021
$483$	−1.00645	−0.0457949
$484$	−2.64195	−0.120089
$485$	−14.1687	−0.643367
$486$	−38.2630	−1.73565
$487$	−12.6010	−0.571007	−0.285504	−	0.958378i	$-0.592161\pi$
$487$	−12.6010	−0.571007	−0.285504	+	0.958378i	$0.592161\pi$
$488$	−77.8609	−3.52460
$489$	24.0061	1.08559
$490$	−4.95872	−0.224012
$491$	−28.6359	−1.29232	−0.646159	−	0.763202i	$-0.723626\pi$
$491$	−28.6359	−1.29232	−0.646159	+	0.763202i	$0.723626\pi$
$492$	−7.52560	−0.339280
$493$	−24.3451	−1.09645
$494$	−13.3758	−0.601806
$495$	−55.9891	−2.51652
$496$	−75.7468	−3.40113
$497$	−37.7041	−1.69126
$498$	−16.1289	−0.722755
$499$	0.0697231	0.00312123	0.00156062	−	0.999999i	$-0.499503\pi$
$499$	0.0697231	0.00312123	0.00156062	+	0.999999i	$0.499503\pi$
$500$	−37.5734	−1.68034
$501$	62.7262	2.80240
$502$	−50.7879	−2.26677
$503$	−42.0619	−1.87545	−0.937723	−	0.347383i	$-0.887070\pi$
$503$	−42.0619	−1.87545	−0.937723	+	0.347383i	$0.887070\pi$
$504$	−94.4229	−4.20593
$505$	50.2212	2.23482
$506$	−1.18947	−0.0528786
$507$	−36.8655	−1.63725
$508$	22.7138	1.00776
$509$	−28.0786	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$509$	−28.0786	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$510$	−84.2276	−3.72966
$511$	−11.9892	−0.530370
$512$	39.7824	1.75815
$513$	−5.51340	−0.243422
$514$	47.9713	2.11593
$515$	−38.1666	−1.68182
$516$	14.8093	0.651942
$517$	−9.45144	−0.415674
$518$	10.5944	0.465492
$519$	−33.9683	−1.49104
$520$	133.798	5.86742
$521$	−3.66228	−0.160447	−0.0802237	−	0.996777i	$-0.525563\pi$
$521$	−3.66228	−0.160447	−0.0802237	+	0.996777i	$0.525563\pi$
$522$	96.8640	4.23962
$523$	6.91382	0.302320	0.151160	−	0.988509i	$-0.451699\pi$
$523$	6.91382	0.302320	0.151160	+	0.988509i	$0.451699\pi$
$524$	89.0667	3.89089
$525$	−51.6875	−2.25583
$526$	−11.3904	−0.496644
$527$	−24.9121	−1.08519
$528$	−90.4655	−3.93700
$529$	−22.9803	−0.999143
$530$	131.121	5.69552
$531$	−38.3665	−1.66497
$532$	12.3572	0.535754
$533$	2.80183	0.121361
$534$	−31.1956	−1.34997
$535$	−17.8269	−0.770724
$536$	3.73279	0.161232
$537$	25.2213	1.08838
$538$	12.1762	0.524954
$539$	−1.75164	−0.0754484
$540$	93.6830	4.03148
$541$	5.08815	0.218757	0.109378	−	0.994000i	$-0.465114\pi$
$541$	5.08815	0.218757	0.109378	+	0.994000i	$0.465114\pi$
$542$	−56.5681	−2.42981
$543$	52.3016	2.24448
$544$	−35.8419	−1.53671
$545$	61.8511	2.64941
$546$	95.8725	4.10297
$547$	−40.9404	−1.75049	−0.875243	−	0.483683i	$-0.839299\pi$
$547$	−40.9404	−1.75049	−0.875243	+	0.483683i	$0.839299\pi$
$548$	−89.8465	−3.83805
$549$	51.4465	2.19568
$550$	−61.0871	−2.60476
$551$	−7.46266	−0.317920
$552$	2.96976	0.126402
$553$	0.987257	0.0419824
$554$	36.4165	1.54719
$555$	−15.6845	−0.665769
$556$	15.6814	0.665039
$557$	12.2491	0.519012	0.259506	−	0.965741i	$-0.416440\pi$
$557$	12.2491	0.519012	0.259506	+	0.965741i	$0.416440\pi$
$558$	99.1201	4.19609
$559$	−5.51360	−0.233200
$560$	−88.0869	−3.72235
$561$	−29.7529	−1.25617
$562$	−52.5083	−2.21493
$563$	37.2561	1.57016	0.785079	−	0.619396i	$-0.212623\pi$
$563$	37.2561	1.57016	0.785079	+	0.619396i	$0.212623\pi$
$564$	40.0846	1.68787
$565$	39.3312	1.65467
$566$	4.87301	0.204828
$567$	1.74259	0.0731820
$568$	111.255	4.66816
$569$	16.2595	0.681635	0.340817	−	0.940129i	$-0.389296\pi$
$569$	16.2595	0.681635	0.340817	+	0.940129i	$0.389296\pi$
$570$	−25.8189	−1.08143
$571$	−1.39643	−0.0584389	−0.0292195	−	0.999573i	$-0.509302\pi$
$571$	−1.39643	−0.0584389	−0.0292195	+	0.999573i	$0.509302\pi$
$572$	80.2853	3.35690
$573$	12.3022	0.513934
$574$	−3.65314	−0.152479
$575$	1.01258	0.0422273
$576$	44.3133	1.84639
$577$	−33.4098	−1.39087	−0.695434	−	0.718590i	$-0.744788\pi$
$577$	−33.4098	−1.39087	−0.695434	+	0.718590i	$0.744788\pi$
$578$	16.6549	0.692754
$579$	61.2351	2.54484
$580$	126.805	5.26528
$581$	−5.54764	−0.230155
$582$	29.9575	1.24178
$583$	46.3176	1.91828
$584$	35.3769	1.46391
$585$	−88.4067	−3.65517
$586$	−84.6092	−3.49517
$587$	−15.5792	−0.643023	−0.321512	−	0.946906i	$-0.604191\pi$
$587$	−15.5792	−0.643023	−0.321512	+	0.946906i	$0.604191\pi$
$588$	7.42889	0.306362
$589$	−7.63648	−0.314656
$590$	−70.8839	−2.91824
$591$	2.69781	0.110973
$592$	−15.7851	−0.648763
$593$	44.1209	1.81183	0.905913	−	0.423463i	$-0.139186\pi$
$593$	44.1209	1.81183	0.905913	+	0.423463i	$0.139186\pi$
$594$	46.7044	1.91630
$595$	−28.9706	−1.18768
$596$	51.4748	2.10849
$597$	−12.8979	−0.527874
$598$	−1.87818	−0.0768043
$599$	−6.58108	−0.268895	−0.134448	−	0.990921i	$-0.542926\pi$
$599$	−6.58108	−0.268895	−0.134448	+	0.990921i	$0.542926\pi$
$600$	152.516	6.22646
$601$	38.6449	1.57636	0.788179	−	0.615447i	$-0.211024\pi$
$601$	38.6449	1.57636	0.788179	+	0.615447i	$0.211024\pi$
$602$	7.18884	0.292995
$603$	−2.46643	−0.100441
$604$	44.2119	1.79896
$605$	1.89864	0.0771908
$606$	−106.185	−4.31348
$607$	−16.0203	−0.650243	−0.325121	−	0.945672i	$-0.605405\pi$
$607$	−16.0203	−0.650243	−0.325121	+	0.945672i	$0.605405\pi$
$608$	−10.9869	−0.445576
$609$	53.4894	2.16750
$610$	95.0497	3.84845
$611$	−14.9238	−0.603752
$612$	78.5970	3.17710
$613$	−1.19280	−0.0481765	−0.0240883	−	0.999710i	$-0.507668\pi$
$613$	−1.19280	−0.0481765	−0.0240883	+	0.999710i	$0.507668\pi$
$614$	42.8333	1.72861
$615$	5.40829	0.218083
$616$	−61.6236	−2.48289
$617$	38.5792	1.55314	0.776570	−	0.630031i	$-0.216958\pi$
$617$	38.5792	1.55314	0.776570	+	0.630031i	$0.216958\pi$
$618$	80.6975	3.24613
$619$	1.00325	0.0403242	0.0201621	−	0.999797i	$-0.493582\pi$
$619$	1.00325	0.0403242	0.0201621	+	0.999797i	$0.493582\pi$
$620$	129.758	5.21121
$621$	−0.774169	−0.0310663
$622$	68.1163	2.73122
$623$	−10.7299	−0.429885
$624$	−142.845	−5.71836
$625$	−9.05406	−0.362162
$626$	−14.0864	−0.563005
$627$	−9.12035	−0.364232
$628$	106.776	4.26084
$629$	−5.19150	−0.206999
$630$	115.268	4.59238
$631$	24.5959	0.979146	0.489573	−	0.871962i	$-0.337153\pi$
$631$	24.5959	0.979146	0.489573	+	0.871962i	$0.337153\pi$
$632$	−2.91314	−0.115878
$633$	−2.82043	−0.112102
$634$	−81.3228	−3.22974
$635$	−16.3233	−0.647772
$636$	−196.438	−7.78928
$637$	−2.76583	−0.109586
$638$	63.2168	2.50278
$639$	−73.5116	−2.90808
$640$	5.08451	0.200983
$641$	6.50912	0.257095	0.128547	−	0.991703i	$-0.458969\pi$
$641$	6.50912	0.257095	0.128547	+	0.991703i	$0.458969\pi$
$642$	37.6923	1.48760
$643$	−16.5049	−0.650890	−0.325445	−	0.945561i	$-0.605514\pi$
$643$	−16.5049	−0.650890	−0.325445	+	0.945561i	$0.605514\pi$
$644$	1.73515	0.0683746
$645$	−10.6427	−0.419056
$646$	−8.54593	−0.336235
$647$	−6.52260	−0.256430	−0.128215	−	0.991746i	$-0.540925\pi$
$647$	−6.52260	−0.256430	−0.128215	+	0.991746i	$0.540925\pi$
$648$	−5.14194	−0.201994
$649$	−25.0393	−0.982879
$650$	−96.4563	−3.78333
$651$	54.7353	2.14525
$652$	−41.3874	−1.62085
$653$	4.85288	0.189908	0.0949539	−	0.995482i	$-0.469730\pi$
$653$	4.85288	0.189908	0.0949539	+	0.995482i	$0.469730\pi$
$654$	−130.775	−5.11370
$655$	−64.0079	−2.50100
$656$	5.44297	0.212512
$657$	−23.3753	−0.911955
$658$	19.4582	0.758560
$659$	34.6302	1.34900	0.674500	−	0.738275i	$-0.264359\pi$
$659$	34.6302	1.34900	0.674500	+	0.738275i	$0.264359\pi$
$660$	154.972	6.03228
$661$	−9.97032	−0.387801	−0.193900	−	0.981021i	$-0.562114\pi$
$661$	−9.97032	−0.387801	−0.193900	+	0.981021i	$0.562114\pi$
$662$	62.4038	2.42539
$663$	−46.9797	−1.82454
$664$	16.3697	0.635266
$665$	−8.88055	−0.344373
$666$	20.6559	0.800400
$667$	−1.04788	−0.0405739
$668$	−108.142	−4.18415
$669$	42.7192	1.65162
$670$	−4.55685	−0.176046
$671$	33.5757	1.29618
$672$	78.7496	3.03783
$673$	38.2539	1.47458	0.737290	−	0.675576i	$-0.236105\pi$
$673$	38.2539	1.47458	0.737290	+	0.675576i	$0.236105\pi$
$674$	−75.1091	−2.89310
$675$	−39.7585	−1.53031
$676$	63.5575	2.44452
$677$	−11.2916	−0.433973	−0.216987	−	0.976175i	$-0.569623\pi$
$677$	−11.2916	−0.433973	−0.216987	+	0.976175i	$0.569623\pi$
$678$	−83.1598	−3.19373
$679$	10.3041	0.395434
$680$	85.4847	3.27819
$681$	7.00792	0.268544
$682$	64.6892	2.47708
$683$	9.43449	0.361001	0.180500	−	0.983575i	$-0.442228\pi$
$683$	9.43449	0.361001	0.180500	+	0.983575i	$0.442228\pi$
$684$	24.0929	0.921214
$685$	64.5683	2.46703
$686$	50.2077	1.91694
$687$	−35.7585	−1.36427
$688$	−10.7110	−0.408352
$689$	73.1353	2.78623
$690$	−3.62538	−0.138016
$691$	44.7149	1.70104	0.850518	−	0.525946i	$-0.176289\pi$
$691$	44.7149	1.70104	0.850518	+	0.525946i	$0.176289\pi$
$692$	58.5626	2.22622
$693$	40.7177	1.54674
$694$	40.4656	1.53605
$695$	−11.2695	−0.427475
$696$	−157.834	−5.98266
$697$	1.79012	0.0678056
$698$	74.6097	2.82402
$699$	48.5357	1.83579
$700$	89.1112	3.36809
$701$	−14.7181	−0.555896	−0.277948	−	0.960596i	$-0.589654\pi$
$701$	−14.7181	−0.555896	−0.277948	+	0.960596i	$0.589654\pi$
$702$	73.7461	2.78337
$703$	−1.59139	−0.0600202
$704$	28.9204	1.08998
$705$	−28.8069	−1.08493
$706$	−68.5281	−2.57909
$707$	−36.5230	−1.37359
$708$	106.195	3.99103
$709$	−12.4198	−0.466434	−0.233217	−	0.972425i	$-0.574925\pi$
$709$	−12.4198	−0.466434	−0.233217	+	0.972425i	$0.574925\pi$
$710$	−135.816	−5.09709
$711$	1.92485	0.0721876
$712$	31.6612	1.18655
$713$	−1.07228	−0.0401573
$714$	61.2539	2.29237
$715$	−57.6972	−2.15775
$716$	−43.4824	−1.62501
$717$	−11.9556	−0.446491
$718$	27.3101	1.01920
$719$	39.8503	1.48616	0.743082	−	0.669200i	$-0.233363\pi$
$719$	39.8503	1.48616	0.743082	+	0.669200i	$0.233363\pi$
$720$	−171.743	−6.40047
$721$	27.7564	1.03370
$722$	−2.61964	−0.0974930
$723$	30.7334	1.14299
$724$	−90.1699	−3.35114
$725$	−53.8152	−1.99864
$726$	−4.01439	−0.148988
$727$	−2.18563	−0.0810604	−0.0405302	−	0.999178i	$-0.512905\pi$
$727$	−2.18563	−0.0810604	−0.0405302	+	0.999178i	$0.512905\pi$
$728$	−97.3035	−3.60631
$729$	−43.2529	−1.60196
$730$	−43.1868	−1.59842
$731$	−3.52269	−0.130291
$732$	−142.398	−5.26320
$733$	14.0231	0.517953	0.258977	−	0.965884i	$-0.416615\pi$
$733$	14.0231	0.517953	0.258977	+	0.965884i	$0.416615\pi$
$734$	81.3376	3.00223
$735$	−5.33879	−0.196924
$736$	−1.54273	−0.0568658
$737$	−1.60968	−0.0592933
$738$	−7.12251	−0.262183
$739$	−11.5138	−0.423542	−0.211771	−	0.977319i	$-0.567923\pi$
$739$	−11.5138	−0.423542	−0.211771	+	0.977319i	$0.567923\pi$
$740$	27.0406	0.994034
$741$	−14.4010	−0.529034
$742$	−95.3566	−3.50065
$743$	−9.91626	−0.363792	−0.181896	−	0.983318i	$-0.558223\pi$
$743$	−9.91626	−0.363792	−0.181896	+	0.983318i	$0.558223\pi$
$744$	−161.510	−5.92123
$745$	−36.9925	−1.35530
$746$	−51.0116	−1.86767
$747$	−10.8162	−0.395745
$748$	51.2951	1.87553
$749$	12.9645	0.473712
$750$	−57.0921	−2.08471
$751$	−44.2693	−1.61541	−0.807705	−	0.589587i	$-0.799290\pi$
$751$	−44.2693	−1.61541	−0.807705	+	0.589587i	$0.799290\pi$
$752$	−28.9916	−1.05722
$753$	−54.6806	−1.99267
$754$	99.8190	3.63520
$755$	−31.7730	−1.15634
$756$	−68.1303	−2.47788
$757$	−44.0438	−1.60080	−0.800399	−	0.599468i	$-0.795379\pi$
$757$	−44.0438	−1.60080	−0.800399	+	0.599468i	$0.795379\pi$
$758$	−18.3233	−0.665532
$759$	−1.28064	−0.0464844
$760$	26.2042	0.950526
$761$	−5.44608	−0.197420	−0.0987102	−	0.995116i	$-0.531472\pi$
$761$	−5.44608	−0.197420	−0.0987102	+	0.995116i	$0.531472\pi$
$762$	34.5132	1.25028
$763$	−44.9807	−1.62841
$764$	−21.2095	−0.767334
$765$	−56.4839	−2.04218
$766$	24.4419	0.883122
$767$	−39.5370	−1.42760
$768$	39.6985	1.43250
$769$	0.447380	0.0161330	0.00806648	−	0.999967i	$-0.497432\pi$
$769$	0.447380	0.0161330	0.00806648	+	0.999967i	$0.497432\pi$
$770$	75.2278	2.71102
$771$	51.6482	1.86006
$772$	−105.572	−3.79960
$773$	44.1238	1.58702	0.793511	−	0.608556i	$-0.208251\pi$
$773$	44.1238	1.58702	0.793511	+	0.608556i	$0.208251\pi$
$774$	14.0161	0.503797
$775$	−55.0686	−1.97812
$776$	−30.4047	−1.09146
$777$	11.4064	0.409203
$778$	−45.4632	−1.62994
$779$	0.548737	0.0196606
$780$	244.700	8.76168
$781$	−47.9762	−1.71672
$782$	−1.19998	−0.0429114
$783$	41.1446	1.47039
$784$	−5.37303	−0.191894
$785$	−76.7351	−2.73879
$786$	135.335	4.82724
$787$	21.7535	0.775427	0.387713	−	0.921780i	$-0.373265\pi$
$787$	21.7535	0.775427	0.387713	+	0.921780i	$0.373265\pi$
$788$	−4.65113	−0.165690
$789$	−12.2634	−0.436588
$790$	3.55625	0.126526
$791$	−28.6033	−1.01702
$792$	−120.147	−4.26925
$793$	53.0159	1.88265
$794$	87.1561	3.09306
$795$	141.171	5.00680
$796$	22.2364	0.788148
$797$	−36.7946	−1.30333	−0.651666	−	0.758506i	$-0.725930\pi$
$797$	−36.7946	−1.30333	−0.651666	+	0.758506i	$0.725930\pi$
$798$	18.7766	0.664683
$799$	−9.53496	−0.337323
$800$	−79.2291	−2.80117
$801$	−20.9201	−0.739176
$802$	31.5132	1.11277
$803$	−15.2555	−0.538355
$804$	6.82683	0.240764
$805$	−1.24697	−0.0439499
$806$	102.144	3.59787
$807$	13.1095	0.461476
$808$	107.770	3.79134
$809$	14.1327	0.496878	0.248439	−	0.968648i	$-0.420082\pi$
$809$	14.1327	0.496878	0.248439	+	0.968648i	$0.420082\pi$
$810$	6.27709	0.220554
$811$	−28.7826	−1.01070	−0.505348	−	0.862916i	$-0.668636\pi$
$811$	−28.7826	−1.01070	−0.505348	+	0.862916i	$0.668636\pi$
$812$	−92.2178	−3.23621
$813$	−60.9038	−2.13599
$814$	13.4807	0.472500
$815$	29.7431	1.04186
$816$	−91.2649	−3.19491
$817$	−1.07983	−0.0377786
$818$	−58.3666	−2.04074
$819$	64.2931	2.24658
$820$	−9.32409	−0.325611
$821$	43.5511	1.51994	0.759972	−	0.649956i	$-0.225212\pi$
$821$	43.5511	1.51994	0.759972	+	0.649956i	$0.225212\pi$
$822$	−136.520	−4.76168
$823$	55.1594	1.92274	0.961368	−	0.275267i	$-0.0887663\pi$
$823$	55.1594	1.92274	0.961368	+	0.275267i	$0.0887663\pi$
$824$	−81.9020	−2.85319
$825$	−65.7692	−2.28979
$826$	51.5498	1.79365
$827$	13.1756	0.458160	0.229080	−	0.973408i	$-0.426428\pi$
$827$	13.1756	0.458160	0.229080	+	0.973408i	$0.426428\pi$
$828$	3.38302	0.117568
$829$	20.5821	0.714846	0.357423	−	0.933943i	$-0.383655\pi$
$829$	20.5821	0.714846	0.357423	+	0.933943i	$0.383655\pi$
$830$	−19.9835	−0.693637
$831$	39.2077	1.36010
$832$	45.6652	1.58316
$833$	−1.76712	−0.0612269
$834$	23.8276	0.825082
$835$	77.7167	2.68950
$836$	15.7238	0.543820
$837$	42.1029	1.45529
$838$	54.5369	1.88395
$839$	−22.6837	−0.783128	−0.391564	−	0.920151i	$-0.628066\pi$
$839$	−22.6837	−0.783128	−0.391564	+	0.920151i	$0.628066\pi$
$840$	−187.822	−6.48046
$841$	26.6913	0.920389
$842$	81.6647	2.81435
$843$	−56.5328	−1.94709
$844$	4.86252	0.167375
$845$	−45.6757	−1.57129
$846$	37.9376	1.30432
$847$	−1.38077	−0.0474440
$848$	142.076	4.87891
$849$	5.24651	0.180060
$850$	−61.6269	−2.11379
$851$	−0.223456	−0.00765997
$852$	203.472	6.97085
$853$	24.1198	0.825846	0.412923	−	0.910766i	$-0.364508\pi$
$853$	24.1198	0.825846	0.412923	+	0.910766i	$0.364508\pi$
$854$	−69.1242	−2.36538
$855$	−17.3144	−0.592139
$856$	−38.2548	−1.30752
$857$	55.2831	1.88843	0.944217	−	0.329324i	$-0.106821\pi$
$857$	55.2831	1.88843	0.944217	+	0.329324i	$0.106821\pi$
$858$	121.992	4.16474
$859$	−2.41234	−0.0823081	−0.0411540	−	0.999153i	$-0.513103\pi$
$859$	−2.41234	−0.0823081	−0.0411540	+	0.999153i	$0.513103\pi$
$860$	18.3484	0.625676
$861$	−3.93314	−0.134041
$862$	38.6433	1.31619
$863$	−19.5846	−0.666669	−0.333334	−	0.942809i	$-0.608174\pi$
$863$	−19.5846	−0.666669	−0.333334	+	0.942809i	$0.608174\pi$
$864$	60.5749	2.06080
$865$	−42.0861	−1.43097
$866$	10.1759	0.345792
$867$	17.9315	0.608985
$868$	−94.3657	−3.20298
$869$	1.25622	0.0426145
$870$	192.677	6.53237
$871$	−2.54168	−0.0861215
$872$	132.727	4.49469
$873$	20.0898	0.679938
$874$	−0.367839	−0.0124424
$875$	−19.6372	−0.663858
$876$	64.7002	2.18602
$877$	34.4715	1.16402	0.582010	−	0.813182i	$-0.302266\pi$
$877$	34.4715	1.16402	0.582010	+	0.813182i	$0.302266\pi$
$878$	−15.6179	−0.527078
$879$	−91.0942	−3.07253
$880$	−112.085	−3.77839
$881$	−45.2599	−1.52485	−0.762423	−	0.647079i	$-0.775990\pi$
$881$	−45.2599	−1.52485	−0.762423	+	0.647079i	$0.775990\pi$
$882$	7.03099	0.236746
$883$	−47.2362	−1.58962	−0.794812	−	0.606856i	$-0.792431\pi$
$883$	−47.2362	−1.58962	−0.794812	+	0.606856i	$0.792431\pi$
$884$	80.9948	2.72415
$885$	−76.3169	−2.56536
$886$	28.6690	0.963155
$887$	−32.0304	−1.07547	−0.537737	−	0.843112i	$-0.680721\pi$
$887$	−32.0304	−1.07547	−0.537737	+	0.843112i	$0.680721\pi$
$888$	−33.6574	−1.12947
$889$	11.8710	0.398142
$890$	−38.6509	−1.29558
$891$	2.21734	0.0742838
$892$	−73.6495	−2.46597
$893$	−2.92282	−0.0978083
$894$	78.2150	2.61590
$895$	31.2487	1.04453
$896$	−3.69767	−0.123530
$897$	−2.02213	−0.0675170
$898$	−52.6389	−1.75658
$899$	56.9884	1.90067
$900$	173.740	5.79133
$901$	46.7269	1.55670
$902$	−4.64840	−0.154775
$903$	7.73983	0.257566
$904$	84.4009	2.80713
$905$	64.8008	2.15405
$906$	67.1792	2.23188
$907$	−33.6567	−1.11755	−0.558776	−	0.829318i	$-0.688729\pi$
$907$	−33.6567	−1.11755	−0.558776	+	0.829318i	$0.688729\pi$
$908$	−12.0819	−0.400952
$909$	−71.2089	−2.36185
$910$	118.784	3.93767
$911$	−56.6691	−1.87753	−0.938765	−	0.344558i	$-0.888029\pi$
$911$	−56.6691	−1.87753	−0.938765	+	0.344558i	$0.888029\pi$
$912$	−27.9760	−0.926379
$913$	−7.05904	−0.233620
$914$	49.2093	1.62770
$915$	102.335	3.38309
$916$	61.6490	2.03694
$917$	46.5493	1.53719
$918$	47.1171	1.55510
$919$	−35.5443	−1.17250	−0.586249	−	0.810131i	$-0.699396\pi$
$919$	−35.5443	−1.17250	−0.586249	+	0.810131i	$0.699396\pi$
$920$	3.67949	0.121309
$921$	46.1163	1.51958
$922$	−14.0017	−0.461120
$923$	−75.7543	−2.49348
$924$	−112.702	−3.70763
$925$	−11.4759	−0.377325
$926$	−12.4100	−0.407819
$927$	54.1166	1.77742
$928$	81.9912	2.69149
$929$	14.3698	0.471457	0.235729	−	0.971819i	$-0.424252\pi$
$929$	14.3698	0.471457	0.235729	+	0.971819i	$0.424252\pi$
$930$	197.165	6.46530
$931$	−0.541686	−0.0177530
$932$	−83.6773	−2.74094
$933$	73.3372	2.40095
$934$	−87.4260	−2.86066
$935$	−36.8633	−1.20556
$936$	−189.712	−6.20094
$937$	9.49677	0.310246	0.155123	−	0.987895i	$-0.450423\pi$
$937$	9.49677	0.310246	0.155123	+	0.987895i	$0.450423\pi$
$938$	3.31394	0.108204
$939$	−15.1660	−0.494925
$940$	49.6642	1.61987
$941$	−28.5965	−0.932219	−0.466109	−	0.884727i	$-0.654345\pi$
$941$	−28.5965	−0.932219	−0.466109	+	0.884727i	$0.654345\pi$
$942$	162.245	5.28622
$943$	0.0770515	0.00250914
$944$	−76.8063	−2.49983
$945$	48.9620	1.59273
$946$	9.14736	0.297406
$947$	50.6715	1.64660	0.823302	−	0.567604i	$-0.192129\pi$
$947$	50.6715	1.64660	0.823302	+	0.567604i	$0.192129\pi$
$948$	−5.32779	−0.173039
$949$	−24.0884	−0.781941
$950$	−18.8909	−0.612902
$951$	−87.5559	−2.83919
$952$	−62.1681	−2.01488
$953$	−6.59431	−0.213611	−0.106805	−	0.994280i	$-0.534062\pi$
$953$	−6.59431	−0.213611	−0.106805	+	0.994280i	$0.534062\pi$
$954$	−185.916	−6.01927
$955$	15.2423	0.493228
$956$	20.6119	0.666638
$957$	68.0621	2.20013
$958$	−72.7929	−2.35183
$959$	−46.9569	−1.51632
$960$	88.1459	2.84490
$961$	27.3158	0.881154
$962$	21.2860	0.686290
$963$	25.2768	0.814534
$964$	−52.9855	−1.70655
$965$	75.8692	2.44232
$966$	2.63653	0.0848290
$967$	47.3508	1.52270	0.761350	−	0.648341i	$-0.224537\pi$
$967$	47.3508	1.52270	0.761350	+	0.648341i	$0.224537\pi$
$968$	4.07431	0.130953
$969$	−9.20094	−0.295577
$970$	37.1169	1.19175
$971$	43.5047	1.39613	0.698066	−	0.716033i	$-0.254044\pi$
$971$	43.5047	1.39613	0.698066	+	0.716033i	$0.254044\pi$
$972$	71.0230	2.27806
$973$	8.19564	0.262740
$974$	33.0102	1.05771
$975$	−103.849	−3.32584
$976$	102.991	3.29667
$977$	−56.9925	−1.82335	−0.911676	−	0.410910i	$-0.865211\pi$
$977$	−56.9925	−1.82335	−0.911676	+	0.410910i	$0.865211\pi$
$978$	−62.8873	−2.01091
$979$	−13.6532	−0.436357
$980$	9.20427	0.294020
$981$	−87.6989	−2.80001
$982$	75.0157	2.39385
$983$	−40.8934	−1.30430	−0.652149	−	0.758091i	$-0.726132\pi$
$983$	−40.8934	−1.30430	−0.652149	+	0.758091i	$0.726132\pi$
$984$	11.6057	0.369975
$985$	3.34254	0.106502
$986$	63.7754	2.03102
$987$	20.9496	0.666833
$988$	24.8279	0.789880
$989$	−0.151626	−0.00482142
$990$	146.671	4.66153
$991$	−27.7482	−0.881451	−0.440726	−	0.897642i	$-0.645279\pi$
$991$	−27.7482	−0.881451	−0.440726	+	0.897642i	$0.645279\pi$
$992$	83.9009	2.66386
$993$	67.1868	2.13211
$994$	98.7713	3.13283
$995$	−15.9802	−0.506607
$996$	29.9382	0.948628
$997$	50.9146	1.61248	0.806241	−	0.591587i	$-0.201498\pi$
$997$	50.9146	1.61248	0.806241	+	0.591587i	$0.201498\pi$
$998$	−0.182650	−0.00578167
$999$	8.77393	0.277595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.6	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.6	✓	71	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.61964 q^{2} -2.82043 q^{3} +4.86252 q^{4} -3.49446 q^{5} +7.38851 q^{6} +2.54132 q^{7} -7.49878 q^{8} +4.95481 q^{9} +O(q^{10})\)	\(q-2.61964 q^{2} -2.82043 q^{3} +4.86252 q^{4} -3.49446 q^{5} +7.38851 q^{6} +2.54132 q^{7} -7.49878 q^{8} +4.95481 q^{9} +9.15424 q^{10} +3.23368 q^{11} -13.7144 q^{12} +5.10597 q^{13} -6.65735 q^{14} +9.85587 q^{15} +9.91908 q^{16} +3.26225 q^{17} -12.9798 q^{18} +1.00000 q^{19} -16.9919 q^{20} -7.16761 q^{21} -8.47108 q^{22} +0.140416 q^{23} +21.1498 q^{24} +7.21126 q^{25} -13.3758 q^{26} -5.51340 q^{27} +12.3572 q^{28} -7.46266 q^{29} -25.8189 q^{30} -7.63648 q^{31} -10.9869 q^{32} -9.12035 q^{33} -8.54593 q^{34} -8.88055 q^{35} +24.0929 q^{36} -1.59139 q^{37} -2.61964 q^{38} -14.4010 q^{39} +26.2042 q^{40} +0.548737 q^{41} +18.7766 q^{42} -1.07983 q^{43} +15.7238 q^{44} -17.3144 q^{45} -0.367839 q^{46} -2.92282 q^{47} -27.9760 q^{48} -0.541686 q^{49} -18.8909 q^{50} -9.20094 q^{51} +24.8279 q^{52} +14.3235 q^{53} +14.4431 q^{54} -11.3000 q^{55} -19.0568 q^{56} -2.82043 q^{57} +19.5495 q^{58} -7.74329 q^{59} +47.9244 q^{60} +10.3831 q^{61} +20.0048 q^{62} +12.5918 q^{63} +8.94349 q^{64} -17.8426 q^{65} +23.8921 q^{66} -0.497786 q^{67} +15.8628 q^{68} -0.396033 q^{69} +23.2639 q^{70} -14.8364 q^{71} -37.1550 q^{72} -4.71769 q^{73} +4.16886 q^{74} -20.3388 q^{75} +4.86252 q^{76} +8.21781 q^{77} +37.7255 q^{78} +0.388482 q^{79} -34.6618 q^{80} +0.685703 q^{81} -1.43750 q^{82} -2.18298 q^{83} -34.8527 q^{84} -11.3998 q^{85} +2.82878 q^{86} +21.0479 q^{87} -24.2486 q^{88} -4.22218 q^{89} +45.3575 q^{90} +12.9759 q^{91} +0.682776 q^{92} +21.5381 q^{93} +7.65673 q^{94} -3.49446 q^{95} +30.9877 q^{96} +4.05461 q^{97} +1.41902 q^{98} +16.0223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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