LMFDB - Embedded newform 8047.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55403 q^{2} +1.62979 q^{3} +4.52309 q^{4} +1.02400 q^{5} -4.16254 q^{6} -2.90866 q^{7} -6.44404 q^{8} -0.343777 q^{9} +O(q^{10})$	\(q-2.55403 q^{2} +1.62979 q^{3} +4.52309 q^{4} +1.02400 q^{5} -4.16254 q^{6} -2.90866 q^{7} -6.44404 q^{8} -0.343777 q^{9} -2.61533 q^{10} +3.97579 q^{11} +7.37169 q^{12} +1.00000 q^{13} +7.42882 q^{14} +1.66891 q^{15} +7.41213 q^{16} +1.98444 q^{17} +0.878017 q^{18} -4.42735 q^{19} +4.63164 q^{20} -4.74052 q^{21} -10.1543 q^{22} +0.308540 q^{23} -10.5025 q^{24} -3.95142 q^{25} -2.55403 q^{26} -5.44966 q^{27} -13.1561 q^{28} +8.91343 q^{29} -4.26245 q^{30} +6.21590 q^{31} -6.04274 q^{32} +6.47971 q^{33} -5.06833 q^{34} -2.97847 q^{35} -1.55493 q^{36} +3.18018 q^{37} +11.3076 q^{38} +1.62979 q^{39} -6.59870 q^{40} -0.0267494 q^{41} +12.1074 q^{42} -9.51525 q^{43} +17.9828 q^{44} -0.352027 q^{45} -0.788021 q^{46} -4.23608 q^{47} +12.0802 q^{48} +1.46032 q^{49} +10.0921 q^{50} +3.23423 q^{51} +4.52309 q^{52} -14.1988 q^{53} +13.9186 q^{54} +4.07121 q^{55} +18.7435 q^{56} -7.21565 q^{57} -22.7652 q^{58} -6.82188 q^{59} +7.54861 q^{60} -5.60653 q^{61} -15.8756 q^{62} +0.999930 q^{63} +0.609101 q^{64} +1.02400 q^{65} -16.5494 q^{66} +2.47014 q^{67} +8.97580 q^{68} +0.502856 q^{69} +7.60711 q^{70} -2.65237 q^{71} +2.21531 q^{72} +2.32901 q^{73} -8.12229 q^{74} -6.44000 q^{75} -20.0253 q^{76} -11.5642 q^{77} -4.16254 q^{78} -12.0907 q^{79} +7.59003 q^{80} -7.85049 q^{81} +0.0683188 q^{82} +10.0470 q^{83} -21.4418 q^{84} +2.03207 q^{85} +24.3023 q^{86} +14.5270 q^{87} -25.6202 q^{88} +15.5551 q^{89} +0.899090 q^{90} -2.90866 q^{91} +1.39555 q^{92} +10.1306 q^{93} +10.8191 q^{94} -4.53360 q^{95} -9.84842 q^{96} -2.41387 q^{97} -3.72969 q^{98} -1.36678 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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 $

$2$	−2.55403	−1.80597	−0.902987	−	0.429668i	$-0.858631\pi$
$2$	−2.55403	−1.80597	−0.902987	+	0.429668i	$0.858631\pi$
$3$	1.62979	0.940961	0.470481	−	0.882410i	$-0.344081\pi$
$3$	1.62979	0.940961	0.470481	+	0.882410i	$0.344081\pi$
$4$	4.52309	2.26154
$5$	1.02400	0.457947	0.228973	−	0.973433i	$-0.426463\pi$
$5$	1.02400	0.457947	0.228973	+	0.973433i	$0.426463\pi$
$6$	−4.16254	−1.69935
$7$	−2.90866	−1.09937	−0.549685	−	0.835372i	$-0.685252\pi$
$7$	−2.90866	−1.09937	−0.549685	+	0.835372i	$0.685252\pi$
$8$	−6.44404	−2.27831
$9$	−0.343777	−0.114592
$10$	−2.61533	−0.827040
$11$	3.97579	1.19875	0.599373	−	0.800470i	$-0.295417\pi$
$11$	3.97579	1.19875	0.599373	+	0.800470i	$0.295417\pi$
$12$	7.37169	2.12802
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.42882	1.98544
$15$	1.66891	0.430910
$16$	7.41213	1.85303
$17$	1.98444	0.481298	0.240649	−	0.970612i	$-0.422640\pi$
$17$	1.98444	0.481298	0.240649	+	0.970612i	$0.422640\pi$
$18$	0.878017	0.206951
$19$	−4.42735	−1.01570	−0.507851	−	0.861445i	$-0.669560\pi$
$19$	−4.42735	−1.01570	−0.507851	+	0.861445i	$0.669560\pi$
$20$	4.63164	1.03567
$21$	−4.74052	−1.03447
$22$	−10.1543	−2.16490
$23$	0.308540	0.0643350	0.0321675	−	0.999482i	$-0.489759\pi$
$23$	0.308540	0.0643350	0.0321675	+	0.999482i	$0.489759\pi$
$24$	−10.5025	−2.14380
$25$	−3.95142	−0.790285
$26$	−2.55403	−0.500887
$27$	−5.44966	−1.04879
$28$	−13.1561	−2.48627
$29$	8.91343	1.65518	0.827591	−	0.561331i	$-0.189710\pi$
$29$	8.91343	1.65518	0.827591	+	0.561331i	$0.189710\pi$
$30$	−4.26245	−0.778213
$31$	6.21590	1.11641	0.558204	−	0.829704i	$-0.311491\pi$
$31$	6.21590	1.11641	0.558204	+	0.829704i	$0.311491\pi$
$32$	−6.04274	−1.06822
$33$	6.47971	1.12797
$34$	−5.06833	−0.869211
$35$	−2.97847	−0.503453
$36$	−1.55493	−0.259155
$37$	3.18018	0.522819	0.261409	−	0.965228i	$-0.415813\pi$
$37$	3.18018	0.522819	0.261409	+	0.965228i	$0.415813\pi$
$38$	11.3076	1.83433
$39$	1.62979	0.260976
$40$	−6.59870	−1.04335
$41$	−0.0267494	−0.00417755	−0.00208878	−	0.999998i	$-0.500665\pi$
$41$	−0.0267494	−0.00417755	−0.00208878	+	0.999998i	$0.500665\pi$
$42$	12.1074	1.86822
$43$	−9.51525	−1.45106	−0.725531	−	0.688190i	$-0.758406\pi$
$43$	−9.51525	−1.45106	−0.725531	+	0.688190i	$0.758406\pi$
$44$	17.9828	2.71101
$45$	−0.352027	−0.0524771
$46$	−0.788021	−0.116187
$47$	−4.23608	−0.617896	−0.308948	−	0.951079i	$-0.599977\pi$
$47$	−4.23608	−0.617896	−0.308948	+	0.951079i	$0.599977\pi$
$48$	12.0802	1.74363
$49$	1.46032	0.208616
$50$	10.0921	1.42723
$51$	3.23423	0.452882
$52$	4.52309	0.627239
$53$	−14.1988	−1.95036	−0.975180	−	0.221415i	$-0.928932\pi$
$53$	−14.1988	−1.95036	−0.975180	+	0.221415i	$0.928932\pi$
$54$	13.9186	1.89408
$55$	4.07121	0.548962
$56$	18.7435	2.50471
$57$	−7.21565	−0.955737
$58$	−22.7652	−2.98922
$59$	−6.82188	−0.888133	−0.444067	−	0.895994i	$-0.646465\pi$
$59$	−6.82188	−0.888133	−0.444067	+	0.895994i	$0.646465\pi$
$60$	7.54861	0.974522
$61$	−5.60653	−0.717842	−0.358921	−	0.933368i	$-0.616855\pi$
$61$	−5.60653	−0.717842	−0.358921	+	0.933368i	$0.616855\pi$
$62$	−15.8756	−2.01620
$63$	0.999930	0.125979
$64$	0.609101	0.0761376
$65$	1.02400	0.127012
$66$	−16.5494	−2.03709
$67$	2.47014	0.301776	0.150888	−	0.988551i	$-0.451787\pi$
$67$	2.47014	0.301776	0.150888	+	0.988551i	$0.451787\pi$
$68$	8.97580	1.08848
$69$	0.502856	0.0605367
$70$	7.60711	0.909224
$71$	−2.65237	−0.314779	−0.157389	−	0.987537i	$-0.550308\pi$
$71$	−2.65237	−0.314779	−0.157389	+	0.987537i	$0.550308\pi$
$72$	2.21531	0.261077
$73$	2.32901	0.272590	0.136295	−	0.990668i	$-0.456481\pi$
$73$	2.32901	0.272590	0.136295	+	0.990668i	$0.456481\pi$
$74$	−8.12229	−0.944198
$75$	−6.44000	−0.743627
$76$	−20.0253	−2.29706
$77$	−11.5642	−1.31787
$78$	−4.16254	−0.471315
$79$	−12.0907	−1.36031	−0.680157	−	0.733067i	$-0.738088\pi$
$79$	−12.0907	−1.36031	−0.680157	+	0.733067i	$0.738088\pi$
$80$	7.59003	0.848591
$81$	−7.85049	−0.872276
$82$	0.0683188	0.00754455
$83$	10.0470	1.10280	0.551399	−	0.834242i	$-0.314094\pi$
$83$	10.0470	1.10280	0.551399	+	0.834242i	$0.314094\pi$
$84$	−21.4418	−2.33949
$85$	2.03207	0.220409
$86$	24.3023	2.62058
$87$	14.5270	1.55746
$88$	−25.6202	−2.73112
$89$	15.5551	1.64884	0.824418	−	0.565981i	$-0.191503\pi$
$89$	15.5551	1.64884	0.824418	+	0.565981i	$0.191503\pi$
$90$	0.899090	0.0947724
$91$	−2.90866	−0.304911
$92$	1.39555	0.145496
$93$	10.1306	1.05050
$94$	10.8191	1.11590
$95$	−4.53360	−0.465138
$96$	−9.84842	−1.00515
$97$	−2.41387	−0.245092	−0.122546	−	0.992463i	$-0.539106\pi$
$97$	−2.41387	−0.245092	−0.122546	+	0.992463i	$0.539106\pi$
$98$	−3.72969	−0.376756
$99$	−1.36678	−0.137367
$100$	−17.8726	−1.78726
$101$	−10.6593	−1.06064	−0.530318	−	0.847799i	$-0.677927\pi$
$101$	−10.6593	−1.06064	−0.530318	+	0.847799i	$0.677927\pi$
$102$	−8.26033	−0.817894
$103$	−6.09693	−0.600748	−0.300374	−	0.953821i	$-0.597112\pi$
$103$	−6.09693	−0.600748	−0.300374	+	0.953821i	$0.597112\pi$
$104$	−6.44404	−0.631891
$105$	−4.85429	−0.473730
$106$	36.2643	3.52230
$107$	−7.31454	−0.707123	−0.353562	−	0.935411i	$-0.615030\pi$
$107$	−7.31454	−0.707123	−0.353562	+	0.935411i	$0.615030\pi$
$108$	−24.6493	−2.37188
$109$	−6.67058	−0.638926	−0.319463	−	0.947599i	$-0.603503\pi$
$109$	−6.67058	−0.638926	−0.319463	+	0.947599i	$0.603503\pi$
$110$	−10.3980	−0.991411
$111$	5.18304	0.491952
$112$	−21.5594	−2.03717
$113$	−2.69215	−0.253256	−0.126628	−	0.991950i	$-0.540415\pi$
$113$	−2.69215	−0.253256	−0.126628	+	0.991950i	$0.540415\pi$
$114$	18.4290	1.72604
$115$	0.315945	0.0294620
$116$	40.3162	3.74327
$117$	−0.343777	−0.0317822
$118$	17.4233	1.60395
$119$	−5.77207	−0.529125
$120$	−10.7545	−0.981749
$121$	4.80690	0.436991
$122$	14.3193	1.29640
$123$	−0.0435959	−0.00393091
$124$	28.1150	2.52480
$125$	−9.16626	−0.819855
$126$	−2.55385	−0.227515
$127$	19.6413	1.74288	0.871441	−	0.490501i	$-0.163186\pi$
$127$	19.6413	1.74288	0.871441	+	0.490501i	$0.163186\pi$
$128$	10.5298	0.930713
$129$	−15.5079	−1.36539
$130$	−2.61533	−0.229380
$131$	7.20969	0.629914	0.314957	−	0.949106i	$-0.398010\pi$
$131$	7.20969	0.629914	0.314957	+	0.949106i	$0.398010\pi$
$132$	29.3083	2.55096
$133$	12.8777	1.11663
$134$	−6.30882	−0.544999
$135$	−5.58046	−0.480289
$136$	−12.7878	−1.09655
$137$	6.56121	0.560562	0.280281	−	0.959918i	$-0.409572\pi$
$137$	6.56121	0.560562	0.280281	+	0.959918i	$0.409572\pi$
$138$	−1.28431	−0.109328
$139$	17.1211	1.45219	0.726096	−	0.687594i	$-0.241333\pi$
$139$	17.1211	1.45219	0.726096	+	0.687594i	$0.241333\pi$
$140$	−13.4719	−1.13858
$141$	−6.90394	−0.581416
$142$	6.77425	0.568482
$143$	3.97579	0.332472
$144$	−2.54812	−0.212343
$145$	9.12736	0.757986
$146$	−5.94836	−0.492290
$147$	2.38001	0.196300
$148$	14.3842	1.18238
$149$	11.7777	0.964864	0.482432	−	0.875933i	$-0.339754\pi$
$149$	11.7777	0.964864	0.482432	+	0.875933i	$0.339754\pi$
$150$	16.4480	1.34297
$151$	6.67775	0.543428	0.271714	−	0.962378i	$-0.412410\pi$
$151$	6.67775	0.543428	0.271714	+	0.962378i	$0.412410\pi$
$152$	28.5300	2.31409
$153$	−0.682205	−0.0551530
$154$	29.5354	2.38003
$155$	6.36508	0.511255
$156$	7.37169	0.590208
$157$	−21.3365	−1.70284	−0.851421	−	0.524484i	$-0.824258\pi$
$157$	−21.3365	−1.70284	−0.851421	+	0.524484i	$0.824258\pi$
$158$	30.8801	2.45669
$159$	−23.1411	−1.83521
$160$	−6.18777	−0.489186
$161$	−0.897438	−0.0707280
$162$	20.0504	1.57531
$163$	3.18815	0.249715	0.124857	−	0.992175i	$-0.460153\pi$
$163$	3.18815	0.249715	0.124857	+	0.992175i	$0.460153\pi$
$164$	−0.120990	−0.00944771
$165$	6.63523	0.516552
$166$	−25.6603	−1.99162
$167$	3.88570	0.300685	0.150342	−	0.988634i	$-0.451962\pi$
$167$	3.88570	0.300685	0.150342	+	0.988634i	$0.451962\pi$
$168$	30.5481	2.35684
$169$	1.00000	0.0769231
$170$	−5.18997	−0.398053
$171$	1.52202	0.116392
$172$	−43.0383	−3.28164
$173$	−16.1138	−1.22511	−0.612553	−	0.790429i	$-0.709857\pi$
$173$	−16.1138	−1.22511	−0.612553	+	0.790429i	$0.709857\pi$
$174$	−37.1026	−2.81274
$175$	11.4934	0.868816
$176$	29.4691	2.22132
$177$	−11.1182	−0.835699
$178$	−39.7282	−2.97776
$179$	−14.3389	−1.07174	−0.535868	−	0.844302i	$-0.680016\pi$
$179$	−14.3389	−1.07174	−0.535868	+	0.844302i	$0.680016\pi$
$180$	−1.59225	−0.118679
$181$	15.1000	1.12237	0.561187	−	0.827689i	$-0.310345\pi$
$181$	15.1000	1.12237	0.561187	+	0.827689i	$0.310345\pi$
$182$	7.42882	0.550661
$183$	−9.13748	−0.675462
$184$	−1.98824	−0.146575
$185$	3.25651	0.239423
$186$	−25.8739	−1.89717
$187$	7.88972	0.576954
$188$	−19.1602	−1.39740
$189$	15.8512	1.15301
$190$	11.5790	0.840027
$191$	1.14122	0.0825754	0.0412877	−	0.999147i	$-0.486854\pi$
$191$	1.14122	0.0825754	0.0412877	+	0.999147i	$0.486854\pi$
$192$	0.992708	0.0716426
$193$	−2.92073	−0.210239	−0.105119	−	0.994460i	$-0.533523\pi$
$193$	−2.92073	−0.210239	−0.105119	+	0.994460i	$0.533523\pi$
$194$	6.16511	0.442629
$195$	1.66891	0.119513
$196$	6.60513	0.471795
$197$	−7.08505	−0.504789	−0.252394	−	0.967624i	$-0.581218\pi$
$197$	−7.08505	−0.504789	−0.252394	+	0.967624i	$0.581218\pi$
$198$	3.49081	0.248081
$199$	−0.196138	−0.0139038	−0.00695191	−	0.999976i	$-0.502213\pi$
$199$	−0.196138	−0.0139038	−0.00695191	+	0.999976i	$0.502213\pi$
$200$	25.4631	1.80052
$201$	4.02582	0.283959
$202$	27.2241	1.91548
$203$	−25.9262	−1.81966
$204$	14.6287	1.02421
$205$	−0.0273914	−0.00191310
$206$	15.5718	1.08494
$207$	−0.106069	−0.00737229
$208$	7.41213	0.513939
$209$	−17.6022	−1.21757
$210$	12.3980	0.855544
$211$	23.3760	1.60927	0.804634	−	0.593770i	$-0.202361\pi$
$211$	23.3760	1.60927	0.804634	+	0.593770i	$0.202361\pi$
$212$	−64.2225	−4.41082
$213$	−4.32282	−0.296195
$214$	18.6816	1.27705
$215$	−9.74361	−0.664509
$216$	35.1179	2.38947
$217$	−18.0799	−1.22735
$218$	17.0369	1.15388
$219$	3.79580	0.256496
$220$	18.4144	1.24150
$221$	1.98444	0.133488
$222$	−13.2377	−0.888453
$223$	−26.1888	−1.75373	−0.876866	−	0.480735i	$-0.840370\pi$
$223$	−26.1888	−1.75373	−0.876866	+	0.480735i	$0.840370\pi$
$224$	17.5763	1.17437
$225$	1.35841	0.0905605
$226$	6.87584	0.457374
$227$	−6.63089	−0.440108	−0.220054	−	0.975488i	$-0.570623\pi$
$227$	−6.63089	−0.440108	−0.220054	+	0.975488i	$0.570623\pi$
$228$	−32.6370	−2.16144
$229$	0.611183	0.0403881	0.0201941	−	0.999796i	$-0.493572\pi$
$229$	0.611183	0.0403881	0.0201941	+	0.999796i	$0.493572\pi$
$230$	−0.806933	−0.0532076
$231$	−18.8473	−1.24006
$232$	−57.4386	−3.77103
$233$	−5.30818	−0.347750	−0.173875	−	0.984768i	$-0.555629\pi$
$233$	−5.30818	−0.347750	−0.173875	+	0.984768i	$0.555629\pi$
$234$	0.878017	0.0573978
$235$	−4.33775	−0.282964
$236$	−30.8559	−2.00855
$237$	−19.7054	−1.28000
$238$	14.7421	0.955586
$239$	−8.62869	−0.558144	−0.279072	−	0.960270i	$-0.590027\pi$
$239$	−8.62869	−0.558144	−0.279072	+	0.960270i	$0.590027\pi$
$240$	12.3702	0.798491
$241$	−18.2713	−1.17696	−0.588480	−	0.808512i	$-0.700273\pi$
$241$	−18.2713	−1.17696	−0.588480	+	0.808512i	$0.700273\pi$
$242$	−12.2770	−0.789195
$243$	3.55432	0.228010
$244$	−25.3588	−1.62343
$245$	1.49536	0.0955352
$246$	0.111345	0.00709913
$247$	−4.42735	−0.281705
$248$	−40.0555	−2.54353
$249$	16.3745	1.03769
$250$	23.4109	1.48064
$251$	−5.12680	−0.323601	−0.161801	−	0.986823i	$-0.551730\pi$
$251$	−5.12680	−0.323601	−0.161801	+	0.986823i	$0.551730\pi$
$252$	4.52277	0.284908
$253$	1.22669	0.0771213
$254$	−50.1645	−3.14760
$255$	3.31185	0.207396
$256$	−28.1117	−1.75698
$257$	−2.73524	−0.170620	−0.0853098	−	0.996354i	$-0.527188\pi$
$257$	−2.73524	−0.170620	−0.0853098	+	0.996354i	$0.527188\pi$
$258$	39.6076	2.46586
$259$	−9.25008	−0.574772
$260$	4.63164	0.287242
$261$	−3.06423	−0.189671
$262$	−18.4138	−1.13761
$263$	12.1182	0.747241	0.373621	−	0.927582i	$-0.378116\pi$
$263$	12.1182	0.747241	0.373621	+	0.927582i	$0.378116\pi$
$264$	−41.7555	−2.56988
$265$	−14.5396	−0.893161
$266$	−32.8900	−2.01661
$267$	25.3516	1.55149
$268$	11.1727	0.682479
$269$	6.00156	0.365922	0.182961	−	0.983120i	$-0.441432\pi$
$269$	6.00156	0.365922	0.182961	+	0.983120i	$0.441432\pi$
$270$	14.2527	0.867390
$271$	−4.34502	−0.263941	−0.131971	−	0.991254i	$-0.542130\pi$
$271$	−4.34502	−0.263941	−0.131971	+	0.991254i	$0.542130\pi$
$272$	14.7089	0.891861
$273$	−4.74052	−0.286909
$274$	−16.7575	−1.01236
$275$	−15.7100	−0.947350
$276$	2.27446	0.136906
$277$	0.625737	0.0375969	0.0187984	−	0.999823i	$-0.494016\pi$
$277$	0.625737	0.0375969	0.0187984	+	0.999823i	$0.494016\pi$
$278$	−43.7278	−2.62262
$279$	−2.13688	−0.127932
$280$	19.1934	1.14702
$281$	−4.69727	−0.280216	−0.140108	−	0.990136i	$-0.544745\pi$
$281$	−4.69727	−0.280216	−0.140108	+	0.990136i	$0.544745\pi$
$282$	17.6329	1.05002
$283$	−2.01970	−0.120059	−0.0600293	−	0.998197i	$-0.519119\pi$
$283$	−2.01970	−0.120059	−0.0600293	+	0.998197i	$0.519119\pi$
$284$	−11.9969	−0.711886
$285$	−7.38883	−0.437677
$286$	−10.1543	−0.600436
$287$	0.0778049	0.00459268
$288$	2.07735	0.122409
$289$	−13.0620	−0.768352
$290$	−23.3116	−1.36890
$291$	−3.93411	−0.230622
$292$	10.5343	0.616473
$293$	−30.0046	−1.75289	−0.876445	−	0.481502i	$-0.840091\pi$
$293$	−30.0046	−1.75289	−0.876445	+	0.481502i	$0.840091\pi$
$294$	−6.07863	−0.354513
$295$	−6.98561	−0.406718
$296$	−20.4932	−1.19115
$297$	−21.6667	−1.25723
$298$	−30.0805	−1.74252
$299$	0.308540	0.0178433
$300$	−29.1287	−1.68174
$301$	27.6766	1.59525
$302$	−17.0552	−0.981417
$303$	−17.3724	−0.998017
$304$	−32.8161	−1.88213
$305$	−5.74109	−0.328734
$306$	1.74237	0.0996048
$307$	−1.51115	−0.0862457	−0.0431228	−	0.999070i	$-0.513731\pi$
$307$	−1.51115	−0.0862457	−0.0431228	+	0.999070i	$0.513731\pi$
$308$	−52.3060	−2.98041
$309$	−9.93673	−0.565281
$310$	−16.2566	−0.923314
$311$	−21.5447	−1.22169	−0.610845	−	0.791750i	$-0.709170\pi$
$311$	−21.5447	−1.22169	−0.610845	+	0.791750i	$0.709170\pi$
$312$	−10.5025	−0.594584
$313$	−14.5264	−0.821083	−0.410542	−	0.911842i	$-0.634660\pi$
$313$	−14.5264	−0.821083	−0.410542	+	0.911842i	$0.634660\pi$
$314$	54.4942	3.07529
$315$	1.02393	0.0576918
$316$	−54.6874	−3.07641
$317$	−14.5019	−0.814508	−0.407254	−	0.913315i	$-0.633514\pi$
$317$	−14.5019	−0.814508	−0.407254	+	0.913315i	$0.633514\pi$
$318$	59.1033	3.31435
$319$	35.4379	1.98414
$320$	0.623720	0.0348670
$321$	−11.9212	−0.665376
$322$	2.29209	0.127733
$323$	−8.78581	−0.488855
$324$	−35.5084	−1.97269
$325$	−3.95142	−0.219186
$326$	−8.14263	−0.450979
$327$	−10.8717	−0.601205
$328$	0.172374	0.00951777
$329$	12.3213	0.679297
$330$	−16.9466	−0.932879
$331$	26.7247	1.46892	0.734461	−	0.678651i	$-0.237435\pi$
$331$	26.7247	1.46892	0.734461	+	0.678651i	$0.237435\pi$
$332$	45.4433	2.49402
$333$	−1.09327	−0.0599110
$334$	−9.92421	−0.543029
$335$	2.52942	0.138197
$336$	−35.1373	−1.91690
$337$	−16.4810	−0.897776	−0.448888	−	0.893588i	$-0.648180\pi$
$337$	−16.4810	−0.897776	−0.448888	+	0.893588i	$0.648180\pi$
$338$	−2.55403	−0.138921
$339$	−4.38765	−0.238304
$340$	9.19122	0.498464
$341$	24.7131	1.33829
$342$	−3.88728	−0.210200
$343$	16.1131	0.870024
$344$	61.3167	3.30597
$345$	0.514924	0.0277226
$346$	41.1551	2.21251
$347$	−0.946071	−0.0507877	−0.0253939	−	0.999678i	$-0.508084\pi$
$347$	−0.946071	−0.0507877	−0.0253939	+	0.999678i	$0.508084\pi$
$348$	65.7071	3.52227
$349$	25.3789	1.35850	0.679250	−	0.733907i	$-0.262305\pi$
$349$	25.3789	1.35850	0.679250	+	0.733907i	$0.262305\pi$
$350$	−29.3544	−1.56906
$351$	−5.44966	−0.290881
$352$	−24.0247	−1.28052
$353$	−11.7032	−0.622897	−0.311449	−	0.950263i	$-0.600814\pi$
$353$	−11.7032	−0.622897	−0.311449	+	0.950263i	$0.600814\pi$
$354$	28.3964	1.50925
$355$	−2.71603	−0.144152
$356$	70.3570	3.72891
$357$	−9.40728	−0.497886
$358$	36.6219	1.93553
$359$	−4.76066	−0.251258	−0.125629	−	0.992077i	$-0.540095\pi$
$359$	−4.76066	−0.251258	−0.125629	+	0.992077i	$0.540095\pi$
$360$	2.26848	0.119559
$361$	0.601391	0.0316521
$362$	−38.5659	−2.02698
$363$	7.83425	0.411192
$364$	−13.1561	−0.689568
$365$	2.38490	0.124832
$366$	23.3374	1.21987
$367$	10.6744	0.557199	0.278600	−	0.960407i	$-0.410130\pi$
$367$	10.6744	0.557199	0.278600	+	0.960407i	$0.410130\pi$
$368$	2.28694	0.119215
$369$	0.00919581	0.000478715 0
$370$	−8.31723	−0.432392
$371$	41.2996	2.14417
$372$	45.8217	2.37574
$373$	17.7186	0.917434	0.458717	−	0.888582i	$-0.348309\pi$
$373$	17.7186	0.917434	0.458717	+	0.888582i	$0.348309\pi$
$374$	−20.1506	−1.04196
$375$	−14.9391	−0.771452
$376$	27.2975	1.40776
$377$	8.91343	0.459065
$378$	−40.4846	−2.08230
$379$	−22.9984	−1.18135	−0.590674	−	0.806910i	$-0.701138\pi$
$379$	−22.9984	−1.18135	−0.590674	+	0.806910i	$0.701138\pi$
$380$	−20.5059	−1.05193
$381$	32.0112	1.63998
$382$	−2.91470	−0.149129
$383$	−0.696736	−0.0356015	−0.0178008	−	0.999842i	$-0.505666\pi$
$383$	−0.696736	−0.0356015	−0.0178008	+	0.999842i	$0.505666\pi$
$384$	17.1614	0.875765
$385$	−11.8418	−0.603513
$386$	7.45965	0.379686
$387$	3.27112	0.166280
$388$	−10.9182	−0.554285
$389$	2.43948	0.123686	0.0618432	−	0.998086i	$-0.480302\pi$
$389$	2.43948	0.123686	0.0618432	+	0.998086i	$0.480302\pi$
$390$	−4.26245	−0.215837
$391$	0.612279	0.0309643
$392$	−9.41033	−0.475294
$393$	11.7503	0.592724
$394$	18.0954	0.911636
$395$	−12.3809	−0.622951
$396$	−6.18208	−0.310661
$397$	−34.0916	−1.71101	−0.855504	−	0.517797i	$-0.826752\pi$
$397$	−34.0916	−1.71101	−0.855504	+	0.517797i	$0.826752\pi$
$398$	0.500942	0.0251100
$399$	20.9879	1.05071
$400$	−29.2885	−1.46442
$401$	−30.2271	−1.50947	−0.754735	−	0.656030i	$-0.772234\pi$
$401$	−30.2271	−1.50947	−0.754735	+	0.656030i	$0.772234\pi$
$402$	−10.2821	−0.512823
$403$	6.21590	0.309636
$404$	−48.2127	−2.39867
$405$	−8.03890	−0.399456
$406$	66.2163	3.28626
$407$	12.6437	0.626727
$408$	−20.8415	−1.03181
$409$	−9.89835	−0.489442	−0.244721	−	0.969594i	$-0.578696\pi$
$409$	−9.89835	−0.489442	−0.244721	+	0.969594i	$0.578696\pi$
$410$	0.0699585	0.00345500
$411$	10.6934	0.527467
$412$	−27.5769	−1.35862
$413$	19.8425	0.976388
$414$	0.270903	0.0133142
$415$	10.2881	0.505022
$416$	−6.04274	−0.296270
$417$	27.9038	1.36646
$418$	44.9566	2.19890
$419$	−11.0171	−0.538221	−0.269111	−	0.963109i	$-0.586730\pi$
$419$	−11.0171	−0.538221	−0.269111	+	0.963109i	$0.586730\pi$
$420$	−21.9564	−1.07136
$421$	34.5975	1.68618	0.843089	−	0.537774i	$-0.180735\pi$
$421$	34.5975	1.68618	0.843089	+	0.537774i	$0.180735\pi$
$422$	−59.7030	−2.90630
$423$	1.45627	0.0708061
$424$	91.4979	4.44353
$425$	−7.84137	−0.380362
$426$	11.0406	0.534920
$427$	16.3075	0.789175
$428$	−33.0843	−1.59919
$429$	6.47971	0.312843
$430$	24.8855	1.20009
$431$	−19.9265	−0.959825	−0.479913	−	0.877316i	$-0.659332\pi$
$431$	−19.9265	−0.959825	−0.479913	+	0.877316i	$0.659332\pi$
$432$	−40.3936	−1.94344
$433$	−4.63653	−0.222817	−0.111409	−	0.993775i	$-0.535536\pi$
$433$	−4.63653	−0.222817	−0.111409	+	0.993775i	$0.535536\pi$
$434$	46.1768	2.21656
$435$	14.8757	0.713235
$436$	−30.1716	−1.44496
$437$	−1.36601	−0.0653452
$438$	−9.69459	−0.463226
$439$	14.5172	0.692868	0.346434	−	0.938074i	$-0.387392\pi$
$439$	14.5172	0.692868	0.346434	+	0.938074i	$0.387392\pi$
$440$	−26.2351	−1.25071
$441$	−0.502022	−0.0239058
$442$	−5.06833	−0.241076
$443$	−9.38896	−0.446083	−0.223041	−	0.974809i	$-0.571599\pi$
$443$	−9.38896	−0.446083	−0.223041	+	0.974809i	$0.571599\pi$
$444$	23.4433	1.11257
$445$	15.9284	0.755079
$446$	66.8871	3.16719
$447$	19.1952	0.907899
$448$	−1.77167	−0.0837035
$449$	−31.0801	−1.46676	−0.733379	−	0.679820i	$-0.762058\pi$
$449$	−31.0801	−1.46676	−0.733379	+	0.679820i	$0.762058\pi$
$450$	−3.46942	−0.163550
$451$	−0.106350	−0.00500782
$452$	−12.1768	−0.572750
$453$	10.8834	0.511344
$454$	16.9355	0.794823
$455$	−2.97847	−0.139633
$456$	46.4980	2.17747
$457$	14.0870	0.658963	0.329482	−	0.944162i	$-0.393126\pi$
$457$	14.0870	0.658963	0.329482	+	0.944162i	$0.393126\pi$
$458$	−1.56098	−0.0729399
$459$	−10.8145	−0.504779
$460$	1.42905	0.0666296
$461$	37.0414	1.72519	0.862594	−	0.505897i	$-0.168838\pi$
$461$	37.0414	1.72519	0.862594	+	0.505897i	$0.168838\pi$
$462$	48.1366	2.23952
$463$	22.0189	1.02330	0.511652	−	0.859193i	$-0.329034\pi$
$463$	22.0189	1.02330	0.511652	+	0.859193i	$0.329034\pi$
$464$	66.0675	3.06711
$465$	10.3738	0.481071
$466$	13.5573	0.628028
$467$	8.09255	0.374478	0.187239	−	0.982314i	$-0.440046\pi$
$467$	8.09255	0.374478	0.187239	+	0.982314i	$0.440046\pi$
$468$	−1.55493	−0.0718767
$469$	−7.18480	−0.331763
$470$	11.0788	0.511025
$471$	−34.7741	−1.60231
$472$	43.9605	2.02345
$473$	−37.8306	−1.73945
$474$	50.3282	2.31165
$475$	17.4943	0.802694
$476$	−26.1076	−1.19664
$477$	4.88123	0.223496
$478$	22.0380	1.00799
$479$	16.1781	0.739196	0.369598	−	0.929192i	$-0.379495\pi$
$479$	16.1781	0.739196	0.369598	+	0.929192i	$0.379495\pi$
$480$	−10.0848	−0.460305
$481$	3.18018	0.145004
$482$	46.6656	2.12556
$483$	−1.46264	−0.0665523
$484$	21.7420	0.988274
$485$	−2.47181	−0.112239
$486$	−9.07785	−0.411780
$487$	−19.6696	−0.891316	−0.445658	−	0.895203i	$-0.647030\pi$
$487$	−19.6696	−0.891316	−0.445658	+	0.895203i	$0.647030\pi$
$488$	36.1287	1.63547
$489$	5.19602	0.234972
$490$	−3.81921	−0.172534
$491$	−28.7346	−1.29678	−0.648388	−	0.761310i	$-0.724557\pi$
$491$	−28.7346	−1.29678	−0.648388	+	0.761310i	$0.724557\pi$
$492$	−0.197188	−0.00888993
$493$	17.6882	0.796636
$494$	11.3076	0.508752
$495$	−1.39959	−0.0629067
$496$	46.0730	2.06874
$497$	7.71486	0.346059
$498$	−41.8209	−1.87404
$499$	10.9592	0.490601	0.245301	−	0.969447i	$-0.421113\pi$
$499$	10.9592	0.490601	0.245301	+	0.969447i	$0.421113\pi$
$500$	−41.4598	−1.85414
$501$	6.33289	0.282933
$502$	13.0940	0.584415
$503$	0.948787	0.0423043	0.0211522	−	0.999776i	$-0.493267\pi$
$503$	0.948787	0.0423043	0.0211522	+	0.999776i	$0.493267\pi$
$504$	−6.44359	−0.287020
$505$	−10.9151	−0.485715
$506$	−3.13300	−0.139279
$507$	1.62979	0.0723816
$508$	88.8392	3.94160
$509$	−31.3466	−1.38941	−0.694707	−	0.719293i	$-0.744466\pi$
$509$	−31.3466	−1.38941	−0.694707	+	0.719293i	$0.744466\pi$
$510$	−8.45858	−0.374552
$511$	−6.77429	−0.299677
$512$	50.7386	2.24235
$513$	24.1275	1.06526
$514$	6.98590	0.308135
$515$	−6.24326	−0.275111
$516$	−70.1434	−3.08789
$517$	−16.8418	−0.740700
$518$	23.6250	1.03802
$519$	−26.2621	−1.15278
$520$	−6.59870	−0.289372
$521$	2.83178	0.124063	0.0620313	−	0.998074i	$-0.480242\pi$
$521$	2.83178	0.124063	0.0620313	+	0.998074i	$0.480242\pi$
$522$	7.82614	0.342541
$523$	−33.6100	−1.46966	−0.734831	−	0.678250i	$-0.762739\pi$
$523$	−33.6100	−1.46966	−0.734831	+	0.678250i	$0.762739\pi$
$524$	32.6101	1.42458
$525$	18.7318	0.817522
$526$	−30.9503	−1.34950
$527$	12.3351	0.537325
$528$	48.0285	2.09017
$529$	−22.9048	−0.995861
$530$	37.1346	1.61303
$531$	2.34520	0.101773
$532$	58.2467	2.52532
$533$	−0.0267494	−0.00115864
$534$	−64.7487	−2.80195
$535$	−7.49009	−0.323825
$536$	−15.9177	−0.687540
$537$	−23.3694	−1.00846
$538$	−15.3282	−0.660845
$539$	5.80591	0.250078
$540$	−25.2409	−1.08619
$541$	−35.9301	−1.54475	−0.772377	−	0.635164i	$-0.780932\pi$
$541$	−35.9301	−1.54475	−0.772377	+	0.635164i	$0.780932\pi$
$542$	11.0973	0.476671
$543$	24.6099	1.05611
$544$	−11.9915	−0.514130
$545$	−6.83068	−0.292594
$546$	12.1074	0.518150
$547$	0.307514	0.0131483	0.00657417	−	0.999978i	$-0.497907\pi$
$547$	0.307514	0.0131483	0.00657417	+	0.999978i	$0.497907\pi$
$548$	29.6769	1.26773
$549$	1.92739	0.0822592
$550$	40.1239	1.71089
$551$	−39.4628	−1.68117
$552$	−3.24042	−0.137922
$553$	35.1678	1.49549
$554$	−1.59815	−0.0678990
$555$	5.30743	0.225288
$556$	77.4401	3.28419
$557$	−26.3182	−1.11514	−0.557570	−	0.830130i	$-0.688266\pi$
$557$	−26.3182	−1.11514	−0.557570	+	0.830130i	$0.688266\pi$
$558$	5.45766	0.231041
$559$	−9.51525	−0.402452
$560$	−22.0768	−0.932916
$561$	12.8586	0.542891
$562$	11.9970	0.506062
$563$	6.77145	0.285383	0.142691	−	0.989767i	$-0.454424\pi$
$563$	6.77145	0.285383	0.142691	+	0.989767i	$0.454424\pi$
$564$	−31.2271	−1.31490
$565$	−2.75676	−0.115978
$566$	5.15838	0.216823
$567$	22.8344	0.958955
$568$	17.0920	0.717165
$569$	41.5018	1.73984	0.869922	−	0.493189i	$-0.164169\pi$
$569$	41.5018	1.73984	0.869922	+	0.493189i	$0.164169\pi$
$570$	18.8713	0.790433
$571$	3.86062	0.161562	0.0807810	−	0.996732i	$-0.474259\pi$
$571$	3.86062	0.161562	0.0807810	+	0.996732i	$0.474259\pi$
$572$	17.9828	0.751900
$573$	1.85994	0.0777003
$574$	−0.198716	−0.00829426
$575$	−1.21917	−0.0508430
$576$	−0.209395	−0.00872478
$577$	24.5035	1.02009	0.510047	−	0.860147i	$-0.329628\pi$
$577$	24.5035	1.02009	0.510047	+	0.860147i	$0.329628\pi$
$578$	33.3608	1.38762
$579$	−4.76019	−0.197827
$580$	41.2838	1.71422
$581$	−29.2232	−1.21238
$582$	10.0479	0.416497
$583$	−56.4516	−2.33798
$584$	−15.0082	−0.621045
$585$	−0.352027	−0.0145545
$586$	76.6328	3.16567
$587$	18.7607	0.774336	0.387168	−	0.922009i	$-0.373453\pi$
$587$	18.7607	0.774336	0.387168	+	0.922009i	$0.373453\pi$
$588$	10.7650	0.443941
$589$	−27.5199	−1.13394
$590$	17.8415	0.734522
$591$	−11.5472	−0.474987
$592$	23.5719	0.968801
$593$	33.0962	1.35910	0.679550	−	0.733629i	$-0.262175\pi$
$593$	33.0962	1.35910	0.679550	+	0.733629i	$0.262175\pi$
$594$	55.3375	2.27052
$595$	−5.91060	−0.242311
$596$	53.2714	2.18208
$597$	−0.319664	−0.0130830
$598$	−0.788021	−0.0322246
$599$	−29.2262	−1.19415	−0.597075	−	0.802185i	$-0.703671\pi$
$599$	−29.2262	−1.19415	−0.597075	+	0.802185i	$0.703671\pi$
$600$	41.4996	1.69422
$601$	−0.652714	−0.0266248	−0.0133124	−	0.999911i	$-0.504238\pi$
$601$	−0.652714	−0.0266248	−0.0133124	+	0.999911i	$0.504238\pi$
$602$	−70.6870	−2.88099
$603$	−0.849177	−0.0345811
$604$	30.2041	1.22899
$605$	4.92227	0.200119
$606$	44.3696	1.80239
$607$	−10.4935	−0.425918	−0.212959	−	0.977061i	$-0.568310\pi$
$607$	−10.4935	−0.425918	−0.212959	+	0.977061i	$0.568310\pi$
$608$	26.7533	1.08499
$609$	−42.2543	−1.71223
$610$	14.6629	0.593685
$611$	−4.23608	−0.171374
$612$	−3.08567	−0.124731
$613$	−10.5606	−0.426539	−0.213270	−	0.976993i	$-0.568411\pi$
$613$	−10.5606	−0.426539	−0.213270	+	0.976993i	$0.568411\pi$
$614$	3.85952	0.155757
$615$	−0.0446423	−0.00180015
$616$	74.5204	3.00251
$617$	−18.6433	−0.750551	−0.375276	−	0.926913i	$-0.622452\pi$
$617$	−18.6433	−0.750551	−0.375276	+	0.926913i	$0.622452\pi$
$618$	25.3787	1.02088
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	28.7898	1.15623
$621$	−1.68144	−0.0674738
$622$	55.0260	2.20634
$623$	−45.2445	−1.81268
$624$	12.0802	0.483596
$625$	10.3709	0.414835
$626$	37.1010	1.48286
$627$	−28.6879	−1.14569
$628$	−96.5070	−3.85105
$629$	6.31089	0.251632
$630$	−2.61515	−0.104190
$631$	−25.0624	−0.997719	−0.498859	−	0.866683i	$-0.666248\pi$
$631$	−25.0624	−0.997719	−0.498859	+	0.866683i	$0.666248\pi$
$632$	77.9132	3.09922
$633$	38.0980	1.51426
$634$	37.0383	1.47098
$635$	20.1127	0.798147
$636$	−104.669	−4.15041
$637$	1.46032	0.0578598
$638$	−90.5097	−3.58331
$639$	0.911824	0.0360712
$640$	10.7825	0.426217
$641$	6.74162	0.266278	0.133139	−	0.991097i	$-0.457494\pi$
$641$	6.74162	0.266278	0.133139	+	0.991097i	$0.457494\pi$
$642$	30.4471	1.20165
$643$	−19.3974	−0.764960	−0.382480	−	0.923964i	$-0.624930\pi$
$643$	−19.3974	−0.764960	−0.382480	+	0.923964i	$0.624930\pi$
$644$	−4.05919	−0.159954
$645$	−15.8801	−0.625277
$646$	22.4392	0.882860
$647$	−24.4001	−0.959267	−0.479634	−	0.877469i	$-0.659230\pi$
$647$	−24.4001	−0.959267	−0.479634	+	0.877469i	$0.659230\pi$
$648$	50.5889	1.98732
$649$	−27.1224	−1.06465
$650$	10.0921	0.395843
$651$	−29.4665	−1.15489
$652$	14.4203	0.564741
$653$	−21.8357	−0.854499	−0.427249	−	0.904134i	$-0.640517\pi$
$653$	−21.8357	−0.854499	−0.427249	+	0.904134i	$0.640517\pi$
$654$	27.7666	1.08576
$655$	7.38273	0.288467
$656$	−0.198270	−0.00774114
$657$	−0.800658	−0.0312366
$658$	−31.4691	−1.22679
$659$	−1.10163	−0.0429135	−0.0214568	−	0.999770i	$-0.506830\pi$
$659$	−1.10163	−0.0429135	−0.0214568	+	0.999770i	$0.506830\pi$
$660$	30.0117	1.16820
$661$	10.5964	0.412152	0.206076	−	0.978536i	$-0.433931\pi$
$661$	10.5964	0.412152	0.206076	+	0.978536i	$0.433931\pi$
$662$	−68.2557	−2.65283
$663$	3.23423	0.125607
$664$	−64.7431	−2.51252
$665$	13.1867	0.511359
$666$	2.79225	0.108198
$667$	2.75015	0.106486
$668$	17.5754	0.680011
$669$	−42.6823	−1.65019
$670$	−6.46024	−0.249581
$671$	−22.2904	−0.860511
$672$	28.6457	1.10503
$673$	−39.9817	−1.54118	−0.770591	−	0.637330i	$-0.780039\pi$
$673$	−39.9817	−1.54118	−0.770591	+	0.637330i	$0.780039\pi$
$674$	42.0930	1.62136
$675$	21.5339	0.828841
$676$	4.52309	0.173965
$677$	−32.2859	−1.24085	−0.620424	−	0.784266i	$-0.713040\pi$
$677$	−32.2859	−1.24085	−0.620424	+	0.784266i	$0.713040\pi$
$678$	11.2062	0.430371
$679$	7.02114	0.269447
$680$	−13.0947	−0.502160
$681$	−10.8070	−0.414124
$682$	−63.1181	−2.41692
$683$	35.2089	1.34723	0.673616	−	0.739082i	$-0.264740\pi$
$683$	35.2089	1.34723	0.673616	+	0.739082i	$0.264740\pi$
$684$	6.88422	0.263225
$685$	6.71868	0.256708
$686$	−41.1533	−1.57124
$687$	0.996101	0.0380036
$688$	−70.5283	−2.68886
$689$	−14.1988	−0.540932
$690$	−1.31513	−0.0500663
$691$	32.7784	1.24695	0.623475	−	0.781843i	$-0.285720\pi$
$691$	32.7784	1.24695	0.623475	+	0.781843i	$0.285720\pi$
$692$	−72.8839	−2.77063
$693$	3.97551	0.151017
$694$	2.41630	0.0917213
$695$	17.5320	0.665026
$696$	−93.6129	−3.54839
$697$	−0.0530826	−0.00201065
$698$	−64.8185	−2.45342
$699$	−8.65123	−0.327220
$700$	51.9854	1.96486
$701$	18.6055	0.702721	0.351361	−	0.936240i	$-0.385719\pi$
$701$	18.6055	0.702721	0.351361	+	0.936240i	$0.385719\pi$
$702$	13.9186	0.525324
$703$	−14.0798	−0.531029
$704$	2.42166	0.0912697
$705$	−7.06963	−0.266258
$706$	29.8903	1.12494
$707$	31.0042	1.16603
$708$	−50.2888	−1.88997
$709$	6.90762	0.259421	0.129711	−	0.991552i	$-0.458595\pi$
$709$	6.90762	0.259421	0.129711	+	0.991552i	$0.458595\pi$
$710$	6.93683	0.260335
$711$	4.15651	0.155881
$712$	−100.238	−3.75657
$713$	1.91785	0.0718241
$714$	24.0265	0.899169
$715$	4.07121	0.152255
$716$	−64.8559	−2.42378
$717$	−14.0630	−0.525192
$718$	12.1589	0.453766
$719$	−3.87798	−0.144624	−0.0723122	−	0.997382i	$-0.523038\pi$
$719$	−3.87798	−0.144624	−0.0723122	+	0.997382i	$0.523038\pi$
$720$	−2.60927	−0.0972419
$721$	17.7339	0.660445
$722$	−1.53597	−0.0571629
$723$	−29.7785	−1.10747
$724$	68.2986	2.53830
$725$	−35.2207	−1.30807
$726$	−20.0089	−0.742601
$727$	32.5422	1.20692	0.603462	−	0.797392i	$-0.293788\pi$
$727$	32.5422	1.20692	0.603462	+	0.797392i	$0.293788\pi$
$728$	18.7435	0.694682
$729$	29.3443	1.08682
$730$	−6.09112	−0.225443
$731$	−18.8824	−0.698393
$732$	−41.3296	−1.52759
$733$	11.9408	0.441043	0.220521	−	0.975382i	$-0.429224\pi$
$733$	11.9408	0.441043	0.220521	+	0.975382i	$0.429224\pi$
$734$	−27.2628	−1.00629
$735$	2.43713	0.0898949
$736$	−1.86443	−0.0687237
$737$	9.82076	0.361752
$738$	−0.0234864	−0.000864547 0
$739$	−41.1678	−1.51438	−0.757191	−	0.653193i	$-0.773429\pi$
$739$	−41.1678	−1.51438	−0.757191	+	0.653193i	$0.773429\pi$
$740$	14.7295	0.541466
$741$	−7.21565	−0.265074
$742$	−105.481	−3.87231
$743$	−9.84809	−0.361291	−0.180646	−	0.983548i	$-0.557819\pi$
$743$	−9.84809	−0.361291	−0.180646	+	0.983548i	$0.557819\pi$
$744$	−65.2822	−2.39336
$745$	12.0603	0.441856
$746$	−45.2539	−1.65686
$747$	−3.45391	−0.126372
$748$	35.6859	1.30481
$749$	21.2755	0.777391
$750$	38.1550	1.39322
$751$	−5.66009	−0.206540	−0.103270	−	0.994653i	$-0.532931\pi$
$751$	−5.66009	−0.206540	−0.103270	+	0.994653i	$0.532931\pi$
$752$	−31.3984	−1.14498
$753$	−8.35563	−0.304496
$754$	−22.7652	−0.829060
$755$	6.83802	0.248861
$756$	71.6964	2.60757
$757$	45.5787	1.65659	0.828293	−	0.560295i	$-0.189312\pi$
$757$	45.5787	1.65659	0.828293	+	0.560295i	$0.189312\pi$
$758$	58.7386	2.13348
$759$	1.99925	0.0725681
$760$	29.2147	1.05973
$761$	−18.5609	−0.672832	−0.336416	−	0.941714i	$-0.609215\pi$
$761$	−18.5609	−0.672832	−0.336416	+	0.941714i	$0.609215\pi$
$762$	−81.7577	−2.96177
$763$	19.4025	0.702417
$764$	5.16181	0.186748
$765$	−0.698578	−0.0252571
$766$	1.77949	0.0642955
$767$	−6.82188	−0.246324
$768$	−45.8163	−1.65325
$769$	−18.2706	−0.658854	−0.329427	−	0.944181i	$-0.606855\pi$
$769$	−18.2706	−0.658854	−0.329427	+	0.944181i	$0.606855\pi$
$770$	30.2443	1.08993
$771$	−4.45787	−0.160546
$772$	−13.2107	−0.475464
$773$	−39.8456	−1.43315	−0.716574	−	0.697511i	$-0.754291\pi$
$773$	−39.8456	−1.43315	−0.716574	+	0.697511i	$0.754291\pi$
$774$	−8.35455	−0.300298
$775$	−24.5616	−0.882280
$776$	15.5551	0.558396
$777$	−15.0757	−0.540838
$778$	−6.23050	−0.223374
$779$	0.118429	0.00424315
$780$	7.54861	0.270284
$781$	−10.5453	−0.377340
$782$	−1.56378	−0.0559207
$783$	−48.5752	−1.73594
$784$	10.8240	0.386573
$785$	−21.8486	−0.779811
$786$	−30.0107	−1.07044
$787$	5.40363	0.192619	0.0963093	−	0.995351i	$-0.469296\pi$
$787$	5.40363	0.192619	0.0963093	+	0.995351i	$0.469296\pi$
$788$	−32.0463	−1.14160
$789$	19.7502	0.703125
$790$	31.6213	1.12503
$791$	7.83056	0.278422
$792$	8.80761	0.312965
$793$	−5.60653	−0.199094
$794$	87.0710	3.09003
$795$	−23.6965	−0.840430
$796$	−0.887147	−0.0314441
$797$	−19.3230	−0.684455	−0.342228	−	0.939617i	$-0.611181\pi$
$797$	−19.3230	−0.684455	−0.342228	+	0.939617i	$0.611181\pi$
$798$	−53.6038	−1.89755
$799$	−8.40626	−0.297392
$800$	23.8774	0.844195
$801$	−5.34748	−0.188944
$802$	77.2010	2.72606
$803$	9.25964	0.326766
$804$	18.2091	0.642186
$805$	−0.918977	−0.0323897
$806$	−15.8756	−0.559194
$807$	9.78131	0.344318
$808$	68.6887	2.41646
$809$	14.8754	0.522991	0.261495	−	0.965205i	$-0.415784\pi$
$809$	14.8754	0.522991	0.261495	+	0.965205i	$0.415784\pi$
$810$	20.5316	0.721408
$811$	47.0386	1.65175	0.825875	−	0.563853i	$-0.190682\pi$
$811$	47.0386	1.65175	0.825875	+	0.563853i	$0.190682\pi$
$812$	−117.266	−4.11524
$813$	−7.08148	−0.248358
$814$	−32.2925	−1.13185
$815$	3.26466	0.114356
$816$	23.9725	0.839206
$817$	42.1273	1.47385
$818$	25.2807	0.883920
$819$	0.999930	0.0349404
$820$	−0.123894	−0.00432655
$821$	−28.9893	−1.01173	−0.505867	−	0.862612i	$-0.668827\pi$
$821$	−28.9893	−1.01173	−0.505867	+	0.862612i	$0.668827\pi$
$822$	−27.3113	−0.952592
$823$	−1.09719	−0.0382455	−0.0191228	−	0.999817i	$-0.506087\pi$
$823$	−1.09719	−0.0382455	−0.0191228	+	0.999817i	$0.506087\pi$
$824$	39.2889	1.36869
$825$	−25.6041	−0.891420
$826$	−50.6785	−1.76333
$827$	15.4497	0.537239	0.268620	−	0.963246i	$-0.413433\pi$
$827$	15.4497	0.537239	0.268620	+	0.963246i	$0.413433\pi$
$828$	−0.479758	−0.0166727
$829$	18.4367	0.640332	0.320166	−	0.947361i	$-0.396261\pi$
$829$	18.4367	0.640332	0.320166	+	0.947361i	$0.396261\pi$
$830$	−26.2761	−0.912058
$831$	1.01982	0.0353772
$832$	0.609101	0.0211168
$833$	2.89791	0.100407
$834$	−71.2673	−2.46778
$835$	3.97896	0.137698
$836$	−79.6162	−2.75359
$837$	−33.8745	−1.17088
$838$	28.1381	0.972013
$839$	−10.7781	−0.372102	−0.186051	−	0.982540i	$-0.559569\pi$
$839$	−10.7781	−0.372102	−0.186051	+	0.982540i	$0.559569\pi$
$840$	31.2813	1.07931
$841$	50.4493	1.73963
$842$	−88.3631	−3.04519
$843$	−7.65558	−0.263672
$844$	105.732	3.63943
$845$	1.02400	0.0352267
$846$	−3.71935	−0.127874
$847$	−13.9817	−0.480415
$848$	−105.244	−3.61408
$849$	−3.29169	−0.112971
$850$	20.0271	0.686924
$851$	0.981213	0.0336356
$852$	−19.5525	−0.669857
$853$	53.0279	1.81564	0.907820	−	0.419361i	$-0.137746\pi$
$853$	53.0279	1.81564	0.907820	+	0.419361i	$0.137746\pi$
$854$	−41.6499	−1.42523
$855$	1.55855	0.0533012
$856$	47.1352	1.61105
$857$	−46.8574	−1.60062	−0.800309	−	0.599587i	$-0.795331\pi$
$857$	−46.8574	−1.60062	−0.800309	+	0.599587i	$0.795331\pi$
$858$	−16.5494	−0.564987
$859$	−48.4914	−1.65450	−0.827252	−	0.561831i	$-0.810097\pi$
$859$	−48.4914	−1.65450	−0.827252	+	0.561831i	$0.810097\pi$
$860$	−44.0712	−1.50282
$861$	0.126806	0.00432153
$862$	50.8929	1.73342
$863$	−28.7922	−0.980098	−0.490049	−	0.871695i	$-0.663021\pi$
$863$	−28.7922	−0.980098	−0.490049	+	0.871695i	$0.663021\pi$
$864$	32.9309	1.12033
$865$	−16.5005	−0.561034
$866$	11.8418	0.402402
$867$	−21.2883	−0.722990
$868$	−81.7771	−2.77570
$869$	−48.0702	−1.63067
$870$	−37.9930	−1.28808
$871$	2.47014	0.0836975
$872$	42.9855	1.45567
$873$	0.829833	0.0280856
$874$	3.48884	0.118012
$875$	26.6616	0.901325
$876$	17.1687	0.580077
$877$	7.73754	0.261278	0.130639	−	0.991430i	$-0.458297\pi$
$877$	7.73754	0.261278	0.130639	+	0.991430i	$0.458297\pi$
$878$	−37.0774	−1.25130
$879$	−48.9013	−1.64940
$880$	30.1763	1.01724
$881$	−5.25275	−0.176970	−0.0884849	−	0.996078i	$-0.528203\pi$
$881$	−5.25275	−0.176970	−0.0884849	+	0.996078i	$0.528203\pi$
$882$	1.28218	0.0431733
$883$	−50.0178	−1.68323	−0.841616	−	0.540077i	$-0.818395\pi$
$883$	−50.0178	−1.68323	−0.841616	+	0.540077i	$0.818395\pi$
$884$	8.97580	0.301889
$885$	−11.3851	−0.382706
$886$	23.9797	0.805614
$887$	17.6175	0.591539	0.295770	−	0.955259i	$-0.404424\pi$
$887$	17.6175	0.591539	0.295770	+	0.955259i	$0.404424\pi$
$888$	−33.3997	−1.12082
$889$	−57.1298	−1.91607
$890$	−40.6817	−1.36365
$891$	−31.2119	−1.04564
$892$	−118.454	−3.96614
$893$	18.7546	0.627599
$894$	−49.0251	−1.63964
$895$	−14.6830	−0.490798
$896$	−30.6277	−1.02320
$897$	0.502856	0.0167899
$898$	79.3795	2.64893
$899$	55.4050	1.84786
$900$	6.14419	0.204806
$901$	−28.1767	−0.938704
$902$	0.271621	0.00904400
$903$	45.1072	1.50107
$904$	17.3483	0.576997
$905$	15.4624	0.513988
$906$	−27.7964	−0.923475
$907$	9.67109	0.321123	0.160562	−	0.987026i	$-0.448669\pi$
$907$	9.67109	0.321123	0.160562	+	0.987026i	$0.448669\pi$
$908$	−29.9921	−0.995322
$909$	3.66440	0.121541
$910$	7.60711	0.252173
$911$	14.7174	0.487608	0.243804	−	0.969825i	$-0.421605\pi$
$911$	14.7174	0.487608	0.243804	+	0.969825i	$0.421605\pi$
$912$	−53.4834	−1.77101
$913$	39.9446	1.32197
$914$	−35.9787	−1.19007
$915$	−9.35678	−0.309326
$916$	2.76443	0.0913394
$917$	−20.9706	−0.692509
$918$	27.6207	0.911618
$919$	−13.7418	−0.453302	−0.226651	−	0.973976i	$-0.572778\pi$
$919$	−13.7418	−0.453302	−0.226651	+	0.973976i	$0.572778\pi$
$920$	−2.03596	−0.0671237
$921$	−2.46285	−0.0811538
$922$	−94.6049	−3.11565
$923$	−2.65237	−0.0873039
$924$	−85.2479	−2.80445
$925$	−12.5663	−0.413176
$926$	−56.2370	−1.84806
$927$	2.09598	0.0688411
$928$	−53.8616	−1.76809
$929$	−25.3935	−0.833133	−0.416567	−	0.909105i	$-0.636767\pi$
$929$	−25.3935	−0.833133	−0.416567	+	0.909105i	$0.636767\pi$
$930$	−26.4949	−0.868803
$931$	−6.46532	−0.211892
$932$	−24.0094	−0.786453
$933$	−35.1135	−1.14956
$934$	−20.6686	−0.676298
$935$	8.07908	0.264214
$936$	2.21531	0.0724097
$937$	40.5030	1.32317	0.661587	−	0.749868i	$-0.269883\pi$
$937$	40.5030	1.32317	0.661587	+	0.749868i	$0.269883\pi$
$938$	18.3502	0.599156
$939$	−23.6751	−0.772607
$940$	−19.6200	−0.639934
$941$	18.2501	0.594935	0.297468	−	0.954732i	$-0.403858\pi$
$941$	18.2501	0.594935	0.297468	+	0.954732i	$0.403858\pi$
$942$	88.8143	2.89373
$943$	−0.00825325	−0.000268763 0
$944$	−50.5647	−1.64574
$945$	16.2317	0.528016
$946$	96.6206	3.14141
$947$	−32.2871	−1.04919	−0.524594	−	0.851352i	$-0.675783\pi$
$947$	−32.2871	−1.04919	−0.524594	+	0.851352i	$0.675783\pi$
$948$	−89.1291	−2.89478
$949$	2.32901	0.0756028
$950$	−44.6811	−1.44965
$951$	−23.6351	−0.766420
$952$	37.1955	1.20551
$953$	9.14062	0.296094	0.148047	−	0.988980i	$-0.452701\pi$
$953$	9.14062	0.296094	0.148047	+	0.988980i	$0.452701\pi$
$954$	−12.4668	−0.403628
$955$	1.16860	0.0378152
$956$	−39.0283	−1.26227
$957$	57.7565	1.86700
$958$	−41.3194	−1.33497
$959$	−19.0843	−0.616266
$960$	1.01653	0.0328085
$961$	7.63736	0.246366
$962$	−8.12229	−0.261873
$963$	2.51457	0.0810308
$964$	−82.6428	−2.66174
$965$	−2.99083	−0.0962783
$966$	3.73562	0.120192
$967$	57.2053	1.83960	0.919800	−	0.392387i	$-0.128351\pi$
$967$	57.2053	1.83960	0.919800	+	0.392387i	$0.128351\pi$
$968$	−30.9759	−0.995603
$969$	−14.3190	−0.459994
$970$	6.31308	0.202701
$971$	−5.65710	−0.181545	−0.0907725	−	0.995872i	$-0.528934\pi$
$971$	−5.65710	−0.181545	−0.0907725	+	0.995872i	$0.528934\pi$
$972$	16.0765	0.515654
$973$	−49.7994	−1.59650
$974$	50.2369	1.60969
$975$	−6.44000	−0.206245
$976$	−41.5563	−1.33019
$977$	21.4501	0.686248	0.343124	−	0.939290i	$-0.388515\pi$
$977$	21.4501	0.686248	0.343124	+	0.939290i	$0.388515\pi$
$978$	−13.2708	−0.424354
$979$	61.8438	1.97653
$980$	6.76365	0.216057
$981$	2.29319	0.0732160
$982$	73.3892	2.34194
$983$	34.6664	1.10569	0.552843	−	0.833285i	$-0.313543\pi$
$983$	34.6664	1.10569	0.552843	+	0.833285i	$0.313543\pi$
$984$	0.280934	0.00895586
$985$	−7.25509	−0.231166
$986$	−45.1762	−1.43870
$987$	20.0812	0.639192
$988$	−20.0253	−0.637089
$989$	−2.93583	−0.0933540
$990$	3.57459	0.113608
$991$	10.4081	0.330626	0.165313	−	0.986241i	$-0.447137\pi$
$991$	10.4081	0.330626	0.165313	+	0.986241i	$0.447137\pi$
$992$	−37.5611	−1.19256
$993$	43.5557	1.38220
$994$	−19.7040	−0.624973
$995$	−0.200845	−0.00636721
$996$	74.0631	2.34678
$997$	−3.70648	−0.117385	−0.0586927	−	0.998276i	$-0.518693\pi$
$997$	−3.70648	−0.117385	−0.0586927	+	0.998276i	$0.518693\pi$
$998$	−27.9902	−0.886013
$999$	−17.3309	−0.548326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.9	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.9	✓	142	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55403 q^{2} +1.62979 q^{3} +4.52309 q^{4} +1.02400 q^{5} -4.16254 q^{6} -2.90866 q^{7} -6.44404 q^{8} -0.343777 q^{9} +O(q^{10})\)	\(q-2.55403 q^{2} +1.62979 q^{3} +4.52309 q^{4} +1.02400 q^{5} -4.16254 q^{6} -2.90866 q^{7} -6.44404 q^{8} -0.343777 q^{9} -2.61533 q^{10} +3.97579 q^{11} +7.37169 q^{12} +1.00000 q^{13} +7.42882 q^{14} +1.66891 q^{15} +7.41213 q^{16} +1.98444 q^{17} +0.878017 q^{18} -4.42735 q^{19} +4.63164 q^{20} -4.74052 q^{21} -10.1543 q^{22} +0.308540 q^{23} -10.5025 q^{24} -3.95142 q^{25} -2.55403 q^{26} -5.44966 q^{27} -13.1561 q^{28} +8.91343 q^{29} -4.26245 q^{30} +6.21590 q^{31} -6.04274 q^{32} +6.47971 q^{33} -5.06833 q^{34} -2.97847 q^{35} -1.55493 q^{36} +3.18018 q^{37} +11.3076 q^{38} +1.62979 q^{39} -6.59870 q^{40} -0.0267494 q^{41} +12.1074 q^{42} -9.51525 q^{43} +17.9828 q^{44} -0.352027 q^{45} -0.788021 q^{46} -4.23608 q^{47} +12.0802 q^{48} +1.46032 q^{49} +10.0921 q^{50} +3.23423 q^{51} +4.52309 q^{52} -14.1988 q^{53} +13.9186 q^{54} +4.07121 q^{55} +18.7435 q^{56} -7.21565 q^{57} -22.7652 q^{58} -6.82188 q^{59} +7.54861 q^{60} -5.60653 q^{61} -15.8756 q^{62} +0.999930 q^{63} +0.609101 q^{64} +1.02400 q^{65} -16.5494 q^{66} +2.47014 q^{67} +8.97580 q^{68} +0.502856 q^{69} +7.60711 q^{70} -2.65237 q^{71} +2.21531 q^{72} +2.32901 q^{73} -8.12229 q^{74} -6.44000 q^{75} -20.0253 q^{76} -11.5642 q^{77} -4.16254 q^{78} -12.0907 q^{79} +7.59003 q^{80} -7.85049 q^{81} +0.0683188 q^{82} +10.0470 q^{83} -21.4418 q^{84} +2.03207 q^{85} +24.3023 q^{86} +14.5270 q^{87} -25.6202 q^{88} +15.5551 q^{89} +0.899090 q^{90} -2.90866 q^{91} +1.39555 q^{92} +10.1306 q^{93} +10.8191 q^{94} -4.53360 q^{95} -9.84842 q^{96} -2.41387 q^{97} -3.72969 q^{98} -1.36678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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