LMFDB - Embedded newform 6038.2.a.c.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$1$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.49224 q^{3} +1.00000 q^{4} -4.20825 q^{5} +1.49224 q^{6} -4.82054 q^{7} -1.00000 q^{8} -0.773233 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.49224 q^{3} +1.00000 q^{4} -4.20825 q^{5} +1.49224 q^{6} -4.82054 q^{7} -1.00000 q^{8} -0.773233 q^{9} +4.20825 q^{10} -0.202803 q^{11} -1.49224 q^{12} -5.19057 q^{13} +4.82054 q^{14} +6.27970 q^{15} +1.00000 q^{16} -2.30293 q^{17} +0.773233 q^{18} -7.33375 q^{19} -4.20825 q^{20} +7.19338 q^{21} +0.202803 q^{22} +4.80985 q^{23} +1.49224 q^{24} +12.7094 q^{25} +5.19057 q^{26} +5.63055 q^{27} -4.82054 q^{28} -8.07116 q^{29} -6.27970 q^{30} -4.76051 q^{31} -1.00000 q^{32} +0.302630 q^{33} +2.30293 q^{34} +20.2860 q^{35} -0.773233 q^{36} +6.39037 q^{37} +7.33375 q^{38} +7.74555 q^{39} +4.20825 q^{40} +4.70034 q^{41} -7.19338 q^{42} +10.5654 q^{43} -0.202803 q^{44} +3.25396 q^{45} -4.80985 q^{46} -7.79198 q^{47} -1.49224 q^{48} +16.2376 q^{49} -12.7094 q^{50} +3.43651 q^{51} -5.19057 q^{52} -13.1120 q^{53} -5.63055 q^{54} +0.853446 q^{55} +4.82054 q^{56} +10.9437 q^{57} +8.07116 q^{58} +5.39597 q^{59} +6.27970 q^{60} -1.37038 q^{61} +4.76051 q^{62} +3.72740 q^{63} +1.00000 q^{64} +21.8432 q^{65} -0.302630 q^{66} -6.23098 q^{67} -2.30293 q^{68} -7.17743 q^{69} -20.2860 q^{70} +2.78076 q^{71} +0.773233 q^{72} +9.79915 q^{73} -6.39037 q^{74} -18.9653 q^{75} -7.33375 q^{76} +0.977621 q^{77} -7.74555 q^{78} -1.25548 q^{79} -4.20825 q^{80} -6.08241 q^{81} -4.70034 q^{82} -15.0322 q^{83} +7.19338 q^{84} +9.69130 q^{85} -10.5654 q^{86} +12.0441 q^{87} +0.202803 q^{88} +14.5569 q^{89} -3.25396 q^{90} +25.0213 q^{91} +4.80985 q^{92} +7.10379 q^{93} +7.79198 q^{94} +30.8622 q^{95} +1.49224 q^{96} -13.8747 q^{97} -16.2376 q^{98} +0.156814 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) $ 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.49224	−0.861543	−0.430771	−	0.902461i	$-0.641758\pi$
$3$	−1.49224	−0.861543	−0.430771	+	0.902461i	$0.641758\pi$
$4$	1.00000	0.500000
$5$	−4.20825	−1.88199	−0.940993	−	0.338426i	$-0.890105\pi$
$5$	−4.20825	−1.88199	−0.940993	+	0.338426i	$0.890105\pi$
$6$	1.49224	0.609203
$7$	−4.82054	−1.82199	−0.910996	−	0.412415i	$-0.864685\pi$
$7$	−4.82054	−1.82199	−0.910996	+	0.412415i	$0.864685\pi$
$8$	−1.00000	−0.353553
$9$	−0.773233	−0.257744
$10$	4.20825	1.33076
$11$	−0.202803	−0.0611475	−0.0305737	−	0.999533i	$-0.509733\pi$
$11$	−0.202803	−0.0611475	−0.0305737	+	0.999533i	$0.509733\pi$
$12$	−1.49224	−0.430771
$13$	−5.19057	−1.43960	−0.719802	−	0.694179i	$-0.755767\pi$
$13$	−5.19057	−1.43960	−0.719802	+	0.694179i	$0.755767\pi$
$14$	4.82054	1.28834
$15$	6.27970	1.62141
$16$	1.00000	0.250000
$17$	−2.30293	−0.558543	−0.279271	−	0.960212i	$-0.590093\pi$
$17$	−2.30293	−0.558543	−0.279271	+	0.960212i	$0.590093\pi$
$18$	0.773233	0.182253
$19$	−7.33375	−1.68248	−0.841239	−	0.540664i	$-0.818173\pi$
$19$	−7.33375	−1.68248	−0.841239	+	0.540664i	$0.818173\pi$
$20$	−4.20825	−0.940993
$21$	7.19338	1.56972
$22$	0.202803	0.0432378
$23$	4.80985	1.00292	0.501461	−	0.865180i	$-0.332796\pi$
$23$	4.80985	1.00292	0.501461	+	0.865180i	$0.332796\pi$
$24$	1.49224	0.304601
$25$	12.7094	2.54187
$26$	5.19057	1.01795
$27$	5.63055	1.08360
$28$	−4.82054	−0.910996
$29$	−8.07116	−1.49878	−0.749389	−	0.662130i	$-0.769653\pi$
$29$	−8.07116	−1.49878	−0.749389	+	0.662130i	$0.769653\pi$
$30$	−6.27970	−1.14651
$31$	−4.76051	−0.855012	−0.427506	−	0.904013i	$-0.640608\pi$
$31$	−4.76051	−0.855012	−0.427506	+	0.904013i	$0.640608\pi$
$32$	−1.00000	−0.176777
$33$	0.302630	0.0526811
$34$	2.30293	0.394949
$35$	20.2860	3.42896
$36$	−0.773233	−0.128872
$37$	6.39037	1.05057	0.525286	−	0.850926i	$-0.323959\pi$
$37$	6.39037	1.05057	0.525286	+	0.850926i	$0.323959\pi$
$38$	7.33375	1.18969
$39$	7.74555	1.24028
$40$	4.20825	0.665382
$41$	4.70034	0.734070	0.367035	−	0.930207i	$-0.380373\pi$
$41$	4.70034	0.734070	0.367035	+	0.930207i	$0.380373\pi$
$42$	−7.19338	−1.10996
$43$	10.5654	1.61121	0.805606	−	0.592451i	$-0.201840\pi$
$43$	10.5654	1.61121	0.805606	+	0.592451i	$0.201840\pi$
$44$	−0.202803	−0.0305737
$45$	3.25396	0.485071
$46$	−4.80985	−0.709174
$47$	−7.79198	−1.13658	−0.568289	−	0.822829i	$-0.692394\pi$
$47$	−7.79198	−1.13658	−0.568289	+	0.822829i	$0.692394\pi$
$48$	−1.49224	−0.215386
$49$	16.2376	2.31966
$50$	−12.7094	−1.79737
$51$	3.43651	0.481208
$52$	−5.19057	−0.719802
$53$	−13.1120	−1.80108	−0.900538	−	0.434778i	$-0.856827\pi$
$53$	−13.1120	−1.80108	−0.900538	+	0.434778i	$0.856827\pi$
$54$	−5.63055	−0.766221
$55$	0.853446	0.115079
$56$	4.82054	0.644172
$57$	10.9437	1.44953
$58$	8.07116	1.05980
$59$	5.39597	0.702495	0.351248	−	0.936283i	$-0.385757\pi$
$59$	5.39597	0.702495	0.351248	+	0.936283i	$0.385757\pi$
$60$	6.27970	0.810705
$61$	−1.37038	−0.175459	−0.0877296	−	0.996144i	$-0.527961\pi$
$61$	−1.37038	−0.175459	−0.0877296	+	0.996144i	$0.527961\pi$
$62$	4.76051	0.604585
$63$	3.72740	0.469608
$64$	1.00000	0.125000
$65$	21.8432	2.70931
$66$	−0.302630	−0.0372512
$67$	−6.23098	−0.761236	−0.380618	−	0.924732i	$-0.624289\pi$
$67$	−6.23098	−0.761236	−0.380618	+	0.924732i	$0.624289\pi$
$68$	−2.30293	−0.279271
$69$	−7.17743	−0.864061
$70$	−20.2860	−2.42464
$71$	2.78076	0.330016	0.165008	−	0.986292i	$-0.447235\pi$
$71$	2.78076	0.330016	0.165008	+	0.986292i	$0.447235\pi$
$72$	0.773233	0.0911264
$73$	9.79915	1.14690	0.573452	−	0.819239i	$-0.305604\pi$
$73$	9.79915	1.14690	0.573452	+	0.819239i	$0.305604\pi$
$74$	−6.39037	−0.742866
$75$	−18.9653	−2.18993
$76$	−7.33375	−0.841239
$77$	0.977621	0.111410
$78$	−7.74555	−0.877011
$79$	−1.25548	−0.141252	−0.0706262	−	0.997503i	$-0.522500\pi$
$79$	−1.25548	−0.141252	−0.0706262	+	0.997503i	$0.522500\pi$
$80$	−4.20825	−0.470496
$81$	−6.08241	−0.675823
$82$	−4.70034	−0.519066
$83$	−15.0322	−1.65000	−0.824999	−	0.565134i	$-0.808824\pi$
$83$	−15.0322	−1.65000	−0.824999	+	0.565134i	$0.808824\pi$
$84$	7.19338	0.784862
$85$	9.69130	1.05117
$86$	−10.5654	−1.13930
$87$	12.0441	1.29126
$88$	0.202803	0.0216189
$89$	14.5569	1.54303	0.771513	−	0.636213i	$-0.219500\pi$
$89$	14.5569	1.54303	0.771513	+	0.636213i	$0.219500\pi$
$90$	−3.25396	−0.342997
$91$	25.0213	2.62295
$92$	4.80985	0.501461
$93$	7.10379	0.736629
$94$	7.79198	0.803682
$95$	30.8622	3.16640
$96$	1.49224	0.152301
$97$	−13.8747	−1.40876	−0.704380	−	0.709823i	$-0.748775\pi$
$97$	−13.8747	−1.40876	−0.704380	+	0.709823i	$0.748775\pi$
$98$	−16.2376	−1.64024
$99$	0.156814	0.0157604
$100$	12.7094	1.27094
$101$	15.6783	1.56005	0.780023	−	0.625751i	$-0.215207\pi$
$101$	15.6783	1.56005	0.780023	+	0.625751i	$0.215207\pi$
$102$	−3.43651	−0.340266
$103$	−15.7398	−1.55089	−0.775445	−	0.631415i	$-0.782475\pi$
$103$	−15.7398	−1.55089	−0.775445	+	0.631415i	$0.782475\pi$
$104$	5.19057	0.508977
$105$	−30.2715	−2.95420
$106$	13.1120	1.27355
$107$	1.30199	0.125868	0.0629342	−	0.998018i	$-0.479954\pi$
$107$	1.30199	0.125868	0.0629342	+	0.998018i	$0.479954\pi$
$108$	5.63055	0.541800
$109$	−1.76120	−0.168693	−0.0843463	−	0.996436i	$-0.526880\pi$
$109$	−1.76120	−0.168693	−0.0843463	+	0.996436i	$0.526880\pi$
$110$	−0.853446	−0.0813729
$111$	−9.53594	−0.905112
$112$	−4.82054	−0.455498
$113$	−3.06425	−0.288260	−0.144130	−	0.989559i	$-0.546038\pi$
$113$	−3.06425	−0.288260	−0.144130	+	0.989559i	$0.546038\pi$
$114$	−10.9437	−1.02497
$115$	−20.2410	−1.88749
$116$	−8.07116	−0.749389
$117$	4.01352	0.371050
$118$	−5.39597	−0.496739
$119$	11.1014	1.01766
$120$	−6.27970	−0.573255
$121$	−10.9589	−0.996261
$122$	1.37038	0.124068
$123$	−7.01402	−0.632433
$124$	−4.76051	−0.427506
$125$	−32.4429	−2.90178
$126$	−3.72740	−0.332063
$127$	8.28792	0.735434	0.367717	−	0.929938i	$-0.380140\pi$
$127$	8.28792	0.735434	0.367717	+	0.929938i	$0.380140\pi$
$128$	−1.00000	−0.0883883
$129$	−15.7661	−1.38813
$130$	−21.8432	−1.91577
$131$	14.5467	1.27095	0.635475	−	0.772121i	$-0.280804\pi$
$131$	14.5467	1.27095	0.635475	+	0.772121i	$0.280804\pi$
$132$	0.302630	0.0263406
$133$	35.3526	3.06546
$134$	6.23098	0.538275
$135$	−23.6948	−2.03932
$136$	2.30293	0.197475
$137$	17.0858	1.45974	0.729870	−	0.683586i	$-0.239581\pi$
$137$	17.0858	1.45974	0.729870	+	0.683586i	$0.239581\pi$
$138$	7.17743	0.610983
$139$	5.59992	0.474979	0.237490	−	0.971390i	$-0.423675\pi$
$139$	5.59992	0.474979	0.237490	+	0.971390i	$0.423675\pi$
$140$	20.2860	1.71448
$141$	11.6275	0.979210
$142$	−2.78076	−0.233356
$143$	1.05266	0.0880282
$144$	−0.773233	−0.0644361
$145$	33.9655	2.82068
$146$	−9.79915	−0.810984
$147$	−24.2303	−1.99848
$148$	6.39037	0.525286
$149$	−2.97043	−0.243347	−0.121674	−	0.992570i	$-0.538826\pi$
$149$	−2.97043	−0.243347	−0.121674	+	0.992570i	$0.538826\pi$
$150$	18.9653	1.54851
$151$	3.47314	0.282640	0.141320	−	0.989964i	$-0.454865\pi$
$151$	3.47314	0.282640	0.141320	+	0.989964i	$0.454865\pi$
$152$	7.33375	0.594845
$153$	1.78070	0.143961
$154$	−0.977621	−0.0787789
$155$	20.0334	1.60912
$156$	7.74555	0.620140
$157$	−7.56643	−0.603867	−0.301934	−	0.953329i	$-0.597632\pi$
$157$	−7.56643	−0.603867	−0.301934	+	0.953329i	$0.597632\pi$
$158$	1.25548	0.0998806
$159$	19.5662	1.55170
$160$	4.20825	0.332691
$161$	−23.1861	−1.82732
$162$	6.08241	0.477879
$163$	−1.00488	−0.0787085	−0.0393542	−	0.999225i	$-0.512530\pi$
$163$	−1.00488	−0.0787085	−0.0393542	+	0.999225i	$0.512530\pi$
$164$	4.70034	0.367035
$165$	−1.27354	−0.0991452
$166$	15.0322	1.16672
$167$	6.40314	0.495490	0.247745	−	0.968825i	$-0.420310\pi$
$167$	6.40314	0.495490	0.247745	+	0.968825i	$0.420310\pi$
$168$	−7.19338	−0.554981
$169$	13.9420	1.07246
$170$	−9.69130	−0.743289
$171$	5.67070	0.433649
$172$	10.5654	0.805606
$173$	−10.2750	−0.781194	−0.390597	−	0.920562i	$-0.627731\pi$
$173$	−10.2750	−0.781194	−0.390597	+	0.920562i	$0.627731\pi$
$174$	−12.0441	−0.913059
$175$	−61.2659	−4.63127
$176$	−0.202803	−0.0152869
$177$	−8.05206	−0.605230
$178$	−14.5569	−1.09108
$179$	−4.90855	−0.366882	−0.183441	−	0.983031i	$-0.558724\pi$
$179$	−4.90855	−0.366882	−0.183441	+	0.983031i	$0.558724\pi$
$180$	3.25396	0.242536
$181$	9.91293	0.736823	0.368411	−	0.929663i	$-0.379902\pi$
$181$	9.91293	0.736823	0.368411	+	0.929663i	$0.379902\pi$
$182$	−25.0213	−1.85470
$183$	2.04493	0.151166
$184$	−4.80985	−0.354587
$185$	−26.8923	−1.97716
$186$	−7.10379	−0.520875
$187$	0.467042	0.0341535
$188$	−7.79198	−0.568289
$189$	−27.1423	−1.97431
$190$	−30.8622	−2.23898
$191$	3.44608	0.249349	0.124675	−	0.992198i	$-0.460211\pi$
$191$	3.44608	0.249349	0.124675	+	0.992198i	$0.460211\pi$
$192$	−1.49224	−0.107693
$193$	−6.24147	−0.449271	−0.224636	−	0.974443i	$-0.572119\pi$
$193$	−6.24147	−0.449271	−0.224636	+	0.974443i	$0.572119\pi$
$194$	13.8747	0.996144
$195$	−32.5952	−2.33419
$196$	16.2376	1.15983
$197$	11.8321	0.842999	0.421500	−	0.906829i	$-0.361504\pi$
$197$	11.8321	0.842999	0.421500	+	0.906829i	$0.361504\pi$
$198$	−0.156814	−0.0111443
$199$	−11.1640	−0.791396	−0.395698	−	0.918381i	$-0.629497\pi$
$199$	−11.1640	−0.791396	−0.395698	+	0.918381i	$0.629497\pi$
$200$	−12.7094	−0.898687
$201$	9.29809	0.655837
$202$	−15.6783	−1.10312
$203$	38.9074	2.73076
$204$	3.43651	0.240604
$205$	−19.7802	−1.38151
$206$	15.7398	1.09665
$207$	−3.71914	−0.258498
$208$	−5.19057	−0.359901
$209$	1.48731	0.102879
$210$	30.2715	2.08893
$211$	−14.8609	−1.02306	−0.511532	−	0.859264i	$-0.670922\pi$
$211$	−14.8609	−1.02306	−0.511532	+	0.859264i	$0.670922\pi$
$212$	−13.1120	−0.900538
$213$	−4.14955	−0.284323
$214$	−1.30199	−0.0890024
$215$	−44.4619	−3.03228
$216$	−5.63055	−0.383111
$217$	22.9482	1.55783
$218$	1.76120	0.119284
$219$	−14.6226	−0.988106
$220$	0.853446	0.0575393
$221$	11.9535	0.804081
$222$	9.53594	0.640011
$223$	−12.4880	−0.836258	−0.418129	−	0.908388i	$-0.637314\pi$
$223$	−12.4880	−0.836258	−0.418129	+	0.908388i	$0.637314\pi$
$224$	4.82054	0.322086
$225$	−9.82730	−0.655153
$226$	3.06425	0.203831
$227$	−8.55243	−0.567645	−0.283822	−	0.958877i	$-0.591603\pi$
$227$	−8.55243	−0.567645	−0.283822	+	0.958877i	$0.591603\pi$
$228$	10.9437	0.724763
$229$	−7.02375	−0.464142	−0.232071	−	0.972699i	$-0.574550\pi$
$229$	−7.02375	−0.464142	−0.232071	+	0.972699i	$0.574550\pi$
$230$	20.2410	1.33465
$231$	−1.45884	−0.0959846
$232$	8.07116	0.529898
$233$	−28.9320	−1.89540	−0.947700	−	0.319163i	$-0.896598\pi$
$233$	−28.9320	−1.89540	−0.947700	+	0.319163i	$0.896598\pi$
$234$	−4.01352	−0.262372
$235$	32.7906	2.13902
$236$	5.39597	0.351248
$237$	1.87347	0.121695
$238$	−11.1014	−0.719595
$239$	24.3573	1.57554	0.787770	−	0.615969i	$-0.211235\pi$
$239$	24.3573	1.57554	0.787770	+	0.615969i	$0.211235\pi$
$240$	6.27970	0.405353
$241$	3.86961	0.249263	0.124632	−	0.992203i	$-0.460225\pi$
$241$	3.86961	0.249263	0.124632	+	0.992203i	$0.460225\pi$
$242$	10.9589	0.704463
$243$	−7.81527	−0.501350
$244$	−1.37038	−0.0877296
$245$	−68.3318	−4.36556
$246$	7.01402	0.447198
$247$	38.0663	2.42210
$248$	4.76051	0.302292
$249$	22.4316	1.42154
$250$	32.4429	2.05187
$251$	−7.73037	−0.487937	−0.243968	−	0.969783i	$-0.578449\pi$
$251$	−7.73037	−0.487937	−0.243968	+	0.969783i	$0.578449\pi$
$252$	3.72740	0.234804
$253$	−0.975453	−0.0613262
$254$	−8.28792	−0.520030
$255$	−14.4617	−0.905627
$256$	1.00000	0.0625000
$257$	15.0552	0.939121	0.469560	−	0.882900i	$-0.344412\pi$
$257$	15.0552	0.939121	0.469560	+	0.882900i	$0.344412\pi$
$258$	15.7661	0.981555
$259$	−30.8050	−1.91413
$260$	21.8432	1.35466
$261$	6.24089	0.386302
$262$	−14.5467	−0.898698
$263$	19.3963	1.19602	0.598012	−	0.801487i	$-0.295957\pi$
$263$	19.3963	1.19602	0.598012	+	0.801487i	$0.295957\pi$
$264$	−0.302630	−0.0186256
$265$	55.1786	3.38960
$266$	−35.3526	−2.16761
$267$	−21.7223	−1.32938
$268$	−6.23098	−0.380618
$269$	16.8323	1.02629	0.513143	−	0.858303i	$-0.328481\pi$
$269$	16.8323	1.02629	0.513143	+	0.858303i	$0.328481\pi$
$270$	23.6948	1.44202
$271$	5.73370	0.348297	0.174149	−	0.984719i	$-0.444283\pi$
$271$	5.73370	0.348297	0.174149	+	0.984719i	$0.444283\pi$
$272$	−2.30293	−0.139636
$273$	−37.3377	−2.25978
$274$	−17.0858	−1.03219
$275$	−2.57750	−0.155429
$276$	−7.17743	−0.432030
$277$	−2.25689	−0.135604	−0.0678018	−	0.997699i	$-0.521599\pi$
$277$	−2.25689	−0.135604	−0.0678018	+	0.997699i	$0.521599\pi$
$278$	−5.59992	−0.335861
$279$	3.68098	0.220375
$280$	−20.2860	−1.21232
$281$	23.4981	1.40178	0.700889	−	0.713270i	$-0.252787\pi$
$281$	23.4981	1.40178	0.700889	+	0.713270i	$0.252787\pi$
$282$	−11.6275	−0.692406
$283$	26.1340	1.55351	0.776753	−	0.629805i	$-0.216865\pi$
$283$	26.1340	1.55351	0.776753	+	0.629805i	$0.216865\pi$
$284$	2.78076	0.165008
$285$	−46.0537	−2.72799
$286$	−1.05266	−0.0622453
$287$	−22.6582	−1.33747
$288$	0.773233	0.0455632
$289$	−11.6965	−0.688030
$290$	−33.9655	−1.99452
$291$	20.7043	1.21371
$292$	9.79915	0.573452
$293$	−27.0372	−1.57953	−0.789765	−	0.613410i	$-0.789798\pi$
$293$	−27.0372	−1.57953	−0.789765	+	0.613410i	$0.789798\pi$
$294$	24.2303	1.41314
$295$	−22.7076	−1.32209
$296$	−6.39037	−0.371433
$297$	−1.14189	−0.0662594
$298$	2.97043	0.172072
$299$	−24.9658	−1.44381
$300$	−18.9653	−1.09496
$301$	−50.9311	−2.93562
$302$	−3.47314	−0.199857
$303$	−23.3957	−1.34405
$304$	−7.33375	−0.420619
$305$	5.76690	0.330212
$306$	−1.78070	−0.101796
$307$	−24.5036	−1.39850	−0.699248	−	0.714879i	$-0.746482\pi$
$307$	−24.5036	−1.39850	−0.699248	+	0.714879i	$0.746482\pi$
$308$	0.977621	0.0557051
$309$	23.4875	1.33616
$310$	−20.0334	−1.13782
$311$	24.2556	1.37541	0.687706	−	0.725990i	$-0.258618\pi$
$311$	24.2556	1.37541	0.687706	+	0.725990i	$0.258618\pi$
$312$	−7.74555	−0.438505
$313$	6.44923	0.364532	0.182266	−	0.983249i	$-0.441657\pi$
$313$	6.44923	0.364532	0.182266	+	0.983249i	$0.441657\pi$
$314$	7.56643	0.426998
$315$	−15.6858	−0.883796
$316$	−1.25548	−0.0706262
$317$	−27.3127	−1.53403	−0.767016	−	0.641628i	$-0.778259\pi$
$317$	−27.3127	−1.53403	−0.767016	+	0.641628i	$0.778259\pi$
$318$	−19.5662	−1.09722
$319$	1.63686	0.0916465
$320$	−4.20825	−0.235248
$321$	−1.94288	−0.108441
$322$	23.1861	1.29211
$323$	16.8891	0.939735
$324$	−6.08241	−0.337912
$325$	−65.9687	−3.65929
$326$	1.00488	0.0556553
$327$	2.62813	0.145336
$328$	−4.70034	−0.259533
$329$	37.5615	2.07084
$330$	1.27354	0.0701062
$331$	14.9465	0.821535	0.410768	−	0.911740i	$-0.365261\pi$
$331$	14.9465	0.821535	0.410768	+	0.911740i	$0.365261\pi$
$332$	−15.0322	−0.824999
$333$	−4.94125	−0.270779
$334$	−6.40314	−0.350364
$335$	26.2215	1.43263
$336$	7.19338	0.392431
$337$	3.59091	0.195609	0.0978046	−	0.995206i	$-0.468818\pi$
$337$	3.59091	0.195609	0.0978046	+	0.995206i	$0.468818\pi$
$338$	−13.9420	−0.758344
$339$	4.57258	0.248349
$340$	9.69130	0.525585
$341$	0.965446	0.0522818
$342$	−5.67070	−0.306636
$343$	−44.5302	−2.40440
$344$	−10.5654	−0.569650
$345$	30.2044	1.62615
$346$	10.2750	0.552388
$347$	14.1662	0.760481	0.380241	−	0.924888i	$-0.375841\pi$
$347$	14.1662	0.760481	0.380241	+	0.924888i	$0.375841\pi$
$348$	12.0441	0.645630
$349$	4.16283	0.222831	0.111416	−	0.993774i	$-0.464461\pi$
$349$	4.16283	0.222831	0.111416	+	0.993774i	$0.464461\pi$
$350$	61.2659	3.27480
$351$	−29.2258	−1.55996
$352$	0.202803	0.0108094
$353$	7.96830	0.424110	0.212055	−	0.977258i	$-0.431984\pi$
$353$	7.96830	0.424110	0.212055	+	0.977258i	$0.431984\pi$
$354$	8.05206	0.427962
$355$	−11.7021	−0.621085
$356$	14.5569	0.771513
$357$	−16.5659	−0.876758
$358$	4.90855	0.259425
$359$	−2.49269	−0.131559	−0.0657796	−	0.997834i	$-0.520953\pi$
$359$	−2.49269	−0.131559	−0.0657796	+	0.997834i	$0.520953\pi$
$360$	−3.25396	−0.171499
$361$	34.7839	1.83073
$362$	−9.91293	−0.521012
$363$	16.3532	0.858321
$364$	25.0213	1.31147
$365$	−41.2373	−2.15846
$366$	−2.04493	−0.106890
$367$	−11.8678	−0.619493	−0.309747	−	0.950819i	$-0.600244\pi$
$367$	−11.8678	−0.619493	−0.309747	+	0.950819i	$0.600244\pi$
$368$	4.80985	0.250731
$369$	−3.63446	−0.189203
$370$	26.8923	1.39806
$371$	63.2070	3.28155
$372$	7.10379	0.368315
$373$	8.13910	0.421426	0.210713	−	0.977548i	$-0.432421\pi$
$373$	8.13910	0.421426	0.210713	+	0.977548i	$0.432421\pi$
$374$	−0.467042	−0.0241502
$375$	48.4124	2.50000
$376$	7.79198	0.401841
$377$	41.8939	2.15765
$378$	27.1423	1.39605
$379$	1.76865	0.0908495	0.0454247	−	0.998968i	$-0.485536\pi$
$379$	1.76865	0.0908495	0.0454247	+	0.998968i	$0.485536\pi$
$380$	30.8622	1.58320
$381$	−12.3675	−0.633607
$382$	−3.44608	−0.176317
$383$	−14.9406	−0.763431	−0.381715	−	0.924280i	$-0.624667\pi$
$383$	−14.9406	−0.763431	−0.381715	+	0.924280i	$0.624667\pi$
$384$	1.49224	0.0761503
$385$	−4.11407	−0.209672
$386$	6.24147	0.317683
$387$	−8.16954	−0.415281
$388$	−13.8747	−0.704380
$389$	−29.0731	−1.47407	−0.737033	−	0.675857i	$-0.763774\pi$
$389$	−29.0731	−1.47407	−0.737033	+	0.675857i	$0.763774\pi$
$390$	32.5952	1.65052
$391$	−11.0767	−0.560175
$392$	−16.2376	−0.820122
$393$	−21.7071	−1.09498
$394$	−11.8321	−0.596090
$395$	5.28337	0.265835
$396$	0.156814	0.00788021
$397$	−24.5180	−1.23052	−0.615261	−	0.788324i	$-0.710949\pi$
$397$	−24.5180	−1.23052	−0.615261	+	0.788324i	$0.710949\pi$
$398$	11.1640	0.559602
$399$	−52.7544	−2.64102
$400$	12.7094	0.635468
$401$	30.0778	1.50201	0.751006	−	0.660295i	$-0.229569\pi$
$401$	30.0778	1.50201	0.751006	+	0.660295i	$0.229569\pi$
$402$	−9.29809	−0.463747
$403$	24.7097	1.23088
$404$	15.6783	0.780023
$405$	25.5963	1.27189
$406$	−38.9074	−1.93094
$407$	−1.29599	−0.0642398
$408$	−3.43651	−0.170133
$409$	−10.0926	−0.499045	−0.249522	−	0.968369i	$-0.580274\pi$
$409$	−10.0926	−0.499045	−0.249522	+	0.968369i	$0.580274\pi$
$410$	19.7802	0.976875
$411$	−25.4961	−1.25763
$412$	−15.7398	−0.775445
$413$	−26.0115	−1.27994
$414$	3.71914	0.182786
$415$	63.2592	3.10527
$416$	5.19057	0.254488
$417$	−8.35640	−0.409215
$418$	−1.48731	−0.0727466
$419$	−30.9316	−1.51111	−0.755554	−	0.655086i	$-0.772632\pi$
$419$	−30.9316	−1.51111	−0.755554	+	0.655086i	$0.772632\pi$
$420$	−30.2715	−1.47710
$421$	−30.8111	−1.50164	−0.750822	−	0.660505i	$-0.770342\pi$
$421$	−30.8111	−1.50164	−0.750822	+	0.660505i	$0.770342\pi$
$422$	14.8609	0.723416
$423$	6.02502	0.292947
$424$	13.1120	0.636776
$425$	−29.2688	−1.41974
$426$	4.14955	0.201046
$427$	6.60597	0.319685
$428$	1.30199	0.0629342
$429$	−1.57082	−0.0758400
$430$	44.4619	2.14415
$431$	32.4231	1.56177	0.780883	−	0.624677i	$-0.214769\pi$
$431$	32.4231	1.56177	0.780883	+	0.624677i	$0.214769\pi$
$432$	5.63055	0.270900
$433$	23.8894	1.14805	0.574026	−	0.818837i	$-0.305381\pi$
$433$	23.8894	1.14805	0.574026	+	0.818837i	$0.305381\pi$
$434$	−22.9482	−1.10155
$435$	−50.6845	−2.43013
$436$	−1.76120	−0.0843463
$437$	−35.2742	−1.68739
$438$	14.6226	0.698697
$439$	23.2197	1.10822	0.554108	−	0.832445i	$-0.313059\pi$
$439$	23.2197	1.10822	0.554108	+	0.832445i	$0.313059\pi$
$440$	−0.853446	−0.0406865
$441$	−12.5555	−0.597879
$442$	−11.9535	−0.568571
$443$	34.8343	1.65503	0.827514	−	0.561444i	$-0.189754\pi$
$443$	34.8343	1.65503	0.827514	+	0.561444i	$0.189754\pi$
$444$	−9.53594	−0.452556
$445$	−61.2590	−2.90395
$446$	12.4880	0.591324
$447$	4.43258	0.209654
$448$	−4.82054	−0.227749
$449$	−21.4195	−1.01085	−0.505425	−	0.862870i	$-0.668664\pi$
$449$	−21.4195	−1.01085	−0.505425	+	0.862870i	$0.668664\pi$
$450$	9.82730	0.463263
$451$	−0.953245	−0.0448865
$452$	−3.06425	−0.144130
$453$	−5.18274	−0.243506
$454$	8.55243	0.401386
$455$	−105.296	−4.93635
$456$	−10.9437	−0.512485
$457$	−17.1091	−0.800332	−0.400166	−	0.916443i	$-0.631048\pi$
$457$	−17.1091	−0.800332	−0.400166	+	0.916443i	$0.631048\pi$
$458$	7.02375	0.328198
$459$	−12.9668	−0.605237
$460$	−20.2410	−0.943743
$461$	6.64079	0.309293	0.154646	−	0.987970i	$-0.450576\pi$
$461$	6.64079	0.309293	0.154646	+	0.987970i	$0.450576\pi$
$462$	1.45884	0.0678714
$463$	1.20351	0.0559319	0.0279659	−	0.999609i	$-0.491097\pi$
$463$	1.20351	0.0559319	0.0279659	+	0.999609i	$0.491097\pi$
$464$	−8.07116	−0.374694
$465$	−29.8945	−1.38633
$466$	28.9320	1.34025
$467$	22.9727	1.06305	0.531525	−	0.847043i	$-0.321619\pi$
$467$	22.9727	1.06305	0.531525	+	0.847043i	$0.321619\pi$
$468$	4.01352	0.185525
$469$	30.0367	1.38697
$470$	−32.7906	−1.51252
$471$	11.2909	0.520257
$472$	−5.39597	−0.248370
$473$	−2.14270	−0.0985216
$474$	−1.87347	−0.0860514
$475$	−93.2072	−4.27664
$476$	11.1014	0.508830
$477$	10.1387	0.464217
$478$	−24.3573	−1.11408
$479$	−14.9465	−0.682922	−0.341461	−	0.939896i	$-0.610922\pi$
$479$	−14.9465	−0.682922	−0.341461	+	0.939896i	$0.610922\pi$
$480$	−6.27970	−0.286628
$481$	−33.1697	−1.51241
$482$	−3.86961	−0.176256
$483$	34.5991	1.57431
$484$	−10.9589	−0.498130
$485$	58.3881	2.65127
$486$	7.81527	0.354508
$487$	43.2823	1.96131	0.980655	−	0.195746i	$-0.0627128\pi$
$487$	43.2823	1.96131	0.980655	+	0.195746i	$0.0627128\pi$
$488$	1.37038	0.0620342
$489$	1.49952	0.0678107
$490$	68.3318	3.08692
$491$	26.6586	1.20309	0.601543	−	0.798840i	$-0.294553\pi$
$491$	26.6586	1.20309	0.601543	+	0.798840i	$0.294553\pi$
$492$	−7.01402	−0.316216
$493$	18.5873	0.837131
$494$	−38.0663	−1.71268
$495$	−0.659913	−0.0296609
$496$	−4.76051	−0.213753
$497$	−13.4048	−0.601286
$498$	−22.4316	−1.00518
$499$	−8.63794	−0.386688	−0.193344	−	0.981131i	$-0.561933\pi$
$499$	−8.63794	−0.386688	−0.193344	+	0.981131i	$0.561933\pi$
$500$	−32.4429	−1.45089
$501$	−9.55500	−0.426886
$502$	7.73037	0.345023
$503$	−31.9299	−1.42368	−0.711841	−	0.702340i	$-0.752138\pi$
$503$	−31.9299	−1.42368	−0.711841	+	0.702340i	$0.752138\pi$
$504$	−3.72740	−0.166032
$505$	−65.9780	−2.93598
$506$	0.975453	0.0433642
$507$	−20.8047	−0.923970
$508$	8.28792	0.367717
$509$	21.0248	0.931907	0.465953	−	0.884809i	$-0.345711\pi$
$509$	21.0248	0.931907	0.465953	+	0.884809i	$0.345711\pi$
$510$	14.4617	0.640375
$511$	−47.2372	−2.08965
$512$	−1.00000	−0.0441942
$513$	−41.2931	−1.82313
$514$	−15.0552	−0.664059
$515$	66.2371	2.91875
$516$	−15.7661	−0.694064
$517$	1.58024	0.0694988
$518$	30.8050	1.35350
$519$	15.3327	0.673032
$520$	−21.8432	−0.957887
$521$	4.44134	0.194578	0.0972892	−	0.995256i	$-0.468983\pi$
$521$	4.44134	0.194578	0.0972892	+	0.995256i	$0.468983\pi$
$522$	−6.24089	−0.273157
$523$	−5.72087	−0.250156	−0.125078	−	0.992147i	$-0.539918\pi$
$523$	−5.72087	−0.250156	−0.125078	+	0.992147i	$0.539918\pi$
$524$	14.5467	0.635475
$525$	91.4232	3.99003
$526$	−19.3963	−0.845717
$527$	10.9631	0.477561
$528$	0.302630	0.0131703
$529$	0.134645	0.00585411
$530$	−55.1786	−2.39681
$531$	−4.17234	−0.181064
$532$	35.3526	1.53273
$533$	−24.3974	−1.05677
$534$	21.7223	0.940016
$535$	−5.47911	−0.236883
$536$	6.23098	0.269137
$537$	7.32471	0.316085
$538$	−16.8323	−0.725694
$539$	−3.29304	−0.141841
$540$	−23.6948	−1.01966
$541$	24.9535	1.07283	0.536417	−	0.843953i	$-0.319778\pi$
$541$	24.9535	1.07283	0.536417	+	0.843953i	$0.319778\pi$
$542$	−5.73370	−0.246283
$543$	−14.7924	−0.634804
$544$	2.30293	0.0987373
$545$	7.41158	0.317477
$546$	37.3377	1.59791
$547$	−11.1998	−0.478871	−0.239435	−	0.970912i	$-0.576962\pi$
$547$	−11.1998	−0.478871	−0.239435	+	0.970912i	$0.576962\pi$
$548$	17.0858	0.729870
$549$	1.05962	0.0452236
$550$	2.57750	0.109905
$551$	59.1919	2.52166
$552$	7.17743	0.305492
$553$	6.05209	0.257361
$554$	2.25689	0.0958862
$555$	40.1296	1.70341
$556$	5.59992	0.237490
$557$	29.9475	1.26892	0.634458	−	0.772957i	$-0.281223\pi$
$557$	29.9475	1.26892	0.634458	+	0.772957i	$0.281223\pi$
$558$	−3.68098	−0.155828
$559$	−54.8406	−2.31951
$560$	20.2860	0.857241
$561$	−0.696936	−0.0294247
$562$	−23.4981	−0.991207
$563$	6.74259	0.284166	0.142083	−	0.989855i	$-0.454620\pi$
$563$	6.74259	0.284166	0.142083	+	0.989855i	$0.454620\pi$
$564$	11.6275	0.489605
$565$	12.8951	0.542502
$566$	−26.1340	−1.09849
$567$	29.3205	1.23134
$568$	−2.78076	−0.116678
$569$	−7.63864	−0.320228	−0.160114	−	0.987099i	$-0.551186\pi$
$569$	−7.63864	−0.320228	−0.160114	+	0.987099i	$0.551186\pi$
$570$	46.0537	1.92898
$571$	25.2934	1.05850	0.529248	−	0.848467i	$-0.322474\pi$
$571$	25.2934	1.05850	0.529248	+	0.848467i	$0.322474\pi$
$572$	1.05266	0.0440141
$573$	−5.14236	−0.214825
$574$	22.6582	0.945734
$575$	61.1301	2.54930
$576$	−0.773233	−0.0322181
$577$	−19.6781	−0.819208	−0.409604	−	0.912263i	$-0.634333\pi$
$577$	−19.6781	−0.819208	−0.409604	+	0.912263i	$0.634333\pi$
$578$	11.6965	0.486511
$579$	9.31375	0.387066
$580$	33.9655	1.41034
$581$	72.4633	3.00628
$582$	−20.7043	−0.858221
$583$	2.65916	0.110131
$584$	−9.79915	−0.405492
$585$	−16.8899	−0.698311
$586$	27.0372	1.11690
$587$	13.7374	0.567002	0.283501	−	0.958972i	$-0.408504\pi$
$587$	13.7374	0.567002	0.283501	+	0.958972i	$0.408504\pi$
$588$	−24.2303	−0.999241
$589$	34.9123	1.43854
$590$	22.7076	0.934856
$591$	−17.6562	−0.726280
$592$	6.39037	0.262643
$593$	30.9739	1.27194	0.635972	−	0.771712i	$-0.280599\pi$
$593$	30.9739	1.27194	0.635972	+	0.771712i	$0.280599\pi$
$594$	1.14189	0.0468525
$595$	−46.7173	−1.91522
$596$	−2.97043	−0.121674
$597$	16.6593	0.681822
$598$	24.9658	1.02093
$599$	−0.464318	−0.0189715	−0.00948577	−	0.999955i	$-0.503019\pi$
$599$	−0.464318	−0.0189715	−0.00948577	+	0.999955i	$0.503019\pi$
$600$	18.9653	0.774257
$601$	−2.06292	−0.0841482	−0.0420741	−	0.999114i	$-0.513397\pi$
$601$	−2.06292	−0.0841482	−0.0420741	+	0.999114i	$0.513397\pi$
$602$	50.9311	2.07579
$603$	4.81800	0.196204
$604$	3.47314	0.141320
$605$	46.1176	1.87495
$606$	23.3957	0.950384
$607$	39.2326	1.59240	0.796200	−	0.605033i	$-0.206840\pi$
$607$	39.2326	1.59240	0.796200	+	0.605033i	$0.206840\pi$
$608$	7.33375	0.297423
$609$	−58.0589	−2.35267
$610$	−5.76690	−0.233495
$611$	40.4448	1.63622
$612$	1.78070	0.0719807
$613$	−34.6476	−1.39940	−0.699702	−	0.714435i	$-0.746684\pi$
$613$	−34.6476	−1.39940	−0.699702	+	0.714435i	$0.746684\pi$
$614$	24.5036	0.988886
$615$	29.5167	1.19023
$616$	−0.977621	−0.0393895
$617$	−21.0818	−0.848722	−0.424361	−	0.905493i	$-0.639501\pi$
$617$	−21.0818	−0.848722	−0.424361	+	0.905493i	$0.639501\pi$
$618$	−23.4875	−0.944806
$619$	−22.9001	−0.920435	−0.460217	−	0.887806i	$-0.652229\pi$
$619$	−22.9001	−0.920435	−0.460217	+	0.887806i	$0.652229\pi$
$620$	20.0334	0.804560
$621$	27.0821	1.08677
$622$	−24.2556	−0.972563
$623$	−70.1720	−2.81138
$624$	7.74555	0.310070
$625$	72.9808	2.91923
$626$	−6.44923	−0.257763
$627$	−2.21941	−0.0886348
$628$	−7.56643	−0.301934
$629$	−14.7166	−0.586789
$630$	15.6858	0.624938
$631$	−38.4745	−1.53165	−0.765823	−	0.643051i	$-0.777668\pi$
$631$	−38.4745	−1.53165	−0.765823	+	0.643051i	$0.777668\pi$
$632$	1.25548	0.0499403
$633$	22.1759	0.881414
$634$	27.3127	1.08472
$635$	−34.8776	−1.38408
$636$	19.5662	0.775851
$637$	−84.2823	−3.33939
$638$	−1.63686	−0.0648038
$639$	−2.15018	−0.0850597
$640$	4.20825	0.166346
$641$	10.0101	0.395374	0.197687	−	0.980265i	$-0.436657\pi$
$641$	10.0101	0.395374	0.197687	+	0.980265i	$0.436657\pi$
$642$	1.94288	0.0766794
$643$	−45.9168	−1.81078	−0.905391	−	0.424579i	$-0.860422\pi$
$643$	−45.9168	−1.81078	−0.905391	+	0.424579i	$0.860422\pi$
$644$	−23.1861	−0.913659
$645$	66.3477	2.61244
$646$	−16.8891	−0.664493
$647$	9.28077	0.364865	0.182432	−	0.983218i	$-0.441603\pi$
$647$	9.28077	0.364865	0.182432	+	0.983218i	$0.441603\pi$
$648$	6.08241	0.238940
$649$	−1.09432	−0.0429558
$650$	65.9687	2.58751
$651$	−34.2441	−1.34213
$652$	−1.00488	−0.0393542
$653$	23.2614	0.910291	0.455145	−	0.890417i	$-0.349587\pi$
$653$	23.2614	0.910291	0.455145	+	0.890417i	$0.349587\pi$
$654$	−2.62813	−0.102768
$655$	−61.2161	−2.39191
$656$	4.70034	0.183518
$657$	−7.57703	−0.295608
$658$	−37.5615	−1.46430
$659$	19.0910	0.743680	0.371840	−	0.928297i	$-0.378727\pi$
$659$	19.0910	0.743680	0.371840	+	0.928297i	$0.378727\pi$
$660$	−1.27354	−0.0495726
$661$	−28.5350	−1.10988	−0.554942	−	0.831889i	$-0.687260\pi$
$661$	−28.5350	−1.10988	−0.554942	+	0.831889i	$0.687260\pi$
$662$	−14.9465	−0.580913
$663$	−17.8375	−0.692750
$664$	15.0322	0.583362
$665$	−148.773	−5.76915
$666$	4.94125	0.191470
$667$	−38.8211	−1.50316
$668$	6.40314	0.247745
$669$	18.6350	0.720472
$670$	−26.2215	−1.01303
$671$	0.277917	0.0107289
$672$	−7.19338	−0.277491
$673$	−24.0264	−0.926149	−0.463074	−	0.886319i	$-0.653254\pi$
$673$	−24.0264	−0.926149	−0.463074	+	0.886319i	$0.653254\pi$
$674$	−3.59091	−0.138317
$675$	71.5607	2.75437
$676$	13.9420	0.536230
$677$	5.25393	0.201925	0.100962	−	0.994890i	$-0.467808\pi$
$677$	5.25393	0.201925	0.100962	+	0.994890i	$0.467808\pi$
$678$	−4.57258	−0.175609
$679$	66.8835	2.56675
$680$	−9.69130	−0.371645
$681$	12.7622	0.489050
$682$	−0.965446	−0.0369688
$683$	−12.3426	−0.472277	−0.236139	−	0.971719i	$-0.575882\pi$
$683$	−12.3426	−0.472277	−0.236139	+	0.971719i	$0.575882\pi$
$684$	5.67070	0.216825
$685$	−71.9013	−2.74721
$686$	44.5302	1.70017
$687$	10.4811	0.399878
$688$	10.5654	0.402803
$689$	68.0588	2.59284
$690$	−30.2044	−1.14986
$691$	6.60106	0.251116	0.125558	−	0.992086i	$-0.459928\pi$
$691$	6.60106	0.251116	0.125558	+	0.992086i	$0.459928\pi$
$692$	−10.2750	−0.390597
$693$	−0.755929	−0.0287154
$694$	−14.1662	−0.537741
$695$	−23.5659	−0.893904
$696$	−12.0441	−0.456530
$697$	−10.8246	−0.410010
$698$	−4.16283	−0.157566
$699$	43.1734	1.63297
$700$	−61.2659	−2.31563
$701$	−11.2267	−0.424028	−0.212014	−	0.977267i	$-0.568002\pi$
$701$	−11.2267	−0.424028	−0.212014	+	0.977267i	$0.568002\pi$
$702$	29.2258	1.10306
$703$	−46.8654	−1.76756
$704$	−0.202803	−0.00764343
$705$	−48.9313	−1.84286
$706$	−7.96830	−0.299891
$707$	−75.5777	−2.84239
$708$	−8.05206	−0.302615
$709$	−11.1614	−0.419176	−0.209588	−	0.977790i	$-0.567212\pi$
$709$	−11.1614	−0.419176	−0.209588	+	0.977790i	$0.567212\pi$
$710$	11.7021	0.439173
$711$	0.970779	0.0364070
$712$	−14.5569	−0.545542
$713$	−22.8973	−0.857511
$714$	16.5659	0.619961
$715$	−4.42987	−0.165668
$716$	−4.90855	−0.183441
$717$	−36.3468	−1.35740
$718$	2.49269	0.0930264
$719$	−24.2383	−0.903935	−0.451967	−	0.892035i	$-0.649278\pi$
$719$	−24.2383	−0.903935	−0.451967	+	0.892035i	$0.649278\pi$
$720$	3.25396	0.121268
$721$	75.8744	2.82571
$722$	−34.7839	−1.29452
$723$	−5.77437	−0.214751
$724$	9.91293	0.368411
$725$	−102.579	−3.80970
$726$	−16.3532	−0.606925
$727$	−43.3101	−1.60628	−0.803142	−	0.595788i	$-0.796840\pi$
$727$	−43.3101	−1.60628	−0.803142	+	0.595788i	$0.796840\pi$
$728$	−25.0213	−0.927352
$729$	29.9095	1.10776
$730$	41.2373	1.52626
$731$	−24.3314	−0.899931
$732$	2.04493	0.0755828
$733$	39.9757	1.47654	0.738268	−	0.674507i	$-0.235644\pi$
$733$	39.9757	1.47654	0.738268	+	0.674507i	$0.235644\pi$
$734$	11.8678	0.438048
$735$	101.967	3.76112
$736$	−4.80985	−0.177293
$737$	1.26366	0.0465476
$738$	3.63446	0.133786
$739$	−3.77800	−0.138976	−0.0694880	−	0.997583i	$-0.522137\pi$
$739$	−3.77800	−0.138976	−0.0694880	+	0.997583i	$0.522137\pi$
$740$	−26.8923	−0.988580
$741$	−56.8039	−2.08674
$742$	−63.2070	−2.32040
$743$	23.9349	0.878087	0.439043	−	0.898466i	$-0.355317\pi$
$743$	23.9349	0.878087	0.439043	+	0.898466i	$0.355317\pi$
$744$	−7.10379	−0.260438
$745$	12.5003	0.457976
$746$	−8.13910	−0.297993
$747$	11.6234	0.425278
$748$	0.467042	0.0170767
$749$	−6.27631	−0.229331
$750$	−48.4124	−1.76777
$751$	−0.918606	−0.0335204	−0.0167602	−	0.999860i	$-0.505335\pi$
$751$	−0.918606	−0.0335204	−0.0167602	+	0.999860i	$0.505335\pi$
$752$	−7.79198	−0.284144
$753$	11.5355	0.420378
$754$	−41.8939	−1.52569
$755$	−14.6158	−0.531925
$756$	−27.1423	−0.987156
$757$	−9.03399	−0.328346	−0.164173	−	0.986432i	$-0.552496\pi$
$757$	−9.03399	−0.328346	−0.164173	+	0.986432i	$0.552496\pi$
$758$	−1.76865	−0.0642403
$759$	1.45561	0.0528351
$760$	−30.8622	−1.11949
$761$	−39.1063	−1.41760	−0.708802	−	0.705408i	$-0.750764\pi$
$761$	−39.1063	−1.41760	−0.708802	+	0.705408i	$0.750764\pi$
$762$	12.3675	0.448028
$763$	8.48995	0.307357
$764$	3.44608	0.124675
$765$	−7.49364	−0.270933
$766$	14.9406	0.539827
$767$	−28.0081	−1.01132
$768$	−1.49224	−0.0538464
$769$	19.4900	0.702827	0.351413	−	0.936220i	$-0.385701\pi$
$769$	19.4900	0.702827	0.351413	+	0.936220i	$0.385701\pi$
$770$	4.11407	0.148261
$771$	−22.4660	−0.809092
$772$	−6.24147	−0.224636
$773$	49.3191	1.77389	0.886943	−	0.461880i	$-0.152825\pi$
$773$	49.3191	1.77389	0.886943	+	0.461880i	$0.152825\pi$
$774$	8.16954	0.293648
$775$	−60.5029	−2.17333
$776$	13.8747	0.498072
$777$	45.9684	1.64911
$778$	29.0731	1.04232
$779$	−34.4711	−1.23506
$780$	−32.5952	−1.16709
$781$	−0.563947	−0.0201796
$782$	11.0767	0.396104
$783$	−45.4451	−1.62408
$784$	16.2376	0.579914
$785$	31.8414	1.13647
$786$	21.7071	0.774266
$787$	0.0646913	0.00230600	0.00115300	−	0.999999i	$-0.499633\pi$
$787$	0.0646913	0.00230600	0.00115300	+	0.999999i	$0.499633\pi$
$788$	11.8321	0.421500
$789$	−28.9438	−1.03043
$790$	−5.28337	−0.187974
$791$	14.7713	0.525208
$792$	−0.156814	−0.00557215
$793$	7.11305	0.252592
$794$	24.5180	0.870110
$795$	−82.3395	−2.92028
$796$	−11.1640	−0.395698
$797$	27.9488	0.989996	0.494998	−	0.868894i	$-0.335169\pi$
$797$	27.9488	0.989996	0.494998	+	0.868894i	$0.335169\pi$
$798$	52.7544	1.86749
$799$	17.9444	0.634827
$800$	−12.7094	−0.449343
$801$	−11.2559	−0.397707
$802$	−30.0778	−1.06208
$803$	−1.98730	−0.0701303
$804$	9.29809	0.327918
$805$	97.5727	3.43899
$806$	−24.7097	−0.870363
$807$	−25.1178	−0.884189
$808$	−15.6783	−0.551559
$809$	36.7012	1.29034	0.645172	−	0.764037i	$-0.276786\pi$
$809$	36.7012	1.29034	0.645172	+	0.764037i	$0.276786\pi$
$810$	−25.5963	−0.899362
$811$	38.3834	1.34782	0.673912	−	0.738812i	$-0.264613\pi$
$811$	38.3834	1.34782	0.673912	+	0.738812i	$0.264613\pi$
$812$	38.9074	1.36538
$813$	−8.55603	−0.300073
$814$	1.29599	0.0454244
$815$	4.22880	0.148128
$816$	3.43651	0.120302
$817$	−77.4842	−2.71083
$818$	10.0926	0.352878
$819$	−19.3473	−0.676050
$820$	−19.7802	−0.690755
$821$	16.5650	0.578121	0.289061	−	0.957311i	$-0.406657\pi$
$821$	16.5650	0.578121	0.289061	+	0.957311i	$0.406657\pi$
$822$	25.4961	0.889277
$823$	−5.94551	−0.207247	−0.103624	−	0.994617i	$-0.533044\pi$
$823$	−5.94551	−0.207247	−0.103624	+	0.994617i	$0.533044\pi$
$824$	15.7398	0.548323
$825$	3.84623	0.133909
$826$	26.0115	0.905055
$827$	12.0993	0.420733	0.210367	−	0.977623i	$-0.432534\pi$
$827$	12.0993	0.420733	0.210367	+	0.977623i	$0.432534\pi$
$828$	−3.71914	−0.129249
$829$	−26.5656	−0.922662	−0.461331	−	0.887228i	$-0.652628\pi$
$829$	−26.5656	−0.922662	−0.461331	+	0.887228i	$0.652628\pi$
$830$	−63.2592	−2.19576
$831$	3.36781	0.116828
$832$	−5.19057	−0.179951
$833$	−37.3941	−1.29563
$834$	8.35640	0.289358
$835$	−26.9460	−0.932505
$836$	1.48731	0.0514396
$837$	−26.8043	−0.926491
$838$	30.9316	1.06851
$839$	13.4469	0.464239	0.232120	−	0.972687i	$-0.425434\pi$
$839$	13.4469	0.464239	0.232120	+	0.972687i	$0.425434\pi$
$840$	30.2715	1.04447
$841$	36.1437	1.24633
$842$	30.8111	1.06182
$843$	−35.0647	−1.20769
$844$	−14.8609	−0.511532
$845$	−58.6713	−2.01836
$846$	−6.02502	−0.207144
$847$	52.8277	1.81518
$848$	−13.1120	−0.450269
$849$	−38.9981	−1.33841
$850$	29.2688	1.00391
$851$	30.7367	1.05364
$852$	−4.14955	−0.142161
$853$	−50.8726	−1.74184	−0.870922	−	0.491421i	$-0.836478\pi$
$853$	−50.8726	−1.74184	−0.870922	+	0.491421i	$0.836478\pi$
$854$	−6.60597	−0.226052
$855$	−23.8637	−0.816122
$856$	−1.30199	−0.0445012
$857$	−25.0643	−0.856182	−0.428091	−	0.903736i	$-0.640814\pi$
$857$	−25.0643	−0.856182	−0.428091	+	0.903736i	$0.640814\pi$
$858$	1.57082	0.0536270
$859$	20.8303	0.710721	0.355361	−	0.934729i	$-0.384358\pi$
$859$	20.8303	0.710721	0.355361	+	0.934729i	$0.384358\pi$
$860$	−44.4619	−1.51614
$861$	33.8114	1.15229
$862$	−32.4231	−1.10434
$863$	13.2946	0.452553	0.226277	−	0.974063i	$-0.427345\pi$
$863$	13.2946	0.452553	0.226277	+	0.974063i	$0.427345\pi$
$864$	−5.63055	−0.191555
$865$	43.2398	1.47020
$866$	−23.8894	−0.811795
$867$	17.4539	0.592767
$868$	22.9482	0.778913
$869$	0.254615	0.00863723
$870$	50.6845	1.71836
$871$	32.3423	1.09588
$872$	1.76120	0.0596419
$873$	10.7284	0.363100
$874$	35.2742	1.19317
$875$	156.392	5.28702
$876$	−14.6226	−0.494053
$877$	−54.5252	−1.84118	−0.920592	−	0.390526i	$-0.872293\pi$
$877$	−54.5252	−1.84118	−0.920592	+	0.390526i	$0.872293\pi$
$878$	−23.2197	−0.783627
$879$	40.3459	1.36083
$880$	0.853446	0.0287697
$881$	−46.7492	−1.57502	−0.787510	−	0.616302i	$-0.788630\pi$
$881$	−46.7492	−1.57502	−0.787510	+	0.616302i	$0.788630\pi$
$882$	12.5555	0.422764
$883$	−25.5439	−0.859622	−0.429811	−	0.902919i	$-0.641420\pi$
$883$	−25.5439	−0.859622	−0.429811	+	0.902919i	$0.641420\pi$
$884$	11.9535	0.402040
$885$	33.8851	1.13903
$886$	−34.8343	−1.17028
$887$	45.1167	1.51487	0.757436	−	0.652909i	$-0.226452\pi$
$887$	45.1167	1.51487	0.757436	+	0.652909i	$0.226452\pi$
$888$	9.53594	0.320005
$889$	−39.9522	−1.33995
$890$	61.2590	2.05341
$891$	1.23353	0.0413249
$892$	−12.4880	−0.418129
$893$	57.1444	1.91227
$894$	−4.43258	−0.148248
$895$	20.6564	0.690467
$896$	4.82054	0.161043
$897$	37.2549	1.24391
$898$	21.4195	0.714779
$899$	38.4228	1.28147
$900$	−9.82730	−0.327577
$901$	30.1961	1.00598
$902$	0.953245	0.0317396
$903$	76.0011	2.52916
$904$	3.06425	0.101915
$905$	−41.7161	−1.38669
$906$	5.18274	0.172185
$907$	17.3066	0.574657	0.287328	−	0.957832i	$-0.407233\pi$
$907$	17.3066	0.574657	0.287328	+	0.957832i	$0.407233\pi$
$908$	−8.55243	−0.283822
$909$	−12.1230	−0.402093
$910$	105.296	3.49053
$911$	−20.8144	−0.689611	−0.344806	−	0.938674i	$-0.612055\pi$
$911$	−20.8144	−0.689611	−0.344806	+	0.938674i	$0.612055\pi$
$912$	10.9437	0.362381
$913$	3.04858	0.100893
$914$	17.1091	0.565920
$915$	−8.60557	−0.284491
$916$	−7.02375	−0.232071
$917$	−70.1229	−2.31566
$918$	12.9668	0.427967
$919$	24.4650	0.807026	0.403513	−	0.914974i	$-0.367789\pi$
$919$	24.4650	0.807026	0.403513	+	0.914974i	$0.367789\pi$
$920$	20.2410	0.667327
$921$	36.5652	1.20486
$922$	−6.64079	−0.218703
$923$	−14.4337	−0.475092
$924$	−1.45884	−0.0479923
$925$	81.2175	2.67042
$926$	−1.20351	−0.0395498
$927$	12.1706	0.399733
$928$	8.07116	0.264949
$929$	−3.88963	−0.127615	−0.0638074	−	0.997962i	$-0.520324\pi$
$929$	−3.88963	−0.127615	−0.0638074	+	0.997962i	$0.520324\pi$
$930$	29.8945	0.980280
$931$	−119.082	−3.90277
$932$	−28.9320	−0.947700
$933$	−36.1951	−1.18498
$934$	−22.9727	−0.751689
$935$	−1.96543	−0.0642764
$936$	−4.01352	−0.131186
$937$	−33.1132	−1.08176	−0.540881	−	0.841099i	$-0.681909\pi$
$937$	−33.1132	−1.08176	−0.540881	+	0.841099i	$0.681909\pi$
$938$	−30.0367	−0.980733
$939$	−9.62377	−0.314060
$940$	32.7906	1.06951
$941$	31.3296	1.02132	0.510658	−	0.859784i	$-0.329402\pi$
$941$	31.3296	1.02132	0.510658	+	0.859784i	$0.329402\pi$
$942$	−11.2909	−0.367877
$943$	22.6079	0.736216
$944$	5.39597	0.175624
$945$	114.222	3.71563
$946$	2.14270	0.0696653
$947$	−25.0191	−0.813013	−0.406506	−	0.913648i	$-0.633253\pi$
$947$	−25.0191	−0.813013	−0.406506	+	0.913648i	$0.633253\pi$
$948$	1.87347	0.0608475
$949$	−50.8632	−1.65109
$950$	93.2072	3.02404
$951$	40.7569	1.32163
$952$	−11.1014	−0.359797
$953$	−20.0664	−0.650013	−0.325007	−	0.945712i	$-0.605366\pi$
$953$	−20.0664	−0.650013	−0.325007	+	0.945712i	$0.605366\pi$
$954$	−10.1387	−0.328251
$955$	−14.5019	−0.469272
$956$	24.3573	0.787770
$957$	−2.44258	−0.0789573
$958$	14.9465	0.482899
$959$	−82.3628	−2.65963
$960$	6.27970	0.202676
$961$	−8.33759	−0.268954
$962$	33.1697	1.06943
$963$	−1.00674	−0.0324419
$964$	3.86961	0.124632
$965$	26.2657	0.845522
$966$	−34.5991	−1.11321
$967$	−35.9159	−1.15498	−0.577488	−	0.816399i	$-0.695967\pi$
$967$	−35.9159	−1.15498	−0.577488	+	0.816399i	$0.695967\pi$
$968$	10.9589	0.352231
$969$	−25.2025	−0.809622
$970$	−58.3881	−1.87473
$971$	24.2162	0.777134	0.388567	−	0.921420i	$-0.372970\pi$
$971$	24.2162	0.777134	0.388567	+	0.921420i	$0.372970\pi$
$972$	−7.81527	−0.250675
$973$	−26.9946	−0.865408
$974$	−43.2823	−1.38686
$975$	98.4409	3.15263
$976$	−1.37038	−0.0438648
$977$	−0.148166	−0.00474024	−0.00237012	−	0.999997i	$-0.500754\pi$
$977$	−0.148166	−0.00474024	−0.00237012	+	0.999997i	$0.500754\pi$
$978$	−1.49952	−0.0479494
$979$	−2.95218	−0.0943522
$980$	−68.3318	−2.18278
$981$	1.36182	0.0434796
$982$	−26.6586	−0.850711
$983$	−4.77076	−0.152164	−0.0760819	−	0.997102i	$-0.524241\pi$
$983$	−4.77076	−0.152164	−0.0760819	+	0.997102i	$0.524241\pi$
$984$	7.01402	0.223599
$985$	−49.7922	−1.58651
$986$	−18.5873	−0.591941
$987$	−56.0507	−1.78411
$988$	38.0663	1.21105
$989$	50.8181	1.61592
$990$	0.659913	0.0209734
$991$	58.5251	1.85911	0.929555	−	0.368684i	$-0.120192\pi$
$991$	58.5251	1.85911	0.929555	+	0.368684i	$0.120192\pi$
$992$	4.76051	0.151146
$993$	−22.3037	−0.707787
$994$	13.4048	0.425173
$995$	46.9810	1.48940
$996$	22.4316	0.710772
$997$	55.8067	1.76742	0.883709	−	0.468038i	$-0.155039\pi$
$997$	55.8067	1.76742	0.883709	+	0.468038i	$0.155039\pi$
$998$	8.63794	0.273429
$999$	35.9813	1.13840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.c.1.18	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.c.1.18	✓	57	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.49224 q^{3} +1.00000 q^{4} -4.20825 q^{5} +1.49224 q^{6} -4.82054 q^{7} -1.00000 q^{8} -0.773233 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.49224 q^{3} +1.00000 q^{4} -4.20825 q^{5} +1.49224 q^{6} -4.82054 q^{7} -1.00000 q^{8} -0.773233 q^{9} +4.20825 q^{10} -0.202803 q^{11} -1.49224 q^{12} -5.19057 q^{13} +4.82054 q^{14} +6.27970 q^{15} +1.00000 q^{16} -2.30293 q^{17} +0.773233 q^{18} -7.33375 q^{19} -4.20825 q^{20} +7.19338 q^{21} +0.202803 q^{22} +4.80985 q^{23} +1.49224 q^{24} +12.7094 q^{25} +5.19057 q^{26} +5.63055 q^{27} -4.82054 q^{28} -8.07116 q^{29} -6.27970 q^{30} -4.76051 q^{31} -1.00000 q^{32} +0.302630 q^{33} +2.30293 q^{34} +20.2860 q^{35} -0.773233 q^{36} +6.39037 q^{37} +7.33375 q^{38} +7.74555 q^{39} +4.20825 q^{40} +4.70034 q^{41} -7.19338 q^{42} +10.5654 q^{43} -0.202803 q^{44} +3.25396 q^{45} -4.80985 q^{46} -7.79198 q^{47} -1.49224 q^{48} +16.2376 q^{49} -12.7094 q^{50} +3.43651 q^{51} -5.19057 q^{52} -13.1120 q^{53} -5.63055 q^{54} +0.853446 q^{55} +4.82054 q^{56} +10.9437 q^{57} +8.07116 q^{58} +5.39597 q^{59} +6.27970 q^{60} -1.37038 q^{61} +4.76051 q^{62} +3.72740 q^{63} +1.00000 q^{64} +21.8432 q^{65} -0.302630 q^{66} -6.23098 q^{67} -2.30293 q^{68} -7.17743 q^{69} -20.2860 q^{70} +2.78076 q^{71} +0.773233 q^{72} +9.79915 q^{73} -6.39037 q^{74} -18.9653 q^{75} -7.33375 q^{76} +0.977621 q^{77} -7.74555 q^{78} -1.25548 q^{79} -4.20825 q^{80} -6.08241 q^{81} -4.70034 q^{82} -15.0322 q^{83} +7.19338 q^{84} +9.69130 q^{85} -10.5654 q^{86} +12.0441 q^{87} +0.202803 q^{88} +14.5569 q^{89} -3.25396 q^{90} +25.0213 q^{91} +4.80985 q^{92} +7.10379 q^{93} +7.79198 q^{94} +30.8622 q^{95} +1.49224 q^{96} -13.8747 q^{97} -16.2376 q^{98} +0.156814 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * q^99

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