LMFDB - Embedded newform 6038.2.a.e.1.2

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:	$0$
Dimension:	$70$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.00085 q^{3} +1.00000 q^{4} -0.777425 q^{5} -3.00085 q^{6} +3.31557 q^{7} +1.00000 q^{8} +6.00511 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.00085 q^{3} +1.00000 q^{4} -0.777425 q^{5} -3.00085 q^{6} +3.31557 q^{7} +1.00000 q^{8} +6.00511 q^{9} -0.777425 q^{10} +3.00939 q^{11} -3.00085 q^{12} +0.139817 q^{13} +3.31557 q^{14} +2.33294 q^{15} +1.00000 q^{16} +6.00785 q^{17} +6.00511 q^{18} +0.993792 q^{19} -0.777425 q^{20} -9.94955 q^{21} +3.00939 q^{22} +7.61004 q^{23} -3.00085 q^{24} -4.39561 q^{25} +0.139817 q^{26} -9.01789 q^{27} +3.31557 q^{28} +4.89741 q^{29} +2.33294 q^{30} +1.91394 q^{31} +1.00000 q^{32} -9.03073 q^{33} +6.00785 q^{34} -2.57761 q^{35} +6.00511 q^{36} +0.677305 q^{37} +0.993792 q^{38} -0.419571 q^{39} -0.777425 q^{40} +5.92906 q^{41} -9.94955 q^{42} +1.00446 q^{43} +3.00939 q^{44} -4.66852 q^{45} +7.61004 q^{46} +10.2282 q^{47} -3.00085 q^{48} +3.99303 q^{49} -4.39561 q^{50} -18.0287 q^{51} +0.139817 q^{52} +3.06009 q^{53} -9.01789 q^{54} -2.33957 q^{55} +3.31557 q^{56} -2.98222 q^{57} +4.89741 q^{58} -2.62789 q^{59} +2.33294 q^{60} +8.78264 q^{61} +1.91394 q^{62} +19.9104 q^{63} +1.00000 q^{64} -0.108697 q^{65} -9.03073 q^{66} -14.8016 q^{67} +6.00785 q^{68} -22.8366 q^{69} -2.57761 q^{70} -12.3176 q^{71} +6.00511 q^{72} -0.642611 q^{73} +0.677305 q^{74} +13.1906 q^{75} +0.993792 q^{76} +9.97785 q^{77} -0.419571 q^{78} -10.4089 q^{79} -0.777425 q^{80} +9.04603 q^{81} +5.92906 q^{82} -11.3800 q^{83} -9.94955 q^{84} -4.67065 q^{85} +1.00446 q^{86} -14.6964 q^{87} +3.00939 q^{88} +10.7248 q^{89} -4.66852 q^{90} +0.463575 q^{91} +7.61004 q^{92} -5.74344 q^{93} +10.2282 q^{94} -0.772598 q^{95} -3.00085 q^{96} -10.2756 q^{97} +3.99303 q^{98} +18.0717 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) $ 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9 + 18 * q^10 + 41 * q^11 + 25 * q^12 + 41 * q^13 + 50 * q^14 + 13 * q^15 + 70 * q^16 + 40 * q^17 + 89 * q^18 + 55 * q^19 + 18 * q^20 + 2 * q^21 + 41 * q^22 + 41 * q^23 + 25 * q^24 + 104 * q^25 + 41 * q^26 + 82 * q^27 + 50 * q^28 + 11 * q^29 + 13 * q^30 + 78 * q^31 + 70 * q^32 + 45 * q^33 + 40 * q^34 + 25 * q^35 + 89 * q^36 + 46 * q^37 + 55 * q^38 + 19 * q^39 + 18 * q^40 + 51 * q^41 + 2 * q^42 + 68 * q^43 + 41 * q^44 + 37 * q^45 + 41 * q^46 + 69 * q^47 + 25 * q^48 + 126 * q^49 + 104 * q^50 + 36 * q^51 + 41 * q^52 + 23 * q^53 + 82 * q^54 + 42 * q^55 + 50 * q^56 + 14 * q^57 + 11 * q^58 + 89 * q^59 + 13 * q^60 + 32 * q^61 + 78 * q^62 + 106 * q^63 + 70 * q^64 + 18 * q^65 + 45 * q^66 + 90 * q^67 + 40 * q^68 - 12 * q^69 + 25 * q^70 + 54 * q^71 + 89 * q^72 + 94 * q^73 + 46 * q^74 + 72 * q^75 + 55 * q^76 - 16 * q^77 + 19 * q^78 + 54 * q^79 + 18 * q^80 + 102 * q^81 + 51 * q^82 + 60 * q^83 + 2 * q^84 - 5 * q^85 + 68 * q^86 + 9 * q^87 + 41 * q^88 + 77 * q^89 + 37 * q^90 + 54 * q^91 + 41 * q^92 - 2 * q^93 + 69 * q^94 + 39 * q^95 + 25 * q^96 + 139 * q^97 + 126 * q^98 + 60 * 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
$3$	−3.00085	−1.73254	−0.866271	−	0.499574i	$-0.833490\pi$
$3$	−3.00085	−1.73254	−0.866271	+	0.499574i	$0.833490\pi$
$4$	1.00000	0.500000
$5$	−0.777425	−0.347675	−0.173837	−	0.984774i	$-0.555617\pi$
$5$	−0.777425	−0.347675	−0.173837	+	0.984774i	$0.555617\pi$
$6$	−3.00085	−1.22509
$7$	3.31557	1.25317	0.626585	−	0.779353i	$-0.284452\pi$
$7$	3.31557	1.25317	0.626585	+	0.779353i	$0.284452\pi$
$8$	1.00000	0.353553
$9$	6.00511	2.00170
$10$	−0.777425	−0.245843
$11$	3.00939	0.907365	0.453682	−	0.891163i	$-0.350110\pi$
$11$	3.00939	0.907365	0.453682	+	0.891163i	$0.350110\pi$
$12$	−3.00085	−0.866271
$13$	0.139817	0.0387783	0.0193892	−	0.999812i	$-0.493828\pi$
$13$	0.139817	0.0387783	0.0193892	+	0.999812i	$0.493828\pi$
$14$	3.31557	0.886124
$15$	2.33294	0.602361
$16$	1.00000	0.250000
$17$	6.00785	1.45712	0.728559	−	0.684983i	$-0.240190\pi$
$17$	6.00785	1.45712	0.728559	+	0.684983i	$0.240190\pi$
$18$	6.00511	1.41542
$19$	0.993792	0.227991	0.113996	−	0.993481i	$-0.463635\pi$
$19$	0.993792	0.227991	0.113996	+	0.993481i	$0.463635\pi$
$20$	−0.777425	−0.173837
$21$	−9.94955	−2.17117
$22$	3.00939	0.641604
$23$	7.61004	1.58680	0.793401	−	0.608699i	$-0.208308\pi$
$23$	7.61004	1.58680	0.793401	+	0.608699i	$0.208308\pi$
$24$	−3.00085	−0.612546
$25$	−4.39561	−0.879122
$26$	0.139817	0.0274204
$27$	−9.01789	−1.73549
$28$	3.31557	0.626585
$29$	4.89741	0.909426	0.454713	−	0.890638i	$-0.349742\pi$
$29$	4.89741	0.909426	0.454713	+	0.890638i	$0.349742\pi$
$30$	2.33294	0.425934
$31$	1.91394	0.343753	0.171877	−	0.985118i	$-0.445017\pi$
$31$	1.91394	0.343753	0.171877	+	0.985118i	$0.445017\pi$
$32$	1.00000	0.176777
$33$	−9.03073	−1.57205
$34$	6.00785	1.03034
$35$	−2.57761	−0.435695
$36$	6.00511	1.00085
$37$	0.677305	0.111348	0.0556741	−	0.998449i	$-0.482269\pi$
$37$	0.677305	0.111348	0.0556741	+	0.998449i	$0.482269\pi$
$38$	0.993792	0.161214
$39$	−0.419571	−0.0671851
$40$	−0.777425	−0.122922
$41$	5.92906	0.925963	0.462981	−	0.886368i	$-0.346780\pi$
$41$	5.92906	0.925963	0.462981	+	0.886368i	$0.346780\pi$
$42$	−9.94955	−1.53525
$43$	1.00446	0.153179	0.0765894	−	0.997063i	$-0.475597\pi$
$43$	1.00446	0.153179	0.0765894	+	0.997063i	$0.475597\pi$
$44$	3.00939	0.453682
$45$	−4.66852	−0.695942
$46$	7.61004	1.12204
$47$	10.2282	1.49193	0.745965	−	0.665985i	$-0.231989\pi$
$47$	10.2282	1.49193	0.745965	+	0.665985i	$0.231989\pi$
$48$	−3.00085	−0.433136
$49$	3.99303	0.570433
$50$	−4.39561	−0.621633
$51$	−18.0287	−2.52452
$52$	0.139817	0.0193892
$53$	3.06009	0.420335	0.210168	−	0.977665i	$-0.432599\pi$
$53$	3.06009	0.420335	0.210168	+	0.977665i	$0.432599\pi$
$54$	−9.01789	−1.22718
$55$	−2.33957	−0.315468
$56$	3.31557	0.443062
$57$	−2.98222	−0.395005
$58$	4.89741	0.643061
$59$	−2.62789	−0.342122	−0.171061	−	0.985260i	$-0.554720\pi$
$59$	−2.62789	−0.342122	−0.171061	+	0.985260i	$0.554720\pi$
$60$	2.33294	0.301181
$61$	8.78264	1.12450	0.562251	−	0.826967i	$-0.309936\pi$
$61$	8.78264	1.12450	0.562251	+	0.826967i	$0.309936\pi$
$62$	1.91394	0.243070
$63$	19.9104	2.50847
$64$	1.00000	0.125000
$65$	−0.108697	−0.0134823
$66$	−9.03073	−1.11161
$67$	−14.8016	−1.80831	−0.904154	−	0.427206i	$-0.859498\pi$
$67$	−14.8016	−1.80831	−0.904154	+	0.427206i	$0.859498\pi$
$68$	6.00785	0.728559
$69$	−22.8366	−2.74920
$70$	−2.57761	−0.308083
$71$	−12.3176	−1.46183	−0.730916	−	0.682468i	$-0.760907\pi$
$71$	−12.3176	−1.46183	−0.730916	+	0.682468i	$0.760907\pi$
$72$	6.00511	0.707709
$73$	−0.642611	−0.0752119	−0.0376060	−	0.999293i	$-0.511973\pi$
$73$	−0.642611	−0.0752119	−0.0376060	+	0.999293i	$0.511973\pi$
$74$	0.677305	0.0787351
$75$	13.1906	1.52312
$76$	0.993792	0.113996
$77$	9.97785	1.13708
$78$	−0.419571	−0.0475071
$79$	−10.4089	−1.17109	−0.585545	−	0.810640i	$-0.699119\pi$
$79$	−10.4089	−1.17109	−0.585545	+	0.810640i	$0.699119\pi$
$80$	−0.777425	−0.0869187
$81$	9.04603	1.00511
$82$	5.92906	0.654755
$83$	−11.3800	−1.24912	−0.624561	−	0.780976i	$-0.714722\pi$
$83$	−11.3800	−1.24912	−0.624561	+	0.780976i	$0.714722\pi$
$84$	−9.94955	−1.08558
$85$	−4.67065	−0.506603
$86$	1.00446	0.108314
$87$	−14.6964	−1.57562
$88$	3.00939	0.320802
$89$	10.7248	1.13682	0.568412	−	0.822744i	$-0.307558\pi$
$89$	10.7248	1.13682	0.568412	+	0.822744i	$0.307558\pi$
$90$	−4.66852	−0.492105
$91$	0.463575	0.0485958
$92$	7.61004	0.793401
$93$	−5.74344	−0.595567
$94$	10.2282	1.05495
$95$	−0.772598	−0.0792669
$96$	−3.00085	−0.306273
$97$	−10.2756	−1.04333	−0.521663	−	0.853152i	$-0.674688\pi$
$97$	−10.2756	−1.04333	−0.521663	+	0.853152i	$0.674688\pi$
$98$	3.99303	0.403357
$99$	18.0717	1.81628
$100$	−4.39561	−0.439561
$101$	−7.36149	−0.732496	−0.366248	−	0.930517i	$-0.619358\pi$
$101$	−7.36149	−0.732496	−0.366248	+	0.930517i	$0.619358\pi$
$102$	−18.0287	−1.78510
$103$	12.1838	1.20050	0.600251	−	0.799811i	$-0.295067\pi$
$103$	12.1838	1.20050	0.600251	+	0.799811i	$0.295067\pi$
$104$	0.139817	0.0137102
$105$	7.73502	0.754861
$106$	3.06009	0.297222
$107$	−15.9959	−1.54638	−0.773190	−	0.634174i	$-0.781340\pi$
$107$	−15.9959	−1.54638	−0.773190	+	0.634174i	$0.781340\pi$
$108$	−9.01789	−0.867747
$109$	−13.6988	−1.31211	−0.656056	−	0.754713i	$-0.727776\pi$
$109$	−13.6988	−1.31211	−0.656056	+	0.754713i	$0.727776\pi$
$110$	−2.33957	−0.223070
$111$	−2.03249	−0.192916
$112$	3.31557	0.313292
$113$	−14.9492	−1.40631	−0.703154	−	0.711038i	$-0.748225\pi$
$113$	−14.9492	−1.40631	−0.703154	+	0.711038i	$0.748225\pi$
$114$	−2.98222	−0.279311
$115$	−5.91623	−0.551691
$116$	4.89741	0.454713
$117$	0.839619	0.0776228
$118$	−2.62789	−0.241917
$119$	19.9195	1.82602
$120$	2.33294	0.212967
$121$	−1.94358	−0.176689
$122$	8.78264	0.795143
$123$	−17.7922	−1.60427
$124$	1.91394	0.171877
$125$	7.30438	0.653324
$126$	19.9104	1.77376
$127$	7.78309	0.690637	0.345319	−	0.938485i	$-0.387771\pi$
$127$	7.78309	0.690637	0.345319	+	0.938485i	$0.387771\pi$
$128$	1.00000	0.0883883
$129$	−3.01424	−0.265389
$130$	−0.108697	−0.00953340
$131$	0.211885	0.0185125	0.00925625	−	0.999957i	$-0.497054\pi$
$131$	0.211885	0.0185125	0.00925625	+	0.999957i	$0.497054\pi$
$132$	−9.03073	−0.786024
$133$	3.29499	0.285712
$134$	−14.8016	−1.27867
$135$	7.01073	0.603388
$136$	6.00785	0.515169
$137$	5.03539	0.430203	0.215101	−	0.976592i	$-0.430992\pi$
$137$	5.03539	0.430203	0.215101	+	0.976592i	$0.430992\pi$
$138$	−22.8366	−1.94398
$139$	14.5380	1.23309	0.616547	−	0.787318i	$-0.288531\pi$
$139$	14.5380	1.23309	0.616547	+	0.787318i	$0.288531\pi$
$140$	−2.57761	−0.217848
$141$	−30.6932	−2.58483
$142$	−12.3176	−1.03367
$143$	0.420765	0.0351861
$144$	6.00511	0.500426
$145$	−3.80736	−0.316184
$146$	−0.642611	−0.0531828
$147$	−11.9825	−0.988299
$148$	0.677305	0.0556741
$149$	5.37270	0.440148	0.220074	−	0.975483i	$-0.429370\pi$
$149$	5.37270	0.440148	0.220074	+	0.975483i	$0.429370\pi$
$150$	13.1906	1.07701
$151$	13.8193	1.12460	0.562300	−	0.826933i	$-0.309916\pi$
$151$	13.8193	1.12460	0.562300	+	0.826933i	$0.309916\pi$
$152$	0.993792	0.0806071
$153$	36.0778	2.91672
$154$	9.97785	0.804038
$155$	−1.48794	−0.119514
$156$	−0.419571	−0.0335926
$157$	−12.1156	−0.966933	−0.483466	−	0.875363i	$-0.660622\pi$
$157$	−12.1156	−0.966933	−0.483466	+	0.875363i	$0.660622\pi$
$158$	−10.4089	−0.828085
$159$	−9.18287	−0.728249
$160$	−0.777425	−0.0614608
$161$	25.2316	1.98853
$162$	9.04603	0.710723
$163$	−5.15860	−0.404053	−0.202026	−	0.979380i	$-0.564753\pi$
$163$	−5.15860	−0.404053	−0.202026	+	0.979380i	$0.564753\pi$
$164$	5.92906	0.462981
$165$	7.02071	0.546562
$166$	−11.3800	−0.883263
$167$	3.50926	0.271554	0.135777	−	0.990739i	$-0.456647\pi$
$167$	3.50926	0.271554	0.135777	+	0.990739i	$0.456647\pi$
$168$	−9.94955	−0.767624
$169$	−12.9805	−0.998496
$170$	−4.67065	−0.358223
$171$	5.96783	0.456371
$172$	1.00446	0.0765894
$173$	17.7484	1.34938	0.674692	−	0.738099i	$-0.264276\pi$
$173$	17.7484	1.34938	0.674692	+	0.738099i	$0.264276\pi$
$174$	−14.6964	−1.11413
$175$	−14.5740	−1.10169
$176$	3.00939	0.226841
$177$	7.88592	0.592742
$178$	10.7248	0.803856
$179$	−17.5756	−1.31366	−0.656832	−	0.754037i	$-0.728104\pi$
$179$	−17.5756	−1.31366	−0.656832	+	0.754037i	$0.728104\pi$
$180$	−4.66852	−0.347971
$181$	3.47129	0.258019	0.129009	−	0.991643i	$-0.458820\pi$
$181$	3.47129	0.258019	0.129009	+	0.991643i	$0.458820\pi$
$182$	0.463575	0.0343624
$183$	−26.3554	−1.94825
$184$	7.61004	0.561019
$185$	−0.526553	−0.0387130
$186$	−5.74344	−0.421130
$187$	18.0800	1.32214
$188$	10.2282	0.745965
$189$	−29.8995	−2.17487
$190$	−0.772598	−0.0560502
$191$	16.5229	1.19555	0.597777	−	0.801662i	$-0.296051\pi$
$191$	16.5229	1.19555	0.597777	+	0.801662i	$0.296051\pi$
$192$	−3.00085	−0.216568
$193$	8.39480	0.604271	0.302135	−	0.953265i	$-0.402300\pi$
$193$	8.39480	0.604271	0.302135	+	0.953265i	$0.402300\pi$
$194$	−10.2756	−0.737742
$195$	0.326185	0.0233586
$196$	3.99303	0.285216
$197$	−0.800261	−0.0570163	−0.0285081	−	0.999594i	$-0.509076\pi$
$197$	−0.800261	−0.0570163	−0.0285081	+	0.999594i	$0.509076\pi$
$198$	18.0717	1.28430
$199$	21.0376	1.49132	0.745660	−	0.666327i	$-0.232135\pi$
$199$	21.0376	1.49132	0.745660	+	0.666327i	$0.232135\pi$
$200$	−4.39561	−0.310817
$201$	44.4175	3.13297
$202$	−7.36149	−0.517953
$203$	16.2377	1.13966
$204$	−18.0287	−1.26226
$205$	−4.60939	−0.321934
$206$	12.1838	0.848884
$207$	45.6991	3.17631
$208$	0.139817	0.00969459
$209$	2.99071	0.206871
$210$	7.73502	0.533767
$211$	5.57652	0.383903	0.191952	−	0.981404i	$-0.438518\pi$
$211$	5.57652	0.383903	0.191952	+	0.981404i	$0.438518\pi$
$212$	3.06009	0.210168
$213$	36.9633	2.53268
$214$	−15.9959	−1.09346
$215$	−0.780892	−0.0532564
$216$	−9.01789	−0.613590
$217$	6.34580	0.430781
$218$	−13.6988	−0.927803
$219$	1.92838	0.130308
$220$	−2.33957	−0.157734
$221$	0.840002	0.0565046
$222$	−2.03249	−0.136412
$223$	22.7747	1.52511	0.762554	−	0.646925i	$-0.223945\pi$
$223$	22.7747	1.52511	0.762554	+	0.646925i	$0.223945\pi$
$224$	3.31557	0.221531
$225$	−26.3961	−1.75974
$226$	−14.9492	−0.994409
$227$	−6.01708	−0.399368	−0.199684	−	0.979860i	$-0.563992\pi$
$227$	−6.01708	−0.399368	−0.199684	+	0.979860i	$0.563992\pi$
$228$	−2.98222	−0.197502
$229$	3.23187	0.213568	0.106784	−	0.994282i	$-0.465945\pi$
$229$	3.23187	0.213568	0.106784	+	0.994282i	$0.465945\pi$
$230$	−5.91623	−0.390105
$231$	−29.9421	−1.97004
$232$	4.89741	0.321531
$233$	0.873832	0.0572466	0.0286233	−	0.999590i	$-0.490888\pi$
$233$	0.873832	0.0572466	0.0286233	+	0.999590i	$0.490888\pi$
$234$	0.839619	0.0548876
$235$	−7.95162	−0.518707
$236$	−2.62789	−0.171061
$237$	31.2355	2.02896
$238$	19.9195	1.29119
$239$	13.4190	0.868002	0.434001	−	0.900912i	$-0.357101\pi$
$239$	13.4190	0.868002	0.434001	+	0.900912i	$0.357101\pi$
$240$	2.33294	0.150590
$241$	−16.8473	−1.08523	−0.542615	−	0.839982i	$-0.682566\pi$
$241$	−16.8473	−1.08523	−0.542615	+	0.839982i	$0.682566\pi$
$242$	−1.94358	−0.124938
$243$	−0.0921101	−0.00590886
$244$	8.78264	0.562251
$245$	−3.10428	−0.198325
$246$	−17.7922	−1.13439
$247$	0.138949	0.00884113
$248$	1.91394	0.121535
$249$	34.1498	2.16416
$250$	7.30438	0.461969
$251$	25.9714	1.63930	0.819650	−	0.572865i	$-0.194168\pi$
$251$	25.9714	1.63930	0.819650	+	0.572865i	$0.194168\pi$
$252$	19.9104	1.25424
$253$	22.9016	1.43981
$254$	7.78309	0.488354
$255$	14.0159	0.877712
$256$	1.00000	0.0625000
$257$	−27.6770	−1.72644	−0.863221	−	0.504827i	$-0.831556\pi$
$257$	−27.6770	−1.72644	−0.863221	+	0.504827i	$0.831556\pi$
$258$	−3.01424	−0.187658
$259$	2.24565	0.139538
$260$	−0.108697	−0.00674113
$261$	29.4095	1.82040
$262$	0.211885	0.0130903
$263$	11.4081	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$263$	11.4081	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$264$	−9.03073	−0.555803
$265$	−2.37899	−0.146140
$266$	3.29499	0.202029
$267$	−32.1835	−1.96960
$268$	−14.8016	−0.904154
$269$	−5.17255	−0.315376	−0.157688	−	0.987489i	$-0.550404\pi$
$269$	−5.17255	−0.315376	−0.157688	+	0.987489i	$0.550404\pi$
$270$	7.01073	0.426660
$271$	21.0636	1.27952	0.639762	−	0.768573i	$-0.279033\pi$
$271$	21.0636	1.27952	0.639762	+	0.768573i	$0.279033\pi$
$272$	6.00785	0.364280
$273$	−1.39112	−0.0841943
$274$	5.03539	0.304199
$275$	−13.2281	−0.797685
$276$	−22.8366	−1.37460
$277$	−2.80048	−0.168265	−0.0841324	−	0.996455i	$-0.526812\pi$
$277$	−2.80048	−0.168265	−0.0841324	+	0.996455i	$0.526812\pi$
$278$	14.5380	0.871930
$279$	11.4934	0.688092
$280$	−2.57761	−0.154042
$281$	−9.23580	−0.550962	−0.275481	−	0.961307i	$-0.588837\pi$
$281$	−9.23580	−0.550962	−0.275481	+	0.961307i	$0.588837\pi$
$282$	−30.6932	−1.82775
$283$	−10.0099	−0.595026	−0.297513	−	0.954718i	$-0.596157\pi$
$283$	−10.0099	−0.595026	−0.297513	+	0.954718i	$0.596157\pi$
$284$	−12.3176	−0.730916
$285$	2.31845	0.137333
$286$	0.420765	0.0248803
$287$	19.6582	1.16039
$288$	6.00511	0.353855
$289$	19.0943	1.12319
$290$	−3.80736	−0.223576
$291$	30.8354	1.80761
$292$	−0.642611	−0.0376060
$293$	29.6278	1.73088	0.865438	−	0.501017i	$-0.167041\pi$
$293$	29.6278	1.73088	0.865438	+	0.501017i	$0.167041\pi$
$294$	−11.9825	−0.698833
$295$	2.04299	0.118947
$296$	0.677305	0.0393675
$297$	−27.1384	−1.57473
$298$	5.37270	0.311232
$299$	1.06401	0.0615336
$300$	13.1906	0.761558
$301$	3.33036	0.191959
$302$	13.8193	0.795213
$303$	22.0907	1.26908
$304$	0.993792	0.0569979
$305$	−6.82784	−0.390961
$306$	36.0778	2.06243
$307$	−12.8645	−0.734218	−0.367109	−	0.930178i	$-0.619652\pi$
$307$	−12.8645	−0.734218	−0.367109	+	0.930178i	$0.619652\pi$
$308$	9.97785	0.568541
$309$	−36.5617	−2.07992
$310$	−1.48794	−0.0845094
$311$	−13.4177	−0.760849	−0.380424	−	0.924812i	$-0.624222\pi$
$311$	−13.4177	−0.760849	−0.380424	+	0.924812i	$0.624222\pi$
$312$	−0.419571	−0.0237535
$313$	−28.4428	−1.60768	−0.803842	−	0.594843i	$-0.797214\pi$
$313$	−28.4428	−1.60768	−0.803842	+	0.594843i	$0.797214\pi$
$314$	−12.1156	−0.683725
$315$	−15.4788	−0.872133
$316$	−10.4089	−0.585545
$317$	−25.1950	−1.41509	−0.707547	−	0.706666i	$-0.750198\pi$
$317$	−25.1950	−1.41509	−0.707547	+	0.706666i	$0.750198\pi$
$318$	−9.18287	−0.514950
$319$	14.7382	0.825181
$320$	−0.777425	−0.0434594
$321$	48.0013	2.67917
$322$	25.2316	1.40610
$323$	5.97055	0.332210
$324$	9.04603	0.502557
$325$	−0.614583	−0.0340909
$326$	−5.15860	−0.285709
$327$	41.1082	2.27329
$328$	5.92906	0.327377
$329$	33.9122	1.86964
$330$	7.02071	0.386477
$331$	17.4864	0.961137	0.480569	−	0.876957i	$-0.340430\pi$
$331$	17.4864	0.961137	0.480569	+	0.876957i	$0.340430\pi$
$332$	−11.3800	−0.624561
$333$	4.06729	0.222886
$334$	3.50926	0.192018
$335$	11.5072	0.628703
$336$	−9.94955	−0.542792
$337$	−11.1389	−0.606776	−0.303388	−	0.952867i	$-0.598118\pi$
$337$	−11.1389	−0.606776	−0.303388	+	0.952867i	$0.598118\pi$
$338$	−12.9805	−0.706043
$339$	44.8605	2.43649
$340$	−4.67065	−0.253302
$341$	5.75978	0.311910
$342$	5.96783	0.322703
$343$	−9.96983	−0.538320
$344$	1.00446	0.0541569
$345$	17.7537	0.955828
$346$	17.7484	0.954159
$347$	−8.11797	−0.435795	−0.217898	−	0.975972i	$-0.569920\pi$
$347$	−8.11797	−0.435795	−0.217898	+	0.975972i	$0.569920\pi$
$348$	−14.6964	−0.787809
$349$	−19.8922	−1.06480	−0.532402	−	0.846491i	$-0.678711\pi$
$349$	−19.8922	−1.06480	−0.532402	+	0.846491i	$0.678711\pi$
$350$	−14.5740	−0.779012
$351$	−1.26086	−0.0672996
$352$	3.00939	0.160401
$353$	−30.2287	−1.60891	−0.804457	−	0.594011i	$-0.797543\pi$
$353$	−30.2287	−1.60891	−0.804457	+	0.594011i	$0.797543\pi$
$354$	7.88592	0.419132
$355$	9.57601	0.508242
$356$	10.7248	0.568412
$357$	−59.7754	−3.16365
$358$	−17.5756	−0.928901
$359$	−31.3667	−1.65547	−0.827736	−	0.561118i	$-0.810371\pi$
$359$	−31.3667	−1.65547	−0.827736	+	0.561118i	$0.810371\pi$
$360$	−4.66852	−0.246053
$361$	−18.0124	−0.948020
$362$	3.47129	0.182447
$363$	5.83239	0.306121
$364$	0.463575	0.0242979
$365$	0.499581	0.0261493
$366$	−26.3554	−1.37762
$367$	1.06213	0.0554427	0.0277213	−	0.999616i	$-0.491175\pi$
$367$	1.06213	0.0554427	0.0277213	+	0.999616i	$0.491175\pi$
$368$	7.61004	0.396701
$369$	35.6046	1.85350
$370$	−0.526553	−0.0273742
$371$	10.1459	0.526751
$372$	−5.74344	−0.297784
$373$	7.06695	0.365913	0.182956	−	0.983121i	$-0.441433\pi$
$373$	7.06695	0.365913	0.182956	+	0.983121i	$0.441433\pi$
$374$	18.0800	0.934893
$375$	−21.9194	−1.13191
$376$	10.2282	0.527477
$377$	0.684742	0.0352660
$378$	−29.8995	−1.53786
$379$	−26.5215	−1.36232	−0.681160	−	0.732135i	$-0.738524\pi$
$379$	−26.5215	−1.36232	−0.681160	+	0.732135i	$0.738524\pi$
$380$	−0.772598	−0.0396334
$381$	−23.3559	−1.19656
$382$	16.5229	0.845385
$383$	4.44067	0.226908	0.113454	−	0.993543i	$-0.463809\pi$
$383$	4.44067	0.226908	0.113454	+	0.993543i	$0.463809\pi$
$384$	−3.00085	−0.153137
$385$	−7.75703	−0.395335
$386$	8.39480	0.427284
$387$	6.03190	0.306619
$388$	−10.2756	−0.521663
$389$	11.5844	0.587351	0.293675	−	0.955905i	$-0.405122\pi$
$389$	11.5844	0.587351	0.293675	+	0.955905i	$0.405122\pi$
$390$	0.326185	0.0165170
$391$	45.7200	2.31216
$392$	3.99303	0.201679
$393$	−0.635836	−0.0320737
$394$	−0.800261	−0.0403166
$395$	8.09211	0.407158
$396$	18.0717	0.908138
$397$	22.2456	1.11647	0.558236	−	0.829682i	$-0.311478\pi$
$397$	22.2456	1.11647	0.558236	+	0.829682i	$0.311478\pi$
$398$	21.0376	1.05452
$399$	−9.88778	−0.495008
$400$	−4.39561	−0.219781
$401$	−21.3099	−1.06416	−0.532082	−	0.846693i	$-0.678590\pi$
$401$	−21.3099	−1.06416	−0.532082	+	0.846693i	$0.678590\pi$
$402$	44.4175	2.21535
$403$	0.267602	0.0133302
$404$	−7.36149	−0.366248
$405$	−7.03261	−0.349453
$406$	16.2377	0.805864
$407$	2.03827	0.101033
$408$	−18.0287	−0.892552
$409$	17.0025	0.840718	0.420359	−	0.907358i	$-0.361904\pi$
$409$	17.0025	0.840718	0.420359	+	0.907358i	$0.361904\pi$
$410$	−4.60939	−0.227642
$411$	−15.1105	−0.745344
$412$	12.1838	0.600251
$413$	−8.71297	−0.428737
$414$	45.6991	2.24599
$415$	8.84712	0.434288
$416$	0.139817	0.00685511
$417$	−43.6263	−2.13639
$418$	2.99071	0.146280
$419$	−34.1085	−1.66631	−0.833155	−	0.553039i	$-0.813468\pi$
$419$	−34.1085	−1.66631	−0.833155	+	0.553039i	$0.813468\pi$
$420$	7.73502	0.377430
$421$	4.34744	0.211881	0.105941	−	0.994372i	$-0.466215\pi$
$421$	4.34744	0.211881	0.105941	+	0.994372i	$0.466215\pi$
$422$	5.57652	0.271461
$423$	61.4212	2.98640
$424$	3.06009	0.148611
$425$	−26.4082	−1.28098
$426$	36.9633	1.79088
$427$	29.1195	1.40919
$428$	−15.9959	−0.773190
$429$	−1.26265	−0.0609614
$430$	−0.780892	−0.0376580
$431$	−7.16645	−0.345196	−0.172598	−	0.984992i	$-0.555216\pi$
$431$	−7.16645	−0.345196	−0.172598	+	0.984992i	$0.555216\pi$
$432$	−9.01789	−0.433874
$433$	0.164833	0.00792138	0.00396069	−	0.999992i	$-0.498739\pi$
$433$	0.164833	0.00792138	0.00396069	+	0.999992i	$0.498739\pi$
$434$	6.34580	0.304608
$435$	11.4253	0.547803
$436$	−13.6988	−0.656056
$437$	7.56279	0.361777
$438$	1.92838	0.0921415
$439$	−30.4518	−1.45339	−0.726693	−	0.686962i	$-0.758944\pi$
$439$	−30.4518	−1.45339	−0.726693	+	0.686962i	$0.758944\pi$
$440$	−2.33957	−0.111535
$441$	23.9786	1.14184
$442$	0.840002	0.0399548
$443$	22.9721	1.09144	0.545718	−	0.837969i	$-0.316257\pi$
$443$	22.9721	1.09144	0.545718	+	0.837969i	$0.316257\pi$
$444$	−2.03249	−0.0964578
$445$	−8.33771	−0.395245
$446$	22.7747	1.07841
$447$	−16.1227	−0.762576
$448$	3.31557	0.156646
$449$	1.40012	0.0660758	0.0330379	−	0.999454i	$-0.489482\pi$
$449$	1.40012	0.0660758	0.0330379	+	0.999454i	$0.489482\pi$
$450$	−26.3961	−1.24433
$451$	17.8428	0.840186
$452$	−14.9492	−0.703154
$453$	−41.4697	−1.94842
$454$	−6.01708	−0.282396
$455$	−0.360394	−0.0168955
$456$	−2.98222	−0.139655
$457$	6.59174	0.308349	0.154174	−	0.988044i	$-0.450728\pi$
$457$	6.59174	0.308349	0.154174	+	0.988044i	$0.450728\pi$
$458$	3.23187	0.151015
$459$	−54.1782	−2.52882
$460$	−5.91623	−0.275846
$461$	−23.1642	−1.07887	−0.539433	−	0.842029i	$-0.681361\pi$
$461$	−23.1642	−1.07887	−0.539433	+	0.842029i	$0.681361\pi$
$462$	−29.9421	−1.39303
$463$	11.8520	0.550809	0.275405	−	0.961328i	$-0.411188\pi$
$463$	11.8520	0.550809	0.275405	+	0.961328i	$0.411188\pi$
$464$	4.89741	0.227356
$465$	4.46509	0.207064
$466$	0.873832	0.0404795
$467$	36.4818	1.68818	0.844088	−	0.536204i	$-0.180142\pi$
$467$	36.4818	1.68818	0.844088	+	0.536204i	$0.180142\pi$
$468$	0.839619	0.0388114
$469$	−49.0759	−2.26612
$470$	−7.95162	−0.366781
$471$	36.3572	1.67525
$472$	−2.62789	−0.120959
$473$	3.02281	0.138989
$474$	31.2355	1.43469
$475$	−4.36832	−0.200432
$476$	19.9195	0.913008
$477$	18.3762	0.841387
$478$	13.4190	0.613770
$479$	30.5704	1.39680	0.698398	−	0.715710i	$-0.253897\pi$
$479$	30.5704	1.39680	0.698398	+	0.715710i	$0.253897\pi$
$480$	2.33294	0.106483
$481$	0.0946989	0.00431790
$482$	−16.8473	−0.767373
$483$	−75.7164	−3.44522
$484$	−1.94358	−0.0883444
$485$	7.98847	0.362738
$486$	−0.0921101	−0.00417820
$487$	34.6705	1.57107	0.785535	−	0.618817i	$-0.212388\pi$
$487$	34.6705	1.57107	0.785535	+	0.618817i	$0.212388\pi$
$488$	8.78264	0.397571
$489$	15.4802	0.700039
$490$	−3.10428	−0.140237
$491$	40.2008	1.81424	0.907119	−	0.420875i	$-0.138277\pi$
$491$	40.2008	1.81424	0.907119	+	0.420875i	$0.138277\pi$
$492$	−17.7922	−0.802135
$493$	29.4229	1.32514
$494$	0.138949	0.00625162
$495$	−14.0494	−0.631473
$496$	1.91394	0.0859383
$497$	−40.8399	−1.83192
$498$	34.1498	1.53029
$499$	−20.3190	−0.909603	−0.454801	−	0.890593i	$-0.650290\pi$
$499$	−20.3190	−0.909603	−0.454801	+	0.890593i	$0.650290\pi$
$500$	7.30438	0.326662
$501$	−10.5308	−0.470479
$502$	25.9714	1.15916
$503$	27.1946	1.21255	0.606274	−	0.795256i	$-0.292663\pi$
$503$	27.1946	1.21255	0.606274	+	0.795256i	$0.292663\pi$
$504$	19.9104	0.886879
$505$	5.72301	0.254670
$506$	22.9016	1.01810
$507$	38.9524	1.72994
$508$	7.78309	0.345319
$509$	−8.94634	−0.396540	−0.198270	−	0.980147i	$-0.563532\pi$
$509$	−8.94634	−0.396540	−0.198270	+	0.980147i	$0.563532\pi$
$510$	14.0159	0.620636
$511$	−2.13062	−0.0942532
$512$	1.00000	0.0441942
$513$	−8.96191	−0.395678
$514$	−27.6770	−1.22078
$515$	−9.47197	−0.417385
$516$	−3.01424	−0.132694
$517$	30.7805	1.35373
$518$	2.24565	0.0986684
$519$	−53.2602	−2.33787
$520$	−0.108697	−0.00476670
$521$	−18.5317	−0.811888	−0.405944	−	0.913898i	$-0.633057\pi$
$521$	−18.5317	−0.811888	−0.405944	+	0.913898i	$0.633057\pi$
$522$	29.4095	1.28722
$523$	24.4270	1.06812	0.534058	−	0.845448i	$-0.320666\pi$
$523$	24.4270	1.06812	0.534058	+	0.845448i	$0.320666\pi$
$524$	0.211885	0.00925625
$525$	43.7343	1.90872
$526$	11.4081	0.497416
$527$	11.4987	0.500889
$528$	−9.03073	−0.393012
$529$	34.9126	1.51794
$530$	−2.37899	−0.103337
$531$	−15.7808	−0.684828
$532$	3.29499	0.142856
$533$	0.828985	0.0359073
$534$	−32.1835	−1.39271
$535$	12.4356	0.537638
$536$	−14.8016	−0.639334
$537$	52.7419	2.27598
$538$	−5.17255	−0.223004
$539$	12.0166	0.517591
$540$	7.01073	0.301694
$541$	−16.0565	−0.690321	−0.345161	−	0.938544i	$-0.612176\pi$
$541$	−16.0565	−0.690321	−0.345161	+	0.938544i	$0.612176\pi$
$542$	21.0636	0.904760
$543$	−10.4168	−0.447028
$544$	6.00785	0.257585
$545$	10.6498	0.456188
$546$	−1.39112	−0.0595344
$547$	34.8768	1.49122	0.745611	−	0.666381i	$-0.232158\pi$
$547$	34.8768	1.49122	0.745611	+	0.666381i	$0.232158\pi$
$548$	5.03539	0.215101
$549$	52.7407	2.25092
$550$	−13.2281	−0.564048
$551$	4.86700	0.207341
$552$	−22.8366	−0.971990
$553$	−34.5114	−1.46757
$554$	−2.80048	−0.118981
$555$	1.58011	0.0670719
$556$	14.5380	0.616547
$557$	−35.6009	−1.50846	−0.754229	−	0.656611i	$-0.771989\pi$
$557$	−35.6009	−1.50846	−0.754229	+	0.656611i	$0.771989\pi$
$558$	11.4934	0.486555
$559$	0.140441	0.00594002
$560$	−2.57761	−0.108924
$561$	−54.2553	−2.29066
$562$	−9.23580	−0.389589
$563$	22.7288	0.957904	0.478952	−	0.877841i	$-0.341017\pi$
$563$	22.7288	0.957904	0.478952	+	0.877841i	$0.341017\pi$
$564$	−30.6932	−1.29242
$565$	11.6219	0.488938
$566$	−10.0099	−0.420747
$567$	29.9928	1.25958
$568$	−12.3176	−0.516835
$569$	−15.3442	−0.643261	−0.321631	−	0.946865i	$-0.604231\pi$
$569$	−15.3442	−0.643261	−0.321631	+	0.946865i	$0.604231\pi$
$570$	2.31845	0.0971093
$571$	−34.9629	−1.46315	−0.731577	−	0.681759i	$-0.761215\pi$
$571$	−34.9629	−1.46315	−0.731577	+	0.681759i	$0.761215\pi$
$572$	0.420765	0.0175931
$573$	−49.5828	−2.07135
$574$	19.6582	0.820518
$575$	−33.4508	−1.39499
$576$	6.00511	0.250213
$577$	13.9493	0.580716	0.290358	−	0.956918i	$-0.406226\pi$
$577$	13.9493	0.580716	0.290358	+	0.956918i	$0.406226\pi$
$578$	19.0943	0.794218
$579$	−25.1916	−1.04693
$580$	−3.80736	−0.158092
$581$	−37.7314	−1.56536
$582$	30.8354	1.27817
$583$	9.20899	0.381397
$584$	−0.642611	−0.0265914
$585$	−0.652740	−0.0269875
$586$	29.6278	1.22391
$587$	−35.2849	−1.45636	−0.728181	−	0.685385i	$-0.759634\pi$
$587$	−35.2849	−1.45636	−0.728181	+	0.685385i	$0.759634\pi$
$588$	−11.9825	−0.494150
$589$	1.90205	0.0783728
$590$	2.04299	0.0841085
$591$	2.40147	0.0987831
$592$	0.677305	0.0278371
$593$	−4.65721	−0.191249	−0.0956243	−	0.995417i	$-0.530485\pi$
$593$	−4.65721	−0.191249	−0.0956243	+	0.995417i	$0.530485\pi$
$594$	−27.1384	−1.11350
$595$	−15.4859	−0.634860
$596$	5.37270	0.220074
$597$	−63.1309	−2.58377
$598$	1.06401	0.0435108
$599$	−16.7585	−0.684735	−0.342367	−	0.939566i	$-0.611229\pi$
$599$	−16.7585	−0.684735	−0.342367	+	0.939566i	$0.611229\pi$
$600$	13.1906	0.538503
$601$	−7.89707	−0.322128	−0.161064	−	0.986944i	$-0.551493\pi$
$601$	−7.89707	−0.322128	−0.161064	+	0.986944i	$0.551493\pi$
$602$	3.33036	0.135735
$603$	−88.8855	−3.61970
$604$	13.8193	0.562300
$605$	1.51098	0.0614303
$606$	22.0907	0.897375
$607$	29.7391	1.20707	0.603537	−	0.797335i	$-0.293758\pi$
$607$	29.7391	1.20707	0.603537	+	0.797335i	$0.293758\pi$
$608$	0.993792	0.0403036
$609$	−48.7270	−1.97452
$610$	−6.82784	−0.276451
$611$	1.43007	0.0578546
$612$	36.0778	1.45836
$613$	1.05928	0.0427840	0.0213920	−	0.999771i	$-0.493190\pi$
$613$	1.05928	0.0427840	0.0213920	+	0.999771i	$0.493190\pi$
$614$	−12.8645	−0.519171
$615$	13.8321	0.557764
$616$	9.97785	0.402019
$617$	23.8025	0.958254	0.479127	−	0.877745i	$-0.340953\pi$
$617$	23.8025	0.958254	0.479127	+	0.877745i	$0.340953\pi$
$618$	−36.5617	−1.47073
$619$	−19.4850	−0.783170	−0.391585	−	0.920142i	$-0.628073\pi$
$619$	−19.4850	−0.783170	−0.391585	+	0.920142i	$0.628073\pi$
$620$	−1.48794	−0.0597572
$621$	−68.6265	−2.75389
$622$	−13.4177	−0.538001
$623$	35.5588	1.42463
$624$	−0.419571	−0.0167963
$625$	16.2995	0.651978
$626$	−28.4428	−1.13680
$627$	−8.97466	−0.358414
$628$	−12.1156	−0.483466
$629$	4.06915	0.162248
$630$	−15.4788	−0.616691
$631$	19.9984	0.796123	0.398062	−	0.917359i	$-0.369683\pi$
$631$	19.9984	0.796123	0.398062	+	0.917359i	$0.369683\pi$
$632$	−10.4089	−0.414043
$633$	−16.7343	−0.665129
$634$	−25.1950	−1.00062
$635$	−6.05076	−0.240117
$636$	−9.18287	−0.364124
$637$	0.558295	0.0221204
$638$	14.7382	0.583491
$639$	−73.9686	−2.92615
$640$	−0.777425	−0.0307304
$641$	32.6992	1.29154	0.645770	−	0.763532i	$-0.276536\pi$
$641$	32.6992	1.29154	0.645770	+	0.763532i	$0.276536\pi$
$642$	48.0013	1.89446
$643$	32.5649	1.28423	0.642117	−	0.766607i	$-0.278056\pi$
$643$	32.5649	1.28423	0.642117	+	0.766607i	$0.278056\pi$
$644$	25.2316	0.994266
$645$	2.34334	0.0922690
$646$	5.97055	0.234908
$647$	−45.5130	−1.78930	−0.894651	−	0.446765i	$-0.852576\pi$
$647$	−45.5130	−1.78930	−0.894651	+	0.446765i	$0.852576\pi$
$648$	9.04603	0.355362
$649$	−7.90835	−0.310430
$650$	−0.614583	−0.0241059
$651$	−19.0428	−0.746346
$652$	−5.15860	−0.202026
$653$	37.7309	1.47653	0.738263	−	0.674513i	$-0.235646\pi$
$653$	37.7309	1.47653	0.738263	+	0.674513i	$0.235646\pi$
$654$	41.1082	1.60746
$655$	−0.164725	−0.00643633
$656$	5.92906	0.231491
$657$	−3.85895	−0.150552
$658$	33.9122	1.32204
$659$	−11.0477	−0.430356	−0.215178	−	0.976575i	$-0.569033\pi$
$659$	−11.0477	−0.430356	−0.215178	+	0.976575i	$0.569033\pi$
$660$	7.02071	0.273281
$661$	10.5239	0.409334	0.204667	−	0.978832i	$-0.434389\pi$
$661$	10.5239	0.409334	0.204667	+	0.978832i	$0.434389\pi$
$662$	17.4864	0.679627
$663$	−2.52072	−0.0978967
$664$	−11.3800	−0.441631
$665$	−2.56161	−0.0993348
$666$	4.06729	0.157604
$667$	37.2694	1.44308
$668$	3.50926	0.135777
$669$	−68.3435	−2.64231
$670$	11.5072	0.444560
$671$	26.4304	1.02033
$672$	−9.94955	−0.383812
$673$	42.7163	1.64659	0.823296	−	0.567612i	$-0.192132\pi$
$673$	42.7163	1.64659	0.823296	+	0.567612i	$0.192132\pi$
$674$	−11.1389	−0.429055
$675$	39.6392	1.52571
$676$	−12.9805	−0.499248
$677$	−46.5516	−1.78912	−0.894561	−	0.446946i	$-0.852512\pi$
$677$	−46.5516	−1.78912	−0.894561	+	0.446946i	$0.852512\pi$
$678$	44.8605	1.72286
$679$	−34.0694	−1.30746
$680$	−4.67065	−0.179111
$681$	18.0564	0.691922
$682$	5.75978	0.220553
$683$	−17.3165	−0.662599	−0.331299	−	0.943526i	$-0.607487\pi$
$683$	−17.3165	−0.662599	−0.331299	+	0.943526i	$0.607487\pi$
$684$	5.96783	0.228186
$685$	−3.91464	−0.149571
$686$	−9.96983	−0.380650
$687$	−9.69835	−0.370015
$688$	1.00446	0.0382947
$689$	0.427853	0.0162999
$690$	17.7537	0.675873
$691$	22.4970	0.855827	0.427914	−	0.903820i	$-0.359249\pi$
$691$	22.4970	0.855827	0.427914	+	0.903820i	$0.359249\pi$
$692$	17.7484	0.674692
$693$	59.9181	2.27610
$694$	−8.11797	−0.308154
$695$	−11.3022	−0.428716
$696$	−14.6964	−0.557065
$697$	35.6209	1.34924
$698$	−19.8922	−0.752931
$699$	−2.62224	−0.0991822
$700$	−14.5740	−0.550844
$701$	−14.2130	−0.536817	−0.268408	−	0.963305i	$-0.586498\pi$
$701$	−14.2130	−0.536817	−0.268408	+	0.963305i	$0.586498\pi$
$702$	−1.26086	−0.0475880
$703$	0.673100	0.0253864
$704$	3.00939	0.113421
$705$	23.8616	0.898681
$706$	−30.2287	−1.13767
$707$	−24.4076	−0.917941
$708$	7.88592	0.296371
$709$	−39.1154	−1.46901	−0.734505	−	0.678604i	$-0.762585\pi$
$709$	−39.1154	−1.46901	−0.734505	+	0.678604i	$0.762585\pi$
$710$	9.57601	0.359381
$711$	−62.5064	−2.34417
$712$	10.7248	0.401928
$713$	14.5651	0.545468
$714$	−59.7754	−2.23704
$715$	−0.327113	−0.0122333
$716$	−17.5756	−0.656832
$717$	−40.2684	−1.50385
$718$	−31.3667	−1.17060
$719$	14.5573	0.542897	0.271448	−	0.962453i	$-0.412497\pi$
$719$	14.5573	0.542897	0.271448	+	0.962453i	$0.412497\pi$
$720$	−4.66852	−0.173986
$721$	40.3962	1.50443
$722$	−18.0124	−0.670351
$723$	50.5562	1.88021
$724$	3.47129	0.129009
$725$	−21.5271	−0.799496
$726$	5.83239	0.216460
$727$	14.4986	0.537724	0.268862	−	0.963179i	$-0.413352\pi$
$727$	14.4986	0.537724	0.268862	+	0.963179i	$0.413352\pi$
$728$	0.463575	0.0171812
$729$	−26.8617	−0.994877
$730$	0.499581	0.0184903
$731$	6.03465	0.223200
$732$	−26.3554	−0.974124
$733$	4.71960	0.174322	0.0871612	−	0.996194i	$-0.472220\pi$
$733$	4.71960	0.174322	0.0871612	+	0.996194i	$0.472220\pi$
$734$	1.06213	0.0392039
$735$	9.31549	0.343607
$736$	7.61004	0.280510
$737$	−44.5439	−1.64080
$738$	35.6046	1.31062
$739$	−32.9192	−1.21095	−0.605477	−	0.795863i	$-0.707018\pi$
$739$	−32.9192	−1.21095	−0.605477	+	0.795863i	$0.707018\pi$
$740$	−0.526553	−0.0193565
$741$	−0.416966	−0.0153176
$742$	10.1459	0.372469
$743$	−31.7298	−1.16405	−0.582027	−	0.813169i	$-0.697740\pi$
$743$	−31.7298	−1.16405	−0.582027	+	0.813169i	$0.697740\pi$
$744$	−5.74344	−0.210565
$745$	−4.17687	−0.153029
$746$	7.06695	0.258740
$747$	−68.3384	−2.50037
$748$	18.0800	0.661069
$749$	−53.0355	−1.93788
$750$	−21.9194	−0.800382
$751$	50.5875	1.84596	0.922982	−	0.384844i	$-0.125745\pi$
$751$	50.5875	1.84596	0.922982	+	0.384844i	$0.125745\pi$
$752$	10.2282	0.372983
$753$	−77.9363	−2.84016
$754$	0.684742	0.0249368
$755$	−10.7435	−0.390995
$756$	−29.8995	−1.08743
$757$	4.28696	0.155812	0.0779061	−	0.996961i	$-0.475177\pi$
$757$	4.28696	0.155812	0.0779061	+	0.996961i	$0.475177\pi$
$758$	−26.5215	−0.963305
$759$	−68.7242	−2.49453
$760$	−0.772598	−0.0280251
$761$	−16.8236	−0.609853	−0.304927	−	0.952376i	$-0.598632\pi$
$761$	−16.8236	−0.609853	−0.304927	+	0.952376i	$0.598632\pi$
$762$	−23.3559	−0.846095
$763$	−45.4195	−1.64430
$764$	16.5229	0.597777
$765$	−28.0478	−1.01407
$766$	4.44067	0.160448
$767$	−0.367425	−0.0132669
$768$	−3.00085	−0.108284
$769$	36.9410	1.33213	0.666064	−	0.745895i	$-0.267978\pi$
$769$	36.9410	1.33213	0.666064	+	0.745895i	$0.267978\pi$
$770$	−7.75703	−0.279544
$771$	83.0545	2.99113
$772$	8.39480	0.302135
$773$	−31.9788	−1.15020	−0.575098	−	0.818084i	$-0.695036\pi$
$773$	−31.9788	−1.15020	−0.575098	+	0.818084i	$0.695036\pi$
$774$	6.03190	0.216812
$775$	−8.41292	−0.302201
$776$	−10.2756	−0.368871
$777$	−6.73887	−0.241756
$778$	11.5844	0.415320
$779$	5.89225	0.211112
$780$	0.326185	0.0116793
$781$	−37.0685	−1.32641
$782$	45.7200	1.63494
$783$	−44.1643	−1.57830
$784$	3.99303	0.142608
$785$	9.41899	0.336178
$786$	−0.635836	−0.0226795
$787$	10.2481	0.365306	0.182653	−	0.983177i	$-0.441532\pi$
$787$	10.2481	0.365306	0.182653	+	0.983177i	$0.441532\pi$
$788$	−0.800261	−0.0285081
$789$	−34.2339	−1.21876
$790$	8.09211	0.287904
$791$	−49.5653	−1.76234
$792$	18.0717	0.642151
$793$	1.22797	0.0436063
$794$	22.2456	0.789465
$795$	7.13899	0.253194
$796$	21.0376	0.745660
$797$	17.6521	0.625269	0.312634	−	0.949874i	$-0.398789\pi$
$797$	17.6521	0.625269	0.312634	+	0.949874i	$0.398789\pi$
$798$	−9.88778	−0.350023
$799$	61.4493	2.17392
$800$	−4.39561	−0.155408
$801$	64.4035	2.27559
$802$	−21.3099	−0.752477
$803$	−1.93387	−0.0682446
$804$	44.4175	1.56649
$805$	−19.6157	−0.691362
$806$	0.267602	0.00942586
$807$	15.5220	0.546402
$808$	−7.36149	−0.258976
$809$	−35.0246	−1.23140	−0.615700	−	0.787981i	$-0.711126\pi$
$809$	−35.0246	−1.23140	−0.615700	+	0.787981i	$0.711126\pi$
$810$	−7.03261	−0.247101
$811$	−7.97665	−0.280098	−0.140049	−	0.990145i	$-0.544726\pi$
$811$	−7.97665	−0.280098	−0.140049	+	0.990145i	$0.544726\pi$
$812$	16.2377	0.569832
$813$	−63.2088	−2.21683
$814$	2.03827	0.0714415
$815$	4.01042	0.140479
$816$	−18.0287	−0.631130
$817$	0.998224	0.0349234
$818$	17.0025	0.594477
$819$	2.78382	0.0972745
$820$	−4.60939	−0.160967
$821$	3.72687	0.130069	0.0650344	−	0.997883i	$-0.479284\pi$
$821$	3.72687	0.130069	0.0650344	+	0.997883i	$0.479284\pi$
$822$	−15.1105	−0.527038
$823$	−14.5683	−0.507821	−0.253910	−	0.967228i	$-0.581717\pi$
$823$	−14.5683	−0.507821	−0.253910	+	0.967228i	$0.581717\pi$
$824$	12.1838	0.424442
$825$	39.6956	1.38202
$826$	−8.71297	−0.303163
$827$	−2.13266	−0.0741598	−0.0370799	−	0.999312i	$-0.511806\pi$
$827$	−2.13266	−0.0741598	−0.0370799	+	0.999312i	$0.511806\pi$
$828$	45.6991	1.58815
$829$	−41.5885	−1.44443	−0.722215	−	0.691669i	$-0.756876\pi$
$829$	−41.5885	−1.44443	−0.722215	+	0.691669i	$0.756876\pi$
$830$	8.84712	0.307088
$831$	8.40384	0.291526
$832$	0.139817	0.00484729
$833$	23.9895	0.831188
$834$	−43.6263	−1.51066
$835$	−2.72818	−0.0944126
$836$	2.99071	0.103436
$837$	−17.2597	−0.596582
$838$	−34.1085	−1.17826
$839$	−30.6018	−1.05649	−0.528245	−	0.849092i	$-0.677150\pi$
$839$	−30.6018	−1.05649	−0.528245	+	0.849092i	$0.677150\pi$
$840$	7.73502	0.266884
$841$	−5.01540	−0.172945
$842$	4.34744	0.149823
$843$	27.7153	0.954565
$844$	5.57652	0.191952
$845$	10.0913	0.347152
$846$	61.4212	2.11171
$847$	−6.44407	−0.221421
$848$	3.06009	0.105084
$849$	30.0382	1.03091
$850$	−26.4082	−0.905793
$851$	5.15431	0.176688
$852$	36.9633	1.26634
$853$	5.56873	0.190670	0.0953348	−	0.995445i	$-0.469608\pi$
$853$	5.56873	0.190670	0.0953348	+	0.995445i	$0.469608\pi$
$854$	29.1195	0.996448
$855$	−4.63954	−0.158669
$856$	−15.9959	−0.546728
$857$	−16.3891	−0.559841	−0.279920	−	0.960023i	$-0.590308\pi$
$857$	−16.3891	−0.559841	−0.279920	+	0.960023i	$0.590308\pi$
$858$	−1.26265	−0.0431063
$859$	−37.3251	−1.27352	−0.636759	−	0.771063i	$-0.719725\pi$
$859$	−37.3251	−1.27352	−0.636759	+	0.771063i	$0.719725\pi$
$860$	−0.780892	−0.0266282
$861$	−58.9914	−2.01042
$862$	−7.16645	−0.244090
$863$	47.5999	1.62032	0.810160	−	0.586209i	$-0.199380\pi$
$863$	47.5999	1.62032	0.810160	+	0.586209i	$0.199380\pi$
$864$	−9.01789	−0.306795
$865$	−13.7980	−0.469147
$866$	0.164833	0.00560126
$867$	−57.2991	−1.94598
$868$	6.34580	0.215391
$869$	−31.3243	−1.06261
$870$	11.4253	0.387355
$871$	−2.06953	−0.0701232
$872$	−13.6988	−0.463901
$873$	−61.7059	−2.08843
$874$	7.56279	0.255815
$875$	24.2182	0.818725
$876$	1.92838	0.0651539
$877$	12.6922	0.428587	0.214293	−	0.976769i	$-0.431255\pi$
$877$	12.6922	0.428587	0.214293	+	0.976769i	$0.431255\pi$
$878$	−30.4518	−1.02770
$879$	−88.9087	−2.99881
$880$	−2.33957	−0.0788670
$881$	−6.15857	−0.207488	−0.103744	−	0.994604i	$-0.533082\pi$
$881$	−6.15857	−0.207488	−0.103744	+	0.994604i	$0.533082\pi$
$882$	23.9786	0.807401
$883$	15.6481	0.526601	0.263301	−	0.964714i	$-0.415189\pi$
$883$	15.6481	0.526601	0.263301	+	0.964714i	$0.415189\pi$
$884$	0.840002	0.0282523
$885$	−6.13071	−0.206081
$886$	22.9721	0.771762
$887$	−26.9477	−0.904814	−0.452407	−	0.891812i	$-0.649435\pi$
$887$	−26.9477	−0.904814	−0.452407	+	0.891812i	$0.649435\pi$
$888$	−2.03249	−0.0682059
$889$	25.8054	0.865486
$890$	−8.33771	−0.279481
$891$	27.2230	0.912006
$892$	22.7747	0.762554
$893$	10.1647	0.340147
$894$	−16.1227	−0.539223
$895$	13.6637	0.456728
$896$	3.31557	0.110766
$897$	−3.19295	−0.106610
$898$	1.40012	0.0467227
$899$	9.37333	0.312618
$900$	−26.3961	−0.879871
$901$	18.3845	0.612478
$902$	17.8428	0.594101
$903$	−9.99393	−0.332577
$904$	−14.9492	−0.497205
$905$	−2.69866	−0.0897066
$906$	−41.4697	−1.37774
$907$	−29.4836	−0.978986	−0.489493	−	0.872007i	$-0.662818\pi$
$907$	−29.4836	−0.978986	−0.489493	+	0.872007i	$0.662818\pi$
$908$	−6.01708	−0.199684
$909$	−44.2066	−1.46624
$910$	−0.360394	−0.0119470
$911$	−19.3446	−0.640915	−0.320457	−	0.947263i	$-0.603837\pi$
$911$	−19.3446	−0.640915	−0.320457	+	0.947263i	$0.603837\pi$
$912$	−2.98222	−0.0987512
$913$	−34.2470	−1.13341
$914$	6.59174	0.218036
$915$	20.4893	0.677357
$916$	3.23187	0.106784
$917$	0.702521	0.0231993
$918$	−54.1782	−1.78815
$919$	3.71215	0.122452	0.0612262	−	0.998124i	$-0.480499\pi$
$919$	3.71215	0.122452	0.0612262	+	0.998124i	$0.480499\pi$
$920$	−5.91623	−0.195052
$921$	38.6046	1.27206
$922$	−23.1642	−0.762873
$923$	−1.72221	−0.0566874
$924$	−29.9421	−0.985021
$925$	−2.97717	−0.0978887
$926$	11.8520	0.389481
$927$	73.1649	2.40305
$928$	4.89741	0.160765
$929$	4.20728	0.138036	0.0690182	−	0.997615i	$-0.478013\pi$
$929$	4.20728	0.138036	0.0690182	+	0.997615i	$0.478013\pi$
$930$	4.46509	0.146416
$931$	3.96824	0.130054
$932$	0.873832	0.0286233
$933$	40.2646	1.31820
$934$	36.4818	1.19372
$935$	−14.0558	−0.459674
$936$	0.839619	0.0274438
$937$	−42.2486	−1.38020	−0.690100	−	0.723714i	$-0.742434\pi$
$937$	−42.2486	−1.38020	−0.690100	+	0.723714i	$0.742434\pi$
$938$	−49.0759	−1.60239
$939$	85.3528	2.78538
$940$	−7.95162	−0.259353
$941$	33.0334	1.07686	0.538429	−	0.842671i	$-0.319018\pi$
$941$	33.0334	1.07686	0.538429	+	0.842671i	$0.319018\pi$
$942$	36.3572	1.18458
$943$	45.1203	1.46932
$944$	−2.62789	−0.0855306
$945$	23.2446	0.756147
$946$	3.02281	0.0982801
$947$	45.4875	1.47814	0.739072	−	0.673627i	$-0.235264\pi$
$947$	45.4875	1.47814	0.739072	+	0.673627i	$0.235264\pi$
$948$	31.2355	1.01448
$949$	−0.0898481	−0.00291659
$950$	−4.36832	−0.141727
$951$	75.6066	2.45171
$952$	19.9195	0.645594
$953$	2.38750	0.0773386	0.0386693	−	0.999252i	$-0.487688\pi$
$953$	2.38750	0.0773386	0.0386693	+	0.999252i	$0.487688\pi$
$954$	18.3762	0.594950
$955$	−12.8453	−0.415664
$956$	13.4190	0.434001
$957$	−44.2272	−1.42966
$958$	30.5704	0.987684
$959$	16.6952	0.539117
$960$	2.33294	0.0752952
$961$	−27.3368	−0.881834
$962$	0.0946989	0.00305322
$963$	−96.0571	−3.09540
$964$	−16.8473	−0.542615
$965$	−6.52632	−0.210090
$966$	−75.7164	−2.43614
$967$	2.10202	0.0675964	0.0337982	−	0.999429i	$-0.489240\pi$
$967$	2.10202	0.0675964	0.0337982	+	0.999429i	$0.489240\pi$
$968$	−1.94358	−0.0624689
$969$	−17.9167	−0.575569
$970$	7.98847	0.256494
$971$	−30.6327	−0.983049	−0.491524	−	0.870864i	$-0.663560\pi$
$971$	−30.6327	−0.983049	−0.491524	+	0.870864i	$0.663560\pi$
$972$	−0.0921101	−0.00295443
$973$	48.2017	1.54528
$974$	34.6705	1.11091
$975$	1.84427	0.0590639
$976$	8.78264	0.281125
$977$	−40.1926	−1.28587	−0.642937	−	0.765919i	$-0.722284\pi$
$977$	−40.1926	−1.28587	−0.642937	+	0.765919i	$0.722284\pi$
$978$	15.4802	0.495002
$979$	32.2750	1.03151
$980$	−3.10428	−0.0991626
$981$	−82.2631	−2.62646
$982$	40.2008	1.28286
$983$	53.3290	1.70093	0.850465	−	0.526032i	$-0.176321\pi$
$983$	53.3290	1.70093	0.850465	+	0.526032i	$0.176321\pi$
$984$	−17.7922	−0.567195
$985$	0.622143	0.0198231
$986$	29.4229	0.937016
$987$	−101.766	−3.23923
$988$	0.138949	0.00442057
$989$	7.64398	0.243064
$990$	−14.0494	−0.446519
$991$	23.2490	0.738529	0.369265	−	0.929324i	$-0.379610\pi$
$991$	23.2490	0.738529	0.369265	+	0.929324i	$0.379610\pi$
$992$	1.91394	0.0607676
$993$	−52.4740	−1.66521
$994$	−40.8399	−1.29536
$995$	−16.3552	−0.518494
$996$	34.1498	1.08208
$997$	24.9089	0.788872	0.394436	−	0.918923i	$-0.370940\pi$
$997$	24.9089	0.788872	0.394436	+	0.918923i	$0.370940\pi$
$998$	−20.3190	−0.643186
$999$	−6.10786	−0.193244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.e.1.2	✓	70

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.e.1.2	✓	70	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.00085 q^{3} +1.00000 q^{4} -0.777425 q^{5} -3.00085 q^{6} +3.31557 q^{7} +1.00000 q^{8} +6.00511 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.00085 q^{3} +1.00000 q^{4} -0.777425 q^{5} -3.00085 q^{6} +3.31557 q^{7} +1.00000 q^{8} +6.00511 q^{9} -0.777425 q^{10} +3.00939 q^{11} -3.00085 q^{12} +0.139817 q^{13} +3.31557 q^{14} +2.33294 q^{15} +1.00000 q^{16} +6.00785 q^{17} +6.00511 q^{18} +0.993792 q^{19} -0.777425 q^{20} -9.94955 q^{21} +3.00939 q^{22} +7.61004 q^{23} -3.00085 q^{24} -4.39561 q^{25} +0.139817 q^{26} -9.01789 q^{27} +3.31557 q^{28} +4.89741 q^{29} +2.33294 q^{30} +1.91394 q^{31} +1.00000 q^{32} -9.03073 q^{33} +6.00785 q^{34} -2.57761 q^{35} +6.00511 q^{36} +0.677305 q^{37} +0.993792 q^{38} -0.419571 q^{39} -0.777425 q^{40} +5.92906 q^{41} -9.94955 q^{42} +1.00446 q^{43} +3.00939 q^{44} -4.66852 q^{45} +7.61004 q^{46} +10.2282 q^{47} -3.00085 q^{48} +3.99303 q^{49} -4.39561 q^{50} -18.0287 q^{51} +0.139817 q^{52} +3.06009 q^{53} -9.01789 q^{54} -2.33957 q^{55} +3.31557 q^{56} -2.98222 q^{57} +4.89741 q^{58} -2.62789 q^{59} +2.33294 q^{60} +8.78264 q^{61} +1.91394 q^{62} +19.9104 q^{63} +1.00000 q^{64} -0.108697 q^{65} -9.03073 q^{66} -14.8016 q^{67} +6.00785 q^{68} -22.8366 q^{69} -2.57761 q^{70} -12.3176 q^{71} +6.00511 q^{72} -0.642611 q^{73} +0.677305 q^{74} +13.1906 q^{75} +0.993792 q^{76} +9.97785 q^{77} -0.419571 q^{78} -10.4089 q^{79} -0.777425 q^{80} +9.04603 q^{81} +5.92906 q^{82} -11.3800 q^{83} -9.94955 q^{84} -4.67065 q^{85} +1.00446 q^{86} -14.6964 q^{87} +3.00939 q^{88} +10.7248 q^{89} -4.66852 q^{90} +0.463575 q^{91} +7.61004 q^{92} -5.74344 q^{93} +10.2282 q^{94} -0.772598 q^{95} -3.00085 q^{96} -10.2756 q^{97} +3.99303 q^{98} +18.0717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9	\( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 70 * q + 70 * q^2 + 25 * q^3 + 70 * q^4 + 18 * q^5 + 25 * q^6 + 50 * q^7 + 70 * q^8 + 89 * q^9 + 18 * q^10 + 41 * q^11 + 25 * q^12 + 41 * q^13 + 50 * q^14 + 13 * q^15 + 70 * q^16 + 40 * q^17 + 89 * q^18 + 55 * q^19 + 18 * q^20 + 2 * q^21 + 41 * q^22 + 41 * q^23 + 25 * q^24 + 104 * q^25 + 41 * q^26 + 82 * q^27 + 50 * q^28 + 11 * q^29 + 13 * q^30 + 78 * q^31 + 70 * q^32 + 45 * q^33 + 40 * q^34 + 25 * q^35 + 89 * q^36 + 46 * q^37 + 55 * q^38 + 19 * q^39 + 18 * q^40 + 51 * q^41 + 2 * q^42 + 68 * q^43 + 41 * q^44 + 37 * q^45 + 41 * q^46 + 69 * q^47 + 25 * q^48 + 126 * q^49 + 104 * q^50 + 36 * q^51 + 41 * q^52 + 23 * q^53 + 82 * q^54 + 42 * q^55 + 50 * q^56 + 14 * q^57 + 11 * q^58 + 89 * q^59 + 13 * q^60 + 32 * q^61 + 78 * q^62 + 106 * q^63 + 70 * q^64 + 18 * q^65 + 45 * q^66 + 90 * q^67 + 40 * q^68 - 12 * q^69 + 25 * q^70 + 54 * q^71 + 89 * q^72 + 94 * q^73 + 46 * q^74 + 72 * q^75 + 55 * q^76 - 16 * q^77 + 19 * q^78 + 54 * q^79 + 18 * q^80 + 102 * q^81 + 51 * q^82 + 60 * q^83 + 2 * q^84 - 5 * q^85 + 68 * q^86 + 9 * q^87 + 41 * q^88 + 77 * q^89 + 37 * q^90 + 54 * q^91 + 41 * q^92 - 2 * q^93 + 69 * q^94 + 39 * q^95 + 25 * q^96 + 139 * q^97 + 126 * q^98 + 60 * q^99

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