LMFDB - Embedded newform 6046.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6046,2,Mod(1,6046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6046, 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("6046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6046 = 2 \cdot 3023 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6046.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:	$48.2775530621$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.31246 q^{3} +1.00000 q^{4} +3.61669 q^{5} +3.31246 q^{6} -0.0193368 q^{7} -1.00000 q^{8} +7.97240 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.31246 q^{3} +1.00000 q^{4} +3.61669 q^{5} +3.31246 q^{6} -0.0193368 q^{7} -1.00000 q^{8} +7.97240 q^{9} -3.61669 q^{10} +2.03368 q^{11} -3.31246 q^{12} -0.780430 q^{13} +0.0193368 q^{14} -11.9802 q^{15} +1.00000 q^{16} -1.13261 q^{17} -7.97240 q^{18} -0.756752 q^{19} +3.61669 q^{20} +0.0640523 q^{21} -2.03368 q^{22} -5.35525 q^{23} +3.31246 q^{24} +8.08047 q^{25} +0.780430 q^{26} -16.4709 q^{27} -0.0193368 q^{28} -2.21880 q^{29} +11.9802 q^{30} +5.88823 q^{31} -1.00000 q^{32} -6.73647 q^{33} +1.13261 q^{34} -0.0699352 q^{35} +7.97240 q^{36} -7.53339 q^{37} +0.756752 q^{38} +2.58514 q^{39} -3.61669 q^{40} +7.76931 q^{41} -0.0640523 q^{42} -8.96086 q^{43} +2.03368 q^{44} +28.8337 q^{45} +5.35525 q^{46} -3.95577 q^{47} -3.31246 q^{48} -6.99963 q^{49} -8.08047 q^{50} +3.75173 q^{51} -0.780430 q^{52} +8.12618 q^{53} +16.4709 q^{54} +7.35518 q^{55} +0.0193368 q^{56} +2.50671 q^{57} +2.21880 q^{58} +11.8167 q^{59} -11.9802 q^{60} +14.3412 q^{61} -5.88823 q^{62} -0.154160 q^{63} +1.00000 q^{64} -2.82257 q^{65} +6.73647 q^{66} -3.53642 q^{67} -1.13261 q^{68} +17.7390 q^{69} +0.0699352 q^{70} -12.0885 q^{71} -7.97240 q^{72} +5.46864 q^{73} +7.53339 q^{74} -26.7662 q^{75} -0.756752 q^{76} -0.0393247 q^{77} -2.58514 q^{78} +16.5996 q^{79} +3.61669 q^{80} +30.6419 q^{81} -7.76931 q^{82} +11.0251 q^{83} +0.0640523 q^{84} -4.09630 q^{85} +8.96086 q^{86} +7.34970 q^{87} -2.03368 q^{88} +0.573623 q^{89} -28.8337 q^{90} +0.0150910 q^{91} -5.35525 q^{92} -19.5045 q^{93} +3.95577 q^{94} -2.73694 q^{95} +3.31246 q^{96} +15.5701 q^{97} +6.99963 q^{98} +16.2133 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9 - 13 * q^10 + 42 * q^11 - 5 * q^13 + 27 * q^14 + 18 * q^15 + 69 * q^16 + 24 * q^17 - 99 * q^18 + q^19 + 13 * q^20 + 7 * q^21 - 42 * q^22 + 25 * q^23 + 100 * q^25 + 5 * q^26 + 15 * q^27 - 27 * q^28 + 87 * q^29 - 18 * q^30 + 5 * q^31 - 69 * q^32 + 28 * q^33 - 24 * q^34 + 33 * q^35 + 99 * q^36 - 5 * q^37 - q^38 + 22 * q^39 - 13 * q^40 + 47 * q^41 - 7 * q^42 - 23 * q^43 + 42 * q^44 + 14 * q^45 - 25 * q^46 + 13 * q^47 + 106 * q^49 - 100 * q^50 + 2 * q^51 - 5 * q^52 + 51 * q^53 - 15 * q^54 - 11 * q^55 + 27 * q^56 + 52 * q^57 - 87 * q^58 + 73 * q^59 + 18 * q^60 + 4 * q^61 - 5 * q^62 - 86 * q^63 + 69 * q^64 + 70 * q^65 - 28 * q^66 - 24 * q^67 + 24 * q^68 + 56 * q^69 - 33 * q^70 + 84 * q^71 - 99 * q^72 + 27 * q^73 + 5 * q^74 + 27 * q^75 + q^76 + 45 * q^77 - 22 * q^78 + 42 * q^79 + 13 * q^80 + 205 * q^81 - 47 * q^82 + q^83 + 7 * q^84 - 18 * q^85 + 23 * q^86 - q^87 - 42 * q^88 + 94 * q^89 - 14 * q^90 + 6 * q^91 + 25 * q^92 - 13 * q^93 - 13 * q^94 + 86 * q^95 + 35 * q^97 - 106 * q^98 + 83 * 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
$3$	−3.31246	−1.91245	−0.956225	−	0.292632i	$-0.905469\pi$
$3$	−3.31246	−1.91245	−0.956225	+	0.292632i	$0.905469\pi$
$4$	1.00000	0.500000
$5$	3.61669	1.61743	0.808717	−	0.588198i	$-0.200162\pi$
$5$	3.61669	1.61743	0.808717	+	0.588198i	$0.200162\pi$
$6$	3.31246	1.35231
$7$	−0.0193368	−0.00730861	−0.00365431	−	0.999993i	$-0.501163\pi$
$7$	−0.0193368	−0.00730861	−0.00365431	+	0.999993i	$0.501163\pi$
$8$	−1.00000	−0.353553
$9$	7.97240	2.65747
$10$	−3.61669	−1.14370
$11$	2.03368	0.613176	0.306588	−	0.951842i	$-0.400813\pi$
$11$	2.03368	0.613176	0.306588	+	0.951842i	$0.400813\pi$
$12$	−3.31246	−0.956225
$13$	−0.780430	−0.216452	−0.108226	−	0.994126i	$-0.534517\pi$
$13$	−0.780430	−0.216452	−0.108226	+	0.994126i	$0.534517\pi$
$14$	0.0193368	0.00516797
$15$	−11.9802	−3.09326
$16$	1.00000	0.250000
$17$	−1.13261	−0.274698	−0.137349	−	0.990523i	$-0.543858\pi$
$17$	−1.13261	−0.274698	−0.137349	+	0.990523i	$0.543858\pi$
$18$	−7.97240	−1.87911
$19$	−0.756752	−0.173611	−0.0868054	−	0.996225i	$-0.527666\pi$
$19$	−0.756752	−0.173611	−0.0868054	+	0.996225i	$0.527666\pi$
$20$	3.61669	0.808717
$21$	0.0640523	0.0139774
$22$	−2.03368	−0.433581
$23$	−5.35525	−1.11665	−0.558323	−	0.829624i	$-0.688555\pi$
$23$	−5.35525	−1.11665	−0.558323	+	0.829624i	$0.688555\pi$
$24$	3.31246	0.676153
$25$	8.08047	1.61609
$26$	0.780430	0.153055
$27$	−16.4709	−3.16982
$28$	−0.0193368	−0.00365431
$29$	−2.21880	−0.412021	−0.206011	−	0.978550i	$-0.566048\pi$
$29$	−2.21880	−0.412021	−0.206011	+	0.978550i	$0.566048\pi$
$30$	11.9802	2.18727
$31$	5.88823	1.05756	0.528779	−	0.848760i	$-0.322650\pi$
$31$	5.88823	1.05756	0.528779	+	0.848760i	$0.322650\pi$
$32$	−1.00000	−0.176777
$33$	−6.73647	−1.17267
$34$	1.13261	0.194241
$35$	−0.0699352	−0.0118212
$36$	7.97240	1.32873
$37$	−7.53339	−1.23848	−0.619241	−	0.785201i	$-0.712560\pi$
$37$	−7.53339	−1.23848	−0.619241	+	0.785201i	$0.712560\pi$
$38$	0.756752	0.122761
$39$	2.58514	0.413954
$40$	−3.61669	−0.571849
$41$	7.76931	1.21336	0.606681	−	0.794946i	$-0.292501\pi$
$41$	7.76931	1.21336	0.606681	+	0.794946i	$0.292501\pi$
$42$	−0.0640523	−0.00988349
$43$	−8.96086	−1.36652	−0.683259	−	0.730176i	$-0.739438\pi$
$43$	−8.96086	−1.36652	−0.683259	+	0.730176i	$0.739438\pi$
$44$	2.03368	0.306588
$45$	28.8337	4.29828
$46$	5.35525	0.789588
$47$	−3.95577	−0.577008	−0.288504	−	0.957479i	$-0.593158\pi$
$47$	−3.95577	−0.577008	−0.288504	+	0.957479i	$0.593158\pi$
$48$	−3.31246	−0.478113
$49$	−6.99963	−0.999947
$50$	−8.08047	−1.14275
$51$	3.75173	0.525347
$52$	−0.780430	−0.108226
$53$	8.12618	1.11622	0.558108	−	0.829768i	$-0.311527\pi$
$53$	8.12618	1.11622	0.558108	+	0.829768i	$0.311527\pi$
$54$	16.4709	2.24140
$55$	7.35518	0.991772
$56$	0.0193368	0.00258399
$57$	2.50671	0.332022
$58$	2.21880	0.291343
$59$	11.8167	1.53841	0.769203	−	0.639004i	$-0.220653\pi$
$59$	11.8167	1.53841	0.769203	+	0.639004i	$0.220653\pi$
$60$	−11.9802	−1.54663
$61$	14.3412	1.83621	0.918104	−	0.396340i	$-0.129720\pi$
$61$	14.3412	1.83621	0.918104	+	0.396340i	$0.129720\pi$
$62$	−5.88823	−0.747806
$63$	−0.154160	−0.0194224
$64$	1.00000	0.125000
$65$	−2.82257	−0.350097
$66$	6.73647	0.829202
$67$	−3.53642	−0.432042	−0.216021	−	0.976389i	$-0.569308\pi$
$67$	−3.53642	−0.432042	−0.216021	+	0.976389i	$0.569308\pi$
$68$	−1.13261	−0.137349
$69$	17.7390	2.13553
$70$	0.0699352	0.00835885
$71$	−12.0885	−1.43464	−0.717322	−	0.696742i	$-0.754632\pi$
$71$	−12.0885	−1.43464	−0.717322	+	0.696742i	$0.754632\pi$
$72$	−7.97240	−0.939556
$73$	5.46864	0.640056	0.320028	−	0.947408i	$-0.396308\pi$
$73$	5.46864	0.640056	0.320028	+	0.947408i	$0.396308\pi$
$74$	7.53339	0.875739
$75$	−26.7662	−3.09070
$76$	−0.756752	−0.0868054
$77$	−0.0393247	−0.00448147
$78$	−2.58514	−0.292710
$79$	16.5996	1.86760	0.933800	−	0.357796i	$-0.116472\pi$
$79$	16.5996	1.86760	0.933800	+	0.357796i	$0.116472\pi$
$80$	3.61669	0.404359
$81$	30.6419	3.40466
$82$	−7.76931	−0.857976
$83$	11.0251	1.21016	0.605081	−	0.796164i	$-0.293141\pi$
$83$	11.0251	1.21016	0.605081	+	0.796164i	$0.293141\pi$
$84$	0.0640523	0.00698868
$85$	−4.09630	−0.444306
$86$	8.96086	0.966274
$87$	7.34970	0.787970
$88$	−2.03368	−0.216791
$89$	0.573623	0.0608039	0.0304019	−	0.999538i	$-0.490321\pi$
$89$	0.573623	0.0608039	0.0304019	+	0.999538i	$0.490321\pi$
$90$	−28.8337	−3.03934
$91$	0.0150910	0.00158197
$92$	−5.35525	−0.558323
$93$	−19.5045	−2.02253
$94$	3.95577	0.408006
$95$	−2.73694	−0.280804
$96$	3.31246	0.338077
$97$	15.5701	1.58091	0.790453	−	0.612522i	$-0.209845\pi$
$97$	15.5701	1.58091	0.790453	+	0.612522i	$0.209845\pi$
$98$	6.99963	0.707069
$99$	16.2133	1.62949
$100$	8.08047	0.808047
$101$	−5.58710	−0.555937	−0.277968	−	0.960590i	$-0.589661\pi$
$101$	−5.58710	−0.555937	−0.277968	+	0.960590i	$0.589661\pi$
$102$	−3.75173	−0.371476
$103$	−5.65798	−0.557497	−0.278749	−	0.960364i	$-0.589920\pi$
$103$	−5.65798	−0.557497	−0.278749	+	0.960364i	$0.589920\pi$
$104$	0.780430	0.0765274
$105$	0.231658	0.0226075
$106$	−8.12618	−0.789284
$107$	−8.34063	−0.806320	−0.403160	−	0.915130i	$-0.632088\pi$
$107$	−8.34063	−0.806320	−0.403160	+	0.915130i	$0.632088\pi$
$108$	−16.4709	−1.58491
$109$	10.4014	0.996275	0.498137	−	0.867098i	$-0.334018\pi$
$109$	10.4014	0.996275	0.498137	+	0.867098i	$0.334018\pi$
$110$	−7.35518	−0.701289
$111$	24.9541	2.36853
$112$	−0.0193368	−0.00182715
$113$	8.86554	0.834000	0.417000	−	0.908907i	$-0.363082\pi$
$113$	8.86554	0.834000	0.417000	+	0.908907i	$0.363082\pi$
$114$	−2.50671	−0.234775
$115$	−19.3683	−1.80610
$116$	−2.21880	−0.206011
$117$	−6.22190	−0.575215
$118$	−11.8167	−1.08782
$119$	0.0219010	0.00200766
$120$	11.9802	1.09363
$121$	−6.86416	−0.624015
$122$	−14.3412	−1.29840
$123$	−25.7355	−2.32049
$124$	5.88823	0.528779
$125$	11.1411	0.996490
$126$	0.154160	0.0137337
$127$	10.5214	0.933627	0.466814	−	0.884356i	$-0.345402\pi$
$127$	10.5214	0.933627	0.466814	+	0.884356i	$0.345402\pi$
$128$	−1.00000	−0.0883883
$129$	29.6825	2.61340
$130$	2.82257	0.247556
$131$	−9.48565	−0.828765	−0.414383	−	0.910103i	$-0.636002\pi$
$131$	−9.48565	−0.828765	−0.414383	+	0.910103i	$0.636002\pi$
$132$	−6.73647	−0.586334
$133$	0.0146331	0.00126885
$134$	3.53642	0.305500
$135$	−59.5701	−5.12698
$136$	1.13261	0.0971205
$137$	−3.61136	−0.308540	−0.154270	−	0.988029i	$-0.549303\pi$
$137$	−3.61136	−0.308540	−0.154270	+	0.988029i	$0.549303\pi$
$138$	−17.7390	−1.51005
$139$	−13.2821	−1.12658	−0.563288	−	0.826261i	$-0.690464\pi$
$139$	−13.2821	−1.12658	−0.563288	+	0.826261i	$0.690464\pi$
$140$	−0.0699352	−0.00591060
$141$	13.1033	1.10350
$142$	12.0885	1.01445
$143$	−1.58714	−0.132723
$144$	7.97240	0.664366
$145$	−8.02473	−0.666417
$146$	−5.46864	−0.452588
$147$	23.1860	1.91235
$148$	−7.53339	−0.619241
$149$	−4.94633	−0.405219	−0.202609	−	0.979260i	$-0.564942\pi$
$149$	−4.94633	−0.405219	−0.202609	+	0.979260i	$0.564942\pi$
$150$	26.7662	2.18545
$151$	10.5795	0.860944	0.430472	−	0.902604i	$-0.358347\pi$
$151$	10.5795	0.860944	0.430472	+	0.902604i	$0.358347\pi$
$152$	0.756752	0.0613807
$153$	−9.02961	−0.730001
$154$	0.0393247	0.00316888
$155$	21.2959	1.71053
$156$	2.58514	0.206977
$157$	−2.31555	−0.184801	−0.0924006	−	0.995722i	$-0.529454\pi$
$157$	−2.31555	−0.184801	−0.0924006	+	0.995722i	$0.529454\pi$
$158$	−16.5996	−1.32059
$159$	−26.9176	−2.13471
$160$	−3.61669	−0.285925
$161$	0.103553	0.00816114
$162$	−30.6419	−2.40746
$163$	1.12147	0.0878402	0.0439201	−	0.999035i	$-0.486015\pi$
$163$	1.12147	0.0878402	0.0439201	+	0.999035i	$0.486015\pi$
$164$	7.76931	0.606681
$165$	−24.3637	−1.89671
$166$	−11.0251	−0.855714
$167$	17.9544	1.38936	0.694678	−	0.719321i	$-0.255547\pi$
$167$	17.9544	1.38936	0.694678	+	0.719321i	$0.255547\pi$
$168$	−0.0640523	−0.00494174
$169$	−12.3909	−0.953148
$170$	4.09630	0.314172
$171$	−6.03313	−0.461365
$172$	−8.96086	−0.683259
$173$	17.6635	1.34293	0.671466	−	0.741035i	$-0.265665\pi$
$173$	17.6635	1.34293	0.671466	+	0.741035i	$0.265665\pi$
$174$	−7.34970	−0.557179
$175$	−0.156250	−0.0118114
$176$	2.03368	0.153294
$177$	−39.1424	−2.94213
$178$	−0.573623	−0.0429948
$179$	9.64850	0.721163	0.360581	−	0.932728i	$-0.382578\pi$
$179$	9.64850	0.721163	0.360581	+	0.932728i	$0.382578\pi$
$180$	28.8337	2.14914
$181$	13.0005	0.966319	0.483160	−	0.875532i	$-0.339489\pi$
$181$	13.0005	0.966319	0.483160	+	0.875532i	$0.339489\pi$
$182$	−0.0150910	−0.00111862
$183$	−47.5048	−3.51166
$184$	5.35525	0.394794
$185$	−27.2460	−2.00316
$186$	19.5045	1.43014
$187$	−2.30336	−0.168438
$188$	−3.95577	−0.288504
$189$	0.318494	0.0231670
$190$	2.73694	0.198559
$191$	4.68654	0.339106	0.169553	−	0.985521i	$-0.445768\pi$
$191$	4.68654	0.339106	0.169553	+	0.985521i	$0.445768\pi$
$192$	−3.31246	−0.239056
$193$	0.682192	0.0491052	0.0245526	−	0.999699i	$-0.492184\pi$
$193$	0.682192	0.0491052	0.0245526	+	0.999699i	$0.492184\pi$
$194$	−15.5701	−1.11787
$195$	9.34967	0.669544
$196$	−6.99963	−0.499973
$197$	−2.71862	−0.193693	−0.0968467	−	0.995299i	$-0.530876\pi$
$197$	−2.71862	−0.193693	−0.0968467	+	0.995299i	$0.530876\pi$
$198$	−16.2133	−1.15223
$199$	−5.95323	−0.422013	−0.211006	−	0.977485i	$-0.567674\pi$
$199$	−5.95323	−0.422013	−0.211006	+	0.977485i	$0.567674\pi$
$200$	−8.08047	−0.571375
$201$	11.7143	0.826260
$202$	5.58710	0.393107
$203$	0.0429045	0.00301130
$204$	3.75173	0.262673
$205$	28.0992	1.96253
$206$	5.65798	0.394210
$207$	−42.6942	−2.96745
$208$	−0.780430	−0.0541131
$209$	−1.53899	−0.106454
$210$	−0.231658	−0.0159859
$211$	−14.7314	−1.01415	−0.507076	−	0.861901i	$-0.669274\pi$
$211$	−14.7314	−1.01415	−0.507076	+	0.861901i	$0.669274\pi$
$212$	8.12618	0.558108
$213$	40.0428	2.74368
$214$	8.34063	0.570154
$215$	−32.4087	−2.21025
$216$	16.4709	1.12070
$217$	−0.113859	−0.00772928
$218$	−10.4014	−0.704473
$219$	−18.1147	−1.22408
$220$	7.35518	0.495886
$221$	0.883922	0.0594591
$222$	−24.9541	−1.67481
$223$	29.4314	1.97087	0.985437	−	0.170040i	$-0.0543897\pi$
$223$	29.4314	1.97087	0.985437	+	0.170040i	$0.0543897\pi$
$224$	0.0193368	0.00129199
$225$	64.4207	4.29471
$226$	−8.86554	−0.589727
$227$	−13.1280	−0.871337	−0.435669	−	0.900107i	$-0.643488\pi$
$227$	−13.1280	−0.871337	−0.435669	+	0.900107i	$0.643488\pi$
$228$	2.50671	0.166011
$229$	17.4293	1.15176	0.575882	−	0.817533i	$-0.304659\pi$
$229$	17.4293	1.15176	0.575882	+	0.817533i	$0.304659\pi$
$230$	19.3683	1.27711
$231$	0.130262	0.00857058
$232$	2.21880	0.145672
$233$	−8.27944	−0.542404	−0.271202	−	0.962522i	$-0.587421\pi$
$233$	−8.27944	−0.542404	−0.271202	+	0.962522i	$0.587421\pi$
$234$	6.22190	0.406738
$235$	−14.3068	−0.933272
$236$	11.8167	0.769203
$237$	−54.9855	−3.57169
$238$	−0.0219010	−0.00141963
$239$	−25.0288	−1.61898	−0.809488	−	0.587136i	$-0.800255\pi$
$239$	−25.0288	−1.61898	−0.809488	+	0.587136i	$0.800255\pi$
$240$	−11.9802	−0.773316
$241$	−15.1102	−0.973336	−0.486668	−	0.873587i	$-0.661788\pi$
$241$	−15.1102	−0.973336	−0.486668	+	0.873587i	$0.661788\pi$
$242$	6.86416	0.441245
$243$	−52.0876	−3.34142
$244$	14.3412	0.918104
$245$	−25.3155	−1.61735
$246$	25.7355	1.64084
$247$	0.590592	0.0375785
$248$	−5.88823	−0.373903
$249$	−36.5202	−2.31437
$250$	−11.1411	−0.704625
$251$	−10.2624	−0.647760	−0.323880	−	0.946098i	$-0.604987\pi$
$251$	−10.2624	−0.647760	−0.323880	+	0.946098i	$0.604987\pi$
$252$	−0.154160	−0.00971120
$253$	−10.8908	−0.684701
$254$	−10.5214	−0.660174
$255$	13.5688	0.849714
$256$	1.00000	0.0625000
$257$	4.84962	0.302511	0.151256	−	0.988495i	$-0.451668\pi$
$257$	4.84962	0.302511	0.151256	+	0.988495i	$0.451668\pi$
$258$	−29.6825	−1.84795
$259$	0.145671	0.00905159
$260$	−2.82257	−0.175049
$261$	−17.6892	−1.09493
$262$	9.48565	0.586026
$263$	22.2845	1.37412	0.687060	−	0.726601i	$-0.258901\pi$
$263$	22.2845	1.37412	0.687060	+	0.726601i	$0.258901\pi$
$264$	6.73647	0.414601
$265$	29.3899	1.80541
$266$	−0.0146331	−0.000897216 0
$267$	−1.90010	−0.116284
$268$	−3.53642	−0.216021
$269$	−25.0930	−1.52994	−0.764972	−	0.644063i	$-0.777247\pi$
$269$	−25.0930	−1.52994	−0.764972	+	0.644063i	$0.777247\pi$
$270$	59.5701	3.62532
$271$	−14.6268	−0.888512	−0.444256	−	0.895900i	$-0.646532\pi$
$271$	−14.6268	−0.888512	−0.444256	+	0.895900i	$0.646532\pi$
$272$	−1.13261	−0.0686746
$273$	−0.0499883	−0.00302543
$274$	3.61136	0.218171
$275$	16.4330	0.990950
$276$	17.7390	1.06777
$277$	−1.29785	−0.0779801	−0.0389901	−	0.999240i	$-0.512414\pi$
$277$	−1.29785	−0.0779801	−0.0389901	+	0.999240i	$0.512414\pi$
$278$	13.2821	0.796609
$279$	46.9433	2.81042
$280$	0.0699352	0.00417943
$281$	8.85904	0.528486	0.264243	−	0.964456i	$-0.414878\pi$
$281$	8.85904	0.528486	0.264243	+	0.964456i	$0.414878\pi$
$282$	−13.1033	−0.780292
$283$	13.5170	0.803502	0.401751	−	0.915749i	$-0.368402\pi$
$283$	13.5170	0.803502	0.401751	+	0.915749i	$0.368402\pi$
$284$	−12.0885	−0.717322
$285$	9.06601	0.537024
$286$	1.58714	0.0938496
$287$	−0.150233	−0.00886799
$288$	−7.97240	−0.469778
$289$	−15.7172	−0.924541
$290$	8.02473	0.471228
$291$	−51.5754	−3.02341
$292$	5.46864	0.320028
$293$	29.1967	1.70569	0.852844	−	0.522166i	$-0.174876\pi$
$293$	29.1967	1.70569	0.852844	+	0.522166i	$0.174876\pi$
$294$	−23.1860	−1.35223
$295$	42.7375	2.48827
$296$	7.53339	0.437869
$297$	−33.4964	−1.94366
$298$	4.94633	0.286533
$299$	4.17939	0.241701
$300$	−26.7662	−1.54535
$301$	0.173274	0.00998735
$302$	−10.5795	−0.608780
$303$	18.5070	1.06320
$304$	−0.756752	−0.0434027
$305$	51.8679	2.96995
$306$	9.02961	0.516189
$307$	−3.68652	−0.210401	−0.105200	−	0.994451i	$-0.533548\pi$
$307$	−3.68652	−0.210401	−0.105200	+	0.994451i	$0.533548\pi$
$308$	−0.0393247	−0.00224073
$309$	18.7418	1.06619
$310$	−21.2959	−1.20953
$311$	30.8487	1.74927	0.874635	−	0.484781i	$-0.161101\pi$
$311$	30.8487	1.74927	0.874635	+	0.484781i	$0.161101\pi$
$312$	−2.58514	−0.146355
$313$	3.85313	0.217792	0.108896	−	0.994053i	$-0.465269\pi$
$313$	3.85313	0.217792	0.108896	+	0.994053i	$0.465269\pi$
$314$	2.31555	0.130674
$315$	−0.557551	−0.0314144
$316$	16.5996	0.933800
$317$	14.6422	0.822388	0.411194	−	0.911548i	$-0.365112\pi$
$317$	14.6422	0.822388	0.411194	+	0.911548i	$0.365112\pi$
$318$	26.9176	1.50947
$319$	−4.51232	−0.252642
$320$	3.61669	0.202179
$321$	27.6280	1.54205
$322$	−0.103553	−0.00577079
$323$	0.857105	0.0476906
$324$	30.6419	1.70233
$325$	−6.30624	−0.349807
$326$	−1.12147	−0.0621124
$327$	−34.4543	−1.90533
$328$	−7.76931	−0.428988
$329$	0.0764918	0.00421713
$330$	24.3637	1.34118
$331$	17.1140	0.940671	0.470335	−	0.882488i	$-0.344133\pi$
$331$	17.1140	0.940671	0.470335	+	0.882488i	$0.344133\pi$
$332$	11.0251	0.605081
$333$	−60.0592	−3.29122
$334$	−17.9544	−0.982423
$335$	−12.7901	−0.698800
$336$	0.0640523	0.00349434
$337$	22.1501	1.20660	0.603298	−	0.797516i	$-0.293853\pi$
$337$	22.1501	1.20660	0.603298	+	0.797516i	$0.293853\pi$
$338$	12.3909	0.673978
$339$	−29.3668	−1.59498
$340$	−4.09630	−0.222153
$341$	11.9748	0.648469
$342$	6.03313	0.326234
$343$	0.270708	0.0146168
$344$	8.96086	0.483137
$345$	64.1567	3.45408
$346$	−17.6635	−0.949597
$347$	−4.94016	−0.265202	−0.132601	−	0.991170i	$-0.542333\pi$
$347$	−4.94016	−0.265202	−0.132601	+	0.991170i	$0.542333\pi$
$348$	7.34970	0.393985
$349$	15.6229	0.836276	0.418138	−	0.908384i	$-0.362683\pi$
$349$	15.6229	0.836276	0.418138	+	0.908384i	$0.362683\pi$
$350$	0.156250	0.00835192
$351$	12.8544	0.686115
$352$	−2.03368	−0.108395
$353$	−3.50483	−0.186543	−0.0932716	−	0.995641i	$-0.529732\pi$
$353$	−3.50483	−0.186543	−0.0932716	+	0.995641i	$0.529732\pi$
$354$	39.1424	2.08040
$355$	−43.7205	−2.32044
$356$	0.573623	0.0304019
$357$	−0.0725463	−0.00383956
$358$	−9.64850	−0.509939
$359$	12.6368	0.666945	0.333473	−	0.942760i	$-0.391779\pi$
$359$	12.6368	0.666945	0.333473	+	0.942760i	$0.391779\pi$
$360$	−28.8337	−1.51967
$361$	−18.4273	−0.969859
$362$	−13.0005	−0.683291
$363$	22.7373	1.19340
$364$	0.0150910	0.000790983 0
$365$	19.7784	1.03525
$366$	47.5048	2.48312
$367$	8.55054	0.446335	0.223167	−	0.974780i	$-0.428360\pi$
$367$	8.55054	0.446335	0.223167	+	0.974780i	$0.428360\pi$
$368$	−5.35525	−0.279162
$369$	61.9400	3.22447
$370$	27.2460	1.41645
$371$	−0.157134	−0.00815799
$372$	−19.5045	−1.01126
$373$	20.2179	1.04684	0.523421	−	0.852074i	$-0.324655\pi$
$373$	20.2179	1.04684	0.523421	+	0.852074i	$0.324655\pi$
$374$	2.30336	0.119104
$375$	−36.9045	−1.90574
$376$	3.95577	0.204003
$377$	1.73162	0.0891829
$378$	−0.318494	−0.0163815
$379$	4.26728	0.219196	0.109598	−	0.993976i	$-0.465044\pi$
$379$	4.26728	0.219196	0.109598	+	0.993976i	$0.465044\pi$
$380$	−2.73694	−0.140402
$381$	−34.8519	−1.78552
$382$	−4.68654	−0.239784
$383$	10.4343	0.533166	0.266583	−	0.963812i	$-0.414105\pi$
$383$	10.4343	0.533166	0.266583	+	0.963812i	$0.414105\pi$
$384$	3.31246	0.169038
$385$	−0.142225	−0.00724848
$386$	−0.682192	−0.0347226
$387$	−71.4395	−3.63147
$388$	15.5701	0.790453
$389$	21.7433	1.10243	0.551215	−	0.834363i	$-0.314164\pi$
$389$	21.7433	1.10243	0.551215	+	0.834363i	$0.314164\pi$
$390$	−9.34967	−0.473439
$391$	6.06541	0.306741
$392$	6.99963	0.353535
$393$	31.4209	1.58497
$394$	2.71862	0.136962
$395$	60.0356	3.02072
$396$	16.2133	0.814747
$397$	0.0728121	0.00365433	0.00182717	−	0.999998i	$-0.499418\pi$
$397$	0.0728121	0.00365433	0.00182717	+	0.999998i	$0.499418\pi$
$398$	5.95323	0.298408
$399$	−0.0484717	−0.00242662
$400$	8.08047	0.404023
$401$	20.1332	1.00541	0.502703	−	0.864459i	$-0.332339\pi$
$401$	20.1332	1.00541	0.502703	+	0.864459i	$0.332339\pi$
$402$	−11.7143	−0.584254
$403$	−4.59535	−0.228911
$404$	−5.58710	−0.277968
$405$	110.822	5.50681
$406$	−0.0429045	−0.00212931
$407$	−15.3205	−0.759408
$408$	−3.75173	−0.185738
$409$	15.2631	0.754713	0.377357	−	0.926068i	$-0.376833\pi$
$409$	15.2631	0.754713	0.377357	+	0.926068i	$0.376833\pi$
$410$	−28.0992	−1.38772
$411$	11.9625	0.590067
$412$	−5.65798	−0.278749
$413$	−0.228497	−0.0112436
$414$	42.6942	2.09830
$415$	39.8744	1.95736
$416$	0.780430	0.0382637
$417$	43.9965	2.15452
$418$	1.53899	0.0752744
$419$	2.60981	0.127498	0.0637489	−	0.997966i	$-0.479694\pi$
$419$	2.60981	0.127498	0.0637489	+	0.997966i	$0.479694\pi$
$420$	0.231658	0.0113037
$421$	−32.3600	−1.57713	−0.788565	−	0.614952i	$-0.789175\pi$
$421$	−32.3600	−1.57713	−0.788565	+	0.614952i	$0.789175\pi$
$422$	14.7314	0.717114
$423$	−31.5369	−1.53338
$424$	−8.12618	−0.394642
$425$	−9.15202	−0.443938
$426$	−40.0428	−1.94008
$427$	−0.277313	−0.0134201
$428$	−8.34063	−0.403160
$429$	5.25734	0.253827
$430$	32.4087	1.56288
$431$	−27.2964	−1.31482	−0.657411	−	0.753532i	$-0.728348\pi$
$431$	−27.2964	−1.31482	−0.657411	+	0.753532i	$0.728348\pi$
$432$	−16.4709	−0.792455
$433$	0.619573	0.0297748	0.0148874	−	0.999889i	$-0.495261\pi$
$433$	0.619573	0.0297748	0.0148874	+	0.999889i	$0.495261\pi$
$434$	0.113859	0.00546543
$435$	26.5816	1.27449
$436$	10.4014	0.498137
$437$	4.05259	0.193862
$438$	18.1147	0.865552
$439$	0.0872992	0.00416657	0.00208328	−	0.999998i	$-0.499337\pi$
$439$	0.0872992	0.00416657	0.00208328	+	0.999998i	$0.499337\pi$
$440$	−7.35518	−0.350644
$441$	−55.8038	−2.65732
$442$	−0.883922	−0.0420439
$443$	25.3901	1.20632	0.603159	−	0.797621i	$-0.293908\pi$
$443$	25.3901	1.20632	0.603159	+	0.797621i	$0.293908\pi$
$444$	24.9541	1.18427
$445$	2.07462	0.0983463
$446$	−29.4314	−1.39362
$447$	16.3845	0.774961
$448$	−0.0193368	−0.000913577 0
$449$	2.58814	0.122142	0.0610709	−	0.998133i	$-0.480548\pi$
$449$	2.58814	0.122142	0.0610709	+	0.998133i	$0.480548\pi$
$450$	−64.4207	−3.03682
$451$	15.8002	0.744004
$452$	8.86554	0.417000
$453$	−35.0441	−1.64651
$454$	13.1280	0.616129
$455$	0.0545795	0.00255873
$456$	−2.50671	−0.117388
$457$	11.2232	0.525001	0.262501	−	0.964932i	$-0.415453\pi$
$457$	11.2232	0.525001	0.262501	+	0.964932i	$0.415453\pi$
$458$	−17.4293	−0.814420
$459$	18.6551	0.870744
$460$	−19.3683	−0.903051
$461$	−1.76310	−0.0821159	−0.0410580	−	0.999157i	$-0.513073\pi$
$461$	−1.76310	−0.0821159	−0.0410580	+	0.999157i	$0.513073\pi$
$462$	−0.130262	−0.00606032
$463$	3.77691	0.175528	0.0877640	−	0.996141i	$-0.472028\pi$
$463$	3.77691	0.175528	0.0877640	+	0.996141i	$0.472028\pi$
$464$	−2.21880	−0.103005
$465$	−70.5419	−3.27130
$466$	8.27944	0.383537
$467$	10.0404	0.464615	0.232308	−	0.972642i	$-0.425372\pi$
$467$	10.0404	0.464615	0.232308	+	0.972642i	$0.425372\pi$
$468$	−6.22190	−0.287607
$469$	0.0683829	0.00315763
$470$	14.3068	0.659923
$471$	7.67018	0.353423
$472$	−11.8167	−0.543909
$473$	−18.2235	−0.837916
$474$	54.9855	2.52557
$475$	−6.11491	−0.280571
$476$	0.0219010	0.00100383
$477$	64.7851	2.96631
$478$	25.0288	1.14479
$479$	−22.9730	−1.04966	−0.524832	−	0.851206i	$-0.675872\pi$
$479$	−22.9730	−1.04966	−0.524832	+	0.851206i	$0.675872\pi$
$480$	11.9802	0.546817
$481$	5.87928	0.268072
$482$	15.1102	0.688253
$483$	−0.343016	−0.0156078
$484$	−6.86416	−0.312007
$485$	56.3124	2.55701
$486$	52.0876	2.36274
$487$	33.1146	1.50057	0.750283	−	0.661117i	$-0.229917\pi$
$487$	33.1146	1.50057	0.750283	+	0.661117i	$0.229917\pi$
$488$	−14.3412	−0.649198
$489$	−3.71482	−0.167990
$490$	25.3155	1.14364
$491$	11.6804	0.527130	0.263565	−	0.964642i	$-0.415102\pi$
$491$	11.6804	0.527130	0.263565	+	0.964642i	$0.415102\pi$
$492$	−25.7355	−1.16025
$493$	2.51304	0.113182
$494$	−0.590592	−0.0265720
$495$	58.6384	2.63560
$496$	5.88823	0.264389
$497$	0.233753	0.0104853
$498$	36.5202	1.63651
$499$	4.07666	0.182496	0.0912482	−	0.995828i	$-0.470914\pi$
$499$	4.07666	0.182496	0.0912482	+	0.995828i	$0.470914\pi$
$500$	11.1411	0.498245
$501$	−59.4733	−2.65707
$502$	10.2624	0.458035
$503$	−31.5069	−1.40482	−0.702412	−	0.711771i	$-0.747894\pi$
$503$	−31.5069	−1.40482	−0.702412	+	0.711771i	$0.747894\pi$
$504$	0.154160	0.00686685
$505$	−20.2068	−0.899191
$506$	10.8908	0.484157
$507$	41.0445	1.82285
$508$	10.5214	0.466814
$509$	−14.0758	−0.623897	−0.311949	−	0.950099i	$-0.600982\pi$
$509$	−14.0758	−0.623897	−0.311949	+	0.950099i	$0.600982\pi$
$510$	−13.5688	−0.600838
$511$	−0.105746	−0.00467792
$512$	−1.00000	−0.0441942
$513$	12.4644	0.550315
$514$	−4.84962	−0.213908
$515$	−20.4632	−0.901715
$516$	29.6825	1.30670
$517$	−8.04475	−0.353808
$518$	−0.145671	−0.00640044
$519$	−58.5097	−2.56829
$520$	2.82257	0.123778
$521$	−27.6957	−1.21337	−0.606686	−	0.794941i	$-0.707502\pi$
$521$	−27.6957	−1.21337	−0.606686	+	0.794941i	$0.707502\pi$
$522$	17.6892	0.774234
$523$	−26.0505	−1.13911	−0.569555	−	0.821953i	$-0.692884\pi$
$523$	−26.0505	−1.13911	−0.569555	+	0.821953i	$0.692884\pi$
$524$	−9.48565	−0.414383
$525$	0.517573	0.0225887
$526$	−22.2845	−0.971650
$527$	−6.66907	−0.290509
$528$	−6.73647	−0.293167
$529$	5.67867	0.246899
$530$	−29.3899	−1.27661
$531$	94.2076	4.08826
$532$	0.0146331	0.000634427 0
$533$	−6.06340	−0.262635
$534$	1.90010	0.0822255
$535$	−30.1655	−1.30417
$536$	3.53642	0.152750
$537$	−31.9603	−1.37919
$538$	25.0930	1.08183
$539$	−14.2350	−0.613143
$540$	−59.5701	−2.56349
$541$	−19.8742	−0.854459	−0.427230	−	0.904143i	$-0.640510\pi$
$541$	−19.8742	−0.854459	−0.427230	+	0.904143i	$0.640510\pi$
$542$	14.6268	0.628273
$543$	−43.0636	−1.84804
$544$	1.13261	0.0485602
$545$	37.6187	1.61141
$546$	0.0499883	0.00213930
$547$	38.8424	1.66078	0.830390	−	0.557183i	$-0.188118\pi$
$547$	38.8424	1.66078	0.830390	+	0.557183i	$0.188118\pi$
$548$	−3.61136	−0.154270
$549$	114.334	4.87966
$550$	−16.4330	−0.700707
$551$	1.67908	0.0715314
$552$	−17.7390	−0.755024
$553$	−0.320983	−0.0136496
$554$	1.29785	0.0551403
$555$	90.2512	3.83095
$556$	−13.2821	−0.563288
$557$	28.8213	1.22120	0.610599	−	0.791940i	$-0.290929\pi$
$557$	28.8213	1.22120	0.610599	+	0.791940i	$0.290929\pi$
$558$	−46.9433	−1.98727
$559$	6.99332	0.295786
$560$	−0.0699352	−0.00295530
$561$	7.62979	0.322130
$562$	−8.85904	−0.373696
$563$	−38.9150	−1.64007	−0.820035	−	0.572313i	$-0.806046\pi$
$563$	−38.9150	−1.64007	−0.820035	+	0.572313i	$0.806046\pi$
$564$	13.1033	0.551750
$565$	32.0639	1.34894
$566$	−13.5170	−0.568162
$567$	−0.592516	−0.0248833
$568$	12.0885	0.507223
$569$	37.1426	1.55710	0.778549	−	0.627584i	$-0.215956\pi$
$569$	37.1426	1.55710	0.778549	+	0.627584i	$0.215956\pi$
$570$	−9.06601	−0.379733
$571$	28.2540	1.18239	0.591196	−	0.806528i	$-0.298656\pi$
$571$	28.2540	1.18239	0.591196	+	0.806528i	$0.298656\pi$
$572$	−1.58714	−0.0663617
$573$	−15.5240	−0.648524
$574$	0.150233	0.00627062
$575$	−43.2729	−1.80460
$576$	7.97240	0.332183
$577$	25.7734	1.07296	0.536481	−	0.843913i	$-0.319753\pi$
$577$	25.7734	1.07296	0.536481	+	0.843913i	$0.319753\pi$
$578$	15.7172	0.653749
$579$	−2.25973	−0.0939113
$580$	−8.02473	−0.333209
$581$	−0.213190	−0.00884461
$582$	51.5754	2.13787
$583$	16.5260	0.684437
$584$	−5.46864	−0.226294
$585$	−22.5027	−0.930372
$586$	−29.1967	−1.20610
$587$	−20.0472	−0.827436	−0.413718	−	0.910405i	$-0.635770\pi$
$587$	−20.0472	−0.827436	−0.413718	+	0.910405i	$0.635770\pi$
$588$	23.1860	0.956174
$589$	−4.45593	−0.183603
$590$	−42.7375	−1.75947
$591$	9.00532	0.370429
$592$	−7.53339	−0.309620
$593$	17.9043	0.735241	0.367621	−	0.929976i	$-0.380172\pi$
$593$	17.9043	0.735241	0.367621	+	0.929976i	$0.380172\pi$
$594$	33.4964	1.37437
$595$	0.0792093	0.00324726
$596$	−4.94633	−0.202609
$597$	19.7198	0.807079
$598$	−4.17939	−0.170908
$599$	20.0765	0.820302	0.410151	−	0.912018i	$-0.365476\pi$
$599$	20.0765	0.820302	0.410151	+	0.912018i	$0.365476\pi$
$600$	26.7662	1.09273
$601$	−8.62621	−0.351870	−0.175935	−	0.984402i	$-0.556295\pi$
$601$	−8.62621	−0.351870	−0.175935	+	0.984402i	$0.556295\pi$
$602$	−0.173274	−0.00706212
$603$	−28.1937	−1.14814
$604$	10.5795	0.430472
$605$	−24.8256	−1.00930
$606$	−18.5070	−0.751797
$607$	22.2760	0.904154	0.452077	−	0.891979i	$-0.350683\pi$
$607$	22.2760	0.904154	0.452077	+	0.891979i	$0.350683\pi$
$608$	0.756752	0.0306904
$609$	−0.142119	−0.00575897
$610$	−51.8679	−2.10007
$611$	3.08720	0.124895
$612$	−9.02961	−0.365001
$613$	−9.37804	−0.378776	−0.189388	−	0.981902i	$-0.560650\pi$
$613$	−9.37804	−0.378776	−0.189388	+	0.981902i	$0.560650\pi$
$614$	3.68652	0.148776
$615$	−93.0775	−3.75325
$616$	0.0393247	0.00158444
$617$	−6.76494	−0.272346	−0.136173	−	0.990685i	$-0.543480\pi$
$617$	−6.76494	−0.272346	−0.136173	+	0.990685i	$0.543480\pi$
$618$	−18.7418	−0.753907
$619$	32.7318	1.31560	0.657802	−	0.753191i	$-0.271486\pi$
$619$	32.7318	1.31560	0.657802	+	0.753191i	$0.271486\pi$
$620$	21.2959	0.855265
$621$	88.2056	3.53957
$622$	−30.8487	−1.23692
$623$	−0.0110920	−0.000444392 0
$624$	2.58514	0.103489
$625$	−0.108397	−0.00433587
$626$	−3.85313	−0.154002
$627$	5.09784	0.203588
$628$	−2.31555	−0.0924006
$629$	8.53239	0.340209
$630$	0.557551	0.0222134
$631$	15.8784	0.632108	0.316054	−	0.948741i	$-0.397642\pi$
$631$	15.8784	0.632108	0.316054	+	0.948741i	$0.397642\pi$
$632$	−16.5996	−0.660296
$633$	48.7972	1.93952
$634$	−14.6422	−0.581516
$635$	38.0528	1.51008
$636$	−26.9176	−1.06735
$637$	5.46272	0.216441
$638$	4.51232	0.178645
$639$	−96.3745	−3.81252
$640$	−3.61669	−0.142962
$641$	11.5813	0.457432	0.228716	−	0.973493i	$-0.426547\pi$
$641$	11.5813	0.457432	0.228716	+	0.973493i	$0.426547\pi$
$642$	−27.6280	−1.09039
$643$	−23.8893	−0.942104	−0.471052	−	0.882105i	$-0.656126\pi$
$643$	−23.8893	−0.942104	−0.471052	+	0.882105i	$0.656126\pi$
$644$	0.103553	0.00408057
$645$	107.352	4.22700
$646$	−0.857105	−0.0337223
$647$	−13.0783	−0.514161	−0.257080	−	0.966390i	$-0.582761\pi$
$647$	−13.0783	−0.514161	−0.257080	+	0.966390i	$0.582761\pi$
$648$	−30.6419	−1.20373
$649$	24.0314	0.943314
$650$	6.30624	0.247351
$651$	0.377155	0.0147819
$652$	1.12147	0.0439201
$653$	−20.9476	−0.819742	−0.409871	−	0.912143i	$-0.634426\pi$
$653$	−20.9476	−0.819742	−0.409871	+	0.912143i	$0.634426\pi$
$654$	34.4543	1.34727
$655$	−34.3067	−1.34047
$656$	7.76931	0.303340
$657$	43.5982	1.70093
$658$	−0.0764918	−0.00298196
$659$	41.6318	1.62175	0.810873	−	0.585222i	$-0.198993\pi$
$659$	41.6318	1.62175	0.810873	+	0.585222i	$0.198993\pi$
$660$	−24.3637	−0.948357
$661$	−43.5477	−1.69381	−0.846905	−	0.531744i	$-0.821537\pi$
$661$	−43.5477	−1.69381	−0.846905	+	0.531744i	$0.821537\pi$
$662$	−17.1140	−0.665155
$663$	−2.92796	−0.113712
$664$	−11.0251	−0.427857
$665$	0.0529236	0.00205229
$666$	60.0592	2.32725
$667$	11.8822	0.460082
$668$	17.9544	0.694678
$669$	−97.4905	−3.76920
$670$	12.7901	0.494126
$671$	29.1654	1.12592
$672$	−0.0640523	−0.00247087
$673$	−42.3470	−1.63236	−0.816179	−	0.577800i	$-0.803912\pi$
$673$	−42.3470	−1.63236	−0.816179	+	0.577800i	$0.803912\pi$
$674$	−22.1501	−0.853192
$675$	−133.092	−5.12273
$676$	−12.3909	−0.476574
$677$	−41.4453	−1.59287	−0.796436	−	0.604723i	$-0.793284\pi$
$677$	−41.4453	−1.59287	−0.796436	+	0.604723i	$0.793284\pi$
$678$	29.3668	1.12782
$679$	−0.301076	−0.0115542
$680$	4.09630	0.157086
$681$	43.4861	1.66639
$682$	−11.9748	−0.458537
$683$	18.0175	0.689420	0.344710	−	0.938709i	$-0.387977\pi$
$683$	18.0175	0.689420	0.344710	+	0.938709i	$0.387977\pi$
$684$	−6.03313	−0.230682
$685$	−13.0612	−0.499043
$686$	−0.270708	−0.0103357
$687$	−57.7340	−2.20269
$688$	−8.96086	−0.341629
$689$	−6.34191	−0.241607
$690$	−64.1567	−2.44240
$691$	−38.9679	−1.48241	−0.741204	−	0.671280i	$-0.765745\pi$
$691$	−38.9679	−1.48241	−0.741204	+	0.671280i	$0.765745\pi$
$692$	17.6635	0.671466
$693$	−0.313512	−0.0119093
$694$	4.94016	0.187526
$695$	−48.0374	−1.82216
$696$	−7.34970	−0.278590
$697$	−8.79959	−0.333308
$698$	−15.6229	−0.591336
$699$	27.4253	1.03732
$700$	−0.156250	−0.00590570
$701$	−6.58400	−0.248674	−0.124337	−	0.992240i	$-0.539680\pi$
$701$	−6.58400	−0.248674	−0.124337	+	0.992240i	$0.539680\pi$
$702$	−12.8544	−0.485156
$703$	5.70091	0.215014
$704$	2.03368	0.0766470
$705$	47.3907	1.78484
$706$	3.50483	0.131906
$707$	0.108036	0.00406313
$708$	−39.1424	−1.47106
$709$	41.5128	1.55905	0.779524	−	0.626373i	$-0.215461\pi$
$709$	41.5128	1.55905	0.779524	+	0.626373i	$0.215461\pi$
$710$	43.7205	1.64080
$711$	132.339	4.96308
$712$	−0.573623	−0.0214974
$713$	−31.5329	−1.18092
$714$	0.0725463	0.00271498
$715$	−5.74020	−0.214671
$716$	9.64850	0.360581
$717$	82.9068	3.09621
$718$	−12.6368	−0.471602
$719$	29.1564	1.08735	0.543676	−	0.839295i	$-0.317032\pi$
$719$	29.1564	1.08735	0.543676	+	0.839295i	$0.317032\pi$
$720$	28.8337	1.07457
$721$	0.109407	0.00407453
$722$	18.4273	0.685794
$723$	50.0521	1.86146
$724$	13.0005	0.483160
$725$	−17.9290	−0.665865
$726$	−22.7373	−0.843860
$727$	−14.7926	−0.548626	−0.274313	−	0.961640i	$-0.588451\pi$
$727$	−14.7926	−0.548626	−0.274313	+	0.961640i	$0.588451\pi$
$728$	−0.0150910	−0.000559309 0
$729$	80.6122	2.98564
$730$	−19.7784	−0.732031
$731$	10.1492	0.375380
$732$	−47.5048	−1.75583
$733$	−31.2062	−1.15263	−0.576313	−	0.817229i	$-0.695509\pi$
$733$	−31.2062	−1.15263	−0.576313	+	0.817229i	$0.695509\pi$
$734$	−8.55054	−0.315606
$735$	83.8566	3.09310
$736$	5.35525	0.197397
$737$	−7.19193	−0.264918
$738$	−61.9400	−2.28004
$739$	27.9693	1.02887	0.514434	−	0.857530i	$-0.328002\pi$
$739$	27.9693	1.02887	0.514434	+	0.857530i	$0.328002\pi$
$740$	−27.2460	−1.00158
$741$	−1.95631	−0.0718669
$742$	0.157134	0.00576857
$743$	−29.5419	−1.08379	−0.541894	−	0.840447i	$-0.682293\pi$
$743$	−29.5419	−1.08379	−0.541894	+	0.840447i	$0.682293\pi$
$744$	19.5045	0.715071
$745$	−17.8893	−0.655415
$746$	−20.2179	−0.740229
$747$	87.8965	3.21596
$748$	−2.30336	−0.0842192
$749$	0.161281	0.00589308
$750$	36.9045	1.34756
$751$	−48.9370	−1.78574	−0.892868	−	0.450318i	$-0.851311\pi$
$751$	−48.9370	−1.78574	−0.892868	+	0.450318i	$0.851311\pi$
$752$	−3.95577	−0.144252
$753$	33.9939	1.23881
$754$	−1.73162	−0.0630619
$755$	38.2627	1.39252
$756$	0.318494	0.0115835
$757$	7.23652	0.263016	0.131508	−	0.991315i	$-0.458018\pi$
$757$	7.23652	0.263016	0.131508	+	0.991315i	$0.458018\pi$
$758$	−4.26728	−0.154995
$759$	36.0755	1.30946
$760$	2.73694	0.0992793
$761$	28.1975	1.02216	0.511079	−	0.859533i	$-0.329246\pi$
$761$	28.1975	1.02216	0.511079	+	0.859533i	$0.329246\pi$
$762$	34.8519	1.26255
$763$	−0.201130	−0.00728139
$764$	4.68654	0.169553
$765$	−32.6573	−1.18073
$766$	−10.4343	−0.377005
$767$	−9.22212	−0.332992
$768$	−3.31246	−0.119528
$769$	44.0323	1.58784	0.793922	−	0.608019i	$-0.208036\pi$
$769$	44.0323	1.58784	0.793922	+	0.608019i	$0.208036\pi$
$770$	0.142225	0.00512545
$771$	−16.0642	−0.578537
$772$	0.682192	0.0245526
$773$	−3.86573	−0.139041	−0.0695204	−	0.997581i	$-0.522147\pi$
$773$	−3.86573	−0.139041	−0.0695204	+	0.997581i	$0.522147\pi$
$774$	71.4395	2.56784
$775$	47.5797	1.70911
$776$	−15.5701	−0.558935
$777$	−0.482531	−0.0173107
$778$	−21.7433	−0.779536
$779$	−5.87944	−0.210653
$780$	9.34967	0.334772
$781$	−24.5841	−0.879689
$782$	−6.06541	−0.216898
$783$	36.5456	1.30603
$784$	−6.99963	−0.249987
$785$	−8.37464	−0.298904
$786$	−31.4209	−1.12074
$787$	−41.9806	−1.49645	−0.748224	−	0.663447i	$-0.769093\pi$
$787$	−41.9806	−1.49645	−0.748224	+	0.663447i	$0.769093\pi$
$788$	−2.71862	−0.0968467
$789$	−73.8165	−2.62794
$790$	−60.0356	−2.13597
$791$	−0.171431	−0.00609538
$792$	−16.2133	−0.576113
$793$	−11.1923	−0.397451
$794$	−0.0728121	−0.00258400
$795$	−97.3528	−3.45275
$796$	−5.95323	−0.211006
$797$	15.3189	0.542622	0.271311	−	0.962492i	$-0.412543\pi$
$797$	15.3189	0.542622	0.271311	+	0.962492i	$0.412543\pi$
$798$	0.0484717	0.00171588
$799$	4.48034	0.158503
$800$	−8.08047	−0.285688
$801$	4.57315	0.161584
$802$	−20.1332	−0.710929
$803$	11.1214	0.392467
$804$	11.7143	0.413130
$805$	0.374520	0.0132001
$806$	4.59535	0.161864
$807$	83.1194	2.92594
$808$	5.58710	0.196553
$809$	21.5298	0.756949	0.378474	−	0.925612i	$-0.376449\pi$
$809$	21.5298	0.756949	0.378474	+	0.925612i	$0.376449\pi$
$810$	−110.822	−3.89390
$811$	−33.6630	−1.18207	−0.591035	−	0.806646i	$-0.701280\pi$
$811$	−33.6630	−1.18207	−0.591035	+	0.806646i	$0.701280\pi$
$812$	0.0429045	0.00150565
$813$	48.4506	1.69924
$814$	15.3205	0.536982
$815$	4.05601	0.142076
$816$	3.75173	0.131337
$817$	6.78115	0.237242
$818$	−15.2631	−0.533663
$819$	0.120311	0.00420402
$820$	28.0992	0.981266
$821$	−34.9770	−1.22071	−0.610353	−	0.792129i	$-0.708973\pi$
$821$	−34.9770	−1.22071	−0.610353	+	0.792129i	$0.708973\pi$
$822$	−11.9625	−0.417240
$823$	34.5722	1.20511	0.602555	−	0.798077i	$-0.294149\pi$
$823$	34.5722	1.20511	0.602555	+	0.798077i	$0.294149\pi$
$824$	5.65798	0.197105
$825$	−54.4338	−1.89514
$826$	0.228497	0.00795044
$827$	40.3944	1.40465	0.702326	−	0.711856i	$-0.252145\pi$
$827$	40.3944	1.40465	0.702326	+	0.711856i	$0.252145\pi$
$828$	−42.6942	−1.48372
$829$	26.6065	0.924083	0.462041	−	0.886858i	$-0.347117\pi$
$829$	26.6065	0.924083	0.462041	+	0.886858i	$0.347117\pi$
$830$	−39.8744	−1.38406
$831$	4.29907	0.149133
$832$	−0.780430	−0.0270565
$833$	7.92785	0.274684
$834$	−43.9965	−1.52348
$835$	64.9356	2.24719
$836$	−1.53899	−0.0532270
$837$	−96.9843	−3.35227
$838$	−2.60981	−0.0901545
$839$	50.5807	1.74624	0.873119	−	0.487506i	$-0.162093\pi$
$839$	50.5807	1.74624	0.873119	+	0.487506i	$0.162093\pi$
$840$	−0.231658	−0.00799294
$841$	−24.0769	−0.830238
$842$	32.3600	1.11520
$843$	−29.3452	−1.01070
$844$	−14.7314	−0.507076
$845$	−44.8142	−1.54165
$846$	31.5369	1.08426
$847$	0.132731	0.00456068
$848$	8.12618	0.279054
$849$	−44.7745	−1.53666
$850$	9.15202	0.313912
$851$	40.3432	1.38295
$852$	40.0428	1.37184
$853$	20.5622	0.704036	0.352018	−	0.935993i	$-0.385496\pi$
$853$	20.5622	0.704036	0.352018	+	0.935993i	$0.385496\pi$
$854$	0.277313	0.00948947
$855$	−21.8200	−0.746227
$856$	8.34063	0.285077
$857$	−18.8934	−0.645387	−0.322694	−	0.946504i	$-0.604588\pi$
$857$	−18.8934	−0.645387	−0.322694	+	0.946504i	$0.604588\pi$
$858$	−5.25734	−0.179483
$859$	−40.8433	−1.39356	−0.696778	−	0.717287i	$-0.745384\pi$
$859$	−40.8433	−1.39356	−0.696778	+	0.717287i	$0.745384\pi$
$860$	−32.4087	−1.10513
$861$	0.497642	0.0169596
$862$	27.2964	0.929719
$863$	35.4940	1.20823	0.604115	−	0.796897i	$-0.293527\pi$
$863$	35.4940	1.20823	0.604115	+	0.796897i	$0.293527\pi$
$864$	16.4709	0.560350
$865$	63.8835	2.17210
$866$	−0.619573	−0.0210539
$867$	52.0626	1.76814
$868$	−0.113859	−0.00386464
$869$	33.7582	1.14517
$870$	−26.5816	−0.901200
$871$	2.75993	0.0935166
$872$	−10.4014	−0.352236
$873$	124.131	4.20120
$874$	−4.05259	−0.137081
$875$	−0.215433	−0.00728296
$876$	−18.1147	−0.612038
$877$	−29.3686	−0.991708	−0.495854	−	0.868406i	$-0.665145\pi$
$877$	−29.3686	−0.991708	−0.495854	+	0.868406i	$0.665145\pi$
$878$	−0.0872992	−0.00294621
$879$	−96.7128	−3.26204
$880$	7.35518	0.247943
$881$	21.4109	0.721353	0.360676	−	0.932691i	$-0.382546\pi$
$881$	21.4109	0.721353	0.360676	+	0.932691i	$0.382546\pi$
$882$	55.8038	1.87901
$883$	46.2234	1.55554	0.777771	−	0.628548i	$-0.216350\pi$
$883$	46.2234	1.55554	0.777771	+	0.628548i	$0.216350\pi$
$884$	0.883922	0.0297295
$885$	−141.566	−4.75869
$886$	−25.3901	−0.852996
$887$	24.0592	0.807827	0.403914	−	0.914797i	$-0.367650\pi$
$887$	24.0592	0.807827	0.403914	+	0.914797i	$0.367650\pi$
$888$	−24.9541	−0.837404
$889$	−0.203451	−0.00682352
$890$	−2.07462	−0.0695413
$891$	62.3157	2.08766
$892$	29.4314	0.985437
$893$	2.99354	0.100175
$894$	−16.3845	−0.547980
$895$	34.8957	1.16643
$896$	0.0193368	0.000645996 0
$897$	−13.8441	−0.462240
$898$	−2.58814	−0.0863673
$899$	−13.0648	−0.435736
$900$	64.4207	2.14736
$901$	−9.20379	−0.306623
$902$	−15.8002	−0.526091
$903$	−0.573964	−0.0191003
$904$	−8.86554	−0.294863
$905$	47.0188	1.56296
$906$	35.0441	1.16426
$907$	1.42805	0.0474176	0.0237088	−	0.999719i	$-0.492453\pi$
$907$	1.42805	0.0474176	0.0237088	+	0.999719i	$0.492453\pi$
$908$	−13.1280	−0.435669
$909$	−44.5426	−1.47738
$910$	−0.0545795	−0.00180929
$911$	−30.6004	−1.01384	−0.506918	−	0.861994i	$-0.669215\pi$
$911$	−30.6004	−1.01384	−0.506918	+	0.861994i	$0.669215\pi$
$912$	2.50671	0.0830055
$913$	22.4215	0.742042
$914$	−11.2232	−0.371232
$915$	−171.810	−5.67987
$916$	17.4293	0.575882
$917$	0.183422	0.00605713
$918$	−18.6551	−0.615709
$919$	−2.63837	−0.0870319	−0.0435160	−	0.999053i	$-0.513856\pi$
$919$	−2.63837	−0.0870319	−0.0435160	+	0.999053i	$0.513856\pi$
$920$	19.3683	0.638553
$921$	12.2115	0.402381
$922$	1.76310	0.0580647
$923$	9.43424	0.310532
$924$	0.130262	0.00428529
$925$	−60.8733	−2.00150
$926$	−3.77691	−0.124117
$927$	−45.1077	−1.48153
$928$	2.21880	0.0728358
$929$	24.6529	0.808835	0.404417	−	0.914575i	$-0.367474\pi$
$929$	24.6529	0.808835	0.404417	+	0.914575i	$0.367474\pi$
$930$	70.5419	2.31316
$931$	5.29698	0.173602
$932$	−8.27944	−0.271202
$933$	−102.185	−3.34539
$934$	−10.0404	−0.328533
$935$	−8.33055	−0.272438
$936$	6.22190	0.203369
$937$	−51.7457	−1.69046	−0.845229	−	0.534404i	$-0.820536\pi$
$937$	−51.7457	−1.69046	−0.845229	+	0.534404i	$0.820536\pi$
$938$	−0.0683829	−0.00223278
$939$	−12.7633	−0.416515
$940$	−14.3068	−0.466636
$941$	36.2815	1.18274	0.591372	−	0.806399i	$-0.298586\pi$
$941$	36.2815	1.18274	0.591372	+	0.806399i	$0.298586\pi$
$942$	−7.67018	−0.249908
$943$	−41.6065	−1.35490
$944$	11.8167	0.384602
$945$	1.15189	0.0374711
$946$	18.2235	0.592496
$947$	49.5527	1.61025	0.805123	−	0.593108i	$-0.202099\pi$
$947$	49.5527	1.61025	0.805123	+	0.593108i	$0.202099\pi$
$948$	−54.9855	−1.78585
$949$	−4.26789	−0.138542
$950$	6.11491	0.198394
$951$	−48.5017	−1.57278
$952$	−0.0219010	−0.000709816 0
$953$	−12.5555	−0.406712	−0.203356	−	0.979105i	$-0.565185\pi$
$953$	−12.5555	−0.406712	−0.203356	+	0.979105i	$0.565185\pi$
$954$	−64.7851	−2.09749
$955$	16.9498	0.548482
$956$	−25.0288	−0.809488
$957$	14.9469	0.483165
$958$	22.9730	0.742225
$959$	0.0698321	0.00225500
$960$	−11.9802	−0.386658
$961$	3.67128	0.118428
$962$	−5.87928	−0.189556
$963$	−66.4948	−2.14277
$964$	−15.1102	−0.486668
$965$	2.46728	0.0794245
$966$	0.343016	0.0110364
$967$	26.5343	0.853286	0.426643	−	0.904420i	$-0.359696\pi$
$967$	26.5343	0.853286	0.426643	+	0.904420i	$0.359696\pi$
$968$	6.86416	0.220623
$969$	−2.83913	−0.0912059
$970$	−56.3124	−1.80808
$971$	−30.9881	−0.994456	−0.497228	−	0.867620i	$-0.665649\pi$
$971$	−30.9881	−0.994456	−0.497228	+	0.867620i	$0.665649\pi$
$972$	−52.0876	−1.67071
$973$	0.256834	0.00823371
$974$	−33.1146	−1.06106
$975$	20.8892	0.668989
$976$	14.3412	0.459052
$977$	37.4192	1.19715	0.598574	−	0.801068i	$-0.295734\pi$
$977$	37.4192	1.19715	0.598574	+	0.801068i	$0.295734\pi$
$978$	3.71482	0.118787
$979$	1.16656	0.0372835
$980$	−25.3155	−0.808674
$981$	82.9242	2.64757
$982$	−11.6804	−0.372737
$983$	13.3683	0.426384	0.213192	−	0.977010i	$-0.431614\pi$
$983$	13.3683	0.426384	0.213192	+	0.977010i	$0.431614\pi$
$984$	25.7355	0.820418
$985$	−9.83241	−0.313286
$986$	−2.51304	−0.0800314
$987$	−0.253376	−0.00806505
$988$	0.590592	0.0187892
$989$	47.9876	1.52592
$990$	−58.6384	−1.86365
$991$	−6.67645	−0.212084	−0.106042	−	0.994362i	$-0.533818\pi$
$991$	−6.67645	−0.212084	−0.106042	+	0.994362i	$0.533818\pi$
$992$	−5.88823	−0.186952
$993$	−56.6895	−1.79899
$994$	−0.233753	−0.00741420
$995$	−21.5310	−0.682578
$996$	−36.5202	−1.15719
$997$	−13.6340	−0.431793	−0.215896	−	0.976416i	$-0.569267\pi$
$997$	−13.6340	−0.431793	−0.215896	+	0.976416i	$0.569267\pi$
$998$	−4.07666	−0.129044
$999$	124.082	3.92577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6046.2.a.g.1.1	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6046.2.a.g.1.1	✓	69	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:	\( N \)	\(=\)	\( 6046 = 2 \cdot 3023 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.31246 q^{3} +1.00000 q^{4} +3.61669 q^{5} +3.31246 q^{6} -0.0193368 q^{7} -1.00000 q^{8} +7.97240 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.31246 q^{3} +1.00000 q^{4} +3.61669 q^{5} +3.31246 q^{6} -0.0193368 q^{7} -1.00000 q^{8} +7.97240 q^{9} -3.61669 q^{10} +2.03368 q^{11} -3.31246 q^{12} -0.780430 q^{13} +0.0193368 q^{14} -11.9802 q^{15} +1.00000 q^{16} -1.13261 q^{17} -7.97240 q^{18} -0.756752 q^{19} +3.61669 q^{20} +0.0640523 q^{21} -2.03368 q^{22} -5.35525 q^{23} +3.31246 q^{24} +8.08047 q^{25} +0.780430 q^{26} -16.4709 q^{27} -0.0193368 q^{28} -2.21880 q^{29} +11.9802 q^{30} +5.88823 q^{31} -1.00000 q^{32} -6.73647 q^{33} +1.13261 q^{34} -0.0699352 q^{35} +7.97240 q^{36} -7.53339 q^{37} +0.756752 q^{38} +2.58514 q^{39} -3.61669 q^{40} +7.76931 q^{41} -0.0640523 q^{42} -8.96086 q^{43} +2.03368 q^{44} +28.8337 q^{45} +5.35525 q^{46} -3.95577 q^{47} -3.31246 q^{48} -6.99963 q^{49} -8.08047 q^{50} +3.75173 q^{51} -0.780430 q^{52} +8.12618 q^{53} +16.4709 q^{54} +7.35518 q^{55} +0.0193368 q^{56} +2.50671 q^{57} +2.21880 q^{58} +11.8167 q^{59} -11.9802 q^{60} +14.3412 q^{61} -5.88823 q^{62} -0.154160 q^{63} +1.00000 q^{64} -2.82257 q^{65} +6.73647 q^{66} -3.53642 q^{67} -1.13261 q^{68} +17.7390 q^{69} +0.0699352 q^{70} -12.0885 q^{71} -7.97240 q^{72} +5.46864 q^{73} +7.53339 q^{74} -26.7662 q^{75} -0.756752 q^{76} -0.0393247 q^{77} -2.58514 q^{78} +16.5996 q^{79} +3.61669 q^{80} +30.6419 q^{81} -7.76931 q^{82} +11.0251 q^{83} +0.0640523 q^{84} -4.09630 q^{85} +8.96086 q^{86} +7.34970 q^{87} -2.03368 q^{88} +0.573623 q^{89} -28.8337 q^{90} +0.0150910 q^{91} -5.35525 q^{92} -19.5045 q^{93} +3.95577 q^{94} -2.73694 q^{95} +3.31246 q^{96} +15.5701 q^{97} +6.99963 q^{98} +16.2133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9	\( 69 q - 69 q^{2} + 69 q^{4} + 13 q^{5} - 27 q^{7} - 69 q^{8} + 99 q^{9} - 13 q^{10} + 42 q^{11} - 5 q^{13} + 27 q^{14} + 18 q^{15} + 69 q^{16} + 24 q^{17} - 99 q^{18} + q^{19} + 13 q^{20} + 7 q^{21} - 42 q^{22} + 25 q^{23} + 100 q^{25} + 5 q^{26} + 15 q^{27} - 27 q^{28} + 87 q^{29} - 18 q^{30} + 5 q^{31} - 69 q^{32} + 28 q^{33} - 24 q^{34} + 33 q^{35} + 99 q^{36} - 5 q^{37} - q^{38} + 22 q^{39} - 13 q^{40} + 47 q^{41} - 7 q^{42} - 23 q^{43} + 42 q^{44} + 14 q^{45} - 25 q^{46} + 13 q^{47} + 106 q^{49} - 100 q^{50} + 2 q^{51} - 5 q^{52} + 51 q^{53} - 15 q^{54} - 11 q^{55} + 27 q^{56} + 52 q^{57} - 87 q^{58} + 73 q^{59} + 18 q^{60} + 4 q^{61} - 5 q^{62} - 86 q^{63} + 69 q^{64} + 70 q^{65} - 28 q^{66} - 24 q^{67} + 24 q^{68} + 56 q^{69} - 33 q^{70} + 84 q^{71} - 99 q^{72} + 27 q^{73} + 5 q^{74} + 27 q^{75} + q^{76} + 45 q^{77} - 22 q^{78} + 42 q^{79} + 13 q^{80} + 205 q^{81} - 47 q^{82} + q^{83} + 7 q^{84} - 18 q^{85} + 23 q^{86} - q^{87} - 42 q^{88} + 94 q^{89} - 14 q^{90} + 6 q^{91} + 25 q^{92} - 13 q^{93} - 13 q^{94} + 86 q^{95} + 35 q^{97} - 106 q^{98} + 83 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 69 * q^4 + 13 * q^5 - 27 * q^7 - 69 * q^8 + 99 * q^9 - 13 * q^10 + 42 * q^11 - 5 * q^13 + 27 * q^14 + 18 * q^15 + 69 * q^16 + 24 * q^17 - 99 * q^18 + q^19 + 13 * q^20 + 7 * q^21 - 42 * q^22 + 25 * q^23 + 100 * q^25 + 5 * q^26 + 15 * q^27 - 27 * q^28 + 87 * q^29 - 18 * q^30 + 5 * q^31 - 69 * q^32 + 28 * q^33 - 24 * q^34 + 33 * q^35 + 99 * q^36 - 5 * q^37 - q^38 + 22 * q^39 - 13 * q^40 + 47 * q^41 - 7 * q^42 - 23 * q^43 + 42 * q^44 + 14 * q^45 - 25 * q^46 + 13 * q^47 + 106 * q^49 - 100 * q^50 + 2 * q^51 - 5 * q^52 + 51 * q^53 - 15 * q^54 - 11 * q^55 + 27 * q^56 + 52 * q^57 - 87 * q^58 + 73 * q^59 + 18 * q^60 + 4 * q^61 - 5 * q^62 - 86 * q^63 + 69 * q^64 + 70 * q^65 - 28 * q^66 - 24 * q^67 + 24 * q^68 + 56 * q^69 - 33 * q^70 + 84 * q^71 - 99 * q^72 + 27 * q^73 + 5 * q^74 + 27 * q^75 + q^76 + 45 * q^77 - 22 * q^78 + 42 * q^79 + 13 * q^80 + 205 * q^81 - 47 * q^82 + q^83 + 7 * q^84 - 18 * q^85 + 23 * q^86 - q^87 - 42 * q^88 + 94 * q^89 - 14 * q^90 + 6 * q^91 + 25 * q^92 - 13 * q^93 - 13 * q^94 + 86 * q^95 + 35 * q^97 - 106 * q^98 + 83 * q^99

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