LMFDB - Embedded newform 1339.2.a.f.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08251 q^{2} -0.806331 q^{3} -0.828177 q^{4} +1.00289 q^{5} +0.872859 q^{6} -2.77298 q^{7} +3.06152 q^{8} -2.34983 q^{9} +O(q^{10})$	\(q-1.08251 q^{2} -0.806331 q^{3} -0.828177 q^{4} +1.00289 q^{5} +0.872859 q^{6} -2.77298 q^{7} +3.06152 q^{8} -2.34983 q^{9} -1.08564 q^{10} -5.77317 q^{11} +0.667784 q^{12} +1.00000 q^{13} +3.00177 q^{14} -0.808663 q^{15} -1.65777 q^{16} +4.28749 q^{17} +2.54371 q^{18} -4.41238 q^{19} -0.830573 q^{20} +2.23594 q^{21} +6.24950 q^{22} -0.981744 q^{23} -2.46860 q^{24} -3.99421 q^{25} -1.08251 q^{26} +4.31373 q^{27} +2.29652 q^{28} -6.36142 q^{29} +0.875385 q^{30} +5.65263 q^{31} -4.32850 q^{32} +4.65508 q^{33} -4.64124 q^{34} -2.78100 q^{35} +1.94608 q^{36} +10.6046 q^{37} +4.77643 q^{38} -0.806331 q^{39} +3.07038 q^{40} -4.28114 q^{41} -2.42042 q^{42} -3.11446 q^{43} +4.78120 q^{44} -2.35663 q^{45} +1.06275 q^{46} +5.36198 q^{47} +1.33671 q^{48} +0.689415 q^{49} +4.32376 q^{50} -3.45714 q^{51} -0.828177 q^{52} +4.88956 q^{53} -4.66965 q^{54} -5.78987 q^{55} -8.48954 q^{56} +3.55784 q^{57} +6.88629 q^{58} +5.37039 q^{59} +0.669716 q^{60} -4.51504 q^{61} -6.11902 q^{62} +6.51603 q^{63} +8.00117 q^{64} +1.00289 q^{65} -5.03916 q^{66} +4.14539 q^{67} -3.55080 q^{68} +0.791610 q^{69} +3.01046 q^{70} -13.2077 q^{71} -7.19406 q^{72} -0.834149 q^{73} -11.4796 q^{74} +3.22065 q^{75} +3.65423 q^{76} +16.0089 q^{77} +0.872859 q^{78} +3.55519 q^{79} -1.66257 q^{80} +3.57120 q^{81} +4.63437 q^{82} -8.09224 q^{83} -1.85175 q^{84} +4.29990 q^{85} +3.37143 q^{86} +5.12941 q^{87} -17.6747 q^{88} +11.0000 q^{89} +2.55107 q^{90} -2.77298 q^{91} +0.813057 q^{92} -4.55789 q^{93} -5.80439 q^{94} -4.42514 q^{95} +3.49020 q^{96} +14.3314 q^{97} -0.746297 q^{98} +13.5660 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) $ 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * 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 $

$2$	−1.08251	−0.765449	−0.382724	−	0.923863i	$-0.625014\pi$
$2$	−1.08251	−0.765449	−0.382724	+	0.923863i	$0.625014\pi$
$3$	−0.806331	−0.465535	−0.232768	−	0.972532i	$-0.574778\pi$
$3$	−0.806331	−0.465535	−0.232768	+	0.972532i	$0.574778\pi$
$4$	−0.828177	−0.414088
$5$	1.00289	0.448507	0.224254	−	0.974531i	$-0.428006\pi$
$5$	1.00289	0.448507	0.224254	+	0.974531i	$0.428006\pi$
$6$	0.872859	0.356343
$7$	−2.77298	−1.04809	−0.524044	−	0.851691i	$-0.675577\pi$
$7$	−2.77298	−1.04809	−0.524044	+	0.851691i	$0.675577\pi$
$8$	3.06152	1.08241
$9$	−2.34983	−0.783277
$10$	−1.08564	−0.343309
$11$	−5.77317	−1.74068	−0.870338	−	0.492455i	$-0.836100\pi$
$11$	−5.77317	−1.74068	−0.870338	+	0.492455i	$0.836100\pi$
$12$	0.667784	0.192773
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.00177	0.802257
$15$	−0.808663	−0.208796
$16$	−1.65777	−0.414442
$17$	4.28749	1.03987	0.519935	−	0.854206i	$-0.325956\pi$
$17$	4.28749	1.03987	0.519935	+	0.854206i	$0.325956\pi$
$18$	2.54371	0.599558
$19$	−4.41238	−1.01227	−0.506134	−	0.862455i	$-0.668926\pi$
$19$	−4.41238	−1.01227	−0.506134	+	0.862455i	$0.668926\pi$
$20$	−0.830573	−0.185722
$21$	2.23594	0.487922
$22$	6.24950	1.33240
$23$	−0.981744	−0.204708	−0.102354	−	0.994748i	$-0.532637\pi$
$23$	−0.981744	−0.204708	−0.102354	+	0.994748i	$0.532637\pi$
$24$	−2.46860	−0.503901
$25$	−3.99421	−0.798841
$26$	−1.08251	−0.212297
$27$	4.31373	0.830178
$28$	2.29652	0.434001
$29$	−6.36142	−1.18129	−0.590643	−	0.806933i	$-0.701126\pi$
$29$	−6.36142	−1.18129	−0.590643	+	0.806933i	$0.701126\pi$
$30$	0.875385	0.159823
$31$	5.65263	1.01524	0.507622	−	0.861580i	$-0.330525\pi$
$31$	5.65263	1.01524	0.507622	+	0.861580i	$0.330525\pi$
$32$	−4.32850	−0.765178
$33$	4.65508	0.810346
$34$	−4.64124	−0.795967
$35$	−2.78100	−0.470075
$36$	1.94608	0.324346
$37$	10.6046	1.74339	0.871695	−	0.490049i	$-0.163021\pi$
$37$	10.6046	1.74339	0.871695	+	0.490049i	$0.163021\pi$
$38$	4.77643	0.774840
$39$	−0.806331	−0.129116
$40$	3.07038	0.485470
$41$	−4.28114	−0.668602	−0.334301	−	0.942466i	$-0.608500\pi$
$41$	−4.28114	−0.668602	−0.334301	+	0.942466i	$0.608500\pi$
$42$	−2.42042	−0.373479
$43$	−3.11446	−0.474951	−0.237475	−	0.971394i	$-0.576320\pi$
$43$	−3.11446	−0.474951	−0.237475	+	0.971394i	$0.576320\pi$
$44$	4.78120	0.720794
$45$	−2.35663	−0.351306
$46$	1.06275	0.156693
$47$	5.36198	0.782125	0.391063	−	0.920364i	$-0.372108\pi$
$47$	5.36198	0.782125	0.391063	+	0.920364i	$0.372108\pi$
$48$	1.33671	0.192937
$49$	0.689415	0.0984879
$50$	4.32376	0.611472
$51$	−3.45714	−0.484096
$52$	−0.828177	−0.114847
$53$	4.88956	0.671633	0.335817	−	0.941927i	$-0.390988\pi$
$53$	4.88956	0.671633	0.335817	+	0.941927i	$0.390988\pi$
$54$	−4.66965	−0.635459
$55$	−5.78987	−0.780706
$56$	−8.48954	−1.13446
$57$	3.55784	0.471247
$58$	6.88629	0.904214
$59$	5.37039	0.699165	0.349583	−	0.936906i	$-0.386323\pi$
$59$	5.37039	0.699165	0.349583	+	0.936906i	$0.386323\pi$
$60$	0.669716	0.0864600
$61$	−4.51504	−0.578092	−0.289046	−	0.957315i	$-0.593338\pi$
$61$	−4.51504	−0.578092	−0.289046	+	0.957315i	$0.593338\pi$
$62$	−6.11902	−0.777116
$63$	6.51603	0.820943
$64$	8.00117	1.00015
$65$	1.00289	0.124394
$66$	−5.03916	−0.620278
$67$	4.14539	0.506441	0.253220	−	0.967409i	$-0.418510\pi$
$67$	4.14539	0.506441	0.253220	+	0.967409i	$0.418510\pi$
$68$	−3.55080	−0.430598
$69$	0.791610	0.0952986
$70$	3.01046	0.359818
$71$	−13.2077	−1.56747	−0.783734	−	0.621097i	$-0.786687\pi$
$71$	−13.2077	−1.56747	−0.783734	+	0.621097i	$0.786687\pi$
$72$	−7.19406	−0.847828
$73$	−0.834149	−0.0976297	−0.0488148	−	0.998808i	$-0.515544\pi$
$73$	−0.834149	−0.0976297	−0.0488148	+	0.998808i	$0.515544\pi$
$74$	−11.4796	−1.33448
$75$	3.22065	0.371889
$76$	3.65423	0.419169
$77$	16.0089	1.82438
$78$	0.872859	0.0988318
$79$	3.55519	0.399990	0.199995	−	0.979797i	$-0.435907\pi$
$79$	3.55519	0.399990	0.199995	+	0.979797i	$0.435907\pi$
$80$	−1.66257	−0.185881
$81$	3.57120	0.396800
$82$	4.63437	0.511781
$83$	−8.09224	−0.888239	−0.444120	−	0.895967i	$-0.646483\pi$
$83$	−8.09224	−0.888239	−0.444120	+	0.895967i	$0.646483\pi$
$84$	−1.85175	−0.202043
$85$	4.29990	0.466389
$86$	3.37143	0.363550
$87$	5.12941	0.549930
$88$	−17.6747	−1.88413
$89$	11.0000	1.16599	0.582997	−	0.812474i	$-0.301880\pi$
$89$	11.0000	1.16599	0.582997	+	0.812474i	$0.301880\pi$
$90$	2.55107	0.268906
$91$	−2.77298	−0.290687
$92$	0.813057	0.0847671
$93$	−4.55789	−0.472631
$94$	−5.80439	−0.598677
$95$	−4.42514	−0.454010
$96$	3.49020	0.356217
$97$	14.3314	1.45513	0.727565	−	0.686039i	$-0.240652\pi$
$97$	14.3314	1.45513	0.727565	+	0.686039i	$0.240652\pi$
$98$	−0.746297	−0.0753874
$99$	13.5660	1.36343
$100$	3.30791	0.330791
$101$	8.46675	0.842473	0.421237	−	0.906951i	$-0.361596\pi$
$101$	8.46675	0.842473	0.421237	+	0.906951i	$0.361596\pi$
$102$	3.74238	0.370551
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	3.06152	0.300207
$105$	2.24241	0.218837
$106$	−5.29299	−0.514101
$107$	5.76031	0.556870	0.278435	−	0.960455i	$-0.410184\pi$
$107$	5.76031	0.556870	0.278435	+	0.960455i	$0.410184\pi$
$108$	−3.57253	−0.343767
$109$	16.3548	1.56651	0.783255	−	0.621700i	$-0.213558\pi$
$109$	16.3548	1.56651	0.783255	+	0.621700i	$0.213558\pi$
$110$	6.26758	0.597590
$111$	−8.55083	−0.811609
$112$	4.59696	0.434372
$113$	15.5907	1.46665	0.733324	−	0.679880i	$-0.237968\pi$
$113$	15.5907	1.46665	0.733324	+	0.679880i	$0.237968\pi$
$114$	−3.85138	−0.360715
$115$	−0.984584	−0.0918129
$116$	5.26838	0.489157
$117$	−2.34983	−0.217242
$118$	−5.81349	−0.535175
$119$	−11.8891	−1.08987
$120$	−2.47574	−0.226003
$121$	22.3295	2.02995
$122$	4.88757	0.442499
$123$	3.45202	0.311258
$124$	−4.68138	−0.420400
$125$	−9.02023	−0.806794
$126$	−7.05366	−0.628390
$127$	−5.68735	−0.504671	−0.252335	−	0.967640i	$-0.581199\pi$
$127$	−5.68735	−0.504671	−0.252335	+	0.967640i	$0.581199\pi$
$128$	−0.00433453	−0.000383122 0
$129$	2.51128	0.221106
$130$	−1.08564	−0.0952169
$131$	15.7704	1.37786	0.688932	−	0.724826i	$-0.258080\pi$
$131$	15.7704	1.37786	0.688932	+	0.724826i	$0.258080\pi$
$132$	−3.85523	−0.335555
$133$	12.2354	1.06095
$134$	−4.48742	−0.387654
$135$	4.32621	0.372341
$136$	13.1263	1.12557
$137$	14.7546	1.26057	0.630286	−	0.776363i	$-0.282937\pi$
$137$	14.7546	1.26057	0.630286	+	0.776363i	$0.282937\pi$
$138$	−0.856924	−0.0729462
$139$	−3.38393	−0.287021	−0.143511	−	0.989649i	$-0.545839\pi$
$139$	−3.38393	−0.287021	−0.143511	+	0.989649i	$0.545839\pi$
$140$	2.30316	0.194653
$141$	−4.32353	−0.364107
$142$	14.2975	1.19982
$143$	−5.77317	−0.482777
$144$	3.89548	0.324623
$145$	−6.37983	−0.529816
$146$	0.902972	0.0747305
$147$	−0.555897	−0.0458496
$148$	−8.78251	−0.721917
$149$	8.71833	0.714233	0.357117	−	0.934060i	$-0.383760\pi$
$149$	8.71833	0.714233	0.357117	+	0.934060i	$0.383760\pi$
$150$	−3.48638	−0.284662
$151$	−17.1775	−1.39788	−0.698942	−	0.715178i	$-0.746346\pi$
$151$	−17.1775	−1.39788	−0.698942	+	0.715178i	$0.746346\pi$
$152$	−13.5086	−1.09569
$153$	−10.0749	−0.814506
$154$	−17.3297	−1.39647
$155$	5.66899	0.455344
$156$	0.667784	0.0534655
$157$	3.79627	0.302975	0.151488	−	0.988459i	$-0.451594\pi$
$157$	3.79627	0.302975	0.151488	+	0.988459i	$0.451594\pi$
$158$	−3.84852	−0.306172
$159$	−3.94260	−0.312669
$160$	−4.34102	−0.343188
$161$	2.72236	0.214552
$162$	−3.86585	−0.303730
$163$	11.3024	0.885273	0.442636	−	0.896701i	$-0.354043\pi$
$163$	11.3024	0.885273	0.442636	+	0.896701i	$0.354043\pi$
$164$	3.54554	0.276860
$165$	4.66855	0.363446
$166$	8.75992	0.679902
$167$	−8.77210	−0.678806	−0.339403	−	0.940641i	$-0.610225\pi$
$167$	−8.77210	−0.678806	−0.339403	+	0.940641i	$0.610225\pi$
$168$	6.84538	0.528132
$169$	1.00000	0.0769231
$170$	−4.65467	−0.356997
$171$	10.3683	0.792887
$172$	2.57932	0.196672
$173$	−4.90221	−0.372708	−0.186354	−	0.982483i	$-0.559667\pi$
$173$	−4.90221	−0.372708	−0.186354	+	0.982483i	$0.559667\pi$
$174$	−5.55262	−0.420943
$175$	11.0758	0.837255
$176$	9.57058	0.721410
$177$	−4.33031	−0.325486
$178$	−11.9075	−0.892508
$179$	20.6448	1.54307	0.771533	−	0.636189i	$-0.219490\pi$
$179$	20.6448	1.54307	0.771533	+	0.636189i	$0.219490\pi$
$180$	1.95171	0.145472
$181$	−22.4488	−1.66860	−0.834302	−	0.551308i	$-0.814129\pi$
$181$	−22.4488	−1.66860	−0.834302	+	0.551308i	$0.814129\pi$
$182$	3.00177	0.222506
$183$	3.64062	0.269122
$184$	−3.00563	−0.221578
$185$	10.6353	0.781923
$186$	4.93395	0.361775
$187$	−24.7524	−1.81008
$188$	−4.44067	−0.323869
$189$	−11.9619	−0.870100
$190$	4.79025	0.347521
$191$	−8.28981	−0.599830	−0.299915	−	0.953966i	$-0.596958\pi$
$191$	−8.28981	−0.599830	−0.299915	+	0.953966i	$0.596958\pi$
$192$	−6.45159	−0.465603
$193$	0.250633	0.0180410	0.00902049	−	0.999959i	$-0.497129\pi$
$193$	0.250633	0.0180410	0.00902049	+	0.999959i	$0.497129\pi$
$194$	−15.5138	−1.11383
$195$	−0.808663	−0.0579096
$196$	−0.570958	−0.0407827
$197$	−19.1549	−1.36473	−0.682365	−	0.731011i	$-0.739049\pi$
$197$	−19.1549	−1.36473	−0.682365	+	0.731011i	$0.739049\pi$
$198$	−14.6853	−1.04364
$199$	−1.03660	−0.0734827	−0.0367413	−	0.999325i	$-0.511698\pi$
$199$	−1.03660	−0.0734827	−0.0367413	+	0.999325i	$0.511698\pi$
$200$	−12.2284	−0.864675
$201$	−3.34256	−0.235766
$202$	−9.16533	−0.644870
$203$	17.6401	1.23809
$204$	2.86312	0.200459
$205$	−4.29353	−0.299873
$206$	1.08251	0.0754219
$207$	2.30693	0.160343
$208$	−1.65777	−0.114946
$209$	25.4734	1.76203
$210$	−2.42742	−0.167508
$211$	−14.0078	−0.964339	−0.482169	−	0.876078i	$-0.660151\pi$
$211$	−14.0078	−0.964339	−0.482169	+	0.876078i	$0.660151\pi$
$212$	−4.04942	−0.278115
$213$	10.6498	0.729711
$214$	−6.23558	−0.426255
$215$	−3.12347	−0.213019
$216$	13.2066	0.898595
$217$	−15.6746	−1.06406
$218$	−17.7043	−1.19908
$219$	0.672599	0.0454501
$220$	4.79504	0.323281
$221$	4.28749	0.288408
$222$	9.25634	0.621245
$223$	−9.11139	−0.610144	−0.305072	−	0.952329i	$-0.598681\pi$
$223$	−9.11139	−0.610144	−0.305072	+	0.952329i	$0.598681\pi$
$224$	12.0028	0.801973
$225$	9.38571	0.625714
$226$	−16.8770	−1.12264
$227$	17.2449	1.14458	0.572291	−	0.820050i	$-0.306055\pi$
$227$	17.2449	1.14458	0.572291	+	0.820050i	$0.306055\pi$
$228$	−2.94652	−0.195138
$229$	7.84420	0.518359	0.259180	−	0.965829i	$-0.416548\pi$
$229$	7.84420	0.518359	0.259180	+	0.965829i	$0.416548\pi$
$230$	1.06582	0.0702781
$231$	−12.9084	−0.849313
$232$	−19.4756	−1.27864
$233$	−14.0217	−0.918595	−0.459297	−	0.888283i	$-0.651899\pi$
$233$	−14.0217	−0.918595	−0.459297	+	0.888283i	$0.651899\pi$
$234$	2.54371	0.166288
$235$	5.37749	0.350789
$236$	−4.44763	−0.289516
$237$	−2.86666	−0.186209
$238$	12.8701	0.834243
$239$	16.6364	1.07612	0.538061	−	0.842906i	$-0.319157\pi$
$239$	16.6364	1.07612	0.538061	+	0.842906i	$0.319157\pi$
$240$	1.34058	0.0865339
$241$	−22.7628	−1.46628	−0.733139	−	0.680079i	$-0.761946\pi$
$241$	−22.7628	−1.46628	−0.733139	+	0.680079i	$0.761946\pi$
$242$	−24.1718	−1.55382
$243$	−15.8208	−1.01490
$244$	3.73925	0.239381
$245$	0.691410	0.0441726
$246$	−3.73683	−0.238252
$247$	−4.41238	−0.280753
$248$	17.3057	1.09891
$249$	6.52502	0.413507
$250$	9.76447	0.617559
$251$	−18.3275	−1.15682	−0.578411	−	0.815745i	$-0.696327\pi$
$251$	−18.3275	−1.15682	−0.578411	+	0.815745i	$0.696327\pi$
$252$	−5.39643	−0.339943
$253$	5.66777	0.356330
$254$	6.15660	0.386299
$255$	−3.46714	−0.217121
$256$	−15.9977	−0.999853
$257$	11.6836	0.728805	0.364402	−	0.931242i	$-0.381273\pi$
$257$	11.6836	0.728805	0.364402	+	0.931242i	$0.381273\pi$
$258$	−2.71848	−0.169245
$259$	−29.4064	−1.82723
$260$	−0.830573	−0.0515099
$261$	14.9483	0.925274
$262$	−17.0716	−1.05468
$263$	2.77350	0.171021	0.0855106	−	0.996337i	$-0.472748\pi$
$263$	2.77350	0.171021	0.0855106	+	0.996337i	$0.472748\pi$
$264$	14.2516	0.877128
$265$	4.90371	0.301232
$266$	−13.2450	−0.812100
$267$	−8.86961	−0.542811
$268$	−3.43312	−0.209711
$269$	27.6554	1.68618	0.843091	−	0.537771i	$-0.180734\pi$
$269$	27.6554	1.68618	0.843091	+	0.537771i	$0.180734\pi$
$270$	−4.68316	−0.285008
$271$	8.51024	0.516960	0.258480	−	0.966017i	$-0.416778\pi$
$271$	8.51024	0.516960	0.258480	+	0.966017i	$0.416778\pi$
$272$	−7.10767	−0.430966
$273$	2.23594	0.135325
$274$	−15.9720	−0.964904
$275$	23.0592	1.39052
$276$	−0.655593	−0.0394621
$277$	7.34966	0.441598	0.220799	−	0.975319i	$-0.429133\pi$
$277$	7.34966	0.441598	0.220799	+	0.975319i	$0.429133\pi$
$278$	3.66313	0.219700
$279$	−13.2827	−0.795217
$280$	−8.51410	−0.508815
$281$	−12.6402	−0.754052	−0.377026	−	0.926203i	$-0.623053\pi$
$281$	−12.6402	−0.754052	−0.377026	+	0.926203i	$0.623053\pi$
$282$	4.68025	0.278705
$283$	−25.3559	−1.50725	−0.753625	−	0.657305i	$-0.771696\pi$
$283$	−25.3559	−1.50725	−0.753625	+	0.657305i	$0.771696\pi$
$284$	10.9383	0.649070
$285$	3.56813	0.211358
$286$	6.24950	0.369541
$287$	11.8715	0.700754
$288$	10.1712	0.599346
$289$	1.38259	0.0813289
$290$	6.90621	0.405547
$291$	−11.5558	−0.677414
$292$	0.690822	0.0404273
$293$	−4.14278	−0.242024	−0.121012	−	0.992651i	$-0.538614\pi$
$293$	−4.14278	−0.242024	−0.121012	+	0.992651i	$0.538614\pi$
$294$	0.601762	0.0350955
$295$	5.38593	0.313581
$296$	32.4663	1.88707
$297$	−24.9039	−1.44507
$298$	−9.43766	−0.546709
$299$	−0.981744	−0.0567757
$300$	−2.66727	−0.153995
$301$	8.63633	0.497790
$302$	18.5948	1.07001
$303$	−6.82700	−0.392201
$304$	7.31471	0.419527
$305$	−4.52810	−0.259278
$306$	10.9061	0.623462
$307$	21.7934	1.24382	0.621909	−	0.783090i	$-0.286357\pi$
$307$	21.7934	1.24382	0.621909	+	0.783090i	$0.286357\pi$
$308$	−13.2582	−0.755455
$309$	0.806331	0.0458705
$310$	−6.13672	−0.348543
$311$	−26.5852	−1.50751	−0.753754	−	0.657156i	$-0.771759\pi$
$311$	−26.5852	−1.50751	−0.753754	+	0.657156i	$0.771759\pi$
$312$	−2.46860	−0.139757
$313$	−24.8085	−1.40226	−0.701129	−	0.713035i	$-0.747320\pi$
$313$	−24.8085	−1.40226	−0.701129	+	0.713035i	$0.747320\pi$
$314$	−4.10949	−0.231912
$315$	6.53489	0.368199
$316$	−2.94433	−0.165631
$317$	1.45713	0.0818407	0.0409203	−	0.999162i	$-0.486971\pi$
$317$	1.45713	0.0818407	0.0409203	+	0.999162i	$0.486971\pi$
$318$	4.26790	0.239332
$319$	36.7255	2.05624
$320$	8.02432	0.448573
$321$	−4.64471	−0.259243
$322$	−2.94697	−0.164228
$323$	−18.9180	−1.05263
$324$	−2.95758	−0.164310
$325$	−3.99421	−0.221559
$326$	−12.2349	−0.677631
$327$	−13.1874	−0.729266
$328$	−13.1068	−0.723703
$329$	−14.8687	−0.819736
$330$	−5.05374	−0.278199
$331$	19.9198	1.09489	0.547444	−	0.836842i	$-0.315601\pi$
$331$	19.9198	1.09489	0.547444	+	0.836842i	$0.315601\pi$
$332$	6.70181	0.367810
$333$	−24.9191	−1.36556
$334$	9.49587	0.519591
$335$	4.15739	0.227142
$336$	−3.70667	−0.202215
$337$	−10.1465	−0.552717	−0.276359	−	0.961055i	$-0.589128\pi$
$337$	−10.1465	−0.552717	−0.276359	+	0.961055i	$0.589128\pi$
$338$	−1.08251	−0.0588807
$339$	−12.5712	−0.682776
$340$	−3.56108	−0.193126
$341$	−32.6336	−1.76721
$342$	−11.2238	−0.606914
$343$	17.4991	0.944864
$344$	−9.53499	−0.514092
$345$	0.793900	0.0427422
$346$	5.30668	0.285289
$347$	−19.5738	−1.05078	−0.525389	−	0.850862i	$-0.676080\pi$
$347$	−19.5738	−1.05078	−0.525389	+	0.850862i	$0.676080\pi$
$348$	−4.24806	−0.227720
$349$	−32.6398	−1.74717	−0.873583	−	0.486675i	$-0.838210\pi$
$349$	−32.6398	−1.74717	−0.873583	+	0.486675i	$0.838210\pi$
$350$	−11.9897	−0.640876
$351$	4.31373	0.230250
$352$	24.9891	1.33193
$353$	20.0474	1.06701	0.533507	−	0.845796i	$-0.320874\pi$
$353$	20.0474	1.06701	0.533507	+	0.845796i	$0.320874\pi$
$354$	4.68759	0.249143
$355$	−13.2459	−0.703021
$356$	−9.10992	−0.482825
$357$	9.58657	0.507375
$358$	−22.3482	−1.18114
$359$	33.3641	1.76089	0.880445	−	0.474147i	$-0.157244\pi$
$359$	33.3641	1.76089	0.880445	+	0.474147i	$0.157244\pi$
$360$	−7.21488	−0.380257
$361$	0.469081	0.0246885
$362$	24.3010	1.27723
$363$	−18.0049	−0.945014
$364$	2.29652	0.120370
$365$	−0.836562	−0.0437877
$366$	−3.94099	−0.205999
$367$	−2.05532	−0.107287	−0.0536434	−	0.998560i	$-0.517083\pi$
$367$	−2.05532	−0.107287	−0.0536434	+	0.998560i	$0.517083\pi$
$368$	1.62750	0.0848395
$369$	10.0600	0.523701
$370$	−11.5128	−0.598522
$371$	−13.5587	−0.703930
$372$	3.77474	0.195711
$373$	11.8579	0.613980	0.306990	−	0.951713i	$-0.400678\pi$
$373$	11.8579	0.613980	0.306990	+	0.951713i	$0.400678\pi$
$374$	26.7947	1.38552
$375$	7.27329	0.375591
$376$	16.4158	0.846582
$377$	−6.36142	−0.327630
$378$	12.9488	0.666016
$379$	−15.5635	−0.799443	−0.399722	−	0.916637i	$-0.630893\pi$
$379$	−15.5635	−0.799443	−0.399722	+	0.916637i	$0.630893\pi$
$380$	3.66480	0.188000
$381$	4.58588	0.234942
$382$	8.97379	0.459139
$383$	−12.2763	−0.627287	−0.313644	−	0.949541i	$-0.601550\pi$
$383$	−12.2763	−0.627287	−0.313644	+	0.949541i	$0.601550\pi$
$384$	0.00349507	0.000178357 0
$385$	16.0552	0.818248
$386$	−0.271313	−0.0138094
$387$	7.31845	0.372018
$388$	−11.8689	−0.602552
$389$	−9.81478	−0.497629	−0.248815	−	0.968551i	$-0.580041\pi$
$389$	−9.81478	−0.497629	−0.248815	+	0.968551i	$0.580041\pi$
$390$	0.875385	0.0443268
$391$	−4.20922	−0.212869
$392$	2.11066	0.106604
$393$	−12.7161	−0.641444
$394$	20.7353	1.04463
$395$	3.56548	0.179399
$396$	−11.2350	−0.564581
$397$	12.6916	0.636974	0.318487	−	0.947927i	$-0.396825\pi$
$397$	12.6916	0.636974	0.318487	+	0.947927i	$0.396825\pi$
$398$	1.12213	0.0562472
$399$	−9.86580	−0.493908
$400$	6.62147	0.331074
$401$	12.0582	0.602157	0.301079	−	0.953599i	$-0.402653\pi$
$401$	12.0582	0.602157	0.301079	+	0.953599i	$0.402653\pi$
$402$	3.61835	0.180467
$403$	5.65263	0.281578
$404$	−7.01197	−0.348858
$405$	3.58153	0.177968
$406$	−19.0955	−0.947695
$407$	−61.2223	−3.03468
$408$	−10.5841	−0.523991
$409$	−17.6116	−0.870838	−0.435419	−	0.900228i	$-0.643400\pi$
$409$	−17.6116	−0.870838	−0.435419	+	0.900228i	$0.643400\pi$
$410$	4.64778	0.229537
$411$	−11.8971	−0.586841
$412$	0.828177	0.0408013
$413$	−14.8920	−0.732786
$414$	−2.49727	−0.122734
$415$	−8.11566	−0.398382
$416$	−4.32850	−0.212222
$417$	2.72857	0.133618
$418$	−27.5752	−1.34874
$419$	36.5701	1.78657	0.893283	−	0.449495i	$-0.148396\pi$
$419$	36.5701	1.78657	0.893283	+	0.449495i	$0.148396\pi$
$420$	−1.85711	−0.0906177
$421$	6.57956	0.320668	0.160334	−	0.987063i	$-0.448743\pi$
$421$	6.57956	0.320668	0.160334	+	0.987063i	$0.448743\pi$
$422$	15.1636	0.738152
$423$	−12.5997	−0.612621
$424$	14.9695	0.726984
$425$	−17.1251	−0.830691
$426$	−11.5285	−0.558556
$427$	12.5201	0.605891
$428$	−4.77055	−0.230593
$429$	4.65508	0.224749
$430$	3.38118	0.163055
$431$	35.6133	1.71543	0.857716	−	0.514123i	$-0.171883\pi$
$431$	35.6133	1.71543	0.857716	+	0.514123i	$0.171883\pi$
$432$	−7.15117	−0.344061
$433$	−4.01413	−0.192907	−0.0964534	−	0.995338i	$-0.530750\pi$
$433$	−4.01413	−0.192907	−0.0964534	+	0.995338i	$0.530750\pi$
$434$	16.9679	0.814486
$435$	5.14425	0.246648
$436$	−13.5447	−0.648674
$437$	4.33182	0.207219
$438$	−0.728094	−0.0347897
$439$	−13.4808	−0.643403	−0.321701	−	0.946841i	$-0.604255\pi$
$439$	−13.4808	−0.643403	−0.321701	+	0.946841i	$0.604255\pi$
$440$	−17.7258	−0.845046
$441$	−1.62001	−0.0771433
$442$	−4.64124	−0.220761
$443$	41.3234	1.96333	0.981667	−	0.190604i	$-0.0610447\pi$
$443$	41.3234	1.96333	0.981667	+	0.190604i	$0.0610447\pi$
$444$	7.08160	0.336078
$445$	11.0318	0.522957
$446$	9.86316	0.467034
$447$	−7.02985	−0.332501
$448$	−22.1871	−1.04824
$449$	−6.64433	−0.313566	−0.156783	−	0.987633i	$-0.550112\pi$
$449$	−6.64433	−0.313566	−0.156783	+	0.987633i	$0.550112\pi$
$450$	−10.1601	−0.478952
$451$	24.7158	1.16382
$452$	−12.9118	−0.607322
$453$	13.8507	0.650765
$454$	−18.6677	−0.876119
$455$	−2.78100	−0.130375
$456$	10.8924	0.510083
$457$	40.6958	1.90367	0.951835	−	0.306612i	$-0.0991954\pi$
$457$	40.6958	1.90367	0.951835	+	0.306612i	$0.0991954\pi$
$458$	−8.49141	−0.396777
$459$	18.4951	0.863277
$460$	0.815410	0.0380187
$461$	14.8768	0.692884	0.346442	−	0.938071i	$-0.387390\pi$
$461$	14.8768	0.692884	0.346442	+	0.938071i	$0.387390\pi$
$462$	13.9735	0.650106
$463$	−10.1753	−0.472886	−0.236443	−	0.971645i	$-0.575982\pi$
$463$	−10.1753	−0.472886	−0.236443	+	0.971645i	$0.575982\pi$
$464$	10.5458	0.489575
$465$	−4.57108	−0.211979
$466$	15.1786	0.703137
$467$	−23.6913	−1.09630	−0.548152	−	0.836378i	$-0.684669\pi$
$467$	−23.6913	−1.09630	−0.548152	+	0.836378i	$0.684669\pi$
$468$	1.94608	0.0899574
$469$	−11.4951	−0.530794
$470$	−5.82118	−0.268511
$471$	−3.06105	−0.141046
$472$	16.4416	0.756785
$473$	17.9803	0.826735
$474$	3.10318	0.142534
$475$	17.6239	0.808642
$476$	9.84630	0.451304
$477$	−11.4896	−0.526075
$478$	−18.0091	−0.823716
$479$	36.5993	1.67226	0.836132	−	0.548528i	$-0.184812\pi$
$479$	36.5993	1.67226	0.836132	+	0.548528i	$0.184812\pi$
$480$	3.50030	0.159766
$481$	10.6046	0.483529
$482$	24.6409	1.12236
$483$	−2.19512	−0.0998813
$484$	−18.4927	−0.840579
$485$	14.3728	0.652637
$486$	17.1261	0.776856
$487$	−14.1731	−0.642245	−0.321123	−	0.947038i	$-0.604060\pi$
$487$	−14.1731	−0.642245	−0.321123	+	0.947038i	$0.604060\pi$
$488$	−13.8229	−0.625733
$489$	−9.11348	−0.412126
$490$	−0.748457	−0.0338118
$491$	38.7095	1.74693	0.873467	−	0.486883i	$-0.161866\pi$
$491$	38.7095	1.74693	0.873467	+	0.486883i	$0.161866\pi$
$492$	−2.85888	−0.128888
$493$	−27.2745	−1.22838
$494$	4.77643	0.214902
$495$	13.6052	0.611509
$496$	−9.37076	−0.420760
$497$	36.6247	1.64284
$498$	−7.06339	−0.316518
$499$	−17.6708	−0.791053	−0.395527	−	0.918455i	$-0.629438\pi$
$499$	−17.6708	−0.791053	−0.395527	+	0.918455i	$0.629438\pi$
$500$	7.47034	0.334084
$501$	7.07321	0.316008
$502$	19.8397	0.885488
$503$	−4.11089	−0.183295	−0.0916477	−	0.995791i	$-0.529213\pi$
$503$	−4.11089	−0.183295	−0.0916477	+	0.995791i	$0.529213\pi$
$504$	19.9490	0.888599
$505$	8.49125	0.377856
$506$	−6.13541	−0.272752
$507$	−0.806331	−0.0358104
$508$	4.71013	0.208978
$509$	16.1818	0.717246	0.358623	−	0.933482i	$-0.383246\pi$
$509$	16.1818	0.717246	0.358623	+	0.933482i	$0.383246\pi$
$510$	3.75320	0.166195
$511$	2.31308	0.102324
$512$	17.3263	0.765719
$513$	−19.0338	−0.840364
$514$	−12.6476	−0.557863
$515$	−1.00289	−0.0441928
$516$	−2.07979	−0.0915575
$517$	−30.9556	−1.36143
$518$	31.8327	1.39865
$519$	3.95280	0.173509
$520$	3.07038	0.134645
$521$	30.0861	1.31809	0.659047	−	0.752102i	$-0.270960\pi$
$521$	30.0861	1.31809	0.659047	+	0.752102i	$0.270960\pi$
$522$	−16.1816	−0.708250
$523$	−19.5404	−0.854443	−0.427221	−	0.904147i	$-0.640508\pi$
$523$	−19.5404	−0.854443	−0.427221	+	0.904147i	$0.640508\pi$
$524$	−13.0607	−0.570558
$525$	−8.93080	−0.389772
$526$	−3.00233	−0.130908
$527$	24.2356	1.05572
$528$	−7.71705	−0.335842
$529$	−22.0362	−0.958095
$530$	−5.30830	−0.230578
$531$	−12.6195	−0.547640
$532$	−10.1331	−0.439326
$533$	−4.28114	−0.185437
$534$	9.60142	0.415494
$535$	5.77697	0.249760
$536$	12.6912	0.548177
$537$	−16.6466	−0.718352
$538$	−29.9372	−1.29069
$539$	−3.98011	−0.171435
$540$	−3.58287	−0.154182
$541$	23.0310	0.990178	0.495089	−	0.868842i	$-0.335135\pi$
$541$	23.0310	0.990178	0.495089	+	0.868842i	$0.335135\pi$
$542$	−9.21240	−0.395706
$543$	18.1011	0.776794
$544$	−18.5584	−0.795685
$545$	16.4022	0.702592
$546$	−2.42042	−0.103584
$547$	2.20191	0.0941468	0.0470734	−	0.998891i	$-0.485011\pi$
$547$	2.20191	0.0941468	0.0470734	+	0.998891i	$0.485011\pi$
$548$	−12.2194	−0.521989
$549$	10.6096	0.452806
$550$	−24.9618	−1.06437
$551$	28.0690	1.19578
$552$	2.42353	0.103152
$553$	−9.85847	−0.419225
$554$	−7.95606	−0.338021
$555$	−8.57557	−0.364013
$556$	2.80249	0.118852
$557$	−9.14262	−0.387385	−0.193693	−	0.981062i	$-0.562046\pi$
$557$	−9.14262	−0.387385	−0.193693	+	0.981062i	$0.562046\pi$
$558$	14.3787	0.608697
$559$	−3.11446	−0.131728
$560$	4.61026	0.194819
$561$	19.9586	0.842654
$562$	13.6831	0.577188
$563$	−39.0737	−1.64676	−0.823380	−	0.567490i	$-0.807915\pi$
$563$	−39.0737	−1.64676	−0.823380	+	0.567490i	$0.807915\pi$
$564$	3.58065	0.150772
$565$	15.6358	0.657802
$566$	27.4479	1.15372
$567$	−9.90286	−0.415881
$568$	−40.4357	−1.69665
$569$	−24.3140	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$569$	−24.3140	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$570$	−3.86253	−0.161783
$571$	6.36231	0.266254	0.133127	−	0.991099i	$-0.457498\pi$
$571$	6.36231	0.266254	0.133127	+	0.991099i	$0.457498\pi$
$572$	4.78120	0.199912
$573$	6.68433	0.279242
$574$	−12.8510	−0.536391
$575$	3.92129	0.163529
$576$	−18.8014	−0.783392
$577$	15.9541	0.664177	0.332089	−	0.943248i	$-0.392247\pi$
$577$	15.9541	0.664177	0.332089	+	0.943248i	$0.392247\pi$
$578$	−1.49667	−0.0622531
$579$	−0.202093	−0.00839871
$580$	5.28362	0.219391
$581$	22.4396	0.930953
$582$	12.5093	0.518526
$583$	−28.2283	−1.16910
$584$	−2.55377	−0.105676
$585$	−2.35663	−0.0974346
$586$	4.48459	0.185257
$587$	−30.3285	−1.25179	−0.625895	−	0.779907i	$-0.715266\pi$
$587$	−30.3285	−1.25179	−0.625895	+	0.779907i	$0.715266\pi$
$588$	0.460381	0.0189858
$589$	−24.9416	−1.02770
$590$	−5.83031	−0.240030
$591$	15.4452	0.635330
$592$	−17.5800	−0.722535
$593$	11.7299	0.481691	0.240845	−	0.970564i	$-0.422575\pi$
$593$	11.7299	0.481691	0.240845	+	0.970564i	$0.422575\pi$
$594$	26.9587	1.10613
$595$	−11.9235	−0.488817
$596$	−7.22032	−0.295756
$597$	0.835843	0.0342088
$598$	1.06275	0.0434589
$599$	−16.3324	−0.667323	−0.333661	−	0.942693i	$-0.608284\pi$
$599$	−16.3324	−0.667323	−0.333661	+	0.942693i	$0.608284\pi$
$600$	9.86009	0.402537
$601$	−0.531464	−0.0216789	−0.0108394	−	0.999941i	$-0.503450\pi$
$601$	−0.531464	−0.0216789	−0.0108394	+	0.999941i	$0.503450\pi$
$602$	−9.34890	−0.381033
$603$	−9.74098	−0.396683
$604$	14.2260	0.578848
$605$	22.3941	0.910448
$606$	7.39028	0.300210
$607$	0.438298	0.0177900	0.00889498	−	0.999960i	$-0.497169\pi$
$607$	0.438298	0.0177900	0.00889498	+	0.999960i	$0.497169\pi$
$608$	19.0990	0.774566
$609$	−14.2237	−0.576375
$610$	4.90171	0.198464
$611$	5.36198	0.216923
$612$	8.34378	0.337278
$613$	−23.9586	−0.967679	−0.483839	−	0.875157i	$-0.660758\pi$
$613$	−23.9586	−0.967679	−0.483839	+	0.875157i	$0.660758\pi$
$614$	−23.5916	−0.952078
$615$	3.46200	0.139601
$616$	49.0115	1.97473
$617$	40.6738	1.63747	0.818733	−	0.574174i	$-0.194677\pi$
$617$	40.6738	1.63747	0.818733	+	0.574174i	$0.194677\pi$
$618$	−0.872859	−0.0351115
$619$	11.3958	0.458036	0.229018	−	0.973422i	$-0.426449\pi$
$619$	11.3958	0.458036	0.229018	+	0.973422i	$0.426449\pi$
$620$	−4.69492	−0.188553
$621$	−4.23498	−0.169944
$622$	28.7787	1.15392
$623$	−30.5027	−1.22206
$624$	1.33671	0.0535112
$625$	10.9247	0.436988
$626$	26.8554	1.07336
$627$	−20.5400	−0.820288
$628$	−3.14398	−0.125459
$629$	45.4673	1.81290
$630$	−7.07406	−0.281837
$631$	−37.6415	−1.49848	−0.749242	−	0.662296i	$-0.769582\pi$
$631$	−37.6415	−1.49848	−0.749242	+	0.662296i	$0.769582\pi$
$632$	10.8843	0.432954
$633$	11.2949	0.448934
$634$	−1.57736	−0.0626448
$635$	−5.70380	−0.226349
$636$	3.26517	0.129473
$637$	0.689415	0.0273156
$638$	−39.7557	−1.57394
$639$	31.0359	1.22776
$640$	−0.00434707	−0.000171833 0
$641$	−20.7377	−0.819090	−0.409545	−	0.912290i	$-0.634313\pi$
$641$	−20.7377	−0.819090	−0.409545	+	0.912290i	$0.634313\pi$
$642$	5.02794	0.198437
$643$	33.4804	1.32034	0.660168	−	0.751118i	$-0.270485\pi$
$643$	33.4804	1.32034	0.660168	+	0.751118i	$0.270485\pi$
$644$	−2.25459	−0.0888433
$645$	2.51855	0.0991678
$646$	20.4789	0.805732
$647$	15.7143	0.617794	0.308897	−	0.951096i	$-0.400040\pi$
$647$	15.7143	0.617794	0.308897	+	0.951096i	$0.400040\pi$
$648$	10.9333	0.429501
$649$	−31.0042	−1.21702
$650$	4.32376	0.169592
$651$	12.6389	0.495359
$652$	−9.36039	−0.366581
$653$	−16.7978	−0.657350	−0.328675	−	0.944443i	$-0.606602\pi$
$653$	−16.7978	−0.657350	−0.328675	+	0.944443i	$0.606602\pi$
$654$	14.2755	0.558215
$655$	15.8160	0.617983
$656$	7.09715	0.277097
$657$	1.96011	0.0764711
$658$	16.0954	0.627466
$659$	−31.2370	−1.21682	−0.608411	−	0.793622i	$-0.708193\pi$
$659$	−31.2370	−1.21682	−0.608411	+	0.793622i	$0.708193\pi$
$660$	−3.86638	−0.150499
$661$	−35.2183	−1.36983	−0.684917	−	0.728621i	$-0.740162\pi$
$661$	−35.2183	−1.36983	−0.684917	+	0.728621i	$0.740162\pi$
$662$	−21.5633	−0.838081
$663$	−3.45714	−0.134264
$664$	−24.7746	−0.961441
$665$	12.2708	0.475843
$666$	26.9751	1.04526
$667$	6.24528	0.241818
$668$	7.26485	0.281086
$669$	7.34680	0.284044
$670$	−4.50041	−0.173866
$671$	26.0661	1.00627
$672$	−9.67826	−0.373347
$673$	30.3287	1.16908	0.584542	−	0.811363i	$-0.301274\pi$
$673$	30.3287	1.16908	0.584542	+	0.811363i	$0.301274\pi$
$674$	10.9837	0.423077
$675$	−17.2299	−0.663180
$676$	−0.828177	−0.0318530
$677$	43.8259	1.68437	0.842184	−	0.539191i	$-0.181270\pi$
$677$	43.8259	1.68437	0.842184	+	0.539191i	$0.181270\pi$
$678$	13.6085	0.522630
$679$	−39.7406	−1.52510
$680$	13.1642	0.504825
$681$	−13.9051	−0.532843
$682$	35.3261	1.35271
$683$	30.4591	1.16549	0.582743	−	0.812656i	$-0.301979\pi$
$683$	30.4591	1.16549	0.582743	+	0.812656i	$0.301979\pi$
$684$	−8.58682	−0.328325
$685$	14.7973	0.565376
$686$	−18.9429	−0.723245
$687$	−6.32502	−0.241314
$688$	5.16306	0.196840
$689$	4.88956	0.186278
$690$	−0.859403	−0.0327169
$691$	5.16974	0.196666	0.0983331	−	0.995154i	$-0.468649\pi$
$691$	5.16974	0.196666	0.0983331	+	0.995154i	$0.468649\pi$
$692$	4.05990	0.154334
$693$	−37.6182	−1.42900
$694$	21.1888	0.804317
$695$	−3.39372	−0.128731
$696$	15.7038	0.595251
$697$	−18.3554	−0.695259
$698$	35.3328	1.33737
$699$	11.3062	0.427638
$700$	−9.17276	−0.346698
$701$	23.5436	0.889228	0.444614	−	0.895722i	$-0.353341\pi$
$701$	23.5436	0.889228	0.444614	+	0.895722i	$0.353341\pi$
$702$	−4.66965	−0.176245
$703$	−46.7916	−1.76478
$704$	−46.1921	−1.74093
$705$	−4.33604	−0.163305
$706$	−21.7014	−0.816744
$707$	−23.4781	−0.882986
$708$	3.58626	0.134780
$709$	23.6662	0.888805	0.444402	−	0.895827i	$-0.353416\pi$
$709$	23.6662	0.888805	0.444402	+	0.895827i	$0.353416\pi$
$710$	14.3388	0.538126
$711$	−8.35410	−0.313303
$712$	33.6767	1.26209
$713$	−5.54944	−0.207828
$714$	−10.3775	−0.388369
$715$	−5.78987	−0.216529
$716$	−17.0976	−0.638966
$717$	−13.4145	−0.500973
$718$	−36.1169	−1.34787
$719$	−0.278740	−0.0103953	−0.00519763	−	0.999986i	$-0.501654\pi$
$719$	−0.278740	−0.0103953	−0.00519763	+	0.999986i	$0.501654\pi$
$720$	3.90675	0.145596
$721$	2.77298	0.103271
$722$	−0.507783	−0.0188977
$723$	18.3543	0.682604
$724$	18.5915	0.690949
$725$	25.4088	0.943660
$726$	19.4905	0.723359
$727$	23.6474	0.877033	0.438517	−	0.898723i	$-0.355504\pi$
$727$	23.6474	0.877033	0.438517	+	0.898723i	$0.355504\pi$
$728$	−8.48954	−0.314643
$729$	2.04317	0.0756729
$730$	0.905585	0.0335172
$731$	−13.3532	−0.493887
$732$	−3.01507	−0.111440
$733$	7.38187	0.272656	0.136328	−	0.990664i	$-0.456470\pi$
$733$	7.38187	0.272656	0.136328	+	0.990664i	$0.456470\pi$
$734$	2.22490	0.0821225
$735$	−0.557505	−0.0205639
$736$	4.24948	0.156638
$737$	−23.9321	−0.881549
$738$	−10.8900	−0.400866
$739$	27.0051	0.993400	0.496700	−	0.867922i	$-0.334545\pi$
$739$	27.0051	0.993400	0.496700	+	0.867922i	$0.334545\pi$
$740$	−8.80791	−0.323785
$741$	3.55784	0.130700
$742$	14.6774	0.538823
$743$	9.10826	0.334150	0.167075	−	0.985944i	$-0.446568\pi$
$743$	9.10826	0.334150	0.167075	+	0.985944i	$0.446568\pi$
$744$	−13.9541	−0.511582
$745$	8.74355	0.320339
$746$	−12.8363	−0.469970
$747$	19.0154	0.695737
$748$	20.4994	0.749531
$749$	−15.9732	−0.583649
$750$	−7.87339	−0.287495
$751$	32.2788	1.17787	0.588935	−	0.808181i	$-0.299547\pi$
$751$	32.2788	1.17787	0.588935	+	0.808181i	$0.299547\pi$
$752$	−8.88893	−0.324146
$753$	14.7780	0.538542
$754$	6.88629	0.250784
$755$	−17.2272	−0.626962
$756$	9.90656	0.360298
$757$	30.1896	1.09726	0.548630	−	0.836065i	$-0.315150\pi$
$757$	30.1896	1.09726	0.548630	+	0.836065i	$0.315150\pi$
$758$	16.8476	0.611933
$759$	−4.57010	−0.165884
$760$	−13.5477	−0.491426
$761$	41.2784	1.49634	0.748171	−	0.663506i	$-0.230932\pi$
$761$	41.2784	1.49634	0.748171	+	0.663506i	$0.230932\pi$
$762$	−4.96426	−0.179836
$763$	−45.3517	−1.64184
$764$	6.86543	0.248383
$765$	−10.1040	−0.365312
$766$	13.2891	0.480156
$767$	5.37039	0.193914
$768$	12.8994	0.465467
$769$	34.8448	1.25654	0.628268	−	0.777997i	$-0.283764\pi$
$769$	34.8448	1.25654	0.628268	+	0.777997i	$0.283764\pi$
$770$	−17.3799	−0.626327
$771$	−9.42087	−0.339284
$772$	−0.207569	−0.00747056
$773$	22.8335	0.821264	0.410632	−	0.911801i	$-0.365308\pi$
$773$	22.8335	0.821264	0.410632	+	0.911801i	$0.365308\pi$
$774$	−7.92228	−0.284761
$775$	−22.5778	−0.811018
$776$	43.8758	1.57505
$777$	23.7113	0.850638
$778$	10.6246	0.380910
$779$	18.8900	0.676805
$780$	0.669716	0.0239797
$781$	76.2504	2.72845
$782$	4.55651	0.162941
$783$	−27.4415	−0.980678
$784$	−1.14289	−0.0408176
$785$	3.80725	0.135887
$786$	13.7653	0.490993
$787$	−1.34036	−0.0477785	−0.0238893	−	0.999715i	$-0.507605\pi$
$787$	−1.34036	−0.0477785	−0.0238893	+	0.999715i	$0.507605\pi$
$788$	15.8637	0.565119
$789$	−2.23636	−0.0796164
$790$	−3.85966	−0.137320
$791$	−43.2326	−1.53718
$792$	41.5325	1.47579
$793$	−4.51504	−0.160334
$794$	−13.7388	−0.487571
$795$	−3.95401	−0.140234
$796$	0.858489	0.0304283
$797$	32.8619	1.16403	0.582014	−	0.813178i	$-0.302265\pi$
$797$	32.8619	1.16403	0.582014	+	0.813178i	$0.302265\pi$
$798$	10.6798	0.378061
$799$	22.9895	0.813308
$800$	17.2889	0.611255
$801$	−25.8481	−0.913296
$802$	−13.0531	−0.460920
$803$	4.81568	0.169942
$804$	2.76823	0.0976279
$805$	2.73023	0.0962280
$806$	−6.11902	−0.215533
$807$	−22.2994	−0.784977
$808$	25.9212	0.911903
$809$	36.7460	1.29192	0.645961	−	0.763370i	$-0.276457\pi$
$809$	36.7460	1.29192	0.645961	+	0.763370i	$0.276457\pi$
$810$	−3.87704	−0.136225
$811$	16.1201	0.566053	0.283026	−	0.959112i	$-0.408662\pi$
$811$	16.1201	0.566053	0.283026	+	0.959112i	$0.408662\pi$
$812$	−14.6091	−0.512679
$813$	−6.86207	−0.240663
$814$	66.2736	2.32289
$815$	11.3351	0.397051
$816$	5.73113	0.200630
$817$	13.7422	0.480778
$818$	19.0647	0.666582
$819$	6.51603	0.227689
$820$	3.55580	0.124174
$821$	−39.8024	−1.38911	−0.694556	−	0.719439i	$-0.744399\pi$
$821$	−39.8024	−1.38911	−0.694556	+	0.719439i	$0.744399\pi$
$822$	12.8787	0.449197
$823$	−29.0727	−1.01341	−0.506705	−	0.862119i	$-0.669137\pi$
$823$	−29.0727	−1.01341	−0.506705	+	0.862119i	$0.669137\pi$
$824$	−3.06152	−0.106653
$825$	−18.5934	−0.647337
$826$	16.1207	0.560910
$827$	0.448085	0.0155815	0.00779073	−	0.999970i	$-0.497520\pi$
$827$	0.448085	0.0155815	0.00779073	+	0.999970i	$0.497520\pi$
$828$	−1.91055	−0.0663961
$829$	−34.7967	−1.20854	−0.604270	−	0.796780i	$-0.706535\pi$
$829$	−34.7967	−1.20854	−0.604270	+	0.796780i	$0.706535\pi$
$830$	8.78526	0.304941
$831$	−5.92625	−0.205579
$832$	8.00117	0.277391
$833$	2.95586	0.102415
$834$	−2.95370	−0.102278
$835$	−8.79748	−0.304449
$836$	−21.0965	−0.729637
$837$	24.3839	0.842833
$838$	−39.5874	−1.36752
$839$	11.1812	0.386018	0.193009	−	0.981197i	$-0.438175\pi$
$839$	11.1812	0.386018	0.193009	+	0.981197i	$0.438175\pi$
$840$	6.86518	0.236871
$841$	11.4677	0.395437
$842$	−7.12243	−0.245455
$843$	10.1922	0.351038
$844$	11.6010	0.399322
$845$	1.00289	0.0345006
$846$	13.6393	0.468930
$847$	−61.9191	−2.12757
$848$	−8.10577	−0.278353
$849$	20.4452	0.701678
$850$	18.5381	0.635851
$851$	−10.4110	−0.356885
$852$	−8.81991	−0.302165
$853$	−19.8243	−0.678770	−0.339385	−	0.940648i	$-0.610219\pi$
$853$	−19.8243	−0.678770	−0.339385	+	0.940648i	$0.610219\pi$
$854$	−13.5531	−0.463778
$855$	10.3983	0.355616
$856$	17.6353	0.602763
$857$	−5.84930	−0.199808	−0.0999041	−	0.994997i	$-0.531854\pi$
$857$	−5.84930	−0.199808	−0.0999041	+	0.994997i	$0.531854\pi$
$858$	−5.03916	−0.172034
$859$	−49.1450	−1.67680	−0.838402	−	0.545052i	$-0.816510\pi$
$859$	−49.1450	−1.67680	−0.838402	+	0.545052i	$0.816510\pi$
$860$	2.58679	0.0882087
$861$	−9.57237	−0.326225
$862$	−38.5517	−1.31308
$863$	5.03160	0.171278	0.0856388	−	0.996326i	$-0.472707\pi$
$863$	5.03160	0.171278	0.0856388	+	0.996326i	$0.472707\pi$
$864$	−18.6720	−0.635234
$865$	−4.91640	−0.167162
$866$	4.34533	0.147660
$867$	−1.11483	−0.0378615
$868$	12.9814	0.440616
$869$	−20.5247	−0.696253
$870$	−5.56869	−0.188796
$871$	4.14539	0.140461
$872$	50.0708	1.69561
$873$	−33.6763	−1.13977
$874$	−4.68923	−0.158616
$875$	25.0129	0.845591
$876$	−0.557031	−0.0188203
$877$	19.5044	0.658617	0.329308	−	0.944222i	$-0.393184\pi$
$877$	19.5044	0.658617	0.329308	+	0.944222i	$0.393184\pi$
$878$	14.5930	0.492492
$879$	3.34045	0.112670
$880$	9.59827	0.323558
$881$	12.9020	0.434679	0.217340	−	0.976096i	$-0.430262\pi$
$881$	12.9020	0.434679	0.217340	+	0.976096i	$0.430262\pi$
$882$	1.75367	0.0590492
$883$	9.02230	0.303624	0.151812	−	0.988409i	$-0.451489\pi$
$883$	9.02230	0.303624	0.151812	+	0.988409i	$0.451489\pi$
$884$	−3.55080	−0.119426
$885$	−4.34284	−0.145983
$886$	−44.7329	−1.50283
$887$	29.8998	1.00394	0.501969	−	0.864886i	$-0.332609\pi$
$887$	29.8998	1.00394	0.501969	+	0.864886i	$0.332609\pi$
$888$	−26.1786	−0.878496
$889$	15.7709	0.528939
$890$	−11.9420	−0.400297
$891$	−20.6171	−0.690700
$892$	7.54585	0.252654
$893$	−23.6591	−0.791721
$894$	7.60987	0.254512
$895$	20.7046	0.692077
$896$	0.0120196	0.000401546 0
$897$	0.791610	0.0264311
$898$	7.19254	0.240018
$899$	−35.9588	−1.19929
$900$	−7.77303	−0.259101
$901$	20.9640	0.698411
$902$	−26.7550	−0.890844
$903$	−6.96374	−0.231739
$904$	47.7312	1.58752
$905$	−22.5137	−0.748381
$906$	−14.9935	−0.498127
$907$	−24.6415	−0.818207	−0.409104	−	0.912488i	$-0.634159\pi$
$907$	−24.6415	−0.818207	−0.409104	+	0.912488i	$0.634159\pi$
$908$	−14.2818	−0.473958
$909$	−19.8954	−0.659890
$910$	3.01046	0.0997957
$911$	42.3775	1.40403	0.702014	−	0.712163i	$-0.252284\pi$
$911$	42.3775	1.40403	0.702014	+	0.712163i	$0.252284\pi$
$912$	−5.89807	−0.195305
$913$	46.7179	1.54614
$914$	−44.0535	−1.45716
$915$	3.65115	0.120703
$916$	−6.49638	−0.214647
$917$	−43.7309	−1.44412
$918$	−20.0211	−0.660794
$919$	41.8041	1.37899	0.689495	−	0.724290i	$-0.257832\pi$
$919$	41.8041	1.37899	0.689495	+	0.724290i	$0.257832\pi$
$920$	−3.01433	−0.0993794
$921$	−17.5727	−0.579041
$922$	−16.1043	−0.530367
$923$	−13.2077	−0.434737
$924$	10.6905	0.351691
$925$	−42.3571	−1.39269
$926$	11.0148	0.361970
$927$	2.34983	0.0771786
$928$	27.5354	0.903894
$929$	−22.4918	−0.737932	−0.368966	−	0.929443i	$-0.620288\pi$
$929$	−22.4918	−0.737932	−0.368966	+	0.929443i	$0.620288\pi$
$930$	4.94823	0.162259
$931$	−3.04196	−0.0996962
$932$	11.6125	0.380380
$933$	21.4365	0.701798
$934$	25.6461	0.839165
$935$	−24.8240	−0.811832
$936$	−7.19406	−0.235145
$937$	35.3964	1.15635	0.578174	−	0.815913i	$-0.303765\pi$
$937$	35.3964	1.15635	0.578174	+	0.815913i	$0.303765\pi$
$938$	12.4435	0.406296
$939$	20.0038	0.652800
$940$	−4.45352	−0.145258
$941$	−51.2691	−1.67133	−0.835663	−	0.549243i	$-0.814916\pi$
$941$	−51.2691	−1.67133	−0.835663	+	0.549243i	$0.814916\pi$
$942$	3.31361	0.107963
$943$	4.20298	0.136868
$944$	−8.90287	−0.289764
$945$	−11.9965	−0.390246
$946$	−19.4638	−0.632823
$947$	−5.32644	−0.173086	−0.0865430	−	0.996248i	$-0.527582\pi$
$947$	−5.32644	−0.173086	−0.0865430	+	0.996248i	$0.527582\pi$
$948$	2.37410	0.0771072
$949$	−0.834149	−0.0270776
$950$	−19.0781	−0.618974
$951$	−1.17493	−0.0380997
$952$	−36.3988	−1.17969
$953$	−23.2404	−0.752832	−0.376416	−	0.926451i	$-0.622844\pi$
$953$	−23.2404	−0.752832	−0.376416	+	0.926451i	$0.622844\pi$
$954$	12.4376	0.402683
$955$	−8.31380	−0.269028
$956$	−13.7779	−0.445610
$957$	−29.6129	−0.957250
$958$	−39.6190	−1.28003
$959$	−40.9143	−1.32119
$960$	−6.47026	−0.208827
$961$	0.952271	0.0307184
$962$	−11.4796	−0.370117
$963$	−13.5358	−0.436183
$964$	18.8516	0.607169
$965$	0.251359	0.00809152
$966$	2.37623	0.0764540
$967$	−26.6313	−0.856405	−0.428203	−	0.903683i	$-0.640853\pi$
$967$	−26.6313	−0.856405	−0.428203	+	0.903683i	$0.640853\pi$
$968$	68.3622	2.19724
$969$	15.2542	0.490035
$970$	−15.5587	−0.499560
$971$	−55.6117	−1.78466	−0.892332	−	0.451380i	$-0.850932\pi$
$971$	−55.6117	−1.78466	−0.892332	+	0.451380i	$0.850932\pi$
$972$	13.1024	0.420259
$973$	9.38357	0.300823
$974$	15.3425	0.491606
$975$	3.22065	0.103143
$976$	7.48490	0.239586
$977$	−25.5179	−0.816390	−0.408195	−	0.912895i	$-0.633842\pi$
$977$	−25.5179	−0.816390	−0.408195	+	0.912895i	$0.633842\pi$
$978$	9.86541	0.315461
$979$	−63.5046	−2.02962
$980$	−0.572610	−0.0182913
$981$	−38.4311	−1.22701
$982$	−41.9033	−1.33719
$983$	−31.8635	−1.01629	−0.508145	−	0.861272i	$-0.669668\pi$
$983$	−31.8635	−1.01629	−0.508145	+	0.861272i	$0.669668\pi$
$984$	10.5684	0.336909
$985$	−19.2103	−0.612092
$986$	29.5249	0.940265
$987$	11.9891	0.381616
$988$	3.65423	0.116257
$989$	3.05760	0.0972261
$990$	−14.7278	−0.468079
$991$	−1.10556	−0.0351194	−0.0175597	−	0.999846i	$-0.505590\pi$
$991$	−1.10556	−0.0351194	−0.0175597	+	0.999846i	$0.505590\pi$
$992$	−24.4674	−0.776841
$993$	−16.0619	−0.509709
$994$	−39.6466	−1.25751
$995$	−1.03960	−0.0329575
$996$	−5.40387	−0.171228
$997$	32.5066	1.02949	0.514747	−	0.857342i	$-0.327886\pi$
$997$	32.5066	1.02949	0.514747	+	0.857342i	$0.327886\pi$
$998$	19.1288	0.605511
$999$	45.7455	1.44732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.f.1.8	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.f.1.8	✓	28	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-1.08251 q^{2} -0.806331 q^{3} -0.828177 q^{4} +1.00289 q^{5} +0.872859 q^{6} -2.77298 q^{7} +3.06152 q^{8} -2.34983 q^{9} +O(q^{10})\)	\(q-1.08251 q^{2} -0.806331 q^{3} -0.828177 q^{4} +1.00289 q^{5} +0.872859 q^{6} -2.77298 q^{7} +3.06152 q^{8} -2.34983 q^{9} -1.08564 q^{10} -5.77317 q^{11} +0.667784 q^{12} +1.00000 q^{13} +3.00177 q^{14} -0.808663 q^{15} -1.65777 q^{16} +4.28749 q^{17} +2.54371 q^{18} -4.41238 q^{19} -0.830573 q^{20} +2.23594 q^{21} +6.24950 q^{22} -0.981744 q^{23} -2.46860 q^{24} -3.99421 q^{25} -1.08251 q^{26} +4.31373 q^{27} +2.29652 q^{28} -6.36142 q^{29} +0.875385 q^{30} +5.65263 q^{31} -4.32850 q^{32} +4.65508 q^{33} -4.64124 q^{34} -2.78100 q^{35} +1.94608 q^{36} +10.6046 q^{37} +4.77643 q^{38} -0.806331 q^{39} +3.07038 q^{40} -4.28114 q^{41} -2.42042 q^{42} -3.11446 q^{43} +4.78120 q^{44} -2.35663 q^{45} +1.06275 q^{46} +5.36198 q^{47} +1.33671 q^{48} +0.689415 q^{49} +4.32376 q^{50} -3.45714 q^{51} -0.828177 q^{52} +4.88956 q^{53} -4.66965 q^{54} -5.78987 q^{55} -8.48954 q^{56} +3.55784 q^{57} +6.88629 q^{58} +5.37039 q^{59} +0.669716 q^{60} -4.51504 q^{61} -6.11902 q^{62} +6.51603 q^{63} +8.00117 q^{64} +1.00289 q^{65} -5.03916 q^{66} +4.14539 q^{67} -3.55080 q^{68} +0.791610 q^{69} +3.01046 q^{70} -13.2077 q^{71} -7.19406 q^{72} -0.834149 q^{73} -11.4796 q^{74} +3.22065 q^{75} +3.65423 q^{76} +16.0089 q^{77} +0.872859 q^{78} +3.55519 q^{79} -1.66257 q^{80} +3.57120 q^{81} +4.63437 q^{82} -8.09224 q^{83} -1.85175 q^{84} +4.29990 q^{85} +3.37143 q^{86} +5.12941 q^{87} -17.6747 q^{88} +11.0000 q^{89} +2.55107 q^{90} -2.77298 q^{91} +0.813057 q^{92} -4.55789 q^{93} -5.80439 q^{94} -4.42514 q^{95} +3.49020 q^{96} +14.3314 q^{97} -0.746297 q^{98} +13.5660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * q^99

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