LMFDB - Embedded newform 1339.2.a.e.1.4

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:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-1.94515 q^{2} +1.90358 q^{3} +1.78363 q^{4} +1.71445 q^{5} -3.70276 q^{6} -2.09421 q^{7} +0.420881 q^{8} +0.623614 q^{9} +O(q^{10})$	\(q-1.94515 q^{2} +1.90358 q^{3} +1.78363 q^{4} +1.71445 q^{5} -3.70276 q^{6} -2.09421 q^{7} +0.420881 q^{8} +0.623614 q^{9} -3.33487 q^{10} -2.38074 q^{11} +3.39527 q^{12} -1.00000 q^{13} +4.07355 q^{14} +3.26359 q^{15} -4.38593 q^{16} -2.16796 q^{17} -1.21302 q^{18} +0.710140 q^{19} +3.05793 q^{20} -3.98649 q^{21} +4.63090 q^{22} -4.24062 q^{23} +0.801181 q^{24} -2.06067 q^{25} +1.94515 q^{26} -4.52364 q^{27} -3.73528 q^{28} +3.29824 q^{29} -6.34818 q^{30} -2.18797 q^{31} +7.68955 q^{32} -4.53192 q^{33} +4.21701 q^{34} -3.59040 q^{35} +1.11229 q^{36} -1.50046 q^{37} -1.38133 q^{38} -1.90358 q^{39} +0.721579 q^{40} -9.82877 q^{41} +7.75433 q^{42} -8.95903 q^{43} -4.24634 q^{44} +1.06915 q^{45} +8.24865 q^{46} +11.6427 q^{47} -8.34897 q^{48} -2.61430 q^{49} +4.00832 q^{50} -4.12688 q^{51} -1.78363 q^{52} -8.91011 q^{53} +8.79918 q^{54} -4.08165 q^{55} -0.881412 q^{56} +1.35181 q^{57} -6.41559 q^{58} -2.57861 q^{59} +5.82102 q^{60} +4.72930 q^{61} +4.25593 q^{62} -1.30598 q^{63} -6.18550 q^{64} -1.71445 q^{65} +8.81528 q^{66} +9.85342 q^{67} -3.86682 q^{68} -8.07235 q^{69} +6.98389 q^{70} -3.87084 q^{71} +0.262467 q^{72} +9.19408 q^{73} +2.91862 q^{74} -3.92265 q^{75} +1.26662 q^{76} +4.98575 q^{77} +3.70276 q^{78} +2.85248 q^{79} -7.51945 q^{80} -10.4819 q^{81} +19.1185 q^{82} +9.11960 q^{83} -7.11040 q^{84} -3.71685 q^{85} +17.4267 q^{86} +6.27846 q^{87} -1.00201 q^{88} -9.71757 q^{89} -2.07967 q^{90} +2.09421 q^{91} -7.56367 q^{92} -4.16497 q^{93} -22.6469 q^{94} +1.21750 q^{95} +14.6377 q^{96} +5.08808 q^{97} +5.08523 q^{98} -1.48466 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * 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 $

$2$	−1.94515	−1.37543	−0.687716	−	0.725980i	$-0.741387\pi$
$2$	−1.94515	−1.37543	−0.687716	+	0.725980i	$0.741387\pi$
$3$	1.90358	1.09903	0.549516	−	0.835483i	$-0.314812\pi$
$3$	1.90358	1.09903	0.549516	+	0.835483i	$0.314812\pi$
$4$	1.78363	0.891813
$5$	1.71445	0.766724	0.383362	−	0.923598i	$-0.374766\pi$
$5$	1.71445	0.766724	0.383362	+	0.923598i	$0.374766\pi$
$6$	−3.70276	−1.51164
$7$	−2.09421	−0.791535	−0.395768	−	0.918351i	$-0.629521\pi$
$7$	−2.09421	−0.791535	−0.395768	+	0.918351i	$0.629521\pi$
$8$	0.420881	0.148804
$9$	0.623614	0.207871
$10$	−3.33487	−1.05458
$11$	−2.38074	−0.717819	−0.358909	−	0.933372i	$-0.616851\pi$
$11$	−2.38074	−0.717819	−0.358909	+	0.933372i	$0.616851\pi$
$12$	3.39527	0.980131
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.07355	1.08870
$15$	3.26359	0.842654
$16$	−4.38593	−1.09648
$17$	−2.16796	−0.525807	−0.262903	−	0.964822i	$-0.584680\pi$
$17$	−2.16796	−0.525807	−0.262903	+	0.964822i	$0.584680\pi$
$18$	−1.21302	−0.285913
$19$	0.710140	0.162917	0.0814587	−	0.996677i	$-0.474042\pi$
$19$	0.710140	0.162917	0.0814587	+	0.996677i	$0.474042\pi$
$20$	3.05793	0.683775
$21$	−3.98649	−0.869922
$22$	4.63090	0.987311
$23$	−4.24062	−0.884230	−0.442115	−	0.896958i	$-0.645772\pi$
$23$	−4.24062	−0.884230	−0.442115	+	0.896958i	$0.645772\pi$
$24$	0.801181	0.163540
$25$	−2.06067	−0.412134
$26$	1.94515	0.381476
$27$	−4.52364	−0.870575
$28$	−3.73528	−0.705901
$29$	3.29824	0.612468	0.306234	−	0.951956i	$-0.400931\pi$
$29$	3.29824	0.612468	0.306234	+	0.951956i	$0.400931\pi$
$30$	−6.34818	−1.15901
$31$	−2.18797	−0.392971	−0.196485	−	0.980507i	$-0.562953\pi$
$31$	−2.18797	−0.392971	−0.196485	+	0.980507i	$0.562953\pi$
$32$	7.68955	1.35933
$33$	−4.53192	−0.788906
$34$	4.21701	0.723211
$35$	−3.59040	−0.606889
$36$	1.11229	0.185382
$37$	−1.50046	−0.246674	−0.123337	−	0.992365i	$-0.539360\pi$
$37$	−1.50046	−0.246674	−0.123337	+	0.992365i	$0.539360\pi$
$38$	−1.38133	−0.224082
$39$	−1.90358	−0.304817
$40$	0.721579	0.114092
$41$	−9.82877	−1.53500	−0.767498	−	0.641051i	$-0.778499\pi$
$41$	−9.82877	−1.53500	−0.767498	+	0.641051i	$0.778499\pi$
$42$	7.75433	1.19652
$43$	−8.95903	−1.36624	−0.683120	−	0.730306i	$-0.739377\pi$
$43$	−8.95903	−1.36624	−0.683120	+	0.730306i	$0.739377\pi$
$44$	−4.24634	−0.640160
$45$	1.06915	0.159380
$46$	8.24865	1.21620
$47$	11.6427	1.69827	0.849135	−	0.528176i	$-0.177124\pi$
$47$	11.6427	1.69827	0.849135	+	0.528176i	$0.177124\pi$
$48$	−8.34897	−1.20507
$49$	−2.61430	−0.373472
$50$	4.00832	0.566862
$51$	−4.12688	−0.577878
$52$	−1.78363	−0.247344
$53$	−8.91011	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$53$	−8.91011	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$54$	8.79918	1.19742
$55$	−4.08165	−0.550369
$56$	−0.881412	−0.117784
$57$	1.35181	0.179051
$58$	−6.41559	−0.842408
$59$	−2.57861	−0.335707	−0.167854	−	0.985812i	$-0.553684\pi$
$59$	−2.57861	−0.335707	−0.167854	+	0.985812i	$0.553684\pi$
$60$	5.82102	0.751490
$61$	4.72930	0.605525	0.302762	−	0.953066i	$-0.402091\pi$
$61$	4.72930	0.605525	0.302762	+	0.953066i	$0.402091\pi$
$62$	4.25593	0.540504
$63$	−1.30598	−0.164537
$64$	−6.18550	−0.773188
$65$	−1.71445	−0.212651
$66$	8.81528	1.08509
$67$	9.85342	1.20379	0.601894	−	0.798576i	$-0.294413\pi$
$67$	9.85342	1.20379	0.601894	+	0.798576i	$0.294413\pi$
$68$	−3.86682	−0.468921
$69$	−8.07235	−0.971797
$70$	6.98389	0.834735
$71$	−3.87084	−0.459384	−0.229692	−	0.973263i	$-0.573772\pi$
$71$	−3.87084	−0.459384	−0.229692	+	0.973263i	$0.573772\pi$
$72$	0.262467	0.0309321
$73$	9.19408	1.07609	0.538043	−	0.842918i	$-0.319164\pi$
$73$	9.19408	1.07609	0.538043	+	0.842918i	$0.319164\pi$
$74$	2.91862	0.339283
$75$	−3.92265	−0.452948
$76$	1.26662	0.145292
$77$	4.98575	0.568179
$78$	3.70276	0.419254
$79$	2.85248	0.320929	0.160464	−	0.987042i	$-0.448701\pi$
$79$	2.85248	0.320929	0.160464	+	0.987042i	$0.448701\pi$
$80$	−7.51945	−0.840700
$81$	−10.4819	−1.16466
$82$	19.1185	2.11128
$83$	9.11960	1.00101	0.500503	−	0.865735i	$-0.333148\pi$
$83$	9.11960	1.00101	0.500503	+	0.865735i	$0.333148\pi$
$84$	−7.11040	−0.775808
$85$	−3.71685	−0.403149
$86$	17.4267	1.87917
$87$	6.27846	0.673122
$88$	−1.00201	−0.106814
$89$	−9.71757	−1.03006	−0.515030	−	0.857172i	$-0.672219\pi$
$89$	−9.71757	−1.03006	−0.515030	+	0.857172i	$0.672219\pi$
$90$	−2.07967	−0.219216
$91$	2.09421	0.219532
$92$	−7.56367	−0.788567
$93$	−4.16497	−0.431887
$94$	−22.6469	−2.33585
$95$	1.21750	0.124913
$96$	14.6377	1.49395
$97$	5.08808	0.516616	0.258308	−	0.966063i	$-0.416835\pi$
$97$	5.08808	0.516616	0.258308	+	0.966063i	$0.416835\pi$
$98$	5.08523	0.513685
$99$	−1.48466	−0.149214
$100$	−3.67546	−0.367546
$101$	−0.0403308	−0.00401306	−0.00200653	−	0.999998i	$-0.500639\pi$
$101$	−0.0403308	−0.00401306	−0.00200653	+	0.999998i	$0.500639\pi$
$102$	8.02741	0.794832
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−0.420881	−0.0412708
$105$	−6.83462	−0.666991
$106$	17.3315	1.68339
$107$	−13.5582	−1.31072	−0.655361	−	0.755316i	$-0.727484\pi$
$107$	−13.5582	−1.31072	−0.655361	+	0.755316i	$0.727484\pi$
$108$	−8.06848	−0.776390
$109$	12.7877	1.22484	0.612419	−	0.790533i	$-0.290196\pi$
$109$	12.7877	1.22484	0.612419	+	0.790533i	$0.290196\pi$
$110$	7.93943	0.756995
$111$	−2.85624	−0.271103
$112$	9.18504	0.867905
$113$	6.47579	0.609191	0.304596	−	0.952482i	$-0.401479\pi$
$113$	6.47579	0.609191	0.304596	+	0.952482i	$0.401479\pi$
$114$	−2.62948	−0.246273
$115$	−7.27031	−0.677960
$116$	5.88283	0.546207
$117$	−0.623614	−0.0576531
$118$	5.01580	0.461742
$119$	4.54015	0.416194
$120$	1.37358	0.125390
$121$	−5.33210	−0.484736
$122$	−9.19922	−0.832858
$123$	−18.7098	−1.68701
$124$	−3.90251	−0.350456
$125$	−12.1051	−1.08272
$126$	2.54032	0.226310
$127$	−1.40876	−0.125008	−0.0625038	−	0.998045i	$-0.519909\pi$
$127$	−1.40876	−0.125008	−0.0625038	+	0.998045i	$0.519909\pi$
$128$	−3.34735	−0.295866
$129$	−17.0542	−1.50154
$130$	3.33487	0.292487
$131$	12.2443	1.06979	0.534894	−	0.844919i	$-0.320352\pi$
$131$	12.2443	1.06979	0.534894	+	0.844919i	$0.320352\pi$
$132$	−8.08325	−0.703556
$133$	−1.48718	−0.128955
$134$	−19.1664	−1.65573
$135$	−7.75554	−0.667491
$136$	−0.912452	−0.0782421
$137$	−19.3631	−1.65431	−0.827153	−	0.561977i	$-0.810041\pi$
$137$	−19.3631	−1.65431	−0.827153	+	0.561977i	$0.810041\pi$
$138$	15.7020	1.33664
$139$	12.0814	1.02473	0.512364	−	0.858769i	$-0.328770\pi$
$139$	12.0814	1.02473	0.512364	+	0.858769i	$0.328770\pi$
$140$	−6.40394	−0.541232
$141$	22.1629	1.86645
$142$	7.52937	0.631851
$143$	2.38074	0.199087
$144$	−2.73513	−0.227927
$145$	5.65466	0.469594
$146$	−17.8839	−1.48008
$147$	−4.97654	−0.410458
$148$	−2.67626	−0.219987
$149$	−2.22385	−0.182185	−0.0910926	−	0.995842i	$-0.529036\pi$
$149$	−2.22385	−0.182185	−0.0910926	+	0.995842i	$0.529036\pi$
$150$	7.63016	0.623000
$151$	2.95783	0.240705	0.120352	−	0.992731i	$-0.461598\pi$
$151$	2.95783	0.240705	0.120352	+	0.992731i	$0.461598\pi$
$152$	0.298885	0.0242428
$153$	−1.35197	−0.109300
$154$	−9.69805	−0.781491
$155$	−3.75116	−0.301300
$156$	−3.39527	−0.271839
$157$	−6.32588	−0.504860	−0.252430	−	0.967615i	$-0.581230\pi$
$157$	−6.32588	−0.504860	−0.252430	+	0.967615i	$0.581230\pi$
$158$	−5.54851	−0.441416
$159$	−16.9611	−1.34510
$160$	13.1833	1.04223
$161$	8.88072	0.699899
$162$	20.3890	1.60191
$163$	−15.3559	−1.20277	−0.601385	−	0.798960i	$-0.705384\pi$
$163$	−15.3559	−1.20277	−0.601385	+	0.798960i	$0.705384\pi$
$164$	−17.5309	−1.36893
$165$	−7.76974	−0.604873
$166$	−17.7390	−1.37682
$167$	−7.98733	−0.618078	−0.309039	−	0.951049i	$-0.600007\pi$
$167$	−7.98733	−0.618078	−0.309039	+	0.951049i	$0.600007\pi$
$168$	−1.67784	−0.129448
$169$	1.00000	0.0769231
$170$	7.22984	0.554504
$171$	0.442853	0.0338658
$172$	−15.9796	−1.21843
$173$	−17.5595	−1.33502	−0.667512	−	0.744599i	$-0.732641\pi$
$173$	−17.5595	−1.33502	−0.667512	+	0.744599i	$0.732641\pi$
$174$	−12.2126	−0.925833
$175$	4.31547	0.326219
$176$	10.4417	0.787076
$177$	−4.90860	−0.368953
$178$	18.9022	1.41678
$179$	−3.41067	−0.254926	−0.127463	−	0.991843i	$-0.540683\pi$
$179$	−3.41067	−0.254926	−0.127463	+	0.991843i	$0.540683\pi$
$180$	1.90697	0.142137
$181$	−6.42600	−0.477641	−0.238820	−	0.971064i	$-0.576761\pi$
$181$	−6.42600	−0.477641	−0.238820	+	0.971064i	$0.576761\pi$
$182$	−4.07355	−0.301952
$183$	9.00259	0.665491
$184$	−1.78480	−0.131577
$185$	−2.57246	−0.189131
$186$	8.10151	0.594031
$187$	5.16133	0.377434
$188$	20.7663	1.51454
$189$	9.47343	0.689091
$190$	−2.36822	−0.171809
$191$	0.736772	0.0533109	0.0266555	−	0.999645i	$-0.491514\pi$
$191$	0.736772	0.0533109	0.0266555	+	0.999645i	$0.491514\pi$
$192$	−11.7746	−0.849758
$193$	−1.22704	−0.0883246	−0.0441623	−	0.999024i	$-0.514062\pi$
$193$	−1.22704	−0.0883246	−0.0441623	+	0.999024i	$0.514062\pi$
$194$	−9.89710	−0.710571
$195$	−3.26359	−0.233710
$196$	−4.66294	−0.333067
$197$	−14.7117	−1.04817	−0.524083	−	0.851667i	$-0.675592\pi$
$197$	−14.7117	−1.04817	−0.524083	+	0.851667i	$0.675592\pi$
$198$	2.88789	0.205234
$199$	−14.4674	−1.02557	−0.512785	−	0.858517i	$-0.671386\pi$
$199$	−14.4674	−1.02557	−0.512785	+	0.858517i	$0.671386\pi$
$200$	−0.867297	−0.0613272
$201$	18.7568	1.32300
$202$	0.0784496	0.00551969
$203$	−6.90719	−0.484790
$204$	−7.36080	−0.515359
$205$	−16.8509	−1.17692
$206$	1.94515	0.135525
$207$	−2.64451	−0.183806
$208$	4.38593	0.304110
$209$	−1.69066	−0.116945
$210$	13.2944	0.917400
$211$	6.70081	0.461303	0.230651	−	0.973036i	$-0.425914\pi$
$211$	6.70081	0.461303	0.230651	+	0.973036i	$0.425914\pi$
$212$	−15.8923	−1.09149
$213$	−7.36844	−0.504877
$214$	26.3728	1.80281
$215$	−15.3598	−1.04753
$216$	−1.90392	−0.129545
$217$	4.58205	0.311050
$218$	−24.8740	−1.68468
$219$	17.5017	1.18265
$220$	−7.28013	−0.490826
$221$	2.16796	0.145833
$222$	5.55583	0.372883
$223$	26.3101	1.76185	0.880927	−	0.473253i	$-0.156920\pi$
$223$	26.3101	1.76185	0.880927	+	0.473253i	$0.156920\pi$
$224$	−16.1035	−1.07596
$225$	−1.28506	−0.0856708
$226$	−12.5964	−0.837901
$227$	23.3318	1.54858	0.774292	−	0.632829i	$-0.218106\pi$
$227$	23.3318	1.54858	0.774292	+	0.632829i	$0.218106\pi$
$228$	2.41112	0.159680
$229$	−3.67320	−0.242732	−0.121366	−	0.992608i	$-0.538727\pi$
$229$	−3.67320	−0.242732	−0.121366	+	0.992608i	$0.538727\pi$
$230$	14.1419	0.932488
$231$	9.49077	0.624447
$232$	1.38817	0.0911377
$233$	−3.27861	−0.214789	−0.107394	−	0.994217i	$-0.534251\pi$
$233$	−3.27861	−0.214789	−0.107394	+	0.994217i	$0.534251\pi$
$234$	1.21302	0.0792979
$235$	19.9609	1.30210
$236$	−4.59928	−0.299388
$237$	5.42992	0.352711
$238$	−8.83128	−0.572447
$239$	−20.5107	−1.32673	−0.663364	−	0.748297i	$-0.730872\pi$
$239$	−20.5107	−1.32673	−0.663364	+	0.748297i	$0.730872\pi$
$240$	−14.3139	−0.923956
$241$	4.54622	0.292848	0.146424	−	0.989222i	$-0.453224\pi$
$241$	4.54622	0.292848	0.146424	+	0.989222i	$0.453224\pi$
$242$	10.3717	0.666721
$243$	−6.38230	−0.409425
$244$	8.43530	0.540015
$245$	−4.48209	−0.286350
$246$	36.3935	2.32037
$247$	−0.710140	−0.0451851
$248$	−0.920874	−0.0584756
$249$	17.3599	1.10014
$250$	23.5464	1.48920
$251$	3.93554	0.248409	0.124205	−	0.992257i	$-0.460362\pi$
$251$	3.93554	0.248409	0.124205	+	0.992257i	$0.460362\pi$
$252$	−2.32937	−0.146737
$253$	10.0958	0.634717
$254$	2.74026	0.171939
$255$	−7.07531	−0.443073
$256$	18.8821	1.18013
$257$	−6.90377	−0.430646	−0.215323	−	0.976543i	$-0.569080\pi$
$257$	−6.90377	−0.430646	−0.215323	+	0.976543i	$0.569080\pi$
$258$	33.1731	2.06527
$259$	3.14227	0.195251
$260$	−3.05793	−0.189645
$261$	2.05683	0.127314
$262$	−23.8170	−1.47142
$263$	−6.39842	−0.394544	−0.197272	−	0.980349i	$-0.563208\pi$
$263$	−6.39842	−0.394544	−0.197272	+	0.980349i	$0.563208\pi$
$264$	−1.90740	−0.117392
$265$	−15.2759	−0.938392
$266$	2.89279	0.177369
$267$	−18.4982	−1.13207
$268$	17.5748	1.07355
$269$	−15.0089	−0.915108	−0.457554	−	0.889182i	$-0.651274\pi$
$269$	−15.0089	−0.915108	−0.457554	+	0.889182i	$0.651274\pi$
$270$	15.0857	0.918088
$271$	14.2399	0.865014	0.432507	−	0.901631i	$-0.357629\pi$
$271$	14.2399	0.865014	0.432507	+	0.901631i	$0.357629\pi$
$272$	9.50851	0.576538
$273$	3.98649	0.241273
$274$	37.6643	2.27538
$275$	4.90591	0.295838
$276$	−14.3980	−0.866661
$277$	9.98002	0.599641	0.299821	−	0.953996i	$-0.403073\pi$
$277$	9.98002	0.599641	0.299821	+	0.953996i	$0.403073\pi$
$278$	−23.5001	−1.40944
$279$	−1.36445	−0.0816873
$280$	−1.51113	−0.0903075
$281$	11.8198	0.705109	0.352555	−	0.935791i	$-0.385313\pi$
$281$	11.8198	0.705109	0.352555	+	0.935791i	$0.385313\pi$
$282$	−43.1102	−2.56718
$283$	11.5939	0.689187	0.344594	−	0.938752i	$-0.388017\pi$
$283$	11.5939	0.689187	0.344594	+	0.938752i	$0.388017\pi$
$284$	−6.90412	−0.409684
$285$	2.31760	0.137283
$286$	−4.63090	−0.273831
$287$	20.5835	1.21500
$288$	4.79531	0.282566
$289$	−12.3000	−0.723527
$290$	−10.9992	−0.645894
$291$	9.68557	0.567778
$292$	16.3988	0.959667
$293$	−28.6178	−1.67187	−0.835934	−	0.548830i	$-0.815073\pi$
$293$	−28.6178	−1.67187	−0.835934	+	0.548830i	$0.815073\pi$
$294$	9.68013	0.564557
$295$	−4.42090	−0.257395
$296$	−0.631515	−0.0367061
$297$	10.7696	0.624915
$298$	4.32574	0.250583
$299$	4.24062	0.245241
$300$	−6.99654	−0.403945
$301$	18.7621	1.08143
$302$	−5.75344	−0.331073
$303$	−0.0767728	−0.00441048
$304$	−3.11463	−0.178636
$305$	8.10813	0.464270
$306$	2.62979	0.150335
$307$	4.73589	0.270291	0.135146	−	0.990826i	$-0.456850\pi$
$307$	4.73589	0.270291	0.135146	+	0.990826i	$0.456850\pi$
$308$	8.89271	0.506709
$309$	−1.90358	−0.108291
$310$	7.29658	0.414418
$311$	−22.2532	−1.26186	−0.630932	−	0.775838i	$-0.717327\pi$
$311$	−22.2532	−1.26186	−0.630932	+	0.775838i	$0.717327\pi$
$312$	−0.801181	−0.0453579
$313$	3.49080	0.197312	0.0986559	−	0.995122i	$-0.468546\pi$
$313$	3.49080	0.197312	0.0986559	+	0.995122i	$0.468546\pi$
$314$	12.3048	0.694401
$315$	−2.23903	−0.126155
$316$	5.08775	0.286208
$317$	34.0307	1.91136	0.955679	−	0.294412i	$-0.0951238\pi$
$317$	34.0307	1.91136	0.955679	+	0.294412i	$0.0951238\pi$
$318$	32.9920	1.85010
$319$	−7.85224	−0.439641
$320$	−10.6047	−0.592822
$321$	−25.8091	−1.44053
$322$	−17.2744	−0.962663
$323$	−1.53955	−0.0856630
$324$	−18.6959	−1.03866
$325$	2.06067	0.114305
$326$	29.8696	1.65433
$327$	24.3424	1.34614
$328$	−4.13675	−0.228414
$329$	−24.3823	−1.34424
$330$	15.1133	0.831962
$331$	9.25041	0.508448	0.254224	−	0.967145i	$-0.418180\pi$
$331$	9.25041	0.508448	0.254224	+	0.967145i	$0.418180\pi$
$332$	16.2660	0.892710
$333$	−0.935707	−0.0512764
$334$	15.5366	0.850124
$335$	16.8932	0.922973
$336$	17.4844	0.953855
$337$	22.7954	1.24174	0.620872	−	0.783912i	$-0.286779\pi$
$337$	22.7954	1.24174	0.620872	+	0.783912i	$0.286779\pi$
$338$	−1.94515	−0.105802
$339$	12.3272	0.669521
$340$	−6.62947	−0.359533
$341$	5.20897	0.282082
$342$	−0.861418	−0.0465801
$343$	20.1343	1.08715
$344$	−3.77069	−0.203302
$345$	−13.8396	−0.745100
$346$	34.1559	1.83623
$347$	9.06260	0.486506	0.243253	−	0.969963i	$-0.421785\pi$
$347$	9.06260	0.486506	0.243253	+	0.969963i	$0.421785\pi$
$348$	11.1984	0.600299
$349$	26.0030	1.39191	0.695953	−	0.718087i	$-0.254982\pi$
$349$	26.0030	1.39191	0.695953	+	0.718087i	$0.254982\pi$
$350$	−8.39425	−0.448691
$351$	4.52364	0.241454
$352$	−18.3068	−0.975755
$353$	36.2179	1.92769	0.963843	−	0.266470i	$-0.0858572\pi$
$353$	36.2179	1.92769	0.963843	+	0.266470i	$0.0858572\pi$
$354$	9.54798	0.507469
$355$	−6.63635	−0.352221
$356$	−17.3325	−0.918621
$357$	8.64253	0.457411
$358$	6.63428	0.350633
$359$	18.7940	0.991909	0.495954	−	0.868349i	$-0.334818\pi$
$359$	18.7940	0.991909	0.495954	+	0.868349i	$0.334818\pi$
$360$	0.449986	0.0237164
$361$	−18.4957	−0.973458
$362$	12.4996	0.656962
$363$	−10.1501	−0.532740
$364$	3.73528	0.195782
$365$	15.7628	0.825061
$366$	−17.5114	−0.915337
$367$	19.9574	1.04177	0.520883	−	0.853628i	$-0.325603\pi$
$367$	19.9574	1.04177	0.520883	+	0.853628i	$0.325603\pi$
$368$	18.5990	0.969542
$369$	−6.12936	−0.319082
$370$	5.00383	0.260137
$371$	18.6596	0.968758
$372$	−7.42875	−0.385163
$373$	20.5545	1.06427	0.532135	−	0.846660i	$-0.321390\pi$
$373$	20.5545	1.06427	0.532135	+	0.846660i	$0.321390\pi$
$374$	−10.0396	−0.519135
$375$	−23.0431	−1.18994
$376$	4.90021	0.252709
$377$	−3.29824	−0.169868
$378$	−18.4273	−0.947797
$379$	6.70937	0.344637	0.172319	−	0.985041i	$-0.444874\pi$
$379$	6.70937	0.344637	0.172319	+	0.985041i	$0.444874\pi$
$380$	2.17156	0.111399
$381$	−2.68169	−0.137387
$382$	−1.43313	−0.0733255
$383$	8.13985	0.415927	0.207963	−	0.978137i	$-0.433316\pi$
$383$	8.13985	0.415927	0.207963	+	0.978137i	$0.433316\pi$
$384$	−6.37194	−0.325167
$385$	8.54781	0.435637
$386$	2.38679	0.121484
$387$	−5.58697	−0.284002
$388$	9.07523	0.460725
$389$	−31.4794	−1.59607	−0.798035	−	0.602612i	$-0.794127\pi$
$389$	−31.4794	−1.59607	−0.798035	+	0.602612i	$0.794127\pi$
$390$	6.34818	0.321453
$391$	9.19347	0.464934
$392$	−1.10031	−0.0555741
$393$	23.3079	1.17573
$394$	28.6165	1.44168
$395$	4.89042	0.246064
$396$	−2.64808	−0.133071
$397$	23.3034	1.16956	0.584781	−	0.811191i	$-0.301180\pi$
$397$	23.3034	1.16956	0.584781	+	0.811191i	$0.301180\pi$
$398$	28.1414	1.41060
$399$	−2.83096	−0.141725
$400$	9.03795	0.451898
$401$	3.01529	0.150576	0.0752881	−	0.997162i	$-0.476012\pi$
$401$	3.01529	0.150576	0.0752881	+	0.997162i	$0.476012\pi$
$402$	−36.4848	−1.81970
$403$	2.18797	0.108990
$404$	−0.0719350	−0.00357890
$405$	−17.9707	−0.892974
$406$	13.4356	0.666795
$407$	3.57220	0.177067
$408$	−1.73693	−0.0859906
$409$	−27.1689	−1.34342	−0.671709	−	0.740815i	$-0.734439\pi$
$409$	−27.1689	−1.34342	−0.671709	+	0.740815i	$0.734439\pi$
$410$	32.7776	1.61877
$411$	−36.8593	−1.81813
$412$	−1.78363	−0.0878729
$413$	5.40015	0.265724
$414$	5.14397	0.252812
$415$	15.6351	0.767496
$416$	−7.68955	−0.377011
$417$	22.9978	1.12621
$418$	3.28859	0.160850
$419$	−12.6591	−0.618436	−0.309218	−	0.950991i	$-0.600067\pi$
$419$	−12.6591	−0.618436	−0.309218	+	0.950991i	$0.600067\pi$
$420$	−12.1904	−0.594831
$421$	−14.4127	−0.702431	−0.351215	−	0.936295i	$-0.614231\pi$
$421$	−14.4127	−0.702431	−0.351215	+	0.936295i	$0.614231\pi$
$422$	−13.0341	−0.634491
$423$	7.26058	0.353021
$424$	−3.75010	−0.182121
$425$	4.46744	0.216703
$426$	14.3328	0.694425
$427$	−9.90412	−0.479294
$428$	−24.1828	−1.16892
$429$	4.53192	0.218803
$430$	29.8772	1.44080
$431$	−35.3020	−1.70044	−0.850219	−	0.526429i	$-0.823531\pi$
$431$	−35.3020	−1.70044	−0.850219	+	0.526429i	$0.823531\pi$
$432$	19.8404	0.954570
$433$	−19.7247	−0.947909	−0.473955	−	0.880549i	$-0.657174\pi$
$433$	−19.7247	−0.947909	−0.473955	+	0.880549i	$0.657174\pi$
$434$	−8.91280	−0.427828
$435$	10.7641	0.516099
$436$	22.8085	1.09233
$437$	−3.01143	−0.144056
$438$	−34.0434	−1.62666
$439$	12.0424	0.574751	0.287375	−	0.957818i	$-0.407217\pi$
$439$	12.0424	0.574751	0.287375	+	0.957818i	$0.407217\pi$
$440$	−1.71789	−0.0818971
$441$	−1.63032	−0.0776341
$442$	−4.21701	−0.200583
$443$	1.43019	0.0679502	0.0339751	−	0.999423i	$-0.489183\pi$
$443$	1.43019	0.0679502	0.0339751	+	0.999423i	$0.489183\pi$
$444$	−5.09447	−0.241773
$445$	−16.6603	−0.789772
$446$	−51.1772	−2.42331
$447$	−4.23328	−0.200227
$448$	12.9537	0.612005
$449$	−25.3867	−1.19807	−0.599037	−	0.800721i	$-0.704450\pi$
$449$	−25.3867	−1.19807	−0.599037	+	0.800721i	$0.704450\pi$
$450$	2.49964	0.117834
$451$	23.3997	1.10185
$452$	11.5504	0.543285
$453$	5.63046	0.264542
$454$	−45.3839	−2.12997
$455$	3.59040	0.168321
$456$	0.568951	0.0266436
$457$	28.6821	1.34169	0.670845	−	0.741597i	$-0.265931\pi$
$457$	28.6821	1.34169	0.670845	+	0.741597i	$0.265931\pi$
$458$	7.14494	0.333861
$459$	9.80705	0.457754
$460$	−12.9675	−0.604614
$461$	22.4681	1.04644	0.523221	−	0.852197i	$-0.324730\pi$
$461$	22.4681	1.04644	0.523221	+	0.852197i	$0.324730\pi$
$462$	−18.4610	−0.858884
$463$	−11.3585	−0.527873	−0.263936	−	0.964540i	$-0.585021\pi$
$463$	−11.3585	−0.527873	−0.263936	+	0.964540i	$0.585021\pi$
$464$	−14.4659	−0.671560
$465$	−7.14062	−0.331138
$466$	6.37739	0.295427
$467$	30.2587	1.40020	0.700102	−	0.714043i	$-0.253138\pi$
$467$	30.2587	1.40020	0.700102	+	0.714043i	$0.253138\pi$
$468$	−1.11229	−0.0514158
$469$	−20.6351	−0.952840
$470$	−38.8270	−1.79096
$471$	−12.0418	−0.554857
$472$	−1.08529	−0.0499546
$473$	21.3291	0.980713
$474$	−10.5620	−0.485130
$475$	−1.46336	−0.0671438
$476$	8.09792	0.371168
$477$	−5.55647	−0.254413
$478$	39.8965	1.82482
$479$	−30.7846	−1.40658	−0.703292	−	0.710901i	$-0.748287\pi$
$479$	−30.7846	−1.40658	−0.703292	+	0.710901i	$0.748287\pi$
$480$	25.0955	1.14545
$481$	1.50046	0.0684151
$482$	−8.84310	−0.402792
$483$	16.9052	0.769211
$484$	−9.51046	−0.432294
$485$	8.72325	0.396102
$486$	12.4146	0.563136
$487$	−39.5723	−1.79319	−0.896595	−	0.442851i	$-0.853967\pi$
$487$	−39.5723	−1.79319	−0.896595	+	0.442851i	$0.853967\pi$
$488$	1.99047	0.0901045
$489$	−29.2312	−1.32188
$490$	8.71835	0.393855
$491$	−5.27552	−0.238081	−0.119040	−	0.992889i	$-0.537982\pi$
$491$	−5.27552	−0.238081	−0.119040	+	0.992889i	$0.537982\pi$
$492$	−33.3714	−1.50450
$493$	−7.15044	−0.322040
$494$	1.38133	0.0621491
$495$	−2.54537	−0.114406
$496$	9.59627	0.430885
$497$	8.10633	0.363618
$498$	−33.7677	−1.51316
$499$	−7.03515	−0.314936	−0.157468	−	0.987524i	$-0.550333\pi$
$499$	−7.03515	−0.314936	−0.157468	+	0.987524i	$0.550333\pi$
$500$	−21.5911	−0.965581
$501$	−15.2045	−0.679288
$502$	−7.65523	−0.341670
$503$	−34.7588	−1.54982	−0.774909	−	0.632073i	$-0.782204\pi$
$503$	−34.7588	−1.54982	−0.774909	+	0.632073i	$0.782204\pi$
$504$	−0.549660	−0.0244838
$505$	−0.0691450	−0.00307691
$506$	−19.6379	−0.873010
$507$	1.90358	0.0845409
$508$	−2.51271	−0.111483
$509$	14.4338	0.639766	0.319883	−	0.947457i	$-0.396356\pi$
$509$	14.4338	0.639766	0.319883	+	0.947457i	$0.396356\pi$
$510$	13.7626	0.609417
$511$	−19.2543	−0.851759
$512$	−30.0339	−1.32732
$513$	−3.21242	−0.141832
$514$	13.4289	0.592324
$515$	−1.71445	−0.0755476
$516$	−30.4184	−1.33909
$517$	−27.7183	−1.21905
$518$	−6.11220	−0.268555
$519$	−33.4259	−1.46723
$520$	−0.721579	−0.0316433
$521$	−14.6542	−0.642013	−0.321006	−	0.947077i	$-0.604021\pi$
$521$	−14.6542	−0.642013	−0.321006	+	0.947077i	$0.604021\pi$
$522$	−4.00085	−0.175112
$523$	7.19571	0.314647	0.157323	−	0.987547i	$-0.449714\pi$
$523$	7.19571	0.314647	0.157323	+	0.987547i	$0.449714\pi$
$524$	21.8392	0.954050
$525$	8.21483	0.358525
$526$	12.4459	0.542668
$527$	4.74342	0.206627
$528$	19.8767	0.865022
$529$	−5.01718	−0.218138
$530$	29.7140	1.29069
$531$	−1.60806	−0.0697838
$532$	−2.65257	−0.115004
$533$	9.82877	0.425731
$534$	35.9818	1.55708
$535$	−23.2449	−1.00496
$536$	4.14712	0.179128
$537$	−6.49248	−0.280171
$538$	29.1946	1.25867
$539$	6.22397	0.268085
$540$	−13.8330	−0.595277
$541$	−37.7713	−1.62392	−0.811958	−	0.583716i	$-0.801598\pi$
$541$	−37.7713	−1.62392	−0.811958	+	0.583716i	$0.801598\pi$
$542$	−27.6989	−1.18977
$543$	−12.2324	−0.524942
$544$	−16.6706	−0.714746
$545$	21.9238	0.939113
$546$	−7.75433	−0.331855
$547$	−6.29415	−0.269118	−0.134559	−	0.990906i	$-0.542962\pi$
$547$	−6.29415	−0.269118	−0.134559	+	0.990906i	$0.542962\pi$
$548$	−34.5366	−1.47533
$549$	2.94926	0.125871
$550$	−9.54275	−0.406904
$551$	2.34221	0.0997816
$552$	−3.39750	−0.144607
$553$	−5.97367	−0.254026
$554$	−19.4127	−0.824765
$555$	−4.89688	−0.207861
$556$	21.5486	0.913865
$557$	−22.4454	−0.951042	−0.475521	−	0.879704i	$-0.657740\pi$
$557$	−22.4454	−0.951042	−0.475521	+	0.879704i	$0.657740\pi$
$558$	2.65406	0.112355
$559$	8.95903	0.378927
$560$	15.7473	0.665443
$561$	9.82500	0.414812
$562$	−22.9913	−0.969830
$563$	5.85149	0.246611	0.123305	−	0.992369i	$-0.460651\pi$
$563$	5.85149	0.246611	0.123305	+	0.992369i	$0.460651\pi$
$564$	39.5303	1.66453
$565$	11.1024	0.467082
$566$	−22.5520	−0.947930
$567$	21.9513	0.921870
$568$	−1.62916	−0.0683581
$569$	34.7983	1.45882	0.729411	−	0.684076i	$-0.239794\pi$
$569$	34.7983	1.45882	0.729411	+	0.684076i	$0.239794\pi$
$570$	−4.50810	−0.188823
$571$	46.4315	1.94310	0.971548	−	0.236842i	$-0.0761122\pi$
$571$	46.4315	1.94310	0.971548	+	0.236842i	$0.0761122\pi$
$572$	4.24634	0.177548
$573$	1.40250	0.0585904
$574$	−40.0380	−1.67115
$575$	8.73851	0.364421
$576$	−3.85736	−0.160723
$577$	−30.9767	−1.28958	−0.644788	−	0.764361i	$-0.723055\pi$
$577$	−30.9767	−1.28958	−0.644788	+	0.764361i	$0.723055\pi$
$578$	23.9253	0.995163
$579$	−2.33578	−0.0970716
$580$	10.0858	0.418790
$581$	−19.0983	−0.792332
$582$	−18.8399	−0.780940
$583$	21.2126	0.878537
$584$	3.86961	0.160126
$585$	−1.06915	−0.0442040
$586$	55.6660	2.29954
$587$	14.0482	0.579833	0.289916	−	0.957052i	$-0.406372\pi$
$587$	14.0482	0.579833	0.289916	+	0.957052i	$0.406372\pi$
$588$	−8.87628	−0.366052
$589$	−1.55376	−0.0640217
$590$	8.59933	0.354029
$591$	−28.0049	−1.15197
$592$	6.58091	0.270474
$593$	31.6839	1.30110	0.650550	−	0.759463i	$-0.274538\pi$
$593$	31.6839	1.30110	0.650550	+	0.759463i	$0.274538\pi$
$594$	−20.9485	−0.859528
$595$	7.78384	0.319106
$596$	−3.96652	−0.162475
$597$	−27.5399	−1.12713
$598$	−8.24865	−0.337312
$599$	3.06571	0.125262	0.0626309	−	0.998037i	$-0.480051\pi$
$599$	3.06571	0.125262	0.0626309	+	0.998037i	$0.480051\pi$
$600$	−1.65097	−0.0674005
$601$	34.4132	1.40375	0.701873	−	0.712302i	$-0.252348\pi$
$601$	34.4132	1.40375	0.701873	+	0.712302i	$0.252348\pi$
$602$	−36.4951	−1.48743
$603$	6.14473	0.250233
$604$	5.27566	0.214664
$605$	−9.14160	−0.371659
$606$	0.149335	0.00606632
$607$	22.2346	0.902476	0.451238	−	0.892404i	$-0.350983\pi$
$607$	22.2346	0.902476	0.451238	+	0.892404i	$0.350983\pi$
$608$	5.46066	0.221459
$609$	−13.1484	−0.532800
$610$	−15.7716	−0.638572
$611$	−11.6427	−0.471015
$612$	−2.41140	−0.0974752
$613$	−23.3579	−0.943416	−0.471708	−	0.881755i	$-0.656362\pi$
$613$	−23.3579	−0.943416	−0.471708	+	0.881755i	$0.656362\pi$
$614$	−9.21204	−0.371767
$615$	−32.0771	−1.29347
$616$	2.09841	0.0845473
$617$	7.47198	0.300810	0.150405	−	0.988624i	$-0.451942\pi$
$617$	7.47198	0.300810	0.150405	+	0.988624i	$0.451942\pi$
$618$	3.70276	0.148947
$619$	−45.2633	−1.81929	−0.909643	−	0.415392i	$-0.863644\pi$
$619$	−45.2633	−1.81929	−0.909643	+	0.415392i	$0.863644\pi$
$620$	−6.69066	−0.268703
$621$	19.1830	0.769788
$622$	43.2859	1.73561
$623$	20.3506	0.815329
$624$	8.34897	0.334226
$625$	−10.4503	−0.418012
$626$	−6.79015	−0.271389
$627$	−3.21830	−0.128526
$628$	−11.2830	−0.450241
$629$	3.25293	0.129703
$630$	4.35525	0.173517
$631$	−13.6603	−0.543806	−0.271903	−	0.962325i	$-0.587653\pi$
$631$	−13.6603	−0.543806	−0.271903	+	0.962325i	$0.587653\pi$
$632$	1.20055	0.0477555
$633$	12.7555	0.506987
$634$	−66.1951	−2.62894
$635$	−2.41525	−0.0958463
$636$	−30.2523	−1.19958
$637$	2.61430	0.103583
$638$	15.2738	0.604696
$639$	−2.41391	−0.0954927
$640$	−5.73885	−0.226848
$641$	38.0429	1.50260	0.751302	−	0.659959i	$-0.229426\pi$
$641$	38.0429	1.50260	0.751302	+	0.659959i	$0.229426\pi$
$642$	50.2028	1.98135
$643$	10.5697	0.416827	0.208413	−	0.978041i	$-0.433170\pi$
$643$	10.5697	0.416827	0.208413	+	0.978041i	$0.433170\pi$
$644$	15.8399	0.624179
$645$	−29.2386	−1.15127
$646$	2.99467	0.117824
$647$	5.19937	0.204408	0.102204	−	0.994763i	$-0.467410\pi$
$647$	5.19937	0.204408	0.102204	+	0.994763i	$0.467410\pi$
$648$	−4.41166	−0.173306
$649$	6.13900	0.240977
$650$	−4.00832	−0.157219
$651$	8.72230	0.341854
$652$	−27.3892	−1.07264
$653$	18.0390	0.705922	0.352961	−	0.935638i	$-0.385175\pi$
$653$	18.0390	0.705922	0.352961	+	0.935638i	$0.385175\pi$
$654$	−47.3497	−1.85152
$655$	20.9922	0.820232
$656$	43.1083	1.68310
$657$	5.73355	0.223687
$658$	47.4273	1.84891
$659$	−2.70773	−0.105478	−0.0527390	−	0.998608i	$-0.516795\pi$
$659$	−2.70773	−0.105478	−0.0527390	+	0.998608i	$0.516795\pi$
$660$	−13.8583	−0.539434
$661$	14.6175	0.568556	0.284278	−	0.958742i	$-0.408246\pi$
$661$	14.6175	0.568556	0.284278	+	0.958742i	$0.408246\pi$
$662$	−17.9935	−0.699336
$663$	4.12688	0.160275
$664$	3.83827	0.148954
$665$	−2.54969	−0.0988728
$666$	1.82009	0.0705272
$667$	−13.9866	−0.541562
$668$	−14.2464	−0.551210
$669$	50.0833	1.93633
$670$	−32.8598	−1.26949
$671$	−11.2592	−0.434657
$672$	−30.6543	−1.18251
$673$	−39.1025	−1.50729	−0.753645	−	0.657281i	$-0.771706\pi$
$673$	−39.1025	−1.50729	−0.753645	+	0.657281i	$0.771706\pi$
$674$	−44.3406	−1.70793
$675$	9.32173	0.358793
$676$	1.78363	0.0686010
$677$	10.7560	0.413386	0.206693	−	0.978406i	$-0.433730\pi$
$677$	10.7560	0.413386	0.206693	+	0.978406i	$0.433730\pi$
$678$	−23.9783	−0.920880
$679$	−10.6555	−0.408920
$680$	−1.56435	−0.0599901
$681$	44.4139	1.70194
$682$	−10.1323	−0.387984
$683$	28.5266	1.09154	0.545770	−	0.837935i	$-0.316237\pi$
$683$	28.5266	1.09154	0.545770	+	0.837935i	$0.316237\pi$
$684$	0.789884	0.0302020
$685$	−33.1971	−1.26840
$686$	−39.1644	−1.49530
$687$	−6.99222	−0.266770
$688$	39.2937	1.49806
$689$	8.91011	0.339448
$690$	26.9202	1.02483
$691$	−38.6379	−1.46985	−0.734927	−	0.678146i	$-0.762784\pi$
$691$	−38.6379	−1.46985	−0.734927	+	0.678146i	$0.762784\pi$
$692$	−31.3196	−1.19059
$693$	3.10918	0.118108
$694$	−17.6282	−0.669156
$695$	20.7129	0.785683
$696$	2.64249	0.100163
$697$	21.3084	0.807111
$698$	−50.5798	−1.91447
$699$	−6.24109	−0.236060
$700$	7.69717	0.290926
$701$	−25.6735	−0.969675	−0.484838	−	0.874604i	$-0.661121\pi$
$701$	−25.6735	−0.969675	−0.484838	+	0.874604i	$0.661121\pi$
$702$	−8.79918	−0.332104
$703$	−1.06554	−0.0401875
$704$	14.7260	0.555009
$705$	37.9971	1.43105
$706$	−70.4495	−2.65140
$707$	0.0844609	0.00317648
$708$	−8.75510	−0.329037
$709$	29.8687	1.12174	0.560871	−	0.827903i	$-0.310466\pi$
$709$	29.8687	1.12174	0.560871	+	0.827903i	$0.310466\pi$
$710$	12.9087	0.484456
$711$	1.77884	0.0667119
$712$	−4.08994	−0.153277
$713$	9.27833	0.347476
$714$	−16.8110	−0.629138
$715$	4.08165	0.152645
$716$	−6.08336	−0.227346
$717$	−39.0438	−1.45812
$718$	−36.5572	−1.36430
$719$	−18.9554	−0.706917	−0.353459	−	0.935450i	$-0.614994\pi$
$719$	−18.9554	−0.706917	−0.353459	+	0.935450i	$0.614994\pi$
$720$	−4.68923	−0.174757
$721$	2.09421	0.0779923
$722$	35.9770	1.33893
$723$	8.65408	0.321849
$724$	−11.4616	−0.425966
$725$	−6.79658	−0.252419
$726$	19.7434	0.732748
$727$	−42.4438	−1.57415	−0.787077	−	0.616854i	$-0.788407\pi$
$727$	−42.4438	−1.57415	−0.787077	+	0.616854i	$0.788407\pi$
$728$	0.881412	0.0326673
$729$	19.2966	0.714690
$730$	−30.6610	−1.13481
$731$	19.4228	0.718378
$732$	16.0573	0.593493
$733$	6.32747	0.233710	0.116855	−	0.993149i	$-0.462719\pi$
$733$	6.32747	0.233710	0.116855	+	0.993149i	$0.462719\pi$
$734$	−38.8202	−1.43288
$735$	−8.53201	−0.314708
$736$	−32.6084	−1.20196
$737$	−23.4584	−0.864101
$738$	11.9225	0.438875
$739$	−23.8648	−0.877880	−0.438940	−	0.898516i	$-0.644646\pi$
$739$	−23.8648	−0.877880	−0.438940	+	0.898516i	$0.644646\pi$
$740$	−4.58830	−0.168669
$741$	−1.35181	−0.0496599
$742$	−36.2958	−1.33246
$743$	−8.57229	−0.314487	−0.157243	−	0.987560i	$-0.550261\pi$
$743$	−8.57229	−0.314487	−0.157243	+	0.987560i	$0.550261\pi$
$744$	−1.75296	−0.0642665
$745$	−3.81268	−0.139686
$746$	−39.9816	−1.46383
$747$	5.68711	0.208080
$748$	9.20589	0.336601
$749$	28.3937	1.03748
$750$	44.8224	1.63668
$751$	26.1850	0.955504	0.477752	−	0.878495i	$-0.341452\pi$
$751$	26.1850	0.955504	0.477752	+	0.878495i	$0.341452\pi$
$752$	−51.0643	−1.86212
$753$	7.49161	0.273010
$754$	6.41559	0.233642
$755$	5.07105	0.184554
$756$	16.8971	0.614540
$757$	39.5384	1.43705	0.718524	−	0.695502i	$-0.244818\pi$
$757$	39.5384	1.43705	0.718524	+	0.695502i	$0.244818\pi$
$758$	−13.0508	−0.474025
$759$	19.2181	0.697574
$760$	0.512422	0.0185875
$761$	34.0880	1.23569	0.617844	−	0.786300i	$-0.288006\pi$
$761$	34.0880	1.23569	0.617844	+	0.786300i	$0.288006\pi$
$762$	5.21631	0.188967
$763$	−26.7800	−0.969503
$764$	1.31412	0.0475434
$765$	−2.31788	−0.0838030
$766$	−15.8333	−0.572079
$767$	2.57861	0.0931084
$768$	35.9436	1.29700
$769$	−28.3915	−1.02383	−0.511913	−	0.859038i	$-0.671063\pi$
$769$	−28.3915	−1.02383	−0.511913	+	0.859038i	$0.671063\pi$
$770$	−16.6268	−0.599188
$771$	−13.1419	−0.473293
$772$	−2.18859	−0.0787690
$773$	−12.4979	−0.449518	−0.224759	−	0.974414i	$-0.572160\pi$
$773$	−12.4979	−0.449518	−0.224759	+	0.974414i	$0.572160\pi$
$774$	10.8675	0.390625
$775$	4.50868	0.161956
$776$	2.14148	0.0768746
$777$	5.98156	0.214587
$778$	61.2323	2.19528
$779$	−6.97981	−0.250078
$780$	−5.82102	−0.208426
$781$	9.21544	0.329754
$782$	−17.8827	−0.639485
$783$	−14.9201	−0.533199
$784$	11.4662	0.409506
$785$	−10.8454	−0.387088
$786$	−45.3376	−1.61714
$787$	14.3514	0.511573	0.255786	−	0.966733i	$-0.417666\pi$
$787$	14.3514	0.511573	0.255786	+	0.966733i	$0.417666\pi$
$788$	−26.2402	−0.934768
$789$	−12.1799	−0.433616
$790$	−9.51263	−0.338444
$791$	−13.5616	−0.482196
$792$	−0.624865	−0.0222036
$793$	−4.72930	−0.167942
$794$	−45.3286	−1.60865
$795$	−29.0789	−1.03132
$796$	−25.8045	−0.914617
$797$	1.96437	0.0695814	0.0347907	−	0.999395i	$-0.488924\pi$
$797$	1.96437	0.0695814	0.0347907	+	0.999395i	$0.488924\pi$
$798$	5.50666	0.194934
$799$	−25.2410	−0.892961
$800$	−15.8456	−0.560227
$801$	−6.06001	−0.214120
$802$	−5.86520	−0.207107
$803$	−21.8887	−0.772434
$804$	33.4551	1.17987
$805$	15.2255	0.536629
$806$	−4.25593	−0.149909
$807$	−28.5706	−1.00573
$808$	−0.0169745	−0.000597160 0
$809$	−46.8661	−1.64772	−0.823861	−	0.566791i	$-0.808185\pi$
$809$	−46.8661	−1.64772	−0.823861	+	0.566791i	$0.808185\pi$
$810$	34.9559	1.22822
$811$	17.3469	0.609130	0.304565	−	0.952491i	$-0.401489\pi$
$811$	17.3469	0.609130	0.304565	+	0.952491i	$0.401489\pi$
$812$	−12.3198	−0.432342
$813$	27.1068	0.950678
$814$	−6.94848	−0.243544
$815$	−26.3269	−0.922192
$816$	18.1002	0.633634
$817$	−6.36217	−0.222584
$818$	52.8478	1.84778
$819$	1.30598	0.0456345
$820$	−30.0557	−1.04959
$821$	−41.0414	−1.43236	−0.716178	−	0.697917i	$-0.754110\pi$
$821$	−41.0414	−1.43236	−0.716178	+	0.697917i	$0.754110\pi$
$822$	71.6970	2.50072
$823$	16.5619	0.577310	0.288655	−	0.957433i	$-0.406792\pi$
$823$	16.5619	0.577310	0.288655	+	0.957433i	$0.406792\pi$
$824$	−0.420881	−0.0146621
$825$	9.33879	0.325135
$826$	−10.5041	−0.365485
$827$	36.9767	1.28581	0.642903	−	0.765948i	$-0.277730\pi$
$827$	36.9767	1.28581	0.642903	+	0.765948i	$0.277730\pi$
$828$	−4.71681	−0.163920
$829$	4.34572	0.150933	0.0754665	−	0.997148i	$-0.475955\pi$
$829$	4.34572	0.150933	0.0754665	+	0.997148i	$0.475955\pi$
$830$	−30.4126	−1.05564
$831$	18.9978	0.659025
$832$	6.18550	0.214444
$833$	5.66770	0.196374
$834$	−44.7343	−1.54902
$835$	−13.6939	−0.473895
$836$	−3.01550	−0.104293
$837$	9.89758	0.342110
$838$	24.6238	0.850616
$839$	−12.6497	−0.436716	−0.218358	−	0.975869i	$-0.570070\pi$
$839$	−12.6497	−0.436716	−0.218358	+	0.975869i	$0.570070\pi$
$840$	−2.87656	−0.0992509
$841$	−18.1216	−0.624883
$842$	28.0349	0.966145
$843$	22.4999	0.774938
$844$	11.9517	0.411396
$845$	1.71445	0.0589788
$846$	−14.1229	−0.485557
$847$	11.1665	0.383686
$848$	39.0791	1.34198
$849$	22.0700	0.757439
$850$	−8.68987	−0.298060
$851$	6.36287	0.218116
$852$	−13.1425	−0.450256
$853$	−3.42479	−0.117262	−0.0586312	−	0.998280i	$-0.518674\pi$
$853$	−3.42479	−0.117262	−0.0586312	+	0.998280i	$0.518674\pi$
$854$	19.2650	0.659236
$855$	0.759248	0.0259657
$856$	−5.70640	−0.195041
$857$	42.4790	1.45106	0.725528	−	0.688193i	$-0.241596\pi$
$857$	42.4790	1.45106	0.725528	+	0.688193i	$0.241596\pi$
$858$	−8.81528	−0.300949
$859$	40.8551	1.39396	0.696979	−	0.717091i	$-0.254527\pi$
$859$	40.8551	1.39396	0.696979	+	0.717091i	$0.254527\pi$
$860$	−27.3961	−0.934200
$861$	39.1823	1.33533
$862$	68.6679	2.33884
$863$	−1.18913	−0.0404785	−0.0202393	−	0.999795i	$-0.506443\pi$
$863$	−1.18913	−0.0404785	−0.0202393	+	0.999795i	$0.506443\pi$
$864$	−34.7848	−1.18340
$865$	−30.1048	−1.02359
$866$	38.3676	1.30378
$867$	−23.4140	−0.795180
$868$	8.17267	0.277398
$869$	−6.79100	−0.230369
$870$	−20.9378	−0.709859
$871$	−9.85342	−0.333870
$872$	5.38210	0.182261
$873$	3.17300	0.107390
$874$	5.85770	0.198140
$875$	25.3507	0.857009
$876$	31.2164	1.05470
$877$	−7.34633	−0.248068	−0.124034	−	0.992278i	$-0.539583\pi$
$877$	−7.34633	−0.248068	−0.124034	+	0.992278i	$0.539583\pi$
$878$	−23.4243	−0.790531
$879$	−54.4762	−1.83744
$880$	17.9018	0.603470
$881$	−37.7332	−1.27126	−0.635632	−	0.771992i	$-0.719260\pi$
$881$	−37.7332	−1.27126	−0.635632	+	0.771992i	$0.719260\pi$
$882$	3.17122	0.106780
$883$	−30.6373	−1.03103	−0.515513	−	0.856882i	$-0.672399\pi$
$883$	−30.6373	−1.03103	−0.515513	+	0.856882i	$0.672399\pi$
$884$	3.86682	0.130055
$885$	−8.41553	−0.282885
$886$	−2.78193	−0.0934609
$887$	−14.4106	−0.483859	−0.241930	−	0.970294i	$-0.577780\pi$
$887$	−14.4106	−0.483859	−0.241930	+	0.970294i	$0.577780\pi$
$888$	−1.20214	−0.0403411
$889$	2.95024	0.0989479
$890$	32.4068	1.08628
$891$	24.9547	0.836016
$892$	46.9273	1.57124
$893$	8.26798	0.276678
$894$	8.23438	0.275399
$895$	−5.84742	−0.195458
$896$	7.01003	0.234189
$897$	8.07235	0.269528
$898$	49.3811	1.64787
$899$	−7.21644	−0.240682
$900$	−2.29207	−0.0764023
$901$	19.3167	0.643534
$902$	−45.5161	−1.51552
$903$	35.7150	1.18852
$904$	2.72554	0.0906501
$905$	−11.0170	−0.366219
$906$	−10.9521	−0.363860
$907$	24.7695	0.822457	0.411229	−	0.911532i	$-0.365100\pi$
$907$	24.7695	0.822457	0.411229	+	0.911532i	$0.365100\pi$
$908$	41.6151	1.38105
$909$	−0.0251508	−0.000834200 0
$910$	−6.98389	−0.231514
$911$	−12.0070	−0.397810	−0.198905	−	0.980019i	$-0.563738\pi$
$911$	−12.0070	−0.397810	−0.198905	+	0.980019i	$0.563738\pi$
$912$	−5.92894	−0.196327
$913$	−21.7114	−0.718541
$914$	−55.7910	−1.84540
$915$	15.4345	0.510248
$916$	−6.55161	−0.216471
$917$	−25.6420	−0.846774
$918$	−19.0762	−0.629610
$919$	13.4212	0.442724	0.221362	−	0.975192i	$-0.428950\pi$
$919$	13.4212	0.442724	0.221362	+	0.975192i	$0.428950\pi$
$920$	−3.05994	−0.100883
$921$	9.01514	0.297059
$922$	−43.7039	−1.43931
$923$	3.87084	0.127410
$924$	16.9280	0.556890
$925$	3.09195	0.101663
$926$	22.0940	0.726053
$927$	−0.623614	−0.0204822
$928$	25.3620	0.832548
$929$	−42.1574	−1.38314	−0.691569	−	0.722310i	$-0.743080\pi$
$929$	−42.1574	−1.38314	−0.691569	+	0.722310i	$0.743080\pi$
$930$	13.8896	0.455458
$931$	−1.85652	−0.0608451
$932$	−5.84781	−0.191551
$933$	−42.3607	−1.38683
$934$	−58.8578	−1.92589
$935$	8.84883	0.289388
$936$	−0.262467	−0.00857901
$937$	−15.3658	−0.501980	−0.250990	−	0.967990i	$-0.580756\pi$
$937$	−15.3658	−0.501980	−0.250990	+	0.967990i	$0.580756\pi$
$938$	40.1384	1.31057
$939$	6.64502	0.216852
$940$	35.6027	1.16123
$941$	−32.0230	−1.04392	−0.521960	−	0.852970i	$-0.674799\pi$
$941$	−32.0230	−1.04392	−0.521960	+	0.852970i	$0.674799\pi$
$942$	23.4232	0.763168
$943$	41.6800	1.35729
$944$	11.3096	0.368097
$945$	16.2417	0.528342
$946$	−41.4884	−1.34890
$947$	28.2308	0.917379	0.458689	−	0.888597i	$-0.348319\pi$
$947$	28.2308	0.917379	0.458689	+	0.888597i	$0.348319\pi$
$948$	9.68494	0.314552
$949$	−9.19408	−0.298452
$950$	2.84647	0.0923517
$951$	64.7802	2.10064
$952$	1.91086	0.0619314
$953$	−21.0297	−0.681220	−0.340610	−	0.940205i	$-0.610634\pi$
$953$	−21.0297	−0.681220	−0.340610	+	0.940205i	$0.610634\pi$
$954$	10.8082	0.349928
$955$	1.26316	0.0408748
$956$	−36.5834	−1.18319
$957$	−14.9474	−0.483180
$958$	59.8808	1.93466
$959$	40.5504	1.30944
$960$	−20.1869	−0.651530
$961$	−26.2128	−0.845574
$962$	−2.91862	−0.0941002
$963$	−8.45509	−0.272461
$964$	8.10875	0.261165
$965$	−2.10370	−0.0677206
$966$	−32.8831	−1.05800
$967$	14.9668	0.481299	0.240649	−	0.970612i	$-0.422640\pi$
$967$	14.9668	0.481299	0.240649	+	0.970612i	$0.422640\pi$
$968$	−2.24418	−0.0721307
$969$	−2.93066	−0.0941464
$970$	−16.9681	−0.544812
$971$	25.0979	0.805429	0.402714	−	0.915326i	$-0.368067\pi$
$971$	25.0979	0.805429	0.402714	+	0.915326i	$0.368067\pi$
$972$	−11.3836	−0.365130
$973$	−25.3008	−0.811108
$974$	76.9742	2.46641
$975$	3.92265	0.125625
$976$	−20.7424	−0.663947
$977$	5.91802	0.189334	0.0946672	−	0.995509i	$-0.469821\pi$
$977$	5.91802	0.189334	0.0946672	+	0.995509i	$0.469821\pi$
$978$	56.8592	1.81816
$979$	23.1350	0.739397
$980$	−7.99437	−0.255371
$981$	7.97458	0.254609
$982$	10.2617	0.327464
$983$	48.8986	1.55962	0.779811	−	0.626015i	$-0.215315\pi$
$983$	48.8986	1.55962	0.779811	+	0.626015i	$0.215315\pi$
$984$	−7.87462	−0.251034
$985$	−25.2224	−0.803654
$986$	13.9087	0.442944
$987$	−46.4136	−1.47736
$988$	−1.26662	−0.0402967
$989$	37.9918	1.20807
$990$	4.95114	0.157358
$991$	−37.5071	−1.19145	−0.595727	−	0.803187i	$-0.703136\pi$
$991$	−37.5071	−1.19145	−0.595727	+	0.803187i	$0.703136\pi$
$992$	−16.8245	−0.534178
$993$	17.6089	0.558801
$994$	−15.7681	−0.500132
$995$	−24.8037	−0.786330
$996$	30.9635	0.981117
$997$	−5.11215	−0.161904	−0.0809518	−	0.996718i	$-0.525796\pi$
$997$	−5.11215	−0.161904	−0.0809518	+	0.996718i	$0.525796\pi$
$998$	13.6844	0.433174
$999$	6.78754	0.214748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.e.1.4	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.e.1.4	✓	21	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)

\(f(q)\)	\(=\)	\(q-1.94515 q^{2} +1.90358 q^{3} +1.78363 q^{4} +1.71445 q^{5} -3.70276 q^{6} -2.09421 q^{7} +0.420881 q^{8} +0.623614 q^{9} +O(q^{10})\)	\(q-1.94515 q^{2} +1.90358 q^{3} +1.78363 q^{4} +1.71445 q^{5} -3.70276 q^{6} -2.09421 q^{7} +0.420881 q^{8} +0.623614 q^{9} -3.33487 q^{10} -2.38074 q^{11} +3.39527 q^{12} -1.00000 q^{13} +4.07355 q^{14} +3.26359 q^{15} -4.38593 q^{16} -2.16796 q^{17} -1.21302 q^{18} +0.710140 q^{19} +3.05793 q^{20} -3.98649 q^{21} +4.63090 q^{22} -4.24062 q^{23} +0.801181 q^{24} -2.06067 q^{25} +1.94515 q^{26} -4.52364 q^{27} -3.73528 q^{28} +3.29824 q^{29} -6.34818 q^{30} -2.18797 q^{31} +7.68955 q^{32} -4.53192 q^{33} +4.21701 q^{34} -3.59040 q^{35} +1.11229 q^{36} -1.50046 q^{37} -1.38133 q^{38} -1.90358 q^{39} +0.721579 q^{40} -9.82877 q^{41} +7.75433 q^{42} -8.95903 q^{43} -4.24634 q^{44} +1.06915 q^{45} +8.24865 q^{46} +11.6427 q^{47} -8.34897 q^{48} -2.61430 q^{49} +4.00832 q^{50} -4.12688 q^{51} -1.78363 q^{52} -8.91011 q^{53} +8.79918 q^{54} -4.08165 q^{55} -0.881412 q^{56} +1.35181 q^{57} -6.41559 q^{58} -2.57861 q^{59} +5.82102 q^{60} +4.72930 q^{61} +4.25593 q^{62} -1.30598 q^{63} -6.18550 q^{64} -1.71445 q^{65} +8.81528 q^{66} +9.85342 q^{67} -3.86682 q^{68} -8.07235 q^{69} +6.98389 q^{70} -3.87084 q^{71} +0.262467 q^{72} +9.19408 q^{73} +2.91862 q^{74} -3.92265 q^{75} +1.26662 q^{76} +4.98575 q^{77} +3.70276 q^{78} +2.85248 q^{79} -7.51945 q^{80} -10.4819 q^{81} +19.1185 q^{82} +9.11960 q^{83} -7.11040 q^{84} -3.71685 q^{85} +17.4267 q^{86} +6.27846 q^{87} -1.00201 q^{88} -9.71757 q^{89} -2.07967 q^{90} +2.09421 q^{91} -7.56367 q^{92} -4.16497 q^{93} -22.6469 q^{94} +1.21750 q^{95} +14.6377 q^{96} +5.08808 q^{97} +5.08523 q^{98} -1.48466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

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