LMFDB - Embedded newform 4022.2.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.62663 q^{3} +1.00000 q^{4} -2.01799 q^{5} -2.62663 q^{6} +0.869262 q^{7} +1.00000 q^{8} +3.89919 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.62663 q^{3} +1.00000 q^{4} -2.01799 q^{5} -2.62663 q^{6} +0.869262 q^{7} +1.00000 q^{8} +3.89919 q^{9} -2.01799 q^{10} -3.06749 q^{11} -2.62663 q^{12} +4.63284 q^{13} +0.869262 q^{14} +5.30052 q^{15} +1.00000 q^{16} -0.775089 q^{17} +3.89919 q^{18} -4.44299 q^{19} -2.01799 q^{20} -2.28323 q^{21} -3.06749 q^{22} -0.835104 q^{23} -2.62663 q^{24} -0.927713 q^{25} +4.63284 q^{26} -2.36185 q^{27} +0.869262 q^{28} +7.31632 q^{29} +5.30052 q^{30} -1.45586 q^{31} +1.00000 q^{32} +8.05716 q^{33} -0.775089 q^{34} -1.75416 q^{35} +3.89919 q^{36} +9.35814 q^{37} -4.44299 q^{38} -12.1688 q^{39} -2.01799 q^{40} +1.66167 q^{41} -2.28323 q^{42} -9.08328 q^{43} -3.06749 q^{44} -7.86854 q^{45} -0.835104 q^{46} -1.03325 q^{47} -2.62663 q^{48} -6.24438 q^{49} -0.927713 q^{50} +2.03587 q^{51} +4.63284 q^{52} +4.38682 q^{53} -2.36185 q^{54} +6.19016 q^{55} +0.869262 q^{56} +11.6701 q^{57} +7.31632 q^{58} +11.3418 q^{59} +5.30052 q^{60} -0.898436 q^{61} -1.45586 q^{62} +3.38942 q^{63} +1.00000 q^{64} -9.34903 q^{65} +8.05716 q^{66} +4.69714 q^{67} -0.775089 q^{68} +2.19351 q^{69} -1.75416 q^{70} +5.82527 q^{71} +3.89919 q^{72} +0.985216 q^{73} +9.35814 q^{74} +2.43676 q^{75} -4.44299 q^{76} -2.66645 q^{77} -12.1688 q^{78} -8.32160 q^{79} -2.01799 q^{80} -5.49386 q^{81} +1.66167 q^{82} -6.68708 q^{83} -2.28323 q^{84} +1.56412 q^{85} -9.08328 q^{86} -19.2173 q^{87} -3.06749 q^{88} -5.26768 q^{89} -7.86854 q^{90} +4.02715 q^{91} -0.835104 q^{92} +3.82400 q^{93} -1.03325 q^{94} +8.96591 q^{95} -2.62663 q^{96} -6.27376 q^{97} -6.24438 q^{98} -11.9607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) $ 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9 - 13 * q^10 - 29 * q^11 - 14 * q^12 - 23 * q^13 - 29 * q^14 - 14 * q^15 + 31 * q^16 - 19 * q^17 + 27 * q^18 - 36 * q^19 - 13 * q^20 - 29 * q^22 - 24 * q^23 - 14 * q^24 + 4 * q^25 - 23 * q^26 - 65 * q^27 - 29 * q^28 + 18 * q^29 - 14 * q^30 - 41 * q^31 + 31 * q^32 - 18 * q^33 - 19 * q^34 - 28 * q^35 + 27 * q^36 - 31 * q^37 - 36 * q^38 - 31 * q^39 - 13 * q^40 - 51 * q^41 - 14 * q^43 - 29 * q^44 - 34 * q^45 - 24 * q^46 - 64 * q^47 - 14 * q^48 + 8 * q^49 + 4 * q^50 - 17 * q^51 - 23 * q^52 + 3 * q^53 - 65 * q^54 - 50 * q^55 - 29 * q^56 - 19 * q^57 + 18 * q^58 - 58 * q^59 - 14 * q^60 - 14 * q^61 - 41 * q^62 - 66 * q^63 + 31 * q^64 + 6 * q^65 - 18 * q^66 - 55 * q^67 - 19 * q^68 + q^69 - 28 * q^70 - 42 * q^71 + 27 * q^72 - 83 * q^73 - 31 * q^74 - 49 * q^75 - 36 * q^76 + 18 * q^77 - 31 * q^78 - 56 * q^79 - 13 * q^80 + 3 * q^81 - 51 * q^82 - 43 * q^83 + 4 * q^85 - 14 * q^86 - 76 * q^87 - 29 * q^88 + 17 * q^89 - 34 * q^90 - 49 * q^91 - 24 * q^92 - 24 * q^93 - 64 * q^94 - 43 * q^95 - 14 * q^96 - 98 * q^97 + 8 * q^98 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.62663	−1.51649	−0.758243	−	0.651972i	$-0.773942\pi$
$3$	−2.62663	−1.51649	−0.758243	+	0.651972i	$0.773942\pi$
$4$	1.00000	0.500000
$5$	−2.01799	−0.902473	−0.451236	−	0.892404i	$-0.649017\pi$
$5$	−2.01799	−0.902473	−0.451236	+	0.892404i	$0.649017\pi$
$6$	−2.62663	−1.07232
$7$	0.869262	0.328550	0.164275	−	0.986415i	$-0.447472\pi$
$7$	0.869262	0.328550	0.164275	+	0.986415i	$0.447472\pi$
$8$	1.00000	0.353553
$9$	3.89919	1.29973
$10$	−2.01799	−0.638145
$11$	−3.06749	−0.924882	−0.462441	−	0.886650i	$-0.653026\pi$
$11$	−3.06749	−0.924882	−0.462441	+	0.886650i	$0.653026\pi$
$12$	−2.62663	−0.758243
$13$	4.63284	1.28492	0.642460	−	0.766319i	$-0.277914\pi$
$13$	4.63284	1.28492	0.642460	+	0.766319i	$0.277914\pi$
$14$	0.869262	0.232320
$15$	5.30052	1.36859
$16$	1.00000	0.250000
$17$	−0.775089	−0.187987	−0.0939933	−	0.995573i	$-0.529963\pi$
$17$	−0.775089	−0.187987	−0.0939933	+	0.995573i	$0.529963\pi$
$18$	3.89919	0.919049
$19$	−4.44299	−1.01929	−0.509646	−	0.860384i	$-0.670224\pi$
$19$	−4.44299	−1.01929	−0.509646	+	0.860384i	$0.670224\pi$
$20$	−2.01799	−0.451236
$21$	−2.28323	−0.498242
$22$	−3.06749	−0.653991
$23$	−0.835104	−0.174131	−0.0870656	−	0.996203i	$-0.527749\pi$
$23$	−0.835104	−0.174131	−0.0870656	+	0.996203i	$0.527749\pi$
$24$	−2.62663	−0.536159
$25$	−0.927713	−0.185543
$26$	4.63284	0.908575
$27$	−2.36185	−0.454539
$28$	0.869262	0.164275
$29$	7.31632	1.35861	0.679303	−	0.733858i	$-0.262282\pi$
$29$	7.31632	1.35861	0.679303	+	0.733858i	$0.262282\pi$
$30$	5.30052	0.967738
$31$	−1.45586	−0.261480	−0.130740	−	0.991417i	$-0.541735\pi$
$31$	−1.45586	−0.261480	−0.130740	+	0.991417i	$0.541735\pi$
$32$	1.00000	0.176777
$33$	8.05716	1.40257
$34$	−0.775089	−0.132927
$35$	−1.75416	−0.296508
$36$	3.89919	0.649866
$37$	9.35814	1.53847	0.769234	−	0.638967i	$-0.220638\pi$
$37$	9.35814	1.53847	0.769234	+	0.638967i	$0.220638\pi$
$38$	−4.44299	−0.720748
$39$	−12.1688	−1.94856
$40$	−2.01799	−0.319072
$41$	1.66167	0.259509	0.129754	−	0.991546i	$-0.458581\pi$
$41$	1.66167	0.259509	0.129754	+	0.991546i	$0.458581\pi$
$42$	−2.28323	−0.352310
$43$	−9.08328	−1.38519	−0.692593	−	0.721328i	$-0.743532\pi$
$43$	−9.08328	−1.38519	−0.692593	+	0.721328i	$0.743532\pi$
$44$	−3.06749	−0.462441
$45$	−7.86854	−1.17297
$46$	−0.835104	−0.123129
$47$	−1.03325	−0.150715	−0.0753576	−	0.997157i	$-0.524010\pi$
$47$	−1.03325	−0.150715	−0.0753576	+	0.997157i	$0.524010\pi$
$48$	−2.62663	−0.379122
$49$	−6.24438	−0.892055
$50$	−0.927713	−0.131198
$51$	2.03587	0.285079
$52$	4.63284	0.642460
$53$	4.38682	0.602576	0.301288	−	0.953533i	$-0.402583\pi$
$53$	4.38682	0.602576	0.301288	+	0.953533i	$0.402583\pi$
$54$	−2.36185	−0.321408
$55$	6.19016	0.834681
$56$	0.869262	0.116160
$57$	11.6701	1.54574
$58$	7.31632	0.960680
$59$	11.3418	1.47657	0.738287	−	0.674487i	$-0.235635\pi$
$59$	11.3418	1.47657	0.738287	+	0.674487i	$0.235635\pi$
$60$	5.30052	0.684294
$61$	−0.898436	−0.115033	−0.0575165	−	0.998345i	$-0.518318\pi$
$61$	−0.898436	−0.115033	−0.0575165	+	0.998345i	$0.518318\pi$
$62$	−1.45586	−0.184894
$63$	3.38942	0.427027
$64$	1.00000	0.125000
$65$	−9.34903	−1.15960
$66$	8.05716	0.991768
$67$	4.69714	0.573847	0.286923	−	0.957954i	$-0.407368\pi$
$67$	4.69714	0.573847	0.286923	+	0.957954i	$0.407368\pi$
$68$	−0.775089	−0.0939933
$69$	2.19351	0.264068
$70$	−1.75416	−0.209662
$71$	5.82527	0.691332	0.345666	−	0.938358i	$-0.387653\pi$
$71$	5.82527	0.691332	0.345666	+	0.938358i	$0.387653\pi$
$72$	3.89919	0.459525
$73$	0.985216	0.115311	0.0576554	−	0.998337i	$-0.481638\pi$
$73$	0.985216	0.115311	0.0576554	+	0.998337i	$0.481638\pi$
$74$	9.35814	1.08786
$75$	2.43676	0.281373
$76$	−4.44299	−0.509646
$77$	−2.66645	−0.303870
$78$	−12.1688	−1.37784
$79$	−8.32160	−0.936254	−0.468127	−	0.883661i	$-0.655071\pi$
$79$	−8.32160	−0.936254	−0.468127	+	0.883661i	$0.655071\pi$
$80$	−2.01799	−0.225618
$81$	−5.49386	−0.610429
$82$	1.66167	0.183501
$83$	−6.68708	−0.734002	−0.367001	−	0.930220i	$-0.619616\pi$
$83$	−6.68708	−0.734002	−0.367001	+	0.930220i	$0.619616\pi$
$84$	−2.28323	−0.249121
$85$	1.56412	0.169653
$86$	−9.08328	−0.979475
$87$	−19.2173	−2.06031
$88$	−3.06749	−0.326995
$89$	−5.26768	−0.558373	−0.279186	−	0.960237i	$-0.590065\pi$
$89$	−5.26768	−0.558373	−0.279186	+	0.960237i	$0.590065\pi$
$90$	−7.86854	−0.829417
$91$	4.02715	0.422160
$92$	−0.835104	−0.0870656
$93$	3.82400	0.396530
$94$	−1.03325	−0.106572
$95$	8.96591	0.919883
$96$	−2.62663	−0.268079
$97$	−6.27376	−0.637004	−0.318502	−	0.947922i	$-0.603180\pi$
$97$	−6.27376	−0.637004	−0.318502	+	0.947922i	$0.603180\pi$
$98$	−6.24438	−0.630778
$99$	−11.9607	−1.20210
$100$	−0.927713	−0.0927713
$101$	−12.2375	−1.21768	−0.608840	−	0.793293i	$-0.708365\pi$
$101$	−12.2375	−1.21768	−0.608840	+	0.793293i	$0.708365\pi$
$102$	2.03587	0.201582
$103$	−0.130840	−0.0128921	−0.00644603	−	0.999979i	$-0.502052\pi$
$103$	−0.130840	−0.0128921	−0.00644603	+	0.999979i	$0.502052\pi$
$104$	4.63284	0.454288
$105$	4.60754	0.449650
$106$	4.38682	0.426086
$107$	−13.7200	−1.32636	−0.663182	−	0.748458i	$-0.730794\pi$
$107$	−13.7200	−1.32636	−0.663182	+	0.748458i	$0.730794\pi$
$108$	−2.36185	−0.227269
$109$	−16.3980	−1.57065	−0.785323	−	0.619086i	$-0.787503\pi$
$109$	−16.3980	−1.57065	−0.785323	+	0.619086i	$0.787503\pi$
$110$	6.19016	0.590209
$111$	−24.5804	−2.33307
$112$	0.869262	0.0821375
$113$	16.7815	1.57867	0.789337	−	0.613960i	$-0.210424\pi$
$113$	16.7815	1.57867	0.789337	+	0.613960i	$0.210424\pi$
$114$	11.6701	1.09300
$115$	1.68523	0.157149
$116$	7.31632	0.679303
$117$	18.0644	1.67005
$118$	11.3418	1.04410
$119$	−0.673755	−0.0617630
$120$	5.30052	0.483869
$121$	−1.59052	−0.144592
$122$	−0.898436	−0.0813406
$123$	−4.36459	−0.393542
$124$	−1.45586	−0.130740
$125$	11.9621	1.06992
$126$	3.38942	0.301954
$127$	−18.7825	−1.66667	−0.833337	−	0.552765i	$-0.813573\pi$
$127$	−18.7825	−1.66667	−0.833337	+	0.552765i	$0.813573\pi$
$128$	1.00000	0.0883883
$129$	23.8584	2.10062
$130$	−9.34903	−0.819965
$131$	−7.58408	−0.662624	−0.331312	−	0.943521i	$-0.607491\pi$
$131$	−7.58408	−0.662624	−0.331312	+	0.943521i	$0.607491\pi$
$132$	8.05716	0.701286
$133$	−3.86212	−0.334888
$134$	4.69714	0.405771
$135$	4.76620	0.410209
$136$	−0.775089	−0.0664633
$137$	5.47271	0.467565	0.233783	−	0.972289i	$-0.424890\pi$
$137$	5.47271	0.467565	0.233783	+	0.972289i	$0.424890\pi$
$138$	2.19351	0.186724
$139$	−12.4032	−1.05202	−0.526012	−	0.850477i	$-0.676313\pi$
$139$	−12.4032	−1.05202	−0.526012	+	0.850477i	$0.676313\pi$
$140$	−1.75416	−0.148254
$141$	2.71397	0.228558
$142$	5.82527	0.488845
$143$	−14.2112	−1.18840
$144$	3.89919	0.324933
$145$	−14.7643	−1.22611
$146$	0.985216	0.0815371
$147$	16.4017	1.35279
$148$	9.35814	0.769234
$149$	5.67473	0.464892	0.232446	−	0.972609i	$-0.425327\pi$
$149$	5.67473	0.464892	0.232446	+	0.972609i	$0.425327\pi$
$150$	2.43676	0.198961
$151$	−2.77842	−0.226104	−0.113052	−	0.993589i	$-0.536063\pi$
$151$	−2.77842	−0.226104	−0.113052	+	0.993589i	$0.536063\pi$
$152$	−4.44299	−0.360374
$153$	−3.02222	−0.244332
$154$	−2.66645	−0.214869
$155$	2.93791	0.235978
$156$	−12.1688	−0.974282
$157$	7.38851	0.589668	0.294834	−	0.955549i	$-0.404736\pi$
$157$	7.38851	0.589668	0.294834	+	0.955549i	$0.404736\pi$
$158$	−8.32160	−0.662031
$159$	−11.5226	−0.913799
$160$	−2.01799	−0.159536
$161$	−0.725924	−0.0572108
$162$	−5.49386	−0.431639
$163$	19.6105	1.53601	0.768007	−	0.640441i	$-0.221248\pi$
$163$	19.6105	1.53601	0.768007	+	0.640441i	$0.221248\pi$
$164$	1.66167	0.129754
$165$	−16.2593	−1.26578
$166$	−6.68708	−0.519018
$167$	−15.9501	−1.23425	−0.617127	−	0.786863i	$-0.711704\pi$
$167$	−15.9501	−1.23425	−0.617127	+	0.786863i	$0.711704\pi$
$168$	−2.28323	−0.176155
$169$	8.46323	0.651018
$170$	1.56412	0.119963
$171$	−17.3241	−1.32481
$172$	−9.08328	−0.692593
$173$	−21.5598	−1.63916	−0.819581	−	0.572963i	$-0.805794\pi$
$173$	−21.5598	−1.63916	−0.819581	+	0.572963i	$0.805794\pi$
$174$	−19.2173	−1.45686
$175$	−0.806426	−0.0609600
$176$	−3.06749	−0.231221
$177$	−29.7907	−2.23920
$178$	−5.26768	−0.394829
$179$	4.17437	0.312007	0.156003	−	0.987757i	$-0.450139\pi$
$179$	4.17437	0.312007	0.156003	+	0.987757i	$0.450139\pi$
$180$	−7.86854	−0.586486
$181$	−10.1009	−0.750797	−0.375399	−	0.926863i	$-0.622494\pi$
$181$	−10.1009	−0.750797	−0.375399	+	0.926863i	$0.622494\pi$
$182$	4.02715	0.298512
$183$	2.35986	0.174446
$184$	−0.835104	−0.0615647
$185$	−18.8846	−1.38843
$186$	3.82400	0.280389
$187$	2.37758	0.173866
$188$	−1.03325	−0.0753576
$189$	−2.05307	−0.149339
$190$	8.96591	0.650456
$191$	−11.6898	−0.845843	−0.422922	−	0.906166i	$-0.638995\pi$
$191$	−11.6898	−0.845843	−0.422922	+	0.906166i	$0.638995\pi$
$192$	−2.62663	−0.189561
$193$	−20.8039	−1.49750	−0.748750	−	0.662852i	$-0.769346\pi$
$193$	−20.8039	−1.49750	−0.748750	+	0.662852i	$0.769346\pi$
$194$	−6.27376	−0.450430
$195$	24.5565	1.75853
$196$	−6.24438	−0.446027
$197$	4.24154	0.302197	0.151099	−	0.988519i	$-0.451719\pi$
$197$	4.24154	0.302197	0.151099	+	0.988519i	$0.451719\pi$
$198$	−11.9607	−0.850012
$199$	−26.0230	−1.84472	−0.922362	−	0.386326i	$-0.873744\pi$
$199$	−26.0230	−1.84472	−0.922362	+	0.386326i	$0.873744\pi$
$200$	−0.927713	−0.0655992
$201$	−12.3376	−0.870231
$202$	−12.2375	−0.861030
$203$	6.35980	0.446370
$204$	2.03587	0.142540
$205$	−3.35323	−0.234200
$206$	−0.130840	−0.00911606
$207$	−3.25623	−0.226324
$208$	4.63284	0.321230
$209$	13.6288	0.942725
$210$	4.60754	0.317950
$211$	−3.43432	−0.236429	−0.118214	−	0.992988i	$-0.537717\pi$
$211$	−3.43432	−0.236429	−0.118214	+	0.992988i	$0.537717\pi$
$212$	4.38682	0.301288
$213$	−15.3008	−1.04840
$214$	−13.7200	−0.937881
$215$	18.3300	1.25009
$216$	−2.36185	−0.160704
$217$	−1.26552	−0.0859092
$218$	−16.3980	−1.11061
$219$	−2.58780	−0.174867
$220$	6.19016	0.417341
$221$	−3.59087	−0.241548
$222$	−24.5804	−1.64973
$223$	−24.7701	−1.65873	−0.829363	−	0.558710i	$-0.811297\pi$
$223$	−24.7701	−1.65873	−0.829363	+	0.558710i	$0.811297\pi$
$224$	0.869262	0.0580800
$225$	−3.61733	−0.241156
$226$	16.7815	1.11629
$227$	−15.3499	−1.01881	−0.509405	−	0.860527i	$-0.670134\pi$
$227$	−15.3499	−1.01881	−0.509405	+	0.860527i	$0.670134\pi$
$228$	11.6701	0.772871
$229$	−17.2681	−1.14111	−0.570555	−	0.821259i	$-0.693272\pi$
$229$	−17.2681	−1.14111	−0.570555	+	0.821259i	$0.693272\pi$
$230$	1.68523	0.111121
$231$	7.00378	0.460815
$232$	7.31632	0.480340
$233$	−7.61184	−0.498668	−0.249334	−	0.968417i	$-0.580212\pi$
$233$	−7.61184	−0.498668	−0.249334	+	0.968417i	$0.580212\pi$
$234$	18.0644	1.18090
$235$	2.08509	0.136016
$236$	11.3418	0.738287
$237$	21.8578	1.41982
$238$	−0.673755	−0.0436731
$239$	20.7851	1.34448	0.672239	−	0.740334i	$-0.265333\pi$
$239$	20.7851	1.34448	0.672239	+	0.740334i	$0.265333\pi$
$240$	5.30052	0.342147
$241$	−6.31856	−0.407014	−0.203507	−	0.979073i	$-0.565234\pi$
$241$	−6.31856	−0.407014	−0.203507	+	0.979073i	$0.565234\pi$
$242$	−1.59052	−0.102242
$243$	21.5159	1.38025
$244$	−0.898436	−0.0575165
$245$	12.6011	0.805055
$246$	−4.36459	−0.278276
$247$	−20.5837	−1.30971
$248$	−1.45586	−0.0924470
$249$	17.5645	1.11310
$250$	11.9621	0.756548
$251$	1.19661	0.0755295	0.0377648	−	0.999287i	$-0.487976\pi$
$251$	1.19661	0.0755295	0.0377648	+	0.999287i	$0.487976\pi$
$252$	3.38942	0.213513
$253$	2.56167	0.161051
$254$	−18.7825	−1.17852
$255$	−4.10837	−0.257276
$256$	1.00000	0.0625000
$257$	0.713759	0.0445230	0.0222615	−	0.999752i	$-0.492913\pi$
$257$	0.713759	0.0445230	0.0222615	+	0.999752i	$0.492913\pi$
$258$	23.8584	1.48536
$259$	8.13467	0.505464
$260$	−9.34903	−0.579802
$261$	28.5278	1.76582
$262$	−7.58408	−0.468546
$263$	30.3768	1.87312	0.936558	−	0.350513i	$-0.113993\pi$
$263$	30.3768	1.87312	0.936558	+	0.350513i	$0.113993\pi$
$264$	8.05716	0.495884
$265$	−8.85257	−0.543809
$266$	−3.86212	−0.236802
$267$	13.8363	0.846765
$268$	4.69714	0.286923
$269$	−0.769870	−0.0469398	−0.0234699	−	0.999725i	$-0.507471\pi$
$269$	−0.769870	−0.0469398	−0.0234699	+	0.999725i	$0.507471\pi$
$270$	4.76620	0.290062
$271$	6.50686	0.395264	0.197632	−	0.980276i	$-0.436675\pi$
$271$	6.50686	0.395264	0.197632	+	0.980276i	$0.436675\pi$
$272$	−0.775089	−0.0469967
$273$	−10.5778	−0.640201
$274$	5.47271	0.330619
$275$	2.84575	0.171605
$276$	2.19351	0.132034
$277$	−19.1652	−1.15153	−0.575763	−	0.817617i	$-0.695295\pi$
$277$	−19.1652	−1.15153	−0.575763	+	0.817617i	$0.695295\pi$
$278$	−12.4032	−0.743893
$279$	−5.67667	−0.339853
$280$	−1.75416	−0.104831
$281$	19.9604	1.19074	0.595370	−	0.803452i	$-0.297005\pi$
$281$	19.9604	1.19074	0.595370	+	0.803452i	$0.297005\pi$
$282$	2.71397	0.161615
$283$	21.5655	1.28193	0.640967	−	0.767569i	$-0.278534\pi$
$283$	21.5655	1.28193	0.640967	+	0.767569i	$0.278534\pi$
$284$	5.82527	0.345666
$285$	−23.5501	−1.39499
$286$	−14.2112	−0.840325
$287$	1.44442	0.0852617
$288$	3.89919	0.229762
$289$	−16.3992	−0.964661
$290$	−14.7643	−0.866987
$291$	16.4789	0.966009
$292$	0.985216	0.0576554
$293$	9.51601	0.555931	0.277966	−	0.960591i	$-0.410340\pi$
$293$	9.51601	0.555931	0.277966	+	0.960591i	$0.410340\pi$
$294$	16.4017	0.956566
$295$	−22.8876	−1.33257
$296$	9.35814	0.543931
$297$	7.24496	0.420395
$298$	5.67473	0.328728
$299$	−3.86890	−0.223745
$300$	2.43676	0.140686
$301$	−7.89575	−0.455103
$302$	−2.77842	−0.159880
$303$	32.1435	1.84660
$304$	−4.44299	−0.254823
$305$	1.81304	0.103814
$306$	−3.02222	−0.172769
$307$	26.7324	1.52570	0.762848	−	0.646577i	$-0.223800\pi$
$307$	26.7324	1.52570	0.762848	+	0.646577i	$0.223800\pi$
$308$	−2.66645	−0.151935
$309$	0.343669	0.0195506
$310$	2.93791	0.166862
$311$	−3.78382	−0.214561	−0.107280	−	0.994229i	$-0.534214\pi$
$311$	−3.78382	−0.214561	−0.107280	+	0.994229i	$0.534214\pi$
$312$	−12.1688	−0.688921
$313$	−6.50925	−0.367925	−0.183962	−	0.982933i	$-0.558892\pi$
$313$	−6.50925	−0.367925	−0.183962	+	0.982933i	$0.558892\pi$
$314$	7.38851	0.416958
$315$	−6.83982	−0.385380
$316$	−8.32160	−0.468127
$317$	−18.8198	−1.05703	−0.528513	−	0.848925i	$-0.677250\pi$
$317$	−18.8198	−1.05703	−0.528513	+	0.848925i	$0.677250\pi$
$318$	−11.5226	−0.646153
$319$	−22.4427	−1.25655
$320$	−2.01799	−0.112809
$321$	36.0374	2.01141
$322$	−0.725924	−0.0404542
$323$	3.44371	0.191613
$324$	−5.49386	−0.305215
$325$	−4.29795	−0.238407
$326$	19.6105	1.08613
$327$	43.0716	2.38186
$328$	1.66167	0.0917503
$329$	−0.898166	−0.0495175
$330$	−16.2593	−0.895044
$331$	16.9172	0.929854	0.464927	−	0.885349i	$-0.346081\pi$
$331$	16.9172	0.929854	0.464927	+	0.885349i	$0.346081\pi$
$332$	−6.68708	−0.367001
$333$	36.4892	1.99960
$334$	−15.9501	−0.872750
$335$	−9.47878	−0.517881
$336$	−2.28323	−0.124560
$337$	5.19508	0.282994	0.141497	−	0.989939i	$-0.454808\pi$
$337$	5.19508	0.282994	0.141497	+	0.989939i	$0.454808\pi$
$338$	8.46323	0.460339
$339$	−44.0789	−2.39404
$340$	1.56412	0.0848265
$341$	4.46582	0.241838
$342$	−17.3241	−0.936779
$343$	−11.5128	−0.621635
$344$	−9.08328	−0.489737
$345$	−4.42648	−0.238314
$346$	−21.5598	−1.15906
$347$	−16.9331	−0.909016	−0.454508	−	0.890743i	$-0.650185\pi$
$347$	−16.9331	−0.909016	−0.454508	+	0.890743i	$0.650185\pi$
$348$	−19.2173	−1.03015
$349$	−22.8937	−1.22547	−0.612735	−	0.790288i	$-0.709931\pi$
$349$	−22.8937	−1.22547	−0.612735	+	0.790288i	$0.709931\pi$
$350$	−0.806426	−0.0431053
$351$	−10.9421	−0.584046
$352$	−3.06749	−0.163498
$353$	−8.90369	−0.473896	−0.236948	−	0.971522i	$-0.576147\pi$
$353$	−8.90369	−0.473896	−0.236948	+	0.971522i	$0.576147\pi$
$354$	−29.7907	−1.58336
$355$	−11.7553	−0.623908
$356$	−5.26768	−0.279186
$357$	1.76971	0.0936628
$358$	4.17437	0.220622
$359$	11.3467	0.598856	0.299428	−	0.954119i	$-0.403204\pi$
$359$	11.3467	0.598856	0.299428	+	0.954119i	$0.403204\pi$
$360$	−7.86854	−0.414708
$361$	0.740152	0.0389554
$362$	−10.1009	−0.530894
$363$	4.17770	0.219272
$364$	4.02715	0.211080
$365$	−1.98816	−0.104065
$366$	2.35986	0.123352
$367$	−29.6837	−1.54948	−0.774738	−	0.632282i	$-0.782118\pi$
$367$	−29.6837	−1.54948	−0.774738	+	0.632282i	$0.782118\pi$
$368$	−0.835104	−0.0435328
$369$	6.47917	0.337292
$370$	−18.8846	−0.981765
$371$	3.81330	0.197976
$372$	3.82400	0.198265
$373$	19.6595	1.01793	0.508964	−	0.860788i	$-0.330029\pi$
$373$	19.6595	1.01793	0.508964	+	0.860788i	$0.330029\pi$
$374$	2.37758	0.122942
$375$	−31.4200	−1.62252
$376$	−1.03325	−0.0532859
$377$	33.8954	1.74570
$378$	−2.05307	−0.105598
$379$	−6.66140	−0.342173	−0.171087	−	0.985256i	$-0.554728\pi$
$379$	−6.66140	−0.342173	−0.171087	+	0.985256i	$0.554728\pi$
$380$	8.96591	0.459942
$381$	49.3346	2.52749
$382$	−11.6898	−0.598101
$383$	−18.8105	−0.961170	−0.480585	−	0.876948i	$-0.659576\pi$
$383$	−18.8105	−0.961170	−0.480585	+	0.876948i	$0.659576\pi$
$384$	−2.62663	−0.134040
$385$	5.38087	0.274235
$386$	−20.8039	−1.05889
$387$	−35.4175	−1.80037
$388$	−6.27376	−0.318502
$389$	19.2467	0.975849	0.487924	−	0.872886i	$-0.337754\pi$
$389$	19.2467	0.975849	0.487924	+	0.872886i	$0.337754\pi$
$390$	24.5565	1.24347
$391$	0.647280	0.0327343
$392$	−6.24438	−0.315389
$393$	19.9206	1.00486
$394$	4.24154	0.213686
$395$	16.7929	0.844943
$396$	−11.9607	−0.601049
$397$	3.92053	0.196766	0.0983829	−	0.995149i	$-0.468633\pi$
$397$	3.92053	0.196766	0.0983829	+	0.995149i	$0.468633\pi$
$398$	−26.0230	−1.30442
$399$	10.1444	0.507854
$400$	−0.927713	−0.0463857
$401$	14.4986	0.724023	0.362012	−	0.932174i	$-0.382090\pi$
$401$	14.4986	0.724023	0.362012	+	0.932174i	$0.382090\pi$
$402$	−12.3376	−0.615346
$403$	−6.74476	−0.335980
$404$	−12.2375	−0.608840
$405$	11.0866	0.550896
$406$	6.35980	0.315631
$407$	−28.7060	−1.42290
$408$	2.03587	0.100791
$409$	−23.4191	−1.15800	−0.579001	−	0.815327i	$-0.696557\pi$
$409$	−23.4191	−1.15800	−0.579001	+	0.815327i	$0.696557\pi$
$410$	−3.35323	−0.165604
$411$	−14.3748	−0.709056
$412$	−0.130840	−0.00644603
$413$	9.85897	0.485128
$414$	−3.25623	−0.160035
$415$	13.4945	0.662417
$416$	4.63284	0.227144
$417$	32.5786	1.59538
$418$	13.6288	0.666607
$419$	11.8966	0.581187	0.290593	−	0.956847i	$-0.406147\pi$
$419$	11.8966	0.581187	0.290593	+	0.956847i	$0.406147\pi$
$420$	4.60754	0.224825
$421$	6.94450	0.338454	0.169227	−	0.985577i	$-0.445873\pi$
$421$	6.94450	0.338454	0.169227	+	0.985577i	$0.445873\pi$
$422$	−3.43432	−0.167180
$423$	−4.02885	−0.195889
$424$	4.38682	0.213043
$425$	0.719060	0.0348795
$426$	−15.3008	−0.741327
$427$	−0.780976	−0.0377941
$428$	−13.7200	−0.663182
$429$	37.3276	1.80219
$430$	18.3300	0.883950
$431$	15.5617	0.749580	0.374790	−	0.927110i	$-0.377715\pi$
$431$	15.5617	0.749580	0.374790	+	0.927110i	$0.377715\pi$
$432$	−2.36185	−0.113635
$433$	21.6943	1.04256	0.521280	−	0.853386i	$-0.325455\pi$
$433$	21.6943	1.04256	0.521280	+	0.853386i	$0.325455\pi$
$434$	−1.26552	−0.0607469
$435$	38.7803	1.85937
$436$	−16.3980	−0.785323
$437$	3.71036	0.177490
$438$	−2.58780	−0.123650
$439$	20.4553	0.976276	0.488138	−	0.872766i	$-0.337676\pi$
$439$	20.4553	0.976276	0.488138	+	0.872766i	$0.337676\pi$
$440$	6.19016	0.295104
$441$	−24.3481	−1.15943
$442$	−3.59087	−0.170800
$443$	−2.10477	−0.100001	−0.0500004	−	0.998749i	$-0.515922\pi$
$443$	−2.10477	−0.100001	−0.0500004	+	0.998749i	$0.515922\pi$
$444$	−24.5804	−1.16653
$445$	10.6301	0.503916
$446$	−24.7701	−1.17290
$447$	−14.9054	−0.705003
$448$	0.869262	0.0410688
$449$	14.3330	0.676416	0.338208	−	0.941071i	$-0.390179\pi$
$449$	14.3330	0.676416	0.338208	+	0.941071i	$0.390179\pi$
$450$	−3.61733	−0.170523
$451$	−5.09715	−0.240015
$452$	16.7815	0.789337
$453$	7.29788	0.342884
$454$	−15.3499	−0.720407
$455$	−8.12676	−0.380988
$456$	11.6701	0.546502
$457$	12.3206	0.576332	0.288166	−	0.957581i	$-0.406955\pi$
$457$	12.3206	0.576332	0.288166	+	0.957581i	$0.406955\pi$
$458$	−17.2681	−0.806887
$459$	1.83065	0.0854473
$460$	1.68523	0.0785743
$461$	4.23570	0.197276	0.0986381	−	0.995123i	$-0.468551\pi$
$461$	4.23570	0.197276	0.0986381	+	0.995123i	$0.468551\pi$
$462$	7.00378	0.325845
$463$	2.85761	0.132804	0.0664022	−	0.997793i	$-0.478848\pi$
$463$	2.85761	0.132804	0.0664022	+	0.997793i	$0.478848\pi$
$464$	7.31632	0.339652
$465$	−7.71680	−0.357858
$466$	−7.61184	−0.352612
$467$	−2.96500	−0.137204	−0.0686019	−	0.997644i	$-0.521854\pi$
$467$	−2.96500	−0.137204	−0.0686019	+	0.997644i	$0.521854\pi$
$468$	18.0644	0.835025
$469$	4.08304	0.188537
$470$	2.08509	0.0961781
$471$	−19.4069	−0.894223
$472$	11.3418	0.522048
$473$	27.8628	1.28114
$474$	21.8578	1.00396
$475$	4.12182	0.189122
$476$	−0.673755	−0.0308815
$477$	17.1051	0.783187
$478$	20.7851	0.950689
$479$	−29.9074	−1.36650	−0.683251	−	0.730183i	$-0.739435\pi$
$479$	−29.9074	−1.36650	−0.683251	+	0.730183i	$0.739435\pi$
$480$	5.30052	0.241934
$481$	43.3548	1.97681
$482$	−6.31856	−0.287802
$483$	1.90673	0.0867594
$484$	−1.59052	−0.0722962
$485$	12.6604	0.574879
$486$	21.5159	0.975982
$487$	−33.0113	−1.49588	−0.747941	−	0.663765i	$-0.768958\pi$
$487$	−33.0113	−1.49588	−0.747941	+	0.663765i	$0.768958\pi$
$488$	−0.898436	−0.0406703
$489$	−51.5096	−2.32935
$490$	12.6011	0.569260
$491$	15.9388	0.719308	0.359654	−	0.933086i	$-0.382895\pi$
$491$	15.9388	0.719308	0.359654	+	0.933086i	$0.382895\pi$
$492$	−4.36459	−0.196771
$493$	−5.67080	−0.255400
$494$	−20.5837	−0.926103
$495$	24.1367	1.08486
$496$	−1.45586	−0.0653699
$497$	5.06368	0.227137
$498$	17.5645	0.787084
$499$	30.3075	1.35675	0.678375	−	0.734716i	$-0.262685\pi$
$499$	30.3075	1.35675	0.678375	+	0.734716i	$0.262685\pi$
$500$	11.9621	0.534960
$501$	41.8950	1.87173
$502$	1.19661	0.0534074
$503$	−21.0148	−0.937006	−0.468503	−	0.883462i	$-0.655206\pi$
$503$	−21.0148	−0.937006	−0.468503	+	0.883462i	$0.655206\pi$
$504$	3.38942	0.150977
$505$	24.6952	1.09892
$506$	2.56167	0.113880
$507$	−22.2298	−0.987260
$508$	−18.7825	−0.833337
$509$	8.40933	0.372737	0.186368	−	0.982480i	$-0.440328\pi$
$509$	8.40933	0.372737	0.186368	+	0.982480i	$0.440328\pi$
$510$	−4.10837	−0.181922
$511$	0.856411	0.0378854
$512$	1.00000	0.0441942
$513$	10.4937	0.463308
$514$	0.713759	0.0314825
$515$	0.264034	0.0116347
$516$	23.8584	1.05031
$517$	3.16949	0.139394
$518$	8.13467	0.357417
$519$	56.6297	2.48577
$520$	−9.34903	−0.409982
$521$	−30.4088	−1.33223	−0.666116	−	0.745848i	$-0.732045\pi$
$521$	−30.4088	−1.33223	−0.666116	+	0.745848i	$0.732045\pi$
$522$	28.5278	1.24863
$523$	14.4158	0.630359	0.315179	−	0.949032i	$-0.397935\pi$
$523$	14.4158	0.630359	0.315179	+	0.949032i	$0.397935\pi$
$524$	−7.58408	−0.331312
$525$	2.11818	0.0924451
$526$	30.3768	1.32449
$527$	1.12842	0.0491547
$528$	8.05716	0.350643
$529$	−22.3026	−0.969678
$530$	−8.85257	−0.384531
$531$	44.2238	1.91915
$532$	−3.86212	−0.167444
$533$	7.69825	0.333448
$534$	13.8363	0.598753
$535$	27.6869	1.19701
$536$	4.69714	0.202885
$537$	−10.9645	−0.473154
$538$	−0.769870	−0.0331915
$539$	19.1546	0.825046
$540$	4.76620	0.205105
$541$	−25.8777	−1.11257	−0.556284	−	0.830992i	$-0.687773\pi$
$541$	−25.8777	−1.11257	−0.556284	+	0.830992i	$0.687773\pi$
$542$	6.50686	0.279494
$543$	26.5315	1.13857
$544$	−0.775089	−0.0332317
$545$	33.0911	1.41747
$546$	−10.5778	−0.452690
$547$	−23.5624	−1.00745	−0.503727	−	0.863863i	$-0.668038\pi$
$547$	−23.5624	−1.00745	−0.503727	+	0.863863i	$0.668038\pi$
$548$	5.47271	0.233783
$549$	−3.50318	−0.149512
$550$	2.84575	0.121343
$551$	−32.5063	−1.38482
$552$	2.19351	0.0933620
$553$	−7.23365	−0.307606
$554$	−19.1652	−0.814252
$555$	49.6030	2.10553
$556$	−12.4032	−0.526012
$557$	−3.83761	−0.162605	−0.0813023	−	0.996689i	$-0.525908\pi$
$557$	−3.83761	−0.162605	−0.0813023	+	0.996689i	$0.525908\pi$
$558$	−5.67667	−0.240313
$559$	−42.0814	−1.77985
$560$	−1.75416	−0.0741269
$561$	−6.24502	−0.263665
$562$	19.9604	0.841980
$563$	−2.79561	−0.117821	−0.0589105	−	0.998263i	$-0.518763\pi$
$563$	−2.79561	−0.117821	−0.0589105	+	0.998263i	$0.518763\pi$
$564$	2.71397	0.114279
$565$	−33.8650	−1.42471
$566$	21.5655	0.906464
$567$	−4.77561	−0.200557
$568$	5.82527	0.244423
$569$	−23.4932	−0.984886	−0.492443	−	0.870345i	$-0.663896\pi$
$569$	−23.4932	−0.984886	−0.492443	+	0.870345i	$0.663896\pi$
$570$	−23.5501	−0.986407
$571$	−15.6974	−0.656916	−0.328458	−	0.944519i	$-0.606529\pi$
$571$	−15.6974	−0.656916	−0.328458	+	0.944519i	$0.606529\pi$
$572$	−14.2112	−0.594200
$573$	30.7048	1.28271
$574$	1.44442	0.0602891
$575$	0.774737	0.0323088
$576$	3.89919	0.162466
$577$	−24.7883	−1.03195	−0.515976	−	0.856603i	$-0.672571\pi$
$577$	−24.7883	−1.03195	−0.515976	+	0.856603i	$0.672571\pi$
$578$	−16.3992	−0.682118
$579$	54.6443	2.27094
$580$	−14.7643	−0.613053
$581$	−5.81282	−0.241157
$582$	16.4789	0.683071
$583$	−13.4565	−0.557312
$584$	0.985216	0.0407685
$585$	−36.4537	−1.50718
$586$	9.51601	0.393103
$587$	3.28898	0.135751	0.0678753	−	0.997694i	$-0.478378\pi$
$587$	3.28898	0.135751	0.0678753	+	0.997694i	$0.478378\pi$
$588$	16.4017	0.676395
$589$	6.46836	0.266524
$590$	−22.8876	−0.942268
$591$	−11.1410	−0.458278
$592$	9.35814	0.384617
$593$	33.8030	1.38812	0.694062	−	0.719915i	$-0.255819\pi$
$593$	33.8030	1.38812	0.694062	+	0.719915i	$0.255819\pi$
$594$	7.24496	0.297264
$595$	1.35963	0.0557395
$596$	5.67473	0.232446
$597$	68.3530	2.79750
$598$	−3.86890	−0.158211
$599$	−26.9428	−1.10085	−0.550427	−	0.834883i	$-0.685535\pi$
$599$	−26.9428	−1.10085	−0.550427	+	0.834883i	$0.685535\pi$
$600$	2.43676	0.0994804
$601$	15.9044	0.648756	0.324378	−	0.945928i	$-0.394845\pi$
$601$	15.9044	0.648756	0.324378	+	0.945928i	$0.394845\pi$
$602$	−7.89575	−0.321807
$603$	18.3151	0.745847
$604$	−2.77842	−0.113052
$605$	3.20965	0.130491
$606$	32.1435	1.30574
$607$	−31.7959	−1.29055	−0.645277	−	0.763948i	$-0.723258\pi$
$607$	−31.7959	−1.29055	−0.645277	+	0.763948i	$0.723258\pi$
$608$	−4.44299	−0.180187
$609$	−16.7048	−0.676914
$610$	1.81304	0.0734077
$611$	−4.78689	−0.193657
$612$	−3.02222	−0.122166
$613$	37.7191	1.52346	0.761730	−	0.647895i	$-0.224350\pi$
$613$	37.7191	1.52346	0.761730	+	0.647895i	$0.224350\pi$
$614$	26.7324	1.07883
$615$	8.80770	0.355161
$616$	−2.66645	−0.107434
$617$	19.4220	0.781900	0.390950	−	0.920412i	$-0.372147\pi$
$617$	19.4220	0.781900	0.390950	+	0.920412i	$0.372147\pi$
$618$	0.343669	0.0138244
$619$	−10.2017	−0.410041	−0.205021	−	0.978758i	$-0.565726\pi$
$619$	−10.2017	−0.410041	−0.205021	+	0.978758i	$0.565726\pi$
$620$	2.93791	0.117989
$621$	1.97239	0.0791494
$622$	−3.78382	−0.151717
$623$	−4.57899	−0.183453
$624$	−12.1688	−0.487141
$625$	−19.5008	−0.780031
$626$	−6.50925	−0.260162
$627$	−35.7979	−1.42963
$628$	7.38851	0.294834
$629$	−7.25339	−0.289212
$630$	−6.83982	−0.272505
$631$	38.8917	1.54825	0.774127	−	0.633031i	$-0.218189\pi$
$631$	38.8917	1.54825	0.774127	+	0.633031i	$0.218189\pi$
$632$	−8.32160	−0.331016
$633$	9.02071	0.358541
$634$	−18.8198	−0.747430
$635$	37.9028	1.50413
$636$	−11.5226	−0.456899
$637$	−28.9292	−1.14622
$638$	−22.4427	−0.888516
$639$	22.7138	0.898546
$640$	−2.01799	−0.0797681
$641$	25.4677	1.00591	0.502956	−	0.864312i	$-0.332246\pi$
$641$	25.4677	1.00591	0.502956	+	0.864312i	$0.332246\pi$
$642$	36.0374	1.42228
$643$	−37.3387	−1.47250	−0.736248	−	0.676712i	$-0.763404\pi$
$643$	−37.3387	−1.47250	−0.736248	+	0.676712i	$0.763404\pi$
$644$	−0.725924	−0.0286054
$645$	−48.1461	−1.89575
$646$	3.44371	0.135491
$647$	−27.3563	−1.07549	−0.537744	−	0.843108i	$-0.680723\pi$
$647$	−27.3563	−1.07549	−0.537744	+	0.843108i	$0.680723\pi$
$648$	−5.49386	−0.215819
$649$	−34.7908	−1.36566
$650$	−4.29795	−0.168579
$651$	3.32406	0.130280
$652$	19.6105	0.768007
$653$	15.4388	0.604169	0.302084	−	0.953281i	$-0.402318\pi$
$653$	15.4388	0.604169	0.302084	+	0.953281i	$0.402318\pi$
$654$	43.0716	1.68423
$655$	15.3046	0.598000
$656$	1.66167	0.0648772
$657$	3.84155	0.149873
$658$	−0.898166	−0.0350142
$659$	22.7415	0.885883	0.442942	−	0.896550i	$-0.353935\pi$
$659$	22.7415	0.885883	0.442942	+	0.896550i	$0.353935\pi$
$660$	−16.2593	−0.632892
$661$	39.1052	1.52102	0.760508	−	0.649329i	$-0.224950\pi$
$661$	39.1052	1.52102	0.760508	+	0.649329i	$0.224950\pi$
$662$	16.9172	0.657506
$663$	9.43188	0.366304
$664$	−6.68708	−0.259509
$665$	7.79372	0.302228
$666$	36.4892	1.41393
$667$	−6.10989	−0.236576
$668$	−15.9501	−0.617127
$669$	65.0618	2.51544
$670$	−9.47878	−0.366197
$671$	2.75594	0.106392
$672$	−2.28323	−0.0880775
$673$	0.167066	0.00643992	0.00321996	−	0.999995i	$-0.498975\pi$
$673$	0.167066	0.00643992	0.00321996	+	0.999995i	$0.498975\pi$
$674$	5.19508	0.200107
$675$	2.19112	0.0843364
$676$	8.46323	0.325509
$677$	14.2245	0.546693	0.273346	−	0.961916i	$-0.411870\pi$
$677$	14.2245	0.546693	0.273346	+	0.961916i	$0.411870\pi$
$678$	−44.0789	−1.69284
$679$	−5.45354	−0.209288
$680$	1.56412	0.0599814
$681$	40.3186	1.54501
$682$	4.46582	0.171005
$683$	−23.4179	−0.896060	−0.448030	−	0.894019i	$-0.647874\pi$
$683$	−23.4179	−0.896060	−0.448030	+	0.894019i	$0.647874\pi$
$684$	−17.3241	−0.662403
$685$	−11.0439	−0.421965
$686$	−11.5128	−0.439562
$687$	45.3570	1.73048
$688$	−9.08328	−0.346297
$689$	20.3235	0.774262
$690$	−4.42648	−0.168513
$691$	48.6733	1.85162	0.925809	−	0.377991i	$-0.123385\pi$
$691$	48.6733	1.85162	0.925809	+	0.377991i	$0.123385\pi$
$692$	−21.5598	−0.819581
$693$	−10.3970	−0.394950
$694$	−16.9331	−0.642771
$695$	25.0295	0.949423
$696$	−19.2173	−0.728429
$697$	−1.28794	−0.0487842
$698$	−22.8937	−0.866539
$699$	19.9935	0.756224
$700$	−0.806426	−0.0304800
$701$	50.1979	1.89595	0.947975	−	0.318344i	$-0.103127\pi$
$701$	50.1979	1.89595	0.947975	+	0.318344i	$0.103127\pi$
$702$	−10.9421	−0.412983
$703$	−41.5781	−1.56815
$704$	−3.06749	−0.115610
$705$	−5.47677	−0.206267
$706$	−8.90369	−0.335095
$707$	−10.6376	−0.400069
$708$	−29.7907	−1.11960
$709$	21.5133	0.807951	0.403975	−	0.914770i	$-0.367628\pi$
$709$	21.5133	0.807951	0.403975	+	0.914770i	$0.367628\pi$
$710$	−11.7553	−0.441170
$711$	−32.4476	−1.21688
$712$	−5.26768	−0.197415
$713$	1.21579	0.0455318
$714$	1.76971	0.0662296
$715$	28.6781	1.07250
$716$	4.17437	0.156003
$717$	−54.5949	−2.03888
$718$	11.3467	0.423455
$719$	−26.1358	−0.974701	−0.487350	−	0.873206i	$-0.662037\pi$
$719$	−26.1358	−0.974701	−0.487350	+	0.873206i	$0.662037\pi$
$720$	−7.86854	−0.293243
$721$	−0.113734	−0.00423569
$722$	0.740152	0.0275456
$723$	16.5965	0.617231
$724$	−10.1009	−0.375399
$725$	−6.78745	−0.252079
$726$	4.17770	0.155049
$727$	−3.28252	−0.121742	−0.0608709	−	0.998146i	$-0.519388\pi$
$727$	−3.28252	−0.121742	−0.0608709	+	0.998146i	$0.519388\pi$
$728$	4.02715	0.149256
$729$	−40.0328	−1.48270
$730$	−1.98816	−0.0735850
$731$	7.04035	0.260397
$732$	2.35986	0.0872230
$733$	11.8125	0.436306	0.218153	−	0.975915i	$-0.429997\pi$
$733$	11.8125	0.436306	0.218153	+	0.975915i	$0.429997\pi$
$734$	−29.6837	−1.09564
$735$	−33.0985	−1.22086
$736$	−0.835104	−0.0307823
$737$	−14.4084	−0.530741
$738$	6.47917	0.238501
$739$	−17.3528	−0.638332	−0.319166	−	0.947699i	$-0.603403\pi$
$739$	−17.3528	−0.638332	−0.319166	+	0.947699i	$0.603403\pi$
$740$	−18.8846	−0.694213
$741$	54.0657	1.98615
$742$	3.81330	0.139991
$743$	−52.0966	−1.91124	−0.955619	−	0.294606i	$-0.904812\pi$
$743$	−52.0966	−1.91124	−0.955619	+	0.294606i	$0.904812\pi$
$744$	3.82400	0.140195
$745$	−11.4516	−0.419553
$746$	19.6595	0.719784
$747$	−26.0742	−0.954006
$748$	2.37758	0.0869328
$749$	−11.9263	−0.435777
$750$	−31.4200	−1.14729
$751$	−11.5572	−0.421730	−0.210865	−	0.977515i	$-0.567628\pi$
$751$	−11.5572	−0.421730	−0.210865	+	0.977515i	$0.567628\pi$
$752$	−1.03325	−0.0376788
$753$	−3.14306	−0.114539
$754$	33.8954	1.23440
$755$	5.60682	0.204053
$756$	−2.05307	−0.0746694
$757$	4.04044	0.146852	0.0734262	−	0.997301i	$-0.476607\pi$
$757$	4.04044	0.146852	0.0734262	+	0.997301i	$0.476607\pi$
$758$	−6.66140	−0.241953
$759$	−6.72857	−0.244231
$760$	8.96591	0.325228
$761$	−25.4419	−0.922268	−0.461134	−	0.887330i	$-0.652557\pi$
$761$	−25.4419	−0.922268	−0.461134	+	0.887330i	$0.652557\pi$
$762$	49.3346	1.78721
$763$	−14.2542	−0.516036
$764$	−11.6898	−0.422922
$765$	6.09882	0.220503
$766$	−18.8105	−0.679650
$767$	52.5447	1.89728
$768$	−2.62663	−0.0947804
$769$	8.44581	0.304564	0.152282	−	0.988337i	$-0.451338\pi$
$769$	8.44581	0.304564	0.152282	+	0.988337i	$0.451338\pi$
$770$	5.38087	0.193913
$771$	−1.87478	−0.0675186
$772$	−20.8039	−0.748750
$773$	−21.1247	−0.759801	−0.379900	−	0.925027i	$-0.624042\pi$
$773$	−21.1247	−0.759801	−0.379900	+	0.925027i	$0.624042\pi$
$774$	−35.4175	−1.27305
$775$	1.35062	0.0485156
$776$	−6.27376	−0.225215
$777$	−21.3668	−0.766529
$778$	19.2467	0.690029
$779$	−7.38277	−0.264515
$780$	24.5565	0.879263
$781$	−17.8689	−0.639401
$782$	0.647280	0.0231467
$783$	−17.2801	−0.617539
$784$	−6.24438	−0.223014
$785$	−14.9100	−0.532159
$786$	19.9206	0.710543
$787$	17.2896	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$787$	17.2896	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$788$	4.24154	0.151099
$789$	−79.7888	−2.84055
$790$	16.7929	0.597465
$791$	14.5876	0.518674
$792$	−11.9607	−0.425006
$793$	−4.16231	−0.147808
$794$	3.92053	0.139134
$795$	23.2524	0.824679
$796$	−26.0230	−0.922362
$797$	−12.0918	−0.428312	−0.214156	−	0.976799i	$-0.568700\pi$
$797$	−12.0918	−0.428312	−0.214156	+	0.976799i	$0.568700\pi$
$798$	10.1444	0.359107
$799$	0.800862	0.0283325
$800$	−0.927713	−0.0327996
$801$	−20.5397	−0.725735
$802$	14.4986	0.511962
$803$	−3.02214	−0.106649
$804$	−12.3376	−0.435115
$805$	1.46491	0.0516312
$806$	−6.74476	−0.237574
$807$	2.02217	0.0711836
$808$	−12.2375	−0.430515
$809$	8.28065	0.291132	0.145566	−	0.989349i	$-0.453500\pi$
$809$	8.28065	0.291132	0.145566	+	0.989349i	$0.453500\pi$
$810$	11.0866	0.389542
$811$	−17.0409	−0.598388	−0.299194	−	0.954192i	$-0.596718\pi$
$811$	−17.0409	−0.598388	−0.299194	+	0.954192i	$0.596718\pi$
$812$	6.35980	0.223185
$813$	−17.0911	−0.599412
$814$	−28.7060	−1.00614
$815$	−39.5739	−1.38621
$816$	2.03587	0.0712698
$817$	40.3569	1.41191
$818$	−23.4191	−0.818831
$819$	15.7027	0.548695
$820$	−3.35323	−0.117100
$821$	−23.2456	−0.811277	−0.405638	−	0.914034i	$-0.632951\pi$
$821$	−23.2456	−0.811277	−0.405638	+	0.914034i	$0.632951\pi$
$822$	−14.3748	−0.501379
$823$	31.2422	1.08904	0.544518	−	0.838749i	$-0.316713\pi$
$823$	31.2422	1.08904	0.544518	+	0.838749i	$0.316713\pi$
$824$	−0.130840	−0.00455803
$825$	−7.47474	−0.260237
$826$	9.85897	0.343038
$827$	31.8831	1.10869	0.554343	−	0.832288i	$-0.312970\pi$
$827$	31.8831	1.10869	0.554343	+	0.832288i	$0.312970\pi$
$828$	−3.25623	−0.113162
$829$	41.9577	1.45725	0.728626	−	0.684912i	$-0.240159\pi$
$829$	41.9577	1.45725	0.728626	+	0.684912i	$0.240159\pi$
$830$	13.4945	0.468400
$831$	50.3399	1.74627
$832$	4.63284	0.160615
$833$	4.83995	0.167694
$834$	32.5786	1.12810
$835$	32.1871	1.11388
$836$	13.6288	0.471362
$837$	3.43852	0.118853
$838$	11.8966	0.410961
$839$	24.1542	0.833895	0.416947	−	0.908931i	$-0.363100\pi$
$839$	24.1542	0.833895	0.416947	+	0.908931i	$0.363100\pi$
$840$	4.60754	0.158975
$841$	24.5285	0.845811
$842$	6.94450	0.239323
$843$	−52.4287	−1.80574
$844$	−3.43432	−0.118214
$845$	−17.0787	−0.587526
$846$	−4.02885	−0.138515
$847$	−1.38258	−0.0475058
$848$	4.38682	0.150644
$849$	−56.6445	−1.94403
$850$	0.719060	0.0246636
$851$	−7.81501	−0.267895
$852$	−15.3008	−0.524198
$853$	−14.0040	−0.479487	−0.239743	−	0.970836i	$-0.577063\pi$
$853$	−14.0040	−0.479487	−0.239743	+	0.970836i	$0.577063\pi$
$854$	−0.780976	−0.0267245
$855$	34.9598	1.19560
$856$	−13.7200	−0.468940
$857$	−44.2853	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$857$	−44.2853	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$858$	37.3276	1.27434
$859$	39.4789	1.34700	0.673501	−	0.739187i	$-0.264790\pi$
$859$	39.4789	1.34700	0.673501	+	0.739187i	$0.264790\pi$
$860$	18.3300	0.625047
$861$	−3.79397	−0.129298
$862$	15.5617	0.530033
$863$	−41.8207	−1.42359	−0.711796	−	0.702386i	$-0.752118\pi$
$863$	−41.8207	−1.42359	−0.711796	+	0.702386i	$0.752118\pi$
$864$	−2.36185	−0.0803519
$865$	43.5075	1.47930
$866$	21.6943	0.737202
$867$	43.0748	1.46290
$868$	−1.26552	−0.0429546
$869$	25.5264	0.865925
$870$	38.7803	1.31477
$871$	21.7611	0.737347
$872$	−16.3980	−0.555307
$873$	−24.4626	−0.827935
$874$	3.71036	0.125505
$875$	10.3982	0.351522
$876$	−2.58780	−0.0874337
$877$	40.8081	1.37799	0.688996	−	0.724766i	$-0.258052\pi$
$877$	40.8081	1.37799	0.688996	+	0.724766i	$0.258052\pi$
$878$	20.4553	0.690331
$879$	−24.9950	−0.843062
$880$	6.19016	0.208670
$881$	39.4179	1.32802	0.664011	−	0.747723i	$-0.268853\pi$
$881$	39.4179	1.32802	0.664011	+	0.747723i	$0.268853\pi$
$882$	−24.3481	−0.819842
$883$	−32.1634	−1.08238	−0.541192	−	0.840899i	$-0.682027\pi$
$883$	−32.1634	−1.08238	−0.541192	+	0.840899i	$0.682027\pi$
$884$	−3.59087	−0.120774
$885$	60.1173	2.02082
$886$	−2.10477	−0.0707112
$887$	6.25831	0.210133	0.105067	−	0.994465i	$-0.466494\pi$
$887$	6.25831	0.210133	0.105067	+	0.994465i	$0.466494\pi$
$888$	−24.5804	−0.824863
$889$	−16.3269	−0.547586
$890$	10.6301	0.356323
$891$	16.8524	0.564575
$892$	−24.7701	−0.829363
$893$	4.59073	0.153623
$894$	−14.9054	−0.498512
$895$	−8.42383	−0.281578
$896$	0.869262	0.0290400
$897$	10.1622	0.339306
$898$	14.3330	0.478298
$899$	−10.6515	−0.355248
$900$	−3.61733	−0.120578
$901$	−3.40018	−0.113276
$902$	−5.09715	−0.169716
$903$	20.7392	0.690158
$904$	16.7815	0.558146
$905$	20.3836	0.677574
$906$	7.29788	0.242456
$907$	5.58079	0.185307	0.0926535	−	0.995698i	$-0.470465\pi$
$907$	5.58079	0.185307	0.0926535	+	0.995698i	$0.470465\pi$
$908$	−15.3499	−0.509405
$909$	−47.7166	−1.58266
$910$	−8.12676	−0.269399
$911$	−18.1293	−0.600650	−0.300325	−	0.953837i	$-0.597095\pi$
$911$	−18.1293	−0.600650	−0.300325	+	0.953837i	$0.597095\pi$
$912$	11.6701	0.386436
$913$	20.5125	0.678866
$914$	12.3206	0.407528
$915$	−4.76218	−0.157433
$916$	−17.2681	−0.570555
$917$	−6.59255	−0.217705
$918$	1.83065	0.0604204
$919$	−18.3856	−0.606485	−0.303242	−	0.952913i	$-0.598069\pi$
$919$	−18.3856	−0.606485	−0.303242	+	0.952913i	$0.598069\pi$
$920$	1.68523	0.0555604
$921$	−70.2161	−2.31370
$922$	4.23570	0.139495
$923$	26.9875	0.888306
$924$	7.00378	0.230408
$925$	−8.68167	−0.285451
$926$	2.85761	0.0939068
$927$	−0.510171	−0.0167562
$928$	7.31632	0.240170
$929$	5.84001	0.191605	0.0958023	−	0.995400i	$-0.469458\pi$
$929$	5.84001	0.191605	0.0958023	+	0.995400i	$0.469458\pi$
$930$	−7.71680	−0.253044
$931$	27.7437	0.909264
$932$	−7.61184	−0.249334
$933$	9.93870	0.325378
$934$	−2.96500	−0.0970177
$935$	−4.79793	−0.156909
$936$	18.0644	0.590452
$937$	42.9149	1.40197	0.700985	−	0.713176i	$-0.252744\pi$
$937$	42.9149	1.40197	0.700985	+	0.713176i	$0.252744\pi$
$938$	4.08304	0.133316
$939$	17.0974	0.557953
$940$	2.08509	0.0680082
$941$	−52.2286	−1.70260	−0.851302	−	0.524675i	$-0.824187\pi$
$941$	−52.2286	−1.70260	−0.851302	+	0.524675i	$0.824187\pi$
$942$	−19.4069	−0.632311
$943$	−1.38767	−0.0451886
$944$	11.3418	0.369143
$945$	4.14307	0.134774
$946$	27.8628	0.905899
$947$	−46.7622	−1.51957	−0.759784	−	0.650176i	$-0.774695\pi$
$947$	−46.7622	−1.51957	−0.759784	+	0.650176i	$0.774695\pi$
$948$	21.8578	0.709908
$949$	4.56435	0.148165
$950$	4.12182	0.133729
$951$	49.4327	1.60296
$952$	−0.673755	−0.0218365
$953$	28.3752	0.919162	0.459581	−	0.888136i	$-0.348000\pi$
$953$	28.3752	0.919162	0.459581	+	0.888136i	$0.348000\pi$
$954$	17.1051	0.553797
$955$	23.5899	0.763351
$956$	20.7851	0.672239
$957$	58.9488	1.90554
$958$	−29.9074	−0.966263
$959$	4.75722	0.153619
$960$	5.30052	0.171074
$961$	−28.8805	−0.931628
$962$	43.3548	1.39781
$963$	−53.4970	−1.72392
$964$	−6.31856	−0.203507
$965$	41.9822	1.35145
$966$	1.90673	0.0613482
$967$	−2.41319	−0.0776029	−0.0388015	−	0.999247i	$-0.512354\pi$
$967$	−2.41319	−0.0776029	−0.0388015	+	0.999247i	$0.512354\pi$
$968$	−1.59052	−0.0511211
$969$	−9.04536	−0.290579
$970$	12.6604	0.406501
$971$	−52.8134	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$971$	−52.8134	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$972$	21.5159	0.690123
$973$	−10.7816	−0.345642
$974$	−33.0113	−1.05775
$975$	11.2891	0.361542
$976$	−0.898436	−0.0287582
$977$	19.3440	0.618870	0.309435	−	0.950921i	$-0.399860\pi$
$977$	19.3440	0.618870	0.309435	+	0.950921i	$0.399860\pi$
$978$	−51.5096	−1.64710
$979$	16.1585	0.516429
$980$	12.6011	0.402528
$981$	−63.9391	−2.04142
$982$	15.9388	0.508628
$983$	−43.6044	−1.39077	−0.695383	−	0.718639i	$-0.744765\pi$
$983$	−43.6044	−1.39077	−0.695383	+	0.718639i	$0.744765\pi$
$984$	−4.36459	−0.139138
$985$	−8.55939	−0.272725
$986$	−5.67080	−0.180595
$987$	2.35915	0.0750926
$988$	−20.5837	−0.654854
$989$	7.58548	0.241204
$990$	24.1367	0.767113
$991$	1.06141	0.0337167	0.0168583	−	0.999858i	$-0.494634\pi$
$991$	1.06141	0.0337167	0.0168583	+	0.999858i	$0.494634\pi$
$992$	−1.45586	−0.0462235
$993$	−44.4353	−1.41011
$994$	5.06368	0.160610
$995$	52.5143	1.66481
$996$	17.5645	0.556552
$997$	25.6535	0.812455	0.406227	−	0.913772i	$-0.366844\pi$
$997$	25.6535	0.812455	0.406227	+	0.913772i	$0.366844\pi$
$998$	30.3075	0.959367
$999$	−22.1025	−0.699294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.c.1.6	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.c.1.6	✓	31	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.62663 q^{3} +1.00000 q^{4} -2.01799 q^{5} -2.62663 q^{6} +0.869262 q^{7} +1.00000 q^{8} +3.89919 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.62663 q^{3} +1.00000 q^{4} -2.01799 q^{5} -2.62663 q^{6} +0.869262 q^{7} +1.00000 q^{8} +3.89919 q^{9} -2.01799 q^{10} -3.06749 q^{11} -2.62663 q^{12} +4.63284 q^{13} +0.869262 q^{14} +5.30052 q^{15} +1.00000 q^{16} -0.775089 q^{17} +3.89919 q^{18} -4.44299 q^{19} -2.01799 q^{20} -2.28323 q^{21} -3.06749 q^{22} -0.835104 q^{23} -2.62663 q^{24} -0.927713 q^{25} +4.63284 q^{26} -2.36185 q^{27} +0.869262 q^{28} +7.31632 q^{29} +5.30052 q^{30} -1.45586 q^{31} +1.00000 q^{32} +8.05716 q^{33} -0.775089 q^{34} -1.75416 q^{35} +3.89919 q^{36} +9.35814 q^{37} -4.44299 q^{38} -12.1688 q^{39} -2.01799 q^{40} +1.66167 q^{41} -2.28323 q^{42} -9.08328 q^{43} -3.06749 q^{44} -7.86854 q^{45} -0.835104 q^{46} -1.03325 q^{47} -2.62663 q^{48} -6.24438 q^{49} -0.927713 q^{50} +2.03587 q^{51} +4.63284 q^{52} +4.38682 q^{53} -2.36185 q^{54} +6.19016 q^{55} +0.869262 q^{56} +11.6701 q^{57} +7.31632 q^{58} +11.3418 q^{59} +5.30052 q^{60} -0.898436 q^{61} -1.45586 q^{62} +3.38942 q^{63} +1.00000 q^{64} -9.34903 q^{65} +8.05716 q^{66} +4.69714 q^{67} -0.775089 q^{68} +2.19351 q^{69} -1.75416 q^{70} +5.82527 q^{71} +3.89919 q^{72} +0.985216 q^{73} +9.35814 q^{74} +2.43676 q^{75} -4.44299 q^{76} -2.66645 q^{77} -12.1688 q^{78} -8.32160 q^{79} -2.01799 q^{80} -5.49386 q^{81} +1.66167 q^{82} -6.68708 q^{83} -2.28323 q^{84} +1.56412 q^{85} -9.08328 q^{86} -19.2173 q^{87} -3.06749 q^{88} -5.26768 q^{89} -7.86854 q^{90} +4.02715 q^{91} -0.835104 q^{92} +3.82400 q^{93} -1.03325 q^{94} +8.96591 q^{95} -2.62663 q^{96} -6.27376 q^{97} -6.24438 q^{98} -11.9607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9	\( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 31 * q^2 - 14 * q^3 + 31 * q^4 - 13 * q^5 - 14 * q^6 - 29 * q^7 + 31 * q^8 + 27 * q^9 - 13 * q^10 - 29 * q^11 - 14 * q^12 - 23 * q^13 - 29 * q^14 - 14 * q^15 + 31 * q^16 - 19 * q^17 + 27 * q^18 - 36 * q^19 - 13 * q^20 - 29 * q^22 - 24 * q^23 - 14 * q^24 + 4 * q^25 - 23 * q^26 - 65 * q^27 - 29 * q^28 + 18 * q^29 - 14 * q^30 - 41 * q^31 + 31 * q^32 - 18 * q^33 - 19 * q^34 - 28 * q^35 + 27 * q^36 - 31 * q^37 - 36 * q^38 - 31 * q^39 - 13 * q^40 - 51 * q^41 - 14 * q^43 - 29 * q^44 - 34 * q^45 - 24 * q^46 - 64 * q^47 - 14 * q^48 + 8 * q^49 + 4 * q^50 - 17 * q^51 - 23 * q^52 + 3 * q^53 - 65 * q^54 - 50 * q^55 - 29 * q^56 - 19 * q^57 + 18 * q^58 - 58 * q^59 - 14 * q^60 - 14 * q^61 - 41 * q^62 - 66 * q^63 + 31 * q^64 + 6 * q^65 - 18 * q^66 - 55 * q^67 - 19 * q^68 + q^69 - 28 * q^70 - 42 * q^71 + 27 * q^72 - 83 * q^73 - 31 * q^74 - 49 * q^75 - 36 * q^76 + 18 * q^77 - 31 * q^78 - 56 * q^79 - 13 * q^80 + 3 * q^81 - 51 * q^82 - 43 * q^83 + 4 * q^85 - 14 * q^86 - 76 * q^87 - 29 * q^88 + 17 * q^89 - 34 * q^90 - 49 * q^91 - 24 * q^92 - 24 * q^93 - 64 * q^94 - 43 * q^95 - 14 * q^96 - 98 * q^97 + 8 * q^98 - q^99

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