LMFDB - Embedded newform 8049.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, 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("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$1$
Dimension:	$104$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52808 q^{2} -1.00000 q^{3} +4.39119 q^{4} -2.04731 q^{5} +2.52808 q^{6} +3.64200 q^{7} -6.04511 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.52808 q^{2} -1.00000 q^{3} +4.39119 q^{4} -2.04731 q^{5} +2.52808 q^{6} +3.64200 q^{7} -6.04511 q^{8} +1.00000 q^{9} +5.17577 q^{10} -1.87626 q^{11} -4.39119 q^{12} +6.41879 q^{13} -9.20726 q^{14} +2.04731 q^{15} +6.50014 q^{16} +2.88113 q^{17} -2.52808 q^{18} +3.34047 q^{19} -8.99013 q^{20} -3.64200 q^{21} +4.74333 q^{22} -0.154262 q^{23} +6.04511 q^{24} -0.808509 q^{25} -16.2272 q^{26} -1.00000 q^{27} +15.9927 q^{28} +2.55966 q^{29} -5.17577 q^{30} -4.26768 q^{31} -4.34266 q^{32} +1.87626 q^{33} -7.28373 q^{34} -7.45631 q^{35} +4.39119 q^{36} -1.42252 q^{37} -8.44496 q^{38} -6.41879 q^{39} +12.3762 q^{40} +8.40629 q^{41} +9.20726 q^{42} +0.952507 q^{43} -8.23901 q^{44} -2.04731 q^{45} +0.389986 q^{46} -5.20366 q^{47} -6.50014 q^{48} +6.26416 q^{49} +2.04398 q^{50} -2.88113 q^{51} +28.1861 q^{52} -11.8257 q^{53} +2.52808 q^{54} +3.84129 q^{55} -22.0163 q^{56} -3.34047 q^{57} -6.47103 q^{58} -0.532287 q^{59} +8.99013 q^{60} +2.38966 q^{61} +10.7890 q^{62} +3.64200 q^{63} -2.02169 q^{64} -13.1413 q^{65} -4.74333 q^{66} -14.7480 q^{67} +12.6516 q^{68} +0.154262 q^{69} +18.8502 q^{70} -4.50886 q^{71} -6.04511 q^{72} -7.40126 q^{73} +3.59624 q^{74} +0.808509 q^{75} +14.6686 q^{76} -6.83334 q^{77} +16.2272 q^{78} -3.42284 q^{79} -13.3078 q^{80} +1.00000 q^{81} -21.2518 q^{82} -4.20847 q^{83} -15.9927 q^{84} -5.89858 q^{85} -2.40801 q^{86} -2.55966 q^{87} +11.3422 q^{88} -12.6772 q^{89} +5.17577 q^{90} +23.3772 q^{91} -0.677391 q^{92} +4.26768 q^{93} +13.1553 q^{94} -6.83898 q^{95} +4.34266 q^{96} -9.81142 q^{97} -15.8363 q^{98} -1.87626 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) $ 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * 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 $

$2$	−2.52808	−1.78762	−0.893811	−	0.448444i	$-0.851978\pi$
$2$	−2.52808	−1.78762	−0.893811	+	0.448444i	$0.851978\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.39119	2.19559
$5$	−2.04731	−0.915586	−0.457793	−	0.889059i	$-0.651360\pi$
$5$	−2.04731	−0.915586	−0.457793	+	0.889059i	$0.651360\pi$
$6$	2.52808	1.03208
$7$	3.64200	1.37655	0.688273	−	0.725452i	$-0.258369\pi$
$7$	3.64200	1.37655	0.688273	+	0.725452i	$0.258369\pi$
$8$	−6.04511	−2.13727
$9$	1.00000	0.333333
$10$	5.17577	1.63672
$11$	−1.87626	−0.565714	−0.282857	−	0.959162i	$-0.591282\pi$
$11$	−1.87626	−0.565714	−0.282857	+	0.959162i	$0.591282\pi$
$12$	−4.39119	−1.26763
$13$	6.41879	1.78025	0.890126	−	0.455714i	$-0.150616\pi$
$13$	6.41879	1.78025	0.890126	+	0.455714i	$0.150616\pi$
$14$	−9.20726	−2.46075
$15$	2.04731	0.528614
$16$	6.50014	1.62504
$17$	2.88113	0.698778	0.349389	−	0.936978i	$-0.386389\pi$
$17$	2.88113	0.698778	0.349389	+	0.936978i	$0.386389\pi$
$18$	−2.52808	−0.595874
$19$	3.34047	0.766355	0.383178	−	0.923675i	$-0.374830\pi$
$19$	3.34047	0.766355	0.383178	+	0.923675i	$0.374830\pi$
$20$	−8.99013	−2.01025
$21$	−3.64200	−0.794749
$22$	4.74333	1.01128
$23$	−0.154262	−0.0321658	−0.0160829	−	0.999871i	$-0.505120\pi$
$23$	−0.154262	−0.0321658	−0.0160829	+	0.999871i	$0.505120\pi$
$24$	6.04511	1.23395
$25$	−0.808509	−0.161702
$26$	−16.2272	−3.18242
$27$	−1.00000	−0.192450
$28$	15.9927	3.02234
$29$	2.55966	0.475317	0.237659	−	0.971349i	$-0.423620\pi$
$29$	2.55966	0.475317	0.237659	+	0.971349i	$0.423620\pi$
$30$	−5.17577	−0.944962
$31$	−4.26768	−0.766497	−0.383249	−	0.923645i	$-0.625195\pi$
$31$	−4.26768	−0.766497	−0.383249	+	0.923645i	$0.625195\pi$
$32$	−4.34266	−0.767682
$33$	1.87626	0.326615
$34$	−7.28373	−1.24915
$35$	−7.45631	−1.26035
$36$	4.39119	0.731864
$37$	−1.42252	−0.233861	−0.116930	−	0.993140i	$-0.537305\pi$
$37$	−1.42252	−0.233861	−0.116930	+	0.993140i	$0.537305\pi$
$38$	−8.44496	−1.36995
$39$	−6.41879	−1.02783
$40$	12.3762	1.95685
$41$	8.40629	1.31284	0.656421	−	0.754395i	$-0.272070\pi$
$41$	8.40629	1.31284	0.656421	+	0.754395i	$0.272070\pi$
$42$	9.20726	1.42071
$43$	0.952507	0.145256	0.0726280	−	0.997359i	$-0.476861\pi$
$43$	0.952507	0.145256	0.0726280	+	0.997359i	$0.476861\pi$
$44$	−8.23901	−1.24208
$45$	−2.04731	−0.305195
$46$	0.389986	0.0575002
$47$	−5.20366	−0.759032	−0.379516	−	0.925185i	$-0.623910\pi$
$47$	−5.20366	−0.759032	−0.379516	+	0.925185i	$0.623910\pi$
$48$	−6.50014	−0.938215
$49$	6.26416	0.894880
$50$	2.04398	0.289062
$51$	−2.88113	−0.403439
$52$	28.1861	3.90871
$53$	−11.8257	−1.62438	−0.812192	−	0.583390i	$-0.801726\pi$
$53$	−11.8257	−1.62438	−0.812192	+	0.583390i	$0.801726\pi$
$54$	2.52808	0.344028
$55$	3.84129	0.517960
$56$	−22.0163	−2.94205
$57$	−3.34047	−0.442455
$58$	−6.47103	−0.849688
$59$	−0.532287	−0.0692979	−0.0346489	−	0.999400i	$-0.511031\pi$
$59$	−0.532287	−0.0692979	−0.0346489	+	0.999400i	$0.511031\pi$
$60$	8.99013	1.16062
$61$	2.38966	0.305965	0.152982	−	0.988229i	$-0.451112\pi$
$61$	2.38966	0.305965	0.152982	+	0.988229i	$0.451112\pi$
$62$	10.7890	1.37021
$63$	3.64200	0.458849
$64$	−2.02169	−0.252711
$65$	−13.1413	−1.62997
$66$	−4.74333	−0.583864
$67$	−14.7480	−1.80176	−0.900880	−	0.434068i	$-0.857078\pi$
$67$	−14.7480	−1.80176	−0.900880	+	0.434068i	$0.857078\pi$
$68$	12.6516	1.53423
$69$	0.154262	0.0185709
$70$	18.8502	2.25302
$71$	−4.50886	−0.535103	−0.267551	−	0.963544i	$-0.586215\pi$
$71$	−4.50886	−0.535103	−0.267551	+	0.963544i	$0.586215\pi$
$72$	−6.04511	−0.712423
$73$	−7.40126	−0.866252	−0.433126	−	0.901333i	$-0.642590\pi$
$73$	−7.40126	−0.866252	−0.433126	+	0.901333i	$0.642590\pi$
$74$	3.59624	0.418054
$75$	0.808509	0.0933586
$76$	14.6686	1.68260
$77$	−6.83334	−0.778731
$78$	16.2272	1.83737
$79$	−3.42284	−0.385100	−0.192550	−	0.981287i	$-0.561676\pi$
$79$	−3.42284	−0.385100	−0.192550	+	0.981287i	$0.561676\pi$
$80$	−13.3078	−1.48786
$81$	1.00000	0.111111
$82$	−21.2518	−2.34686
$83$	−4.20847	−0.461939	−0.230970	−	0.972961i	$-0.574190\pi$
$83$	−4.20847	−0.461939	−0.230970	+	0.972961i	$0.574190\pi$
$84$	−15.9927	−1.74495
$85$	−5.89858	−0.639791
$86$	−2.40801	−0.259663
$87$	−2.55966	−0.274425
$88$	11.3422	1.20908
$89$	−12.6772	−1.34379	−0.671893	−	0.740648i	$-0.734518\pi$
$89$	−12.6772	−1.34379	−0.671893	+	0.740648i	$0.734518\pi$
$90$	5.17577	0.545574
$91$	23.3772	2.45060
$92$	−0.677391	−0.0706229
$93$	4.26768	0.442538
$94$	13.1553	1.35686
$95$	−6.83898	−0.701664
$96$	4.34266	0.443221
$97$	−9.81142	−0.996199	−0.498099	−	0.867120i	$-0.665969\pi$
$97$	−9.81142	−0.996199	−0.498099	+	0.867120i	$0.665969\pi$
$98$	−15.8363	−1.59971
$99$	−1.87626	−0.188571
$100$	−3.55031	−0.355031
$101$	2.12522	0.211468	0.105734	−	0.994394i	$-0.466281\pi$
$101$	2.12522	0.211468	0.105734	+	0.994394i	$0.466281\pi$
$102$	7.28373	0.721197
$103$	−6.92437	−0.682279	−0.341139	−	0.940013i	$-0.610813\pi$
$103$	−6.92437	−0.682279	−0.341139	+	0.940013i	$0.610813\pi$
$104$	−38.8023	−3.80488
$105$	7.45631	0.727662
$106$	29.8963	2.90379
$107$	1.87707	0.181463	0.0907314	−	0.995875i	$-0.471080\pi$
$107$	1.87707	0.181463	0.0907314	+	0.995875i	$0.471080\pi$
$108$	−4.39119	−0.422542
$109$	12.2726	1.17550	0.587749	−	0.809043i	$-0.300014\pi$
$109$	12.2726	1.17550	0.587749	+	0.809043i	$0.300014\pi$
$110$	−9.71109	−0.925916
$111$	1.42252	0.135019
$112$	23.6735	2.23694
$113$	−2.03854	−0.191770	−0.0958851	−	0.995392i	$-0.530568\pi$
$113$	−2.03854	−0.191770	−0.0958851	+	0.995392i	$0.530568\pi$
$114$	8.44496	0.790943
$115$	0.315822	0.0294505
$116$	11.2400	1.04360
$117$	6.41879	0.593417
$118$	1.34566	0.123878
$119$	10.4931	0.961900
$120$	−12.3762	−1.12979
$121$	−7.47965	−0.679968
$122$	−6.04125	−0.546949
$123$	−8.40629	−0.757969
$124$	−18.7402	−1.68292
$125$	11.8918	1.06364
$126$	−9.20726	−0.820248
$127$	−4.36692	−0.387502	−0.193751	−	0.981051i	$-0.562065\pi$
$127$	−4.36692	−0.387502	−0.193751	+	0.981051i	$0.562065\pi$
$128$	13.7963	1.21943
$129$	−0.952507	−0.0838636
$130$	33.2222	2.91378
$131$	−17.6677	−1.54363	−0.771816	−	0.635846i	$-0.780651\pi$
$131$	−17.6677	−1.54363	−0.771816	+	0.635846i	$0.780651\pi$
$132$	8.23901	0.717113
$133$	12.1660	1.05492
$134$	37.2842	3.22087
$135$	2.04731	0.176205
$136$	−17.4168	−1.49348
$137$	5.09781	0.435535	0.217768	−	0.976001i	$-0.430122\pi$
$137$	5.09781	0.435535	0.217768	+	0.976001i	$0.430122\pi$
$138$	−0.389986	−0.0331978
$139$	−10.4547	−0.886759	−0.443379	−	0.896334i	$-0.646221\pi$
$139$	−10.4547	−0.886759	−0.443379	+	0.896334i	$0.646221\pi$
$140$	−32.7421	−2.76721
$141$	5.20366	0.438228
$142$	11.3987	0.956562
$143$	−12.0433	−1.00711
$144$	6.50014	0.541679
$145$	−5.24043	−0.435194
$146$	18.7110	1.54853
$147$	−6.26416	−0.516659
$148$	−6.24654	−0.513463
$149$	−18.4560	−1.51197	−0.755986	−	0.654588i	$-0.772842\pi$
$149$	−18.4560	−1.51197	−0.755986	+	0.654588i	$0.772842\pi$
$150$	−2.04398	−0.166890
$151$	1.55795	0.126785	0.0633923	−	0.997989i	$-0.479808\pi$
$151$	1.55795	0.126785	0.0633923	+	0.997989i	$0.479808\pi$
$152$	−20.1935	−1.63791
$153$	2.88113	0.232926
$154$	17.2752	1.39208
$155$	8.73727	0.701795
$156$	−28.1861	−2.25669
$157$	−6.41127	−0.511675	−0.255837	−	0.966720i	$-0.582351\pi$
$157$	−6.41127	−0.511675	−0.255837	+	0.966720i	$0.582351\pi$
$158$	8.65321	0.688412
$159$	11.8257	0.937839
$160$	8.89079	0.702879
$161$	−0.561821	−0.0442777
$162$	−2.52808	−0.198625
$163$	−5.61540	−0.439832	−0.219916	−	0.975519i	$-0.570578\pi$
$163$	−5.61540	−0.439832	−0.219916	+	0.975519i	$0.570578\pi$
$164$	36.9136	2.88247
$165$	−3.84129	−0.299044
$166$	10.6393	0.825773
$167$	−7.89894	−0.611238	−0.305619	−	0.952154i	$-0.598863\pi$
$167$	−7.89894	−0.611238	−0.305619	+	0.952154i	$0.598863\pi$
$168$	22.0163	1.69859
$169$	28.2009	2.16930
$170$	14.9121	1.14370
$171$	3.34047	0.255452
$172$	4.18264	0.318923
$173$	−14.5765	−1.10823	−0.554115	−	0.832440i	$-0.686943\pi$
$173$	−14.5765	−1.10823	−0.554115	+	0.832440i	$0.686943\pi$
$174$	6.47103	0.490567
$175$	−2.94459	−0.222590
$176$	−12.1960	−0.919305
$177$	0.532287	0.0400091
$178$	32.0491	2.40218
$179$	10.7927	0.806681	0.403341	−	0.915050i	$-0.367849\pi$
$179$	10.7927	0.806681	0.403341	+	0.915050i	$0.367849\pi$
$180$	−8.99013	−0.670085
$181$	−7.16737	−0.532746	−0.266373	−	0.963870i	$-0.585825\pi$
$181$	−7.16737	−0.532746	−0.266373	+	0.963870i	$0.585825\pi$
$182$	−59.0995	−4.38075
$183$	−2.38966	−0.176649
$184$	0.932528	0.0687469
$185$	2.91234	0.214119
$186$	−10.7890	−0.791090
$187$	−5.40576	−0.395308
$188$	−22.8503	−1.66653
$189$	−3.64200	−0.264916
$190$	17.2895	1.25431
$191$	−12.7427	−0.922028	−0.461014	−	0.887393i	$-0.652514\pi$
$191$	−12.7427	−0.922028	−0.461014	+	0.887393i	$0.652514\pi$
$192$	2.02169	0.145903
$193$	0.125252	0.00901583	0.00450792	−	0.999990i	$-0.498565\pi$
$193$	0.125252	0.00901583	0.00450792	+	0.999990i	$0.498565\pi$
$194$	24.8041	1.78083
$195$	13.1413	0.941066
$196$	27.5071	1.96479
$197$	7.33716	0.522751	0.261376	−	0.965237i	$-0.415824\pi$
$197$	7.33716	0.522751	0.261376	+	0.965237i	$0.415824\pi$
$198$	4.74333	0.337094
$199$	−24.7324	−1.75323	−0.876615	−	0.481192i	$-0.840204\pi$
$199$	−24.7324	−1.75323	−0.876615	+	0.481192i	$0.840204\pi$
$200$	4.88753	0.345600
$201$	14.7480	1.04025
$202$	−5.37273	−0.378024
$203$	9.32229	0.654296
$204$	−12.6516	−0.885789
$205$	−17.2103	−1.20202
$206$	17.5054	1.21966
$207$	−0.154262	−0.0107219
$208$	41.7231	2.89297
$209$	−6.26758	−0.433538
$210$	−18.8502	−1.30078
$211$	−17.9054	−1.23266	−0.616328	−	0.787489i	$-0.711381\pi$
$211$	−17.9054	−1.23266	−0.616328	+	0.787489i	$0.711381\pi$
$212$	−51.9288	−3.56649
$213$	4.50886	0.308942
$214$	−4.74537	−0.324387
$215$	−1.95008	−0.132994
$216$	6.04511	0.411318
$217$	−15.5429	−1.05512
$218$	−31.0260	−2.10135
$219$	7.40126	0.500131
$220$	16.8678	1.13723
$221$	18.4934	1.24400
$222$	−3.59624	−0.241364
$223$	−3.17069	−0.212325	−0.106163	−	0.994349i	$-0.533856\pi$
$223$	−3.17069	−0.212325	−0.106163	+	0.994349i	$0.533856\pi$
$224$	−15.8160	−1.05675
$225$	−0.808509	−0.0539006
$226$	5.15360	0.342813
$227$	−11.1154	−0.737754	−0.368877	−	0.929478i	$-0.620258\pi$
$227$	−11.1154	−0.737754	−0.368877	+	0.929478i	$0.620258\pi$
$228$	−14.6686	−0.971452
$229$	9.04045	0.597410	0.298705	−	0.954346i	$-0.403445\pi$
$229$	9.04045	0.597410	0.298705	+	0.954346i	$0.403445\pi$
$230$	−0.798422	−0.0526464
$231$	6.83334	0.449601
$232$	−15.4734	−1.01588
$233$	8.32865	0.545628	0.272814	−	0.962067i	$-0.412046\pi$
$233$	8.32865	0.545628	0.272814	+	0.962067i	$0.412046\pi$
$234$	−16.2272	−1.06081
$235$	10.6535	0.694960
$236$	−2.33737	−0.152150
$237$	3.42284	0.222337
$238$	−26.5274	−1.71951
$239$	14.1337	0.914234	0.457117	−	0.889407i	$-0.348882\pi$
$239$	14.1337	0.914234	0.457117	+	0.889407i	$0.348882\pi$
$240$	13.3078	0.859017
$241$	−13.4913	−0.869049	−0.434524	−	0.900660i	$-0.643083\pi$
$241$	−13.4913	−0.869049	−0.434524	+	0.900660i	$0.643083\pi$
$242$	18.9091	1.21553
$243$	−1.00000	−0.0641500
$244$	10.4934	0.671774
$245$	−12.8247	−0.819340
$246$	21.2518	1.35496
$247$	21.4417	1.36431
$248$	25.7986	1.63821
$249$	4.20847	0.266701
$250$	−30.0635	−1.90138
$251$	15.7616	0.994865	0.497433	−	0.867503i	$-0.334276\pi$
$251$	15.7616	0.994865	0.497433	+	0.867503i	$0.334276\pi$
$252$	15.9927	1.00745
$253$	0.289435	0.0181966
$254$	11.0399	0.692707
$255$	5.89858	0.369384
$256$	−30.8348	−1.92718
$257$	27.8148	1.73504	0.867520	−	0.497403i	$-0.165713\pi$
$257$	27.8148	1.73504	0.867520	+	0.497403i	$0.165713\pi$
$258$	2.40801	0.149916
$259$	−5.18081	−0.321920
$260$	−57.7058	−3.57876
$261$	2.55966	0.158439
$262$	44.6653	2.75943
$263$	−4.14095	−0.255342	−0.127671	−	0.991817i	$-0.540750\pi$
$263$	−4.14095	−0.255342	−0.127671	+	0.991817i	$0.540750\pi$
$264$	−11.3422	−0.698064
$265$	24.2109	1.48726
$266$	−30.7565	−1.88580
$267$	12.6772	0.775835
$268$	−64.7614	−3.95593
$269$	19.9641	1.21723	0.608615	−	0.793466i	$-0.291725\pi$
$269$	19.9641	1.21723	0.608615	+	0.793466i	$0.291725\pi$
$270$	−5.17577	−0.314987
$271$	2.47793	0.150523	0.0752617	−	0.997164i	$-0.476021\pi$
$271$	2.47793	0.150523	0.0752617	+	0.997164i	$0.476021\pi$
$272$	18.7278	1.13554
$273$	−23.3772	−1.41485
$274$	−12.8877	−0.778573
$275$	1.51697	0.0914769
$276$	0.677391	0.0407742
$277$	6.27473	0.377012	0.188506	−	0.982072i	$-0.439636\pi$
$277$	6.27473	0.377012	0.188506	+	0.982072i	$0.439636\pi$
$278$	26.4304	1.58519
$279$	−4.26768	−0.255499
$280$	45.0742	2.69370
$281$	−9.61796	−0.573759	−0.286880	−	0.957967i	$-0.592618\pi$
$281$	−9.61796	−0.573759	−0.286880	+	0.957967i	$0.592618\pi$
$282$	−13.1553	−0.783385
$283$	2.24629	0.133528	0.0667640	−	0.997769i	$-0.478733\pi$
$283$	2.24629	0.133528	0.0667640	+	0.997769i	$0.478733\pi$
$284$	−19.7992	−1.17487
$285$	6.83898	0.405106
$286$	30.4465	1.80034
$287$	30.6157	1.80719
$288$	−4.34266	−0.255894
$289$	−8.69907	−0.511710
$290$	13.2482	0.777962
$291$	9.81142	0.575156
$292$	−32.5003	−1.90194
$293$	−17.8234	−1.04125	−0.520626	−	0.853785i	$-0.674301\pi$
$293$	−17.8234	−1.04125	−0.520626	+	0.853785i	$0.674301\pi$
$294$	15.8363	0.923592
$295$	1.08976	0.0634482
$296$	8.59928	0.499823
$297$	1.87626	0.108872
$298$	46.6582	2.70283
$299$	−0.990173	−0.0572632
$300$	3.55031	0.204978
$301$	3.46903	0.199952
$302$	−3.93863	−0.226643
$303$	−2.12522	−0.122091
$304$	21.7135	1.24535
$305$	−4.89238	−0.280137
$306$	−7.28373	−0.416383
$307$	−4.97502	−0.283939	−0.141970	−	0.989871i	$-0.545344\pi$
$307$	−4.97502	−0.283939	−0.141970	+	0.989871i	$0.545344\pi$
$308$	−30.0065	−1.70978
$309$	6.92437	0.393914
$310$	−22.0885	−1.25454
$311$	−14.8183	−0.840268	−0.420134	−	0.907462i	$-0.638017\pi$
$311$	−14.8183	−0.840268	−0.420134	+	0.907462i	$0.638017\pi$
$312$	38.8023	2.19675
$313$	14.5307	0.821324	0.410662	−	0.911788i	$-0.365298\pi$
$313$	14.5307	0.821324	0.410662	+	0.911788i	$0.365298\pi$
$314$	16.2082	0.914681
$315$	−7.45631	−0.420116
$316$	−15.0303	−0.845522
$317$	22.9481	1.28890	0.644448	−	0.764648i	$-0.277087\pi$
$317$	22.9481	1.28890	0.644448	+	0.764648i	$0.277087\pi$
$318$	−29.8963	−1.67650
$319$	−4.80259	−0.268893
$320$	4.13903	0.231379
$321$	−1.87707	−0.104768
$322$	1.42033	0.0791517
$323$	9.62433	0.535512
$324$	4.39119	0.243955
$325$	−5.18965	−0.287870
$326$	14.1962	0.786253
$327$	−12.2726	−0.678674
$328$	−50.8169	−2.80590
$329$	−18.9517	−1.04484
$330$	9.71109	0.534578
$331$	25.7649	1.41617	0.708083	−	0.706129i	$-0.249560\pi$
$331$	25.7649	1.41617	0.708083	+	0.706129i	$0.249560\pi$
$332$	−18.4802	−1.01423
$333$	−1.42252	−0.0779535
$334$	19.9691	1.09266
$335$	30.1939	1.64967
$336$	−23.6735	−1.29150
$337$	18.5989	1.01315	0.506573	−	0.862197i	$-0.330912\pi$
$337$	18.5989	1.01315	0.506573	+	0.862197i	$0.330912\pi$
$338$	−71.2940	−3.87788
$339$	2.03854	0.110719
$340$	−25.9018	−1.40472
$341$	8.00727	0.433618
$342$	−8.44496	−0.456651
$343$	−2.67992	−0.144702
$344$	−5.75801	−0.310451
$345$	−0.315822	−0.0170033
$346$	36.8505	1.98110
$347$	2.42974	0.130435	0.0652176	−	0.997871i	$-0.479226\pi$
$347$	2.42974	0.130435	0.0652176	+	0.997871i	$0.479226\pi$
$348$	−11.2400	−0.602525
$349$	−5.35413	−0.286600	−0.143300	−	0.989679i	$-0.545771\pi$
$349$	−5.35413	−0.286600	−0.143300	+	0.989679i	$0.545771\pi$
$350$	7.44416	0.397907
$351$	−6.41879	−0.342610
$352$	8.14797	0.434288
$353$	22.2046	1.18183	0.590915	−	0.806734i	$-0.298767\pi$
$353$	22.2046	1.18183	0.590915	+	0.806734i	$0.298767\pi$
$354$	−1.34566	−0.0715212
$355$	9.23104	0.489933
$356$	−55.6682	−2.95041
$357$	−10.4931	−0.555353
$358$	−27.2847	−1.44204
$359$	−19.3592	−1.02174	−0.510870	−	0.859658i	$-0.670677\pi$
$359$	−19.3592	−1.02174	−0.510870	+	0.859658i	$0.670677\pi$
$360$	12.3762	0.652285
$361$	−7.84129	−0.412700
$362$	18.1197	0.952349
$363$	7.47965	0.392580
$364$	102.654	5.38052
$365$	15.1527	0.793129
$366$	6.04125	0.315781
$367$	24.5616	1.28210	0.641052	−	0.767498i	$-0.278498\pi$
$367$	24.5616	1.28210	0.641052	+	0.767498i	$0.278498\pi$
$368$	−1.00272	−0.0522705
$369$	8.40629	0.437614
$370$	−7.36263	−0.382765
$371$	−43.0692	−2.23604
$372$	18.7402	0.971632
$373$	5.95418	0.308296	0.154148	−	0.988048i	$-0.450737\pi$
$373$	5.95418	0.308296	0.154148	+	0.988048i	$0.450737\pi$
$374$	13.6662	0.706661
$375$	−11.8918	−0.614092
$376$	31.4567	1.62226
$377$	16.4299	0.846184
$378$	9.20726	0.473571
$379$	0.652186	0.0335005	0.0167503	−	0.999860i	$-0.494668\pi$
$379$	0.652186	0.0335005	0.0167503	+	0.999860i	$0.494668\pi$
$380$	−30.0312	−1.54057
$381$	4.36692	0.223724
$382$	32.2145	1.64824
$383$	13.6573	0.697853	0.348927	−	0.937150i	$-0.386546\pi$
$383$	13.6573	0.697853	0.348927	+	0.937150i	$0.386546\pi$
$384$	−13.7963	−0.704040
$385$	13.9900	0.712996
$386$	−0.316647	−0.0161169
$387$	0.952507	0.0484187
$388$	−43.0838	−2.18725
$389$	−17.0621	−0.865085	−0.432543	−	0.901613i	$-0.642383\pi$
$389$	−17.0621	−0.865085	−0.432543	+	0.901613i	$0.642383\pi$
$390$	−33.2222	−1.68227
$391$	−0.444448	−0.0224767
$392$	−37.8675	−1.91260
$393$	17.6677	0.891216
$394$	−18.5489	−0.934481
$395$	7.00762	0.352592
$396$	−8.23901	−0.414026
$397$	4.32470	0.217051	0.108525	−	0.994094i	$-0.465387\pi$
$397$	4.32470	0.217051	0.108525	+	0.994094i	$0.465387\pi$
$398$	62.5254	3.13411
$399$	−12.1660	−0.609060
$400$	−5.25543	−0.262771
$401$	6.19677	0.309452	0.154726	−	0.987957i	$-0.450551\pi$
$401$	6.19677	0.309452	0.154726	+	0.987957i	$0.450551\pi$
$402$	−37.2842	−1.85957
$403$	−27.3933	−1.36456
$404$	9.33225	0.464297
$405$	−2.04731	−0.101732
$406$	−23.5675	−1.16963
$407$	2.66901	0.132298
$408$	17.4168	0.862258
$409$	−14.1335	−0.698854	−0.349427	−	0.936964i	$-0.613624\pi$
$409$	−14.1335	−0.698854	−0.349427	+	0.936964i	$0.613624\pi$
$410$	43.5090	2.14876
$411$	−5.09781	−0.251457
$412$	−30.4062	−1.49801
$413$	−1.93859	−0.0953917
$414$	0.389986	0.0191667
$415$	8.61605	0.422945
$416$	−27.8746	−1.36667
$417$	10.4547	0.511970
$418$	15.8449	0.775001
$419$	−1.59125	−0.0777377	−0.0388688	−	0.999244i	$-0.512375\pi$
$419$	−1.59125	−0.0777377	−0.0388688	+	0.999244i	$0.512375\pi$
$420$	32.7421	1.59765
$421$	−30.0655	−1.46530	−0.732651	−	0.680604i	$-0.761717\pi$
$421$	−30.0655	−1.46530	−0.732651	+	0.680604i	$0.761717\pi$
$422$	45.2662	2.20352
$423$	−5.20366	−0.253011
$424$	71.4876	3.47175
$425$	−2.32942	−0.112994
$426$	−11.3987	−0.552271
$427$	8.70314	0.421175
$428$	8.24255	0.398418
$429$	12.0433	0.581457
$430$	4.92996	0.237744
$431$	−23.4937	−1.13165	−0.565826	−	0.824524i	$-0.691443\pi$
$431$	−23.4937	−1.13165	−0.565826	+	0.824524i	$0.691443\pi$
$432$	−6.50014	−0.312738
$433$	21.0659	1.01236	0.506181	−	0.862427i	$-0.331057\pi$
$433$	21.0659	1.01236	0.506181	+	0.862427i	$0.331057\pi$
$434$	39.2936	1.88615
$435$	5.24043	0.251259
$436$	53.8911	2.58092
$437$	−0.515305	−0.0246504
$438$	−18.7110	−0.894045
$439$	−12.9329	−0.617251	−0.308626	−	0.951184i	$-0.599869\pi$
$439$	−12.9329	−0.617251	−0.308626	+	0.951184i	$0.599869\pi$
$440$	−23.2210	−1.10702
$441$	6.26416	0.298293
$442$	−46.7528	−2.22380
$443$	−5.43375	−0.258165	−0.129083	−	0.991634i	$-0.541203\pi$
$443$	−5.43375	−0.258165	−0.129083	+	0.991634i	$0.541203\pi$
$444$	6.24654	0.296448
$445$	25.9543	1.23035
$446$	8.01575	0.379557
$447$	18.4560	0.872937
$448$	−7.36299	−0.347869
$449$	17.3621	0.819369	0.409685	−	0.912227i	$-0.365639\pi$
$449$	17.3621	0.819369	0.409685	+	0.912227i	$0.365639\pi$
$450$	2.04398	0.0963539
$451$	−15.7724	−0.742692
$452$	−8.95163	−0.421049
$453$	−1.55795	−0.0731991
$454$	28.1006	1.31883
$455$	−47.8605	−2.24374
$456$	20.1935	0.945646
$457$	7.35236	0.343929	0.171965	−	0.985103i	$-0.444989\pi$
$457$	7.35236	0.343929	0.171965	+	0.985103i	$0.444989\pi$
$458$	−22.8550	−1.06794
$459$	−2.88113	−0.134480
$460$	1.38683	0.0646614
$461$	3.36686	0.156810	0.0784051	−	0.996922i	$-0.475017\pi$
$461$	3.36686	0.156810	0.0784051	+	0.996922i	$0.475017\pi$
$462$	−17.2752	−0.803716
$463$	19.3857	0.900928	0.450464	−	0.892794i	$-0.351258\pi$
$463$	19.3857	0.900928	0.450464	+	0.892794i	$0.351258\pi$
$464$	16.6382	0.772408
$465$	−8.73727	−0.405181
$466$	−21.0555	−0.975377
$467$	−1.85809	−0.0859821	−0.0429911	−	0.999075i	$-0.513689\pi$
$467$	−1.85809	−0.0859821	−0.0429911	+	0.999075i	$0.513689\pi$
$468$	28.1861	1.30290
$469$	−53.7124	−2.48021
$470$	−26.9330	−1.24233
$471$	6.41127	0.295416
$472$	3.21773	0.148108
$473$	−1.78715	−0.0821733
$474$	−8.65321	−0.397455
$475$	−2.70080	−0.123921
$476$	46.0771	2.11194
$477$	−11.8257	−0.541462
$478$	−35.7311	−1.63430
$479$	12.2452	0.559498	0.279749	−	0.960073i	$-0.409749\pi$
$479$	12.2452	0.559498	0.279749	+	0.960073i	$0.409749\pi$
$480$	−8.89079	−0.405807
$481$	−9.13085	−0.416331
$482$	34.1070	1.55353
$483$	0.561821	0.0255637
$484$	−32.8445	−1.49293
$485$	20.0870	0.912106
$486$	2.52808	0.114676
$487$	−17.2786	−0.782969	−0.391485	−	0.920185i	$-0.628038\pi$
$487$	−17.2786	−0.782969	−0.391485	+	0.920185i	$0.628038\pi$
$488$	−14.4458	−0.653929
$489$	5.61540	0.253937
$490$	32.4219	1.46467
$491$	−10.3949	−0.469116	−0.234558	−	0.972102i	$-0.575364\pi$
$491$	−10.3949	−0.469116	−0.234558	+	0.972102i	$0.575364\pi$
$492$	−36.9136	−1.66419
$493$	7.37473	0.332141
$494$	−54.2064	−2.43886
$495$	3.84129	0.172653
$496$	−27.7405	−1.24559
$497$	−16.4213	−0.736594
$498$	−10.6393	−0.476760
$499$	−30.5834	−1.36910	−0.684551	−	0.728965i	$-0.740002\pi$
$499$	−30.5834	−1.36910	−0.684551	+	0.728965i	$0.740002\pi$
$500$	52.2193	2.33532
$501$	7.89894	0.352899
$502$	−39.8466	−1.77844
$503$	5.86291	0.261414	0.130707	−	0.991421i	$-0.458275\pi$
$503$	5.86291	0.261414	0.130707	+	0.991421i	$0.458275\pi$
$504$	−22.0163	−0.980683
$505$	−4.35100	−0.193617
$506$	−0.731714	−0.0325287
$507$	−28.2009	−1.25244
$508$	−19.1760	−0.850796
$509$	30.7064	1.36104	0.680518	−	0.732732i	$-0.261755\pi$
$509$	30.7064	1.36104	0.680518	+	0.732732i	$0.261755\pi$
$510$	−14.9121	−0.660318
$511$	−26.9554	−1.19244
$512$	50.3602	2.22563
$513$	−3.34047	−0.147485
$514$	−70.3180	−3.10160
$515$	14.1764	0.624685
$516$	−4.18264	−0.184130
$517$	9.76343	0.429395
$518$	13.0975	0.575471
$519$	14.5765	0.639837
$520$	79.4404	3.48369
$521$	−25.1134	−1.10024	−0.550118	−	0.835087i	$-0.685417\pi$
$521$	−25.1134	−1.10024	−0.550118	+	0.835087i	$0.685417\pi$
$522$	−6.47103	−0.283229
$523$	45.4822	1.98880	0.994398	−	0.105701i	$-0.0337085\pi$
$523$	45.4822	1.98880	0.994398	+	0.105701i	$0.0337085\pi$
$524$	−77.5820	−3.38919
$525$	2.94459	0.128512
$526$	10.4687	0.456455
$527$	−12.2957	−0.535611
$528$	12.1960	0.530761
$529$	−22.9762	−0.998965
$530$	−61.2071	−2.65867
$531$	−0.532287	−0.0230993
$532$	53.4231	2.31618
$533$	53.9582	2.33719
$534$	−32.0491	−1.38690
$535$	−3.84294	−0.166145
$536$	89.1535	3.85085
$537$	−10.7927	−0.465738
$538$	−50.4707	−2.17595
$539$	−11.7532	−0.506246
$540$	8.99013	0.386874
$541$	18.2046	0.782678	0.391339	−	0.920247i	$-0.372012\pi$
$541$	18.2046	0.782678	0.391339	+	0.920247i	$0.372012\pi$
$542$	−6.26440	−0.269079
$543$	7.16737	0.307581
$544$	−12.5118	−0.536439
$545$	−25.1258	−1.07627
$546$	59.0995	2.52923
$547$	11.8635	0.507247	0.253624	−	0.967303i	$-0.418378\pi$
$547$	11.8635	0.507247	0.253624	+	0.967303i	$0.418378\pi$
$548$	22.3854	0.956259
$549$	2.38966	0.101988
$550$	−3.83503	−0.163526
$551$	8.55046	0.364262
$552$	−0.932528	−0.0396910
$553$	−12.4660	−0.530107
$554$	−15.8630	−0.673955
$555$	−2.91234	−0.123622
$556$	−45.9087	−1.94696
$557$	38.7975	1.64390	0.821952	−	0.569557i	$-0.192885\pi$
$557$	38.7975	1.64390	0.821952	+	0.569557i	$0.192885\pi$
$558$	10.7890	0.456736
$559$	6.11395	0.258592
$560$	−48.4671	−2.04811
$561$	5.40576	0.228231
$562$	24.3150	1.02566
$563$	29.0688	1.22511	0.612553	−	0.790430i	$-0.290143\pi$
$563$	29.0688	1.22511	0.612553	+	0.790430i	$0.290143\pi$
$564$	22.8503	0.962170
$565$	4.17354	0.175582
$566$	−5.67879	−0.238697
$567$	3.64200	0.152950
$568$	27.2565	1.14366
$569$	−22.4048	−0.939260	−0.469630	−	0.882863i	$-0.655613\pi$
$569$	−22.4048	−0.939260	−0.469630	+	0.882863i	$0.655613\pi$
$570$	−17.2895	−0.724177
$571$	7.39760	0.309580	0.154790	−	0.987947i	$-0.450530\pi$
$571$	7.39760	0.309580	0.154790	+	0.987947i	$0.450530\pi$
$572$	−52.8845	−2.21121
$573$	12.7427	0.532333
$574$	−77.3989	−3.23057
$575$	0.124722	0.00520126
$576$	−2.02169	−0.0842370
$577$	−43.9999	−1.83174	−0.915870	−	0.401475i	$-0.868498\pi$
$577$	−43.9999	−1.83174	−0.915870	+	0.401475i	$0.868498\pi$
$578$	21.9919	0.914744
$579$	−0.125252	−0.00520529
$580$	−23.0117	−0.955509
$581$	−15.3272	−0.635881
$582$	−24.8041	−1.02816
$583$	22.1881	0.918936
$584$	44.7414	1.85141
$585$	−13.1413	−0.543325
$586$	45.0589	1.86137
$587$	−40.7326	−1.68122	−0.840608	−	0.541644i	$-0.817802\pi$
$587$	−40.7326	−1.68122	−0.840608	+	0.541644i	$0.817802\pi$
$588$	−27.5071	−1.13437
$589$	−14.2560	−0.587409
$590$	−2.75500	−0.113421
$591$	−7.33716	−0.301810
$592$	−9.24657	−0.380032
$593$	−26.1976	−1.07581	−0.537903	−	0.843007i	$-0.680783\pi$
$593$	−26.1976	−1.07581	−0.537903	+	0.843007i	$0.680783\pi$
$594$	−4.74333	−0.194621
$595$	−21.4826	−0.880702
$596$	−81.0436	−3.31968
$597$	24.7324	1.01223
$598$	2.50324	0.102365
$599$	19.8507	0.811076	0.405538	−	0.914078i	$-0.367084\pi$
$599$	19.8507	0.811076	0.405538	+	0.914078i	$0.367084\pi$
$600$	−4.88753	−0.199532
$601$	6.66157	0.271731	0.135866	−	0.990727i	$-0.456618\pi$
$601$	6.66157	0.271731	0.135866	+	0.990727i	$0.456618\pi$
$602$	−8.76999	−0.357438
$603$	−14.7480	−0.600587
$604$	6.84127	0.278367
$605$	15.3132	0.622569
$606$	5.37273	0.218252
$607$	30.8336	1.25150	0.625748	−	0.780025i	$-0.284794\pi$
$607$	30.8336	1.25150	0.625748	+	0.780025i	$0.284794\pi$
$608$	−14.5065	−0.588317
$609$	−9.32229	−0.377758
$610$	12.3683	0.500779
$611$	−33.4012	−1.35127
$612$	12.6516	0.511410
$613$	−11.6847	−0.471940	−0.235970	−	0.971760i	$-0.575827\pi$
$613$	−11.6847	−0.471940	−0.235970	+	0.971760i	$0.575827\pi$
$614$	12.5772	0.507576
$615$	17.2103	0.693986
$616$	41.3083	1.66436
$617$	−36.9622	−1.48804	−0.744020	−	0.668157i	$-0.767083\pi$
$617$	−36.9622	−1.48804	−0.744020	+	0.668157i	$0.767083\pi$
$618$	−17.5054	−0.704169
$619$	6.39050	0.256856	0.128428	−	0.991719i	$-0.459007\pi$
$619$	6.39050	0.256856	0.128428	+	0.991719i	$0.459007\pi$
$620$	38.3670	1.54086
$621$	0.154262	0.00619030
$622$	37.4618	1.50208
$623$	−46.1705	−1.84978
$624$	−41.7231	−1.67026
$625$	−20.3038	−0.812151
$626$	−36.7348	−1.46822
$627$	6.26758	0.250303
$628$	−28.1531	−1.12343
$629$	−4.09846	−0.163416
$630$	18.8502	0.751008
$631$	25.4913	1.01479	0.507396	−	0.861713i	$-0.330608\pi$
$631$	25.4913	1.01479	0.507396	+	0.861713i	$0.330608\pi$
$632$	20.6914	0.823061
$633$	17.9054	0.711675
$634$	−58.0147	−2.30406
$635$	8.94046	0.354791
$636$	51.9288	2.05911
$637$	40.2083	1.59311
$638$	12.1413	0.480680
$639$	−4.50886	−0.178368
$640$	−28.2454	−1.11650
$641$	18.5181	0.731423	0.365711	−	0.930728i	$-0.380826\pi$
$641$	18.5181	0.731423	0.365711	+	0.930728i	$0.380826\pi$
$642$	4.74537	0.187285
$643$	18.3385	0.723200	0.361600	−	0.932333i	$-0.382231\pi$
$643$	18.3385	0.723200	0.361600	+	0.932333i	$0.382231\pi$
$644$	−2.46706	−0.0972157
$645$	1.95008	0.0767844
$646$	−24.3311	−0.957293
$647$	35.4402	1.39330	0.696650	−	0.717411i	$-0.254673\pi$
$647$	35.4402	1.39330	0.696650	+	0.717411i	$0.254673\pi$
$648$	−6.04511	−0.237474
$649$	0.998709	0.0392028
$650$	13.1199	0.514603
$651$	15.5429	0.609173
$652$	−24.6583	−0.965692
$653$	−40.2950	−1.57687	−0.788433	−	0.615121i	$-0.789107\pi$
$653$	−40.2950	−1.57687	−0.788433	+	0.615121i	$0.789107\pi$
$654$	31.0260	1.21321
$655$	36.1713	1.41333
$656$	54.6421	2.13341
$657$	−7.40126	−0.288751
$658$	47.9115	1.86779
$659$	21.4291	0.834759	0.417380	−	0.908732i	$-0.362949\pi$
$659$	21.4291	0.834759	0.417380	+	0.908732i	$0.362949\pi$
$660$	−16.8678	−0.656579
$661$	−46.5801	−1.81176	−0.905878	−	0.423538i	$-0.860788\pi$
$661$	−46.5801	−1.81176	−0.905878	+	0.423538i	$0.860788\pi$
$662$	−65.1357	−2.53157
$663$	−18.4934	−0.718224
$664$	25.4406	0.987288
$665$	−24.9076	−0.965874
$666$	3.59624	0.139351
$667$	−0.394857	−0.0152889
$668$	−34.6857	−1.34203
$669$	3.17069	0.122586
$670$	−76.3325	−2.94898
$671$	−4.48363	−0.173088
$672$	15.8160	0.610115
$673$	40.4829	1.56050	0.780250	−	0.625468i	$-0.215092\pi$
$673$	40.4829	1.56050	0.780250	+	0.625468i	$0.215092\pi$
$674$	−47.0195	−1.81112
$675$	0.808509	0.0311195
$676$	123.835	4.76289
$677$	−23.5346	−0.904509	−0.452254	−	0.891889i	$-0.649380\pi$
$677$	−23.5346	−0.904509	−0.452254	+	0.891889i	$0.649380\pi$
$678$	−5.15360	−0.197923
$679$	−35.7332	−1.37131
$680$	35.6576	1.36741
$681$	11.1154	0.425942
$682$	−20.2430	−0.775145
$683$	1.57897	0.0604176	0.0302088	−	0.999544i	$-0.490383\pi$
$683$	1.57897	0.0604176	0.0302088	+	0.999544i	$0.490383\pi$
$684$	14.6686	0.560868
$685$	−10.4368	−0.398770
$686$	6.77506	0.258673
$687$	−9.04045	−0.344915
$688$	6.19144	0.236046
$689$	−75.9067	−2.89181
$690$	0.798422	0.0303954
$691$	−34.0009	−1.29346	−0.646728	−	0.762721i	$-0.723863\pi$
$691$	−34.0009	−1.29346	−0.646728	+	0.762721i	$0.723863\pi$
$692$	−64.0081	−2.43322
$693$	−6.83334	−0.259577
$694$	−6.14257	−0.233169
$695$	21.4041	0.811904
$696$	15.4734	0.586519
$697$	24.2196	0.917384
$698$	13.5357	0.512332
$699$	−8.32865	−0.315018
$700$	−12.9302	−0.488717
$701$	−3.68029	−0.139003	−0.0695013	−	0.997582i	$-0.522141\pi$
$701$	−3.68029	−0.139003	−0.0695013	+	0.997582i	$0.522141\pi$
$702$	16.2272	0.612457
$703$	−4.75187	−0.179220
$704$	3.79321	0.142962
$705$	−10.6535	−0.401235
$706$	−56.1349	−2.11267
$707$	7.74006	0.291095
$708$	2.33737	0.0878438
$709$	13.7791	0.517485	0.258742	−	0.965946i	$-0.416692\pi$
$709$	13.7791	0.517485	0.258742	+	0.965946i	$0.416692\pi$
$710$	−23.3368	−0.875815
$711$	−3.42284	−0.128367
$712$	76.6353	2.87203
$713$	0.658339	0.0246550
$714$	26.5274	0.992761
$715$	24.6564	0.922099
$716$	47.3926	1.77114
$717$	−14.1337	−0.527833
$718$	48.9416	1.82648
$719$	−8.16709	−0.304581	−0.152291	−	0.988336i	$-0.548665\pi$
$719$	−8.16709	−0.304581	−0.152291	+	0.988336i	$0.548665\pi$
$720$	−13.3078	−0.495954
$721$	−25.2186	−0.939188
$722$	19.8234	0.737751
$723$	13.4913	0.501745
$724$	−31.4732	−1.16969
$725$	−2.06951	−0.0768597
$726$	−18.9091	−0.701784
$727$	15.0426	0.557900	0.278950	−	0.960306i	$-0.410014\pi$
$727$	15.0426	0.557900	0.278950	+	0.960306i	$0.410014\pi$
$728$	−141.318	−5.23759
$729$	1.00000	0.0370370
$730$	−38.3072	−1.41781
$731$	2.74430	0.101502
$732$	−10.4934	−0.387849
$733$	−33.1337	−1.22382	−0.611911	−	0.790927i	$-0.709599\pi$
$733$	−33.1337	−1.22382	−0.611911	+	0.790927i	$0.709599\pi$
$734$	−62.0936	−2.29192
$735$	12.8247	0.473046
$736$	0.669906	0.0246931
$737$	27.6712	1.01928
$738$	−21.2518	−0.782288
$739$	14.4436	0.531318	0.265659	−	0.964067i	$-0.414411\pi$
$739$	14.4436	0.531318	0.265659	+	0.964067i	$0.414411\pi$
$740$	12.7886	0.470119
$741$	−21.4417	−0.787682
$742$	108.882	3.99720
$743$	−7.61546	−0.279384	−0.139692	−	0.990195i	$-0.544611\pi$
$743$	−7.61546	−0.279384	−0.139692	+	0.990195i	$0.544611\pi$
$744$	−25.7986	−0.945822
$745$	37.7851	1.38434
$746$	−15.0526	−0.551116
$747$	−4.20847	−0.153980
$748$	−23.7377	−0.867935
$749$	6.83627	0.249792
$750$	30.0635	1.09776
$751$	−15.9717	−0.582815	−0.291408	−	0.956599i	$-0.594124\pi$
$751$	−15.9717	−0.582815	−0.291408	+	0.956599i	$0.594124\pi$
$752$	−33.8246	−1.23346
$753$	−15.7616	−0.574386
$754$	−41.5362	−1.51266
$755$	−3.18962	−0.116082
$756$	−15.9927	−0.581649
$757$	49.8313	1.81115	0.905574	−	0.424188i	$-0.139440\pi$
$757$	49.8313	1.81115	0.905574	+	0.424188i	$0.139440\pi$
$758$	−1.64878	−0.0598863
$759$	−0.289435	−0.0105058
$760$	41.3424	1.49965
$761$	−9.73387	−0.352853	−0.176426	−	0.984314i	$-0.556454\pi$
$761$	−9.73387	−0.352853	−0.176426	+	0.984314i	$0.556454\pi$
$762$	−11.0399	−0.399934
$763$	44.6967	1.61813
$764$	−55.9555	−2.02440
$765$	−5.89858	−0.213264
$766$	−34.5266	−1.24750
$767$	−3.41664	−0.123368
$768$	30.8348	1.11266
$769$	−9.00130	−0.324595	−0.162298	−	0.986742i	$-0.551890\pi$
$769$	−9.00130	−0.324595	−0.162298	+	0.986742i	$0.551890\pi$
$770$	−35.3678	−1.27457
$771$	−27.8148	−1.00173
$772$	0.550005	0.0197951
$773$	−40.0021	−1.43877	−0.719387	−	0.694609i	$-0.755577\pi$
$773$	−40.0021	−1.43877	−0.719387	+	0.694609i	$0.755577\pi$
$774$	−2.40801	−0.0865543
$775$	3.45046	0.123944
$776$	59.3111	2.12914
$777$	5.18081	0.185861
$778$	43.1345	1.54645
$779$	28.0809	1.00610
$780$	57.7058	2.06620
$781$	8.45979	0.302715
$782$	1.12360	0.0401799
$783$	−2.55966	−0.0914748
$784$	40.7180	1.45421
$785$	13.1259	0.468482
$786$	−44.6653	−1.59316
$787$	−24.7116	−0.880872	−0.440436	−	0.897784i	$-0.645176\pi$
$787$	−24.7116	−0.880872	−0.440436	+	0.897784i	$0.645176\pi$
$788$	32.2188	1.14775
$789$	4.14095	0.147422
$790$	−17.7158	−0.630301
$791$	−7.42438	−0.263981
$792$	11.3422	0.403027
$793$	15.3387	0.544694
$794$	−10.9332	−0.388004
$795$	−24.2109	−0.858672
$796$	−108.604	−3.84938
$797$	25.2137	0.893116	0.446558	−	0.894755i	$-0.352650\pi$
$797$	25.2137	0.893116	0.446558	+	0.894755i	$0.352650\pi$
$798$	30.7565	1.08877
$799$	−14.9925	−0.530395
$800$	3.51108	0.124136
$801$	−12.6772	−0.447929
$802$	−15.6659	−0.553183
$803$	13.8867	0.490051
$804$	64.7614	2.28396
$805$	1.15022	0.0405400
$806$	69.2525	2.43932
$807$	−19.9641	−0.702768
$808$	−12.8472	−0.451963
$809$	4.65761	0.163753	0.0818765	−	0.996642i	$-0.473909\pi$
$809$	4.65761	0.163753	0.0818765	+	0.996642i	$0.473909\pi$
$810$	5.17577	0.181858
$811$	44.2874	1.55514	0.777570	−	0.628796i	$-0.216452\pi$
$811$	44.2874	1.55514	0.777570	+	0.628796i	$0.216452\pi$
$812$	40.9359	1.43657
$813$	−2.47793	−0.0869048
$814$	−6.74748	−0.236499
$815$	11.4965	0.402704
$816$	−18.7278	−0.655604
$817$	3.18182	0.111318
$818$	35.7305	1.24929
$819$	23.3772	0.816867
$820$	−75.5736	−2.63915
$821$	−46.3836	−1.61880	−0.809399	−	0.587260i	$-0.800207\pi$
$821$	−46.3836	−1.61880	−0.809399	+	0.587260i	$0.800207\pi$
$822$	12.8877	0.449509
$823$	−15.4513	−0.538598	−0.269299	−	0.963057i	$-0.586792\pi$
$823$	−15.4513	−0.538598	−0.269299	+	0.963057i	$0.586792\pi$
$824$	41.8586	1.45821
$825$	−1.51697	−0.0528142
$826$	4.90091	0.170524
$827$	4.09079	0.142251	0.0711253	−	0.997467i	$-0.477341\pi$
$827$	4.09079	0.142251	0.0711253	+	0.997467i	$0.477341\pi$
$828$	−0.677391	−0.0235410
$829$	−27.9047	−0.969170	−0.484585	−	0.874744i	$-0.661029\pi$
$829$	−27.9047	−0.969170	−0.484585	+	0.874744i	$0.661029\pi$
$830$	−21.7820	−0.756066
$831$	−6.27473	−0.217668
$832$	−12.9768	−0.449890
$833$	18.0479	0.625322
$834$	−26.4304	−0.915210
$835$	16.1716	0.559641
$836$	−27.5221	−0.951872
$837$	4.26768	0.147513
$838$	4.02281	0.138966
$839$	−44.1548	−1.52439	−0.762197	−	0.647345i	$-0.775879\pi$
$839$	−44.1548	−1.52439	−0.762197	+	0.647345i	$0.775879\pi$
$840$	−45.0742	−1.55521
$841$	−22.4481	−0.774074
$842$	76.0079	2.61941
$843$	9.61796	0.331260
$844$	−78.6258	−2.70641
$845$	−57.7360	−1.98618
$846$	13.1553	0.452288
$847$	−27.2409	−0.936008
$848$	−76.8687	−2.63968
$849$	−2.24629	−0.0770924
$850$	5.88897	0.201990
$851$	0.219440	0.00752230
$852$	19.7992	0.678310
$853$	1.75668	0.0601477	0.0300738	−	0.999548i	$-0.490426\pi$
$853$	1.75668	0.0601477	0.0300738	+	0.999548i	$0.490426\pi$
$854$	−22.0022	−0.752901
$855$	−6.83898	−0.233888
$856$	−11.3471	−0.387835
$857$	−7.51821	−0.256817	−0.128409	−	0.991721i	$-0.540987\pi$
$857$	−7.51821	−0.256817	−0.128409	+	0.991721i	$0.540987\pi$
$858$	−30.4465	−1.03943
$859$	−18.4249	−0.628649	−0.314324	−	0.949316i	$-0.601778\pi$
$859$	−18.4249	−0.628649	−0.314324	+	0.949316i	$0.601778\pi$
$860$	−8.56317	−0.292002
$861$	−30.6157	−1.04338
$862$	59.3940	2.02297
$863$	−27.5649	−0.938319	−0.469159	−	0.883114i	$-0.655443\pi$
$863$	−27.5649	−0.938319	−0.469159	+	0.883114i	$0.655443\pi$
$864$	4.34266	0.147740
$865$	29.8426	1.01468
$866$	−53.2563	−1.80972
$867$	8.69907	0.295436
$868$	−68.2517	−2.31661
$869$	6.42214	0.217856
$870$	−13.2482	−0.449157
$871$	−94.6646	−3.20759
$872$	−74.1890	−2.51236
$873$	−9.81142	−0.332066
$874$	1.30273	0.0440656
$875$	43.3101	1.46415
$876$	32.5003	1.09808
$877$	−44.4373	−1.50054	−0.750271	−	0.661130i	$-0.770077\pi$
$877$	−44.4373	−1.50054	−0.750271	+	0.661130i	$0.770077\pi$
$878$	32.6953	1.10341
$879$	17.8234	0.601167
$880$	24.9689	0.841703
$881$	−28.4854	−0.959696	−0.479848	−	0.877352i	$-0.659308\pi$
$881$	−28.4854	−0.959696	−0.479848	+	0.877352i	$0.659308\pi$
$882$	−15.8363	−0.533236
$883$	8.52061	0.286741	0.143371	−	0.989669i	$-0.454206\pi$
$883$	8.52061	0.286741	0.143371	+	0.989669i	$0.454206\pi$
$884$	81.2079	2.73132
$885$	−1.08976	−0.0366318
$886$	13.7370	0.461502
$887$	−43.2090	−1.45082	−0.725408	−	0.688319i	$-0.758349\pi$
$887$	−43.2090	−1.45082	−0.725408	+	0.688319i	$0.758349\pi$
$888$	−8.59928	−0.288573
$889$	−15.9043	−0.533414
$890$	−65.6145	−2.19940
$891$	−1.87626	−0.0628571
$892$	−13.9231	−0.466179
$893$	−17.3827	−0.581689
$894$	−46.6582	−1.56048
$895$	−22.0959	−0.738586
$896$	50.2462	1.67861
$897$	0.990173	0.0330609
$898$	−43.8928	−1.46472
$899$	−10.9238	−0.364329
$900$	−3.55031	−0.118344
$901$	−34.0714	−1.13508
$902$	39.8738	1.32765
$903$	−3.46903	−0.115442
$904$	12.3232	0.409864
$905$	14.6738	0.487775
$906$	3.93863	0.130852
$907$	−7.73745	−0.256918	−0.128459	−	0.991715i	$-0.541003\pi$
$907$	−7.73745	−0.256918	−0.128459	+	0.991715i	$0.541003\pi$
$908$	−48.8097	−1.61981
$909$	2.12522	0.0704892
$910$	120.995	4.01095
$911$	−37.0500	−1.22752	−0.613761	−	0.789492i	$-0.710344\pi$
$911$	−37.0500	−1.22752	−0.613761	+	0.789492i	$0.710344\pi$
$912$	−21.7135	−0.719006
$913$	7.89617	0.261325
$914$	−18.5874	−0.614815
$915$	4.89238	0.161737
$916$	39.6983	1.31167
$917$	−64.3457	−2.12488
$918$	7.28373	0.240399
$919$	18.4862	0.609802	0.304901	−	0.952384i	$-0.401377\pi$
$919$	18.4862	0.609802	0.304901	+	0.952384i	$0.401377\pi$
$920$	−1.90918	−0.0629437
$921$	4.97502	0.163932
$922$	−8.51168	−0.280317
$923$	−28.9414	−0.952618
$924$	30.0065	0.987140
$925$	1.15012	0.0378157
$926$	−49.0085	−1.61052
$927$	−6.92437	−0.227426
$928$	−11.1157	−0.364892
$929$	19.6890	0.645977	0.322988	−	0.946403i	$-0.395313\pi$
$929$	19.6890	0.645977	0.322988	+	0.946403i	$0.395313\pi$
$930$	22.0885	0.724311
$931$	20.9252	0.685796
$932$	36.5727	1.19798
$933$	14.8183	0.485129
$934$	4.69740	0.153704
$935$	11.0673	0.361939
$936$	−38.8023	−1.26829
$937$	−42.6599	−1.39364	−0.696819	−	0.717247i	$-0.745402\pi$
$937$	−42.6599	−1.39364	−0.696819	+	0.717247i	$0.745402\pi$
$938$	135.789	4.43367
$939$	−14.5307	−0.474192
$940$	46.7816	1.52585
$941$	−11.7829	−0.384113	−0.192057	−	0.981384i	$-0.561516\pi$
$941$	−11.7829	−0.384113	−0.192057	+	0.981384i	$0.561516\pi$
$942$	−16.2082	−0.528091
$943$	−1.29677	−0.0422285
$944$	−3.45994	−0.112612
$945$	7.45631	0.242554
$946$	4.51806	0.146895
$947$	38.9874	1.26692	0.633461	−	0.773775i	$-0.281634\pi$
$947$	38.9874	1.26692	0.633461	+	0.773775i	$0.281634\pi$
$948$	15.0303	0.488162
$949$	−47.5071	−1.54215
$950$	6.82783	0.221524
$951$	−22.9481	−0.744144
$952$	−63.4319	−2.05584
$953$	−9.18081	−0.297396	−0.148698	−	0.988883i	$-0.547508\pi$
$953$	−9.18081	−0.297396	−0.148698	+	0.988883i	$0.547508\pi$
$954$	29.8963	0.967929
$955$	26.0882	0.844196
$956$	62.0638	2.00729
$957$	4.80259	0.155246
$958$	−30.9569	−1.00017
$959$	18.5662	0.599535
$960$	−4.13903	−0.133587
$961$	−12.7869	−0.412482
$962$	23.0835	0.744242
$963$	1.87707	0.0604876
$964$	−59.2426	−1.90808
$965$	−0.256430	−0.00825477
$966$	−1.42033	−0.0456983
$967$	1.11931	0.0359946	0.0179973	−	0.999838i	$-0.494271\pi$
$967$	1.11931	0.0359946	0.0179973	+	0.999838i	$0.494271\pi$
$968$	45.2153	1.45327
$969$	−9.62433	−0.309178
$970$	−50.7817	−1.63050
$971$	−15.4898	−0.497091	−0.248546	−	0.968620i	$-0.579953\pi$
$971$	−15.4898	−0.497091	−0.248546	+	0.968620i	$0.579953\pi$
$972$	−4.39119	−0.140847
$973$	−38.0761	−1.22066
$974$	43.6817	1.39965
$975$	5.18965	0.166202
$976$	15.5331	0.497204
$977$	−23.3753	−0.747842	−0.373921	−	0.927461i	$-0.621987\pi$
$977$	−23.3753	−0.747842	−0.373921	+	0.927461i	$0.621987\pi$
$978$	−14.1962	−0.453944
$979$	23.7858	0.760198
$980$	−56.3156	−1.79894
$981$	12.2726	0.391833
$982$	26.2792	0.838603
$983$	−55.0129	−1.75464	−0.877320	−	0.479906i	$-0.840671\pi$
$983$	−55.0129	−1.75464	−0.877320	+	0.479906i	$0.840671\pi$
$984$	50.8169	1.61998
$985$	−15.0215	−0.478624
$986$	−18.6439	−0.593743
$987$	18.9517	0.603241
$988$	94.1547	2.99546
$989$	−0.146935	−0.00467227
$990$	−9.71109	−0.308639
$991$	−15.0973	−0.479582	−0.239791	−	0.970825i	$-0.577079\pi$
$991$	−15.0973	−0.479582	−0.239791	+	0.970825i	$0.577079\pi$
$992$	18.5331	0.588426
$993$	−25.7649	−0.817624
$994$	41.5142	1.31675
$995$	50.6349	1.60523
$996$	18.4802	0.585566
$997$	22.4421	0.710748	0.355374	−	0.934724i	$-0.384353\pi$
$997$	22.4421	0.710748	0.355374	+	0.934724i	$0.384353\pi$
$998$	77.3173	2.44744
$999$	1.42252	0.0450065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.b.1.8	✓	104

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.b.1.8	✓	104	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52808 q^{2} -1.00000 q^{3} +4.39119 q^{4} -2.04731 q^{5} +2.52808 q^{6} +3.64200 q^{7} -6.04511 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.52808 q^{2} -1.00000 q^{3} +4.39119 q^{4} -2.04731 q^{5} +2.52808 q^{6} +3.64200 q^{7} -6.04511 q^{8} +1.00000 q^{9} +5.17577 q^{10} -1.87626 q^{11} -4.39119 q^{12} +6.41879 q^{13} -9.20726 q^{14} +2.04731 q^{15} +6.50014 q^{16} +2.88113 q^{17} -2.52808 q^{18} +3.34047 q^{19} -8.99013 q^{20} -3.64200 q^{21} +4.74333 q^{22} -0.154262 q^{23} +6.04511 q^{24} -0.808509 q^{25} -16.2272 q^{26} -1.00000 q^{27} +15.9927 q^{28} +2.55966 q^{29} -5.17577 q^{30} -4.26768 q^{31} -4.34266 q^{32} +1.87626 q^{33} -7.28373 q^{34} -7.45631 q^{35} +4.39119 q^{36} -1.42252 q^{37} -8.44496 q^{38} -6.41879 q^{39} +12.3762 q^{40} +8.40629 q^{41} +9.20726 q^{42} +0.952507 q^{43} -8.23901 q^{44} -2.04731 q^{45} +0.389986 q^{46} -5.20366 q^{47} -6.50014 q^{48} +6.26416 q^{49} +2.04398 q^{50} -2.88113 q^{51} +28.1861 q^{52} -11.8257 q^{53} +2.52808 q^{54} +3.84129 q^{55} -22.0163 q^{56} -3.34047 q^{57} -6.47103 q^{58} -0.532287 q^{59} +8.99013 q^{60} +2.38966 q^{61} +10.7890 q^{62} +3.64200 q^{63} -2.02169 q^{64} -13.1413 q^{65} -4.74333 q^{66} -14.7480 q^{67} +12.6516 q^{68} +0.154262 q^{69} +18.8502 q^{70} -4.50886 q^{71} -6.04511 q^{72} -7.40126 q^{73} +3.59624 q^{74} +0.808509 q^{75} +14.6686 q^{76} -6.83334 q^{77} +16.2272 q^{78} -3.42284 q^{79} -13.3078 q^{80} +1.00000 q^{81} -21.2518 q^{82} -4.20847 q^{83} -15.9927 q^{84} -5.89858 q^{85} -2.40801 q^{86} -2.55966 q^{87} +11.3422 q^{88} -12.6772 q^{89} +5.17577 q^{90} +23.3772 q^{91} -0.677391 q^{92} +4.26768 q^{93} +13.1553 q^{94} -6.83898 q^{95} +4.34266 q^{96} -9.81142 q^{97} -15.8363 q^{98} -1.87626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * q^99

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