LMFDB - Embedded newform 1110.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.655762$ of defining polynomial
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.46569 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.46569 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.15417 q^{11} +1.00000 q^{12} -3.73836 q^{13} -3.46569 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.63408 q^{17} +1.00000 q^{18} +5.88150 q^{19} +1.00000 q^{20} -3.46569 q^{21} +4.15417 q^{22} +4.36141 q^{23} +1.00000 q^{24} +1.00000 q^{25} -3.73836 q^{26} +1.00000 q^{27} -3.46569 q^{28} +8.03567 q^{29} +1.00000 q^{30} +0.895717 q^{31} +1.00000 q^{32} +4.15417 q^{33} +4.63408 q^{34} -3.46569 q^{35} +1.00000 q^{36} -1.00000 q^{37} +5.88150 q^{38} -3.73836 q^{39} +1.00000 q^{40} -2.89572 q^{41} -3.46569 q^{42} -10.0357 q^{43} +4.15417 q^{44} +1.00000 q^{45} +4.36141 q^{46} -6.78825 q^{47} +1.00000 q^{48} +5.01103 q^{49} +1.00000 q^{50} +4.63408 q^{51} -3.73836 q^{52} -8.63408 q^{53} +1.00000 q^{54} +4.15417 q^{55} -3.46569 q^{56} +5.88150 q^{57} +8.03567 q^{58} +6.93139 q^{59} +1.00000 q^{60} -2.20724 q^{61} +0.895717 q^{62} -3.46569 q^{63} +1.00000 q^{64} -3.73836 q^{65} +4.15417 q^{66} -6.09978 q^{67} +4.63408 q^{68} +4.36141 q^{69} -3.46569 q^{70} -13.1400 q^{71} +1.00000 q^{72} -7.19302 q^{73} -1.00000 q^{74} +1.00000 q^{75} +5.88150 q^{76} -14.3971 q^{77} -3.73836 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.89572 q^{82} +10.0467 q^{83} -3.46569 q^{84} +4.63408 q^{85} -10.0357 q^{86} +8.03567 q^{87} +4.15417 q^{88} -12.8783 q^{89} +1.00000 q^{90} +12.9560 q^{91} +4.36141 q^{92} +0.895717 q^{93} -6.78825 q^{94} +5.88150 q^{95} +1.00000 q^{96} +7.06092 q^{97} +5.01103 q^{98} +4.15417 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{11} + 4 q^{12} - 3 q^{13} + 4 q^{14} + 4 q^{15} + 4 q^{16} + 6 q^{17} + 4 q^{18} + 3 q^{19} + 4 q^{20} + 4 q^{21} + 2 q^{22} - q^{23} + 4 q^{24} + 4 q^{25} - 3 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} + 4 q^{30} + 3 q^{31} + 4 q^{32} + 2 q^{33} + 6 q^{34} + 4 q^{35} + 4 q^{36} - 4 q^{37} + 3 q^{38} - 3 q^{39} + 4 q^{40} - 11 q^{41} + 4 q^{42} - 5 q^{43} + 2 q^{44} + 4 q^{45} - q^{46} + 4 q^{48} + 14 q^{49} + 4 q^{50} + 6 q^{51} - 3 q^{52} - 22 q^{53} + 4 q^{54} + 2 q^{55} + 4 q^{56} + 3 q^{57} - 3 q^{58} - 8 q^{59} + 4 q^{60} - 5 q^{61} + 3 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{65} + 2 q^{66} + 6 q^{67} + 6 q^{68} - q^{69} + 4 q^{70} - 18 q^{71} + 4 q^{72} - 5 q^{73} - 4 q^{74} + 4 q^{75} + 3 q^{76} - 4 q^{77} - 3 q^{78} + 16 q^{79} + 4 q^{80} + 4 q^{81} - 11 q^{82} - q^{83} + 4 q^{84} + 6 q^{85} - 5 q^{86} - 3 q^{87} + 2 q^{88} - 5 q^{89} + 4 q^{90} - 13 q^{91} - q^{92} + 3 q^{93} + 3 q^{95} + 4 q^{96} + 7 q^{97} + 14 q^{98} + 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9 + 4 * q^10 + 2 * q^11 + 4 * q^12 - 3 * q^13 + 4 * q^14 + 4 * q^15 + 4 * q^16 + 6 * q^17 + 4 * q^18 + 3 * q^19 + 4 * q^20 + 4 * q^21 + 2 * q^22 - q^23 + 4 * q^24 + 4 * q^25 - 3 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 + 4 * q^30 + 3 * q^31 + 4 * q^32 + 2 * q^33 + 6 * q^34 + 4 * q^35 + 4 * q^36 - 4 * q^37 + 3 * q^38 - 3 * q^39 + 4 * q^40 - 11 * q^41 + 4 * q^42 - 5 * q^43 + 2 * q^44 + 4 * q^45 - q^46 + 4 * q^48 + 14 * q^49 + 4 * q^50 + 6 * q^51 - 3 * q^52 - 22 * q^53 + 4 * q^54 + 2 * q^55 + 4 * q^56 + 3 * q^57 - 3 * q^58 - 8 * q^59 + 4 * q^60 - 5 * q^61 + 3 * q^62 + 4 * q^63 + 4 * q^64 - 3 * q^65 + 2 * q^66 + 6 * q^67 + 6 * q^68 - q^69 + 4 * q^70 - 18 * q^71 + 4 * q^72 - 5 * q^73 - 4 * q^74 + 4 * q^75 + 3 * q^76 - 4 * q^77 - 3 * q^78 + 16 * q^79 + 4 * q^80 + 4 * q^81 - 11 * q^82 - q^83 + 4 * q^84 + 6 * q^85 - 5 * q^86 - 3 * q^87 + 2 * q^88 - 5 * q^89 + 4 * q^90 - 13 * q^91 - q^92 + 3 * q^93 + 3 * q^95 + 4 * q^96 + 7 * q^97 + 14 * q^98 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−3.46569	−1.30991	−0.654955	−	0.755668i	$-0.727312\pi$
$7$	−3.46569	−1.30991	−0.654955	+	0.755668i	$0.727312\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.15417	1.25253	0.626265	−	0.779610i	$-0.284583\pi$
$11$	4.15417	1.25253	0.626265	+	0.779610i	$0.284583\pi$
$12$	1.00000	0.288675
$13$	−3.73836	−1.03684	−0.518418	−	0.855127i	$-0.673479\pi$
$13$	−3.73836	−1.03684	−0.518418	+	0.855127i	$0.673479\pi$
$14$	−3.46569	−0.926246
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	4.63408	1.12393	0.561965	−	0.827161i	$-0.310046\pi$
$17$	4.63408	1.12393	0.561965	+	0.827161i	$0.310046\pi$
$18$	1.00000	0.235702
$19$	5.88150	1.34931	0.674654	−	0.738134i	$-0.264293\pi$
$19$	5.88150	1.34931	0.674654	+	0.738134i	$0.264293\pi$
$20$	1.00000	0.223607
$21$	−3.46569	−0.756276
$22$	4.15417	0.885672
$23$	4.36141	0.909417	0.454709	−	0.890640i	$-0.349743\pi$
$23$	4.36141	0.909417	0.454709	+	0.890640i	$0.349743\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−3.73836	−0.733153
$27$	1.00000	0.192450
$28$	−3.46569	−0.654955
$29$	8.03567	1.49219	0.746093	−	0.665841i	$-0.231927\pi$
$29$	8.03567	1.49219	0.746093	+	0.665841i	$0.231927\pi$
$30$	1.00000	0.182574
$31$	0.895717	0.160876	0.0804378	−	0.996760i	$-0.474368\pi$
$31$	0.895717	0.160876	0.0804378	+	0.996760i	$0.474368\pi$
$32$	1.00000	0.176777
$33$	4.15417	0.723148
$34$	4.63408	0.794738
$35$	−3.46569	−0.585809
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	5.88150	0.954105
$39$	−3.73836	−0.598617
$40$	1.00000	0.158114
$41$	−2.89572	−0.452235	−0.226118	−	0.974100i	$-0.572603\pi$
$41$	−2.89572	−0.452235	−0.226118	+	0.974100i	$0.572603\pi$
$42$	−3.46569	−0.534768
$43$	−10.0357	−1.53043	−0.765213	−	0.643778i	$-0.777366\pi$
$43$	−10.0357	−1.53043	−0.765213	+	0.643778i	$0.777366\pi$
$44$	4.15417	0.626265
$45$	1.00000	0.149071
$46$	4.36141	0.643055
$47$	−6.78825	−0.990168	−0.495084	−	0.868845i	$-0.664863\pi$
$47$	−6.78825	−0.990168	−0.495084	+	0.868845i	$0.664863\pi$
$48$	1.00000	0.144338
$49$	5.01103	0.715862
$50$	1.00000	0.141421
$51$	4.63408	0.648901
$52$	−3.73836	−0.518418
$53$	−8.63408	−1.18598	−0.592991	−	0.805209i	$-0.702053\pi$
$53$	−8.63408	−1.18598	−0.592991	+	0.805209i	$0.702053\pi$
$54$	1.00000	0.136083
$55$	4.15417	0.560148
$56$	−3.46569	−0.463123
$57$	5.88150	0.779024
$58$	8.03567	1.05514
$59$	6.93139	0.902390	0.451195	−	0.892425i	$-0.350998\pi$
$59$	6.93139	0.902390	0.451195	+	0.892425i	$0.350998\pi$
$60$	1.00000	0.129099
$61$	−2.20724	−0.282608	−0.141304	−	0.989966i	$-0.545130\pi$
$61$	−2.20724	−0.282608	−0.141304	+	0.989966i	$0.545130\pi$
$62$	0.895717	0.113756
$63$	−3.46569	−0.436636
$64$	1.00000	0.125000
$65$	−3.73836	−0.463687
$66$	4.15417	0.511343
$67$	−6.09978	−0.745206	−0.372603	−	0.927991i	$-0.621535\pi$
$67$	−6.09978	−0.745206	−0.372603	+	0.927991i	$0.621535\pi$
$68$	4.63408	0.561965
$69$	4.36141	0.525052
$70$	−3.46569	−0.414230
$71$	−13.1400	−1.55943	−0.779713	−	0.626137i	$-0.784635\pi$
$71$	−13.1400	−1.55943	−0.779713	+	0.626137i	$0.784635\pi$
$72$	1.00000	0.117851
$73$	−7.19302	−0.841880	−0.420940	−	0.907089i	$-0.638300\pi$
$73$	−7.19302	−0.841880	−0.420940	+	0.907089i	$0.638300\pi$
$74$	−1.00000	−0.116248
$75$	1.00000	0.115470
$76$	5.88150	0.674654
$77$	−14.3971	−1.64070
$78$	−3.73836	−0.423286
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.89572	−0.319778
$83$	10.0467	1.10277	0.551385	−	0.834251i	$-0.314100\pi$
$83$	10.0467	1.10277	0.551385	+	0.834251i	$0.314100\pi$
$84$	−3.46569	−0.378138
$85$	4.63408	0.502637
$86$	−10.0357	−1.08217
$87$	8.03567	0.861514
$88$	4.15417	0.442836
$89$	−12.8783	−1.36510	−0.682549	−	0.730839i	$-0.739129\pi$
$89$	−12.8783	−1.36510	−0.682549	+	0.730839i	$0.739129\pi$
$90$	1.00000	0.105409
$91$	12.9560	1.35816
$92$	4.36141	0.454709
$93$	0.895717	0.0928816
$94$	−6.78825	−0.700155
$95$	5.88150	0.603429
$96$	1.00000	0.102062
$97$	7.06092	0.716928	0.358464	−	0.933544i	$-0.383301\pi$
$97$	7.06092	0.716928	0.358464	+	0.933544i	$0.383301\pi$
$98$	5.01103	0.506191
$99$	4.15417	0.417510
$100$	1.00000	0.100000
$101$	−16.2429	−1.61623	−0.808115	−	0.589025i	$-0.799512\pi$
$101$	−16.2429	−1.61623	−0.808115	+	0.589025i	$0.799512\pi$
$102$	4.63408	0.458842
$103$	6.78825	0.668866	0.334433	−	0.942419i	$-0.391455\pi$
$103$	6.78825	0.668866	0.334433	+	0.942419i	$0.391455\pi$
$104$	−3.73836	−0.366577
$105$	−3.46569	−0.338217
$106$	−8.63408	−0.838616
$107$	−8.66975	−0.838137	−0.419068	−	0.907955i	$-0.637643\pi$
$107$	−8.66975	−0.838137	−0.419068	+	0.907955i	$0.637643\pi$
$108$	1.00000	0.0962250
$109$	−16.5623	−1.58638	−0.793190	−	0.608975i	$-0.791581\pi$
$109$	−16.5623	−1.58638	−0.793190	+	0.608975i	$0.791581\pi$
$110$	4.15417	0.396085
$111$	−1.00000	−0.0949158
$112$	−3.46569	−0.327477
$113$	−16.0357	−1.50851	−0.754254	−	0.656582i	$-0.772002\pi$
$113$	−16.0357	−1.50851	−0.754254	+	0.656582i	$0.772002\pi$
$114$	5.88150	0.550853
$115$	4.36141	0.406704
$116$	8.03567	0.746093
$117$	−3.73836	−0.345612
$118$	6.93139	0.638086
$119$	−16.0603	−1.47225
$120$	1.00000	0.0912871
$121$	6.25713	0.568830
$122$	−2.20724	−0.199834
$123$	−2.89572	−0.261098
$124$	0.895717	0.0804378
$125$	1.00000	0.0894427
$126$	−3.46569	−0.308749
$127$	−3.11532	−0.276440	−0.138220	−	0.990402i	$-0.544138\pi$
$127$	−3.11532	−0.276440	−0.138220	+	0.990402i	$0.544138\pi$
$128$	1.00000	0.0883883
$129$	−10.0357	−0.883591
$130$	−3.73836	−0.327876
$131$	12.0713	1.05468	0.527339	−	0.849655i	$-0.323190\pi$
$131$	12.0713	1.05468	0.527339	+	0.849655i	$0.323190\pi$
$132$	4.15417	0.361574
$133$	−20.3835	−1.76747
$134$	−6.09978	−0.526940
$135$	1.00000	0.0860663
$136$	4.63408	0.397369
$137$	21.6912	1.85320	0.926602	−	0.376043i	$-0.122715\pi$
$137$	21.6912	1.85320	0.926602	+	0.376043i	$0.122715\pi$
$138$	4.36141	0.371268
$139$	5.31285	0.450630	0.225315	−	0.974286i	$-0.427659\pi$
$139$	5.31285	0.450630	0.225315	+	0.974286i	$0.427659\pi$
$140$	−3.46569	−0.292905
$141$	−6.78825	−0.571674
$142$	−13.1400	−1.10268
$143$	−15.5298	−1.29867
$144$	1.00000	0.0833333
$145$	8.03567	0.667326
$146$	−7.19302	−0.595299
$147$	5.01103	0.413303
$148$	−1.00000	−0.0821995
$149$	22.6510	1.85564	0.927822	−	0.373023i	$-0.121679\pi$
$149$	22.6510	1.85564	0.927822	+	0.373023i	$0.121679\pi$
$150$	1.00000	0.0816497
$151$	3.94693	0.321197	0.160598	−	0.987020i	$-0.448658\pi$
$151$	3.94693	0.321197	0.160598	+	0.987020i	$0.448658\pi$
$152$	5.88150	0.477053
$153$	4.63408	0.374643
$154$	−14.3971	−1.16015
$155$	0.895717	0.0719458
$156$	−3.73836	−0.299309
$157$	20.7585	1.65671	0.828354	−	0.560205i	$-0.189278\pi$
$157$	20.7585	1.65671	0.828354	+	0.560205i	$0.189278\pi$
$158$	4.00000	0.318223
$159$	−8.63408	−0.684727
$160$	1.00000	0.0790569
$161$	−15.1153	−1.19125
$162$	1.00000	0.0785674
$163$	19.6652	1.54030	0.770150	−	0.637862i	$-0.220181\pi$
$163$	19.6652	1.54030	0.770150	+	0.637862i	$0.220181\pi$
$164$	−2.89572	−0.226118
$165$	4.15417	0.323402
$166$	10.0467	0.779775
$167$	20.7980	1.60939	0.804697	−	0.593685i	$-0.202328\pi$
$167$	20.7980	1.60939	0.804697	+	0.593685i	$0.202328\pi$
$168$	−3.46569	−0.267384
$169$	0.975364	0.0750280
$170$	4.63408	0.355418
$171$	5.88150	0.449770
$172$	−10.0357	−0.765213
$173$	−5.15735	−0.392106	−0.196053	−	0.980593i	$-0.562813\pi$
$173$	−5.15735	−0.392106	−0.196053	+	0.980593i	$0.562813\pi$
$174$	8.03567	0.609183
$175$	−3.46569	−0.261982
$176$	4.15417	0.313132
$177$	6.93139	0.520995
$178$	−12.8783	−0.965271
$179$	1.66323	0.124315	0.0621576	−	0.998066i	$-0.480202\pi$
$179$	1.66323	0.124315	0.0621576	+	0.998066i	$0.480202\pi$
$180$	1.00000	0.0745356
$181$	−15.8628	−1.17907	−0.589535	−	0.807743i	$-0.700689\pi$
$181$	−15.8628	−1.17907	−0.589535	+	0.807743i	$0.700689\pi$
$182$	12.9560	0.960364
$183$	−2.20724	−0.163164
$184$	4.36141	0.321528
$185$	−1.00000	−0.0735215
$186$	0.895717	0.0656772
$187$	19.2508	1.40776
$188$	−6.78825	−0.495084
$189$	−3.46569	−0.252092
$190$	5.88150	0.426689
$191$	−19.2287	−1.39134	−0.695670	−	0.718362i	$-0.744892\pi$
$191$	−19.2287	−1.39134	−0.695670	+	0.718362i	$0.744892\pi$
$192$	1.00000	0.0721688
$193$	−12.7883	−0.920518	−0.460259	−	0.887785i	$-0.652244\pi$
$193$	−12.7883	−0.920518	−0.460259	+	0.887785i	$0.652244\pi$
$194$	7.06092	0.506945
$195$	−3.73836	−0.267710
$196$	5.01103	0.357931
$197$	−10.3614	−0.738220	−0.369110	−	0.929386i	$-0.620337\pi$
$197$	−10.3614	−0.738220	−0.369110	+	0.929386i	$0.620337\pi$
$198$	4.15417	0.295224
$199$	10.6230	0.753048	0.376524	−	0.926407i	$-0.377119\pi$
$199$	10.6230	0.753048	0.376524	+	0.926407i	$0.377119\pi$
$200$	1.00000	0.0707107
$201$	−6.09978	−0.430245
$202$	−16.2429	−1.14285
$203$	−27.8492	−1.95463
$204$	4.63408	0.324451
$205$	−2.89572	−0.202246
$206$	6.78825	0.472960
$207$	4.36141	0.303139
$208$	−3.73836	−0.259209
$209$	24.4328	1.69005
$210$	−3.46569	−0.239156
$211$	4.06410	0.279785	0.139892	−	0.990167i	$-0.455324\pi$
$211$	4.06410	0.279785	0.139892	+	0.990167i	$0.455324\pi$
$212$	−8.63408	−0.592991
$213$	−13.1400	−0.900335
$214$	−8.66975	−0.592652
$215$	−10.0357	−0.684427
$216$	1.00000	0.0680414
$217$	−3.10428	−0.210732
$218$	−16.5623	−1.12174
$219$	−7.19302	−0.486060
$220$	4.15417	0.280074
$221$	−17.3239	−1.16533
$222$	−1.00000	−0.0671156
$223$	−9.61854	−0.644105	−0.322053	−	0.946722i	$-0.604373\pi$
$223$	−9.61854	−0.644105	−0.322053	+	0.946722i	$0.604373\pi$
$224$	−3.46569	−0.231561
$225$	1.00000	0.0666667
$226$	−16.0357	−1.06668
$227$	−11.6122	−0.770727	−0.385363	−	0.922765i	$-0.625924\pi$
$227$	−11.6122	−0.770727	−0.385363	+	0.922765i	$0.625924\pi$
$228$	5.88150	0.389512
$229$	23.0312	1.52194	0.760971	−	0.648786i	$-0.224723\pi$
$229$	23.0312	1.52194	0.760971	+	0.648786i	$0.224723\pi$
$230$	4.36141	0.287583
$231$	−14.3971	−0.947258
$232$	8.03567	0.527568
$233$	−1.42031	−0.0930478	−0.0465239	−	0.998917i	$-0.514814\pi$
$233$	−1.42031	−0.0930478	−0.0465239	+	0.998917i	$0.514814\pi$
$234$	−3.73836	−0.244384
$235$	−6.78825	−0.442817
$236$	6.93139	0.451195
$237$	4.00000	0.259828
$238$	−16.0603	−1.04104
$239$	−18.4502	−1.19344	−0.596721	−	0.802449i	$-0.703530\pi$
$239$	−18.4502	−1.19344	−0.596721	+	0.802449i	$0.703530\pi$
$240$	1.00000	0.0645497
$241$	−26.8162	−1.72739	−0.863693	−	0.504019i	$-0.831854\pi$
$241$	−26.8162	−1.72739	−0.863693	+	0.504019i	$0.831854\pi$
$242$	6.25713	0.402223
$243$	1.00000	0.0641500
$244$	−2.20724	−0.141304
$245$	5.01103	0.320143
$246$	−2.89572	−0.184624
$247$	−21.9872	−1.39901
$248$	0.895717	0.0568781
$249$	10.0467	0.636684
$250$	1.00000	0.0632456
$251$	17.0532	1.07639	0.538195	−	0.842820i	$-0.319106\pi$
$251$	17.0532	1.07639	0.538195	+	0.842820i	$0.319106\pi$
$252$	−3.46569	−0.218318
$253$	18.1180	1.13907
$254$	−3.11532	−0.195472
$255$	4.63408	0.290197
$256$	1.00000	0.0625000
$257$	19.5014	1.21646	0.608231	−	0.793760i	$-0.291879\pi$
$257$	19.5014	1.21646	0.608231	+	0.793760i	$0.291879\pi$
$258$	−10.0357	−0.624794
$259$	3.46569	0.215348
$260$	−3.73836	−0.231843
$261$	8.03567	0.497395
$262$	12.0713	0.745770
$263$	−21.6555	−1.33534	−0.667669	−	0.744459i	$-0.732708\pi$
$263$	−21.6555	−1.33534	−0.667669	+	0.744459i	$0.732708\pi$
$264$	4.15417	0.255671
$265$	−8.63408	−0.530387
$266$	−20.3835	−1.24979
$267$	−12.8783	−0.788140
$268$	−6.09978	−0.372603
$269$	−16.0811	−0.980479	−0.490239	−	0.871588i	$-0.663091\pi$
$269$	−16.0811	−0.980479	−0.490239	+	0.871588i	$0.663091\pi$
$270$	1.00000	0.0608581
$271$	6.62305	0.402321	0.201161	−	0.979558i	$-0.435529\pi$
$271$	6.62305	0.402321	0.201161	+	0.979558i	$0.435529\pi$
$272$	4.63408	0.280982
$273$	12.9560	0.784134
$274$	21.6912	1.31041
$275$	4.15417	0.250506
$276$	4.36141	0.262526
$277$	−28.0467	−1.68516	−0.842582	−	0.538569i	$-0.818965\pi$
$277$	−28.0467	−1.68516	−0.842582	+	0.538569i	$0.818965\pi$
$278$	5.31285	0.318643
$279$	0.895717	0.0536252
$280$	−3.46569	−0.207115
$281$	−4.26164	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$281$	−4.26164	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$282$	−6.78825	−0.404234
$283$	0.884684	0.0525890	0.0262945	−	0.999654i	$-0.491629\pi$
$283$	0.884684	0.0525890	0.0262945	+	0.999654i	$0.491629\pi$
$284$	−13.1400	−0.779713
$285$	5.88150	0.348390
$286$	−15.5298	−0.918296
$287$	10.0357	0.592387
$288$	1.00000	0.0589256
$289$	4.47471	0.263218
$290$	8.03567	0.471871
$291$	7.06092	0.413919
$292$	−7.19302	−0.420940
$293$	−21.4657	−1.25404	−0.627020	−	0.779003i	$-0.715725\pi$
$293$	−21.4657	−1.25404	−0.627020	+	0.779003i	$0.715725\pi$
$294$	5.01103	0.292249
$295$	6.93139	0.403561
$296$	−1.00000	−0.0581238
$297$	4.15417	0.241049
$298$	22.6510	1.31214
$299$	−16.3045	−0.942916
$300$	1.00000	0.0577350
$301$	34.7806	2.00472
$302$	3.94693	0.227120
$303$	−16.2429	−0.933131
$304$	5.88150	0.337327
$305$	−2.20724	−0.126386
$306$	4.63408	0.264913
$307$	34.1711	1.95025	0.975124	−	0.221659i	$-0.0711471\pi$
$307$	34.1711	1.95025	0.975124	+	0.221659i	$0.0711471\pi$
$308$	−14.3971	−0.820350
$309$	6.78825	0.386170
$310$	0.895717	0.0508733
$311$	−10.1639	−0.576341	−0.288170	−	0.957579i	$-0.593047\pi$
$311$	−10.1639	−0.576341	−0.288170	+	0.957579i	$0.593047\pi$
$312$	−3.73836	−0.211643
$313$	−9.31152	−0.526318	−0.263159	−	0.964752i	$-0.584764\pi$
$313$	−9.31152	−0.526318	−0.263159	+	0.964752i	$0.584764\pi$
$314$	20.7585	1.17147
$315$	−3.46569	−0.195270
$316$	4.00000	0.225018
$317$	16.8673	0.947361	0.473680	−	0.880697i	$-0.342925\pi$
$317$	16.8673	0.947361	0.473680	+	0.880697i	$0.342925\pi$
$318$	−8.63408	−0.484175
$319$	33.3815	1.86901
$320$	1.00000	0.0559017
$321$	−8.66975	−0.483898
$322$	−15.1153	−0.842344
$323$	27.2553	1.51653
$324$	1.00000	0.0555556
$325$	−3.73836	−0.207367
$326$	19.6652	1.08916
$327$	−16.5623	−0.915896
$328$	−2.89572	−0.159889
$329$	23.5260	1.29703
$330$	4.15417	0.228680
$331$	−0.451476	−0.0248154	−0.0124077	−	0.999923i	$-0.503950\pi$
$331$	−0.451476	−0.0248154	−0.0124077	+	0.999923i	$0.503950\pi$
$332$	10.0467	0.551385
$333$	−1.00000	−0.0547997
$334$	20.7980	1.13801
$335$	−6.09978	−0.333266
$336$	−3.46569	−0.189069
$337$	−28.3550	−1.54460	−0.772299	−	0.635259i	$-0.780893\pi$
$337$	−28.3550	−1.54460	−0.772299	+	0.635259i	$0.780893\pi$
$338$	0.975364	0.0530528
$339$	−16.0357	−0.870938
$340$	4.63408	0.251318
$341$	3.72096	0.201501
$342$	5.88150	0.318035
$343$	6.89315	0.372195
$344$	−10.0357	−0.541087
$345$	4.36141	0.234810
$346$	−5.15735	−0.277261
$347$	5.37695	0.288650	0.144325	−	0.989530i	$-0.453899\pi$
$347$	5.37695	0.288650	0.144325	+	0.989530i	$0.453899\pi$
$348$	8.03567	0.430757
$349$	8.22798	0.440434	0.220217	−	0.975451i	$-0.429323\pi$
$349$	8.22798	0.440434	0.220217	+	0.975451i	$0.429323\pi$
$350$	−3.46569	−0.185249
$351$	−3.73836	−0.199539
$352$	4.15417	0.221418
$353$	5.95531	0.316969	0.158485	−	0.987361i	$-0.449339\pi$
$353$	5.95531	0.316969	0.158485	+	0.987361i	$0.449339\pi$
$354$	6.93139	0.368399
$355$	−13.1400	−0.697396
$356$	−12.8783	−0.682549
$357$	−16.0603	−0.850002
$358$	1.66323	0.0879042
$359$	12.0713	0.637101	0.318550	−	0.947906i	$-0.396804\pi$
$359$	12.0713	0.637101	0.318550	+	0.947906i	$0.396804\pi$
$360$	1.00000	0.0527046
$361$	15.5920	0.820634
$362$	−15.8628	−0.833729
$363$	6.25713	0.328414
$364$	12.9560	0.679080
$365$	−7.19302	−0.376500
$366$	−2.20724	−0.115374
$367$	−0.139243	−0.00726845	−0.00363422	−	0.999993i	$-0.501157\pi$
$367$	−0.139243	−0.00726845	−0.00363422	+	0.999993i	$0.501157\pi$
$368$	4.36141	0.227354
$369$	−2.89572	−0.150745
$370$	−1.00000	−0.0519875
$371$	29.9231	1.55353
$372$	0.895717	0.0464408
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	19.2508	0.995433
$375$	1.00000	0.0516398
$376$	−6.78825	−0.350077
$377$	−30.0403	−1.54715
$378$	−3.46569	−0.178256
$379$	−5.33950	−0.274272	−0.137136	−	0.990552i	$-0.543790\pi$
$379$	−5.33950	−0.274272	−0.137136	+	0.990552i	$0.543790\pi$
$380$	5.88150	0.301715
$381$	−3.11532	−0.159602
$382$	−19.2287	−0.983826
$383$	−13.7384	−0.701998	−0.350999	−	0.936376i	$-0.614158\pi$
$383$	−13.7384	−0.701998	−0.350999	+	0.936376i	$0.614158\pi$
$384$	1.00000	0.0510310
$385$	−14.3971	−0.733743
$386$	−12.7883	−0.650905
$387$	−10.0357	−0.510142
$388$	7.06092	0.358464
$389$	−1.61217	−0.0817404	−0.0408702	−	0.999164i	$-0.513013\pi$
$389$	−1.61217	−0.0817404	−0.0408702	+	0.999164i	$0.513013\pi$
$390$	−3.73836	−0.189299
$391$	20.2111	1.02212
$392$	5.01103	0.253095
$393$	12.0713	0.608919
$394$	−10.3614	−0.522000
$395$	4.00000	0.201262
$396$	4.15417	0.208755
$397$	−17.4767	−0.877132	−0.438566	−	0.898699i	$-0.644513\pi$
$397$	−17.4767	−0.877132	−0.438566	+	0.898699i	$0.644513\pi$
$398$	10.6230	0.532485
$399$	−20.3835	−1.02045
$400$	1.00000	0.0500000
$401$	−4.87832	−0.243611	−0.121806	−	0.992554i	$-0.538868\pi$
$401$	−4.87832	−0.243611	−0.121806	+	0.992554i	$0.538868\pi$
$402$	−6.09978	−0.304229
$403$	−3.34852	−0.166802
$404$	−16.2429	−0.808115
$405$	1.00000	0.0496904
$406$	−27.8492	−1.38213
$407$	−4.15417	−0.205915
$408$	4.63408	0.229421
$409$	−13.9626	−0.690404	−0.345202	−	0.938528i	$-0.612190\pi$
$409$	−13.9626	−0.690404	−0.345202	+	0.938528i	$0.612190\pi$
$410$	−2.89572	−0.143009
$411$	21.6912	1.06995
$412$	6.78825	0.334433
$413$	−24.0221	−1.18205
$414$	4.36141	0.214352
$415$	10.0467	0.493173
$416$	−3.73836	−0.183288
$417$	5.31285	0.260171
$418$	24.4328	1.19504
$419$	4.85034	0.236954	0.118477	−	0.992957i	$-0.462199\pi$
$419$	4.85034	0.236954	0.118477	+	0.992957i	$0.462199\pi$
$420$	−3.46569	−0.169109
$421$	28.0280	1.36600	0.683000	−	0.730418i	$-0.260675\pi$
$421$	28.0280	1.36600	0.683000	+	0.730418i	$0.260675\pi$
$422$	4.06410	0.197838
$423$	−6.78825	−0.330056
$424$	−8.63408	−0.419308
$425$	4.63408	0.224786
$426$	−13.1400	−0.636633
$427$	7.64962	0.370191
$428$	−8.66975	−0.419068
$429$	−15.5298	−0.749786
$430$	−10.0357	−0.483963
$431$	−30.4191	−1.46524	−0.732619	−	0.680639i	$-0.761702\pi$
$431$	−30.4191	−1.46524	−0.732619	+	0.680639i	$0.761702\pi$
$432$	1.00000	0.0481125
$433$	9.52980	0.457973	0.228986	−	0.973430i	$-0.426459\pi$
$433$	9.52980	0.457973	0.228986	+	0.973430i	$0.426459\pi$
$434$	−3.10428	−0.149010
$435$	8.03567	0.385281
$436$	−16.5623	−0.793190
$437$	25.6516	1.22708
$438$	−7.19302	−0.343696
$439$	11.9580	0.570722	0.285361	−	0.958420i	$-0.407886\pi$
$439$	11.9580	0.570722	0.285361	+	0.958420i	$0.407886\pi$
$440$	4.15417	0.198042
$441$	5.01103	0.238621
$442$	−17.3239	−0.824013
$443$	18.8252	0.894414	0.447207	−	0.894430i	$-0.352419\pi$
$443$	18.8252	0.894414	0.447207	+	0.894430i	$0.352419\pi$
$444$	−1.00000	−0.0474579
$445$	−12.8783	−0.610491
$446$	−9.61854	−0.455451
$447$	22.6510	1.07136
$448$	−3.46569	−0.163739
$449$	18.8316	0.888719	0.444359	−	0.895849i	$-0.353431\pi$
$449$	18.8316	0.888719	0.444359	+	0.895849i	$0.353431\pi$
$450$	1.00000	0.0471405
$451$	−12.0293	−0.566438
$452$	−16.0357	−0.754254
$453$	3.94693	0.185443
$454$	−11.6122	−0.544986
$455$	12.9560	0.607388
$456$	5.88150	0.275426
$457$	6.58692	0.308123	0.154062	−	0.988061i	$-0.450765\pi$
$457$	6.58692	0.308123	0.154062	+	0.988061i	$0.450765\pi$
$458$	23.0312	1.07618
$459$	4.63408	0.216300
$460$	4.36141	0.203352
$461$	−28.9386	−1.34781	−0.673903	−	0.738820i	$-0.735383\pi$
$461$	−28.9386	−1.34781	−0.673903	+	0.738820i	$0.735383\pi$
$462$	−14.3971	−0.669813
$463$	22.6600	1.05310	0.526551	−	0.850144i	$-0.323485\pi$
$463$	22.6600	1.05310	0.526551	+	0.850144i	$0.323485\pi$
$464$	8.03567	0.373047
$465$	0.895717	0.0415379
$466$	−1.42031	−0.0657948
$467$	−31.6835	−1.46614	−0.733069	−	0.680154i	$-0.761913\pi$
$467$	−31.6835	−1.46614	−0.733069	+	0.680154i	$0.761913\pi$
$468$	−3.73836	−0.172806
$469$	21.1400	0.976152
$470$	−6.78825	−0.313119
$471$	20.7585	0.956501
$472$	6.93139	0.319043
$473$	−41.6899	−1.91690
$474$	4.00000	0.183726
$475$	5.88150	0.269862
$476$	−16.0603	−0.736123
$477$	−8.63408	−0.395327
$478$	−18.4502	−0.843890
$479$	−2.87832	−0.131514	−0.0657568	−	0.997836i	$-0.520946\pi$
$479$	−2.87832	−0.131514	−0.0657568	+	0.997836i	$0.520946\pi$
$480$	1.00000	0.0456435
$481$	3.73836	0.170455
$482$	−26.8162	−1.22145
$483$	−15.1153	−0.687771
$484$	6.25713	0.284415
$485$	7.06092	0.320620
$486$	1.00000	0.0453609
$487$	21.4334	0.971239	0.485619	−	0.874170i	$-0.338594\pi$
$487$	21.4334	0.971239	0.485619	+	0.874170i	$0.338594\pi$
$488$	−2.20724	−0.0999171
$489$	19.6652	0.889293
$490$	5.01103	0.226375
$491$	−1.04989	−0.0473808	−0.0236904	−	0.999719i	$-0.507542\pi$
$491$	−1.04989	−0.0473808	−0.0236904	+	0.999719i	$0.507542\pi$
$492$	−2.89572	−0.130549
$493$	37.2379	1.67711
$494$	−21.9872	−0.989250
$495$	4.15417	0.186716
$496$	0.895717	0.0402189
$497$	45.5391	2.04271
$498$	10.0467	0.450204
$499$	14.1898	0.635224	0.317612	−	0.948221i	$-0.397119\pi$
$499$	14.1898	0.635224	0.317612	+	0.948221i	$0.397119\pi$
$500$	1.00000	0.0447214
$501$	20.7980	0.929184
$502$	17.0532	0.761123
$503$	16.6167	0.740901	0.370451	−	0.928852i	$-0.379203\pi$
$503$	16.6167	0.740901	0.370451	+	0.928852i	$0.379203\pi$
$504$	−3.46569	−0.154374
$505$	−16.2429	−0.722800
$506$	18.1180	0.805445
$507$	0.975364	0.0433174
$508$	−3.11532	−0.138220
$509$	25.0409	1.10992	0.554959	−	0.831878i	$-0.312734\pi$
$509$	25.0409	1.10992	0.554959	+	0.831878i	$0.312734\pi$
$510$	4.63408	0.205201
$511$	24.9288	1.10279
$512$	1.00000	0.0441942
$513$	5.88150	0.259675
$514$	19.5014	0.860168
$515$	6.78825	0.299126
$516$	−10.0357	−0.441796
$517$	−28.1996	−1.24021
$518$	3.46569	0.152274
$519$	−5.15735	−0.226383
$520$	−3.73836	−0.163938
$521$	15.7987	0.692152	0.346076	−	0.938206i	$-0.387514\pi$
$521$	15.7987	0.692152	0.346076	+	0.938206i	$0.387514\pi$
$522$	8.03567	0.351712
$523$	−4.19955	−0.183634	−0.0918168	−	0.995776i	$-0.529267\pi$
$523$	−4.19955	−0.183634	−0.0918168	+	0.995776i	$0.529267\pi$
$524$	12.0713	0.527339
$525$	−3.46569	−0.151255
$526$	−21.6555	−0.944226
$527$	4.15083	0.180813
$528$	4.15417	0.180787
$529$	−3.97809	−0.172960
$530$	−8.63408	−0.375041
$531$	6.93139	0.300797
$532$	−20.3835	−0.883736
$533$	10.8252	0.468893
$534$	−12.8783	−0.557299
$535$	−8.66975	−0.374826
$536$	−6.09978	−0.263470
$537$	1.66323	0.0717735
$538$	−16.0811	−0.693303
$539$	20.8167	0.896638
$540$	1.00000	0.0430331
$541$	29.8531	1.28348	0.641742	−	0.766921i	$-0.278212\pi$
$541$	29.8531	1.28348	0.641742	+	0.766921i	$0.278212\pi$
$542$	6.62305	0.284484
$543$	−15.8628	−0.680737
$544$	4.63408	0.198685
$545$	−16.5623	−0.709450
$546$	12.9560	0.554467
$547$	−28.1912	−1.20537	−0.602684	−	0.797980i	$-0.705902\pi$
$547$	−28.1912	−1.20537	−0.602684	+	0.797980i	$0.705902\pi$
$548$	21.6912	0.926602
$549$	−2.20724	−0.0942028
$550$	4.15417	0.177134
$551$	47.2618	2.01342
$552$	4.36141	0.185634
$553$	−13.8628	−0.589505
$554$	−28.0467	−1.19159
$555$	−1.00000	−0.0424476
$556$	5.31285	0.225315
$557$	13.0596	0.553353	0.276676	−	0.960963i	$-0.410767\pi$
$557$	13.0596	0.553353	0.276676	+	0.960963i	$0.410767\pi$
$558$	0.895717	0.0379187
$559$	37.5170	1.58680
$560$	−3.46569	−0.146452
$561$	19.2508	0.812768
$562$	−4.26164	−0.179766
$563$	−10.3414	−0.435836	−0.217918	−	0.975967i	$-0.569927\pi$
$563$	−10.3414	−0.435836	−0.217918	+	0.975967i	$0.569927\pi$
$564$	−6.78825	−0.285837
$565$	−16.0357	−0.674626
$566$	0.884684	0.0371860
$567$	−3.46569	−0.145545
$568$	−13.1400	−0.551340
$569$	−20.5546	−0.861693	−0.430847	−	0.902425i	$-0.641785\pi$
$569$	−20.5546	−0.861693	−0.430847	+	0.902425i	$0.641785\pi$
$570$	5.88150	0.246349
$571$	4.89837	0.204990	0.102495	−	0.994734i	$-0.467317\pi$
$571$	4.89837	0.204990	0.102495	+	0.994734i	$0.467317\pi$
$572$	−15.5298	−0.649334
$573$	−19.2287	−0.803290
$574$	10.0357	0.418881
$575$	4.36141	0.181883
$576$	1.00000	0.0416667
$577$	−21.0745	−0.877344	−0.438672	−	0.898647i	$-0.644551\pi$
$577$	−21.0745	−0.877344	−0.438672	+	0.898647i	$0.644551\pi$
$578$	4.47471	0.186123
$579$	−12.7883	−0.531462
$580$	8.03567	0.333663
$581$	−34.8188	−1.44453
$582$	7.06092	0.292685
$583$	−35.8674	−1.48548
$584$	−7.19302	−0.297649
$585$	−3.73836	−0.154562
$586$	−21.4657	−0.886740
$587$	−6.78056	−0.279864	−0.139932	−	0.990161i	$-0.544688\pi$
$587$	−6.78056	−0.279864	−0.139932	+	0.990161i	$0.544688\pi$
$588$	5.01103	0.206652
$589$	5.26816	0.217071
$590$	6.93139	0.285361
$591$	−10.3614	−0.426212
$592$	−1.00000	−0.0410997
$593$	−46.5449	−1.91137	−0.955685	−	0.294392i	$-0.904883\pi$
$593$	−46.5449	−1.91137	−0.955685	+	0.294392i	$0.904883\pi$
$594$	4.15417	0.170448
$595$	−16.0603	−0.658408
$596$	22.6510	0.927822
$597$	10.6230	0.434772
$598$	−16.3045	−0.666742
$599$	41.3021	1.68756	0.843778	−	0.536692i	$-0.180326\pi$
$599$	41.3021	1.68756	0.843778	+	0.536692i	$0.180326\pi$
$600$	1.00000	0.0408248
$601$	10.5122	0.428803	0.214402	−	0.976746i	$-0.431220\pi$
$601$	10.5122	0.428803	0.214402	+	0.976746i	$0.431220\pi$
$602$	34.7806	1.41755
$603$	−6.09978	−0.248402
$604$	3.94693	0.160598
$605$	6.25713	0.254388
$606$	−16.2429	−0.659823
$607$	25.0461	1.01659	0.508295	−	0.861183i	$-0.330276\pi$
$607$	25.0461	1.01659	0.508295	+	0.861183i	$0.330276\pi$
$608$	5.88150	0.238526
$609$	−27.8492	−1.12851
$610$	−2.20724	−0.0893686
$611$	25.3770	1.02664
$612$	4.63408	0.187322
$613$	−16.6524	−0.672582	−0.336291	−	0.941758i	$-0.609173\pi$
$613$	−16.6524	−0.672582	−0.336291	+	0.941758i	$0.609173\pi$
$614$	34.1711	1.37903
$615$	−2.89572	−0.116767
$616$	−14.3971	−0.580075
$617$	6.12107	0.246425	0.123212	−	0.992380i	$-0.460680\pi$
$617$	6.12107	0.246425	0.123212	+	0.992380i	$0.460680\pi$
$618$	6.78825	0.273064
$619$	28.6587	1.15189	0.575946	−	0.817488i	$-0.304634\pi$
$619$	28.6587	1.15189	0.575946	+	0.817488i	$0.304634\pi$
$620$	0.895717	0.0359729
$621$	4.36141	0.175017
$622$	−10.1639	−0.407534
$623$	44.6323	1.78816
$624$	−3.73836	−0.149654
$625$	1.00000	0.0400000
$626$	−9.31152	−0.372163
$627$	24.4328	0.975750
$628$	20.7585	0.828354
$629$	−4.63408	−0.184773
$630$	−3.46569	−0.138077
$631$	−24.7301	−0.984488	−0.492244	−	0.870457i	$-0.663823\pi$
$631$	−24.7301	−0.984488	−0.492244	+	0.870457i	$0.663823\pi$
$632$	4.00000	0.159111
$633$	4.06410	0.161534
$634$	16.8673	0.669885
$635$	−3.11532	−0.123628
$636$	−8.63408	−0.342364
$637$	−18.7331	−0.742231
$638$	33.3815	1.32159
$639$	−13.1400	−0.519808
$640$	1.00000	0.0395285
$641$	−4.60308	−0.181811	−0.0909053	−	0.995860i	$-0.528976\pi$
$641$	−4.60308	−0.181811	−0.0909053	+	0.995860i	$0.528976\pi$
$642$	−8.66975	−0.342168
$643$	4.74924	0.187292	0.0936458	−	0.995606i	$-0.470148\pi$
$643$	4.74924	0.187292	0.0936458	+	0.995606i	$0.470148\pi$
$644$	−15.1153	−0.595627
$645$	−10.0357	−0.395154
$646$	27.2553	1.07235
$647$	34.8409	1.36974	0.684868	−	0.728667i	$-0.259860\pi$
$647$	34.8409	1.36974	0.684868	+	0.728667i	$0.259860\pi$
$648$	1.00000	0.0392837
$649$	28.7942	1.13027
$650$	−3.73836	−0.146631
$651$	−3.10428	−0.121666
$652$	19.6652	0.770150
$653$	19.2682	0.754021	0.377011	−	0.926209i	$-0.376952\pi$
$653$	19.2682	0.754021	0.377011	+	0.926209i	$0.376952\pi$
$654$	−16.5623	−0.647637
$655$	12.0713	0.471666
$656$	−2.89572	−0.113059
$657$	−7.19302	−0.280627
$658$	23.5260	0.917139
$659$	−4.03435	−0.157156	−0.0785779	−	0.996908i	$-0.525038\pi$
$659$	−4.03435	−0.157156	−0.0785779	+	0.996908i	$0.525038\pi$
$660$	4.15417	0.161701
$661$	35.8020	1.39254	0.696268	−	0.717781i	$-0.254842\pi$
$661$	35.8020	1.39254	0.696268	+	0.717781i	$0.254842\pi$
$662$	−0.451476	−0.0175471
$663$	−17.3239	−0.672804
$664$	10.0467	0.389888
$665$	−20.3835	−0.790437
$666$	−1.00000	−0.0387492
$667$	35.0469	1.35702
$668$	20.7980	0.804697
$669$	−9.61854	−0.371874
$670$	−6.09978	−0.235655
$671$	−9.16926	−0.353975
$672$	−3.46569	−0.133692
$673$	−46.2152	−1.78146	−0.890732	−	0.454529i	$-0.849808\pi$
$673$	−46.2152	−1.78146	−0.890732	+	0.454529i	$0.849808\pi$
$674$	−28.3550	−1.09220
$675$	1.00000	0.0384900
$676$	0.975364	0.0375140
$677$	−25.9315	−0.996630	−0.498315	−	0.866996i	$-0.666048\pi$
$677$	−25.9315	−0.996630	−0.498315	+	0.866996i	$0.666048\pi$
$678$	−16.0357	−0.615846
$679$	−24.4710	−0.939110
$680$	4.63408	0.177709
$681$	−11.6122	−0.444979
$682$	3.72096	0.142483
$683$	33.8894	1.29674	0.648372	−	0.761324i	$-0.275450\pi$
$683$	33.8894	1.29674	0.648372	+	0.761324i	$0.275450\pi$
$684$	5.88150	0.224885
$685$	21.6912	0.828778
$686$	6.89315	0.263182
$687$	23.0312	0.878694
$688$	−10.0357	−0.382606
$689$	32.2773	1.22967
$690$	4.36141	0.166036
$691$	−17.7987	−0.677093	−0.338547	−	0.940950i	$-0.609935\pi$
$691$	−17.7987	−0.677093	−0.338547	+	0.940950i	$0.609935\pi$
$692$	−5.15735	−0.196053
$693$	−14.3971	−0.546900
$694$	5.37695	0.204106
$695$	5.31285	0.201528
$696$	8.03567	0.304591
$697$	−13.4190	−0.508280
$698$	8.22798	0.311434
$699$	−1.42031	−0.0537212
$700$	−3.46569	−0.130991
$701$	52.6596	1.98893	0.994463	−	0.105092i	$-0.0335136\pi$
$701$	52.6596	1.98893	0.994463	+	0.105092i	$0.0335136\pi$
$702$	−3.73836	−0.141095
$703$	−5.88150	−0.221825
$704$	4.15417	0.156566
$705$	−6.78825	−0.255660
$706$	5.95531	0.224131
$707$	56.2930	2.11711
$708$	6.93139	0.260497
$709$	−20.4689	−0.768725	−0.384362	−	0.923182i	$-0.625579\pi$
$709$	−20.4689	−0.768725	−0.384362	+	0.923182i	$0.625579\pi$
$710$	−13.1400	−0.493134
$711$	4.00000	0.150012
$712$	−12.8783	−0.482635
$713$	3.90659	0.146303
$714$	−16.0603	−0.601042
$715$	−15.5298	−0.580782
$716$	1.66323	0.0621576
$717$	−18.4502	−0.689034
$718$	12.0713	0.450498
$719$	15.2618	0.569169	0.284584	−	0.958651i	$-0.408144\pi$
$719$	15.2618	0.569169	0.284584	+	0.958651i	$0.408144\pi$
$720$	1.00000	0.0372678
$721$	−23.5260	−0.876154
$722$	15.5920	0.580276
$723$	−26.8162	−0.997306
$724$	−15.8628	−0.589535
$725$	8.03567	0.298437
$726$	6.25713	0.232224
$727$	−3.94359	−0.146260	−0.0731298	−	0.997322i	$-0.523299\pi$
$727$	−3.94359	−0.146260	−0.0731298	+	0.997322i	$0.523299\pi$
$728$	12.9560	0.480182
$729$	1.00000	0.0370370
$730$	−7.19302	−0.266226
$731$	−46.5061	−1.72009
$732$	−2.20724	−0.0815820
$733$	39.2728	1.45057	0.725286	−	0.688448i	$-0.241707\pi$
$733$	39.2728	1.45057	0.725286	+	0.688448i	$0.241707\pi$
$734$	−0.139243	−0.00513957
$735$	5.01103	0.184835
$736$	4.36141	0.160764
$737$	−25.3395	−0.933393
$738$	−2.89572	−0.106593
$739$	−9.92688	−0.365166	−0.182583	−	0.983190i	$-0.558446\pi$
$739$	−9.92688	−0.365166	−0.182583	+	0.983190i	$0.558446\pi$
$740$	−1.00000	−0.0367607
$741$	−21.9872	−0.807719
$742$	29.9231	1.09851
$743$	−14.8524	−0.544880	−0.272440	−	0.962173i	$-0.587831\pi$
$743$	−14.8524	−0.544880	−0.272440	+	0.962173i	$0.587831\pi$
$744$	0.895717	0.0328386
$745$	22.6510	0.829869
$746$	−6.00000	−0.219676
$747$	10.0467	0.367590
$748$	19.2508	0.703878
$749$	30.0467	1.09788
$750$	1.00000	0.0365148
$751$	−11.4393	−0.417425	−0.208713	−	0.977977i	$-0.566927\pi$
$751$	−11.4393	−0.417425	−0.208713	+	0.977977i	$0.566927\pi$
$752$	−6.78825	−0.247542
$753$	17.0532	0.621454
$754$	−30.0403	−1.09400
$755$	3.94693	0.143643
$756$	−3.46569	−0.126046
$757$	−41.1646	−1.49615	−0.748076	−	0.663613i	$-0.769022\pi$
$757$	−41.1646	−1.49615	−0.748076	+	0.663613i	$0.769022\pi$
$758$	−5.33950	−0.193939
$759$	18.1180	0.657643
$760$	5.88150	0.213344
$761$	31.2041	1.13115	0.565573	−	0.824698i	$-0.308655\pi$
$761$	31.2041	1.13115	0.565573	+	0.824698i	$0.308655\pi$
$762$	−3.11532	−0.112856
$763$	57.3998	2.07801
$764$	−19.2287	−0.695670
$765$	4.63408	0.167546
$766$	−13.7384	−0.496387
$767$	−25.9120	−0.935630
$768$	1.00000	0.0360844
$769$	−11.2902	−0.407136	−0.203568	−	0.979061i	$-0.565254\pi$
$769$	−11.2902	−0.407136	−0.203568	+	0.979061i	$0.565254\pi$
$770$	−14.3971	−0.518835
$771$	19.5014	0.702324
$772$	−12.7883	−0.460259
$773$	3.18844	0.114680	0.0573400	−	0.998355i	$-0.481738\pi$
$773$	3.18844	0.114680	0.0573400	+	0.998355i	$0.481738\pi$
$774$	−10.0357	−0.360725
$775$	0.895717	0.0321751
$776$	7.06092	0.253472
$777$	3.46569	0.124331
$778$	−1.61217	−0.0577992
$779$	−17.0312	−0.610205
$780$	−3.73836	−0.133855
$781$	−54.5856	−1.95323
$782$	20.2111	0.722749
$783$	8.03567	0.287171
$784$	5.01103	0.178965
$785$	20.7585	0.740902
$786$	12.0713	0.430570
$787$	3.56345	0.127023	0.0635116	−	0.997981i	$-0.479770\pi$
$787$	3.56345	0.127023	0.0635116	+	0.997981i	$0.479770\pi$
$788$	−10.3614	−0.369110
$789$	−21.6555	−0.770957
$790$	4.00000	0.142314
$791$	55.5747	1.97601
$792$	4.15417	0.147612
$793$	8.25147	0.293018
$794$	−17.4767	−0.620226
$795$	−8.63408	−0.306219
$796$	10.6230	0.376524
$797$	16.1517	0.572122	0.286061	−	0.958211i	$-0.407654\pi$
$797$	16.1517	0.572122	0.286061	+	0.958211i	$0.407654\pi$
$798$	−20.3835	−0.721567
$799$	−31.4573	−1.11288
$800$	1.00000	0.0353553
$801$	−12.8783	−0.455033
$802$	−4.87832	−0.172259
$803$	−29.8810	−1.05448
$804$	−6.09978	−0.215122
$805$	−15.1153	−0.532745
$806$	−3.34852	−0.117947
$807$	−16.0811	−0.566080
$808$	−16.2429	−0.571424
$809$	−12.7501	−0.448270	−0.224135	−	0.974558i	$-0.571956\pi$
$809$	−12.7501	−0.448270	−0.224135	+	0.974558i	$0.571956\pi$
$810$	1.00000	0.0351364
$811$	−18.9535	−0.665546	−0.332773	−	0.943007i	$-0.607984\pi$
$811$	−18.9535	−0.665546	−0.332773	+	0.943007i	$0.607984\pi$
$812$	−27.8492	−0.977314
$813$	6.62305	0.232280
$814$	−4.15417	−0.145604
$815$	19.6652	0.688843
$816$	4.63408	0.162225
$817$	−59.0248	−2.06502
$818$	−13.9626	−0.488189
$819$	12.9560	0.452720
$820$	−2.89572	−0.101123
$821$	−3.28689	−0.114713	−0.0573566	−	0.998354i	$-0.518267\pi$
$821$	−3.28689	−0.114713	−0.0573566	+	0.998354i	$0.518267\pi$
$822$	21.6912	0.756568
$823$	42.4612	1.48010	0.740052	−	0.672550i	$-0.234801\pi$
$823$	42.4612	1.48010	0.740052	+	0.672550i	$0.234801\pi$
$824$	6.78825	0.236480
$825$	4.15417	0.144630
$826$	−24.0221	−0.835835
$827$	36.9296	1.28417	0.642084	−	0.766634i	$-0.278070\pi$
$827$	36.9296	1.28417	0.642084	+	0.766634i	$0.278070\pi$
$828$	4.36141	0.151570
$829$	−49.7515	−1.72794	−0.863971	−	0.503542i	$-0.832030\pi$
$829$	−49.7515	−1.72794	−0.863971	+	0.503542i	$0.832030\pi$
$830$	10.0467	0.348726
$831$	−28.0467	−0.972929
$832$	−3.73836	−0.129604
$833$	23.2215	0.804579
$834$	5.31285	0.183969
$835$	20.7980	0.719743
$836$	24.4328	0.845024
$837$	0.895717	0.0309605
$838$	4.85034	0.167552
$839$	42.6569	1.47268	0.736341	−	0.676611i	$-0.236552\pi$
$839$	42.6569	1.47268	0.736341	+	0.676611i	$0.236552\pi$
$840$	−3.46569	−0.119578
$841$	35.5720	1.22662
$842$	28.0280	0.965908
$843$	−4.26164	−0.146779
$844$	4.06410	0.139892
$845$	0.975364	0.0335535
$846$	−6.78825	−0.233385
$847$	−21.6853	−0.745115
$848$	−8.63408	−0.296496
$849$	0.884684	0.0303623
$850$	4.63408	0.158948
$851$	−4.36141	−0.149507
$852$	−13.1400	−0.450167
$853$	−12.0246	−0.411716	−0.205858	−	0.978582i	$-0.565998\pi$
$853$	−12.0246	−0.411716	−0.205858	+	0.978582i	$0.565998\pi$
$854$	7.64962	0.261765
$855$	5.88150	0.201143
$856$	−8.66975	−0.296326
$857$	19.2792	0.658565	0.329282	−	0.944231i	$-0.393193\pi$
$857$	19.2792	0.658565	0.329282	+	0.944231i	$0.393193\pi$
$858$	−15.5298	−0.530179
$859$	−34.0811	−1.16283	−0.581415	−	0.813607i	$-0.697501\pi$
$859$	−34.0811	−1.16283	−0.581415	+	0.813607i	$0.697501\pi$
$860$	−10.0357	−0.342214
$861$	10.0357	0.342015
$862$	−30.4191	−1.03608
$863$	−13.6709	−0.465363	−0.232682	−	0.972553i	$-0.574750\pi$
$863$	−13.6709	−0.465363	−0.232682	+	0.972553i	$0.574750\pi$
$864$	1.00000	0.0340207
$865$	−5.15735	−0.175355
$866$	9.52980	0.323836
$867$	4.47471	0.151969
$868$	−3.10428	−0.105366
$869$	16.6167	0.563682
$870$	8.03567	0.272435
$871$	22.8032	0.772656
$872$	−16.5623	−0.560870
$873$	7.06092	0.238976
$874$	25.6516	0.867680
$875$	−3.46569	−0.117162
$876$	−7.19302	−0.243030
$877$	−54.4931	−1.84010	−0.920050	−	0.391801i	$-0.871852\pi$
$877$	−54.4931	−1.84010	−0.920050	+	0.391801i	$0.871852\pi$
$878$	11.9580	0.403562
$879$	−21.4657	−0.724020
$880$	4.15417	0.140037
$881$	55.4529	1.86826	0.934128	−	0.356939i	$-0.116180\pi$
$881$	55.4529	1.86826	0.934128	+	0.356939i	$0.116180\pi$
$882$	5.01103	0.168730
$883$	−32.5189	−1.09435	−0.547174	−	0.837019i	$-0.684297\pi$
$883$	−32.5189	−1.09435	−0.547174	+	0.837019i	$0.684297\pi$
$884$	−17.3239	−0.582665
$885$	6.93139	0.232996
$886$	18.8252	0.632446
$887$	0.930064	0.0312285	0.0156142	−	0.999878i	$-0.495030\pi$
$887$	0.930064	0.0312285	0.0156142	+	0.999878i	$0.495030\pi$
$888$	−1.00000	−0.0335578
$889$	10.7967	0.362111
$890$	−12.8783	−0.431682
$891$	4.15417	0.139170
$892$	−9.61854	−0.322053
$893$	−39.9251	−1.33604
$894$	22.6510	0.757564
$895$	1.66323	0.0555955
$896$	−3.46569	−0.115781
$897$	−16.3045	−0.544393
$898$	18.8316	0.628419
$899$	7.19769	0.240056
$900$	1.00000	0.0333333
$901$	−40.0110	−1.33296
$902$	−12.0293	−0.400532
$903$	34.7806	1.15742
$904$	−16.0357	−0.533338
$905$	−15.8628	−0.527296
$906$	3.94693	0.131128
$907$	−7.68786	−0.255271	−0.127636	−	0.991821i	$-0.540739\pi$
$907$	−7.68786	−0.255271	−0.127636	+	0.991821i	$0.540739\pi$
$908$	−11.6122	−0.385363
$909$	−16.2429	−0.538743
$910$	12.9560	0.429488
$911$	−49.3021	−1.63345	−0.816725	−	0.577027i	$-0.804213\pi$
$911$	−49.3021	−1.63345	−0.816725	+	0.577027i	$0.804213\pi$
$912$	5.88150	0.194756
$913$	41.7357	1.38125
$914$	6.58692	0.217876
$915$	−2.20724	−0.0729691
$916$	23.0312	0.760971
$917$	−41.8356	−1.38153
$918$	4.63408	0.152947
$919$	−22.5662	−0.744390	−0.372195	−	0.928155i	$-0.621395\pi$
$919$	−22.5662	−0.744390	−0.372195	+	0.928155i	$0.621395\pi$
$920$	4.36141	0.143791
$921$	34.1711	1.12598
$922$	−28.9386	−0.953043
$923$	49.1219	1.61687
$924$	−14.3971	−0.473629
$925$	−1.00000	−0.0328798
$926$	22.6600	0.744655
$927$	6.78825	0.222955
$928$	8.03567	0.263784
$929$	−49.4659	−1.62292	−0.811462	−	0.584405i	$-0.801328\pi$
$929$	−49.4659	−1.62292	−0.811462	+	0.584405i	$0.801328\pi$
$930$	0.895717	0.0293717
$931$	29.4724	0.965919
$932$	−1.42031	−0.0465239
$933$	−10.1639	−0.332750
$934$	−31.6835	−1.03672
$935$	19.2508	0.629567
$936$	−3.73836	−0.122192
$937$	38.1842	1.24742	0.623711	−	0.781655i	$-0.285624\pi$
$937$	38.1842	1.24742	0.623711	+	0.781655i	$0.285624\pi$
$938$	21.1400	0.690244
$939$	−9.31152	−0.303870
$940$	−6.78825	−0.221408
$941$	0.240264	0.00783238	0.00391619	−	0.999992i	$-0.498753\pi$
$941$	0.240264	0.00783238	0.00391619	+	0.999992i	$0.498753\pi$
$942$	20.7585	0.676348
$943$	−12.6294	−0.411270
$944$	6.93139	0.225597
$945$	−3.46569	−0.112739
$946$	−41.6899	−1.35545
$947$	5.29078	0.171927	0.0859636	−	0.996298i	$-0.472603\pi$
$947$	5.29078	0.171927	0.0859636	+	0.996298i	$0.472603\pi$
$948$	4.00000	0.129914
$949$	26.8901	0.872891
$950$	5.88150	0.190821
$951$	16.8673	0.546959
$952$	−16.0603	−0.520518
$953$	−5.28946	−0.171342	−0.0856712	−	0.996323i	$-0.527303\pi$
$953$	−5.28946	−0.171342	−0.0856712	+	0.996323i	$0.527303\pi$
$954$	−8.63408	−0.279539
$955$	−19.2287	−0.622226
$956$	−18.4502	−0.596721
$957$	33.3815	1.07907
$958$	−2.87832	−0.0929942
$959$	−75.1751	−2.42753
$960$	1.00000	0.0322749
$961$	−30.1977	−0.974119
$962$	3.73836	0.120530
$963$	−8.66975	−0.279379
$964$	−26.8162	−0.863693
$965$	−12.7883	−0.411668
$966$	−15.1153	−0.486327
$967$	51.1679	1.64545	0.822725	−	0.568440i	$-0.192453\pi$
$967$	51.1679	1.64545	0.822725	+	0.568440i	$0.192453\pi$
$968$	6.25713	0.201112
$969$	27.2553	0.875568
$970$	7.06092	0.226713
$971$	−15.9354	−0.511393	−0.255696	−	0.966757i	$-0.582305\pi$
$971$	−15.9354	−0.511393	−0.255696	+	0.966757i	$0.582305\pi$
$972$	1.00000	0.0320750
$973$	−18.4127	−0.590284
$974$	21.4334	0.686769
$975$	−3.73836	−0.119723
$976$	−2.20724	−0.0706521
$977$	−15.3090	−0.489780	−0.244890	−	0.969551i	$-0.578752\pi$
$977$	−15.3090	−0.489780	−0.244890	+	0.969551i	$0.578752\pi$
$978$	19.6652	0.628825
$979$	−53.4987	−1.70983
$980$	5.01103	0.160072
$981$	−16.5623	−0.528793
$982$	−1.04989	−0.0335033
$983$	56.8019	1.81170	0.905849	−	0.423601i	$-0.139234\pi$
$983$	56.8019	1.81170	0.905849	+	0.423601i	$0.139234\pi$
$984$	−2.89572	−0.0923121
$985$	−10.3614	−0.330142
$986$	37.2379	1.18590
$987$	23.5260	0.748841
$988$	−21.9872	−0.699506
$989$	−43.7697	−1.39180
$990$	4.15417	0.132028
$991$	54.9232	1.74469	0.872347	−	0.488887i	$-0.162597\pi$
$991$	54.9232	1.74469	0.872347	+	0.488887i	$0.162597\pi$
$992$	0.895717	0.0284391
$993$	−0.451476	−0.0143272
$994$	45.5391	1.44441
$995$	10.6230	0.336773
$996$	10.0467	0.318342
$997$	−44.4612	−1.40810	−0.704050	−	0.710150i	$-0.748627\pi$
$997$	−44.4612	−1.40810	−0.704050	+	0.710150i	$0.748627\pi$
$998$	14.1898	0.449172
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.s.1.1	✓	4
3.2	odd	2		3330.2.a.bj.1.1		4
4.3	odd	2		8880.2.a.cg.1.4		4
5.4	even	2		5550.2.a.cj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.s.1.1	✓	4	1.1	even	1	trivial
3330.2.a.bj.1.1		4	3.2	odd	2
5550.2.a.cj.1.4		4	5.4	even	2
8880.2.a.cg.1.4		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.46569 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.46569 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.15417 q^{11} +1.00000 q^{12} -3.73836 q^{13} -3.46569 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.63408 q^{17} +1.00000 q^{18} +5.88150 q^{19} +1.00000 q^{20} -3.46569 q^{21} +4.15417 q^{22} +4.36141 q^{23} +1.00000 q^{24} +1.00000 q^{25} -3.73836 q^{26} +1.00000 q^{27} -3.46569 q^{28} +8.03567 q^{29} +1.00000 q^{30} +0.895717 q^{31} +1.00000 q^{32} +4.15417 q^{33} +4.63408 q^{34} -3.46569 q^{35} +1.00000 q^{36} -1.00000 q^{37} +5.88150 q^{38} -3.73836 q^{39} +1.00000 q^{40} -2.89572 q^{41} -3.46569 q^{42} -10.0357 q^{43} +4.15417 q^{44} +1.00000 q^{45} +4.36141 q^{46} -6.78825 q^{47} +1.00000 q^{48} +5.01103 q^{49} +1.00000 q^{50} +4.63408 q^{51} -3.73836 q^{52} -8.63408 q^{53} +1.00000 q^{54} +4.15417 q^{55} -3.46569 q^{56} +5.88150 q^{57} +8.03567 q^{58} +6.93139 q^{59} +1.00000 q^{60} -2.20724 q^{61} +0.895717 q^{62} -3.46569 q^{63} +1.00000 q^{64} -3.73836 q^{65} +4.15417 q^{66} -6.09978 q^{67} +4.63408 q^{68} +4.36141 q^{69} -3.46569 q^{70} -13.1400 q^{71} +1.00000 q^{72} -7.19302 q^{73} -1.00000 q^{74} +1.00000 q^{75} +5.88150 q^{76} -14.3971 q^{77} -3.73836 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.89572 q^{82} +10.0467 q^{83} -3.46569 q^{84} +4.63408 q^{85} -10.0357 q^{86} +8.03567 q^{87} +4.15417 q^{88} -12.8783 q^{89} +1.00000 q^{90} +12.9560 q^{91} +4.36141 q^{92} +0.895717 q^{93} -6.78825 q^{94} +5.88150 q^{95} +1.00000 q^{96} +7.06092 q^{97} +5.01103 q^{98} +4.15417 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{11} + 4 q^{12} - 3 q^{13} + 4 q^{14} + 4 q^{15} + 4 q^{16} + 6 q^{17} + 4 q^{18} + 3 q^{19} + 4 q^{20} + 4 q^{21} + 2 q^{22} - q^{23} + 4 q^{24} + 4 q^{25} - 3 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} + 4 q^{30} + 3 q^{31} + 4 q^{32} + 2 q^{33} + 6 q^{34} + 4 q^{35} + 4 q^{36} - 4 q^{37} + 3 q^{38} - 3 q^{39} + 4 q^{40} - 11 q^{41} + 4 q^{42} - 5 q^{43} + 2 q^{44} + 4 q^{45} - q^{46} + 4 q^{48} + 14 q^{49} + 4 q^{50} + 6 q^{51} - 3 q^{52} - 22 q^{53} + 4 q^{54} + 2 q^{55} + 4 q^{56} + 3 q^{57} - 3 q^{58} - 8 q^{59} + 4 q^{60} - 5 q^{61} + 3 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{65} + 2 q^{66} + 6 q^{67} + 6 q^{68} - q^{69} + 4 q^{70} - 18 q^{71} + 4 q^{72} - 5 q^{73} - 4 q^{74} + 4 q^{75} + 3 q^{76} - 4 q^{77} - 3 q^{78} + 16 q^{79} + 4 q^{80} + 4 q^{81} - 11 q^{82} - q^{83} + 4 q^{84} + 6 q^{85} - 5 q^{86} - 3 q^{87} + 2 q^{88} - 5 q^{89} + 4 q^{90} - 13 q^{91} - q^{92} + 3 q^{93} + 3 q^{95} + 4 q^{96} + 7 q^{97} + 14 q^{98} + 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9 + 4 * q^10 + 2 * q^11 + 4 * q^12 - 3 * q^13 + 4 * q^14 + 4 * q^15 + 4 * q^16 + 6 * q^17 + 4 * q^18 + 3 * q^19 + 4 * q^20 + 4 * q^21 + 2 * q^22 - q^23 + 4 * q^24 + 4 * q^25 - 3 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 + 4 * q^30 + 3 * q^31 + 4 * q^32 + 2 * q^33 + 6 * q^34 + 4 * q^35 + 4 * q^36 - 4 * q^37 + 3 * q^38 - 3 * q^39 + 4 * q^40 - 11 * q^41 + 4 * q^42 - 5 * q^43 + 2 * q^44 + 4 * q^45 - q^46 + 4 * q^48 + 14 * q^49 + 4 * q^50 + 6 * q^51 - 3 * q^52 - 22 * q^53 + 4 * q^54 + 2 * q^55 + 4 * q^56 + 3 * q^57 - 3 * q^58 - 8 * q^59 + 4 * q^60 - 5 * q^61 + 3 * q^62 + 4 * q^63 + 4 * q^64 - 3 * q^65 + 2 * q^66 + 6 * q^67 + 6 * q^68 - q^69 + 4 * q^70 - 18 * q^71 + 4 * q^72 - 5 * q^73 - 4 * q^74 + 4 * q^75 + 3 * q^76 - 4 * q^77 - 3 * q^78 + 16 * q^79 + 4 * q^80 + 4 * q^81 - 11 * q^82 - q^83 + 4 * q^84 + 6 * q^85 - 5 * q^86 - 3 * q^87 + 2 * q^88 - 5 * q^89 + 4 * q^90 - 13 * q^91 - q^92 + 3 * q^93 + 3 * q^95 + 4 * q^96 + 7 * q^97 + 14 * q^98 + 2 * q^99

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