LMFDB - Embedded newform 4009.2.a.e.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53574 q^{2} +2.00602 q^{3} +0.358512 q^{4} +1.47205 q^{5} -3.08073 q^{6} -4.88163 q^{7} +2.52091 q^{8} +1.02410 q^{9} +O(q^{10})$	\(q-1.53574 q^{2} +2.00602 q^{3} +0.358512 q^{4} +1.47205 q^{5} -3.08073 q^{6} -4.88163 q^{7} +2.52091 q^{8} +1.02410 q^{9} -2.26069 q^{10} +4.27935 q^{11} +0.719181 q^{12} -4.21981 q^{13} +7.49694 q^{14} +2.95296 q^{15} -4.58849 q^{16} -3.61352 q^{17} -1.57276 q^{18} +1.00000 q^{19} +0.527748 q^{20} -9.79263 q^{21} -6.57199 q^{22} -3.93404 q^{23} +5.05698 q^{24} -2.83307 q^{25} +6.48056 q^{26} -3.96368 q^{27} -1.75012 q^{28} +0.0888231 q^{29} -4.53499 q^{30} +3.73800 q^{31} +2.00494 q^{32} +8.58445 q^{33} +5.54944 q^{34} -7.18601 q^{35} +0.367153 q^{36} +4.88664 q^{37} -1.53574 q^{38} -8.46502 q^{39} +3.71090 q^{40} +7.87714 q^{41} +15.0390 q^{42} +6.60373 q^{43} +1.53420 q^{44} +1.50753 q^{45} +6.04168 q^{46} +2.93809 q^{47} -9.20459 q^{48} +16.8303 q^{49} +4.35087 q^{50} -7.24877 q^{51} -1.51285 q^{52} +9.76937 q^{53} +6.08721 q^{54} +6.29942 q^{55} -12.3061 q^{56} +2.00602 q^{57} -0.136410 q^{58} -6.33595 q^{59} +1.05867 q^{60} +9.65676 q^{61} -5.74061 q^{62} -4.99929 q^{63} +6.09791 q^{64} -6.21178 q^{65} -13.1835 q^{66} -5.52411 q^{67} -1.29549 q^{68} -7.89175 q^{69} +11.0359 q^{70} +0.430221 q^{71} +2.58167 q^{72} +6.78411 q^{73} -7.50463 q^{74} -5.68318 q^{75} +0.358512 q^{76} -20.8902 q^{77} +13.0001 q^{78} +12.2748 q^{79} -6.75450 q^{80} -11.0235 q^{81} -12.0973 q^{82} +8.62368 q^{83} -3.51078 q^{84} -5.31928 q^{85} -10.1416 q^{86} +0.178181 q^{87} +10.7878 q^{88} -0.167478 q^{89} -2.31518 q^{90} +20.5996 q^{91} -1.41040 q^{92} +7.49849 q^{93} -4.51215 q^{94} +1.47205 q^{95} +4.02195 q^{96} -14.7752 q^{97} -25.8471 q^{98} +4.38249 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−1.53574	−1.08594	−0.542968	−	0.839754i	$-0.682699\pi$
$2$	−1.53574	−1.08594	−0.542968	+	0.839754i	$0.682699\pi$
$3$	2.00602	1.15817	0.579087	−	0.815266i	$-0.303409\pi$
$3$	2.00602	1.15817	0.579087	+	0.815266i	$0.303409\pi$
$4$	0.358512	0.179256
$5$	1.47205	0.658321	0.329161	−	0.944274i	$-0.393234\pi$
$5$	1.47205	0.658321	0.329161	+	0.944274i	$0.393234\pi$
$6$	−3.08073	−1.25770
$7$	−4.88163	−1.84508	−0.922542	−	0.385897i	$-0.873892\pi$
$7$	−4.88163	−1.84508	−0.922542	+	0.385897i	$0.873892\pi$
$8$	2.52091	0.891275
$9$	1.02410	0.341367
$10$	−2.26069	−0.714894
$11$	4.27935	1.29027	0.645136	−	0.764067i	$-0.276801\pi$
$11$	4.27935	1.29027	0.645136	+	0.764067i	$0.276801\pi$
$12$	0.719181	0.207610
$13$	−4.21981	−1.17037	−0.585183	−	0.810901i	$-0.698977\pi$
$13$	−4.21981	−1.17037	−0.585183	+	0.810901i	$0.698977\pi$
$14$	7.49694	2.00364
$15$	2.95296	0.762451
$16$	−4.58849	−1.14712
$17$	−3.61352	−0.876407	−0.438203	−	0.898876i	$-0.644385\pi$
$17$	−3.61352	−0.876407	−0.438203	+	0.898876i	$0.644385\pi$
$18$	−1.57276	−0.370703
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.527748	0.118008
$21$	−9.79263	−2.13693
$22$	−6.57199	−1.40115
$23$	−3.93404	−0.820305	−0.410152	−	0.912017i	$-0.634524\pi$
$23$	−3.93404	−0.820305	−0.410152	+	0.912017i	$0.634524\pi$
$24$	5.05698	1.03225
$25$	−2.83307	−0.566613
$26$	6.48056	1.27094
$27$	−3.96368	−0.762811
$28$	−1.75012	−0.330742
$29$	0.0888231	0.0164940	0.00824702	−	0.999966i	$-0.497375\pi$
$29$	0.0888231	0.0164940	0.00824702	+	0.999966i	$0.497375\pi$
$30$	−4.53499	−0.827972
$31$	3.73800	0.671364	0.335682	−	0.941975i	$-0.391033\pi$
$31$	3.73800	0.671364	0.335682	+	0.941975i	$0.391033\pi$
$32$	2.00494	0.354427
$33$	8.58445	1.49436
$34$	5.54944	0.951721
$35$	−7.18601	−1.21466
$36$	0.367153	0.0611922
$37$	4.88664	0.803359	0.401679	−	0.915780i	$-0.368427\pi$
$37$	4.88664	0.803359	0.401679	+	0.915780i	$0.368427\pi$
$38$	−1.53574	−0.249131
$39$	−8.46502	−1.35549
$40$	3.71090	0.586745
$41$	7.87714	1.23020	0.615101	−	0.788448i	$-0.289115\pi$
$41$	7.87714	1.23020	0.615101	+	0.788448i	$0.289115\pi$
$42$	15.0390	2.32057
$43$	6.60373	1.00706	0.503529	−	0.863978i	$-0.332035\pi$
$43$	6.60373	1.00706	0.503529	+	0.863978i	$0.332035\pi$
$44$	1.53420	0.231289
$45$	1.50753	0.224729
$46$	6.04168	0.890798
$47$	2.93809	0.428564	0.214282	−	0.976772i	$-0.431259\pi$
$47$	2.93809	0.428564	0.214282	+	0.976772i	$0.431259\pi$
$48$	−9.20459	−1.32857
$49$	16.8303	2.40433
$50$	4.35087	0.615305
$51$	−7.24877	−1.01503
$52$	−1.51285	−0.209795
$53$	9.76937	1.34193	0.670963	−	0.741491i	$-0.265881\pi$
$53$	9.76937	1.34193	0.670963	+	0.741491i	$0.265881\pi$
$54$	6.08721	0.828364
$55$	6.29942	0.849414
$56$	−12.3061	−1.64448
$57$	2.00602	0.265703
$58$	−0.136410	−0.0179115
$59$	−6.33595	−0.824871	−0.412435	−	0.910987i	$-0.635322\pi$
$59$	−6.33595	−0.824871	−0.412435	+	0.910987i	$0.635322\pi$
$60$	1.05867	0.136674
$61$	9.65676	1.23642	0.618211	−	0.786012i	$-0.287858\pi$
$61$	9.65676	1.23642	0.618211	+	0.786012i	$0.287858\pi$
$62$	−5.74061	−0.729058
$63$	−4.99929	−0.629851
$64$	6.09791	0.762239
$65$	−6.21178	−0.770477
$66$	−13.1835	−1.62278
$67$	−5.52411	−0.674878	−0.337439	−	0.941347i	$-0.609561\pi$
$67$	−5.52411	−0.674878	−0.337439	+	0.941347i	$0.609561\pi$
$68$	−1.29549	−0.157101
$69$	−7.89175	−0.950056
$70$	11.0359	1.31904
$71$	0.430221	0.0510579	0.0255289	−	0.999674i	$-0.491873\pi$
$71$	0.430221	0.0510579	0.0255289	+	0.999674i	$0.491873\pi$
$72$	2.58167	0.304252
$73$	6.78411	0.794021	0.397010	−	0.917814i	$-0.370048\pi$
$73$	6.78411	0.794021	0.397010	+	0.917814i	$0.370048\pi$
$74$	−7.50463	−0.872396
$75$	−5.68318	−0.656237
$76$	0.358512	0.0411241
$77$	−20.8902	−2.38066
$78$	13.0001	1.47197
$79$	12.2748	1.38102	0.690511	−	0.723322i	$-0.257386\pi$
$79$	12.2748	1.38102	0.690511	+	0.723322i	$0.257386\pi$
$80$	−6.75450	−0.755176
$81$	−11.0235	−1.22484
$82$	−12.0973	−1.33592
$83$	8.62368	0.946572	0.473286	−	0.880909i	$-0.343068\pi$
$83$	8.62368	0.946572	0.473286	+	0.880909i	$0.343068\pi$
$84$	−3.51078	−0.383057
$85$	−5.31928	−0.576957
$86$	−10.1416	−1.09360
$87$	0.178181	0.0191030
$88$	10.7878	1.14999
$89$	−0.167478	−0.0177527	−0.00887634	−	0.999961i	$-0.502825\pi$
$89$	−0.167478	−0.0177527	−0.00887634	+	0.999961i	$0.502825\pi$
$90$	−2.31518	−0.244042
$91$	20.5996	2.15942
$92$	−1.41040	−0.147045
$93$	7.49849	0.777557
$94$	−4.51215	−0.465393
$95$	1.47205	0.151029
$96$	4.02195	0.410488
$97$	−14.7752	−1.50019	−0.750095	−	0.661330i	$-0.769992\pi$
$97$	−14.7752	−1.50019	−0.750095	+	0.661330i	$0.769992\pi$
$98$	−25.8471	−2.61095
$99$	4.38249	0.440457
$100$	−1.01569	−0.101569
$101$	−0.141822	−0.0141118	−0.00705592	−	0.999975i	$-0.502246\pi$
$101$	−0.141822	−0.0141118	−0.00705592	+	0.999975i	$0.502246\pi$
$102$	11.1323	1.10226
$103$	9.94263	0.979677	0.489838	−	0.871813i	$-0.337056\pi$
$103$	9.94263	0.979677	0.489838	+	0.871813i	$0.337056\pi$
$104$	−10.6378	−1.04312
$105$	−14.4153	−1.40679
$106$	−15.0033	−1.45725
$107$	5.62464	0.543754	0.271877	−	0.962332i	$-0.412356\pi$
$107$	5.62464	0.543754	0.271877	+	0.962332i	$0.412356\pi$
$108$	−1.42103	−0.136738
$109$	0.703976	0.0674286	0.0337143	−	0.999432i	$-0.489266\pi$
$109$	0.703976	0.0674286	0.0337143	+	0.999432i	$0.489266\pi$
$110$	−9.67430	−0.922409
$111$	9.80268	0.930429
$112$	22.3993	2.11654
$113$	−6.44914	−0.606684	−0.303342	−	0.952882i	$-0.598102\pi$
$113$	−6.44914	−0.606684	−0.303342	+	0.952882i	$0.598102\pi$
$114$	−3.08073	−0.288537
$115$	−5.79111	−0.540024
$116$	0.0318442	0.00295666
$117$	−4.32152	−0.399525
$118$	9.73040	0.895756
$119$	17.6399	1.61704
$120$	7.44413	0.679553
$121$	7.31284	0.664804
$122$	−14.8303	−1.34267
$123$	15.8017	1.42479
$124$	1.34012	0.120346
$125$	−11.5307	−1.03133
$126$	7.67763	0.683978
$127$	18.8137	1.66944	0.834722	−	0.550672i	$-0.185628\pi$
$127$	18.8137	1.66944	0.834722	+	0.550672i	$0.185628\pi$
$128$	−13.3747	−1.18217
$129$	13.2472	1.16635
$130$	9.53971	0.836688
$131$	6.03744	0.527493	0.263747	−	0.964592i	$-0.415042\pi$
$131$	6.03744	0.527493	0.263747	+	0.964592i	$0.415042\pi$
$132$	3.07763	0.267873
$133$	−4.88163	−0.423291
$134$	8.48362	0.732874
$135$	−5.83474	−0.502175
$136$	−9.10934	−0.781119
$137$	11.5872	0.989961	0.494981	−	0.868904i	$-0.335175\pi$
$137$	11.5872	0.989961	0.494981	+	0.868904i	$0.335175\pi$
$138$	12.1197	1.03170
$139$	−9.00976	−0.764198	−0.382099	−	0.924121i	$-0.624799\pi$
$139$	−9.00976	−0.764198	−0.382099	+	0.924121i	$0.624799\pi$
$140$	−2.57627	−0.217735
$141$	5.89385	0.496352
$142$	−0.660710	−0.0554456
$143$	−18.0581	−1.51009
$144$	−4.69909	−0.391591
$145$	0.130752	0.0108584
$146$	−10.4187	−0.862255
$147$	33.7619	2.78464
$148$	1.75192	0.144007
$149$	16.0740	1.31683	0.658417	−	0.752653i	$-0.271226\pi$
$149$	16.0740	1.31683	0.658417	+	0.752653i	$0.271226\pi$
$150$	8.72791	0.712631
$151$	9.21645	0.750024	0.375012	−	0.927020i	$-0.377639\pi$
$151$	9.21645	0.750024	0.375012	+	0.927020i	$0.377639\pi$
$152$	2.52091	0.204473
$153$	−3.70061	−0.299177
$154$	32.0820	2.58524
$155$	5.50252	0.441973
$156$	−3.03481	−0.242979
$157$	1.39311	0.111182	0.0555911	−	0.998454i	$-0.482296\pi$
$157$	1.39311	0.111182	0.0555911	+	0.998454i	$0.482296\pi$
$158$	−18.8510	−1.49970
$159$	19.5975	1.55418
$160$	2.95138	0.233327
$161$	19.2045	1.51353
$162$	16.9293	1.33009
$163$	−20.5818	−1.61209	−0.806045	−	0.591854i	$-0.798396\pi$
$163$	−20.5818	−1.61209	−0.806045	+	0.591854i	$0.798396\pi$
$164$	2.82405	0.220521
$165$	12.6367	0.983769
$166$	−13.2438	−1.02792
$167$	−15.7123	−1.21585	−0.607927	−	0.793993i	$-0.707999\pi$
$167$	−15.7123	−1.21585	−0.607927	+	0.793993i	$0.707999\pi$
$168$	−24.6863	−1.90459
$169$	4.80683	0.369756
$170$	8.16906	0.626538
$171$	1.02410	0.0783151
$172$	2.36752	0.180521
$173$	−1.07972	−0.0820899	−0.0410450	−	0.999157i	$-0.513069\pi$
$173$	−1.07972	−0.0820899	−0.0410450	+	0.999157i	$0.513069\pi$
$174$	−0.273640	−0.0207446
$175$	13.8300	1.04545
$176$	−19.6358	−1.48010
$177$	−12.7100	−0.955344
$178$	0.257204	0.0192783
$179$	−11.7559	−0.878675	−0.439338	−	0.898322i	$-0.644787\pi$
$179$	−11.7559	−0.878675	−0.439338	+	0.898322i	$0.644787\pi$
$180$	0.540468	0.0402841
$181$	17.8402	1.32605	0.663027	−	0.748596i	$-0.269272\pi$
$181$	17.8402	1.32605	0.663027	+	0.748596i	$0.269272\pi$
$182$	−31.6357	−2.34499
$183$	19.3716	1.43199
$184$	−9.91735	−0.731117
$185$	7.19338	0.528868
$186$	−11.5158	−0.844377
$187$	−15.4635	−1.13080
$188$	1.05334	0.0768227
$189$	19.3492	1.40745
$190$	−2.26069	−0.164008
$191$	3.14805	0.227785	0.113893	−	0.993493i	$-0.463668\pi$
$191$	3.14805	0.227785	0.113893	+	0.993493i	$0.463668\pi$
$192$	12.2325	0.882805
$193$	18.7989	1.35317	0.676586	−	0.736364i	$-0.263459\pi$
$193$	18.7989	1.35317	0.676586	+	0.736364i	$0.263459\pi$
$194$	22.6909	1.62911
$195$	−12.4609	−0.892346
$196$	6.03388	0.430991
$197$	−9.11632	−0.649511	−0.324755	−	0.945798i	$-0.605282\pi$
$197$	−9.11632	−0.649511	−0.324755	+	0.945798i	$0.605282\pi$
$198$	−6.73039	−0.478308
$199$	10.7486	0.761949	0.380975	−	0.924586i	$-0.375589\pi$
$199$	10.7486	0.761949	0.380975	+	0.924586i	$0.375589\pi$
$200$	−7.14190	−0.505008
$201$	−11.0815	−0.781626
$202$	0.217803	0.0153245
$203$	−0.433602	−0.0304329
$204$	−2.59877	−0.181950
$205$	11.5955	0.809868
$206$	−15.2693	−1.06387
$207$	−4.02886	−0.280025
$208$	19.3626	1.34255
$209$	4.27935	0.296009
$210$	22.1382	1.52768
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	3.50244	0.240548
$213$	0.863031	0.0591339
$214$	−8.63800	−0.590482
$215$	9.72102	0.662968
$216$	−9.99208	−0.679875
$217$	−18.2475	−1.23872
$218$	−1.08113	−0.0732231
$219$	13.6090	0.919614
$220$	2.25842	0.152263
$221$	15.2484	1.02572
$222$	−15.0544	−1.01039
$223$	6.27635	0.420295	0.210148	−	0.977670i	$-0.432606\pi$
$223$	6.27635	0.420295	0.210148	+	0.977670i	$0.432606\pi$
$224$	−9.78739	−0.653947
$225$	−2.90135	−0.193423
$226$	9.90423	0.658819
$227$	−17.5153	−1.16253	−0.581267	−	0.813713i	$-0.697443\pi$
$227$	−17.5153	−1.16253	−0.581267	+	0.813713i	$0.697443\pi$
$228$	0.719181	0.0476289
$229$	4.54019	0.300024	0.150012	−	0.988684i	$-0.452069\pi$
$229$	4.54019	0.300024	0.150012	+	0.988684i	$0.452069\pi$
$230$	8.89367	0.586431
$231$	−41.9061	−2.75722
$232$	0.223915	0.0147007
$233$	4.10062	0.268641	0.134320	−	0.990938i	$-0.457115\pi$
$233$	4.10062	0.268641	0.134320	+	0.990938i	$0.457115\pi$
$234$	6.63675	0.433858
$235$	4.32502	0.282133
$236$	−2.27151	−0.147863
$237$	24.6234	1.59946
$238$	−27.0903	−1.75600
$239$	17.4541	1.12901	0.564507	−	0.825429i	$-0.309066\pi$
$239$	17.4541	1.12901	0.564507	+	0.825429i	$0.309066\pi$
$240$	−13.5496	−0.874625
$241$	−10.3230	−0.664964	−0.332482	−	0.943110i	$-0.607886\pi$
$241$	−10.3230	−0.664964	−0.332482	+	0.943110i	$0.607886\pi$
$242$	−11.2307	−0.721934
$243$	−10.2223	−0.655762
$244$	3.46207	0.221636
$245$	24.7751	1.58282
$246$	−24.2673	−1.54723
$247$	−4.21981	−0.268500
$248$	9.42314	0.598370
$249$	17.2992	1.09629
$250$	17.7082	1.11996
$251$	20.2847	1.28036	0.640180	−	0.768225i	$-0.278860\pi$
$251$	20.2847	1.28036	0.640180	+	0.768225i	$0.278860\pi$
$252$	−1.79231	−0.112905
$253$	−16.8351	−1.05842
$254$	−28.8930	−1.81291
$255$	−10.6706	−0.668217
$256$	8.34433	0.521521
$257$	−10.7336	−0.669542	−0.334771	−	0.942300i	$-0.608659\pi$
$257$	−10.7336	−0.669542	−0.334771	+	0.942300i	$0.608659\pi$
$258$	−20.3443	−1.26658
$259$	−23.8548	−1.48226
$260$	−2.22700	−0.138113
$261$	0.0909640	0.00563053
$262$	−9.27196	−0.572824
$263$	13.5423	0.835056	0.417528	−	0.908664i	$-0.362897\pi$
$263$	13.5423	0.835056	0.417528	+	0.908664i	$0.362897\pi$
$264$	21.6406	1.33189
$265$	14.3810	0.883419
$266$	7.49694	0.459667
$267$	−0.335964	−0.0205607
$268$	−1.98046	−0.120976
$269$	1.73151	0.105572	0.0527861	−	0.998606i	$-0.483190\pi$
$269$	1.73151	0.105572	0.0527861	+	0.998606i	$0.483190\pi$
$270$	8.96068	0.545329
$271$	6.43808	0.391086	0.195543	−	0.980695i	$-0.437353\pi$
$271$	6.43808	0.391086	0.195543	+	0.980695i	$0.437353\pi$
$272$	16.5806	1.00535
$273$	41.3231	2.50099
$274$	−17.7950	−1.07503
$275$	−12.1237	−0.731086
$276$	−2.82929	−0.170303
$277$	−14.6774	−0.881881	−0.440941	−	0.897536i	$-0.645355\pi$
$277$	−14.6774	−0.881881	−0.440941	+	0.897536i	$0.645355\pi$
$278$	13.8367	0.829870
$279$	3.82809	0.229182
$280$	−18.1153	−1.08259
$281$	−11.2810	−0.672968	−0.336484	−	0.941689i	$-0.609238\pi$
$281$	−11.2810	−0.672968	−0.336484	+	0.941689i	$0.609238\pi$
$282$	−9.05146	−0.539006
$283$	5.23002	0.310893	0.155446	−	0.987844i	$-0.450318\pi$
$283$	5.23002	0.310893	0.155446	+	0.987844i	$0.450318\pi$
$284$	0.154240	0.00915243
$285$	2.95296	0.174918
$286$	27.7326	1.63986
$287$	−38.4533	−2.26983
$288$	2.05326	0.120990
$289$	−3.94250	−0.231911
$290$	−0.200802	−0.0117915
$291$	−29.6392	−1.73748
$292$	2.43219	0.142333
$293$	−4.03732	−0.235862	−0.117931	−	0.993022i	$-0.537626\pi$
$293$	−4.03732	−0.235862	−0.117931	+	0.993022i	$0.537626\pi$
$294$	−51.8497	−3.02394
$295$	−9.32684	−0.543030
$296$	12.3188	0.716014
$297$	−16.9620	−0.984234
$298$	−24.6856	−1.43000
$299$	16.6009	0.960056
$300$	−2.03749	−0.117634
$301$	−32.2370	−1.85811
$302$	−14.1541	−0.814478
$303$	−0.284498	−0.0163440
$304$	−4.58849	−0.263168
$305$	14.2152	0.813963
$306$	5.68319	0.324887
$307$	−22.0404	−1.25791	−0.628955	−	0.777442i	$-0.716517\pi$
$307$	−22.0404	−1.25791	−0.628955	+	0.777442i	$0.716517\pi$
$308$	−7.48939	−0.426748
$309$	19.9451	1.13464
$310$	−8.45047	−0.479955
$311$	12.5735	0.712975	0.356488	−	0.934300i	$-0.383974\pi$
$311$	12.5735	0.712975	0.356488	+	0.934300i	$0.383974\pi$
$312$	−21.3395	−1.20811
$313$	17.8786	1.01056	0.505280	−	0.862955i	$-0.331389\pi$
$313$	17.8786	1.01056	0.505280	+	0.862955i	$0.331389\pi$
$314$	−2.13946	−0.120737
$315$	−7.35921	−0.414645
$316$	4.40066	0.247557
$317$	−6.17814	−0.346999	−0.173499	−	0.984834i	$-0.555507\pi$
$317$	−6.17814	−0.346999	−0.173499	+	0.984834i	$0.555507\pi$
$318$	−30.0968	−1.68774
$319$	0.380105	0.0212818
$320$	8.97643	0.501798
$321$	11.2831	0.629762
$322$	−29.4933	−1.64360
$323$	−3.61352	−0.201061
$324$	−3.95206	−0.219559
$325$	11.9550	0.663145
$326$	31.6084	1.75063
$327$	1.41219	0.0780941
$328$	19.8575	1.09645
$329$	−14.3427	−0.790737
$330$	−19.4068	−1.06831
$331$	−27.4284	−1.50760	−0.753801	−	0.657103i	$-0.771782\pi$
$331$	−27.4284	−1.50760	−0.753801	+	0.657103i	$0.771782\pi$
$332$	3.09169	0.169679
$333$	5.00442	0.274241
$334$	24.1301	1.32034
$335$	−8.13177	−0.444286
$336$	44.9334	2.45132
$337$	20.3760	1.10995	0.554977	−	0.831866i	$-0.312727\pi$
$337$	20.3760	1.10995	0.554977	+	0.831866i	$0.312727\pi$
$338$	−7.38206	−0.401531
$339$	−12.9371	−0.702645
$340$	−1.90703	−0.103423
$341$	15.9962	0.866243
$342$	−1.57276	−0.0850451
$343$	−47.9881	−2.59111
$344$	16.6474	0.897566
$345$	−11.6171	−0.625442
$346$	1.65818	0.0891444
$347$	−17.2807	−0.927677	−0.463838	−	0.885920i	$-0.653528\pi$
$347$	−17.2807	−0.927677	−0.463838	+	0.885920i	$0.653528\pi$
$348$	0.0638799	0.00342432
$349$	−1.14966	−0.0615402	−0.0307701	−	0.999526i	$-0.509796\pi$
$349$	−1.14966	−0.0615402	−0.0307701	+	0.999526i	$0.509796\pi$
$350$	−21.2393	−1.13529
$351$	16.7260	0.892768
$352$	8.57985	0.457307
$353$	34.0723	1.81349	0.906744	−	0.421683i	$-0.138560\pi$
$353$	34.0723	1.81349	0.906744	+	0.421683i	$0.138560\pi$
$354$	19.5194	1.03744
$355$	0.633308	0.0336125
$356$	−0.0600430	−0.00318227
$357$	35.3859	1.87282
$358$	18.0540	0.954185
$359$	3.91681	0.206721	0.103361	−	0.994644i	$-0.467040\pi$
$359$	3.91681	0.206721	0.103361	+	0.994644i	$0.467040\pi$
$360$	3.80034	0.200296
$361$	1.00000	0.0526316
$362$	−27.3980	−1.44001
$363$	14.6697	0.769958
$364$	7.38520	0.387089
$365$	9.98656	0.522721
$366$	−29.7499	−1.55505
$367$	−28.6057	−1.49320	−0.746602	−	0.665271i	$-0.768316\pi$
$367$	−28.6057	−1.49320	−0.746602	+	0.665271i	$0.768316\pi$
$368$	18.0513	0.940990
$369$	8.06700	0.419951
$370$	−11.0472	−0.574317
$371$	−47.6905	−2.47597
$372$	2.68830	0.139382
$373$	−5.34067	−0.276530	−0.138265	−	0.990395i	$-0.544152\pi$
$373$	−5.34067	−0.276530	−0.138265	+	0.990395i	$0.544152\pi$
$374$	23.7480	1.22798
$375$	−23.1307	−1.19447
$376$	7.40665	0.381969
$377$	−0.374817	−0.0193041
$378$	−29.7155	−1.52840
$379$	15.0287	0.771974	0.385987	−	0.922504i	$-0.373861\pi$
$379$	15.0287	0.771974	0.385987	+	0.922504i	$0.373861\pi$
$380$	0.527748	0.0270729
$381$	37.7406	1.93351
$382$	−4.83460	−0.247360
$383$	20.5231	1.04868	0.524341	−	0.851508i	$-0.324312\pi$
$383$	20.5231	1.04868	0.524341	+	0.851508i	$0.324312\pi$
$384$	−26.8299	−1.36916
$385$	−30.7515	−1.56724
$386$	−28.8703	−1.46946
$387$	6.76289	0.343777
$388$	−5.29707	−0.268918
$389$	−1.21872	−0.0617913	−0.0308957	−	0.999523i	$-0.509836\pi$
$389$	−1.21872	−0.0617913	−0.0308957	+	0.999523i	$0.509836\pi$
$390$	19.1368	0.969030
$391$	14.2157	0.718920
$392$	42.4277	2.14292
$393$	12.1112	0.610929
$394$	14.0003	0.705327
$395$	18.0691	0.909156
$396$	1.57118	0.0789546
$397$	7.27388	0.365066	0.182533	−	0.983200i	$-0.441570\pi$
$397$	7.27388	0.365066	0.182533	+	0.983200i	$0.441570\pi$
$398$	−16.5071	−0.827428
$399$	−9.79263	−0.490245
$400$	12.9995	0.649975
$401$	−9.46397	−0.472608	−0.236304	−	0.971679i	$-0.575936\pi$
$401$	−9.46397	−0.472608	−0.236304	+	0.971679i	$0.575936\pi$
$402$	17.0183	0.848795
$403$	−15.7737	−0.785742
$404$	−0.0508450	−0.00252963
$405$	−16.2272	−0.806335
$406$	0.665902	0.0330482
$407$	20.9116	1.03655
$408$	−18.2735	−0.904672
$409$	7.96227	0.393709	0.196855	−	0.980433i	$-0.436927\pi$
$409$	7.96227	0.393709	0.196855	+	0.980433i	$0.436927\pi$
$410$	−17.8078	−0.879465
$411$	23.2441	1.14655
$412$	3.56455	0.175613
$413$	30.9298	1.52196
$414$	6.18730	0.304089
$415$	12.6945	0.623148
$416$	−8.46048	−0.414809
$417$	−18.0737	−0.885074
$418$	−6.57199	−0.321447
$419$	−14.9554	−0.730620	−0.365310	−	0.930886i	$-0.619037\pi$
$419$	−14.9554	−0.730620	−0.365310	+	0.930886i	$0.619037\pi$
$420$	−5.16804	−0.252175
$421$	31.6025	1.54021	0.770105	−	0.637917i	$-0.220204\pi$
$421$	31.6025	1.54021	0.770105	+	0.637917i	$0.220204\pi$
$422$	1.53574	0.0747589
$423$	3.00890	0.146298
$424$	24.6277	1.19603
$425$	10.2373	0.496584
$426$	−1.32540	−0.0642156
$427$	−47.1408	−2.28130
$428$	2.01650	0.0974712
$429$	−36.2248	−1.74895
$430$	−14.9290	−0.719941
$431$	19.5661	0.942465	0.471233	−	0.882009i	$-0.343809\pi$
$431$	19.5661	0.942465	0.471233	+	0.882009i	$0.343809\pi$
$432$	18.1873	0.875039
$433$	−35.5257	−1.70726	−0.853629	−	0.520882i	$-0.825603\pi$
$433$	−35.5257	−1.70726	−0.853629	+	0.520882i	$0.825603\pi$
$434$	28.0236	1.34517
$435$	0.262291	0.0125759
$436$	0.252384	0.0120870
$437$	−3.93404	−0.188191
$438$	−20.9000	−0.998642
$439$	40.1189	1.91477	0.957386	−	0.288813i	$-0.0932605\pi$
$439$	40.1189	1.91477	0.957386	+	0.288813i	$0.0932605\pi$
$440$	15.8803	0.757061
$441$	17.2360	0.820761
$442$	−23.4176	−1.11386
$443$	−6.01366	−0.285718	−0.142859	−	0.989743i	$-0.545629\pi$
$443$	−6.01366	−0.285718	−0.142859	+	0.989743i	$0.545629\pi$
$444$	3.51438	0.166785
$445$	−0.246537	−0.0116870
$446$	−9.63887	−0.456414
$447$	32.2447	1.52512
$448$	−29.7677	−1.40639
$449$	10.3655	0.489180	0.244590	−	0.969627i	$-0.421347\pi$
$449$	10.3655	0.489180	0.244590	+	0.969627i	$0.421347\pi$
$450$	4.45573	0.210045
$451$	33.7090	1.58730
$452$	−2.31209	−0.108752
$453$	18.4884	0.868659
$454$	26.8991	1.26244
$455$	30.3236	1.42159
$456$	5.05698	0.236815
$457$	40.5359	1.89619	0.948094	−	0.317990i	$-0.103008\pi$
$457$	40.5359	1.89619	0.948094	+	0.317990i	$0.103008\pi$
$458$	−6.97258	−0.325807
$459$	14.3228	0.668533
$460$	−2.07618	−0.0968025
$461$	37.0218	1.72428	0.862139	−	0.506672i	$-0.169124\pi$
$461$	37.0218	1.72428	0.862139	+	0.506672i	$0.169124\pi$
$462$	64.3571	2.99416
$463$	0.909026	0.0422460	0.0211230	−	0.999777i	$-0.493276\pi$
$463$	0.909026	0.0422460	0.0211230	+	0.999777i	$0.493276\pi$
$464$	−0.407564	−0.0189207
$465$	11.0382	0.511882
$466$	−6.29751	−0.291726
$467$	−13.2011	−0.610874	−0.305437	−	0.952212i	$-0.598803\pi$
$467$	−13.2011	−0.610874	−0.305437	+	0.952212i	$0.598803\pi$
$468$	−1.54932	−0.0716172
$469$	26.9667	1.24521
$470$	−6.64212	−0.306378
$471$	2.79460	0.128768
$472$	−15.9723	−0.735187
$473$	28.2597	1.29938
$474$	−37.8153	−1.73692
$475$	−2.83307	−0.129990
$476$	6.32410	0.289865
$477$	10.0048	0.458090
$478$	−26.8051	−1.22604
$479$	25.8755	1.18228	0.591142	−	0.806568i	$-0.298677\pi$
$479$	25.8755	1.18228	0.591142	+	0.806568i	$0.298677\pi$
$480$	5.92051	0.270233
$481$	−20.6207	−0.940224
$482$	15.8535	0.722108
$483$	38.5246	1.75293
$484$	2.62174	0.119170
$485$	−21.7498	−0.987607
$486$	15.6989	0.712115
$487$	−7.16353	−0.324610	−0.162305	−	0.986741i	$-0.551893\pi$
$487$	−7.16353	−0.324610	−0.162305	+	0.986741i	$0.551893\pi$
$488$	24.3438	1.10199
$489$	−41.2874	−1.86708
$490$	−38.0482	−1.71884
$491$	−41.4786	−1.87190	−0.935951	−	0.352129i	$-0.885458\pi$
$491$	−41.4786	−1.87190	−0.935951	+	0.352129i	$0.885458\pi$
$492$	5.66509	0.255402
$493$	−0.320964	−0.0144555
$494$	6.48056	0.291574
$495$	6.45125	0.289962
$496$	−17.1518	−0.770138
$497$	−2.10018	−0.0942061
$498$	−26.5672	−1.19051
$499$	26.8722	1.20296	0.601482	−	0.798886i	$-0.294577\pi$
$499$	26.8722	1.20296	0.601482	+	0.798886i	$0.294577\pi$
$500$	−4.13388	−0.184873
$501$	−31.5192	−1.40817
$502$	−31.1521	−1.39039
$503$	35.1047	1.56524	0.782620	−	0.622500i	$-0.213883\pi$
$503$	35.1047	1.56524	0.782620	+	0.622500i	$0.213883\pi$
$504$	−12.6027	−0.561371
$505$	−0.208769	−0.00929012
$506$	25.8545	1.14937
$507$	9.64258	0.428242
$508$	6.74493	0.299258
$509$	−42.2821	−1.87412	−0.937062	−	0.349164i	$-0.886466\pi$
$509$	−42.2821	−1.87412	−0.937062	+	0.349164i	$0.886466\pi$
$510$	16.3873	0.725640
$511$	−33.1176	−1.46503
$512$	13.9347	0.615831
$513$	−3.96368	−0.175001
$514$	16.4840	0.727080
$515$	14.6361	0.644942
$516$	4.74927	0.209075
$517$	12.5731	0.552965
$518$	36.6349	1.60964
$519$	−2.16595	−0.0950744
$520$	−15.6593	−0.686707
$521$	25.9128	1.13526	0.567630	−	0.823284i	$-0.307860\pi$
$521$	25.9128	1.13526	0.567630	+	0.823284i	$0.307860\pi$
$522$	−0.139697	−0.00611439
$523$	23.4648	1.02604	0.513021	−	0.858376i	$-0.328526\pi$
$523$	23.4648	1.02604	0.513021	+	0.858376i	$0.328526\pi$
$524$	2.16449	0.0945563
$525$	27.7432	1.21081
$526$	−20.7976	−0.906816
$527$	−13.5073	−0.588388
$528$	−39.3897	−1.71422
$529$	−7.52331	−0.327100
$530$	−22.0856	−0.959336
$531$	−6.48866	−0.281584
$532$	−1.75012	−0.0758775
$533$	−33.2401	−1.43979
$534$	0.515956	0.0223276
$535$	8.27975	0.357965
$536$	−13.9258	−0.601502
$537$	−23.5825	−1.01766
$538$	−2.65916	−0.114645
$539$	72.0229	3.10225
$540$	−2.09183	−0.0900178
$541$	−43.2915	−1.86125	−0.930625	−	0.365975i	$-0.880735\pi$
$541$	−43.2915	−1.86125	−0.930625	+	0.365975i	$0.880735\pi$
$542$	−9.88725	−0.424694
$543$	35.7878	1.53580
$544$	−7.24489	−0.310622
$545$	1.03629	0.0443897
$546$	−63.4617	−2.71591
$547$	19.5022	0.833852	0.416926	−	0.908940i	$-0.363107\pi$
$547$	19.5022	0.833852	0.416926	+	0.908940i	$0.363107\pi$
$548$	4.15415	0.177457
$549$	9.88951	0.422074
$550$	18.6189	0.793912
$551$	0.0888231	0.00378399
$552$	−19.8944	−0.846761
$553$	−59.9210	−2.54810
$554$	22.5408	0.957666
$555$	14.4300	0.612521
$556$	−3.23011	−0.136987
$557$	32.0467	1.35786	0.678931	−	0.734202i	$-0.262444\pi$
$557$	32.0467	1.35786	0.678931	+	0.734202i	$0.262444\pi$
$558$	−5.87897	−0.248877
$559$	−27.8665	−1.17863
$560$	32.9730	1.39336
$561$	−31.0200	−1.30967
$562$	17.3247	0.730800
$563$	−36.5044	−1.53848	−0.769239	−	0.638961i	$-0.779364\pi$
$563$	−36.5044	−1.53848	−0.769239	+	0.638961i	$0.779364\pi$
$564$	2.11302	0.0889741
$565$	−9.49346	−0.399393
$566$	−8.03198	−0.337609
$567$	53.8128	2.25992
$568$	1.08455	0.0455066
$569$	−1.08710	−0.0455735	−0.0227868	−	0.999740i	$-0.507254\pi$
$569$	−1.08710	−0.0455735	−0.0227868	+	0.999740i	$0.507254\pi$
$570$	−4.53499	−0.189950
$571$	31.7497	1.32868	0.664342	−	0.747429i	$-0.268712\pi$
$571$	31.7497	1.32868	0.664342	+	0.747429i	$0.268712\pi$
$572$	−6.47403	−0.270693
$573$	6.31505	0.263815
$574$	59.0544	2.46489
$575$	11.1454	0.464795
$576$	6.24488	0.260203
$577$	−4.11240	−0.171202	−0.0856008	−	0.996330i	$-0.527281\pi$
$577$	−4.11240	−0.171202	−0.0856008	+	0.996330i	$0.527281\pi$
$578$	6.05467	0.251841
$579$	37.7108	1.56721
$580$	0.0468762	0.00194643
$581$	−42.0976	−1.74650
$582$	45.5183	1.88679
$583$	41.8066	1.73145
$584$	17.1021	0.707691
$585$	−6.36150	−0.263016
$586$	6.20029	0.256131
$587$	27.0772	1.11760	0.558798	−	0.829304i	$-0.311263\pi$
$587$	27.0772	1.11760	0.558798	+	0.829304i	$0.311263\pi$
$588$	12.1041	0.499163
$589$	3.73800	0.154022
$590$	14.3236	0.589695
$591$	−18.2875	−0.752247
$592$	−22.4223	−0.921552
$593$	−41.4809	−1.70342	−0.851708	−	0.524016i	$-0.824433\pi$
$593$	−41.4809	−1.70342	−0.851708	+	0.524016i	$0.824433\pi$
$594$	26.0493	1.06882
$595$	25.9668	1.06453
$596$	5.76273	0.236050
$597$	21.5619	0.882470
$598$	−25.4948	−1.04256
$599$	−46.1464	−1.88549	−0.942745	−	0.333514i	$-0.891765\pi$
$599$	−46.1464	−1.88549	−0.942745	+	0.333514i	$0.891765\pi$
$600$	−14.3268	−0.584888
$601$	15.4986	0.632201	0.316100	−	0.948726i	$-0.397626\pi$
$601$	15.4986	0.632201	0.316100	+	0.948726i	$0.397626\pi$
$602$	49.5077	2.01779
$603$	−5.65725	−0.230381
$604$	3.30421	0.134446
$605$	10.7649	0.437654
$606$	0.436916	0.0177485
$607$	−36.0731	−1.46416	−0.732080	−	0.681219i	$-0.761450\pi$
$607$	−36.0731	−1.46416	−0.732080	+	0.681219i	$0.761450\pi$
$608$	2.00494	0.0813111
$609$	−0.869813	−0.0352466
$610$	−21.8310	−0.883911
$611$	−12.3982	−0.501577
$612$	−1.32671	−0.0536292
$613$	−6.57977	−0.265754	−0.132877	−	0.991133i	$-0.542422\pi$
$613$	−6.57977	−0.265754	−0.132877	+	0.991133i	$0.542422\pi$
$614$	33.8484	1.36601
$615$	23.2609	0.937969
$616$	−52.6623	−2.12182
$617$	7.90342	0.318180	0.159090	−	0.987264i	$-0.449144\pi$
$617$	7.90342	0.318180	0.159090	+	0.987264i	$0.449144\pi$
$618$	−30.6306	−1.23214
$619$	−18.7316	−0.752886	−0.376443	−	0.926440i	$-0.622853\pi$
$619$	−18.7316	−0.752886	−0.376443	+	0.926440i	$0.622853\pi$
$620$	1.97272	0.0792264
$621$	15.5933	0.625738
$622$	−19.3096	−0.774245
$623$	0.817568	0.0327552
$624$	38.8417	1.55491
$625$	−2.80840	−0.112336
$626$	−27.4570	−1.09740
$627$	8.58445	0.342830
$628$	0.499446	0.0199301
$629$	−17.6580	−0.704069
$630$	11.3019	0.450277
$631$	−30.7281	−1.22327	−0.611634	−	0.791141i	$-0.709487\pi$
$631$	−30.7281	−1.22327	−0.611634	+	0.791141i	$0.709487\pi$
$632$	30.9436	1.23087
$633$	−2.00602	−0.0797320
$634$	9.48804	0.376818
$635$	27.6947	1.09903
$636$	7.02595	0.278597
$637$	−71.0209	−2.81395
$638$	−0.583745	−0.0231107
$639$	0.440591	0.0174295
$640$	−19.6883	−0.778247
$641$	25.7140	1.01564	0.507820	−	0.861463i	$-0.330451\pi$
$641$	25.7140	1.01564	0.507820	+	0.861463i	$0.330451\pi$
$642$	−17.3280	−0.683881
$643$	9.38776	0.370217	0.185109	−	0.982718i	$-0.440736\pi$
$643$	9.38776	0.370217	0.185109	+	0.982718i	$0.440736\pi$
$644$	6.88506	0.271309
$645$	19.5005	0.767833
$646$	5.54944	0.218340
$647$	31.4018	1.23453	0.617267	−	0.786754i	$-0.288240\pi$
$647$	31.4018	1.23453	0.617267	+	0.786754i	$0.288240\pi$
$648$	−27.7893	−1.09167
$649$	−27.1138	−1.06431
$650$	−18.3598	−0.720132
$651$	−36.6048	−1.43466
$652$	−7.37882	−0.288977
$653$	−43.1364	−1.68806	−0.844028	−	0.536299i	$-0.819822\pi$
$653$	−43.1364	−1.68806	−0.844028	+	0.536299i	$0.819822\pi$
$654$	−2.16876	−0.0848052
$655$	8.88741	0.347260
$656$	−36.1442	−1.41119
$657$	6.94763	0.271053
$658$	22.0267	0.858689
$659$	21.8339	0.850528	0.425264	−	0.905069i	$-0.360181\pi$
$659$	21.8339	0.850528	0.425264	+	0.905069i	$0.360181\pi$
$660$	4.53042	0.176347
$661$	3.60798	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$661$	3.60798	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$662$	42.1230	1.63716
$663$	30.5885	1.18796
$664$	21.7395	0.843656
$665$	−7.18601	−0.278662
$666$	−7.68551	−0.297808
$667$	−0.349434	−0.0135301
$668$	−5.63305	−0.217949
$669$	12.5905	0.486775
$670$	12.4883	0.482466
$671$	41.3247	1.59532
$672$	−19.6337	−0.757385
$673$	2.91281	0.112280	0.0561402	−	0.998423i	$-0.482121\pi$
$673$	2.91281	0.112280	0.0561402	+	0.998423i	$0.482121\pi$
$674$	−31.2924	−1.20534
$675$	11.2294	0.432219
$676$	1.72331	0.0662810
$677$	12.9879	0.499164	0.249582	−	0.968354i	$-0.419707\pi$
$677$	12.9879	0.499164	0.249582	+	0.968354i	$0.419707\pi$
$678$	19.8680	0.763027
$679$	72.1269	2.76798
$680$	−13.4094	−0.514227
$681$	−35.1361	−1.34642
$682$	−24.5661	−0.940684
$683$	6.10046	0.233427	0.116714	−	0.993166i	$-0.462764\pi$
$683$	6.10046	0.233427	0.116714	+	0.993166i	$0.462764\pi$
$684$	0.367153	0.0140384
$685$	17.0569	0.651712
$686$	73.6974	2.81378
$687$	9.10770	0.347480
$688$	−30.3012	−1.15522
$689$	−41.2249	−1.57055
$690$	17.8408	0.679189
$691$	−36.3770	−1.38385	−0.691923	−	0.721972i	$-0.743236\pi$
$691$	−36.3770	−1.38385	−0.691923	+	0.721972i	$0.743236\pi$
$692$	−0.387094	−0.0147151
$693$	−21.3937	−0.812680
$694$	26.5387	1.00740
$695$	−13.2628	−0.503088
$696$	0.449177	0.0170260
$697$	−28.4642	−1.07816
$698$	1.76559	0.0668286
$699$	8.22591	0.311133
$700$	4.95822	0.187403
$701$	−0.751456	−0.0283821	−0.0141911	−	0.999899i	$-0.504517\pi$
$701$	−0.751456	−0.0283821	−0.0141911	+	0.999899i	$0.504517\pi$
$702$	−25.6869	−0.969489
$703$	4.88664	0.184303
$704$	26.0951	0.983496
$705$	8.67605	0.326759
$706$	−52.3264	−1.96933
$707$	0.692324	0.0260375
$708$	−4.55670	−0.171251
$709$	−23.0597	−0.866024	−0.433012	−	0.901388i	$-0.642549\pi$
$709$	−23.0597	−0.866024	−0.433012	+	0.901388i	$0.642549\pi$
$710$	−0.972599	−0.0365010
$711$	12.5706	0.471436
$712$	−0.422197	−0.0158225
$713$	−14.7054	−0.550723
$714$	−54.3436	−2.03376
$715$	−26.5824	−0.994125
$716$	−4.21462	−0.157508
$717$	35.0133	1.30759
$718$	−6.01521	−0.224486
$719$	−8.79812	−0.328115	−0.164057	−	0.986451i	$-0.552458\pi$
$719$	−8.79812	−0.328115	−0.164057	+	0.986451i	$0.552458\pi$
$720$	−6.91729	−0.257792
$721$	−48.5363	−1.80759
$722$	−1.53574	−0.0571545
$723$	−20.7082	−0.770144
$724$	6.39593	0.237703
$725$	−0.251642	−0.00934574
$726$	−22.5289	−0.836125
$727$	16.8871	0.626310	0.313155	−	0.949702i	$-0.398614\pi$
$727$	16.8871	0.626310	0.313155	+	0.949702i	$0.398614\pi$
$728$	51.9296	1.92464
$729$	12.5644	0.465349
$730$	−15.3368	−0.567641
$731$	−23.8627	−0.882593
$732$	6.94496	0.256693
$733$	11.2004	0.413695	0.206848	−	0.978373i	$-0.433680\pi$
$733$	11.2004	0.413695	0.206848	+	0.978373i	$0.433680\pi$
$734$	43.9310	1.62152
$735$	49.6993	1.83319
$736$	−7.88752	−0.290738
$737$	−23.6396	−0.870776
$738$	−12.3888	−0.456040
$739$	45.6500	1.67926	0.839632	−	0.543156i	$-0.182771\pi$
$739$	45.6500	1.67926	0.839632	+	0.543156i	$0.182771\pi$
$740$	2.57891	0.0948028
$741$	−8.46502	−0.310970
$742$	73.2404	2.68874
$743$	−3.23921	−0.118835	−0.0594176	−	0.998233i	$-0.518924\pi$
$743$	−3.23921	−0.118835	−0.0594176	+	0.998233i	$0.518924\pi$
$744$	18.9030	0.693017
$745$	23.6618	0.866900
$746$	8.20191	0.300293
$747$	8.83153	0.323129
$748$	−5.54385	−0.202703
$749$	−27.4574	−1.00327
$750$	35.5229	1.29711
$751$	−18.0882	−0.660050	−0.330025	−	0.943972i	$-0.607057\pi$
$751$	−18.0882	−0.660050	−0.330025	+	0.943972i	$0.607057\pi$
$752$	−13.4814	−0.491616
$753$	40.6915	1.48288
$754$	0.575623	0.0209630
$755$	13.5671	0.493757
$756$	6.93694	0.252294
$757$	33.3156	1.21088	0.605438	−	0.795892i	$-0.292998\pi$
$757$	33.3156	1.21088	0.605438	+	0.795892i	$0.292998\pi$
$758$	−23.0803	−0.838314
$759$	−33.7716	−1.22583
$760$	3.71090	0.134609
$761$	19.4472	0.704961	0.352480	−	0.935819i	$-0.385338\pi$
$761$	19.4472	0.704961	0.352480	+	0.935819i	$0.385338\pi$
$762$	−57.9599	−2.09966
$763$	−3.43655	−0.124411
$764$	1.12861	0.0408318
$765$	−5.44749	−0.196954
$766$	−31.5183	−1.13880
$767$	26.7365	0.965400
$768$	16.7389	0.604012
$769$	−44.4046	−1.60127	−0.800636	−	0.599151i	$-0.795505\pi$
$769$	−44.4046	−1.60127	−0.800636	+	0.599151i	$0.795505\pi$
$770$	47.2264	1.70192
$771$	−21.5317	−0.775446
$772$	6.73962	0.242564
$773$	−38.9430	−1.40068	−0.700340	−	0.713809i	$-0.746968\pi$
$773$	−38.9430	−1.40068	−0.700340	+	0.713809i	$0.746968\pi$
$774$	−10.3861	−0.373320
$775$	−10.5900	−0.380404
$776$	−37.2468	−1.33708
$777$	−47.8531	−1.71672
$778$	1.87164	0.0671014
$779$	7.87714	0.282228
$780$	−4.46739	−0.159958
$781$	1.84107	0.0658786
$782$	−21.8317	−0.780701
$783$	−0.352067	−0.0125818
$784$	−77.2259	−2.75807
$785$	2.05073	0.0731936
$786$	−18.5997	−0.663430
$787$	22.9207	0.817034	0.408517	−	0.912751i	$-0.366046\pi$
$787$	22.9207	0.817034	0.408517	+	0.912751i	$0.366046\pi$
$788$	−3.26831	−0.116429
$789$	27.1661	0.967140
$790$	−27.7496	−0.987285
$791$	31.4823	1.11938
$792$	11.0479	0.392568
$793$	−40.7497	−1.44707
$794$	−11.1708	−0.396438
$795$	28.8486	1.02315
$796$	3.85351	0.136584
$797$	−19.0593	−0.675117	−0.337558	−	0.941305i	$-0.609601\pi$
$797$	−19.0593	−0.675117	−0.337558	+	0.941305i	$0.609601\pi$
$798$	15.0390	0.532374
$799$	−10.6168	−0.375597
$800$	−5.68013	−0.200823
$801$	−0.171515	−0.00606018
$802$	14.5342	0.513222
$803$	29.0316	1.02450
$804$	−3.97284	−0.140111
$805$	28.2701	0.996389
$806$	24.2243	0.853265
$807$	3.47344	0.122271
$808$	−0.357520	−0.0125775
$809$	14.0874	0.495285	0.247643	−	0.968851i	$-0.420344\pi$
$809$	14.0874	0.495285	0.247643	+	0.968851i	$0.420344\pi$
$810$	24.9208	0.875628
$811$	−23.5328	−0.826347	−0.413174	−	0.910652i	$-0.635580\pi$
$811$	−23.5328	−0.826347	−0.413174	+	0.910652i	$0.635580\pi$
$812$	−0.155451	−0.00545528
$813$	12.9149	0.452945
$814$	−32.1150	−1.12563
$815$	−30.2974	−1.06127
$816$	33.2610	1.16437
$817$	6.60373	0.231035
$818$	−12.2280	−0.427543
$819$	21.0961	0.737157
$820$	4.15714	0.145174
$821$	−2.09164	−0.0729986	−0.0364993	−	0.999334i	$-0.511621\pi$
$821$	−2.09164	−0.0729986	−0.0364993	+	0.999334i	$0.511621\pi$
$822$	−35.6970	−1.24508
$823$	−3.65827	−0.127519	−0.0637596	−	0.997965i	$-0.520309\pi$
$823$	−3.65827	−0.127519	−0.0637596	+	0.997965i	$0.520309\pi$
$824$	25.0645	0.873162
$825$	−24.3203	−0.846724
$826$	−47.5003	−1.65275
$827$	−21.3459	−0.742270	−0.371135	−	0.928579i	$-0.621031\pi$
$827$	−21.3459	−0.742270	−0.371135	+	0.928579i	$0.621031\pi$
$828$	−1.44440	−0.0501962
$829$	56.8573	1.97473	0.987367	−	0.158448i	$-0.0506490\pi$
$829$	56.8573	1.97473	0.987367	+	0.158448i	$0.0506490\pi$
$830$	−19.4955	−0.676699
$831$	−29.4432	−1.02137
$832$	−25.7320	−0.892098
$833$	−60.8167	−2.10717
$834$	27.7566	0.961133
$835$	−23.1293	−0.800423
$836$	1.53420	0.0530614
$837$	−14.8162	−0.512124
$838$	22.9677	0.793406
$839$	36.2399	1.25114	0.625570	−	0.780168i	$-0.284866\pi$
$839$	36.2399	1.25114	0.625570	+	0.780168i	$0.284866\pi$
$840$	−36.3395	−1.25383
$841$	−28.9921	−0.999728
$842$	−48.5333	−1.67257
$843$	−22.6299	−0.779414
$844$	−0.358512	−0.0123405
$845$	7.07590	0.243418
$846$	−4.62091	−0.158870
$847$	−35.6986	−1.22662
$848$	−44.8267	−1.53936
$849$	10.4915	0.360068
$850$	−15.7219	−0.539258
$851$	−19.2243	−0.658999
$852$	0.309407	0.0106001
$853$	−8.85313	−0.303126	−0.151563	−	0.988448i	$-0.548431\pi$
$853$	−8.85313	−0.303126	−0.151563	+	0.988448i	$0.548431\pi$
$854$	72.3962	2.47735
$855$	1.50753	0.0515565
$856$	14.1792	0.484634
$857$	−22.4249	−0.766020	−0.383010	−	0.923744i	$-0.625112\pi$
$857$	−22.4249	−0.766020	−0.383010	+	0.923744i	$0.625112\pi$
$858$	55.6320	1.89925
$859$	15.7193	0.536336	0.268168	−	0.963372i	$-0.413582\pi$
$859$	15.7193	0.536336	0.268168	+	0.963372i	$0.413582\pi$
$860$	3.48510	0.118841
$861$	−77.1379	−2.62885
$862$	−30.0485	−1.02346
$863$	54.4747	1.85434	0.927170	−	0.374640i	$-0.122234\pi$
$863$	54.4747	1.85434	0.927170	+	0.374640i	$0.122234\pi$
$864$	−7.94695	−0.270361
$865$	−1.58941	−0.0540415
$866$	54.5584	1.85397
$867$	−7.90871	−0.268594
$868$	−6.54196	−0.222049
$869$	52.5281	1.78190
$870$	−0.402812	−0.0136566
$871$	23.3107	0.789854
$872$	1.77466	0.0600975
$873$	−15.1313	−0.512116
$874$	6.04168	0.204363
$875$	56.2885	1.90290
$876$	4.87901	0.164846
$877$	8.14358	0.274989	0.137495	−	0.990503i	$-0.456095\pi$
$877$	8.14358	0.274989	0.137495	+	0.990503i	$0.456095\pi$
$878$	−61.6124	−2.07932
$879$	−8.09892	−0.273170
$880$	−28.9049	−0.974382
$881$	39.3891	1.32705	0.663526	−	0.748153i	$-0.269059\pi$
$881$	39.3891	1.32705	0.663526	+	0.748153i	$0.269059\pi$
$882$	−26.4701	−0.891294
$883$	−10.8737	−0.365930	−0.182965	−	0.983119i	$-0.558570\pi$
$883$	−10.8737	−0.365930	−0.182965	+	0.983119i	$0.558570\pi$
$884$	5.46672	0.183866
$885$	−18.7098	−0.628923
$886$	9.23545	0.310271
$887$	−1.51350	−0.0508185	−0.0254092	−	0.999677i	$-0.508089\pi$
$887$	−1.51350	−0.0508185	−0.0254092	+	0.999677i	$0.508089\pi$
$888$	24.7116	0.829269
$889$	−91.8415	−3.08026
$890$	0.378617	0.0126913
$891$	−47.1735	−1.58037
$892$	2.25015	0.0753405
$893$	2.93809	0.0983194
$894$	−49.5197	−1.65619
$895$	−17.3052	−0.578451
$896$	65.2904	2.18120
$897$	33.3017	1.11191
$898$	−15.9188	−0.531218
$899$	0.332021	0.0110735
$900$	−1.04017	−0.0346723
$901$	−35.3018	−1.17607
$902$	−51.7685	−1.72370
$903$	−64.6679	−2.15201
$904$	−16.2577	−0.540722
$905$	26.2617	0.872969
$906$	−28.3934	−0.943307
$907$	29.1807	0.968930	0.484465	−	0.874811i	$-0.339014\pi$
$907$	29.1807	0.968930	0.484465	+	0.874811i	$0.339014\pi$
$908$	−6.27946	−0.208391
$909$	−0.145240	−0.00481732
$910$	−46.5694	−1.54376
$911$	17.8183	0.590348	0.295174	−	0.955444i	$-0.404622\pi$
$911$	17.8183	0.590348	0.295174	+	0.955444i	$0.404622\pi$
$912$	−9.20459	−0.304795
$913$	36.9037	1.22134
$914$	−62.2528	−2.05914
$915$	28.5160	0.942711
$916$	1.62771	0.0537812
$917$	−29.4725	−0.973269
$918$	−21.9962	−0.725984
$919$	4.07301	0.134356	0.0671780	−	0.997741i	$-0.478600\pi$
$919$	4.07301	0.134356	0.0671780	+	0.997741i	$0.478600\pi$
$920$	−14.5988	−0.481310
$921$	−44.2133	−1.45688
$922$	−56.8561	−1.87245
$923$	−1.81545	−0.0597564
$924$	−15.0238	−0.494248
$925$	−13.8442	−0.455194
$926$	−1.39603	−0.0458765
$927$	10.1823	0.334430
$928$	0.178085	0.00584593
$929$	3.81445	0.125148	0.0625739	−	0.998040i	$-0.480069\pi$
$929$	3.81445	0.125148	0.0625739	+	0.998040i	$0.480069\pi$
$930$	−16.9518	−0.555871
$931$	16.8303	0.551592
$932$	1.47012	0.0481554
$933$	25.2226	0.825749
$934$	20.2735	0.663370
$935$	−22.7631	−0.744432
$936$	−10.8942	−0.356086
$937$	47.4794	1.55108	0.775542	−	0.631296i	$-0.217477\pi$
$937$	47.4794	1.55108	0.775542	+	0.631296i	$0.217477\pi$
$938$	−41.4139	−1.35221
$939$	35.8649	1.17041
$940$	1.55057	0.0505740
$941$	58.4872	1.90663	0.953314	−	0.301980i	$-0.0976475\pi$
$941$	58.4872	1.90663	0.953314	+	0.301980i	$0.0976475\pi$
$942$	−4.29179	−0.139834
$943$	−30.9890	−1.00914
$944$	29.0725	0.946228
$945$	28.4831	0.926554
$946$	−43.3996	−1.41104
$947$	21.7542	0.706918	0.353459	−	0.935450i	$-0.385005\pi$
$947$	21.7542	0.706918	0.353459	+	0.935450i	$0.385005\pi$
$948$	8.82780	0.286714
$949$	−28.6277	−0.929294
$950$	4.35087	0.141161
$951$	−12.3934	−0.401885
$952$	44.4684	1.44123
$953$	−10.3628	−0.335684	−0.167842	−	0.985814i	$-0.553680\pi$
$953$	−10.3628	−0.335684	−0.167842	+	0.985814i	$0.553680\pi$
$954$	−15.3649	−0.497456
$955$	4.63409	0.149956
$956$	6.25751	0.202382
$957$	0.762498	0.0246480
$958$	−39.7382	−1.28388
$959$	−56.5644	−1.82656
$960$	18.0069	0.581169
$961$	−17.0274	−0.549270
$962$	31.6682	1.02102
$963$	5.76020	0.185620
$964$	−3.70093	−0.119199
$965$	27.6729	0.890822
$966$	−59.1640	−1.90357
$967$	12.5957	0.405050	0.202525	−	0.979277i	$-0.435085\pi$
$967$	12.5957	0.405050	0.202525	+	0.979277i	$0.435085\pi$
$968$	18.4350	0.592523
$969$	−7.24877	−0.232864
$970$	33.4021	1.07248
$971$	−50.9772	−1.63594	−0.817968	−	0.575263i	$-0.804900\pi$
$971$	−50.9772	−1.63594	−0.817968	+	0.575263i	$0.804900\pi$
$972$	−3.66482	−0.117549
$973$	43.9823	1.41001
$974$	11.0013	0.352506
$975$	23.9820	0.768037
$976$	−44.3100	−1.41833
$977$	−46.6367	−1.49204	−0.746021	−	0.665923i	$-0.768038\pi$
$977$	−46.6367	−1.49204	−0.746021	+	0.665923i	$0.768038\pi$
$978$	63.4069	2.02753
$979$	−0.716699	−0.0229058
$980$	8.88217	0.283731
$981$	0.720943	0.0230179
$982$	63.7005	2.03277
$983$	−35.2102	−1.12303	−0.561515	−	0.827466i	$-0.689781\pi$
$983$	−35.2102	−1.12303	−0.561515	+	0.827466i	$0.689781\pi$
$984$	39.8345	1.26988
$985$	−13.4197	−0.427587
$986$	0.492919	0.0156977
$987$	−28.7716	−0.915811
$988$	−1.51285	−0.0481303
$989$	−25.9793	−0.826095
$990$	−9.90748	−0.314880
$991$	16.2722	0.516905	0.258452	−	0.966024i	$-0.416788\pi$
$991$	16.2722	0.516905	0.258452	+	0.966024i	$0.416788\pi$
$992$	7.49447	0.237950
$993$	−55.0218	−1.74607
$994$	3.22534	0.102302
$995$	15.8225	0.501607
$996$	6.20199	0.196517
$997$	8.50252	0.269277	0.134639	−	0.990895i	$-0.457013\pi$
$997$	8.50252	0.269277	0.134639	+	0.990895i	$0.457013\pi$
$998$	−41.2688	−1.30634
$999$	−19.3691	−0.612811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.18	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.18	✓	82	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53574 q^{2} +2.00602 q^{3} +0.358512 q^{4} +1.47205 q^{5} -3.08073 q^{6} -4.88163 q^{7} +2.52091 q^{8} +1.02410 q^{9} +O(q^{10})\)	\(q-1.53574 q^{2} +2.00602 q^{3} +0.358512 q^{4} +1.47205 q^{5} -3.08073 q^{6} -4.88163 q^{7} +2.52091 q^{8} +1.02410 q^{9} -2.26069 q^{10} +4.27935 q^{11} +0.719181 q^{12} -4.21981 q^{13} +7.49694 q^{14} +2.95296 q^{15} -4.58849 q^{16} -3.61352 q^{17} -1.57276 q^{18} +1.00000 q^{19} +0.527748 q^{20} -9.79263 q^{21} -6.57199 q^{22} -3.93404 q^{23} +5.05698 q^{24} -2.83307 q^{25} +6.48056 q^{26} -3.96368 q^{27} -1.75012 q^{28} +0.0888231 q^{29} -4.53499 q^{30} +3.73800 q^{31} +2.00494 q^{32} +8.58445 q^{33} +5.54944 q^{34} -7.18601 q^{35} +0.367153 q^{36} +4.88664 q^{37} -1.53574 q^{38} -8.46502 q^{39} +3.71090 q^{40} +7.87714 q^{41} +15.0390 q^{42} +6.60373 q^{43} +1.53420 q^{44} +1.50753 q^{45} +6.04168 q^{46} +2.93809 q^{47} -9.20459 q^{48} +16.8303 q^{49} +4.35087 q^{50} -7.24877 q^{51} -1.51285 q^{52} +9.76937 q^{53} +6.08721 q^{54} +6.29942 q^{55} -12.3061 q^{56} +2.00602 q^{57} -0.136410 q^{58} -6.33595 q^{59} +1.05867 q^{60} +9.65676 q^{61} -5.74061 q^{62} -4.99929 q^{63} +6.09791 q^{64} -6.21178 q^{65} -13.1835 q^{66} -5.52411 q^{67} -1.29549 q^{68} -7.89175 q^{69} +11.0359 q^{70} +0.430221 q^{71} +2.58167 q^{72} +6.78411 q^{73} -7.50463 q^{74} -5.68318 q^{75} +0.358512 q^{76} -20.8902 q^{77} +13.0001 q^{78} +12.2748 q^{79} -6.75450 q^{80} -11.0235 q^{81} -12.0973 q^{82} +8.62368 q^{83} -3.51078 q^{84} -5.31928 q^{85} -10.1416 q^{86} +0.178181 q^{87} +10.7878 q^{88} -0.167478 q^{89} -2.31518 q^{90} +20.5996 q^{91} -1.41040 q^{92} +7.49849 q^{93} -4.51215 q^{94} +1.47205 q^{95} +4.02195 q^{96} -14.7752 q^{97} -25.8471 q^{98} +4.38249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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