LMFDB - Embedded newform 4009.2.a.d.1.16

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:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-1.81819 q^{2} +1.88652 q^{3} +1.30583 q^{4} -3.78374 q^{5} -3.43005 q^{6} +2.14600 q^{7} +1.26214 q^{8} +0.558942 q^{9} +O(q^{10})$	\(q-1.81819 q^{2} +1.88652 q^{3} +1.30583 q^{4} -3.78374 q^{5} -3.43005 q^{6} +2.14600 q^{7} +1.26214 q^{8} +0.558942 q^{9} +6.87957 q^{10} -3.52159 q^{11} +2.46346 q^{12} +6.24903 q^{13} -3.90185 q^{14} -7.13809 q^{15} -4.90647 q^{16} -3.30498 q^{17} -1.01626 q^{18} -1.00000 q^{19} -4.94091 q^{20} +4.04847 q^{21} +6.40292 q^{22} -0.0872254 q^{23} +2.38105 q^{24} +9.31670 q^{25} -11.3619 q^{26} -4.60509 q^{27} +2.80231 q^{28} -3.48115 q^{29} +12.9784 q^{30} +8.80857 q^{31} +6.39663 q^{32} -6.64353 q^{33} +6.00910 q^{34} -8.11993 q^{35} +0.729881 q^{36} +4.61409 q^{37} +1.81819 q^{38} +11.7889 q^{39} -4.77562 q^{40} -3.33867 q^{41} -7.36090 q^{42} -6.44056 q^{43} -4.59858 q^{44} -2.11489 q^{45} +0.158593 q^{46} +12.1176 q^{47} -9.25613 q^{48} -2.39466 q^{49} -16.9396 q^{50} -6.23490 q^{51} +8.16015 q^{52} +4.45724 q^{53} +8.37295 q^{54} +13.3248 q^{55} +2.70856 q^{56} -1.88652 q^{57} +6.32940 q^{58} -1.45966 q^{59} -9.32111 q^{60} -8.89957 q^{61} -16.0157 q^{62} +1.19949 q^{63} -1.81737 q^{64} -23.6447 q^{65} +12.0792 q^{66} +7.80761 q^{67} -4.31574 q^{68} -0.164552 q^{69} +14.7636 q^{70} -15.2885 q^{71} +0.705464 q^{72} -10.5444 q^{73} -8.38931 q^{74} +17.5761 q^{75} -1.30583 q^{76} -7.55734 q^{77} -21.4345 q^{78} -5.94661 q^{79} +18.5648 q^{80} -10.3644 q^{81} +6.07035 q^{82} -10.8558 q^{83} +5.28660 q^{84} +12.5052 q^{85} +11.7102 q^{86} -6.56724 q^{87} -4.44474 q^{88} +15.9439 q^{89} +3.84528 q^{90} +13.4104 q^{91} -0.113901 q^{92} +16.6175 q^{93} -22.0322 q^{94} +3.78374 q^{95} +12.0673 q^{96} -4.28346 q^{97} +4.35396 q^{98} -1.96836 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * 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 $

$2$	−1.81819	−1.28566	−0.642828	−	0.766010i	$-0.722239\pi$
$2$	−1.81819	−1.28566	−0.642828	+	0.766010i	$0.722239\pi$
$3$	1.88652	1.08918	0.544590	−	0.838702i	$-0.316685\pi$
$3$	1.88652	1.08918	0.544590	+	0.838702i	$0.316685\pi$
$4$	1.30583	0.652913
$5$	−3.78374	−1.69214	−0.846070	−	0.533071i	$-0.821038\pi$
$5$	−3.78374	−1.69214	−0.846070	+	0.533071i	$0.821038\pi$
$6$	−3.43005	−1.40031
$7$	2.14600	0.811113	0.405557	−	0.914070i	$-0.367078\pi$
$7$	2.14600	0.811113	0.405557	+	0.914070i	$0.367078\pi$
$8$	1.26214	0.446234
$9$	0.558942	0.186314
$10$	6.87957	2.17551
$11$	−3.52159	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$11$	−3.52159	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$12$	2.46346	0.711140
$13$	6.24903	1.73317	0.866584	−	0.499031i	$-0.166311\pi$
$13$	6.24903	1.73317	0.866584	+	0.499031i	$0.166311\pi$
$14$	−3.90185	−1.04281
$15$	−7.13809	−1.84305
$16$	−4.90647	−1.22662
$17$	−3.30498	−0.801576	−0.400788	−	0.916171i	$-0.631264\pi$
$17$	−3.30498	−0.801576	−0.400788	+	0.916171i	$0.631264\pi$
$18$	−1.01626	−0.239536
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.94091	−1.10482
$21$	4.04847	0.883449
$22$	6.40292	1.36511
$23$	−0.0872254	−0.0181878	−0.00909388	−	0.999959i	$-0.502895\pi$
$23$	−0.0872254	−0.0181878	−0.00909388	+	0.999959i	$0.502895\pi$
$24$	2.38105	0.486030
$25$	9.31670	1.86334
$26$	−11.3619	−2.22826
$27$	−4.60509	−0.886251
$28$	2.80231	0.529587
$29$	−3.48115	−0.646433	−0.323217	−	0.946325i	$-0.604764\pi$
$29$	−3.48115	−0.646433	−0.323217	+	0.946325i	$0.604764\pi$
$30$	12.9784	2.36953
$31$	8.80857	1.58207	0.791033	−	0.611774i	$-0.209544\pi$
$31$	8.80857	1.58207	0.791033	+	0.611774i	$0.209544\pi$
$32$	6.39663	1.13077
$33$	−6.64353	−1.15649
$34$	6.00910	1.03055
$35$	−8.11993	−1.37252
$36$	0.729881	0.121647
$37$	4.61409	0.758552	0.379276	−	0.925284i	$-0.376173\pi$
$37$	4.61409	0.758552	0.379276	+	0.925284i	$0.376173\pi$
$38$	1.81819	0.294950
$39$	11.7889	1.88773
$40$	−4.77562	−0.755091
$41$	−3.33867	−0.521413	−0.260706	−	0.965418i	$-0.583955\pi$
$41$	−3.33867	−0.521413	−0.260706	+	0.965418i	$0.583955\pi$
$42$	−7.36090	−1.13581
$43$	−6.44056	−0.982177	−0.491088	−	0.871110i	$-0.663401\pi$
$43$	−6.44056	−0.982177	−0.491088	+	0.871110i	$0.663401\pi$
$44$	−4.59858	−0.693262
$45$	−2.11489	−0.315269
$46$	0.158593	0.0233832
$47$	12.1176	1.76754	0.883768	−	0.467925i	$-0.154998\pi$
$47$	12.1176	1.76754	0.883768	+	0.467925i	$0.154998\pi$
$48$	−9.25613	−1.33601
$49$	−2.39466	−0.342095
$50$	−16.9396	−2.39562
$51$	−6.23490	−0.873061
$52$	8.16015	1.13161
$53$	4.45724	0.612249	0.306125	−	0.951991i	$-0.400968\pi$
$53$	4.45724	0.612249	0.306125	+	0.951991i	$0.400968\pi$
$54$	8.37295	1.13941
$55$	13.3248	1.79671
$56$	2.70856	0.361947
$57$	−1.88652	−0.249875
$58$	6.32940	0.831091
$59$	−1.45966	−0.190031	−0.0950157	−	0.995476i	$-0.530290\pi$
$59$	−1.45966	−0.190031	−0.0950157	+	0.995476i	$0.530290\pi$
$60$	−9.32111	−1.20335
$61$	−8.89957	−1.13947	−0.569736	−	0.821827i	$-0.692955\pi$
$61$	−8.89957	−1.13947	−0.569736	+	0.821827i	$0.692955\pi$
$62$	−16.0157	−2.03399
$63$	1.19949	0.151122
$64$	−1.81737	−0.227171
$65$	−23.6447	−2.93277
$66$	12.0792	1.48685
$67$	7.80761	0.953851	0.476925	−	0.878944i	$-0.341751\pi$
$67$	7.80761	0.953851	0.476925	+	0.878944i	$0.341751\pi$
$68$	−4.31574	−0.523360
$69$	−0.164552	−0.0198098
$70$	14.7636	1.76459
$71$	−15.2885	−1.81441	−0.907205	−	0.420690i	$-0.861788\pi$
$71$	−15.2885	−1.81441	−0.907205	+	0.420690i	$0.861788\pi$
$72$	0.705464	0.0831397
$73$	−10.5444	−1.23413	−0.617066	−	0.786911i	$-0.711679\pi$
$73$	−10.5444	−1.23413	−0.617066	+	0.786911i	$0.711679\pi$
$74$	−8.38931	−0.975238
$75$	17.5761	2.02951
$76$	−1.30583	−0.149789
$77$	−7.55734	−0.861239
$78$	−21.4345	−2.42698
$79$	−5.94661	−0.669046	−0.334523	−	0.942388i	$-0.608575\pi$
$79$	−5.94661	−0.669046	−0.334523	+	0.942388i	$0.608575\pi$
$80$	18.5648	2.07561
$81$	−10.3644	−1.15160
$82$	6.07035	0.670358
$83$	−10.8558	−1.19157	−0.595787	−	0.803142i	$-0.703160\pi$
$83$	−10.8558	−1.19157	−0.595787	+	0.803142i	$0.703160\pi$
$84$	5.28660	0.576815
$85$	12.5052	1.35638
$86$	11.7102	1.26274
$87$	−6.56724	−0.704082
$88$	−4.44474	−0.473811
$89$	15.9439	1.69005	0.845026	−	0.534725i	$-0.179585\pi$
$89$	15.9439	1.69005	0.845026	+	0.534725i	$0.179585\pi$
$90$	3.84528	0.405328
$91$	13.4104	1.40580
$92$	−0.113901	−0.0118750
$93$	16.6175	1.72315
$94$	−22.0322	−2.27245
$95$	3.78374	0.388204
$96$	12.0673	1.23162
$97$	−4.28346	−0.434920	−0.217460	−	0.976069i	$-0.569777\pi$
$97$	−4.28346	−0.434920	−0.217460	+	0.976069i	$0.569777\pi$
$98$	4.35396	0.439817
$99$	−1.96836	−0.197828
$100$	12.1660	1.21660
$101$	11.8193	1.17606	0.588031	−	0.808838i	$-0.299903\pi$
$101$	11.8193	1.17606	0.588031	+	0.808838i	$0.299903\pi$
$102$	11.3363	1.12246
$103$	13.8456	1.36425	0.682123	−	0.731237i	$-0.261057\pi$
$103$	13.8456	1.36425	0.682123	+	0.731237i	$0.261057\pi$
$104$	7.88716	0.773399
$105$	−15.3184	−1.49492
$106$	−8.10413	−0.787143
$107$	−7.33755	−0.709348	−0.354674	−	0.934990i	$-0.615408\pi$
$107$	−7.33755	−0.709348	−0.354674	+	0.934990i	$0.615408\pi$
$108$	−6.01346	−0.578645
$109$	16.3469	1.56575	0.782875	−	0.622179i	$-0.213752\pi$
$109$	16.3469	1.56575	0.782875	+	0.622179i	$0.213752\pi$
$110$	−24.2270	−2.30996
$111$	8.70456	0.826200
$112$	−10.5293	−0.994926
$113$	1.73388	0.163110	0.0815549	−	0.996669i	$-0.474011\pi$
$113$	1.73388	0.163110	0.0815549	+	0.996669i	$0.474011\pi$
$114$	3.43005	0.321254
$115$	0.330039	0.0307763
$116$	−4.54578	−0.422065
$117$	3.49284	0.322913
$118$	2.65394	0.244315
$119$	−7.09251	−0.650169
$120$	−9.00928	−0.822431
$121$	1.40157	0.127416
$122$	16.1811	1.46497
$123$	−6.29846	−0.567913
$124$	11.5025	1.03295
$125$	−16.3333	−1.46089
$126$	−2.18091	−0.194291
$127$	−7.80232	−0.692344	−0.346172	−	0.938171i	$-0.612519\pi$
$127$	−7.80232	−0.692344	−0.346172	+	0.938171i	$0.612519\pi$
$128$	−9.48893	−0.838711
$129$	−12.1502	−1.06977
$130$	42.9907	3.77053
$131$	11.5203	1.00653	0.503266	−	0.864132i	$-0.332132\pi$
$131$	11.5203	1.00653	0.503266	+	0.864132i	$0.332132\pi$
$132$	−8.67530	−0.755088
$133$	−2.14600	−0.186082
$134$	−14.1957	−1.22632
$135$	17.4245	1.49966
$136$	−4.17136	−0.357691
$137$	−18.7466	−1.60163	−0.800817	−	0.598909i	$-0.795601\pi$
$137$	−18.7466	−1.60163	−0.800817	+	0.598909i	$0.795601\pi$
$138$	0.299188	0.0254685
$139$	−18.6995	−1.58607	−0.793034	−	0.609177i	$-0.791500\pi$
$139$	−18.6995	−1.58607	−0.793034	+	0.609177i	$0.791500\pi$
$140$	−10.6032	−0.896135
$141$	22.8601	1.92517
$142$	27.7974	2.33271
$143$	−22.0065	−1.84028
$144$	−2.74243	−0.228536
$145$	13.1718	1.09386
$146$	19.1718	1.58667
$147$	−4.51757	−0.372603
$148$	6.02521	0.495269
$149$	6.53410	0.535295	0.267647	−	0.963517i	$-0.413754\pi$
$149$	6.53410	0.535295	0.267647	+	0.963517i	$0.413754\pi$
$150$	−31.9568	−2.60926
$151$	−10.2148	−0.831265	−0.415633	−	0.909533i	$-0.636440\pi$
$151$	−10.2148	−0.831265	−0.415633	+	0.909533i	$0.636440\pi$
$152$	−1.26214	−0.102373
$153$	−1.84729	−0.149345
$154$	13.7407	1.10726
$155$	−33.3293	−2.67708
$156$	15.3942	1.23253
$157$	20.3842	1.62684	0.813419	−	0.581678i	$-0.197604\pi$
$157$	20.3842	1.62684	0.813419	+	0.581678i	$0.197604\pi$
$158$	10.8121	0.860164
$159$	8.40866	0.666850
$160$	−24.2032	−1.91343
$161$	−0.187186	−0.0147523
$162$	18.8445	1.48056
$163$	−7.16401	−0.561129	−0.280564	−	0.959835i	$-0.590522\pi$
$163$	−7.16401	−0.561129	−0.280564	+	0.959835i	$0.590522\pi$
$164$	−4.35973	−0.340437
$165$	25.1374	1.95694
$166$	19.7379	1.53196
$167$	−11.2907	−0.873703	−0.436851	−	0.899534i	$-0.643906\pi$
$167$	−11.2907	−0.873703	−0.436851	+	0.899534i	$0.643906\pi$
$168$	5.10974	0.394225
$169$	26.0504	2.00387
$170$	−22.7369	−1.74384
$171$	−0.558942	−0.0427433
$172$	−8.41026	−0.641276
$173$	−11.6872	−0.888560	−0.444280	−	0.895888i	$-0.646540\pi$
$173$	−11.6872	−0.888560	−0.444280	+	0.895888i	$0.646540\pi$
$174$	11.9405	0.905208
$175$	19.9937	1.51138
$176$	17.2786	1.30242
$177$	−2.75367	−0.206978
$178$	−28.9891	−2.17283
$179$	−20.6659	−1.54465	−0.772323	−	0.635230i	$-0.780905\pi$
$179$	−20.6659	−1.54465	−0.772323	+	0.635230i	$0.780905\pi$
$180$	−2.76168	−0.205844
$181$	−17.6451	−1.31155	−0.655776	−	0.754955i	$-0.727659\pi$
$181$	−17.6451	−1.31155	−0.655776	+	0.754955i	$0.727659\pi$
$182$	−24.3828	−1.80737
$183$	−16.7892	−1.24109
$184$	−0.110091	−0.00811600
$185$	−17.4585	−1.28358
$186$	−30.2138	−2.21538
$187$	11.6388	0.851112
$188$	15.8235	1.15405
$189$	−9.88255	−0.718850
$190$	−6.87957	−0.499097
$191$	23.3851	1.69209	0.846044	−	0.533113i	$-0.178978\pi$
$191$	23.3851	1.69209	0.846044	+	0.533113i	$0.178978\pi$
$192$	−3.42849	−0.247430
$193$	−25.7480	−1.85339	−0.926693	−	0.375820i	$-0.877361\pi$
$193$	−25.7480	−1.85339	−0.926693	+	0.375820i	$0.877361\pi$
$194$	7.78816	0.559157
$195$	−44.6061	−3.19431
$196$	−3.12702	−0.223358
$197$	3.15279	0.224627	0.112313	−	0.993673i	$-0.464174\pi$
$197$	3.15279	0.224627	0.112313	+	0.993673i	$0.464174\pi$
$198$	3.57886	0.254339
$199$	−19.5289	−1.38437	−0.692183	−	0.721722i	$-0.743351\pi$
$199$	−19.5289	−1.38437	−0.692183	+	0.721722i	$0.743351\pi$
$200$	11.7590	0.831487
$201$	14.7292	1.03892
$202$	−21.4897	−1.51201
$203$	−7.47056	−0.524331
$204$	−8.14170	−0.570033
$205$	12.6327	0.882304
$206$	−25.1740	−1.75395
$207$	−0.0487539	−0.00338863
$208$	−30.6607	−2.12593
$209$	3.52159	0.243593
$210$	27.8518	1.92195
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	5.82039	0.399746
$213$	−28.8420	−1.97622
$214$	13.3411	0.911978
$215$	24.3694	1.66198
$216$	−5.81228	−0.395476
$217$	18.9032	1.28323
$218$	−29.7218	−2.01302
$219$	−19.8922	−1.34419
$220$	17.3998	1.17310
$221$	−20.6529	−1.38927
$222$	−15.8266	−1.06221
$223$	−20.1782	−1.35123	−0.675616	−	0.737253i	$-0.736122\pi$
$223$	−20.1782	−1.35123	−0.675616	+	0.737253i	$0.736122\pi$
$224$	13.7272	0.917187
$225$	5.20749	0.347166
$226$	−3.15253	−0.209703
$227$	−5.97763	−0.396749	−0.198375	−	0.980126i	$-0.563566\pi$
$227$	−5.97763	−0.396749	−0.198375	+	0.980126i	$0.563566\pi$
$228$	−2.46346	−0.163147
$229$	−20.8444	−1.37744	−0.688718	−	0.725029i	$-0.741826\pi$
$229$	−20.8444	−1.37744	−0.688718	+	0.725029i	$0.741826\pi$
$230$	−0.600074	−0.0395677
$231$	−14.2570	−0.938044
$232$	−4.39370	−0.288461
$233$	17.2137	1.12771	0.563853	−	0.825875i	$-0.309318\pi$
$233$	17.2137	1.12771	0.563853	+	0.825875i	$0.309318\pi$
$234$	−6.35066	−0.415156
$235$	−45.8499	−2.99092
$236$	−1.90606	−0.124074
$237$	−11.2184	−0.728712
$238$	12.8956	0.835895
$239$	−7.56793	−0.489529	−0.244764	−	0.969583i	$-0.578711\pi$
$239$	−7.56793	−0.489529	−0.244764	+	0.969583i	$0.578711\pi$
$240$	35.0228	2.26071
$241$	−17.2128	−1.10877	−0.554386	−	0.832260i	$-0.687047\pi$
$241$	−17.2128	−1.10877	−0.554386	+	0.832260i	$0.687047\pi$
$242$	−2.54833	−0.163813
$243$	−5.73734	−0.368050
$244$	−11.6213	−0.743977
$245$	9.06079	0.578873
$246$	11.4518	0.730141
$247$	−6.24903	−0.397616
$248$	11.1177	0.705972
$249$	−20.4796	−1.29784
$250$	29.6971	1.87821
$251$	20.6586	1.30396	0.651979	−	0.758237i	$-0.273939\pi$
$251$	20.6586	1.30396	0.651979	+	0.758237i	$0.273939\pi$
$252$	1.56633	0.0986694
$253$	0.307172	0.0193117
$254$	14.1861	0.890116
$255$	23.5913	1.47734
$256$	20.8874	1.30547
$257$	−12.2927	−0.766801	−0.383400	−	0.923582i	$-0.625247\pi$
$257$	−12.2927	−0.766801	−0.383400	+	0.923582i	$0.625247\pi$
$258$	22.0915	1.37535
$259$	9.90186	0.615272
$260$	−30.8759	−1.91484
$261$	−1.94576	−0.120439
$262$	−20.9461	−1.29405
$263$	−11.6250	−0.716829	−0.358415	−	0.933562i	$-0.616683\pi$
$263$	−11.6250	−0.716829	−0.358415	+	0.933562i	$0.616683\pi$
$264$	−8.38507	−0.516066
$265$	−16.8651	−1.03601
$266$	3.90185	0.239238
$267$	30.0785	1.84077
$268$	10.1954	0.622782
$269$	4.58603	0.279615	0.139808	−	0.990179i	$-0.455352\pi$
$269$	4.58603	0.279615	0.139808	+	0.990179i	$0.455352\pi$
$270$	−31.6811	−1.92805
$271$	−13.9745	−0.848891	−0.424446	−	0.905453i	$-0.639531\pi$
$271$	−13.9745	−0.848891	−0.424446	+	0.905453i	$0.639531\pi$
$272$	16.2158	0.983228
$273$	25.2990	1.53117
$274$	34.0850	2.05915
$275$	−32.8096	−1.97849
$276$	−0.214877	−0.0129341
$277$	24.6601	1.48168	0.740840	−	0.671682i	$-0.234428\pi$
$277$	24.6601	1.48168	0.740840	+	0.671682i	$0.234428\pi$
$278$	33.9992	2.03914
$279$	4.92348	0.294761
$280$	−10.2485	−0.612465
$281$	30.7544	1.83465	0.917327	−	0.398135i	$-0.130342\pi$
$281$	30.7544	1.83465	0.917327	+	0.398135i	$0.130342\pi$
$282$	−41.5640	−2.47510
$283$	−14.4297	−0.857757	−0.428878	−	0.903362i	$-0.641091\pi$
$283$	−14.4297	−0.857757	−0.428878	+	0.903362i	$0.641091\pi$
$284$	−19.9641	−1.18465
$285$	7.13809	0.422824
$286$	40.0121	2.36596
$287$	−7.16480	−0.422925
$288$	3.57534	0.210679
$289$	−6.07708	−0.357475
$290$	−23.9488	−1.40632
$291$	−8.08082	−0.473706
$292$	−13.7692	−0.805782
$293$	−9.80388	−0.572749	−0.286374	−	0.958118i	$-0.592450\pi$
$293$	−9.80388	−0.572749	−0.286374	+	0.958118i	$0.592450\pi$
$294$	8.21382	0.479040
$295$	5.52297	0.321560
$296$	5.82364	0.338492
$297$	16.2172	0.941020
$298$	−11.8803	−0.688205
$299$	−0.545074	−0.0315225
$300$	22.9513	1.32510
$301$	−13.8215	−0.796657
$302$	18.5724	1.06872
$303$	22.2972	1.28094
$304$	4.90647	0.281405
$305$	33.6737	1.92815
$306$	3.35874	0.192006
$307$	−15.6236	−0.891686	−0.445843	−	0.895111i	$-0.647096\pi$
$307$	−15.6236	−0.891686	−0.445843	+	0.895111i	$0.647096\pi$
$308$	−9.86858	−0.562314
$309$	26.1199	1.48591
$310$	60.5992	3.44180
$311$	2.66454	0.151092	0.0755460	−	0.997142i	$-0.475930\pi$
$311$	2.66454	0.151092	0.0755460	+	0.997142i	$0.475930\pi$
$312$	14.8792	0.842371
$313$	−23.4143	−1.32345	−0.661727	−	0.749745i	$-0.730176\pi$
$313$	−23.4143	−1.32345	−0.661727	+	0.749745i	$0.730176\pi$
$314$	−37.0625	−2.09156
$315$	−4.53857	−0.255719
$316$	−7.76525	−0.436829
$317$	−14.2156	−0.798429	−0.399215	−	0.916858i	$-0.630717\pi$
$317$	−14.2156	−0.798429	−0.399215	+	0.916858i	$0.630717\pi$
$318$	−15.2886	−0.857340
$319$	12.2592	0.686382
$320$	6.87644	0.384405
$321$	−13.8424	−0.772608
$322$	0.340341	0.0189664
$323$	3.30498	0.183894
$324$	−13.5341	−0.751896
$325$	58.2203	3.22948
$326$	13.0256	0.721419
$327$	30.8387	1.70538
$328$	−4.21388	−0.232672
$329$	26.0045	1.43367
$330$	−45.7046	−2.51596
$331$	29.3083	1.61093	0.805465	−	0.592644i	$-0.201916\pi$
$331$	29.3083	1.61093	0.805465	+	0.592644i	$0.201916\pi$
$332$	−14.1757	−0.777995
$333$	2.57901	0.141329
$334$	20.5287	1.12328
$335$	−29.5420	−1.61405
$336$	−19.8637	−1.08365
$337$	8.76941	0.477700	0.238850	−	0.971056i	$-0.423230\pi$
$337$	8.76941	0.477700	0.238850	+	0.971056i	$0.423230\pi$
$338$	−47.3646	−2.57629
$339$	3.27099	0.177656
$340$	16.3296	0.885599
$341$	−31.0201	−1.67983
$342$	1.01626	0.0549533
$343$	−20.1610	−1.08859
$344$	−8.12890	−0.438281
$345$	0.622623	0.0335209
$346$	21.2496	1.14238
$347$	−35.5073	−1.90613	−0.953067	−	0.302760i	$-0.902092\pi$
$347$	−35.5073	−1.90613	−0.953067	+	0.302760i	$0.902092\pi$
$348$	−8.57568	−0.459705
$349$	−15.4167	−0.825237	−0.412618	−	0.910904i	$-0.635386\pi$
$349$	−15.4167	−0.825237	−0.412618	+	0.910904i	$0.635386\pi$
$350$	−36.3524	−1.94312
$351$	−28.7774	−1.53602
$352$	−22.5263	−1.20065
$353$	−20.1960	−1.07492	−0.537461	−	0.843289i	$-0.680617\pi$
$353$	−20.1960	−1.07492	−0.537461	+	0.843289i	$0.680617\pi$
$354$	5.00670	0.266103
$355$	57.8477	3.07024
$356$	20.8200	1.10346
$357$	−13.3801	−0.708152
$358$	37.5747	1.98588
$359$	21.3925	1.12905	0.564526	−	0.825415i	$-0.309059\pi$
$359$	21.3925	1.12905	0.564526	+	0.825415i	$0.309059\pi$
$360$	−2.66929	−0.140684
$361$	1.00000	0.0526316
$362$	32.0823	1.68621
$363$	2.64408	0.138778
$364$	17.5117	0.917863
$365$	39.8974	2.08833
$366$	30.5260	1.59562
$367$	8.09352	0.422478	0.211239	−	0.977434i	$-0.432250\pi$
$367$	8.09352	0.422478	0.211239	+	0.977434i	$0.432250\pi$
$368$	0.427969	0.0223094
$369$	−1.86612	−0.0971465
$370$	31.7430	1.65024
$371$	9.56526	0.496604
$372$	21.6996	1.12507
$373$	−8.76341	−0.453752	−0.226876	−	0.973924i	$-0.572851\pi$
$373$	−8.76341	−0.453752	−0.226876	+	0.973924i	$0.572851\pi$
$374$	−21.1616	−1.09424
$375$	−30.8130	−1.59118
$376$	15.2941	0.788736
$377$	−21.7538	−1.12038
$378$	17.9684	0.924194
$379$	−14.3162	−0.735371	−0.367686	−	0.929950i	$-0.619850\pi$
$379$	−14.3162	−0.735371	−0.367686	+	0.929950i	$0.619850\pi$
$380$	4.94091	0.253463
$381$	−14.7192	−0.754087
$382$	−42.5187	−2.17544
$383$	16.6973	0.853191	0.426595	−	0.904443i	$-0.359713\pi$
$383$	16.6973	0.853191	0.426595	+	0.904443i	$0.359713\pi$
$384$	−17.9010	−0.913508
$385$	28.5950	1.45734
$386$	46.8149	2.38282
$387$	−3.59990	−0.182993
$388$	−5.59346	−0.283965
$389$	3.40403	0.172591	0.0862956	−	0.996270i	$-0.472497\pi$
$389$	3.40403	0.172591	0.0862956	+	0.996270i	$0.472497\pi$
$390$	81.1025	4.10679
$391$	0.288279	0.0145789
$392$	−3.02241	−0.152655
$393$	21.7332	1.09629
$394$	−5.73238	−0.288793
$395$	22.5005	1.13212
$396$	−2.57034	−0.129164
$397$	14.0888	0.707096	0.353548	−	0.935416i	$-0.384975\pi$
$397$	14.0888	0.707096	0.353548	+	0.935416i	$0.384975\pi$
$398$	35.5073	1.77982
$399$	−4.04847	−0.202677
$400$	−45.7121	−2.28561
$401$	−33.0010	−1.64799	−0.823995	−	0.566596i	$-0.808260\pi$
$401$	−33.0010	−1.64799	−0.823995	+	0.566596i	$0.808260\pi$
$402$	−26.7805	−1.33569
$403$	55.0450	2.74199
$404$	15.4339	0.767866
$405$	39.2163	1.94867
$406$	13.5829	0.674109
$407$	−16.2489	−0.805429
$408$	−7.86933	−0.389590
$409$	−37.2366	−1.84123	−0.920615	−	0.390471i	$-0.872312\pi$
$409$	−37.2366	−1.84123	−0.920615	+	0.390471i	$0.872312\pi$
$410$	−22.9686	−1.13434
$411$	−35.3658	−1.74447
$412$	18.0799	0.890735
$413$	−3.13243	−0.154137
$414$	0.0886441	0.00435662
$415$	41.0754	2.01631
$416$	39.9727	1.95982
$417$	−35.2768	−1.72751
$418$	−6.40292	−0.313177
$419$	37.5041	1.83219	0.916096	−	0.400958i	$-0.131323\pi$
$419$	37.5041	1.83219	0.916096	+	0.400958i	$0.131323\pi$
$420$	−20.0031	−0.976053
$421$	−4.93622	−0.240577	−0.120288	−	0.992739i	$-0.538382\pi$
$421$	−4.93622	−0.240577	−0.120288	+	0.992739i	$0.538382\pi$
$422$	1.81819	0.0885083
$423$	6.77304	0.329317
$424$	5.62567	0.273207
$425$	−30.7916	−1.49361
$426$	52.4403	2.54074
$427$	−19.0985	−0.924242
$428$	−9.58157	−0.463143
$429$	−41.5156	−2.00439
$430$	−44.3083	−2.13674
$431$	−18.7461	−0.902968	−0.451484	−	0.892279i	$-0.649105\pi$
$431$	−18.7461	−0.902968	−0.451484	+	0.892279i	$0.649105\pi$
$432$	22.5948	1.08709
$433$	−40.2284	−1.93326	−0.966628	−	0.256186i	$-0.917534\pi$
$433$	−40.2284	−1.93326	−0.966628	+	0.256186i	$0.917534\pi$
$434$	−34.3697	−1.64980
$435$	24.8488	1.19141
$436$	21.3462	1.02230
$437$	0.0872254	0.00417256
$438$	36.1679	1.72817
$439$	−4.00275	−0.191041	−0.0955204	−	0.995427i	$-0.530452\pi$
$439$	−4.00275	−0.191041	−0.0955204	+	0.995427i	$0.530452\pi$
$440$	16.8177	0.801755
$441$	−1.33848	−0.0637371
$442$	37.5510	1.78612
$443$	−17.3114	−0.822491	−0.411245	−	0.911525i	$-0.634906\pi$
$443$	−17.3114	−0.822491	−0.411245	+	0.911525i	$0.634906\pi$
$444$	11.3666	0.539437
$445$	−60.3277	−2.85981
$446$	36.6879	1.73722
$447$	12.3267	0.583032
$448$	−3.90007	−0.184261
$449$	−25.2442	−1.19135	−0.595675	−	0.803226i	$-0.703115\pi$
$449$	−25.2442	−1.19135	−0.595675	+	0.803226i	$0.703115\pi$
$450$	−9.46823	−0.446337
$451$	11.7574	0.553635
$452$	2.26415	0.106497
$453$	−19.2703	−0.905398
$454$	10.8685	0.510083
$455$	−50.7417	−2.37881
$456$	−2.38105	−0.111503
$457$	6.40939	0.299819	0.149909	−	0.988700i	$-0.452102\pi$
$457$	6.40939	0.299819	0.149909	+	0.988700i	$0.452102\pi$
$458$	37.8991	1.77091
$459$	15.2198	0.710398
$460$	0.430973	0.0200942
$461$	13.0392	0.607296	0.303648	−	0.952784i	$-0.401795\pi$
$461$	13.0392	0.607296	0.303648	+	0.952784i	$0.401795\pi$
$462$	25.9221	1.20600
$463$	3.34924	0.155652	0.0778261	−	0.996967i	$-0.475202\pi$
$463$	3.34924	0.155652	0.0778261	+	0.996967i	$0.475202\pi$
$464$	17.0802	0.792926
$465$	−62.8763	−2.91582
$466$	−31.2978	−1.44984
$467$	2.15818	0.0998687	0.0499343	−	0.998753i	$-0.484099\pi$
$467$	2.15818	0.0998687	0.0499343	+	0.998753i	$0.484099\pi$
$468$	4.56105	0.210834
$469$	16.7552	0.773681
$470$	83.3641	3.84530
$471$	38.4552	1.77192
$472$	−1.84230	−0.0847985
$473$	22.6810	1.04287
$474$	20.3972	0.936874
$475$	−9.31670	−0.427480
$476$	−9.26159	−0.424504
$477$	2.49134	0.114071
$478$	13.7600	0.629366
$479$	−15.5214	−0.709192	−0.354596	−	0.935020i	$-0.615382\pi$
$479$	−15.5214	−0.709192	−0.354596	+	0.935020i	$0.615382\pi$
$480$	−45.6597	−2.08407
$481$	28.8336	1.31470
$482$	31.2961	1.42550
$483$	−0.353130	−0.0160680
$484$	1.83021	0.0831913
$485$	16.2075	0.735945
$486$	10.4316	0.473186
$487$	−5.32544	−0.241319	−0.120659	−	0.992694i	$-0.538501\pi$
$487$	−5.32544	−0.241319	−0.120659	+	0.992694i	$0.538501\pi$
$488$	−11.2325	−0.508472
$489$	−13.5150	−0.611171
$490$	−16.4743	−0.744232
$491$	−19.2453	−0.868526	−0.434263	−	0.900786i	$-0.642991\pi$
$491$	−19.2453	−0.868526	−0.434263	+	0.900786i	$0.642991\pi$
$492$	−8.22469	−0.370798
$493$	11.5051	0.518166
$494$	11.3619	0.511198
$495$	7.44777	0.334752
$496$	−43.2190	−1.94059
$497$	−32.8092	−1.47169
$498$	37.2358	1.66858
$499$	41.3501	1.85108	0.925542	−	0.378644i	$-0.123610\pi$
$499$	41.3501	1.85108	0.925542	+	0.378644i	$0.123610\pi$
$500$	−21.3284	−0.953837
$501$	−21.3001	−0.951620
$502$	−37.5613	−1.67644
$503$	−25.2654	−1.12653	−0.563264	−	0.826277i	$-0.690455\pi$
$503$	−25.2654	−1.12653	−0.563264	+	0.826277i	$0.690455\pi$
$504$	1.51393	0.0674357
$505$	−44.7211	−1.99006
$506$	−0.558498	−0.0248283
$507$	49.1444	2.18258
$508$	−10.1885	−0.452040
$509$	14.2475	0.631511	0.315756	−	0.948841i	$-0.397742\pi$
$509$	14.2475	0.631511	0.315756	+	0.948841i	$0.397742\pi$
$510$	−42.8935	−1.89936
$511$	−22.6284	−1.00102
$512$	−18.9995	−0.839669
$513$	4.60509	0.203320
$514$	22.3506	0.985842
$515$	−52.3881	−2.30850
$516$	−15.8661	−0.698465
$517$	−42.6732	−1.87677
$518$	−18.0035	−0.791029
$519$	−22.0481	−0.967802
$520$	−29.8430	−1.30870
$521$	33.8855	1.48455	0.742275	−	0.670095i	$-0.233747\pi$
$521$	33.8855	1.48455	0.742275	+	0.670095i	$0.233747\pi$
$522$	3.53777	0.154844
$523$	19.3119	0.844449	0.422225	−	0.906491i	$-0.361249\pi$
$523$	19.3119	0.844449	0.422225	+	0.906491i	$0.361249\pi$
$524$	15.0435	0.657178
$525$	37.7184	1.64617
$526$	21.1365	0.921597
$527$	−29.1122	−1.26815
$528$	32.5963	1.41857
$529$	−22.9924	−0.999669
$530$	30.6639	1.33196
$531$	−0.815864	−0.0354055
$532$	−2.80231	−0.121496
$533$	−20.8634	−0.903696
$534$	−54.6884	−2.36660
$535$	27.7634	1.20032
$536$	9.85430	0.425641
$537$	−38.9866	−1.68240
$538$	−8.33829	−0.359489
$539$	8.43302	0.363236
$540$	22.7534	0.979149
$541$	13.9238	0.598633	0.299316	−	0.954154i	$-0.403241\pi$
$541$	13.9238	0.598633	0.299316	+	0.954154i	$0.403241\pi$
$542$	25.4084	1.09138
$543$	−33.2878	−1.42852
$544$	−21.1408	−0.906402
$545$	−61.8525	−2.64947
$546$	−45.9985	−1.96855
$547$	3.43145	0.146718	0.0733591	−	0.997306i	$-0.476628\pi$
$547$	3.43145	0.146718	0.0733591	+	0.997306i	$0.476628\pi$
$548$	−24.4799	−1.04573
$549$	−4.97434	−0.212300
$550$	59.6541	2.54366
$551$	3.48115	0.148302
$552$	−0.207688	−0.00883979
$553$	−12.7615	−0.542673
$554$	−44.8368	−1.90493
$555$	−32.9358	−1.39805
$556$	−24.4183	−1.03556
$557$	−41.7985	−1.77106	−0.885530	−	0.464582i	$-0.846205\pi$
$557$	−41.7985	−1.77106	−0.885530	+	0.464582i	$0.846205\pi$
$558$	−8.95183	−0.378961
$559$	−40.2473	−1.70228
$560$	39.8402	1.68355
$561$	21.9568	0.927015
$562$	−55.9174	−2.35873
$563$	−27.1922	−1.14601	−0.573007	−	0.819551i	$-0.694223\pi$
$563$	−27.1922	−1.14601	−0.573007	+	0.819551i	$0.694223\pi$
$564$	29.8513	1.25697
$565$	−6.56056	−0.276005
$566$	26.2360	1.10278
$567$	−22.2421	−0.934079
$568$	−19.2962	−0.809652
$569$	−2.74307	−0.114995	−0.0574977	−	0.998346i	$-0.518312\pi$
$569$	−2.74307	−0.114995	−0.0574977	+	0.998346i	$0.518312\pi$
$570$	−12.9784	−0.543606
$571$	20.2442	0.847192	0.423596	−	0.905851i	$-0.360768\pi$
$571$	20.2442	0.847192	0.423596	+	0.905851i	$0.360768\pi$
$572$	−28.7367	−1.20154
$573$	44.1164	1.84299
$574$	13.0270	0.543736
$575$	−0.812654	−0.0338900
$576$	−1.01580	−0.0423251
$577$	−4.21073	−0.175295	−0.0876475	−	0.996152i	$-0.527935\pi$
$577$	−4.21073	−0.175295	−0.0876475	+	0.996152i	$0.527935\pi$
$578$	11.0493	0.459590
$579$	−48.5741	−2.01867
$580$	17.2000	0.714193
$581$	−23.2965	−0.966502
$582$	14.6925	0.609023
$583$	−15.6966	−0.650085
$584$	−13.3086	−0.550713
$585$	−13.2160	−0.546415
$586$	17.8254	0.736359
$587$	17.0714	0.704614	0.352307	−	0.935885i	$-0.385397\pi$
$587$	17.0714	0.704614	0.352307	+	0.935885i	$0.385397\pi$
$588$	−5.89917	−0.243278
$589$	−8.80857	−0.362951
$590$	−10.0418	−0.413416
$591$	5.94779	0.244659
$592$	−22.6389	−0.930453
$593$	−25.5945	−1.05104	−0.525520	−	0.850781i	$-0.676129\pi$
$593$	−25.5945	−1.05104	−0.525520	+	0.850781i	$0.676129\pi$
$594$	−29.4861	−1.20983
$595$	26.8362	1.10018
$596$	8.53241	0.349501
$597$	−36.8416	−1.50782
$598$	0.991050	0.0405271
$599$	33.9046	1.38530	0.692652	−	0.721272i	$-0.256442\pi$
$599$	33.9046	1.38530	0.692652	+	0.721272i	$0.256442\pi$
$600$	22.1835	0.905639
$601$	−11.6557	−0.475446	−0.237723	−	0.971333i	$-0.576401\pi$
$601$	−11.6557	−0.475446	−0.237723	+	0.971333i	$0.576401\pi$
$602$	25.1301	1.02423
$603$	4.36400	0.177716
$604$	−13.3387	−0.542744
$605$	−5.30318	−0.215605
$606$	−40.5407	−1.64685
$607$	−24.4378	−0.991900	−0.495950	−	0.868351i	$-0.665180\pi$
$607$	−24.4378	−0.991900	−0.495950	+	0.868351i	$0.665180\pi$
$608$	−6.39663	−0.259418
$609$	−14.0933	−0.571091
$610$	−61.2252	−2.47894
$611$	75.7233	3.06344
$612$	−2.41225	−0.0975092
$613$	15.7222	0.635016	0.317508	−	0.948256i	$-0.397154\pi$
$613$	15.7222	0.635016	0.317508	+	0.948256i	$0.397154\pi$
$614$	28.4067	1.14640
$615$	23.8317	0.960988
$616$	−9.53843	−0.384314
$617$	20.5305	0.826529	0.413264	−	0.910611i	$-0.364389\pi$
$617$	20.5305	0.826529	0.413264	+	0.910611i	$0.364389\pi$
$618$	−47.4911	−1.91037
$619$	2.77815	0.111663	0.0558317	−	0.998440i	$-0.482219\pi$
$619$	2.77815	0.111663	0.0558317	+	0.998440i	$0.482219\pi$
$620$	−43.5223	−1.74790
$621$	0.401681	0.0161189
$622$	−4.84464	−0.194253
$623$	34.2157	1.37082
$624$	−57.8418	−2.31553
$625$	15.2174	0.608698
$626$	42.5717	1.70151
$627$	6.64353	0.265317
$628$	26.6183	1.06218
$629$	−15.2495	−0.608038
$630$	8.25199	0.328767
$631$	−25.7245	−1.02407	−0.512037	−	0.858963i	$-0.671109\pi$
$631$	−25.7245	−1.02407	−0.512037	+	0.858963i	$0.671109\pi$
$632$	−7.50547	−0.298552
$633$	−1.88652	−0.0749823
$634$	25.8468	1.02651
$635$	29.5220	1.17154
$636$	10.9803	0.435395
$637$	−14.9643	−0.592908
$638$	−22.2895	−0.882451
$639$	−8.54537	−0.338050
$640$	35.9037	1.41922
$641$	2.04248	0.0806733	0.0403366	−	0.999186i	$-0.487157\pi$
$641$	2.04248	0.0806733	0.0403366	+	0.999186i	$0.487157\pi$
$642$	25.1682	0.993308
$643$	21.4216	0.844785	0.422393	−	0.906413i	$-0.361190\pi$
$643$	21.4216	0.844785	0.422393	+	0.906413i	$0.361190\pi$
$644$	−0.244433	−0.00963200
$645$	45.9733	1.81020
$646$	−6.00910	−0.236425
$647$	47.7010	1.87532	0.937660	−	0.347554i	$-0.112988\pi$
$647$	47.7010	1.87532	0.937660	+	0.347554i	$0.112988\pi$
$648$	−13.0814	−0.513884
$649$	5.14031	0.201775
$650$	−105.856	−4.15201
$651$	35.6612	1.39767
$652$	−9.35496	−0.366368
$653$	−25.5713	−1.00068	−0.500340	−	0.865829i	$-0.666792\pi$
$653$	−25.5713	−1.00068	−0.500340	+	0.865829i	$0.666792\pi$
$654$	−56.0707	−2.19254
$655$	−43.5898	−1.70319
$656$	16.3811	0.639574
$657$	−5.89373	−0.229936
$658$	−47.2811	−1.84321
$659$	17.7872	0.692891	0.346445	−	0.938070i	$-0.387389\pi$
$659$	17.7872	0.692891	0.346445	+	0.938070i	$0.387389\pi$
$660$	32.8251	1.27771
$661$	23.7433	0.923508	0.461754	−	0.887008i	$-0.347220\pi$
$661$	23.7433	0.923508	0.461754	+	0.887008i	$0.347220\pi$
$662$	−53.2881	−2.07110
$663$	−38.9621	−1.51316
$664$	−13.7015	−0.531721
$665$	8.11993	0.314877
$666$	−4.68914	−0.181700
$667$	0.303645	0.0117572
$668$	−14.7437	−0.570452
$669$	−38.0665	−1.47174
$670$	53.7130	2.07511
$671$	31.3406	1.20989
$672$	25.8966	0.998982
$673$	34.2055	1.31853	0.659263	−	0.751912i	$-0.270868\pi$
$673$	34.2055	1.31853	0.659263	+	0.751912i	$0.270868\pi$
$674$	−15.9445	−0.614158
$675$	−42.9043	−1.65139
$676$	34.0172	1.30836
$677$	30.8872	1.18709	0.593546	−	0.804800i	$-0.297727\pi$
$677$	30.8872	1.18709	0.593546	+	0.804800i	$0.297727\pi$
$678$	−5.94730	−0.228405
$679$	−9.19233	−0.352769
$680$	15.7833	0.605263
$681$	−11.2769	−0.432131
$682$	56.4006	2.15969
$683$	−8.81030	−0.337117	−0.168558	−	0.985692i	$-0.553911\pi$
$683$	−8.81030	−0.337117	−0.168558	+	0.985692i	$0.553911\pi$
$684$	−0.729881	−0.0279077
$685$	70.9325	2.71019
$686$	36.6566	1.39955
$687$	−39.3233	−1.50028
$688$	31.6004	1.20476
$689$	27.8534	1.06113
$690$	−1.13205	−0.0430964
$691$	−44.7478	−1.70229	−0.851143	−	0.524934i	$-0.824090\pi$
$691$	−44.7478	−1.70229	−0.851143	+	0.524934i	$0.824090\pi$
$692$	−15.2614	−0.580153
$693$	−4.22411	−0.160461
$694$	64.5592	2.45063
$695$	70.7540	2.68385
$696$	−8.28879	−0.314186
$697$	11.0343	0.417952
$698$	28.0305	1.06097
$699$	32.4739	1.22828
$700$	26.1083	0.986801
$701$	−26.9443	−1.01767	−0.508836	−	0.860863i	$-0.669924\pi$
$701$	−26.9443	−1.01767	−0.508836	+	0.860863i	$0.669924\pi$
$702$	52.3228	1.97480
$703$	−4.61409	−0.174024
$704$	6.40001	0.241209
$705$	−86.4966	−3.25765
$706$	36.7202	1.38198
$707$	25.3642	0.953920
$708$	−3.59581	−0.135139
$709$	25.5134	0.958174	0.479087	−	0.877767i	$-0.340968\pi$
$709$	25.5134	0.958174	0.479087	+	0.877767i	$0.340968\pi$
$710$	−105.178	−3.94727
$711$	−3.32381	−0.124653
$712$	20.1235	0.754159
$713$	−0.768331	−0.0287742
$714$	24.3277	0.910440
$715$	83.2669	3.11401
$716$	−26.9861	−1.00852
$717$	−14.2770	−0.533185
$718$	−38.8957	−1.45157
$719$	−36.2519	−1.35197	−0.675984	−	0.736916i	$-0.736281\pi$
$719$	−36.2519	−1.35197	−0.675984	+	0.736916i	$0.736281\pi$
$720$	10.3767	0.386715
$721$	29.7127	1.10656
$722$	−1.81819	−0.0676661
$723$	−32.4722	−1.20765
$724$	−23.0415	−0.856330
$725$	−32.4328	−1.20453
$726$	−4.80746	−0.178421
$727$	50.7667	1.88283	0.941417	−	0.337244i	$-0.109495\pi$
$727$	50.7667	1.88283	0.941417	+	0.337244i	$0.109495\pi$
$728$	16.9259	0.627315
$729$	20.2697	0.750728
$730$	−72.5412	−2.68487
$731$	21.2860	0.787290
$732$	−21.9238	−0.810325
$733$	−15.4256	−0.569758	−0.284879	−	0.958564i	$-0.591953\pi$
$733$	−15.4256	−0.569758	−0.284879	+	0.958564i	$0.591953\pi$
$734$	−14.7156	−0.543162
$735$	17.0933	0.630497
$736$	−0.557949	−0.0205663
$737$	−27.4952	−1.01280
$738$	3.39297	0.124897
$739$	4.15378	0.152799	0.0763996	−	0.997077i	$-0.475658\pi$
$739$	4.15378	0.152799	0.0763996	+	0.997077i	$0.475658\pi$
$740$	−22.7978	−0.838065
$741$	−11.7889	−0.433076
$742$	−17.3915	−0.638462
$743$	−24.4746	−0.897887	−0.448944	−	0.893560i	$-0.648200\pi$
$743$	−24.4746	−0.897887	−0.448944	+	0.893560i	$0.648200\pi$
$744$	20.9736	0.768931
$745$	−24.7234	−0.905794
$746$	15.9336	0.583369
$747$	−6.06774	−0.222007
$748$	15.1982	0.555703
$749$	−15.7464	−0.575362
$750$	56.0240	2.04571
$751$	−1.63410	−0.0596291	−0.0298145	−	0.999555i	$-0.509492\pi$
$751$	−1.63410	−0.0596291	−0.0298145	+	0.999555i	$0.509492\pi$
$752$	−59.4547	−2.16809
$753$	38.9728	1.42025
$754$	39.5526	1.44042
$755$	38.6500	1.40662
$756$	−12.9049	−0.469347
$757$	12.8912	0.468539	0.234269	−	0.972172i	$-0.424730\pi$
$757$	12.8912	0.468539	0.234269	+	0.972172i	$0.424730\pi$
$758$	26.0295	0.945435
$759$	0.579485	0.0210340
$760$	4.77562	0.173230
$761$	−25.5146	−0.924905	−0.462452	−	0.886644i	$-0.653030\pi$
$761$	−25.5146	−0.924905	−0.462452	+	0.886644i	$0.653030\pi$
$762$	26.7623	0.969497
$763$	35.0805	1.27000
$764$	30.5369	1.10479
$765$	6.98968	0.252713
$766$	−30.3589	−1.09691
$767$	−9.12145	−0.329356
$768$	39.4045	1.42189
$769$	4.82978	0.174166	0.0870832	−	0.996201i	$-0.472245\pi$
$769$	4.82978	0.174166	0.0870832	+	0.996201i	$0.472245\pi$
$770$	−51.9913	−1.87364
$771$	−23.1905	−0.835184
$772$	−33.6225	−1.21010
$773$	17.5314	0.630561	0.315280	−	0.948999i	$-0.397901\pi$
$773$	17.5314	0.630561	0.315280	+	0.948999i	$0.397901\pi$
$774$	6.54531	0.235266
$775$	82.0668	2.94793
$776$	−5.40633	−0.194076
$777$	18.6800	0.670142
$778$	−6.18918	−0.221893
$779$	3.33867	0.119620
$780$	−58.2479	−2.08561
$781$	53.8397	1.92654
$782$	−0.524146	−0.0187434
$783$	16.0310	0.572902
$784$	11.7494	0.419620
$785$	−77.1286	−2.75284
$786$	−39.5151	−1.40946
$787$	5.85732	0.208791	0.104395	−	0.994536i	$-0.466709\pi$
$787$	5.85732	0.208791	0.104395	+	0.994536i	$0.466709\pi$
$788$	4.11700	0.146662
$789$	−21.9308	−0.780757
$790$	−40.9102	−1.45552
$791$	3.72092	0.132301
$792$	−2.48435	−0.0882776
$793$	−55.6137	−1.97490
$794$	−25.6161	−0.909083
$795$	−31.8162	−1.12840
$796$	−25.5013	−0.903871
$797$	11.2661	0.399067	0.199534	−	0.979891i	$-0.436057\pi$
$797$	11.2661	0.399067	0.199534	+	0.979891i	$0.436057\pi$
$798$	7.36090	0.260573
$799$	−40.0485	−1.41682
$800$	59.5955	2.10702
$801$	8.91172	0.314880
$802$	60.0022	2.11875
$803$	37.1331	1.31040
$804$	19.2337	0.678322
$805$	0.708264	0.0249630
$806$	−100.082	−3.52525
$807$	8.65162	0.304552
$808$	14.9176	0.524799
$809$	50.6521	1.78083	0.890417	−	0.455145i	$-0.150413\pi$
$809$	50.6521	1.78083	0.890417	+	0.455145i	$0.150413\pi$
$810$	−71.3027	−2.50532
$811$	23.0634	0.809865	0.404932	−	0.914347i	$-0.367295\pi$
$811$	23.0634	0.809865	0.404932	+	0.914347i	$0.367295\pi$
$812$	−9.75526	−0.342342
$813$	−26.3631	−0.924596
$814$	29.5437	1.03551
$815$	27.1068	0.949509
$816$	30.5914	1.07091
$817$	6.44056	0.225327
$818$	67.7033	2.36719
$819$	7.49566	0.261919
$820$	16.4961	0.576068
$821$	13.7686	0.480526	0.240263	−	0.970708i	$-0.422766\pi$
$821$	13.7686	0.480526	0.240263	+	0.970708i	$0.422766\pi$
$822$	64.3019	2.24279
$823$	31.4555	1.09647	0.548235	−	0.836324i	$-0.315300\pi$
$823$	31.4555	1.09647	0.548235	+	0.836324i	$0.315300\pi$
$824$	17.4751	0.608774
$825$	−61.8958	−2.15493
$826$	5.69537	0.198167
$827$	−0.989139	−0.0343957	−0.0171979	−	0.999852i	$-0.505475\pi$
$827$	−0.989139	−0.0343957	−0.0171979	+	0.999852i	$0.505475\pi$
$828$	−0.0636642	−0.00221248
$829$	18.0885	0.628241	0.314121	−	0.949383i	$-0.398290\pi$
$829$	18.0885	0.628241	0.314121	+	0.949383i	$0.398290\pi$
$830$	−74.6830	−2.59228
$831$	46.5216	1.61382
$832$	−11.3568	−0.393725
$833$	7.91433	0.274215
$834$	64.1401	2.22099
$835$	42.7212	1.47843
$836$	4.59858	0.159045
$837$	−40.5643	−1.40211
$838$	−68.1896	−2.35557
$839$	−27.6926	−0.956054	−0.478027	−	0.878345i	$-0.658648\pi$
$839$	−27.6926	−0.956054	−0.478027	+	0.878345i	$0.658648\pi$
$840$	−19.3340	−0.667085
$841$	−16.8816	−0.582124
$842$	8.97500	0.309299
$843$	58.0187	1.99827
$844$	−1.30583	−0.0449484
$845$	−98.5678	−3.39084
$846$	−12.3147	−0.423388
$847$	3.00778	0.103348
$848$	−21.8693	−0.750996
$849$	−27.2219	−0.934252
$850$	55.9850	1.92027
$851$	−0.402466	−0.0137964
$852$	−37.6626	−1.29030
$853$	−27.9079	−0.955547	−0.477773	−	0.878483i	$-0.658556\pi$
$853$	−27.9079	−0.955547	−0.477773	+	0.878483i	$0.658556\pi$
$854$	34.7248	1.18826
$855$	2.11489	0.0723278
$856$	−9.26103	−0.316535
$857$	10.0300	0.342620	0.171310	−	0.985217i	$-0.445200\pi$
$857$	10.0300	0.342620	0.171310	+	0.985217i	$0.445200\pi$
$858$	75.4834	2.57696
$859$	−26.9727	−0.920298	−0.460149	−	0.887842i	$-0.652204\pi$
$859$	−26.9727	−0.920298	−0.460149	+	0.887842i	$0.652204\pi$
$860$	31.8222	1.08513
$861$	−13.5165	−0.460642
$862$	34.0840	1.16091
$863$	−22.7753	−0.775280	−0.387640	−	0.921811i	$-0.626710\pi$
$863$	−22.7753	−0.775280	−0.387640	+	0.921811i	$0.626710\pi$
$864$	−29.4571	−1.00215
$865$	44.2213	1.50357
$866$	73.1431	2.48550
$867$	−11.4645	−0.389355
$868$	24.6843	0.837841
$869$	20.9415	0.710392
$870$	−45.1798	−1.53174
$871$	48.7899	1.65318
$872$	20.6321	0.698691
$873$	−2.39421	−0.0810316
$874$	−0.158593	−0.00536448
$875$	−35.0513	−1.18495
$876$	−25.9758	−0.877642
$877$	40.4512	1.36594	0.682970	−	0.730447i	$-0.260688\pi$
$877$	40.4512	1.36594	0.682970	+	0.730447i	$0.260688\pi$
$878$	7.27777	0.245613
$879$	−18.4952	−0.623827
$880$	−65.3776	−2.20388
$881$	−40.3640	−1.35990	−0.679948	−	0.733260i	$-0.737998\pi$
$881$	−40.3640	−1.35990	−0.679948	+	0.733260i	$0.737998\pi$
$882$	2.43361	0.0819440
$883$	−43.6647	−1.46943	−0.734717	−	0.678373i	$-0.762685\pi$
$883$	−43.6647	−1.46943	−0.734717	+	0.678373i	$0.762685\pi$
$884$	−26.9692	−0.907071
$885$	10.4192	0.350237
$886$	31.4755	1.05744
$887$	46.1981	1.55118	0.775590	−	0.631238i	$-0.217453\pi$
$887$	46.1981	1.55118	0.775590	+	0.631238i	$0.217453\pi$
$888$	10.9864	0.368679
$889$	−16.7438	−0.561569
$890$	109.687	3.67673
$891$	36.4992	1.22277
$892$	−26.3492	−0.882238
$893$	−12.1176	−0.405501
$894$	−22.4123	−0.749580
$895$	78.1946	2.61376
$896$	−20.3633	−0.680290
$897$	−1.02829	−0.0343336
$898$	45.8989	1.53167
$899$	−30.6639	−1.02270
$900$	6.80008	0.226669
$901$	−14.7311	−0.490765
$902$	−21.3773	−0.711785
$903$	−26.0744	−0.867703
$904$	2.18840	0.0727852
$905$	66.7646	2.21933
$906$	35.0371	1.16403
$907$	39.4830	1.31101	0.655506	−	0.755190i	$-0.272456\pi$
$907$	39.4830	1.31101	0.655506	+	0.755190i	$0.272456\pi$
$908$	−7.80575	−0.259043
$909$	6.60629	0.219117
$910$	92.2581	3.05833
$911$	−5.38896	−0.178544	−0.0892721	−	0.996007i	$-0.528454\pi$
$911$	−5.38896	−0.178544	−0.0892721	+	0.996007i	$0.528454\pi$
$912$	9.25613	0.306501
$913$	38.2295	1.26521
$914$	−11.6535	−0.385464
$915$	63.5259	2.10010
$916$	−27.2192	−0.899346
$917$	24.7226	0.816411
$918$	−27.6725	−0.913328
$919$	−0.567009	−0.0187039	−0.00935195	−	0.999956i	$-0.502977\pi$
$919$	−0.567009	−0.0187039	−0.00935195	+	0.999956i	$0.502977\pi$
$920$	0.416555	0.0137334
$921$	−29.4742	−0.971207
$922$	−23.7078	−0.780775
$923$	−95.5382	−3.14468
$924$	−18.6172	−0.612462
$925$	42.9881	1.41344
$926$	−6.08956	−0.200115
$927$	7.73888	0.254178
$928$	−22.2676	−0.730970
$929$	20.4719	0.671663	0.335831	−	0.941922i	$-0.390983\pi$
$929$	20.4719	0.671663	0.335831	+	0.941922i	$0.390983\pi$
$930$	114.321	3.74874
$931$	2.39466	0.0784820
$932$	22.4781	0.736294
$933$	5.02669	0.164566
$934$	−3.92399	−0.128397
$935$	−44.0382	−1.44020
$936$	4.40846	0.144095
$937$	23.7189	0.774862	0.387431	−	0.921899i	$-0.373363\pi$
$937$	23.7189	0.774862	0.387431	+	0.921899i	$0.373363\pi$
$938$	−30.4641	−0.994689
$939$	−44.1714	−1.44148
$940$	−59.8721	−1.95281
$941$	23.1721	0.755389	0.377695	−	0.925930i	$-0.376717\pi$
$941$	23.1721	0.755389	0.377695	+	0.925930i	$0.376717\pi$
$942$	−69.9189	−2.27808
$943$	0.291217	0.00948333
$944$	7.16177	0.233096
$945$	37.3930	1.21640
$946$	−41.2384	−1.34078
$947$	−32.0394	−1.04114	−0.520570	−	0.853819i	$-0.674281\pi$
$947$	−32.0394	−1.04114	−0.520570	+	0.853819i	$0.674281\pi$
$948$	−14.6493	−0.475786
$949$	−65.8925	−2.13896
$950$	16.9396	0.549592
$951$	−26.8180	−0.869633
$952$	−8.95175	−0.290128
$953$	4.37101	0.141591	0.0707955	−	0.997491i	$-0.477446\pi$
$953$	4.37101	0.141591	0.0707955	+	0.997491i	$0.477446\pi$
$954$	−4.52974	−0.146656
$955$	−88.4832	−2.86325
$956$	−9.88241	−0.319620
$957$	23.1271	0.747593
$958$	28.2210	0.911778
$959$	−40.2304	−1.29911
$960$	12.9725	0.418686
$961$	46.5908	1.50293
$962$	−52.4251	−1.69025
$963$	−4.10126	−0.132161
$964$	−22.4769	−0.723932
$965$	97.4240	3.13619
$966$	0.642058	0.0206579
$967$	45.9585	1.47793	0.738963	−	0.673746i	$-0.235316\pi$
$967$	45.9585	1.47793	0.738963	+	0.673746i	$0.235316\pi$
$968$	1.76898	0.0568572
$969$	6.23490	0.200294
$970$	−29.4684	−0.946173
$971$	12.1510	0.389944	0.194972	−	0.980809i	$-0.437538\pi$
$971$	12.1510	0.389944	0.194972	+	0.980809i	$0.437538\pi$
$972$	−7.49197	−0.240305
$973$	−40.1291	−1.28648
$974$	9.68267	0.310253
$975$	109.834	3.51749
$976$	43.6655	1.39770
$977$	3.10719	0.0994078	0.0497039	−	0.998764i	$-0.484172\pi$
$977$	3.10719	0.0994078	0.0497039	+	0.998764i	$0.484172\pi$
$978$	24.5729	0.785755
$979$	−56.1479	−1.79449
$980$	11.8318	0.377954
$981$	9.13697	0.291721
$982$	34.9916	1.11663
$983$	7.46232	0.238011	0.119006	−	0.992894i	$-0.462029\pi$
$983$	7.46232	0.238011	0.119006	+	0.992894i	$0.462029\pi$
$984$	−7.94954	−0.253422
$985$	−11.9293	−0.380101
$986$	−20.9186	−0.666183
$987$	49.0578	1.56153
$988$	−8.16015	−0.259609
$989$	0.561781	0.0178636
$990$	−13.5415	−0.430377
$991$	−25.2984	−0.803630	−0.401815	−	0.915721i	$-0.631620\pi$
$991$	−25.2984	−0.803630	−0.401815	+	0.915721i	$0.631620\pi$
$992$	56.3451	1.78896
$993$	55.2906	1.75459
$994$	59.6534	1.89209
$995$	73.8923	2.34254
$996$	−26.7427	−0.847376
$997$	−10.5323	−0.333560	−0.166780	−	0.985994i	$-0.553337\pi$
$997$	−10.5323	−0.333560	−0.166780	+	0.985994i	$0.553337\pi$
$998$	−75.1825	−2.37986
$999$	−21.2483	−0.672268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.16	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.16	✓	75	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)

\(f(q)\)	\(=\)	\(q-1.81819 q^{2} +1.88652 q^{3} +1.30583 q^{4} -3.78374 q^{5} -3.43005 q^{6} +2.14600 q^{7} +1.26214 q^{8} +0.558942 q^{9} +O(q^{10})\)	\(q-1.81819 q^{2} +1.88652 q^{3} +1.30583 q^{4} -3.78374 q^{5} -3.43005 q^{6} +2.14600 q^{7} +1.26214 q^{8} +0.558942 q^{9} +6.87957 q^{10} -3.52159 q^{11} +2.46346 q^{12} +6.24903 q^{13} -3.90185 q^{14} -7.13809 q^{15} -4.90647 q^{16} -3.30498 q^{17} -1.01626 q^{18} -1.00000 q^{19} -4.94091 q^{20} +4.04847 q^{21} +6.40292 q^{22} -0.0872254 q^{23} +2.38105 q^{24} +9.31670 q^{25} -11.3619 q^{26} -4.60509 q^{27} +2.80231 q^{28} -3.48115 q^{29} +12.9784 q^{30} +8.80857 q^{31} +6.39663 q^{32} -6.64353 q^{33} +6.00910 q^{34} -8.11993 q^{35} +0.729881 q^{36} +4.61409 q^{37} +1.81819 q^{38} +11.7889 q^{39} -4.77562 q^{40} -3.33867 q^{41} -7.36090 q^{42} -6.44056 q^{43} -4.59858 q^{44} -2.11489 q^{45} +0.158593 q^{46} +12.1176 q^{47} -9.25613 q^{48} -2.39466 q^{49} -16.9396 q^{50} -6.23490 q^{51} +8.16015 q^{52} +4.45724 q^{53} +8.37295 q^{54} +13.3248 q^{55} +2.70856 q^{56} -1.88652 q^{57} +6.32940 q^{58} -1.45966 q^{59} -9.32111 q^{60} -8.89957 q^{61} -16.0157 q^{62} +1.19949 q^{63} -1.81737 q^{64} -23.6447 q^{65} +12.0792 q^{66} +7.80761 q^{67} -4.31574 q^{68} -0.164552 q^{69} +14.7636 q^{70} -15.2885 q^{71} +0.705464 q^{72} -10.5444 q^{73} -8.38931 q^{74} +17.5761 q^{75} -1.30583 q^{76} -7.55734 q^{77} -21.4345 q^{78} -5.94661 q^{79} +18.5648 q^{80} -10.3644 q^{81} +6.07035 q^{82} -10.8558 q^{83} +5.28660 q^{84} +12.5052 q^{85} +11.7102 q^{86} -6.56724 q^{87} -4.44474 q^{88} +15.9439 q^{89} +3.84528 q^{90} +13.4104 q^{91} -0.113901 q^{92} +16.6175 q^{93} -22.0322 q^{94} +3.78374 q^{95} +12.0673 q^{96} -4.28346 q^{97} +4.35396 q^{98} -1.96836 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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