LMFDB - Embedded newform 8043.2.a.t.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.869058 q^{2} -1.00000 q^{3} -1.24474 q^{4} -3.05181 q^{5} +0.869058 q^{6} +1.00000 q^{7} +2.81987 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.869058 q^{2} -1.00000 q^{3} -1.24474 q^{4} -3.05181 q^{5} +0.869058 q^{6} +1.00000 q^{7} +2.81987 q^{8} +1.00000 q^{9} +2.65220 q^{10} +0.712082 q^{11} +1.24474 q^{12} -4.02159 q^{13} -0.869058 q^{14} +3.05181 q^{15} +0.0388458 q^{16} +1.21971 q^{17} -0.869058 q^{18} -3.38040 q^{19} +3.79870 q^{20} -1.00000 q^{21} -0.618841 q^{22} +2.85470 q^{23} -2.81987 q^{24} +4.31353 q^{25} +3.49499 q^{26} -1.00000 q^{27} -1.24474 q^{28} -3.20901 q^{29} -2.65220 q^{30} -5.18946 q^{31} -5.67349 q^{32} -0.712082 q^{33} -1.06000 q^{34} -3.05181 q^{35} -1.24474 q^{36} +10.3804 q^{37} +2.93776 q^{38} +4.02159 q^{39} -8.60569 q^{40} -2.02565 q^{41} +0.869058 q^{42} +5.92118 q^{43} -0.886355 q^{44} -3.05181 q^{45} -2.48090 q^{46} -4.02786 q^{47} -0.0388458 q^{48} +1.00000 q^{49} -3.74871 q^{50} -1.21971 q^{51} +5.00582 q^{52} +1.52434 q^{53} +0.869058 q^{54} -2.17314 q^{55} +2.81987 q^{56} +3.38040 q^{57} +2.78882 q^{58} +4.36113 q^{59} -3.79870 q^{60} -1.75357 q^{61} +4.50994 q^{62} +1.00000 q^{63} +4.85291 q^{64} +12.2731 q^{65} +0.618841 q^{66} +3.24868 q^{67} -1.51822 q^{68} -2.85470 q^{69} +2.65220 q^{70} +11.4733 q^{71} +2.81987 q^{72} +6.96468 q^{73} -9.02120 q^{74} -4.31353 q^{75} +4.20770 q^{76} +0.712082 q^{77} -3.49499 q^{78} -11.0316 q^{79} -0.118550 q^{80} +1.00000 q^{81} +1.76040 q^{82} -12.8639 q^{83} +1.24474 q^{84} -3.72231 q^{85} -5.14586 q^{86} +3.20901 q^{87} +2.00798 q^{88} +0.646252 q^{89} +2.65220 q^{90} -4.02159 q^{91} -3.55335 q^{92} +5.18946 q^{93} +3.50044 q^{94} +10.3163 q^{95} +5.67349 q^{96} -18.1231 q^{97} -0.869058 q^{98} +0.712082 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	−0.869058	−0.614517	−0.307259	−	0.951626i	$-0.599412\pi$
$2$	−0.869058	−0.614517	−0.307259	+	0.951626i	$0.599412\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.24474	−0.622369
$5$	−3.05181	−1.36481	−0.682405	−	0.730974i	$-0.739066\pi$
$5$	−3.05181	−1.36481	−0.682405	+	0.730974i	$0.739066\pi$
$6$	0.869058	0.354792
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.81987	0.996973
$9$	1.00000	0.333333
$10$	2.65220	0.838699
$11$	0.712082	0.214701	0.107350	−	0.994221i	$-0.465763\pi$
$11$	0.712082	0.214701	0.107350	+	0.994221i	$0.465763\pi$
$12$	1.24474	0.359325
$13$	−4.02159	−1.11539	−0.557694	−	0.830047i	$-0.688314\pi$
$13$	−4.02159	−1.11539	−0.557694	+	0.830047i	$0.688314\pi$
$14$	−0.869058	−0.232266
$15$	3.05181	0.787973
$16$	0.0388458	0.00971145
$17$	1.21971	0.295823	0.147911	−	0.989001i	$-0.452745\pi$
$17$	1.21971	0.295823	0.147911	+	0.989001i	$0.452745\pi$
$18$	−0.869058	−0.204839
$19$	−3.38040	−0.775516	−0.387758	−	0.921761i	$-0.626750\pi$
$19$	−3.38040	−0.775516	−0.387758	+	0.921761i	$0.626750\pi$
$20$	3.79870	0.849415
$21$	−1.00000	−0.218218
$22$	−0.618841	−0.131937
$23$	2.85470	0.595245	0.297623	−	0.954684i	$-0.403806\pi$
$23$	2.85470	0.595245	0.297623	+	0.954684i	$0.403806\pi$
$24$	−2.81987	−0.575603
$25$	4.31353	0.862706
$26$	3.49499	0.685425
$27$	−1.00000	−0.192450
$28$	−1.24474	−0.235233
$29$	−3.20901	−0.595898	−0.297949	−	0.954582i	$-0.596303\pi$
$29$	−3.20901	−0.595898	−0.297949	+	0.954582i	$0.596303\pi$
$30$	−2.65220	−0.484223
$31$	−5.18946	−0.932054	−0.466027	−	0.884770i	$-0.654315\pi$
$31$	−5.18946	−0.932054	−0.466027	+	0.884770i	$0.654315\pi$
$32$	−5.67349	−1.00294
$33$	−0.712082	−0.123958
$34$	−1.06000	−0.181788
$35$	−3.05181	−0.515850
$36$	−1.24474	−0.207456
$37$	10.3804	1.70653	0.853266	−	0.521476i	$-0.174618\pi$
$37$	10.3804	1.70653	0.853266	+	0.521476i	$0.174618\pi$
$38$	2.93776	0.476568
$39$	4.02159	0.643969
$40$	−8.60569	−1.36068
$41$	−2.02565	−0.316353	−0.158176	−	0.987411i	$-0.550561\pi$
$41$	−2.02565	−0.316353	−0.158176	+	0.987411i	$0.550561\pi$
$42$	0.869058	0.134099
$43$	5.92118	0.902972	0.451486	−	0.892278i	$-0.350894\pi$
$43$	5.92118	0.902972	0.451486	+	0.892278i	$0.350894\pi$
$44$	−0.886355	−0.133623
$45$	−3.05181	−0.454937
$46$	−2.48090	−0.365789
$47$	−4.02786	−0.587524	−0.293762	−	0.955879i	$-0.594907\pi$
$47$	−4.02786	−0.587524	−0.293762	+	0.955879i	$0.594907\pi$
$48$	−0.0388458	−0.00560691
$49$	1.00000	0.142857
$50$	−3.74871	−0.530148
$51$	−1.21971	−0.170793
$52$	5.00582	0.694182
$53$	1.52434	0.209385	0.104692	−	0.994505i	$-0.466614\pi$
$53$	1.52434	0.209385	0.104692	+	0.994505i	$0.466614\pi$
$54$	0.869058	0.118264
$55$	−2.17314	−0.293026
$56$	2.81987	0.376821
$57$	3.38040	0.447744
$58$	2.78882	0.366190
$59$	4.36113	0.567771	0.283885	−	0.958858i	$-0.408377\pi$
$59$	4.36113	0.567771	0.283885	+	0.958858i	$0.408377\pi$
$60$	−3.79870	−0.490410
$61$	−1.75357	−0.224522	−0.112261	−	0.993679i	$-0.535809\pi$
$61$	−1.75357	−0.224522	−0.112261	+	0.993679i	$0.535809\pi$
$62$	4.50994	0.572763
$63$	1.00000	0.125988
$64$	4.85291	0.606613
$65$	12.2731	1.52229
$66$	0.618841	0.0761741
$67$	3.24868	0.396890	0.198445	−	0.980112i	$-0.436411\pi$
$67$	3.24868	0.396890	0.198445	+	0.980112i	$0.436411\pi$
$68$	−1.51822	−0.184111
$69$	−2.85470	−0.343665
$70$	2.65220	0.316998
$71$	11.4733	1.36162	0.680812	−	0.732458i	$-0.261627\pi$
$71$	11.4733	1.36162	0.680812	+	0.732458i	$0.261627\pi$
$72$	2.81987	0.332324
$73$	6.96468	0.815154	0.407577	−	0.913171i	$-0.366374\pi$
$73$	6.96468	0.815154	0.407577	+	0.913171i	$0.366374\pi$
$74$	−9.02120	−1.04869
$75$	−4.31353	−0.498084
$76$	4.20770	0.482657
$77$	0.712082	0.0811493
$78$	−3.49499	−0.395730
$79$	−11.0316	−1.24116	−0.620578	−	0.784145i	$-0.713102\pi$
$79$	−11.0316	−1.24116	−0.620578	+	0.784145i	$0.713102\pi$
$80$	−0.118550	−0.0132543
$81$	1.00000	0.111111
$82$	1.76040	0.194404
$83$	−12.8639	−1.41199	−0.705996	−	0.708216i	$-0.749500\pi$
$83$	−12.8639	−1.41199	−0.705996	+	0.708216i	$0.749500\pi$
$84$	1.24474	0.135812
$85$	−3.72231	−0.403742
$86$	−5.14586	−0.554892
$87$	3.20901	0.344042
$88$	2.00798	0.214051
$89$	0.646252	0.0685026	0.0342513	−	0.999413i	$-0.489095\pi$
$89$	0.646252	0.0685026	0.0342513	+	0.999413i	$0.489095\pi$
$90$	2.65220	0.279566
$91$	−4.02159	−0.421577
$92$	−3.55335	−0.370462
$93$	5.18946	0.538122
$94$	3.50044	0.361043
$95$	10.3163	1.05843
$96$	5.67349	0.579048
$97$	−18.1231	−1.84012	−0.920061	−	0.391776i	$-0.871861\pi$
$97$	−18.1231	−1.84012	−0.920061	+	0.391776i	$0.871861\pi$
$98$	−0.869058	−0.0877882
$99$	0.712082	0.0715670
$100$	−5.36921	−0.536921
$101$	−1.30201	−0.129555	−0.0647773	−	0.997900i	$-0.520634\pi$
$101$	−1.30201	−0.129555	−0.0647773	+	0.997900i	$0.520634\pi$
$102$	1.06000	0.104955
$103$	−12.0119	−1.18357	−0.591783	−	0.806098i	$-0.701576\pi$
$103$	−12.0119	−1.18357	−0.591783	+	0.806098i	$0.701576\pi$
$104$	−11.3403	−1.11201
$105$	3.05181	0.297826
$106$	−1.32474	−0.128670
$107$	−14.4434	−1.39629	−0.698146	−	0.715955i	$-0.745991\pi$
$107$	−14.4434	−1.39629	−0.698146	+	0.715955i	$0.745991\pi$
$108$	1.24474	0.119775
$109$	−1.06043	−0.101571	−0.0507854	−	0.998710i	$-0.516172\pi$
$109$	−1.06043	−0.101571	−0.0507854	+	0.998710i	$0.516172\pi$
$110$	1.88858	0.180069
$111$	−10.3804	−0.985267
$112$	0.0388458	0.00367058
$113$	−11.4615	−1.07821	−0.539104	−	0.842239i	$-0.681237\pi$
$113$	−11.4615	−1.07821	−0.539104	+	0.842239i	$0.681237\pi$
$114$	−2.93776	−0.275147
$115$	−8.71199	−0.812397
$116$	3.99437	0.370868
$117$	−4.02159	−0.371796
$118$	−3.79008	−0.348905
$119$	1.21971	0.111810
$120$	8.60569	0.785588
$121$	−10.4929	−0.953904
$122$	1.52396	0.137973
$123$	2.02565	0.182646
$124$	6.45951	0.580081
$125$	2.09497	0.187380
$126$	−0.869058	−0.0774219
$127$	−13.0003	−1.15359	−0.576797	−	0.816888i	$-0.695698\pi$
$127$	−13.0003	−1.15359	−0.576797	+	0.816888i	$0.695698\pi$
$128$	7.12953	0.630167
$129$	−5.92118	−0.521331
$130$	−10.6661	−0.935475
$131$	−21.7017	−1.89609	−0.948045	−	0.318135i	$-0.896943\pi$
$131$	−21.7017	−1.89609	−0.948045	+	0.318135i	$0.896943\pi$
$132$	0.886355	0.0771473
$133$	−3.38040	−0.293117
$134$	−2.82329	−0.243895
$135$	3.05181	0.262658
$136$	3.43941	0.294927
$137$	−12.2984	−1.05073	−0.525363	−	0.850878i	$-0.676070\pi$
$137$	−12.2984	−1.05073	−0.525363	+	0.850878i	$0.676070\pi$
$138$	2.48090	0.211188
$139$	10.5398	0.893973	0.446986	−	0.894541i	$-0.352497\pi$
$139$	10.5398	0.893973	0.446986	+	0.894541i	$0.352497\pi$
$140$	3.79870	0.321049
$141$	4.02786	0.339207
$142$	−9.97093	−0.836742
$143$	−2.86370	−0.239475
$144$	0.0388458	0.00323715
$145$	9.79328	0.813288
$146$	−6.05271	−0.500926
$147$	−1.00000	−0.0824786
$148$	−12.9209	−1.06209
$149$	18.4051	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$149$	18.4051	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$150$	3.74871	0.306081
$151$	6.75993	0.550115	0.275058	−	0.961428i	$-0.411303\pi$
$151$	6.75993	0.550115	0.275058	+	0.961428i	$0.411303\pi$
$152$	−9.53226	−0.773169
$153$	1.21971	0.0986075
$154$	−0.618841	−0.0498676
$155$	15.8372	1.27208
$156$	−5.00582	−0.400786
$157$	−0.695158	−0.0554796	−0.0277398	−	0.999615i	$-0.508831\pi$
$157$	−0.695158	−0.0554796	−0.0277398	+	0.999615i	$0.508831\pi$
$158$	9.58713	0.762711
$159$	−1.52434	−0.120888
$160$	17.3144	1.36882
$161$	2.85470	0.224982
$162$	−0.869058	−0.0682797
$163$	15.8268	1.23965	0.619826	−	0.784739i	$-0.287203\pi$
$163$	15.8268	1.23965	0.619826	+	0.784739i	$0.287203\pi$
$164$	2.52140	0.196888
$165$	2.17314	0.169179
$166$	11.1794	0.867693
$167$	−20.4671	−1.58379	−0.791895	−	0.610657i	$-0.790906\pi$
$167$	−20.4671	−1.58379	−0.791895	+	0.610657i	$0.790906\pi$
$168$	−2.81987	−0.217557
$169$	3.17317	0.244090
$170$	3.23491	0.248106
$171$	−3.38040	−0.258505
$172$	−7.37032	−0.561982
$173$	−15.3906	−1.17013	−0.585064	−	0.810987i	$-0.698931\pi$
$173$	−15.3906	−1.17013	−0.585064	+	0.810987i	$0.698931\pi$
$174$	−2.78882	−0.211420
$175$	4.31353	0.326072
$176$	0.0276614	0.00208506
$177$	−4.36113	−0.327803
$178$	−0.561631	−0.0420960
$179$	7.40804	0.553703	0.276852	−	0.960913i	$-0.410709\pi$
$179$	7.40804	0.553703	0.276852	+	0.960913i	$0.410709\pi$
$180$	3.79870	0.283138
$181$	−3.80490	−0.282816	−0.141408	−	0.989951i	$-0.545163\pi$
$181$	−3.80490	−0.282816	−0.141408	+	0.989951i	$0.545163\pi$
$182$	3.49499	0.259066
$183$	1.75357	0.129628
$184$	8.04986	0.593444
$185$	−31.6791	−2.32909
$186$	−4.50994	−0.330685
$187$	0.868532	0.0635134
$188$	5.01363	0.365656
$189$	−1.00000	−0.0727393
$190$	−8.96548	−0.650425
$191$	−11.5600	−0.836451	−0.418226	−	0.908343i	$-0.637348\pi$
$191$	−11.5600	−0.836451	−0.418226	+	0.908343i	$0.637348\pi$
$192$	−4.85291	−0.350228
$193$	8.47034	0.609709	0.304854	−	0.952399i	$-0.401392\pi$
$193$	8.47034	0.609709	0.304854	+	0.952399i	$0.401392\pi$
$194$	15.7500	1.13079
$195$	−12.2731	−0.878896
$196$	−1.24474	−0.0889098
$197$	8.76026	0.624143	0.312071	−	0.950059i	$-0.398977\pi$
$197$	8.76026	0.624143	0.312071	+	0.950059i	$0.398977\pi$
$198$	−0.618841	−0.0439791
$199$	22.6094	1.60274	0.801371	−	0.598168i	$-0.204104\pi$
$199$	22.6094	1.60274	0.801371	+	0.598168i	$0.204104\pi$
$200$	12.1636	0.860095
$201$	−3.24868	−0.229144
$202$	1.13152	0.0796135
$203$	−3.20901	−0.225228
$204$	1.51822	0.106296
$205$	6.18188	0.431761
$206$	10.4390	0.727321
$207$	2.85470	0.198415
$208$	−0.156222	−0.0108320
$209$	−2.40712	−0.166504
$210$	−2.65220	−0.183019
$211$	−11.0757	−0.762483	−0.381242	−	0.924475i	$-0.624503\pi$
$211$	−11.0757	−0.762483	−0.381242	+	0.924475i	$0.624503\pi$
$212$	−1.89741	−0.130314
$213$	−11.4733	−0.786134
$214$	12.5521	0.858045
$215$	−18.0703	−1.23239
$216$	−2.81987	−0.191868
$217$	−5.18946	−0.352283
$218$	0.921576	0.0624170
$219$	−6.96468	−0.470629
$220$	2.70499	0.182370
$221$	−4.90516	−0.329957
$222$	9.02120	0.605463
$223$	12.2137	0.817891	0.408946	−	0.912559i	$-0.365897\pi$
$223$	12.2137	0.817891	0.408946	+	0.912559i	$0.365897\pi$
$224$	−5.67349	−0.379076
$225$	4.31353	0.287569
$226$	9.96072	0.662577
$227$	5.02074	0.333238	0.166619	−	0.986021i	$-0.446715\pi$
$227$	5.02074	0.333238	0.166619	+	0.986021i	$0.446715\pi$
$228$	−4.20770	−0.278662
$229$	−17.1485	−1.13320	−0.566601	−	0.823992i	$-0.691742\pi$
$229$	−17.1485	−1.13320	−0.566601	+	0.823992i	$0.691742\pi$
$230$	7.57123	0.499232
$231$	−0.712082	−0.0468516
$232$	−9.04898	−0.594095
$233$	−26.1749	−1.71477	−0.857386	−	0.514674i	$-0.827913\pi$
$233$	−26.1749	−1.71477	−0.857386	+	0.514674i	$0.827913\pi$
$234$	3.49499	0.228475
$235$	12.2923	0.801858
$236$	−5.42846	−0.353363
$237$	11.0316	0.716581
$238$	−1.06000	−0.0687094
$239$	−3.38308	−0.218833	−0.109417	−	0.993996i	$-0.534898\pi$
$239$	−3.38308	−0.218833	−0.109417	+	0.993996i	$0.534898\pi$
$240$	0.118550	0.00765236
$241$	13.9845	0.900819	0.450409	−	0.892822i	$-0.351278\pi$
$241$	13.9845	0.900819	0.450409	+	0.892822i	$0.351278\pi$
$242$	9.11898	0.586190
$243$	−1.00000	−0.0641500
$244$	2.18274	0.139735
$245$	−3.05181	−0.194973
$246$	−1.76040	−0.112239
$247$	13.5946	0.865001
$248$	−14.6336	−0.929233
$249$	12.8639	0.815214
$250$	−1.82065	−0.115148
$251$	12.9214	0.815589	0.407794	−	0.913074i	$-0.366298\pi$
$251$	12.9214	0.815589	0.407794	+	0.913074i	$0.366298\pi$
$252$	−1.24474	−0.0784111
$253$	2.03278	0.127800
$254$	11.2981	0.708903
$255$	3.72231	0.233100
$256$	−15.9018	−0.993862
$257$	4.45689	0.278013	0.139007	−	0.990291i	$-0.455609\pi$
$257$	4.45689	0.278013	0.139007	+	0.990291i	$0.455609\pi$
$258$	5.14586	0.320367
$259$	10.3804	0.645008
$260$	−15.2768	−0.947427
$261$	−3.20901	−0.198633
$262$	18.8601	1.16518
$263$	−26.0734	−1.60776	−0.803878	−	0.594795i	$-0.797233\pi$
$263$	−26.0734	−1.60776	−0.803878	+	0.594795i	$0.797233\pi$
$264$	−2.00798	−0.123582
$265$	−4.65200	−0.285770
$266$	2.93776	0.180126
$267$	−0.646252	−0.0395500
$268$	−4.04375	−0.247012
$269$	24.1132	1.47020	0.735102	−	0.677956i	$-0.237134\pi$
$269$	24.1132	1.47020	0.735102	+	0.677956i	$0.237134\pi$
$270$	−2.65220	−0.161408
$271$	17.6735	1.07359	0.536793	−	0.843714i	$-0.319636\pi$
$271$	17.6735	1.07359	0.536793	+	0.843714i	$0.319636\pi$
$272$	0.0473805	0.00287287
$273$	4.02159	0.243398
$274$	10.6881	0.645689
$275$	3.07159	0.185224
$276$	3.55335	0.213886
$277$	13.5147	0.812021	0.406010	−	0.913868i	$-0.366920\pi$
$277$	13.5147	0.812021	0.406010	+	0.913868i	$0.366920\pi$
$278$	−9.15969	−0.549361
$279$	−5.18946	−0.310685
$280$	−8.60569	−0.514288
$281$	11.0748	0.660667	0.330333	−	0.943864i	$-0.392839\pi$
$281$	11.0748	0.660667	0.330333	+	0.943864i	$0.392839\pi$
$282$	−3.50044	−0.208448
$283$	14.9421	0.888217	0.444109	−	0.895973i	$-0.353520\pi$
$283$	14.9421	0.888217	0.444109	+	0.895973i	$0.353520\pi$
$284$	−14.2812	−0.847433
$285$	−10.3163	−0.611086
$286$	2.48872	0.147161
$287$	−2.02565	−0.119570
$288$	−5.67349	−0.334314
$289$	−15.5123	−0.912489
$290$	−8.51093	−0.499779
$291$	18.1231	1.06239
$292$	−8.66919	−0.507326
$293$	−19.3031	−1.12770	−0.563851	−	0.825877i	$-0.690681\pi$
$293$	−19.3031	−1.12770	−0.563851	+	0.825877i	$0.690681\pi$
$294$	0.869058	0.0506845
$295$	−13.3093	−0.774899
$296$	29.2714	1.70137
$297$	−0.712082	−0.0413192
$298$	−15.9951	−0.926574
$299$	−11.4804	−0.663929
$300$	5.36921	0.309992
$301$	5.92118	0.341291
$302$	−5.87477	−0.338055
$303$	1.30201	0.0747983
$304$	−0.131314	−0.00753138
$305$	5.35156	0.306430
$306$	−1.06000	−0.0605960
$307$	25.4155	1.45054	0.725270	−	0.688465i	$-0.241715\pi$
$307$	25.4155	1.45054	0.725270	+	0.688465i	$0.241715\pi$
$308$	−0.886355	−0.0505048
$309$	12.0119	0.683332
$310$	−13.7635	−0.781713
$311$	−24.6491	−1.39772	−0.698861	−	0.715257i	$-0.746310\pi$
$311$	−24.6491	−1.39772	−0.698861	+	0.715257i	$0.746310\pi$
$312$	11.3403	0.642020
$313$	−8.72404	−0.493112	−0.246556	−	0.969129i	$-0.579299\pi$
$313$	−8.72404	−0.493112	−0.246556	+	0.969129i	$0.579299\pi$
$314$	0.604133	0.0340932
$315$	−3.05181	−0.171950
$316$	13.7315	0.772456
$317$	11.9952	0.673716	0.336858	−	0.941555i	$-0.390636\pi$
$317$	11.9952	0.673716	0.336858	+	0.941555i	$0.390636\pi$
$318$	1.32474	0.0742879
$319$	−2.28508	−0.127940
$320$	−14.8101	−0.827912
$321$	14.4434	0.806150
$322$	−2.48090	−0.138255
$323$	−4.12310	−0.229415
$324$	−1.24474	−0.0691521
$325$	−17.3472	−0.962252
$326$	−13.7544	−0.761787
$327$	1.06043	0.0586419
$328$	−5.71205	−0.315395
$329$	−4.02786	−0.222063
$330$	−1.88858	−0.103963
$331$	28.5858	1.57122	0.785610	−	0.618722i	$-0.212349\pi$
$331$	28.5858	1.57122	0.785610	+	0.618722i	$0.212349\pi$
$332$	16.0121	0.878780
$333$	10.3804	0.568844
$334$	17.7871	0.973267
$335$	−9.91435	−0.541679
$336$	−0.0388458	−0.00211921
$337$	29.0305	1.58139	0.790697	−	0.612208i	$-0.209718\pi$
$337$	29.0305	1.58139	0.790697	+	0.612208i	$0.209718\pi$
$338$	−2.75767	−0.149997
$339$	11.4615	0.622503
$340$	4.63330	0.251276
$341$	−3.69532	−0.200113
$342$	2.93776	0.158856
$343$	1.00000	0.0539949
$344$	16.6969	0.900239
$345$	8.71199	0.469038
$346$	13.3754	0.719064
$347$	32.8486	1.76340	0.881701	−	0.471808i	$-0.156398\pi$
$347$	32.8486	1.76340	0.881701	+	0.471808i	$0.156398\pi$
$348$	−3.99437	−0.214121
$349$	25.5987	1.37027	0.685134	−	0.728417i	$-0.259744\pi$
$349$	25.5987	1.37027	0.685134	+	0.728417i	$0.259744\pi$
$350$	−3.74871	−0.200377
$351$	4.02159	0.214656
$352$	−4.03999	−0.215332
$353$	−25.0781	−1.33477	−0.667385	−	0.744713i	$-0.732587\pi$
$353$	−25.0781	−1.33477	−0.667385	+	0.744713i	$0.732587\pi$
$354$	3.79008	0.201440
$355$	−35.0142	−1.85836
$356$	−0.804414	−0.0426339
$357$	−1.21971	−0.0645538
$358$	−6.43802	−0.340260
$359$	13.8151	0.729135	0.364568	−	0.931177i	$-0.381217\pi$
$359$	13.8151	0.729135	0.364568	+	0.931177i	$0.381217\pi$
$360$	−8.60569	−0.453560
$361$	−7.57292	−0.398575
$362$	3.30668	0.173795
$363$	10.4929	0.550736
$364$	5.00582	0.262376
$365$	−21.2549	−1.11253
$366$	−1.52396	−0.0796585
$367$	27.5782	1.43957	0.719784	−	0.694198i	$-0.244241\pi$
$367$	27.5782	1.43957	0.719784	+	0.694198i	$0.244241\pi$
$368$	0.110893	0.00578069
$369$	−2.02565	−0.105451
$370$	27.5310	1.43127
$371$	1.52434	0.0791399
$372$	−6.45951	−0.334910
$373$	−7.20808	−0.373220	−0.186610	−	0.982434i	$-0.559750\pi$
$373$	−7.20808	−0.373220	−0.186610	+	0.982434i	$0.559750\pi$
$374$	−0.754805	−0.0390301
$375$	−2.09497	−0.108184
$376$	−11.3580	−0.585745
$377$	12.9053	0.664657
$378$	0.869058	0.0446995
$379$	−21.5284	−1.10584	−0.552920	−	0.833234i	$-0.686486\pi$
$379$	−21.5284	−1.10584	−0.552920	+	0.833234i	$0.686486\pi$
$380$	−12.8411	−0.658735
$381$	13.0003	0.666027
$382$	10.0463	0.514014
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−7.12953	−0.363827
$385$	−2.17314	−0.110753
$386$	−7.36122	−0.374676
$387$	5.92118	0.300991
$388$	22.5585	1.14523
$389$	−12.8543	−0.651737	−0.325869	−	0.945415i	$-0.605657\pi$
$389$	−12.8543	−0.651737	−0.325869	+	0.945415i	$0.605657\pi$
$390$	10.6661	0.540097
$391$	3.48190	0.176087
$392$	2.81987	0.142425
$393$	21.7017	1.09471
$394$	−7.61318	−0.383547
$395$	33.6664	1.69394
$396$	−0.886355	−0.0445410
$397$	−2.78599	−0.139825	−0.0699123	−	0.997553i	$-0.522272\pi$
$397$	−2.78599	−0.139825	−0.0699123	+	0.997553i	$0.522272\pi$
$398$	−19.6489	−0.984912
$399$	3.38040	0.169231
$400$	0.167563	0.00837813
$401$	−3.30807	−0.165197	−0.0825986	−	0.996583i	$-0.526322\pi$
$401$	−3.30807	−0.165197	−0.0825986	+	0.996583i	$0.526322\pi$
$402$	2.82329	0.140813
$403$	20.8699	1.03960
$404$	1.62066	0.0806307
$405$	−3.05181	−0.151646
$406$	2.78882	0.138407
$407$	7.39172	0.366394
$408$	−3.43941	−0.170276
$409$	10.3338	0.510972	0.255486	−	0.966813i	$-0.417765\pi$
$409$	10.3338	0.510972	0.255486	+	0.966813i	$0.417765\pi$
$410$	−5.37242	−0.265325
$411$	12.2984	0.606637
$412$	14.9516	0.736614
$413$	4.36113	0.214597
$414$	−2.48090	−0.121930
$415$	39.2580	1.92710
$416$	22.8164	1.11867
$417$	−10.5398	−0.516135
$418$	2.09193	0.102320
$419$	17.3773	0.848937	0.424469	−	0.905443i	$-0.360461\pi$
$419$	17.3773	0.848937	0.424469	+	0.905443i	$0.360461\pi$
$420$	−3.79870	−0.185358
$421$	3.35597	0.163560	0.0817801	−	0.996650i	$-0.473939\pi$
$421$	3.35597	0.163560	0.0817801	+	0.996650i	$0.473939\pi$
$422$	9.62544	0.468559
$423$	−4.02786	−0.195841
$424$	4.29844	0.208751
$425$	5.26125	0.255208
$426$	9.97093	0.483093
$427$	−1.75357	−0.0848613
$428$	17.9782	0.869009
$429$	2.86370	0.138261
$430$	15.7042	0.757322
$431$	−1.68530	−0.0811778	−0.0405889	−	0.999176i	$-0.512923\pi$
$431$	−1.68530	−0.0811778	−0.0405889	+	0.999176i	$0.512923\pi$
$432$	−0.0388458	−0.00186897
$433$	29.4616	1.41583	0.707917	−	0.706296i	$-0.249635\pi$
$433$	29.4616	1.41583	0.707917	+	0.706296i	$0.249635\pi$
$434$	4.50994	0.216484
$435$	−9.79328	−0.469552
$436$	1.31996	0.0632145
$437$	−9.65001	−0.461622
$438$	6.05271	0.289210
$439$	−4.58842	−0.218994	−0.109497	−	0.993987i	$-0.534924\pi$
$439$	−4.58842	−0.218994	−0.109497	+	0.993987i	$0.534924\pi$
$440$	−6.12796	−0.292139
$441$	1.00000	0.0476190
$442$	4.26287	0.202764
$443$	−26.7679	−1.27178	−0.635891	−	0.771779i	$-0.719367\pi$
$443$	−26.7679	−1.27178	−0.635891	+	0.771779i	$0.719367\pi$
$444$	12.9209	0.613199
$445$	−1.97224	−0.0934930
$446$	−10.6144	−0.502608
$447$	−18.4051	−0.870534
$448$	4.85291	0.229278
$449$	4.15361	0.196021	0.0980104	−	0.995185i	$-0.468752\pi$
$449$	4.15361	0.196021	0.0980104	+	0.995185i	$0.468752\pi$
$450$	−3.74871	−0.176716
$451$	−1.44243	−0.0679212
$452$	14.2666	0.671043
$453$	−6.75993	−0.317609
$454$	−4.36332	−0.204781
$455$	12.2731	0.575372
$456$	9.53226	0.446389
$457$	−37.4012	−1.74955	−0.874777	−	0.484525i	$-0.838992\pi$
$457$	−37.4012	−1.74955	−0.874777	+	0.484525i	$0.838992\pi$
$458$	14.9030	0.696372
$459$	−1.21971	−0.0569311
$460$	10.8441	0.505610
$461$	−30.6301	−1.42659	−0.713293	−	0.700866i	$-0.752797\pi$
$461$	−30.6301	−1.42659	−0.713293	+	0.700866i	$0.752797\pi$
$462$	0.618841	0.0287911
$463$	13.2222	0.614488	0.307244	−	0.951631i	$-0.400593\pi$
$463$	13.2222	0.614488	0.307244	+	0.951631i	$0.400593\pi$
$464$	−0.124657	−0.00578703
$465$	−15.8372	−0.734434
$466$	22.7475	1.05376
$467$	3.84353	0.177857	0.0889286	−	0.996038i	$-0.471656\pi$
$467$	3.84353	0.177857	0.0889286	+	0.996038i	$0.471656\pi$
$468$	5.00582	0.231394
$469$	3.24868	0.150010
$470$	−10.6827	−0.492756
$471$	0.695158	0.0320312
$472$	12.2978	0.566052
$473$	4.21637	0.193869
$474$	−9.58713	−0.440351
$475$	−14.5814	−0.669042
$476$	−1.51822	−0.0695873
$477$	1.52434	0.0697949
$478$	2.94009	0.134477
$479$	14.0308	0.641085	0.320542	−	0.947234i	$-0.396135\pi$
$479$	14.0308	0.641085	0.320542	+	0.947234i	$0.396135\pi$
$480$	−17.3144	−0.790291
$481$	−41.7458	−1.90344
$482$	−12.1533	−0.553569
$483$	−2.85470	−0.129893
$484$	13.0610	0.593680
$485$	55.3082	2.51142
$486$	0.869058	0.0394213
$487$	21.7892	0.987363	0.493681	−	0.869643i	$-0.335651\pi$
$487$	21.7892	0.987363	0.493681	+	0.869643i	$0.335651\pi$
$488$	−4.94484	−0.223842
$489$	−15.8268	−0.715713
$490$	2.65220	0.119814
$491$	−29.1929	−1.31746	−0.658730	−	0.752380i	$-0.728906\pi$
$491$	−29.1929	−1.31746	−0.658730	+	0.752380i	$0.728906\pi$
$492$	−2.52140	−0.113673
$493$	−3.91405	−0.176280
$494$	−11.8145	−0.531558
$495$	−2.17314	−0.0976753
$496$	−0.201589	−0.00905159
$497$	11.4733	0.514646
$498$	−11.1794	−0.500963
$499$	15.3039	0.685095	0.342547	−	0.939501i	$-0.388710\pi$
$499$	15.3039	0.685095	0.342547	+	0.939501i	$0.388710\pi$
$500$	−2.60769	−0.116619
$501$	20.4671	0.914402
$502$	−11.2294	−0.501193
$503$	−27.8533	−1.24192	−0.620958	−	0.783844i	$-0.713256\pi$
$503$	−27.8533	−1.24192	−0.620958	+	0.783844i	$0.713256\pi$
$504$	2.81987	0.125607
$505$	3.97348	0.176817
$506$	−1.76660	−0.0785351
$507$	−3.17317	−0.140925
$508$	16.1820	0.717960
$509$	31.6478	1.40276	0.701382	−	0.712786i	$-0.252567\pi$
$509$	31.6478	1.40276	0.701382	+	0.712786i	$0.252567\pi$
$510$	−3.23491	−0.143244
$511$	6.96468	0.308099
$512$	−0.439471	−0.0194221
$513$	3.38040	0.149248
$514$	−3.87330	−0.170844
$515$	36.6579	1.61534
$516$	7.37032	0.324460
$517$	−2.86817	−0.126142
$518$	−9.02120	−0.396369
$519$	15.3906	0.675574
$520$	34.6085	1.51768
$521$	5.61037	0.245795	0.122897	−	0.992419i	$-0.460781\pi$
$521$	5.61037	0.245795	0.122897	+	0.992419i	$0.460781\pi$
$522$	2.78882	0.122063
$523$	−2.61486	−0.114340	−0.0571700	−	0.998364i	$-0.518208\pi$
$523$	−2.61486	−0.114340	−0.0571700	+	0.998364i	$0.518208\pi$
$524$	27.0130	1.18007
$525$	−4.31353	−0.188258
$526$	22.6593	0.987993
$527$	−6.32962	−0.275723
$528$	−0.0276614	−0.00120381
$529$	−14.8507	−0.645683
$530$	4.04286	0.175611
$531$	4.36113	0.189257
$532$	4.20770	0.182427
$533$	8.14631	0.352856
$534$	0.561631	0.0243042
$535$	44.0784	1.90567
$536$	9.16085	0.395688
$537$	−7.40804	−0.319681
$538$	−20.9557	−0.903466
$539$	0.712082	0.0306716
$540$	−3.79870	−0.163470
$541$	26.5094	1.13973	0.569863	−	0.821740i	$-0.306996\pi$
$541$	26.5094	1.13973	0.569863	+	0.821740i	$0.306996\pi$
$542$	−15.3593	−0.659737
$543$	3.80490	0.163284
$544$	−6.92000	−0.296693
$545$	3.23623	0.138625
$546$	−3.49499	−0.149572
$547$	6.33115	0.270700	0.135350	−	0.990798i	$-0.456784\pi$
$547$	6.33115	0.270700	0.135350	+	0.990798i	$0.456784\pi$
$548$	15.3083	0.653939
$549$	−1.75357	−0.0748406
$550$	−2.66939	−0.113823
$551$	10.8477	0.462129
$552$	−8.04986	−0.342625
$553$	−11.0316	−0.469113
$554$	−11.7451	−0.499001
$555$	31.6791	1.34470
$556$	−13.1193	−0.556381
$557$	29.3754	1.24468	0.622339	−	0.782748i	$-0.286183\pi$
$557$	29.3754	1.24468	0.622339	+	0.782748i	$0.286183\pi$
$558$	4.50994	0.190921
$559$	−23.8126	−1.00716
$560$	−0.118550	−0.00500965
$561$	−0.868532	−0.0366695
$562$	−9.62464	−0.405991
$563$	41.2453	1.73828	0.869140	−	0.494565i	$-0.164673\pi$
$563$	41.2453	1.73828	0.869140	+	0.494565i	$0.164673\pi$
$564$	−5.01363	−0.211112
$565$	34.9783	1.47155
$566$	−12.9856	−0.545825
$567$	1.00000	0.0419961
$568$	32.3530	1.35750
$569$	20.4837	0.858720	0.429360	−	0.903133i	$-0.358739\pi$
$569$	20.4837	0.858720	0.429360	+	0.903133i	$0.358739\pi$
$570$	8.96548	0.375523
$571$	−6.02327	−0.252066	−0.126033	−	0.992026i	$-0.540225\pi$
$571$	−6.02327	−0.252066	−0.126033	+	0.992026i	$0.540225\pi$
$572$	3.56456	0.149042
$573$	11.5600	0.482925
$574$	1.76040	0.0734779
$575$	12.3138	0.513522
$576$	4.85291	0.202204
$577$	1.08710	0.0452566	0.0226283	−	0.999744i	$-0.492797\pi$
$577$	1.08710	0.0452566	0.0226283	+	0.999744i	$0.492797\pi$
$578$	13.4811	0.560740
$579$	−8.47034	−0.352015
$580$	−12.1901	−0.506165
$581$	−12.8639	−0.533683
$582$	−15.7500	−0.652860
$583$	1.08546	0.0449551
$584$	19.6395	0.812687
$585$	12.2731	0.507431
$586$	16.7756	0.692992
$587$	−19.6433	−0.810765	−0.405382	−	0.914147i	$-0.632862\pi$
$587$	−19.6433	−0.810765	−0.405382	+	0.914147i	$0.632862\pi$
$588$	1.24474	0.0513321
$589$	17.5424	0.722823
$590$	11.5666	0.476189
$591$	−8.76026	−0.360349
$592$	0.403236	0.0165729
$593$	20.3632	0.836216	0.418108	−	0.908397i	$-0.362693\pi$
$593$	20.3632	0.836216	0.418108	+	0.908397i	$0.362693\pi$
$594$	0.618841	0.0253914
$595$	−3.72231	−0.152600
$596$	−22.9096	−0.938413
$597$	−22.6094	−0.925343
$598$	9.97715	0.407996
$599$	−31.0351	−1.26806	−0.634030	−	0.773308i	$-0.718601\pi$
$599$	−31.0351	−1.26806	−0.634030	+	0.773308i	$0.718601\pi$
$600$	−12.1636	−0.496576
$601$	−35.8022	−1.46040	−0.730201	−	0.683232i	$-0.760574\pi$
$601$	−35.8022	−1.46040	−0.730201	+	0.683232i	$0.760574\pi$
$602$	−5.14586	−0.209729
$603$	3.24868	0.132297
$604$	−8.41433	−0.342374
$605$	32.0224	1.30190
$606$	−1.13152	−0.0459649
$607$	−27.9748	−1.13546	−0.567731	−	0.823214i	$-0.692179\pi$
$607$	−27.9748	−1.13546	−0.567731	+	0.823214i	$0.692179\pi$
$608$	19.1786	0.777797
$609$	3.20901	0.130036
$610$	−4.65082	−0.188306
$611$	16.1984	0.655317
$612$	−1.51822	−0.0613702
$613$	11.4650	0.463068	0.231534	−	0.972827i	$-0.425626\pi$
$613$	11.4650	0.463068	0.231534	+	0.972827i	$0.425626\pi$
$614$	−22.0876	−0.891381
$615$	−6.18188	−0.249277
$616$	2.00798	0.0809037
$617$	34.2539	1.37901	0.689504	−	0.724281i	$-0.257828\pi$
$617$	34.2539	1.37901	0.689504	+	0.724281i	$0.257828\pi$
$618$	−10.4390	−0.419919
$619$	41.3180	1.66071	0.830355	−	0.557235i	$-0.188138\pi$
$619$	41.3180	1.66071	0.830355	+	0.557235i	$0.188138\pi$
$620$	−19.7132	−0.791701
$621$	−2.85470	−0.114555
$622$	21.4215	0.858925
$623$	0.646252	0.0258916
$624$	0.156222	0.00625387
$625$	−27.9611	−1.11844
$626$	7.58170	0.303026
$627$	2.40712	0.0961311
$628$	0.865289	0.0345288
$629$	12.6611	0.504831
$630$	2.65220	0.105666
$631$	−2.97622	−0.118481	−0.0592407	−	0.998244i	$-0.518868\pi$
$631$	−2.97622	−0.118481	−0.0592407	+	0.998244i	$0.518868\pi$
$632$	−31.1077	−1.23740
$633$	11.0757	0.440220
$634$	−10.4245	−0.414010
$635$	39.6745	1.57444
$636$	1.89741	0.0752371
$637$	−4.02159	−0.159341
$638$	1.98587	0.0786212
$639$	11.4733	0.453875
$640$	−21.7579	−0.860058
$641$	23.5025	0.928293	0.464147	−	0.885758i	$-0.346361\pi$
$641$	23.5025	0.928293	0.464147	+	0.885758i	$0.346361\pi$
$642$	−12.5521	−0.495393
$643$	36.0438	1.42143	0.710715	−	0.703480i	$-0.248372\pi$
$643$	36.0438	1.42143	0.710715	+	0.703480i	$0.248372\pi$
$644$	−3.55335	−0.140022
$645$	18.0703	0.711518
$646$	3.58321	0.140980
$647$	23.0074	0.904514	0.452257	−	0.891888i	$-0.350619\pi$
$647$	23.0074	0.904514	0.452257	+	0.891888i	$0.350619\pi$
$648$	2.81987	0.110775
$649$	3.10548	0.121901
$650$	15.0758	0.591320
$651$	5.18946	0.203391
$652$	−19.7002	−0.771521
$653$	−6.42806	−0.251549	−0.125775	−	0.992059i	$-0.540142\pi$
$653$	−6.42806	−0.251549	−0.125775	+	0.992059i	$0.540142\pi$
$654$	−0.921576	−0.0360365
$655$	66.2296	2.58780
$656$	−0.0786878	−0.00307224
$657$	6.96468	0.271718
$658$	3.50044	0.136462
$659$	−26.7353	−1.04146	−0.520731	−	0.853721i	$-0.674340\pi$
$659$	−26.7353	−1.04146	−0.520731	+	0.853721i	$0.674340\pi$
$660$	−2.70499	−0.105291
$661$	11.7644	0.457580	0.228790	−	0.973476i	$-0.426523\pi$
$661$	11.7644	0.457580	0.228790	+	0.973476i	$0.426523\pi$
$662$	−24.8428	−0.965542
$663$	4.90516	0.190501
$664$	−36.2744	−1.40772
$665$	10.3163	0.400050
$666$	−9.02120	−0.349564
$667$	−9.16075	−0.354706
$668$	25.4762	0.985702
$669$	−12.2137	−0.472210
$670$	8.61615	0.332871
$671$	−1.24869	−0.0482050
$672$	5.67349	0.218860
$673$	16.8939	0.651212	0.325606	−	0.945505i	$-0.394432\pi$
$673$	16.8939	0.651212	0.325606	+	0.945505i	$0.394432\pi$
$674$	−25.2292	−0.971794
$675$	−4.31353	−0.166028
$676$	−3.94976	−0.151914
$677$	−12.5088	−0.480753	−0.240377	−	0.970680i	$-0.577271\pi$
$677$	−12.5088	−0.480753	−0.240377	+	0.970680i	$0.577271\pi$
$678$	−9.96072	−0.382539
$679$	−18.1231	−0.695500
$680$	−10.4964	−0.402520
$681$	−5.02074	−0.192395
$682$	3.21145	0.122973
$683$	24.1673	0.924737	0.462368	−	0.886688i	$-0.347000\pi$
$683$	24.1673	0.924737	0.462368	+	0.886688i	$0.347000\pi$
$684$	4.20770	0.160886
$685$	37.5324	1.43404
$686$	−0.869058	−0.0331808
$687$	17.1485	0.654254
$688$	0.230013	0.00876917
$689$	−6.13028	−0.233545
$690$	−7.57123	−0.288232
$691$	−26.1032	−0.993014	−0.496507	−	0.868033i	$-0.665384\pi$
$691$	−26.1032	−0.993014	−0.496507	+	0.868033i	$0.665384\pi$
$692$	19.1573	0.728251
$693$	0.712082	0.0270498
$694$	−28.5473	−1.08364
$695$	−32.1654	−1.22010
$696$	9.04898	0.343001
$697$	−2.47070	−0.0935843
$698$	−22.2468	−0.842053
$699$	26.1749	0.990024
$700$	−5.36921	−0.202937
$701$	2.19236	0.0828042	0.0414021	−	0.999143i	$-0.486818\pi$
$701$	2.19236	0.0828042	0.0414021	+	0.999143i	$0.486818\pi$
$702$	−3.49499	−0.131910
$703$	−35.0900	−1.32344
$704$	3.45567	0.130240
$705$	−12.2923	−0.462953
$706$	21.7943	0.820239
$707$	−1.30201	−0.0489670
$708$	5.42846	0.204014
$709$	−1.02856	−0.0386284	−0.0193142	−	0.999813i	$-0.506148\pi$
$709$	−1.02856	−0.0386284	−0.0193142	+	0.999813i	$0.506148\pi$
$710$	30.4294	1.14199
$711$	−11.0316	−0.413718
$712$	1.82235	0.0682953
$713$	−14.8143	−0.554801
$714$	1.06000	0.0396694
$715$	8.73946	0.326837
$716$	−9.22107	−0.344608
$717$	3.38308	0.126343
$718$	−12.0062	−0.448066
$719$	13.3528	0.497975	0.248987	−	0.968507i	$-0.419902\pi$
$719$	13.3528	0.497975	0.248987	+	0.968507i	$0.419902\pi$
$720$	−0.118550	−0.00441809
$721$	−12.0119	−0.447346
$722$	6.58131	0.244931
$723$	−13.9845	−0.520088
$724$	4.73611	0.176016
$725$	−13.8422	−0.514085
$726$	−9.11898	−0.338437
$727$	−1.80430	−0.0669177	−0.0334588	−	0.999440i	$-0.510652\pi$
$727$	−1.80430	−0.0669177	−0.0334588	+	0.999440i	$0.510652\pi$
$728$	−11.3403	−0.420301
$729$	1.00000	0.0370370
$730$	18.4717	0.683669
$731$	7.22212	0.267120
$732$	−2.18274	−0.0806762
$733$	42.9866	1.58775	0.793873	−	0.608083i	$-0.208061\pi$
$733$	42.9866	1.58775	0.793873	+	0.608083i	$0.208061\pi$
$734$	−23.9670	−0.884639
$735$	3.05181	0.112568
$736$	−16.1961	−0.596996
$737$	2.31333	0.0852125
$738$	1.76040	0.0648014
$739$	0.550689	0.0202574	0.0101287	−	0.999949i	$-0.496776\pi$
$739$	0.550689	0.0202574	0.0101287	+	0.999949i	$0.496776\pi$
$740$	39.4321	1.44955
$741$	−13.5946	−0.499409
$742$	−1.32474	−0.0486328
$743$	−45.6409	−1.67440	−0.837200	−	0.546896i	$-0.815809\pi$
$743$	−45.6409	−1.67440	−0.837200	+	0.546896i	$0.815809\pi$
$744$	14.6336	0.536493
$745$	−56.1690	−2.05787
$746$	6.26425	0.229350
$747$	−12.8639	−0.470664
$748$	−1.08109	−0.0395287
$749$	−14.4434	−0.527749
$750$	1.82065	0.0664808
$751$	−7.06296	−0.257731	−0.128865	−	0.991662i	$-0.541134\pi$
$751$	−7.06296	−0.257731	−0.128865	+	0.991662i	$0.541134\pi$
$752$	−0.156465	−0.00570570
$753$	−12.9214	−0.470880
$754$	−11.2155	−0.408443
$755$	−20.6300	−0.750803
$756$	1.24474	0.0452707
$757$	42.9605	1.56143	0.780713	−	0.624889i	$-0.214856\pi$
$757$	42.9605	1.56143	0.780713	+	0.624889i	$0.214856\pi$
$758$	18.7094	0.679558
$759$	−2.03278	−0.0737852
$760$	29.0906	1.05523
$761$	−23.6023	−0.855583	−0.427791	−	0.903877i	$-0.640708\pi$
$761$	−23.6023	−0.855583	−0.427791	+	0.903877i	$0.640708\pi$
$762$	−11.2981	−0.409285
$763$	−1.06043	−0.0383901
$764$	14.3891	0.520581
$765$	−3.72231	−0.134581
$766$	0.869058	0.0314004
$767$	−17.5387	−0.633284
$768$	15.9018	0.573806
$769$	−52.8325	−1.90519	−0.952594	−	0.304245i	$-0.901596\pi$
$769$	−52.8325	−1.90519	−0.952594	+	0.304245i	$0.901596\pi$
$770$	1.88858	0.0680598
$771$	−4.45689	−0.160511
$772$	−10.5434	−0.379463
$773$	−53.0699	−1.90879	−0.954395	−	0.298546i	$-0.903499\pi$
$773$	−53.0699	−1.90879	−0.954395	+	0.298546i	$0.903499\pi$
$774$	−5.14586	−0.184964
$775$	−22.3849	−0.804089
$776$	−51.1047	−1.83455
$777$	−10.3804	−0.372396
$778$	11.1711	0.400504
$779$	6.84748	0.245337
$780$	15.2768	0.546997
$781$	8.16990	0.292342
$782$	−3.02597	−0.108209
$783$	3.20901	0.114681
$784$	0.0388458	0.00138735
$785$	2.12149	0.0757191
$786$	−18.8601	−0.672717
$787$	−39.0547	−1.39215	−0.696075	−	0.717969i	$-0.745072\pi$
$787$	−39.0547	−1.39215	−0.696075	+	0.717969i	$0.745072\pi$
$788$	−10.9042	−0.388447
$789$	26.0734	0.928238
$790$	−29.2581	−1.04096
$791$	−11.4615	−0.407524
$792$	2.00798	0.0713503
$793$	7.05214	0.250429
$794$	2.42118	0.0859246
$795$	4.65200	0.164989
$796$	−28.1428	−0.997496
$797$	36.2988	1.28577	0.642884	−	0.765963i	$-0.277738\pi$
$797$	36.2988	1.28577	0.642884	+	0.765963i	$0.277738\pi$
$798$	−2.93776	−0.103996
$799$	−4.91281	−0.173803
$800$	−24.4728	−0.865244
$801$	0.646252	0.0228342
$802$	2.87491	0.101517
$803$	4.95942	0.175014
$804$	4.04375	0.142612
$805$	−8.71199	−0.307057
$806$	−18.1371	−0.638853
$807$	−24.1132	−0.848823
$808$	−3.67149	−0.129162
$809$	6.16707	0.216823	0.108411	−	0.994106i	$-0.465424\pi$
$809$	6.16707	0.216823	0.108411	+	0.994106i	$0.465424\pi$
$810$	2.65220	0.0931888
$811$	25.4290	0.892933	0.446467	−	0.894800i	$-0.352682\pi$
$811$	25.4290	0.892933	0.446467	+	0.894800i	$0.352682\pi$
$812$	3.99437	0.140175
$813$	−17.6735	−0.619835
$814$	−6.42384	−0.225155
$815$	−48.3004	−1.69189
$816$	−0.0473805	−0.00165865
$817$	−20.0159	−0.700269
$818$	−8.98065	−0.314001
$819$	−4.02159	−0.140526
$820$	−7.69482	−0.268715
$821$	−13.6947	−0.477947	−0.238974	−	0.971026i	$-0.576811\pi$
$821$	−13.6947	−0.477947	−0.238974	+	0.971026i	$0.576811\pi$
$822$	−10.6881	−0.372789
$823$	25.2952	0.881735	0.440867	−	0.897572i	$-0.354671\pi$
$823$	25.2952	0.881735	0.440867	+	0.897572i	$0.354671\pi$
$824$	−33.8719	−1.17998
$825$	−3.07159	−0.106939
$826$	−3.79008	−0.131874
$827$	−35.3224	−1.22828	−0.614140	−	0.789197i	$-0.710497\pi$
$827$	−35.3224	−1.22828	−0.614140	+	0.789197i	$0.710497\pi$
$828$	−3.55335	−0.123487
$829$	−34.1660	−1.18663	−0.593317	−	0.804969i	$-0.702182\pi$
$829$	−34.1660	−1.18663	−0.593317	+	0.804969i	$0.702182\pi$
$830$	−34.1175	−1.18424
$831$	−13.5147	−0.468820
$832$	−19.5164	−0.676609
$833$	1.21971	0.0422604
$834$	9.15969	0.317174
$835$	62.4616	2.16157
$836$	2.99623	0.103627
$837$	5.18946	0.179374
$838$	−15.1019	−0.521686
$839$	42.2105	1.45727	0.728635	−	0.684902i	$-0.240155\pi$
$839$	42.2105	1.45727	0.728635	+	0.684902i	$0.240155\pi$
$840$	8.60569	0.296925
$841$	−18.7023	−0.644905
$842$	−2.91654	−0.100511
$843$	−11.0748	−0.381436
$844$	13.7863	0.474546
$845$	−9.68389	−0.333136
$846$	3.50044	0.120348
$847$	−10.4929	−0.360542
$848$	0.0592143	0.00203343
$849$	−14.9421	−0.512812
$850$	−4.57233	−0.156830
$851$	29.6330	1.01581
$852$	14.2812	0.489265
$853$	39.8269	1.36365	0.681823	−	0.731517i	$-0.261187\pi$
$853$	39.8269	1.36365	0.681823	+	0.731517i	$0.261187\pi$
$854$	1.52396	0.0521487
$855$	10.3163	0.352811
$856$	−40.7283	−1.39207
$857$	−28.8073	−0.984037	−0.492019	−	0.870585i	$-0.663741\pi$
$857$	−28.8073	−0.984037	−0.492019	+	0.870585i	$0.663741\pi$
$858$	−2.48872	−0.0849636
$859$	7.89364	0.269328	0.134664	−	0.990891i	$-0.457005\pi$
$859$	7.89364	0.269328	0.134664	+	0.990891i	$0.457005\pi$
$860$	22.4928	0.766998
$861$	2.02565	0.0690338
$862$	1.46462	0.0498852
$863$	22.9224	0.780287	0.390143	−	0.920754i	$-0.372425\pi$
$863$	22.9224	0.780287	0.390143	+	0.920754i	$0.372425\pi$
$864$	5.67349	0.193016
$865$	46.9693	1.59700
$866$	−25.6038	−0.870054
$867$	15.5123	0.526826
$868$	6.45951	0.219250
$869$	−7.85543	−0.266477
$870$	8.51093	0.288548
$871$	−13.0649	−0.442686
$872$	−2.99027	−0.101263
$873$	−18.1231	−0.613374
$874$	8.38642	0.283675
$875$	2.09497	0.0708230
$876$	8.66919	0.292905
$877$	−20.0714	−0.677764	−0.338882	−	0.940829i	$-0.610049\pi$
$877$	−20.0714	−0.677764	−0.338882	+	0.940829i	$0.610049\pi$
$878$	3.98761	0.134575
$879$	19.3031	0.651079
$880$	−0.0844173	−0.00284571
$881$	−23.1525	−0.780026	−0.390013	−	0.920809i	$-0.627530\pi$
$881$	−23.1525	−0.780026	−0.390013	+	0.920809i	$0.627530\pi$
$882$	−0.869058	−0.0292627
$883$	−37.3045	−1.25540	−0.627699	−	0.778456i	$-0.716003\pi$
$883$	−37.3045	−1.25540	−0.627699	+	0.778456i	$0.716003\pi$
$884$	6.10564	0.205355
$885$	13.3093	0.447388
$886$	23.2629	0.781532
$887$	20.3074	0.681855	0.340927	−	0.940090i	$-0.389259\pi$
$887$	20.3074	0.681855	0.340927	+	0.940090i	$0.389259\pi$
$888$	−29.2714	−0.982285
$889$	−13.0003	−0.436017
$890$	1.71399	0.0574531
$891$	0.712082	0.0238557
$892$	−15.2029	−0.509030
$893$	13.6158	0.455634
$894$	15.9951	0.534958
$895$	−22.6079	−0.755700
$896$	7.12953	0.238181
$897$	11.4804	0.383320
$898$	−3.60973	−0.120458
$899$	16.6530	0.555409
$900$	−5.36921	−0.178974
$901$	1.85925	0.0619407
$902$	1.25355	0.0417387
$903$	−5.92118	−0.197045
$904$	−32.3199	−1.07494
$905$	11.6118	0.385990
$906$	5.87477	0.195176
$907$	57.0805	1.89533	0.947664	−	0.319269i	$-0.103437\pi$
$907$	57.0805	1.89533	0.947664	+	0.319269i	$0.103437\pi$
$908$	−6.24950	−0.207397
$909$	−1.30201	−0.0431848
$910$	−10.6661	−0.353576
$911$	33.3581	1.10520	0.552602	−	0.833445i	$-0.313635\pi$
$911$	33.3581	1.10520	0.552602	+	0.833445i	$0.313635\pi$
$912$	0.131314	0.00434825
$913$	−9.16012	−0.303156
$914$	32.5038	1.07513
$915$	−5.35156	−0.176917
$916$	21.3453	0.705269
$917$	−21.7017	−0.716655
$918$	1.06000	0.0349851
$919$	43.1144	1.42221	0.711107	−	0.703084i	$-0.248194\pi$
$919$	43.1144	1.42221	0.711107	+	0.703084i	$0.248194\pi$
$920$	−24.5666	−0.809938
$921$	−25.4155	−0.837469
$922$	26.6193	0.876661
$923$	−46.1407	−1.51874
$924$	0.886355	0.0291589
$925$	44.7763	1.47224
$926$	−11.4909	−0.377614
$927$	−12.0119	−0.394522
$928$	18.2063	0.597651
$929$	31.2176	1.02422	0.512108	−	0.858921i	$-0.328865\pi$
$929$	31.2176	1.02422	0.512108	+	0.858921i	$0.328865\pi$
$930$	13.7635	0.451322
$931$	−3.38040	−0.110788
$932$	32.5808	1.06722
$933$	24.6491	0.806976
$934$	−3.34025	−0.109296
$935$	−2.65059	−0.0866837
$936$	−11.3403	−0.370671
$937$	59.2654	1.93612	0.968058	−	0.250727i	$-0.0806695\pi$
$937$	59.2654	1.93612	0.968058	+	0.250727i	$0.0806695\pi$
$938$	−2.82329	−0.0921838
$939$	8.72404	0.284698
$940$	−15.3006	−0.499051
$941$	−4.70864	−0.153497	−0.0767486	−	0.997050i	$-0.524454\pi$
$941$	−4.70864	−0.153497	−0.0767486	+	0.997050i	$0.524454\pi$
$942$	−0.604133	−0.0196837
$943$	−5.78260	−0.188307
$944$	0.169411	0.00551387
$945$	3.05181	0.0992753
$946$	−3.66427	−0.119136
$947$	−25.5318	−0.829672	−0.414836	−	0.909896i	$-0.636161\pi$
$947$	−25.5318	−0.829672	−0.414836	+	0.909896i	$0.636161\pi$
$948$	−13.7315	−0.445978
$949$	−28.0091	−0.909213
$950$	12.6721	0.411138
$951$	−11.9952	−0.388970
$952$	3.43941	0.111472
$953$	−23.9119	−0.774581	−0.387290	−	0.921958i	$-0.626589\pi$
$953$	−23.9119	−0.774581	−0.387290	+	0.921958i	$0.626589\pi$
$954$	−1.32474	−0.0428901
$955$	35.2789	1.14160
$956$	4.21104	0.136195
$957$	2.28508	0.0738661
$958$	−12.1936	−0.393958
$959$	−12.2984	−0.397137
$960$	14.8101	0.477995
$961$	−4.06953	−0.131275
$962$	36.2795	1.16970
$963$	−14.4434	−0.465431
$964$	−17.4070	−0.560641
$965$	−25.8499	−0.832136
$966$	2.48090	0.0798216
$967$	8.91273	0.286614	0.143307	−	0.989678i	$-0.454226\pi$
$967$	8.91273	0.286614	0.143307	+	0.989678i	$0.454226\pi$
$968$	−29.5887	−0.951016
$969$	4.12310	0.132453
$970$	−48.0661	−1.54331
$971$	−17.1131	−0.549185	−0.274593	−	0.961561i	$-0.588543\pi$
$971$	−17.1131	−0.549185	−0.274593	+	0.961561i	$0.588543\pi$
$972$	1.24474	0.0399250
$973$	10.5398	0.337890
$974$	−18.9361	−0.606751
$975$	17.3472	0.555556
$976$	−0.0681189	−0.00218043
$977$	11.8882	0.380336	0.190168	−	0.981752i	$-0.439097\pi$
$977$	11.8882	0.380336	0.190168	+	0.981752i	$0.439097\pi$
$978$	13.7544	0.439818
$979$	0.460185	0.0147076
$980$	3.79870	0.121345
$981$	−1.06043	−0.0338569
$982$	25.3704	0.809601
$983$	43.8613	1.39896	0.699480	−	0.714653i	$-0.253415\pi$
$983$	43.8613	1.39896	0.699480	+	0.714653i	$0.253415\pi$
$984$	5.71205	0.182093
$985$	−26.7346	−0.851836
$986$	3.40154	0.108327
$987$	4.02786	0.128208
$988$	−16.9217	−0.538349
$989$	16.9032	0.537490
$990$	1.88858	0.0600231
$991$	13.6109	0.432365	0.216183	−	0.976353i	$-0.430639\pi$
$991$	13.6109	0.432365	0.216183	+	0.976353i	$0.430639\pi$
$992$	29.4423	0.934795
$993$	−28.5858	−0.907144
$994$	−9.97093	−0.316259
$995$	−68.9997	−2.18744
$996$	−16.0121	−0.507364
$997$	4.27269	0.135317	0.0676587	−	0.997709i	$-0.478447\pi$
$997$	4.27269	0.135317	0.0676587	+	0.997709i	$0.478447\pi$
$998$	−13.2999	−0.421002
$999$	−10.3804	−0.328422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.19	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.19	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.869058 q^{2} -1.00000 q^{3} -1.24474 q^{4} -3.05181 q^{5} +0.869058 q^{6} +1.00000 q^{7} +2.81987 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.869058 q^{2} -1.00000 q^{3} -1.24474 q^{4} -3.05181 q^{5} +0.869058 q^{6} +1.00000 q^{7} +2.81987 q^{8} +1.00000 q^{9} +2.65220 q^{10} +0.712082 q^{11} +1.24474 q^{12} -4.02159 q^{13} -0.869058 q^{14} +3.05181 q^{15} +0.0388458 q^{16} +1.21971 q^{17} -0.869058 q^{18} -3.38040 q^{19} +3.79870 q^{20} -1.00000 q^{21} -0.618841 q^{22} +2.85470 q^{23} -2.81987 q^{24} +4.31353 q^{25} +3.49499 q^{26} -1.00000 q^{27} -1.24474 q^{28} -3.20901 q^{29} -2.65220 q^{30} -5.18946 q^{31} -5.67349 q^{32} -0.712082 q^{33} -1.06000 q^{34} -3.05181 q^{35} -1.24474 q^{36} +10.3804 q^{37} +2.93776 q^{38} +4.02159 q^{39} -8.60569 q^{40} -2.02565 q^{41} +0.869058 q^{42} +5.92118 q^{43} -0.886355 q^{44} -3.05181 q^{45} -2.48090 q^{46} -4.02786 q^{47} -0.0388458 q^{48} +1.00000 q^{49} -3.74871 q^{50} -1.21971 q^{51} +5.00582 q^{52} +1.52434 q^{53} +0.869058 q^{54} -2.17314 q^{55} +2.81987 q^{56} +3.38040 q^{57} +2.78882 q^{58} +4.36113 q^{59} -3.79870 q^{60} -1.75357 q^{61} +4.50994 q^{62} +1.00000 q^{63} +4.85291 q^{64} +12.2731 q^{65} +0.618841 q^{66} +3.24868 q^{67} -1.51822 q^{68} -2.85470 q^{69} +2.65220 q^{70} +11.4733 q^{71} +2.81987 q^{72} +6.96468 q^{73} -9.02120 q^{74} -4.31353 q^{75} +4.20770 q^{76} +0.712082 q^{77} -3.49499 q^{78} -11.0316 q^{79} -0.118550 q^{80} +1.00000 q^{81} +1.76040 q^{82} -12.8639 q^{83} +1.24474 q^{84} -3.72231 q^{85} -5.14586 q^{86} +3.20901 q^{87} +2.00798 q^{88} +0.646252 q^{89} +2.65220 q^{90} -4.02159 q^{91} -3.55335 q^{92} +5.18946 q^{93} +3.50044 q^{94} +10.3163 q^{95} +5.67349 q^{96} -18.1231 q^{97} -0.869058 q^{98} +0.712082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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