LMFDB - Embedded newform 4031.2.a.c.1.57

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.57
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.25443 q^{2} +1.42218 q^{3} +3.08248 q^{4} -0.571069 q^{5} +3.20620 q^{6} -4.05014 q^{7} +2.44037 q^{8} -0.977417 q^{9} +O(q^{10})$	\(q+2.25443 q^{2} +1.42218 q^{3} +3.08248 q^{4} -0.571069 q^{5} +3.20620 q^{6} -4.05014 q^{7} +2.44037 q^{8} -0.977417 q^{9} -1.28744 q^{10} -3.49510 q^{11} +4.38382 q^{12} +2.86422 q^{13} -9.13078 q^{14} -0.812160 q^{15} -0.663295 q^{16} -0.197488 q^{17} -2.20352 q^{18} +2.37312 q^{19} -1.76030 q^{20} -5.76001 q^{21} -7.87947 q^{22} -0.387551 q^{23} +3.47064 q^{24} -4.67388 q^{25} +6.45720 q^{26} -5.65658 q^{27} -12.4845 q^{28} -1.00000 q^{29} -1.83096 q^{30} -4.31304 q^{31} -6.37610 q^{32} -4.97064 q^{33} -0.445224 q^{34} +2.31291 q^{35} -3.01286 q^{36} +0.179213 q^{37} +5.35005 q^{38} +4.07343 q^{39} -1.39362 q^{40} +0.176515 q^{41} -12.9856 q^{42} +0.927631 q^{43} -10.7736 q^{44} +0.558172 q^{45} -0.873709 q^{46} -6.38689 q^{47} -0.943321 q^{48} +9.40364 q^{49} -10.5370 q^{50} -0.280863 q^{51} +8.82890 q^{52} -3.30060 q^{53} -12.7524 q^{54} +1.99594 q^{55} -9.88385 q^{56} +3.37500 q^{57} -2.25443 q^{58} +6.86458 q^{59} -2.50346 q^{60} -8.63770 q^{61} -9.72346 q^{62} +3.95868 q^{63} -13.0479 q^{64} -1.63567 q^{65} -11.2060 q^{66} -2.19974 q^{67} -0.608753 q^{68} -0.551166 q^{69} +5.21430 q^{70} +1.85139 q^{71} -2.38526 q^{72} -3.66211 q^{73} +0.404024 q^{74} -6.64708 q^{75} +7.31509 q^{76} +14.1556 q^{77} +9.18328 q^{78} +11.2774 q^{79} +0.378787 q^{80} -5.11241 q^{81} +0.397942 q^{82} -2.65834 q^{83} -17.7551 q^{84} +0.112779 q^{85} +2.09128 q^{86} -1.42218 q^{87} -8.52934 q^{88} -6.67647 q^{89} +1.25836 q^{90} -11.6005 q^{91} -1.19462 q^{92} -6.13390 q^{93} -14.3988 q^{94} -1.35521 q^{95} -9.06793 q^{96} +2.33871 q^{97} +21.1999 q^{98} +3.41617 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.25443	1.59413	0.797063	−	0.603896i	$-0.206386\pi$
$2$	2.25443	1.59413	0.797063	+	0.603896i	$0.206386\pi$
$3$	1.42218	0.821093	0.410547	−	0.911840i	$-0.365338\pi$
$3$	1.42218	0.821093	0.410547	+	0.911840i	$0.365338\pi$
$4$	3.08248	1.54124
$5$	−0.571069	−0.255390	−0.127695	−	0.991814i	$-0.540758\pi$
$5$	−0.571069	−0.255390	−0.127695	+	0.991814i	$0.540758\pi$
$6$	3.20620	1.30893
$7$	−4.05014	−1.53081	−0.765405	−	0.643549i	$-0.777461\pi$
$7$	−4.05014	−1.53081	−0.765405	+	0.643549i	$0.777461\pi$
$8$	2.44037	0.862801
$9$	−0.977417	−0.325806
$10$	−1.28744	−0.407123
$11$	−3.49510	−1.05381	−0.526906	−	0.849924i	$-0.676648\pi$
$11$	−3.49510	−1.05381	−0.526906	+	0.849924i	$0.676648\pi$
$12$	4.38382	1.26550
$13$	2.86422	0.794393	0.397196	−	0.917734i	$-0.369983\pi$
$13$	2.86422	0.794393	0.397196	+	0.917734i	$0.369983\pi$
$14$	−9.13078	−2.44030
$15$	−0.812160	−0.209699
$16$	−0.663295	−0.165824
$17$	−0.197488	−0.0478979	−0.0239490	−	0.999713i	$-0.507624\pi$
$17$	−0.197488	−0.0478979	−0.0239490	+	0.999713i	$0.507624\pi$
$18$	−2.20352	−0.519375
$19$	2.37312	0.544431	0.272216	−	0.962236i	$-0.412244\pi$
$19$	2.37312	0.544431	0.272216	+	0.962236i	$0.412244\pi$
$20$	−1.76030	−0.393616
$21$	−5.76001	−1.25694
$22$	−7.87947	−1.67991
$23$	−0.387551	−0.0808100	−0.0404050	−	0.999183i	$-0.512865\pi$
$23$	−0.387551	−0.0808100	−0.0404050	+	0.999183i	$0.512865\pi$
$24$	3.47064	0.708441
$25$	−4.67388	−0.934776
$26$	6.45720	1.26636
$27$	−5.65658	−1.08861
$28$	−12.4845	−2.35934
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−1.83096	−0.334286
$31$	−4.31304	−0.774645	−0.387322	−	0.921944i	$-0.626600\pi$
$31$	−4.31304	−0.774645	−0.387322	+	0.921944i	$0.626600\pi$
$32$	−6.37610	−1.12715
$33$	−4.97064	−0.865278
$34$	−0.445224	−0.0763553
$35$	2.31291	0.390953
$36$	−3.01286	−0.502144
$37$	0.179213	0.0294624	0.0147312	−	0.999891i	$-0.495311\pi$
$37$	0.179213	0.0294624	0.0147312	+	0.999891i	$0.495311\pi$
$38$	5.35005	0.867892
$39$	4.07343	0.652271
$40$	−1.39362	−0.220351
$41$	0.176515	0.0275670	0.0137835	−	0.999905i	$-0.495612\pi$
$41$	0.176515	0.0275670	0.0137835	+	0.999905i	$0.495612\pi$
$42$	−12.9856	−2.00372
$43$	0.927631	0.141462	0.0707312	−	0.997495i	$-0.477467\pi$
$43$	0.927631	0.141462	0.0707312	+	0.997495i	$0.477467\pi$
$44$	−10.7736	−1.62417
$45$	0.558172	0.0832074
$46$	−0.873709	−0.128821
$47$	−6.38689	−0.931624	−0.465812	−	0.884884i	$-0.654238\pi$
$47$	−6.38689	−0.931624	−0.465812	+	0.884884i	$0.654238\pi$
$48$	−0.943321	−0.136157
$49$	9.40364	1.34338
$50$	−10.5370	−1.49015
$51$	−0.280863	−0.0393287
$52$	8.82890	1.22435
$53$	−3.30060	−0.453372	−0.226686	−	0.973968i	$-0.572789\pi$
$53$	−3.30060	−0.453372	−0.226686	+	0.973968i	$0.572789\pi$
$54$	−12.7524	−1.73538
$55$	1.99594	0.269133
$56$	−9.88385	−1.32078
$57$	3.37500	0.447029
$58$	−2.25443	−0.296022
$59$	6.86458	0.893693	0.446846	−	0.894611i	$-0.352547\pi$
$59$	6.86458	0.893693	0.446846	+	0.894611i	$0.352547\pi$
$60$	−2.50346	−0.323196
$61$	−8.63770	−1.10594	−0.552972	−	0.833200i	$-0.686506\pi$
$61$	−8.63770	−1.10594	−0.552972	+	0.833200i	$0.686506\pi$
$62$	−9.72346	−1.23488
$63$	3.95868	0.498746
$64$	−13.0479	−1.63099
$65$	−1.63567	−0.202880
$66$	−11.2060	−1.37936
$67$	−2.19974	−0.268741	−0.134370	−	0.990931i	$-0.542901\pi$
$67$	−2.19974	−0.268741	−0.134370	+	0.990931i	$0.542901\pi$
$68$	−0.608753	−0.0738221
$69$	−0.551166	−0.0663526
$70$	5.21430	0.623228
$71$	1.85139	0.219719	0.109860	−	0.993947i	$-0.464960\pi$
$71$	1.85139	0.219719	0.109860	+	0.993947i	$0.464960\pi$
$72$	−2.38526	−0.281106
$73$	−3.66211	−0.428618	−0.214309	−	0.976766i	$-0.568750\pi$
$73$	−3.66211	−0.428618	−0.214309	+	0.976766i	$0.568750\pi$
$74$	0.404024	0.0469668
$75$	−6.64708	−0.767539
$76$	7.31509	0.839098
$77$	14.1556	1.61319
$78$	9.18328	1.03980
$79$	11.2774	1.26881	0.634405	−	0.773000i	$-0.281245\pi$
$79$	11.2774	1.26881	0.634405	+	0.773000i	$0.281245\pi$
$80$	0.378787	0.0423496
$81$	−5.11241	−0.568045
$82$	0.397942	0.0439453
$83$	−2.65834	−0.291791	−0.145895	−	0.989300i	$-0.546606\pi$
$83$	−2.65834	−0.291791	−0.145895	+	0.989300i	$0.546606\pi$
$84$	−17.7551	−1.93724
$85$	0.112779	0.0122326
$86$	2.09128	0.225509
$87$	−1.42218	−0.152473
$88$	−8.52934	−0.909230
$89$	−6.67647	−0.707704	−0.353852	−	0.935301i	$-0.615128\pi$
$89$	−6.67647	−0.707704	−0.353852	+	0.935301i	$0.615128\pi$
$90$	1.25836	0.132643
$91$	−11.6005	−1.21606
$92$	−1.19462	−0.124548
$93$	−6.13390	−0.636056
$94$	−14.3988	−1.48513
$95$	−1.35521	−0.139042
$96$	−9.06793	−0.925492
$97$	2.33871	0.237460	0.118730	−	0.992927i	$-0.462118\pi$
$97$	2.33871	0.237460	0.118730	+	0.992927i	$0.462118\pi$
$98$	21.1999	2.14151
$99$	3.41617	0.343338
$100$	−14.4071	−1.44071
$101$	16.1041	1.60241	0.801207	−	0.598388i	$-0.204192\pi$
$101$	16.1041	1.60241	0.801207	+	0.598388i	$0.204192\pi$
$102$	−0.633187	−0.0626949
$103$	−4.30780	−0.424460	−0.212230	−	0.977220i	$-0.568073\pi$
$103$	−4.30780	−0.424460	−0.212230	+	0.977220i	$0.568073\pi$
$104$	6.98977	0.685403
$105$	3.28936	0.321009
$106$	−7.44098	−0.722732
$107$	14.6365	1.41496	0.707480	−	0.706734i	$-0.249832\pi$
$107$	14.6365	1.41496	0.707480	+	0.706734i	$0.249832\pi$
$108$	−17.4363	−1.67781
$109$	16.5421	1.58445	0.792224	−	0.610231i	$-0.208923\pi$
$109$	16.5421	1.58445	0.792224	+	0.610231i	$0.208923\pi$
$110$	4.49972	0.429031
$111$	0.254872	0.0241914
$112$	2.68644	0.253844
$113$	−16.6701	−1.56819	−0.784094	−	0.620643i	$-0.786872\pi$
$113$	−16.6701	−1.56819	−0.784094	+	0.620643i	$0.786872\pi$
$114$	7.60871	0.712621
$115$	0.221318	0.0206380
$116$	−3.08248	−0.286201
$117$	−2.79954	−0.258818
$118$	15.4758	1.42466
$119$	0.799855	0.0733226
$120$	−1.98197	−0.180928
$121$	1.21571	0.110519
$122$	−19.4731	−1.76301
$123$	0.251035	0.0226351
$124$	−13.2948	−1.19391
$125$	5.52445	0.494122
$126$	8.92458	0.795064
$127$	7.18313	0.637400	0.318700	−	0.947856i	$-0.396754\pi$
$127$	7.18313	0.637400	0.318700	+	0.947856i	$0.396754\pi$
$128$	−16.6635	−1.47286
$129$	1.31925	0.116154
$130$	−3.68751	−0.323416
$131$	−12.2928	−1.07403	−0.537015	−	0.843572i	$-0.680448\pi$
$131$	−12.2928	−1.07403	−0.537015	+	0.843572i	$0.680448\pi$
$132$	−15.3219	−1.33360
$133$	−9.61148	−0.833421
$134$	−4.95916	−0.428406
$135$	3.23030	0.278020
$136$	−0.481945	−0.0413264
$137$	5.45584	0.466124	0.233062	−	0.972462i	$-0.425126\pi$
$137$	5.45584	0.466124	0.233062	+	0.972462i	$0.425126\pi$
$138$	−1.24257	−0.105774
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	7.12948	0.602551
$141$	−9.08328	−0.764950
$142$	4.17383	0.350260
$143$	−10.0107	−0.837140
$144$	0.648315	0.0540263
$145$	0.571069	0.0474247
$146$	−8.25599	−0.683271
$147$	13.3736	1.10304
$148$	0.552419	0.0454086
$149$	−9.84994	−0.806939	−0.403469	−	0.914993i	$-0.632196\pi$
$149$	−9.84994	−0.806939	−0.403469	+	0.914993i	$0.632196\pi$
$150$	−14.9854	−1.22355
$151$	−7.56650	−0.615753	−0.307877	−	0.951426i	$-0.599618\pi$
$151$	−7.56650	−0.615753	−0.307877	+	0.951426i	$0.599618\pi$
$152$	5.79130	0.469736
$153$	0.193028	0.0156054
$154$	31.9130	2.57162
$155$	2.46304	0.197836
$156$	12.5562	1.00530
$157$	−2.89669	−0.231181	−0.115591	−	0.993297i	$-0.536876\pi$
$157$	−2.89669	−0.231181	−0.115591	+	0.993297i	$0.536876\pi$
$158$	25.4242	2.02264
$159$	−4.69403	−0.372260
$160$	3.64119	0.287861
$161$	1.56964	0.123705
$162$	−11.5256	−0.905536
$163$	−6.46240	−0.506174	−0.253087	−	0.967444i	$-0.581446\pi$
$163$	−6.46240	−0.506174	−0.253087	+	0.967444i	$0.581446\pi$
$164$	0.544103	0.0424873
$165$	2.83858	0.220983
$166$	−5.99305	−0.465151
$167$	13.0782	1.01202	0.506010	−	0.862528i	$-0.331120\pi$
$167$	13.0782	1.01202	0.506010	+	0.862528i	$0.331120\pi$
$168$	−14.0566	−1.08449
$169$	−4.79623	−0.368940
$170$	0.254254	0.0195004
$171$	−2.31953	−0.177379
$172$	2.85940	0.218027
$173$	−11.0573	−0.840673	−0.420337	−	0.907368i	$-0.638088\pi$
$173$	−11.0573	−0.840673	−0.420337	+	0.907368i	$0.638088\pi$
$174$	−3.20620	−0.243062
$175$	18.9299	1.43096
$176$	2.31828	0.174747
$177$	9.76264	0.733805
$178$	−15.0517	−1.12817
$179$	8.60372	0.643072	0.321536	−	0.946897i	$-0.395801\pi$
$179$	8.60372	0.643072	0.321536	+	0.946897i	$0.395801\pi$
$180$	1.72055	0.128242
$181$	−15.1734	−1.12783	−0.563914	−	0.825833i	$-0.690705\pi$
$181$	−15.1734	−1.12783	−0.563914	+	0.825833i	$0.690705\pi$
$182$	−26.1526	−1.93856
$183$	−12.2843	−0.908083
$184$	−0.945769	−0.0697230
$185$	−0.102343	−0.00752439
$186$	−13.8285	−1.01395
$187$	0.690241	0.0504754
$188$	−19.6874	−1.43585
$189$	22.9100	1.66645
$190$	−3.05524	−0.221651
$191$	1.86009	0.134591	0.0672957	−	0.997733i	$-0.478563\pi$
$191$	1.86009	0.134591	0.0672957	+	0.997733i	$0.478563\pi$
$192$	−18.5564	−1.33919
$193$	20.3472	1.46463	0.732313	−	0.680969i	$-0.238441\pi$
$193$	20.3472	1.46463	0.732313	+	0.680969i	$0.238441\pi$
$194$	5.27246	0.378540
$195$	−2.32621	−0.166583
$196$	28.9865	2.07046
$197$	−4.81644	−0.343157	−0.171579	−	0.985170i	$-0.554887\pi$
$197$	−4.81644	−0.343157	−0.171579	+	0.985170i	$0.554887\pi$
$198$	7.70153	0.547324
$199$	4.31025	0.305546	0.152773	−	0.988261i	$-0.451180\pi$
$199$	4.31025	0.305546	0.152773	+	0.988261i	$0.451180\pi$
$200$	−11.4060	−0.806526
$201$	−3.12841	−0.220661
$202$	36.3055	2.55445
$203$	4.05014	0.284264
$204$	−0.865753	−0.0606148
$205$	−0.100802	−0.00704033
$206$	−9.71165	−0.676642
$207$	0.378799	0.0263284
$208$	−1.89982	−0.131729
$209$	−8.29429	−0.573728
$210$	7.41565	0.511728
$211$	18.2783	1.25833	0.629165	−	0.777272i	$-0.283397\pi$
$211$	18.2783	1.25833	0.629165	+	0.777272i	$0.283397\pi$
$212$	−10.1740	−0.698754
$213$	2.63300	0.180410
$214$	32.9969	2.25562
$215$	−0.529741	−0.0361280
$216$	−13.8042	−0.939254
$217$	17.4684	1.18583
$218$	37.2931	2.52581
$219$	−5.20817	−0.351935
$220$	6.15244	0.414797
$221$	−0.565650	−0.0380498
$222$	0.574592	0.0385641
$223$	−17.1413	−1.14787	−0.573934	−	0.818901i	$-0.694584\pi$
$223$	−17.1413	−1.14787	−0.573934	+	0.818901i	$0.694584\pi$
$224$	25.8241	1.72544
$225$	4.56833	0.304555
$226$	−37.5816	−2.49989
$227$	22.6300	1.50200	0.751002	−	0.660300i	$-0.229571\pi$
$227$	22.6300	1.50200	0.751002	+	0.660300i	$0.229571\pi$
$228$	10.4033	0.688978
$229$	−17.1683	−1.13451	−0.567257	−	0.823541i	$-0.691995\pi$
$229$	−17.1683	−1.13451	−0.567257	+	0.823541i	$0.691995\pi$
$230$	0.498948	0.0328996
$231$	20.1318	1.32458
$232$	−2.44037	−0.160218
$233$	14.1415	0.926439	0.463219	−	0.886244i	$-0.346694\pi$
$233$	14.1415	0.926439	0.463219	+	0.886244i	$0.346694\pi$
$234$	−6.31138	−0.412588
$235$	3.64735	0.237927
$236$	21.1599	1.37739
$237$	16.0385	1.04181
$238$	1.80322	0.116885
$239$	−15.5495	−1.00581	−0.502906	−	0.864341i	$-0.667736\pi$
$239$	−15.5495	−1.00581	−0.502906	+	0.864341i	$0.667736\pi$
$240$	0.538701	0.0347730
$241$	−4.71348	−0.303622	−0.151811	−	0.988410i	$-0.548511\pi$
$241$	−4.71348	−0.303622	−0.151811	+	0.988410i	$0.548511\pi$
$242$	2.74074	0.176182
$243$	9.69902	0.622192
$244$	−26.6255	−1.70452
$245$	−5.37012	−0.343085
$246$	0.565943	0.0360832
$247$	6.79715	0.432492
$248$	−10.5254	−0.668364
$249$	−3.78063	−0.239587
$250$	12.4545	0.787692
$251$	−21.5282	−1.35885	−0.679424	−	0.733746i	$-0.737770\pi$
$251$	−21.5282	−1.35885	−0.679424	+	0.733746i	$0.737770\pi$
$252$	12.2025	0.768687
$253$	1.35453	0.0851586
$254$	16.1939	1.01610
$255$	0.160392	0.0100441
$256$	−11.4709	−0.716929
$257$	3.09526	0.193077	0.0965385	−	0.995329i	$-0.469223\pi$
$257$	3.09526	0.193077	0.0965385	+	0.995329i	$0.469223\pi$
$258$	2.97417	0.185164
$259$	−0.725837	−0.0451013
$260$	−5.04191	−0.312686
$261$	0.977417	0.0605006
$262$	−27.7134	−1.71214
$263$	7.96540	0.491167	0.245584	−	0.969375i	$-0.421020\pi$
$263$	7.96540	0.491167	0.245584	+	0.969375i	$0.421020\pi$
$264$	−12.1302	−0.746563
$265$	1.88487	0.115786
$266$	−21.6684	−1.32858
$267$	−9.49511	−0.581091
$268$	−6.78064	−0.414193
$269$	−12.9155	−0.787474	−0.393737	−	0.919223i	$-0.628818\pi$
$269$	−12.9155	−0.787474	−0.393737	+	0.919223i	$0.628818\pi$
$270$	7.28249	0.443199
$271$	4.11864	0.250190	0.125095	−	0.992145i	$-0.460076\pi$
$271$	4.11864	0.250190	0.125095	+	0.992145i	$0.460076\pi$
$272$	0.130993	0.00794261
$273$	−16.4980	−0.998502
$274$	12.2998	0.743060
$275$	16.3357	0.985078
$276$	−1.69896	−0.102265
$277$	−16.6856	−1.00254	−0.501271	−	0.865290i	$-0.667134\pi$
$277$	−16.6856	−1.00254	−0.501271	+	0.865290i	$0.667134\pi$
$278$	−2.25443	−0.135212
$279$	4.21564	0.252384
$280$	5.64435	0.337315
$281$	22.0564	1.31577	0.657887	−	0.753117i	$-0.271451\pi$
$281$	22.0564	1.31577	0.657887	+	0.753117i	$0.271451\pi$
$282$	−20.4777	−1.21943
$283$	0.533412	0.0317081	0.0158540	−	0.999874i	$-0.494953\pi$
$283$	0.533412	0.0317081	0.0158540	+	0.999874i	$0.494953\pi$
$284$	5.70686	0.338640
$285$	−1.92735	−0.114167
$286$	−22.5686	−1.33451
$287$	−0.714911	−0.0421999
$288$	6.23210	0.367230
$289$	−16.9610	−0.997706
$290$	1.28744	0.0756009
$291$	3.32605	0.194976
$292$	−11.2884	−0.660602
$293$	−0.689313	−0.0402701	−0.0201350	−	0.999797i	$-0.506410\pi$
$293$	−0.689313	−0.0402701	−0.0201350	+	0.999797i	$0.506410\pi$
$294$	30.1500	1.75838
$295$	−3.92015	−0.228240
$296$	0.437346	0.0254202
$297$	19.7703	1.14719
$298$	−22.2060	−1.28636
$299$	−1.11003	−0.0641949
$300$	−20.4895	−1.18296
$301$	−3.75704	−0.216552
$302$	−17.0582	−0.981588
$303$	22.9028	1.31573
$304$	−1.57408	−0.0902796
$305$	4.93272	0.282447
$306$	0.435170	0.0248770
$307$	27.5734	1.57370	0.786848	−	0.617147i	$-0.211712\pi$
$307$	27.5734	1.57370	0.786848	+	0.617147i	$0.211712\pi$
$308$	43.6344	2.48630
$309$	−6.12644	−0.348521
$310$	5.55276	0.315376
$311$	33.3481	1.89100	0.945499	−	0.325624i	$-0.105575\pi$
$311$	33.3481	1.89100	0.945499	+	0.325624i	$0.105575\pi$
$312$	9.94068	0.562780
$313$	5.09455	0.287961	0.143981	−	0.989581i	$-0.454010\pi$
$313$	5.09455	0.287961	0.143981	+	0.989581i	$0.454010\pi$
$314$	−6.53040	−0.368532
$315$	−2.26068	−0.127375
$316$	34.7624	1.95554
$317$	13.3601	0.750376	0.375188	−	0.926949i	$-0.377578\pi$
$317$	13.3601	0.750376	0.375188	+	0.926949i	$0.377578\pi$
$318$	−10.5824	−0.593430
$319$	3.49510	0.195688
$320$	7.45125	0.416537
$321$	20.8156	1.16181
$322$	3.53865	0.197201
$323$	−0.468664	−0.0260771
$324$	−15.7589	−0.875493
$325$	−13.3870	−0.742579
$326$	−14.5690	−0.806905
$327$	23.5258	1.30098
$328$	0.430762	0.0237849
$329$	25.8678	1.42614
$330$	6.39939	0.352275
$331$	−13.6666	−0.751184	−0.375592	−	0.926785i	$-0.622561\pi$
$331$	−13.6666	−0.751184	−0.375592	+	0.926785i	$0.622561\pi$
$332$	−8.19427	−0.449719
$333$	−0.175166	−0.00959902
$334$	29.4839	1.61329
$335$	1.25620	0.0686336
$336$	3.82058	0.208430
$337$	−9.61701	−0.523872	−0.261936	−	0.965085i	$-0.584361\pi$
$337$	−9.61701	−0.523872	−0.261936	+	0.965085i	$0.584361\pi$
$338$	−10.8128	−0.588138
$339$	−23.7077	−1.28763
$340$	0.347640	0.0188534
$341$	15.0745	0.816330
$342$	−5.22923	−0.282764
$343$	−9.73510	−0.525646
$344$	2.26376	0.122054
$345$	0.314754	0.0169458
$346$	−24.9280	−1.34014
$347$	8.59887	0.461612	0.230806	−	0.973000i	$-0.425864\pi$
$347$	8.59887	0.461612	0.230806	+	0.973000i	$0.425864\pi$
$348$	−4.38382	−0.234998
$349$	−18.4978	−0.990162	−0.495081	−	0.868847i	$-0.664862\pi$
$349$	−18.4978	−0.990162	−0.495081	+	0.868847i	$0.664862\pi$
$350$	42.6762	2.28114
$351$	−16.2017	−0.864784
$352$	22.2851	1.18780
$353$	−1.25110	−0.0665893	−0.0332946	−	0.999446i	$-0.510600\pi$
$353$	−1.25110	−0.0665893	−0.0332946	+	0.999446i	$0.510600\pi$
$354$	22.0092	1.16978
$355$	−1.05727	−0.0561140
$356$	−20.5801	−1.09074
$357$	1.13753	0.0602047
$358$	19.3965	1.02514
$359$	−4.14614	−0.218825	−0.109413	−	0.993996i	$-0.534897\pi$
$359$	−4.14614	−0.218825	−0.109413	+	0.993996i	$0.534897\pi$
$360$	1.36215	0.0717914
$361$	−13.3683	−0.703594
$362$	−34.2074	−1.79790
$363$	1.72896	0.0907466
$364$	−35.7583	−1.87424
$365$	2.09132	0.109465
$366$	−27.6942	−1.44760
$367$	4.18528	0.218470	0.109235	−	0.994016i	$-0.465160\pi$
$367$	4.18528	0.218470	0.109235	+	0.994016i	$0.465160\pi$
$368$	0.257061	0.0134002
$369$	−0.172529	−0.00898149
$370$	−0.230725	−0.0119948
$371$	13.3679	0.694026
$372$	−18.9076	−0.980313
$373$	6.92483	0.358554	0.179277	−	0.983799i	$-0.442624\pi$
$373$	6.92483	0.358554	0.179277	+	0.983799i	$0.442624\pi$
$374$	1.55610	0.0804642
$375$	7.85674	0.405720
$376$	−15.5864	−0.803806
$377$	−2.86422	−0.147515
$378$	51.6490	2.65654
$379$	−6.30762	−0.324001	−0.162000	−	0.986791i	$-0.551795\pi$
$379$	−6.30762	−0.324001	−0.162000	+	0.986791i	$0.551795\pi$
$380$	−4.17742	−0.214297
$381$	10.2157	0.523365
$382$	4.19345	0.214556
$383$	−11.4421	−0.584664	−0.292332	−	0.956317i	$-0.594431\pi$
$383$	−11.4421	−0.584664	−0.292332	+	0.956317i	$0.594431\pi$
$384$	−23.6984	−1.20935
$385$	−8.08384	−0.411991
$386$	45.8715	2.33480
$387$	−0.906682	−0.0460892
$388$	7.20900	0.365982
$389$	−19.8750	−1.00770	−0.503852	−	0.863790i	$-0.668084\pi$
$389$	−19.8750	−1.00770	−0.503852	+	0.863790i	$0.668084\pi$
$390$	−5.24428	−0.265554
$391$	0.0765368	0.00387063
$392$	22.9484	1.15907
$393$	−17.4826	−0.881880
$394$	−10.8584	−0.547036
$395$	−6.44019	−0.324041
$396$	10.5303	0.529165
$397$	17.6593	0.886295	0.443148	−	0.896449i	$-0.353862\pi$
$397$	17.6593	0.886295	0.443148	+	0.896449i	$0.353862\pi$
$398$	9.71718	0.487078
$399$	−13.6692	−0.684316
$400$	3.10016	0.155008
$401$	−9.61590	−0.480195	−0.240098	−	0.970749i	$-0.577179\pi$
$401$	−9.61590	−0.480195	−0.240098	+	0.970749i	$0.577179\pi$
$402$	−7.05280	−0.351762
$403$	−12.3535	−0.615372
$404$	49.6403	2.46970
$405$	2.91953	0.145073
$406$	9.13078	0.453153
$407$	−0.626366	−0.0310478
$408$	−0.685410	−0.0339328
$409$	−38.2921	−1.89342	−0.946711	−	0.322083i	$-0.895617\pi$
$409$	−38.2921	−1.89342	−0.946711	+	0.322083i	$0.895617\pi$
$410$	−0.227252	−0.0112232
$411$	7.75916	0.382731
$412$	−13.2787	−0.654194
$413$	−27.8025	−1.36807
$414$	0.853978	0.0419707
$415$	1.51809	0.0745203
$416$	−18.2626	−0.895396
$417$	−1.42218	−0.0696442
$418$	−18.6989	−0.914595
$419$	−15.2303	−0.744048	−0.372024	−	0.928223i	$-0.621336\pi$
$419$	−15.2303	−0.744048	−0.372024	+	0.928223i	$0.621336\pi$
$420$	10.1394	0.494751
$421$	2.71972	0.132551	0.0662754	−	0.997801i	$-0.478888\pi$
$421$	2.71972	0.132551	0.0662754	+	0.997801i	$0.478888\pi$
$422$	41.2072	2.00594
$423$	6.24265	0.303528
$424$	−8.05468	−0.391170
$425$	0.923036	0.0447738
$426$	5.93592	0.287596
$427$	34.9839	1.69299
$428$	45.1165	2.18079
$429$	−14.2370	−0.687370
$430$	−1.19427	−0.0575926
$431$	−6.95572	−0.335045	−0.167523	−	0.985868i	$-0.553577\pi$
$431$	−6.95572	−0.335045	−0.167523	+	0.985868i	$0.553577\pi$
$432$	3.75198	0.180517
$433$	−37.6212	−1.80796	−0.903980	−	0.427576i	$-0.859368\pi$
$433$	−37.6212	−1.80796	−0.903980	+	0.427576i	$0.859368\pi$
$434$	39.3814	1.89037
$435$	0.812160	0.0389401
$436$	50.9907	2.44201
$437$	−0.919706	−0.0439955
$438$	−11.7415	−0.561029
$439$	12.6178	0.602214	0.301107	−	0.953590i	$-0.402644\pi$
$439$	12.6178	0.602214	0.301107	+	0.953590i	$0.402644\pi$
$440$	4.87084	0.232208
$441$	−9.19128	−0.437680
$442$	−1.27522	−0.0606561
$443$	−0.230415	−0.0109474	−0.00547368	−	0.999985i	$-0.501742\pi$
$443$	−0.230415	−0.0109474	−0.00547368	+	0.999985i	$0.501742\pi$
$444$	0.785637	0.0372847
$445$	3.81272	0.180740
$446$	−38.6440	−1.82985
$447$	−14.0083	−0.662572
$448$	52.8459	2.49673
$449$	−0.603078	−0.0284610	−0.0142305	−	0.999899i	$-0.504530\pi$
$449$	−0.603078	−0.0284610	−0.0142305	+	0.999899i	$0.504530\pi$
$450$	10.2990	0.485500
$451$	−0.616937	−0.0290505
$452$	−51.3851	−2.41695
$453$	−10.7609	−0.505591
$454$	51.0178	2.39438
$455$	6.62468	0.310570
$456$	8.23624	0.385697
$457$	−3.35523	−0.156951	−0.0784754	−	0.996916i	$-0.525005\pi$
$457$	−3.35523	−0.156951	−0.0784754	+	0.996916i	$0.525005\pi$
$458$	−38.7048	−1.80856
$459$	1.11711	0.0521422
$460$	0.682209	0.0318081
$461$	3.66884	0.170875	0.0854375	−	0.996344i	$-0.472771\pi$
$461$	3.66884	0.170875	0.0854375	+	0.996344i	$0.472771\pi$
$462$	45.3858	2.11154
$463$	6.03992	0.280699	0.140349	−	0.990102i	$-0.455177\pi$
$463$	6.03992	0.280699	0.140349	+	0.990102i	$0.455177\pi$
$464$	0.663295	0.0307927
$465$	3.50288	0.162442
$466$	31.8810	1.47686
$467$	−16.2684	−0.752811	−0.376406	−	0.926455i	$-0.622840\pi$
$467$	−16.2684	−0.752811	−0.376406	+	0.926455i	$0.622840\pi$
$468$	−8.62951	−0.398899
$469$	8.90924	0.411391
$470$	8.22272	0.379286
$471$	−4.11960	−0.189821
$472$	16.7521	0.771079
$473$	−3.24216	−0.149075
$474$	36.1577	1.66078
$475$	−11.0917	−0.508921
$476$	2.46553	0.113008
$477$	3.22606	0.147711
$478$	−35.0553	−1.60339
$479$	13.6886	0.625448	0.312724	−	0.949844i	$-0.398759\pi$
$479$	13.6886	0.625448	0.312724	+	0.949844i	$0.398759\pi$
$480$	5.17841	0.236361
$481$	0.513305	0.0234047
$482$	−10.6262	−0.484012
$483$	2.23230	0.101573
$484$	3.74740	0.170336
$485$	−1.33556	−0.0606447
$486$	21.8658	0.991853
$487$	−35.7832	−1.62149	−0.810745	−	0.585400i	$-0.800938\pi$
$487$	−35.7832	−1.62149	−0.810745	+	0.585400i	$0.800938\pi$
$488$	−21.0792	−0.954210
$489$	−9.19066	−0.415616
$490$	−12.1066	−0.546920
$491$	−3.21810	−0.145231	−0.0726153	−	0.997360i	$-0.523135\pi$
$491$	−3.21810	−0.145231	−0.0726153	+	0.997360i	$0.523135\pi$
$492$	0.773811	0.0348861
$493$	0.197488	0.00889442
$494$	15.3237	0.689447
$495$	−1.95087	−0.0876849
$496$	2.86082	0.128454
$497$	−7.49838	−0.336348
$498$	−8.52318	−0.381933
$499$	1.64912	0.0738249	0.0369125	−	0.999319i	$-0.488248\pi$
$499$	1.64912	0.0738249	0.0369125	+	0.999319i	$0.488248\pi$
$500$	17.0290	0.761559
$501$	18.5995	0.830963
$502$	−48.5339	−2.16617
$503$	−5.35533	−0.238782	−0.119391	−	0.992847i	$-0.538094\pi$
$503$	−5.35533	−0.238782	−0.119391	+	0.992847i	$0.538094\pi$
$504$	9.66064	0.430319
$505$	−9.19652	−0.409240
$506$	3.05370	0.135754
$507$	−6.82108	−0.302935
$508$	22.1418	0.982384
$509$	8.61616	0.381904	0.190952	−	0.981599i	$-0.438842\pi$
$509$	8.61616	0.381904	0.190952	+	0.981599i	$0.438842\pi$
$510$	0.361593	0.0160116
$511$	14.8321	0.656132
$512$	7.46660	0.329980
$513$	−13.4238	−0.592674
$514$	6.97806	0.307789
$515$	2.46005	0.108403
$516$	4.06657	0.179021
$517$	22.3228	0.981756
$518$	−1.63635	−0.0718972
$519$	−15.7255	−0.690271
$520$	−3.99164	−0.175045
$521$	−25.5670	−1.12011	−0.560056	−	0.828455i	$-0.689220\pi$
$521$	−25.5670	−1.12011	−0.560056	+	0.828455i	$0.689220\pi$
$522$	2.20352	0.0964455
$523$	−22.4160	−0.980183	−0.490092	−	0.871671i	$-0.663037\pi$
$523$	−22.4160	−0.980183	−0.490092	+	0.871671i	$0.663037\pi$
$524$	−37.8924	−1.65534
$525$	26.9216	1.17496
$526$	17.9575	0.782983
$527$	0.851774	0.0371039
$528$	3.29700	0.143484
$529$	−22.8498	−0.993470
$530$	4.24931	0.184578
$531$	−6.70956	−0.291170
$532$	−29.6271	−1.28450
$533$	0.505579	0.0218990
$534$	−21.4061	−0.926333
$535$	−8.35842	−0.361366
$536$	−5.36817	−0.231870
$537$	12.2360	0.528022
$538$	−29.1172	−1.25533
$539$	−32.8667	−1.41567
$540$	9.95731	0.428495
$541$	6.19360	0.266283	0.133142	−	0.991097i	$-0.457493\pi$
$541$	6.19360	0.266283	0.133142	+	0.991097i	$0.457493\pi$
$542$	9.28521	0.398834
$543$	−21.5792	−0.926052
$544$	1.25920	0.0539879
$545$	−9.44668	−0.404651
$546$	−37.1936	−1.59174
$547$	3.82597	0.163587	0.0817933	−	0.996649i	$-0.473935\pi$
$547$	3.82597	0.163587	0.0817933	+	0.996649i	$0.473935\pi$
$548$	16.8175	0.718407
$549$	8.44263	0.360323
$550$	36.8277	1.57034
$551$	−2.37312	−0.101098
$552$	−1.34505	−0.0572491
$553$	−45.6752	−1.94231
$554$	−37.6167	−1.59818
$555$	−0.145549	−0.00617823
$556$	−3.08248	−0.130726
$557$	−41.0466	−1.73920	−0.869600	−	0.493756i	$-0.835624\pi$
$557$	−41.0466	−1.73920	−0.869600	+	0.493756i	$0.835624\pi$
$558$	9.50388	0.402331
$559$	2.65694	0.112377
$560$	−1.53414	−0.0648292
$561$	0.981644	0.0414450
$562$	49.7247	2.09751
$563$	−20.1852	−0.850704	−0.425352	−	0.905028i	$-0.639850\pi$
$563$	−20.1852	−0.850704	−0.425352	+	0.905028i	$0.639850\pi$
$564$	−27.9990	−1.17897
$565$	9.51975	0.400499
$566$	1.20254	0.0505466
$567$	20.7060	0.869569
$568$	4.51807	0.189574
$569$	−0.807042	−0.0338330	−0.0169165	−	0.999857i	$-0.505385\pi$
$569$	−0.807042	−0.0338330	−0.0169165	+	0.999857i	$0.505385\pi$
$570$	−4.34509	−0.181996
$571$	8.06198	0.337383	0.168692	−	0.985669i	$-0.446046\pi$
$571$	8.06198	0.337383	0.168692	+	0.985669i	$0.446046\pi$
$572$	−30.8579	−1.29023
$573$	2.64537	0.110512
$574$	−1.61172	−0.0672719
$575$	1.81137	0.0755393
$576$	12.7532	0.531385
$577$	−23.4416	−0.975887	−0.487944	−	0.872875i	$-0.662253\pi$
$577$	−23.4416	−0.975887	−0.487944	+	0.872875i	$0.662253\pi$
$578$	−38.2375	−1.59047
$579$	28.9373	1.20259
$580$	1.76030	0.0730927
$581$	10.7667	0.446676
$582$	7.49836	0.310817
$583$	11.5359	0.477768
$584$	−8.93691	−0.369812
$585$	1.59873	0.0660993
$586$	−1.55401	−0.0641956
$587$	−19.2866	−0.796042	−0.398021	−	0.917376i	$-0.630303\pi$
$587$	−19.2866	−0.796042	−0.398021	+	0.917376i	$0.630303\pi$
$588$	41.2239	1.70004
$589$	−10.2354	−0.421741
$590$	−8.83772	−0.363843
$591$	−6.84983	−0.281764
$592$	−0.118871	−0.00488556
$593$	−2.08744	−0.0857209	−0.0428604	−	0.999081i	$-0.513647\pi$
$593$	−2.08744	−0.0857209	−0.0428604	+	0.999081i	$0.513647\pi$
$594$	44.5709	1.82877
$595$	−0.456772	−0.0187258
$596$	−30.3622	−1.24368
$597$	6.12994	0.250882
$598$	−2.50250	−0.102335
$599$	22.4536	0.917431	0.458716	−	0.888583i	$-0.348310\pi$
$599$	22.4536	0.917431	0.458716	+	0.888583i	$0.348310\pi$
$600$	−16.2213	−0.662233
$601$	4.06318	0.165741	0.0828703	−	0.996560i	$-0.473591\pi$
$601$	4.06318	0.165741	0.0828703	+	0.996560i	$0.473591\pi$
$602$	−8.46999	−0.345211
$603$	2.15006	0.0875572
$604$	−23.3236	−0.949022
$605$	−0.694255	−0.0282255
$606$	51.6328	2.09744
$607$	27.0484	1.09786	0.548930	−	0.835869i	$-0.315035\pi$
$607$	27.0484	1.09786	0.548930	+	0.835869i	$0.315035\pi$
$608$	−15.1313	−0.613653
$609$	5.76001	0.233407
$610$	11.1205	0.450255
$611$	−18.2935	−0.740075
$612$	0.595005	0.0240517
$613$	−22.2722	−0.899566	−0.449783	−	0.893138i	$-0.648499\pi$
$613$	−22.2722	−0.899566	−0.449783	+	0.893138i	$0.648499\pi$
$614$	62.1624	2.50867
$615$	−0.143358	−0.00578077
$616$	34.5450	1.39186
$617$	−45.7579	−1.84214	−0.921071	−	0.389394i	$-0.872684\pi$
$617$	−45.7579	−1.84214	−0.921071	+	0.389394i	$0.872684\pi$
$618$	−13.8117	−0.555587
$619$	17.4205	0.700191	0.350095	−	0.936714i	$-0.386149\pi$
$619$	17.4205	0.700191	0.350095	+	0.936714i	$0.386149\pi$
$620$	7.59226	0.304913
$621$	2.19222	0.0879706
$622$	75.1812	3.01449
$623$	27.0406	1.08336
$624$	−2.70188	−0.108162
$625$	20.2146	0.808583
$626$	11.4853	0.459046
$627$	−11.7959	−0.471084
$628$	−8.92898	−0.356305
$629$	−0.0353924	−0.00141119
$630$	−5.09654	−0.203051
$631$	−27.5529	−1.09686	−0.548432	−	0.836195i	$-0.684775\pi$
$631$	−27.5529	−1.09686	−0.548432	+	0.836195i	$0.684775\pi$
$632$	27.5211	1.09473
$633$	25.9949	1.03321
$634$	30.1194	1.19619
$635$	−4.10206	−0.162785
$636$	−14.4692	−0.573742
$637$	26.9341	1.06717
$638$	7.87947	0.311951
$639$	−1.80958	−0.0715857
$640$	9.51597	0.376152
$641$	−24.8347	−0.980911	−0.490455	−	0.871466i	$-0.663170\pi$
$641$	−24.8347	−0.980911	−0.490455	+	0.871466i	$0.663170\pi$
$642$	46.9274	1.85208
$643$	−19.6140	−0.773500	−0.386750	−	0.922185i	$-0.626402\pi$
$643$	−19.6140	−0.773500	−0.386750	+	0.922185i	$0.626402\pi$
$644$	4.83837	0.190659
$645$	−0.753384	−0.0296645
$646$	−1.05657	−0.0415702
$647$	−8.39772	−0.330148	−0.165074	−	0.986281i	$-0.552786\pi$
$647$	−8.39772	−0.330148	−0.165074	+	0.986281i	$0.552786\pi$
$648$	−12.4762	−0.490110
$649$	−23.9924	−0.941784
$650$	−30.1802	−1.18376
$651$	24.8432	0.973680
$652$	−19.9202	−0.780134
$653$	6.16214	0.241143	0.120572	−	0.992705i	$-0.461527\pi$
$653$	6.16214	0.241143	0.120572	+	0.992705i	$0.461527\pi$
$654$	53.0374	2.07392
$655$	7.02006	0.274296
$656$	−0.117081	−0.00457126
$657$	3.57941	0.139646
$658$	58.3173	2.27344
$659$	−22.7630	−0.886722	−0.443361	−	0.896343i	$-0.646214\pi$
$659$	−22.7630	−0.886722	−0.443361	+	0.896343i	$0.646214\pi$
$660$	8.74985	0.340587
$661$	−13.1631	−0.511984	−0.255992	−	0.966679i	$-0.582402\pi$
$661$	−13.1631	−0.511984	−0.255992	+	0.966679i	$0.582402\pi$
$662$	−30.8104	−1.19748
$663$	−0.804454	−0.0312424
$664$	−6.48734	−0.251757
$665$	5.48881	0.212847
$666$	−0.394899	−0.0153020
$667$	0.387551	0.0150060
$668$	40.3132	1.55976
$669$	−24.3780	−0.942507
$670$	2.83202	0.109411
$671$	30.1896	1.16546
$672$	36.7264	1.41675
$673$	41.7201	1.60819	0.804095	−	0.594501i	$-0.202650\pi$
$673$	41.7201	1.60819	0.804095	+	0.594501i	$0.202650\pi$
$674$	−21.6809	−0.835118
$675$	26.4382	1.01761
$676$	−14.7843	−0.568625
$677$	−43.1235	−1.65737	−0.828685	−	0.559716i	$-0.810910\pi$
$677$	−43.1235	−1.65737	−0.828685	+	0.559716i	$0.810910\pi$
$678$	−53.4476	−2.05264
$679$	−9.47209	−0.363505
$680$	0.275223	0.0105543
$681$	32.1838	1.23329
$682$	33.9845	1.30133
$683$	−11.5939	−0.443628	−0.221814	−	0.975089i	$-0.571198\pi$
$683$	−11.5939	−0.443628	−0.221814	+	0.975089i	$0.571198\pi$
$684$	−7.14989	−0.273383
$685$	−3.11566	−0.119043
$686$	−21.9471	−0.837946
$687$	−24.4164	−0.931542
$688$	−0.615292	−0.0234578
$689$	−9.45364	−0.360155
$690$	0.709591	0.0270137
$691$	22.1221	0.841566	0.420783	−	0.907161i	$-0.361755\pi$
$691$	22.1221	0.841566	0.420783	+	0.907161i	$0.361755\pi$
$692$	−34.0839	−1.29568
$693$	−13.8360	−0.525585
$694$	19.3856	0.735867
$695$	0.571069	0.0216619
$696$	−3.47064	−0.131554
$697$	−0.0348596	−0.00132040
$698$	−41.7020	−1.57844
$699$	20.1117	0.760693
$700$	58.3509	2.20546
$701$	15.7267	0.593989	0.296995	−	0.954879i	$-0.404016\pi$
$701$	15.7267	0.593989	0.296995	+	0.954879i	$0.404016\pi$
$702$	−36.5257	−1.37857
$703$	0.425294	0.0160403
$704$	45.6037	1.71875
$705$	5.18717	0.195360
$706$	−2.82052	−0.106152
$707$	−65.2237	−2.45299
$708$	30.0931	1.13097
$709$	46.7156	1.75444	0.877220	−	0.480088i	$-0.159395\pi$
$709$	46.7156	1.75444	0.877220	+	0.480088i	$0.159395\pi$
$710$	−2.38354	−0.0894528
$711$	−11.0228	−0.413386
$712$	−16.2931	−0.610608
$713$	1.67152	0.0625991
$714$	2.56450	0.0959739
$715$	5.71682	0.213797
$716$	26.5208	0.991127
$717$	−22.1141	−0.825866
$718$	−9.34721	−0.348835
$719$	−25.8445	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$719$	−25.8445	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$720$	−0.370232	−0.0137977
$721$	17.4472	0.649767
$722$	−30.1379	−1.12162
$723$	−6.70340	−0.249302
$724$	−46.7716	−1.73825
$725$	4.67388	0.173584
$726$	3.89782	0.144662
$727$	32.3860	1.20113	0.600565	−	0.799576i	$-0.294942\pi$
$727$	32.3860	1.20113	0.600565	+	0.799576i	$0.294942\pi$
$728$	−28.3095	−1.04922
$729$	29.1309	1.07892
$730$	4.71474	0.174500
$731$	−0.183196	−0.00677575
$732$	−37.8661	−1.39957
$733$	40.1298	1.48223	0.741114	−	0.671380i	$-0.234298\pi$
$733$	40.1298	1.48223	0.741114	+	0.671380i	$0.234298\pi$
$734$	9.43544	0.348269
$735$	−7.63726	−0.281705
$736$	2.47106	0.0910847
$737$	7.68830	0.283202
$738$	−0.388955	−0.0143176
$739$	−18.0357	−0.663453	−0.331726	−	0.943376i	$-0.607631\pi$
$739$	−18.0357	−0.663453	−0.331726	+	0.943376i	$0.607631\pi$
$740$	−0.315469	−0.0115969
$741$	9.66674	0.355117
$742$	30.1370	1.10636
$743$	−7.45557	−0.273518	−0.136759	−	0.990604i	$-0.543669\pi$
$743$	−7.45557	−0.273518	−0.136759	+	0.990604i	$0.543669\pi$
$744$	−14.9690	−0.548790
$745$	5.62499	0.206084
$746$	15.6116	0.571580
$747$	2.59831	0.0950671
$748$	2.12765	0.0777946
$749$	−59.2797	−2.16603
$750$	17.7125	0.646769
$751$	−18.5337	−0.676303	−0.338152	−	0.941092i	$-0.609802\pi$
$751$	−18.5337	−0.676303	−0.338152	+	0.941092i	$0.609802\pi$
$752$	4.23639	0.154485
$753$	−30.6169	−1.11574
$754$	−6.45720	−0.235157
$755$	4.32099	0.157257
$756$	70.6194	2.56840
$757$	−30.9108	−1.12347	−0.561736	−	0.827316i	$-0.689866\pi$
$757$	−30.9108	−1.12347	−0.561736	+	0.827316i	$0.689866\pi$
$758$	−14.2201	−0.516498
$759$	1.92638	0.0699231
$760$	−3.30723	−0.119966
$761$	−2.60649	−0.0944852	−0.0472426	−	0.998883i	$-0.515043\pi$
$761$	−2.60649	−0.0944852	−0.0472426	+	0.998883i	$0.515043\pi$
$762$	23.0306	0.834309
$763$	−66.9979	−2.42549
$764$	5.73368	0.207437
$765$	−0.110232	−0.00398546
$766$	−25.7955	−0.932029
$767$	19.6617	0.709943
$768$	−16.3136	−0.588666
$769$	−3.32732	−0.119986	−0.0599932	−	0.998199i	$-0.519108\pi$
$769$	−3.32732	−0.119986	−0.0599932	+	0.998199i	$0.519108\pi$
$770$	−18.2245	−0.656765
$771$	4.40200	0.158534
$772$	62.7198	2.25734
$773$	−8.16093	−0.293528	−0.146764	−	0.989172i	$-0.546886\pi$
$773$	−8.16093	−0.293528	−0.146764	+	0.989172i	$0.546886\pi$
$774$	−2.04405	−0.0734720
$775$	20.1586	0.724119
$776$	5.70731	0.204880
$777$	−1.03227	−0.0370324
$778$	−44.8069	−1.60641
$779$	0.418892	0.0150084
$780$	−7.17048	−0.256744
$781$	−6.47078	−0.231543
$782$	0.172547	0.00617028
$783$	5.65658	0.202150
$784$	−6.23739	−0.222764
$785$	1.65421	0.0590412
$786$	−39.4133	−1.40583
$787$	−2.85074	−0.101618	−0.0508090	−	0.998708i	$-0.516180\pi$
$787$	−2.85074	−0.101618	−0.0508090	+	0.998708i	$0.516180\pi$
$788$	−14.8466	−0.528887
$789$	11.3282	0.403294
$790$	−14.5190	−0.516562
$791$	67.5161	2.40060
$792$	8.33672	0.296232
$793$	−24.7403	−0.878554
$794$	39.8118	1.41287
$795$	2.68061	0.0950715
$796$	13.2863	0.470919
$797$	−4.22956	−0.149819	−0.0749094	−	0.997190i	$-0.523867\pi$
$797$	−4.22956	−0.149819	−0.0749094	+	0.997190i	$0.523867\pi$
$798$	−30.8163	−1.09089
$799$	1.26134	0.0446228
$800$	29.8011	1.05363
$801$	6.52569	0.230574
$802$	−21.6784	−0.765492
$803$	12.7994	0.451683
$804$	−9.64325	−0.340091
$805$	−0.896371	−0.0315929
$806$	−27.8502	−0.980980
$807$	−18.3682	−0.646590
$808$	39.2999	1.38256
$809$	33.4971	1.17770	0.588848	−	0.808244i	$-0.299582\pi$
$809$	33.4971	1.17770	0.588848	+	0.808244i	$0.299582\pi$
$810$	6.58190	0.231264
$811$	−23.2447	−0.816232	−0.408116	−	0.912930i	$-0.633814\pi$
$811$	−23.2447	−0.816232	−0.408116	+	0.912930i	$0.633814\pi$
$812$	12.4845	0.438119
$813$	5.85743	0.205429
$814$	−1.41210	−0.0494942
$815$	3.69047	0.129272
$816$	0.186295	0.00652162
$817$	2.20138	0.0770165
$818$	−86.3271	−3.01835
$819$	11.3385	0.396200
$820$	−0.310720	−0.0108508
$821$	40.5462	1.41507	0.707536	−	0.706677i	$-0.249807\pi$
$821$	40.5462	1.41507	0.707536	+	0.706677i	$0.249807\pi$
$822$	17.4925	0.610121
$823$	38.6467	1.34714	0.673569	−	0.739124i	$-0.264760\pi$
$823$	38.6467	1.34714	0.673569	+	0.739124i	$0.264760\pi$
$824$	−10.5126	−0.366225
$825$	23.2322	0.808841
$826$	−62.6790	−2.18088
$827$	−9.34834	−0.325074	−0.162537	−	0.986702i	$-0.551968\pi$
$827$	−9.34834	−0.325074	−0.162537	+	0.986702i	$0.551968\pi$
$828$	1.16764	0.0405783
$829$	24.4200	0.848142	0.424071	−	0.905629i	$-0.360601\pi$
$829$	24.4200	0.848142	0.424071	+	0.905629i	$0.360601\pi$
$830$	3.42244	0.118795
$831$	−23.7299	−0.823181
$832$	−37.3721	−1.29564
$833$	−1.85711	−0.0643450
$834$	−3.20620	−0.111022
$835$	−7.46853	−0.258459
$836$	−25.5670	−0.884252
$837$	24.3971	0.843286
$838$	−34.3357	−1.18611
$839$	32.6743	1.12804	0.564021	−	0.825760i	$-0.309254\pi$
$839$	32.6743	1.12804	0.564021	+	0.825760i	$0.309254\pi$
$840$	8.02726	0.276967
$841$	1.00000	0.0344828
$842$	6.13142	0.211303
$843$	31.3680	1.08037
$844$	56.3424	1.93938
$845$	2.73897	0.0942236
$846$	14.0737	0.483862
$847$	−4.92380	−0.169184
$848$	2.18927	0.0751797
$849$	0.758606	0.0260353
$850$	2.08093	0.0713752
$851$	−0.0694542	−0.00238086
$852$	8.11615	0.278055
$853$	−37.0105	−1.26722	−0.633608	−	0.773654i	$-0.718427\pi$
$853$	−37.0105	−1.26722	−0.633608	+	0.773654i	$0.718427\pi$
$854$	78.8689	2.69884
$855$	1.32461	0.0453007
$856$	35.7184	1.22083
$857$	−11.7476	−0.401289	−0.200645	−	0.979664i	$-0.564304\pi$
$857$	−11.7476	−0.401289	−0.200645	+	0.979664i	$0.564304\pi$
$858$	−32.0965	−1.09575
$859$	37.6954	1.28615	0.643076	−	0.765802i	$-0.277658\pi$
$859$	37.6954	1.28615	0.643076	+	0.765802i	$0.277658\pi$
$860$	−1.63291	−0.0556819
$861$	−1.01673	−0.0346500
$862$	−15.6812	−0.534104
$863$	41.8975	1.42621	0.713103	−	0.701059i	$-0.247289\pi$
$863$	41.8975	1.42621	0.713103	+	0.701059i	$0.247289\pi$
$864$	36.0669	1.22702
$865$	6.31449	0.214699
$866$	−84.8145	−2.88211
$867$	−24.1215	−0.819210
$868$	53.8460	1.82765
$869$	−39.4158	−1.33709
$870$	1.83096	0.0620754
$871$	−6.30054	−0.213486
$872$	40.3689	1.36706
$873$	−2.28589	−0.0773656
$874$	−2.07342	−0.0701344
$875$	−22.3748	−0.756406
$876$	−16.0541	−0.542416
$877$	−56.3993	−1.90447	−0.952235	−	0.305365i	$-0.901221\pi$
$877$	−56.3993	−1.90447	−0.952235	+	0.305365i	$0.901221\pi$
$878$	28.4460	0.960005
$879$	−0.980323	−0.0330655
$880$	−1.32390	−0.0446285
$881$	12.7495	0.429542	0.214771	−	0.976664i	$-0.431100\pi$
$881$	12.7495	0.429542	0.214771	+	0.976664i	$0.431100\pi$
$882$	−20.7211	−0.697717
$883$	−38.5557	−1.29750	−0.648752	−	0.761000i	$-0.724709\pi$
$883$	−38.5557	−1.29750	−0.648752	+	0.761000i	$0.724709\pi$
$884$	−1.74360	−0.0586437
$885$	−5.57514	−0.187406
$886$	−0.519456	−0.0174515
$887$	−0.988306	−0.0331841	−0.0165920	−	0.999862i	$-0.505282\pi$
$887$	−0.988306	−0.0331841	−0.0165920	+	0.999862i	$0.505282\pi$
$888$	0.621982	0.0208724
$889$	−29.0927	−0.975737
$890$	8.59553	0.288123
$891$	17.8684	0.598613
$892$	−52.8377	−1.76914
$893$	−15.1569	−0.507205
$894$	−31.5809	−1.05622
$895$	−4.91331	−0.164234
$896$	67.4893	2.25466
$897$	−1.57866	−0.0527100
$898$	−1.35960	−0.0453705
$899$	4.31304	0.143848
$900$	14.0818	0.469392
$901$	0.651829	0.0217156
$902$	−1.39085	−0.0463101
$903$	−5.34316	−0.177809
$904$	−40.6811	−1.35303
$905$	8.66504	0.288036
$906$	−24.2597	−0.805976
$907$	−16.6280	−0.552125	−0.276062	−	0.961140i	$-0.589030\pi$
$907$	−16.6280	−0.552125	−0.276062	+	0.961140i	$0.589030\pi$
$908$	69.7564	2.31495
$909$	−15.7404	−0.522075
$910$	14.9349	0.495088
$911$	7.30840	0.242138	0.121069	−	0.992644i	$-0.461368\pi$
$911$	7.30840	0.242138	0.121069	+	0.992644i	$0.461368\pi$
$912$	−2.23862	−0.0741280
$913$	9.29116	0.307493
$914$	−7.56414	−0.250199
$915$	7.01519	0.231915
$916$	−52.9209	−1.74856
$917$	49.7878	1.64414
$918$	2.51845	0.0831212
$919$	−7.76023	−0.255986	−0.127993	−	0.991775i	$-0.540854\pi$
$919$	−7.76023	−0.255986	−0.127993	+	0.991775i	$0.540854\pi$
$920$	0.540099	0.0178065
$921$	39.2142	1.29215
$922$	8.27116	0.272396
$923$	5.30279	0.174543
$924$	62.0558	2.04149
$925$	−0.837619	−0.0275408
$926$	13.6166	0.447469
$927$	4.21051	0.138291
$928$	6.37610	0.209306
$929$	−24.8324	−0.814726	−0.407363	−	0.913266i	$-0.633552\pi$
$929$	−24.8324	−0.814726	−0.407363	+	0.913266i	$0.633552\pi$
$930$	7.89700	0.258953
$931$	22.3160	0.731377
$932$	43.5908	1.42786
$933$	47.4269	1.55269
$934$	−36.6760	−1.20008
$935$	−0.394175	−0.0128909
$936$	−6.83192	−0.223308
$937$	−28.4100	−0.928116	−0.464058	−	0.885805i	$-0.653607\pi$
$937$	−28.4100	−0.928116	−0.464058	+	0.885805i	$0.653607\pi$
$938$	20.0853	0.655809
$939$	7.24535	0.236443
$940$	11.2429	0.366702
$941$	12.4903	0.407173	0.203587	−	0.979057i	$-0.434740\pi$
$941$	12.4903	0.407173	0.203587	+	0.979057i	$0.434740\pi$
$942$	−9.28737	−0.302599
$943$	−0.0684087	−0.00222769
$944$	−4.55324	−0.148195
$945$	−13.0832	−0.425595
$946$	−7.30924	−0.237644
$947$	4.61661	0.150020	0.0750098	−	0.997183i	$-0.476101\pi$
$947$	4.61661	0.150020	0.0750098	+	0.997183i	$0.476101\pi$
$948$	49.4383	1.60568
$949$	−10.4891	−0.340491
$950$	−25.0055	−0.811285
$951$	19.0004	0.616129
$952$	1.95194	0.0632629
$953$	−22.2722	−0.721467	−0.360733	−	0.932669i	$-0.617474\pi$
$953$	−22.2722	−0.721467	−0.360733	+	0.932669i	$0.617474\pi$
$954$	7.27294	0.235470
$955$	−1.06224	−0.0343732
$956$	−47.9309	−1.55020
$957$	4.97064	0.160678
$958$	30.8600	0.997043
$959$	−22.0969	−0.713546
$960$	10.5970	0.342016
$961$	−12.3977	−0.399926
$962$	1.15721	0.0373101
$963$	−14.3059	−0.461002
$964$	−14.5292	−0.467954
$965$	−11.6197	−0.374050
$966$	5.03258	0.161920
$967$	22.7974	0.733114	0.366557	−	0.930396i	$-0.380537\pi$
$967$	22.7974	0.733114	0.366557	+	0.930396i	$0.380537\pi$
$968$	2.96679	0.0953562
$969$	−0.666522	−0.0214118
$970$	−3.01094	−0.0966753
$971$	52.0070	1.66899	0.834493	−	0.551019i	$-0.185761\pi$
$971$	52.0070	1.66899	0.834493	+	0.551019i	$0.185761\pi$
$972$	29.8970	0.958946
$973$	4.05014	0.129842
$974$	−80.6708	−2.58486
$975$	−19.0387	−0.609727
$976$	5.72934	0.183392
$977$	5.18044	0.165737	0.0828685	−	0.996560i	$-0.473592\pi$
$977$	5.18044	0.165737	0.0828685	+	0.996560i	$0.473592\pi$
$978$	−20.7197	−0.662544
$979$	23.3349	0.745787
$980$	−16.5533	−0.528775
$981$	−16.1685	−0.516222
$982$	−7.25499	−0.231516
$983$	61.0964	1.94867	0.974336	−	0.225098i	$-0.0722702\pi$
$983$	61.0964	1.94867	0.974336	+	0.225098i	$0.0722702\pi$
$984$	0.612620	0.0195296
$985$	2.75052	0.0876388
$986$	0.445224	0.0141788
$987$	36.7886	1.17099
$988$	20.9520	0.666573
$989$	−0.359505	−0.0114316
$990$	−4.39810	−0.139781
$991$	−17.6607	−0.561012	−0.280506	−	0.959852i	$-0.590502\pi$
$991$	−17.6607	−0.561012	−0.280506	+	0.959852i	$0.590502\pi$
$992$	27.5003	0.873137
$993$	−19.4363	−0.616792
$994$	−16.9046	−0.536181
$995$	−2.46145	−0.0780332
$996$	−11.6537	−0.369261
$997$	−34.9275	−1.10616	−0.553082	−	0.833127i	$-0.686549\pi$
$997$	−34.9275	−1.10616	−0.553082	+	0.833127i	$0.686549\pi$
$998$	3.71784	0.117686
$999$	−1.01373	−0.0320731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.57	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.57	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q+2.25443 q^{2} +1.42218 q^{3} +3.08248 q^{4} -0.571069 q^{5} +3.20620 q^{6} -4.05014 q^{7} +2.44037 q^{8} -0.977417 q^{9} +O(q^{10})\)	\(q+2.25443 q^{2} +1.42218 q^{3} +3.08248 q^{4} -0.571069 q^{5} +3.20620 q^{6} -4.05014 q^{7} +2.44037 q^{8} -0.977417 q^{9} -1.28744 q^{10} -3.49510 q^{11} +4.38382 q^{12} +2.86422 q^{13} -9.13078 q^{14} -0.812160 q^{15} -0.663295 q^{16} -0.197488 q^{17} -2.20352 q^{18} +2.37312 q^{19} -1.76030 q^{20} -5.76001 q^{21} -7.87947 q^{22} -0.387551 q^{23} +3.47064 q^{24} -4.67388 q^{25} +6.45720 q^{26} -5.65658 q^{27} -12.4845 q^{28} -1.00000 q^{29} -1.83096 q^{30} -4.31304 q^{31} -6.37610 q^{32} -4.97064 q^{33} -0.445224 q^{34} +2.31291 q^{35} -3.01286 q^{36} +0.179213 q^{37} +5.35005 q^{38} +4.07343 q^{39} -1.39362 q^{40} +0.176515 q^{41} -12.9856 q^{42} +0.927631 q^{43} -10.7736 q^{44} +0.558172 q^{45} -0.873709 q^{46} -6.38689 q^{47} -0.943321 q^{48} +9.40364 q^{49} -10.5370 q^{50} -0.280863 q^{51} +8.82890 q^{52} -3.30060 q^{53} -12.7524 q^{54} +1.99594 q^{55} -9.88385 q^{56} +3.37500 q^{57} -2.25443 q^{58} +6.86458 q^{59} -2.50346 q^{60} -8.63770 q^{61} -9.72346 q^{62} +3.95868 q^{63} -13.0479 q^{64} -1.63567 q^{65} -11.2060 q^{66} -2.19974 q^{67} -0.608753 q^{68} -0.551166 q^{69} +5.21430 q^{70} +1.85139 q^{71} -2.38526 q^{72} -3.66211 q^{73} +0.404024 q^{74} -6.64708 q^{75} +7.31509 q^{76} +14.1556 q^{77} +9.18328 q^{78} +11.2774 q^{79} +0.378787 q^{80} -5.11241 q^{81} +0.397942 q^{82} -2.65834 q^{83} -17.7551 q^{84} +0.112779 q^{85} +2.09128 q^{86} -1.42218 q^{87} -8.52934 q^{88} -6.67647 q^{89} +1.25836 q^{90} -11.6005 q^{91} -1.19462 q^{92} -6.13390 q^{93} -14.3988 q^{94} -1.35521 q^{95} -9.06793 q^{96} +2.33871 q^{97} +21.1999 q^{98} +3.41617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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