LMFDB - Embedded newform 4006.2.a.h.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.57856 q^{3} +1.00000 q^{4} +1.66904 q^{5} +1.57856 q^{6} +0.263990 q^{7} -1.00000 q^{8} -0.508157 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.57856 q^{3} +1.00000 q^{4} +1.66904 q^{5} +1.57856 q^{6} +0.263990 q^{7} -1.00000 q^{8} -0.508157 q^{9} -1.66904 q^{10} +1.56051 q^{11} -1.57856 q^{12} +6.57360 q^{13} -0.263990 q^{14} -2.63467 q^{15} +1.00000 q^{16} +5.46624 q^{17} +0.508157 q^{18} -5.25199 q^{19} +1.66904 q^{20} -0.416723 q^{21} -1.56051 q^{22} +0.430395 q^{23} +1.57856 q^{24} -2.21431 q^{25} -6.57360 q^{26} +5.53783 q^{27} +0.263990 q^{28} +4.27702 q^{29} +2.63467 q^{30} +7.78485 q^{31} -1.00000 q^{32} -2.46336 q^{33} -5.46624 q^{34} +0.440609 q^{35} -0.508157 q^{36} +8.47373 q^{37} +5.25199 q^{38} -10.3768 q^{39} -1.66904 q^{40} -5.07748 q^{41} +0.416723 q^{42} -2.93970 q^{43} +1.56051 q^{44} -0.848134 q^{45} -0.430395 q^{46} -0.00890887 q^{47} -1.57856 q^{48} -6.93031 q^{49} +2.21431 q^{50} -8.62877 q^{51} +6.57360 q^{52} +2.51186 q^{53} -5.53783 q^{54} +2.60456 q^{55} -0.263990 q^{56} +8.29057 q^{57} -4.27702 q^{58} -4.97914 q^{59} -2.63467 q^{60} +5.35666 q^{61} -7.78485 q^{62} -0.134148 q^{63} +1.00000 q^{64} +10.9716 q^{65} +2.46336 q^{66} -12.7223 q^{67} +5.46624 q^{68} -0.679404 q^{69} -0.440609 q^{70} -2.49503 q^{71} +0.508157 q^{72} +16.4892 q^{73} -8.47373 q^{74} +3.49541 q^{75} -5.25199 q^{76} +0.411960 q^{77} +10.3768 q^{78} +11.7942 q^{79} +1.66904 q^{80} -7.21730 q^{81} +5.07748 q^{82} +0.564350 q^{83} -0.416723 q^{84} +9.12337 q^{85} +2.93970 q^{86} -6.75152 q^{87} -1.56051 q^{88} +6.71047 q^{89} +0.848134 q^{90} +1.73536 q^{91} +0.430395 q^{92} -12.2888 q^{93} +0.00890887 q^{94} -8.76578 q^{95} +1.57856 q^{96} -9.01960 q^{97} +6.93031 q^{98} -0.792986 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) $ 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.57856	−0.911380	−0.455690	−	0.890138i	$-0.650608\pi$
$3$	−1.57856	−0.911380	−0.455690	+	0.890138i	$0.650608\pi$
$4$	1.00000	0.500000
$5$	1.66904	0.746417	0.373209	−	0.927747i	$-0.378258\pi$
$5$	1.66904	0.746417	0.373209	+	0.927747i	$0.378258\pi$
$6$	1.57856	0.644443
$7$	0.263990	0.0997788	0.0498894	−	0.998755i	$-0.484113\pi$
$7$	0.263990	0.0997788	0.0498894	+	0.998755i	$0.484113\pi$
$8$	−1.00000	−0.353553
$9$	−0.508157	−0.169386
$10$	−1.66904	−0.527797
$11$	1.56051	0.470513	0.235256	−	0.971933i	$-0.424407\pi$
$11$	1.56051	0.470513	0.235256	+	0.971933i	$0.424407\pi$
$12$	−1.57856	−0.455690
$13$	6.57360	1.82319	0.911595	−	0.411090i	$-0.134852\pi$
$13$	6.57360	1.82319	0.911595	+	0.411090i	$0.134852\pi$
$14$	−0.263990	−0.0705542
$15$	−2.63467	−0.680270
$16$	1.00000	0.250000
$17$	5.46624	1.32576	0.662879	−	0.748726i	$-0.269334\pi$
$17$	5.46624	1.32576	0.662879	+	0.748726i	$0.269334\pi$
$18$	0.508157	0.119774
$19$	−5.25199	−1.20489	−0.602445	−	0.798161i	$-0.705807\pi$
$19$	−5.25199	−1.20489	−0.602445	+	0.798161i	$0.705807\pi$
$20$	1.66904	0.373209
$21$	−0.416723	−0.0909364
$22$	−1.56051	−0.332703
$23$	0.430395	0.0897437	0.0448718	−	0.998993i	$-0.485712\pi$
$23$	0.430395	0.0897437	0.0448718	+	0.998993i	$0.485712\pi$
$24$	1.57856	0.322222
$25$	−2.21431	−0.442861
$26$	−6.57360	−1.28919
$27$	5.53783	1.06576
$28$	0.263990	0.0498894
$29$	4.27702	0.794223	0.397111	−	0.917770i	$-0.370013\pi$
$29$	4.27702	0.794223	0.397111	+	0.917770i	$0.370013\pi$
$30$	2.63467	0.481023
$31$	7.78485	1.39820	0.699100	−	0.715024i	$-0.253584\pi$
$31$	7.78485	1.39820	0.699100	+	0.715024i	$0.253584\pi$
$32$	−1.00000	−0.176777
$33$	−2.46336	−0.428816
$34$	−5.46624	−0.937453
$35$	0.440609	0.0744766
$36$	−0.508157	−0.0846929
$37$	8.47373	1.39307	0.696537	−	0.717521i	$-0.254723\pi$
$37$	8.47373	1.39307	0.696537	+	0.717521i	$0.254723\pi$
$38$	5.25199	0.851985
$39$	−10.3768	−1.66162
$40$	−1.66904	−0.263898
$41$	−5.07748	−0.792969	−0.396484	−	0.918041i	$-0.629770\pi$
$41$	−5.07748	−0.792969	−0.396484	+	0.918041i	$0.629770\pi$
$42$	0.416723	0.0643018
$43$	−2.93970	−0.448300	−0.224150	−	0.974555i	$-0.571960\pi$
$43$	−2.93970	−0.448300	−0.224150	+	0.974555i	$0.571960\pi$
$44$	1.56051	0.235256
$45$	−0.848134	−0.126432
$46$	−0.430395	−0.0634584
$47$	−0.00890887	−0.00129949	−0.000649746	−	1.00000i	$-0.500207\pi$
$47$	−0.00890887	−0.00129949	−0.000649746		1.00000i	$0.500207\pi$
$48$	−1.57856	−0.227845
$49$	−6.93031	−0.990044
$50$	2.21431	0.313150
$51$	−8.62877	−1.20827
$52$	6.57360	0.911595
$53$	2.51186	0.345031	0.172515	−	0.985007i	$-0.444811\pi$
$53$	2.51186	0.345031	0.172515	+	0.985007i	$0.444811\pi$
$54$	−5.53783	−0.753603
$55$	2.60456	0.351199
$56$	−0.263990	−0.0352771
$57$	8.29057	1.09811
$58$	−4.27702	−0.561600
$59$	−4.97914	−0.648229	−0.324114	−	0.946018i	$-0.605066\pi$
$59$	−4.97914	−0.648229	−0.324114	+	0.946018i	$0.605066\pi$
$60$	−2.63467	−0.340135
$61$	5.35666	0.685850	0.342925	−	0.939363i	$-0.388582\pi$
$61$	5.35666	0.685850	0.342925	+	0.939363i	$0.388582\pi$
$62$	−7.78485	−0.988677
$63$	−0.134148	−0.0169011
$64$	1.00000	0.125000
$65$	10.9716	1.36086
$66$	2.46336	0.303219
$67$	−12.7223	−1.55427	−0.777137	−	0.629331i	$-0.783329\pi$
$67$	−12.7223	−1.55427	−0.777137	+	0.629331i	$0.783329\pi$
$68$	5.46624	0.662879
$69$	−0.679404	−0.0817906
$70$	−0.440609	−0.0526629
$71$	−2.49503	−0.296106	−0.148053	−	0.988979i	$-0.547301\pi$
$71$	−2.49503	−0.296106	−0.148053	+	0.988979i	$0.547301\pi$
$72$	0.508157	0.0598869
$73$	16.4892	1.92991	0.964957	−	0.262408i	$-0.0845166\pi$
$73$	16.4892	1.92991	0.964957	+	0.262408i	$0.0845166\pi$
$74$	−8.47373	−0.985051
$75$	3.49541	0.403615
$76$	−5.25199	−0.602445
$77$	0.411960	0.0469472
$78$	10.3768	1.17494
$79$	11.7942	1.32696	0.663478	−	0.748196i	$-0.269080\pi$
$79$	11.7942	1.32696	0.663478	+	0.748196i	$0.269080\pi$
$80$	1.66904	0.186604
$81$	−7.21730	−0.801923
$82$	5.07748	0.560714
$83$	0.564350	0.0619454	0.0309727	−	0.999520i	$-0.490139\pi$
$83$	0.564350	0.0619454	0.0309727	+	0.999520i	$0.490139\pi$
$84$	−0.416723	−0.0454682
$85$	9.12337	0.989569
$86$	2.93970	0.316996
$87$	−6.75152	−0.723839
$88$	−1.56051	−0.166351
$89$	6.71047	0.711309	0.355654	−	0.934618i	$-0.384258\pi$
$89$	6.71047	0.711309	0.355654	+	0.934618i	$0.384258\pi$
$90$	0.848134	0.0894012
$91$	1.73536	0.181916
$92$	0.430395	0.0448718
$93$	−12.2888	−1.27429
$94$	0.00890887	0.000918880 0
$95$	−8.76578	−0.899350
$96$	1.57856	0.161111
$97$	−9.01960	−0.915802	−0.457901	−	0.889003i	$-0.651399\pi$
$97$	−9.01960	−0.915802	−0.457901	+	0.889003i	$0.651399\pi$
$98$	6.93031	0.700067
$99$	−0.792986	−0.0796981
$100$	−2.21431	−0.221431
$101$	−4.29765	−0.427632	−0.213816	−	0.976874i	$-0.568589\pi$
$101$	−4.29765	−0.427632	−0.213816	+	0.976874i	$0.568589\pi$
$102$	8.62877	0.854376
$103$	3.90916	0.385181	0.192591	−	0.981279i	$-0.438311\pi$
$103$	3.90916	0.385181	0.192591	+	0.981279i	$0.438311\pi$
$104$	−6.57360	−0.644595
$105$	−0.695527	−0.0678765
$106$	−2.51186	−0.243974
$107$	1.57415	0.152179	0.0760894	−	0.997101i	$-0.475757\pi$
$107$	1.57415	0.152179	0.0760894	+	0.997101i	$0.475757\pi$
$108$	5.53783	0.532878
$109$	−13.8902	−1.33044	−0.665218	−	0.746649i	$-0.731661\pi$
$109$	−13.8902	−1.33044	−0.665218	+	0.746649i	$0.731661\pi$
$110$	−2.60456	−0.248335
$111$	−13.3763	−1.26962
$112$	0.263990	0.0249447
$113$	8.04705	0.757003	0.378501	−	0.925601i	$-0.376440\pi$
$113$	8.04705	0.757003	0.378501	+	0.925601i	$0.376440\pi$
$114$	−8.29057	−0.776483
$115$	0.718347	0.0669862
$116$	4.27702	0.397111
$117$	−3.34042	−0.308822
$118$	4.97914	0.458367
$119$	1.44303	0.132283
$120$	2.63467	0.240512
$121$	−8.56480	−0.778618
$122$	−5.35666	−0.484969
$123$	8.01509	0.722696
$124$	7.78485	0.699100
$125$	−12.0410	−1.07698
$126$	0.134148	0.0119509
$127$	−14.3275	−1.27136	−0.635679	−	0.771954i	$-0.719280\pi$
$127$	−14.3275	−1.27136	−0.635679	+	0.771954i	$0.719280\pi$
$128$	−1.00000	−0.0883883
$129$	4.64048	0.408571
$130$	−10.9716	−0.962273
$131$	−4.85016	−0.423760	−0.211880	−	0.977296i	$-0.567959\pi$
$131$	−4.85016	−0.423760	−0.211880	+	0.977296i	$0.567959\pi$
$132$	−2.46336	−0.214408
$133$	−1.38647	−0.120222
$134$	12.7223	1.09904
$135$	9.24285	0.795498
$136$	−5.46624	−0.468726
$137$	11.9342	1.01960	0.509802	−	0.860292i	$-0.329719\pi$
$137$	11.9342	1.01960	0.509802	+	0.860292i	$0.329719\pi$
$138$	0.679404	0.0578347
$139$	7.98987	0.677692	0.338846	−	0.940842i	$-0.389963\pi$
$139$	7.98987	0.677692	0.338846	+	0.940842i	$0.389963\pi$
$140$	0.440609	0.0372383
$141$	0.0140632	0.00118433
$142$	2.49503	0.209379
$143$	10.2582	0.857834
$144$	−0.508157	−0.0423464
$145$	7.13852	0.592821
$146$	−16.4892	−1.36466
$147$	10.9399	0.902307
$148$	8.47373	0.696537
$149$	14.8679	1.21802	0.609011	−	0.793162i	$-0.291566\pi$
$149$	14.8679	1.21802	0.609011	+	0.793162i	$0.291566\pi$
$150$	−3.49541	−0.285399
$151$	−7.99720	−0.650803	−0.325401	−	0.945576i	$-0.605499\pi$
$151$	−7.99720	−0.650803	−0.325401	+	0.945576i	$0.605499\pi$
$152$	5.25199	0.425993
$153$	−2.77771	−0.224565
$154$	−0.411960	−0.0331967
$155$	12.9932	1.04364
$156$	−10.3768	−0.830810
$157$	−3.78478	−0.302058	−0.151029	−	0.988529i	$-0.548259\pi$
$157$	−3.78478	−0.302058	−0.151029	+	0.988529i	$0.548259\pi$
$158$	−11.7942	−0.938300
$159$	−3.96512	−0.314454
$160$	−1.66904	−0.131949
$161$	0.113620	0.00895451
$162$	7.21730	0.567045
$163$	9.56634	0.749294	0.374647	−	0.927168i	$-0.377764\pi$
$163$	9.56634	0.749294	0.374647	+	0.927168i	$0.377764\pi$
$164$	−5.07748	−0.396484
$165$	−4.11145	−0.320076
$166$	−0.564350	−0.0438020
$167$	−20.3882	−1.57769	−0.788843	−	0.614595i	$-0.789319\pi$
$167$	−20.3882	−1.57769	−0.788843	+	0.614595i	$0.789319\pi$
$168$	0.416723	0.0321509
$169$	30.2123	2.32402
$170$	−9.12337	−0.699731
$171$	2.66884	0.204091
$172$	−2.93970	−0.224150
$173$	−11.0372	−0.839144	−0.419572	−	0.907722i	$-0.637820\pi$
$173$	−11.0372	−0.839144	−0.419572	+	0.907722i	$0.637820\pi$
$174$	6.75152	0.511832
$175$	−0.584555	−0.0441882
$176$	1.56051	0.117628
$177$	7.85986	0.590783
$178$	−6.71047	−0.502971
$179$	24.6178	1.84002	0.920010	−	0.391896i	$-0.128181\pi$
$179$	24.6178	1.84002	0.920010	+	0.391896i	$0.128181\pi$
$180$	−0.848134	−0.0632162
$181$	−1.68090	−0.124940	−0.0624700	−	0.998047i	$-0.519898\pi$
$181$	−1.68090	−0.124940	−0.0624700	+	0.998047i	$0.519898\pi$
$182$	−1.73536	−0.128634
$183$	−8.45579	−0.625070
$184$	−0.430395	−0.0317292
$185$	14.1430	1.03981
$186$	12.2888	0.901061
$187$	8.53015	0.623786
$188$	−0.00890887	−0.000649746 0
$189$	1.46193	0.106340
$190$	8.76578	0.635936
$191$	−11.9401	−0.863955	−0.431977	−	0.901884i	$-0.642184\pi$
$191$	−11.9401	−0.863955	−0.431977	+	0.901884i	$0.642184\pi$
$192$	−1.57856	−0.113923
$193$	1.05892	0.0762229	0.0381114	−	0.999273i	$-0.487866\pi$
$193$	1.05892	0.0762229	0.0381114	+	0.999273i	$0.487866\pi$
$194$	9.01960	0.647569
$195$	−17.3193	−1.24026
$196$	−6.93031	−0.495022
$197$	−9.94657	−0.708664	−0.354332	−	0.935120i	$-0.615292\pi$
$197$	−9.94657	−0.708664	−0.354332	+	0.935120i	$0.615292\pi$
$198$	0.792986	0.0563551
$199$	−1.71787	−0.121776	−0.0608881	−	0.998145i	$-0.519393\pi$
$199$	−1.71787	−0.121776	−0.0608881	+	0.998145i	$0.519393\pi$
$200$	2.21431	0.156575
$201$	20.0828	1.41653
$202$	4.29765	0.302381
$203$	1.12909	0.0792466
$204$	−8.62877	−0.604135
$205$	−8.47451	−0.591886
$206$	−3.90916	−0.272364
$207$	−0.218709	−0.0152013
$208$	6.57360	0.455797
$209$	−8.19580	−0.566916
$210$	0.695527	0.0479959
$211$	−10.3341	−0.711430	−0.355715	−	0.934594i	$-0.615763\pi$
$211$	−10.3341	−0.711430	−0.355715	+	0.934594i	$0.615763\pi$
$212$	2.51186	0.172515
$213$	3.93856	0.269865
$214$	−1.57415	−0.107607
$215$	−4.90647	−0.334618
$216$	−5.53783	−0.376801
$217$	2.05512	0.139511
$218$	13.8902	0.940760
$219$	−26.0291	−1.75889
$220$	2.60456	0.175599
$221$	35.9329	2.41711
$222$	13.3763	0.897757
$223$	0.285537	0.0191210	0.00956048	−	0.999954i	$-0.496957\pi$
$223$	0.285537	0.0191210	0.00956048	+	0.999954i	$0.496957\pi$
$224$	−0.263990	−0.0176386
$225$	1.12522	0.0750144
$226$	−8.04705	−0.535282
$227$	−12.8590	−0.853482	−0.426741	−	0.904374i	$-0.640338\pi$
$227$	−12.8590	−0.853482	−0.426741	+	0.904374i	$0.640338\pi$
$228$	8.29057	0.549056
$229$	14.1197	0.933057	0.466528	−	0.884506i	$-0.345505\pi$
$229$	14.1197	0.933057	0.466528	+	0.884506i	$0.345505\pi$
$230$	−0.718347	−0.0473664
$231$	−0.650302	−0.0427867
$232$	−4.27702	−0.280800
$233$	30.1449	1.97486	0.987429	−	0.158064i	$-0.0505253\pi$
$233$	30.1449	1.97486	0.987429	+	0.158064i	$0.0505253\pi$
$234$	3.34042	0.218370
$235$	−0.0148693	−0.000969963 0
$236$	−4.97914	−0.324114
$237$	−18.6179	−1.20936
$238$	−1.44303	−0.0935379
$239$	13.6801	0.884889	0.442444	−	0.896796i	$-0.354111\pi$
$239$	13.6801	0.884889	0.442444	+	0.896796i	$0.354111\pi$
$240$	−2.63467	−0.170067
$241$	13.8194	0.890185	0.445092	−	0.895485i	$-0.353171\pi$
$241$	13.8194	0.890185	0.445092	+	0.895485i	$0.353171\pi$
$242$	8.56480	0.550566
$243$	−5.22055	−0.334899
$244$	5.35666	0.342925
$245$	−11.5670	−0.738986
$246$	−8.01509	−0.511024
$247$	−34.5245	−2.19674
$248$	−7.78485	−0.494339
$249$	−0.890859	−0.0564559
$250$	12.0410	0.761537
$251$	10.1659	0.641665	0.320833	−	0.947136i	$-0.396037\pi$
$251$	10.1659	0.641665	0.320833	+	0.947136i	$0.396037\pi$
$252$	−0.134148	−0.00845055
$253$	0.671638	0.0422255
$254$	14.3275	0.898985
$255$	−14.4018	−0.901873
$256$	1.00000	0.0625000
$257$	−24.1182	−1.50445	−0.752225	−	0.658906i	$-0.771020\pi$
$257$	−24.1182	−1.50445	−0.752225	+	0.658906i	$0.771020\pi$
$258$	−4.64048	−0.288904
$259$	2.23698	0.138999
$260$	10.9716	0.680430
$261$	−2.17340	−0.134530
$262$	4.85016	0.299644
$263$	24.5523	1.51396	0.756981	−	0.653437i	$-0.226673\pi$
$263$	24.5523	1.51396	0.756981	+	0.653437i	$0.226673\pi$
$264$	2.46336	0.151609
$265$	4.19240	0.257537
$266$	1.38647	0.0850100
$267$	−10.5929	−0.648273
$268$	−12.7223	−0.777137
$269$	−3.27958	−0.199960	−0.0999799	−	0.994989i	$-0.531878\pi$
$269$	−3.27958	−0.199960	−0.0999799	+	0.994989i	$0.531878\pi$
$270$	−9.24285	−0.562502
$271$	−11.4277	−0.694181	−0.347091	−	0.937832i	$-0.612830\pi$
$271$	−11.4277	−0.694181	−0.347091	+	0.937832i	$0.612830\pi$
$272$	5.46624	0.331440
$273$	−2.73937	−0.165794
$274$	−11.9342	−0.720969
$275$	−3.45546	−0.208372
$276$	−0.679404	−0.0408953
$277$	3.76300	0.226097	0.113048	−	0.993589i	$-0.463938\pi$
$277$	3.76300	0.226097	0.113048	+	0.993589i	$0.463938\pi$
$278$	−7.98987	−0.479201
$279$	−3.95593	−0.236835
$280$	−0.440609	−0.0263314
$281$	14.8797	0.887646	0.443823	−	0.896114i	$-0.353622\pi$
$281$	14.8797	0.887646	0.443823	+	0.896114i	$0.353622\pi$
$282$	−0.0140632	−0.000837449 0
$283$	−3.42369	−0.203518	−0.101759	−	0.994809i	$-0.532447\pi$
$283$	−3.42369	−0.203518	−0.101759	+	0.994809i	$0.532447\pi$
$284$	−2.49503	−0.148053
$285$	13.8373	0.819650
$286$	−10.2582	−0.606580
$287$	−1.34040	−0.0791215
$288$	0.508157	0.0299434
$289$	12.8798	0.757635
$290$	−7.13852	−0.419188
$291$	14.2380	0.834644
$292$	16.4892	0.964957
$293$	14.3078	0.835870	0.417935	−	0.908477i	$-0.362754\pi$
$293$	14.3078	0.835870	0.417935	+	0.908477i	$0.362754\pi$
$294$	−10.9399	−0.638027
$295$	−8.31038	−0.483849
$296$	−8.47373	−0.492526
$297$	8.64186	0.501451
$298$	−14.8679	−0.861272
$299$	2.82925	0.163620
$300$	3.49541	0.201808
$301$	−0.776050	−0.0447308
$302$	7.99720	0.460187
$303$	6.78408	0.389735
$304$	−5.25199	−0.301222
$305$	8.94047	0.511930
$306$	2.77771	0.158791
$307$	12.6071	0.719528	0.359764	−	0.933043i	$-0.382857\pi$
$307$	12.6071	0.719528	0.359764	+	0.933043i	$0.382857\pi$
$308$	0.411960	0.0234736
$309$	−6.17084	−0.351047
$310$	−12.9932	−0.737965
$311$	29.7238	1.68548	0.842741	−	0.538319i	$-0.180940\pi$
$311$	29.7238	1.68548	0.842741	+	0.538319i	$0.180940\pi$
$312$	10.3768	0.587471
$313$	−16.0132	−0.905118	−0.452559	−	0.891734i	$-0.649489\pi$
$313$	−16.0132	−0.905118	−0.452559	+	0.891734i	$0.649489\pi$
$314$	3.78478	0.213587
$315$	−0.223899	−0.0126153
$316$	11.7942	0.663478
$317$	20.1467	1.13155	0.565776	−	0.824559i	$-0.308577\pi$
$317$	20.1467	1.13155	0.565776	+	0.824559i	$0.308577\pi$
$318$	3.96512	0.222353
$319$	6.67435	0.373692
$320$	1.66904	0.0933021
$321$	−2.48488	−0.138693
$322$	−0.113620	−0.00633180
$323$	−28.7086	−1.59739
$324$	−7.21730	−0.400961
$325$	−14.5560	−0.807421
$326$	−9.56634	−0.529831
$327$	21.9264	1.21253
$328$	5.07748	0.280357
$329$	−0.00235185	−0.000129662 0
$330$	4.11145	0.226328
$331$	−5.16024	−0.283633	−0.141816	−	0.989893i	$-0.545294\pi$
$331$	−5.16024	−0.283633	−0.141816	+	0.989893i	$0.545294\pi$
$332$	0.564350	0.0309727
$333$	−4.30599	−0.235967
$334$	20.3882	1.11559
$335$	−21.2340	−1.16014
$336$	−0.416723	−0.0227341
$337$	14.8235	0.807489	0.403745	−	0.914872i	$-0.367708\pi$
$337$	14.8235	0.807489	0.403745	+	0.914872i	$0.367708\pi$
$338$	−30.2123	−1.64333
$339$	−12.7027	−0.689918
$340$	9.12337	0.494784
$341$	12.1484	0.657871
$342$	−2.66884	−0.144314
$343$	−3.67746	−0.198564
$344$	2.93970	0.158498
$345$	−1.13395	−0.0610499
$346$	11.0372	0.593365
$347$	14.8433	0.796831	0.398416	−	0.917205i	$-0.369560\pi$
$347$	14.8433	0.796831	0.398416	+	0.917205i	$0.369560\pi$
$348$	−6.75152	−0.361920
$349$	−0.615364	−0.0329397	−0.0164698	−	0.999864i	$-0.505243\pi$
$349$	−0.615364	−0.0329397	−0.0164698	+	0.999864i	$0.505243\pi$
$350$	0.584555	0.0312458
$351$	36.4035	1.94307
$352$	−1.56051	−0.0831757
$353$	25.1695	1.33964	0.669819	−	0.742524i	$-0.266372\pi$
$353$	25.1695	1.33964	0.669819	+	0.742524i	$0.266372\pi$
$354$	−7.85986	−0.417747
$355$	−4.16431	−0.221019
$356$	6.71047	0.355654
$357$	−2.27791	−0.120560
$358$	−24.6178	−1.30109
$359$	27.2134	1.43627	0.718134	−	0.695904i	$-0.244996\pi$
$359$	27.2134	1.43627	0.718134	+	0.695904i	$0.244996\pi$
$360$	0.848134	0.0447006
$361$	8.58339	0.451758
$362$	1.68090	0.0883459
$363$	13.5200	0.709617
$364$	1.73536	0.0909578
$365$	27.5211	1.44052
$366$	8.45579	0.441991
$367$	2.26804	0.118391	0.0591955	−	0.998246i	$-0.481146\pi$
$367$	2.26804	0.118391	0.0591955	+	0.998246i	$0.481146\pi$
$368$	0.430395	0.0224359
$369$	2.58016	0.134318
$370$	−14.1430	−0.735259
$371$	0.663106	0.0344267
$372$	−12.2888	−0.637146
$373$	−3.11492	−0.161284	−0.0806422	−	0.996743i	$-0.525697\pi$
$373$	−3.11492	−0.161284	−0.0806422	+	0.996743i	$0.525697\pi$
$374$	−8.53015	−0.441083
$375$	19.0073	0.981535
$376$	0.00890887	0.000459440 0
$377$	28.1154	1.44802
$378$	−1.46193	−0.0751936
$379$	26.5307	1.36279	0.681396	−	0.731915i	$-0.261373\pi$
$379$	26.5307	1.36279	0.681396	+	0.731915i	$0.261373\pi$
$380$	−8.76578	−0.449675
$381$	22.6167	1.15869
$382$	11.9401	0.610908
$383$	−2.01425	−0.102924	−0.0514618	−	0.998675i	$-0.516388\pi$
$383$	−2.01425	−0.102924	−0.0514618	+	0.998675i	$0.516388\pi$
$384$	1.57856	0.0805554
$385$	0.687577	0.0350422
$386$	−1.05892	−0.0538977
$387$	1.49383	0.0759355
$388$	−9.01960	−0.457901
$389$	31.0352	1.57354	0.786772	−	0.617243i	$-0.211751\pi$
$389$	31.0352	1.57354	0.786772	+	0.617243i	$0.211751\pi$
$390$	17.3193	0.876997
$391$	2.35265	0.118978
$392$	6.93031	0.350033
$393$	7.65625	0.386207
$394$	9.94657	0.501101
$395$	19.6851	0.990463
$396$	−0.792986	−0.0398491
$397$	−20.1299	−1.01029	−0.505146	−	0.863034i	$-0.668561\pi$
$397$	−20.1299	−1.01029	−0.505146	+	0.863034i	$0.668561\pi$
$398$	1.71787	0.0861088
$399$	2.18862	0.109568
$400$	−2.21431	−0.110715
$401$	2.88270	0.143955	0.0719776	−	0.997406i	$-0.477069\pi$
$401$	2.88270	0.143955	0.0719776	+	0.997406i	$0.477069\pi$
$402$	−20.0828	−1.00164
$403$	51.1745	2.54918
$404$	−4.29765	−0.213816
$405$	−12.0460	−0.598569
$406$	−1.12909	−0.0560358
$407$	13.2234	0.655459
$408$	8.62877	0.427188
$409$	28.3360	1.40112	0.700562	−	0.713592i	$-0.252933\pi$
$409$	28.3360	1.40112	0.700562	+	0.713592i	$0.252933\pi$
$410$	8.47451	0.418526
$411$	−18.8387	−0.929247
$412$	3.90916	0.192591
$413$	−1.31444	−0.0646795
$414$	0.218709	0.0107489
$415$	0.941922	0.0462371
$416$	−6.57360	−0.322297
$417$	−12.6125	−0.617635
$418$	8.19580	0.400870
$419$	−13.4666	−0.657888	−0.328944	−	0.944349i	$-0.606693\pi$
$419$	−13.4666	−0.657888	−0.328944	+	0.944349i	$0.606693\pi$
$420$	−0.695527	−0.0339382
$421$	−3.63317	−0.177070	−0.0885350	−	0.996073i	$-0.528219\pi$
$421$	−3.63317	−0.177070	−0.0885350	+	0.996073i	$0.528219\pi$
$422$	10.3341	0.503057
$423$	0.00452711	0.000220115 0
$424$	−2.51186	−0.121987
$425$	−12.1039	−0.587127
$426$	−3.93856	−0.190824
$427$	1.41410	0.0684332
$428$	1.57415	0.0760894
$429$	−16.1932	−0.781813
$430$	4.90647	0.236611
$431$	34.8711	1.67968	0.839842	−	0.542832i	$-0.182648\pi$
$431$	34.8711	1.67968	0.839842	+	0.542832i	$0.182648\pi$
$432$	5.53783	0.266439
$433$	−11.1405	−0.535379	−0.267690	−	0.963505i	$-0.586260\pi$
$433$	−11.1405	−0.535379	−0.267690	+	0.963505i	$0.586260\pi$
$434$	−2.05512	−0.0986490
$435$	−11.2686	−0.540286
$436$	−13.8902	−0.665218
$437$	−2.26043	−0.108131
$438$	26.0291	1.24372
$439$	27.8695	1.33014	0.665069	−	0.746782i	$-0.268402\pi$
$439$	27.8695	1.33014	0.665069	+	0.746782i	$0.268402\pi$
$440$	−2.60456	−0.124168
$441$	3.52169	0.167699
$442$	−35.9329	−1.70915
$443$	−24.0905	−1.14457	−0.572287	−	0.820054i	$-0.693944\pi$
$443$	−24.0905	−1.14457	−0.572287	+	0.820054i	$0.693944\pi$
$444$	−13.3763	−0.634810
$445$	11.2000	0.530933
$446$	−0.285537	−0.0135206
$447$	−23.4698	−1.11008
$448$	0.263990	0.0124723
$449$	−25.1702	−1.18785	−0.593927	−	0.804519i	$-0.702423\pi$
$449$	−25.1702	−1.18785	−0.593927	+	0.804519i	$0.702423\pi$
$450$	−1.12522	−0.0530432
$451$	−7.92348	−0.373102
$452$	8.04705	0.378501
$453$	12.6240	0.593129
$454$	12.8590	0.603503
$455$	2.89639	0.135785
$456$	−8.29057	−0.388241
$457$	14.1219	0.660595	0.330298	−	0.943877i	$-0.392851\pi$
$457$	14.1219	0.660595	0.330298	+	0.943877i	$0.392851\pi$
$458$	−14.1197	−0.659771
$459$	30.2711	1.41293
$460$	0.718347	0.0334931
$461$	−0.832657	−0.0387807	−0.0193904	−	0.999812i	$-0.506173\pi$
$461$	−0.832657	−0.0387807	−0.0193904	+	0.999812i	$0.506173\pi$
$462$	0.650302	0.0302548
$463$	18.5368	0.861479	0.430739	−	0.902476i	$-0.358253\pi$
$463$	18.5368	0.861479	0.430739	+	0.902476i	$0.358253\pi$
$464$	4.27702	0.198556
$465$	−20.5105	−0.951154
$466$	−30.1449	−1.39644
$467$	10.6243	0.491632	0.245816	−	0.969316i	$-0.420944\pi$
$467$	10.6243	0.491632	0.245816	+	0.969316i	$0.420944\pi$
$468$	−3.34042	−0.154411
$469$	−3.35855	−0.155084
$470$	0.0148693	0.000685868 0
$471$	5.97449	0.275290
$472$	4.97914	0.229183
$473$	−4.58744	−0.210931
$474$	18.6179	0.855148
$475$	11.6295	0.533599
$476$	1.44303	0.0661413
$477$	−1.27642	−0.0584433
$478$	−13.6801	−0.625711
$479$	−37.4630	−1.71173	−0.855863	−	0.517203i	$-0.826973\pi$
$479$	−37.4630	−1.71173	−0.855863	+	0.517203i	$0.826973\pi$
$480$	2.63467	0.120256
$481$	55.7030	2.53984
$482$	−13.8194	−0.629456
$483$	−0.179356	−0.00816097
$484$	−8.56480	−0.389309
$485$	−15.0541	−0.683570
$486$	5.22055	0.236809
$487$	−9.37558	−0.424848	−0.212424	−	0.977178i	$-0.568136\pi$
$487$	−9.37558	−0.424848	−0.212424	+	0.977178i	$0.568136\pi$
$488$	−5.35666	−0.242485
$489$	−15.1010	−0.682892
$490$	11.5670	0.522542
$491$	3.87368	0.174817	0.0874083	−	0.996173i	$-0.472142\pi$
$491$	3.87368	0.174817	0.0874083	+	0.996173i	$0.472142\pi$
$492$	8.01509	0.361348
$493$	23.3792	1.05295
$494$	34.5245	1.55333
$495$	−1.32353	−0.0594881
$496$	7.78485	0.349550
$497$	−0.658664	−0.0295451
$498$	0.890859	0.0399203
$499$	−23.0631	−1.03245	−0.516223	−	0.856454i	$-0.672663\pi$
$499$	−23.0631	−1.03245	−0.516223	+	0.856454i	$0.672663\pi$
$500$	−12.0410	−0.538488
$501$	32.1839	1.43787
$502$	−10.1659	−0.453726
$503$	18.6273	0.830552	0.415276	−	0.909695i	$-0.363685\pi$
$503$	18.6273	0.830552	0.415276	+	0.909695i	$0.363685\pi$
$504$	0.134148	0.00597544
$505$	−7.17294	−0.319192
$506$	−0.671638	−0.0298580
$507$	−47.6918	−2.11807
$508$	−14.3275	−0.635679
$509$	33.2941	1.47574	0.737869	−	0.674944i	$-0.235832\pi$
$509$	33.2941	1.47574	0.737869	+	0.674944i	$0.235832\pi$
$510$	14.4018	0.637721
$511$	4.35298	0.192564
$512$	−1.00000	−0.0441942
$513$	−29.0846	−1.28412
$514$	24.1182	1.06381
$515$	6.52454	0.287506
$516$	4.64048	0.204286
$517$	−0.0139024	−0.000611428 0
$518$	−2.23698	−0.0982872
$519$	17.4229	0.764780
$520$	−10.9716	−0.481137
$521$	20.5613	0.900805	0.450403	−	0.892826i	$-0.351280\pi$
$521$	20.5613	0.900805	0.450403	+	0.892826i	$0.351280\pi$
$522$	2.17340	0.0951271
$523$	−13.2217	−0.578145	−0.289073	−	0.957307i	$-0.593347\pi$
$523$	−13.2217	−0.578145	−0.289073	+	0.957307i	$0.593347\pi$
$524$	−4.85016	−0.211880
$525$	0.922753	0.0402722
$526$	−24.5523	−1.07053
$527$	42.5539	1.85368
$528$	−2.46336	−0.107204
$529$	−22.8148	−0.991946
$530$	−4.19240	−0.182106
$531$	2.53018	0.109801
$532$	−1.38647	−0.0601112
$533$	−33.3773	−1.44573
$534$	10.5929	0.458398
$535$	2.62732	0.113589
$536$	12.7223	0.549519
$537$	−38.8606	−1.67696
$538$	3.27958	0.141393
$539$	−10.8148	−0.465828
$540$	9.24285	0.397749
$541$	−40.2937	−1.73236	−0.866181	−	0.499730i	$-0.833433\pi$
$541$	−40.2937	−1.73236	−0.866181	+	0.499730i	$0.833433\pi$
$542$	11.4277	0.490860
$543$	2.65339	0.113868
$544$	−5.46624	−0.234363
$545$	−23.1832	−0.993060
$546$	2.73937	0.117234
$547$	−18.8394	−0.805516	−0.402758	−	0.915306i	$-0.631948\pi$
$547$	−18.8394	−0.805516	−0.402758	+	0.915306i	$0.631948\pi$
$548$	11.9342	0.509802
$549$	−2.72202	−0.116173
$550$	3.45546	0.147341
$551$	−22.4629	−0.956950
$552$	0.679404	0.0289174
$553$	3.11356	0.132402
$554$	−3.76300	−0.159875
$555$	−22.3255	−0.947666
$556$	7.98987	0.338846
$557$	−18.5143	−0.784477	−0.392238	−	0.919864i	$-0.628299\pi$
$557$	−18.5143	−0.784477	−0.392238	+	0.919864i	$0.628299\pi$
$558$	3.95593	0.167468
$559$	−19.3244	−0.817335
$560$	0.440609	0.0186191
$561$	−13.4653	−0.568507
$562$	−14.8797	−0.627661
$563$	38.6817	1.63024	0.815120	−	0.579292i	$-0.196671\pi$
$563$	38.6817	1.63024	0.815120	+	0.579292i	$0.196671\pi$
$564$	0.0140632	0.000592166 0
$565$	13.4308	0.565040
$566$	3.42369	0.143909
$567$	−1.90529	−0.0800149
$568$	2.49503	0.104689
$569$	−14.5480	−0.609884	−0.304942	−	0.952371i	$-0.598637\pi$
$569$	−14.5480	−0.609884	−0.304942	+	0.952371i	$0.598637\pi$
$570$	−13.8373	−0.579580
$571$	−12.0486	−0.504219	−0.252110	−	0.967699i	$-0.581124\pi$
$571$	−12.0486	−0.504219	−0.252110	+	0.967699i	$0.581124\pi$
$572$	10.2582	0.428917
$573$	18.8481	0.787392
$574$	1.34040	0.0559473
$575$	−0.953028	−0.0397440
$576$	−0.508157	−0.0211732
$577$	−18.0187	−0.750128	−0.375064	−	0.926999i	$-0.622379\pi$
$577$	−18.0187	−0.750128	−0.375064	+	0.926999i	$0.622379\pi$
$578$	−12.8798	−0.535729
$579$	−1.67157	−0.0694680
$580$	7.13852	0.296411
$581$	0.148983	0.00618084
$582$	−14.2380	−0.590182
$583$	3.91980	0.162341
$584$	−16.4892	−0.682328
$585$	−5.57530	−0.230510
$586$	−14.3078	−0.591050
$587$	25.5683	1.05532	0.527659	−	0.849456i	$-0.323070\pi$
$587$	25.5683	1.05532	0.527659	+	0.849456i	$0.323070\pi$
$588$	10.9399	0.451153
$589$	−40.8860	−1.68468
$590$	8.31038	0.342133
$591$	15.7012	0.645862
$592$	8.47373	0.348268
$593$	−11.8112	−0.485029	−0.242514	−	0.970148i	$-0.577972\pi$
$593$	−11.8112	−0.485029	−0.242514	+	0.970148i	$0.577972\pi$
$594$	−8.64186	−0.354580
$595$	2.40848	0.0987379
$596$	14.8679	0.609011
$597$	2.71175	0.110984
$598$	−2.82925	−0.115697
$599$	17.6847	0.722579	0.361290	−	0.932454i	$-0.382337\pi$
$599$	17.6847	0.722579	0.361290	+	0.932454i	$0.382337\pi$
$600$	−3.49541	−0.142700
$601$	36.1926	1.47633	0.738163	−	0.674622i	$-0.235694\pi$
$601$	36.1926	1.47633	0.738163	+	0.674622i	$0.235694\pi$
$602$	0.776050	0.0316294
$603$	6.46492	0.263272
$604$	−7.99720	−0.325401
$605$	−14.2950	−0.581174
$606$	−6.78408	−0.275585
$607$	−41.2888	−1.67586	−0.837930	−	0.545777i	$-0.816234\pi$
$607$	−41.2888	−1.67586	−0.837930	+	0.545777i	$0.816234\pi$
$608$	5.25199	0.212996
$609$	−1.78233	−0.0722238
$610$	−8.94047	−0.361989
$611$	−0.0585634	−0.00236922
$612$	−2.77771	−0.112282
$613$	16.6156	0.671096	0.335548	−	0.942023i	$-0.391078\pi$
$613$	16.6156	0.671096	0.335548	+	0.942023i	$0.391078\pi$
$614$	−12.6071	−0.508783
$615$	13.3775	0.539433
$616$	−0.411960	−0.0165983
$617$	−15.7333	−0.633400	−0.316700	−	0.948526i	$-0.602575\pi$
$617$	−15.7333	−0.633400	−0.316700	+	0.948526i	$0.602575\pi$
$618$	6.17084	0.248227
$619$	−23.6284	−0.949705	−0.474852	−	0.880065i	$-0.657499\pi$
$619$	−23.6284	−0.949705	−0.474852	+	0.880065i	$0.657499\pi$
$620$	12.9932	0.521820
$621$	2.38346	0.0956448
$622$	−29.7238	−1.19182
$623$	1.77150	0.0709735
$624$	−10.3768	−0.415405
$625$	−9.02531	−0.361012
$626$	16.0132	0.640015
$627$	12.9375	0.516676
$628$	−3.78478	−0.151029
$629$	46.3195	1.84688
$630$	0.223899	0.00892034
$631$	−44.4769	−1.77060	−0.885299	−	0.465023i	$-0.846046\pi$
$631$	−44.4769	−1.77060	−0.885299	+	0.465023i	$0.846046\pi$
$632$	−11.7942	−0.469150
$633$	16.3130	0.648383
$634$	−20.1467	−0.800128
$635$	−23.9131	−0.948963
$636$	−3.96512	−0.157227
$637$	−45.5571	−1.80504
$638$	−6.67435	−0.264240
$639$	1.26787	0.0501562
$640$	−1.66904	−0.0659746
$641$	13.9157	0.549637	0.274818	−	0.961496i	$-0.411382\pi$
$641$	13.9157	0.549637	0.274818	+	0.961496i	$0.411382\pi$
$642$	2.48488	0.0980706
$643$	2.53824	0.100098	0.0500492	−	0.998747i	$-0.484062\pi$
$643$	2.53824	0.100098	0.0500492	+	0.998747i	$0.484062\pi$
$644$	0.113620	0.00447726
$645$	7.74514	0.304965
$646$	28.7086	1.12953
$647$	−14.0119	−0.550866	−0.275433	−	0.961320i	$-0.588821\pi$
$647$	−14.0119	−0.550866	−0.275433	+	0.961320i	$0.588821\pi$
$648$	7.21730	0.283523
$649$	−7.77002	−0.305000
$650$	14.5560	0.570933
$651$	−3.24413	−0.127147
$652$	9.56634	0.374647
$653$	−17.4558	−0.683100	−0.341550	−	0.939864i	$-0.610952\pi$
$653$	−17.4558	−0.683100	−0.341550	+	0.939864i	$0.610952\pi$
$654$	−21.9264	−0.857391
$655$	−8.09510	−0.316302
$656$	−5.07748	−0.198242
$657$	−8.37910	−0.326900
$658$	0.00235185	9.16847e−5 0
$659$	−16.2765	−0.634043	−0.317022	−	0.948418i	$-0.602683\pi$
$659$	−16.2765	−0.634043	−0.317022	+	0.948418i	$0.602683\pi$
$660$	−4.11145	−0.160038
$661$	−6.33939	−0.246574	−0.123287	−	0.992371i	$-0.539344\pi$
$661$	−6.33939	−0.246574	−0.123287	+	0.992371i	$0.539344\pi$
$662$	5.16024	0.200559
$663$	−56.7221	−2.20291
$664$	−0.564350	−0.0219010
$665$	−2.31408	−0.0897360
$666$	4.30599	0.166854
$667$	1.84081	0.0712765
$668$	−20.3882	−0.788843
$669$	−0.450736	−0.0174265
$670$	21.2340	0.820341
$671$	8.35914	0.322701
$672$	0.416723	0.0160754
$673$	21.0310	0.810685	0.405343	−	0.914165i	$-0.367152\pi$
$673$	21.0310	0.810685	0.405343	+	0.914165i	$0.367152\pi$
$674$	−14.8235	−0.570981
$675$	−12.2625	−0.471982
$676$	30.2123	1.16201
$677$	11.8736	0.456339	0.228170	−	0.973621i	$-0.426726\pi$
$677$	11.8736	0.456339	0.228170	+	0.973621i	$0.426726\pi$
$678$	12.7027	0.487845
$679$	−2.38108	−0.0913776
$680$	−9.12337	−0.349865
$681$	20.2987	0.777847
$682$	−12.1484	−0.465185
$683$	−12.7240	−0.486872	−0.243436	−	0.969917i	$-0.578275\pi$
$683$	−12.7240	−0.486872	−0.243436	+	0.969917i	$0.578275\pi$
$684$	2.66884	0.102045
$685$	19.9186	0.761050
$686$	3.67746	0.140406
$687$	−22.2888	−0.850370
$688$	−2.93970	−0.112075
$689$	16.5120	0.629056
$690$	1.13395	0.0431688
$691$	36.9857	1.40700	0.703501	−	0.710694i	$-0.251619\pi$
$691$	36.9857	1.40700	0.703501	+	0.710694i	$0.251619\pi$
$692$	−11.0372	−0.419572
$693$	−0.209340	−0.00795218
$694$	−14.8433	−0.563445
$695$	13.3354	0.505841
$696$	6.75152	0.255916
$697$	−27.7547	−1.05129
$698$	0.615364	0.0232919
$699$	−47.5854	−1.79985
$700$	−0.584555	−0.0220941
$701$	10.5275	0.397617	0.198808	−	0.980038i	$-0.436293\pi$
$701$	10.5275	0.397617	0.198808	+	0.980038i	$0.436293\pi$
$702$	−36.4035	−1.37396
$703$	−44.5040	−1.67850
$704$	1.56051	0.0588141
$705$	0.0234720	0.000884006 0
$706$	−25.1695	−0.947268
$707$	−1.13454	−0.0426686
$708$	7.85986	0.295391
$709$	23.0568	0.865917	0.432959	−	0.901414i	$-0.357470\pi$
$709$	23.0568	0.865917	0.432959	+	0.901414i	$0.357470\pi$
$710$	4.16431	0.156284
$711$	−5.99333	−0.224767
$712$	−6.71047	−0.251486
$713$	3.35056	0.125480
$714$	2.27791	0.0852486
$715$	17.1213	0.640302
$716$	24.6178	0.920010
$717$	−21.5947	−0.806471
$718$	−27.2134	−1.01560
$719$	15.6802	0.584773	0.292387	−	0.956300i	$-0.405551\pi$
$719$	15.6802	0.584773	0.292387	+	0.956300i	$0.405551\pi$
$720$	−0.848134	−0.0316081
$721$	1.03198	0.0384329
$722$	−8.58339	−0.319441
$723$	−21.8147	−0.811297
$724$	−1.68090	−0.0624700
$725$	−9.47064	−0.351731
$726$	−13.5200	−0.501775
$727$	−38.6550	−1.43363	−0.716817	−	0.697261i	$-0.754402\pi$
$727$	−38.6550	−1.43363	−0.716817	+	0.697261i	$0.754402\pi$
$728$	−1.73536	−0.0643169
$729$	29.8929	1.10714
$730$	−27.5211	−1.01860
$731$	−16.0691	−0.594337
$732$	−8.45579	−0.312535
$733$	−4.96676	−0.183451	−0.0917257	−	0.995784i	$-0.529238\pi$
$733$	−4.96676	−0.183451	−0.0917257	+	0.995784i	$0.529238\pi$
$734$	−2.26804	−0.0837150
$735$	18.2591	0.673497
$736$	−0.430395	−0.0158646
$737$	−19.8533	−0.731306
$738$	−2.58016	−0.0949769
$739$	48.2243	1.77396	0.886980	−	0.461808i	$-0.152799\pi$
$739$	48.2243	1.77396	0.886980	+	0.461808i	$0.152799\pi$
$740$	14.1430	0.519907
$741$	54.4989	2.00207
$742$	−0.663106	−0.0243434
$743$	−16.5908	−0.608659	−0.304330	−	0.952567i	$-0.598432\pi$
$743$	−16.5908	−0.608659	−0.304330	+	0.952567i	$0.598432\pi$
$744$	12.2888	0.450530
$745$	24.8150	0.909153
$746$	3.11492	0.114045
$747$	−0.286778	−0.0104927
$748$	8.53015	0.311893
$749$	0.415559	0.0151842
$750$	−19.0073	−0.694050
$751$	−24.9790	−0.911496	−0.455748	−	0.890109i	$-0.650628\pi$
$751$	−24.9790	−0.911496	−0.455748	+	0.890109i	$0.650628\pi$
$752$	−0.00890887	−0.000324873 0
$753$	−16.0474	−0.584801
$754$	−28.1154	−1.02390
$755$	−13.3476	−0.485770
$756$	1.46193	0.0531699
$757$	−35.2424	−1.28091	−0.640454	−	0.767997i	$-0.721254\pi$
$757$	−35.2424	−1.28091	−0.640454	+	0.767997i	$0.721254\pi$
$758$	−26.5307	−0.963639
$759$	−1.06022	−0.0384835
$760$	8.76578	0.317968
$761$	−7.91159	−0.286795	−0.143397	−	0.989665i	$-0.545803\pi$
$761$	−7.91159	−0.286795	−0.143397	+	0.989665i	$0.545803\pi$
$762$	−22.6167	−0.819318
$763$	−3.66686	−0.132749
$764$	−11.9401	−0.431977
$765$	−4.63611	−0.167619
$766$	2.01425	0.0727779
$767$	−32.7309	−1.18184
$768$	−1.57856	−0.0569613
$769$	35.7737	1.29003	0.645016	−	0.764169i	$-0.276851\pi$
$769$	35.7737	1.29003	0.645016	+	0.764169i	$0.276851\pi$
$770$	−0.687577	−0.0247786
$771$	38.0719	1.37113
$772$	1.05892	0.0381114
$773$	18.9038	0.679922	0.339961	−	0.940439i	$-0.389586\pi$
$773$	18.9038	0.679922	0.339961	+	0.940439i	$0.389586\pi$
$774$	−1.49383	−0.0536945
$775$	−17.2381	−0.619209
$776$	9.01960	0.323785
$777$	−3.53120	−0.126681
$778$	−31.0352	−1.11266
$779$	26.6669	0.955440
$780$	−17.3193	−0.620131
$781$	−3.89354	−0.139322
$782$	−2.35265	−0.0841304
$783$	23.6854	0.846447
$784$	−6.93031	−0.247511
$785$	−6.31694	−0.225461
$786$	−7.65625	−0.273089
$787$	15.3774	0.548144	0.274072	−	0.961709i	$-0.411629\pi$
$787$	15.3774	0.548144	0.274072	+	0.961709i	$0.411629\pi$
$788$	−9.94657	−0.354332
$789$	−38.7573	−1.37980
$790$	−19.6851	−0.700363
$791$	2.12434	0.0755328
$792$	0.792986	0.0281775
$793$	35.2125	1.25043
$794$	20.1299	0.714384
$795$	−6.61794	−0.234714
$796$	−1.71787	−0.0608881
$797$	−31.4455	−1.11386	−0.556928	−	0.830561i	$-0.688020\pi$
$797$	−31.4455	−1.11386	−0.556928	+	0.830561i	$0.688020\pi$
$798$	−2.18862	−0.0774765
$799$	−0.0486980	−0.00172281
$800$	2.21431	0.0782876
$801$	−3.40997	−0.120486
$802$	−2.88270	−0.101792
$803$	25.7316	0.908049
$804$	20.0828	0.708267
$805$	0.189636	0.00668380
$806$	−51.1745	−1.80255
$807$	5.17701	0.182239
$808$	4.29765	0.151191
$809$	−25.0800	−0.881767	−0.440884	−	0.897564i	$-0.645335\pi$
$809$	−25.0800	−0.881767	−0.440884	+	0.897564i	$0.645335\pi$
$810$	12.0460	0.423252
$811$	−37.0823	−1.30214	−0.651069	−	0.759019i	$-0.725679\pi$
$811$	−37.0823	−1.30214	−0.651069	+	0.759019i	$0.725679\pi$
$812$	1.12909	0.0396233
$813$	18.0392	0.632663
$814$	−13.2234	−0.463479
$815$	15.9666	0.559286
$816$	−8.62877	−0.302068
$817$	15.4393	0.540151
$818$	−28.3360	−0.990744
$819$	−0.881838	−0.0308139
$820$	−8.47451	−0.295943
$821$	−12.6180	−0.440371	−0.220186	−	0.975458i	$-0.570666\pi$
$821$	−12.6180	−0.440371	−0.220186	+	0.975458i	$0.570666\pi$
$822$	18.8387	0.657077
$823$	−17.5675	−0.612363	−0.306182	−	0.951973i	$-0.599051\pi$
$823$	−17.5675	−0.612363	−0.306182	+	0.951973i	$0.599051\pi$
$824$	−3.90916	−0.136182
$825$	5.45464	0.189906
$826$	1.31444	0.0457353
$827$	−34.2619	−1.19140	−0.595702	−	0.803206i	$-0.703126\pi$
$827$	−34.2619	−1.19140	−0.595702	+	0.803206i	$0.703126\pi$
$828$	−0.218709	−0.00760065
$829$	−13.4237	−0.466223	−0.233112	−	0.972450i	$-0.574891\pi$
$829$	−13.4237	−0.466223	−0.233112	+	0.972450i	$0.574891\pi$
$830$	−0.941922	−0.0326946
$831$	−5.94011	−0.206060
$832$	6.57360	0.227899
$833$	−37.8827	−1.31256
$834$	12.6125	0.436734
$835$	−34.0287	−1.17761
$836$	−8.19580	−0.283458
$837$	43.1112	1.49014
$838$	13.4666	0.465197
$839$	−30.5010	−1.05301	−0.526506	−	0.850172i	$-0.676498\pi$
$839$	−30.5010	−1.05301	−0.526506	+	0.850172i	$0.676498\pi$
$840$	0.695527	0.0239980
$841$	−10.7071	−0.369210
$842$	3.63317	0.125207
$843$	−23.4884	−0.808983
$844$	−10.3341	−0.355715
$845$	50.4255	1.73469
$846$	−0.00452711	−0.000155645 0
$847$	−2.26102	−0.0776895
$848$	2.51186	0.0862577
$849$	5.40450	0.185482
$850$	12.1039	0.415162
$851$	3.64706	0.125019
$852$	3.93856	0.134933
$853$	−33.3529	−1.14198	−0.570990	−	0.820957i	$-0.693441\pi$
$853$	−33.3529	−1.14198	−0.570990	+	0.820957i	$0.693441\pi$
$854$	−1.41410	−0.0483896
$855$	4.45439	0.152337
$856$	−1.57415	−0.0538033
$857$	47.0687	1.60784	0.803919	−	0.594739i	$-0.202745\pi$
$857$	47.0687	1.60784	0.803919	+	0.594739i	$0.202745\pi$
$858$	16.1932	0.552825
$859$	−15.1915	−0.518329	−0.259164	−	0.965833i	$-0.583447\pi$
$859$	−15.1915	−0.518329	−0.259164	+	0.965833i	$0.583447\pi$
$860$	−4.90647	−0.167309
$861$	2.11590	0.0721098
$862$	−34.8711	−1.18772
$863$	−25.5330	−0.869153	−0.434577	−	0.900635i	$-0.643102\pi$
$863$	−25.5330	−0.869153	−0.434577	+	0.900635i	$0.643102\pi$
$864$	−5.53783	−0.188401
$865$	−18.4216	−0.626352
$866$	11.1405	0.378570
$867$	−20.3315	−0.690494
$868$	2.05512	0.0697554
$869$	18.4051	0.624350
$870$	11.2686	0.382040
$871$	−83.6312	−2.83374
$872$	13.8902	0.470380
$873$	4.58337	0.155124
$874$	2.26043	0.0764603
$875$	−3.17869	−0.107459
$876$	−26.0291	−0.879443
$877$	56.4378	1.90577	0.952885	−	0.303331i	$-0.0980987\pi$
$877$	56.4378	1.90577	0.952885	+	0.303331i	$0.0980987\pi$
$878$	−27.8695	−0.940550
$879$	−22.5857	−0.761796
$880$	2.60456	0.0877997
$881$	−57.7369	−1.94521	−0.972603	−	0.232471i	$-0.925319\pi$
$881$	−57.7369	−1.94521	−0.972603	+	0.232471i	$0.925319\pi$
$882$	−3.52169	−0.118581
$883$	5.69762	0.191740	0.0958701	−	0.995394i	$-0.469437\pi$
$883$	5.69762	0.191740	0.0958701	+	0.995394i	$0.469437\pi$
$884$	35.9329	1.20855
$885$	13.1184	0.440970
$886$	24.0905	0.809335
$887$	23.4519	0.787439	0.393720	−	0.919231i	$-0.371188\pi$
$887$	23.4519	0.787439	0.393720	+	0.919231i	$0.371188\pi$
$888$	13.3763	0.448878
$889$	−3.78231	−0.126854
$890$	−11.2000	−0.375426
$891$	−11.2627	−0.377315
$892$	0.285537	0.00956048
$893$	0.0467893	0.00156574
$894$	23.4698	0.784946
$895$	41.0880	1.37342
$896$	−0.263990	−0.00881928
$897$	−4.46613	−0.149120
$898$	25.1702	0.839940
$899$	33.2960	1.11048
$900$	1.12522	0.0375072
$901$	13.7304	0.457427
$902$	7.92348	0.263823
$903$	1.22504	0.0407668
$904$	−8.04705	−0.267641
$905$	−2.80548	−0.0932574
$906$	−12.6240	−0.419405
$907$	6.19585	0.205730	0.102865	−	0.994695i	$-0.467199\pi$
$907$	6.19585	0.205730	0.102865	+	0.994695i	$0.467199\pi$
$908$	−12.8590	−0.426741
$909$	2.18388	0.0724348
$910$	−2.89639	−0.0960144
$911$	−12.6468	−0.419008	−0.209504	−	0.977808i	$-0.567185\pi$
$911$	−12.6468	−0.419008	−0.209504	+	0.977808i	$0.567185\pi$
$912$	8.29057	0.274528
$913$	0.880676	0.0291461
$914$	−14.1219	−0.467111
$915$	−14.1130	−0.466563
$916$	14.1197	0.466528
$917$	−1.28039	−0.0422823
$918$	−30.2711	−0.999095
$919$	5.19805	0.171468	0.0857340	−	0.996318i	$-0.472676\pi$
$919$	5.19805	0.171468	0.0857340	+	0.996318i	$0.472676\pi$
$920$	−0.718347	−0.0236832
$921$	−19.9011	−0.655764
$922$	0.832657	0.0274221
$923$	−16.4014	−0.539858
$924$	−0.650302	−0.0213934
$925$	−18.7634	−0.616938
$926$	−18.5368	−0.609158
$927$	−1.98647	−0.0652442
$928$	−4.27702	−0.140400
$929$	−26.2681	−0.861830	−0.430915	−	0.902393i	$-0.641809\pi$
$929$	−26.2681	−0.861830	−0.430915	+	0.902393i	$0.641809\pi$
$930$	20.5105	0.672567
$931$	36.3979	1.19289
$932$	30.1449	0.987429
$933$	−46.9207	−1.53612
$934$	−10.6243	−0.347636
$935$	14.2372	0.465605
$936$	3.34042	0.109185
$937$	−44.8911	−1.46653	−0.733264	−	0.679944i	$-0.762004\pi$
$937$	−44.8911	−1.46653	−0.733264	+	0.679944i	$0.762004\pi$
$938$	3.35855	0.109661
$939$	25.2777	0.824907
$940$	−0.0148693	−0.000484982 0
$941$	42.7985	1.39519	0.697596	−	0.716491i	$-0.254253\pi$
$941$	42.7985	1.39519	0.697596	+	0.716491i	$0.254253\pi$
$942$	−5.97449	−0.194659
$943$	−2.18532	−0.0711639
$944$	−4.97914	−0.162057
$945$	2.44002	0.0793738
$946$	4.58744	0.149150
$947$	2.03282	0.0660578	0.0330289	−	0.999454i	$-0.489485\pi$
$947$	2.03282	0.0660578	0.0330289	+	0.999454i	$0.489485\pi$
$948$	−18.6179	−0.604681
$949$	108.393	3.51860
$950$	−11.6295	−0.377311
$951$	−31.8027	−1.03127
$952$	−1.44303	−0.0467689
$953$	−20.3249	−0.658389	−0.329194	−	0.944262i	$-0.606777\pi$
$953$	−20.3249	−0.658389	−0.329194	+	0.944262i	$0.606777\pi$
$954$	1.27642	0.0413256
$955$	−19.9285	−0.644871
$956$	13.6801	0.442444
$957$	−10.5358	−0.340576
$958$	37.4630	1.21037
$959$	3.15050	0.101735
$960$	−2.63467	−0.0850337
$961$	29.6039	0.954965
$962$	−55.7030	−1.79594
$963$	−0.799915	−0.0257769
$964$	13.8194	0.445092
$965$	1.76738	0.0568941
$966$	0.179356	0.00577068
$967$	−34.1130	−1.09700	−0.548500	−	0.836151i	$-0.684801\pi$
$967$	−34.1130	−1.09700	−0.548500	+	0.836151i	$0.684801\pi$
$968$	8.56480	0.275283
$969$	45.3182	1.45583
$970$	15.0541	0.483357
$971$	−25.2153	−0.809198	−0.404599	−	0.914494i	$-0.632589\pi$
$971$	−25.2153	−0.809198	−0.404599	+	0.914494i	$0.632589\pi$
$972$	−5.22055	−0.167449
$973$	2.10924	0.0676193
$974$	9.37558	0.300413
$975$	22.9774	0.735867
$976$	5.35666	0.171462
$977$	26.6367	0.852184	0.426092	−	0.904680i	$-0.359890\pi$
$977$	26.6367	0.852184	0.426092	+	0.904680i	$0.359890\pi$
$978$	15.1010	0.482877
$979$	10.4718	0.334680
$980$	−11.5670	−0.369493
$981$	7.05838	0.225357
$982$	−3.87368	−0.123614
$983$	−24.7687	−0.790000	−0.395000	−	0.918681i	$-0.629255\pi$
$983$	−24.7687	−0.790000	−0.395000	+	0.918681i	$0.629255\pi$
$984$	−8.01509	−0.255512
$985$	−16.6012	−0.528959
$986$	−23.3792	−0.744546
$987$	0.00371253	0.000118171 0
$988$	−34.5245	−1.09837
$989$	−1.26523	−0.0402320
$990$	1.32353	0.0420644
$991$	−36.0327	−1.14462	−0.572308	−	0.820039i	$-0.693952\pi$
$991$	−36.0327	−1.14462	−0.572308	+	0.820039i	$0.693952\pi$
$992$	−7.78485	−0.247169
$993$	8.14574	0.258497
$994$	0.658664	0.0208915
$995$	−2.86719	−0.0908959
$996$	−0.890859	−0.0282279
$997$	−44.7526	−1.41733	−0.708664	−	0.705546i	$-0.750702\pi$
$997$	−44.7526	−1.41733	−0.708664	+	0.705546i	$0.750702\pi$
$998$	23.0631	0.730050
$999$	46.9261	1.48467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.h.1.14	✓	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.h.1.14	✓	42	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.57856 q^{3} +1.00000 q^{4} +1.66904 q^{5} +1.57856 q^{6} +0.263990 q^{7} -1.00000 q^{8} -0.508157 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.57856 q^{3} +1.00000 q^{4} +1.66904 q^{5} +1.57856 q^{6} +0.263990 q^{7} -1.00000 q^{8} -0.508157 q^{9} -1.66904 q^{10} +1.56051 q^{11} -1.57856 q^{12} +6.57360 q^{13} -0.263990 q^{14} -2.63467 q^{15} +1.00000 q^{16} +5.46624 q^{17} +0.508157 q^{18} -5.25199 q^{19} +1.66904 q^{20} -0.416723 q^{21} -1.56051 q^{22} +0.430395 q^{23} +1.57856 q^{24} -2.21431 q^{25} -6.57360 q^{26} +5.53783 q^{27} +0.263990 q^{28} +4.27702 q^{29} +2.63467 q^{30} +7.78485 q^{31} -1.00000 q^{32} -2.46336 q^{33} -5.46624 q^{34} +0.440609 q^{35} -0.508157 q^{36} +8.47373 q^{37} +5.25199 q^{38} -10.3768 q^{39} -1.66904 q^{40} -5.07748 q^{41} +0.416723 q^{42} -2.93970 q^{43} +1.56051 q^{44} -0.848134 q^{45} -0.430395 q^{46} -0.00890887 q^{47} -1.57856 q^{48} -6.93031 q^{49} +2.21431 q^{50} -8.62877 q^{51} +6.57360 q^{52} +2.51186 q^{53} -5.53783 q^{54} +2.60456 q^{55} -0.263990 q^{56} +8.29057 q^{57} -4.27702 q^{58} -4.97914 q^{59} -2.63467 q^{60} +5.35666 q^{61} -7.78485 q^{62} -0.134148 q^{63} +1.00000 q^{64} +10.9716 q^{65} +2.46336 q^{66} -12.7223 q^{67} +5.46624 q^{68} -0.679404 q^{69} -0.440609 q^{70} -2.49503 q^{71} +0.508157 q^{72} +16.4892 q^{73} -8.47373 q^{74} +3.49541 q^{75} -5.25199 q^{76} +0.411960 q^{77} +10.3768 q^{78} +11.7942 q^{79} +1.66904 q^{80} -7.21730 q^{81} +5.07748 q^{82} +0.564350 q^{83} -0.416723 q^{84} +9.12337 q^{85} +2.93970 q^{86} -6.75152 q^{87} -1.56051 q^{88} +6.71047 q^{89} +0.848134 q^{90} +1.73536 q^{91} +0.430395 q^{92} -12.2888 q^{93} +0.00890887 q^{94} -8.76578 q^{95} +1.57856 q^{96} -9.01960 q^{97} +6.93031 q^{98} -0.792986 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * q^99

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