LMFDB - Embedded newform 8007.2.a.f.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.803950 q^{2} -1.00000 q^{3} -1.35366 q^{4} -0.904757 q^{5} -0.803950 q^{6} +1.61084 q^{7} -2.69618 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.803950 q^{2} -1.00000 q^{3} -1.35366 q^{4} -0.904757 q^{5} -0.803950 q^{6} +1.61084 q^{7} -2.69618 q^{8} +1.00000 q^{9} -0.727380 q^{10} -1.65375 q^{11} +1.35366 q^{12} -0.0916358 q^{13} +1.29503 q^{14} +0.904757 q^{15} +0.539733 q^{16} -1.00000 q^{17} +0.803950 q^{18} -5.68127 q^{19} +1.22474 q^{20} -1.61084 q^{21} -1.32954 q^{22} +5.14537 q^{23} +2.69618 q^{24} -4.18141 q^{25} -0.0736706 q^{26} -1.00000 q^{27} -2.18053 q^{28} +7.70133 q^{29} +0.727380 q^{30} +4.00116 q^{31} +5.82628 q^{32} +1.65375 q^{33} -0.803950 q^{34} -1.45742 q^{35} -1.35366 q^{36} +9.54524 q^{37} -4.56746 q^{38} +0.0916358 q^{39} +2.43939 q^{40} +0.791309 q^{41} -1.29503 q^{42} -1.21613 q^{43} +2.23863 q^{44} -0.904757 q^{45} +4.13662 q^{46} -0.249380 q^{47} -0.539733 q^{48} -4.40521 q^{49} -3.36165 q^{50} +1.00000 q^{51} +0.124044 q^{52} -2.79921 q^{53} -0.803950 q^{54} +1.49625 q^{55} -4.34310 q^{56} +5.68127 q^{57} +6.19149 q^{58} -3.55467 q^{59} -1.22474 q^{60} -1.36280 q^{61} +3.21674 q^{62} +1.61084 q^{63} +3.60457 q^{64} +0.0829082 q^{65} +1.32954 q^{66} +8.35596 q^{67} +1.35366 q^{68} -5.14537 q^{69} -1.17169 q^{70} -10.6303 q^{71} -2.69618 q^{72} +6.03964 q^{73} +7.67390 q^{74} +4.18141 q^{75} +7.69053 q^{76} -2.66393 q^{77} +0.0736706 q^{78} +4.64140 q^{79} -0.488328 q^{80} +1.00000 q^{81} +0.636173 q^{82} -13.3886 q^{83} +2.18053 q^{84} +0.904757 q^{85} -0.977709 q^{86} -7.70133 q^{87} +4.45882 q^{88} -10.9910 q^{89} -0.727380 q^{90} -0.147610 q^{91} -6.96510 q^{92} -4.00116 q^{93} -0.200489 q^{94} +5.14017 q^{95} -5.82628 q^{96} -0.693595 q^{97} -3.54157 q^{98} -1.65375 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.803950	0.568479	0.284239	−	0.958753i	$-0.408259\pi$
$2$	0.803950	0.568479	0.284239	+	0.958753i	$0.408259\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.35366	−0.676832
$5$	−0.904757	−0.404620	−0.202310	−	0.979322i	$-0.564845\pi$
$5$	−0.904757	−0.404620	−0.202310	+	0.979322i	$0.564845\pi$
$6$	−0.803950	−0.328211
$7$	1.61084	0.608839	0.304419	−	0.952538i	$-0.401538\pi$
$7$	1.61084	0.608839	0.304419	+	0.952538i	$0.401538\pi$
$8$	−2.69618	−0.953243
$9$	1.00000	0.333333
$10$	−0.727380	−0.230018
$11$	−1.65375	−0.498625	−0.249313	−	0.968423i	$-0.580205\pi$
$11$	−1.65375	−0.498625	−0.249313	+	0.968423i	$0.580205\pi$
$12$	1.35366	0.390769
$13$	−0.0916358	−0.0254152	−0.0127076	−	0.999919i	$-0.504045\pi$
$13$	−0.0916358	−0.0254152	−0.0127076	+	0.999919i	$0.504045\pi$
$14$	1.29503	0.346112
$15$	0.904757	0.233607
$16$	0.539733	0.134933
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.803950	0.189493
$19$	−5.68127	−1.30337	−0.651687	−	0.758488i	$-0.725938\pi$
$19$	−5.68127	−1.30337	−0.651687	+	0.758488i	$0.725938\pi$
$20$	1.22474	0.273860
$21$	−1.61084	−0.351513
$22$	−1.32954	−0.283458
$23$	5.14537	1.07288	0.536442	−	0.843937i	$-0.319768\pi$
$23$	5.14537	1.07288	0.536442	+	0.843937i	$0.319768\pi$
$24$	2.69618	0.550355
$25$	−4.18141	−0.836283
$26$	−0.0736706	−0.0144480
$27$	−1.00000	−0.192450
$28$	−2.18053	−0.412082
$29$	7.70133	1.43010	0.715051	−	0.699072i	$-0.246404\pi$
$29$	7.70133	1.43010	0.715051	+	0.699072i	$0.246404\pi$
$30$	0.727380	0.132801
$31$	4.00116	0.718630	0.359315	−	0.933216i	$-0.383010\pi$
$31$	4.00116	0.718630	0.359315	+	0.933216i	$0.383010\pi$
$32$	5.82628	1.02995
$33$	1.65375	0.287882
$34$	−0.803950	−0.137876
$35$	−1.45742	−0.246348
$36$	−1.35366	−0.225611
$37$	9.54524	1.56923	0.784613	−	0.619985i	$-0.212861\pi$
$37$	9.54524	1.56923	0.784613	+	0.619985i	$0.212861\pi$
$38$	−4.56746	−0.740940
$39$	0.0916358	0.0146735
$40$	2.43939	0.385701
$41$	0.791309	0.123582	0.0617908	−	0.998089i	$-0.480319\pi$
$41$	0.791309	0.123582	0.0617908	+	0.998089i	$0.480319\pi$
$42$	−1.29503	−0.199828
$43$	−1.21613	−0.185458	−0.0927291	−	0.995691i	$-0.529559\pi$
$43$	−1.21613	−0.185458	−0.0927291	+	0.995691i	$0.529559\pi$
$44$	2.23863	0.337486
$45$	−0.904757	−0.134873
$46$	4.13662	0.609912
$47$	−0.249380	−0.0363758	−0.0181879	−	0.999835i	$-0.505790\pi$
$47$	−0.249380	−0.0363758	−0.0181879	+	0.999835i	$0.505790\pi$
$48$	−0.539733	−0.0779038
$49$	−4.40521	−0.629315
$50$	−3.36165	−0.475409
$51$	1.00000	0.140028
$52$	0.124044	0.0172018
$53$	−2.79921	−0.384502	−0.192251	−	0.981346i	$-0.561579\pi$
$53$	−2.79921	−0.384502	−0.192251	+	0.981346i	$0.561579\pi$
$54$	−0.803950	−0.109404
$55$	1.49625	0.201754
$56$	−4.34310	−0.580372
$57$	5.68127	0.752503
$58$	6.19149	0.812982
$59$	−3.55467	−0.462779	−0.231389	−	0.972861i	$-0.574327\pi$
$59$	−3.55467	−0.462779	−0.231389	+	0.972861i	$0.574327\pi$
$60$	−1.22474	−0.158113
$61$	−1.36280	−0.174488	−0.0872442	−	0.996187i	$-0.527806\pi$
$61$	−1.36280	−0.174488	−0.0872442	+	0.996187i	$0.527806\pi$
$62$	3.21674	0.408526
$63$	1.61084	0.202946
$64$	3.60457	0.450571
$65$	0.0829082	0.0102835
$66$	1.32954	0.163655
$67$	8.35596	1.02084	0.510421	−	0.859924i	$-0.329489\pi$
$67$	8.35596	1.02084	0.510421	+	0.859924i	$0.329489\pi$
$68$	1.35366	0.164156
$69$	−5.14537	−0.619430
$70$	−1.17169	−0.140044
$71$	−10.6303	−1.26158	−0.630791	−	0.775953i	$-0.717269\pi$
$71$	−10.6303	−1.26158	−0.630791	+	0.775953i	$0.717269\pi$
$72$	−2.69618	−0.317748
$73$	6.03964	0.706886	0.353443	−	0.935456i	$-0.385011\pi$
$73$	6.03964	0.706886	0.353443	+	0.935456i	$0.385011\pi$
$74$	7.67390	0.892072
$75$	4.18141	0.482828
$76$	7.69053	0.882164
$77$	−2.66393	−0.303583
$78$	0.0736706	0.00834156
$79$	4.64140	0.522198	0.261099	−	0.965312i	$-0.415915\pi$
$79$	4.64140	0.522198	0.261099	+	0.965312i	$0.415915\pi$
$80$	−0.488328	−0.0545967
$81$	1.00000	0.111111
$82$	0.636173	0.0702536
$83$	−13.3886	−1.46959	−0.734795	−	0.678289i	$-0.762722\pi$
$83$	−13.3886	−1.46959	−0.734795	+	0.678289i	$0.762722\pi$
$84$	2.18053	0.237915
$85$	0.904757	0.0981347
$86$	−0.977709	−0.105429
$87$	−7.70133	−0.825670
$88$	4.45882	0.475311
$89$	−10.9910	−1.16504	−0.582521	−	0.812816i	$-0.697934\pi$
$89$	−10.9910	−1.16504	−0.582521	+	0.812816i	$0.697934\pi$
$90$	−0.727380	−0.0766726
$91$	−0.147610	−0.0154738
$92$	−6.96510	−0.726162
$93$	−4.00116	−0.414901
$94$	−0.200489	−0.0206789
$95$	5.14017	0.527371
$96$	−5.82628	−0.594642
$97$	−0.693595	−0.0704239	−0.0352120	−	0.999380i	$-0.511211\pi$
$97$	−0.693595	−0.0704239	−0.0352120	+	0.999380i	$0.511211\pi$
$98$	−3.54157	−0.357752
$99$	−1.65375	−0.166208
$100$	5.66023	0.566023
$101$	8.68888	0.864576	0.432288	−	0.901736i	$-0.357706\pi$
$101$	8.68888	0.864576	0.432288	+	0.901736i	$0.357706\pi$
$102$	0.803950	0.0796029
$103$	5.02979	0.495600	0.247800	−	0.968811i	$-0.420292\pi$
$103$	5.02979	0.495600	0.247800	+	0.968811i	$0.420292\pi$
$104$	0.247067	0.0242269
$105$	1.45742	0.142229
$106$	−2.25043	−0.218581
$107$	8.73839	0.844772	0.422386	−	0.906416i	$-0.361193\pi$
$107$	8.73839	0.844772	0.422386	+	0.906416i	$0.361193\pi$
$108$	1.35366	0.130256
$109$	−15.6798	−1.50185	−0.750926	−	0.660386i	$-0.770392\pi$
$109$	−15.6798	−1.50185	−0.750926	+	0.660386i	$0.770392\pi$
$110$	1.20291	0.114693
$111$	−9.54524	−0.905994
$112$	0.869422	0.0821526
$113$	8.44997	0.794906	0.397453	−	0.917622i	$-0.369894\pi$
$113$	8.44997	0.794906	0.397453	+	0.917622i	$0.369894\pi$
$114$	4.56746	0.427782
$115$	−4.65531	−0.434110
$116$	−10.4250	−0.967938
$117$	−0.0916358	−0.00847173
$118$	−2.85778	−0.263080
$119$	−1.61084	−0.147665
$120$	−2.43939	−0.222685
$121$	−8.26510	−0.751373
$122$	−1.09562	−0.0991930
$123$	−0.791309	−0.0713499
$124$	−5.41623	−0.486392
$125$	8.30695	0.742996
$126$	1.29503	0.115371
$127$	11.0619	0.981583	0.490792	−	0.871277i	$-0.336708\pi$
$127$	11.0619	0.981583	0.490792	+	0.871277i	$0.336708\pi$
$128$	−8.75466	−0.773810
$129$	1.21613	0.107074
$130$	0.0666540	0.00584595
$131$	16.0262	1.40021	0.700106	−	0.714039i	$-0.253136\pi$
$131$	16.0262	1.40021	0.700106	+	0.714039i	$0.253136\pi$
$132$	−2.23863	−0.194847
$133$	−9.15160	−0.793544
$134$	6.71777	0.580327
$135$	0.904757	0.0778691
$136$	2.69618	0.231195
$137$	−5.73989	−0.490392	−0.245196	−	0.969474i	$-0.578852\pi$
$137$	−5.73989	−0.490392	−0.245196	+	0.969474i	$0.578852\pi$
$138$	−4.13662	−0.352133
$139$	5.11048	0.433465	0.216732	−	0.976231i	$-0.430460\pi$
$139$	5.11048	0.433465	0.216732	+	0.976231i	$0.430460\pi$
$140$	1.97285	0.166736
$141$	0.249380	0.0210016
$142$	−8.54621	−0.717182
$143$	0.151543	0.0126727
$144$	0.539733	0.0449778
$145$	−6.96784	−0.578647
$146$	4.85557	0.401850
$147$	4.40521	0.363335
$148$	−12.9210	−1.06210
$149$	−14.4213	−1.18144	−0.590720	−	0.806876i	$-0.701156\pi$
$149$	−14.4213	−1.18144	−0.590720	+	0.806876i	$0.701156\pi$
$150$	3.36165	0.274478
$151$	−9.67344	−0.787213	−0.393607	−	0.919279i	$-0.628773\pi$
$151$	−9.67344	−0.787213	−0.393607	+	0.919279i	$0.628773\pi$
$152$	15.3177	1.24243
$153$	−1.00000	−0.0808452
$154$	−2.14166	−0.172580
$155$	−3.62008	−0.290772
$156$	−0.124044	−0.00993147
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	3.73145	0.296858
$159$	2.79921	0.221992
$160$	−5.27137	−0.416738
$161$	8.28835	0.653213
$162$	0.803950	0.0631643
$163$	−11.9782	−0.938201	−0.469101	−	0.883145i	$-0.655422\pi$
$163$	−11.9782	−0.938201	−0.469101	+	0.883145i	$0.655422\pi$
$164$	−1.07117	−0.0836440
$165$	−1.49625	−0.116483
$166$	−10.7638	−0.835430
$167$	18.4715	1.42937	0.714684	−	0.699447i	$-0.246571\pi$
$167$	18.4715	1.42937	0.714684	+	0.699447i	$0.246571\pi$
$168$	4.34310	0.335078
$169$	−12.9916	−0.999354
$170$	0.727380	0.0557875
$171$	−5.68127	−0.434458
$172$	1.64623	0.125524
$173$	23.5239	1.78849	0.894243	−	0.447581i	$-0.147714\pi$
$173$	23.5239	1.78849	0.894243	+	0.447581i	$0.147714\pi$
$174$	−6.19149	−0.469376
$175$	−6.73557	−0.509161
$176$	−0.892586	−0.0672812
$177$	3.55467	0.267185
$178$	−8.83620	−0.662302
$179$	−26.1494	−1.95450	−0.977248	−	0.212101i	$-0.931969\pi$
$179$	−26.1494	−1.95450	−0.977248	+	0.212101i	$0.931969\pi$
$180$	1.22474	0.0912865
$181$	−15.3791	−1.14312	−0.571558	−	0.820562i	$-0.693661\pi$
$181$	−15.3791	−1.14312	−0.571558	+	0.820562i	$0.693661\pi$
$182$	−0.118671	−0.00879650
$183$	1.36280	0.100741
$184$	−13.8728	−1.02272
$185$	−8.63612	−0.634940
$186$	−3.21674	−0.235863
$187$	1.65375	0.120934
$188$	0.337577	0.0246203
$189$	−1.61084	−0.117171
$190$	4.13244	0.299799
$191$	6.93828	0.502036	0.251018	−	0.967982i	$-0.419235\pi$
$191$	6.93828	0.502036	0.251018	+	0.967982i	$0.419235\pi$
$192$	−3.60457	−0.260138
$193$	0.330525	0.0237917	0.0118959	−	0.999929i	$-0.496213\pi$
$193$	0.330525	0.0237917	0.0118959	+	0.999929i	$0.496213\pi$
$194$	−0.557616	−0.0400345
$195$	−0.0829082	−0.00593718
$196$	5.96317	0.425941
$197$	−15.7318	−1.12084	−0.560421	−	0.828208i	$-0.689361\pi$
$197$	−15.7318	−1.12084	−0.560421	+	0.828208i	$0.689361\pi$
$198$	−1.32954	−0.0944860
$199$	−23.6125	−1.67385	−0.836923	−	0.547320i	$-0.815648\pi$
$199$	−23.6125	−1.67385	−0.836923	+	0.547320i	$0.815648\pi$
$200$	11.2738	0.797181
$201$	−8.35596	−0.589384
$202$	6.98543	0.491493
$203$	12.4056	0.870701
$204$	−1.35366	−0.0947754
$205$	−0.715942	−0.0500036
$206$	4.04370	0.281738
$207$	5.14537	0.357628
$208$	−0.0494589	−0.00342936
$209$	9.39542	0.649895
$210$	1.17169	0.0808543
$211$	27.8021	1.91397	0.956986	−	0.290134i	$-0.0936998\pi$
$211$	27.8021	1.91397	0.956986	+	0.290134i	$0.0936998\pi$
$212$	3.78919	0.260243
$213$	10.6303	0.728374
$214$	7.02523	0.480235
$215$	1.10030	0.0750401
$216$	2.69618	0.183452
$217$	6.44522	0.437530
$218$	−12.6058	−0.853771
$219$	−6.03964	−0.408121
$220$	−2.02541	−0.136553
$221$	0.0916358	0.00616409
$222$	−7.67390	−0.515038
$223$	−8.02098	−0.537124	−0.268562	−	0.963262i	$-0.586548\pi$
$223$	−8.02098	−0.537124	−0.268562	+	0.963262i	$0.586548\pi$
$224$	9.38518	0.627074
$225$	−4.18141	−0.278761
$226$	6.79336	0.451887
$227$	6.55387	0.434996	0.217498	−	0.976061i	$-0.430210\pi$
$227$	6.55387	0.434996	0.217498	+	0.976061i	$0.430210\pi$
$228$	−7.69053	−0.509318
$229$	10.1111	0.668157	0.334079	−	0.942545i	$-0.391575\pi$
$229$	10.1111	0.668157	0.334079	+	0.942545i	$0.391575\pi$
$230$	−3.74264	−0.246782
$231$	2.66393	0.175273
$232$	−20.7642	−1.36323
$233$	5.50557	0.360682	0.180341	−	0.983604i	$-0.442280\pi$
$233$	5.50557	0.360682	0.180341	+	0.983604i	$0.442280\pi$
$234$	−0.0736706	−0.00481600
$235$	0.225629	0.0147184
$236$	4.81183	0.313223
$237$	−4.64140	−0.301491
$238$	−1.29503	−0.0839445
$239$	−23.7783	−1.53809	−0.769046	−	0.639193i	$-0.779269\pi$
$239$	−23.7783	−1.53809	−0.769046	+	0.639193i	$0.779269\pi$
$240$	0.488328	0.0315214
$241$	−2.65562	−0.171063	−0.0855317	−	0.996335i	$-0.527259\pi$
$241$	−2.65562	−0.171063	−0.0855317	+	0.996335i	$0.527259\pi$
$242$	−6.64473	−0.427139
$243$	−1.00000	−0.0641500
$244$	1.84477	0.118099
$245$	3.98564	0.254633
$246$	−0.636173	−0.0405609
$247$	0.520608	0.0331255
$248$	−10.7879	−0.685030
$249$	13.3886	0.848468
$250$	6.67838	0.422378
$251$	24.7174	1.56015	0.780074	−	0.625687i	$-0.215181\pi$
$251$	24.7174	1.56015	0.780074	+	0.625687i	$0.215181\pi$
$252$	−2.18053	−0.137361
$253$	−8.50917	−0.534967
$254$	8.89321	0.558009
$255$	−0.904757	−0.0566581
$256$	−14.2475	−0.890466
$257$	−13.6334	−0.850427	−0.425214	−	0.905093i	$-0.639801\pi$
$257$	−13.6334	−0.850427	−0.425214	+	0.905093i	$0.639801\pi$
$258$	0.977709	0.0608695
$259$	15.3758	0.955406
$260$	−0.112230	−0.00696019
$261$	7.70133	0.476701
$262$	12.8842	0.795990
$263$	−8.51665	−0.525159	−0.262580	−	0.964910i	$-0.584573\pi$
$263$	−8.51665	−0.525159	−0.262580	+	0.964910i	$0.584573\pi$
$264$	−4.45882	−0.274421
$265$	2.53261	0.155577
$266$	−7.35743	−0.451113
$267$	10.9910	0.672637
$268$	−11.3112	−0.690939
$269$	−12.3629	−0.753782	−0.376891	−	0.926258i	$-0.623007\pi$
$269$	−12.3629	−0.753782	−0.376891	+	0.926258i	$0.623007\pi$
$270$	0.727380	0.0442669
$271$	2.18769	0.132892	0.0664462	−	0.997790i	$-0.478834\pi$
$271$	2.18769	0.132892	0.0664462	+	0.997790i	$0.478834\pi$
$272$	−0.539733	−0.0327261
$273$	0.147610	0.00893378
$274$	−4.61459	−0.278777
$275$	6.91503	0.416992
$276$	6.96510	0.419250
$277$	−29.8676	−1.79457	−0.897285	−	0.441453i	$-0.854463\pi$
$277$	−29.8676	−1.79457	−0.897285	+	0.441453i	$0.854463\pi$
$278$	4.10857	0.246416
$279$	4.00116	0.239543
$280$	3.92945	0.234830
$281$	−15.2973	−0.912563	−0.456281	−	0.889836i	$-0.650819\pi$
$281$	−15.2973	−0.912563	−0.456281	+	0.889836i	$0.650819\pi$
$282$	0.200489	0.0119390
$283$	−8.20474	−0.487721	−0.243861	−	0.969810i	$-0.578414\pi$
$283$	−8.20474	−0.487721	−0.243861	+	0.969810i	$0.578414\pi$
$284$	14.3898	0.853878
$285$	−5.14017	−0.304478
$286$	0.121833	0.00720414
$287$	1.27467	0.0752413
$288$	5.82628	0.343317
$289$	1.00000	0.0588235
$290$	−5.60180	−0.328949
$291$	0.693595	0.0406593
$292$	−8.17564	−0.478443
$293$	20.2152	1.18098	0.590491	−	0.807044i	$-0.298934\pi$
$293$	20.2152	1.18098	0.590491	+	0.807044i	$0.298934\pi$
$294$	3.54157	0.206548
$295$	3.21612	0.187249
$296$	−25.7357	−1.49586
$297$	1.65375	0.0959605
$298$	−11.5940	−0.671624
$299$	−0.471500	−0.0272675
$300$	−5.66023	−0.326793
$301$	−1.95899	−0.112914
$302$	−7.77696	−0.447514
$303$	−8.68888	−0.499163
$304$	−3.06637	−0.175868
$305$	1.23300	0.0706015
$306$	−0.803950	−0.0459588
$307$	−18.2955	−1.04418	−0.522090	−	0.852890i	$-0.674847\pi$
$307$	−18.2955	−1.04418	−0.522090	+	0.852890i	$0.674847\pi$
$308$	3.60606	0.205474
$309$	−5.02979	−0.286135
$310$	−2.91037	−0.165298
$311$	−12.0568	−0.683678	−0.341839	−	0.939759i	$-0.611050\pi$
$311$	−12.0568	−0.683678	−0.341839	+	0.939759i	$0.611050\pi$
$312$	−0.247067	−0.0139874
$313$	−3.26567	−0.184586	−0.0922932	−	0.995732i	$-0.529420\pi$
$313$	−3.26567	−0.184586	−0.0922932	+	0.995732i	$0.529420\pi$
$314$	−0.803950	−0.0453695
$315$	−1.45742	−0.0821161
$316$	−6.28289	−0.353440
$317$	−7.25078	−0.407244	−0.203622	−	0.979050i	$-0.565271\pi$
$317$	−7.25078	−0.407244	−0.203622	+	0.979050i	$0.565271\pi$
$318$	2.25043	0.126198
$319$	−12.7361	−0.713085
$320$	−3.26126	−0.182310
$321$	−8.73839	−0.487729
$322$	6.66342	0.371338
$323$	5.68127	0.316114
$324$	−1.35366	−0.0752035
$325$	0.383167	0.0212543
$326$	−9.62984	−0.533347
$327$	15.6798	0.867095
$328$	−2.13351	−0.117803
$329$	−0.401711	−0.0221470
$330$	−1.20291	−0.0662179
$331$	−2.46217	−0.135333	−0.0676666	−	0.997708i	$-0.521555\pi$
$331$	−2.46217	−0.135333	−0.0676666	+	0.997708i	$0.521555\pi$
$332$	18.1237	0.994665
$333$	9.54524	0.523076
$334$	14.8502	0.812565
$335$	−7.56011	−0.413053
$336$	−0.869422	−0.0474308
$337$	−22.8291	−1.24358	−0.621791	−	0.783183i	$-0.713595\pi$
$337$	−22.8291	−1.24358	−0.621791	+	0.783183i	$0.713595\pi$
$338$	−10.4446	−0.568112
$339$	−8.44997	−0.458939
$340$	−1.22474	−0.0664207
$341$	−6.61694	−0.358327
$342$	−4.56746	−0.246980
$343$	−18.3719	−0.991990
$344$	3.27891	0.176787
$345$	4.65531	0.250633
$346$	18.9120	1.01672
$347$	−22.1051	−1.18667	−0.593333	−	0.804957i	$-0.702188\pi$
$347$	−22.1051	−1.18667	−0.593333	+	0.804957i	$0.702188\pi$
$348$	10.4250	0.558840
$349$	1.63160	0.0873374	0.0436687	−	0.999046i	$-0.486095\pi$
$349$	1.63160	0.0873374	0.0436687	+	0.999046i	$0.486095\pi$
$350$	−5.41507	−0.289447
$351$	0.0916358	0.00489116
$352$	−9.63523	−0.513559
$353$	−5.25543	−0.279718	−0.139859	−	0.990171i	$-0.544665\pi$
$353$	−5.25543	−0.279718	−0.139859	+	0.990171i	$0.544665\pi$
$354$	2.85778	0.151889
$355$	9.61782	0.510461
$356$	14.8781	0.788537
$357$	1.61084	0.0852545
$358$	−21.0228	−1.11109
$359$	−36.6056	−1.93197	−0.965985	−	0.258598i	$-0.916739\pi$
$359$	−36.6056	−1.93197	−0.965985	+	0.258598i	$0.916739\pi$
$360$	2.43939	0.128567
$361$	13.2768	0.698781
$362$	−12.3640	−0.649837
$363$	8.26510	0.433805
$364$	0.199815	0.0104731
$365$	−5.46440	−0.286020
$366$	1.09562	0.0572691
$367$	−28.3117	−1.47786	−0.738929	−	0.673783i	$-0.764668\pi$
$367$	−28.3117	−1.47786	−0.738929	+	0.673783i	$0.764668\pi$
$368$	2.77713	0.144768
$369$	0.791309	0.0411939
$370$	−6.94301	−0.360950
$371$	−4.50907	−0.234099
$372$	5.41623	0.280819
$373$	−14.8154	−0.767111	−0.383555	−	0.923518i	$-0.625301\pi$
$373$	−14.8154	−0.767111	−0.383555	+	0.923518i	$0.625301\pi$
$374$	1.32954	0.0687487
$375$	−8.30695	−0.428969
$376$	0.672374	0.0346750
$377$	−0.705718	−0.0363463
$378$	−1.29503	−0.0666093
$379$	−10.5079	−0.539755	−0.269878	−	0.962895i	$-0.586983\pi$
$379$	−10.5079	−0.539755	−0.269878	+	0.962895i	$0.586983\pi$
$380$	−6.95806	−0.356941
$381$	−11.0619	−0.566717
$382$	5.57803	0.285397
$383$	−5.65256	−0.288832	−0.144416	−	0.989517i	$-0.546130\pi$
$383$	−5.65256	−0.288832	−0.144416	+	0.989517i	$0.546130\pi$
$384$	8.75466	0.446759
$385$	2.41021	0.122835
$386$	0.265726	0.0135251
$387$	−1.21613	−0.0618194
$388$	0.938895	0.0476652
$389$	−15.4207	−0.781859	−0.390930	−	0.920421i	$-0.627846\pi$
$389$	−15.4207	−0.781859	−0.390930	+	0.920421i	$0.627846\pi$
$390$	−0.0666540	−0.00337516
$391$	−5.14537	−0.260212
$392$	11.8772	0.599891
$393$	−16.0262	−0.808412
$394$	−12.6476	−0.637175
$395$	−4.19934	−0.211292
$396$	2.23863	0.112495
$397$	11.7381	0.589116	0.294558	−	0.955634i	$-0.404828\pi$
$397$	11.7381	0.589116	0.294558	+	0.955634i	$0.404828\pi$
$398$	−18.9833	−0.951546
$399$	9.15160	0.458153
$400$	−2.25685	−0.112842
$401$	−9.53240	−0.476025	−0.238013	−	0.971262i	$-0.576496\pi$
$401$	−9.53240	−0.476025	−0.238013	+	0.971262i	$0.576496\pi$
$402$	−6.71777	−0.335052
$403$	−0.366650	−0.0182641
$404$	−11.7618	−0.585173
$405$	−0.904757	−0.0449578
$406$	9.97347	0.494975
$407$	−15.7855	−0.782457
$408$	−2.69618	−0.133481
$409$	−24.1096	−1.19214	−0.596072	−	0.802931i	$-0.703273\pi$
$409$	−24.1096	−1.19214	−0.596072	+	0.802931i	$0.703273\pi$
$410$	−0.575582	−0.0284260
$411$	5.73989	0.283128
$412$	−6.80865	−0.335438
$413$	−5.72599	−0.281758
$414$	4.13662	0.203304
$415$	12.1134	0.594625
$416$	−0.533896	−0.0261764
$417$	−5.11048	−0.250261
$418$	7.55345	0.369452
$419$	22.7111	1.10951	0.554755	−	0.832014i	$-0.312812\pi$
$419$	22.7111	1.10951	0.554755	+	0.832014i	$0.312812\pi$
$420$	−1.97285	−0.0962653
$421$	4.02437	0.196136	0.0980678	−	0.995180i	$-0.468734\pi$
$421$	4.02437	0.196136	0.0980678	+	0.995180i	$0.468734\pi$
$422$	22.3515	1.08805
$423$	−0.249380	−0.0121253
$424$	7.54718	0.366524
$425$	4.18141	0.202828
$426$	8.54621	0.414065
$427$	−2.19524	−0.106235
$428$	−11.8288	−0.571768
$429$	−0.151543	−0.00731657
$430$	0.884590	0.0426587
$431$	−32.7346	−1.57677	−0.788385	−	0.615182i	$-0.789082\pi$
$431$	−32.7346	−1.57677	−0.788385	+	0.615182i	$0.789082\pi$
$432$	−0.539733	−0.0259679
$433$	−9.99326	−0.480245	−0.240123	−	0.970743i	$-0.577188\pi$
$433$	−9.99326	−0.480245	−0.240123	+	0.970743i	$0.577188\pi$
$434$	5.18164	0.248727
$435$	6.96784	0.334082
$436$	21.2252	1.01650
$437$	−29.2322	−1.39837
$438$	−4.85557	−0.232008
$439$	−17.1750	−0.819716	−0.409858	−	0.912149i	$-0.634422\pi$
$439$	−17.1750	−0.819716	−0.409858	+	0.912149i	$0.634422\pi$
$440$	−4.03415	−0.192320
$441$	−4.40521	−0.209772
$442$	0.0736706	0.00350415
$443$	7.84031	0.372505	0.186252	−	0.982502i	$-0.440366\pi$
$443$	7.84031	0.372505	0.186252	+	0.982502i	$0.440366\pi$
$444$	12.9210	0.613205
$445$	9.94417	0.471399
$446$	−6.44847	−0.305344
$447$	14.4213	0.682105
$448$	5.80637	0.274325
$449$	−3.45648	−0.163121	−0.0815607	−	0.996668i	$-0.525990\pi$
$449$	−3.45648	−0.163121	−0.0815607	+	0.996668i	$0.525990\pi$
$450$	−3.36165	−0.158470
$451$	−1.30863	−0.0616210
$452$	−11.4384	−0.538018
$453$	9.67344	0.454498
$454$	5.26899	0.247286
$455$	0.133551	0.00626099
$456$	−15.3177	−0.717318
$457$	−9.24640	−0.432528	−0.216264	−	0.976335i	$-0.569387\pi$
$457$	−9.24640	−0.432528	−0.216264	+	0.976335i	$0.569387\pi$
$458$	8.12879	0.379833
$459$	1.00000	0.0466760
$460$	6.30172	0.293819
$461$	36.9374	1.72035	0.860174	−	0.510001i	$-0.170355\pi$
$461$	36.9374	1.72035	0.860174	+	0.510001i	$0.170355\pi$
$462$	2.14166	0.0996392
$463$	−10.2255	−0.475219	−0.237609	−	0.971361i	$-0.576364\pi$
$463$	−10.2255	−0.475219	−0.237609	+	0.971361i	$0.576364\pi$
$464$	4.15667	0.192968
$465$	3.62008	0.167877
$466$	4.42621	0.205040
$467$	−7.02356	−0.325012	−0.162506	−	0.986708i	$-0.551958\pi$
$467$	−7.02356	−0.325012	−0.162506	+	0.986708i	$0.551958\pi$
$468$	0.124044	0.00573394
$469$	13.4601	0.621529
$470$	0.181394	0.00836709
$471$	1.00000	0.0460776
$472$	9.58403	0.441141
$473$	2.01118	0.0924742
$474$	−3.73145	−0.171391
$475$	23.7557	1.08999
$476$	2.18053	0.0999444
$477$	−2.79921	−0.128167
$478$	−19.1166	−0.874373
$479$	17.0977	0.781215	0.390608	−	0.920557i	$-0.372265\pi$
$479$	17.0977	0.781215	0.390608	+	0.920557i	$0.372265\pi$
$480$	5.27137	0.240604
$481$	−0.874685	−0.0398822
$482$	−2.13498	−0.0972459
$483$	−8.28835	−0.377133
$484$	11.1882	0.508553
$485$	0.627536	0.0284949
$486$	−0.803950	−0.0364679
$487$	33.9184	1.53699	0.768494	−	0.639857i	$-0.221006\pi$
$487$	33.9184	1.53699	0.768494	+	0.639857i	$0.221006\pi$
$488$	3.67435	0.166330
$489$	11.9782	0.541671
$490$	3.20426	0.144754
$491$	−2.45805	−0.110930	−0.0554651	−	0.998461i	$-0.517664\pi$
$491$	−2.45805	−0.110930	−0.0554651	+	0.998461i	$0.517664\pi$
$492$	1.07117	0.0482919
$493$	−7.70133	−0.346851
$494$	0.418543	0.0188311
$495$	1.49625	0.0672512
$496$	2.15956	0.0969672
$497$	−17.1236	−0.768100
$498$	10.7638	0.482336
$499$	−6.31577	−0.282733	−0.141366	−	0.989957i	$-0.545150\pi$
$499$	−6.31577	−0.282733	−0.141366	+	0.989957i	$0.545150\pi$
$500$	−11.2448	−0.502884
$501$	−18.4715	−0.825246
$502$	19.8716	0.886911
$503$	−1.23700	−0.0551552	−0.0275776	−	0.999620i	$-0.508779\pi$
$503$	−1.23700	−0.0551552	−0.0275776	+	0.999620i	$0.508779\pi$
$504$	−4.34310	−0.193457
$505$	−7.86133	−0.349825
$506$	−6.84095	−0.304117
$507$	12.9916	0.576977
$508$	−14.9741	−0.664367
$509$	−30.6654	−1.35922	−0.679609	−	0.733574i	$-0.737851\pi$
$509$	−30.6654	−1.35922	−0.679609	+	0.733574i	$0.737851\pi$
$510$	−0.727380	−0.0322089
$511$	9.72886	0.430380
$512$	6.05507	0.267599
$513$	5.68127	0.250834
$514$	−10.9606	−0.483450
$515$	−4.55074	−0.200530
$516$	−1.64623	−0.0724714
$517$	0.412413	0.0181379
$518$	12.3614	0.543128
$519$	−23.5239	−1.03258
$520$	−0.223535	−0.00980267
$521$	19.8837	0.871119	0.435560	−	0.900160i	$-0.356550\pi$
$521$	19.8837	0.871119	0.435560	+	0.900160i	$0.356550\pi$
$522$	6.19149	0.270994
$523$	−35.7865	−1.56483	−0.782416	−	0.622756i	$-0.786013\pi$
$523$	−35.7865	−1.56483	−0.782416	+	0.622756i	$0.786013\pi$
$524$	−21.6940	−0.947708
$525$	6.73557	0.293964
$526$	−6.84696	−0.298542
$527$	−4.00116	−0.174293
$528$	0.892586	0.0388448
$529$	3.47482	0.151079
$530$	2.03609	0.0884422
$531$	−3.55467	−0.154260
$532$	12.3882	0.537096
$533$	−0.0725122	−0.00314085
$534$	8.83620	0.382380
$535$	−7.90612	−0.341811
$536$	−22.5292	−0.973111
$537$	26.1494	1.12843
$538$	−9.93919	−0.428509
$539$	7.28513	0.313793
$540$	−1.22474	−0.0527043
$541$	−10.0756	−0.433183	−0.216592	−	0.976262i	$-0.569494\pi$
$541$	−10.0756	−0.433183	−0.216592	+	0.976262i	$0.569494\pi$
$542$	1.75879	0.0755466
$543$	15.3791	0.659978
$544$	−5.82628	−0.249800
$545$	14.1864	0.607679
$546$	0.118671	0.00507866
$547$	−9.16606	−0.391912	−0.195956	−	0.980613i	$-0.562781\pi$
$547$	−9.16606	−0.391912	−0.195956	+	0.980613i	$0.562781\pi$
$548$	7.76988	0.331913
$549$	−1.36280	−0.0581628
$550$	5.55934	0.237051
$551$	−43.7534	−1.86396
$552$	13.8728	0.590467
$553$	7.47653	0.317934
$554$	−24.0121	−1.02017
$555$	8.63612	0.366583
$556$	−6.91787	−0.293383
$557$	8.31472	0.352306	0.176153	−	0.984363i	$-0.443635\pi$
$557$	8.31472	0.352306	0.176153	+	0.984363i	$0.443635\pi$
$558$	3.21674	0.136175
$559$	0.111441	0.00471346
$560$	−0.786616	−0.0332406
$561$	−1.65375	−0.0698215
$562$	−12.2983	−0.518773
$563$	−5.47764	−0.230855	−0.115427	−	0.993316i	$-0.536824\pi$
$563$	−5.47764	−0.230855	−0.115427	+	0.993316i	$0.536824\pi$
$564$	−0.337577	−0.0142146
$565$	−7.64517	−0.321635
$566$	−6.59621	−0.277259
$567$	1.61084	0.0676488
$568$	28.6611	1.20259
$569$	9.02752	0.378453	0.189227	−	0.981933i	$-0.439402\pi$
$569$	9.02752	0.378453	0.189227	+	0.981933i	$0.439402\pi$
$570$	−4.13244	−0.173089
$571$	−15.3469	−0.642247	−0.321124	−	0.947037i	$-0.604061\pi$
$571$	−15.3469	−0.642247	−0.321124	+	0.947037i	$0.604061\pi$
$572$	−0.205138	−0.00857726
$573$	−6.93828	−0.289851
$574$	1.02477	0.0427731
$575$	−21.5149	−0.897234
$576$	3.60457	0.150190
$577$	20.2804	0.844285	0.422142	−	0.906529i	$-0.361278\pi$
$577$	20.2804	0.844285	0.422142	+	0.906529i	$0.361278\pi$
$578$	0.803950	0.0334399
$579$	−0.330525	−0.0137361
$580$	9.43211	0.391647
$581$	−21.5668	−0.894743
$582$	0.557616	0.0231139
$583$	4.62921	0.191722
$584$	−16.2839	−0.673834
$585$	0.0829082	0.00342783
$586$	16.2520	0.671364
$587$	32.5848	1.34492	0.672459	−	0.740134i	$-0.265238\pi$
$587$	32.5848	1.34492	0.672459	+	0.740134i	$0.265238\pi$
$588$	−5.96317	−0.245917
$589$	−22.7317	−0.936644
$590$	2.58560	0.106447
$591$	15.7318	0.647119
$592$	5.15188	0.211741
$593$	33.6310	1.38106	0.690530	−	0.723304i	$-0.257378\pi$
$593$	33.6310	1.38106	0.690530	+	0.723304i	$0.257378\pi$
$594$	1.32954	0.0545515
$595$	1.45742	0.0597482
$596$	19.5216	0.799637
$597$	23.6125	0.966396
$598$	−0.379063	−0.0155010
$599$	5.60394	0.228971	0.114485	−	0.993425i	$-0.463478\pi$
$599$	5.60394	0.228971	0.114485	+	0.993425i	$0.463478\pi$
$600$	−11.2738	−0.460253
$601$	3.29883	0.134562	0.0672811	−	0.997734i	$-0.478568\pi$
$601$	3.29883	0.134562	0.0672811	+	0.997734i	$0.478568\pi$
$602$	−1.57493	−0.0641893
$603$	8.35596	0.340281
$604$	13.0946	0.532811
$605$	7.47791	0.304020
$606$	−6.98543	−0.283764
$607$	30.6868	1.24554	0.622769	−	0.782406i	$-0.286008\pi$
$607$	30.6868	1.24554	0.622769	+	0.782406i	$0.286008\pi$
$608$	−33.1007	−1.34241
$609$	−12.4056	−0.502700
$610$	0.991272	0.0401354
$611$	0.0228522	0.000924499 0
$612$	1.35366	0.0547186
$613$	40.5002	1.63579	0.817894	−	0.575369i	$-0.195141\pi$
$613$	40.5002	1.63579	0.817894	+	0.575369i	$0.195141\pi$
$614$	−14.7087	−0.593594
$615$	0.715942	0.0288696
$616$	7.18242	0.289388
$617$	7.41310	0.298440	0.149220	−	0.988804i	$-0.452324\pi$
$617$	7.41310	0.298440	0.149220	+	0.988804i	$0.452324\pi$
$618$	−4.04370	−0.162662
$619$	2.32291	0.0933656	0.0466828	−	0.998910i	$-0.485135\pi$
$619$	2.32291	0.0933656	0.0466828	+	0.998910i	$0.485135\pi$
$620$	4.90038	0.196804
$621$	−5.14537	−0.206477
$622$	−9.69306	−0.388656
$623$	−17.7047	−0.709323
$624$	0.0494589	0.00197994
$625$	13.3913	0.535652
$626$	−2.62543	−0.104933
$627$	−9.39542	−0.375217
$628$	1.35366	0.0540171
$629$	−9.54524	−0.380593
$630$	−1.17169	−0.0466812
$631$	−23.7123	−0.943972	−0.471986	−	0.881606i	$-0.656463\pi$
$631$	−23.7123	−0.943972	−0.471986	+	0.881606i	$0.656463\pi$
$632$	−12.5140	−0.497782
$633$	−27.8021	−1.10503
$634$	−5.82927	−0.231510
$635$	−10.0083	−0.397168
$636$	−3.78919	−0.150251
$637$	0.403675	0.0159942
$638$	−10.2392	−0.405374
$639$	−10.6303	−0.420527
$640$	7.92084	0.313099
$641$	−13.8543	−0.547213	−0.273606	−	0.961842i	$-0.588217\pi$
$641$	−13.8543	−0.547213	−0.273606	+	0.961842i	$0.588217\pi$
$642$	−7.02523	−0.277264
$643$	18.6936	0.737203	0.368602	−	0.929587i	$-0.379837\pi$
$643$	18.6936	0.737203	0.368602	+	0.929587i	$0.379837\pi$
$644$	−11.2196	−0.442115
$645$	−1.10030	−0.0433244
$646$	4.56746	0.179704
$647$	10.4629	0.411340	0.205670	−	0.978621i	$-0.434063\pi$
$647$	10.4629	0.411340	0.205670	+	0.978621i	$0.434063\pi$
$648$	−2.69618	−0.105916
$649$	5.87855	0.230753
$650$	0.308047	0.0120826
$651$	−6.44522	−0.252608
$652$	16.2144	0.635004
$653$	25.2592	0.988469	0.494235	−	0.869328i	$-0.335448\pi$
$653$	25.2592	0.988469	0.494235	+	0.869328i	$0.335448\pi$
$654$	12.6058	0.492925
$655$	−14.4998	−0.566553
$656$	0.427096	0.0166753
$657$	6.03964	0.235629
$658$	−0.322955	−0.0125901
$659$	25.9584	1.01119	0.505597	−	0.862770i	$-0.331272\pi$
$659$	25.9584	1.01119	0.505597	+	0.862770i	$0.331272\pi$
$660$	2.02541	0.0788391
$661$	−33.8501	−1.31662	−0.658309	−	0.752748i	$-0.728728\pi$
$661$	−33.8501	−1.31662	−0.658309	+	0.752748i	$0.728728\pi$
$662$	−1.97947	−0.0769341
$663$	−0.0916358	−0.00355884
$664$	36.0981	1.40088
$665$	8.27997	0.321084
$666$	7.67390	0.297357
$667$	39.6262	1.53433
$668$	−25.0042	−0.967442
$669$	8.02098	0.310109
$670$	−6.07796	−0.234812
$671$	2.25373	0.0870044
$672$	−9.38518	−0.362041
$673$	−36.7361	−1.41607	−0.708036	−	0.706176i	$-0.750419\pi$
$673$	−36.7361	−1.41607	−0.708036	+	0.706176i	$0.750419\pi$
$674$	−18.3535	−0.706950
$675$	4.18141	0.160943
$676$	17.5863	0.676395
$677$	−27.9614	−1.07465	−0.537323	−	0.843377i	$-0.680564\pi$
$677$	−27.9614	−1.07465	−0.537323	+	0.843377i	$0.680564\pi$
$678$	−6.79336	−0.260897
$679$	−1.11727	−0.0428768
$680$	−2.43939	−0.0935463
$681$	−6.55387	−0.251145
$682$	−5.31969	−0.203702
$683$	28.4644	1.08916	0.544579	−	0.838709i	$-0.316689\pi$
$683$	28.4644	1.08916	0.544579	+	0.838709i	$0.316689\pi$
$684$	7.69053	0.294055
$685$	5.19321	0.198422
$686$	−14.7701	−0.563925
$687$	−10.1111	−0.385761
$688$	−0.656386	−0.0250245
$689$	0.256508	0.00977218
$690$	3.74264	0.142480
$691$	7.99042	0.303970	0.151985	−	0.988383i	$-0.451433\pi$
$691$	7.99042	0.303970	0.151985	+	0.988383i	$0.451433\pi$
$692$	−31.8434	−1.21050
$693$	−2.66393	−0.101194
$694$	−17.7714	−0.674595
$695$	−4.62374	−0.175389
$696$	20.7642	0.787064
$697$	−0.791309	−0.0299730
$698$	1.31172	0.0496494
$699$	−5.50557	−0.208240
$700$	9.11770	0.344617
$701$	−21.2227	−0.801571	−0.400785	−	0.916172i	$-0.631263\pi$
$701$	−21.2227	−0.801571	−0.400785	+	0.916172i	$0.631263\pi$
$702$	0.0736706	0.00278052
$703$	−54.2291	−2.04529
$704$	−5.96107	−0.224666
$705$	−0.225629	−0.00849767
$706$	−4.22511	−0.159014
$707$	13.9964	0.526388
$708$	−4.81183	−0.180840
$709$	11.6853	0.438851	0.219425	−	0.975629i	$-0.429582\pi$
$709$	11.6853	0.438851	0.219425	+	0.975629i	$0.429582\pi$
$710$	7.73225	0.290186
$711$	4.64140	0.174066
$712$	29.6337	1.11057
$713$	20.5875	0.771007
$714$	1.29503	0.0484654
$715$	−0.137110	−0.00512761
$716$	35.3975	1.32286
$717$	23.7783	0.888018
$718$	−29.4291	−1.09828
$719$	42.7654	1.59488	0.797441	−	0.603397i	$-0.206187\pi$
$719$	42.7654	1.59488	0.797441	+	0.603397i	$0.206187\pi$
$720$	−0.488328	−0.0181989
$721$	8.10217	0.301741
$722$	10.6739	0.397242
$723$	2.65562	0.0987635
$724$	20.8181	0.773697
$725$	−32.2025	−1.19597
$726$	6.64473	0.246609
$727$	−13.7752	−0.510893	−0.255447	−	0.966823i	$-0.582223\pi$
$727$	−13.7752	−0.510893	−0.255447	+	0.966823i	$0.582223\pi$
$728$	0.397984	0.0147503
$729$	1.00000	0.0370370
$730$	−4.39311	−0.162596
$731$	1.21613	0.0449802
$732$	−1.84477	−0.0681847
$733$	12.4182	0.458676	0.229338	−	0.973347i	$-0.426344\pi$
$733$	12.4182	0.458676	0.229338	+	0.973347i	$0.426344\pi$
$734$	−22.7612	−0.840131
$735$	−3.98564	−0.147013
$736$	29.9783	1.10502
$737$	−13.8187	−0.509018
$738$	0.636173	0.0234179
$739$	16.7363	0.615654	0.307827	−	0.951442i	$-0.400398\pi$
$739$	16.7363	0.615654	0.307827	+	0.951442i	$0.400398\pi$
$740$	11.6904	0.429748
$741$	−0.520608	−0.0191250
$742$	−3.62507	−0.133081
$743$	26.6018	0.975927	0.487963	−	0.872864i	$-0.337740\pi$
$743$	26.6018	0.975927	0.487963	+	0.872864i	$0.337740\pi$
$744$	10.7879	0.395502
$745$	13.0478	0.478034
$746$	−11.9108	−0.436086
$747$	−13.3886	−0.489863
$748$	−2.23863	−0.0818523
$749$	14.0761	0.514330
$750$	−6.67838	−0.243860
$751$	−23.4297	−0.854961	−0.427480	−	0.904025i	$-0.640599\pi$
$751$	−23.4297	−0.854961	−0.427480	+	0.904025i	$0.640599\pi$
$752$	−0.134599	−0.00490831
$753$	−24.7174	−0.900752
$754$	−0.567362	−0.0206621
$755$	8.75211	0.318522
$756$	2.18053	0.0793051
$757$	26.5306	0.964271	0.482135	−	0.876097i	$-0.339861\pi$
$757$	26.5306	0.964271	0.482135	+	0.876097i	$0.339861\pi$
$758$	−8.44784	−0.306840
$759$	8.50917	0.308863
$760$	−13.8588	−0.502712
$761$	−13.2481	−0.480242	−0.240121	−	0.970743i	$-0.577187\pi$
$761$	−13.2481	−0.480242	−0.240121	+	0.970743i	$0.577187\pi$
$762$	−8.89321	−0.322167
$763$	−25.2576	−0.914386
$764$	−9.39209	−0.339794
$765$	0.904757	0.0327116
$766$	−4.54438	−0.164195
$767$	0.325735	0.0117616
$768$	14.2475	0.514111
$769$	11.5443	0.416299	0.208150	−	0.978097i	$-0.433256\pi$
$769$	11.5443	0.416299	0.208150	+	0.978097i	$0.433256\pi$
$770$	1.93769	0.0698294
$771$	13.6334	0.490995
$772$	−0.447420	−0.0161030
$773$	17.4946	0.629237	0.314618	−	0.949218i	$-0.398123\pi$
$773$	17.4946	0.629237	0.314618	+	0.949218i	$0.398123\pi$
$774$	−0.977709	−0.0351430
$775$	−16.7305	−0.600978
$776$	1.87006	0.0671312
$777$	−15.3758	−0.551604
$778$	−12.3975	−0.444470
$779$	−4.49564	−0.161073
$780$	0.112230	0.00401847
$781$	17.5799	0.629057
$782$	−4.13662	−0.147925
$783$	−7.70133	−0.275223
$784$	−2.37764	−0.0849156
$785$	0.904757	0.0322922
$786$	−12.8842	−0.459565
$787$	12.7278	0.453696	0.226848	−	0.973930i	$-0.427158\pi$
$787$	12.7278	0.453696	0.226848	+	0.973930i	$0.427158\pi$
$788$	21.2955	0.758622
$789$	8.51665	0.303201
$790$	−3.37606	−0.120115
$791$	13.6115	0.483970
$792$	4.45882	0.158437
$793$	0.124881	0.00443466
$794$	9.43682	0.334900
$795$	−2.53261	−0.0898224
$796$	31.9634	1.13291
$797$	−4.20025	−0.148781	−0.0743903	−	0.997229i	$-0.523701\pi$
$797$	−4.20025	−0.148781	−0.0743903	+	0.997229i	$0.523701\pi$
$798$	7.35743	0.260450
$799$	0.249380	0.00882244
$800$	−24.3621	−0.861330
$801$	−10.9910	−0.388347
$802$	−7.66357	−0.270610
$803$	−9.98807	−0.352471
$804$	11.3112	0.398914
$805$	−7.49894	−0.264303
$806$	−0.294768	−0.0103828
$807$	12.3629	0.435196
$808$	−23.4268	−0.824152
$809$	35.9815	1.26504	0.632521	−	0.774543i	$-0.282020\pi$
$809$	35.9815	1.26504	0.632521	+	0.774543i	$0.282020\pi$
$810$	−0.727380	−0.0255575
$811$	−17.8320	−0.626167	−0.313083	−	0.949726i	$-0.601362\pi$
$811$	−17.8320	−0.626167	−0.313083	+	0.949726i	$0.601362\pi$
$812$	−16.7930	−0.589318
$813$	−2.18769	−0.0767255
$814$	−12.6907	−0.444810
$815$	10.8373	0.379615
$816$	0.539733	0.0188944
$817$	6.90917	0.241721
$818$	−19.3829	−0.677708
$819$	−0.147610	−0.00515792
$820$	0.969145	0.0338440
$821$	5.28531	0.184459	0.0922293	−	0.995738i	$-0.470601\pi$
$821$	5.28531	0.184459	0.0922293	+	0.995738i	$0.470601\pi$
$822$	4.61459	0.160952
$823$	−25.4428	−0.886881	−0.443440	−	0.896304i	$-0.646242\pi$
$823$	−25.4428	−0.886881	−0.443440	+	0.896304i	$0.646242\pi$
$824$	−13.5612	−0.472427
$825$	−6.91503	−0.240750
$826$	−4.60341	−0.160173
$827$	−27.1093	−0.942681	−0.471341	−	0.881951i	$-0.656230\pi$
$827$	−27.1093	−0.942681	−0.471341	+	0.881951i	$0.656230\pi$
$828$	−6.96510	−0.242054
$829$	−37.8778	−1.31555	−0.657775	−	0.753215i	$-0.728502\pi$
$829$	−37.8778	−1.31555	−0.657775	+	0.753215i	$0.728502\pi$
$830$	9.73860	0.338032
$831$	29.8676	1.03609
$832$	−0.330308	−0.0114514
$833$	4.40521	0.152631
$834$	−4.10857	−0.142268
$835$	−16.7122	−0.578351
$836$	−12.7182	−0.439870
$837$	−4.00116	−0.138300
$838$	18.2586	0.630733
$839$	26.5124	0.915309	0.457654	−	0.889130i	$-0.348690\pi$
$839$	26.5124	0.915309	0.457654	+	0.889130i	$0.348690\pi$
$840$	−3.92945	−0.135579
$841$	30.3105	1.04519
$842$	3.23539	0.111499
$843$	15.2973	0.526868
$844$	−37.6346	−1.29544
$845$	11.7542	0.404358
$846$	−0.200489	−0.00689297
$847$	−13.3137	−0.457465
$848$	−1.51083	−0.0518821
$849$	8.20474	0.281586
$850$	3.36165	0.115304
$851$	49.1138	1.68360
$852$	−14.3898	−0.492987
$853$	−33.7541	−1.15572	−0.577858	−	0.816137i	$-0.696111\pi$
$853$	−33.7541	−1.15572	−0.577858	+	0.816137i	$0.696111\pi$
$854$	−1.76487	−0.0603925
$855$	5.14017	0.175790
$856$	−23.5603	−0.805273
$857$	−42.5808	−1.45453	−0.727267	−	0.686355i	$-0.759210\pi$
$857$	−42.5808	−1.45453	−0.727267	+	0.686355i	$0.759210\pi$
$858$	−0.121833	−0.00415931
$859$	−15.9109	−0.542872	−0.271436	−	0.962457i	$-0.587498\pi$
$859$	−15.9109	−0.542872	−0.271436	+	0.962457i	$0.587498\pi$
$860$	−1.48944	−0.0507895
$861$	−1.27467	−0.0434406
$862$	−26.3170	−0.896360
$863$	33.0354	1.12454	0.562269	−	0.826954i	$-0.309928\pi$
$863$	33.0354	1.12454	0.562269	+	0.826954i	$0.309928\pi$
$864$	−5.82628	−0.198214
$865$	−21.2834	−0.723657
$866$	−8.03408	−0.273009
$867$	−1.00000	−0.0339618
$868$	−8.72466	−0.296134
$869$	−7.67573	−0.260381
$870$	5.60180	0.189919
$871$	−0.765705	−0.0259449
$872$	42.2755	1.43163
$873$	−0.693595	−0.0234746
$874$	−23.5013	−0.794942
$875$	13.3811	0.452365
$876$	8.17564	0.276229
$877$	25.9924	0.877702	0.438851	−	0.898560i	$-0.355386\pi$
$877$	25.9924	0.877702	0.438851	+	0.898560i	$0.355386\pi$
$878$	−13.8078	−0.465991
$879$	−20.2152	−0.681841
$880$	0.807573	0.0272233
$881$	52.4284	1.76636	0.883179	−	0.469037i	$-0.155399\pi$
$881$	52.4284	1.76636	0.883179	+	0.469037i	$0.155399\pi$
$882$	−3.54157	−0.119251
$883$	−42.5118	−1.43064	−0.715318	−	0.698799i	$-0.753718\pi$
$883$	−42.5118	−1.43064	−0.715318	+	0.698799i	$0.753718\pi$
$884$	−0.124044	−0.00417205
$885$	−3.21612	−0.108109
$886$	6.30322	0.211761
$887$	−23.5832	−0.791845	−0.395922	−	0.918284i	$-0.629575\pi$
$887$	−23.5832	−0.791845	−0.395922	+	0.918284i	$0.629575\pi$
$888$	25.7357	0.863632
$889$	17.8189	0.597626
$890$	7.99462	0.267980
$891$	−1.65375	−0.0554028
$892$	10.8577	0.363543
$893$	1.41680	0.0474113
$894$	11.5940	0.387762
$895$	23.6588	0.790828
$896$	−14.1023	−0.471125
$897$	0.471500	0.0157429
$898$	−2.77884	−0.0927311
$899$	30.8143	1.02771
$900$	5.66023	0.188674
$901$	2.79921	0.0932553
$902$	−1.05207	−0.0350302
$903$	1.95899	0.0651910
$904$	−22.7826	−0.757739
$905$	13.9143	0.462527
$906$	7.77696	0.258372
$907$	−10.9562	−0.363796	−0.181898	−	0.983317i	$-0.558224\pi$
$907$	−10.9562	−0.363796	−0.181898	+	0.983317i	$0.558224\pi$
$908$	−8.87174	−0.294419
$909$	8.68888	0.288192
$910$	0.107369	0.00355924
$911$	−52.1336	−1.72726	−0.863632	−	0.504122i	$-0.831816\pi$
$911$	−52.1336	−1.72726	−0.863632	+	0.504122i	$0.831816\pi$
$912$	3.06637	0.101538
$913$	22.1414	0.732775
$914$	−7.43365	−0.245883
$915$	−1.23300	−0.0407618
$916$	−13.6870	−0.452230
$917$	25.8155	0.852503
$918$	0.803950	0.0265343
$919$	−28.1686	−0.929197	−0.464599	−	0.885521i	$-0.653801\pi$
$919$	−28.1686	−0.929197	−0.464599	+	0.885521i	$0.653801\pi$
$920$	12.5516	0.413812
$921$	18.2955	0.602858
$922$	29.6959	0.977981
$923$	0.974113	0.0320633
$924$	−3.60606	−0.118631
$925$	−39.9126	−1.31232
$926$	−8.22078	−0.270152
$927$	5.02979	0.165200
$928$	44.8701	1.47293
$929$	20.4835	0.672041	0.336020	−	0.941855i	$-0.390919\pi$
$929$	20.4835	0.672041	0.336020	+	0.941855i	$0.390919\pi$
$930$	2.91037	0.0954347
$931$	25.0272	0.820233
$932$	−7.45269	−0.244121
$933$	12.0568	0.394721
$934$	−5.64659	−0.184762
$935$	−1.49625	−0.0489325
$936$	0.247067	0.00807562
$937$	4.62838	0.151203	0.0756013	−	0.997138i	$-0.475912\pi$
$937$	4.62838	0.151203	0.0756013	+	0.997138i	$0.475912\pi$
$938$	10.8212	0.353326
$939$	3.26567	0.106571
$940$	−0.305425	−0.00996187
$941$	7.94391	0.258964	0.129482	−	0.991582i	$-0.458669\pi$
$941$	7.94391	0.258964	0.129482	+	0.991582i	$0.458669\pi$
$942$	0.803950	0.0261941
$943$	4.07158	0.132589
$944$	−1.91857	−0.0624443
$945$	1.45742	0.0474097
$946$	1.61689	0.0525696
$947$	49.7703	1.61732	0.808659	−	0.588277i	$-0.200194\pi$
$947$	49.7703	1.61732	0.808659	+	0.588277i	$0.200194\pi$
$948$	6.28289	0.204059
$949$	−0.553447	−0.0179656
$950$	19.0984	0.619635
$951$	7.25078	0.235123
$952$	4.34310	0.140761
$953$	20.1753	0.653541	0.326771	−	0.945104i	$-0.394040\pi$
$953$	20.1753	0.653541	0.326771	+	0.945104i	$0.394040\pi$
$954$	−2.25043	−0.0728603
$955$	−6.27746	−0.203134
$956$	32.1879	1.04103
$957$	12.7361	0.411700
$958$	13.7457	0.444104
$959$	−9.24602	−0.298569
$960$	3.26126	0.105257
$961$	−14.9907	−0.483570
$962$	−0.703203	−0.0226722
$963$	8.73839	0.281591
$964$	3.59481	0.115781
$965$	−0.299045	−0.00962659
$966$	−6.66342	−0.214392
$967$	6.69093	0.215166	0.107583	−	0.994196i	$-0.465689\pi$
$967$	6.69093	0.215166	0.107583	+	0.994196i	$0.465689\pi$
$968$	22.2842	0.716241
$969$	−5.68127	−0.182509
$970$	0.504507	0.0161988
$971$	30.3676	0.974542	0.487271	−	0.873251i	$-0.337992\pi$
$971$	30.3676	0.974542	0.487271	+	0.873251i	$0.337992\pi$
$972$	1.35366	0.0434188
$973$	8.23214	0.263910
$974$	27.2687	0.873745
$975$	−0.383167	−0.0122712
$976$	−0.735548	−0.0235443
$977$	−26.4301	−0.845574	−0.422787	−	0.906229i	$-0.638948\pi$
$977$	−26.4301	−0.845574	−0.422787	+	0.906229i	$0.638948\pi$
$978$	9.62984	0.307928
$979$	18.1764	0.580920
$980$	−5.39522	−0.172344
$981$	−15.6798	−0.500617
$982$	−1.97615	−0.0630615
$983$	54.7375	1.74585	0.872927	−	0.487850i	$-0.162219\pi$
$983$	54.7375	1.74585	0.872927	+	0.487850i	$0.162219\pi$
$984$	2.13351	0.0680138
$985$	14.2334	0.453515
$986$	−6.19149	−0.197177
$987$	0.401711	0.0127866
$988$	−0.704728	−0.0224204
$989$	−6.25744	−0.198975
$990$	1.20291	0.0382309
$991$	23.2925	0.739910	0.369955	−	0.929050i	$-0.379373\pi$
$991$	23.2925	0.739910	0.369955	+	0.929050i	$0.379373\pi$
$992$	23.3119	0.740153
$993$	2.46217	0.0781347
$994$	−13.7665	−0.436648
$995$	21.3636	0.677272
$996$	−18.1237	−0.574270
$997$	−18.5796	−0.588421	−0.294210	−	0.955741i	$-0.595057\pi$
$997$	−18.5796	−0.588421	−0.294210	+	0.955741i	$0.595057\pi$
$998$	−5.07757	−0.160728
$999$	−9.54524	−0.301998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.30	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.30	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+0.803950 q^{2} -1.00000 q^{3} -1.35366 q^{4} -0.904757 q^{5} -0.803950 q^{6} +1.61084 q^{7} -2.69618 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.803950 q^{2} -1.00000 q^{3} -1.35366 q^{4} -0.904757 q^{5} -0.803950 q^{6} +1.61084 q^{7} -2.69618 q^{8} +1.00000 q^{9} -0.727380 q^{10} -1.65375 q^{11} +1.35366 q^{12} -0.0916358 q^{13} +1.29503 q^{14} +0.904757 q^{15} +0.539733 q^{16} -1.00000 q^{17} +0.803950 q^{18} -5.68127 q^{19} +1.22474 q^{20} -1.61084 q^{21} -1.32954 q^{22} +5.14537 q^{23} +2.69618 q^{24} -4.18141 q^{25} -0.0736706 q^{26} -1.00000 q^{27} -2.18053 q^{28} +7.70133 q^{29} +0.727380 q^{30} +4.00116 q^{31} +5.82628 q^{32} +1.65375 q^{33} -0.803950 q^{34} -1.45742 q^{35} -1.35366 q^{36} +9.54524 q^{37} -4.56746 q^{38} +0.0916358 q^{39} +2.43939 q^{40} +0.791309 q^{41} -1.29503 q^{42} -1.21613 q^{43} +2.23863 q^{44} -0.904757 q^{45} +4.13662 q^{46} -0.249380 q^{47} -0.539733 q^{48} -4.40521 q^{49} -3.36165 q^{50} +1.00000 q^{51} +0.124044 q^{52} -2.79921 q^{53} -0.803950 q^{54} +1.49625 q^{55} -4.34310 q^{56} +5.68127 q^{57} +6.19149 q^{58} -3.55467 q^{59} -1.22474 q^{60} -1.36280 q^{61} +3.21674 q^{62} +1.61084 q^{63} +3.60457 q^{64} +0.0829082 q^{65} +1.32954 q^{66} +8.35596 q^{67} +1.35366 q^{68} -5.14537 q^{69} -1.17169 q^{70} -10.6303 q^{71} -2.69618 q^{72} +6.03964 q^{73} +7.67390 q^{74} +4.18141 q^{75} +7.69053 q^{76} -2.66393 q^{77} +0.0736706 q^{78} +4.64140 q^{79} -0.488328 q^{80} +1.00000 q^{81} +0.636173 q^{82} -13.3886 q^{83} +2.18053 q^{84} +0.904757 q^{85} -0.977709 q^{86} -7.70133 q^{87} +4.45882 q^{88} -10.9910 q^{89} -0.727380 q^{90} -0.147610 q^{91} -6.96510 q^{92} -4.00116 q^{93} -0.200489 q^{94} +5.14017 q^{95} -5.82628 q^{96} -0.693595 q^{97} -3.54157 q^{98} -1.65375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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