LMFDB - Embedded newform 8042.2.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.58298 q^{3} +1.00000 q^{4} +0.255702 q^{5} +2.58298 q^{6} -2.29966 q^{7} -1.00000 q^{8} +3.67181 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.58298 q^{3} +1.00000 q^{4} +0.255702 q^{5} +2.58298 q^{6} -2.29966 q^{7} -1.00000 q^{8} +3.67181 q^{9} -0.255702 q^{10} +4.75397 q^{11} -2.58298 q^{12} +0.531542 q^{13} +2.29966 q^{14} -0.660474 q^{15} +1.00000 q^{16} -3.54643 q^{17} -3.67181 q^{18} -4.26077 q^{19} +0.255702 q^{20} +5.93998 q^{21} -4.75397 q^{22} +2.75131 q^{23} +2.58298 q^{24} -4.93462 q^{25} -0.531542 q^{26} -1.73528 q^{27} -2.29966 q^{28} -5.42753 q^{29} +0.660474 q^{30} -9.45140 q^{31} -1.00000 q^{32} -12.2794 q^{33} +3.54643 q^{34} -0.588027 q^{35} +3.67181 q^{36} +4.06786 q^{37} +4.26077 q^{38} -1.37297 q^{39} -0.255702 q^{40} +8.17919 q^{41} -5.93998 q^{42} +2.34699 q^{43} +4.75397 q^{44} +0.938889 q^{45} -2.75131 q^{46} +5.42015 q^{47} -2.58298 q^{48} -1.71157 q^{49} +4.93462 q^{50} +9.16037 q^{51} +0.531542 q^{52} +7.32698 q^{53} +1.73528 q^{54} +1.21560 q^{55} +2.29966 q^{56} +11.0055 q^{57} +5.42753 q^{58} -4.92228 q^{59} -0.660474 q^{60} -8.97884 q^{61} +9.45140 q^{62} -8.44391 q^{63} +1.00000 q^{64} +0.135916 q^{65} +12.2794 q^{66} -2.67472 q^{67} -3.54643 q^{68} -7.10659 q^{69} +0.588027 q^{70} +9.63127 q^{71} -3.67181 q^{72} +11.6776 q^{73} -4.06786 q^{74} +12.7460 q^{75} -4.26077 q^{76} -10.9325 q^{77} +1.37297 q^{78} +8.60030 q^{79} +0.255702 q^{80} -6.53324 q^{81} -8.17919 q^{82} +4.68449 q^{83} +5.93998 q^{84} -0.906828 q^{85} -2.34699 q^{86} +14.0192 q^{87} -4.75397 q^{88} -10.3194 q^{89} -0.938889 q^{90} -1.22236 q^{91} +2.75131 q^{92} +24.4128 q^{93} -5.42015 q^{94} -1.08949 q^{95} +2.58298 q^{96} -0.192932 q^{97} +1.71157 q^{98} +17.4557 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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
$3$	−2.58298	−1.49129	−0.745644	−	0.666345i	$-0.767858\pi$
$3$	−2.58298	−1.49129	−0.745644	+	0.666345i	$0.767858\pi$
$4$	1.00000	0.500000
$5$	0.255702	0.114353	0.0571767	−	0.998364i	$-0.481790\pi$
$5$	0.255702	0.114353	0.0571767	+	0.998364i	$0.481790\pi$
$6$	2.58298	1.05450
$7$	−2.29966	−0.869189	−0.434595	−	0.900626i	$-0.643108\pi$
$7$	−2.29966	−0.869189	−0.434595	+	0.900626i	$0.643108\pi$
$8$	−1.00000	−0.353553
$9$	3.67181	1.22394
$10$	−0.255702	−0.0808601
$11$	4.75397	1.43337	0.716687	−	0.697395i	$-0.245657\pi$
$11$	4.75397	1.43337	0.716687	+	0.697395i	$0.245657\pi$
$12$	−2.58298	−0.745644
$13$	0.531542	0.147423	0.0737116	−	0.997280i	$-0.476516\pi$
$13$	0.531542	0.147423	0.0737116	+	0.997280i	$0.476516\pi$
$14$	2.29966	0.614609
$15$	−0.660474	−0.170534
$16$	1.00000	0.250000
$17$	−3.54643	−0.860135	−0.430067	−	0.902797i	$-0.641510\pi$
$17$	−3.54643	−0.860135	−0.430067	+	0.902797i	$0.641510\pi$
$18$	−3.67181	−0.865454
$19$	−4.26077	−0.977488	−0.488744	−	0.872427i	$-0.662545\pi$
$19$	−4.26077	−0.977488	−0.488744	+	0.872427i	$0.662545\pi$
$20$	0.255702	0.0571767
$21$	5.93998	1.29621
$22$	−4.75397	−1.01355
$23$	2.75131	0.573688	0.286844	−	0.957977i	$-0.407394\pi$
$23$	2.75131	0.573688	0.286844	+	0.957977i	$0.407394\pi$
$24$	2.58298	0.527250
$25$	−4.93462	−0.986923
$26$	−0.531542	−0.104244
$27$	−1.73528	−0.333954
$28$	−2.29966	−0.434595
$29$	−5.42753	−1.00787	−0.503934	−	0.863742i	$-0.668114\pi$
$29$	−5.42753	−1.00787	−0.503934	+	0.863742i	$0.668114\pi$
$30$	0.660474	0.120586
$31$	−9.45140	−1.69752	−0.848761	−	0.528777i	$-0.822651\pi$
$31$	−9.45140	−1.69752	−0.848761	+	0.528777i	$0.822651\pi$
$32$	−1.00000	−0.176777
$33$	−12.2794	−2.13757
$34$	3.54643	0.608207
$35$	−0.588027	−0.0993947
$36$	3.67181	0.611968
$37$	4.06786	0.668752	0.334376	−	0.942440i	$-0.391474\pi$
$37$	4.06786	0.668752	0.334376	+	0.942440i	$0.391474\pi$
$38$	4.26077	0.691188
$39$	−1.37297	−0.219850
$40$	−0.255702	−0.0404300
$41$	8.17919	1.27738	0.638688	−	0.769466i	$-0.279478\pi$
$41$	8.17919	1.27738	0.638688	+	0.769466i	$0.279478\pi$
$42$	−5.93998	−0.916559
$43$	2.34699	0.357912	0.178956	−	0.983857i	$-0.442728\pi$
$43$	2.34699	0.357912	0.178956	+	0.983857i	$0.442728\pi$
$44$	4.75397	0.716687
$45$	0.938889	0.139961
$46$	−2.75131	−0.405659
$47$	5.42015	0.790611	0.395305	−	0.918550i	$-0.370639\pi$
$47$	5.42015	0.790611	0.395305	+	0.918550i	$0.370639\pi$
$48$	−2.58298	−0.372822
$49$	−1.71157	−0.244510
$50$	4.93462	0.697860
$51$	9.16037	1.28271
$52$	0.531542	0.0737116
$53$	7.32698	1.00644	0.503219	−	0.864159i	$-0.332149\pi$
$53$	7.32698	1.00644	0.503219	+	0.864159i	$0.332149\pi$
$54$	1.73528	0.236141
$55$	1.21560	0.163911
$56$	2.29966	0.307305
$57$	11.0055	1.45772
$58$	5.42753	0.712670
$59$	−4.92228	−0.640827	−0.320413	−	0.947278i	$-0.603822\pi$
$59$	−4.92228	−0.640827	−0.320413	+	0.947278i	$0.603822\pi$
$60$	−0.660474	−0.0852669
$61$	−8.97884	−1.14962	−0.574812	−	0.818286i	$-0.694925\pi$
$61$	−8.97884	−1.14962	−0.574812	+	0.818286i	$0.694925\pi$
$62$	9.45140	1.20033
$63$	−8.44391	−1.06383
$64$	1.00000	0.125000
$65$	0.135916	0.0168583
$66$	12.2794	1.51149
$67$	−2.67472	−0.326769	−0.163385	−	0.986562i	$-0.552241\pi$
$67$	−2.67472	−0.326769	−0.163385	+	0.986562i	$0.552241\pi$
$68$	−3.54643	−0.430067
$69$	−7.10659	−0.855533
$70$	0.588027	0.0702827
$71$	9.63127	1.14302	0.571511	−	0.820594i	$-0.306357\pi$
$71$	9.63127	1.14302	0.571511	+	0.820594i	$0.306357\pi$
$72$	−3.67181	−0.432727
$73$	11.6776	1.36676	0.683380	−	0.730063i	$-0.260509\pi$
$73$	11.6776	1.36676	0.683380	+	0.730063i	$0.260509\pi$
$74$	−4.06786	−0.472879
$75$	12.7460	1.47179
$76$	−4.26077	−0.488744
$77$	−10.9325	−1.24587
$78$	1.37297	0.155458
$79$	8.60030	0.967610	0.483805	−	0.875176i	$-0.339254\pi$
$79$	8.60030	0.967610	0.483805	+	0.875176i	$0.339254\pi$
$80$	0.255702	0.0285883
$81$	−6.53324	−0.725915
$82$	−8.17919	−0.903241
$83$	4.68449	0.514189	0.257095	−	0.966386i	$-0.417235\pi$
$83$	4.68449	0.514189	0.257095	+	0.966386i	$0.417235\pi$
$84$	5.93998	0.648105
$85$	−0.906828	−0.0983593
$86$	−2.34699	−0.253082
$87$	14.0192	1.50302
$88$	−4.75397	−0.506774
$89$	−10.3194	−1.09386	−0.546929	−	0.837179i	$-0.684203\pi$
$89$	−10.3194	−1.09386	−0.546929	+	0.837179i	$0.684203\pi$
$90$	−0.938889	−0.0989676
$91$	−1.22236	−0.128139
$92$	2.75131	0.286844
$93$	24.4128	2.53149
$94$	−5.42015	−0.559046
$95$	−1.08949	−0.111779
$96$	2.58298	0.263625
$97$	−0.192932	−0.0195893	−0.00979464	−	0.999952i	$-0.503118\pi$
$97$	−0.192932	−0.0195893	−0.00979464	+	0.999952i	$0.503118\pi$
$98$	1.71157	0.172895
$99$	17.4557	1.75436
$100$	−4.93462	−0.493462
$101$	−18.4028	−1.83115	−0.915574	−	0.402150i	$-0.868263\pi$
$101$	−18.4028	−1.83115	−0.915574	+	0.402150i	$0.868263\pi$
$102$	−9.16037	−0.907012
$103$	−14.8940	−1.46755	−0.733777	−	0.679391i	$-0.762244\pi$
$103$	−14.8940	−1.46755	−0.733777	+	0.679391i	$0.762244\pi$
$104$	−0.531542	−0.0521220
$105$	1.51886	0.148226
$106$	−7.32698	−0.711660
$107$	9.96286	0.963146	0.481573	−	0.876406i	$-0.340066\pi$
$107$	9.96286	0.963146	0.481573	+	0.876406i	$0.340066\pi$
$108$	−1.73528	−0.166977
$109$	−3.23819	−0.310162	−0.155081	−	0.987902i	$-0.549564\pi$
$109$	−3.23819	−0.310162	−0.155081	+	0.987902i	$0.549564\pi$
$110$	−1.21560	−0.115903
$111$	−10.5072	−0.997301
$112$	−2.29966	−0.217297
$113$	12.6432	1.18937	0.594687	−	0.803958i	$-0.297276\pi$
$113$	12.6432	1.18937	0.594687	+	0.803958i	$0.297276\pi$
$114$	−11.0055	−1.03076
$115$	0.703515	0.0656032
$116$	−5.42753	−0.503934
$117$	1.95172	0.180437
$118$	4.92228	0.453133
$119$	8.15557	0.747620
$120$	0.660474	0.0602928
$121$	11.6002	1.05456
$122$	8.97884	0.812906
$123$	−21.1267	−1.90493
$124$	−9.45140	−0.848761
$125$	−2.54030	−0.227211
$126$	8.44391	0.752243
$127$	−6.10413	−0.541654	−0.270827	−	0.962628i	$-0.587297\pi$
$127$	−6.10413	−0.541654	−0.270827	+	0.962628i	$0.587297\pi$
$128$	−1.00000	−0.0883883
$129$	−6.06223	−0.533750
$130$	−0.135916	−0.0119207
$131$	−9.12062	−0.796873	−0.398436	−	0.917196i	$-0.630447\pi$
$131$	−9.12062	−0.796873	−0.398436	+	0.917196i	$0.630447\pi$
$132$	−12.2794	−1.06879
$133$	9.79832	0.849622
$134$	2.67472	0.231061
$135$	−0.443714	−0.0381888
$136$	3.54643	0.304104
$137$	−7.38745	−0.631153	−0.315576	−	0.948900i	$-0.602198\pi$
$137$	−7.38745	−0.631153	−0.315576	+	0.948900i	$0.602198\pi$
$138$	7.10659	0.604953
$139$	1.21366	0.102942	0.0514708	−	0.998675i	$-0.483609\pi$
$139$	1.21366	0.102942	0.0514708	+	0.998675i	$0.483609\pi$
$140$	−0.588027	−0.0496974
$141$	−14.0002	−1.17903
$142$	−9.63127	−0.808239
$143$	2.52693	0.211313
$144$	3.67181	0.305984
$145$	−1.38783	−0.115253
$146$	−11.6776	−0.966446
$147$	4.42097	0.364635
$148$	4.06786	0.334376
$149$	−4.77978	−0.391575	−0.195788	−	0.980646i	$-0.562726\pi$
$149$	−4.77978	−0.391575	−0.195788	+	0.980646i	$0.562726\pi$
$150$	−12.7460	−1.04071
$151$	−17.5057	−1.42459	−0.712296	−	0.701879i	$-0.752345\pi$
$151$	−17.5057	−1.42459	−0.712296	+	0.701879i	$0.752345\pi$
$152$	4.26077	0.345594
$153$	−13.0218	−1.05275
$154$	10.9325	0.880966
$155$	−2.41674	−0.194117
$156$	−1.37297	−0.109925
$157$	6.56075	0.523605	0.261802	−	0.965122i	$-0.415683\pi$
$157$	6.56075	0.523605	0.261802	+	0.965122i	$0.415683\pi$
$158$	−8.60030	−0.684203
$159$	−18.9255	−1.50089
$160$	−0.255702	−0.0202150
$161$	−6.32707	−0.498643
$162$	6.53324	0.513300
$163$	−11.2432	−0.880638	−0.440319	−	0.897841i	$-0.645135\pi$
$163$	−11.2432	−0.880638	−0.440319	+	0.897841i	$0.645135\pi$
$164$	8.17919	0.638688
$165$	−3.13987	−0.244439
$166$	−4.68449	−0.363587
$167$	4.60273	0.356170	0.178085	−	0.984015i	$-0.443010\pi$
$167$	4.60273	0.356170	0.178085	+	0.984015i	$0.443010\pi$
$168$	−5.93998	−0.458280
$169$	−12.7175	−0.978266
$170$	0.906828	0.0695506
$171$	−15.6447	−1.19638
$172$	2.34699	0.178956
$173$	−12.4098	−0.943503	−0.471751	−	0.881732i	$-0.656378\pi$
$173$	−12.4098	−0.943503	−0.471751	+	0.881732i	$0.656378\pi$
$174$	−14.0192	−1.06280
$175$	11.3479	0.857823
$176$	4.75397	0.358344
$177$	12.7142	0.955656
$178$	10.3194	0.773474
$179$	3.64873	0.272719	0.136360	−	0.990659i	$-0.456460\pi$
$179$	3.64873	0.272719	0.136360	+	0.990659i	$0.456460\pi$
$180$	0.938889	0.0699807
$181$	−10.9963	−0.817346	−0.408673	−	0.912681i	$-0.634008\pi$
$181$	−10.9963	−0.817346	−0.408673	+	0.912681i	$0.634008\pi$
$182$	1.22236	0.0906077
$183$	23.1922	1.71442
$184$	−2.75131	−0.202829
$185$	1.04016	0.0764740
$186$	−24.4128	−1.79004
$187$	−16.8596	−1.23290
$188$	5.42015	0.395305
$189$	3.99054	0.290269
$190$	1.08949	0.0790397
$191$	18.1965	1.31666	0.658328	−	0.752732i	$-0.271264\pi$
$191$	18.1965	1.31666	0.658328	+	0.752732i	$0.271264\pi$
$192$	−2.58298	−0.186411
$193$	−2.80343	−0.201795	−0.100898	−	0.994897i	$-0.532171\pi$
$193$	−2.80343	−0.201795	−0.100898	+	0.994897i	$0.532171\pi$
$194$	0.192932	0.0138517
$195$	−0.351070	−0.0251406
$196$	−1.71157	−0.122255
$197$	4.29682	0.306135	0.153068	−	0.988216i	$-0.451085\pi$
$197$	4.29682	0.306135	0.153068	+	0.988216i	$0.451085\pi$
$198$	−17.4557	−1.24052
$199$	−7.31217	−0.518346	−0.259173	−	0.965831i	$-0.583450\pi$
$199$	−7.31217	−0.518346	−0.259173	+	0.965831i	$0.583450\pi$
$200$	4.93462	0.348930
$201$	6.90877	0.487307
$202$	18.4028	1.29482
$203$	12.4815	0.876027
$204$	9.16037	0.641354
$205$	2.09144	0.146072
$206$	14.8940	1.03772
$207$	10.1023	0.702158
$208$	0.531542	0.0368558
$209$	−20.2556	−1.40111
$210$	−1.51886	−0.104812
$211$	15.3743	1.05841	0.529204	−	0.848494i	$-0.322490\pi$
$211$	15.3743	1.05841	0.529204	+	0.848494i	$0.322490\pi$
$212$	7.32698	0.503219
$213$	−24.8774	−1.70457
$214$	−9.96286	−0.681047
$215$	0.600129	0.0409285
$216$	1.73528	0.118071
$217$	21.7350	1.47547
$218$	3.23819	0.219318
$219$	−30.1631	−2.03823
$220$	1.21560	0.0819556
$221$	−1.88508	−0.126804
$222$	10.5072	0.705198
$223$	1.80779	0.121059	0.0605294	−	0.998166i	$-0.480721\pi$
$223$	1.80779	0.121059	0.0605294	+	0.998166i	$0.480721\pi$
$224$	2.29966	0.153652
$225$	−18.1190	−1.20793
$226$	−12.6432	−0.841014
$227$	11.3254	0.751695	0.375847	−	0.926682i	$-0.377352\pi$
$227$	11.3254	0.751695	0.375847	+	0.926682i	$0.377352\pi$
$228$	11.0055	0.728858
$229$	−17.0130	−1.12425	−0.562125	−	0.827052i	$-0.690016\pi$
$229$	−17.0130	−1.12425	−0.562125	+	0.827052i	$0.690016\pi$
$230$	−0.703515	−0.0463884
$231$	28.2385	1.85796
$232$	5.42753	0.356335
$233$	−6.58144	−0.431164	−0.215582	−	0.976486i	$-0.569165\pi$
$233$	−6.58144	−0.431164	−0.215582	+	0.976486i	$0.569165\pi$
$234$	−1.95172	−0.127588
$235$	1.38594	0.0904090
$236$	−4.92228	−0.320413
$237$	−22.2145	−1.44298
$238$	−8.15557	−0.528647
$239$	5.75546	0.372290	0.186145	−	0.982522i	$-0.440401\pi$
$239$	5.75546	0.372290	0.186145	+	0.982522i	$0.440401\pi$
$240$	−0.660474	−0.0426334
$241$	1.37750	0.0887326	0.0443663	−	0.999015i	$-0.485873\pi$
$241$	1.37750	0.0887326	0.0443663	+	0.999015i	$0.485873\pi$
$242$	−11.6002	−0.745689
$243$	22.0811	1.41650
$244$	−8.97884	−0.574812
$245$	−0.437653	−0.0279606
$246$	21.1267	1.34699
$247$	−2.26478	−0.144104
$248$	9.45140	0.600165
$249$	−12.1000	−0.766804
$250$	2.54030	0.160663
$251$	29.5871	1.86752	0.933761	−	0.357896i	$-0.116506\pi$
$251$	29.5871	1.86752	0.933761	+	0.357896i	$0.116506\pi$
$252$	−8.44391	−0.531916
$253$	13.0796	0.822310
$254$	6.10413	0.383007
$255$	2.34232	0.146682
$256$	1.00000	0.0625000
$257$	1.01590	0.0633699	0.0316849	−	0.999498i	$-0.489913\pi$
$257$	1.01590	0.0633699	0.0316849	+	0.999498i	$0.489913\pi$
$258$	6.06223	0.377418
$259$	−9.35468	−0.581272
$260$	0.135916	0.00842917
$261$	−19.9289	−1.23357
$262$	9.12062	0.563474
$263$	6.08254	0.375065	0.187533	−	0.982258i	$-0.439951\pi$
$263$	6.08254	0.375065	0.187533	+	0.982258i	$0.439951\pi$
$264$	12.2794	0.755746
$265$	1.87352	0.115090
$266$	−9.79832	−0.600773
$267$	26.6549	1.63126
$268$	−2.67472	−0.163385
$269$	0.519321	0.0316635	0.0158318	−	0.999875i	$-0.494960\pi$
$269$	0.519321	0.0316635	0.0158318	+	0.999875i	$0.494960\pi$
$270$	0.443714	0.0270036
$271$	13.3601	0.811566	0.405783	−	0.913970i	$-0.366999\pi$
$271$	13.3601	0.811566	0.405783	+	0.913970i	$0.366999\pi$
$272$	−3.54643	−0.215034
$273$	3.15735	0.191092
$274$	7.38745	0.446292
$275$	−23.4590	−1.41463
$276$	−7.10659	−0.427767
$277$	22.9534	1.37913	0.689567	−	0.724222i	$-0.257801\pi$
$277$	22.9534	1.37913	0.689567	+	0.724222i	$0.257801\pi$
$278$	−1.21366	−0.0727906
$279$	−34.7038	−2.07766
$280$	0.588027	0.0351413
$281$	12.8812	0.768426	0.384213	−	0.923245i	$-0.374473\pi$
$281$	12.8812	0.768426	0.384213	+	0.923245i	$0.374473\pi$
$282$	14.0002	0.833698
$283$	−6.13463	−0.364666	−0.182333	−	0.983237i	$-0.558365\pi$
$283$	−6.13463	−0.364666	−0.182333	+	0.983237i	$0.558365\pi$
$284$	9.63127	0.571511
$285$	2.81413	0.166695
$286$	−2.52693	−0.149421
$287$	−18.8093	−1.11028
$288$	−3.67181	−0.216364
$289$	−4.42285	−0.260168
$290$	1.38783	0.0814962
$291$	0.498340	0.0292132
$292$	11.6776	0.683380
$293$	−19.8943	−1.16223	−0.581117	−	0.813820i	$-0.697384\pi$
$293$	−19.8943	−1.16223	−0.581117	+	0.813820i	$0.697384\pi$
$294$	−4.42097	−0.257836
$295$	−1.25864	−0.0732807
$296$	−4.06786	−0.236439
$297$	−8.24945	−0.478682
$298$	4.77978	0.276885
$299$	1.46244	0.0845749
$300$	12.7460	0.735893
$301$	−5.39727	−0.311093
$302$	17.5057	1.00734
$303$	47.5342	2.73077
$304$	−4.26077	−0.244372
$305$	−2.29591	−0.131463
$306$	13.0218	0.744407
$307$	3.79527	0.216607	0.108304	−	0.994118i	$-0.465458\pi$
$307$	3.79527	0.216607	0.108304	+	0.994118i	$0.465458\pi$
$308$	−10.9325	−0.622937
$309$	38.4711	2.18854
$310$	2.41674	0.137262
$311$	−10.9816	−0.622708	−0.311354	−	0.950294i	$-0.600782\pi$
$311$	−10.9816	−0.622708	−0.311354	+	0.950294i	$0.600782\pi$
$312$	1.37297	0.0777288
$313$	19.8895	1.12422	0.562110	−	0.827062i	$-0.309990\pi$
$313$	19.8895	1.12422	0.562110	+	0.827062i	$0.309990\pi$
$314$	−6.56075	−0.370244
$315$	−2.15912	−0.121653
$316$	8.60030	0.483805
$317$	−17.8387	−1.00192	−0.500962	−	0.865469i	$-0.667020\pi$
$317$	−17.8387	−1.00192	−0.500962	+	0.865469i	$0.667020\pi$
$318$	18.9255	1.06129
$319$	−25.8023	−1.44465
$320$	0.255702	0.0142942
$321$	−25.7339	−1.43633
$322$	6.32707	0.352594
$323$	15.1105	0.840772
$324$	−6.53324	−0.362958
$325$	−2.62296	−0.145495
$326$	11.2432	0.622705
$327$	8.36420	0.462541
$328$	−8.17919	−0.451620
$329$	−12.4645	−0.687190
$330$	3.13987	0.172844
$331$	20.8272	1.14477	0.572383	−	0.819987i	$-0.306019\pi$
$331$	20.8272	1.14477	0.572383	+	0.819987i	$0.306019\pi$
$332$	4.68449	0.257095
$333$	14.9364	0.818510
$334$	−4.60273	−0.251850
$335$	−0.683932	−0.0373672
$336$	5.93998	0.324053
$337$	−12.4668	−0.679109	−0.339555	−	0.940586i	$-0.610276\pi$
$337$	−12.4668	−0.679109	−0.339555	+	0.940586i	$0.610276\pi$
$338$	12.7175	0.691739
$339$	−32.6572	−1.77370
$340$	−0.906828	−0.0491797
$341$	−44.9316	−2.43318
$342$	15.6447	0.845971
$343$	20.0336	1.08171
$344$	−2.34699	−0.126541
$345$	−1.81717	−0.0978331
$346$	12.4098	0.667157
$347$	−1.21299	−0.0651169	−0.0325585	−	0.999470i	$-0.510366\pi$
$347$	−1.21299	−0.0651169	−0.0325585	+	0.999470i	$0.510366\pi$
$348$	14.0192	0.751510
$349$	21.3604	1.14339	0.571697	−	0.820465i	$-0.306285\pi$
$349$	21.3604	1.14339	0.571697	+	0.820465i	$0.306285\pi$
$350$	−11.3479	−0.606572
$351$	−0.922373	−0.0492326
$352$	−4.75397	−0.253387
$353$	−11.5593	−0.615242	−0.307621	−	0.951509i	$-0.599533\pi$
$353$	−11.5593	−0.615242	−0.307621	+	0.951509i	$0.599533\pi$
$354$	−12.7142	−0.675751
$355$	2.46274	0.130708
$356$	−10.3194	−0.546929
$357$	−21.0657	−1.11492
$358$	−3.64873	−0.192841
$359$	−1.36694	−0.0721443	−0.0360721	−	0.999349i	$-0.511485\pi$
$359$	−1.36694	−0.0721443	−0.0360721	+	0.999349i	$0.511485\pi$
$360$	−0.938889	−0.0494838
$361$	−0.845829	−0.0445173
$362$	10.9963	0.577951
$363$	−29.9631	−1.57266
$364$	−1.22236	−0.0640693
$365$	2.98599	0.156294
$366$	−23.1922	−1.21228
$367$	19.4295	1.01421	0.507106	−	0.861884i	$-0.330715\pi$
$367$	19.4295	1.01421	0.507106	+	0.861884i	$0.330715\pi$
$368$	2.75131	0.143422
$369$	30.0324	1.56343
$370$	−1.04016	−0.0540753
$371$	−16.8496	−0.874785
$372$	24.4128	1.26575
$373$	−7.76266	−0.401935	−0.200968	−	0.979598i	$-0.564409\pi$
$373$	−7.76266	−0.401935	−0.200968	+	0.979598i	$0.564409\pi$
$374$	16.8596	0.871789
$375$	6.56156	0.338837
$376$	−5.42015	−0.279523
$377$	−2.88496	−0.148583
$378$	−3.99054	−0.205251
$379$	−6.97878	−0.358476	−0.179238	−	0.983806i	$-0.557363\pi$
$379$	−6.97878	−0.358476	−0.179238	+	0.983806i	$0.557363\pi$
$380$	−1.08949	−0.0558895
$381$	15.7669	0.807761
$382$	−18.1965	−0.931016
$383$	−27.7128	−1.41606	−0.708029	−	0.706183i	$-0.750416\pi$
$383$	−27.7128	−1.41606	−0.708029	+	0.706183i	$0.750416\pi$
$384$	2.58298	0.131812
$385$	−2.79546	−0.142470
$386$	2.80343	0.142691
$387$	8.61770	0.438062
$388$	−0.192932	−0.00979464
$389$	−31.3993	−1.59201	−0.796003	−	0.605293i	$-0.793056\pi$
$389$	−31.3993	−1.59201	−0.796003	+	0.605293i	$0.793056\pi$
$390$	0.351070	0.0177771
$391$	−9.75732	−0.493449
$392$	1.71157	0.0864475
$393$	23.5584	1.18837
$394$	−4.29682	−0.216470
$395$	2.19911	0.110649
$396$	17.4557	0.877180
$397$	11.9343	0.598966	0.299483	−	0.954102i	$-0.403186\pi$
$397$	11.9343	0.598966	0.299483	+	0.954102i	$0.403186\pi$
$398$	7.31217	0.366526
$399$	−25.3089	−1.26703
$400$	−4.93462	−0.246731
$401$	37.1352	1.85444	0.927221	−	0.374515i	$-0.122191\pi$
$401$	37.1352	1.85444	0.927221	+	0.374515i	$0.122191\pi$
$402$	−6.90877	−0.344578
$403$	−5.02382	−0.250254
$404$	−18.4028	−0.915574
$405$	−1.67056	−0.0830109
$406$	−12.4815	−0.619445
$407$	19.3385	0.958572
$408$	−9.16037	−0.453506
$409$	17.2827	0.854573	0.427286	−	0.904116i	$-0.359470\pi$
$409$	17.2827	0.854573	0.427286	+	0.904116i	$0.359470\pi$
$410$	−2.09144	−0.103289
$411$	19.0817	0.941230
$412$	−14.8940	−0.733777
$413$	11.3196	0.556999
$414$	−10.1023	−0.496501
$415$	1.19783	0.0587993
$416$	−0.531542	−0.0260610
$417$	−3.13487	−0.153515
$418$	20.2556	0.990732
$419$	−19.2029	−0.938125	−0.469062	−	0.883165i	$-0.655408\pi$
$419$	−19.2029	−0.938125	−0.469062	+	0.883165i	$0.655408\pi$
$420$	1.51886	0.0741130
$421$	−15.1998	−0.740792	−0.370396	−	0.928874i	$-0.620778\pi$
$421$	−15.1998	−0.740792	−0.370396	+	0.928874i	$0.620778\pi$
$422$	−15.3743	−0.748408
$423$	19.9018	0.967657
$424$	−7.32698	−0.355830
$425$	17.5003	0.848887
$426$	24.8774	1.20532
$427$	20.6483	0.999240
$428$	9.96286	0.481573
$429$	−6.52703	−0.315128
$430$	−0.600129	−0.0289408
$431$	20.5384	0.989299	0.494650	−	0.869092i	$-0.335296\pi$
$431$	20.5384	0.989299	0.494650	+	0.869092i	$0.335296\pi$
$432$	−1.73528	−0.0834886
$433$	17.2265	0.827851	0.413925	−	0.910311i	$-0.364157\pi$
$433$	17.2265	0.827851	0.413925	+	0.910311i	$0.364157\pi$
$434$	−21.7350	−1.04331
$435$	3.58474	0.171875
$436$	−3.23819	−0.155081
$437$	−11.7227	−0.560773
$438$	30.1631	1.44125
$439$	−16.1789	−0.772176	−0.386088	−	0.922462i	$-0.626174\pi$
$439$	−16.1789	−0.772176	−0.386088	+	0.922462i	$0.626174\pi$
$440$	−1.21560	−0.0579514
$441$	−6.28457	−0.299265
$442$	1.88508	0.0896639
$443$	18.6495	0.886063	0.443032	−	0.896506i	$-0.353903\pi$
$443$	18.6495	0.886063	0.443032	+	0.896506i	$0.353903\pi$
$444$	−10.5072	−0.498650
$445$	−2.63870	−0.125086
$446$	−1.80779	−0.0856015
$447$	12.3461	0.583951
$448$	−2.29966	−0.108649
$449$	5.64613	0.266457	0.133229	−	0.991085i	$-0.457466\pi$
$449$	5.64613	0.266457	0.133229	+	0.991085i	$0.457466\pi$
$450$	18.1190	0.854137
$451$	38.8836	1.83096
$452$	12.6432	0.594687
$453$	45.2169	2.12448
$454$	−11.3254	−0.531529
$455$	−0.312561	−0.0146531
$456$	−11.0055	−0.515380
$457$	−13.8930	−0.649889	−0.324944	−	0.945733i	$-0.605346\pi$
$457$	−13.8930	−0.649889	−0.324944	+	0.945733i	$0.605346\pi$
$458$	17.0130	0.794965
$459$	6.15404	0.287246
$460$	0.703515	0.0328016
$461$	−3.85963	−0.179761	−0.0898805	−	0.995953i	$-0.528649\pi$
$461$	−3.85963	−0.179761	−0.0898805	+	0.995953i	$0.528649\pi$
$462$	−28.2385	−1.31377
$463$	24.0415	1.11730	0.558651	−	0.829403i	$-0.311319\pi$
$463$	24.0415	1.11730	0.558651	+	0.829403i	$0.311319\pi$
$464$	−5.42753	−0.251967
$465$	6.24241	0.289485
$466$	6.58144	0.304879
$467$	−25.9190	−1.19939	−0.599695	−	0.800229i	$-0.704711\pi$
$467$	−25.9190	−1.19939	−0.599695	+	0.800229i	$0.704711\pi$
$468$	1.95172	0.0902184
$469$	6.15095	0.284024
$470$	−1.38594	−0.0639288
$471$	−16.9463	−0.780845
$472$	4.92228	0.226566
$473$	11.1575	0.513022
$474$	22.2145	1.02034
$475$	21.0253	0.964706
$476$	8.15557	0.373810
$477$	26.9033	1.23182
$478$	−5.75546	−0.263249
$479$	−4.75090	−0.217074	−0.108537	−	0.994092i	$-0.534617\pi$
$479$	−4.75090	−0.217074	−0.108537	+	0.994092i	$0.534617\pi$
$480$	0.660474	0.0301464
$481$	2.16224	0.0985896
$482$	−1.37750	−0.0627434
$483$	16.3427	0.743620
$484$	11.6002	0.527281
$485$	−0.0493331	−0.00224010
$486$	−22.0811	−1.00162
$487$	−2.52316	−0.114335	−0.0571677	−	0.998365i	$-0.518207\pi$
$487$	−2.52316	−0.114335	−0.0571677	+	0.998365i	$0.518207\pi$
$488$	8.97884	0.406453
$489$	29.0411	1.31328
$490$	0.437653	0.0197711
$491$	−28.9562	−1.30677	−0.653387	−	0.757024i	$-0.726652\pi$
$491$	−28.9562	−1.30677	−0.653387	+	0.757024i	$0.726652\pi$
$492$	−21.1267	−0.952467
$493$	19.2483	0.866902
$494$	2.26478	0.101897
$495$	4.46345	0.200617
$496$	−9.45140	−0.424380
$497$	−22.1486	−0.993502
$498$	12.1000	0.542212
$499$	−1.73702	−0.0777598	−0.0388799	−	0.999244i	$-0.512379\pi$
$499$	−1.73702	−0.0777598	−0.0388799	+	0.999244i	$0.512379\pi$
$500$	−2.54030	−0.113606
$501$	−11.8888	−0.531152
$502$	−29.5871	−1.32054
$503$	30.5444	1.36191	0.680954	−	0.732327i	$-0.261566\pi$
$503$	30.5444	1.36191	0.680954	+	0.732327i	$0.261566\pi$
$504$	8.44391	0.376122
$505$	−4.70563	−0.209398
$506$	−13.0796	−0.581461
$507$	32.8490	1.45888
$508$	−6.10413	−0.270827
$509$	28.3632	1.25718	0.628588	−	0.777739i	$-0.283633\pi$
$509$	28.3632	1.25718	0.628588	+	0.777739i	$0.283633\pi$
$510$	−2.34232	−0.103720
$511$	−26.8545	−1.18797
$512$	−1.00000	−0.0441942
$513$	7.39362	0.326436
$514$	−1.01590	−0.0448093
$515$	−3.80844	−0.167820
$516$	−6.06223	−0.266875
$517$	25.7672	1.13324
$518$	9.35468	0.411021
$519$	32.0544	1.40703
$520$	−0.135916	−0.00596033
$521$	0.00673097	0.000294889 0	0.000147445	−	1.00000i	$-0.499953\pi$
$521$	0.00673097	0.000294889 0	0.000147445		1.00000i	$0.499953\pi$
$522$	19.9289	0.872263
$523$	−4.00317	−0.175046	−0.0875232	−	0.996162i	$-0.527895\pi$
$523$	−4.00317	−0.175046	−0.0875232	+	0.996162i	$0.527895\pi$
$524$	−9.12062	−0.398436
$525$	−29.3115	−1.27926
$526$	−6.08254	−0.265211
$527$	33.5187	1.46010
$528$	−12.2794	−0.534393
$529$	−15.4303	−0.670882
$530$	−1.87352	−0.0813807
$531$	−18.0737	−0.784331
$532$	9.79832	0.424811
$533$	4.34758	0.188315
$534$	−26.6549	−1.15347
$535$	2.54752	0.110139
$536$	2.67472	0.115530
$537$	−9.42462	−0.406702
$538$	−0.519321	−0.0223895
$539$	−8.13676	−0.350475
$540$	−0.443714	−0.0190944
$541$	32.0008	1.37582	0.687910	−	0.725796i	$-0.258528\pi$
$541$	32.0008	1.37582	0.687910	+	0.725796i	$0.258528\pi$
$542$	−13.3601	−0.573864
$543$	28.4032	1.21890
$544$	3.54643	0.152052
$545$	−0.828011	−0.0354681
$546$	−3.15735	−0.135122
$547$	4.36019	0.186428	0.0932142	−	0.995646i	$-0.470286\pi$
$547$	4.36019	0.186428	0.0932142	+	0.995646i	$0.470286\pi$
$548$	−7.38745	−0.315576
$549$	−32.9686	−1.40707
$550$	23.4590	1.00030
$551$	23.1255	0.985178
$552$	7.10659	0.302477
$553$	−19.7778	−0.841036
$554$	−22.9534	−0.975194
$555$	−2.68672	−0.114045
$556$	1.21366	0.0514708
$557$	30.6946	1.30057	0.650286	−	0.759690i	$-0.274649\pi$
$557$	30.6946	1.30057	0.650286	+	0.759690i	$0.274649\pi$
$558$	34.7038	1.46913
$559$	1.24752	0.0527646
$560$	−0.588027	−0.0248487
$561$	43.5481	1.83860
$562$	−12.8812	−0.543359
$563$	33.4323	1.40900	0.704502	−	0.709702i	$-0.251170\pi$
$563$	33.4323	1.40900	0.704502	+	0.709702i	$0.251170\pi$
$564$	−14.0002	−0.589514
$565$	3.23289	0.136009
$566$	6.13463	0.257858
$567$	15.0242	0.630958
$568$	−9.63127	−0.404119
$569$	21.9206	0.918960	0.459480	−	0.888188i	$-0.348036\pi$
$569$	21.9206	0.918960	0.459480	+	0.888188i	$0.348036\pi$
$570$	−2.81413	−0.117871
$571$	38.7483	1.62157	0.810783	−	0.585347i	$-0.199042\pi$
$571$	38.7483	1.62157	0.810783	+	0.585347i	$0.199042\pi$
$572$	2.52693	0.105656
$573$	−47.0014	−1.96351
$574$	18.8093	0.785087
$575$	−13.5767	−0.566186
$576$	3.67181	0.152992
$577$	−12.7225	−0.529643	−0.264821	−	0.964298i	$-0.585313\pi$
$577$	−12.7225	−0.529643	−0.264821	+	0.964298i	$0.585313\pi$
$578$	4.42285	0.183966
$579$	7.24121	0.300934
$580$	−1.38783	−0.0576265
$581$	−10.7727	−0.446928
$582$	−0.498340	−0.0206569
$583$	34.8322	1.44260
$584$	−11.6776	−0.483223
$585$	0.499059	0.0206336
$586$	19.8943	0.821824
$587$	11.2868	0.465857	0.232928	−	0.972494i	$-0.425169\pi$
$587$	11.2868	0.465857	0.232928	+	0.972494i	$0.425169\pi$
$588$	4.42097	0.182318
$589$	40.2703	1.65931
$590$	1.25864	0.0518173
$591$	−11.0986	−0.456536
$592$	4.06786	0.167188
$593$	−41.8544	−1.71876	−0.859378	−	0.511341i	$-0.829149\pi$
$593$	−41.8544	−1.71876	−0.859378	+	0.511341i	$0.829149\pi$
$594$	8.24945	0.338479
$595$	2.08540	0.0854929
$596$	−4.77978	−0.195788
$597$	18.8872	0.773003
$598$	−1.46244	−0.0598035
$599$	16.3718	0.668934	0.334467	−	0.942407i	$-0.391444\pi$
$599$	16.3718	0.668934	0.334467	+	0.942407i	$0.391444\pi$
$600$	−12.7460	−0.520355
$601$	−4.07972	−0.166415	−0.0832076	−	0.996532i	$-0.526516\pi$
$601$	−4.07972	−0.166415	−0.0832076	+	0.996532i	$0.526516\pi$
$602$	5.39727	0.219976
$603$	−9.82108	−0.399945
$604$	−17.5057	−0.712296
$605$	2.96619	0.120593
$606$	−47.5342	−1.93094
$607$	−0.0803433	−0.00326103	−0.00163052	−	0.999999i	$-0.500519\pi$
$607$	−0.0803433	−0.00326103	−0.00163052	+	0.999999i	$0.500519\pi$
$608$	4.26077	0.172797
$609$	−32.2394	−1.30641
$610$	2.29591	0.0929586
$611$	2.88104	0.116554
$612$	−13.0218	−0.526375
$613$	−9.45182	−0.381755	−0.190878	−	0.981614i	$-0.561133\pi$
$613$	−9.45182	−0.381755	−0.190878	+	0.981614i	$0.561133\pi$
$614$	−3.79527	−0.153165
$615$	−5.40215	−0.217836
$616$	10.9325	0.440483
$617$	4.30842	0.173451	0.0867253	−	0.996232i	$-0.472360\pi$
$617$	4.30842	0.173451	0.0867253	+	0.996232i	$0.472360\pi$
$618$	−38.4711	−1.54753
$619$	8.85654	0.355974	0.177987	−	0.984033i	$-0.443041\pi$
$619$	8.85654	0.355974	0.177987	+	0.984033i	$0.443041\pi$
$620$	−2.41674	−0.0970587
$621$	−4.77429	−0.191586
$622$	10.9816	0.440321
$623$	23.7312	0.950769
$624$	−1.37297	−0.0549626
$625$	24.0235	0.960941
$626$	−19.8895	−0.794944
$627$	52.3198	2.08945
$628$	6.56075	0.261802
$629$	−14.4264	−0.575217
$630$	2.15912	0.0860216
$631$	36.2875	1.44458	0.722292	−	0.691588i	$-0.243089\pi$
$631$	36.2875	1.44458	0.722292	+	0.691588i	$0.243089\pi$
$632$	−8.60030	−0.342102
$633$	−39.7115	−1.57839
$634$	17.8387	0.708467
$635$	−1.56084	−0.0619399
$636$	−18.9255	−0.750444
$637$	−0.909773	−0.0360465
$638$	25.8023	1.02152
$639$	35.3642	1.39899
$640$	−0.255702	−0.0101075
$641$	−14.7369	−0.582072	−0.291036	−	0.956712i	$-0.594000\pi$
$641$	−14.7369	−0.582072	−0.291036	+	0.956712i	$0.594000\pi$
$642$	25.7339	1.01564
$643$	−14.3162	−0.564577	−0.282289	−	0.959330i	$-0.591094\pi$
$643$	−14.3162	−0.564577	−0.282289	+	0.959330i	$0.591094\pi$
$644$	−6.32707	−0.249322
$645$	−1.55013	−0.0610361
$646$	−15.1105	−0.594515
$647$	43.7432	1.71972	0.859861	−	0.510529i	$-0.170550\pi$
$647$	43.7432	1.71972	0.859861	+	0.510529i	$0.170550\pi$
$648$	6.53324	0.256650
$649$	−23.4004	−0.918545
$650$	2.62296	0.102881
$651$	−56.1411	−2.20035
$652$	−11.2432	−0.440319
$653$	43.5572	1.70452	0.852262	−	0.523116i	$-0.175230\pi$
$653$	43.5572	1.70452	0.852262	+	0.523116i	$0.175230\pi$
$654$	−8.36420	−0.327066
$655$	−2.33216	−0.0911251
$656$	8.17919	0.319344
$657$	42.8780	1.67283
$658$	12.4645	0.485917
$659$	−36.1525	−1.40830	−0.704151	−	0.710050i	$-0.748672\pi$
$659$	−36.1525	−1.40830	−0.704151	+	0.710050i	$0.748672\pi$
$660$	−3.13987	−0.122219
$661$	17.4547	0.678911	0.339455	−	0.940622i	$-0.389757\pi$
$661$	17.4547	0.678911	0.339455	+	0.940622i	$0.389757\pi$
$662$	−20.8272	−0.809471
$663$	4.86912	0.189101
$664$	−4.68449	−0.181793
$665$	2.50545	0.0971571
$666$	−14.9364	−0.578774
$667$	−14.9328	−0.578201
$668$	4.60273	0.178085
$669$	−4.66950	−0.180533
$670$	0.683932	0.0264226
$671$	−42.6851	−1.64784
$672$	−5.93998	−0.229140
$673$	32.5652	1.25530	0.627649	−	0.778497i	$-0.284017\pi$
$673$	32.5652	1.25530	0.627649	+	0.778497i	$0.284017\pi$
$674$	12.4668	0.480203
$675$	8.56293	0.329587
$676$	−12.7175	−0.489133
$677$	18.7680	0.721314	0.360657	−	0.932699i	$-0.382553\pi$
$677$	18.7680	0.721314	0.360657	+	0.932699i	$0.382553\pi$
$678$	32.6572	1.25419
$679$	0.443677	0.0170268
$680$	0.906828	0.0347753
$681$	−29.2534	−1.12099
$682$	44.9316	1.72052
$683$	−6.68495	−0.255793	−0.127896	−	0.991788i	$-0.540822\pi$
$683$	−6.68495	−0.255793	−0.127896	+	0.991788i	$0.540822\pi$
$684$	−15.6447	−0.598192
$685$	−1.88899	−0.0721744
$686$	−20.0336	−0.764888
$687$	43.9443	1.67658
$688$	2.34699	0.0894781
$689$	3.89460	0.148372
$690$	1.81717	0.0691785
$691$	25.2257	0.959631	0.479815	−	0.877369i	$-0.340704\pi$
$691$	25.2257	0.959631	0.479815	+	0.877369i	$0.340704\pi$
$692$	−12.4098	−0.471751
$693$	−40.1421	−1.52487
$694$	1.21299	0.0460446
$695$	0.310336	0.0117717
$696$	−14.0192	−0.531398
$697$	−29.0069	−1.09872
$698$	−21.3604	−0.808502
$699$	16.9998	0.642990
$700$	11.3479	0.428911
$701$	5.00180	0.188915	0.0944577	−	0.995529i	$-0.469888\pi$
$701$	5.00180	0.188915	0.0944577	+	0.995529i	$0.469888\pi$
$702$	0.922373	0.0348127
$703$	−17.3322	−0.653697
$704$	4.75397	0.179172
$705$	−3.57987	−0.134826
$706$	11.5593	0.435042
$707$	42.3202	1.59161
$708$	12.7142	0.477828
$709$	−21.0843	−0.791836	−0.395918	−	0.918286i	$-0.629574\pi$
$709$	−21.0843	−0.791836	−0.395918	+	0.918286i	$0.629574\pi$
$710$	−2.46274	−0.0924248
$711$	31.5787	1.18429
$712$	10.3194	0.386737
$713$	−26.0037	−0.973848
$714$	21.0657	0.788364
$715$	0.646142	0.0241643
$716$	3.64873	0.136360
$717$	−14.8663	−0.555191
$718$	1.36694	0.0510137
$719$	10.8353	0.404088	0.202044	−	0.979376i	$-0.435242\pi$
$719$	10.8353	0.404088	0.202044	+	0.979376i	$0.435242\pi$
$720$	0.938889	0.0349903
$721$	34.2512	1.27558
$722$	0.845829	0.0314785
$723$	−3.55806	−0.132326
$724$	−10.9963	−0.408673
$725$	26.7828	0.994688
$726$	29.9631	1.11204
$727$	30.5210	1.13196	0.565981	−	0.824418i	$-0.308497\pi$
$727$	30.5210	1.13196	0.565981	+	0.824418i	$0.308497\pi$
$728$	1.22236	0.0453039
$729$	−37.4354	−1.38650
$730$	−2.98599	−0.110516
$731$	−8.32342	−0.307853
$732$	23.1922	0.857209
$733$	−24.1276	−0.891174	−0.445587	−	0.895239i	$-0.647005\pi$
$733$	−24.1276	−0.891174	−0.445587	+	0.895239i	$0.647005\pi$
$734$	−19.4295	−0.717157
$735$	1.13045	0.0416973
$736$	−2.75131	−0.101415
$737$	−12.7155	−0.468383
$738$	−30.0324	−1.10551
$739$	37.1991	1.36839	0.684195	−	0.729299i	$-0.260154\pi$
$739$	37.1991	1.36839	0.684195	+	0.729299i	$0.260154\pi$
$740$	1.04016	0.0382370
$741$	5.84989	0.214901
$742$	16.8496	0.618567
$743$	1.62668	0.0596772	0.0298386	−	0.999555i	$-0.490501\pi$
$743$	1.62668	0.0596772	0.0298386	+	0.999555i	$0.490501\pi$
$744$	−24.4128	−0.895018
$745$	−1.22220	−0.0447779
$746$	7.76266	0.284211
$747$	17.2006	0.629335
$748$	−16.8596	−0.616448
$749$	−22.9112	−0.837156
$750$	−6.56156	−0.239594
$751$	13.2496	0.483485	0.241743	−	0.970340i	$-0.422281\pi$
$751$	13.2496	0.483485	0.241743	+	0.970340i	$0.422281\pi$
$752$	5.42015	0.197653
$753$	−76.4231	−2.78501
$754$	2.88496	0.105064
$755$	−4.47624	−0.162907
$756$	3.99054	0.145135
$757$	−41.4521	−1.50660	−0.753301	−	0.657675i	$-0.771540\pi$
$757$	−41.4521	−1.50660	−0.753301	+	0.657675i	$0.771540\pi$
$758$	6.97878	0.253481
$759$	−33.7845	−1.22630
$760$	1.08949	0.0395199
$761$	−9.70505	−0.351808	−0.175904	−	0.984407i	$-0.556285\pi$
$761$	−9.70505	−0.351808	−0.175904	+	0.984407i	$0.556285\pi$
$762$	−15.7669	−0.571173
$763$	7.44673	0.269590
$764$	18.1965	0.658328
$765$	−3.32970	−0.120386
$766$	27.7128	1.00130
$767$	−2.61640	−0.0944727
$768$	−2.58298	−0.0932054
$769$	−3.80266	−0.137127	−0.0685637	−	0.997647i	$-0.521842\pi$
$769$	−3.80266	−0.137127	−0.0685637	+	0.997647i	$0.521842\pi$
$770$	2.79546	0.100741
$771$	−2.62404	−0.0945027
$772$	−2.80343	−0.100898
$773$	−52.5078	−1.88857	−0.944287	−	0.329123i	$-0.893247\pi$
$773$	−52.5078	−1.88857	−0.944287	+	0.329123i	$0.893247\pi$
$774$	−8.61770	−0.309757
$775$	46.6390	1.67532
$776$	0.192932	0.00692585
$777$	24.1630	0.866843
$778$	31.3993	1.12572
$779$	−34.8497	−1.24862
$780$	−0.351070	−0.0125703
$781$	45.7867	1.63838
$782$	9.75732	0.348921
$783$	9.41827	0.336582
$784$	−1.71157	−0.0611276
$785$	1.67760	0.0598760
$786$	−23.5584	−0.840302
$787$	−30.2766	−1.07924	−0.539621	−	0.841908i	$-0.681433\pi$
$787$	−30.2766	−1.07924	−0.539621	+	0.841908i	$0.681433\pi$
$788$	4.29682	0.153068
$789$	−15.7111	−0.559330
$790$	−2.19911	−0.0782410
$791$	−29.0751	−1.03379
$792$	−17.4557	−0.620260
$793$	−4.77263	−0.169481
$794$	−11.9343	−0.423533
$795$	−4.83928	−0.171632
$796$	−7.31217	−0.259173
$797$	16.8208	0.595822	0.297911	−	0.954594i	$-0.403710\pi$
$797$	16.8208	0.595822	0.297911	+	0.954594i	$0.403710\pi$
$798$	25.3089	0.895925
$799$	−19.2222	−0.680032
$800$	4.93462	0.174465
$801$	−37.8910	−1.33881
$802$	−37.1352	−1.31129
$803$	55.5149	1.95908
$804$	6.90877	0.243653
$805$	−1.61784	−0.0570215
$806$	5.02382	0.176956
$807$	−1.34140	−0.0472194
$808$	18.4028	0.647408
$809$	−1.63854	−0.0576080	−0.0288040	−	0.999585i	$-0.509170\pi$
$809$	−1.63854	−0.0576080	−0.0288040	+	0.999585i	$0.509170\pi$
$810$	1.67056	0.0586975
$811$	−25.6862	−0.901964	−0.450982	−	0.892533i	$-0.648926\pi$
$811$	−25.6862	−0.901964	−0.450982	+	0.892533i	$0.648926\pi$
$812$	12.4815	0.438014
$813$	−34.5088	−1.21028
$814$	−19.3385	−0.677813
$815$	−2.87492	−0.100704
$816$	9.16037	0.320677
$817$	−9.99998	−0.349855
$818$	−17.2827	−0.604274
$819$	−4.48829	−0.156834
$820$	2.09144	0.0730361
$821$	−42.6847	−1.48971	−0.744853	−	0.667229i	$-0.767480\pi$
$821$	−42.6847	−1.48971	−0.744853	+	0.667229i	$0.767480\pi$
$822$	−19.0817	−0.665550
$823$	40.5864	1.41475	0.707377	−	0.706836i	$-0.249878\pi$
$823$	40.5864	1.41475	0.707377	+	0.706836i	$0.249878\pi$
$824$	14.8940	0.518859
$825$	60.5942	2.10962
$826$	−11.3196	−0.393858
$827$	44.6760	1.55354	0.776768	−	0.629787i	$-0.216858\pi$
$827$	44.6760	1.55354	0.776768	+	0.629787i	$0.216858\pi$
$828$	10.1023	0.351079
$829$	10.8787	0.377833	0.188916	−	0.981993i	$-0.439502\pi$
$829$	10.8787	0.377833	0.188916	+	0.981993i	$0.439502\pi$
$830$	−1.19783	−0.0415774
$831$	−59.2882	−2.05668
$832$	0.531542	0.0184279
$833$	6.06997	0.210312
$834$	3.13487	0.108552
$835$	1.17693	0.0407293
$836$	−20.2556	−0.700553
$837$	16.4008	0.566895
$838$	19.2029	0.663354
$839$	18.9961	0.655817	0.327909	−	0.944709i	$-0.393656\pi$
$839$	18.9961	0.655817	0.327909	+	0.944709i	$0.393656\pi$
$840$	−1.51886	−0.0524058
$841$	0.458096	0.0157964
$842$	15.1998	0.523819
$843$	−33.2718	−1.14594
$844$	15.3743	0.529204
$845$	−3.25188	−0.111868
$846$	−19.9018	−0.684237
$847$	−26.6765	−0.916615
$848$	7.32698	0.251610
$849$	15.8457	0.543822
$850$	−17.5003	−0.600254
$851$	11.1919	0.383655
$852$	−24.8774	−0.852287
$853$	51.7180	1.77079	0.885395	−	0.464840i	$-0.153888\pi$
$853$	51.7180	1.77079	0.885395	+	0.464840i	$0.153888\pi$
$854$	−20.6483	−0.706569
$855$	−4.00039	−0.136811
$856$	−9.96286	−0.340524
$857$	6.16112	0.210460	0.105230	−	0.994448i	$-0.466442\pi$
$857$	6.16112	0.210460	0.105230	+	0.994448i	$0.466442\pi$
$858$	6.52703	0.222829
$859$	38.8222	1.32459	0.662297	−	0.749241i	$-0.269582\pi$
$859$	38.8222	1.32459	0.662297	+	0.749241i	$0.269582\pi$
$860$	0.600129	0.0204642
$861$	48.5843	1.65575
$862$	−20.5384	−0.699540
$863$	−8.12238	−0.276489	−0.138245	−	0.990398i	$-0.544146\pi$
$863$	−8.12238	−0.276489	−0.138245	+	0.990398i	$0.544146\pi$
$864$	1.73528	0.0590353
$865$	−3.17322	−0.107893
$866$	−17.2265	−0.585379
$867$	11.4242	0.387985
$868$	21.7350	0.737734
$869$	40.8856	1.38695
$870$	−3.58474	−0.121534
$871$	−1.42173	−0.0481734
$872$	3.23819	0.109659
$873$	−0.708410	−0.0239760
$874$	11.7227	0.396526
$875$	5.84182	0.197490
$876$	−30.1631	−1.01912
$877$	−6.79906	−0.229588	−0.114794	−	0.993389i	$-0.536621\pi$
$877$	−6.79906	−0.229588	−0.114794	+	0.993389i	$0.536621\pi$
$878$	16.1789	0.546011
$879$	51.3866	1.73323
$880$	1.21560	0.0409778
$881$	38.6729	1.30292	0.651462	−	0.758681i	$-0.274156\pi$
$881$	38.6729	1.30292	0.651462	+	0.758681i	$0.274156\pi$
$882$	6.28457	0.211613
$883$	17.4252	0.586406	0.293203	−	0.956050i	$-0.405279\pi$
$883$	17.4252	0.586406	0.293203	+	0.956050i	$0.405279\pi$
$884$	−1.88508	−0.0634019
$885$	3.25104	0.109283
$886$	−18.6495	−0.626541
$887$	17.4779	0.586851	0.293425	−	0.955982i	$-0.405205\pi$
$887$	17.4779	0.586851	0.293425	+	0.955982i	$0.405205\pi$
$888$	10.5072	0.352599
$889$	14.0374	0.470800
$890$	2.63870	0.0884494
$891$	−31.0588	−1.04051
$892$	1.80779	0.0605294
$893$	−23.0940	−0.772812
$894$	−12.3461	−0.412916
$895$	0.932988	0.0311863
$896$	2.29966	0.0768262
$897$	−3.77745	−0.126125
$898$	−5.64613	−0.188414
$899$	51.2978	1.71088
$900$	−18.1190	−0.603966
$901$	−25.9846	−0.865673
$902$	−38.8836	−1.29468
$903$	13.9411	0.463930
$904$	−12.6432	−0.420507
$905$	−2.81177	−0.0934663
$906$	−45.2169	−1.50223
$907$	7.99327	0.265412	0.132706	−	0.991155i	$-0.457633\pi$
$907$	7.99327	0.265412	0.132706	+	0.991155i	$0.457633\pi$
$908$	11.3254	0.375847
$909$	−67.5716	−2.24121
$910$	0.312561	0.0103613
$911$	−13.5268	−0.448161	−0.224081	−	0.974571i	$-0.571938\pi$
$911$	−13.5268	−0.448161	−0.224081	+	0.974571i	$0.571938\pi$
$912$	11.0055	0.364429
$913$	22.2699	0.737026
$914$	13.8930	0.459541
$915$	5.93030	0.196050
$916$	−17.0130	−0.562125
$917$	20.9743	0.692633
$918$	−6.15404	−0.203113
$919$	−3.70903	−0.122350	−0.0611748	−	0.998127i	$-0.519485\pi$
$919$	−3.70903	−0.122350	−0.0611748	+	0.998127i	$0.519485\pi$
$920$	−0.703515	−0.0231942
$921$	−9.80312	−0.323024
$922$	3.85963	0.127110
$923$	5.11943	0.168508
$924$	28.2385	0.928978
$925$	−20.0733	−0.660007
$926$	−24.0415	−0.790052
$927$	−54.6881	−1.79619
$928$	5.42753	0.178167
$929$	11.3952	0.373863	0.186931	−	0.982373i	$-0.440146\pi$
$929$	11.3952	0.373863	0.186931	+	0.982373i	$0.440146\pi$
$930$	−6.24241	−0.204697
$931$	7.29262	0.239006
$932$	−6.58144	−0.215582
$933$	28.3652	0.928636
$934$	25.9190	0.848097
$935$	−4.31103	−0.140986
$936$	−1.95172	−0.0637940
$937$	42.6986	1.39490	0.697451	−	0.716632i	$-0.254317\pi$
$937$	42.6986	1.39490	0.697451	+	0.716632i	$0.254317\pi$
$938$	−6.15095	−0.200836
$939$	−51.3743	−1.67654
$940$	1.38594	0.0452045
$941$	52.0091	1.69545	0.847724	−	0.530438i	$-0.177973\pi$
$941$	52.0091	1.69545	0.847724	+	0.530438i	$0.177973\pi$
$942$	16.9463	0.552141
$943$	22.5035	0.732815
$944$	−4.92228	−0.160207
$945$	1.02039	0.0331933
$946$	−11.1575	−0.362762
$947$	20.8192	0.676533	0.338266	−	0.941050i	$-0.390159\pi$
$947$	20.8192	0.676533	0.338266	+	0.941050i	$0.390159\pi$
$948$	−22.2145	−0.721492
$949$	6.20714	0.201492
$950$	−21.0253	−0.682150
$951$	46.0772	1.49416
$952$	−8.15557	−0.264324
$953$	9.93499	0.321826	0.160913	−	0.986969i	$-0.448556\pi$
$953$	9.93499	0.321826	0.160913	+	0.986969i	$0.448556\pi$
$954$	−26.9033	−0.871027
$955$	4.65289	0.150564
$956$	5.75546	0.186145
$957$	66.6469	2.15439
$958$	4.75090	0.153495
$959$	16.9886	0.548591
$960$	−0.660474	−0.0213167
$961$	58.3290	1.88158
$962$	−2.16224	−0.0697133
$963$	36.5817	1.17883
$964$	1.37750	0.0443663
$965$	−0.716842	−0.0230760
$966$	−16.3427	−0.525819
$967$	31.7654	1.02151	0.510753	−	0.859728i	$-0.329367\pi$
$967$	31.7654	1.02151	0.510753	+	0.859728i	$0.329367\pi$
$968$	−11.6002	−0.372844
$969$	−39.0302	−1.25383
$970$	0.0493331	0.00158399
$971$	0.510723	0.0163899	0.00819495	−	0.999966i	$-0.497391\pi$
$971$	0.510723	0.0163899	0.00819495	+	0.999966i	$0.497391\pi$
$972$	22.0811	0.708251
$973$	−2.79101	−0.0894756
$974$	2.52316	0.0808474
$975$	6.77506	0.216975
$976$	−8.97884	−0.287406
$977$	19.6854	0.629793	0.314896	−	0.949126i	$-0.398030\pi$
$977$	19.6854	0.629793	0.314896	+	0.949126i	$0.398030\pi$
$978$	−29.0411	−0.928632
$979$	−49.0582	−1.56791
$980$	−0.437653	−0.0139803
$981$	−11.8900	−0.379619
$982$	28.9562	0.924028
$983$	1.32746	0.0423395	0.0211698	−	0.999776i	$-0.493261\pi$
$983$	1.32746	0.0423395	0.0211698	+	0.999776i	$0.493261\pi$
$984$	21.1267	0.673496
$985$	1.09870	0.0350076
$986$	−19.2483	−0.612992
$987$	32.1956	1.02480
$988$	−2.26478	−0.0720522
$989$	6.45729	0.205330
$990$	−4.46345	−0.141858
$991$	16.3980	0.520900	0.260450	−	0.965487i	$-0.416129\pi$
$991$	16.3980	0.520900	0.260450	+	0.965487i	$0.416129\pi$
$992$	9.45140	0.300082
$993$	−53.7963	−1.70717
$994$	22.1486	0.702512
$995$	−1.86974	−0.0592746
$996$	−12.1000	−0.383402
$997$	−7.87930	−0.249540	−0.124770	−	0.992186i	$-0.539819\pi$
$997$	−7.87930	−0.249540	−0.124770	+	0.992186i	$0.539819\pi$
$998$	1.73702	0.0549845
$999$	−7.05886	−0.223333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.8	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.8	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.58298 q^{3} +1.00000 q^{4} +0.255702 q^{5} +2.58298 q^{6} -2.29966 q^{7} -1.00000 q^{8} +3.67181 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.58298 q^{3} +1.00000 q^{4} +0.255702 q^{5} +2.58298 q^{6} -2.29966 q^{7} -1.00000 q^{8} +3.67181 q^{9} -0.255702 q^{10} +4.75397 q^{11} -2.58298 q^{12} +0.531542 q^{13} +2.29966 q^{14} -0.660474 q^{15} +1.00000 q^{16} -3.54643 q^{17} -3.67181 q^{18} -4.26077 q^{19} +0.255702 q^{20} +5.93998 q^{21} -4.75397 q^{22} +2.75131 q^{23} +2.58298 q^{24} -4.93462 q^{25} -0.531542 q^{26} -1.73528 q^{27} -2.29966 q^{28} -5.42753 q^{29} +0.660474 q^{30} -9.45140 q^{31} -1.00000 q^{32} -12.2794 q^{33} +3.54643 q^{34} -0.588027 q^{35} +3.67181 q^{36} +4.06786 q^{37} +4.26077 q^{38} -1.37297 q^{39} -0.255702 q^{40} +8.17919 q^{41} -5.93998 q^{42} +2.34699 q^{43} +4.75397 q^{44} +0.938889 q^{45} -2.75131 q^{46} +5.42015 q^{47} -2.58298 q^{48} -1.71157 q^{49} +4.93462 q^{50} +9.16037 q^{51} +0.531542 q^{52} +7.32698 q^{53} +1.73528 q^{54} +1.21560 q^{55} +2.29966 q^{56} +11.0055 q^{57} +5.42753 q^{58} -4.92228 q^{59} -0.660474 q^{60} -8.97884 q^{61} +9.45140 q^{62} -8.44391 q^{63} +1.00000 q^{64} +0.135916 q^{65} +12.2794 q^{66} -2.67472 q^{67} -3.54643 q^{68} -7.10659 q^{69} +0.588027 q^{70} +9.63127 q^{71} -3.67181 q^{72} +11.6776 q^{73} -4.06786 q^{74} +12.7460 q^{75} -4.26077 q^{76} -10.9325 q^{77} +1.37297 q^{78} +8.60030 q^{79} +0.255702 q^{80} -6.53324 q^{81} -8.17919 q^{82} +4.68449 q^{83} +5.93998 q^{84} -0.906828 q^{85} -2.34699 q^{86} +14.0192 q^{87} -4.75397 q^{88} -10.3194 q^{89} -0.938889 q^{90} -1.22236 q^{91} +2.75131 q^{92} +24.4128 q^{93} -5.42015 q^{94} -1.08949 q^{95} +2.58298 q^{96} -0.192932 q^{97} +1.71157 q^{98} +17.4557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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