LMFDB - Embedded newform 4002.2.a.bj.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 26x^{6} + 4x^{5} + 209x^{4} + 113x^{3} - 436x^{2} - 360x - 16 $ x^8 - x^7 - 26x^6 + 4x^5 + 209x^4 + 113x^3 - 436x^2 - 360x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.869754$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.869754 q^{5} -1.00000 q^{6} +3.97135 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.869754 q^{5} -1.00000 q^{6} +3.97135 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.869754 q^{10} +3.41188 q^{11} -1.00000 q^{12} +3.88600 q^{13} +3.97135 q^{14} +0.869754 q^{15} +1.00000 q^{16} -1.10357 q^{17} +1.00000 q^{18} +3.97135 q^{19} -0.869754 q^{20} -3.97135 q^{21} +3.41188 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.24353 q^{25} +3.88600 q^{26} -1.00000 q^{27} +3.97135 q^{28} +1.00000 q^{29} +0.869754 q^{30} +10.7142 q^{31} +1.00000 q^{32} -3.41188 q^{33} -1.10357 q^{34} -3.45410 q^{35} +1.00000 q^{36} -10.8290 q^{37} +3.97135 q^{38} -3.88600 q^{39} -0.869754 q^{40} -12.2743 q^{41} -3.97135 q^{42} +8.48596 q^{43} +3.41188 q^{44} -0.869754 q^{45} +1.00000 q^{46} +6.30315 q^{47} -1.00000 q^{48} +8.77162 q^{49} -4.24353 q^{50} +1.10357 q^{51} +3.88600 q^{52} +3.72896 q^{53} -1.00000 q^{54} -2.96750 q^{55} +3.97135 q^{56} -3.97135 q^{57} +1.00000 q^{58} -9.89660 q^{59} +0.869754 q^{60} +8.38725 q^{61} +10.7142 q^{62} +3.97135 q^{63} +1.00000 q^{64} -3.37987 q^{65} -3.41188 q^{66} -11.7200 q^{67} -1.10357 q^{68} -1.00000 q^{69} -3.45410 q^{70} -3.08559 q^{71} +1.00000 q^{72} -14.2293 q^{73} -10.8290 q^{74} +4.24353 q^{75} +3.97135 q^{76} +13.5498 q^{77} -3.88600 q^{78} +0.168878 q^{79} -0.869754 q^{80} +1.00000 q^{81} -12.2743 q^{82} -7.07491 q^{83} -3.97135 q^{84} +0.959831 q^{85} +8.48596 q^{86} -1.00000 q^{87} +3.41188 q^{88} +6.49903 q^{89} -0.869754 q^{90} +15.4327 q^{91} +1.00000 q^{92} -10.7142 q^{93} +6.30315 q^{94} -3.45410 q^{95} -1.00000 q^{96} -0.491941 q^{97} +8.77162 q^{98} +3.41188 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.869754	−0.388966	−0.194483	−	0.980906i	$-0.562303\pi$
$5$	−0.869754	−0.388966	−0.194483	+	0.980906i	$0.562303\pi$
$6$	−1.00000	−0.408248
$7$	3.97135	1.50103	0.750515	−	0.660854i	$-0.229806\pi$
$7$	3.97135	1.50103	0.750515	+	0.660854i	$0.229806\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.869754	−0.275041
$11$	3.41188	1.02872	0.514360	−	0.857574i	$-0.328029\pi$
$11$	3.41188	1.02872	0.514360	+	0.857574i	$0.328029\pi$
$12$	−1.00000	−0.288675
$13$	3.88600	1.07778	0.538892	−	0.842375i	$-0.318843\pi$
$13$	3.88600	1.07778	0.538892	+	0.842375i	$0.318843\pi$
$14$	3.97135	1.06139
$15$	0.869754	0.224570
$16$	1.00000	0.250000
$17$	−1.10357	−0.267654	−0.133827	−	0.991005i	$-0.542727\pi$
$17$	−1.10357	−0.267654	−0.133827	+	0.991005i	$0.542727\pi$
$18$	1.00000	0.235702
$19$	3.97135	0.911090	0.455545	−	0.890213i	$-0.349444\pi$
$19$	3.97135	0.911090	0.455545	+	0.890213i	$0.349444\pi$
$20$	−0.869754	−0.194483
$21$	−3.97135	−0.866620
$22$	3.41188	0.727415
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−4.24353	−0.848705
$26$	3.88600	0.762108
$27$	−1.00000	−0.192450
$28$	3.97135	0.750515
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.869754	0.158795
$31$	10.7142	1.92433	0.962163	−	0.272473i	$-0.0878416\pi$
$31$	10.7142	1.92433	0.962163	+	0.272473i	$0.0878416\pi$
$32$	1.00000	0.176777
$33$	−3.41188	−0.593932
$34$	−1.10357	−0.189260
$35$	−3.45410	−0.583849
$36$	1.00000	0.166667
$37$	−10.8290	−1.78028	−0.890140	−	0.455686i	$-0.849394\pi$
$37$	−10.8290	−1.78028	−0.890140	+	0.455686i	$0.849394\pi$
$38$	3.97135	0.644238
$39$	−3.88600	−0.622259
$40$	−0.869754	−0.137520
$41$	−12.2743	−1.91692	−0.958458	−	0.285233i	$-0.907929\pi$
$41$	−12.2743	−1.91692	−0.958458	+	0.285233i	$0.907929\pi$
$42$	−3.97135	−0.612793
$43$	8.48596	1.29410	0.647049	−	0.762449i	$-0.276003\pi$
$43$	8.48596	1.29410	0.647049	+	0.762449i	$0.276003\pi$
$44$	3.41188	0.514360
$45$	−0.869754	−0.129655
$46$	1.00000	0.147442
$47$	6.30315	0.919409	0.459704	−	0.888072i	$-0.347955\pi$
$47$	6.30315	0.919409	0.459704	+	0.888072i	$0.347955\pi$
$48$	−1.00000	−0.144338
$49$	8.77162	1.25309
$50$	−4.24353	−0.600125
$51$	1.10357	0.154530
$52$	3.88600	0.538892
$53$	3.72896	0.512212	0.256106	−	0.966649i	$-0.417560\pi$
$53$	3.72896	0.512212	0.256106	+	0.966649i	$0.417560\pi$
$54$	−1.00000	−0.136083
$55$	−2.96750	−0.400137
$56$	3.97135	0.530694
$57$	−3.97135	−0.526018
$58$	1.00000	0.131306
$59$	−9.89660	−1.28843	−0.644214	−	0.764846i	$-0.722815\pi$
$59$	−9.89660	−1.28843	−0.644214	+	0.764846i	$0.722815\pi$
$60$	0.869754	0.112285
$61$	8.38725	1.07388	0.536939	−	0.843621i	$-0.319581\pi$
$61$	8.38725	1.07388	0.536939	+	0.843621i	$0.319581\pi$
$62$	10.7142	1.36070
$63$	3.97135	0.500343
$64$	1.00000	0.125000
$65$	−3.37987	−0.419221
$66$	−3.41188	−0.419974
$67$	−11.7200	−1.43183	−0.715914	−	0.698188i	$-0.753990\pi$
$67$	−11.7200	−1.43183	−0.715914	+	0.698188i	$0.753990\pi$
$68$	−1.10357	−0.133827
$69$	−1.00000	−0.120386
$70$	−3.45410	−0.412844
$71$	−3.08559	−0.366192	−0.183096	−	0.983095i	$-0.558612\pi$
$71$	−3.08559	−0.366192	−0.183096	+	0.983095i	$0.558612\pi$
$72$	1.00000	0.117851
$73$	−14.2293	−1.66542	−0.832708	−	0.553712i	$-0.813211\pi$
$73$	−14.2293	−1.66542	−0.832708	+	0.553712i	$0.813211\pi$
$74$	−10.8290	−1.25885
$75$	4.24353	0.490000
$76$	3.97135	0.455545
$77$	13.5498	1.54414
$78$	−3.88600	−0.440003
$79$	0.168878	0.0190003	0.00950015	−	0.999955i	$-0.496976\pi$
$79$	0.168878	0.0190003	0.00950015	+	0.999955i	$0.496976\pi$
$80$	−0.869754	−0.0972415
$81$	1.00000	0.111111
$82$	−12.2743	−1.35546
$83$	−7.07491	−0.776573	−0.388286	−	0.921539i	$-0.626933\pi$
$83$	−7.07491	−0.776573	−0.388286	+	0.921539i	$0.626933\pi$
$84$	−3.97135	−0.433310
$85$	0.959831	0.104108
$86$	8.48596	0.915065
$87$	−1.00000	−0.107211
$88$	3.41188	0.363708
$89$	6.49903	0.688896	0.344448	−	0.938805i	$-0.388066\pi$
$89$	6.49903	0.688896	0.344448	+	0.938805i	$0.388066\pi$
$90$	−0.869754	−0.0916802
$91$	15.4327	1.61778
$92$	1.00000	0.104257
$93$	−10.7142	−1.11101
$94$	6.30315	0.650120
$95$	−3.45410	−0.354383
$96$	−1.00000	−0.102062
$97$	−0.491941	−0.0499491	−0.0249745	−	0.999688i	$-0.507950\pi$
$97$	−0.491941	−0.0499491	−0.0249745	+	0.999688i	$0.507950\pi$
$98$	8.77162	0.886067
$99$	3.41188	0.342907
$100$	−4.24353	−0.424353
$101$	−9.09579	−0.905065	−0.452533	−	0.891748i	$-0.649479\pi$
$101$	−9.09579	−0.905065	−0.452533	+	0.891748i	$0.649479\pi$
$102$	1.10357	0.109269
$103$	5.11596	0.504091	0.252045	−	0.967715i	$-0.418897\pi$
$103$	5.11596	0.504091	0.252045	+	0.967715i	$0.418897\pi$
$104$	3.88600	0.381054
$105$	3.45410	0.337086
$106$	3.72896	0.362189
$107$	14.2956	1.38201	0.691006	−	0.722849i	$-0.257168\pi$
$107$	14.2956	1.38201	0.691006	+	0.722849i	$0.257168\pi$
$108$	−1.00000	−0.0962250
$109$	4.26840	0.408839	0.204420	−	0.978883i	$-0.434469\pi$
$109$	4.26840	0.408839	0.204420	+	0.978883i	$0.434469\pi$
$110$	−2.96750	−0.282940
$111$	10.8290	1.02785
$112$	3.97135	0.375257
$113$	−3.74994	−0.352764	−0.176382	−	0.984322i	$-0.556439\pi$
$113$	−3.74994	−0.352764	−0.176382	+	0.984322i	$0.556439\pi$
$114$	−3.97135	−0.371951
$115$	−0.869754	−0.0811050
$116$	1.00000	0.0928477
$117$	3.88600	0.359261
$118$	−9.89660	−0.911056
$119$	−4.38264	−0.401756
$120$	0.869754	0.0793974
$121$	0.640933	0.0582666
$122$	8.38725	0.759346
$123$	12.2743	1.10673
$124$	10.7142	0.962163
$125$	8.03960	0.719084
$126$	3.97135	0.353796
$127$	20.9534	1.85931	0.929657	−	0.368425i	$-0.120103\pi$
$127$	20.9534	1.85931	0.929657	+	0.368425i	$0.120103\pi$
$128$	1.00000	0.0883883
$129$	−8.48596	−0.747147
$130$	−3.37987	−0.296434
$131$	−17.2489	−1.50704	−0.753522	−	0.657423i	$-0.771646\pi$
$131$	−17.2489	−1.50704	−0.753522	+	0.657423i	$0.771646\pi$
$132$	−3.41188	−0.296966
$133$	15.7716	1.36757
$134$	−11.7200	−1.01246
$135$	0.869754	0.0748565
$136$	−1.10357	−0.0946299
$137$	5.00919	0.427964	0.213982	−	0.976838i	$-0.431357\pi$
$137$	5.00919	0.427964	0.213982	+	0.976838i	$0.431357\pi$
$138$	−1.00000	−0.0851257
$139$	−1.82292	−0.154618	−0.0773089	−	0.997007i	$-0.524633\pi$
$139$	−1.82292	−0.154618	−0.0773089	+	0.997007i	$0.524633\pi$
$140$	−3.45410	−0.291925
$141$	−6.30315	−0.530821
$142$	−3.08559	−0.258937
$143$	13.2586	1.10874
$144$	1.00000	0.0833333
$145$	−0.869754	−0.0722292
$146$	−14.2293	−1.17763
$147$	−8.77162	−0.723471
$148$	−10.8290	−0.890140
$149$	−17.4286	−1.42780	−0.713902	−	0.700246i	$-0.753074\pi$
$149$	−17.4286	−1.42780	−0.713902	+	0.700246i	$0.753074\pi$
$150$	4.24353	0.346483
$151$	19.6072	1.59561	0.797805	−	0.602915i	$-0.205994\pi$
$151$	19.6072	1.59561	0.797805	+	0.602915i	$0.205994\pi$
$152$	3.97135	0.322119
$153$	−1.10357	−0.0892180
$154$	13.5498	1.09187
$155$	−9.31872	−0.748498
$156$	−3.88600	−0.311129
$157$	−7.35307	−0.586839	−0.293419	−	0.955984i	$-0.594793\pi$
$157$	−7.35307	−0.586839	−0.293419	+	0.955984i	$0.594793\pi$
$158$	0.168878	0.0134352
$159$	−3.72896	−0.295726
$160$	−0.869754	−0.0687601
$161$	3.97135	0.312986
$162$	1.00000	0.0785674
$163$	15.8239	1.23942	0.619711	−	0.784830i	$-0.287250\pi$
$163$	15.8239	1.23942	0.619711	+	0.784830i	$0.287250\pi$
$164$	−12.2743	−0.958458
$165$	2.96750	0.231019
$166$	−7.07491	−0.549120
$167$	19.4468	1.50484	0.752421	−	0.658683i	$-0.228886\pi$
$167$	19.4468	1.50484	0.752421	+	0.658683i	$0.228886\pi$
$168$	−3.97135	−0.306396
$169$	2.10103	0.161618
$170$	0.959831	0.0736157
$171$	3.97135	0.303697
$172$	8.48596	0.647049
$173$	8.51012	0.647012	0.323506	−	0.946226i	$-0.395138\pi$
$173$	8.51012	0.647012	0.323506	+	0.946226i	$0.395138\pi$
$174$	−1.00000	−0.0758098
$175$	−16.8525	−1.27393
$176$	3.41188	0.257180
$177$	9.89660	0.743874
$178$	6.49903	0.487123
$179$	12.6383	0.944632	0.472316	−	0.881429i	$-0.343418\pi$
$179$	12.6383	0.944632	0.472316	+	0.881429i	$0.343418\pi$
$180$	−0.869754	−0.0648277
$181$	17.6425	1.31135	0.655677	−	0.755042i	$-0.272383\pi$
$181$	17.6425	1.31135	0.655677	+	0.755042i	$0.272383\pi$
$182$	15.4327	1.14395
$183$	−8.38725	−0.620003
$184$	1.00000	0.0737210
$185$	9.41859	0.692469
$186$	−10.7142	−0.785603
$187$	−3.76523	−0.275341
$188$	6.30315	0.459704
$189$	−3.97135	−0.288873
$190$	−3.45410	−0.250587
$191$	−21.0605	−1.52388	−0.761941	−	0.647646i	$-0.775754\pi$
$191$	−21.0605	−1.52388	−0.761941	+	0.647646i	$0.775754\pi$
$192$	−1.00000	−0.0721688
$193$	−7.16232	−0.515555	−0.257778	−	0.966204i	$-0.582990\pi$
$193$	−7.16232	−0.515555	−0.257778	+	0.966204i	$0.582990\pi$
$194$	−0.491941	−0.0353193
$195$	3.37987	0.242037
$196$	8.77162	0.626544
$197$	13.1123	0.934211	0.467106	−	0.884202i	$-0.345297\pi$
$197$	13.1123	0.934211	0.467106	+	0.884202i	$0.345297\pi$
$198$	3.41188	0.242472
$199$	−15.9337	−1.12951	−0.564756	−	0.825258i	$-0.691030\pi$
$199$	−15.9337	−1.12951	−0.564756	+	0.825258i	$0.691030\pi$
$200$	−4.24353	−0.300063
$201$	11.7200	0.826666
$202$	−9.09579	−0.639978
$203$	3.97135	0.278734
$204$	1.10357	0.0772650
$205$	10.6756	0.745615
$206$	5.11596	0.356446
$207$	1.00000	0.0695048
$208$	3.88600	0.269446
$209$	13.5498	0.937257
$210$	3.45410	0.238355
$211$	−10.2020	−0.702331	−0.351166	−	0.936313i	$-0.614215\pi$
$211$	−10.2020	−0.702331	−0.351166	+	0.936313i	$0.614215\pi$
$212$	3.72896	0.256106
$213$	3.08559	0.211421
$214$	14.2956	0.977230
$215$	−7.38070	−0.503360
$216$	−1.00000	−0.0680414
$217$	42.5498	2.88847
$218$	4.26840	0.289093
$219$	14.2293	0.961528
$220$	−2.96750	−0.200069
$221$	−4.28846	−0.288473
$222$	10.8290	0.726797
$223$	−10.4267	−0.698225	−0.349113	−	0.937081i	$-0.613517\pi$
$223$	−10.4267	−0.698225	−0.349113	+	0.937081i	$0.613517\pi$
$224$	3.97135	0.265347
$225$	−4.24353	−0.282902
$226$	−3.74994	−0.249442
$227$	−8.26175	−0.548351	−0.274176	−	0.961680i	$-0.588405\pi$
$227$	−8.26175	−0.548351	−0.274176	+	0.961680i	$0.588405\pi$
$228$	−3.97135	−0.263009
$229$	26.8246	1.77262	0.886309	−	0.463095i	$-0.153261\pi$
$229$	26.8246	1.77262	0.886309	+	0.463095i	$0.153261\pi$
$230$	−0.869754	−0.0573499
$231$	−13.5498	−0.891510
$232$	1.00000	0.0656532
$233$	−2.00939	−0.131640	−0.0658199	−	0.997832i	$-0.520966\pi$
$233$	−2.00939	−0.131640	−0.0658199	+	0.997832i	$0.520966\pi$
$234$	3.88600	0.254036
$235$	−5.48219	−0.357619
$236$	−9.89660	−0.644214
$237$	−0.168878	−0.0109698
$238$	−4.38264	−0.284085
$239$	24.0485	1.55557	0.777783	−	0.628533i	$-0.216344\pi$
$239$	24.0485	1.55557	0.777783	+	0.628533i	$0.216344\pi$
$240$	0.869754	0.0561424
$241$	7.05790	0.454639	0.227320	−	0.973820i	$-0.427004\pi$
$241$	7.05790	0.454639	0.227320	+	0.973820i	$0.427004\pi$
$242$	0.640933	0.0412007
$243$	−1.00000	−0.0641500
$244$	8.38725	0.536939
$245$	−7.62915	−0.487409
$246$	12.2743	0.782578
$247$	15.4327	0.981958
$248$	10.7142	0.680352
$249$	7.07491	0.448355
$250$	8.03960	0.508469
$251$	−4.30776	−0.271903	−0.135952	−	0.990715i	$-0.543409\pi$
$251$	−4.30776	−0.271903	−0.135952	+	0.990715i	$0.543409\pi$
$252$	3.97135	0.250172
$253$	3.41188	0.214503
$254$	20.9534	1.31473
$255$	−0.959831	−0.0601069
$256$	1.00000	0.0625000
$257$	−13.7037	−0.854816	−0.427408	−	0.904059i	$-0.640573\pi$
$257$	−13.7037	−0.854816	−0.427408	+	0.904059i	$0.640573\pi$
$258$	−8.48596	−0.528313
$259$	−43.0058	−2.67225
$260$	−3.37987	−0.209611
$261$	1.00000	0.0618984
$262$	−17.2489	−1.06564
$263$	5.27982	0.325568	0.162784	−	0.986662i	$-0.447953\pi$
$263$	5.27982	0.325568	0.162784	+	0.986662i	$0.447953\pi$
$264$	−3.41188	−0.209987
$265$	−3.24328	−0.199233
$266$	15.7716	0.967020
$267$	−6.49903	−0.397734
$268$	−11.7200	−0.715914
$269$	12.4184	0.757161	0.378580	−	0.925568i	$-0.376412\pi$
$269$	12.4184	0.757161	0.378580	+	0.925568i	$0.376412\pi$
$270$	0.869754	0.0529316
$271$	7.61564	0.462617	0.231309	−	0.972880i	$-0.425699\pi$
$271$	7.61564	0.462617	0.231309	+	0.972880i	$0.425699\pi$
$272$	−1.10357	−0.0669135
$273$	−15.4327	−0.934028
$274$	5.00919	0.302616
$275$	−14.4784	−0.873081
$276$	−1.00000	−0.0601929
$277$	3.08343	0.185266	0.0926328	−	0.995700i	$-0.470472\pi$
$277$	3.08343	0.185266	0.0926328	+	0.995700i	$0.470472\pi$
$278$	−1.82292	−0.109331
$279$	10.7142	0.641442
$280$	−3.45410	−0.206422
$281$	−16.2254	−0.967923	−0.483962	−	0.875089i	$-0.660803\pi$
$281$	−16.2254	−0.967923	−0.483962	+	0.875089i	$0.660803\pi$
$282$	−6.30315	−0.375347
$283$	−4.77190	−0.283660	−0.141830	−	0.989891i	$-0.545299\pi$
$283$	−4.77190	−0.283660	−0.141830	+	0.989891i	$0.545299\pi$
$284$	−3.08559	−0.183096
$285$	3.45410	0.204603
$286$	13.2586	0.783997
$287$	−48.7454	−2.87735
$288$	1.00000	0.0589256
$289$	−15.7821	−0.928361
$290$	−0.869754	−0.0510737
$291$	0.491941	0.0288381
$292$	−14.2293	−0.832708
$293$	−12.7599	−0.745439	−0.372720	−	0.927944i	$-0.621575\pi$
$293$	−12.7599	−0.745439	−0.372720	+	0.927944i	$0.621575\pi$
$294$	−8.77162	−0.511571
$295$	8.60761	0.501154
$296$	−10.8290	−0.629424
$297$	−3.41188	−0.197977
$298$	−17.4286	−1.00961
$299$	3.88600	0.224733
$300$	4.24353	0.245000
$301$	33.7007	1.94248
$302$	19.6072	1.12827
$303$	9.09579	0.522540
$304$	3.97135	0.227773
$305$	−7.29485	−0.417702
$306$	−1.10357	−0.0630866
$307$	1.10710	0.0631858	0.0315929	−	0.999501i	$-0.489942\pi$
$307$	1.10710	0.0631858	0.0315929	+	0.999501i	$0.489942\pi$
$308$	13.5498	0.772070
$309$	−5.11596	−0.291037
$310$	−9.31872	−0.529268
$311$	−28.1662	−1.59716	−0.798579	−	0.601890i	$-0.794414\pi$
$311$	−28.1662	−1.59716	−0.798579	+	0.601890i	$0.794414\pi$
$312$	−3.88600	−0.220002
$313$	30.7562	1.73844	0.869221	−	0.494424i	$-0.164621\pi$
$313$	30.7562	1.73844	0.869221	+	0.494424i	$0.164621\pi$
$314$	−7.35307	−0.414958
$315$	−3.45410	−0.194616
$316$	0.168878	0.00950015
$317$	−4.57743	−0.257094	−0.128547	−	0.991703i	$-0.541031\pi$
$317$	−4.57743	−0.257094	−0.128547	+	0.991703i	$0.541031\pi$
$318$	−3.72896	−0.209110
$319$	3.41188	0.191029
$320$	−0.869754	−0.0486208
$321$	−14.2956	−0.797905
$322$	3.97135	0.221315
$323$	−4.38264	−0.243857
$324$	1.00000	0.0555556
$325$	−16.4904	−0.914721
$326$	15.8239	0.876404
$327$	−4.26840	−0.236043
$328$	−12.2743	−0.677732
$329$	25.0320	1.38006
$330$	2.96750	0.163355
$331$	−3.85345	−0.211805	−0.105903	−	0.994377i	$-0.533773\pi$
$331$	−3.85345	−0.211805	−0.105903	+	0.994377i	$0.533773\pi$
$332$	−7.07491	−0.388286
$333$	−10.8290	−0.593427
$334$	19.4468	1.06408
$335$	10.1935	0.556932
$336$	−3.97135	−0.216655
$337$	−18.9026	−1.02969	−0.514844	−	0.857284i	$-0.672150\pi$
$337$	−18.9026	−1.02969	−0.514844	+	0.857284i	$0.672150\pi$
$338$	2.10103	0.114281
$339$	3.74994	0.203669
$340$	0.959831	0.0520541
$341$	36.5556	1.97960
$342$	3.97135	0.214746
$343$	7.03571	0.379893
$344$	8.48596	0.457533
$345$	0.869754	0.0468260
$346$	8.51012	0.457507
$347$	21.3126	1.14412	0.572060	−	0.820212i	$-0.306145\pi$
$347$	21.3126	1.14412	0.572060	+	0.820212i	$0.306145\pi$
$348$	−1.00000	−0.0536056
$349$	−31.6871	−1.69617	−0.848086	−	0.529859i	$-0.822245\pi$
$349$	−31.6871	−1.69617	−0.848086	+	0.529859i	$0.822245\pi$
$350$	−16.8525	−0.900806
$351$	−3.88600	−0.207420
$352$	3.41188	0.181854
$353$	−30.9852	−1.64918	−0.824588	−	0.565733i	$-0.808593\pi$
$353$	−30.9852	−1.64918	−0.824588	+	0.565733i	$0.808593\pi$
$354$	9.89660	0.525998
$355$	2.68371	0.142436
$356$	6.49903	0.344448
$357$	4.38264	0.231954
$358$	12.6383	0.667956
$359$	2.10902	0.111310	0.0556549	−	0.998450i	$-0.482275\pi$
$359$	2.10902	0.111310	0.0556549	+	0.998450i	$0.482275\pi$
$360$	−0.869754	−0.0458401
$361$	−3.22838	−0.169915
$362$	17.6425	0.927267
$363$	−0.640933	−0.0336402
$364$	15.4327	0.808892
$365$	12.3760	0.647790
$366$	−8.38725	−0.438409
$367$	−22.6023	−1.17983	−0.589915	−	0.807466i	$-0.700839\pi$
$367$	−22.6023	−1.17983	−0.589915	+	0.807466i	$0.700839\pi$
$368$	1.00000	0.0521286
$369$	−12.2743	−0.638972
$370$	9.41859	0.489649
$371$	14.8090	0.768845
$372$	−10.7142	−0.555505
$373$	−1.00960	−0.0522752	−0.0261376	−	0.999658i	$-0.508321\pi$
$373$	−1.00960	−0.0522752	−0.0261376	+	0.999658i	$0.508321\pi$
$374$	−3.76523	−0.194696
$375$	−8.03960	−0.415163
$376$	6.30315	0.325060
$377$	3.88600	0.200139
$378$	−3.97135	−0.204264
$379$	28.5480	1.46641	0.733207	−	0.680006i	$-0.238023\pi$
$379$	28.5480	1.46641	0.733207	+	0.680006i	$0.238023\pi$
$380$	−3.45410	−0.177192
$381$	−20.9534	−1.07348
$382$	−21.0605	−1.07755
$383$	−13.3023	−0.679714	−0.339857	−	0.940477i	$-0.610379\pi$
$383$	−13.3023	−0.679714	−0.339857	+	0.940477i	$0.610379\pi$
$384$	−1.00000	−0.0510310
$385$	−11.7850	−0.600618
$386$	−7.16232	−0.364553
$387$	8.48596	0.431366
$388$	−0.491941	−0.0249745
$389$	10.4749	0.531100	0.265550	−	0.964097i	$-0.414446\pi$
$389$	10.4749	0.531100	0.265550	+	0.964097i	$0.414446\pi$
$390$	3.37987	0.171146
$391$	−1.10357	−0.0558097
$392$	8.77162	0.443034
$393$	17.2489	0.870092
$394$	13.1123	0.660587
$395$	−0.146883	−0.00739047
$396$	3.41188	0.171453
$397$	20.4609	1.02690	0.513452	−	0.858118i	$-0.328367\pi$
$397$	20.4609	1.02690	0.513452	+	0.858118i	$0.328367\pi$
$398$	−15.9337	−0.798686
$399$	−15.7716	−0.789568
$400$	−4.24353	−0.212176
$401$	−3.62643	−0.181095	−0.0905476	−	0.995892i	$-0.528862\pi$
$401$	−3.62643	−0.181095	−0.0905476	+	0.995892i	$0.528862\pi$
$402$	11.7200	0.584541
$403$	41.6354	2.07401
$404$	−9.09579	−0.452533
$405$	−0.869754	−0.0432184
$406$	3.97135	0.197095
$407$	−36.9473	−1.83141
$408$	1.10357	0.0546346
$409$	15.7385	0.778220	0.389110	−	0.921191i	$-0.372783\pi$
$409$	15.7385	0.778220	0.389110	+	0.921191i	$0.372783\pi$
$410$	10.6756	0.527230
$411$	−5.00919	−0.247085
$412$	5.11596	0.252045
$413$	−39.3029	−1.93397
$414$	1.00000	0.0491473
$415$	6.15344	0.302060
$416$	3.88600	0.190527
$417$	1.82292	0.0892686
$418$	13.5498	0.662741
$419$	−16.5985	−0.810888	−0.405444	−	0.914120i	$-0.632883\pi$
$419$	−16.5985	−0.810888	−0.405444	+	0.914120i	$0.632883\pi$
$420$	3.45410	0.168543
$421$	5.12375	0.249716	0.124858	−	0.992175i	$-0.460152\pi$
$421$	5.12375	0.249716	0.124858	+	0.992175i	$0.460152\pi$
$422$	−10.2020	−0.496623
$423$	6.30315	0.306470
$424$	3.72896	0.181094
$425$	4.68301	0.227159
$426$	3.08559	0.149497
$427$	33.3087	1.61192
$428$	14.2956	0.691006
$429$	−13.2586	−0.640130
$430$	−7.38070	−0.355929
$431$	17.2532	0.831056	0.415528	−	0.909580i	$-0.363597\pi$
$431$	17.2532	0.831056	0.415528	+	0.909580i	$0.363597\pi$
$432$	−1.00000	−0.0481125
$433$	1.16706	0.0560855	0.0280427	−	0.999607i	$-0.491073\pi$
$433$	1.16706	0.0560855	0.0280427	+	0.999607i	$0.491073\pi$
$434$	42.5498	2.04246
$435$	0.869754	0.0417015
$436$	4.26840	0.204420
$437$	3.97135	0.189975
$438$	14.2293	0.679903
$439$	−18.9365	−0.903791	−0.451896	−	0.892071i	$-0.649252\pi$
$439$	−18.9365	−0.903791	−0.451896	+	0.892071i	$0.649252\pi$
$440$	−2.96750	−0.141470
$441$	8.77162	0.417696
$442$	−4.28846	−0.203981
$443$	20.7698	0.986803	0.493401	−	0.869802i	$-0.335753\pi$
$443$	20.7698	0.986803	0.493401	+	0.869802i	$0.335753\pi$
$444$	10.8290	0.513923
$445$	−5.65256	−0.267957
$446$	−10.4267	−0.493720
$447$	17.4286	0.824343
$448$	3.97135	0.187629
$449$	−14.9893	−0.707389	−0.353694	−	0.935361i	$-0.615075\pi$
$449$	−14.9893	−0.707389	−0.353694	+	0.935361i	$0.615075\pi$
$450$	−4.24353	−0.200042
$451$	−41.8783	−1.97197
$452$	−3.74994	−0.176382
$453$	−19.6072	−0.921226
$454$	−8.26175	−0.387743
$455$	−13.4226	−0.629263
$456$	−3.97135	−0.185975
$457$	−23.7100	−1.10911	−0.554554	−	0.832148i	$-0.687111\pi$
$457$	−23.7100	−1.10911	−0.554554	+	0.832148i	$0.687111\pi$
$458$	26.8246	1.25343
$459$	1.10357	0.0515100
$460$	−0.869754	−0.0405525
$461$	22.5922	1.05222	0.526112	−	0.850415i	$-0.323649\pi$
$461$	22.5922	1.05222	0.526112	+	0.850415i	$0.323649\pi$
$462$	−13.5498	−0.630392
$463$	−29.9269	−1.39082	−0.695411	−	0.718612i	$-0.744778\pi$
$463$	−29.9269	−1.39082	−0.695411	+	0.718612i	$0.744778\pi$
$464$	1.00000	0.0464238
$465$	9.31872	0.432145
$466$	−2.00939	−0.0930834
$467$	−37.7312	−1.74599	−0.872996	−	0.487727i	$-0.837826\pi$
$467$	−37.7312	−1.74599	−0.872996	+	0.487727i	$0.837826\pi$
$468$	3.88600	0.179631
$469$	−46.5443	−2.14922
$470$	−5.48219	−0.252875
$471$	7.35307	0.338811
$472$	−9.89660	−0.455528
$473$	28.9531	1.33126
$474$	−0.168878	−0.00775684
$475$	−16.8525	−0.773247
$476$	−4.38264	−0.200878
$477$	3.72896	0.170737
$478$	24.0485	1.09995
$479$	13.8725	0.633850	0.316925	−	0.948451i	$-0.397350\pi$
$479$	13.8725	0.633850	0.316925	+	0.948451i	$0.397350\pi$
$480$	0.869754	0.0396987
$481$	−42.0816	−1.91876
$482$	7.05790	0.321478
$483$	−3.97135	−0.180703
$484$	0.640933	0.0291333
$485$	0.427868	0.0194285
$486$	−1.00000	−0.0453609
$487$	−13.2357	−0.599767	−0.299884	−	0.953976i	$-0.596948\pi$
$487$	−13.2357	−0.599767	−0.299884	+	0.953976i	$0.596948\pi$
$488$	8.38725	0.379673
$489$	−15.8239	−0.715581
$490$	−7.62915	−0.344650
$491$	−9.13444	−0.412231	−0.206116	−	0.978528i	$-0.566082\pi$
$491$	−9.13444	−0.412231	−0.206116	+	0.978528i	$0.566082\pi$
$492$	12.2743	0.553366
$493$	−1.10357	−0.0497021
$494$	15.4327	0.694349
$495$	−2.96750	−0.133379
$496$	10.7142	0.481082
$497$	−12.2540	−0.549665
$498$	7.07491	0.317035
$499$	−14.8193	−0.663405	−0.331702	−	0.943384i	$-0.607623\pi$
$499$	−14.8193	−0.663405	−0.331702	+	0.943384i	$0.607623\pi$
$500$	8.03960	0.359542
$501$	−19.4468	−0.868821
$502$	−4.30776	−0.192265
$503$	24.9943	1.11444	0.557221	−	0.830364i	$-0.311868\pi$
$503$	24.9943	1.11444	0.557221	+	0.830364i	$0.311868\pi$
$504$	3.97135	0.176898
$505$	7.91111	0.352040
$506$	3.41188	0.151677
$507$	−2.10103	−0.0933100
$508$	20.9534	0.929657
$509$	−36.3559	−1.61145	−0.805723	−	0.592292i	$-0.798223\pi$
$509$	−36.3559	−1.61145	−0.805723	+	0.592292i	$0.798223\pi$
$510$	−0.959831	−0.0425020
$511$	−56.5096	−2.49984
$512$	1.00000	0.0441942
$513$	−3.97135	−0.175339
$514$	−13.7037	−0.604446
$515$	−4.44963	−0.196074
$516$	−8.48596	−0.373574
$517$	21.5056	0.945815
$518$	−43.0058	−1.88957
$519$	−8.51012	−0.373553
$520$	−3.37987	−0.148217
$521$	28.6153	1.25366	0.626829	−	0.779157i	$-0.284352\pi$
$521$	28.6153	1.25366	0.626829	+	0.779157i	$0.284352\pi$
$522$	1.00000	0.0437688
$523$	20.2550	0.885688	0.442844	−	0.896599i	$-0.353970\pi$
$523$	20.2550	0.885688	0.442844	+	0.896599i	$0.353970\pi$
$524$	−17.2489	−0.753522
$525$	16.8525	0.735505
$526$	5.27982	0.230211
$527$	−11.8238	−0.515053
$528$	−3.41188	−0.148483
$529$	1.00000	0.0434783
$530$	−3.24328	−0.140879
$531$	−9.89660	−0.429476
$532$	15.7716	0.683786
$533$	−47.6978	−2.06602
$534$	−6.49903	−0.281241
$535$	−12.4337	−0.537556
$536$	−11.7200	−0.506228
$537$	−12.6383	−0.545383
$538$	12.4184	0.535393
$539$	29.9277	1.28908
$540$	0.869754	0.0374283
$541$	−0.743839	−0.0319801	−0.0159901	−	0.999872i	$-0.505090\pi$
$541$	−0.743839	−0.0319801	−0.0159901	+	0.999872i	$0.505090\pi$
$542$	7.61564	0.327120
$543$	−17.6425	−0.757110
$544$	−1.10357	−0.0473150
$545$	−3.71246	−0.159024
$546$	−15.4327	−0.660458
$547$	−24.6338	−1.05326	−0.526632	−	0.850093i	$-0.676545\pi$
$547$	−24.6338	−1.05326	−0.526632	+	0.850093i	$0.676545\pi$
$548$	5.00919	0.213982
$549$	8.38725	0.357959
$550$	−14.4784	−0.617361
$551$	3.97135	0.169185
$552$	−1.00000	−0.0425628
$553$	0.670675	0.0285200
$554$	3.08343	0.131003
$555$	−9.41859	−0.399797
$556$	−1.82292	−0.0773089
$557$	−27.5147	−1.16584	−0.582919	−	0.812531i	$-0.698089\pi$
$557$	−27.5147	−1.16584	−0.582919	+	0.812531i	$0.698089\pi$
$558$	10.7142	0.453568
$559$	32.9765	1.39476
$560$	−3.45410	−0.145962
$561$	3.76523	0.158968
$562$	−16.2254	−0.684425
$563$	22.6939	0.956436	0.478218	−	0.878241i	$-0.341283\pi$
$563$	22.6939	0.956436	0.478218	+	0.878241i	$0.341283\pi$
$564$	−6.30315	−0.265411
$565$	3.26153	0.137213
$566$	−4.77190	−0.200578
$567$	3.97135	0.166781
$568$	−3.08559	−0.129468
$569$	−21.7374	−0.911281	−0.455641	−	0.890164i	$-0.650590\pi$
$569$	−21.7374	−0.911281	−0.455641	+	0.890164i	$0.650590\pi$
$570$	3.45410	0.144676
$571$	−26.5786	−1.11228	−0.556140	−	0.831089i	$-0.687718\pi$
$571$	−26.5786	−1.11228	−0.556140	+	0.831089i	$0.687718\pi$
$572$	13.2586	0.554369
$573$	21.0605	0.879814
$574$	−48.7454	−2.03459
$575$	−4.24353	−0.176967
$576$	1.00000	0.0416667
$577$	−28.3585	−1.18058	−0.590289	−	0.807192i	$-0.700986\pi$
$577$	−28.3585	−1.18058	−0.590289	+	0.807192i	$0.700986\pi$
$578$	−15.7821	−0.656451
$579$	7.16232	0.297656
$580$	−0.869754	−0.0361146
$581$	−28.0970	−1.16566
$582$	0.491941	0.0203916
$583$	12.7228	0.526923
$584$	−14.2293	−0.588814
$585$	−3.37987	−0.139740
$586$	−12.7599	−0.527105
$587$	−22.0034	−0.908179	−0.454089	−	0.890956i	$-0.650035\pi$
$587$	−22.0034	−0.908179	−0.454089	+	0.890956i	$0.650035\pi$
$588$	−8.77162	−0.361735
$589$	42.5498	1.75324
$590$	8.60761	0.354370
$591$	−13.1123	−0.539367
$592$	−10.8290	−0.445070
$593$	−5.31579	−0.218293	−0.109147	−	0.994026i	$-0.534812\pi$
$593$	−5.31579	−0.218293	−0.109147	+	0.994026i	$0.534812\pi$
$594$	−3.41188	−0.139991
$595$	3.81182	0.156270
$596$	−17.4286	−0.713902
$597$	15.9337	0.652124
$598$	3.88600	0.158911
$599$	1.93302	0.0789810	0.0394905	−	0.999220i	$-0.487427\pi$
$599$	1.93302	0.0789810	0.0394905	+	0.999220i	$0.487427\pi$
$600$	4.24353	0.173241
$601$	−24.9439	−1.01748	−0.508740	−	0.860920i	$-0.669889\pi$
$601$	−24.9439	−1.01748	−0.508740	+	0.860920i	$0.669889\pi$
$602$	33.7007	1.37354
$603$	−11.7200	−0.477276
$604$	19.6072	0.797805
$605$	−0.557454	−0.0226637
$606$	9.09579	0.369491
$607$	6.60672	0.268158	0.134079	−	0.990971i	$-0.457192\pi$
$607$	6.60672	0.268158	0.134079	+	0.990971i	$0.457192\pi$
$608$	3.97135	0.161059
$609$	−3.97135	−0.160927
$610$	−7.29485	−0.295360
$611$	24.4941	0.990924
$612$	−1.10357	−0.0446090
$613$	20.8250	0.841116	0.420558	−	0.907266i	$-0.361834\pi$
$613$	20.8250	0.841116	0.420558	+	0.907266i	$0.361834\pi$
$614$	1.10710	0.0446791
$615$	−10.6756	−0.430481
$616$	13.5498	0.545936
$617$	40.7541	1.64070	0.820349	−	0.571864i	$-0.193779\pi$
$617$	40.7541	1.64070	0.820349	+	0.571864i	$0.193779\pi$
$618$	−5.11596	−0.205794
$619$	39.1010	1.57160	0.785800	−	0.618480i	$-0.212251\pi$
$619$	39.1010	1.57160	0.785800	+	0.618480i	$0.212251\pi$
$620$	−9.31872	−0.374249
$621$	−1.00000	−0.0401286
$622$	−28.1662	−1.12936
$623$	25.8099	1.03405
$624$	−3.88600	−0.155565
$625$	14.2252	0.569006
$626$	30.7562	1.22926
$627$	−13.5498	−0.541126
$628$	−7.35307	−0.293419
$629$	11.9505	0.476499
$630$	−3.45410	−0.137615
$631$	18.0567	0.718826	0.359413	−	0.933179i	$-0.382977\pi$
$631$	18.0567	0.718826	0.359413	+	0.933179i	$0.382977\pi$
$632$	0.168878	0.00671762
$633$	10.2020	0.405491
$634$	−4.57743	−0.181793
$635$	−18.2243	−0.723210
$636$	−3.72896	−0.147863
$637$	34.0865	1.35056
$638$	3.41188	0.135078
$639$	−3.08559	−0.122064
$640$	−0.869754	−0.0343801
$641$	2.88089	0.113788	0.0568941	−	0.998380i	$-0.481880\pi$
$641$	2.88089	0.113788	0.0568941	+	0.998380i	$0.481880\pi$
$642$	−14.2956	−0.564204
$643$	−16.4349	−0.648129	−0.324065	−	0.946035i	$-0.605049\pi$
$643$	−16.4349	−0.648129	−0.324065	+	0.946035i	$0.605049\pi$
$644$	3.97135	0.156493
$645$	7.38070	0.290615
$646$	−4.38264	−0.172433
$647$	−2.83959	−0.111636	−0.0558179	−	0.998441i	$-0.517777\pi$
$647$	−2.83959	−0.111636	−0.0558179	+	0.998441i	$0.517777\pi$
$648$	1.00000	0.0392837
$649$	−33.7660	−1.32543
$650$	−16.4904	−0.646805
$651$	−42.5498	−1.66766
$652$	15.8239	0.619711
$653$	−28.9064	−1.13120	−0.565598	−	0.824681i	$-0.691355\pi$
$653$	−28.9064	−1.13120	−0.565598	+	0.824681i	$0.691355\pi$
$654$	−4.26840	−0.166908
$655$	15.0023	0.586189
$656$	−12.2743	−0.479229
$657$	−14.2293	−0.555139
$658$	25.0320	0.975849
$659$	1.14344	0.0445420	0.0222710	−	0.999752i	$-0.492910\pi$
$659$	1.14344	0.0445420	0.0222710	+	0.999752i	$0.492910\pi$
$660$	2.96750	0.115510
$661$	3.19109	0.124119	0.0620595	−	0.998072i	$-0.480233\pi$
$661$	3.19109	0.124119	0.0620595	+	0.998072i	$0.480233\pi$
$662$	−3.85345	−0.149769
$663$	4.28846	0.166550
$664$	−7.07491	−0.274560
$665$	−13.7174	−0.531939
$666$	−10.8290	−0.419616
$667$	1.00000	0.0387202
$668$	19.4468	0.752421
$669$	10.4267	0.403121
$670$	10.1935	0.393811
$671$	28.6163	1.10472
$672$	−3.97135	−0.153198
$673$	−1.17878	−0.0454388	−0.0227194	−	0.999742i	$-0.507232\pi$
$673$	−1.17878	−0.0454388	−0.0227194	+	0.999742i	$0.507232\pi$
$674$	−18.9026	−0.728099
$675$	4.24353	0.163333
$676$	2.10103	0.0808088
$677$	−5.70017	−0.219075	−0.109538	−	0.993983i	$-0.534937\pi$
$677$	−5.70017	−0.219075	−0.109538	+	0.993983i	$0.534937\pi$
$678$	3.74994	0.144015
$679$	−1.95367	−0.0749750
$680$	0.959831	0.0368078
$681$	8.26175	0.316591
$682$	36.5556	1.39979
$683$	37.1446	1.42130	0.710649	−	0.703547i	$-0.248402\pi$
$683$	37.1446	1.42130	0.710649	+	0.703547i	$0.248402\pi$
$684$	3.97135	0.151848
$685$	−4.35677	−0.166464
$686$	7.03571	0.268625
$687$	−26.8246	−1.02342
$688$	8.48596	0.323524
$689$	14.4908	0.552054
$690$	0.869754	0.0331110
$691$	−14.5747	−0.554450	−0.277225	−	0.960805i	$-0.589415\pi$
$691$	−14.5747	−0.554450	−0.277225	+	0.960805i	$0.589415\pi$
$692$	8.51012	0.323506
$693$	13.5498	0.514713
$694$	21.3126	0.809015
$695$	1.58549	0.0601410
$696$	−1.00000	−0.0379049
$697$	13.5454	0.513070
$698$	−31.6871	−1.19937
$699$	2.00939	0.0760023
$700$	−16.8525	−0.636966
$701$	28.1619	1.06366	0.531830	−	0.846851i	$-0.321505\pi$
$701$	28.1619	1.06366	0.531830	+	0.846851i	$0.321505\pi$
$702$	−3.88600	−0.146668
$703$	−43.0058	−1.62200
$704$	3.41188	0.128590
$705$	5.48219	0.206471
$706$	−30.9852	−1.16614
$707$	−36.1226	−1.35853
$708$	9.89660	0.371937
$709$	36.8117	1.38249	0.691247	−	0.722619i	$-0.257062\pi$
$709$	36.8117	1.38249	0.691247	+	0.722619i	$0.257062\pi$
$710$	2.68371	0.100718
$711$	0.168878	0.00633343
$712$	6.49903	0.243561
$713$	10.7142	0.401250
$714$	4.38264	0.164016
$715$	−11.5317	−0.431262
$716$	12.6383	0.472316
$717$	−24.0485	−0.898106
$718$	2.10902	0.0787079
$719$	−36.2805	−1.35304	−0.676518	−	0.736426i	$-0.736512\pi$
$719$	−36.2805	−1.35304	−0.676518	+	0.736426i	$0.736512\pi$
$720$	−0.869754	−0.0324138
$721$	20.3173	0.756655
$722$	−3.22838	−0.120148
$723$	−7.05790	−0.262486
$724$	17.6425	0.655677
$725$	−4.24353	−0.157601
$726$	−0.640933	−0.0237872
$727$	−41.5726	−1.54184	−0.770922	−	0.636930i	$-0.780204\pi$
$727$	−41.5726	−1.54184	−0.770922	+	0.636930i	$0.780204\pi$
$728$	15.4327	0.571973
$729$	1.00000	0.0370370
$730$	12.3760	0.458057
$731$	−9.36481	−0.346370
$732$	−8.38725	−0.310002
$733$	18.6006	0.687028	0.343514	−	0.939147i	$-0.388383\pi$
$733$	18.6006	0.687028	0.343514	+	0.939147i	$0.388383\pi$
$734$	−22.6023	−0.834265
$735$	7.62915	0.281406
$736$	1.00000	0.0368605
$737$	−39.9873	−1.47295
$738$	−12.2743	−0.451822
$739$	30.2426	1.11249	0.556246	−	0.831018i	$-0.312241\pi$
$739$	30.2426	1.11249	0.556246	+	0.831018i	$0.312241\pi$
$740$	9.41859	0.346234
$741$	−15.4327	−0.566934
$742$	14.8090	0.543655
$743$	17.0003	0.623681	0.311840	−	0.950135i	$-0.399055\pi$
$743$	17.0003	0.623681	0.311840	+	0.950135i	$0.399055\pi$
$744$	−10.7142	−0.392802
$745$	15.1586	0.555367
$746$	−1.00960	−0.0369641
$747$	−7.07491	−0.258858
$748$	−3.76523	−0.137671
$749$	56.7730	2.07444
$750$	−8.03960	−0.293565
$751$	−21.4712	−0.783495	−0.391748	−	0.920073i	$-0.628129\pi$
$751$	−21.4712	−0.783495	−0.391748	+	0.920073i	$0.628129\pi$
$752$	6.30315	0.229852
$753$	4.30776	0.156983
$754$	3.88600	0.141520
$755$	−17.0534	−0.620638
$756$	−3.97135	−0.144437
$757$	22.1135	0.803728	0.401864	−	0.915699i	$-0.368362\pi$
$757$	22.1135	0.803728	0.401864	+	0.915699i	$0.368362\pi$
$758$	28.5480	1.03691
$759$	−3.41188	−0.123843
$760$	−3.45410	−0.125293
$761$	−52.4118	−1.89993	−0.949963	−	0.312363i	$-0.898880\pi$
$761$	−52.4118	−1.89993	−0.949963	+	0.312363i	$0.898880\pi$
$762$	−20.9534	−0.759062
$763$	16.9513	0.613679
$764$	−21.0605	−0.761941
$765$	0.959831	0.0347028
$766$	−13.3023	−0.480630
$767$	−38.4582	−1.38865
$768$	−1.00000	−0.0360844
$769$	−45.3061	−1.63378	−0.816890	−	0.576794i	$-0.804304\pi$
$769$	−45.3061	−1.63378	−0.816890	+	0.576794i	$0.804304\pi$
$770$	−11.7850	−0.424701
$771$	13.7037	0.493528
$772$	−7.16232	−0.257778
$773$	7.27520	0.261671	0.130835	−	0.991404i	$-0.458234\pi$
$773$	7.27520	0.261671	0.130835	+	0.991404i	$0.458234\pi$
$774$	8.48596	0.305022
$775$	−45.4660	−1.63319
$776$	−0.491941	−0.0176597
$777$	43.0058	1.54283
$778$	10.4749	0.375544
$779$	−48.7454	−1.74648
$780$	3.37987	0.121019
$781$	−10.5277	−0.376709
$782$	−1.10357	−0.0394634
$783$	−1.00000	−0.0357371
$784$	8.77162	0.313272
$785$	6.39536	0.228260
$786$	17.2489	0.615248
$787$	−49.0563	−1.74867	−0.874334	−	0.485325i	$-0.838701\pi$
$787$	−49.0563	−1.74867	−0.874334	+	0.485325i	$0.838701\pi$
$788$	13.1123	0.467106
$789$	−5.27982	−0.187967
$790$	−0.146883	−0.00522585
$791$	−14.8923	−0.529510
$792$	3.41188	0.121236
$793$	32.5929	1.15741
$794$	20.4609	0.726130
$795$	3.24328	0.115027
$796$	−15.9337	−0.564756
$797$	−41.4125	−1.46691	−0.733454	−	0.679739i	$-0.762093\pi$
$797$	−41.4125	−1.46691	−0.733454	+	0.679739i	$0.762093\pi$
$798$	−15.7716	−0.558309
$799$	−6.95594	−0.246083
$800$	−4.24353	−0.150031
$801$	6.49903	0.229632
$802$	−3.62643	−0.128054
$803$	−48.5488	−1.71325
$804$	11.7200	0.413333
$805$	−3.45410	−0.121741
$806$	41.6354	1.46655
$807$	−12.4184	−0.437147
$808$	−9.09579	−0.319989
$809$	0.109614	0.00385383	0.00192691	−	0.999998i	$-0.499387\pi$
$809$	0.109614	0.00385383	0.00192691	+	0.999998i	$0.499387\pi$
$810$	−0.869754	−0.0305601
$811$	−10.0041	−0.351292	−0.175646	−	0.984453i	$-0.556201\pi$
$811$	−10.0041	−0.351292	−0.175646	+	0.984453i	$0.556201\pi$
$812$	3.97135	0.139367
$813$	−7.61564	−0.267092
$814$	−36.9473	−1.29500
$815$	−13.7629	−0.482093
$816$	1.10357	0.0386325
$817$	33.7007	1.17904
$818$	15.7385	0.550284
$819$	15.4327	0.539262
$820$	10.6756	0.372808
$821$	1.95182	0.0681191	0.0340596	−	0.999420i	$-0.489156\pi$
$821$	1.95182	0.0681191	0.0340596	+	0.999420i	$0.489156\pi$
$822$	−5.00919	−0.174716
$823$	49.0305	1.70910	0.854548	−	0.519372i	$-0.173834\pi$
$823$	49.0305	1.70910	0.854548	+	0.519372i	$0.173834\pi$
$824$	5.11596	0.178223
$825$	14.4784	0.504074
$826$	−39.3029	−1.36752
$827$	−30.3703	−1.05608	−0.528039	−	0.849220i	$-0.677073\pi$
$827$	−30.3703	−1.05608	−0.528039	+	0.849220i	$0.677073\pi$
$828$	1.00000	0.0347524
$829$	−14.4696	−0.502551	−0.251276	−	0.967916i	$-0.580850\pi$
$829$	−14.4696	−0.502551	−0.251276	+	0.967916i	$0.580850\pi$
$830$	6.15344	0.213589
$831$	−3.08343	−0.106963
$832$	3.88600	0.134723
$833$	−9.68005	−0.335394
$834$	1.82292	0.0631224
$835$	−16.9140	−0.585332
$836$	13.5498	0.468629
$837$	−10.7142	−0.370337
$838$	−16.5985	−0.573384
$839$	−25.8736	−0.893257	−0.446628	−	0.894720i	$-0.647375\pi$
$839$	−25.8736	−0.893257	−0.446628	+	0.894720i	$0.647375\pi$
$840$	3.45410	0.119178
$841$	1.00000	0.0344828
$842$	5.12375	0.176576
$843$	16.2254	0.558831
$844$	−10.2020	−0.351166
$845$	−1.82738	−0.0628638
$846$	6.30315	0.216707
$847$	2.54537	0.0874599
$848$	3.72896	0.128053
$849$	4.77190	0.163771
$850$	4.68301	0.160626
$851$	−10.8290	−0.371214
$852$	3.08559	0.105711
$853$	5.34374	0.182966	0.0914830	−	0.995807i	$-0.470839\pi$
$853$	5.34374	0.182966	0.0914830	+	0.995807i	$0.470839\pi$
$854$	33.3087	1.13980
$855$	−3.45410	−0.118128
$856$	14.2956	0.488615
$857$	38.2959	1.30816	0.654081	−	0.756424i	$-0.273055\pi$
$857$	38.2959	1.30816	0.654081	+	0.756424i	$0.273055\pi$
$858$	−13.2586	−0.452641
$859$	45.1669	1.54107	0.770537	−	0.637396i	$-0.219988\pi$
$859$	45.1669	1.54107	0.770537	+	0.637396i	$0.219988\pi$
$860$	−7.38070	−0.251680
$861$	48.7454	1.66124
$862$	17.2532	0.587645
$863$	4.48316	0.152608	0.0763042	−	0.997085i	$-0.475688\pi$
$863$	4.48316	0.152608	0.0763042	+	0.997085i	$0.475688\pi$
$864$	−1.00000	−0.0340207
$865$	−7.40171	−0.251666
$866$	1.16706	0.0396584
$867$	15.7821	0.535990
$868$	42.5498	1.44424
$869$	0.576193	0.0195460
$870$	0.869754	0.0294874
$871$	−45.5440	−1.54320
$872$	4.26840	0.144546
$873$	−0.491941	−0.0166497
$874$	3.97135	0.134333
$875$	31.9281	1.07937
$876$	14.2293	0.480764
$877$	22.7218	0.767260	0.383630	−	0.923487i	$-0.374674\pi$
$877$	22.7218	0.767260	0.383630	+	0.923487i	$0.374674\pi$
$878$	−18.9365	−0.639077
$879$	12.7599	0.430379
$880$	−2.96750	−0.100034
$881$	−25.5531	−0.860906	−0.430453	−	0.902613i	$-0.641646\pi$
$881$	−25.5531	−0.860906	−0.430453	+	0.902613i	$0.641646\pi$
$882$	8.77162	0.295356
$883$	14.1858	0.477390	0.238695	−	0.971095i	$-0.423280\pi$
$883$	14.1858	0.477390	0.238695	+	0.971095i	$0.423280\pi$
$884$	−4.28846	−0.144236
$885$	−8.60761	−0.289342
$886$	20.7698	0.697775
$887$	25.7836	0.865729	0.432864	−	0.901459i	$-0.357503\pi$
$887$	25.7836	0.865729	0.432864	+	0.901459i	$0.357503\pi$
$888$	10.8290	0.363398
$889$	83.2133	2.79089
$890$	−5.65256	−0.189474
$891$	3.41188	0.114302
$892$	−10.4267	−0.349113
$893$	25.0320	0.837664
$894$	17.4286	0.582899
$895$	−10.9922	−0.367430
$896$	3.97135	0.132673
$897$	−3.88600	−0.129750
$898$	−14.9893	−0.500199
$899$	10.7142	0.357339
$900$	−4.24353	−0.141451
$901$	−4.11515	−0.137095
$902$	−41.8783	−1.39439
$903$	−33.7007	−1.12149
$904$	−3.74994	−0.124721
$905$	−15.3446	−0.510072
$906$	−19.6072	−0.651405
$907$	−21.7571	−0.722431	−0.361216	−	0.932482i	$-0.617638\pi$
$907$	−21.7571	−0.722431	−0.361216	+	0.932482i	$0.617638\pi$
$908$	−8.26175	−0.274176
$909$	−9.09579	−0.301688
$910$	−13.4226	−0.444956
$911$	9.73815	0.322639	0.161320	−	0.986902i	$-0.448425\pi$
$911$	9.73815	0.322639	0.161320	+	0.986902i	$0.448425\pi$
$912$	−3.97135	−0.131505
$913$	−24.1388	−0.798877
$914$	−23.7100	−0.784257
$915$	7.29485	0.241160
$916$	26.8246	0.886309
$917$	−68.5014	−2.26212
$918$	1.10357	0.0364231
$919$	21.0956	0.695879	0.347939	−	0.937517i	$-0.386881\pi$
$919$	21.0956	0.695879	0.347939	+	0.937517i	$0.386881\pi$
$920$	−0.869754	−0.0286750
$921$	−1.10710	−0.0364803
$922$	22.5922	0.744035
$923$	−11.9906	−0.394676
$924$	−13.5498	−0.445755
$925$	45.9533	1.51093
$926$	−29.9269	−0.983460
$927$	5.11596	0.168030
$928$	1.00000	0.0328266
$929$	49.4089	1.62105	0.810527	−	0.585701i	$-0.199181\pi$
$929$	49.4089	1.62105	0.810527	+	0.585701i	$0.199181\pi$
$930$	9.31872	0.305573
$931$	34.8352	1.14168
$932$	−2.00939	−0.0658199
$933$	28.1662	0.922119
$934$	−37.7312	−1.23460
$935$	3.27483	0.107098
$936$	3.88600	0.127018
$937$	−11.0242	−0.360145	−0.180072	−	0.983653i	$-0.557633\pi$
$937$	−11.0242	−0.360145	−0.180072	+	0.983653i	$0.557633\pi$
$938$	−46.5443	−1.51973
$939$	−30.7562	−1.00369
$940$	−5.48219	−0.178809
$941$	17.0192	0.554810	0.277405	−	0.960753i	$-0.410526\pi$
$941$	17.0192	0.554810	0.277405	+	0.960753i	$0.410526\pi$
$942$	7.35307	0.239576
$943$	−12.2743	−0.399705
$944$	−9.89660	−0.322107
$945$	3.45410	0.112362
$946$	28.9531	0.941346
$947$	29.5904	0.961559	0.480779	−	0.876842i	$-0.340354\pi$
$947$	29.5904	0.961559	0.480779	+	0.876842i	$0.340354\pi$
$948$	−0.168878	−0.00548491
$949$	−55.2952	−1.79496
$950$	−16.8525	−0.546768
$951$	4.57743	0.148433
$952$	−4.38264	−0.142042
$953$	−43.2397	−1.40067	−0.700336	−	0.713813i	$-0.746966\pi$
$953$	−43.2397	−1.40067	−0.700336	+	0.713813i	$0.746966\pi$
$954$	3.72896	0.120730
$955$	18.3174	0.592739
$956$	24.0485	0.777783
$957$	−3.41188	−0.110290
$958$	13.8725	0.448200
$959$	19.8933	0.642387
$960$	0.869754	0.0280712
$961$	83.7940	2.70303
$962$	−42.0816	−1.35677
$963$	14.2956	0.460671
$964$	7.05790	0.227320
$965$	6.22946	0.200533
$966$	−3.97135	−0.127776
$967$	37.0535	1.19156	0.595780	−	0.803148i	$-0.296843\pi$
$967$	37.0535	1.19156	0.595780	+	0.803148i	$0.296843\pi$
$968$	0.640933	0.0206004
$969$	4.38264	0.140791
$970$	0.427868	0.0137380
$971$	4.20394	0.134911	0.0674555	−	0.997722i	$-0.478512\pi$
$971$	4.20394	0.134911	0.0674555	+	0.997722i	$0.478512\pi$
$972$	−1.00000	−0.0320750
$973$	−7.23944	−0.232086
$974$	−13.2357	−0.424099
$975$	16.4904	0.528114
$976$	8.38725	0.268469
$977$	−55.5661	−1.77772	−0.888859	−	0.458180i	$-0.848501\pi$
$977$	−55.5661	−1.77772	−0.888859	+	0.458180i	$0.848501\pi$
$978$	−15.8239	−0.505992
$979$	22.1739	0.708681
$980$	−7.62915	−0.243704
$981$	4.26840	0.136280
$982$	−9.13444	−0.291492
$983$	9.14713	0.291748	0.145874	−	0.989303i	$-0.453401\pi$
$983$	9.14713	0.291748	0.145874	+	0.989303i	$0.453401\pi$
$984$	12.2743	0.391289
$985$	−11.4045	−0.363376
$986$	−1.10357	−0.0351447
$987$	−25.0320	−0.796778
$988$	15.4327	0.490979
$989$	8.48596	0.269838
$990$	−2.96750	−0.0943133
$991$	−14.5033	−0.460711	−0.230355	−	0.973107i	$-0.573989\pi$
$991$	−14.5033	−0.460711	−0.230355	+	0.973107i	$0.573989\pi$
$992$	10.7142	0.340176
$993$	3.85345	0.122286
$994$	−12.2540	−0.388672
$995$	13.8584	0.439342
$996$	7.07491	0.224177
$997$	3.70874	0.117457	0.0587284	−	0.998274i	$-0.481295\pi$
$997$	3.70874	0.117457	0.0587284	+	0.998274i	$0.481295\pi$
$998$	−14.8193	−0.469098
$999$	10.8290	0.342615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bj.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bj.1.4	✓	8	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.869754 q^{5} -1.00000 q^{6} +3.97135 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.869754 q^{5} -1.00000 q^{6} +3.97135 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.869754 q^{10} +3.41188 q^{11} -1.00000 q^{12} +3.88600 q^{13} +3.97135 q^{14} +0.869754 q^{15} +1.00000 q^{16} -1.10357 q^{17} +1.00000 q^{18} +3.97135 q^{19} -0.869754 q^{20} -3.97135 q^{21} +3.41188 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.24353 q^{25} +3.88600 q^{26} -1.00000 q^{27} +3.97135 q^{28} +1.00000 q^{29} +0.869754 q^{30} +10.7142 q^{31} +1.00000 q^{32} -3.41188 q^{33} -1.10357 q^{34} -3.45410 q^{35} +1.00000 q^{36} -10.8290 q^{37} +3.97135 q^{38} -3.88600 q^{39} -0.869754 q^{40} -12.2743 q^{41} -3.97135 q^{42} +8.48596 q^{43} +3.41188 q^{44} -0.869754 q^{45} +1.00000 q^{46} +6.30315 q^{47} -1.00000 q^{48} +8.77162 q^{49} -4.24353 q^{50} +1.10357 q^{51} +3.88600 q^{52} +3.72896 q^{53} -1.00000 q^{54} -2.96750 q^{55} +3.97135 q^{56} -3.97135 q^{57} +1.00000 q^{58} -9.89660 q^{59} +0.869754 q^{60} +8.38725 q^{61} +10.7142 q^{62} +3.97135 q^{63} +1.00000 q^{64} -3.37987 q^{65} -3.41188 q^{66} -11.7200 q^{67} -1.10357 q^{68} -1.00000 q^{69} -3.45410 q^{70} -3.08559 q^{71} +1.00000 q^{72} -14.2293 q^{73} -10.8290 q^{74} +4.24353 q^{75} +3.97135 q^{76} +13.5498 q^{77} -3.88600 q^{78} +0.168878 q^{79} -0.869754 q^{80} +1.00000 q^{81} -12.2743 q^{82} -7.07491 q^{83} -3.97135 q^{84} +0.959831 q^{85} +8.48596 q^{86} -1.00000 q^{87} +3.41188 q^{88} +6.49903 q^{89} -0.869754 q^{90} +15.4327 q^{91} +1.00000 q^{92} -10.7142 q^{93} +6.30315 q^{94} -3.45410 q^{95} -1.00000 q^{96} -0.491941 q^{97} +8.77162 q^{98} +3.41188 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * q^99

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