LMFDB - Embedded newform 4002.2.a.bj.1.2

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			1.2
Root			$-2.26152$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.26152 q^{5} -1.00000 q^{6} +3.32565 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.26152 q^{5} -1.00000 q^{6} +3.32565 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.26152 q^{10} -5.42976 q^{11} -1.00000 q^{12} +5.07555 q^{13} +3.32565 q^{14} +2.26152 q^{15} +1.00000 q^{16} -0.475759 q^{17} +1.00000 q^{18} +3.32565 q^{19} -2.26152 q^{20} -3.32565 q^{21} -5.42976 q^{22} +1.00000 q^{23} -1.00000 q^{24} +0.114494 q^{25} +5.07555 q^{26} -1.00000 q^{27} +3.32565 q^{28} +1.00000 q^{29} +2.26152 q^{30} -10.7446 q^{31} +1.00000 q^{32} +5.42976 q^{33} -0.475759 q^{34} -7.52105 q^{35} +1.00000 q^{36} +10.2383 q^{37} +3.32565 q^{38} -5.07555 q^{39} -2.26152 q^{40} +7.67552 q^{41} -3.32565 q^{42} -3.21017 q^{43} -5.42976 q^{44} -2.26152 q^{45} +1.00000 q^{46} +10.9185 q^{47} -1.00000 q^{48} +4.05997 q^{49} +0.114494 q^{50} +0.475759 q^{51} +5.07555 q^{52} -8.38154 q^{53} -1.00000 q^{54} +12.2795 q^{55} +3.32565 q^{56} -3.32565 q^{57} +1.00000 q^{58} +8.47422 q^{59} +2.26152 q^{60} +13.3343 q^{61} -10.7446 q^{62} +3.32565 q^{63} +1.00000 q^{64} -11.4785 q^{65} +5.42976 q^{66} -6.27220 q^{67} -0.475759 q^{68} -1.00000 q^{69} -7.52105 q^{70} -9.51825 q^{71} +1.00000 q^{72} -4.26658 q^{73} +10.2383 q^{74} -0.114494 q^{75} +3.32565 q^{76} -18.0575 q^{77} -5.07555 q^{78} +11.3630 q^{79} -2.26152 q^{80} +1.00000 q^{81} +7.67552 q^{82} -5.80141 q^{83} -3.32565 q^{84} +1.07594 q^{85} -3.21017 q^{86} -1.00000 q^{87} -5.42976 q^{88} -15.0725 q^{89} -2.26152 q^{90} +16.8795 q^{91} +1.00000 q^{92} +10.7446 q^{93} +10.9185 q^{94} -7.52105 q^{95} -1.00000 q^{96} +12.8116 q^{97} +4.05997 q^{98} -5.42976 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.26152	−1.01138	−0.505692	−	0.862714i	$-0.668763\pi$
$5$	−2.26152	−1.01138	−0.505692	+	0.862714i	$0.668763\pi$
$6$	−1.00000	−0.408248
$7$	3.32565	1.25698	0.628489	−	0.777818i	$-0.283674\pi$
$7$	3.32565	1.25698	0.628489	+	0.777818i	$0.283674\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.26152	−0.715157
$11$	−5.42976	−1.63713	−0.818567	−	0.574411i	$-0.805231\pi$
$11$	−5.42976	−1.63713	−0.818567	+	0.574411i	$0.805231\pi$
$12$	−1.00000	−0.288675
$13$	5.07555	1.40770	0.703852	−	0.710347i	$-0.251462\pi$
$13$	5.07555	1.40770	0.703852	+	0.710347i	$0.251462\pi$
$14$	3.32565	0.888818
$15$	2.26152	0.583923
$16$	1.00000	0.250000
$17$	−0.475759	−0.115388	−0.0576942	−	0.998334i	$-0.518375\pi$
$17$	−0.475759	−0.115388	−0.0576942	+	0.998334i	$0.518375\pi$
$18$	1.00000	0.235702
$19$	3.32565	0.762957	0.381479	−	0.924378i	$-0.375415\pi$
$19$	3.32565	0.762957	0.381479	+	0.924378i	$0.375415\pi$
$20$	−2.26152	−0.505692
$21$	−3.32565	−0.725717
$22$	−5.42976	−1.15763
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	0.114494	0.0228988
$26$	5.07555	0.995397
$27$	−1.00000	−0.192450
$28$	3.32565	0.628489
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	2.26152	0.412896
$31$	−10.7446	−1.92979	−0.964895	−	0.262636i	$-0.915408\pi$
$31$	−10.7446	−1.92979	−0.964895	+	0.262636i	$0.915408\pi$
$32$	1.00000	0.176777
$33$	5.42976	0.945200
$34$	−0.475759	−0.0815920
$35$	−7.52105	−1.27129
$36$	1.00000	0.166667
$37$	10.2383	1.68317	0.841586	−	0.540123i	$-0.181622\pi$
$37$	10.2383	1.68317	0.841586	+	0.540123i	$0.181622\pi$
$38$	3.32565	0.539492
$39$	−5.07555	−0.812738
$40$	−2.26152	−0.357578
$41$	7.67552	1.19871	0.599357	−	0.800482i	$-0.295423\pi$
$41$	7.67552	1.19871	0.599357	+	0.800482i	$0.295423\pi$
$42$	−3.32565	−0.513159
$43$	−3.21017	−0.489547	−0.244773	−	0.969580i	$-0.578714\pi$
$43$	−3.21017	−0.489547	−0.244773	+	0.969580i	$0.578714\pi$
$44$	−5.42976	−0.818567
$45$	−2.26152	−0.337128
$46$	1.00000	0.147442
$47$	10.9185	1.59262	0.796310	−	0.604888i	$-0.206782\pi$
$47$	10.9185	1.59262	0.796310	+	0.604888i	$0.206782\pi$
$48$	−1.00000	−0.144338
$49$	4.05997	0.579996
$50$	0.114494	0.0161919
$51$	0.475759	0.0666196
$52$	5.07555	0.703852
$53$	−8.38154	−1.15129	−0.575646	−	0.817699i	$-0.695249\pi$
$53$	−8.38154	−1.15129	−0.575646	+	0.817699i	$0.695249\pi$
$54$	−1.00000	−0.136083
$55$	12.2795	1.65577
$56$	3.32565	0.444409
$57$	−3.32565	−0.440494
$58$	1.00000	0.131306
$59$	8.47422	1.10325	0.551625	−	0.834092i	$-0.314008\pi$
$59$	8.47422	1.10325	0.551625	+	0.834092i	$0.314008\pi$
$60$	2.26152	0.291962
$61$	13.3343	1.70728	0.853639	−	0.520865i	$-0.174390\pi$
$61$	13.3343	1.70728	0.853639	+	0.520865i	$0.174390\pi$
$62$	−10.7446	−1.36457
$63$	3.32565	0.418993
$64$	1.00000	0.125000
$65$	−11.4785	−1.42373
$66$	5.42976	0.668357
$67$	−6.27220	−0.766271	−0.383136	−	0.923692i	$-0.625156\pi$
$67$	−6.27220	−0.766271	−0.383136	+	0.923692i	$0.625156\pi$
$68$	−0.475759	−0.0576942
$69$	−1.00000	−0.120386
$70$	−7.52105	−0.898937
$71$	−9.51825	−1.12961	−0.564804	−	0.825225i	$-0.691048\pi$
$71$	−9.51825	−1.12961	−0.564804	+	0.825225i	$0.691048\pi$
$72$	1.00000	0.117851
$73$	−4.26658	−0.499365	−0.249682	−	0.968328i	$-0.580326\pi$
$73$	−4.26658	−0.499365	−0.249682	+	0.968328i	$0.580326\pi$
$74$	10.2383	1.19018
$75$	−0.114494	−0.0132206
$76$	3.32565	0.381479
$77$	−18.0575	−2.05784
$78$	−5.07555	−0.574693
$79$	11.3630	1.27843	0.639217	−	0.769026i	$-0.279259\pi$
$79$	11.3630	1.27843	0.639217	+	0.769026i	$0.279259\pi$
$80$	−2.26152	−0.252846
$81$	1.00000	0.111111
$82$	7.67552	0.847619
$83$	−5.80141	−0.636788	−0.318394	−	0.947959i	$-0.603143\pi$
$83$	−5.80141	−0.636788	−0.318394	+	0.947959i	$0.603143\pi$
$84$	−3.32565	−0.362859
$85$	1.07594	0.116702
$86$	−3.21017	−0.346162
$87$	−1.00000	−0.107211
$88$	−5.42976	−0.578814
$89$	−15.0725	−1.59769	−0.798843	−	0.601540i	$-0.794554\pi$
$89$	−15.0725	−1.59769	−0.798843	+	0.601540i	$0.794554\pi$
$90$	−2.26152	−0.238386
$91$	16.8795	1.76945
$92$	1.00000	0.104257
$93$	10.7446	1.11416
$94$	10.9185	1.12615
$95$	−7.52105	−0.771643
$96$	−1.00000	−0.102062
$97$	12.8116	1.30082	0.650409	−	0.759584i	$-0.274598\pi$
$97$	12.8116	1.30082	0.650409	+	0.759584i	$0.274598\pi$
$98$	4.05997	0.410119
$99$	−5.42976	−0.545711
$100$	0.114494	0.0114494
$101$	11.1227	1.10675	0.553373	−	0.832934i	$-0.313341\pi$
$101$	11.1227	1.10675	0.553373	+	0.832934i	$0.313341\pi$
$102$	0.475759	0.0471071
$103$	−10.3154	−1.01641	−0.508204	−	0.861237i	$-0.669690\pi$
$103$	−10.3154	−1.01641	−0.508204	+	0.861237i	$0.669690\pi$
$104$	5.07555	0.497698
$105$	7.52105	0.733979
$106$	−8.38154	−0.814087
$107$	13.5989	1.31466	0.657329	−	0.753604i	$-0.271686\pi$
$107$	13.5989	1.31466	0.657329	+	0.753604i	$0.271686\pi$
$108$	−1.00000	−0.0962250
$109$	10.0411	0.961763	0.480882	−	0.876785i	$-0.340317\pi$
$109$	10.0411	0.961763	0.480882	+	0.876785i	$0.340317\pi$
$110$	12.2795	1.17081
$111$	−10.2383	−0.971780
$112$	3.32565	0.314245
$113$	9.67975	0.910594	0.455297	−	0.890340i	$-0.349533\pi$
$113$	9.67975	0.910594	0.455297	+	0.890340i	$0.349533\pi$
$114$	−3.32565	−0.311476
$115$	−2.26152	−0.210888
$116$	1.00000	0.0928477
$117$	5.07555	0.469235
$118$	8.47422	0.780116
$119$	−1.58221	−0.145041
$120$	2.26152	0.206448
$121$	18.4823	1.68021
$122$	13.3343	1.20723
$123$	−7.67552	−0.692078
$124$	−10.7446	−0.964895
$125$	11.0487	0.988225
$126$	3.32565	0.296273
$127$	15.5574	1.38050	0.690249	−	0.723572i	$-0.257501\pi$
$127$	15.5574	1.38050	0.690249	+	0.723572i	$0.257501\pi$
$128$	1.00000	0.0883883
$129$	3.21017	0.282640
$130$	−11.4785	−1.00673
$131$	−7.51117	−0.656254	−0.328127	−	0.944634i	$-0.606417\pi$
$131$	−7.51117	−0.656254	−0.328127	+	0.944634i	$0.606417\pi$
$132$	5.42976	0.472600
$133$	11.0600	0.959021
$134$	−6.27220	−0.541836
$135$	2.26152	0.194641
$136$	−0.475759	−0.0407960
$137$	−13.0070	−1.11126	−0.555630	−	0.831430i	$-0.687523\pi$
$137$	−13.0070	−1.11126	−0.555630	+	0.831430i	$0.687523\pi$
$138$	−1.00000	−0.0851257
$139$	10.5726	0.896753	0.448377	−	0.893845i	$-0.352002\pi$
$139$	10.5726	0.896753	0.448377	+	0.893845i	$0.352002\pi$
$140$	−7.52105	−0.635645
$141$	−10.9185	−0.919500
$142$	−9.51825	−0.798753
$143$	−27.5590	−2.30460
$144$	1.00000	0.0833333
$145$	−2.26152	−0.187809
$146$	−4.26658	−0.353104
$147$	−4.05997	−0.334861
$148$	10.2383	0.841586
$149$	12.7765	1.04669	0.523347	−	0.852120i	$-0.324683\pi$
$149$	12.7765	1.04669	0.523347	+	0.852120i	$0.324683\pi$
$150$	−0.114494	−0.00934841
$151$	−10.9044	−0.887387	−0.443694	−	0.896178i	$-0.646332\pi$
$151$	−10.9044	−0.887387	−0.443694	+	0.896178i	$0.646332\pi$
$152$	3.32565	0.269746
$153$	−0.475759	−0.0384628
$154$	−18.0575	−1.45511
$155$	24.2992	1.95176
$156$	−5.07555	−0.406369
$157$	14.8916	1.18848	0.594240	−	0.804287i	$-0.297453\pi$
$157$	14.8916	1.18848	0.594240	+	0.804287i	$0.297453\pi$
$158$	11.3630	0.903989
$159$	8.38154	0.664699
$160$	−2.26152	−0.178789
$161$	3.32565	0.262098
$162$	1.00000	0.0785674
$163$	13.5964	1.06495	0.532475	−	0.846446i	$-0.321262\pi$
$163$	13.5964	1.06495	0.532475	+	0.846446i	$0.321262\pi$
$164$	7.67552	0.599357
$165$	−12.2795	−0.955960
$166$	−5.80141	−0.450277
$167$	10.8182	0.837135	0.418567	−	0.908186i	$-0.362532\pi$
$167$	10.8182	0.837135	0.418567	+	0.908186i	$0.362532\pi$
$168$	−3.32565	−0.256580
$169$	12.7612	0.981629
$170$	1.07594	0.0825208
$171$	3.32565	0.254319
$172$	−3.21017	−0.244773
$173$	15.2170	1.15692	0.578462	−	0.815710i	$-0.303653\pi$
$173$	15.2170	1.15692	0.578462	+	0.815710i	$0.303653\pi$
$174$	−1.00000	−0.0758098
$175$	0.380768	0.0287833
$176$	−5.42976	−0.409283
$177$	−8.47422	−0.636962
$178$	−15.0725	−1.12973
$179$	4.92868	0.368387	0.184193	−	0.982890i	$-0.441033\pi$
$179$	4.92868	0.368387	0.184193	+	0.982890i	$0.441033\pi$
$180$	−2.26152	−0.168564
$181$	−0.148726	−0.0110547	−0.00552736	−	0.999985i	$-0.501759\pi$
$181$	−0.148726	−0.0110547	−0.00552736	+	0.999985i	$0.501759\pi$
$182$	16.8795	1.25119
$183$	−13.3343	−0.985698
$184$	1.00000	0.0737210
$185$	−23.1543	−1.70233
$186$	10.7446	0.787833
$187$	2.58326	0.188906
$188$	10.9185	0.796310
$189$	−3.32565	−0.241906
$190$	−7.52105	−0.545634
$191$	10.5269	0.761697	0.380848	−	0.924637i	$-0.375632\pi$
$191$	10.5269	0.761697	0.380848	+	0.924637i	$0.375632\pi$
$192$	−1.00000	−0.0721688
$193$	−7.02932	−0.505981	−0.252991	−	0.967469i	$-0.581414\pi$
$193$	−7.02932	−0.505981	−0.252991	+	0.967469i	$0.581414\pi$
$194$	12.8116	0.919817
$195$	11.4785	0.821991
$196$	4.05997	0.289998
$197$	20.5809	1.46633	0.733163	−	0.680052i	$-0.238043\pi$
$197$	20.5809	1.46633	0.733163	+	0.680052i	$0.238043\pi$
$198$	−5.42976	−0.385876
$199$	5.93277	0.420563	0.210282	−	0.977641i	$-0.432562\pi$
$199$	5.93277	0.420563	0.210282	+	0.977641i	$0.432562\pi$
$200$	0.114494	0.00809596
$201$	6.27220	0.442407
$202$	11.1227	0.782587
$203$	3.32565	0.233415
$204$	0.475759	0.0333098
$205$	−17.3584	−1.21236
$206$	−10.3154	−0.718708
$207$	1.00000	0.0695048
$208$	5.07555	0.351926
$209$	−18.0575	−1.24906
$210$	7.52105	0.519002
$211$	11.3274	0.779810	0.389905	−	0.920855i	$-0.372508\pi$
$211$	11.3274	0.779810	0.389905	+	0.920855i	$0.372508\pi$
$212$	−8.38154	−0.575646
$213$	9.51825	0.652179
$214$	13.5989	0.929603
$215$	7.25988	0.495120
$216$	−1.00000	−0.0680414
$217$	−35.7329	−2.42571
$218$	10.0411	0.680069
$219$	4.26658	0.288308
$220$	12.2795	0.827886
$221$	−2.41474	−0.162433
$222$	−10.2383	−0.687152
$223$	12.7244	0.852090	0.426045	−	0.904702i	$-0.359907\pi$
$223$	12.7244	0.852090	0.426045	+	0.904702i	$0.359907\pi$
$224$	3.32565	0.222205
$225$	0.114494	0.00763294
$226$	9.67975	0.643887
$227$	−5.11116	−0.339240	−0.169620	−	0.985510i	$-0.554254\pi$
$227$	−5.11116	−0.339240	−0.169620	+	0.985510i	$0.554254\pi$
$228$	−3.32565	−0.220247
$229$	−10.9399	−0.722930	−0.361465	−	0.932386i	$-0.617723\pi$
$229$	−10.9399	−0.722930	−0.361465	+	0.932386i	$0.617723\pi$
$230$	−2.26152	−0.149121
$231$	18.0575	1.18810
$232$	1.00000	0.0656532
$233$	−20.6365	−1.35194	−0.675970	−	0.736929i	$-0.736275\pi$
$233$	−20.6365	−1.35194	−0.675970	+	0.736929i	$0.736275\pi$
$234$	5.07555	0.331799
$235$	−24.6924	−1.61075
$236$	8.47422	0.551625
$237$	−11.3630	−0.738104
$238$	−1.58221	−0.102559
$239$	−1.29606	−0.0838351	−0.0419175	−	0.999121i	$-0.513347\pi$
$239$	−1.29606	−0.0838351	−0.0419175	+	0.999121i	$0.513347\pi$
$240$	2.26152	0.145981
$241$	−17.9590	−1.15684	−0.578420	−	0.815739i	$-0.696331\pi$
$241$	−17.9590	−1.15684	−0.578420	+	0.815739i	$0.696331\pi$
$242$	18.4823	1.18809
$243$	−1.00000	−0.0641500
$244$	13.3343	0.853639
$245$	−9.18172	−0.586599
$246$	−7.67552	−0.489373
$247$	16.8795	1.07402
$248$	−10.7446	−0.682284
$249$	5.80141	0.367650
$250$	11.0487	0.698781
$251$	−16.6705	−1.05223	−0.526117	−	0.850412i	$-0.676353\pi$
$251$	−16.6705	−1.05223	−0.526117	+	0.850412i	$0.676353\pi$
$252$	3.32565	0.209496
$253$	−5.42976	−0.341366
$254$	15.5574	0.976160
$255$	−1.07594	−0.0673780
$256$	1.00000	0.0625000
$257$	−17.6203	−1.09912	−0.549562	−	0.835453i	$-0.685205\pi$
$257$	−17.6203	−1.09912	−0.549562	+	0.835453i	$0.685205\pi$
$258$	3.21017	0.199857
$259$	34.0492	2.11571
$260$	−11.4785	−0.711865
$261$	1.00000	0.0618984
$262$	−7.51117	−0.464042
$263$	−24.3782	−1.50323	−0.751614	−	0.659604i	$-0.770724\pi$
$263$	−24.3782	−1.50323	−0.751614	+	0.659604i	$0.770724\pi$
$264$	5.42976	0.334179
$265$	18.9551	1.16440
$266$	11.0600	0.678130
$267$	15.0725	0.922424
$268$	−6.27220	−0.383136
$269$	8.45618	0.515582	0.257791	−	0.966201i	$-0.417005\pi$
$269$	8.45618	0.515582	0.257791	+	0.966201i	$0.417005\pi$
$270$	2.26152	0.137632
$271$	7.22535	0.438909	0.219455	−	0.975623i	$-0.429572\pi$
$271$	7.22535	0.438909	0.219455	+	0.975623i	$0.429572\pi$
$272$	−0.475759	−0.0288471
$273$	−16.8795	−1.02159
$274$	−13.0070	−0.785779
$275$	−0.621676	−0.0374884
$276$	−1.00000	−0.0601929
$277$	14.1385	0.849500	0.424750	−	0.905311i	$-0.360362\pi$
$277$	14.1385	0.849500	0.424750	+	0.905311i	$0.360362\pi$
$278$	10.5726	0.634100
$279$	−10.7446	−0.643263
$280$	−7.52105	−0.449469
$281$	23.7945	1.41946	0.709731	−	0.704473i	$-0.248817\pi$
$281$	23.7945	1.41946	0.709731	+	0.704473i	$0.248817\pi$
$282$	−10.9185	−0.650185
$283$	−24.7981	−1.47409	−0.737047	−	0.675842i	$-0.763780\pi$
$283$	−24.7981	−1.47409	−0.737047	+	0.675842i	$0.763780\pi$
$284$	−9.51825	−0.564804
$285$	7.52105	0.445508
$286$	−27.5590	−1.62960
$287$	25.5261	1.50676
$288$	1.00000	0.0589256
$289$	−16.7737	−0.986686
$290$	−2.26152	−0.132801
$291$	−12.8116	−0.751028
$292$	−4.26658	−0.249682
$293$	−18.5890	−1.08598	−0.542991	−	0.839738i	$-0.682708\pi$
$293$	−18.5890	−1.08598	−0.542991	+	0.839738i	$0.682708\pi$
$294$	−4.05997	−0.236782
$295$	−19.1647	−1.11581
$296$	10.2383	0.595091
$297$	5.42976	0.315067
$298$	12.7765	0.740124
$299$	5.07555	0.293526
$300$	−0.114494	−0.00661032
$301$	−10.6759	−0.615350
$302$	−10.9044	−0.627478
$303$	−11.1227	−0.638980
$304$	3.32565	0.190739
$305$	−30.1558	−1.72672
$306$	−0.475759	−0.0271973
$307$	19.6049	1.11891	0.559456	−	0.828860i	$-0.311010\pi$
$307$	19.6049	1.11891	0.559456	+	0.828860i	$0.311010\pi$
$308$	−18.0575	−1.02892
$309$	10.3154	0.586823
$310$	24.2992	1.38010
$311$	30.3066	1.71853	0.859266	−	0.511529i	$-0.170921\pi$
$311$	30.3066	1.71853	0.859266	+	0.511529i	$0.170921\pi$
$312$	−5.07555	−0.287346
$313$	−21.3348	−1.20591	−0.602957	−	0.797774i	$-0.706011\pi$
$313$	−21.3348	−1.20591	−0.602957	+	0.797774i	$0.706011\pi$
$314$	14.8916	0.840383
$315$	−7.52105	−0.423763
$316$	11.3630	0.639217
$317$	−10.9052	−0.612495	−0.306248	−	0.951952i	$-0.599074\pi$
$317$	−10.9052	−0.612495	−0.306248	+	0.951952i	$0.599074\pi$
$318$	8.38154	0.470013
$319$	−5.42976	−0.304008
$320$	−2.26152	−0.126423
$321$	−13.5989	−0.759018
$322$	3.32565	0.185331
$323$	−1.58221	−0.0880364
$324$	1.00000	0.0555556
$325$	0.581120	0.0322348
$326$	13.5964	0.753034
$327$	−10.0411	−0.555274
$328$	7.67552	0.423810
$329$	36.3110	2.00189
$330$	−12.2795	−0.675966
$331$	−5.41872	−0.297840	−0.148920	−	0.988849i	$-0.547580\pi$
$331$	−5.41872	−0.297840	−0.148920	+	0.988849i	$0.547580\pi$
$332$	−5.80141	−0.318394
$333$	10.2383	0.561057
$334$	10.8182	0.591944
$335$	14.1847	0.774995
$336$	−3.32565	−0.181429
$337$	11.5349	0.628344	0.314172	−	0.949366i	$-0.398273\pi$
$337$	11.5349	0.628344	0.314172	+	0.949366i	$0.398273\pi$
$338$	12.7612	0.694117
$339$	−9.67975	−0.525732
$340$	1.07594	0.0583510
$341$	58.3407	3.15932
$342$	3.32565	0.179831
$343$	−9.77752	−0.527937
$344$	−3.21017	−0.173081
$345$	2.26152	0.121756
$346$	15.2170	0.818068
$347$	14.5031	0.778567	0.389283	−	0.921118i	$-0.372723\pi$
$347$	14.5031	0.778567	0.389283	+	0.921118i	$0.372723\pi$
$348$	−1.00000	−0.0536056
$349$	20.5327	1.09909	0.549544	−	0.835465i	$-0.314802\pi$
$349$	20.5327	1.09909	0.549544	+	0.835465i	$0.314802\pi$
$350$	0.380768	0.0203529
$351$	−5.07555	−0.270913
$352$	−5.42976	−0.289407
$353$	3.20547	0.170610	0.0853050	−	0.996355i	$-0.472814\pi$
$353$	3.20547	0.170610	0.0853050	+	0.996355i	$0.472814\pi$
$354$	−8.47422	−0.450400
$355$	21.5257	1.14247
$356$	−15.0725	−0.798843
$357$	1.58221	0.0837394
$358$	4.92868	0.260489
$359$	24.7598	1.30677	0.653385	−	0.757026i	$-0.273348\pi$
$359$	24.7598	1.30677	0.653385	+	0.757026i	$0.273348\pi$
$360$	−2.26152	−0.119193
$361$	−7.94003	−0.417896
$362$	−0.148726	−0.00781687
$363$	−18.4823	−0.970068
$364$	16.8795	0.884727
$365$	9.64897	0.505050
$366$	−13.3343	−0.696994
$367$	−8.73921	−0.456183	−0.228091	−	0.973640i	$-0.573249\pi$
$367$	−8.73921	−0.456183	−0.228091	+	0.973640i	$0.573249\pi$
$368$	1.00000	0.0521286
$369$	7.67552	0.399572
$370$	−23.1543	−1.20373
$371$	−27.8741	−1.44715
$372$	10.7446	0.557082
$373$	−21.9187	−1.13491	−0.567454	−	0.823405i	$-0.692071\pi$
$373$	−21.9187	−1.13491	−0.567454	+	0.823405i	$0.692071\pi$
$374$	2.58326	0.133577
$375$	−11.0487	−0.570552
$376$	10.9185	0.563076
$377$	5.07555	0.261404
$378$	−3.32565	−0.171053
$379$	33.3114	1.71109	0.855547	−	0.517725i	$-0.173221\pi$
$379$	33.3114	1.71109	0.855547	+	0.517725i	$0.173221\pi$
$380$	−7.52105	−0.385822
$381$	−15.5574	−0.797031
$382$	10.5269	0.538601
$383$	−26.3568	−1.34677	−0.673385	−	0.739292i	$-0.735160\pi$
$383$	−26.3568	−1.34677	−0.673385	+	0.739292i	$0.735160\pi$
$384$	−1.00000	−0.0510310
$385$	40.8375	2.08127
$386$	−7.02932	−0.357783
$387$	−3.21017	−0.163182
$388$	12.8116	0.650409
$389$	14.3999	0.730102	0.365051	−	0.930988i	$-0.381052\pi$
$389$	14.3999	0.730102	0.365051	+	0.930988i	$0.381052\pi$
$390$	11.4785	0.581235
$391$	−0.475759	−0.0240602
$392$	4.05997	0.205059
$393$	7.51117	0.378889
$394$	20.5809	1.03685
$395$	−25.6976	−1.29299
$396$	−5.42976	−0.272856
$397$	−30.6067	−1.53611	−0.768054	−	0.640385i	$-0.778775\pi$
$397$	−30.6067	−1.53611	−0.768054	+	0.640385i	$0.778775\pi$
$398$	5.93277	0.297383
$399$	−11.0600	−0.553691
$400$	0.114494	0.00572471
$401$	−12.1268	−0.605586	−0.302793	−	0.953056i	$-0.597919\pi$
$401$	−12.1268	−0.605586	−0.302793	+	0.953056i	$0.597919\pi$
$402$	6.27220	0.312829
$403$	−54.5348	−2.71657
$404$	11.1227	0.553373
$405$	−2.26152	−0.112376
$406$	3.32565	0.165049
$407$	−55.5917	−2.75558
$408$	0.475759	0.0235536
$409$	18.9080	0.934938	0.467469	−	0.884009i	$-0.345166\pi$
$409$	18.9080	0.934938	0.467469	+	0.884009i	$0.345166\pi$
$410$	−17.3584	−0.857269
$411$	13.0070	0.641586
$412$	−10.3154	−0.508204
$413$	28.1823	1.38676
$414$	1.00000	0.0491473
$415$	13.1200	0.644037
$416$	5.07555	0.248849
$417$	−10.5726	−0.517741
$418$	−18.0575	−0.883221
$419$	−4.48063	−0.218893	−0.109447	−	0.993993i	$-0.534908\pi$
$419$	−4.48063	−0.218893	−0.109447	+	0.993993i	$0.534908\pi$
$420$	7.52105	0.366990
$421$	−8.53845	−0.416138	−0.208069	−	0.978114i	$-0.566718\pi$
$421$	−8.53845	−0.416138	−0.208069	+	0.978114i	$0.566718\pi$
$422$	11.3274	0.551409
$423$	10.9185	0.530874
$424$	−8.38154	−0.407043
$425$	−0.0544716	−0.00264226
$426$	9.51825	0.461161
$427$	44.3452	2.14601
$428$	13.5989	0.657329
$429$	27.5590	1.33056
$430$	7.25988	0.350103
$431$	−20.6797	−0.996106	−0.498053	−	0.867147i	$-0.665952\pi$
$431$	−20.6797	−0.996106	−0.498053	+	0.867147i	$0.665952\pi$
$432$	−1.00000	−0.0481125
$433$	−9.84704	−0.473219	−0.236609	−	0.971605i	$-0.576036\pi$
$433$	−9.84704	−0.473219	−0.236609	+	0.971605i	$0.576036\pi$
$434$	−35.7329	−1.71523
$435$	2.26152	0.108432
$436$	10.0411	0.480882
$437$	3.32565	0.159088
$438$	4.26658	0.203865
$439$	−13.4088	−0.639966	−0.319983	−	0.947423i	$-0.603677\pi$
$439$	−13.4088	−0.639966	−0.319983	+	0.947423i	$0.603677\pi$
$440$	12.2795	0.585404
$441$	4.05997	0.193332
$442$	−2.41474	−0.114857
$443$	4.02228	0.191104	0.0955521	−	0.995424i	$-0.469538\pi$
$443$	4.02228	0.191104	0.0955521	+	0.995424i	$0.469538\pi$
$444$	−10.2383	−0.485890
$445$	34.0869	1.61587
$446$	12.7244	0.602519
$447$	−12.7765	−0.604309
$448$	3.32565	0.157122
$449$	16.0556	0.757711	0.378855	−	0.925456i	$-0.376318\pi$
$449$	16.0556	0.757711	0.378855	+	0.925456i	$0.376318\pi$
$450$	0.114494	0.00539731
$451$	−41.6762	−1.96246
$452$	9.67975	0.455297
$453$	10.9044	0.512333
$454$	−5.11116	−0.239879
$455$	−38.1734	−1.78960
$456$	−3.32565	−0.155738
$457$	17.0167	0.796008	0.398004	−	0.917384i	$-0.369703\pi$
$457$	17.0167	0.796008	0.398004	+	0.917384i	$0.369703\pi$
$458$	−10.9399	−0.511189
$459$	0.475759	0.0222065
$460$	−2.26152	−0.105444
$461$	−27.7056	−1.29038	−0.645189	−	0.764023i	$-0.723221\pi$
$461$	−27.7056	−1.29038	−0.645189	+	0.764023i	$0.723221\pi$
$462$	18.0575	0.840111
$463$	26.7779	1.24447	0.622236	−	0.782829i	$-0.286224\pi$
$463$	26.7779	1.24447	0.622236	+	0.782829i	$0.286224\pi$
$464$	1.00000	0.0464238
$465$	−24.2992	−1.12685
$466$	−20.6365	−0.955966
$467$	−4.74293	−0.219477	−0.109738	−	0.993961i	$-0.535001\pi$
$467$	−4.74293	−0.219477	−0.109738	+	0.993961i	$0.535001\pi$
$468$	5.07555	0.234617
$469$	−20.8592	−0.963187
$470$	−24.6924	−1.13897
$471$	−14.8916	−0.686170
$472$	8.47422	0.390058
$473$	17.4305	0.801454
$474$	−11.3630	−0.521919
$475$	0.380768	0.0174708
$476$	−1.58221	−0.0725204
$477$	−8.38154	−0.383764
$478$	−1.29606	−0.0592803
$479$	12.2155	0.558138	0.279069	−	0.960271i	$-0.409974\pi$
$479$	12.2155	0.558138	0.279069	+	0.960271i	$0.409974\pi$
$480$	2.26152	0.103224
$481$	51.9652	2.36941
$482$	−17.9590	−0.818009
$483$	−3.32565	−0.151322
$484$	18.4823	0.840104
$485$	−28.9737	−1.31563
$486$	−1.00000	−0.0453609
$487$	−3.51808	−0.159419	−0.0797097	−	0.996818i	$-0.525399\pi$
$487$	−3.51808	−0.159419	−0.0797097	+	0.996818i	$0.525399\pi$
$488$	13.3343	0.603614
$489$	−13.5964	−0.614849
$490$	−9.18172	−0.414788
$491$	22.4101	1.01136	0.505678	−	0.862722i	$-0.331242\pi$
$491$	22.4101	1.01136	0.505678	+	0.862722i	$0.331242\pi$
$492$	−7.67552	−0.346039
$493$	−0.475759	−0.0214271
$494$	16.8795	0.759445
$495$	12.2795	0.551924
$496$	−10.7446	−0.482448
$497$	−31.6544	−1.41989
$498$	5.80141	0.259967
$499$	−32.9006	−1.47283	−0.736417	−	0.676528i	$-0.763484\pi$
$499$	−32.9006	−1.47283	−0.736417	+	0.676528i	$0.763484\pi$
$500$	11.0487	0.494113
$501$	−10.8182	−0.483320
$502$	−16.6705	−0.744042
$503$	−38.8276	−1.73123	−0.865617	−	0.500706i	$-0.833074\pi$
$503$	−38.8276	−1.73123	−0.865617	+	0.500706i	$0.833074\pi$
$504$	3.32565	0.148136
$505$	−25.1541	−1.11934
$506$	−5.42976	−0.241382
$507$	−12.7612	−0.566744
$508$	15.5574	0.690249
$509$	9.38831	0.416130	0.208065	−	0.978115i	$-0.433283\pi$
$509$	9.38831	0.416130	0.208065	+	0.978115i	$0.433283\pi$
$510$	−1.07594	−0.0476434
$511$	−14.1892	−0.627691
$512$	1.00000	0.0441942
$513$	−3.32565	−0.146831
$514$	−17.6203	−0.777198
$515$	23.3285	1.02798
$516$	3.21017	0.141320
$517$	−59.2846	−2.60733
$518$	34.0492	1.49603
$519$	−15.2170	−0.667950
$520$	−11.4785	−0.503364
$521$	−33.2284	−1.45576	−0.727882	−	0.685703i	$-0.759495\pi$
$521$	−33.2284	−1.45576	−0.727882	+	0.685703i	$0.759495\pi$
$522$	1.00000	0.0437688
$523$	−42.5475	−1.86047	−0.930237	−	0.366960i	$-0.880399\pi$
$523$	−42.5475	−1.86047	−0.930237	+	0.366960i	$0.880399\pi$
$524$	−7.51117	−0.328127
$525$	−0.380768	−0.0166181
$526$	−24.3782	−1.06294
$527$	5.11185	0.222675
$528$	5.42976	0.236300
$529$	1.00000	0.0434783
$530$	18.9551	0.823355
$531$	8.47422	0.367750
$532$	11.0600	0.479511
$533$	38.9575	1.68743
$534$	15.0725	0.652252
$535$	−30.7543	−1.32962
$536$	−6.27220	−0.270918
$537$	−4.92868	−0.212688
$538$	8.45618	0.364572
$539$	−22.0447	−0.949531
$540$	2.26152	0.0973205
$541$	29.7144	1.27752	0.638761	−	0.769405i	$-0.279447\pi$
$541$	29.7144	1.27752	0.638761	+	0.769405i	$0.279447\pi$
$542$	7.22535	0.310356
$543$	0.148726	0.00638244
$544$	−0.475759	−0.0203980
$545$	−22.7082	−0.972713
$546$	−16.8795	−0.722376
$547$	−27.3757	−1.17050	−0.585250	−	0.810853i	$-0.699004\pi$
$547$	−27.3757	−1.17050	−0.585250	+	0.810853i	$0.699004\pi$
$548$	−13.0070	−0.555630
$549$	13.3343	0.569093
$550$	−0.621676	−0.0265083
$551$	3.32565	0.141678
$552$	−1.00000	−0.0425628
$553$	37.7893	1.60696
$554$	14.1385	0.600687
$555$	23.1543	0.982843
$556$	10.5726	0.448377
$557$	25.9964	1.10150	0.550752	−	0.834669i	$-0.314341\pi$
$557$	25.9964	1.10150	0.550752	+	0.834669i	$0.314341\pi$
$558$	−10.7446	−0.454856
$559$	−16.2934	−0.689137
$560$	−7.52105	−0.317822
$561$	−2.58326	−0.109065
$562$	23.7945	1.00371
$563$	−27.6383	−1.16482	−0.582408	−	0.812897i	$-0.697889\pi$
$563$	−27.6383	−1.16482	−0.582408	+	0.812897i	$0.697889\pi$
$564$	−10.9185	−0.459750
$565$	−21.8910	−0.920961
$566$	−24.7981	−1.04234
$567$	3.32565	0.139664
$568$	−9.51825	−0.399377
$569$	37.2581	1.56194	0.780970	−	0.624569i	$-0.214725\pi$
$569$	37.2581	1.56194	0.780970	+	0.624569i	$0.214725\pi$
$570$	7.52105	0.315022
$571$	−26.3765	−1.10382	−0.551911	−	0.833903i	$-0.686101\pi$
$571$	−26.3765	−1.10382	−0.551911	+	0.833903i	$0.686101\pi$
$572$	−27.5590	−1.15230
$573$	−10.5269	−0.439766
$574$	25.5261	1.06544
$575$	0.114494	0.00477474
$576$	1.00000	0.0416667
$577$	4.33482	0.180461	0.0902305	−	0.995921i	$-0.471240\pi$
$577$	4.33482	0.180461	0.0902305	+	0.995921i	$0.471240\pi$
$578$	−16.7737	−0.697692
$579$	7.02932	0.292128
$580$	−2.26152	−0.0939047
$581$	−19.2935	−0.800429
$582$	−12.8116	−0.531057
$583$	45.5097	1.88482
$584$	−4.26658	−0.176552
$585$	−11.4785	−0.474577
$586$	−18.5890	−0.767906
$587$	−13.4523	−0.555235	−0.277617	−	0.960692i	$-0.589545\pi$
$587$	−13.4523	−0.555235	−0.277617	+	0.960692i	$0.589545\pi$
$588$	−4.05997	−0.167430
$589$	−35.7329	−1.47235
$590$	−19.1647	−0.788997
$591$	−20.5809	−0.846584
$592$	10.2383	0.420793
$593$	−12.8445	−0.527459	−0.263730	−	0.964597i	$-0.584953\pi$
$593$	−12.8445	−0.527459	−0.263730	+	0.964597i	$0.584953\pi$
$594$	5.42976	0.222786
$595$	3.57820	0.146692
$596$	12.7765	0.523347
$597$	−5.93277	−0.242812
$598$	5.07555	0.207555
$599$	−10.3493	−0.422862	−0.211431	−	0.977393i	$-0.567812\pi$
$599$	−10.3493	−0.422862	−0.211431	+	0.977393i	$0.567812\pi$
$600$	−0.114494	−0.00467420
$601$	−1.08778	−0.0443713	−0.0221856	−	0.999754i	$-0.507062\pi$
$601$	−1.08778	−0.0443713	−0.0221856	+	0.999754i	$0.507062\pi$
$602$	−10.6759	−0.435118
$603$	−6.27220	−0.255424
$604$	−10.9044	−0.443694
$605$	−41.7981	−1.69934
$606$	−11.1227	−0.451827
$607$	−1.88557	−0.0765328	−0.0382664	−	0.999268i	$-0.512184\pi$
$607$	−1.88557	−0.0765328	−0.0382664	+	0.999268i	$0.512184\pi$
$608$	3.32565	0.134873
$609$	−3.32565	−0.134762
$610$	−30.1558	−1.22097
$611$	55.4171	2.24194
$612$	−0.475759	−0.0192314
$613$	−22.0389	−0.890143	−0.445072	−	0.895495i	$-0.646822\pi$
$613$	−22.0389	−0.890143	−0.445072	+	0.895495i	$0.646822\pi$
$614$	19.6049	0.791191
$615$	17.3584	0.699957
$616$	−18.0575	−0.727557
$617$	5.88198	0.236800	0.118400	−	0.992966i	$-0.462224\pi$
$617$	5.88198	0.236800	0.118400	+	0.992966i	$0.462224\pi$
$618$	10.3154	0.414946
$619$	−41.6908	−1.67569	−0.837847	−	0.545904i	$-0.816186\pi$
$619$	−41.6908	−1.67569	−0.837847	+	0.545904i	$0.816186\pi$
$620$	24.2992	0.975880
$621$	−1.00000	−0.0401286
$622$	30.3066	1.21519
$623$	−50.1260	−2.00826
$624$	−5.07555	−0.203185
$625$	−25.5594	−1.02237
$626$	−21.3348	−0.852710
$627$	18.0575	0.721147
$628$	14.8916	0.594240
$629$	−4.87098	−0.194219
$630$	−7.52105	−0.299646
$631$	−7.84823	−0.312433	−0.156217	−	0.987723i	$-0.549930\pi$
$631$	−7.84823	−0.312433	−0.156217	+	0.987723i	$0.549930\pi$
$632$	11.3630	0.451995
$633$	−11.3274	−0.450224
$634$	−10.9052	−0.433099
$635$	−35.1835	−1.39621
$636$	8.38154	0.332350
$637$	20.6066	0.816462
$638$	−5.42976	−0.214966
$639$	−9.51825	−0.376536
$640$	−2.26152	−0.0893946
$641$	−24.5930	−0.971364	−0.485682	−	0.874136i	$-0.661429\pi$
$641$	−24.5930	−0.971364	−0.485682	+	0.874136i	$0.661429\pi$
$642$	−13.5989	−0.536707
$643$	−29.1898	−1.15113	−0.575567	−	0.817754i	$-0.695219\pi$
$643$	−29.1898	−1.15113	−0.575567	+	0.817754i	$0.695219\pi$
$644$	3.32565	0.131049
$645$	−7.25988	−0.285858
$646$	−1.58221	−0.0622512
$647$	−5.44896	−0.214221	−0.107110	−	0.994247i	$-0.534160\pi$
$647$	−5.44896	−0.214221	−0.107110	+	0.994247i	$0.534160\pi$
$648$	1.00000	0.0392837
$649$	−46.0130	−1.80617
$650$	0.581120	0.0227934
$651$	35.7329	1.40048
$652$	13.5964	0.532475
$653$	0.781257	0.0305729	0.0152865	−	0.999883i	$-0.495134\pi$
$653$	0.781257	0.0305729	0.0152865	+	0.999883i	$0.495134\pi$
$654$	−10.0411	−0.392638
$655$	16.9867	0.663725
$656$	7.67552	0.299679
$657$	−4.26658	−0.166455
$658$	36.3110	1.41555
$659$	−29.9409	−1.16633	−0.583165	−	0.812354i	$-0.698186\pi$
$659$	−29.9409	−1.16633	−0.583165	+	0.812354i	$0.698186\pi$
$660$	−12.2795	−0.477980
$661$	45.2682	1.76073	0.880364	−	0.474298i	$-0.157298\pi$
$661$	45.2682	1.76073	0.880364	+	0.474298i	$0.157298\pi$
$662$	−5.41872	−0.210604
$663$	2.41474	0.0937806
$664$	−5.80141	−0.225138
$665$	−25.0124	−0.969939
$666$	10.2383	0.396728
$667$	1.00000	0.0387202
$668$	10.8182	0.418567
$669$	−12.7244	−0.491954
$670$	14.1847	0.548004
$671$	−72.4019	−2.79504
$672$	−3.32565	−0.128290
$673$	−21.3304	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$673$	−21.3304	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$674$	11.5349	0.444307
$675$	−0.114494	−0.00440688
$676$	12.7612	0.490815
$677$	28.5225	1.09621	0.548105	−	0.836410i	$-0.315350\pi$
$677$	28.5225	1.09621	0.548105	+	0.836410i	$0.315350\pi$
$678$	−9.67975	−0.371748
$679$	42.6068	1.63510
$680$	1.07594	0.0412604
$681$	5.11116	0.195860
$682$	58.3407	2.23398
$683$	12.8811	0.492882	0.246441	−	0.969158i	$-0.420739\pi$
$683$	12.8811	0.492882	0.246441	+	0.969158i	$0.420739\pi$
$684$	3.32565	0.127160
$685$	29.4156	1.12391
$686$	−9.77752	−0.373307
$687$	10.9399	0.417384
$688$	−3.21017	−0.122387
$689$	−42.5409	−1.62068
$690$	2.26152	0.0860948
$691$	20.5790	0.782863	0.391431	−	0.920207i	$-0.371980\pi$
$691$	20.5790	0.782863	0.391431	+	0.920207i	$0.371980\pi$
$692$	15.2170	0.578462
$693$	−18.0575	−0.685948
$694$	14.5031	0.550530
$695$	−23.9101	−0.906962
$696$	−1.00000	−0.0379049
$697$	−3.65170	−0.138318
$698$	20.5327	0.777172
$699$	20.6365	0.780543
$700$	0.380768	0.0143917
$701$	21.3176	0.805155	0.402577	−	0.915386i	$-0.368114\pi$
$701$	21.3176	0.805155	0.402577	+	0.915386i	$0.368114\pi$
$702$	−5.07555	−0.191564
$703$	34.0492	1.28419
$704$	−5.42976	−0.204642
$705$	24.6924	0.929968
$706$	3.20547	0.120639
$707$	36.9901	1.39116
$708$	−8.47422	−0.318481
$709$	14.6272	0.549337	0.274668	−	0.961539i	$-0.411432\pi$
$709$	14.6272	0.549337	0.274668	+	0.961539i	$0.411432\pi$
$710$	21.5257	0.807847
$711$	11.3630	0.426145
$712$	−15.0725	−0.564867
$713$	−10.7446	−0.402389
$714$	1.58221	0.0592127
$715$	62.3254	2.33084
$716$	4.92868	0.184193
$717$	1.29606	0.0484022
$718$	24.7598	0.924026
$719$	37.9533	1.41542	0.707709	−	0.706504i	$-0.249729\pi$
$719$	37.9533	1.41542	0.707709	+	0.706504i	$0.249729\pi$
$720$	−2.26152	−0.0842821
$721$	−34.3055	−1.27760
$722$	−7.94003	−0.295497
$723$	17.9590	0.667902
$724$	−0.148726	−0.00552736
$725$	0.114494	0.00425221
$726$	−18.4823	−0.685942
$727$	−14.4469	−0.535805	−0.267902	−	0.963446i	$-0.586330\pi$
$727$	−14.4469	−0.535805	−0.267902	+	0.963446i	$0.586330\pi$
$728$	16.8795	0.625596
$729$	1.00000	0.0370370
$730$	9.64897	0.357124
$731$	1.52727	0.0564880
$732$	−13.3343	−0.492849
$733$	−26.7577	−0.988318	−0.494159	−	0.869372i	$-0.664524\pi$
$733$	−26.7577	−0.988318	−0.494159	+	0.869372i	$0.664524\pi$
$734$	−8.73921	−0.322570
$735$	9.18172	0.338673
$736$	1.00000	0.0368605
$737$	34.0565	1.25449
$738$	7.67552	0.282540
$739$	−48.8184	−1.79581	−0.897907	−	0.440185i	$-0.854913\pi$
$739$	−48.8184	−1.79581	−0.897907	+	0.440185i	$0.854913\pi$
$740$	−23.1543	−0.851167
$741$	−16.8795	−0.620084
$742$	−27.8741	−1.02329
$743$	8.07099	0.296096	0.148048	−	0.988980i	$-0.452701\pi$
$743$	8.07099	0.296096	0.148048	+	0.988980i	$0.452701\pi$
$744$	10.7446	0.393917
$745$	−28.8944	−1.05861
$746$	−21.9187	−0.802501
$747$	−5.80141	−0.212263
$748$	2.58326	0.0944532
$749$	45.2253	1.65250
$750$	−11.0487	−0.403441
$751$	−38.7219	−1.41298	−0.706491	−	0.707722i	$-0.749723\pi$
$751$	−38.7219	−1.41298	−0.706491	+	0.707722i	$0.749723\pi$
$752$	10.9185	0.398155
$753$	16.6705	0.607508
$754$	5.07555	0.184841
$755$	24.6606	0.897490
$756$	−3.32565	−0.120953
$757$	−36.5139	−1.32712	−0.663560	−	0.748123i	$-0.730955\pi$
$757$	−36.5139	−1.32712	−0.663560	+	0.748123i	$0.730955\pi$
$758$	33.3114	1.20993
$759$	5.42976	0.197088
$760$	−7.52105	−0.272817
$761$	−11.1352	−0.403649	−0.201825	−	0.979422i	$-0.564687\pi$
$761$	−11.1352	−0.403649	−0.201825	+	0.979422i	$0.564687\pi$
$762$	−15.5574	−0.563586
$763$	33.3932	1.20892
$764$	10.5269	0.380848
$765$	1.07594	0.0389007
$766$	−26.3568	−0.952310
$767$	43.0113	1.55305
$768$	−1.00000	−0.0360844
$769$	30.4924	1.09958	0.549792	−	0.835301i	$-0.314707\pi$
$769$	30.4924	1.09958	0.549792	+	0.835301i	$0.314707\pi$
$770$	40.8375	1.47168
$771$	17.6203	0.634579
$772$	−7.02932	−0.252991
$773$	23.9836	0.862629	0.431314	−	0.902202i	$-0.358050\pi$
$773$	23.9836	0.862629	0.431314	+	0.902202i	$0.358050\pi$
$774$	−3.21017	−0.115387
$775$	−1.23020	−0.0441899
$776$	12.8116	0.459909
$777$	−34.0492	−1.22151
$778$	14.3999	0.516260
$779$	25.5261	0.914568
$780$	11.4785	0.410995
$781$	51.6818	1.84932
$782$	−0.475759	−0.0170131
$783$	−1.00000	−0.0357371
$784$	4.05997	0.144999
$785$	−33.6778	−1.20201
$786$	7.51117	0.267915
$787$	23.5295	0.838737	0.419368	−	0.907816i	$-0.362252\pi$
$787$	23.5295	0.838737	0.419368	+	0.907816i	$0.362252\pi$
$788$	20.5809	0.733163
$789$	24.3782	0.867889
$790$	−25.6976	−0.914281
$791$	32.1915	1.14460
$792$	−5.42976	−0.192938
$793$	67.6787	2.40334
$794$	−30.6067	−1.08619
$795$	−18.9551	−0.672266
$796$	5.93277	0.210282
$797$	23.6907	0.839169	0.419584	−	0.907716i	$-0.362176\pi$
$797$	23.6907	0.839169	0.419584	+	0.907716i	$0.362176\pi$
$798$	−11.0600	−0.391519
$799$	−5.19455	−0.183770
$800$	0.114494	0.00404798
$801$	−15.0725	−0.532562
$802$	−12.1268	−0.428214
$803$	23.1665	0.817527
$804$	6.27220	0.221203
$805$	−7.52105	−0.265082
$806$	−54.5348	−1.92091
$807$	−8.45618	−0.297672
$808$	11.1227	0.391293
$809$	16.4816	0.579463	0.289732	−	0.957108i	$-0.406434\pi$
$809$	16.4816	0.579463	0.289732	+	0.957108i	$0.406434\pi$
$810$	−2.26152	−0.0794619
$811$	−26.4127	−0.927477	−0.463738	−	0.885972i	$-0.653492\pi$
$811$	−26.4127	−0.927477	−0.463738	+	0.885972i	$0.653492\pi$
$812$	3.32565	0.116708
$813$	−7.22535	−0.253404
$814$	−55.5917	−1.94849
$815$	−30.7485	−1.07707
$816$	0.475759	0.0166549
$817$	−10.6759	−0.373503
$818$	18.9080	0.661101
$819$	16.8795	0.589818
$820$	−17.3584	−0.606181
$821$	43.3040	1.51132	0.755660	−	0.654964i	$-0.227316\pi$
$821$	43.3040	1.51132	0.755660	+	0.654964i	$0.227316\pi$
$822$	13.0070	0.453670
$823$	46.1512	1.60873	0.804366	−	0.594135i	$-0.202505\pi$
$823$	46.1512	1.60873	0.804366	+	0.594135i	$0.202505\pi$
$824$	−10.3154	−0.359354
$825$	0.621676	0.0216440
$826$	28.1823	0.980589
$827$	−3.66221	−0.127348	−0.0636738	−	0.997971i	$-0.520282\pi$
$827$	−3.66221	−0.127348	−0.0636738	+	0.997971i	$0.520282\pi$
$828$	1.00000	0.0347524
$829$	1.79537	0.0623559	0.0311779	−	0.999514i	$-0.490074\pi$
$829$	1.79537	0.0623559	0.0311779	+	0.999514i	$0.490074\pi$
$830$	13.1200	0.455403
$831$	−14.1385	−0.490459
$832$	5.07555	0.175963
$833$	−1.93157	−0.0669248
$834$	−10.5726	−0.366098
$835$	−24.4656	−0.846665
$836$	−18.0575	−0.624532
$837$	10.7446	0.371388
$838$	−4.48063	−0.154781
$839$	−20.9770	−0.724206	−0.362103	−	0.932138i	$-0.617941\pi$
$839$	−20.9770	−0.724206	−0.362103	+	0.932138i	$0.617941\pi$
$840$	7.52105	0.259501
$841$	1.00000	0.0344828
$842$	−8.53845	−0.294254
$843$	−23.7945	−0.819526
$844$	11.3274	0.389905
$845$	−28.8597	−0.992805
$846$	10.9185	0.375384
$847$	61.4657	2.11199
$848$	−8.38154	−0.287823
$849$	24.7981	0.851068
$850$	−0.0544716	−0.00186836
$851$	10.2383	0.350966
$852$	9.51825	0.326090
$853$	44.9562	1.53927	0.769635	−	0.638484i	$-0.220438\pi$
$853$	44.9562	1.53927	0.769635	+	0.638484i	$0.220438\pi$
$854$	44.3452	1.51746
$855$	−7.52105	−0.257214
$856$	13.5989	0.464802
$857$	−28.5878	−0.976539	−0.488270	−	0.872693i	$-0.662372\pi$
$857$	−28.5878	−0.976539	−0.488270	+	0.872693i	$0.662372\pi$
$858$	27.5590	0.940849
$859$	−52.9099	−1.80526	−0.902632	−	0.430414i	$-0.858368\pi$
$859$	−52.9099	−1.80526	−0.902632	+	0.430414i	$0.858368\pi$
$860$	7.25988	0.247560
$861$	−25.5261	−0.869928
$862$	−20.6797	−0.704354
$863$	−21.4526	−0.730255	−0.365127	−	0.930958i	$-0.618975\pi$
$863$	−21.4526	−0.730255	−0.365127	+	0.930958i	$0.618975\pi$
$864$	−1.00000	−0.0340207
$865$	−34.4135	−1.17009
$866$	−9.84704	−0.334616
$867$	16.7737	0.569663
$868$	−35.7329	−1.21285
$869$	−61.6982	−2.09297
$870$	2.26152	0.0766729
$871$	−31.8349	−1.07868
$872$	10.0411	0.340035
$873$	12.8116	0.433606
$874$	3.32565	0.112492
$875$	36.7441	1.24218
$876$	4.26658	0.144154
$877$	23.0122	0.777068	0.388534	−	0.921434i	$-0.372982\pi$
$877$	23.0122	0.777068	0.388534	+	0.921434i	$0.372982\pi$
$878$	−13.4088	−0.452524
$879$	18.5890	0.626993
$880$	12.2795	0.413943
$881$	−38.0521	−1.28201	−0.641004	−	0.767538i	$-0.721482\pi$
$881$	−38.0521	−1.28201	−0.641004	+	0.767538i	$0.721482\pi$
$882$	4.05997	0.136706
$883$	−24.7884	−0.834197	−0.417098	−	0.908861i	$-0.636953\pi$
$883$	−24.7884	−0.834197	−0.417098	+	0.908861i	$0.636953\pi$
$884$	−2.41474	−0.0812164
$885$	19.1647	0.644213
$886$	4.02228	0.135131
$887$	44.8102	1.50458	0.752289	−	0.658833i	$-0.228950\pi$
$887$	44.8102	1.50458	0.752289	+	0.658833i	$0.228950\pi$
$888$	−10.2383	−0.343576
$889$	51.7386	1.73526
$890$	34.0869	1.14260
$891$	−5.42976	−0.181904
$892$	12.7244	0.426045
$893$	36.3110	1.21510
$894$	−12.7765	−0.427311
$895$	−11.1463	−0.372581
$896$	3.32565	0.111102
$897$	−5.07555	−0.169468
$898$	16.0556	0.535782
$899$	−10.7446	−0.358353
$900$	0.114494	0.00381647
$901$	3.98759	0.132846
$902$	−41.6762	−1.38767
$903$	10.6759	0.355272
$904$	9.67975	0.321944
$905$	0.336348	0.0111806
$906$	10.9044	0.362274
$907$	−30.3654	−1.00827	−0.504133	−	0.863626i	$-0.668188\pi$
$907$	−30.3654	−1.00827	−0.504133	+	0.863626i	$0.668188\pi$
$908$	−5.11116	−0.169620
$909$	11.1227	0.368915
$910$	−38.1734	−1.26544
$911$	3.13192	0.103765	0.0518825	−	0.998653i	$-0.483478\pi$
$911$	3.13192	0.103765	0.0518825	+	0.998653i	$0.483478\pi$
$912$	−3.32565	−0.110123
$913$	31.5003	1.04251
$914$	17.0167	0.562863
$915$	30.1558	0.996919
$916$	−10.9399	−0.361465
$917$	−24.9796	−0.824898
$918$	0.475759	0.0157024
$919$	8.64405	0.285141	0.142570	−	0.989785i	$-0.454463\pi$
$919$	8.64405	0.285141	0.142570	+	0.989785i	$0.454463\pi$
$920$	−2.26152	−0.0745603
$921$	−19.6049	−0.646005
$922$	−27.7056	−0.912435
$923$	−48.3103	−1.59015
$924$	18.0575	0.594048
$925$	1.17223	0.0385427
$926$	26.7779	0.879975
$927$	−10.3154	−0.338802
$928$	1.00000	0.0328266
$929$	−52.1329	−1.71042	−0.855212	−	0.518279i	$-0.826573\pi$
$929$	−52.1329	−1.71042	−0.855212	+	0.518279i	$0.826573\pi$
$930$	−24.2992	−0.796803
$931$	13.5021	0.442512
$932$	−20.6365	−0.675970
$933$	−30.3066	−0.992195
$934$	−4.74293	−0.155193
$935$	−5.84210	−0.191057
$936$	5.07555	0.165899
$937$	17.7798	0.580841	0.290421	−	0.956899i	$-0.406205\pi$
$937$	17.7798	0.580841	0.290421	+	0.956899i	$0.406205\pi$
$938$	−20.8592	−0.681076
$939$	21.3348	0.696235
$940$	−24.6924	−0.805376
$941$	1.79043	0.0583664	0.0291832	−	0.999574i	$-0.490709\pi$
$941$	1.79043	0.0583664	0.0291832	+	0.999574i	$0.490709\pi$
$942$	−14.8916	−0.485195
$943$	7.67552	0.249949
$944$	8.47422	0.275813
$945$	7.52105	0.244660
$946$	17.4305	0.566713
$947$	9.80867	0.318739	0.159369	−	0.987219i	$-0.449054\pi$
$947$	9.80867	0.318739	0.159369	+	0.987219i	$0.449054\pi$
$948$	−11.3630	−0.369052
$949$	−21.6552	−0.702958
$950$	0.380768	0.0123537
$951$	10.9052	0.353624
$952$	−1.58221	−0.0512797
$953$	−42.3654	−1.37235	−0.686175	−	0.727437i	$-0.740711\pi$
$953$	−42.3654	−1.37235	−0.686175	+	0.727437i	$0.740711\pi$
$954$	−8.38154	−0.271362
$955$	−23.8068	−0.770368
$956$	−1.29606	−0.0419175
$957$	5.42976	0.175519
$958$	12.2155	0.394664
$959$	−43.2567	−1.39683
$960$	2.26152	0.0729904
$961$	84.4468	2.72409
$962$	51.9652	1.67542
$963$	13.5989	0.438219
$964$	−17.9590	−0.578420
$965$	15.8970	0.511742
$966$	−3.32565	−0.107001
$967$	−33.3307	−1.07184	−0.535922	−	0.844267i	$-0.680036\pi$
$967$	−33.3307	−1.07184	−0.535922	+	0.844267i	$0.680036\pi$
$968$	18.4823	0.594043
$969$	1.58221	0.0508279
$970$	−28.9737	−0.930289
$971$	−58.0574	−1.86315	−0.931575	−	0.363548i	$-0.881565\pi$
$971$	−58.0574	−1.86315	−0.931575	+	0.363548i	$0.881565\pi$
$972$	−1.00000	−0.0320750
$973$	35.1607	1.12720
$974$	−3.51808	−0.112727
$975$	−0.581120	−0.0186107
$976$	13.3343	0.426820
$977$	−13.4850	−0.431424	−0.215712	−	0.976457i	$-0.569207\pi$
$977$	−13.4850	−0.431424	−0.215712	+	0.976457i	$0.569207\pi$
$978$	−13.5964	−0.434764
$979$	81.8402	2.61563
$980$	−9.18172	−0.293299
$981$	10.0411	0.320588
$982$	22.4101	0.715136
$983$	41.5855	1.32637	0.663186	−	0.748454i	$-0.269204\pi$
$983$	41.5855	1.32637	0.663186	+	0.748454i	$0.269204\pi$
$984$	−7.67552	−0.244687
$985$	−46.5442	−1.48302
$986$	−0.475759	−0.0151512
$987$	−36.3110	−1.15579
$988$	16.8795	0.537009
$989$	−3.21017	−0.102078
$990$	12.2795	0.390269
$991$	7.79185	0.247516	0.123758	−	0.992312i	$-0.460505\pi$
$991$	7.79185	0.247516	0.123758	+	0.992312i	$0.460505\pi$
$992$	−10.7446	−0.341142
$993$	5.41872	0.171958
$994$	−31.6544	−1.00402
$995$	−13.4171	−0.425351
$996$	5.80141	0.183825
$997$	13.3866	0.423958	0.211979	−	0.977274i	$-0.432009\pi$
$997$	13.3866	0.423958	0.211979	+	0.977274i	$0.432009\pi$
$998$	−32.9006	−1.04145
$999$	−10.2383	−0.323927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bj.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bj.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.26152 q^{5} -1.00000 q^{6} +3.32565 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.26152 q^{5} -1.00000 q^{6} +3.32565 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.26152 q^{10} -5.42976 q^{11} -1.00000 q^{12} +5.07555 q^{13} +3.32565 q^{14} +2.26152 q^{15} +1.00000 q^{16} -0.475759 q^{17} +1.00000 q^{18} +3.32565 q^{19} -2.26152 q^{20} -3.32565 q^{21} -5.42976 q^{22} +1.00000 q^{23} -1.00000 q^{24} +0.114494 q^{25} +5.07555 q^{26} -1.00000 q^{27} +3.32565 q^{28} +1.00000 q^{29} +2.26152 q^{30} -10.7446 q^{31} +1.00000 q^{32} +5.42976 q^{33} -0.475759 q^{34} -7.52105 q^{35} +1.00000 q^{36} +10.2383 q^{37} +3.32565 q^{38} -5.07555 q^{39} -2.26152 q^{40} +7.67552 q^{41} -3.32565 q^{42} -3.21017 q^{43} -5.42976 q^{44} -2.26152 q^{45} +1.00000 q^{46} +10.9185 q^{47} -1.00000 q^{48} +4.05997 q^{49} +0.114494 q^{50} +0.475759 q^{51} +5.07555 q^{52} -8.38154 q^{53} -1.00000 q^{54} +12.2795 q^{55} +3.32565 q^{56} -3.32565 q^{57} +1.00000 q^{58} +8.47422 q^{59} +2.26152 q^{60} +13.3343 q^{61} -10.7446 q^{62} +3.32565 q^{63} +1.00000 q^{64} -11.4785 q^{65} +5.42976 q^{66} -6.27220 q^{67} -0.475759 q^{68} -1.00000 q^{69} -7.52105 q^{70} -9.51825 q^{71} +1.00000 q^{72} -4.26658 q^{73} +10.2383 q^{74} -0.114494 q^{75} +3.32565 q^{76} -18.0575 q^{77} -5.07555 q^{78} +11.3630 q^{79} -2.26152 q^{80} +1.00000 q^{81} +7.67552 q^{82} -5.80141 q^{83} -3.32565 q^{84} +1.07594 q^{85} -3.21017 q^{86} -1.00000 q^{87} -5.42976 q^{88} -15.0725 q^{89} -2.26152 q^{90} +16.8795 q^{91} +1.00000 q^{92} +10.7446 q^{93} +10.9185 q^{94} -7.52105 q^{95} -1.00000 q^{96} +12.8116 q^{97} +4.05997 q^{98} -5.42976 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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