LMFDB - Embedded newform 4002.2.a.bj.1.1

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			1.1
Root			$-3.17046$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.17046 q^{5} -1.00000 q^{6} -4.05963 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.17046 q^{5} -1.00000 q^{6} -4.05963 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.17046 q^{10} +2.38421 q^{11} -1.00000 q^{12} +0.293951 q^{13} -4.05963 q^{14} +3.17046 q^{15} +1.00000 q^{16} -4.81470 q^{17} +1.00000 q^{18} -4.05963 q^{19} -3.17046 q^{20} +4.05963 q^{21} +2.38421 q^{22} +1.00000 q^{23} -1.00000 q^{24} +5.05182 q^{25} +0.293951 q^{26} -1.00000 q^{27} -4.05963 q^{28} +1.00000 q^{29} +3.17046 q^{30} -9.51309 q^{31} +1.00000 q^{32} -2.38421 q^{33} -4.81470 q^{34} +12.8709 q^{35} +1.00000 q^{36} -2.46083 q^{37} -4.05963 q^{38} -0.293951 q^{39} -3.17046 q^{40} +0.0864292 q^{41} +4.05963 q^{42} -5.12858 q^{43} +2.38421 q^{44} -3.17046 q^{45} +1.00000 q^{46} +2.56454 q^{47} -1.00000 q^{48} +9.48060 q^{49} +5.05182 q^{50} +4.81470 q^{51} +0.293951 q^{52} +3.57515 q^{53} -1.00000 q^{54} -7.55905 q^{55} -4.05963 q^{56} +4.05963 q^{57} +1.00000 q^{58} -7.42377 q^{59} +3.17046 q^{60} +12.3791 q^{61} -9.51309 q^{62} -4.05963 q^{63} +1.00000 q^{64} -0.931959 q^{65} -2.38421 q^{66} +7.08697 q^{67} -4.81470 q^{68} -1.00000 q^{69} +12.8709 q^{70} +7.95622 q^{71} +1.00000 q^{72} +14.6003 q^{73} -2.46083 q^{74} -5.05182 q^{75} -4.05963 q^{76} -9.67902 q^{77} -0.293951 q^{78} -1.22253 q^{79} -3.17046 q^{80} +1.00000 q^{81} +0.0864292 q^{82} -2.75507 q^{83} +4.05963 q^{84} +15.2648 q^{85} -5.12858 q^{86} -1.00000 q^{87} +2.38421 q^{88} -0.550988 q^{89} -3.17046 q^{90} -1.19333 q^{91} +1.00000 q^{92} +9.51309 q^{93} +2.56454 q^{94} +12.8709 q^{95} -1.00000 q^{96} +8.84877 q^{97} +9.48060 q^{98} +2.38421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.17046	−1.41787	−0.708936	−	0.705272i	$-0.750825\pi$
$5$	−3.17046	−1.41787	−0.708936	+	0.705272i	$0.750825\pi$
$6$	−1.00000	−0.408248
$7$	−4.05963	−1.53440	−0.767198	−	0.641410i	$-0.778350\pi$
$7$	−4.05963	−1.53440	−0.767198	+	0.641410i	$0.778350\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.17046	−1.00259
$11$	2.38421	0.718867	0.359433	−	0.933171i	$-0.382970\pi$
$11$	2.38421	0.718867	0.359433	+	0.933171i	$0.382970\pi$
$12$	−1.00000	−0.288675
$13$	0.293951	0.0815272	0.0407636	−	0.999169i	$-0.487021\pi$
$13$	0.293951	0.0815272	0.0407636	+	0.999169i	$0.487021\pi$
$14$	−4.05963	−1.08498
$15$	3.17046	0.818609
$16$	1.00000	0.250000
$17$	−4.81470	−1.16774	−0.583869	−	0.811848i	$-0.698462\pi$
$17$	−4.81470	−1.16774	−0.583869	+	0.811848i	$0.698462\pi$
$18$	1.00000	0.235702
$19$	−4.05963	−0.931343	−0.465672	−	0.884958i	$-0.654187\pi$
$19$	−4.05963	−0.931343	−0.465672	+	0.884958i	$0.654187\pi$
$20$	−3.17046	−0.708936
$21$	4.05963	0.885884
$22$	2.38421	0.508316
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	5.05182	1.01036
$26$	0.293951	0.0576485
$27$	−1.00000	−0.192450
$28$	−4.05963	−0.767198
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	3.17046	0.578844
$31$	−9.51309	−1.70860	−0.854300	−	0.519780i	$-0.826014\pi$
$31$	−9.51309	−1.70860	−0.854300	+	0.519780i	$0.826014\pi$
$32$	1.00000	0.176777
$33$	−2.38421	−0.415038
$34$	−4.81470	−0.825715
$35$	12.8709	2.17558
$36$	1.00000	0.166667
$37$	−2.46083	−0.404557	−0.202279	−	0.979328i	$-0.564835\pi$
$37$	−2.46083	−0.404557	−0.202279	+	0.979328i	$0.564835\pi$
$38$	−4.05963	−0.658559
$39$	−0.293951	−0.0470698
$40$	−3.17046	−0.501294
$41$	0.0864292	0.0134980	0.00674899	−	0.999977i	$-0.497852\pi$
$41$	0.0864292	0.0134980	0.00674899	+	0.999977i	$0.497852\pi$
$42$	4.05963	0.626415
$43$	−5.12858	−0.782101	−0.391050	−	0.920369i	$-0.627888\pi$
$43$	−5.12858	−0.782101	−0.391050	+	0.920369i	$0.627888\pi$
$44$	2.38421	0.359433
$45$	−3.17046	−0.472624
$46$	1.00000	0.147442
$47$	2.56454	0.374076	0.187038	−	0.982353i	$-0.440111\pi$
$47$	2.56454	0.374076	0.187038	+	0.982353i	$0.440111\pi$
$48$	−1.00000	−0.144338
$49$	9.48060	1.35437
$50$	5.05182	0.714435
$51$	4.81470	0.674193
$52$	0.293951	0.0407636
$53$	3.57515	0.491084	0.245542	−	0.969386i	$-0.421034\pi$
$53$	3.57515	0.491084	0.245542	+	0.969386i	$0.421034\pi$
$54$	−1.00000	−0.136083
$55$	−7.55905	−1.01926
$56$	−4.05963	−0.542491
$57$	4.05963	0.537711
$58$	1.00000	0.131306
$59$	−7.42377	−0.966493	−0.483246	−	0.875484i	$-0.660542\pi$
$59$	−7.42377	−0.966493	−0.483246	+	0.875484i	$0.660542\pi$
$60$	3.17046	0.409305
$61$	12.3791	1.58498	0.792490	−	0.609885i	$-0.208784\pi$
$61$	12.3791	1.58498	0.792490	+	0.609885i	$0.208784\pi$
$62$	−9.51309	−1.20816
$63$	−4.05963	−0.511465
$64$	1.00000	0.125000
$65$	−0.931959	−0.115595
$66$	−2.38421	−0.293476
$67$	7.08697	0.865811	0.432906	−	0.901439i	$-0.357488\pi$
$67$	7.08697	0.865811	0.432906	+	0.901439i	$0.357488\pi$
$68$	−4.81470	−0.583869
$69$	−1.00000	−0.120386
$70$	12.8709	1.53837
$71$	7.95622	0.944229	0.472115	−	0.881537i	$-0.343491\pi$
$71$	7.95622	0.944229	0.472115	+	0.881537i	$0.343491\pi$
$72$	1.00000	0.117851
$73$	14.6003	1.70884	0.854418	−	0.519586i	$-0.173914\pi$
$73$	14.6003	1.70884	0.854418	+	0.519586i	$0.173914\pi$
$74$	−2.46083	−0.286065
$75$	−5.05182	−0.583334
$76$	−4.05963	−0.465672
$77$	−9.67902	−1.10303
$78$	−0.293951	−0.0332834
$79$	−1.22253	−0.137545	−0.0687726	−	0.997632i	$-0.521908\pi$
$79$	−1.22253	−0.137545	−0.0687726	+	0.997632i	$0.521908\pi$
$80$	−3.17046	−0.354468
$81$	1.00000	0.111111
$82$	0.0864292	0.00954451
$83$	−2.75507	−0.302408	−0.151204	−	0.988503i	$-0.548315\pi$
$83$	−2.75507	−0.302408	−0.151204	+	0.988503i	$0.548315\pi$
$84$	4.05963	0.442942
$85$	15.2648	1.65570
$86$	−5.12858	−0.553029
$87$	−1.00000	−0.107211
$88$	2.38421	0.254158
$89$	−0.550988	−0.0584046	−0.0292023	−	0.999574i	$-0.509297\pi$
$89$	−0.550988	−0.0584046	−0.0292023	+	0.999574i	$0.509297\pi$
$90$	−3.17046	−0.334196
$91$	−1.19333	−0.125095
$92$	1.00000	0.104257
$93$	9.51309	0.986461
$94$	2.56454	0.264512
$95$	12.8709	1.32053
$96$	−1.00000	−0.102062
$97$	8.84877	0.898456	0.449228	−	0.893417i	$-0.351699\pi$
$97$	8.84877	0.898456	0.449228	+	0.893417i	$0.351699\pi$
$98$	9.48060	0.957685
$99$	2.38421	0.239622
$100$	5.05182	0.505182
$101$	−7.06630	−0.703123	−0.351562	−	0.936165i	$-0.614349\pi$
$101$	−7.06630	−0.703123	−0.351562	+	0.936165i	$0.614349\pi$
$102$	4.81470	0.476727
$103$	15.5569	1.53286	0.766432	−	0.642325i	$-0.222030\pi$
$103$	15.5569	1.53286	0.766432	+	0.642325i	$0.222030\pi$
$104$	0.293951	0.0288242
$105$	−12.8709	−1.25607
$106$	3.57515	0.347249
$107$	8.40050	0.812107	0.406053	−	0.913849i	$-0.366905\pi$
$107$	8.40050	0.812107	0.406053	+	0.913849i	$0.366905\pi$
$108$	−1.00000	−0.0962250
$109$	−9.45536	−0.905659	−0.452830	−	0.891597i	$-0.649585\pi$
$109$	−9.45536	−0.905659	−0.452830	+	0.891597i	$0.649585\pi$
$110$	−7.55905	−0.720727
$111$	2.46083	0.233571
$112$	−4.05963	−0.383599
$113$	−1.03830	−0.0976747	−0.0488374	−	0.998807i	$-0.515552\pi$
$113$	−1.03830	−0.0976747	−0.0488374	+	0.998807i	$0.515552\pi$
$114$	4.05963	0.380219
$115$	−3.17046	−0.295647
$116$	1.00000	0.0928477
$117$	0.293951	0.0271757
$118$	−7.42377	−0.683413
$119$	19.5459	1.79177
$120$	3.17046	0.289422
$121$	−5.31553	−0.483230
$122$	12.3791	1.12075
$123$	−0.0864292	−0.00779306
$124$	−9.51309	−0.854300
$125$	−0.164284	−0.0146940
$126$	−4.05963	−0.361661
$127$	10.8850	0.965886	0.482943	−	0.875652i	$-0.339568\pi$
$127$	10.8850	0.965886	0.482943	+	0.875652i	$0.339568\pi$
$128$	1.00000	0.0883883
$129$	5.12858	0.451546
$130$	−0.931959	−0.0817382
$131$	16.3045	1.42453	0.712265	−	0.701911i	$-0.247670\pi$
$131$	16.3045	1.42453	0.712265	+	0.701911i	$0.247670\pi$
$132$	−2.38421	−0.207519
$133$	16.4806	1.42905
$134$	7.08697	0.612221
$135$	3.17046	0.272870
$136$	−4.81470	−0.412857
$137$	3.96250	0.338539	0.169269	−	0.985570i	$-0.445859\pi$
$137$	3.96250	0.338539	0.169269	+	0.985570i	$0.445859\pi$
$138$	−1.00000	−0.0851257
$139$	1.85916	0.157692	0.0788458	−	0.996887i	$-0.474877\pi$
$139$	1.85916	0.157692	0.0788458	+	0.996887i	$0.474877\pi$
$140$	12.8709	1.08779
$141$	−2.56454	−0.215973
$142$	7.95622	0.667671
$143$	0.700841	0.0586072
$144$	1.00000	0.0833333
$145$	−3.17046	−0.263292
$146$	14.6003	1.20833
$147$	−9.48060	−0.781947
$148$	−2.46083	−0.202279
$149$	−19.4743	−1.59540	−0.797698	−	0.603057i	$-0.793949\pi$
$149$	−19.4743	−1.59540	−0.797698	+	0.603057i	$0.793949\pi$
$150$	−5.05182	−0.412479
$151$	10.1207	0.823611	0.411805	−	0.911272i	$-0.364898\pi$
$151$	10.1207	0.823611	0.411805	+	0.911272i	$0.364898\pi$
$152$	−4.05963	−0.329280
$153$	−4.81470	−0.389246
$154$	−9.67902	−0.779958
$155$	30.1609	2.42258
$156$	−0.293951	−0.0235349
$157$	1.15127	0.0918817	0.0459408	−	0.998944i	$-0.485371\pi$
$157$	1.15127	0.0918817	0.0459408	+	0.998944i	$0.485371\pi$
$158$	−1.22253	−0.0972591
$159$	−3.57515	−0.283528
$160$	−3.17046	−0.250647
$161$	−4.05963	−0.319944
$162$	1.00000	0.0785674
$163$	18.6429	1.46022	0.730110	−	0.683329i	$-0.239469\pi$
$163$	18.6429	1.46022	0.730110	+	0.683329i	$0.239469\pi$
$164$	0.0864292	0.00674899
$165$	7.55905	0.588471
$166$	−2.75507	−0.213835
$167$	−12.8315	−0.992935	−0.496467	−	0.868055i	$-0.665370\pi$
$167$	−12.8315	−0.992935	−0.496467	+	0.868055i	$0.665370\pi$
$168$	4.05963	0.313207
$169$	−12.9136	−0.993353
$170$	15.2648	1.17076
$171$	−4.05963	−0.310448
$172$	−5.12858	−0.391050
$173$	3.76150	0.285981	0.142991	−	0.989724i	$-0.454328\pi$
$173$	3.76150	0.285981	0.142991	+	0.989724i	$0.454328\pi$
$174$	−1.00000	−0.0758098
$175$	−20.5085	−1.55030
$176$	2.38421	0.179717
$177$	7.42377	0.558005
$178$	−0.550988	−0.0412983
$179$	6.58132	0.491911	0.245956	−	0.969281i	$-0.420898\pi$
$179$	6.58132	0.491911	0.245956	+	0.969281i	$0.420898\pi$
$180$	−3.17046	−0.236312
$181$	−7.14349	−0.530971	−0.265486	−	0.964115i	$-0.585532\pi$
$181$	−7.14349	−0.530971	−0.265486	+	0.964115i	$0.585532\pi$
$182$	−1.19333	−0.0884556
$183$	−12.3791	−0.915089
$184$	1.00000	0.0737210
$185$	7.80195	0.573611
$186$	9.51309	0.697533
$187$	−11.4793	−0.839448
$188$	2.56454	0.187038
$189$	4.05963	0.295295
$190$	12.8709	0.933753
$191$	−7.95298	−0.575457	−0.287729	−	0.957712i	$-0.592900\pi$
$191$	−7.95298	−0.575457	−0.287729	+	0.957712i	$0.592900\pi$
$192$	−1.00000	−0.0721688
$193$	26.4698	1.90534	0.952670	−	0.304007i	$-0.0983246\pi$
$193$	26.4698	1.90534	0.952670	+	0.304007i	$0.0983246\pi$
$194$	8.84877	0.635305
$195$	0.931959	0.0667390
$196$	9.48060	0.677186
$197$	11.6841	0.832461	0.416230	−	0.909259i	$-0.363351\pi$
$197$	11.6841	0.832461	0.416230	+	0.909259i	$0.363351\pi$
$198$	2.38421	0.169439
$199$	−17.0525	−1.20882	−0.604408	−	0.796675i	$-0.706590\pi$
$199$	−17.0525	−1.20882	−0.604408	+	0.796675i	$0.706590\pi$
$200$	5.05182	0.357217
$201$	−7.08697	−0.499876
$202$	−7.06630	−0.497183
$203$	−4.05963	−0.284930
$204$	4.81470	0.337097
$205$	−0.274020	−0.0191384
$206$	15.5569	1.08390
$207$	1.00000	0.0695048
$208$	0.293951	0.0203818
$209$	−9.67902	−0.669512
$210$	−12.8709	−0.888176
$211$	11.3890	0.784053	0.392027	−	0.919954i	$-0.371774\pi$
$211$	11.3890	0.784053	0.392027	+	0.919954i	$0.371774\pi$
$212$	3.57515	0.245542
$213$	−7.95622	−0.545151
$214$	8.40050	0.574246
$215$	16.2600	1.10892
$216$	−1.00000	−0.0680414
$217$	38.6196	2.62167
$218$	−9.45536	−0.640398
$219$	−14.6003	−0.986597
$220$	−7.55905	−0.509631
$221$	−1.41529	−0.0952024
$222$	2.46083	0.165160
$223$	5.85359	0.391985	0.195993	−	0.980605i	$-0.437207\pi$
$223$	5.85359	0.391985	0.195993	+	0.980605i	$0.437207\pi$
$224$	−4.05963	−0.271245
$225$	5.05182	0.336788
$226$	−1.03830	−0.0690665
$227$	−7.11647	−0.472337	−0.236168	−	0.971712i	$-0.575892\pi$
$227$	−7.11647	−0.472337	−0.236168	+	0.971712i	$0.575892\pi$
$228$	4.05963	0.268856
$229$	15.6553	1.03453	0.517266	−	0.855824i	$-0.326950\pi$
$229$	15.6553	1.03453	0.517266	+	0.855824i	$0.326950\pi$
$230$	−3.17046	−0.209054
$231$	9.67902	0.636833
$232$	1.00000	0.0656532
$233$	19.9822	1.30908	0.654539	−	0.756029i	$-0.272863\pi$
$233$	19.9822	1.30908	0.654539	+	0.756029i	$0.272863\pi$
$234$	0.293951	0.0192162
$235$	−8.13076	−0.530392
$236$	−7.42377	−0.483246
$237$	1.22253	0.0794117
$238$	19.5459	1.26697
$239$	−10.8759	−0.703501	−0.351751	−	0.936094i	$-0.614413\pi$
$239$	−10.8759	−0.703501	−0.351751	+	0.936094i	$0.614413\pi$
$240$	3.17046	0.204652
$241$	24.8928	1.60348	0.801742	−	0.597670i	$-0.203907\pi$
$241$	24.8928	1.60348	0.801742	+	0.597670i	$0.203907\pi$
$242$	−5.31553	−0.341695
$243$	−1.00000	−0.0641500
$244$	12.3791	0.792490
$245$	−30.0579	−1.92033
$246$	−0.0864292	−0.00551052
$247$	−1.19333	−0.0759298
$248$	−9.51309	−0.604082
$249$	2.75507	0.174596
$250$	−0.164284	−0.0103902
$251$	−28.4895	−1.79824	−0.899122	−	0.437698i	$-0.855794\pi$
$251$	−28.4895	−1.79824	−0.899122	+	0.437698i	$0.855794\pi$
$252$	−4.05963	−0.255733
$253$	2.38421	0.149894
$254$	10.8850	0.682985
$255$	−15.2648	−0.955920
$256$	1.00000	0.0625000
$257$	−0.224549	−0.0140070	−0.00700350	−	0.999975i	$-0.502229\pi$
$257$	−0.224549	−0.0140070	−0.00700350	+	0.999975i	$0.502229\pi$
$258$	5.12858	0.319291
$259$	9.99005	0.620751
$260$	−0.931959	−0.0577976
$261$	1.00000	0.0618984
$262$	16.3045	1.00729
$263$	−21.8772	−1.34900	−0.674502	−	0.738273i	$-0.735642\pi$
$263$	−21.8772	−1.34900	−0.674502	+	0.738273i	$0.735642\pi$
$264$	−2.38421	−0.146738
$265$	−11.3349	−0.696295
$266$	16.4806	1.01049
$267$	0.550988	0.0337199
$268$	7.08697	0.432906
$269$	−21.4223	−1.30614	−0.653070	−	0.757298i	$-0.726519\pi$
$269$	−21.4223	−1.30614	−0.653070	+	0.757298i	$0.726519\pi$
$270$	3.17046	0.192948
$271$	−12.9825	−0.788633	−0.394317	−	0.918975i	$-0.629019\pi$
$271$	−12.9825	−0.788633	−0.394317	+	0.918975i	$0.629019\pi$
$272$	−4.81470	−0.291934
$273$	1.19333	0.0722237
$274$	3.96250	0.239383
$275$	12.0446	0.726317
$276$	−1.00000	−0.0601929
$277$	18.1731	1.09192	0.545958	−	0.837813i	$-0.316166\pi$
$277$	18.1731	1.09192	0.545958	+	0.837813i	$0.316166\pi$
$278$	1.85916	0.111505
$279$	−9.51309	−0.569534
$280$	12.8709	0.769183
$281$	4.85606	0.289688	0.144844	−	0.989455i	$-0.453732\pi$
$281$	4.85606	0.289688	0.144844	+	0.989455i	$0.453732\pi$
$282$	−2.56454	−0.152716
$283$	−22.7269	−1.35098	−0.675488	−	0.737371i	$-0.736067\pi$
$283$	−22.7269	−1.35098	−0.675488	+	0.737371i	$0.736067\pi$
$284$	7.95622	0.472115
$285$	−12.8709	−0.762406
$286$	0.700841	0.0414416
$287$	−0.350871	−0.0207112
$288$	1.00000	0.0589256
$289$	6.18137	0.363610
$290$	−3.17046	−0.186176
$291$	−8.84877	−0.518724
$292$	14.6003	0.854418
$293$	20.8539	1.21830	0.609149	−	0.793056i	$-0.291511\pi$
$293$	20.8539	1.21830	0.609149	+	0.793056i	$0.291511\pi$
$294$	−9.48060	−0.552920
$295$	23.5368	1.37036
$296$	−2.46083	−0.143033
$297$	−2.38421	−0.138346
$298$	−19.4743	−1.12812
$299$	0.293951	0.0169996
$300$	−5.05182	−0.291667
$301$	20.8201	1.20005
$302$	10.1207	0.582381
$303$	7.06630	0.405949
$304$	−4.05963	−0.232836
$305$	−39.2474	−2.24730
$306$	−4.81470	−0.275238
$307$	−14.9464	−0.853034	−0.426517	−	0.904480i	$-0.640259\pi$
$307$	−14.9464	−0.853034	−0.426517	+	0.904480i	$0.640259\pi$
$308$	−9.67902	−0.551513
$309$	−15.5569	−0.885000
$310$	30.1609	1.71302
$311$	−11.7851	−0.668274	−0.334137	−	0.942525i	$-0.608445\pi$
$311$	−11.7851	−0.668274	−0.334137	+	0.942525i	$0.608445\pi$
$312$	−0.293951	−0.0166417
$313$	−18.4565	−1.04322	−0.521612	−	0.853183i	$-0.674669\pi$
$313$	−18.4565	−1.04322	−0.521612	+	0.853183i	$0.674669\pi$
$314$	1.15127	0.0649702
$315$	12.8709	0.725193
$316$	−1.22253	−0.0687726
$317$	−5.85907	−0.329078	−0.164539	−	0.986371i	$-0.552614\pi$
$317$	−5.85907	−0.329078	−0.164539	+	0.986371i	$0.552614\pi$
$318$	−3.57515	−0.200484
$319$	2.38421	0.133490
$320$	−3.17046	−0.177234
$321$	−8.40050	−0.468870
$322$	−4.05963	−0.226234
$323$	19.5459	1.08756
$324$	1.00000	0.0555556
$325$	1.48499	0.0823721
$326$	18.6429	1.03253
$327$	9.45536	0.522883
$328$	0.0864292	0.00477225
$329$	−10.4111	−0.573981
$330$	7.55905	0.416112
$331$	15.9458	0.876458	0.438229	−	0.898863i	$-0.355606\pi$
$331$	15.9458	0.876458	0.438229	+	0.898863i	$0.355606\pi$
$332$	−2.75507	−0.151204
$333$	−2.46083	−0.134852
$334$	−12.8315	−0.702111
$335$	−22.4690	−1.22761
$336$	4.05963	0.221471
$337$	−7.04160	−0.383581	−0.191790	−	0.981436i	$-0.561429\pi$
$337$	−7.04160	−0.383581	−0.191790	+	0.981436i	$0.561429\pi$
$338$	−12.9136	−0.702407
$339$	1.03830	0.0563925
$340$	15.2648	0.827851
$341$	−22.6812	−1.22826
$342$	−4.05963	−0.219520
$343$	−10.0703	−0.543747
$344$	−5.12858	−0.276514
$345$	3.17046	0.170692
$346$	3.76150	0.202219
$347$	−3.18389	−0.170920	−0.0854601	−	0.996342i	$-0.527236\pi$
$347$	−3.18389	−0.170920	−0.0854601	+	0.996342i	$0.527236\pi$
$348$	−1.00000	−0.0536056
$349$	−17.6171	−0.943023	−0.471511	−	0.881860i	$-0.656291\pi$
$349$	−17.6171	−0.943023	−0.471511	+	0.881860i	$0.656291\pi$
$350$	−20.5085	−1.09623
$351$	−0.293951	−0.0156899
$352$	2.38421	0.127079
$353$	23.7099	1.26195	0.630977	−	0.775802i	$-0.282654\pi$
$353$	23.7099	1.26195	0.630977	+	0.775802i	$0.282654\pi$
$354$	7.42377	0.394569
$355$	−25.2249	−1.33880
$356$	−0.550988	−0.0292023
$357$	−19.5459	−1.03448
$358$	6.58132	0.347834
$359$	−37.5276	−1.98063	−0.990316	−	0.138835i	$-0.955664\pi$
$359$	−37.5276	−1.98063	−0.990316	+	0.138835i	$0.955664\pi$
$360$	−3.17046	−0.167098
$361$	−2.51940	−0.132600
$362$	−7.14349	−0.375453
$363$	5.31553	0.278993
$364$	−1.19333	−0.0625476
$365$	−46.2897	−2.42291
$366$	−12.3791	−0.647065
$367$	−4.03811	−0.210788	−0.105394	−	0.994431i	$-0.533610\pi$
$367$	−4.03811	−0.210788	−0.105394	+	0.994431i	$0.533610\pi$
$368$	1.00000	0.0521286
$369$	0.0864292	0.00449932
$370$	7.80195	0.405604
$371$	−14.5138	−0.753518
$372$	9.51309	0.493231
$373$	−35.8490	−1.85619	−0.928096	−	0.372341i	$-0.878555\pi$
$373$	−35.8490	−1.85619	−0.928096	+	0.372341i	$0.878555\pi$
$374$	−11.4793	−0.593579
$375$	0.164284	0.00848358
$376$	2.56454	0.132256
$377$	0.293951	0.0151392
$378$	4.05963	0.208805
$379$	7.66532	0.393741	0.196871	−	0.980429i	$-0.436922\pi$
$379$	7.66532	0.393741	0.196871	+	0.980429i	$0.436922\pi$
$380$	12.8709	0.660263
$381$	−10.8850	−0.557655
$382$	−7.95298	−0.406910
$383$	2.81503	0.143841	0.0719207	−	0.997410i	$-0.477087\pi$
$383$	2.81503	0.143841	0.0719207	+	0.997410i	$0.477087\pi$
$384$	−1.00000	−0.0510310
$385$	30.6869	1.56395
$386$	26.4698	1.34728
$387$	−5.12858	−0.260700
$388$	8.84877	0.449228
$389$	−7.11768	−0.360881	−0.180440	−	0.983586i	$-0.557752\pi$
$389$	−7.11768	−0.360881	−0.180440	+	0.983586i	$0.557752\pi$
$390$	0.931959	0.0471916
$391$	−4.81470	−0.243490
$392$	9.48060	0.478843
$393$	−16.3045	−0.822452
$394$	11.6841	0.588639
$395$	3.87598	0.195022
$396$	2.38421	0.119811
$397$	31.1581	1.56378	0.781891	−	0.623415i	$-0.214255\pi$
$397$	31.1581	1.56378	0.781891	+	0.623415i	$0.214255\pi$
$398$	−17.0525	−0.854763
$399$	−16.4806	−0.825062
$400$	5.05182	0.252591
$401$	24.3982	1.21839	0.609193	−	0.793022i	$-0.291494\pi$
$401$	24.3982	1.21839	0.609193	+	0.793022i	$0.291494\pi$
$402$	−7.08697	−0.353466
$403$	−2.79638	−0.139298
$404$	−7.06630	−0.351562
$405$	−3.17046	−0.157541
$406$	−4.05963	−0.201476
$407$	−5.86713	−0.290823
$408$	4.81470	0.238363
$409$	2.02539	0.100149	0.0500746	−	0.998745i	$-0.484054\pi$
$409$	2.02539	0.100149	0.0500746	+	0.998745i	$0.484054\pi$
$410$	−0.274020	−0.0135329
$411$	−3.96250	−0.195456
$412$	15.5569	0.766432
$413$	30.1378	1.48298
$414$	1.00000	0.0491473
$415$	8.73485	0.428777
$416$	0.293951	0.0144121
$417$	−1.85916	−0.0910433
$418$	−9.67902	−0.473416
$419$	−31.9885	−1.56274	−0.781371	−	0.624067i	$-0.785479\pi$
$419$	−31.9885	−1.56274	−0.781371	+	0.624067i	$0.785479\pi$
$420$	−12.8709	−0.628035
$421$	23.8338	1.16159	0.580794	−	0.814050i	$-0.302742\pi$
$421$	23.8338	1.16159	0.580794	+	0.814050i	$0.302742\pi$
$422$	11.3890	0.554409
$423$	2.56454	0.124692
$424$	3.57515	0.173625
$425$	−24.3230	−1.17984
$426$	−7.95622	−0.385480
$427$	−50.2545	−2.43199
$428$	8.40050	0.406053
$429$	−0.700841	−0.0338369
$430$	16.2600	0.784125
$431$	−0.809784	−0.0390059	−0.0195030	−	0.999810i	$-0.506208\pi$
$431$	−0.809784	−0.0390059	−0.0195030	+	0.999810i	$0.506208\pi$
$432$	−1.00000	−0.0481125
$433$	−28.6544	−1.37704	−0.688521	−	0.725217i	$-0.741740\pi$
$433$	−28.6544	−1.37704	−0.688521	+	0.725217i	$0.741740\pi$
$434$	38.6196	1.85380
$435$	3.17046	0.152012
$436$	−9.45536	−0.452830
$437$	−4.05963	−0.194198
$438$	−14.6003	−0.697630
$439$	37.2593	1.77829	0.889144	−	0.457628i	$-0.151301\pi$
$439$	37.2593	1.77829	0.889144	+	0.457628i	$0.151301\pi$
$440$	−7.55905	−0.360364
$441$	9.48060	0.451457
$442$	−1.41529	−0.0673183
$443$	19.4470	0.923956	0.461978	−	0.886891i	$-0.347140\pi$
$443$	19.4470	0.923956	0.461978	+	0.886891i	$0.347140\pi$
$444$	2.46083	0.116786
$445$	1.74688	0.0828103
$446$	5.85359	0.277175
$447$	19.4743	0.921103
$448$	−4.05963	−0.191800
$449$	8.77796	0.414258	0.207129	−	0.978314i	$-0.433588\pi$
$449$	8.77796	0.414258	0.207129	+	0.978314i	$0.433588\pi$
$450$	5.05182	0.238145
$451$	0.206066	0.00970325
$452$	−1.03830	−0.0488374
$453$	−10.1207	−0.475512
$454$	−7.11647	−0.333992
$455$	3.78341	0.177369
$456$	4.05963	0.190110
$457$	18.4645	0.863734	0.431867	−	0.901937i	$-0.357855\pi$
$457$	18.4645	0.863734	0.431867	+	0.901937i	$0.357855\pi$
$458$	15.6553	0.731525
$459$	4.81470	0.224731
$460$	−3.17046	−0.147823
$461$	9.35761	0.435827	0.217914	−	0.975968i	$-0.430075\pi$
$461$	9.35761	0.435827	0.217914	+	0.975968i	$0.430075\pi$
$462$	9.67902	0.450309
$463$	−25.0879	−1.16593	−0.582967	−	0.812496i	$-0.698108\pi$
$463$	−25.0879	−1.16593	−0.582967	+	0.812496i	$0.698108\pi$
$464$	1.00000	0.0464238
$465$	−30.1609	−1.39868
$466$	19.9822	0.925657
$467$	24.9417	1.15417	0.577083	−	0.816685i	$-0.304191\pi$
$467$	24.9417	1.15417	0.577083	+	0.816685i	$0.304191\pi$
$468$	0.293951	0.0135879
$469$	−28.7705	−1.32850
$470$	−8.13076	−0.375044
$471$	−1.15127	−0.0530479
$472$	−7.42377	−0.341707
$473$	−12.2276	−0.562227
$474$	1.22253	0.0561526
$475$	−20.5085	−0.940995
$476$	19.5459	0.895886
$477$	3.57515	0.163695
$478$	−10.8759	−0.497450
$479$	0.303388	0.0138622	0.00693108	−	0.999976i	$-0.497794\pi$
$479$	0.303388	0.0138622	0.00693108	+	0.999976i	$0.497794\pi$
$480$	3.17046	0.144711
$481$	−0.723362	−0.0329825
$482$	24.8928	1.13383
$483$	4.05963	0.184720
$484$	−5.31553	−0.241615
$485$	−28.0547	−1.27390
$486$	−1.00000	−0.0453609
$487$	11.3504	0.514338	0.257169	−	0.966366i	$-0.417210\pi$
$487$	11.3504	0.514338	0.257169	+	0.966366i	$0.417210\pi$
$488$	12.3791	0.560375
$489$	−18.6429	−0.843059
$490$	−30.0579	−1.35788
$491$	−5.42852	−0.244985	−0.122493	−	0.992469i	$-0.539089\pi$
$491$	−5.42852	−0.244985	−0.122493	+	0.992469i	$0.539089\pi$
$492$	−0.0864292	−0.00389653
$493$	−4.81470	−0.216843
$494$	−1.19333	−0.0536905
$495$	−7.55905	−0.339754
$496$	−9.51309	−0.427150
$497$	−32.2993	−1.44882
$498$	2.75507	0.123458
$499$	29.4610	1.31886	0.659428	−	0.751768i	$-0.270799\pi$
$499$	29.4610	1.31886	0.659428	+	0.751768i	$0.270799\pi$
$500$	−0.164284	−0.00734700
$501$	12.8315	0.573271
$502$	−28.4895	−1.27155
$503$	−15.7124	−0.700580	−0.350290	−	0.936641i	$-0.613917\pi$
$503$	−15.7124	−0.700580	−0.350290	+	0.936641i	$0.613917\pi$
$504$	−4.05963	−0.180830
$505$	22.4034	0.996940
$506$	2.38421	0.105991
$507$	12.9136	0.573513
$508$	10.8850	0.482943
$509$	−30.4241	−1.34853	−0.674263	−	0.738492i	$-0.735538\pi$
$509$	−30.4241	−1.34853	−0.674263	+	0.738492i	$0.735538\pi$
$510$	−15.2648	−0.675938
$511$	−59.2719	−2.62203
$512$	1.00000	0.0441942
$513$	4.05963	0.179237
$514$	−0.224549	−0.00990445
$515$	−49.3224	−2.17341
$516$	5.12858	0.225773
$517$	6.11440	0.268911
$518$	9.99005	0.438938
$519$	−3.76150	−0.165111
$520$	−0.931959	−0.0408691
$521$	11.9135	0.521940	0.260970	−	0.965347i	$-0.415958\pi$
$521$	11.9135	0.521940	0.260970	+	0.965347i	$0.415958\pi$
$522$	1.00000	0.0437688
$523$	6.52292	0.285227	0.142614	−	0.989778i	$-0.454449\pi$
$523$	6.52292	0.285227	0.142614	+	0.989778i	$0.454449\pi$
$524$	16.3045	0.712265
$525$	20.5085	0.895065
$526$	−21.8772	−0.953890
$527$	45.8027	1.99520
$528$	−2.38421	−0.103760
$529$	1.00000	0.0434783
$530$	−11.3349	−0.492355
$531$	−7.42377	−0.322164
$532$	16.4806	0.714525
$533$	0.0254059	0.00110045
$534$	0.550988	0.0238436
$535$	−26.6334	−1.15146
$536$	7.08697	0.306111
$537$	−6.58132	−0.284005
$538$	−21.4223	−0.923580
$539$	22.6038	0.973613
$540$	3.17046	0.136435
$541$	28.3567	1.21915	0.609575	−	0.792729i	$-0.291340\pi$
$541$	28.3567	1.21915	0.609575	+	0.792729i	$0.291340\pi$
$542$	−12.9825	−0.557648
$543$	7.14349	0.306556
$544$	−4.81470	−0.206429
$545$	29.9778	1.28411
$546$	1.19333	0.0510699
$547$	42.0953	1.79987	0.899933	−	0.436029i	$-0.143616\pi$
$547$	42.0953	1.79987	0.899933	+	0.436029i	$0.143616\pi$
$548$	3.96250	0.169269
$549$	12.3791	0.528327
$550$	12.0446	0.513584
$551$	−4.05963	−0.172946
$552$	−1.00000	−0.0425628
$553$	4.96301	0.211049
$554$	18.1731	0.772101
$555$	−7.80195	−0.331175
$556$	1.85916	0.0788458
$557$	9.83697	0.416806	0.208403	−	0.978043i	$-0.433173\pi$
$557$	9.83697	0.416806	0.208403	+	0.978043i	$0.433173\pi$
$558$	−9.51309	−0.402721
$559$	−1.50755	−0.0637625
$560$	12.8709	0.543895
$561$	11.4793	0.484655
$562$	4.85606	0.204840
$563$	15.0782	0.635470	0.317735	−	0.948179i	$-0.397078\pi$
$563$	15.0782	0.635470	0.317735	+	0.948179i	$0.397078\pi$
$564$	−2.56454	−0.107986
$565$	3.29188	0.138490
$566$	−22.7269	−0.955284
$567$	−4.05963	−0.170488
$568$	7.95622	0.333835
$569$	9.87623	0.414033	0.207017	−	0.978337i	$-0.433625\pi$
$569$	9.87623	0.414033	0.207017	+	0.978337i	$0.433625\pi$
$570$	−12.8709	−0.539103
$571$	9.38051	0.392562	0.196281	−	0.980548i	$-0.437114\pi$
$571$	9.38051	0.392562	0.196281	+	0.980548i	$0.437114\pi$
$572$	0.700841	0.0293036
$573$	7.95298	0.332240
$574$	−0.350871	−0.0146451
$575$	5.05182	0.210675
$576$	1.00000	0.0416667
$577$	−4.58130	−0.190722	−0.0953611	−	0.995443i	$-0.530401\pi$
$577$	−4.58130	−0.190722	−0.0953611	+	0.995443i	$0.530401\pi$
$578$	6.18137	0.257111
$579$	−26.4698	−1.10005
$580$	−3.17046	−0.131646
$581$	11.1846	0.464014
$582$	−8.84877	−0.366793
$583$	8.52391	0.353024
$584$	14.6003	0.604165
$585$	−0.931959	−0.0385318
$586$	20.8539	0.861466
$587$	−20.2991	−0.837834	−0.418917	−	0.908025i	$-0.637590\pi$
$587$	−20.2991	−0.837834	−0.418917	+	0.908025i	$0.637590\pi$
$588$	−9.48060	−0.390973
$589$	38.6196	1.59129
$590$	23.5368	0.968993
$591$	−11.6841	−0.480621
$592$	−2.46083	−0.101139
$593$	11.5972	0.476240	0.238120	−	0.971236i	$-0.423469\pi$
$593$	11.5972	0.476240	0.238120	+	0.971236i	$0.423469\pi$
$594$	−2.38421	−0.0978254
$595$	−61.9696	−2.54050
$596$	−19.4743	−0.797698
$597$	17.0525	0.697911
$598$	0.293951	0.0120205
$599$	10.1703	0.415547	0.207773	−	0.978177i	$-0.433378\pi$
$599$	10.1703	0.415547	0.207773	+	0.978177i	$0.433378\pi$
$600$	−5.05182	−0.206240
$601$	−11.1940	−0.456613	−0.228306	−	0.973589i	$-0.573319\pi$
$601$	−11.1940	−0.456613	−0.228306	+	0.973589i	$0.573319\pi$
$602$	20.8201	0.848565
$603$	7.08697	0.288604
$604$	10.1207	0.411805
$605$	16.8527	0.685159
$606$	7.06630	0.287049
$607$	−15.3926	−0.624767	−0.312383	−	0.949956i	$-0.601127\pi$
$607$	−15.3926	−0.624767	−0.312383	+	0.949956i	$0.601127\pi$
$608$	−4.05963	−0.164640
$609$	4.05963	0.164505
$610$	−39.2474	−1.58908
$611$	0.753847	0.0304974
$612$	−4.81470	−0.194623
$613$	15.5968	0.629950	0.314975	−	0.949100i	$-0.398004\pi$
$613$	15.5968	0.629950	0.314975	+	0.949100i	$0.398004\pi$
$614$	−14.9464	−0.603186
$615$	0.274020	0.0110496
$616$	−9.67902	−0.389979
$617$	1.60825	0.0647457	0.0323729	−	0.999476i	$-0.489694\pi$
$617$	1.60825	0.0647457	0.0323729	+	0.999476i	$0.489694\pi$
$618$	−15.5569	−0.625789
$619$	25.8271	1.03808	0.519039	−	0.854750i	$-0.326290\pi$
$619$	25.8271	1.03808	0.519039	+	0.854750i	$0.326290\pi$
$620$	30.1609	1.21129
$621$	−1.00000	−0.0401286
$622$	−11.7851	−0.472541
$623$	2.23681	0.0896158
$624$	−0.293951	−0.0117674
$625$	−24.7382	−0.989529
$626$	−18.4565	−0.737671
$627$	9.67902	0.386543
$628$	1.15127	0.0459408
$629$	11.8482	0.472417
$630$	12.8709	0.512789
$631$	24.8024	0.987367	0.493684	−	0.869642i	$-0.335650\pi$
$631$	24.8024	0.987367	0.493684	+	0.869642i	$0.335650\pi$
$632$	−1.22253	−0.0486296
$633$	−11.3890	−0.452673
$634$	−5.85907	−0.232694
$635$	−34.5104	−1.36950
$636$	−3.57515	−0.141764
$637$	2.78683	0.110418
$638$	2.38421	0.0943919
$639$	7.95622	0.314743
$640$	−3.17046	−0.125323
$641$	−11.1097	−0.438809	−0.219404	−	0.975634i	$-0.570411\pi$
$641$	−11.1097	−0.438809	−0.219404	+	0.975634i	$0.570411\pi$
$642$	−8.40050	−0.331541
$643$	−21.7519	−0.857810	−0.428905	−	0.903350i	$-0.641100\pi$
$643$	−21.7519	−0.857810	−0.428905	+	0.903350i	$0.641100\pi$
$644$	−4.05963	−0.159972
$645$	−16.2600	−0.640235
$646$	19.5459	0.769024
$647$	0.219713	0.00863782	0.00431891	−	0.999991i	$-0.498625\pi$
$647$	0.219713	0.00863782	0.00431891	+	0.999991i	$0.498625\pi$
$648$	1.00000	0.0392837
$649$	−17.6998	−0.694780
$650$	1.48499	0.0582459
$651$	−38.6196	−1.51362
$652$	18.6429	0.730110
$653$	−12.8545	−0.503035	−0.251517	−	0.967853i	$-0.580930\pi$
$653$	−12.8545	−0.503035	−0.251517	+	0.967853i	$0.580930\pi$
$654$	9.45536	0.369734
$655$	−51.6927	−2.01980
$656$	0.0864292	0.00337449
$657$	14.6003	0.569612
$658$	−10.4111	−0.405866
$659$	−10.0094	−0.389911	−0.194955	−	0.980812i	$-0.562456\pi$
$659$	−10.0094	−0.389911	−0.194955	+	0.980812i	$0.562456\pi$
$660$	7.55905	0.294236
$661$	40.4133	1.57189	0.785947	−	0.618293i	$-0.212176\pi$
$661$	40.4133	1.57189	0.785947	+	0.618293i	$0.212176\pi$
$662$	15.9458	0.619750
$663$	1.41529	0.0549651
$664$	−2.75507	−0.106918
$665$	−52.2511	−2.02621
$666$	−2.46083	−0.0953551
$667$	1.00000	0.0387202
$668$	−12.8315	−0.496467
$669$	−5.85359	−0.226313
$670$	−22.4690	−0.868052
$671$	29.5144	1.13939
$672$	4.05963	0.156604
$673$	30.7904	1.18688	0.593442	−	0.804877i	$-0.297769\pi$
$673$	30.7904	1.18688	0.593442	+	0.804877i	$0.297769\pi$
$674$	−7.04160	−0.271232
$675$	−5.05182	−0.194445
$676$	−12.9136	−0.496677
$677$	17.5459	0.674344	0.337172	−	0.941443i	$-0.390530\pi$
$677$	17.5459	0.674344	0.337172	+	0.941443i	$0.390530\pi$
$678$	1.03830	0.0398755
$679$	−35.9227	−1.37859
$680$	15.2648	0.585379
$681$	7.11647	0.272704
$682$	−22.6812	−0.868509
$683$	40.9251	1.56595	0.782977	−	0.622051i	$-0.213700\pi$
$683$	40.9251	1.56595	0.782977	+	0.622051i	$0.213700\pi$
$684$	−4.05963	−0.155224
$685$	−12.5629	−0.480005
$686$	−10.0703	−0.384487
$687$	−15.6553	−0.597288
$688$	−5.12858	−0.195525
$689$	1.05092	0.0400367
$690$	3.17046	0.120697
$691$	32.3186	1.22946	0.614728	−	0.788739i	$-0.289266\pi$
$691$	32.3186	1.22946	0.614728	+	0.788739i	$0.289266\pi$
$692$	3.76150	0.142991
$693$	−9.67902	−0.367676
$694$	−3.18389	−0.120859
$695$	−5.89438	−0.223587
$696$	−1.00000	−0.0379049
$697$	−0.416131	−0.0157621
$698$	−17.6171	−0.666818
$699$	−19.9822	−0.755796
$700$	−20.5085	−0.775149
$701$	1.30420	0.0492591	0.0246295	−	0.999697i	$-0.492159\pi$
$701$	1.30420	0.0492591	0.0246295	+	0.999697i	$0.492159\pi$
$702$	−0.293951	−0.0110945
$703$	9.99005	0.376782
$704$	2.38421	0.0898584
$705$	8.13076	0.306222
$706$	23.7099	0.892336
$707$	28.6866	1.07887
$708$	7.42377	0.279002
$709$	−25.0792	−0.941870	−0.470935	−	0.882168i	$-0.656083\pi$
$709$	−25.0792	−0.941870	−0.470935	+	0.882168i	$0.656083\pi$
$710$	−25.2249	−0.946673
$711$	−1.22253	−0.0458484
$712$	−0.550988	−0.0206491
$713$	−9.51309	−0.356268
$714$	−19.5459	−0.731488
$715$	−2.22199	−0.0830976
$716$	6.58132	0.245956
$717$	10.8759	0.406167
$718$	−37.5276	−1.40052
$719$	1.78248	0.0664752	0.0332376	−	0.999447i	$-0.489418\pi$
$719$	1.78248	0.0664752	0.0332376	+	0.999447i	$0.489418\pi$
$720$	−3.17046	−0.118156
$721$	−63.1552	−2.35202
$722$	−2.51940	−0.0937623
$723$	−24.8928	−0.925772
$724$	−7.14349	−0.265486
$725$	5.05182	0.187620
$726$	5.31553	0.197278
$727$	22.6816	0.841213	0.420606	−	0.907243i	$-0.361817\pi$
$727$	22.6816	0.841213	0.420606	+	0.907243i	$0.361817\pi$
$728$	−1.19333	−0.0442278
$729$	1.00000	0.0370370
$730$	−46.2897	−1.71326
$731$	24.6926	0.913288
$732$	−12.3791	−0.457544
$733$	42.0778	1.55418	0.777090	−	0.629390i	$-0.216695\pi$
$733$	42.0778	1.55418	0.777090	+	0.629390i	$0.216695\pi$
$734$	−4.03811	−0.149049
$735$	30.0579	1.10870
$736$	1.00000	0.0368605
$737$	16.8968	0.622403
$738$	0.0864292	0.00318150
$739$	−15.9487	−0.586684	−0.293342	−	0.956008i	$-0.594767\pi$
$739$	−15.9487	−0.586684	−0.293342	+	0.956008i	$0.594767\pi$
$740$	7.80195	0.286806
$741$	1.19333	0.0438381
$742$	−14.5138	−0.532818
$743$	−45.1624	−1.65685	−0.828425	−	0.560100i	$-0.810763\pi$
$743$	−45.1624	−1.65685	−0.828425	+	0.560100i	$0.810763\pi$
$744$	9.51309	0.348767
$745$	61.7425	2.26207
$746$	−35.8490	−1.31253
$747$	−2.75507	−0.100803
$748$	−11.4793	−0.419724
$749$	−34.1029	−1.24609
$750$	0.164284	0.00599880
$751$	−11.3286	−0.413388	−0.206694	−	0.978406i	$-0.566270\pi$
$751$	−11.3286	−0.413388	−0.206694	+	0.978406i	$0.566270\pi$
$752$	2.56454	0.0935190
$753$	28.4895	1.03822
$754$	0.293951	0.0107051
$755$	−32.0873	−1.16778
$756$	4.05963	0.147647
$757$	−29.7784	−1.08232	−0.541158	−	0.840921i	$-0.682014\pi$
$757$	−29.7784	−1.08232	−0.541158	+	0.840921i	$0.682014\pi$
$758$	7.66532	0.278417
$759$	−2.38421	−0.0865414
$760$	12.8709	0.466877
$761$	32.0083	1.16030	0.580150	−	0.814510i	$-0.302994\pi$
$761$	32.0083	1.16030	0.580150	+	0.814510i	$0.302994\pi$
$762$	−10.8850	−0.394321
$763$	38.3853	1.38964
$764$	−7.95298	−0.287729
$765$	15.2648	0.551901
$766$	2.81503	0.101711
$767$	−2.18222	−0.0787955
$768$	−1.00000	−0.0360844
$769$	−20.7143	−0.746975	−0.373488	−	0.927635i	$-0.621838\pi$
$769$	−20.7143	−0.746975	−0.373488	+	0.927635i	$0.621838\pi$
$770$	30.6869	1.10588
$771$	0.224549	0.00808695
$772$	26.4698	0.952670
$773$	21.1528	0.760815	0.380408	−	0.924819i	$-0.375784\pi$
$773$	21.1528	0.760815	0.380408	+	0.924819i	$0.375784\pi$
$774$	−5.12858	−0.184343
$775$	−48.0584	−1.72631
$776$	8.84877	0.317652
$777$	−9.99005	−0.358391
$778$	−7.11768	−0.255181
$779$	−0.350871	−0.0125712
$780$	0.931959	0.0333695
$781$	18.9693	0.678775
$782$	−4.81470	−0.172173
$783$	−1.00000	−0.0357371
$784$	9.48060	0.338593
$785$	−3.65007	−0.130277
$786$	−16.3045	−0.581562
$787$	11.1704	0.398183	0.199092	−	0.979981i	$-0.436201\pi$
$787$	11.1704	0.398183	0.199092	+	0.979981i	$0.436201\pi$
$788$	11.6841	0.416230
$789$	21.8772	0.778848
$790$	3.87598	0.137901
$791$	4.21510	0.149872
$792$	2.38421	0.0847193
$793$	3.63884	0.129219
$794$	31.1581	1.10576
$795$	11.3349	0.402006
$796$	−17.0525	−0.604408
$797$	27.7121	0.981614	0.490807	−	0.871268i	$-0.336702\pi$
$797$	27.7121	0.981614	0.490807	+	0.871268i	$0.336702\pi$
$798$	−16.4806	−0.583407
$799$	−12.3475	−0.436822
$800$	5.05182	0.178609
$801$	−0.550988	−0.0194682
$802$	24.3982	0.861529
$803$	34.8102	1.22843
$804$	−7.08697	−0.249938
$805$	12.8709	0.453640
$806$	−2.79638	−0.0984982
$807$	21.4223	0.754100
$808$	−7.06630	−0.248592
$809$	2.21695	0.0779439	0.0389720	−	0.999240i	$-0.487592\pi$
$809$	2.21695	0.0779439	0.0389720	+	0.999240i	$0.487592\pi$
$810$	−3.17046	−0.111399
$811$	−11.0125	−0.386700	−0.193350	−	0.981130i	$-0.561935\pi$
$811$	−11.0125	−0.386700	−0.193350	+	0.981130i	$0.561935\pi$
$812$	−4.05963	−0.142465
$813$	12.9825	0.455318
$814$	−5.86713	−0.205643
$815$	−59.1064	−2.07041
$816$	4.81470	0.168548
$817$	20.8201	0.728404
$818$	2.02539	0.0708162
$819$	−1.19333	−0.0416984
$820$	−0.274020	−0.00956921
$821$	−16.1404	−0.563303	−0.281651	−	0.959517i	$-0.590882\pi$
$821$	−16.1404	−0.563303	−0.281651	+	0.959517i	$0.590882\pi$
$822$	−3.96250	−0.138208
$823$	34.4125	1.19954	0.599771	−	0.800171i	$-0.295258\pi$
$823$	34.4125	1.19954	0.599771	+	0.800171i	$0.295258\pi$
$824$	15.5569	0.541949
$825$	−12.0446	−0.419339
$826$	30.1378	1.04863
$827$	−2.90923	−0.101164	−0.0505820	−	0.998720i	$-0.516108\pi$
$827$	−2.90923	−0.101164	−0.0505820	+	0.998720i	$0.516108\pi$
$828$	1.00000	0.0347524
$829$	45.1976	1.56978	0.784888	−	0.619637i	$-0.212720\pi$
$829$	45.1976	1.56978	0.784888	+	0.619637i	$0.212720\pi$
$830$	8.73485	0.303191
$831$	−18.1731	−0.630418
$832$	0.293951	0.0101909
$833$	−45.6463	−1.58155
$834$	−1.85916	−0.0643773
$835$	40.6819	1.40785
$836$	−9.67902	−0.334756
$837$	9.51309	0.328820
$838$	−31.9885	−1.10503
$839$	−16.2206	−0.559996	−0.279998	−	0.960001i	$-0.590334\pi$
$839$	−16.2206	−0.559996	−0.279998	+	0.960001i	$0.590334\pi$
$840$	−12.8709	−0.444088
$841$	1.00000	0.0344828
$842$	23.8338	0.821367
$843$	−4.85606	−0.167251
$844$	11.3890	0.392027
$845$	40.9420	1.40845
$846$	2.56454	0.0881705
$847$	21.5791	0.741467
$848$	3.57515	0.122771
$849$	22.7269	0.779987
$850$	−24.3230	−0.834272
$851$	−2.46083	−0.0843561
$852$	−7.95622	−0.272576
$853$	31.2695	1.07065	0.535323	−	0.844647i	$-0.320190\pi$
$853$	31.2695	1.07065	0.535323	+	0.844647i	$0.320190\pi$
$854$	−50.2545	−1.71967
$855$	12.8709	0.440175
$856$	8.40050	0.287123
$857$	−42.4846	−1.45125	−0.725623	−	0.688093i	$-0.758448\pi$
$857$	−42.4846	−1.45125	−0.725623	+	0.688093i	$0.758448\pi$
$858$	−0.700841	−0.0239263
$859$	−26.8892	−0.917446	−0.458723	−	0.888579i	$-0.651693\pi$
$859$	−26.8892	−0.917446	−0.458723	+	0.888579i	$0.651693\pi$
$860$	16.2600	0.554460
$861$	0.350871	0.0119576
$862$	−0.809784	−0.0275814
$863$	−33.7655	−1.14939	−0.574695	−	0.818368i	$-0.694879\pi$
$863$	−33.7655	−1.14939	−0.574695	+	0.818368i	$0.694879\pi$
$864$	−1.00000	−0.0340207
$865$	−11.9257	−0.405485
$866$	−28.6544	−0.973715
$867$	−6.18137	−0.209930
$868$	38.6196	1.31084
$869$	−2.91477	−0.0988767
$870$	3.17046	0.107489
$871$	2.08322	0.0705872
$872$	−9.45536	−0.320199
$873$	8.84877	0.299485
$874$	−4.05963	−0.137319
$875$	0.666932	0.0225464
$876$	−14.6003	−0.493299
$877$	8.49590	0.286886	0.143443	−	0.989659i	$-0.454183\pi$
$877$	8.49590	0.286886	0.143443	+	0.989659i	$0.454183\pi$
$878$	37.2593	1.25744
$879$	−20.8539	−0.703384
$880$	−7.55905	−0.254815
$881$	36.4843	1.22919	0.614593	−	0.788844i	$-0.289320\pi$
$881$	36.4843	1.22919	0.614593	+	0.788844i	$0.289320\pi$
$882$	9.48060	0.319228
$883$	−26.6516	−0.896896	−0.448448	−	0.893809i	$-0.648023\pi$
$883$	−26.6516	−0.896896	−0.448448	+	0.893809i	$0.648023\pi$
$884$	−1.41529	−0.0476012
$885$	−23.5368	−0.791180
$886$	19.4470	0.653336
$887$	−25.1504	−0.844469	−0.422235	−	0.906487i	$-0.638754\pi$
$887$	−25.1504	−0.844469	−0.422235	+	0.906487i	$0.638754\pi$
$888$	2.46083	0.0825799
$889$	−44.1890	−1.48205
$890$	1.74688	0.0585557
$891$	2.38421	0.0798741
$892$	5.85359	0.195993
$893$	−10.4111	−0.348393
$894$	19.4743	0.651318
$895$	−20.8658	−0.697468
$896$	−4.05963	−0.135623
$897$	−0.293951	−0.00981473
$898$	8.77796	0.292924
$899$	−9.51309	−0.317279
$900$	5.05182	0.168394
$901$	−17.2133	−0.573457
$902$	0.206066	0.00686123
$903$	−20.8201	−0.692851
$904$	−1.03830	−0.0345332
$905$	22.6481	0.752850
$906$	−10.1207	−0.336238
$907$	12.6380	0.419638	0.209819	−	0.977740i	$-0.432712\pi$
$907$	12.6380	0.419638	0.209819	+	0.977740i	$0.432712\pi$
$908$	−7.11647	−0.236168
$909$	−7.06630	−0.234374
$910$	3.78341	0.125419
$911$	−44.6885	−1.48059	−0.740297	−	0.672279i	$-0.765315\pi$
$911$	−44.6885	−1.48059	−0.740297	+	0.672279i	$0.765315\pi$
$912$	4.05963	0.134428
$913$	−6.56868	−0.217391
$914$	18.4645	0.610752
$915$	39.2474	1.29748
$916$	15.6553	0.517266
$917$	−66.1902	−2.18579
$918$	4.81470	0.158909
$919$	−13.8839	−0.457987	−0.228994	−	0.973428i	$-0.573544\pi$
$919$	−13.8839	−0.457987	−0.228994	+	0.973428i	$0.573544\pi$
$920$	−3.17046	−0.104527
$921$	14.9464	0.492499
$922$	9.35761	0.308176
$923$	2.33874	0.0769804
$924$	9.67902	0.318416
$925$	−12.4316	−0.408750
$926$	−25.0879	−0.824439
$927$	15.5569	0.510955
$928$	1.00000	0.0328266
$929$	−12.5477	−0.411676	−0.205838	−	0.978586i	$-0.565992\pi$
$929$	−12.5477	−0.411676	−0.205838	+	0.978586i	$0.565992\pi$
$930$	−30.1609	−0.989014
$931$	−38.4877	−1.26138
$932$	19.9822	0.654539
$933$	11.7851	0.385828
$934$	24.9417	0.816119
$935$	36.3946	1.19023
$936$	0.293951	0.00960808
$937$	−9.79742	−0.320068	−0.160034	−	0.987112i	$-0.551160\pi$
$937$	−9.79742	−0.320068	−0.160034	+	0.987112i	$0.551160\pi$
$938$	−28.7705	−0.939390
$939$	18.4565	0.602306
$940$	−8.13076	−0.265196
$941$	33.6451	1.09680	0.548400	−	0.836216i	$-0.315237\pi$
$941$	33.6451	1.09680	0.548400	+	0.836216i	$0.315237\pi$
$942$	−1.15127	−0.0375105
$943$	0.0864292	0.00281452
$944$	−7.42377	−0.241623
$945$	−12.8709	−0.418690
$946$	−12.2276	−0.397554
$947$	31.0362	1.00854	0.504271	−	0.863546i	$-0.331761\pi$
$947$	31.0362	1.00854	0.504271	+	0.863546i	$0.331761\pi$
$948$	1.22253	0.0397059
$949$	4.29177	0.139317
$950$	−20.5085	−0.665384
$951$	5.85907	0.189993
$952$	19.5459	0.633487
$953$	−35.5274	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$953$	−35.5274	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$954$	3.57515	0.115750
$955$	25.2146	0.815925
$956$	−10.8759	−0.351751
$957$	−2.38421	−0.0770706
$958$	0.303388	0.00980203
$959$	−16.0863	−0.519453
$960$	3.17046	0.102326
$961$	59.4988	1.91932
$962$	−0.723362	−0.0233221
$963$	8.40050	0.270702
$964$	24.8928	0.801742
$965$	−83.9215	−2.70153
$966$	4.05963	0.130616
$967$	−51.9511	−1.67064	−0.835318	−	0.549768i	$-0.814716\pi$
$967$	−51.9511	−1.67064	−0.835318	+	0.549768i	$0.814716\pi$
$968$	−5.31553	−0.170848
$969$	−19.5459	−0.627905
$970$	−28.0547	−0.900781
$971$	−30.2755	−0.971586	−0.485793	−	0.874074i	$-0.661469\pi$
$971$	−30.2755	−0.971586	−0.485793	+	0.874074i	$0.661469\pi$
$972$	−1.00000	−0.0320750
$973$	−7.54749	−0.241961
$974$	11.3504	0.363692
$975$	−1.48499	−0.0475576
$976$	12.3791	0.396245
$977$	36.2839	1.16082	0.580412	−	0.814323i	$-0.302892\pi$
$977$	36.2839	1.16082	0.580412	+	0.814323i	$0.302892\pi$
$978$	−18.6429	−0.596133
$979$	−1.31367	−0.0419851
$980$	−30.0579	−0.960164
$981$	−9.45536	−0.301886
$982$	−5.42852	−0.173231
$983$	−3.09383	−0.0986780	−0.0493390	−	0.998782i	$-0.515711\pi$
$983$	−3.09383	−0.0986780	−0.0493390	+	0.998782i	$0.515711\pi$
$984$	−0.0864292	−0.00275526
$985$	−37.0441	−1.18032
$986$	−4.81470	−0.153331
$987$	10.4111	0.331388
$988$	−1.19333	−0.0379649
$989$	−5.12858	−0.163079
$990$	−7.55905	−0.240242
$991$	−41.6515	−1.32310	−0.661552	−	0.749899i	$-0.730102\pi$
$991$	−41.6515	−1.32310	−0.661552	+	0.749899i	$0.730102\pi$
$992$	−9.51309	−0.302041
$993$	−15.9458	−0.506023
$994$	−32.2993	−1.02447
$995$	54.0641	1.71395
$996$	2.75507	0.0872978
$997$	−38.8830	−1.23144	−0.615719	−	0.787966i	$-0.711134\pi$
$997$	−38.8830	−1.23144	−0.615719	+	0.787966i	$0.711134\pi$
$998$	29.4610	0.932572
$999$	2.46083	0.0778571

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bj.1.1	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.17046 q^{5} -1.00000 q^{6} -4.05963 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.17046 q^{5} -1.00000 q^{6} -4.05963 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.17046 q^{10} +2.38421 q^{11} -1.00000 q^{12} +0.293951 q^{13} -4.05963 q^{14} +3.17046 q^{15} +1.00000 q^{16} -4.81470 q^{17} +1.00000 q^{18} -4.05963 q^{19} -3.17046 q^{20} +4.05963 q^{21} +2.38421 q^{22} +1.00000 q^{23} -1.00000 q^{24} +5.05182 q^{25} +0.293951 q^{26} -1.00000 q^{27} -4.05963 q^{28} +1.00000 q^{29} +3.17046 q^{30} -9.51309 q^{31} +1.00000 q^{32} -2.38421 q^{33} -4.81470 q^{34} +12.8709 q^{35} +1.00000 q^{36} -2.46083 q^{37} -4.05963 q^{38} -0.293951 q^{39} -3.17046 q^{40} +0.0864292 q^{41} +4.05963 q^{42} -5.12858 q^{43} +2.38421 q^{44} -3.17046 q^{45} +1.00000 q^{46} +2.56454 q^{47} -1.00000 q^{48} +9.48060 q^{49} +5.05182 q^{50} +4.81470 q^{51} +0.293951 q^{52} +3.57515 q^{53} -1.00000 q^{54} -7.55905 q^{55} -4.05963 q^{56} +4.05963 q^{57} +1.00000 q^{58} -7.42377 q^{59} +3.17046 q^{60} +12.3791 q^{61} -9.51309 q^{62} -4.05963 q^{63} +1.00000 q^{64} -0.931959 q^{65} -2.38421 q^{66} +7.08697 q^{67} -4.81470 q^{68} -1.00000 q^{69} +12.8709 q^{70} +7.95622 q^{71} +1.00000 q^{72} +14.6003 q^{73} -2.46083 q^{74} -5.05182 q^{75} -4.05963 q^{76} -9.67902 q^{77} -0.293951 q^{78} -1.22253 q^{79} -3.17046 q^{80} +1.00000 q^{81} +0.0864292 q^{82} -2.75507 q^{83} +4.05963 q^{84} +15.2648 q^{85} -5.12858 q^{86} -1.00000 q^{87} +2.38421 q^{88} -0.550988 q^{89} -3.17046 q^{90} -1.19333 q^{91} +1.00000 q^{92} +9.51309 q^{93} +2.56454 q^{94} +12.8709 q^{95} -1.00000 q^{96} +8.84877 q^{97} +9.48060 q^{98} +2.38421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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