LMFDB - Embedded newform 4002.2.a.bd.1.3

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:	$4$
Coefficient field:	4.4.19796.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 8 $ x^4 - x^3 - 7*x^2 + x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.82082$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.38424 q^{5} +1.00000 q^{6} -3.20506 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.38424 q^{5} +1.00000 q^{6} -3.20506 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.38424 q^{10} +5.82082 q^{11} +1.00000 q^{12} -0.957038 q^{13} -3.20506 q^{14} +1.38424 q^{15} +1.00000 q^{16} -3.20506 q^{17} +1.00000 q^{18} +3.20506 q^{19} +1.38424 q^{20} -3.20506 q^{21} +5.82082 q^{22} -1.00000 q^{23} +1.00000 q^{24} -3.08388 q^{25} -0.957038 q^{26} +1.00000 q^{27} -3.20506 q^{28} +1.00000 q^{29} +1.38424 q^{30} +5.72552 q^{31} +1.00000 q^{32} +5.82082 q^{33} -3.20506 q^{34} -4.43658 q^{35} +1.00000 q^{36} +5.22009 q^{37} +3.20506 q^{38} -0.957038 q^{39} +1.38424 q^{40} -0.0838762 q^{41} -3.20506 q^{42} +9.91408 q^{43} +5.82082 q^{44} +1.38424 q^{45} -1.00000 q^{46} +8.60433 q^{47} +1.00000 q^{48} +3.27243 q^{49} -3.08388 q^{50} -3.20506 q^{51} -0.957038 q^{52} +12.9814 q^{53} +1.00000 q^{54} +8.05742 q^{55} -3.20506 q^{56} +3.20506 q^{57} +1.00000 q^{58} +1.55777 q^{59} +1.38424 q^{60} -11.1342 q^{61} +5.72552 q^{62} -3.20506 q^{63} +1.00000 q^{64} -1.32477 q^{65} +5.82082 q^{66} +8.39510 q^{67} -3.20506 q^{68} -1.00000 q^{69} -4.43658 q^{70} -11.0947 q^{71} +1.00000 q^{72} +9.74993 q^{73} +5.22009 q^{74} -3.08388 q^{75} +3.20506 q^{76} -18.6561 q^{77} -0.957038 q^{78} -8.33190 q^{79} +1.38424 q^{80} +1.00000 q^{81} -0.0838762 q^{82} -10.5449 q^{83} -3.20506 q^{84} -4.43658 q^{85} +9.91408 q^{86} +1.00000 q^{87} +5.82082 q^{88} -8.98145 q^{89} +1.38424 q^{90} +3.06737 q^{91} -1.00000 q^{92} +5.72552 q^{93} +8.60433 q^{94} +4.43658 q^{95} +1.00000 q^{96} +7.71987 q^{97} +3.27243 q^{98} +5.82082 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9 - q^10 + 15 * q^11 + 4 * q^12 + 11 * q^13 + 2 * q^14 - q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^19 - q^20 + 2 * q^21 + 15 * q^22 - 4 * q^23 + 4 * q^24 - q^25 + 11 * q^26 + 4 * q^27 + 2 * q^28 + 4 * q^29 - q^30 - 5 * q^31 + 4 * q^32 + 15 * q^33 + 2 * q^34 - 16 * q^35 + 4 * q^36 + 3 * q^37 - 2 * q^38 + 11 * q^39 - q^40 + 11 * q^41 + 2 * q^42 + 10 * q^43 + 15 * q^44 - q^45 - 4 * q^46 + 10 * q^47 + 4 * q^48 - q^50 + 2 * q^51 + 11 * q^52 + 24 * q^53 + 4 * q^54 - 7 * q^55 + 2 * q^56 - 2 * q^57 + 4 * q^58 + q^59 - q^60 + 3 * q^61 - 5 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^65 + 15 * q^66 + 7 * q^67 + 2 * q^68 - 4 * q^69 - 16 * q^70 - 13 * q^71 + 4 * q^72 - 2 * q^73 + 3 * q^74 - q^75 - 2 * q^76 - 4 * q^77 + 11 * q^78 - 22 * q^79 - q^80 + 4 * q^81 + 11 * q^82 - 16 * q^83 + 2 * q^84 - 16 * q^85 + 10 * q^86 + 4 * q^87 + 15 * q^88 - 8 * q^89 - q^90 + 14 * q^91 - 4 * q^92 - 5 * q^93 + 10 * q^94 + 16 * q^95 + 4 * q^96 - 4 * q^97 + 15 * 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
$5$	1.38424	0.619051	0.309526	−	0.950891i	$-0.399830\pi$
$5$	1.38424	0.619051	0.309526	+	0.950891i	$0.399830\pi$
$6$	1.00000	0.408248
$7$	−3.20506	−1.21140	−0.605700	−	0.795693i	$-0.707107\pi$
$7$	−3.20506	−1.21140	−0.605700	+	0.795693i	$0.707107\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.38424	0.437736
$11$	5.82082	1.75504	0.877522	−	0.479536i	$-0.159195\pi$
$11$	5.82082	1.75504	0.877522	+	0.479536i	$0.159195\pi$
$12$	1.00000	0.288675
$13$	−0.957038	−0.265435	−0.132717	−	0.991154i	$-0.542370\pi$
$13$	−0.957038	−0.265435	−0.132717	+	0.991154i	$0.542370\pi$
$14$	−3.20506	−0.856589
$15$	1.38424	0.357410
$16$	1.00000	0.250000
$17$	−3.20506	−0.777342	−0.388671	−	0.921377i	$-0.627066\pi$
$17$	−3.20506	−0.777342	−0.388671	+	0.921377i	$0.627066\pi$
$18$	1.00000	0.235702
$19$	3.20506	0.735292	0.367646	−	0.929966i	$-0.380164\pi$
$19$	3.20506	0.735292	0.367646	+	0.929966i	$0.380164\pi$
$20$	1.38424	0.309526
$21$	−3.20506	−0.699402
$22$	5.82082	1.24100
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	−3.08388	−0.616775
$26$	−0.957038	−0.187691
$27$	1.00000	0.192450
$28$	−3.20506	−0.605700
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	1.38424	0.252727
$31$	5.72552	1.02833	0.514167	−	0.857690i	$-0.328101\pi$
$31$	5.72552	1.02833	0.514167	+	0.857690i	$0.328101\pi$
$32$	1.00000	0.176777
$33$	5.82082	1.01328
$34$	−3.20506	−0.549664
$35$	−4.43658	−0.749919
$36$	1.00000	0.166667
$37$	5.22009	0.858178	0.429089	−	0.903262i	$-0.358835\pi$
$37$	5.22009	0.858178	0.429089	+	0.903262i	$0.358835\pi$
$38$	3.20506	0.519930
$39$	−0.957038	−0.153249
$40$	1.38424	0.218868
$41$	−0.0838762	−0.0130993	−0.00654963	−	0.999979i	$-0.502085\pi$
$41$	−0.0838762	−0.0130993	−0.00654963	+	0.999979i	$0.502085\pi$
$42$	−3.20506	−0.494552
$43$	9.91408	1.51188	0.755941	−	0.654639i	$-0.227180\pi$
$43$	9.91408	1.51188	0.755941	+	0.654639i	$0.227180\pi$
$44$	5.82082	0.877522
$45$	1.38424	0.206350
$46$	−1.00000	−0.147442
$47$	8.60433	1.25507	0.627536	−	0.778588i	$-0.284064\pi$
$47$	8.60433	1.25507	0.627536	+	0.778588i	$0.284064\pi$
$48$	1.00000	0.144338
$49$	3.27243	0.467490
$50$	−3.08388	−0.436126
$51$	−3.20506	−0.448799
$52$	−0.957038	−0.132717
$53$	12.9814	1.78314	0.891569	−	0.452884i	$-0.149605\pi$
$53$	12.9814	1.78314	0.891569	+	0.452884i	$0.149605\pi$
$54$	1.00000	0.136083
$55$	8.05742	1.08646
$56$	−3.20506	−0.428295
$57$	3.20506	0.424521
$58$	1.00000	0.131306
$59$	1.55777	0.202804	0.101402	−	0.994846i	$-0.467667\pi$
$59$	1.55777	0.202804	0.101402	+	0.994846i	$0.467667\pi$
$60$	1.38424	0.178705
$61$	−11.1342	−1.42558	−0.712792	−	0.701375i	$-0.752570\pi$
$61$	−11.1342	−1.42558	−0.712792	+	0.701375i	$0.752570\pi$
$62$	5.72552	0.727142
$63$	−3.20506	−0.403800
$64$	1.00000	0.125000
$65$	−1.32477	−0.164318
$66$	5.82082	0.716494
$67$	8.39510	1.02562	0.512812	−	0.858501i	$-0.328604\pi$
$67$	8.39510	1.02562	0.512812	+	0.858501i	$0.328604\pi$
$68$	−3.20506	−0.388671
$69$	−1.00000	−0.120386
$70$	−4.43658	−0.530273
$71$	−11.0947	−1.31670	−0.658351	−	0.752711i	$-0.728746\pi$
$71$	−11.0947	−1.31670	−0.658351	+	0.752711i	$0.728746\pi$
$72$	1.00000	0.117851
$73$	9.74993	1.14114	0.570571	−	0.821248i	$-0.306722\pi$
$73$	9.74993	1.14114	0.570571	+	0.821248i	$0.306722\pi$
$74$	5.22009	0.606823
$75$	−3.08388	−0.356095
$76$	3.20506	0.367646
$77$	−18.6561	−2.12606
$78$	−0.957038	−0.108363
$79$	−8.33190	−0.937412	−0.468706	−	0.883354i	$-0.655280\pi$
$79$	−8.33190	−0.937412	−0.468706	+	0.883354i	$0.655280\pi$
$80$	1.38424	0.154763
$81$	1.00000	0.111111
$82$	−0.0838762	−0.00926258
$83$	−10.5449	−1.15745	−0.578725	−	0.815523i	$-0.696449\pi$
$83$	−10.5449	−1.15745	−0.578725	+	0.815523i	$0.696449\pi$
$84$	−3.20506	−0.349701
$85$	−4.43658	−0.481215
$86$	9.91408	1.06906
$87$	1.00000	0.107211
$88$	5.82082	0.620502
$89$	−8.98145	−0.952031	−0.476016	−	0.879437i	$-0.657919\pi$
$89$	−8.98145	−0.952031	−0.476016	+	0.879437i	$0.657919\pi$
$90$	1.38424	0.145912
$91$	3.06737	0.321548
$92$	−1.00000	−0.104257
$93$	5.72552	0.593709
$94$	8.60433	0.887469
$95$	4.43658	0.455184
$96$	1.00000	0.102062
$97$	7.71987	0.783834	0.391917	−	0.920001i	$-0.371812\pi$
$97$	7.71987	0.783834	0.391917	+	0.920001i	$0.371812\pi$
$98$	3.27243	0.330566
$99$	5.82082	0.585015
$100$	−3.08388	−0.308388
$101$	2.68461	0.267128	0.133564	−	0.991040i	$-0.457358\pi$
$101$	2.68461	0.267128	0.133564	+	0.991040i	$0.457358\pi$
$102$	−3.20506	−0.317349
$103$	−17.6490	−1.73901	−0.869503	−	0.493928i	$-0.835561\pi$
$103$	−17.6490	−1.73901	−0.869503	+	0.493928i	$0.835561\pi$
$104$	−0.957038	−0.0938453
$105$	−4.43658	−0.432966
$106$	12.9814	1.26087
$107$	11.9923	1.15934	0.579670	−	0.814851i	$-0.303182\pi$
$107$	11.9923	1.15934	0.579670	+	0.814851i	$0.303182\pi$
$108$	1.00000	0.0962250
$109$	−1.91408	−0.183335	−0.0916676	−	0.995790i	$-0.529220\pi$
$109$	−1.91408	−0.183335	−0.0916676	+	0.995790i	$0.529220\pi$
$110$	8.05742	0.768245
$111$	5.22009	0.495469
$112$	−3.20506	−0.302850
$113$	−9.72347	−0.914707	−0.457354	−	0.889285i	$-0.651203\pi$
$113$	−9.72347	−0.914707	−0.457354	+	0.889285i	$0.651203\pi$
$114$	3.20506	0.300182
$115$	−1.38424	−0.129081
$116$	1.00000	0.0928477
$117$	−0.957038	−0.0884782
$118$	1.55777	0.143404
$119$	10.2724	0.941672
$120$	1.38424	0.126363
$121$	22.8820	2.08018
$122$	−11.1342	−1.00804
$123$	−0.0838762	−0.00756286
$124$	5.72552	0.514167
$125$	−11.1900	−1.00087
$126$	−3.20506	−0.285530
$127$	17.0088	1.50929	0.754644	−	0.656135i	$-0.227810\pi$
$127$	17.0088	1.50929	0.754644	+	0.656135i	$0.227810\pi$
$128$	1.00000	0.0883883
$129$	9.91408	0.872886
$130$	−1.32477	−0.116190
$131$	−3.17501	−0.277402	−0.138701	−	0.990334i	$-0.544293\pi$
$131$	−3.17501	−0.277402	−0.138701	+	0.990334i	$0.544293\pi$
$132$	5.82082	0.506638
$133$	−10.2724	−0.890733
$134$	8.39510	0.725226
$135$	1.38424	0.119137
$136$	−3.20506	−0.274832
$137$	−16.0253	−1.36914	−0.684568	−	0.728949i	$-0.740009\pi$
$137$	−16.0253	−1.36914	−0.684568	+	0.728949i	$0.740009\pi$
$138$	−1.00000	−0.0851257
$139$	4.46664	0.378855	0.189428	−	0.981895i	$-0.439337\pi$
$139$	4.46664	0.378855	0.189428	+	0.981895i	$0.439337\pi$
$140$	−4.43658	−0.374960
$141$	8.60433	0.724616
$142$	−11.0947	−0.931049
$143$	−5.57075	−0.465850
$144$	1.00000	0.0833333
$145$	1.38424	0.114955
$146$	9.74993	0.806910
$147$	3.27243	0.269906
$148$	5.22009	0.429089
$149$	−1.79076	−0.146705	−0.0733526	−	0.997306i	$-0.523370\pi$
$149$	−1.79076	−0.146705	−0.0733526	+	0.997306i	$0.523370\pi$
$150$	−3.08388	−0.251797
$151$	22.5666	1.83644	0.918221	−	0.396068i	$-0.129626\pi$
$151$	22.5666	1.83644	0.918221	+	0.396068i	$0.129626\pi$
$152$	3.20506	0.259965
$153$	−3.20506	−0.259114
$154$	−18.6561	−1.50335
$155$	7.92550	0.636592
$156$	−0.957038	−0.0766244
$157$	4.46960	0.356713	0.178356	−	0.983966i	$-0.442922\pi$
$157$	4.46960	0.356713	0.178356	+	0.983966i	$0.442922\pi$
$158$	−8.33190	−0.662850
$159$	12.9814	1.02950
$160$	1.38424	0.109434
$161$	3.20506	0.252594
$162$	1.00000	0.0785674
$163$	−20.5128	−1.60668	−0.803342	−	0.595518i	$-0.796947\pi$
$163$	−20.5128	−1.60668	−0.803342	+	0.595518i	$0.796947\pi$
$164$	−0.0838762	−0.00654963
$165$	8.05742	0.627269
$166$	−10.5449	−0.818440
$167$	−7.56137	−0.585117	−0.292558	−	0.956248i	$-0.594507\pi$
$167$	−7.56137	−0.585117	−0.292558	+	0.956248i	$0.594507\pi$
$168$	−3.20506	−0.247276
$169$	−12.0841	−0.929544
$170$	−4.43658	−0.340270
$171$	3.20506	0.245097
$172$	9.91408	0.755941
$173$	6.57428	0.499833	0.249916	−	0.968267i	$-0.419597\pi$
$173$	6.57428	0.499833	0.249916	+	0.968267i	$0.419597\pi$
$174$	1.00000	0.0758098
$175$	9.88402	0.747162
$176$	5.82082	0.438761
$177$	1.55777	0.117089
$178$	−8.98145	−0.673188
$179$	1.94413	0.145311	0.0726557	−	0.997357i	$-0.476853\pi$
$179$	1.94413	0.145311	0.0726557	+	0.997357i	$0.476853\pi$
$180$	1.38424	0.103175
$181$	15.4323	1.14707	0.573536	−	0.819180i	$-0.305571\pi$
$181$	15.4323	1.14707	0.573536	+	0.819180i	$0.305571\pi$
$182$	3.06737	0.227368
$183$	−11.1342	−0.823061
$184$	−1.00000	−0.0737210
$185$	7.22587	0.531256
$186$	5.72552	0.419816
$187$	−18.6561	−1.36427
$188$	8.60433	0.627536
$189$	−3.20506	−0.233134
$190$	4.43658	0.321863
$191$	−18.1186	−1.31101	−0.655507	−	0.755189i	$-0.727545\pi$
$191$	−18.1186	−1.31101	−0.655507	+	0.755189i	$0.727545\pi$
$192$	1.00000	0.0721688
$193$	12.8955	0.928240	0.464120	−	0.885772i	$-0.346371\pi$
$193$	12.8955	0.928240	0.464120	+	0.885772i	$0.346371\pi$
$194$	7.71987	0.554254
$195$	−1.32477	−0.0948689
$196$	3.27243	0.233745
$197$	1.86231	0.132684	0.0663419	−	0.997797i	$-0.478867\pi$
$197$	1.86231	0.132684	0.0663419	+	0.997797i	$0.478867\pi$
$198$	5.82082	0.413668
$199$	1.85088	0.131205	0.0656027	−	0.997846i	$-0.479103\pi$
$199$	1.85088	0.131205	0.0656027	+	0.997846i	$0.479103\pi$
$200$	−3.08388	−0.218063
$201$	8.39510	0.592145
$202$	2.68461	0.188888
$203$	−3.20506	−0.224951
$204$	−3.20506	−0.224399
$205$	−0.116105	−0.00810912
$206$	−17.6490	−1.22966
$207$	−1.00000	−0.0695048
$208$	−0.957038	−0.0663587
$209$	18.6561	1.29047
$210$	−4.43658	−0.306153
$211$	2.54987	0.175540	0.0877701	−	0.996141i	$-0.472026\pi$
$211$	2.54987	0.175540	0.0877701	+	0.996141i	$0.472026\pi$
$212$	12.9814	0.891569
$213$	−11.0947	−0.760199
$214$	11.9923	0.819777
$215$	13.7235	0.935933
$216$	1.00000	0.0680414
$217$	−18.3507	−1.24572
$218$	−1.91408	−0.129638
$219$	9.74993	0.658839
$220$	8.05742	0.543231
$221$	3.06737	0.206334
$222$	5.22009	0.350350
$223$	−26.6561	−1.78502	−0.892512	−	0.451023i	$-0.851059\pi$
$223$	−26.6561	−1.78502	−0.892512	+	0.451023i	$0.851059\pi$
$224$	−3.20506	−0.214147
$225$	−3.08388	−0.205592
$226$	−9.72347	−0.646796
$227$	16.0217	1.06340	0.531699	−	0.846933i	$-0.321554\pi$
$227$	16.0217	1.06340	0.531699	+	0.846933i	$0.321554\pi$
$228$	3.20506	0.212261
$229$	0.634515	0.0419299	0.0209650	−	0.999780i	$-0.493326\pi$
$229$	0.634515	0.0419299	0.0209650	+	0.999780i	$0.493326\pi$
$230$	−1.38424	−0.0912742
$231$	−18.6561	−1.22748
$232$	1.00000	0.0656532
$233$	6.49605	0.425570	0.212785	−	0.977099i	$-0.431747\pi$
$233$	6.49605	0.425570	0.212785	+	0.977099i	$0.431747\pi$
$234$	−0.957038	−0.0625636
$235$	11.9105	0.776954
$236$	1.55777	0.101402
$237$	−8.33190	−0.541215
$238$	10.2724	0.665863
$239$	−16.3328	−1.05648	−0.528241	−	0.849095i	$-0.677148\pi$
$239$	−16.3328	−1.05648	−0.528241	+	0.849095i	$0.677148\pi$
$240$	1.38424	0.0893524
$241$	2.82499	0.181974	0.0909869	−	0.995852i	$-0.470998\pi$
$241$	2.82499	0.181974	0.0909869	+	0.995852i	$0.470998\pi$
$242$	22.8820	1.47091
$243$	1.00000	0.0641500
$244$	−11.1342	−0.712792
$245$	4.52984	0.289401
$246$	−0.0838762	−0.00534775
$247$	−3.06737	−0.195172
$248$	5.72552	0.363571
$249$	−10.5449	−0.668254
$250$	−11.1900	−0.707720
$251$	−10.9212	−0.689340	−0.344670	−	0.938724i	$-0.612009\pi$
$251$	−10.9212	−0.689340	−0.344670	+	0.938724i	$0.612009\pi$
$252$	−3.20506	−0.201900
$253$	−5.82082	−0.365952
$254$	17.0088	1.06723
$255$	−4.43658	−0.277829
$256$	1.00000	0.0625000
$257$	8.96565	0.559262	0.279631	−	0.960108i	$-0.409788\pi$
$257$	8.96565	0.559262	0.279631	+	0.960108i	$0.409788\pi$
$258$	9.91408	0.617223
$259$	−16.7307	−1.03960
$260$	−1.32477	−0.0821589
$261$	1.00000	0.0618984
$262$	−3.17501	−0.196153
$263$	18.9473	1.16834	0.584170	−	0.811631i	$-0.301420\pi$
$263$	18.9473	1.16834	0.584170	+	0.811631i	$0.301420\pi$
$264$	5.82082	0.358247
$265$	17.9695	1.10385
$266$	−10.2724	−0.629843
$267$	−8.98145	−0.549656
$268$	8.39510	0.512812
$269$	−25.7291	−1.56873	−0.784366	−	0.620298i	$-0.787012\pi$
$269$	−25.7291	−1.56873	−0.784366	+	0.620298i	$0.787012\pi$
$270$	1.38424	0.0842422
$271$	−25.6914	−1.56064	−0.780320	−	0.625381i	$-0.784944\pi$
$271$	−25.6914	−1.56064	−0.780320	+	0.625381i	$0.784944\pi$
$272$	−3.20506	−0.194336
$273$	3.06737	0.185646
$274$	−16.0253	−0.968125
$275$	−17.9507	−1.08247
$276$	−1.00000	−0.0601929
$277$	−19.5578	−1.17511	−0.587556	−	0.809183i	$-0.699910\pi$
$277$	−19.5578	−1.17511	−0.587556	+	0.809183i	$0.699910\pi$
$278$	4.46664	0.267891
$279$	5.72552	0.342778
$280$	−4.43658	−0.265136
$281$	31.4246	1.87463	0.937317	−	0.348478i	$-0.113301\pi$
$281$	31.4246	1.87463	0.937317	+	0.348478i	$0.113301\pi$
$282$	8.60433	0.512381
$283$	1.78781	0.106274	0.0531370	−	0.998587i	$-0.483078\pi$
$283$	1.78781	0.106274	0.0531370	+	0.998587i	$0.483078\pi$
$284$	−11.0947	−0.658351
$285$	4.43658	0.262800
$286$	−5.57075	−0.329405
$287$	0.268829	0.0158685
$288$	1.00000	0.0589256
$289$	−6.72757	−0.395739
$290$	1.38424	0.0812854
$291$	7.71987	0.452547
$292$	9.74993	0.570571
$293$	32.5966	1.90432	0.952158	−	0.305607i	$-0.0988592\pi$
$293$	32.5966	1.90432	0.952158	+	0.305607i	$0.0988592\pi$
$294$	3.27243	0.190852
$295$	2.15633	0.125546
$296$	5.22009	0.303412
$297$	5.82082	0.337758
$298$	−1.79076	−0.103736
$299$	0.957038	0.0553470
$300$	−3.08388	−0.178048
$301$	−31.7752	−1.83149
$302$	22.5666	1.29856
$303$	2.68461	0.154227
$304$	3.20506	0.183823
$305$	−15.4124	−0.882510
$306$	−3.20506	−0.183221
$307$	−27.6095	−1.57576	−0.787880	−	0.615829i	$-0.788821\pi$
$307$	−27.6095	−1.57576	−0.787880	+	0.615829i	$0.788821\pi$
$308$	−18.6561	−1.06303
$309$	−17.6490	−1.00402
$310$	7.92550	0.450138
$311$	−9.53040	−0.540420	−0.270210	−	0.962801i	$-0.587093\pi$
$311$	−9.53040	−0.540420	−0.270210	+	0.962801i	$0.587093\pi$
$312$	−0.957038	−0.0541816
$313$	31.7716	1.79584	0.897920	−	0.440159i	$-0.145078\pi$
$313$	31.7716	1.79584	0.897920	+	0.440159i	$0.145078\pi$
$314$	4.46960	0.252234
$315$	−4.43658	−0.249973
$316$	−8.33190	−0.468706
$317$	6.76939	0.380207	0.190103	−	0.981764i	$-0.439118\pi$
$317$	6.76939	0.380207	0.190103	+	0.981764i	$0.439118\pi$
$318$	12.9814	0.727963
$319$	5.82082	0.325903
$320$	1.38424	0.0773814
$321$	11.9923	0.669345
$322$	3.20506	0.178611
$323$	−10.2724	−0.571573
$324$	1.00000	0.0555556
$325$	2.95139	0.163714
$326$	−20.5128	−1.13610
$327$	−1.91408	−0.105849
$328$	−0.0838762	−0.00463129
$329$	−27.5774	−1.52039
$330$	8.05742	0.443546
$331$	−2.44428	−0.134350	−0.0671749	−	0.997741i	$-0.521399\pi$
$331$	−2.44428	−0.134350	−0.0671749	+	0.997741i	$0.521399\pi$
$332$	−10.5449	−0.578725
$333$	5.22009	0.286059
$334$	−7.56137	−0.413740
$335$	11.6208	0.634914
$336$	−3.20506	−0.174851
$337$	−5.00057	−0.272398	−0.136199	−	0.990681i	$-0.543489\pi$
$337$	−5.00057	−0.272398	−0.136199	+	0.990681i	$0.543489\pi$
$338$	−12.0841	−0.657287
$339$	−9.72347	−0.528107
$340$	−4.43658	−0.240607
$341$	33.3272	1.80477
$342$	3.20506	0.173310
$343$	11.9471	0.645082
$344$	9.91408	0.534531
$345$	−1.38424	−0.0745250
$346$	6.57428	0.353435
$347$	−23.2833	−1.24991	−0.624956	−	0.780660i	$-0.714883\pi$
$347$	−23.2833	−1.24991	−0.624956	+	0.780660i	$0.714883\pi$
$348$	1.00000	0.0536056
$349$	−33.1730	−1.77571	−0.887854	−	0.460125i	$-0.847804\pi$
$349$	−33.1730	−1.77571	−0.887854	+	0.460125i	$0.847804\pi$
$350$	9.88402	0.528323
$351$	−0.957038	−0.0510829
$352$	5.82082	0.310251
$353$	4.70901	0.250635	0.125318	−	0.992117i	$-0.460005\pi$
$353$	4.70901	0.250635	0.125318	+	0.992117i	$0.460005\pi$
$354$	1.55777	0.0827945
$355$	−15.3578	−0.815107
$356$	−8.98145	−0.476016
$357$	10.2724	0.543675
$358$	1.94413	0.102751
$359$	−30.5401	−1.61185	−0.805923	−	0.592020i	$-0.798331\pi$
$359$	−30.5401	−1.61185	−0.805923	+	0.592020i	$0.798331\pi$
$360$	1.38424	0.0729559
$361$	−8.72757	−0.459346
$362$	15.4323	0.811103
$363$	22.8820	1.20099
$364$	3.06737	0.160774
$365$	13.4963	0.706426
$366$	−11.1342	−0.581992
$367$	−5.61159	−0.292922	−0.146461	−	0.989216i	$-0.546788\pi$
$367$	−5.61159	−0.292922	−0.146461	+	0.989216i	$0.546788\pi$
$368$	−1.00000	−0.0521286
$369$	−0.0838762	−0.00436642
$370$	7.22587	0.375655
$371$	−41.6064	−2.16009
$372$	5.72552	0.296854
$373$	−6.43954	−0.333427	−0.166713	−	0.986005i	$-0.553315\pi$
$373$	−6.43954	−0.333427	−0.166713	+	0.986005i	$0.553315\pi$
$374$	−18.6561	−0.964684
$375$	−11.1900	−0.577851
$376$	8.60433	0.443735
$377$	−0.957038	−0.0492900
$378$	−3.20506	−0.164851
$379$	7.91047	0.406334	0.203167	−	0.979144i	$-0.434877\pi$
$379$	7.91047	0.406334	0.203167	+	0.979144i	$0.434877\pi$
$380$	4.43658	0.227592
$381$	17.0088	0.871388
$382$	−18.1186	−0.927027
$383$	22.1901	1.13386	0.566931	−	0.823765i	$-0.308131\pi$
$383$	22.1901	1.13386	0.566931	+	0.823765i	$0.308131\pi$
$384$	1.00000	0.0510310
$385$	−25.8245	−1.31614
$386$	12.8955	0.656365
$387$	9.91408	0.503961
$388$	7.71987	0.391917
$389$	−27.4734	−1.39296	−0.696478	−	0.717578i	$-0.745251\pi$
$389$	−27.4734	−1.39296	−0.696478	+	0.717578i	$0.745251\pi$
$390$	−1.32477	−0.0670824
$391$	3.20506	0.162087
$392$	3.27243	0.165283
$393$	−3.17501	−0.160158
$394$	1.86231	0.0938216
$395$	−11.5334	−0.580306
$396$	5.82082	0.292507
$397$	−24.8173	−1.24554	−0.622772	−	0.782403i	$-0.713994\pi$
$397$	−24.8173	−1.24554	−0.622772	+	0.782403i	$0.713994\pi$
$398$	1.85088	0.0927762
$399$	−10.2724	−0.514265
$400$	−3.08388	−0.154194
$401$	14.7270	0.735431	0.367716	−	0.929938i	$-0.380140\pi$
$401$	14.7270	0.735431	0.367716	+	0.929938i	$0.380140\pi$
$402$	8.39510	0.418709
$403$	−5.47954	−0.272955
$404$	2.68461	0.133564
$405$	1.38424	0.0687835
$406$	−3.20506	−0.159065
$407$	30.3852	1.50614
$408$	−3.20506	−0.158674
$409$	−33.6328	−1.66304	−0.831518	−	0.555498i	$-0.812528\pi$
$409$	−33.6328	−1.66304	−0.831518	+	0.555498i	$0.812528\pi$
$410$	−0.116105	−0.00573401
$411$	−16.0253	−0.790471
$412$	−17.6490	−0.869503
$413$	−4.99275	−0.245677
$414$	−1.00000	−0.0491473
$415$	−14.5966	−0.716521
$416$	−0.957038	−0.0469227
$417$	4.46664	0.218732
$418$	18.6561	0.912500
$419$	−21.7493	−1.06252	−0.531261	−	0.847208i	$-0.678282\pi$
$419$	−21.7493	−1.06252	−0.531261	+	0.847208i	$0.678282\pi$
$420$	−4.43658	−0.216483
$421$	−28.0969	−1.36936	−0.684679	−	0.728845i	$-0.740057\pi$
$421$	−28.0969	−1.36936	−0.684679	+	0.728845i	$0.740057\pi$
$422$	2.54987	0.124126
$423$	8.60433	0.418357
$424$	12.9814	0.630435
$425$	9.88402	0.479445
$426$	−11.0947	−0.537542
$427$	35.6857	1.72695
$428$	11.9923	0.579670
$429$	−5.57075	−0.268958
$430$	13.7235	0.661805
$431$	40.4872	1.95020	0.975100	−	0.221764i	$-0.0711816\pi$
$431$	40.4872	1.95020	0.975100	+	0.221764i	$0.0711816\pi$
$432$	1.00000	0.0481125
$433$	16.3495	0.785708	0.392854	−	0.919601i	$-0.371488\pi$
$433$	16.3495	0.785708	0.392854	+	0.919601i	$0.371488\pi$
$434$	−18.3507	−0.880860
$435$	1.38424	0.0663693
$436$	−1.91408	−0.0916676
$437$	−3.20506	−0.153319
$438$	9.74993	0.465870
$439$	−31.0038	−1.47973	−0.739865	−	0.672755i	$-0.765111\pi$
$439$	−31.0038	−1.47973	−0.739865	+	0.672755i	$0.765111\pi$
$440$	8.05742	0.384123
$441$	3.27243	0.155830
$442$	3.06737	0.145900
$443$	−21.7904	−1.03529	−0.517647	−	0.855595i	$-0.673192\pi$
$443$	−21.7904	−1.03529	−0.517647	+	0.855595i	$0.673192\pi$
$444$	5.22009	0.247735
$445$	−12.4325	−0.589356
$446$	−26.6561	−1.26220
$447$	−1.79076	−0.0847003
$448$	−3.20506	−0.151425
$449$	37.3848	1.76430	0.882149	−	0.470971i	$-0.156096\pi$
$449$	37.3848	1.76430	0.882149	+	0.470971i	$0.156096\pi$
$450$	−3.08388	−0.145375
$451$	−0.488229	−0.0229898
$452$	−9.72347	−0.457354
$453$	22.5666	1.06027
$454$	16.0217	0.751936
$455$	4.24598	0.199055
$456$	3.20506	0.150091
$457$	14.1490	0.661862	0.330931	−	0.943655i	$-0.392637\pi$
$457$	14.1490	0.661862	0.330931	+	0.943655i	$0.392637\pi$
$458$	0.634515	0.0296489
$459$	−3.20506	−0.149600
$460$	−1.38424	−0.0645406
$461$	29.4784	1.37295	0.686473	−	0.727155i	$-0.259158\pi$
$461$	29.4784	1.37295	0.686473	+	0.727155i	$0.259158\pi$
$462$	−18.6561	−0.867961
$463$	−29.7300	−1.38167	−0.690836	−	0.723012i	$-0.742757\pi$
$463$	−29.7300	−1.38167	−0.690836	+	0.723012i	$0.742757\pi$
$464$	1.00000	0.0464238
$465$	7.92550	0.367536
$466$	6.49605	0.300924
$467$	2.86583	0.132615	0.0663074	−	0.997799i	$-0.478878\pi$
$467$	2.86583	0.132615	0.0663074	+	0.997799i	$0.478878\pi$
$468$	−0.957038	−0.0442391
$469$	−26.9068	−1.24244
$470$	11.9105	0.549389
$471$	4.46960	0.205948
$472$	1.55777	0.0717021
$473$	57.7081	2.65342
$474$	−8.33190	−0.382697
$475$	−9.88402	−0.453510
$476$	10.2724	0.470836
$477$	12.9814	0.594380
$478$	−16.3328	−0.747045
$479$	−25.1500	−1.14913	−0.574566	−	0.818458i	$-0.694829\pi$
$479$	−25.1500	−1.14913	−0.574566	+	0.818458i	$0.694829\pi$
$480$	1.38424	0.0631817
$481$	−4.99583	−0.227790
$482$	2.82499	0.128675
$483$	3.20506	0.145835
$484$	22.8820	1.04009
$485$	10.6862	0.485234
$486$	1.00000	0.0453609
$487$	−35.2798	−1.59868	−0.799341	−	0.600878i	$-0.794818\pi$
$487$	−35.2798	−1.59868	−0.799341	+	0.600878i	$0.794818\pi$
$488$	−11.1342	−0.504020
$489$	−20.5128	−0.927619
$490$	4.52984	0.204637
$491$	20.1630	0.909944	0.454972	−	0.890506i	$-0.349649\pi$
$491$	20.1630	0.909944	0.454972	+	0.890506i	$0.349649\pi$
$492$	−0.0838762	−0.00378143
$493$	−3.20506	−0.144349
$494$	−3.06737	−0.138007
$495$	8.05742	0.362154
$496$	5.72552	0.257083
$497$	35.5593	1.59505
$498$	−10.5449	−0.472527
$499$	−38.7343	−1.73399	−0.866993	−	0.498320i	$-0.833951\pi$
$499$	−38.7343	−1.73399	−0.866993	+	0.498320i	$0.833951\pi$
$500$	−11.1900	−0.500434
$501$	−7.56137	−0.337817
$502$	−10.9212	−0.487437
$503$	−22.2460	−0.991899	−0.495950	−	0.868351i	$-0.665180\pi$
$503$	−22.2460	−0.991899	−0.495950	+	0.868351i	$0.665180\pi$
$504$	−3.20506	−0.142765
$505$	3.71614	0.165366
$506$	−5.82082	−0.258767
$507$	−12.0841	−0.536673
$508$	17.0088	0.754644
$509$	20.4583	0.906798	0.453399	−	0.891308i	$-0.350211\pi$
$509$	20.4583	0.906798	0.453399	+	0.891308i	$0.350211\pi$
$510$	−4.43658	−0.196455
$511$	−31.2491	−1.38238
$512$	1.00000	0.0441942
$513$	3.20506	0.141507
$514$	8.96565	0.395458
$515$	−24.4304	−1.07653
$516$	9.91408	0.436443
$517$	50.0843	2.20270
$518$	−16.7307	−0.735106
$519$	6.57428	0.288579
$520$	−1.32477	−0.0580951
$521$	−22.4325	−0.982785	−0.491393	−	0.870938i	$-0.663512\pi$
$521$	−22.4325	−0.982785	−0.491393	+	0.870938i	$0.663512\pi$
$522$	1.00000	0.0437688
$523$	35.3761	1.54689	0.773443	−	0.633865i	$-0.218533\pi$
$523$	35.3761	1.54689	0.773443	+	0.633865i	$0.218533\pi$
$524$	−3.17501	−0.138701
$525$	9.88402	0.431374
$526$	18.9473	0.826141
$527$	−18.3507	−0.799367
$528$	5.82082	0.253319
$529$	1.00000	0.0434783
$530$	17.9695	0.780543
$531$	1.55777	0.0676014
$532$	−10.2724	−0.445366
$533$	0.0802728	0.00347700
$534$	−8.98145	−0.388665
$535$	16.6002	0.717691
$536$	8.39510	0.362613
$537$	1.94413	0.0838956
$538$	−25.7291	−1.10926
$539$	19.0482	0.820466
$540$	1.38424	0.0595683
$541$	−14.9579	−0.643092	−0.321546	−	0.946894i	$-0.604203\pi$
$541$	−14.9579	−0.643092	−0.321546	+	0.946894i	$0.604203\pi$
$542$	−25.6914	−1.10354
$543$	15.4323	0.662263
$544$	−3.20506	−0.137416
$545$	−2.64954	−0.113494
$546$	3.06737	0.131271
$547$	25.7309	1.10018	0.550088	−	0.835107i	$-0.314594\pi$
$547$	25.7309	1.10018	0.550088	+	0.835107i	$0.314594\pi$
$548$	−16.0253	−0.684568
$549$	−11.1342	−0.475195
$550$	−17.9507	−0.765420
$551$	3.20506	0.136540
$552$	−1.00000	−0.0425628
$553$	26.7043	1.13558
$554$	−19.5578	−0.830930
$555$	7.22587	0.306721
$556$	4.46664	0.189428
$557$	29.8461	1.26462	0.632311	−	0.774715i	$-0.282107\pi$
$557$	29.8461	1.26462	0.632311	+	0.774715i	$0.282107\pi$
$558$	5.72552	0.242381
$559$	−9.48815	−0.401306
$560$	−4.43658	−0.187480
$561$	−18.6561	−0.787661
$562$	31.4246	1.32557
$563$	−28.2563	−1.19086	−0.595430	−	0.803407i	$-0.703018\pi$
$563$	−28.2563	−1.19086	−0.595430	+	0.803407i	$0.703018\pi$
$564$	8.60433	0.362308
$565$	−13.4596	−0.566251
$566$	1.78781	0.0751471
$567$	−3.20506	−0.134600
$568$	−11.0947	−0.465525
$569$	26.4354	1.10823	0.554116	−	0.832440i	$-0.313056\pi$
$569$	26.4354	1.10823	0.554116	+	0.832440i	$0.313056\pi$
$570$	4.43658	0.185828
$571$	11.5895	0.485006	0.242503	−	0.970151i	$-0.422032\pi$
$571$	11.5895	0.485006	0.242503	+	0.970151i	$0.422032\pi$
$572$	−5.57075	−0.232925
$573$	−18.1186	−0.756914
$574$	0.268829	0.0112207
$575$	3.08388	0.128607
$576$	1.00000	0.0416667
$577$	−3.63059	−0.151143	−0.0755716	−	0.997140i	$-0.524078\pi$
$577$	−3.63059	−0.151143	−0.0755716	+	0.997140i	$0.524078\pi$
$578$	−6.72757	−0.279830
$579$	12.8955	0.535920
$580$	1.38424	0.0574775
$581$	33.7970	1.40213
$582$	7.71987	0.319999
$583$	75.5627	3.12949
$584$	9.74993	0.403455
$585$	−1.32477	−0.0547726
$586$	32.5966	1.34655
$587$	−17.8653	−0.737378	−0.368689	−	0.929553i	$-0.620193\pi$
$587$	−17.8653	−0.737378	−0.368689	+	0.929553i	$0.620193\pi$
$588$	3.27243	0.134953
$589$	18.3507	0.756126
$590$	2.15633	0.0887746
$591$	1.86231	0.0766050
$592$	5.22009	0.214544
$593$	−23.5858	−0.968552	−0.484276	−	0.874915i	$-0.660917\pi$
$593$	−23.5858	−0.968552	−0.484276	+	0.874915i	$0.660917\pi$
$594$	5.82082	0.238831
$595$	14.2195	0.582944
$596$	−1.79076	−0.0733526
$597$	1.85088	0.0757515
$598$	0.957038	0.0391362
$599$	−14.8467	−0.606620	−0.303310	−	0.952892i	$-0.598092\pi$
$599$	−14.8467	−0.606620	−0.303310	+	0.952892i	$0.598092\pi$
$600$	−3.08388	−0.125899
$601$	−25.2726	−1.03089	−0.515446	−	0.856922i	$-0.672374\pi$
$601$	−25.2726	−1.03089	−0.515446	+	0.856922i	$0.672374\pi$
$602$	−31.7752	−1.29506
$603$	8.39510	0.341875
$604$	22.5666	0.918221
$605$	31.6742	1.28774
$606$	2.68461	0.109055
$607$	35.3152	1.43340	0.716699	−	0.697382i	$-0.245652\pi$
$607$	35.3152	1.43340	0.716699	+	0.697382i	$0.245652\pi$
$608$	3.20506	0.129982
$609$	−3.20506	−0.129876
$610$	−15.4124	−0.624029
$611$	−8.23468	−0.333139
$612$	−3.20506	−0.129557
$613$	19.0626	0.769932	0.384966	−	0.922931i	$-0.374213\pi$
$613$	19.0626	0.769932	0.384966	+	0.922931i	$0.374213\pi$
$614$	−27.6095	−1.11423
$615$	−0.116105	−0.00468180
$616$	−18.6561	−0.751676
$617$	18.2496	0.734700	0.367350	−	0.930083i	$-0.380265\pi$
$617$	18.2496	0.734700	0.367350	+	0.930083i	$0.380265\pi$
$618$	−17.6490	−0.709946
$619$	30.5184	1.22664	0.613319	−	0.789835i	$-0.289834\pi$
$619$	30.5184	1.22664	0.613319	+	0.789835i	$0.289834\pi$
$620$	7.92550	0.318296
$621$	−1.00000	−0.0401286
$622$	−9.53040	−0.382134
$623$	28.7861	1.15329
$624$	−0.957038	−0.0383122
$625$	−0.0703264	−0.00281305
$626$	31.7716	1.26985
$627$	18.6561	0.745053
$628$	4.46960	0.178356
$629$	−16.7307	−0.667098
$630$	−4.43658	−0.176758
$631$	−5.79733	−0.230788	−0.115394	−	0.993320i	$-0.536813\pi$
$631$	−5.79733	−0.230788	−0.115394	+	0.993320i	$0.536813\pi$
$632$	−8.33190	−0.331425
$633$	2.54987	0.101348
$634$	6.76939	0.268847
$635$	23.5443	0.934327
$636$	12.9814	0.514748
$637$	−3.13184	−0.124088
$638$	5.82082	0.230449
$639$	−11.0947	−0.438901
$640$	1.38424	0.0547169
$641$	32.5261	1.28470	0.642352	−	0.766410i	$-0.277959\pi$
$641$	32.5261	1.28470	0.642352	+	0.766410i	$0.277959\pi$
$642$	11.9923	0.473298
$643$	−38.1342	−1.50387	−0.751934	−	0.659238i	$-0.770879\pi$
$643$	−38.1342	−1.50387	−0.751934	+	0.659238i	$0.770879\pi$
$644$	3.20506	0.126297
$645$	13.7235	0.540361
$646$	−10.2724	−0.404163
$647$	−17.9355	−0.705118	−0.352559	−	0.935790i	$-0.614688\pi$
$647$	−17.9355	−0.705118	−0.352559	+	0.935790i	$0.614688\pi$
$648$	1.00000	0.0392837
$649$	9.06749	0.355930
$650$	2.95139	0.115763
$651$	−18.3507	−0.719219
$652$	−20.5128	−0.803342
$653$	−35.8356	−1.40236	−0.701178	−	0.712986i	$-0.747342\pi$
$653$	−35.8356	−1.40236	−0.701178	+	0.712986i	$0.747342\pi$
$654$	−1.91408	−0.0748463
$655$	−4.39497	−0.171726
$656$	−0.0838762	−0.00327482
$657$	9.74993	0.380381
$658$	−27.5774	−1.07508
$659$	−8.00065	−0.311661	−0.155830	−	0.987784i	$-0.549805\pi$
$659$	−8.00065	−0.311661	−0.155830	+	0.987784i	$0.549805\pi$
$660$	8.05742	0.313635
$661$	−11.3681	−0.442167	−0.221083	−	0.975255i	$-0.570959\pi$
$661$	−11.3681	−0.442167	−0.221083	+	0.975255i	$0.570959\pi$
$662$	−2.44428	−0.0949996
$663$	3.06737	0.119127
$664$	−10.5449	−0.409220
$665$	−14.2195	−0.551409
$666$	5.22009	0.202274
$667$	−1.00000	−0.0387202
$668$	−7.56137	−0.292558
$669$	−26.6561	−1.03058
$670$	11.6208	0.448952
$671$	−64.8100	−2.50196
$672$	−3.20506	−0.123638
$673$	22.7673	0.877617	0.438808	−	0.898581i	$-0.355401\pi$
$673$	22.7673	0.877617	0.438808	+	0.898581i	$0.355401\pi$
$674$	−5.00057	−0.192615
$675$	−3.08388	−0.118698
$676$	−12.0841	−0.464772
$677$	−14.8090	−0.569154	−0.284577	−	0.958653i	$-0.591853\pi$
$677$	−14.8090	−0.569154	−0.284577	+	0.958653i	$0.591853\pi$
$678$	−9.72347	−0.373428
$679$	−24.7427	−0.949537
$680$	−4.43658	−0.170135
$681$	16.0217	0.613953
$682$	33.3272	1.27617
$683$	20.9600	0.802012	0.401006	−	0.916075i	$-0.368661\pi$
$683$	20.9600	0.802012	0.401006	+	0.916075i	$0.368661\pi$
$684$	3.20506	0.122549
$685$	−22.1829	−0.847565
$686$	11.9471	0.456142
$687$	0.634515	0.0242083
$688$	9.91408	0.377971
$689$	−12.4237	−0.473307
$690$	−1.38424	−0.0526972
$691$	−5.01495	−0.190778	−0.0953889	−	0.995440i	$-0.530409\pi$
$691$	−5.01495	−0.190778	−0.0953889	+	0.995440i	$0.530409\pi$
$692$	6.57428	0.249916
$693$	−18.6561	−0.708687
$694$	−23.2833	−0.883822
$695$	6.18291	0.234531
$696$	1.00000	0.0379049
$697$	0.268829	0.0101826
$698$	−33.1730	−1.25562
$699$	6.49605	0.245703
$700$	9.88402	0.373581
$701$	−3.76905	−0.142355	−0.0711775	−	0.997464i	$-0.522676\pi$
$701$	−3.76905	−0.142355	−0.0711775	+	0.997464i	$0.522676\pi$
$702$	−0.957038	−0.0361211
$703$	16.7307	0.631011
$704$	5.82082	0.219380
$705$	11.9105	0.448574
$706$	4.70901	0.177226
$707$	−8.60433	−0.323599
$708$	1.55777	0.0585445
$709$	−38.5544	−1.44794	−0.723970	−	0.689831i	$-0.757685\pi$
$709$	−38.5544	−1.44794	−0.723970	+	0.689831i	$0.757685\pi$
$710$	−15.3578	−0.576367
$711$	−8.33190	−0.312471
$712$	−8.98145	−0.336594
$713$	−5.72552	−0.214422
$714$	10.2724	0.384436
$715$	−7.71126	−0.288385
$716$	1.94413	0.0726557
$717$	−16.3328	−0.609960
$718$	−30.5401	−1.13975
$719$	47.2959	1.76384	0.881920	−	0.471399i	$-0.156251\pi$
$719$	47.2959	1.76384	0.881920	+	0.471399i	$0.156251\pi$
$720$	1.38424	0.0515876
$721$	56.5661	2.10663
$722$	−8.72757	−0.324806
$723$	2.82499	0.105063
$724$	15.4323	0.573536
$725$	−3.08388	−0.114532
$726$	22.8820	0.849230
$727$	1.65295	0.0613044	0.0306522	−	0.999530i	$-0.490242\pi$
$727$	1.65295	0.0613044	0.0306522	+	0.999530i	$0.490242\pi$
$728$	3.06737	0.113684
$729$	1.00000	0.0370370
$730$	13.4963	0.499519
$731$	−31.7752	−1.17525
$732$	−11.1342	−0.411531
$733$	4.22735	0.156141	0.0780703	−	0.996948i	$-0.475124\pi$
$733$	4.22735	0.156141	0.0780703	+	0.996948i	$0.475124\pi$
$734$	−5.61159	−0.207127
$735$	4.52984	0.167085
$736$	−1.00000	−0.0368605
$737$	48.8664	1.80002
$738$	−0.0838762	−0.00308753
$739$	20.1550	0.741415	0.370708	−	0.928750i	$-0.379115\pi$
$739$	20.1550	0.741415	0.370708	+	0.928750i	$0.379115\pi$
$740$	7.22587	0.265628
$741$	−3.06737	−0.112683
$742$	−41.6064	−1.52742
$743$	−20.0969	−0.737282	−0.368641	−	0.929572i	$-0.620177\pi$
$743$	−20.0969	−0.737282	−0.368641	+	0.929572i	$0.620177\pi$
$744$	5.72552	0.209908
$745$	−2.47885	−0.0908181
$746$	−6.43954	−0.235768
$747$	−10.5449	−0.385816
$748$	−18.6561	−0.682135
$749$	−38.4361	−1.40442
$750$	−11.1900	−0.408602
$751$	−22.8625	−0.834265	−0.417132	−	0.908846i	$-0.636965\pi$
$751$	−22.8625	−0.834265	−0.417132	+	0.908846i	$0.636965\pi$
$752$	8.60433	0.313768
$753$	−10.9212	−0.397991
$754$	−0.957038	−0.0348533
$755$	31.2376	1.13685
$756$	−3.20506	−0.116567
$757$	−29.6291	−1.07689	−0.538444	−	0.842662i	$-0.680988\pi$
$757$	−29.6291	−1.07689	−0.538444	+	0.842662i	$0.680988\pi$
$758$	7.91047	0.287321
$759$	−5.82082	−0.211282
$760$	4.43658	0.160932
$761$	−43.2120	−1.56643	−0.783217	−	0.621748i	$-0.786423\pi$
$761$	−43.2120	−1.56643	−0.783217	+	0.621748i	$0.786423\pi$
$762$	17.0088	0.616164
$763$	6.13474	0.222092
$764$	−18.1186	−0.655507
$765$	−4.43658	−0.160405
$766$	22.1901	0.801761
$767$	−1.49084	−0.0538313
$768$	1.00000	0.0360844
$769$	−6.09686	−0.219858	−0.109929	−	0.993939i	$-0.535062\pi$
$769$	−6.09686	−0.219858	−0.109929	+	0.993939i	$0.535062\pi$
$770$	−25.8245	−0.930652
$771$	8.96565	0.322890
$772$	12.8955	0.464120
$773$	−54.0206	−1.94299	−0.971493	−	0.237070i	$-0.923813\pi$
$773$	−54.0206	−1.94299	−0.971493	+	0.237070i	$0.923813\pi$
$774$	9.91408	0.356354
$775$	−17.6568	−0.634251
$776$	7.71987	0.277127
$777$	−16.7307	−0.600211
$778$	−27.4734	−0.984969
$779$	−0.268829	−0.00963178
$780$	−1.32477	−0.0474344
$781$	−64.5805	−2.31087
$782$	3.20506	0.114613
$783$	1.00000	0.0357371
$784$	3.27243	0.116873
$785$	6.18700	0.220823
$786$	−3.17501	−0.113249
$787$	−28.3080	−1.00907	−0.504536	−	0.863390i	$-0.668337\pi$
$787$	−28.3080	−1.00907	−0.504536	+	0.863390i	$0.668337\pi$
$788$	1.86231	0.0663419
$789$	18.9473	0.674542
$790$	−11.5334	−0.410339
$791$	31.1644	1.10808
$792$	5.82082	0.206834
$793$	10.6558	0.378399
$794$	−24.8173	−0.880733
$795$	17.9695	0.637311
$796$	1.85088	0.0656027
$797$	−17.2711	−0.611774	−0.305887	−	0.952068i	$-0.598953\pi$
$797$	−17.2711	−0.611774	−0.305887	+	0.952068i	$0.598953\pi$
$798$	−10.2724	−0.363640
$799$	−27.5774	−0.975620
$800$	−3.08388	−0.109031
$801$	−8.98145	−0.317344
$802$	14.7270	0.520028
$803$	56.7526	2.00276
$804$	8.39510	0.296072
$805$	4.43658	0.156369
$806$	−5.47954	−0.193009
$807$	−25.7291	−0.905708
$808$	2.68461	0.0944441
$809$	34.0834	1.19831	0.599154	−	0.800634i	$-0.295504\pi$
$809$	34.0834	1.19831	0.599154	+	0.800634i	$0.295504\pi$
$810$	1.38424	0.0486373
$811$	17.5510	0.616298	0.308149	−	0.951338i	$-0.400290\pi$
$811$	17.5510	0.616298	0.308149	+	0.951338i	$0.400290\pi$
$812$	−3.20506	−0.112476
$813$	−25.6914	−0.901036
$814$	30.3852	1.06500
$815$	−28.3946	−0.994620
$816$	−3.20506	−0.112200
$817$	31.7752	1.11168
$818$	−33.6328	−1.17594
$819$	3.06737	0.107183
$820$	−0.116105	−0.00405456
$821$	−44.2294	−1.54362	−0.771809	−	0.635854i	$-0.780648\pi$
$821$	−44.2294	−1.54362	−0.771809	+	0.635854i	$0.780648\pi$
$822$	−16.0253	−0.558947
$823$	−20.2326	−0.705265	−0.352633	−	0.935762i	$-0.614713\pi$
$823$	−20.2326	−0.705265	−0.352633	+	0.935762i	$0.614713\pi$
$824$	−17.6490	−0.614831
$825$	−17.9507	−0.624963
$826$	−4.99275	−0.173720
$827$	9.67247	0.336345	0.168172	−	0.985758i	$-0.446213\pi$
$827$	9.67247	0.336345	0.168172	+	0.985758i	$0.446213\pi$
$828$	−1.00000	−0.0347524
$829$	−45.1555	−1.56832	−0.784158	−	0.620561i	$-0.786905\pi$
$829$	−45.1555	−1.56832	−0.784158	+	0.620561i	$0.786905\pi$
$830$	−14.5966	−0.506657
$831$	−19.5578	−0.678451
$832$	−0.957038	−0.0331793
$833$	−10.4884	−0.363400
$834$	4.46664	0.154667
$835$	−10.4668	−0.362217
$836$	18.6561	0.645235
$837$	5.72552	0.197903
$838$	−21.7493	−0.751316
$839$	−11.1148	−0.383724	−0.191862	−	0.981422i	$-0.561453\pi$
$839$	−11.1148	−0.383724	−0.191862	+	0.981422i	$0.561453\pi$
$840$	−4.43658	−0.153077
$841$	1.00000	0.0344828
$842$	−28.0969	−0.968282
$843$	31.4246	1.08232
$844$	2.54987	0.0877701
$845$	−16.7273	−0.575436
$846$	8.60433	0.295823
$847$	−73.3382	−2.51993
$848$	12.9814	0.445785
$849$	1.78781	0.0613574
$850$	9.88402	0.339019
$851$	−5.22009	−0.178942
$852$	−11.0947	−0.380099
$853$	35.4809	1.21484	0.607421	−	0.794380i	$-0.292204\pi$
$853$	35.4809	1.21484	0.607421	+	0.794380i	$0.292204\pi$
$854$	35.6857	1.22114
$855$	4.43658	0.151728
$856$	11.9923	0.409888
$857$	15.4368	0.527310	0.263655	−	0.964617i	$-0.415072\pi$
$857$	15.4368	0.527310	0.263655	+	0.964617i	$0.415072\pi$
$858$	−5.57075	−0.190182
$859$	−40.5107	−1.38221	−0.691104	−	0.722755i	$-0.742875\pi$
$859$	−40.5107	−1.38221	−0.691104	+	0.722755i	$0.742875\pi$
$860$	13.7235	0.467967
$861$	0.268829	0.00916165
$862$	40.4872	1.37900
$863$	−31.4531	−1.07068	−0.535338	−	0.844638i	$-0.679816\pi$
$863$	−31.4531	−1.07068	−0.535338	+	0.844638i	$0.679816\pi$
$864$	1.00000	0.0340207
$865$	9.10038	0.309422
$866$	16.3495	0.555579
$867$	−6.72757	−0.228480
$868$	−18.3507	−0.622862
$869$	−48.4985	−1.64520
$870$	1.38424	0.0469302
$871$	−8.03443	−0.272236
$872$	−1.91408	−0.0648188
$873$	7.71987	0.261278
$874$	−3.20506	−0.108413
$875$	35.8648	1.21245
$876$	9.74993	0.329420
$877$	14.9179	0.503741	0.251871	−	0.967761i	$-0.418954\pi$
$877$	14.9179	0.503741	0.251871	+	0.967761i	$0.418954\pi$
$878$	−31.0038	−1.04633
$879$	32.5966	1.09946
$880$	8.05742	0.271616
$881$	52.4024	1.76548	0.882741	−	0.469859i	$-0.155695\pi$
$881$	52.4024	1.76548	0.882741	+	0.469859i	$0.155695\pi$
$882$	3.27243	0.110189
$883$	−30.6749	−1.03229	−0.516146	−	0.856501i	$-0.672634\pi$
$883$	−30.6749	−1.03229	−0.516146	+	0.856501i	$0.672634\pi$
$884$	3.06737	0.103167
$885$	2.15633	0.0724841
$886$	−21.7904	−0.732063
$887$	−9.82520	−0.329898	−0.164949	−	0.986302i	$-0.552746\pi$
$887$	−9.82520	−0.329898	−0.164949	+	0.986302i	$0.552746\pi$
$888$	5.22009	0.175175
$889$	−54.5143	−1.82835
$890$	−12.4325	−0.416738
$891$	5.82082	0.195005
$892$	−26.6561	−0.892512
$893$	27.5774	0.922844
$894$	−1.79076	−0.0598921
$895$	2.69115	0.0899552
$896$	−3.20506	−0.107074
$897$	0.957038	0.0319546
$898$	37.3848	1.24755
$899$	5.72552	0.190957
$900$	−3.08388	−0.102796
$901$	−41.6064	−1.38611
$902$	−0.488229	−0.0162562
$903$	−31.7752	−1.05741
$904$	−9.72347	−0.323398
$905$	21.3620	0.710097
$906$	22.5666	0.749724
$907$	14.7685	0.490379	0.245190	−	0.969475i	$-0.421150\pi$
$907$	14.7685	0.490379	0.245190	+	0.969475i	$0.421150\pi$
$908$	16.0217	0.531699
$909$	2.68461	0.0890428
$910$	4.24598	0.140753
$911$	−26.5780	−0.880569	−0.440284	−	0.897858i	$-0.645122\pi$
$911$	−26.5780	−0.880569	−0.440284	+	0.897858i	$0.645122\pi$
$912$	3.20506	0.106130
$913$	−61.3798	−2.03137
$914$	14.1490	0.468007
$915$	−15.4124	−0.509517
$916$	0.634515	0.0209650
$917$	10.1761	0.336044
$918$	−3.20506	−0.105783
$919$	3.16808	0.104505	0.0522526	−	0.998634i	$-0.483360\pi$
$919$	3.16808	0.104505	0.0522526	+	0.998634i	$0.483360\pi$
$920$	−1.38424	−0.0456371
$921$	−27.6095	−0.909765
$922$	29.4784	0.970820
$923$	10.6181	0.349499
$924$	−18.6561	−0.613741
$925$	−16.0981	−0.529303
$926$	−29.7300	−0.976990
$927$	−17.6490	−0.579668
$928$	1.00000	0.0328266
$929$	−0.630142	−0.0206743	−0.0103371	−	0.999947i	$-0.503290\pi$
$929$	−0.630142	−0.0206743	−0.0103371	+	0.999947i	$0.503290\pi$
$930$	7.92550	0.259887
$931$	10.4884	0.343742
$932$	6.49605	0.212785
$933$	−9.53040	−0.312011
$934$	2.86583	0.0937729
$935$	−25.8245	−0.844553
$936$	−0.957038	−0.0312818
$937$	49.0404	1.60208	0.801040	−	0.598610i	$-0.204280\pi$
$937$	49.0404	1.60208	0.801040	+	0.598610i	$0.204280\pi$
$938$	−26.9068	−0.878539
$939$	31.7716	1.03683
$940$	11.9105	0.388477
$941$	12.4109	0.404584	0.202292	−	0.979325i	$-0.435161\pi$
$941$	12.4109	0.404584	0.202292	+	0.979325i	$0.435161\pi$
$942$	4.46960	0.145627
$943$	0.0838762	0.00273139
$944$	1.55777	0.0507010
$945$	−4.43658	−0.144322
$946$	57.7081	1.87625
$947$	11.5822	0.376370	0.188185	−	0.982134i	$-0.439740\pi$
$947$	11.5822	0.376370	0.188185	+	0.982134i	$0.439740\pi$
$948$	−8.33190	−0.270608
$949$	−9.33105	−0.302899
$950$	−9.88402	−0.320680
$951$	6.76939	0.219513
$952$	10.2724	0.332931
$953$	21.4088	0.693499	0.346749	−	0.937958i	$-0.387285\pi$
$953$	21.4088	0.693499	0.346749	+	0.937958i	$0.387285\pi$
$954$	12.9814	0.420290
$955$	−25.0805	−0.811585
$956$	−16.3328	−0.528241
$957$	5.82082	0.188160
$958$	−25.1500	−0.812559
$959$	51.3622	1.65857
$960$	1.38424	0.0446762
$961$	1.78159	0.0574706
$962$	−4.99583	−0.161072
$963$	11.9923	0.386447
$964$	2.82499	0.0909869
$965$	17.8505	0.574628
$966$	3.20506	0.103121
$967$	29.4440	0.946855	0.473427	−	0.880833i	$-0.343017\pi$
$967$	29.4440	0.946855	0.473427	+	0.880833i	$0.343017\pi$
$968$	22.8820	0.735454
$969$	−10.2724	−0.329998
$970$	10.6862	0.343112
$971$	−14.4098	−0.462431	−0.231216	−	0.972903i	$-0.574270\pi$
$971$	−14.4098	−0.462431	−0.231216	+	0.972903i	$0.574270\pi$
$972$	1.00000	0.0320750
$973$	−14.3159	−0.458945
$974$	−35.2798	−1.13044
$975$	2.95139	0.0945201
$976$	−11.1342	−0.356396
$977$	26.6734	0.853359	0.426680	−	0.904403i	$-0.359683\pi$
$977$	26.6734	0.853359	0.426680	+	0.904403i	$0.359683\pi$
$978$	−20.5128	−0.655926
$979$	−52.2794	−1.67086
$980$	4.52984	0.144700
$981$	−1.91408	−0.0611118
$982$	20.1630	0.643428
$983$	21.8545	0.697050	0.348525	−	0.937300i	$-0.386683\pi$
$983$	21.8545	0.697050	0.348525	+	0.937300i	$0.386683\pi$
$984$	−0.0838762	−0.00267388
$985$	2.57788	0.0821381
$986$	−3.20506	−0.102070
$987$	−27.5774	−0.877799
$988$	−3.06737	−0.0975860
$989$	−9.91408	−0.315249
$990$	8.05742	0.256082
$991$	61.6004	1.95680	0.978401	−	0.206715i	$-0.0662772\pi$
$991$	61.6004	1.95680	0.978401	+	0.206715i	$0.0662772\pi$
$992$	5.72552	0.181785
$993$	−2.44428	−0.0775668
$994$	35.5593	1.12787
$995$	2.56206	0.0812229
$996$	−10.5449	−0.334127
$997$	−5.21636	−0.165204	−0.0826020	−	0.996583i	$-0.526323\pi$
$997$	−5.21636	−0.165204	−0.0826020	+	0.996583i	$0.526323\pi$
$998$	−38.7343	−1.22611
$999$	5.22009	0.165156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bd.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bd.1.3	✓	4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.38424 q^{5} +1.00000 q^{6} -3.20506 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.38424 q^{5} +1.00000 q^{6} -3.20506 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.38424 q^{10} +5.82082 q^{11} +1.00000 q^{12} -0.957038 q^{13} -3.20506 q^{14} +1.38424 q^{15} +1.00000 q^{16} -3.20506 q^{17} +1.00000 q^{18} +3.20506 q^{19} +1.38424 q^{20} -3.20506 q^{21} +5.82082 q^{22} -1.00000 q^{23} +1.00000 q^{24} -3.08388 q^{25} -0.957038 q^{26} +1.00000 q^{27} -3.20506 q^{28} +1.00000 q^{29} +1.38424 q^{30} +5.72552 q^{31} +1.00000 q^{32} +5.82082 q^{33} -3.20506 q^{34} -4.43658 q^{35} +1.00000 q^{36} +5.22009 q^{37} +3.20506 q^{38} -0.957038 q^{39} +1.38424 q^{40} -0.0838762 q^{41} -3.20506 q^{42} +9.91408 q^{43} +5.82082 q^{44} +1.38424 q^{45} -1.00000 q^{46} +8.60433 q^{47} +1.00000 q^{48} +3.27243 q^{49} -3.08388 q^{50} -3.20506 q^{51} -0.957038 q^{52} +12.9814 q^{53} +1.00000 q^{54} +8.05742 q^{55} -3.20506 q^{56} +3.20506 q^{57} +1.00000 q^{58} +1.55777 q^{59} +1.38424 q^{60} -11.1342 q^{61} +5.72552 q^{62} -3.20506 q^{63} +1.00000 q^{64} -1.32477 q^{65} +5.82082 q^{66} +8.39510 q^{67} -3.20506 q^{68} -1.00000 q^{69} -4.43658 q^{70} -11.0947 q^{71} +1.00000 q^{72} +9.74993 q^{73} +5.22009 q^{74} -3.08388 q^{75} +3.20506 q^{76} -18.6561 q^{77} -0.957038 q^{78} -8.33190 q^{79} +1.38424 q^{80} +1.00000 q^{81} -0.0838762 q^{82} -10.5449 q^{83} -3.20506 q^{84} -4.43658 q^{85} +9.91408 q^{86} +1.00000 q^{87} +5.82082 q^{88} -8.98145 q^{89} +1.38424 q^{90} +3.06737 q^{91} -1.00000 q^{92} +5.72552 q^{93} +8.60433 q^{94} +4.43658 q^{95} +1.00000 q^{96} +7.71987 q^{97} +3.27243 q^{98} +5.82082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9 - q^10 + 15 * q^11 + 4 * q^12 + 11 * q^13 + 2 * q^14 - q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^19 - q^20 + 2 * q^21 + 15 * q^22 - 4 * q^23 + 4 * q^24 - q^25 + 11 * q^26 + 4 * q^27 + 2 * q^28 + 4 * q^29 - q^30 - 5 * q^31 + 4 * q^32 + 15 * q^33 + 2 * q^34 - 16 * q^35 + 4 * q^36 + 3 * q^37 - 2 * q^38 + 11 * q^39 - q^40 + 11 * q^41 + 2 * q^42 + 10 * q^43 + 15 * q^44 - q^45 - 4 * q^46 + 10 * q^47 + 4 * q^48 - q^50 + 2 * q^51 + 11 * q^52 + 24 * q^53 + 4 * q^54 - 7 * q^55 + 2 * q^56 - 2 * q^57 + 4 * q^58 + q^59 - q^60 + 3 * q^61 - 5 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^65 + 15 * q^66 + 7 * q^67 + 2 * q^68 - 4 * q^69 - 16 * q^70 - 13 * q^71 + 4 * q^72 - 2 * q^73 + 3 * q^74 - q^75 - 2 * q^76 - 4 * q^77 + 11 * q^78 - 22 * q^79 - q^80 + 4 * q^81 + 11 * q^82 - 16 * q^83 + 2 * q^84 - 16 * q^85 + 10 * q^86 + 4 * q^87 + 15 * q^88 - 8 * q^89 - q^90 + 14 * q^91 - 4 * q^92 - 5 * q^93 + 10 * q^94 + 16 * q^95 + 4 * q^96 - 4 * q^97 + 15 * q^99

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