LMFDB - Embedded newform 4002.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 $ x^8 - 2x^7 - 30x^6 + 52x^5 + 267x^4 - 352x^3 - 632x^2 + 240*x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.42323$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.42323 q^{5} -1.00000 q^{6} -1.25167 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.42323 q^{5} -1.00000 q^{6} -1.25167 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.42323 q^{10} -2.48043 q^{11} -1.00000 q^{12} -5.96144 q^{13} -1.25167 q^{14} +3.42323 q^{15} +1.00000 q^{16} -5.90366 q^{17} +1.00000 q^{18} +0.0494308 q^{19} -3.42323 q^{20} +1.25167 q^{21} -2.48043 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.71852 q^{25} -5.96144 q^{26} -1.00000 q^{27} -1.25167 q^{28} -1.00000 q^{29} +3.42323 q^{30} +1.65549 q^{31} +1.00000 q^{32} +2.48043 q^{33} -5.90366 q^{34} +4.28477 q^{35} +1.00000 q^{36} +11.1995 q^{37} +0.0494308 q^{38} +5.96144 q^{39} -3.42323 q^{40} -6.37459 q^{41} +1.25167 q^{42} +9.44108 q^{43} -2.48043 q^{44} -3.42323 q^{45} -1.00000 q^{46} -0.286731 q^{47} -1.00000 q^{48} -5.43332 q^{49} +6.71852 q^{50} +5.90366 q^{51} -5.96144 q^{52} +1.42752 q^{53} -1.00000 q^{54} +8.49109 q^{55} -1.25167 q^{56} -0.0494308 q^{57} -1.00000 q^{58} +1.26100 q^{59} +3.42323 q^{60} +4.48043 q^{61} +1.65549 q^{62} -1.25167 q^{63} +1.00000 q^{64} +20.4074 q^{65} +2.48043 q^{66} -8.37070 q^{67} -5.90366 q^{68} +1.00000 q^{69} +4.28477 q^{70} -5.41935 q^{71} +1.00000 q^{72} -4.43236 q^{73} +11.1995 q^{74} -6.71852 q^{75} +0.0494308 q^{76} +3.10469 q^{77} +5.96144 q^{78} +12.1466 q^{79} -3.42323 q^{80} +1.00000 q^{81} -6.37459 q^{82} +11.8542 q^{83} +1.25167 q^{84} +20.2096 q^{85} +9.44108 q^{86} +1.00000 q^{87} -2.48043 q^{88} -0.796776 q^{89} -3.42323 q^{90} +7.46177 q^{91} -1.00000 q^{92} -1.65549 q^{93} -0.286731 q^{94} -0.169213 q^{95} -1.00000 q^{96} -0.811353 q^{97} -5.43332 q^{98} -2.48043 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * 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
$5$	−3.42323	−1.53092	−0.765458	−	0.643486i	$-0.777488\pi$
$5$	−3.42323	−1.53092	−0.765458	+	0.643486i	$0.777488\pi$
$6$	−1.00000	−0.408248
$7$	−1.25167	−0.473088	−0.236544	−	0.971621i	$-0.576015\pi$
$7$	−1.25167	−0.473088	−0.236544	+	0.971621i	$0.576015\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.42323	−1.08252
$11$	−2.48043	−0.747878	−0.373939	−	0.927453i	$-0.621993\pi$
$11$	−2.48043	−0.747878	−0.373939	+	0.927453i	$0.621993\pi$
$12$	−1.00000	−0.288675
$13$	−5.96144	−1.65341	−0.826703	−	0.562639i	$-0.809786\pi$
$13$	−5.96144	−1.65341	−0.826703	+	0.562639i	$0.809786\pi$
$14$	−1.25167	−0.334524
$15$	3.42323	0.883875
$16$	1.00000	0.250000
$17$	−5.90366	−1.43185	−0.715924	−	0.698178i	$-0.753994\pi$
$17$	−5.90366	−1.43185	−0.715924	+	0.698178i	$0.753994\pi$
$18$	1.00000	0.235702
$19$	0.0494308	0.0113402	0.00567011	−	0.999984i	$-0.498195\pi$
$19$	0.0494308	0.0113402	0.00567011	+	0.999984i	$0.498195\pi$
$20$	−3.42323	−0.765458
$21$	1.25167	0.273137
$22$	−2.48043	−0.528830
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	6.71852	1.34370
$26$	−5.96144	−1.16913
$27$	−1.00000	−0.192450
$28$	−1.25167	−0.236544
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.42323	0.624994
$31$	1.65549	0.297335	0.148668	−	0.988887i	$-0.452502\pi$
$31$	1.65549	0.297335	0.148668	+	0.988887i	$0.452502\pi$
$32$	1.00000	0.176777
$33$	2.48043	0.431788
$34$	−5.90366	−1.01247
$35$	4.28477	0.724258
$36$	1.00000	0.166667
$37$	11.1995	1.84119	0.920595	−	0.390518i	$-0.127704\pi$
$37$	11.1995	1.84119	0.920595	+	0.390518i	$0.127704\pi$
$38$	0.0494308	0.00801874
$39$	5.96144	0.954594
$40$	−3.42323	−0.541261
$41$	−6.37459	−0.995543	−0.497772	−	0.867308i	$-0.665848\pi$
$41$	−6.37459	−0.995543	−0.497772	+	0.867308i	$0.665848\pi$
$42$	1.25167	0.193137
$43$	9.44108	1.43975	0.719876	−	0.694103i	$-0.244199\pi$
$43$	9.44108	1.43975	0.719876	+	0.694103i	$0.244199\pi$
$44$	−2.48043	−0.373939
$45$	−3.42323	−0.510305
$46$	−1.00000	−0.147442
$47$	−0.286731	−0.0418240	−0.0209120	−	0.999781i	$-0.506657\pi$
$47$	−0.286731	−0.0418240	−0.0209120	+	0.999781i	$0.506657\pi$
$48$	−1.00000	−0.144338
$49$	−5.43332	−0.776188
$50$	6.71852	0.950142
$51$	5.90366	0.826678
$52$	−5.96144	−0.826703
$53$	1.42752	0.196085	0.0980425	−	0.995182i	$-0.468742\pi$
$53$	1.42752	0.196085	0.0980425	+	0.995182i	$0.468742\pi$
$54$	−1.00000	−0.136083
$55$	8.49109	1.14494
$56$	−1.25167	−0.167262
$57$	−0.0494308	−0.00654727
$58$	−1.00000	−0.131306
$59$	1.26100	0.164168	0.0820841	−	0.996625i	$-0.473842\pi$
$59$	1.26100	0.164168	0.0820841	+	0.996625i	$0.473842\pi$
$60$	3.42323	0.441937
$61$	4.48043	0.573660	0.286830	−	0.957981i	$-0.407398\pi$
$61$	4.48043	0.573660	0.286830	+	0.957981i	$0.407398\pi$
$62$	1.65549	0.210248
$63$	−1.25167	−0.157696
$64$	1.00000	0.125000
$65$	20.4074	2.53123
$66$	2.48043	0.305320
$67$	−8.37070	−1.02264	−0.511322	−	0.859389i	$-0.670844\pi$
$67$	−8.37070	−1.02264	−0.511322	+	0.859389i	$0.670844\pi$
$68$	−5.90366	−0.715924
$69$	1.00000	0.120386
$70$	4.28477	0.512127
$71$	−5.41935	−0.643158	−0.321579	−	0.946883i	$-0.604214\pi$
$71$	−5.41935	−0.643158	−0.321579	+	0.946883i	$0.604214\pi$
$72$	1.00000	0.117851
$73$	−4.43236	−0.518769	−0.259384	−	0.965774i	$-0.583520\pi$
$73$	−4.43236	−0.518769	−0.259384	+	0.965774i	$0.583520\pi$
$74$	11.1995	1.30192
$75$	−6.71852	−0.775788
$76$	0.0494308	0.00567011
$77$	3.10469	0.353812
$78$	5.96144	0.675000
$79$	12.1466	1.36660	0.683299	−	0.730139i	$-0.260545\pi$
$79$	12.1466	1.36660	0.683299	+	0.730139i	$0.260545\pi$
$80$	−3.42323	−0.382729
$81$	1.00000	0.111111
$82$	−6.37459	−0.703955
$83$	11.8542	1.30117	0.650586	−	0.759433i	$-0.274524\pi$
$83$	11.8542	1.30117	0.650586	+	0.759433i	$0.274524\pi$
$84$	1.25167	0.136569
$85$	20.2096	2.19204
$86$	9.44108	1.01806
$87$	1.00000	0.107211
$88$	−2.48043	−0.264415
$89$	−0.796776	−0.0844580	−0.0422290	−	0.999108i	$-0.513446\pi$
$89$	−0.796776	−0.0844580	−0.0422290	+	0.999108i	$0.513446\pi$
$90$	−3.42323	−0.360840
$91$	7.46177	0.782206
$92$	−1.00000	−0.104257
$93$	−1.65549	−0.171667
$94$	−0.286731	−0.0295740
$95$	−0.169213	−0.0173609
$96$	−1.00000	−0.102062
$97$	−0.811353	−0.0823804	−0.0411902	−	0.999151i	$-0.513115\pi$
$97$	−0.811353	−0.0823804	−0.0411902	+	0.999151i	$0.513115\pi$
$98$	−5.43332	−0.548848
$99$	−2.48043	−0.249293
$100$	6.71852	0.671852
$101$	0.311563	0.0310016	0.0155008	−	0.999880i	$-0.495066\pi$
$101$	0.311563	0.0310016	0.0155008	+	0.999880i	$0.495066\pi$
$102$	5.90366	0.584550
$103$	−16.0371	−1.58018	−0.790090	−	0.612991i	$-0.789966\pi$
$103$	−16.0371	−1.58018	−0.790090	+	0.612991i	$0.789966\pi$
$104$	−5.96144	−0.584567
$105$	−4.28477	−0.418150
$106$	1.42752	0.138653
$107$	7.79784	0.753846	0.376923	−	0.926245i	$-0.376982\pi$
$107$	7.79784	0.753846	0.376923	+	0.926245i	$0.376982\pi$
$108$	−1.00000	−0.0962250
$109$	−9.71695	−0.930715	−0.465358	−	0.885123i	$-0.654074\pi$
$109$	−9.71695	−0.930715	−0.465358	+	0.885123i	$0.654074\pi$
$110$	8.49109	0.809594
$111$	−11.1995	−1.06301
$112$	−1.25167	−0.118272
$113$	−2.63452	−0.247835	−0.123917	−	0.992293i	$-0.539546\pi$
$113$	−2.63452	−0.247835	−0.123917	+	0.992293i	$0.539546\pi$
$114$	−0.0494308	−0.00462962
$115$	3.42323	0.319218
$116$	−1.00000	−0.0928477
$117$	−5.96144	−0.551135
$118$	1.26100	0.116085
$119$	7.38945	0.677390
$120$	3.42323	0.312497
$121$	−4.84746	−0.440678
$122$	4.48043	0.405639
$123$	6.37459	0.574777
$124$	1.65549	0.148668
$125$	−5.88289	−0.526181
$126$	−1.25167	−0.111508
$127$	16.8037	1.49109	0.745543	−	0.666458i	$-0.232190\pi$
$127$	16.8037	1.49109	0.745543	+	0.666458i	$0.232190\pi$
$128$	1.00000	0.0883883
$129$	−9.44108	−0.831241
$130$	20.4074	1.78985
$131$	−3.20126	−0.279695	−0.139848	−	0.990173i	$-0.544661\pi$
$131$	−3.20126	−0.279695	−0.139848	+	0.990173i	$0.544661\pi$
$132$	2.48043	0.215894
$133$	−0.0618712	−0.00536491
$134$	−8.37070	−0.723119
$135$	3.42323	0.294625
$136$	−5.90366	−0.506235
$137$	14.1466	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$137$	14.1466	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$138$	1.00000	0.0851257
$139$	6.46531	0.548380	0.274190	−	0.961675i	$-0.411590\pi$
$139$	6.46531	0.548380	0.274190	+	0.961675i	$0.411590\pi$
$140$	4.28477	0.362129
$141$	0.286731	0.0241471
$142$	−5.41935	−0.454782
$143$	14.7869	1.23655
$144$	1.00000	0.0833333
$145$	3.42323	0.284284
$146$	−4.43236	−0.366825
$147$	5.43332	0.448132
$148$	11.1995	0.920595
$149$	0.0280400	0.00229713	0.00114856	−	0.999999i	$-0.499634\pi$
$149$	0.0280400	0.00229713	0.00114856	+	0.999999i	$0.499634\pi$
$150$	−6.71852	−0.548565
$151$	−2.83192	−0.230458	−0.115229	−	0.993339i	$-0.536760\pi$
$151$	−2.83192	−0.230458	−0.115229	+	0.993339i	$0.536760\pi$
$152$	0.0494308	0.00400937
$153$	−5.90366	−0.477283
$154$	3.10469	0.250183
$155$	−5.66714	−0.455195
$156$	5.96144	0.477297
$157$	−12.1009	−0.965754	−0.482877	−	0.875688i	$-0.660408\pi$
$157$	−12.1009	−0.965754	−0.482877	+	0.875688i	$0.660408\pi$
$158$	12.1466	0.966330
$159$	−1.42752	−0.113210
$160$	−3.42323	−0.270630
$161$	1.25167	0.0986456
$162$	1.00000	0.0785674
$163$	5.14348	0.402869	0.201434	−	0.979502i	$-0.435440\pi$
$163$	5.14348	0.402869	0.201434	+	0.979502i	$0.435440\pi$
$164$	−6.37459	−0.497772
$165$	−8.49109	−0.661031
$166$	11.8542	0.920067
$167$	17.0100	1.31627	0.658136	−	0.752899i	$-0.271345\pi$
$167$	17.0100	1.31627	0.658136	+	0.752899i	$0.271345\pi$
$168$	1.25167	0.0965686
$169$	22.5388	1.73375
$170$	20.2096	1.55001
$171$	0.0494308	0.00378007
$172$	9.44108	0.719876
$173$	−15.5687	−1.18366	−0.591832	−	0.806062i	$-0.701595\pi$
$173$	−15.5687	−1.18366	−0.591832	+	0.806062i	$0.701595\pi$
$174$	1.00000	0.0758098
$175$	−8.40938	−0.635690
$176$	−2.48043	−0.186970
$177$	−1.26100	−0.0947826
$178$	−0.796776	−0.0597209
$179$	−15.5798	−1.16449	−0.582243	−	0.813015i	$-0.697825\pi$
$179$	−15.5798	−1.16449	−0.582243	+	0.813015i	$0.697825\pi$
$180$	−3.42323	−0.255153
$181$	14.8151	1.10120	0.550599	−	0.834770i	$-0.314399\pi$
$181$	14.8151	1.10120	0.550599	+	0.834770i	$0.314399\pi$
$182$	7.46177	0.553103
$183$	−4.48043	−0.331203
$184$	−1.00000	−0.0737210
$185$	−38.3386	−2.81871
$186$	−1.65549	−0.121387
$187$	14.6436	1.07085
$188$	−0.286731	−0.0209120
$189$	1.25167	0.0910458
$190$	−0.169213	−0.0122760
$191$	3.44769	0.249466	0.124733	−	0.992190i	$-0.460193\pi$
$191$	3.44769	0.249466	0.124733	+	0.992190i	$0.460193\pi$
$192$	−1.00000	−0.0721688
$193$	24.1237	1.73646	0.868232	−	0.496159i	$-0.165257\pi$
$193$	24.1237	1.73646	0.868232	+	0.496159i	$0.165257\pi$
$194$	−0.811353	−0.0582517
$195$	−20.4074	−1.46140
$196$	−5.43332	−0.388094
$197$	19.0298	1.35581	0.677907	−	0.735147i	$-0.262887\pi$
$197$	19.0298	1.35581	0.677907	+	0.735147i	$0.262887\pi$
$198$	−2.48043	−0.176277
$199$	−2.13847	−0.151592	−0.0757960	−	0.997123i	$-0.524150\pi$
$199$	−2.13847	−0.151592	−0.0757960	+	0.997123i	$0.524150\pi$
$200$	6.71852	0.475071
$201$	8.37070	0.590424
$202$	0.311563	0.0219215
$203$	1.25167	0.0878502
$204$	5.90366	0.413339
$205$	21.8217	1.52409
$206$	−16.0371	−1.11736
$207$	−1.00000	−0.0695048
$208$	−5.96144	−0.413351
$209$	−0.122610	−0.00848110
$210$	−4.28477	−0.295677
$211$	−13.7573	−0.947094	−0.473547	−	0.880768i	$-0.657027\pi$
$211$	−13.7573	−0.947094	−0.473547	+	0.880768i	$0.657027\pi$
$212$	1.42752	0.0980425
$213$	5.41935	0.371328
$214$	7.79784	0.533050
$215$	−32.3190	−2.20414
$216$	−1.00000	−0.0680414
$217$	−2.07213	−0.140666
$218$	−9.71695	−0.658115
$219$	4.43236	0.299511
$220$	8.49109	0.572469
$221$	35.1943	2.36743
$222$	−11.1995	−0.751663
$223$	−1.26560	−0.0847510	−0.0423755	−	0.999102i	$-0.513493\pi$
$223$	−1.26560	−0.0847510	−0.0423755	+	0.999102i	$0.513493\pi$
$224$	−1.25167	−0.0836309
$225$	6.71852	0.447901
$226$	−2.63452	−0.175246
$227$	15.7506	1.04540	0.522701	−	0.852516i	$-0.324924\pi$
$227$	15.7506	1.04540	0.522701	+	0.852516i	$0.324924\pi$
$228$	−0.0494308	−0.00327364
$229$	28.2236	1.86506	0.932532	−	0.361086i	$-0.117594\pi$
$229$	28.2236	1.86506	0.932532	+	0.361086i	$0.117594\pi$
$230$	3.42323	0.225721
$231$	−3.10469	−0.204273
$232$	−1.00000	−0.0656532
$233$	5.63781	0.369345	0.184672	−	0.982800i	$-0.440878\pi$
$233$	5.63781	0.369345	0.184672	+	0.982800i	$0.440878\pi$
$234$	−5.96144	−0.389711
$235$	0.981547	0.0640290
$236$	1.26100	0.0820841
$237$	−12.1466	−0.789005
$238$	7.38945	0.478987
$239$	−22.9285	−1.48312	−0.741561	−	0.670885i	$-0.765914\pi$
$239$	−22.9285	−1.48312	−0.741561	+	0.670885i	$0.765914\pi$
$240$	3.42323	0.220969
$241$	15.0573	0.969924	0.484962	−	0.874535i	$-0.338833\pi$
$241$	15.0573	0.969924	0.484962	+	0.874535i	$0.338833\pi$
$242$	−4.84746	−0.311607
$243$	−1.00000	−0.0641500
$244$	4.48043	0.286830
$245$	18.5995	1.18828
$246$	6.37459	0.406429
$247$	−0.294679	−0.0187500
$248$	1.65549	0.105124
$249$	−11.8542	−0.751231
$250$	−5.88289	−0.372066
$251$	11.7565	0.742066	0.371033	−	0.928620i	$-0.379004\pi$
$251$	11.7565	0.742066	0.371033	+	0.928620i	$0.379004\pi$
$252$	−1.25167	−0.0788480
$253$	2.48043	0.155943
$254$	16.8037	1.05436
$255$	−20.2096	−1.26557
$256$	1.00000	0.0625000
$257$	−9.62315	−0.600276	−0.300138	−	0.953896i	$-0.597033\pi$
$257$	−9.62315	−0.600276	−0.300138	+	0.953896i	$0.597033\pi$
$258$	−9.44108	−0.587776
$259$	−14.0181	−0.871045
$260$	20.4074	1.26561
$261$	−1.00000	−0.0618984
$262$	−3.20126	−0.197774
$263$	−1.50326	−0.0926949	−0.0463475	−	0.998925i	$-0.514758\pi$
$263$	−1.50326	−0.0926949	−0.0463475	+	0.998925i	$0.514758\pi$
$264$	2.48043	0.152660
$265$	−4.88673	−0.300190
$266$	−0.0618712	−0.00379357
$267$	0.796776	0.0487619
$268$	−8.37070	−0.511322
$269$	−12.9944	−0.792285	−0.396142	−	0.918189i	$-0.629651\pi$
$269$	−12.9944	−0.792285	−0.396142	+	0.918189i	$0.629651\pi$
$270$	3.42323	0.208331
$271$	−15.7996	−0.959760	−0.479880	−	0.877334i	$-0.659320\pi$
$271$	−15.7996	−0.959760	−0.479880	+	0.877334i	$0.659320\pi$
$272$	−5.90366	−0.357962
$273$	−7.46177	−0.451607
$274$	14.1466	0.854626
$275$	−16.6648	−1.00493
$276$	1.00000	0.0601929
$277$	−18.5873	−1.11680	−0.558401	−	0.829571i	$-0.688585\pi$
$277$	−18.5873	−1.11680	−0.558401	+	0.829571i	$0.688585\pi$
$278$	6.46531	0.387763
$279$	1.65549	0.0991118
$280$	4.28477	0.256064
$281$	−29.8134	−1.77852	−0.889259	−	0.457405i	$-0.848779\pi$
$281$	−29.8134	−1.77852	−0.889259	+	0.457405i	$0.848779\pi$
$282$	0.286731	0.0170746
$283$	25.8779	1.53828	0.769141	−	0.639079i	$-0.220684\pi$
$283$	25.8779	1.53828	0.769141	+	0.639079i	$0.220684\pi$
$284$	−5.41935	−0.321579
$285$	0.169213	0.0100233
$286$	14.7869	0.874370
$287$	7.97890	0.470979
$288$	1.00000	0.0589256
$289$	17.8532	1.05019
$290$	3.42323	0.201019
$291$	0.811353	0.0475623
$292$	−4.43236	−0.259384
$293$	−21.9314	−1.28125	−0.640623	−	0.767856i	$-0.721324\pi$
$293$	−21.9314	−1.28125	−0.640623	+	0.767856i	$0.721324\pi$
$294$	5.43332	0.316877
$295$	−4.31670	−0.251328
$296$	11.1995	0.650959
$297$	2.48043	0.143929
$298$	0.0280400	0.00162431
$299$	5.96144	0.344759
$300$	−6.71852	−0.387894
$301$	−11.8171	−0.681129
$302$	−2.83192	−0.162959
$303$	−0.311563	−0.0178988
$304$	0.0494308	0.00283505
$305$	−15.3376	−0.878226
$306$	−5.90366	−0.337490
$307$	19.0530	1.08741	0.543705	−	0.839276i	$-0.317021\pi$
$307$	19.0530	1.08741	0.543705	+	0.839276i	$0.317021\pi$
$308$	3.10469	0.176906
$309$	16.0371	0.912317
$310$	−5.66714	−0.321872
$311$	29.8045	1.69006	0.845029	−	0.534721i	$-0.179583\pi$
$311$	29.8045	1.69006	0.845029	+	0.534721i	$0.179583\pi$
$312$	5.96144	0.337500
$313$	29.0561	1.64235	0.821173	−	0.570679i	$-0.193320\pi$
$313$	29.0561	1.64235	0.821173	+	0.570679i	$0.193320\pi$
$314$	−12.1009	−0.682891
$315$	4.28477	0.241419
$316$	12.1466	0.683299
$317$	−24.7475	−1.38996	−0.694980	−	0.719029i	$-0.744587\pi$
$317$	−24.7475	−1.38996	−0.694980	+	0.719029i	$0.744587\pi$
$318$	−1.42752	−0.0800514
$319$	2.48043	0.138877
$320$	−3.42323	−0.191364
$321$	−7.79784	−0.435233
$322$	1.25167	0.0697530
$323$	−0.291823	−0.0162375
$324$	1.00000	0.0555556
$325$	−40.0520	−2.22169
$326$	5.14348	0.284871
$327$	9.71695	0.537349
$328$	−6.37459	−0.351978
$329$	0.358893	0.0197864
$330$	−8.49109	−0.467419
$331$	−2.82184	−0.155102	−0.0775512	−	0.996988i	$-0.524710\pi$
$331$	−2.82184	−0.155102	−0.0775512	+	0.996988i	$0.524710\pi$
$332$	11.8542	0.650586
$333$	11.1995	0.613730
$334$	17.0100	0.930744
$335$	28.6549	1.56558
$336$	1.25167	0.0682843
$337$	9.01515	0.491086	0.245543	−	0.969386i	$-0.421034\pi$
$337$	9.01515	0.491086	0.245543	+	0.969386i	$0.421034\pi$
$338$	22.5388	1.22595
$339$	2.63452	0.143088
$340$	20.2096	1.09602
$341$	−4.10634	−0.222371
$342$	0.0494308	0.00267291
$343$	15.5624	0.840293
$344$	9.44108	0.509029
$345$	−3.42323	−0.184301
$346$	−15.5687	−0.836976
$347$	−2.56587	−0.137743	−0.0688716	−	0.997626i	$-0.521940\pi$
$347$	−2.56587	−0.137743	−0.0688716	+	0.997626i	$0.521940\pi$
$348$	1.00000	0.0536056
$349$	9.48440	0.507688	0.253844	−	0.967245i	$-0.418305\pi$
$349$	9.48440	0.507688	0.253844	+	0.967245i	$0.418305\pi$
$350$	−8.40938	−0.449501
$351$	5.96144	0.318198
$352$	−2.48043	−0.132207
$353$	−18.1416	−0.965582	−0.482791	−	0.875736i	$-0.660377\pi$
$353$	−18.1416	−0.965582	−0.482791	+	0.875736i	$0.660377\pi$
$354$	−1.26100	−0.0670214
$355$	18.5517	0.984621
$356$	−0.796776	−0.0422290
$357$	−7.38945	−0.391091
$358$	−15.5798	−0.823416
$359$	−17.1102	−0.903042	−0.451521	−	0.892260i	$-0.649118\pi$
$359$	−17.1102	−0.903042	−0.451521	+	0.892260i	$0.649118\pi$
$360$	−3.42323	−0.180420
$361$	−18.9976	−0.999871
$362$	14.8151	0.778664
$363$	4.84746	0.254426
$364$	7.46177	0.391103
$365$	15.1730	0.794192
$366$	−4.48043	−0.234196
$367$	−12.4083	−0.647707	−0.323853	−	0.946107i	$-0.604978\pi$
$367$	−12.4083	−0.647707	−0.323853	+	0.946107i	$0.604978\pi$
$368$	−1.00000	−0.0521286
$369$	−6.37459	−0.331848
$370$	−38.3386	−1.99313
$371$	−1.78679	−0.0927654
$372$	−1.65549	−0.0858333
$373$	27.2148	1.40913	0.704565	−	0.709640i	$-0.251142\pi$
$373$	27.2148	1.40913	0.704565	+	0.709640i	$0.251142\pi$
$374$	14.6436	0.757204
$375$	5.88289	0.303791
$376$	−0.286731	−0.0147870
$377$	5.96144	0.307030
$378$	1.25167	0.0643791
$379$	−11.3839	−0.584751	−0.292375	−	0.956304i	$-0.594446\pi$
$379$	−11.3839	−0.584751	−0.292375	+	0.956304i	$0.594446\pi$
$380$	−0.169213	−0.00868045
$381$	−16.8037	−0.860878
$382$	3.44769	0.176399
$383$	−35.7156	−1.82498	−0.912490	−	0.409100i	$-0.865843\pi$
$383$	−35.7156	−1.82498	−0.912490	+	0.409100i	$0.865843\pi$
$384$	−1.00000	−0.0510310
$385$	−10.6281	−0.541656
$386$	24.1237	1.22786
$387$	9.44108	0.479917
$388$	−0.811353	−0.0411902
$389$	6.85337	0.347480	0.173740	−	0.984792i	$-0.444415\pi$
$389$	6.85337	0.347480	0.173740	+	0.984792i	$0.444415\pi$
$390$	−20.4074	−1.03337
$391$	5.90366	0.298561
$392$	−5.43332	−0.274424
$393$	3.20126	0.161482
$394$	19.0298	0.958706
$395$	−41.5806	−2.09215
$396$	−2.48043	−0.124646
$397$	−5.16414	−0.259181	−0.129590	−	0.991568i	$-0.541366\pi$
$397$	−5.16414	−0.259181	−0.129590	+	0.991568i	$0.541366\pi$
$398$	−2.13847	−0.107192
$399$	0.0618712	0.00309743
$400$	6.71852	0.335926
$401$	3.17994	0.158799	0.0793994	−	0.996843i	$-0.474700\pi$
$401$	3.17994	0.158799	0.0793994	+	0.996843i	$0.474700\pi$
$402$	8.37070	0.417493
$403$	−9.86912	−0.491616
$404$	0.311563	0.0155008
$405$	−3.42323	−0.170102
$406$	1.25167	0.0621195
$407$	−27.7797	−1.37699
$408$	5.90366	0.292275
$409$	7.97642	0.394409	0.197204	−	0.980362i	$-0.436814\pi$
$409$	7.97642	0.394409	0.197204	+	0.980362i	$0.436814\pi$
$410$	21.8217	1.07770
$411$	−14.1466	−0.697800
$412$	−16.0371	−0.790090
$413$	−1.57836	−0.0776660
$414$	−1.00000	−0.0491473
$415$	−40.5798	−1.99198
$416$	−5.96144	−0.292284
$417$	−6.46531	−0.316608
$418$	−0.122610	−0.00599704
$419$	−10.4857	−0.512260	−0.256130	−	0.966642i	$-0.582448\pi$
$419$	−10.4857	−0.512260	−0.256130	+	0.966642i	$0.582448\pi$
$420$	−4.28477	−0.209075
$421$	15.7041	0.765373	0.382686	−	0.923878i	$-0.374999\pi$
$421$	15.7041	0.765373	0.382686	+	0.923878i	$0.374999\pi$
$422$	−13.7573	−0.669697
$423$	−0.286731	−0.0139413
$424$	1.42752	0.0693265
$425$	−39.6639	−1.92398
$426$	5.41935	0.262568
$427$	−5.60803	−0.271392
$428$	7.79784	0.376923
$429$	−14.7869	−0.713920
$430$	−32.3190	−1.55856
$431$	−12.8101	−0.617040	−0.308520	−	0.951218i	$-0.599834\pi$
$431$	−12.8101	−0.617040	−0.308520	+	0.951218i	$0.599834\pi$
$432$	−1.00000	−0.0481125
$433$	21.3978	1.02831	0.514156	−	0.857696i	$-0.328105\pi$
$433$	21.3978	1.02831	0.514156	+	0.857696i	$0.328105\pi$
$434$	−2.07213	−0.0994657
$435$	−3.42323	−0.164131
$436$	−9.71695	−0.465358
$437$	−0.0494308	−0.00236460
$438$	4.43236	0.211787
$439$	−28.2476	−1.34818	−0.674091	−	0.738648i	$-0.735465\pi$
$439$	−28.2476	−1.34818	−0.674091	+	0.738648i	$0.735465\pi$
$440$	8.49109	0.404797
$441$	−5.43332	−0.258729
$442$	35.1943	1.67402
$443$	−3.60019	−0.171050	−0.0855251	−	0.996336i	$-0.527257\pi$
$443$	−3.60019	−0.171050	−0.0855251	+	0.996336i	$0.527257\pi$
$444$	−11.1995	−0.531506
$445$	2.72755	0.129298
$446$	−1.26560	−0.0599280
$447$	−0.0280400	−0.00132625
$448$	−1.25167	−0.0591360
$449$	26.4204	1.24685	0.623427	−	0.781881i	$-0.285740\pi$
$449$	26.4204	1.24685	0.623427	+	0.781881i	$0.285740\pi$
$450$	6.71852	0.316714
$451$	15.8117	0.744545
$452$	−2.63452	−0.123917
$453$	2.83192	0.133055
$454$	15.7506	0.739211
$455$	−25.5434	−1.19749
$456$	−0.0494308	−0.00231481
$457$	−7.19372	−0.336508	−0.168254	−	0.985744i	$-0.553813\pi$
$457$	−7.19372	−0.336508	−0.168254	+	0.985744i	$0.553813\pi$
$458$	28.2236	1.31880
$459$	5.90366	0.275559
$460$	3.42323	0.159609
$461$	−29.3838	−1.36854	−0.684270	−	0.729229i	$-0.739879\pi$
$461$	−29.3838	−1.36854	−0.684270	+	0.729229i	$0.739879\pi$
$462$	−3.10469	−0.144443
$463$	−20.4254	−0.949247	−0.474623	−	0.880189i	$-0.657416\pi$
$463$	−20.4254	−0.949247	−0.474623	+	0.880189i	$0.657416\pi$
$464$	−1.00000	−0.0464238
$465$	5.66714	0.262807
$466$	5.63781	0.261166
$467$	41.6099	1.92548	0.962739	−	0.270431i	$-0.0871663\pi$
$467$	41.6099	1.92548	0.962739	+	0.270431i	$0.0871663\pi$
$468$	−5.96144	−0.275568
$469$	10.4774	0.483801
$470$	0.981547	0.0452754
$471$	12.1009	0.557578
$472$	1.26100	0.0580423
$473$	−23.4180	−1.07676
$474$	−12.1466	−0.557911
$475$	0.332102	0.0152379
$476$	7.38945	0.338695
$477$	1.42752	0.0653617
$478$	−22.9285	−1.04873
$479$	−25.8977	−1.18330	−0.591649	−	0.806196i	$-0.701523\pi$
$479$	−25.8977	−1.18330	−0.591649	+	0.806196i	$0.701523\pi$
$480$	3.42323	0.156248
$481$	−66.7653	−3.04424
$482$	15.0573	0.685840
$483$	−1.25167	−0.0569531
$484$	−4.84746	−0.220339
$485$	2.77745	0.126117
$486$	−1.00000	−0.0453609
$487$	−16.7191	−0.757617	−0.378808	−	0.925475i	$-0.623666\pi$
$487$	−16.7191	−0.757617	−0.378808	+	0.925475i	$0.623666\pi$
$488$	4.48043	0.202820
$489$	−5.14348	−0.232596
$490$	18.5995	0.840240
$491$	2.14894	0.0969804	0.0484902	−	0.998824i	$-0.484559\pi$
$491$	2.14894	0.0969804	0.0484902	+	0.998824i	$0.484559\pi$
$492$	6.37459	0.287389
$493$	5.90366	0.265888
$494$	−0.294679	−0.0132582
$495$	8.49109	0.381646
$496$	1.65549	0.0743338
$497$	6.78325	0.304270
$498$	−11.8542	−0.531201
$499$	−29.2331	−1.30865	−0.654327	−	0.756211i	$-0.727048\pi$
$499$	−29.2331	−1.30865	−0.654327	+	0.756211i	$0.727048\pi$
$500$	−5.88289	−0.263091
$501$	−17.0100	−0.759950
$502$	11.7565	0.524720
$503$	−21.1768	−0.944226	−0.472113	−	0.881538i	$-0.656509\pi$
$503$	−21.1768	−0.944226	−0.472113	+	0.881538i	$0.656509\pi$
$504$	−1.25167	−0.0557539
$505$	−1.06655	−0.0474609
$506$	2.48043	0.110269
$507$	−22.5388	−1.00098
$508$	16.8037	0.745543
$509$	32.7617	1.45214	0.726069	−	0.687622i	$-0.241345\pi$
$509$	32.7617	1.45214	0.726069	+	0.687622i	$0.241345\pi$
$510$	−20.2096	−0.894897
$511$	5.54787	0.245423
$512$	1.00000	0.0441942
$513$	−0.0494308	−0.00218242
$514$	−9.62315	−0.424459
$515$	54.8986	2.41912
$516$	−9.44108	−0.415621
$517$	0.711216	0.0312793
$518$	−14.0181	−0.615922
$519$	15.5687	0.683388
$520$	20.4074	0.894923
$521$	−23.1467	−1.01408	−0.507038	−	0.861923i	$-0.669260\pi$
$521$	−23.1467	−1.01408	−0.507038	+	0.861923i	$0.669260\pi$
$522$	−1.00000	−0.0437688
$523$	14.7123	0.643322	0.321661	−	0.946855i	$-0.395759\pi$
$523$	14.7123	0.643322	0.321661	+	0.946855i	$0.395759\pi$
$524$	−3.20126	−0.139848
$525$	8.40938	0.367016
$526$	−1.50326	−0.0655452
$527$	−9.77347	−0.425739
$528$	2.48043	0.107947
$529$	1.00000	0.0434783
$530$	−4.88673	−0.212266
$531$	1.26100	0.0547228
$532$	−0.0618712	−0.00268246
$533$	38.0017	1.64604
$534$	0.796776	0.0344799
$535$	−26.6938	−1.15407
$536$	−8.37070	−0.361559
$537$	15.5798	0.672316
$538$	−12.9944	−0.560230
$539$	13.4770	0.580494
$540$	3.42323	0.147312
$541$	5.69481	0.244839	0.122419	−	0.992478i	$-0.460935\pi$
$541$	5.69481	0.244839	0.122419	+	0.992478i	$0.460935\pi$
$542$	−15.7996	−0.678653
$543$	−14.8151	−0.635777
$544$	−5.90366	−0.253117
$545$	33.2634	1.42485
$546$	−7.46177	−0.319334
$547$	14.6491	0.626349	0.313174	−	0.949696i	$-0.398608\pi$
$547$	14.6491	0.626349	0.313174	+	0.949696i	$0.398608\pi$
$548$	14.1466	0.604312
$549$	4.48043	0.191220
$550$	−16.6648	−0.710590
$551$	−0.0494308	−0.00210582
$552$	1.00000	0.0425628
$553$	−15.2035	−0.646521
$554$	−18.5873	−0.789699
$555$	38.3386	1.62738
$556$	6.46531	0.274190
$557$	−1.69393	−0.0717740	−0.0358870	−	0.999356i	$-0.511426\pi$
$557$	−1.69393	−0.0717740	−0.0358870	+	0.999356i	$0.511426\pi$
$558$	1.65549	0.0700826
$559$	−56.2825	−2.38049
$560$	4.28477	0.181064
$561$	−14.6436	−0.618255
$562$	−29.8134	−1.25760
$563$	−17.9267	−0.755520	−0.377760	−	0.925904i	$-0.623306\pi$
$563$	−17.9267	−0.755520	−0.377760	+	0.925904i	$0.623306\pi$
$564$	0.286731	0.0120736
$565$	9.01858	0.379414
$566$	25.8779	1.08773
$567$	−1.25167	−0.0525653
$568$	−5.41935	−0.227391
$569$	−30.0759	−1.26085	−0.630424	−	0.776251i	$-0.717119\pi$
$569$	−30.0759	−1.26085	−0.630424	+	0.776251i	$0.717119\pi$
$570$	0.169213	0.00708756
$571$	−26.1559	−1.09459	−0.547295	−	0.836940i	$-0.684342\pi$
$571$	−26.1559	−1.09459	−0.547295	+	0.836940i	$0.684342\pi$
$572$	14.7869	0.618273
$573$	−3.44769	−0.144029
$574$	7.97890	0.333033
$575$	−6.71852	−0.280182
$576$	1.00000	0.0416667
$577$	−1.35438	−0.0563834	−0.0281917	−	0.999603i	$-0.508975\pi$
$577$	−1.35438	−0.0563834	−0.0281917	+	0.999603i	$0.508975\pi$
$578$	17.8532	0.742597
$579$	−24.1237	−1.00255
$580$	3.42323	0.142142
$581$	−14.8376	−0.615568
$582$	0.811353	0.0336317
$583$	−3.54087	−0.146648
$584$	−4.43236	−0.183413
$585$	20.4074	0.843742
$586$	−21.9314	−0.905977
$587$	−18.3243	−0.756327	−0.378163	−	0.925739i	$-0.623444\pi$
$587$	−18.3243	−0.756327	−0.378163	+	0.925739i	$0.623444\pi$
$588$	5.43332	0.224066
$589$	0.0818324	0.00337184
$590$	−4.31670	−0.177716
$591$	−19.0298	−0.782780
$592$	11.1995	0.460298
$593$	45.0673	1.85069	0.925346	−	0.379125i	$-0.123775\pi$
$593$	45.0673	1.85069	0.925346	+	0.379125i	$0.123775\pi$
$594$	2.48043	0.101773
$595$	−25.2958	−1.03703
$596$	0.0280400	0.00114856
$597$	2.13847	0.0875217
$598$	5.96144	0.243781
$599$	−12.4545	−0.508877	−0.254438	−	0.967089i	$-0.581891\pi$
$599$	−12.4545	−0.508877	−0.254438	+	0.967089i	$0.581891\pi$
$600$	−6.71852	−0.274282
$601$	−15.2181	−0.620757	−0.310379	−	0.950613i	$-0.600456\pi$
$601$	−15.2181	−0.620757	−0.310379	+	0.950613i	$0.600456\pi$
$602$	−11.8171	−0.481631
$603$	−8.37070	−0.340881
$604$	−2.83192	−0.115229
$605$	16.5940	0.674641
$606$	−0.311563	−0.0126564
$607$	−31.9059	−1.29502	−0.647510	−	0.762057i	$-0.724189\pi$
$607$	−31.9059	−1.29502	−0.647510	+	0.762057i	$0.724189\pi$
$608$	0.0494308	0.00200468
$609$	−1.25167	−0.0507203
$610$	−15.3376	−0.621000
$611$	1.70933	0.0691521
$612$	−5.90366	−0.238641
$613$	13.3171	0.537873	0.268936	−	0.963158i	$-0.413328\pi$
$613$	13.3171	0.537873	0.268936	+	0.963158i	$0.413328\pi$
$614$	19.0530	0.768915
$615$	−21.8217	−0.879936
$616$	3.10469	0.125091
$617$	45.9054	1.84808	0.924041	−	0.382293i	$-0.124865\pi$
$617$	45.9054	1.84808	0.924041	+	0.382293i	$0.124865\pi$
$618$	16.0371	0.645105
$619$	37.2120	1.49568	0.747839	−	0.663880i	$-0.231092\pi$
$619$	37.2120	1.49568	0.747839	+	0.663880i	$0.231092\pi$
$620$	−5.66714	−0.227598
$621$	1.00000	0.0401286
$622$	29.8045	1.19505
$623$	0.997302	0.0399561
$624$	5.96144	0.238649
$625$	−13.4541	−0.538164
$626$	29.0561	1.16131
$627$	0.122610	0.00489656
$628$	−12.1009	−0.482877
$629$	−66.1182	−2.63631
$630$	4.28477	0.170709
$631$	−12.2235	−0.486612	−0.243306	−	0.969950i	$-0.578232\pi$
$631$	−12.2235	−0.486612	−0.243306	+	0.969950i	$0.578232\pi$
$632$	12.1466	0.483165
$633$	13.7573	0.546805
$634$	−24.7475	−0.982850
$635$	−57.5229	−2.28273
$636$	−1.42752	−0.0566049
$637$	32.3904	1.28335
$638$	2.48043	0.0982012
$639$	−5.41935	−0.214386
$640$	−3.42323	−0.135315
$641$	−5.18516	−0.204802	−0.102401	−	0.994743i	$-0.532652\pi$
$641$	−5.18516	−0.204802	−0.102401	+	0.994743i	$0.532652\pi$
$642$	−7.79784	−0.307756
$643$	25.7268	1.01457	0.507284	−	0.861779i	$-0.330650\pi$
$643$	25.7268	1.01457	0.507284	+	0.861779i	$0.330650\pi$
$644$	1.25167	0.0493228
$645$	32.3190	1.27256
$646$	−0.291823	−0.0114816
$647$	31.0162	1.21937	0.609687	−	0.792642i	$-0.291295\pi$
$647$	31.0162	1.21937	0.609687	+	0.792642i	$0.291295\pi$
$648$	1.00000	0.0392837
$649$	−3.12783	−0.122778
$650$	−40.0520	−1.57097
$651$	2.07213	0.0812134
$652$	5.14348	0.201434
$653$	36.3135	1.42106	0.710529	−	0.703668i	$-0.248456\pi$
$653$	36.3135	1.42106	0.710529	+	0.703668i	$0.248456\pi$
$654$	9.71695	0.379963
$655$	10.9587	0.428190
$656$	−6.37459	−0.248886
$657$	−4.43236	−0.172923
$658$	0.358893	0.0139911
$659$	18.4974	0.720557	0.360279	−	0.932845i	$-0.382682\pi$
$659$	18.4974	0.720557	0.360279	+	0.932845i	$0.382682\pi$
$660$	−8.49109	−0.330515
$661$	−32.0191	−1.24540	−0.622699	−	0.782461i	$-0.713964\pi$
$661$	−32.0191	−1.24540	−0.622699	+	0.782461i	$0.713964\pi$
$662$	−2.82184	−0.109674
$663$	−35.1943	−1.36683
$664$	11.8542	0.460033
$665$	0.211800	0.00821323
$666$	11.1995	0.433973
$667$	1.00000	0.0387202
$668$	17.0100	0.658136
$669$	1.26560	0.0489310
$670$	28.6549	1.10703
$671$	−11.1134	−0.429028
$672$	1.25167	0.0482843
$673$	24.3732	0.939518	0.469759	−	0.882795i	$-0.344341\pi$
$673$	24.3732	0.939518	0.469759	+	0.882795i	$0.344341\pi$
$674$	9.01515	0.347250
$675$	−6.71852	−0.258596
$676$	22.5388	0.866875
$677$	23.2758	0.894562	0.447281	−	0.894393i	$-0.352392\pi$
$677$	23.2758	0.894562	0.447281	+	0.894393i	$0.352392\pi$
$678$	2.63452	0.101178
$679$	1.01555	0.0389732
$680$	20.2096	0.775003
$681$	−15.7506	−0.603563
$682$	−4.10634	−0.157240
$683$	28.5651	1.09301	0.546506	−	0.837455i	$-0.315957\pi$
$683$	28.5651	1.09301	0.546506	+	0.837455i	$0.315957\pi$
$684$	0.0494308	0.00189004
$685$	−48.4270	−1.85030
$686$	15.5624	0.594177
$687$	−28.2236	−1.07680
$688$	9.44108	0.359938
$689$	−8.51008	−0.324208
$690$	−3.42323	−0.130320
$691$	38.4394	1.46230	0.731152	−	0.682215i	$-0.238983\pi$
$691$	38.4394	1.46230	0.731152	+	0.682215i	$0.238983\pi$
$692$	−15.5687	−0.591832
$693$	3.10469	0.117937
$694$	−2.56587	−0.0973992
$695$	−22.1323	−0.839524
$696$	1.00000	0.0379049
$697$	37.6334	1.42547
$698$	9.48440	0.358990
$699$	−5.63781	−0.213241
$700$	−8.40938	−0.317845
$701$	−40.0888	−1.51413	−0.757066	−	0.653338i	$-0.773368\pi$
$701$	−40.0888	−1.51413	−0.757066	+	0.653338i	$0.773368\pi$
$702$	5.96144	0.225000
$703$	0.553602	0.0208795
$704$	−2.48043	−0.0934848
$705$	−0.981547	−0.0369672
$706$	−18.1416	−0.682770
$707$	−0.389974	−0.0146665
$708$	−1.26100	−0.0473913
$709$	46.7043	1.75402	0.877008	−	0.480475i	$-0.159536\pi$
$709$	46.7043	1.75402	0.877008	+	0.480475i	$0.159536\pi$
$710$	18.5517	0.696232
$711$	12.1466	0.455533
$712$	−0.796776	−0.0298604
$713$	−1.65549	−0.0619987
$714$	−7.38945	−0.276543
$715$	−50.6191	−1.89305
$716$	−15.5798	−0.582243
$717$	22.9285	0.856281
$718$	−17.1102	−0.638547
$719$	7.71868	0.287858	0.143929	−	0.989588i	$-0.454026\pi$
$719$	7.71868	0.287858	0.143929	+	0.989588i	$0.454026\pi$
$720$	−3.42323	−0.127576
$721$	20.0732	0.747563
$722$	−18.9976	−0.707016
$723$	−15.0573	−0.559986
$724$	14.8151	0.550599
$725$	−6.71852	−0.249519
$726$	4.84746	0.179906
$727$	−9.09828	−0.337437	−0.168718	−	0.985664i	$-0.553963\pi$
$727$	−9.09828	−0.337437	−0.168718	+	0.985664i	$0.553963\pi$
$728$	7.46177	0.276552
$729$	1.00000	0.0370370
$730$	15.1730	0.561578
$731$	−55.7370	−2.06151
$732$	−4.48043	−0.165602
$733$	20.7658	0.767002	0.383501	−	0.923541i	$-0.374718\pi$
$733$	20.7658	0.767002	0.383501	+	0.923541i	$0.374718\pi$
$734$	−12.4083	−0.457998
$735$	−18.5995	−0.686053
$736$	−1.00000	−0.0368605
$737$	20.7630	0.764813
$738$	−6.37459	−0.234652
$739$	7.72286	0.284090	0.142045	−	0.989860i	$-0.454632\pi$
$739$	7.72286	0.284090	0.142045	+	0.989860i	$0.454632\pi$
$740$	−38.3386	−1.40935
$741$	0.294679	0.0108253
$742$	−1.78679	−0.0655951
$743$	18.3274	0.672368	0.336184	−	0.941796i	$-0.390864\pi$
$743$	18.3274	0.672368	0.336184	+	0.941796i	$0.390864\pi$
$744$	−1.65549	−0.0606933
$745$	−0.0959875	−0.00351671
$746$	27.2148	0.996405
$747$	11.8542	0.433724
$748$	14.6436	0.535424
$749$	−9.76035	−0.356635
$750$	5.88289	0.214813
$751$	8.13997	0.297032	0.148516	−	0.988910i	$-0.452550\pi$
$751$	8.13997	0.297032	0.148516	+	0.988910i	$0.452550\pi$
$752$	−0.286731	−0.0104560
$753$	−11.7565	−0.428432
$754$	5.96144	0.217103
$755$	9.69432	0.352812
$756$	1.25167	0.0455229
$757$	8.06059	0.292967	0.146484	−	0.989213i	$-0.453204\pi$
$757$	8.06059	0.292967	0.146484	+	0.989213i	$0.453204\pi$
$758$	−11.3839	−0.413481
$759$	−2.48043	−0.0900340
$760$	−0.169213	−0.00613801
$761$	20.6510	0.748599	0.374299	−	0.927308i	$-0.377883\pi$
$761$	20.6510	0.748599	0.374299	+	0.927308i	$0.377883\pi$
$762$	−16.8037	−0.608733
$763$	12.1624	0.440310
$764$	3.44769	0.124733
$765$	20.2096	0.730680
$766$	−35.7156	−1.29046
$767$	−7.51738	−0.271437
$768$	−1.00000	−0.0360844
$769$	−3.88096	−0.139951	−0.0699755	−	0.997549i	$-0.522292\pi$
$769$	−3.88096	−0.139951	−0.0699755	+	0.997549i	$0.522292\pi$
$770$	−10.6281	−0.383009
$771$	9.62315	0.346569
$772$	24.1237	0.868232
$773$	−16.9976	−0.611359	−0.305680	−	0.952134i	$-0.598884\pi$
$773$	−16.9976	−0.611359	−0.305680	+	0.952134i	$0.598884\pi$
$774$	9.44108	0.339353
$775$	11.1225	0.399531
$776$	−0.811353	−0.0291259
$777$	14.0181	0.502898
$778$	6.85337	0.245705
$779$	−0.315101	−0.0112897
$780$	−20.4074	−0.730702
$781$	13.4423	0.481004
$782$	5.90366	0.211115
$783$	1.00000	0.0357371
$784$	−5.43332	−0.194047
$785$	41.4241	1.47849
$786$	3.20126	0.114185
$787$	22.4425	0.799989	0.399994	−	0.916518i	$-0.369012\pi$
$787$	22.4425	0.799989	0.399994	+	0.916518i	$0.369012\pi$
$788$	19.0298	0.677907
$789$	1.50326	0.0535174
$790$	−41.5806	−1.47937
$791$	3.29756	0.117248
$792$	−2.48043	−0.0881383
$793$	−26.7098	−0.948494
$794$	−5.16414	−0.183268
$795$	4.88673	0.173315
$796$	−2.13847	−0.0757960
$797$	28.7644	1.01889	0.509443	−	0.860504i	$-0.329851\pi$
$797$	28.7644	1.01889	0.509443	+	0.860504i	$0.329851\pi$
$798$	0.0618712	0.00219022
$799$	1.69276	0.0598857
$800$	6.71852	0.237535
$801$	−0.796776	−0.0281527
$802$	3.17994	0.112288
$803$	10.9942	0.387976
$804$	8.37070	0.295212
$805$	−4.28477	−0.151018
$806$	−9.86912	−0.347625
$807$	12.9944	0.457426
$808$	0.311563	0.0109607
$809$	30.8036	1.08300	0.541499	−	0.840701i	$-0.317857\pi$
$809$	30.8036	1.08300	0.541499	+	0.840701i	$0.317857\pi$
$810$	−3.42323	−0.120280
$811$	0.602248	0.0211478	0.0105739	−	0.999944i	$-0.496634\pi$
$811$	0.602248	0.0211478	0.0105739	+	0.999944i	$0.496634\pi$
$812$	1.25167	0.0439251
$813$	15.7996	0.554118
$814$	−27.7797	−0.973676
$815$	−17.6073	−0.616758
$816$	5.90366	0.206670
$817$	0.466681	0.0163271
$818$	7.97642	0.278889
$819$	7.46177	0.260735
$820$	21.8217	0.762047
$821$	−2.10468	−0.0734537	−0.0367269	−	0.999325i	$-0.511693\pi$
$821$	−2.10468	−0.0734537	−0.0367269	+	0.999325i	$0.511693\pi$
$822$	−14.1466	−0.493419
$823$	5.93359	0.206832	0.103416	−	0.994638i	$-0.467023\pi$
$823$	5.93359	0.206832	0.103416	+	0.994638i	$0.467023\pi$
$824$	−16.0371	−0.558678
$825$	16.6648	0.580195
$826$	−1.57836	−0.0549182
$827$	26.0644	0.906349	0.453174	−	0.891422i	$-0.350291\pi$
$827$	26.0644	0.906349	0.453174	+	0.891422i	$0.350291\pi$
$828$	−1.00000	−0.0347524
$829$	15.2371	0.529205	0.264602	−	0.964358i	$-0.414759\pi$
$829$	15.2371	0.529205	0.264602	+	0.964358i	$0.414759\pi$
$830$	−40.5798	−1.40855
$831$	18.5873	0.644786
$832$	−5.96144	−0.206676
$833$	32.0765	1.11138
$834$	−6.46531	−0.223875
$835$	−58.2291	−2.01510
$836$	−0.122610	−0.00424055
$837$	−1.65549	−0.0572222
$838$	−10.4857	−0.362223
$839$	7.48575	0.258437	0.129218	−	0.991616i	$-0.458753\pi$
$839$	7.48575	0.258437	0.129218	+	0.991616i	$0.458753\pi$
$840$	−4.28477	−0.147838
$841$	1.00000	0.0344828
$842$	15.7041	0.541200
$843$	29.8134	1.02683
$844$	−13.7573	−0.473547
$845$	−77.1554	−2.65423
$846$	−0.286731	−0.00985801
$847$	6.06743	0.208479
$848$	1.42752	0.0490213
$849$	−25.8779	−0.888128
$850$	−39.6639	−1.36046
$851$	−11.1995	−0.383915
$852$	5.41935	0.185664
$853$	28.8071	0.986336	0.493168	−	0.869934i	$-0.335839\pi$
$853$	28.8071	0.986336	0.493168	+	0.869934i	$0.335839\pi$
$854$	−5.60803	−0.191903
$855$	−0.169213	−0.00578697
$856$	7.79784	0.266525
$857$	3.15892	0.107907	0.0539533	−	0.998543i	$-0.482818\pi$
$857$	3.15892	0.107907	0.0539533	+	0.998543i	$0.482818\pi$
$858$	−14.7869	−0.504818
$859$	25.7180	0.877486	0.438743	−	0.898613i	$-0.355424\pi$
$859$	25.7180	0.877486	0.438743	+	0.898613i	$0.355424\pi$
$860$	−32.3190	−1.10207
$861$	−7.97890	−0.271920
$862$	−12.8101	−0.436313
$863$	−7.72221	−0.262867	−0.131434	−	0.991325i	$-0.541958\pi$
$863$	−7.72221	−0.262867	−0.131434	+	0.991325i	$0.541958\pi$
$864$	−1.00000	−0.0340207
$865$	53.2951	1.81209
$866$	21.3978	0.727127
$867$	−17.8532	−0.606328
$868$	−2.07213	−0.0703328
$869$	−30.1288	−1.02205
$870$	−3.42323	−0.116058
$871$	49.9015	1.69085
$872$	−9.71695	−0.329057
$873$	−0.811353	−0.0274601
$874$	−0.0494308	−0.00167202
$875$	7.36345	0.248930
$876$	4.43236	0.149756
$877$	−25.7601	−0.869857	−0.434929	−	0.900465i	$-0.643226\pi$
$877$	−25.7601	−0.869857	−0.434929	+	0.900465i	$0.643226\pi$
$878$	−28.2476	−0.953309
$879$	21.9314	0.739727
$880$	8.49109	0.286235
$881$	27.5117	0.926893	0.463447	−	0.886125i	$-0.346613\pi$
$881$	27.5117	0.926893	0.463447	+	0.886125i	$0.346613\pi$
$882$	−5.43332	−0.182949
$883$	−36.2104	−1.21858	−0.609288	−	0.792949i	$-0.708545\pi$
$883$	−36.2104	−1.21858	−0.609288	+	0.792949i	$0.708545\pi$
$884$	35.1943	1.18371
$885$	4.31670	0.145104
$886$	−3.60019	−0.120951
$887$	−37.2658	−1.25126	−0.625632	−	0.780118i	$-0.715159\pi$
$887$	−37.2658	−1.25126	−0.625632	+	0.780118i	$0.715159\pi$
$888$	−11.1995	−0.375831
$889$	−21.0327	−0.705414
$890$	2.72755	0.0914276
$891$	−2.48043	−0.0830976
$892$	−1.26560	−0.0423755
$893$	−0.0141733	−0.000474293 0
$894$	−0.0280400	−0.000937798 0
$895$	53.3331	1.78273
$896$	−1.25167	−0.0418154
$897$	−5.96144	−0.199047
$898$	26.4204	0.881659
$899$	−1.65549	−0.0552138
$900$	6.71852	0.223951
$901$	−8.42760	−0.280764
$902$	15.8117	0.526473
$903$	11.8171	0.393250
$904$	−2.63452	−0.0876229
$905$	−50.7155	−1.68584
$906$	2.83192	0.0940842
$907$	10.5076	0.348899	0.174449	−	0.984666i	$-0.444185\pi$
$907$	10.5076	0.348899	0.174449	+	0.984666i	$0.444185\pi$
$908$	15.7506	0.522701
$909$	0.311563	0.0103339
$910$	−25.5434	−0.846755
$911$	−11.1668	−0.369971	−0.184986	−	0.982741i	$-0.559224\pi$
$911$	−11.1668	−0.369971	−0.184986	+	0.982741i	$0.559224\pi$
$912$	−0.0494308	−0.00163682
$913$	−29.4036	−0.973117
$914$	−7.19372	−0.237947
$915$	15.3376	0.507044
$916$	28.2236	0.932532
$917$	4.00693	0.132320
$918$	5.90366	0.194850
$919$	41.0557	1.35430	0.677151	−	0.735844i	$-0.263215\pi$
$919$	41.0557	1.35430	0.677151	+	0.735844i	$0.263215\pi$
$920$	3.42323	0.112861
$921$	−19.0530	−0.627817
$922$	−29.3838	−0.967704
$923$	32.3071	1.06340
$924$	−3.10469	−0.102137
$925$	75.2442	2.47401
$926$	−20.4254	−0.671219
$927$	−16.0371	−0.526726
$928$	−1.00000	−0.0328266
$929$	0.615142	0.0201822	0.0100911	−	0.999949i	$-0.496788\pi$
$929$	0.615142	0.0201822	0.0100911	+	0.999949i	$0.496788\pi$
$930$	5.66714	0.185833
$931$	−0.268573	−0.00880214
$932$	5.63781	0.184672
$933$	−29.8045	−0.975755
$934$	41.6099	1.36152
$935$	−50.1285	−1.63938
$936$	−5.96144	−0.194856
$937$	20.0072	0.653608	0.326804	−	0.945092i	$-0.394028\pi$
$937$	20.0072	0.653608	0.326804	+	0.945092i	$0.394028\pi$
$938$	10.4774	0.342099
$939$	−29.0561	−0.948209
$940$	0.981547	0.0320145
$941$	−37.6403	−1.22704	−0.613520	−	0.789679i	$-0.710247\pi$
$941$	−37.6403	−1.22704	−0.613520	+	0.789679i	$0.710247\pi$
$942$	12.1009	0.394268
$943$	6.37459	0.207585
$944$	1.26100	0.0410421
$945$	−4.28477	−0.139383
$946$	−23.4180	−0.761384
$947$	12.1147	0.393675	0.196837	−	0.980436i	$-0.436933\pi$
$947$	12.1147	0.393675	0.196837	+	0.980436i	$0.436933\pi$
$948$	−12.1466	−0.394503
$949$	26.4233	0.857736
$950$	0.332102	0.0107748
$951$	24.7475	0.802493
$952$	7.38945	0.239494
$953$	−41.1265	−1.33222	−0.666110	−	0.745854i	$-0.732042\pi$
$953$	−41.1265	−1.33222	−0.666110	+	0.745854i	$0.732042\pi$
$954$	1.42752	0.0462177
$955$	−11.8022	−0.381912
$956$	−22.9285	−0.741561
$957$	−2.48043	−0.0801810
$958$	−25.8977	−0.836718
$959$	−17.7069	−0.571785
$960$	3.42323	0.110484
$961$	−28.2593	−0.911592
$962$	−66.7653	−2.15260
$963$	7.79784	0.251282
$964$	15.0573	0.484962
$965$	−82.5811	−2.65838
$966$	−1.25167	−0.0402719
$967$	−36.3071	−1.16756	−0.583779	−	0.811912i	$-0.698427\pi$
$967$	−36.3071	−1.16756	−0.583779	+	0.811912i	$0.698427\pi$
$968$	−4.84746	−0.155803
$969$	0.291823	0.00937471
$970$	2.77745	0.0891785
$971$	5.75572	0.184710	0.0923549	−	0.995726i	$-0.470561\pi$
$971$	5.75572	0.184710	0.0923549	+	0.995726i	$0.470561\pi$
$972$	−1.00000	−0.0320750
$973$	−8.09245	−0.259432
$974$	−16.7191	−0.535716
$975$	40.0520	1.28269
$976$	4.48043	0.143415
$977$	9.58165	0.306544	0.153272	−	0.988184i	$-0.451019\pi$
$977$	9.58165	0.306544	0.153272	+	0.988184i	$0.451019\pi$
$978$	−5.14348	−0.164470
$979$	1.97635	0.0631643
$980$	18.5995	0.594139
$981$	−9.71695	−0.310238
$982$	2.14894	0.0685755
$983$	−50.2321	−1.60215	−0.801077	−	0.598561i	$-0.795739\pi$
$983$	−50.2321	−1.60215	−0.801077	+	0.598561i	$0.795739\pi$
$984$	6.37459	0.203214
$985$	−65.1433	−2.07564
$986$	5.90366	0.188011
$987$	−0.358893	−0.0114237
$988$	−0.294679	−0.00937499
$989$	−9.44108	−0.300209
$990$	8.49109	0.269865
$991$	−27.3240	−0.867975	−0.433987	−	0.900919i	$-0.642894\pi$
$991$	−27.3240	−0.867975	−0.433987	+	0.900919i	$0.642894\pi$
$992$	1.65549	0.0525620
$993$	2.82184	0.0895484
$994$	6.78325	0.215152
$995$	7.32047	0.232075
$996$	−11.8542	−0.375616
$997$	17.4923	0.553987	0.276994	−	0.960872i	$-0.410662\pi$
$997$	17.4923	0.553987	0.276994	+	0.960872i	$0.410662\pi$
$998$	−29.2331	−0.925359
$999$	−11.1995	−0.354337

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.2	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.42323 q^{5} -1.00000 q^{6} -1.25167 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.42323 q^{5} -1.00000 q^{6} -1.25167 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.42323 q^{10} -2.48043 q^{11} -1.00000 q^{12} -5.96144 q^{13} -1.25167 q^{14} +3.42323 q^{15} +1.00000 q^{16} -5.90366 q^{17} +1.00000 q^{18} +0.0494308 q^{19} -3.42323 q^{20} +1.25167 q^{21} -2.48043 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.71852 q^{25} -5.96144 q^{26} -1.00000 q^{27} -1.25167 q^{28} -1.00000 q^{29} +3.42323 q^{30} +1.65549 q^{31} +1.00000 q^{32} +2.48043 q^{33} -5.90366 q^{34} +4.28477 q^{35} +1.00000 q^{36} +11.1995 q^{37} +0.0494308 q^{38} +5.96144 q^{39} -3.42323 q^{40} -6.37459 q^{41} +1.25167 q^{42} +9.44108 q^{43} -2.48043 q^{44} -3.42323 q^{45} -1.00000 q^{46} -0.286731 q^{47} -1.00000 q^{48} -5.43332 q^{49} +6.71852 q^{50} +5.90366 q^{51} -5.96144 q^{52} +1.42752 q^{53} -1.00000 q^{54} +8.49109 q^{55} -1.25167 q^{56} -0.0494308 q^{57} -1.00000 q^{58} +1.26100 q^{59} +3.42323 q^{60} +4.48043 q^{61} +1.65549 q^{62} -1.25167 q^{63} +1.00000 q^{64} +20.4074 q^{65} +2.48043 q^{66} -8.37070 q^{67} -5.90366 q^{68} +1.00000 q^{69} +4.28477 q^{70} -5.41935 q^{71} +1.00000 q^{72} -4.43236 q^{73} +11.1995 q^{74} -6.71852 q^{75} +0.0494308 q^{76} +3.10469 q^{77} +5.96144 q^{78} +12.1466 q^{79} -3.42323 q^{80} +1.00000 q^{81} -6.37459 q^{82} +11.8542 q^{83} +1.25167 q^{84} +20.2096 q^{85} +9.44108 q^{86} +1.00000 q^{87} -2.48043 q^{88} -0.796776 q^{89} -3.42323 q^{90} +7.46177 q^{91} -1.00000 q^{92} -1.65549 q^{93} -0.286731 q^{94} -0.169213 q^{95} -1.00000 q^{96} -0.811353 q^{97} -5.43332 q^{98} -2.48043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

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