LMFDB - Embedded newform 8002.2.a.d.1.53

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.53
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.38511 q^{3} +1.00000 q^{4} -2.30035 q^{5} +1.38511 q^{6} +4.50043 q^{7} +1.00000 q^{8} -1.08147 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.38511 q^{3} +1.00000 q^{4} -2.30035 q^{5} +1.38511 q^{6} +4.50043 q^{7} +1.00000 q^{8} -1.08147 q^{9} -2.30035 q^{10} +1.18933 q^{11} +1.38511 q^{12} -2.03559 q^{13} +4.50043 q^{14} -3.18624 q^{15} +1.00000 q^{16} -8.01806 q^{17} -1.08147 q^{18} +5.58952 q^{19} -2.30035 q^{20} +6.23359 q^{21} +1.18933 q^{22} -2.78093 q^{23} +1.38511 q^{24} +0.291614 q^{25} -2.03559 q^{26} -5.65329 q^{27} +4.50043 q^{28} -10.2852 q^{29} -3.18624 q^{30} -5.31283 q^{31} +1.00000 q^{32} +1.64735 q^{33} -8.01806 q^{34} -10.3526 q^{35} -1.08147 q^{36} -8.81821 q^{37} +5.58952 q^{38} -2.81952 q^{39} -2.30035 q^{40} -10.2154 q^{41} +6.23359 q^{42} +3.51955 q^{43} +1.18933 q^{44} +2.48775 q^{45} -2.78093 q^{46} +2.66940 q^{47} +1.38511 q^{48} +13.2538 q^{49} +0.291614 q^{50} -11.1059 q^{51} -2.03559 q^{52} -10.1017 q^{53} -5.65329 q^{54} -2.73587 q^{55} +4.50043 q^{56} +7.74211 q^{57} -10.2852 q^{58} +10.9174 q^{59} -3.18624 q^{60} -7.81373 q^{61} -5.31283 q^{62} -4.86706 q^{63} +1.00000 q^{64} +4.68257 q^{65} +1.64735 q^{66} -12.9014 q^{67} -8.01806 q^{68} -3.85189 q^{69} -10.3526 q^{70} +8.82436 q^{71} -1.08147 q^{72} -6.50614 q^{73} -8.81821 q^{74} +0.403918 q^{75} +5.58952 q^{76} +5.35247 q^{77} -2.81952 q^{78} +9.31181 q^{79} -2.30035 q^{80} -4.58603 q^{81} -10.2154 q^{82} -3.80888 q^{83} +6.23359 q^{84} +18.4444 q^{85} +3.51955 q^{86} -14.2462 q^{87} +1.18933 q^{88} +3.94750 q^{89} +2.48775 q^{90} -9.16102 q^{91} -2.78093 q^{92} -7.35886 q^{93} +2.66940 q^{94} -12.8579 q^{95} +1.38511 q^{96} -2.83972 q^{97} +13.2538 q^{98} -1.28622 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	1.38511	0.799694	0.399847	−	0.916582i	$-0.369063\pi$
$3$	1.38511	0.799694	0.399847	+	0.916582i	$0.369063\pi$
$4$	1.00000	0.500000
$5$	−2.30035	−1.02875	−0.514374	−	0.857566i	$-0.671976\pi$
$5$	−2.30035	−1.02875	−0.514374	+	0.857566i	$0.671976\pi$
$6$	1.38511	0.565469
$7$	4.50043	1.70100	0.850501	−	0.525974i	$-0.176299\pi$
$7$	4.50043	1.70100	0.850501	+	0.525974i	$0.176299\pi$
$8$	1.00000	0.353553
$9$	−1.08147	−0.360489
$10$	−2.30035	−0.727435
$11$	1.18933	0.358595	0.179298	−	0.983795i	$-0.442617\pi$
$11$	1.18933	0.358595	0.179298	+	0.983795i	$0.442617\pi$
$12$	1.38511	0.399847
$13$	−2.03559	−0.564571	−0.282286	−	0.959330i	$-0.591093\pi$
$13$	−2.03559	−0.564571	−0.282286	+	0.959330i	$0.591093\pi$
$14$	4.50043	1.20279
$15$	−3.18624	−0.822684
$16$	1.00000	0.250000
$17$	−8.01806	−1.94467	−0.972333	−	0.233599i	$-0.924950\pi$
$17$	−8.01806	−1.94467	−0.972333	+	0.233599i	$0.924950\pi$
$18$	−1.08147	−0.254904
$19$	5.58952	1.28232	0.641162	−	0.767406i	$-0.278453\pi$
$19$	5.58952	1.28232	0.641162	+	0.767406i	$0.278453\pi$
$20$	−2.30035	−0.514374
$21$	6.23359	1.36028
$22$	1.18933	0.253565
$23$	−2.78093	−0.579863	−0.289932	−	0.957047i	$-0.593633\pi$
$23$	−2.78093	−0.579863	−0.289932	+	0.957047i	$0.593633\pi$
$24$	1.38511	0.282735
$25$	0.291614	0.0583228
$26$	−2.03559	−0.399212
$27$	−5.65329	−1.08798
$28$	4.50043	0.850501
$29$	−10.2852	−1.90992	−0.954958	−	0.296742i	$-0.904100\pi$
$29$	−10.2852	−1.90992	−0.954958	+	0.296742i	$0.904100\pi$
$30$	−3.18624	−0.581726
$31$	−5.31283	−0.954213	−0.477106	−	0.878846i	$-0.658314\pi$
$31$	−5.31283	−0.954213	−0.477106	+	0.878846i	$0.658314\pi$
$32$	1.00000	0.176777
$33$	1.64735	0.286767
$34$	−8.01806	−1.37509
$35$	−10.3526	−1.74990
$36$	−1.08147	−0.180244
$37$	−8.81821	−1.44971	−0.724853	−	0.688904i	$-0.758092\pi$
$37$	−8.81821	−1.44971	−0.724853	+	0.688904i	$0.758092\pi$
$38$	5.58952	0.906740
$39$	−2.81952	−0.451484
$40$	−2.30035	−0.363717
$41$	−10.2154	−1.59537	−0.797685	−	0.603074i	$-0.793942\pi$
$41$	−10.2154	−1.59537	−0.797685	+	0.603074i	$0.793942\pi$
$42$	6.23359	0.961864
$43$	3.51955	0.536726	0.268363	−	0.963318i	$-0.413517\pi$
$43$	3.51955	0.536726	0.268363	+	0.963318i	$0.413517\pi$
$44$	1.18933	0.179298
$45$	2.48775	0.370852
$46$	−2.78093	−0.410025
$47$	2.66940	0.389372	0.194686	−	0.980866i	$-0.437631\pi$
$47$	2.66940	0.389372	0.194686	+	0.980866i	$0.437631\pi$
$48$	1.38511	0.199924
$49$	13.2538	1.89341
$50$	0.291614	0.0412405
$51$	−11.1059	−1.55514
$52$	−2.03559	−0.282286
$53$	−10.1017	−1.38757	−0.693785	−	0.720182i	$-0.744058\pi$
$53$	−10.1017	−1.38757	−0.693785	+	0.720182i	$0.744058\pi$
$54$	−5.65329	−0.769315
$55$	−2.73587	−0.368904
$56$	4.50043	0.601395
$57$	7.74211	1.02547
$58$	−10.2852	−1.35051
$59$	10.9174	1.42132	0.710661	−	0.703535i	$-0.248396\pi$
$59$	10.9174	1.42132	0.710661	+	0.703535i	$0.248396\pi$
$60$	−3.18624	−0.411342
$61$	−7.81373	−1.00045	−0.500223	−	0.865897i	$-0.666749\pi$
$61$	−7.81373	−1.00045	−0.500223	+	0.865897i	$0.666749\pi$
$62$	−5.31283	−0.674730
$63$	−4.86706	−0.613192
$64$	1.00000	0.125000
$65$	4.68257	0.580802
$66$	1.64735	0.202775
$67$	−12.9014	−1.57615	−0.788077	−	0.615576i	$-0.788923\pi$
$67$	−12.9014	−1.57615	−0.788077	+	0.615576i	$0.788923\pi$
$68$	−8.01806	−0.972333
$69$	−3.85189	−0.463713
$70$	−10.3526	−1.23737
$71$	8.82436	1.04726	0.523629	−	0.851946i	$-0.324578\pi$
$71$	8.82436	1.04726	0.523629	+	0.851946i	$0.324578\pi$
$72$	−1.08147	−0.127452
$73$	−6.50614	−0.761486	−0.380743	−	0.924681i	$-0.624332\pi$
$73$	−6.50614	−0.761486	−0.380743	+	0.924681i	$0.624332\pi$
$74$	−8.81821	−1.02510
$75$	0.403918	0.0466404
$76$	5.58952	0.641162
$77$	5.35247	0.609971
$78$	−2.81952	−0.319248
$79$	9.31181	1.04766	0.523830	−	0.851823i	$-0.324503\pi$
$79$	9.31181	1.04766	0.523830	+	0.851823i	$0.324503\pi$
$80$	−2.30035	−0.257187
$81$	−4.58603	−0.509559
$82$	−10.2154	−1.12810
$83$	−3.80888	−0.418079	−0.209040	−	0.977907i	$-0.567034\pi$
$83$	−3.80888	−0.418079	−0.209040	+	0.977907i	$0.567034\pi$
$84$	6.23359	0.680141
$85$	18.4444	2.00057
$86$	3.51955	0.379523
$87$	−14.2462	−1.52735
$88$	1.18933	0.126783
$89$	3.94750	0.418434	0.209217	−	0.977869i	$-0.432908\pi$
$89$	3.94750	0.418434	0.209217	+	0.977869i	$0.432908\pi$
$90$	2.48775	0.262232
$91$	−9.16102	−0.960336
$92$	−2.78093	−0.289932
$93$	−7.35886	−0.763078
$94$	2.66940	0.275327
$95$	−12.8579	−1.31919
$96$	1.38511	0.141367
$97$	−2.83972	−0.288330	−0.144165	−	0.989554i	$-0.546050\pi$
$97$	−2.83972	−0.288330	−0.144165	+	0.989554i	$0.546050\pi$
$98$	13.2538	1.33884
$99$	−1.28622	−0.129270
$100$	0.291614	0.0291614
$101$	15.5427	1.54655	0.773277	−	0.634068i	$-0.218616\pi$
$101$	15.5427	1.54655	0.773277	+	0.634068i	$0.218616\pi$
$102$	−11.1059	−1.09965
$103$	20.2265	1.99297	0.996487	−	0.0837522i	$-0.0266904\pi$
$103$	20.2265	1.99297	0.996487	+	0.0837522i	$0.0266904\pi$
$104$	−2.03559	−0.199606
$105$	−14.3395	−1.39939
$106$	−10.1017	−0.981161
$107$	15.6946	1.51726	0.758629	−	0.651522i	$-0.225869\pi$
$107$	15.6946	1.51726	0.758629	+	0.651522i	$0.225869\pi$
$108$	−5.65329	−0.543988
$109$	15.9385	1.52663	0.763317	−	0.646024i	$-0.223569\pi$
$109$	15.9385	1.52663	0.763317	+	0.646024i	$0.223569\pi$
$110$	−2.73587	−0.260855
$111$	−12.2142	−1.15932
$112$	4.50043	0.425250
$113$	8.24144	0.775290	0.387645	−	0.921809i	$-0.373289\pi$
$113$	8.24144	0.775290	0.387645	+	0.921809i	$0.373289\pi$
$114$	7.74211	0.725115
$115$	6.39710	0.596533
$116$	−10.2852	−0.954958
$117$	2.20142	0.203522
$118$	10.9174	1.00503
$119$	−36.0847	−3.30788
$120$	−3.18624	−0.290863
$121$	−9.58550	−0.871409
$122$	−7.81373	−0.707422
$123$	−14.1494	−1.27581
$124$	−5.31283	−0.477106
$125$	10.8309	0.968749
$126$	−4.86706	−0.433592
$127$	−14.7325	−1.30730	−0.653651	−	0.756797i	$-0.726763\pi$
$127$	−14.7325	−1.30730	−0.653651	+	0.756797i	$0.726763\pi$
$128$	1.00000	0.0883883
$129$	4.87497	0.429217
$130$	4.68257	0.410689
$131$	2.22143	0.194088	0.0970438	−	0.995280i	$-0.469061\pi$
$131$	2.22143	0.194088	0.0970438	+	0.995280i	$0.469061\pi$
$132$	1.64735	0.143383
$133$	25.1552	2.18123
$134$	−12.9014	−1.11451
$135$	13.0045	1.11925
$136$	−8.01806	−0.687543
$137$	−8.41558	−0.718991	−0.359496	−	0.933147i	$-0.617051\pi$
$137$	−8.41558	−0.718991	−0.359496	+	0.933147i	$0.617051\pi$
$138$	−3.85189	−0.327895
$139$	−13.9707	−1.18498	−0.592490	−	0.805578i	$-0.701855\pi$
$139$	−13.9707	−1.18498	−0.592490	+	0.805578i	$0.701855\pi$
$140$	−10.3526	−0.874951
$141$	3.69741	0.311378
$142$	8.82436	0.740523
$143$	−2.42098	−0.202453
$144$	−1.08147	−0.0901222
$145$	23.6596	1.96482
$146$	−6.50614	−0.538452
$147$	18.3581	1.51415
$148$	−8.81821	−0.724853
$149$	−11.8394	−0.969920	−0.484960	−	0.874536i	$-0.661166\pi$
$149$	−11.8394	−0.969920	−0.484960	+	0.874536i	$0.661166\pi$
$150$	0.403918	0.0329798
$151$	11.1715	0.909125	0.454563	−	0.890715i	$-0.349796\pi$
$151$	11.1715	0.909125	0.454563	+	0.890715i	$0.349796\pi$
$152$	5.58952	0.453370
$153$	8.67127	0.701030
$154$	5.35247	0.431315
$155$	12.2214	0.981644
$156$	−2.81952	−0.225742
$157$	0.685186	0.0546838	0.0273419	−	0.999626i	$-0.491296\pi$
$157$	0.685186	0.0546838	0.0273419	+	0.999626i	$0.491296\pi$
$158$	9.31181	0.740808
$159$	−13.9919	−1.10963
$160$	−2.30035	−0.181859
$161$	−12.5154	−0.986348
$162$	−4.58603	−0.360313
$163$	5.56358	0.435773	0.217887	−	0.975974i	$-0.430084\pi$
$163$	5.56358	0.435773	0.217887	+	0.975974i	$0.430084\pi$
$164$	−10.2154	−0.797685
$165$	−3.78948	−0.295011
$166$	−3.80888	−0.295627
$167$	−4.44356	−0.343853	−0.171926	−	0.985110i	$-0.554999\pi$
$167$	−4.44356	−0.343853	−0.171926	+	0.985110i	$0.554999\pi$
$168$	6.23359	0.480932
$169$	−8.85637	−0.681259
$170$	18.4444	1.41462
$171$	−6.04488	−0.462263
$172$	3.51955	0.268363
$173$	0.210202	0.0159814	0.00799070	−	0.999968i	$-0.497456\pi$
$173$	0.210202	0.0159814	0.00799070	+	0.999968i	$0.497456\pi$
$174$	−14.2462	−1.08000
$175$	1.31239	0.0992072
$176$	1.18933	0.0896488
$177$	15.1218	1.13662
$178$	3.94750	0.295878
$179$	0.964079	0.0720587	0.0360293	−	0.999351i	$-0.488529\pi$
$179$	0.964079	0.0720587	0.0360293	+	0.999351i	$0.488529\pi$
$180$	2.48775	0.185426
$181$	10.2162	0.759366	0.379683	−	0.925117i	$-0.376033\pi$
$181$	10.2162	0.759366	0.379683	+	0.925117i	$0.376033\pi$
$182$	−9.16102	−0.679060
$183$	−10.8229	−0.800051
$184$	−2.78093	−0.205013
$185$	20.2850	1.49138
$186$	−7.35886	−0.539578
$187$	−9.53609	−0.697348
$188$	2.66940	0.194686
$189$	−25.4422	−1.85065
$190$	−12.8579	−0.932807
$191$	13.4591	0.973870	0.486935	−	0.873438i	$-0.338115\pi$
$191$	13.4591	0.973870	0.486935	+	0.873438i	$0.338115\pi$
$192$	1.38511	0.0999618
$193$	−13.5891	−0.978168	−0.489084	−	0.872237i	$-0.662669\pi$
$193$	−13.5891	−0.978168	−0.489084	+	0.872237i	$0.662669\pi$
$194$	−2.83972	−0.203880
$195$	6.48588	0.464464
$196$	13.2538	0.946703
$197$	−13.1219	−0.934899	−0.467450	−	0.884020i	$-0.654827\pi$
$197$	−13.1219	−0.934899	−0.467450	+	0.884020i	$0.654827\pi$
$198$	−1.28622	−0.0914074
$199$	−2.84692	−0.201813	−0.100906	−	0.994896i	$-0.532174\pi$
$199$	−2.84692	−0.201813	−0.100906	+	0.994896i	$0.532174\pi$
$200$	0.291614	0.0206202
$201$	−17.8699	−1.26044
$202$	15.5427	1.09358
$203$	−46.2878	−3.24877
$204$	−11.1059	−0.777569
$205$	23.4989	1.64123
$206$	20.2265	1.40924
$207$	3.00748	0.209034
$208$	−2.03559	−0.141143
$209$	6.64776	0.459835
$210$	−14.3395	−0.989516
$211$	22.4665	1.54666	0.773330	−	0.634004i	$-0.218590\pi$
$211$	22.4665	1.54666	0.773330	+	0.634004i	$0.218590\pi$
$212$	−10.1017	−0.693785
$213$	12.2227	0.837487
$214$	15.6946	1.07286
$215$	−8.09619	−0.552156
$216$	−5.65329	−0.384657
$217$	−23.9100	−1.62312
$218$	15.9385	1.07949
$219$	−9.01173	−0.608956
$220$	−2.73587	−0.184452
$221$	16.3215	1.09790
$222$	−12.2142	−0.819764
$223$	14.9837	1.00338	0.501691	−	0.865047i	$-0.332711\pi$
$223$	14.9837	1.00338	0.501691	+	0.865047i	$0.332711\pi$
$224$	4.50043	0.300697
$225$	−0.315371	−0.0210247
$226$	8.24144	0.548213
$227$	−22.5345	−1.49567	−0.747833	−	0.663887i	$-0.768906\pi$
$227$	−22.5345	−1.49567	−0.747833	+	0.663887i	$0.768906\pi$
$228$	7.74211	0.512734
$229$	−2.65074	−0.175166	−0.0875831	−	0.996157i	$-0.527914\pi$
$229$	−2.65074	−0.175166	−0.0875831	+	0.996157i	$0.527914\pi$
$230$	6.39710	0.421813
$231$	7.41377	0.487791
$232$	−10.2852	−0.675257
$233$	1.59672	0.104605	0.0523025	−	0.998631i	$-0.483344\pi$
$233$	1.59672	0.104605	0.0523025	+	0.998631i	$0.483344\pi$
$234$	2.20142	0.143911
$235$	−6.14055	−0.400565
$236$	10.9174	0.710661
$237$	12.8979	0.837808
$238$	−36.0847	−2.33902
$239$	−24.0872	−1.55807	−0.779035	−	0.626981i	$-0.784290\pi$
$239$	−24.0872	−1.55807	−0.779035	+	0.626981i	$0.784290\pi$
$240$	−3.18624	−0.205671
$241$	0.445480	0.0286959	0.0143479	−	0.999897i	$-0.495433\pi$
$241$	0.445480	0.0286959	0.0143479	+	0.999897i	$0.495433\pi$
$242$	−9.58550	−0.616180
$243$	10.6077	0.680484
$244$	−7.81373	−0.500223
$245$	−30.4885	−1.94784
$246$	−14.1494	−0.902133
$247$	−11.3780	−0.723963
$248$	−5.31283	−0.337365
$249$	−5.27573	−0.334336
$250$	10.8309	0.685009
$251$	−21.0089	−1.32607	−0.663036	−	0.748587i	$-0.730732\pi$
$251$	−21.0089	−1.32607	−0.663036	+	0.748587i	$0.730732\pi$
$252$	−4.86706	−0.306596
$253$	−3.30743	−0.207936
$254$	−14.7325	−0.924401
$255$	25.5475	1.59985
$256$	1.00000	0.0625000
$257$	−18.9953	−1.18489	−0.592446	−	0.805610i	$-0.701838\pi$
$257$	−18.9953	−1.18489	−0.592446	+	0.805610i	$0.701838\pi$
$258$	4.87497	0.303502
$259$	−39.6857	−2.46595
$260$	4.68257	0.290401
$261$	11.1231	0.688503
$262$	2.22143	0.137241
$263$	−15.0682	−0.929142	−0.464571	−	0.885536i	$-0.653792\pi$
$263$	−15.0682	−0.929142	−0.464571	+	0.885536i	$0.653792\pi$
$264$	1.64735	0.101387
$265$	23.2374	1.42746
$266$	25.1552	1.54237
$267$	5.46773	0.334620
$268$	−12.9014	−0.788077
$269$	−10.2893	−0.627348	−0.313674	−	0.949531i	$-0.601560\pi$
$269$	−10.2893	−0.627348	−0.313674	+	0.949531i	$0.601560\pi$
$270$	13.0045	0.791431
$271$	10.5884	0.643201	0.321601	−	0.946875i	$-0.395779\pi$
$271$	10.5884	0.643201	0.321601	+	0.946875i	$0.395779\pi$
$272$	−8.01806	−0.486167
$273$	−12.6890	−0.767976
$274$	−8.41558	−0.508403
$275$	0.346824	0.0209143
$276$	−3.85189	−0.231857
$277$	8.70305	0.522916	0.261458	−	0.965215i	$-0.415797\pi$
$277$	8.70305	0.522916	0.261458	+	0.965215i	$0.415797\pi$
$278$	−13.9707	−0.837907
$279$	5.74565	0.343983
$280$	−10.3526	−0.618684
$281$	−14.8072	−0.883323	−0.441661	−	0.897182i	$-0.645611\pi$
$281$	−14.8072	−0.883323	−0.441661	+	0.897182i	$0.645611\pi$
$282$	3.69741	0.220178
$283$	13.4922	0.802030	0.401015	−	0.916071i	$-0.368658\pi$
$283$	13.4922	0.802030	0.401015	+	0.916071i	$0.368658\pi$
$284$	8.82436	0.523629
$285$	−17.8096	−1.05495
$286$	−2.42098	−0.143156
$287$	−45.9734	−2.71373
$288$	−1.08147	−0.0637260
$289$	47.2894	2.78173
$290$	23.6596	1.38934
$291$	−3.93333	−0.230576
$292$	−6.50614	−0.380743
$293$	−27.5762	−1.61102	−0.805511	−	0.592581i	$-0.798109\pi$
$293$	−27.5762	−1.61102	−0.805511	+	0.592581i	$0.798109\pi$
$294$	18.3581	1.07066
$295$	−25.1138	−1.46218
$296$	−8.81821	−0.512548
$297$	−6.72360	−0.390143
$298$	−11.8394	−0.685837
$299$	5.66082	0.327374
$300$	0.403918	0.0233202
$301$	15.8395	0.912972
$302$	11.1715	0.642849
$303$	21.5283	1.23677
$304$	5.58952	0.320581
$305$	17.9743	1.02921
$306$	8.67127	0.495703
$307$	18.6949	1.06697	0.533487	−	0.845808i	$-0.320881\pi$
$307$	18.6949	1.06697	0.533487	+	0.845808i	$0.320881\pi$
$308$	5.35247	0.304986
$309$	28.0159	1.59377
$310$	12.2214	0.694127
$311$	−1.93125	−0.109511	−0.0547556	−	0.998500i	$-0.517438\pi$
$311$	−1.93125	−0.109511	−0.0547556	+	0.998500i	$0.517438\pi$
$312$	−2.81952	−0.159624
$313$	−20.0601	−1.13387	−0.566933	−	0.823764i	$-0.691870\pi$
$313$	−20.0601	−1.13387	−0.566933	+	0.823764i	$0.691870\pi$
$314$	0.685186	0.0386673
$315$	11.1959	0.630820
$316$	9.31181	0.523830
$317$	14.9580	0.840124	0.420062	−	0.907496i	$-0.362008\pi$
$317$	14.9580	0.840124	0.420062	+	0.907496i	$0.362008\pi$
$318$	−13.9919	−0.784629
$319$	−12.2325	−0.684887
$320$	−2.30035	−0.128594
$321$	21.7388	1.21334
$322$	−12.5154	−0.697453
$323$	−44.8171	−2.49369
$324$	−4.58603	−0.254780
$325$	−0.593607	−0.0329274
$326$	5.56358	0.308138
$327$	22.0766	1.22084
$328$	−10.2154	−0.564048
$329$	12.0134	0.662322
$330$	−3.78948	−0.208604
$331$	27.8593	1.53129	0.765644	−	0.643265i	$-0.222421\pi$
$331$	27.8593	1.53129	0.765644	+	0.643265i	$0.222421\pi$
$332$	−3.80888	−0.209040
$333$	9.53660	0.522603
$334$	−4.44356	−0.243141
$335$	29.6777	1.62147
$336$	6.23359	0.340070
$337$	2.55237	0.139037	0.0695183	−	0.997581i	$-0.477854\pi$
$337$	2.55237	0.139037	0.0695183	+	0.997581i	$0.477854\pi$
$338$	−8.85637	−0.481723
$339$	11.4153	0.619995
$340$	18.4444	1.00029
$341$	−6.31869	−0.342176
$342$	−6.04488	−0.326870
$343$	28.1450	1.51969
$344$	3.51955	0.189761
$345$	8.86070	0.477044
$346$	0.210202	0.0113006
$347$	−15.0204	−0.806339	−0.403169	−	0.915125i	$-0.632091\pi$
$347$	−15.0204	−0.806339	−0.403169	+	0.915125i	$0.632091\pi$
$348$	−14.2462	−0.763674
$349$	22.8845	1.22498	0.612490	−	0.790478i	$-0.290168\pi$
$349$	22.8845	1.22498	0.612490	+	0.790478i	$0.290168\pi$
$350$	1.31239	0.0701501
$351$	11.5078	0.614239
$352$	1.18933	0.0633913
$353$	17.3204	0.921870	0.460935	−	0.887434i	$-0.347514\pi$
$353$	17.3204	0.921870	0.460935	+	0.887434i	$0.347514\pi$
$354$	15.1218	0.803714
$355$	−20.2991	−1.07737
$356$	3.94750	0.209217
$357$	−49.9813	−2.64529
$358$	0.964079	0.0509532
$359$	29.4820	1.55600	0.778001	−	0.628262i	$-0.216234\pi$
$359$	29.4820	1.55600	0.778001	+	0.628262i	$0.216234\pi$
$360$	2.48775	0.131116
$361$	12.2427	0.644354
$362$	10.2162	0.536953
$363$	−13.2770	−0.696861
$364$	−9.16102	−0.480168
$365$	14.9664	0.783378
$366$	−10.8229	−0.565721
$367$	11.4451	0.597430	0.298715	−	0.954342i	$-0.403442\pi$
$367$	11.4451	0.597430	0.298715	+	0.954342i	$0.403442\pi$
$368$	−2.78093	−0.144966
$369$	11.0476	0.575113
$370$	20.2850	1.05457
$371$	−45.4618	−2.36026
$372$	−7.35886	−0.381539
$373$	−12.5117	−0.647831	−0.323915	−	0.946086i	$-0.604999\pi$
$373$	−12.5117	−0.647831	−0.323915	+	0.946086i	$0.604999\pi$
$374$	−9.53609	−0.493100
$375$	15.0021	0.774703
$376$	2.66940	0.137664
$377$	20.9365	1.07828
$378$	−25.4422	−1.30861
$379$	32.4393	1.66630	0.833148	−	0.553050i	$-0.186536\pi$
$379$	32.4393	1.66630	0.833148	+	0.553050i	$0.186536\pi$
$380$	−12.8579	−0.659594
$381$	−20.4062	−1.04544
$382$	13.4591	0.688630
$383$	−5.93388	−0.303207	−0.151603	−	0.988441i	$-0.548444\pi$
$383$	−5.93388	−0.303207	−0.151603	+	0.988441i	$0.548444\pi$
$384$	1.38511	0.0706837
$385$	−12.3126	−0.627507
$386$	−13.5891	−0.691669
$387$	−3.80627	−0.193484
$388$	−2.83972	−0.144165
$389$	−25.8128	−1.30876	−0.654380	−	0.756166i	$-0.727070\pi$
$389$	−25.8128	−1.30876	−0.654380	+	0.756166i	$0.727070\pi$
$390$	6.48588	0.328425
$391$	22.2976	1.12764
$392$	13.2538	0.669420
$393$	3.07693	0.155211
$394$	−13.1219	−0.661074
$395$	−21.4204	−1.07778
$396$	−1.28622	−0.0646348
$397$	−18.0917	−0.907998	−0.453999	−	0.891002i	$-0.650003\pi$
$397$	−18.0917	−0.907998	−0.453999	+	0.891002i	$0.650003\pi$
$398$	−2.84692	−0.142703
$399$	34.8428	1.74432
$400$	0.291614	0.0145807
$401$	16.8070	0.839303	0.419652	−	0.907685i	$-0.362152\pi$
$401$	16.8070	0.839303	0.419652	+	0.907685i	$0.362152\pi$
$402$	−17.8699	−0.891267
$403$	10.8147	0.538721
$404$	15.5427	0.773277
$405$	10.5495	0.524208
$406$	−46.2878	−2.29723
$407$	−10.4877	−0.519858
$408$	−11.1059	−0.549825
$409$	13.2533	0.655335	0.327668	−	0.944793i	$-0.393737\pi$
$409$	13.2533	0.655335	0.327668	+	0.944793i	$0.393737\pi$
$410$	23.4989	1.16053
$411$	−11.6565	−0.574973
$412$	20.2265	0.996487
$413$	49.1329	2.41767
$414$	3.00748	0.147809
$415$	8.76176	0.430098
$416$	−2.03559	−0.0998030
$417$	−19.3510	−0.947622
$418$	6.64776	0.325153
$419$	−2.41199	−0.117834	−0.0589168	−	0.998263i	$-0.518765\pi$
$419$	−2.41199	−0.117834	−0.0589168	+	0.998263i	$0.518765\pi$
$420$	−14.3395	−0.699694
$421$	−19.8574	−0.967789	−0.483894	−	0.875126i	$-0.660778\pi$
$421$	−19.8574	−0.967789	−0.483894	+	0.875126i	$0.660778\pi$
$422$	22.4665	1.09365
$423$	−2.88686	−0.140364
$424$	−10.1017	−0.490580
$425$	−2.33818	−0.113418
$426$	12.2227	0.592192
$427$	−35.1651	−1.70176
$428$	15.6946	0.758629
$429$	−3.35333	−0.161900
$430$	−8.09619	−0.390433
$431$	−22.9885	−1.10732	−0.553659	−	0.832744i	$-0.686769\pi$
$431$	−22.9885	−1.10732	−0.553659	+	0.832744i	$0.686769\pi$
$432$	−5.65329	−0.271994
$433$	−12.4063	−0.596207	−0.298103	−	0.954534i	$-0.596354\pi$
$433$	−12.4063	−0.596207	−0.298103	+	0.954534i	$0.596354\pi$
$434$	−23.9100	−1.14772
$435$	32.7712	1.57126
$436$	15.9385	0.763317
$437$	−15.5440	−0.743572
$438$	−9.01173	−0.430597
$439$	−10.8042	−0.515659	−0.257829	−	0.966190i	$-0.583007\pi$
$439$	−10.8042	−0.515659	−0.257829	+	0.966190i	$0.583007\pi$
$440$	−2.73587	−0.130427
$441$	−14.3336	−0.682552
$442$	16.3215	0.776334
$443$	2.91122	0.138316	0.0691580	−	0.997606i	$-0.477969\pi$
$443$	2.91122	0.138316	0.0691580	+	0.997606i	$0.477969\pi$
$444$	−12.2142	−0.579661
$445$	−9.08064	−0.430464
$446$	14.9837	0.709499
$447$	−16.3989	−0.775640
$448$	4.50043	0.212625
$449$	23.3874	1.10372	0.551859	−	0.833937i	$-0.313919\pi$
$449$	23.3874	1.10372	0.551859	+	0.833937i	$0.313919\pi$
$450$	−0.315371	−0.0148667
$451$	−12.1494	−0.572092
$452$	8.24144	0.387645
$453$	15.4738	0.727022
$454$	−22.5345	−1.05760
$455$	21.0736	0.987944
$456$	7.74211	0.362557
$457$	30.7341	1.43768	0.718841	−	0.695174i	$-0.244673\pi$
$457$	30.7341	1.43768	0.718841	+	0.695174i	$0.244673\pi$
$458$	−2.65074	−0.123861
$459$	45.3284	2.11575
$460$	6.39710	0.298267
$461$	7.71167	0.359168	0.179584	−	0.983743i	$-0.442525\pi$
$461$	7.71167	0.359168	0.179584	+	0.983743i	$0.442525\pi$
$462$	7.41377	0.344920
$463$	−36.4773	−1.69524	−0.847621	−	0.530602i	$-0.821966\pi$
$463$	−36.4773	−1.69524	−0.847621	+	0.530602i	$0.821966\pi$
$464$	−10.2852	−0.477479
$465$	16.9280	0.785016
$466$	1.59672	0.0739668
$467$	−32.2385	−1.49182	−0.745909	−	0.666048i	$-0.767985\pi$
$467$	−32.2385	−1.49182	−0.745909	+	0.666048i	$0.767985\pi$
$468$	2.20142	0.101761
$469$	−58.0617	−2.68104
$470$	−6.14055	−0.283242
$471$	0.949059	0.0437303
$472$	10.9174	0.502513
$473$	4.18589	0.192467
$474$	12.8979	0.592420
$475$	1.62998	0.0747887
$476$	−36.0847	−1.65394
$477$	10.9246	0.500204
$478$	−24.0872	−1.10172
$479$	13.2524	0.605519	0.302760	−	0.953067i	$-0.402092\pi$
$479$	13.2524	0.605519	0.302760	+	0.953067i	$0.402092\pi$
$480$	−3.18624	−0.145431
$481$	17.9503	0.818462
$482$	0.445480	0.0202911
$483$	−17.3352	−0.788777
$484$	−9.58550	−0.435705
$485$	6.53236	0.296619
$486$	10.6077	0.481175
$487$	13.3585	0.605331	0.302666	−	0.953097i	$-0.402123\pi$
$487$	13.3585	0.605331	0.302666	+	0.953097i	$0.402123\pi$
$488$	−7.81373	−0.353711
$489$	7.70617	0.348485
$490$	−30.4885	−1.37733
$491$	−33.1217	−1.49476	−0.747381	−	0.664395i	$-0.768689\pi$
$491$	−33.1217	−1.49476	−0.747381	+	0.664395i	$0.768689\pi$
$492$	−14.1494	−0.637904
$493$	82.4675	3.71415
$494$	−11.3780	−0.511919
$495$	2.95875	0.132986
$496$	−5.31283	−0.238553
$497$	39.7134	1.78139
$498$	−5.27573	−0.236411
$499$	−15.1028	−0.676094	−0.338047	−	0.941129i	$-0.609766\pi$
$499$	−15.1028	−0.676094	−0.338047	+	0.941129i	$0.609766\pi$
$500$	10.8309	0.484374
$501$	−6.15482	−0.274977
$502$	−21.0089	−0.937675
$503$	−43.7923	−1.95260	−0.976301	−	0.216418i	$-0.930563\pi$
$503$	−43.7923	−1.95260	−0.976301	+	0.216418i	$0.930563\pi$
$504$	−4.86706	−0.216796
$505$	−35.7536	−1.59101
$506$	−3.30743	−0.147033
$507$	−12.2671	−0.544799
$508$	−14.7325	−0.653651
$509$	8.90799	0.394840	0.197420	−	0.980319i	$-0.436744\pi$
$509$	8.90799	0.394840	0.197420	+	0.980319i	$0.436744\pi$
$510$	25.5475	1.13126
$511$	−29.2804	−1.29529
$512$	1.00000	0.0441942
$513$	−31.5992	−1.39514
$514$	−18.9953	−0.837845
$515$	−46.5280	−2.05027
$516$	4.87497	0.214608
$517$	3.17478	0.139627
$518$	−39.6857	−1.74369
$519$	0.291154	0.0127802
$520$	4.68257	0.205344
$521$	23.3944	1.02493	0.512464	−	0.858709i	$-0.328733\pi$
$521$	23.3944	1.02493	0.512464	+	0.858709i	$0.328733\pi$
$522$	11.1231	0.486845
$523$	−30.0183	−1.31261	−0.656305	−	0.754496i	$-0.727881\pi$
$523$	−30.0183	−1.31261	−0.656305	+	0.754496i	$0.727881\pi$
$524$	2.22143	0.0970438
$525$	1.81780	0.0793355
$526$	−15.0682	−0.657003
$527$	42.5986	1.85562
$528$	1.64735	0.0716917
$529$	−15.2665	−0.663759
$530$	23.2374	1.00937
$531$	−11.8068	−0.512371
$532$	25.1552	1.09062
$533$	20.7943	0.900700
$534$	5.46773	0.236612
$535$	−36.1032	−1.56088
$536$	−12.9014	−0.557255
$537$	1.33536	0.0576249
$538$	−10.2893	−0.443602
$539$	15.7631	0.678967
$540$	13.0045	0.559626
$541$	1.14043	0.0490307	0.0245154	−	0.999699i	$-0.492196\pi$
$541$	1.14043	0.0490307	0.0245154	+	0.999699i	$0.492196\pi$
$542$	10.5884	0.454812
$543$	14.1506	0.607261
$544$	−8.01806	−0.343772
$545$	−36.6642	−1.57052
$546$	−12.6890	−0.543041
$547$	−34.3050	−1.46678	−0.733389	−	0.679810i	$-0.762062\pi$
$547$	−34.3050	−1.46678	−0.733389	+	0.679810i	$0.762062\pi$
$548$	−8.41558	−0.359496
$549$	8.45029	0.360649
$550$	0.346824	0.0147886
$551$	−57.4894	−2.44913
$552$	−3.85189	−0.163947
$553$	41.9071	1.78207
$554$	8.70305	0.369757
$555$	28.0970	1.19265
$556$	−13.9707	−0.592490
$557$	28.6679	1.21470	0.607350	−	0.794435i	$-0.292233\pi$
$557$	28.6679	1.21470	0.607350	+	0.794435i	$0.292233\pi$
$558$	5.74565	0.243233
$559$	−7.16436	−0.303020
$560$	−10.3526	−0.437476
$561$	−13.2085	−0.557665
$562$	−14.8072	−0.624604
$563$	−43.6324	−1.83889	−0.919443	−	0.393222i	$-0.871360\pi$
$563$	−43.6324	−1.83889	−0.919443	+	0.393222i	$0.871360\pi$
$564$	3.69741	0.155689
$565$	−18.9582	−0.797578
$566$	13.4922	0.567121
$567$	−20.6391	−0.866761
$568$	8.82436	0.370262
$569$	30.1041	1.26203	0.631014	−	0.775772i	$-0.282639\pi$
$569$	30.1041	1.26203	0.631014	+	0.775772i	$0.282639\pi$
$570$	−17.8096	−0.745961
$571$	38.7287	1.62075	0.810373	−	0.585914i	$-0.199264\pi$
$571$	38.7287	1.62075	0.810373	+	0.585914i	$0.199264\pi$
$572$	−2.42098	−0.101226
$573$	18.6424	0.778798
$574$	−45.9734	−1.91889
$575$	−0.810957	−0.0338193
$576$	−1.08147	−0.0450611
$577$	21.6922	0.903058	0.451529	−	0.892256i	$-0.350879\pi$
$577$	21.6922	0.903058	0.451529	+	0.892256i	$0.350879\pi$
$578$	47.2894	1.96698
$579$	−18.8225	−0.782236
$580$	23.6596	0.982411
$581$	−17.1416	−0.711153
$582$	−3.93333	−0.163042
$583$	−12.0142	−0.497576
$584$	−6.50614	−0.269226
$585$	−5.06404	−0.209372
$586$	−27.5762	−1.13916
$587$	23.3507	0.963786	0.481893	−	0.876230i	$-0.339949\pi$
$587$	23.3507	0.963786	0.481893	+	0.876230i	$0.339949\pi$
$588$	18.3581	0.757073
$589$	−29.6962	−1.22361
$590$	−25.1138	−1.03392
$591$	−18.1753	−0.747634
$592$	−8.81821	−0.362426
$593$	−42.5219	−1.74616	−0.873082	−	0.487574i	$-0.837882\pi$
$593$	−42.5219	−1.74616	−0.873082	+	0.487574i	$0.837882\pi$
$594$	−6.72360	−0.275873
$595$	83.0075	3.40298
$596$	−11.8394	−0.484960
$597$	−3.94330	−0.161389
$598$	5.66082	0.231488
$599$	12.9236	0.528046	0.264023	−	0.964516i	$-0.414951\pi$
$599$	12.9236	0.528046	0.264023	+	0.964516i	$0.414951\pi$
$600$	0.403918	0.0164899
$601$	−28.0080	−1.14247	−0.571235	−	0.820786i	$-0.693536\pi$
$601$	−28.0080	−1.14247	−0.571235	+	0.820786i	$0.693536\pi$
$602$	15.8395	0.645568
$603$	13.9524	0.568186
$604$	11.1715	0.454563
$605$	22.0500	0.896461
$606$	21.5283	0.874529
$607$	−27.6253	−1.12128	−0.560638	−	0.828061i	$-0.689444\pi$
$607$	−27.6253	−1.12128	−0.560638	+	0.828061i	$0.689444\pi$
$608$	5.58952	0.226685
$609$	−64.1138	−2.59802
$610$	17.9743	0.727759
$611$	−5.43380	−0.219828
$612$	8.67127	0.350515
$613$	−14.5829	−0.588999	−0.294500	−	0.955652i	$-0.595153\pi$
$613$	−14.5829	−0.588999	−0.294500	+	0.955652i	$0.595153\pi$
$614$	18.6949	0.754465
$615$	32.5486	1.31249
$616$	5.35247	0.215657
$617$	−12.3228	−0.496096	−0.248048	−	0.968748i	$-0.579789\pi$
$617$	−12.3228	−0.496096	−0.248048	+	0.968748i	$0.579789\pi$
$618$	28.0159	1.12697
$619$	−10.7288	−0.431227	−0.215614	−	0.976479i	$-0.569175\pi$
$619$	−10.7288	−0.431227	−0.215614	+	0.976479i	$0.569175\pi$
$620$	12.2214	0.490822
$621$	15.7214	0.630877
$622$	−1.93125	−0.0774361
$623$	17.7654	0.711758
$624$	−2.81952	−0.112871
$625$	−26.3730	−1.05492
$626$	−20.0601	−0.801764
$627$	9.20789	0.367728
$628$	0.685186	0.0273419
$629$	70.7050	2.81919
$630$	11.1959	0.446057
$631$	13.2839	0.528824	0.264412	−	0.964410i	$-0.414822\pi$
$631$	13.2839	0.528824	0.264412	+	0.964410i	$0.414822\pi$
$632$	9.31181	0.370404
$633$	31.1186	1.23685
$634$	14.9580	0.594057
$635$	33.8900	1.34488
$636$	−13.9919	−0.554816
$637$	−26.9794	−1.06896
$638$	−12.2325	−0.484288
$639$	−9.54324	−0.377525
$640$	−2.30035	−0.0909294
$641$	−29.5189	−1.16593	−0.582963	−	0.812499i	$-0.698107\pi$
$641$	−29.5189	−1.16593	−0.582963	+	0.812499i	$0.698107\pi$
$642$	21.7388	0.857963
$643$	13.7325	0.541556	0.270778	−	0.962642i	$-0.412719\pi$
$643$	13.7325	0.541556	0.270778	+	0.962642i	$0.412719\pi$
$644$	−12.5154	−0.493174
$645$	−11.2141	−0.441556
$646$	−44.8171	−1.76331
$647$	45.5724	1.79164	0.895818	−	0.444421i	$-0.146591\pi$
$647$	45.5724	1.79164	0.895818	+	0.444421i	$0.146591\pi$
$648$	−4.58603	−0.180156
$649$	12.9843	0.509679
$650$	−0.593607	−0.0232832
$651$	−33.1180	−1.29800
$652$	5.56358	0.217887
$653$	−47.5854	−1.86216	−0.931081	−	0.364813i	$-0.881133\pi$
$653$	−47.5854	−1.86216	−0.931081	+	0.364813i	$0.881133\pi$
$654$	22.0766	0.863265
$655$	−5.11008	−0.199667
$656$	−10.2154	−0.398843
$657$	7.03617	0.274507
$658$	12.0134	0.468332
$659$	2.95394	0.115069	0.0575346	−	0.998344i	$-0.481676\pi$
$659$	2.95394	0.115069	0.0575346	+	0.998344i	$0.481676\pi$
$660$	−3.78948	−0.147505
$661$	−30.9410	−1.20347	−0.601733	−	0.798698i	$-0.705523\pi$
$661$	−30.9410	−1.20347	−0.601733	+	0.798698i	$0.705523\pi$
$662$	27.8593	1.08278
$663$	22.6071	0.877986
$664$	−3.80888	−0.147813
$665$	−57.8658	−2.24394
$666$	9.53660	0.369536
$667$	28.6024	1.10749
$668$	−4.44356	−0.171926
$669$	20.7541	0.802400
$670$	29.6777	1.14655
$671$	−9.29307	−0.358755
$672$	6.23359	0.240466
$673$	6.36079	0.245190	0.122595	−	0.992457i	$-0.460878\pi$
$673$	6.36079	0.245190	0.122595	+	0.992457i	$0.460878\pi$
$674$	2.55237	0.0983137
$675$	−1.64858	−0.0634538
$676$	−8.85637	−0.340630
$677$	−22.5626	−0.867151	−0.433576	−	0.901117i	$-0.642748\pi$
$677$	−22.5626	−0.867151	−0.433576	+	0.901117i	$0.642748\pi$
$678$	11.4153	0.438403
$679$	−12.7800	−0.490450
$680$	18.4444	0.707309
$681$	−31.2128	−1.19608
$682$	−6.31869	−0.241955
$683$	−31.1936	−1.19359	−0.596794	−	0.802394i	$-0.703559\pi$
$683$	−31.1936	−1.19359	−0.596794	+	0.802394i	$0.703559\pi$
$684$	−6.04488	−0.231132
$685$	19.3588	0.739661
$686$	28.1450	1.07458
$687$	−3.67158	−0.140079
$688$	3.51955	0.134181
$689$	20.5629	0.783382
$690$	8.86070	0.337321
$691$	10.5575	0.401627	0.200813	−	0.979630i	$-0.435642\pi$
$691$	10.5575	0.401627	0.200813	+	0.979630i	$0.435642\pi$
$692$	0.210202	0.00799070
$693$	−5.78852	−0.219888
$694$	−15.0204	−0.570167
$695$	32.1375	1.21905
$696$	−14.2462	−0.539999
$697$	81.9073	3.10246
$698$	22.8845	0.866192
$699$	2.21164	0.0836520
$700$	1.31239	0.0496036
$701$	23.9135	0.903199	0.451600	−	0.892221i	$-0.350854\pi$
$701$	23.9135	0.903199	0.451600	+	0.892221i	$0.350854\pi$
$702$	11.5078	0.434333
$703$	−49.2896	−1.85899
$704$	1.18933	0.0448244
$705$	−8.50534	−0.320330
$706$	17.3204	0.651860
$707$	69.9487	2.63069
$708$	15.1218	0.568312
$709$	39.8708	1.49738	0.748689	−	0.662921i	$-0.230684\pi$
$709$	39.8708	1.49738	0.748689	+	0.662921i	$0.230684\pi$
$710$	−20.2991	−0.761812
$711$	−10.0704	−0.377670
$712$	3.94750	0.147939
$713$	14.7746	0.553313
$714$	−49.9813	−1.87050
$715$	5.56910	0.208273
$716$	0.964079	0.0360293
$717$	−33.3634	−1.24598
$718$	29.4820	1.10026
$719$	19.2423	0.717618	0.358809	−	0.933411i	$-0.383183\pi$
$719$	19.2423	0.717618	0.358809	+	0.933411i	$0.383183\pi$
$720$	2.48775	0.0927130
$721$	91.0278	3.39005
$722$	12.2427	0.455627
$723$	0.617039	0.0229479
$724$	10.2162	0.379683
$725$	−2.99931	−0.111392
$726$	−13.2770	−0.492755
$727$	−19.8560	−0.736420	−0.368210	−	0.929743i	$-0.620029\pi$
$727$	−19.8560	−0.736420	−0.368210	+	0.929743i	$0.620029\pi$
$728$	−9.16102	−0.339530
$729$	28.4509	1.05374
$730$	14.9664	0.553932
$731$	−28.2200	−1.04375
$732$	−10.8229	−0.400025
$733$	10.9737	0.405322	0.202661	−	0.979249i	$-0.435041\pi$
$733$	10.9737	0.405322	0.202661	+	0.979249i	$0.435041\pi$
$734$	11.4451	0.422447
$735$	−42.2300	−1.55768
$736$	−2.78093	−0.102506
$737$	−15.3440	−0.565202
$738$	11.0476	0.406666
$739$	−27.1901	−1.00020	−0.500102	−	0.865966i	$-0.666704\pi$
$739$	−27.1901	−1.00020	−0.500102	+	0.865966i	$0.666704\pi$
$740$	20.2850	0.745691
$741$	−15.7598	−0.578949
$742$	−45.4618	−1.66896
$743$	3.76686	0.138193	0.0690964	−	0.997610i	$-0.477988\pi$
$743$	3.76686	0.138193	0.0690964	+	0.997610i	$0.477988\pi$
$744$	−7.35886	−0.269789
$745$	27.2347	0.997804
$746$	−12.5117	−0.458086
$747$	4.11918	0.150713
$748$	−9.53609	−0.348674
$749$	70.6326	2.58086
$750$	15.0021	0.547798
$751$	36.8554	1.34487	0.672436	−	0.740155i	$-0.265248\pi$
$751$	36.8554	1.34487	0.672436	+	0.740155i	$0.265248\pi$
$752$	2.66940	0.0973429
$753$	−29.0997	−1.06045
$754$	20.9365	0.762461
$755$	−25.6984	−0.935261
$756$	−25.4422	−0.925324
$757$	10.1010	0.367128	0.183564	−	0.983008i	$-0.441237\pi$
$757$	10.1010	0.367128	0.183564	+	0.983008i	$0.441237\pi$
$758$	32.4393	1.17825
$759$	−4.58115	−0.166285
$760$	−12.8579	−0.466403
$761$	−34.5993	−1.25422	−0.627112	−	0.778929i	$-0.715763\pi$
$761$	−34.5993	−1.25422	−0.627112	+	0.778929i	$0.715763\pi$
$762$	−20.4062	−0.739239
$763$	71.7302	2.59681
$764$	13.4591	0.486935
$765$	−19.9470	−0.721184
$766$	−5.93388	−0.214400
$767$	−22.2233	−0.802437
$768$	1.38511	0.0499809
$769$	−20.3242	−0.732908	−0.366454	−	0.930436i	$-0.619428\pi$
$769$	−20.3242	−0.732908	−0.366454	+	0.930436i	$0.619428\pi$
$770$	−12.3126	−0.443714
$771$	−26.3106	−0.947552
$772$	−13.5891	−0.489084
$773$	15.4458	0.555546	0.277773	−	0.960647i	$-0.410404\pi$
$773$	15.4458	0.555546	0.277773	+	0.960647i	$0.410404\pi$
$774$	−3.80627	−0.136814
$775$	−1.54930	−0.0556524
$776$	−2.83972	−0.101940
$777$	−54.9692	−1.97201
$778$	−25.8128	−0.925433
$779$	−57.0989	−2.04578
$780$	6.48588	0.232232
$781$	10.4950	0.375542
$782$	22.2976	0.797362
$783$	58.1452	2.07794
$784$	13.2538	0.473352
$785$	−1.57617	−0.0562558
$786$	3.07693	0.109751
$787$	−26.7444	−0.953335	−0.476667	−	0.879084i	$-0.658155\pi$
$787$	−26.7444	−0.953335	−0.476667	+	0.879084i	$0.658155\pi$
$788$	−13.1219	−0.467450
$789$	−20.8711	−0.743030
$790$	−21.4204	−0.762105
$791$	37.0900	1.31877
$792$	−1.28622	−0.0457037
$793$	15.9056	0.564823
$794$	−18.0917	−0.642052
$795$	32.1864	1.14153
$796$	−2.84692	−0.100906
$797$	7.34468	0.260162	0.130081	−	0.991503i	$-0.458476\pi$
$797$	7.34468	0.260162	0.130081	+	0.991503i	$0.458476\pi$
$798$	34.8428	1.23342
$799$	−21.4034	−0.757198
$800$	0.291614	0.0103101
$801$	−4.26909	−0.150841
$802$	16.8070	0.593477
$803$	−7.73792	−0.273065
$804$	−17.8699	−0.630221
$805$	28.7897	1.01470
$806$	10.8147	0.380933
$807$	−14.2518	−0.501686
$808$	15.5427	0.546789
$809$	11.8327	0.416014	0.208007	−	0.978127i	$-0.433302\pi$
$809$	11.8327	0.416014	0.208007	+	0.978127i	$0.433302\pi$
$810$	10.5495	0.370671
$811$	−7.63644	−0.268152	−0.134076	−	0.990971i	$-0.542807\pi$
$811$	−7.63644	−0.268152	−0.134076	+	0.990971i	$0.542807\pi$
$812$	−46.2878	−1.62438
$813$	14.6662	0.514364
$814$	−10.4877	−0.367595
$815$	−12.7982	−0.448301
$816$	−11.1059	−0.388785
$817$	19.6726	0.688256
$818$	13.2533	0.463392
$819$	9.90734	0.346190
$820$	23.4989	0.820617
$821$	−1.71650	−0.0599061	−0.0299531	−	0.999551i	$-0.509536\pi$
$821$	−1.71650	−0.0599061	−0.0299531	+	0.999551i	$0.509536\pi$
$822$	−11.6565	−0.406567
$823$	−51.0041	−1.77789	−0.888946	−	0.458012i	$-0.848562\pi$
$823$	−51.0041	−1.77789	−0.888946	+	0.458012i	$0.848562\pi$
$824$	20.2265	0.704622
$825$	0.480390	0.0167250
$826$	49.1329	1.70955
$827$	31.4220	1.09265	0.546325	−	0.837574i	$-0.316026\pi$
$827$	31.4220	1.09265	0.546325	+	0.837574i	$0.316026\pi$
$828$	3.00748	0.104517
$829$	−36.9780	−1.28430	−0.642149	−	0.766580i	$-0.721957\pi$
$829$	−36.9780	−1.28430	−0.642149	+	0.766580i	$0.721957\pi$
$830$	8.76176	0.304125
$831$	12.0547	0.418173
$832$	−2.03559	−0.0705714
$833$	−106.270	−3.68204
$834$	−19.3510	−0.670070
$835$	10.2217	0.353738
$836$	6.64776	0.229918
$837$	30.0349	1.03816
$838$	−2.41199	−0.0833209
$839$	55.9928	1.93309	0.966544	−	0.256502i	$-0.0825701\pi$
$839$	55.9928	1.93309	0.966544	+	0.256502i	$0.0825701\pi$
$840$	−14.3395	−0.494758
$841$	76.7855	2.64778
$842$	−19.8574	−0.684330
$843$	−20.5096	−0.706388
$844$	22.4665	0.773330
$845$	20.3728	0.700844
$846$	−2.88686	−0.0992524
$847$	−43.1389	−1.48227
$848$	−10.1017	−0.346893
$849$	18.6882	0.641379
$850$	−2.33818	−0.0801989
$851$	24.5228	0.840631
$852$	12.2227	0.418743
$853$	16.4152	0.562046	0.281023	−	0.959701i	$-0.409326\pi$
$853$	16.4152	0.562046	0.281023	+	0.959701i	$0.409326\pi$
$854$	−35.1651	−1.20333
$855$	13.9053	0.475553
$856$	15.6946	0.536432
$857$	0.771640	0.0263587	0.0131794	−	0.999913i	$-0.495805\pi$
$857$	0.771640	0.0263587	0.0131794	+	0.999913i	$0.495805\pi$
$858$	−3.35333	−0.114481
$859$	−4.35220	−0.148495	−0.0742476	−	0.997240i	$-0.523656\pi$
$859$	−4.35220	−0.148495	−0.0742476	+	0.997240i	$0.523656\pi$
$860$	−8.09619	−0.276078
$861$	−63.6784	−2.17015
$862$	−22.9885	−0.782992
$863$	33.1641	1.12892	0.564459	−	0.825461i	$-0.309085\pi$
$863$	33.1641	1.12892	0.564459	+	0.825461i	$0.309085\pi$
$864$	−5.65329	−0.192329
$865$	−0.483540	−0.0164408
$866$	−12.4063	−0.421582
$867$	65.5010	2.22453
$868$	−23.9100	−0.811559
$869$	11.0748	0.375686
$870$	32.7712	1.11105
$871$	26.2619	0.889852
$872$	15.9385	0.539747
$873$	3.07107	0.103940
$874$	−15.5440	−0.525785
$875$	48.7439	1.64784
$876$	−9.01173	−0.304478
$877$	−18.3641	−0.620112	−0.310056	−	0.950718i	$-0.600348\pi$
$877$	−18.3641	−0.620112	−0.310056	+	0.950718i	$0.600348\pi$
$878$	−10.8042	−0.364626
$879$	−38.1962	−1.28832
$880$	−2.73587	−0.0922261
$881$	0.460443	0.0155127	0.00775635	−	0.999970i	$-0.497531\pi$
$881$	0.460443	0.0155127	0.00775635	+	0.999970i	$0.497531\pi$
$882$	−14.3336	−0.482637
$883$	37.7823	1.27148	0.635738	−	0.771905i	$-0.280696\pi$
$883$	37.7823	1.27148	0.635738	+	0.771905i	$0.280696\pi$
$884$	16.3215	0.548951
$885$	−34.7854	−1.16930
$886$	2.91122	0.0978042
$887$	−19.8038	−0.664945	−0.332472	−	0.943113i	$-0.607883\pi$
$887$	−19.8038	−0.664945	−0.332472	+	0.943113i	$0.607883\pi$
$888$	−12.2142	−0.409882
$889$	−66.3027	−2.22372
$890$	−9.08064	−0.304384
$891$	−5.45429	−0.182725
$892$	14.9837	0.501691
$893$	14.9206	0.499300
$894$	−16.3989	−0.548460
$895$	−2.21772	−0.0741302
$896$	4.50043	0.150349
$897$	7.84087	0.261799
$898$	23.3874	0.780446
$899$	54.6436	1.82247
$900$	−0.315371	−0.0105124
$901$	80.9958	2.69836
$902$	−12.1494	−0.404530
$903$	21.9394	0.730098
$904$	8.24144	0.274106
$905$	−23.5009	−0.781197
$906$	15.4738	0.514083
$907$	24.7967	0.823361	0.411680	−	0.911328i	$-0.364942\pi$
$907$	24.7967	0.823361	0.411680	+	0.911328i	$0.364942\pi$
$908$	−22.5345	−0.747833
$909$	−16.8089	−0.557515
$910$	21.0736	0.698582
$911$	8.93859	0.296149	0.148074	−	0.988976i	$-0.452692\pi$
$911$	8.93859	0.296149	0.148074	+	0.988976i	$0.452692\pi$
$912$	7.74211	0.256367
$913$	−4.53000	−0.149921
$914$	30.7341	1.01660
$915$	24.8964	0.823051
$916$	−2.65074	−0.0875831
$917$	9.99740	0.330143
$918$	45.3284	1.49606
$919$	−1.22997	−0.0405730	−0.0202865	−	0.999794i	$-0.506458\pi$
$919$	−1.22997	−0.0405730	−0.0202865	+	0.999794i	$0.506458\pi$
$920$	6.39710	0.210906
$921$	25.8945	0.853254
$922$	7.71167	0.253970
$923$	−17.9628	−0.591252
$924$	7.41377	0.243895
$925$	−2.57152	−0.0845509
$926$	−36.4773	−1.19872
$927$	−21.8742	−0.718444
$928$	−10.2852	−0.337629
$929$	−30.2904	−0.993797	−0.496898	−	0.867809i	$-0.665528\pi$
$929$	−30.2904	−0.993797	−0.496898	+	0.867809i	$0.665528\pi$
$930$	16.9280	0.555090
$931$	74.0826	2.42796
$932$	1.59672	0.0523025
$933$	−2.67500	−0.0875755
$934$	−32.2385	−1.05487
$935$	21.9364	0.717396
$936$	2.20142	0.0719557
$937$	−2.35672	−0.0769908	−0.0384954	−	0.999259i	$-0.512256\pi$
$937$	−2.35672	−0.0769908	−0.0384954	+	0.999259i	$0.512256\pi$
$938$	−58.0617	−1.89578
$939$	−27.7855	−0.906746
$940$	−6.14055	−0.200283
$941$	2.85840	0.0931812	0.0465906	−	0.998914i	$-0.485164\pi$
$941$	2.85840	0.0931812	0.0465906	+	0.998914i	$0.485164\pi$
$942$	0.949059	0.0309220
$943$	28.4081	0.925096
$944$	10.9174	0.355330
$945$	58.5260	1.90385
$946$	4.18589	0.136095
$947$	9.09537	0.295560	0.147780	−	0.989020i	$-0.452787\pi$
$947$	9.09537	0.295560	0.147780	+	0.989020i	$0.452787\pi$
$948$	12.8979	0.418904
$949$	13.2438	0.429913
$950$	1.62998	0.0528836
$951$	20.7185	0.671842
$952$	−36.0847	−1.16951
$953$	10.9130	0.353507	0.176753	−	0.984255i	$-0.443440\pi$
$953$	10.9130	0.353507	0.176753	+	0.984255i	$0.443440\pi$
$954$	10.9246	0.353697
$955$	−30.9608	−1.00187
$956$	−24.0872	−0.779035
$957$	−16.9433	−0.547700
$958$	13.2524	0.428167
$959$	−37.8737	−1.22301
$960$	−3.18624	−0.102836
$961$	−2.77383	−0.0894784
$962$	17.9503	0.578740
$963$	−16.9732	−0.546955
$964$	0.445480	0.0143479
$965$	31.2598	1.00629
$966$	−17.3352	−0.557750
$967$	36.3507	1.16896	0.584480	−	0.811408i	$-0.301298\pi$
$967$	36.3507	1.16896	0.584480	+	0.811408i	$0.301298\pi$
$968$	−9.58550	−0.308090
$969$	−62.0767	−1.99419
$970$	6.53236	0.209741
$971$	−11.9710	−0.384167	−0.192084	−	0.981379i	$-0.561524\pi$
$971$	−11.9710	−0.384167	−0.192084	+	0.981379i	$0.561524\pi$
$972$	10.6077	0.340242
$973$	−62.8741	−2.01565
$974$	13.3585	0.428034
$975$	−0.822212	−0.0263318
$976$	−7.81373	−0.250111
$977$	−41.7781	−1.33660	−0.668300	−	0.743892i	$-0.732978\pi$
$977$	−41.7781	−1.33660	−0.668300	+	0.743892i	$0.732978\pi$
$978$	7.70617	0.246416
$979$	4.69487	0.150049
$980$	−30.4885	−0.973919
$981$	−17.2370	−0.550335
$982$	−33.1217	−1.05696
$983$	1.37764	0.0439400	0.0219700	−	0.999759i	$-0.493006\pi$
$983$	1.37764	0.0439400	0.0219700	+	0.999759i	$0.493006\pi$
$984$	−14.1494	−0.451066
$985$	30.1851	0.961776
$986$	82.4675	2.62630
$987$	16.6399	0.529655
$988$	−11.3780	−0.361981
$989$	−9.78760	−0.311228
$990$	2.95875	0.0940352
$991$	−10.5819	−0.336145	−0.168073	−	0.985775i	$-0.553754\pi$
$991$	−10.5819	−0.336145	−0.168073	+	0.985775i	$0.553754\pi$
$992$	−5.31283	−0.168683
$993$	38.5883	1.22456
$994$	39.7134	1.25963
$995$	6.54891	0.207614
$996$	−5.27573	−0.167168
$997$	−26.7134	−0.846021	−0.423011	−	0.906125i	$-0.639027\pi$
$997$	−26.7134	−0.846021	−0.423011	+	0.906125i	$0.639027\pi$
$998$	−15.1028	−0.478071
$999$	49.8519	1.57724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.53	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.53	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.38511 q^{3} +1.00000 q^{4} -2.30035 q^{5} +1.38511 q^{6} +4.50043 q^{7} +1.00000 q^{8} -1.08147 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.38511 q^{3} +1.00000 q^{4} -2.30035 q^{5} +1.38511 q^{6} +4.50043 q^{7} +1.00000 q^{8} -1.08147 q^{9} -2.30035 q^{10} +1.18933 q^{11} +1.38511 q^{12} -2.03559 q^{13} +4.50043 q^{14} -3.18624 q^{15} +1.00000 q^{16} -8.01806 q^{17} -1.08147 q^{18} +5.58952 q^{19} -2.30035 q^{20} +6.23359 q^{21} +1.18933 q^{22} -2.78093 q^{23} +1.38511 q^{24} +0.291614 q^{25} -2.03559 q^{26} -5.65329 q^{27} +4.50043 q^{28} -10.2852 q^{29} -3.18624 q^{30} -5.31283 q^{31} +1.00000 q^{32} +1.64735 q^{33} -8.01806 q^{34} -10.3526 q^{35} -1.08147 q^{36} -8.81821 q^{37} +5.58952 q^{38} -2.81952 q^{39} -2.30035 q^{40} -10.2154 q^{41} +6.23359 q^{42} +3.51955 q^{43} +1.18933 q^{44} +2.48775 q^{45} -2.78093 q^{46} +2.66940 q^{47} +1.38511 q^{48} +13.2538 q^{49} +0.291614 q^{50} -11.1059 q^{51} -2.03559 q^{52} -10.1017 q^{53} -5.65329 q^{54} -2.73587 q^{55} +4.50043 q^{56} +7.74211 q^{57} -10.2852 q^{58} +10.9174 q^{59} -3.18624 q^{60} -7.81373 q^{61} -5.31283 q^{62} -4.86706 q^{63} +1.00000 q^{64} +4.68257 q^{65} +1.64735 q^{66} -12.9014 q^{67} -8.01806 q^{68} -3.85189 q^{69} -10.3526 q^{70} +8.82436 q^{71} -1.08147 q^{72} -6.50614 q^{73} -8.81821 q^{74} +0.403918 q^{75} +5.58952 q^{76} +5.35247 q^{77} -2.81952 q^{78} +9.31181 q^{79} -2.30035 q^{80} -4.58603 q^{81} -10.2154 q^{82} -3.80888 q^{83} +6.23359 q^{84} +18.4444 q^{85} +3.51955 q^{86} -14.2462 q^{87} +1.18933 q^{88} +3.94750 q^{89} +2.48775 q^{90} -9.16102 q^{91} -2.78093 q^{92} -7.35886 q^{93} +2.66940 q^{94} -12.8579 q^{95} +1.38511 q^{96} -2.83972 q^{97} +13.2538 q^{98} -1.28622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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