LMFDB - Embedded newform 8002.2.a.f.1.19

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:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.15966 q^{3} +1.00000 q^{4} +1.78112 q^{5} +2.15966 q^{6} +1.44604 q^{7} -1.00000 q^{8} +1.66414 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.15966 q^{3} +1.00000 q^{4} +1.78112 q^{5} +2.15966 q^{6} +1.44604 q^{7} -1.00000 q^{8} +1.66414 q^{9} -1.78112 q^{10} +6.18258 q^{11} -2.15966 q^{12} -1.68666 q^{13} -1.44604 q^{14} -3.84662 q^{15} +1.00000 q^{16} -4.28862 q^{17} -1.66414 q^{18} -4.15338 q^{19} +1.78112 q^{20} -3.12296 q^{21} -6.18258 q^{22} +7.47677 q^{23} +2.15966 q^{24} -1.82761 q^{25} +1.68666 q^{26} +2.88500 q^{27} +1.44604 q^{28} +5.50721 q^{29} +3.84662 q^{30} -5.07464 q^{31} -1.00000 q^{32} -13.3523 q^{33} +4.28862 q^{34} +2.57558 q^{35} +1.66414 q^{36} -4.34885 q^{37} +4.15338 q^{38} +3.64261 q^{39} -1.78112 q^{40} +0.694655 q^{41} +3.12296 q^{42} -2.07175 q^{43} +6.18258 q^{44} +2.96404 q^{45} -7.47677 q^{46} +1.39301 q^{47} -2.15966 q^{48} -4.90896 q^{49} +1.82761 q^{50} +9.26197 q^{51} -1.68666 q^{52} +2.76446 q^{53} -2.88500 q^{54} +11.0119 q^{55} -1.44604 q^{56} +8.96990 q^{57} -5.50721 q^{58} +3.47643 q^{59} -3.84662 q^{60} -11.3027 q^{61} +5.07464 q^{62} +2.40642 q^{63} +1.00000 q^{64} -3.00415 q^{65} +13.3523 q^{66} -12.0699 q^{67} -4.28862 q^{68} -16.1473 q^{69} -2.57558 q^{70} +6.64580 q^{71} -1.66414 q^{72} -5.80574 q^{73} +4.34885 q^{74} +3.94701 q^{75} -4.15338 q^{76} +8.94027 q^{77} -3.64261 q^{78} -15.0772 q^{79} +1.78112 q^{80} -11.2231 q^{81} -0.694655 q^{82} -10.1392 q^{83} -3.12296 q^{84} -7.63855 q^{85} +2.07175 q^{86} -11.8937 q^{87} -6.18258 q^{88} +13.1908 q^{89} -2.96404 q^{90} -2.43898 q^{91} +7.47677 q^{92} +10.9595 q^{93} -1.39301 q^{94} -7.39768 q^{95} +2.15966 q^{96} -15.7722 q^{97} +4.90896 q^{98} +10.2887 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−2.15966	−1.24688	−0.623441	−	0.781870i	$-0.714266\pi$
$3$	−2.15966	−1.24688	−0.623441	+	0.781870i	$0.714266\pi$
$4$	1.00000	0.500000
$5$	1.78112	0.796542	0.398271	−	0.917268i	$-0.369610\pi$
$5$	1.78112	0.796542	0.398271	+	0.917268i	$0.369610\pi$
$6$	2.15966	0.881679
$7$	1.44604	0.546552	0.273276	−	0.961936i	$-0.411893\pi$
$7$	1.44604	0.546552	0.273276	+	0.961936i	$0.411893\pi$
$8$	−1.00000	−0.353553
$9$	1.66414	0.554714
$10$	−1.78112	−0.563240
$11$	6.18258	1.86412	0.932059	−	0.362306i	$-0.118011\pi$
$11$	6.18258	1.86412	0.932059	+	0.362306i	$0.118011\pi$
$12$	−2.15966	−0.623441
$13$	−1.68666	−0.467795	−0.233898	−	0.972261i	$-0.575148\pi$
$13$	−1.68666	−0.467795	−0.233898	+	0.972261i	$0.575148\pi$
$14$	−1.44604	−0.386471
$15$	−3.84662	−0.993193
$16$	1.00000	0.250000
$17$	−4.28862	−1.04014	−0.520071	−	0.854123i	$-0.674095\pi$
$17$	−4.28862	−1.04014	−0.520071	+	0.854123i	$0.674095\pi$
$18$	−1.66414	−0.392242
$19$	−4.15338	−0.952851	−0.476426	−	0.879215i	$-0.658068\pi$
$19$	−4.15338	−0.952851	−0.476426	+	0.879215i	$0.658068\pi$
$20$	1.78112	0.398271
$21$	−3.12296	−0.681486
$22$	−6.18258	−1.31813
$23$	7.47677	1.55901	0.779507	−	0.626393i	$-0.215470\pi$
$23$	7.47677	1.55901	0.779507	+	0.626393i	$0.215470\pi$
$24$	2.15966	0.440839
$25$	−1.82761	−0.365521
$26$	1.68666	0.330781
$27$	2.88500	0.555219
$28$	1.44604	0.273276
$29$	5.50721	1.02266	0.511332	−	0.859383i	$-0.329152\pi$
$29$	5.50721	1.02266	0.511332	+	0.859383i	$0.329152\pi$
$30$	3.84662	0.702294
$31$	−5.07464	−0.911432	−0.455716	−	0.890125i	$-0.650617\pi$
$31$	−5.07464	−0.911432	−0.455716	+	0.890125i	$0.650617\pi$
$32$	−1.00000	−0.176777
$33$	−13.3523	−2.32433
$34$	4.28862	0.735492
$35$	2.57558	0.435352
$36$	1.66414	0.277357
$37$	−4.34885	−0.714947	−0.357474	−	0.933923i	$-0.616362\pi$
$37$	−4.34885	−0.714947	−0.357474	+	0.933923i	$0.616362\pi$
$38$	4.15338	0.673768
$39$	3.64261	0.583285
$40$	−1.78112	−0.281620
$41$	0.694655	0.108487	0.0542434	−	0.998528i	$-0.482725\pi$
$41$	0.694655	0.108487	0.0542434	+	0.998528i	$0.482725\pi$
$42$	3.12296	0.481883
$43$	−2.07175	−0.315939	−0.157970	−	0.987444i	$-0.550495\pi$
$43$	−2.07175	−0.315939	−0.157970	+	0.987444i	$0.550495\pi$
$44$	6.18258	0.932059
$45$	2.96404	0.441853
$46$	−7.47677	−1.10239
$47$	1.39301	0.203191	0.101596	−	0.994826i	$-0.467605\pi$
$47$	1.39301	0.203191	0.101596	+	0.994826i	$0.467605\pi$
$48$	−2.15966	−0.311720
$49$	−4.90896	−0.701281
$50$	1.82761	0.258463
$51$	9.26197	1.29693
$52$	−1.68666	−0.233898
$53$	2.76446	0.379728	0.189864	−	0.981810i	$-0.439195\pi$
$53$	2.76446	0.379728	0.189864	+	0.981810i	$0.439195\pi$
$54$	−2.88500	−0.392599
$55$	11.0119	1.48485
$56$	−1.44604	−0.193235
$57$	8.96990	1.18809
$58$	−5.50721	−0.723133
$59$	3.47643	0.452592	0.226296	−	0.974059i	$-0.427338\pi$
$59$	3.47643	0.452592	0.226296	+	0.974059i	$0.427338\pi$
$60$	−3.84662	−0.496597
$61$	−11.3027	−1.44716	−0.723578	−	0.690242i	$-0.757504\pi$
$61$	−11.3027	−1.44716	−0.723578	+	0.690242i	$0.757504\pi$
$62$	5.07464	0.644480
$63$	2.40642	0.303180
$64$	1.00000	0.125000
$65$	−3.00415	−0.372618
$66$	13.3523	1.64355
$67$	−12.0699	−1.47458	−0.737289	−	0.675577i	$-0.763895\pi$
$67$	−12.0699	−1.47458	−0.737289	+	0.675577i	$0.763895\pi$
$68$	−4.28862	−0.520071
$69$	−16.1473	−1.94391
$70$	−2.57558	−0.307840
$71$	6.64580	0.788711	0.394355	−	0.918958i	$-0.370968\pi$
$71$	6.64580	0.788711	0.394355	+	0.918958i	$0.370968\pi$
$72$	−1.66414	−0.196121
$73$	−5.80574	−0.679511	−0.339755	−	0.940514i	$-0.610344\pi$
$73$	−5.80574	−0.679511	−0.339755	+	0.940514i	$0.610344\pi$
$74$	4.34885	0.505544
$75$	3.94701	0.455762
$76$	−4.15338	−0.476426
$77$	8.94027	1.01884
$78$	−3.64261	−0.412445
$79$	−15.0772	−1.69631	−0.848157	−	0.529746i	$-0.822287\pi$
$79$	−15.0772	−1.69631	−0.848157	+	0.529746i	$0.822287\pi$
$80$	1.78112	0.199135
$81$	−11.2231	−1.24701
$82$	−0.694655	−0.0767118
$83$	−10.1392	−1.11292	−0.556459	−	0.830875i	$-0.687840\pi$
$83$	−10.1392	−1.11292	−0.556459	+	0.830875i	$0.687840\pi$
$84$	−3.12296	−0.340743
$85$	−7.63855	−0.828517
$86$	2.07175	0.223403
$87$	−11.8937	−1.27514
$88$	−6.18258	−0.659065
$89$	13.1908	1.39822	0.699110	−	0.715014i	$-0.253580\pi$
$89$	13.1908	1.39822	0.699110	+	0.715014i	$0.253580\pi$
$90$	−2.96404	−0.312437
$91$	−2.43898	−0.255674
$92$	7.47677	0.779507
$93$	10.9595	1.13645
$94$	−1.39301	−0.143678
$95$	−7.39768	−0.758986
$96$	2.15966	0.220420
$97$	−15.7722	−1.60142	−0.800711	−	0.599051i	$-0.795545\pi$
$97$	−15.7722	−1.60142	−0.800711	+	0.599051i	$0.795545\pi$
$98$	4.90896	0.495880
$99$	10.2887	1.03405
$100$	−1.82761	−0.182761
$101$	−16.6377	−1.65551	−0.827756	−	0.561088i	$-0.810383\pi$
$101$	−16.6377	−1.65551	−0.827756	+	0.561088i	$0.810383\pi$
$102$	−9.26197	−0.917071
$103$	−15.2978	−1.50733	−0.753667	−	0.657256i	$-0.771717\pi$
$103$	−15.2978	−1.50733	−0.753667	+	0.657256i	$0.771717\pi$
$104$	1.68666	0.165391
$105$	−5.56237	−0.542832
$106$	−2.76446	−0.268509
$107$	3.45700	0.334201	0.167100	−	0.985940i	$-0.446560\pi$
$107$	3.45700	0.334201	0.167100	+	0.985940i	$0.446560\pi$
$108$	2.88500	0.277609
$109$	9.18393	0.879661	0.439830	−	0.898081i	$-0.355039\pi$
$109$	9.18393	0.879661	0.439830	+	0.898081i	$0.355039\pi$
$110$	−11.0119	−1.04995
$111$	9.39206	0.891455
$112$	1.44604	0.136638
$113$	−4.60468	−0.433172	−0.216586	−	0.976264i	$-0.569492\pi$
$113$	−4.60468	−0.433172	−0.216586	+	0.976264i	$0.569492\pi$
$114$	−8.96990	−0.840109
$115$	13.3170	1.24182
$116$	5.50721	0.511332
$117$	−2.80684	−0.259493
$118$	−3.47643	−0.320031
$119$	−6.20152	−0.568492
$120$	3.84662	0.351147
$121$	27.2243	2.47494
$122$	11.3027	1.02329
$123$	−1.50022	−0.135270
$124$	−5.07464	−0.455716
$125$	−12.1608	−1.08769
$126$	−2.40642	−0.214381
$127$	15.4192	1.36823	0.684116	−	0.729373i	$-0.260188\pi$
$127$	15.4192	1.36823	0.684116	+	0.729373i	$0.260188\pi$
$128$	−1.00000	−0.0883883
$129$	4.47428	0.393939
$130$	3.00415	0.263481
$131$	−16.9087	−1.47732	−0.738661	−	0.674078i	$-0.764541\pi$
$131$	−16.9087	−1.47732	−0.738661	+	0.674078i	$0.764541\pi$
$132$	−13.3523	−1.16217
$133$	−6.00596	−0.520783
$134$	12.0699	1.04268
$135$	5.13854	0.442255
$136$	4.28862	0.367746
$137$	3.58254	0.306077	0.153038	−	0.988220i	$-0.451094\pi$
$137$	3.58254	0.306077	0.153038	+	0.988220i	$0.451094\pi$
$138$	16.1473	1.37455
$139$	−8.07207	−0.684664	−0.342332	−	0.939579i	$-0.611217\pi$
$139$	−8.07207	−0.684664	−0.342332	+	0.939579i	$0.611217\pi$
$140$	2.57558	0.217676
$141$	−3.00843	−0.253355
$142$	−6.64580	−0.557703
$143$	−10.4279	−0.872025
$144$	1.66414	0.138679
$145$	9.80902	0.814594
$146$	5.80574	0.480487
$147$	10.6017	0.874414
$148$	−4.34885	−0.357474
$149$	10.6182	0.869873	0.434937	−	0.900461i	$-0.356771\pi$
$149$	10.6182	0.869873	0.434937	+	0.900461i	$0.356771\pi$
$150$	−3.94701	−0.322272
$151$	17.7275	1.44264	0.721322	−	0.692599i	$-0.243535\pi$
$151$	17.7275	1.44264	0.721322	+	0.692599i	$0.243535\pi$
$152$	4.15338	0.336884
$153$	−7.13687	−0.576982
$154$	−8.94027	−0.720427
$155$	−9.03855	−0.725994
$156$	3.64261	0.291643
$157$	14.4824	1.15582	0.577911	−	0.816100i	$-0.303868\pi$
$157$	14.4824	1.15582	0.577911	+	0.816100i	$0.303868\pi$
$158$	15.0772	1.19947
$159$	−5.97031	−0.473476
$160$	−1.78112	−0.140810
$161$	10.8117	0.852083
$162$	11.2231	0.881767
$163$	0.695189	0.0544514	0.0272257	−	0.999629i	$-0.491333\pi$
$163$	0.695189	0.0544514	0.0272257	+	0.999629i	$0.491333\pi$
$164$	0.694655	0.0542434
$165$	−23.7820	−1.85143
$166$	10.1392	0.786952
$167$	−8.61594	−0.666722	−0.333361	−	0.942799i	$-0.608183\pi$
$167$	−8.61594	−0.666722	−0.333361	+	0.942799i	$0.608183\pi$
$168$	3.12296	0.240942
$169$	−10.1552	−0.781168
$170$	7.63855	0.585850
$171$	−6.91182	−0.528560
$172$	−2.07175	−0.157970
$173$	−20.7867	−1.58038	−0.790190	−	0.612862i	$-0.790018\pi$
$173$	−20.7867	−1.58038	−0.790190	+	0.612862i	$0.790018\pi$
$174$	11.8937	0.901661
$175$	−2.64279	−0.199776
$176$	6.18258	0.466030
$177$	−7.50791	−0.564329
$178$	−13.1908	−0.988691
$179$	19.3778	1.44837	0.724184	−	0.689607i	$-0.242217\pi$
$179$	19.3778	1.44837	0.724184	+	0.689607i	$0.242217\pi$
$180$	2.96404	0.220926
$181$	−8.50245	−0.631982	−0.315991	−	0.948762i	$-0.602337\pi$
$181$	−8.50245	−0.631982	−0.315991	+	0.948762i	$0.602337\pi$
$182$	2.43898	0.180789
$183$	24.4099	1.80443
$184$	−7.47677	−0.551195
$185$	−7.74584	−0.569485
$186$	−10.9595	−0.803590
$187$	−26.5147	−1.93895
$188$	1.39301	0.101596
$189$	4.17183	0.303456
$190$	7.39768	0.536684
$191$	−1.53432	−0.111019	−0.0555097	−	0.998458i	$-0.517678\pi$
$191$	−1.53432	−0.111019	−0.0555097	+	0.998458i	$0.517678\pi$
$192$	−2.15966	−0.155860
$193$	6.81269	0.490388	0.245194	−	0.969474i	$-0.421148\pi$
$193$	6.81269	0.490388	0.245194	+	0.969474i	$0.421148\pi$
$194$	15.7722	1.13238
$195$	6.48794	0.464611
$196$	−4.90896	−0.350640
$197$	−10.6478	−0.758626	−0.379313	−	0.925268i	$-0.623840\pi$
$197$	−10.6478	−0.758626	−0.379313	+	0.925268i	$0.623840\pi$
$198$	−10.2887	−0.731186
$199$	13.3760	0.948198	0.474099	−	0.880472i	$-0.342774\pi$
$199$	13.3760	0.948198	0.474099	+	0.880472i	$0.342774\pi$
$200$	1.82761	0.129231
$201$	26.0670	1.83863
$202$	16.6377	1.17062
$203$	7.96366	0.558939
$204$	9.26197	0.648467
$205$	1.23726	0.0864143
$206$	15.2978	1.06585
$207$	12.4424	0.864807
$208$	−1.68666	−0.116949
$209$	−25.6786	−1.77623
$210$	5.56237	0.383840
$211$	−20.4942	−1.41088	−0.705439	−	0.708771i	$-0.749250\pi$
$211$	−20.4942	−1.41088	−0.705439	+	0.708771i	$0.749250\pi$
$212$	2.76446	0.189864
$213$	−14.3527	−0.983429
$214$	−3.45700	−0.236316
$215$	−3.69004	−0.251659
$216$	−2.88500	−0.196299
$217$	−7.33814	−0.498145
$218$	−9.18393	−0.622014
$219$	12.5384	0.847269
$220$	11.0119	0.742424
$221$	7.23344	0.486574
$222$	−9.39206	−0.630354
$223$	18.6198	1.24687	0.623436	−	0.781875i	$-0.285736\pi$
$223$	18.6198	1.24687	0.623436	+	0.781875i	$0.285736\pi$
$224$	−1.44604	−0.0966177
$225$	−3.04140	−0.202760
$226$	4.60468	0.306299
$227$	13.4849	0.895026	0.447513	−	0.894277i	$-0.352310\pi$
$227$	13.4849	0.895026	0.447513	+	0.894277i	$0.352310\pi$
$228$	8.96990	0.594046
$229$	26.1013	1.72482	0.862411	−	0.506209i	$-0.168953\pi$
$229$	26.1013	1.72482	0.862411	+	0.506209i	$0.168953\pi$
$230$	−13.3170	−0.878099
$231$	−19.3080	−1.27037
$232$	−5.50721	−0.361566
$233$	−24.6980	−1.61802	−0.809011	−	0.587793i	$-0.799997\pi$
$233$	−24.6980	−1.61802	−0.809011	+	0.587793i	$0.799997\pi$
$234$	2.80684	0.183489
$235$	2.48112	0.161850
$236$	3.47643	0.226296
$237$	32.5616	2.11510
$238$	6.20152	0.401985
$239$	−25.2984	−1.63642	−0.818208	−	0.574922i	$-0.805032\pi$
$239$	−25.2984	−1.63642	−0.818208	+	0.574922i	$0.805032\pi$
$240$	−3.84662	−0.248298
$241$	8.91901	0.574524	0.287262	−	0.957852i	$-0.407255\pi$
$241$	8.91901	0.574524	0.287262	+	0.957852i	$0.407255\pi$
$242$	−27.2243	−1.75004
$243$	15.5830	0.999651
$244$	−11.3027	−0.723578
$245$	−8.74346	−0.558599
$246$	1.50022	0.0956505
$247$	7.00534	0.445739
$248$	5.07464	0.322240
$249$	21.8972	1.38768
$250$	12.1608	0.769116
$251$	10.9496	0.691133	0.345567	−	0.938394i	$-0.387687\pi$
$251$	10.9496	0.691133	0.345567	+	0.938394i	$0.387687\pi$
$252$	2.40642	0.151590
$253$	46.2257	2.90619
$254$	−15.4192	−0.967486
$255$	16.4967	1.03306
$256$	1.00000	0.0625000
$257$	14.2156	0.886743	0.443371	−	0.896338i	$-0.353782\pi$
$257$	14.2156	0.886743	0.443371	+	0.896338i	$0.353782\pi$
$258$	−4.47428	−0.278557
$259$	−6.28862	−0.390756
$260$	−3.00415	−0.186309
$261$	9.16479	0.567286
$262$	16.9087	1.04462
$263$	−24.9830	−1.54052	−0.770259	−	0.637731i	$-0.779873\pi$
$263$	−24.9830	−1.54052	−0.770259	+	0.637731i	$0.779873\pi$
$264$	13.3523	0.821776
$265$	4.92385	0.302469
$266$	6.00596	0.368249
$267$	−28.4877	−1.74342
$268$	−12.0699	−0.737289
$269$	20.6747	1.26056	0.630279	−	0.776369i	$-0.282940\pi$
$269$	20.6747	1.26056	0.630279	+	0.776369i	$0.282940\pi$
$270$	−5.13854	−0.312721
$271$	−21.5763	−1.31067	−0.655334	−	0.755339i	$-0.727472\pi$
$271$	−21.5763	−1.31067	−0.655334	+	0.755339i	$0.727472\pi$
$272$	−4.28862	−0.260036
$273$	5.26737	0.318796
$274$	−3.58254	−0.216429
$275$	−11.2993	−0.681375
$276$	−16.1473	−0.971953
$277$	−12.9067	−0.775488	−0.387744	−	0.921767i	$-0.626746\pi$
$277$	−12.9067	−0.775488	−0.387744	+	0.921767i	$0.626746\pi$
$278$	8.07207	0.484130
$279$	−8.44492	−0.505584
$280$	−2.57558	−0.153920
$281$	32.1346	1.91699	0.958494	−	0.285113i	$-0.0920311\pi$
$281$	32.1346	1.91699	0.958494	+	0.285113i	$0.0920311\pi$
$282$	3.00843	0.179149
$283$	−1.36066	−0.0808826	−0.0404413	−	0.999182i	$-0.512876\pi$
$283$	−1.36066	−0.0808826	−0.0404413	+	0.999182i	$0.512876\pi$
$284$	6.64580	0.394355
$285$	15.9765	0.946366
$286$	10.4279	0.616615
$287$	1.00450	0.0592937
$288$	−1.66414	−0.0980605
$289$	1.39224	0.0818964
$290$	−9.80902	−0.576005
$291$	34.0626	1.99678
$292$	−5.80574	−0.339755
$293$	9.02286	0.527121	0.263561	−	0.964643i	$-0.415103\pi$
$293$	9.02286	0.527121	0.263561	+	0.964643i	$0.415103\pi$
$294$	−10.6017	−0.618304
$295$	6.19194	0.360508
$296$	4.34885	0.252772
$297$	17.8368	1.03499
$298$	−10.6182	−0.615093
$299$	−12.6108	−0.729299
$300$	3.94701	0.227881
$301$	−2.99584	−0.172677
$302$	−17.7275	−1.02010
$303$	35.9318	2.06423
$304$	−4.15338	−0.238213
$305$	−20.1314	−1.15272
$306$	7.13687	0.407988
$307$	−23.1230	−1.31970	−0.659849	−	0.751398i	$-0.729380\pi$
$307$	−23.1230	−1.31970	−0.659849	+	0.751398i	$0.729380\pi$
$308$	8.94027	0.509419
$309$	33.0380	1.87947
$310$	9.03855	0.513355
$311$	6.33887	0.359445	0.179722	−	0.983717i	$-0.442480\pi$
$311$	6.33887	0.359445	0.179722	+	0.983717i	$0.442480\pi$
$312$	−3.64261	−0.206222
$313$	−31.5895	−1.78554	−0.892771	−	0.450511i	$-0.851242\pi$
$313$	−31.5895	−1.78554	−0.892771	+	0.450511i	$0.851242\pi$
$314$	−14.4824	−0.817289
$315$	4.28612	0.241496
$316$	−15.0772	−0.848157
$317$	2.48373	0.139500	0.0697500	−	0.997564i	$-0.477780\pi$
$317$	2.48373	0.139500	0.0697500	+	0.997564i	$0.477780\pi$
$318$	5.97031	0.334798
$319$	34.0488	1.90637
$320$	1.78112	0.0995677
$321$	−7.46595	−0.416709
$322$	−10.8117	−0.602514
$323$	17.8123	0.991101
$324$	−11.2231	−0.623503
$325$	3.08255	0.170989
$326$	−0.695189	−0.0385030
$327$	−19.8342	−1.09683
$328$	−0.694655	−0.0383559
$329$	2.01435	0.111055
$330$	23.7820	1.30916
$331$	29.2976	1.61034	0.805171	−	0.593043i	$-0.202074\pi$
$331$	29.2976	1.61034	0.805171	+	0.593043i	$0.202074\pi$
$332$	−10.1392	−0.556459
$333$	−7.23711	−0.396591
$334$	8.61594	0.471443
$335$	−21.4980	−1.17456
$336$	−3.12296	−0.170371
$337$	29.8909	1.62826	0.814129	−	0.580683i	$-0.197215\pi$
$337$	29.8909	1.62826	0.814129	+	0.580683i	$0.197215\pi$
$338$	10.1552	0.552369
$339$	9.94455	0.540114
$340$	−7.63855	−0.414258
$341$	−31.3744	−1.69902
$342$	6.91182	0.373748
$343$	−17.2209	−0.929839
$344$	2.07175	0.111701
$345$	−28.7603	−1.54840
$346$	20.7867	1.11750
$347$	−24.4217	−1.31103	−0.655513	−	0.755184i	$-0.727547\pi$
$347$	−24.4217	−1.31103	−0.655513	+	0.755184i	$0.727547\pi$
$348$	−11.8937	−0.637570
$349$	19.0967	1.02222	0.511112	−	0.859514i	$-0.329234\pi$
$349$	19.0967	1.02222	0.511112	+	0.859514i	$0.329234\pi$
$350$	2.64279	0.141263
$351$	−4.86601	−0.259729
$352$	−6.18258	−0.329533
$353$	29.3239	1.56075	0.780377	−	0.625310i	$-0.215027\pi$
$353$	29.3239	1.56075	0.780377	+	0.625310i	$0.215027\pi$
$354$	7.50791	0.399041
$355$	11.8370	0.628241
$356$	13.1908	0.699110
$357$	13.3932	0.708843
$358$	−19.3778	−1.02415
$359$	2.29972	0.121375	0.0606874	−	0.998157i	$-0.480671\pi$
$359$	2.29972	0.121375	0.0606874	+	0.998157i	$0.480671\pi$
$360$	−2.96404	−0.156219
$361$	−1.74941	−0.0920745
$362$	8.50245	0.446879
$363$	−58.7953	−3.08595
$364$	−2.43898	−0.127837
$365$	−10.3407	−0.541259
$366$	−24.4099	−1.27593
$367$	−8.29895	−0.433202	−0.216601	−	0.976260i	$-0.569497\pi$
$367$	−8.29895	−0.433202	−0.216601	+	0.976260i	$0.569497\pi$
$368$	7.47677	0.389754
$369$	1.15600	0.0601792
$370$	7.74584	0.402687
$371$	3.99753	0.207541
$372$	10.9595	0.568224
$373$	22.0345	1.14090	0.570452	−	0.821331i	$-0.306768\pi$
$373$	22.0345	1.14090	0.570452	+	0.821331i	$0.306768\pi$
$374$	26.5147	1.37104
$375$	26.2632	1.35623
$376$	−1.39301	−0.0718389
$377$	−9.28879	−0.478397
$378$	−4.17183	−0.214576
$379$	−21.6985	−1.11458	−0.557288	−	0.830319i	$-0.688158\pi$
$379$	−21.6985	−1.11458	−0.557288	+	0.830319i	$0.688158\pi$
$380$	−7.39768	−0.379493
$381$	−33.3002	−1.70602
$382$	1.53432	0.0785026
$383$	1.06925	0.0546361	0.0273181	−	0.999627i	$-0.491303\pi$
$383$	1.06925	0.0546361	0.0273181	+	0.999627i	$0.491303\pi$
$384$	2.15966	0.110210
$385$	15.9237	0.811547
$386$	−6.81269	−0.346757
$387$	−3.44769	−0.175256
$388$	−15.7722	−0.800711
$389$	−17.4658	−0.885551	−0.442776	−	0.896632i	$-0.646006\pi$
$389$	−17.4658	−0.885551	−0.442776	+	0.896632i	$0.646006\pi$
$390$	−6.48794	−0.328530
$391$	−32.0650	−1.62160
$392$	4.90896	0.247940
$393$	36.5171	1.84204
$394$	10.6478	0.536429
$395$	−26.8543	−1.35118
$396$	10.2887	0.517026
$397$	27.9209	1.40131	0.700655	−	0.713501i	$-0.252891\pi$
$397$	27.9209	1.40131	0.700655	+	0.713501i	$0.252891\pi$
$398$	−13.3760	−0.670477
$399$	12.9709	0.649355
$400$	−1.82761	−0.0913803
$401$	13.3301	0.665671	0.332836	−	0.942985i	$-0.391995\pi$
$401$	13.3301	0.665671	0.332836	+	0.942985i	$0.391995\pi$
$402$	−26.0670	−1.30010
$403$	8.55919	0.426364
$404$	−16.6377	−0.827756
$405$	−19.9896	−0.993293
$406$	−7.96366	−0.395230
$407$	−26.8871	−1.33275
$408$	−9.26197	−0.458536
$409$	−24.1403	−1.19366	−0.596830	−	0.802367i	$-0.703574\pi$
$409$	−24.1403	−1.19366	−0.596830	+	0.802367i	$0.703574\pi$
$410$	−1.23726	−0.0611041
$411$	−7.73707	−0.381642
$412$	−15.2978	−0.753667
$413$	5.02705	0.247365
$414$	−12.4424	−0.611511
$415$	−18.0591	−0.886486
$416$	1.68666	0.0826953
$417$	17.4329	0.853695
$418$	25.6786	1.25598
$419$	−33.6063	−1.64177	−0.820887	−	0.571090i	$-0.806521\pi$
$419$	−33.6063	−1.64177	−0.820887	+	0.571090i	$0.806521\pi$
$420$	−5.56237	−0.271416
$421$	−17.3779	−0.846946	−0.423473	−	0.905909i	$-0.639189\pi$
$421$	−17.3779	−0.846946	−0.423473	+	0.905909i	$0.639189\pi$
$422$	20.4942	0.997642
$423$	2.31816	0.112713
$424$	−2.76446	−0.134254
$425$	7.83790	0.380194
$426$	14.3527	0.695389
$427$	−16.3441	−0.790947
$428$	3.45700	0.167100
$429$	22.5208	1.08731
$430$	3.69004	0.177950
$431$	3.41427	0.164459	0.0822297	−	0.996613i	$-0.473796\pi$
$431$	3.41427	0.164459	0.0822297	+	0.996613i	$0.473796\pi$
$432$	2.88500	0.138805
$433$	−35.1727	−1.69029	−0.845147	−	0.534535i	$-0.820487\pi$
$433$	−35.1727	−1.69029	−0.845147	+	0.534535i	$0.820487\pi$
$434$	7.33814	0.352242
$435$	−21.1842	−1.01570
$436$	9.18393	0.439830
$437$	−31.0539	−1.48551
$438$	−12.5384	−0.599110
$439$	20.1234	0.960437	0.480218	−	0.877149i	$-0.340557\pi$
$439$	20.1234	0.960437	0.480218	+	0.877149i	$0.340557\pi$
$440$	−11.0119	−0.524973
$441$	−8.16922	−0.389010
$442$	−7.23344	−0.344059
$443$	−16.6001	−0.788693	−0.394346	−	0.918962i	$-0.629029\pi$
$443$	−16.6001	−0.788693	−0.394346	+	0.918962i	$0.629029\pi$
$444$	9.39206	0.445727
$445$	23.4944	1.11374
$446$	−18.6198	−0.881671
$447$	−22.9316	−1.08463
$448$	1.44604	0.0683190
$449$	18.7511	0.884920	0.442460	−	0.896788i	$-0.354106\pi$
$449$	18.7511	0.884920	0.442460	+	0.896788i	$0.354106\pi$
$450$	3.04140	0.143373
$451$	4.29476	0.202232
$452$	−4.60468	−0.216586
$453$	−38.2855	−1.79881
$454$	−13.4849	−0.632879
$455$	−4.34412	−0.203655
$456$	−8.96990	−0.420054
$457$	−21.8443	−1.02183	−0.510916	−	0.859631i	$-0.670694\pi$
$457$	−21.8443	−1.02183	−0.510916	+	0.859631i	$0.670694\pi$
$458$	−26.1013	−1.21963
$459$	−12.3727	−0.577507
$460$	13.3170	0.620910
$461$	12.1972	0.568079	0.284040	−	0.958813i	$-0.408325\pi$
$461$	12.1972	0.568079	0.284040	+	0.958813i	$0.408325\pi$
$462$	19.3080	0.898287
$463$	25.1165	1.16726	0.583632	−	0.812018i	$-0.301631\pi$
$463$	25.1165	1.16726	0.583632	+	0.812018i	$0.301631\pi$
$464$	5.50721	0.255666
$465$	19.5202	0.905229
$466$	24.6980	1.14411
$467$	−7.59357	−0.351388	−0.175694	−	0.984445i	$-0.556217\pi$
$467$	−7.59357	−0.351388	−0.175694	+	0.984445i	$0.556217\pi$
$468$	−2.80684	−0.129746
$469$	−17.4536	−0.805934
$470$	−2.48112	−0.114445
$471$	−31.2771	−1.44117
$472$	−3.47643	−0.160015
$473$	−12.8088	−0.588948
$474$	−32.5616	−1.49560
$475$	7.59075	0.348287
$476$	−6.20152	−0.284246
$477$	4.60046	0.210641
$478$	25.2984	1.15712
$479$	18.2647	0.834535	0.417267	−	0.908784i	$-0.362988\pi$
$479$	18.2647	0.834535	0.417267	+	0.908784i	$0.362988\pi$
$480$	3.84662	0.175573
$481$	7.33504	0.334449
$482$	−8.91901	−0.406250
$483$	−23.3497	−1.06245
$484$	27.2243	1.23747
$485$	−28.0922	−1.27560
$486$	−15.5830	−0.706860
$487$	−34.5720	−1.56661	−0.783304	−	0.621639i	$-0.786467\pi$
$487$	−34.5720	−1.56661	−0.783304	+	0.621639i	$0.786467\pi$
$488$	11.3027	0.511647
$489$	−1.50137	−0.0678945
$490$	8.74346	0.394989
$491$	35.1210	1.58499	0.792495	−	0.609879i	$-0.208782\pi$
$491$	35.1210	1.58499	0.792495	+	0.609879i	$0.208782\pi$
$492$	−1.50022	−0.0676351
$493$	−23.6183	−1.06372
$494$	−7.00534	−0.315185
$495$	18.3254	0.823666
$496$	−5.07464	−0.227858
$497$	9.61009	0.431072
$498$	−21.8972	−0.981236
$499$	−15.5197	−0.694756	−0.347378	−	0.937725i	$-0.612928\pi$
$499$	−15.5197	−0.694756	−0.347378	+	0.937725i	$0.612928\pi$
$500$	−12.1608	−0.543847
$501$	18.6075	0.831323
$502$	−10.9496	−0.488705
$503$	29.5863	1.31919	0.659593	−	0.751623i	$-0.270729\pi$
$503$	29.5863	1.31919	0.659593	+	0.751623i	$0.270729\pi$
$504$	−2.40642	−0.107190
$505$	−29.6338	−1.31868
$506$	−46.2257	−2.05498
$507$	21.9318	0.974024
$508$	15.4192	0.684116
$509$	−12.9821	−0.575423	−0.287711	−	0.957717i	$-0.592894\pi$
$509$	−12.9821	−0.575423	−0.287711	+	0.957717i	$0.592894\pi$
$510$	−16.4967	−0.730486
$511$	−8.39534	−0.371388
$512$	−1.00000	−0.0441942
$513$	−11.9825	−0.529041
$514$	−14.2156	−0.627022
$515$	−27.2472	−1.20066
$516$	4.47428	0.196969
$517$	8.61238	0.378772
$518$	6.28862	0.276306
$519$	44.8922	1.97055
$520$	3.00415	0.131740
$521$	−28.0823	−1.23031	−0.615154	−	0.788407i	$-0.710906\pi$
$521$	−28.0823	−1.23031	−0.615154	+	0.788407i	$0.710906\pi$
$522$	−9.16479	−0.401132
$523$	−30.3208	−1.32583	−0.662917	−	0.748693i	$-0.730682\pi$
$523$	−30.3208	−1.32583	−0.662917	+	0.748693i	$0.730682\pi$
$524$	−16.9087	−0.738661
$525$	5.70754	0.249098
$526$	24.9830	1.08931
$527$	21.7632	0.948019
$528$	−13.3523	−0.581084
$529$	32.9021	1.43053
$530$	−4.92385	−0.213878
$531$	5.78527	0.251059
$532$	−6.00596	−0.260391
$533$	−1.17165	−0.0507496
$534$	28.4877	1.23278
$535$	6.15733	0.266205
$536$	12.0699	0.521342
$537$	−41.8496	−1.80594
$538$	−20.6747	−0.891349
$539$	−30.3501	−1.30727
$540$	5.13854	0.221127
$541$	−30.4172	−1.30774	−0.653868	−	0.756609i	$-0.726855\pi$
$541$	−30.4172	−1.30774	−0.653868	+	0.756609i	$0.726855\pi$
$542$	21.5763	0.926782
$543$	18.3624	0.788007
$544$	4.28862	0.183873
$545$	16.3577	0.700686
$546$	−5.26737	−0.225423
$547$	−18.0374	−0.771224	−0.385612	−	0.922661i	$-0.626010\pi$
$547$	−18.0374	−0.771224	−0.385612	+	0.922661i	$0.626010\pi$
$548$	3.58254	0.153038
$549$	−18.8092	−0.802758
$550$	11.2993	0.481805
$551$	−22.8736	−0.974447
$552$	16.1473	0.687275
$553$	−21.8022	−0.927124
$554$	12.9067	0.548353
$555$	16.7284	0.710081
$556$	−8.07207	−0.342332
$557$	20.1888	0.855428	0.427714	−	0.903914i	$-0.359319\pi$
$557$	20.1888	0.855428	0.427714	+	0.903914i	$0.359319\pi$
$558$	8.44492	0.357502
$559$	3.49434	0.147795
$560$	2.57558	0.108838
$561$	57.2628	2.41764
$562$	−32.1346	−1.35552
$563$	−36.8136	−1.55151	−0.775754	−	0.631035i	$-0.782630\pi$
$563$	−36.8136	−1.55151	−0.775754	+	0.631035i	$0.782630\pi$
$564$	−3.00843	−0.126678
$565$	−8.20149	−0.345039
$566$	1.36066	0.0571926
$567$	−16.2290	−0.681554
$568$	−6.64580	−0.278851
$569$	5.77836	0.242241	0.121121	−	0.992638i	$-0.461351\pi$
$569$	5.77836	0.242241	0.121121	+	0.992638i	$0.461351\pi$
$570$	−15.9765	−0.669182
$571$	−0.884043	−0.0369960	−0.0184980	−	0.999829i	$-0.505888\pi$
$571$	−0.884043	−0.0369960	−0.0184980	+	0.999829i	$0.505888\pi$
$572$	−10.4279	−0.436013
$573$	3.31361	0.138428
$574$	−1.00450	−0.0419270
$575$	−13.6646	−0.569853
$576$	1.66414	0.0693393
$577$	−37.3391	−1.55445	−0.777224	−	0.629224i	$-0.783373\pi$
$577$	−37.3391	−1.55445	−0.777224	+	0.629224i	$0.783373\pi$
$578$	−1.39224	−0.0579095
$579$	−14.7131	−0.611456
$580$	9.80902	0.407297
$581$	−14.6617	−0.608268
$582$	−34.0626	−1.41194
$583$	17.0915	0.707859
$584$	5.80574	0.240243
$585$	−4.99933	−0.206697
$586$	−9.02286	−0.372731
$587$	−14.7579	−0.609126	−0.304563	−	0.952492i	$-0.598510\pi$
$587$	−14.7579	−0.609126	−0.304563	+	0.952492i	$0.598510\pi$
$588$	10.6017	0.437207
$589$	21.0769	0.868459
$590$	−6.19194	−0.254918
$591$	22.9957	0.945917
$592$	−4.34885	−0.178737
$593$	9.17408	0.376734	0.188367	−	0.982099i	$-0.439681\pi$
$593$	9.17408	0.376734	0.188367	+	0.982099i	$0.439681\pi$
$594$	−17.8368	−0.731851
$595$	−11.0457	−0.452828
$596$	10.6182	0.434937
$597$	−28.8876	−1.18229
$598$	12.6108	0.515692
$599$	6.98536	0.285414	0.142707	−	0.989765i	$-0.454419\pi$
$599$	6.98536	0.285414	0.142707	+	0.989765i	$0.454419\pi$
$600$	−3.94701	−0.161136
$601$	4.94507	0.201714	0.100857	−	0.994901i	$-0.467842\pi$
$601$	4.94507	0.201714	0.100857	+	0.994901i	$0.467842\pi$
$602$	2.99584	0.122101
$603$	−20.0861	−0.817970
$604$	17.7275	0.721322
$605$	48.4898	1.97139
$606$	−35.9318	−1.45963
$607$	8.99469	0.365083	0.182542	−	0.983198i	$-0.441568\pi$
$607$	8.99469	0.365083	0.182542	+	0.983198i	$0.441568\pi$
$608$	4.15338	0.168442
$609$	−17.1988	−0.696931
$610$	20.1314	0.815096
$611$	−2.34953	−0.0950518
$612$	−7.13687	−0.288491
$613$	−11.9952	−0.484483	−0.242241	−	0.970216i	$-0.577883\pi$
$613$	−11.9952	−0.484483	−0.242241	+	0.970216i	$0.577883\pi$
$614$	23.1230	0.933168
$615$	−2.67207	−0.107748
$616$	−8.94027	−0.360214
$617$	9.24769	0.372298	0.186149	−	0.982522i	$-0.440399\pi$
$617$	9.24769	0.372298	0.186149	+	0.982522i	$0.440399\pi$
$618$	−33.0380	−1.32898
$619$	−22.2645	−0.894887	−0.447444	−	0.894312i	$-0.647665\pi$
$619$	−22.2645	−0.894887	−0.447444	+	0.894312i	$0.647665\pi$
$620$	−9.03855	−0.362997
$621$	21.5705	0.865594
$622$	−6.33887	−0.254166
$623$	19.0744	0.764201
$624$	3.64261	0.145821
$625$	−12.5218	−0.500873
$626$	31.5895	1.26257
$627$	55.4572	2.21475
$628$	14.4824	0.577911
$629$	18.6506	0.743647
$630$	−4.28612	−0.170763
$631$	9.21470	0.366831	0.183416	−	0.983035i	$-0.441285\pi$
$631$	9.21470	0.366831	0.183416	+	0.983035i	$0.441285\pi$
$632$	15.0772	0.599737
$633$	44.2605	1.75920
$634$	−2.48373	−0.0986415
$635$	27.4634	1.08985
$636$	−5.97031	−0.236738
$637$	8.27975	0.328056
$638$	−34.0488	−1.34800
$639$	11.0595	0.437509
$640$	−1.78112	−0.0704050
$641$	37.9658	1.49956	0.749780	−	0.661687i	$-0.230159\pi$
$641$	37.9658	1.49956	0.749780	+	0.661687i	$0.230159\pi$
$642$	7.46595	0.294657
$643$	−41.7093	−1.64486	−0.822428	−	0.568870i	$-0.807381\pi$
$643$	−41.7093	−1.64486	−0.822428	+	0.568870i	$0.807381\pi$
$644$	10.8117	0.426041
$645$	7.96924	0.313789
$646$	−17.8123	−0.700814
$647$	34.7237	1.36513	0.682565	−	0.730825i	$-0.260865\pi$
$647$	34.7237	1.36513	0.682565	+	0.730825i	$0.260865\pi$
$648$	11.2231	0.440883
$649$	21.4933	0.843685
$650$	−3.08255	−0.120908
$651$	15.8479	0.621128
$652$	0.695189	0.0272257
$653$	−12.3344	−0.482683	−0.241341	−	0.970440i	$-0.577587\pi$
$653$	−12.3344	−0.482683	−0.241341	+	0.970440i	$0.577587\pi$
$654$	19.8342	0.775578
$655$	−30.1165	−1.17675
$656$	0.694655	0.0271217
$657$	−9.66158	−0.376934
$658$	−2.01435	−0.0785274
$659$	−18.7714	−0.731229	−0.365615	−	0.930766i	$-0.619141\pi$
$659$	−18.7714	−0.731229	−0.365615	+	0.930766i	$0.619141\pi$
$660$	−23.7820	−0.925715
$661$	−35.9093	−1.39671	−0.698356	−	0.715751i	$-0.746085\pi$
$661$	−35.9093	−1.39671	−0.698356	+	0.715751i	$0.746085\pi$
$662$	−29.2976	−1.13868
$663$	−15.6218	−0.606700
$664$	10.1392	0.393476
$665$	−10.6973	−0.414825
$666$	7.23711	0.280432
$667$	41.1762	1.59435
$668$	−8.61594	−0.333361
$669$	−40.2124	−1.55470
$670$	21.4980	0.830542
$671$	−69.8796	−2.69767
$672$	3.12296	0.120471
$673$	−1.67091	−0.0644088	−0.0322044	−	0.999481i	$-0.510253\pi$
$673$	−1.67091	−0.0644088	−0.0322044	+	0.999481i	$0.510253\pi$
$674$	−29.8909	−1.15135
$675$	−5.27265	−0.202944
$676$	−10.1552	−0.390584
$677$	−1.04571	−0.0401897	−0.0200949	−	0.999798i	$-0.506397\pi$
$677$	−1.04571	−0.0401897	−0.0200949	+	0.999798i	$0.506397\pi$
$678$	−9.94455	−0.381918
$679$	−22.8072	−0.875261
$680$	7.63855	0.292925
$681$	−29.1229	−1.11599
$682$	31.3744	1.20139
$683$	−20.4638	−0.783025	−0.391512	−	0.920173i	$-0.628048\pi$
$683$	−20.4638	−0.783025	−0.391512	+	0.920173i	$0.628048\pi$
$684$	−6.91182	−0.264280
$685$	6.38094	0.243803
$686$	17.2209	0.657495
$687$	−56.3700	−2.15065
$688$	−2.07175	−0.0789848
$689$	−4.66271	−0.177635
$690$	28.7603	1.09489
$691$	43.3866	1.65050	0.825252	−	0.564765i	$-0.191033\pi$
$691$	43.3866	1.65050	0.825252	+	0.564765i	$0.191033\pi$
$692$	−20.7867	−0.790190
$693$	14.8779	0.565164
$694$	24.4217	0.927035
$695$	−14.3773	−0.545363
$696$	11.8937	0.450830
$697$	−2.97911	−0.112842
$698$	−19.0967	−0.722821
$699$	53.3394	2.01748
$700$	−2.64279	−0.0998882
$701$	12.7085	0.479994	0.239997	−	0.970774i	$-0.422854\pi$
$701$	12.7085	0.479994	0.239997	+	0.970774i	$0.422854\pi$
$702$	4.86601	0.183656
$703$	18.0625	0.681238
$704$	6.18258	0.233015
$705$	−5.35837	−0.201808
$706$	−29.3239	−1.10362
$707$	−24.0588	−0.904824
$708$	−7.50791	−0.282164
$709$	52.0638	1.95530	0.977649	−	0.210243i	$-0.0674256\pi$
$709$	52.0638	1.95530	0.977649	+	0.210243i	$0.0674256\pi$
$710$	−11.8370	−0.444234
$711$	−25.0905	−0.940969
$712$	−13.1908	−0.494346
$713$	−37.9419	−1.42094
$714$	−13.3932	−0.501227
$715$	−18.5734	−0.694605
$716$	19.3778	0.724184
$717$	54.6360	2.04042
$718$	−2.29972	−0.0858249
$719$	47.1568	1.75865	0.879325	−	0.476221i	$-0.157994\pi$
$719$	47.1568	1.75865	0.879325	+	0.476221i	$0.157994\pi$
$720$	2.96404	0.110463
$721$	−22.1212	−0.823837
$722$	1.74941	0.0651065
$723$	−19.2621	−0.716364
$724$	−8.50245	−0.315991
$725$	−10.0650	−0.373805
$726$	58.7953	2.18210
$727$	−39.4809	−1.46427	−0.732134	−	0.681161i	$-0.761475\pi$
$727$	−39.4809	−1.46427	−0.732134	+	0.681161i	$0.761475\pi$
$728$	2.43898	0.0903946
$729$	0.0151246	0.000560170 0
$730$	10.3407	0.382728
$731$	8.88495	0.328622
$732$	24.4099	0.902216
$733$	9.65195	0.356503	0.178251	−	0.983985i	$-0.442956\pi$
$733$	9.65195	0.356503	0.178251	+	0.983985i	$0.442956\pi$
$734$	8.29895	0.306320
$735$	18.8829	0.696507
$736$	−7.47677	−0.275597
$737$	−74.6234	−2.74879
$738$	−1.15600	−0.0425531
$739$	−0.231272	−0.00850749	−0.00425374	−	0.999991i	$-0.501354\pi$
$739$	−0.231272	−0.00850749	−0.00425374	+	0.999991i	$0.501354\pi$
$740$	−7.74584	−0.284743
$741$	−15.1292	−0.555784
$742$	−3.99753	−0.146754
$743$	−31.1025	−1.14104	−0.570519	−	0.821284i	$-0.693258\pi$
$743$	−31.1025	−1.14104	−0.570519	+	0.821284i	$0.693258\pi$
$744$	−10.9595	−0.401795
$745$	18.9122	0.692891
$746$	−22.0345	−0.806741
$747$	−16.8730	−0.617352
$748$	−26.5147	−0.969474
$749$	4.99896	0.182658
$750$	−26.2632	−0.958997
$751$	−32.7668	−1.19568	−0.597839	−	0.801616i	$-0.703974\pi$
$751$	−32.7668	−1.19568	−0.597839	+	0.801616i	$0.703974\pi$
$752$	1.39301	0.0507978
$753$	−23.6475	−0.861762
$754$	9.28879	0.338278
$755$	31.5749	1.14913
$756$	4.17183	0.151728
$757$	−0.237080	−0.00861681	−0.00430841	−	0.999991i	$-0.501371\pi$
$757$	−0.237080	−0.00861681	−0.00430841	+	0.999991i	$0.501371\pi$
$758$	21.6985	0.788124
$759$	−99.8320	−3.62367
$760$	7.39768	0.268342
$761$	−0.997083	−0.0361442	−0.0180721	−	0.999837i	$-0.505753\pi$
$761$	−0.997083	−0.0361442	−0.0180721	+	0.999837i	$0.505753\pi$
$762$	33.3002	1.20634
$763$	13.2803	0.480781
$764$	−1.53432	−0.0555097
$765$	−12.7116	−0.459590
$766$	−1.06925	−0.0386336
$767$	−5.86355	−0.211720
$768$	−2.15966	−0.0779301
$769$	30.6281	1.10448	0.552240	−	0.833685i	$-0.313773\pi$
$769$	30.6281	1.10448	0.552240	+	0.833685i	$0.313773\pi$
$770$	−15.9237	−0.573850
$771$	−30.7008	−1.10566
$772$	6.81269	0.245194
$773$	17.6553	0.635017	0.317509	−	0.948255i	$-0.397154\pi$
$773$	17.6553	0.635017	0.317509	+	0.948255i	$0.397154\pi$
$774$	3.44769	0.123925
$775$	9.27444	0.333148
$776$	15.7722	0.566188
$777$	13.5813	0.487227
$778$	17.4658	0.626179
$779$	−2.88517	−0.103372
$780$	6.48794	0.232306
$781$	41.0882	1.47025
$782$	32.0650	1.14664
$783$	15.8883	0.567802
$784$	−4.90896	−0.175320
$785$	25.7949	0.920660
$786$	−36.5171	−1.30252
$787$	19.0387	0.678656	0.339328	−	0.940668i	$-0.389800\pi$
$787$	19.0387	0.678656	0.339328	+	0.940668i	$0.389800\pi$
$788$	−10.6478	−0.379313
$789$	53.9549	1.92084
$790$	26.8543	0.955432
$791$	−6.65855	−0.236751
$792$	−10.2887	−0.365593
$793$	19.0637	0.676973
$794$	−27.9209	−0.990875
$795$	−10.6338	−0.377144
$796$	13.3760	0.474099
$797$	−11.4025	−0.403898	−0.201949	−	0.979396i	$-0.564728\pi$
$797$	−11.4025	−0.403898	−0.201949	+	0.979396i	$0.564728\pi$
$798$	−12.9709	−0.459163
$799$	−5.97408	−0.211348
$800$	1.82761	0.0646156
$801$	21.9514	0.775613
$802$	−13.3301	−0.470701
$803$	−35.8945	−1.26669
$804$	26.0670	0.919313
$805$	19.2570	0.678719
$806$	−8.55919	−0.301485
$807$	−44.6504	−1.57177
$808$	16.6377	0.585312
$809$	−11.6943	−0.411150	−0.205575	−	0.978641i	$-0.565906\pi$
$809$	−11.6943	−0.411150	−0.205575	+	0.978641i	$0.565906\pi$
$810$	19.9896	0.702364
$811$	−6.50290	−0.228348	−0.114174	−	0.993461i	$-0.536422\pi$
$811$	−6.50290	−0.228348	−0.114174	+	0.993461i	$0.536422\pi$
$812$	7.96366	0.279470
$813$	46.5976	1.63425
$814$	26.8871	0.942394
$815$	1.23822	0.0433728
$816$	9.26197	0.324234
$817$	8.60477	0.301043
$818$	24.1403	0.844046
$819$	−4.05881	−0.141826
$820$	1.23726	0.0432071
$821$	−26.1685	−0.913287	−0.456644	−	0.889650i	$-0.650949\pi$
$821$	−26.1685	−0.913287	−0.456644	+	0.889650i	$0.650949\pi$
$822$	7.73707	0.269861
$823$	−26.6677	−0.929579	−0.464789	−	0.885421i	$-0.653870\pi$
$823$	−26.6677	−0.929579	−0.464789	+	0.885421i	$0.653870\pi$
$824$	15.2978	0.532923
$825$	24.4027	0.849594
$826$	−5.02705	−0.174914
$827$	−16.8505	−0.585951	−0.292975	−	0.956120i	$-0.594645\pi$
$827$	−16.8505	−0.585951	−0.292975	+	0.956120i	$0.594645\pi$
$828$	12.4424	0.432404
$829$	41.6329	1.44597	0.722985	−	0.690864i	$-0.242770\pi$
$829$	41.6329	1.44597	0.722985	+	0.690864i	$0.242770\pi$
$830$	18.0591	0.626840
$831$	27.8741	0.966942
$832$	−1.68666	−0.0584744
$833$	21.0527	0.729432
$834$	−17.4329	−0.603653
$835$	−15.3460	−0.531072
$836$	−25.6786	−0.888114
$837$	−14.6403	−0.506044
$838$	33.6063	1.16091
$839$	−35.3287	−1.21968	−0.609841	−	0.792524i	$-0.708767\pi$
$839$	−35.3287	−1.21968	−0.609841	+	0.792524i	$0.708767\pi$
$840$	5.56237	0.191920
$841$	1.32940	0.0458415
$842$	17.3779	0.598881
$843$	−69.3998	−2.39026
$844$	−20.4942	−0.705439
$845$	−18.0876	−0.622233
$846$	−2.31816	−0.0797001
$847$	39.3675	1.35268
$848$	2.76446	0.0949321
$849$	2.93856	0.100851
$850$	−7.83790	−0.268838
$851$	−32.5154	−1.11461
$852$	−14.3527	−0.491715
$853$	−16.6697	−0.570760	−0.285380	−	0.958415i	$-0.592120\pi$
$853$	−16.6697	−0.570760	−0.285380	+	0.958415i	$0.592120\pi$
$854$	16.3441	0.559284
$855$	−12.3108	−0.421020
$856$	−3.45700	−0.118158
$857$	−27.0505	−0.924028	−0.462014	−	0.886873i	$-0.652873\pi$
$857$	−27.0505	−0.924028	−0.462014	+	0.886873i	$0.652873\pi$
$858$	−22.5208	−0.768846
$859$	6.20652	0.211764	0.105882	−	0.994379i	$-0.466233\pi$
$859$	6.20652	0.211764	0.105882	+	0.994379i	$0.466233\pi$
$860$	−3.69004	−0.125829
$861$	−2.16938	−0.0739323
$862$	−3.41427	−0.116290
$863$	14.8010	0.503831	0.251915	−	0.967749i	$-0.418940\pi$
$863$	14.8010	0.503831	0.251915	+	0.967749i	$0.418940\pi$
$864$	−2.88500	−0.0981497
$865$	−37.0236	−1.25884
$866$	35.1727	1.19522
$867$	−3.00677	−0.102115
$868$	−7.33814	−0.249073
$869$	−93.2158	−3.16213
$870$	21.1842	0.718210
$871$	20.3579	0.689801
$872$	−9.18393	−0.311007
$873$	−26.2472	−0.888332
$874$	31.0539	1.05041
$875$	−17.5850	−0.594482
$876$	12.5384	0.423635
$877$	5.06409	0.171002	0.0855011	−	0.996338i	$-0.472751\pi$
$877$	5.06409	0.171002	0.0855011	+	0.996338i	$0.472751\pi$
$878$	−20.1234	−0.679131
$879$	−19.4863	−0.657258
$880$	11.0119	0.371212
$881$	−41.0760	−1.38388	−0.691942	−	0.721953i	$-0.743245\pi$
$881$	−41.0760	−1.38388	−0.691942	+	0.721953i	$0.743245\pi$
$882$	8.16922	0.275072
$883$	−18.4972	−0.622481	−0.311240	−	0.950331i	$-0.600744\pi$
$883$	−18.4972	−0.622481	−0.311240	+	0.950331i	$0.600744\pi$
$884$	7.23344	0.243287
$885$	−13.3725	−0.449511
$886$	16.6001	0.557690
$887$	−37.2305	−1.25008	−0.625039	−	0.780594i	$-0.714917\pi$
$887$	−37.2305	−1.25008	−0.625039	+	0.780594i	$0.714917\pi$
$888$	−9.39206	−0.315177
$889$	22.2968	0.747810
$890$	−23.4944	−0.787534
$891$	−69.3875	−2.32457
$892$	18.6198	0.623436
$893$	−5.78569	−0.193611
$894$	22.9316	0.766949
$895$	34.5143	1.15368
$896$	−1.44604	−0.0483088
$897$	27.2350	0.909350
$898$	−18.7511	−0.625733
$899$	−27.9471	−0.932089
$900$	−3.04140	−0.101380
$901$	−11.8557	−0.394972
$902$	−4.29476	−0.143000
$903$	6.47000	0.215308
$904$	4.60468	0.153149
$905$	−15.1439	−0.503400
$906$	38.2855	1.27195
$907$	17.0062	0.564680	0.282340	−	0.959314i	$-0.408889\pi$
$907$	17.0062	0.564680	0.282340	+	0.959314i	$0.408889\pi$
$908$	13.4849	0.447513
$909$	−27.6875	−0.918336
$910$	4.34412	0.144006
$911$	20.7588	0.687770	0.343885	−	0.939012i	$-0.388257\pi$
$911$	20.7588	0.687770	0.343885	+	0.939012i	$0.388257\pi$
$912$	8.96990	0.297023
$913$	−62.6862	−2.07461
$914$	21.8443	0.722544
$915$	43.4770	1.43731
$916$	26.1013	0.862411
$917$	−24.4507	−0.807433
$918$	12.3727	0.408359
$919$	−45.5199	−1.50156	−0.750782	−	0.660550i	$-0.770323\pi$
$919$	−45.5199	−1.50156	−0.750782	+	0.660550i	$0.770323\pi$
$920$	−13.3170	−0.439050
$921$	49.9378	1.64551
$922$	−12.1972	−0.401693
$923$	−11.2092	−0.368955
$924$	−19.3080	−0.635185
$925$	7.94799	0.261328
$926$	−25.1165	−0.825380
$927$	−25.4577	−0.836140
$928$	−5.50721	−0.180783
$929$	−31.9294	−1.04757	−0.523786	−	0.851850i	$-0.675481\pi$
$929$	−31.9294	−1.04757	−0.523786	+	0.851850i	$0.675481\pi$
$930$	−19.5202	−0.640093
$931$	20.3888	0.668216
$932$	−24.6980	−0.809011
$933$	−13.6898	−0.448185
$934$	7.59357	0.248469
$935$	−47.2259	−1.54445
$936$	2.80684	0.0917445
$937$	−45.3932	−1.48293	−0.741465	−	0.670991i	$-0.765869\pi$
$937$	−45.3932	−1.48293	−0.741465	+	0.670991i	$0.765869\pi$
$938$	17.4536	0.569882
$939$	68.2226	2.22636
$940$	2.48112	0.0809251
$941$	−32.0190	−1.04379	−0.521894	−	0.853010i	$-0.674774\pi$
$941$	−32.0190	−1.04379	−0.521894	+	0.853010i	$0.674774\pi$
$942$	31.2771	1.01906
$943$	5.19377	0.169133
$944$	3.47643	0.113148
$945$	7.43054	0.241715
$946$	12.8088	0.416449
$947$	20.4282	0.663827	0.331914	−	0.943310i	$-0.392306\pi$
$947$	20.4282	0.663827	0.331914	+	0.943310i	$0.392306\pi$
$948$	32.5616	1.05755
$949$	9.79231	0.317872
$950$	−7.59075	−0.246276
$951$	−5.36402	−0.173940
$952$	6.20152	0.200992
$953$	−20.1209	−0.651781	−0.325891	−	0.945407i	$-0.605664\pi$
$953$	−20.1209	−0.651781	−0.325891	+	0.945407i	$0.605664\pi$
$954$	−4.60046	−0.148945
$955$	−2.73281	−0.0884316
$956$	−25.2984	−0.818208
$957$	−73.5339	−2.37701
$958$	−18.2647	−0.590105
$959$	5.18050	0.167287
$960$	−3.84662	−0.124149
$961$	−5.24803	−0.169291
$962$	−7.33504	−0.236491
$963$	5.75294	0.185386
$964$	8.91901	0.287262
$965$	12.1342	0.390615
$966$	23.3497	0.751263
$967$	−16.2995	−0.524157	−0.262078	−	0.965047i	$-0.584408\pi$
$967$	−16.2995	−0.524157	−0.262078	+	0.965047i	$0.584408\pi$
$968$	−27.2243	−0.875022
$969$	−38.4685	−1.23579
$970$	28.0922	0.901985
$971$	21.2158	0.680846	0.340423	−	0.940272i	$-0.389430\pi$
$971$	21.2158	0.680846	0.340423	+	0.940272i	$0.389430\pi$
$972$	15.5830	0.499825
$973$	−11.6725	−0.374204
$974$	34.5720	1.10776
$975$	−6.65727	−0.213203
$976$	−11.3027	−0.361789
$977$	−13.3862	−0.428262	−0.214131	−	0.976805i	$-0.568692\pi$
$977$	−13.3862	−0.428262	−0.214131	+	0.976805i	$0.568692\pi$
$978$	1.50137	0.0480086
$979$	81.5531	2.60645
$980$	−8.74346	−0.279300
$981$	15.2834	0.487960
$982$	−35.1210	−1.12076
$983$	−29.3750	−0.936916	−0.468458	−	0.883486i	$-0.655190\pi$
$983$	−29.3750	−0.936916	−0.468458	+	0.883486i	$0.655190\pi$
$984$	1.50022	0.0478253
$985$	−18.9651	−0.604277
$986$	23.6183	0.752161
$987$	−4.35031	−0.138472
$988$	7.00534	0.222870
$989$	−15.4900	−0.492553
$990$	−18.3254	−0.582420
$991$	−27.2433	−0.865412	−0.432706	−	0.901535i	$-0.642441\pi$
$991$	−27.2433	−0.865412	−0.432706	+	0.901535i	$0.642441\pi$
$992$	5.07464	0.161120
$993$	−63.2729	−2.00791
$994$	−9.61009	−0.304814
$995$	23.8242	0.755279
$996$	21.8972	0.693839
$997$	−10.2583	−0.324883	−0.162441	−	0.986718i	$-0.551937\pi$
$997$	−10.2583	−0.324883	−0.162441	+	0.986718i	$0.551937\pi$
$998$	15.5197	0.491266
$999$	−12.5465	−0.396952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.19	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.19	✓	89	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.15966 q^{3} +1.00000 q^{4} +1.78112 q^{5} +2.15966 q^{6} +1.44604 q^{7} -1.00000 q^{8} +1.66414 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.15966 q^{3} +1.00000 q^{4} +1.78112 q^{5} +2.15966 q^{6} +1.44604 q^{7} -1.00000 q^{8} +1.66414 q^{9} -1.78112 q^{10} +6.18258 q^{11} -2.15966 q^{12} -1.68666 q^{13} -1.44604 q^{14} -3.84662 q^{15} +1.00000 q^{16} -4.28862 q^{17} -1.66414 q^{18} -4.15338 q^{19} +1.78112 q^{20} -3.12296 q^{21} -6.18258 q^{22} +7.47677 q^{23} +2.15966 q^{24} -1.82761 q^{25} +1.68666 q^{26} +2.88500 q^{27} +1.44604 q^{28} +5.50721 q^{29} +3.84662 q^{30} -5.07464 q^{31} -1.00000 q^{32} -13.3523 q^{33} +4.28862 q^{34} +2.57558 q^{35} +1.66414 q^{36} -4.34885 q^{37} +4.15338 q^{38} +3.64261 q^{39} -1.78112 q^{40} +0.694655 q^{41} +3.12296 q^{42} -2.07175 q^{43} +6.18258 q^{44} +2.96404 q^{45} -7.47677 q^{46} +1.39301 q^{47} -2.15966 q^{48} -4.90896 q^{49} +1.82761 q^{50} +9.26197 q^{51} -1.68666 q^{52} +2.76446 q^{53} -2.88500 q^{54} +11.0119 q^{55} -1.44604 q^{56} +8.96990 q^{57} -5.50721 q^{58} +3.47643 q^{59} -3.84662 q^{60} -11.3027 q^{61} +5.07464 q^{62} +2.40642 q^{63} +1.00000 q^{64} -3.00415 q^{65} +13.3523 q^{66} -12.0699 q^{67} -4.28862 q^{68} -16.1473 q^{69} -2.57558 q^{70} +6.64580 q^{71} -1.66414 q^{72} -5.80574 q^{73} +4.34885 q^{74} +3.94701 q^{75} -4.15338 q^{76} +8.94027 q^{77} -3.64261 q^{78} -15.0772 q^{79} +1.78112 q^{80} -11.2231 q^{81} -0.694655 q^{82} -10.1392 q^{83} -3.12296 q^{84} -7.63855 q^{85} +2.07175 q^{86} -11.8937 q^{87} -6.18258 q^{88} +13.1908 q^{89} -2.96404 q^{90} -2.43898 q^{91} +7.47677 q^{92} +10.9595 q^{93} -1.39301 q^{94} -7.39768 q^{95} +2.15966 q^{96} -15.7722 q^{97} +4.90896 q^{98} +10.2887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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