LMFDB - Embedded newform 8002.2.a.f.1.11

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			1.11
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.78190 q^{3} +1.00000 q^{4} +1.30300 q^{5} +2.78190 q^{6} +4.15781 q^{7} -1.00000 q^{8} +4.73899 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.78190 q^{3} +1.00000 q^{4} +1.30300 q^{5} +2.78190 q^{6} +4.15781 q^{7} -1.00000 q^{8} +4.73899 q^{9} -1.30300 q^{10} -4.57523 q^{11} -2.78190 q^{12} -5.94146 q^{13} -4.15781 q^{14} -3.62482 q^{15} +1.00000 q^{16} -7.78721 q^{17} -4.73899 q^{18} +6.95913 q^{19} +1.30300 q^{20} -11.5666 q^{21} +4.57523 q^{22} -0.662571 q^{23} +2.78190 q^{24} -3.30219 q^{25} +5.94146 q^{26} -4.83771 q^{27} +4.15781 q^{28} +6.26025 q^{29} +3.62482 q^{30} -5.36113 q^{31} -1.00000 q^{32} +12.7278 q^{33} +7.78721 q^{34} +5.41764 q^{35} +4.73899 q^{36} +1.72452 q^{37} -6.95913 q^{38} +16.5286 q^{39} -1.30300 q^{40} +8.22216 q^{41} +11.5666 q^{42} +10.3059 q^{43} -4.57523 q^{44} +6.17491 q^{45} +0.662571 q^{46} +6.74493 q^{47} -2.78190 q^{48} +10.2874 q^{49} +3.30219 q^{50} +21.6633 q^{51} -5.94146 q^{52} -2.23194 q^{53} +4.83771 q^{54} -5.96153 q^{55} -4.15781 q^{56} -19.3596 q^{57} -6.26025 q^{58} +0.421952 q^{59} -3.62482 q^{60} -1.06149 q^{61} +5.36113 q^{62} +19.7038 q^{63} +1.00000 q^{64} -7.74173 q^{65} -12.7278 q^{66} -10.0857 q^{67} -7.78721 q^{68} +1.84321 q^{69} -5.41764 q^{70} +13.3355 q^{71} -4.73899 q^{72} +3.36187 q^{73} -1.72452 q^{74} +9.18637 q^{75} +6.95913 q^{76} -19.0229 q^{77} -16.5286 q^{78} -9.16363 q^{79} +1.30300 q^{80} -0.758940 q^{81} -8.22216 q^{82} -15.0781 q^{83} -11.5666 q^{84} -10.1467 q^{85} -10.3059 q^{86} -17.4154 q^{87} +4.57523 q^{88} -7.98806 q^{89} -6.17491 q^{90} -24.7035 q^{91} -0.662571 q^{92} +14.9141 q^{93} -6.74493 q^{94} +9.06776 q^{95} +2.78190 q^{96} +11.2352 q^{97} -10.2874 q^{98} -21.6820 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.78190	−1.60613	−0.803067	−	0.595889i	$-0.796800\pi$
$3$	−2.78190	−1.60613	−0.803067	+	0.595889i	$0.796800\pi$
$4$	1.00000	0.500000
$5$	1.30300	0.582720	0.291360	−	0.956614i	$-0.405892\pi$
$5$	1.30300	0.582720	0.291360	+	0.956614i	$0.405892\pi$
$6$	2.78190	1.13571
$7$	4.15781	1.57151	0.785753	−	0.618541i	$-0.212276\pi$
$7$	4.15781	1.57151	0.785753	+	0.618541i	$0.212276\pi$
$8$	−1.00000	−0.353553
$9$	4.73899	1.57966
$10$	−1.30300	−0.412045
$11$	−4.57523	−1.37948	−0.689742	−	0.724055i	$-0.742276\pi$
$11$	−4.57523	−1.37948	−0.689742	+	0.724055i	$0.742276\pi$
$12$	−2.78190	−0.803067
$13$	−5.94146	−1.64786	−0.823932	−	0.566688i	$-0.808224\pi$
$13$	−5.94146	−1.64786	−0.823932	+	0.566688i	$0.808224\pi$
$14$	−4.15781	−1.11122
$15$	−3.62482	−0.935926
$16$	1.00000	0.250000
$17$	−7.78721	−1.88868	−0.944338	−	0.328977i	$-0.893296\pi$
$17$	−7.78721	−1.88868	−0.944338	+	0.328977i	$0.893296\pi$
$18$	−4.73899	−1.11699
$19$	6.95913	1.59653	0.798267	−	0.602304i	$-0.205750\pi$
$19$	6.95913	1.59653	0.798267	+	0.602304i	$0.205750\pi$
$20$	1.30300	0.291360
$21$	−11.5666	−2.52405
$22$	4.57523	0.975442
$23$	−0.662571	−0.138156	−0.0690778	−	0.997611i	$-0.522006\pi$
$23$	−0.662571	−0.138156	−0.0690778	+	0.997611i	$0.522006\pi$
$24$	2.78190	0.567854
$25$	−3.30219	−0.660438
$26$	5.94146	1.16522
$27$	−4.83771	−0.931017
$28$	4.15781	0.785753
$29$	6.26025	1.16250	0.581250	−	0.813725i	$-0.302564\pi$
$29$	6.26025	1.16250	0.581250	+	0.813725i	$0.302564\pi$
$30$	3.62482	0.661799
$31$	−5.36113	−0.962887	−0.481443	−	0.876477i	$-0.659887\pi$
$31$	−5.36113	−0.962887	−0.481443	+	0.876477i	$0.659887\pi$
$32$	−1.00000	−0.176777
$33$	12.7278	2.21563
$34$	7.78721	1.33550
$35$	5.41764	0.915748
$36$	4.73899	0.789832
$37$	1.72452	0.283509	0.141754	−	0.989902i	$-0.454726\pi$
$37$	1.72452	0.283509	0.141754	+	0.989902i	$0.454726\pi$
$38$	−6.95913	−1.12892
$39$	16.5286	2.64669
$40$	−1.30300	−0.206023
$41$	8.22216	1.28409	0.642043	−	0.766669i	$-0.278087\pi$
$41$	8.22216	1.28409	0.642043	+	0.766669i	$0.278087\pi$
$42$	11.5666	1.78477
$43$	10.3059	1.57163	0.785817	−	0.618459i	$-0.212243\pi$
$43$	10.3059	1.57163	0.785817	+	0.618459i	$0.212243\pi$
$44$	−4.57523	−0.689742
$45$	6.17491	0.920501
$46$	0.662571	0.0976907
$47$	6.74493	0.983849	0.491924	−	0.870638i	$-0.336294\pi$
$47$	6.74493	0.983849	0.491924	+	0.870638i	$0.336294\pi$
$48$	−2.78190	−0.401533
$49$	10.2874	1.46963
$50$	3.30219	0.467000
$51$	21.6633	3.03347
$52$	−5.94146	−0.823932
$53$	−2.23194	−0.306580	−0.153290	−	0.988181i	$-0.548987\pi$
$53$	−2.23194	−0.306580	−0.153290	+	0.988181i	$0.548987\pi$
$54$	4.83771	0.658328
$55$	−5.96153	−0.803852
$56$	−4.15781	−0.555611
$57$	−19.3596	−2.56425
$58$	−6.26025	−0.822011
$59$	0.421952	0.0549335	0.0274667	−	0.999623i	$-0.491256\pi$
$59$	0.421952	0.0549335	0.0274667	+	0.999623i	$0.491256\pi$
$60$	−3.62482	−0.467963
$61$	−1.06149	−0.135910	−0.0679548	−	0.997688i	$-0.521647\pi$
$61$	−1.06149	−0.135910	−0.0679548	+	0.997688i	$0.521647\pi$
$62$	5.36113	0.680864
$63$	19.7038	2.48245
$64$	1.00000	0.125000
$65$	−7.74173	−0.960244
$66$	−12.7278	−1.56669
$67$	−10.0857	−1.23216	−0.616080	−	0.787684i	$-0.711280\pi$
$67$	−10.0857	−1.23216	−0.616080	+	0.787684i	$0.711280\pi$
$68$	−7.78721	−0.944338
$69$	1.84321	0.221896
$70$	−5.41764	−0.647531
$71$	13.3355	1.58263	0.791315	−	0.611409i	$-0.209397\pi$
$71$	13.3355	1.58263	0.791315	+	0.611409i	$0.209397\pi$
$72$	−4.73899	−0.558495
$73$	3.36187	0.393477	0.196738	−	0.980456i	$-0.436965\pi$
$73$	3.36187	0.393477	0.196738	+	0.980456i	$0.436965\pi$
$74$	−1.72452	−0.200471
$75$	9.18637	1.06075
$76$	6.95913	0.798267
$77$	−19.0229	−2.16787
$78$	−16.5286	−1.87149
$79$	−9.16363	−1.03099	−0.515494	−	0.856893i	$-0.672392\pi$
$79$	−9.16363	−1.03099	−0.515494	+	0.856893i	$0.672392\pi$
$80$	1.30300	0.145680
$81$	−0.758940	−0.0843266
$82$	−8.22216	−0.907985
$83$	−15.0781	−1.65504	−0.827519	−	0.561438i	$-0.810248\pi$
$83$	−15.0781	−1.65504	−0.827519	+	0.561438i	$0.810248\pi$
$84$	−11.5666	−1.26202
$85$	−10.1467	−1.10057
$86$	−10.3059	−1.11131
$87$	−17.4154	−1.86713
$88$	4.57523	0.487721
$89$	−7.98806	−0.846733	−0.423366	−	0.905959i	$-0.639152\pi$
$89$	−7.98806	−0.846733	−0.423366	+	0.905959i	$0.639152\pi$
$90$	−6.17491	−0.650893
$91$	−24.7035	−2.58963
$92$	−0.662571	−0.0690778
$93$	14.9141	1.54652
$94$	−6.74493	−0.695686
$95$	9.06776	0.930332
$96$	2.78190	0.283927
$97$	11.2352	1.14076	0.570381	−	0.821380i	$-0.306796\pi$
$97$	11.2352	1.14076	0.570381	+	0.821380i	$0.306796\pi$
$98$	−10.2874	−1.03919
$99$	−21.6820	−2.17912
$100$	−3.30219	−0.330219
$101$	2.95061	0.293596	0.146798	−	0.989166i	$-0.453103\pi$
$101$	2.95061	0.293596	0.146798	+	0.989166i	$0.453103\pi$
$102$	−21.6633	−2.14498
$103$	1.37882	0.135859	0.0679295	−	0.997690i	$-0.478361\pi$
$103$	1.37882	0.135859	0.0679295	+	0.997690i	$0.478361\pi$
$104$	5.94146	0.582608
$105$	−15.0713	−1.47081
$106$	2.23194	0.216785
$107$	10.5950	1.02426	0.512129	−	0.858909i	$-0.328857\pi$
$107$	10.5950	1.02426	0.512129	+	0.858909i	$0.328857\pi$
$108$	−4.83771	−0.465508
$109$	15.6549	1.49947	0.749733	−	0.661740i	$-0.230182\pi$
$109$	15.6549	1.49947	0.749733	+	0.661740i	$0.230182\pi$
$110$	5.96153	0.568409
$111$	−4.79744	−0.455352
$112$	4.15781	0.392876
$113$	−8.12771	−0.764590	−0.382295	−	0.924040i	$-0.624866\pi$
$113$	−8.12771	−0.764590	−0.382295	+	0.924040i	$0.624866\pi$
$114$	19.3596	1.81320
$115$	−0.863330	−0.0805060
$116$	6.26025	0.581250
$117$	−28.1565	−2.60307
$118$	−0.421952	−0.0388438
$119$	−32.3778	−2.96807
$120$	3.62482	0.330900
$121$	9.93272	0.902975
$122$	1.06149	0.0961026
$123$	−22.8733	−2.06241
$124$	−5.36113	−0.481443
$125$	−10.8178	−0.967570
$126$	−19.7038	−1.75536
$127$	−7.30114	−0.647871	−0.323936	−	0.946079i	$-0.605006\pi$
$127$	−7.30114	−0.647871	−0.323936	+	0.946079i	$0.605006\pi$
$128$	−1.00000	−0.0883883
$129$	−28.6700	−2.52425
$130$	7.74173	0.678995
$131$	17.7116	1.54747	0.773734	−	0.633511i	$-0.218387\pi$
$131$	17.7116	1.54747	0.773734	+	0.633511i	$0.218387\pi$
$132$	12.7278	1.10782
$133$	28.9348	2.50896
$134$	10.0857	0.871269
$135$	−6.30354	−0.542522
$136$	7.78721	0.667748
$137$	−9.77336	−0.834994	−0.417497	−	0.908678i	$-0.637093\pi$
$137$	−9.77336	−0.834994	−0.417497	+	0.908678i	$0.637093\pi$
$138$	−1.84321	−0.156904
$139$	10.3473	0.877644	0.438822	−	0.898574i	$-0.355396\pi$
$139$	10.3473	0.877644	0.438822	+	0.898574i	$0.355396\pi$
$140$	5.41764	0.457874
$141$	−18.7637	−1.58019
$142$	−13.3355	−1.11909
$143$	27.1835	2.27320
$144$	4.73899	0.394916
$145$	8.15711	0.677411
$146$	−3.36187	−0.278230
$147$	−28.6186	−2.36042
$148$	1.72452	0.141754
$149$	−8.31812	−0.681447	−0.340724	−	0.940164i	$-0.610672\pi$
$149$	−8.31812	−0.681447	−0.340724	+	0.940164i	$0.610672\pi$
$150$	−9.18637	−0.750064
$151$	6.05716	0.492925	0.246462	−	0.969152i	$-0.420732\pi$
$151$	6.05716	0.492925	0.246462	+	0.969152i	$0.420732\pi$
$152$	−6.95913	−0.564460
$153$	−36.9035	−2.98347
$154$	19.0229	1.53291
$155$	−6.98555	−0.561093
$156$	16.5286	1.32335
$157$	9.42635	0.752305	0.376152	−	0.926558i	$-0.377247\pi$
$157$	9.42635	0.752305	0.376152	+	0.926558i	$0.377247\pi$
$158$	9.16363	0.729019
$159$	6.20904	0.492409
$160$	−1.30300	−0.103011
$161$	−2.75485	−0.217112
$162$	0.758940	0.0596279
$163$	−13.0293	−1.02053	−0.510265	−	0.860017i	$-0.670453\pi$
$163$	−13.0293	−1.02053	−0.510265	+	0.860017i	$0.670453\pi$
$164$	8.22216	0.642043
$165$	16.5844	1.29109
$166$	15.0781	1.17029
$167$	−2.19217	−0.169635	−0.0848177	−	0.996396i	$-0.527031\pi$
$167$	−2.19217	−0.169635	−0.0848177	+	0.996396i	$0.527031\pi$
$168$	11.5666	0.892385
$169$	22.3010	1.71546
$170$	10.1467	0.778220
$171$	32.9793	2.52199
$172$	10.3059	0.785817
$173$	−11.4956	−0.873994	−0.436997	−	0.899463i	$-0.643958\pi$
$173$	−11.4956	−0.873994	−0.436997	+	0.899463i	$0.643958\pi$
$174$	17.4154	1.32026
$175$	−13.7299	−1.03788
$176$	−4.57523	−0.344871
$177$	−1.17383	−0.0882305
$178$	7.98806	0.598731
$179$	0.170590	0.0127505	0.00637524	−	0.999980i	$-0.497971\pi$
$179$	0.170590	0.0127505	0.00637524	+	0.999980i	$0.497971\pi$
$180$	6.17491	0.460251
$181$	20.2608	1.50598	0.752988	−	0.658034i	$-0.228612\pi$
$181$	20.2608	1.50598	0.752988	+	0.658034i	$0.228612\pi$
$182$	24.7035	1.83114
$183$	2.95296	0.218289
$184$	0.662571	0.0488454
$185$	2.24705	0.165206
$186$	−14.9141	−1.09356
$187$	35.6283	2.60540
$188$	6.74493	0.491924
$189$	−20.1143	−1.46310
$190$	−9.06776	−0.657844
$191$	−24.1072	−1.74433	−0.872167	−	0.489209i	$-0.837286\pi$
$191$	−24.1072	−1.74433	−0.872167	+	0.489209i	$0.837286\pi$
$192$	−2.78190	−0.200767
$193$	12.0775	0.869357	0.434679	−	0.900586i	$-0.356862\pi$
$193$	12.0775	0.869357	0.434679	+	0.900586i	$0.356862\pi$
$194$	−11.2352	−0.806641
$195$	21.5368	1.54228
$196$	10.2874	0.734815
$197$	21.0926	1.50279	0.751393	−	0.659855i	$-0.229382\pi$
$197$	21.0926	1.50279	0.751393	+	0.659855i	$0.229382\pi$
$198$	21.6820	1.54087
$199$	−20.7131	−1.46832	−0.734158	−	0.678979i	$-0.762423\pi$
$199$	−20.7131	−1.46832	−0.734158	+	0.678979i	$0.762423\pi$
$200$	3.30219	0.233500
$201$	28.0573	1.97901
$202$	−2.95061	−0.207604
$203$	26.0289	1.82687
$204$	21.6633	1.51673
$205$	10.7135	0.748262
$206$	−1.37882	−0.0960668
$207$	−3.13992	−0.218239
$208$	−5.94146	−0.411966
$209$	−31.8396	−2.20239
$210$	15.0713	1.04002
$211$	−11.1026	−0.764333	−0.382167	−	0.924093i	$-0.624822\pi$
$211$	−11.1026	−0.764333	−0.382167	+	0.924093i	$0.624822\pi$
$212$	−2.23194	−0.153290
$213$	−37.0980	−2.54191
$214$	−10.5950	−0.724259
$215$	13.4286	0.915823
$216$	4.83771	0.329164
$217$	−22.2906	−1.51318
$218$	−15.6549	−1.06028
$219$	−9.35239	−0.631976
$220$	−5.96153	−0.401926
$221$	46.2674	3.11228
$222$	4.79744	0.321983
$223$	−25.5562	−1.71137	−0.855684	−	0.517498i	$-0.826863\pi$
$223$	−25.5562	−1.71137	−0.855684	+	0.517498i	$0.826863\pi$
$224$	−4.15781	−0.277806
$225$	−15.6490	−1.04327
$226$	8.12771	0.540647
$227$	−22.0997	−1.46681	−0.733403	−	0.679794i	$-0.762069\pi$
$227$	−22.0997	−1.46681	−0.733403	+	0.679794i	$0.762069\pi$
$228$	−19.3596	−1.28212
$229$	−15.3567	−1.01480	−0.507398	−	0.861712i	$-0.669393\pi$
$229$	−15.3567	−1.01480	−0.507398	+	0.861712i	$0.669393\pi$
$230$	0.863330	0.0569263
$231$	52.9200	3.48188
$232$	−6.26025	−0.411006
$233$	−21.5889	−1.41434	−0.707169	−	0.707044i	$-0.750028\pi$
$233$	−21.5889	−1.41434	−0.707169	+	0.707044i	$0.750028\pi$
$234$	28.1565	1.84065
$235$	8.78865	0.573308
$236$	0.421952	0.0274667
$237$	25.4923	1.65590
$238$	32.3778	2.09874
$239$	−20.5579	−1.32978	−0.664891	−	0.746940i	$-0.731522\pi$
$239$	−20.5579	−1.32978	−0.664891	+	0.746940i	$0.731522\pi$
$240$	−3.62482	−0.233981
$241$	4.13741	0.266514	0.133257	−	0.991082i	$-0.457456\pi$
$241$	4.13741	0.266514	0.133257	+	0.991082i	$0.457456\pi$
$242$	−9.93272	−0.638500
$243$	16.6244	1.06646
$244$	−1.06149	−0.0679548
$245$	13.4045	0.856383
$246$	22.8733	1.45835
$247$	−41.3474	−2.63087
$248$	5.36113	0.340432
$249$	41.9459	2.65821
$250$	10.8178	0.684175
$251$	−17.0131	−1.07386	−0.536928	−	0.843628i	$-0.680415\pi$
$251$	−17.0131	−1.07386	−0.536928	+	0.843628i	$0.680415\pi$
$252$	19.7038	1.24123
$253$	3.03141	0.190583
$254$	7.30114	0.458114
$255$	28.2273	1.76766
$256$	1.00000	0.0625000
$257$	−22.7984	−1.42212	−0.711061	−	0.703130i	$-0.751785\pi$
$257$	−22.7984	−1.42212	−0.711061	+	0.703130i	$0.751785\pi$
$258$	28.6700	1.78492
$259$	7.17021	0.445535
$260$	−7.74173	−0.480122
$261$	29.6673	1.83636
$262$	−17.7116	−1.09422
$263$	−13.0334	−0.803671	−0.401835	−	0.915712i	$-0.631628\pi$
$263$	−13.0334	−0.803671	−0.401835	+	0.915712i	$0.631628\pi$
$264$	−12.7278	−0.783345
$265$	−2.90822	−0.178651
$266$	−28.9348	−1.77410
$267$	22.2220	1.35997
$268$	−10.0857	−0.616080
$269$	−10.8392	−0.660878	−0.330439	−	0.943827i	$-0.607197\pi$
$269$	−10.8392	−0.660878	−0.330439	+	0.943827i	$0.607197\pi$
$270$	6.30354	0.383621
$271$	12.3380	0.749479	0.374739	−	0.927130i	$-0.377732\pi$
$271$	12.3380	0.749479	0.374739	+	0.927130i	$0.377732\pi$
$272$	−7.78721	−0.472169
$273$	68.7227	4.15929
$274$	9.77336	0.590430
$275$	15.1083	0.911063
$276$	1.84321	0.110948
$277$	−2.78406	−0.167278	−0.0836389	−	0.996496i	$-0.526654\pi$
$277$	−2.78406	−0.167278	−0.0836389	+	0.996496i	$0.526654\pi$
$278$	−10.3473	−0.620588
$279$	−25.4063	−1.52104
$280$	−5.41764	−0.323766
$281$	−15.0499	−0.897801	−0.448900	−	0.893582i	$-0.648184\pi$
$281$	−15.0499	−0.897801	−0.448900	+	0.893582i	$0.648184\pi$
$282$	18.7637	1.11736
$283$	26.0884	1.55079	0.775397	−	0.631474i	$-0.217550\pi$
$283$	26.0884	1.55079	0.775397	+	0.631474i	$0.217550\pi$
$284$	13.3355	0.791315
$285$	−25.2256	−1.49424
$286$	−27.1835	−1.60740
$287$	34.1862	2.01795
$288$	−4.73899	−0.279248
$289$	43.6407	2.56710
$290$	−8.15711	−0.479002
$291$	−31.2553	−1.83222
$292$	3.36187	0.196738
$293$	−9.84849	−0.575355	−0.287677	−	0.957727i	$-0.592883\pi$
$293$	−9.84849	−0.575355	−0.287677	+	0.957727i	$0.592883\pi$
$294$	28.6186	1.66907
$295$	0.549804	0.0320108
$296$	−1.72452	−0.100235
$297$	22.1336	1.28432
$298$	8.31812	0.481856
$299$	3.93664	0.227662
$300$	9.18637	0.530375
$301$	42.8500	2.46983
$302$	−6.05716	−0.348550
$303$	−8.20831	−0.471555
$304$	6.95913	0.399134
$305$	−1.38312	−0.0791972
$306$	36.9035	2.10963
$307$	−28.9115	−1.65006	−0.825032	−	0.565086i	$-0.808843\pi$
$307$	−28.9115	−1.65006	−0.825032	+	0.565086i	$0.808843\pi$
$308$	−19.0229	−1.08393
$309$	−3.83574	−0.218207
$310$	6.98555	0.396753
$311$	−23.3300	−1.32293	−0.661463	−	0.749978i	$-0.730064\pi$
$311$	−23.3300	−1.32293	−0.661463	+	0.749978i	$0.730064\pi$
$312$	−16.5286	−0.935746
$313$	27.4349	1.55071	0.775356	−	0.631524i	$-0.217570\pi$
$313$	27.4349	1.55071	0.775356	+	0.631524i	$0.217570\pi$
$314$	−9.42635	−0.531960
$315$	25.6741	1.44657
$316$	−9.16363	−0.515494
$317$	6.85699	0.385127	0.192564	−	0.981284i	$-0.438320\pi$
$317$	6.85699	0.385127	0.192564	+	0.981284i	$0.438320\pi$
$318$	−6.20904	−0.348186
$319$	−28.6421	−1.60365
$320$	1.30300	0.0728400
$321$	−29.4743	−1.64509
$322$	2.75485	0.153522
$323$	−54.1922	−3.01534
$324$	−0.758940	−0.0421633
$325$	19.6198	1.08831
$326$	13.0293	0.721624
$327$	−43.5504	−2.40834
$328$	−8.22216	−0.453993
$329$	28.0441	1.54612
$330$	−16.5844	−0.912941
$331$	11.2367	0.617627	0.308814	−	0.951123i	$-0.400068\pi$
$331$	11.2367	0.617627	0.308814	+	0.951123i	$0.400068\pi$
$332$	−15.0781	−0.827519
$333$	8.17246	0.447848
$334$	2.19217	0.119950
$335$	−13.1416	−0.718004
$336$	−11.5666	−0.631012
$337$	−17.3269	−0.943855	−0.471927	−	0.881637i	$-0.656442\pi$
$337$	−17.3269	−0.943855	−0.471927	+	0.881637i	$0.656442\pi$
$338$	−22.3010	−1.21301
$339$	22.6105	1.22803
$340$	−10.1467	−0.550285
$341$	24.5284	1.32829
$342$	−32.9793	−1.78331
$343$	13.6684	0.738026
$344$	−10.3059	−0.555657
$345$	2.40170	0.129303
$346$	11.4956	0.618007
$347$	5.36505	0.288011	0.144005	−	0.989577i	$-0.454002\pi$
$347$	5.36505	0.288011	0.144005	+	0.989577i	$0.454002\pi$
$348$	−17.4154	−0.933564
$349$	−19.2764	−1.03184	−0.515921	−	0.856636i	$-0.672550\pi$
$349$	−19.2764	−1.03184	−0.515921	+	0.856636i	$0.672550\pi$
$350$	13.7299	0.733893
$351$	28.7430	1.53419
$352$	4.57523	0.243861
$353$	−20.1960	−1.07493	−0.537463	−	0.843287i	$-0.680617\pi$
$353$	−20.1960	−1.07493	−0.537463	+	0.843287i	$0.680617\pi$
$354$	1.17383	0.0623884
$355$	17.3761	0.922230
$356$	−7.98806	−0.423366
$357$	90.0718	4.76711
$358$	−0.170590	−0.00901595
$359$	−25.4827	−1.34492	−0.672462	−	0.740131i	$-0.734763\pi$
$359$	−25.4827	−1.34492	−0.672462	+	0.740131i	$0.734763\pi$
$360$	−6.17491	−0.325446
$361$	29.4295	1.54892
$362$	−20.2608	−1.06489
$363$	−27.6319	−1.45030
$364$	−24.7035	−1.29481
$365$	4.38052	0.229287
$366$	−2.95296	−0.154354
$367$	7.82843	0.408641	0.204320	−	0.978904i	$-0.434502\pi$
$367$	7.82843	0.408641	0.204320	+	0.978904i	$0.434502\pi$
$368$	−0.662571	−0.0345389
$369$	38.9647	2.02842
$370$	−2.24705	−0.116818
$371$	−9.27999	−0.481793
$372$	14.9141	0.773262
$373$	6.18145	0.320063	0.160032	−	0.987112i	$-0.448840\pi$
$373$	6.18145	0.320063	0.160032	+	0.987112i	$0.448840\pi$
$374$	−35.6283	−1.84229
$375$	30.0940	1.55405
$376$	−6.74493	−0.347843
$377$	−37.1950	−1.91564
$378$	20.1143	1.03457
$379$	−2.45514	−0.126112	−0.0630561	−	0.998010i	$-0.520085\pi$
$379$	−2.45514	−0.126112	−0.0630561	+	0.998010i	$0.520085\pi$
$380$	9.06776	0.465166
$381$	20.3111	1.04057
$382$	24.1072	1.23343
$383$	23.7818	1.21519	0.607596	−	0.794247i	$-0.292134\pi$
$383$	23.7818	1.21519	0.607596	+	0.794247i	$0.292134\pi$
$384$	2.78190	0.141963
$385$	−24.7869	−1.26326
$386$	−12.0775	−0.614729
$387$	48.8396	2.48265
$388$	11.2352	0.570381
$389$	−4.11225	−0.208499	−0.104250	−	0.994551i	$-0.533244\pi$
$389$	−4.11225	−0.208499	−0.104250	+	0.994551i	$0.533244\pi$
$390$	−21.5368	−1.09056
$391$	5.15958	0.260931
$392$	−10.2874	−0.519593
$393$	−49.2719	−2.48544
$394$	−21.0926	−1.06263
$395$	−11.9402	−0.600777
$396$	−21.6820	−1.08956
$397$	2.38660	0.119780	0.0598901	−	0.998205i	$-0.480925\pi$
$397$	2.38660	0.119780	0.0598901	+	0.998205i	$0.480925\pi$
$398$	20.7131	1.03826
$399$	−80.4937	−4.02973
$400$	−3.30219	−0.165109
$401$	20.4630	1.02187	0.510936	−	0.859619i	$-0.329299\pi$
$401$	20.4630	1.02187	0.510936	+	0.859619i	$0.329299\pi$
$402$	−28.0573	−1.39937
$403$	31.8529	1.58671
$404$	2.95061	0.146798
$405$	−0.988899	−0.0491388
$406$	−26.0289	−1.29179
$407$	−7.89005	−0.391095
$408$	−21.6633	−1.07249
$409$	4.18560	0.206965	0.103482	−	0.994631i	$-0.467001\pi$
$409$	4.18560	0.206965	0.103482	+	0.994631i	$0.467001\pi$
$410$	−10.7135	−0.529101
$411$	27.1885	1.34111
$412$	1.37882	0.0679295
$413$	1.75440	0.0863283
$414$	3.13992	0.154318
$415$	−19.6468	−0.964424
$416$	5.94146	0.291304
$417$	−28.7851	−1.40961
$418$	31.8396	1.55733
$419$	24.6505	1.20426	0.602129	−	0.798399i	$-0.294319\pi$
$419$	24.6505	1.20426	0.602129	+	0.798399i	$0.294319\pi$
$420$	−15.0713	−0.735406
$421$	7.72683	0.376583	0.188291	−	0.982113i	$-0.439705\pi$
$421$	7.72683	0.376583	0.188291	+	0.982113i	$0.439705\pi$
$422$	11.1026	0.540465
$423$	31.9641	1.55415
$424$	2.23194	0.108393
$425$	25.7148	1.24735
$426$	37.0980	1.79740
$427$	−4.41347	−0.213583
$428$	10.5950	0.512129
$429$	−75.6220	−3.65107
$430$	−13.4286	−0.647584
$431$	21.9308	1.05637	0.528184	−	0.849130i	$-0.322873\pi$
$431$	21.9308	1.05637	0.528184	+	0.849130i	$0.322873\pi$
$432$	−4.83771	−0.232754
$433$	−28.8745	−1.38762	−0.693811	−	0.720157i	$-0.744070\pi$
$433$	−28.8745	−1.38762	−0.693811	+	0.720157i	$0.744070\pi$
$434$	22.2906	1.06998
$435$	−22.6923	−1.08801
$436$	15.6549	0.749733
$437$	−4.61092	−0.220570
$438$	9.35239	0.446874
$439$	−0.967477	−0.0461752	−0.0230876	−	0.999733i	$-0.507350\pi$
$439$	−0.967477	−0.0461752	−0.0230876	+	0.999733i	$0.507350\pi$
$440$	5.96153	0.284205
$441$	48.7519	2.32152
$442$	−46.2674	−2.20072
$443$	−2.79407	−0.132750	−0.0663752	−	0.997795i	$-0.521143\pi$
$443$	−2.79407	−0.132750	−0.0663752	+	0.997795i	$0.521143\pi$
$444$	−4.79744	−0.227676
$445$	−10.4085	−0.493408
$446$	25.5562	1.21012
$447$	23.1402	1.09449
$448$	4.15781	0.196438
$449$	4.13801	0.195285	0.0976425	−	0.995222i	$-0.468870\pi$
$449$	4.13801	0.195285	0.0976425	+	0.995222i	$0.468870\pi$
$450$	15.6490	0.737703
$451$	−37.6183	−1.77137
$452$	−8.12771	−0.382295
$453$	−16.8504	−0.791703
$454$	22.0997	1.03719
$455$	−32.1887	−1.50903
$456$	19.3596	0.906598
$457$	30.4298	1.42345	0.711723	−	0.702460i	$-0.247915\pi$
$457$	30.4298	1.42345	0.711723	+	0.702460i	$0.247915\pi$
$458$	15.3567	0.717570
$459$	37.6722	1.75839
$460$	−0.863330	−0.0402530
$461$	0.399993	0.0186296	0.00931478	−	0.999957i	$-0.497035\pi$
$461$	0.399993	0.0186296	0.00931478	+	0.999957i	$0.497035\pi$
$462$	−52.9200	−2.46206
$463$	−9.08075	−0.422018	−0.211009	−	0.977484i	$-0.567675\pi$
$463$	−9.08075	−0.422018	−0.211009	+	0.977484i	$0.567675\pi$
$464$	6.26025	0.290625
$465$	19.4331	0.901190
$466$	21.5889	1.00009
$467$	−5.21980	−0.241544	−0.120772	−	0.992680i	$-0.538537\pi$
$467$	−5.21980	−0.241544	−0.120772	+	0.992680i	$0.538537\pi$
$468$	−28.1565	−1.30154
$469$	−41.9343	−1.93635
$470$	−8.78865	−0.405390
$471$	−26.2232	−1.20830
$472$	−0.421952	−0.0194219
$473$	−47.1518	−2.16804
$474$	−25.4923	−1.17090
$475$	−22.9804	−1.05441
$476$	−32.3778	−1.48403
$477$	−10.5771	−0.484294
$478$	20.5579	0.940298
$479$	−33.2661	−1.51996	−0.759982	−	0.649944i	$-0.774792\pi$
$479$	−33.2661	−1.51996	−0.759982	+	0.649944i	$0.774792\pi$
$480$	3.62482	0.165450
$481$	−10.2461	−0.467184
$482$	−4.13741	−0.188454
$483$	7.66371	0.348711
$484$	9.93272	0.451487
$485$	14.6395	0.664745
$486$	−16.6244	−0.754099
$487$	−5.61983	−0.254659	−0.127329	−	0.991860i	$-0.540641\pi$
$487$	−5.61983	−0.254659	−0.127329	+	0.991860i	$0.540641\pi$
$488$	1.06149	0.0480513
$489$	36.2461	1.63911
$490$	−13.4045	−0.605554
$491$	13.9532	0.629698	0.314849	−	0.949142i	$-0.398046\pi$
$491$	13.9532	0.629698	0.314849	+	0.949142i	$0.398046\pi$
$492$	−22.8733	−1.03121
$493$	−48.7499	−2.19558
$494$	41.3474	1.86031
$495$	−28.2516	−1.26982
$496$	−5.36113	−0.240722
$497$	55.4464	2.48711
$498$	−41.9459	−1.87964
$499$	17.2033	0.770127	0.385064	−	0.922890i	$-0.374180\pi$
$499$	17.2033	0.770127	0.385064	+	0.922890i	$0.374180\pi$
$500$	−10.8178	−0.483785
$501$	6.09841	0.272457
$502$	17.0131	0.759330
$503$	3.10386	0.138394	0.0691970	−	0.997603i	$-0.477956\pi$
$503$	3.10386	0.138394	0.0691970	+	0.997603i	$0.477956\pi$
$504$	−19.7038	−0.877679
$505$	3.84464	0.171084
$506$	−3.03141	−0.134763
$507$	−62.0391	−2.75525
$508$	−7.30114	−0.323936
$509$	−20.2878	−0.899241	−0.449620	−	0.893220i	$-0.648441\pi$
$509$	−20.2878	−0.899241	−0.449620	+	0.893220i	$0.648441\pi$
$510$	−28.2273	−1.24992
$511$	13.9780	0.618351
$512$	−1.00000	−0.0441942
$513$	−33.6662	−1.48640
$514$	22.7984	1.00559
$515$	1.79660	0.0791677
$516$	−28.6700	−1.26213
$517$	−30.8596	−1.35720
$518$	−7.17021	−0.315041
$519$	31.9796	1.40375
$520$	7.74173	0.339497
$521$	27.8643	1.22076	0.610378	−	0.792110i	$-0.291017\pi$
$521$	27.8643	1.22076	0.610378	+	0.792110i	$0.291017\pi$
$522$	−29.6673	−1.29850
$523$	−32.6237	−1.42653	−0.713267	−	0.700892i	$-0.752785\pi$
$523$	−32.6237	−1.42653	−0.713267	+	0.700892i	$0.752785\pi$
$524$	17.7116	0.773734
$525$	38.1952	1.66698
$526$	13.0334	0.568281
$527$	41.7482	1.81858
$528$	12.7278	0.553909
$529$	−22.5610	−0.980913
$530$	2.90822	0.126325
$531$	1.99963	0.0867764
$532$	28.9348	1.25448
$533$	−48.8516	−2.11600
$534$	−22.2220	−0.961641
$535$	13.8053	0.596855
$536$	10.0857	0.435634
$537$	−0.474565	−0.0204790
$538$	10.8392	0.467311
$539$	−47.0673	−2.02733
$540$	−6.30354	−0.271261
$541$	22.5463	0.969342	0.484671	−	0.874697i	$-0.338939\pi$
$541$	22.5463	0.969342	0.484671	+	0.874697i	$0.338939\pi$
$542$	−12.3380	−0.529962
$543$	−56.3637	−2.41880
$544$	7.78721	0.333874
$545$	20.3983	0.873769
$546$	−68.7227	−2.94106
$547$	36.8591	1.57598	0.787990	−	0.615688i	$-0.211122\pi$
$547$	36.8591	1.57598	0.787990	+	0.615688i	$0.211122\pi$
$548$	−9.77336	−0.417497
$549$	−5.03038	−0.214692
$550$	−15.1083	−0.644219
$551$	43.5659	1.85597
$552$	−1.84321	−0.0784521
$553$	−38.1006	−1.62020
$554$	2.78406	0.118283
$555$	−6.25106	−0.265343
$556$	10.3473	0.438822
$557$	−26.6993	−1.13128	−0.565642	−	0.824651i	$-0.691372\pi$
$557$	−26.6993	−1.13128	−0.565642	+	0.824651i	$0.691372\pi$
$558$	25.4063	1.07554
$559$	−61.2321	−2.58984
$560$	5.41764	0.228937
$561$	−99.1144	−4.18462
$562$	15.0499	0.634841
$563$	29.4192	1.23987	0.619935	−	0.784653i	$-0.287159\pi$
$563$	29.4192	1.23987	0.619935	+	0.784653i	$0.287159\pi$
$564$	−18.7637	−0.790096
$565$	−10.5904	−0.445542
$566$	−26.0884	−1.09658
$567$	−3.15553	−0.132520
$568$	−13.3355	−0.559544
$569$	−37.1619	−1.55791	−0.778953	−	0.627082i	$-0.784249\pi$
$569$	−37.1619	−1.55791	−0.778953	+	0.627082i	$0.784249\pi$
$570$	25.2256	1.05659
$571$	9.64427	0.403600	0.201800	−	0.979427i	$-0.435321\pi$
$571$	9.64427	0.403600	0.201800	+	0.979427i	$0.435321\pi$
$572$	27.1835	1.13660
$573$	67.0638	2.80163
$574$	−34.1862	−1.42690
$575$	2.18793	0.0912431
$576$	4.73899	0.197458
$577$	−5.85278	−0.243654	−0.121827	−	0.992551i	$-0.538875\pi$
$577$	−5.85278	−0.243654	−0.121827	+	0.992551i	$0.538875\pi$
$578$	−43.6407	−1.81521
$579$	−33.5984	−1.39630
$580$	8.15711	0.338706
$581$	−62.6920	−2.60090
$582$	31.2553	1.29557
$583$	10.2116	0.422923
$584$	−3.36187	−0.139115
$585$	−36.6880	−1.51686
$586$	9.84849	0.406837
$587$	−25.7244	−1.06176	−0.530880	−	0.847447i	$-0.678138\pi$
$587$	−25.7244	−1.06176	−0.530880	+	0.847447i	$0.678138\pi$
$588$	−28.6186	−1.18021
$589$	−37.3088	−1.53728
$590$	−0.549804	−0.0226351
$591$	−58.6776	−2.41367
$592$	1.72452	0.0708771
$593$	9.92841	0.407711	0.203855	−	0.979001i	$-0.434653\pi$
$593$	9.92841	0.407711	0.203855	+	0.979001i	$0.434653\pi$
$594$	−22.1336	−0.908153
$595$	−42.1883	−1.72955
$596$	−8.31812	−0.340724
$597$	57.6220	2.35831
$598$	−3.93664	−0.160981
$599$	−18.8198	−0.768956	−0.384478	−	0.923134i	$-0.625619\pi$
$599$	−18.8198	−0.768956	−0.384478	+	0.923134i	$0.625619\pi$
$600$	−9.18637	−0.375032
$601$	39.9022	1.62765	0.813823	−	0.581113i	$-0.197382\pi$
$601$	39.9022	1.62765	0.813823	+	0.581113i	$0.197382\pi$
$602$	−42.8500	−1.74644
$603$	−47.7959	−1.94640
$604$	6.05716	0.246462
$605$	12.9423	0.526181
$606$	8.20831	0.333440
$607$	−22.9993	−0.933514	−0.466757	−	0.884386i	$-0.654578\pi$
$607$	−22.9993	−0.933514	−0.466757	+	0.884386i	$0.654578\pi$
$608$	−6.95913	−0.282230
$609$	−72.4100	−2.93420
$610$	1.38312	0.0560009
$611$	−40.0747	−1.62125
$612$	−36.9035	−1.49174
$613$	14.6363	0.591153	0.295576	−	0.955319i	$-0.404488\pi$
$613$	14.6363	0.591153	0.295576	+	0.955319i	$0.404488\pi$
$614$	28.9115	1.16677
$615$	−29.8039	−1.20181
$616$	19.0229	0.766456
$617$	−8.73630	−0.351710	−0.175855	−	0.984416i	$-0.556269\pi$
$617$	−8.73630	−0.351710	−0.175855	+	0.984416i	$0.556269\pi$
$618$	3.83574	0.154296
$619$	−45.7346	−1.83823	−0.919114	−	0.393991i	$-0.871094\pi$
$619$	−45.7346	−1.83823	−0.919114	+	0.393991i	$0.871094\pi$
$620$	−6.98555	−0.280547
$621$	3.20532	0.128625
$622$	23.3300	0.935450
$623$	−33.2129	−1.33065
$624$	16.5286	0.661673
$625$	2.41538	0.0966154
$626$	−27.4349	−1.09652
$627$	88.5748	3.53734
$628$	9.42635	0.376152
$629$	−13.4292	−0.535456
$630$	−25.6741	−1.02288
$631$	−12.5599	−0.500000	−0.250000	−	0.968246i	$-0.580431\pi$
$631$	−12.5599	−0.500000	−0.250000	+	0.968246i	$0.580431\pi$
$632$	9.16363	0.364509
$633$	30.8863	1.22762
$634$	−6.85699	−0.272326
$635$	−9.51339	−0.377527
$636$	6.20904	0.246205
$637$	−61.1223	−2.42175
$638$	28.6421	1.13395
$639$	63.1967	2.50002
$640$	−1.30300	−0.0515056
$641$	19.2127	0.758855	0.379428	−	0.925221i	$-0.376121\pi$
$641$	19.2127	0.758855	0.379428	+	0.925221i	$0.376121\pi$
$642$	29.4743	1.16326
$643$	−19.5626	−0.771474	−0.385737	−	0.922609i	$-0.626053\pi$
$643$	−19.5626	−0.771474	−0.385737	+	0.922609i	$0.626053\pi$
$644$	−2.75485	−0.108556
$645$	−37.3571	−1.47093
$646$	54.1922	2.13216
$647$	−23.0227	−0.905114	−0.452557	−	0.891735i	$-0.649488\pi$
$647$	−23.0227	−0.905114	−0.452557	+	0.891735i	$0.649488\pi$
$648$	0.758940	0.0298140
$649$	−1.93053	−0.0757798
$650$	−19.6198	−0.769553
$651$	62.0102	2.43037
$652$	−13.0293	−0.510265
$653$	−13.9970	−0.547746	−0.273873	−	0.961766i	$-0.588305\pi$
$653$	−13.9970	−0.547746	−0.273873	+	0.961766i	$0.588305\pi$
$654$	43.5504	1.70295
$655$	23.0782	0.901740
$656$	8.22216	0.321021
$657$	15.9319	0.621561
$658$	−28.0441	−1.09327
$659$	8.11076	0.315950	0.157975	−	0.987443i	$-0.449503\pi$
$659$	8.11076	0.315950	0.157975	+	0.987443i	$0.449503\pi$
$660$	16.5844	0.645547
$661$	6.63306	0.257996	0.128998	−	0.991645i	$-0.458824\pi$
$661$	6.63306	0.257996	0.128998	+	0.991645i	$0.458824\pi$
$662$	−11.2367	−0.436728
$663$	−128.712	−4.99874
$664$	15.0781	0.585144
$665$	37.7020	1.46202
$666$	−8.17246	−0.316676
$667$	−4.14786	−0.160606
$668$	−2.19217	−0.0848177
$669$	71.0948	2.74869
$670$	13.1416	0.507705
$671$	4.85655	0.187485
$672$	11.5666	0.446193
$673$	36.1397	1.39308	0.696541	−	0.717517i	$-0.254722\pi$
$673$	36.1397	1.39308	0.696541	+	0.717517i	$0.254722\pi$
$674$	17.3269	0.667406
$675$	15.9750	0.614878
$676$	22.3010	0.857729
$677$	−25.7094	−0.988093	−0.494046	−	0.869436i	$-0.664483\pi$
$677$	−25.7094	−0.988093	−0.494046	+	0.869436i	$0.664483\pi$
$678$	−22.6105	−0.868351
$679$	46.7139	1.79271
$680$	10.1467	0.389110
$681$	61.4792	2.35589
$682$	−24.5284	−0.939240
$683$	8.48817	0.324791	0.162395	−	0.986726i	$-0.448078\pi$
$683$	8.48817	0.324791	0.162395	+	0.986726i	$0.448078\pi$
$684$	32.9793	1.26099
$685$	−12.7347	−0.486568
$686$	−13.6684	−0.521863
$687$	42.7208	1.62990
$688$	10.3059	0.392909
$689$	13.2610	0.505203
$690$	−2.40170	−0.0914312
$691$	−28.0565	−1.06732	−0.533659	−	0.845700i	$-0.679183\pi$
$691$	−28.0565	−1.06732	−0.533659	+	0.845700i	$0.679183\pi$
$692$	−11.4956	−0.436997
$693$	−90.1496	−3.42450
$694$	−5.36505	−0.203654
$695$	13.4825	0.511421
$696$	17.4154	0.660130
$697$	−64.0277	−2.42522
$698$	19.2764	0.729623
$699$	60.0584	2.27162
$700$	−13.7299	−0.518941
$701$	−4.87450	−0.184107	−0.0920536	−	0.995754i	$-0.529343\pi$
$701$	−4.87450	−0.184107	−0.0920536	+	0.995754i	$0.529343\pi$
$702$	−28.7430	−1.08484
$703$	12.0011	0.452631
$704$	−4.57523	−0.172435
$705$	−24.4492	−0.920809
$706$	20.1960	0.760087
$707$	12.2681	0.461388
$708$	−1.17383	−0.0441152
$709$	−25.7483	−0.966996	−0.483498	−	0.875346i	$-0.660634\pi$
$709$	−25.7483	−0.966996	−0.483498	+	0.875346i	$0.660634\pi$
$710$	−17.3761	−0.652115
$711$	−43.4263	−1.62861
$712$	7.98806	0.299365
$713$	3.55213	0.133028
$714$	−90.0718	−3.37085
$715$	35.4202	1.32464
$716$	0.170590	0.00637524
$717$	57.1902	2.13581
$718$	25.4827	0.951006
$719$	−34.3375	−1.28057	−0.640287	−	0.768136i	$-0.721185\pi$
$719$	−34.3375	−1.28057	−0.640287	+	0.768136i	$0.721185\pi$
$720$	6.17491	0.230125
$721$	5.73287	0.213503
$722$	−29.4295	−1.09525
$723$	−11.5099	−0.428057
$724$	20.2608	0.752988
$725$	−20.6725	−0.767758
$726$	27.6319	1.02552
$727$	−11.3073	−0.419363	−0.209681	−	0.977770i	$-0.567243\pi$
$727$	−11.3073	−0.419363	−0.209681	+	0.977770i	$0.567243\pi$
$728$	24.7035	0.915572
$729$	−43.9707	−1.62854
$730$	−4.38052	−0.162130
$731$	−80.2542	−2.96831
$732$	2.95296	0.109144
$733$	16.0029	0.591081	0.295541	−	0.955330i	$-0.404500\pi$
$733$	16.0029	0.591081	0.295541	+	0.955330i	$0.404500\pi$
$734$	−7.82843	−0.288953
$735$	−37.2901	−1.37546
$736$	0.662571	0.0244227
$737$	46.1442	1.69974
$738$	−38.9647	−1.43431
$739$	39.7820	1.46340	0.731702	−	0.681625i	$-0.238726\pi$
$739$	39.7820	1.46340	0.731702	+	0.681625i	$0.238726\pi$
$740$	2.24705	0.0826030
$741$	115.025	4.22553
$742$	9.27999	0.340679
$743$	−28.2019	−1.03463	−0.517313	−	0.855796i	$-0.673068\pi$
$743$	−28.2019	−1.03463	−0.517313	+	0.855796i	$0.673068\pi$
$744$	−14.9141	−0.546779
$745$	−10.8385	−0.397093
$746$	−6.18145	−0.226319
$747$	−71.4550	−2.61440
$748$	35.6283	1.30270
$749$	44.0520	1.60963
$750$	−30.0940	−1.09888
$751$	−28.0573	−1.02382	−0.511912	−	0.859038i	$-0.671063\pi$
$751$	−28.0573	−1.02382	−0.511912	+	0.859038i	$0.671063\pi$
$752$	6.74493	0.245962
$753$	47.3287	1.72475
$754$	37.1950	1.35456
$755$	7.89249	0.287237
$756$	−20.1143	−0.731549
$757$	1.55680	0.0565830	0.0282915	−	0.999600i	$-0.490993\pi$
$757$	1.55680	0.0565830	0.0282915	+	0.999600i	$0.490993\pi$
$758$	2.45514	0.0891748
$759$	−8.43310	−0.306102
$760$	−9.06776	−0.328922
$761$	−22.9359	−0.831426	−0.415713	−	0.909496i	$-0.636468\pi$
$761$	−22.9359	−0.831426	−0.415713	+	0.909496i	$0.636468\pi$
$762$	−20.3111	−0.735792
$763$	65.0901	2.35642
$764$	−24.1072	−0.872167
$765$	−48.0853	−1.73853
$766$	−23.7818	−0.859270
$767$	−2.50701	−0.0905229
$768$	−2.78190	−0.100383
$769$	−36.0709	−1.30075	−0.650375	−	0.759614i	$-0.725388\pi$
$769$	−36.0709	−1.30075	−0.650375	+	0.759614i	$0.725388\pi$
$770$	24.7869	0.893259
$771$	63.4229	2.28412
$772$	12.0775	0.434679
$773$	−18.9780	−0.682591	−0.341295	−	0.939956i	$-0.610866\pi$
$773$	−18.9780	−0.682591	−0.341295	+	0.939956i	$0.610866\pi$
$774$	−48.8396	−1.75550
$775$	17.7034	0.635927
$776$	−11.2352	−0.403320
$777$	−19.9468	−0.715589
$778$	4.11225	0.147431
$779$	57.2191	2.05009
$780$	21.5368	0.771140
$781$	−61.0128	−2.18321
$782$	−5.15958	−0.184506
$783$	−30.2852	−1.08231
$784$	10.2874	0.367408
$785$	12.2825	0.438383
$786$	49.2719	1.75747
$787$	−33.3049	−1.18719	−0.593596	−	0.804763i	$-0.702292\pi$
$787$	−33.3049	−1.18719	−0.593596	+	0.804763i	$0.702292\pi$
$788$	21.0926	0.751393
$789$	36.2575	1.29080
$790$	11.9402	0.424814
$791$	−33.7935	−1.20156
$792$	21.6820	0.770435
$793$	6.30679	0.223961
$794$	−2.38660	−0.0846974
$795$	8.09039	0.286937
$796$	−20.7131	−0.734158
$797$	19.1895	0.679729	0.339864	−	0.940474i	$-0.389619\pi$
$797$	19.1895	0.679729	0.339864	+	0.940474i	$0.389619\pi$
$798$	80.4937	2.84945
$799$	−52.5242	−1.85817
$800$	3.30219	0.116750
$801$	−37.8554	−1.33755
$802$	−20.4630	−0.722572
$803$	−15.3813	−0.542795
$804$	28.0573	0.989506
$805$	−3.58957	−0.126516
$806$	−31.8529	−1.12197
$807$	30.1536	1.06146
$808$	−2.95061	−0.103802
$809$	9.88895	0.347677	0.173838	−	0.984774i	$-0.444383\pi$
$809$	9.88895	0.347677	0.173838	+	0.984774i	$0.444383\pi$
$810$	0.988899	0.0347464
$811$	−1.67143	−0.0586917	−0.0293458	−	0.999569i	$-0.509342\pi$
$811$	−1.67143	−0.0586917	−0.0293458	+	0.999569i	$0.509342\pi$
$812$	26.0289	0.913437
$813$	−34.3231	−1.20376
$814$	7.89005	0.276546
$815$	−16.9771	−0.594683
$816$	21.6633	0.758366
$817$	71.7201	2.50917
$818$	−4.18560	−0.146346
$819$	−117.070	−4.09074
$820$	10.7135	0.374131
$821$	30.4944	1.06426	0.532131	−	0.846662i	$-0.321392\pi$
$821$	30.4944	1.06426	0.532131	+	0.846662i	$0.321392\pi$
$822$	−27.1885	−0.948309
$823$	−31.9179	−1.11259	−0.556295	−	0.830985i	$-0.687777\pi$
$823$	−31.9179	−1.11259	−0.556295	+	0.830985i	$0.687777\pi$
$824$	−1.37882	−0.0480334
$825$	−42.0297	−1.46329
$826$	−1.75440	−0.0610433
$827$	−20.6465	−0.717950	−0.358975	−	0.933347i	$-0.616874\pi$
$827$	−20.6465	−0.717950	−0.358975	+	0.933347i	$0.616874\pi$
$828$	−3.13992	−0.109120
$829$	9.74510	0.338461	0.169231	−	0.985576i	$-0.445872\pi$
$829$	9.74510	0.338461	0.169231	+	0.985576i	$0.445872\pi$
$830$	19.6468	0.681950
$831$	7.74498	0.268671
$832$	−5.94146	−0.205983
$833$	−80.1102	−2.77566
$834$	28.7851	0.996747
$835$	−2.85640	−0.0988499
$836$	−31.8396	−1.10120
$837$	25.9356	0.896464
$838$	−24.6505	−0.851539
$839$	−32.8480	−1.13404	−0.567020	−	0.823704i	$-0.691904\pi$
$839$	−32.8480	−1.13404	−0.567020	+	0.823704i	$0.691904\pi$
$840$	15.0713	0.520011
$841$	10.1907	0.351404
$842$	−7.72683	−0.266284
$843$	41.8673	1.44199
$844$	−11.1026	−0.382167
$845$	29.0582	0.999632
$846$	−31.9641	−1.09895
$847$	41.2984	1.41903
$848$	−2.23194	−0.0766451
$849$	−72.5754	−2.49078
$850$	−25.7148	−0.882011
$851$	−1.14261	−0.0391683
$852$	−37.0980	−1.27096
$853$	−41.6317	−1.42544	−0.712721	−	0.701447i	$-0.752538\pi$
$853$	−41.6317	−1.42544	−0.712721	+	0.701447i	$0.752538\pi$
$854$	4.41347	0.151026
$855$	42.9720	1.46961
$856$	−10.5950	−0.362130
$857$	48.9436	1.67188	0.835941	−	0.548820i	$-0.184923\pi$
$857$	48.9436	1.67188	0.835941	+	0.548820i	$0.184923\pi$
$858$	75.6220	2.58169
$859$	−32.0514	−1.09358	−0.546790	−	0.837270i	$-0.684150\pi$
$859$	−32.0514	−1.09358	−0.546790	+	0.837270i	$0.684150\pi$
$860$	13.4286	0.457911
$861$	−95.1027	−3.24109
$862$	−21.9308	−0.746965
$863$	12.6359	0.430132	0.215066	−	0.976600i	$-0.431003\pi$
$863$	12.6359	0.430132	0.215066	+	0.976600i	$0.431003\pi$
$864$	4.83771	0.164582
$865$	−14.9788	−0.509293
$866$	28.8745	0.981197
$867$	−121.404	−4.12310
$868$	−22.2906	−0.756591
$869$	41.9257	1.42223
$870$	22.6923	0.769341
$871$	59.9236	2.03043
$872$	−15.6549	−0.530141
$873$	53.2435	1.80202
$874$	4.61092	0.155967
$875$	−44.9782	−1.52054
$876$	−9.35239	−0.315988
$877$	53.1701	1.79543	0.897713	−	0.440580i	$-0.145227\pi$
$877$	53.1701	1.79543	0.897713	+	0.440580i	$0.145227\pi$
$878$	0.967477	0.0326508
$879$	27.3975	0.924096
$880$	−5.96153	−0.200963
$881$	−2.08239	−0.0701573	−0.0350787	−	0.999385i	$-0.511168\pi$
$881$	−2.08239	−0.0701573	−0.0350787	+	0.999385i	$0.511168\pi$
$882$	−48.7519	−1.64156
$883$	27.9513	0.940637	0.470319	−	0.882497i	$-0.344139\pi$
$883$	27.9513	0.940637	0.470319	+	0.882497i	$0.344139\pi$
$884$	46.2674	1.55614
$885$	−1.52950	−0.0514136
$886$	2.79407	0.0938687
$887$	38.4939	1.29250	0.646249	−	0.763127i	$-0.276337\pi$
$887$	38.4939	1.29250	0.646249	+	0.763127i	$0.276337\pi$
$888$	4.79744	0.160991
$889$	−30.3568	−1.01813
$890$	10.4085	0.348892
$891$	3.47232	0.116327
$892$	−25.5562	−0.855684
$893$	46.9388	1.57075
$894$	−23.1402	−0.773925
$895$	0.222279	0.00742996
$896$	−4.15781	−0.138903
$897$	−10.9513	−0.365655
$898$	−4.13801	−0.138087
$899$	−33.5620	−1.11936
$900$	−15.6490	−0.521635
$901$	17.3806	0.579031
$902$	37.6183	1.25255
$903$	−119.205	−3.96688
$904$	8.12771	0.270324
$905$	26.3999	0.877562
$906$	16.8504	0.559818
$907$	1.47539	0.0489896	0.0244948	−	0.999700i	$-0.492202\pi$
$907$	1.47539	0.0489896	0.0244948	+	0.999700i	$0.492202\pi$
$908$	−22.0997	−0.733403
$909$	13.9829	0.463783
$910$	32.1887	1.06704
$911$	4.48059	0.148449	0.0742243	−	0.997242i	$-0.476352\pi$
$911$	4.48059	0.148449	0.0742243	+	0.997242i	$0.476352\pi$
$912$	−19.3596	−0.641062
$913$	68.9858	2.28310
$914$	−30.4298	−1.00653
$915$	3.84771	0.127201
$916$	−15.3567	−0.507398
$917$	73.6414	2.43185
$918$	−37.6722	−1.24337
$919$	−10.6987	−0.352919	−0.176459	−	0.984308i	$-0.556464\pi$
$919$	−10.6987	−0.352919	−0.176459	+	0.984308i	$0.556464\pi$
$920$	0.863330	0.0284632
$921$	80.4289	2.65022
$922$	−0.399993	−0.0131731
$923$	−79.2322	−2.60796
$924$	52.9200	1.74094
$925$	−5.69467	−0.187240
$926$	9.08075	0.298412
$927$	6.53420	0.214611
$928$	−6.26025	−0.205503
$929$	−5.60099	−0.183762	−0.0918812	−	0.995770i	$-0.529288\pi$
$929$	−5.60099	−0.183762	−0.0918812	+	0.995770i	$0.529288\pi$
$930$	−19.4331	−0.637238
$931$	71.5914	2.34631
$932$	−21.5889	−0.707169
$933$	64.9020	2.12479
$934$	5.21980	0.170797
$935$	46.4237	1.51822
$936$	28.1565	0.920325
$937$	−5.71248	−0.186619	−0.0933093	−	0.995637i	$-0.529745\pi$
$937$	−5.71248	−0.186619	−0.0933093	+	0.995637i	$0.529745\pi$
$938$	41.9343	1.36920
$939$	−76.3213	−2.49065
$940$	8.78865	0.286654
$941$	42.9556	1.40031	0.700156	−	0.713990i	$-0.253114\pi$
$941$	42.9556	1.40031	0.700156	+	0.713990i	$0.253114\pi$
$942$	26.2232	0.854398
$943$	−5.44776	−0.177403
$944$	0.421952	0.0137334
$945$	−26.2089	−0.852576
$946$	47.1518	1.53304
$947$	48.6781	1.58183	0.790914	−	0.611928i	$-0.209606\pi$
$947$	48.6781	1.58183	0.790914	+	0.611928i	$0.209606\pi$
$948$	25.4923	0.827952
$949$	−19.9744	−0.648396
$950$	22.9804	0.745581
$951$	−19.0755	−0.618566
$952$	32.3778	1.04937
$953$	−46.4483	−1.50461	−0.752304	−	0.658816i	$-0.771058\pi$
$953$	−46.4483	−1.50461	−0.752304	+	0.658816i	$0.771058\pi$
$954$	10.5771	0.342448
$955$	−31.4117	−1.01646
$956$	−20.5579	−0.664891
$957$	79.6795	2.57567
$958$	33.2661	1.07478
$959$	−40.6358	−1.31220
$960$	−3.62482	−0.116991
$961$	−2.25832	−0.0728490
$962$	10.2461	0.330349
$963$	50.2096	1.61798
$964$	4.13741	0.133257
$965$	15.7370	0.506592
$966$	−7.66371	−0.246576
$967$	29.4686	0.947646	0.473823	−	0.880620i	$-0.342874\pi$
$967$	29.4686	0.947646	0.473823	+	0.880620i	$0.342874\pi$
$968$	−9.93272	−0.319250
$969$	150.758	4.84303
$970$	−14.6395	−0.470045
$971$	−54.5840	−1.75168	−0.875842	−	0.482597i	$-0.839693\pi$
$971$	−54.5840	−1.75168	−0.875842	+	0.482597i	$0.839693\pi$
$972$	16.6244	0.533228
$973$	43.0220	1.37922
$974$	5.61983	0.180071
$975$	−54.5805	−1.74797
$976$	−1.06149	−0.0339774
$977$	−31.3228	−1.00211	−0.501053	−	0.865417i	$-0.667054\pi$
$977$	−31.3228	−1.00211	−0.501053	+	0.865417i	$0.667054\pi$
$978$	−36.2461	−1.15902
$979$	36.5472	1.16805
$980$	13.4045	0.428191
$981$	74.1884	2.36865
$982$	−13.9532	−0.445264
$983$	37.8367	1.20680	0.603402	−	0.797437i	$-0.293811\pi$
$983$	37.8367	1.20680	0.603402	+	0.797437i	$0.293811\pi$
$984$	22.8733	0.729173
$985$	27.4837	0.875703
$986$	48.7499	1.55251
$987$	−78.0161	−2.48328
$988$	−41.3474	−1.31544
$989$	−6.82839	−0.217130
$990$	28.2516	0.897896
$991$	−20.7191	−0.658163	−0.329082	−	0.944301i	$-0.606739\pi$
$991$	−20.7191	−0.658163	−0.329082	+	0.944301i	$0.606739\pi$
$992$	5.36113	0.170216
$993$	−31.2595	−0.991991
$994$	−55.4464	−1.75865
$995$	−26.9893	−0.855617
$996$	41.9459	1.32911
$997$	−46.5670	−1.47479	−0.737396	−	0.675461i	$-0.763945\pi$
$997$	−46.5670	−1.47479	−0.737396	+	0.675461i	$0.763945\pi$
$998$	−17.2033	−0.544562
$999$	−8.34270	−0.263951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.11	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.78190 q^{3} +1.00000 q^{4} +1.30300 q^{5} +2.78190 q^{6} +4.15781 q^{7} -1.00000 q^{8} +4.73899 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.78190 q^{3} +1.00000 q^{4} +1.30300 q^{5} +2.78190 q^{6} +4.15781 q^{7} -1.00000 q^{8} +4.73899 q^{9} -1.30300 q^{10} -4.57523 q^{11} -2.78190 q^{12} -5.94146 q^{13} -4.15781 q^{14} -3.62482 q^{15} +1.00000 q^{16} -7.78721 q^{17} -4.73899 q^{18} +6.95913 q^{19} +1.30300 q^{20} -11.5666 q^{21} +4.57523 q^{22} -0.662571 q^{23} +2.78190 q^{24} -3.30219 q^{25} +5.94146 q^{26} -4.83771 q^{27} +4.15781 q^{28} +6.26025 q^{29} +3.62482 q^{30} -5.36113 q^{31} -1.00000 q^{32} +12.7278 q^{33} +7.78721 q^{34} +5.41764 q^{35} +4.73899 q^{36} +1.72452 q^{37} -6.95913 q^{38} +16.5286 q^{39} -1.30300 q^{40} +8.22216 q^{41} +11.5666 q^{42} +10.3059 q^{43} -4.57523 q^{44} +6.17491 q^{45} +0.662571 q^{46} +6.74493 q^{47} -2.78190 q^{48} +10.2874 q^{49} +3.30219 q^{50} +21.6633 q^{51} -5.94146 q^{52} -2.23194 q^{53} +4.83771 q^{54} -5.96153 q^{55} -4.15781 q^{56} -19.3596 q^{57} -6.26025 q^{58} +0.421952 q^{59} -3.62482 q^{60} -1.06149 q^{61} +5.36113 q^{62} +19.7038 q^{63} +1.00000 q^{64} -7.74173 q^{65} -12.7278 q^{66} -10.0857 q^{67} -7.78721 q^{68} +1.84321 q^{69} -5.41764 q^{70} +13.3355 q^{71} -4.73899 q^{72} +3.36187 q^{73} -1.72452 q^{74} +9.18637 q^{75} +6.95913 q^{76} -19.0229 q^{77} -16.5286 q^{78} -9.16363 q^{79} +1.30300 q^{80} -0.758940 q^{81} -8.22216 q^{82} -15.0781 q^{83} -11.5666 q^{84} -10.1467 q^{85} -10.3059 q^{86} -17.4154 q^{87} +4.57523 q^{88} -7.98806 q^{89} -6.17491 q^{90} -24.7035 q^{91} -0.662571 q^{92} +14.9141 q^{93} -6.74493 q^{94} +9.06776 q^{95} +2.78190 q^{96} +11.2352 q^{97} -10.2874 q^{98} -21.6820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 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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