LMFDB - Embedded newform 8002.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.40667 q^{3} +1.00000 q^{4} +2.18421 q^{5} -3.40667 q^{6} -1.24057 q^{7} +1.00000 q^{8} +8.60538 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.40667 q^{3} +1.00000 q^{4} +2.18421 q^{5} -3.40667 q^{6} -1.24057 q^{7} +1.00000 q^{8} +8.60538 q^{9} +2.18421 q^{10} +3.12896 q^{11} -3.40667 q^{12} -1.99513 q^{13} -1.24057 q^{14} -7.44089 q^{15} +1.00000 q^{16} -2.59036 q^{17} +8.60538 q^{18} +6.38500 q^{19} +2.18421 q^{20} +4.22621 q^{21} +3.12896 q^{22} -2.47389 q^{23} -3.40667 q^{24} -0.229210 q^{25} -1.99513 q^{26} -19.0957 q^{27} -1.24057 q^{28} -7.09981 q^{29} -7.44089 q^{30} +1.66000 q^{31} +1.00000 q^{32} -10.6593 q^{33} -2.59036 q^{34} -2.70967 q^{35} +8.60538 q^{36} -7.59708 q^{37} +6.38500 q^{38} +6.79676 q^{39} +2.18421 q^{40} +10.4866 q^{41} +4.22621 q^{42} -7.73770 q^{43} +3.12896 q^{44} +18.7960 q^{45} -2.47389 q^{46} +2.02672 q^{47} -3.40667 q^{48} -5.46098 q^{49} -0.229210 q^{50} +8.82448 q^{51} -1.99513 q^{52} -12.3391 q^{53} -19.0957 q^{54} +6.83431 q^{55} -1.24057 q^{56} -21.7516 q^{57} -7.09981 q^{58} -5.00543 q^{59} -7.44089 q^{60} -10.1642 q^{61} +1.66000 q^{62} -10.6756 q^{63} +1.00000 q^{64} -4.35780 q^{65} -10.6593 q^{66} +5.74299 q^{67} -2.59036 q^{68} +8.42774 q^{69} -2.70967 q^{70} +3.73012 q^{71} +8.60538 q^{72} +0.446822 q^{73} -7.59708 q^{74} +0.780844 q^{75} +6.38500 q^{76} -3.88169 q^{77} +6.79676 q^{78} +5.38480 q^{79} +2.18421 q^{80} +39.2365 q^{81} +10.4866 q^{82} -4.50553 q^{83} +4.22621 q^{84} -5.65789 q^{85} -7.73770 q^{86} +24.1867 q^{87} +3.12896 q^{88} -12.6524 q^{89} +18.7960 q^{90} +2.47510 q^{91} -2.47389 q^{92} -5.65508 q^{93} +2.02672 q^{94} +13.9462 q^{95} -3.40667 q^{96} -14.8116 q^{97} -5.46098 q^{98} +26.9259 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.40667	−1.96684	−0.983420	−	0.181342i	$-0.941956\pi$
$3$	−3.40667	−1.96684	−0.983420	+	0.181342i	$0.941956\pi$
$4$	1.00000	0.500000
$5$	2.18421	0.976810	0.488405	−	0.872617i	$-0.337579\pi$
$5$	2.18421	0.976810	0.488405	+	0.872617i	$0.337579\pi$
$6$	−3.40667	−1.39077
$7$	−1.24057	−0.468892	−0.234446	−	0.972129i	$-0.575328\pi$
$7$	−1.24057	−0.468892	−0.234446	+	0.972129i	$0.575328\pi$
$8$	1.00000	0.353553
$9$	8.60538	2.86846
$10$	2.18421	0.690709
$11$	3.12896	0.943416	0.471708	−	0.881755i	$-0.343638\pi$
$11$	3.12896	0.943416	0.471708	+	0.881755i	$0.343638\pi$
$12$	−3.40667	−0.983420
$13$	−1.99513	−0.553351	−0.276675	−	0.960963i	$-0.589233\pi$
$13$	−1.99513	−0.553351	−0.276675	+	0.960963i	$0.589233\pi$
$14$	−1.24057	−0.331556
$15$	−7.44089	−1.92123
$16$	1.00000	0.250000
$17$	−2.59036	−0.628254	−0.314127	−	0.949381i	$-0.601712\pi$
$17$	−2.59036	−0.628254	−0.314127	+	0.949381i	$0.601712\pi$
$18$	8.60538	2.02831
$19$	6.38500	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$19$	6.38500	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$20$	2.18421	0.488405
$21$	4.22621	0.922235
$22$	3.12896	0.667096
$23$	−2.47389	−0.515843	−0.257921	−	0.966166i	$-0.583038\pi$
$23$	−2.47389	−0.515843	−0.257921	+	0.966166i	$0.583038\pi$
$24$	−3.40667	−0.695383
$25$	−0.229210	−0.0458421
$26$	−1.99513	−0.391278
$27$	−19.0957	−3.67496
$28$	−1.24057	−0.234446
$29$	−7.09981	−1.31840	−0.659201	−	0.751967i	$-0.729105\pi$
$29$	−7.09981	−1.31840	−0.659201	+	0.751967i	$0.729105\pi$
$30$	−7.44089	−1.35851
$31$	1.66000	0.298145	0.149073	−	0.988826i	$-0.452371\pi$
$31$	1.66000	0.298145	0.149073	+	0.988826i	$0.452371\pi$
$32$	1.00000	0.176777
$33$	−10.6593	−1.85555
$34$	−2.59036	−0.444242
$35$	−2.70967	−0.458018
$36$	8.60538	1.43423
$37$	−7.59708	−1.24895	−0.624476	−	0.781044i	$-0.714688\pi$
$37$	−7.59708	−1.24895	−0.624476	+	0.781044i	$0.714688\pi$
$38$	6.38500	1.03578
$39$	6.79676	1.08835
$40$	2.18421	0.345355
$41$	10.4866	1.63773	0.818864	−	0.573988i	$-0.194604\pi$
$41$	10.4866	1.63773	0.818864	+	0.573988i	$0.194604\pi$
$42$	4.22621	0.652119
$43$	−7.73770	−1.17999	−0.589994	−	0.807408i	$-0.700870\pi$
$43$	−7.73770	−1.17999	−0.589994	+	0.807408i	$0.700870\pi$
$44$	3.12896	0.471708
$45$	18.7960	2.80194
$46$	−2.47389	−0.364756
$47$	2.02672	0.295628	0.147814	−	0.989015i	$-0.452776\pi$
$47$	2.02672	0.295628	0.147814	+	0.989015i	$0.452776\pi$
$48$	−3.40667	−0.491710
$49$	−5.46098	−0.780141
$50$	−0.229210	−0.0324152
$51$	8.82448	1.23567
$52$	−1.99513	−0.276675
$53$	−12.3391	−1.69491	−0.847455	−	0.530867i	$-0.821866\pi$
$53$	−12.3391	−1.69491	−0.847455	+	0.530867i	$0.821866\pi$
$54$	−19.0957	−2.59859
$55$	6.83431	0.921538
$56$	−1.24057	−0.165778
$57$	−21.7516	−2.88107
$58$	−7.09981	−0.932251
$59$	−5.00543	−0.651652	−0.325826	−	0.945430i	$-0.605642\pi$
$59$	−5.00543	−0.651652	−0.325826	+	0.945430i	$0.605642\pi$
$60$	−7.44089	−0.960615
$61$	−10.1642	−1.30140	−0.650699	−	0.759336i	$-0.725524\pi$
$61$	−10.1642	−1.30140	−0.650699	+	0.759336i	$0.725524\pi$
$62$	1.66000	0.210821
$63$	−10.6756	−1.34500
$64$	1.00000	0.125000
$65$	−4.35780	−0.540518
$66$	−10.6593	−1.31207
$67$	5.74299	0.701618	0.350809	−	0.936447i	$-0.385907\pi$
$67$	5.74299	0.701618	0.350809	+	0.936447i	$0.385907\pi$
$68$	−2.59036	−0.314127
$69$	8.42774	1.01458
$70$	−2.70967	−0.323868
$71$	3.73012	0.442684	0.221342	−	0.975196i	$-0.428956\pi$
$71$	3.73012	0.442684	0.221342	+	0.975196i	$0.428956\pi$
$72$	8.60538	1.01415
$73$	0.446822	0.0522965	0.0261483	−	0.999658i	$-0.491676\pi$
$73$	0.446822	0.0522965	0.0261483	+	0.999658i	$0.491676\pi$
$74$	−7.59708	−0.883143
$75$	0.780844	0.0901640
$76$	6.38500	0.732410
$77$	−3.88169	−0.442360
$78$	6.79676	0.769581
$79$	5.38480	0.605837	0.302919	−	0.953016i	$-0.402039\pi$
$79$	5.38480	0.605837	0.302919	+	0.953016i	$0.402039\pi$
$80$	2.18421	0.244203
$81$	39.2365	4.35961
$82$	10.4866	1.15805
$83$	−4.50553	−0.494546	−0.247273	−	0.968946i	$-0.579534\pi$
$83$	−4.50553	−0.494546	−0.247273	+	0.968946i	$0.579534\pi$
$84$	4.22621	0.461117
$85$	−5.65789	−0.613684
$86$	−7.73770	−0.834378
$87$	24.1867	2.59309
$88$	3.12896	0.333548
$89$	−12.6524	−1.34115	−0.670574	−	0.741843i	$-0.733952\pi$
$89$	−12.6524	−1.34115	−0.670574	+	0.741843i	$0.733952\pi$
$90$	18.7960	1.98127
$91$	2.47510	0.259461
$92$	−2.47389	−0.257921
$93$	−5.65508	−0.586404
$94$	2.02672	0.209040
$95$	13.9462	1.43085
$96$	−3.40667	−0.347692
$97$	−14.8116	−1.50389	−0.751943	−	0.659228i	$-0.770883\pi$
$97$	−14.8116	−1.50389	−0.751943	+	0.659228i	$0.770883\pi$
$98$	−5.46098	−0.551643
$99$	26.9259	2.70615
$100$	−0.229210	−0.0229210
$101$	1.04857	0.104336	0.0521682	−	0.998638i	$-0.483387\pi$
$101$	1.04857	0.104336	0.0521682	+	0.998638i	$0.483387\pi$
$102$	8.82448	0.873754
$103$	0.178606	0.0175985	0.00879927	−	0.999961i	$-0.497199\pi$
$103$	0.178606	0.0175985	0.00879927	+	0.999961i	$0.497199\pi$
$104$	−1.99513	−0.195639
$105$	9.23095	0.900848
$106$	−12.3391	−1.19848
$107$	−0.325185	−0.0314368	−0.0157184	−	0.999876i	$-0.505004\pi$
$107$	−0.325185	−0.0314368	−0.0157184	+	0.999876i	$0.505004\pi$
$108$	−19.0957	−1.83748
$109$	14.0632	1.34701	0.673504	−	0.739183i	$-0.264788\pi$
$109$	14.0632	1.34701	0.673504	+	0.739183i	$0.264788\pi$
$110$	6.83431	0.651626
$111$	25.8807	2.45649
$112$	−1.24057	−0.117223
$113$	11.2841	1.06152	0.530760	−	0.847522i	$-0.321907\pi$
$113$	11.2841	1.06152	0.530760	+	0.847522i	$0.321907\pi$
$114$	−21.7516	−2.03722
$115$	−5.40351	−0.503880
$116$	−7.09981	−0.659201
$117$	−17.1689	−1.58726
$118$	−5.00543	−0.460787
$119$	3.21352	0.294583
$120$	−7.44089	−0.679257
$121$	−1.20962	−0.109966
$122$	−10.1642	−0.920227
$123$	−35.7243	−3.22115
$124$	1.66000	0.149073
$125$	−11.4217	−1.02159
$126$	−10.6756	−0.951057
$127$	13.3360	1.18338	0.591690	−	0.806166i	$-0.298461\pi$
$127$	13.3360	1.18338	0.591690	+	0.806166i	$0.298461\pi$
$128$	1.00000	0.0883883
$129$	26.3598	2.32085
$130$	−4.35780	−0.382204
$131$	−4.62266	−0.403883	−0.201942	−	0.979398i	$-0.564725\pi$
$131$	−4.62266	−0.403883	−0.201942	+	0.979398i	$0.564725\pi$
$132$	−10.6593	−0.927775
$133$	−7.92104	−0.686841
$134$	5.74299	0.496119
$135$	−41.7090	−3.58974
$136$	−2.59036	−0.222121
$137$	2.86310	0.244611	0.122306	−	0.992492i	$-0.460971\pi$
$137$	2.86310	0.244611	0.122306	+	0.992492i	$0.460971\pi$
$138$	8.42774	0.717417
$139$	5.34512	0.453367	0.226684	−	0.973968i	$-0.427212\pi$
$139$	5.34512	0.453367	0.226684	+	0.973968i	$0.427212\pi$
$140$	−2.70967	−0.229009
$141$	−6.90436	−0.581452
$142$	3.73012	0.313025
$143$	−6.24269	−0.522040
$144$	8.60538	0.717115
$145$	−15.5075	−1.28783
$146$	0.446822	0.0369792
$147$	18.6038	1.53441
$148$	−7.59708	−0.624476
$149$	−19.8374	−1.62515	−0.812573	−	0.582860i	$-0.801934\pi$
$149$	−19.8374	−1.62515	−0.812573	+	0.582860i	$0.801934\pi$
$150$	0.780844	0.0637556
$151$	−19.1586	−1.55911	−0.779553	−	0.626336i	$-0.784554\pi$
$151$	−19.1586	−1.55911	−0.779553	+	0.626336i	$0.784554\pi$
$152$	6.38500	0.517892
$153$	−22.2910	−1.80212
$154$	−3.88169	−0.312796
$155$	3.62580	0.291231
$156$	6.79676	0.544176
$157$	−1.99863	−0.159508	−0.0797540	−	0.996815i	$-0.525413\pi$
$157$	−1.99863	−0.159508	−0.0797540	+	0.996815i	$0.525413\pi$
$158$	5.38480	0.428392
$159$	42.0353	3.33362
$160$	2.18421	0.172677
$161$	3.06904	0.241874
$162$	39.2365	3.08271
$163$	20.0518	1.57058	0.785290	−	0.619128i	$-0.212514\pi$
$163$	20.0518	1.57058	0.785290	+	0.619128i	$0.212514\pi$
$164$	10.4866	0.818864
$165$	−23.2822	−1.81252
$166$	−4.50553	−0.349697
$167$	5.91510	0.457724	0.228862	−	0.973459i	$-0.426500\pi$
$167$	5.91510	0.457724	0.228862	+	0.973459i	$0.426500\pi$
$168$	4.22621	0.326059
$169$	−9.01944	−0.693803
$170$	−5.65789	−0.433940
$171$	54.9454	4.20178
$172$	−7.73770	−0.589994
$173$	6.42835	0.488738	0.244369	−	0.969682i	$-0.421419\pi$
$173$	6.42835	0.488738	0.244369	+	0.969682i	$0.421419\pi$
$174$	24.1867	1.83359
$175$	0.284352	0.0214950
$176$	3.12896	0.235854
$177$	17.0518	1.28170
$178$	−12.6524	−0.948335
$179$	6.81874	0.509656	0.254828	−	0.966986i	$-0.417981\pi$
$179$	6.81874	0.509656	0.254828	+	0.966986i	$0.417981\pi$
$180$	18.7960	1.40097
$181$	3.76552	0.279889	0.139944	−	0.990159i	$-0.455308\pi$
$181$	3.76552	0.279889	0.139944	+	0.990159i	$0.455308\pi$
$182$	2.47510	0.183467
$183$	34.6262	2.55964
$184$	−2.47389	−0.182378
$185$	−16.5936	−1.21999
$186$	−5.65508	−0.414650
$187$	−8.10511	−0.592705
$188$	2.02672	0.147814
$189$	23.6895	1.72316
$190$	13.9462	1.01176
$191$	7.37336	0.533517	0.266759	−	0.963763i	$-0.414047\pi$
$191$	7.37336	0.533517	0.266759	+	0.963763i	$0.414047\pi$
$192$	−3.40667	−0.245855
$193$	15.6558	1.12693	0.563466	−	0.826140i	$-0.309468\pi$
$193$	15.6558	1.12693	0.563466	+	0.826140i	$0.309468\pi$
$194$	−14.8116	−1.06341
$195$	14.8456	1.06311
$196$	−5.46098	−0.390070
$197$	1.29758	0.0924485	0.0462242	−	0.998931i	$-0.485281\pi$
$197$	1.29758	0.0924485	0.0462242	+	0.998931i	$0.485281\pi$
$198$	26.9259	1.91354
$199$	−23.4585	−1.66293	−0.831463	−	0.555580i	$-0.812496\pi$
$199$	−23.4585	−1.66293	−0.831463	+	0.555580i	$0.812496\pi$
$200$	−0.229210	−0.0162076
$201$	−19.5645	−1.37997
$202$	1.04857	0.0737769
$203$	8.80782	0.618187
$204$	8.82448	0.617837
$205$	22.9049	1.59975
$206$	0.178606	0.0124440
$207$	−21.2888	−1.47967
$208$	−1.99513	−0.138338
$209$	19.9784	1.38193
$210$	9.23095	0.636996
$211$	−1.71669	−0.118182	−0.0590909	−	0.998253i	$-0.518820\pi$
$211$	−1.71669	−0.118182	−0.0590909	+	0.998253i	$0.518820\pi$
$212$	−12.3391	−0.847455
$213$	−12.7073	−0.870688
$214$	−0.325185	−0.0222292
$215$	−16.9008	−1.15262
$216$	−19.0957	−1.29930
$217$	−2.05935	−0.139798
$218$	14.0632	0.952479
$219$	−1.52217	−0.102859
$220$	6.83431	0.460769
$221$	5.16811	0.347644
$222$	25.8807	1.73700
$223$	−12.2684	−0.821554	−0.410777	−	0.911736i	$-0.634742\pi$
$223$	−12.2684	−0.821554	−0.410777	+	0.911736i	$0.634742\pi$
$224$	−1.24057	−0.0828891
$225$	−1.97244	−0.131496
$226$	11.2841	0.750608
$227$	−11.2272	−0.745173	−0.372586	−	0.927998i	$-0.621529\pi$
$227$	−11.2272	−0.745173	−0.372586	+	0.927998i	$0.621529\pi$
$228$	−21.7516	−1.44053
$229$	21.3645	1.41180	0.705902	−	0.708309i	$-0.250542\pi$
$229$	21.3645	1.41180	0.705902	+	0.708309i	$0.250542\pi$
$230$	−5.40351	−0.356297
$231$	13.2236	0.870051
$232$	−7.09981	−0.466125
$233$	−16.5906	−1.08689	−0.543443	−	0.839446i	$-0.682880\pi$
$233$	−16.5906	−1.08689	−0.543443	+	0.839446i	$0.682880\pi$
$234$	−17.1689	−1.12237
$235$	4.42679	0.288772
$236$	−5.00543	−0.325826
$237$	−18.3442	−1.19159
$238$	3.21352	0.208301
$239$	17.3345	1.12127	0.560636	−	0.828062i	$-0.310557\pi$
$239$	17.3345	1.12127	0.560636	+	0.828062i	$0.310557\pi$
$240$	−7.44089	−0.480307
$241$	−19.4544	−1.25317	−0.626584	−	0.779354i	$-0.715547\pi$
$241$	−19.4544	−1.25317	−0.626584	+	0.779354i	$0.715547\pi$
$242$	−1.20962	−0.0777576
$243$	−76.3786	−4.89969
$244$	−10.1642	−0.650699
$245$	−11.9280	−0.762049
$246$	−35.7243	−2.27770
$247$	−12.7389	−0.810558
$248$	1.66000	0.105410
$249$	15.3488	0.972692
$250$	−11.4217	−0.722373
$251$	−3.45087	−0.217817	−0.108908	−	0.994052i	$-0.534736\pi$
$251$	−3.45087	−0.217817	−0.108908	+	0.994052i	$0.534736\pi$
$252$	−10.6756	−0.672499
$253$	−7.74071	−0.486654
$254$	13.3360	0.836776
$255$	19.2746	1.20702
$256$	1.00000	0.0625000
$257$	−20.1091	−1.25437	−0.627185	−	0.778871i	$-0.715793\pi$
$257$	−20.1091	−1.25437	−0.627185	+	0.778871i	$0.715793\pi$
$258$	26.3598	1.64109
$259$	9.42471	0.585623
$260$	−4.35780	−0.270259
$261$	−61.0966	−3.78178
$262$	−4.62266	−0.285589
$263$	15.8997	0.980415	0.490208	−	0.871606i	$-0.336921\pi$
$263$	15.8997	0.980415	0.490208	+	0.871606i	$0.336921\pi$
$264$	−10.6593	−0.656036
$265$	−26.9513	−1.65561
$266$	−7.92104	−0.485670
$267$	43.1024	2.63782
$268$	5.74299	0.350809
$269$	21.0403	1.28285	0.641425	−	0.767186i	$-0.278344\pi$
$269$	21.0403	1.28285	0.641425	+	0.767186i	$0.278344\pi$
$270$	−41.7090	−2.53833
$271$	9.09976	0.552771	0.276386	−	0.961047i	$-0.410863\pi$
$271$	9.09976	0.552771	0.276386	+	0.961047i	$0.410863\pi$
$272$	−2.59036	−0.157063
$273$	−8.43186	−0.510319
$274$	2.86310	0.172966
$275$	−0.717190	−0.0432482
$276$	8.42774	0.507290
$277$	−17.5912	−1.05695	−0.528477	−	0.848948i	$-0.677237\pi$
$277$	−17.5912	−1.05695	−0.528477	+	0.848948i	$0.677237\pi$
$278$	5.34512	0.320579
$279$	14.2850	0.855218
$280$	−2.70967	−0.161934
$281$	−4.98580	−0.297428	−0.148714	−	0.988880i	$-0.547513\pi$
$281$	−4.98580	−0.297428	−0.148714	+	0.988880i	$0.547513\pi$
$282$	−6.90436	−0.411149
$283$	−3.98301	−0.236765	−0.118383	−	0.992968i	$-0.537771\pi$
$283$	−3.98301	−0.236765	−0.118383	+	0.992968i	$0.537771\pi$
$284$	3.73012	0.221342
$285$	−47.5101	−2.81425
$286$	−6.24269	−0.369138
$287$	−13.0093	−0.767916
$288$	8.60538	0.507077
$289$	−10.2901	−0.605298
$290$	−15.5075	−0.910632
$291$	50.4581	2.95790
$292$	0.446822	0.0261483
$293$	−12.2987	−0.718497	−0.359249	−	0.933242i	$-0.616967\pi$
$293$	−12.2987	−0.718497	−0.359249	+	0.933242i	$0.616967\pi$
$294$	18.6038	1.08499
$295$	−10.9329	−0.636540
$296$	−7.59708	−0.441571
$297$	−59.7496	−3.46702
$298$	−19.8374	−1.14915
$299$	4.93575	0.285442
$300$	0.780844	0.0450820
$301$	9.59916	0.553286
$302$	−19.1586	−1.10245
$303$	−3.57212	−0.205213
$304$	6.38500	0.366205
$305$	−22.2009	−1.27122
$306$	−22.2910	−1.27429
$307$	−4.59629	−0.262324	−0.131162	−	0.991361i	$-0.541871\pi$
$307$	−4.59629	−0.262324	−0.131162	+	0.991361i	$0.541871\pi$
$308$	−3.88169	−0.221180
$309$	−0.608450	−0.0346135
$310$	3.62580	0.205932
$311$	−11.0806	−0.628323	−0.314162	−	0.949370i	$-0.601723\pi$
$311$	−11.0806	−0.628323	−0.314162	+	0.949370i	$0.601723\pi$
$312$	6.79676	0.384791
$313$	1.73003	0.0977869	0.0488935	−	0.998804i	$-0.484431\pi$
$313$	1.73003	0.0977869	0.0488935	+	0.998804i	$0.484431\pi$
$314$	−1.99863	−0.112789
$315$	−23.3178	−1.31381
$316$	5.38480	0.302919
$317$	−14.5987	−0.819946	−0.409973	−	0.912098i	$-0.634462\pi$
$317$	−14.5987	−0.819946	−0.409973	+	0.912098i	$0.634462\pi$
$318$	42.0353	2.35722
$319$	−22.2150	−1.24380
$320$	2.18421	0.122101
$321$	1.10780	0.0618312
$322$	3.06904	0.171031
$323$	−16.5394	−0.920278
$324$	39.2365	2.17980
$325$	0.457305	0.0253667
$326$	20.0518	1.11057
$327$	−47.9086	−2.64935
$328$	10.4866	0.579024
$329$	−2.51429	−0.138617
$330$	−23.2822	−1.28164
$331$	−0.810115	−0.0445280	−0.0222640	−	0.999752i	$-0.507087\pi$
$331$	−0.810115	−0.0445280	−0.0222640	+	0.999752i	$0.507087\pi$
$332$	−4.50553	−0.247273
$333$	−65.3758	−3.58257
$334$	5.91510	0.323660
$335$	12.5439	0.685347
$336$	4.22621	0.230559
$337$	35.1262	1.91345	0.956724	−	0.290998i	$-0.0939874\pi$
$337$	35.1262	1.91345	0.956724	+	0.290998i	$0.0939874\pi$
$338$	−9.01944	−0.490593
$339$	−38.4412	−2.08784
$340$	−5.65789	−0.306842
$341$	5.19408	0.281275
$342$	54.9454	2.97110
$343$	15.4587	0.834693
$344$	−7.73770	−0.417189
$345$	18.4080	0.991052
$346$	6.42835	0.345590
$347$	−10.8891	−0.584559	−0.292279	−	0.956333i	$-0.594414\pi$
$347$	−10.8891	−0.584559	−0.292279	+	0.956333i	$0.594414\pi$
$348$	24.1867	1.29654
$349$	−23.8404	−1.27615	−0.638074	−	0.769975i	$-0.720269\pi$
$349$	−23.8404	−1.27615	−0.638074	+	0.769975i	$0.720269\pi$
$350$	0.284352	0.0151992
$351$	38.0984	2.03354
$352$	3.12896	0.166774
$353$	0.117951	0.00627791	0.00313896	−	0.999995i	$-0.499001\pi$
$353$	0.117951	0.00627791	0.00313896	+	0.999995i	$0.499001\pi$
$354$	17.0518	0.906295
$355$	8.14738	0.432418
$356$	−12.6524	−0.670574
$357$	−10.9474	−0.579397
$358$	6.81874	0.360381
$359$	−18.7826	−0.991310	−0.495655	−	0.868520i	$-0.665072\pi$
$359$	−18.7826	−0.991310	−0.495655	+	0.868520i	$0.665072\pi$
$360$	18.7960	0.990636
$361$	21.7682	1.14570
$362$	3.76552	0.197911
$363$	4.12079	0.216285
$364$	2.47510	0.129731
$365$	0.975954	0.0510838
$366$	34.6262	1.80994
$367$	−27.4418	−1.43245	−0.716224	−	0.697870i	$-0.754131\pi$
$367$	−27.4418	−1.43245	−0.716224	+	0.697870i	$0.754131\pi$
$368$	−2.47389	−0.128961
$369$	90.2410	4.69776
$370$	−16.5936	−0.862663
$371$	15.3076	0.794729
$372$	−5.65508	−0.293202
$373$	−14.4700	−0.749228	−0.374614	−	0.927181i	$-0.622225\pi$
$373$	−14.4700	−0.749228	−0.374614	+	0.927181i	$0.622225\pi$
$374$	−8.10511	−0.419105
$375$	38.9100	2.00930
$376$	2.02672	0.104520
$377$	14.1651	0.729538
$378$	23.6895	1.21846
$379$	4.00097	0.205516	0.102758	−	0.994706i	$-0.467233\pi$
$379$	4.00097	0.205516	0.102758	+	0.994706i	$0.467233\pi$
$380$	13.9462	0.715425
$381$	−45.4313	−2.32752
$382$	7.37336	0.377254
$383$	5.83486	0.298147	0.149074	−	0.988826i	$-0.452371\pi$
$383$	5.83486	0.298147	0.149074	+	0.988826i	$0.452371\pi$
$384$	−3.40667	−0.173846
$385$	−8.47845	−0.432102
$386$	15.6558	0.796861
$387$	−66.5859	−3.38475
$388$	−14.8116	−0.751943
$389$	1.54890	0.0785326	0.0392663	−	0.999229i	$-0.487498\pi$
$389$	1.54890	0.0785326	0.0392663	+	0.999229i	$0.487498\pi$
$390$	14.8456	0.751735
$391$	6.40827	0.324080
$392$	−5.46098	−0.275821
$393$	15.7479	0.794374
$394$	1.29758	0.0653709
$395$	11.7616	0.591788
$396$	26.9259	1.35308
$397$	8.79677	0.441497	0.220749	−	0.975331i	$-0.429150\pi$
$397$	8.79677	0.441497	0.220749	+	0.975331i	$0.429150\pi$
$398$	−23.4585	−1.17587
$399$	26.9844	1.35091
$400$	−0.229210	−0.0114605
$401$	−21.7856	−1.08792	−0.543961	−	0.839110i	$-0.683076\pi$
$401$	−21.7856	−1.08792	−0.543961	+	0.839110i	$0.683076\pi$
$402$	−19.5645	−0.975786
$403$	−3.31193	−0.164979
$404$	1.04857	0.0521682
$405$	85.7008	4.25851
$406$	8.80782	0.437125
$407$	−23.7709	−1.17828
$408$	8.82448	0.436877
$409$	−1.33713	−0.0661167	−0.0330584	−	0.999453i	$-0.510525\pi$
$409$	−1.33713	−0.0661167	−0.0330584	+	0.999453i	$0.510525\pi$
$410$	22.9049	1.13119
$411$	−9.75363	−0.481111
$412$	0.178606	0.00879927
$413$	6.20959	0.305554
$414$	−21.2888	−1.04629
$415$	−9.84103	−0.483077
$416$	−1.99513	−0.0978195
$417$	−18.2090	−0.891701
$418$	19.9784	0.977175
$419$	−37.1393	−1.81438	−0.907188	−	0.420726i	$-0.861775\pi$
$419$	−37.1393	−1.81438	−0.907188	+	0.420726i	$0.861775\pi$
$420$	9.23095	0.450424
$421$	22.0995	1.07706	0.538531	−	0.842605i	$-0.318979\pi$
$421$	22.0995	1.07706	0.538531	+	0.842605i	$0.318979\pi$
$422$	−1.71669	−0.0835671
$423$	17.4407	0.847996
$424$	−12.3391	−0.599241
$425$	0.593736	0.0288004
$426$	−12.7073	−0.615670
$427$	12.6095	0.610214
$428$	−0.325185	−0.0157184
$429$	21.2668	1.02677
$430$	−16.9008	−0.815028
$431$	16.6790	0.803398	0.401699	−	0.915772i	$-0.368420\pi$
$431$	16.6790	0.803398	0.401699	+	0.915772i	$0.368420\pi$
$432$	−19.0957	−0.918741
$433$	15.0939	0.725368	0.362684	−	0.931912i	$-0.381861\pi$
$433$	15.0939	0.725368	0.362684	+	0.931912i	$0.381861\pi$
$434$	−2.05935	−0.0988520
$435$	52.8289	2.53295
$436$	14.0632	0.673504
$437$	−15.7958	−0.755616
$438$	−1.52217	−0.0727322
$439$	10.2442	0.488931	0.244465	−	0.969658i	$-0.421388\pi$
$439$	10.2442	0.488931	0.244465	+	0.969658i	$0.421388\pi$
$440$	6.83431	0.325813
$441$	−46.9939	−2.23780
$442$	5.16811	0.245822
$443$	0.727793	0.0345785	0.0172892	−	0.999851i	$-0.494496\pi$
$443$	0.727793	0.0345785	0.0172892	+	0.999851i	$0.494496\pi$
$444$	25.8807	1.22825
$445$	−27.6355	−1.31005
$446$	−12.2684	−0.580926
$447$	67.5795	3.19640
$448$	−1.24057	−0.0586114
$449$	32.5788	1.53749	0.768743	−	0.639557i	$-0.220882\pi$
$449$	32.5788	1.53749	0.768743	+	0.639557i	$0.220882\pi$
$450$	−1.97244	−0.0929819
$451$	32.8120	1.54506
$452$	11.2841	0.530760
$453$	65.2670	3.06651
$454$	−11.2272	−0.526917
$455$	5.40616	0.253445
$456$	−21.7516	−1.01861
$457$	−22.4254	−1.04902	−0.524509	−	0.851405i	$-0.675751\pi$
$457$	−22.4254	−1.04902	−0.524509	+	0.851405i	$0.675751\pi$
$458$	21.3645	0.998297
$459$	49.4646	2.30881
$460$	−5.40351	−0.251940
$461$	1.07914	0.0502608	0.0251304	−	0.999684i	$-0.492000\pi$
$461$	1.07914	0.0502608	0.0251304	+	0.999684i	$0.492000\pi$
$462$	13.2236	0.615219
$463$	−27.2368	−1.26580	−0.632900	−	0.774234i	$-0.718135\pi$
$463$	−27.2368	−1.26580	−0.632900	+	0.774234i	$0.718135\pi$
$464$	−7.09981	−0.329600
$465$	−12.3519	−0.572806
$466$	−16.5906	−0.768545
$467$	13.6387	0.631123	0.315561	−	0.948905i	$-0.397807\pi$
$467$	13.6387	0.631123	0.315561	+	0.948905i	$0.397807\pi$
$468$	−17.1689	−0.793632
$469$	−7.12458	−0.328983
$470$	4.42679	0.204193
$471$	6.80866	0.313727
$472$	−5.00543	−0.230394
$473$	−24.2109	−1.11322
$474$	−18.3442	−0.842578
$475$	−1.46351	−0.0671504
$476$	3.21352	0.147291
$477$	−106.183	−4.86178
$478$	17.3345	0.792859
$479$	−38.8629	−1.77569	−0.887845	−	0.460142i	$-0.847799\pi$
$479$	−38.8629	−1.77569	−0.887845	+	0.460142i	$0.847799\pi$
$480$	−7.44089	−0.339629
$481$	15.1572	0.691109
$482$	−19.4544	−0.886124
$483$	−10.4552	−0.475728
$484$	−1.20962	−0.0549829
$485$	−32.3516	−1.46901
$486$	−76.3786	−3.46460
$487$	−34.2190	−1.55061	−0.775305	−	0.631587i	$-0.782404\pi$
$487$	−34.2190	−1.55061	−0.775305	+	0.631587i	$0.782404\pi$
$488$	−10.1642	−0.460114
$489$	−68.3099	−3.08908
$490$	−11.9280	−0.538850
$491$	28.9422	1.30614	0.653072	−	0.757296i	$-0.273480\pi$
$491$	28.9422	1.30614	0.653072	+	0.757296i	$0.273480\pi$
$492$	−35.7243	−1.61057
$493$	18.3910	0.828290
$494$	−12.7389	−0.573151
$495$	58.8119	2.64340
$496$	1.66000	0.0745363
$497$	−4.62748	−0.207571
$498$	15.3488	0.687797
$499$	−32.0218	−1.43349	−0.716746	−	0.697334i	$-0.754369\pi$
$499$	−32.0218	−1.43349	−0.716746	+	0.697334i	$0.754369\pi$
$500$	−11.4217	−0.510795
$501$	−20.1508	−0.900270
$502$	−3.45087	−0.154020
$503$	22.1446	0.987380	0.493690	−	0.869638i	$-0.335648\pi$
$503$	22.1446	0.987380	0.493690	+	0.869638i	$0.335648\pi$
$504$	−10.6756	−0.475528
$505$	2.29029	0.101917
$506$	−7.74071	−0.344117
$507$	30.7262	1.36460
$508$	13.3360	0.591690
$509$	−3.27128	−0.144997	−0.0724984	−	0.997369i	$-0.523097\pi$
$509$	−3.27128	−0.144997	−0.0724984	+	0.997369i	$0.523097\pi$
$510$	19.2746	0.853491
$511$	−0.554314	−0.0245214
$512$	1.00000	0.0441942
$513$	−121.926	−5.38316
$514$	−20.1091	−0.886973
$515$	0.390113	0.0171904
$516$	26.3598	1.16042
$517$	6.34152	0.278900
$518$	9.42471	0.414098
$519$	−21.8993	−0.961271
$520$	−4.35780	−0.191102
$521$	−34.2898	−1.50226	−0.751132	−	0.660152i	$-0.770492\pi$
$521$	−34.2898	−1.50226	−0.751132	+	0.660152i	$0.770492\pi$
$522$	−61.0966	−2.67412
$523$	−35.5228	−1.55331	−0.776653	−	0.629929i	$-0.783084\pi$
$523$	−35.5228	−1.55331	−0.776653	+	0.629929i	$0.783084\pi$
$524$	−4.62266	−0.201942
$525$	−0.968691	−0.0422772
$526$	15.8997	0.693258
$527$	−4.30000	−0.187311
$528$	−10.6593	−0.463887
$529$	−16.8798	−0.733906
$530$	−26.9513	−1.17069
$531$	−43.0737	−1.86924
$532$	−7.92104	−0.343421
$533$	−20.9221	−0.906237
$534$	43.1024	1.86522
$535$	−0.710273	−0.0307078
$536$	5.74299	0.248059
$537$	−23.2292	−1.00241
$538$	21.0403	0.907111
$539$	−17.0872	−0.735997
$540$	−41.7090	−1.79487
$541$	10.3266	0.443976	0.221988	−	0.975049i	$-0.428745\pi$
$541$	10.3266	0.443976	0.221988	+	0.975049i	$0.428745\pi$
$542$	9.09976	0.390868
$543$	−12.8279	−0.550497
$544$	−2.59036	−0.111061
$545$	30.7170	1.31577
$546$	−8.43186	−0.360850
$547$	6.79187	0.290399	0.145200	−	0.989402i	$-0.453618\pi$
$547$	6.79187	0.290399	0.145200	+	0.989402i	$0.453618\pi$
$548$	2.86310	0.122306
$549$	−87.4672	−3.73301
$550$	−0.717190	−0.0305811
$551$	−45.3323	−1.93122
$552$	8.42774	0.358708
$553$	−6.68022	−0.284072
$554$	−17.5912	−0.747379
$555$	56.5290	2.39952
$556$	5.34512	0.226684
$557$	22.2839	0.944199	0.472099	−	0.881545i	$-0.343496\pi$
$557$	22.2839	0.944199	0.472099	+	0.881545i	$0.343496\pi$
$558$	14.2850	0.604731
$559$	15.4377	0.652947
$560$	−2.70967	−0.114505
$561$	27.6114	1.16576
$562$	−4.98580	−0.210313
$563$	−9.61350	−0.405161	−0.202580	−	0.979266i	$-0.564933\pi$
$563$	−9.61350	−0.405161	−0.202580	+	0.979266i	$0.564933\pi$
$564$	−6.90436	−0.290726
$565$	24.6469	1.03690
$566$	−3.98301	−0.167418
$567$	−48.6756	−2.04418
$568$	3.73012	0.156512
$569$	23.3837	0.980298	0.490149	−	0.871639i	$-0.336942\pi$
$569$	23.3837	0.980298	0.490149	+	0.871639i	$0.336942\pi$
$570$	−47.5101	−1.98998
$571$	−17.3851	−0.727546	−0.363773	−	0.931488i	$-0.618512\pi$
$571$	−17.3851	−0.727546	−0.363773	+	0.931488i	$0.618512\pi$
$572$	−6.24269	−0.261020
$573$	−25.1186	−1.04934
$574$	−13.0093	−0.542999
$575$	0.567042	0.0236473
$576$	8.60538	0.358558
$577$	−33.3301	−1.38755	−0.693774	−	0.720192i	$-0.744054\pi$
$577$	−33.3301	−1.38755	−0.693774	+	0.720192i	$0.744054\pi$
$578$	−10.2901	−0.428010
$579$	−53.3342	−2.21649
$580$	−15.5075	−0.643914
$581$	5.58942	0.231888
$582$	50.4581	2.09155
$583$	−38.6086	−1.59901
$584$	0.446822	0.0184896
$585$	−37.5005	−1.55046
$586$	−12.2987	−0.508054
$587$	−17.6014	−0.726486	−0.363243	−	0.931694i	$-0.618331\pi$
$587$	−17.6014	−0.726486	−0.363243	+	0.931694i	$0.618331\pi$
$588$	18.6038	0.767206
$589$	10.5991	0.436729
$590$	−10.9329	−0.450102
$591$	−4.42041	−0.181831
$592$	−7.59708	−0.312238
$593$	7.78790	0.319811	0.159905	−	0.987132i	$-0.448881\pi$
$593$	7.78790	0.319811	0.159905	+	0.987132i	$0.448881\pi$
$594$	−59.7496	−2.45155
$595$	7.01901	0.287751
$596$	−19.8374	−0.812573
$597$	79.9151	3.27071
$598$	4.93575	0.201838
$599$	35.4160	1.44706	0.723529	−	0.690294i	$-0.242519\pi$
$599$	35.4160	1.44706	0.723529	+	0.690294i	$0.242519\pi$
$600$	0.780844	0.0318778
$601$	2.75677	0.112451	0.0562254	−	0.998418i	$-0.482093\pi$
$601$	2.75677	0.112451	0.0562254	+	0.998418i	$0.482093\pi$
$602$	9.59916	0.391233
$603$	49.4206	2.01256
$604$	−19.1586	−0.779553
$605$	−2.64208	−0.107416
$606$	−3.57212	−0.145107
$607$	−33.1773	−1.34662	−0.673312	−	0.739358i	$-0.735129\pi$
$607$	−33.1773	−1.34662	−0.673312	+	0.739358i	$0.735129\pi$
$608$	6.38500	0.258946
$609$	−30.0053	−1.21588
$610$	−22.2009	−0.898887
$611$	−4.04358	−0.163586
$612$	−22.2910	−0.901060
$613$	16.3866	0.661848	0.330924	−	0.943657i	$-0.392640\pi$
$613$	16.3866	0.661848	0.330924	+	0.943657i	$0.392640\pi$
$614$	−4.59629	−0.185491
$615$	−78.0294	−3.14645
$616$	−3.88169	−0.156398
$617$	29.2740	1.17853	0.589264	−	0.807940i	$-0.299418\pi$
$617$	29.2740	1.17853	0.589264	+	0.807940i	$0.299418\pi$
$618$	−0.608450	−0.0244754
$619$	−31.7136	−1.27468	−0.637338	−	0.770584i	$-0.719965\pi$
$619$	−31.7136	−1.27468	−0.637338	+	0.770584i	$0.719965\pi$
$620$	3.62580	0.145616
$621$	47.2407	1.89570
$622$	−11.0806	−0.444292
$623$	15.6962	0.628853
$624$	6.79676	0.272088
$625$	−23.8014	−0.952056
$626$	1.73003	0.0691458
$627$	−68.0597	−2.71804
$628$	−1.99863	−0.0797540
$629$	19.6791	0.784659
$630$	−23.3178	−0.929002
$631$	27.3071	1.08708	0.543539	−	0.839384i	$-0.317084\pi$
$631$	27.3071	1.08708	0.543539	+	0.839384i	$0.317084\pi$
$632$	5.38480	0.214196
$633$	5.84819	0.232445
$634$	−14.5987	−0.579790
$635$	29.1287	1.15594
$636$	42.0353	1.66681
$637$	10.8954	0.431691
$638$	−22.2150	−0.879500
$639$	32.0991	1.26982
$640$	2.18421	0.0863386
$641$	−43.2563	−1.70852	−0.854261	−	0.519845i	$-0.825990\pi$
$641$	−43.2563	−1.70852	−0.854261	+	0.519845i	$0.825990\pi$
$642$	1.10780	0.0437212
$643$	16.2752	0.641833	0.320916	−	0.947108i	$-0.396009\pi$
$643$	16.2752	0.641833	0.320916	+	0.947108i	$0.396009\pi$
$644$	3.06904	0.120937
$645$	57.5754	2.26703
$646$	−16.5394	−0.650735
$647$	18.5191	0.728062	0.364031	−	0.931387i	$-0.381400\pi$
$647$	18.5191	0.728062	0.364031	+	0.931387i	$0.381400\pi$
$648$	39.2365	1.54135
$649$	−15.6618	−0.614779
$650$	0.457305	0.0179370
$651$	7.01552	0.274960
$652$	20.0518	0.785290
$653$	−29.4032	−1.15064	−0.575319	−	0.817929i	$-0.695122\pi$
$653$	−29.4032	−1.15064	−0.575319	+	0.817929i	$0.695122\pi$
$654$	−47.9086	−1.87337
$655$	−10.0969	−0.394517
$656$	10.4866	0.409432
$657$	3.84507	0.150011
$658$	−2.51429	−0.0980172
$659$	−27.4789	−1.07042	−0.535212	−	0.844718i	$-0.679768\pi$
$659$	−27.4789	−1.07042	−0.535212	+	0.844718i	$0.679768\pi$
$660$	−23.2822	−0.906260
$661$	31.7819	1.23617	0.618086	−	0.786111i	$-0.287908\pi$
$661$	31.7819	1.23617	0.618086	+	0.786111i	$0.287908\pi$
$662$	−0.810115	−0.0314860
$663$	−17.6060	−0.683761
$664$	−4.50553	−0.174848
$665$	−17.3012	−0.670914
$666$	−65.3758	−2.53326
$667$	17.5642	0.680088
$668$	5.91510	0.228862
$669$	41.7944	1.61586
$670$	12.5439	0.484614
$671$	−31.8035	−1.22776
$672$	4.22621	0.163030
$673$	37.5834	1.44873	0.724366	−	0.689416i	$-0.242133\pi$
$673$	37.5834	1.44873	0.724366	+	0.689416i	$0.242133\pi$
$674$	35.1262	1.35301
$675$	4.37693	0.168468
$676$	−9.01944	−0.346902
$677$	−25.2287	−0.969617	−0.484808	−	0.874620i	$-0.661111\pi$
$677$	−25.2287	−0.969617	−0.484808	+	0.874620i	$0.661111\pi$
$678$	−38.4412	−1.47633
$679$	18.3748	0.705159
$680$	−5.65789	−0.216970
$681$	38.2472	1.46564
$682$	5.19408	0.198892
$683$	45.8812	1.75560	0.877798	−	0.479031i	$-0.159012\pi$
$683$	45.8812	1.75560	0.877798	+	0.479031i	$0.159012\pi$
$684$	54.9454	2.10089
$685$	6.25362	0.238939
$686$	15.4587	0.590217
$687$	−72.7817	−2.77679
$688$	−7.73770	−0.294997
$689$	24.6182	0.937880
$690$	18.4080	0.700780
$691$	−8.25520	−0.314043	−0.157021	−	0.987595i	$-0.550189\pi$
$691$	−8.25520	−0.314043	−0.157021	+	0.987595i	$0.550189\pi$
$692$	6.42835	0.244369
$693$	−33.4034	−1.26889
$694$	−10.8891	−0.413345
$695$	11.6749	0.442854
$696$	24.1867	0.916794
$697$	−27.1639	−1.02891
$698$	−23.8404	−0.902373
$699$	56.5187	2.13773
$700$	0.284352	0.0107475
$701$	38.7440	1.46334	0.731671	−	0.681658i	$-0.238741\pi$
$701$	38.7440	1.46334	0.731671	+	0.681658i	$0.238741\pi$
$702$	38.0984	1.43793
$703$	−48.5074	−1.82949
$704$	3.12896	0.117927
$705$	−15.0806	−0.567968
$706$	0.117951	0.00443916
$707$	−1.30082	−0.0489224
$708$	17.0518	0.640848
$709$	−32.0807	−1.20481	−0.602407	−	0.798189i	$-0.705792\pi$
$709$	−32.0807	−1.20481	−0.602407	+	0.798189i	$0.705792\pi$
$710$	8.14738	0.305766
$711$	46.3383	1.73782
$712$	−12.6524	−0.474168
$713$	−4.10667	−0.153796
$714$	−10.9474	−0.409696
$715$	−13.6354	−0.509934
$716$	6.81874	0.254828
$717$	−59.0527	−2.20536
$718$	−18.7826	−0.700962
$719$	31.4735	1.17377	0.586883	−	0.809672i	$-0.300355\pi$
$719$	31.4735	1.17377	0.586883	+	0.809672i	$0.300355\pi$
$720$	18.7960	0.700485
$721$	−0.221573	−0.00825181
$722$	21.7682	0.810129
$723$	66.2747	2.46478
$724$	3.76552	0.139944
$725$	1.62735	0.0604383
$726$	4.12079	0.152937
$727$	33.8918	1.25698	0.628489	−	0.777818i	$-0.283674\pi$
$727$	33.8918	1.25698	0.628489	+	0.777818i	$0.283674\pi$
$728$	2.47510	0.0917335
$729$	142.487	5.27730
$730$	0.975954	0.0361217
$731$	20.0434	0.741332
$732$	34.6262	1.27982
$733$	0.336785	0.0124395	0.00621973	−	0.999981i	$-0.498020\pi$
$733$	0.336785	0.0124395	0.00621973	+	0.999981i	$0.498020\pi$
$734$	−27.4418	−1.01289
$735$	40.6346	1.49883
$736$	−2.47389	−0.0911890
$737$	17.9696	0.661918
$738$	90.2410	3.32182
$739$	−14.5796	−0.536320	−0.268160	−	0.963374i	$-0.586416\pi$
$739$	−14.5796	−0.536320	−0.268160	+	0.963374i	$0.586416\pi$
$740$	−16.5936	−0.609995
$741$	43.3973	1.59424
$742$	15.3076	0.561958
$743$	−43.0797	−1.58044	−0.790220	−	0.612824i	$-0.790034\pi$
$743$	−43.0797	−1.58044	−0.790220	+	0.612824i	$0.790034\pi$
$744$	−5.65508	−0.207325
$745$	−43.3292	−1.58746
$746$	−14.4700	−0.529784
$747$	−38.7718	−1.41859
$748$	−8.10511	−0.296352
$749$	0.403415	0.0147405
$750$	38.9100	1.42079
$751$	13.5188	0.493309	0.246655	−	0.969103i	$-0.420669\pi$
$751$	13.5188	0.493309	0.246655	+	0.969103i	$0.420669\pi$
$752$	2.02672	0.0739069
$753$	11.7560	0.428411
$754$	14.1651	0.515861
$755$	−41.8465	−1.52295
$756$	23.6895	0.861580
$757$	28.2087	1.02526	0.512632	−	0.858609i	$-0.328671\pi$
$757$	28.2087	1.02526	0.512632	+	0.858609i	$0.328671\pi$
$758$	4.00097	0.145322
$759$	26.3700	0.957171
$760$	13.9462	0.505882
$761$	33.2311	1.20462	0.602312	−	0.798260i	$-0.294246\pi$
$761$	33.2311	1.20462	0.602312	+	0.798260i	$0.294246\pi$
$762$	−45.4313	−1.64580
$763$	−17.4464	−0.631601
$764$	7.37336	0.266759
$765$	−48.6883	−1.76033
$766$	5.83486	0.210822
$767$	9.98651	0.360592
$768$	−3.40667	−0.122928
$769$	−37.0884	−1.33744	−0.668721	−	0.743514i	$-0.733158\pi$
$769$	−37.0884	−1.33744	−0.668721	+	0.743514i	$0.733158\pi$
$770$	−8.47845	−0.305542
$771$	68.5049	2.46714
$772$	15.6558	0.563466
$773$	3.38354	0.121698	0.0608488	−	0.998147i	$-0.480619\pi$
$773$	3.38354	0.121698	0.0608488	+	0.998147i	$0.480619\pi$
$774$	−66.5859	−2.39338
$775$	−0.380490	−0.0136676
$776$	−14.8116	−0.531704
$777$	−32.1069	−1.15183
$778$	1.54890	0.0555309
$779$	66.9567	2.39897
$780$	14.8456	0.531557
$781$	11.6714	0.417635
$782$	6.40827	0.229159
$783$	135.576	4.84508
$784$	−5.46098	−0.195035
$785$	−4.36543	−0.155809
$786$	15.7479	0.561707
$787$	23.9531	0.853835	0.426917	−	0.904291i	$-0.359600\pi$
$787$	23.9531	0.853835	0.426917	+	0.904291i	$0.359600\pi$
$788$	1.29758	0.0462242
$789$	−54.1649	−1.92832
$790$	11.7616	0.418457
$791$	−13.9987	−0.497738
$792$	26.9259	0.956769
$793$	20.2790	0.720129
$794$	8.79677	0.312186
$795$	91.8141	3.25631
$796$	−23.4585	−0.831463
$797$	29.2232	1.03514	0.517570	−	0.855641i	$-0.326837\pi$
$797$	29.2232	1.03514	0.517570	+	0.855641i	$0.326837\pi$
$798$	26.9844	0.955236
$799$	−5.24993	−0.185729
$800$	−0.229210	−0.00810381
$801$	−108.878	−3.84703
$802$	−21.7856	−0.769277
$803$	1.39809	0.0493374
$804$	−19.5645	−0.689985
$805$	6.70344	0.236265
$806$	−3.31193	−0.116658
$807$	−71.6773	−2.52316
$808$	1.04857	0.0368885
$809$	18.1604	0.638487	0.319244	−	0.947673i	$-0.396571\pi$
$809$	18.1604	0.638487	0.319244	+	0.947673i	$0.396571\pi$
$810$	85.7008	3.01122
$811$	−48.8565	−1.71558	−0.857792	−	0.513997i	$-0.828164\pi$
$811$	−48.8565	−1.71558	−0.857792	+	0.513997i	$0.828164\pi$
$812$	8.80782	0.309094
$813$	−30.9999	−1.08721
$814$	−23.7709	−0.833171
$815$	43.7975	1.53416
$816$	8.82448	0.308919
$817$	−49.4052	−1.72847
$818$	−1.33713	−0.0467516
$819$	21.2992	0.744255
$820$	22.9049	0.799874
$821$	11.2135	0.391352	0.195676	−	0.980669i	$-0.437310\pi$
$821$	11.2135	0.391352	0.195676	+	0.980669i	$0.437310\pi$
$822$	−9.75363	−0.340197
$823$	−5.85760	−0.204183	−0.102092	−	0.994775i	$-0.532553\pi$
$823$	−5.85760	−0.204183	−0.102092	+	0.994775i	$0.532553\pi$
$824$	0.178606	0.00622202
$825$	2.44323	0.0850622
$826$	6.20959	0.216059
$827$	−20.4737	−0.711940	−0.355970	−	0.934497i	$-0.615849\pi$
$827$	−20.4737	−0.711940	−0.355970	+	0.934497i	$0.615849\pi$
$828$	−21.2888	−0.739837
$829$	−32.3365	−1.12309	−0.561547	−	0.827445i	$-0.689794\pi$
$829$	−32.3365	−1.12309	−0.561547	+	0.827445i	$0.689794\pi$
$830$	−9.84103	−0.341587
$831$	59.9274	2.07886
$832$	−1.99513	−0.0691688
$833$	14.1459	0.490126
$834$	−18.2090	−0.630528
$835$	12.9198	0.447109
$836$	19.9784	0.690967
$837$	−31.6989	−1.09567
$838$	−37.1393	−1.28296
$839$	−8.11697	−0.280229	−0.140114	−	0.990135i	$-0.544747\pi$
$839$	−8.11697	−0.280229	−0.140114	+	0.990135i	$0.544747\pi$
$840$	9.23095	0.318498
$841$	21.4073	0.738183
$842$	22.0995	0.761599
$843$	16.9850	0.584994
$844$	−1.71669	−0.0590909
$845$	−19.7004	−0.677714
$846$	17.4407	0.599624
$847$	1.50062	0.0515621
$848$	−12.3391	−0.423728
$849$	13.5688	0.465680
$850$	0.593736	0.0203650
$851$	18.7944	0.644263
$852$	−12.7073	−0.435344
$853$	−29.4219	−1.00739	−0.503693	−	0.863883i	$-0.668026\pi$
$853$	−29.4219	−1.00739	−0.503693	+	0.863883i	$0.668026\pi$
$854$	12.6095	0.431487
$855$	120.012	4.10434
$856$	−0.325185	−0.0111146
$857$	39.6284	1.35368	0.676840	−	0.736130i	$-0.263349\pi$
$857$	39.6284	1.35368	0.676840	+	0.736130i	$0.263349\pi$
$858$	21.2668	0.726035
$859$	−36.7906	−1.25528	−0.627640	−	0.778504i	$-0.715979\pi$
$859$	−36.7906	−1.25528	−0.627640	+	0.778504i	$0.715979\pi$
$860$	−16.9008	−0.576312
$861$	44.3185	1.51037
$862$	16.6790	0.568088
$863$	37.6883	1.28293	0.641463	−	0.767154i	$-0.278328\pi$
$863$	37.6883	1.28293	0.641463	+	0.767154i	$0.278328\pi$
$864$	−19.0957	−0.649648
$865$	14.0409	0.477405
$866$	15.0939	0.512912
$867$	35.0548	1.19052
$868$	−2.05935	−0.0698989
$869$	16.8488	0.571557
$870$	52.8289	1.79107
$871$	−11.4580	−0.388241
$872$	14.0632	0.476240
$873$	−127.459	−4.31384
$874$	−15.7958	−0.534301
$875$	14.1694	0.479015
$876$	−1.52217	−0.0514295
$877$	−7.17712	−0.242354	−0.121177	−	0.992631i	$-0.538667\pi$
$877$	−7.17712	−0.242354	−0.121177	+	0.992631i	$0.538667\pi$
$878$	10.2442	0.345726
$879$	41.8976	1.41317
$880$	6.83431	0.230385
$881$	−48.3501	−1.62896	−0.814479	−	0.580193i	$-0.802977\pi$
$881$	−48.3501	−1.62896	−0.814479	+	0.580193i	$0.802977\pi$
$882$	−46.9939	−1.58237
$883$	12.5406	0.422024	0.211012	−	0.977483i	$-0.432324\pi$
$883$	12.5406	0.422024	0.211012	+	0.977483i	$0.432324\pi$
$884$	5.16811	0.173822
$885$	37.2449	1.25197
$886$	0.727793	0.0244507
$887$	11.8974	0.399475	0.199738	−	0.979849i	$-0.435991\pi$
$887$	11.8974	0.399475	0.199738	+	0.979849i	$0.435991\pi$
$888$	25.8807	0.868500
$889$	−16.5443	−0.554877
$890$	−27.6355	−0.926343
$891$	122.769	4.11292
$892$	−12.2684	−0.410777
$893$	12.9406	0.433041
$894$	67.5795	2.26020
$895$	14.8936	0.497837
$896$	−1.24057	−0.0414446
$897$	−16.8145	−0.561419
$898$	32.5788	1.08717
$899$	−11.7857	−0.393075
$900$	−1.97244	−0.0657481
$901$	31.9627	1.06483
$902$	32.8120	1.09252
$903$	−32.7011	−1.08823
$904$	11.2841	0.375304
$905$	8.22470	0.273398
$906$	65.2670	2.16835
$907$	−32.3048	−1.07266	−0.536331	−	0.844007i	$-0.680190\pi$
$907$	−32.3048	−1.07266	−0.536331	+	0.844007i	$0.680190\pi$
$908$	−11.2272	−0.372586
$909$	9.02332	0.299285
$910$	5.40616	0.179212
$911$	55.5945	1.84193	0.920964	−	0.389647i	$-0.127403\pi$
$911$	55.5945	1.84193	0.920964	+	0.389647i	$0.127403\pi$
$912$	−21.7516	−0.720266
$913$	−14.0976	−0.466562
$914$	−22.4254	−0.741768
$915$	75.6310	2.50028
$916$	21.3645	0.705902
$917$	5.73473	0.189378
$918$	49.4646	1.63257
$919$	−14.5392	−0.479605	−0.239803	−	0.970822i	$-0.577083\pi$
$919$	−14.5392	−0.479605	−0.239803	+	0.970822i	$0.577083\pi$
$920$	−5.40351	−0.178149
$921$	15.6580	0.515950
$922$	1.07914	0.0355397
$923$	−7.44209	−0.244959
$924$	13.2236	0.435026
$925$	1.74133	0.0572546
$926$	−27.2368	−0.895055
$927$	1.53697	0.0504807
$928$	−7.09981	−0.233063
$929$	−35.8937	−1.17763	−0.588817	−	0.808266i	$-0.700406\pi$
$929$	−35.8937	−1.17763	−0.588817	+	0.808266i	$0.700406\pi$
$930$	−12.3519	−0.405035
$931$	−34.8684	−1.14277
$932$	−16.5906	−0.543443
$933$	37.7479	1.23581
$934$	13.6387	0.446271
$935$	−17.7033	−0.578960
$936$	−17.1689	−0.561183
$937$	−8.98337	−0.293474	−0.146737	−	0.989176i	$-0.546877\pi$
$937$	−8.98337	−0.293474	−0.146737	+	0.989176i	$0.546877\pi$
$938$	−7.12458	−0.232626
$939$	−5.89363	−0.192331
$940$	4.42679	0.144386
$941$	−25.5842	−0.834020	−0.417010	−	0.908902i	$-0.636922\pi$
$941$	−25.5842	−0.834020	−0.417010	+	0.908902i	$0.636922\pi$
$942$	6.80866	0.221838
$943$	−25.9427	−0.844810
$944$	−5.00543	−0.162913
$945$	51.7430	1.68320
$946$	−24.2109	−0.787165
$947$	27.9479	0.908186	0.454093	−	0.890954i	$-0.349963\pi$
$947$	27.9479	0.908186	0.454093	+	0.890954i	$0.349963\pi$
$948$	−18.3442	−0.595793
$949$	−0.891469	−0.0289383
$950$	−1.46351	−0.0474825
$951$	49.7330	1.61270
$952$	3.21352	0.104151
$953$	18.7833	0.608452	0.304226	−	0.952600i	$-0.401602\pi$
$953$	18.7833	0.608452	0.304226	+	0.952600i	$0.401602\pi$
$954$	−106.183	−3.43780
$955$	16.1050	0.521145
$956$	17.3345	0.560636
$957$	75.6791	2.44636
$958$	−38.8629	−1.25560
$959$	−3.55188	−0.114696
$960$	−7.44089	−0.240154
$961$	−28.2444	−0.911109
$962$	15.1572	0.488688
$963$	−2.79834	−0.0901752
$964$	−19.4544	−0.626584
$965$	34.1957	1.10080
$966$	−10.4552	−0.336391
$967$	56.6176	1.82070	0.910350	−	0.413839i	$-0.135812\pi$
$967$	56.6176	1.82070	0.910350	+	0.413839i	$0.135812\pi$
$968$	−1.20962	−0.0388788
$969$	56.3443	1.81004
$970$	−32.3516	−1.03875
$971$	40.3249	1.29409	0.647044	−	0.762452i	$-0.276005\pi$
$971$	40.3249	1.29409	0.647044	+	0.762452i	$0.276005\pi$
$972$	−76.3786	−2.44984
$973$	−6.63100	−0.212580
$974$	−34.2190	−1.09645
$975$	−1.55789	−0.0498923
$976$	−10.1642	−0.325349
$977$	−33.1556	−1.06074	−0.530371	−	0.847766i	$-0.677947\pi$
$977$	−33.1556	−1.06074	−0.530371	+	0.847766i	$0.677947\pi$
$978$	−68.3099	−2.18431
$979$	−39.5887	−1.26526
$980$	−11.9280	−0.381025
$981$	121.019	3.86384
$982$	28.9422	0.923583
$983$	6.88230	0.219511	0.109756	−	0.993959i	$-0.464993\pi$
$983$	6.88230	0.219511	0.109756	+	0.993959i	$0.464993\pi$
$984$	−35.7243	−1.13885
$985$	2.83418	0.0903046
$986$	18.3910	0.585690
$987$	8.56535	0.272638
$988$	−12.7389	−0.405279
$989$	19.1423	0.608688
$990$	58.8119	1.86916
$991$	46.2279	1.46848	0.734239	−	0.678891i	$-0.237539\pi$
$991$	46.2279	1.46848	0.734239	+	0.678891i	$0.237539\pi$
$992$	1.66000	0.0527052
$993$	2.75979	0.0875794
$994$	−4.62748	−0.146775
$995$	−51.2383	−1.62436
$996$	15.3488	0.486346
$997$	−42.9040	−1.35878	−0.679392	−	0.733775i	$-0.737756\pi$
$997$	−42.9040	−1.35878	−0.679392	+	0.733775i	$0.737756\pi$
$998$	−32.0218	−1.01363
$999$	145.071	4.58986

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.2	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.2	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.40667 q^{3} +1.00000 q^{4} +2.18421 q^{5} -3.40667 q^{6} -1.24057 q^{7} +1.00000 q^{8} +8.60538 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.40667 q^{3} +1.00000 q^{4} +2.18421 q^{5} -3.40667 q^{6} -1.24057 q^{7} +1.00000 q^{8} +8.60538 q^{9} +2.18421 q^{10} +3.12896 q^{11} -3.40667 q^{12} -1.99513 q^{13} -1.24057 q^{14} -7.44089 q^{15} +1.00000 q^{16} -2.59036 q^{17} +8.60538 q^{18} +6.38500 q^{19} +2.18421 q^{20} +4.22621 q^{21} +3.12896 q^{22} -2.47389 q^{23} -3.40667 q^{24} -0.229210 q^{25} -1.99513 q^{26} -19.0957 q^{27} -1.24057 q^{28} -7.09981 q^{29} -7.44089 q^{30} +1.66000 q^{31} +1.00000 q^{32} -10.6593 q^{33} -2.59036 q^{34} -2.70967 q^{35} +8.60538 q^{36} -7.59708 q^{37} +6.38500 q^{38} +6.79676 q^{39} +2.18421 q^{40} +10.4866 q^{41} +4.22621 q^{42} -7.73770 q^{43} +3.12896 q^{44} +18.7960 q^{45} -2.47389 q^{46} +2.02672 q^{47} -3.40667 q^{48} -5.46098 q^{49} -0.229210 q^{50} +8.82448 q^{51} -1.99513 q^{52} -12.3391 q^{53} -19.0957 q^{54} +6.83431 q^{55} -1.24057 q^{56} -21.7516 q^{57} -7.09981 q^{58} -5.00543 q^{59} -7.44089 q^{60} -10.1642 q^{61} +1.66000 q^{62} -10.6756 q^{63} +1.00000 q^{64} -4.35780 q^{65} -10.6593 q^{66} +5.74299 q^{67} -2.59036 q^{68} +8.42774 q^{69} -2.70967 q^{70} +3.73012 q^{71} +8.60538 q^{72} +0.446822 q^{73} -7.59708 q^{74} +0.780844 q^{75} +6.38500 q^{76} -3.88169 q^{77} +6.79676 q^{78} +5.38480 q^{79} +2.18421 q^{80} +39.2365 q^{81} +10.4866 q^{82} -4.50553 q^{83} +4.22621 q^{84} -5.65789 q^{85} -7.73770 q^{86} +24.1867 q^{87} +3.12896 q^{88} -12.6524 q^{89} +18.7960 q^{90} +2.47510 q^{91} -2.47389 q^{92} -5.65508 q^{93} +2.02672 q^{94} +13.9462 q^{95} -3.40667 q^{96} -14.8116 q^{97} -5.46098 q^{98} +26.9259 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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