LMFDB - Embedded newform 8002.2.a.d.1.39

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			1.39
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.0377834 q^{3} +1.00000 q^{4} -1.82359 q^{5} -0.0377834 q^{6} -4.49307 q^{7} +1.00000 q^{8} -2.99857 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.0377834 q^{3} +1.00000 q^{4} -1.82359 q^{5} -0.0377834 q^{6} -4.49307 q^{7} +1.00000 q^{8} -2.99857 q^{9} -1.82359 q^{10} +2.94797 q^{11} -0.0377834 q^{12} +1.02878 q^{13} -4.49307 q^{14} +0.0689015 q^{15} +1.00000 q^{16} +5.14957 q^{17} -2.99857 q^{18} +0.924142 q^{19} -1.82359 q^{20} +0.169764 q^{21} +2.94797 q^{22} +0.320712 q^{23} -0.0377834 q^{24} -1.67451 q^{25} +1.02878 q^{26} +0.226646 q^{27} -4.49307 q^{28} +0.862102 q^{29} +0.0689015 q^{30} +7.56634 q^{31} +1.00000 q^{32} -0.111384 q^{33} +5.14957 q^{34} +8.19353 q^{35} -2.99857 q^{36} -2.65226 q^{37} +0.924142 q^{38} -0.0388706 q^{39} -1.82359 q^{40} +3.90986 q^{41} +0.169764 q^{42} -0.456090 q^{43} +2.94797 q^{44} +5.46817 q^{45} +0.320712 q^{46} +7.41479 q^{47} -0.0377834 q^{48} +13.1877 q^{49} -1.67451 q^{50} -0.194568 q^{51} +1.02878 q^{52} -12.1425 q^{53} +0.226646 q^{54} -5.37590 q^{55} -4.49307 q^{56} -0.0349172 q^{57} +0.862102 q^{58} -3.89724 q^{59} +0.0689015 q^{60} -2.25350 q^{61} +7.56634 q^{62} +13.4728 q^{63} +1.00000 q^{64} -1.87607 q^{65} -0.111384 q^{66} -15.3437 q^{67} +5.14957 q^{68} -0.0121176 q^{69} +8.19353 q^{70} -7.28376 q^{71} -2.99857 q^{72} -1.35637 q^{73} -2.65226 q^{74} +0.0632689 q^{75} +0.924142 q^{76} -13.2455 q^{77} -0.0388706 q^{78} +9.40766 q^{79} -1.82359 q^{80} +8.98715 q^{81} +3.90986 q^{82} -15.5768 q^{83} +0.169764 q^{84} -9.39071 q^{85} -0.456090 q^{86} -0.0325731 q^{87} +2.94797 q^{88} +5.43141 q^{89} +5.46817 q^{90} -4.62236 q^{91} +0.320712 q^{92} -0.285882 q^{93} +7.41479 q^{94} -1.68526 q^{95} -0.0377834 q^{96} +6.98364 q^{97} +13.1877 q^{98} -8.83971 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.0377834	−0.0218143	−0.0109071	−	0.999941i	$-0.503472\pi$
$3$	−0.0377834	−0.0218143	−0.0109071	+	0.999941i	$0.503472\pi$
$4$	1.00000	0.500000
$5$	−1.82359	−0.815535	−0.407767	−	0.913086i	$-0.633693\pi$
$5$	−1.82359	−0.815535	−0.407767	+	0.913086i	$0.633693\pi$
$6$	−0.0377834	−0.0154250
$7$	−4.49307	−1.69822	−0.849111	−	0.528215i	$-0.822862\pi$
$7$	−4.49307	−1.69822	−0.849111	+	0.528215i	$0.822862\pi$
$8$	1.00000	0.353553
$9$	−2.99857	−0.999524
$10$	−1.82359	−0.576670
$11$	2.94797	0.888848	0.444424	−	0.895817i	$-0.353408\pi$
$11$	2.94797	0.888848	0.444424	+	0.895817i	$0.353408\pi$
$12$	−0.0377834	−0.0109071
$13$	1.02878	0.285331	0.142665	−	0.989771i	$-0.454433\pi$
$13$	1.02878	0.285331	0.142665	+	0.989771i	$0.454433\pi$
$14$	−4.49307	−1.20082
$15$	0.0689015	0.0177903
$16$	1.00000	0.250000
$17$	5.14957	1.24895	0.624477	−	0.781043i	$-0.285312\pi$
$17$	5.14957	1.24895	0.624477	+	0.781043i	$0.285312\pi$
$18$	−2.99857	−0.706770
$19$	0.924142	0.212013	0.106006	−	0.994365i	$-0.466194\pi$
$19$	0.924142	0.212013	0.106006	+	0.994365i	$0.466194\pi$
$20$	−1.82359	−0.407767
$21$	0.169764	0.0370454
$22$	2.94797	0.628510
$23$	0.320712	0.0668730	0.0334365	−	0.999441i	$-0.489355\pi$
$23$	0.320712	0.0668730	0.0334365	+	0.999441i	$0.489355\pi$
$24$	−0.0377834	−0.00771250
$25$	−1.67451	−0.334903
$26$	1.02878	0.201759
$27$	0.226646	0.0436181
$28$	−4.49307	−0.849111
$29$	0.862102	0.160088	0.0800441	−	0.996791i	$-0.474494\pi$
$29$	0.862102	0.160088	0.0800441	+	0.996791i	$0.474494\pi$
$30$	0.0689015	0.0125796
$31$	7.56634	1.35895	0.679477	−	0.733697i	$-0.262207\pi$
$31$	7.56634	1.35895	0.679477	+	0.733697i	$0.262207\pi$
$32$	1.00000	0.176777
$33$	−0.111384	−0.0193895
$34$	5.14957	0.883144
$35$	8.19353	1.38496
$36$	−2.99857	−0.499762
$37$	−2.65226	−0.436029	−0.218015	−	0.975946i	$-0.569958\pi$
$37$	−2.65226	−0.436029	−0.218015	+	0.975946i	$0.569958\pi$
$38$	0.924142	0.149916
$39$	−0.0388706	−0.00622428
$40$	−1.82359	−0.288335
$41$	3.90986	0.610617	0.305309	−	0.952253i	$-0.401240\pi$
$41$	3.90986	0.610617	0.305309	+	0.952253i	$0.401240\pi$
$42$	0.169764	0.0261951
$43$	−0.456090	−0.0695531	−0.0347765	−	0.999395i	$-0.511072\pi$
$43$	−0.456090	−0.0695531	−0.0347765	+	0.999395i	$0.511072\pi$
$44$	2.94797	0.444424
$45$	5.46817	0.815147
$46$	0.320712	0.0472864
$47$	7.41479	1.08156	0.540779	−	0.841165i	$-0.318130\pi$
$47$	7.41479	1.08156	0.540779	+	0.841165i	$0.318130\pi$
$48$	−0.0377834	−0.00545356
$49$	13.1877	1.88396
$50$	−1.67451	−0.236812
$51$	−0.194568	−0.0272450
$52$	1.02878	0.142665
$53$	−12.1425	−1.66790	−0.833949	−	0.551841i	$-0.813925\pi$
$53$	−12.1425	−1.66790	−0.833949	+	0.551841i	$0.813925\pi$
$54$	0.226646	0.0308427
$55$	−5.37590	−0.724886
$56$	−4.49307	−0.600412
$57$	−0.0349172	−0.00462490
$58$	0.862102	0.113200
$59$	−3.89724	−0.507378	−0.253689	−	0.967286i	$-0.581644\pi$
$59$	−3.89724	−0.507378	−0.253689	+	0.967286i	$0.581644\pi$
$60$	0.0689015	0.00889514
$61$	−2.25350	−0.288531	−0.144266	−	0.989539i	$-0.546082\pi$
$61$	−2.25350	−0.288531	−0.144266	+	0.989539i	$0.546082\pi$
$62$	7.56634	0.960926
$63$	13.4728	1.69741
$64$	1.00000	0.125000
$65$	−1.87607	−0.232697
$66$	−0.111384	−0.0137105
$67$	−15.3437	−1.87453	−0.937264	−	0.348622i	$-0.886650\pi$
$67$	−15.3437	−1.87453	−0.937264	+	0.348622i	$0.886650\pi$
$68$	5.14957	0.624477
$69$	−0.0121176	−0.00145879
$70$	8.19353	0.979314
$71$	−7.28376	−0.864423	−0.432211	−	0.901772i	$-0.642267\pi$
$71$	−7.28376	−0.864423	−0.432211	+	0.901772i	$0.642267\pi$
$72$	−2.99857	−0.353385
$73$	−1.35637	−0.158751	−0.0793755	−	0.996845i	$-0.525293\pi$
$73$	−1.35637	−0.158751	−0.0793755	+	0.996845i	$0.525293\pi$
$74$	−2.65226	−0.308319
$75$	0.0632689	0.00730566
$76$	0.924142	0.106006
$77$	−13.2455	−1.50946
$78$	−0.0388706	−0.00440123
$79$	9.40766	1.05844	0.529222	−	0.848483i	$-0.322484\pi$
$79$	9.40766	1.05844	0.529222	+	0.848483i	$0.322484\pi$
$80$	−1.82359	−0.203884
$81$	8.98715	0.998573
$82$	3.90986	0.431772
$83$	−15.5768	−1.70978	−0.854890	−	0.518809i	$-0.826376\pi$
$83$	−15.5768	−1.70978	−0.854890	+	0.518809i	$0.826376\pi$
$84$	0.169764	0.0185227
$85$	−9.39071	−1.01857
$86$	−0.456090	−0.0491814
$87$	−0.0325731	−0.00349221
$88$	2.94797	0.314255
$89$	5.43141	0.575728	0.287864	−	0.957671i	$-0.407055\pi$
$89$	5.43141	0.575728	0.287864	+	0.957671i	$0.407055\pi$
$90$	5.46817	0.576396
$91$	−4.62236	−0.484555
$92$	0.320712	0.0334365
$93$	−0.285882	−0.0296446
$94$	7.41479	0.764777
$95$	−1.68526	−0.172904
$96$	−0.0377834	−0.00385625
$97$	6.98364	0.709081	0.354541	−	0.935041i	$-0.384637\pi$
$97$	6.98364	0.709081	0.354541	+	0.935041i	$0.384637\pi$
$98$	13.1877	1.33216
$99$	−8.83971	−0.888425
$100$	−1.67451	−0.167451
$101$	−14.3009	−1.42299	−0.711497	−	0.702689i	$-0.751982\pi$
$101$	−14.3009	−1.42299	−0.711497	+	0.702689i	$0.751982\pi$
$102$	−0.194568	−0.0192651
$103$	2.22338	0.219076	0.109538	−	0.993983i	$-0.465063\pi$
$103$	2.22338	0.219076	0.109538	+	0.993983i	$0.465063\pi$
$104$	1.02878	0.100880
$105$	−0.309579	−0.0302118
$106$	−12.1425	−1.17938
$107$	−11.7096	−1.13201	−0.566003	−	0.824403i	$-0.691511\pi$
$107$	−11.7096	−1.13201	−0.566003	+	0.824403i	$0.691511\pi$
$108$	0.226646	0.0218091
$109$	−16.7727	−1.60653	−0.803265	−	0.595622i	$-0.796905\pi$
$109$	−16.7727	−1.60653	−0.803265	+	0.595622i	$0.796905\pi$
$110$	−5.37590	−0.512572
$111$	0.100211	0.00951165
$112$	−4.49307	−0.424555
$113$	16.4564	1.54809	0.774043	−	0.633133i	$-0.218231\pi$
$113$	16.4564	1.54809	0.774043	+	0.633133i	$0.218231\pi$
$114$	−0.0349172	−0.00327030
$115$	−0.584847	−0.0545373
$116$	0.862102	0.0800441
$117$	−3.08486	−0.285195
$118$	−3.89724	−0.358770
$119$	−23.1374	−2.12100
$120$	0.0689015	0.00628982
$121$	−2.30945	−0.209950
$122$	−2.25350	−0.204023
$123$	−0.147728	−0.0133202
$124$	7.56634	0.679477
$125$	12.1716	1.08866
$126$	13.4728	1.20025
$127$	3.60195	0.319622	0.159811	−	0.987148i	$-0.448912\pi$
$127$	3.60195	0.319622	0.159811	+	0.987148i	$0.448912\pi$
$128$	1.00000	0.0883883
$129$	0.0172326	0.00151725
$130$	−1.87607	−0.164542
$131$	−9.57362	−0.836451	−0.418226	−	0.908343i	$-0.637348\pi$
$131$	−9.57362	−0.836451	−0.418226	+	0.908343i	$0.637348\pi$
$132$	−0.111384	−0.00969477
$133$	−4.15224	−0.360045
$134$	−15.3437	−1.32549
$135$	−0.413311	−0.0355721
$136$	5.14957	0.441572
$137$	−8.67464	−0.741124	−0.370562	−	0.928808i	$-0.620835\pi$
$137$	−8.67464	−0.741124	−0.370562	+	0.928808i	$0.620835\pi$
$138$	−0.0121176	−0.00103152
$139$	−16.3559	−1.38729	−0.693646	−	0.720316i	$-0.743997\pi$
$139$	−16.3559	−1.38729	−0.693646	+	0.720316i	$0.743997\pi$
$140$	8.19353	0.692479
$141$	−0.280156	−0.0235934
$142$	−7.28376	−0.611239
$143$	3.03280	0.253616
$144$	−2.99857	−0.249881
$145$	−1.57212	−0.130558
$146$	−1.35637	−0.112254
$147$	−0.498276	−0.0410971
$148$	−2.65226	−0.218015
$149$	9.69180	0.793983	0.396991	−	0.917822i	$-0.370054\pi$
$149$	9.69180	0.793983	0.396991	+	0.917822i	$0.370054\pi$
$150$	0.0632689	0.00516588
$151$	−20.4748	−1.66621	−0.833107	−	0.553112i	$-0.813440\pi$
$151$	−20.4748	−1.66621	−0.833107	+	0.553112i	$0.813440\pi$
$152$	0.924142	0.0749578
$153$	−15.4414	−1.24836
$154$	−13.2455	−1.06735
$155$	−13.7979	−1.10827
$156$	−0.0388706	−0.00311214
$157$	18.6092	1.48517	0.742587	−	0.669749i	$-0.233598\pi$
$157$	18.6092	1.48517	0.742587	+	0.669749i	$0.233598\pi$
$158$	9.40766	0.748433
$159$	0.458784	0.0363840
$160$	−1.82359	−0.144168
$161$	−1.44098	−0.113565
$162$	8.98715	0.706097
$163$	−7.24772	−0.567686	−0.283843	−	0.958871i	$-0.591609\pi$
$163$	−7.24772	−0.567686	−0.283843	+	0.958871i	$0.591609\pi$
$164$	3.90986	0.305309
$165$	0.203120	0.0158129
$166$	−15.5768	−1.20900
$167$	16.5763	1.28272	0.641358	−	0.767242i	$-0.278371\pi$
$167$	16.5763	1.28272	0.641358	+	0.767242i	$0.278371\pi$
$168$	0.169764	0.0130975
$169$	−11.9416	−0.918586
$170$	−9.39071	−0.720235
$171$	−2.77111	−0.211912
$172$	−0.456090	−0.0347765
$173$	−8.73022	−0.663746	−0.331873	−	0.943324i	$-0.607681\pi$
$173$	−8.73022	−0.663746	−0.331873	+	0.943324i	$0.607681\pi$
$174$	−0.0325731	−0.00246936
$175$	7.52372	0.568740
$176$	2.94797	0.222212
$177$	0.147251	0.0110681
$178$	5.43141	0.407101
$179$	11.7391	0.877419	0.438710	−	0.898629i	$-0.355436\pi$
$179$	11.7391	0.877419	0.438710	+	0.898629i	$0.355436\pi$
$180$	5.46817	0.407573
$181$	−1.64171	−0.122028	−0.0610138	−	0.998137i	$-0.519433\pi$
$181$	−1.64171	−0.122028	−0.0610138	+	0.998137i	$0.519433\pi$
$182$	−4.62236	−0.342632
$183$	0.0851450	0.00629410
$184$	0.320712	0.0236432
$185$	4.83664	0.355597
$186$	−0.285882	−0.0209619
$187$	15.1808	1.11013
$188$	7.41479	0.540779
$189$	−1.01834	−0.0740733
$190$	−1.68526	−0.122261
$191$	7.21088	0.521761	0.260880	−	0.965371i	$-0.415987\pi$
$191$	7.21088	0.521761	0.260880	+	0.965371i	$0.415987\pi$
$192$	−0.0377834	−0.00272678
$193$	−26.4571	−1.90443	−0.952213	−	0.305435i	$-0.901198\pi$
$193$	−26.4571	−1.90443	−0.952213	+	0.305435i	$0.901198\pi$
$194$	6.98364	0.501396
$195$	0.0708841	0.00507612
$196$	13.1877	0.941978
$197$	23.8317	1.69794	0.848971	−	0.528440i	$-0.177223\pi$
$197$	23.8317	1.69794	0.848971	+	0.528440i	$0.177223\pi$
$198$	−8.83971	−0.628211
$199$	−4.40209	−0.312056	−0.156028	−	0.987753i	$-0.549869\pi$
$199$	−4.40209	−0.312056	−0.156028	+	0.987753i	$0.549869\pi$
$200$	−1.67451	−0.118406
$201$	0.579736	0.0408914
$202$	−14.3009	−1.00621
$203$	−3.87349	−0.271865
$204$	−0.194568	−0.0136225
$205$	−7.12998	−0.497980
$206$	2.22338	0.154910
$207$	−0.961677	−0.0668412
$208$	1.02878	0.0713327
$209$	2.72435	0.188447
$210$	−0.309579	−0.0213630
$211$	−14.1619	−0.974945	−0.487472	−	0.873138i	$-0.662081\pi$
$211$	−14.1619	−0.974945	−0.487472	+	0.873138i	$0.662081\pi$
$212$	−12.1425	−0.833949
$213$	0.275205	0.0188567
$214$	−11.7096	−0.800450
$215$	0.831722	0.0567229
$216$	0.226646	0.0154213
$217$	−33.9961	−2.30781
$218$	−16.7727	−1.13599
$219$	0.0512482	0.00346304
$220$	−5.37590	−0.362443
$221$	5.29775	0.356365
$222$	0.100211	0.00672575
$223$	−6.30457	−0.422185	−0.211093	−	0.977466i	$-0.567702\pi$
$223$	−6.30457	−0.422185	−0.211093	+	0.977466i	$0.567702\pi$
$224$	−4.49307	−0.300206
$225$	5.02115	0.334744
$226$	16.4564	1.09466
$227$	−21.3486	−1.41696	−0.708480	−	0.705731i	$-0.750619\pi$
$227$	−21.3486	−1.41696	−0.708480	+	0.705731i	$0.750619\pi$
$228$	−0.0349172	−0.00231245
$229$	−24.4463	−1.61546	−0.807729	−	0.589554i	$-0.799304\pi$
$229$	−24.4463	−1.61546	−0.807729	+	0.589554i	$0.799304\pi$
$230$	−0.584847	−0.0385637
$231$	0.500458	0.0329278
$232$	0.862102	0.0565998
$233$	−13.6707	−0.895599	−0.447799	−	0.894134i	$-0.647792\pi$
$233$	−13.6707	−0.895599	−0.447799	+	0.894134i	$0.647792\pi$
$234$	−3.08486	−0.201663
$235$	−13.5215	−0.882048
$236$	−3.89724	−0.253689
$237$	−0.355453	−0.0230892
$238$	−23.1374	−1.49977
$239$	25.3886	1.64225	0.821127	−	0.570745i	$-0.193346\pi$
$239$	25.3886	1.64225	0.821127	+	0.570745i	$0.193346\pi$
$240$	0.0689015	0.00444757
$241$	13.6569	0.879716	0.439858	−	0.898067i	$-0.355029\pi$
$241$	13.6569	0.879716	0.439858	+	0.898067i	$0.355029\pi$
$242$	−2.30945	−0.148457
$243$	−1.01950	−0.0654013
$244$	−2.25350	−0.144266
$245$	−24.0490	−1.53643
$246$	−0.147728	−0.00941878
$247$	0.950735	0.0604938
$248$	7.56634	0.480463
$249$	0.588546	0.0372976
$250$	12.1716	0.769799
$251$	−0.103352	−0.00652350	−0.00326175	−	0.999995i	$-0.501038\pi$
$251$	−0.103352	−0.00652350	−0.00326175	+	0.999995i	$0.501038\pi$
$252$	13.4728	0.848707
$253$	0.945450	0.0594399
$254$	3.60195	0.226007
$255$	0.354813	0.0222193
$256$	1.00000	0.0625000
$257$	−6.90889	−0.430965	−0.215482	−	0.976508i	$-0.569132\pi$
$257$	−6.90889	−0.430965	−0.215482	+	0.976508i	$0.569132\pi$
$258$	0.0172326	0.00107286
$259$	11.9168	0.740474
$260$	−1.87607	−0.116349
$261$	−2.58507	−0.160012
$262$	−9.57362	−0.591460
$263$	28.3357	1.74726	0.873628	−	0.486594i	$-0.161761\pi$
$263$	28.3357	1.74726	0.873628	+	0.486594i	$0.161761\pi$
$264$	−0.111384	−0.00685524
$265$	22.1429	1.36023
$266$	−4.15224	−0.254590
$267$	−0.205217	−0.0125591
$268$	−15.3437	−0.937264
$269$	16.7044	1.01848	0.509241	−	0.860624i	$-0.329926\pi$
$269$	16.7044	1.01848	0.509241	+	0.860624i	$0.329926\pi$
$270$	−0.413311	−0.0251533
$271$	−0.224835	−0.0136578	−0.00682888	−	0.999977i	$-0.502174\pi$
$271$	−0.224835	−0.0136578	−0.00682888	+	0.999977i	$0.502174\pi$
$272$	5.14957	0.312239
$273$	0.174649	0.0105702
$274$	−8.67464	−0.524054
$275$	−4.93643	−0.297678
$276$	−0.0121176	−0.000729393 0
$277$	−12.2290	−0.734771	−0.367386	−	0.930069i	$-0.619747\pi$
$277$	−12.2290	−0.734771	−0.367386	+	0.930069i	$0.619747\pi$
$278$	−16.3559	−0.980964
$279$	−22.6882	−1.35831
$280$	8.19353	0.489657
$281$	−25.4001	−1.51524	−0.757621	−	0.652695i	$-0.773638\pi$
$281$	−25.4001	−1.51524	−0.757621	+	0.652695i	$0.773638\pi$
$282$	−0.280156	−0.0166830
$283$	−15.0402	−0.894044	−0.447022	−	0.894523i	$-0.647515\pi$
$283$	−15.0402	−0.894044	−0.447022	+	0.894523i	$0.647515\pi$
$284$	−7.28376	−0.432211
$285$	0.0636748	0.00377177
$286$	3.03280	0.179333
$287$	−17.5673	−1.03696
$288$	−2.99857	−0.176693
$289$	9.51807	0.559887
$290$	−1.57212	−0.0923181
$291$	−0.263866	−0.0154681
$292$	−1.35637	−0.0793755
$293$	−28.8577	−1.68588	−0.842941	−	0.538006i	$-0.819178\pi$
$293$	−28.8577	−1.68588	−0.842941	+	0.538006i	$0.819178\pi$
$294$	−0.498276	−0.0290600
$295$	7.10698	0.413784
$296$	−2.65226	−0.154160
$297$	0.668148	0.0387699
$298$	9.69180	0.561431
$299$	0.329940	0.0190809
$300$	0.0632689	0.00365283
$301$	2.04924	0.118117
$302$	−20.4748	−1.17819
$303$	0.540337	0.0310416
$304$	0.924142	0.0530032
$305$	4.10947	0.235307
$306$	−15.4414	−0.882724
$307$	−0.131909	−0.00752847	−0.00376423	−	0.999993i	$-0.501198\pi$
$307$	−0.131909	−0.00752847	−0.00376423	+	0.999993i	$0.501198\pi$
$308$	−13.2455	−0.754730
$309$	−0.0840070	−0.00477899
$310$	−13.7979	−0.783669
$311$	26.0942	1.47967	0.739833	−	0.672790i	$-0.234904\pi$
$311$	26.0942	1.47967	0.739833	+	0.672790i	$0.234904\pi$
$312$	−0.0388706	−0.00220062
$313$	−1.72927	−0.0977440	−0.0488720	−	0.998805i	$-0.515563\pi$
$313$	−1.72927	−0.0977440	−0.0488720	+	0.998805i	$0.515563\pi$
$314$	18.6092	1.05018
$315$	−24.5689	−1.38430
$316$	9.40766	0.529222
$317$	−19.1740	−1.07692	−0.538459	−	0.842652i	$-0.680993\pi$
$317$	−19.1740	−1.07692	−0.538459	+	0.842652i	$0.680993\pi$
$318$	0.458784	0.0257273
$319$	2.54145	0.142294
$320$	−1.82359	−0.101942
$321$	0.442427	0.0246939
$322$	−1.44098	−0.0803027
$323$	4.75894	0.264794
$324$	8.98715	0.499286
$325$	−1.72270	−0.0955582
$326$	−7.24772	−0.401414
$327$	0.633728	0.0350452
$328$	3.90986	0.215886
$329$	−33.3152	−1.83672
$330$	0.203120	0.0111814
$331$	−7.69020	−0.422692	−0.211346	−	0.977411i	$-0.567785\pi$
$331$	−7.69020	−0.422692	−0.211346	+	0.977411i	$0.567785\pi$
$332$	−15.5768	−0.854890
$333$	7.95300	0.435822
$334$	16.5763	0.907017
$335$	27.9806	1.52874
$336$	0.169764	0.00926136
$337$	−21.9129	−1.19367	−0.596835	−	0.802364i	$-0.703575\pi$
$337$	−21.9129	−1.19367	−0.596835	+	0.802364i	$0.703575\pi$
$338$	−11.9416	−0.649539
$339$	−0.621778	−0.0337704
$340$	−9.39071	−0.509283
$341$	22.3054	1.20790
$342$	−2.77111	−0.149844
$343$	−27.8018	−1.50115
$344$	−0.456090	−0.0245907
$345$	0.0220975	0.00118969
$346$	−8.73022	−0.469339
$347$	−26.4690	−1.42093	−0.710464	−	0.703733i	$-0.751515\pi$
$347$	−26.4690	−1.42093	−0.710464	+	0.703733i	$0.751515\pi$
$348$	−0.0325731	−0.00174610
$349$	−20.3844	−1.09115	−0.545576	−	0.838061i	$-0.683689\pi$
$349$	−20.3844	−1.09115	−0.545576	+	0.838061i	$0.683689\pi$
$350$	7.52372	0.402160
$351$	0.233168	0.0124456
$352$	2.94797	0.157128
$353$	5.40597	0.287731	0.143865	−	0.989597i	$-0.454047\pi$
$353$	5.40597	0.287731	0.143865	+	0.989597i	$0.454047\pi$
$354$	0.147251	0.00782631
$355$	13.2826	0.704967
$356$	5.43141	0.287864
$357$	0.874209	0.0462681
$358$	11.7391	0.620429
$359$	22.6610	1.19600	0.598002	−	0.801494i	$-0.295961\pi$
$359$	22.6610	1.19600	0.598002	+	0.801494i	$0.295961\pi$
$360$	5.46817	0.288198
$361$	−18.1460	−0.955051
$362$	−1.64171	−0.0862866
$363$	0.0872589	0.00457991
$364$	−4.62236	−0.242278
$365$	2.47346	0.129467
$366$	0.0851450	0.00445060
$367$	−9.85446	−0.514399	−0.257199	−	0.966358i	$-0.582800\pi$
$367$	−9.85446	−0.514399	−0.257199	+	0.966358i	$0.582800\pi$
$368$	0.320712	0.0167183
$369$	−11.7240	−0.610327
$370$	4.83664	0.251445
$371$	54.5571	2.83246
$372$	−0.285882	−0.0148223
$373$	10.0253	0.519092	0.259546	−	0.965731i	$-0.416427\pi$
$373$	10.0253	0.519092	0.259546	+	0.965731i	$0.416427\pi$
$374$	15.1808	0.784980
$375$	−0.459884	−0.0237483
$376$	7.41479	0.382388
$377$	0.886909	0.0456781
$378$	−1.01834	−0.0523777
$379$	−3.53539	−0.181601	−0.0908003	−	0.995869i	$-0.528942\pi$
$379$	−3.53539	−0.181601	−0.0908003	+	0.995869i	$0.528942\pi$
$380$	−1.68526	−0.0864519
$381$	−0.136094	−0.00697231
$382$	7.21088	0.368941
$383$	−8.20074	−0.419038	−0.209519	−	0.977805i	$-0.567190\pi$
$383$	−8.20074	−0.419038	−0.209519	+	0.977805i	$0.567190\pi$
$384$	−0.0377834	−0.00192813
$385$	24.1543	1.23102
$386$	−26.4571	−1.34663
$387$	1.36762	0.0695200
$388$	6.98364	0.354541
$389$	24.3269	1.23342	0.616711	−	0.787190i	$-0.288465\pi$
$389$	24.3269	1.23342	0.616711	+	0.787190i	$0.288465\pi$
$390$	0.0708841	0.00358936
$391$	1.65153	0.0835214
$392$	13.1877	0.666079
$393$	0.361724	0.0182466
$394$	23.8317	1.20063
$395$	−17.1557	−0.863198
$396$	−8.83971	−0.444212
$397$	8.66286	0.434776	0.217388	−	0.976085i	$-0.430246\pi$
$397$	8.66286	0.434776	0.217388	+	0.976085i	$0.430246\pi$
$398$	−4.40209	−0.220657
$399$	0.156886	0.00785411
$400$	−1.67451	−0.0837257
$401$	15.8309	0.790558	0.395279	−	0.918561i	$-0.370648\pi$
$401$	15.8309	0.790558	0.395279	+	0.918561i	$0.370648\pi$
$402$	0.579736	0.0289146
$403$	7.78406	0.387752
$404$	−14.3009	−0.711497
$405$	−16.3889	−0.814371
$406$	−3.87349	−0.192238
$407$	−7.81880	−0.387563
$408$	−0.194568	−0.00963257
$409$	16.7629	0.828870	0.414435	−	0.910079i	$-0.363979\pi$
$409$	16.7629	0.828870	0.414435	+	0.910079i	$0.363979\pi$
$410$	−7.12998	−0.352125
$411$	0.327757	0.0161671
$412$	2.22338	0.109538
$413$	17.5106	0.861640
$414$	−0.961677	−0.0472639
$415$	28.4058	1.39439
$416$	1.02878	0.0504399
$417$	0.617983	0.0302627
$418$	2.72435	0.133252
$419$	21.9696	1.07329	0.536643	−	0.843809i	$-0.319692\pi$
$419$	21.9696	1.07329	0.536643	+	0.843809i	$0.319692\pi$
$420$	−0.309579	−0.0151059
$421$	−1.87813	−0.0915347	−0.0457673	−	0.998952i	$-0.514573\pi$
$421$	−1.87813	−0.0915347	−0.0457673	+	0.998952i	$0.514573\pi$
$422$	−14.1619	−0.689390
$423$	−22.2338	−1.08104
$424$	−12.1425	−0.589691
$425$	−8.62303	−0.418279
$426$	0.275205	0.0133337
$427$	10.1251	0.489990
$428$	−11.7096	−0.566003
$429$	−0.114590	−0.00553244
$430$	0.831722	0.0401092
$431$	−9.03987	−0.435435	−0.217718	−	0.976012i	$-0.569861\pi$
$431$	−9.03987	−0.435435	−0.217718	+	0.976012i	$0.569861\pi$
$432$	0.226646	0.0109045
$433$	−21.0738	−1.01274	−0.506372	−	0.862315i	$-0.669014\pi$
$433$	−21.0738	−1.01274	−0.506372	+	0.862315i	$0.669014\pi$
$434$	−33.9961	−1.63187
$435$	0.0594001	0.00284802
$436$	−16.7727	−0.803265
$437$	0.296383	0.0141779
$438$	0.0512482	0.00244874
$439$	36.9824	1.76507	0.882537	−	0.470242i	$-0.155833\pi$
$439$	36.9824	1.76507	0.882537	+	0.470242i	$0.155833\pi$
$440$	−5.37590	−0.256286
$441$	−39.5443	−1.88306
$442$	5.29775	0.251988
$443$	−33.2110	−1.57790	−0.788951	−	0.614456i	$-0.789375\pi$
$443$	−33.2110	−1.57790	−0.788951	+	0.614456i	$0.789375\pi$
$444$	0.100211	0.00475583
$445$	−9.90467	−0.469526
$446$	−6.30457	−0.298530
$447$	−0.366189	−0.0173201
$448$	−4.49307	−0.212278
$449$	22.7352	1.07294	0.536471	−	0.843919i	$-0.319757\pi$
$449$	22.7352	1.07294	0.536471	+	0.843919i	$0.319757\pi$
$450$	5.02115	0.236699
$451$	11.5262	0.542746
$452$	16.4564	0.774043
$453$	0.773607	0.0363472
$454$	−21.3486	−1.00194
$455$	8.42930	0.395172
$456$	−0.0349172	−0.00163515
$457$	4.52284	0.211569	0.105785	−	0.994389i	$-0.466265\pi$
$457$	4.52284	0.211569	0.105785	+	0.994389i	$0.466265\pi$
$458$	−24.4463	−1.14230
$459$	1.16713	0.0544771
$460$	−0.584847	−0.0272686
$461$	−14.1256	−0.657894	−0.328947	−	0.944348i	$-0.606694\pi$
$461$	−14.1256	−0.657894	−0.328947	+	0.944348i	$0.606694\pi$
$462$	0.500458	0.0232834
$463$	−30.0023	−1.39433	−0.697164	−	0.716912i	$-0.745555\pi$
$463$	−30.0023	−1.39433	−0.697164	+	0.716912i	$0.745555\pi$
$464$	0.862102	0.0400221
$465$	0.521332	0.0241762
$466$	−13.6707	−0.633284
$467$	35.5460	1.64487	0.822436	−	0.568857i	$-0.192614\pi$
$467$	35.5460	1.64487	0.822436	+	0.568857i	$0.192614\pi$
$468$	−3.08486	−0.142598
$469$	68.9402	3.18336
$470$	−13.5215	−0.623702
$471$	−0.703118	−0.0323980
$472$	−3.89724	−0.179385
$473$	−1.34454	−0.0618221
$474$	−0.355453	−0.0163265
$475$	−1.54749	−0.0710037
$476$	−23.1374	−1.06050
$477$	36.4101	1.66710
$478$	25.3886	1.16125
$479$	−4.77037	−0.217964	−0.108982	−	0.994044i	$-0.534759\pi$
$479$	−4.77037	−0.217964	−0.108982	+	0.994044i	$0.534759\pi$
$480$	0.0689015	0.00314491
$481$	−2.72858	−0.124413
$482$	13.6569	0.622053
$483$	0.0544452	0.00247734
$484$	−2.30945	−0.104975
$485$	−12.7353	−0.578281
$486$	−1.01950	−0.0462457
$487$	27.8040	1.25992	0.629960	−	0.776627i	$-0.283071\pi$
$487$	27.8040	1.25992	0.629960	+	0.776627i	$0.283071\pi$
$488$	−2.25350	−0.102011
$489$	0.273844	0.0123836
$490$	−24.0490	−1.08642
$491$	−36.6377	−1.65344	−0.826718	−	0.562617i	$-0.809795\pi$
$491$	−36.6377	−1.65344	−0.826718	+	0.562617i	$0.809795\pi$
$492$	−0.147728	−0.00666008
$493$	4.43945	0.199943
$494$	0.950735	0.0427756
$495$	16.1200	0.724541
$496$	7.56634	0.339739
$497$	32.7264	1.46798
$498$	0.588546	0.0263734
$499$	35.2001	1.57577	0.787886	−	0.615821i	$-0.211176\pi$
$499$	35.2001	1.57577	0.787886	+	0.615821i	$0.211176\pi$
$500$	12.1716	0.544330
$501$	−0.626310	−0.0279815
$502$	−0.103352	−0.00461281
$503$	−37.9983	−1.69426	−0.847130	−	0.531386i	$-0.821671\pi$
$503$	−37.9983	−1.69426	−0.847130	+	0.531386i	$0.821671\pi$
$504$	13.4728	0.600126
$505$	26.0790	1.16050
$506$	0.945450	0.0420304
$507$	0.451195	0.0200383
$508$	3.60195	0.159811
$509$	−35.5358	−1.57510	−0.787549	−	0.616252i	$-0.788650\pi$
$509$	−35.5358	−1.57510	−0.787549	+	0.616252i	$0.788650\pi$
$510$	0.354813	0.0157114
$511$	6.09426	0.269594
$512$	1.00000	0.0441942
$513$	0.209454	0.00924760
$514$	−6.90889	−0.304738
$515$	−4.05454	−0.178664
$516$	0.0172326	0.000758624 0
$517$	21.8586	0.961340
$518$	11.9168	0.523594
$519$	0.329857	0.0144791
$520$	−1.87607	−0.0822709
$521$	31.3429	1.37316	0.686578	−	0.727056i	$-0.259112\pi$
$521$	31.3429	1.37316	0.686578	+	0.727056i	$0.259112\pi$
$522$	−2.58507	−0.113146
$523$	−26.6164	−1.16386	−0.581928	−	0.813241i	$-0.697701\pi$
$523$	−26.6164	−1.16386	−0.581928	+	0.813241i	$0.697701\pi$
$524$	−9.57362	−0.418226
$525$	−0.284272	−0.0124066
$526$	28.3357	1.23550
$527$	38.9634	1.69727
$528$	−0.111384	−0.00484739
$529$	−22.8971	−0.995528
$530$	22.1429	0.961827
$531$	11.6862	0.507137
$532$	−4.15224	−0.180022
$533$	4.02237	0.174228
$534$	−0.205217	−0.00888061
$535$	21.3535	0.923191
$536$	−15.3437	−0.662745
$537$	−0.443542	−0.0191403
$538$	16.7044	0.720176
$539$	38.8770	1.67455
$540$	−0.413311	−0.0177861
$541$	13.9405	0.599348	0.299674	−	0.954042i	$-0.403122\pi$
$541$	13.9405	0.599348	0.299674	+	0.954042i	$0.403122\pi$
$542$	−0.224835	−0.00965750
$543$	0.0620295	0.00266194
$544$	5.14957	0.220786
$545$	30.5865	1.31018
$546$	0.174649	0.00747427
$547$	27.5973	1.17997	0.589987	−	0.807413i	$-0.299133\pi$
$547$	27.5973	1.17997	0.589987	+	0.807413i	$0.299133\pi$
$548$	−8.67464	−0.370562
$549$	6.75729	0.288394
$550$	−4.93643	−0.210490
$551$	0.796705	0.0339408
$552$	−0.0121176	−0.000515759 0
$553$	−42.2693	−1.79747
$554$	−12.2290	−0.519562
$555$	−0.182745	−0.00775708
$556$	−16.3559	−0.693646
$557$	12.3568	0.523572	0.261786	−	0.965126i	$-0.415688\pi$
$557$	12.3568	0.523572	0.261786	+	0.965126i	$0.415688\pi$
$558$	−22.6882	−0.960469
$559$	−0.469214	−0.0198456
$560$	8.19353	0.346240
$561$	−0.573582	−0.0242167
$562$	−25.4001	−1.07144
$563$	3.45606	0.145656	0.0728278	−	0.997345i	$-0.476798\pi$
$563$	3.45606	0.145656	0.0728278	+	0.997345i	$0.476798\pi$
$564$	−0.280156	−0.0117967
$565$	−30.0097	−1.26252
$566$	−15.0402	−0.632185
$567$	−40.3799	−1.69580
$568$	−7.28376	−0.305620
$569$	−23.9714	−1.00494	−0.502468	−	0.864596i	$-0.667574\pi$
$569$	−23.9714	−1.00494	−0.502468	+	0.864596i	$0.667574\pi$
$570$	0.0636748	0.00266704
$571$	−23.0899	−0.966282	−0.483141	−	0.875543i	$-0.660504\pi$
$571$	−23.0899	−0.966282	−0.483141	+	0.875543i	$0.660504\pi$
$572$	3.03280	0.126808
$573$	−0.272452	−0.0113818
$574$	−17.5673	−0.733244
$575$	−0.537037	−0.0223960
$576$	−2.99857	−0.124941
$577$	−8.78931	−0.365904	−0.182952	−	0.983122i	$-0.558565\pi$
$577$	−8.78931	−0.365904	−0.182952	+	0.983122i	$0.558565\pi$
$578$	9.51807	0.395900
$579$	0.999640	0.0415436
$580$	−1.57212	−0.0652788
$581$	69.9879	2.90359
$582$	−0.263866	−0.0109376
$583$	−35.7957	−1.48251
$584$	−1.35637	−0.0561270
$585$	5.62552	0.232587
$586$	−28.8577	−1.19210
$587$	22.7542	0.939166	0.469583	−	0.882888i	$-0.344404\pi$
$587$	22.7542	0.939166	0.469583	+	0.882888i	$0.344404\pi$
$588$	−0.498276	−0.0205486
$589$	6.99237	0.288116
$590$	7.10698	0.292590
$591$	−0.900444	−0.0370393
$592$	−2.65226	−0.109007
$593$	13.7129	0.563122	0.281561	−	0.959543i	$-0.409148\pi$
$593$	13.7129	0.563122	0.281561	+	0.959543i	$0.409148\pi$
$594$	0.668148	0.0274144
$595$	42.1931	1.72975
$596$	9.69180	0.396991
$597$	0.166326	0.00680726
$598$	0.329940	0.0134923
$599$	34.2896	1.40103	0.700517	−	0.713636i	$-0.252953\pi$
$599$	34.2896	1.40103	0.700517	+	0.713636i	$0.252953\pi$
$600$	0.0632689	0.00258294
$601$	−36.6194	−1.49373	−0.746867	−	0.664973i	$-0.768443\pi$
$601$	−36.6194	−1.49373	−0.746867	+	0.664973i	$0.768443\pi$
$602$	2.04924	0.0835210
$603$	46.0091	1.87364
$604$	−20.4748	−0.833107
$605$	4.21149	0.171222
$606$	0.540337	0.0219497
$607$	−4.43835	−0.180147	−0.0900736	−	0.995935i	$-0.528710\pi$
$607$	−4.43835	−0.180147	−0.0900736	+	0.995935i	$0.528710\pi$
$608$	0.924142	0.0374789
$609$	0.146353	0.00593054
$610$	4.10947	0.166387
$611$	7.62815	0.308602
$612$	−15.4414	−0.624180
$613$	24.7473	0.999536	0.499768	−	0.866159i	$-0.333419\pi$
$613$	24.7473	0.999536	0.499768	+	0.866159i	$0.333419\pi$
$614$	−0.131909	−0.00532343
$615$	0.269395	0.0108631
$616$	−13.2455	−0.533675
$617$	−2.25272	−0.0906911	−0.0453456	−	0.998971i	$-0.514439\pi$
$617$	−2.25272	−0.0906911	−0.0453456	+	0.998971i	$0.514439\pi$
$618$	−0.0840070	−0.00337926
$619$	1.47588	0.0593205	0.0296602	−	0.999560i	$-0.490557\pi$
$619$	1.47588	0.0593205	0.0296602	+	0.999560i	$0.490557\pi$
$620$	−13.7979	−0.554137
$621$	0.0726882	0.00291688
$622$	26.0942	1.04628
$623$	−24.4037	−0.977714
$624$	−0.0388706	−0.00155607
$625$	−13.8234	−0.552937
$626$	−1.72927	−0.0691154
$627$	−0.102935	−0.00411083
$628$	18.6092	0.742587
$629$	−13.6580	−0.544580
$630$	−24.5689	−0.978848
$631$	32.9470	1.31160	0.655799	−	0.754935i	$-0.272332\pi$
$631$	32.9470	1.31160	0.655799	+	0.754935i	$0.272332\pi$
$632$	9.40766	0.374216
$633$	0.535084	0.0212677
$634$	−19.1740	−0.761496
$635$	−6.56849	−0.260663
$636$	0.458784	0.0181920
$637$	13.5672	0.537551
$638$	2.54145	0.100617
$639$	21.8409	0.864011
$640$	−1.82359	−0.0720838
$641$	16.1229	0.636817	0.318409	−	0.947954i	$-0.396852\pi$
$641$	16.1229	0.636817	0.318409	+	0.947954i	$0.396852\pi$
$642$	0.442427	0.0174612
$643$	−17.4973	−0.690026	−0.345013	−	0.938598i	$-0.612125\pi$
$643$	−17.4973	−0.690026	−0.345013	+	0.938598i	$0.612125\pi$
$644$	−1.44098	−0.0567826
$645$	−0.0314253	−0.00123737
$646$	4.75894	0.187238
$647$	−21.4974	−0.845151	−0.422575	−	0.906328i	$-0.638874\pi$
$647$	−21.4974	−0.845151	−0.422575	+	0.906328i	$0.638874\pi$
$648$	8.98715	0.353049
$649$	−11.4890	−0.450982
$650$	−1.72270	−0.0675698
$651$	1.28449	0.0503431
$652$	−7.24772	−0.283843
$653$	5.41020	0.211718	0.105859	−	0.994381i	$-0.466241\pi$
$653$	5.41020	0.211718	0.105859	+	0.994381i	$0.466241\pi$
$654$	0.633728	0.0247807
$655$	17.4584	0.682155
$656$	3.90986	0.152654
$657$	4.06717	0.158675
$658$	−33.3152	−1.29876
$659$	26.9747	1.05078	0.525392	−	0.850860i	$-0.323919\pi$
$659$	26.9747	1.05078	0.525392	+	0.850860i	$0.323919\pi$
$660$	0.203120	0.00790643
$661$	28.0869	1.09245	0.546227	−	0.837637i	$-0.316064\pi$
$661$	28.0869	1.09245	0.546227	+	0.837637i	$0.316064\pi$
$662$	−7.69020	−0.298888
$663$	−0.200167	−0.00777384
$664$	−15.5768	−0.604499
$665$	7.57198	0.293629
$666$	7.95300	0.308172
$667$	0.276486	0.0107056
$668$	16.5763	0.641358
$669$	0.238208	0.00920966
$670$	27.9806	1.08098
$671$	−6.64327	−0.256460
$672$	0.169764	0.00654877
$673$	14.4046	0.555258	0.277629	−	0.960688i	$-0.410451\pi$
$673$	14.4046	0.555258	0.277629	+	0.960688i	$0.410451\pi$
$674$	−21.9129	−0.844053
$675$	−0.379523	−0.0146078
$676$	−11.9416	−0.459293
$677$	−3.10980	−0.119520	−0.0597598	−	0.998213i	$-0.519033\pi$
$677$	−3.10980	−0.119520	−0.0597598	+	0.998213i	$0.519033\pi$
$678$	−0.621778	−0.0238792
$679$	−31.3780	−1.20418
$680$	−9.39071	−0.360117
$681$	0.806624	0.0309099
$682$	22.3054	0.854117
$683$	−39.1509	−1.49807	−0.749034	−	0.662531i	$-0.769482\pi$
$683$	−39.1509	−1.49807	−0.749034	+	0.662531i	$0.769482\pi$
$684$	−2.77111	−0.105956
$685$	15.8190	0.604413
$686$	−27.8018	−1.06148
$687$	0.923665	0.0352400
$688$	−0.456090	−0.0173883
$689$	−12.4919	−0.475903
$690$	0.0220975	0.000841238 0
$691$	−23.1580	−0.880973	−0.440487	−	0.897759i	$-0.645194\pi$
$691$	−23.1580	−0.880973	−0.440487	+	0.897759i	$0.645194\pi$
$692$	−8.73022	−0.331873
$693$	39.7175	1.50874
$694$	−26.4690	−1.00475
$695$	29.8265	1.13139
$696$	−0.0325731	−0.00123468
$697$	20.1341	0.762633
$698$	−20.3844	−0.771561
$699$	0.516526	0.0195368
$700$	7.52372	0.284370
$701$	−38.6929	−1.46141	−0.730705	−	0.682694i	$-0.760808\pi$
$701$	−38.6929	−1.46141	−0.730705	+	0.682694i	$0.760808\pi$
$702$	0.233168	0.00880037
$703$	−2.45107	−0.0924437
$704$	2.94797	0.111106
$705$	0.510890	0.0192412
$706$	5.40597	0.203456
$707$	64.2550	2.41656
$708$	0.147251	0.00553404
$709$	−48.2105	−1.81058	−0.905292	−	0.424790i	$-0.860348\pi$
$709$	−48.2105	−1.81058	−0.905292	+	0.424790i	$0.860348\pi$
$710$	13.2826	0.498487
$711$	−28.2095	−1.05794
$712$	5.43141	0.203551
$713$	2.42661	0.0908774
$714$	0.874209	0.0327165
$715$	−5.53059	−0.206832
$716$	11.7391	0.438710
$717$	−0.959269	−0.0358246
$718$	22.6610	0.845703
$719$	14.9606	0.557937	0.278969	−	0.960300i	$-0.410007\pi$
$719$	14.9606	0.557937	0.278969	+	0.960300i	$0.410007\pi$
$720$	5.46817	0.203787
$721$	−9.98982	−0.372040
$722$	−18.1460	−0.675323
$723$	−0.516003	−0.0191904
$724$	−1.64171	−0.0610138
$725$	−1.44360	−0.0536140
$726$	0.0872589	0.00323848
$727$	35.6098	1.32069	0.660346	−	0.750961i	$-0.270409\pi$
$727$	35.6098	1.32069	0.660346	+	0.750961i	$0.270409\pi$
$728$	−4.62236	−0.171316
$729$	−26.9229	−0.997146
$730$	2.47346	0.0915470
$731$	−2.34867	−0.0868686
$732$	0.0851450	0.00314705
$733$	16.2142	0.598885	0.299442	−	0.954114i	$-0.403199\pi$
$733$	16.2142	0.598885	0.299442	+	0.954114i	$0.403199\pi$
$734$	−9.85446	−0.363735
$735$	0.908652	0.0335161
$736$	0.320712	0.0118216
$737$	−45.2327	−1.66617
$738$	−11.7240	−0.431566
$739$	36.8109	1.35411	0.677055	−	0.735933i	$-0.263256\pi$
$739$	36.8109	1.35411	0.677055	+	0.735933i	$0.263256\pi$
$740$	4.83664	0.177798
$741$	−0.0359220	−0.00131963
$742$	54.5571	2.00285
$743$	−42.4165	−1.55611	−0.778054	−	0.628197i	$-0.783793\pi$
$743$	−42.4165	−1.55611	−0.778054	+	0.628197i	$0.783793\pi$
$744$	−0.285882	−0.0104809
$745$	−17.6739	−0.647521
$746$	10.0253	0.367053
$747$	46.7083	1.70897
$748$	15.1808	0.555065
$749$	52.6119	1.92240
$750$	−0.459884	−0.0167926
$751$	−2.08442	−0.0760614	−0.0380307	−	0.999277i	$-0.512108\pi$
$751$	−2.08442	−0.0760614	−0.0380307	+	0.999277i	$0.512108\pi$
$752$	7.41479	0.270389
$753$	0.00390498	0.000142305 0
$754$	0.886909	0.0322993
$755$	37.3376	1.35886
$756$	−1.01834	−0.0370366
$757$	9.06435	0.329449	0.164725	−	0.986340i	$-0.447326\pi$
$757$	9.06435	0.329449	0.164725	+	0.986340i	$0.447326\pi$
$758$	−3.53539	−0.128411
$759$	−0.0357223	−0.00129664
$760$	−1.68526	−0.0611307
$761$	−27.0023	−0.978834	−0.489417	−	0.872050i	$-0.662790\pi$
$761$	−27.0023	−0.978834	−0.489417	+	0.872050i	$0.662790\pi$
$762$	−0.136094	−0.00493017
$763$	75.3608	2.72824
$764$	7.21088	0.260880
$765$	28.1587	1.01808
$766$	−8.20074	−0.296305
$767$	−4.00939	−0.144771
$768$	−0.0377834	−0.00136339
$769$	24.3762	0.879030	0.439515	−	0.898235i	$-0.355150\pi$
$769$	24.3762	0.879030	0.439515	+	0.898235i	$0.355150\pi$
$770$	24.1543	0.870461
$771$	0.261041	0.00940118
$772$	−26.4571	−0.952213
$773$	−15.4317	−0.555040	−0.277520	−	0.960720i	$-0.589513\pi$
$773$	−15.4317	−0.555040	−0.277520	+	0.960720i	$0.589513\pi$
$774$	1.36762	0.0491580
$775$	−12.6699	−0.455118
$776$	6.98364	0.250698
$777$	−0.450257	−0.0161529
$778$	24.3269	0.872161
$779$	3.61327	0.129459
$780$	0.0708841	0.00253806
$781$	−21.4723	−0.768340
$782$	1.65153	0.0590585
$783$	0.195392	0.00698275
$784$	13.1877	0.470989
$785$	−33.9356	−1.21121
$786$	0.361724	0.0129023
$787$	5.47057	0.195005	0.0975023	−	0.995235i	$-0.468915\pi$
$787$	5.47057	0.195005	0.0975023	+	0.995235i	$0.468915\pi$
$788$	23.8317	0.848971
$789$	−1.07062	−0.0381151
$790$	−17.1557	−0.610373
$791$	−73.9397	−2.62899
$792$	−8.83971	−0.314106
$793$	−2.31835	−0.0823269
$794$	8.66286	0.307433
$795$	−0.836635	−0.0296724
$796$	−4.40209	−0.156028
$797$	−17.4494	−0.618089	−0.309044	−	0.951048i	$-0.600009\pi$
$797$	−17.4494	−0.618089	−0.309044	+	0.951048i	$0.600009\pi$
$798$	0.156886	0.00555369
$799$	38.1830	1.35082
$800$	−1.67451	−0.0592030
$801$	−16.2865	−0.575454
$802$	15.8309	0.559009
$803$	−3.99854	−0.141105
$804$	0.579736	0.0204457
$805$	2.62776	0.0926164
$806$	7.78406	0.274182
$807$	−0.631147	−0.0222174
$808$	−14.3009	−0.503104
$809$	−3.38334	−0.118952	−0.0594760	−	0.998230i	$-0.518943\pi$
$809$	−3.38334	−0.118952	−0.0594760	+	0.998230i	$0.518943\pi$
$810$	−16.3889	−0.575847
$811$	29.8356	1.04767	0.523836	−	0.851819i	$-0.324501\pi$
$811$	29.8356	1.04767	0.523836	+	0.851819i	$0.324501\pi$
$812$	−3.87349	−0.135933
$813$	0.00849504	0.000297934 0
$814$	−7.81880	−0.274049
$815$	13.2169	0.462967
$816$	−0.194568	−0.00681125
$817$	−0.421492	−0.0147461
$818$	16.7629	0.586100
$819$	13.8605	0.484325
$820$	−7.12998	−0.248990
$821$	−20.1358	−0.702744	−0.351372	−	0.936236i	$-0.614285\pi$
$821$	−20.1358	−0.702744	−0.351372	+	0.936236i	$0.614285\pi$
$822$	0.327757	0.0114318
$823$	−1.21520	−0.0423593	−0.0211797	−	0.999776i	$-0.506742\pi$
$823$	−1.21520	−0.0423593	−0.0211797	+	0.999776i	$0.506742\pi$
$824$	2.22338	0.0774552
$825$	0.186515	0.00649362
$826$	17.5106	0.609272
$827$	−32.8738	−1.14313	−0.571567	−	0.820556i	$-0.693664\pi$
$827$	−32.8738	−1.14313	−0.571567	+	0.820556i	$0.693664\pi$
$828$	−0.961677	−0.0334206
$829$	7.21413	0.250557	0.125279	−	0.992122i	$-0.460018\pi$
$829$	7.21413	0.250557	0.125279	+	0.992122i	$0.460018\pi$
$830$	28.4058	0.985980
$831$	0.462054	0.0160285
$832$	1.02878	0.0356664
$833$	67.9110	2.35298
$834$	0.617983	0.0213990
$835$	−30.2285	−1.04610
$836$	2.72435	0.0942235
$837$	1.71488	0.0592751
$838$	21.9696	0.758928
$839$	−27.2669	−0.941359	−0.470680	−	0.882304i	$-0.655991\pi$
$839$	−27.2669	−0.941359	−0.470680	+	0.882304i	$0.655991\pi$
$840$	−0.309579	−0.0106815
$841$	−28.2568	−0.974372
$842$	−1.87813	−0.0647248
$843$	0.959701	0.0330539
$844$	−14.1619	−0.487472
$845$	21.7766	0.749139
$846$	−22.2338	−0.764413
$847$	10.3765	0.356542
$848$	−12.1425	−0.416975
$849$	0.568268	0.0195029
$850$	−8.62303	−0.295768
$851$	−0.850612	−0.0291586
$852$	0.275205	0.00942837
$853$	−13.6699	−0.468050	−0.234025	−	0.972231i	$-0.575190\pi$
$853$	−13.6699	−0.468050	−0.234025	+	0.972231i	$0.575190\pi$
$854$	10.1251	0.346475
$855$	5.05337	0.172822
$856$	−11.7096	−0.400225
$857$	53.0060	1.81065	0.905326	−	0.424718i	$-0.139627\pi$
$857$	53.0060	1.81065	0.905326	+	0.424718i	$0.139627\pi$
$858$	−0.114590	−0.00391202
$859$	−49.6428	−1.69379	−0.846896	−	0.531759i	$-0.821531\pi$
$859$	−49.6428	−1.69379	−0.846896	+	0.531759i	$0.821531\pi$
$860$	0.831722	0.0283615
$861$	0.663752	0.0226206
$862$	−9.03987	−0.307899
$863$	−30.1326	−1.02572	−0.512862	−	0.858471i	$-0.671415\pi$
$863$	−30.1326	−1.02572	−0.512862	+	0.858471i	$0.671415\pi$
$864$	0.226646	0.00771067
$865$	15.9203	0.541308
$866$	−21.0738	−0.716118
$867$	−0.359625	−0.0122135
$868$	−33.9961	−1.15390
$869$	27.7335	0.940795
$870$	0.0594001	0.00201385
$871$	−15.7852	−0.534861
$872$	−16.7727	−0.567994
$873$	−20.9410	−0.708744
$874$	0.296383	0.0100253
$875$	−54.6878	−1.84879
$876$	0.0512482	0.00173152
$877$	23.4961	0.793405	0.396703	−	0.917947i	$-0.370154\pi$
$877$	23.4961	0.793405	0.396703	+	0.917947i	$0.370154\pi$
$878$	36.9824	1.24810
$879$	1.09034	0.0367763
$880$	−5.37590	−0.181222
$881$	−19.7011	−0.663748	−0.331874	−	0.943324i	$-0.607681\pi$
$881$	−19.7011	−0.663748	−0.331874	+	0.943324i	$0.607681\pi$
$882$	−39.5443	−1.33152
$883$	45.3162	1.52501	0.762506	−	0.646981i	$-0.223969\pi$
$883$	45.3162	1.52501	0.762506	+	0.646981i	$0.223969\pi$
$884$	5.29775	0.178183
$885$	−0.268526	−0.00902640
$886$	−33.2110	−1.11574
$887$	−31.3761	−1.05351	−0.526753	−	0.850018i	$-0.676591\pi$
$887$	−31.3761	−1.05351	−0.526753	+	0.850018i	$0.676591\pi$
$888$	0.100211	0.00336288
$889$	−16.1838	−0.542788
$890$	−9.90467	−0.332005
$891$	26.4939	0.887579
$892$	−6.30457	−0.211093
$893$	6.85232	0.229304
$894$	−0.366189	−0.0122472
$895$	−21.4073	−0.715566
$896$	−4.49307	−0.150103
$897$	−0.0124663	−0.000416237 0
$898$	22.7352	0.758685
$899$	6.52296	0.217553
$900$	5.02115	0.167372
$901$	−62.5286	−2.08313
$902$	11.5262	0.383779
$903$	−0.0774274	−0.00257662
$904$	16.4564	0.547331
$905$	2.99381	0.0995178
$906$	0.773607	0.0257014
$907$	−4.10728	−0.136380	−0.0681899	−	0.997672i	$-0.521722\pi$
$907$	−4.10728	−0.136380	−0.0681899	+	0.997672i	$0.521722\pi$
$908$	−21.3486	−0.708480
$909$	42.8823	1.42232
$910$	8.42930	0.279428
$911$	−9.11676	−0.302052	−0.151026	−	0.988530i	$-0.548258\pi$
$911$	−9.11676	−0.302052	−0.151026	+	0.988530i	$0.548258\pi$
$912$	−0.0349172	−0.00115623
$913$	−45.9201	−1.51973
$914$	4.52284	0.149602
$915$	−0.155270	−0.00513306
$916$	−24.4463	−0.807729
$917$	43.0150	1.42048
$918$	1.16713	0.0385211
$919$	23.6252	0.779323	0.389662	−	0.920958i	$-0.372592\pi$
$919$	23.6252	0.779323	0.389662	+	0.920958i	$0.372592\pi$
$920$	−0.584847	−0.0192818
$921$	0.00498398	0.000164228 0
$922$	−14.1256	−0.465201
$923$	−7.49335	−0.246647
$924$	0.500458	0.0164639
$925$	4.44125	0.146027
$926$	−30.0023	−0.985938
$927$	−6.66698	−0.218972
$928$	0.862102	0.0282999
$929$	−14.2110	−0.466247	−0.233124	−	0.972447i	$-0.574895\pi$
$929$	−14.2110	−0.466247	−0.233124	+	0.972447i	$0.574895\pi$
$930$	0.521332	0.0170951
$931$	12.1873	0.399423
$932$	−13.6707	−0.447799
$933$	−0.985928	−0.0322778
$934$	35.5460	1.16310
$935$	−27.6836	−0.905350
$936$	−3.08486	−0.100832
$937$	−26.2628	−0.857969	−0.428985	−	0.903312i	$-0.641129\pi$
$937$	−26.2628	−0.857969	−0.428985	+	0.903312i	$0.641129\pi$
$938$	68.9402	2.25098
$939$	0.0653376	0.00213221
$940$	−13.5215	−0.441024
$941$	32.4071	1.05644	0.528220	−	0.849108i	$-0.322860\pi$
$941$	32.4071	1.05644	0.528220	+	0.849108i	$0.322860\pi$
$942$	−0.703118	−0.0229088
$943$	1.25394	0.0408338
$944$	−3.89724	−0.126845
$945$	1.85703	0.0604093
$946$	−1.34454	−0.0437148
$947$	−18.3451	−0.596135	−0.298068	−	0.954545i	$-0.596342\pi$
$947$	−18.3451	−0.596135	−0.298068	+	0.954545i	$0.596342\pi$
$948$	−0.355453	−0.0115446
$949$	−1.39540	−0.0452966
$950$	−1.54749	−0.0502072
$951$	0.724458	0.0234922
$952$	−23.1374	−0.749887
$953$	−10.3656	−0.335776	−0.167888	−	0.985806i	$-0.553695\pi$
$953$	−10.3656	−0.335776	−0.167888	+	0.985806i	$0.553695\pi$
$954$	36.4101	1.17882
$955$	−13.1497	−0.425514
$956$	25.3886	0.821127
$957$	−0.0960248	−0.00310404
$958$	−4.77037	−0.154124
$959$	38.9758	1.25859
$960$	0.0689015	0.00222379
$961$	26.2495	0.846758
$962$	−2.72858	−0.0879730
$963$	35.1120	1.13147
$964$	13.6569	0.439858
$965$	48.2470	1.55313
$966$	0.0544452	0.00175174
$967$	1.65278	0.0531499	0.0265750	−	0.999647i	$-0.491540\pi$
$967$	1.65278	0.0531499	0.0265750	+	0.999647i	$0.491540\pi$
$968$	−2.30945	−0.0742286
$969$	−0.179809	−0.00577629
$970$	−12.7353	−0.408906
$971$	−35.6275	−1.14334	−0.571671	−	0.820483i	$-0.693705\pi$
$971$	−35.6275	−1.14334	−0.571671	+	0.820483i	$0.693705\pi$
$972$	−1.01950	−0.0327006
$973$	73.4884	2.35593
$974$	27.8040	0.890898
$975$	0.0650894	0.00208453
$976$	−2.25350	−0.0721328
$977$	−45.7044	−1.46221	−0.731107	−	0.682263i	$-0.760996\pi$
$977$	−45.7044	−1.46221	−0.731107	+	0.682263i	$0.760996\pi$
$978$	0.273844	0.00875656
$979$	16.0116	0.511734
$980$	−24.0490	−0.768216
$981$	50.2940	1.60576
$982$	−36.6377	−1.16916
$983$	28.8714	0.920854	0.460427	−	0.887697i	$-0.347696\pi$
$983$	28.8714	0.920854	0.460427	+	0.887697i	$0.347696\pi$
$984$	−0.147728	−0.00470939
$985$	−43.4594	−1.38473
$986$	4.43945	0.141381
$987$	1.25876	0.0400668
$988$	0.950735	0.0302469
$989$	−0.146273	−0.00465122
$990$	16.1200	0.512328
$991$	−19.0766	−0.605987	−0.302994	−	0.952993i	$-0.597986\pi$
$991$	−19.0766	−0.605987	−0.302994	+	0.952993i	$0.597986\pi$
$992$	7.56634	0.240232
$993$	0.290562	0.00922071
$994$	32.7264	1.03802
$995$	8.02760	0.254492
$996$	0.588546	0.0186488
$997$	−47.0741	−1.49085	−0.745426	−	0.666589i	$-0.767754\pi$
$997$	−47.0741	−1.49085	−0.745426	+	0.666589i	$0.767754\pi$
$998$	35.2001	1.11424
$999$	−0.601126	−0.0190188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.39	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.39	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.0377834 q^{3} +1.00000 q^{4} -1.82359 q^{5} -0.0377834 q^{6} -4.49307 q^{7} +1.00000 q^{8} -2.99857 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.0377834 q^{3} +1.00000 q^{4} -1.82359 q^{5} -0.0377834 q^{6} -4.49307 q^{7} +1.00000 q^{8} -2.99857 q^{9} -1.82359 q^{10} +2.94797 q^{11} -0.0377834 q^{12} +1.02878 q^{13} -4.49307 q^{14} +0.0689015 q^{15} +1.00000 q^{16} +5.14957 q^{17} -2.99857 q^{18} +0.924142 q^{19} -1.82359 q^{20} +0.169764 q^{21} +2.94797 q^{22} +0.320712 q^{23} -0.0377834 q^{24} -1.67451 q^{25} +1.02878 q^{26} +0.226646 q^{27} -4.49307 q^{28} +0.862102 q^{29} +0.0689015 q^{30} +7.56634 q^{31} +1.00000 q^{32} -0.111384 q^{33} +5.14957 q^{34} +8.19353 q^{35} -2.99857 q^{36} -2.65226 q^{37} +0.924142 q^{38} -0.0388706 q^{39} -1.82359 q^{40} +3.90986 q^{41} +0.169764 q^{42} -0.456090 q^{43} +2.94797 q^{44} +5.46817 q^{45} +0.320712 q^{46} +7.41479 q^{47} -0.0377834 q^{48} +13.1877 q^{49} -1.67451 q^{50} -0.194568 q^{51} +1.02878 q^{52} -12.1425 q^{53} +0.226646 q^{54} -5.37590 q^{55} -4.49307 q^{56} -0.0349172 q^{57} +0.862102 q^{58} -3.89724 q^{59} +0.0689015 q^{60} -2.25350 q^{61} +7.56634 q^{62} +13.4728 q^{63} +1.00000 q^{64} -1.87607 q^{65} -0.111384 q^{66} -15.3437 q^{67} +5.14957 q^{68} -0.0121176 q^{69} +8.19353 q^{70} -7.28376 q^{71} -2.99857 q^{72} -1.35637 q^{73} -2.65226 q^{74} +0.0632689 q^{75} +0.924142 q^{76} -13.2455 q^{77} -0.0388706 q^{78} +9.40766 q^{79} -1.82359 q^{80} +8.98715 q^{81} +3.90986 q^{82} -15.5768 q^{83} +0.169764 q^{84} -9.39071 q^{85} -0.456090 q^{86} -0.0325731 q^{87} +2.94797 q^{88} +5.43141 q^{89} +5.46817 q^{90} -4.62236 q^{91} +0.320712 q^{92} -0.285882 q^{93} +7.41479 q^{94} -1.68526 q^{95} -0.0377834 q^{96} +6.98364 q^{97} +13.1877 q^{98} -8.83971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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