LMFDB - Embedded newform 8042.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.16571 q^{3} +1.00000 q^{4} -2.70471 q^{5} +3.16571 q^{6} +0.275737 q^{7} -1.00000 q^{8} +7.02173 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.16571 q^{3} +1.00000 q^{4} -2.70471 q^{5} +3.16571 q^{6} +0.275737 q^{7} -1.00000 q^{8} +7.02173 q^{9} +2.70471 q^{10} -2.70248 q^{11} -3.16571 q^{12} +4.89080 q^{13} -0.275737 q^{14} +8.56235 q^{15} +1.00000 q^{16} +2.22302 q^{17} -7.02173 q^{18} -6.65008 q^{19} -2.70471 q^{20} -0.872905 q^{21} +2.70248 q^{22} +5.45696 q^{23} +3.16571 q^{24} +2.31548 q^{25} -4.89080 q^{26} -12.7317 q^{27} +0.275737 q^{28} +2.02565 q^{29} -8.56235 q^{30} -5.14376 q^{31} -1.00000 q^{32} +8.55528 q^{33} -2.22302 q^{34} -0.745791 q^{35} +7.02173 q^{36} +0.980793 q^{37} +6.65008 q^{38} -15.4828 q^{39} +2.70471 q^{40} +5.27398 q^{41} +0.872905 q^{42} -8.64462 q^{43} -2.70248 q^{44} -18.9918 q^{45} -5.45696 q^{46} -7.83525 q^{47} -3.16571 q^{48} -6.92397 q^{49} -2.31548 q^{50} -7.03745 q^{51} +4.89080 q^{52} +7.98384 q^{53} +12.7317 q^{54} +7.30945 q^{55} -0.275737 q^{56} +21.0522 q^{57} -2.02565 q^{58} +3.72815 q^{59} +8.56235 q^{60} -6.34553 q^{61} +5.14376 q^{62} +1.93615 q^{63} +1.00000 q^{64} -13.2282 q^{65} -8.55528 q^{66} -7.36126 q^{67} +2.22302 q^{68} -17.2752 q^{69} +0.745791 q^{70} -15.1361 q^{71} -7.02173 q^{72} +14.5547 q^{73} -0.980793 q^{74} -7.33015 q^{75} -6.65008 q^{76} -0.745175 q^{77} +15.4828 q^{78} +17.0116 q^{79} -2.70471 q^{80} +19.2395 q^{81} -5.27398 q^{82} +0.128923 q^{83} -0.872905 q^{84} -6.01264 q^{85} +8.64462 q^{86} -6.41263 q^{87} +2.70248 q^{88} +9.38881 q^{89} +18.9918 q^{90} +1.34857 q^{91} +5.45696 q^{92} +16.2837 q^{93} +7.83525 q^{94} +17.9866 q^{95} +3.16571 q^{96} -2.06603 q^{97} +6.92397 q^{98} -18.9761 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) $ 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * 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
$3$	−3.16571	−1.82772	−0.913862	−	0.406024i	$-0.866915\pi$
$3$	−3.16571	−1.82772	−0.913862	+	0.406024i	$0.866915\pi$
$4$	1.00000	0.500000
$5$	−2.70471	−1.20959	−0.604793	−	0.796383i	$-0.706744\pi$
$5$	−2.70471	−1.20959	−0.604793	+	0.796383i	$0.706744\pi$
$6$	3.16571	1.29240
$7$	0.275737	0.104219	0.0521094	−	0.998641i	$-0.483406\pi$
$7$	0.275737	0.104219	0.0521094	+	0.998641i	$0.483406\pi$
$8$	−1.00000	−0.353553
$9$	7.02173	2.34058
$10$	2.70471	0.855306
$11$	−2.70248	−0.814829	−0.407415	−	0.913243i	$-0.633570\pi$
$11$	−2.70248	−0.814829	−0.407415	+	0.913243i	$0.633570\pi$
$12$	−3.16571	−0.913862
$13$	4.89080	1.35646	0.678231	−	0.734849i	$-0.262747\pi$
$13$	4.89080	1.35646	0.678231	+	0.734849i	$0.262747\pi$
$14$	−0.275737	−0.0736939
$15$	8.56235	2.21079
$16$	1.00000	0.250000
$17$	2.22302	0.539162	0.269581	−	0.962978i	$-0.413115\pi$
$17$	2.22302	0.539162	0.269581	+	0.962978i	$0.413115\pi$
$18$	−7.02173	−1.65504
$19$	−6.65008	−1.52563	−0.762816	−	0.646616i	$-0.776184\pi$
$19$	−6.65008	−1.52563	−0.762816	+	0.646616i	$0.776184\pi$
$20$	−2.70471	−0.604793
$21$	−0.872905	−0.190483
$22$	2.70248	0.576171
$23$	5.45696	1.13786	0.568928	−	0.822388i	$-0.307358\pi$
$23$	5.45696	1.13786	0.568928	+	0.822388i	$0.307358\pi$
$24$	3.16571	0.646198
$25$	2.31548	0.463097
$26$	−4.89080	−0.959164
$27$	−12.7317	−2.45021
$28$	0.275737	0.0521094
$29$	2.02565	0.376154	0.188077	−	0.982154i	$-0.439775\pi$
$29$	2.02565	0.376154	0.188077	+	0.982154i	$0.439775\pi$
$30$	−8.56235	−1.56326
$31$	−5.14376	−0.923846	−0.461923	−	0.886920i	$-0.652840\pi$
$31$	−5.14376	−0.923846	−0.461923	+	0.886920i	$0.652840\pi$
$32$	−1.00000	−0.176777
$33$	8.55528	1.48928
$34$	−2.22302	−0.381245
$35$	−0.745791	−0.126062
$36$	7.02173	1.17029
$37$	0.980793	0.161241	0.0806207	−	0.996745i	$-0.474310\pi$
$37$	0.980793	0.161241	0.0806207	+	0.996745i	$0.474310\pi$
$38$	6.65008	1.07878
$39$	−15.4828	−2.47924
$40$	2.70471	0.427653
$41$	5.27398	0.823657	0.411828	−	0.911261i	$-0.364890\pi$
$41$	5.27398	0.823657	0.411828	+	0.911261i	$0.364890\pi$
$42$	0.872905	0.134692
$43$	−8.64462	−1.31829	−0.659146	−	0.752015i	$-0.729082\pi$
$43$	−8.64462	−1.31829	−0.659146	+	0.752015i	$0.729082\pi$
$44$	−2.70248	−0.407415
$45$	−18.9918	−2.83113
$46$	−5.45696	−0.804585
$47$	−7.83525	−1.14289	−0.571444	−	0.820641i	$-0.693617\pi$
$47$	−7.83525	−1.14289	−0.571444	+	0.820641i	$0.693617\pi$
$48$	−3.16571	−0.456931
$49$	−6.92397	−0.989138
$50$	−2.31548	−0.327459
$51$	−7.03745	−0.985440
$52$	4.89080	0.678231
$53$	7.98384	1.09666	0.548332	−	0.836260i	$-0.315263\pi$
$53$	7.98384	1.09666	0.548332	+	0.836260i	$0.315263\pi$
$54$	12.7317	1.73256
$55$	7.30945	0.985606
$56$	−0.275737	−0.0368469
$57$	21.0522	2.78844
$58$	−2.02565	−0.265981
$59$	3.72815	0.485364	0.242682	−	0.970106i	$-0.421973\pi$
$59$	3.72815	0.485364	0.242682	+	0.970106i	$0.421973\pi$
$60$	8.56235	1.10539
$61$	−6.34553	−0.812461	−0.406231	−	0.913771i	$-0.633157\pi$
$61$	−6.34553	−0.812461	−0.406231	+	0.913771i	$0.633157\pi$
$62$	5.14376	0.653258
$63$	1.93615	0.243932
$64$	1.00000	0.125000
$65$	−13.2282	−1.64076
$66$	−8.55528	−1.05308
$67$	−7.36126	−0.899321	−0.449660	−	0.893200i	$-0.648455\pi$
$67$	−7.36126	−0.899321	−0.449660	+	0.893200i	$0.648455\pi$
$68$	2.22302	0.269581
$69$	−17.2752	−2.07969
$70$	0.745791	0.0891390
$71$	−15.1361	−1.79632	−0.898159	−	0.439670i	$-0.855095\pi$
$71$	−15.1361	−1.79632	−0.898159	+	0.439670i	$0.855095\pi$
$72$	−7.02173	−0.827519
$73$	14.5547	1.70349	0.851747	−	0.523953i	$-0.175543\pi$
$73$	14.5547	1.70349	0.851747	+	0.523953i	$0.175543\pi$
$74$	−0.980793	−0.114015
$75$	−7.33015	−0.846413
$76$	−6.65008	−0.762816
$77$	−0.745175	−0.0849206
$78$	15.4828	1.75309
$79$	17.0116	1.91395	0.956977	−	0.290162i	$-0.0937093\pi$
$79$	17.0116	1.91395	0.956977	+	0.290162i	$0.0937093\pi$
$80$	−2.70471	−0.302396
$81$	19.2395	2.13773
$82$	−5.27398	−0.582413
$83$	0.128923	0.0141512	0.00707559	−	0.999975i	$-0.497748\pi$
$83$	0.128923	0.0141512	0.00707559	+	0.999975i	$0.497748\pi$
$84$	−0.872905	−0.0952417
$85$	−6.01264	−0.652162
$86$	8.64462	0.932173
$87$	−6.41263	−0.687506
$88$	2.70248	0.288086
$89$	9.38881	0.995212	0.497606	−	0.867403i	$-0.334213\pi$
$89$	9.38881	0.995212	0.497606	+	0.867403i	$0.334213\pi$
$90$	18.9918	2.00191
$91$	1.34857	0.141369
$92$	5.45696	0.568928
$93$	16.2837	1.68854
$94$	7.83525	0.808144
$95$	17.9866	1.84538
$96$	3.16571	0.323099
$97$	−2.06603	−0.209774	−0.104887	−	0.994484i	$-0.533448\pi$
$97$	−2.06603	−0.209774	−0.104887	+	0.994484i	$0.533448\pi$
$98$	6.92397	0.699426
$99$	−18.9761	−1.90717
$100$	2.31548	0.231548
$101$	11.3409	1.12847	0.564233	−	0.825616i	$-0.309172\pi$
$101$	11.3409	1.12847	0.564233	+	0.825616i	$0.309172\pi$
$102$	7.03745	0.696811
$103$	−3.41079	−0.336075	−0.168037	−	0.985781i	$-0.553743\pi$
$103$	−3.41079	−0.336075	−0.168037	+	0.985781i	$0.553743\pi$
$104$	−4.89080	−0.479582
$105$	2.36096	0.230406
$106$	−7.98384	−0.775459
$107$	−4.06130	−0.392621	−0.196310	−	0.980542i	$-0.562896\pi$
$107$	−4.06130	−0.392621	−0.196310	+	0.980542i	$0.562896\pi$
$108$	−12.7317	−1.22510
$109$	−12.3032	−1.17843	−0.589215	−	0.807976i	$-0.700563\pi$
$109$	−12.3032	−1.17843	−0.589215	+	0.807976i	$0.700563\pi$
$110$	−7.30945	−0.696928
$111$	−3.10491	−0.294705
$112$	0.275737	0.0260547
$113$	−9.38748	−0.883100	−0.441550	−	0.897237i	$-0.645571\pi$
$113$	−9.38748	−0.883100	−0.441550	+	0.897237i	$0.645571\pi$
$114$	−21.0522	−1.97172
$115$	−14.7595	−1.37633
$116$	2.02565	0.188077
$117$	34.3419	3.17491
$118$	−3.72815	−0.343204
$119$	0.612970	0.0561909
$120$	−8.56235	−0.781632
$121$	−3.69658	−0.336053
$122$	6.34553	0.574497
$123$	−16.6959	−1.50542
$124$	−5.14376	−0.461923
$125$	7.26085	0.649431
$126$	−1.93615	−0.172486
$127$	4.30871	0.382336	0.191168	−	0.981557i	$-0.438773\pi$
$127$	4.30871	0.382336	0.191168	+	0.981557i	$0.438773\pi$
$128$	−1.00000	−0.0883883
$129$	27.3664	2.40947
$130$	13.2282	1.16019
$131$	18.2135	1.59132	0.795661	−	0.605742i	$-0.207123\pi$
$131$	18.2135	1.59132	0.795661	+	0.605742i	$0.207123\pi$
$132$	8.55528	0.744642
$133$	−1.83367	−0.159000
$134$	7.36126	0.635916
$135$	34.4355	2.96373
$136$	−2.22302	−0.190623
$137$	3.04668	0.260295	0.130148	−	0.991495i	$-0.458455\pi$
$137$	3.04668	0.260295	0.130148	+	0.991495i	$0.458455\pi$
$138$	17.2752	1.47056
$139$	12.3343	1.04618	0.523090	−	0.852278i	$-0.324779\pi$
$139$	12.3343	1.04618	0.523090	+	0.852278i	$0.324779\pi$
$140$	−0.745791	−0.0630308
$141$	24.8041	2.08889
$142$	15.1361	1.27019
$143$	−13.2173	−1.10529
$144$	7.02173	0.585144
$145$	−5.47881	−0.454990
$146$	−14.5547	−1.20455
$147$	21.9193	1.80787
$148$	0.980793	0.0806207
$149$	−3.65683	−0.299579	−0.149789	−	0.988718i	$-0.547860\pi$
$149$	−3.65683	−0.299579	−0.149789	+	0.988718i	$0.547860\pi$
$150$	7.33015	0.598504
$151$	−3.60217	−0.293140	−0.146570	−	0.989200i	$-0.546823\pi$
$151$	−3.60217	−0.293140	−0.146570	+	0.989200i	$0.546823\pi$
$152$	6.65008	0.539392
$153$	15.6095	1.26195
$154$	0.745175	0.0600479
$155$	13.9124	1.11747
$156$	−15.4828	−1.23962
$157$	8.34857	0.666288	0.333144	−	0.942876i	$-0.391890\pi$
$157$	8.34857	0.666288	0.333144	+	0.942876i	$0.391890\pi$
$158$	−17.0116	−1.35337
$159$	−25.2745	−2.00440
$160$	2.70471	0.213826
$161$	1.50469	0.118586
$162$	−19.2395	−1.51160
$163$	2.81946	0.220837	0.110419	−	0.993885i	$-0.464781\pi$
$163$	2.81946	0.220837	0.110419	+	0.993885i	$0.464781\pi$
$164$	5.27398	0.411828
$165$	−23.1396	−1.80142
$166$	−0.128923	−0.0100064
$167$	4.56269	0.353072	0.176536	−	0.984294i	$-0.443511\pi$
$167$	4.56269	0.353072	0.176536	+	0.984294i	$0.443511\pi$
$168$	0.872905	0.0673461
$169$	10.9199	0.839991
$170$	6.01264	0.461148
$171$	−46.6951	−3.57086
$172$	−8.64462	−0.659146
$173$	−4.18227	−0.317972	−0.158986	−	0.987281i	$-0.550822\pi$
$173$	−4.18227	−0.317972	−0.158986	+	0.987281i	$0.550822\pi$
$174$	6.41263	0.486140
$175$	0.638465	0.0482634
$176$	−2.70248	−0.203707
$177$	−11.8023	−0.887112
$178$	−9.38881	−0.703721
$179$	18.5168	1.38401	0.692006	−	0.721892i	$-0.256727\pi$
$179$	18.5168	1.38401	0.692006	+	0.721892i	$0.256727\pi$
$180$	−18.9918	−1.41556
$181$	−20.4392	−1.51924	−0.759618	−	0.650370i	$-0.774614\pi$
$181$	−20.4392	−1.51924	−0.759618	+	0.650370i	$0.774614\pi$
$182$	−1.34857	−0.0999630
$183$	20.0881	1.48496
$184$	−5.45696	−0.402293
$185$	−2.65277	−0.195035
$186$	−16.2837	−1.19398
$187$	−6.00768	−0.439325
$188$	−7.83525	−0.571444
$189$	−3.51059	−0.255358
$190$	−17.9866	−1.30488
$191$	18.0485	1.30594	0.652972	−	0.757382i	$-0.273522\pi$
$191$	18.0485	1.30594	0.652972	+	0.757382i	$0.273522\pi$
$192$	−3.16571	−0.228466
$193$	16.8287	1.21135	0.605677	−	0.795711i	$-0.292902\pi$
$193$	16.8287	1.21135	0.605677	+	0.795711i	$0.292902\pi$
$194$	2.06603	0.148333
$195$	41.8767	2.99885
$196$	−6.92397	−0.494569
$197$	−20.7034	−1.47505	−0.737527	−	0.675318i	$-0.764006\pi$
$197$	−20.7034	−1.47505	−0.737527	+	0.675318i	$0.764006\pi$
$198$	18.9761	1.34857
$199$	8.36206	0.592771	0.296385	−	0.955068i	$-0.404219\pi$
$199$	8.36206	0.592771	0.296385	+	0.955068i	$0.404219\pi$
$200$	−2.31548	−0.163729
$201$	23.3036	1.64371
$202$	−11.3409	−0.797946
$203$	0.558547	0.0392023
$204$	−7.03745	−0.492720
$205$	−14.2646	−0.996283
$206$	3.41079	0.237641
$207$	38.3173	2.66324
$208$	4.89080	0.339116
$209$	17.9717	1.24313
$210$	−2.36096	−0.162922
$211$	−6.43052	−0.442695	−0.221348	−	0.975195i	$-0.571046\pi$
$211$	−6.43052	−0.442695	−0.221348	+	0.975195i	$0.571046\pi$
$212$	7.98384	0.548332
$213$	47.9164	3.28318
$214$	4.06130	0.277625
$215$	23.3812	1.59459
$216$	12.7317	0.866279
$217$	−1.41833	−0.0962822
$218$	12.3032	0.833276
$219$	−46.0759	−3.11352
$220$	7.30945	0.492803
$221$	10.8723	0.731353
$222$	3.10491	0.208388
$223$	8.12471	0.544070	0.272035	−	0.962287i	$-0.412303\pi$
$223$	8.12471	0.544070	0.272035	+	0.962287i	$0.412303\pi$
$224$	−0.275737	−0.0184235
$225$	16.2587	1.08391
$226$	9.38748	0.624446
$227$	−15.7849	−1.04768	−0.523840	−	0.851817i	$-0.675501\pi$
$227$	−15.7849	−1.04768	−0.523840	+	0.851817i	$0.675501\pi$
$228$	21.0522	1.39422
$229$	26.1380	1.72725	0.863625	−	0.504134i	$-0.168188\pi$
$229$	26.1380	1.72725	0.863625	+	0.504134i	$0.168188\pi$
$230$	14.7595	0.973214
$231$	2.35901	0.155211
$232$	−2.02565	−0.132990
$233$	3.89323	0.255054	0.127527	−	0.991835i	$-0.459296\pi$
$233$	3.89323	0.255054	0.127527	+	0.991835i	$0.459296\pi$
$234$	−34.3419	−2.24500
$235$	21.1921	1.38242
$236$	3.72815	0.242682
$237$	−53.8538	−3.49818
$238$	−0.612970	−0.0397329
$239$	4.77813	0.309072	0.154536	−	0.987987i	$-0.450612\pi$
$239$	4.77813	0.309072	0.154536	+	0.987987i	$0.450612\pi$
$240$	8.56235	0.552697
$241$	23.2558	1.49804	0.749019	−	0.662548i	$-0.230525\pi$
$241$	23.2558	1.49804	0.749019	+	0.662548i	$0.230525\pi$
$242$	3.69658	0.237625
$243$	−22.7119	−1.45697
$244$	−6.34553	−0.406231
$245$	18.7274	1.19645
$246$	16.6959	1.06449
$247$	−32.5242	−2.06946
$248$	5.14376	0.326629
$249$	−0.408134	−0.0258644
$250$	−7.26085	−0.459217
$251$	−14.2770	−0.901156	−0.450578	−	0.892737i	$-0.648782\pi$
$251$	−14.2770	−0.901156	−0.450578	+	0.892737i	$0.648782\pi$
$252$	1.93615	0.121966
$253$	−14.7473	−0.927158
$254$	−4.30871	−0.270352
$255$	19.0343	1.19197
$256$	1.00000	0.0625000
$257$	−3.14564	−0.196219	−0.0981097	−	0.995176i	$-0.531280\pi$
$257$	−3.14564	−0.196219	−0.0981097	+	0.995176i	$0.531280\pi$
$258$	−27.3664	−1.70376
$259$	0.270441	0.0168044
$260$	−13.2282	−0.820379
$261$	14.2236	0.880417
$262$	−18.2135	−1.12524
$263$	−3.00582	−0.185347	−0.0926735	−	0.995697i	$-0.529541\pi$
$263$	−3.00582	−0.185347	−0.0926735	+	0.995697i	$0.529541\pi$
$264$	−8.55528	−0.526541
$265$	−21.5940	−1.32651
$266$	1.83367	0.112430
$267$	−29.7223	−1.81897
$268$	−7.36126	−0.449660
$269$	30.8103	1.87854	0.939268	−	0.343185i	$-0.111506\pi$
$269$	30.8103	1.87854	0.939268	+	0.343185i	$0.111506\pi$
$270$	−34.4355	−2.09568
$271$	4.76079	0.289197	0.144599	−	0.989490i	$-0.453811\pi$
$271$	4.76079	0.289197	0.144599	+	0.989490i	$0.453811\pi$
$272$	2.22302	0.134790
$273$	−4.26920	−0.258384
$274$	−3.04668	−0.184057
$275$	−6.25755	−0.377345
$276$	−17.2752	−1.03984
$277$	−8.83978	−0.531131	−0.265566	−	0.964093i	$-0.585559\pi$
$277$	−8.83978	−0.531131	−0.265566	+	0.964093i	$0.585559\pi$
$278$	−12.3343	−0.739760
$279$	−36.1181	−2.16233
$280$	0.745791	0.0445695
$281$	−2.87952	−0.171778	−0.0858890	−	0.996305i	$-0.527373\pi$
$281$	−2.87952	−0.171778	−0.0858890	+	0.996305i	$0.527373\pi$
$282$	−24.8041	−1.47707
$283$	28.5113	1.69482	0.847410	−	0.530939i	$-0.178161\pi$
$283$	28.5113	1.69482	0.847410	+	0.530939i	$0.178161\pi$
$284$	−15.1361	−0.898159
$285$	−56.9403	−3.37285
$286$	13.2173	0.781555
$287$	1.45423	0.0858406
$288$	−7.02173	−0.413760
$289$	−12.0582	−0.709304
$290$	5.47881	0.321727
$291$	6.54047	0.383409
$292$	14.5547	0.851747
$293$	31.3367	1.83071	0.915355	−	0.402647i	$-0.131910\pi$
$293$	31.3367	1.83071	0.915355	+	0.402647i	$0.131910\pi$
$294$	−21.9193	−1.27836
$295$	−10.0836	−0.587090
$296$	−0.980793	−0.0570075
$297$	34.4071	1.99650
$298$	3.65683	0.211834
$299$	26.6889	1.54346
$300$	−7.33015	−0.423206
$301$	−2.38364	−0.137391
$302$	3.60217	0.207281
$303$	−35.9021	−2.06252
$304$	−6.65008	−0.381408
$305$	17.1628	0.982741
$306$	−15.6095	−0.892334
$307$	−23.3973	−1.33535	−0.667677	−	0.744451i	$-0.732711\pi$
$307$	−23.3973	−1.33535	−0.667677	+	0.744451i	$0.732711\pi$
$308$	−0.745175	−0.0424603
$309$	10.7976	0.614252
$310$	−13.9124	−0.790171
$311$	20.8357	1.18148	0.590742	−	0.806860i	$-0.298835\pi$
$311$	20.8357	1.18148	0.590742	+	0.806860i	$0.298835\pi$
$312$	15.4828	0.876544
$313$	−25.1080	−1.41919	−0.709594	−	0.704611i	$-0.751122\pi$
$313$	−25.1080	−1.41919	−0.709594	+	0.704611i	$0.751122\pi$
$314$	−8.34857	−0.471137
$315$	−5.23674	−0.295057
$316$	17.0116	0.956977
$317$	−31.9058	−1.79201	−0.896004	−	0.444046i	$-0.853543\pi$
$317$	−31.9058	−1.79201	−0.896004	+	0.444046i	$0.853543\pi$
$318$	25.2745	1.41733
$319$	−5.47429	−0.306501
$320$	−2.70471	−0.151198
$321$	12.8569	0.717603
$322$	−1.50469	−0.0838530
$323$	−14.7833	−0.822563
$324$	19.2395	1.06886
$325$	11.3246	0.628173
$326$	−2.81946	−0.156156
$327$	38.9483	2.15385
$328$	−5.27398	−0.291207
$329$	−2.16047	−0.119111
$330$	23.1396	1.27379
$331$	−23.6289	−1.29876	−0.649382	−	0.760462i	$-0.724972\pi$
$331$	−23.6289	−1.29876	−0.649382	+	0.760462i	$0.724972\pi$
$332$	0.128923	0.00707559
$333$	6.88687	0.377398
$334$	−4.56269	−0.249659
$335$	19.9101	1.08781
$336$	−0.872905	−0.0476209
$337$	4.52657	0.246578	0.123289	−	0.992371i	$-0.460656\pi$
$337$	4.52657	0.246578	0.123289	+	0.992371i	$0.460656\pi$
$338$	−10.9199	−0.593963
$339$	29.7181	1.61406
$340$	−6.01264	−0.326081
$341$	13.9009	0.752777
$342$	46.6951	2.52498
$343$	−3.83936	−0.207306
$344$	8.64462	0.466086
$345$	46.7244	2.51556
$346$	4.18227	0.224840
$347$	−12.5273	−0.672500	−0.336250	−	0.941773i	$-0.609159\pi$
$347$	−12.5273	−0.672500	−0.336250	+	0.941773i	$0.609159\pi$
$348$	−6.41263	−0.343753
$349$	−25.6735	−1.37427	−0.687135	−	0.726529i	$-0.741132\pi$
$349$	−25.6735	−1.37427	−0.687135	+	0.726529i	$0.741132\pi$
$350$	−0.638465	−0.0341274
$351$	−62.2679	−3.32361
$352$	2.70248	0.144043
$353$	−4.89003	−0.260270	−0.130135	−	0.991496i	$-0.541541\pi$
$353$	−4.89003	−0.260270	−0.130135	+	0.991496i	$0.541541\pi$
$354$	11.8023	0.627283
$355$	40.9387	2.17280
$356$	9.38881	0.497606
$357$	−1.94049	−0.102701
$358$	−18.5168	−0.978644
$359$	23.1862	1.22372	0.611860	−	0.790966i	$-0.290422\pi$
$359$	23.1862	1.22372	0.611860	+	0.790966i	$0.290422\pi$
$360$	18.9918	1.00096
$361$	25.2235	1.32755
$362$	20.4392	1.07426
$363$	11.7023	0.614213
$364$	1.34857	0.0706845
$365$	−39.3662	−2.06052
$366$	−20.0881	−1.05002
$367$	−11.8704	−0.619630	−0.309815	−	0.950797i	$-0.600267\pi$
$367$	−11.8704	−0.619630	−0.309815	+	0.950797i	$0.600267\pi$
$368$	5.45696	0.284464
$369$	37.0324	1.92783
$370$	2.65277	0.137911
$371$	2.20144	0.114293
$372$	16.2837	0.844268
$373$	−12.1024	−0.626641	−0.313320	−	0.949647i	$-0.601441\pi$
$373$	−12.1024	−0.626641	−0.313320	+	0.949647i	$0.601441\pi$
$374$	6.00768	0.310650
$375$	−22.9858	−1.18698
$376$	7.83525	0.404072
$377$	9.90704	0.510239
$378$	3.51059	0.180565
$379$	−7.20856	−0.370279	−0.185139	−	0.982712i	$-0.559274\pi$
$379$	−7.20856	−0.370279	−0.185139	+	0.982712i	$0.559274\pi$
$380$	17.9866	0.922691
$381$	−13.6401	−0.698805
$382$	−18.0485	−0.923442
$383$	−3.89914	−0.199236	−0.0996182	−	0.995026i	$-0.531762\pi$
$383$	−3.89914	−0.199236	−0.0996182	+	0.995026i	$0.531762\pi$
$384$	3.16571	0.161550
$385$	2.01549	0.102719
$386$	−16.8287	−0.856556
$387$	−60.7002	−3.08556
$388$	−2.06603	−0.104887
$389$	2.06103	0.104498	0.0522492	−	0.998634i	$-0.483361\pi$
$389$	2.06103	0.104498	0.0522492	+	0.998634i	$0.483361\pi$
$390$	−41.8767	−2.12051
$391$	12.1309	0.613488
$392$	6.92397	0.349713
$393$	−57.6588	−2.90850
$394$	20.7034	1.04302
$395$	−46.0115	−2.31509
$396$	−18.9761	−0.953586
$397$	−35.3695	−1.77515	−0.887573	−	0.460668i	$-0.847610\pi$
$397$	−35.3695	−1.77515	−0.887573	+	0.460668i	$0.847610\pi$
$398$	−8.36206	−0.419152
$399$	5.80488	0.290608
$400$	2.31548	0.115774
$401$	17.8689	0.892330	0.446165	−	0.894951i	$-0.352789\pi$
$401$	17.8689	0.892330	0.446165	+	0.894951i	$0.352789\pi$
$402$	−23.3036	−1.16228
$403$	−25.1571	−1.25316
$404$	11.3409	0.564233
$405$	−52.0375	−2.58576
$406$	−0.558547	−0.0277202
$407$	−2.65058	−0.131384
$408$	7.03745	0.348406
$409$	10.6376	0.525994	0.262997	−	0.964797i	$-0.415289\pi$
$409$	10.6376	0.525994	0.262997	+	0.964797i	$0.415289\pi$
$410$	14.2646	0.704478
$411$	−9.64491	−0.475748
$412$	−3.41079	−0.168037
$413$	1.02799	0.0505841
$414$	−38.3173	−1.88319
$415$	−0.348701	−0.0171170
$416$	−4.89080	−0.239791
$417$	−39.0468	−1.91213
$418$	−17.9717	−0.879026
$419$	−25.1908	−1.23065	−0.615325	−	0.788273i	$-0.710975\pi$
$419$	−25.1908	−1.23065	−0.615325	+	0.788273i	$0.710975\pi$
$420$	2.36096	0.115203
$421$	−10.8888	−0.530689	−0.265344	−	0.964154i	$-0.585486\pi$
$421$	−10.8888	−0.530689	−0.265344	+	0.964154i	$0.585486\pi$
$422$	6.43052	0.313033
$423$	−55.0170	−2.67502
$424$	−7.98384	−0.387730
$425$	5.14737	0.249684
$426$	−47.9164	−2.32156
$427$	−1.74970	−0.0846738
$428$	−4.06130	−0.196310
$429$	41.8421	2.02016
$430$	−23.3812	−1.12754
$431$	37.4181	1.80237	0.901184	−	0.433437i	$-0.142699\pi$
$431$	37.4181	1.80237	0.901184	+	0.433437i	$0.142699\pi$
$432$	−12.7317	−0.612552
$433$	−14.7028	−0.706570	−0.353285	−	0.935516i	$-0.614935\pi$
$433$	−14.7028	−0.706570	−0.353285	+	0.935516i	$0.614935\pi$
$434$	1.41833	0.0680818
$435$	17.3443	0.831597
$436$	−12.3032	−0.589215
$437$	−36.2892	−1.73595
$438$	46.0759	2.20159
$439$	29.3013	1.39848	0.699238	−	0.714889i	$-0.253523\pi$
$439$	29.3013	1.39848	0.699238	+	0.714889i	$0.253523\pi$
$440$	−7.30945	−0.348464
$441$	−48.6183	−2.31516
$442$	−10.8723	−0.517145
$443$	−18.9430	−0.900011	−0.450006	−	0.893026i	$-0.648578\pi$
$443$	−18.9430	−0.900011	−0.450006	+	0.893026i	$0.648578\pi$
$444$	−3.10491	−0.147352
$445$	−25.3941	−1.20379
$446$	−8.12471	−0.384716
$447$	11.5765	0.547548
$448$	0.275737	0.0130274
$449$	−15.6074	−0.736558	−0.368279	−	0.929715i	$-0.620053\pi$
$449$	−15.6074	−0.736558	−0.368279	+	0.929715i	$0.620053\pi$
$450$	−16.2587	−0.766443
$451$	−14.2528	−0.671140
$452$	−9.38748	−0.441550
$453$	11.4034	0.535779
$454$	15.7849	0.740822
$455$	−3.64751	−0.170998
$456$	−21.0522	−0.985861
$457$	−7.71799	−0.361032	−0.180516	−	0.983572i	$-0.557777\pi$
$457$	−7.71799	−0.361032	−0.180516	+	0.983572i	$0.557777\pi$
$458$	−26.1380	−1.22135
$459$	−28.3027	−1.32106
$460$	−14.7595	−0.688166
$461$	37.1286	1.72925	0.864625	−	0.502417i	$-0.167556\pi$
$461$	37.1286	1.72925	0.864625	+	0.502417i	$0.167556\pi$
$462$	−2.35901	−0.109751
$463$	−20.0192	−0.930373	−0.465186	−	0.885213i	$-0.654013\pi$
$463$	−20.0192	−0.930373	−0.465186	+	0.885213i	$0.654013\pi$
$464$	2.02565	0.0940385
$465$	−44.0427	−2.04243
$466$	−3.89323	−0.180350
$467$	27.1451	1.25612	0.628062	−	0.778163i	$-0.283848\pi$
$467$	27.1451	1.25612	0.628062	+	0.778163i	$0.283848\pi$
$468$	34.3419	1.58745
$469$	−2.02977	−0.0937262
$470$	−21.1921	−0.977519
$471$	−26.4292	−1.21779
$472$	−3.72815	−0.171602
$473$	23.3619	1.07418
$474$	53.8538	2.47359
$475$	−15.3981	−0.706515
$476$	0.612970	0.0280954
$477$	56.0604	2.56683
$478$	−4.77813	−0.218547
$479$	20.1347	0.919979	0.459989	−	0.887924i	$-0.347853\pi$
$479$	20.1347	0.919979	0.459989	+	0.887924i	$0.347853\pi$
$480$	−8.56235	−0.390816
$481$	4.79686	0.218718
$482$	−23.2558	−1.05927
$483$	−4.76341	−0.216743
$484$	−3.69658	−0.168027
$485$	5.58803	0.253739
$486$	22.7119	1.03023
$487$	−12.0068	−0.544080	−0.272040	−	0.962286i	$-0.587698\pi$
$487$	−12.0068	−0.544080	−0.272040	+	0.962286i	$0.587698\pi$
$488$	6.34553	0.287248
$489$	−8.92560	−0.403630
$490$	−18.7274	−0.846016
$491$	−6.69018	−0.301923	−0.150962	−	0.988540i	$-0.548237\pi$
$491$	−6.69018	−0.301923	−0.150962	+	0.988540i	$0.548237\pi$
$492$	−16.6959	−0.752709
$493$	4.50307	0.202808
$494$	32.5242	1.46333
$495$	51.3250	2.30689
$496$	−5.14376	−0.230962
$497$	−4.17357	−0.187210
$498$	0.408134	0.0182889
$499$	−16.0578	−0.718846	−0.359423	−	0.933175i	$-0.617026\pi$
$499$	−16.0578	−0.718846	−0.359423	+	0.933175i	$0.617026\pi$
$500$	7.26085	0.324715
$501$	−14.4442	−0.645318
$502$	14.2770	0.637213
$503$	31.3215	1.39655	0.698277	−	0.715827i	$-0.253950\pi$
$503$	31.3215	1.39655	0.698277	+	0.715827i	$0.253950\pi$
$504$	−1.93615	−0.0862431
$505$	−30.6740	−1.36498
$506$	14.7473	0.655600
$507$	−34.5692	−1.53527
$508$	4.30871	0.191168
$509$	−26.6494	−1.18121	−0.590606	−	0.806960i	$-0.701111\pi$
$509$	−26.6494	−1.18121	−0.590606	+	0.806960i	$0.701111\pi$
$510$	−19.0343	−0.842852
$511$	4.01326	0.177536
$512$	−1.00000	−0.0441942
$513$	84.6664	3.73811
$514$	3.14564	0.138748
$515$	9.22520	0.406511
$516$	27.3664	1.20474
$517$	21.1746	0.931259
$518$	−0.270441	−0.0118825
$519$	13.2399	0.581165
$520$	13.2282	0.580095
$521$	−21.9412	−0.961263	−0.480631	−	0.876923i	$-0.659592\pi$
$521$	−21.9412	−0.961263	−0.480631	+	0.876923i	$0.659592\pi$
$522$	−14.2236	−0.622549
$523$	5.86208	0.256331	0.128165	−	0.991753i	$-0.459091\pi$
$523$	5.86208	0.256331	0.128165	+	0.991753i	$0.459091\pi$
$524$	18.2135	0.795661
$525$	−2.02120	−0.0882122
$526$	3.00582	0.131060
$527$	−11.4347	−0.498103
$528$	8.55528	0.372321
$529$	6.77843	0.294714
$530$	21.5940	0.937984
$531$	26.1781	1.13603
$532$	−1.83367	−0.0794998
$533$	25.7939	1.11726
$534$	29.7223	1.28621
$535$	10.9847	0.474908
$536$	7.36126	0.317958
$537$	−58.6189	−2.52959
$538$	−30.8103	−1.32833
$539$	18.7119	0.805979
$540$	34.4355	1.48187
$541$	−8.12900	−0.349493	−0.174746	−	0.984613i	$-0.555911\pi$
$541$	−8.12900	−0.349493	−0.174746	+	0.984613i	$0.555911\pi$
$542$	−4.76079	−0.204493
$543$	64.7047	2.77675
$544$	−2.22302	−0.0953113
$545$	33.2766	1.42541
$546$	4.26920	0.182705
$547$	27.3125	1.16780	0.583900	−	0.811826i	$-0.301526\pi$
$547$	27.3125	1.16780	0.583900	+	0.811826i	$0.301526\pi$
$548$	3.04668	0.130148
$549$	−44.5566	−1.90163
$550$	6.25755	0.266823
$551$	−13.4707	−0.573872
$552$	17.2752	0.735280
$553$	4.69073	0.199470
$554$	8.83978	0.375566
$555$	8.39789	0.356471
$556$	12.3343	0.523090
$557$	−6.33880	−0.268584	−0.134292	−	0.990942i	$-0.542876\pi$
$557$	−6.33880	−0.268584	−0.134292	+	0.990942i	$0.542876\pi$
$558$	36.1181	1.52900
$559$	−42.2790	−1.78821
$560$	−0.745791	−0.0315154
$561$	19.0186	0.802965
$562$	2.87952	0.121465
$563$	9.61507	0.405227	0.202613	−	0.979259i	$-0.435057\pi$
$563$	9.61507	0.405227	0.202613	+	0.979259i	$0.435057\pi$
$564$	24.8041	1.04444
$565$	25.3905	1.06818
$566$	−28.5113	−1.19842
$567$	5.30506	0.222791
$568$	15.1361	0.635095
$569$	−15.5935	−0.653714	−0.326857	−	0.945074i	$-0.605989\pi$
$569$	−15.5935	−0.653714	−0.326857	+	0.945074i	$0.605989\pi$
$570$	56.9403	2.38497
$571$	18.4528	0.772226	0.386113	−	0.922452i	$-0.373817\pi$
$571$	18.4528	0.772226	0.386113	+	0.922452i	$0.373817\pi$
$572$	−13.2173	−0.552643
$573$	−57.1364	−2.38691
$574$	−1.45423	−0.0606984
$575$	12.6355	0.526937
$576$	7.02173	0.292572
$577$	−0.645089	−0.0268554	−0.0134277	−	0.999910i	$-0.504274\pi$
$577$	−0.645089	−0.0268554	−0.0134277	+	0.999910i	$0.504274\pi$
$578$	12.0582	0.501554
$579$	−53.2747	−2.21402
$580$	−5.47881	−0.227495
$581$	0.0355490	0.00147482
$582$	−6.54047	−0.271111
$583$	−21.5762	−0.893595
$584$	−14.5547	−0.602276
$585$	−92.8849	−3.84032
$586$	−31.3367	−1.29451
$587$	−20.2852	−0.837258	−0.418629	−	0.908157i	$-0.637489\pi$
$587$	−20.2852	−0.837258	−0.418629	+	0.908157i	$0.637489\pi$
$588$	21.9193	0.903936
$589$	34.2064	1.40945
$590$	10.0836	0.415135
$591$	65.5409	2.69599
$592$	0.980793	0.0403104
$593$	23.5665	0.967760	0.483880	−	0.875134i	$-0.339227\pi$
$593$	23.5665	0.967760	0.483880	+	0.875134i	$0.339227\pi$
$594$	−34.4071	−1.41174
$595$	−1.65791	−0.0679676
$596$	−3.65683	−0.149789
$597$	−26.4719	−1.08342
$598$	−26.6889	−1.09139
$599$	4.44056	0.181436	0.0907182	−	0.995877i	$-0.471084\pi$
$599$	4.44056	0.181436	0.0907182	+	0.995877i	$0.471084\pi$
$600$	7.33015	0.299252
$601$	−27.8500	−1.13602	−0.568012	−	0.823021i	$-0.692287\pi$
$601$	−27.8500	−1.13602	−0.568012	+	0.823021i	$0.692287\pi$
$602$	2.38364	0.0971500
$603$	−51.6888	−2.10493
$604$	−3.60217	−0.146570
$605$	9.99820	0.406485
$606$	35.9021	1.45842
$607$	34.0100	1.38042	0.690211	−	0.723608i	$-0.257518\pi$
$607$	34.0100	1.38042	0.690211	+	0.723608i	$0.257518\pi$
$608$	6.65008	0.269696
$609$	−1.76820	−0.0716511
$610$	−17.1628	−0.694903
$611$	−38.3206	−1.55029
$612$	15.6095	0.630975
$613$	−43.2129	−1.74535	−0.872676	−	0.488300i	$-0.837617\pi$
$613$	−43.2129	−1.74535	−0.872676	+	0.488300i	$0.837617\pi$
$614$	23.3973	0.944238
$615$	45.1576	1.82093
$616$	0.745175	0.0300240
$617$	20.9445	0.843193	0.421596	−	0.906784i	$-0.361470\pi$
$617$	20.9445	0.843193	0.421596	+	0.906784i	$0.361470\pi$
$618$	−10.7976	−0.434342
$619$	−8.37913	−0.336786	−0.168393	−	0.985720i	$-0.553858\pi$
$619$	−8.37913	−0.336786	−0.168393	+	0.985720i	$0.553858\pi$
$620$	13.9124	0.558735
$621$	−69.4761	−2.78798
$622$	−20.8357	−0.835436
$623$	2.58885	0.103720
$624$	−15.4828	−0.619810
$625$	−31.2160	−1.24864
$626$	25.1080	1.00352
$627$	−56.8933	−2.27210
$628$	8.34857	0.333144
$629$	2.18032	0.0869352
$630$	5.23674	0.208637
$631$	−19.4020	−0.772380	−0.386190	−	0.922419i	$-0.626209\pi$
$631$	−19.4020	−0.772380	−0.386190	+	0.922419i	$0.626209\pi$
$632$	−17.0116	−0.676685
$633$	20.3572	0.809125
$634$	31.9058	1.26714
$635$	−11.6538	−0.462468
$636$	−25.2745	−1.00220
$637$	−33.8637	−1.34173
$638$	5.47429	0.216729
$639$	−106.281	−4.20442
$640$	2.70471	0.106913
$641$	42.1152	1.66345	0.831726	−	0.555187i	$-0.187353\pi$
$641$	42.1152	1.66345	0.831726	+	0.555187i	$0.187353\pi$
$642$	−12.8569	−0.507422
$643$	−5.69405	−0.224551	−0.112276	−	0.993677i	$-0.535814\pi$
$643$	−5.69405	−0.224551	−0.112276	+	0.993677i	$0.535814\pi$
$644$	1.50469	0.0592930
$645$	−74.0182	−2.91446
$646$	14.7833	0.581640
$647$	−21.3492	−0.839325	−0.419662	−	0.907680i	$-0.637851\pi$
$647$	−21.3492	−0.839325	−0.419662	+	0.907680i	$0.637851\pi$
$648$	−19.2395	−0.755800
$649$	−10.0753	−0.395489
$650$	−11.3246	−0.444185
$651$	4.49001	0.175977
$652$	2.81946	0.110419
$653$	−24.0658	−0.941769	−0.470884	−	0.882195i	$-0.656065\pi$
$653$	−24.0658	−0.941769	−0.470884	+	0.882195i	$0.656065\pi$
$654$	−38.9483	−1.52300
$655$	−49.2624	−1.92484
$656$	5.27398	0.205914
$657$	102.199	3.98716
$658$	2.16047	0.0842239
$659$	27.7530	1.08110	0.540552	−	0.841311i	$-0.318215\pi$
$659$	27.7530	1.08110	0.540552	+	0.841311i	$0.318215\pi$
$660$	−23.1396	−0.900708
$661$	−12.5498	−0.488131	−0.244065	−	0.969759i	$-0.578481\pi$
$661$	−12.5498	−0.488131	−0.244065	+	0.969759i	$0.578481\pi$
$662$	23.6289	0.918365
$663$	−34.4187	−1.33671
$664$	−0.128923	−0.00500319
$665$	4.95956	0.192324
$666$	−6.88687	−0.266861
$667$	11.0539	0.428009
$668$	4.56269	0.176536
$669$	−25.7205	−0.994411
$670$	−19.9101	−0.769194
$671$	17.1487	0.662017
$672$	0.872905	0.0336730
$673$	22.9492	0.884626	0.442313	−	0.896861i	$-0.354158\pi$
$673$	22.9492	0.884626	0.442313	+	0.896861i	$0.354158\pi$
$674$	−4.52657	−0.174357
$675$	−29.4799	−1.13468
$676$	10.9199	0.419995
$677$	−37.7184	−1.44963	−0.724817	−	0.688941i	$-0.758076\pi$
$677$	−37.7184	−1.44963	−0.724817	+	0.688941i	$0.758076\pi$
$678$	−29.7181	−1.14132
$679$	−0.569682	−0.0218624
$680$	6.01264	0.230574
$681$	49.9704	1.91487
$682$	−13.9009	−0.532294
$683$	30.3003	1.15941	0.579704	−	0.814827i	$-0.303168\pi$
$683$	30.3003	1.15941	0.579704	+	0.814827i	$0.303168\pi$
$684$	−46.6951	−1.78543
$685$	−8.24040	−0.314849
$686$	3.83936	0.146587
$687$	−82.7455	−3.15694
$688$	−8.64462	−0.329573
$689$	39.0473	1.48758
$690$	−46.7244	−1.77877
$691$	−15.0232	−0.571509	−0.285755	−	0.958303i	$-0.592244\pi$
$691$	−15.0232	−0.571509	−0.285755	+	0.958303i	$0.592244\pi$
$692$	−4.18227	−0.158986
$693$	−5.23242	−0.198763
$694$	12.5273	0.475530
$695$	−33.3607	−1.26544
$696$	6.41263	0.243070
$697$	11.7242	0.444084
$698$	25.6735	0.971756
$699$	−12.3248	−0.466169
$700$	0.638465	0.0241317
$701$	7.47433	0.282302	0.141151	−	0.989988i	$-0.454920\pi$
$701$	7.47433	0.282302	0.141151	+	0.989988i	$0.454920\pi$
$702$	62.2679	2.35015
$703$	−6.52235	−0.245995
$704$	−2.70248	−0.101854
$705$	−67.0881	−2.52669
$706$	4.89003	0.184039
$707$	3.12712	0.117607
$708$	−11.8023	−0.443556
$709$	−29.2075	−1.09691	−0.548455	−	0.836180i	$-0.684784\pi$
$709$	−29.2075	−1.09691	−0.548455	+	0.836180i	$0.684784\pi$
$710$	−40.9387	−1.53640
$711$	119.451	4.47976
$712$	−9.38881	−0.351861
$713$	−28.0693	−1.05120
$714$	1.94049	0.0726209
$715$	35.7490	1.33694
$716$	18.5168	0.692006
$717$	−15.1262	−0.564898
$718$	−23.1862	−0.865301
$719$	12.9542	0.483111	0.241556	−	0.970387i	$-0.422342\pi$
$719$	12.9542	0.483111	0.241556	+	0.970387i	$0.422342\pi$
$720$	−18.9918	−0.707782
$721$	−0.940481	−0.0350253
$722$	−25.2235	−0.938722
$723$	−73.6212	−2.73800
$724$	−20.4392	−0.759618
$725$	4.69036	0.174196
$726$	−11.7023	−0.434314
$727$	−19.5358	−0.724541	−0.362271	−	0.932073i	$-0.617998\pi$
$727$	−19.5358	−0.724541	−0.362271	+	0.932073i	$0.617998\pi$
$728$	−1.34857	−0.0499815
$729$	14.1807	0.525211
$730$	39.3662	1.45701
$731$	−19.2172	−0.710773
$732$	20.0881	0.742478
$733$	−16.1237	−0.595542	−0.297771	−	0.954637i	$-0.596243\pi$
$733$	−16.1237	−0.595542	−0.297771	+	0.954637i	$0.596243\pi$
$734$	11.8704	0.438145
$735$	−59.2854	−2.18678
$736$	−5.45696	−0.201146
$737$	19.8937	0.732793
$738$	−37.0324	−1.36318
$739$	−20.2612	−0.745321	−0.372660	−	0.927968i	$-0.621554\pi$
$739$	−20.2612	−0.745321	−0.372660	+	0.927968i	$0.621554\pi$
$740$	−2.65277	−0.0975176
$741$	102.962	3.78241
$742$	−2.20144	−0.0808175
$743$	−47.9745	−1.76001	−0.880006	−	0.474962i	$-0.842462\pi$
$743$	−47.9745	−1.76001	−0.880006	+	0.474962i	$0.842462\pi$
$744$	−16.2837	−0.596988
$745$	9.89067	0.362366
$746$	12.1024	0.443102
$747$	0.905265	0.0331219
$748$	−6.00768	−0.219663
$749$	−1.11985	−0.0409185
$750$	22.9858	0.839322
$751$	−36.3547	−1.32660	−0.663301	−	0.748353i	$-0.730845\pi$
$751$	−36.3547	−1.32660	−0.663301	+	0.748353i	$0.730845\pi$
$752$	−7.83525	−0.285722
$753$	45.1969	1.64707
$754$	−9.90704	−0.360793
$755$	9.74283	0.354578
$756$	−3.51059	−0.127679
$757$	33.2919	1.21001	0.605007	−	0.796220i	$-0.293170\pi$
$757$	33.2919	1.21001	0.605007	+	0.796220i	$0.293170\pi$
$758$	7.20856	0.261827
$759$	46.6859	1.69459
$760$	−17.9866	−0.652441
$761$	40.4771	1.46729	0.733646	−	0.679531i	$-0.237817\pi$
$761$	40.4771	1.46729	0.733646	+	0.679531i	$0.237817\pi$
$762$	13.6401	0.494129
$763$	−3.39244	−0.122815
$764$	18.0485	0.652972
$765$	−42.2192	−1.52644
$766$	3.89914	0.140881
$767$	18.2336	0.658379
$768$	−3.16571	−0.114233
$769$	−23.8721	−0.860848	−0.430424	−	0.902627i	$-0.641636\pi$
$769$	−23.8721	−0.860848	−0.430424	+	0.902627i	$0.641636\pi$
$770$	−2.01549	−0.0726331
$771$	9.95818	0.358635
$772$	16.8287	0.605677
$773$	−30.3172	−1.09043	−0.545217	−	0.838295i	$-0.683553\pi$
$773$	−30.3172	−1.09043	−0.545217	+	0.838295i	$0.683553\pi$
$774$	60.7002	2.18182
$775$	−11.9103	−0.427830
$776$	2.06603	0.0741663
$777$	−0.856139	−0.0307138
$778$	−2.06103	−0.0738916
$779$	−35.0723	−1.25660
$780$	41.8767	1.49943
$781$	40.9049	1.46369
$782$	−12.1309	−0.433802
$783$	−25.7899	−0.921655
$784$	−6.92397	−0.247285
$785$	−22.5805	−0.805932
$786$	57.6588	2.05662
$787$	−4.33360	−0.154476	−0.0772380	−	0.997013i	$-0.524610\pi$
$787$	−4.33360	−0.154476	−0.0772380	+	0.997013i	$0.524610\pi$
$788$	−20.7034	−0.737527
$789$	9.51557	0.338763
$790$	46.0115	1.63702
$791$	−2.58848	−0.0920357
$792$	18.9761	0.674287
$793$	−31.0347	−1.10207
$794$	35.3695	1.25522
$795$	68.3604	2.42449
$796$	8.36206	0.296385
$797$	−39.4479	−1.39732	−0.698658	−	0.715456i	$-0.746219\pi$
$797$	−39.4479	−1.39732	−0.698658	+	0.715456i	$0.746219\pi$
$798$	−5.80488	−0.205491
$799$	−17.4179	−0.616202
$800$	−2.31548	−0.0818647
$801$	65.9257	2.32937
$802$	−17.8689	−0.630973
$803$	−39.3337	−1.38806
$804$	23.3036	0.821855
$805$	−4.06975	−0.143440
$806$	25.1571	0.886120
$807$	−97.5365	−3.43345
$808$	−11.3409	−0.398973
$809$	−14.2668	−0.501595	−0.250797	−	0.968040i	$-0.580693\pi$
$809$	−14.2668	−0.501595	−0.250797	+	0.968040i	$0.580693\pi$
$810$	52.0375	1.82841
$811$	−8.57439	−0.301088	−0.150544	−	0.988603i	$-0.548102\pi$
$811$	−8.57439	−0.301088	−0.150544	+	0.988603i	$0.548102\pi$
$812$	0.558547	0.0196012
$813$	−15.0713	−0.528573
$814$	2.65058	0.0929027
$815$	−7.62584	−0.267121
$816$	−7.03745	−0.246360
$817$	57.4873	2.01123
$818$	−10.6376	−0.371934
$819$	9.46933	0.330885
$820$	−14.2646	−0.498141
$821$	−37.4224	−1.30605	−0.653026	−	0.757336i	$-0.726501\pi$
$821$	−37.4224	−1.30605	−0.653026	+	0.757336i	$0.726501\pi$
$822$	9.64491	0.336405
$823$	−34.4417	−1.20056	−0.600282	−	0.799789i	$-0.704945\pi$
$823$	−34.4417	−1.20056	−0.600282	+	0.799789i	$0.704945\pi$
$824$	3.41079	0.118820
$825$	19.8096	0.689682
$826$	−1.02799	−0.0357684
$827$	45.9884	1.59917	0.799586	−	0.600551i	$-0.205052\pi$
$827$	45.9884	1.59917	0.799586	+	0.600551i	$0.205052\pi$
$828$	38.3173	1.33162
$829$	17.8993	0.621668	0.310834	−	0.950464i	$-0.399392\pi$
$829$	17.8993	0.621668	0.310834	+	0.950464i	$0.399392\pi$
$830$	0.348701	0.0121036
$831$	27.9842	0.970762
$832$	4.89080	0.169558
$833$	−15.3921	−0.533306
$834$	39.0468	1.35208
$835$	−12.3408	−0.427070
$836$	17.9717	0.621565
$837$	65.4885	2.26361
$838$	25.1908	0.870201
$839$	−1.73700	−0.0599680	−0.0299840	−	0.999550i	$-0.509546\pi$
$839$	−1.73700	−0.0599680	−0.0299840	+	0.999550i	$0.509546\pi$
$840$	−2.36096	−0.0814608
$841$	−24.8967	−0.858508
$842$	10.8888	0.375254
$843$	9.11574	0.313963
$844$	−6.43052	−0.221348
$845$	−29.5352	−1.01604
$846$	55.0170	1.89152
$847$	−1.01929	−0.0350231
$848$	7.98384	0.274166
$849$	−90.2585	−3.09766
$850$	−5.14737	−0.176553
$851$	5.35215	0.183469
$852$	47.9164	1.64159
$853$	−28.2758	−0.968147	−0.484073	−	0.875027i	$-0.660843\pi$
$853$	−28.2758	−0.968147	−0.484073	+	0.875027i	$0.660843\pi$
$854$	1.74970	0.0598734
$855$	126.297	4.31926
$856$	4.06130	0.138812
$857$	28.2392	0.964632	0.482316	−	0.875997i	$-0.339796\pi$
$857$	28.2392	0.964632	0.482316	+	0.875997i	$0.339796\pi$
$858$	−41.8421	−1.42847
$859$	36.1572	1.23367	0.616834	−	0.787094i	$-0.288415\pi$
$859$	36.1572	1.23367	0.616834	+	0.787094i	$0.288415\pi$
$860$	23.3812	0.797293
$861$	−4.60368	−0.156893
$862$	−37.4181	−1.27447
$863$	−9.20745	−0.313425	−0.156713	−	0.987644i	$-0.550090\pi$
$863$	−9.20745	−0.313425	−0.156713	+	0.987644i	$0.550090\pi$
$864$	12.7317	0.433140
$865$	11.3118	0.384614
$866$	14.7028	0.499621
$867$	38.1727	1.29641
$868$	−1.41833	−0.0481411
$869$	−45.9736	−1.55955
$870$	−17.3443	−0.588028
$871$	−36.0024	−1.21989
$872$	12.3032	0.416638
$873$	−14.5071	−0.490992
$874$	36.2892	1.22750
$875$	2.00209	0.0676829
$876$	−46.0759	−1.55676
$877$	17.5164	0.591488	0.295744	−	0.955267i	$-0.404432\pi$
$877$	17.5164	0.591488	0.295744	+	0.955267i	$0.404432\pi$
$878$	−29.3013	−0.988871
$879$	−99.2030	−3.34603
$880$	7.30945	0.246401
$881$	3.35625	0.113075	0.0565374	−	0.998400i	$-0.481994\pi$
$881$	3.35625	0.113075	0.0565374	+	0.998400i	$0.481994\pi$
$882$	48.6183	1.63706
$883$	−4.29516	−0.144543	−0.0722717	−	0.997385i	$-0.523025\pi$
$883$	−4.29516	−0.144543	−0.0722717	+	0.997385i	$0.523025\pi$
$884$	10.8723	0.365677
$885$	31.9218	1.07304
$886$	18.9430	0.636404
$887$	−35.5435	−1.19343	−0.596716	−	0.802452i	$-0.703528\pi$
$887$	−35.5435	−1.19343	−0.596716	+	0.802452i	$0.703528\pi$
$888$	3.10491	0.104194
$889$	1.18807	0.0398466
$890$	25.3941	0.851211
$891$	−51.9945	−1.74188
$892$	8.12471	0.272035
$893$	52.1050	1.74363
$894$	−11.5765	−0.387175
$895$	−50.0827	−1.67408
$896$	−0.275737	−0.00921173
$897$	−84.4893	−2.82102
$898$	15.6074	0.520825
$899$	−10.4195	−0.347508
$900$	16.2587	0.541957
$901$	17.7483	0.591280
$902$	14.2528	0.474567
$903$	7.54593	0.251113
$904$	9.38748	0.312223
$905$	55.2823	1.83765
$906$	−11.4034	−0.378853
$907$	22.5875	0.750005	0.375002	−	0.927024i	$-0.377642\pi$
$907$	22.5875	0.750005	0.375002	+	0.927024i	$0.377642\pi$
$908$	−15.7849	−0.523840
$909$	79.6330	2.64126
$910$	3.64751	0.120914
$911$	−16.1027	−0.533508	−0.266754	−	0.963765i	$-0.585951\pi$
$911$	−16.1027	−0.533508	−0.266754	+	0.963765i	$0.585951\pi$
$912$	21.0522	0.697109
$913$	−0.348413	−0.0115308
$914$	7.71799	0.255288
$915$	−54.3326	−1.79618
$916$	26.1380	0.863625
$917$	5.02215	0.165846
$918$	28.3027	0.934130
$919$	−12.7611	−0.420949	−0.210474	−	0.977599i	$-0.567501\pi$
$919$	−12.7611	−0.420949	−0.210474	+	0.977599i	$0.567501\pi$
$920$	14.7595	0.486607
$921$	74.0691	2.44066
$922$	−37.1286	−1.22276
$923$	−74.0273	−2.43664
$924$	2.35901	0.0776057
$925$	2.27101	0.0746703
$926$	20.0192	0.657873
$927$	−23.9496	−0.786609
$928$	−2.02565	−0.0664952
$929$	52.3157	1.71642	0.858212	−	0.513295i	$-0.171575\pi$
$929$	52.3157	1.71642	0.858212	+	0.513295i	$0.171575\pi$
$930$	44.0427	1.44422
$931$	46.0449	1.50906
$932$	3.89323	0.127527
$933$	−65.9599	−2.15943
$934$	−27.1451	−0.888214
$935$	16.2491	0.531401
$936$	−34.3419	−1.12250
$937$	10.7805	0.352184	0.176092	−	0.984374i	$-0.443654\pi$
$937$	10.7805	0.352184	0.176092	+	0.984374i	$0.443654\pi$
$938$	2.02977	0.0662744
$939$	79.4847	2.59389
$940$	21.1921	0.691211
$941$	25.2966	0.824646	0.412323	−	0.911038i	$-0.364717\pi$
$941$	25.2966	0.824646	0.412323	+	0.911038i	$0.364717\pi$
$942$	26.4292	0.861109
$943$	28.7799	0.937202
$944$	3.72815	0.121341
$945$	9.49515	0.308877
$946$	−23.3619	−0.759562
$947$	49.0314	1.59330	0.796652	−	0.604438i	$-0.206602\pi$
$947$	49.0314	1.59330	0.796652	+	0.604438i	$0.206602\pi$
$948$	−53.8538	−1.74909
$949$	71.1839	2.31073
$950$	15.3981	0.499581
$951$	101.005	3.27530
$952$	−0.612970	−0.0198665
$953$	−22.6264	−0.732940	−0.366470	−	0.930430i	$-0.619434\pi$
$953$	−22.6264	−0.732940	−0.366470	+	0.930430i	$0.619434\pi$
$954$	−56.0604	−1.81502
$955$	−48.8160	−1.57965
$956$	4.77813	0.154536
$957$	17.3300	0.560200
$958$	−20.1347	−0.650523
$959$	0.840083	0.0271277
$960$	8.56235	0.276349
$961$	−4.54175	−0.146508
$962$	−4.79686	−0.154657
$963$	−28.5174	−0.918959
$964$	23.2558	0.749019
$965$	−45.5167	−1.46524
$966$	4.76341	0.153260
$967$	−22.6931	−0.729762	−0.364881	−	0.931054i	$-0.618890\pi$
$967$	−22.6931	−0.729762	−0.364881	+	0.931054i	$0.618890\pi$
$968$	3.69658	0.118813
$969$	46.7996	1.50342
$970$	−5.58803	−0.179421
$971$	−41.4686	−1.33079	−0.665395	−	0.746491i	$-0.731737\pi$
$971$	−41.4686	−1.33079	−0.665395	+	0.746491i	$0.731737\pi$
$972$	−22.7119	−0.728484
$973$	3.40102	0.109032
$974$	12.0068	0.384723
$975$	−35.8503	−1.14813
$976$	−6.34553	−0.203115
$977$	15.9392	0.509942	0.254971	−	0.966949i	$-0.417934\pi$
$977$	15.9392	0.509942	0.254971	+	0.966949i	$0.417934\pi$
$978$	8.92560	0.285409
$979$	−25.3731	−0.810928
$980$	18.7274	0.598224
$981$	−86.3896	−2.75821
$982$	6.69018	0.213492
$983$	−54.0463	−1.72381	−0.861904	−	0.507071i	$-0.830728\pi$
$983$	−54.0463	−1.72381	−0.861904	+	0.507071i	$0.830728\pi$
$984$	16.6959	0.532245
$985$	55.9967	1.78420
$986$	−4.50307	−0.143407
$987$	6.83943	0.217701
$988$	−32.5242	−1.03473
$989$	−47.1733	−1.50002
$990$	−51.3250	−1.63122
$991$	55.5685	1.76519	0.882596	−	0.470132i	$-0.155794\pi$
$991$	55.5685	1.76519	0.882596	+	0.470132i	$0.155794\pi$
$992$	5.14376	0.163314
$993$	74.8024	2.37378
$994$	4.17357	0.132378
$995$	−22.6170	−0.717007
$996$	−0.408134	−0.0129322
$997$	26.3416	0.834248	0.417124	−	0.908850i	$-0.363038\pi$
$997$	26.3416	0.834248	0.417124	+	0.908850i	$0.363038\pi$
$998$	16.0578	0.508301
$999$	−12.4871	−0.395075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.b.1.5	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.b.1.5	✓	82	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.16571 q^{3} +1.00000 q^{4} -2.70471 q^{5} +3.16571 q^{6} +0.275737 q^{7} -1.00000 q^{8} +7.02173 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.16571 q^{3} +1.00000 q^{4} -2.70471 q^{5} +3.16571 q^{6} +0.275737 q^{7} -1.00000 q^{8} +7.02173 q^{9} +2.70471 q^{10} -2.70248 q^{11} -3.16571 q^{12} +4.89080 q^{13} -0.275737 q^{14} +8.56235 q^{15} +1.00000 q^{16} +2.22302 q^{17} -7.02173 q^{18} -6.65008 q^{19} -2.70471 q^{20} -0.872905 q^{21} +2.70248 q^{22} +5.45696 q^{23} +3.16571 q^{24} +2.31548 q^{25} -4.89080 q^{26} -12.7317 q^{27} +0.275737 q^{28} +2.02565 q^{29} -8.56235 q^{30} -5.14376 q^{31} -1.00000 q^{32} +8.55528 q^{33} -2.22302 q^{34} -0.745791 q^{35} +7.02173 q^{36} +0.980793 q^{37} +6.65008 q^{38} -15.4828 q^{39} +2.70471 q^{40} +5.27398 q^{41} +0.872905 q^{42} -8.64462 q^{43} -2.70248 q^{44} -18.9918 q^{45} -5.45696 q^{46} -7.83525 q^{47} -3.16571 q^{48} -6.92397 q^{49} -2.31548 q^{50} -7.03745 q^{51} +4.89080 q^{52} +7.98384 q^{53} +12.7317 q^{54} +7.30945 q^{55} -0.275737 q^{56} +21.0522 q^{57} -2.02565 q^{58} +3.72815 q^{59} +8.56235 q^{60} -6.34553 q^{61} +5.14376 q^{62} +1.93615 q^{63} +1.00000 q^{64} -13.2282 q^{65} -8.55528 q^{66} -7.36126 q^{67} +2.22302 q^{68} -17.2752 q^{69} +0.745791 q^{70} -15.1361 q^{71} -7.02173 q^{72} +14.5547 q^{73} -0.980793 q^{74} -7.33015 q^{75} -6.65008 q^{76} -0.745175 q^{77} +15.4828 q^{78} +17.0116 q^{79} -2.70471 q^{80} +19.2395 q^{81} -5.27398 q^{82} +0.128923 q^{83} -0.872905 q^{84} -6.01264 q^{85} +8.64462 q^{86} -6.41263 q^{87} +2.70248 q^{88} +9.38881 q^{89} +18.9918 q^{90} +1.34857 q^{91} +5.45696 q^{92} +16.2837 q^{93} +7.83525 q^{94} +17.9866 q^{95} +3.16571 q^{96} -2.06603 q^{97} +6.92397 q^{98} -18.9761 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9}+O(q^{10}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9	\( 82 q - 82 q^{2} - 13 q^{3} + 82 q^{4} + 3 q^{5} + 13 q^{6} - 37 q^{7} - 82 q^{8} + 91 q^{9} - 3 q^{10} - 16 q^{11} - 13 q^{12} - 42 q^{13} + 37 q^{14} - 9 q^{15} + 82 q^{16} + 3 q^{17} - 91 q^{18} - 42 q^{19} + 3 q^{20} - q^{21} + 16 q^{22} - 6 q^{23} + 13 q^{24} + 53 q^{25} + 42 q^{26} - 49 q^{27} - 37 q^{28} + 15 q^{29} + 9 q^{30} - 40 q^{31} - 82 q^{32} - 37 q^{33} - 3 q^{34} - 42 q^{35} + 91 q^{36} - 72 q^{37} + 42 q^{38} - 14 q^{39} - 3 q^{40} + 8 q^{41} + q^{42} - 93 q^{43} - 16 q^{44} - 11 q^{45} + 6 q^{46} + 7 q^{47} - 13 q^{48} + 61 q^{49} - 53 q^{50} - 70 q^{51} - 42 q^{52} + 18 q^{53} + 49 q^{54} - 62 q^{55} + 37 q^{56} - 51 q^{57} - 15 q^{58} - 47 q^{59} - 9 q^{60} - 14 q^{61} + 40 q^{62} - 100 q^{63} + 82 q^{64} + q^{65} + 37 q^{66} - 150 q^{67} + 3 q^{68} + 31 q^{69} + 42 q^{70} + 7 q^{71} - 91 q^{72} - 78 q^{73} + 72 q^{74} - 49 q^{75} - 42 q^{76} + 29 q^{77} + 14 q^{78} - 59 q^{79} + 3 q^{80} + 122 q^{81} - 8 q^{82} - 52 q^{83} - q^{84} - 108 q^{85} + 93 q^{86} - 49 q^{87} + 16 q^{88} + 38 q^{89} + 11 q^{90} - 69 q^{91} - 6 q^{92} - 63 q^{93} - 7 q^{94} + 5 q^{95} + 13 q^{96} - 74 q^{97} - 61 q^{98} - 89 q^{99}+O(q^{100}) \) 82 * q - 82 * q^2 - 13 * q^3 + 82 * q^4 + 3 * q^5 + 13 * q^6 - 37 * q^7 - 82 * q^8 + 91 * q^9 - 3 * q^10 - 16 * q^11 - 13 * q^12 - 42 * q^13 + 37 * q^14 - 9 * q^15 + 82 * q^16 + 3 * q^17 - 91 * q^18 - 42 * q^19 + 3 * q^20 - q^21 + 16 * q^22 - 6 * q^23 + 13 * q^24 + 53 * q^25 + 42 * q^26 - 49 * q^27 - 37 * q^28 + 15 * q^29 + 9 * q^30 - 40 * q^31 - 82 * q^32 - 37 * q^33 - 3 * q^34 - 42 * q^35 + 91 * q^36 - 72 * q^37 + 42 * q^38 - 14 * q^39 - 3 * q^40 + 8 * q^41 + q^42 - 93 * q^43 - 16 * q^44 - 11 * q^45 + 6 * q^46 + 7 * q^47 - 13 * q^48 + 61 * q^49 - 53 * q^50 - 70 * q^51 - 42 * q^52 + 18 * q^53 + 49 * q^54 - 62 * q^55 + 37 * q^56 - 51 * q^57 - 15 * q^58 - 47 * q^59 - 9 * q^60 - 14 * q^61 + 40 * q^62 - 100 * q^63 + 82 * q^64 + q^65 + 37 * q^66 - 150 * q^67 + 3 * q^68 + 31 * q^69 + 42 * q^70 + 7 * q^71 - 91 * q^72 - 78 * q^73 + 72 * q^74 - 49 * q^75 - 42 * q^76 + 29 * q^77 + 14 * q^78 - 59 * q^79 + 3 * q^80 + 122 * q^81 - 8 * q^82 - 52 * q^83 - q^84 - 108 * q^85 + 93 * q^86 - 49 * q^87 + 16 * q^88 + 38 * q^89 + 11 * q^90 - 69 * q^91 - 6 * q^92 - 63 * q^93 - 7 * q^94 + 5 * q^95 + 13 * q^96 - 74 * q^97 - 61 * q^98 - 89 * q^99

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