LMFDB - Embedded newform 8002.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.65847 q^{3} +1.00000 q^{4} -0.802626 q^{5} +1.65847 q^{6} -0.0977002 q^{7} -1.00000 q^{8} -0.249472 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.65847 q^{3} +1.00000 q^{4} -0.802626 q^{5} +1.65847 q^{6} -0.0977002 q^{7} -1.00000 q^{8} -0.249472 q^{9} +0.802626 q^{10} -4.28508 q^{11} -1.65847 q^{12} -5.88772 q^{13} +0.0977002 q^{14} +1.33113 q^{15} +1.00000 q^{16} +4.10249 q^{17} +0.249472 q^{18} +4.88102 q^{19} -0.802626 q^{20} +0.162033 q^{21} +4.28508 q^{22} -7.15578 q^{23} +1.65847 q^{24} -4.35579 q^{25} +5.88772 q^{26} +5.38916 q^{27} -0.0977002 q^{28} +6.43182 q^{29} -1.33113 q^{30} -4.30233 q^{31} -1.00000 q^{32} +7.10668 q^{33} -4.10249 q^{34} +0.0784167 q^{35} -0.249472 q^{36} -0.738173 q^{37} -4.88102 q^{38} +9.76462 q^{39} +0.802626 q^{40} -9.15637 q^{41} -0.162033 q^{42} -3.11395 q^{43} -4.28508 q^{44} +0.200233 q^{45} +7.15578 q^{46} +7.49518 q^{47} -1.65847 q^{48} -6.99045 q^{49} +4.35579 q^{50} -6.80386 q^{51} -5.88772 q^{52} -8.00141 q^{53} -5.38916 q^{54} +3.43932 q^{55} +0.0977002 q^{56} -8.09504 q^{57} -6.43182 q^{58} -8.80842 q^{59} +1.33113 q^{60} -2.70949 q^{61} +4.30233 q^{62} +0.0243734 q^{63} +1.00000 q^{64} +4.72564 q^{65} -7.10668 q^{66} -10.2707 q^{67} +4.10249 q^{68} +11.8677 q^{69} -0.0784167 q^{70} -10.3343 q^{71} +0.249472 q^{72} +0.0428290 q^{73} +0.738173 q^{74} +7.22396 q^{75} +4.88102 q^{76} +0.418653 q^{77} -9.76462 q^{78} +3.97141 q^{79} -0.802626 q^{80} -8.18935 q^{81} +9.15637 q^{82} +6.86346 q^{83} +0.162033 q^{84} -3.29276 q^{85} +3.11395 q^{86} -10.6670 q^{87} +4.28508 q^{88} -5.35914 q^{89} -0.200233 q^{90} +0.575231 q^{91} -7.15578 q^{92} +7.13529 q^{93} -7.49518 q^{94} -3.91764 q^{95} +1.65847 q^{96} -4.07812 q^{97} +6.99045 q^{98} +1.06901 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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
$3$	−1.65847	−0.957519	−0.478760	−	0.877946i	$-0.658913\pi$
$3$	−1.65847	−0.957519	−0.478760	+	0.877946i	$0.658913\pi$
$4$	1.00000	0.500000
$5$	−0.802626	−0.358945	−0.179473	−	0.983763i	$-0.557439\pi$
$5$	−0.802626	−0.358945	−0.179473	+	0.983763i	$0.557439\pi$
$6$	1.65847	0.677068
$7$	−0.0977002	−0.0369272	−0.0184636	−	0.999830i	$-0.505877\pi$
$7$	−0.0977002	−0.0369272	−0.0184636	+	0.999830i	$0.505877\pi$
$8$	−1.00000	−0.353553
$9$	−0.249472	−0.0831572
$10$	0.802626	0.253813
$11$	−4.28508	−1.29200	−0.646000	−	0.763338i	$-0.723559\pi$
$11$	−4.28508	−1.29200	−0.646000	+	0.763338i	$0.723559\pi$
$12$	−1.65847	−0.478760
$13$	−5.88772	−1.63296	−0.816480	−	0.577374i	$-0.804078\pi$
$13$	−5.88772	−1.63296	−0.816480	+	0.577374i	$0.804078\pi$
$14$	0.0977002	0.0261115
$15$	1.33113	0.343697
$16$	1.00000	0.250000
$17$	4.10249	0.994999	0.497500	−	0.867464i	$-0.334252\pi$
$17$	4.10249	0.994999	0.497500	+	0.867464i	$0.334252\pi$
$18$	0.249472	0.0588011
$19$	4.88102	1.11978	0.559892	−	0.828566i	$-0.310843\pi$
$19$	4.88102	1.11978	0.559892	+	0.828566i	$0.310843\pi$
$20$	−0.802626	−0.179473
$21$	0.162033	0.0353585
$22$	4.28508	0.913582
$23$	−7.15578	−1.49208	−0.746042	−	0.665899i	$-0.768048\pi$
$23$	−7.15578	−1.49208	−0.746042	+	0.665899i	$0.768048\pi$
$24$	1.65847	0.338534
$25$	−4.35579	−0.871158
$26$	5.88772	1.15468
$27$	5.38916	1.03714
$28$	−0.0977002	−0.0184636
$29$	6.43182	1.19436	0.597179	−	0.802108i	$-0.296288\pi$
$29$	6.43182	1.19436	0.597179	+	0.802108i	$0.296288\pi$
$30$	−1.33113	−0.243031
$31$	−4.30233	−0.772721	−0.386360	−	0.922348i	$-0.626268\pi$
$31$	−4.30233	−0.772721	−0.386360	+	0.922348i	$0.626268\pi$
$32$	−1.00000	−0.176777
$33$	7.10668	1.23711
$34$	−4.10249	−0.703571
$35$	0.0784167	0.0132548
$36$	−0.249472	−0.0415786
$37$	−0.738173	−0.121355	−0.0606775	−	0.998157i	$-0.519326\pi$
$37$	−0.738173	−0.121355	−0.0606775	+	0.998157i	$0.519326\pi$
$38$	−4.88102	−0.791806
$39$	9.76462	1.56359
$40$	0.802626	0.126906
$41$	−9.15637	−1.42999	−0.714993	−	0.699132i	$-0.753570\pi$
$41$	−9.15637	−1.42999	−0.714993	+	0.699132i	$0.753570\pi$
$42$	−0.162033	−0.0250022
$43$	−3.11395	−0.474873	−0.237437	−	0.971403i	$-0.576307\pi$
$43$	−3.11395	−0.474873	−0.237437	+	0.971403i	$0.576307\pi$
$44$	−4.28508	−0.646000
$45$	0.200233	0.0298489
$46$	7.15578	1.05506
$47$	7.49518	1.09328	0.546642	−	0.837366i	$-0.315906\pi$
$47$	7.49518	1.09328	0.546642	+	0.837366i	$0.315906\pi$
$48$	−1.65847	−0.239380
$49$	−6.99045	−0.998636
$50$	4.35579	0.616002
$51$	−6.80386	−0.952731
$52$	−5.88772	−0.816480
$53$	−8.00141	−1.09908	−0.549539	−	0.835468i	$-0.685197\pi$
$53$	−8.00141	−1.09908	−0.549539	+	0.835468i	$0.685197\pi$
$54$	−5.38916	−0.733371
$55$	3.43932	0.463757
$56$	0.0977002	0.0130557
$57$	−8.09504	−1.07221
$58$	−6.43182	−0.844539
$59$	−8.80842	−1.14676	−0.573379	−	0.819290i	$-0.694368\pi$
$59$	−8.80842	−1.14676	−0.573379	+	0.819290i	$0.694368\pi$
$60$	1.33113	0.171849
$61$	−2.70949	−0.346915	−0.173458	−	0.984841i	$-0.555494\pi$
$61$	−2.70949	−0.346915	−0.173458	+	0.984841i	$0.555494\pi$
$62$	4.30233	0.546396
$63$	0.0243734	0.00307076
$64$	1.00000	0.125000
$65$	4.72564	0.586143
$66$	−7.10668	−0.874772
$67$	−10.2707	−1.25476	−0.627380	−	0.778713i	$-0.715873\pi$
$67$	−10.2707	−1.25476	−0.627380	+	0.778713i	$0.715873\pi$
$68$	4.10249	0.497500
$69$	11.8677	1.42870
$70$	−0.0784167	−0.00937259
$71$	−10.3343	−1.22646	−0.613229	−	0.789905i	$-0.710130\pi$
$71$	−10.3343	−1.22646	−0.613229	+	0.789905i	$0.710130\pi$
$72$	0.249472	0.0294005
$73$	0.0428290	0.00501276	0.00250638	−	0.999997i	$-0.499202\pi$
$73$	0.0428290	0.00501276	0.00250638	+	0.999997i	$0.499202\pi$
$74$	0.738173	0.0858109
$75$	7.22396	0.834151
$76$	4.88102	0.559892
$77$	0.418653	0.0477099
$78$	−9.76462	−1.10563
$79$	3.97141	0.446818	0.223409	−	0.974725i	$-0.428281\pi$
$79$	3.97141	0.446818	0.223409	+	0.974725i	$0.428281\pi$
$80$	−0.802626	−0.0897363
$81$	−8.18935	−0.909928
$82$	9.15637	1.01115
$83$	6.86346	0.753362	0.376681	−	0.926343i	$-0.377065\pi$
$83$	6.86346	0.753362	0.376681	+	0.926343i	$0.377065\pi$
$84$	0.162033	0.0176792
$85$	−3.29276	−0.357150
$86$	3.11395	0.335786
$87$	−10.6670	−1.14362
$88$	4.28508	0.456791
$89$	−5.35914	−0.568067	−0.284034	−	0.958814i	$-0.591673\pi$
$89$	−5.35914	−0.568067	−0.284034	+	0.958814i	$0.591673\pi$
$90$	−0.200233	−0.0211064
$91$	0.575231	0.0603006
$92$	−7.15578	−0.746042
$93$	7.13529	0.739895
$94$	−7.49518	−0.773069
$95$	−3.91764	−0.401941
$96$	1.65847	0.169267
$97$	−4.07812	−0.414071	−0.207035	−	0.978333i	$-0.566381\pi$
$97$	−4.07812	−0.414071	−0.207035	+	0.978333i	$0.566381\pi$
$98$	6.99045	0.706143
$99$	1.06901	0.107439
$100$	−4.35579	−0.435579
$101$	−5.49159	−0.546434	−0.273217	−	0.961952i	$-0.588088\pi$
$101$	−5.49159	−0.546434	−0.273217	+	0.961952i	$0.588088\pi$
$102$	6.80386	0.673682
$103$	−17.9063	−1.76436	−0.882180	−	0.470913i	$-0.843925\pi$
$103$	−17.9063	−1.76436	−0.882180	+	0.470913i	$0.843925\pi$
$104$	5.88772	0.577339
$105$	−0.130052	−0.0126918
$106$	8.00141	0.777166
$107$	−11.2606	−1.08860	−0.544301	−	0.838890i	$-0.683205\pi$
$107$	−11.2606	−1.08860	−0.544301	+	0.838890i	$0.683205\pi$
$108$	5.38916	0.518572
$109$	−6.84529	−0.655660	−0.327830	−	0.944737i	$-0.606317\pi$
$109$	−6.84529	−0.655660	−0.327830	+	0.944737i	$0.606317\pi$
$110$	−3.43932	−0.327926
$111$	1.22424	0.116200
$112$	−0.0977002	−0.00923180
$113$	10.7483	1.01112	0.505558	−	0.862792i	$-0.331286\pi$
$113$	10.7483	1.01112	0.505558	+	0.862792i	$0.331286\pi$
$114$	8.09504	0.758170
$115$	5.74342	0.535577
$116$	6.43182	0.597179
$117$	1.46882	0.135792
$118$	8.80842	0.810881
$119$	−0.400814	−0.0367425
$120$	−1.33113	−0.121515
$121$	7.36190	0.669263
$122$	2.70949	0.245306
$123$	15.1856	1.36924
$124$	−4.30233	−0.386360
$125$	7.50920	0.671644
$126$	−0.0243734	−0.00217136
$127$	−7.85976	−0.697441	−0.348721	−	0.937227i	$-0.613384\pi$
$127$	−7.85976	−0.697441	−0.348721	+	0.937227i	$0.613384\pi$
$128$	−1.00000	−0.0883883
$129$	5.16440	0.454700
$130$	−4.72564	−0.414466
$131$	−20.3039	−1.77396	−0.886981	−	0.461806i	$-0.847202\pi$
$131$	−20.3039	−1.77396	−0.886981	+	0.461806i	$0.847202\pi$
$132$	7.10668	0.618557
$133$	−0.476877	−0.0413505
$134$	10.2707	0.887250
$135$	−4.32548	−0.372278
$136$	−4.10249	−0.351785
$137$	11.7342	1.00252	0.501262	−	0.865296i	$-0.332869\pi$
$137$	11.7342	1.00252	0.501262	+	0.865296i	$0.332869\pi$
$138$	−11.8677	−1.01024
$139$	3.26959	0.277323	0.138661	−	0.990340i	$-0.455720\pi$
$139$	3.26959	0.277323	0.138661	+	0.990340i	$0.455720\pi$
$140$	0.0784167	0.00662742
$141$	−12.4305	−1.04684
$142$	10.3343	0.867237
$143$	25.2293	2.10978
$144$	−0.249472	−0.0207893
$145$	−5.16235	−0.428710
$146$	−0.0428290	−0.00354455
$147$	11.5935	0.956213
$148$	−0.738173	−0.0606775
$149$	4.82312	0.395126	0.197563	−	0.980290i	$-0.436697\pi$
$149$	4.82312	0.395126	0.197563	+	0.980290i	$0.436697\pi$
$150$	−7.22396	−0.589834
$151$	12.8253	1.04371	0.521854	−	0.853035i	$-0.325241\pi$
$151$	12.8253	1.04371	0.521854	+	0.853035i	$0.325241\pi$
$152$	−4.88102	−0.395903
$153$	−1.02345	−0.0827414
$154$	−0.418653	−0.0337360
$155$	3.45316	0.277365
$156$	9.76462	0.781795
$157$	−0.792102	−0.0632166	−0.0316083	−	0.999500i	$-0.510063\pi$
$157$	−0.792102	−0.0632166	−0.0316083	+	0.999500i	$0.510063\pi$
$158$	−3.97141	−0.315948
$159$	13.2701	1.05239
$160$	0.802626	0.0634532
$161$	0.699121	0.0550985
$162$	8.18935	0.643416
$163$	11.2416	0.880513	0.440256	−	0.897872i	$-0.354888\pi$
$163$	11.2416	0.880513	0.440256	+	0.897872i	$0.354888\pi$
$164$	−9.15637	−0.714993
$165$	−5.70401	−0.444056
$166$	−6.86346	−0.532708
$167$	−11.1276	−0.861082	−0.430541	−	0.902571i	$-0.641677\pi$
$167$	−11.1276	−0.861082	−0.430541	+	0.902571i	$0.641677\pi$
$168$	−0.162033	−0.0125011
$169$	21.6653	1.66656
$170$	3.29276	0.252543
$171$	−1.21768	−0.0931181
$172$	−3.11395	−0.237437
$173$	−3.93491	−0.299166	−0.149583	−	0.988749i	$-0.547793\pi$
$173$	−3.93491	−0.299166	−0.149583	+	0.988749i	$0.547793\pi$
$174$	10.6670	0.808662
$175$	0.425562	0.0321694
$176$	−4.28508	−0.323000
$177$	14.6085	1.09804
$178$	5.35914	0.401684
$179$	−9.10842	−0.680795	−0.340398	−	0.940282i	$-0.610562\pi$
$179$	−9.10842	−0.680795	−0.340398	+	0.940282i	$0.610562\pi$
$180$	0.200233	0.0149245
$181$	−9.30241	−0.691443	−0.345721	−	0.938337i	$-0.612366\pi$
$181$	−9.30241	−0.691443	−0.345721	+	0.938337i	$0.612366\pi$
$182$	−0.575231	−0.0426390
$183$	4.49362	0.332178
$184$	7.15578	0.527531
$185$	0.592477	0.0435598
$186$	−7.13529	−0.523185
$187$	−17.5795	−1.28554
$188$	7.49518	0.546642
$189$	−0.526522	−0.0382988
$190$	3.91764	0.284215
$191$	13.3414	0.965349	0.482674	−	0.875800i	$-0.339665\pi$
$191$	13.3414	0.965349	0.482674	+	0.875800i	$0.339665\pi$
$192$	−1.65847	−0.119690
$193$	4.90829	0.353307	0.176653	−	0.984273i	$-0.443473\pi$
$193$	4.90829	0.353307	0.176653	+	0.984273i	$0.443473\pi$
$194$	4.07812	0.292792
$195$	−7.83734	−0.561244
$196$	−6.99045	−0.499318
$197$	1.64123	0.116933	0.0584663	−	0.998289i	$-0.481379\pi$
$197$	1.64123	0.116933	0.0584663	+	0.998289i	$0.481379\pi$
$198$	−1.06901	−0.0759710
$199$	−4.52532	−0.320791	−0.160396	−	0.987053i	$-0.551277\pi$
$199$	−4.52532	−0.320791	−0.160396	+	0.987053i	$0.551277\pi$
$200$	4.35579	0.308001
$201$	17.0336	1.20146
$202$	5.49159	0.386387
$203$	−0.628390	−0.0441043
$204$	−6.80386	−0.476365
$205$	7.34915	0.513287
$206$	17.9063	1.24759
$207$	1.78517	0.124078
$208$	−5.88772	−0.408240
$209$	−20.9156	−1.44676
$210$	0.130052	0.00897444
$211$	−0.227580	−0.0156672	−0.00783361	−	0.999969i	$-0.502494\pi$
$211$	−0.227580	−0.0156672	−0.00783361	+	0.999969i	$0.502494\pi$
$212$	−8.00141	−0.549539
$213$	17.1392	1.17436
$214$	11.2606	0.769758
$215$	2.49934	0.170454
$216$	−5.38916	−0.366686
$217$	0.420338	0.0285344
$218$	6.84529	0.463622
$219$	−0.0710307	−0.00479981
$220$	3.43932	0.231879
$221$	−24.1543	−1.62479
$222$	−1.22424	−0.0821656
$223$	13.0588	0.874483	0.437241	−	0.899344i	$-0.355956\pi$
$223$	13.0588	0.874483	0.437241	+	0.899344i	$0.355956\pi$
$224$	0.0977002	0.00652787
$225$	1.08665	0.0724431
$226$	−10.7483	−0.714968
$227$	12.3681	0.820898	0.410449	−	0.911884i	$-0.365372\pi$
$227$	12.3681	0.820898	0.410449	+	0.911884i	$0.365372\pi$
$228$	−8.09504	−0.536107
$229$	−27.7355	−1.83281	−0.916406	−	0.400251i	$-0.868923\pi$
$229$	−27.7355	−1.83281	−0.916406	+	0.400251i	$0.868923\pi$
$230$	−5.74342	−0.378710
$231$	−0.694324	−0.0456832
$232$	−6.43182	−0.422270
$233$	16.0290	1.05010	0.525048	−	0.851073i	$-0.324047\pi$
$233$	16.0290	1.05010	0.525048	+	0.851073i	$0.324047\pi$
$234$	−1.46882	−0.0960198
$235$	−6.01583	−0.392429
$236$	−8.80842	−0.573379
$237$	−6.58646	−0.427837
$238$	0.400814	0.0259809
$239$	19.1279	1.23728	0.618640	−	0.785674i	$-0.287684\pi$
$239$	19.1279	1.23728	0.618640	+	0.785674i	$0.287684\pi$
$240$	1.33113	0.0859243
$241$	−26.5054	−1.70737	−0.853683	−	0.520794i	$-0.825636\pi$
$241$	−26.5054	−1.70737	−0.853683	+	0.520794i	$0.825636\pi$
$242$	−7.36190	−0.473241
$243$	−2.58567	−0.165871
$244$	−2.70949	−0.173458
$245$	5.61072	0.358456
$246$	−15.1856	−0.968198
$247$	−28.7381	−1.82856
$248$	4.30233	0.273198
$249$	−11.3828	−0.721359
$250$	−7.50920	−0.474924
$251$	10.3539	0.653535	0.326767	−	0.945105i	$-0.394041\pi$
$251$	10.3539	0.653535	0.326767	+	0.945105i	$0.394041\pi$
$252$	0.0243734	0.00153538
$253$	30.6631	1.92777
$254$	7.85976	0.493165
$255$	5.46096	0.341978
$256$	1.00000	0.0625000
$257$	1.73524	0.108242	0.0541208	−	0.998534i	$-0.482764\pi$
$257$	1.73524	0.108242	0.0541208	+	0.998534i	$0.482764\pi$
$258$	−5.16440	−0.321521
$259$	0.0721197	0.00448130
$260$	4.72564	0.293072
$261$	−1.60456	−0.0993196
$262$	20.3039	1.25438
$263$	8.81690	0.543673	0.271837	−	0.962343i	$-0.412369\pi$
$263$	8.81690	0.543673	0.271837	+	0.962343i	$0.412369\pi$
$264$	−7.10668	−0.437386
$265$	6.42214	0.394509
$266$	0.476877	0.0292392
$267$	8.88798	0.543935
$268$	−10.2707	−0.627380
$269$	3.26925	0.199330	0.0996649	−	0.995021i	$-0.468223\pi$
$269$	3.26925	0.199330	0.0996649	+	0.995021i	$0.468223\pi$
$270$	4.32548	0.263240
$271$	−23.0568	−1.40060	−0.700301	−	0.713848i	$-0.746951\pi$
$271$	−23.0568	−1.40060	−0.700301	+	0.713848i	$0.746951\pi$
$272$	4.10249	0.248750
$273$	−0.954005	−0.0577390
$274$	−11.7342	−0.708891
$275$	18.6649	1.12554
$276$	11.8677	0.714349
$277$	−20.6326	−1.23969	−0.619846	−	0.784723i	$-0.712805\pi$
$277$	−20.6326	−1.23969	−0.619846	+	0.784723i	$0.712805\pi$
$278$	−3.26959	−0.196097
$279$	1.07331	0.0642573
$280$	−0.0784167	−0.00468630
$281$	−28.5470	−1.70297	−0.851486	−	0.524377i	$-0.824298\pi$
$281$	−28.5470	−1.70297	−0.851486	+	0.524377i	$0.824298\pi$
$282$	12.4305	0.740228
$283$	12.6676	0.753012	0.376506	−	0.926414i	$-0.377126\pi$
$283$	12.6676	0.753012	0.376506	+	0.926414i	$0.377126\pi$
$284$	−10.3343	−0.613229
$285$	6.49729	0.384866
$286$	−25.2293	−1.49184
$287$	0.894579	0.0528053
$288$	0.249472	0.0147003
$289$	−0.169596	−0.00997623
$290$	5.16235	0.303143
$291$	6.76345	0.396480
$292$	0.0428290	0.00250638
$293$	−16.3219	−0.953533	−0.476767	−	0.879030i	$-0.658191\pi$
$293$	−16.3219	−0.953533	−0.476767	+	0.879030i	$0.658191\pi$
$294$	−11.5935	−0.676145
$295$	7.06987	0.411624
$296$	0.738173	0.0429055
$297$	−23.0930	−1.33999
$298$	−4.82312	−0.279396
$299$	42.1312	2.43651
$300$	7.22396	0.417075
$301$	0.304234	0.0175357
$302$	−12.8253	−0.738013
$303$	9.10765	0.523221
$304$	4.88102	0.279946
$305$	2.17471	0.124524
$306$	1.02345	0.0585070
$307$	11.0823	0.632502	0.316251	−	0.948676i	$-0.397576\pi$
$307$	11.0823	0.632502	0.316251	+	0.948676i	$0.397576\pi$
$308$	0.418653	0.0238550
$309$	29.6971	1.68941
$310$	−3.45316	−0.196126
$311$	29.0299	1.64614	0.823068	−	0.567943i	$-0.192260\pi$
$311$	29.0299	1.64614	0.823068	+	0.567943i	$0.192260\pi$
$312$	−9.76462	−0.552813
$313$	−6.07292	−0.343262	−0.171631	−	0.985161i	$-0.554904\pi$
$313$	−6.07292	−0.343262	−0.171631	+	0.985161i	$0.554904\pi$
$314$	0.792102	0.0447009
$315$	−0.0195628	−0.00110224
$316$	3.97141	0.223409
$317$	−9.81771	−0.551417	−0.275709	−	0.961241i	$-0.588912\pi$
$317$	−9.81771	−0.551417	−0.275709	+	0.961241i	$0.588912\pi$
$318$	−13.2701	−0.744151
$319$	−27.5608	−1.54311
$320$	−0.802626	−0.0448682
$321$	18.6754	1.04236
$322$	−0.699121	−0.0389605
$323$	20.0243	1.11418
$324$	−8.18935	−0.454964
$325$	25.6457	1.42257
$326$	−11.2416	−0.622617
$327$	11.3527	0.627807
$328$	9.15637	0.505576
$329$	−0.732281	−0.0403719
$330$	5.70401	0.313995
$331$	−31.5144	−1.73219	−0.866094	−	0.499882i	$-0.833377\pi$
$331$	−31.5144	−1.73219	−0.866094	+	0.499882i	$0.833377\pi$
$332$	6.86346	0.376681
$333$	0.184153	0.0100915
$334$	11.1276	0.608877
$335$	8.24350	0.450391
$336$	0.162033	0.00883962
$337$	2.34590	0.127790	0.0638948	−	0.997957i	$-0.479648\pi$
$337$	2.34590	0.127790	0.0638948	+	0.997957i	$0.479648\pi$
$338$	−21.6653	−1.17843
$339$	−17.8258	−0.968164
$340$	−3.29276	−0.178575
$341$	18.4358	0.998355
$342$	1.21768	0.0658444
$343$	1.36687	0.0738040
$344$	3.11395	0.167893
$345$	−9.52530	−0.512825
$346$	3.93491	0.211542
$347$	−16.3252	−0.876380	−0.438190	−	0.898882i	$-0.644380\pi$
$347$	−16.3252	−0.876380	−0.438190	+	0.898882i	$0.644380\pi$
$348$	−10.6670	−0.571811
$349$	33.6684	1.80223	0.901114	−	0.433581i	$-0.142750\pi$
$349$	33.6684	1.80223	0.901114	+	0.433581i	$0.142750\pi$
$350$	−0.425562	−0.0227472
$351$	−31.7299	−1.69361
$352$	4.28508	0.228395
$353$	−12.2203	−0.650419	−0.325209	−	0.945642i	$-0.605435\pi$
$353$	−12.2203	−0.650419	−0.325209	+	0.945642i	$0.605435\pi$
$354$	−14.6085	−0.776434
$355$	8.29460	0.440232
$356$	−5.35914	−0.284034
$357$	0.664738	0.0351817
$358$	9.10842	0.481395
$359$	−0.786973	−0.0415348	−0.0207674	−	0.999784i	$-0.506611\pi$
$359$	−0.786973	−0.0415348	−0.0207674	+	0.999784i	$0.506611\pi$
$360$	−0.200233	−0.0105532
$361$	4.82438	0.253915
$362$	9.30241	0.488924
$363$	−12.2095	−0.640833
$364$	0.575231	0.0301503
$365$	−0.0343757	−0.00179931
$366$	−4.49362	−0.234885
$367$	15.6163	0.815164	0.407582	−	0.913169i	$-0.366372\pi$
$367$	15.6163	0.815164	0.407582	+	0.913169i	$0.366372\pi$
$368$	−7.15578	−0.373021
$369$	2.28426	0.118914
$370$	−0.592477	−0.0308014
$371$	0.781739	0.0405859
$372$	7.13529	0.369947
$373$	2.46615	0.127692	0.0638461	−	0.997960i	$-0.479663\pi$
$373$	2.46615	0.127692	0.0638461	+	0.997960i	$0.479663\pi$
$374$	17.5795	0.909013
$375$	−12.4538	−0.643112
$376$	−7.49518	−0.386535
$377$	−37.8688	−1.95034
$378$	0.526522	0.0270813
$379$	−2.86090	−0.146954	−0.0734772	−	0.997297i	$-0.523410\pi$
$379$	−2.86090	−0.146954	−0.0734772	+	0.997297i	$0.523410\pi$
$380$	−3.91764	−0.200971
$381$	13.0352	0.667813
$382$	−13.3414	−0.682605
$383$	27.7756	1.41927	0.709634	−	0.704570i	$-0.248860\pi$
$383$	27.7756	1.41927	0.709634	+	0.704570i	$0.248860\pi$
$384$	1.65847	0.0846335
$385$	−0.336022	−0.0171253
$386$	−4.90829	−0.249826
$387$	0.776843	0.0394891
$388$	−4.07812	−0.207035
$389$	−19.4925	−0.988311	−0.494156	−	0.869374i	$-0.664523\pi$
$389$	−19.4925	−0.988311	−0.494156	+	0.869374i	$0.664523\pi$
$390$	7.83734	0.396859
$391$	−29.3565	−1.48462
$392$	6.99045	0.353071
$393$	33.6735	1.69860
$394$	−1.64123	−0.0826838
$395$	−3.18755	−0.160383
$396$	1.06901	0.0537196
$397$	−5.27141	−0.264564	−0.132282	−	0.991212i	$-0.542231\pi$
$397$	−5.27141	−0.264564	−0.132282	+	0.991212i	$0.542231\pi$
$398$	4.52532	0.226834
$399$	0.790887	0.0395939
$400$	−4.35579	−0.217790
$401$	17.5397	0.875889	0.437944	−	0.899002i	$-0.355707\pi$
$401$	17.5397	0.875889	0.437944	+	0.899002i	$0.355707\pi$
$402$	−17.0336	−0.849559
$403$	25.3309	1.26182
$404$	−5.49159	−0.273217
$405$	6.57299	0.326614
$406$	0.628390	0.0311865
$407$	3.16313	0.156791
$408$	6.80386	0.336841
$409$	−19.6791	−0.973067	−0.486533	−	0.873662i	$-0.661739\pi$
$409$	−19.6791	−0.973067	−0.486533	+	0.873662i	$0.661739\pi$
$410$	−7.34915	−0.362948
$411$	−19.4609	−0.959935
$412$	−17.9063	−0.882180
$413$	0.860584	0.0423466
$414$	−1.78517	−0.0877361
$415$	−5.50879	−0.270416
$416$	5.88772	0.288669
$417$	−5.42252	−0.265542
$418$	20.9156	1.02301
$419$	39.5113	1.93025	0.965126	−	0.261786i	$-0.0843116\pi$
$419$	39.5113	1.93025	0.965126	+	0.261786i	$0.0843116\pi$
$420$	−0.130052	−0.00634588
$421$	−6.83947	−0.333335	−0.166668	−	0.986013i	$-0.553301\pi$
$421$	−6.83947	−0.333335	−0.166668	+	0.986013i	$0.553301\pi$
$422$	0.227580	0.0110784
$423$	−1.86984	−0.0909146
$424$	8.00141	0.388583
$425$	−17.8696	−0.866802
$426$	−17.1392	−0.830396
$427$	0.264718	0.0128106
$428$	−11.2606	−0.544301
$429$	−41.8422	−2.02016
$430$	−2.49934	−0.120529
$431$	−15.8031	−0.761210	−0.380605	−	0.924738i	$-0.624284\pi$
$431$	−15.8031	−0.761210	−0.380605	+	0.924738i	$0.624284\pi$
$432$	5.38916	0.259286
$433$	−22.3223	−1.07274	−0.536370	−	0.843983i	$-0.680205\pi$
$433$	−22.3223	−1.07274	−0.536370	+	0.843983i	$0.680205\pi$
$434$	−0.420338	−0.0201769
$435$	8.56160	0.410498
$436$	−6.84529	−0.327830
$437$	−34.9275	−1.67081
$438$	0.0710307	0.00339398
$439$	−7.48827	−0.357396	−0.178698	−	0.983904i	$-0.557188\pi$
$439$	−7.48827	−0.357396	−0.178698	+	0.983904i	$0.557188\pi$
$440$	−3.43932	−0.163963
$441$	1.74392	0.0830439
$442$	24.1543	1.14890
$443$	−8.30280	−0.394478	−0.197239	−	0.980355i	$-0.563197\pi$
$443$	−8.30280	−0.394478	−0.197239	+	0.980355i	$0.563197\pi$
$444$	1.22424	0.0580998
$445$	4.30138	0.203905
$446$	−13.0588	−0.618353
$447$	−7.99901	−0.378340
$448$	−0.0977002	−0.00461590
$449$	22.7332	1.07285	0.536423	−	0.843949i	$-0.319775\pi$
$449$	22.7332	1.07285	0.536423	+	0.843949i	$0.319775\pi$
$450$	−1.08665	−0.0512250
$451$	39.2358	1.84754
$452$	10.7483	0.505558
$453$	−21.2704	−0.999371
$454$	−12.3681	−0.580462
$455$	−0.461696	−0.0216446
$456$	8.09504	0.379085
$457$	15.6497	0.732064	0.366032	−	0.930602i	$-0.380716\pi$
$457$	15.6497	0.732064	0.366032	+	0.930602i	$0.380716\pi$
$458$	27.7355	1.29599
$459$	22.1089	1.03196
$460$	5.74342	0.267788
$461$	−21.2303	−0.988792	−0.494396	−	0.869237i	$-0.664611\pi$
$461$	−21.2303	−0.988792	−0.494396	+	0.869237i	$0.664611\pi$
$462$	0.694324	0.0323029
$463$	26.7309	1.24229	0.621145	−	0.783696i	$-0.286668\pi$
$463$	26.7309	1.24229	0.621145	+	0.783696i	$0.286668\pi$
$464$	6.43182	0.298590
$465$	−5.72697	−0.265582
$466$	−16.0290	−0.742530
$467$	9.39671	0.434828	0.217414	−	0.976079i	$-0.430238\pi$
$467$	9.39671	0.434828	0.217414	+	0.976079i	$0.430238\pi$
$468$	1.46882	0.0678962
$469$	1.00345	0.0463348
$470$	6.01583	0.277490
$471$	1.31368	0.0605311
$472$	8.80842	0.405440
$473$	13.3435	0.613536
$474$	6.58646	0.302526
$475$	−21.2607	−0.975509
$476$	−0.400814	−0.0183713
$477$	1.99613	0.0913963
$478$	−19.1279	−0.874890
$479$	−10.0603	−0.459667	−0.229833	−	0.973230i	$-0.573818\pi$
$479$	−10.0603	−0.459667	−0.229833	+	0.973230i	$0.573818\pi$
$480$	−1.33113	−0.0607576
$481$	4.34616	0.198168
$482$	26.5054	1.20729
$483$	−1.15947	−0.0527578
$484$	7.36190	0.334632
$485$	3.27321	0.148629
$486$	2.58567	0.117288
$487$	−11.1698	−0.506150	−0.253075	−	0.967447i	$-0.581442\pi$
$487$	−11.1698	−0.506150	−0.253075	+	0.967447i	$0.581442\pi$
$488$	2.70949	0.122653
$489$	−18.6439	−0.843108
$490$	−5.61072	−0.253467
$491$	16.0088	0.722467	0.361233	−	0.932475i	$-0.382356\pi$
$491$	16.0088	0.722467	0.361233	+	0.932475i	$0.382356\pi$
$492$	15.1856	0.684619
$493$	26.3865	1.18839
$494$	28.7381	1.29299
$495$	−0.858012	−0.0385648
$496$	−4.30233	−0.193180
$497$	1.00967	0.0452897
$498$	11.3828	0.510078
$499$	7.49915	0.335708	0.167854	−	0.985812i	$-0.446316\pi$
$499$	7.49915	0.335708	0.167854	+	0.985812i	$0.446316\pi$
$500$	7.50920	0.335822
$501$	18.4549	0.824503
$502$	−10.3539	−0.462119
$503$	8.50531	0.379233	0.189616	−	0.981858i	$-0.439276\pi$
$503$	8.50531	0.379233	0.189616	+	0.981858i	$0.439276\pi$
$504$	−0.0243734	−0.00108568
$505$	4.40770	0.196140
$506$	−30.6631	−1.36314
$507$	−35.9312	−1.59576
$508$	−7.85976	−0.348721
$509$	−14.9416	−0.662273	−0.331137	−	0.943583i	$-0.607432\pi$
$509$	−14.9416	−0.662273	−0.331137	+	0.943583i	$0.607432\pi$
$510$	−5.46096	−0.241815
$511$	−0.00418440	−0.000185107 0
$512$	−1.00000	−0.0441942
$513$	26.3046	1.16138
$514$	−1.73524	−0.0765383
$515$	14.3721	0.633309
$516$	5.16440	0.227350
$517$	−32.1174	−1.41252
$518$	−0.0721197	−0.00316876
$519$	6.52594	0.286457
$520$	−4.72564	−0.207233
$521$	18.1638	0.795768	0.397884	−	0.917436i	$-0.369745\pi$
$521$	18.1638	0.795768	0.397884	+	0.917436i	$0.369745\pi$
$522$	1.60456	0.0702296
$523$	−3.89656	−0.170385	−0.0851924	−	0.996365i	$-0.527150\pi$
$523$	−3.89656	−0.170385	−0.0851924	+	0.996365i	$0.527150\pi$
$524$	−20.3039	−0.886981
$525$	−0.705782	−0.0308028
$526$	−8.81690	−0.384435
$527$	−17.6502	−0.768857
$528$	7.10668	0.309279
$529$	28.2052	1.22631
$530$	−6.42214	−0.278960
$531$	2.19745	0.0953613
$532$	−0.476877	−0.0206752
$533$	53.9102	2.33511
$534$	−8.88798	−0.384620
$535$	9.03805	0.390749
$536$	10.2707	0.443625
$537$	15.1061	0.651875
$538$	−3.26925	−0.140947
$539$	29.9546	1.29024
$540$	−4.32548	−0.186139
$541$	29.6054	1.27283	0.636417	−	0.771345i	$-0.280416\pi$
$541$	29.6054	1.27283	0.636417	+	0.771345i	$0.280416\pi$
$542$	23.0568	0.990375
$543$	15.4278	0.662070
$544$	−4.10249	−0.175893
$545$	5.49421	0.235346
$546$	0.954005	0.0408276
$547$	27.6682	1.18301	0.591503	−	0.806303i	$-0.298535\pi$
$547$	27.6682	1.18301	0.591503	+	0.806303i	$0.298535\pi$
$548$	11.7342	0.501262
$549$	0.675942	0.0288485
$550$	−18.6649	−0.795874
$551$	31.3939	1.33742
$552$	−11.8677	−0.505121
$553$	−0.388007	−0.0164997
$554$	20.6326	0.876595
$555$	−0.982607	−0.0417093
$556$	3.26959	0.138661
$557$	30.9459	1.31122	0.655610	−	0.755100i	$-0.272412\pi$
$557$	30.9459	1.31122	0.655610	+	0.755100i	$0.272412\pi$
$558$	−1.07331	−0.0454368
$559$	18.3341	0.775449
$560$	0.0784167	0.00331371
$561$	29.1551	1.23093
$562$	28.5470	1.20418
$563$	28.5625	1.20377	0.601883	−	0.798584i	$-0.294417\pi$
$563$	28.5625	1.20377	0.601883	+	0.798584i	$0.294417\pi$
$564$	−12.4305	−0.523420
$565$	−8.62688	−0.362936
$566$	−12.6676	−0.532460
$567$	0.800101	0.0336011
$568$	10.3343	0.433619
$569$	41.8278	1.75351	0.876757	−	0.480934i	$-0.159702\pi$
$569$	41.8278	1.75351	0.876757	+	0.480934i	$0.159702\pi$
$570$	−6.49729	−0.272142
$571$	−30.5524	−1.27858	−0.639289	−	0.768967i	$-0.720771\pi$
$571$	−30.5524	−1.27858	−0.639289	+	0.768967i	$0.720771\pi$
$572$	25.2293	1.05489
$573$	−22.1263	−0.924340
$574$	−0.894579	−0.0373390
$575$	31.1691	1.29984
$576$	−0.249472	−0.0103947
$577$	−15.9307	−0.663203	−0.331602	−	0.943419i	$-0.607589\pi$
$577$	−15.9307	−0.663203	−0.331602	+	0.943419i	$0.607589\pi$
$578$	0.169596	0.00705426
$579$	−8.14026	−0.338298
$580$	−5.16235	−0.214355
$581$	−0.670561	−0.0278196
$582$	−6.76345	−0.280354
$583$	34.2867	1.42001
$584$	−0.0428290	−0.00177228
$585$	−1.17891	−0.0487421
$586$	16.3219	0.674250
$587$	6.09090	0.251398	0.125699	−	0.992068i	$-0.459883\pi$
$587$	6.09090	0.251398	0.125699	+	0.992068i	$0.459883\pi$
$588$	11.5935	0.478107
$589$	−20.9998	−0.865280
$590$	−7.06987	−0.291062
$591$	−2.72193	−0.111965
$592$	−0.738173	−0.0303387
$593$	−42.0285	−1.72590	−0.862952	−	0.505285i	$-0.831387\pi$
$593$	−42.0285	−1.72590	−0.862952	+	0.505285i	$0.831387\pi$
$594$	23.0930	0.947516
$595$	0.321704	0.0131886
$596$	4.82312	0.197563
$597$	7.50512	0.307164
$598$	−42.1312	−1.72287
$599$	15.7289	0.642666	0.321333	−	0.946966i	$-0.395869\pi$
$599$	15.7289	0.642666	0.321333	+	0.946966i	$0.395869\pi$
$600$	−7.22396	−0.294917
$601$	−41.2478	−1.68253	−0.841266	−	0.540621i	$-0.818189\pi$
$601$	−41.2478	−1.68253	−0.841266	+	0.540621i	$0.818189\pi$
$602$	−0.304234	−0.0123996
$603$	2.56224	0.104342
$604$	12.8253	0.521854
$605$	−5.90885	−0.240229
$606$	−9.10765	−0.369973
$607$	−5.31367	−0.215675	−0.107838	−	0.994169i	$-0.534393\pi$
$607$	−5.31367	−0.215675	−0.107838	+	0.994169i	$0.534393\pi$
$608$	−4.88102	−0.197952
$609$	1.04217	0.0422307
$610$	−2.17471	−0.0880515
$611$	−44.1295	−1.78529
$612$	−1.02345	−0.0413707
$613$	−7.97335	−0.322041	−0.161020	−	0.986951i	$-0.551478\pi$
$613$	−7.97335	−0.322041	−0.161020	+	0.986951i	$0.551478\pi$
$614$	−11.0823	−0.447246
$615$	−12.1883	−0.491482
$616$	−0.418653	−0.0168680
$617$	−4.90797	−0.197587	−0.0987937	−	0.995108i	$-0.531498\pi$
$617$	−4.90797	−0.197587	−0.0987937	+	0.995108i	$0.531498\pi$
$618$	−29.6971	−1.19459
$619$	−21.0560	−0.846311	−0.423155	−	0.906057i	$-0.639078\pi$
$619$	−21.0560	−0.846311	−0.423155	+	0.906057i	$0.639078\pi$
$620$	3.45316	0.138682
$621$	−38.5636	−1.54751
$622$	−29.0299	−1.16399
$623$	0.523589	0.0209771
$624$	9.76462	0.390898
$625$	15.7519	0.630075
$626$	6.07292	0.242723
$627$	34.6879	1.38530
$628$	−0.792102	−0.0316083
$629$	−3.02835	−0.120748
$630$	0.0195628	0.000779399 0
$631$	30.3314	1.20747	0.603737	−	0.797184i	$-0.293678\pi$
$631$	30.3314	1.20747	0.603737	+	0.797184i	$0.293678\pi$
$632$	−3.97141	−0.157974
$633$	0.377434	0.0150017
$634$	9.81771	0.389911
$635$	6.30845	0.250343
$636$	13.2701	0.526194
$637$	41.1578	1.63073
$638$	27.5608	1.09114
$639$	2.57812	0.101989
$640$	0.802626	0.0317266
$641$	45.8922	1.81263	0.906317	−	0.422599i	$-0.138882\pi$
$641$	45.8922	1.81263	0.906317	+	0.422599i	$0.138882\pi$
$642$	−18.6754	−0.737058
$643$	−11.0864	−0.437204	−0.218602	−	0.975814i	$-0.570150\pi$
$643$	−11.0864	−0.437204	−0.218602	+	0.975814i	$0.570150\pi$
$644$	0.699121	0.0275492
$645$	−4.14508	−0.163212
$646$	−20.0243	−0.787847
$647$	33.8818	1.33203	0.666015	−	0.745938i	$-0.267999\pi$
$647$	33.8818	1.33203	0.666015	+	0.745938i	$0.267999\pi$
$648$	8.18935	0.321708
$649$	37.7448	1.48161
$650$	−25.6457	−1.00591
$651$	−0.697119	−0.0273222
$652$	11.2416	0.440256
$653$	7.10147	0.277902	0.138951	−	0.990299i	$-0.455627\pi$
$653$	7.10147	0.277902	0.138951	+	0.990299i	$0.455627\pi$
$654$	−11.3527	−0.443927
$655$	16.2965	0.636755
$656$	−9.15637	−0.357496
$657$	−0.0106846	−0.000416847 0
$658$	0.732281	0.0285473
$659$	12.3793	0.482228	0.241114	−	0.970497i	$-0.422487\pi$
$659$	12.3793	0.482228	0.241114	+	0.970497i	$0.422487\pi$
$660$	−5.70401	−0.222028
$661$	36.6294	1.42472	0.712359	−	0.701815i	$-0.247627\pi$
$661$	36.6294	1.42472	0.712359	+	0.701815i	$0.247627\pi$
$662$	31.5144	1.22484
$663$	40.0592	1.55577
$664$	−6.86346	−0.266354
$665$	0.382754	0.0148426
$666$	−0.184153	−0.00713580
$667$	−46.0247	−1.78208
$668$	−11.1276	−0.430541
$669$	−21.6577	−0.837334
$670$	−8.24350	−0.318474
$671$	11.6104	0.448214
$672$	−0.162033	−0.00625056
$673$	43.8692	1.69103	0.845517	−	0.533948i	$-0.179292\pi$
$673$	43.8692	1.69103	0.845517	+	0.533948i	$0.179292\pi$
$674$	−2.34590	−0.0903609
$675$	−23.4740	−0.903516
$676$	21.6653	0.833279
$677$	−0.191652	−0.00736577	−0.00368288	−	0.999993i	$-0.501172\pi$
$677$	−0.191652	−0.00736577	−0.00368288	+	0.999993i	$0.501172\pi$
$678$	17.8258	0.684595
$679$	0.398433	0.0152905
$680$	3.29276	0.126272
$681$	−20.5121	−0.786025
$682$	−18.4358	−0.705944
$683$	23.0823	0.883221	0.441611	−	0.897207i	$-0.354407\pi$
$683$	23.0823	0.883221	0.441611	+	0.897207i	$0.354407\pi$
$684$	−1.21768	−0.0465591
$685$	−9.41820	−0.359851
$686$	−1.36687	−0.0521873
$687$	45.9985	1.75495
$688$	−3.11395	−0.118718
$689$	47.1101	1.79475
$690$	9.52530	0.362622
$691$	−14.5510	−0.553544	−0.276772	−	0.960936i	$-0.589265\pi$
$691$	−14.5510	−0.553544	−0.276772	+	0.960936i	$0.589265\pi$
$692$	−3.93491	−0.149583
$693$	−0.104442	−0.00396743
$694$	16.3252	0.619694
$695$	−2.62426	−0.0995438
$696$	10.6670	0.404331
$697$	−37.5639	−1.42283
$698$	−33.6684	−1.27437
$699$	−26.5837	−1.00549
$700$	0.425562	0.0160847
$701$	−4.85468	−0.183359	−0.0916794	−	0.995789i	$-0.529224\pi$
$701$	−4.85468	−0.183359	−0.0916794	+	0.995789i	$0.529224\pi$
$702$	31.7299	1.19757
$703$	−3.60304	−0.135891
$704$	−4.28508	−0.161500
$705$	9.97708	0.375759
$706$	12.2203	0.459915
$707$	0.536530	0.0201783
$708$	14.6085	0.549022
$709$	51.6131	1.93837	0.969186	−	0.246330i	$-0.0792248\pi$
$709$	51.6131	1.93837	0.969186	+	0.246330i	$0.0792248\pi$
$710$	−8.29460	−0.311291
$711$	−0.990754	−0.0371562
$712$	5.35914	0.200842
$713$	30.7865	1.15296
$714$	−0.664738	−0.0248772
$715$	−20.2497	−0.757297
$716$	−9.10842	−0.340398
$717$	−31.7231	−1.18472
$718$	0.786973	0.0293696
$719$	22.4510	0.837280	0.418640	−	0.908152i	$-0.362507\pi$
$719$	22.4510	0.837280	0.418640	+	0.908152i	$0.362507\pi$
$720$	0.200233	0.00746223
$721$	1.74945	0.0651529
$722$	−4.82438	−0.179545
$723$	43.9585	1.63483
$724$	−9.30241	−0.345721
$725$	−28.0157	−1.04048
$726$	12.2095	0.453137
$727$	29.2295	1.08406	0.542031	−	0.840359i	$-0.317656\pi$
$727$	29.2295	1.08406	0.542031	+	0.840359i	$0.317656\pi$
$728$	−0.575231	−0.0213195
$729$	28.8563	1.06875
$730$	0.0343757	0.00127230
$731$	−12.7749	−0.472498
$732$	4.49362	0.166089
$733$	49.8269	1.84040	0.920200	−	0.391450i	$-0.128026\pi$
$733$	49.8269	1.84040	0.920200	+	0.391450i	$0.128026\pi$
$734$	−15.6163	−0.576408
$735$	−9.30522	−0.343228
$736$	7.15578	0.263766
$737$	44.0106	1.62115
$738$	−2.28426	−0.0840846
$739$	−53.0377	−1.95102	−0.975511	−	0.219949i	$-0.929411\pi$
$739$	−53.0377	−1.95102	−0.975511	+	0.219949i	$0.929411\pi$
$740$	0.592477	0.0217799
$741$	47.6613	1.75088
$742$	−0.781739	−0.0286986
$743$	−14.1535	−0.519240	−0.259620	−	0.965711i	$-0.583597\pi$
$743$	−14.1535	−0.519240	−0.259620	+	0.965711i	$0.583597\pi$
$744$	−7.13529	−0.261592
$745$	−3.87116	−0.141828
$746$	−2.46615	−0.0902921
$747$	−1.71224	−0.0626475
$748$	−17.5795	−0.642769
$749$	1.10016	0.0401990
$750$	12.4538	0.454749
$751$	−38.0657	−1.38904	−0.694519	−	0.719474i	$-0.744383\pi$
$751$	−38.0657	−1.38904	−0.694519	+	0.719474i	$0.744383\pi$
$752$	7.49518	0.273321
$753$	−17.1717	−0.625772
$754$	37.8688	1.37910
$755$	−10.2939	−0.374634
$756$	−0.526522	−0.0191494
$757$	0.760696	0.0276480	0.0138240	−	0.999904i	$-0.495600\pi$
$757$	0.760696	0.0276480	0.0138240	+	0.999904i	$0.495600\pi$
$758$	2.86090	0.103912
$759$	−50.8539	−1.84588
$760$	3.91764	0.142108
$761$	47.3777	1.71744	0.858720	−	0.512446i	$-0.171260\pi$
$761$	47.3777	1.71744	0.858720	+	0.512446i	$0.171260\pi$
$762$	−13.0352	−0.472215
$763$	0.668786	0.0242117
$764$	13.3414	0.482674
$765$	0.821452	0.0296996
$766$	−27.7756	−1.00357
$767$	51.8615	1.87261
$768$	−1.65847	−0.0598449
$769$	−31.3581	−1.13080	−0.565401	−	0.824816i	$-0.691279\pi$
$769$	−31.3581	−1.13080	−0.565401	+	0.824816i	$0.691279\pi$
$770$	0.336022	0.0121094
$771$	−2.87785	−0.103643
$772$	4.90829	0.176653
$773$	12.3595	0.444540	0.222270	−	0.974985i	$-0.428653\pi$
$773$	12.3595	0.444540	0.222270	+	0.974985i	$0.428653\pi$
$774$	−0.776843	−0.0279230
$775$	18.7400	0.673162
$776$	4.07812	0.146396
$777$	−0.119608	−0.00429093
$778$	19.4925	0.698841
$779$	−44.6925	−1.60127
$780$	−7.83734	−0.280622
$781$	44.2834	1.58458
$782$	29.3565	1.04979
$783$	34.6621	1.23872
$784$	−6.99045	−0.249659
$785$	0.635762	0.0226913
$786$	−33.6735	−1.20109
$787$	−52.5595	−1.87354	−0.936771	−	0.349942i	$-0.886201\pi$
$787$	−52.5595	−1.87354	−0.936771	+	0.349942i	$0.886201\pi$
$788$	1.64123	0.0584663
$789$	−14.6226	−0.520577
$790$	3.18755	0.113408
$791$	−1.05011	−0.0373377
$792$	−1.06901	−0.0379855
$793$	15.9527	0.566499
$794$	5.27141	0.187075
$795$	−10.6509	−0.377750
$796$	−4.52532	−0.160396
$797$	−22.6091	−0.800856	−0.400428	−	0.916328i	$-0.631139\pi$
$797$	−22.6091	−0.800856	−0.400428	+	0.916328i	$0.631139\pi$
$798$	−0.790887	−0.0279971
$799$	30.7489	1.08782
$800$	4.35579	0.154000
$801$	1.33695	0.0472389
$802$	−17.5397	−0.619347
$803$	−0.183526	−0.00647648
$804$	17.0336	0.600729
$805$	−0.561133	−0.0197773
$806$	−25.3309	−0.892243
$807$	−5.42196	−0.190862
$808$	5.49159	0.193194
$809$	−39.0578	−1.37320	−0.686599	−	0.727036i	$-0.740897\pi$
$809$	−39.0578	−1.37320	−0.686599	+	0.727036i	$0.740897\pi$
$810$	−6.57299	−0.230951
$811$	21.7603	0.764109	0.382054	−	0.924140i	$-0.375217\pi$
$811$	21.7603	0.764109	0.382054	+	0.924140i	$0.375217\pi$
$812$	−0.628390	−0.0220522
$813$	38.2391	1.34110
$814$	−3.16313	−0.110868
$815$	−9.02283	−0.316056
$816$	−6.80386	−0.238183
$817$	−15.1993	−0.531755
$818$	19.6791	0.688062
$819$	−0.143504	−0.00501444
$820$	7.34915	0.256643
$821$	−45.5228	−1.58876	−0.794379	−	0.607423i	$-0.792203\pi$
$821$	−45.5228	−1.58876	−0.794379	+	0.607423i	$0.792203\pi$
$822$	19.4609	0.678777
$823$	−17.6042	−0.613643	−0.306822	−	0.951767i	$-0.599266\pi$
$823$	−17.6042	−0.613643	−0.306822	+	0.951767i	$0.599266\pi$
$824$	17.9063	0.623795
$825$	−30.9552	−1.07772
$826$	−0.860584	−0.0299436
$827$	50.3123	1.74953	0.874765	−	0.484547i	$-0.161015\pi$
$827$	50.3123	1.74953	0.874765	+	0.484547i	$0.161015\pi$
$828$	1.78517	0.0620388
$829$	47.1262	1.63676	0.818381	−	0.574676i	$-0.194872\pi$
$829$	47.1262	1.63676	0.818381	+	0.574676i	$0.194872\pi$
$830$	5.50879	0.191213
$831$	34.2186	1.18703
$832$	−5.88772	−0.204120
$833$	−28.6783	−0.993643
$834$	5.42252	0.187767
$835$	8.93133	0.309082
$836$	−20.9156	−0.723380
$837$	−23.1859	−0.801423
$838$	−39.5113	−1.36489
$839$	9.19789	0.317546	0.158773	−	0.987315i	$-0.449246\pi$
$839$	9.19789	0.317546	0.158773	+	0.987315i	$0.449246\pi$
$840$	0.130052	0.00448722
$841$	12.3683	0.426493
$842$	6.83947	0.235704
$843$	47.3444	1.63063
$844$	−0.227580	−0.00783361
$845$	−17.3891	−0.598203
$846$	1.86984	0.0642863
$847$	−0.719259	−0.0247140
$848$	−8.00141	−0.274770
$849$	−21.0089	−0.721023
$850$	17.8696	0.612921
$851$	5.28221	0.181072
$852$	17.1392	0.587179
$853$	−21.1509	−0.724193	−0.362097	−	0.932141i	$-0.617939\pi$
$853$	−21.1509	−0.724193	−0.362097	+	0.932141i	$0.617939\pi$
$854$	−0.264718	−0.00905847
$855$	0.977340	0.0334243
$856$	11.2606	0.384879
$857$	17.7553	0.606511	0.303255	−	0.952909i	$-0.401926\pi$
$857$	17.7553	0.606511	0.303255	+	0.952909i	$0.401926\pi$
$858$	41.8422	1.42847
$859$	−47.0523	−1.60540	−0.802702	−	0.596381i	$-0.796605\pi$
$859$	−47.0523	−1.60540	−0.802702	+	0.596381i	$0.796605\pi$
$860$	2.49934	0.0852268
$861$	−1.48363	−0.0505621
$862$	15.8031	0.538257
$863$	−4.78579	−0.162910	−0.0814551	−	0.996677i	$-0.525957\pi$
$863$	−4.78579	−0.162910	−0.0814551	+	0.996677i	$0.525957\pi$
$864$	−5.38916	−0.183343
$865$	3.15826	0.107384
$866$	22.3223	0.758542
$867$	0.281270	0.00955243
$868$	0.420338	0.0142672
$869$	−17.0178	−0.577289
$870$	−8.56160	−0.290266
$871$	60.4708	2.04897
$872$	6.84529	0.231811
$873$	1.01738	0.0344330
$874$	34.9275	1.18144
$875$	−0.733651	−0.0248019
$876$	−0.0710307	−0.00239991
$877$	−20.8922	−0.705480	−0.352740	−	0.935721i	$-0.614750\pi$
$877$	−20.8922	−0.705480	−0.352740	+	0.935721i	$0.614750\pi$
$878$	7.48827	0.252717
$879$	27.0693	0.913026
$880$	3.43932	0.115939
$881$	−47.5976	−1.60360	−0.801802	−	0.597590i	$-0.796125\pi$
$881$	−47.5976	−1.60360	−0.801802	+	0.597590i	$0.796125\pi$
$882$	−1.74392	−0.0587209
$883$	−35.4578	−1.19325	−0.596625	−	0.802520i	$-0.703492\pi$
$883$	−35.4578	−1.19325	−0.596625	+	0.802520i	$0.703492\pi$
$884$	−24.1543	−0.812397
$885$	−11.7252	−0.394138
$886$	8.30280	0.278938
$887$	37.9632	1.27468	0.637340	−	0.770582i	$-0.280035\pi$
$887$	37.9632	1.27468	0.637340	+	0.770582i	$0.280035\pi$
$888$	−1.22424	−0.0410828
$889$	0.767900	0.0257546
$890$	−4.30138	−0.144183
$891$	35.0920	1.17563
$892$	13.0588	0.437241
$893$	36.5842	1.22424
$894$	7.99901	0.267527
$895$	7.31066	0.244368
$896$	0.0977002	0.00326393
$897$	−69.8735	−2.33301
$898$	−22.7332	−0.758616
$899$	−27.6718	−0.922906
$900$	1.08665	0.0362216
$901$	−32.8257	−1.09358
$902$	−39.2358	−1.30641
$903$	−0.504563	−0.0167908
$904$	−10.7483	−0.357484
$905$	7.46636	0.248190
$906$	21.2704	0.706662
$907$	21.0148	0.697786	0.348893	−	0.937163i	$-0.386558\pi$
$907$	21.0148	0.697786	0.348893	+	0.937163i	$0.386558\pi$
$908$	12.3681	0.410449
$909$	1.37000	0.0454399
$910$	0.461696	0.0153051
$911$	−31.4707	−1.04267	−0.521335	−	0.853352i	$-0.674566\pi$
$911$	−31.4707	−1.04267	−0.521335	+	0.853352i	$0.674566\pi$
$912$	−8.09504	−0.268053
$913$	−29.4104	−0.973344
$914$	−15.6497	−0.517647
$915$	−3.60670	−0.119234
$916$	−27.7355	−0.916406
$917$	1.98370	0.0655074
$918$	−22.1089	−0.729704
$919$	32.3594	1.06744	0.533719	−	0.845662i	$-0.320794\pi$
$919$	32.3594	1.06744	0.533719	+	0.845662i	$0.320794\pi$
$920$	−5.74342	−0.189355
$921$	−18.3797	−0.605632
$922$	21.2303	0.699181
$923$	60.8456	2.00276
$924$	−0.694324	−0.0228416
$925$	3.21533	0.105719
$926$	−26.7309	−0.878432
$927$	4.46711	0.146719
$928$	−6.43182	−0.211135
$929$	20.2891	0.665664	0.332832	−	0.942986i	$-0.391996\pi$
$929$	20.2891	0.665664	0.332832	+	0.942986i	$0.391996\pi$
$930$	5.72697	0.187795
$931$	−34.1206	−1.11826
$932$	16.0290	0.525048
$933$	−48.1453	−1.57621
$934$	−9.39671	−0.307470
$935$	14.1098	0.461438
$936$	−1.46882	−0.0480099
$937$	−25.6601	−0.838280	−0.419140	−	0.907922i	$-0.637668\pi$
$937$	−25.6601	−0.838280	−0.419140	+	0.907922i	$0.637668\pi$
$938$	−1.00345	−0.0327636
$939$	10.0718	0.328680
$940$	−6.01583	−0.196215
$941$	−25.2864	−0.824313	−0.412157	−	0.911113i	$-0.635224\pi$
$941$	−25.2864	−0.824313	−0.412157	+	0.911113i	$0.635224\pi$
$942$	−1.31368	−0.0428019
$943$	65.5210	2.13366
$944$	−8.80842	−0.286690
$945$	0.422600	0.0137472
$946$	−13.3435	−0.433835
$947$	−9.96278	−0.323747	−0.161873	−	0.986812i	$-0.551754\pi$
$947$	−9.96278	−0.323747	−0.161873	+	0.986812i	$0.551754\pi$
$948$	−6.58646	−0.213918
$949$	−0.252165	−0.00818563
$950$	21.2607	0.689789
$951$	16.2824	0.527993
$952$	0.400814	0.0129904
$953$	−45.2388	−1.46543	−0.732715	−	0.680536i	$-0.761747\pi$
$953$	−45.2388	−1.46543	−0.732715	+	0.680536i	$0.761747\pi$
$954$	−1.99613	−0.0646270
$955$	−10.7081	−0.346508
$956$	19.1279	0.618640
$957$	45.7089	1.47756
$958$	10.0603	0.325033
$959$	−1.14644	−0.0370204
$960$	1.33113	0.0429621
$961$	−12.4900	−0.402903
$962$	−4.34616	−0.140126
$963$	2.80920	0.0905252
$964$	−26.5054	−0.853683
$965$	−3.93952	−0.126818
$966$	1.15947	0.0373054
$967$	45.5246	1.46397	0.731987	−	0.681319i	$-0.238593\pi$
$967$	45.5246	1.46397	0.731987	+	0.681319i	$0.238593\pi$
$968$	−7.36190	−0.236620
$969$	−33.2098	−1.06685
$970$	−3.27321	−0.105096
$971$	−12.0528	−0.386792	−0.193396	−	0.981121i	$-0.561950\pi$
$971$	−12.0528	−0.386792	−0.193396	+	0.981121i	$0.561950\pi$
$972$	−2.58567	−0.0829353
$973$	−0.319440	−0.0102408
$974$	11.1698	0.357902
$975$	−42.5326	−1.36213
$976$	−2.70949	−0.0867288
$977$	40.3367	1.29049	0.645243	−	0.763977i	$-0.276756\pi$
$977$	40.3367	1.29049	0.645243	+	0.763977i	$0.276756\pi$
$978$	18.6439	0.596167
$979$	22.9643	0.733943
$980$	5.61072	0.179228
$981$	1.70771	0.0545229
$982$	−16.0088	−0.510861
$983$	0.920216	0.0293503	0.0146752	−	0.999892i	$-0.495329\pi$
$983$	0.920216	0.0293503	0.0146752	+	0.999892i	$0.495329\pi$
$984$	−15.1856	−0.484099
$985$	−1.31729	−0.0419724
$986$	−26.3865	−0.840316
$987$	1.21447	0.0386569
$988$	−28.7381	−0.914281
$989$	22.2828	0.708550
$990$	0.858012	0.0272694
$991$	28.7618	0.913650	0.456825	−	0.889557i	$-0.348987\pi$
$991$	28.7618	0.913650	0.456825	+	0.889557i	$0.348987\pi$
$992$	4.30233	0.136599
$993$	52.2657	1.65860
$994$	−1.00967	−0.0320246
$995$	3.63214	0.115147
$996$	−11.3828	−0.360679
$997$	−46.0569	−1.45864	−0.729318	−	0.684175i	$-0.760163\pi$
$997$	−46.0569	−1.45864	−0.729318	+	0.684175i	$0.760163\pi$
$998$	−7.49915	−0.237381
$999$	−3.97813	−0.125863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.19	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.19	✓	77	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.65847 q^{3} +1.00000 q^{4} -0.802626 q^{5} +1.65847 q^{6} -0.0977002 q^{7} -1.00000 q^{8} -0.249472 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.65847 q^{3} +1.00000 q^{4} -0.802626 q^{5} +1.65847 q^{6} -0.0977002 q^{7} -1.00000 q^{8} -0.249472 q^{9} +0.802626 q^{10} -4.28508 q^{11} -1.65847 q^{12} -5.88772 q^{13} +0.0977002 q^{14} +1.33113 q^{15} +1.00000 q^{16} +4.10249 q^{17} +0.249472 q^{18} +4.88102 q^{19} -0.802626 q^{20} +0.162033 q^{21} +4.28508 q^{22} -7.15578 q^{23} +1.65847 q^{24} -4.35579 q^{25} +5.88772 q^{26} +5.38916 q^{27} -0.0977002 q^{28} +6.43182 q^{29} -1.33113 q^{30} -4.30233 q^{31} -1.00000 q^{32} +7.10668 q^{33} -4.10249 q^{34} +0.0784167 q^{35} -0.249472 q^{36} -0.738173 q^{37} -4.88102 q^{38} +9.76462 q^{39} +0.802626 q^{40} -9.15637 q^{41} -0.162033 q^{42} -3.11395 q^{43} -4.28508 q^{44} +0.200233 q^{45} +7.15578 q^{46} +7.49518 q^{47} -1.65847 q^{48} -6.99045 q^{49} +4.35579 q^{50} -6.80386 q^{51} -5.88772 q^{52} -8.00141 q^{53} -5.38916 q^{54} +3.43932 q^{55} +0.0977002 q^{56} -8.09504 q^{57} -6.43182 q^{58} -8.80842 q^{59} +1.33113 q^{60} -2.70949 q^{61} +4.30233 q^{62} +0.0243734 q^{63} +1.00000 q^{64} +4.72564 q^{65} -7.10668 q^{66} -10.2707 q^{67} +4.10249 q^{68} +11.8677 q^{69} -0.0784167 q^{70} -10.3343 q^{71} +0.249472 q^{72} +0.0428290 q^{73} +0.738173 q^{74} +7.22396 q^{75} +4.88102 q^{76} +0.418653 q^{77} -9.76462 q^{78} +3.97141 q^{79} -0.802626 q^{80} -8.18935 q^{81} +9.15637 q^{82} +6.86346 q^{83} +0.162033 q^{84} -3.29276 q^{85} +3.11395 q^{86} -10.6670 q^{87} +4.28508 q^{88} -5.35914 q^{89} -0.200233 q^{90} +0.575231 q^{91} -7.15578 q^{92} +7.13529 q^{93} -7.49518 q^{94} -3.91764 q^{95} +1.65847 q^{96} -4.07812 q^{97} +6.99045 q^{98} +1.06901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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