LMFDB - Embedded newform 8002.2.a.e.1.16

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			1.16
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.85869 q^{3} +1.00000 q^{4} -0.435894 q^{5} +1.85869 q^{6} +1.03466 q^{7} -1.00000 q^{8} +0.454718 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.85869 q^{3} +1.00000 q^{4} -0.435894 q^{5} +1.85869 q^{6} +1.03466 q^{7} -1.00000 q^{8} +0.454718 q^{9} +0.435894 q^{10} +0.520763 q^{11} -1.85869 q^{12} +4.30262 q^{13} -1.03466 q^{14} +0.810191 q^{15} +1.00000 q^{16} -6.53111 q^{17} -0.454718 q^{18} -4.45215 q^{19} -0.435894 q^{20} -1.92311 q^{21} -0.520763 q^{22} -3.22721 q^{23} +1.85869 q^{24} -4.81000 q^{25} -4.30262 q^{26} +4.73088 q^{27} +1.03466 q^{28} +3.89653 q^{29} -0.810191 q^{30} -7.29886 q^{31} -1.00000 q^{32} -0.967935 q^{33} +6.53111 q^{34} -0.451002 q^{35} +0.454718 q^{36} -4.64766 q^{37} +4.45215 q^{38} -7.99722 q^{39} +0.435894 q^{40} -1.79580 q^{41} +1.92311 q^{42} -5.48331 q^{43} +0.520763 q^{44} -0.198209 q^{45} +3.22721 q^{46} +7.14256 q^{47} -1.85869 q^{48} -5.92948 q^{49} +4.81000 q^{50} +12.1393 q^{51} +4.30262 q^{52} +5.49063 q^{53} -4.73088 q^{54} -0.226997 q^{55} -1.03466 q^{56} +8.27516 q^{57} -3.89653 q^{58} -2.34835 q^{59} +0.810191 q^{60} +12.0427 q^{61} +7.29886 q^{62} +0.470478 q^{63} +1.00000 q^{64} -1.87549 q^{65} +0.967935 q^{66} +7.04743 q^{67} -6.53111 q^{68} +5.99836 q^{69} +0.451002 q^{70} -1.51568 q^{71} -0.454718 q^{72} +1.54224 q^{73} +4.64766 q^{74} +8.94028 q^{75} -4.45215 q^{76} +0.538813 q^{77} +7.99722 q^{78} +8.77247 q^{79} -0.435894 q^{80} -10.1574 q^{81} +1.79580 q^{82} +0.586340 q^{83} -1.92311 q^{84} +2.84687 q^{85} +5.48331 q^{86} -7.24243 q^{87} -0.520763 q^{88} -3.03939 q^{89} +0.198209 q^{90} +4.45175 q^{91} -3.22721 q^{92} +13.5663 q^{93} -7.14256 q^{94} +1.94067 q^{95} +1.85869 q^{96} -17.0736 q^{97} +5.92948 q^{98} +0.236800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.85869	−1.07311	−0.536557	−	0.843864i	$-0.680275\pi$
$3$	−1.85869	−1.07311	−0.536557	+	0.843864i	$0.680275\pi$
$4$	1.00000	0.500000
$5$	−0.435894	−0.194938	−0.0974689	−	0.995239i	$-0.531075\pi$
$5$	−0.435894	−0.194938	−0.0974689	+	0.995239i	$0.531075\pi$
$6$	1.85869	0.758806
$7$	1.03466	0.391065	0.195532	−	0.980697i	$-0.437357\pi$
$7$	1.03466	0.391065	0.195532	+	0.980697i	$0.437357\pi$
$8$	−1.00000	−0.353553
$9$	0.454718	0.151573
$10$	0.435894	0.137842
$11$	0.520763	0.157016	0.0785080	−	0.996913i	$-0.474984\pi$
$11$	0.520763	0.157016	0.0785080	+	0.996913i	$0.474984\pi$
$12$	−1.85869	−0.536557
$13$	4.30262	1.19333	0.596666	−	0.802490i	$-0.296492\pi$
$13$	4.30262	1.19333	0.596666	+	0.802490i	$0.296492\pi$
$14$	−1.03466	−0.276525
$15$	0.810191	0.209190
$16$	1.00000	0.250000
$17$	−6.53111	−1.58403	−0.792013	−	0.610504i	$-0.790967\pi$
$17$	−6.53111	−1.58403	−0.792013	+	0.610504i	$0.790967\pi$
$18$	−0.454718	−0.107178
$19$	−4.45215	−1.02139	−0.510697	−	0.859761i	$-0.670613\pi$
$19$	−4.45215	−1.02139	−0.510697	+	0.859761i	$0.670613\pi$
$20$	−0.435894	−0.0974689
$21$	−1.92311	−0.419657
$22$	−0.520763	−0.111027
$23$	−3.22721	−0.672919	−0.336459	−	0.941698i	$-0.609229\pi$
$23$	−3.22721	−0.672919	−0.336459	+	0.941698i	$0.609229\pi$
$24$	1.85869	0.379403
$25$	−4.81000	−0.961999
$26$	−4.30262	−0.843813
$27$	4.73088	0.910459
$28$	1.03466	0.195532
$29$	3.89653	0.723568	0.361784	−	0.932262i	$-0.382168\pi$
$29$	3.89653	0.723568	0.361784	+	0.932262i	$0.382168\pi$
$30$	−0.810191	−0.147920
$31$	−7.29886	−1.31091	−0.655457	−	0.755232i	$-0.727524\pi$
$31$	−7.29886	−1.31091	−0.655457	+	0.755232i	$0.727524\pi$
$32$	−1.00000	−0.176777
$33$	−0.967935	−0.168496
$34$	6.53111	1.12008
$35$	−0.451002	−0.0762333
$36$	0.454718	0.0757863
$37$	−4.64766	−0.764070	−0.382035	−	0.924148i	$-0.624777\pi$
$37$	−4.64766	−0.764070	−0.382035	+	0.924148i	$0.624777\pi$
$38$	4.45215	0.722235
$39$	−7.99722	−1.28058
$40$	0.435894	0.0689209
$41$	−1.79580	−0.280457	−0.140228	−	0.990119i	$-0.544784\pi$
$41$	−1.79580	−0.280457	−0.140228	+	0.990119i	$0.544784\pi$
$42$	1.92311	0.296742
$43$	−5.48331	−0.836198	−0.418099	−	0.908402i	$-0.637303\pi$
$43$	−5.48331	−0.836198	−0.418099	+	0.908402i	$0.637303\pi$
$44$	0.520763	0.0785080
$45$	−0.198209	−0.0295472
$46$	3.22721	0.475825
$47$	7.14256	1.04185	0.520925	−	0.853603i	$-0.325587\pi$
$47$	7.14256	1.04185	0.520925	+	0.853603i	$0.325587\pi$
$48$	−1.85869	−0.268278
$49$	−5.92948	−0.847068
$50$	4.81000	0.680236
$51$	12.1393	1.69984
$52$	4.30262	0.596666
$53$	5.49063	0.754196	0.377098	−	0.926173i	$-0.376922\pi$
$53$	5.49063	0.754196	0.377098	+	0.926173i	$0.376922\pi$
$54$	−4.73088	−0.643792
$55$	−0.226997	−0.0306083
$56$	−1.03466	−0.138262
$57$	8.27516	1.09607
$58$	−3.89653	−0.511640
$59$	−2.34835	−0.305729	−0.152865	−	0.988247i	$-0.548850\pi$
$59$	−2.34835	−0.305729	−0.152865	+	0.988247i	$0.548850\pi$
$60$	0.810191	0.104595
$61$	12.0427	1.54190	0.770952	−	0.636893i	$-0.219781\pi$
$61$	12.0427	1.54190	0.770952	+	0.636893i	$0.219781\pi$
$62$	7.29886	0.926957
$63$	0.470478	0.0592747
$64$	1.00000	0.125000
$65$	−1.87549	−0.232625
$66$	0.967935	0.119145
$67$	7.04743	0.860980	0.430490	−	0.902595i	$-0.358341\pi$
$67$	7.04743	0.860980	0.430490	+	0.902595i	$0.358341\pi$
$68$	−6.53111	−0.792013
$69$	5.99836	0.722118
$70$	0.451002	0.0539051
$71$	−1.51568	−0.179879	−0.0899393	−	0.995947i	$-0.528667\pi$
$71$	−1.51568	−0.179879	−0.0899393	+	0.995947i	$0.528667\pi$
$72$	−0.454718	−0.0535890
$73$	1.54224	0.180506	0.0902528	−	0.995919i	$-0.471233\pi$
$73$	1.54224	0.180506	0.0902528	+	0.995919i	$0.471233\pi$
$74$	4.64766	0.540279
$75$	8.94028	1.03233
$76$	−4.45215	−0.510697
$77$	0.538813	0.0614034
$78$	7.99722	0.905507
$79$	8.77247	0.986980	0.493490	−	0.869751i	$-0.335721\pi$
$79$	8.77247	0.986980	0.493490	+	0.869751i	$0.335721\pi$
$80$	−0.435894	−0.0487344
$81$	−10.1574	−1.12860
$82$	1.79580	0.198313
$83$	0.586340	0.0643592	0.0321796	−	0.999482i	$-0.489755\pi$
$83$	0.586340	0.0643592	0.0321796	+	0.999482i	$0.489755\pi$
$84$	−1.92311	−0.209828
$85$	2.84687	0.308786
$86$	5.48331	0.591281
$87$	−7.24243	−0.776470
$88$	−0.520763	−0.0555135
$89$	−3.03939	−0.322175	−0.161087	−	0.986940i	$-0.551500\pi$
$89$	−3.03939	−0.322175	−0.161087	+	0.986940i	$0.551500\pi$
$90$	0.198209	0.0208930
$91$	4.45175	0.466670
$92$	−3.22721	−0.336459
$93$	13.5663	1.40676
$94$	−7.14256	−0.736699
$95$	1.94067	0.199108
$96$	1.85869	0.189701
$97$	−17.0736	−1.73356	−0.866782	−	0.498687i	$-0.833816\pi$
$97$	−17.0736	−1.73356	−0.866782	+	0.498687i	$0.833816\pi$
$98$	5.92948	0.598968
$99$	0.236800	0.0237993
$100$	−4.81000	−0.481000
$101$	14.0107	1.39412	0.697058	−	0.717015i	$-0.254492\pi$
$101$	14.0107	1.39412	0.697058	+	0.717015i	$0.254492\pi$
$102$	−12.1393	−1.20197
$103$	−10.7851	−1.06269	−0.531346	−	0.847155i	$-0.678314\pi$
$103$	−10.7851	−1.06269	−0.531346	+	0.847155i	$0.678314\pi$
$104$	−4.30262	−0.421906
$105$	0.838272	0.0818070
$106$	−5.49063	−0.533297
$107$	−0.570891	−0.0551901	−0.0275951	−	0.999619i	$-0.508785\pi$
$107$	−0.570891	−0.0551901	−0.0275951	+	0.999619i	$0.508785\pi$
$108$	4.73088	0.455229
$109$	−6.25788	−0.599396	−0.299698	−	0.954034i	$-0.596886\pi$
$109$	−6.25788	−0.599396	−0.299698	+	0.954034i	$0.596886\pi$
$110$	0.226997	0.0216434
$111$	8.63854	0.819934
$112$	1.03466	0.0977662
$113$	−6.51293	−0.612685	−0.306342	−	0.951921i	$-0.599105\pi$
$113$	−6.51293	−0.612685	−0.306342	+	0.951921i	$0.599105\pi$
$114$	−8.27516	−0.775040
$115$	1.40672	0.131177
$116$	3.89653	0.361784
$117$	1.95648	0.180876
$118$	2.34835	0.216183
$119$	−6.75747	−0.619457
$120$	−0.810191	−0.0739600
$121$	−10.7288	−0.975346
$122$	−12.0427	−1.09029
$123$	3.33783	0.300962
$124$	−7.29886	−0.655457
$125$	4.27612	0.382468
$126$	−0.470478	−0.0419135
$127$	4.57387	0.405866	0.202933	−	0.979193i	$-0.434953\pi$
$127$	4.57387	0.405866	0.202933	+	0.979193i	$0.434953\pi$
$128$	−1.00000	−0.0883883
$129$	10.1918	0.897335
$130$	1.87549	0.164491
$131$	19.1978	1.67732	0.838661	−	0.544653i	$-0.183339\pi$
$131$	19.1978	1.67732	0.838661	+	0.544653i	$0.183339\pi$
$132$	−0.967935	−0.0842480
$133$	−4.60647	−0.399431
$134$	−7.04743	−0.608805
$135$	−2.06216	−0.177483
$136$	6.53111	0.560038
$137$	−2.72558	−0.232862	−0.116431	−	0.993199i	$-0.537145\pi$
$137$	−2.72558	−0.232862	−0.116431	+	0.993199i	$0.537145\pi$
$138$	−5.99836	−0.510615
$139$	1.00198	0.0849866	0.0424933	−	0.999097i	$-0.486470\pi$
$139$	1.00198	0.0849866	0.0424933	+	0.999097i	$0.486470\pi$
$140$	−0.451002	−0.0381166
$141$	−13.2758	−1.11802
$142$	1.51568	0.127193
$143$	2.24064	0.187372
$144$	0.454718	0.0378931
$145$	−1.69848	−0.141051
$146$	−1.54224	−0.127637
$147$	11.0210	0.909000
$148$	−4.64766	−0.382035
$149$	−7.18757	−0.588829	−0.294414	−	0.955678i	$-0.595125\pi$
$149$	−7.18757	−0.588829	−0.294414	+	0.955678i	$0.595125\pi$
$150$	−8.94028	−0.729971
$151$	−2.78209	−0.226403	−0.113202	−	0.993572i	$-0.536111\pi$
$151$	−2.78209	−0.226403	−0.113202	+	0.993572i	$0.536111\pi$
$152$	4.45215	0.361117
$153$	−2.96981	−0.240095
$154$	−0.538813	−0.0434188
$155$	3.18153	0.255547
$156$	−7.99722	−0.640290
$157$	0.0507138	0.00404740	0.00202370	−	0.999998i	$-0.499356\pi$
$157$	0.0507138	0.00404740	0.00202370	+	0.999998i	$0.499356\pi$
$158$	−8.77247	−0.697900
$159$	−10.2054	−0.809338
$160$	0.435894	0.0344605
$161$	−3.33906	−0.263155
$162$	10.1574	0.798040
$163$	4.43377	0.347279	0.173640	−	0.984809i	$-0.444447\pi$
$163$	4.43377	0.347279	0.173640	+	0.984809i	$0.444447\pi$
$164$	−1.79580	−0.140228
$165$	0.421917	0.0328462
$166$	−0.586340	−0.0455088
$167$	−6.52642	−0.505030	−0.252515	−	0.967593i	$-0.581258\pi$
$167$	−6.52642	−0.505030	−0.252515	+	0.967593i	$0.581258\pi$
$168$	1.92311	0.148371
$169$	5.51251	0.424039
$170$	−2.84687	−0.218345
$171$	−2.02447	−0.154815
$172$	−5.48331	−0.418099
$173$	−9.43170	−0.717079	−0.358539	−	0.933515i	$-0.616725\pi$
$173$	−9.43170	−0.717079	−0.358539	+	0.933515i	$0.616725\pi$
$174$	7.24243	0.549048
$175$	−4.97671	−0.376204
$176$	0.520763	0.0392540
$177$	4.36485	0.328082
$178$	3.03939	0.227812
$179$	17.4937	1.30754	0.653771	−	0.756692i	$-0.273186\pi$
$179$	17.4937	1.30754	0.653771	+	0.756692i	$0.273186\pi$
$180$	−0.198209	−0.0147736
$181$	8.94210	0.664661	0.332331	−	0.943163i	$-0.392165\pi$
$181$	8.94210	0.664661	0.332331	+	0.943163i	$0.392165\pi$
$182$	−4.45175	−0.329985
$183$	−22.3835	−1.65464
$184$	3.22721	0.237913
$185$	2.02589	0.148946
$186$	−13.5663	−0.994730
$187$	−3.40116	−0.248717
$188$	7.14256	0.520925
$189$	4.89486	0.356048
$190$	−1.94067	−0.140791
$191$	7.85435	0.568321	0.284160	−	0.958777i	$-0.408285\pi$
$191$	7.85435	0.568321	0.284160	+	0.958777i	$0.408285\pi$
$192$	−1.85869	−0.134139
$193$	14.8251	1.06714	0.533568	−	0.845757i	$-0.320851\pi$
$193$	14.8251	1.06714	0.533568	+	0.845757i	$0.320851\pi$
$194$	17.0736	1.22582
$195$	3.48594	0.249633
$196$	−5.92948	−0.423534
$197$	9.35429	0.666466	0.333233	−	0.942845i	$-0.391860\pi$
$197$	9.35429	0.666466	0.333233	+	0.942845i	$0.391860\pi$
$198$	−0.236800	−0.0168287
$199$	−19.9525	−1.41439	−0.707196	−	0.707018i	$-0.750040\pi$
$199$	−19.9525	−1.41439	−0.707196	+	0.707018i	$0.750040\pi$
$200$	4.81000	0.340118
$201$	−13.0990	−0.923929
$202$	−14.0107	−0.985788
$203$	4.03159	0.282962
$204$	12.1393	0.849920
$205$	0.782779	0.0546716
$206$	10.7851	0.751437
$207$	−1.46747	−0.101996
$208$	4.30262	0.298333
$209$	−2.31852	−0.160375
$210$	−0.838272	−0.0578463
$211$	−6.18616	−0.425873	−0.212936	−	0.977066i	$-0.568303\pi$
$211$	−6.18616	−0.425873	−0.212936	+	0.977066i	$0.568303\pi$
$212$	5.49063	0.377098
$213$	2.81718	0.193030
$214$	0.570891	0.0390253
$215$	2.39014	0.163007
$216$	−4.73088	−0.321896
$217$	−7.55184	−0.512653
$218$	6.25788	0.423837
$219$	−2.86654	−0.193703
$220$	−0.226997	−0.0153042
$221$	−28.1008	−1.89027
$222$	−8.63854	−0.579781
$223$	9.93719	0.665444	0.332722	−	0.943025i	$-0.392033\pi$
$223$	9.93719	0.665444	0.332722	+	0.943025i	$0.392033\pi$
$224$	−1.03466	−0.0691311
$225$	−2.18719	−0.145813
$226$	6.51293	0.433234
$227$	−6.46212	−0.428906	−0.214453	−	0.976734i	$-0.568797\pi$
$227$	−6.46212	−0.428906	−0.214453	+	0.976734i	$0.568797\pi$
$228$	8.27516	0.548036
$229$	−17.3861	−1.14891	−0.574454	−	0.818537i	$-0.694786\pi$
$229$	−17.3861	−1.14891	−0.574454	+	0.818537i	$0.694786\pi$
$230$	−1.40672	−0.0927564
$231$	−1.00148	−0.0658928
$232$	−3.89653	−0.255820
$233$	19.3064	1.26481	0.632404	−	0.774639i	$-0.282068\pi$
$233$	19.3064	1.26481	0.632404	+	0.774639i	$0.282068\pi$
$234$	−1.95648	−0.127899
$235$	−3.11340	−0.203096
$236$	−2.34835	−0.152865
$237$	−16.3053	−1.05914
$238$	6.75747	0.438022
$239$	−11.9735	−0.774503	−0.387252	−	0.921974i	$-0.626576\pi$
$239$	−11.9735	−0.774503	−0.387252	+	0.921974i	$0.626576\pi$
$240$	0.810191	0.0522976
$241$	−12.1999	−0.785862	−0.392931	−	0.919568i	$-0.628539\pi$
$241$	−12.1999	−0.785862	−0.392931	+	0.919568i	$0.628539\pi$
$242$	10.7288	0.689674
$243$	4.68675	0.300655
$244$	12.0427	0.770952
$245$	2.58462	0.165126
$246$	−3.33783	−0.212812
$247$	−19.1559	−1.21886
$248$	7.29886	0.463478
$249$	−1.08982	−0.0690647
$250$	−4.27612	−0.270446
$251$	13.0091	0.821126	0.410563	−	0.911832i	$-0.365332\pi$
$251$	13.0091	0.821126	0.410563	+	0.911832i	$0.365332\pi$
$252$	0.470478	0.0296373
$253$	−1.68061	−0.105659
$254$	−4.57387	−0.286990
$255$	−5.29144	−0.331363
$256$	1.00000	0.0625000
$257$	18.4616	1.15160	0.575802	−	0.817589i	$-0.304690\pi$
$257$	18.4616	1.15160	0.575802	+	0.817589i	$0.304690\pi$
$258$	−10.1918	−0.634512
$259$	−4.80875	−0.298801
$260$	−1.87549	−0.116313
$261$	1.77182	0.109673
$262$	−19.1978	−1.18605
$263$	−0.0843656	−0.00520220	−0.00260110	−	0.999997i	$-0.500828\pi$
$263$	−0.0843656	−0.00520220	−0.00260110	+	0.999997i	$0.500828\pi$
$264$	0.967935	0.0595723
$265$	−2.39333	−0.147021
$266$	4.60647	0.282441
$267$	5.64927	0.345730
$268$	7.04743	0.430490
$269$	8.92589	0.544221	0.272110	−	0.962266i	$-0.412278\pi$
$269$	8.92589	0.544221	0.272110	+	0.962266i	$0.412278\pi$
$270$	2.06216	0.125499
$271$	−1.85248	−0.112530	−0.0562650	−	0.998416i	$-0.517919\pi$
$271$	−1.85248	−0.112530	−0.0562650	+	0.998416i	$0.517919\pi$
$272$	−6.53111	−0.396006
$273$	−8.27440	−0.500790
$274$	2.72558	0.164659
$275$	−2.50487	−0.151049
$276$	5.99836	0.361059
$277$	−29.7486	−1.78742	−0.893710	−	0.448646i	$-0.851906\pi$
$277$	−29.7486	−1.78742	−0.893710	+	0.448646i	$0.851906\pi$
$278$	−1.00198	−0.0600946
$279$	−3.31892	−0.198699
$280$	0.451002	0.0269525
$281$	0.765756	0.0456812	0.0228406	−	0.999739i	$-0.492729\pi$
$281$	0.765756	0.0456812	0.0228406	+	0.999739i	$0.492729\pi$
$282$	13.2758	0.790561
$283$	−10.6552	−0.633384	−0.316692	−	0.948528i	$-0.602572\pi$
$283$	−10.6552	−0.633384	−0.316692	+	0.948528i	$0.602572\pi$
$284$	−1.51568	−0.0899393
$285$	−3.60709	−0.213666
$286$	−2.24064	−0.132492
$287$	−1.85804	−0.109677
$288$	−0.454718	−0.0267945
$289$	25.6553	1.50914
$290$	1.69848	0.0997379
$291$	31.7345	1.86031
$292$	1.54224	0.0902528
$293$	16.9540	0.990462	0.495231	−	0.868761i	$-0.335083\pi$
$293$	16.9540	0.990462	0.495231	+	0.868761i	$0.335083\pi$
$294$	−11.0210	−0.642760
$295$	1.02363	0.0595981
$296$	4.64766	0.270140
$297$	2.46367	0.142957
$298$	7.18757	0.416365
$299$	−13.8854	−0.803015
$300$	8.94028	0.516167
$301$	−5.67337	−0.327007
$302$	2.78209	0.160091
$303$	−26.0415	−1.49604
$304$	−4.45215	−0.255348
$305$	−5.24932	−0.300575
$306$	2.96981	0.169773
$307$	14.7678	0.842844	0.421422	−	0.906865i	$-0.361531\pi$
$307$	14.7678	0.842844	0.421422	+	0.906865i	$0.361531\pi$
$308$	0.538813	0.0307017
$309$	20.0462	1.14039
$310$	−3.18153	−0.180699
$311$	−31.5195	−1.78731	−0.893655	−	0.448756i	$-0.851867\pi$
$311$	−31.5195	−1.78731	−0.893655	+	0.448756i	$0.851867\pi$
$312$	7.99722	0.452753
$313$	19.1415	1.08194	0.540972	−	0.841041i	$-0.318056\pi$
$313$	19.1415	1.08194	0.540972	+	0.841041i	$0.318056\pi$
$314$	−0.0507138	−0.00286194
$315$	−0.205079	−0.0115549
$316$	8.77247	0.493490
$317$	20.3279	1.14173	0.570865	−	0.821044i	$-0.306608\pi$
$317$	20.3279	1.14173	0.570865	+	0.821044i	$0.306608\pi$
$318$	10.2054	0.572289
$319$	2.02917	0.113612
$320$	−0.435894	−0.0243672
$321$	1.06111	0.0592253
$322$	3.33906	0.186079
$323$	29.0775	1.61791
$324$	−10.1574	−0.564299
$325$	−20.6956	−1.14798
$326$	−4.43377	−0.245564
$327$	11.6314	0.643220
$328$	1.79580	0.0991565
$329$	7.39012	0.407431
$330$	−0.421917	−0.0232258
$331$	−12.0119	−0.660233	−0.330117	−	0.943940i	$-0.607088\pi$
$331$	−12.0119	−0.660233	−0.330117	+	0.943940i	$0.607088\pi$
$332$	0.586340	0.0321796
$333$	−2.11337	−0.115812
$334$	6.52642	0.357110
$335$	−3.07193	−0.167838
$336$	−1.92311	−0.104914
$337$	−26.8853	−1.46454	−0.732269	−	0.681016i	$-0.761538\pi$
$337$	−26.8853	−1.46454	−0.732269	+	0.681016i	$0.761538\pi$
$338$	−5.51251	−0.299841
$339$	12.1055	0.657480
$340$	2.84687	0.154393
$341$	−3.80098	−0.205834
$342$	2.02447	0.109471
$343$	−13.3776	−0.722323
$344$	5.48331	0.295641
$345$	−2.61465	−0.140768
$346$	9.43170	0.507051
$347$	6.88088	0.369385	0.184693	−	0.982796i	$-0.440871\pi$
$347$	6.88088	0.369385	0.184693	+	0.982796i	$0.440871\pi$
$348$	−7.24243	−0.388235
$349$	−13.0921	−0.700806	−0.350403	−	0.936599i	$-0.613955\pi$
$349$	−13.0921	−0.700806	−0.350403	+	0.936599i	$0.613955\pi$
$350$	4.97671	0.266016
$351$	20.3552	1.08648
$352$	−0.520763	−0.0277568
$353$	5.86331	0.312072	0.156036	−	0.987751i	$-0.450128\pi$
$353$	5.86331	0.312072	0.156036	+	0.987751i	$0.450128\pi$
$354$	−4.36485	−0.231989
$355$	0.660678	0.0350651
$356$	−3.03939	−0.161087
$357$	12.5600	0.664747
$358$	−17.4937	−0.924572
$359$	1.49982	0.0791574	0.0395787	−	0.999216i	$-0.487398\pi$
$359$	1.49982	0.0791574	0.0395787	+	0.999216i	$0.487398\pi$
$360$	0.198209	0.0104465
$361$	0.821668	0.0432457
$362$	−8.94210	−0.469987
$363$	19.9415	1.04666
$364$	4.45175	0.233335
$365$	−0.672253	−0.0351873
$366$	22.3835	1.17001
$367$	21.9337	1.14493	0.572465	−	0.819929i	$-0.305987\pi$
$367$	21.9337	1.14493	0.572465	+	0.819929i	$0.305987\pi$
$368$	−3.22721	−0.168230
$369$	−0.816582	−0.0425096
$370$	−2.02589	−0.105321
$371$	5.68094	0.294940
$372$	13.5663	0.703380
$373$	−28.3045	−1.46555	−0.732776	−	0.680470i	$-0.761776\pi$
$373$	−28.3045	−1.46555	−0.732776	+	0.680470i	$0.761776\pi$
$374$	3.40116	0.175870
$375$	−7.94797	−0.410431
$376$	−7.14256	−0.368349
$377$	16.7653	0.863456
$378$	−4.89486	−0.251764
$379$	30.2959	1.55620	0.778098	−	0.628143i	$-0.216185\pi$
$379$	30.2959	1.55620	0.778098	+	0.628143i	$0.216185\pi$
$380$	1.94067	0.0995541
$381$	−8.50140	−0.435540
$382$	−7.85435	−0.401863
$383$	19.9051	1.01710	0.508551	−	0.861032i	$-0.330181\pi$
$383$	19.9051	1.01710	0.508551	+	0.861032i	$0.330181\pi$
$384$	1.85869	0.0948507
$385$	−0.234865	−0.0119698
$386$	−14.8251	−0.754579
$387$	−2.49336	−0.126745
$388$	−17.0736	−0.866782
$389$	−9.62178	−0.487844	−0.243922	−	0.969795i	$-0.578434\pi$
$389$	−9.62178	−0.487844	−0.243922	+	0.969795i	$0.578434\pi$
$390$	−3.48594	−0.176517
$391$	21.0772	1.06592
$392$	5.92948	0.299484
$393$	−35.6828	−1.79996
$394$	−9.35429	−0.471263
$395$	−3.82387	−0.192400
$396$	0.236800	0.0118997
$397$	20.4751	1.02761	0.513807	−	0.857906i	$-0.328235\pi$
$397$	20.4751	1.02761	0.513807	+	0.857906i	$0.328235\pi$
$398$	19.9525	1.00013
$399$	8.56198	0.428635
$400$	−4.81000	−0.240500
$401$	35.9833	1.79692	0.898459	−	0.439057i	$-0.144687\pi$
$401$	35.9833	1.79692	0.898459	+	0.439057i	$0.144687\pi$
$402$	13.0990	0.653317
$403$	−31.4042	−1.56436
$404$	14.0107	0.697058
$405$	4.42754	0.220006
$406$	−4.03159	−0.200084
$407$	−2.42033	−0.119971
$408$	−12.1393	−0.600984
$409$	4.96671	0.245588	0.122794	−	0.992432i	$-0.460815\pi$
$409$	4.96671	0.245588	0.122794	+	0.992432i	$0.460815\pi$
$410$	−0.782779	−0.0386587
$411$	5.06601	0.249888
$412$	−10.7851	−0.531346
$413$	−2.42974	−0.119560
$414$	1.46747	0.0721221
$415$	−0.255582	−0.0125460
$416$	−4.30262	−0.210953
$417$	−1.86236	−0.0912003
$418$	2.31852	0.113402
$419$	−18.1640	−0.887368	−0.443684	−	0.896183i	$-0.646329\pi$
$419$	−18.1640	−0.887368	−0.443684	+	0.896183i	$0.646329\pi$
$420$	0.838272	0.0409035
$421$	30.2360	1.47361	0.736805	−	0.676105i	$-0.236334\pi$
$421$	30.2360	1.47361	0.736805	+	0.676105i	$0.236334\pi$
$422$	6.18616	0.301137
$423$	3.24785	0.157916
$424$	−5.49063	−0.266649
$425$	31.4146	1.52383
$426$	−2.81718	−0.136493
$427$	12.4601	0.602985
$428$	−0.570891	−0.0275951
$429$	−4.16465	−0.201071
$430$	−2.39014	−0.115263
$431$	27.3266	1.31628	0.658139	−	0.752897i	$-0.271344\pi$
$431$	27.3266	1.31628	0.658139	+	0.752897i	$0.271344\pi$
$432$	4.73088	0.227615
$433$	11.8044	0.567283	0.283641	−	0.958930i	$-0.408457\pi$
$433$	11.8044	0.567283	0.283641	+	0.958930i	$0.408457\pi$
$434$	7.55184	0.362500
$435$	3.15693	0.151363
$436$	−6.25788	−0.299698
$437$	14.3680	0.687315
$438$	2.86654	0.136969
$439$	2.00990	0.0959274	0.0479637	−	0.998849i	$-0.484727\pi$
$439$	2.00990	0.0959274	0.0479637	+	0.998849i	$0.484727\pi$
$440$	0.226997	0.0108217
$441$	−2.69624	−0.128392
$442$	28.1008	1.33662
$443$	23.5392	1.11838	0.559190	−	0.829040i	$-0.311112\pi$
$443$	23.5392	1.11838	0.559190	+	0.829040i	$0.311112\pi$
$444$	8.63854	0.409967
$445$	1.32485	0.0628040
$446$	−9.93719	−0.470540
$447$	13.3594	0.631880
$448$	1.03466	0.0488831
$449$	−11.6494	−0.549767	−0.274883	−	0.961478i	$-0.588639\pi$
$449$	−11.6494	−0.549767	−0.274883	+	0.961478i	$0.588639\pi$
$450$	2.18719	0.103105
$451$	−0.935186	−0.0440362
$452$	−6.51293	−0.306342
$453$	5.17103	0.242956
$454$	6.46212	0.303282
$455$	−1.94049	−0.0909716
$456$	−8.27516	−0.387520
$457$	11.7228	0.548370	0.274185	−	0.961677i	$-0.411592\pi$
$457$	11.7228	0.548370	0.274185	+	0.961677i	$0.411592\pi$
$458$	17.3861	0.812401
$459$	−30.8979	−1.44219
$460$	1.40672	0.0655886
$461$	0.0193888	0.000903027 0	0.000451513	−	1.00000i	$-0.499856\pi$
$461$	0.0193888	0.000903027 0	0.000451513		1.00000i	$0.499856\pi$
$462$	1.00148	0.0465933
$463$	32.6526	1.51749	0.758746	−	0.651386i	$-0.225812\pi$
$463$	32.6526	1.51749	0.758746	+	0.651386i	$0.225812\pi$
$464$	3.89653	0.180892
$465$	−5.91347	−0.274231
$466$	−19.3064	−0.894354
$467$	−27.5535	−1.27502	−0.637511	−	0.770441i	$-0.720036\pi$
$467$	−27.5535	−1.27502	−0.637511	+	0.770441i	$0.720036\pi$
$468$	1.95648	0.0904381
$469$	7.29169	0.336699
$470$	3.11340	0.143610
$471$	−0.0942611	−0.00434332
$472$	2.34835	0.108092
$473$	−2.85551	−0.131296
$474$	16.3053	0.748926
$475$	21.4148	0.982580
$476$	−6.75747	−0.309728
$477$	2.49669	0.114315
$478$	11.9735	0.547656
$479$	−9.91352	−0.452960	−0.226480	−	0.974016i	$-0.572722\pi$
$479$	−9.91352	−0.452960	−0.226480	+	0.974016i	$0.572722\pi$
$480$	−0.810191	−0.0369800
$481$	−19.9971	−0.911789
$482$	12.1999	0.555689
$483$	6.20627	0.282395
$484$	−10.7288	−0.487673
$485$	7.44230	0.337937
$486$	−4.68675	−0.212595
$487$	26.6498	1.20762	0.603810	−	0.797129i	$-0.293649\pi$
$487$	26.6498	1.20762	0.603810	+	0.797129i	$0.293649\pi$
$488$	−12.0427	−0.545146
$489$	−8.24099	−0.372670
$490$	−2.58462	−0.116761
$491$	−4.95333	−0.223541	−0.111770	−	0.993734i	$-0.535652\pi$
$491$	−4.95333	−0.223541	−0.111770	+	0.993734i	$0.535652\pi$
$492$	3.33783	0.150481
$493$	−25.4487	−1.14615
$494$	19.1559	0.861865
$495$	−0.103220	−0.00463938
$496$	−7.29886	−0.327729
$497$	−1.56822	−0.0703442
$498$	1.08982	0.0488361
$499$	27.3781	1.22561	0.612806	−	0.790233i	$-0.290041\pi$
$499$	27.3781	1.22561	0.612806	+	0.790233i	$0.290041\pi$
$500$	4.27612	0.191234
$501$	12.1306	0.541954
$502$	−13.0091	−0.580624
$503$	8.13098	0.362542	0.181271	−	0.983433i	$-0.441979\pi$
$503$	8.13098	0.362542	0.181271	+	0.983433i	$0.441979\pi$
$504$	−0.470478	−0.0209568
$505$	−6.10718	−0.271766
$506$	1.68061	0.0747122
$507$	−10.2460	−0.455042
$508$	4.57387	0.202933
$509$	42.7146	1.89329	0.946645	−	0.322277i	$-0.104448\pi$
$509$	42.7146	1.89329	0.946645	+	0.322277i	$0.104448\pi$
$510$	5.29144	0.234309
$511$	1.59569	0.0705893
$512$	−1.00000	−0.0441942
$513$	−21.0626	−0.929937
$514$	−18.4616	−0.814307
$515$	4.70118	0.207159
$516$	10.1918	0.448668
$517$	3.71958	0.163587
$518$	4.80875	0.211284
$519$	17.5306	0.769507
$520$	1.87549	0.0822455
$521$	−25.7134	−1.12652	−0.563262	−	0.826278i	$-0.690454\pi$
$521$	−25.7134	−1.12652	−0.563262	+	0.826278i	$0.690454\pi$
$522$	−1.77182	−0.0775506
$523$	−30.1438	−1.31810	−0.659048	−	0.752101i	$-0.729041\pi$
$523$	−30.1438	−1.31810	−0.659048	+	0.752101i	$0.729041\pi$
$524$	19.1978	0.838661
$525$	9.25015	0.403710
$526$	0.0843656	0.00367851
$527$	47.6697	2.07652
$528$	−0.967935	−0.0421240
$529$	−12.5851	−0.547180
$530$	2.39333	0.103960
$531$	−1.06784	−0.0463401
$532$	−4.60647	−0.199716
$533$	−7.72664	−0.334678
$534$	−5.64927	−0.244468
$535$	0.248848	0.0107586
$536$	−7.04743	−0.304402
$537$	−32.5154	−1.40314
$538$	−8.92589	−0.384822
$539$	−3.08785	−0.133003
$540$	−2.06216	−0.0887414
$541$	−20.0448	−0.861791	−0.430896	−	0.902402i	$-0.641802\pi$
$541$	−20.0448	−0.861791	−0.430896	+	0.902402i	$0.641802\pi$
$542$	1.85248	0.0795708
$543$	−16.6206	−0.713257
$544$	6.53111	0.280019
$545$	2.72777	0.116845
$546$	8.27440	0.354112
$547$	3.85550	0.164849	0.0824247	−	0.996597i	$-0.473734\pi$
$547$	3.85550	0.164849	0.0824247	+	0.996597i	$0.473734\pi$
$548$	−2.72558	−0.116431
$549$	5.47601	0.233710
$550$	2.50487	0.106808
$551$	−17.3480	−0.739048
$552$	−5.99836	−0.255307
$553$	9.07653	0.385973
$554$	29.7486	1.26390
$555$	−3.76549	−0.159836
$556$	1.00198	0.0424933
$557$	−24.7677	−1.04944	−0.524721	−	0.851274i	$-0.675830\pi$
$557$	−24.7677	−1.04944	−0.524721	+	0.851274i	$0.675830\pi$
$558$	3.31892	0.140501
$559$	−23.5926	−0.997861
$560$	−0.451002	−0.0190583
$561$	6.32169	0.266902
$562$	−0.765756	−0.0323015
$563$	14.2251	0.599517	0.299758	−	0.954015i	$-0.403094\pi$
$563$	14.2251	0.599517	0.299758	+	0.954015i	$0.403094\pi$
$564$	−13.2758	−0.559011
$565$	2.83895	0.119435
$566$	10.6552	0.447870
$567$	−10.5094	−0.441355
$568$	1.51568	0.0635967
$569$	−5.37623	−0.225383	−0.112692	−	0.993630i	$-0.535947\pi$
$569$	−5.37623	−0.225383	−0.112692	+	0.993630i	$0.535947\pi$
$570$	3.60709	0.151085
$571$	−8.49024	−0.355305	−0.177653	−	0.984093i	$-0.556850\pi$
$571$	−8.49024	−0.355305	−0.177653	+	0.984093i	$0.556850\pi$
$572$	2.24064	0.0936860
$573$	−14.5988	−0.609872
$574$	1.85804	0.0775532
$575$	15.5228	0.647347
$576$	0.454718	0.0189466
$577$	3.62440	0.150886	0.0754430	−	0.997150i	$-0.475963\pi$
$577$	3.62440	0.150886	0.0754430	+	0.997150i	$0.475963\pi$
$578$	−25.6553	−1.06712
$579$	−27.5553	−1.14516
$580$	−1.69848	−0.0705254
$581$	0.606663	0.0251686
$582$	−31.7345	−1.31544
$583$	2.85932	0.118421
$584$	−1.54224	−0.0638183
$585$	−0.852816	−0.0352596
$586$	−16.9540	−0.700363
$587$	28.0307	1.15695	0.578476	−	0.815699i	$-0.303648\pi$
$587$	28.0307	1.15695	0.578476	+	0.815699i	$0.303648\pi$
$588$	11.0210	0.454500
$589$	32.4957	1.33896
$590$	−1.02363	−0.0421422
$591$	−17.3867	−0.715194
$592$	−4.64766	−0.191018
$593$	22.0893	0.907101	0.453550	−	0.891231i	$-0.350157\pi$
$593$	22.0893	0.907101	0.453550	+	0.891231i	$0.350157\pi$
$594$	−2.46367	−0.101086
$595$	2.94554	0.120756
$596$	−7.18757	−0.294414
$597$	37.0854	1.51780
$598$	13.8854	0.567817
$599$	40.1422	1.64016	0.820082	−	0.572246i	$-0.193928\pi$
$599$	40.1422	1.64016	0.820082	+	0.572246i	$0.193928\pi$
$600$	−8.94028	−0.364985
$601$	25.2196	1.02873	0.514365	−	0.857571i	$-0.328028\pi$
$601$	25.2196	1.02873	0.514365	+	0.857571i	$0.328028\pi$
$602$	5.67337	0.231229
$603$	3.20459	0.130501
$604$	−2.78209	−0.113202
$605$	4.67662	0.190132
$606$	26.0415	1.05786
$607$	37.9757	1.54138	0.770692	−	0.637208i	$-0.219911\pi$
$607$	37.9757	1.54138	0.770692	+	0.637208i	$0.219911\pi$
$608$	4.45215	0.180559
$609$	−7.49346	−0.303650
$610$	5.24932	0.212539
$611$	30.7317	1.24327
$612$	−2.96981	−0.120047
$613$	−31.3025	−1.26429	−0.632147	−	0.774848i	$-0.717826\pi$
$613$	−31.3025	−1.26429	−0.632147	+	0.774848i	$0.717826\pi$
$614$	−14.7678	−0.595980
$615$	−1.45494	−0.0586689
$616$	−0.538813	−0.0217094
$617$	4.83656	0.194713	0.0973564	−	0.995250i	$-0.468961\pi$
$617$	4.83656	0.194713	0.0973564	+	0.995250i	$0.468961\pi$
$618$	−20.0462	−0.806377
$619$	−25.3804	−1.02012	−0.510061	−	0.860138i	$-0.670377\pi$
$619$	−25.3804	−1.02012	−0.510061	+	0.860138i	$0.670377\pi$
$620$	3.18153	0.127773
$621$	−15.2675	−0.612665
$622$	31.5195	1.26382
$623$	−3.14473	−0.125991
$624$	−7.99722	−0.320145
$625$	22.1860	0.887442
$626$	−19.1415	−0.765050
$627$	4.30940	0.172101
$628$	0.0507138	0.00202370
$629$	30.3543	1.21031
$630$	0.205079	0.00817053
$631$	−27.4582	−1.09309	−0.546546	−	0.837429i	$-0.684058\pi$
$631$	−27.4582	−1.09309	−0.546546	+	0.837429i	$0.684058\pi$
$632$	−8.77247	−0.348950
$633$	11.4981	0.457010
$634$	−20.3279	−0.807325
$635$	−1.99372	−0.0791185
$636$	−10.2054	−0.404669
$637$	−25.5123	−1.01083
$638$	−2.02917	−0.0803356
$639$	−0.689209	−0.0272647
$640$	0.435894	0.0172302
$641$	−40.0826	−1.58317	−0.791583	−	0.611062i	$-0.790743\pi$
$641$	−40.0826	−1.58317	−0.791583	+	0.611062i	$0.790743\pi$
$642$	−1.06111	−0.0418786
$643$	−3.76213	−0.148364	−0.0741819	−	0.997245i	$-0.523635\pi$
$643$	−3.76213	−0.148364	−0.0741819	+	0.997245i	$0.523635\pi$
$644$	−3.33906	−0.131577
$645$	−4.44253	−0.174924
$646$	−29.0775	−1.14404
$647$	−4.96817	−0.195319	−0.0976595	−	0.995220i	$-0.531136\pi$
$647$	−4.96817	−0.195319	−0.0976595	+	0.995220i	$0.531136\pi$
$648$	10.1574	0.399020
$649$	−1.22293	−0.0480043
$650$	20.6956	0.811747
$651$	14.0365	0.550134
$652$	4.43377	0.173640
$653$	−22.2952	−0.872478	−0.436239	−	0.899831i	$-0.643690\pi$
$653$	−22.2952	−0.872478	−0.436239	+	0.899831i	$0.643690\pi$
$654$	−11.6314	−0.454825
$655$	−8.36822	−0.326974
$656$	−1.79580	−0.0701142
$657$	0.701284	0.0273597
$658$	−7.39012	−0.288097
$659$	−9.12111	−0.355308	−0.177654	−	0.984093i	$-0.556851\pi$
$659$	−9.12111	−0.355308	−0.177654	+	0.984093i	$0.556851\pi$
$660$	0.421917	0.0164231
$661$	−3.40820	−0.132564	−0.0662819	−	0.997801i	$-0.521114\pi$
$661$	−3.40820	−0.132564	−0.0662819	+	0.997801i	$0.521114\pi$
$662$	12.0119	0.466856
$663$	52.2307	2.02847
$664$	−0.586340	−0.0227544
$665$	2.00793	0.0778642
$666$	2.11337	0.0818915
$667$	−12.5749	−0.486902
$668$	−6.52642	−0.252515
$669$	−18.4701	−0.714097
$670$	3.07193	0.118679
$671$	6.27137	0.242104
$672$	1.92311	0.0741856
$673$	22.5940	0.870933	0.435467	−	0.900205i	$-0.356583\pi$
$673$	22.5940	0.870933	0.435467	+	0.900205i	$0.356583\pi$
$674$	26.8853	1.03558
$675$	−22.7555	−0.875861
$676$	5.51251	0.212020
$677$	0.214420	0.00824082	0.00412041	−	0.999992i	$-0.498688\pi$
$677$	0.214420	0.00824082	0.00412041	+	0.999992i	$0.498688\pi$
$678$	−12.1055	−0.464909
$679$	−17.6654	−0.677936
$680$	−2.84687	−0.109173
$681$	12.0111	0.460265
$682$	3.80098	0.145547
$683$	18.3807	0.703317	0.351659	−	0.936128i	$-0.385618\pi$
$683$	18.3807	0.703317	0.351659	+	0.936128i	$0.385618\pi$
$684$	−2.02447	−0.0774077
$685$	1.18807	0.0453937
$686$	13.3776	0.510760
$687$	32.3154	1.23291
$688$	−5.48331	−0.209049
$689$	23.6241	0.900006
$690$	2.61465	0.0995381
$691$	40.6511	1.54644	0.773221	−	0.634136i	$-0.218644\pi$
$691$	40.6511	1.54644	0.773221	+	0.634136i	$0.218644\pi$
$692$	−9.43170	−0.358539
$693$	0.245008	0.00930707
$694$	−6.88088	−0.261195
$695$	−0.436756	−0.0165671
$696$	7.24243	0.274524
$697$	11.7286	0.444251
$698$	13.0921	0.495544
$699$	−35.8846	−1.35728
$700$	−4.97671	−0.188102
$701$	−9.53820	−0.360253	−0.180126	−	0.983643i	$-0.557651\pi$
$701$	−9.53820	−0.360253	−0.180126	+	0.983643i	$0.557651\pi$
$702$	−20.3552	−0.768257
$703$	20.6921	0.780417
$704$	0.520763	0.0196270
$705$	5.78684	0.217945
$706$	−5.86331	−0.220669
$707$	14.4963	0.545189
$708$	4.36485	0.164041
$709$	11.4919	0.431586	0.215793	−	0.976439i	$-0.430766\pi$
$709$	11.4919	0.431586	0.215793	+	0.976439i	$0.430766\pi$
$710$	−0.660678	−0.0247948
$711$	3.98900	0.149599
$712$	3.03939	0.113906
$713$	23.5549	0.882139
$714$	−12.5600	−0.470047
$715$	−0.976683	−0.0365259
$716$	17.4937	0.653771
$717$	22.2550	0.831130
$718$	−1.49982	−0.0559728
$719$	12.8652	0.479790	0.239895	−	0.970799i	$-0.422887\pi$
$719$	12.8652	0.479790	0.239895	+	0.970799i	$0.422887\pi$
$720$	−0.198209	−0.00738680
$721$	−11.1590	−0.415581
$722$	−0.821668	−0.0305793
$723$	22.6757	0.843319
$724$	8.94210	0.332331
$725$	−18.7423	−0.696072
$726$	−19.9415	−0.740098
$727$	26.1465	0.969721	0.484861	−	0.874591i	$-0.338870\pi$
$727$	26.1465	0.969721	0.484861	+	0.874591i	$0.338870\pi$
$728$	−4.45175	−0.164993
$729$	21.7610	0.805961
$730$	0.672253	0.0248812
$731$	35.8121	1.32456
$732$	−22.3835	−0.827319
$733$	−32.1623	−1.18794	−0.593971	−	0.804487i	$-0.702440\pi$
$733$	−32.1623	−1.18794	−0.593971	+	0.804487i	$0.702440\pi$
$734$	−21.9337	−0.809588
$735$	−4.80401	−0.177199
$736$	3.22721	0.118956
$737$	3.67004	0.135188
$738$	0.816582	0.0300588
$739$	27.3003	1.00426	0.502130	−	0.864792i	$-0.332550\pi$
$739$	27.3003	1.00426	0.502130	+	0.864792i	$0.332550\pi$
$740$	2.02589	0.0744731
$741$	35.6048	1.30798
$742$	−5.68094	−0.208554
$743$	21.3312	0.782565	0.391283	−	0.920271i	$-0.372032\pi$
$743$	21.3312	0.782565	0.391283	+	0.920271i	$0.372032\pi$
$744$	−13.5663	−0.497365
$745$	3.13302	0.114785
$746$	28.3045	1.03630
$747$	0.266619	0.00975509
$748$	−3.40116	−0.124359
$749$	−0.590678	−0.0215829
$750$	7.94797	0.290219
$751$	−30.4559	−1.11135	−0.555676	−	0.831399i	$-0.687540\pi$
$751$	−30.4559	−1.11135	−0.555676	+	0.831399i	$0.687540\pi$
$752$	7.14256	0.260462
$753$	−24.1798	−0.881162
$754$	−16.7653	−0.610556
$755$	1.21270	0.0441345
$756$	4.89486	0.178024
$757$	−19.2823	−0.700827	−0.350413	−	0.936595i	$-0.613959\pi$
$757$	−19.2823	−0.700827	−0.350413	+	0.936595i	$0.613959\pi$
$758$	−30.2959	−1.10040
$759$	3.12373	0.113384
$760$	−1.94067	−0.0703954
$761$	−11.3371	−0.410971	−0.205485	−	0.978660i	$-0.565877\pi$
$761$	−11.3371	−0.410971	−0.205485	+	0.978660i	$0.565877\pi$
$762$	8.50140	0.307973
$763$	−6.47478	−0.234403
$764$	7.85435	0.284160
$765$	1.29452	0.0468036
$766$	−19.9051	−0.719199
$767$	−10.1040	−0.364836
$768$	−1.85869	−0.0670696
$769$	5.01036	0.180678	0.0903391	−	0.995911i	$-0.471205\pi$
$769$	5.01036	0.180678	0.0903391	+	0.995911i	$0.471205\pi$
$770$	0.234865	0.00846396
$771$	−34.3144	−1.23580
$772$	14.8251	0.533568
$773$	27.8337	1.00111	0.500555	−	0.865705i	$-0.333129\pi$
$773$	27.8337	1.00111	0.500555	+	0.865705i	$0.333129\pi$
$774$	2.49336	0.0896220
$775$	35.1075	1.26110
$776$	17.0736	0.612908
$777$	8.93795	0.320647
$778$	9.62178	0.344958
$779$	7.99518	0.286457
$780$	3.48594	0.124817
$781$	−0.789312	−0.0282438
$782$	−21.0772	−0.753720
$783$	18.4340	0.658779
$784$	−5.92948	−0.211767
$785$	−0.0221058	−0.000788991 0
$786$	35.6828	1.27276
$787$	38.9035	1.38676	0.693380	−	0.720572i	$-0.256121\pi$
$787$	38.9035	1.38676	0.693380	+	0.720572i	$0.256121\pi$
$788$	9.35429	0.333233
$789$	0.156809	0.00558256
$790$	3.82387	0.136047
$791$	−6.73867	−0.239599
$792$	−0.236800	−0.00841433
$793$	51.8149	1.84000
$794$	−20.4751	−0.726633
$795$	4.44846	0.157771
$796$	−19.9525	−0.707196
$797$	15.8799	0.562494	0.281247	−	0.959635i	$-0.409252\pi$
$797$	15.8799	0.562494	0.281247	+	0.959635i	$0.409252\pi$
$798$	−8.56198	−0.303091
$799$	−46.6488	−1.65032
$800$	4.81000	0.170059
$801$	−1.38206	−0.0488328
$802$	−35.9833	−1.27061
$803$	0.803141	0.0283422
$804$	−13.0990	−0.461965
$805$	1.45548	0.0512988
$806$	31.4042	1.10617
$807$	−16.5904	−0.584011
$808$	−14.0107	−0.492894
$809$	6.03265	0.212097	0.106048	−	0.994361i	$-0.466180\pi$
$809$	6.03265	0.212097	0.106048	+	0.994361i	$0.466180\pi$
$810$	−4.42754	−0.155568
$811$	43.7916	1.53773	0.768866	−	0.639410i	$-0.220821\pi$
$811$	43.7916	1.53773	0.768866	+	0.639410i	$0.220821\pi$
$812$	4.03159	0.141481
$813$	3.44318	0.120758
$814$	2.42033	0.0848324
$815$	−1.93265	−0.0676979
$816$	12.1393	0.424960
$817$	24.4126	0.854087
$818$	−4.96671	−0.173657
$819$	2.02429	0.0707343
$820$	0.782779	0.0273358
$821$	−7.84406	−0.273760	−0.136880	−	0.990588i	$-0.543707\pi$
$821$	−7.84406	−0.273760	−0.136880	+	0.990588i	$0.543707\pi$
$822$	−5.06601	−0.176697
$823$	−7.37256	−0.256991	−0.128496	−	0.991710i	$-0.541015\pi$
$823$	−7.37256	−0.256991	−0.128496	+	0.991710i	$0.541015\pi$
$824$	10.7851	0.375718
$825$	4.65577	0.162093
$826$	2.42974	0.0845416
$827$	29.0652	1.01070	0.505348	−	0.862916i	$-0.331364\pi$
$827$	29.0652	1.01070	0.505348	+	0.862916i	$0.331364\pi$
$828$	−1.46747	−0.0509980
$829$	10.0430	0.348808	0.174404	−	0.984674i	$-0.444200\pi$
$829$	10.0430	0.348808	0.174404	+	0.984674i	$0.444200\pi$
$830$	0.255582	0.00887139
$831$	55.2933	1.91810
$832$	4.30262	0.149166
$833$	38.7261	1.34178
$834$	1.86236	0.0644883
$835$	2.84483	0.0984494
$836$	−2.31852	−0.0801876
$837$	−34.5301	−1.19353
$838$	18.1640	0.627464
$839$	−23.4626	−0.810018	−0.405009	−	0.914313i	$-0.632732\pi$
$839$	−23.4626	−0.810018	−0.405009	+	0.914313i	$0.632732\pi$
$840$	−0.838272	−0.0289231
$841$	−13.8170	−0.476450
$842$	−30.2360	−1.04200
$843$	−1.42330	−0.0490211
$844$	−6.18616	−0.212936
$845$	−2.40287	−0.0826612
$846$	−3.24785	−0.111663
$847$	−11.1007	−0.381423
$848$	5.49063	0.188549
$849$	19.8046	0.679693
$850$	−31.4146	−1.07751
$851$	14.9989	0.514157
$852$	2.81718	0.0965151
$853$	39.3787	1.34830	0.674150	−	0.738595i	$-0.264510\pi$
$853$	39.3787	1.34830	0.674150	+	0.738595i	$0.264510\pi$
$854$	−12.4601	−0.426374
$855$	0.882456	0.0301794
$856$	0.570891	0.0195127
$857$	9.10463	0.311008	0.155504	−	0.987835i	$-0.450300\pi$
$857$	9.10463	0.311008	0.155504	+	0.987835i	$0.450300\pi$
$858$	4.16465	0.142179
$859$	43.7039	1.49116	0.745579	−	0.666417i	$-0.232173\pi$
$859$	43.7039	1.49116	0.745579	+	0.666417i	$0.232173\pi$
$860$	2.39014	0.0815033
$861$	3.45352	0.117696
$862$	−27.3266	−0.930749
$863$	53.8352	1.83257	0.916285	−	0.400527i	$-0.131173\pi$
$863$	53.8352	1.83257	0.916285	+	0.400527i	$0.131173\pi$
$864$	−4.73088	−0.160948
$865$	4.11122	0.139786
$866$	−11.8044	−0.401130
$867$	−47.6853	−1.61948
$868$	−7.55184	−0.256326
$869$	4.56838	0.154972
$870$	−3.15693	−0.107030
$871$	30.3224	1.02743
$872$	6.25788	0.211919
$873$	−7.76368	−0.262761
$874$	−14.3680	−0.486005
$875$	4.42433	0.149570
$876$	−2.86654	−0.0968514
$877$	42.5724	1.43757	0.718784	−	0.695234i	$-0.244699\pi$
$877$	42.5724	1.43757	0.718784	+	0.695234i	$0.244699\pi$
$878$	−2.00990	−0.0678309
$879$	−31.5121	−1.06288
$880$	−0.226997	−0.00765208
$881$	31.0494	1.04608	0.523040	−	0.852308i	$-0.324798\pi$
$881$	31.0494	1.04608	0.523040	+	0.852308i	$0.324798\pi$
$882$	2.69624	0.0907871
$883$	43.4054	1.46071	0.730354	−	0.683068i	$-0.239355\pi$
$883$	43.4054	1.46071	0.730354	+	0.683068i	$0.239355\pi$
$884$	−28.1008	−0.945134
$885$	−1.90261	−0.0639556
$886$	−23.5392	−0.790814
$887$	51.6225	1.73332	0.866658	−	0.498903i	$-0.166264\pi$
$887$	51.6225	1.73332	0.866658	+	0.498903i	$0.166264\pi$
$888$	−8.63854	−0.289890
$889$	4.73240	0.158720
$890$	−1.32485	−0.0444091
$891$	−5.28959	−0.177208
$892$	9.93719	0.332722
$893$	−31.7998	−1.06414
$894$	−13.3594	−0.446807
$895$	−7.62541	−0.254889
$896$	−1.03466	−0.0345656
$897$	25.8087	0.861726
$898$	11.6494	0.388744
$899$	−28.4403	−0.948536
$900$	−2.18719	−0.0729063
$901$	−35.8599	−1.19467
$902$	0.935186	0.0311383
$903$	10.5450	0.350916
$904$	6.51293	0.216617
$905$	−3.89781	−0.129568
$906$	−5.17103	−0.171796
$907$	13.8163	0.458764	0.229382	−	0.973336i	$-0.426329\pi$
$907$	13.8163	0.458764	0.229382	+	0.973336i	$0.426329\pi$
$908$	−6.46212	−0.214453
$909$	6.37091	0.211310
$910$	1.94049	0.0643266
$911$	−37.9538	−1.25747	−0.628733	−	0.777621i	$-0.716426\pi$
$911$	−37.9538	−1.25747	−0.628733	+	0.777621i	$0.716426\pi$
$912$	8.27516	0.274018
$913$	0.305344	0.0101054
$914$	−11.7228	−0.387756
$915$	9.75685	0.322552
$916$	−17.3861	−0.574454
$917$	19.8632	0.655942
$918$	30.8979	1.01978
$919$	28.7653	0.948881	0.474440	−	0.880288i	$-0.342650\pi$
$919$	28.7653	0.948881	0.474440	+	0.880288i	$0.342650\pi$
$920$	−1.40672	−0.0463782
$921$	−27.4487	−0.904467
$922$	−0.0193888	−0.000638536 0
$923$	−6.52141	−0.214655
$924$	−1.00148	−0.0329464
$925$	22.3552	0.735035
$926$	−32.6526	−1.07303
$927$	−4.90420	−0.161075
$928$	−3.89653	−0.127910
$929$	−44.0018	−1.44365	−0.721826	−	0.692075i	$-0.756697\pi$
$929$	−44.0018	−1.44365	−0.721826	+	0.692075i	$0.756697\pi$
$930$	5.91347	0.193910
$931$	26.3989	0.865191
$932$	19.3064	0.632404
$933$	58.5850	1.91799
$934$	27.5535	0.901577
$935$	1.48254	0.0484844
$936$	−1.95648	−0.0639494
$937$	30.9671	1.01165	0.505826	−	0.862636i	$-0.331188\pi$
$937$	30.9671	1.01165	0.505826	+	0.862636i	$0.331188\pi$
$938$	−7.29169	−0.238082
$939$	−35.5781	−1.16105
$940$	−3.11340	−0.101548
$941$	−44.0721	−1.43671	−0.718355	−	0.695677i	$-0.755104\pi$
$941$	−44.0721	−1.43671	−0.718355	+	0.695677i	$0.755104\pi$
$942$	0.0942611	0.00307119
$943$	5.79542	0.188725
$944$	−2.34835	−0.0764323
$945$	−2.13364	−0.0694073
$946$	2.85551	0.0928405
$947$	−1.91844	−0.0623408	−0.0311704	−	0.999514i	$-0.509923\pi$
$947$	−1.91844	−0.0623408	−0.0311704	+	0.999514i	$0.509923\pi$
$948$	−16.3053	−0.529571
$949$	6.63567	0.215403
$950$	−21.4148	−0.694789
$951$	−37.7832	−1.22521
$952$	6.75747	0.219011
$953$	−36.0234	−1.16691	−0.583457	−	0.812144i	$-0.698300\pi$
$953$	−36.0234	−1.16691	−0.583457	+	0.812144i	$0.698300\pi$
$954$	−2.49669	−0.0808333
$955$	−3.42366	−0.110787
$956$	−11.9735	−0.387252
$957$	−3.77159	−0.121918
$958$	9.91352	0.320291
$959$	−2.82005	−0.0910643
$960$	0.810191	0.0261488
$961$	22.2734	0.718497
$962$	19.9971	0.644732
$963$	−0.259594	−0.00836531
$964$	−12.1999	−0.392931
$965$	−6.46218	−0.208025
$966$	−6.20627	−0.199683
$967$	11.7727	0.378585	0.189292	−	0.981921i	$-0.439381\pi$
$967$	11.7727	0.378585	0.189292	+	0.981921i	$0.439381\pi$
$968$	10.7288	0.344837
$969$	−54.0459	−1.73621
$970$	−7.44230	−0.238958
$971$	−4.77578	−0.153262	−0.0766311	−	0.997060i	$-0.524416\pi$
$971$	−4.77578	−0.153262	−0.0766311	+	0.997060i	$0.524416\pi$
$972$	4.68675	0.150328
$973$	1.03671	0.0332353
$974$	−26.6498	−0.853916
$975$	38.4666	1.23192
$976$	12.0427	0.385476
$977$	−7.85800	−0.251400	−0.125700	−	0.992068i	$-0.540118\pi$
$977$	−7.85800	−0.251400	−0.125700	+	0.992068i	$0.540118\pi$
$978$	8.24099	0.263518
$979$	−1.58280	−0.0505865
$980$	2.58462	0.0825628
$981$	−2.84557	−0.0908520
$982$	4.95333	0.158067
$983$	−21.7065	−0.692331	−0.346166	−	0.938173i	$-0.612516\pi$
$983$	−21.7065	−0.692331	−0.346166	+	0.938173i	$0.612516\pi$
$984$	−3.33783	−0.106406
$985$	−4.07748	−0.129919
$986$	25.4487	0.810451
$987$	−13.7359	−0.437219
$988$	−19.1559	−0.609431
$989$	17.6958	0.562693
$990$	0.103220	0.00328054
$991$	39.5243	1.25553	0.627765	−	0.778403i	$-0.283970\pi$
$991$	39.5243	1.25553	0.627765	+	0.778403i	$0.283970\pi$
$992$	7.29886	0.231739
$993$	22.3264	0.708505
$994$	1.56822	0.0497409
$995$	8.69716	0.275718
$996$	−1.08982	−0.0345324
$997$	8.52245	0.269909	0.134954	−	0.990852i	$-0.456911\pi$
$997$	8.52245	0.269909	0.134954	+	0.990852i	$0.456911\pi$
$998$	−27.3781	−0.866639
$999$	−21.9875	−0.695654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.16	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.85869 q^{3} +1.00000 q^{4} -0.435894 q^{5} +1.85869 q^{6} +1.03466 q^{7} -1.00000 q^{8} +0.454718 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.85869 q^{3} +1.00000 q^{4} -0.435894 q^{5} +1.85869 q^{6} +1.03466 q^{7} -1.00000 q^{8} +0.454718 q^{9} +0.435894 q^{10} +0.520763 q^{11} -1.85869 q^{12} +4.30262 q^{13} -1.03466 q^{14} +0.810191 q^{15} +1.00000 q^{16} -6.53111 q^{17} -0.454718 q^{18} -4.45215 q^{19} -0.435894 q^{20} -1.92311 q^{21} -0.520763 q^{22} -3.22721 q^{23} +1.85869 q^{24} -4.81000 q^{25} -4.30262 q^{26} +4.73088 q^{27} +1.03466 q^{28} +3.89653 q^{29} -0.810191 q^{30} -7.29886 q^{31} -1.00000 q^{32} -0.967935 q^{33} +6.53111 q^{34} -0.451002 q^{35} +0.454718 q^{36} -4.64766 q^{37} +4.45215 q^{38} -7.99722 q^{39} +0.435894 q^{40} -1.79580 q^{41} +1.92311 q^{42} -5.48331 q^{43} +0.520763 q^{44} -0.198209 q^{45} +3.22721 q^{46} +7.14256 q^{47} -1.85869 q^{48} -5.92948 q^{49} +4.81000 q^{50} +12.1393 q^{51} +4.30262 q^{52} +5.49063 q^{53} -4.73088 q^{54} -0.226997 q^{55} -1.03466 q^{56} +8.27516 q^{57} -3.89653 q^{58} -2.34835 q^{59} +0.810191 q^{60} +12.0427 q^{61} +7.29886 q^{62} +0.470478 q^{63} +1.00000 q^{64} -1.87549 q^{65} +0.967935 q^{66} +7.04743 q^{67} -6.53111 q^{68} +5.99836 q^{69} +0.451002 q^{70} -1.51568 q^{71} -0.454718 q^{72} +1.54224 q^{73} +4.64766 q^{74} +8.94028 q^{75} -4.45215 q^{76} +0.538813 q^{77} +7.99722 q^{78} +8.77247 q^{79} -0.435894 q^{80} -10.1574 q^{81} +1.79580 q^{82} +0.586340 q^{83} -1.92311 q^{84} +2.84687 q^{85} +5.48331 q^{86} -7.24243 q^{87} -0.520763 q^{88} -3.03939 q^{89} +0.198209 q^{90} +4.45175 q^{91} -3.22721 q^{92} +13.5663 q^{93} -7.14256 q^{94} +1.94067 q^{95} +1.85869 q^{96} -17.0736 q^{97} +5.92948 q^{98} +0.236800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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