LMFDB - Embedded newform 8002.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.99019 q^{3} +1.00000 q^{4} +2.92742 q^{5} +2.99019 q^{6} +4.68595 q^{7} -1.00000 q^{8} +5.94125 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.99019 q^{3} +1.00000 q^{4} +2.92742 q^{5} +2.99019 q^{6} +4.68595 q^{7} -1.00000 q^{8} +5.94125 q^{9} -2.92742 q^{10} -5.17836 q^{11} -2.99019 q^{12} +2.06018 q^{13} -4.68595 q^{14} -8.75354 q^{15} +1.00000 q^{16} +1.67078 q^{17} -5.94125 q^{18} -5.52131 q^{19} +2.92742 q^{20} -14.0119 q^{21} +5.17836 q^{22} -2.70870 q^{23} +2.99019 q^{24} +3.56977 q^{25} -2.06018 q^{26} -8.79489 q^{27} +4.68595 q^{28} +4.56083 q^{29} +8.75354 q^{30} +2.62806 q^{31} -1.00000 q^{32} +15.4843 q^{33} -1.67078 q^{34} +13.7177 q^{35} +5.94125 q^{36} +4.38044 q^{37} +5.52131 q^{38} -6.16034 q^{39} -2.92742 q^{40} -7.11619 q^{41} +14.0119 q^{42} -1.92574 q^{43} -5.17836 q^{44} +17.3925 q^{45} +2.70870 q^{46} -10.8832 q^{47} -2.99019 q^{48} +14.9582 q^{49} -3.56977 q^{50} -4.99595 q^{51} +2.06018 q^{52} +10.4325 q^{53} +8.79489 q^{54} -15.1592 q^{55} -4.68595 q^{56} +16.5098 q^{57} -4.56083 q^{58} -10.5100 q^{59} -8.75354 q^{60} -0.322451 q^{61} -2.62806 q^{62} +27.8404 q^{63} +1.00000 q^{64} +6.03101 q^{65} -15.4843 q^{66} +4.07640 q^{67} +1.67078 q^{68} +8.09955 q^{69} -13.7177 q^{70} -13.1116 q^{71} -5.94125 q^{72} -15.8300 q^{73} -4.38044 q^{74} -10.6743 q^{75} -5.52131 q^{76} -24.2655 q^{77} +6.16034 q^{78} -7.73672 q^{79} +2.92742 q^{80} +8.47467 q^{81} +7.11619 q^{82} +4.62573 q^{83} -14.0119 q^{84} +4.89107 q^{85} +1.92574 q^{86} -13.6377 q^{87} +5.17836 q^{88} -17.7547 q^{89} -17.3925 q^{90} +9.65391 q^{91} -2.70870 q^{92} -7.85841 q^{93} +10.8832 q^{94} -16.1632 q^{95} +2.99019 q^{96} -15.2132 q^{97} -14.9582 q^{98} -30.7659 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−2.99019	−1.72639	−0.863194	−	0.504872i	$-0.831540\pi$
$3$	−2.99019	−1.72639	−0.863194	+	0.504872i	$0.831540\pi$
$4$	1.00000	0.500000
$5$	2.92742	1.30918	0.654590	−	0.755984i	$-0.272841\pi$
$5$	2.92742	1.30918	0.654590	+	0.755984i	$0.272841\pi$
$6$	2.99019	1.22074
$7$	4.68595	1.77112	0.885562	−	0.464522i	$-0.153774\pi$
$7$	4.68595	1.77112	0.885562	+	0.464522i	$0.153774\pi$
$8$	−1.00000	−0.353553
$9$	5.94125	1.98042
$10$	−2.92742	−0.925730
$11$	−5.17836	−1.56133	−0.780666	−	0.624948i	$-0.785120\pi$
$11$	−5.17836	−1.56133	−0.780666	+	0.624948i	$0.785120\pi$
$12$	−2.99019	−0.863194
$13$	2.06018	0.571392	0.285696	−	0.958320i	$-0.407775\pi$
$13$	2.06018	0.571392	0.285696	+	0.958320i	$0.407775\pi$
$14$	−4.68595	−1.25237
$15$	−8.75354	−2.26015
$16$	1.00000	0.250000
$17$	1.67078	0.405224	0.202612	−	0.979259i	$-0.435057\pi$
$17$	1.67078	0.405224	0.202612	+	0.979259i	$0.435057\pi$
$18$	−5.94125	−1.40037
$19$	−5.52131	−1.26668	−0.633338	−	0.773876i	$-0.718316\pi$
$19$	−5.52131	−1.26668	−0.633338	+	0.773876i	$0.718316\pi$
$20$	2.92742	0.654590
$21$	−14.0119	−3.05765
$22$	5.17836	1.10403
$23$	−2.70870	−0.564804	−0.282402	−	0.959296i	$-0.591131\pi$
$23$	−2.70870	−0.564804	−0.282402	+	0.959296i	$0.591131\pi$
$24$	2.99019	0.610370
$25$	3.56977	0.713954
$26$	−2.06018	−0.404035
$27$	−8.79489	−1.69258
$28$	4.68595	0.885562
$29$	4.56083	0.846924	0.423462	−	0.905914i	$-0.360815\pi$
$29$	4.56083	0.846924	0.423462	+	0.905914i	$0.360815\pi$
$30$	8.75354	1.59817
$31$	2.62806	0.472014	0.236007	−	0.971751i	$-0.424161\pi$
$31$	2.62806	0.472014	0.236007	+	0.971751i	$0.424161\pi$
$32$	−1.00000	−0.176777
$33$	15.4843	2.69547
$34$	−1.67078	−0.286536
$35$	13.7177	2.31872
$36$	5.94125	0.990208
$37$	4.38044	0.720139	0.360070	−	0.932925i	$-0.382753\pi$
$37$	4.38044	0.720139	0.360070	+	0.932925i	$0.382753\pi$
$38$	5.52131	0.895675
$39$	−6.16034	−0.986444
$40$	−2.92742	−0.462865
$41$	−7.11619	−1.11136	−0.555681	−	0.831395i	$-0.687543\pi$
$41$	−7.11619	−1.11136	−0.555681	+	0.831395i	$0.687543\pi$
$42$	14.0119	2.16208
$43$	−1.92574	−0.293673	−0.146836	−	0.989161i	$-0.546909\pi$
$43$	−1.92574	−0.293673	−0.146836	+	0.989161i	$0.546909\pi$
$44$	−5.17836	−0.780666
$45$	17.3925	2.59272
$46$	2.70870	0.399377
$47$	−10.8832	−1.58748	−0.793742	−	0.608254i	$-0.791870\pi$
$47$	−10.8832	−1.58748	−0.793742	+	0.608254i	$0.791870\pi$
$48$	−2.99019	−0.431597
$49$	14.9582	2.13688
$50$	−3.56977	−0.504842
$51$	−4.99595	−0.699573
$52$	2.06018	0.285696
$53$	10.4325	1.43301	0.716506	−	0.697581i	$-0.245740\pi$
$53$	10.4325	1.43301	0.716506	+	0.697581i	$0.245740\pi$
$54$	8.79489	1.19683
$55$	−15.1592	−2.04407
$56$	−4.68595	−0.626187
$57$	16.5098	2.18677
$58$	−4.56083	−0.598866
$59$	−10.5100	−1.36828	−0.684142	−	0.729349i	$-0.739823\pi$
$59$	−10.5100	−1.36828	−0.684142	+	0.729349i	$0.739823\pi$
$60$	−8.75354	−1.13008
$61$	−0.322451	−0.0412856	−0.0206428	−	0.999787i	$-0.506571\pi$
$61$	−0.322451	−0.0412856	−0.0206428	+	0.999787i	$0.506571\pi$
$62$	−2.62806	−0.333764
$63$	27.8404	3.50756
$64$	1.00000	0.125000
$65$	6.03101	0.748055
$66$	−15.4843	−1.90598
$67$	4.07640	0.498012	0.249006	−	0.968502i	$-0.419896\pi$
$67$	4.07640	0.498012	0.249006	+	0.968502i	$0.419896\pi$
$68$	1.67078	0.202612
$69$	8.09955	0.975071
$70$	−13.7177	−1.63958
$71$	−13.1116	−1.55606	−0.778032	−	0.628224i	$-0.783782\pi$
$71$	−13.1116	−1.55606	−0.778032	+	0.628224i	$0.783782\pi$
$72$	−5.94125	−0.700183
$73$	−15.8300	−1.85276	−0.926380	−	0.376590i	$-0.877096\pi$
$73$	−15.8300	−1.85276	−0.926380	+	0.376590i	$0.877096\pi$
$74$	−4.38044	−0.509215
$75$	−10.6743	−1.23256
$76$	−5.52131	−0.633338
$77$	−24.2655	−2.76531
$78$	6.16034	0.697521
$79$	−7.73672	−0.870449	−0.435225	−	0.900322i	$-0.643331\pi$
$79$	−7.73672	−0.870449	−0.435225	+	0.900322i	$0.643331\pi$
$80$	2.92742	0.327295
$81$	8.47467	0.941630
$82$	7.11619	0.785852
$83$	4.62573	0.507739	0.253870	−	0.967238i	$-0.418297\pi$
$83$	4.62573	0.507739	0.253870	+	0.967238i	$0.418297\pi$
$84$	−14.0119	−1.52882
$85$	4.89107	0.530511
$86$	1.92574	0.207658
$87$	−13.6377	−1.46212
$88$	5.17836	0.552015
$89$	−17.7547	−1.88199	−0.940995	−	0.338421i	$-0.890107\pi$
$89$	−17.7547	−1.88199	−0.940995	+	0.338421i	$0.890107\pi$
$90$	−17.3925	−1.83333
$91$	9.65391	1.01201
$92$	−2.70870	−0.282402
$93$	−7.85841	−0.814880
$94$	10.8832	1.12252
$95$	−16.1632	−1.65831
$96$	2.99019	0.305185
$97$	−15.2132	−1.54467	−0.772333	−	0.635218i	$-0.780910\pi$
$97$	−15.2132	−1.54467	−0.772333	+	0.635218i	$0.780910\pi$
$98$	−14.9582	−1.51100
$99$	−30.7659	−3.09209
$100$	3.56977	0.356977
$101$	−7.79148	−0.775282	−0.387641	−	0.921811i	$-0.626710\pi$
$101$	−7.79148	−0.775282	−0.387641	+	0.921811i	$0.626710\pi$
$102$	4.99595	0.494673
$103$	−9.65313	−0.951151	−0.475576	−	0.879675i	$-0.657760\pi$
$103$	−9.65313	−0.951151	−0.475576	+	0.879675i	$0.657760\pi$
$104$	−2.06018	−0.202017
$105$	−41.0187	−4.00301
$106$	−10.4325	−1.01329
$107$	13.7320	1.32752	0.663761	−	0.747944i	$-0.268959\pi$
$107$	13.7320	1.32752	0.663761	+	0.747944i	$0.268959\pi$
$108$	−8.79489	−0.846289
$109$	6.89391	0.660317	0.330159	−	0.943925i	$-0.392898\pi$
$109$	6.89391	0.660317	0.330159	+	0.943925i	$0.392898\pi$
$110$	15.1592	1.44537
$111$	−13.0983	−1.24324
$112$	4.68595	0.442781
$113$	9.76257	0.918385	0.459193	−	0.888337i	$-0.348139\pi$
$113$	9.76257	0.918385	0.459193	+	0.888337i	$0.348139\pi$
$114$	−16.5098	−1.54628
$115$	−7.92951	−0.739430
$116$	4.56083	0.423462
$117$	12.2400	1.13159
$118$	10.5100	0.967523
$119$	7.82919	0.717701
$120$	8.75354	0.799085
$121$	15.8154	1.43776
$122$	0.322451	0.0291934
$123$	21.2788	1.91864
$124$	2.62806	0.236007
$125$	−4.18688	−0.374486
$126$	−27.8404	−2.48022
$127$	12.9876	1.15246	0.576231	−	0.817287i	$-0.304523\pi$
$127$	12.9876	1.15246	0.576231	+	0.817287i	$0.304523\pi$
$128$	−1.00000	−0.0883883
$129$	5.75834	0.506993
$130$	−6.03101	−0.528955
$131$	−5.32795	−0.465505	−0.232753	−	0.972536i	$-0.574773\pi$
$131$	−5.32795	−0.465505	−0.232753	+	0.972536i	$0.574773\pi$
$132$	15.4843	1.34773
$133$	−25.8726	−2.24344
$134$	−4.07640	−0.352147
$135$	−25.7463	−2.21589
$136$	−1.67078	−0.143268
$137$	−0.684963	−0.0585203	−0.0292602	−	0.999572i	$-0.509315\pi$
$137$	−0.684963	−0.0585203	−0.0292602	+	0.999572i	$0.509315\pi$
$138$	−8.09955	−0.689479
$139$	−21.8613	−1.85425	−0.927124	−	0.374754i	$-0.877727\pi$
$139$	−21.8613	−1.85425	−0.927124	+	0.374754i	$0.877727\pi$
$140$	13.7177	1.15936
$141$	32.5430	2.74061
$142$	13.1116	1.10030
$143$	−10.6684	−0.892132
$144$	5.94125	0.495104
$145$	13.3514	1.10878
$146$	15.8300	1.31010
$147$	−44.7277	−3.68908
$148$	4.38044	0.360070
$149$	2.96041	0.242526	0.121263	−	0.992620i	$-0.461306\pi$
$149$	2.96041	0.242526	0.121263	+	0.992620i	$0.461306\pi$
$150$	10.6743	0.871553
$151$	10.1721	0.827797	0.413899	−	0.910323i	$-0.364167\pi$
$151$	10.1721	0.827797	0.413899	+	0.910323i	$0.364167\pi$
$152$	5.52131	0.447837
$153$	9.92651	0.802511
$154$	24.2655	1.95537
$155$	7.69344	0.617952
$156$	−6.16034	−0.493222
$157$	5.80418	0.463224	0.231612	−	0.972808i	$-0.425600\pi$
$157$	5.80418	0.463224	0.231612	+	0.972808i	$0.425600\pi$
$158$	7.73672	0.615501
$159$	−31.1951	−2.47393
$160$	−2.92742	−0.231433
$161$	−12.6929	−1.00034
$162$	−8.47467	−0.665833
$163$	5.21845	0.408741	0.204370	−	0.978894i	$-0.434485\pi$
$163$	5.21845	0.408741	0.204370	+	0.978894i	$0.434485\pi$
$164$	−7.11619	−0.555681
$165$	45.3289	3.52885
$166$	−4.62573	−0.359026
$167$	−2.67211	−0.206774	−0.103387	−	0.994641i	$-0.532968\pi$
$167$	−2.67211	−0.206774	−0.103387	+	0.994641i	$0.532968\pi$
$168$	14.0119	1.08104
$169$	−8.75565	−0.673512
$170$	−4.89107	−0.375128
$171$	−32.8035	−2.50854
$172$	−1.92574	−0.146836
$173$	−20.3028	−1.54359	−0.771797	−	0.635869i	$-0.780642\pi$
$173$	−20.3028	−1.54359	−0.771797	+	0.635869i	$0.780642\pi$
$174$	13.6377	1.03387
$175$	16.7278	1.26450
$176$	−5.17836	−0.390333
$177$	31.4269	2.36219
$178$	17.7547	1.33077
$179$	−3.64166	−0.272190	−0.136095	−	0.990696i	$-0.543455\pi$
$179$	−3.64166	−0.272190	−0.136095	+	0.990696i	$0.543455\pi$
$180$	17.3925	1.29636
$181$	13.5131	1.00442	0.502212	−	0.864745i	$-0.332520\pi$
$181$	13.5131	1.00442	0.502212	+	0.864745i	$0.332520\pi$
$182$	−9.65391	−0.715596
$183$	0.964191	0.0712750
$184$	2.70870	0.199688
$185$	12.8234	0.942792
$186$	7.85841	0.576207
$187$	−8.65189	−0.632689
$188$	−10.8832	−0.793742
$189$	−41.2124	−2.99776
$190$	16.1632	1.17260
$191$	15.8814	1.14914	0.574568	−	0.818457i	$-0.305170\pi$
$191$	15.8814	1.14914	0.574568	+	0.818457i	$0.305170\pi$
$192$	−2.99019	−0.215799
$193$	−19.9696	−1.43744	−0.718721	−	0.695299i	$-0.755272\pi$
$193$	−19.9696	−1.43744	−0.718721	+	0.695299i	$0.755272\pi$
$194$	15.2132	1.09224
$195$	−18.0339	−1.29143
$196$	14.9582	1.06844
$197$	−25.3449	−1.80575	−0.902876	−	0.429900i	$-0.858549\pi$
$197$	−25.3449	−1.80575	−0.902876	+	0.429900i	$0.858549\pi$
$198$	30.7659	2.18644
$199$	−12.2240	−0.866535	−0.433267	−	0.901265i	$-0.642639\pi$
$199$	−12.2240	−0.866535	−0.433267	+	0.901265i	$0.642639\pi$
$200$	−3.56977	−0.252421
$201$	−12.1892	−0.859761
$202$	7.79148	0.548207
$203$	21.3718	1.50001
$204$	−4.99595	−0.349787
$205$	−20.8321	−1.45497
$206$	9.65313	0.672566
$207$	−16.0931	−1.11855
$208$	2.06018	0.142848
$209$	28.5913	1.97770
$210$	41.0187	2.83056
$211$	10.6980	0.736480	0.368240	−	0.929731i	$-0.379960\pi$
$211$	10.6980	0.736480	0.368240	+	0.929731i	$0.379960\pi$
$212$	10.4325	0.716506
$213$	39.2063	2.68637
$214$	−13.7320	−0.938700
$215$	−5.63745	−0.384471
$216$	8.79489	0.598417
$217$	12.3150	0.835995
$218$	−6.89391	−0.466915
$219$	47.3347	3.19858
$220$	−15.1592	−1.02203
$221$	3.44211	0.231541
$222$	13.0983	0.879103
$223$	10.7899	0.722542	0.361271	−	0.932461i	$-0.382343\pi$
$223$	10.7899	0.722542	0.361271	+	0.932461i	$0.382343\pi$
$224$	−4.68595	−0.313093
$225$	21.2089	1.41393
$226$	−9.76257	−0.649396
$227$	−6.94563	−0.460998	−0.230499	−	0.973073i	$-0.574036\pi$
$227$	−6.94563	−0.460998	−0.230499	+	0.973073i	$0.574036\pi$
$228$	16.5098	1.09339
$229$	4.15825	0.274785	0.137393	−	0.990517i	$-0.456128\pi$
$229$	4.15825	0.274785	0.137393	+	0.990517i	$0.456128\pi$
$230$	7.92951	0.522856
$231$	72.5586	4.77400
$232$	−4.56083	−0.299433
$233$	18.3145	1.19982	0.599910	−	0.800067i	$-0.295203\pi$
$233$	18.3145	1.19982	0.599910	+	0.800067i	$0.295203\pi$
$234$	−12.2400	−0.800157
$235$	−31.8598	−2.07830
$236$	−10.5100	−0.684142
$237$	23.1343	1.50273
$238$	−7.82919	−0.507491
$239$	−2.85638	−0.184764	−0.0923820	−	0.995724i	$-0.529448\pi$
$239$	−2.85638	−0.184764	−0.0923820	+	0.995724i	$0.529448\pi$
$240$	−8.75354	−0.565038
$241$	−7.80775	−0.502941	−0.251471	−	0.967865i	$-0.580914\pi$
$241$	−7.80775	−0.502941	−0.251471	+	0.967865i	$0.580914\pi$
$242$	−15.8154	−1.01665
$243$	1.04378	0.0669585
$244$	−0.322451	−0.0206428
$245$	43.7888	2.79756
$246$	−21.2788	−1.35669
$247$	−11.3749	−0.723768
$248$	−2.62806	−0.166882
$249$	−13.8318	−0.876555
$250$	4.18688	0.264802
$251$	17.0013	1.07311	0.536557	−	0.843864i	$-0.319725\pi$
$251$	17.0013	1.07311	0.536557	+	0.843864i	$0.319725\pi$
$252$	27.8404	1.75378
$253$	14.0266	0.881847
$254$	−12.9876	−0.814914
$255$	−14.6252	−0.915867
$256$	1.00000	0.0625000
$257$	20.5889	1.28430	0.642149	−	0.766580i	$-0.278043\pi$
$257$	20.5889	1.28430	0.642149	+	0.766580i	$0.278043\pi$
$258$	−5.75834	−0.358499
$259$	20.5265	1.27546
$260$	6.03101	0.374027
$261$	27.0970	1.67726
$262$	5.32795	0.329162
$263$	−23.2881	−1.43600	−0.718002	−	0.696042i	$-0.754943\pi$
$263$	−23.2881	−1.43600	−0.718002	+	0.696042i	$0.754943\pi$
$264$	−15.4843	−0.952991
$265$	30.5402	1.87607
$266$	25.8726	1.58635
$267$	53.0898	3.24904
$268$	4.07640	0.249006
$269$	−19.9471	−1.21620	−0.608098	−	0.793862i	$-0.708067\pi$
$269$	−19.9471	−1.21620	−0.608098	+	0.793862i	$0.708067\pi$
$270$	25.7463	1.56687
$271$	19.4836	1.18354	0.591771	−	0.806106i	$-0.298429\pi$
$271$	19.4836	1.18354	0.591771	+	0.806106i	$0.298429\pi$
$272$	1.67078	0.101306
$273$	−28.8671	−1.74711
$274$	0.684963	0.0413801
$275$	−18.4855	−1.11472
$276$	8.09955	0.487535
$277$	−17.9525	−1.07866	−0.539330	−	0.842094i	$-0.681323\pi$
$277$	−17.9525	−1.07866	−0.539330	+	0.842094i	$0.681323\pi$
$278$	21.8613	1.31115
$279$	15.6140	0.934784
$280$	−13.7177	−0.819792
$281$	−26.7488	−1.59570	−0.797851	−	0.602855i	$-0.794030\pi$
$281$	−26.7488	−1.59570	−0.797851	+	0.602855i	$0.794030\pi$
$282$	−32.5430	−1.93791
$283$	3.51865	0.209162	0.104581	−	0.994516i	$-0.466650\pi$
$283$	3.51865	0.209162	0.104581	+	0.994516i	$0.466650\pi$
$284$	−13.1116	−0.778032
$285$	48.3310	2.86288
$286$	10.6684	0.630833
$287$	−33.3461	−1.96836
$288$	−5.94125	−0.350091
$289$	−14.2085	−0.835794
$290$	−13.3514	−0.784023
$291$	45.4904	2.66669
$292$	−15.8300	−0.926380
$293$	−12.0285	−0.702711	−0.351356	−	0.936242i	$-0.614279\pi$
$293$	−12.0285	−0.702711	−0.351356	+	0.936242i	$0.614279\pi$
$294$	44.7277	2.60858
$295$	−30.7671	−1.79133
$296$	−4.38044	−0.254608
$297$	45.5431	2.64268
$298$	−2.96041	−0.171492
$299$	−5.58042	−0.322724
$300$	−10.6743	−0.616281
$301$	−9.02394	−0.520131
$302$	−10.1721	−0.585341
$303$	23.2980	1.33844
$304$	−5.52131	−0.316669
$305$	−0.943949	−0.0540504
$306$	−9.92651	−0.567461
$307$	12.1061	0.690931	0.345465	−	0.938432i	$-0.387721\pi$
$307$	12.1061	0.690931	0.345465	+	0.938432i	$0.387721\pi$
$308$	−24.2655	−1.38266
$309$	28.8647	1.64206
$310$	−7.69344	−0.436958
$311$	−27.5187	−1.56044	−0.780222	−	0.625503i	$-0.784894\pi$
$311$	−27.5187	−1.56044	−0.780222	+	0.625503i	$0.784894\pi$
$312$	6.16034	0.348760
$313$	23.6046	1.33421	0.667106	−	0.744963i	$-0.267533\pi$
$313$	23.6046	1.33421	0.667106	+	0.744963i	$0.267533\pi$
$314$	−5.80418	−0.327549
$315$	81.5005	4.59203
$316$	−7.73672	−0.435225
$317$	−11.5331	−0.647763	−0.323882	−	0.946098i	$-0.604988\pi$
$317$	−11.5331	−0.647763	−0.323882	+	0.946098i	$0.604988\pi$
$318$	31.1951	1.74934
$319$	−23.6176	−1.32233
$320$	2.92742	0.163648
$321$	−41.0613	−2.29182
$322$	12.6929	0.707346
$323$	−9.22489	−0.513287
$324$	8.47467	0.470815
$325$	7.35437	0.407947
$326$	−5.21845	−0.289023
$327$	−20.6141	−1.13996
$328$	7.11619	0.392926
$329$	−50.9984	−2.81163
$330$	−45.3289	−2.49528
$331$	34.5706	1.90017	0.950087	−	0.311985i	$-0.100994\pi$
$331$	34.5706	1.90017	0.950087	+	0.311985i	$0.100994\pi$
$332$	4.62573	0.253870
$333$	26.0252	1.42617
$334$	2.67211	0.146211
$335$	11.9333	0.651987
$336$	−14.0119	−0.764412
$337$	4.47468	0.243752	0.121876	−	0.992545i	$-0.461109\pi$
$337$	4.47468	0.243752	0.121876	+	0.992545i	$0.461109\pi$
$338$	8.75565	0.476245
$339$	−29.1920	−1.58549
$340$	4.89107	0.265255
$341$	−13.6090	−0.736971
$342$	32.8035	1.77381
$343$	37.2915	2.01355
$344$	1.92574	0.103829
$345$	23.7107	1.27654
$346$	20.3028	1.09149
$347$	−11.2646	−0.604717	−0.302358	−	0.953194i	$-0.597774\pi$
$347$	−11.2646	−0.604717	−0.302358	+	0.953194i	$0.597774\pi$
$348$	−13.6377	−0.731060
$349$	−17.2848	−0.925237	−0.462618	−	0.886558i	$-0.653090\pi$
$349$	−17.2848	−0.925237	−0.462618	+	0.886558i	$0.653090\pi$
$350$	−16.7278	−0.894137
$351$	−18.1191	−0.967125
$352$	5.17836	0.276007
$353$	16.7501	0.891517	0.445759	−	0.895153i	$-0.352934\pi$
$353$	16.7501	0.891517	0.445759	+	0.895153i	$0.352934\pi$
$354$	−31.4269	−1.67032
$355$	−38.3832	−2.03717
$356$	−17.7547	−0.940995
$357$	−23.4108	−1.23903
$358$	3.64166	0.192468
$359$	−0.0884378	−0.00466757	−0.00233378	−	0.999997i	$-0.500743\pi$
$359$	−0.0884378	−0.00466757	−0.00233378	+	0.999997i	$0.500743\pi$
$360$	−17.3925	−0.916666
$361$	11.4849	0.604467
$362$	−13.5131	−0.710235
$363$	−47.2910	−2.48213
$364$	9.65391	0.506003
$365$	−46.3410	−2.42560
$366$	−0.964191	−0.0503991
$367$	6.60464	0.344759	0.172380	−	0.985031i	$-0.444854\pi$
$367$	6.60464	0.344759	0.172380	+	0.985031i	$0.444854\pi$
$368$	−2.70870	−0.141201
$369$	−42.2791	−2.20096
$370$	−12.8234	−0.666655
$371$	48.8861	2.53804
$372$	−7.85841	−0.407440
$373$	22.3665	1.15809	0.579046	−	0.815295i	$-0.303425\pi$
$373$	22.3665	1.15809	0.579046	+	0.815295i	$0.303425\pi$
$374$	8.65189	0.447379
$375$	12.5196	0.646508
$376$	10.8832	0.561261
$377$	9.39613	0.483925
$378$	41.2124	2.11974
$379$	15.8626	0.814806	0.407403	−	0.913249i	$-0.366434\pi$
$379$	15.8626	0.814806	0.407403	+	0.913249i	$0.366434\pi$
$380$	−16.1632	−0.829153
$381$	−38.8354	−1.98960
$382$	−15.8814	−0.812562
$383$	−25.2229	−1.28883	−0.644415	−	0.764676i	$-0.722899\pi$
$383$	−25.2229	−1.28883	−0.644415	+	0.764676i	$0.722899\pi$
$384$	2.99019	0.152593
$385$	−71.0353	−3.62029
$386$	19.9696	1.01642
$387$	−11.4413	−0.581594
$388$	−15.2132	−0.772333
$389$	30.7020	1.55665	0.778327	−	0.627859i	$-0.216069\pi$
$389$	30.7020	1.55665	0.778327	+	0.627859i	$0.216069\pi$
$390$	18.0339	0.913181
$391$	−4.52565	−0.228872
$392$	−14.9582	−0.755501
$393$	15.9316	0.803643
$394$	25.3449	1.27686
$395$	−22.6486	−1.13958
$396$	−30.7659	−1.54604
$397$	−14.8403	−0.744815	−0.372408	−	0.928069i	$-0.621468\pi$
$397$	−14.8403	−0.744815	−0.372408	+	0.928069i	$0.621468\pi$
$398$	12.2240	0.612733
$399$	77.3640	3.87305
$400$	3.56977	0.178488
$401$	−1.11915	−0.0558876	−0.0279438	−	0.999609i	$-0.508896\pi$
$401$	−1.11915	−0.0558876	−0.0279438	+	0.999609i	$0.508896\pi$
$402$	12.1892	0.607943
$403$	5.41429	0.269705
$404$	−7.79148	−0.387641
$405$	24.8089	1.23276
$406$	−21.3718	−1.06067
$407$	−22.6835	−1.12438
$408$	4.99595	0.247336
$409$	25.2216	1.24713	0.623563	−	0.781773i	$-0.285684\pi$
$409$	25.2216	1.24713	0.623563	+	0.781773i	$0.285684\pi$
$410$	20.8321	1.02882
$411$	2.04817	0.101029
$412$	−9.65313	−0.475576
$413$	−49.2493	−2.42340
$414$	16.0931	0.790932
$415$	13.5414	0.664723
$416$	−2.06018	−0.101009
$417$	65.3694	3.20115
$418$	−28.5913	−1.39845
$419$	8.71709	0.425858	0.212929	−	0.977068i	$-0.431700\pi$
$419$	8.71709	0.425858	0.212929	+	0.977068i	$0.431700\pi$
$420$	−41.0187	−2.00151
$421$	15.1977	0.740691	0.370345	−	0.928894i	$-0.379239\pi$
$421$	15.1977	0.740691	0.370345	+	0.928894i	$0.379239\pi$
$422$	−10.6980	−0.520770
$423$	−64.6600	−3.14388
$424$	−10.4325	−0.506646
$425$	5.96430	0.289311
$426$	−39.2063	−1.89955
$427$	−1.51099	−0.0731220
$428$	13.7320	0.663761
$429$	31.9004	1.54017
$430$	5.63745	0.271862
$431$	24.9038	1.19957	0.599787	−	0.800160i	$-0.295252\pi$
$431$	24.9038	1.19957	0.599787	+	0.800160i	$0.295252\pi$
$432$	−8.79489	−0.423144
$433$	−30.6843	−1.47460	−0.737298	−	0.675568i	$-0.763898\pi$
$433$	−30.6843	−1.47460	−0.737298	+	0.675568i	$0.763898\pi$
$434$	−12.3150	−0.591138
$435$	−39.9234	−1.91418
$436$	6.89391	0.330159
$437$	14.9556	0.715423
$438$	−47.3347	−2.26174
$439$	−11.8829	−0.567141	−0.283571	−	0.958951i	$-0.591519\pi$
$439$	−11.8829	−0.567141	−0.283571	+	0.958951i	$0.591519\pi$
$440$	15.1592	0.722687
$441$	88.8701	4.23191
$442$	−3.44211	−0.163724
$443$	8.67085	0.411965	0.205982	−	0.978556i	$-0.433961\pi$
$443$	8.67085	0.411965	0.205982	+	0.978556i	$0.433961\pi$
$444$	−13.0983	−0.621620
$445$	−51.9753	−2.46386
$446$	−10.7899	−0.510915
$447$	−8.85218	−0.418694
$448$	4.68595	0.221390
$449$	−31.3266	−1.47840	−0.739198	−	0.673489i	$-0.764795\pi$
$449$	−31.3266	−1.47840	−0.739198	+	0.673489i	$0.764795\pi$
$450$	−21.2089	−0.999796
$451$	36.8502	1.73521
$452$	9.76257	0.459193
$453$	−30.4166	−1.42910
$454$	6.94563	0.325975
$455$	28.2610	1.32490
$456$	−16.5098	−0.773141
$457$	0.900942	0.0421443	0.0210721	−	0.999778i	$-0.493292\pi$
$457$	0.900942	0.0421443	0.0210721	+	0.999778i	$0.493292\pi$
$458$	−4.15825	−0.194302
$459$	−14.6943	−0.685872
$460$	−7.92951	−0.369715
$461$	−25.5950	−1.19208	−0.596039	−	0.802955i	$-0.703260\pi$
$461$	−25.5950	−1.19208	−0.596039	+	0.802955i	$0.703260\pi$
$462$	−72.5586	−3.37573
$463$	−3.85361	−0.179092	−0.0895461	−	0.995983i	$-0.528542\pi$
$463$	−3.85361	−0.179092	−0.0895461	+	0.995983i	$0.528542\pi$
$464$	4.56083	0.211731
$465$	−23.0049	−1.06682
$466$	−18.3145	−0.848401
$467$	24.2665	1.12292	0.561460	−	0.827504i	$-0.310240\pi$
$467$	24.2665	1.12292	0.561460	+	0.827504i	$0.310240\pi$
$468$	12.2400	0.565796
$469$	19.1018	0.882040
$470$	31.8598	1.46958
$471$	−17.3556	−0.799704
$472$	10.5100	0.483762
$473$	9.97218	0.458521
$474$	−23.1343	−1.06259
$475$	−19.7098	−0.904348
$476$	7.82919	0.358851
$477$	61.9820	2.83796
$478$	2.85638	0.130648
$479$	10.7624	0.491745	0.245873	−	0.969302i	$-0.420926\pi$
$479$	10.7624	0.491745	0.245873	+	0.969302i	$0.420926\pi$
$480$	8.75354	0.399543
$481$	9.02449	0.411481
$482$	7.80775	0.355633
$483$	37.9541	1.72697
$484$	15.8154	0.718880
$485$	−44.5354	−2.02225
$486$	−1.04378	−0.0473468
$487$	−36.5062	−1.65425	−0.827126	−	0.562017i	$-0.810026\pi$
$487$	−36.5062	−1.65425	−0.827126	+	0.562017i	$0.810026\pi$
$488$	0.322451	0.0145967
$489$	−15.6042	−0.705645
$490$	−43.7888	−1.97817
$491$	31.6993	1.43057	0.715285	−	0.698833i	$-0.246297\pi$
$491$	31.6993	1.43057	0.715285	+	0.698833i	$0.246297\pi$
$492$	21.2788	0.959321
$493$	7.62013	0.343194
$494$	11.3749	0.511781
$495$	−90.0646	−4.04810
$496$	2.62806	0.118004
$497$	−61.4405	−2.75598
$498$	13.8318	0.619818
$499$	−4.78448	−0.214183	−0.107091	−	0.994249i	$-0.534154\pi$
$499$	−4.78448	−0.214183	−0.107091	+	0.994249i	$0.534154\pi$
$500$	−4.18688	−0.187243
$501$	7.99012	0.356972
$502$	−17.0013	−0.758806
$503$	31.4245	1.40115	0.700574	−	0.713579i	$-0.252927\pi$
$503$	31.4245	1.40115	0.700574	+	0.713579i	$0.252927\pi$
$504$	−27.8404	−1.24011
$505$	−22.8089	−1.01498
$506$	−14.0266	−0.623560
$507$	26.1811	1.16274
$508$	12.9876	0.576231
$509$	10.7446	0.476244	0.238122	−	0.971235i	$-0.423468\pi$
$509$	10.7446	0.476244	0.238122	+	0.971235i	$0.423468\pi$
$510$	14.6252	0.647616
$511$	−74.1786	−3.28147
$512$	−1.00000	−0.0441942
$513$	48.5593	2.14395
$514$	−20.5889	−0.908136
$515$	−28.2587	−1.24523
$516$	5.75834	0.253497
$517$	56.3573	2.47859
$518$	−20.5265	−0.901883
$519$	60.7093	2.66484
$520$	−6.03101	−0.264477
$521$	12.2185	0.535301	0.267650	−	0.963516i	$-0.413753\pi$
$521$	12.2185	0.535301	0.267650	+	0.963516i	$0.413753\pi$
$522$	−27.0970	−1.18600
$523$	12.1246	0.530170	0.265085	−	0.964225i	$-0.414600\pi$
$523$	12.1246	0.530170	0.265085	+	0.964225i	$0.414600\pi$
$524$	−5.32795	−0.232753
$525$	−50.0192	−2.18302
$526$	23.2881	1.01541
$527$	4.39091	0.191271
$528$	15.4843	0.673867
$529$	−15.6629	−0.680997
$530$	−30.5402	−1.32658
$531$	−62.4425	−2.70977
$532$	−25.8726	−1.12172
$533$	−14.6606	−0.635023
$534$	−53.0898	−2.29742
$535$	40.1993	1.73797
$536$	−4.07640	−0.176074
$537$	10.8893	0.469906
$538$	19.9471	0.859980
$539$	−77.4586	−3.33638
$540$	−25.7463	−1.10794
$541$	43.3051	1.86183	0.930916	−	0.365233i	$-0.119011\pi$
$541$	43.3051	1.86183	0.930916	+	0.365233i	$0.119011\pi$
$542$	−19.4836	−0.836891
$543$	−40.4069	−1.73403
$544$	−1.67078	−0.0716341
$545$	20.1814	0.864475
$546$	28.8671	1.23540
$547$	24.6240	1.05284	0.526422	−	0.850223i	$-0.323533\pi$
$547$	24.6240	1.05284	0.526422	+	0.850223i	$0.323533\pi$
$548$	−0.684963	−0.0292602
$549$	−1.91576	−0.0817627
$550$	18.4855	0.788226
$551$	−25.1817	−1.07278
$552$	−8.09955	−0.344740
$553$	−36.2539	−1.54167
$554$	17.9525	0.762728
$555$	−38.3443	−1.62763
$556$	−21.8613	−0.927124
$557$	13.1198	0.555904	0.277952	−	0.960595i	$-0.410344\pi$
$557$	13.1198	0.555904	0.277952	+	0.960595i	$0.410344\pi$
$558$	−15.6140	−0.660992
$559$	−3.96738	−0.167802
$560$	13.7177	0.579680
$561$	25.8708	1.09227
$562$	26.7488	1.12833
$563$	0.618306	0.0260585	0.0130292	−	0.999915i	$-0.495853\pi$
$563$	0.618306	0.0260585	0.0130292	+	0.999915i	$0.495853\pi$
$564$	32.5430	1.37031
$565$	28.5791	1.20233
$566$	−3.51865	−0.147900
$567$	39.7119	1.66774
$568$	13.1116	0.550152
$569$	31.4567	1.31873	0.659366	−	0.751822i	$-0.270825\pi$
$569$	31.4567	1.31873	0.659366	+	0.751822i	$0.270825\pi$
$570$	−48.3310	−2.02436
$571$	−31.2906	−1.30947	−0.654736	−	0.755858i	$-0.727220\pi$
$571$	−31.2906	−1.30947	−0.654736	+	0.755858i	$0.727220\pi$
$572$	−10.6684	−0.446066
$573$	−47.4883	−1.98385
$574$	33.3461	1.39184
$575$	−9.66945	−0.403244
$576$	5.94125	0.247552
$577$	13.2958	0.553511	0.276755	−	0.960940i	$-0.410741\pi$
$577$	13.2958	0.553511	0.276755	+	0.960940i	$0.410741\pi$
$578$	14.2085	0.590996
$579$	59.7129	2.48158
$580$	13.3514	0.554388
$581$	21.6759	0.899269
$582$	−45.4904	−1.88564
$583$	−54.0231	−2.23741
$584$	15.8300	0.655050
$585$	35.8317	1.48146
$586$	12.0285	0.496892
$587$	−21.1160	−0.871553	−0.435776	−	0.900055i	$-0.643526\pi$
$587$	−21.1160	−0.871553	−0.435776	+	0.900055i	$0.643526\pi$
$588$	−44.7277	−1.84454
$589$	−14.5104	−0.597889
$590$	30.7671	1.26666
$591$	75.7863	3.11743
$592$	4.38044	0.180035
$593$	−19.3578	−0.794929	−0.397464	−	0.917618i	$-0.630110\pi$
$593$	−19.3578	−0.794929	−0.397464	+	0.917618i	$0.630110\pi$
$594$	−45.5431	−1.86865
$595$	22.9193	0.939600
$596$	2.96041	0.121263
$597$	36.5520	1.49598
$598$	5.58042	0.228200
$599$	10.2701	0.419624	0.209812	−	0.977742i	$-0.432715\pi$
$599$	10.2701	0.419624	0.209812	+	0.977742i	$0.432715\pi$
$600$	10.6743	0.435776
$601$	20.8754	0.851524	0.425762	−	0.904835i	$-0.360006\pi$
$601$	20.8754	0.851524	0.425762	+	0.904835i	$0.360006\pi$
$602$	9.02394	0.367788
$603$	24.2189	0.986270
$604$	10.1721	0.413899
$605$	46.2982	1.88229
$606$	−23.2980	−0.946418
$607$	−9.45756	−0.383871	−0.191935	−	0.981408i	$-0.561476\pi$
$607$	−9.45756	−0.383871	−0.191935	+	0.981408i	$0.561476\pi$
$608$	5.52131	0.223919
$609$	−63.9058	−2.58959
$610$	0.943949	0.0382194
$611$	−22.4215	−0.907075
$612$	9.92651	0.401256
$613$	21.3189	0.861061	0.430531	−	0.902576i	$-0.358326\pi$
$613$	21.3189	0.861061	0.430531	+	0.902576i	$0.358326\pi$
$614$	−12.1061	−0.488562
$615$	62.2919	2.51185
$616$	24.2655	0.977686
$617$	−4.74353	−0.190967	−0.0954836	−	0.995431i	$-0.530440\pi$
$617$	−4.74353	−0.190967	−0.0954836	+	0.995431i	$0.530440\pi$
$618$	−28.8647	−1.16111
$619$	−6.18721	−0.248685	−0.124343	−	0.992239i	$-0.539682\pi$
$619$	−6.18721	−0.248685	−0.124343	+	0.992239i	$0.539682\pi$
$620$	7.69344	0.308976
$621$	23.8228	0.955975
$622$	27.5187	1.10340
$623$	−83.1975	−3.33324
$624$	−6.16034	−0.246611
$625$	−30.1056	−1.20422
$626$	−23.6046	−0.943430
$627$	−85.4935	−3.41428
$628$	5.80418	0.231612
$629$	7.31874	0.291817
$630$	−81.5005	−3.24706
$631$	−38.9655	−1.55119	−0.775596	−	0.631230i	$-0.782550\pi$
$631$	−38.9655	−1.55119	−0.775596	+	0.631230i	$0.782550\pi$
$632$	7.73672	0.307750
$633$	−31.9890	−1.27145
$634$	11.5331	0.458038
$635$	38.0201	1.50878
$636$	−31.1951	−1.23697
$637$	30.8165	1.22099
$638$	23.6176	0.935029
$639$	−77.8994	−3.08165
$640$	−2.92742	−0.115716
$641$	9.93974	0.392596	0.196298	−	0.980544i	$-0.437108\pi$
$641$	9.93974	0.392596	0.196298	+	0.980544i	$0.437108\pi$
$642$	41.0613	1.62056
$643$	−32.6551	−1.28779	−0.643896	−	0.765113i	$-0.722683\pi$
$643$	−32.6551	−1.28779	−0.643896	+	0.765113i	$0.722683\pi$
$644$	−12.6929	−0.500169
$645$	16.8571	0.663746
$646$	9.22489	0.362949
$647$	−30.2991	−1.19118	−0.595590	−	0.803289i	$-0.703082\pi$
$647$	−30.2991	−1.19118	−0.595590	+	0.803289i	$0.703082\pi$
$648$	−8.47467	−0.332917
$649$	54.4245	2.13635
$650$	−7.35437	−0.288462
$651$	−36.8242	−1.44325
$652$	5.21845	0.204370
$653$	−15.5697	−0.609288	−0.304644	−	0.952466i	$-0.598538\pi$
$653$	−15.5697	−0.609288	−0.304644	+	0.952466i	$0.598538\pi$
$654$	20.6141	0.806076
$655$	−15.5971	−0.609430
$656$	−7.11619	−0.277841
$657$	−94.0499	−3.66923
$658$	50.9984	1.98812
$659$	−27.4767	−1.07034	−0.535170	−	0.844744i	$-0.679753\pi$
$659$	−27.4767	−1.07034	−0.535170	+	0.844744i	$0.679753\pi$
$660$	45.3289	1.76443
$661$	28.8340	1.12151	0.560756	−	0.827981i	$-0.310511\pi$
$661$	28.8340	1.12151	0.560756	+	0.827981i	$0.310511\pi$
$662$	−34.5706	−1.34363
$663$	−10.2926	−0.399730
$664$	−4.62573	−0.179513
$665$	−75.7399	−2.93707
$666$	−26.0252	−1.00846
$667$	−12.3539	−0.478346
$668$	−2.67211	−0.103387
$669$	−32.2637	−1.24739
$670$	−11.9333	−0.461025
$671$	1.66977	0.0644606
$672$	14.0119	0.540521
$673$	−20.9455	−0.807391	−0.403696	−	0.914893i	$-0.632275\pi$
$673$	−20.9455	−0.807391	−0.403696	+	0.914893i	$0.632275\pi$
$674$	−4.47468	−0.172358
$675$	−31.3957	−1.20842
$676$	−8.75565	−0.336756
$677$	−18.9656	−0.728908	−0.364454	−	0.931221i	$-0.618744\pi$
$677$	−18.9656	−0.728908	−0.364454	+	0.931221i	$0.618744\pi$
$678$	29.1920	1.12111
$679$	−71.2883	−2.73580
$680$	−4.89107	−0.187564
$681$	20.7688	0.795861
$682$	13.6090	0.521117
$683$	16.7744	0.641853	0.320926	−	0.947104i	$-0.396006\pi$
$683$	16.7744	0.641853	0.320926	+	0.947104i	$0.396006\pi$
$684$	−32.8035	−1.25427
$685$	−2.00517	−0.0766136
$686$	−37.2915	−1.42380
$687$	−12.4340	−0.474386
$688$	−1.92574	−0.0734182
$689$	21.4928	0.818811
$690$	−23.7107	−0.902653
$691$	27.7057	1.05398	0.526988	−	0.849873i	$-0.323321\pi$
$691$	27.7057	1.05398	0.526988	+	0.849873i	$0.323321\pi$
$692$	−20.3028	−0.771797
$693$	−144.167	−5.47647
$694$	11.2646	0.427599
$695$	−63.9970	−2.42755
$696$	13.6377	0.516937
$697$	−11.8896	−0.450350
$698$	17.2848	0.654241
$699$	−54.7638	−2.07136
$700$	16.7278	0.632250
$701$	33.3897	1.26111	0.630556	−	0.776144i	$-0.282827\pi$
$701$	33.3897	1.26111	0.630556	+	0.776144i	$0.282827\pi$
$702$	18.1191	0.683860
$703$	−24.1857	−0.912183
$704$	−5.17836	−0.195167
$705$	95.2669	3.58796
$706$	−16.7501	−0.630398
$707$	−36.5105	−1.37312
$708$	31.4269	1.18109
$709$	−22.0682	−0.828788	−0.414394	−	0.910097i	$-0.636007\pi$
$709$	−22.0682	−0.828788	−0.414394	+	0.910097i	$0.636007\pi$
$710$	38.3832	1.44050
$711$	−45.9658	−1.72385
$712$	17.7547	0.665384
$713$	−7.11865	−0.266595
$714$	23.4108	0.876127
$715$	−31.2307	−1.16796
$716$	−3.64166	−0.136095
$717$	8.54113	0.318974
$718$	0.0884378	0.00330047
$719$	−47.4048	−1.76790	−0.883950	−	0.467581i	$-0.845126\pi$
$719$	−47.4048	−1.76790	−0.883950	+	0.467581i	$0.845126\pi$
$720$	17.3925	0.648180
$721$	−45.2341	−1.68461
$722$	−11.4849	−0.427422
$723$	23.3467	0.868272
$724$	13.5131	0.502212
$725$	16.2811	0.604665
$726$	47.2910	1.75513
$727$	−1.22608	−0.0454728	−0.0227364	−	0.999741i	$-0.507238\pi$
$727$	−1.22608	−0.0454728	−0.0227364	+	0.999741i	$0.507238\pi$
$728$	−9.65391	−0.357798
$729$	−28.5451	−1.05723
$730$	46.3410	1.71516
$731$	−3.21749	−0.119003
$732$	0.964191	0.0356375
$733$	1.76984	0.0653705	0.0326853	−	0.999466i	$-0.489594\pi$
$733$	1.76984	0.0653705	0.0326853	+	0.999466i	$0.489594\pi$
$734$	−6.60464	−0.243782
$735$	−130.937	−4.82968
$736$	2.70870	0.0998442
$737$	−21.1091	−0.777562
$738$	42.2791	1.55631
$739$	−2.85599	−0.105059	−0.0525297	−	0.998619i	$-0.516728\pi$
$739$	−2.85599	−0.105059	−0.0525297	+	0.998619i	$0.516728\pi$
$740$	12.8234	0.471396
$741$	34.0131	1.24950
$742$	−48.8861	−1.79467
$743$	−35.0324	−1.28521	−0.642607	−	0.766196i	$-0.722147\pi$
$743$	−35.0324	−1.28521	−0.642607	+	0.766196i	$0.722147\pi$
$744$	7.85841	0.288103
$745$	8.66635	0.317510
$746$	−22.3665	−0.818895
$747$	27.4826	1.00554
$748$	−8.65189	−0.316344
$749$	64.3475	2.35121
$750$	−12.5196	−0.457150
$751$	6.49797	0.237114	0.118557	−	0.992947i	$-0.462173\pi$
$751$	6.49797	0.237114	0.118557	+	0.992947i	$0.462173\pi$
$752$	−10.8832	−0.396871
$753$	−50.8372	−1.85261
$754$	−9.39613	−0.342187
$755$	29.7781	1.08374
$756$	−41.2124	−1.49888
$757$	29.9114	1.08715	0.543574	−	0.839361i	$-0.317071\pi$
$757$	29.9114	1.08715	0.543574	+	0.839361i	$0.317071\pi$
$758$	−15.8626	−0.576155
$759$	−41.9423	−1.52241
$760$	16.1632	0.586300
$761$	47.2800	1.71390	0.856949	−	0.515400i	$-0.172357\pi$
$761$	47.2800	1.71390	0.856949	+	0.515400i	$0.172357\pi$
$762$	38.8354	1.40686
$763$	32.3046	1.16950
$764$	15.8814	0.574568
$765$	29.0590	1.05063
$766$	25.2229	0.911341
$767$	−21.6525	−0.781826
$768$	−2.99019	−0.107899
$769$	−48.7094	−1.75651	−0.878253	−	0.478197i	$-0.841291\pi$
$769$	−48.7094	−1.75651	−0.878253	+	0.478197i	$0.841291\pi$
$770$	71.0353	2.55994
$771$	−61.5647	−2.21720
$772$	−19.9696	−0.718721
$773$	−37.2024	−1.33808	−0.669038	−	0.743228i	$-0.733294\pi$
$773$	−37.2024	−1.33808	−0.669038	+	0.743228i	$0.733294\pi$
$774$	11.4413	0.411249
$775$	9.38158	0.336996
$776$	15.2132	0.546122
$777$	−61.3782	−2.20193
$778$	−30.7020	−1.10072
$779$	39.2907	1.40774
$780$	−18.0339	−0.645716
$781$	67.8967	2.42953
$782$	4.52565	0.161837
$783$	−40.1120	−1.43348
$784$	14.9582	0.534220
$785$	16.9912	0.606444
$786$	−15.9316	−0.568261
$787$	16.9762	0.605136	0.302568	−	0.953128i	$-0.402156\pi$
$787$	16.9762	0.605136	0.302568	+	0.953128i	$0.402156\pi$
$788$	−25.3449	−0.902876
$789$	69.6358	2.47910
$790$	22.6486	0.805801
$791$	45.7469	1.62657
$792$	30.7659	1.09322
$793$	−0.664308	−0.0235903
$794$	14.8403	0.526664
$795$	−91.3211	−3.23883
$796$	−12.2240	−0.433267
$797$	−5.16314	−0.182888	−0.0914440	−	0.995810i	$-0.529148\pi$
$797$	−5.16314	−0.182888	−0.0914440	+	0.995810i	$0.529148\pi$
$798$	−77.3640	−2.73866
$799$	−18.1835	−0.643286
$800$	−3.56977	−0.126210
$801$	−105.485	−3.72712
$802$	1.11915	0.0395185
$803$	81.9733	2.89277
$804$	−12.1892	−0.429881
$805$	−37.1573	−1.30962
$806$	−5.41429	−0.190710
$807$	59.6456	2.09963
$808$	7.79148	0.274103
$809$	34.0048	1.19555	0.597773	−	0.801666i	$-0.296053\pi$
$809$	34.0048	1.19555	0.597773	+	0.801666i	$0.296053\pi$
$810$	−24.8089	−0.871696
$811$	36.0050	1.26431	0.632153	−	0.774844i	$-0.282171\pi$
$811$	36.0050	1.26431	0.632153	+	0.774844i	$0.282171\pi$
$812$	21.3718	0.750004
$813$	−58.2596	−2.04325
$814$	22.6835	0.795055
$815$	15.2766	0.535115
$816$	−4.99595	−0.174893
$817$	10.6326	0.371988
$818$	−25.2216	−0.881851
$819$	57.3563	2.00419
$820$	−20.8321	−0.727487
$821$	−5.93184	−0.207023	−0.103511	−	0.994628i	$-0.533008\pi$
$821$	−5.93184	−0.207023	−0.103511	+	0.994628i	$0.533008\pi$
$822$	−2.04817	−0.0714381
$823$	18.2383	0.635748	0.317874	−	0.948133i	$-0.397031\pi$
$823$	18.2383	0.635748	0.317874	+	0.948133i	$0.397031\pi$
$824$	9.65313	0.336283
$825$	55.2753	1.92444
$826$	49.2493	1.71360
$827$	−47.6306	−1.65628	−0.828139	−	0.560523i	$-0.810600\pi$
$827$	−47.6306	−1.65628	−0.828139	+	0.560523i	$0.810600\pi$
$828$	−16.0931	−0.559273
$829$	−5.09451	−0.176940	−0.0884699	−	0.996079i	$-0.528198\pi$
$829$	−5.09451	−0.176940	−0.0884699	+	0.996079i	$0.528198\pi$
$830$	−13.5414	−0.470030
$831$	53.6814	1.86219
$832$	2.06018	0.0714239
$833$	24.9918	0.865914
$834$	−65.3694	−2.26356
$835$	−7.82237	−0.270704
$836$	28.5913	0.988851
$837$	−23.1135	−0.798921
$838$	−8.71709	−0.301127
$839$	7.06592	0.243943	0.121971	−	0.992534i	$-0.461078\pi$
$839$	7.06592	0.243943	0.121971	+	0.992534i	$0.461078\pi$
$840$	41.0187	1.41528
$841$	−8.19887	−0.282720
$842$	−15.1977	−0.523747
$843$	79.9842	2.75480
$844$	10.6980	0.368240
$845$	−25.6314	−0.881748
$846$	64.6600	2.22306
$847$	74.1100	2.54645
$848$	10.4325	0.358253
$849$	−10.5214	−0.361095
$850$	−5.96430	−0.204574
$851$	−11.8653	−0.406737
$852$	39.2063	1.34319
$853$	−56.4322	−1.93220	−0.966101	−	0.258165i	$-0.916882\pi$
$853$	−56.4322	−1.93220	−0.966101	+	0.258165i	$0.916882\pi$
$854$	1.51099	0.0517050
$855$	−96.0294	−3.28414
$856$	−13.7320	−0.469350
$857$	−41.0055	−1.40072	−0.700360	−	0.713789i	$-0.746977\pi$
$857$	−41.0055	−1.40072	−0.700360	+	0.713789i	$0.746977\pi$
$858$	−31.9004	−1.08906
$859$	−29.8437	−1.01825	−0.509127	−	0.860691i	$-0.670032\pi$
$859$	−29.8437	−1.01825	−0.509127	+	0.860691i	$0.670032\pi$
$860$	−5.63745	−0.192235
$861$	99.7114	3.39815
$862$	−24.9038	−0.848227
$863$	28.5931	0.973320	0.486660	−	0.873591i	$-0.338215\pi$
$863$	28.5931	0.973320	0.486660	+	0.873591i	$0.338215\pi$
$864$	8.79489	0.299208
$865$	−59.4348	−2.02084
$866$	30.6843	1.04270
$867$	42.4861	1.44290
$868$	12.3150	0.417998
$869$	40.0635	1.35906
$870$	39.9234	1.35353
$871$	8.39813	0.284560
$872$	−6.89391	−0.233457
$873$	−90.3854	−3.05908
$874$	−14.9556	−0.505881
$875$	−19.6195	−0.663261
$876$	47.3347	1.59929
$877$	−8.31713	−0.280849	−0.140425	−	0.990091i	$-0.544847\pi$
$877$	−8.31713	−0.280849	−0.140425	+	0.990091i	$0.544847\pi$
$878$	11.8829	0.401029
$879$	35.9675	1.21315
$880$	−15.1592	−0.511017
$881$	−24.3677	−0.820970	−0.410485	−	0.911867i	$-0.634641\pi$
$881$	−24.3677	−0.820970	−0.410485	+	0.911867i	$0.634641\pi$
$882$	−88.8701	−2.99241
$883$	−38.3964	−1.29214	−0.646071	−	0.763277i	$-0.723589\pi$
$883$	−38.3964	−1.29214	−0.646071	+	0.763277i	$0.723589\pi$
$884$	3.44211	0.115771
$885$	91.9996	3.09253
$886$	−8.67085	−0.291303
$887$	51.9478	1.74424	0.872118	−	0.489296i	$-0.162746\pi$
$887$	51.9478	1.74424	0.872118	+	0.489296i	$0.162746\pi$
$888$	13.0983	0.439552
$889$	60.8593	2.04115
$890$	51.9753	1.74222
$891$	−43.8849	−1.47020
$892$	10.7899	0.361271
$893$	60.0898	2.01083
$894$	8.85218	0.296061
$895$	−10.6606	−0.356346
$896$	−4.68595	−0.156547
$897$	16.6865	0.557147
$898$	31.3266	1.04538
$899$	11.9861	0.399760
$900$	21.2089	0.706963
$901$	17.4304	0.580690
$902$	−36.8502	−1.22698
$903$	26.9833	0.897948
$904$	−9.76257	−0.324698
$905$	39.5586	1.31497
$906$	30.4166	1.01053
$907$	−4.42741	−0.147010	−0.0735049	−	0.997295i	$-0.523418\pi$
$907$	−4.42741	−0.147010	−0.0735049	+	0.997295i	$0.523418\pi$
$908$	−6.94563	−0.230499
$909$	−46.2911	−1.53538
$910$	−28.2610	−0.936844
$911$	10.4728	0.346980	0.173490	−	0.984836i	$-0.444496\pi$
$911$	10.4728	0.346980	0.173490	+	0.984836i	$0.444496\pi$
$912$	16.5098	0.546693
$913$	−23.9537	−0.792750
$914$	−0.900942	−0.0298005
$915$	2.82259	0.0933119
$916$	4.15825	0.137393
$917$	−24.9665	−0.824467
$918$	14.6943	0.484985
$919$	−43.5249	−1.43575	−0.717877	−	0.696170i	$-0.754886\pi$
$919$	−43.5249	−1.43575	−0.717877	+	0.696170i	$0.754886\pi$
$920$	7.92951	0.261428
$921$	−36.1995	−1.19281
$922$	25.5950	0.842926
$923$	−27.0123	−0.889122
$924$	72.5586	2.38700
$925$	15.6371	0.514146
$926$	3.85361	0.126637
$927$	−57.3516	−1.88367
$928$	−4.56083	−0.149716
$929$	−14.9816	−0.491530	−0.245765	−	0.969329i	$-0.579039\pi$
$929$	−14.9816	−0.491530	−0.245765	+	0.969329i	$0.579039\pi$
$930$	23.0049	0.754359
$931$	−82.5886	−2.70673
$932$	18.3145	0.599910
$933$	82.2862	2.69393
$934$	−24.2665	−0.794025
$935$	−25.3277	−0.828304
$936$	−12.2400	−0.400078
$937$	19.1960	0.627105	0.313553	−	0.949571i	$-0.398481\pi$
$937$	19.1960	0.627105	0.313553	+	0.949571i	$0.398481\pi$
$938$	−19.1018	−0.623697
$939$	−70.5823	−2.30337
$940$	−31.8598	−1.03915
$941$	−19.4556	−0.634234	−0.317117	−	0.948386i	$-0.602715\pi$
$941$	−19.4556	−0.634234	−0.317117	+	0.948386i	$0.602715\pi$
$942$	17.3556	0.565476
$943$	19.2757	0.627702
$944$	−10.5100	−0.342071
$945$	−120.646	−3.92462
$946$	−9.97218	−0.324223
$947$	4.83054	0.156972	0.0784858	−	0.996915i	$-0.474991\pi$
$947$	4.83054	0.156972	0.0784858	+	0.996915i	$0.474991\pi$
$948$	23.1343	0.751367
$949$	−32.6126	−1.05865
$950$	19.7098	0.639470
$951$	34.4862	1.11829
$952$	−7.82919	−0.253746
$953$	−38.9117	−1.26047	−0.630236	−	0.776403i	$-0.717042\pi$
$953$	−38.9117	−1.26047	−0.630236	+	0.776403i	$0.717042\pi$
$954$	−61.9820	−2.00674
$955$	46.4914	1.50443
$956$	−2.85638	−0.0923820
$957$	70.6211	2.28285
$958$	−10.7624	−0.347716
$959$	−3.20970	−0.103647
$960$	−8.75354	−0.282519
$961$	−24.0933	−0.777203
$962$	−9.02449	−0.290961
$963$	81.5852	2.62905
$964$	−7.80775	−0.251471
$965$	−58.4593	−1.88187
$966$	−37.9541	−1.22115
$967$	−21.1346	−0.679644	−0.339822	−	0.940490i	$-0.610367\pi$
$967$	−21.1346	−0.679644	−0.339822	+	0.940490i	$0.610367\pi$
$968$	−15.8154	−0.508325
$969$	27.5842	0.886132
$970$	44.5354	1.42994
$971$	11.1363	0.357381	0.178691	−	0.983905i	$-0.442814\pi$
$971$	11.1363	0.357381	0.178691	+	0.983905i	$0.442814\pi$
$972$	1.04378	0.0334793
$973$	−102.441	−3.28410
$974$	36.5062	1.16973
$975$	−21.9910	−0.704275
$976$	−0.322451	−0.0103214
$977$	36.2416	1.15947	0.579735	−	0.814805i	$-0.303156\pi$
$977$	36.2416	1.15947	0.579735	+	0.814805i	$0.303156\pi$
$978$	15.6042	0.498966
$979$	91.9399	2.93841
$980$	43.7888	1.39878
$981$	40.9584	1.30770
$982$	−31.6993	−1.01157
$983$	−19.3120	−0.615958	−0.307979	−	0.951393i	$-0.599653\pi$
$983$	−19.3120	−0.615958	−0.307979	+	0.951393i	$0.599653\pi$
$984$	−21.2788	−0.678343
$985$	−74.1952	−2.36406
$986$	−7.62013	−0.242674
$987$	152.495	4.85397
$988$	−11.3749	−0.361884
$989$	5.21627	0.165868
$990$	90.0646	2.86244
$991$	−13.6888	−0.434838	−0.217419	−	0.976078i	$-0.569764\pi$
$991$	−13.6888	−0.434838	−0.217419	+	0.976078i	$0.569764\pi$
$992$	−2.62806	−0.0834411
$993$	−103.373	−3.28044
$994$	61.4405	1.94877
$995$	−35.7847	−1.13445
$996$	−13.8318	−0.438278
$997$	37.8299	1.19808	0.599042	−	0.800718i	$-0.295548\pi$
$997$	37.8299	1.19808	0.599042	+	0.800718i	$0.295548\pi$
$998$	4.78448	0.151450
$999$	−38.5255	−1.21889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.7	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.7	✓	89	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.99019 q^{3} +1.00000 q^{4} +2.92742 q^{5} +2.99019 q^{6} +4.68595 q^{7} -1.00000 q^{8} +5.94125 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.99019 q^{3} +1.00000 q^{4} +2.92742 q^{5} +2.99019 q^{6} +4.68595 q^{7} -1.00000 q^{8} +5.94125 q^{9} -2.92742 q^{10} -5.17836 q^{11} -2.99019 q^{12} +2.06018 q^{13} -4.68595 q^{14} -8.75354 q^{15} +1.00000 q^{16} +1.67078 q^{17} -5.94125 q^{18} -5.52131 q^{19} +2.92742 q^{20} -14.0119 q^{21} +5.17836 q^{22} -2.70870 q^{23} +2.99019 q^{24} +3.56977 q^{25} -2.06018 q^{26} -8.79489 q^{27} +4.68595 q^{28} +4.56083 q^{29} +8.75354 q^{30} +2.62806 q^{31} -1.00000 q^{32} +15.4843 q^{33} -1.67078 q^{34} +13.7177 q^{35} +5.94125 q^{36} +4.38044 q^{37} +5.52131 q^{38} -6.16034 q^{39} -2.92742 q^{40} -7.11619 q^{41} +14.0119 q^{42} -1.92574 q^{43} -5.17836 q^{44} +17.3925 q^{45} +2.70870 q^{46} -10.8832 q^{47} -2.99019 q^{48} +14.9582 q^{49} -3.56977 q^{50} -4.99595 q^{51} +2.06018 q^{52} +10.4325 q^{53} +8.79489 q^{54} -15.1592 q^{55} -4.68595 q^{56} +16.5098 q^{57} -4.56083 q^{58} -10.5100 q^{59} -8.75354 q^{60} -0.322451 q^{61} -2.62806 q^{62} +27.8404 q^{63} +1.00000 q^{64} +6.03101 q^{65} -15.4843 q^{66} +4.07640 q^{67} +1.67078 q^{68} +8.09955 q^{69} -13.7177 q^{70} -13.1116 q^{71} -5.94125 q^{72} -15.8300 q^{73} -4.38044 q^{74} -10.6743 q^{75} -5.52131 q^{76} -24.2655 q^{77} +6.16034 q^{78} -7.73672 q^{79} +2.92742 q^{80} +8.47467 q^{81} +7.11619 q^{82} +4.62573 q^{83} -14.0119 q^{84} +4.89107 q^{85} +1.92574 q^{86} -13.6377 q^{87} +5.17836 q^{88} -17.7547 q^{89} -17.3925 q^{90} +9.65391 q^{91} -2.70870 q^{92} -7.85841 q^{93} +10.8832 q^{94} -16.1632 q^{95} +2.99019 q^{96} -15.2132 q^{97} -14.9582 q^{98} -30.7659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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