LMFDB - Embedded newform 8006.2.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, 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("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.08249 q^{3} +1.00000 q^{4} +1.32186 q^{5} -3.08249 q^{6} +1.26521 q^{7} +1.00000 q^{8} +6.50177 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.08249 q^{3} +1.00000 q^{4} +1.32186 q^{5} -3.08249 q^{6} +1.26521 q^{7} +1.00000 q^{8} +6.50177 q^{9} +1.32186 q^{10} -5.42729 q^{11} -3.08249 q^{12} +5.98048 q^{13} +1.26521 q^{14} -4.07461 q^{15} +1.00000 q^{16} +2.43614 q^{17} +6.50177 q^{18} -4.15394 q^{19} +1.32186 q^{20} -3.90000 q^{21} -5.42729 q^{22} -1.30378 q^{23} -3.08249 q^{24} -3.25270 q^{25} +5.98048 q^{26} -10.7942 q^{27} +1.26521 q^{28} +5.25549 q^{29} -4.07461 q^{30} -2.93179 q^{31} +1.00000 q^{32} +16.7296 q^{33} +2.43614 q^{34} +1.67242 q^{35} +6.50177 q^{36} -3.81769 q^{37} -4.15394 q^{38} -18.4348 q^{39} +1.32186 q^{40} -9.08722 q^{41} -3.90000 q^{42} -8.55560 q^{43} -5.42729 q^{44} +8.59440 q^{45} -1.30378 q^{46} +8.90702 q^{47} -3.08249 q^{48} -5.39925 q^{49} -3.25270 q^{50} -7.50940 q^{51} +5.98048 q^{52} +0.438153 q^{53} -10.7942 q^{54} -7.17410 q^{55} +1.26521 q^{56} +12.8045 q^{57} +5.25549 q^{58} -3.69253 q^{59} -4.07461 q^{60} +5.58864 q^{61} -2.93179 q^{62} +8.22610 q^{63} +1.00000 q^{64} +7.90534 q^{65} +16.7296 q^{66} -6.55966 q^{67} +2.43614 q^{68} +4.01888 q^{69} +1.67242 q^{70} +5.54033 q^{71} +6.50177 q^{72} -12.0119 q^{73} -3.81769 q^{74} +10.0264 q^{75} -4.15394 q^{76} -6.86666 q^{77} -18.4348 q^{78} +7.82284 q^{79} +1.32186 q^{80} +13.7677 q^{81} -9.08722 q^{82} -16.4524 q^{83} -3.90000 q^{84} +3.22023 q^{85} -8.55560 q^{86} -16.2000 q^{87} -5.42729 q^{88} -10.6250 q^{89} +8.59440 q^{90} +7.56656 q^{91} -1.30378 q^{92} +9.03723 q^{93} +8.90702 q^{94} -5.49091 q^{95} -3.08249 q^{96} +0.454316 q^{97} -5.39925 q^{98} -35.2870 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.08249	−1.77968	−0.889839	−	0.456274i	$-0.849184\pi$
$3$	−3.08249	−1.77968	−0.889839	+	0.456274i	$0.849184\pi$
$4$	1.00000	0.500000
$5$	1.32186	0.591152	0.295576	−	0.955319i	$-0.404488\pi$
$5$	1.32186	0.591152	0.295576	+	0.955319i	$0.404488\pi$
$6$	−3.08249	−1.25842
$7$	1.26521	0.478204	0.239102	−	0.970994i	$-0.423147\pi$
$7$	1.26521	0.478204	0.239102	+	0.970994i	$0.423147\pi$
$8$	1.00000	0.353553
$9$	6.50177	2.16726
$10$	1.32186	0.418008
$11$	−5.42729	−1.63639	−0.818195	−	0.574940i	$-0.805025\pi$
$11$	−5.42729	−1.63639	−0.818195	+	0.574940i	$0.805025\pi$
$12$	−3.08249	−0.889839
$13$	5.98048	1.65869	0.829344	−	0.558739i	$-0.188714\pi$
$13$	5.98048	1.65869	0.829344	+	0.558739i	$0.188714\pi$
$14$	1.26521	0.338141
$15$	−4.07461	−1.05206
$16$	1.00000	0.250000
$17$	2.43614	0.590852	0.295426	−	0.955366i	$-0.404538\pi$
$17$	2.43614	0.590852	0.295426	+	0.955366i	$0.404538\pi$
$18$	6.50177	1.53248
$19$	−4.15394	−0.952979	−0.476489	−	0.879180i	$-0.658091\pi$
$19$	−4.15394	−0.952979	−0.476489	+	0.879180i	$0.658091\pi$
$20$	1.32186	0.295576
$21$	−3.90000	−0.851049
$22$	−5.42729	−1.15710
$23$	−1.30378	−0.271856	−0.135928	−	0.990719i	$-0.543402\pi$
$23$	−1.30378	−0.271856	−0.135928	+	0.990719i	$0.543402\pi$
$24$	−3.08249	−0.629211
$25$	−3.25270	−0.650539
$26$	5.98048	1.17287
$27$	−10.7942	−2.07734
$28$	1.26521	0.239102
$29$	5.25549	0.975920	0.487960	−	0.872866i	$-0.337741\pi$
$29$	5.25549	0.975920	0.487960	+	0.872866i	$0.337741\pi$
$30$	−4.07461	−0.743919
$31$	−2.93179	−0.526565	−0.263283	−	0.964719i	$-0.584805\pi$
$31$	−2.93179	−0.526565	−0.263283	+	0.964719i	$0.584805\pi$
$32$	1.00000	0.176777
$33$	16.7296	2.91225
$34$	2.43614	0.417795
$35$	1.67242	0.282691
$36$	6.50177	1.08363
$37$	−3.81769	−0.627624	−0.313812	−	0.949485i	$-0.601606\pi$
$37$	−3.81769	−0.627624	−0.313812	+	0.949485i	$0.601606\pi$
$38$	−4.15394	−0.673858
$39$	−18.4348	−2.95193
$40$	1.32186	0.209004
$41$	−9.08722	−1.41919	−0.709593	−	0.704612i	$-0.751121\pi$
$41$	−9.08722	−1.41919	−0.709593	+	0.704612i	$0.751121\pi$
$42$	−3.90000	−0.601783
$43$	−8.55560	−1.30472	−0.652358	−	0.757911i	$-0.726220\pi$
$43$	−8.55560	−1.30472	−0.652358	+	0.757911i	$0.726220\pi$
$44$	−5.42729	−0.818195
$45$	8.59440	1.28118
$46$	−1.30378	−0.192231
$47$	8.90702	1.29922	0.649611	−	0.760266i	$-0.274932\pi$
$47$	8.90702	1.29922	0.649611	+	0.760266i	$0.274932\pi$
$48$	−3.08249	−0.444920
$49$	−5.39925	−0.771321
$50$	−3.25270	−0.460001
$51$	−7.50940	−1.05153
$52$	5.98048	0.829344
$53$	0.438153	0.0601850	0.0300925	−	0.999547i	$-0.490420\pi$
$53$	0.438153	0.0601850	0.0300925	+	0.999547i	$0.490420\pi$
$54$	−10.7942	−1.46890
$55$	−7.17410	−0.967355
$56$	1.26521	0.169071
$57$	12.8045	1.69600
$58$	5.25549	0.690080
$59$	−3.69253	−0.480726	−0.240363	−	0.970683i	$-0.577267\pi$
$59$	−3.69253	−0.480726	−0.240363	+	0.970683i	$0.577267\pi$
$60$	−4.07461	−0.526030
$61$	5.58864	0.715552	0.357776	−	0.933807i	$-0.383535\pi$
$61$	5.58864	0.715552	0.357776	+	0.933807i	$0.383535\pi$
$62$	−2.93179	−0.372338
$63$	8.22610	1.03639
$64$	1.00000	0.125000
$65$	7.90534	0.980537
$66$	16.7296	2.05927
$67$	−6.55966	−0.801390	−0.400695	−	0.916212i	$-0.631231\pi$
$67$	−6.55966	−0.801390	−0.400695	+	0.916212i	$0.631231\pi$
$68$	2.43614	0.295426
$69$	4.01888	0.483816
$70$	1.67242	0.199893
$71$	5.54033	0.657516	0.328758	−	0.944414i	$-0.393370\pi$
$71$	5.54033	0.657516	0.328758	+	0.944414i	$0.393370\pi$
$72$	6.50177	0.766241
$73$	−12.0119	−1.40588	−0.702942	−	0.711247i	$-0.748131\pi$
$73$	−12.0119	−1.40588	−0.702942	+	0.711247i	$0.748131\pi$
$74$	−3.81769	−0.443797
$75$	10.0264	1.15775
$76$	−4.15394	−0.476489
$77$	−6.86666	−0.782528
$78$	−18.4348	−2.08733
$79$	7.82284	0.880138	0.440069	−	0.897964i	$-0.354954\pi$
$79$	7.82284	0.880138	0.440069	+	0.897964i	$0.354954\pi$
$80$	1.32186	0.147788
$81$	13.7677	1.52974
$82$	−9.08722	−1.00352
$83$	−16.4524	−1.80589	−0.902943	−	0.429761i	$-0.858598\pi$
$83$	−16.4524	−1.80589	−0.902943	+	0.429761i	$0.858598\pi$
$84$	−3.90000	−0.425525
$85$	3.22023	0.349283
$86$	−8.55560	−0.922574
$87$	−16.2000	−1.73682
$88$	−5.42729	−0.578551
$89$	−10.6250	−1.12625	−0.563123	−	0.826373i	$-0.690400\pi$
$89$	−10.6250	−1.12625	−0.563123	+	0.826373i	$0.690400\pi$
$90$	8.59440	0.905930
$91$	7.56656	0.793191
$92$	−1.30378	−0.135928
$93$	9.03723	0.937117
$94$	8.90702	0.918689
$95$	−5.49091	−0.563355
$96$	−3.08249	−0.314606
$97$	0.454316	0.0461288	0.0230644	−	0.999734i	$-0.492658\pi$
$97$	0.454316	0.0461288	0.0230644	+	0.999734i	$0.492658\pi$
$98$	−5.39925	−0.545406
$99$	−35.2870	−3.54648
$100$	−3.25270	−0.325270
$101$	16.5270	1.64450	0.822251	−	0.569125i	$-0.192718\pi$
$101$	16.5270	1.64450	0.822251	+	0.569125i	$0.192718\pi$
$102$	−7.50940	−0.743541
$103$	2.38916	0.235411	0.117706	−	0.993049i	$-0.462446\pi$
$103$	2.38916	0.235411	0.117706	+	0.993049i	$0.462446\pi$
$104$	5.98048	0.586435
$105$	−5.15524	−0.503100
$106$	0.438153	0.0425572
$107$	10.8517	1.04907	0.524537	−	0.851388i	$-0.324238\pi$
$107$	10.8517	1.04907	0.524537	+	0.851388i	$0.324238\pi$
$108$	−10.7942	−1.03867
$109$	9.20979	0.882138	0.441069	−	0.897473i	$-0.354599\pi$
$109$	9.20979	0.882138	0.441069	+	0.897473i	$0.354599\pi$
$110$	−7.17410	−0.684024
$111$	11.7680	1.11697
$112$	1.26521	0.119551
$113$	−3.59737	−0.338412	−0.169206	−	0.985581i	$-0.554120\pi$
$113$	−3.59737	−0.338412	−0.169206	+	0.985581i	$0.554120\pi$
$114$	12.8045	1.19925
$115$	−1.72340	−0.160708
$116$	5.25549	0.487960
$117$	38.8837	3.59480
$118$	−3.69253	−0.339925
$119$	3.08223	0.282548
$120$	−4.07461	−0.371960
$121$	18.4555	1.67777
$122$	5.58864	0.505972
$123$	28.0113	2.52569
$124$	−2.93179	−0.263283
$125$	−10.9089	−0.975720
$126$	8.22610	0.732839
$127$	−13.1675	−1.16843	−0.584215	−	0.811599i	$-0.698598\pi$
$127$	−13.1675	−1.16843	−0.584215	+	0.811599i	$0.698598\pi$
$128$	1.00000	0.0883883
$129$	26.3726	2.32198
$130$	7.90534	0.693344
$131$	−18.1171	−1.58290	−0.791450	−	0.611233i	$-0.790674\pi$
$131$	−18.1171	−1.58290	−0.791450	+	0.611233i	$0.790674\pi$
$132$	16.7296	1.45612
$133$	−5.25560	−0.455718
$134$	−6.55966	−0.566668
$135$	−14.2684	−1.22802
$136$	2.43614	0.208898
$137$	−18.9995	−1.62324	−0.811619	−	0.584187i	$-0.801413\pi$
$137$	−18.9995	−1.62324	−0.811619	+	0.584187i	$0.801413\pi$
$138$	4.01888	0.342110
$139$	5.37661	0.456038	0.228019	−	0.973657i	$-0.426775\pi$
$139$	5.37661	0.456038	0.228019	+	0.973657i	$0.426775\pi$
$140$	1.67242	0.141346
$141$	−27.4558	−2.31220
$142$	5.54033	0.464934
$143$	−32.4578	−2.71426
$144$	6.50177	0.541814
$145$	6.94700	0.576917
$146$	−12.0119	−0.994110
$147$	16.6431	1.37270
$148$	−3.81769	−0.313812
$149$	5.30832	0.434875	0.217437	−	0.976074i	$-0.430230\pi$
$149$	5.30832	0.434875	0.217437	+	0.976074i	$0.430230\pi$
$150$	10.0264	0.818654
$151$	15.5495	1.26540	0.632700	−	0.774397i	$-0.281947\pi$
$151$	15.5495	1.26540	0.632700	+	0.774397i	$0.281947\pi$
$152$	−4.15394	−0.336929
$153$	15.8392	1.28053
$154$	−6.86666	−0.553331
$155$	−3.87541	−0.311280
$156$	−18.4348	−1.47597
$157$	8.36186	0.667349	0.333674	−	0.942688i	$-0.391711\pi$
$157$	8.36186	0.667349	0.333674	+	0.942688i	$0.391711\pi$
$158$	7.82284	0.622352
$159$	−1.35061	−0.107110
$160$	1.32186	0.104502
$161$	−1.64955	−0.130003
$162$	13.7677	1.08169
$163$	22.2462	1.74246	0.871229	−	0.490876i	$-0.163323\pi$
$163$	22.2462	1.74246	0.871229	+	0.490876i	$0.163323\pi$
$164$	−9.08722	−0.709593
$165$	22.1141	1.72158
$166$	−16.4524	−1.27695
$167$	−17.1407	−1.32639	−0.663195	−	0.748447i	$-0.730800\pi$
$167$	−17.1407	−1.32639	−0.663195	+	0.748447i	$0.730800\pi$
$168$	−3.90000	−0.300891
$169$	22.7662	1.75125
$170$	3.22023	0.246981
$171$	−27.0079	−2.06535
$172$	−8.55560	−0.652358
$173$	−17.7937	−1.35283	−0.676415	−	0.736520i	$-0.736468\pi$
$173$	−17.7937	−1.35283	−0.676415	+	0.736520i	$0.736468\pi$
$174$	−16.2000	−1.22812
$175$	−4.11534	−0.311090
$176$	−5.42729	−0.409098
$177$	11.3822	0.855538
$178$	−10.6250	−0.796377
$179$	22.4311	1.67658	0.838289	−	0.545227i	$-0.183556\pi$
$179$	22.4311	1.67658	0.838289	+	0.545227i	$0.183556\pi$
$180$	8.59440	0.640589
$181$	5.88737	0.437605	0.218802	−	0.975769i	$-0.429785\pi$
$181$	5.88737	0.437605	0.218802	+	0.975769i	$0.429785\pi$
$182$	7.56656	0.560871
$183$	−17.2270	−1.27345
$184$	−1.30378	−0.0961156
$185$	−5.04643	−0.371021
$186$	9.03723	0.662642
$187$	−13.2217	−0.966864
$188$	8.90702	0.649611
$189$	−13.6569	−0.993393
$190$	−5.49091	−0.398352
$191$	−15.6529	−1.13260	−0.566301	−	0.824199i	$-0.691626\pi$
$191$	−15.6529	−1.13260	−0.566301	+	0.824199i	$0.691626\pi$
$192$	−3.08249	−0.222460
$193$	−13.9134	−1.00151	−0.500756	−	0.865589i	$-0.666944\pi$
$193$	−13.9134	−1.00151	−0.500756	+	0.865589i	$0.666944\pi$
$194$	0.454316	0.0326180
$195$	−24.3682	−1.74504
$196$	−5.39925	−0.385660
$197$	22.6959	1.61702	0.808508	−	0.588485i	$-0.200275\pi$
$197$	22.6959	1.61702	0.808508	+	0.588485i	$0.200275\pi$
$198$	−35.2870	−2.50774
$199$	−8.75610	−0.620704	−0.310352	−	0.950622i	$-0.600447\pi$
$199$	−8.75610	−0.620704	−0.310352	+	0.950622i	$0.600447\pi$
$200$	−3.25270	−0.230000
$201$	20.2201	1.42622
$202$	16.5270	1.16284
$203$	6.64929	0.466689
$204$	−7.50940	−0.525763
$205$	−12.0120	−0.838954
$206$	2.38916	0.166461
$207$	−8.47685	−0.589182
$208$	5.98048	0.414672
$209$	22.5446	1.55945
$210$	−5.15524	−0.355745
$211$	−21.6540	−1.49072	−0.745360	−	0.666662i	$-0.767723\pi$
$211$	−21.6540	−1.49072	−0.745360	+	0.666662i	$0.767723\pi$
$212$	0.438153	0.0300925
$213$	−17.0780	−1.17017
$214$	10.8517	0.741807
$215$	−11.3093	−0.771285
$216$	−10.7942	−0.734451
$217$	−3.70933	−0.251806
$218$	9.20979	0.623766
$219$	37.0265	2.50202
$220$	−7.17410	−0.483678
$221$	14.5693	0.980039
$222$	11.7680	0.789816
$223$	−17.7375	−1.18779	−0.593894	−	0.804543i	$-0.702410\pi$
$223$	−17.7375	−1.18779	−0.593894	+	0.804543i	$0.702410\pi$
$224$	1.26521	0.0845353
$225$	−21.1483	−1.40989
$226$	−3.59737	−0.239293
$227$	28.4512	1.88837	0.944185	−	0.329417i	$-0.106852\pi$
$227$	28.4512	1.88837	0.944185	+	0.329417i	$0.106852\pi$
$228$	12.8045	0.847998
$229$	−21.9042	−1.44747	−0.723735	−	0.690078i	$-0.757576\pi$
$229$	−21.9042	−1.44747	−0.723735	+	0.690078i	$0.757576\pi$
$230$	−1.72340	−0.113638
$231$	21.1664	1.39265
$232$	5.25549	0.345040
$233$	−17.6475	−1.15613	−0.578064	−	0.815992i	$-0.696191\pi$
$233$	−17.6475	−1.15613	−0.578064	+	0.815992i	$0.696191\pi$
$234$	38.8837	2.54191
$235$	11.7738	0.768038
$236$	−3.69253	−0.240363
$237$	−24.1139	−1.56636
$238$	3.08223	0.199791
$239$	5.35528	0.346404	0.173202	−	0.984886i	$-0.444589\pi$
$239$	5.35528	0.346404	0.173202	+	0.984886i	$0.444589\pi$
$240$	−4.07461	−0.263015
$241$	−5.95584	−0.383650	−0.191825	−	0.981429i	$-0.561441\pi$
$241$	−5.95584	−0.383650	−0.191825	+	0.981429i	$0.561441\pi$
$242$	18.4555	1.18637
$243$	−10.0563	−0.645112
$244$	5.58864	0.357776
$245$	−7.13703	−0.455968
$246$	28.0113	1.78594
$247$	−24.8426	−1.58069
$248$	−2.93179	−0.186169
$249$	50.7144	3.21390
$250$	−10.9089	−0.689938
$251$	−16.6497	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$251$	−16.6497	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$252$	8.22610	0.518195
$253$	7.07597	0.444863
$254$	−13.1675	−0.826205
$255$	−9.92635	−0.621612
$256$	1.00000	0.0625000
$257$	−11.9681	−0.746550	−0.373275	−	0.927721i	$-0.621765\pi$
$257$	−11.9681	−0.746550	−0.373275	+	0.927721i	$0.621765\pi$
$258$	26.3726	1.64188
$259$	−4.83017	−0.300132
$260$	7.90534	0.490268
$261$	34.1700	2.11507
$262$	−18.1171	−1.11928
$263$	−16.1809	−0.997755	−0.498877	−	0.866673i	$-0.666254\pi$
$263$	−16.1809	−0.997755	−0.498877	+	0.866673i	$0.666254\pi$
$264$	16.7296	1.02964
$265$	0.579176	0.0355785
$266$	−5.25560	−0.322241
$267$	32.7515	2.00436
$268$	−6.55966	−0.400695
$269$	31.3862	1.91365	0.956826	−	0.290662i	$-0.0938756\pi$
$269$	31.3862	1.91365	0.956826	+	0.290662i	$0.0938756\pi$
$270$	−14.2684	−0.868345
$271$	2.35963	0.143337	0.0716686	−	0.997429i	$-0.477168\pi$
$271$	2.35963	0.143337	0.0716686	+	0.997429i	$0.477168\pi$
$272$	2.43614	0.147713
$273$	−23.3239	−1.41163
$274$	−18.9995	−1.14780
$275$	17.6533	1.06454
$276$	4.01888	0.241908
$277$	26.4512	1.58930	0.794649	−	0.607069i	$-0.207655\pi$
$277$	26.4512	1.58930	0.794649	+	0.607069i	$0.207655\pi$
$278$	5.37661	0.322468
$279$	−19.0618	−1.14120
$280$	1.67242	0.0999464
$281$	−11.8247	−0.705402	−0.352701	−	0.935736i	$-0.614737\pi$
$281$	−11.8247	−0.705402	−0.352701	+	0.935736i	$0.614737\pi$
$282$	−27.4558	−1.63497
$283$	16.7125	0.993453	0.496727	−	0.867907i	$-0.334535\pi$
$283$	16.7125	0.993453	0.496727	+	0.867907i	$0.334535\pi$
$284$	5.54033	0.328758
$285$	16.9257	1.00259
$286$	−32.4578	−1.91927
$287$	−11.4972	−0.678660
$288$	6.50177	0.383120
$289$	−11.0652	−0.650894
$290$	6.94700	0.407942
$291$	−1.40043	−0.0820944
$292$	−12.0119	−0.702942
$293$	19.5074	1.13963	0.569816	−	0.821772i	$-0.307015\pi$
$293$	19.5074	1.13963	0.569816	+	0.821772i	$0.307015\pi$
$294$	16.6431	0.970648
$295$	−4.88099	−0.284182
$296$	−3.81769	−0.221899
$297$	58.5832	3.39934
$298$	5.30832	0.307503
$299$	−7.79721	−0.450924
$300$	10.0264	0.578876
$301$	−10.8246	−0.623920
$302$	15.5495	0.894772
$303$	−50.9445	−2.92669
$304$	−4.15394	−0.238245
$305$	7.38738	0.423000
$306$	15.8392	0.905470
$307$	5.38550	0.307367	0.153683	−	0.988120i	$-0.450886\pi$
$307$	5.38550	0.307367	0.153683	+	0.988120i	$0.450886\pi$
$308$	−6.86666	−0.391264
$309$	−7.36458	−0.418956
$310$	−3.87541	−0.220108
$311$	9.96482	0.565053	0.282527	−	0.959259i	$-0.408827\pi$
$311$	9.96482	0.565053	0.282527	+	0.959259i	$0.408827\pi$
$312$	−18.4348	−1.04367
$313$	17.3393	0.980077	0.490039	−	0.871701i	$-0.336983\pi$
$313$	17.3393	0.980077	0.490039	+	0.871701i	$0.336983\pi$
$314$	8.36186	0.471887
$315$	10.8737	0.612664
$316$	7.82284	0.440069
$317$	−12.8285	−0.720521	−0.360260	−	0.932852i	$-0.617312\pi$
$317$	−12.8285	−0.720521	−0.360260	+	0.932852i	$0.617312\pi$
$318$	−1.35061	−0.0757382
$319$	−28.5231	−1.59699
$320$	1.32186	0.0738940
$321$	−33.4503	−1.86701
$322$	−1.64955	−0.0919257
$323$	−10.1196	−0.563069
$324$	13.7677	0.764872
$325$	−19.4527	−1.07904
$326$	22.2462	1.23210
$327$	−28.3891	−1.56992
$328$	−9.08722	−0.501758
$329$	11.2692	0.621294
$330$	22.1141	1.21734
$331$	−10.1561	−0.558232	−0.279116	−	0.960257i	$-0.590041\pi$
$331$	−10.1561	−0.558232	−0.279116	+	0.960257i	$0.590041\pi$
$332$	−16.4524	−0.902943
$333$	−24.8217	−1.36022
$334$	−17.1407	−0.937899
$335$	−8.67092	−0.473743
$336$	−3.90000	−0.212762
$337$	−18.6408	−1.01543	−0.507713	−	0.861526i	$-0.669509\pi$
$337$	−18.6408	−1.01543	−0.507713	+	0.861526i	$0.669509\pi$
$338$	22.7662	1.23832
$339$	11.0889	0.602264
$340$	3.22023	0.174642
$341$	15.9117	0.861667
$342$	−27.0079	−1.46042
$343$	−15.6876	−0.847053
$344$	−8.55560	−0.461287
$345$	5.31238	0.286009
$346$	−17.7937	−0.956596
$347$	−36.4774	−1.95821	−0.979104	−	0.203360i	$-0.934814\pi$
$347$	−36.4774	−1.95821	−0.979104	+	0.203360i	$0.934814\pi$
$348$	−16.2000	−0.868412
$349$	−23.4849	−1.25712	−0.628559	−	0.777762i	$-0.716355\pi$
$349$	−23.4849	−1.25712	−0.628559	+	0.777762i	$0.716355\pi$
$350$	−4.11534	−0.219974
$351$	−64.5544	−3.44566
$352$	−5.42729	−0.289276
$353$	−28.5682	−1.52053	−0.760267	−	0.649611i	$-0.774932\pi$
$353$	−28.5682	−1.52053	−0.760267	+	0.649611i	$0.774932\pi$
$354$	11.3822	0.604957
$355$	7.32352	0.388692
$356$	−10.6250	−0.563123
$357$	−9.50096	−0.502844
$358$	22.4311	1.18552
$359$	−2.31067	−0.121952	−0.0609762	−	0.998139i	$-0.519421\pi$
$359$	−2.31067	−0.121952	−0.0609762	+	0.998139i	$0.519421\pi$
$360$	8.59440	0.452965
$361$	−1.74480	−0.0918316
$362$	5.88737	0.309433
$363$	−56.8890	−2.98590
$364$	7.56656	0.396596
$365$	−15.8780	−0.831091
$366$	−17.2270	−0.900467
$367$	10.8231	0.564963	0.282482	−	0.959273i	$-0.408842\pi$
$367$	10.8231	0.564963	0.282482	+	0.959273i	$0.408842\pi$
$368$	−1.30378	−0.0679640
$369$	−59.0830	−3.07574
$370$	−5.04643	−0.262352
$371$	0.554355	0.0287807
$372$	9.03723	0.468559
$373$	−18.6184	−0.964026	−0.482013	−	0.876164i	$-0.660094\pi$
$373$	−18.6184	−0.964026	−0.482013	+	0.876164i	$0.660094\pi$
$374$	−13.2217	−0.683676
$375$	33.6265	1.73647
$376$	8.90702	0.459345
$377$	31.4304	1.61875
$378$	−13.6569	−0.702435
$379$	−11.0353	−0.566845	−0.283423	−	0.958995i	$-0.591470\pi$
$379$	−11.0353	−0.566845	−0.283423	+	0.958995i	$0.591470\pi$
$380$	−5.49091	−0.281678
$381$	40.5889	2.07943
$382$	−15.6529	−0.800870
$383$	7.73048	0.395009	0.197505	−	0.980302i	$-0.436716\pi$
$383$	7.73048	0.395009	0.197505	+	0.980302i	$0.436716\pi$
$384$	−3.08249	−0.157303
$385$	−9.07673	−0.462593
$386$	−13.9134	−0.708175
$387$	−55.6265	−2.82765
$388$	0.454316	0.0230644
$389$	−27.7130	−1.40510	−0.702552	−	0.711632i	$-0.747956\pi$
$389$	−27.7130	−1.40510	−0.702552	+	0.711632i	$0.747956\pi$
$390$	−24.3682	−1.23393
$391$	−3.17619	−0.160627
$392$	−5.39925	−0.272703
$393$	55.8459	2.81705
$394$	22.6959	1.14340
$395$	10.3407	0.520296
$396$	−35.2870	−1.77324
$397$	−14.3587	−0.720643	−0.360321	−	0.932828i	$-0.617333\pi$
$397$	−14.3587	−0.720643	−0.360321	+	0.932828i	$0.617333\pi$
$398$	−8.75610	−0.438904
$399$	16.2004	0.811032
$400$	−3.25270	−0.162635
$401$	−14.1123	−0.704734	−0.352367	−	0.935862i	$-0.614623\pi$
$401$	−14.1123	−0.704734	−0.352367	+	0.935862i	$0.614623\pi$
$402$	20.2201	1.00849
$403$	−17.5335	−0.873408
$404$	16.5270	0.822251
$405$	18.1989	0.904311
$406$	6.64929	0.329999
$407$	20.7197	1.02704
$408$	−7.50940	−0.371771
$409$	0.263925	0.0130503	0.00652513	−	0.999979i	$-0.497923\pi$
$409$	0.263925	0.0130503	0.00652513	+	0.999979i	$0.497923\pi$
$410$	−12.0120	−0.593230
$411$	58.5659	2.88884
$412$	2.38916	0.117706
$413$	−4.67182	−0.229885
$414$	−8.47685	−0.416614
$415$	−21.7477	−1.06755
$416$	5.98048	0.293217
$417$	−16.5734	−0.811601
$418$	22.5446	1.10269
$419$	27.7849	1.35738	0.678690	−	0.734425i	$-0.262548\pi$
$419$	27.7849	1.35738	0.678690	+	0.734425i	$0.262548\pi$
$420$	−5.15524	−0.251550
$421$	−30.0583	−1.46495	−0.732476	−	0.680793i	$-0.761635\pi$
$421$	−30.0583	−1.46495	−0.732476	+	0.680793i	$0.761635\pi$
$422$	−21.6540	−1.05410
$423$	57.9114	2.81575
$424$	0.438153	0.0212786
$425$	−7.92404	−0.384372
$426$	−17.0780	−0.827433
$427$	7.07080	0.342180
$428$	10.8517	0.524537
$429$	100.051	4.83051
$430$	−11.3093	−0.545381
$431$	−24.7808	−1.19365	−0.596824	−	0.802372i	$-0.703571\pi$
$431$	−24.7808	−1.19365	−0.596824	+	0.802372i	$0.703571\pi$
$432$	−10.7942	−0.519335
$433$	−24.1168	−1.15898	−0.579489	−	0.814980i	$-0.696748\pi$
$433$	−24.1168	−1.15898	−0.579489	+	0.814980i	$0.696748\pi$
$434$	−3.70933	−0.178054
$435$	−21.4141	−1.02673
$436$	9.20979	0.441069
$437$	5.41580	0.259073
$438$	37.0265	1.76920
$439$	12.5456	0.598770	0.299385	−	0.954132i	$-0.403218\pi$
$439$	12.5456	0.598770	0.299385	+	0.954132i	$0.403218\pi$
$440$	−7.17410	−0.342012
$441$	−35.1047	−1.67165
$442$	14.5693	0.692992
$443$	5.08143	0.241426	0.120713	−	0.992687i	$-0.461482\pi$
$443$	5.08143	0.241426	0.120713	+	0.992687i	$0.461482\pi$
$444$	11.7680	0.558485
$445$	−14.0447	−0.665783
$446$	−17.7375	−0.839893
$447$	−16.3629	−0.773937
$448$	1.26521	0.0597755
$449$	0.505644	0.0238628	0.0119314	−	0.999929i	$-0.496202\pi$
$449$	0.505644	0.0238628	0.0119314	+	0.999929i	$0.496202\pi$
$450$	−21.1483	−0.996940
$451$	49.3190	2.32234
$452$	−3.59737	−0.169206
$453$	−47.9312	−2.25200
$454$	28.4512	1.33528
$455$	10.0019	0.468896
$456$	12.8045	0.599625
$457$	−4.20294	−0.196605	−0.0983026	−	0.995157i	$-0.531341\pi$
$457$	−4.20294	−0.196605	−0.0983026	+	0.995157i	$0.531341\pi$
$458$	−21.9042	−1.02352
$459$	−26.2962	−1.22740
$460$	−1.72340	−0.0803541
$461$	−20.0780	−0.935125	−0.467562	−	0.883960i	$-0.654868\pi$
$461$	−20.0780	−0.935125	−0.467562	+	0.883960i	$0.654868\pi$
$462$	21.1664	0.984752
$463$	24.9166	1.15797	0.578985	−	0.815338i	$-0.303449\pi$
$463$	24.9166	1.15797	0.578985	+	0.815338i	$0.303449\pi$
$464$	5.25549	0.243980
$465$	11.9459	0.553979
$466$	−17.6475	−0.817506
$467$	−2.96641	−0.137269	−0.0686345	−	0.997642i	$-0.521864\pi$
$467$	−2.96641	−0.137269	−0.0686345	+	0.997642i	$0.521864\pi$
$468$	38.8837	1.79740
$469$	−8.29933	−0.383228
$470$	11.7738	0.543085
$471$	−25.7754	−1.18767
$472$	−3.69253	−0.169962
$473$	46.4337	2.13502
$474$	−24.1139	−1.10759
$475$	13.5115	0.619950
$476$	3.08223	0.141274
$477$	2.84877	0.130436
$478$	5.35528	0.244945
$479$	32.0133	1.46273	0.731364	−	0.681988i	$-0.238884\pi$
$479$	32.0133	1.46273	0.731364	+	0.681988i	$0.238884\pi$
$480$	−4.07461	−0.185980
$481$	−22.8316	−1.04103
$482$	−5.95584	−0.271281
$483$	5.08472	0.231363
$484$	18.4555	0.838887
$485$	0.600540	0.0272691
$486$	−10.0563	−0.456163
$487$	−2.10114	−0.0952117	−0.0476058	−	0.998866i	$-0.515159\pi$
$487$	−2.10114	−0.0952117	−0.0476058	+	0.998866i	$0.515159\pi$
$488$	5.58864	0.252986
$489$	−68.5738	−3.10102
$490$	−7.13703	−0.322418
$491$	−25.9735	−1.17217	−0.586083	−	0.810251i	$-0.699331\pi$
$491$	−25.9735	−1.17217	−0.586083	+	0.810251i	$0.699331\pi$
$492$	28.0113	1.26285
$493$	12.8031	0.576624
$494$	−24.8426	−1.11772
$495$	−46.6443	−2.09651
$496$	−2.93179	−0.131641
$497$	7.00967	0.314427
$498$	50.7144	2.27257
$499$	1.51789	0.0679499	0.0339750	−	0.999423i	$-0.489183\pi$
$499$	1.51789	0.0679499	0.0339750	+	0.999423i	$0.489183\pi$
$500$	−10.9089	−0.487860
$501$	52.8362	2.36055
$502$	−16.6497	−0.743113
$503$	2.25441	0.100519	0.0502595	−	0.998736i	$-0.483995\pi$
$503$	2.25441	0.100519	0.0502595	+	0.998736i	$0.483995\pi$
$504$	8.22610	0.366419
$505$	21.8464	0.972151
$506$	7.07597	0.314565
$507$	−70.1766	−3.11665
$508$	−13.1675	−0.584215
$509$	0.929901	0.0412171	0.0206086	−	0.999788i	$-0.493440\pi$
$509$	0.929901	0.0412171	0.0206086	+	0.999788i	$0.493440\pi$
$510$	−9.92635	−0.439546
$511$	−15.1975	−0.672299
$512$	1.00000	0.0441942
$513$	44.8384	1.97966
$514$	−11.9681	−0.527890
$515$	3.15813	0.139164
$516$	26.3726	1.16099
$517$	−48.3410	−2.12604
$518$	−4.83017	−0.212226
$519$	54.8490	2.40760
$520$	7.90534	0.346672
$521$	14.0004	0.613371	0.306685	−	0.951811i	$-0.400780\pi$
$521$	14.0004	0.613371	0.306685	+	0.951811i	$0.400780\pi$
$522$	34.1700	1.49558
$523$	−12.2402	−0.535228	−0.267614	−	0.963526i	$-0.586235\pi$
$523$	−12.2402	−0.535228	−0.267614	+	0.963526i	$0.586235\pi$
$524$	−18.1171	−0.791450
$525$	12.6855	0.553641
$526$	−16.1809	−0.705519
$527$	−7.14227	−0.311122
$528$	16.7296	0.728062
$529$	−21.3002	−0.926094
$530$	0.579176	0.0251578
$531$	−24.0080	−1.04186
$532$	−5.25560	−0.227859
$533$	−54.3460	−2.35399
$534$	32.7515	1.41729
$535$	14.3444	0.620162
$536$	−6.55966	−0.283334
$537$	−69.1436	−2.98377
$538$	31.3862	1.35316
$539$	29.3033	1.26218
$540$	−14.2684	−0.614012
$541$	−10.0031	−0.430067	−0.215034	−	0.976607i	$-0.568986\pi$
$541$	−10.0031	−0.430067	−0.215034	+	0.976607i	$0.568986\pi$
$542$	2.35963	0.101355
$543$	−18.1478	−0.778796
$544$	2.43614	0.104449
$545$	12.1740	0.521478
$546$	−23.3239	−0.998170
$547$	11.5517	0.493917	0.246958	−	0.969026i	$-0.420569\pi$
$547$	11.5517	0.493917	0.246958	+	0.969026i	$0.420569\pi$
$548$	−18.9995	−0.811619
$549$	36.3361	1.55079
$550$	17.6533	0.752741
$551$	−21.8310	−0.930031
$552$	4.01888	0.171055
$553$	9.89753	0.420886
$554$	26.4512	1.12380
$555$	15.5556	0.660298
$556$	5.37661	0.228019
$557$	41.8869	1.77481	0.887403	−	0.460995i	$-0.152507\pi$
$557$	41.8869	1.77481	0.887403	+	0.460995i	$0.152507\pi$
$558$	−19.0618	−0.806952
$559$	−51.1666	−2.16412
$560$	1.67242	0.0706728
$561$	40.7557	1.72071
$562$	−11.8247	−0.498795
$563$	0.974428	0.0410672	0.0205336	−	0.999789i	$-0.493463\pi$
$563$	0.974428	0.0410672	0.0205336	+	0.999789i	$0.493463\pi$
$564$	−27.4558	−1.15610
$565$	−4.75520	−0.200053
$566$	16.7125	0.702478
$567$	17.4190	0.731530
$568$	5.54033	0.232467
$569$	−5.32637	−0.223293	−0.111646	−	0.993748i	$-0.535612\pi$
$569$	−5.32637	−0.223293	−0.111646	+	0.993748i	$0.535612\pi$
$570$	16.9257	0.708939
$571$	−37.8102	−1.58231	−0.791154	−	0.611617i	$-0.790519\pi$
$571$	−37.8102	−1.58231	−0.791154	+	0.611617i	$0.790519\pi$
$572$	−32.4578	−1.35713
$573$	48.2499	2.01567
$574$	−11.4972	−0.479885
$575$	4.24079	0.176853
$576$	6.50177	0.270907
$577$	−39.4832	−1.64371	−0.821854	−	0.569698i	$-0.807060\pi$
$577$	−39.4832	−1.64371	−0.821854	+	0.569698i	$0.807060\pi$
$578$	−11.0652	−0.460252
$579$	42.8881	1.78237
$580$	6.94700	0.288459
$581$	−20.8157	−0.863581
$582$	−1.40043	−0.0580495
$583$	−2.37799	−0.0984862
$584$	−12.0119	−0.497055
$585$	51.3987	2.12507
$586$	19.5074	0.805841
$587$	−32.1743	−1.32798	−0.663989	−	0.747743i	$-0.731138\pi$
$587$	−32.1743	−1.32798	−0.663989	+	0.747743i	$0.731138\pi$
$588$	16.6431	0.686352
$589$	12.1785	0.501806
$590$	−4.88099	−0.200947
$591$	−69.9600	−2.87777
$592$	−3.81769	−0.156906
$593$	13.1873	0.541539	0.270770	−	0.962644i	$-0.412722\pi$
$593$	13.1873	0.541539	0.270770	+	0.962644i	$0.412722\pi$
$594$	58.5832	2.40370
$595$	4.07427	0.167029
$596$	5.30832	0.217437
$597$	26.9906	1.10465
$598$	−7.79721	−0.318852
$599$	32.7181	1.33682	0.668412	−	0.743791i	$-0.266974\pi$
$599$	32.7181	1.33682	0.668412	+	0.743791i	$0.266974\pi$
$600$	10.0264	0.409327
$601$	3.98267	0.162456	0.0812282	−	0.996696i	$-0.474116\pi$
$601$	3.98267	0.162456	0.0812282	+	0.996696i	$0.474116\pi$
$602$	−10.8246	−0.441178
$603$	−42.6494	−1.73682
$604$	15.5495	0.632700
$605$	24.3955	0.991819
$606$	−50.9445	−2.06948
$607$	−18.8579	−0.765417	−0.382708	−	0.923869i	$-0.625009\pi$
$607$	−18.8579	−0.765417	−0.382708	+	0.923869i	$0.625009\pi$
$608$	−4.15394	−0.168464
$609$	−20.4964	−0.830556
$610$	7.38738	0.299106
$611$	53.2683	2.15501
$612$	15.8392	0.640264
$613$	−9.42844	−0.380811	−0.190405	−	0.981706i	$-0.560980\pi$
$613$	−9.42844	−0.380811	−0.190405	+	0.981706i	$0.560980\pi$
$614$	5.38550	0.217341
$615$	37.0269	1.49307
$616$	−6.86666	−0.276666
$617$	−37.5211	−1.51054	−0.755271	−	0.655413i	$-0.772495\pi$
$617$	−37.5211	−1.51054	−0.755271	+	0.655413i	$0.772495\pi$
$618$	−7.36458	−0.296247
$619$	−11.3646	−0.456780	−0.228390	−	0.973570i	$-0.573346\pi$
$619$	−11.3646	−0.456780	−0.228390	+	0.973570i	$0.573346\pi$
$620$	−3.87541	−0.155640
$621$	14.0732	0.564738
$622$	9.96482	0.399553
$623$	−13.4428	−0.538576
$624$	−18.4348	−0.737983
$625$	1.84352	0.0737408
$626$	17.3393	0.693019
$627$	−69.4937	−2.77531
$628$	8.36186	0.333674
$629$	−9.30044	−0.370833
$630$	10.8737	0.433219
$631$	3.93938	0.156824	0.0784120	−	0.996921i	$-0.475015\pi$
$631$	3.93938	0.156824	0.0784120	+	0.996921i	$0.475015\pi$
$632$	7.82284	0.311176
$633$	66.7482	2.65300
$634$	−12.8285	−0.509485
$635$	−17.4056	−0.690720
$636$	−1.35061	−0.0535550
$637$	−32.2901	−1.27938
$638$	−28.5231	−1.12924
$639$	36.0219	1.42501
$640$	1.32186	0.0522509
$641$	1.10430	0.0436172	0.0218086	−	0.999762i	$-0.493058\pi$
$641$	1.10430	0.0436172	0.0218086	+	0.999762i	$0.493058\pi$
$642$	−33.4503	−1.32018
$643$	−49.7421	−1.96164	−0.980819	−	0.194920i	$-0.937555\pi$
$643$	−49.7421	−1.96164	−0.980819	+	0.194920i	$0.937555\pi$
$644$	−1.64955	−0.0650013
$645$	34.8607	1.37264
$646$	−10.1196	−0.398150
$647$	5.59457	0.219945	0.109973	−	0.993935i	$-0.464924\pi$
$647$	5.59457	0.219945	0.109973	+	0.993935i	$0.464924\pi$
$648$	13.7677	0.540846
$649$	20.0404	0.786656
$650$	−19.4527	−0.762998
$651$	11.4340	0.448133
$652$	22.2462	0.871229
$653$	19.0147	0.744103	0.372052	−	0.928212i	$-0.378654\pi$
$653$	19.0147	0.744103	0.372052	+	0.928212i	$0.378654\pi$
$654$	−28.3891	−1.11010
$655$	−23.9482	−0.935735
$656$	−9.08722	−0.354796
$657$	−78.0984	−3.04691
$658$	11.2692	0.439321
$659$	9.25412	0.360489	0.180245	−	0.983622i	$-0.442311\pi$
$659$	9.25412	0.360489	0.180245	+	0.983622i	$0.442311\pi$
$660$	22.1141	0.860791
$661$	11.8626	0.461401	0.230700	−	0.973025i	$-0.425898\pi$
$661$	11.8626	0.461401	0.230700	+	0.973025i	$0.425898\pi$
$662$	−10.1561	−0.394729
$663$	−44.9098	−1.74415
$664$	−16.4524	−0.638477
$665$	−6.94714	−0.269399
$666$	−24.8217	−0.961822
$667$	−6.85198	−0.265310
$668$	−17.1407	−0.663195
$669$	54.6756	2.11388
$670$	−8.67092	−0.334987
$671$	−30.3312	−1.17092
$672$	−3.90000	−0.150446
$673$	10.1356	0.390698	0.195349	−	0.980734i	$-0.437416\pi$
$673$	10.1356	0.390698	0.195349	+	0.980734i	$0.437416\pi$
$674$	−18.6408	−0.718015
$675$	35.1102	1.35139
$676$	22.7662	0.875623
$677$	3.04915	0.117189	0.0585943	−	0.998282i	$-0.481338\pi$
$677$	3.04915	0.117189	0.0585943	+	0.998282i	$0.481338\pi$
$678$	11.0889	0.425865
$679$	0.574804	0.0220590
$680$	3.22023	0.123490
$681$	−87.7005	−3.36069
$682$	15.9117	0.609290
$683$	18.0397	0.690269	0.345134	−	0.938553i	$-0.387833\pi$
$683$	18.0397	0.690269	0.345134	+	0.938553i	$0.387833\pi$
$684$	−27.0079	−1.03267
$685$	−25.1146	−0.959580
$686$	−15.6876	−0.598957
$687$	67.5196	2.57603
$688$	−8.55560	−0.326179
$689$	2.62037	0.0998281
$690$	5.31238	0.202239
$691$	5.45195	0.207402	0.103701	−	0.994609i	$-0.466931\pi$
$691$	5.45195	0.207402	0.103701	+	0.994609i	$0.466931\pi$
$692$	−17.7937	−0.676415
$693$	−44.6454	−1.69594
$694$	−36.4774	−1.38466
$695$	7.10710	0.269588
$696$	−16.2000	−0.614060
$697$	−22.1378	−0.838528
$698$	−23.4849	−0.888916
$699$	54.3984	2.05754
$700$	−4.11534	−0.155545
$701$	−22.0082	−0.831239	−0.415620	−	0.909538i	$-0.636435\pi$
$701$	−22.0082	−0.831239	−0.415620	+	0.909538i	$0.636435\pi$
$702$	−64.5544	−2.43645
$703$	15.8584	0.598112
$704$	−5.42729	−0.204549
$705$	−36.2927	−1.36686
$706$	−28.5682	−1.07518
$707$	20.9102	0.786407
$708$	11.3822	0.427769
$709$	10.0221	0.376389	0.188195	−	0.982132i	$-0.439736\pi$
$709$	10.0221	0.376389	0.188195	+	0.982132i	$0.439736\pi$
$710$	7.32352	0.274847
$711$	50.8623	1.90749
$712$	−10.6250	−0.398188
$713$	3.82240	0.143150
$714$	−9.50096	−0.355564
$715$	−42.9046	−1.60454
$716$	22.4311	0.838289
$717$	−16.5076	−0.616488
$718$	−2.31067	−0.0862334
$719$	38.6862	1.44275	0.721376	−	0.692544i	$-0.243510\pi$
$719$	38.6862	1.44275	0.721376	+	0.692544i	$0.243510\pi$
$720$	8.59440	0.320294
$721$	3.02279	0.112575
$722$	−1.74480	−0.0649348
$723$	18.3588	0.682773
$724$	5.88737	0.218802
$725$	−17.0945	−0.634875
$726$	−56.8890	−2.11135
$727$	−18.0590	−0.669771	−0.334886	−	0.942259i	$-0.608698\pi$
$727$	−18.0590	−0.669771	−0.334886	+	0.942259i	$0.608698\pi$
$728$	7.56656	0.280435
$729$	−10.3046	−0.381653
$730$	−15.8780	−0.587670
$731$	−20.8427	−0.770894
$732$	−17.2270	−0.636727
$733$	−37.2732	−1.37672	−0.688358	−	0.725371i	$-0.741668\pi$
$733$	−37.2732	−1.37672	−0.688358	+	0.725371i	$0.741668\pi$
$734$	10.8231	0.399489
$735$	21.9998	0.811476
$736$	−1.30378	−0.0480578
$737$	35.6012	1.31139
$738$	−59.0830	−2.17488
$739$	−23.6296	−0.869227	−0.434614	−	0.900617i	$-0.643115\pi$
$739$	−23.6296	−0.869227	−0.434614	+	0.900617i	$0.643115\pi$
$740$	−5.04643	−0.185511
$741$	76.5770	2.81313
$742$	0.554355	0.0203510
$743$	20.7589	0.761570	0.380785	−	0.924664i	$-0.375654\pi$
$743$	20.7589	0.761570	0.380785	+	0.924664i	$0.375654\pi$
$744$	9.03723	0.331321
$745$	7.01684	0.257077
$746$	−18.6184	−0.681669
$747$	−106.970	−3.91382
$748$	−13.2217	−0.483432
$749$	13.7297	0.501671
$750$	33.6265	1.22787
$751$	−7.73120	−0.282116	−0.141058	−	0.990001i	$-0.545050\pi$
$751$	−7.73120	−0.282116	−0.141058	+	0.990001i	$0.545050\pi$
$752$	8.90702	0.324806
$753$	51.3227	1.87030
$754$	31.4304	1.14463
$755$	20.5542	0.748043
$756$	−13.6569	−0.496696
$757$	8.07471	0.293480	0.146740	−	0.989175i	$-0.453122\pi$
$757$	8.07471	0.293480	0.146740	+	0.989175i	$0.453122\pi$
$758$	−11.0353	−0.400820
$759$	−21.8116	−0.791713
$760$	−5.49091	−0.199176
$761$	38.6060	1.39947	0.699734	−	0.714404i	$-0.253302\pi$
$761$	38.6060	1.39947	0.699734	+	0.714404i	$0.253302\pi$
$762$	40.5889	1.47038
$763$	11.6523	0.421842
$764$	−15.6529	−0.566301
$765$	20.9372	0.756986
$766$	7.73048	0.279314
$767$	−22.0831	−0.797375
$768$	−3.08249	−0.111230
$769$	16.8325	0.606996	0.303498	−	0.952832i	$-0.401845\pi$
$769$	16.8325	0.606996	0.303498	+	0.952832i	$0.401845\pi$
$770$	−9.07673	−0.327103
$771$	36.8916	1.32862
$772$	−13.9134	−0.500756
$773$	16.9180	0.608499	0.304250	−	0.952592i	$-0.401594\pi$
$773$	16.9180	0.608499	0.304250	+	0.952592i	$0.401594\pi$
$774$	−55.6265	−1.99945
$775$	9.53623	0.342552
$776$	0.454316	0.0163090
$777$	14.8890	0.534139
$778$	−27.7130	−0.993559
$779$	37.7478	1.35245
$780$	−24.3682	−0.872520
$781$	−30.0690	−1.07595
$782$	−3.17619	−0.113580
$783$	−56.7287	−2.02732
$784$	−5.39925	−0.192830
$785$	11.0532	0.394505
$786$	55.8459	1.99196
$787$	39.4323	1.40561	0.702804	−	0.711383i	$-0.251931\pi$
$787$	39.4323	1.40561	0.702804	+	0.711383i	$0.251931\pi$
$788$	22.6959	0.808508
$789$	49.8774	1.77568
$790$	10.3407	0.367905
$791$	−4.55142	−0.161830
$792$	−35.2870	−1.25387
$793$	33.4228	1.18688
$794$	−14.3587	−0.509571
$795$	−1.78531	−0.0633183
$796$	−8.75610	−0.310352
$797$	−1.39104	−0.0492730	−0.0246365	−	0.999696i	$-0.507843\pi$
$797$	−1.39104	−0.0492730	−0.0246365	+	0.999696i	$0.507843\pi$
$798$	16.2004	0.573486
$799$	21.6988	0.767648
$800$	−3.25270	−0.115000
$801$	−69.0812	−2.44087
$802$	−14.1123	−0.498322
$803$	65.1920	2.30057
$804$	20.2201	0.713108
$805$	−2.18047	−0.0768513
$806$	−17.5335	−0.617592
$807$	−96.7478	−3.40568
$808$	16.5270	0.581419
$809$	−0.251194	−0.00883149	−0.00441575	−	0.999990i	$-0.501406\pi$
$809$	−0.251194	−0.00883149	−0.00441575	+	0.999990i	$0.501406\pi$
$810$	18.1989	0.639445
$811$	2.26935	0.0796876	0.0398438	−	0.999206i	$-0.487314\pi$
$811$	2.26935	0.0796876	0.0398438	+	0.999206i	$0.487314\pi$
$812$	6.64929	0.233344
$813$	−7.27354	−0.255094
$814$	20.7197	0.726225
$815$	29.4063	1.03006
$816$	−7.50940	−0.262882
$817$	35.5394	1.24337
$818$	0.263925	0.00922792
$819$	49.1960	1.71905
$820$	−12.0120	−0.419477
$821$	27.5826	0.962640	0.481320	−	0.876545i	$-0.340158\pi$
$821$	27.5826	0.962640	0.481320	+	0.876545i	$0.340158\pi$
$822$	58.5659	2.04272
$823$	−2.12998	−0.0742465	−0.0371233	−	0.999311i	$-0.511819\pi$
$823$	−2.12998	−0.0742465	−0.0371233	+	0.999311i	$0.511819\pi$
$824$	2.38916	0.0832304
$825$	−54.4163	−1.89453
$826$	−4.67182	−0.162553
$827$	6.61999	0.230200	0.115100	−	0.993354i	$-0.463281\pi$
$827$	6.61999	0.230200	0.115100	+	0.993354i	$0.463281\pi$
$828$	−8.47685	−0.294591
$829$	−18.8776	−0.655645	−0.327823	−	0.944739i	$-0.606315\pi$
$829$	−18.8776	−0.655645	−0.327823	+	0.944739i	$0.606315\pi$
$830$	−21.7477	−0.754874
$831$	−81.5356	−2.82844
$832$	5.98048	0.207336
$833$	−13.1533	−0.455736
$834$	−16.5734	−0.573889
$835$	−22.6576	−0.784098
$836$	22.5446	0.779723
$837$	31.6463	1.09386
$838$	27.7849	0.959812
$839$	33.0703	1.14171	0.570857	−	0.821050i	$-0.306611\pi$
$839$	33.0703	1.14171	0.570857	+	0.821050i	$0.306611\pi$
$840$	−5.15524	−0.177873
$841$	−1.37980	−0.0475793
$842$	−30.0583	−1.03588
$843$	36.4496	1.25539
$844$	−21.6540	−0.745360
$845$	30.0936	1.03525
$846$	57.9114	1.99104
$847$	23.3501	0.802318
$848$	0.438153	0.0150462
$849$	−51.5161	−1.76803
$850$	−7.92404	−0.271792
$851$	4.97741	0.170623
$852$	−17.0780	−0.585084
$853$	11.5409	0.395154	0.197577	−	0.980287i	$-0.436693\pi$
$853$	11.5409	0.395154	0.197577	+	0.980287i	$0.436693\pi$
$854$	7.07080	0.241958
$855$	−35.7006	−1.22094
$856$	10.8517	0.370904
$857$	29.3779	1.00353	0.501766	−	0.865004i	$-0.332684\pi$
$857$	29.3779	1.00353	0.501766	+	0.865004i	$0.332684\pi$
$858$	100.051	3.41569
$859$	2.86655	0.0978053	0.0489027	−	0.998804i	$-0.484428\pi$
$859$	2.86655	0.0978053	0.0489027	+	0.998804i	$0.484428\pi$
$860$	−11.3093	−0.385643
$861$	35.4402	1.20780
$862$	−24.7808	−0.844037
$863$	29.4288	1.00177	0.500884	−	0.865515i	$-0.333008\pi$
$863$	29.4288	1.00177	0.500884	+	0.865515i	$0.333008\pi$
$864$	−10.7942	−0.367226
$865$	−23.5207	−0.799729
$866$	−24.1168	−0.819521
$867$	34.1084	1.15838
$868$	−3.70933	−0.125903
$869$	−42.4569	−1.44025
$870$	−21.4141	−0.726006
$871$	−39.2299	−1.32926
$872$	9.20979	0.311883
$873$	2.95386	0.0999729
$874$	5.41580	0.183192
$875$	−13.8020	−0.466593
$876$	37.0265	1.25101
$877$	54.0132	1.82390	0.911948	−	0.410306i	$-0.134578\pi$
$877$	54.0132	1.82390	0.911948	+	0.410306i	$0.134578\pi$
$878$	12.5456	0.423394
$879$	−60.1313	−2.02818
$880$	−7.17410	−0.241839
$881$	−39.5685	−1.33310	−0.666548	−	0.745462i	$-0.732229\pi$
$881$	−39.5685	−1.33310	−0.666548	+	0.745462i	$0.732229\pi$
$882$	−35.1047	−1.18204
$883$	−13.0660	−0.439705	−0.219853	−	0.975533i	$-0.570558\pi$
$883$	−13.0660	−0.439705	−0.219853	+	0.975533i	$0.570558\pi$
$884$	14.5693	0.490019
$885$	15.0456	0.505753
$886$	5.08143	0.170714
$887$	33.2692	1.11707	0.558534	−	0.829481i	$-0.311364\pi$
$887$	33.2692	1.11707	0.558534	+	0.829481i	$0.311364\pi$
$888$	11.7680	0.394908
$889$	−16.6597	−0.558748
$890$	−14.0447	−0.470780
$891$	−74.7213	−2.50326
$892$	−17.7375	−0.593894
$893$	−36.9992	−1.23813
$894$	−16.3629	−0.547256
$895$	29.6506	0.991112
$896$	1.26521	0.0422677
$897$	24.0349	0.802500
$898$	0.505644	0.0168735
$899$	−15.4080	−0.513886
$900$	−21.1483	−0.704943
$901$	1.06740	0.0355604
$902$	49.3190	1.64214
$903$	33.3668	1.11038
$904$	−3.59737	−0.119647
$905$	7.78226	0.258691
$906$	−47.9312	−1.59241
$907$	9.91523	0.329230	0.164615	−	0.986358i	$-0.447362\pi$
$907$	9.91523	0.329230	0.164615	+	0.986358i	$0.447362\pi$
$908$	28.4512	0.944185
$909$	107.455	3.56406
$910$	10.0019	0.331560
$911$	−3.10272	−0.102798	−0.0513988	−	0.998678i	$-0.516368\pi$
$911$	−3.10272	−0.102798	−0.0513988	+	0.998678i	$0.516368\pi$
$912$	12.8045	0.423999
$913$	89.2920	2.95513
$914$	−4.20294	−0.139021
$915$	−22.7716	−0.752804
$916$	−21.9042	−0.723735
$917$	−22.9219	−0.756949
$918$	−26.2962	−0.867904
$919$	−22.5379	−0.743458	−0.371729	−	0.928341i	$-0.621235\pi$
$919$	−22.5379	−0.743458	−0.371729	+	0.928341i	$0.621235\pi$
$920$	−1.72340	−0.0568189
$921$	−16.6008	−0.547014
$922$	−20.0780	−0.661233
$923$	33.1339	1.09061
$924$	21.1664	0.696325
$925$	12.4178	0.408294
$926$	24.9166	0.818809
$927$	15.5338	0.510196
$928$	5.25549	0.172520
$929$	11.2086	0.367743	0.183871	−	0.982950i	$-0.441137\pi$
$929$	11.2086	0.367743	0.183871	+	0.982950i	$0.441137\pi$
$930$	11.9459	0.391722
$931$	22.4281	0.735052
$932$	−17.6475	−0.578064
$933$	−30.7165	−1.00561
$934$	−2.96641	−0.0970639
$935$	−17.4771	−0.571564
$936$	38.8837	1.27095
$937$	47.0445	1.53688	0.768438	−	0.639924i	$-0.221034\pi$
$937$	47.0445	1.53688	0.768438	+	0.639924i	$0.221034\pi$
$938$	−8.29933	−0.270983
$939$	−53.4484	−1.74422
$940$	11.7738	0.384019
$941$	−48.5707	−1.58336	−0.791680	−	0.610936i	$-0.790793\pi$
$941$	−48.5707	−1.58336	−0.791680	+	0.610936i	$0.790793\pi$
$942$	−25.7754	−0.839807
$943$	11.8477	0.385814
$944$	−3.69253	−0.120182
$945$	−18.0524	−0.587246
$946$	46.4337	1.50969
$947$	1.44903	0.0470870	0.0235435	−	0.999723i	$-0.492505\pi$
$947$	1.44903	0.0470870	0.0235435	+	0.999723i	$0.492505\pi$
$948$	−24.1139	−0.783182
$949$	−71.8368	−2.33192
$950$	13.5115	0.438371
$951$	39.5438	1.28230
$952$	3.08223	0.0998957
$953$	43.8238	1.41959	0.709796	−	0.704407i	$-0.248787\pi$
$953$	43.8238	1.41959	0.709796	+	0.704407i	$0.248787\pi$
$954$	2.84877	0.0922324
$955$	−20.6908	−0.669540
$956$	5.35528	0.173202
$957$	87.9223	2.84212
$958$	32.0133	1.03430
$959$	−24.0383	−0.776239
$960$	−4.07461	−0.131508
$961$	−22.4046	−0.722729
$962$	−22.8316	−0.736121
$963$	70.5553	2.27361
$964$	−5.95584	−0.191825
$965$	−18.3916	−0.592045
$966$	5.08472	0.163598
$967$	22.2343	0.715008	0.357504	−	0.933912i	$-0.383628\pi$
$967$	22.2343	0.715008	0.357504	+	0.933912i	$0.383628\pi$
$968$	18.4555	0.593183
$969$	31.1936	1.00208
$970$	0.600540	0.0192822
$971$	3.44039	0.110407	0.0552037	−	0.998475i	$-0.482419\pi$
$971$	3.44039	0.110407	0.0552037	+	0.998475i	$0.482419\pi$
$972$	−10.0563	−0.322556
$973$	6.80253	0.218079
$974$	−2.10114	−0.0673248
$975$	59.9628	1.92035
$976$	5.58864	0.178888
$977$	−59.0172	−1.88813	−0.944063	−	0.329765i	$-0.893031\pi$
$977$	−59.0172	−1.88813	−0.944063	+	0.329765i	$0.893031\pi$
$978$	−68.5738	−2.19275
$979$	57.6649	1.84298
$980$	−7.13703	−0.227984
$981$	59.8800	1.91182
$982$	−25.9735	−0.828847
$983$	17.8267	0.568584	0.284292	−	0.958738i	$-0.408241\pi$
$983$	17.8267	0.568584	0.284292	+	0.958738i	$0.408241\pi$
$984$	28.0113	0.892968
$985$	30.0007	0.955903
$986$	12.8031	0.407735
$987$	−34.7374	−1.10570
$988$	−24.8426	−0.790347
$989$	11.1546	0.354695
$990$	−46.6443	−1.48245
$991$	49.7702	1.58100	0.790501	−	0.612461i	$-0.209820\pi$
$991$	49.7702	1.58100	0.790501	+	0.612461i	$0.209820\pi$
$992$	−2.93179	−0.0930845
$993$	31.3062	0.993473
$994$	7.00967	0.222333
$995$	−11.5743	−0.366930
$996$	50.7144	1.60695
$997$	45.8830	1.45313	0.726565	−	0.687098i	$-0.241116\pi$
$997$	45.8830	1.45313	0.726565	+	0.687098i	$0.241116\pi$
$998$	1.51789	0.0480478
$999$	41.2088	1.30379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.a.1.4	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.a.1.4	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.08249 q^{3} +1.00000 q^{4} +1.32186 q^{5} -3.08249 q^{6} +1.26521 q^{7} +1.00000 q^{8} +6.50177 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.08249 q^{3} +1.00000 q^{4} +1.32186 q^{5} -3.08249 q^{6} +1.26521 q^{7} +1.00000 q^{8} +6.50177 q^{9} +1.32186 q^{10} -5.42729 q^{11} -3.08249 q^{12} +5.98048 q^{13} +1.26521 q^{14} -4.07461 q^{15} +1.00000 q^{16} +2.43614 q^{17} +6.50177 q^{18} -4.15394 q^{19} +1.32186 q^{20} -3.90000 q^{21} -5.42729 q^{22} -1.30378 q^{23} -3.08249 q^{24} -3.25270 q^{25} +5.98048 q^{26} -10.7942 q^{27} +1.26521 q^{28} +5.25549 q^{29} -4.07461 q^{30} -2.93179 q^{31} +1.00000 q^{32} +16.7296 q^{33} +2.43614 q^{34} +1.67242 q^{35} +6.50177 q^{36} -3.81769 q^{37} -4.15394 q^{38} -18.4348 q^{39} +1.32186 q^{40} -9.08722 q^{41} -3.90000 q^{42} -8.55560 q^{43} -5.42729 q^{44} +8.59440 q^{45} -1.30378 q^{46} +8.90702 q^{47} -3.08249 q^{48} -5.39925 q^{49} -3.25270 q^{50} -7.50940 q^{51} +5.98048 q^{52} +0.438153 q^{53} -10.7942 q^{54} -7.17410 q^{55} +1.26521 q^{56} +12.8045 q^{57} +5.25549 q^{58} -3.69253 q^{59} -4.07461 q^{60} +5.58864 q^{61} -2.93179 q^{62} +8.22610 q^{63} +1.00000 q^{64} +7.90534 q^{65} +16.7296 q^{66} -6.55966 q^{67} +2.43614 q^{68} +4.01888 q^{69} +1.67242 q^{70} +5.54033 q^{71} +6.50177 q^{72} -12.0119 q^{73} -3.81769 q^{74} +10.0264 q^{75} -4.15394 q^{76} -6.86666 q^{77} -18.4348 q^{78} +7.82284 q^{79} +1.32186 q^{80} +13.7677 q^{81} -9.08722 q^{82} -16.4524 q^{83} -3.90000 q^{84} +3.22023 q^{85} -8.55560 q^{86} -16.2000 q^{87} -5.42729 q^{88} -10.6250 q^{89} +8.59440 q^{90} +7.56656 q^{91} -1.30378 q^{92} +9.03723 q^{93} +8.90702 q^{94} -5.49091 q^{95} -3.08249 q^{96} +0.454316 q^{97} -5.39925 q^{98} -35.2870 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

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