LMFDB - Embedded newform 8002.2.a.d.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.04234 q^{3} +1.00000 q^{4} -4.37946 q^{5} -2.04234 q^{6} -3.34038 q^{7} +1.00000 q^{8} +1.17114 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.04234 q^{3} +1.00000 q^{4} -4.37946 q^{5} -2.04234 q^{6} -3.34038 q^{7} +1.00000 q^{8} +1.17114 q^{9} -4.37946 q^{10} -6.47165 q^{11} -2.04234 q^{12} +2.30446 q^{13} -3.34038 q^{14} +8.94433 q^{15} +1.00000 q^{16} -1.94500 q^{17} +1.17114 q^{18} +0.497927 q^{19} -4.37946 q^{20} +6.82218 q^{21} -6.47165 q^{22} +9.13451 q^{23} -2.04234 q^{24} +14.1797 q^{25} +2.30446 q^{26} +3.73514 q^{27} -3.34038 q^{28} -8.73359 q^{29} +8.94433 q^{30} +0.749179 q^{31} +1.00000 q^{32} +13.2173 q^{33} -1.94500 q^{34} +14.6291 q^{35} +1.17114 q^{36} +4.43414 q^{37} +0.497927 q^{38} -4.70648 q^{39} -4.37946 q^{40} +6.22772 q^{41} +6.82218 q^{42} +3.60194 q^{43} -6.47165 q^{44} -5.12897 q^{45} +9.13451 q^{46} -7.36820 q^{47} -2.04234 q^{48} +4.15813 q^{49} +14.1797 q^{50} +3.97235 q^{51} +2.30446 q^{52} +1.69540 q^{53} +3.73514 q^{54} +28.3423 q^{55} -3.34038 q^{56} -1.01694 q^{57} -8.73359 q^{58} -13.4470 q^{59} +8.94433 q^{60} -3.81707 q^{61} +0.749179 q^{62} -3.91206 q^{63} +1.00000 q^{64} -10.0923 q^{65} +13.2173 q^{66} +11.9649 q^{67} -1.94500 q^{68} -18.6557 q^{69} +14.6291 q^{70} +1.43259 q^{71} +1.17114 q^{72} +1.41227 q^{73} +4.43414 q^{74} -28.9597 q^{75} +0.497927 q^{76} +21.6178 q^{77} -4.70648 q^{78} +12.5869 q^{79} -4.37946 q^{80} -11.1419 q^{81} +6.22772 q^{82} -5.82838 q^{83} +6.82218 q^{84} +8.51805 q^{85} +3.60194 q^{86} +17.8369 q^{87} -6.47165 q^{88} +15.3847 q^{89} -5.12897 q^{90} -7.69776 q^{91} +9.13451 q^{92} -1.53008 q^{93} -7.36820 q^{94} -2.18065 q^{95} -2.04234 q^{96} -9.69510 q^{97} +4.15813 q^{98} -7.57922 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	−2.04234	−1.17914	−0.589572	−	0.807716i	$-0.700704\pi$
$3$	−2.04234	−1.17914	−0.589572	+	0.807716i	$0.700704\pi$
$4$	1.00000	0.500000
$5$	−4.37946	−1.95855	−0.979277	−	0.202527i	$-0.935085\pi$
$5$	−4.37946	−1.95855	−0.979277	+	0.202527i	$0.935085\pi$
$6$	−2.04234	−0.833781
$7$	−3.34038	−1.26254	−0.631272	−	0.775561i	$-0.717467\pi$
$7$	−3.34038	−1.26254	−0.631272	+	0.775561i	$0.717467\pi$
$8$	1.00000	0.353553
$9$	1.17114	0.390381
$10$	−4.37946	−1.38491
$11$	−6.47165	−1.95127	−0.975637	−	0.219389i	$-0.929594\pi$
$11$	−6.47165	−1.95127	−0.975637	+	0.219389i	$0.929594\pi$
$12$	−2.04234	−0.589572
$13$	2.30446	0.639141	0.319571	−	0.947562i	$-0.396461\pi$
$13$	2.30446	0.639141	0.319571	+	0.947562i	$0.396461\pi$
$14$	−3.34038	−0.892754
$15$	8.94433	2.30942
$16$	1.00000	0.250000
$17$	−1.94500	−0.471732	−0.235866	−	0.971786i	$-0.575793\pi$
$17$	−1.94500	−0.471732	−0.235866	+	0.971786i	$0.575793\pi$
$18$	1.17114	0.276041
$19$	0.497927	0.114232	0.0571162	−	0.998368i	$-0.481809\pi$
$19$	0.497927	0.114232	0.0571162	+	0.998368i	$0.481809\pi$
$20$	−4.37946	−0.979277
$21$	6.82218	1.48872
$22$	−6.47165	−1.37976
$23$	9.13451	1.90468	0.952338	−	0.305044i	$-0.0986711\pi$
$23$	9.13451	1.90468	0.952338	+	0.305044i	$0.0986711\pi$
$24$	−2.04234	−0.416890
$25$	14.1797	2.83593
$26$	2.30446	0.451941
$27$	3.73514	0.718829
$28$	−3.34038	−0.631272
$29$	−8.73359	−1.62179	−0.810894	−	0.585193i	$-0.801019\pi$
$29$	−8.73359	−1.62179	−0.810894	+	0.585193i	$0.801019\pi$
$30$	8.94433	1.63300
$31$	0.749179	0.134556	0.0672782	−	0.997734i	$-0.478568\pi$
$31$	0.749179	0.134556	0.0672782	+	0.997734i	$0.478568\pi$
$32$	1.00000	0.176777
$33$	13.2173	2.30083
$34$	−1.94500	−0.333565
$35$	14.6291	2.47276
$36$	1.17114	0.195190
$37$	4.43414	0.728968	0.364484	−	0.931210i	$-0.381245\pi$
$37$	4.43414	0.728968	0.364484	+	0.931210i	$0.381245\pi$
$38$	0.497927	0.0807744
$39$	−4.70648	−0.753640
$40$	−4.37946	−0.692453
$41$	6.22772	0.972607	0.486303	−	0.873790i	$-0.338345\pi$
$41$	6.22772	0.972607	0.486303	+	0.873790i	$0.338345\pi$
$42$	6.82218	1.05269
$43$	3.60194	0.549291	0.274645	−	0.961546i	$-0.411440\pi$
$43$	3.60194	0.549291	0.274645	+	0.961546i	$0.411440\pi$
$44$	−6.47165	−0.975637
$45$	−5.12897	−0.764582
$46$	9.13451	1.34681
$47$	−7.36820	−1.07476	−0.537382	−	0.843339i	$-0.680586\pi$
$47$	−7.36820	−1.07476	−0.537382	+	0.843339i	$0.680586\pi$
$48$	−2.04234	−0.294786
$49$	4.15813	0.594019
$50$	14.1797	2.00531
$51$	3.97235	0.556240
$52$	2.30446	0.319571
$53$	1.69540	0.232881	0.116440	−	0.993198i	$-0.462852\pi$
$53$	1.69540	0.232881	0.116440	+	0.993198i	$0.462852\pi$
$54$	3.73514	0.508289
$55$	28.3423	3.82168
$56$	−3.34038	−0.446377
$57$	−1.01694	−0.134696
$58$	−8.73359	−1.14678
$59$	−13.4470	−1.75065	−0.875323	−	0.483539i	$-0.839351\pi$
$59$	−13.4470	−1.75065	−0.875323	+	0.483539i	$0.839351\pi$
$60$	8.94433	1.15471
$61$	−3.81707	−0.488726	−0.244363	−	0.969684i	$-0.578579\pi$
$61$	−3.81707	−0.488726	−0.244363	+	0.969684i	$0.578579\pi$
$62$	0.749179	0.0951458
$63$	−3.91206	−0.492873
$64$	1.00000	0.125000
$65$	−10.0923	−1.25179
$66$	13.2173	1.62694
$67$	11.9649	1.46174	0.730871	−	0.682515i	$-0.239114\pi$
$67$	11.9649	1.46174	0.730871	+	0.682515i	$0.239114\pi$
$68$	−1.94500	−0.235866
$69$	−18.6557	−2.24589
$70$	14.6291	1.74851
$71$	1.43259	0.170018	0.0850089	−	0.996380i	$-0.472908\pi$
$71$	1.43259	0.170018	0.0850089	+	0.996380i	$0.472908\pi$
$72$	1.17114	0.138021
$73$	1.41227	0.165293	0.0826466	−	0.996579i	$-0.473663\pi$
$73$	1.41227	0.165293	0.0826466	+	0.996579i	$0.473663\pi$
$74$	4.43414	0.515458
$75$	−28.9597	−3.34397
$76$	0.497927	0.0571162
$77$	21.6178	2.46357
$78$	−4.70648	−0.532904
$79$	12.5869	1.41614	0.708070	−	0.706142i	$-0.249566\pi$
$79$	12.5869	1.41614	0.708070	+	0.706142i	$0.249566\pi$
$80$	−4.37946	−0.489638
$81$	−11.1419	−1.23798
$82$	6.22772	0.687737
$83$	−5.82838	−0.639748	−0.319874	−	0.947460i	$-0.603641\pi$
$83$	−5.82838	−0.639748	−0.319874	+	0.947460i	$0.603641\pi$
$84$	6.82218	0.744361
$85$	8.51805	0.923913
$86$	3.60194	0.388407
$87$	17.8369	1.91232
$88$	−6.47165	−0.689880
$89$	15.3847	1.63077	0.815386	−	0.578918i	$-0.196525\pi$
$89$	15.3847	1.63077	0.815386	+	0.578918i	$0.196525\pi$
$90$	−5.12897	−0.540641
$91$	−7.69776	−0.806945
$92$	9.13451	0.952338
$93$	−1.53008	−0.158661
$94$	−7.36820	−0.759972
$95$	−2.18065	−0.223730
$96$	−2.04234	−0.208445
$97$	−9.69510	−0.984388	−0.492194	−	0.870486i	$-0.663805\pi$
$97$	−9.69510	−0.984388	−0.492194	+	0.870486i	$0.663805\pi$
$98$	4.15813	0.420035
$99$	−7.57922	−0.761741
$100$	14.1797	1.41797
$101$	−7.02597	−0.699110	−0.349555	−	0.936916i	$-0.613667\pi$
$101$	−7.02597	−0.699110	−0.349555	+	0.936916i	$0.613667\pi$
$102$	3.97235	0.393321
$103$	18.7509	1.84758	0.923789	−	0.382901i	$-0.125075\pi$
$103$	18.7509	1.84758	0.923789	+	0.382901i	$0.125075\pi$
$104$	2.30446	0.225971
$105$	−29.8775	−2.91574
$106$	1.69540	0.164672
$107$	6.44341	0.622908	0.311454	−	0.950261i	$-0.399184\pi$
$107$	6.44341	0.622908	0.311454	+	0.950261i	$0.399184\pi$
$108$	3.73514	0.359414
$109$	−7.54936	−0.723097	−0.361549	−	0.932353i	$-0.617752\pi$
$109$	−7.54936	−0.723097	−0.361549	+	0.932353i	$0.617752\pi$
$110$	28.3423	2.70233
$111$	−9.05601	−0.859558
$112$	−3.34038	−0.315636
$113$	−9.24288	−0.869497	−0.434748	−	0.900552i	$-0.643163\pi$
$113$	−9.24288	−0.869497	−0.434748	+	0.900552i	$0.643163\pi$
$114$	−1.01694	−0.0952447
$115$	−40.0042	−3.73041
$116$	−8.73359	−0.810894
$117$	2.69885	0.249509
$118$	−13.4470	−1.23789
$119$	6.49704	0.595583
$120$	8.94433	0.816502
$121$	30.8822	2.80747
$122$	−3.81707	−0.345581
$123$	−12.7191	−1.14684
$124$	0.749179	0.0672782
$125$	−40.2019	−3.59577
$126$	−3.91206	−0.348514
$127$	−1.60403	−0.142334	−0.0711672	−	0.997464i	$-0.522672\pi$
$127$	−1.60403	−0.142334	−0.0711672	+	0.997464i	$0.522672\pi$
$128$	1.00000	0.0883883
$129$	−7.35637	−0.647693
$130$	−10.0923	−0.885151
$131$	−21.5751	−1.88502	−0.942511	−	0.334175i	$-0.891542\pi$
$131$	−21.5751	−1.88502	−0.942511	+	0.334175i	$0.891542\pi$
$132$	13.2173	1.15042
$133$	−1.66327	−0.144223
$134$	11.9649	1.03361
$135$	−16.3579	−1.40786
$136$	−1.94500	−0.166783
$137$	−10.2572	−0.876328	−0.438164	−	0.898895i	$-0.644371\pi$
$137$	−10.2572	−0.876328	−0.438164	+	0.898895i	$0.644371\pi$
$138$	−18.6557	−1.58808
$139$	9.99255	0.847557	0.423779	−	0.905766i	$-0.360703\pi$
$139$	9.99255	0.847557	0.423779	+	0.905766i	$0.360703\pi$
$140$	14.6291	1.23638
$141$	15.0484	1.26730
$142$	1.43259	0.120221
$143$	−14.9136	−1.24714
$144$	1.17114	0.0975952
$145$	38.2484	3.17636
$146$	1.41227	0.116880
$147$	−8.49231	−0.700434
$148$	4.43414	0.364484
$149$	11.5562	0.946718	0.473359	−	0.880870i	$-0.343041\pi$
$149$	11.5562	0.946718	0.473359	+	0.880870i	$0.343041\pi$
$150$	−28.9597	−2.36455
$151$	−16.9180	−1.37677	−0.688383	−	0.725348i	$-0.741679\pi$
$151$	−16.9180	−1.37677	−0.688383	+	0.725348i	$0.741679\pi$
$152$	0.497927	0.0403872
$153$	−2.27787	−0.184155
$154$	21.6178	1.74201
$155$	−3.28100	−0.263536
$156$	−4.70648	−0.376820
$157$	20.2628	1.61715	0.808574	−	0.588395i	$-0.200240\pi$
$157$	20.2628	1.61715	0.808574	+	0.588395i	$0.200240\pi$
$158$	12.5869	1.00136
$159$	−3.46258	−0.274600
$160$	−4.37946	−0.346227
$161$	−30.5127	−2.40474
$162$	−11.1419	−0.875387
$163$	−5.60012	−0.438635	−0.219318	−	0.975654i	$-0.570383\pi$
$163$	−5.60012	−0.438635	−0.219318	+	0.975654i	$0.570383\pi$
$164$	6.22772	0.486303
$165$	−57.8846	−4.50631
$166$	−5.82838	−0.452370
$167$	−1.79995	−0.139284	−0.0696420	−	0.997572i	$-0.522186\pi$
$167$	−1.79995	−0.139284	−0.0696420	+	0.997572i	$0.522186\pi$
$168$	6.82218	0.526343
$169$	−7.68948	−0.591498
$170$	8.51805	0.653305
$171$	0.583144	0.0445941
$172$	3.60194	0.274645
$173$	−4.24598	−0.322816	−0.161408	−	0.986888i	$-0.551604\pi$
$173$	−4.24598	−0.322816	−0.161408	+	0.986888i	$0.551604\pi$
$174$	17.8369	1.35222
$175$	−47.3654	−3.58049
$176$	−6.47165	−0.487819
$177$	27.4632	2.06426
$178$	15.3847	1.15313
$179$	9.28123	0.693712	0.346856	−	0.937918i	$-0.387249\pi$
$179$	9.28123	0.693712	0.346856	+	0.937918i	$0.387249\pi$
$180$	−5.12897	−0.382291
$181$	−25.2680	−1.87816	−0.939078	−	0.343704i	$-0.888318\pi$
$181$	−25.2680	−1.87816	−0.939078	+	0.343704i	$0.888318\pi$
$182$	−7.69776	−0.570596
$183$	7.79575	0.576278
$184$	9.13451	0.673405
$185$	−19.4191	−1.42772
$186$	−1.53008	−0.112191
$187$	12.5874	0.920479
$188$	−7.36820	−0.537382
$189$	−12.4768	−0.907553
$190$	−2.18065	−0.158201
$191$	12.8689	0.931162	0.465581	−	0.885005i	$-0.345845\pi$
$191$	12.8689	0.931162	0.465581	+	0.885005i	$0.345845\pi$
$192$	−2.04234	−0.147393
$193$	−18.5606	−1.33602	−0.668010	−	0.744152i	$-0.732854\pi$
$193$	−18.5606	−1.33602	−0.668010	+	0.744152i	$0.732854\pi$
$194$	−9.69510	−0.696067
$195$	20.6118	1.47604
$196$	4.15813	0.297009
$197$	1.00470	0.0715819	0.0357909	−	0.999359i	$-0.488605\pi$
$197$	1.00470	0.0715819	0.0357909	+	0.999359i	$0.488605\pi$
$198$	−7.57922	−0.538632
$199$	−3.89656	−0.276220	−0.138110	−	0.990417i	$-0.544103\pi$
$199$	−3.89656	−0.276220	−0.138110	+	0.990417i	$0.544103\pi$
$200$	14.1797	1.00265
$201$	−24.4363	−1.72361
$202$	−7.02597	−0.494345
$203$	29.1735	2.04758
$204$	3.97235	0.278120
$205$	−27.2741	−1.90490
$206$	18.7509	1.30644
$207$	10.6978	0.743549
$208$	2.30446	0.159785
$209$	−3.22241	−0.222899
$210$	−29.8775	−2.06174
$211$	4.21378	0.290088	0.145044	−	0.989425i	$-0.453668\pi$
$211$	4.21378	0.290088	0.145044	+	0.989425i	$0.453668\pi$
$212$	1.69540	0.116440
$213$	−2.92584	−0.200475
$214$	6.44341	0.440462
$215$	−15.7745	−1.07581
$216$	3.73514	0.254144
$217$	−2.50254	−0.169884
$218$	−7.54936	−0.511307
$219$	−2.88432	−0.194905
$220$	28.3423	1.91084
$221$	−4.48217	−0.301504
$222$	−9.05601	−0.607800
$223$	18.8200	1.26028	0.630140	−	0.776482i	$-0.282998\pi$
$223$	18.8200	1.26028	0.630140	+	0.776482i	$0.282998\pi$
$224$	−3.34038	−0.223188
$225$	16.6064	1.10709
$226$	−9.24288	−0.614827
$227$	2.50659	0.166368	0.0831842	−	0.996534i	$-0.473491\pi$
$227$	2.50659	0.166368	0.0831842	+	0.996534i	$0.473491\pi$
$228$	−1.01694	−0.0673482
$229$	4.64419	0.306897	0.153448	−	0.988157i	$-0.450962\pi$
$229$	4.64419	0.306897	0.153448	+	0.988157i	$0.450962\pi$
$230$	−40.0042	−2.63780
$231$	−44.1507	−2.90491
$232$	−8.73359	−0.573389
$233$	18.2942	1.19849	0.599247	−	0.800564i	$-0.295467\pi$
$233$	18.2942	1.19849	0.599247	+	0.800564i	$0.295467\pi$
$234$	2.69885	0.176429
$235$	32.2687	2.10498
$236$	−13.4470	−0.875323
$237$	−25.7067	−1.66983
$238$	6.49704	0.421141
$239$	14.5962	0.944153	0.472076	−	0.881558i	$-0.343505\pi$
$239$	14.5962	0.944153	0.472076	+	0.881558i	$0.343505\pi$
$240$	8.94433	0.577354
$241$	24.1313	1.55443	0.777215	−	0.629235i	$-0.216632\pi$
$241$	24.1313	1.55443	0.777215	+	0.629235i	$0.216632\pi$
$242$	30.8822	1.98518
$243$	11.5500	0.740932
$244$	−3.81707	−0.244363
$245$	−18.2104	−1.16342
$246$	−12.7191	−0.810941
$247$	1.14745	0.0730106
$248$	0.749179	0.0475729
$249$	11.9035	0.754355
$250$	−40.2019	−2.54259
$251$	9.69901	0.612196	0.306098	−	0.952000i	$-0.400976\pi$
$251$	9.69901	0.612196	0.306098	+	0.952000i	$0.400976\pi$
$252$	−3.91206	−0.246437
$253$	−59.1153	−3.71655
$254$	−1.60403	−0.100646
$255$	−17.3967	−1.08943
$256$	1.00000	0.0625000
$257$	25.4077	1.58489	0.792444	−	0.609945i	$-0.208808\pi$
$257$	25.4077	1.58489	0.792444	+	0.609945i	$0.208808\pi$
$258$	−7.35637	−0.457988
$259$	−14.8117	−0.920355
$260$	−10.0923	−0.625896
$261$	−10.2283	−0.633115
$262$	−21.5751	−1.33291
$263$	8.23917	0.508049	0.254025	−	0.967198i	$-0.418246\pi$
$263$	8.23917	0.508049	0.254025	+	0.967198i	$0.418246\pi$
$264$	13.2173	0.813468
$265$	−7.42493	−0.456110
$266$	−1.66327	−0.101981
$267$	−31.4207	−1.92292
$268$	11.9649	0.730871
$269$	17.6001	1.07310	0.536549	−	0.843869i	$-0.319728\pi$
$269$	17.6001	1.07310	0.536549	+	0.843869i	$0.319728\pi$
$270$	−16.3579	−0.995511
$271$	−22.6100	−1.37346	−0.686730	−	0.726912i	$-0.740955\pi$
$271$	−22.6100	−1.37346	−0.686730	+	0.726912i	$0.740955\pi$
$272$	−1.94500	−0.117933
$273$	15.7214	0.951504
$274$	−10.2572	−0.619657
$275$	−91.7657	−5.53368
$276$	−18.6557	−1.12294
$277$	5.92295	0.355876	0.177938	−	0.984042i	$-0.443057\pi$
$277$	5.92295	0.355876	0.177938	+	0.984042i	$0.443057\pi$
$278$	9.99255	0.599313
$279$	0.877395	0.0525283
$280$	14.6291	0.874253
$281$	−5.70819	−0.340522	−0.170261	−	0.985399i	$-0.554461\pi$
$281$	−5.70819	−0.340522	−0.170261	+	0.985399i	$0.554461\pi$
$282$	15.0484	0.896117
$283$	21.3512	1.26920	0.634598	−	0.772842i	$-0.281166\pi$
$283$	21.3512	1.26920	0.634598	+	0.772842i	$0.281166\pi$
$284$	1.43259	0.0850089
$285$	4.45363	0.263810
$286$	−14.9136	−0.881862
$287$	−20.8030	−1.22796
$288$	1.17114	0.0690103
$289$	−13.2170	−0.777469
$290$	38.2484	2.24602
$291$	19.8007	1.16074
$292$	1.41227	0.0826466
$293$	−28.4207	−1.66035	−0.830176	−	0.557501i	$-0.811760\pi$
$293$	−28.4207	−1.66035	−0.830176	+	0.557501i	$0.811760\pi$
$294$	−8.49231	−0.495281
$295$	58.8904	3.42873
$296$	4.43414	0.257729
$297$	−24.1725	−1.40263
$298$	11.5562	0.669431
$299$	21.0501	1.21736
$300$	−28.9597	−1.67199
$301$	−12.0318	−0.693504
$302$	−16.9180	−0.973520
$303$	14.3494	0.824351
$304$	0.497927	0.0285581
$305$	16.7167	0.957196
$306$	−2.27787	−0.130217
$307$	25.6219	1.46232	0.731159	−	0.682207i	$-0.238980\pi$
$307$	25.6219	1.46232	0.731159	+	0.682207i	$0.238980\pi$
$308$	21.6178	1.23179
$309$	−38.2956	−2.17856
$310$	−3.28100	−0.186348
$311$	−6.99098	−0.396422	−0.198211	−	0.980159i	$-0.563513\pi$
$311$	−6.99098	−0.396422	−0.198211	+	0.980159i	$0.563513\pi$
$312$	−4.70648	−0.266452
$313$	−19.5500	−1.10503	−0.552517	−	0.833502i	$-0.686333\pi$
$313$	−19.5500	−1.10503	−0.552517	+	0.833502i	$0.686333\pi$
$314$	20.2628	1.14350
$315$	17.1327	0.965319
$316$	12.5869	0.708070
$317$	−5.35610	−0.300829	−0.150414	−	0.988623i	$-0.548061\pi$
$317$	−5.35610	−0.300829	−0.150414	+	0.988623i	$0.548061\pi$
$318$	−3.46258	−0.194172
$319$	56.5207	3.16455
$320$	−4.37946	−0.244819
$321$	−13.1596	−0.734498
$322$	−30.5127	−1.70041
$323$	−0.968469	−0.0538871
$324$	−11.1419	−0.618992
$325$	32.6764	1.81256
$326$	−5.60012	−0.310162
$327$	15.4183	0.852636
$328$	6.22772	0.343868
$329$	24.6126	1.35694
$330$	−57.8846	−3.18644
$331$	−15.1757	−0.834133	−0.417066	−	0.908876i	$-0.636942\pi$
$331$	−15.1757	−0.834133	−0.417066	+	0.908876i	$0.636942\pi$
$332$	−5.82838	−0.319874
$333$	5.19301	0.284575
$334$	−1.79995	−0.0984886
$335$	−52.3997	−2.86290
$336$	6.82218	0.372181
$337$	3.37094	0.183627	0.0918134	−	0.995776i	$-0.470734\pi$
$337$	3.37094	0.183627	0.0918134	+	0.995776i	$0.470734\pi$
$338$	−7.68948	−0.418252
$339$	18.8771	1.02526
$340$	8.51805	0.461956
$341$	−4.84842	−0.262557
$342$	0.583144	0.0315328
$343$	9.49292	0.512569
$344$	3.60194	0.194204
$345$	81.7021	4.39869
$346$	−4.24598	−0.228265
$347$	12.5778	0.675214	0.337607	−	0.941287i	$-0.390382\pi$
$347$	12.5778	0.675214	0.337607	+	0.941287i	$0.390382\pi$
$348$	17.8369	0.956161
$349$	−12.6349	−0.676330	−0.338165	−	0.941087i	$-0.609806\pi$
$349$	−12.6349	−0.676330	−0.338165	+	0.941087i	$0.609806\pi$
$350$	−47.3654	−2.53179
$351$	8.60748	0.459433
$352$	−6.47165	−0.344940
$353$	32.8797	1.75001	0.875005	−	0.484113i	$-0.160858\pi$
$353$	32.8797	1.75001	0.875005	+	0.484113i	$0.160858\pi$
$354$	27.4632	1.45965
$355$	−6.27399	−0.332989
$356$	15.3847	0.815386
$357$	−13.2692	−0.702278
$358$	9.28123	0.490528
$359$	14.2910	0.754249	0.377125	−	0.926163i	$-0.376913\pi$
$359$	14.2910	0.754249	0.377125	+	0.926163i	$0.376913\pi$
$360$	−5.12897	−0.270321
$361$	−18.7521	−0.986951
$362$	−25.2680	−1.32806
$363$	−63.0719	−3.31042
$364$	−7.69776	−0.403472
$365$	−6.18496	−0.323736
$366$	7.79575	0.407490
$367$	−5.15525	−0.269102	−0.134551	−	0.990907i	$-0.542959\pi$
$367$	−5.15525	−0.269102	−0.134551	+	0.990907i	$0.542959\pi$
$368$	9.13451	0.476169
$369$	7.29355	0.379687
$370$	−19.4191	−1.00955
$371$	−5.66327	−0.294023
$372$	−1.53008	−0.0793307
$373$	−5.87609	−0.304252	−0.152126	−	0.988361i	$-0.548612\pi$
$373$	−5.87609	−0.304252	−0.152126	+	0.988361i	$0.548612\pi$
$374$	12.5874	0.650877
$375$	82.1059	4.23993
$376$	−7.36820	−0.379986
$377$	−20.1262	−1.03655
$378$	−12.4768	−0.641737
$379$	3.68032	0.189045	0.0945227	−	0.995523i	$-0.469867\pi$
$379$	3.68032	0.189045	0.0945227	+	0.995523i	$0.469867\pi$
$380$	−2.18065	−0.111865
$381$	3.27596	0.167833
$382$	12.8689	0.658431
$383$	−16.3448	−0.835182	−0.417591	−	0.908635i	$-0.637125\pi$
$383$	−16.3448	−0.835182	−0.417591	+	0.908635i	$0.637125\pi$
$384$	−2.04234	−0.104223
$385$	−94.6741	−4.82504
$386$	−18.5606	−0.944709
$387$	4.21838	0.214433
$388$	−9.69510	−0.492194
$389$	−11.4399	−0.580027	−0.290013	−	0.957023i	$-0.593660\pi$
$389$	−11.4399	−0.580027	−0.290013	+	0.957023i	$0.593660\pi$
$390$	20.6118	1.04372
$391$	−17.7666	−0.898497
$392$	4.15813	0.210017
$393$	44.0636	2.22271
$394$	1.00470	0.0506160
$395$	−55.1239	−2.77359
$396$	−7.57922	−0.380870
$397$	13.7570	0.690446	0.345223	−	0.938521i	$-0.387803\pi$
$397$	13.7570	0.690446	0.345223	+	0.938521i	$0.387803\pi$
$398$	−3.89656	−0.195317
$399$	3.39695	0.170060
$400$	14.1797	0.708983
$401$	−7.04530	−0.351825	−0.175913	−	0.984406i	$-0.556288\pi$
$401$	−7.04530	−0.351825	−0.175913	+	0.984406i	$0.556288\pi$
$402$	−24.4363	−1.21877
$403$	1.72645	0.0860006
$404$	−7.02597	−0.349555
$405$	48.7953	2.42466
$406$	29.1735	1.44786
$407$	−28.6962	−1.42242
$408$	3.97235	0.196661
$409$	−28.1091	−1.38990	−0.694952	−	0.719056i	$-0.744574\pi$
$409$	−28.1091	−1.38990	−0.694952	+	0.719056i	$0.744574\pi$
$410$	−27.2741	−1.34697
$411$	20.9486	1.03332
$412$	18.7509	0.923789
$413$	44.9180	2.21027
$414$	10.6978	0.525769
$415$	25.5252	1.25298
$416$	2.30446	0.112985
$417$	−20.4082	−0.999392
$418$	−3.22241	−0.157613
$419$	−7.71131	−0.376722	−0.188361	−	0.982100i	$-0.560318\pi$
$419$	−7.71131	−0.376722	−0.188361	+	0.982100i	$0.560318\pi$
$420$	−29.8775	−1.45787
$421$	−26.3788	−1.28562	−0.642812	−	0.766024i	$-0.722233\pi$
$421$	−26.3788	−1.28562	−0.642812	+	0.766024i	$0.722233\pi$
$422$	4.21378	0.205124
$423$	−8.62922	−0.419567
$424$	1.69540	0.0823358
$425$	−27.5795	−1.33780
$426$	−2.92584	−0.141758
$427$	12.7505	0.617038
$428$	6.44341	0.311454
$429$	30.4587	1.47056
$430$	−15.7745	−0.760716
$431$	17.6942	0.852301	0.426150	−	0.904652i	$-0.359869\pi$
$431$	17.6942	0.852301	0.426150	+	0.904652i	$0.359869\pi$
$432$	3.73514	0.179707
$433$	2.40132	0.115400	0.0576999	−	0.998334i	$-0.481623\pi$
$433$	2.40132	0.115400	0.0576999	+	0.998334i	$0.481623\pi$
$434$	−2.50254	−0.120126
$435$	−78.1162	−3.74538
$436$	−7.54936	−0.361549
$437$	4.54832	0.217576
$438$	−2.88432	−0.137818
$439$	22.4515	1.07155	0.535775	−	0.844361i	$-0.320020\pi$
$439$	22.4515	1.07155	0.535775	+	0.844361i	$0.320020\pi$
$440$	28.3423	1.35117
$441$	4.86977	0.231894
$442$	−4.48217	−0.213195
$443$	19.7224	0.937041	0.468521	−	0.883453i	$-0.344787\pi$
$443$	19.7224	0.937041	0.468521	+	0.883453i	$0.344787\pi$
$444$	−9.05601	−0.429779
$445$	−67.3765	−3.19395
$446$	18.8200	0.891152
$447$	−23.6016	−1.11632
$448$	−3.34038	−0.157818
$449$	4.72949	0.223198	0.111599	−	0.993753i	$-0.464403\pi$
$449$	4.72949	0.223198	0.111599	+	0.993753i	$0.464403\pi$
$450$	16.6064	0.782834
$451$	−40.3036	−1.89782
$452$	−9.24288	−0.434748
$453$	34.5522	1.62340
$454$	2.50659	0.117640
$455$	33.7120	1.58044
$456$	−1.01694	−0.0476224
$457$	8.44358	0.394974	0.197487	−	0.980306i	$-0.436722\pi$
$457$	8.44358	0.394974	0.197487	+	0.980306i	$0.436722\pi$
$458$	4.64419	0.217009
$459$	−7.26486	−0.339095
$460$	−40.0042	−1.86521
$461$	−3.02639	−0.140953	−0.0704765	−	0.997513i	$-0.522452\pi$
$461$	−3.02639	−0.140953	−0.0704765	+	0.997513i	$0.522452\pi$
$462$	−44.1507	−2.05408
$463$	13.8911	0.645576	0.322788	−	0.946471i	$-0.395380\pi$
$463$	13.8911	0.645576	0.322788	+	0.946471i	$0.395380\pi$
$464$	−8.73359	−0.405447
$465$	6.70090	0.310747
$466$	18.2942	0.847464
$467$	−18.8979	−0.874491	−0.437245	−	0.899342i	$-0.644046\pi$
$467$	−18.8979	−0.874491	−0.437245	+	0.899342i	$0.644046\pi$
$468$	2.69885	0.124754
$469$	−39.9672	−1.84552
$470$	32.2687	1.48845
$471$	−41.3835	−1.90685
$472$	−13.4470	−0.618947
$473$	−23.3105	−1.07182
$474$	−25.7067	−1.18075
$475$	7.06044	0.323955
$476$	6.49704	0.297791
$477$	1.98555	0.0909123
$478$	14.5962	0.667617
$479$	−15.6279	−0.714056	−0.357028	−	0.934094i	$-0.616210\pi$
$479$	−15.6279	−0.714056	−0.357028	+	0.934094i	$0.616210\pi$
$480$	8.94433	0.408251
$481$	10.2183	0.465914
$482$	24.1313	1.09915
$483$	62.3173	2.83553
$484$	30.8822	1.40374
$485$	42.4593	1.92798
$486$	11.5500	0.523918
$487$	27.5791	1.24973	0.624864	−	0.780734i	$-0.285155\pi$
$487$	27.5791	1.24973	0.624864	+	0.780734i	$0.285155\pi$
$488$	−3.81707	−0.172791
$489$	11.4373	0.517214
$490$	−18.2104	−0.822660
$491$	19.0538	0.859887	0.429944	−	0.902856i	$-0.358533\pi$
$491$	19.0538	0.859887	0.429944	+	0.902856i	$0.358533\pi$
$492$	−12.7191	−0.573422
$493$	16.9869	0.765049
$494$	1.14745	0.0516263
$495$	33.1929	1.49191
$496$	0.749179	0.0336391
$497$	−4.78541	−0.214655
$498$	11.9035	0.533410
$499$	−15.5409	−0.695706	−0.347853	−	0.937549i	$-0.613089\pi$
$499$	−15.5409	−0.695706	−0.347853	+	0.937549i	$0.613089\pi$
$500$	−40.2019	−1.79789
$501$	3.67610	0.164236
$502$	9.69901	0.432888
$503$	−26.3741	−1.17596	−0.587981	−	0.808875i	$-0.700077\pi$
$503$	−26.3741	−1.17596	−0.587981	+	0.808875i	$0.700077\pi$
$504$	−3.91206	−0.174257
$505$	30.7699	1.36924
$506$	−59.1153	−2.62800
$507$	15.7045	0.697462
$508$	−1.60403	−0.0711672
$509$	20.7416	0.919357	0.459679	−	0.888085i	$-0.347965\pi$
$509$	20.7416	0.919357	0.459679	+	0.888085i	$0.347965\pi$
$510$	−17.3967	−0.770341
$511$	−4.71750	−0.208690
$512$	1.00000	0.0441942
$513$	1.85983	0.0821135
$514$	25.4077	1.12068
$515$	−82.1187	−3.61858
$516$	−7.35637	−0.323846
$517$	47.6844	2.09716
$518$	−14.8117	−0.650789
$519$	8.67173	0.380647
$520$	−10.0923	−0.442576
$521$	17.1214	0.750103	0.375051	−	0.927004i	$-0.377625\pi$
$521$	17.1214	0.750103	0.375051	+	0.927004i	$0.377625\pi$
$522$	−10.2283	−0.447680
$523$	2.44924	0.107098	0.0535490	−	0.998565i	$-0.482947\pi$
$523$	2.44924	0.107098	0.0535490	+	0.998565i	$0.482947\pi$
$524$	−21.5751	−0.942511
$525$	96.7362	4.22191
$526$	8.23917	0.359245
$527$	−1.45715	−0.0634746
$528$	13.2173	0.575209
$529$	60.4392	2.62779
$530$	−7.42493	−0.322518
$531$	−15.7483	−0.683419
$532$	−1.66327	−0.0721117
$533$	14.3515	0.621633
$534$	−31.4207	−1.35971
$535$	−28.2186	−1.22000
$536$	11.9649	0.516804
$537$	−18.9554	−0.817986
$538$	17.6001	0.758794
$539$	−26.9100	−1.15909
$540$	−16.3579	−0.703932
$541$	−32.0947	−1.37986	−0.689929	−	0.723877i	$-0.742358\pi$
$541$	−32.0947	−1.37986	−0.689929	+	0.723877i	$0.742358\pi$
$542$	−22.6100	−0.971183
$543$	51.6058	2.21462
$544$	−1.94500	−0.0833913
$545$	33.0621	1.41622
$546$	15.7214	0.672815
$547$	−26.3157	−1.12518	−0.562589	−	0.826737i	$-0.690195\pi$
$547$	−26.3157	−1.12518	−0.562589	+	0.826737i	$0.690195\pi$
$548$	−10.2572	−0.438164
$549$	−4.47034	−0.190789
$550$	−91.7657	−3.91290
$551$	−4.34869	−0.185261
$552$	−18.6557	−0.794041
$553$	−42.0451	−1.78794
$554$	5.92295	0.251642
$555$	39.6604	1.68349
$556$	9.99255	0.423779
$557$	25.5775	1.08375	0.541877	−	0.840458i	$-0.317714\pi$
$557$	25.5775	1.08375	0.541877	+	0.840458i	$0.317714\pi$
$558$	0.877395	0.0371431
$559$	8.30051	0.351074
$560$	14.6291	0.618190
$561$	−25.7076	−1.08538
$562$	−5.70819	−0.240785
$563$	8.63895	0.364088	0.182044	−	0.983290i	$-0.441729\pi$
$563$	8.63895	0.364088	0.182044	+	0.983290i	$0.441729\pi$
$564$	15.0484	0.633650
$565$	40.4788	1.70296
$566$	21.3512	0.897457
$567$	37.2180	1.56301
$568$	1.43259	0.0601103
$569$	−29.6846	−1.24444	−0.622222	−	0.782841i	$-0.713770\pi$
$569$	−29.6846	−1.24444	−0.622222	+	0.782841i	$0.713770\pi$
$570$	4.45363	0.186542
$571$	38.0107	1.59070	0.795350	−	0.606150i	$-0.207287\pi$
$571$	38.0107	1.59070	0.795350	+	0.606150i	$0.207287\pi$
$572$	−14.9136	−0.623570
$573$	−26.2827	−1.09797
$574$	−20.8030	−0.868298
$575$	129.524	5.40153
$576$	1.17114	0.0487976
$577$	−30.6860	−1.27748	−0.638738	−	0.769424i	$-0.720543\pi$
$577$	−30.6860	−1.27748	−0.638738	+	0.769424i	$0.720543\pi$
$578$	−13.2170	−0.549753
$579$	37.9070	1.57536
$580$	38.2484	1.58818
$581$	19.4690	0.807711
$582$	19.8007	0.820764
$583$	−10.9720	−0.454415
$584$	1.41227	0.0584400
$585$	−11.8195	−0.488676
$586$	−28.4207	−1.17405
$587$	−34.8650	−1.43903	−0.719516	−	0.694476i	$-0.755636\pi$
$587$	−34.8650	−1.43903	−0.719516	+	0.694476i	$0.755636\pi$
$588$	−8.49231	−0.350217
$589$	0.373036	0.0153707
$590$	58.8904	2.42448
$591$	−2.05194	−0.0844053
$592$	4.43414	0.182242
$593$	−32.9455	−1.35291	−0.676456	−	0.736483i	$-0.736485\pi$
$593$	−32.9455	−1.35291	−0.676456	+	0.736483i	$0.736485\pi$
$594$	−24.1725	−0.991811
$595$	−28.4535	−1.16648
$596$	11.5562	0.473359
$597$	7.95809	0.325703
$598$	21.0501	0.860802
$599$	−12.0595	−0.492736	−0.246368	−	0.969176i	$-0.579237\pi$
$599$	−12.0595	−0.492736	−0.246368	+	0.969176i	$0.579237\pi$
$600$	−28.9597	−1.18227
$601$	−31.0655	−1.26719	−0.633594	−	0.773665i	$-0.718421\pi$
$601$	−31.0655	−1.26719	−0.633594	+	0.773665i	$0.718421\pi$
$602$	−12.0318	−0.490381
$603$	14.0126	0.570636
$604$	−16.9180	−0.688383
$605$	−135.247	−5.49859
$606$	14.3494	0.582904
$607$	18.2721	0.741640	0.370820	−	0.928705i	$-0.379077\pi$
$607$	18.2721	0.741640	0.370820	+	0.928705i	$0.379077\pi$
$608$	0.497927	0.0201936
$609$	−59.5822	−2.41439
$610$	16.7167	0.676840
$611$	−16.9797	−0.686926
$612$	−2.27787	−0.0920776
$613$	39.7285	1.60462	0.802309	−	0.596909i	$-0.203605\pi$
$613$	39.7285	1.60462	0.802309	+	0.596909i	$0.203605\pi$
$614$	25.6219	1.03402
$615$	55.7028	2.24615
$616$	21.6178	0.871004
$617$	−28.2368	−1.13677	−0.568385	−	0.822763i	$-0.692431\pi$
$617$	−28.2368	−1.13677	−0.568385	+	0.822763i	$0.692431\pi$
$618$	−38.2956	−1.54048
$619$	−30.5843	−1.22929	−0.614644	−	0.788805i	$-0.710700\pi$
$619$	−30.5843	−1.22929	−0.614644	+	0.788805i	$0.710700\pi$
$620$	−3.28100	−0.131768
$621$	34.1187	1.36914
$622$	−6.99098	−0.280313
$623$	−51.3906	−2.05892
$624$	−4.70648	−0.188410
$625$	105.164	4.20658
$626$	−19.5500	−0.781377
$627$	6.58125	0.262830
$628$	20.2628	0.808574
$629$	−8.62441	−0.343878
$630$	17.1327	0.682583
$631$	−0.331266	−0.0131875	−0.00659374	−	0.999978i	$-0.502099\pi$
$631$	−0.331266	−0.0131875	−0.00659374	+	0.999978i	$0.502099\pi$
$632$	12.5869	0.500681
$633$	−8.60596	−0.342056
$634$	−5.35610	−0.212718
$635$	7.02477	0.278769
$636$	−3.46258	−0.137300
$637$	9.58224	0.379662
$638$	56.5207	2.23768
$639$	1.67777	0.0663717
$640$	−4.37946	−0.173113
$641$	−46.8651	−1.85106	−0.925531	−	0.378673i	$-0.876381\pi$
$641$	−46.8651	−1.85106	−0.925531	+	0.378673i	$0.876381\pi$
$642$	−13.1596	−0.519368
$643$	−37.4873	−1.47836	−0.739178	−	0.673510i	$-0.764786\pi$
$643$	−37.4873	−1.47836	−0.739178	+	0.673510i	$0.764786\pi$
$644$	−30.5127	−1.20237
$645$	32.2169	1.26854
$646$	−0.968469	−0.0381039
$647$	−36.0169	−1.41597	−0.707984	−	0.706228i	$-0.750395\pi$
$647$	−36.0169	−1.41597	−0.707984	+	0.706228i	$0.750395\pi$
$648$	−11.1419	−0.437693
$649$	87.0240	3.41599
$650$	32.6764	1.28167
$651$	5.11103	0.200317
$652$	−5.60012	−0.219318
$653$	−48.2071	−1.88649	−0.943245	−	0.332098i	$-0.892244\pi$
$653$	−48.2071	−1.88649	−0.943245	+	0.332098i	$0.892244\pi$
$654$	15.4183	0.602905
$655$	94.4871	3.69192
$656$	6.22772	0.243152
$657$	1.65397	0.0645273
$658$	24.6126	0.959499
$659$	−5.71394	−0.222584	−0.111292	−	0.993788i	$-0.535499\pi$
$659$	−5.71394	−0.222584	−0.111292	+	0.993788i	$0.535499\pi$
$660$	−57.8846	−2.25315
$661$	−2.24297	−0.0872412	−0.0436206	−	0.999048i	$-0.513889\pi$
$661$	−2.24297	−0.0872412	−0.0436206	+	0.999048i	$0.513889\pi$
$662$	−15.1757	−0.589821
$663$	9.15411	0.355516
$664$	−5.82838	−0.226185
$665$	7.28420	0.282469
$666$	5.19301	0.201225
$667$	−79.7771	−3.08898
$668$	−1.79995	−0.0696420
$669$	−38.4368	−1.48605
$670$	−52.3997	−2.02438
$671$	24.7027	0.953639
$672$	6.82218	0.263171
$673$	30.7983	1.18719	0.593595	−	0.804764i	$-0.297708\pi$
$673$	30.7983	1.18719	0.593595	+	0.804764i	$0.297708\pi$
$674$	3.37094	0.129844
$675$	52.9631	2.03855
$676$	−7.68948	−0.295749
$677$	23.5843	0.906418	0.453209	−	0.891404i	$-0.350279\pi$
$677$	23.5843	0.906418	0.453209	+	0.891404i	$0.350279\pi$
$678$	18.8771	0.724970
$679$	32.3853	1.24283
$680$	8.51805	0.326652
$681$	−5.11931	−0.196172
$682$	−4.84842	−0.185656
$683$	44.9004	1.71806	0.859032	−	0.511922i	$-0.171066\pi$
$683$	44.9004	1.71806	0.859032	+	0.511922i	$0.171066\pi$
$684$	0.583144	0.0222971
$685$	44.9208	1.71633
$686$	9.49292	0.362441
$687$	−9.48501	−0.361876
$688$	3.60194	0.137323
$689$	3.90697	0.148844
$690$	81.7021	3.11035
$691$	5.98698	0.227755	0.113878	−	0.993495i	$-0.463673\pi$
$691$	5.98698	0.227755	0.113878	+	0.993495i	$0.463673\pi$
$692$	−4.24598	−0.161408
$693$	25.3175	0.961731
$694$	12.5778	0.477449
$695$	−43.7620	−1.65999
$696$	17.8369	0.676108
$697$	−12.1129	−0.458810
$698$	−12.6349	−0.478238
$699$	−37.3630	−1.41320
$700$	−47.3654	−1.79025
$701$	44.3162	1.67380	0.836899	−	0.547357i	$-0.184366\pi$
$701$	44.3162	1.67380	0.836899	+	0.547357i	$0.184366\pi$
$702$	8.60748	0.324868
$703$	2.20788	0.0832717
$704$	−6.47165	−0.243909
$705$	−65.9037	−2.48208
$706$	32.8797	1.23744
$707$	23.4694	0.882657
$708$	27.4632	1.03213
$709$	−6.36586	−0.239075	−0.119537	−	0.992830i	$-0.538141\pi$
$709$	−6.36586	−0.239075	−0.119537	+	0.992830i	$0.538141\pi$
$710$	−6.27399	−0.235459
$711$	14.7411	0.552834
$712$	15.3847	0.576565
$713$	6.84338	0.256287
$714$	−13.2692	−0.496586
$715$	65.3136	2.44259
$716$	9.28123	0.346856
$717$	−29.8105	−1.11329
$718$	14.2910	0.533335
$719$	−12.3566	−0.460822	−0.230411	−	0.973093i	$-0.574007\pi$
$719$	−12.3566	−0.460822	−0.230411	+	0.973093i	$0.574007\pi$
$720$	−5.12897	−0.191145
$721$	−62.6350	−2.33265
$722$	−18.7521	−0.697880
$723$	−49.2842	−1.83290
$724$	−25.2680	−0.939078
$725$	−123.839	−4.59928
$726$	−63.0719	−2.34082
$727$	−49.2605	−1.82697	−0.913485	−	0.406873i	$-0.866619\pi$
$727$	−49.2605	−1.82697	−0.913485	+	0.406873i	$0.866619\pi$
$728$	−7.69776	−0.285298
$729$	9.83657	0.364317
$730$	−6.18496	−0.228916
$731$	−7.00578	−0.259118
$732$	7.79575	0.288139
$733$	−38.1606	−1.40949	−0.704747	−	0.709458i	$-0.748940\pi$
$733$	−38.1606	−1.40949	−0.704747	+	0.709458i	$0.748940\pi$
$734$	−5.15525	−0.190284
$735$	37.1917	1.37184
$736$	9.13451	0.336702
$737$	−77.4325	−2.85226
$738$	7.29355	0.268479
$739$	−3.94406	−0.145085	−0.0725423	−	0.997365i	$-0.523111\pi$
$739$	−3.94406	−0.145085	−0.0725423	+	0.997365i	$0.523111\pi$
$740$	−19.4191	−0.713861
$741$	−2.34348	−0.0860900
$742$	−5.66327	−0.207905
$743$	5.13388	0.188344	0.0941719	−	0.995556i	$-0.469980\pi$
$743$	5.13388	0.188344	0.0941719	+	0.995556i	$0.469980\pi$
$744$	−1.53008	−0.0560953
$745$	−50.6097	−1.85420
$746$	−5.87609	−0.215139
$747$	−6.82587	−0.249746
$748$	12.5874	0.460240
$749$	−21.5234	−0.786449
$750$	82.1059	2.99808
$751$	−9.80426	−0.357762	−0.178881	−	0.983871i	$-0.557248\pi$
$751$	−9.80426	−0.357762	−0.178881	+	0.983871i	$0.557248\pi$
$752$	−7.36820	−0.268691
$753$	−19.8086	−0.721867
$754$	−20.1262	−0.732953
$755$	74.0916	2.69647
$756$	−12.4768	−0.453777
$757$	44.1375	1.60421	0.802103	−	0.597186i	$-0.203715\pi$
$757$	44.1375	1.60421	0.802103	+	0.597186i	$0.203715\pi$
$758$	3.68032	0.133675
$759$	120.733	4.38235
$760$	−2.18065	−0.0791005
$761$	−9.97185	−0.361479	−0.180740	−	0.983531i	$-0.557849\pi$
$761$	−9.97185	−0.361479	−0.180740	+	0.983531i	$0.557849\pi$
$762$	3.27596	0.118676
$763$	25.2177	0.912942
$764$	12.8689	0.465581
$765$	9.97586	0.360678
$766$	−16.3448	−0.590563
$767$	−30.9880	−1.11891
$768$	−2.04234	−0.0736965
$769$	−32.6905	−1.17885	−0.589424	−	0.807824i	$-0.700645\pi$
$769$	−32.6905	−1.17885	−0.589424	+	0.807824i	$0.700645\pi$
$770$	−94.6741	−3.41182
$771$	−51.8911	−1.86881
$772$	−18.5606	−0.668010
$773$	−8.10816	−0.291630	−0.145815	−	0.989312i	$-0.546580\pi$
$773$	−8.10816	−0.291630	−0.145815	+	0.989312i	$0.546580\pi$
$774$	4.21838	0.151627
$775$	10.6231	0.381593
$776$	−9.69510	−0.348034
$777$	30.2505	1.08523
$778$	−11.4399	−0.410141
$779$	3.10095	0.111103
$780$	20.6118	0.738022
$781$	−9.27125	−0.331751
$782$	−17.7666	−0.635334
$783$	−32.6212	−1.16579
$784$	4.15813	0.148505
$785$	−88.7401	−3.16727
$786$	44.0636	1.57170
$787$	23.2504	0.828786	0.414393	−	0.910098i	$-0.363994\pi$
$787$	23.2504	0.828786	0.414393	+	0.910098i	$0.363994\pi$
$788$	1.00470	0.0357909
$789$	−16.8272	−0.599063
$790$	−55.1239	−1.96122
$791$	30.8747	1.09778
$792$	−7.57922	−0.269316
$793$	−8.79628	−0.312365
$794$	13.7570	0.488219
$795$	15.1642	0.537819
$796$	−3.89656	−0.138110
$797$	8.72336	0.308997	0.154499	−	0.987993i	$-0.450624\pi$
$797$	8.72336	0.308997	0.154499	+	0.987993i	$0.450624\pi$
$798$	3.39695	0.120251
$799$	14.3312	0.507000
$800$	14.1797	0.501327
$801$	18.0177	0.636622
$802$	−7.04530	−0.248778
$803$	−9.13969	−0.322533
$804$	−24.4363	−0.861803
$805$	133.629	4.70981
$806$	1.72645	0.0608116
$807$	−35.9454	−1.26534
$808$	−7.02597	−0.247173
$809$	14.3716	0.505277	0.252639	−	0.967561i	$-0.418702\pi$
$809$	14.3716	0.505277	0.252639	+	0.967561i	$0.418702\pi$
$810$	48.7953	1.71449
$811$	−2.24050	−0.0786747	−0.0393373	−	0.999226i	$-0.512525\pi$
$811$	−2.24050	−0.0786747	−0.0393373	+	0.999226i	$0.512525\pi$
$812$	29.1735	1.02379
$813$	46.1773	1.61951
$814$	−28.6962	−1.00580
$815$	24.5255	0.859091
$816$	3.97235	0.139060
$817$	1.79350	0.0627467
$818$	−28.1091	−0.982810
$819$	−9.01518	−0.315016
$820$	−27.2741	−0.952451
$821$	15.1105	0.527358	0.263679	−	0.964610i	$-0.415064\pi$
$821$	15.1105	0.527358	0.263679	+	0.964610i	$0.415064\pi$
$822$	20.9486	0.730665
$823$	−49.7631	−1.73463	−0.867316	−	0.497757i	$-0.834157\pi$
$823$	−49.7631	−1.73463	−0.867316	+	0.497757i	$0.834157\pi$
$824$	18.7509	0.653218
$825$	187.417	6.52501
$826$	44.9180	1.56290
$827$	−6.10476	−0.212283	−0.106142	−	0.994351i	$-0.533850\pi$
$827$	−6.10476	−0.212283	−0.106142	+	0.994351i	$0.533850\pi$
$828$	10.6978	0.371775
$829$	−38.8927	−1.35080	−0.675400	−	0.737451i	$-0.736029\pi$
$829$	−38.8927	−1.35080	−0.675400	+	0.737451i	$0.736029\pi$
$830$	25.5252	0.885992
$831$	−12.0967	−0.419629
$832$	2.30446	0.0798927
$833$	−8.08757	−0.280218
$834$	−20.4082	−0.706677
$835$	7.88279	0.272795
$836$	−3.22241	−0.111449
$837$	2.79829	0.0967230
$838$	−7.71131	−0.266383
$839$	−31.4935	−1.08728	−0.543639	−	0.839319i	$-0.682954\pi$
$839$	−31.4935	−1.08728	−0.543639	+	0.839319i	$0.682954\pi$
$840$	−29.8775	−1.03087
$841$	47.2757	1.63020
$842$	−26.3788	−0.909074
$843$	11.6580	0.401524
$844$	4.21378	0.145044
$845$	33.6757	1.15848
$846$	−8.62922	−0.296679
$847$	−103.158	−3.54456
$848$	1.69540	0.0582202
$849$	−43.6063	−1.49657
$850$	−27.5795	−0.945968
$851$	40.5037	1.38845
$852$	−2.92584	−0.100238
$853$	12.3807	0.423906	0.211953	−	0.977280i	$-0.432018\pi$
$853$	12.3807	0.423906	0.211953	+	0.977280i	$0.432018\pi$
$854$	12.7505	0.436312
$855$	−2.55385	−0.0873400
$856$	6.44341	0.220231
$857$	26.6706	0.911050	0.455525	−	0.890223i	$-0.349452\pi$
$857$	26.6706	0.911050	0.455525	+	0.890223i	$0.349452\pi$
$858$	30.4587	1.03984
$859$	−16.1083	−0.549608	−0.274804	−	0.961500i	$-0.588613\pi$
$859$	−16.1083	−0.549608	−0.274804	+	0.961500i	$0.588613\pi$
$860$	−15.7745	−0.537907
$861$	42.4867	1.44794
$862$	17.6942	0.602668
$863$	24.2202	0.824464	0.412232	−	0.911079i	$-0.364749\pi$
$863$	24.2202	0.824464	0.412232	+	0.911079i	$0.364749\pi$
$864$	3.73514	0.127072
$865$	18.5951	0.632253
$866$	2.40132	0.0816000
$867$	26.9935	0.916748
$868$	−2.50254	−0.0849418
$869$	−81.4581	−2.76328
$870$	−78.1162	−2.64839
$871$	27.5726	0.934260
$872$	−7.54936	−0.255653
$873$	−11.3543	−0.384286
$874$	4.54832	0.153849
$875$	134.290	4.53982
$876$	−2.88432	−0.0974523
$877$	10.6456	0.359476	0.179738	−	0.983715i	$-0.442475\pi$
$877$	10.6456	0.359476	0.179738	+	0.983715i	$0.442475\pi$
$878$	22.4515	0.757700
$879$	58.0446	1.95780
$880$	28.3423	0.955419
$881$	46.2810	1.55925	0.779623	−	0.626249i	$-0.215410\pi$
$881$	46.2810	1.55925	0.779623	+	0.626249i	$0.215410\pi$
$882$	4.86977	0.163974
$883$	−38.8860	−1.30862	−0.654309	−	0.756228i	$-0.727040\pi$
$883$	−38.8860	−1.30862	−0.654309	+	0.756228i	$0.727040\pi$
$884$	−4.48217	−0.150752
$885$	−120.274	−4.04297
$886$	19.7224	0.662588
$887$	2.99547	0.100578	0.0502890	−	0.998735i	$-0.483986\pi$
$887$	2.99547	0.100578	0.0502890	+	0.998735i	$0.483986\pi$
$888$	−9.05601	−0.303900
$889$	5.35806	0.179703
$890$	−67.3765	−2.25847
$891$	72.1061	2.41565
$892$	18.8200	0.630140
$893$	−3.66883	−0.122773
$894$	−23.6016	−0.789355
$895$	−40.6468	−1.35867
$896$	−3.34038	−0.111594
$897$	−42.9914	−1.43544
$898$	4.72949	0.157825
$899$	−6.54302	−0.218222
$900$	16.6064	0.553547
$901$	−3.29755	−0.109857
$902$	−40.3036	−1.34196
$903$	24.5731	0.817741
$904$	−9.24288	−0.307414
$905$	110.660	3.67847
$906$	34.5522	1.14792
$907$	40.0753	1.33068	0.665339	−	0.746542i	$-0.268287\pi$
$907$	40.0753	1.33068	0.665339	+	0.746542i	$0.268287\pi$
$908$	2.50659	0.0831842
$909$	−8.22841	−0.272919
$910$	33.7120	1.11754
$911$	−10.6188	−0.351818	−0.175909	−	0.984406i	$-0.556286\pi$
$911$	−10.6188	−0.351818	−0.175909	+	0.984406i	$0.556286\pi$
$912$	−1.01694	−0.0336741
$913$	37.7192	1.24832
$914$	8.44358	0.279289
$915$	−34.1412	−1.12867
$916$	4.64419	0.153448
$917$	72.0689	2.37992
$918$	−7.26486	−0.239776
$919$	49.7138	1.63991	0.819953	−	0.572431i	$-0.194000\pi$
$919$	49.7138	1.63991	0.819953	+	0.572431i	$0.194000\pi$
$920$	−40.0042	−1.31890
$921$	−52.3286	−1.72428
$922$	−3.02639	−0.0996689
$923$	3.30135	0.108665
$924$	−44.1507	−1.45245
$925$	62.8746	2.06730
$926$	13.8911	0.456491
$927$	21.9600	0.721260
$928$	−8.73359	−0.286694
$929$	−14.7596	−0.484247	−0.242123	−	0.970245i	$-0.577844\pi$
$929$	−14.7596	−0.484247	−0.242123	+	0.970245i	$0.577844\pi$
$930$	6.70090	0.219731
$931$	2.07045	0.0678561
$932$	18.2942	0.599247
$933$	14.2779	0.467439
$934$	−18.8979	−0.618358
$935$	−55.1258	−1.80281
$936$	2.69885	0.0882146
$937$	7.00193	0.228743	0.114372	−	0.993438i	$-0.463515\pi$
$937$	7.00193	0.228743	0.114372	+	0.993438i	$0.463515\pi$
$938$	−39.9672	−1.30498
$939$	39.9278	1.30299
$940$	32.2687	1.05249
$941$	43.3572	1.41340	0.706701	−	0.707512i	$-0.250182\pi$
$941$	43.3572	1.41340	0.706701	+	0.707512i	$0.250182\pi$
$942$	−41.3835	−1.34835
$943$	56.8872	1.85250
$944$	−13.4470	−0.437661
$945$	54.6416	1.77749
$946$	−23.3105	−0.757889
$947$	−45.4850	−1.47806	−0.739031	−	0.673671i	$-0.764716\pi$
$947$	−45.4850	−1.47806	−0.739031	+	0.673671i	$0.764716\pi$
$948$	−25.7067	−0.834916
$949$	3.25451	0.105646
$950$	7.06044	0.229071
$951$	10.9390	0.354720
$952$	6.49704	0.210570
$953$	−34.6300	−1.12178	−0.560888	−	0.827892i	$-0.689540\pi$
$953$	−34.6300	−1.12178	−0.560888	+	0.827892i	$0.689540\pi$
$954$	1.98555	0.0642847
$955$	−56.3589	−1.82373
$956$	14.5962	0.472076
$957$	−115.434	−3.73146
$958$	−15.6279	−0.504914
$959$	34.2628	1.10640
$960$	8.94433	0.288677
$961$	−30.4387	−0.981895
$962$	10.2183	0.329451
$963$	7.54615	0.243171
$964$	24.1313	0.777215
$965$	81.2853	2.61667
$966$	62.3173	2.00503
$967$	41.4357	1.33248	0.666242	−	0.745736i	$-0.267902\pi$
$967$	41.4357	1.33248	0.666242	+	0.745736i	$0.267902\pi$
$968$	30.8822	0.992592
$969$	1.97794	0.0635406
$970$	42.4593	1.36329
$971$	38.0429	1.22085	0.610427	−	0.792072i	$-0.290998\pi$
$971$	38.0429	1.22085	0.610427	+	0.792072i	$0.290998\pi$
$972$	11.5500	0.370466
$973$	−33.3789	−1.07008
$974$	27.5791	0.883691
$975$	−66.7363	−2.13727
$976$	−3.81707	−0.122181
$977$	−15.1038	−0.483212	−0.241606	−	0.970374i	$-0.577674\pi$
$977$	−15.1038	−0.483212	−0.241606	+	0.970374i	$0.577674\pi$
$978$	11.4373	0.365726
$979$	−99.5642	−3.18208
$980$	−18.2104	−0.581709
$981$	−8.84138	−0.282283
$982$	19.0538	0.608032
$983$	−0.502919	−0.0160406	−0.00802031	−	0.999968i	$-0.502553\pi$
$983$	−0.502919	−0.0160406	−0.00802031	+	0.999968i	$0.502553\pi$
$984$	−12.7191	−0.405470
$985$	−4.40004	−0.140197
$986$	16.9869	0.540972
$987$	−50.2672	−1.60002
$988$	1.14745	0.0365053
$989$	32.9019	1.04622
$990$	33.1929	1.05494
$991$	12.1409	0.385668	0.192834	−	0.981231i	$-0.438232\pi$
$991$	12.1409	0.385668	0.192834	+	0.981231i	$0.438232\pi$
$992$	0.749179	0.0237864
$993$	30.9939	0.983563
$994$	−4.78541	−0.151784
$995$	17.0648	0.540991
$996$	11.9035	0.377178
$997$	−48.3450	−1.53110	−0.765550	−	0.643376i	$-0.777533\pi$
$997$	−48.3450	−1.53110	−0.765550	+	0.643376i	$0.777533\pi$
$998$	−15.5409	−0.491939
$999$	16.5621	0.524003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.16	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.16	✓	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.04234 q^{3} +1.00000 q^{4} -4.37946 q^{5} -2.04234 q^{6} -3.34038 q^{7} +1.00000 q^{8} +1.17114 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.04234 q^{3} +1.00000 q^{4} -4.37946 q^{5} -2.04234 q^{6} -3.34038 q^{7} +1.00000 q^{8} +1.17114 q^{9} -4.37946 q^{10} -6.47165 q^{11} -2.04234 q^{12} +2.30446 q^{13} -3.34038 q^{14} +8.94433 q^{15} +1.00000 q^{16} -1.94500 q^{17} +1.17114 q^{18} +0.497927 q^{19} -4.37946 q^{20} +6.82218 q^{21} -6.47165 q^{22} +9.13451 q^{23} -2.04234 q^{24} +14.1797 q^{25} +2.30446 q^{26} +3.73514 q^{27} -3.34038 q^{28} -8.73359 q^{29} +8.94433 q^{30} +0.749179 q^{31} +1.00000 q^{32} +13.2173 q^{33} -1.94500 q^{34} +14.6291 q^{35} +1.17114 q^{36} +4.43414 q^{37} +0.497927 q^{38} -4.70648 q^{39} -4.37946 q^{40} +6.22772 q^{41} +6.82218 q^{42} +3.60194 q^{43} -6.47165 q^{44} -5.12897 q^{45} +9.13451 q^{46} -7.36820 q^{47} -2.04234 q^{48} +4.15813 q^{49} +14.1797 q^{50} +3.97235 q^{51} +2.30446 q^{52} +1.69540 q^{53} +3.73514 q^{54} +28.3423 q^{55} -3.34038 q^{56} -1.01694 q^{57} -8.73359 q^{58} -13.4470 q^{59} +8.94433 q^{60} -3.81707 q^{61} +0.749179 q^{62} -3.91206 q^{63} +1.00000 q^{64} -10.0923 q^{65} +13.2173 q^{66} +11.9649 q^{67} -1.94500 q^{68} -18.6557 q^{69} +14.6291 q^{70} +1.43259 q^{71} +1.17114 q^{72} +1.41227 q^{73} +4.43414 q^{74} -28.9597 q^{75} +0.497927 q^{76} +21.6178 q^{77} -4.70648 q^{78} +12.5869 q^{79} -4.37946 q^{80} -11.1419 q^{81} +6.22772 q^{82} -5.82838 q^{83} +6.82218 q^{84} +8.51805 q^{85} +3.60194 q^{86} +17.8369 q^{87} -6.47165 q^{88} +15.3847 q^{89} -5.12897 q^{90} -7.69776 q^{91} +9.13451 q^{92} -1.53008 q^{93} -7.36820 q^{94} -2.18065 q^{95} -2.04234 q^{96} -9.69510 q^{97} +4.15813 q^{98} -7.57922 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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