LMFDB - Embedded newform 8002.2.a.d.1.8

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			1.8
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.80104 q^{3} +1.00000 q^{4} +0.676703 q^{5} -2.80104 q^{6} +3.76033 q^{7} +1.00000 q^{8} +4.84585 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.80104 q^{3} +1.00000 q^{4} +0.676703 q^{5} -2.80104 q^{6} +3.76033 q^{7} +1.00000 q^{8} +4.84585 q^{9} +0.676703 q^{10} -0.631840 q^{11} -2.80104 q^{12} -6.22087 q^{13} +3.76033 q^{14} -1.89548 q^{15} +1.00000 q^{16} +3.39684 q^{17} +4.84585 q^{18} +1.27370 q^{19} +0.676703 q^{20} -10.5329 q^{21} -0.631840 q^{22} +3.84216 q^{23} -2.80104 q^{24} -4.54207 q^{25} -6.22087 q^{26} -5.17032 q^{27} +3.76033 q^{28} +4.19303 q^{29} -1.89548 q^{30} -9.96120 q^{31} +1.00000 q^{32} +1.76981 q^{33} +3.39684 q^{34} +2.54463 q^{35} +4.84585 q^{36} -9.06735 q^{37} +1.27370 q^{38} +17.4249 q^{39} +0.676703 q^{40} +0.532380 q^{41} -10.5329 q^{42} -7.96327 q^{43} -0.631840 q^{44} +3.27920 q^{45} +3.84216 q^{46} -3.81367 q^{47} -2.80104 q^{48} +7.14011 q^{49} -4.54207 q^{50} -9.51470 q^{51} -6.22087 q^{52} +3.38497 q^{53} -5.17032 q^{54} -0.427568 q^{55} +3.76033 q^{56} -3.56770 q^{57} +4.19303 q^{58} -6.10929 q^{59} -1.89548 q^{60} -2.32597 q^{61} -9.96120 q^{62} +18.2220 q^{63} +1.00000 q^{64} -4.20968 q^{65} +1.76981 q^{66} +3.52438 q^{67} +3.39684 q^{68} -10.7621 q^{69} +2.54463 q^{70} -6.46908 q^{71} +4.84585 q^{72} +1.69845 q^{73} -9.06735 q^{74} +12.7226 q^{75} +1.27370 q^{76} -2.37593 q^{77} +17.4249 q^{78} -8.68079 q^{79} +0.676703 q^{80} -0.0552674 q^{81} +0.532380 q^{82} -12.7963 q^{83} -10.5329 q^{84} +2.29865 q^{85} -7.96327 q^{86} -11.7449 q^{87} -0.631840 q^{88} +4.85184 q^{89} +3.27920 q^{90} -23.3926 q^{91} +3.84216 q^{92} +27.9018 q^{93} -3.81367 q^{94} +0.861920 q^{95} -2.80104 q^{96} +2.77992 q^{97} +7.14011 q^{98} -3.06180 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.80104	−1.61718	−0.808592	−	0.588370i	$-0.799770\pi$
$3$	−2.80104	−1.61718	−0.808592	+	0.588370i	$0.799770\pi$
$4$	1.00000	0.500000
$5$	0.676703	0.302631	0.151315	−	0.988486i	$-0.451649\pi$
$5$	0.676703	0.302631	0.151315	+	0.988486i	$0.451649\pi$
$6$	−2.80104	−1.14352
$7$	3.76033	1.42127	0.710636	−	0.703559i	$-0.248407\pi$
$7$	3.76033	1.42127	0.710636	+	0.703559i	$0.248407\pi$
$8$	1.00000	0.353553
$9$	4.84585	1.61528
$10$	0.676703	0.213992
$11$	−0.631840	−0.190507	−0.0952534	−	0.995453i	$-0.530366\pi$
$11$	−0.631840	−0.190507	−0.0952534	+	0.995453i	$0.530366\pi$
$12$	−2.80104	−0.808592
$13$	−6.22087	−1.72536	−0.862680	−	0.505750i	$-0.831216\pi$
$13$	−6.22087	−1.72536	−0.862680	+	0.505750i	$0.831216\pi$
$14$	3.76033	1.00499
$15$	−1.89548	−0.489410
$16$	1.00000	0.250000
$17$	3.39684	0.823855	0.411927	−	0.911217i	$-0.364856\pi$
$17$	3.39684	0.823855	0.411927	+	0.911217i	$0.364856\pi$
$18$	4.84585	1.14218
$19$	1.27370	0.292208	0.146104	−	0.989269i	$-0.453327\pi$
$19$	1.27370	0.292208	0.146104	+	0.989269i	$0.453327\pi$
$20$	0.676703	0.151315
$21$	−10.5329	−2.29846
$22$	−0.631840	−0.134709
$23$	3.84216	0.801147	0.400573	−	0.916265i	$-0.368811\pi$
$23$	3.84216	0.801147	0.400573	+	0.916265i	$0.368811\pi$
$24$	−2.80104	−0.571761
$25$	−4.54207	−0.908415
$26$	−6.22087	−1.22001
$27$	−5.17032	−0.995028
$28$	3.76033	0.710636
$29$	4.19303	0.778627	0.389313	−	0.921105i	$-0.372712\pi$
$29$	4.19303	0.778627	0.389313	+	0.921105i	$0.372712\pi$
$30$	−1.89548	−0.346065
$31$	−9.96120	−1.78908	−0.894542	−	0.446984i	$-0.852498\pi$
$31$	−9.96120	−1.78908	−0.894542	+	0.446984i	$0.852498\pi$
$32$	1.00000	0.176777
$33$	1.76981	0.308085
$34$	3.39684	0.582553
$35$	2.54463	0.430121
$36$	4.84585	0.807642
$37$	−9.06735	−1.49066	−0.745332	−	0.666694i	$-0.767709\pi$
$37$	−9.06735	−1.49066	−0.745332	+	0.666694i	$0.767709\pi$
$38$	1.27370	0.206622
$39$	17.4249	2.79022
$40$	0.676703	0.106996
$41$	0.532380	0.0831438	0.0415719	−	0.999136i	$-0.486763\pi$
$41$	0.532380	0.0831438	0.0415719	+	0.999136i	$0.486763\pi$
$42$	−10.5329	−1.62526
$43$	−7.96327	−1.21439	−0.607194	−	0.794554i	$-0.707705\pi$
$43$	−7.96327	−1.21439	−0.607194	+	0.794554i	$0.707705\pi$
$44$	−0.631840	−0.0952534
$45$	3.27920	0.488835
$46$	3.84216	0.566496
$47$	−3.81367	−0.556281	−0.278140	−	0.960540i	$-0.589718\pi$
$47$	−3.81367	−0.556281	−0.278140	+	0.960540i	$0.589718\pi$
$48$	−2.80104	−0.404296
$49$	7.14011	1.02002
$50$	−4.54207	−0.642346
$51$	−9.51470	−1.33233
$52$	−6.22087	−0.862680
$53$	3.38497	0.464961	0.232481	−	0.972601i	$-0.425316\pi$
$53$	3.38497	0.464961	0.232481	+	0.972601i	$0.425316\pi$
$54$	−5.17032	−0.703591
$55$	−0.427568	−0.0576532
$56$	3.76033	0.502496
$57$	−3.56770	−0.472554
$58$	4.19303	0.550572
$59$	−6.10929	−0.795362	−0.397681	−	0.917524i	$-0.630185\pi$
$59$	−6.10929	−0.795362	−0.397681	+	0.917524i	$0.630185\pi$
$60$	−1.89548	−0.244705
$61$	−2.32597	−0.297810	−0.148905	−	0.988851i	$-0.547575\pi$
$61$	−2.32597	−0.297810	−0.148905	+	0.988851i	$0.547575\pi$
$62$	−9.96120	−1.26507
$63$	18.2220	2.29576
$64$	1.00000	0.125000
$65$	−4.20968	−0.522147
$66$	1.76981	0.217849
$67$	3.52438	0.430571	0.215286	−	0.976551i	$-0.430932\pi$
$67$	3.52438	0.430571	0.215286	+	0.976551i	$0.430932\pi$
$68$	3.39684	0.411927
$69$	−10.7621	−1.29560
$70$	2.54463	0.304141
$71$	−6.46908	−0.767738	−0.383869	−	0.923388i	$-0.625409\pi$
$71$	−6.46908	−0.767738	−0.383869	+	0.923388i	$0.625409\pi$
$72$	4.84585	0.571089
$73$	1.69845	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$73$	1.69845	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$74$	−9.06735	−1.05406
$75$	12.7226	1.46907
$76$	1.27370	0.146104
$77$	−2.37593	−0.270762
$78$	17.4249	1.97299
$79$	−8.68079	−0.976665	−0.488332	−	0.872658i	$-0.662395\pi$
$79$	−8.68079	−0.976665	−0.488332	+	0.872658i	$0.662395\pi$
$80$	0.676703	0.0756577
$81$	−0.0552674	−0.00614083
$82$	0.532380	0.0587915
$83$	−12.7963	−1.40458	−0.702290	−	0.711891i	$-0.747839\pi$
$83$	−12.7963	−1.40458	−0.702290	+	0.711891i	$0.747839\pi$
$84$	−10.5329	−1.14923
$85$	2.29865	0.249324
$86$	−7.96327	−0.858702
$87$	−11.7449	−1.25918
$88$	−0.631840	−0.0673544
$89$	4.85184	0.514294	0.257147	−	0.966372i	$-0.417217\pi$
$89$	4.85184	0.514294	0.257147	+	0.966372i	$0.417217\pi$
$90$	3.27920	0.345658
$91$	−23.3926	−2.45221
$92$	3.84216	0.400573
$93$	27.9018	2.89328
$94$	−3.81367	−0.393350
$95$	0.861920	0.0884311
$96$	−2.80104	−0.285880
$97$	2.77992	0.282258	0.141129	−	0.989991i	$-0.454927\pi$
$97$	2.77992	0.282258	0.141129	+	0.989991i	$0.454927\pi$
$98$	7.14011	0.721260
$99$	−3.06180	−0.307723
$100$	−4.54207	−0.454207
$101$	−5.96969	−0.594006	−0.297003	−	0.954877i	$-0.595987\pi$
$101$	−5.96969	−0.594006	−0.297003	+	0.954877i	$0.595987\pi$
$102$	−9.51470	−0.942096
$103$	5.47500	0.539468	0.269734	−	0.962935i	$-0.413064\pi$
$103$	5.47500	0.539468	0.269734	+	0.962935i	$0.413064\pi$
$104$	−6.22087	−0.610007
$105$	−7.12762	−0.695585
$106$	3.38497	0.328777
$107$	14.9969	1.44981	0.724904	−	0.688850i	$-0.241884\pi$
$107$	14.9969	1.44981	0.724904	+	0.688850i	$0.241884\pi$
$108$	−5.17032	−0.497514
$109$	−12.1876	−1.16736	−0.583682	−	0.811982i	$-0.698389\pi$
$109$	−12.1876	−1.16736	−0.583682	+	0.811982i	$0.698389\pi$
$110$	−0.427568	−0.0407670
$111$	25.3981	2.41068
$112$	3.76033	0.355318
$113$	15.6649	1.47363	0.736817	−	0.676092i	$-0.236328\pi$
$113$	15.6649	1.47363	0.736817	+	0.676092i	$0.236328\pi$
$114$	−3.56770	−0.334146
$115$	2.60000	0.242452
$116$	4.19303	0.389313
$117$	−30.1454	−2.78695
$118$	−6.10929	−0.562406
$119$	12.7733	1.17092
$120$	−1.89548	−0.173032
$121$	−10.6008	−0.963707
$122$	−2.32597	−0.210584
$123$	−1.49122	−0.134459
$124$	−9.96120	−0.894542
$125$	−6.45715	−0.577545
$126$	18.2220	1.62335
$127$	4.99299	0.443056	0.221528	−	0.975154i	$-0.428896\pi$
$127$	4.99299	0.443056	0.221528	+	0.975154i	$0.428896\pi$
$128$	1.00000	0.0883883
$129$	22.3055	1.96389
$130$	−4.20968	−0.369214
$131$	−3.86571	−0.337749	−0.168874	−	0.985638i	$-0.554013\pi$
$131$	−3.86571	−0.337749	−0.168874	+	0.985638i	$0.554013\pi$
$132$	1.76981	0.154042
$133$	4.78956	0.415307
$134$	3.52438	0.304460
$135$	−3.49877	−0.301126
$136$	3.39684	0.291277
$137$	−9.58553	−0.818947	−0.409474	−	0.912322i	$-0.634288\pi$
$137$	−9.58553	−0.818947	−0.409474	+	0.912322i	$0.634288\pi$
$138$	−10.7621	−0.916129
$139$	−4.56613	−0.387294	−0.193647	−	0.981071i	$-0.562032\pi$
$139$	−4.56613	−0.387294	−0.193647	+	0.981071i	$0.562032\pi$
$140$	2.54463	0.215060
$141$	10.6823	0.899608
$142$	−6.46908	−0.542873
$143$	3.93060	0.328693
$144$	4.84585	0.403821
$145$	2.83744	0.235636
$146$	1.69845	0.140565
$147$	−19.9998	−1.64955
$148$	−9.06735	−0.745332
$149$	−0.673255	−0.0551552	−0.0275776	−	0.999620i	$-0.508779\pi$
$149$	−0.673255	−0.0551552	−0.0275776	+	0.999620i	$0.508779\pi$
$150$	12.7226	1.03879
$151$	19.3660	1.57598	0.787990	−	0.615689i	$-0.211122\pi$
$151$	19.3660	1.57598	0.787990	+	0.615689i	$0.211122\pi$
$152$	1.27370	0.103311
$153$	16.4606	1.33076
$154$	−2.37593	−0.191458
$155$	−6.74077	−0.541432
$156$	17.4249	1.39511
$157$	−16.9998	−1.35673	−0.678366	−	0.734724i	$-0.737312\pi$
$157$	−16.9998	−1.35673	−0.678366	+	0.734724i	$0.737312\pi$
$158$	−8.68079	−0.690606
$159$	−9.48145	−0.751928
$160$	0.676703	0.0534981
$161$	14.4478	1.13865
$162$	−0.0552674	−0.00434222
$163$	−13.0919	−1.02544	−0.512720	−	0.858556i	$-0.671362\pi$
$163$	−13.0919	−1.02544	−0.512720	+	0.858556i	$0.671362\pi$
$164$	0.532380	0.0415719
$165$	1.19764	0.0932359
$166$	−12.7963	−0.993188
$167$	−4.43489	−0.343182	−0.171591	−	0.985168i	$-0.554891\pi$
$167$	−4.43489	−0.343182	−0.171591	+	0.985168i	$0.554891\pi$
$168$	−10.5329	−0.812628
$169$	25.6993	1.97687
$170$	2.29865	0.176299
$171$	6.17219	0.471999
$172$	−7.96327	−0.607194
$173$	14.9456	1.13629	0.568146	−	0.822928i	$-0.307661\pi$
$173$	14.9456	1.13629	0.568146	+	0.822928i	$0.307661\pi$
$174$	−11.7449	−0.890376
$175$	−17.0797	−1.29110
$176$	−0.631840	−0.0476267
$177$	17.1124	1.28625
$178$	4.85184	0.363661
$179$	4.46588	0.333795	0.166898	−	0.985974i	$-0.446625\pi$
$179$	4.46588	0.333795	0.166898	+	0.985974i	$0.446625\pi$
$180$	3.27920	0.244417
$181$	8.75756	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$181$	8.75756	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$182$	−23.3926	−1.73397
$183$	6.51516	0.481614
$184$	3.84216	0.283248
$185$	−6.13590	−0.451121
$186$	27.9018	2.04586
$187$	−2.14626	−0.156950
$188$	−3.81367	−0.278140
$189$	−19.4421	−1.41421
$190$	0.861920	0.0625303
$191$	16.4938	1.19345	0.596724	−	0.802447i	$-0.296469\pi$
$191$	16.4938	1.19345	0.596724	+	0.802447i	$0.296469\pi$
$192$	−2.80104	−0.202148
$193$	10.3176	0.742678	0.371339	−	0.928497i	$-0.378899\pi$
$193$	10.3176	0.742678	0.371339	+	0.928497i	$0.378899\pi$
$194$	2.77992	0.199587
$195$	11.7915	0.844408
$196$	7.14011	0.510008
$197$	−21.1641	−1.50788	−0.753939	−	0.656944i	$-0.771849\pi$
$197$	−21.1641	−1.50788	−0.753939	+	0.656944i	$0.771849\pi$
$198$	−3.06180	−0.217593
$199$	−0.760069	−0.0538799	−0.0269399	−	0.999637i	$-0.508576\pi$
$199$	−0.760069	−0.0538799	−0.0269399	+	0.999637i	$0.508576\pi$
$200$	−4.54207	−0.321173
$201$	−9.87194	−0.696313
$202$	−5.96969	−0.420026
$203$	15.7672	1.10664
$204$	−9.51470	−0.666163
$205$	0.360263	0.0251619
$206$	5.47500	0.381462
$207$	18.6186	1.29408
$208$	−6.22087	−0.431340
$209$	−0.804778	−0.0556676
$210$	−7.12762	−0.491853
$211$	20.5980	1.41802	0.709012	−	0.705196i	$-0.249141\pi$
$211$	20.5980	1.41802	0.709012	+	0.705196i	$0.249141\pi$
$212$	3.38497	0.232481
$213$	18.1202	1.24157
$214$	14.9969	1.02517
$215$	−5.38877	−0.367511
$216$	−5.17032	−0.351796
$217$	−37.4574	−2.54278
$218$	−12.1876	−0.825452
$219$	−4.75744	−0.321478
$220$	−0.427568	−0.0288266
$221$	−21.1313	−1.42145
$222$	25.3981	1.70461
$223$	2.76363	0.185066	0.0925332	−	0.995710i	$-0.470504\pi$
$223$	2.76363	0.185066	0.0925332	+	0.995710i	$0.470504\pi$
$224$	3.76033	0.251248
$225$	−22.0102	−1.46735
$226$	15.6649	1.04202
$227$	3.60460	0.239246	0.119623	−	0.992819i	$-0.461831\pi$
$227$	3.60460	0.239246	0.119623	+	0.992819i	$0.461831\pi$
$228$	−3.56770	−0.236277
$229$	−19.7873	−1.30758	−0.653791	−	0.756676i	$-0.726822\pi$
$229$	−19.7873	−1.30758	−0.653791	+	0.756676i	$0.726822\pi$
$230$	2.60000	0.171439
$231$	6.65508	0.437872
$232$	4.19303	0.275286
$233$	−1.24894	−0.0818207	−0.0409103	−	0.999163i	$-0.513026\pi$
$233$	−1.24894	−0.0818207	−0.0409103	+	0.999163i	$0.513026\pi$
$234$	−30.1454	−1.97067
$235$	−2.58072	−0.168348
$236$	−6.10929	−0.397681
$237$	24.3153	1.57945
$238$	12.7733	0.827967
$239$	6.22948	0.402952	0.201476	−	0.979493i	$-0.435426\pi$
$239$	6.22948	0.402952	0.201476	+	0.979493i	$0.435426\pi$
$240$	−1.89548	−0.122352
$241$	−13.0332	−0.839542	−0.419771	−	0.907630i	$-0.637890\pi$
$241$	−13.0332	−0.839542	−0.419771	+	0.907630i	$0.637890\pi$
$242$	−10.6008	−0.681444
$243$	15.6658	1.00496
$244$	−2.32597	−0.148905
$245$	4.83174	0.308688
$246$	−1.49122	−0.0950767
$247$	−7.92356	−0.504164
$248$	−9.96120	−0.632537
$249$	35.8431	2.27146
$250$	−6.45715	−0.408386
$251$	13.6041	0.858683	0.429342	−	0.903142i	$-0.358746\pi$
$251$	13.6041	0.858683	0.429342	+	0.903142i	$0.358746\pi$
$252$	18.2220	1.14788
$253$	−2.42763	−0.152624
$254$	4.99299	0.313288
$255$	−6.43863	−0.403203
$256$	1.00000	0.0625000
$257$	3.05899	0.190815	0.0954074	−	0.995438i	$-0.469585\pi$
$257$	3.05899	0.190815	0.0954074	+	0.995438i	$0.469585\pi$
$258$	22.3055	1.38868
$259$	−34.0963	−2.11864
$260$	−4.20968	−0.261074
$261$	20.3188	1.25770
$262$	−3.86571	−0.238824
$263$	−12.8608	−0.793030	−0.396515	−	0.918028i	$-0.629780\pi$
$263$	−12.8608	−0.793030	−0.396515	+	0.918028i	$0.629780\pi$
$264$	1.76981	0.108924
$265$	2.29062	0.140712
$266$	4.78956	0.293667
$267$	−13.5902	−0.831709
$268$	3.52438	0.215286
$269$	−0.356613	−0.0217431	−0.0108715	−	0.999941i	$-0.503461\pi$
$269$	−0.356613	−0.0217431	−0.0108715	+	0.999941i	$0.503461\pi$
$270$	−3.49877	−0.212928
$271$	−15.6107	−0.948285	−0.474142	−	0.880448i	$-0.657242\pi$
$271$	−15.6107	−0.948285	−0.474142	+	0.880448i	$0.657242\pi$
$272$	3.39684	0.205964
$273$	65.5236	3.96567
$274$	−9.58553	−0.579083
$275$	2.86986	0.173059
$276$	−10.7621	−0.647801
$277$	−17.5079	−1.05195	−0.525975	−	0.850500i	$-0.676300\pi$
$277$	−17.5079	−1.05195	−0.525975	+	0.850500i	$0.676300\pi$
$278$	−4.56613	−0.273858
$279$	−48.2705	−2.88988
$280$	2.54463	0.152071
$281$	7.79420	0.464963	0.232482	−	0.972601i	$-0.425315\pi$
$281$	7.79420	0.464963	0.232482	+	0.972601i	$0.425315\pi$
$282$	10.6823	0.636119
$283$	−15.3517	−0.912566	−0.456283	−	0.889835i	$-0.650819\pi$
$283$	−15.3517	−0.912566	−0.456283	+	0.889835i	$0.650819\pi$
$284$	−6.46908	−0.383869
$285$	−2.41428	−0.143009
$286$	3.93060	0.232421
$287$	2.00193	0.118170
$288$	4.84585	0.285545
$289$	−5.46147	−0.321263
$290$	2.83744	0.166620
$291$	−7.78668	−0.456463
$292$	1.69845	0.0993944
$293$	−5.71102	−0.333641	−0.166821	−	0.985987i	$-0.553350\pi$
$293$	−5.71102	−0.333641	−0.166821	+	0.985987i	$0.553350\pi$
$294$	−19.9998	−1.16641
$295$	−4.13418	−0.240701
$296$	−9.06735	−0.527029
$297$	3.26681	0.189560
$298$	−0.673255	−0.0390006
$299$	−23.9016	−1.38227
$300$	12.7226	0.734537
$301$	−29.9446	−1.72598
$302$	19.3660	1.11439
$303$	16.7214	0.960617
$304$	1.27370	0.0730520
$305$	−1.57399	−0.0901266
$306$	16.4606	0.940989
$307$	−6.30793	−0.360013	−0.180006	−	0.983665i	$-0.557612\pi$
$307$	−6.30793	−0.360013	−0.180006	+	0.983665i	$0.557612\pi$
$308$	−2.37593	−0.135381
$309$	−15.3357	−0.872419
$310$	−6.74077	−0.382850
$311$	18.4919	1.04858	0.524291	−	0.851539i	$-0.324330\pi$
$311$	18.4919	1.04858	0.524291	+	0.851539i	$0.324330\pi$
$312$	17.4249	0.986493
$313$	−33.9583	−1.91944	−0.959719	−	0.280961i	$-0.909347\pi$
$313$	−33.9583	−1.91944	−0.959719	+	0.280961i	$0.909347\pi$
$314$	−16.9998	−0.959355
$315$	12.3309	0.694768
$316$	−8.68079	−0.488332
$317$	8.09258	0.454525	0.227262	−	0.973834i	$-0.427023\pi$
$317$	8.09258	0.454525	0.227262	+	0.973834i	$0.427023\pi$
$318$	−9.48145	−0.531693
$319$	−2.64932	−0.148334
$320$	0.676703	0.0378289
$321$	−42.0071	−2.34461
$322$	14.4478	0.805146
$323$	4.32657	0.240737
$324$	−0.0552674	−0.00307041
$325$	28.2557	1.56734
$326$	−13.0919	−0.725096
$327$	34.1381	1.88784
$328$	0.532380	0.0293958
$329$	−14.3407	−0.790626
$330$	1.19764	0.0659277
$331$	−9.85515	−0.541688	−0.270844	−	0.962623i	$-0.587303\pi$
$331$	−9.85515	−0.541688	−0.270844	+	0.962623i	$0.587303\pi$
$332$	−12.7963	−0.702290
$333$	−43.9390	−2.40784
$334$	−4.43489	−0.242666
$335$	2.38496	0.130304
$336$	−10.5329	−0.574615
$337$	−5.81069	−0.316529	−0.158264	−	0.987397i	$-0.550590\pi$
$337$	−5.81069	−0.316529	−0.158264	+	0.987397i	$0.550590\pi$
$338$	25.6993	1.39786
$339$	−43.8782	−2.38314
$340$	2.29865	0.124662
$341$	6.29388	0.340833
$342$	6.17219	0.333754
$343$	0.526871	0.0284484
$344$	−7.96327	−0.429351
$345$	−7.28273	−0.392089
$346$	14.9456	0.803480
$347$	−22.5232	−1.20911	−0.604555	−	0.796563i	$-0.706649\pi$
$347$	−22.5232	−1.20911	−0.604555	+	0.796563i	$0.706649\pi$
$348$	−11.7449	−0.629591
$349$	−5.10939	−0.273500	−0.136750	−	0.990606i	$-0.543666\pi$
$349$	−5.10939	−0.273500	−0.136750	+	0.990606i	$0.543666\pi$
$350$	−17.0797	−0.912949
$351$	32.1639	1.71678
$352$	−0.631840	−0.0336772
$353$	−21.5775	−1.14845	−0.574226	−	0.818697i	$-0.694697\pi$
$353$	−21.5775	−1.14845	−0.574226	+	0.818697i	$0.694697\pi$
$354$	17.1124	0.909514
$355$	−4.37764	−0.232341
$356$	4.85184	0.257147
$357$	−35.7785	−1.89360
$358$	4.46588	0.236029
$359$	−9.88203	−0.521553	−0.260777	−	0.965399i	$-0.583979\pi$
$359$	−9.88203	−0.521553	−0.260777	+	0.965399i	$0.583979\pi$
$360$	3.27920	0.172829
$361$	−17.3777	−0.914615
$362$	8.75756	0.460287
$363$	29.6933	1.55849
$364$	−23.3926	−1.22610
$365$	1.14935	0.0601596
$366$	6.51516	0.340553
$367$	−1.69860	−0.0886663	−0.0443331	−	0.999017i	$-0.514116\pi$
$367$	−1.69860	−0.0886663	−0.0443331	+	0.999017i	$0.514116\pi$
$368$	3.84216	0.200287
$369$	2.57983	0.134301
$370$	−6.13590	−0.318990
$371$	12.7286	0.660837
$372$	27.9018	1.44664
$373$	11.9796	0.620282	0.310141	−	0.950691i	$-0.399624\pi$
$373$	11.9796	0.620282	0.310141	+	0.950691i	$0.399624\pi$
$374$	−2.14626	−0.110980
$375$	18.0868	0.933997
$376$	−3.81367	−0.196675
$377$	−26.0843	−1.34341
$378$	−19.4421	−0.999995
$379$	−2.65510	−0.136383	−0.0681917	−	0.997672i	$-0.521723\pi$
$379$	−2.65510	−0.136383	−0.0681917	+	0.997672i	$0.521723\pi$
$380$	0.861920	0.0442156
$381$	−13.9856	−0.716504
$382$	16.4938	0.843895
$383$	−20.1176	−1.02796	−0.513981	−	0.857802i	$-0.671830\pi$
$383$	−20.1176	−1.02796	−0.513981	+	0.857802i	$0.671830\pi$
$384$	−2.80104	−0.142940
$385$	−1.60780	−0.0819410
$386$	10.3176	0.525153
$387$	−38.5888	−1.96158
$388$	2.77992	0.141129
$389$	9.23455	0.468210	0.234105	−	0.972211i	$-0.424784\pi$
$389$	9.23455	0.468210	0.234105	+	0.972211i	$0.424784\pi$
$390$	11.7915	0.597087
$391$	13.0512	0.660029
$392$	7.14011	0.360630
$393$	10.8280	0.546202
$394$	−21.1641	−1.06623
$395$	−5.87432	−0.295569
$396$	−3.06180	−0.153861
$397$	−15.7963	−0.792795	−0.396397	−	0.918079i	$-0.629740\pi$
$397$	−15.7963	−0.792795	−0.396397	+	0.918079i	$0.629740\pi$
$398$	−0.760069	−0.0380988
$399$	−13.4158	−0.671628
$400$	−4.54207	−0.227104
$401$	18.3253	0.915121	0.457560	−	0.889179i	$-0.348723\pi$
$401$	18.3253	0.915121	0.457560	+	0.889179i	$0.348723\pi$
$402$	−9.87194	−0.492367
$403$	61.9674	3.08681
$404$	−5.96969	−0.297003
$405$	−0.0373996	−0.00185840
$406$	15.7672	0.782513
$407$	5.72911	0.283982
$408$	−9.51470	−0.471048
$409$	−32.7999	−1.62185	−0.810926	−	0.585149i	$-0.801036\pi$
$409$	−32.7999	−1.62185	−0.810926	+	0.585149i	$0.801036\pi$
$410$	0.360263	0.0177921
$411$	26.8495	1.32439
$412$	5.47500	0.269734
$413$	−22.9730	−1.13043
$414$	18.6186	0.915052
$415$	−8.65932	−0.425069
$416$	−6.22087	−0.305003
$417$	12.7899	0.626326
$418$	−0.804778	−0.0393630
$419$	29.1546	1.42429	0.712147	−	0.702030i	$-0.247723\pi$
$419$	29.1546	1.42429	0.712147	+	0.702030i	$0.247723\pi$
$420$	−7.12762	−0.347792
$421$	−14.9129	−0.726811	−0.363405	−	0.931631i	$-0.618386\pi$
$421$	−14.9129	−0.726811	−0.363405	+	0.931631i	$0.618386\pi$
$422$	20.5980	1.00269
$423$	−18.4805	−0.898551
$424$	3.38497	0.164389
$425$	−15.4287	−0.748402
$426$	18.1202	0.877925
$427$	−8.74644	−0.423270
$428$	14.9969	0.724904
$429$	−11.0098	−0.531557
$430$	−5.38877	−0.259870
$431$	10.7514	0.517879	0.258939	−	0.965894i	$-0.416627\pi$
$431$	10.7514	0.517879	0.258939	+	0.965894i	$0.416627\pi$
$432$	−5.17032	−0.248757
$433$	30.9398	1.48687	0.743435	−	0.668808i	$-0.233195\pi$
$433$	30.9398	1.48687	0.743435	+	0.668808i	$0.233195\pi$
$434$	−37.4574	−1.79801
$435$	−7.94779	−0.381067
$436$	−12.1876	−0.583682
$437$	4.89378	0.234101
$438$	−4.75744	−0.227319
$439$	11.9559	0.570622	0.285311	−	0.958435i	$-0.407903\pi$
$439$	11.9559	0.570622	0.285311	+	0.958435i	$0.407903\pi$
$440$	−0.427568	−0.0203835
$441$	34.5999	1.64762
$442$	−21.1313	−1.00511
$443$	−3.58332	−0.170249	−0.0851244	−	0.996370i	$-0.527129\pi$
$443$	−3.58332	−0.170249	−0.0851244	+	0.996370i	$0.527129\pi$
$444$	25.3981	1.20534
$445$	3.28326	0.155641
$446$	2.76363	0.130862
$447$	1.88582	0.0891961
$448$	3.76033	0.177659
$449$	−10.7821	−0.508837	−0.254419	−	0.967094i	$-0.581884\pi$
$449$	−10.7821	−0.508837	−0.254419	+	0.967094i	$0.581884\pi$
$450$	−22.0102	−1.03757
$451$	−0.336379	−0.0158395
$452$	15.6649	0.736817
$453$	−54.2449	−2.54865
$454$	3.60460	0.169172
$455$	−15.8298	−0.742113
$456$	−3.56770	−0.167073
$457$	−14.8925	−0.696642	−0.348321	−	0.937375i	$-0.613248\pi$
$457$	−14.8925	−0.696642	−0.348321	+	0.937375i	$0.613248\pi$
$458$	−19.7873	−0.924599
$459$	−17.5627	−0.819759
$460$	2.60000	0.121226
$461$	−6.81045	−0.317194	−0.158597	−	0.987343i	$-0.550697\pi$
$461$	−6.81045	−0.317194	−0.158597	+	0.987343i	$0.550697\pi$
$462$	6.65508	0.309623
$463$	10.9756	0.510079	0.255040	−	0.966931i	$-0.417911\pi$
$463$	10.9756	0.510079	0.255040	+	0.966931i	$0.417911\pi$
$464$	4.19303	0.194657
$465$	18.8812	0.875595
$466$	−1.24894	−0.0578560
$467$	−6.92170	−0.320298	−0.160149	−	0.987093i	$-0.551197\pi$
$467$	−6.92170	−0.320298	−0.160149	+	0.987093i	$0.551197\pi$
$468$	−30.1454	−1.39347
$469$	13.2528	0.611959
$470$	−2.58072	−0.119040
$471$	47.6172	2.19409
$472$	−6.10929	−0.281203
$473$	5.03151	0.231349
$474$	24.3153	1.11684
$475$	−5.78526	−0.265446
$476$	12.7733	0.585461
$477$	16.4031	0.751045
$478$	6.22948	0.284930
$479$	−30.6547	−1.40065	−0.700324	−	0.713825i	$-0.746961\pi$
$479$	−30.6547	−1.40065	−0.700324	+	0.713825i	$0.746961\pi$
$480$	−1.89548	−0.0865162
$481$	56.4068	2.57193
$482$	−13.0332	−0.593646
$483$	−40.4690	−1.84140
$484$	−10.6008	−0.481854
$485$	1.88118	0.0854200
$486$	15.6658	0.710613
$487$	27.5206	1.24708	0.623539	−	0.781792i	$-0.285694\pi$
$487$	27.5206	1.24708	0.623539	+	0.781792i	$0.285694\pi$
$488$	−2.32597	−0.105292
$489$	36.6711	1.65833
$490$	4.83174	0.218276
$491$	−5.76173	−0.260023	−0.130012	−	0.991512i	$-0.541501\pi$
$491$	−5.76173	−0.260023	−0.130012	+	0.991512i	$0.541501\pi$
$492$	−1.49122	−0.0672294
$493$	14.2431	0.641475
$494$	−7.92356	−0.356498
$495$	−2.07193	−0.0931264
$496$	−9.96120	−0.447271
$497$	−24.3259	−1.09117
$498$	35.8431	1.60617
$499$	41.1456	1.84193	0.920965	−	0.389645i	$-0.127402\pi$
$499$	41.1456	1.84193	0.920965	+	0.389645i	$0.127402\pi$
$500$	−6.45715	−0.288773
$501$	12.4223	0.554988
$502$	13.6041	0.607181
$503$	29.9578	1.33575	0.667877	−	0.744272i	$-0.267203\pi$
$503$	29.9578	1.33575	0.667877	+	0.744272i	$0.267203\pi$
$504$	18.2220	0.811674
$505$	−4.03971	−0.179765
$506$	−2.42763	−0.107921
$507$	−71.9848	−3.19696
$508$	4.99299	0.221528
$509$	6.97677	0.309240	0.154620	−	0.987974i	$-0.450585\pi$
$509$	6.97677	0.309240	0.154620	+	0.987974i	$0.450585\pi$
$510$	−6.43863	−0.285107
$511$	6.38675	0.282533
$512$	1.00000	0.0441942
$513$	−6.58546	−0.290755
$514$	3.05899	0.134926
$515$	3.70495	0.163260
$516$	22.3055	0.981944
$517$	2.40963	0.105975
$518$	−34.0963	−1.49810
$519$	−41.8633	−1.83759
$520$	−4.20968	−0.184607
$521$	−38.9300	−1.70556	−0.852778	−	0.522274i	$-0.825084\pi$
$521$	−38.9300	−1.70556	−0.852778	+	0.522274i	$0.825084\pi$
$522$	20.3188	0.889331
$523$	−20.7987	−0.909465	−0.454732	−	0.890628i	$-0.650265\pi$
$523$	−20.7987	−0.909465	−0.454732	+	0.890628i	$0.650265\pi$
$524$	−3.86571	−0.168874
$525$	47.8410	2.08795
$526$	−12.8608	−0.560757
$527$	−33.8366	−1.47395
$528$	1.76981	0.0770212
$529$	−8.23777	−0.358164
$530$	2.29062	0.0994981
$531$	−29.6047	−1.28474
$532$	4.78956	0.207654
$533$	−3.31187	−0.143453
$534$	−13.5902	−0.588107
$535$	10.1485	0.438757
$536$	3.52438	0.152230
$537$	−12.5091	−0.539809
$538$	−0.356613	−0.0153747
$539$	−4.51141	−0.194320
$540$	−3.49877	−0.150563
$541$	−23.4349	−1.00755	−0.503773	−	0.863836i	$-0.668055\pi$
$541$	−23.4349	−1.00755	−0.503773	+	0.863836i	$0.668055\pi$
$542$	−15.6107	−0.670538
$543$	−24.5303	−1.05270
$544$	3.39684	0.145638
$545$	−8.24742	−0.353281
$546$	65.5236	2.80415
$547$	−34.1967	−1.46214	−0.731072	−	0.682301i	$-0.760980\pi$
$547$	−34.1967	−1.46214	−0.731072	+	0.682301i	$0.760980\pi$
$548$	−9.58553	−0.409474
$549$	−11.2713	−0.481049
$550$	2.86986	0.122371
$551$	5.34069	0.227521
$552$	−10.7621	−0.458064
$553$	−32.6427	−1.38811
$554$	−17.5079	−0.743841
$555$	17.1869	0.729545
$556$	−4.56613	−0.193647
$557$	−8.65534	−0.366739	−0.183369	−	0.983044i	$-0.558700\pi$
$557$	−8.65534	−0.366739	−0.183369	+	0.983044i	$0.558700\pi$
$558$	−48.2705	−2.04345
$559$	49.5385	2.09526
$560$	2.54463	0.107530
$561$	6.01177	0.253817
$562$	7.79420	0.328779
$563$	−24.9699	−1.05236	−0.526178	−	0.850374i	$-0.676375\pi$
$563$	−24.9699	−1.05236	−0.526178	+	0.850374i	$0.676375\pi$
$564$	10.6823	0.449804
$565$	10.6005	0.445967
$566$	−15.3517	−0.645281
$567$	−0.207824	−0.00872779
$568$	−6.46908	−0.271436
$569$	−23.0538	−0.966464	−0.483232	−	0.875492i	$-0.660537\pi$
$569$	−23.0538	−0.966464	−0.483232	+	0.875492i	$0.660537\pi$
$570$	−2.41428	−0.101123
$571$	41.2184	1.72494	0.862469	−	0.506110i	$-0.168917\pi$
$571$	41.2184	1.72494	0.862469	+	0.506110i	$0.168917\pi$
$572$	3.93060	0.164346
$573$	−46.1998	−1.93002
$574$	2.00193	0.0835588
$575$	−17.4514	−0.727773
$576$	4.84585	0.201911
$577$	−9.49822	−0.395416	−0.197708	−	0.980261i	$-0.563350\pi$
$577$	−9.49822	−0.395416	−0.197708	+	0.980261i	$0.563350\pi$
$578$	−5.46147	−0.227167
$579$	−28.9001	−1.20105
$580$	2.83744	0.117818
$581$	−48.1185	−1.99629
$582$	−7.78668	−0.322768
$583$	−2.13876	−0.0885783
$584$	1.69845	0.0702824
$585$	−20.3995	−0.843416
$586$	−5.71102	−0.235920
$587$	35.6897	1.47307	0.736536	−	0.676399i	$-0.236460\pi$
$587$	35.6897	1.47307	0.736536	+	0.676399i	$0.236460\pi$
$588$	−19.9998	−0.824777
$589$	−12.6876	−0.522785
$590$	−4.13418	−0.170201
$591$	59.2816	2.43852
$592$	−9.06735	−0.372666
$593$	−5.75974	−0.236524	−0.118262	−	0.992982i	$-0.537732\pi$
$593$	−5.75974	−0.236524	−0.118262	+	0.992982i	$0.537732\pi$
$594$	3.26681	0.134039
$595$	8.64370	0.354357
$596$	−0.673255	−0.0275776
$597$	2.12899	0.0871337
$598$	−23.9016	−0.977410
$599$	3.16499	0.129318	0.0646589	−	0.997907i	$-0.479404\pi$
$599$	3.16499	0.129318	0.0646589	+	0.997907i	$0.479404\pi$
$600$	12.7226	0.519396
$601$	16.7751	0.684272	0.342136	−	0.939650i	$-0.388850\pi$
$601$	16.7751	0.684272	0.342136	+	0.939650i	$0.388850\pi$
$602$	−29.9446	−1.22045
$603$	17.0786	0.695495
$604$	19.3660	0.787990
$605$	−7.17358	−0.291647
$606$	16.7214	0.679259
$607$	−46.2442	−1.87699	−0.938497	−	0.345287i	$-0.887782\pi$
$607$	−46.2442	−1.87699	−0.938497	+	0.345287i	$0.887782\pi$
$608$	1.27370	0.0516556
$609$	−44.1646	−1.78964
$610$	−1.57399	−0.0637291
$611$	23.7243	0.959784
$612$	16.4606	0.665380
$613$	−3.63707	−0.146900	−0.0734500	−	0.997299i	$-0.523401\pi$
$613$	−3.63707	−0.146900	−0.0734500	+	0.997299i	$0.523401\pi$
$614$	−6.30793	−0.254567
$615$	−1.00911	−0.0406914
$616$	−2.37593	−0.0957289
$617$	−29.5745	−1.19062	−0.595312	−	0.803495i	$-0.702971\pi$
$617$	−29.5745	−1.19062	−0.595312	+	0.803495i	$0.702971\pi$
$618$	−15.3357	−0.616894
$619$	−10.9005	−0.438129	−0.219065	−	0.975710i	$-0.570301\pi$
$619$	−10.9005	−0.438129	−0.219065	+	0.975710i	$0.570301\pi$
$620$	−6.74077	−0.270716
$621$	−19.8652	−0.797163
$622$	18.4919	0.741460
$623$	18.2446	0.730953
$624$	17.4249	0.697556
$625$	18.3408	0.733632
$626$	−33.9583	−1.35725
$627$	2.25422	0.0900248
$628$	−16.9998	−0.678366
$629$	−30.8003	−1.22809
$630$	12.3309	0.491275
$631$	2.39342	0.0952807	0.0476404	−	0.998865i	$-0.484830\pi$
$631$	2.39342	0.0952807	0.0476404	+	0.998865i	$0.484830\pi$
$632$	−8.68079	−0.345303
$633$	−57.6959	−2.29321
$634$	8.09258	0.321397
$635$	3.37877	0.134083
$636$	−9.48145	−0.375964
$637$	−44.4177	−1.75989
$638$	−2.64932	−0.104888
$639$	−31.3482	−1.24012
$640$	0.676703	0.0267490
$641$	−34.1887	−1.35037	−0.675186	−	0.737647i	$-0.735937\pi$
$641$	−34.1887	−1.35037	−0.675186	+	0.737647i	$0.735937\pi$
$642$	−42.0071	−1.65789
$643$	0.527584	0.0208059	0.0104029	−	0.999946i	$-0.496689\pi$
$643$	0.527584	0.0208059	0.0104029	+	0.999946i	$0.496689\pi$
$644$	14.4478	0.569324
$645$	15.0942	0.594333
$646$	4.32657	0.170227
$647$	−33.4190	−1.31384	−0.656918	−	0.753962i	$-0.728140\pi$
$647$	−33.4190	−1.31384	−0.656918	+	0.753962i	$0.728140\pi$
$648$	−0.0552674	−0.00217111
$649$	3.86009	0.151522
$650$	28.2557	1.10828
$651$	104.920	4.11214
$652$	−13.0919	−0.512720
$653$	6.97891	0.273106	0.136553	−	0.990633i	$-0.456398\pi$
$653$	6.97891	0.273106	0.136553	+	0.990633i	$0.456398\pi$
$654$	34.1381	1.33491
$655$	−2.61594	−0.102213
$656$	0.532380	0.0207859
$657$	8.23045	0.321100
$658$	−14.3407	−0.559057
$659$	2.15208	0.0838332	0.0419166	−	0.999121i	$-0.486654\pi$
$659$	2.15208	0.0838332	0.0419166	+	0.999121i	$0.486654\pi$
$660$	1.19764	0.0466180
$661$	42.0362	1.63502	0.817509	−	0.575916i	$-0.195354\pi$
$661$	42.0362	1.63502	0.817509	+	0.575916i	$0.195354\pi$
$662$	−9.85515	−0.383031
$663$	59.1898	2.29874
$664$	−12.7963	−0.496594
$665$	3.24111	0.125685
$666$	−43.9390	−1.70260
$667$	16.1103	0.623794
$668$	−4.43489	−0.171591
$669$	−7.74106	−0.299286
$670$	2.38496	0.0921389
$671$	1.46964	0.0567349
$672$	−10.5329	−0.406314
$673$	5.86954	0.226254	0.113127	−	0.993581i	$-0.463913\pi$
$673$	5.86954	0.226254	0.113127	+	0.993581i	$0.463913\pi$
$674$	−5.81069	−0.223820
$675$	23.4840	0.903898
$676$	25.6993	0.988433
$677$	−36.6203	−1.40743	−0.703716	−	0.710481i	$-0.748477\pi$
$677$	−36.6203	−1.40743	−0.703716	+	0.710481i	$0.748477\pi$
$678$	−43.8782	−1.68513
$679$	10.4534	0.401166
$680$	2.29865	0.0881493
$681$	−10.0966	−0.386904
$682$	6.29388	0.241005
$683$	−14.6062	−0.558892	−0.279446	−	0.960161i	$-0.590151\pi$
$683$	−14.6062	−0.558892	−0.279446	+	0.960161i	$0.590151\pi$
$684$	6.17219	0.235999
$685$	−6.48656	−0.247839
$686$	0.526871	0.0201160
$687$	55.4251	2.11460
$688$	−7.96327	−0.303597
$689$	−21.0575	−0.802226
$690$	−7.28273	−0.277249
$691$	−9.52870	−0.362489	−0.181244	−	0.983438i	$-0.558012\pi$
$691$	−9.52870	−0.362489	−0.181244	+	0.983438i	$0.558012\pi$
$692$	14.9456	0.568146
$693$	−11.5134	−0.437358
$694$	−22.5232	−0.854970
$695$	−3.08992	−0.117207
$696$	−11.7449	−0.445188
$697$	1.80841	0.0684984
$698$	−5.10939	−0.193393
$699$	3.49833	0.132319
$700$	−17.0797	−0.645552
$701$	5.03585	0.190201	0.0951007	−	0.995468i	$-0.469683\pi$
$701$	5.03585	0.190201	0.0951007	+	0.995468i	$0.469683\pi$
$702$	32.1639	1.21395
$703$	−11.5491	−0.435584
$704$	−0.631840	−0.0238134
$705$	7.22871	0.272249
$706$	−21.5775	−0.812078
$707$	−22.4480	−0.844245
$708$	17.1124	0.643123
$709$	38.2526	1.43661	0.718303	−	0.695730i	$-0.244919\pi$
$709$	38.2526	1.43661	0.718303	+	0.695730i	$0.244919\pi$
$710$	−4.37764	−0.164290
$711$	−42.0658	−1.57759
$712$	4.85184	0.181831
$713$	−38.2726	−1.43332
$714$	−35.7785	−1.33898
$715$	2.65985	0.0994726
$716$	4.46588	0.166898
$717$	−17.4491	−0.651647
$718$	−9.88203	−0.368794
$719$	38.6953	1.44309	0.721545	−	0.692368i	$-0.243432\pi$
$719$	38.6953	1.44309	0.721545	+	0.692368i	$0.243432\pi$
$720$	3.27920	0.122209
$721$	20.5878	0.766731
$722$	−17.3777	−0.646730
$723$	36.5066	1.35769
$724$	8.75756	0.325472
$725$	−19.0451	−0.707316
$726$	29.6933	1.10202
$727$	−2.53127	−0.0938796	−0.0469398	−	0.998898i	$-0.514947\pi$
$727$	−2.53127	−0.0938796	−0.0469398	+	0.998898i	$0.514947\pi$
$728$	−23.3926	−0.866986
$729$	−43.7147	−1.61906
$730$	1.14935	0.0425393
$731$	−27.0500	−1.00048
$732$	6.51516	0.240807
$733$	−22.6497	−0.836587	−0.418293	−	0.908312i	$-0.637372\pi$
$733$	−22.6497	−0.836587	−0.418293	+	0.908312i	$0.637372\pi$
$734$	−1.69860	−0.0626965
$735$	−13.5339	−0.499206
$736$	3.84216	0.141624
$737$	−2.22684	−0.0820267
$738$	2.57983	0.0949650
$739$	−31.6763	−1.16523	−0.582616	−	0.812748i	$-0.697971\pi$
$739$	−31.6763	−1.16523	−0.582616	+	0.812748i	$0.697971\pi$
$740$	−6.13590	−0.225560
$741$	22.1942	0.815326
$742$	12.7286	0.467282
$743$	44.8864	1.64672	0.823361	−	0.567517i	$-0.192096\pi$
$743$	44.8864	1.64672	0.823361	+	0.567517i	$0.192096\pi$
$744$	27.9018	1.02293
$745$	−0.455594	−0.0166917
$746$	11.9796	0.438606
$747$	−62.0091	−2.26880
$748$	−2.14626	−0.0784750
$749$	56.3935	2.06057
$750$	18.0868	0.660435
$751$	41.5031	1.51447	0.757235	−	0.653143i	$-0.226550\pi$
$751$	41.5031	1.51447	0.757235	+	0.653143i	$0.226550\pi$
$752$	−3.81367	−0.139070
$753$	−38.1057	−1.38865
$754$	−26.0843	−0.949935
$755$	13.1050	0.476940
$756$	−19.4421	−0.707103
$757$	41.9409	1.52437	0.762184	−	0.647360i	$-0.224127\pi$
$757$	41.9409	1.52437	0.762184	+	0.647360i	$0.224127\pi$
$758$	−2.65510	−0.0964376
$759$	6.79991	0.246821
$760$	0.861920	0.0312651
$761$	−19.0384	−0.690143	−0.345071	−	0.938576i	$-0.612145\pi$
$761$	−19.0384	−0.690143	−0.345071	+	0.938576i	$0.612145\pi$
$762$	−13.9856	−0.506645
$763$	−45.8296	−1.65914
$764$	16.4938	0.596724
$765$	11.1389	0.402729
$766$	−20.1176	−0.726879
$767$	38.0051	1.37229
$768$	−2.80104	−0.101074
$769$	15.7182	0.566812	0.283406	−	0.959000i	$-0.408536\pi$
$769$	15.7182	0.566812	0.283406	+	0.959000i	$0.408536\pi$
$770$	−1.60780	−0.0579410
$771$	−8.56838	−0.308583
$772$	10.3176	0.371339
$773$	−0.540300	−0.0194332	−0.00971662	−	0.999953i	$-0.503093\pi$
$773$	−0.540300	−0.0194332	−0.00971662	+	0.999953i	$0.503093\pi$
$774$	−38.5888	−1.38705
$775$	45.2445	1.62523
$776$	2.77992	0.0997933
$777$	95.5052	3.42623
$778$	9.23455	0.331075
$779$	0.678095	0.0242953
$780$	11.7915	0.422204
$781$	4.08742	0.146259
$782$	13.0512	0.466711
$783$	−21.6793	−0.774755
$784$	7.14011	0.255004
$785$	−11.5038	−0.410589
$786$	10.8280	0.386223
$787$	12.6465	0.450799	0.225400	−	0.974266i	$-0.427631\pi$
$787$	12.6465	0.450799	0.225400	+	0.974266i	$0.427631\pi$
$788$	−21.1641	−0.753939
$789$	36.0236	1.28247
$790$	−5.87432	−0.208999
$791$	58.9054	2.09444
$792$	−3.06180	−0.108796
$793$	14.4696	0.513830
$794$	−15.7963	−0.560590
$795$	−6.41613	−0.227557
$796$	−0.760069	−0.0269399
$797$	−54.4649	−1.92924	−0.964622	−	0.263636i	$-0.915078\pi$
$797$	−54.4649	−1.92924	−0.964622	+	0.263636i	$0.915078\pi$
$798$	−13.4158	−0.474913
$799$	−12.9544	−0.458294
$800$	−4.54207	−0.160587
$801$	23.5113	0.830732
$802$	18.3253	0.647088
$803$	−1.07315	−0.0378706
$804$	−9.87194	−0.348156
$805$	9.77688	0.344590
$806$	61.9674	2.18271
$807$	0.998889	0.0351626
$808$	−5.96969	−0.210013
$809$	1.35956	0.0477996	0.0238998	−	0.999714i	$-0.492392\pi$
$809$	1.35956	0.0477996	0.0238998	+	0.999714i	$0.492392\pi$
$810$	−0.0373996	−0.00131409
$811$	−9.33605	−0.327833	−0.163917	−	0.986474i	$-0.552413\pi$
$811$	−9.33605	−0.327833	−0.163917	+	0.986474i	$0.552413\pi$
$812$	15.7672	0.553320
$813$	43.7264	1.53355
$814$	5.72911	0.200805
$815$	−8.85936	−0.310330
$816$	−9.51470	−0.333081
$817$	−10.1429	−0.354854
$818$	−32.7999	−1.14682
$819$	−113.357	−3.96101
$820$	0.360263	0.0125809
$821$	−18.6988	−0.652593	−0.326296	−	0.945267i	$-0.605801\pi$
$821$	−18.6988	−0.652593	−0.326296	+	0.945267i	$0.605801\pi$
$822$	26.8495	0.936484
$823$	23.0774	0.804426	0.402213	−	0.915546i	$-0.368241\pi$
$823$	23.0774	0.804426	0.402213	+	0.915546i	$0.368241\pi$
$824$	5.47500	0.190731
$825$	−8.03861	−0.279869
$826$	−22.9730	−0.799332
$827$	20.6856	0.719309	0.359655	−	0.933086i	$-0.382895\pi$
$827$	20.6856	0.719309	0.359655	+	0.933086i	$0.382895\pi$
$828$	18.6186	0.647040
$829$	−45.9765	−1.59683	−0.798416	−	0.602107i	$-0.794328\pi$
$829$	−45.9765	−1.59683	−0.798416	+	0.602107i	$0.794328\pi$
$830$	−8.65932	−0.300569
$831$	49.0405	1.70120
$832$	−6.22087	−0.215670
$833$	24.2538	0.840345
$834$	12.7899	0.442879
$835$	−3.00110	−0.103857
$836$	−0.804778	−0.0278338
$837$	51.5026	1.78019
$838$	29.1546	1.00713
$839$	45.6270	1.57522	0.787610	−	0.616174i	$-0.211318\pi$
$839$	45.6270	1.57522	0.787610	+	0.616174i	$0.211318\pi$
$840$	−7.12762	−0.245926
$841$	−11.4185	−0.393741
$842$	−14.9129	−0.513933
$843$	−21.8319	−0.751931
$844$	20.5980	0.709012
$845$	17.3908	0.598261
$846$	−18.4805	−0.635372
$847$	−39.8625	−1.36969
$848$	3.38497	0.116240
$849$	43.0009	1.47579
$850$	−15.4287	−0.529200
$851$	−34.8382	−1.19424
$852$	18.1202	0.620787
$853$	−13.2743	−0.454502	−0.227251	−	0.973836i	$-0.572974\pi$
$853$	−13.2743	−0.454502	−0.227251	+	0.973836i	$0.572974\pi$
$854$	−8.74644	−0.299297
$855$	4.17674	0.142841
$856$	14.9969	0.512585
$857$	24.7967	0.847039	0.423519	−	0.905887i	$-0.360795\pi$
$857$	24.7967	0.847039	0.423519	+	0.905887i	$0.360795\pi$
$858$	−11.0098	−0.375868
$859$	1.35142	0.0461097	0.0230548	−	0.999734i	$-0.492661\pi$
$859$	1.35142	0.0461097	0.0230548	+	0.999734i	$0.492661\pi$
$860$	−5.38877	−0.183756
$861$	−5.60749	−0.191103
$862$	10.7514	0.366196
$863$	33.6053	1.14394	0.571968	−	0.820276i	$-0.306180\pi$
$863$	33.6053	1.14394	0.571968	+	0.820276i	$0.306180\pi$
$864$	−5.17032	−0.175898
$865$	10.1137	0.343877
$866$	30.9398	1.05138
$867$	15.2978	0.519542
$868$	−37.4574	−1.27139
$869$	5.48487	0.186061
$870$	−7.94779	−0.269455
$871$	−21.9247	−0.742890
$872$	−12.1876	−0.412726
$873$	13.4711	0.455927
$874$	4.89378	0.165535
$875$	−24.2810	−0.820849
$876$	−4.75744	−0.160739
$877$	−1.06643	−0.0360108	−0.0180054	−	0.999838i	$-0.505732\pi$
$877$	−1.06643	−0.0360108	−0.0180054	+	0.999838i	$0.505732\pi$
$878$	11.9559	0.403491
$879$	15.9968	0.539559
$880$	−0.427568	−0.0144133
$881$	55.4004	1.86649	0.933244	−	0.359244i	$-0.116965\pi$
$881$	55.4004	1.86649	0.933244	+	0.359244i	$0.116965\pi$
$882$	34.5999	1.16504
$883$	−47.3965	−1.59502	−0.797510	−	0.603306i	$-0.793850\pi$
$883$	−47.3965	−1.59502	−0.797510	+	0.603306i	$0.793850\pi$
$884$	−21.1313	−0.710723
$885$	11.5800	0.389258
$886$	−3.58332	−0.120384
$887$	22.8628	0.767657	0.383828	−	0.923404i	$-0.374605\pi$
$887$	22.8628	0.767657	0.383828	+	0.923404i	$0.374605\pi$
$888$	25.3981	0.852303
$889$	18.7753	0.629704
$890$	3.28326	0.110055
$891$	0.0349202	0.00116987
$892$	2.76363	0.0925332
$893$	−4.85749	−0.162550
$894$	1.88582	0.0630712
$895$	3.02207	0.101017
$896$	3.76033	0.125624
$897$	66.9495	2.23538
$898$	−10.7821	−0.359802
$899$	−41.7676	−1.39303
$900$	−22.0102	−0.733674
$901$	11.4982	0.383061
$902$	−0.336379	−0.0112002
$903$	83.8761	2.79122
$904$	15.6649	0.521008
$905$	5.92627	0.196996
$906$	−54.2449	−1.80217
$907$	20.5832	0.683453	0.341726	−	0.939799i	$-0.388988\pi$
$907$	20.5832	0.683453	0.341726	+	0.939799i	$0.388988\pi$
$908$	3.60460	0.119623
$909$	−28.9282	−0.959489
$910$	−15.8298	−0.524753
$911$	57.8412	1.91636	0.958182	−	0.286159i	$-0.0923786\pi$
$911$	57.8412	1.91636	0.958182	+	0.286159i	$0.0923786\pi$
$912$	−3.56770	−0.118139
$913$	8.08523	0.267582
$914$	−14.8925	−0.492601
$915$	4.40883	0.145751
$916$	−19.7873	−0.653791
$917$	−14.5364	−0.480033
$918$	−17.5627	−0.579657
$919$	39.1324	1.29086	0.645430	−	0.763820i	$-0.276678\pi$
$919$	39.1324	1.29086	0.645430	+	0.763820i	$0.276678\pi$
$920$	2.60000	0.0857196
$921$	17.6688	0.582207
$922$	−6.81045	−0.224290
$923$	40.2433	1.32462
$924$	6.65508	0.218936
$925$	41.1846	1.35414
$926$	10.9756	0.360681
$927$	26.5311	0.871395
$928$	4.19303	0.137643
$929$	10.1071	0.331602	0.165801	−	0.986159i	$-0.446979\pi$
$929$	10.1071	0.331602	0.165801	+	0.986159i	$0.446979\pi$
$930$	18.8812	0.619139
$931$	9.09440	0.298057
$932$	−1.24894	−0.0409103
$933$	−51.7968	−1.69575
$934$	−6.92170	−0.226485
$935$	−1.45238	−0.0474979
$936$	−30.1454	−0.985334
$937$	53.9203	1.76150	0.880750	−	0.473582i	$-0.157039\pi$
$937$	53.9203	1.76150	0.880750	+	0.473582i	$0.157039\pi$
$938$	13.2528	0.432720
$939$	95.1188	3.10409
$940$	−2.58072	−0.0841738
$941$	35.1814	1.14688	0.573440	−	0.819248i	$-0.305609\pi$
$941$	35.1814	1.14688	0.573440	+	0.819248i	$0.305609\pi$
$942$	47.6172	1.55145
$943$	2.04549	0.0666104
$944$	−6.10929	−0.198840
$945$	−13.1565	−0.427982
$946$	5.03151	0.163589
$947$	−31.7472	−1.03164	−0.515822	−	0.856696i	$-0.672513\pi$
$947$	−31.7472	−1.03164	−0.515822	+	0.856696i	$0.672513\pi$
$948$	24.3153	0.789724
$949$	−10.5659	−0.342982
$950$	−5.78526	−0.187699
$951$	−22.6677	−0.735050
$952$	12.7733	0.413984
$953$	42.7912	1.38614	0.693071	−	0.720869i	$-0.256257\pi$
$953$	42.7912	1.38614	0.693071	+	0.720869i	$0.256257\pi$
$954$	16.4031	0.531069
$955$	11.1614	0.361174
$956$	6.22948	0.201476
$957$	7.42088	0.239883
$958$	−30.6547	−0.990408
$959$	−36.0448	−1.16395
$960$	−1.89548	−0.0611762
$961$	68.2255	2.20082
$962$	56.4068	1.81863
$963$	72.6729	2.34185
$964$	−13.0332	−0.419771
$965$	6.98196	0.224757
$966$	−40.4690	−1.30207
$967$	−7.16842	−0.230521	−0.115260	−	0.993335i	$-0.536770\pi$
$967$	−7.16842	−0.230521	−0.115260	+	0.993335i	$0.536770\pi$
$968$	−10.6008	−0.340722
$969$	−12.1189	−0.389316
$970$	1.88118	0.0604011
$971$	−55.7743	−1.78988	−0.894941	−	0.446185i	$-0.852782\pi$
$971$	−55.7743	−1.78988	−0.894941	+	0.446185i	$0.852782\pi$
$972$	15.6658	0.502479
$973$	−17.1702	−0.550451
$974$	27.5206	0.881817
$975$	−79.1454	−2.53468
$976$	−2.32597	−0.0744526
$977$	−44.4161	−1.42100	−0.710499	−	0.703698i	$-0.751531\pi$
$977$	−44.4161	−1.42100	−0.710499	+	0.703698i	$0.751531\pi$
$978$	36.6711	1.17261
$979$	−3.06559	−0.0979766
$980$	4.83174	0.154344
$981$	−59.0595	−1.88563
$982$	−5.76173	−0.183864
$983$	−1.42424	−0.0454262	−0.0227131	−	0.999742i	$-0.507230\pi$
$983$	−1.42424	−0.0454262	−0.0227131	+	0.999742i	$0.507230\pi$
$984$	−1.49122	−0.0475384
$985$	−14.3218	−0.456331
$986$	14.2431	0.453592
$987$	40.1688	1.27859
$988$	−7.92356	−0.252082
$989$	−30.5962	−0.972903
$990$	−2.07193	−0.0658503
$991$	49.3353	1.56719	0.783594	−	0.621273i	$-0.213384\pi$
$991$	49.3353	1.56719	0.783594	+	0.621273i	$0.213384\pi$
$992$	−9.96120	−0.316268
$993$	27.6047	0.876009
$994$	−24.3259	−0.771570
$995$	−0.514341	−0.0163057
$996$	35.8431	1.13573
$997$	34.3648	1.08834	0.544172	−	0.838974i	$-0.316844\pi$
$997$	34.3648	1.08834	0.544172	+	0.838974i	$0.316844\pi$
$998$	41.1456	1.30244
$999$	46.8811	1.48325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.8	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.80104 q^{3} +1.00000 q^{4} +0.676703 q^{5} -2.80104 q^{6} +3.76033 q^{7} +1.00000 q^{8} +4.84585 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.80104 q^{3} +1.00000 q^{4} +0.676703 q^{5} -2.80104 q^{6} +3.76033 q^{7} +1.00000 q^{8} +4.84585 q^{9} +0.676703 q^{10} -0.631840 q^{11} -2.80104 q^{12} -6.22087 q^{13} +3.76033 q^{14} -1.89548 q^{15} +1.00000 q^{16} +3.39684 q^{17} +4.84585 q^{18} +1.27370 q^{19} +0.676703 q^{20} -10.5329 q^{21} -0.631840 q^{22} +3.84216 q^{23} -2.80104 q^{24} -4.54207 q^{25} -6.22087 q^{26} -5.17032 q^{27} +3.76033 q^{28} +4.19303 q^{29} -1.89548 q^{30} -9.96120 q^{31} +1.00000 q^{32} +1.76981 q^{33} +3.39684 q^{34} +2.54463 q^{35} +4.84585 q^{36} -9.06735 q^{37} +1.27370 q^{38} +17.4249 q^{39} +0.676703 q^{40} +0.532380 q^{41} -10.5329 q^{42} -7.96327 q^{43} -0.631840 q^{44} +3.27920 q^{45} +3.84216 q^{46} -3.81367 q^{47} -2.80104 q^{48} +7.14011 q^{49} -4.54207 q^{50} -9.51470 q^{51} -6.22087 q^{52} +3.38497 q^{53} -5.17032 q^{54} -0.427568 q^{55} +3.76033 q^{56} -3.56770 q^{57} +4.19303 q^{58} -6.10929 q^{59} -1.89548 q^{60} -2.32597 q^{61} -9.96120 q^{62} +18.2220 q^{63} +1.00000 q^{64} -4.20968 q^{65} +1.76981 q^{66} +3.52438 q^{67} +3.39684 q^{68} -10.7621 q^{69} +2.54463 q^{70} -6.46908 q^{71} +4.84585 q^{72} +1.69845 q^{73} -9.06735 q^{74} +12.7226 q^{75} +1.27370 q^{76} -2.37593 q^{77} +17.4249 q^{78} -8.68079 q^{79} +0.676703 q^{80} -0.0552674 q^{81} +0.532380 q^{82} -12.7963 q^{83} -10.5329 q^{84} +2.29865 q^{85} -7.96327 q^{86} -11.7449 q^{87} -0.631840 q^{88} +4.85184 q^{89} +3.27920 q^{90} -23.3926 q^{91} +3.84216 q^{92} +27.9018 q^{93} -3.81367 q^{94} +0.861920 q^{95} -2.80104 q^{96} +2.77992 q^{97} +7.14011 q^{98} -3.06180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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