LMFDB - Embedded newform 8047.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$156$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42132 q^{2} -1.26535 q^{3} +3.86281 q^{4} +0.549340 q^{5} +3.06382 q^{6} +1.75655 q^{7} -4.51045 q^{8} -1.39889 q^{9} +O(q^{10})$	\(q-2.42132 q^{2} -1.26535 q^{3} +3.86281 q^{4} +0.549340 q^{5} +3.06382 q^{6} +1.75655 q^{7} -4.51045 q^{8} -1.39889 q^{9} -1.33013 q^{10} -3.52910 q^{11} -4.88781 q^{12} -1.00000 q^{13} -4.25318 q^{14} -0.695109 q^{15} +3.19566 q^{16} -2.99475 q^{17} +3.38715 q^{18} -4.75227 q^{19} +2.12200 q^{20} -2.22266 q^{21} +8.54509 q^{22} -1.18010 q^{23} +5.70731 q^{24} -4.69823 q^{25} +2.42132 q^{26} +5.56614 q^{27} +6.78522 q^{28} +4.51235 q^{29} +1.68308 q^{30} +5.82642 q^{31} +1.28319 q^{32} +4.46555 q^{33} +7.25127 q^{34} +0.964945 q^{35} -5.40362 q^{36} -9.40626 q^{37} +11.5068 q^{38} +1.26535 q^{39} -2.47777 q^{40} -4.36940 q^{41} +5.38177 q^{42} -7.04977 q^{43} -13.6322 q^{44} -0.768464 q^{45} +2.85741 q^{46} -3.38871 q^{47} -4.04363 q^{48} -3.91453 q^{49} +11.3759 q^{50} +3.78942 q^{51} -3.86281 q^{52} -2.11399 q^{53} -13.4774 q^{54} -1.93868 q^{55} -7.92285 q^{56} +6.01329 q^{57} -10.9259 q^{58} -4.73356 q^{59} -2.68507 q^{60} -13.7870 q^{61} -14.1076 q^{62} -2.45721 q^{63} -9.49834 q^{64} -0.549340 q^{65} -10.8125 q^{66} -0.196773 q^{67} -11.5682 q^{68} +1.49325 q^{69} -2.33644 q^{70} +5.05804 q^{71} +6.30961 q^{72} -4.20137 q^{73} +22.7756 q^{74} +5.94491 q^{75} -18.3571 q^{76} -6.19904 q^{77} -3.06382 q^{78} +2.54414 q^{79} +1.75550 q^{80} -2.84646 q^{81} +10.5797 q^{82} -14.8909 q^{83} -8.58568 q^{84} -1.64514 q^{85} +17.0698 q^{86} -5.70971 q^{87} +15.9178 q^{88} +14.2220 q^{89} +1.86070 q^{90} -1.75655 q^{91} -4.55851 q^{92} -7.37246 q^{93} +8.20516 q^{94} -2.61061 q^{95} -1.62369 q^{96} -1.66006 q^{97} +9.47834 q^{98} +4.93680 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * 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 $

$2$	−2.42132	−1.71213	−0.856067	−	0.516865i	$-0.827099\pi$
$2$	−2.42132	−1.71213	−0.856067	+	0.516865i	$0.827099\pi$
$3$	−1.26535	−0.730551	−0.365276	−	0.930899i	$-0.619025\pi$
$3$	−1.26535	−0.730551	−0.365276	+	0.930899i	$0.619025\pi$
$4$	3.86281	1.93140
$5$	0.549340	0.245672	0.122836	−	0.992427i	$-0.460801\pi$
$5$	0.549340	0.245672	0.122836	+	0.992427i	$0.460801\pi$
$6$	3.06382	1.25080
$7$	1.75655	0.663914	0.331957	−	0.943295i	$-0.392291\pi$
$7$	1.75655	0.663914	0.331957	+	0.943295i	$0.392291\pi$
$8$	−4.51045	−1.59469
$9$	−1.39889	−0.466295
$10$	−1.33013	−0.420624
$11$	−3.52910	−1.06406	−0.532031	−	0.846725i	$-0.678571\pi$
$11$	−3.52910	−1.06406	−0.532031	+	0.846725i	$0.678571\pi$
$12$	−4.88781	−1.41099
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.25318	−1.13671
$15$	−0.695109	−0.179476
$16$	3.19566	0.798914
$17$	−2.99475	−0.726335	−0.363167	−	0.931724i	$-0.618305\pi$
$17$	−2.99475	−0.726335	−0.363167	+	0.931724i	$0.618305\pi$
$18$	3.38715	0.798360
$19$	−4.75227	−1.09025	−0.545123	−	0.838356i	$-0.683517\pi$
$19$	−4.75227	−1.09025	−0.545123	+	0.838356i	$0.683517\pi$
$20$	2.12200	0.474493
$21$	−2.22266	−0.485023
$22$	8.54509	1.82182
$23$	−1.18010	−0.246069	−0.123034	−	0.992402i	$-0.539263\pi$
$23$	−1.18010	−0.246069	−0.123034	+	0.992402i	$0.539263\pi$
$24$	5.70731	1.16500
$25$	−4.69823	−0.939645
$26$	2.42132	0.474861
$27$	5.56614	1.07120
$28$	6.78522	1.28229
$29$	4.51235	0.837923	0.418961	−	0.908004i	$-0.362394\pi$
$29$	4.51235	0.837923	0.418961	+	0.908004i	$0.362394\pi$
$30$	1.68308	0.307287
$31$	5.82642	1.04646	0.523228	−	0.852193i	$-0.324728\pi$
$31$	5.82642	1.04646	0.523228	+	0.852193i	$0.324728\pi$
$32$	1.28319	0.226839
$33$	4.46555	0.777352
$34$	7.25127	1.24358
$35$	0.964945	0.163105
$36$	−5.40362	−0.900604
$37$	−9.40626	−1.54638	−0.773190	−	0.634175i	$-0.781340\pi$
$37$	−9.40626	−1.54638	−0.773190	+	0.634175i	$0.781340\pi$
$38$	11.5068	1.86665
$39$	1.26535	0.202618
$40$	−2.47777	−0.391771
$41$	−4.36940	−0.682386	−0.341193	−	0.939993i	$-0.610831\pi$
$41$	−4.36940	−0.682386	−0.341193	+	0.939993i	$0.610831\pi$
$42$	5.38177	0.830425
$43$	−7.04977	−1.07508	−0.537540	−	0.843238i	$-0.680646\pi$
$43$	−7.04977	−1.07508	−0.537540	+	0.843238i	$0.680646\pi$
$44$	−13.6322	−2.05513
$45$	−0.768464	−0.114556
$46$	2.85741	0.421302
$47$	−3.38871	−0.494294	−0.247147	−	0.968978i	$-0.579493\pi$
$47$	−3.38871	−0.494294	−0.247147	+	0.968978i	$0.579493\pi$
$48$	−4.04363	−0.583648
$49$	−3.91453	−0.559218
$50$	11.3759	1.60880
$51$	3.78942	0.530625
$52$	−3.86281	−0.535675
$53$	−2.11399	−0.290379	−0.145190	−	0.989404i	$-0.546379\pi$
$53$	−2.11399	−0.290379	−0.145190	+	0.989404i	$0.546379\pi$
$54$	−13.4774	−1.83404
$55$	−1.93868	−0.261411
$56$	−7.92285	−1.05873
$57$	6.01329	0.796480
$58$	−10.9259	−1.43464
$59$	−4.73356	−0.616257	−0.308128	−	0.951345i	$-0.599703\pi$
$59$	−4.73356	−0.616257	−0.308128	+	0.951345i	$0.599703\pi$
$60$	−2.68507	−0.346641
$61$	−13.7870	−1.76525	−0.882625	−	0.470079i	$-0.844226\pi$
$61$	−13.7870	−1.76525	−0.882625	+	0.470079i	$0.844226\pi$
$62$	−14.1076	−1.79167
$63$	−2.45721	−0.309580
$64$	−9.49834	−1.18729
$65$	−0.549340	−0.0681373
$66$	−10.8125	−1.33093
$67$	−0.196773	−0.0240396	−0.0120198	−	0.999928i	$-0.503826\pi$
$67$	−0.196773	−0.0240396	−0.0120198	+	0.999928i	$0.503826\pi$
$68$	−11.5682	−1.40284
$69$	1.49325	0.179766
$70$	−2.33644	−0.279258
$71$	5.05804	0.600278	0.300139	−	0.953895i	$-0.402967\pi$
$71$	5.05804	0.600278	0.300139	+	0.953895i	$0.402967\pi$
$72$	6.30961	0.743595
$73$	−4.20137	−0.491733	−0.245867	−	0.969304i	$-0.579072\pi$
$73$	−4.20137	−0.491733	−0.245867	+	0.969304i	$0.579072\pi$
$74$	22.7756	2.64761
$75$	5.94491	0.686459
$76$	−18.3571	−2.10570
$77$	−6.19904	−0.706446
$78$	−3.06382	−0.346910
$79$	2.54414	0.286238	0.143119	−	0.989705i	$-0.454287\pi$
$79$	2.54414	0.286238	0.143119	+	0.989705i	$0.454287\pi$
$80$	1.75550	0.196271
$81$	−2.84646	−0.316274
$82$	10.5797	1.16834
$83$	−14.8909	−1.63449	−0.817246	−	0.576288i	$-0.804500\pi$
$83$	−14.8909	−1.63449	−0.817246	+	0.576288i	$0.804500\pi$
$84$	−8.58568	−0.936775
$85$	−1.64514	−0.178440
$86$	17.0698	1.84068
$87$	−5.70971	−0.612145
$88$	15.9178	1.69685
$89$	14.2220	1.50752	0.753762	−	0.657147i	$-0.228237\pi$
$89$	14.2220	1.50752	0.753762	+	0.657147i	$0.228237\pi$
$90$	1.86070	0.196135
$91$	−1.75655	−0.184137
$92$	−4.55851	−0.475258
$93$	−7.37246	−0.764489
$94$	8.20516	0.846298
$95$	−2.61061	−0.267843
$96$	−1.62369	−0.165717
$97$	−1.66006	−0.168553	−0.0842766	−	0.996442i	$-0.526858\pi$
$97$	−1.66006	−0.168553	−0.0842766	+	0.996442i	$0.526858\pi$
$98$	9.47834	0.957456
$99$	4.93680	0.496167
$100$	−18.1483	−1.81483
$101$	−3.39778	−0.338092	−0.169046	−	0.985608i	$-0.554069\pi$
$101$	−3.39778	−0.338092	−0.169046	+	0.985608i	$0.554069\pi$
$102$	−9.17540	−0.908500
$103$	16.2164	1.59785	0.798927	−	0.601429i	$-0.205402\pi$
$103$	16.2164	1.59785	0.798927	+	0.601429i	$0.205402\pi$
$104$	4.51045	0.442286
$105$	−1.22099	−0.119157
$106$	5.11866	0.497168
$107$	11.6293	1.12425	0.562126	−	0.827052i	$-0.309984\pi$
$107$	11.6293	1.12425	0.562126	+	0.827052i	$0.309984\pi$
$108$	21.5009	2.06893
$109$	−7.97066	−0.763451	−0.381725	−	0.924276i	$-0.624670\pi$
$109$	−7.97066	−0.763451	−0.381725	+	0.924276i	$0.624670\pi$
$110$	4.69416	0.447571
$111$	11.9022	1.12971
$112$	5.61333	0.530410
$113$	19.2689	1.81267	0.906334	−	0.422563i	$-0.138869\pi$
$113$	19.2689	1.81267	0.906334	+	0.422563i	$0.138869\pi$
$114$	−14.5601	−1.36368
$115$	−0.648278	−0.0604523
$116$	17.4303	1.61837
$117$	1.39889	0.129327
$118$	11.4615	1.05511
$119$	−5.26044	−0.482224
$120$	3.13526	0.286208
$121$	1.45453	0.132230
$122$	33.3829	3.02234
$123$	5.52883	0.498518
$124$	22.5063	2.02113
$125$	−5.32763	−0.476517
$126$	5.94971	0.530042
$127$	−15.3946	−1.36605	−0.683026	−	0.730394i	$-0.739336\pi$
$127$	−15.3946	−1.36605	−0.683026	+	0.730394i	$0.739336\pi$
$128$	20.4322	1.80596
$129$	8.92043	0.785400
$130$	1.33013	0.116660
$131$	12.1244	1.05932	0.529658	−	0.848211i	$-0.322320\pi$
$131$	12.1244	1.05932	0.529658	+	0.848211i	$0.322320\pi$
$132$	17.2495	1.50138
$133$	−8.34760	−0.723829
$134$	0.476450	0.0411590
$135$	3.05770	0.263165
$136$	13.5077	1.15828
$137$	0.657101	0.0561399	0.0280700	−	0.999606i	$-0.491064\pi$
$137$	0.657101	0.0561399	0.0280700	+	0.999606i	$0.491064\pi$
$138$	−3.61563	−0.307783
$139$	−7.29948	−0.619134	−0.309567	−	0.950878i	$-0.600184\pi$
$139$	−7.29948	−0.619134	−0.309567	+	0.950878i	$0.600184\pi$
$140$	3.72739	0.315022
$141$	4.28791	0.361107
$142$	−12.2471	−1.02776
$143$	3.52910	0.295118
$144$	−4.47036	−0.372530
$145$	2.47882	0.205855
$146$	10.1729	0.841913
$147$	4.95325	0.408537
$148$	−36.3346	−2.98668
$149$	2.25678	0.184883	0.0924414	−	0.995718i	$-0.470533\pi$
$149$	2.25678	0.184883	0.0924414	+	0.995718i	$0.470533\pi$
$150$	−14.3945	−1.17531
$151$	−10.0637	−0.818972	−0.409486	−	0.912316i	$-0.634292\pi$
$151$	−10.0637	−0.818972	−0.409486	+	0.912316i	$0.634292\pi$
$152$	21.4349	1.73860
$153$	4.18932	0.338686
$154$	15.0099	1.20953
$155$	3.20069	0.257085
$156$	4.88781	0.391338
$157$	−10.8141	−0.863062	−0.431531	−	0.902098i	$-0.642026\pi$
$157$	−10.8141	−0.863062	−0.431531	+	0.902098i	$0.642026\pi$
$158$	−6.16019	−0.490078
$159$	2.67495	0.212137
$160$	0.704909	0.0557280
$161$	−2.07291	−0.163368
$162$	6.89221	0.541503
$163$	1.96451	0.153872	0.0769361	−	0.997036i	$-0.475486\pi$
$163$	1.96451	0.153872	0.0769361	+	0.997036i	$0.475486\pi$
$164$	−16.8782	−1.31796
$165$	2.45311	0.190974
$166$	36.0558	2.79847
$167$	16.7953	1.29966	0.649831	−	0.760078i	$-0.274840\pi$
$167$	16.7953	1.29966	0.649831	+	0.760078i	$0.274840\pi$
$168$	10.0252	0.773460
$169$	1.00000	0.0769231
$170$	3.98341	0.305514
$171$	6.64788	0.508376
$172$	−27.2319	−2.07641
$173$	−23.3542	−1.77559	−0.887794	−	0.460242i	$-0.847763\pi$
$173$	−23.3542	−1.77559	−0.887794	+	0.460242i	$0.847763\pi$
$174$	13.8251	1.04808
$175$	−8.25267	−0.623844
$176$	−11.2778	−0.850095
$177$	5.98962	0.450207
$178$	−34.4360	−2.58108
$179$	−22.4549	−1.67836	−0.839179	−	0.543856i	$-0.816964\pi$
$179$	−22.4549	−1.67836	−0.839179	+	0.543856i	$0.816964\pi$
$180$	−2.96843	−0.221254
$181$	16.1355	1.19934	0.599672	−	0.800246i	$-0.295298\pi$
$181$	16.1355	1.19934	0.599672	+	0.800246i	$0.295298\pi$
$182$	4.25318	0.315267
$183$	17.4454	1.28960
$184$	5.32280	0.392402
$185$	−5.16724	−0.379903
$186$	17.8511	1.30891
$187$	10.5688	0.772866
$188$	−13.0899	−0.954681
$189$	9.77720	0.711187
$190$	6.32114	0.458583
$191$	9.04267	0.654305	0.327152	−	0.944972i	$-0.393911\pi$
$191$	9.04267	0.654305	0.327152	+	0.944972i	$0.393911\pi$
$192$	12.0187	0.867377
$193$	0.434416	0.0312699	0.0156350	−	0.999878i	$-0.495023\pi$
$193$	0.434416	0.0312699	0.0156350	+	0.999878i	$0.495023\pi$
$194$	4.01953	0.288586
$195$	0.695109	0.0497778
$196$	−15.1211	−1.08008
$197$	−5.24261	−0.373521	−0.186760	−	0.982405i	$-0.559799\pi$
$197$	−5.24261	−0.373521	−0.186760	+	0.982405i	$0.559799\pi$
$198$	−11.9536	−0.849505
$199$	11.7332	0.831746	0.415873	−	0.909423i	$-0.363476\pi$
$199$	11.7332	0.831746	0.415873	+	0.909423i	$0.363476\pi$
$200$	21.1911	1.49844
$201$	0.248987	0.0175622
$202$	8.22713	0.578859
$203$	7.92618	0.556309
$204$	14.6378	1.02485
$205$	−2.40029	−0.167643
$206$	−39.2652	−2.73574
$207$	1.65083	0.114741
$208$	−3.19566	−0.221579
$209$	16.7712	1.16009
$210$	2.95642	0.204012
$211$	−12.2988	−0.846681	−0.423340	−	0.905971i	$-0.639143\pi$
$211$	−12.2988	−0.846681	−0.423340	+	0.905971i	$0.639143\pi$
$212$	−8.16595	−0.560840
$213$	−6.40019	−0.438534
$214$	−28.1584	−1.92487
$215$	−3.87272	−0.264117
$216$	−25.1058	−1.70823
$217$	10.2344	0.694756
$218$	19.2995	1.30713
$219$	5.31621	0.359236
$220$	−7.48873	−0.504890
$221$	2.99475	0.201449
$222$	−28.8191	−1.93421
$223$	14.1754	0.949258	0.474629	−	0.880186i	$-0.342582\pi$
$223$	14.1754	0.949258	0.474629	+	0.880186i	$0.342582\pi$
$224$	2.25399	0.150601
$225$	6.57228	0.438152
$226$	−46.6563	−3.10353
$227$	−6.76237	−0.448834	−0.224417	−	0.974493i	$-0.572048\pi$
$227$	−6.76237	−0.448834	−0.224417	+	0.974493i	$0.572048\pi$
$228$	23.2282	1.53832
$229$	18.7095	1.23636	0.618179	−	0.786037i	$-0.287871\pi$
$229$	18.7095	1.23636	0.618179	+	0.786037i	$0.287871\pi$
$230$	1.56969	0.103502
$231$	7.84397	0.516095
$232$	−20.3528	−1.33622
$233$	24.9926	1.63732	0.818659	−	0.574280i	$-0.194718\pi$
$233$	24.9926	1.63732	0.818659	+	0.574280i	$0.194718\pi$
$234$	−3.38715	−0.221425
$235$	−1.86156	−0.121435
$236$	−18.2848	−1.19024
$237$	−3.21923	−0.209112
$238$	12.7372	0.825632
$239$	−17.2530	−1.11601	−0.558003	−	0.829839i	$-0.688432\pi$
$239$	−17.2530	−1.11601	−0.558003	+	0.829839i	$0.688432\pi$
$240$	−2.22133	−0.143386
$241$	−20.9034	−1.34650	−0.673252	−	0.739413i	$-0.735103\pi$
$241$	−20.9034	−1.34650	−0.673252	+	0.739413i	$0.735103\pi$
$242$	−3.52189	−0.226395
$243$	−13.0966	−0.840149
$244$	−53.2566	−3.40941
$245$	−2.15041	−0.137385
$246$	−13.3871	−0.853529
$247$	4.75227	0.302380
$248$	−26.2798	−1.66877
$249$	18.8423	1.19408
$250$	12.8999	0.815862
$251$	10.6113	0.669777	0.334888	−	0.942258i	$-0.391301\pi$
$251$	10.6113	0.669777	0.334888	+	0.942258i	$0.391301\pi$
$252$	−9.49174	−0.597923
$253$	4.16470	0.261832
$254$	37.2754	2.33886
$255$	2.08168	0.130360
$256$	−30.4762	−1.90476
$257$	−18.3144	−1.14242	−0.571209	−	0.820805i	$-0.693525\pi$
$257$	−18.3144	−1.14242	−0.571209	+	0.820805i	$0.693525\pi$
$258$	−21.5993	−1.34471
$259$	−16.5226	−1.02666
$260$	−2.12200	−0.131601
$261$	−6.31226	−0.390719
$262$	−29.3571	−1.81369
$263$	−7.72346	−0.476249	−0.238124	−	0.971235i	$-0.576533\pi$
$263$	−7.72346	−0.476249	−0.238124	+	0.971235i	$0.576533\pi$
$264$	−20.1417	−1.23963
$265$	−1.16130	−0.0713382
$266$	20.2122	1.23929
$267$	−17.9958	−1.10132
$268$	−0.760095	−0.0464302
$269$	26.7359	1.63012	0.815059	−	0.579378i	$-0.196705\pi$
$269$	26.7359	1.63012	0.815059	+	0.579378i	$0.196705\pi$
$270$	−7.40369	−0.450574
$271$	−7.96967	−0.484123	−0.242061	−	0.970261i	$-0.577824\pi$
$271$	−7.96967	−0.484123	−0.242061	+	0.970261i	$0.577824\pi$
$272$	−9.57021	−0.580279
$273$	2.22266	0.134521
$274$	−1.59105	−0.0961190
$275$	16.5805	0.999841
$276$	5.76812	0.347200
$277$	7.27869	0.437334	0.218667	−	0.975799i	$-0.429829\pi$
$277$	7.27869	0.437334	0.218667	+	0.975799i	$0.429829\pi$
$278$	17.6744	1.06004
$279$	−8.15049	−0.487957
$280$	−4.35234	−0.260102
$281$	17.1472	1.02291	0.511457	−	0.859309i	$-0.329106\pi$
$281$	17.1472	1.02291	0.511457	+	0.859309i	$0.329106\pi$
$282$	−10.3824	−0.618264
$283$	−25.0330	−1.48806	−0.744029	−	0.668147i	$-0.767088\pi$
$283$	−25.0330	−1.48806	−0.744029	+	0.668147i	$0.767088\pi$
$284$	19.5382	1.15938
$285$	3.30334	0.195673
$286$	−8.54509	−0.505281
$287$	−7.67508	−0.453046
$288$	−1.79504	−0.105774
$289$	−8.03144	−0.472438
$290$	−6.00202	−0.352451
$291$	2.10056	0.123137
$292$	−16.2291	−0.949735
$293$	2.10747	0.123120	0.0615598	−	0.998103i	$-0.480392\pi$
$293$	2.10747	0.123120	0.0615598	+	0.998103i	$0.480392\pi$
$294$	−11.9934	−0.699471
$295$	−2.60033	−0.151397
$296$	42.4265	2.46599
$297$	−19.6434	−1.13983
$298$	−5.46440	−0.316544
$299$	1.18010	0.0682471
$300$	22.9640	1.32583
$301$	−12.3833	−0.713760
$302$	24.3674	1.40219
$303$	4.29939	0.246993
$304$	−15.1866	−0.871012
$305$	−7.57378	−0.433673
$306$	−10.1437	−0.579876
$307$	29.9788	1.71098	0.855490	−	0.517820i	$-0.173256\pi$
$307$	29.9788	1.71098	0.855490	+	0.517820i	$0.173256\pi$
$308$	−23.9457	−1.36443
$309$	−20.5195	−1.16731
$310$	−7.74989	−0.440164
$311$	7.22846	0.409888	0.204944	−	0.978774i	$-0.434299\pi$
$311$	7.22846	0.409888	0.204944	+	0.978774i	$0.434299\pi$
$312$	−5.70731	−0.323113
$313$	2.67575	0.151242	0.0756212	−	0.997137i	$-0.475906\pi$
$313$	2.67575	0.151242	0.0756212	+	0.997137i	$0.475906\pi$
$314$	26.1845	1.47768
$315$	−1.34985	−0.0760553
$316$	9.82752	0.552841
$317$	13.4402	0.754878	0.377439	−	0.926034i	$-0.376805\pi$
$317$	13.4402	0.754878	0.377439	+	0.926034i	$0.376805\pi$
$318$	−6.47691	−0.363207
$319$	−15.9245	−0.891603
$320$	−5.21782	−0.291685
$321$	−14.7152	−0.821323
$322$	5.01919	0.279709
$323$	14.2319	0.791883
$324$	−10.9953	−0.610852
$325$	4.69823	0.260611
$326$	−4.75671	−0.263450
$327$	10.0857	0.557740
$328$	19.7080	1.08819
$329$	−5.95244	−0.328169
$330$	−5.93976	−0.326973
$331$	10.3235	0.567432	0.283716	−	0.958908i	$-0.408433\pi$
$331$	10.3235	0.567432	0.283716	+	0.958908i	$0.408433\pi$
$332$	−57.5208	−3.15686
$333$	13.1583	0.721069
$334$	−40.6670	−2.22520
$335$	−0.108095	−0.00590587
$336$	−7.10284	−0.387492
$337$	34.7926	1.89527	0.947637	−	0.319351i	$-0.103465\pi$
$337$	34.7926	1.89527	0.947637	+	0.319351i	$0.103465\pi$
$338$	−2.42132	−0.131703
$339$	−24.3820	−1.32425
$340$	−6.35485	−0.344640
$341$	−20.5620	−1.11349
$342$	−16.0967	−0.870408
$343$	−19.1719	−1.03519
$344$	31.7977	1.71441
$345$	0.820300	0.0441635
$346$	56.5481	3.04004
$347$	10.3450	0.555349	0.277675	−	0.960675i	$-0.410436\pi$
$347$	10.3450	0.555349	0.277675	+	0.960675i	$0.410436\pi$
$348$	−22.0555	−1.18230
$349$	−6.80597	−0.364315	−0.182158	−	0.983269i	$-0.558308\pi$
$349$	−6.80597	−0.364315	−0.182158	+	0.983269i	$0.558308\pi$
$350$	19.9824	1.06810
$351$	−5.56614	−0.297098
$352$	−4.52851	−0.241370
$353$	−16.6702	−0.887264	−0.443632	−	0.896209i	$-0.646310\pi$
$353$	−16.6702	−0.887264	−0.443632	+	0.896209i	$0.646310\pi$
$354$	−14.5028	−0.770815
$355$	2.77858	0.147472
$356$	54.9367	2.91164
$357$	6.65631	0.352289
$358$	54.3705	2.87357
$359$	0.159022	0.00839284	0.00419642	−	0.999991i	$-0.498664\pi$
$359$	0.159022	0.00839284	0.00419642	+	0.999991i	$0.498664\pi$
$360$	3.46612	0.182681
$361$	3.58405	0.188634
$362$	−39.0693	−2.05344
$363$	−1.84049	−0.0966007
$364$	−6.78522	−0.355642
$365$	−2.30798	−0.120805
$366$	−42.2411	−2.20798
$367$	−35.9653	−1.87737	−0.938687	−	0.344770i	$-0.887957\pi$
$367$	−35.9653	−1.87737	−0.938687	+	0.344770i	$0.887957\pi$
$368$	−3.77120	−0.196588
$369$	6.11229	0.318193
$370$	12.5116	0.650445
$371$	−3.71334	−0.192787
$372$	−28.4784	−1.47654
$373$	−3.08857	−0.159920	−0.0799600	−	0.996798i	$-0.525479\pi$
$373$	−3.08857	−0.159920	−0.0799600	+	0.996798i	$0.525479\pi$
$374$	−25.5904	−1.32325
$375$	6.74132	0.348120
$376$	15.2846	0.788244
$377$	−4.51235	−0.232398
$378$	−23.6738	−1.21765
$379$	0.113628	0.00583669	0.00291834	−	0.999996i	$-0.499071\pi$
$379$	0.113628	0.00583669	0.00291834	+	0.999996i	$0.499071\pi$
$380$	−10.0843	−0.517313
$381$	19.4796	0.997971
$382$	−21.8952	−1.12026
$383$	12.3769	0.632428	0.316214	−	0.948688i	$-0.397588\pi$
$383$	12.3769	0.632428	0.316214	+	0.948688i	$0.397588\pi$
$384$	−25.8539	−1.31935
$385$	−3.40538	−0.173554
$386$	−1.05186	−0.0535383
$387$	9.86182	0.501304
$388$	−6.41248	−0.325544
$389$	−32.0766	−1.62635	−0.813175	−	0.582020i	$-0.802263\pi$
$389$	−32.0766	−1.62635	−0.813175	+	0.582020i	$0.802263\pi$
$390$	−1.68308	−0.0852262
$391$	3.53412	0.178728
$392$	17.6563	0.891778
$393$	−15.3417	−0.773884
$394$	12.6941	0.639518
$395$	1.39760	0.0703209
$396$	19.0699	0.958299
$397$	14.1660	0.710970	0.355485	−	0.934682i	$-0.384316\pi$
$397$	14.1660	0.710970	0.355485	+	0.934682i	$0.384316\pi$
$398$	−28.4099	−1.42406
$399$	10.5627	0.528794
$400$	−15.0139	−0.750696
$401$	−36.9466	−1.84503	−0.922514	−	0.385965i	$-0.873869\pi$
$401$	−36.9466	−1.84503	−0.922514	+	0.385965i	$0.873869\pi$
$402$	−0.602877	−0.0300688
$403$	−5.82642	−0.290234
$404$	−13.1250	−0.652992
$405$	−1.56368	−0.0776997
$406$	−19.1918	−0.952475
$407$	33.1956	1.64545
$408$	−17.0920	−0.846180
$409$	23.2609	1.15018	0.575089	−	0.818091i	$-0.304967\pi$
$409$	23.2609	1.15018	0.575089	+	0.818091i	$0.304967\pi$
$410$	5.81187	0.287028
$411$	−0.831463	−0.0410131
$412$	62.6410	3.08610
$413$	−8.31474	−0.409142
$414$	−3.99719	−0.196451
$415$	−8.18019	−0.401550
$416$	−1.28319	−0.0629137
$417$	9.23641	0.452309
$418$	−40.6085	−1.98623
$419$	−17.1812	−0.839357	−0.419678	−	0.907673i	$-0.637857\pi$
$419$	−17.1812	−0.839357	−0.419678	+	0.907673i	$0.637857\pi$
$420$	−4.71646	−0.230140
$421$	−18.5415	−0.903660	−0.451830	−	0.892104i	$-0.649229\pi$
$421$	−18.5415	−0.903660	−0.451830	+	0.892104i	$0.649229\pi$
$422$	29.7792	1.44963
$423$	4.74042	0.230487
$424$	9.53507	0.463064
$425$	14.0700	0.682497
$426$	15.4969	0.750829
$427$	−24.2176	−1.17197
$428$	44.9219	2.17138
$429$	−4.46555	−0.215599
$430$	9.37711	0.452204
$431$	18.9822	0.914339	0.457169	−	0.889380i	$-0.348863\pi$
$431$	18.9822	0.914339	0.457169	+	0.889380i	$0.348863\pi$
$432$	17.7875	0.855800
$433$	−7.12599	−0.342453	−0.171227	−	0.985232i	$-0.554773\pi$
$433$	−7.12599	−0.342453	−0.171227	+	0.985232i	$0.554773\pi$
$434$	−24.7808	−1.18952
$435$	−3.13658	−0.150387
$436$	−30.7891	−1.47453
$437$	5.60817	0.268275
$438$	−12.8723	−0.615060
$439$	−34.1562	−1.63019	−0.815094	−	0.579329i	$-0.803315\pi$
$439$	−34.1562	−1.63019	−0.815094	+	0.579329i	$0.803315\pi$
$440$	8.74431	0.416869
$441$	5.47598	0.260761
$442$	−7.25127	−0.344908
$443$	−25.0104	−1.18828	−0.594141	−	0.804361i	$-0.702508\pi$
$443$	−25.0104	−1.18828	−0.594141	+	0.804361i	$0.702508\pi$
$444$	45.9760	2.18192
$445$	7.81270	0.370357
$446$	−34.3233	−1.62526
$447$	−2.85562	−0.135066
$448$	−16.6843	−0.788260
$449$	14.1031	0.665566	0.332783	−	0.943004i	$-0.392012\pi$
$449$	14.1031	0.665566	0.332783	+	0.943004i	$0.392012\pi$
$450$	−15.9136	−0.750175
$451$	15.4200	0.726102
$452$	74.4321	3.50099
$453$	12.7341	0.598301
$454$	16.3739	0.768465
$455$	−0.964945	−0.0452373
$456$	−27.1227	−1.27014
$457$	36.5744	1.71088	0.855440	−	0.517903i	$-0.173287\pi$
$457$	36.5744	1.71088	0.855440	+	0.517903i	$0.173287\pi$
$458$	−45.3017	−2.11681
$459$	−16.6692	−0.778052
$460$	−2.50417	−0.116758
$461$	−27.6783	−1.28911	−0.644555	−	0.764558i	$-0.722957\pi$
$461$	−27.6783	−1.28911	−0.644555	+	0.764558i	$0.722957\pi$
$462$	−18.9928	−0.883624
$463$	31.2093	1.45042	0.725209	−	0.688529i	$-0.241743\pi$
$463$	31.2093	1.45042	0.725209	+	0.688529i	$0.241743\pi$
$464$	14.4199	0.669428
$465$	−4.04999	−0.187814
$466$	−60.5151	−2.80331
$467$	−1.82487	−0.0844448	−0.0422224	−	0.999108i	$-0.513444\pi$
$467$	−1.82487	−0.0844448	−0.0422224	+	0.999108i	$0.513444\pi$
$468$	5.40362	0.249783
$469$	−0.345641	−0.0159602
$470$	4.50743	0.207912
$471$	13.6837	0.630511
$472$	21.3505	0.982736
$473$	24.8793	1.14395
$474$	7.79480	0.358027
$475$	22.3272	1.02444
$476$	−20.3201	−0.931368
$477$	2.95724	0.135402
$478$	41.7752	1.91075
$479$	−22.4141	−1.02413	−0.512064	−	0.858947i	$-0.671119\pi$
$479$	−22.4141	−1.02413	−0.512064	+	0.858947i	$0.671119\pi$
$480$	−0.891958	−0.0407121
$481$	9.40626	0.428889
$482$	50.6138	2.30539
$483$	2.62296	0.119349
$484$	5.61856	0.255389
$485$	−0.911936	−0.0414089
$486$	31.7112	1.43845
$487$	7.83101	0.354857	0.177428	−	0.984134i	$-0.443222\pi$
$487$	7.83101	0.354857	0.177428	+	0.984134i	$0.443222\pi$
$488$	62.1858	2.81502
$489$	−2.48580	−0.112412
$490$	5.20683	0.235221
$491$	13.0598	0.589379	0.294690	−	0.955593i	$-0.404784\pi$
$491$	13.0598	0.589379	0.294690	+	0.955593i	$0.404784\pi$
$492$	21.3568	0.962839
$493$	−13.5134	−0.608612
$494$	−11.5068	−0.517714
$495$	2.71199	0.121895
$496$	18.6192	0.836028
$497$	8.88470	0.398533
$498$	−45.6232	−2.04443
$499$	−41.3816	−1.85249	−0.926246	−	0.376919i	$-0.876984\pi$
$499$	−41.3816	−1.85249	−0.926246	+	0.376919i	$0.876984\pi$
$500$	−20.5796	−0.920347
$501$	−21.2520	−0.949470
$502$	−25.6933	−1.14675
$503$	33.5506	1.49595	0.747974	−	0.663728i	$-0.231027\pi$
$503$	33.5506	1.49595	0.747974	+	0.663728i	$0.231027\pi$
$504$	11.0832	0.493683
$505$	−1.86654	−0.0830599
$506$	−10.0841	−0.448292
$507$	−1.26535	−0.0561962
$508$	−59.4664	−2.63840
$509$	8.64628	0.383240	0.191620	−	0.981469i	$-0.438626\pi$
$509$	8.64628	0.383240	0.191620	+	0.981469i	$0.438626\pi$
$510$	−5.04042	−0.223194
$511$	−7.37992	−0.326468
$512$	32.9284	1.45524
$513$	−26.4518	−1.16787
$514$	44.3450	1.95597
$515$	8.90834	0.392549
$516$	34.4579	1.51692
$517$	11.9591	0.525960
$518$	40.0065	1.75778
$519$	29.5513	1.29716
$520$	2.47777	0.108658
$521$	17.9573	0.786722	0.393361	−	0.919384i	$-0.371312\pi$
$521$	17.9573	0.786722	0.393361	+	0.919384i	$0.371312\pi$
$522$	15.2840	0.668964
$523$	9.60629	0.420054	0.210027	−	0.977696i	$-0.432645\pi$
$523$	9.60629	0.420054	0.210027	+	0.977696i	$0.432645\pi$
$524$	46.8343	2.04597
$525$	10.4425	0.455750
$526$	18.7010	0.815402
$527$	−17.4487	−0.760077
$528$	14.2704	0.621038
$529$	−21.6074	−0.939450
$530$	2.81189	0.122141
$531$	6.62171	0.287358
$532$	−32.2452	−1.39801
$533$	4.36940	0.189260
$534$	43.5736	1.88561
$535$	6.38847	0.276198
$536$	0.887534	0.0383357
$537$	28.4133	1.22613
$538$	−64.7363	−2.79098
$539$	13.8147	0.595043
$540$	11.8113	0.508278
$541$	−11.8057	−0.507568	−0.253784	−	0.967261i	$-0.581675\pi$
$541$	−11.8057	−0.507568	−0.253784	+	0.967261i	$0.581675\pi$
$542$	19.2971	0.828883
$543$	−20.4171	−0.876182
$544$	−3.84285	−0.164761
$545$	−4.37861	−0.187559
$546$	−5.38177	−0.230318
$547$	−11.0823	−0.473845	−0.236922	−	0.971529i	$-0.576139\pi$
$547$	−11.0823	−0.473845	−0.236922	+	0.971529i	$0.576139\pi$
$548$	2.53825	0.108429
$549$	19.2865	0.823127
$550$	−40.1467	−1.71186
$551$	−21.4439	−0.913541
$552$	−6.73522	−0.286670
$553$	4.46891	0.190038
$554$	−17.6241	−0.748775
$555$	6.53837	0.277538
$556$	−28.1965	−1.19580
$557$	−1.22878	−0.0520649	−0.0260325	−	0.999661i	$-0.508287\pi$
$557$	−1.22878	−0.0520649	−0.0260325	+	0.999661i	$0.508287\pi$
$558$	19.7350	0.835448
$559$	7.04977	0.298173
$560$	3.08363	0.130307
$561$	−13.3732	−0.564618
$562$	−41.5189	−1.75137
$563$	28.3020	1.19279	0.596394	−	0.802692i	$-0.296599\pi$
$563$	28.3020	1.19279	0.596394	+	0.802692i	$0.296599\pi$
$564$	16.5634	0.697443
$565$	10.5852	0.445322
$566$	60.6130	2.54776
$567$	−4.99996	−0.209979
$568$	−22.8140	−0.957256
$569$	−32.7317	−1.37219	−0.686093	−	0.727514i	$-0.740676\pi$
$569$	−32.7317	−1.37219	−0.686093	+	0.727514i	$0.740676\pi$
$570$	−7.99846	−0.335019
$571$	−28.5560	−1.19503	−0.597515	−	0.801858i	$-0.703845\pi$
$571$	−28.5560	−1.19503	−0.597515	+	0.801858i	$0.703845\pi$
$572$	13.6322	0.569992
$573$	−11.4422	−0.478003
$574$	18.5838	0.775675
$575$	5.54439	0.231217
$576$	13.2871	0.553629
$577$	−45.5878	−1.89784	−0.948922	−	0.315511i	$-0.897824\pi$
$577$	−45.5878	−1.89784	−0.948922	+	0.315511i	$0.897824\pi$
$578$	19.4467	0.808877
$579$	−0.549688	−0.0228443
$580$	9.57519	0.397588
$581$	−26.1567	−1.08516
$582$	−5.08612	−0.210827
$583$	7.46049	0.308982
$584$	18.9501	0.784160
$585$	0.768464	0.0317721
$586$	−5.10286	−0.210797
$587$	−39.2746	−1.62104	−0.810519	−	0.585713i	$-0.800815\pi$
$587$	−39.2746	−1.62104	−0.810519	+	0.585713i	$0.800815\pi$
$588$	19.1335	0.789050
$589$	−27.6887	−1.14089
$590$	6.29625	0.259213
$591$	6.63375	0.272876
$592$	−30.0592	−1.23542
$593$	30.8313	1.26609	0.633046	−	0.774115i	$-0.281805\pi$
$593$	30.8313	1.26609	0.633046	+	0.774115i	$0.281805\pi$
$594$	47.5631	1.95154
$595$	−2.88977	−0.118469
$596$	8.71751	0.357083
$597$	−14.8467	−0.607633
$598$	−2.85741	−0.116848
$599$	11.7691	0.480873	0.240437	−	0.970665i	$-0.422709\pi$
$599$	11.7691	0.480873	0.240437	+	0.970665i	$0.422709\pi$
$600$	−26.8142	−1.09469
$601$	4.79387	0.195546	0.0977730	−	0.995209i	$-0.468828\pi$
$601$	4.79387	0.195546	0.0977730	+	0.995209i	$0.468828\pi$
$602$	29.9839	1.22205
$603$	0.275263	0.0112096
$604$	−38.8741	−1.58176
$605$	0.799032	0.0324853
$606$	−10.4102	−0.422886
$607$	−1.53490	−0.0622997	−0.0311498	−	0.999515i	$-0.509917\pi$
$607$	−1.53490	−0.0622997	−0.0311498	+	0.999515i	$0.509917\pi$
$608$	−6.09808	−0.247310
$609$	−10.0294	−0.406412
$610$	18.3386	0.742507
$611$	3.38871	0.137093
$612$	16.1825	0.654140
$613$	42.0834	1.69973	0.849866	−	0.526999i	$-0.176683\pi$
$613$	42.0834	1.69973	0.849866	+	0.526999i	$0.176683\pi$
$614$	−72.5883	−2.92943
$615$	3.03721	0.122472
$616$	27.9605	1.12656
$617$	16.2719	0.655083	0.327541	−	0.944837i	$-0.393780\pi$
$617$	16.2719	0.655083	0.327541	+	0.944837i	$0.393780\pi$
$618$	49.6843	1.99860
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	12.3636	0.496535
$621$	−6.56862	−0.263590
$622$	−17.5024	−0.701784
$623$	24.9816	1.00087
$624$	4.04363	0.161875
$625$	20.5644	0.822578
$626$	−6.47886	−0.258947
$627$	−21.2215	−0.847505
$628$	−41.7729	−1.66692
$629$	28.1694	1.12319
$630$	3.26842	0.130217
$631$	−4.99586	−0.198882	−0.0994411	−	0.995043i	$-0.531705\pi$
$631$	−4.99586	−0.198882	−0.0994411	+	0.995043i	$0.531705\pi$
$632$	−11.4752	−0.456460
$633$	15.5622	0.618544
$634$	−32.5431	−1.29245
$635$	−8.45689	−0.335601
$636$	10.3328	0.409722
$637$	3.91453	0.155099
$638$	38.5584	1.52654
$639$	−7.07561	−0.279907
$640$	11.2242	0.443676
$641$	−4.21623	−0.166531	−0.0832656	−	0.996527i	$-0.526535\pi$
$641$	−4.21623	−0.166531	−0.0832656	+	0.996527i	$0.526535\pi$
$642$	35.6303	1.40622
$643$	15.5200	0.612051	0.306025	−	0.952023i	$-0.401001\pi$
$643$	15.5200	0.612051	0.306025	+	0.952023i	$0.401001\pi$
$644$	−8.00726	−0.315530
$645$	4.90035	0.192951
$646$	−34.4600	−1.35581
$647$	−40.0386	−1.57408	−0.787039	−	0.616903i	$-0.788387\pi$
$647$	−40.0386	−1.57408	−0.787039	+	0.616903i	$0.788387\pi$
$648$	12.8388	0.504357
$649$	16.7052	0.655736
$650$	−11.3759	−0.446200
$651$	−12.9501	−0.507555
$652$	7.58852	0.297189
$653$	15.8135	0.618829	0.309415	−	0.950927i	$-0.399867\pi$
$653$	15.8135	0.618829	0.309415	+	0.950927i	$0.399867\pi$
$654$	−24.4207	−0.954925
$655$	6.66044	0.260245
$656$	−13.9631	−0.545168
$657$	5.87723	0.229293
$658$	14.4128	0.561869
$659$	13.4012	0.522036	0.261018	−	0.965334i	$-0.415942\pi$
$659$	13.4012	0.522036	0.261018	+	0.965334i	$0.415942\pi$
$660$	9.47587	0.368848
$661$	11.6179	0.451882	0.225941	−	0.974141i	$-0.427454\pi$
$661$	11.6179	0.451882	0.225941	+	0.974141i	$0.427454\pi$
$662$	−24.9966	−0.971520
$663$	−3.78942	−0.147169
$664$	67.1649	2.60650
$665$	−4.58568	−0.177825
$666$	−31.8604	−1.23457
$667$	−5.32504	−0.206186
$668$	64.8772	2.51017
$669$	−17.9369	−0.693481
$670$	0.261733	0.0101116
$671$	48.6558	1.87834
$672$	−2.85209	−0.110022
$673$	16.4594	0.634463	0.317231	−	0.948348i	$-0.397247\pi$
$673$	16.4594	0.634463	0.317231	+	0.948348i	$0.397247\pi$
$674$	−84.2441	−3.24496
$675$	−26.1510	−1.00655
$676$	3.86281	0.148569
$677$	38.5264	1.48069	0.740346	−	0.672226i	$-0.234662\pi$
$677$	38.5264	1.48069	0.740346	+	0.672226i	$0.234662\pi$
$678$	59.0366	2.26729
$679$	−2.91598	−0.111905
$680$	7.42033	0.284557
$681$	8.55678	0.327897
$682$	49.7872	1.90645
$683$	−4.66274	−0.178415	−0.0892074	−	0.996013i	$-0.528433\pi$
$683$	−4.66274	−0.178415	−0.0892074	+	0.996013i	$0.528433\pi$
$684$	25.6795	0.981879
$685$	0.360972	0.0137920
$686$	46.4214	1.77238
$687$	−23.6741	−0.903223
$688$	−22.5286	−0.858896
$689$	2.11399	0.0805367
$690$	−1.98621	−0.0756138
$691$	7.65108	0.291061	0.145530	−	0.989354i	$-0.453511\pi$
$691$	7.65108	0.291061	0.145530	+	0.989354i	$0.453511\pi$
$692$	−90.2128	−3.42937
$693$	8.67175	0.329412
$694$	−25.0486	−0.950832
$695$	−4.00990	−0.152104
$696$	25.7534	0.976180
$697$	13.0853	0.495641
$698$	16.4795	0.623757
$699$	−31.6244	−1.19614
$700$	−31.8785	−1.20489
$701$	41.3459	1.56161	0.780806	−	0.624774i	$-0.214809\pi$
$701$	41.3459	1.56161	0.780806	+	0.624774i	$0.214809\pi$
$702$	13.4774	0.508672
$703$	44.7011	1.68593
$704$	33.5206	1.26335
$705$	2.35552	0.0887141
$706$	40.3639	1.51911
$707$	−5.96838	−0.224464
$708$	23.1367	0.869531
$709$	28.3740	1.06561	0.532804	−	0.846239i	$-0.321138\pi$
$709$	28.3740	1.06561	0.532804	+	0.846239i	$0.321138\pi$
$710$	−6.72785	−0.252492
$711$	−3.55896	−0.133472
$712$	−64.1475	−2.40403
$713$	−6.87577	−0.257500
$714$	−16.1171	−0.603166
$715$	1.93868	0.0725024
$716$	−86.7389	−3.24158
$717$	21.8312	0.815300
$718$	−0.385043	−0.0143697
$719$	34.7968	1.29770	0.648852	−	0.760915i	$-0.275250\pi$
$719$	34.7968	1.29770	0.648852	+	0.760915i	$0.275250\pi$
$720$	−2.45575	−0.0915203
$721$	28.4850	1.06084
$722$	−8.67815	−0.322967
$723$	26.4501	0.983689
$724$	62.3284	2.31642
$725$	−21.2000	−0.787350
$726$	4.45642	0.165393
$727$	17.8224	0.660997	0.330499	−	0.943806i	$-0.392783\pi$
$727$	17.8224	0.660997	0.330499	+	0.943806i	$0.392783\pi$
$728$	7.92285	0.293640
$729$	25.1112	0.930046
$730$	5.58837	0.206835
$731$	21.1123	0.780868
$732$	67.3884	2.49075
$733$	−30.2868	−1.11867	−0.559333	−	0.828943i	$-0.688943\pi$
$733$	−30.2868	−1.11867	−0.559333	+	0.828943i	$0.688943\pi$
$734$	87.0837	3.21432
$735$	2.72102	0.100366
$736$	−1.51430	−0.0558178
$737$	0.694430	0.0255797
$738$	−14.7998	−0.544790
$739$	−7.94458	−0.292246	−0.146123	−	0.989266i	$-0.546680\pi$
$739$	−7.94458	−0.292246	−0.146123	+	0.989266i	$0.546680\pi$
$740$	−19.9600	−0.733746
$741$	−6.01329	−0.220904
$742$	8.99119	0.330077
$743$	11.7107	0.429624	0.214812	−	0.976655i	$-0.431086\pi$
$743$	11.7107	0.429624	0.214812	+	0.976655i	$0.431086\pi$
$744$	33.2532	1.21912
$745$	1.23974	0.0454206
$746$	7.47843	0.273805
$747$	20.8307	0.762156
$748$	40.8252	1.49272
$749$	20.4275	0.746406
$750$	−16.3229	−0.596029
$751$	−11.7563	−0.428995	−0.214497	−	0.976725i	$-0.568811\pi$
$751$	−11.7563	−0.428995	−0.214497	+	0.976725i	$0.568811\pi$
$752$	−10.8292	−0.394899
$753$	−13.4270	−0.489306
$754$	10.9259	0.397897
$755$	−5.52839	−0.201199
$756$	37.7674	1.37359
$757$	34.7641	1.26352	0.631762	−	0.775163i	$-0.282332\pi$
$757$	34.7641	1.26352	0.631762	+	0.775163i	$0.282332\pi$
$758$	−0.275130	−0.00999319
$759$	−5.26981	−0.191282
$760$	11.7750	0.427126
$761$	2.71729	0.0985016	0.0492508	−	0.998786i	$-0.484317\pi$
$761$	2.71729	0.0985016	0.0492508	+	0.998786i	$0.484317\pi$
$762$	−47.1664	−1.70866
$763$	−14.0009	−0.506866
$764$	34.9301	1.26373
$765$	2.30136	0.0832059
$766$	−29.9684	−1.08280
$767$	4.73356	0.170919
$768$	38.5631	1.39153
$769$	−3.17783	−0.114596	−0.0572978	−	0.998357i	$-0.518248\pi$
$769$	−3.17783	−0.114596	−0.0572978	+	0.998357i	$0.518248\pi$
$770$	8.24553	0.297148
$771$	23.1741	0.834595
$772$	1.67806	0.0603948
$773$	40.8528	1.46937	0.734686	−	0.678407i	$-0.237330\pi$
$773$	40.8528	1.46937	0.734686	+	0.678407i	$0.237330\pi$
$774$	−23.8786	−0.858300
$775$	−27.3738	−0.983296
$776$	7.48761	0.268790
$777$	20.9069	0.750030
$778$	77.6679	2.78453
$779$	20.7646	0.743968
$780$	2.68507	0.0961409
$781$	−17.8503	−0.638734
$782$	−8.55725	−0.306007
$783$	25.1164	0.897586
$784$	−12.5095	−0.446767
$785$	−5.94064	−0.212031
$786$	37.1471	1.32499
$787$	−23.2292	−0.828031	−0.414016	−	0.910270i	$-0.635874\pi$
$787$	−23.2292	−0.828031	−0.414016	+	0.910270i	$0.635874\pi$
$788$	−20.2512	−0.721419
$789$	9.77289	0.347924
$790$	−3.38404	−0.120399
$791$	33.8468	1.20346
$792$	−22.2672	−0.791231
$793$	13.7870	0.489592
$794$	−34.3004	−1.21728
$795$	1.46946	0.0521162
$796$	45.3232	1.60644
$797$	−15.0893	−0.534489	−0.267245	−	0.963629i	$-0.586113\pi$
$797$	−15.0893	−0.534489	−0.267245	+	0.963629i	$0.586113\pi$
$798$	−25.5756	−0.905366
$799$	10.1484	0.359023
$800$	−6.02873	−0.213148
$801$	−19.8949	−0.702951
$802$	89.4598	3.15893
$803$	14.8270	0.523235
$804$	0.961787	0.0339196
$805$	−1.13873	−0.0401351
$806$	14.1076	0.496920
$807$	−33.8303	−1.19088
$808$	15.3255	0.539151
$809$	−52.3589	−1.84084	−0.920420	−	0.390932i	$-0.872153\pi$
$809$	−52.3589	−1.84084	−0.920420	+	0.390932i	$0.872153\pi$
$810$	3.78617	0.133032
$811$	42.4263	1.48979	0.744895	−	0.667182i	$-0.232500\pi$
$811$	42.4263	1.48979	0.744895	+	0.667182i	$0.232500\pi$
$812$	30.6173	1.07446
$813$	10.0844	0.353676
$814$	−80.3773	−2.81722
$815$	1.07918	0.0378022
$816$	12.1097	0.423923
$817$	33.5024	1.17210
$818$	−56.3222	−1.96926
$819$	2.45721	0.0858620
$820$	−9.27185	−0.323787
$821$	10.1772	0.355188	0.177594	−	0.984104i	$-0.443169\pi$
$821$	10.1772	0.355188	0.177594	+	0.984104i	$0.443169\pi$
$822$	2.01324	0.0702199
$823$	−15.6626	−0.545962	−0.272981	−	0.962019i	$-0.588010\pi$
$823$	−15.6626	−0.545962	−0.272981	+	0.962019i	$0.588010\pi$
$824$	−73.1435	−2.54807
$825$	−20.9802	−0.730435
$826$	20.1327	0.700505
$827$	26.3857	0.917521	0.458760	−	0.888560i	$-0.348294\pi$
$827$	26.3857	0.917521	0.458760	+	0.888560i	$0.348294\pi$
$828$	6.37683	0.221610
$829$	37.5809	1.30524	0.652619	−	0.757686i	$-0.273670\pi$
$829$	37.5809	1.30524	0.652619	+	0.757686i	$0.273670\pi$
$830$	19.8069	0.687507
$831$	−9.21011	−0.319495
$832$	9.49834	0.329296
$833$	11.7230	0.406180
$834$	−22.3643	−0.774413
$835$	9.22636	0.319291
$836$	64.7840	2.24060
$837$	32.4306	1.12097
$838$	41.6012	1.43709
$839$	−27.4142	−0.946443	−0.473221	−	0.880944i	$-0.656909\pi$
$839$	−27.4142	−0.946443	−0.473221	+	0.880944i	$0.656909\pi$
$840$	5.50724	0.190018
$841$	−8.63867	−0.297885
$842$	44.8951	1.54719
$843$	−21.6972	−0.747292
$844$	−47.5077	−1.63528
$845$	0.549340	0.0188979
$846$	−11.4781	−0.394625
$847$	2.55496	0.0877893
$848$	−6.75560	−0.231988
$849$	31.6756	1.08710
$850$	−34.0681	−1.16853
$851$	11.1004	0.380515
$852$	−24.7227	−0.846986
$853$	6.80357	0.232950	0.116475	−	0.993194i	$-0.462841\pi$
$853$	6.80357	0.232950	0.116475	+	0.993194i	$0.462841\pi$
$854$	58.6387	2.00658
$855$	3.65195	0.124894
$856$	−52.4536	−1.79283
$857$	10.8698	0.371307	0.185653	−	0.982615i	$-0.440560\pi$
$857$	10.8698	0.371307	0.185653	+	0.982615i	$0.440560\pi$
$858$	10.8125	0.369134
$859$	−18.7324	−0.639142	−0.319571	−	0.947562i	$-0.603539\pi$
$859$	−18.7324	−0.639142	−0.319571	+	0.947562i	$0.603539\pi$
$860$	−14.9596	−0.510117
$861$	9.71167	0.330973
$862$	−45.9620	−1.56547
$863$	−4.84444	−0.164907	−0.0824534	−	0.996595i	$-0.526276\pi$
$863$	−4.84444	−0.164907	−0.0824534	+	0.996595i	$0.526276\pi$
$864$	7.14242	0.242990
$865$	−12.8294	−0.436213
$866$	17.2543	0.586326
$867$	10.1626	0.345140
$868$	39.5335	1.34185
$869$	−8.97852	−0.304576
$870$	7.59466	0.257483
$871$	0.196773	0.00666739
$872$	35.9513	1.21746
$873$	2.32223	0.0785956
$874$	−13.5792	−0.459323
$875$	−9.35825	−0.316367
$876$	20.5355	0.693830
$877$	31.1538	1.05199	0.525994	−	0.850488i	$-0.323693\pi$
$877$	31.1538	1.05199	0.525994	+	0.850488i	$0.323693\pi$
$878$	82.7033	2.79110
$879$	−2.66669	−0.0899452
$880$	−6.19534	−0.208845
$881$	−33.3740	−1.12440	−0.562199	−	0.827002i	$-0.690045\pi$
$881$	−33.3740	−1.12440	−0.562199	+	0.827002i	$0.690045\pi$
$882$	−13.2591	−0.446457
$883$	−6.04640	−0.203478	−0.101739	−	0.994811i	$-0.532441\pi$
$883$	−6.04640	−0.203478	−0.101739	+	0.994811i	$0.532441\pi$
$884$	11.5682	0.389079
$885$	3.29034	0.110603
$886$	60.5584	2.03450
$887$	5.70909	0.191692	0.0958462	−	0.995396i	$-0.469444\pi$
$887$	5.70909	0.191692	0.0958462	+	0.995396i	$0.469444\pi$
$888$	−53.6844	−1.80153
$889$	−27.0414	−0.906941
$890$	−18.9171	−0.634101
$891$	10.0454	0.336535
$892$	54.7570	1.83340
$893$	16.1041	0.538902
$894$	6.91438	0.231252
$895$	−12.3354	−0.412326
$896$	35.8901	1.19901
$897$	−1.49325	−0.0498580
$898$	−34.1481	−1.13954
$899$	26.2908	0.876849
$900$	25.3874	0.846248
$901$	6.33089	0.210913
$902$	−37.3369	−1.24318
$903$	15.6692	0.521438
$904$	−86.9116	−2.89064
$905$	8.86390	0.294646
$906$	−30.8334	−1.02437
$907$	25.3777	0.842654	0.421327	−	0.906909i	$-0.361565\pi$
$907$	25.3777	0.842654	0.421327	+	0.906909i	$0.361565\pi$
$908$	−26.1217	−0.866880
$909$	4.75311	0.157651
$910$	2.33644	0.0774523
$911$	16.9033	0.560032	0.280016	−	0.959995i	$-0.409660\pi$
$911$	16.9033	0.560032	0.280016	+	0.959995i	$0.409660\pi$
$912$	19.2164	0.636319
$913$	52.5516	1.73920
$914$	−88.5585	−2.92925
$915$	9.58349	0.316820
$916$	72.2711	2.38791
$917$	21.2972	0.703295
$918$	40.3616	1.33213
$919$	−32.1906	−1.06187	−0.530936	−	0.847412i	$-0.678159\pi$
$919$	−32.1906	−1.06187	−0.530936	+	0.847412i	$0.678159\pi$
$920$	2.92403	0.0964024
$921$	−37.9337	−1.24996
$922$	67.0182	2.20713
$923$	−5.05804	−0.166487
$924$	30.2997	0.996788
$925$	44.1927	1.45305
$926$	−75.5677	−2.48331
$927$	−22.6849	−0.745071
$928$	5.79022	0.190073
$929$	7.83996	0.257221	0.128610	−	0.991695i	$-0.458948\pi$
$929$	7.83996	0.257221	0.128610	+	0.991695i	$0.458948\pi$
$930$	9.80634	0.321563
$931$	18.6029	0.609685
$932$	96.5414	3.16232
$933$	−9.14654	−0.299444
$934$	4.41859	0.144581
$935$	5.80586	0.189872
$936$	−6.30961	−0.206236
$937$	41.1512	1.34435	0.672176	−	0.740391i	$-0.265360\pi$
$937$	41.1512	1.34435	0.672176	+	0.740391i	$0.265360\pi$
$938$	0.836910	0.0273261
$939$	−3.38576	−0.110490
$940$	−7.19083	−0.234539
$941$	39.3486	1.28273	0.641364	−	0.767237i	$-0.278369\pi$
$941$	39.3486	1.28273	0.641364	+	0.767237i	$0.278369\pi$
$942$	−33.1326	−1.07952
$943$	5.15635	0.167914
$944$	−15.1268	−0.492336
$945$	5.37101	0.174719
$946$	−60.2409	−1.95860
$947$	−22.8805	−0.743515	−0.371758	−	0.928330i	$-0.621245\pi$
$947$	−22.8805	−0.743515	−0.371758	+	0.928330i	$0.621245\pi$
$948$	−12.4353	−0.403879
$949$	4.20137	0.136382
$950$	−54.0614	−1.75398
$951$	−17.0066	−0.551477
$952$	23.7270	0.768996
$953$	−61.5408	−1.99350	−0.996751	−	0.0805456i	$-0.974334\pi$
$953$	−61.5408	−1.99350	−0.996751	+	0.0805456i	$0.974334\pi$
$954$	−7.16042	−0.231827
$955$	4.96751	0.160745
$956$	−66.6452	−2.15546
$957$	20.1501	0.651361
$958$	54.2718	1.75344
$959$	1.15423	0.0372721
$960$	6.60238	0.213091
$961$	2.94712	0.0950685
$962$	−22.7756	−0.734315
$963$	−16.2681	−0.524233
$964$	−80.7456	−2.60064
$965$	0.238642	0.00768216
$966$	−6.35104	−0.204341
$967$	32.5475	1.04666	0.523328	−	0.852131i	$-0.324690\pi$
$967$	32.5475	1.04666	0.523328	+	0.852131i	$0.324690\pi$
$968$	−6.56059	−0.210865
$969$	−18.0083	−0.578511
$970$	2.20809	0.0708976
$971$	−7.71174	−0.247482	−0.123741	−	0.992315i	$-0.539489\pi$
$971$	−7.71174	−0.247482	−0.123741	+	0.992315i	$0.539489\pi$
$972$	−50.5897	−1.62267
$973$	−12.8219	−0.411051
$974$	−18.9614	−0.607562
$975$	−5.94491	−0.190389
$976$	−44.0586	−1.41028
$977$	−13.7733	−0.440648	−0.220324	−	0.975427i	$-0.570711\pi$
$977$	−13.7733	−0.440648	−0.220324	+	0.975427i	$0.570711\pi$
$978$	6.01891	0.192464
$979$	−50.1907	−1.60410
$980$	−8.30661	−0.265345
$981$	11.1500	0.355993
$982$	−31.6219	−1.00910
$983$	11.4734	0.365945	0.182973	−	0.983118i	$-0.441428\pi$
$983$	11.4734	0.365945	0.182973	+	0.983118i	$0.441428\pi$
$984$	−24.9375	−0.794980
$985$	−2.87998	−0.0917638
$986$	32.7203	1.04203
$987$	7.53194	0.239744
$988$	18.3571	0.584017
$989$	8.31946	0.264543
$990$	−6.56659	−0.208700
$991$	−59.7371	−1.89761	−0.948805	−	0.315861i	$-0.897707\pi$
$991$	−59.7371	−1.89761	−0.948805	+	0.315861i	$0.897707\pi$
$992$	7.47641	0.237376
$993$	−13.0629	−0.414538
$994$	−21.5127	−0.682342
$995$	6.44554	0.204337
$996$	72.7840	2.30625
$997$	0.190195	0.00602353	0.00301177	−	0.999995i	$-0.499041\pi$
$997$	0.190195	0.00602353	0.00301177	+	0.999995i	$0.499041\pi$
$998$	100.198	3.17172
$999$	−52.3565	−1.65649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.13	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.13	✓	156	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42132 q^{2} -1.26535 q^{3} +3.86281 q^{4} +0.549340 q^{5} +3.06382 q^{6} +1.75655 q^{7} -4.51045 q^{8} -1.39889 q^{9} +O(q^{10})\)	\(q-2.42132 q^{2} -1.26535 q^{3} +3.86281 q^{4} +0.549340 q^{5} +3.06382 q^{6} +1.75655 q^{7} -4.51045 q^{8} -1.39889 q^{9} -1.33013 q^{10} -3.52910 q^{11} -4.88781 q^{12} -1.00000 q^{13} -4.25318 q^{14} -0.695109 q^{15} +3.19566 q^{16} -2.99475 q^{17} +3.38715 q^{18} -4.75227 q^{19} +2.12200 q^{20} -2.22266 q^{21} +8.54509 q^{22} -1.18010 q^{23} +5.70731 q^{24} -4.69823 q^{25} +2.42132 q^{26} +5.56614 q^{27} +6.78522 q^{28} +4.51235 q^{29} +1.68308 q^{30} +5.82642 q^{31} +1.28319 q^{32} +4.46555 q^{33} +7.25127 q^{34} +0.964945 q^{35} -5.40362 q^{36} -9.40626 q^{37} +11.5068 q^{38} +1.26535 q^{39} -2.47777 q^{40} -4.36940 q^{41} +5.38177 q^{42} -7.04977 q^{43} -13.6322 q^{44} -0.768464 q^{45} +2.85741 q^{46} -3.38871 q^{47} -4.04363 q^{48} -3.91453 q^{49} +11.3759 q^{50} +3.78942 q^{51} -3.86281 q^{52} -2.11399 q^{53} -13.4774 q^{54} -1.93868 q^{55} -7.92285 q^{56} +6.01329 q^{57} -10.9259 q^{58} -4.73356 q^{59} -2.68507 q^{60} -13.7870 q^{61} -14.1076 q^{62} -2.45721 q^{63} -9.49834 q^{64} -0.549340 q^{65} -10.8125 q^{66} -0.196773 q^{67} -11.5682 q^{68} +1.49325 q^{69} -2.33644 q^{70} +5.05804 q^{71} +6.30961 q^{72} -4.20137 q^{73} +22.7756 q^{74} +5.94491 q^{75} -18.3571 q^{76} -6.19904 q^{77} -3.06382 q^{78} +2.54414 q^{79} +1.75550 q^{80} -2.84646 q^{81} +10.5797 q^{82} -14.8909 q^{83} -8.58568 q^{84} -1.64514 q^{85} +17.0698 q^{86} -5.70971 q^{87} +15.9178 q^{88} +14.2220 q^{89} +1.86070 q^{90} -1.75655 q^{91} -4.55851 q^{92} -7.37246 q^{93} +8.20516 q^{94} -2.61061 q^{95} -1.62369 q^{96} -1.66006 q^{97} +9.47834 q^{98} +4.93680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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