LMFDB - Embedded newform 8047.2.a.d.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.61991 q^{2} -3.07843 q^{3} +4.86391 q^{4} -2.25474 q^{5} +8.06519 q^{6} +3.02795 q^{7} -7.50318 q^{8} +6.47672 q^{9} +O(q^{10})$	\(q-2.61991 q^{2} -3.07843 q^{3} +4.86391 q^{4} -2.25474 q^{5} +8.06519 q^{6} +3.02795 q^{7} -7.50318 q^{8} +6.47672 q^{9} +5.90721 q^{10} +1.57362 q^{11} -14.9732 q^{12} -1.00000 q^{13} -7.93295 q^{14} +6.94106 q^{15} +9.92982 q^{16} +5.17171 q^{17} -16.9684 q^{18} -6.55233 q^{19} -10.9669 q^{20} -9.32132 q^{21} -4.12275 q^{22} +3.43105 q^{23} +23.0980 q^{24} +0.0838553 q^{25} +2.61991 q^{26} -10.7028 q^{27} +14.7277 q^{28} +1.57815 q^{29} -18.1849 q^{30} -6.53929 q^{31} -11.0088 q^{32} -4.84429 q^{33} -13.5494 q^{34} -6.82724 q^{35} +31.5022 q^{36} +0.847251 q^{37} +17.1665 q^{38} +3.07843 q^{39} +16.9177 q^{40} -0.805164 q^{41} +24.4210 q^{42} -6.26875 q^{43} +7.65397 q^{44} -14.6033 q^{45} -8.98904 q^{46} +2.01191 q^{47} -30.5682 q^{48} +2.16848 q^{49} -0.219693 q^{50} -15.9207 q^{51} -4.86391 q^{52} -7.36518 q^{53} +28.0404 q^{54} -3.54811 q^{55} -22.7193 q^{56} +20.1709 q^{57} -4.13460 q^{58} +4.93760 q^{59} +33.7607 q^{60} -3.87440 q^{61} +17.1323 q^{62} +19.6112 q^{63} +8.98249 q^{64} +2.25474 q^{65} +12.6916 q^{66} -7.83467 q^{67} +25.1547 q^{68} -10.5623 q^{69} +17.8867 q^{70} -0.564401 q^{71} -48.5960 q^{72} +11.8849 q^{73} -2.21972 q^{74} -0.258143 q^{75} -31.8700 q^{76} +4.76485 q^{77} -8.06519 q^{78} -9.83308 q^{79} -22.3892 q^{80} +13.5177 q^{81} +2.10945 q^{82} +9.55569 q^{83} -45.3381 q^{84} -11.6609 q^{85} +16.4235 q^{86} -4.85821 q^{87} -11.8072 q^{88} +16.7313 q^{89} +38.2593 q^{90} -3.02795 q^{91} +16.6883 q^{92} +20.1307 q^{93} -5.27101 q^{94} +14.7738 q^{95} +33.8899 q^{96} +18.6813 q^{97} -5.68121 q^{98} +10.1919 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.61991	−1.85255	−0.926277	−	0.376843i	$-0.877010\pi$
$2$	−2.61991	−1.85255	−0.926277	+	0.376843i	$0.877010\pi$
$3$	−3.07843	−1.77733	−0.888666	−	0.458556i	$-0.848367\pi$
$3$	−3.07843	−1.77733	−0.888666	+	0.458556i	$0.848367\pi$
$4$	4.86391	2.43196
$5$	−2.25474	−1.00835	−0.504175	−	0.863601i	$-0.668203\pi$
$5$	−2.25474	−1.00835	−0.504175	+	0.863601i	$0.668203\pi$
$6$	8.06519	3.29260
$7$	3.02795	1.14446	0.572229	−	0.820094i	$-0.306079\pi$
$7$	3.02795	1.14446	0.572229	+	0.820094i	$0.306079\pi$
$8$	−7.50318	−2.65278
$9$	6.47672	2.15891
$10$	5.90721	1.86802
$11$	1.57362	0.474466	0.237233	−	0.971453i	$-0.423760\pi$
$11$	1.57362	0.474466	0.237233	+	0.971453i	$0.423760\pi$
$12$	−14.9732	−4.32239
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−7.93295	−2.12017
$15$	6.94106	1.79217
$16$	9.92982	2.48246
$17$	5.17171	1.25432	0.627162	−	0.778889i	$-0.284217\pi$
$17$	5.17171	1.25432	0.627162	+	0.778889i	$0.284217\pi$
$18$	−16.9684	−3.99949
$19$	−6.55233	−1.50321	−0.751604	−	0.659615i	$-0.770719\pi$
$19$	−6.55233	−1.50321	−0.751604	+	0.659615i	$0.770719\pi$
$20$	−10.9669	−2.45226
$21$	−9.32132	−2.03408
$22$	−4.12275	−0.878973
$23$	3.43105	0.715424	0.357712	−	0.933832i	$-0.383557\pi$
$23$	3.43105	0.715424	0.357712	+	0.933832i	$0.383557\pi$
$24$	23.0980	4.71486
$25$	0.0838553	0.0167711
$26$	2.61991	0.513806
$27$	−10.7028	−2.05976
$28$	14.7277	2.78327
$29$	1.57815	0.293054	0.146527	−	0.989207i	$-0.453190\pi$
$29$	1.57815	0.293054	0.146527	+	0.989207i	$0.453190\pi$
$30$	−18.1849	−3.32010
$31$	−6.53929	−1.17449	−0.587246	−	0.809409i	$-0.699788\pi$
$31$	−6.53929	−1.17449	−0.587246	+	0.809409i	$0.699788\pi$
$32$	−11.0088	−1.94611
$33$	−4.84429	−0.843282
$34$	−13.5494	−2.32370
$35$	−6.82724	−1.15401
$36$	31.5022	5.25037
$37$	0.847251	0.139287	0.0696436	−	0.997572i	$-0.477814\pi$
$37$	0.847251	0.139287	0.0696436	+	0.997572i	$0.477814\pi$
$38$	17.1665	2.78477
$39$	3.07843	0.492943
$40$	16.9177	2.67493
$41$	−0.805164	−0.125745	−0.0628727	−	0.998022i	$-0.520026\pi$
$41$	−0.805164	−0.125745	−0.0628727	+	0.998022i	$0.520026\pi$
$42$	24.4210	3.76824
$43$	−6.26875	−0.955975	−0.477988	−	0.878367i	$-0.658634\pi$
$43$	−6.26875	−0.955975	−0.477988	+	0.878367i	$0.658634\pi$
$44$	7.65397	1.15388
$45$	−14.6033	−2.17693
$46$	−8.98904	−1.32536
$47$	2.01191	0.293467	0.146733	−	0.989176i	$-0.453124\pi$
$47$	2.01191	0.293467	0.146733	+	0.989176i	$0.453124\pi$
$48$	−30.5682	−4.41215
$49$	2.16848	0.309783
$50$	−0.219693	−0.0310693
$51$	−15.9207	−2.22935
$52$	−4.86391	−0.674503
$53$	−7.36518	−1.01169	−0.505843	−	0.862626i	$-0.668818\pi$
$53$	−7.36518	−1.01169	−0.505843	+	0.862626i	$0.668818\pi$
$54$	28.0404	3.81582
$55$	−3.54811	−0.478428
$56$	−22.7193	−3.03599
$57$	20.1709	2.67170
$58$	−4.13460	−0.542899
$59$	4.93760	0.642821	0.321410	−	0.946940i	$-0.395843\pi$
$59$	4.93760	0.642821	0.321410	+	0.946940i	$0.395843\pi$
$60$	33.7607	4.35849
$61$	−3.87440	−0.496067	−0.248033	−	0.968751i	$-0.579784\pi$
$61$	−3.87440	−0.496067	−0.248033	+	0.968751i	$0.579784\pi$
$62$	17.1323	2.17581
$63$	19.6112	2.47078
$64$	8.98249	1.12281
$65$	2.25474	0.279666
$66$	12.6916	1.56223
$67$	−7.83467	−0.957157	−0.478578	−	0.878045i	$-0.658848\pi$
$67$	−7.83467	−0.957157	−0.478578	+	0.878045i	$0.658848\pi$
$68$	25.1547	3.05046
$69$	−10.5623	−1.27155
$70$	17.8867	2.13787
$71$	−0.564401	−0.0669820	−0.0334910	−	0.999439i	$-0.510663\pi$
$71$	−0.564401	−0.0669820	−0.0334910	+	0.999439i	$0.510663\pi$
$72$	−48.5960	−5.72709
$73$	11.8849	1.39102	0.695511	−	0.718515i	$-0.255178\pi$
$73$	11.8849	1.39102	0.695511	+	0.718515i	$0.255178\pi$
$74$	−2.21972	−0.258037
$75$	−0.258143	−0.0298077
$76$	−31.8700	−3.65574
$77$	4.76485	0.543006
$78$	−8.06519	−0.913203
$79$	−9.83308	−1.10631	−0.553154	−	0.833079i	$-0.686576\pi$
$79$	−9.83308	−1.10631	−0.553154	+	0.833079i	$0.686576\pi$
$80$	−22.3892	−2.50319
$81$	13.5177	1.50197
$82$	2.10945	0.232950
$83$	9.55569	1.04887	0.524436	−	0.851450i	$-0.324276\pi$
$83$	9.55569	1.04887	0.524436	+	0.851450i	$0.324276\pi$
$84$	−45.3381	−4.94679
$85$	−11.6609	−1.26480
$86$	16.4235	1.77100
$87$	−4.85821	−0.520855
$88$	−11.8072	−1.25865
$89$	16.7313	1.77351	0.886757	−	0.462236i	$-0.152953\pi$
$89$	16.7313	1.77351	0.886757	+	0.462236i	$0.152953\pi$
$90$	38.2593	4.03289
$91$	−3.02795	−0.317415
$92$	16.6883	1.73988
$93$	20.1307	2.08746
$94$	−5.27101	−0.543663
$95$	14.7738	1.51576
$96$	33.8899	3.45887
$97$	18.6813	1.89679	0.948397	−	0.317086i	$-0.102704\pi$
$97$	18.6813	1.89679	0.948397	+	0.317086i	$0.102704\pi$
$98$	−5.68121	−0.573889
$99$	10.1919	1.02433
$100$	0.407865	0.0407865
$101$	0.551517	0.0548780	0.0274390	−	0.999623i	$-0.491265\pi$
$101$	0.551517	0.0548780	0.0274390	+	0.999623i	$0.491265\pi$
$102$	41.7108	4.12999
$103$	1.82567	0.179889	0.0899444	−	0.995947i	$-0.471331\pi$
$103$	1.82567	0.179889	0.0899444	+	0.995947i	$0.471331\pi$
$104$	7.50318	0.735748
$105$	21.0172	2.05107
$106$	19.2961	1.87420
$107$	−6.19674	−0.599061	−0.299531	−	0.954087i	$-0.596830\pi$
$107$	−6.19674	−0.599061	−0.299531	+	0.954087i	$0.596830\pi$
$108$	−52.0576	−5.00925
$109$	−8.43558	−0.807982	−0.403991	−	0.914763i	$-0.632377\pi$
$109$	−8.43558	−0.807982	−0.403991	+	0.914763i	$0.632377\pi$
$110$	9.29573	0.886313
$111$	−2.60820	−0.247559
$112$	30.0670	2.84106
$113$	5.86461	0.551696	0.275848	−	0.961201i	$-0.411041\pi$
$113$	5.86461	0.551696	0.275848	+	0.961201i	$0.411041\pi$
$114$	−52.8458	−4.94946
$115$	−7.73614	−0.721398
$116$	7.67596	0.712695
$117$	−6.47672	−0.598773
$118$	−12.9361	−1.19086
$119$	15.6597	1.43552
$120$	−52.0800	−4.75423
$121$	−8.52371	−0.774882
$122$	10.1506	0.918990
$123$	2.47864	0.223491
$124$	−31.8066	−2.85631
$125$	11.0846	0.991440
$126$	−51.3795	−4.57725
$127$	1.27582	0.113211	0.0566054	−	0.998397i	$-0.481972\pi$
$127$	1.27582	0.113211	0.0566054	+	0.998397i	$0.481972\pi$
$128$	−1.51561	−0.133962
$129$	19.2979	1.69908
$130$	−5.90721	−0.518097
$131$	3.78260	0.330487	0.165244	−	0.986253i	$-0.447159\pi$
$131$	3.78260	0.330487	0.165244	+	0.986253i	$0.447159\pi$
$132$	−23.5622	−2.05083
$133$	−19.8401	−1.72036
$134$	20.5261	1.77319
$135$	24.1321	2.07696
$136$	−38.8043	−3.32744
$137$	−7.75773	−0.662787	−0.331394	−	0.943493i	$-0.607519\pi$
$137$	−7.75773	−0.662787	−0.331394	+	0.943493i	$0.607519\pi$
$138$	27.6721	2.35561
$139$	7.82348	0.663579	0.331789	−	0.943353i	$-0.392348\pi$
$139$	7.82348	0.663579	0.331789	+	0.943353i	$0.392348\pi$
$140$	−33.2071	−2.80651
$141$	−6.19351	−0.521588
$142$	1.47868	0.124088
$143$	−1.57362	−0.131593
$144$	64.3127	5.35939
$145$	−3.55831	−0.295502
$146$	−31.1373	−2.57694
$147$	−6.67550	−0.550586
$148$	4.12095	0.338740
$149$	6.39729	0.524086	0.262043	−	0.965056i	$-0.415604\pi$
$149$	6.39729	0.524086	0.262043	+	0.965056i	$0.415604\pi$
$150$	0.676310	0.0552205
$151$	14.1925	1.15497	0.577485	−	0.816401i	$-0.304034\pi$
$151$	14.1925	1.15497	0.577485	+	0.816401i	$0.304034\pi$
$152$	49.1633	3.98767
$153$	33.4957	2.70797
$154$	−12.4835	−1.00595
$155$	14.7444	1.18430
$156$	14.9732	1.19882
$157$	−10.9188	−0.871416	−0.435708	−	0.900088i	$-0.643502\pi$
$157$	−10.9188	−0.871416	−0.435708	+	0.900088i	$0.643502\pi$
$158$	25.7618	2.04949
$159$	22.6732	1.79810
$160$	24.8221	1.96236
$161$	10.3891	0.818772
$162$	−35.4152	−2.78248
$163$	15.3412	1.20162	0.600809	−	0.799393i	$-0.294845\pi$
$163$	15.3412	1.20162	0.600809	+	0.799393i	$0.294845\pi$
$164$	−3.91625	−0.305808
$165$	10.9226	0.850324
$166$	−25.0350	−1.94309
$167$	−15.3893	−1.19086	−0.595430	−	0.803407i	$-0.703018\pi$
$167$	−15.3893	−1.19086	−0.595430	+	0.803407i	$0.703018\pi$
$168$	69.9396	5.39596
$169$	1.00000	0.0769231
$170$	30.5504	2.34311
$171$	−42.4376	−3.24528
$172$	−30.4907	−2.32489
$173$	−5.00464	−0.380496	−0.190248	−	0.981736i	$-0.560929\pi$
$173$	−5.00464	−0.380496	−0.190248	+	0.981736i	$0.560929\pi$
$174$	12.7281	0.964911
$175$	0.253910	0.0191938
$176$	15.6258	1.17784
$177$	−15.2001	−1.14251
$178$	−43.8344	−3.28553
$179$	−6.46718	−0.483380	−0.241690	−	0.970354i	$-0.577702\pi$
$179$	−6.46718	−0.483380	−0.241690	+	0.970354i	$0.577702\pi$
$180$	−71.0293	−5.29421
$181$	−6.53302	−0.485596	−0.242798	−	0.970077i	$-0.578065\pi$
$181$	−6.53302	−0.485596	−0.242798	+	0.970077i	$0.578065\pi$
$182$	7.93295	0.588029
$183$	11.9271	0.881675
$184$	−25.7438	−1.89786
$185$	−1.91033	−0.140450
$186$	−52.7407	−3.86713
$187$	8.13833	0.595133
$188$	9.78574	0.713698
$189$	−32.4076	−2.35731
$190$	−38.7060	−2.80803
$191$	−11.2208	−0.811909	−0.405954	−	0.913893i	$-0.633061\pi$
$191$	−11.2208	−0.811909	−0.405954	+	0.913893i	$0.633061\pi$
$192$	−27.6519	−1.99561
$193$	16.8161	1.21045	0.605224	−	0.796055i	$-0.293083\pi$
$193$	16.8161	1.21045	0.605224	+	0.796055i	$0.293083\pi$
$194$	−48.9431	−3.51391
$195$	−6.94106	−0.497059
$196$	10.5473	0.753378
$197$	−10.9966	−0.783472	−0.391736	−	0.920078i	$-0.628125\pi$
$197$	−10.9966	−0.783472	−0.391736	+	0.920078i	$0.628125\pi$
$198$	−26.7019	−1.89762
$199$	−14.1540	−1.00335	−0.501674	−	0.865057i	$-0.667282\pi$
$199$	−14.1540	−1.00335	−0.501674	+	0.865057i	$0.667282\pi$
$200$	−0.629182	−0.0444899
$201$	24.1185	1.70118
$202$	−1.44492	−0.101665
$203$	4.77855	0.335388
$204$	−77.4370	−5.42168
$205$	1.81544	0.126796
$206$	−4.78309	−0.333254
$207$	22.2220	1.54453
$208$	−9.92982	−0.688509
$209$	−10.3109	−0.713220
$210$	−55.0630	−3.79971
$211$	−2.59640	−0.178744	−0.0893718	−	0.995998i	$-0.528486\pi$
$211$	−2.59640	−0.178744	−0.0893718	+	0.995998i	$0.528486\pi$
$212$	−35.8236	−2.46038
$213$	1.73747	0.119049
$214$	16.2349	1.10979
$215$	14.1344	0.963958
$216$	80.3053	5.46408
$217$	−19.8007	−1.34416
$218$	22.1004	1.49683
$219$	−36.5868	−2.47231
$220$	−17.2577	−1.16352
$221$	−5.17171	−0.347887
$222$	6.83324	0.458617
$223$	7.22389	0.483747	0.241874	−	0.970308i	$-0.422238\pi$
$223$	7.22389	0.483747	0.241874	+	0.970308i	$0.422238\pi$
$224$	−33.3342	−2.22724
$225$	0.543107	0.0362072
$226$	−15.3647	−1.02205
$227$	8.53446	0.566452	0.283226	−	0.959053i	$-0.408595\pi$
$227$	8.53446	0.566452	0.283226	+	0.959053i	$0.408595\pi$
$228$	98.1094	6.49745
$229$	25.7230	1.69982	0.849912	−	0.526924i	$-0.176655\pi$
$229$	25.7230	1.69982	0.849912	+	0.526924i	$0.176655\pi$
$230$	20.2680	1.33643
$231$	−14.6683	−0.965101
$232$	−11.8411	−0.777408
$233$	10.1977	0.668071	0.334036	−	0.942560i	$-0.391589\pi$
$233$	10.1977	0.668071	0.334036	+	0.942560i	$0.391589\pi$
$234$	16.9684	1.10926
$235$	−4.53633	−0.295917
$236$	24.0161	1.56331
$237$	30.2704	1.96628
$238$	−41.0269	−2.65938
$239$	−14.0894	−0.911365	−0.455682	−	0.890142i	$-0.650605\pi$
$239$	−14.0894	−0.911365	−0.455682	+	0.890142i	$0.650605\pi$
$240$	68.9234	4.44899
$241$	−11.6050	−0.747542	−0.373771	−	0.927521i	$-0.621935\pi$
$241$	−11.6050	−0.747542	−0.373771	+	0.927521i	$0.621935\pi$
$242$	22.3313	1.43551
$243$	−9.50486	−0.609737
$244$	−18.8448	−1.20641
$245$	−4.88936	−0.312369
$246$	−6.49380	−0.414030
$247$	6.55233	0.416915
$248$	49.0655	3.11566
$249$	−29.4165	−1.86419
$250$	−29.0407	−1.83670
$251$	−18.2857	−1.15419	−0.577093	−	0.816679i	$-0.695813\pi$
$251$	−18.2857	−1.15419	−0.577093	+	0.816679i	$0.695813\pi$
$252$	95.3870	6.00882
$253$	5.39919	0.339444
$254$	−3.34253	−0.209729
$255$	35.8971	2.24796
$256$	−13.9942	−0.874639
$257$	15.6321	0.975102	0.487551	−	0.873094i	$-0.337890\pi$
$257$	15.6321	0.975102	0.487551	+	0.873094i	$0.337890\pi$
$258$	−50.5587	−3.14765
$259$	2.56543	0.159408
$260$	10.9669	0.680136
$261$	10.2212	0.632677
$262$	−9.91005	−0.612245
$263$	20.0689	1.23750	0.618750	−	0.785588i	$-0.287639\pi$
$263$	20.0689	1.23750	0.618750	+	0.785588i	$0.287639\pi$
$264$	36.3476	2.23704
$265$	16.6066	1.02013
$266$	51.9793	3.18705
$267$	−51.5061	−3.15212
$268$	−38.1071	−2.32776
$269$	−27.5889	−1.68212	−0.841062	−	0.540939i	$-0.818069\pi$
$269$	−27.5889	−1.68212	−0.841062	+	0.540939i	$0.818069\pi$
$270$	−63.2238	−3.84768
$271$	−8.05246	−0.489152	−0.244576	−	0.969630i	$-0.578649\pi$
$271$	−8.05246	−0.489152	−0.244576	+	0.969630i	$0.578649\pi$
$272$	51.3541	3.11380
$273$	9.32132	0.564152
$274$	20.3245	1.22785
$275$	0.131957	0.00795729
$276$	−51.3739	−3.09234
$277$	−4.02097	−0.241596	−0.120798	−	0.992677i	$-0.538545\pi$
$277$	−4.02097	−0.241596	−0.120798	+	0.992677i	$0.538545\pi$
$278$	−20.4968	−1.22932
$279$	−42.3532	−2.53562
$280$	51.2260	3.06134
$281$	4.56424	0.272279	0.136140	−	0.990690i	$-0.456530\pi$
$281$	4.56424	0.272279	0.136140	+	0.990690i	$0.456530\pi$
$282$	16.2264	0.966269
$283$	8.88667	0.528257	0.264129	−	0.964487i	$-0.414916\pi$
$283$	8.88667	0.528257	0.264129	+	0.964487i	$0.414916\pi$
$284$	−2.74520	−0.162897
$285$	−45.4801	−2.69401
$286$	4.12275	0.243783
$287$	−2.43800	−0.143910
$288$	−71.3011	−4.20146
$289$	9.74656	0.573327
$290$	9.32244	0.547433
$291$	−57.5089	−3.37123
$292$	57.8071	3.38291
$293$	−10.8329	−0.632867	−0.316434	−	0.948615i	$-0.602485\pi$
$293$	−10.8329	−0.632867	−0.316434	+	0.948615i	$0.602485\pi$
$294$	17.4892	1.01999
$295$	−11.1330	−0.648189
$296$	−6.35708	−0.369498
$297$	−16.8422	−0.977285
$298$	−16.7603	−0.970898
$299$	−3.43105	−0.198423
$300$	−1.25558	−0.0724911
$301$	−18.9815	−1.09407
$302$	−37.1830	−2.13964
$303$	−1.69781	−0.0975364
$304$	−65.0635	−3.73165
$305$	8.73578	0.500209
$306$	−87.7556	−5.01665
$307$	17.1805	0.980545	0.490272	−	0.871569i	$-0.336897\pi$
$307$	17.1805	0.980545	0.490272	+	0.871569i	$0.336897\pi$
$308$	23.1758	1.32057
$309$	−5.62020	−0.319722
$310$	−38.6290	−2.19398
$311$	−1.59138	−0.0902391	−0.0451195	−	0.998982i	$-0.514367\pi$
$311$	−1.59138	−0.0902391	−0.0451195	+	0.998982i	$0.514367\pi$
$312$	−23.0980	−1.30767
$313$	−8.74323	−0.494196	−0.247098	−	0.968990i	$-0.579477\pi$
$313$	−8.74323	−0.494196	−0.247098	+	0.968990i	$0.579477\pi$
$314$	28.6063	1.61435
$315$	−44.2181	−2.49141
$316$	−47.8272	−2.69049
$317$	−11.6098	−0.652071	−0.326035	−	0.945358i	$-0.605713\pi$
$317$	−11.6098	−0.652071	−0.326035	+	0.945358i	$0.605713\pi$
$318$	−59.4016	−3.33108
$319$	2.48341	0.139044
$320$	−20.2532	−1.13219
$321$	19.0762	1.06473
$322$	−27.2184	−1.51682
$323$	−33.8867	−1.88551
$324$	65.7490	3.65272
$325$	−0.0838553	−0.00465146
$326$	−40.1926	−2.22606
$327$	25.9683	1.43605
$328$	6.04129	0.333575
$329$	6.09195	0.335860
$330$	−28.6162	−1.57527
$331$	11.8930	0.653696	0.326848	−	0.945077i	$-0.394013\pi$
$331$	11.8930	0.653696	0.326848	+	0.945077i	$0.394013\pi$
$332$	46.4780	2.55081
$333$	5.48740	0.300708
$334$	40.3186	2.20613
$335$	17.6651	0.965150
$336$	−92.5591	−5.04951
$337$	−33.0568	−1.80072	−0.900359	−	0.435148i	$-0.856696\pi$
$337$	−33.0568	−1.80072	−0.900359	+	0.435148i	$0.856696\pi$
$338$	−2.61991	−0.142504
$339$	−18.0538	−0.980547
$340$	−56.7174	−3.07593
$341$	−10.2904	−0.557256
$342$	111.183	6.01206
$343$	−14.6296	−0.789924
$344$	47.0356	2.53599
$345$	23.8151	1.28216
$346$	13.1117	0.704889
$347$	23.7221	1.27347	0.636734	−	0.771084i	$-0.280285\pi$
$347$	23.7221	1.27347	0.636734	+	0.771084i	$0.280285\pi$
$348$	−23.6299	−1.26670
$349$	−14.1717	−0.758594	−0.379297	−	0.925275i	$-0.623834\pi$
$349$	−14.1717	−0.758594	−0.379297	+	0.925275i	$0.623834\pi$
$350$	−0.665220	−0.0355575
$351$	10.7028	0.571275
$352$	−17.3238	−0.923360
$353$	10.1994	0.542861	0.271430	−	0.962458i	$-0.412503\pi$
$353$	10.1994	0.542861	0.271430	+	0.962458i	$0.412503\pi$
$354$	39.8227	2.11655
$355$	1.27258	0.0675414
$356$	81.3796	4.31311
$357$	−48.2072	−2.55139
$358$	16.9434	0.895487
$359$	−19.0503	−1.00544	−0.502718	−	0.864451i	$-0.667666\pi$
$359$	−19.0503	−1.00544	−0.502718	+	0.864451i	$0.667666\pi$
$360$	109.571	5.77492
$361$	23.9330	1.25963
$362$	17.1159	0.899593
$363$	26.2396	1.37722
$364$	−14.7277	−0.771940
$365$	−26.7974	−1.40264
$366$	−31.2478	−1.63335
$367$	−4.61851	−0.241084	−0.120542	−	0.992708i	$-0.538463\pi$
$367$	−4.61851	−0.241084	−0.120542	+	0.992708i	$0.538463\pi$
$368$	34.0697	1.77601
$369$	−5.21482	−0.271473
$370$	5.00489	0.260192
$371$	−22.3014	−1.15783
$372$	97.9142	5.07661
$373$	−18.3760	−0.951472	−0.475736	−	0.879588i	$-0.657818\pi$
$373$	−18.3760	−0.951472	−0.475736	+	0.879588i	$0.657818\pi$
$374$	−21.3217	−1.10252
$375$	−34.1232	−1.76212
$376$	−15.0957	−0.778502
$377$	−1.57815	−0.0812786
$378$	84.9049	4.36704
$379$	32.8221	1.68596	0.842979	−	0.537946i	$-0.180800\pi$
$379$	32.8221	1.68596	0.842979	+	0.537946i	$0.180800\pi$
$380$	71.8585	3.68626
$381$	−3.92752	−0.201213
$382$	29.3975	1.50410
$383$	−6.60847	−0.337677	−0.168838	−	0.985644i	$-0.554002\pi$
$383$	−6.60847	−0.337677	−0.168838	+	0.985644i	$0.554002\pi$
$384$	4.66570	0.238095
$385$	−10.7435	−0.547540
$386$	−44.0566	−2.24242
$387$	−40.6009	−2.06386
$388$	90.8640	4.61292
$389$	−11.1150	−0.563553	−0.281777	−	0.959480i	$-0.590924\pi$
$389$	−11.1150	−0.563553	−0.281777	+	0.959480i	$0.590924\pi$
$390$	18.1849	0.920829
$391$	17.7444	0.897373
$392$	−16.2705	−0.821784
$393$	−11.6445	−0.587385
$394$	28.8100	1.45142
$395$	22.1710	1.11555
$396$	49.5726	2.49112
$397$	20.8097	1.04441	0.522203	−	0.852821i	$-0.325110\pi$
$397$	20.8097	1.04441	0.522203	+	0.852821i	$0.325110\pi$
$398$	37.0820	1.85876
$399$	61.0764	3.05764
$400$	0.832669	0.0416334
$401$	27.0504	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$401$	27.0504	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$402$	−63.1881	−3.15154
$403$	6.53929	0.325745
$404$	2.68253	0.133461
$405$	−30.4790	−1.51451
$406$	−12.5193	−0.621325
$407$	1.33325	0.0660869
$408$	119.456	5.91396
$409$	−20.9713	−1.03696	−0.518482	−	0.855088i	$-0.673503\pi$
$409$	−20.9713	−1.03696	−0.518482	+	0.855088i	$0.673503\pi$
$410$	−4.75627	−0.234896
$411$	23.8816	1.17799
$412$	8.87991	0.437482
$413$	14.9508	0.735681
$414$	−58.2195	−2.86133
$415$	−21.5456	−1.05763
$416$	11.0088	0.539753
$417$	−24.0840	−1.17940
$418$	27.0136	1.32128
$419$	−0.763132	−0.0372814	−0.0186407	−	0.999826i	$-0.505934\pi$
$419$	−0.763132	−0.0372814	−0.0186407	+	0.999826i	$0.505934\pi$
$420$	102.226	4.98810
$421$	−17.6142	−0.858465	−0.429232	−	0.903194i	$-0.641216\pi$
$421$	−17.6142	−0.858465	−0.429232	+	0.903194i	$0.641216\pi$
$422$	6.80233	0.331132
$423$	13.0306	0.633567
$424$	55.2623	2.68378
$425$	0.433675	0.0210363
$426$	−4.55200	−0.220545
$427$	−11.7315	−0.567727
$428$	−30.1404	−1.45689
$429$	4.84429	0.233884
$430$	−37.0308	−1.78578
$431$	−26.2555	−1.26468	−0.632342	−	0.774690i	$-0.717906\pi$
$431$	−26.2555	−1.26468	−0.632342	+	0.774690i	$0.717906\pi$
$432$	−106.277	−5.11326
$433$	33.4997	1.60989	0.804947	−	0.593346i	$-0.202194\pi$
$433$	33.4997	1.60989	0.804947	+	0.593346i	$0.202194\pi$
$434$	51.8759	2.49012
$435$	10.9540	0.525204
$436$	−41.0299	−1.96498
$437$	−22.4814	−1.07543
$438$	95.8540	4.58008
$439$	−38.7855	−1.85113	−0.925566	−	0.378586i	$-0.876410\pi$
$439$	−38.7855	−1.85113	−0.925566	+	0.378586i	$0.876410\pi$
$440$	26.6222	1.26916
$441$	14.0446	0.668791
$442$	13.5494	0.644479
$443$	6.40189	0.304163	0.152082	−	0.988368i	$-0.451402\pi$
$443$	6.40189	0.304163	0.152082	+	0.988368i	$0.451402\pi$
$444$	−12.6861	−0.602054
$445$	−37.7247	−1.78832
$446$	−18.9259	−0.896168
$447$	−19.6936	−0.931474
$448$	27.1985	1.28501
$449$	−22.6788	−1.07028	−0.535139	−	0.844764i	$-0.679741\pi$
$449$	−22.6788	−1.07028	−0.535139	+	0.844764i	$0.679741\pi$
$450$	−1.42289	−0.0670757
$451$	−1.26703	−0.0596619
$452$	28.5250	1.34170
$453$	−43.6906	−2.05276
$454$	−22.3595	−1.04938
$455$	6.82724	0.320066
$456$	−151.346	−7.08742
$457$	29.9117	1.39921	0.699605	−	0.714529i	$-0.253359\pi$
$457$	29.9117	1.39921	0.699605	+	0.714529i	$0.253359\pi$
$458$	−67.3919	−3.14902
$459$	−55.3519	−2.58360
$460$	−37.6279	−1.75441
$461$	40.4632	1.88456	0.942279	−	0.334828i	$-0.108678\pi$
$461$	40.4632	1.88456	0.942279	+	0.334828i	$0.108678\pi$
$462$	38.4295	1.78790
$463$	−27.0111	−1.25531	−0.627656	−	0.778491i	$-0.715986\pi$
$463$	−27.0111	−1.25531	−0.627656	+	0.778491i	$0.715986\pi$
$464$	15.6707	0.727494
$465$	−45.3896	−2.10489
$466$	−26.7169	−1.23764
$467$	40.2238	1.86133	0.930667	−	0.365868i	$-0.119228\pi$
$467$	40.2238	1.86133	0.930667	+	0.365868i	$0.119228\pi$
$468$	−31.5022	−1.45619
$469$	−23.7230	−1.09543
$470$	11.8848	0.548203
$471$	33.6128	1.54880
$472$	−37.0477	−1.70526
$473$	−9.86466	−0.453577
$474$	−79.3057	−3.64263
$475$	−0.549448	−0.0252104
$476$	76.1673	3.49112
$477$	−47.7022	−2.18413
$478$	36.9128	1.68835
$479$	−11.5761	−0.528924	−0.264462	−	0.964396i	$-0.585194\pi$
$479$	−11.5761	−0.528924	−0.264462	+	0.964396i	$0.585194\pi$
$480$	−76.4130	−3.48776
$481$	−0.847251	−0.0386313
$482$	30.4040	1.38486
$483$	−31.9820	−1.45523
$484$	−41.4586	−1.88448
$485$	−42.1214	−1.91263
$486$	24.9018	1.12957
$487$	16.4261	0.744339	0.372169	−	0.928165i	$-0.378614\pi$
$487$	16.4261	0.744339	0.372169	+	0.928165i	$0.378614\pi$
$488$	29.0704	1.31595
$489$	−47.2268	−2.13567
$490$	12.8097	0.578681
$491$	35.4185	1.59841	0.799207	−	0.601056i	$-0.205253\pi$
$491$	35.4185	1.59841	0.799207	+	0.601056i	$0.205253\pi$
$492$	12.0559	0.543521
$493$	8.16171	0.367585
$494$	−17.1665	−0.772357
$495$	−22.9801	−1.03288
$496$	−64.9340	−2.91562
$497$	−1.70898	−0.0766581
$498$	77.0685	3.45352
$499$	−15.2283	−0.681712	−0.340856	−	0.940115i	$-0.610717\pi$
$499$	−15.2283	−0.681712	−0.340856	+	0.940115i	$0.610717\pi$
$500$	53.9147	2.41114
$501$	47.3749	2.11655
$502$	47.9069	2.13819
$503$	3.90364	0.174055	0.0870274	−	0.996206i	$-0.472263\pi$
$503$	3.90364	0.174055	0.0870274	+	0.996206i	$0.472263\pi$
$504$	−147.146	−6.55442
$505$	−1.24353	−0.0553363
$506$	−14.1454	−0.628839
$507$	−3.07843	−0.136718
$508$	6.20548	0.275324
$509$	−14.2478	−0.631521	−0.315761	−	0.948839i	$-0.602260\pi$
$509$	−14.2478	−0.631521	−0.315761	+	0.948839i	$0.602260\pi$
$510$	−94.0471	−4.16448
$511$	35.9869	1.59197
$512$	39.6948	1.75428
$513$	70.1284	3.09625
$514$	−40.9546	−1.80643
$515$	−4.11642	−0.181391
$516$	93.8633	4.13210
$517$	3.16599	0.139240
$518$	−6.72119	−0.295312
$519$	15.4064	0.676267
$520$	−16.9177	−0.741892
$521$	−0.170959	−0.00748986	−0.00374493	−	0.999993i	$-0.501192\pi$
$521$	−0.170959	−0.00748986	−0.00374493	+	0.999993i	$0.501192\pi$
$522$	−26.7786	−1.17207
$523$	18.2808	0.799365	0.399682	−	0.916654i	$-0.369120\pi$
$523$	18.2808	0.799365	0.399682	+	0.916654i	$0.369120\pi$
$524$	18.3982	0.803730
$525$	−0.781643	−0.0341137
$526$	−52.5786	−2.29254
$527$	−33.8193	−1.47319
$528$	−48.1029	−2.09341
$529$	−11.2279	−0.488168
$530$	−43.5077	−1.88985
$531$	31.9795	1.38779
$532$	−96.5006	−4.18383
$533$	0.805164	0.0348755
$534$	134.941	5.83947
$535$	13.9720	0.604064
$536$	58.7850	2.53912
$537$	19.9087	0.859126
$538$	72.2803	3.11623
$539$	3.41237	0.146981
$540$	117.376	5.05108
$541$	6.78725	0.291807	0.145903	−	0.989299i	$-0.453391\pi$
$541$	6.78725	0.291807	0.145903	+	0.989299i	$0.453391\pi$
$542$	21.0967	0.906181
$543$	20.1114	0.863065
$544$	−56.9345	−2.44105
$545$	19.0200	0.814729
$546$	−24.4210	−1.04512
$547$	−44.3216	−1.89506	−0.947528	−	0.319673i	$-0.896427\pi$
$547$	−44.3216	−1.89506	−0.947528	+	0.319673i	$0.896427\pi$
$548$	−37.7329	−1.61187
$549$	−25.0934	−1.07096
$550$	−0.345715	−0.0147413
$551$	−10.3405	−0.440521
$552$	79.2505	3.37313
$553$	−29.7741	−1.26612
$554$	10.5346	0.447570
$555$	5.88081	0.249627
$556$	38.0527	1.61380
$557$	3.63907	0.154193	0.0770963	−	0.997024i	$-0.475435\pi$
$557$	3.63907	0.154193	0.0770963	+	0.997024i	$0.475435\pi$
$558$	110.961	4.69737
$559$	6.26875	0.265140
$560$	−67.7933	−2.86479
$561$	−25.0532	−1.05775
$562$	−11.9579	−0.504412
$563$	−22.2428	−0.937424	−0.468712	−	0.883351i	$-0.655282\pi$
$563$	−22.2428	−0.937424	−0.468712	+	0.883351i	$0.655282\pi$
$564$	−30.1247	−1.26848
$565$	−13.2232	−0.556303
$566$	−23.2822	−0.978625
$567$	40.9310	1.71894
$568$	4.23480	0.177688
$569$	44.4038	1.86151	0.930753	−	0.365649i	$-0.119153\pi$
$569$	44.4038	1.86151	0.930753	+	0.365649i	$0.119153\pi$
$570$	119.154	4.99080
$571$	27.0038	1.13007	0.565037	−	0.825065i	$-0.308862\pi$
$571$	27.0038	1.13007	0.565037	+	0.825065i	$0.308862\pi$
$572$	−7.65397	−0.320029
$573$	34.5424	1.44303
$574$	6.38732	0.266602
$575$	0.287712	0.0119984
$576$	58.1771	2.42404
$577$	−0.120802	−0.00502907	−0.00251454	−	0.999997i	$-0.500800\pi$
$577$	−0.120802	−0.00502907	−0.00251454	+	0.999997i	$0.500800\pi$
$578$	−25.5351	−1.06212
$579$	−51.7671	−2.15137
$580$	−17.3073	−0.718647
$581$	28.9341	1.20039
$582$	150.668	6.24539
$583$	−11.5900	−0.480010
$584$	−89.1746	−3.69007
$585$	14.6033	0.603773
$586$	28.3813	1.17242
$587$	−27.2212	−1.12354	−0.561770	−	0.827293i	$-0.689879\pi$
$587$	−27.2212	−1.12354	−0.561770	+	0.827293i	$0.689879\pi$
$588$	−32.4691	−1.33900
$589$	42.8476	1.76551
$590$	29.1675	1.20081
$591$	33.8521	1.39249
$592$	8.41305	0.345774
$593$	12.3229	0.506043	0.253021	−	0.967461i	$-0.418576\pi$
$593$	12.3229	0.506043	0.253021	+	0.967461i	$0.418576\pi$
$594$	44.1251	1.81047
$595$	−35.3085	−1.44751
$596$	31.1158	1.27455
$597$	43.5719	1.78328
$598$	8.98904	0.367589
$599$	−39.2004	−1.60168	−0.800842	−	0.598876i	$-0.795614\pi$
$599$	−39.2004	−1.60168	−0.800842	+	0.598876i	$0.795614\pi$
$600$	1.93689	0.0790733
$601$	−6.08185	−0.248084	−0.124042	−	0.992277i	$-0.539586\pi$
$601$	−6.08185	−0.248084	−0.124042	+	0.992277i	$0.539586\pi$
$602$	49.7297	2.02683
$603$	−50.7429	−2.06641
$604$	69.0311	2.80884
$605$	19.2187	0.781353
$606$	4.44809	0.180691
$607$	32.0378	1.30037	0.650187	−	0.759775i	$-0.274691\pi$
$607$	32.0378	1.30037	0.650187	+	0.759775i	$0.274691\pi$
$608$	72.1335	2.92540
$609$	−14.7104	−0.596096
$610$	−22.8869	−0.926665
$611$	−2.01191	−0.0813930
$612$	162.920	6.58566
$613$	−33.9888	−1.37280	−0.686398	−	0.727226i	$-0.740809\pi$
$613$	−33.9888	−1.37280	−0.686398	+	0.727226i	$0.740809\pi$
$614$	−45.0114	−1.81651
$615$	−5.58869	−0.225358
$616$	−35.7516	−1.44047
$617$	15.9797	0.643318	0.321659	−	0.946856i	$-0.395760\pi$
$617$	15.9797	0.643318	0.321659	+	0.946856i	$0.395760\pi$
$618$	14.7244	0.592302
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	71.7155	2.88016
$621$	−36.7220	−1.47360
$622$	4.16928	0.167173
$623$	50.6615	2.02971
$624$	30.5682	1.22371
$625$	−25.4122	−1.01649
$626$	22.9064	0.915525
$627$	31.7414	1.26763
$628$	−53.1082	−2.11925
$629$	4.38173	0.174711
$630$	115.847	4.61547
$631$	17.4728	0.695580	0.347790	−	0.937572i	$-0.386932\pi$
$631$	17.4728	0.695580	0.347790	+	0.937572i	$0.386932\pi$
$632$	73.7794	2.93479
$633$	7.99283	0.317686
$634$	30.4166	1.20800
$635$	−2.87664	−0.114156
$636$	110.280	4.37290
$637$	−2.16848	−0.0859182
$638$	−6.50630	−0.257587
$639$	−3.65546	−0.144608
$640$	3.41731	0.135081
$641$	−4.26776	−0.168566	−0.0842832	−	0.996442i	$-0.526860\pi$
$641$	−4.26776	−0.168566	−0.0842832	+	0.996442i	$0.526860\pi$
$642$	−49.9779	−1.97247
$643$	−8.02988	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$643$	−8.02988	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$644$	50.5315	1.99122
$645$	−43.5117	−1.71327
$646$	88.7801	3.49301
$647$	37.4313	1.47158	0.735788	−	0.677212i	$-0.236812\pi$
$647$	37.4313	1.47158	0.735788	+	0.677212i	$0.236812\pi$
$648$	−101.426	−3.98439
$649$	7.76993	0.304996
$650$	0.219693	0.00861708
$651$	60.9549	2.38901
$652$	74.6184	2.92228
$653$	6.26870	0.245313	0.122657	−	0.992449i	$-0.460859\pi$
$653$	6.26870	0.245313	0.122657	+	0.992449i	$0.460859\pi$
$654$	−68.0346	−2.66036
$655$	−8.52878	−0.333247
$656$	−7.99513	−0.312158
$657$	76.9752	3.00309
$658$	−15.9603	−0.622199
$659$	43.7792	1.70540	0.852698	−	0.522405i	$-0.174965\pi$
$659$	43.7792	1.70540	0.852698	+	0.522405i	$0.174965\pi$
$660$	53.1266	2.06795
$661$	3.38447	0.131641	0.0658203	−	0.997831i	$-0.479034\pi$
$661$	3.38447	0.131641	0.0658203	+	0.997831i	$0.479034\pi$
$662$	−31.1584	−1.21101
$663$	15.9207	0.618310
$664$	−71.6981	−2.78243
$665$	44.7343	1.73472
$666$	−14.3765	−0.557077
$667$	5.41470	0.209658
$668$	−74.8523	−2.89612
$669$	−22.2382	−0.859779
$670$	−46.2810	−1.78799
$671$	−6.09686	−0.235367
$672$	102.617	3.95853
$673$	43.4676	1.67555	0.837777	−	0.546013i	$-0.183855\pi$
$673$	43.4676	1.67555	0.837777	+	0.546013i	$0.183855\pi$
$674$	86.6057	3.33593
$675$	−0.897489	−0.0345444
$676$	4.86391	0.187074
$677$	12.9021	0.495869	0.247934	−	0.968777i	$-0.420248\pi$
$677$	12.9021	0.495869	0.247934	+	0.968777i	$0.420248\pi$
$678$	47.2992	1.81652
$679$	56.5659	2.17080
$680$	87.4936	3.35523
$681$	−26.2727	−1.00677
$682$	26.9599	1.03235
$683$	−41.9457	−1.60501	−0.802504	−	0.596647i	$-0.796499\pi$
$683$	−41.9457	−1.60501	−0.802504	+	0.596647i	$0.796499\pi$
$684$	−206.413	−7.89239
$685$	17.4917	0.668322
$686$	38.3282	1.46338
$687$	−79.1864	−3.02115
$688$	−62.2476	−2.37317
$689$	7.36518	0.280591
$690$	−62.3934	−2.37528
$691$	−4.59172	−0.174677	−0.0873387	−	0.996179i	$-0.527836\pi$
$691$	−4.59172	−0.174677	−0.0873387	+	0.996179i	$0.527836\pi$
$692$	−24.3422	−0.925350
$693$	30.8606	1.17230
$694$	−62.1496	−2.35917
$695$	−17.6399	−0.669120
$696$	36.4520	1.38171
$697$	−4.16407	−0.157726
$698$	37.1285	1.40534
$699$	−31.3928	−1.18738
$700$	1.23499	0.0466784
$701$	23.3247	0.880961	0.440481	−	0.897762i	$-0.354808\pi$
$701$	23.3247	0.880961	0.440481	+	0.897762i	$0.354808\pi$
$702$	−28.0404	−1.05832
$703$	−5.55147	−0.209377
$704$	14.1351	0.532735
$705$	13.9648	0.525943
$706$	−26.7215	−1.00568
$707$	1.66997	0.0628056
$708$	−73.9317	−2.77852
$709$	34.3154	1.28874	0.644370	−	0.764714i	$-0.277120\pi$
$709$	34.3154	1.28874	0.644370	+	0.764714i	$0.277120\pi$
$710$	−3.33403	−0.125124
$711$	−63.6861	−2.38841
$712$	−125.538	−4.70474
$713$	−22.4367	−0.840260
$714$	126.298	4.72659
$715$	3.54811	0.132692
$716$	−31.4558	−1.17556
$717$	43.3731	1.61980
$718$	49.9100	1.86262
$719$	28.6663	1.06907	0.534537	−	0.845145i	$-0.320486\pi$
$719$	28.6663	1.06907	0.534537	+	0.845145i	$0.320486\pi$
$720$	−145.008	−5.40414
$721$	5.52804	0.205875
$722$	−62.7023	−2.33354
$723$	35.7251	1.32863
$724$	−31.7761	−1.18095
$725$	0.132336	0.00491483
$726$	−68.7454	−2.55138
$727$	−12.5789	−0.466525	−0.233263	−	0.972414i	$-0.574940\pi$
$727$	−12.5789	−0.466525	−0.233263	+	0.972414i	$0.574940\pi$
$728$	22.7193	0.842032
$729$	−11.2932	−0.418265
$730$	70.2066	2.59846
$731$	−32.4201	−1.19910
$732$	58.0123	2.14419
$733$	9.64377	0.356201	0.178100	−	0.984012i	$-0.443005\pi$
$733$	9.64377	0.356201	0.178100	+	0.984012i	$0.443005\pi$
$734$	12.1001	0.446622
$735$	15.0515	0.555184
$736$	−37.7719	−1.39229
$737$	−12.3288	−0.454138
$738$	13.6623	0.502918
$739$	−6.94496	−0.255475	−0.127737	−	0.991808i	$-0.540771\pi$
$739$	−6.94496	−0.255475	−0.127737	+	0.991808i	$0.540771\pi$
$740$	−9.29168	−0.341569
$741$	−20.1709	−0.740996
$742$	58.4276	2.14494
$743$	−11.9683	−0.439074	−0.219537	−	0.975604i	$-0.570455\pi$
$743$	−11.9683	−0.439074	−0.219537	+	0.975604i	$0.570455\pi$
$744$	−151.045	−5.53757
$745$	−14.4242	−0.528463
$746$	48.1434	1.76265
$747$	61.8895	2.26442
$748$	39.5841	1.44734
$749$	−18.7634	−0.685600
$750$	89.3997	3.26442
$751$	1.52014	0.0554707	0.0277353	−	0.999615i	$-0.491170\pi$
$751$	1.52014	0.0554707	0.0277353	+	0.999615i	$0.491170\pi$
$752$	19.9779	0.728518
$753$	56.2913	2.05137
$754$	4.13460	0.150573
$755$	−32.0004	−1.16461
$756$	−157.628	−5.73287
$757$	0.104780	0.00380828	0.00190414	−	0.999998i	$-0.499394\pi$
$757$	0.104780	0.00380828	0.00190414	+	0.999998i	$0.499394\pi$
$758$	−85.9909	−3.12333
$759$	−16.6210	−0.603305
$760$	−110.851	−4.02097
$761$	−31.3178	−1.13527	−0.567635	−	0.823280i	$-0.692142\pi$
$761$	−31.3178	−1.13527	−0.567635	+	0.823280i	$0.692142\pi$
$762$	10.2897	0.372758
$763$	−25.5425	−0.924701
$764$	−54.5770	−1.97453
$765$	−75.5241	−2.73058
$766$	17.3136	0.625565
$767$	−4.93760	−0.178286
$768$	43.0802	1.55452
$769$	24.8190	0.894995	0.447497	−	0.894285i	$-0.352315\pi$
$769$	24.8190	0.894995	0.447497	+	0.894285i	$0.352315\pi$
$770$	28.1470	1.01435
$771$	−48.1222	−1.73308
$772$	81.7920	2.94376
$773$	43.2999	1.55739	0.778694	−	0.627404i	$-0.215882\pi$
$773$	43.2999	1.55739	0.778694	+	0.627404i	$0.215882\pi$
$774$	106.371	3.82341
$775$	−0.548355	−0.0196975
$776$	−140.169	−5.03177
$777$	−7.89750	−0.283321
$778$	29.1203	1.04401
$779$	5.27570	0.189022
$780$	−33.7607	−1.20883
$781$	−0.888155	−0.0317807
$782$	−46.4887	−1.66243
$783$	−16.8906	−0.603621
$784$	21.5326	0.769021
$785$	24.6191	0.878693
$786$	30.5074	1.08816
$787$	−44.6138	−1.59031	−0.795155	−	0.606407i	$-0.792610\pi$
$787$	−44.6138	−1.59031	−0.795155	+	0.606407i	$0.792610\pi$
$788$	−53.4863	−1.90537
$789$	−61.7806	−2.19945
$790$	−58.0861	−2.06661
$791$	17.7577	0.631393
$792$	−76.4719	−2.71731
$793$	3.87440	0.137584
$794$	−54.5194	−1.93482
$795$	−51.1221	−1.81312
$796$	−68.8436	−2.44010
$797$	38.6604	1.36942	0.684712	−	0.728814i	$-0.259928\pi$
$797$	38.6604	1.36942	0.684712	+	0.728814i	$0.259928\pi$
$798$	−160.014	−5.66445
$799$	10.4050	0.368102
$800$	−0.923150	−0.0326383
$801$	108.364	3.82885
$802$	−70.8695	−2.50249
$803$	18.7024	0.659992
$804$	117.310	4.13721
$805$	−23.4246	−0.825610
$806$	−17.1323	−0.603461
$807$	84.9304	2.98969
$808$	−4.13814	−0.145579
$809$	−49.6228	−1.74464	−0.872322	−	0.488932i	$-0.837387\pi$
$809$	−49.6228	−1.74464	−0.872322	+	0.488932i	$0.837387\pi$
$810$	79.8520	2.80571
$811$	30.0356	1.05469	0.527346	−	0.849650i	$-0.323187\pi$
$811$	30.0356	1.05469	0.527346	+	0.849650i	$0.323187\pi$
$812$	23.2424	0.815649
$813$	24.7889	0.869385
$814$	−3.49300	−0.122430
$815$	−34.5905	−1.21165
$816$	−158.090	−5.53426
$817$	41.0749	1.43703
$818$	54.9429	1.92103
$819$	−19.6112	−0.685270
$820$	8.83012	0.308361
$821$	36.0092	1.25673	0.628364	−	0.777919i	$-0.283725\pi$
$821$	36.0092	1.25673	0.628364	+	0.777919i	$0.283725\pi$
$822$	−62.5676	−2.18229
$823$	10.2217	0.356307	0.178153	−	0.984003i	$-0.442988\pi$
$823$	10.2217	0.356307	0.178153	+	0.984003i	$0.442988\pi$
$824$	−13.6984	−0.477205
$825$	−0.406220	−0.0141427
$826$	−39.1697	−1.36289
$827$	26.1343	0.908778	0.454389	−	0.890803i	$-0.349858\pi$
$827$	26.1343	0.908778	0.454389	+	0.890803i	$0.349858\pi$
$828$	108.086	3.75624
$829$	−55.8341	−1.93920	−0.969599	−	0.244701i	$-0.921310\pi$
$829$	−55.8341	−1.93920	−0.969599	+	0.244701i	$0.921310\pi$
$830$	56.4475	1.95932
$831$	12.3783	0.429397
$832$	−8.98249	−0.311412
$833$	11.2147	0.388568
$834$	63.0979	2.18490
$835$	34.6989	1.20081
$836$	−50.1513	−1.73452
$837$	69.9889	2.41917
$838$	1.99933	0.0690658
$839$	49.7062	1.71605	0.858024	−	0.513610i	$-0.171692\pi$
$839$	49.7062	1.71605	0.858024	+	0.513610i	$0.171692\pi$
$840$	−157.696	−5.44102
$841$	−26.5095	−0.914119
$842$	46.1476	1.59035
$843$	−14.0507	−0.483931
$844$	−12.6287	−0.434696
$845$	−2.25474	−0.0775654
$846$	−34.1388	−1.17372
$847$	−25.8094	−0.886820
$848$	−73.1349	−2.51146
$849$	−27.3570	−0.938888
$850$	−1.13619	−0.0389710
$851$	2.90696	0.0996494
$852$	8.45089	0.289523
$853$	41.6718	1.42682	0.713408	−	0.700749i	$-0.247151\pi$
$853$	41.6718	1.42682	0.713408	+	0.700749i	$0.247151\pi$
$854$	30.7354	1.05175
$855$	95.6858	3.27238
$856$	46.4953	1.58918
$857$	11.9968	0.409804	0.204902	−	0.978783i	$-0.434312\pi$
$857$	11.9968	0.409804	0.204902	+	0.978783i	$0.434312\pi$
$858$	−12.6916	−0.433284
$859$	13.4387	0.458522	0.229261	−	0.973365i	$-0.426369\pi$
$859$	13.4387	0.458522	0.229261	+	0.973365i	$0.426369\pi$
$860$	68.7485	2.34430
$861$	7.50519	0.255776
$862$	68.7870	2.34289
$863$	−14.2319	−0.484460	−0.242230	−	0.970219i	$-0.577879\pi$
$863$	−14.2319	−0.484460	−0.242230	+	0.970219i	$0.577879\pi$
$864$	117.826	4.00851
$865$	11.2842	0.383673
$866$	−87.7662	−2.98242
$867$	−30.0041	−1.01899
$868$	−96.3086	−3.26893
$869$	−15.4736	−0.524905
$870$	−28.6985	−0.972969
$871$	7.83467	0.265468
$872$	63.2937	2.14339
$873$	120.993	4.09500
$874$	58.8992	1.99229
$875$	33.5637	1.13466
$876$	−177.955	−6.01254
$877$	−26.3854	−0.890970	−0.445485	−	0.895289i	$-0.646969\pi$
$877$	−26.3854	−0.890970	−0.445485	+	0.895289i	$0.646969\pi$
$878$	101.614	3.42932
$879$	33.3484	1.12481
$880$	−35.2321	−1.18768
$881$	36.8854	1.24270	0.621351	−	0.783533i	$-0.286584\pi$
$881$	36.8854	1.24270	0.621351	+	0.783533i	$0.286584\pi$
$882$	−36.7956	−1.23897
$883$	10.4031	0.350091	0.175046	−	0.984560i	$-0.443993\pi$
$883$	10.4031	0.350091	0.175046	+	0.984560i	$0.443993\pi$
$884$	−25.1547	−0.846045
$885$	34.2722	1.15205
$886$	−16.7724	−0.563479
$887$	−47.6956	−1.60146	−0.800730	−	0.599025i	$-0.795555\pi$
$887$	−47.6956	−1.60146	−0.800730	+	0.599025i	$0.795555\pi$
$888$	19.5698	0.656720
$889$	3.86312	0.129565
$890$	98.8353	3.31297
$891$	21.2718	0.712633
$892$	35.1364	1.17645
$893$	−13.1827	−0.441142
$894$	51.5954	1.72561
$895$	14.5818	0.487416
$896$	−4.58919	−0.153314
$897$	10.5623	0.352663
$898$	59.4164	1.98275
$899$	−10.3200	−0.344190
$900$	2.64163	0.0880542
$901$	−38.0906	−1.26898
$902$	3.31949	0.110527
$903$	58.4330	1.94453
$904$	−44.0033	−1.46353
$905$	14.7303	0.489651
$906$	114.465	3.80286
$907$	−11.6537	−0.386953	−0.193477	−	0.981105i	$-0.561976\pi$
$907$	−11.6537	−0.386953	−0.193477	+	0.981105i	$0.561976\pi$
$908$	41.5109	1.37759
$909$	3.57202	0.118476
$910$	−17.8867	−0.592940
$911$	−24.9507	−0.826655	−0.413327	−	0.910583i	$-0.635633\pi$
$911$	−24.9507	−0.826655	−0.413327	+	0.910583i	$0.635633\pi$
$912$	200.293	6.63237
$913$	15.0371	0.497654
$914$	−78.3659	−2.59211
$915$	−26.8925	−0.889037
$916$	125.114	4.13390
$917$	11.4535	0.378228
$918$	145.017	4.78627
$919$	40.5180	1.33657	0.668283	−	0.743908i	$-0.267030\pi$
$919$	40.5180	1.33657	0.668283	+	0.743908i	$0.267030\pi$
$920$	58.0457	1.91371
$921$	−52.8890	−1.74275
$922$	−106.010	−3.49125
$923$	0.564401	0.0185775
$924$	−71.3451	−2.34708
$925$	0.0710465	0.00233599
$926$	70.7666	2.32553
$927$	11.8244	0.388363
$928$	−17.3736	−0.570315
$929$	3.21121	0.105356	0.0526781	−	0.998612i	$-0.483224\pi$
$929$	3.21121	0.105356	0.0526781	+	0.998612i	$0.483224\pi$
$930$	118.917	3.89943
$931$	−14.2086	−0.465668
$932$	49.6005	1.62472
$933$	4.89896	0.160385
$934$	−105.383	−3.44822
$935$	−18.3498	−0.600103
$936$	48.5960	1.58841
$937$	−27.9478	−0.913014	−0.456507	−	0.889720i	$-0.650900\pi$
$937$	−27.9478	−0.913014	−0.456507	+	0.889720i	$0.650900\pi$
$938$	62.1520	2.02933
$939$	26.9154	0.878350
$940$	−22.0643	−0.719658
$941$	1.75190	0.0571102	0.0285551	−	0.999592i	$-0.490909\pi$
$941$	1.75190	0.0571102	0.0285551	+	0.999592i	$0.490909\pi$
$942$	−88.0624	−2.86923
$943$	−2.76256	−0.0899614
$944$	49.0295	1.59577
$945$	73.0708	2.37699
$946$	25.8445	0.840277
$947$	−39.1647	−1.27268	−0.636340	−	0.771408i	$-0.719553\pi$
$947$	−39.1647	−1.27268	−0.636340	+	0.771408i	$0.719553\pi$
$948$	147.233	4.78190
$949$	−11.8849	−0.385800
$950$	1.43950	0.0467036
$951$	35.7399	1.15895
$952$	−117.497	−3.80811
$953$	−20.7747	−0.672959	−0.336480	−	0.941691i	$-0.609236\pi$
$953$	−20.7747	−0.672959	−0.336480	+	0.941691i	$0.609236\pi$
$954$	124.975	4.04623
$955$	25.3000	0.818689
$956$	−68.5294	−2.21640
$957$	−7.64499	−0.247128
$958$	30.3282	0.979860
$959$	−23.4900	−0.758532
$960$	62.3480	2.01227
$961$	11.7624	0.379431
$962$	2.21972	0.0715666
$963$	−40.1345	−1.29332
$964$	−56.4456	−1.81799
$965$	−37.9159	−1.22056
$966$	83.7898	2.69589
$967$	1.95654	0.0629180	0.0314590	−	0.999505i	$-0.489985\pi$
$967$	1.95654	0.0629180	0.0314590	+	0.999505i	$0.489985\pi$
$968$	63.9549	2.05559
$969$	104.318	3.35117
$970$	110.354	3.54326
$971$	−43.4425	−1.39414	−0.697068	−	0.717005i	$-0.745513\pi$
$971$	−43.4425	−1.39414	−0.697068	+	0.717005i	$0.745513\pi$
$972$	−46.2308	−1.48285
$973$	23.6891	0.759438
$974$	−43.0349	−1.37893
$975$	0.258143	0.00826718
$976$	−38.4721	−1.23146
$977$	22.2890	0.713087	0.356544	−	0.934279i	$-0.383955\pi$
$977$	22.2890	0.713087	0.356544	+	0.934279i	$0.383955\pi$
$978$	123.730	3.95645
$979$	26.3288	0.841471
$980$	−23.7814	−0.759669
$981$	−54.6349	−1.74436
$982$	−92.7932	−2.96115
$983$	39.7994	1.26940	0.634701	−	0.772758i	$-0.281123\pi$
$983$	39.7994	1.26940	0.634701	+	0.772758i	$0.281123\pi$
$984$	−18.5977	−0.592873
$985$	24.7944	0.790015
$986$	−21.3829	−0.680971
$987$	−18.7536	−0.596935
$988$	31.8700	1.01392
$989$	−21.5084	−0.683928
$990$	60.2058	1.91347
$991$	36.0335	1.14464	0.572320	−	0.820030i	$-0.306043\pi$
$991$	36.0335	1.14464	0.572320	+	0.820030i	$0.306043\pi$
$992$	71.9900	2.28569
$993$	−36.6116	−1.16183
$994$	4.47736	0.142013
$995$	31.9135	1.01173
$996$	−143.079	−4.53364
$997$	−25.7806	−0.816479	−0.408239	−	0.912875i	$-0.633857\pi$
$997$	−25.7806	−0.816479	−0.408239	+	0.912875i	$0.633857\pi$
$998$	39.8967	1.26291
$999$	−9.06797	−0.286898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.6	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-2.61991 q^{2} -3.07843 q^{3} +4.86391 q^{4} -2.25474 q^{5} +8.06519 q^{6} +3.02795 q^{7} -7.50318 q^{8} +6.47672 q^{9} +O(q^{10})\)	\(q-2.61991 q^{2} -3.07843 q^{3} +4.86391 q^{4} -2.25474 q^{5} +8.06519 q^{6} +3.02795 q^{7} -7.50318 q^{8} +6.47672 q^{9} +5.90721 q^{10} +1.57362 q^{11} -14.9732 q^{12} -1.00000 q^{13} -7.93295 q^{14} +6.94106 q^{15} +9.92982 q^{16} +5.17171 q^{17} -16.9684 q^{18} -6.55233 q^{19} -10.9669 q^{20} -9.32132 q^{21} -4.12275 q^{22} +3.43105 q^{23} +23.0980 q^{24} +0.0838553 q^{25} +2.61991 q^{26} -10.7028 q^{27} +14.7277 q^{28} +1.57815 q^{29} -18.1849 q^{30} -6.53929 q^{31} -11.0088 q^{32} -4.84429 q^{33} -13.5494 q^{34} -6.82724 q^{35} +31.5022 q^{36} +0.847251 q^{37} +17.1665 q^{38} +3.07843 q^{39} +16.9177 q^{40} -0.805164 q^{41} +24.4210 q^{42} -6.26875 q^{43} +7.65397 q^{44} -14.6033 q^{45} -8.98904 q^{46} +2.01191 q^{47} -30.5682 q^{48} +2.16848 q^{49} -0.219693 q^{50} -15.9207 q^{51} -4.86391 q^{52} -7.36518 q^{53} +28.0404 q^{54} -3.54811 q^{55} -22.7193 q^{56} +20.1709 q^{57} -4.13460 q^{58} +4.93760 q^{59} +33.7607 q^{60} -3.87440 q^{61} +17.1323 q^{62} +19.6112 q^{63} +8.98249 q^{64} +2.25474 q^{65} +12.6916 q^{66} -7.83467 q^{67} +25.1547 q^{68} -10.5623 q^{69} +17.8867 q^{70} -0.564401 q^{71} -48.5960 q^{72} +11.8849 q^{73} -2.21972 q^{74} -0.258143 q^{75} -31.8700 q^{76} +4.76485 q^{77} -8.06519 q^{78} -9.83308 q^{79} -22.3892 q^{80} +13.5177 q^{81} +2.10945 q^{82} +9.55569 q^{83} -45.3381 q^{84} -11.6609 q^{85} +16.4235 q^{86} -4.85821 q^{87} -11.8072 q^{88} +16.7313 q^{89} +38.2593 q^{90} -3.02795 q^{91} +16.6883 q^{92} +20.1307 q^{93} -5.27101 q^{94} +14.7738 q^{95} +33.8899 q^{96} +18.6813 q^{97} -5.68121 q^{98} +10.1919 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 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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