LMFDB - Embedded newform 8009.2.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9521869788$
Analytic rank:	$1$
Dimension:	$306$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72646 q^{2} +1.03232 q^{3} +5.43357 q^{4} +3.59624 q^{5} -2.81456 q^{6} +2.40174 q^{7} -9.36147 q^{8} -1.93433 q^{9} +O(q^{10})$	\(q-2.72646 q^{2} +1.03232 q^{3} +5.43357 q^{4} +3.59624 q^{5} -2.81456 q^{6} +2.40174 q^{7} -9.36147 q^{8} -1.93433 q^{9} -9.80500 q^{10} +0.0963853 q^{11} +5.60915 q^{12} +0.179886 q^{13} -6.54824 q^{14} +3.71245 q^{15} +14.6565 q^{16} +0.968171 q^{17} +5.27386 q^{18} -6.43552 q^{19} +19.5404 q^{20} +2.47935 q^{21} -0.262790 q^{22} -4.26098 q^{23} -9.66399 q^{24} +7.93296 q^{25} -0.490452 q^{26} -5.09378 q^{27} +13.0500 q^{28} -0.333223 q^{29} -10.1218 q^{30} -6.42741 q^{31} -21.2374 q^{32} +0.0995000 q^{33} -2.63968 q^{34} +8.63723 q^{35} -10.5103 q^{36} +1.85766 q^{37} +17.5462 q^{38} +0.185699 q^{39} -33.6661 q^{40} -0.876241 q^{41} -6.75984 q^{42} -6.17836 q^{43} +0.523716 q^{44} -6.95630 q^{45} +11.6174 q^{46} -3.29158 q^{47} +15.1301 q^{48} -1.23165 q^{49} -21.6289 q^{50} +0.999457 q^{51} +0.977424 q^{52} -5.24865 q^{53} +13.8880 q^{54} +0.346625 q^{55} -22.4838 q^{56} -6.64349 q^{57} +0.908519 q^{58} -5.23317 q^{59} +20.1719 q^{60} -9.59241 q^{61} +17.5241 q^{62} -4.64574 q^{63} +28.5899 q^{64} +0.646914 q^{65} -0.271282 q^{66} -11.9620 q^{67} +5.26062 q^{68} -4.39867 q^{69} -23.5490 q^{70} +13.8241 q^{71} +18.1081 q^{72} +8.49684 q^{73} -5.06484 q^{74} +8.18931 q^{75} -34.9679 q^{76} +0.231492 q^{77} -0.506301 q^{78} -2.64825 q^{79} +52.7084 q^{80} +0.544592 q^{81} +2.38903 q^{82} -16.8441 q^{83} +13.4717 q^{84} +3.48178 q^{85} +16.8450 q^{86} -0.343991 q^{87} -0.902308 q^{88} +9.05830 q^{89} +18.9661 q^{90} +0.432040 q^{91} -23.1523 q^{92} -6.63511 q^{93} +8.97434 q^{94} -23.1437 q^{95} -21.9237 q^{96} +7.02071 q^{97} +3.35804 q^{98} -0.186440 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) $ 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * 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 $

$2$	−2.72646	−1.92790	−0.963948	−	0.266090i	$-0.914268\pi$
$2$	−2.72646	−1.92790	−0.963948	+	0.266090i	$0.914268\pi$
$3$	1.03232	0.596007	0.298004	−	0.954565i	$-0.403679\pi$
$3$	1.03232	0.596007	0.298004	+	0.954565i	$0.403679\pi$
$4$	5.43357	2.71678
$5$	3.59624	1.60829	0.804144	−	0.594434i	$-0.202624\pi$
$5$	3.59624	1.60829	0.804144	+	0.594434i	$0.202624\pi$
$6$	−2.81456	−1.14904
$7$	2.40174	0.907772	0.453886	−	0.891060i	$-0.350037\pi$
$7$	2.40174	0.907772	0.453886	+	0.891060i	$0.350037\pi$
$8$	−9.36147	−3.30978
$9$	−1.93433	−0.644775
$10$	−9.80500	−3.10061
$11$	0.0963853	0.0290613	0.0145306	−	0.999894i	$-0.495375\pi$
$11$	0.0963853	0.0290613	0.0145306	+	0.999894i	$0.495375\pi$
$12$	5.60915	1.61922
$13$	0.179886	0.0498914	0.0249457	−	0.999689i	$-0.492059\pi$
$13$	0.179886	0.0498914	0.0249457	+	0.999689i	$0.492059\pi$
$14$	−6.54824	−1.75009
$15$	3.71245	0.958552
$16$	14.6565	3.66413
$17$	0.968171	0.234816	0.117408	−	0.993084i	$-0.462541\pi$
$17$	0.968171	0.234816	0.117408	+	0.993084i	$0.462541\pi$
$18$	5.27386	1.24306
$19$	−6.43552	−1.47641	−0.738205	−	0.674576i	$-0.764326\pi$
$19$	−6.43552	−1.47641	−0.738205	+	0.674576i	$0.764326\pi$
$20$	19.5404	4.36937
$21$	2.47935	0.541039
$22$	−0.262790	−0.0560271
$23$	−4.26098	−0.888476	−0.444238	−	0.895909i	$-0.646525\pi$
$23$	−4.26098	−0.888476	−0.444238	+	0.895909i	$0.646525\pi$
$24$	−9.66399	−1.97265
$25$	7.93296	1.58659
$26$	−0.490452	−0.0961855
$27$	−5.09378	−0.980298
$28$	13.0500	2.46622
$29$	−0.333223	−0.0618780	−0.0309390	−	0.999521i	$-0.509850\pi$
$29$	−0.333223	−0.0618780	−0.0309390	+	0.999521i	$0.509850\pi$
$30$	−10.1218	−1.84799
$31$	−6.42741	−1.15440	−0.577198	−	0.816604i	$-0.695854\pi$
$31$	−6.42741	−1.15440	−0.577198	+	0.816604i	$0.695854\pi$
$32$	−21.2374	−3.75428
$33$	0.0995000	0.0173207
$34$	−2.63968	−0.452701
$35$	8.63723	1.45996
$36$	−10.5103	−1.75171
$37$	1.85766	0.305398	0.152699	−	0.988273i	$-0.451203\pi$
$37$	1.85766	0.305398	0.152699	+	0.988273i	$0.451203\pi$
$38$	17.5462	2.84637
$39$	0.185699	0.0297357
$40$	−33.6661	−5.32308
$41$	−0.876241	−0.136846	−0.0684229	−	0.997656i	$-0.521797\pi$
$41$	−0.876241	−0.136846	−0.0684229	+	0.997656i	$0.521797\pi$
$42$	−6.75984	−1.04307
$43$	−6.17836	−0.942190	−0.471095	−	0.882082i	$-0.656141\pi$
$43$	−6.17836	−0.942190	−0.471095	+	0.882082i	$0.656141\pi$
$44$	0.523716	0.0789531
$45$	−6.95630	−1.03698
$46$	11.6174	1.71289
$47$	−3.29158	−0.480126	−0.240063	−	0.970757i	$-0.577168\pi$
$47$	−3.29158	−0.480126	−0.240063	+	0.970757i	$0.577168\pi$
$48$	15.1301	2.18385
$49$	−1.23165	−0.175950
$50$	−21.6289	−3.05878
$51$	0.999457	0.139952
$52$	0.977424	0.135544
$53$	−5.24865	−0.720958	−0.360479	−	0.932767i	$-0.617387\pi$
$53$	−5.24865	−0.720958	−0.360479	+	0.932767i	$0.617387\pi$
$54$	13.8880	1.88991
$55$	0.346625	0.0467389
$56$	−22.4838	−3.00453
$57$	−6.64349	−0.879952
$58$	0.908519	0.119294
$59$	−5.23317	−0.681301	−0.340651	−	0.940190i	$-0.610647\pi$
$59$	−5.23317	−0.681301	−0.340651	+	0.940190i	$0.610647\pi$
$60$	20.1719	2.60418
$61$	−9.59241	−1.22818	−0.614091	−	0.789235i	$-0.710477\pi$
$61$	−9.59241	−1.22818	−0.614091	+	0.789235i	$0.710477\pi$
$62$	17.5241	2.22556
$63$	−4.64574	−0.585309
$64$	28.5899	3.57373
$65$	0.646914	0.0802398
$66$	−0.271282	−0.0333926
$67$	−11.9620	−1.46139	−0.730693	−	0.682706i	$-0.760803\pi$
$67$	−11.9620	−1.46139	−0.730693	+	0.682706i	$0.760803\pi$
$68$	5.26062	0.637944
$69$	−4.39867	−0.529538
$70$	−23.5490	−2.81465
$71$	13.8241	1.64061	0.820307	−	0.571923i	$-0.193802\pi$
$71$	13.8241	1.64061	0.820307	+	0.571923i	$0.193802\pi$
$72$	18.1081	2.13406
$73$	8.49684	0.994480	0.497240	−	0.867613i	$-0.334347\pi$
$73$	8.49684	0.994480	0.497240	+	0.867613i	$0.334347\pi$
$74$	−5.06484	−0.588776
$75$	8.18931	0.945620
$76$	−34.9679	−4.01109
$77$	0.231492	0.0263810
$78$	−0.506301	−0.0573273
$79$	−2.64825	−0.297951	−0.148976	−	0.988841i	$-0.547598\pi$
$79$	−2.64825	−0.297951	−0.148976	+	0.988841i	$0.547598\pi$
$80$	52.7084	5.89298
$81$	0.544592	0.0605102
$82$	2.38903	0.263825
$83$	−16.8441	−1.84888	−0.924439	−	0.381330i	$-0.875466\pi$
$83$	−16.8441	−1.84888	−0.924439	+	0.381330i	$0.875466\pi$
$84$	13.4717	1.46989
$85$	3.48178	0.377652
$86$	16.8450	1.81645
$87$	−0.343991	−0.0368797
$88$	−0.902308	−0.0961864
$89$	9.05830	0.960178	0.480089	−	0.877220i	$-0.340604\pi$
$89$	9.05830	0.960178	0.480089	+	0.877220i	$0.340604\pi$
$90$	18.9661	1.99920
$91$	0.432040	0.0452901
$92$	−23.1523	−2.41380
$93$	−6.63511	−0.688029
$94$	8.97434	0.925632
$95$	−23.1437	−2.37449
$96$	−21.9237	−2.23758
$97$	7.02071	0.712845	0.356422	−	0.934325i	$-0.383996\pi$
$97$	7.02071	0.712845	0.356422	+	0.934325i	$0.383996\pi$
$98$	3.35804	0.339214
$99$	−0.186440	−0.0187380
$100$	43.1042	4.31042
$101$	16.9629	1.68787	0.843934	−	0.536447i	$-0.180234\pi$
$101$	16.9629	1.68787	0.843934	+	0.536447i	$0.180234\pi$
$102$	−2.72498	−0.269813
$103$	−0.416716	−0.0410602	−0.0205301	−	0.999789i	$-0.506535\pi$
$103$	−0.416716	−0.0410602	−0.0205301	+	0.999789i	$0.506535\pi$
$104$	−1.68400	−0.165130
$105$	8.91635	0.870146
$106$	14.3102	1.38993
$107$	−4.36039	−0.421534	−0.210767	−	0.977536i	$-0.567596\pi$
$107$	−4.36039	−0.421534	−0.210767	+	0.977536i	$0.567596\pi$
$108$	−27.6774	−2.66326
$109$	−4.88573	−0.467968	−0.233984	−	0.972240i	$-0.575176\pi$
$109$	−4.88573	−0.467968	−0.233984	+	0.972240i	$0.575176\pi$
$110$	−0.945057	−0.0901077
$111$	1.91769	0.182020
$112$	35.2011	3.32619
$113$	8.18673	0.770143	0.385071	−	0.922887i	$-0.374177\pi$
$113$	8.18673	0.770143	0.385071	+	0.922887i	$0.374177\pi$
$114$	18.1132	1.69646
$115$	−15.3235	−1.42893
$116$	−1.81059	−0.168109
$117$	−0.347958	−0.0321688
$118$	14.2680	1.31348
$119$	2.32529	0.213159
$120$	−34.7540	−3.17260
$121$	−10.9907	−0.999155
$122$	26.1533	2.36781
$123$	−0.904557	−0.0815612
$124$	−34.9238	−3.13625
$125$	10.5476	0.943408
$126$	12.6664	1.12841
$127$	3.73989	0.331861	0.165931	−	0.986137i	$-0.446937\pi$
$127$	3.73989	0.331861	0.165931	+	0.986137i	$0.446937\pi$
$128$	−35.4742	−3.13551
$129$	−6.37801	−0.561552
$130$	−1.76378	−0.154694
$131$	5.88527	0.514198	0.257099	−	0.966385i	$-0.417233\pi$
$131$	5.88527	0.514198	0.257099	+	0.966385i	$0.417233\pi$
$132$	0.540640	0.0470567
$133$	−15.4564	−1.34024
$134$	32.6138	2.81740
$135$	−18.3185	−1.57660
$136$	−9.06350	−0.777189
$137$	5.95430	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$137$	5.95430	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$138$	11.9928	1.02089
$139$	−15.7362	−1.33472	−0.667362	−	0.744734i	$-0.732577\pi$
$139$	−15.7362	−1.33472	−0.667362	+	0.744734i	$0.732577\pi$
$140$	46.9310	3.96639
$141$	−3.39794	−0.286158
$142$	−37.6907	−3.16293
$143$	0.0173384	0.00144991
$144$	−28.3505	−2.36254
$145$	−1.19835	−0.0995177
$146$	−23.1663	−1.91725
$147$	−1.27145	−0.104868
$148$	10.0937	0.829701
$149$	5.68984	0.466130	0.233065	−	0.972461i	$-0.425124\pi$
$149$	5.68984	0.466130	0.233065	+	0.972461i	$0.425124\pi$
$150$	−22.3278	−1.82306
$151$	10.0600	0.818670	0.409335	−	0.912384i	$-0.365761\pi$
$151$	10.0600	0.818670	0.409335	+	0.912384i	$0.365761\pi$
$152$	60.2460	4.88659
$153$	−1.87276	−0.151403
$154$	−0.631154	−0.0508598
$155$	−23.1145	−1.85660
$156$	1.00901	0.0807854
$157$	0.139515	0.0111345	0.00556724	−	0.999985i	$-0.498228\pi$
$157$	0.139515	0.0111345	0.00556724	+	0.999985i	$0.498228\pi$
$158$	7.22033	0.574419
$159$	−5.41826	−0.429696
$160$	−76.3749	−6.03797
$161$	−10.2338	−0.806533
$162$	−1.48481	−0.116657
$163$	−4.76178	−0.372972	−0.186486	−	0.982458i	$-0.559710\pi$
$163$	−4.76178	−0.372972	−0.186486	+	0.982458i	$0.559710\pi$
$164$	−4.76112	−0.371781
$165$	0.357826	0.0278567
$166$	45.9246	3.56444
$167$	−12.0574	−0.933026	−0.466513	−	0.884514i	$-0.654490\pi$
$167$	−12.0574	−0.933026	−0.466513	+	0.884514i	$0.654490\pi$
$168$	−23.2104	−1.79072
$169$	−12.9676	−0.997511
$170$	−9.49291	−0.728073
$171$	12.4484	0.951953
$172$	−33.5705	−2.55973
$173$	14.8389	1.12818	0.564091	−	0.825713i	$-0.309227\pi$
$173$	14.8389	1.12818	0.564091	+	0.825713i	$0.309227\pi$
$174$	0.937878	0.0711003
$175$	19.0529	1.44026
$176$	1.41267	0.106484
$177$	−5.40228	−0.406060
$178$	−24.6971	−1.85112
$179$	−16.3521	−1.22221	−0.611107	−	0.791548i	$-0.709276\pi$
$179$	−16.3521	−1.22221	−0.611107	+	0.791548i	$0.709276\pi$
$180$	−37.7975	−2.81726
$181$	12.8465	0.954875	0.477438	−	0.878666i	$-0.341566\pi$
$181$	12.8465	0.954875	0.477438	+	0.878666i	$0.341566\pi$
$182$	−1.17794	−0.0873145
$183$	−9.90239	−0.732006
$184$	39.8890	2.94066
$185$	6.68061	0.491168
$186$	18.0903	1.32645
$187$	0.0933174	0.00682404
$188$	−17.8850	−1.30440
$189$	−12.2339	−0.889887
$190$	63.1003	4.57778
$191$	−8.76958	−0.634544	−0.317272	−	0.948335i	$-0.602767\pi$
$191$	−8.76958	−0.634544	−0.317272	+	0.948335i	$0.602767\pi$
$192$	29.5138	2.12997
$193$	−14.0795	−1.01347	−0.506734	−	0.862102i	$-0.669147\pi$
$193$	−14.0795	−1.01347	−0.506734	+	0.862102i	$0.669147\pi$
$194$	−19.1417	−1.37429
$195$	0.667819	0.0478235
$196$	−6.69226	−0.478019
$197$	−14.7302	−1.04948	−0.524741	−	0.851262i	$-0.675838\pi$
$197$	−14.7302	−1.04948	−0.524741	+	0.851262i	$0.675838\pi$
$198$	0.508322	0.0361249
$199$	−18.0055	−1.27638	−0.638188	−	0.769881i	$-0.720316\pi$
$199$	−18.0055	−1.27638	−0.638188	+	0.769881i	$0.720316\pi$
$200$	−74.2642	−5.25127
$201$	−12.3485	−0.870997
$202$	−46.2485	−3.25404
$203$	−0.800315	−0.0561711
$204$	5.43062	0.380219
$205$	−3.15118	−0.220088
$206$	1.13616	0.0791598
$207$	8.24212	0.572867
$208$	2.63651	0.182809
$209$	−0.620290	−0.0429063
$210$	−24.3100	−1.67755
$211$	18.6898	1.28666	0.643329	−	0.765590i	$-0.277553\pi$
$211$	18.6898	1.28666	0.643329	+	0.765590i	$0.277553\pi$
$212$	−28.5189	−1.95869
$213$	14.2708	0.977819
$214$	11.8884	0.812674
$215$	−22.2189	−1.51531
$216$	47.6853	3.24457
$217$	−15.4370	−1.04793
$218$	13.3207	0.902194
$219$	8.77141	0.592717
$220$	1.88341	0.126979
$221$	0.174161	0.0117153
$222$	−5.22851	−0.350915
$223$	3.30063	0.221026	0.110513	−	0.993875i	$-0.464751\pi$
$223$	3.30063	0.221026	0.110513	+	0.993875i	$0.464751\pi$
$224$	−51.0067	−3.40803
$225$	−15.3449	−1.02299
$226$	−22.3208	−1.48475
$227$	−23.3067	−1.54692	−0.773461	−	0.633844i	$-0.781476\pi$
$227$	−23.3067	−1.54692	−0.773461	+	0.633844i	$0.781476\pi$
$228$	−36.0978	−2.39064
$229$	−22.6377	−1.49594	−0.747972	−	0.663731i	$-0.768972\pi$
$229$	−22.6377	−1.49594	−0.747972	+	0.663731i	$0.768972\pi$
$230$	41.7789	2.75482
$231$	0.238973	0.0157233
$232$	3.11946	0.204803
$233$	−24.8237	−1.62626	−0.813128	−	0.582084i	$-0.802237\pi$
$233$	−24.8237	−1.62626	−0.813128	+	0.582084i	$0.802237\pi$
$234$	0.948694	0.0620180
$235$	−11.8373	−0.772180
$236$	−28.4348	−1.85095
$237$	−2.73383	−0.177581
$238$	−6.33981	−0.410949
$239$	−6.13336	−0.396734	−0.198367	−	0.980128i	$-0.563564\pi$
$239$	−6.13336	−0.396734	−0.198367	+	0.980128i	$0.563564\pi$
$240$	54.4117	3.51226
$241$	−15.6458	−1.00784	−0.503919	−	0.863751i	$-0.668109\pi$
$241$	−15.6458	−1.00784	−0.503919	+	0.863751i	$0.668109\pi$
$242$	29.9657	1.92627
$243$	15.8435	1.01636
$244$	−52.1210	−3.33671
$245$	−4.42932	−0.282979
$246$	2.46624	0.157241
$247$	−1.15766	−0.0736603
$248$	60.1700	3.82080
$249$	−17.3884	−1.10194
$250$	−28.7576	−1.81879
$251$	27.0625	1.70817	0.854084	−	0.520134i	$-0.174118\pi$
$251$	27.0625	1.70817	0.854084	+	0.520134i	$0.174118\pi$
$252$	−25.2430	−1.59016
$253$	−0.410696	−0.0258202
$254$	−10.1966	−0.639794
$255$	3.59429	0.225083
$256$	39.5392	2.47120
$257$	12.8113	0.799145	0.399572	−	0.916702i	$-0.369159\pi$
$257$	12.8113	0.799145	0.399572	+	0.916702i	$0.369159\pi$
$258$	17.3894	1.08261
$259$	4.46162	0.277232
$260$	3.51505	0.217994
$261$	0.644562	0.0398974
$262$	−16.0459	−0.991321
$263$	−28.8730	−1.78039	−0.890194	−	0.455582i	$-0.849431\pi$
$263$	−28.8730	−1.78039	−0.890194	+	0.455582i	$0.849431\pi$
$264$	−0.931466	−0.0573278
$265$	−18.8754	−1.15951
$266$	42.1413	2.58385
$267$	9.35102	0.572273
$268$	−64.9961	−3.97027
$269$	20.2908	1.23715	0.618577	−	0.785724i	$-0.287709\pi$
$269$	20.2908	1.23715	0.618577	+	0.785724i	$0.287709\pi$
$270$	49.9445	3.03953
$271$	−12.6756	−0.769989	−0.384995	−	0.922919i	$-0.625797\pi$
$271$	−12.6756	−0.769989	−0.384995	+	0.922919i	$0.625797\pi$
$272$	14.1900	0.860396
$273$	0.446001	0.0269932
$274$	−16.2341	−0.980740
$275$	0.764620	0.0461083
$276$	−23.9005	−1.43864
$277$	−15.4109	−0.925950	−0.462975	−	0.886371i	$-0.653218\pi$
$277$	−15.4109	−0.925950	−0.462975	+	0.886371i	$0.653218\pi$
$278$	42.9040	2.57321
$279$	12.4327	0.744326
$280$	−80.8572	−4.83214
$281$	18.7827	1.12048	0.560240	−	0.828330i	$-0.310709\pi$
$281$	18.7827	1.12048	0.560240	+	0.828330i	$0.310709\pi$
$282$	9.26434	0.551684
$283$	3.95454	0.235073	0.117536	−	0.993069i	$-0.462500\pi$
$283$	3.95454	0.235073	0.117536	+	0.993069i	$0.462500\pi$
$284$	75.1140	4.45720
$285$	−23.8916	−1.41522
$286$	−0.0472723	−0.00279527
$287$	−2.10450	−0.124225
$288$	41.0801	2.42067
$289$	−16.0626	−0.944862
$290$	3.26725	0.191860
$291$	7.24758	0.424861
$292$	46.1681	2.70179
$293$	−2.56741	−0.149990	−0.0749950	−	0.997184i	$-0.523894\pi$
$293$	−2.56741	−0.149990	−0.0749950	+	0.997184i	$0.523894\pi$
$294$	3.46656	0.202174
$295$	−18.8198	−1.09573
$296$	−17.3905	−1.01080
$297$	−0.490965	−0.0284887
$298$	−15.5131	−0.898651
$299$	−0.766491	−0.0443273
$300$	44.4972	2.56905
$301$	−14.8388	−0.855294
$302$	−27.4281	−1.57831
$303$	17.5110	1.00598
$304$	−94.3224	−5.40976
$305$	−34.4966	−1.97527
$306$	5.10599	0.291890
$307$	22.3558	1.27591	0.637956	−	0.770073i	$-0.279780\pi$
$307$	22.3558	1.27591	0.637956	+	0.770073i	$0.279780\pi$
$308$	1.25783	0.0716714
$309$	−0.430182	−0.0244722
$310$	63.0207	3.57934
$311$	3.45802	0.196086	0.0980432	−	0.995182i	$-0.468742\pi$
$311$	3.45802	0.196086	0.0980432	+	0.995182i	$0.468742\pi$
$312$	−1.73842	−0.0984185
$313$	16.4113	0.927621	0.463810	−	0.885934i	$-0.346482\pi$
$313$	16.4113	0.927621	0.463810	+	0.885934i	$0.346482\pi$
$314$	−0.380381	−0.0214661
$315$	−16.7072	−0.941345
$316$	−14.3894	−0.809469
$317$	8.25806	0.463819	0.231910	−	0.972737i	$-0.425503\pi$
$317$	8.25806	0.463819	0.231910	+	0.972737i	$0.425503\pi$
$318$	14.7727	0.828410
$319$	−0.0321178	−0.00179825
$320$	102.816	5.74760
$321$	−4.50129	−0.251238
$322$	27.9019	1.55491
$323$	−6.23069	−0.346685
$324$	2.95908	0.164393
$325$	1.42703	0.0791573
$326$	12.9828	0.719051
$327$	−5.04361	−0.278913
$328$	8.20291	0.452930
$329$	−7.90550	−0.435845
$330$	−0.975597	−0.0537048
$331$	−2.39204	−0.131478	−0.0657391	−	0.997837i	$-0.520941\pi$
$331$	−2.39204	−0.131478	−0.0657391	+	0.997837i	$0.520941\pi$
$332$	−91.5234	−5.02300
$333$	−3.59333	−0.196913
$334$	32.8739	1.79878
$335$	−43.0181	−2.35033
$336$	36.3387	1.98244
$337$	19.5241	1.06354	0.531772	−	0.846888i	$-0.321526\pi$
$337$	19.5241	1.06354	0.531772	+	0.846888i	$0.321526\pi$
$338$	35.3557	1.92310
$339$	8.45128	0.459011
$340$	18.9185	1.02600
$341$	−0.619508	−0.0335482
$342$	−33.9400	−1.83527
$343$	−19.7703	−1.06749
$344$	57.8385	3.11844
$345$	−15.8187	−0.851650
$346$	−40.4577	−2.17502
$347$	30.3732	1.63052	0.815258	−	0.579098i	$-0.196595\pi$
$347$	30.3732	1.63052	0.815258	+	0.579098i	$0.196595\pi$
$348$	−1.86910	−0.100194
$349$	0.133065	0.00712278	0.00356139	−	0.999994i	$-0.498866\pi$
$349$	0.133065	0.00712278	0.00356139	+	0.999994i	$0.498866\pi$
$350$	−51.9469	−2.77668
$351$	−0.916300	−0.0489085
$352$	−2.04697	−0.109104
$353$	31.5690	1.68025	0.840125	−	0.542393i	$-0.182481\pi$
$353$	31.5690	1.68025	0.840125	+	0.542393i	$0.182481\pi$
$354$	14.7291	0.782842
$355$	49.7147	2.63858
$356$	49.2189	2.60860
$357$	2.40043	0.127044
$358$	44.5833	2.35630
$359$	−6.69735	−0.353472	−0.176736	−	0.984258i	$-0.556554\pi$
$359$	−6.69735	−0.353472	−0.176736	+	0.984258i	$0.556554\pi$
$360$	65.1212	3.43219
$361$	22.4160	1.17979
$362$	−35.0255	−1.84090
$363$	−11.3459	−0.595504
$364$	2.34752	0.123043
$365$	30.5567	1.59941
$366$	26.9985	1.41123
$367$	9.64158	0.503287	0.251643	−	0.967820i	$-0.419029\pi$
$367$	9.64158	0.503287	0.251643	+	0.967820i	$0.419029\pi$
$368$	−62.4511	−3.25549
$369$	1.69494	0.0882348
$370$	−18.2144	−0.946921
$371$	−12.6059	−0.654465
$372$	−36.0523	−1.86923
$373$	24.5489	1.27109	0.635547	−	0.772062i	$-0.280775\pi$
$373$	24.5489	1.27109	0.635547	+	0.772062i	$0.280775\pi$
$374$	−0.254426	−0.0131560
$375$	10.8885	0.562278
$376$	30.8140	1.58911
$377$	−0.0599423	−0.00308718
$378$	33.3553	1.71561
$379$	12.6878	0.651729	0.325864	−	0.945417i	$-0.394345\pi$
$379$	12.6878	0.651729	0.325864	+	0.945417i	$0.394345\pi$
$380$	−125.753	−6.45099
$381$	3.86074	0.197792
$382$	23.9099	1.22334
$383$	19.0772	0.974801	0.487400	−	0.873179i	$-0.337945\pi$
$383$	19.0772	0.974801	0.487400	+	0.873179i	$0.337945\pi$
$384$	−36.6206	−1.86879
$385$	0.832502	0.0424282
$386$	38.3873	1.95386
$387$	11.9510	0.607501
$388$	38.1475	1.93664
$389$	−6.58022	−0.333630	−0.166815	−	0.985988i	$-0.553348\pi$
$389$	−6.58022	−0.333630	−0.166815	+	0.985988i	$0.553348\pi$
$390$	−1.82078	−0.0921988
$391$	−4.12536	−0.208628
$392$	11.5301	0.582357
$393$	6.07545	0.306466
$394$	40.1612	2.02329
$395$	−9.52374	−0.479191
$396$	−1.01304	−0.0509070
$397$	38.1650	1.91545	0.957724	−	0.287688i	$-0.0928866\pi$
$397$	38.1650	1.91545	0.957724	+	0.287688i	$0.0928866\pi$
$398$	49.0912	2.46072
$399$	−15.9559	−0.798795
$400$	116.270	5.81348
$401$	−3.37387	−0.168483	−0.0842414	−	0.996445i	$-0.526847\pi$
$401$	−3.37387	−0.168483	−0.0842414	+	0.996445i	$0.526847\pi$
$402$	33.6677	1.67919
$403$	−1.15620	−0.0575945
$404$	92.1689	4.58557
$405$	1.95848	0.0973179
$406$	2.18202	0.108292
$407$	0.179051	0.00887525
$408$	−9.35639	−0.463210
$409$	−9.47071	−0.468297	−0.234148	−	0.972201i	$-0.575230\pi$
$409$	−9.47071	−0.468297	−0.234148	+	0.972201i	$0.575230\pi$
$410$	8.59154	0.424306
$411$	6.14671	0.303195
$412$	−2.26425	−0.111552
$413$	−12.5687	−0.618466
$414$	−22.4718	−1.10443
$415$	−60.5754	−2.97353
$416$	−3.82032	−0.187307
$417$	−16.2447	−0.795505
$418$	1.69119	0.0827190
$419$	−17.7274	−0.866042	−0.433021	−	0.901384i	$-0.642552\pi$
$419$	−17.7274	−0.866042	−0.433021	+	0.901384i	$0.642552\pi$
$420$	48.4476	2.36400
$421$	−5.71347	−0.278458	−0.139229	−	0.990260i	$-0.544462\pi$
$421$	−5.71347	−0.278458	−0.139229	+	0.990260i	$0.544462\pi$
$422$	−50.9569	−2.48054
$423$	6.36698	0.309573
$424$	49.1351	2.38621
$425$	7.68045	0.372557
$426$	−38.9087	−1.88513
$427$	−23.0385	−1.11491
$428$	−23.6924	−1.14522
$429$	0.0178987	0.000864156 0
$430$	60.5788	2.92137
$431$	0.712408	0.0343155	0.0171577	−	0.999853i	$-0.494538\pi$
$431$	0.712408	0.0343155	0.0171577	+	0.999853i	$0.494538\pi$
$432$	−74.6571	−3.59194
$433$	−12.3697	−0.594451	−0.297226	−	0.954807i	$-0.596061\pi$
$433$	−12.3697	−0.594451	−0.297226	+	0.954807i	$0.596061\pi$
$434$	42.0882	2.02030
$435$	−1.23708	−0.0593133
$436$	−26.5470	−1.27137
$437$	27.4216	1.31175
$438$	−23.9149	−1.14270
$439$	−12.4337	−0.593430	−0.296715	−	0.954966i	$-0.595891\pi$
$439$	−12.4337	−0.593430	−0.296715	+	0.954966i	$0.595891\pi$
$440$	−3.24492	−0.154695
$441$	2.38241	0.113448
$442$	−0.474841	−0.0225859
$443$	17.6056	0.836469	0.418234	−	0.908339i	$-0.362649\pi$
$443$	17.6056	0.836469	0.418234	+	0.908339i	$0.362649\pi$
$444$	10.4199	0.494508
$445$	32.5758	1.54424
$446$	−8.99902	−0.426116
$447$	5.87371	0.277817
$448$	68.6654	3.24414
$449$	13.7417	0.648512	0.324256	−	0.945969i	$-0.394886\pi$
$449$	13.7417	0.648512	0.324256	+	0.945969i	$0.394886\pi$
$450$	41.8373	1.97223
$451$	−0.0844567	−0.00397691
$452$	44.4831	2.09231
$453$	10.3851	0.487934
$454$	63.5448	2.98231
$455$	1.55372	0.0728395
$456$	62.1928	2.91245
$457$	−38.5545	−1.80350	−0.901751	−	0.432256i	$-0.857718\pi$
$457$	−38.5545	−1.80350	−0.901751	+	0.432256i	$0.857718\pi$
$458$	61.7208	2.88402
$459$	−4.93165	−0.230190
$460$	−83.2613	−3.88208
$461$	−19.0727	−0.888304	−0.444152	−	0.895951i	$-0.646495\pi$
$461$	−19.0727	−0.888304	−0.444152	+	0.895951i	$0.646495\pi$
$462$	−0.651549	−0.0303128
$463$	−0.651556	−0.0302804	−0.0151402	−	0.999885i	$-0.504819\pi$
$463$	−0.651556	−0.0302804	−0.0151402	+	0.999885i	$0.504819\pi$
$464$	−4.88389	−0.226729
$465$	−23.8615	−1.10655
$466$	67.6808	3.13525
$467$	−20.6935	−0.957581	−0.478791	−	0.877929i	$-0.658925\pi$
$467$	−20.6935	−0.957581	−0.478791	+	0.877929i	$0.658925\pi$
$468$	−1.89066	−0.0873956
$469$	−28.7295	−1.32661
$470$	32.2739	1.48868
$471$	0.144023	0.00663623
$472$	48.9902	2.25496
$473$	−0.595502	−0.0273812
$474$	7.45366	0.342358
$475$	−51.0527	−2.34246
$476$	12.6346	0.579108
$477$	10.1526	0.464856
$478$	16.7223	0.764862
$479$	24.0832	1.10039	0.550194	−	0.835037i	$-0.314554\pi$
$479$	24.0832	1.10039	0.550194	+	0.835037i	$0.314554\pi$
$480$	−78.8430	−3.59867
$481$	0.334168	0.0152368
$482$	42.6577	1.94301
$483$	−10.5645	−0.480700
$484$	−59.7188	−2.71449
$485$	25.2482	1.14646
$486$	−43.1967	−1.95944
$487$	−3.46605	−0.157062	−0.0785308	−	0.996912i	$-0.525023\pi$
$487$	−3.46605	−0.157062	−0.0785308	+	0.996912i	$0.525023\pi$
$488$	89.7991	4.06502
$489$	−4.91566	−0.222294
$490$	12.0763	0.545553
$491$	−35.7174	−1.61191	−0.805953	−	0.591980i	$-0.798346\pi$
$491$	−35.7174	−1.61191	−0.805953	+	0.591980i	$0.798346\pi$
$492$	−4.91497	−0.221584
$493$	−0.322617	−0.0145299
$494$	3.15631	0.142009
$495$	−0.670485	−0.0301361
$496$	−94.2035	−4.22986
$497$	33.2018	1.48930
$498$	47.4087	2.12444
$499$	7.66512	0.343138	0.171569	−	0.985172i	$-0.445116\pi$
$499$	7.66512	0.343138	0.171569	+	0.985172i	$0.445116\pi$
$500$	57.3112	2.56303
$501$	−12.4470	−0.556090
$502$	−73.7847	−3.29317
$503$	26.2455	1.17023	0.585114	−	0.810951i	$-0.301050\pi$
$503$	26.2455	1.17023	0.585114	+	0.810951i	$0.301050\pi$
$504$	43.4910	1.93724
$505$	61.0026	2.71458
$506$	1.11974	0.0497787
$507$	−13.3867	−0.594524
$508$	20.3209	0.901595
$509$	36.6420	1.62413	0.812064	−	0.583568i	$-0.198344\pi$
$509$	36.6420	1.62413	0.812064	+	0.583568i	$0.198344\pi$
$510$	−9.79968	−0.433937
$511$	20.4072	0.902761
$512$	−36.8535	−1.62871
$513$	32.7811	1.44732
$514$	−34.9294	−1.54067
$515$	−1.49861	−0.0660367
$516$	−34.6553	−1.52562
$517$	−0.317259	−0.0139531
$518$	−12.1644	−0.534474
$519$	15.3184	0.672405
$520$	−6.05607	−0.265576
$521$	11.9718	0.524494	0.262247	−	0.965001i	$-0.415536\pi$
$521$	11.9718	0.524494	0.262247	+	0.965001i	$0.415536\pi$
$522$	−1.75737	−0.0769181
$523$	0.389678	0.0170394	0.00851972	−	0.999964i	$-0.497288\pi$
$523$	0.389678	0.0170394	0.00851972	+	0.999964i	$0.497288\pi$
$524$	31.9780	1.39697
$525$	19.6686	0.858407
$526$	78.7211	3.43240
$527$	−6.22283	−0.271071
$528$	1.45832	0.0634654
$529$	−4.84405	−0.210611
$530$	51.4630	2.23541
$531$	10.1227	0.439286
$532$	−83.9836	−3.64115
$533$	−0.157624	−0.00682744
$534$	−25.4951	−1.10328
$535$	−15.6810	−0.677949
$536$	111.982	4.83687
$537$	−16.8805	−0.728449
$538$	−55.3221	−2.38511
$539$	−0.118713	−0.00511333
$540$	−99.5346	−4.28329
$541$	−12.8765	−0.553603	−0.276801	−	0.960927i	$-0.589274\pi$
$541$	−12.8765	−0.553603	−0.276801	+	0.960927i	$0.589274\pi$
$542$	34.5595	1.48446
$543$	13.2617	0.569113
$544$	−20.5614	−0.881565
$545$	−17.5703	−0.752628
$546$	−1.21600	−0.0520401
$547$	−21.8082	−0.932453	−0.466227	−	0.884665i	$-0.654387\pi$
$547$	−21.8082	−0.932453	−0.466227	+	0.884665i	$0.654387\pi$
$548$	32.3531	1.38205
$549$	18.5549	0.791902
$550$	−2.08470	−0.0888921
$551$	2.14447	0.0913573
$552$	41.1781	1.75265
$553$	−6.36040	−0.270472
$554$	42.0171	1.78514
$555$	6.89649	0.292740
$556$	−85.5035	−3.62616
$557$	−17.3906	−0.736863	−0.368432	−	0.929655i	$-0.620105\pi$
$557$	−17.3906	−0.736863	−0.368432	+	0.929655i	$0.620105\pi$
$558$	−33.8972	−1.43498
$559$	−1.11140	−0.0470072
$560$	126.592	5.34948
$561$	0.0963329	0.00406718
$562$	−51.2101	−2.16017
$563$	−27.7306	−1.16870	−0.584352	−	0.811500i	$-0.698651\pi$
$563$	−27.7306	−1.16870	−0.584352	+	0.811500i	$0.698651\pi$
$564$	−18.4630	−0.777430
$565$	29.4415	1.23861
$566$	−10.7819	−0.453196
$567$	1.30797	0.0549295
$568$	−129.414	−5.43008
$569$	−16.5538	−0.693970	−0.346985	−	0.937871i	$-0.612794\pi$
$569$	−16.5538	−0.693970	−0.346985	+	0.937871i	$0.612794\pi$
$570$	65.1394	2.72839
$571$	−14.1225	−0.591007	−0.295504	−	0.955342i	$-0.595487\pi$
$571$	−14.1225	−0.591007	−0.295504	+	0.955342i	$0.595487\pi$
$572$	0.0942092	0.00393909
$573$	−9.05297	−0.378193
$574$	5.73784	0.239493
$575$	−33.8022	−1.40965
$576$	−55.3021	−2.30426
$577$	−34.2097	−1.42417	−0.712084	−	0.702095i	$-0.752248\pi$
$577$	−34.2097	−1.42417	−0.712084	+	0.702095i	$0.752248\pi$
$578$	43.7941	1.82159
$579$	−14.5345	−0.604034
$580$	−6.51132	−0.270368
$581$	−40.4551	−1.67836
$582$	−19.7602	−0.819087
$583$	−0.505893	−0.0209519
$584$	−79.5429	−3.29151
$585$	−1.25134	−0.0517366
$586$	6.99995	0.289165
$587$	40.2560	1.66154	0.830772	−	0.556613i	$-0.187900\pi$
$587$	40.2560	1.66154	0.830772	+	0.556613i	$0.187900\pi$
$588$	−6.90852	−0.284903
$589$	41.3637	1.70436
$590$	51.3112	2.11245
$591$	−15.2062	−0.625499
$592$	27.2269	1.11902
$593$	3.95858	0.162559	0.0812797	−	0.996691i	$-0.474099\pi$
$593$	3.95858	0.162559	0.0812797	+	0.996691i	$0.474099\pi$
$594$	1.33860	0.0549232
$595$	8.36232	0.342822
$596$	30.9162	1.26638
$597$	−18.5873	−0.760729
$598$	2.08981	0.0854585
$599$	−31.0776	−1.26979	−0.634897	−	0.772596i	$-0.718958\pi$
$599$	−31.0776	−1.26979	−0.634897	+	0.772596i	$0.718958\pi$
$600$	−76.6640	−3.12979
$601$	−20.4870	−0.835683	−0.417841	−	0.908520i	$-0.637213\pi$
$601$	−20.4870	−0.835683	−0.417841	+	0.908520i	$0.637213\pi$
$602$	40.4573	1.64892
$603$	23.1383	0.942266
$604$	54.6616	2.22415
$605$	−39.5253	−1.60693
$606$	−47.7431	−1.93943
$607$	−10.6697	−0.433072	−0.216536	−	0.976275i	$-0.569476\pi$
$607$	−10.6697	−0.433072	−0.216536	+	0.976275i	$0.569476\pi$
$608$	136.674	5.54286
$609$	−0.826177	−0.0334784
$610$	94.0536	3.80812
$611$	−0.592109	−0.0239542
$612$	−10.1758	−0.411330
$613$	18.1091	0.731421	0.365710	−	0.930729i	$-0.380826\pi$
$613$	18.1091	0.731421	0.365710	+	0.930729i	$0.380826\pi$
$614$	−60.9520	−2.45982
$615$	−3.25301	−0.131174
$616$	−2.16711	−0.0873153
$617$	21.1802	0.852683	0.426341	−	0.904562i	$-0.359802\pi$
$617$	21.1802	0.852683	0.426341	+	0.904562i	$0.359802\pi$
$618$	1.17287	0.0471799
$619$	−3.47633	−0.139726	−0.0698628	−	0.997557i	$-0.522256\pi$
$619$	−3.47633	−0.139726	−0.0698628	+	0.997557i	$0.522256\pi$
$620$	−125.594	−5.04399
$621$	21.7045	0.870971
$622$	−9.42815	−0.378034
$623$	21.7557	0.871622
$624$	2.72170	0.108955
$625$	−1.73300	−0.0693198
$626$	−44.7447	−1.78836
$627$	−0.640334	−0.0255725
$628$	0.758062	0.0302500
$629$	1.79854	0.0717123
$630$	45.5515	1.81482
$631$	−8.68604	−0.345786	−0.172893	−	0.984941i	$-0.555311\pi$
$631$	−8.68604	−0.345786	−0.172893	+	0.984941i	$0.555311\pi$
$632$	24.7915	0.986153
$633$	19.2937	0.766857
$634$	−22.5153	−0.894195
$635$	13.4495	0.533729
$636$	−29.4405	−1.16739
$637$	−0.221557	−0.00877841
$638$	0.0875678	0.00346684
$639$	−26.7402	−1.05783
$640$	−127.574	−5.04280
$641$	−10.7135	−0.423157	−0.211578	−	0.977361i	$-0.567860\pi$
$641$	−10.7135	−0.423157	−0.211578	+	0.977361i	$0.567860\pi$
$642$	12.2726	0.484360
$643$	−26.6038	−1.04915	−0.524576	−	0.851363i	$-0.675776\pi$
$643$	−26.6038	−1.04915	−0.524576	+	0.851363i	$0.675776\pi$
$644$	−55.6058	−2.19118
$645$	−22.9369	−0.903138
$646$	16.9877	0.668372
$647$	10.7793	0.423776	0.211888	−	0.977294i	$-0.432039\pi$
$647$	10.7793	0.423776	0.211888	+	0.977294i	$0.432039\pi$
$648$	−5.09818	−0.200276
$649$	−0.504401	−0.0197995
$650$	−3.89073	−0.152607
$651$	−15.9358	−0.624573
$652$	−25.8735	−1.01328
$653$	29.5169	1.15509	0.577544	−	0.816360i	$-0.304011\pi$
$653$	29.5169	1.15509	0.577544	+	0.816360i	$0.304011\pi$
$654$	13.7512	0.537714
$655$	21.1648	0.826979
$656$	−12.8426	−0.501421
$657$	−16.4356	−0.641216
$658$	21.5540	0.840263
$659$	24.6988	0.962129	0.481065	−	0.876685i	$-0.340250\pi$
$659$	24.6988	0.962129	0.481065	+	0.876685i	$0.340250\pi$
$660$	1.94427	0.0756807
$661$	−11.1797	−0.434839	−0.217420	−	0.976078i	$-0.569764\pi$
$661$	−11.1797	−0.434839	−0.217420	+	0.976078i	$0.569764\pi$
$662$	6.52179	0.253476
$663$	0.179789	0.00698241
$664$	157.685	6.11938
$665$	−55.5851	−2.15550
$666$	9.79705	0.379628
$667$	1.41986	0.0549771
$668$	−65.5145	−2.53483
$669$	3.40729	0.131733
$670$	117.287	4.53119
$671$	−0.924567	−0.0356925
$672$	−52.6550	−2.03121
$673$	−22.8646	−0.881366	−0.440683	−	0.897663i	$-0.645264\pi$
$673$	−22.8646	−0.881366	−0.440683	+	0.897663i	$0.645264\pi$
$674$	−53.2315	−2.05040
$675$	−40.4087	−1.55533
$676$	−70.4606	−2.71002
$677$	18.9724	0.729170	0.364585	−	0.931170i	$-0.381211\pi$
$677$	18.9724	0.729170	0.364585	+	0.931170i	$0.381211\pi$
$678$	−23.0421	−0.884925
$679$	16.8619	0.647100
$680$	−32.5946	−1.24994
$681$	−24.0599	−0.921977
$682$	1.68906	0.0646775
$683$	19.0252	0.727980	0.363990	−	0.931403i	$-0.381414\pi$
$683$	19.0252	0.727980	0.363990	+	0.931403i	$0.381414\pi$
$684$	67.6392	2.58625
$685$	21.4131	0.818152
$686$	53.9028	2.05802
$687$	−23.3693	−0.891593
$688$	−90.5532	−3.45231
$689$	−0.944160	−0.0359696
$690$	43.1290	1.64189
$691$	39.8892	1.51746	0.758728	−	0.651407i	$-0.225821\pi$
$691$	39.8892	1.51746	0.758728	+	0.651407i	$0.225821\pi$
$692$	80.6283	3.06503
$693$	−0.447781	−0.0170098
$694$	−82.8111	−3.14347
$695$	−56.5910	−2.14662
$696$	3.22027	0.122064
$697$	−0.848351	−0.0321336
$698$	−0.362795	−0.0137320
$699$	−25.6259	−0.969261
$700$	103.525	3.91288
$701$	−35.7296	−1.34949	−0.674745	−	0.738051i	$-0.735746\pi$
$701$	−35.7296	−1.34949	−0.674745	+	0.738051i	$0.735746\pi$
$702$	2.49825	0.0942905
$703$	−11.9550	−0.450893
$704$	2.75564	0.103857
$705$	−12.2198	−0.460225
$706$	−86.0716	−3.23935
$707$	40.7404	1.53220
$708$	−29.3537	−1.10318
$709$	21.1242	0.793336	0.396668	−	0.917962i	$-0.370167\pi$
$709$	21.1242	0.793336	0.396668	+	0.917962i	$0.370167\pi$
$710$	−135.545	−5.08691
$711$	5.12257	0.192111
$712$	−84.7990	−3.17798
$713$	27.3871	1.02565
$714$	−6.54468	−0.244929
$715$	0.0623530	0.00233187
$716$	−88.8503	−3.32049
$717$	−6.33156	−0.236457
$718$	18.2600	0.681458
$719$	8.43728	0.314657	0.157329	−	0.987546i	$-0.449712\pi$
$719$	8.43728	0.314657	0.157329	+	0.987546i	$0.449712\pi$
$720$	−101.955	−3.79965
$721$	−1.00084	−0.0372733
$722$	−61.1162	−2.27451
$723$	−16.1514	−0.600679
$724$	69.8025	2.59419
$725$	−2.64345	−0.0981751
$726$	30.9340	1.14807
$727$	19.7017	0.730697	0.365348	−	0.930871i	$-0.380950\pi$
$727$	19.7017	0.730697	0.365348	+	0.930871i	$0.380950\pi$
$728$	−4.04453	−0.149900
$729$	14.7217	0.545249
$730$	−83.3115	−3.08350
$731$	−5.98170	−0.221241
$732$	−53.8053	−1.98870
$733$	36.2700	1.33966	0.669832	−	0.742513i	$-0.266366\pi$
$733$	36.2700	1.33966	0.669832	+	0.742513i	$0.266366\pi$
$734$	−26.2874	−0.970284
$735$	−4.57245	−0.168657
$736$	90.4922	3.33559
$737$	−1.15296	−0.0424697
$738$	−4.62117	−0.170108
$739$	32.6500	1.20105	0.600524	−	0.799606i	$-0.294959\pi$
$739$	32.6500	1.20105	0.600524	+	0.799606i	$0.294959\pi$
$740$	36.2995	1.33440
$741$	−1.19507	−0.0439021
$742$	34.3694	1.26174
$743$	−15.0113	−0.550710	−0.275355	−	0.961343i	$-0.588795\pi$
$743$	−15.0113	−0.550710	−0.275355	+	0.961343i	$0.588795\pi$
$744$	62.1144	2.27723
$745$	20.4621	0.749672
$746$	−66.9315	−2.45054
$747$	32.5819	1.19211
$748$	0.507046	0.0185394
$749$	−10.4725	−0.382657
$750$	−29.6869	−1.08401
$751$	2.45968	0.0897549	0.0448775	−	0.998992i	$-0.485710\pi$
$751$	2.45968	0.0897549	0.0448775	+	0.998992i	$0.485710\pi$
$752$	−48.2430	−1.75924
$753$	27.9370	1.01808
$754$	0.163430	0.00595177
$755$	36.1782	1.31666
$756$	−66.4739	−2.41763
$757$	−26.7026	−0.970521	−0.485260	−	0.874370i	$-0.661275\pi$
$757$	−26.7026	−0.970521	−0.485260	+	0.874370i	$0.661275\pi$
$758$	−34.5928	−1.25647
$759$	−0.423967	−0.0153890
$760$	216.659	7.85905
$761$	8.08214	0.292977	0.146489	−	0.989212i	$-0.453203\pi$
$761$	8.08214	0.292977	0.146489	+	0.989212i	$0.453203\pi$
$762$	−10.5261	−0.381322
$763$	−11.7343	−0.424808
$764$	−47.6501	−1.72392
$765$	−6.73489	−0.243500
$766$	−52.0132	−1.87931
$767$	−0.941375	−0.0339911
$768$	40.8169	1.47285
$769$	−34.0289	−1.22711	−0.613556	−	0.789651i	$-0.710262\pi$
$769$	−34.0289	−1.22711	−0.613556	+	0.789651i	$0.710262\pi$
$770$	−2.26978	−0.0817972
$771$	13.2253	0.476296
$772$	−76.5022	−2.75337
$773$	50.5643	1.81867	0.909336	−	0.416064i	$-0.136591\pi$
$773$	50.5643	1.81867	0.909336	+	0.416064i	$0.136591\pi$
$774$	−32.5838	−1.17120
$775$	−50.9884	−1.83156
$776$	−65.7242	−2.35936
$777$	4.60580	0.165232
$778$	17.9407	0.643205
$779$	5.63907	0.202041
$780$	3.62864	0.129926
$781$	1.33244	0.0476783
$782$	11.2476	0.402214
$783$	1.69737	0.0606589
$784$	−18.0517	−0.644704
$785$	0.501728	0.0179074
$786$	−16.5645	−0.590834
$787$	−1.09566	−0.0390561	−0.0195281	−	0.999809i	$-0.506216\pi$
$787$	−1.09566	−0.0390561	−0.0195281	+	0.999809i	$0.506216\pi$
$788$	−80.0374	−2.85121
$789$	−29.8061	−1.06112
$790$	25.9661	0.923831
$791$	19.6624	0.699114
$792$	1.74536	0.0620186
$793$	−1.72554	−0.0612758
$794$	−104.055	−3.69279
$795$	−19.4854	−0.691075
$796$	−97.8340	−3.46764
$797$	−16.1310	−0.571388	−0.285694	−	0.958321i	$-0.592224\pi$
$797$	−16.1310	−0.571388	−0.285694	+	0.958321i	$0.592224\pi$
$798$	43.5031	1.53999
$799$	−3.18681	−0.112741
$800$	−168.476	−5.95651
$801$	−17.5217	−0.619099
$802$	9.19870	0.324817
$803$	0.818970	0.0289008
$804$	−67.0965	−2.36631
$805$	−36.8031	−1.29714
$806$	3.15234	0.111036
$807$	20.9465	0.737353
$808$	−158.797	−5.58647
$809$	54.7477	1.92483	0.962413	−	0.271592i	$-0.0875500\pi$
$809$	54.7477	1.92483	0.962413	+	0.271592i	$0.0875500\pi$
$810$	−5.33972	−0.187619
$811$	11.6104	0.407696	0.203848	−	0.979003i	$-0.434655\pi$
$811$	11.6104	0.407696	0.203848	+	0.979003i	$0.434655\pi$
$812$	−4.34857	−0.152605
$813$	−13.0852	−0.458919
$814$	−0.488176	−0.0171106
$815$	−17.1245	−0.599846
$816$	14.6486	0.512802
$817$	39.7610	1.39106
$818$	25.8215	0.902827
$819$	−0.835705	−0.0292019
$820$	−17.1221	−0.597930
$821$	2.70425	0.0943789	0.0471894	−	0.998886i	$-0.484974\pi$
$821$	2.70425	0.0943789	0.0471894	+	0.998886i	$0.484974\pi$
$822$	−16.7587	−0.584528
$823$	29.5819	1.03116	0.515581	−	0.856841i	$-0.327576\pi$
$823$	29.5819	1.03116	0.515581	+	0.856841i	$0.327576\pi$
$824$	3.90107	0.135900
$825$	0.789329	0.0274809
$826$	34.2681	1.19234
$827$	17.1203	0.595331	0.297666	−	0.954670i	$-0.403792\pi$
$827$	17.1203	0.595331	0.297666	+	0.954670i	$0.403792\pi$
$828$	44.7841	1.55636
$829$	−41.6357	−1.44607	−0.723033	−	0.690813i	$-0.757253\pi$
$829$	−41.6357	−1.44607	−0.723033	+	0.690813i	$0.757253\pi$
$830$	165.156	5.73265
$831$	−15.9089	−0.551873
$832$	5.14292	0.178299
$833$	−1.19245	−0.0413159
$834$	44.2904	1.53365
$835$	−43.3612	−1.50057
$836$	−3.37039	−0.116567
$837$	32.7398	1.13165
$838$	48.3331	1.66964
$839$	4.21211	0.145418	0.0727091	−	0.997353i	$-0.476836\pi$
$839$	4.21211	0.145418	0.0727091	+	0.997353i	$0.476836\pi$
$840$	−83.4701	−2.87999
$841$	−28.8890	−0.996171
$842$	15.5775	0.536837
$843$	19.3896	0.667815
$844$	101.552	3.49557
$845$	−46.6348	−1.60428
$846$	−17.3593	−0.596825
$847$	−26.3968	−0.907005
$848$	−76.9270	−2.64168
$849$	4.08233	0.140105
$850$	−20.9404	−0.718251
$851$	−7.91547	−0.271339
$852$	77.5413	2.65652
$853$	−35.7392	−1.22369	−0.611844	−	0.790978i	$-0.709572\pi$
$853$	−35.7392	−1.22369	−0.611844	+	0.790978i	$0.709572\pi$
$854$	62.8134	2.14943
$855$	44.7675	1.53101
$856$	40.8196	1.39519
$857$	16.8767	0.576497	0.288249	−	0.957556i	$-0.406927\pi$
$857$	16.8767	0.576497	0.288249	+	0.957556i	$0.406927\pi$
$858$	−0.0487999	−0.00166600
$859$	54.6003	1.86294	0.931469	−	0.363822i	$-0.118528\pi$
$859$	54.6003	1.86294	0.931469	+	0.363822i	$0.118528\pi$
$860$	−120.728	−4.11678
$861$	−2.17251	−0.0740389
$862$	−1.94235	−0.0661566
$863$	24.9278	0.848554	0.424277	−	0.905533i	$-0.360528\pi$
$863$	24.9278	0.848554	0.424277	+	0.905533i	$0.360528\pi$
$864$	108.179	3.68032
$865$	53.3643	1.81444
$866$	33.7255	1.14604
$867$	−16.5817	−0.563144
$868$	−83.8778	−2.84700
$869$	−0.255252	−0.00865883
$870$	3.37284	0.114350
$871$	−2.15179	−0.0729107
$872$	45.7377	1.54887
$873$	−13.5803	−0.459625
$874$	−74.7639	−2.52893
$875$	25.3326	0.856399
$876$	47.6601	1.61028
$877$	−6.80508	−0.229791	−0.114896	−	0.993378i	$-0.536653\pi$
$877$	−6.80508	−0.229791	−0.114896	+	0.993378i	$0.536653\pi$
$878$	33.9000	1.14407
$879$	−2.65038	−0.0893951
$880$	5.08031	0.171257
$881$	−15.4863	−0.521746	−0.260873	−	0.965373i	$-0.584010\pi$
$881$	−15.4863	−0.521746	−0.260873	+	0.965373i	$0.584010\pi$
$882$	−6.49555	−0.218717
$883$	15.6439	0.526460	0.263230	−	0.964733i	$-0.415212\pi$
$883$	15.6439	0.526460	0.263230	+	0.964733i	$0.415212\pi$
$884$	0.946313	0.0318279
$885$	−19.4279	−0.653062
$886$	−48.0010	−1.61263
$887$	26.8271	0.900765	0.450382	−	0.892836i	$-0.351288\pi$
$887$	26.8271	0.900765	0.450382	+	0.892836i	$0.351288\pi$
$888$	−17.9524	−0.602445
$889$	8.98223	0.301254
$890$	−88.8166	−2.97714
$891$	0.0524907	0.00175850
$892$	17.9342	0.600481
$893$	21.1830	0.708862
$894$	−16.0144	−0.535602
$895$	−58.8062	−1.96567
$896$	−85.1998	−2.84633
$897$	−0.791261	−0.0264194
$898$	−37.4662	−1.25026
$899$	2.14176	0.0714318
$900$	−83.3776	−2.77925
$901$	−5.08159	−0.169292
$902$	0.230268	0.00766707
$903$	−15.3183	−0.509761
$904$	−76.6398	−2.54900
$905$	46.1992	1.53571
$906$	−28.3145	−0.940685
$907$	−4.78208	−0.158786	−0.0793931	−	0.996843i	$-0.525298\pi$
$907$	−4.78208	−0.158786	−0.0793931	+	0.996843i	$0.525298\pi$
$908$	−126.639	−4.20265
$909$	−32.8117	−1.08830
$910$	−4.23615	−0.140427
$911$	27.4761	0.910322	0.455161	−	0.890409i	$-0.349582\pi$
$911$	27.4761	0.910322	0.455161	+	0.890409i	$0.349582\pi$
$912$	−97.3704	−3.22426
$913$	−1.62352	−0.0537307
$914$	105.117	3.47696
$915$	−35.6114	−1.17728
$916$	−123.004	−4.06416
$917$	14.1349	0.466775
$918$	13.4459	0.443782
$919$	−21.4743	−0.708373	−0.354186	−	0.935175i	$-0.615242\pi$
$919$	−21.4743	−0.708373	−0.354186	+	0.935175i	$0.615242\pi$
$920$	143.451	4.72943
$921$	23.0782	0.760452
$922$	52.0009	1.71256
$923$	2.48676	0.0818526
$924$	1.29848	0.0427167
$925$	14.7368	0.484542
$926$	1.77644	0.0583774
$927$	0.806064	0.0264746
$928$	7.07680	0.232307
$929$	−19.2294	−0.630897	−0.315448	−	0.948943i	$-0.602155\pi$
$929$	−19.2294	−0.630897	−0.315448	+	0.948943i	$0.602155\pi$
$930$	65.0573	2.13331
$931$	7.92632	0.259775
$932$	−134.881	−4.41819
$933$	3.56977	0.116869
$934$	56.4199	1.84612
$935$	0.335592	0.0109750
$936$	3.25740	0.106472
$937$	36.9470	1.20701	0.603504	−	0.797360i	$-0.293771\pi$
$937$	36.9470	1.20701	0.603504	+	0.797360i	$0.293771\pi$
$938$	78.3298	2.55756
$939$	16.9416	0.552869
$940$	−64.3188	−2.09785
$941$	−16.3877	−0.534225	−0.267112	−	0.963665i	$-0.586070\pi$
$941$	−16.3877	−0.534225	−0.267112	+	0.963665i	$0.586070\pi$
$942$	−0.392673	−0.0127940
$943$	3.73365	0.121584
$944$	−76.7001	−2.49638
$945$	−43.9962	−1.43119
$946$	1.62361	0.0527882
$947$	0.262106	0.00851731	0.00425866	−	0.999991i	$-0.498644\pi$
$947$	0.262106	0.00851731	0.00425866	+	0.999991i	$0.498644\pi$
$948$	−14.8544	−0.482449
$949$	1.52846	0.0496160
$950$	139.193	4.51602
$951$	8.52492	0.276440
$952$	−21.7682	−0.705510
$953$	−14.1647	−0.458840	−0.229420	−	0.973328i	$-0.573683\pi$
$953$	−14.1647	−0.458840	−0.229420	+	0.973328i	$0.573683\pi$
$954$	−27.6806	−0.896193
$955$	−31.5375	−1.02053
$956$	−33.3260	−1.07784
$957$	−0.0331557	−0.00107177
$958$	−65.6617	−2.12143
$959$	14.3007	0.461793
$960$	106.139	3.42561
$961$	10.3116	0.332632
$962$	−0.911095	−0.0293749
$963$	8.43440	0.271795
$964$	−85.0128	−2.73808
$965$	−50.6335	−1.62995
$966$	28.8036	0.926739
$967$	2.75850	0.0887073	0.0443537	−	0.999016i	$-0.485877\pi$
$967$	2.75850	0.0887073	0.0443537	+	0.999016i	$0.485877\pi$
$968$	102.889	3.30699
$969$	−6.43203	−0.206627
$970$	−68.8380	−2.21026
$971$	−14.6955	−0.471603	−0.235801	−	0.971801i	$-0.575771\pi$
$971$	−14.6955	−0.471603	−0.235801	+	0.971801i	$0.575771\pi$
$972$	86.0869	2.76124
$973$	−37.7941	−1.21162
$974$	9.45003	0.302799
$975$	1.47314	0.0471784
$976$	−140.591	−4.50022
$977$	−6.56338	−0.209981	−0.104991	−	0.994473i	$-0.533481\pi$
$977$	−6.56338	−0.209981	−0.104991	+	0.994473i	$0.533481\pi$
$978$	13.4023	0.428559
$979$	0.873086	0.0279040
$980$	−24.0670	−0.768792
$981$	9.45060	0.301734
$982$	97.3820	3.10759
$983$	17.9101	0.571244	0.285622	−	0.958342i	$-0.407800\pi$
$983$	17.9101	0.571244	0.285622	+	0.958342i	$0.407800\pi$
$984$	8.46799	0.269950
$985$	−52.9733	−1.68787
$986$	0.879601	0.0280122
$987$	−8.16097	−0.259767
$988$	−6.29023	−0.200119
$989$	26.3258	0.837113
$990$	1.82805	0.0580992
$991$	38.9047	1.23585	0.617924	−	0.786238i	$-0.287974\pi$
$991$	38.9047	1.23585	0.617924	+	0.786238i	$0.287974\pi$
$992$	136.502	4.33393
$993$	−2.46934	−0.0783620
$994$	−90.5233	−2.87122
$995$	−64.7521	−2.05278
$996$	−94.4810	−2.99375
$997$	13.0342	0.412798	0.206399	−	0.978468i	$-0.433826\pi$
$997$	13.0342	0.412798	0.206399	+	0.978468i	$0.433826\pi$
$998$	−20.8986	−0.661535
$999$	−9.46253	−0.299381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.a.1.3	✓	306

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.a.1.3	✓	306	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72646 q^{2} +1.03232 q^{3} +5.43357 q^{4} +3.59624 q^{5} -2.81456 q^{6} +2.40174 q^{7} -9.36147 q^{8} -1.93433 q^{9} +O(q^{10})\)	\(q-2.72646 q^{2} +1.03232 q^{3} +5.43357 q^{4} +3.59624 q^{5} -2.81456 q^{6} +2.40174 q^{7} -9.36147 q^{8} -1.93433 q^{9} -9.80500 q^{10} +0.0963853 q^{11} +5.60915 q^{12} +0.179886 q^{13} -6.54824 q^{14} +3.71245 q^{15} +14.6565 q^{16} +0.968171 q^{17} +5.27386 q^{18} -6.43552 q^{19} +19.5404 q^{20} +2.47935 q^{21} -0.262790 q^{22} -4.26098 q^{23} -9.66399 q^{24} +7.93296 q^{25} -0.490452 q^{26} -5.09378 q^{27} +13.0500 q^{28} -0.333223 q^{29} -10.1218 q^{30} -6.42741 q^{31} -21.2374 q^{32} +0.0995000 q^{33} -2.63968 q^{34} +8.63723 q^{35} -10.5103 q^{36} +1.85766 q^{37} +17.5462 q^{38} +0.185699 q^{39} -33.6661 q^{40} -0.876241 q^{41} -6.75984 q^{42} -6.17836 q^{43} +0.523716 q^{44} -6.95630 q^{45} +11.6174 q^{46} -3.29158 q^{47} +15.1301 q^{48} -1.23165 q^{49} -21.6289 q^{50} +0.999457 q^{51} +0.977424 q^{52} -5.24865 q^{53} +13.8880 q^{54} +0.346625 q^{55} -22.4838 q^{56} -6.64349 q^{57} +0.908519 q^{58} -5.23317 q^{59} +20.1719 q^{60} -9.59241 q^{61} +17.5241 q^{62} -4.64574 q^{63} +28.5899 q^{64} +0.646914 q^{65} -0.271282 q^{66} -11.9620 q^{67} +5.26062 q^{68} -4.39867 q^{69} -23.5490 q^{70} +13.8241 q^{71} +18.1081 q^{72} +8.49684 q^{73} -5.06484 q^{74} +8.18931 q^{75} -34.9679 q^{76} +0.231492 q^{77} -0.506301 q^{78} -2.64825 q^{79} +52.7084 q^{80} +0.544592 q^{81} +2.38903 q^{82} -16.8441 q^{83} +13.4717 q^{84} +3.48178 q^{85} +16.8450 q^{86} -0.343991 q^{87} -0.902308 q^{88} +9.05830 q^{89} +18.9661 q^{90} +0.432040 q^{91} -23.1523 q^{92} -6.63511 q^{93} +8.97434 q^{94} -23.1437 q^{95} -21.9237 q^{96} +7.02071 q^{97} +3.35804 q^{98} -0.186440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9	\( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 306 * q - 13 * q^2 - 25 * q^3 + 253 * q^4 - 25 * q^5 - 49 * q^6 - 102 * q^7 - 33 * q^8 + 251 * q^9 - 61 * q^10 - 43 * q^11 - 50 * q^12 - 89 * q^13 - 40 * q^14 - 61 * q^15 + 151 * q^16 - 52 * q^17 - 57 * q^18 - 185 * q^19 - 66 * q^20 - 63 * q^21 - 55 * q^22 - 62 * q^23 - 131 * q^24 + 209 * q^25 - 57 * q^26 - 88 * q^27 - 182 * q^28 - 67 * q^29 - 68 * q^30 - 240 * q^31 - 64 * q^32 - 52 * q^33 - 128 * q^34 - 99 * q^35 + 106 * q^36 - 49 * q^37 - 45 * q^38 - 190 * q^39 - 158 * q^40 - 72 * q^41 - 36 * q^42 - 141 * q^43 - 80 * q^44 - 100 * q^45 - 91 * q^46 - 105 * q^47 - 85 * q^48 + 116 * q^49 - 51 * q^50 - 145 * q^51 - 237 * q^52 - 48 * q^53 - 156 * q^54 - 420 * q^55 - 116 * q^56 - 35 * q^57 - 43 * q^58 - 139 * q^59 - 73 * q^60 - 233 * q^61 - 58 * q^62 - 252 * q^63 - 3 * q^64 - 45 * q^65 - 127 * q^66 - 108 * q^67 - 85 * q^68 - 164 * q^69 - 56 * q^70 - 131 * q^71 - 117 * q^72 - 118 * q^73 - 47 * q^74 - 112 * q^75 - 389 * q^76 - 36 * q^77 + 9 * q^78 - 382 * q^79 - 119 * q^80 + 102 * q^81 - 131 * q^82 - 59 * q^83 - 144 * q^84 - 140 * q^85 - 38 * q^86 - 301 * q^87 - 131 * q^88 - 98 * q^89 - 138 * q^90 - 176 * q^91 - 97 * q^92 - 60 * q^93 - 342 * q^94 - 154 * q^95 - 243 * q^96 - 109 * q^97 - 21 * q^98 - 173 * q^99

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