LMFDB - Embedded newform 8047.2.a.d.1.10

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			1.10
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)

$f(q)$	$=$	$q-2.48921 q^{2} +1.66409 q^{3} +4.19617 q^{4} +0.253243 q^{5} -4.14227 q^{6} -2.12862 q^{7} -5.46674 q^{8} -0.230804 q^{9} +O(q^{10})$	\(q-2.48921 q^{2} +1.66409 q^{3} +4.19617 q^{4} +0.253243 q^{5} -4.14227 q^{6} -2.12862 q^{7} -5.46674 q^{8} -0.230804 q^{9} -0.630376 q^{10} -4.54541 q^{11} +6.98281 q^{12} -1.00000 q^{13} +5.29859 q^{14} +0.421419 q^{15} +5.21552 q^{16} -4.33851 q^{17} +0.574521 q^{18} +2.44687 q^{19} +1.06265 q^{20} -3.54222 q^{21} +11.3145 q^{22} -0.598129 q^{23} -9.09715 q^{24} -4.93587 q^{25} +2.48921 q^{26} -5.37635 q^{27} -8.93206 q^{28} -7.79231 q^{29} -1.04900 q^{30} -0.603090 q^{31} -2.04906 q^{32} -7.56397 q^{33} +10.7995 q^{34} -0.539059 q^{35} -0.968496 q^{36} -9.24695 q^{37} -6.09078 q^{38} -1.66409 q^{39} -1.38441 q^{40} -0.533142 q^{41} +8.81733 q^{42} +0.273423 q^{43} -19.0733 q^{44} -0.0584496 q^{45} +1.48887 q^{46} +9.67283 q^{47} +8.67910 q^{48} -2.46897 q^{49} +12.2864 q^{50} -7.21968 q^{51} -4.19617 q^{52} +5.45235 q^{53} +13.3829 q^{54} -1.15109 q^{55} +11.6366 q^{56} +4.07181 q^{57} +19.3967 q^{58} +11.1015 q^{59} +1.76835 q^{60} +1.34093 q^{61} +1.50122 q^{62} +0.491295 q^{63} -5.33050 q^{64} -0.253243 q^{65} +18.8283 q^{66} -11.0532 q^{67} -18.2052 q^{68} -0.995341 q^{69} +1.34183 q^{70} +11.1252 q^{71} +1.26175 q^{72} +4.97818 q^{73} +23.0176 q^{74} -8.21373 q^{75} +10.2675 q^{76} +9.67546 q^{77} +4.14227 q^{78} -15.1693 q^{79} +1.32080 q^{80} -8.25432 q^{81} +1.32710 q^{82} +2.66066 q^{83} -14.8638 q^{84} -1.09870 q^{85} -0.680607 q^{86} -12.9671 q^{87} +24.8486 q^{88} +14.4890 q^{89} +0.145494 q^{90} +2.12862 q^{91} -2.50985 q^{92} -1.00360 q^{93} -24.0777 q^{94} +0.619653 q^{95} -3.40982 q^{96} -11.4098 q^{97} +6.14579 q^{98} +1.04910 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.48921	−1.76014	−0.880069	−	0.474846i	$-0.842504\pi$
$2$	−2.48921	−1.76014	−0.880069	+	0.474846i	$0.842504\pi$
$3$	1.66409	0.960763	0.480381	−	0.877060i	$-0.340498\pi$
$3$	1.66409	0.960763	0.480381	+	0.877060i	$0.340498\pi$
$4$	4.19617	2.09809
$5$	0.253243	0.113254	0.0566269	−	0.998395i	$-0.481965\pi$
$5$	0.253243	0.113254	0.0566269	+	0.998395i	$0.481965\pi$
$6$	−4.14227	−1.69108
$7$	−2.12862	−0.804543	−0.402272	−	0.915520i	$-0.631779\pi$
$7$	−2.12862	−0.804543	−0.402272	+	0.915520i	$0.631779\pi$
$8$	−5.46674	−1.93278
$9$	−0.230804	−0.0769348
$10$	−0.630376	−0.199342
$11$	−4.54541	−1.37049	−0.685247	−	0.728311i	$-0.740306\pi$
$11$	−4.54541	−1.37049	−0.685247	+	0.728311i	$0.740306\pi$
$12$	6.98281	2.01576
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	5.29859	1.41611
$15$	0.421419	0.108810
$16$	5.21552	1.30388
$17$	−4.33851	−1.05224	−0.526122	−	0.850409i	$-0.676355\pi$
$17$	−4.33851	−1.05224	−0.526122	+	0.850409i	$0.676355\pi$
$18$	0.574521	0.135416
$19$	2.44687	0.561350	0.280675	−	0.959803i	$-0.409442\pi$
$19$	2.44687	0.561350	0.280675	+	0.959803i	$0.409442\pi$
$20$	1.06265	0.237616
$21$	−3.54222	−0.772975
$22$	11.3145	2.41226
$23$	−0.598129	−0.124719	−0.0623593	−	0.998054i	$-0.519862\pi$
$23$	−0.598129	−0.124719	−0.0623593	+	0.998054i	$0.519862\pi$
$24$	−9.09715	−1.85695
$25$	−4.93587	−0.987174
$26$	2.48921	0.488175
$27$	−5.37635	−1.03468
$28$	−8.93206	−1.68800
$29$	−7.79231	−1.44700	−0.723498	−	0.690327i	$-0.757467\pi$
$29$	−7.79231	−1.44700	−0.723498	+	0.690327i	$0.757467\pi$
$30$	−1.04900	−0.191521
$31$	−0.603090	−0.108318	−0.0541591	−	0.998532i	$-0.517248\pi$
$31$	−0.603090	−0.108318	−0.0541591	+	0.998532i	$0.517248\pi$
$32$	−2.04906	−0.362226
$33$	−7.56397	−1.31672
$34$	10.7995	1.85210
$35$	−0.539059	−0.0911175
$36$	−0.968496	−0.161416
$37$	−9.24695	−1.52019	−0.760095	−	0.649812i	$-0.774848\pi$
$37$	−9.24695	−1.52019	−0.760095	+	0.649812i	$0.774848\pi$
$38$	−6.09078	−0.988054
$39$	−1.66409	−0.266468
$40$	−1.38441	−0.218895
$41$	−0.533142	−0.0832628	−0.0416314	−	0.999133i	$-0.513256\pi$
$41$	−0.533142	−0.0832628	−0.0416314	+	0.999133i	$0.513256\pi$
$42$	8.81733	1.36054
$43$	0.273423	0.0416966	0.0208483	−	0.999783i	$-0.493363\pi$
$43$	0.273423	0.0416966	0.0208483	+	0.999783i	$0.493363\pi$
$44$	−19.0733	−2.87541
$45$	−0.0584496	−0.00871316
$46$	1.48887	0.219522
$47$	9.67283	1.41093	0.705463	−	0.708746i	$-0.250739\pi$
$47$	9.67283	1.41093	0.705463	+	0.708746i	$0.250739\pi$
$48$	8.67910	1.25272
$49$	−2.46897	−0.352710
$50$	12.2864	1.73756
$51$	−7.21968	−1.01096
$52$	−4.19617	−0.581905
$53$	5.45235	0.748938	0.374469	−	0.927239i	$-0.377825\pi$
$53$	5.45235	0.748938	0.374469	+	0.927239i	$0.377825\pi$
$54$	13.3829	1.82118
$55$	−1.15109	−0.155213
$56$	11.6366	1.55501
$57$	4.07181	0.539325
$58$	19.3967	2.54691
$59$	11.1015	1.44530	0.722649	−	0.691215i	$-0.242924\pi$
$59$	11.1015	1.44530	0.722649	+	0.691215i	$0.242924\pi$
$60$	1.76835	0.228293
$61$	1.34093	0.171689	0.0858443	−	0.996309i	$-0.472641\pi$
$61$	1.34093	0.171689	0.0858443	+	0.996309i	$0.472641\pi$
$62$	1.50122	0.190655
$63$	0.491295	0.0618974
$64$	−5.33050	−0.666312
$65$	−0.253243	−0.0314109
$66$	18.8283	2.31761
$67$	−11.0532	−1.35037	−0.675183	−	0.737650i	$-0.735935\pi$
$67$	−11.0532	−1.35037	−0.675183	+	0.737650i	$0.735935\pi$
$68$	−18.2052	−2.20770
$69$	−0.995341	−0.119825
$70$	1.34183	0.160379
$71$	11.1252	1.32032	0.660162	−	0.751123i	$-0.270488\pi$
$71$	11.1252	1.32032	0.660162	+	0.751123i	$0.270488\pi$
$72$	1.26175	0.148698
$73$	4.97818	0.582652	0.291326	−	0.956624i	$-0.405904\pi$
$73$	4.97818	0.582652	0.291326	+	0.956624i	$0.405904\pi$
$74$	23.0176	2.67574
$75$	−8.21373	−0.948440
$76$	10.2675	1.17776
$77$	9.67546	1.10262
$78$	4.14227	0.469020
$79$	−15.1693	−1.70668	−0.853342	−	0.521352i	$-0.825428\pi$
$79$	−15.1693	−1.70668	−0.853342	+	0.521352i	$0.825428\pi$
$80$	1.32080	0.147669
$81$	−8.25432	−0.917146
$82$	1.32710	0.146554
$83$	2.66066	0.292045	0.146022	−	0.989281i	$-0.453353\pi$
$83$	2.66066	0.292045	0.146022	+	0.989281i	$0.453353\pi$
$84$	−14.8638	−1.62177
$85$	−1.09870	−0.119171
$86$	−0.680607	−0.0733917
$87$	−12.9671	−1.39022
$88$	24.8486	2.64887
$89$	14.4890	1.53583	0.767913	−	0.640554i	$-0.221295\pi$
$89$	14.4890	1.53583	0.767913	+	0.640554i	$0.221295\pi$
$90$	0.145494	0.0153364
$91$	2.12862	0.223140
$92$	−2.50985	−0.261670
$93$	−1.00360	−0.104068
$94$	−24.0777	−2.48343
$95$	0.619653	0.0635750
$96$	−3.40982	−0.348014
$97$	−11.4098	−1.15849	−0.579246	−	0.815153i	$-0.696653\pi$
$97$	−11.4098	−1.15849	−0.579246	+	0.815153i	$0.696653\pi$
$98$	6.14579	0.620819
$99$	1.04910	0.105439
$100$	−20.7118	−2.07118
$101$	−18.8932	−1.87994	−0.939970	−	0.341257i	$-0.889147\pi$
$101$	−18.8932	−1.87994	−0.939970	+	0.341257i	$0.889147\pi$
$102$	17.9713	1.77942
$103$	−1.88328	−0.185565	−0.0927825	−	0.995686i	$-0.529576\pi$
$103$	−1.88328	−0.185565	−0.0927825	+	0.995686i	$0.529576\pi$
$104$	5.46674	0.536058
$105$	−0.897042	−0.0875423
$106$	−13.5721	−1.31824
$107$	1.92788	0.186375	0.0931877	−	0.995649i	$-0.470294\pi$
$107$	1.92788	0.186375	0.0931877	+	0.995649i	$0.470294\pi$
$108$	−22.5601	−2.17085
$109$	10.5472	1.01024	0.505120	−	0.863049i	$-0.331448\pi$
$109$	10.5472	1.01024	0.505120	+	0.863049i	$0.331448\pi$
$110$	2.86532	0.273197
$111$	−15.3878	−1.46054
$112$	−11.1019	−1.04903
$113$	7.16384	0.673917	0.336959	−	0.941519i	$-0.390602\pi$
$113$	7.16384	0.673917	0.336959	+	0.941519i	$0.390602\pi$
$114$	−10.1356	−0.949286
$115$	−0.151472	−0.0141248
$116$	−32.6979	−3.03592
$117$	0.230804	0.0213379
$118$	−27.6341	−2.54392
$119$	9.23505	0.846576
$120$	−2.30379	−0.210306
$121$	9.66076	0.878251
$122$	−3.33786	−0.302196
$123$	−0.887197	−0.0799958
$124$	−2.53067	−0.227261
$125$	−2.51619	−0.225055
$126$	−1.22294	−0.108948
$127$	12.3444	1.09539	0.547696	−	0.836677i	$-0.315505\pi$
$127$	12.3444	1.09539	0.547696	+	0.836677i	$0.315505\pi$
$128$	17.3669	1.53503
$129$	0.455000	0.0400605
$130$	0.630376	0.0552876
$131$	2.07249	0.181075	0.0905373	−	0.995893i	$-0.471142\pi$
$131$	2.07249	0.181075	0.0905373	+	0.995893i	$0.471142\pi$
$132$	−31.7397	−2.76259
$133$	−5.20846	−0.451631
$134$	27.5138	2.37683
$135$	−1.36152	−0.117181
$136$	23.7175	2.03376
$137$	19.7444	1.68687	0.843437	−	0.537228i	$-0.180528\pi$
$137$	19.7444	1.68687	0.843437	+	0.537228i	$0.180528\pi$
$138$	2.47761	0.210908
$139$	−8.36652	−0.709639	−0.354819	−	0.934935i	$-0.615458\pi$
$139$	−8.36652	−0.709639	−0.354819	+	0.934935i	$0.615458\pi$
$140$	−2.26198	−0.191172
$141$	16.0965	1.35557
$142$	−27.6931	−2.32395
$143$	4.54541	0.380106
$144$	−1.20377	−0.100314
$145$	−1.97335	−0.163878
$146$	−12.3917	−1.02555
$147$	−4.10859	−0.338871
$148$	−38.8018	−3.18949
$149$	−19.9726	−1.63622	−0.818108	−	0.575064i	$-0.804977\pi$
$149$	−19.9726	−1.63622	−0.818108	+	0.575064i	$0.804977\pi$
$150$	20.4457	1.66938
$151$	7.86106	0.639724	0.319862	−	0.947464i	$-0.396363\pi$
$151$	7.86106	0.639724	0.319862	+	0.947464i	$0.396363\pi$
$152$	−13.3764	−1.08497
$153$	1.00135	0.0809542
$154$	−24.0843	−1.94077
$155$	−0.152728	−0.0122674
$156$	−6.98281	−0.559072
$157$	−18.5345	−1.47922	−0.739608	−	0.673037i	$-0.764989\pi$
$157$	−18.5345	−1.47922	−0.739608	+	0.673037i	$0.764989\pi$
$158$	37.7597	3.00400
$159$	9.07321	0.719552
$160$	−0.518911	−0.0410235
$161$	1.27319	0.100341
$162$	20.5467	1.61430
$163$	17.2077	1.34781	0.673907	−	0.738816i	$-0.264615\pi$
$163$	17.2077	1.34781	0.673907	+	0.738816i	$0.264615\pi$
$164$	−2.23716	−0.174693
$165$	−1.91552	−0.149123
$166$	−6.62293	−0.514039
$167$	0.967344	0.0748554	0.0374277	−	0.999299i	$-0.488084\pi$
$167$	0.967344	0.0748554	0.0374277	+	0.999299i	$0.488084\pi$
$168$	19.3644	1.49399
$169$	1.00000	0.0769231
$170$	2.73489	0.209757
$171$	−0.564748	−0.0431874
$172$	1.14733	0.0874830
$173$	2.95895	0.224965	0.112482	−	0.993654i	$-0.464120\pi$
$173$	2.95895	0.224965	0.112482	+	0.993654i	$0.464120\pi$
$174$	32.2779	2.44698
$175$	10.5066	0.794224
$176$	−23.7067	−1.78696
$177$	18.4740	1.38859
$178$	−36.0661	−2.70327
$179$	−23.5180	−1.75782	−0.878908	−	0.476991i	$-0.841727\pi$
$179$	−23.5180	−1.75782	−0.878908	+	0.476991i	$0.841727\pi$
$180$	−0.245265	−0.0182810
$181$	22.4314	1.66731	0.833657	−	0.552282i	$-0.186243\pi$
$181$	22.4314	1.66731	0.833657	+	0.552282i	$0.186243\pi$
$182$	−5.29859	−0.392757
$183$	2.23143	0.164952
$184$	3.26982	0.241054
$185$	−2.34173	−0.172167
$186$	2.49816	0.183174
$187$	19.7203	1.44209
$188$	40.5889	2.96025
$189$	11.4442	0.832444
$190$	−1.54245	−0.111901
$191$	19.3948	1.40336	0.701680	−	0.712492i	$-0.252434\pi$
$191$	19.3948	1.40336	0.701680	+	0.712492i	$0.252434\pi$
$192$	−8.87043	−0.640168
$193$	−5.66752	−0.407957	−0.203979	−	0.978975i	$-0.565387\pi$
$193$	−5.66752	−0.407957	−0.203979	+	0.978975i	$0.565387\pi$
$194$	28.4015	2.03911
$195$	−0.421419	−0.0301785
$196$	−10.3602	−0.740017
$197$	1.02683	0.0731587	0.0365794	−	0.999331i	$-0.488354\pi$
$197$	1.02683	0.0731587	0.0365794	+	0.999331i	$0.488354\pi$
$198$	−2.61143	−0.185587
$199$	13.4025	0.950079	0.475040	−	0.879964i	$-0.342434\pi$
$199$	13.4025	0.950079	0.475040	+	0.879964i	$0.342434\pi$
$200$	26.9831	1.90799
$201$	−18.3936	−1.29738
$202$	47.0291	3.30895
$203$	16.5869	1.16417
$204$	−30.2950	−2.12108
$205$	−0.135015	−0.00942983
$206$	4.68788	0.326620
$207$	0.138051	0.00959520
$208$	−5.21552	−0.361632
$209$	−11.1220	−0.769327
$210$	2.23293	0.154087
$211$	3.32985	0.229236	0.114618	−	0.993410i	$-0.463436\pi$
$211$	3.32985	0.229236	0.114618	+	0.993410i	$0.463436\pi$
$212$	22.8790	1.57134
$213$	18.5134	1.26852
$214$	−4.79891	−0.328047
$215$	0.0692424	0.00472229
$216$	29.3911	1.99981
$217$	1.28375	0.0871467
$218$	−26.2543	−1.77816
$219$	8.28414	0.559791
$220$	−4.83019	−0.325651
$221$	4.33851	0.291840
$222$	38.3034	2.57076
$223$	3.85778	0.258336	0.129168	−	0.991623i	$-0.458769\pi$
$223$	3.85778	0.258336	0.129168	+	0.991623i	$0.458769\pi$
$224$	4.36168	0.291427
$225$	1.13922	0.0759480
$226$	−17.8323	−1.18619
$227$	−9.91185	−0.657872	−0.328936	−	0.944352i	$-0.606690\pi$
$227$	−9.91185	−0.657872	−0.328936	+	0.944352i	$0.606690\pi$
$228$	17.0860	1.13155
$229$	−24.1350	−1.59489	−0.797443	−	0.603394i	$-0.793815\pi$
$229$	−24.1350	−1.59489	−0.797443	+	0.603394i	$0.793815\pi$
$230$	0.377046	0.0248617
$231$	16.1008	1.05936
$232$	42.5985	2.79673
$233$	−20.0088	−1.31082	−0.655409	−	0.755274i	$-0.727504\pi$
$233$	−20.0088	−1.31082	−0.655409	+	0.755274i	$0.727504\pi$
$234$	−0.574521	−0.0375576
$235$	2.44958	0.159793
$236$	46.5840	3.03236
$237$	−25.2431	−1.63972
$238$	−22.9880	−1.49009
$239$	2.55707	0.165403	0.0827015	−	0.996574i	$-0.473645\pi$
$239$	2.55707	0.165403	0.0827015	+	0.996574i	$0.473645\pi$
$240$	2.19792	0.141875
$241$	8.87486	0.571680	0.285840	−	0.958277i	$-0.407727\pi$
$241$	8.87486	0.571680	0.285840	+	0.958277i	$0.407727\pi$
$242$	−24.0477	−1.54584
$243$	2.39312	0.153519
$244$	5.62678	0.360218
$245$	−0.625250	−0.0399458
$246$	2.20842	0.140804
$247$	−2.44687	−0.155691
$248$	3.29694	0.209356
$249$	4.42757	0.280586
$250$	6.26333	0.396128
$251$	1.28371	0.0810271	0.0405136	−	0.999179i	$-0.487101\pi$
$251$	1.28371	0.0810271	0.0405136	+	0.999179i	$0.487101\pi$
$252$	2.06156	0.129866
$253$	2.71874	0.170926
$254$	−30.7279	−1.92804
$255$	−1.82833	−0.114495
$256$	−32.5688	−2.03555
$257$	25.8429	1.61203	0.806016	−	0.591893i	$-0.201619\pi$
$257$	25.8429	1.61203	0.806016	+	0.591893i	$0.201619\pi$
$258$	−1.13259	−0.0705120
$259$	19.6833	1.22306
$260$	−1.06265	−0.0659029
$261$	1.79850	0.111324
$262$	−5.15887	−0.318716
$263$	30.4813	1.87955	0.939777	−	0.341787i	$-0.111032\pi$
$263$	30.4813	1.87955	0.939777	+	0.341787i	$0.111032\pi$
$264$	41.3503	2.54493
$265$	1.38077	0.0848201
$266$	12.9650	0.794932
$267$	24.1109	1.47556
$268$	−46.3812	−2.83318
$269$	18.4073	1.12231	0.561155	−	0.827711i	$-0.310357\pi$
$269$	18.4073	1.12231	0.561155	+	0.827711i	$0.310357\pi$
$270$	3.38912	0.206255
$271$	−10.3151	−0.626596	−0.313298	−	0.949655i	$-0.601434\pi$
$271$	−10.3151	−0.626596	−0.313298	+	0.949655i	$0.601434\pi$
$272$	−22.6276	−1.37200
$273$	3.54222	0.214385
$274$	−49.1479	−2.96913
$275$	22.4355	1.35291
$276$	−4.17662	−0.251403
$277$	−22.8497	−1.37291	−0.686453	−	0.727174i	$-0.740833\pi$
$277$	−22.8497	−1.37291	−0.686453	+	0.727174i	$0.740833\pi$
$278$	20.8260	1.24906
$279$	0.139196	0.00833344
$280$	2.94689	0.176111
$281$	3.57684	0.213376	0.106688	−	0.994293i	$-0.465975\pi$
$281$	3.57684	0.213376	0.106688	+	0.994293i	$0.465975\pi$
$282$	−40.0675	−2.38598
$283$	33.5444	1.99401	0.997003	−	0.0773586i	$-0.0246486\pi$
$283$	33.5444	1.99401	0.997003	+	0.0773586i	$0.0246486\pi$
$284$	46.6835	2.77015
$285$	1.03116	0.0610805
$286$	−11.3145	−0.669040
$287$	1.13486	0.0669886
$288$	0.472933	0.0278678
$289$	1.82271	0.107218
$290$	4.91208	0.288447
$291$	−18.9870	−1.11304
$292$	20.8893	1.22245
$293$	12.5390	0.732535	0.366267	−	0.930510i	$-0.380636\pi$
$293$	12.5390	0.732535	0.366267	+	0.930510i	$0.380636\pi$
$294$	10.2272	0.596460
$295$	2.81139	0.163685
$296$	50.5507	2.93820
$297$	24.4377	1.41802
$298$	49.7159	2.87997
$299$	0.598129	0.0345907
$300$	−34.4662	−1.98991
$301$	−0.582013	−0.0335467
$302$	−19.5678	−1.12600
$303$	−31.4399	−1.80618
$304$	12.7617	0.731934
$305$	0.339581	0.0194444
$306$	−2.49257	−0.142491
$307$	−8.17801	−0.466744	−0.233372	−	0.972388i	$-0.574976\pi$
$307$	−8.17801	−0.466744	−0.233372	+	0.972388i	$0.574976\pi$
$308$	40.5999	2.31339
$309$	−3.13395	−0.178284
$310$	0.380173	0.0215924
$311$	3.85163	0.218406	0.109203	−	0.994019i	$-0.465170\pi$
$311$	3.85163	0.218406	0.109203	+	0.994019i	$0.465170\pi$
$312$	9.09715	0.515025
$313$	10.0930	0.570491	0.285246	−	0.958454i	$-0.407925\pi$
$313$	10.0930	0.570491	0.285246	+	0.958454i	$0.407925\pi$
$314$	46.1364	2.60363
$315$	0.124417	0.00701011
$316$	−63.6531	−3.58077
$317$	−20.3448	−1.14268	−0.571338	−	0.820715i	$-0.693576\pi$
$317$	−20.3448	−1.14268	−0.571338	+	0.820715i	$0.693576\pi$
$318$	−22.5851	−1.26651
$319$	35.4193	1.98310
$320$	−1.34991	−0.0754624
$321$	3.20817	0.179063
$322$	−3.16924	−0.176615
$323$	−10.6158	−0.590678
$324$	−34.6365	−1.92425
$325$	4.93587	0.273793
$326$	−42.8337	−2.37234
$327$	17.5515	0.970601
$328$	2.91455	0.160929
$329$	−20.5898	−1.13515
$330$	4.76814	0.262478
$331$	−31.1237	−1.71072	−0.855358	−	0.518037i	$-0.826663\pi$
$331$	−31.1237	−1.71072	−0.855358	+	0.518037i	$0.826663\pi$
$332$	11.1646	0.612735
$333$	2.13424	0.116956
$334$	−2.40792	−0.131756
$335$	−2.79915	−0.152934
$336$	−18.4745	−1.00787
$337$	−3.28775	−0.179095	−0.0895476	−	0.995983i	$-0.528542\pi$
$337$	−3.28775	−0.179095	−0.0895476	+	0.995983i	$0.528542\pi$
$338$	−2.48921	−0.135395
$339$	11.9213	0.647475
$340$	−4.61033	−0.250030
$341$	2.74129	0.148449
$342$	1.40578	0.0760158
$343$	20.1559	1.08831
$344$	−1.49473	−0.0805905
$345$	−0.252063	−0.0135706
$346$	−7.36545	−0.395969
$347$	−17.0724	−0.916495	−0.458248	−	0.888825i	$-0.651523\pi$
$347$	−17.0724	−0.916495	−0.458248	+	0.888825i	$0.651523\pi$
$348$	−54.4122	−2.91680
$349$	5.55935	0.297585	0.148793	−	0.988868i	$-0.452461\pi$
$349$	5.55935	0.297585	0.148793	+	0.988868i	$0.452461\pi$
$350$	−26.1531	−1.39794
$351$	5.37635	0.286968
$352$	9.31383	0.496429
$353$	−21.0828	−1.12212	−0.561061	−	0.827774i	$-0.689607\pi$
$353$	−21.0828	−1.12212	−0.561061	+	0.827774i	$0.689607\pi$
$354$	−45.9856	−2.44411
$355$	2.81739	0.149532
$356$	60.7982	3.22230
$357$	15.3680	0.813359
$358$	58.5412	3.09400
$359$	−1.93919	−0.102347	−0.0511733	−	0.998690i	$-0.516296\pi$
$359$	−1.93919	−0.102347	−0.0511733	+	0.998690i	$0.516296\pi$
$360$	0.319529	0.0168407
$361$	−13.0128	−0.684886
$362$	−55.8366	−2.93470
$363$	16.0764	0.843791
$364$	8.93206	0.468167
$365$	1.26069	0.0659875
$366$	−5.55450	−0.290338
$367$	16.8378	0.878927	0.439464	−	0.898260i	$-0.355169\pi$
$367$	16.8378	0.878927	0.439464	+	0.898260i	$0.355169\pi$
$368$	−3.11956	−0.162618
$369$	0.123052	0.00640581
$370$	5.82905	0.303038
$371$	−11.6060	−0.602553
$372$	−4.21126	−0.218344
$373$	14.2034	0.735423	0.367712	−	0.929940i	$-0.380141\pi$
$373$	14.2034	0.735423	0.367712	+	0.929940i	$0.380141\pi$
$374$	−49.0881	−2.53828
$375$	−4.18717	−0.216224
$376$	−52.8788	−2.72702
$377$	7.79231	0.401324
$378$	−28.4871	−1.46522
$379$	32.7130	1.68036	0.840178	−	0.542311i	$-0.182450\pi$
$379$	32.7130	1.68036	0.840178	+	0.542311i	$0.182450\pi$
$380$	2.60017	0.133386
$381$	20.5423	1.05241
$382$	−48.2778	−2.47011
$383$	−8.19492	−0.418741	−0.209370	−	0.977836i	$-0.567141\pi$
$383$	−8.19492	−0.418741	−0.209370	+	0.977836i	$0.567141\pi$
$384$	28.9000	1.47480
$385$	2.45024	0.124876
$386$	14.1077	0.718061
$387$	−0.0631072	−0.00320792
$388$	−47.8776	−2.43062
$389$	19.7444	1.00108	0.500541	−	0.865713i	$-0.333134\pi$
$389$	19.7444	1.00108	0.500541	+	0.865713i	$0.333134\pi$
$390$	1.04900	0.0531183
$391$	2.59499	0.131234
$392$	13.4972	0.681713
$393$	3.44882	0.173970
$394$	−2.55600	−0.128769
$395$	−3.84153	−0.193288
$396$	4.40221	0.221219
$397$	−7.01867	−0.352257	−0.176128	−	0.984367i	$-0.556357\pi$
$397$	−7.01867	−0.352257	−0.176128	+	0.984367i	$0.556357\pi$
$398$	−33.3617	−1.67227
$399$	−8.66734	−0.433910
$400$	−25.7431	−1.28716
$401$	2.32688	0.116199	0.0580994	−	0.998311i	$-0.481496\pi$
$401$	2.32688	0.116199	0.0580994	+	0.998311i	$0.481496\pi$
$402$	45.7855	2.28357
$403$	0.603090	0.0300421
$404$	−79.2790	−3.94428
$405$	−2.09035	−0.103870
$406$	−41.2882	−2.04910
$407$	42.0312	2.08341
$408$	39.4681	1.95396
$409$	14.4174	0.712893	0.356447	−	0.934316i	$-0.383988\pi$
$409$	14.4174	0.712893	0.356447	+	0.934316i	$0.383988\pi$
$410$	0.336080	0.0165978
$411$	32.8564	1.62069
$412$	−7.90257	−0.389332
$413$	−23.6310	−1.16280
$414$	−0.343638	−0.0168889
$415$	0.673793	0.0330752
$416$	2.04906	0.100464
$417$	−13.9226	−0.681794
$418$	27.6851	1.35412
$419$	28.9194	1.41280	0.706402	−	0.707811i	$-0.250317\pi$
$419$	28.9194	1.41280	0.706402	+	0.707811i	$0.250317\pi$
$420$	−3.76414	−0.183671
$421$	−31.6526	−1.54266	−0.771328	−	0.636438i	$-0.780407\pi$
$421$	−31.6526	−1.54266	−0.771328	+	0.636438i	$0.780407\pi$
$422$	−8.28870	−0.403487
$423$	−2.23253	−0.108549
$424$	−29.8066	−1.44754
$425$	21.4143	1.03875
$426$	−46.0838	−2.23277
$427$	−2.85433	−0.138131
$428$	8.08973	0.391032
$429$	7.56397	0.365192
$430$	−0.172359	−0.00831189
$431$	−1.33755	−0.0644273	−0.0322136	−	0.999481i	$-0.510256\pi$
$431$	−1.33755	−0.0644273	−0.0322136	+	0.999481i	$0.510256\pi$
$432$	−28.0405	−1.34910
$433$	8.51561	0.409234	0.204617	−	0.978842i	$-0.434405\pi$
$433$	8.51561	0.409234	0.204617	+	0.978842i	$0.434405\pi$
$434$	−3.19553	−0.153390
$435$	−3.28383	−0.157448
$436$	44.2580	2.11957
$437$	−1.46354	−0.0700108
$438$	−20.6210	−0.985309
$439$	−3.49140	−0.166636	−0.0833178	−	0.996523i	$-0.526552\pi$
$439$	−3.49140	−0.166636	−0.0833178	+	0.996523i	$0.526552\pi$
$440$	6.29273	0.299994
$441$	0.569850	0.0271357
$442$	−10.7995	−0.513679
$443$	4.10417	0.194995	0.0974975	−	0.995236i	$-0.468916\pi$
$443$	4.10417	0.194995	0.0974975	+	0.995236i	$0.468916\pi$
$444$	−64.5697	−3.06434
$445$	3.66923	0.173938
$446$	−9.60284	−0.454708
$447$	−33.2362	−1.57202
$448$	11.3466	0.536077
$449$	−19.4027	−0.915671	−0.457836	−	0.889037i	$-0.651375\pi$
$449$	−19.4027	−0.915671	−0.457836	+	0.889037i	$0.651375\pi$
$450$	−2.83576	−0.133679
$451$	2.42335	0.114111
$452$	30.0607	1.41394
$453$	13.0815	0.614623
$454$	24.6727	1.15795
$455$	0.539059	0.0252715
$456$	−22.2595	−1.04240
$457$	4.12246	0.192840	0.0964202	−	0.995341i	$-0.469261\pi$
$457$	4.12246	0.192840	0.0964202	+	0.995341i	$0.469261\pi$
$458$	60.0771	2.80722
$459$	23.3254	1.08873
$460$	−0.635603	−0.0296351
$461$	−2.79055	−0.129969	−0.0649844	−	0.997886i	$-0.520700\pi$
$461$	−2.79055	−0.129969	−0.0649844	+	0.997886i	$0.520700\pi$
$462$	−40.0784	−1.86461
$463$	12.0199	0.558612	0.279306	−	0.960202i	$-0.409896\pi$
$463$	12.0199	0.558612	0.279306	+	0.960202i	$0.409896\pi$
$464$	−40.6410	−1.88671
$465$	−0.254154	−0.0117861
$466$	49.8060	2.30722
$467$	−23.9053	−1.10621	−0.553104	−	0.833113i	$-0.686557\pi$
$467$	−23.9053	−1.10621	−0.553104	+	0.833113i	$0.686557\pi$
$468$	0.968496	0.0447687
$469$	23.5281	1.08643
$470$	−6.09751	−0.281257
$471$	−30.8431	−1.42118
$472$	−60.6892	−2.79345
$473$	−1.24282	−0.0571449
$474$	62.8355	2.88613
$475$	−12.0774	−0.554150
$476$	38.7519	1.77619
$477$	−1.25843	−0.0576194
$478$	−6.36508	−0.291132
$479$	29.9598	1.36890	0.684450	−	0.729060i	$-0.260042\pi$
$479$	29.9598	1.36890	0.684450	+	0.729060i	$0.260042\pi$
$480$	−0.863514	−0.0394139
$481$	9.24695	0.421625
$482$	−22.0914	−1.00624
$483$	2.11870	0.0964043
$484$	40.5382	1.84265
$485$	−2.88946	−0.131204
$486$	−5.95699	−0.270215
$487$	−24.6612	−1.11751	−0.558754	−	0.829334i	$-0.688720\pi$
$487$	−24.6612	−1.11751	−0.558754	+	0.829334i	$0.688720\pi$
$488$	−7.33052	−0.331837
$489$	28.6352	1.29493
$490$	1.55638	0.0703101
$491$	−25.3253	−1.14292	−0.571458	−	0.820632i	$-0.693622\pi$
$491$	−25.3253	−1.14292	−0.571458	+	0.820632i	$0.693622\pi$
$492$	−3.72283	−0.167838
$493$	33.8070	1.52259
$494$	6.09078	0.274037
$495$	0.265678	0.0119413
$496$	−3.14543	−0.141234
$497$	−23.6814	−1.06226
$498$	−11.0212	−0.493870
$499$	−15.5226	−0.694886	−0.347443	−	0.937701i	$-0.612950\pi$
$499$	−15.5226	−0.694886	−0.347443	+	0.937701i	$0.612950\pi$
$500$	−10.5584	−0.472185
$501$	1.60975	0.0719182
$502$	−3.19543	−0.142619
$503$	−38.4823	−1.71584	−0.857921	−	0.513781i	$-0.828244\pi$
$503$	−38.4823	−1.71584	−0.857921	+	0.513781i	$0.828244\pi$
$504$	−2.68578	−0.119634
$505$	−4.78456	−0.212910
$506$	−6.76752	−0.300853
$507$	1.66409	0.0739048
$508$	51.7994	2.29823
$509$	−8.55327	−0.379117	−0.189559	−	0.981869i	$-0.560706\pi$
$509$	−8.55327	−0.379117	−0.189559	+	0.981869i	$0.560706\pi$
$510$	4.55111	0.201526
$511$	−10.5967	−0.468769
$512$	46.3369	2.04782
$513$	−13.1552	−0.580817
$514$	−64.3283	−2.83740
$515$	−0.476928	−0.0210159
$516$	1.90926	0.0840504
$517$	−43.9670	−1.93367
$518$	−48.9958	−2.15275
$519$	4.92396	0.216138
$520$	1.38441	0.0607106
$521$	14.0442	0.615288	0.307644	−	0.951502i	$-0.400459\pi$
$521$	14.0442	0.615288	0.307644	+	0.951502i	$0.400459\pi$
$522$	−4.47685	−0.195946
$523$	−8.70857	−0.380799	−0.190400	−	0.981707i	$-0.560978\pi$
$523$	−8.70857	−0.380799	−0.190400	+	0.981707i	$0.560978\pi$
$524$	8.69654	0.379910
$525$	17.4839	0.763061
$526$	−75.8743	−3.30828
$527$	2.61652	0.113977
$528$	−39.4501	−1.71684
$529$	−22.6422	−0.984445
$530$	−3.43703	−0.149295
$531$	−2.56229	−0.111194
$532$	−21.8556	−0.947560
$533$	0.533142	0.0230930
$534$	−60.0172	−2.59720
$535$	0.488223	0.0211077
$536$	60.4251	2.60997
$537$	−39.1360	−1.68884
$538$	−45.8195	−1.97542
$539$	11.2225	0.483387
$540$	−5.71319	−0.245856
$541$	−14.5329	−0.624818	−0.312409	−	0.949948i	$-0.601136\pi$
$541$	−14.5329	−0.624818	−0.312409	+	0.949948i	$0.601136\pi$
$542$	25.6764	1.10290
$543$	37.3279	1.60189
$544$	8.88989	0.381151
$545$	2.67101	0.114414
$546$	−8.81733	−0.377347
$547$	−35.2122	−1.50557	−0.752783	−	0.658269i	$-0.771289\pi$
$547$	−35.2122	−1.50557	−0.752783	+	0.658269i	$0.771289\pi$
$548$	82.8508	3.53921
$549$	−0.309493	−0.0132088
$550$	−55.8468	−2.38132
$551$	−19.0668	−0.812272
$552$	5.44127	0.231596
$553$	32.2898	1.37310
$554$	56.8778	2.41651
$555$	−3.89685	−0.165412
$556$	−35.1074	−1.48888
$557$	−31.2134	−1.32255	−0.661277	−	0.750142i	$-0.729985\pi$
$557$	−31.2134	−1.32255	−0.661277	+	0.750142i	$0.729985\pi$
$558$	−0.346488	−0.0146680
$559$	−0.273423	−0.0115645
$560$	−2.81147	−0.118806
$561$	32.8164	1.38551
$562$	−8.90351	−0.375572
$563$	−21.6711	−0.913327	−0.456663	−	0.889640i	$-0.650956\pi$
$563$	−21.6711	−0.913327	−0.456663	+	0.889640i	$0.650956\pi$
$564$	67.5435	2.84409
$565$	1.81419	0.0763237
$566$	−83.4991	−3.50973
$567$	17.5703	0.737884
$568$	−60.8188	−2.55190
$569$	−26.2106	−1.09881	−0.549404	−	0.835557i	$-0.685145\pi$
$569$	−26.2106	−1.09881	−0.549404	+	0.835557i	$0.685145\pi$
$570$	−2.56677	−0.107510
$571$	−20.0447	−0.838843	−0.419421	−	0.907792i	$-0.637767\pi$
$571$	−20.0447	−0.838843	−0.419421	+	0.907792i	$0.637767\pi$
$572$	19.0733	0.797496
$573$	32.2747	1.34830
$574$	−2.82490	−0.117909
$575$	2.95229	0.123119
$576$	1.23030	0.0512626
$577$	−4.85958	−0.202307	−0.101153	−	0.994871i	$-0.532253\pi$
$577$	−4.85958	−0.202307	−0.101153	+	0.994871i	$0.532253\pi$
$578$	−4.53710	−0.188719
$579$	−9.43127	−0.391950
$580$	−8.28051	−0.343830
$581$	−5.66353	−0.234963
$582$	47.2626	1.95910
$583$	−24.7832	−1.02641
$584$	−27.2144	−1.12614
$585$	0.0584496	0.00241660
$586$	−31.2122	−1.28936
$587$	34.3513	1.41783	0.708916	−	0.705293i	$-0.249185\pi$
$587$	34.3513	1.41783	0.708916	+	0.705293i	$0.249185\pi$
$588$	−17.2404	−0.710981
$589$	−1.47568	−0.0608045
$590$	−6.99814	−0.288109
$591$	1.70874	0.0702882
$592$	−48.2277	−1.98215
$593$	10.3836	0.426403	0.213202	−	0.977008i	$-0.431611\pi$
$593$	10.3836	0.426403	0.213202	+	0.977008i	$0.431611\pi$
$594$	−60.8306	−2.49591
$595$	2.33871	0.0958779
$596$	−83.8084	−3.43292
$597$	22.3030	0.912801
$598$	−1.48887	−0.0608844
$599$	−18.4895	−0.755459	−0.377730	−	0.925916i	$-0.623295\pi$
$599$	−18.4895	−0.755459	−0.377730	+	0.925916i	$0.623295\pi$
$600$	44.9023	1.83313
$601$	12.2851	0.501120	0.250560	−	0.968101i	$-0.419385\pi$
$601$	12.2851	0.501120	0.250560	+	0.968101i	$0.419385\pi$
$602$	1.44875	0.0590468
$603$	2.55113	0.103890
$604$	32.9864	1.34220
$605$	2.44652	0.0994652
$606$	78.2606	3.17912
$607$	41.0076	1.66445	0.832223	−	0.554441i	$-0.187068\pi$
$607$	41.0076	1.66445	0.832223	+	0.554441i	$0.187068\pi$
$608$	−5.01379	−0.203336
$609$	27.6021	1.11849
$610$	−0.845290	−0.0342248
$611$	−9.67283	−0.391321
$612$	4.20183	0.169849
$613$	33.2505	1.34298	0.671488	−	0.741015i	$-0.265655\pi$
$613$	33.2505	1.34298	0.671488	+	0.741015i	$0.265655\pi$
$614$	20.3568	0.821533
$615$	−0.224676	−0.00905983
$616$	−52.8932	−2.13113
$617$	20.9831	0.844748	0.422374	−	0.906422i	$-0.361197\pi$
$617$	20.9831	0.844748	0.422374	+	0.906422i	$0.361197\pi$
$618$	7.80106	0.313805
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−0.640875	−0.0257382
$621$	3.21575	0.129044
$622$	−9.58753	−0.384425
$623$	−30.8415	−1.23564
$624$	−8.67910	−0.347442
$625$	24.0421	0.961685
$626$	−25.1237	−1.00414
$627$	−18.5081	−0.739141
$628$	−77.7741	−3.10353
$629$	40.1180	1.59961
$630$	−0.309701	−0.0123388
$631$	−2.17697	−0.0866637	−0.0433318	−	0.999061i	$-0.513797\pi$
$631$	−2.17697	−0.0866637	−0.0433318	+	0.999061i	$0.513797\pi$
$632$	82.9268	3.29865
$633$	5.54117	0.220242
$634$	50.6425	2.01127
$635$	3.12615	0.124057
$636$	38.0728	1.50968
$637$	2.46897	0.0978242
$638$	−88.1660	−3.49053
$639$	−2.56776	−0.101579
$640$	4.39804	0.173848
$641$	−19.9299	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$641$	−19.9299	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$642$	−7.98581	−0.315175
$643$	7.07584	0.279044	0.139522	−	0.990219i	$-0.455443\pi$
$643$	7.07584	0.279044	0.139522	+	0.990219i	$0.455443\pi$
$644$	5.34253	0.210525
$645$	0.115226	0.00453700
$646$	26.4249	1.03967
$647$	−6.38090	−0.250859	−0.125429	−	0.992103i	$-0.540031\pi$
$647$	−6.38090	−0.250859	−0.125429	+	0.992103i	$0.540031\pi$
$648$	45.1242	1.77265
$649$	−50.4611	−1.98077
$650$	−12.2864	−0.481913
$651$	2.13628	0.0837273
$652$	72.2067	2.82783
$653$	49.4568	1.93540	0.967698	−	0.252114i	$-0.0811257\pi$
$653$	49.4568	1.93540	0.967698	+	0.252114i	$0.0811257\pi$
$654$	−43.6894	−1.70839
$655$	0.524845	0.0205074
$656$	−2.78062	−0.108565
$657$	−1.14899	−0.0448262
$658$	51.2523	1.99802
$659$	−2.07448	−0.0808103	−0.0404052	−	0.999183i	$-0.512865\pi$
$659$	−2.07448	−0.0808103	−0.0404052	+	0.999183i	$0.512865\pi$
$660$	−8.03787	−0.312874
$661$	17.1411	0.666710	0.333355	−	0.942801i	$-0.391819\pi$
$661$	17.1411	0.666710	0.333355	+	0.942801i	$0.391819\pi$
$662$	77.4736	3.01110
$663$	7.21968	0.280389
$664$	−14.5451	−0.564460
$665$	−1.31901	−0.0511489
$666$	−5.31257	−0.205858
$667$	4.66081	0.180467
$668$	4.05914	0.157053
$669$	6.41970	0.248200
$670$	6.96768	0.269185
$671$	−6.09508	−0.235298
$672$	7.25822	0.279992
$673$	−7.97335	−0.307350	−0.153675	−	0.988121i	$-0.549111\pi$
$673$	−7.97335	−0.307350	−0.153675	+	0.988121i	$0.549111\pi$
$674$	8.18391	0.315232
$675$	26.5370	1.02141
$676$	4.19617	0.161391
$677$	−33.8209	−1.29984	−0.649922	−	0.760001i	$-0.725199\pi$
$677$	−33.8209	−1.29984	−0.649922	+	0.760001i	$0.725199\pi$
$678$	−29.6746	−1.13965
$679$	24.2872	0.932057
$680$	6.00630	0.230331
$681$	−16.4942	−0.632059
$682$	−6.82366	−0.261291
$683$	14.6859	0.561939	0.280969	−	0.959717i	$-0.409344\pi$
$683$	14.6859	0.561939	0.280969	+	0.959717i	$0.409344\pi$
$684$	−2.36978	−0.0906109
$685$	5.00012	0.191045
$686$	−50.1722	−1.91558
$687$	−40.1628	−1.53231
$688$	1.42604	0.0543674
$689$	−5.45235	−0.207718
$690$	0.627438	0.0238862
$691$	−49.1224	−1.86870	−0.934352	−	0.356352i	$-0.884020\pi$
$691$	−49.1224	−1.86870	−0.934352	+	0.356352i	$0.884020\pi$
$692$	12.4163	0.471996
$693$	−2.23314	−0.0848299
$694$	42.4969	1.61316
$695$	−2.11876	−0.0803692
$696$	70.8878	2.68699
$697$	2.31305	0.0876129
$698$	−13.8384	−0.523791
$699$	−33.2964	−1.25939
$700$	44.0875	1.66635
$701$	−18.3318	−0.692383	−0.346191	−	0.938164i	$-0.612525\pi$
$701$	−18.3318	−0.692383	−0.346191	+	0.938164i	$0.612525\pi$
$702$	−13.3829	−0.505104
$703$	−22.6261	−0.853359
$704$	24.2293	0.913176
$705$	4.07632	0.153523
$706$	52.4795	1.97509
$707$	40.2164	1.51249
$708$	77.5200	2.91338
$709$	−24.3257	−0.913571	−0.456786	−	0.889577i	$-0.650999\pi$
$709$	−24.3257	−0.913571	−0.456786	+	0.889577i	$0.650999\pi$
$710$	−7.01308	−0.263196
$711$	3.50115	0.131303
$712$	−79.2073	−2.96842
$713$	0.360726	0.0135093
$714$	−38.2541	−1.43162
$715$	1.15109	0.0430485
$716$	−98.6855	−3.68805
$717$	4.25519	0.158913
$718$	4.82706	0.180144
$719$	27.1891	1.01398	0.506992	−	0.861951i	$-0.330757\pi$
$719$	27.1891	1.01398	0.506992	+	0.861951i	$0.330757\pi$
$720$	−0.304846	−0.0113609
$721$	4.00879	0.149295
$722$	32.3917	1.20549
$723$	14.7686	0.549249
$724$	94.1261	3.49817
$725$	38.4618	1.42844
$726$	−40.0175	−1.48519
$727$	−47.0188	−1.74383	−0.871916	−	0.489655i	$-0.837123\pi$
$727$	−47.0188	−1.74383	−0.871916	+	0.489655i	$0.837123\pi$
$728$	−11.6366	−0.431282
$729$	28.7453	1.06464
$730$	−3.13812	−0.116147
$731$	−1.18625	−0.0438750
$732$	9.36347	0.346084
$733$	43.8669	1.62026	0.810131	−	0.586248i	$-0.199396\pi$
$733$	43.8669	1.62026	0.810131	+	0.586248i	$0.199396\pi$
$734$	−41.9129	−1.54703
$735$	−1.04047	−0.0383784
$736$	1.22560	0.0451763
$737$	50.2414	1.85067
$738$	−0.306302	−0.0112751
$739$	−16.5270	−0.607956	−0.303978	−	0.952679i	$-0.598315\pi$
$739$	−16.5270	−0.607956	−0.303978	+	0.952679i	$0.598315\pi$
$740$	−9.82629	−0.361222
$741$	−4.07181	−0.149582
$742$	28.8898	1.06058
$743$	−14.4054	−0.528484	−0.264242	−	0.964456i	$-0.585122\pi$
$743$	−14.4054	−0.528484	−0.264242	+	0.964456i	$0.585122\pi$
$744$	5.48640	0.201141
$745$	−5.05792	−0.185308
$746$	−35.3552	−1.29445
$747$	−0.614091	−0.0224684
$748$	82.7499	3.02564
$749$	−4.10373	−0.149947
$750$	10.4227	0.380585
$751$	54.0129	1.97096	0.985480	−	0.169794i	$-0.0543103\pi$
$751$	54.0129	1.97096	0.985480	+	0.169794i	$0.0543103\pi$
$752$	50.4489	1.83968
$753$	2.13621	0.0778479
$754$	−19.3967	−0.706386
$755$	1.99076	0.0724511
$756$	48.0219	1.74654
$757$	−6.75078	−0.245361	−0.122681	−	0.992446i	$-0.539149\pi$
$757$	−6.75078	−0.245361	−0.122681	+	0.992446i	$0.539149\pi$
$758$	−81.4297	−2.95766
$759$	4.52423	0.164219
$760$	−3.38748	−0.122877
$761$	43.5483	1.57863	0.789313	−	0.613991i	$-0.210437\pi$
$761$	43.5483	1.57863	0.789313	+	0.613991i	$0.210437\pi$
$762$	−51.1341	−1.85239
$763$	−22.4510	−0.812782
$764$	81.3840	2.94437
$765$	0.253585	0.00916837
$766$	20.3989	0.737042
$767$	−11.1015	−0.400853
$768$	−54.1974	−1.95568
$769$	−28.0864	−1.01282	−0.506411	−	0.862292i	$-0.669028\pi$
$769$	−28.0864	−1.01282	−0.506411	+	0.862292i	$0.669028\pi$
$770$	−6.09917	−0.219799
$771$	43.0048	1.54878
$772$	−23.7819	−0.855929
$773$	−31.4579	−1.13146	−0.565732	−	0.824589i	$-0.691406\pi$
$773$	−31.4579	−1.13146	−0.565732	+	0.824589i	$0.691406\pi$
$774$	0.157087	0.00564638
$775$	2.97677	0.106929
$776$	62.3746	2.23912
$777$	32.7547	1.17507
$778$	−49.1481	−1.76204
$779$	−1.30453	−0.0467396
$780$	−1.76835	−0.0633170
$781$	−50.5688	−1.80949
$782$	−6.45948	−0.230991
$783$	41.8942	1.49718
$784$	−12.8770	−0.459892
$785$	−4.69374	−0.167527
$786$	−8.58483	−0.306211
$787$	−43.1848	−1.53937	−0.769686	−	0.638423i	$-0.779587\pi$
$787$	−43.1848	−1.53937	−0.769686	+	0.638423i	$0.779587\pi$
$788$	4.30876	0.153493
$789$	50.7236	1.80581
$790$	9.56238	0.340214
$791$	−15.2491	−0.542196
$792$	−5.73516	−0.203790
$793$	−1.34093	−0.0476179
$794$	17.4709	0.620021
$795$	2.29773	0.0814920
$796$	56.2393	1.99335
$797$	29.9550	1.06106	0.530531	−	0.847666i	$-0.321993\pi$
$797$	29.9550	1.06106	0.530531	+	0.847666i	$0.321993\pi$
$798$	21.5748	0.763741
$799$	−41.9657	−1.48464
$800$	10.1139	0.357580
$801$	−3.34412	−0.118158
$802$	−5.79210	−0.204526
$803$	−22.6279	−0.798521
$804$	−77.1826	−2.72202
$805$	0.322427	0.0113640
$806$	−1.50122	−0.0528782
$807$	30.6313	1.07827
$808$	103.284	3.63352
$809$	25.4097	0.893358	0.446679	−	0.894694i	$-0.352607\pi$
$809$	25.4097	0.893358	0.446679	+	0.894694i	$0.352607\pi$
$810$	5.20332	0.182826
$811$	−7.35993	−0.258442	−0.129221	−	0.991616i	$-0.541248\pi$
$811$	−7.35993	−0.258442	−0.129221	+	0.991616i	$0.541248\pi$
$812$	69.6014	2.44253
$813$	−17.1652	−0.602011
$814$	−104.625	−3.66709
$815$	4.35774	0.152645
$816$	−37.6544	−1.31817
$817$	0.669030	0.0234064
$818$	−35.8879	−1.25479
$819$	−0.491295	−0.0171672
$820$	−0.566545	−0.0197846
$821$	41.8014	1.45888	0.729439	−	0.684046i	$-0.239781\pi$
$821$	41.8014	1.45888	0.729439	+	0.684046i	$0.239781\pi$
$822$	−81.7865	−2.85263
$823$	35.7352	1.24565	0.622825	−	0.782361i	$-0.285985\pi$
$823$	35.7352	1.24565	0.622825	+	0.782361i	$0.285985\pi$
$824$	10.2954	0.358657
$825$	37.3348	1.29983
$826$	58.8225	2.04670
$827$	−5.30617	−0.184514	−0.0922568	−	0.995735i	$-0.529408\pi$
$827$	−5.30617	−0.184514	−0.0922568	+	0.995735i	$0.529408\pi$
$828$	0.579285	0.0201316
$829$	−49.4644	−1.71797	−0.858985	−	0.512002i	$-0.828904\pi$
$829$	−49.4644	−1.71797	−0.858985	+	0.512002i	$0.828904\pi$
$830$	−1.67721	−0.0582169
$831$	−38.0240	−1.31904
$832$	5.33050	0.184802
$833$	10.7117	0.371137
$834$	34.6564	1.20005
$835$	0.244973	0.00847765
$836$	−46.6700	−1.61411
$837$	3.24242	0.112075
$838$	−71.9864	−2.48673
$839$	19.1039	0.659539	0.329769	−	0.944062i	$-0.393029\pi$
$839$	19.1039	0.659539	0.329769	+	0.944062i	$0.393029\pi$
$840$	4.90390	0.169200
$841$	31.7201	1.09380
$842$	78.7901	2.71529
$843$	5.95218	0.205004
$844$	13.9726	0.480957
$845$	0.253243	0.00871183
$846$	5.55724	0.191062
$847$	−20.5641	−0.706591
$848$	28.4369	0.976527
$849$	55.8209	1.91577
$850$	−53.3048	−1.82834
$851$	5.53087	0.189596
$852$	77.6855	2.66146
$853$	−13.6633	−0.467823	−0.233912	−	0.972258i	$-0.575153\pi$
$853$	−13.6633	−0.467823	−0.233912	+	0.972258i	$0.575153\pi$
$854$	7.10504	0.243129
$855$	−0.143019	−0.00489113
$856$	−10.5392	−0.360224
$857$	4.17854	0.142736	0.0713681	−	0.997450i	$-0.477264\pi$
$857$	4.17854	0.142736	0.0713681	+	0.997450i	$0.477264\pi$
$858$	−18.8283	−0.642789
$859$	22.3677	0.763175	0.381587	−	0.924333i	$-0.375378\pi$
$859$	22.3677	0.763175	0.381587	+	0.924333i	$0.375378\pi$
$860$	0.290553	0.00990778
$861$	1.88851	0.0643601
$862$	3.32943	0.113401
$863$	36.7292	1.25028	0.625138	−	0.780514i	$-0.285043\pi$
$863$	36.7292	1.25028	0.625138	+	0.780514i	$0.285043\pi$
$864$	11.0165	0.374788
$865$	0.749333	0.0254781
$866$	−21.1971	−0.720308
$867$	3.03315	0.103011
$868$	5.38684	0.182841
$869$	68.9509	2.33900
$870$	8.17415	0.277130
$871$	11.0532	0.374524
$872$	−57.6589	−1.95258
$873$	2.63344	0.0891284
$874$	3.64307	0.123229
$875$	5.35602	0.181066
$876$	34.7617	1.17449
$877$	56.3100	1.90145	0.950727	−	0.310028i	$-0.100338\pi$
$877$	56.3100	1.90145	0.950727	+	0.310028i	$0.100338\pi$
$878$	8.69084	0.293302
$879$	20.8660	0.703792
$880$	−6.00356	−0.202380
$881$	−22.8289	−0.769124	−0.384562	−	0.923099i	$-0.625648\pi$
$881$	−22.8289	−0.769124	−0.384562	+	0.923099i	$0.625648\pi$
$882$	−1.41848	−0.0477626
$883$	18.4401	0.620559	0.310279	−	0.950645i	$-0.399577\pi$
$883$	18.4401	0.620559	0.310279	+	0.950645i	$0.399577\pi$
$884$	18.2052	0.612306
$885$	4.67840	0.157263
$886$	−10.2161	−0.343218
$887$	−14.7530	−0.495356	−0.247678	−	0.968842i	$-0.579668\pi$
$887$	−14.7530	−0.495356	−0.247678	+	0.968842i	$0.579668\pi$
$888$	84.1209	2.82291
$889$	−26.2767	−0.881291
$890$	−9.13348	−0.306155
$891$	37.5193	1.25694
$892$	16.1879	0.542012
$893$	23.6681	0.792024
$894$	82.7318	2.76697
$895$	−5.95577	−0.199079
$896$	−36.9675	−1.23500
$897$	0.995341	0.0332335
$898$	48.2975	1.61171
$899$	4.69947	0.156736
$900$	4.78037	0.159346
$901$	−23.6551	−0.788066
$902$	−6.03223	−0.200851
$903$	−0.968523	−0.0322304
$904$	−39.1629	−1.30254
$905$	5.68060	0.188830
$906$	−32.5626	−1.08182
$907$	40.2439	1.33628	0.668138	−	0.744037i	$-0.267091\pi$
$907$	40.2439	1.33628	0.668138	+	0.744037i	$0.267091\pi$
$908$	−41.5918	−1.38027
$909$	4.36063	0.144633
$910$	−1.34183	−0.0444813
$911$	−41.5812	−1.37765	−0.688824	−	0.724929i	$-0.741873\pi$
$911$	−41.5812	−1.37765	−0.688824	+	0.724929i	$0.741873\pi$
$912$	21.2366	0.703215
$913$	−12.0938	−0.400245
$914$	−10.2617	−0.339426
$915$	0.565094	0.0186814
$916$	−101.275	−3.34621
$917$	−4.41155	−0.145682
$918$	−58.0618	−1.91632
$919$	−10.3506	−0.341434	−0.170717	−	0.985320i	$-0.554608\pi$
$919$	−10.3506	−0.341434	−0.170717	+	0.985320i	$0.554608\pi$
$920$	0.828058	0.0273003
$921$	−13.6089	−0.448430
$922$	6.94627	0.228763
$923$	−11.1252	−0.366192
$924$	67.5619	2.22262
$925$	45.6417	1.50069
$926$	−29.9201	−0.983234
$927$	0.434669	0.0142764
$928$	15.9669	0.524140
$929$	−3.35952	−0.110222	−0.0551112	−	0.998480i	$-0.517551\pi$
$929$	−3.35952	−0.110222	−0.0551112	+	0.998480i	$0.517551\pi$
$930$	0.632643	0.0207452
$931$	−6.04125	−0.197994
$932$	−83.9603	−2.75021
$933$	6.40947	0.209836
$934$	59.5054	1.94708
$935$	4.99404	0.163322
$936$	−1.26175	−0.0412415
$937$	−42.3339	−1.38299	−0.691494	−	0.722383i	$-0.743047\pi$
$937$	−42.3339	−1.38299	−0.691494	+	0.722383i	$0.743047\pi$
$938$	−58.5665	−1.91226
$939$	16.7957	0.548107
$940$	10.2788	0.335259
$941$	−20.4438	−0.666447	−0.333224	−	0.942848i	$-0.608136\pi$
$941$	−20.4438	−0.666447	−0.333224	+	0.942848i	$0.608136\pi$
$942$	76.7751	2.50147
$943$	0.318888	0.0103844
$944$	57.9004	1.88450
$945$	2.89817	0.0942774
$946$	3.09364	0.100583
$947$	36.1649	1.17520	0.587601	−	0.809151i	$-0.300073\pi$
$947$	36.1649	1.17520	0.587601	+	0.809151i	$0.300073\pi$
$948$	−105.925	−3.44027
$949$	−4.97818	−0.161599
$950$	30.0633	0.975381
$951$	−33.8556	−1.09784
$952$	−50.4856	−1.63625
$953$	45.6930	1.48014	0.740071	−	0.672529i	$-0.234792\pi$
$953$	45.6930	1.48014	0.740071	+	0.672529i	$0.234792\pi$
$954$	3.13249	0.101418
$955$	4.91160	0.158936
$956$	10.7299	0.347030
$957$	58.9408	1.90529
$958$	−74.5764	−2.40945
$959$	−42.0283	−1.35716
$960$	−2.24638	−0.0725014
$961$	−30.6363	−0.988267
$962$	−23.0176	−0.742118
$963$	−0.444964	−0.0143388
$964$	37.2404	1.19943
$965$	−1.43526	−0.0462027
$966$	−5.27390	−0.169685
$967$	47.1287	1.51556	0.757778	−	0.652512i	$-0.226285\pi$
$967$	47.1287	1.51556	0.757778	+	0.652512i	$0.226285\pi$
$968$	−52.8129	−1.69747
$969$	−17.6656	−0.567501
$970$	7.19248	0.230937
$971$	39.8774	1.27973	0.639863	−	0.768489i	$-0.278991\pi$
$971$	39.8774	1.27973	0.639863	+	0.768489i	$0.278991\pi$
$972$	10.0420	0.322096
$973$	17.8091	0.570935
$974$	61.3870	1.96697
$975$	8.21373	0.263050
$976$	6.99366	0.223862
$977$	52.6219	1.68352	0.841762	−	0.539849i	$-0.181519\pi$
$977$	52.6219	1.68352	0.841762	+	0.539849i	$0.181519\pi$
$978$	−71.2791	−2.27926
$979$	−65.8582	−2.10484
$980$	−2.62366	−0.0838097
$981$	−2.43435	−0.0777227
$982$	63.0401	2.01169
$983$	3.28391	0.104741	0.0523703	−	0.998628i	$-0.483322\pi$
$983$	3.28391	0.104741	0.0523703	+	0.998628i	$0.483322\pi$
$984$	4.85007	0.154615
$985$	0.260038	0.00828550
$986$	−84.1529	−2.67997
$987$	−34.2632	−1.09061
$988$	−10.2675	−0.326652
$989$	−0.163542	−0.00520033
$990$	−0.661328	−0.0210184
$991$	27.4795	0.872914	0.436457	−	0.899725i	$-0.356233\pi$
$991$	27.4795	0.872914	0.436457	+	0.899725i	$0.356233\pi$
$992$	1.23577	0.0392357
$993$	−51.7927	−1.64359
$994$	58.9481	1.86972
$995$	3.39410	0.107600
$996$	18.5789	0.588693
$997$	−7.29781	−0.231124	−0.115562	−	0.993300i	$-0.536867\pi$
$997$	−7.29781	−0.231124	−0.115562	+	0.993300i	$0.536867\pi$
$998$	38.6390	1.22310
$999$	49.7149	1.57291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.10	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.10	✓	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)

\(f(q)\)	\(=\)	\(q-2.48921 q^{2} +1.66409 q^{3} +4.19617 q^{4} +0.253243 q^{5} -4.14227 q^{6} -2.12862 q^{7} -5.46674 q^{8} -0.230804 q^{9} +O(q^{10})\)	\(q-2.48921 q^{2} +1.66409 q^{3} +4.19617 q^{4} +0.253243 q^{5} -4.14227 q^{6} -2.12862 q^{7} -5.46674 q^{8} -0.230804 q^{9} -0.630376 q^{10} -4.54541 q^{11} +6.98281 q^{12} -1.00000 q^{13} +5.29859 q^{14} +0.421419 q^{15} +5.21552 q^{16} -4.33851 q^{17} +0.574521 q^{18} +2.44687 q^{19} +1.06265 q^{20} -3.54222 q^{21} +11.3145 q^{22} -0.598129 q^{23} -9.09715 q^{24} -4.93587 q^{25} +2.48921 q^{26} -5.37635 q^{27} -8.93206 q^{28} -7.79231 q^{29} -1.04900 q^{30} -0.603090 q^{31} -2.04906 q^{32} -7.56397 q^{33} +10.7995 q^{34} -0.539059 q^{35} -0.968496 q^{36} -9.24695 q^{37} -6.09078 q^{38} -1.66409 q^{39} -1.38441 q^{40} -0.533142 q^{41} +8.81733 q^{42} +0.273423 q^{43} -19.0733 q^{44} -0.0584496 q^{45} +1.48887 q^{46} +9.67283 q^{47} +8.67910 q^{48} -2.46897 q^{49} +12.2864 q^{50} -7.21968 q^{51} -4.19617 q^{52} +5.45235 q^{53} +13.3829 q^{54} -1.15109 q^{55} +11.6366 q^{56} +4.07181 q^{57} +19.3967 q^{58} +11.1015 q^{59} +1.76835 q^{60} +1.34093 q^{61} +1.50122 q^{62} +0.491295 q^{63} -5.33050 q^{64} -0.253243 q^{65} +18.8283 q^{66} -11.0532 q^{67} -18.2052 q^{68} -0.995341 q^{69} +1.34183 q^{70} +11.1252 q^{71} +1.26175 q^{72} +4.97818 q^{73} +23.0176 q^{74} -8.21373 q^{75} +10.2675 q^{76} +9.67546 q^{77} +4.14227 q^{78} -15.1693 q^{79} +1.32080 q^{80} -8.25432 q^{81} +1.32710 q^{82} +2.66066 q^{83} -14.8638 q^{84} -1.09870 q^{85} -0.680607 q^{86} -12.9671 q^{87} +24.8486 q^{88} +14.4890 q^{89} +0.145494 q^{90} +2.12862 q^{91} -2.50985 q^{92} -1.00360 q^{93} -24.0777 q^{94} +0.619653 q^{95} -3.40982 q^{96} -11.4098 q^{97} +6.14579 q^{98} +1.04910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 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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