LMFDB - Embedded newform 8047.2.a.b.1.16

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:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-2.41564 q^{2} +2.60884 q^{3} +3.83533 q^{4} +1.90553 q^{5} -6.30203 q^{6} -0.0638029 q^{7} -4.43349 q^{8} +3.80607 q^{9} +O(q^{10})$	\(q-2.41564 q^{2} +2.60884 q^{3} +3.83533 q^{4} +1.90553 q^{5} -6.30203 q^{6} -0.0638029 q^{7} -4.43349 q^{8} +3.80607 q^{9} -4.60307 q^{10} -0.824160 q^{11} +10.0058 q^{12} +1.00000 q^{13} +0.154125 q^{14} +4.97122 q^{15} +3.03908 q^{16} +1.70186 q^{17} -9.19410 q^{18} -3.21690 q^{19} +7.30832 q^{20} -0.166452 q^{21} +1.99088 q^{22} -7.15699 q^{23} -11.5663 q^{24} -1.36897 q^{25} -2.41564 q^{26} +2.10290 q^{27} -0.244705 q^{28} -9.58079 q^{29} -12.0087 q^{30} +1.24948 q^{31} +1.52566 q^{32} -2.15010 q^{33} -4.11107 q^{34} -0.121578 q^{35} +14.5975 q^{36} +2.14765 q^{37} +7.77088 q^{38} +2.60884 q^{39} -8.44814 q^{40} -4.26465 q^{41} +0.402088 q^{42} +10.6940 q^{43} -3.16092 q^{44} +7.25257 q^{45} +17.2887 q^{46} -1.09153 q^{47} +7.92849 q^{48} -6.99593 q^{49} +3.30693 q^{50} +4.43988 q^{51} +3.83533 q^{52} -3.55456 q^{53} -5.07986 q^{54} -1.57046 q^{55} +0.282870 q^{56} -8.39239 q^{57} +23.1438 q^{58} -11.8914 q^{59} +19.0663 q^{60} -12.6040 q^{61} -3.01829 q^{62} -0.242838 q^{63} -9.76360 q^{64} +1.90553 q^{65} +5.19388 q^{66} -13.0800 q^{67} +6.52717 q^{68} -18.6715 q^{69} +0.293690 q^{70} +8.31218 q^{71} -16.8742 q^{72} -5.91273 q^{73} -5.18795 q^{74} -3.57142 q^{75} -12.3379 q^{76} +0.0525838 q^{77} -6.30203 q^{78} +7.96634 q^{79} +5.79105 q^{80} -5.93205 q^{81} +10.3019 q^{82} -5.46635 q^{83} -0.638398 q^{84} +3.24293 q^{85} -25.8329 q^{86} -24.9948 q^{87} +3.65391 q^{88} +7.57667 q^{89} -17.5196 q^{90} -0.0638029 q^{91} -27.4494 q^{92} +3.25969 q^{93} +2.63674 q^{94} -6.12989 q^{95} +3.98020 q^{96} +5.99226 q^{97} +16.8997 q^{98} -3.13681 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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 $

$2$	−2.41564	−1.70812	−0.854058	−	0.520177i	$-0.825866\pi$
$2$	−2.41564	−1.70812	−0.854058	+	0.520177i	$0.825866\pi$
$3$	2.60884	1.50622	0.753108	−	0.657896i	$-0.228554\pi$
$3$	2.60884	1.50622	0.753108	+	0.657896i	$0.228554\pi$
$4$	3.83533	1.91766
$5$	1.90553	0.852178	0.426089	−	0.904681i	$-0.359891\pi$
$5$	1.90553	0.852178	0.426089	+	0.904681i	$0.359891\pi$
$6$	−6.30203	−2.57279
$7$	−0.0638029	−0.0241152	−0.0120576	−	0.999927i	$-0.503838\pi$
$7$	−0.0638029	−0.0241152	−0.0120576	+	0.999927i	$0.503838\pi$
$8$	−4.43349	−1.56748
$9$	3.80607	1.26869
$10$	−4.60307	−1.45562
$11$	−0.824160	−0.248494	−0.124247	−	0.992251i	$-0.539651\pi$
$11$	−0.824160	−0.248494	−0.124247	+	0.992251i	$0.539651\pi$
$12$	10.0058	2.88842
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.154125	0.0411917
$15$	4.97122	1.28356
$16$	3.03908	0.759770
$17$	1.70186	0.412761	0.206380	−	0.978472i	$-0.433832\pi$
$17$	1.70186	0.412761	0.206380	+	0.978472i	$0.433832\pi$
$18$	−9.19410	−2.16707
$19$	−3.21690	−0.738008	−0.369004	−	0.929428i	$-0.620301\pi$
$19$	−3.21690	−0.738008	−0.369004	+	0.929428i	$0.620301\pi$
$20$	7.30832	1.63419
$21$	−0.166452	−0.0363228
$22$	1.99088	0.424456
$23$	−7.15699	−1.49234	−0.746168	−	0.665758i	$-0.768108\pi$
$23$	−7.15699	−1.49234	−0.746168	+	0.665758i	$0.768108\pi$
$24$	−11.5663	−2.36096
$25$	−1.36897	−0.273793
$26$	−2.41564	−0.473746
$27$	2.10290	0.404704
$28$	−0.244705	−0.0462449
$29$	−9.58079	−1.77911	−0.889554	−	0.456829i	$-0.848985\pi$
$29$	−9.58079	−1.77911	−0.889554	+	0.456829i	$0.848985\pi$
$30$	−12.0087	−2.19248
$31$	1.24948	0.224413	0.112206	−	0.993685i	$-0.464208\pi$
$31$	1.24948	0.224413	0.112206	+	0.993685i	$0.464208\pi$
$32$	1.52566	0.269701
$33$	−2.15010	−0.374285
$34$	−4.11107	−0.705043
$35$	−0.121578	−0.0205505
$36$	14.5975	2.43292
$37$	2.14765	0.353071	0.176536	−	0.984294i	$-0.443511\pi$
$37$	2.14765	0.353071	0.176536	+	0.984294i	$0.443511\pi$
$38$	7.77088	1.26060
$39$	2.60884	0.417749
$40$	−8.44814	−1.33577
$41$	−4.26465	−0.666026	−0.333013	−	0.942922i	$-0.608065\pi$
$41$	−4.26465	−0.666026	−0.333013	+	0.942922i	$0.608065\pi$
$42$	0.402088	0.0620436
$43$	10.6940	1.63082	0.815412	−	0.578882i	$-0.196511\pi$
$43$	10.6940	1.63082	0.815412	+	0.578882i	$0.196511\pi$
$44$	−3.16092	−0.476527
$45$	7.25257	1.08115
$46$	17.2887	2.54908
$47$	−1.09153	−0.159216	−0.0796078	−	0.996826i	$-0.525367\pi$
$47$	−1.09153	−0.159216	−0.0796078	+	0.996826i	$0.525367\pi$
$48$	7.92849	1.14438
$49$	−6.99593	−0.999418
$50$	3.30693	0.467671
$51$	4.43988	0.621707
$52$	3.83533	0.531864
$53$	−3.55456	−0.488256	−0.244128	−	0.969743i	$-0.578502\pi$
$53$	−3.55456	−0.488256	−0.244128	+	0.969743i	$0.578502\pi$
$54$	−5.07986	−0.691282
$55$	−1.57046	−0.211761
$56$	0.282870	0.0378001
$57$	−8.39239	−1.11160
$58$	23.1438	3.03893
$59$	−11.8914	−1.54813	−0.774064	−	0.633107i	$-0.781779\pi$
$59$	−11.8914	−1.54813	−0.774064	+	0.633107i	$0.781779\pi$
$60$	19.0663	2.46144
$61$	−12.6040	−1.61377	−0.806887	−	0.590706i	$-0.798849\pi$
$61$	−12.6040	−1.61377	−0.806887	+	0.590706i	$0.798849\pi$
$62$	−3.01829	−0.383323
$63$	−0.242838	−0.0305947
$64$	−9.76360	−1.22045
$65$	1.90553	0.236352
$66$	5.19388	0.639323
$67$	−13.0800	−1.59798	−0.798990	−	0.601345i	$-0.794632\pi$
$67$	−13.0800	−1.59798	−0.798990	+	0.601345i	$0.794632\pi$
$68$	6.52717	0.791536
$69$	−18.6715	−2.24778
$70$	0.293690	0.0351026
$71$	8.31218	0.986474	0.493237	−	0.869895i	$-0.335814\pi$
$71$	8.31218	0.986474	0.493237	+	0.869895i	$0.335814\pi$
$72$	−16.8742	−1.98864
$73$	−5.91273	−0.692033	−0.346016	−	0.938228i	$-0.612466\pi$
$73$	−5.91273	−0.692033	−0.346016	+	0.938228i	$0.612466\pi$
$74$	−5.18795	−0.603087
$75$	−3.57142	−0.412392
$76$	−12.3379	−1.41525
$77$	0.0525838	0.00599248
$78$	−6.30203	−0.713565
$79$	7.96634	0.896283	0.448142	−	0.893963i	$-0.352086\pi$
$79$	7.96634	0.896283	0.448142	+	0.893963i	$0.352086\pi$
$80$	5.79105	0.647459
$81$	−5.93205	−0.659117
$82$	10.3019	1.13765
$83$	−5.46635	−0.600010	−0.300005	−	0.953938i	$-0.596988\pi$
$83$	−5.46635	−0.600010	−0.300005	+	0.953938i	$0.596988\pi$
$84$	−0.638398	−0.0696549
$85$	3.24293	0.351745
$86$	−25.8329	−2.78564
$87$	−24.9948	−2.67972
$88$	3.65391	0.389508
$89$	7.57667	0.803126	0.401563	−	0.915831i	$-0.368467\pi$
$89$	7.57667	0.803126	0.401563	+	0.915831i	$0.368467\pi$
$90$	−17.5196	−1.84673
$91$	−0.0638029	−0.00668837
$92$	−27.4494	−2.86180
$93$	3.25969	0.338014
$94$	2.63674	0.271959
$95$	−6.12989	−0.628914
$96$	3.98020	0.406228
$97$	5.99226	0.608422	0.304211	−	0.952605i	$-0.401607\pi$
$97$	5.99226	0.608422	0.304211	+	0.952605i	$0.401607\pi$
$98$	16.8997	1.70712
$99$	−3.13681	−0.315261
$100$	−5.25043	−0.525043
$101$	−6.14079	−0.611032	−0.305516	−	0.952187i	$-0.598829\pi$
$101$	−6.14079	−0.611032	−0.305516	+	0.952187i	$0.598829\pi$
$102$	−10.7252	−1.06195
$103$	−0.100735	−0.00992568	−0.00496284	−	0.999988i	$-0.501580\pi$
$103$	−0.100735	−0.00992568	−0.00496284	+	0.999988i	$0.501580\pi$
$104$	−4.43349	−0.434740
$105$	−0.317179	−0.0309535
$106$	8.58655	0.833999
$107$	1.09078	0.105450	0.0527249	−	0.998609i	$-0.483209\pi$
$107$	1.09078	0.105450	0.0527249	+	0.998609i	$0.483209\pi$
$108$	8.06532	0.776086
$109$	8.76954	0.839970	0.419985	−	0.907531i	$-0.362035\pi$
$109$	8.76954	0.839970	0.419985	+	0.907531i	$0.362035\pi$
$110$	3.79367	0.361712
$111$	5.60288	0.531802
$112$	−0.193902	−0.0183220
$113$	3.23054	0.303904	0.151952	−	0.988388i	$-0.451444\pi$
$113$	3.23054	0.303904	0.151952	+	0.988388i	$0.451444\pi$
$114$	20.2730	1.89874
$115$	−13.6378	−1.27174
$116$	−36.7455	−3.41173
$117$	3.80607	0.351871
$118$	28.7254	2.64439
$119$	−0.108583	−0.00995382
$120$	−22.0399	−2.01196
$121$	−10.3208	−0.938251
$122$	30.4467	2.75652
$123$	−11.1258	−1.00318
$124$	4.79215	0.430348
$125$	−12.1362	−1.08550
$126$	0.586610	0.0522594
$127$	−8.72529	−0.774244	−0.387122	−	0.922028i	$-0.626531\pi$
$127$	−8.72529	−0.774244	−0.387122	+	0.922028i	$0.626531\pi$
$128$	20.5341	1.81497
$129$	27.8990	2.45637
$130$	−4.60307	−0.403716
$131$	−9.11276	−0.796186	−0.398093	−	0.917345i	$-0.630328\pi$
$131$	−9.11276	−0.796186	−0.398093	+	0.917345i	$0.630328\pi$
$132$	−8.24635	−0.717753
$133$	0.205248	0.0177972
$134$	31.5967	2.72954
$135$	4.00714	0.344880
$136$	−7.54517	−0.646993
$137$	10.5666	0.902763	0.451382	−	0.892331i	$-0.350931\pi$
$137$	10.5666	0.902763	0.451382	+	0.892331i	$0.350931\pi$
$138$	45.1036	3.83947
$139$	16.6761	1.41445	0.707225	−	0.706989i	$-0.249947\pi$
$139$	16.6761	1.41445	0.707225	+	0.706989i	$0.249947\pi$
$140$	−0.466292	−0.0394089
$141$	−2.84762	−0.239813
$142$	−20.0792	−1.68501
$143$	−0.824160	−0.0689197
$144$	11.5669	0.963912
$145$	−18.2565	−1.51612
$146$	14.2830	1.18207
$147$	−18.2513	−1.50534
$148$	8.23694	0.677072
$149$	7.20135	0.589958	0.294979	−	0.955504i	$-0.404687\pi$
$149$	7.20135	0.589958	0.294979	+	0.955504i	$0.404687\pi$
$150$	8.62726	0.704413
$151$	−12.2610	−0.997784	−0.498892	−	0.866664i	$-0.666260\pi$
$151$	−12.2610	−0.997784	−0.498892	+	0.866664i	$0.666260\pi$
$152$	14.2621	1.15681
$153$	6.47738	0.523665
$154$	−0.127024	−0.0102359
$155$	2.38091	0.191239
$156$	10.0058	0.801103
$157$	11.5322	0.920371	0.460186	−	0.887823i	$-0.347783\pi$
$157$	11.5322	0.920371	0.460186	+	0.887823i	$0.347783\pi$
$158$	−19.2438	−1.53096
$159$	−9.27330	−0.735420
$160$	2.90718	0.229833
$161$	0.456637	0.0359880
$162$	14.3297	1.12585
$163$	13.2058	1.03436	0.517180	−	0.855877i	$-0.326982\pi$
$163$	13.2058	1.03436	0.517180	+	0.855877i	$0.326982\pi$
$164$	−16.3563	−1.27721
$165$	−4.09708	−0.318958
$166$	13.2048	1.02489
$167$	2.55278	0.197540	0.0987700	−	0.995110i	$-0.468509\pi$
$167$	2.55278	0.197540	0.0987700	+	0.995110i	$0.468509\pi$
$168$	0.737964	0.0569351
$169$	1.00000	0.0769231
$170$	−7.83377	−0.600822
$171$	−12.2437	−0.936302
$172$	41.0151	3.12737
$173$	−4.61463	−0.350844	−0.175422	−	0.984493i	$-0.556129\pi$
$173$	−4.61463	−0.350844	−0.175422	+	0.984493i	$0.556129\pi$
$174$	60.3785	4.57728
$175$	0.0873440	0.00660259
$176$	−2.50469	−0.188798
$177$	−31.0228	−2.33182
$178$	−18.3025	−1.37183
$179$	−6.35735	−0.475170	−0.237585	−	0.971367i	$-0.576356\pi$
$179$	−6.35735	−0.475170	−0.237585	+	0.971367i	$0.576356\pi$
$180$	27.8160	2.07328
$181$	−21.4995	−1.59804	−0.799021	−	0.601303i	$-0.794649\pi$
$181$	−21.4995	−1.59804	−0.799021	+	0.601303i	$0.794649\pi$
$182$	0.154125	0.0114245
$183$	−32.8818	−2.43069
$184$	31.7305	2.33920
$185$	4.09240	0.300879
$186$	−7.87424	−0.577367
$187$	−1.40260	−0.102568
$188$	−4.18636	−0.305322
$189$	−0.134171	−0.00975954
$190$	14.8076	1.07426
$191$	3.29487	0.238408	0.119204	−	0.992870i	$-0.461966\pi$
$191$	3.29487	0.238408	0.119204	+	0.992870i	$0.461966\pi$
$192$	−25.4717	−1.83826
$193$	8.75016	0.629851	0.314925	−	0.949116i	$-0.398021\pi$
$193$	8.75016	0.629851	0.314925	+	0.949116i	$0.398021\pi$
$194$	−14.4752	−1.03926
$195$	4.97122	0.355997
$196$	−26.8317	−1.91655
$197$	12.0653	0.859620	0.429810	−	0.902919i	$-0.358581\pi$
$197$	12.0653	0.859620	0.429810	+	0.902919i	$0.358581\pi$
$198$	7.57741	0.538503
$199$	−3.69665	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$199$	−3.69665	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$200$	6.06930	0.429164
$201$	−34.1238	−2.40690
$202$	14.8340	1.04371
$203$	0.611283	0.0429036
$204$	17.0284	1.19222
$205$	−8.12641	−0.567573
$206$	0.243339	0.0169542
$207$	−27.2400	−1.89331
$208$	3.03908	0.210722
$209$	2.65124	0.183390
$210$	0.766190	0.0528721
$211$	−0.242385	−0.0166865	−0.00834324	−	0.999965i	$-0.502656\pi$
$211$	−0.242385	−0.0166865	−0.00834324	+	0.999965i	$0.502656\pi$
$212$	−13.6329	−0.936312
$213$	21.6852	1.48584
$214$	−2.63494	−0.180121
$215$	20.3778	1.38975
$216$	−9.32321	−0.634364
$217$	−0.0797202	−0.00541176
$218$	−21.1841	−1.43477
$219$	−15.4254	−1.04235
$220$	−6.02323	−0.406086
$221$	1.70186	0.114479
$222$	−13.5346	−0.908380
$223$	−27.7076	−1.85544	−0.927718	−	0.373283i	$-0.878232\pi$
$223$	−27.7076	−1.85544	−0.927718	+	0.373283i	$0.878232\pi$
$224$	−0.0973414	−0.00650390
$225$	−5.21037	−0.347358
$226$	−7.80384	−0.519104
$227$	−14.4105	−0.956456	−0.478228	−	0.878236i	$-0.658721\pi$
$227$	−14.4105	−0.956456	−0.478228	+	0.878236i	$0.658721\pi$
$228$	−32.1876	−2.13167
$229$	0.0125556	0.000829698 0	0.000414849	−	1.00000i	$-0.499868\pi$
$229$	0.0125556	0.000829698 0	0.000414849		1.00000i	$0.499868\pi$
$230$	32.9441	2.17227
$231$	0.137183	0.00902598
$232$	42.4764	2.78871
$233$	18.0524	1.18265	0.591327	−	0.806432i	$-0.298604\pi$
$233$	18.0524	1.18265	0.591327	+	0.806432i	$0.298604\pi$
$234$	−9.19410	−0.601037
$235$	−2.07993	−0.135680
$236$	−45.6074	−2.96879
$237$	20.7829	1.35000
$238$	0.262299	0.0170023
$239$	−23.6749	−1.53140	−0.765701	−	0.643197i	$-0.777608\pi$
$239$	−23.6749	−1.53140	−0.765701	+	0.643197i	$0.777608\pi$
$240$	15.1079	0.975214
$241$	22.9441	1.47796	0.738978	−	0.673729i	$-0.235309\pi$
$241$	22.9441	1.47796	0.738978	+	0.673729i	$0.235309\pi$
$242$	24.9313	1.60264
$243$	−21.7845	−1.39748
$244$	−48.3404	−3.09468
$245$	−13.3309	−0.851682
$246$	26.8760	1.71355
$247$	−3.21690	−0.204686
$248$	−5.53954	−0.351761
$249$	−14.2609	−0.903746
$250$	29.3168	1.85416
$251$	−26.7989	−1.69153	−0.845764	−	0.533557i	$-0.820855\pi$
$251$	−26.7989	−1.69153	−0.845764	+	0.533557i	$0.820855\pi$
$252$	−0.931364	−0.0586704
$253$	5.89850	0.370836
$254$	21.0772	1.32250
$255$	8.46031	0.529805
$256$	−30.0757	−1.87973
$257$	−5.06781	−0.316122	−0.158061	−	0.987429i	$-0.550524\pi$
$257$	−5.06781	−0.316122	−0.158061	+	0.987429i	$0.550524\pi$
$258$	−67.3941	−4.19577
$259$	−0.137026	−0.00851440
$260$	7.30832	0.453243
$261$	−36.4651	−2.25714
$262$	22.0132	1.35998
$263$	−6.68257	−0.412065	−0.206033	−	0.978545i	$-0.566055\pi$
$263$	−6.68257	−0.412065	−0.206033	+	0.978545i	$0.566055\pi$
$264$	9.53248	0.586683
$265$	−6.77331	−0.416081
$266$	−0.495805	−0.0303998
$267$	19.7664	1.20968
$268$	−50.1662	−3.06439
$269$	−31.7006	−1.93282	−0.966410	−	0.257005i	$-0.917264\pi$
$269$	−31.7006	−1.93282	−0.966410	+	0.257005i	$0.917264\pi$
$270$	−9.67982	−0.589095
$271$	29.6669	1.80213	0.901067	−	0.433680i	$-0.142785\pi$
$271$	29.6669	1.80213	0.901067	+	0.433680i	$0.142785\pi$
$272$	5.17208	0.313603
$273$	−0.166452	−0.0100741
$274$	−25.5251	−1.54203
$275$	1.12825	0.0680358
$276$	−71.6112	−4.31049
$277$	22.7214	1.36520	0.682599	−	0.730793i	$-0.260850\pi$
$277$	22.7214	1.36520	0.682599	+	0.730793i	$0.260850\pi$
$278$	−40.2835	−2.41605
$279$	4.75559	0.284710
$280$	0.539016	0.0322124
$281$	−6.85002	−0.408638	−0.204319	−	0.978904i	$-0.565498\pi$
$281$	−6.85002	−0.408638	−0.204319	+	0.978904i	$0.565498\pi$
$282$	6.87884	0.409629
$283$	−9.68222	−0.575548	−0.287774	−	0.957698i	$-0.592915\pi$
$283$	−9.68222	−0.575548	−0.287774	+	0.957698i	$0.592915\pi$
$284$	31.8799	1.89173
$285$	−15.9919	−0.947280
$286$	1.99088	0.117723
$287$	0.272097	0.0160614
$288$	5.80676	0.342166
$289$	−14.1037	−0.829629
$290$	44.1011	2.58970
$291$	15.6329	0.916415
$292$	−22.6773	−1.32709
$293$	22.0201	1.28643	0.643214	−	0.765687i	$-0.277601\pi$
$293$	22.0201	1.28643	0.643214	+	0.765687i	$0.277601\pi$
$294$	44.0886	2.57130
$295$	−22.6594	−1.31928
$296$	−9.52159	−0.553431
$297$	−1.73313	−0.100566
$298$	−17.3959	−1.00772
$299$	−7.15699	−0.413899
$300$	−13.6976	−0.790828
$301$	−0.682310	−0.0393277
$302$	29.6181	1.70433
$303$	−16.0204	−0.920346
$304$	−9.77642	−0.560716
$305$	−24.0172	−1.37522
$306$	−15.6470	−0.894481
$307$	23.2828	1.32882	0.664410	−	0.747368i	$-0.268683\pi$
$307$	23.2828	1.32882	0.664410	+	0.747368i	$0.268683\pi$
$308$	0.201676	0.0114916
$309$	−0.262801	−0.0149502
$310$	−5.75143	−0.326659
$311$	30.1986	1.71240	0.856202	−	0.516642i	$-0.172818\pi$
$311$	30.1986	1.71240	0.856202	+	0.516642i	$0.172818\pi$
$312$	−11.5663	−0.654812
$313$	−4.73934	−0.267883	−0.133942	−	0.990989i	$-0.542763\pi$
$313$	−4.73934	−0.267883	−0.133942	+	0.990989i	$0.542763\pi$
$314$	−27.8577	−1.57210
$315$	−0.462735	−0.0260722
$316$	30.5535	1.71877
$317$	27.4554	1.54205	0.771023	−	0.636807i	$-0.219745\pi$
$317$	27.4554	1.54205	0.771023	+	0.636807i	$0.219745\pi$
$318$	22.4010	1.25618
$319$	7.89611	0.442097
$320$	−18.6048	−1.04004
$321$	2.84568	0.158830
$322$	−1.10307	−0.0614718
$323$	−5.47470	−0.304620
$324$	−22.7514	−1.26396
$325$	−1.36897	−0.0759365
$326$	−31.9005	−1.76681
$327$	22.8784	1.26518
$328$	18.9073	1.04398
$329$	0.0696426	0.00383952
$330$	9.89709	0.544817
$331$	15.5537	0.854910	0.427455	−	0.904037i	$-0.359410\pi$
$331$	15.5537	0.854910	0.427455	+	0.904037i	$0.359410\pi$
$332$	−20.9653	−1.15062
$333$	8.17410	0.447938
$334$	−6.16660	−0.337421
$335$	−24.9244	−1.36176
$336$	−0.505861	−0.0275970
$337$	−31.9426	−1.74002	−0.870011	−	0.493032i	$-0.835889\pi$
$337$	−31.9426	−1.74002	−0.870011	+	0.493032i	$0.835889\pi$
$338$	−2.41564	−0.131394
$339$	8.42799	0.457745
$340$	12.4377	0.674529
$341$	−1.02977	−0.0557651
$342$	29.5765	1.59931
$343$	0.892981	0.0482165
$344$	−47.4119	−2.55628
$345$	−35.5790	−1.91551
$346$	11.1473	0.599282
$347$	3.66600	0.196801	0.0984007	−	0.995147i	$-0.468627\pi$
$347$	3.66600	0.196801	0.0984007	+	0.995147i	$0.468627\pi$
$348$	−95.8632	−5.13881
$349$	−4.73828	−0.253634	−0.126817	−	0.991926i	$-0.540476\pi$
$349$	−4.73828	−0.253634	−0.126817	+	0.991926i	$0.540476\pi$
$350$	−0.210992	−0.0112780
$351$	2.10290	0.112245
$352$	−1.25739	−0.0670189
$353$	10.9188	0.581148	0.290574	−	0.956853i	$-0.406154\pi$
$353$	10.9188	0.581148	0.290574	+	0.956853i	$0.406154\pi$
$354$	74.9400	3.98302
$355$	15.8391	0.840651
$356$	29.0590	1.54012
$357$	−0.283277	−0.0149926
$358$	15.3571	0.811646
$359$	−2.64298	−0.139491	−0.0697457	−	0.997565i	$-0.522219\pi$
$359$	−2.64298	−0.139491	−0.0697457	+	0.997565i	$0.522219\pi$
$360$	−32.1542	−1.69468
$361$	−8.65155	−0.455345
$362$	51.9350	2.72964
$363$	−26.9253	−1.41321
$364$	−0.244705	−0.0128260
$365$	−11.2669	−0.589735
$366$	79.4307	4.15191
$367$	18.9646	0.989945	0.494973	−	0.868909i	$-0.335178\pi$
$367$	18.9646	0.989945	0.494973	+	0.868909i	$0.335178\pi$
$368$	−21.7507	−1.13383
$369$	−16.2315	−0.844980
$370$	−9.88578	−0.513937
$371$	0.226791	0.0117744
$372$	12.5020	0.648197
$373$	11.4200	0.591306	0.295653	−	0.955295i	$-0.404463\pi$
$373$	11.4200	0.591306	0.295653	+	0.955295i	$0.404463\pi$
$374$	3.38818	0.175199
$375$	−31.6616	−1.63500
$376$	4.83928	0.249567
$377$	−9.58079	−0.493436
$378$	0.324110	0.0166704
$379$	25.9715	1.33407	0.667034	−	0.745027i	$-0.267564\pi$
$379$	25.9715	1.33407	0.667034	+	0.745027i	$0.267564\pi$
$380$	−23.5101	−1.20604
$381$	−22.7629	−1.16618
$382$	−7.95922	−0.407229
$383$	−18.3084	−0.935517	−0.467759	−	0.883856i	$-0.654938\pi$
$383$	−18.3084	−0.935517	−0.467759	+	0.883856i	$0.654938\pi$
$384$	53.5701	2.73374
$385$	0.100200	0.00510666
$386$	−21.1373	−1.07586
$387$	40.7022	2.06901
$388$	22.9823	1.16675
$389$	−21.2237	−1.07609	−0.538043	−	0.842917i	$-0.680836\pi$
$389$	−21.2237	−1.07609	−0.538043	+	0.842917i	$0.680836\pi$
$390$	−12.0087	−0.608084
$391$	−12.1802	−0.615977
$392$	31.0164	1.56657
$393$	−23.7738	−1.19923
$394$	−29.1456	−1.46833
$395$	15.1801	0.763792
$396$	−12.0307	−0.604565
$397$	1.87919	0.0943137	0.0471568	−	0.998887i	$-0.484984\pi$
$397$	1.87919	0.0943137	0.0471568	+	0.998887i	$0.484984\pi$
$398$	8.92978	0.447610
$399$	0.535459	0.0268065
$400$	−4.16039	−0.208020
$401$	7.36040	0.367561	0.183780	−	0.982967i	$-0.441166\pi$
$401$	7.36040	0.367561	0.183780	+	0.982967i	$0.441166\pi$
$402$	82.4308	4.11127
$403$	1.24948	0.0622408
$404$	−23.5519	−1.17175
$405$	−11.3037	−0.561685
$406$	−1.47664	−0.0732844
$407$	−1.77001	−0.0877359
$408$	−19.6842	−0.974511
$409$	18.7092	0.925112	0.462556	−	0.886590i	$-0.346933\pi$
$409$	18.7092	0.925112	0.462556	+	0.886590i	$0.346933\pi$
$410$	19.6305	0.969481
$411$	27.5665	1.35976
$412$	−0.386350	−0.0190341
$413$	0.758706	0.0373335
$414$	65.8021	3.23399
$415$	−10.4163	−0.511315
$416$	1.52566	0.0748015
$417$	43.5054	2.13047
$418$	−6.40445	−0.313252
$419$	18.3118	0.894590	0.447295	−	0.894386i	$-0.352387\pi$
$419$	18.3118	0.894590	0.447295	+	0.894386i	$0.352387\pi$
$420$	−1.21648	−0.0593583
$421$	24.2019	1.17953	0.589764	−	0.807576i	$-0.299221\pi$
$421$	24.2019	1.17953	0.589764	+	0.807576i	$0.299221\pi$
$422$	0.585516	0.0285025
$423$	−4.15442	−0.201995
$424$	15.7591	0.765331
$425$	−2.32978	−0.113011
$426$	−52.3836	−2.53799
$427$	0.804171	0.0389166
$428$	4.18351	0.202217
$429$	−2.15010	−0.103808
$430$	−49.2254	−2.37386
$431$	13.2485	0.638157	0.319079	−	0.947728i	$-0.396627\pi$
$431$	13.2485	0.638157	0.319079	+	0.947728i	$0.396627\pi$
$432$	6.39089	0.307482
$433$	−18.3409	−0.881405	−0.440703	−	0.897653i	$-0.645271\pi$
$433$	−18.3409	−0.881405	−0.440703	+	0.897653i	$0.645271\pi$
$434$	0.192576	0.00924392
$435$	−47.6283	−2.28360
$436$	33.6341	1.61078
$437$	23.0233	1.10135
$438$	37.2622	1.78046
$439$	35.7212	1.70488	0.852440	−	0.522825i	$-0.175122\pi$
$439$	35.7212	1.70488	0.852440	+	0.522825i	$0.175122\pi$
$440$	6.96262	0.331930
$441$	−26.6270	−1.26795
$442$	−4.11107	−0.195544
$443$	16.8973	0.802816	0.401408	−	0.915899i	$-0.368521\pi$
$443$	16.8973	0.802816	0.401408	+	0.915899i	$0.368521\pi$
$444$	21.4889	1.01982
$445$	14.4376	0.684406
$446$	66.9315	3.16930
$447$	18.7872	0.888604
$448$	0.622947	0.0294315
$449$	−21.7834	−1.02802	−0.514012	−	0.857783i	$-0.671841\pi$
$449$	−21.7834	−1.02802	−0.514012	+	0.857783i	$0.671841\pi$
$450$	12.5864	0.593329
$451$	3.51475	0.165503
$452$	12.3902	0.582786
$453$	−31.9870	−1.50288
$454$	34.8105	1.63374
$455$	−0.121578	−0.00569968
$456$	37.2076	1.74241
$457$	7.03014	0.328856	0.164428	−	0.986389i	$-0.447422\pi$
$457$	7.03014	0.328856	0.164428	+	0.986389i	$0.447422\pi$
$458$	−0.0303299	−0.00141722
$459$	3.57884	0.167046
$460$	−52.3056	−2.43876
$461$	−2.44003	−0.113644	−0.0568219	−	0.998384i	$-0.518097\pi$
$461$	−2.44003	−0.113644	−0.0568219	+	0.998384i	$0.518097\pi$
$462$	−0.331385	−0.0154174
$463$	−12.6006	−0.585601	−0.292801	−	0.956173i	$-0.594587\pi$
$463$	−12.6006	−0.585601	−0.292801	+	0.956173i	$0.594587\pi$
$464$	−29.1168	−1.35171
$465$	6.21143	0.288048
$466$	−43.6082	−2.02011
$467$	−33.4221	−1.54659	−0.773295	−	0.634046i	$-0.781393\pi$
$467$	−33.4221	−1.54659	−0.773295	+	0.634046i	$0.781393\pi$
$468$	14.5975	0.674770
$469$	0.834544	0.0385357
$470$	5.02438	0.231757
$471$	30.0858	1.38628
$472$	52.7205	2.42666
$473$	−8.81358	−0.405249
$474$	−50.2041	−2.30595
$475$	4.40382	0.202061
$476$	−0.416453	−0.0190881
$477$	−13.5289	−0.619446
$478$	57.1901	2.61581
$479$	8.98724	0.410637	0.205319	−	0.978695i	$-0.434177\pi$
$479$	8.98724	0.410637	0.205319	+	0.978695i	$0.434177\pi$
$480$	7.58439	0.346178
$481$	2.14765	0.0979244
$482$	−55.4246	−2.52452
$483$	1.19129	0.0542058
$484$	−39.5835	−1.79925
$485$	11.4184	0.518484
$486$	52.6236	2.38705
$487$	15.7344	0.712995	0.356498	−	0.934296i	$-0.383971\pi$
$487$	15.7344	0.712995	0.356498	+	0.934296i	$0.383971\pi$
$488$	55.8797	2.52955
$489$	34.4519	1.55797
$490$	32.2028	1.45477
$491$	−8.82120	−0.398095	−0.199047	−	0.979990i	$-0.563785\pi$
$491$	−8.82120	−0.398095	−0.199047	+	0.979990i	$0.563785\pi$
$492$	−42.6711	−1.92376
$493$	−16.3051	−0.734346
$494$	7.77088	0.349628
$495$	−5.97727	−0.268658
$496$	3.79726	0.170502
$497$	−0.530341	−0.0237891
$498$	34.4491	1.54370
$499$	−2.48821	−0.111387	−0.0556937	−	0.998448i	$-0.517737\pi$
$499$	−2.48821	−0.111387	−0.0556937	+	0.998448i	$0.517737\pi$
$500$	−46.5464	−2.08162
$501$	6.65980	0.297538
$502$	64.7364	2.88933
$503$	−26.3041	−1.17284	−0.586421	−	0.810007i	$-0.699464\pi$
$503$	−26.3041	−1.17284	−0.586421	+	0.810007i	$0.699464\pi$
$504$	1.07662	0.0479566
$505$	−11.7014	−0.520708
$506$	−14.2487	−0.633431
$507$	2.60884	0.115863
$508$	−33.4643	−1.48474
$509$	−32.1102	−1.42326	−0.711630	−	0.702554i	$-0.752043\pi$
$509$	−32.1102	−1.42326	−0.711630	+	0.702554i	$0.752043\pi$
$510$	−20.4371	−0.904969
$511$	0.377250	0.0166885
$512$	31.5841	1.39583
$513$	−6.76483	−0.298675
$514$	12.2420	0.539973
$515$	−0.191953	−0.00845845
$516$	107.002	4.71050
$517$	0.899593	0.0395640
$518$	0.331007	0.0145436
$519$	−12.0388	−0.528446
$520$	−8.44814	−0.370476
$521$	−21.9410	−0.961252	−0.480626	−	0.876926i	$-0.659590\pi$
$521$	−21.9410	−0.961252	−0.480626	+	0.876926i	$0.659590\pi$
$522$	88.0867	3.85545
$523$	23.8296	1.04200	0.520999	−	0.853557i	$-0.325560\pi$
$523$	23.8296	1.04200	0.520999	+	0.853557i	$0.325560\pi$
$524$	−34.9504	−1.52682
$525$	0.227867	0.00994493
$526$	16.1427	0.703855
$527$	2.12643	0.0926287
$528$	−6.53434	−0.284371
$529$	28.2225	1.22707
$530$	16.3619	0.710716
$531$	−45.2595	−1.96409
$532$	0.787192	0.0341291
$533$	−4.26465	−0.184722
$534$	−47.7484	−2.06628
$535$	2.07852	0.0898621
$536$	57.9902	2.50480
$537$	−16.5853	−0.715709
$538$	76.5773	3.30148
$539$	5.76576	0.248349
$540$	15.3687	0.661363
$541$	22.7653	0.978757	0.489378	−	0.872072i	$-0.337224\pi$
$541$	22.7653	0.978757	0.489378	+	0.872072i	$0.337224\pi$
$542$	−71.6645	−3.07826
$543$	−56.0888	−2.40700
$544$	2.59645	0.111322
$545$	16.7106	0.715804
$546$	0.402088	0.0172078
$547$	43.5915	1.86384	0.931919	−	0.362667i	$-0.118134\pi$
$547$	43.5915	1.86384	0.931919	+	0.362667i	$0.118134\pi$
$548$	40.5263	1.73120
$549$	−47.9716	−2.04738
$550$	−2.72544	−0.116213
$551$	30.8205	1.31300
$552$	82.7798	3.52334
$553$	−0.508276	−0.0216141
$554$	−54.8868	−2.33192
$555$	10.6764	0.453190
$556$	63.9584	2.71244
$557$	−12.4726	−0.528479	−0.264240	−	0.964457i	$-0.585121\pi$
$557$	−12.4726	−0.528479	−0.264240	+	0.964457i	$0.585121\pi$
$558$	−11.4878	−0.486318
$559$	10.6940	0.452309
$560$	−0.369486	−0.0156136
$561$	−3.65917	−0.154490
$562$	16.5472	0.698001
$563$	−34.5936	−1.45795	−0.728973	−	0.684543i	$-0.760002\pi$
$563$	−34.5936	−1.45795	−0.728973	+	0.684543i	$0.760002\pi$
$564$	−10.9216	−0.459881
$565$	6.15589	0.258980
$566$	23.3888	0.983104
$567$	0.378482	0.0158948
$568$	−36.8520	−1.54627
$569$	12.0678	0.505908	0.252954	−	0.967478i	$-0.418598\pi$
$569$	12.0678	0.505908	0.252954	+	0.967478i	$0.418598\pi$
$570$	38.6308	1.61807
$571$	−35.3249	−1.47830	−0.739151	−	0.673540i	$-0.764773\pi$
$571$	−35.3249	−1.47830	−0.739151	+	0.673540i	$0.764773\pi$
$572$	−3.16092	−0.132165
$573$	8.59580	0.359095
$574$	−0.657289	−0.0274347
$575$	9.79767	0.408591
$576$	−37.1609	−1.54837
$577$	−12.0527	−0.501759	−0.250879	−	0.968018i	$-0.580720\pi$
$577$	−12.0527	−0.501759	−0.250879	+	0.968018i	$0.580720\pi$
$578$	34.0695	1.41710
$579$	22.8278	0.948692
$580$	−70.0195	−2.90740
$581$	0.348769	0.0144694
$582$	−37.7634	−1.56534
$583$	2.92953	0.121329
$584$	26.2141	1.08475
$585$	7.25257	0.299857
$586$	−53.1927	−2.19737
$587$	−24.2889	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$587$	−24.2889	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$588$	−69.9997	−2.88674
$589$	−4.01944	−0.165618
$590$	54.7370	2.25349
$591$	31.4766	1.29477
$592$	6.52688	0.268253
$593$	−23.8610	−0.979853	−0.489927	−	0.871764i	$-0.662976\pi$
$593$	−23.8610	−0.979853	−0.489927	+	0.871764i	$0.662976\pi$
$594$	4.18662	0.171779
$595$	−0.206909	−0.00848243
$596$	27.6195	1.13134
$597$	−9.64398	−0.394702
$598$	17.2887	0.706989
$599$	43.7740	1.78856	0.894278	−	0.447511i	$-0.147690\pi$
$599$	43.7740	1.78856	0.894278	+	0.447511i	$0.147690\pi$
$600$	15.8339	0.646414
$601$	14.2337	0.580603	0.290302	−	0.956935i	$-0.406244\pi$
$601$	14.2337	0.580603	0.290302	+	0.956935i	$0.406244\pi$
$602$	1.64822	0.0671763
$603$	−49.7835	−2.02734
$604$	−47.0249	−1.91341
$605$	−19.6665	−0.799557
$606$	38.6995	1.57206
$607$	−7.01723	−0.284821	−0.142410	−	0.989808i	$-0.545485\pi$
$607$	−7.01723	−0.284821	−0.142410	+	0.989808i	$0.545485\pi$
$608$	−4.90789	−0.199041
$609$	1.59474	0.0646222
$610$	58.0170	2.34904
$611$	−1.09153	−0.0441584
$612$	24.8429	1.00421
$613$	38.3264	1.54799	0.773995	−	0.633192i	$-0.218256\pi$
$613$	38.3264	1.54799	0.773995	+	0.633192i	$0.218256\pi$
$614$	−56.2429	−2.26978
$615$	−21.2005	−0.854888
$616$	−0.233130	−0.00939308
$617$	−7.24688	−0.291748	−0.145874	−	0.989303i	$-0.546599\pi$
$617$	−7.24688	−0.291748	−0.145874	+	0.989303i	$0.546599\pi$
$618$	0.634833	0.0255367
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	9.13157	0.366733
$621$	−15.0505	−0.603954
$622$	−72.9489	−2.92499
$623$	−0.483414	−0.0193676
$624$	7.92849	0.317393
$625$	−16.2811	−0.651244
$626$	11.4485	0.457576
$627$	6.91667	0.276225
$628$	44.2298	1.76496
$629$	3.65499	0.145734
$630$	1.11780	0.0445343
$631$	12.8226	0.510459	0.255229	−	0.966881i	$-0.417849\pi$
$631$	12.8226	0.510459	0.255229	+	0.966881i	$0.417849\pi$
$632$	−35.3187	−1.40490
$633$	−0.632345	−0.0251335
$634$	−66.3223	−2.63400
$635$	−16.6263	−0.659794
$636$	−35.5661	−1.41029
$637$	−6.99593	−0.277189
$638$	−19.0742	−0.755153
$639$	31.6367	1.25153
$640$	39.1282	1.54668
$641$	33.5471	1.32503	0.662515	−	0.749049i	$-0.269489\pi$
$641$	33.5471	1.32503	0.662515	+	0.749049i	$0.269489\pi$
$642$	−6.87415	−0.271301
$643$	26.8806	1.06007	0.530034	−	0.847977i	$-0.322179\pi$
$643$	26.8806	1.06007	0.530034	+	0.847977i	$0.322179\pi$
$644$	1.75135	0.0690129
$645$	53.1624	2.09327
$646$	13.2249	0.520327
$647$	−21.3863	−0.840784	−0.420392	−	0.907343i	$-0.638107\pi$
$647$	−21.3863	−0.840784	−0.420392	+	0.907343i	$0.638107\pi$
$648$	26.2997	1.03315
$649$	9.80042	0.384700
$650$	3.30693	0.129708
$651$	−0.207978	−0.00815129
$652$	50.6486	1.98355
$653$	13.6991	0.536087	0.268044	−	0.963407i	$-0.413623\pi$
$653$	13.6991	0.536087	0.268044	+	0.963407i	$0.413623\pi$
$654$	−55.2660	−2.16107
$655$	−17.3646	−0.678492
$656$	−12.9606	−0.506027
$657$	−22.5042	−0.877974
$658$	−0.168232	−0.00655835
$659$	31.4603	1.22552	0.612760	−	0.790269i	$-0.290059\pi$
$659$	31.4603	1.22552	0.612760	+	0.790269i	$0.290059\pi$
$660$	−15.7137	−0.611653
$661$	4.31046	0.167658	0.0838288	−	0.996480i	$-0.473285\pi$
$661$	4.31046	0.167658	0.0838288	+	0.996480i	$0.473285\pi$
$662$	−37.5722	−1.46029
$663$	4.43988	0.172431
$664$	24.2350	0.940502
$665$	0.391105	0.0151664
$666$	−19.7457	−0.765130
$667$	68.5696	2.65503
$668$	9.79074	0.378815
$669$	−72.2847	−2.79469
$670$	60.2083	2.32605
$671$	10.3877	0.401013
$672$	−0.253949	−0.00979628
$673$	0.398047	0.0153436	0.00767180	−	0.999971i	$-0.497558\pi$
$673$	0.398047	0.0153436	0.00767180	+	0.999971i	$0.497558\pi$
$674$	77.1618	2.97216
$675$	−2.87880	−0.110805
$676$	3.83533	0.147513
$677$	−33.2389	−1.27747	−0.638737	−	0.769425i	$-0.720543\pi$
$677$	−33.2389	−1.27747	−0.638737	+	0.769425i	$0.720543\pi$
$678$	−20.3590	−0.781883
$679$	−0.382324	−0.0146722
$680$	−14.3775	−0.551353
$681$	−37.5947	−1.44063
$682$	2.48755	0.0952533
$683$	−12.5541	−0.480368	−0.240184	−	0.970727i	$-0.577208\pi$
$683$	−12.5541	−0.480368	−0.240184	+	0.970727i	$0.577208\pi$
$684$	−46.9587	−1.79551
$685$	20.1349	0.769315
$686$	−2.15712	−0.0823594
$687$	0.0327556	0.00124971
$688$	32.5000	1.23905
$689$	−3.55456	−0.135418
$690$	85.9461	3.27191
$691$	−32.7263	−1.24497	−0.622483	−	0.782633i	$-0.713876\pi$
$691$	−32.7263	−1.24497	−0.622483	+	0.782633i	$0.713876\pi$
$692$	−17.6986	−0.672800
$693$	0.200138	0.00760260
$694$	−8.85575	−0.336160
$695$	31.7768	1.20536
$696$	110.814	4.20040
$697$	−7.25782	−0.274909
$698$	11.4460	0.433237
$699$	47.0960	1.78133
$700$	0.334993	0.0126615
$701$	50.0423	1.89007	0.945035	−	0.326968i	$-0.106027\pi$
$701$	50.0423	1.89007	0.945035	+	0.326968i	$0.106027\pi$
$702$	−5.07986	−0.191727
$703$	−6.90877	−0.260569
$704$	8.04677	0.303274
$705$	−5.42622	−0.204363
$706$	−26.3759	−0.992669
$707$	0.391801	0.0147352
$708$	−118.983	−4.47164
$709$	−37.0940	−1.39309	−0.696547	−	0.717511i	$-0.745281\pi$
$709$	−37.0940	−1.39309	−0.696547	+	0.717511i	$0.745281\pi$
$710$	−38.2616	−1.43593
$711$	30.3204	1.13710
$712$	−33.5911	−1.25888
$713$	−8.94249	−0.334899
$714$	0.684296	0.0256091
$715$	−1.57046	−0.0587318
$716$	−24.3825	−0.911217
$717$	−61.7641	−2.30662
$718$	6.38450	0.238268
$719$	−35.2256	−1.31369	−0.656847	−	0.754023i	$-0.728110\pi$
$719$	−35.2256	−1.31369	−0.656847	+	0.754023i	$0.728110\pi$
$720$	22.0411	0.821424
$721$	0.00642717	0.000239360 0
$722$	20.8991	0.777782
$723$	59.8575	2.22612
$724$	−82.4575	−3.06451
$725$	13.1158	0.487108
$726$	65.0418	2.41393
$727$	−3.97574	−0.147452	−0.0737261	−	0.997279i	$-0.523489\pi$
$727$	−3.97574	−0.147452	−0.0737261	+	0.997279i	$0.523489\pi$
$728$	0.282870	0.0104839
$729$	−39.0362	−1.44579
$730$	27.2167	1.00734
$731$	18.1997	0.673140
$732$	−126.113	−4.66125
$733$	34.4535	1.27257	0.636284	−	0.771455i	$-0.280471\pi$
$733$	34.4535	1.27257	0.636284	+	0.771455i	$0.280471\pi$
$734$	−45.8117	−1.69094
$735$	−34.7783	−1.28282
$736$	−10.9191	−0.402484
$737$	10.7800	0.397088
$738$	39.2096	1.44333
$739$	34.4317	1.26659	0.633295	−	0.773911i	$-0.281702\pi$
$739$	34.4317	1.26659	0.633295	+	0.773911i	$0.281702\pi$
$740$	15.6957	0.576986
$741$	−8.39239	−0.308302
$742$	−0.547847	−0.0201121
$743$	3.83161	0.140568	0.0702840	−	0.997527i	$-0.477609\pi$
$743$	3.83161	0.140568	0.0702840	+	0.997527i	$0.477609\pi$
$744$	−14.4518	−0.529829
$745$	13.7224	0.502749
$746$	−27.5867	−1.01002
$747$	−20.8053	−0.761227
$748$	−5.37944	−0.196692
$749$	−0.0695951	−0.00254295
$750$	76.4830	2.79276
$751$	33.3080	1.21542	0.607712	−	0.794157i	$-0.292087\pi$
$751$	33.3080	1.21542	0.607712	+	0.794157i	$0.292087\pi$
$752$	−3.31724	−0.120967
$753$	−69.9140	−2.54781
$754$	23.1438	0.842846
$755$	−23.3636	−0.850290
$756$	−0.514591	−0.0187155
$757$	26.2595	0.954418	0.477209	−	0.878790i	$-0.341649\pi$
$757$	26.2595	0.954418	0.477209	+	0.878790i	$0.341649\pi$
$758$	−62.7379	−2.27874
$759$	15.3883	0.558559
$760$	27.1768	0.985807
$761$	−13.9212	−0.504644	−0.252322	−	0.967643i	$-0.581194\pi$
$761$	−13.9212	−0.504644	−0.252322	+	0.967643i	$0.581194\pi$
$762$	54.9871	1.99197
$763$	−0.559523	−0.0202561
$764$	12.6369	0.457187
$765$	12.3428	0.446256
$766$	44.2266	1.59797
$767$	−11.8914	−0.429374
$768$	−78.4629	−2.83129
$769$	−35.3297	−1.27402	−0.637010	−	0.770855i	$-0.719829\pi$
$769$	−35.3297	−1.27402	−0.637010	+	0.770855i	$0.719829\pi$
$770$	−0.242047	−0.00872277
$771$	−13.2211	−0.476148
$772$	33.5597	1.20784
$773$	−38.5076	−1.38502	−0.692511	−	0.721407i	$-0.743496\pi$
$773$	−38.5076	−1.38502	−0.692511	+	0.721407i	$0.743496\pi$
$774$	−98.3219	−3.53411
$775$	−1.71049	−0.0614426
$776$	−26.5666	−0.953687
$777$	−0.357480	−0.0128245
$778$	51.2689	1.83808
$779$	13.7190	0.491532
$780$	19.0663	0.682682
$781$	−6.85056	−0.245132
$782$	29.4229	1.05216
$783$	−20.1475	−0.720012
$784$	−21.2612	−0.759328
$785$	21.9750	0.784320
$786$	57.4289	2.04842
$787$	−15.0643	−0.536983	−0.268491	−	0.963282i	$-0.586525\pi$
$787$	−15.0643	−0.536983	−0.268491	+	0.963282i	$0.586525\pi$
$788$	46.2745	1.64846
$789$	−17.4338	−0.620659
$790$	−36.6696	−1.30465
$791$	−0.206118	−0.00732872
$792$	13.9070	0.494164
$793$	−12.6040	−0.447580
$794$	−4.53944	−0.161099
$795$	−17.6705	−0.626709
$796$	−14.1779	−0.502521
$797$	−10.7848	−0.382018	−0.191009	−	0.981588i	$-0.561176\pi$
$797$	−10.7848	−0.382018	−0.191009	+	0.981588i	$0.561176\pi$
$798$	−1.29348	−0.0457886
$799$	−1.85762	−0.0657179
$800$	−2.08857	−0.0738422
$801$	28.8373	1.01892
$802$	−17.7801	−0.627837
$803$	4.87304	0.171966
$804$	−130.876	−4.61563
$805$	0.870134	0.0306682
$806$	−3.01829	−0.106315
$807$	−82.7019	−2.91125
$808$	27.2252	0.957778
$809$	47.5574	1.67203	0.836014	−	0.548708i	$-0.184880\pi$
$809$	47.5574	1.67203	0.836014	+	0.548708i	$0.184880\pi$
$810$	27.3057	0.959424
$811$	−7.70241	−0.270468	−0.135234	−	0.990814i	$-0.543179\pi$
$811$	−7.70241	−0.270468	−0.135234	+	0.990814i	$0.543179\pi$
$812$	2.34447	0.0822747
$813$	77.3962	2.71440
$814$	4.27570	0.149863
$815$	25.1640	0.881458
$816$	13.4931	0.472354
$817$	−34.4016	−1.20356
$818$	−45.1948	−1.58020
$819$	−0.242838	−0.00848546
$820$	−31.1674	−1.08841
$821$	30.2128	1.05443	0.527216	−	0.849731i	$-0.323236\pi$
$821$	30.2128	1.05443	0.527216	+	0.849731i	$0.323236\pi$
$822$	−66.5909	−2.32262
$823$	21.9309	0.764463	0.382231	−	0.924067i	$-0.375156\pi$
$823$	21.9309	0.764463	0.382231	+	0.924067i	$0.375156\pi$
$824$	0.446607	0.0155583
$825$	2.94342	0.102477
$826$	−1.83276	−0.0637700
$827$	44.9479	1.56299	0.781496	−	0.623911i	$-0.214457\pi$
$827$	44.9479	1.56299	0.781496	+	0.623911i	$0.214457\pi$
$828$	−104.474	−3.63073
$829$	−14.0334	−0.487402	−0.243701	−	0.969850i	$-0.578361\pi$
$829$	−14.0334	−0.487402	−0.243701	+	0.969850i	$0.578361\pi$
$830$	25.1620	0.873387
$831$	59.2766	2.05628
$832$	−9.76360	−0.338492
$833$	−11.9061	−0.412521
$834$	−105.093	−3.63909
$835$	4.86439	0.168339
$836$	10.1684	0.351681
$837$	2.62753	0.0908207
$838$	−44.2348	−1.52806
$839$	14.2638	0.492442	0.246221	−	0.969214i	$-0.420811\pi$
$839$	14.2638	0.492442	0.246221	+	0.969214i	$0.420811\pi$
$840$	1.40621	0.0485188
$841$	62.7916	2.16523
$842$	−58.4631	−2.01477
$843$	−17.8706	−0.615497
$844$	−0.929627	−0.0319991
$845$	1.90553	0.0655521
$846$	10.0356	0.345031
$847$	0.658495	0.0226262
$848$	−10.8026	−0.370963
$849$	−25.2594	−0.866900
$850$	5.62792	0.193036
$851$	−15.3707	−0.526901
$852$	83.1697	2.84935
$853$	−31.0187	−1.06206	−0.531030	−	0.847353i	$-0.678195\pi$
$853$	−31.0187	−1.06206	−0.531030	+	0.847353i	$0.678195\pi$
$854$	−1.94259	−0.0664740
$855$	−23.3308	−0.797896
$856$	−4.83598	−0.165290
$857$	−5.04435	−0.172312	−0.0861559	−	0.996282i	$-0.527458\pi$
$857$	−5.04435	−0.172312	−0.0861559	+	0.996282i	$0.527458\pi$
$858$	5.19388	0.177316
$859$	21.1328	0.721041	0.360520	−	0.932751i	$-0.382599\pi$
$859$	21.1328	0.721041	0.360520	+	0.932751i	$0.382599\pi$
$860$	78.1553	2.66508
$861$	0.709859	0.0241919
$862$	−32.0036	−1.09005
$863$	33.1210	1.12745	0.563726	−	0.825962i	$-0.309367\pi$
$863$	33.1210	1.12745	0.563726	+	0.825962i	$0.309367\pi$
$864$	3.20831	0.109149
$865$	−8.79330	−0.298981
$866$	44.3049	1.50554
$867$	−36.7943	−1.24960
$868$	−0.305753	−0.0103779
$869$	−6.56554	−0.222721
$870$	115.053	3.90066
$871$	−13.0800	−0.443200
$872$	−38.8797	−1.31663
$873$	22.8069	0.771898
$874$	−55.6161	−1.88124
$875$	0.774328	0.0261771
$876$	−59.1614	−1.99888
$877$	−51.9908	−1.75561	−0.877803	−	0.479022i	$-0.840992\pi$
$877$	−51.9908	−1.75561	−0.877803	+	0.479022i	$0.840992\pi$
$878$	−86.2897	−2.91214
$879$	57.4470	1.93764
$880$	−4.77275	−0.160889
$881$	22.7015	0.764834	0.382417	−	0.923990i	$-0.375092\pi$
$881$	22.7015	0.764834	0.382417	+	0.923990i	$0.375092\pi$
$882$	64.3212	2.16581
$883$	42.8551	1.44219	0.721095	−	0.692836i	$-0.243639\pi$
$883$	42.8551	1.44219	0.721095	+	0.692836i	$0.243639\pi$
$884$	6.52717	0.219533
$885$	−59.1148	−1.98712
$886$	−40.8179	−1.37130
$887$	−31.9804	−1.07380	−0.536898	−	0.843647i	$-0.680404\pi$
$887$	−31.9804	−1.07380	−0.536898	+	0.843647i	$0.680404\pi$
$888$	−24.8403	−0.833587
$889$	0.556699	0.0186711
$890$	−34.8760	−1.16905
$891$	4.88896	0.163786
$892$	−106.268	−3.55810
$893$	3.51133	0.117502
$894$	−45.3832	−1.51784
$895$	−12.1141	−0.404930
$896$	−1.31013	−0.0437685
$897$	−18.6715	−0.623422
$898$	52.6210	1.75599
$899$	−11.9710	−0.399254
$900$	−19.9835	−0.666116
$901$	−6.04935	−0.201533
$902$	−8.49039	−0.282699
$903$	−1.78004	−0.0592360
$904$	−14.3226	−0.476363
$905$	−40.9678	−1.36182
$906$	77.2691	2.56709
$907$	19.4615	0.646209	0.323105	−	0.946363i	$-0.395273\pi$
$907$	19.4615	0.646209	0.323105	+	0.946363i	$0.395273\pi$
$908$	−55.2689	−1.83416
$909$	−23.3723	−0.775209
$910$	0.293690	0.00973571
$911$	−22.6628	−0.750851	−0.375425	−	0.926853i	$-0.622503\pi$
$911$	−22.6628	−0.750851	−0.375425	+	0.926853i	$0.622503\pi$
$912$	−25.5051	−0.844560
$913$	4.50515	0.149099
$914$	−16.9823	−0.561724
$915$	−62.6572	−2.07138
$916$	0.0481549	0.00159108
$917$	0.581421	0.0192002
$918$	−8.64520	−0.285334
$919$	37.0287	1.22146	0.610732	−	0.791838i	$-0.290875\pi$
$919$	37.0287	1.22146	0.610732	+	0.791838i	$0.290875\pi$
$920$	60.4633	1.99342
$921$	60.7412	2.00149
$922$	5.89425	0.194117
$923$	8.31218	0.273599
$924$	0.526142	0.0173088
$925$	−2.94006	−0.0966685
$926$	30.4386	1.00028
$927$	−0.383403	−0.0125926
$928$	−14.6170	−0.479827
$929$	2.38453	0.0782338	0.0391169	−	0.999235i	$-0.487546\pi$
$929$	2.38453	0.0782338	0.0391169	+	0.999235i	$0.487546\pi$
$930$	−15.0046	−0.492020
$931$	22.5052	0.737578
$932$	69.2370	2.26793
$933$	78.7833	2.57925
$934$	80.7359	2.64176
$935$	−2.67270	−0.0874065
$936$	−16.8742	−0.551550
$937$	35.7596	1.16822	0.584108	−	0.811676i	$-0.301444\pi$
$937$	35.7596	1.16822	0.584108	+	0.811676i	$0.301444\pi$
$938$	−2.01596	−0.0658234
$939$	−12.3642	−0.403490
$940$	−7.97723	−0.260188
$941$	−37.9116	−1.23588	−0.617941	−	0.786225i	$-0.712033\pi$
$941$	−37.9116	−1.23588	−0.617941	+	0.786225i	$0.712033\pi$
$942$	−72.6764	−2.36793
$943$	30.5221	0.993935
$944$	−36.1389	−1.17622
$945$	−0.255667	−0.00831686
$946$	21.2905	0.692213
$947$	−15.9966	−0.519818	−0.259909	−	0.965633i	$-0.583693\pi$
$947$	−15.9966	−0.519818	−0.259909	+	0.965633i	$0.583693\pi$
$948$	79.7093	2.58884
$949$	−5.91273	−0.191935
$950$	−10.6381	−0.345144
$951$	71.6267	2.32266
$952$	0.481404	0.0156024
$953$	−7.55234	−0.244644	−0.122322	−	0.992490i	$-0.539034\pi$
$953$	−7.55234	−0.244644	−0.122322	+	0.992490i	$0.539034\pi$
$954$	32.6810	1.05809
$955$	6.27846	0.203166
$956$	−90.8010	−2.93671
$957$	20.5997	0.665894
$958$	−21.7100	−0.701417
$959$	−0.674178	−0.0217704
$960$	−48.5371	−1.56653
$961$	−29.4388	−0.949639
$962$	−5.18795	−0.167266
$963$	4.15159	0.133783
$964$	87.9980	2.83422
$965$	16.6737	0.536745
$966$	−2.87774	−0.0925898
$967$	−53.1304	−1.70856	−0.854280	−	0.519813i	$-0.826002\pi$
$967$	−53.1304	−1.70856	−0.854280	+	0.519813i	$0.826002\pi$
$968$	45.7570	1.47069
$969$	−14.2826	−0.458825
$970$	−27.5828	−0.885630
$971$	−36.8637	−1.18301	−0.591506	−	0.806301i	$-0.701466\pi$
$971$	−36.8637	−1.18301	−0.591506	+	0.806301i	$0.701466\pi$
$972$	−83.5507	−2.67989
$973$	−1.06399	−0.0341098
$974$	−38.0088	−1.21788
$975$	−3.57142	−0.114377
$976$	−38.3045	−1.22610
$977$	−17.3645	−0.555538	−0.277769	−	0.960648i	$-0.589595\pi$
$977$	−17.3645	−0.555538	−0.277769	+	0.960648i	$0.589595\pi$
$978$	−83.2235	−2.66119
$979$	−6.24439	−0.199572
$980$	−51.1285	−1.63324
$981$	33.3775	1.06566
$982$	21.3089	0.679993
$983$	8.82194	0.281376	0.140688	−	0.990054i	$-0.455069\pi$
$983$	8.82194	0.281376	0.140688	+	0.990054i	$0.455069\pi$
$984$	49.3262	1.57246
$985$	22.9908	0.732549
$986$	39.3874	1.25435
$987$	0.181687	0.00578315
$988$	−12.3379	−0.392520
$989$	−76.5370	−2.43374
$990$	14.4390	0.458900
$991$	−50.2499	−1.59624	−0.798121	−	0.602497i	$-0.794173\pi$
$991$	−50.2499	−1.59624	−0.798121	+	0.602497i	$0.794173\pi$
$992$	1.90627	0.0605242
$993$	40.5772	1.28768
$994$	1.28111	0.0406345
$995$	−7.04407	−0.223312
$996$	−54.6951	−1.73308
$997$	40.8540	1.29386	0.646930	−	0.762550i	$-0.276053\pi$
$997$	40.8540	1.29386	0.646930	+	0.762550i	$0.276053\pi$
$998$	6.01062	0.190263
$999$	4.51630	0.142889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.16	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.16	✓	142	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)

\(f(q)\)	\(=\)	\(q-2.41564 q^{2} +2.60884 q^{3} +3.83533 q^{4} +1.90553 q^{5} -6.30203 q^{6} -0.0638029 q^{7} -4.43349 q^{8} +3.80607 q^{9} +O(q^{10})\)	\(q-2.41564 q^{2} +2.60884 q^{3} +3.83533 q^{4} +1.90553 q^{5} -6.30203 q^{6} -0.0638029 q^{7} -4.43349 q^{8} +3.80607 q^{9} -4.60307 q^{10} -0.824160 q^{11} +10.0058 q^{12} +1.00000 q^{13} +0.154125 q^{14} +4.97122 q^{15} +3.03908 q^{16} +1.70186 q^{17} -9.19410 q^{18} -3.21690 q^{19} +7.30832 q^{20} -0.166452 q^{21} +1.99088 q^{22} -7.15699 q^{23} -11.5663 q^{24} -1.36897 q^{25} -2.41564 q^{26} +2.10290 q^{27} -0.244705 q^{28} -9.58079 q^{29} -12.0087 q^{30} +1.24948 q^{31} +1.52566 q^{32} -2.15010 q^{33} -4.11107 q^{34} -0.121578 q^{35} +14.5975 q^{36} +2.14765 q^{37} +7.77088 q^{38} +2.60884 q^{39} -8.44814 q^{40} -4.26465 q^{41} +0.402088 q^{42} +10.6940 q^{43} -3.16092 q^{44} +7.25257 q^{45} +17.2887 q^{46} -1.09153 q^{47} +7.92849 q^{48} -6.99593 q^{49} +3.30693 q^{50} +4.43988 q^{51} +3.83533 q^{52} -3.55456 q^{53} -5.07986 q^{54} -1.57046 q^{55} +0.282870 q^{56} -8.39239 q^{57} +23.1438 q^{58} -11.8914 q^{59} +19.0663 q^{60} -12.6040 q^{61} -3.01829 q^{62} -0.242838 q^{63} -9.76360 q^{64} +1.90553 q^{65} +5.19388 q^{66} -13.0800 q^{67} +6.52717 q^{68} -18.6715 q^{69} +0.293690 q^{70} +8.31218 q^{71} -16.8742 q^{72} -5.91273 q^{73} -5.18795 q^{74} -3.57142 q^{75} -12.3379 q^{76} +0.0525838 q^{77} -6.30203 q^{78} +7.96634 q^{79} +5.79105 q^{80} -5.93205 q^{81} +10.3019 q^{82} -5.46635 q^{83} -0.638398 q^{84} +3.24293 q^{85} -25.8329 q^{86} -24.9948 q^{87} +3.65391 q^{88} +7.57667 q^{89} -17.5196 q^{90} -0.0638029 q^{91} -27.4494 q^{92} +3.25969 q^{93} +2.63674 q^{94} -6.12989 q^{95} +3.98020 q^{96} +5.99226 q^{97} +16.8997 q^{98} -3.13681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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