LMFDB - Embedded newform 8047.2.a.d.1.20

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			1.20
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)

$f(q)$	$=$	$q-2.15517 q^{2} +0.169019 q^{3} +2.64475 q^{4} +4.16569 q^{5} -0.364265 q^{6} +2.92802 q^{7} -1.38954 q^{8} -2.97143 q^{9} +O(q^{10})$	\(q-2.15517 q^{2} +0.169019 q^{3} +2.64475 q^{4} +4.16569 q^{5} -0.364265 q^{6} +2.92802 q^{7} -1.38954 q^{8} -2.97143 q^{9} -8.97775 q^{10} +6.34599 q^{11} +0.447014 q^{12} -1.00000 q^{13} -6.31038 q^{14} +0.704082 q^{15} -2.29480 q^{16} +6.79024 q^{17} +6.40394 q^{18} -4.12839 q^{19} +11.0172 q^{20} +0.494893 q^{21} -13.6767 q^{22} -0.0320136 q^{23} -0.234859 q^{24} +12.3529 q^{25} +2.15517 q^{26} -1.00929 q^{27} +7.74388 q^{28} -8.35544 q^{29} -1.51742 q^{30} +7.76654 q^{31} +7.72477 q^{32} +1.07260 q^{33} -14.6341 q^{34} +12.1972 q^{35} -7.85869 q^{36} +9.15506 q^{37} +8.89738 q^{38} -0.169019 q^{39} -5.78839 q^{40} -2.99925 q^{41} -1.06658 q^{42} +0.0382617 q^{43} +16.7835 q^{44} -12.3781 q^{45} +0.0689946 q^{46} -10.4689 q^{47} -0.387867 q^{48} +1.57331 q^{49} -26.6227 q^{50} +1.14768 q^{51} -2.64475 q^{52} -7.40689 q^{53} +2.17519 q^{54} +26.4354 q^{55} -4.06860 q^{56} -0.697779 q^{57} +18.0074 q^{58} +10.4330 q^{59} +1.86212 q^{60} -1.95694 q^{61} -16.7382 q^{62} -8.70042 q^{63} -12.0586 q^{64} -4.16569 q^{65} -2.31162 q^{66} -2.25396 q^{67} +17.9585 q^{68} -0.00541092 q^{69} -26.2871 q^{70} +15.5069 q^{71} +4.12892 q^{72} -10.2844 q^{73} -19.7307 q^{74} +2.08789 q^{75} -10.9186 q^{76} +18.5812 q^{77} +0.364265 q^{78} +1.56344 q^{79} -9.55943 q^{80} +8.74371 q^{81} +6.46388 q^{82} -10.6009 q^{83} +1.30887 q^{84} +28.2860 q^{85} -0.0824603 q^{86} -1.41223 q^{87} -8.81801 q^{88} -10.4734 q^{89} +26.6768 q^{90} -2.92802 q^{91} -0.0846678 q^{92} +1.31270 q^{93} +22.5623 q^{94} -17.1976 q^{95} +1.30564 q^{96} +4.75774 q^{97} -3.39076 q^{98} -18.8567 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.15517	−1.52393	−0.761967	−	0.647616i	$-0.775766\pi$
$2$	−2.15517	−1.52393	−0.761967	+	0.647616i	$0.775766\pi$
$3$	0.169019	0.0975834	0.0487917	−	0.998809i	$-0.484463\pi$
$3$	0.169019	0.0975834	0.0487917	+	0.998809i	$0.484463\pi$
$4$	2.64475	1.32237
$5$	4.16569	1.86295	0.931476	−	0.363803i	$-0.118522\pi$
$5$	4.16569	1.86295	0.931476	+	0.363803i	$0.118522\pi$
$6$	−0.364265	−0.148711
$7$	2.92802	1.10669	0.553344	−	0.832953i	$-0.313351\pi$
$7$	2.92802	1.10669	0.553344	+	0.832953i	$0.313351\pi$
$8$	−1.38954	−0.491277
$9$	−2.97143	−0.990477
$10$	−8.97775	−2.83902
$11$	6.34599	1.91339	0.956694	−	0.291096i	$-0.0940199\pi$
$11$	6.34599	1.91339	0.956694	+	0.291096i	$0.0940199\pi$
$12$	0.447014	0.129042
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−6.31038	−1.68652
$15$	0.704082	0.181793
$16$	−2.29480	−0.573701
$17$	6.79024	1.64687	0.823437	−	0.567407i	$-0.192053\pi$
$17$	6.79024	1.64687	0.823437	+	0.567407i	$0.192053\pi$
$18$	6.40394	1.50942
$19$	−4.12839	−0.947118	−0.473559	−	0.880762i	$-0.657031\pi$
$19$	−4.12839	−0.947118	−0.473559	+	0.880762i	$0.657031\pi$
$20$	11.0172	2.46352
$21$	0.494893	0.107994
$22$	−13.6767	−2.91588
$23$	−0.0320136	−0.00667529	−0.00333764	−	0.999994i	$-0.501062\pi$
$23$	−0.0320136	−0.00667529	−0.00333764	+	0.999994i	$0.501062\pi$
$24$	−0.234859	−0.0479405
$25$	12.3529	2.47059
$26$	2.15517	0.422663
$27$	−1.00929	−0.194238
$28$	7.74388	1.46346
$29$	−8.35544	−1.55157	−0.775784	−	0.630999i	$-0.782645\pi$
$29$	−8.35544	−1.55157	−0.775784	+	0.630999i	$0.782645\pi$
$30$	−1.51742	−0.277041
$31$	7.76654	1.39491	0.697456	−	0.716628i	$-0.254315\pi$
$31$	7.76654	1.39491	0.697456	+	0.716628i	$0.254315\pi$
$32$	7.72477	1.36556
$33$	1.07260	0.186715
$34$	−14.6341	−2.50973
$35$	12.1972	2.06171
$36$	−7.85869	−1.30978
$37$	9.15506	1.50508	0.752541	−	0.658545i	$-0.228828\pi$
$37$	9.15506	1.50508	0.752541	+	0.658545i	$0.228828\pi$
$38$	8.89738	1.44335
$39$	−0.169019	−0.0270648
$40$	−5.78839	−0.915225
$41$	−2.99925	−0.468404	−0.234202	−	0.972188i	$-0.575248\pi$
$41$	−2.99925	−0.468404	−0.234202	+	0.972188i	$0.575248\pi$
$42$	−1.06658	−0.164576
$43$	0.0382617	0.00583485	0.00291742	−	0.999996i	$-0.499071\pi$
$43$	0.0382617	0.00583485	0.00291742	+	0.999996i	$0.499071\pi$
$44$	16.7835	2.53021
$45$	−12.3781	−1.84521
$46$	0.0689946	0.0101727
$47$	−10.4689	−1.52705	−0.763524	−	0.645779i	$-0.776533\pi$
$47$	−10.4689	−1.52705	−0.763524	+	0.645779i	$0.776533\pi$
$48$	−0.387867	−0.0559837
$49$	1.57331	0.224759
$50$	−26.6227	−3.76501
$51$	1.14768	0.160708
$52$	−2.64475	−0.366761
$53$	−7.40689	−1.01741	−0.508707	−	0.860940i	$-0.669876\pi$
$53$	−7.40689	−1.01741	−0.508707	+	0.860940i	$0.669876\pi$
$54$	2.17519	0.296005
$55$	26.4354	3.56455
$56$	−4.06860	−0.543690
$57$	−0.697779	−0.0924231
$58$	18.0074	2.36449
$59$	10.4330	1.35826	0.679129	−	0.734019i	$-0.262357\pi$
$59$	10.4330	1.35826	0.679129	+	0.734019i	$0.262357\pi$
$60$	1.86212	0.240399
$61$	−1.95694	−0.250560	−0.125280	−	0.992121i	$-0.539983\pi$
$61$	−1.95694	−0.250560	−0.125280	+	0.992121i	$0.539983\pi$
$62$	−16.7382	−2.12575
$63$	−8.70042	−1.09615
$64$	−12.0586	−1.50732
$65$	−4.16569	−0.516690
$66$	−2.31162	−0.284541
$67$	−2.25396	−0.275365	−0.137682	−	0.990476i	$-0.543965\pi$
$67$	−2.25396	−0.275365	−0.137682	+	0.990476i	$0.543965\pi$
$68$	17.9585	2.17778
$69$	−0.00541092	−0.000651398 0
$70$	−26.2871	−3.14191
$71$	15.5069	1.84033	0.920165	−	0.391531i	$-0.128054\pi$
$71$	15.5069	1.84033	0.920165	+	0.391531i	$0.128054\pi$
$72$	4.12892	0.486598
$73$	−10.2844	−1.20369	−0.601846	−	0.798612i	$-0.705568\pi$
$73$	−10.2844	−1.20369	−0.601846	+	0.798612i	$0.705568\pi$
$74$	−19.7307	−2.29365
$75$	2.08789	0.241089
$76$	−10.9186	−1.25244
$77$	18.5812	2.11752
$78$	0.364265	0.0412449
$79$	1.56344	0.175900	0.0879501	−	0.996125i	$-0.471968\pi$
$79$	1.56344	0.175900	0.0879501	+	0.996125i	$0.471968\pi$
$80$	−9.55943	−1.06878
$81$	8.74371	0.971523
$82$	6.46388	0.713816
$83$	−10.6009	−1.16360	−0.581800	−	0.813332i	$-0.697651\pi$
$83$	−10.6009	−1.16360	−0.581800	+	0.813332i	$0.697651\pi$
$84$	1.30887	0.142809
$85$	28.2860	3.06805
$86$	−0.0824603	−0.00889192
$87$	−1.41223	−0.151407
$88$	−8.81801	−0.940003
$89$	−10.4734	−1.11018	−0.555091	−	0.831789i	$-0.687317\pi$
$89$	−10.4734	−1.11018	−0.555091	+	0.831789i	$0.687317\pi$
$90$	26.6768	2.81198
$91$	−2.92802	−0.306940
$92$	−0.0846678	−0.00882723
$93$	1.31270	0.136120
$94$	22.5623	2.32712
$95$	−17.1976	−1.76444
$96$	1.30564	0.133256
$97$	4.75774	0.483076	0.241538	−	0.970391i	$-0.422348\pi$
$97$	4.75774	0.483076	0.241538	+	0.970391i	$0.422348\pi$
$98$	−3.39076	−0.342518
$99$	−18.8567	−1.89517
$100$	32.6704	3.26704
$101$	−2.92908	−0.291455	−0.145727	−	0.989325i	$-0.546552\pi$
$101$	−2.92908	−0.291455	−0.145727	+	0.989325i	$0.546552\pi$
$102$	−2.47345	−0.244908
$103$	9.89987	0.975463	0.487732	−	0.872994i	$-0.337824\pi$
$103$	9.89987	0.975463	0.487732	+	0.872994i	$0.337824\pi$
$104$	1.38954	0.136256
$105$	2.06157	0.201188
$106$	15.9631	1.55047
$107$	−14.9015	−1.44058	−0.720289	−	0.693674i	$-0.755991\pi$
$107$	−14.9015	−1.44058	−0.720289	+	0.693674i	$0.755991\pi$
$108$	−2.66931	−0.256855
$109$	5.22161	0.500140	0.250070	−	0.968228i	$-0.419546\pi$
$109$	5.22161	0.500140	0.250070	+	0.968228i	$0.419546\pi$
$110$	−56.9727	−5.43214
$111$	1.54738	0.146871
$112$	−6.71924	−0.634908
$113$	11.7930	1.10939	0.554695	−	0.832053i	$-0.312835\pi$
$113$	11.7930	1.10939	0.554695	+	0.832053i	$0.312835\pi$
$114$	1.50383	0.140847
$115$	−0.133358	−0.0124357
$116$	−22.0980	−2.05175
$117$	2.97143	0.274709
$118$	−22.4848	−2.06990
$119$	19.8820	1.82258
$120$	−0.978351	−0.0893108
$121$	29.2716	2.66105
$122$	4.21752	0.381837
$123$	−0.506931	−0.0457084
$124$	20.5405	1.84460
$125$	30.6301	2.73964
$126$	18.7509	1.67046
$127$	21.7566	1.93059	0.965293	−	0.261168i	$-0.0841077\pi$
$127$	21.7566	1.93059	0.965293	+	0.261168i	$0.0841077\pi$
$128$	10.5387	0.931497
$129$	0.00646697	0.000569385 0
$130$	8.97775	0.787401
$131$	12.9169	1.12856	0.564278	−	0.825585i	$-0.309154\pi$
$131$	12.9169	1.12856	0.564278	+	0.825585i	$0.309154\pi$
$132$	2.83675	0.246907
$133$	−12.0880	−1.04816
$134$	4.85766	0.419638
$135$	−4.20438	−0.361855
$136$	−9.43531	−0.809071
$137$	4.48312	0.383019	0.191509	−	0.981491i	$-0.438662\pi$
$137$	4.48312	0.383019	0.191509	+	0.981491i	$0.438662\pi$
$138$	0.0116614	0.000992687 0
$139$	−6.20356	−0.526179	−0.263089	−	0.964771i	$-0.584741\pi$
$139$	−6.20356	−0.526179	−0.263089	+	0.964771i	$0.584741\pi$
$140$	32.2586	2.72635
$141$	−1.76945	−0.149015
$142$	−33.4200	−2.80454
$143$	−6.34599	−0.530678
$144$	6.81885	0.568238
$145$	−34.8062	−2.89049
$146$	22.1645	1.83435
$147$	0.265921	0.0219328
$148$	24.2128	1.99028
$149$	14.4980	1.18772	0.593860	−	0.804569i	$-0.297603\pi$
$149$	14.4980	1.18772	0.593860	+	0.804569i	$0.297603\pi$
$150$	−4.49975	−0.367403
$151$	−18.8373	−1.53295	−0.766477	−	0.642272i	$-0.777992\pi$
$151$	−18.8373	−1.53295	−0.766477	+	0.642272i	$0.777992\pi$
$152$	5.73657	0.465297
$153$	−20.1767	−1.63119
$154$	−40.0456	−3.22697
$155$	32.3530	2.59865
$156$	−0.447014	−0.0357898
$157$	11.2088	0.894562	0.447281	−	0.894393i	$-0.352392\pi$
$157$	11.2088	0.894562	0.447281	+	0.894393i	$0.352392\pi$
$158$	−3.36947	−0.268060
$159$	−1.25191	−0.0992828
$160$	32.1790	2.54397
$161$	−0.0937364	−0.00738746
$162$	−18.8442	−1.48054
$163$	−15.1411	−1.18594	−0.592970	−	0.805225i	$-0.702045\pi$
$163$	−15.1411	−1.18594	−0.592970	+	0.805225i	$0.702045\pi$
$164$	−7.93225	−0.619405
$165$	4.46810	0.347841
$166$	22.8467	1.77325
$167$	2.25793	0.174724	0.0873620	−	0.996177i	$-0.472156\pi$
$167$	2.25793	0.174724	0.0873620	+	0.996177i	$0.472156\pi$
$168$	−0.687673	−0.0530552
$169$	1.00000	0.0769231
$170$	−60.9611	−4.67550
$171$	12.2672	0.938099
$172$	0.101192	0.00771585
$173$	−9.04907	−0.687988	−0.343994	−	0.938972i	$-0.611780\pi$
$173$	−9.04907	−0.687988	−0.343994	+	0.938972i	$0.611780\pi$
$174$	3.04360	0.230735
$175$	36.1697	2.73417
$176$	−14.5628	−1.09771
$177$	1.76338	0.132544
$178$	22.5720	1.69185
$179$	1.54777	0.115686	0.0578428	−	0.998326i	$-0.481578\pi$
$179$	1.54777	0.115686	0.0578428	+	0.998326i	$0.481578\pi$
$180$	−32.7368	−2.44006
$181$	6.30197	0.468422	0.234211	−	0.972186i	$-0.424749\pi$
$181$	6.30197	0.468422	0.234211	+	0.972186i	$0.424749\pi$
$182$	6.31038	0.467756
$183$	−0.330760	−0.0244505
$184$	0.0444841	0.00327941
$185$	38.1371	2.80390
$186$	−2.82908	−0.207438
$187$	43.0908	3.15111
$188$	−27.6877	−2.01933
$189$	−2.95522	−0.214961
$190$	37.0637	2.68888
$191$	4.99770	0.361621	0.180810	−	0.983518i	$-0.442128\pi$
$191$	4.99770	0.361621	0.180810	+	0.983518i	$0.442128\pi$
$192$	−2.03813	−0.147090
$193$	−20.8170	−1.49844	−0.749220	−	0.662322i	$-0.769571\pi$
$193$	−20.8170	−1.49844	−0.749220	+	0.662322i	$0.769571\pi$
$194$	−10.2537	−0.736175
$195$	−0.704082	−0.0504204
$196$	4.16102	0.297216
$197$	2.68167	0.191061	0.0955305	−	0.995427i	$-0.469545\pi$
$197$	2.68167	0.191061	0.0955305	+	0.995427i	$0.469545\pi$
$198$	40.6393	2.88811
$199$	7.85539	0.556854	0.278427	−	0.960457i	$-0.410187\pi$
$199$	7.85539	0.556854	0.278427	+	0.960457i	$0.410187\pi$
$200$	−17.1649	−1.21374
$201$	−0.380963	−0.0268711
$202$	6.31267	0.444158
$203$	−24.4649	−1.71710
$204$	3.03533	0.212516
$205$	−12.4939	−0.872613
$206$	−21.3359	−1.48654
$207$	0.0951261	0.00661172
$208$	2.29480	0.159116
$209$	−26.1987	−1.81220
$210$	−4.44303	−0.306598
$211$	13.6318	0.938450	0.469225	−	0.883079i	$-0.344533\pi$
$211$	13.6318	0.938450	0.469225	+	0.883079i	$0.344533\pi$
$212$	−19.5893	−1.34540
$213$	2.62097	0.179586
$214$	32.1151	2.19534
$215$	0.159386	0.0108700
$216$	1.40245	0.0954244
$217$	22.7406	1.54373
$218$	−11.2535	−0.762180
$219$	−1.73826	−0.117460
$220$	69.9150	4.71367
$221$	−6.79024	−0.456761
$222$	−3.33487	−0.223822
$223$	1.80560	0.120912	0.0604560	−	0.998171i	$-0.480745\pi$
$223$	1.80560	0.120912	0.0604560	+	0.998171i	$0.480745\pi$
$224$	22.6183	1.51125
$225$	−36.7059	−2.44706
$226$	−25.4159	−1.69064
$227$	2.15492	0.143027	0.0715136	−	0.997440i	$-0.477217\pi$
$227$	2.15492	0.143027	0.0715136	+	0.997440i	$0.477217\pi$
$228$	−1.84545	−0.122218
$229$	20.3125	1.34228	0.671142	−	0.741328i	$-0.265804\pi$
$229$	20.3125	1.34228	0.671142	+	0.741328i	$0.265804\pi$
$230$	0.287410	0.0189512
$231$	3.14058	0.206635
$232$	11.6102	0.762249
$233$	10.7266	0.702721	0.351360	−	0.936240i	$-0.385719\pi$
$233$	10.7266	0.702721	0.351360	+	0.936240i	$0.385719\pi$
$234$	−6.40394	−0.418638
$235$	−43.6102	−2.84482
$236$	27.5926	1.79613
$237$	0.264251	0.0171650
$238$	−42.8490	−2.77749
$239$	3.82423	0.247369	0.123684	−	0.992322i	$-0.460529\pi$
$239$	3.82423	0.247369	0.123684	+	0.992322i	$0.460529\pi$
$240$	−1.61573	−0.104295
$241$	−13.6936	−0.882079	−0.441040	−	0.897488i	$-0.645390\pi$
$241$	−13.6936	−0.882079	−0.441040	+	0.897488i	$0.645390\pi$
$242$	−63.0852	−4.05527
$243$	4.50572	0.289042
$244$	−5.17560	−0.331334
$245$	6.55393	0.418715
$246$	1.09252	0.0696566
$247$	4.12839	0.262683
$248$	−10.7919	−0.685288
$249$	−1.79176	−0.113548
$250$	−66.0130	−4.17503
$251$	−11.0309	−0.696267	−0.348134	−	0.937445i	$-0.613184\pi$
$251$	−11.0309	−0.696267	−0.348134	+	0.937445i	$0.613184\pi$
$252$	−23.0104	−1.44952
$253$	−0.203158	−0.0127724
$254$	−46.8891	−2.94209
$255$	4.78089	0.299391
$256$	1.40448	0.0877800
$257$	2.19806	0.137111	0.0685556	−	0.997647i	$-0.478161\pi$
$257$	2.19806	0.137111	0.0685556	+	0.997647i	$0.478161\pi$
$258$	−0.0139374	−0.000867704 0
$259$	26.8062	1.66566
$260$	−11.0172	−0.683257
$261$	24.8276	1.53679
$262$	−27.8381	−1.71985
$263$	19.6095	1.20917	0.604586	−	0.796540i	$-0.293338\pi$
$263$	19.6095	1.20917	0.604586	+	0.796540i	$0.293338\pi$
$264$	−1.49041	−0.0917287
$265$	−30.8548	−1.89539
$266$	26.0517	1.59733
$267$	−1.77022	−0.108335
$268$	−5.96115	−0.364135
$269$	−28.5427	−1.74028	−0.870139	−	0.492807i	$-0.835971\pi$
$269$	−28.5427	−1.74028	−0.870139	+	0.492807i	$0.835971\pi$
$270$	9.06114	0.551444
$271$	25.1654	1.52869	0.764344	−	0.644808i	$-0.223063\pi$
$271$	25.1654	1.52869	0.764344	+	0.644808i	$0.223063\pi$
$272$	−15.5823	−0.944814
$273$	−0.494893	−0.0299523
$274$	−9.66188	−0.583695
$275$	78.3917	4.72720
$276$	−0.0143105	−0.000861391 0
$277$	15.0001	0.901269	0.450634	−	0.892709i	$-0.351198\pi$
$277$	15.0001	0.901269	0.450634	+	0.892709i	$0.351198\pi$
$278$	13.3697	0.801861
$279$	−23.0778	−1.38163
$280$	−16.9485	−1.01287
$281$	−11.0500	−0.659186	−0.329593	−	0.944123i	$-0.606912\pi$
$281$	−11.0500	−0.659186	−0.329593	+	0.944123i	$0.606912\pi$
$282$	3.81346	0.227089
$283$	7.66716	0.455766	0.227883	−	0.973689i	$-0.426820\pi$
$283$	7.66716	0.455766	0.227883	+	0.973689i	$0.426820\pi$
$284$	41.0118	2.43360
$285$	−2.90673	−0.172180
$286$	13.6767	0.808718
$287$	−8.78186	−0.518377
$288$	−22.9536	−1.35256
$289$	29.1073	1.71220
$290$	75.0131	4.40492
$291$	0.804151	0.0471402
$292$	−27.1995	−1.59173
$293$	−7.18314	−0.419644	−0.209822	−	0.977740i	$-0.567288\pi$
$293$	−7.18314	−0.419644	−0.209822	+	0.977740i	$0.567288\pi$
$294$	−0.573104	−0.0334241
$295$	43.4605	2.53037
$296$	−12.7213	−0.739412
$297$	−6.40493	−0.371652
$298$	−31.2455	−1.81001
$299$	0.0320136	0.00185139
$300$	5.52194	0.318809
$301$	0.112031	0.00645736
$302$	40.5974	2.33612
$303$	−0.495072	−0.0284412
$304$	9.47385	0.543363
$305$	−8.15198	−0.466781
$306$	43.4842	2.48583
$307$	−21.2442	−1.21247	−0.606235	−	0.795285i	$-0.707321\pi$
$307$	−21.2442	−1.21247	−0.606235	+	0.795285i	$0.707321\pi$
$308$	49.1426	2.80016
$309$	1.67327	0.0951891
$310$	−69.7261	−3.96018
$311$	15.7894	0.895333	0.447667	−	0.894200i	$-0.352255\pi$
$311$	15.7894	0.895333	0.447667	+	0.894200i	$0.352255\pi$
$312$	0.234859	0.0132963
$313$	−7.90200	−0.446648	−0.223324	−	0.974744i	$-0.571691\pi$
$313$	−7.90200	−0.446648	−0.223324	+	0.974744i	$0.571691\pi$
$314$	−24.1569	−1.36325
$315$	−36.2432	−2.04207
$316$	4.13489	0.232606
$317$	−15.5260	−0.872029	−0.436014	−	0.899940i	$-0.643610\pi$
$317$	−15.5260	−0.872029	−0.436014	+	0.899940i	$0.643610\pi$
$318$	2.69807	0.151300
$319$	−53.0236	−2.96875
$320$	−50.2322	−2.80807
$321$	−2.51864	−0.140577
$322$	0.202018	0.0112580
$323$	−28.0328	−1.55979
$324$	23.1249	1.28472
$325$	−12.3529	−0.685218
$326$	32.6315	1.80729
$327$	0.882554	0.0488054
$328$	4.16757	0.230116
$329$	−30.6532	−1.68997
$330$	−9.62950	−0.530087
$331$	4.48725	0.246641	0.123321	−	0.992367i	$-0.460646\pi$
$331$	4.48725	0.246641	0.123321	+	0.992367i	$0.460646\pi$
$332$	−28.0367	−1.53871
$333$	−27.2036	−1.49075
$334$	−4.86622	−0.266268
$335$	−9.38929	−0.512992
$336$	−1.13568	−0.0619565
$337$	−14.8917	−0.811202	−0.405601	−	0.914050i	$-0.632938\pi$
$337$	−14.8917	−0.811202	−0.405601	+	0.914050i	$0.632938\pi$
$338$	−2.15517	−0.117226
$339$	1.99324	0.108258
$340$	74.8094	4.05711
$341$	49.2864	2.66901
$342$	−26.4380	−1.42960
$343$	−15.8895	−0.857950
$344$	−0.0531661	−0.00286652
$345$	−0.0225402	−0.00121352
$346$	19.5023	1.04845
$347$	−9.43647	−0.506576	−0.253288	−	0.967391i	$-0.581512\pi$
$347$	−9.43647	−0.506576	−0.253288	+	0.967391i	$0.581512\pi$
$348$	−3.73500	−0.200217
$349$	−2.87679	−0.153991	−0.0769955	−	0.997031i	$-0.524533\pi$
$349$	−2.87679	−0.153991	−0.0769955	+	0.997031i	$0.524533\pi$
$350$	−77.9518	−4.16670
$351$	1.00929	0.0538718
$352$	49.0213	2.61284
$353$	−15.4877	−0.824328	−0.412164	−	0.911110i	$-0.635227\pi$
$353$	−15.4877	−0.824328	−0.412164	+	0.911110i	$0.635227\pi$
$354$	−3.80037	−0.201988
$355$	64.5969	3.42845
$356$	−27.6996	−1.46808
$357$	3.36044	0.177853
$358$	−3.33570	−0.176297
$359$	16.8386	0.888706	0.444353	−	0.895852i	$-0.353434\pi$
$359$	16.8386	0.888706	0.444353	+	0.895852i	$0.353434\pi$
$360$	17.1998	0.906509
$361$	−1.95637	−0.102967
$362$	−13.5818	−0.713844
$363$	4.94747	0.259675
$364$	−7.74388	−0.405890
$365$	−42.8414	−2.24242
$366$	0.712844	0.0372609
$367$	−12.1766	−0.635613	−0.317806	−	0.948156i	$-0.602946\pi$
$367$	−12.1766	−0.635613	−0.317806	+	0.948156i	$0.602946\pi$
$368$	0.0734648	0.00382962
$369$	8.91206	0.463943
$370$	−82.1919	−4.27295
$371$	−21.6875	−1.12596
$372$	3.47175	0.180002
$373$	1.44792	0.0749705	0.0374852	−	0.999297i	$-0.488065\pi$
$373$	1.44792	0.0749705	0.0374852	+	0.999297i	$0.488065\pi$
$374$	−92.8679	−4.80208
$375$	5.17708	0.267343
$376$	14.5470	0.750204
$377$	8.35544	0.430327
$378$	6.36899	0.327586
$379$	−21.9793	−1.12900	−0.564501	−	0.825432i	$-0.690931\pi$
$379$	−21.9793	−1.12900	−0.564501	+	0.825432i	$0.690931\pi$
$380$	−45.4833	−2.33324
$381$	3.67729	0.188393
$382$	−10.7709	−0.551086
$383$	12.7382	0.650892	0.325446	−	0.945561i	$-0.394486\pi$
$383$	12.7382	0.650892	0.325446	+	0.945561i	$0.394486\pi$
$384$	1.78124	0.0908987
$385$	77.4034	3.94484
$386$	44.8641	2.28352
$387$	−0.113692	−0.00577929
$388$	12.5830	0.638807
$389$	−34.3576	−1.74200	−0.871000	−	0.491283i	$-0.836528\pi$
$389$	−34.3576	−1.74200	−0.871000	+	0.491283i	$0.836528\pi$
$390$	1.51742	0.0768373
$391$	−0.217380	−0.0109934
$392$	−2.18618	−0.110419
$393$	2.18321	0.110128
$394$	−5.77944	−0.291164
$395$	6.51278	0.327694
$396$	−49.8712	−2.50612
$397$	−10.1633	−0.510082	−0.255041	−	0.966930i	$-0.582089\pi$
$397$	−10.1633	−0.510082	−0.255041	+	0.966930i	$0.582089\pi$
$398$	−16.9297	−0.848608
$399$	−2.04311	−0.102284
$400$	−28.3476	−1.41738
$401$	−2.12312	−0.106024	−0.0530118	−	0.998594i	$-0.516882\pi$
$401$	−2.12312	−0.106024	−0.0530118	+	0.998594i	$0.516882\pi$
$402$	0.821039	0.0409497
$403$	−7.76654	−0.386879
$404$	−7.74669	−0.385412
$405$	36.4235	1.80990
$406$	52.7260	2.61675
$407$	58.0979	2.87981
$408$	−1.59475	−0.0789519
$409$	−1.38291	−0.0683806	−0.0341903	−	0.999415i	$-0.510885\pi$
$409$	−1.38291	−0.0683806	−0.0341903	+	0.999415i	$0.510885\pi$
$410$	26.9265	1.32981
$411$	0.757735	0.0373763
$412$	26.1827	1.28993
$413$	30.5480	1.50317
$414$	−0.205013	−0.0100758
$415$	−44.1600	−2.16773
$416$	−7.72477	−0.378738
$417$	−1.04852	−0.0513463
$418$	56.4627	2.76168
$419$	−12.4249	−0.606997	−0.303498	−	0.952832i	$-0.598155\pi$
$419$	−12.4249	−0.606997	−0.303498	+	0.952832i	$0.598155\pi$
$420$	5.45233	0.266046
$421$	−24.9924	−1.21806	−0.609029	−	0.793148i	$-0.708441\pi$
$421$	−24.9924	−1.21806	−0.609029	+	0.793148i	$0.708441\pi$
$422$	−29.3788	−1.43014
$423$	31.1077	1.51251
$424$	10.2922	0.499832
$425$	83.8795	4.06875
$426$	−5.64862	−0.273677
$427$	−5.72995	−0.277292
$428$	−39.4106	−1.90498
$429$	−1.07260	−0.0517854
$430$	−0.343504	−0.0165652
$431$	−3.72274	−0.179318	−0.0896590	−	0.995973i	$-0.528578\pi$
$431$	−3.72274	−0.179318	−0.0896590	+	0.995973i	$0.528578\pi$
$432$	2.31612	0.111434
$433$	−3.75204	−0.180311	−0.0901557	−	0.995928i	$-0.528736\pi$
$433$	−3.75204	−0.180311	−0.0901557	+	0.995928i	$0.528736\pi$
$434$	−49.0098	−2.35255
$435$	−5.88292	−0.282064
$436$	13.8099	0.661372
$437$	0.132165	0.00632229
$438$	3.74623	0.179002
$439$	−34.7370	−1.65791	−0.828953	−	0.559319i	$-0.811063\pi$
$439$	−34.7370	−1.65791	−0.828953	+	0.559319i	$0.811063\pi$
$440$	−36.7331	−1.75118
$441$	−4.67500	−0.222619
$442$	14.6341	0.696073
$443$	−37.3240	−1.77331	−0.886657	−	0.462427i	$-0.846979\pi$
$443$	−37.3240	−1.77331	−0.886657	+	0.462427i	$0.846979\pi$
$444$	4.09244	0.194219
$445$	−43.6291	−2.06822
$446$	−3.89137	−0.184262
$447$	2.45044	0.115902
$448$	−35.3077	−1.66813
$449$	19.4252	0.916731	0.458365	−	0.888764i	$-0.348435\pi$
$449$	19.4252	0.916731	0.458365	+	0.888764i	$0.348435\pi$
$450$	79.1075	3.72916
$451$	−19.0332	−0.896238
$452$	31.1895	1.46703
$453$	−3.18386	−0.149591
$454$	−4.64422	−0.217964
$455$	−12.1972	−0.571815
$456$	0.969592	0.0454053
$457$	1.66790	0.0780210	0.0390105	−	0.999239i	$-0.487579\pi$
$457$	1.66790	0.0780210	0.0390105	+	0.999239i	$0.487579\pi$
$458$	−43.7767	−2.04555
$459$	−6.85331	−0.319885
$460$	−0.352700	−0.0164447
$461$	−2.03580	−0.0948169	−0.0474084	−	0.998876i	$-0.515096\pi$
$461$	−2.03580	−0.0948169	−0.0474084	+	0.998876i	$0.515096\pi$
$462$	−6.76849	−0.314898
$463$	−0.203643	−0.00946411	−0.00473206	−	0.999989i	$-0.501506\pi$
$463$	−0.203643	−0.00946411	−0.00473206	+	0.999989i	$0.501506\pi$
$464$	19.1741	0.890136
$465$	5.46828	0.253586
$466$	−23.1176	−1.07090
$467$	35.2720	1.63220	0.816098	−	0.577914i	$-0.196133\pi$
$467$	35.2720	1.63220	0.816098	+	0.577914i	$0.196133\pi$
$468$	7.85869	0.363268
$469$	−6.59964	−0.304743
$470$	93.9874	4.33532
$471$	1.89451	0.0872944
$472$	−14.4971	−0.667281
$473$	0.242808	0.0111643
$474$	−0.569505	−0.0261583
$475$	−50.9978	−2.33994
$476$	52.5828	2.41013
$477$	22.0091	1.00773
$478$	−8.24185	−0.376973
$479$	17.2807	0.789574	0.394787	−	0.918773i	$-0.370818\pi$
$479$	17.2807	0.789574	0.394787	+	0.918773i	$0.370818\pi$
$480$	5.43887	0.248249
$481$	−9.15506	−0.417435
$482$	29.5119	1.34423
$483$	−0.0158433	−0.000720894 0
$484$	77.4159	3.51891
$485$	19.8193	0.899946
$486$	−9.71059	−0.440481
$487$	−4.91487	−0.222714	−0.111357	−	0.993780i	$-0.535520\pi$
$487$	−4.91487	−0.222714	−0.111357	+	0.993780i	$0.535520\pi$
$488$	2.71924	0.123094
$489$	−2.55913	−0.115728
$490$	−14.1248	−0.638095
$491$	−37.9116	−1.71093	−0.855464	−	0.517862i	$-0.826728\pi$
$491$	−37.9116	−1.71093	−0.855464	+	0.517862i	$0.826728\pi$
$492$	−1.34071	−0.0604437
$493$	−56.7355	−2.55524
$494$	−8.89738	−0.400312
$495$	−78.5510	−3.53061
$496$	−17.8227	−0.800262
$497$	45.4045	2.03667
$498$	3.86154	0.173040
$499$	−7.44343	−0.333214	−0.166607	−	0.986023i	$-0.553281\pi$
$499$	−7.44343	−0.333214	−0.166607	+	0.986023i	$0.553281\pi$
$500$	81.0088	3.62283
$501$	0.381634	0.0170502
$502$	23.7735	1.06107
$503$	10.1886	0.454286	0.227143	−	0.973861i	$-0.427062\pi$
$503$	10.1886	0.454286	0.227143	+	0.973861i	$0.427062\pi$
$504$	12.0896	0.538513
$505$	−12.2016	−0.542966
$506$	0.437839	0.0194643
$507$	0.169019	0.00750642
$508$	57.5407	2.55296
$509$	−22.5978	−1.00163	−0.500816	−	0.865554i	$-0.666967\pi$
$509$	−22.5978	−1.00163	−0.500816	+	0.865554i	$0.666967\pi$
$510$	−10.3036	−0.456252
$511$	−30.1128	−1.33211
$512$	−24.1043	−1.06527
$513$	4.16674	0.183966
$514$	−4.73719	−0.208948
$515$	41.2398	1.81724
$516$	0.0171035	0.000752939 0
$517$	−66.4357	−2.92184
$518$	−57.7719	−2.53835
$519$	−1.52947	−0.0671363
$520$	5.78839	0.253838
$521$	22.7998	0.998877	0.499439	−	0.866349i	$-0.333540\pi$
$521$	22.7998	0.998877	0.499439	+	0.866349i	$0.333540\pi$
$522$	−53.5077	−2.34197
$523$	−9.94592	−0.434905	−0.217452	−	0.976071i	$-0.569775\pi$
$523$	−9.94592	−0.434905	−0.217452	+	0.976071i	$0.569775\pi$
$524$	34.1620	1.49237
$525$	6.11338	0.266810
$526$	−42.2617	−1.84270
$527$	52.7367	2.29725
$528$	−2.46140	−0.107119
$529$	−22.9990	−0.999955
$530$	66.4972	2.88845
$531$	−31.0009	−1.34532
$532$	−31.9698	−1.38607
$533$	2.99925	0.129912
$534$	3.81511	0.165096
$535$	−62.0748	−2.68373
$536$	3.13197	0.135280
$537$	0.261603	0.0112890
$538$	61.5142	2.65207
$539$	9.98423	0.430051
$540$	−11.1195	−0.478508
$541$	−20.2854	−0.872139	−0.436070	−	0.899913i	$-0.643630\pi$
$541$	−20.2854	−0.872139	−0.436070	+	0.899913i	$0.643630\pi$
$542$	−54.2356	−2.32962
$543$	1.06516	0.0457102
$544$	52.4530	2.24890
$545$	21.7516	0.931737
$546$	1.06658	0.0456453
$547$	−12.5886	−0.538251	−0.269126	−	0.963105i	$-0.586735\pi$
$547$	−12.5886	−0.538251	−0.269126	+	0.963105i	$0.586735\pi$
$548$	11.8567	0.506494
$549$	5.81490	0.248174
$550$	−168.947	−7.20393
$551$	34.4946	1.46952
$552$	0.00751868	0.000320016 0
$553$	4.57777	0.194667
$554$	−32.3277	−1.37347
$555$	6.44591	0.273614
$556$	−16.4068	−0.695805
$557$	−14.2195	−0.602500	−0.301250	−	0.953545i	$-0.597404\pi$
$557$	−14.2195	−0.602500	−0.301250	+	0.953545i	$0.597404\pi$
$558$	49.7364	2.10551
$559$	−0.0382617	−0.00161830
$560$	−27.9902	−1.18280
$561$	7.28318	0.307496
$562$	23.8146	1.00456
$563$	−17.3167	−0.729811	−0.364905	−	0.931045i	$-0.618899\pi$
$563$	−17.3167	−0.729811	−0.364905	+	0.931045i	$0.618899\pi$
$564$	−4.67975	−0.197053
$565$	49.1259	2.06674
$566$	−16.5240	−0.694557
$567$	25.6018	1.07517
$568$	−21.5475	−0.904111
$569$	14.1437	0.592935	0.296468	−	0.955043i	$-0.404191\pi$
$569$	14.1437	0.592935	0.296468	+	0.955043i	$0.404191\pi$
$570$	6.26449	0.262391
$571$	−8.10461	−0.339167	−0.169584	−	0.985516i	$-0.554242\pi$
$571$	−8.10461	−0.339167	−0.169584	+	0.985516i	$0.554242\pi$
$572$	−16.7835	−0.701755
$573$	0.844709	0.0352882
$574$	18.9264	0.789972
$575$	−0.395462	−0.0164919
$576$	35.8312	1.49297
$577$	−0.452686	−0.0188456	−0.00942278	−	0.999956i	$-0.502999\pi$
$577$	−0.452686	−0.0188456	−0.00942278	+	0.999956i	$0.502999\pi$
$578$	−62.7312	−2.60927
$579$	−3.51848	−0.146223
$580$	−92.0535	−3.82232
$581$	−31.0396	−1.28774
$582$	−1.73308	−0.0718385
$583$	−47.0040	−1.94671
$584$	14.2905	0.591346
$585$	12.3781	0.511770
$586$	15.4809	0.639509
$587$	−3.81843	−0.157603	−0.0788017	−	0.996890i	$-0.525109\pi$
$587$	−3.81843	−0.157603	−0.0788017	+	0.996890i	$0.525109\pi$
$588$	0.703293	0.0290033
$589$	−32.0633	−1.32115
$590$	−93.6648	−3.85612
$591$	0.453254	0.0186444
$592$	−21.0091	−0.863467
$593$	34.0670	1.39897	0.699483	−	0.714649i	$-0.253414\pi$
$593$	34.0670	1.39897	0.699483	+	0.714649i	$0.253414\pi$
$594$	13.8037	0.566373
$595$	82.8221	3.39537
$596$	38.3434	1.57061
$597$	1.32771	0.0543397
$598$	−0.0689946	−0.00282140
$599$	−47.0108	−1.92081	−0.960405	−	0.278607i	$-0.910127\pi$
$599$	−47.0108	−1.92081	−0.960405	+	0.278607i	$0.910127\pi$
$600$	−2.90121	−0.118441
$601$	−28.1162	−1.14689	−0.573443	−	0.819246i	$-0.694392\pi$
$601$	−28.1162	−1.14689	−0.573443	+	0.819246i	$0.694392\pi$
$602$	−0.241446	−0.00984059
$603$	6.69749	0.272743
$604$	−49.8198	−2.02714
$605$	121.936	4.95741
$606$	1.06696	0.0433424
$607$	−7.03611	−0.285587	−0.142793	−	0.989753i	$-0.545608\pi$
$607$	−7.03611	−0.285587	−0.142793	+	0.989753i	$0.545608\pi$
$608$	−31.8909	−1.29335
$609$	−4.13505	−0.167561
$610$	17.5689	0.711343
$611$	10.4689	0.423527
$612$	−53.3624	−2.15705
$613$	35.5973	1.43776	0.718880	−	0.695134i	$-0.244655\pi$
$613$	35.5973	1.43776	0.718880	+	0.695134i	$0.244655\pi$
$614$	45.7848	1.84773
$615$	−2.11172	−0.0851526
$616$	−25.8193	−1.04029
$617$	37.6717	1.51660	0.758302	−	0.651903i	$-0.226029\pi$
$617$	37.6717	1.51660	0.758302	+	0.651903i	$0.226029\pi$
$618$	−3.60618	−0.145062
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	85.5655	3.43639
$621$	0.0323109	0.00129659
$622$	−34.0288	−1.36443
$623$	−30.6665	−1.22863
$624$	0.387867	0.0155271
$625$	65.8306	2.63322
$626$	17.0301	0.680661
$627$	−4.42810	−0.176841
$628$	29.6445	1.18295
$629$	62.1650	2.47868
$630$	78.1102	3.11199
$631$	10.3351	0.411434	0.205717	−	0.978612i	$-0.434047\pi$
$631$	10.3351	0.411434	0.205717	+	0.978612i	$0.434047\pi$
$632$	−2.17246	−0.0864157
$633$	2.30404	0.0915772
$634$	33.4612	1.32891
$635$	90.6312	3.59659
$636$	−3.31098	−0.131289
$637$	−1.57331	−0.0623370
$638$	114.275	4.52418
$639$	−46.0777	−1.82281
$640$	43.9009	1.73533
$641$	−20.3865	−0.805217	−0.402609	−	0.915372i	$-0.631896\pi$
$641$	−20.3865	−0.805217	−0.402609	+	0.915372i	$0.631896\pi$
$642$	5.42808	0.214229
$643$	2.89549	0.114187	0.0570934	−	0.998369i	$-0.481817\pi$
$643$	2.89549	0.114187	0.0570934	+	0.998369i	$0.481817\pi$
$644$	−0.247909	−0.00976899
$645$	0.0269394	0.00106074
$646$	60.4153	2.37701
$647$	48.7025	1.91469	0.957346	−	0.288944i	$-0.0933040\pi$
$647$	48.7025	1.91469	0.957346	+	0.288944i	$0.0933040\pi$
$648$	−12.1497	−0.477287
$649$	66.2076	2.59888
$650$	26.6227	1.04423
$651$	3.84361	0.150643
$652$	−40.0443	−1.56826
$653$	−2.91064	−0.113902	−0.0569511	−	0.998377i	$-0.518138\pi$
$653$	−2.91064	−0.113902	−0.0569511	+	0.998377i	$0.518138\pi$
$654$	−1.90205	−0.0743761
$655$	53.8079	2.10245
$656$	6.88268	0.268724
$657$	30.5593	1.19223
$658$	66.0629	2.57540
$659$	35.2559	1.37337	0.686687	−	0.726953i	$-0.259064\pi$
$659$	35.2559	1.37337	0.686687	+	0.726953i	$0.259064\pi$
$660$	11.8170	0.459976
$661$	−19.2147	−0.747365	−0.373682	−	0.927557i	$-0.621905\pi$
$661$	−19.2147	−0.747365	−0.373682	+	0.927557i	$0.621905\pi$
$662$	−9.67077	−0.375865
$663$	−1.14768	−0.0445723
$664$	14.7304	0.571649
$665$	−50.3549	−1.95268
$666$	58.6284	2.27180
$667$	0.267488	0.0103572
$668$	5.97166	0.231050
$669$	0.305182	0.0117990
$670$	20.2355	0.781765
$671$	−12.4187	−0.479418
$672$	3.82293	0.147473
$673$	36.9862	1.42572	0.712858	−	0.701309i	$-0.247401\pi$
$673$	36.9862	1.42572	0.712858	+	0.701309i	$0.247401\pi$
$674$	32.0941	1.23622
$675$	−12.4677	−0.479881
$676$	2.64475	0.101721
$677$	−0.985877	−0.0378903	−0.0189452	−	0.999821i	$-0.506031\pi$
$677$	−0.985877	−0.0378903	−0.0189452	+	0.999821i	$0.506031\pi$
$678$	−4.29578	−0.164978
$679$	13.9308	0.534614
$680$	−39.3045	−1.50726
$681$	0.364224	0.0139571
$682$	−106.220	−4.06739
$683$	5.31096	0.203218	0.101609	−	0.994824i	$-0.467601\pi$
$683$	5.31096	0.203218	0.101609	+	0.994824i	$0.467601\pi$
$684$	32.4438	1.24052
$685$	18.6753	0.713546
$686$	34.2444	1.30746
$687$	3.43320	0.130985
$688$	−0.0878030	−0.00334746
$689$	7.40689	0.282180
$690$	0.0485779	0.00184933
$691$	−13.5125	−0.514041	−0.257021	−	0.966406i	$-0.582741\pi$
$691$	−13.5125	−0.514041	−0.257021	+	0.966406i	$0.582741\pi$
$692$	−23.9325	−0.909778
$693$	−55.2128	−2.09736
$694$	20.3372	0.771989
$695$	−25.8421	−0.980246
$696$	1.96235	0.0743829
$697$	−20.3656	−0.771402
$698$	6.19997	0.234672
$699$	1.81300	0.0685739
$700$	95.6598	3.61560
$701$	45.5785	1.72148	0.860739	−	0.509047i	$-0.170002\pi$
$701$	45.5785	1.72148	0.860739	+	0.509047i	$0.170002\pi$
$702$	−2.17519	−0.0820971
$703$	−37.7957	−1.42549
$704$	−76.5235	−2.88409
$705$	−7.37098	−0.277607
$706$	33.3786	1.25622
$707$	−8.57642	−0.322550
$708$	4.66369	0.175272
$709$	36.6410	1.37608	0.688041	−	0.725672i	$-0.258471\pi$
$709$	36.6410	1.37608	0.688041	+	0.725672i	$0.258471\pi$
$710$	−139.217	−5.22472
$711$	−4.64564	−0.174225
$712$	14.5533	0.545407
$713$	−0.248635	−0.00931144
$714$	−7.24231	−0.271037
$715$	−26.4354	−0.988628
$716$	4.09345	0.152980
$717$	0.646369	0.0241391
$718$	−36.2899	−1.35433
$719$	25.2331	0.941035	0.470518	−	0.882391i	$-0.344067\pi$
$719$	25.2331	0.941035	0.470518	+	0.882391i	$0.344067\pi$
$720$	28.4052	1.05860
$721$	28.9870	1.07953
$722$	4.21630	0.156914
$723$	−2.31448	−0.0860763
$724$	16.6671	0.619429
$725$	−103.214	−3.83329
$726$	−10.6626	−0.395727
$727$	−14.6409	−0.543002	−0.271501	−	0.962438i	$-0.587520\pi$
$727$	−14.6409	−0.543002	−0.271501	+	0.962438i	$0.587520\pi$
$728$	4.06860	0.150793
$729$	−25.4696	−0.943317
$730$	92.3304	3.41730
$731$	0.259806	0.00960926
$732$	−0.874778	−0.0323327
$733$	9.95818	0.367814	0.183907	−	0.982944i	$-0.441126\pi$
$733$	9.95818	0.367814	0.183907	+	0.982944i	$0.441126\pi$
$734$	26.2426	0.968632
$735$	1.10774	0.0408597
$736$	−0.247297	−0.00911550
$737$	−14.3036	−0.526880
$738$	−19.2070	−0.707019
$739$	−28.8277	−1.06044	−0.530221	−	0.847859i	$-0.677891\pi$
$739$	−28.8277	−1.06044	−0.530221	+	0.847859i	$0.677891\pi$
$740$	100.863	3.70780
$741$	0.697779	0.0256335
$742$	46.7403	1.71589
$743$	−4.95783	−0.181885	−0.0909425	−	0.995856i	$-0.528988\pi$
$743$	−4.95783	−0.181885	−0.0909425	+	0.995856i	$0.528988\pi$
$744$	−1.82404	−0.0668727
$745$	60.3940	2.21266
$746$	−3.12051	−0.114250
$747$	31.4998	1.15252
$748$	113.964	4.16695
$749$	−43.6318	−1.59427
$750$	−11.1575	−0.407413
$751$	52.7986	1.92665	0.963324	−	0.268339i	$-0.0864748\pi$
$751$	52.7986	1.92665	0.963324	+	0.268339i	$0.0864748\pi$
$752$	24.0241	0.876069
$753$	−1.86445	−0.0679442
$754$	−18.0074	−0.655790
$755$	−78.4701	−2.85582
$756$	−7.81581	−0.284258
$757$	18.7267	0.680635	0.340318	−	0.940311i	$-0.389465\pi$
$757$	18.7267	0.680635	0.340318	+	0.940311i	$0.389465\pi$
$758$	47.3691	1.72052
$759$	−0.0343376	−0.00124638
$760$	23.8968	0.866826
$761$	48.3711	1.75345	0.876725	−	0.480991i	$-0.159723\pi$
$761$	48.3711	1.75345	0.876725	+	0.480991i	$0.159723\pi$
$762$	−7.92518	−0.287099
$763$	15.2890	0.553499
$764$	13.2177	0.478198
$765$	−84.0500	−3.03883
$766$	−27.4530	−0.991917
$767$	−10.4330	−0.376713
$768$	0.237385	0.00856588
$769$	−32.0512	−1.15580	−0.577898	−	0.816109i	$-0.696127\pi$
$769$	−32.0512	−1.15580	−0.577898	+	0.816109i	$0.696127\pi$
$770$	−166.817	−6.01168
$771$	0.371515	0.0133798
$772$	−55.0557	−1.98150
$773$	38.4945	1.38455	0.692275	−	0.721634i	$-0.256609\pi$
$773$	38.4945	1.38455	0.692275	+	0.721634i	$0.256609\pi$
$774$	0.245025	0.00880725
$775$	95.9397	3.44625
$776$	−6.61107	−0.237324
$777$	4.53077	0.162541
$778$	74.0464	2.65469
$779$	12.3821	0.443634
$780$	−1.86212	−0.0666746
$781$	98.4066	3.52126
$782$	0.468490	0.0167532
$783$	8.43305	0.301373
$784$	−3.61045	−0.128945
$785$	46.6925	1.66653
$786$	−4.70519	−0.167828
$787$	46.4709	1.65651	0.828254	−	0.560353i	$-0.189334\pi$
$787$	46.4709	1.65651	0.828254	+	0.560353i	$0.189334\pi$
$788$	7.09234	0.252654
$789$	3.31438	0.117995
$790$	−14.0361	−0.499384
$791$	34.5301	1.22775
$792$	26.2021	0.931051
$793$	1.95694	0.0694928
$794$	21.9036	0.777331
$795$	−5.21506	−0.184959
$796$	20.7755	0.736369
$797$	12.8595	0.455508	0.227754	−	0.973719i	$-0.426862\pi$
$797$	12.8595	0.455508	0.227754	+	0.973719i	$0.426862\pi$
$798$	4.40325	0.155873
$799$	−71.0865	−2.51486
$800$	95.4237	3.37374
$801$	31.1211	1.09961
$802$	4.57568	0.161573
$803$	−65.2644	−2.30313
$804$	−1.00755	−0.0355336
$805$	−0.390477	−0.0137625
$806$	16.7382	0.589578
$807$	−4.82427	−0.169822
$808$	4.07008	0.143185
$809$	1.58649	0.0557779	0.0278889	−	0.999611i	$-0.491122\pi$
$809$	1.58649	0.0557779	0.0278889	+	0.999611i	$0.491122\pi$
$810$	−78.4989	−2.75817
$811$	−13.1238	−0.460840	−0.230420	−	0.973091i	$-0.574010\pi$
$811$	−13.1238	−0.460840	−0.230420	+	0.973091i	$0.574010\pi$
$812$	−64.7036	−2.27065
$813$	4.25344	0.149175
$814$	−125.211	−4.38863
$815$	−63.0729	−2.20935
$816$	−2.63371	−0.0921982
$817$	−0.157959	−0.00552629
$818$	2.98041	0.104208
$819$	8.70042	0.304017
$820$	−33.0433	−1.15392
$821$	−20.9815	−0.732261	−0.366131	−	0.930563i	$-0.619318\pi$
$821$	−20.9815	−0.732261	−0.366131	+	0.930563i	$0.619318\pi$
$822$	−1.63305	−0.0569590
$823$	−11.5632	−0.403069	−0.201534	−	0.979481i	$-0.564593\pi$
$823$	−11.5632	−0.403069	−0.201534	+	0.979481i	$0.564593\pi$
$824$	−13.7563	−0.479222
$825$	13.2497	0.461296
$826$	−65.8361	−2.29073
$827$	−46.9785	−1.63360	−0.816801	−	0.576919i	$-0.804255\pi$
$827$	−46.9785	−1.63360	−0.816801	+	0.576919i	$0.804255\pi$
$828$	0.251585	0.00874317
$829$	−20.2720	−0.704077	−0.352038	−	0.935986i	$-0.614511\pi$
$829$	−20.2720	−0.704077	−0.352038	+	0.935986i	$0.614511\pi$
$830$	95.1722	3.30348
$831$	2.53531	0.0879489
$832$	12.0586	0.418055
$833$	10.6832	0.370150
$834$	2.25974	0.0782484
$835$	9.40584	0.325502
$836$	−69.2891	−2.39641
$837$	−7.83868	−0.270944
$838$	26.7778	0.925023
$839$	17.1132	0.590815	0.295407	−	0.955371i	$-0.404545\pi$
$839$	17.1132	0.590815	0.295407	+	0.955371i	$0.404545\pi$
$840$	−2.86463	−0.0988392
$841$	40.8135	1.40736
$842$	53.8629	1.85624
$843$	−1.86766	−0.0643257
$844$	36.0526	1.24098
$845$	4.16569	0.143304
$846$	−67.0423	−2.30496
$847$	85.7078	2.94496
$848$	16.9974	0.583691
$849$	1.29590	0.0444752
$850$	−180.774	−6.20051
$851$	−0.293086	−0.0100469
$852$	6.93180	0.237479
$853$	33.7274	1.15481	0.577403	−	0.816459i	$-0.304066\pi$
$853$	33.7274	1.15481	0.577403	+	0.816459i	$0.304066\pi$
$854$	12.3490	0.422574
$855$	51.1015	1.74763
$856$	20.7062	0.707722
$857$	−7.81831	−0.267068	−0.133534	−	0.991044i	$-0.542633\pi$
$857$	−7.81831	−0.267068	−0.133534	+	0.991044i	$0.542633\pi$
$858$	2.31162	0.0789175
$859$	−0.297274	−0.0101429	−0.00507143	−	0.999987i	$-0.501614\pi$
$859$	−0.297274	−0.0101429	−0.00507143	+	0.999987i	$0.501614\pi$
$860$	0.421536	0.0143743
$861$	−1.48431	−0.0505850
$862$	8.02312	0.273269
$863$	26.8907	0.915368	0.457684	−	0.889115i	$-0.348679\pi$
$863$	26.8907	0.915368	0.457684	+	0.889115i	$0.348679\pi$
$864$	−7.79652	−0.265243
$865$	−37.6956	−1.28169
$866$	8.08627	0.274783
$867$	4.91971	0.167082
$868$	60.1432	2.04139
$869$	9.92155	0.336565
$870$	12.6787	0.429848
$871$	2.25396	0.0763725
$872$	−7.25564	−0.245707
$873$	−14.1373	−0.478475
$874$	−0.284837	−0.00963475
$875$	89.6855	3.03192
$876$	−4.59725	−0.155327
$877$	−28.2458	−0.953792	−0.476896	−	0.878960i	$-0.658238\pi$
$877$	−28.2458	−0.953792	−0.476896	+	0.878960i	$0.658238\pi$
$878$	74.8640	2.52654
$879$	−1.21409	−0.0409503
$880$	−60.6641	−2.04499
$881$	−42.1010	−1.41842	−0.709209	−	0.704998i	$-0.750948\pi$
$881$	−42.1010	−1.41842	−0.709209	+	0.704998i	$0.750948\pi$
$882$	10.0754	0.339256
$883$	−19.6968	−0.662850	−0.331425	−	0.943482i	$-0.607529\pi$
$883$	−19.6968	−0.662850	−0.331425	+	0.943482i	$0.607529\pi$
$884$	−17.9585	−0.604009
$885$	7.34568	0.246922
$886$	80.4394	2.70241
$887$	−48.4582	−1.62707	−0.813534	−	0.581517i	$-0.802459\pi$
$887$	−48.4582	−1.62707	−0.813534	+	0.581517i	$0.802459\pi$
$888$	−2.15015	−0.0721544
$889$	63.7038	2.13656
$890$	94.0280	3.15183
$891$	55.4875	1.85890
$892$	4.77536	0.159891
$893$	43.2198	1.44630
$894$	−5.28110	−0.176627
$895$	6.44751	0.215517
$896$	30.8575	1.03088
$897$	0.00541092	0.000180665 0
$898$	−41.8645	−1.39704
$899$	−64.8929	−2.16430
$900$	−97.0780	−3.23593
$901$	−50.2945	−1.67555
$902$	41.0197	1.36581
$903$	0.0189354	0.000630131 0
$904$	−16.3868	−0.545018
$905$	26.2521	0.872648
$906$	6.86176	0.227967
$907$	20.8881	0.693578	0.346789	−	0.937943i	$-0.387272\pi$
$907$	20.8881	0.693578	0.346789	+	0.937943i	$0.387272\pi$
$908$	5.69922	0.189135
$909$	8.70358	0.288679
$910$	26.2871	0.871408
$911$	−58.9994	−1.95474	−0.977369	−	0.211541i	$-0.932152\pi$
$911$	−58.9994	−1.95474	−0.977369	+	0.211541i	$0.932152\pi$
$912$	1.60127	0.0530232
$913$	−67.2731	−2.22642
$914$	−3.59460	−0.118899
$915$	−1.37784	−0.0455501
$916$	53.7213	1.77500
$917$	37.8210	1.24896
$918$	14.7700	0.487484
$919$	12.7014	0.418979	0.209490	−	0.977811i	$-0.432820\pi$
$919$	12.7014	0.418979	0.209490	+	0.977811i	$0.432820\pi$
$920$	0.185307	0.00610939
$921$	−3.59068	−0.118317
$922$	4.38750	0.144495
$923$	−15.5069	−0.510416
$924$	8.30605	0.273249
$925$	113.092	3.71844
$926$	0.438886	0.0144227
$927$	−29.4168	−0.966175
$928$	−64.5439	−2.11876
$929$	52.9091	1.73589	0.867945	−	0.496660i	$-0.165441\pi$
$929$	52.9091	1.73589	0.867945	+	0.496660i	$0.165441\pi$
$930$	−11.7851	−0.386448
$931$	−6.49526	−0.212874
$932$	28.3691	0.929260
$933$	2.66871	0.0873697
$934$	−76.0171	−2.48736
$935$	179.503	5.87037
$936$	−4.12892	−0.134958
$937$	−5.23113	−0.170894	−0.0854468	−	0.996343i	$-0.527232\pi$
$937$	−5.23113	−0.170894	−0.0854468	+	0.996343i	$0.527232\pi$
$938$	14.2233	0.464408
$939$	−1.33559	−0.0435854
$940$	−115.338	−3.76191
$941$	−8.16837	−0.266281	−0.133141	−	0.991097i	$-0.542506\pi$
$941$	−8.16837	−0.266281	−0.133141	+	0.991097i	$0.542506\pi$
$942$	−4.08299	−0.133031
$943$	0.0960166	0.00312673
$944$	−23.9417	−0.779234
$945$	−12.3105	−0.400461
$946$	−0.523292	−0.0170137
$947$	−5.15266	−0.167439	−0.0837194	−	0.996489i	$-0.526680\pi$
$947$	−5.15266	−0.167439	−0.0837194	+	0.996489i	$0.526680\pi$
$948$	0.698878	0.0226985
$949$	10.2844	0.333844
$950$	109.909	3.56591
$951$	−2.62420	−0.0850956
$952$	−27.6268	−0.895390
$953$	50.5105	1.63620	0.818098	−	0.575079i	$-0.195029\pi$
$953$	50.5105	1.63620	0.818098	+	0.575079i	$0.195029\pi$
$954$	−47.4332	−1.53571
$955$	20.8189	0.673683
$956$	10.1141	0.327114
$957$	−8.96201	−0.289701
$958$	−37.2427	−1.20326
$959$	13.1267	0.423882
$960$	−8.49022	−0.274021
$961$	29.3192	0.945779
$962$	19.7307	0.636143
$963$	44.2787	1.42686
$964$	−36.2160	−1.16644
$965$	−86.7170	−2.79152
$966$	0.0341449	0.00109860
$967$	19.6672	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$967$	19.6672	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$968$	−40.6740	−1.30731
$969$	−4.73809	−0.152209
$970$	−42.7138	−1.37146
$971$	−37.1809	−1.19319	−0.596596	−	0.802542i	$-0.703480\pi$
$971$	−37.1809	−1.19319	−0.596596	+	0.802542i	$0.703480\pi$
$972$	11.9165	0.382222
$973$	−18.1641	−0.582316
$974$	10.5924	0.339402
$975$	−2.08789	−0.0668660
$976$	4.49078	0.143746
$977$	−21.0197	−0.672480	−0.336240	−	0.941776i	$-0.609155\pi$
$977$	−21.0197	−0.672480	−0.336240	+	0.941776i	$0.609155\pi$
$978$	5.51536	0.176362
$979$	−66.4644	−2.12421
$980$	17.3335	0.553698
$981$	−15.5157	−0.495377
$982$	81.7059	2.60734
$983$	−18.3302	−0.584643	−0.292321	−	0.956320i	$-0.594428\pi$
$983$	−18.3302	−0.584643	−0.292321	+	0.956320i	$0.594428\pi$
$984$	0.704401	0.0224555
$985$	11.1710	0.355937
$986$	122.274	3.89401
$987$	−5.18099	−0.164913
$988$	10.9186	0.347366
$989$	−0.00122489	−3.89493e−5 0
$990$	169.291	5.38041
$991$	−45.2664	−1.43793	−0.718967	−	0.695044i	$-0.755385\pi$
$991$	−45.2664	−1.43793	−0.718967	+	0.695044i	$0.755385\pi$
$992$	59.9947	1.90483
$993$	0.758432	0.0240681
$994$	−97.8544	−3.10375
$995$	32.7231	1.03739
$996$	−4.73875	−0.150153
$997$	−31.9508	−1.01189	−0.505946	−	0.862565i	$-0.668856\pi$
$997$	−31.9508	−1.01189	−0.505946	+	0.862565i	$0.668856\pi$
$998$	16.0418	0.507796
$999$	−9.24010	−0.292344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.20	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.20	✓	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)

\(f(q)\)	\(=\)	\(q-2.15517 q^{2} +0.169019 q^{3} +2.64475 q^{4} +4.16569 q^{5} -0.364265 q^{6} +2.92802 q^{7} -1.38954 q^{8} -2.97143 q^{9} +O(q^{10})\)	\(q-2.15517 q^{2} +0.169019 q^{3} +2.64475 q^{4} +4.16569 q^{5} -0.364265 q^{6} +2.92802 q^{7} -1.38954 q^{8} -2.97143 q^{9} -8.97775 q^{10} +6.34599 q^{11} +0.447014 q^{12} -1.00000 q^{13} -6.31038 q^{14} +0.704082 q^{15} -2.29480 q^{16} +6.79024 q^{17} +6.40394 q^{18} -4.12839 q^{19} +11.0172 q^{20} +0.494893 q^{21} -13.6767 q^{22} -0.0320136 q^{23} -0.234859 q^{24} +12.3529 q^{25} +2.15517 q^{26} -1.00929 q^{27} +7.74388 q^{28} -8.35544 q^{29} -1.51742 q^{30} +7.76654 q^{31} +7.72477 q^{32} +1.07260 q^{33} -14.6341 q^{34} +12.1972 q^{35} -7.85869 q^{36} +9.15506 q^{37} +8.89738 q^{38} -0.169019 q^{39} -5.78839 q^{40} -2.99925 q^{41} -1.06658 q^{42} +0.0382617 q^{43} +16.7835 q^{44} -12.3781 q^{45} +0.0689946 q^{46} -10.4689 q^{47} -0.387867 q^{48} +1.57331 q^{49} -26.6227 q^{50} +1.14768 q^{51} -2.64475 q^{52} -7.40689 q^{53} +2.17519 q^{54} +26.4354 q^{55} -4.06860 q^{56} -0.697779 q^{57} +18.0074 q^{58} +10.4330 q^{59} +1.86212 q^{60} -1.95694 q^{61} -16.7382 q^{62} -8.70042 q^{63} -12.0586 q^{64} -4.16569 q^{65} -2.31162 q^{66} -2.25396 q^{67} +17.9585 q^{68} -0.00541092 q^{69} -26.2871 q^{70} +15.5069 q^{71} +4.12892 q^{72} -10.2844 q^{73} -19.7307 q^{74} +2.08789 q^{75} -10.9186 q^{76} +18.5812 q^{77} +0.364265 q^{78} +1.56344 q^{79} -9.55943 q^{80} +8.74371 q^{81} +6.46388 q^{82} -10.6009 q^{83} +1.30887 q^{84} +28.2860 q^{85} -0.0824603 q^{86} -1.41223 q^{87} -8.81801 q^{88} -10.4734 q^{89} +26.6768 q^{90} -2.92802 q^{91} -0.0846678 q^{92} +1.31270 q^{93} +22.5623 q^{94} -17.1976 q^{95} +1.30564 q^{96} +4.75774 q^{97} -3.39076 q^{98} -18.8567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 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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