LMFDB - Embedded newform 8047.2.a.b.1.17

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			1.17
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)

$f(q)$	$=$	$q-2.27884 q^{2} -1.20174 q^{3} +3.19313 q^{4} -1.58295 q^{5} +2.73859 q^{6} +1.25281 q^{7} -2.71895 q^{8} -1.55581 q^{9} +O(q^{10})$	\(q-2.27884 q^{2} -1.20174 q^{3} +3.19313 q^{4} -1.58295 q^{5} +2.73859 q^{6} +1.25281 q^{7} -2.71895 q^{8} -1.55581 q^{9} +3.60730 q^{10} -5.51479 q^{11} -3.83732 q^{12} +1.00000 q^{13} -2.85495 q^{14} +1.90230 q^{15} -0.190201 q^{16} -0.404259 q^{17} +3.54545 q^{18} -2.11797 q^{19} -5.05456 q^{20} -1.50555 q^{21} +12.5673 q^{22} +1.39164 q^{23} +3.26748 q^{24} -2.49426 q^{25} -2.27884 q^{26} +5.47492 q^{27} +4.00036 q^{28} -6.67488 q^{29} -4.33505 q^{30} +1.02871 q^{31} +5.87133 q^{32} +6.62737 q^{33} +0.921243 q^{34} -1.98313 q^{35} -4.96789 q^{36} -2.15141 q^{37} +4.82652 q^{38} -1.20174 q^{39} +4.30396 q^{40} -0.133329 q^{41} +3.43092 q^{42} +4.96077 q^{43} -17.6094 q^{44} +2.46277 q^{45} -3.17133 q^{46} +10.2848 q^{47} +0.228573 q^{48} -5.43048 q^{49} +5.68404 q^{50} +0.485817 q^{51} +3.19313 q^{52} +12.7026 q^{53} -12.4765 q^{54} +8.72965 q^{55} -3.40631 q^{56} +2.54526 q^{57} +15.2110 q^{58} -7.36247 q^{59} +6.07429 q^{60} +4.42081 q^{61} -2.34426 q^{62} -1.94913 q^{63} -12.9994 q^{64} -1.58295 q^{65} -15.1027 q^{66} +6.49853 q^{67} -1.29085 q^{68} -1.67240 q^{69} +4.51924 q^{70} -9.06311 q^{71} +4.23016 q^{72} +6.63616 q^{73} +4.90273 q^{74} +2.99747 q^{75} -6.76294 q^{76} -6.90896 q^{77} +2.73859 q^{78} +0.656280 q^{79} +0.301079 q^{80} -1.91203 q^{81} +0.303836 q^{82} +3.15024 q^{83} -4.80742 q^{84} +0.639923 q^{85} -11.3048 q^{86} +8.02150 q^{87} +14.9944 q^{88} +9.20804 q^{89} -5.61227 q^{90} +1.25281 q^{91} +4.44368 q^{92} -1.23624 q^{93} -23.4374 q^{94} +3.35264 q^{95} -7.05584 q^{96} +5.02123 q^{97} +12.3752 q^{98} +8.57996 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.27884	−1.61139	−0.805693	−	0.592334i	$-0.798207\pi$
$2$	−2.27884	−1.61139	−0.805693	+	0.592334i	$0.798207\pi$
$3$	−1.20174	−0.693828	−0.346914	−	0.937897i	$-0.612770\pi$
$3$	−1.20174	−0.693828	−0.346914	+	0.937897i	$0.612770\pi$
$4$	3.19313	1.59656
$5$	−1.58295	−0.707917	−0.353959	−	0.935261i	$-0.615165\pi$
$5$	−1.58295	−0.707917	−0.353959	+	0.935261i	$0.615165\pi$
$6$	2.73859	1.11802
$7$	1.25281	0.473516	0.236758	−	0.971569i	$-0.423915\pi$
$7$	1.25281	0.473516	0.236758	+	0.971569i	$0.423915\pi$
$8$	−2.71895	−0.961292
$9$	−1.55581	−0.518603
$10$	3.60730	1.14073
$11$	−5.51479	−1.66277	−0.831386	−	0.555695i	$-0.812452\pi$
$11$	−5.51479	−1.66277	−0.831386	+	0.555695i	$0.812452\pi$
$12$	−3.83732	−1.10774
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.85495	−0.763017
$15$	1.90230	0.491173
$16$	−0.190201	−0.0475503
$17$	−0.404259	−0.0980473	−0.0490236	−	0.998798i	$-0.515611\pi$
$17$	−0.404259	−0.0980473	−0.0490236	+	0.998798i	$0.515611\pi$
$18$	3.54545	0.835669
$19$	−2.11797	−0.485895	−0.242948	−	0.970039i	$-0.578114\pi$
$19$	−2.11797	−0.485895	−0.242948	+	0.970039i	$0.578114\pi$
$20$	−5.05456	−1.13023
$21$	−1.50555	−0.328538
$22$	12.5673	2.67937
$23$	1.39164	0.290177	0.145089	−	0.989419i	$-0.453653\pi$
$23$	1.39164	0.290177	0.145089	+	0.989419i	$0.453653\pi$
$24$	3.26748	0.666971
$25$	−2.49426	−0.498853
$26$	−2.27884	−0.446918
$27$	5.47492	1.05365
$28$	4.00036	0.755998
$29$	−6.67488	−1.23949	−0.619747	−	0.784802i	$-0.712765\pi$
$29$	−6.67488	−1.23949	−0.619747	+	0.784802i	$0.712765\pi$
$30$	−4.33505	−0.791469
$31$	1.02871	0.184761	0.0923807	−	0.995724i	$-0.470552\pi$
$31$	1.02871	0.184761	0.0923807	+	0.995724i	$0.470552\pi$
$32$	5.87133	1.03791
$33$	6.62737	1.15368
$34$	0.921243	0.157992
$35$	−1.98313	−0.335210
$36$	−4.96789	−0.827982
$37$	−2.15141	−0.353690	−0.176845	−	0.984239i	$-0.556589\pi$
$37$	−2.15141	−0.353690	−0.176845	+	0.984239i	$0.556589\pi$
$38$	4.82652	0.782965
$39$	−1.20174	−0.192433
$40$	4.30396	0.680516
$41$	−0.133329	−0.0208225	−0.0104112	−	0.999946i	$-0.503314\pi$
$41$	−0.133329	−0.0208225	−0.0104112	+	0.999946i	$0.503314\pi$
$42$	3.43092	0.529402
$43$	4.96077	0.756510	0.378255	−	0.925701i	$-0.376524\pi$
$43$	4.96077	0.756510	0.378255	+	0.925701i	$0.376524\pi$
$44$	−17.6094	−2.65472
$45$	2.46277	0.367128
$46$	−3.17133	−0.467587
$47$	10.2848	1.50019	0.750096	−	0.661329i	$-0.230007\pi$
$47$	10.2848	1.50019	0.750096	+	0.661329i	$0.230007\pi$
$48$	0.228573	0.0329917
$49$	−5.43048	−0.775783
$50$	5.68404	0.803844
$51$	0.485817	0.0680279
$52$	3.19313	0.442807
$53$	12.7026	1.74484	0.872418	−	0.488760i	$-0.162551\pi$
$53$	12.7026	1.74484	0.872418	+	0.488760i	$0.162551\pi$
$54$	−12.4765	−1.69783
$55$	8.72965	1.17711
$56$	−3.40631	−0.455187
$57$	2.54526	0.337128
$58$	15.2110	1.99730
$59$	−7.36247	−0.958512	−0.479256	−	0.877675i	$-0.659093\pi$
$59$	−7.36247	−0.958512	−0.479256	+	0.877675i	$0.659093\pi$
$60$	6.07429	0.784188
$61$	4.42081	0.566027	0.283014	−	0.959116i	$-0.408666\pi$
$61$	4.42081	0.566027	0.283014	+	0.959116i	$0.408666\pi$
$62$	−2.34426	−0.297722
$63$	−1.94913	−0.245567
$64$	−12.9994	−1.62493
$65$	−1.58295	−0.196341
$66$	−15.1027	−1.85902
$67$	6.49853	0.793922	0.396961	−	0.917836i	$-0.370065\pi$
$67$	6.49853	0.793922	0.396961	+	0.917836i	$0.370065\pi$
$68$	−1.29085	−0.156539
$69$	−1.67240	−0.201333
$70$	4.51924	0.540153
$71$	−9.06311	−1.07559	−0.537797	−	0.843075i	$-0.680743\pi$
$71$	−9.06311	−1.07559	−0.537797	+	0.843075i	$0.680743\pi$
$72$	4.23016	0.498529
$73$	6.63616	0.776704	0.388352	−	0.921511i	$-0.373045\pi$
$73$	6.63616	0.776704	0.388352	+	0.921511i	$0.373045\pi$
$74$	4.90273	0.569930
$75$	2.99747	0.346118
$76$	−6.76294	−0.775762
$77$	−6.90896	−0.787349
$78$	2.73859	0.310084
$79$	0.656280	0.0738373	0.0369187	−	0.999318i	$-0.488246\pi$
$79$	0.656280	0.0738373	0.0369187	+	0.999318i	$0.488246\pi$
$80$	0.301079	0.0336617
$81$	−1.91203	−0.212448
$82$	0.303836	0.0335531
$83$	3.15024	0.345783	0.172892	−	0.984941i	$-0.444689\pi$
$83$	3.15024	0.345783	0.172892	+	0.984941i	$0.444689\pi$
$84$	−4.80742	−0.524532
$85$	0.639923	0.0694094
$86$	−11.3048	−1.21903
$87$	8.02150	0.859996
$88$	14.9944	1.59841
$89$	9.20804	0.976050	0.488025	−	0.872830i	$-0.337717\pi$
$89$	9.20804	0.976050	0.488025	+	0.872830i	$0.337717\pi$
$90$	−5.61227	−0.591585
$91$	1.25281	0.131330
$92$	4.44368	0.463286
$93$	−1.23624	−0.128193
$94$	−23.4374	−2.41739
$95$	3.35264	0.343974
$96$	−7.05584	−0.720134
$97$	5.02123	0.509828	0.254914	−	0.966964i	$-0.417953\pi$
$97$	5.02123	0.509828	0.254914	+	0.966964i	$0.417953\pi$
$98$	12.3752	1.25008
$99$	8.57996	0.862319
$100$	−7.96450	−0.796450
$101$	5.63421	0.560625	0.280313	−	0.959909i	$-0.409562\pi$
$101$	5.63421	0.560625	0.280313	+	0.959909i	$0.409562\pi$
$102$	−1.10710	−0.109619
$103$	13.2046	1.30108	0.650542	−	0.759470i	$-0.274542\pi$
$103$	13.2046	1.30108	0.650542	+	0.759470i	$0.274542\pi$
$104$	−2.71895	−0.266615
$105$	2.38322	0.232578
$106$	−28.9472	−2.81160
$107$	−9.98320	−0.965112	−0.482556	−	0.875865i	$-0.660292\pi$
$107$	−9.98320	−0.965112	−0.482556	+	0.875865i	$0.660292\pi$
$108$	17.4821	1.68222
$109$	6.28771	0.602254	0.301127	−	0.953584i	$-0.402637\pi$
$109$	6.28771	0.602254	0.301127	+	0.953584i	$0.402637\pi$
$110$	−19.8935	−1.89677
$111$	2.58545	0.245400
$112$	−0.238285	−0.0225158
$113$	−18.2093	−1.71299	−0.856493	−	0.516159i	$-0.827361\pi$
$113$	−18.2093	−1.71299	−0.856493	+	0.516159i	$0.827361\pi$
$114$	−5.80024	−0.543243
$115$	−2.20290	−0.205421
$116$	−21.3137	−1.97893
$117$	−1.55581	−0.143835
$118$	16.7779	1.54453
$119$	−0.506458	−0.0464269
$120$	−5.17226	−0.472161
$121$	19.4129	1.76481
$122$	−10.0743	−0.912088
$123$	0.160227	0.0144472
$124$	3.28479	0.294983
$125$	11.8631	1.06106
$126$	4.44175	0.395703
$127$	1.71002	0.151740	0.0758698	−	0.997118i	$-0.475827\pi$
$127$	1.71002	0.151740	0.0758698	+	0.997118i	$0.475827\pi$
$128$	17.8810	1.58047
$129$	−5.96158	−0.524888
$130$	3.60730	0.316381
$131$	−21.6601	−1.89245	−0.946224	−	0.323512i	$-0.895136\pi$
$131$	−21.6601	−1.89245	−0.946224	+	0.323512i	$0.895136\pi$
$132$	21.1620	1.84192
$133$	−2.65340	−0.230079
$134$	−14.8091	−1.27931
$135$	−8.66653	−0.745896
$136$	1.09916	0.0942521
$137$	−2.59661	−0.221844	−0.110922	−	0.993829i	$-0.535380\pi$
$137$	−2.59661	−0.221844	−0.110922	+	0.993829i	$0.535380\pi$
$138$	3.81113	0.324425
$139$	4.16822	0.353544	0.176772	−	0.984252i	$-0.443434\pi$
$139$	4.16822	0.353544	0.176772	+	0.984252i	$0.443434\pi$
$140$	−6.33238	−0.535184
$141$	−12.3597	−1.04087
$142$	20.6534	1.73320
$143$	−5.51479	−0.461170
$144$	0.295917	0.0246597
$145$	10.5660	0.877460
$146$	−15.1228	−1.25157
$147$	6.52605	0.538260
$148$	−6.86972	−0.564688
$149$	1.07574	0.0881284	0.0440642	−	0.999029i	$-0.485969\pi$
$149$	1.07574	0.0881284	0.0440642	+	0.999029i	$0.485969\pi$
$150$	−6.83076	−0.557729
$151$	−3.68162	−0.299606	−0.149803	−	0.988716i	$-0.547864\pi$
$151$	−3.68162	−0.299606	−0.149803	+	0.988716i	$0.547864\pi$
$152$	5.75864	0.467088
$153$	0.628950	0.0508476
$154$	15.7444	1.26872
$155$	−1.62839	−0.130796
$156$	−3.83732	−0.307232
$157$	−4.02538	−0.321261	−0.160630	−	0.987015i	$-0.551353\pi$
$157$	−4.02538	−0.321261	−0.160630	+	0.987015i	$0.551353\pi$
$158$	−1.49556	−0.118980
$159$	−15.2653	−1.21062
$160$	−9.29403	−0.734758
$161$	1.74346	0.137404
$162$	4.35721	0.342335
$163$	7.44504	0.583141	0.291570	−	0.956549i	$-0.405822\pi$
$163$	7.44504	0.583141	0.291570	+	0.956549i	$0.405822\pi$
$164$	−0.425736	−0.0332444
$165$	−10.4908	−0.816708
$166$	−7.17889	−0.557190
$167$	−15.0207	−1.16234	−0.581170	−	0.813783i	$-0.697405\pi$
$167$	−15.0207	−1.16234	−0.581170	+	0.813783i	$0.697405\pi$
$168$	4.09351	0.315822
$169$	1.00000	0.0769231
$170$	−1.45828	−0.111845
$171$	3.29516	0.251987
$172$	15.8404	1.20782
$173$	17.6121	1.33903	0.669513	−	0.742800i	$-0.266503\pi$
$173$	17.6121	1.33903	0.669513	+	0.742800i	$0.266503\pi$
$174$	−18.2797	−1.38578
$175$	−3.12483	−0.236215
$176$	1.04892	0.0790652
$177$	8.84781	0.665042
$178$	−20.9837	−1.57279
$179$	−5.90517	−0.441373	−0.220687	−	0.975345i	$-0.570830\pi$
$179$	−5.90517	−0.441373	−0.220687	+	0.975345i	$0.570830\pi$
$180$	7.86394	0.586143
$181$	24.5488	1.82470	0.912350	−	0.409412i	$-0.134266\pi$
$181$	24.5488	1.82470	0.912350	+	0.409412i	$0.134266\pi$
$182$	−2.85495	−0.211623
$183$	−5.31269	−0.392725
$184$	−3.78380	−0.278945
$185$	3.40558	0.250383
$186$	2.81721	0.206568
$187$	2.22941	0.163030
$188$	32.8406	2.39515
$189$	6.85901	0.498920
$190$	−7.64015	−0.554274
$191$	−23.8207	−1.72361	−0.861803	−	0.507243i	$-0.830665\pi$
$191$	−23.8207	−1.72361	−0.861803	+	0.507243i	$0.830665\pi$
$192$	15.6220	1.12742
$193$	12.6553	0.910950	0.455475	−	0.890249i	$-0.349470\pi$
$193$	12.6553	0.910950	0.455475	+	0.890249i	$0.349470\pi$
$194$	−11.4426	−0.821530
$195$	1.90230	0.136227
$196$	−17.3402	−1.23859
$197$	−23.4933	−1.67383	−0.836913	−	0.547336i	$-0.815642\pi$
$197$	−23.4933	−1.67383	−0.836913	+	0.547336i	$0.815642\pi$
$198$	−19.5524	−1.38953
$199$	−16.1734	−1.14650	−0.573252	−	0.819379i	$-0.694318\pi$
$199$	−16.1734	−1.14650	−0.573252	+	0.819379i	$0.694318\pi$
$200$	6.78177	0.479544
$201$	−7.80958	−0.550845
$202$	−12.8395	−0.903383
$203$	−8.36233	−0.586920
$204$	1.55127	0.108611
$205$	0.211053	0.0147406
$206$	−30.0911	−2.09655
$207$	−2.16513	−0.150487
$208$	−0.190201	−0.0131881
$209$	11.6802	0.807933
$210$	−5.43098	−0.374773
$211$	14.3619	0.988711	0.494355	−	0.869260i	$-0.335404\pi$
$211$	14.3619	0.988711	0.494355	+	0.869260i	$0.335404\pi$
$212$	40.5610	2.78574
$213$	10.8915	0.746277
$214$	22.7501	1.55517
$215$	−7.85266	−0.535547
$216$	−14.8860	−1.01286
$217$	1.28877	0.0874875
$218$	−14.3287	−0.970463
$219$	−7.97497	−0.538899
$220$	27.8749	1.87932
$221$	−0.404259	−0.0271934
$222$	−5.89183	−0.395434
$223$	−13.7175	−0.918592	−0.459296	−	0.888283i	$-0.651898\pi$
$223$	−13.7175	−0.918592	−0.459296	+	0.888283i	$0.651898\pi$
$224$	7.35563	0.491469
$225$	3.88060	0.258707
$226$	41.4961	2.76028
$227$	28.2378	1.87421	0.937104	−	0.349050i	$-0.113496\pi$
$227$	28.2378	1.87421	0.937104	+	0.349050i	$0.113496\pi$
$228$	8.12733	0.538246
$229$	28.8508	1.90652	0.953259	−	0.302155i	$-0.0977059\pi$
$229$	28.8508	1.90652	0.953259	+	0.302155i	$0.0977059\pi$
$230$	5.02006	0.331013
$231$	8.30281	0.546284
$232$	18.1486	1.19152
$233$	−4.88368	−0.319941	−0.159970	−	0.987122i	$-0.551140\pi$
$233$	−4.88368	−0.319941	−0.159970	+	0.987122i	$0.551140\pi$
$234$	3.54545	0.231773
$235$	−16.2803	−1.06201
$236$	−23.5093	−1.53032
$237$	−0.788682	−0.0512304
$238$	1.15414	0.0748117
$239$	2.21674	0.143389	0.0716944	−	0.997427i	$-0.477159\pi$
$239$	2.21674	0.143389	0.0716944	+	0.997427i	$0.477159\pi$
$240$	−0.361820	−0.0233554
$241$	−0.279201	−0.0179849	−0.00899245	−	0.999960i	$-0.502862\pi$
$241$	−0.279201	−0.0179849	−0.00899245	+	0.999960i	$0.502862\pi$
$242$	−44.2390	−2.84379
$243$	−14.1270	−0.906247
$244$	14.1162	0.903698
$245$	8.59618	0.549190
$246$	−0.365133	−0.0232800
$247$	−2.11797	−0.134763
$248$	−2.79700	−0.177610
$249$	−3.78578	−0.239914
$250$	−27.0340	−1.70978
$251$	−20.5887	−1.29954	−0.649772	−	0.760129i	$-0.725136\pi$
$251$	−20.5887	−1.29954	−0.649772	+	0.760129i	$0.725136\pi$
$252$	−6.22380	−0.392063
$253$	−7.67461	−0.482498
$254$	−3.89686	−0.244511
$255$	−0.769024	−0.0481581
$256$	−14.7492	−0.921822
$257$	−15.6804	−0.978115	−0.489058	−	0.872252i	$-0.662659\pi$
$257$	−15.6804	−0.978115	−0.489058	+	0.872252i	$0.662659\pi$
$258$	13.5855	0.845796
$259$	−2.69530	−0.167478
$260$	−5.05456	−0.313471
$261$	10.3848	0.642806
$262$	49.3599	3.04946
$263$	12.6084	0.777467	0.388733	−	0.921350i	$-0.372913\pi$
$263$	12.6084	0.777467	0.388733	+	0.921350i	$0.372913\pi$
$264$	−18.0195	−1.10902
$265$	−20.1076	−1.23520
$266$	6.04669	0.370746
$267$	−11.0657	−0.677211
$268$	20.7506	1.26755
$269$	22.0833	1.34644	0.673222	−	0.739441i	$-0.264910\pi$
$269$	22.0833	1.34644	0.673222	+	0.739441i	$0.264910\pi$
$270$	19.7497	1.20193
$271$	12.1474	0.737904	0.368952	−	0.929448i	$-0.379717\pi$
$271$	12.1474	0.737904	0.368952	+	0.929448i	$0.379717\pi$
$272$	0.0768905	0.00466217
$273$	−1.50555	−0.0911202
$274$	5.91728	0.357476
$275$	13.7553	0.829479
$276$	−5.34017	−0.321441
$277$	−13.1346	−0.789180	−0.394590	−	0.918857i	$-0.629113\pi$
$277$	−13.1346	−0.789180	−0.394590	+	0.918857i	$0.629113\pi$
$278$	−9.49873	−0.569696
$279$	−1.60047	−0.0958178
$280$	5.39202	0.322235
$281$	−6.30905	−0.376366	−0.188183	−	0.982134i	$-0.560260\pi$
$281$	−6.30905	−0.376366	−0.188183	+	0.982134i	$0.560260\pi$
$282$	28.1658	1.67725
$283$	20.4007	1.21270	0.606348	−	0.795199i	$-0.292634\pi$
$283$	20.4007	1.21270	0.606348	+	0.795199i	$0.292634\pi$
$284$	−28.9397	−1.71725
$285$	−4.02902	−0.238659
$286$	12.5673	0.743122
$287$	−0.167035	−0.00985978
$288$	−9.13467	−0.538266
$289$	−16.8366	−0.990387
$290$	−24.0783	−1.41393
$291$	−6.03423	−0.353733
$292$	21.1901	1.24006
$293$	−20.1893	−1.17947	−0.589737	−	0.807595i	$-0.700769\pi$
$293$	−20.1893	−1.17947	−0.589737	+	0.807595i	$0.700769\pi$
$294$	−14.8718	−0.867344
$295$	11.6544	0.678547
$296$	5.84957	0.339999
$297$	−30.1930	−1.75198
$298$	−2.45145	−0.142009
$299$	1.39164	0.0804807
$300$	9.57130	0.552599
$301$	6.21488	0.358220
$302$	8.38984	0.482781
$303$	−6.77089	−0.388977
$304$	0.402840	0.0231045
$305$	−6.99793	−0.400701
$306$	−1.43328	−0.0819351
$307$	33.5707	1.91598	0.957990	−	0.286801i	$-0.0925919\pi$
$307$	33.5707	1.91598	0.957990	+	0.286801i	$0.0925919\pi$
$308$	−22.0612	−1.25705
$309$	−15.8685	−0.902729
$310$	3.71086	0.210762
$311$	32.3521	1.83452	0.917260	−	0.398290i	$-0.130396\pi$
$311$	32.3521	1.83452	0.917260	+	0.398290i	$0.130396\pi$
$312$	3.26748	0.184985
$313$	24.4579	1.38244	0.691221	−	0.722643i	$-0.257073\pi$
$313$	24.4579	1.38244	0.691221	+	0.722643i	$0.257073\pi$
$314$	9.17322	0.517675
$315$	3.08537	0.173841
$316$	2.09559	0.117886
$317$	17.5639	0.986484	0.493242	−	0.869892i	$-0.335812\pi$
$317$	17.5639	0.986484	0.493242	+	0.869892i	$0.335812\pi$
$318$	34.7872	1.95077
$319$	36.8106	2.06100
$320$	20.5775	1.15032
$321$	11.9973	0.669622
$322$	−3.97306	−0.221410
$323$	0.856209	0.0476407
$324$	−6.10535	−0.339186
$325$	−2.49426	−0.138357
$326$	−16.9661	−0.939665
$327$	−7.55623	−0.417860
$328$	0.362514	0.0200165
$329$	12.8848	0.710364
$330$	23.9069	1.31603
$331$	9.17431	0.504266	0.252133	−	0.967693i	$-0.418868\pi$
$331$	9.17431	0.504266	0.252133	+	0.967693i	$0.418868\pi$
$332$	10.0591	0.552065
$333$	3.34718	0.183425
$334$	34.2299	1.87298
$335$	−10.2869	−0.562031
$336$	0.286358	0.0156221
$337$	2.86123	0.155861	0.0779307	−	0.996959i	$-0.475169\pi$
$337$	2.86123	0.155861	0.0779307	+	0.996959i	$0.475169\pi$
$338$	−2.27884	−0.123953
$339$	21.8829	1.18852
$340$	2.04335	0.110816
$341$	−5.67311	−0.307216
$342$	−7.50914	−0.406048
$343$	−15.5730	−0.840861
$344$	−13.4881	−0.727227
$345$	2.64732	0.142527
$346$	−40.1353	−2.15769
$347$	3.29251	0.176751	0.0883755	−	0.996087i	$-0.471832\pi$
$347$	3.29251	0.176751	0.0883755	+	0.996087i	$0.471832\pi$
$348$	25.6137	1.37304
$349$	−12.5920	−0.674035	−0.337018	−	0.941498i	$-0.609418\pi$
$349$	−12.5920	−0.674035	−0.337018	+	0.941498i	$0.609418\pi$
$350$	7.12099	0.380633
$351$	5.47492	0.292230
$352$	−32.3791	−1.72581
$353$	11.4564	0.609765	0.304883	−	0.952390i	$-0.401383\pi$
$353$	11.4564	0.609765	0.304883	+	0.952390i	$0.401383\pi$
$354$	−20.1628	−1.07164
$355$	14.3465	0.761431
$356$	29.4024	1.55833
$357$	0.608634	0.0322123
$358$	13.4570	0.711223
$359$	24.4939	1.29274	0.646370	−	0.763025i	$-0.276286\pi$
$359$	24.4939	1.29274	0.646370	+	0.763025i	$0.276286\pi$
$360$	−6.69614	−0.352918
$361$	−14.5142	−0.763906
$362$	−55.9429	−2.94029
$363$	−23.3294	−1.22447
$364$	4.00036	0.209676
$365$	−10.5047	−0.549842
$366$	12.1068	0.632832
$367$	18.0764	0.943579	0.471790	−	0.881711i	$-0.343608\pi$
$367$	18.0764	0.943579	0.471790	+	0.881711i	$0.343608\pi$
$368$	−0.264692	−0.0137980
$369$	0.207435	0.0107986
$370$	−7.76078	−0.403464
$371$	15.9139	0.826208
$372$	−3.94748	−0.204667
$373$	−20.7881	−1.07637	−0.538183	−	0.842828i	$-0.680889\pi$
$373$	−20.7881	−1.07637	−0.538183	+	0.842828i	$0.680889\pi$
$374$	−5.08046	−0.262705
$375$	−14.2564	−0.736196
$376$	−27.9638	−1.44212
$377$	−6.67488	−0.343774
$378$	−15.6306	−0.803952
$379$	27.8653	1.43134	0.715671	−	0.698437i	$-0.246121\pi$
$379$	27.8653	1.43134	0.715671	+	0.698437i	$0.246121\pi$
$380$	10.7054	0.549176
$381$	−2.05501	−0.105281
$382$	54.2837	2.77739
$383$	−37.3464	−1.90831	−0.954155	−	0.299312i	$-0.903243\pi$
$383$	−37.3464	−1.90831	−0.954155	+	0.299312i	$0.903243\pi$
$384$	−21.4884	−1.09658
$385$	10.9365	0.557378
$386$	−28.8395	−1.46789
$387$	−7.71801	−0.392328
$388$	16.0334	0.813973
$389$	−0.000727062 0	−3.68635e−5 0	−1.84317e−5	−	1.00000i	$-0.500006\pi$
$389$	−0.000727062 0	−3.68635e−5 0	−1.84317e−5		1.00000i	$0.500006\pi$
$390$	−4.33505	−0.219514
$391$	−0.562584	−0.0284511
$392$	14.7652	0.745754
$393$	26.0299	1.31303
$394$	53.5375	2.69718
$395$	−1.03886	−0.0522707
$396$	27.3969	1.37675
$397$	8.66853	0.435061	0.217531	−	0.976053i	$-0.430200\pi$
$397$	8.66853	0.435061	0.217531	+	0.976053i	$0.430200\pi$
$398$	36.8567	1.84746
$399$	3.18871	0.159635
$400$	0.474412	0.0237206
$401$	−32.0752	−1.60176	−0.800879	−	0.598826i	$-0.795634\pi$
$401$	−32.0752	−1.60176	−0.800879	+	0.598826i	$0.795634\pi$
$402$	17.7968	0.887623
$403$	1.02871	0.0512436
$404$	17.9908	0.895073
$405$	3.02665	0.150395
$406$	19.0564	0.945755
$407$	11.8646	0.588105
$408$	−1.32091	−0.0653947
$409$	31.6512	1.56505	0.782525	−	0.622619i	$-0.213931\pi$
$409$	31.6512	1.56505	0.782525	+	0.622619i	$0.213931\pi$
$410$	−0.480958	−0.0237528
$411$	3.12047	0.153921
$412$	42.1638	2.07726
$413$	−9.22374	−0.453871
$414$	4.93399	0.242492
$415$	−4.98667	−0.244786
$416$	5.87133	0.287866
$417$	−5.00914	−0.245299
$418$	−26.6172	−1.30189
$419$	−30.3678	−1.48357	−0.741783	−	0.670640i	$-0.766019\pi$
$419$	−30.3678	−1.48357	−0.741783	+	0.670640i	$0.766019\pi$
$420$	7.60991	0.371326
$421$	−35.9963	−1.75435	−0.877175	−	0.480170i	$-0.840575\pi$
$421$	−35.9963	−1.75435	−0.877175	+	0.480170i	$0.840575\pi$
$422$	−32.7284	−1.59319
$423$	−16.0012	−0.778004
$424$	−34.5377	−1.67730
$425$	1.00833	0.0489112
$426$	−24.8201	−1.20254
$427$	5.53842	0.268023
$428$	−31.8776	−1.54086
$429$	6.62737	0.319972
$430$	17.8950	0.862972
$431$	32.9212	1.58576	0.792879	−	0.609380i	$-0.208581\pi$
$431$	32.9212	1.58576	0.792879	+	0.609380i	$0.208581\pi$
$432$	−1.04134	−0.0501013
$433$	6.85005	0.329192	0.164596	−	0.986361i	$-0.447368\pi$
$433$	6.85005	0.329192	0.164596	+	0.986361i	$0.447368\pi$
$434$	−2.93691	−0.140976
$435$	−12.6977	−0.608806
$436$	20.0775	0.961536
$437$	−2.94745	−0.140996
$438$	18.1737	0.868373
$439$	−32.5582	−1.55392	−0.776959	−	0.629552i	$-0.783239\pi$
$439$	−32.5582	−1.55392	−0.776959	+	0.629552i	$0.783239\pi$
$440$	−23.7354	−1.13154
$441$	8.44879	0.402323
$442$	0.921243	0.0438191
$443$	25.8566	1.22849	0.614243	−	0.789117i	$-0.289462\pi$
$443$	25.8566	1.22849	0.614243	+	0.789117i	$0.289462\pi$
$444$	8.25565	0.391796
$445$	−14.5759	−0.690963
$446$	31.2601	1.48021
$447$	−1.29277	−0.0611459
$448$	−16.2858	−0.769430
$449$	−37.5642	−1.77277	−0.886383	−	0.462953i	$-0.846790\pi$
$449$	−37.5642	−1.77277	−0.886383	+	0.462953i	$0.846790\pi$
$450$	−8.84328	−0.416876
$451$	0.735282	0.0346231
$452$	−58.1445	−2.73489
$453$	4.42437	0.207875
$454$	−64.3495	−3.02007
$455$	−1.98313	−0.0929706
$456$	−6.92042	−0.324078
$457$	−17.3206	−0.810222	−0.405111	−	0.914267i	$-0.632767\pi$
$457$	−17.3206	−0.810222	−0.405111	+	0.914267i	$0.632767\pi$
$458$	−65.7466	−3.07213
$459$	−2.21329	−0.103307
$460$	−7.03414	−0.327968
$461$	15.0484	0.700873	0.350436	−	0.936587i	$-0.386033\pi$
$461$	15.0484	0.700873	0.350436	+	0.936587i	$0.386033\pi$
$462$	−18.9208	−0.880275
$463$	−25.7021	−1.19448	−0.597240	−	0.802063i	$-0.703736\pi$
$463$	−25.7021	−1.19448	−0.597240	+	0.802063i	$0.703736\pi$
$464$	1.26957	0.0589383
$465$	1.95691	0.0907498
$466$	11.1291	0.515547
$467$	11.4233	0.528607	0.264304	−	0.964440i	$-0.414858\pi$
$467$	11.4233	0.528607	0.264304	+	0.964440i	$0.414858\pi$
$468$	−4.96789	−0.229641
$469$	8.14139	0.375935
$470$	37.1003	1.71131
$471$	4.83748	0.222900
$472$	20.0182	0.921410
$473$	−27.3576	−1.25790
$474$	1.79728	0.0825519
$475$	5.28278	0.242390
$476$	−1.61718	−0.0741235
$477$	−19.7628	−0.904877
$478$	−5.05159	−0.231054
$479$	−20.6460	−0.943342	−0.471671	−	0.881775i	$-0.656349\pi$
$479$	−20.6460	−0.943342	−0.471671	+	0.881775i	$0.656349\pi$
$480$	11.1691	0.509795
$481$	−2.15141	−0.0980959
$482$	0.636255	0.0289806
$483$	−2.09519	−0.0953344
$484$	61.9879	2.81763
$485$	−7.94836	−0.360916
$486$	32.1932	1.46031
$487$	19.0480	0.863148	0.431574	−	0.902077i	$-0.357958\pi$
$487$	19.0480	0.863148	0.431574	+	0.902077i	$0.357958\pi$
$488$	−12.0200	−0.544118
$489$	−8.94704	−0.404599
$490$	−19.5894	−0.884957
$491$	2.09167	0.0943958	0.0471979	−	0.998886i	$-0.484971\pi$
$491$	2.09167	0.0943958	0.0471979	+	0.998886i	$0.484971\pi$
$492$	0.511626	0.0230659
$493$	2.69838	0.121529
$494$	4.82652	0.217155
$495$	−13.5817	−0.610450
$496$	−0.195661	−0.00878545
$497$	−11.3543	−0.509311
$498$	8.62720	0.386594
$499$	−20.4098	−0.913668	−0.456834	−	0.889552i	$-0.651017\pi$
$499$	−20.4098	−0.913668	−0.456834	+	0.889552i	$0.651017\pi$
$500$	37.8802	1.69406
$501$	18.0511	0.806463
$502$	46.9183	2.09407
$503$	−18.3337	−0.817457	−0.408729	−	0.912656i	$-0.634028\pi$
$503$	−18.3337	−0.817457	−0.408729	+	0.912656i	$0.634028\pi$
$504$	5.29957	0.236062
$505$	−8.91869	−0.396876
$506$	17.4892	0.777491
$507$	−1.20174	−0.0533714
$508$	5.46030	0.242262
$509$	17.2555	0.764836	0.382418	−	0.923990i	$-0.375092\pi$
$509$	17.2555	0.764836	0.382418	+	0.923990i	$0.375092\pi$
$510$	1.75248	0.0776013
$511$	8.31382	0.367782
$512$	−2.15103	−0.0950628
$513$	−11.5957	−0.511963
$514$	35.7331	1.57612
$515$	−20.9022	−0.921061
$516$	−19.0361	−0.838016
$517$	−56.7185	−2.49448
$518$	6.14216	0.269871
$519$	−21.1653	−0.929054
$520$	4.30396	0.188741
$521$	0.707893	0.0310134	0.0155067	−	0.999880i	$-0.495064\pi$
$521$	0.707893	0.0310134	0.0155067	+	0.999880i	$0.495064\pi$
$522$	−23.6654	−1.03581
$523$	1.61847	0.0707706	0.0353853	−	0.999374i	$-0.488734\pi$
$523$	1.61847	0.0707706	0.0353853	+	0.999374i	$0.488734\pi$
$524$	−69.1633	−3.02141
$525$	3.75525	0.163892
$526$	−28.7326	−1.25280
$527$	−0.415865	−0.0181154
$528$	−1.26053	−0.0548576
$529$	−21.0633	−0.915797
$530$	45.8221	1.99038
$531$	11.4546	0.497087
$532$	−8.47265	−0.367336
$533$	−0.133329	−0.00577512
$534$	25.2170	1.09125
$535$	15.8029	0.683220
$536$	−17.6691	−0.763191
$537$	7.09651	0.306237
$538$	−50.3244	−2.16964
$539$	29.9480	1.28995
$540$	−27.6733	−1.19087
$541$	−16.6794	−0.717103	−0.358552	−	0.933510i	$-0.616729\pi$
$541$	−16.6794	−0.717103	−0.358552	+	0.933510i	$0.616729\pi$
$542$	−27.6821	−1.18905
$543$	−29.5014	−1.26603
$544$	−2.37354	−0.101765
$545$	−9.95315	−0.426346
$546$	3.43092	0.146830
$547$	25.9764	1.11067	0.555335	−	0.831627i	$-0.312590\pi$
$547$	25.9764	1.11067	0.555335	+	0.831627i	$0.312590\pi$
$548$	−8.29132	−0.354188
$549$	−6.87794	−0.293543
$550$	−31.3463	−1.33661
$551$	14.1372	0.602265
$552$	4.54716	0.193540
$553$	0.822192	0.0349631
$554$	29.9316	1.27167
$555$	−4.09264	−0.173723
$556$	13.3097	0.564455
$557$	37.3125	1.58098	0.790491	−	0.612474i	$-0.209826\pi$
$557$	37.3125	1.58098	0.790491	+	0.612474i	$0.209826\pi$
$558$	3.64723	0.154399
$559$	4.96077	0.209818
$560$	0.377193	0.0159393
$561$	−2.67918	−0.113115
$562$	14.3773	0.606471
$563$	−16.3966	−0.691035	−0.345517	−	0.938412i	$-0.612297\pi$
$563$	−16.3966	−0.691035	−0.345517	+	0.938412i	$0.612297\pi$
$564$	−39.4661	−1.66182
$565$	28.8244	1.21265
$566$	−46.4900	−1.95412
$567$	−2.39540	−0.100597
$568$	24.6421	1.03396
$569$	21.4142	0.897732	0.448866	−	0.893599i	$-0.351828\pi$
$569$	21.4142	0.897732	0.448866	+	0.893599i	$0.351828\pi$
$570$	9.18150	0.384571
$571$	−8.83454	−0.369714	−0.184857	−	0.982765i	$-0.559182\pi$
$571$	−8.83454	−0.369714	−0.184857	+	0.982765i	$0.559182\pi$
$572$	−17.6094	−0.736287
$573$	28.6264	1.19589
$574$	0.380647	0.0158879
$575$	−3.47112	−0.144756
$576$	20.2246	0.842693
$577$	−5.84845	−0.243474	−0.121737	−	0.992562i	$-0.538846\pi$
$577$	−5.84845	−0.243474	−0.121737	+	0.992562i	$0.538846\pi$
$578$	38.3679	1.59589
$579$	−15.2085	−0.632042
$580$	33.7386	1.40092
$581$	3.94663	0.163734
$582$	13.7511	0.570000
$583$	−70.0522	−2.90126
$584$	−18.0434	−0.746639
$585$	2.46277	0.101823
$586$	46.0083	1.90059
$587$	36.2760	1.49727	0.748635	−	0.662983i	$-0.230710\pi$
$587$	36.2760	1.49727	0.748635	+	0.662983i	$0.230710\pi$
$588$	20.8385	0.859365
$589$	−2.17877	−0.0897747
$590$	−26.5586	−1.09340
$591$	28.2329	1.16135
$592$	0.409201	0.0168180
$593$	46.7426	1.91949	0.959743	−	0.280878i	$-0.0906258\pi$
$593$	46.7426	1.91949	0.959743	+	0.280878i	$0.0906258\pi$
$594$	68.8052	2.82311
$595$	0.801699	0.0328664
$596$	3.43499	0.140703
$597$	19.4363	0.795476
$598$	−3.17133	−0.129685
$599$	−14.5359	−0.593921	−0.296960	−	0.954890i	$-0.595973\pi$
$599$	−14.5359	−0.593921	−0.296960	+	0.954890i	$0.595973\pi$
$600$	−8.14996	−0.332721
$601$	−16.3889	−0.668516	−0.334258	−	0.942482i	$-0.608486\pi$
$601$	−16.3889	−0.668516	−0.334258	+	0.942482i	$0.608486\pi$
$602$	−14.1627	−0.577230
$603$	−10.1105	−0.411730
$604$	−11.7559	−0.478340
$605$	−30.7297	−1.24934
$606$	15.4298	0.626792
$607$	−24.7393	−1.00414	−0.502068	−	0.864828i	$-0.667427\pi$
$607$	−24.7393	−1.00414	−0.502068	+	0.864828i	$0.667427\pi$
$608$	−12.4353	−0.504318
$609$	10.0494	0.407222
$610$	15.9472	0.645683
$611$	10.2848	0.416078
$612$	2.00832	0.0811814
$613$	15.9430	0.643930	0.321965	−	0.946752i	$-0.395657\pi$
$613$	15.9430	0.643930	0.321965	+	0.946752i	$0.395657\pi$
$614$	−76.5023	−3.08738
$615$	−0.253632	−0.0102274
$616$	18.7851	0.756872
$617$	−7.85966	−0.316418	−0.158209	−	0.987406i	$-0.550572\pi$
$617$	−7.85966	−0.316418	−0.158209	+	0.987406i	$0.550572\pi$
$618$	36.1619	1.45464
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−5.19967	−0.208824
$621$	7.61912	0.305745
$622$	−73.7254	−2.95612
$623$	11.5359	0.462175
$624$	0.228573	0.00915025
$625$	−6.30732	−0.252293
$626$	−55.7357	−2.22765
$627$	−14.0366	−0.560566
$628$	−12.8536	−0.512913
$629$	0.869728	0.0346783
$630$	−7.03108	−0.280125
$631$	−5.15456	−0.205200	−0.102600	−	0.994723i	$-0.532716\pi$
$631$	−5.15456	−0.205200	−0.102600	+	0.994723i	$0.532716\pi$
$632$	−1.78439	−0.0709792
$633$	−17.2593	−0.685995
$634$	−40.0253	−1.58961
$635$	−2.70688	−0.107419
$636$	−48.7440	−1.93282
$637$	−5.43048	−0.215163
$638$	−83.8855	−3.32106
$639$	14.1005	0.557806
$640$	−28.3048	−1.11884
$641$	−18.6613	−0.737077	−0.368538	−	0.929613i	$-0.620142\pi$
$641$	−18.6613	−0.737077	−0.368538	+	0.929613i	$0.620142\pi$
$642$	−27.3399	−1.07902
$643$	−0.874004	−0.0344674	−0.0172337	−	0.999851i	$-0.505486\pi$
$643$	−0.874004	−0.0344674	−0.0172337	+	0.999851i	$0.505486\pi$
$644$	5.56707	0.219373
$645$	9.43689	0.371577
$646$	−1.95117	−0.0767676
$647$	−15.2046	−0.597755	−0.298878	−	0.954291i	$-0.596612\pi$
$647$	−15.2046	−0.597755	−0.298878	+	0.954291i	$0.596612\pi$
$648$	5.19870	0.204224
$649$	40.6025	1.59379
$650$	5.68404	0.222946
$651$	−1.54877	−0.0607012
$652$	23.7730	0.931021
$653$	16.2132	0.634472	0.317236	−	0.948347i	$-0.397245\pi$
$653$	16.2132	0.634472	0.317236	+	0.948347i	$0.397245\pi$
$654$	17.2195	0.673334
$655$	34.2868	1.33970
$656$	0.0253593	0.000990115 0
$657$	−10.3246	−0.402801
$658$	−29.3625	−1.14467
$659$	−24.0335	−0.936213	−0.468106	−	0.883672i	$-0.655064\pi$
$659$	−24.0335	−0.936213	−0.468106	+	0.883672i	$0.655064\pi$
$660$	−33.4985	−1.30393
$661$	−22.5613	−0.877535	−0.438767	−	0.898601i	$-0.644585\pi$
$661$	−22.5613	−0.877535	−0.438767	+	0.898601i	$0.644585\pi$
$662$	−20.9068	−0.812566
$663$	0.485817	0.0188676
$664$	−8.56532	−0.332399
$665$	4.20021	0.162877
$666$	−7.62771	−0.295568
$667$	−9.28904	−0.359673
$668$	−47.9631	−1.85575
$669$	16.4849	0.637345
$670$	23.4421	0.905649
$671$	−24.3799	−0.941174
$672$	−8.83959	−0.340995
$673$	−42.7731	−1.64878	−0.824390	−	0.566022i	$-0.808482\pi$
$673$	−42.7731	−1.64878	−0.824390	+	0.566022i	$0.808482\pi$
$674$	−6.52030	−0.251153
$675$	−13.6559	−0.525616
$676$	3.19313	0.122813
$677$	−46.9698	−1.80520	−0.902598	−	0.430485i	$-0.858343\pi$
$677$	−46.9698	−1.80520	−0.902598	+	0.430485i	$0.858343\pi$
$678$	−49.8677	−1.91516
$679$	6.29062	0.241412
$680$	−1.73992	−0.0667227
$681$	−33.9346	−1.30038
$682$	12.9281	0.495043
$683$	11.4193	0.436946	0.218473	−	0.975843i	$-0.429892\pi$
$683$	11.4193	0.436946	0.218473	+	0.975843i	$0.429892\pi$
$684$	10.5218	0.402313
$685$	4.11032	0.157047
$686$	35.4884	1.35495
$687$	−34.6714	−1.32279
$688$	−0.943543	−0.0359722
$689$	12.7026	0.483930
$690$	−6.03284	−0.229666
$691$	39.8037	1.51420	0.757101	−	0.653298i	$-0.226615\pi$
$691$	39.8037	1.51420	0.757101	+	0.653298i	$0.226615\pi$
$692$	56.2378	2.13784
$693$	10.7490	0.408322
$694$	−7.50310	−0.284814
$695$	−6.59810	−0.250280
$696$	−21.8100	−0.826707
$697$	0.0538995	0.00204159
$698$	28.6952	1.08613
$699$	5.86894	0.221984
$700$	−9.97797	−0.377132
$701$	18.3469	0.692953	0.346477	−	0.938059i	$-0.387378\pi$
$701$	18.3469	0.692953	0.346477	+	0.938059i	$0.387378\pi$
$702$	−12.4765	−0.470895
$703$	4.55662	0.171856
$704$	71.6892	2.70189
$705$	19.5648	0.736853
$706$	−26.1074	−0.982566
$707$	7.05857	0.265465
$708$	28.2522	1.06178
$709$	−19.3493	−0.726678	−0.363339	−	0.931657i	$-0.618363\pi$
$709$	−19.3493	−0.726678	−0.363339	+	0.931657i	$0.618363\pi$
$710$	−32.6933	−1.22696
$711$	−1.02105	−0.0382923
$712$	−25.0362	−0.938269
$713$	1.43159	0.0536135
$714$	−1.38698	−0.0519064
$715$	8.72965	0.326470
$716$	−18.8560	−0.704680
$717$	−2.66395	−0.0994871
$718$	−55.8178	−2.08310
$719$	−1.81047	−0.0675190	−0.0337595	−	0.999430i	$-0.510748\pi$
$719$	−1.81047	−0.0675190	−0.0337595	+	0.999430i	$0.510748\pi$
$720$	−0.468422	−0.0174570
$721$	16.5428	0.616084
$722$	33.0756	1.23095
$723$	0.335528	0.0124784
$724$	78.3875	2.91325
$725$	16.6489	0.618325
$726$	53.1640	1.97310
$727$	7.14913	0.265146	0.132573	−	0.991173i	$-0.457676\pi$
$727$	7.14913	0.265146	0.132573	+	0.991173i	$0.457676\pi$
$728$	−3.40631	−0.126246
$729$	22.7131	0.841227
$730$	23.9386	0.886008
$731$	−2.00544	−0.0741738
$732$	−16.9641	−0.627011
$733$	−24.6905	−0.911966	−0.455983	−	0.889989i	$-0.650712\pi$
$733$	−24.6905	−0.911966	−0.455983	+	0.889989i	$0.650712\pi$
$734$	−41.1932	−1.52047
$735$	−10.3304	−0.381043
$736$	8.17078	0.301179
$737$	−35.8380	−1.32011
$738$	−0.472711	−0.0174007
$739$	−49.8940	−1.83538	−0.917690	−	0.397297i	$-0.869948\pi$
$739$	−49.8940	−1.83538	−0.917690	+	0.397297i	$0.869948\pi$
$740$	10.8744	0.399752
$741$	2.54526	0.0935024
$742$	−36.2652	−1.33134
$743$	−3.11518	−0.114285	−0.0571424	−	0.998366i	$-0.518199\pi$
$743$	−3.11518	−0.114285	−0.0571424	+	0.998366i	$0.518199\pi$
$744$	3.36128	0.123231
$745$	−1.70285	−0.0623876
$746$	47.3727	1.73444
$747$	−4.90117	−0.179324
$748$	7.11877	0.260288
$749$	−12.5070	−0.456996
$750$	32.4880	1.18629
$751$	−34.0670	−1.24312	−0.621561	−	0.783366i	$-0.713501\pi$
$751$	−34.0670	−1.24312	−0.621561	+	0.783366i	$0.713501\pi$
$752$	−1.95618	−0.0713345
$753$	24.7423	0.901660
$754$	15.2110	0.553952
$755$	5.82783	0.212096
$756$	21.9017	0.796556
$757$	−47.0231	−1.70908	−0.854541	−	0.519384i	$-0.826161\pi$
$757$	−47.0231	−1.70908	−0.854541	+	0.519384i	$0.826161\pi$
$758$	−63.5006	−2.30644
$759$	9.22292	0.334771
$760$	−9.11565	−0.330659
$761$	11.1877	0.405554	0.202777	−	0.979225i	$-0.435003\pi$
$761$	11.1877	0.405554	0.202777	+	0.979225i	$0.435003\pi$
$762$	4.68304	0.169648
$763$	7.87728	0.285177
$764$	−76.0625	−2.75184
$765$	−0.995598	−0.0359959
$766$	85.1065	3.07502
$767$	−7.36247	−0.265843
$768$	17.7247	0.639586
$769$	−3.91063	−0.141021	−0.0705104	−	0.997511i	$-0.522463\pi$
$769$	−3.91063	−0.141021	−0.0705104	+	0.997511i	$0.522463\pi$
$770$	−24.9227	−0.898151
$771$	18.8438	0.678643
$772$	40.4100	1.45439
$773$	32.6096	1.17289	0.586443	−	0.809990i	$-0.300528\pi$
$773$	32.6096	1.17289	0.586443	+	0.809990i	$0.300528\pi$
$774$	17.5881	0.632192
$775$	−2.56587	−0.0921688
$776$	−13.6524	−0.490094
$777$	3.23906	0.116201
$778$	0.00165686	5.94013e−5 0
$779$	0.282387	0.0101176
$780$	6.07429	0.217495
$781$	49.9812	1.78847
$782$	1.28204	0.0458457
$783$	−36.5444	−1.30599
$784$	1.03288	0.0368887
$785$	6.37199	0.227426
$786$	−59.3180	−2.11580
$787$	41.1731	1.46766	0.733832	−	0.679331i	$-0.237730\pi$
$787$	41.1731	1.46766	0.733832	+	0.679331i	$0.237730\pi$
$788$	−75.0170	−2.67237
$789$	−15.1521	−0.539428
$790$	2.36740	0.0842283
$791$	−22.8127	−0.811126
$792$	−23.3284	−0.828940
$793$	4.42081	0.156988
$794$	−19.7542	−0.701051
$795$	24.1642	0.857016
$796$	−51.6438	−1.83046
$797$	−10.6317	−0.376595	−0.188297	−	0.982112i	$-0.560297\pi$
$797$	−10.6317	−0.376595	−0.188297	+	0.982112i	$0.560297\pi$
$798$	−7.26658	−0.257234
$799$	−4.15772	−0.147090
$800$	−14.6446	−0.517767
$801$	−14.3260	−0.506183
$802$	73.0943	2.58105
$803$	−36.5970	−1.29148
$804$	−24.9370	−0.879458
$805$	−2.75980	−0.0972703
$806$	−2.34426	−0.0825732
$807$	−26.5385	−0.934200
$808$	−15.3191	−0.538925
$809$	−56.6132	−1.99042	−0.995208	−	0.0977821i	$-0.968825\pi$
$809$	−56.6132	−1.99042	−0.995208	+	0.0977821i	$0.968825\pi$
$810$	−6.89726	−0.242345
$811$	−48.6660	−1.70890	−0.854448	−	0.519537i	$-0.826104\pi$
$811$	−48.6660	−1.70890	−0.854448	+	0.519537i	$0.826104\pi$
$812$	−26.7020	−0.937055
$813$	−14.5981	−0.511978
$814$	−27.0375	−0.947664
$815$	−11.7851	−0.412816
$816$	−0.0924028	−0.00323475
$817$	−10.5068	−0.367585
$818$	−72.1281	−2.52190
$819$	−1.94913	−0.0681080
$820$	0.673920	0.0235343
$821$	15.6395	0.545821	0.272911	−	0.962039i	$-0.412014\pi$
$821$	15.6395	0.545821	0.272911	+	0.962039i	$0.412014\pi$
$822$	−7.11106	−0.248027
$823$	−7.91483	−0.275894	−0.137947	−	0.990440i	$-0.544050\pi$
$823$	−7.91483	−0.275894	−0.137947	+	0.990440i	$0.544050\pi$
$824$	−35.9025	−1.25072
$825$	−16.5304	−0.575515
$826$	21.0195	0.731360
$827$	−22.8187	−0.793484	−0.396742	−	0.917930i	$-0.629859\pi$
$827$	−22.8187	−0.793484	−0.396742	+	0.917930i	$0.629859\pi$
$828$	−6.91353	−0.240262
$829$	−30.7783	−1.06897	−0.534487	−	0.845177i	$-0.679495\pi$
$829$	−30.7783	−1.06897	−0.534487	+	0.845177i	$0.679495\pi$
$830$	11.3638	0.394445
$831$	15.7844	0.547555
$832$	−12.9994	−0.450674
$833$	2.19532	0.0760634
$834$	11.4150	0.395271
$835$	23.7771	0.822840
$836$	37.2962	1.28992
$837$	5.63209	0.194674
$838$	69.2035	2.39060
$839$	−19.6920	−0.679842	−0.339921	−	0.940454i	$-0.610400\pi$
$839$	−19.6920	−0.679842	−0.339921	+	0.940454i	$0.610400\pi$
$840$	−6.47983	−0.223576
$841$	15.5540	0.536346
$842$	82.0298	2.82694
$843$	7.58187	0.261133
$844$	45.8592	1.57854
$845$	−1.58295	−0.0544552
$846$	36.4642	1.25366
$847$	24.3206	0.835666
$848$	−2.41605	−0.0829674
$849$	−24.5164	−0.841402
$850$	−2.29782	−0.0788147
$851$	−2.99399	−0.102633
$852$	34.7781	1.19148
$853$	15.8191	0.541634	0.270817	−	0.962631i	$-0.412706\pi$
$853$	15.8191	0.541634	0.270817	+	0.962631i	$0.412706\pi$
$854$	−12.6212	−0.431888
$855$	−5.21607	−0.178386
$856$	27.1438	0.927755
$857$	−54.2269	−1.85236	−0.926178	−	0.377087i	$-0.876926\pi$
$857$	−54.2269	−1.85236	−0.926178	+	0.377087i	$0.876926\pi$
$858$	−15.1027	−0.515599
$859$	−22.9581	−0.783322	−0.391661	−	0.920110i	$-0.628099\pi$
$859$	−22.9581	−0.783322	−0.391661	+	0.920110i	$0.628099\pi$
$860$	−25.0745	−0.855034
$861$	0.200734	0.00684099
$862$	−75.0222	−2.55527
$863$	46.0740	1.56838	0.784188	−	0.620523i	$-0.213080\pi$
$863$	46.0740	1.56838	0.784188	+	0.620523i	$0.213080\pi$
$864$	32.1451	1.09360
$865$	−27.8792	−0.947920
$866$	−15.6102	−0.530456
$867$	20.2333	0.687158
$868$	4.11521	0.139679
$869$	−3.61925	−0.122775
$870$	28.9360	0.981021
$871$	6.49853	0.220194
$872$	−17.0960	−0.578942
$873$	−7.81207	−0.264399
$874$	6.71678	0.227199
$875$	14.8621	0.502431
$876$	−25.4651	−0.860385
$877$	−9.66009	−0.326198	−0.163099	−	0.986610i	$-0.552149\pi$
$877$	−9.66009	−0.326198	−0.163099	+	0.986610i	$0.552149\pi$
$878$	74.1950	2.50396
$879$	24.2624	0.818352
$880$	−1.66039	−0.0559717
$881$	46.8334	1.57786	0.788928	−	0.614485i	$-0.210636\pi$
$881$	46.8334	1.57786	0.788928	+	0.614485i	$0.210636\pi$
$882$	−19.2535	−0.648298
$883$	36.1406	1.21623	0.608114	−	0.793850i	$-0.291926\pi$
$883$	36.1406	1.21623	0.608114	+	0.793850i	$0.291926\pi$
$884$	−1.29085	−0.0434160
$885$	−14.0057	−0.470795
$886$	−58.9232	−1.97956
$887$	−22.7884	−0.765158	−0.382579	−	0.923923i	$-0.624964\pi$
$887$	−22.7884	−0.765158	−0.382579	+	0.923923i	$0.624964\pi$
$888$	−7.02969	−0.235901
$889$	2.14232	0.0718511
$890$	33.2161	1.11341
$891$	10.5444	0.353252
$892$	−43.8017	−1.46659
$893$	−21.7829	−0.728936
$894$	2.94602	0.0985297
$895$	9.34760	0.312456
$896$	22.4014	0.748379
$897$	−1.67240	−0.0558397
$898$	85.6030	2.85661
$899$	−6.86650	−0.229011
$900$	12.3912	0.413041
$901$	−5.13514	−0.171076
$902$	−1.67559	−0.0557911
$903$	−7.46870	−0.248543
$904$	49.5101	1.64668
$905$	−38.8596	−1.29174
$906$	−10.0824	−0.334967
$907$	22.7419	0.755133	0.377566	−	0.925983i	$-0.376761\pi$
$907$	22.7419	0.755133	0.377566	+	0.925983i	$0.376761\pi$
$908$	90.1668	2.99229
$909$	−8.76576	−0.290742
$910$	4.51924	0.149811
$911$	14.4095	0.477409	0.238704	−	0.971092i	$-0.423277\pi$
$911$	14.4095	0.477409	0.238704	+	0.971092i	$0.423277\pi$
$912$	−0.484111	−0.0160305
$913$	−17.3729	−0.574959
$914$	39.4709	1.30558
$915$	8.40973	0.278017
$916$	92.1244	3.04388
$917$	−27.1358	−0.896104
$918$	5.04373	0.166468
$919$	17.3943	0.573784	0.286892	−	0.957963i	$-0.407378\pi$
$919$	17.3943	0.573784	0.286892	+	0.957963i	$0.407378\pi$
$920$	5.98957	0.197470
$921$	−40.3434	−1.32936
$922$	−34.2929	−1.12938
$923$	−9.06311	−0.298316
$924$	26.5119	0.872177
$925$	5.36619	0.176439
$926$	58.5711	1.92477
$927$	−20.5438	−0.674747
$928$	−39.1904	−1.28649
$929$	22.2307	0.729364	0.364682	−	0.931132i	$-0.381178\pi$
$929$	22.2307	0.729364	0.364682	+	0.931132i	$0.381178\pi$
$930$	−4.45950	−0.146233
$931$	11.5016	0.376949
$932$	−15.5942	−0.510805
$933$	−38.8790	−1.27284
$934$	−26.0319	−0.851790
$935$	−3.52904	−0.115412
$936$	4.23016	0.138267
$937$	−5.81591	−0.189997	−0.0949987	−	0.995477i	$-0.530285\pi$
$937$	−5.81591	−0.189997	−0.0949987	+	0.995477i	$0.530285\pi$
$938$	−18.5530	−0.605775
$939$	−29.3922	−0.959177
$940$	−51.9851	−1.69557
$941$	−54.4176	−1.77396	−0.886982	−	0.461805i	$-0.847202\pi$
$941$	−54.4176	−1.77396	−0.886982	+	0.461805i	$0.847202\pi$
$942$	−11.0239	−0.359177
$943$	−0.185546	−0.00604221
$944$	1.40035	0.0455775
$945$	−10.8575	−0.353194
$946$	62.3437	2.02697
$947$	27.7906	0.903073	0.451536	−	0.892253i	$-0.350876\pi$
$947$	27.7906	0.903073	0.451536	+	0.892253i	$0.350876\pi$
$948$	−2.51836	−0.0817925
$949$	6.63616	0.215419
$950$	−12.0386	−0.390584
$951$	−21.1073	−0.684450
$952$	1.37703	0.0446299
$953$	−36.6055	−1.18577	−0.592884	−	0.805288i	$-0.702011\pi$
$953$	−36.6055	−1.18577	−0.592884	+	0.805288i	$0.702011\pi$
$954$	45.0364	1.45811
$955$	37.7070	1.22017
$956$	7.07831	0.228929
$957$	−44.2369	−1.42998
$958$	47.0491	1.52009
$959$	−3.25305	−0.105047
$960$	−24.7289	−0.798121
$961$	−29.9418	−0.965863
$962$	4.90273	0.158070
$963$	15.5320	0.500510
$964$	−0.891523	−0.0287140
$965$	−20.0328	−0.644877
$966$	4.77460	0.153620
$967$	−47.8012	−1.53718	−0.768591	−	0.639740i	$-0.779042\pi$
$967$	−47.8012	−1.53718	−0.768591	+	0.639740i	$0.779042\pi$
$968$	−52.7826	−1.69650
$969$	−1.02894	−0.0330545
$970$	18.1131	0.581575
$971$	−5.84160	−0.187466	−0.0937330	−	0.995597i	$-0.529880\pi$
$971$	−5.84160	−0.187466	−0.0937330	+	0.995597i	$0.529880\pi$
$972$	−45.1092	−1.44688
$973$	5.22197	0.167409
$974$	−43.4075	−1.39086
$975$	2.99747	0.0959959
$976$	−0.840843	−0.0269147
$977$	−46.9628	−1.50247	−0.751237	−	0.660033i	$-0.770542\pi$
$977$	−46.9628	−1.50247	−0.751237	+	0.660033i	$0.770542\pi$
$978$	20.3889	0.651965
$979$	−50.7804	−1.62295
$980$	27.4487	0.876816
$981$	−9.78248	−0.312331
$982$	−4.76659	−0.152108
$983$	12.1197	0.386557	0.193279	−	0.981144i	$-0.438088\pi$
$983$	12.1197	0.386557	0.193279	+	0.981144i	$0.438088\pi$
$984$	−0.435650	−0.0138880
$985$	37.1887	1.18493
$986$	−6.14919	−0.195830
$987$	−15.4843	−0.492870
$988$	−6.76294	−0.215158
$989$	6.90361	0.219522
$990$	30.9505	0.983671
$991$	−28.7982	−0.914803	−0.457402	−	0.889260i	$-0.651220\pi$
$991$	−28.7982	−0.914803	−0.457402	+	0.889260i	$0.651220\pi$
$992$	6.03988	0.191766
$993$	−11.0252	−0.349874
$994$	25.8747	0.820696
$995$	25.6017	0.811630
$996$	−12.0885	−0.383038
$997$	−6.09938	−0.193169	−0.0965846	−	0.995325i	$-0.530792\pi$
$997$	−6.09938	−0.193169	−0.0965846	+	0.995325i	$0.530792\pi$
$998$	46.5107	1.47227
$999$	−11.7788	−0.372665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.17	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-2.27884 q^{2} -1.20174 q^{3} +3.19313 q^{4} -1.58295 q^{5} +2.73859 q^{6} +1.25281 q^{7} -2.71895 q^{8} -1.55581 q^{9} +O(q^{10})\)	\(q-2.27884 q^{2} -1.20174 q^{3} +3.19313 q^{4} -1.58295 q^{5} +2.73859 q^{6} +1.25281 q^{7} -2.71895 q^{8} -1.55581 q^{9} +3.60730 q^{10} -5.51479 q^{11} -3.83732 q^{12} +1.00000 q^{13} -2.85495 q^{14} +1.90230 q^{15} -0.190201 q^{16} -0.404259 q^{17} +3.54545 q^{18} -2.11797 q^{19} -5.05456 q^{20} -1.50555 q^{21} +12.5673 q^{22} +1.39164 q^{23} +3.26748 q^{24} -2.49426 q^{25} -2.27884 q^{26} +5.47492 q^{27} +4.00036 q^{28} -6.67488 q^{29} -4.33505 q^{30} +1.02871 q^{31} +5.87133 q^{32} +6.62737 q^{33} +0.921243 q^{34} -1.98313 q^{35} -4.96789 q^{36} -2.15141 q^{37} +4.82652 q^{38} -1.20174 q^{39} +4.30396 q^{40} -0.133329 q^{41} +3.43092 q^{42} +4.96077 q^{43} -17.6094 q^{44} +2.46277 q^{45} -3.17133 q^{46} +10.2848 q^{47} +0.228573 q^{48} -5.43048 q^{49} +5.68404 q^{50} +0.485817 q^{51} +3.19313 q^{52} +12.7026 q^{53} -12.4765 q^{54} +8.72965 q^{55} -3.40631 q^{56} +2.54526 q^{57} +15.2110 q^{58} -7.36247 q^{59} +6.07429 q^{60} +4.42081 q^{61} -2.34426 q^{62} -1.94913 q^{63} -12.9994 q^{64} -1.58295 q^{65} -15.1027 q^{66} +6.49853 q^{67} -1.29085 q^{68} -1.67240 q^{69} +4.51924 q^{70} -9.06311 q^{71} +4.23016 q^{72} +6.63616 q^{73} +4.90273 q^{74} +2.99747 q^{75} -6.76294 q^{76} -6.90896 q^{77} +2.73859 q^{78} +0.656280 q^{79} +0.301079 q^{80} -1.91203 q^{81} +0.303836 q^{82} +3.15024 q^{83} -4.80742 q^{84} +0.639923 q^{85} -11.3048 q^{86} +8.02150 q^{87} +14.9944 q^{88} +9.20804 q^{89} -5.61227 q^{90} +1.25281 q^{91} +4.44368 q^{92} -1.23624 q^{93} -23.4374 q^{94} +3.35264 q^{95} -7.05584 q^{96} +5.02123 q^{97} +12.3752 q^{98} +8.57996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 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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