LMFDB - Embedded newform 8027.2.a.e.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8027,2,Mod(1,8027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8027, 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("8027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8027 = 23 \cdot 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8027.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:	$64.0959177025$
Analytic rank:	$0$
Dimension:	$169$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60952 q^{2} -0.136825 q^{3} +4.80961 q^{4} -0.260304 q^{5} +0.357049 q^{6} -0.298256 q^{7} -7.33174 q^{8} -2.98128 q^{9} +O(q^{10})$	\(q-2.60952 q^{2} -0.136825 q^{3} +4.80961 q^{4} -0.260304 q^{5} +0.357049 q^{6} -0.298256 q^{7} -7.33174 q^{8} -2.98128 q^{9} +0.679270 q^{10} +2.08679 q^{11} -0.658077 q^{12} +4.31483 q^{13} +0.778305 q^{14} +0.0356162 q^{15} +9.51313 q^{16} -6.32899 q^{17} +7.77972 q^{18} +3.88371 q^{19} -1.25196 q^{20} +0.0408089 q^{21} -5.44553 q^{22} -1.00000 q^{23} +1.00317 q^{24} -4.93224 q^{25} -11.2596 q^{26} +0.818390 q^{27} -1.43449 q^{28} +3.72117 q^{29} -0.0929414 q^{30} -7.63496 q^{31} -10.1613 q^{32} -0.285526 q^{33} +16.5156 q^{34} +0.0776373 q^{35} -14.3388 q^{36} -7.39885 q^{37} -10.1346 q^{38} -0.590378 q^{39} +1.90848 q^{40} +6.64095 q^{41} -0.106492 q^{42} +9.67676 q^{43} +10.0367 q^{44} +0.776040 q^{45} +2.60952 q^{46} -6.57083 q^{47} -1.30164 q^{48} -6.91104 q^{49} +12.8708 q^{50} +0.865966 q^{51} +20.7526 q^{52} +5.94329 q^{53} -2.13561 q^{54} -0.543201 q^{55} +2.18673 q^{56} -0.531389 q^{57} -9.71048 q^{58} +1.82863 q^{59} +0.171300 q^{60} -7.98593 q^{61} +19.9236 q^{62} +0.889184 q^{63} +7.48976 q^{64} -1.12317 q^{65} +0.745087 q^{66} +9.29569 q^{67} -30.4400 q^{68} +0.136825 q^{69} -0.202596 q^{70} -14.4262 q^{71} +21.8580 q^{72} +7.91292 q^{73} +19.3075 q^{74} +0.674856 q^{75} +18.6791 q^{76} -0.622398 q^{77} +1.54060 q^{78} +10.6953 q^{79} -2.47631 q^{80} +8.83186 q^{81} -17.3297 q^{82} +8.26818 q^{83} +0.196275 q^{84} +1.64746 q^{85} -25.2517 q^{86} -0.509150 q^{87} -15.2998 q^{88} -10.3377 q^{89} -2.02509 q^{90} -1.28692 q^{91} -4.80961 q^{92} +1.04466 q^{93} +17.1467 q^{94} -1.01095 q^{95} +1.39032 q^{96} +7.40816 q^{97} +18.0345 q^{98} -6.22131 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 169 q + 6 q^{2} + 2 q^{3} + 186 q^{4} + 28 q^{5} + 5 q^{6} + 38 q^{7} + 18 q^{8} + 185 q^{9}+O(q^{10}) $ 169 * q + 6 * q^2 + 2 * q^3 + 186 * q^4 + 28 * q^5 + 5 * q^6 + 38 * q^7 + 18 * q^8 + 185 * q^9	\( 169 q + 6 q^{2} + 2 q^{3} + 186 q^{4} + 28 q^{5} + 5 q^{6} + 38 q^{7} + 18 q^{8} + 185 q^{9} + 28 q^{10} + 17 q^{11} + 10 q^{12} + 91 q^{13} + 20 q^{14} + 29 q^{15} + 200 q^{16} + 16 q^{17} + 31 q^{18} + 30 q^{19} + 45 q^{20} + 49 q^{21} + 76 q^{22} - 169 q^{23} + 3 q^{24} + 241 q^{25} - 15 q^{26} + 14 q^{27} + 118 q^{28} + 23 q^{29} + 52 q^{30} + 45 q^{31} + 42 q^{32} + 62 q^{33} + 65 q^{34} - 16 q^{35} + 199 q^{36} + 226 q^{37} + 49 q^{38} + 17 q^{39} + 95 q^{40} + 19 q^{41} + 32 q^{42} + 71 q^{43} + 46 q^{44} + 127 q^{45} - 6 q^{46} + 27 q^{47} + 5 q^{48} + 239 q^{49} + 24 q^{50} + 39 q^{51} + 154 q^{52} + 111 q^{53} + 22 q^{54} + 47 q^{55} + 39 q^{56} + 122 q^{57} + 146 q^{58} - 73 q^{59} + 109 q^{60} + 125 q^{61} + 14 q^{62} + 109 q^{63} + 260 q^{64} + 73 q^{65} + 26 q^{66} + 152 q^{67} + 40 q^{68} - 2 q^{69} + 76 q^{70} - 79 q^{71} + 126 q^{72} + 106 q^{73} + 23 q^{74} + 12 q^{75} + 122 q^{76} + 63 q^{77} + 101 q^{78} + 82 q^{79} + 134 q^{80} + 225 q^{81} + 125 q^{82} + 28 q^{83} + 73 q^{84} + 197 q^{85} + 97 q^{86} + 18 q^{87} + 183 q^{88} + 54 q^{89} + 52 q^{90} + 106 q^{91} - 186 q^{92} + 194 q^{93} + q^{94} + 18 q^{95} - 39 q^{96} + 239 q^{97} + 5 q^{98} + 85 q^{99}+O(q^{100}) \) 169 * q + 6 * q^2 + 2 * q^3 + 186 * q^4 + 28 * q^5 + 5 * q^6 + 38 * q^7 + 18 * q^8 + 185 * q^9 + 28 * q^10 + 17 * q^11 + 10 * q^12 + 91 * q^13 + 20 * q^14 + 29 * q^15 + 200 * q^16 + 16 * q^17 + 31 * q^18 + 30 * q^19 + 45 * q^20 + 49 * q^21 + 76 * q^22 - 169 * q^23 + 3 * q^24 + 241 * q^25 - 15 * q^26 + 14 * q^27 + 118 * q^28 + 23 * q^29 + 52 * q^30 + 45 * q^31 + 42 * q^32 + 62 * q^33 + 65 * q^34 - 16 * q^35 + 199 * q^36 + 226 * q^37 + 49 * q^38 + 17 * q^39 + 95 * q^40 + 19 * q^41 + 32 * q^42 + 71 * q^43 + 46 * q^44 + 127 * q^45 - 6 * q^46 + 27 * q^47 + 5 * q^48 + 239 * q^49 + 24 * q^50 + 39 * q^51 + 154 * q^52 + 111 * q^53 + 22 * q^54 + 47 * q^55 + 39 * q^56 + 122 * q^57 + 146 * q^58 - 73 * q^59 + 109 * q^60 + 125 * q^61 + 14 * q^62 + 109 * q^63 + 260 * q^64 + 73 * q^65 + 26 * q^66 + 152 * q^67 + 40 * q^68 - 2 * q^69 + 76 * q^70 - 79 * q^71 + 126 * q^72 + 106 * q^73 + 23 * q^74 + 12 * q^75 + 122 * q^76 + 63 * q^77 + 101 * q^78 + 82 * q^79 + 134 * q^80 + 225 * q^81 + 125 * q^82 + 28 * q^83 + 73 * q^84 + 197 * q^85 + 97 * q^86 + 18 * q^87 + 183 * q^88 + 54 * q^89 + 52 * q^90 + 106 * q^91 - 186 * q^92 + 194 * q^93 + q^94 + 18 * q^95 - 39 * q^96 + 239 * q^97 + 5 * q^98 + 85 * 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 $

$2$	−2.60952	−1.84521	−0.922606	−	0.385744i	$-0.873945\pi$
$2$	−2.60952	−1.84521	−0.922606	+	0.385744i	$0.873945\pi$
$3$	−0.136825	−0.0789961	−0.0394981	−	0.999220i	$-0.512576\pi$
$3$	−0.136825	−0.0789961	−0.0394981	+	0.999220i	$0.512576\pi$
$4$	4.80961	2.40481
$5$	−0.260304	−0.116412	−0.0582058	−	0.998305i	$-0.518538\pi$
$5$	−0.260304	−0.116412	−0.0582058	+	0.998305i	$0.518538\pi$
$6$	0.357049	0.145765
$7$	−0.298256	−0.112730	−0.0563650	−	0.998410i	$-0.517951\pi$
$7$	−0.298256	−0.112730	−0.0563650	+	0.998410i	$0.517951\pi$
$8$	−7.33174	−2.59216
$9$	−2.98128	−0.993760
$10$	0.679270	0.214804
$11$	2.08679	0.629191	0.314596	−	0.949226i	$-0.398131\pi$
$11$	2.08679	0.629191	0.314596	+	0.949226i	$0.398131\pi$
$12$	−0.658077	−0.189970
$13$	4.31483	1.19672	0.598359	−	0.801228i	$-0.295820\pi$
$13$	4.31483	1.19672	0.598359	+	0.801228i	$0.295820\pi$
$14$	0.778305	0.208011
$15$	0.0356162	0.00919607
$16$	9.51313	2.37828
$17$	−6.32899	−1.53501	−0.767503	−	0.641045i	$-0.778501\pi$
$17$	−6.32899	−1.53501	−0.767503	+	0.641045i	$0.778501\pi$
$18$	7.77972	1.83370
$19$	3.88371	0.890983	0.445492	−	0.895286i	$-0.353029\pi$
$19$	3.88371	0.890983	0.445492	+	0.895286i	$0.353029\pi$
$20$	−1.25196	−0.279947
$21$	0.0408089	0.00890524
$22$	−5.44553	−1.16099
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00317	0.204771
$25$	−4.93224	−0.986448
$26$	−11.2596	−2.20820
$27$	0.818390	0.157499
$28$	−1.43449	−0.271094
$29$	3.72117	0.691004	0.345502	−	0.938418i	$-0.387709\pi$
$29$	3.72117	0.691004	0.345502	+	0.938418i	$0.387709\pi$
$30$	−0.0929414	−0.0169687
$31$	−7.63496	−1.37128	−0.685640	−	0.727941i	$-0.740477\pi$
$31$	−7.63496	−1.37128	−0.685640	+	0.727941i	$0.740477\pi$
$32$	−10.1613	−1.79627
$33$	−0.285526	−0.0497037
$34$	16.5156	2.83241
$35$	0.0776373	0.0131231
$36$	−14.3388	−2.38980
$37$	−7.39885	−1.21636	−0.608182	−	0.793798i	$-0.708101\pi$
$37$	−7.39885	−1.21636	−0.608182	+	0.793798i	$0.708101\pi$
$38$	−10.1346	−1.64405
$39$	−0.590378	−0.0945361
$40$	1.90848	0.301758
$41$	6.64095	1.03714	0.518571	−	0.855034i	$-0.326464\pi$
$41$	6.64095	1.03714	0.518571	+	0.855034i	$0.326464\pi$
$42$	−0.106492	−0.0164321
$43$	9.67676	1.47569	0.737846	−	0.674969i	$-0.235843\pi$
$43$	9.67676	1.47569	0.737846	+	0.674969i	$0.235843\pi$
$44$	10.0367	1.51308
$45$	0.776040	0.115685
$46$	2.60952	0.384753
$47$	−6.57083	−0.958454	−0.479227	−	0.877691i	$-0.659083\pi$
$47$	−6.57083	−0.958454	−0.479227	+	0.877691i	$0.659083\pi$
$48$	−1.30164	−0.187875
$49$	−6.91104	−0.987292
$50$	12.8708	1.82021
$51$	0.865966	0.121260
$52$	20.7526	2.87787
$53$	5.94329	0.816373	0.408187	−	0.912899i	$-0.366161\pi$
$53$	5.94329	0.816373	0.408187	+	0.912899i	$0.366161\pi$
$54$	−2.13561	−0.290620
$55$	−0.543201	−0.0732452
$56$	2.18673	0.292215
$57$	−0.531389	−0.0703842
$58$	−9.71048	−1.27505
$59$	1.82863	0.238067	0.119034	−	0.992890i	$-0.462020\pi$
$59$	1.82863	0.238067	0.119034	+	0.992890i	$0.462020\pi$
$60$	0.171300	0.0221148
$61$	−7.98593	−1.02249	−0.511247	−	0.859434i	$-0.670816\pi$
$61$	−7.98593	−1.02249	−0.511247	+	0.859434i	$0.670816\pi$
$62$	19.9236	2.53030
$63$	0.889184	0.112027
$64$	7.48976	0.936220
$65$	−1.12317	−0.139312
$66$	0.745087	0.0917138
$67$	9.29569	1.13565	0.567825	−	0.823149i	$-0.307785\pi$
$67$	9.29569	1.13565	0.567825	+	0.823149i	$0.307785\pi$
$68$	−30.4400	−3.69139
$69$	0.136825	0.0164718
$70$	−0.202596	−0.0242149
$71$	−14.4262	−1.71207	−0.856036	−	0.516916i	$-0.827080\pi$
$71$	−14.4262	−1.71207	−0.856036	+	0.516916i	$0.827080\pi$
$72$	21.8580	2.57599
$73$	7.91292	0.926137	0.463068	−	0.886323i	$-0.346748\pi$
$73$	7.91292	0.926137	0.463068	+	0.886323i	$0.346748\pi$
$74$	19.3075	2.24445
$75$	0.674856	0.0779256
$76$	18.6791	2.14264
$77$	−0.622398	−0.0709288
$78$	1.54060	0.174439
$79$	10.6953	1.20332	0.601659	−	0.798753i	$-0.294507\pi$
$79$	10.6953	1.20332	0.601659	+	0.798753i	$0.294507\pi$
$80$	−2.47631	−0.276860
$81$	8.83186	0.981318
$82$	−17.3297	−1.91375
$83$	8.26818	0.907551	0.453775	−	0.891116i	$-0.350077\pi$
$83$	8.26818	0.907551	0.453775	+	0.891116i	$0.350077\pi$
$84$	0.196275	0.0214154
$85$	1.64746	0.178693
$86$	−25.2517	−2.72296
$87$	−0.509150	−0.0545867
$88$	−15.2998	−1.63097
$89$	−10.3377	−1.09579	−0.547896	−	0.836547i	$-0.684571\pi$
$89$	−10.3377	−1.09579	−0.547896	+	0.836547i	$0.684571\pi$
$90$	−2.02509	−0.213464
$91$	−1.28692	−0.134906
$92$	−4.80961	−0.501437
$93$	1.04466	0.108326
$94$	17.1467	1.76855
$95$	−1.01095	−0.103721
$96$	1.39032	0.141899
$97$	7.40816	0.752184	0.376092	−	0.926582i	$-0.377268\pi$
$97$	7.40816	0.752184	0.376092	+	0.926582i	$0.377268\pi$
$98$	18.0345	1.82176
$99$	−6.22131	−0.625265
$100$	−23.7222	−2.37222
$101$	−4.24724	−0.422616	−0.211308	−	0.977420i	$-0.567772\pi$
$101$	−4.24724	−0.422616	−0.211308	+	0.977420i	$0.567772\pi$
$102$	−2.25976	−0.223749
$103$	18.6646	1.83908	0.919538	−	0.393001i	$-0.128563\pi$
$103$	18.6646	1.83908	0.919538	+	0.393001i	$0.128563\pi$
$104$	−31.6352	−3.10209
$105$	−0.0106227	−0.00103667
$106$	−15.5091	−1.50638
$107$	−10.5947	−1.02423	−0.512116	−	0.858916i	$-0.671138\pi$
$107$	−10.5947	−1.02423	−0.512116	+	0.858916i	$0.671138\pi$
$108$	3.93614	0.378755
$109$	−5.08462	−0.487018	−0.243509	−	0.969899i	$-0.578299\pi$
$109$	−5.08462	−0.487018	−0.243509	+	0.969899i	$0.578299\pi$
$110$	1.41749	0.135153
$111$	1.01235	0.0960880
$112$	−2.83735	−0.268104
$113$	−18.0297	−1.69609	−0.848047	−	0.529921i	$-0.822221\pi$
$113$	−18.0297	−1.69609	−0.848047	+	0.529921i	$0.822221\pi$
$114$	1.38667	0.129874
$115$	0.260304	0.0242735
$116$	17.8974	1.66173
$117$	−12.8637	−1.18925
$118$	−4.77185	−0.439284
$119$	1.88766	0.173041
$120$	−0.261129	−0.0238377
$121$	−6.64530	−0.604118
$122$	20.8395	1.88672
$123$	−0.908651	−0.0819303
$124$	−36.7212	−3.29766
$125$	2.58541	0.231246
$126$	−2.32035	−0.206713
$127$	19.8982	1.76568	0.882840	−	0.469675i	$-0.155629\pi$
$127$	19.8982	1.76568	0.882840	+	0.469675i	$0.155629\pi$
$128$	0.777808	0.0687492
$129$	−1.32403	−0.116574
$130$	2.93093	0.257060
$131$	−20.7478	−1.81274	−0.906370	−	0.422484i	$-0.861158\pi$
$131$	−20.7478	−1.81274	−0.906370	+	0.422484i	$0.861158\pi$
$132$	−1.37327	−0.119528
$133$	−1.15834	−0.100441
$134$	−24.2573	−2.09551
$135$	−0.213031	−0.0183348
$136$	46.4025	3.97899
$137$	−9.50901	−0.812410	−0.406205	−	0.913782i	$-0.633148\pi$
$137$	−9.50901	−0.812410	−0.406205	+	0.913782i	$0.633148\pi$
$138$	−0.357049	−0.0303940
$139$	7.72833	0.655508	0.327754	−	0.944763i	$-0.393708\pi$
$139$	7.72833	0.655508	0.327754	+	0.944763i	$0.393708\pi$
$140$	0.373405	0.0315585
$141$	0.899056	0.0757141
$142$	37.6454	3.15914
$143$	9.00414	0.752964
$144$	−28.3613	−2.36344
$145$	−0.968637	−0.0804409
$146$	−20.6489	−1.70892
$147$	0.945606	0.0779923
$148$	−35.5856	−2.92512
$149$	12.1700	0.997003	0.498501	−	0.866889i	$-0.333884\pi$
$149$	12.1700	0.997003	0.498501	+	0.866889i	$0.333884\pi$
$150$	−1.76105	−0.143789
$151$	−6.62530	−0.539159	−0.269580	−	0.962978i	$-0.586885\pi$
$151$	−6.62530	−0.539159	−0.269580	+	0.962978i	$0.586885\pi$
$152$	−28.4743	−2.30957
$153$	18.8685	1.52543
$154$	1.62416	0.130879
$155$	1.98741	0.159633
$156$	−2.83949	−0.227341
$157$	−2.95154	−0.235559	−0.117779	−	0.993040i	$-0.537578\pi$
$157$	−2.95154	−0.235559	−0.117779	+	0.993040i	$0.537578\pi$
$158$	−27.9097	−2.22038
$159$	−0.813192	−0.0644903
$160$	2.64502	0.209107
$161$	0.298256	0.0235058
$162$	−23.0469	−1.81074
$163$	15.8807	1.24387	0.621935	−	0.783069i	$-0.286347\pi$
$163$	15.8807	1.24387	0.621935	+	0.783069i	$0.286347\pi$
$164$	31.9404	2.49413
$165$	0.0743236	0.00578609
$166$	−21.5760	−1.67462
$167$	−2.56946	−0.198831	−0.0994153	−	0.995046i	$-0.531697\pi$
$167$	−2.56946	−0.198831	−0.0994153	+	0.995046i	$0.531697\pi$
$168$	−0.299201	−0.0230838
$169$	5.61773	0.432133
$170$	−4.29909	−0.329725
$171$	−11.5784	−0.885423
$172$	46.5414	3.54875
$173$	−9.63925	−0.732859	−0.366429	−	0.930446i	$-0.619420\pi$
$173$	−9.63925	−0.732859	−0.366429	+	0.930446i	$0.619420\pi$
$174$	1.32864	0.100724
$175$	1.47107	0.111202
$176$	19.8519	1.49640
$177$	−0.250203	−0.0188064
$178$	26.9764	2.02197
$179$	15.8280	1.18304	0.591520	−	0.806290i	$-0.298528\pi$
$179$	15.8280	1.18304	0.591520	+	0.806290i	$0.298528\pi$
$180$	3.73245	0.278200
$181$	3.15910	0.234814	0.117407	−	0.993084i	$-0.462542\pi$
$181$	3.15910	0.234814	0.117407	+	0.993084i	$0.462542\pi$
$182$	3.35825	0.248930
$183$	1.09268	0.0807730
$184$	7.33174	0.540503
$185$	1.92595	0.141599
$186$	−2.72605	−0.199884
$187$	−13.2073	−0.965812
$188$	−31.6031	−2.30489
$189$	−0.244090	−0.0177549
$190$	2.63808	0.191387
$191$	3.07979	0.222846	0.111423	−	0.993773i	$-0.464459\pi$
$191$	3.07979	0.222846	0.111423	+	0.993773i	$0.464459\pi$
$192$	−1.02479	−0.0739578
$193$	16.7524	1.20586	0.602931	−	0.797793i	$-0.293999\pi$
$193$	16.7524	1.20586	0.602931	+	0.797793i	$0.293999\pi$
$194$	−19.3318	−1.38794
$195$	0.153678	0.0110051
$196$	−33.2394	−2.37424
$197$	24.0333	1.71230	0.856152	−	0.516724i	$-0.172849\pi$
$197$	24.0333	1.71230	0.856152	+	0.516724i	$0.172849\pi$
$198$	16.2346	1.15375
$199$	−4.35089	−0.308427	−0.154213	−	0.988038i	$-0.549284\pi$
$199$	−4.35089	−0.308427	−0.154213	+	0.988038i	$0.549284\pi$
$200$	36.1619	2.55703
$201$	−1.27189	−0.0897119
$202$	11.0833	0.779816
$203$	−1.10986	−0.0778969
$204$	4.16496	0.291606
$205$	−1.72867	−0.120735
$206$	−48.7057	−3.39348
$207$	2.98128	0.207213
$208$	41.0475	2.84613
$209$	8.10448	0.560599
$210$	0.0277203	0.00191288
$211$	9.68264	0.666580	0.333290	−	0.942824i	$-0.391841\pi$
$211$	9.68264	0.666580	0.333290	+	0.942824i	$0.391841\pi$
$212$	28.5849	1.96322
$213$	1.97387	0.135247
$214$	27.6472	1.88992
$215$	−2.51890	−0.171788
$216$	−6.00023	−0.408264
$217$	2.27717	0.154584
$218$	13.2684	0.898652
$219$	−1.08269	−0.0731612
$220$	−2.61258	−0.176140
$221$	−27.3085	−1.83697
$222$	−2.64175	−0.177303
$223$	−20.7072	−1.38666	−0.693330	−	0.720620i	$-0.743857\pi$
$223$	−20.7072	−1.38666	−0.693330	+	0.720620i	$0.743857\pi$
$224$	3.03065	0.202494
$225$	14.7044	0.980293
$226$	47.0490	3.12965
$227$	−20.8655	−1.38489	−0.692446	−	0.721470i	$-0.743467\pi$
$227$	−20.8655	−1.38489	−0.692446	+	0.721470i	$0.743467\pi$
$228$	−2.55578	−0.169260
$229$	4.39978	0.290746	0.145373	−	0.989377i	$-0.453562\pi$
$229$	4.39978	0.290746	0.145373	+	0.989377i	$0.453562\pi$
$230$	−0.679270	−0.0447897
$231$	0.0851598	0.00560310
$232$	−27.2827	−1.79120
$233$	−3.82988	−0.250904	−0.125452	−	0.992100i	$-0.540038\pi$
$233$	−3.82988	−0.250904	−0.125452	+	0.992100i	$0.540038\pi$
$234$	33.5681	2.19442
$235$	1.71041	0.111575
$236$	8.79499	0.572505
$237$	−1.46339	−0.0950574
$238$	−4.92589	−0.319298
$239$	11.1425	0.720750	0.360375	−	0.932808i	$-0.382649\pi$
$239$	11.1425	0.720750	0.360375	+	0.932808i	$0.382649\pi$
$240$	0.338822	0.0218709
$241$	−12.4826	−0.804073	−0.402036	−	0.915624i	$-0.631697\pi$
$241$	−12.4826	−0.804073	−0.402036	+	0.915624i	$0.631697\pi$
$242$	17.3411	1.11473
$243$	−3.66359	−0.235020
$244$	−38.4092	−2.45890
$245$	1.79897	0.114932
$246$	2.37114	0.151179
$247$	16.7575	1.06626
$248$	55.9776	3.55458
$249$	−1.13130	−0.0716930
$250$	−6.74667	−0.426697
$251$	−4.30270	−0.271584	−0.135792	−	0.990737i	$-0.543358\pi$
$251$	−4.30270	−0.271584	−0.135792	+	0.990737i	$0.543358\pi$
$252$	4.27663	0.269402
$253$	−2.08679	−0.131195
$254$	−51.9248	−3.25805
$255$	−0.225415	−0.0141160
$256$	−17.0092	−1.06308
$257$	−8.04651	−0.501927	−0.250964	−	0.967997i	$-0.580747\pi$
$257$	−8.04651	−0.501927	−0.250964	+	0.967997i	$0.580747\pi$
$258$	3.45508	0.215104
$259$	2.20675	0.137121
$260$	−5.40200	−0.335018
$261$	−11.0938	−0.686692
$262$	54.1418	3.34489
$263$	29.2368	1.80282	0.901410	−	0.432967i	$-0.142533\pi$
$263$	29.2368	1.80282	0.901410	+	0.432967i	$0.142533\pi$
$264$	2.09340	0.128840
$265$	−1.54706	−0.0950353
$266$	3.02271	0.185334
$267$	1.41446	0.0865633
$268$	44.7087	2.73102
$269$	27.8204	1.69624	0.848118	−	0.529807i	$-0.177736\pi$
$269$	27.8204	1.69624	0.848118	+	0.529807i	$0.177736\pi$
$270$	0.555908	0.0338315
$271$	10.8746	0.660587	0.330294	−	0.943878i	$-0.392852\pi$
$271$	10.8746	0.660587	0.330294	+	0.943878i	$0.392852\pi$
$272$	−60.2085	−3.65068
$273$	0.176084	0.0106571
$274$	24.8140	1.49907
$275$	−10.2926	−0.620665
$276$	0.658077	0.0396116
$277$	14.3497	0.862191	0.431096	−	0.902306i	$-0.358127\pi$
$277$	14.3497	0.862191	0.431096	+	0.902306i	$0.358127\pi$
$278$	−20.1672	−1.20955
$279$	22.7619	1.36272
$280$	−0.569216	−0.0340172
$281$	14.2456	0.849821	0.424910	−	0.905235i	$-0.360306\pi$
$281$	14.2456	0.849821	0.424910	+	0.905235i	$0.360306\pi$
$282$	−2.34611	−0.139709
$283$	10.4238	0.619630	0.309815	−	0.950797i	$-0.399733\pi$
$283$	10.4238	0.619630	0.309815	+	0.950797i	$0.399733\pi$
$284$	−69.3843	−4.11720
$285$	0.138323	0.00819354
$286$	−23.4965	−1.38938
$287$	−1.98070	−0.116917
$288$	30.2935	1.78506
$289$	23.0561	1.35624
$290$	2.52768	0.148430
$291$	−1.01362	−0.0594197
$292$	38.0580	2.22718
$293$	−12.8396	−0.750098	−0.375049	−	0.927005i	$-0.622374\pi$
$293$	−12.8396	−0.750098	−0.375049	+	0.927005i	$0.622374\pi$
$294$	−2.46758	−0.143912
$295$	−0.476000	−0.0277138
$296$	54.2465	3.15301
$297$	1.70781	0.0990972
$298$	−31.7578	−1.83968
$299$	−4.31483	−0.249533
$300$	3.24579	0.187396
$301$	−2.88615	−0.166355
$302$	17.2889	0.994863
$303$	0.581130	0.0333850
$304$	36.9462	2.11901
$305$	2.07877	0.119030
$306$	−49.2378	−2.81473
$307$	16.0332	0.915061	0.457531	−	0.889194i	$-0.348734\pi$
$307$	16.0332	0.915061	0.457531	+	0.889194i	$0.348734\pi$
$308$	−2.99349	−0.170570
$309$	−2.55379	−0.145280
$310$	−5.18620	−0.294556
$311$	3.71371	0.210585	0.105293	−	0.994441i	$-0.466422\pi$
$311$	3.71371	0.210585	0.105293	+	0.994441i	$0.466422\pi$
$312$	4.32850	0.245053
$313$	6.21531	0.351310	0.175655	−	0.984452i	$-0.443796\pi$
$313$	6.21531	0.351310	0.175655	+	0.984452i	$0.443796\pi$
$314$	7.70211	0.434655
$315$	−0.231458	−0.0130412
$316$	51.4403	2.89374
$317$	−9.08883	−0.510479	−0.255240	−	0.966878i	$-0.582154\pi$
$317$	−9.08883	−0.510479	−0.255240	+	0.966878i	$0.582154\pi$
$318$	2.12204	0.118998
$319$	7.76531	0.434774
$320$	−1.94962	−0.108987
$321$	1.44963	0.0809103
$322$	−0.778305	−0.0433733
$323$	−24.5799	−1.36766
$324$	42.4778	2.35988
$325$	−21.2818	−1.18050
$326$	−41.4410	−2.29520
$327$	0.695705	0.0384726
$328$	−48.6898	−2.68844
$329$	1.95979	0.108047
$330$	−0.193949	−0.0106766
$331$	−11.2157	−0.616469	−0.308234	−	0.951310i	$-0.599738\pi$
$331$	−11.2157	−0.616469	−0.308234	+	0.951310i	$0.599738\pi$
$332$	39.7667	2.18248
$333$	22.0580	1.20877
$334$	6.70506	0.366885
$335$	−2.41971	−0.132203
$336$	0.388221	0.0211792
$337$	−6.31497	−0.343999	−0.171999	−	0.985097i	$-0.555023\pi$
$337$	−6.31497	−0.343999	−0.171999	+	0.985097i	$0.555023\pi$
$338$	−14.6596	−0.797376
$339$	2.46692	0.133985
$340$	7.92366	0.429721
$341$	−15.9326	−0.862797
$342$	30.2141	1.63379
$343$	4.14905	0.224028
$344$	−70.9475	−3.82523
$345$	−0.0356162	−0.00191751
$346$	25.1539	1.35228
$347$	10.0761	0.540911	0.270456	−	0.962732i	$-0.412826\pi$
$347$	10.0761	0.540911	0.270456	+	0.962732i	$0.412826\pi$
$348$	−2.44882	−0.131270
$349$	1.00000	0.0535288
$349$	1.00000	0.0535288
$350$	−3.83879	−0.205192
$351$	3.53121	0.188482
$352$	−21.2044	−1.13020
$353$	−16.1363	−0.858849	−0.429424	−	0.903103i	$-0.641284\pi$
$353$	−16.1363	−0.858849	−0.429424	+	0.903103i	$0.641284\pi$
$354$	0.652910	0.0347018
$355$	3.75520	0.199305
$356$	−49.7202	−2.63516
$357$	−0.258279	−0.0136696
$358$	−41.3035	−2.18296
$359$	1.33750	0.0705905	0.0352953	−	0.999377i	$-0.488763\pi$
$359$	1.33750	0.0705905	0.0352953	+	0.999377i	$0.488763\pi$
$360$	−5.68972	−0.299875
$361$	−3.91683	−0.206149
$362$	−8.24375	−0.433282
$363$	0.909246	0.0477230
$364$	−6.18959	−0.324423
$365$	−2.05977	−0.107813
$366$	−2.85137	−0.149043
$367$	1.25479	0.0654996	0.0327498	−	0.999464i	$-0.489574\pi$
$367$	1.25479	0.0654996	0.0327498	+	0.999464i	$0.489574\pi$
$368$	−9.51313	−0.495906
$369$	−19.7985	−1.03067
$370$	−5.02582	−0.261280
$371$	−1.77262	−0.0920298
$372$	5.02439	0.260502
$373$	12.3302	0.638431	0.319216	−	0.947682i	$-0.396581\pi$
$373$	12.3302	0.638431	0.319216	+	0.947682i	$0.396581\pi$
$374$	34.4647	1.78213
$375$	−0.353749	−0.0182675
$376$	48.1756	2.48447
$377$	16.0562	0.826937
$378$	0.636958	0.0327616
$379$	−17.0521	−0.875905	−0.437953	−	0.898998i	$-0.644296\pi$
$379$	−17.0521	−0.875905	−0.437953	+	0.898998i	$0.644296\pi$
$380$	−4.86225	−0.249428
$381$	−2.72258	−0.139482
$382$	−8.03678	−0.411197
$383$	−20.5393	−1.04951	−0.524755	−	0.851253i	$-0.675843\pi$
$383$	−20.5393	−1.04951	−0.524755	+	0.851253i	$0.675843\pi$
$384$	−0.106424	−0.00543092
$385$	0.162013	0.00825694
$386$	−43.7157	−2.22507
$387$	−28.8491	−1.46648
$388$	35.6304	1.80886
$389$	−24.0823	−1.22102	−0.610511	−	0.792008i	$-0.709036\pi$
$389$	−24.0823	−1.22102	−0.610511	+	0.792008i	$0.709036\pi$
$390$	−0.401026	−0.0203067
$391$	6.32899	0.320071
$392$	50.6700	2.55922
$393$	2.83882	0.143200
$394$	−62.7155	−3.15956
$395$	−2.78404	−0.140080
$396$	−29.9221	−1.50364
$397$	28.5107	1.43091	0.715457	−	0.698657i	$-0.246219\pi$
$397$	28.5107	1.43091	0.715457	+	0.698657i	$0.246219\pi$
$398$	11.3538	0.569113
$399$	0.158490	0.00793442
$400$	−46.9211	−2.34605
$401$	−9.20397	−0.459624	−0.229812	−	0.973235i	$-0.573811\pi$
$401$	−9.20397	−0.459624	−0.229812	+	0.973235i	$0.573811\pi$
$402$	3.31902	0.165537
$403$	−32.9435	−1.64103
$404$	−20.4276	−1.01631
$405$	−2.29897	−0.114237
$406$	2.89621	0.143736
$407$	−15.4399	−0.765325
$408$	−6.34904	−0.314324
$409$	−35.6556	−1.76306	−0.881528	−	0.472131i	$-0.843485\pi$
$409$	−35.6556	−1.76306	−0.881528	+	0.472131i	$0.843485\pi$
$410$	4.51100	0.222782
$411$	1.30107	0.0641772
$412$	89.7694	4.42262
$413$	−0.545399	−0.0268373
$414$	−7.77972	−0.382352
$415$	−2.15224	−0.105649
$416$	−43.8440	−2.14963
$417$	−1.05743	−0.0517826
$418$	−21.1488	−1.03442
$419$	−15.0993	−0.737648	−0.368824	−	0.929499i	$-0.620240\pi$
$419$	−15.0993	−0.737648	−0.368824	+	0.929499i	$0.620240\pi$
$420$	−0.0510913	−0.00249300
$421$	−21.3173	−1.03894	−0.519470	−	0.854489i	$-0.673871\pi$
$421$	−21.3173	−1.03894	−0.519470	+	0.854489i	$0.673871\pi$
$422$	−25.2671	−1.22998
$423$	19.5895	0.952473
$424$	−43.5747	−2.11617
$425$	31.2161	1.51420
$426$	−5.15085	−0.249560
$427$	2.38185	0.115266
$428$	−50.9565	−2.46308
$429$	−1.23199	−0.0594813
$430$	6.57313	0.316985
$431$	−30.7429	−1.48083	−0.740417	−	0.672148i	$-0.765372\pi$
$431$	−30.7429	−1.48083	−0.740417	+	0.672148i	$0.765372\pi$
$432$	7.78546	0.374578
$433$	−15.1283	−0.727018	−0.363509	−	0.931591i	$-0.618421\pi$
$433$	−15.1283	−0.727018	−0.363509	+	0.931591i	$0.618421\pi$
$434$	−5.94233	−0.285241
$435$	0.132534	0.00635452
$436$	−24.4550	−1.17118
$437$	−3.88371	−0.185783
$438$	2.82530	0.134998
$439$	9.52592	0.454647	0.227324	−	0.973819i	$-0.427002\pi$
$439$	9.52592	0.454647	0.227324	+	0.973819i	$0.427002\pi$
$440$	3.98261	0.189863
$441$	20.6037	0.981131
$442$	71.2622	3.38959
$443$	−22.2898	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$443$	−22.2898	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$444$	4.86901	0.231073
$445$	2.69094	0.127563
$446$	54.0360	2.55868
$447$	−1.66516	−0.0787594
$448$	−2.23386	−0.105540
$449$	3.80839	0.179729	0.0898646	−	0.995954i	$-0.471357\pi$
$449$	3.80839	0.179729	0.0898646	+	0.995954i	$0.471357\pi$
$450$	−38.3714	−1.80885
$451$	13.8583	0.652561
$452$	−86.7159	−4.07877
$453$	0.906509	0.0425915
$454$	54.4490	2.55542
$455$	0.334991	0.0157046
$456$	3.89601	0.182447
$457$	21.9729	1.02785	0.513924	−	0.857836i	$-0.328191\pi$
$457$	21.9729	1.02785	0.513924	+	0.857836i	$0.328191\pi$
$458$	−11.4813	−0.536488
$459$	−5.17959	−0.241762
$460$	1.25196	0.0583730
$461$	6.78400	0.315962	0.157981	−	0.987442i	$-0.449501\pi$
$461$	6.78400	0.315962	0.157981	+	0.987442i	$0.449501\pi$
$462$	−0.222226	−0.0103389
$463$	5.73723	0.266631	0.133316	−	0.991074i	$-0.457438\pi$
$463$	5.73723	0.266631	0.133316	+	0.991074i	$0.457438\pi$
$464$	35.4000	1.64340
$465$	−0.271928	−0.0126104
$466$	9.99417	0.462971
$467$	14.4325	0.667858	0.333929	−	0.942598i	$-0.391625\pi$
$467$	14.4325	0.667858	0.333929	+	0.942598i	$0.391625\pi$
$468$	−61.8694	−2.85991
$469$	−2.77249	−0.128022
$470$	−4.46337	−0.205880
$471$	0.403845	0.0186082
$472$	−13.4070	−0.617109
$473$	20.1934	0.928493
$474$	3.81875	0.175401
$475$	−19.1554	−0.878909
$476$	9.07890	0.416131
$477$	−17.7186	−0.811279
$478$	−29.0767	−1.32994
$479$	6.52365	0.298073	0.149037	−	0.988832i	$-0.452383\pi$
$479$	6.52365	0.298073	0.149037	+	0.988832i	$0.452383\pi$
$480$	−0.361905	−0.0165186
$481$	−31.9248	−1.45564
$482$	32.5735	1.48368
$483$	−0.0408089	−0.00185687
$484$	−31.9613	−1.45279
$485$	−1.92838	−0.0875630
$486$	9.56023	0.433661
$487$	3.71651	0.168411	0.0842055	−	0.996448i	$-0.473165\pi$
$487$	3.71651	0.168411	0.0842055	+	0.996448i	$0.473165\pi$
$488$	58.5508	2.65047
$489$	−2.17288	−0.0982610
$490$	−4.69446	−0.212074
$491$	40.0857	1.80904	0.904520	−	0.426431i	$-0.140229\pi$
$491$	40.0857	1.80904	0.904520	+	0.426431i	$0.140229\pi$
$492$	−4.37026	−0.197026
$493$	−23.5513	−1.06070
$494$	−43.7291	−1.96747
$495$	1.61943	0.0727881
$496$	−72.6324	−3.26129
$497$	4.30269	0.193002
$498$	2.95214	0.132289
$499$	34.0832	1.52577	0.762887	−	0.646532i	$-0.223781\pi$
$499$	34.0832	1.52577	0.762887	+	0.646532i	$0.223781\pi$
$500$	12.4348	0.556101
$501$	0.351567	0.0157069
$502$	11.2280	0.501129
$503$	13.2431	0.590482	0.295241	−	0.955423i	$-0.404600\pi$
$503$	13.2431	0.590482	0.295241	+	0.955423i	$0.404600\pi$
$504$	−6.51927	−0.290391
$505$	1.10557	0.0491974
$506$	5.44553	0.242083
$507$	−0.768647	−0.0341368
$508$	95.7026	4.24611
$509$	−32.8669	−1.45680	−0.728400	−	0.685153i	$-0.759736\pi$
$509$	−32.8669	−1.45680	−0.728400	+	0.685153i	$0.759736\pi$
$510$	0.588225	0.0260470
$511$	−2.36007	−0.104403
$512$	42.8304	1.89285
$513$	3.17839	0.140329
$514$	20.9975	0.926162
$515$	−4.85847	−0.214090
$516$	−6.36805	−0.280338
$517$	−13.7119	−0.603051
$518$	−5.75856	−0.253017
$519$	1.31889	0.0578930
$520$	8.23478	0.361119
$521$	−14.9212	−0.653709	−0.326854	−	0.945075i	$-0.605989\pi$
$521$	−14.9212	−0.653709	−0.326854	+	0.945075i	$0.605989\pi$
$522$	28.9497	1.26709
$523$	−11.0695	−0.484037	−0.242019	−	0.970272i	$-0.577809\pi$
$523$	−11.0695	−0.484037	−0.242019	+	0.970272i	$0.577809\pi$
$524$	−99.7887	−4.35929
$525$	−0.201280	−0.00878456
$526$	−76.2941	−3.32658
$527$	48.3216	2.10492
$528$	−2.71625	−0.118209
$529$	1.00000	0.0434783
$530$	4.03710	0.175360
$531$	−5.45165	−0.236581
$532$	−5.57115	−0.241540
$533$	28.6546	1.24117
$534$	−3.69105	−0.159728
$535$	2.75785	0.119232
$536$	−68.1536	−2.94379
$537$	−2.16567	−0.0934556
$538$	−72.5979	−3.12992
$539$	−14.4219	−0.621196
$540$	−1.02459	−0.0440915
$541$	30.3889	1.30652	0.653261	−	0.757133i	$-0.273401\pi$
$541$	30.3889	1.30652	0.653261	+	0.757133i	$0.273401\pi$
$542$	−28.3776	−1.21892
$543$	−0.432245	−0.0185494
$544$	64.3105	2.75729
$545$	1.32355	0.0566946
$546$	−0.459494	−0.0196645
$547$	13.5171	0.577949	0.288975	−	0.957337i	$-0.406686\pi$
$547$	13.5171	0.577949	0.288975	+	0.957337i	$0.406686\pi$
$548$	−45.7346	−1.95369
$549$	23.8083	1.01611
$550$	26.8587	1.14526
$551$	14.4519	0.615673
$552$	−1.00317	−0.0426977
$553$	−3.18994	−0.135650
$554$	−37.4459	−1.59093
$555$	−0.263519	−0.0111858
$556$	37.1702	1.57637
$557$	8.23668	0.349000	0.174500	−	0.984657i	$-0.444169\pi$
$557$	8.23668	0.349000	0.174500	+	0.984657i	$0.444169\pi$
$558$	−59.3978	−2.51451
$559$	41.7535	1.76599
$560$	0.738574	0.0312104
$561$	1.80709	0.0762954
$562$	−37.1742	−1.56810
$563$	43.8534	1.84820	0.924100	−	0.382150i	$-0.124816\pi$
$563$	43.8534	1.84820	0.924100	+	0.382150i	$0.124816\pi$
$564$	4.32411	0.182078
$565$	4.69321	0.197445
$566$	−27.2011	−1.14335
$567$	−2.63415	−0.110624
$568$	105.769	4.43797
$569$	7.72464	0.323834	0.161917	−	0.986804i	$-0.448232\pi$
$569$	7.72464	0.323834	0.161917	+	0.986804i	$0.448232\pi$
$570$	−0.360957	−0.0151188
$571$	−33.8653	−1.41722	−0.708610	−	0.705600i	$-0.750678\pi$
$571$	−33.8653	−1.41722	−0.708610	+	0.705600i	$0.750678\pi$
$572$	43.3064	1.81073
$573$	−0.421393	−0.0176039
$574$	5.16869	0.215737
$575$	4.93224	0.205689
$576$	−22.3291	−0.930378
$577$	−7.68826	−0.320067	−0.160033	−	0.987112i	$-0.551160\pi$
$577$	−7.68826	−0.320067	−0.160033	+	0.987112i	$0.551160\pi$
$578$	−60.1655	−2.50255
$579$	−2.29215	−0.0952585
$580$	−4.65877	−0.193445
$581$	−2.46603	−0.102308
$582$	2.64507	0.109642
$583$	12.4024	0.513655
$584$	−58.0155	−2.40070
$585$	3.34848	0.138442
$586$	33.5052	1.38409
$587$	20.5113	0.846594	0.423297	−	0.905991i	$-0.360873\pi$
$587$	20.5113	0.846594	0.423297	+	0.905991i	$0.360873\pi$
$588$	4.54800	0.187556
$589$	−29.6519	−1.22179
$590$	1.24213	0.0511378
$591$	−3.28837	−0.135265
$592$	−70.3863	−2.89286
$593$	28.9987	1.19083	0.595416	−	0.803418i	$-0.296987\pi$
$593$	28.9987	1.19083	0.595416	+	0.803418i	$0.296987\pi$
$594$	−4.45657	−0.182855
$595$	−0.491365	−0.0201440
$596$	58.5328	2.39760
$597$	0.595313	0.0243645
$598$	11.2596	0.460441
$599$	14.7230	0.601566	0.300783	−	0.953693i	$-0.402752\pi$
$599$	14.7230	0.601566	0.300783	+	0.953693i	$0.402752\pi$
$600$	−4.94787	−0.201996
$601$	28.5244	1.16354	0.581768	−	0.813355i	$-0.302361\pi$
$601$	28.5244	1.16354	0.581768	+	0.813355i	$0.302361\pi$
$602$	7.53147	0.306960
$603$	−27.7130	−1.12856
$604$	−31.8651	−1.29657
$605$	1.72980	0.0703264
$606$	−1.51647	−0.0616024
$607$	19.2723	0.782239	0.391120	−	0.920340i	$-0.372088\pi$
$607$	19.2723	0.782239	0.391120	+	0.920340i	$0.372088\pi$
$608$	−39.4633	−1.60045
$609$	0.151857	0.00615356
$610$	−5.42460	−0.219636
$611$	−28.3520	−1.14700
$612$	90.7501	3.66835
$613$	−13.5189	−0.546023	−0.273011	−	0.962011i	$-0.588020\pi$
$613$	−13.5189	−0.546023	−0.273011	+	0.962011i	$0.588020\pi$
$614$	−41.8389	−1.68848
$615$	0.236526	0.00953763
$616$	4.56326	0.183859
$617$	4.17915	0.168246	0.0841231	−	0.996455i	$-0.473191\pi$
$617$	4.17915	0.168246	0.0841231	+	0.996455i	$0.473191\pi$
$618$	6.66417	0.268072
$619$	−0.137385	−0.00552196	−0.00276098	−	0.999996i	$-0.500879\pi$
$619$	−0.137385	−0.00552196	−0.00276098	+	0.999996i	$0.500879\pi$
$620$	9.55868	0.383886
$621$	−0.818390	−0.0328409
$622$	−9.69102	−0.388574
$623$	3.08327	0.123529
$624$	−5.61634	−0.224834
$625$	23.9882	0.959529
$626$	−16.2190	−0.648241
$627$	−1.10890	−0.0442851
$628$	−14.1958	−0.566472
$629$	46.8273	1.86713
$630$	0.603996	0.0240638
$631$	1.90570	0.0758648	0.0379324	−	0.999280i	$-0.487923\pi$
$631$	1.90570	0.0758648	0.0379324	+	0.999280i	$0.487923\pi$
$632$	−78.4153	−3.11919
$633$	−1.32483	−0.0526573
$634$	23.7175	0.941943
$635$	−5.17959	−0.205546
$636$	−3.91114	−0.155087
$637$	−29.8200	−1.18151
$638$	−20.2637	−0.802250
$639$	43.0085	1.70139
$640$	−0.202467	−0.00800320
$641$	31.0166	1.22508	0.612542	−	0.790438i	$-0.290147\pi$
$641$	31.0166	1.22508	0.612542	+	0.790438i	$0.290147\pi$
$642$	−3.78284	−0.149297
$643$	−13.1214	−0.517457	−0.258729	−	0.965950i	$-0.583304\pi$
$643$	−13.1214	−0.517457	−0.258729	+	0.965950i	$0.583304\pi$
$644$	1.43449	0.0565270
$645$	0.344650	0.0135706
$646$	64.1419	2.52363
$647$	28.3537	1.11470	0.557350	−	0.830278i	$-0.311818\pi$
$647$	28.3537	1.11470	0.557350	+	0.830278i	$0.311818\pi$
$648$	−64.7529	−2.54374
$649$	3.81597	0.149790
$650$	55.5353	2.17827
$651$	−0.311575	−0.0122116
$652$	76.3799	2.99127
$653$	−44.8367	−1.75460	−0.877298	−	0.479947i	$-0.840656\pi$
$653$	−44.8367	−1.75460	−0.877298	+	0.479947i	$0.840656\pi$
$654$	−1.81546	−0.0709900
$655$	5.40073	0.211024
$656$	63.1763	2.46662
$657$	−23.5906	−0.920357
$658$	−5.11411	−0.199369
$659$	3.77121	0.146905	0.0734527	−	0.997299i	$-0.476598\pi$
$659$	3.77121	0.146905	0.0734527	+	0.997299i	$0.476598\pi$
$660$	0.357468	0.0139144
$661$	17.4040	0.676938	0.338469	−	0.940977i	$-0.390091\pi$
$661$	17.4040	0.676938	0.338469	+	0.940977i	$0.390091\pi$
$662$	29.2675	1.13752
$663$	3.73649	0.145113
$664$	−60.6202	−2.35252
$665$	0.301520	0.0116925
$666$	−57.5610	−2.23044
$667$	−3.72117	−0.144084
$668$	−12.3581	−0.478149
$669$	2.83328	0.109541
$670$	6.31429	0.243942
$671$	−16.6650	−0.643344
$672$	−0.414670	−0.0159962
$673$	6.27365	0.241831	0.120916	−	0.992663i	$-0.461417\pi$
$673$	6.27365	0.241831	0.120916	+	0.992663i	$0.461417\pi$
$674$	16.4791	0.634750
$675$	−4.03650	−0.155365
$676$	27.0191	1.03920
$677$	−33.6701	−1.29405	−0.647024	−	0.762470i	$-0.723987\pi$
$677$	−33.6701	−1.29405	−0.647024	+	0.762470i	$0.723987\pi$
$678$	−6.43749	−0.247230
$679$	−2.20953	−0.0847938
$680$	−12.0788	−0.463200
$681$	2.85493	0.109401
$682$	41.5764	1.59204
$683$	49.9070	1.90964	0.954819	−	0.297188i	$-0.0960489\pi$
$683$	49.9070	1.90964	0.954819	+	0.297188i	$0.0960489\pi$
$684$	−55.6876	−2.12927
$685$	2.47524	0.0945739
$686$	−10.8270	−0.413378
$687$	−0.602002	−0.0229678
$688$	92.0563	3.50961
$689$	25.6442	0.976968
$690$	0.0929414	0.00353822
$691$	40.7728	1.55107	0.775536	−	0.631304i	$-0.217480\pi$
$691$	40.7728	1.55107	0.775536	+	0.631304i	$0.217480\pi$
$692$	−46.3611	−1.76238
$693$	1.85554	0.0704862
$694$	−26.2937	−0.998095
$695$	−2.01172	−0.0763088
$696$	3.73296	0.141498
$697$	−42.0305	−1.59202
$698$	−2.60952	−0.0987719
$699$	0.524025	0.0198204
$700$	7.07527	0.267420
$701$	20.0813	0.758459	0.379229	−	0.925303i	$-0.376189\pi$
$701$	20.0813	0.758459	0.379229	+	0.925303i	$0.376189\pi$
$702$	−9.21478	−0.347790
$703$	−28.7350	−1.08376
$704$	15.6296	0.589061
$705$	−0.234028	−0.00881401
$706$	42.1081	1.58476
$707$	1.26676	0.0476415
$708$	−1.20338	−0.0452257
$709$	−12.7076	−0.477243	−0.238621	−	0.971113i	$-0.576695\pi$
$709$	−12.7076	−0.477243	−0.238621	+	0.971113i	$0.576695\pi$
$710$	−9.79927	−0.367760
$711$	−31.8857	−1.19581
$712$	75.7932	2.84047
$713$	7.63496	0.285931
$714$	0.673986	0.0252233
$715$	−2.34382	−0.0876538
$716$	76.1265	2.84498
$717$	−1.52458	−0.0569364
$718$	−3.49024	−0.130254
$719$	7.55718	0.281835	0.140918	−	0.990021i	$-0.454995\pi$
$719$	7.55718	0.281835	0.140918	+	0.990021i	$0.454995\pi$
$720$	7.38257	0.275132
$721$	−5.56682	−0.207319
$722$	10.2211	0.380389
$723$	1.70793	0.0635187
$724$	15.1941	0.564683
$725$	−18.3537	−0.681640
$726$	−2.37270	−0.0880591
$727$	34.8265	1.29164	0.645822	−	0.763488i	$-0.276515\pi$
$727$	34.8265	1.29164	0.645822	+	0.763488i	$0.276515\pi$
$728$	9.43538	0.349699
$729$	−25.9943	−0.962752
$730$	5.37501	0.198938
$731$	−61.2441	−2.26520
$732$	5.25535	0.194243
$733$	17.9429	0.662736	0.331368	−	0.943502i	$-0.392490\pi$
$733$	17.9429	0.662736	0.331368	+	0.943502i	$0.392490\pi$
$734$	−3.27441	−0.120861
$735$	−0.246145	−0.00907920
$736$	10.1613	0.374549
$737$	19.3982	0.714541
$738$	51.6647	1.90180
$739$	−31.0275	−1.14137	−0.570683	−	0.821170i	$-0.693322\pi$
$739$	−31.0275	−1.14137	−0.570683	+	0.821170i	$0.693322\pi$
$740$	9.26308	0.340518
$741$	−2.29285	−0.0842300
$742$	4.62569	0.169814
$743$	26.5476	0.973937	0.486969	−	0.873419i	$-0.338103\pi$
$743$	26.5476	0.973937	0.486969	+	0.873419i	$0.338103\pi$
$744$	−7.65915	−0.280798
$745$	−3.16789	−0.116063
$746$	−32.1758	−1.17804
$747$	−24.6498	−0.901887
$748$	−63.5219	−2.32259
$749$	3.15994	0.115462
$750$	0.923116	0.0337074
$751$	38.7832	1.41522	0.707609	−	0.706604i	$-0.249774\pi$
$751$	38.7832	1.41522	0.707609	+	0.706604i	$0.249774\pi$
$752$	−62.5092	−2.27947
$753$	0.588718	0.0214541
$754$	−41.8990	−1.52587
$755$	1.72459	0.0627644
$756$	−1.17398	−0.0426971
$757$	−41.2666	−1.49986	−0.749931	−	0.661516i	$-0.769913\pi$
$757$	−41.2666	−1.49986	−0.749931	+	0.661516i	$0.769913\pi$
$758$	44.4977	1.61623
$759$	0.285526	0.0103639
$760$	7.41199	0.268861
$761$	−33.1733	−1.20253	−0.601266	−	0.799049i	$-0.705337\pi$
$761$	−33.1733	−1.20253	−0.601266	+	0.799049i	$0.705337\pi$
$762$	7.10463	0.257374
$763$	1.51652	0.0549016
$764$	14.8126	0.535900
$765$	−4.91155	−0.177577
$766$	53.5978	1.93657
$767$	7.89021	0.284899
$768$	2.32729	0.0839790
$769$	37.9314	1.36784	0.683921	−	0.729556i	$-0.260273\pi$
$769$	37.9314	1.36784	0.683921	+	0.729556i	$0.260273\pi$
$770$	−0.422776	−0.0152358
$771$	1.10097	0.0396503
$772$	80.5724	2.89986
$773$	11.7123	0.421262	0.210631	−	0.977566i	$-0.432448\pi$
$773$	11.7123	0.421262	0.210631	+	0.977566i	$0.432448\pi$
$774$	75.2824	2.70597
$775$	37.6575	1.35270
$776$	−54.3147	−1.94978
$777$	−0.301939	−0.0108320
$778$	62.8433	2.25304
$779$	25.7915	0.924077
$780$	0.739130	0.0264651
$781$	−30.1044	−1.07722
$782$	−16.5156	−0.590598
$783$	3.04537	0.108833
$784$	−65.7457	−2.34806
$785$	0.768299	0.0274218
$786$	−7.40797	−0.264233
$787$	−17.4866	−0.623330	−0.311665	−	0.950192i	$-0.600887\pi$
$787$	−17.4866	−0.623330	−0.311665	+	0.950192i	$0.600887\pi$
$788$	115.591	4.11776
$789$	−4.00034	−0.142416
$790$	7.26501	0.258477
$791$	5.37747	0.191201
$792$	45.6130	1.62079
$793$	−34.4579	−1.22364
$794$	−74.3994	−2.64034
$795$	0.211677	0.00750742
$796$	−20.9261	−0.741706
$797$	37.3579	1.32328	0.661642	−	0.749820i	$-0.269860\pi$
$797$	37.3579	1.32328	0.661642	+	0.749820i	$0.269860\pi$
$798$	−0.413583	−0.0146407
$799$	41.5867	1.47123
$800$	50.1178	1.77193
$801$	30.8195	1.08895
$802$	24.0180	0.848104
$803$	16.5126	0.582717
$804$	−6.11728	−0.215740
$805$	−0.0776373	−0.00273635
$806$	85.9669	3.02805
$807$	−3.80653	−0.133996
$808$	31.1397	1.09549
$809$	43.8028	1.54002	0.770012	−	0.638029i	$-0.220250\pi$
$809$	43.8028	1.54002	0.770012	+	0.638029i	$0.220250\pi$
$810$	5.99922	0.210791
$811$	22.4437	0.788106	0.394053	−	0.919088i	$-0.371073\pi$
$811$	22.4437	0.788106	0.394053	+	0.919088i	$0.371073\pi$
$812$	−5.33800	−0.187327
$813$	−1.48793	−0.0521838
$814$	40.2907	1.41219
$815$	−4.13381	−0.144801
$816$	8.23805	0.288390
$817$	37.5817	1.31482
$818$	93.0441	3.25321
$819$	3.83667	0.134064
$820$	−8.31422	−0.290345
$821$	41.6125	1.45229	0.726143	−	0.687543i	$-0.241311\pi$
$821$	41.6125	1.45229	0.726143	+	0.687543i	$0.241311\pi$
$822$	−3.39518	−0.118421
$823$	−20.1468	−0.702273	−0.351137	−	0.936324i	$-0.614205\pi$
$823$	−20.1468	−0.702273	−0.351137	+	0.936324i	$0.614205\pi$
$824$	−136.844	−4.76718
$825$	1.40828	0.0490301
$826$	1.42323	0.0495205
$827$	31.0385	1.07931	0.539657	−	0.841885i	$-0.318554\pi$
$827$	31.0385	1.07931	0.539657	+	0.841885i	$0.318554\pi$
$828$	14.3388	0.498307
$829$	−33.0259	−1.14704	−0.573518	−	0.819193i	$-0.694422\pi$
$829$	−33.0259	−1.14704	−0.573518	+	0.819193i	$0.694422\pi$
$830$	5.61633	0.194946
$831$	−1.96341	−0.0681098
$832$	32.3170	1.12039
$833$	43.7399	1.51550
$834$	2.75939	0.0955499
$835$	0.668841	0.0231462
$836$	38.9794	1.34813
$837$	−6.24838	−0.215976
$838$	39.4019	1.36112
$839$	−5.60288	−0.193433	−0.0967164	−	0.995312i	$-0.530834\pi$
$839$	−5.60288	−0.193433	−0.0967164	+	0.995312i	$0.530834\pi$
$840$	0.0778832	0.00268723
$841$	−15.1529	−0.522513
$842$	55.6279	1.91706
$843$	−1.94916	−0.0671325
$844$	46.5697	1.60300
$845$	−1.46232	−0.0503053
$846$	−51.1192	−1.75751
$847$	1.98200	0.0681023
$848$	56.5393	1.94157
$849$	−1.42624	−0.0489484
$850$	−81.4592	−2.79403
$851$	7.39885	0.253629
$852$	9.49353	0.325243
$853$	38.9521	1.33369	0.666847	−	0.745195i	$-0.267643\pi$
$853$	38.9521	1.33369	0.666847	+	0.745195i	$0.267643\pi$
$854$	−6.21549	−0.212690
$855$	3.01391	0.103074
$856$	77.6779	2.65498
$857$	39.7943	1.35935	0.679674	−	0.733515i	$-0.262121\pi$
$857$	39.7943	1.35935	0.679674	+	0.733515i	$0.262121\pi$
$858$	3.21492	0.109756
$859$	−2.57300	−0.0877895	−0.0438948	−	0.999036i	$-0.513977\pi$
$859$	−2.57300	−0.0877895	−0.0438948	+	0.999036i	$0.513977\pi$
$860$	−12.1149	−0.413116
$861$	0.271010	0.00923601
$862$	80.2243	2.73245
$863$	26.9018	0.915749	0.457875	−	0.889017i	$-0.348611\pi$
$863$	26.9018	0.915749	0.457875	+	0.889017i	$0.348611\pi$
$864$	−8.31587	−0.282912
$865$	2.50914	0.0853133
$866$	39.4776	1.34150
$867$	−3.15466	−0.107138
$868$	10.9523	0.371745
$869$	22.3189	0.757117
$870$	−0.345851	−0.0117254
$871$	40.1093	1.35905
$872$	37.2791	1.26243
$873$	−22.0858	−0.747490
$874$	10.1346	0.342809
$875$	−0.771112	−0.0260683
$876$	−5.20730	−0.175939
$877$	−21.1257	−0.713365	−0.356683	−	0.934226i	$-0.616092\pi$
$877$	−21.1257	−0.713365	−0.356683	+	0.934226i	$0.616092\pi$
$878$	−24.8581	−0.838920
$879$	1.75678	0.0592548
$880$	−5.16754	−0.174198
$881$	19.8646	0.669255	0.334628	−	0.942350i	$-0.391389\pi$
$881$	19.8646	0.669255	0.334628	+	0.942350i	$0.391389\pi$
$882$	−53.7660	−1.81039
$883$	29.8703	1.00522	0.502608	−	0.864514i	$-0.332374\pi$
$883$	29.8703	1.00522	0.502608	+	0.864514i	$0.332374\pi$
$884$	−131.343	−4.41755
$885$	0.0651288	0.00218928
$886$	58.1657	1.95411
$887$	−29.9755	−1.00648	−0.503239	−	0.864147i	$-0.667858\pi$
$887$	−29.9755	−1.00648	−0.503239	+	0.864147i	$0.667858\pi$
$888$	−7.42229	−0.249076
$889$	−5.93475	−0.199045
$890$	−7.02207	−0.235380
$891$	18.4303	0.617437
$892$	−99.5938	−3.33465
$893$	−25.5192	−0.853966
$894$	4.34527	0.145328
$895$	−4.12009	−0.137720
$896$	−0.231986	−0.00775010
$897$	0.590378	0.0197121
$898$	−9.93809	−0.331638
$899$	−28.4110	−0.947559
$900$	70.7224	2.35741
$901$	−37.6150	−1.25314
$902$	−36.1635	−1.20411
$903$	0.394898	0.0131414
$904$	132.189	4.39655
$905$	−0.822328	−0.0273351
$906$	−2.36556	−0.0785903
$907$	−38.7338	−1.28613	−0.643067	−	0.765810i	$-0.722339\pi$
$907$	−38.7338	−1.28613	−0.643067	+	0.765810i	$0.722339\pi$
$908$	−100.355	−3.33039
$909$	12.6622	0.419979
$910$	−0.874167	−0.0289784
$911$	8.27882	0.274290	0.137145	−	0.990551i	$-0.456207\pi$
$911$	8.27882	0.274290	0.137145	+	0.990551i	$0.456207\pi$
$912$	−5.05518	−0.167394
$913$	17.2540	0.571023
$914$	−57.3388	−1.89660
$915$	−0.284429	−0.00940292
$916$	21.1613	0.699187
$917$	6.18814	0.204350
$918$	13.5162	0.446103
$919$	52.2273	1.72282	0.861409	−	0.507911i	$-0.169582\pi$
$919$	52.2273	1.72282	0.861409	+	0.507911i	$0.169582\pi$
$920$	−1.90848	−0.0629209
$921$	−2.19374	−0.0722863
$922$	−17.7030	−0.583017
$923$	−62.2465	−2.04887
$924$	0.409585	0.0134744
$925$	36.4929	1.19988
$926$	−14.9714	−0.491991
$927$	−55.6443	−1.82760
$928$	−37.8118	−1.24123
$929$	8.35754	0.274202	0.137101	−	0.990557i	$-0.456221\pi$
$929$	8.35754	0.274202	0.137101	+	0.990557i	$0.456221\pi$
$930$	0.709603	0.0232688
$931$	−26.8405	−0.879660
$932$	−18.4202	−0.603375
$933$	−0.508130	−0.0166354
$934$	−37.6620	−1.23234
$935$	3.43791	0.112432
$936$	94.3134	3.08273
$937$	5.85226	0.191185	0.0955925	−	0.995421i	$-0.469525\pi$
$937$	5.85226	0.191185	0.0955925	+	0.995421i	$0.469525\pi$
$938$	7.23489	0.236227
$939$	−0.850412	−0.0277521
$940$	8.22643	0.268316
$941$	1.58719	0.0517411	0.0258705	−	0.999665i	$-0.491764\pi$
$941$	1.58719	0.0517411	0.0258705	+	0.999665i	$0.491764\pi$
$942$	−1.05384	−0.0343361
$943$	−6.64095	−0.216259
$944$	17.3960	0.566191
$945$	0.0635376	0.00206688
$946$	−52.6951	−1.71327
$947$	−58.9933	−1.91702	−0.958512	−	0.285054i	$-0.907989\pi$
$947$	−58.9933	−1.91702	−0.958512	+	0.285054i	$0.907989\pi$
$948$	−7.03834	−0.228595
$949$	34.1429	1.10832
$950$	49.9864	1.62177
$951$	1.24358	0.0403259
$952$	−13.8398	−0.448551
$953$	44.0341	1.42640	0.713202	−	0.700958i	$-0.247244\pi$
$953$	44.0341	1.42640	0.713202	+	0.700958i	$0.247244\pi$
$954$	46.2371	1.49698
$955$	−0.801682	−0.0259418
$956$	53.5912	1.73326
$957$	−1.06249	−0.0343455
$958$	−17.0236	−0.550008
$959$	2.83612	0.0915830
$960$	0.266757	0.00860954
$961$	27.2926	0.880406
$962$	83.3084	2.68597
$963$	31.5859	1.01784
$964$	−60.0363	−1.93364
$965$	−4.36071	−0.140376
$966$	0.106492	0.00342632
$967$	−51.3463	−1.65119	−0.825593	−	0.564267i	$-0.809159\pi$
$967$	−51.3463	−1.65119	−0.825593	+	0.564267i	$0.809159\pi$
$968$	48.7216	1.56597
$969$	3.36316	0.108040
$970$	5.03214	0.161572
$971$	18.1384	0.582089	0.291045	−	0.956709i	$-0.405997\pi$
$971$	18.1384	0.582089	0.291045	+	0.956709i	$0.405997\pi$
$972$	−17.6205	−0.565176
$973$	−2.30502	−0.0738955
$974$	−9.69831	−0.310754
$975$	2.91188	0.0932550
$976$	−75.9712	−2.43178
$977$	44.2395	1.41535	0.707673	−	0.706540i	$-0.249745\pi$
$977$	44.2395	1.41535	0.707673	+	0.706540i	$0.249745\pi$
$978$	5.67018	0.181312
$979$	−21.5726	−0.689462
$980$	8.65237	0.276390
$981$	15.1587	0.483979
$982$	−104.604	−3.33806
$983$	10.1158	0.322643	0.161322	−	0.986902i	$-0.448424\pi$
$983$	10.1158	0.322643	0.161322	+	0.986902i	$0.448424\pi$
$984$	6.66199	0.212377
$985$	−6.25598	−0.199332
$986$	61.4575	1.95721
$987$	−0.268149	−0.00853526
$988$	80.5971	2.56414
$989$	−9.67676	−0.307703
$990$	−4.22595	−0.134309
$991$	−13.1995	−0.419295	−0.209648	−	0.977777i	$-0.567232\pi$
$991$	−13.1995	−0.419295	−0.209648	+	0.977777i	$0.567232\pi$
$992$	77.5808	2.46319
$993$	1.53459	0.0486987
$994$	−11.2280	−0.356130
$995$	1.13256	0.0359045
$996$	−5.44110	−0.172408
$997$	−10.6528	−0.337377	−0.168689	−	0.985669i	$-0.553953\pi$
$997$	−10.6528	−0.337377	−0.168689	+	0.985669i	$0.553953\pi$
$998$	−88.9409	−2.81538
$999$	−6.05515	−0.191576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8027.2.a.e.1.9	✓	169

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8027.2.a.e.1.9	✓	169	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:	\( N \)	\(=\)	\( 8027 = 23 \cdot 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60952 q^{2} -0.136825 q^{3} +4.80961 q^{4} -0.260304 q^{5} +0.357049 q^{6} -0.298256 q^{7} -7.33174 q^{8} -2.98128 q^{9} +O(q^{10})\)	\(q-2.60952 q^{2} -0.136825 q^{3} +4.80961 q^{4} -0.260304 q^{5} +0.357049 q^{6} -0.298256 q^{7} -7.33174 q^{8} -2.98128 q^{9} +0.679270 q^{10} +2.08679 q^{11} -0.658077 q^{12} +4.31483 q^{13} +0.778305 q^{14} +0.0356162 q^{15} +9.51313 q^{16} -6.32899 q^{17} +7.77972 q^{18} +3.88371 q^{19} -1.25196 q^{20} +0.0408089 q^{21} -5.44553 q^{22} -1.00000 q^{23} +1.00317 q^{24} -4.93224 q^{25} -11.2596 q^{26} +0.818390 q^{27} -1.43449 q^{28} +3.72117 q^{29} -0.0929414 q^{30} -7.63496 q^{31} -10.1613 q^{32} -0.285526 q^{33} +16.5156 q^{34} +0.0776373 q^{35} -14.3388 q^{36} -7.39885 q^{37} -10.1346 q^{38} -0.590378 q^{39} +1.90848 q^{40} +6.64095 q^{41} -0.106492 q^{42} +9.67676 q^{43} +10.0367 q^{44} +0.776040 q^{45} +2.60952 q^{46} -6.57083 q^{47} -1.30164 q^{48} -6.91104 q^{49} +12.8708 q^{50} +0.865966 q^{51} +20.7526 q^{52} +5.94329 q^{53} -2.13561 q^{54} -0.543201 q^{55} +2.18673 q^{56} -0.531389 q^{57} -9.71048 q^{58} +1.82863 q^{59} +0.171300 q^{60} -7.98593 q^{61} +19.9236 q^{62} +0.889184 q^{63} +7.48976 q^{64} -1.12317 q^{65} +0.745087 q^{66} +9.29569 q^{67} -30.4400 q^{68} +0.136825 q^{69} -0.202596 q^{70} -14.4262 q^{71} +21.8580 q^{72} +7.91292 q^{73} +19.3075 q^{74} +0.674856 q^{75} +18.6791 q^{76} -0.622398 q^{77} +1.54060 q^{78} +10.6953 q^{79} -2.47631 q^{80} +8.83186 q^{81} -17.3297 q^{82} +8.26818 q^{83} +0.196275 q^{84} +1.64746 q^{85} -25.2517 q^{86} -0.509150 q^{87} -15.2998 q^{88} -10.3377 q^{89} -2.02509 q^{90} -1.28692 q^{91} -4.80961 q^{92} +1.04466 q^{93} +17.1467 q^{94} -1.01095 q^{95} +1.39032 q^{96} +7.40816 q^{97} +18.0345 q^{98} -6.22131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 169 q + 6 q^{2} + 2 q^{3} + 186 q^{4} + 28 q^{5} + 5 q^{6} + 38 q^{7} + 18 q^{8} + 185 q^{9}+O(q^{10}) \) 169 * q + 6 * q^2 + 2 * q^3 + 186 * q^4 + 28 * q^5 + 5 * q^6 + 38 * q^7 + 18 * q^8 + 185 * q^9	\( 169 q + 6 q^{2} + 2 q^{3} + 186 q^{4} + 28 q^{5} + 5 q^{6} + 38 q^{7} + 18 q^{8} + 185 q^{9} + 28 q^{10} + 17 q^{11} + 10 q^{12} + 91 q^{13} + 20 q^{14} + 29 q^{15} + 200 q^{16} + 16 q^{17} + 31 q^{18} + 30 q^{19} + 45 q^{20} + 49 q^{21} + 76 q^{22} - 169 q^{23} + 3 q^{24} + 241 q^{25} - 15 q^{26} + 14 q^{27} + 118 q^{28} + 23 q^{29} + 52 q^{30} + 45 q^{31} + 42 q^{32} + 62 q^{33} + 65 q^{34} - 16 q^{35} + 199 q^{36} + 226 q^{37} + 49 q^{38} + 17 q^{39} + 95 q^{40} + 19 q^{41} + 32 q^{42} + 71 q^{43} + 46 q^{44} + 127 q^{45} - 6 q^{46} + 27 q^{47} + 5 q^{48} + 239 q^{49} + 24 q^{50} + 39 q^{51} + 154 q^{52} + 111 q^{53} + 22 q^{54} + 47 q^{55} + 39 q^{56} + 122 q^{57} + 146 q^{58} - 73 q^{59} + 109 q^{60} + 125 q^{61} + 14 q^{62} + 109 q^{63} + 260 q^{64} + 73 q^{65} + 26 q^{66} + 152 q^{67} + 40 q^{68} - 2 q^{69} + 76 q^{70} - 79 q^{71} + 126 q^{72} + 106 q^{73} + 23 q^{74} + 12 q^{75} + 122 q^{76} + 63 q^{77} + 101 q^{78} + 82 q^{79} + 134 q^{80} + 225 q^{81} + 125 q^{82} + 28 q^{83} + 73 q^{84} + 197 q^{85} + 97 q^{86} + 18 q^{87} + 183 q^{88} + 54 q^{89} + 52 q^{90} + 106 q^{91} - 186 q^{92} + 194 q^{93} + q^{94} + 18 q^{95} - 39 q^{96} + 239 q^{97} + 5 q^{98} + 85 q^{99}+O(q^{100}) \) 169 * q + 6 * q^2 + 2 * q^3 + 186 * q^4 + 28 * q^5 + 5 * q^6 + 38 * q^7 + 18 * q^8 + 185 * q^9 + 28 * q^10 + 17 * q^11 + 10 * q^12 + 91 * q^13 + 20 * q^14 + 29 * q^15 + 200 * q^16 + 16 * q^17 + 31 * q^18 + 30 * q^19 + 45 * q^20 + 49 * q^21 + 76 * q^22 - 169 * q^23 + 3 * q^24 + 241 * q^25 - 15 * q^26 + 14 * q^27 + 118 * q^28 + 23 * q^29 + 52 * q^30 + 45 * q^31 + 42 * q^32 + 62 * q^33 + 65 * q^34 - 16 * q^35 + 199 * q^36 + 226 * q^37 + 49 * q^38 + 17 * q^39 + 95 * q^40 + 19 * q^41 + 32 * q^42 + 71 * q^43 + 46 * q^44 + 127 * q^45 - 6 * q^46 + 27 * q^47 + 5 * q^48 + 239 * q^49 + 24 * q^50 + 39 * q^51 + 154 * q^52 + 111 * q^53 + 22 * q^54 + 47 * q^55 + 39 * q^56 + 122 * q^57 + 146 * q^58 - 73 * q^59 + 109 * q^60 + 125 * q^61 + 14 * q^62 + 109 * q^63 + 260 * q^64 + 73 * q^65 + 26 * q^66 + 152 * q^67 + 40 * q^68 - 2 * q^69 + 76 * q^70 - 79 * q^71 + 126 * q^72 + 106 * q^73 + 23 * q^74 + 12 * q^75 + 122 * q^76 + 63 * q^77 + 101 * q^78 + 82 * q^79 + 134 * q^80 + 225 * q^81 + 125 * q^82 + 28 * q^83 + 73 * q^84 + 197 * q^85 + 97 * q^86 + 18 * q^87 + 183 * q^88 + 54 * q^89 + 52 * q^90 + 106 * q^91 - 186 * q^92 + 194 * q^93 + q^94 + 18 * q^95 - 39 * q^96 + 239 * q^97 + 5 * q^98 + 85 * q^99

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