LMFDB - Embedded newform 547.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [547,2,Mod(1,547)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(547, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("547.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	547.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:	$4.36781699056$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	547.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.41421 q^{3} -1.82843 q^{4} +1.41421 q^{5} -0.585786 q^{6} +2.00000 q^{7} -1.58579 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} -1.41421 q^{3} -1.82843 q^{4} +1.41421 q^{5} -0.585786 q^{6} +2.00000 q^{7} -1.58579 q^{8} -1.00000 q^{9} +0.585786 q^{10} -3.58579 q^{11} +2.58579 q^{12} +3.00000 q^{13} +0.828427 q^{14} -2.00000 q^{15} +3.00000 q^{16} -7.41421 q^{17} -0.414214 q^{18} +3.24264 q^{19} -2.58579 q^{20} -2.82843 q^{21} -1.48528 q^{22} -9.07107 q^{23} +2.24264 q^{24} -3.00000 q^{25} +1.24264 q^{26} +5.65685 q^{27} -3.65685 q^{28} -9.82843 q^{29} -0.828427 q^{30} -0.828427 q^{31} +4.41421 q^{32} +5.07107 q^{33} -3.07107 q^{34} +2.82843 q^{35} +1.82843 q^{36} -0.585786 q^{37} +1.34315 q^{38} -4.24264 q^{39} -2.24264 q^{40} -3.07107 q^{41} -1.17157 q^{42} -8.00000 q^{43} +6.55635 q^{44} -1.41421 q^{45} -3.75736 q^{46} +9.58579 q^{47} -4.24264 q^{48} -3.00000 q^{49} -1.24264 q^{50} +10.4853 q^{51} -5.48528 q^{52} -1.34315 q^{53} +2.34315 q^{54} -5.07107 q^{55} -3.17157 q^{56} -4.58579 q^{57} -4.07107 q^{58} -1.75736 q^{59} +3.65685 q^{60} +0.585786 q^{61} -0.343146 q^{62} -2.00000 q^{63} -4.17157 q^{64} +4.24264 q^{65} +2.10051 q^{66} +4.89949 q^{67} +13.5563 q^{68} +12.8284 q^{69} +1.17157 q^{70} +1.41421 q^{71} +1.58579 q^{72} +4.65685 q^{73} -0.242641 q^{74} +4.24264 q^{75} -5.92893 q^{76} -7.17157 q^{77} -1.75736 q^{78} +3.07107 q^{79} +4.24264 q^{80} -5.00000 q^{81} -1.27208 q^{82} +15.0711 q^{83} +5.17157 q^{84} -10.4853 q^{85} -3.31371 q^{86} +13.8995 q^{87} +5.68629 q^{88} -4.24264 q^{89} -0.585786 q^{90} +6.00000 q^{91} +16.5858 q^{92} +1.17157 q^{93} +3.97056 q^{94} +4.58579 q^{95} -6.24264 q^{96} +0.514719 q^{97} -1.24264 q^{98} +3.58579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} + 4 q^{10} - 10 q^{11} + 8 q^{12} + 6 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 12 q^{17} + 2 q^{18} - 2 q^{19} - 8 q^{20} + 14 q^{22} - 4 q^{23} - 4 q^{24} - 6 q^{25} - 6 q^{26} + 4 q^{28} - 14 q^{29} + 4 q^{30} + 4 q^{31} + 6 q^{32} - 4 q^{33} + 8 q^{34} - 2 q^{36} - 4 q^{37} + 14 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{42} - 16 q^{43} - 18 q^{44} - 16 q^{46} + 22 q^{47} - 6 q^{49} + 6 q^{50} + 4 q^{51} + 6 q^{52} - 14 q^{53} + 16 q^{54} + 4 q^{55} - 12 q^{56} - 12 q^{57} + 6 q^{58} - 12 q^{59} - 4 q^{60} + 4 q^{61} - 12 q^{62} - 4 q^{63} - 14 q^{64} + 24 q^{66} - 10 q^{67} - 4 q^{68} + 20 q^{69} + 8 q^{70} + 6 q^{72} - 2 q^{73} + 8 q^{74} - 26 q^{76} - 20 q^{77} - 12 q^{78} - 8 q^{79} - 10 q^{81} - 28 q^{82} + 16 q^{83} + 16 q^{84} - 4 q^{85} + 16 q^{86} + 8 q^{87} + 34 q^{88} - 4 q^{90} + 12 q^{91} + 36 q^{92} + 8 q^{93} - 26 q^{94} + 12 q^{95} - 4 q^{96} + 18 q^{97} + 6 q^{98} + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9 + 4 * q^10 - 10 * q^11 + 8 * q^12 + 6 * q^13 - 4 * q^14 - 4 * q^15 + 6 * q^16 - 12 * q^17 + 2 * q^18 - 2 * q^19 - 8 * q^20 + 14 * q^22 - 4 * q^23 - 4 * q^24 - 6 * q^25 - 6 * q^26 + 4 * q^28 - 14 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 - 4 * q^33 + 8 * q^34 - 2 * q^36 - 4 * q^37 + 14 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^42 - 16 * q^43 - 18 * q^44 - 16 * q^46 + 22 * q^47 - 6 * q^49 + 6 * q^50 + 4 * q^51 + 6 * q^52 - 14 * q^53 + 16 * q^54 + 4 * q^55 - 12 * q^56 - 12 * q^57 + 6 * q^58 - 12 * q^59 - 4 * q^60 + 4 * q^61 - 12 * q^62 - 4 * q^63 - 14 * q^64 + 24 * q^66 - 10 * q^67 - 4 * q^68 + 20 * q^69 + 8 * q^70 + 6 * q^72 - 2 * q^73 + 8 * q^74 - 26 * q^76 - 20 * q^77 - 12 * q^78 - 8 * q^79 - 10 * q^81 - 28 * q^82 + 16 * q^83 + 16 * q^84 - 4 * q^85 + 16 * q^86 + 8 * q^87 + 34 * q^88 - 4 * q^90 + 12 * q^91 + 36 * q^92 + 8 * q^93 - 26 * q^94 + 12 * q^95 - 4 * q^96 + 18 * q^97 + 6 * q^98 + 10 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	−1.82843	−0.914214
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	−0.585786	−0.239146
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.58579	−0.560660
$9$	−1.00000	−0.333333
$10$	0.585786	0.185242
$11$	−3.58579	−1.08116	−0.540578	−	0.841294i	$-0.681794\pi$
$11$	−3.58579	−1.08116	−0.540578	+	0.841294i	$0.681794\pi$
$12$	2.58579	0.746452
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0.828427	0.221406
$15$	−2.00000	−0.516398
$16$	3.00000	0.750000
$17$	−7.41421	−1.79821	−0.899105	−	0.437732i	$-0.855782\pi$
$17$	−7.41421	−1.79821	−0.899105	+	0.437732i	$0.855782\pi$
$18$	−0.414214	−0.0976311
$19$	3.24264	0.743913	0.371956	−	0.928250i	$-0.378687\pi$
$19$	3.24264	0.743913	0.371956	+	0.928250i	$0.378687\pi$
$20$	−2.58579	−0.578199
$21$	−2.82843	−0.617213
$22$	−1.48528	−0.316663
$23$	−9.07107	−1.89145	−0.945724	−	0.324970i	$-0.894646\pi$
$23$	−9.07107	−1.89145	−0.945724	+	0.324970i	$0.894646\pi$
$24$	2.24264	0.457777
$25$	−3.00000	−0.600000
$26$	1.24264	0.243702
$27$	5.65685	1.08866
$28$	−3.65685	−0.691080
$29$	−9.82843	−1.82509	−0.912547	−	0.408973i	$-0.865887\pi$
$29$	−9.82843	−1.82509	−0.912547	+	0.408973i	$0.865887\pi$
$30$	−0.828427	−0.151249
$31$	−0.828427	−0.148790	−0.0743950	−	0.997229i	$-0.523703\pi$
$31$	−0.828427	−0.148790	−0.0743950	+	0.997229i	$0.523703\pi$
$32$	4.41421	0.780330
$33$	5.07107	0.882760
$34$	−3.07107	−0.526684
$35$	2.82843	0.478091
$36$	1.82843	0.304738
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	1.34315	0.217887
$39$	−4.24264	−0.679366
$40$	−2.24264	−0.354593
$41$	−3.07107	−0.479620	−0.239810	−	0.970820i	$-0.577085\pi$
$41$	−3.07107	−0.479620	−0.239810	+	0.970820i	$0.577085\pi$
$42$	−1.17157	−0.180778
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	6.55635	0.988407
$45$	−1.41421	−0.210819
$46$	−3.75736	−0.553992
$47$	9.58579	1.39823	0.699115	−	0.715009i	$-0.253577\pi$
$47$	9.58579	1.39823	0.699115	+	0.715009i	$0.253577\pi$
$48$	−4.24264	−0.612372
$49$	−3.00000	−0.428571
$50$	−1.24264	−0.175736
$51$	10.4853	1.46823
$52$	−5.48528	−0.760672
$53$	−1.34315	−0.184495	−0.0922476	−	0.995736i	$-0.529405\pi$
$53$	−1.34315	−0.184495	−0.0922476	+	0.995736i	$0.529405\pi$
$54$	2.34315	0.318862
$55$	−5.07107	−0.683783
$56$	−3.17157	−0.423819
$57$	−4.58579	−0.607402
$58$	−4.07107	−0.534557
$59$	−1.75736	−0.228789	−0.114394	−	0.993435i	$-0.536493\pi$
$59$	−1.75736	−0.228789	−0.114394	+	0.993435i	$0.536493\pi$
$60$	3.65685	0.472098
$61$	0.585786	0.0750023	0.0375011	−	0.999297i	$-0.488060\pi$
$61$	0.585786	0.0750023	0.0375011	+	0.999297i	$0.488060\pi$
$62$	−0.343146	−0.0435796
$63$	−2.00000	−0.251976
$64$	−4.17157	−0.521447
$65$	4.24264	0.526235
$66$	2.10051	0.258554
$67$	4.89949	0.598569	0.299284	−	0.954164i	$-0.403252\pi$
$67$	4.89949	0.598569	0.299284	+	0.954164i	$0.403252\pi$
$68$	13.5563	1.64395
$69$	12.8284	1.54436
$70$	1.17157	0.140030
$71$	1.41421	0.167836	0.0839181	−	0.996473i	$-0.473257\pi$
$71$	1.41421	0.167836	0.0839181	+	0.996473i	$0.473257\pi$
$72$	1.58579	0.186887
$73$	4.65685	0.545044	0.272522	−	0.962150i	$-0.412142\pi$
$73$	4.65685	0.545044	0.272522	+	0.962150i	$0.412142\pi$
$74$	−0.242641	−0.0282064
$75$	4.24264	0.489898
$76$	−5.92893	−0.680095
$77$	−7.17157	−0.817277
$78$	−1.75736	−0.198982
$79$	3.07107	0.345522	0.172761	−	0.984964i	$-0.444731\pi$
$79$	3.07107	0.345522	0.172761	+	0.984964i	$0.444731\pi$
$80$	4.24264	0.474342
$81$	−5.00000	−0.555556
$82$	−1.27208	−0.140478
$83$	15.0711	1.65426	0.827132	−	0.562007i	$-0.189971\pi$
$83$	15.0711	1.65426	0.827132	+	0.562007i	$0.189971\pi$
$84$	5.17157	0.564265
$85$	−10.4853	−1.13729
$86$	−3.31371	−0.357326
$87$	13.8995	1.49018
$88$	5.68629	0.606161
$89$	−4.24264	−0.449719	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	−4.24264	−0.449719	−0.224860	+	0.974391i	$0.572192\pi$
$90$	−0.585786	−0.0617473
$91$	6.00000	0.628971
$92$	16.5858	1.72919
$93$	1.17157	0.121486
$94$	3.97056	0.409532
$95$	4.58579	0.470492
$96$	−6.24264	−0.637137
$97$	0.514719	0.0522618	0.0261309	−	0.999659i	$-0.491681\pi$
$97$	0.514719	0.0522618	0.0261309	+	0.999659i	$0.491681\pi$
$98$	−1.24264	−0.125526
$99$	3.58579	0.360385
$100$	5.48528	0.548528
$101$	15.6569	1.55792	0.778958	−	0.627077i	$-0.215749\pi$
$101$	15.6569	1.55792	0.778958	+	0.627077i	$0.215749\pi$
$102$	4.34315	0.430036
$103$	14.2426	1.40337	0.701685	−	0.712488i	$-0.252432\pi$
$103$	14.2426	1.40337	0.701685	+	0.712488i	$0.252432\pi$
$104$	−4.75736	−0.466497
$105$	−4.00000	−0.390360
$106$	−0.556349	−0.0540374
$107$	−10.2426	−0.990193	−0.495097	−	0.868838i	$-0.664867\pi$
$107$	−10.2426	−0.990193	−0.495097	+	0.868838i	$0.664867\pi$
$108$	−10.3431	−0.995270
$109$	1.75736	0.168324	0.0841622	−	0.996452i	$-0.473179\pi$
$109$	1.75736	0.168324	0.0841622	+	0.996452i	$0.473179\pi$
$110$	−2.10051	−0.200275
$111$	0.828427	0.0786308
$112$	6.00000	0.566947
$113$	−20.6569	−1.94323	−0.971617	−	0.236561i	$-0.923980\pi$
$113$	−20.6569	−1.94323	−0.971617	+	0.236561i	$0.923980\pi$
$114$	−1.89949	−0.177904
$115$	−12.8284	−1.19626
$116$	17.9706	1.66852
$117$	−3.00000	−0.277350
$118$	−0.727922	−0.0670107
$119$	−14.8284	−1.35932
$120$	3.17157	0.289524
$121$	1.85786	0.168897
$122$	0.242641	0.0219677
$123$	4.34315	0.391608
$124$	1.51472	0.136026
$125$	−11.3137	−1.01193
$126$	−0.828427	−0.0738022
$127$	15.7279	1.39563	0.697814	−	0.716279i	$-0.254156\pi$
$127$	15.7279	1.39563	0.697814	+	0.716279i	$0.254156\pi$
$128$	−10.5563	−0.933058
$129$	11.3137	0.996116
$130$	1.75736	0.154131
$131$	−10.8995	−0.952293	−0.476147	−	0.879366i	$-0.657967\pi$
$131$	−10.8995	−0.952293	−0.476147	+	0.879366i	$0.657967\pi$
$132$	−9.27208	−0.807031
$133$	6.48528	0.562345
$134$	2.02944	0.175317
$135$	8.00000	0.688530
$136$	11.7574	1.00819
$137$	1.65685	0.141555	0.0707773	−	0.997492i	$-0.477452\pi$
$137$	1.65685	0.141555	0.0707773	+	0.997492i	$0.477452\pi$
$138$	5.31371	0.452333
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−5.17157	−0.437078
$141$	−13.5563	−1.14165
$142$	0.585786	0.0491581
$143$	−10.7574	−0.899576
$144$	−3.00000	−0.250000
$145$	−13.8995	−1.15429
$146$	1.92893	0.159640
$147$	4.24264	0.349927
$148$	1.07107	0.0880412
$149$	21.3137	1.74609	0.873044	−	0.487642i	$-0.162143\pi$
$149$	21.3137	1.74609	0.873044	+	0.487642i	$0.162143\pi$
$150$	1.75736	0.143488
$151$	−18.9706	−1.54380	−0.771901	−	0.635742i	$-0.780694\pi$
$151$	−18.9706	−1.54380	−0.771901	+	0.635742i	$0.780694\pi$
$152$	−5.14214	−0.417082
$153$	7.41421	0.599404
$154$	−2.97056	−0.239375
$155$	−1.17157	−0.0941030
$156$	7.75736	0.621086
$157$	1.65685	0.132231	0.0661157	−	0.997812i	$-0.478939\pi$
$157$	1.65685	0.132231	0.0661157	+	0.997812i	$0.478939\pi$
$158$	1.27208	0.101201
$159$	1.89949	0.150640
$160$	6.24264	0.493524
$161$	−18.1421	−1.42980
$162$	−2.07107	−0.162718
$163$	−18.1421	−1.42100	−0.710501	−	0.703696i	$-0.751532\pi$
$163$	−18.1421	−1.42100	−0.710501	+	0.703696i	$0.751532\pi$
$164$	5.61522	0.438475
$165$	7.17157	0.558306
$166$	6.24264	0.484523
$167$	21.3137	1.64930	0.824652	−	0.565640i	$-0.191371\pi$
$167$	21.3137	1.64930	0.824652	+	0.565640i	$0.191371\pi$
$168$	4.48528	0.346047
$169$	−4.00000	−0.307692
$170$	−4.34315	−0.333104
$171$	−3.24264	−0.247971
$172$	14.6274	1.11533
$173$	−14.4853	−1.10130	−0.550648	−	0.834738i	$-0.685619\pi$
$173$	−14.4853	−1.10130	−0.550648	+	0.834738i	$0.685619\pi$
$174$	5.75736	0.436464
$175$	−6.00000	−0.453557
$176$	−10.7574	−0.810866
$177$	2.48528	0.186805
$178$	−1.75736	−0.131720
$179$	−7.65685	−0.572300	−0.286150	−	0.958185i	$-0.592376\pi$
$179$	−7.65685	−0.572300	−0.286150	+	0.958185i	$0.592376\pi$
$180$	2.58579	0.192733
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	2.48528	0.184221
$183$	−0.828427	−0.0612391
$184$	14.3848	1.06046
$185$	−0.828427	−0.0609072
$186$	0.485281	0.0355826
$187$	26.5858	1.94415
$188$	−17.5269	−1.27828
$189$	11.3137	0.822951
$190$	1.89949	0.137804
$191$	21.7279	1.57218	0.786089	−	0.618114i	$-0.212103\pi$
$191$	21.7279	1.57218	0.786089	+	0.618114i	$0.212103\pi$
$192$	5.89949	0.425759
$193$	−20.4853	−1.47456	−0.737281	−	0.675586i	$-0.763891\pi$
$193$	−20.4853	−1.47456	−0.737281	+	0.675586i	$0.763891\pi$
$194$	0.213203	0.0153071
$195$	−6.00000	−0.429669
$196$	5.48528	0.391806
$197$	−10.9706	−0.781620	−0.390810	−	0.920471i	$-0.627805\pi$
$197$	−10.9706	−0.781620	−0.390810	+	0.920471i	$0.627805\pi$
$198$	1.48528	0.105554
$199$	12.0711	0.855695	0.427848	−	0.903851i	$-0.359272\pi$
$199$	12.0711	0.855695	0.427848	+	0.903851i	$0.359272\pi$
$200$	4.75736	0.336396
$201$	−6.92893	−0.488729
$202$	6.48528	0.456303
$203$	−19.6569	−1.37964
$204$	−19.1716	−1.34228
$205$	−4.34315	−0.303338
$206$	5.89949	0.411037
$207$	9.07107	0.630483
$208$	9.00000	0.624038
$209$	−11.6274	−0.804285
$210$	−1.65685	−0.114334
$211$	−10.9706	−0.755245	−0.377622	−	0.925960i	$-0.623258\pi$
$211$	−10.9706	−0.755245	−0.377622	+	0.925960i	$0.623258\pi$
$212$	2.45584	0.168668
$213$	−2.00000	−0.137038
$214$	−4.24264	−0.290021
$215$	−11.3137	−0.771589
$216$	−8.97056	−0.610369
$217$	−1.65685	−0.112475
$218$	0.727922	0.0493011
$219$	−6.58579	−0.445026
$220$	9.27208	0.625123
$221$	−22.2426	−1.49620
$222$	0.343146	0.0230304
$223$	−21.2132	−1.42054	−0.710271	−	0.703929i	$-0.751427\pi$
$223$	−21.2132	−1.42054	−0.710271	+	0.703929i	$0.751427\pi$
$224$	8.82843	0.589874
$225$	3.00000	0.200000
$226$	−8.55635	−0.569160
$227$	−9.17157	−0.608739	−0.304369	−	0.952554i	$-0.598446\pi$
$227$	−9.17157	−0.608739	−0.304369	+	0.952554i	$0.598446\pi$
$228$	8.38478	0.555295
$229$	20.4853	1.35371	0.676853	−	0.736118i	$-0.263343\pi$
$229$	20.4853	1.35371	0.676853	+	0.736118i	$0.263343\pi$
$230$	−5.31371	−0.350376
$231$	10.1421	0.667304
$232$	15.5858	1.02326
$233$	17.1421	1.12302	0.561509	−	0.827470i	$-0.310221\pi$
$233$	17.1421	1.12302	0.561509	+	0.827470i	$0.310221\pi$
$234$	−1.24264	−0.0812340
$235$	13.5563	0.884319
$236$	3.21320	0.209162
$237$	−4.34315	−0.282118
$238$	−6.14214	−0.398136
$239$	9.72792	0.629247	0.314623	−	0.949217i	$-0.398122\pi$
$239$	9.72792	0.629247	0.314623	+	0.949217i	$0.398122\pi$
$240$	−6.00000	−0.387298
$241$	−5.51472	−0.355234	−0.177617	−	0.984100i	$-0.556839\pi$
$241$	−5.51472	−0.355234	−0.177617	+	0.984100i	$0.556839\pi$
$242$	0.769553	0.0494687
$243$	−9.89949	−0.635053
$244$	−1.07107	−0.0685681
$245$	−4.24264	−0.271052
$246$	1.79899	0.114699
$247$	9.72792	0.618973
$248$	1.31371	0.0834206
$249$	−21.3137	−1.35070
$250$	−4.68629	−0.296387
$251$	−0.828427	−0.0522899	−0.0261449	−	0.999658i	$-0.508323\pi$
$251$	−0.828427	−0.0522899	−0.0261449	+	0.999658i	$0.508323\pi$
$252$	3.65685	0.230360
$253$	32.5269	2.04495
$254$	6.51472	0.408770
$255$	14.8284	0.928592
$256$	3.97056	0.248160
$257$	19.4558	1.21362	0.606811	−	0.794846i	$-0.292448\pi$
$257$	19.4558	1.21362	0.606811	+	0.794846i	$0.292448\pi$
$258$	4.68629	0.291756
$259$	−1.17157	−0.0727980
$260$	−7.75736	−0.481091
$261$	9.82843	0.608364
$262$	−4.51472	−0.278920
$263$	−2.07107	−0.127708	−0.0638538	−	0.997959i	$-0.520339\pi$
$263$	−2.07107	−0.127708	−0.0638538	+	0.997959i	$0.520339\pi$
$264$	−8.04163	−0.494928
$265$	−1.89949	−0.116685
$266$	2.68629	0.164707
$267$	6.00000	0.367194
$268$	−8.95837	−0.547220
$269$	−8.48528	−0.517357	−0.258678	−	0.965964i	$-0.583287\pi$
$269$	−8.48528	−0.517357	−0.258678	+	0.965964i	$0.583287\pi$
$270$	3.31371	0.201666
$271$	24.2426	1.47264	0.736318	−	0.676636i	$-0.236563\pi$
$271$	24.2426	1.47264	0.736318	+	0.676636i	$0.236563\pi$
$272$	−22.2426	−1.34866
$273$	−8.48528	−0.513553
$274$	0.686292	0.0414604
$275$	10.7574	0.648693
$276$	−23.4558	−1.41188
$277$	−13.8284	−0.830870	−0.415435	−	0.909623i	$-0.636371\pi$
$277$	−13.8284	−0.830870	−0.415435	+	0.909623i	$0.636371\pi$
$278$	−1.65685	−0.0993715
$279$	0.828427	0.0495966
$280$	−4.48528	−0.268047
$281$	−16.8284	−1.00390	−0.501950	−	0.864897i	$-0.667384\pi$
$281$	−16.8284	−1.00390	−0.501950	+	0.864897i	$0.667384\pi$
$282$	−5.61522	−0.334382
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	−2.58579	−0.153438
$285$	−6.48528	−0.384155
$286$	−4.45584	−0.263480
$287$	−6.14214	−0.362559
$288$	−4.41421	−0.260110
$289$	37.9706	2.23356
$290$	−5.75736	−0.338084
$291$	−0.727922	−0.0426715
$292$	−8.51472	−0.498286
$293$	2.34315	0.136888	0.0684440	−	0.997655i	$-0.478197\pi$
$293$	2.34315	0.136888	0.0684440	+	0.997655i	$0.478197\pi$
$294$	1.75736	0.102491
$295$	−2.48528	−0.144699
$296$	0.928932	0.0539931
$297$	−20.2843	−1.17701
$298$	8.82843	0.511417
$299$	−27.2132	−1.57378
$300$	−7.75736	−0.447871
$301$	−16.0000	−0.922225
$302$	−7.85786	−0.452169
$303$	−22.1421	−1.27203
$304$	9.72792	0.557935
$305$	0.828427	0.0474356
$306$	3.07107	0.175561
$307$	−20.6274	−1.17727	−0.588634	−	0.808399i	$-0.700334\pi$
$307$	−20.6274	−1.17727	−0.588634	+	0.808399i	$0.700334\pi$
$308$	13.1127	0.747165
$309$	−20.1421	−1.14585
$310$	−0.485281	−0.0275621
$311$	12.4142	0.703945	0.351973	−	0.936010i	$-0.385511\pi$
$311$	12.4142	0.703945	0.351973	+	0.936010i	$0.385511\pi$
$312$	6.72792	0.380894
$313$	−30.7990	−1.74086	−0.870431	−	0.492291i	$-0.836160\pi$
$313$	−30.7990	−1.74086	−0.870431	+	0.492291i	$0.836160\pi$
$314$	0.686292	0.0387297
$315$	−2.82843	−0.159364
$316$	−5.61522	−0.315881
$317$	23.6274	1.32705	0.663524	−	0.748155i	$-0.269060\pi$
$317$	23.6274	1.32705	0.663524	+	0.748155i	$0.269060\pi$
$318$	0.786797	0.0441214
$319$	35.2426	1.97321
$320$	−5.89949	−0.329792
$321$	14.4853	0.808490
$322$	−7.51472	−0.418779
$323$	−24.0416	−1.33771
$324$	9.14214	0.507896
$325$	−9.00000	−0.499230
$326$	−7.51472	−0.416202
$327$	−2.48528	−0.137436
$328$	4.87006	0.268904
$329$	19.1716	1.05696
$330$	2.97056	0.163524
$331$	29.6985	1.63238	0.816188	−	0.577786i	$-0.196083\pi$
$331$	29.6985	1.63238	0.816188	+	0.577786i	$0.196083\pi$
$332$	−27.5563	−1.51235
$333$	0.585786	0.0321009
$334$	8.82843	0.483070
$335$	6.92893	0.378568
$336$	−8.48528	−0.462910
$337$	−1.31371	−0.0715623	−0.0357811	−	0.999360i	$-0.511392\pi$
$337$	−1.31371	−0.0715623	−0.0357811	+	0.999360i	$0.511392\pi$
$338$	−1.65685	−0.0901210
$339$	29.2132	1.58664
$340$	19.1716	1.03972
$341$	2.97056	0.160865
$342$	−1.34315	−0.0726290
$343$	−20.0000	−1.07990
$344$	12.6863	0.683999
$345$	18.1421	0.976740
$346$	−6.00000	−0.322562
$347$	−28.6274	−1.53680	−0.768400	−	0.639970i	$-0.778947\pi$
$347$	−28.6274	−1.53680	−0.768400	+	0.639970i	$0.778947\pi$
$348$	−25.4142	−1.36234
$349$	−13.4853	−0.721851	−0.360925	−	0.932595i	$-0.617539\pi$
$349$	−13.4853	−0.721851	−0.360925	+	0.932595i	$0.617539\pi$
$350$	−2.48528	−0.132844
$351$	16.9706	0.905822
$352$	−15.8284	−0.843658
$353$	−18.1716	−0.967175	−0.483588	−	0.875296i	$-0.660667\pi$
$353$	−18.1716	−0.967175	−0.483588	+	0.875296i	$0.660667\pi$
$354$	1.02944	0.0547140
$355$	2.00000	0.106149
$356$	7.75736	0.411139
$357$	20.9706	1.10988
$358$	−3.17157	−0.167623
$359$	10.9289	0.576807	0.288403	−	0.957509i	$-0.406876\pi$
$359$	10.9289	0.576807	0.288403	+	0.957509i	$0.406876\pi$
$360$	2.24264	0.118198
$361$	−8.48528	−0.446594
$362$	−2.07107	−0.108853
$363$	−2.62742	−0.137904
$364$	−10.9706	−0.575014
$365$	6.58579	0.344716
$366$	−0.343146	−0.0179365
$367$	8.75736	0.457130	0.228565	−	0.973529i	$-0.426597\pi$
$367$	8.75736	0.457130	0.228565	+	0.973529i	$0.426597\pi$
$368$	−27.2132	−1.41859
$369$	3.07107	0.159873
$370$	−0.343146	−0.0178393
$371$	−2.68629	−0.139465
$372$	−2.14214	−0.111065
$373$	−15.7574	−0.815885	−0.407943	−	0.913008i	$-0.633754\pi$
$373$	−15.7574	−0.815885	−0.407943	+	0.913008i	$0.633754\pi$
$374$	11.0122	0.569427
$375$	16.0000	0.826236
$376$	−15.2010	−0.783932
$377$	−29.4853	−1.51857
$378$	4.68629	0.241037
$379$	18.2132	0.935549	0.467775	−	0.883848i	$-0.345056\pi$
$379$	18.2132	0.935549	0.467775	+	0.883848i	$0.345056\pi$
$380$	−8.38478	−0.430130
$381$	−22.2426	−1.13953
$382$	9.00000	0.460480
$383$	−10.4142	−0.532141	−0.266071	−	0.963954i	$-0.585725\pi$
$383$	−10.4142	−0.532141	−0.266071	+	0.963954i	$0.585725\pi$
$384$	14.9289	0.761839
$385$	−10.1421	−0.516891
$386$	−8.48528	−0.431889
$387$	8.00000	0.406663
$388$	−0.941125	−0.0477784
$389$	−15.0711	−0.764133	−0.382067	−	0.924135i	$-0.624788\pi$
$389$	−15.0711	−0.764133	−0.382067	+	0.924135i	$0.624788\pi$
$390$	−2.48528	−0.125847
$391$	67.2548	3.40122
$392$	4.75736	0.240283
$393$	15.4142	0.777544
$394$	−4.54416	−0.228931
$395$	4.34315	0.218527
$396$	−6.55635	−0.329469
$397$	−10.2426	−0.514063	−0.257032	−	0.966403i	$-0.582744\pi$
$397$	−10.2426	−0.514063	−0.257032	+	0.966403i	$0.582744\pi$
$398$	5.00000	0.250627
$399$	−9.17157	−0.459153
$400$	−9.00000	−0.450000
$401$	−7.68629	−0.383835	−0.191918	−	0.981411i	$-0.561471\pi$
$401$	−7.68629	−0.383835	−0.191918	+	0.981411i	$0.561471\pi$
$402$	−2.87006	−0.143145
$403$	−2.48528	−0.123801
$404$	−28.6274	−1.42427
$405$	−7.07107	−0.351364
$406$	−8.14214	−0.404087
$407$	2.10051	0.104118
$408$	−16.6274	−0.823180
$409$	36.9706	1.82808	0.914038	−	0.405628i	$-0.132947\pi$
$409$	36.9706	1.82808	0.914038	+	0.405628i	$0.132947\pi$
$410$	−1.79899	−0.0888458
$411$	−2.34315	−0.115579
$412$	−26.0416	−1.28298
$413$	−3.51472	−0.172948
$414$	3.75736	0.184664
$415$	21.3137	1.04625
$416$	13.2426	0.649274
$417$	5.65685	0.277017
$418$	−4.81623	−0.235570
$419$	13.9289	0.680473	0.340236	−	0.940340i	$-0.389493\pi$
$419$	13.9289	0.680473	0.340236	+	0.940340i	$0.389493\pi$
$420$	7.31371	0.356872
$421$	−12.1005	−0.589743	−0.294871	−	0.955537i	$-0.595277\pi$
$421$	−12.1005	−0.589743	−0.294871	+	0.955537i	$0.595277\pi$
$422$	−4.54416	−0.221206
$423$	−9.58579	−0.466077
$424$	2.12994	0.103439
$425$	22.2426	1.07893
$426$	−0.828427	−0.0401374
$427$	1.17157	0.0566964
$428$	18.7279	0.905248
$429$	15.2132	0.734500
$430$	−4.68629	−0.225993
$431$	33.0711	1.59298	0.796489	−	0.604653i	$-0.206688\pi$
$431$	33.0711	1.59298	0.796489	+	0.604653i	$0.206688\pi$
$432$	16.9706	0.816497
$433$	17.5563	0.843704	0.421852	−	0.906665i	$-0.361380\pi$
$433$	17.5563	0.843704	0.421852	+	0.906665i	$0.361380\pi$
$434$	−0.686292	−0.0329430
$435$	19.6569	0.942474
$436$	−3.21320	−0.153885
$437$	−29.4142	−1.40707
$438$	−2.72792	−0.130345
$439$	29.7279	1.41884	0.709418	−	0.704788i	$-0.248958\pi$
$439$	29.7279	1.41884	0.709418	+	0.704788i	$0.248958\pi$
$440$	8.04163	0.383370
$441$	3.00000	0.142857
$442$	−9.21320	−0.438227
$443$	8.48528	0.403148	0.201574	−	0.979473i	$-0.435394\pi$
$443$	8.48528	0.403148	0.201574	+	0.979473i	$0.435394\pi$
$444$	−1.51472	−0.0718854
$445$	−6.00000	−0.284427
$446$	−8.78680	−0.416067
$447$	−30.1421	−1.42567
$448$	−8.34315	−0.394177
$449$	−0.313708	−0.0148048	−0.00740241	−	0.999973i	$-0.502356\pi$
$449$	−0.313708	−0.0148048	−0.00740241	+	0.999973i	$0.502356\pi$
$450$	1.24264	0.0585786
$451$	11.0122	0.518544
$452$	37.7696	1.77653
$453$	26.8284	1.26051
$454$	−3.79899	−0.178295
$455$	8.48528	0.397796
$456$	7.27208	0.340546
$457$	−39.2132	−1.83432	−0.917158	−	0.398523i	$-0.869523\pi$
$457$	−39.2132	−1.83432	−0.917158	+	0.398523i	$0.869523\pi$
$458$	8.48528	0.396491
$459$	−41.9411	−1.95764
$460$	23.4558	1.09363
$461$	−2.48528	−0.115751	−0.0578755	−	0.998324i	$-0.518433\pi$
$461$	−2.48528	−0.115751	−0.0578755	+	0.998324i	$0.518433\pi$
$462$	4.20101	0.195449
$463$	−33.0711	−1.53694	−0.768471	−	0.639884i	$-0.778982\pi$
$463$	−33.0711	−1.53694	−0.768471	+	0.639884i	$0.778982\pi$
$464$	−29.4853	−1.36882
$465$	1.65685	0.0768348
$466$	7.10051	0.328925
$467$	−32.4142	−1.49995	−0.749975	−	0.661466i	$-0.769935\pi$
$467$	−32.4142	−1.49995	−0.749975	+	0.661466i	$0.769935\pi$
$468$	5.48528	0.253557
$469$	9.79899	0.452475
$470$	5.61522	0.259011
$471$	−2.34315	−0.107966
$472$	2.78680	0.128273
$473$	28.6863	1.31900
$474$	−1.79899	−0.0826303
$475$	−9.72792	−0.446348
$476$	27.1127	1.24271
$477$	1.34315	0.0614984
$478$	4.02944	0.184302
$479$	−15.3137	−0.699701	−0.349851	−	0.936806i	$-0.613768\pi$
$479$	−15.3137	−0.699701	−0.349851	+	0.936806i	$0.613768\pi$
$480$	−8.82843	−0.402961
$481$	−1.75736	−0.0801287
$482$	−2.28427	−0.104046
$483$	25.6569	1.16743
$484$	−3.39697	−0.154408
$485$	0.727922	0.0330532
$486$	−4.10051	−0.186003
$487$	20.9706	0.950267	0.475133	−	0.879914i	$-0.342400\pi$
$487$	20.9706	0.950267	0.475133	+	0.879914i	$0.342400\pi$
$488$	−0.928932	−0.0420508
$489$	25.6569	1.16024
$490$	−1.75736	−0.0793894
$491$	−37.4558	−1.69036	−0.845179	−	0.534483i	$-0.820506\pi$
$491$	−37.4558	−1.69036	−0.845179	+	0.534483i	$0.820506\pi$
$492$	−7.94113	−0.358014
$493$	72.8701	3.28190
$494$	4.02944	0.181293
$495$	5.07107	0.227928
$496$	−2.48528	−0.111592
$497$	2.82843	0.126872
$498$	−8.82843	−0.395611
$499$	−16.7574	−0.750162	−0.375081	−	0.926992i	$-0.622385\pi$
$499$	−16.7574	−0.750162	−0.375081	+	0.926992i	$0.622385\pi$
$500$	20.6863	0.925119
$501$	−30.1421	−1.34665
$502$	−0.343146	−0.0153153
$503$	15.0711	0.671986	0.335993	−	0.941865i	$-0.390928\pi$
$503$	15.0711	0.671986	0.335993	+	0.941865i	$0.390928\pi$
$504$	3.17157	0.141273
$505$	22.1421	0.985312
$506$	13.4731	0.598952
$507$	5.65685	0.251230
$508$	−28.7574	−1.27590
$509$	−10.1716	−0.450847	−0.225424	−	0.974261i	$-0.572377\pi$
$509$	−10.1716	−0.450847	−0.225424	+	0.974261i	$0.572377\pi$
$510$	6.14214	0.271978
$511$	9.31371	0.412014
$512$	22.7574	1.00574
$513$	18.3431	0.809870
$514$	8.05887	0.355462
$515$	20.1421	0.887569
$516$	−20.6863	−0.910663
$517$	−34.3726	−1.51170
$518$	−0.485281	−0.0213220
$519$	20.4853	0.899204
$520$	−6.72792	−0.295039
$521$	25.9706	1.13779	0.568896	−	0.822410i	$-0.307371\pi$
$521$	25.9706	1.13779	0.568896	+	0.822410i	$0.307371\pi$
$522$	4.07107	0.178186
$523$	−23.6985	−1.03626	−0.518131	−	0.855301i	$-0.673372\pi$
$523$	−23.6985	−1.03626	−0.518131	+	0.855301i	$0.673372\pi$
$524$	19.9289	0.870599
$525$	8.48528	0.370328
$526$	−0.857864	−0.0374047
$527$	6.14214	0.267556
$528$	15.2132	0.662070
$529$	59.2843	2.57758
$530$	−0.786797	−0.0341763
$531$	1.75736	0.0762629
$532$	−11.8579	−0.514104
$533$	−9.21320	−0.399068
$534$	2.48528	0.107549
$535$	−14.4853	−0.626253
$536$	−7.76955	−0.335594
$537$	10.8284	0.467281
$538$	−3.51472	−0.151530
$539$	10.7574	0.463352
$540$	−14.6274	−0.629464
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	10.0416	0.431325
$543$	7.07107	0.303449
$544$	−32.7279	−1.40320
$545$	2.48528	0.106458
$546$	−3.51472	−0.150416
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−3.02944	−0.129411
$549$	−0.585786	−0.0250008
$550$	4.45584	0.189998
$551$	−31.8701	−1.35771
$552$	−20.3431	−0.865862
$553$	6.14214	0.261190
$554$	−5.72792	−0.243356
$555$	1.17157	0.0497305
$556$	7.31371	0.310170
$557$	25.6569	1.08712	0.543558	−	0.839372i	$-0.317077\pi$
$557$	25.6569	1.08712	0.543558	+	0.839372i	$0.317077\pi$
$558$	0.343146	0.0145265
$559$	−24.0000	−1.01509
$560$	8.48528	0.358569
$561$	−37.5980	−1.58739
$562$	−6.97056	−0.294035
$563$	−13.3137	−0.561106	−0.280553	−	0.959839i	$-0.590518\pi$
$563$	−13.3137	−0.561106	−0.280553	+	0.959839i	$0.590518\pi$
$564$	24.7868	1.04371
$565$	−29.2132	−1.22901
$566$	2.48528	0.104464
$567$	−10.0000	−0.419961
$568$	−2.24264	−0.0940991
$569$	−24.7279	−1.03665	−0.518324	−	0.855184i	$-0.673444\pi$
$569$	−24.7279	−1.03665	−0.518324	+	0.855184i	$0.673444\pi$
$570$	−2.68629	−0.112516
$571$	−7.24264	−0.303095	−0.151548	−	0.988450i	$-0.548426\pi$
$571$	−7.24264	−0.303095	−0.151548	+	0.988450i	$0.548426\pi$
$572$	19.6690	0.822404
$573$	−30.7279	−1.28368
$574$	−2.54416	−0.106191
$575$	27.2132	1.13487
$576$	4.17157	0.173816
$577$	−10.8284	−0.450793	−0.225397	−	0.974267i	$-0.572368\pi$
$577$	−10.8284	−0.450793	−0.225397	+	0.974267i	$0.572368\pi$
$578$	15.7279	0.654195
$579$	28.9706	1.20398
$580$	25.4142	1.05527
$581$	30.1421	1.25051
$582$	−0.301515	−0.0124982
$583$	4.81623	0.199468
$584$	−7.38478	−0.305584
$585$	−4.24264	−0.175412
$586$	0.970563	0.0400936
$587$	33.0416	1.36377	0.681887	−	0.731458i	$-0.261160\pi$
$587$	33.0416	1.36377	0.681887	+	0.731458i	$0.261160\pi$
$588$	−7.75736	−0.319908
$589$	−2.68629	−0.110687
$590$	−1.02944	−0.0423813
$591$	15.5147	0.638190
$592$	−1.75736	−0.0722270
$593$	45.7696	1.87953	0.939765	−	0.341821i	$-0.111044\pi$
$593$	45.7696	1.87953	0.939765	+	0.341821i	$0.111044\pi$
$594$	−8.40202	−0.344739
$595$	−20.9706	−0.859709
$596$	−38.9706	−1.59630
$597$	−17.0711	−0.698672
$598$	−11.2721	−0.460950
$599$	−30.3431	−1.23979	−0.619894	−	0.784686i	$-0.712824\pi$
$599$	−30.3431	−1.23979	−0.619894	+	0.784686i	$0.712824\pi$
$600$	−6.72792	−0.274666
$601$	−14.7990	−0.603663	−0.301832	−	0.953361i	$-0.597598\pi$
$601$	−14.7990	−0.603663	−0.301832	+	0.953361i	$0.597598\pi$
$602$	−6.62742	−0.270113
$603$	−4.89949	−0.199523
$604$	34.6863	1.41137
$605$	2.62742	0.106820
$606$	−9.17157	−0.372570
$607$	1.04163	0.0422785	0.0211392	−	0.999777i	$-0.493271\pi$
$607$	1.04163	0.0422785	0.0211392	+	0.999777i	$0.493271\pi$
$608$	14.3137	0.580498
$609$	27.7990	1.12647
$610$	0.343146	0.0138936
$611$	28.7574	1.16340
$612$	−13.5563	−0.547983
$613$	−4.02944	−0.162747	−0.0813737	−	0.996684i	$-0.525931\pi$
$613$	−4.02944	−0.162747	−0.0813737	+	0.996684i	$0.525931\pi$
$614$	−8.54416	−0.344814
$615$	6.14214	0.247675
$616$	11.3726	0.458214
$617$	14.1421	0.569341	0.284670	−	0.958625i	$-0.408116\pi$
$617$	14.1421	0.569341	0.284670	+	0.958625i	$0.408116\pi$
$618$	−8.34315	−0.335611
$619$	−23.7990	−0.956562	−0.478281	−	0.878207i	$-0.658740\pi$
$619$	−23.7990	−0.956562	−0.478281	+	0.878207i	$0.658740\pi$
$620$	2.14214	0.0860302
$621$	−51.3137	−2.05915
$622$	5.14214	0.206181
$623$	−8.48528	−0.339956
$624$	−12.7279	−0.509525
$625$	−1.00000	−0.0400000
$626$	−12.7574	−0.509887
$627$	16.4437	0.656696
$628$	−3.02944	−0.120888
$629$	4.34315	0.173173
$630$	−1.17157	−0.0466766
$631$	−8.41421	−0.334965	−0.167482	−	0.985875i	$-0.553564\pi$
$631$	−8.41421	−0.334965	−0.167482	+	0.985875i	$0.553564\pi$
$632$	−4.87006	−0.193720
$633$	15.5147	0.616655
$634$	9.78680	0.388683
$635$	22.2426	0.882672
$636$	−3.47309	−0.137717
$637$	−9.00000	−0.356593
$638$	14.5980	0.577940
$639$	−1.41421	−0.0559454
$640$	−14.9289	−0.590118
$641$	1.79899	0.0710558	0.0355279	−	0.999369i	$-0.488689\pi$
$641$	1.79899	0.0710558	0.0355279	+	0.999369i	$0.488689\pi$
$642$	6.00000	0.236801
$643$	24.9706	0.984743	0.492371	−	0.870385i	$-0.336130\pi$
$643$	24.9706	0.984743	0.492371	+	0.870385i	$0.336130\pi$
$644$	33.1716	1.30714
$645$	16.0000	0.629999
$646$	−9.95837	−0.391807
$647$	−22.8284	−0.897478	−0.448739	−	0.893663i	$-0.648127\pi$
$647$	−22.8284	−0.897478	−0.448739	+	0.893663i	$0.648127\pi$
$648$	7.92893	0.311478
$649$	6.30152	0.247356
$650$	−3.72792	−0.146221
$651$	2.34315	0.0918351
$652$	33.1716	1.29910
$653$	−12.7279	−0.498082	−0.249041	−	0.968493i	$-0.580115\pi$
$653$	−12.7279	−0.498082	−0.249041	+	0.968493i	$0.580115\pi$
$654$	−1.02944	−0.0402542
$655$	−15.4142	−0.602283
$656$	−9.21320	−0.359715
$657$	−4.65685	−0.181681
$658$	7.94113	0.309577
$659$	−12.1005	−0.471369	−0.235684	−	0.971830i	$-0.575733\pi$
$659$	−12.1005	−0.471369	−0.235684	+	0.971830i	$0.575733\pi$
$660$	−13.1127	−0.510411
$661$	−30.1127	−1.17125	−0.585624	−	0.810583i	$-0.699151\pi$
$661$	−30.1127	−1.17125	−0.585624	+	0.810583i	$0.699151\pi$
$662$	12.3015	0.478112
$663$	31.4558	1.22164
$664$	−23.8995	−0.927480
$665$	9.17157	0.355658
$666$	0.242641	0.00940214
$667$	89.1543	3.45207
$668$	−38.9706	−1.50782
$669$	30.0000	1.15987
$670$	2.87006	0.110880
$671$	−2.10051	−0.0810891
$672$	−12.4853	−0.481630
$673$	12.4558	0.480138	0.240069	−	0.970756i	$-0.422830\pi$
$673$	12.4558	0.480138	0.240069	+	0.970756i	$0.422830\pi$
$674$	−0.544156	−0.0209601
$675$	−16.9706	−0.653197
$676$	7.31371	0.281296
$677$	−34.1127	−1.31106	−0.655529	−	0.755170i	$-0.727554\pi$
$677$	−34.1127	−1.31106	−0.655529	+	0.755170i	$0.727554\pi$
$678$	12.1005	0.464717
$679$	1.02944	0.0395062
$680$	16.6274	0.637632
$681$	12.9706	0.497033
$682$	1.23045	0.0471163
$683$	−1.24264	−0.0475483	−0.0237742	−	0.999717i	$-0.507568\pi$
$683$	−1.24264	−0.0475483	−0.0237742	+	0.999717i	$0.507568\pi$
$684$	5.92893	0.226698
$685$	2.34315	0.0895270
$686$	−8.28427	−0.316295
$687$	−28.9706	−1.10530
$688$	−24.0000	−0.914991
$689$	−4.02944	−0.153509
$690$	7.51472	0.286080
$691$	8.34315	0.317388	0.158694	−	0.987328i	$-0.449272\pi$
$691$	8.34315	0.317388	0.158694	+	0.987328i	$0.449272\pi$
$692$	26.4853	1.00682
$693$	7.17157	0.272426
$694$	−11.8579	−0.450118
$695$	−5.65685	−0.214577
$696$	−22.0416	−0.835486
$697$	22.7696	0.862458
$698$	−5.58579	−0.211425
$699$	−24.2426	−0.916941
$700$	10.9706	0.414648
$701$	28.4558	1.07476	0.537381	−	0.843339i	$-0.319414\pi$
$701$	28.4558	1.07476	0.537381	+	0.843339i	$0.319414\pi$
$702$	7.02944	0.265309
$703$	−1.89949	−0.0716408
$704$	14.9584	0.563765
$705$	−19.1716	−0.722043
$706$	−7.52691	−0.283279
$707$	31.3137	1.17767
$708$	−4.54416	−0.170780
$709$	−10.3848	−0.390008	−0.195004	−	0.980802i	$-0.562472\pi$
$709$	−10.3848	−0.390008	−0.195004	+	0.980802i	$0.562472\pi$
$710$	0.828427	0.0310903
$711$	−3.07107	−0.115174
$712$	6.72792	0.252140
$713$	7.51472	0.281428
$714$	8.68629	0.325076
$715$	−15.2132	−0.568942
$716$	14.0000	0.523205
$717$	−13.7574	−0.513778
$718$	4.52691	0.168943
$719$	32.4853	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$719$	32.4853	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$720$	−4.24264	−0.158114
$721$	28.4853	1.06085
$722$	−3.51472	−0.130804
$723$	7.79899	0.290048
$724$	9.14214	0.339765
$725$	29.4853	1.09506
$726$	−1.08831	−0.0403910
$727$	−24.7279	−0.917108	−0.458554	−	0.888667i	$-0.651632\pi$
$727$	−24.7279	−0.917108	−0.458554	+	0.888667i	$0.651632\pi$
$728$	−9.51472	−0.352639
$729$	29.0000	1.07407
$730$	2.72792	0.100965
$731$	59.3137	2.19380
$732$	1.51472	0.0559856
$733$	−21.9411	−0.810414	−0.405207	−	0.914225i	$-0.632801\pi$
$733$	−21.9411	−0.810414	−0.405207	+	0.914225i	$0.632801\pi$
$734$	3.62742	0.133890
$735$	6.00000	0.221313
$736$	−40.0416	−1.47595
$737$	−17.5685	−0.647146
$738$	1.27208	0.0468258
$739$	−18.7279	−0.688918	−0.344459	−	0.938801i	$-0.611938\pi$
$739$	−18.7279	−0.688918	−0.344459	+	0.938801i	$0.611938\pi$
$740$	1.51472	0.0556822
$741$	−13.7574	−0.505389
$742$	−1.11270	−0.0408484
$743$	−50.1421	−1.83954	−0.919768	−	0.392462i	$-0.871623\pi$
$743$	−50.1421	−1.83954	−0.919768	+	0.392462i	$0.871623\pi$
$744$	−1.85786	−0.0681126
$745$	30.1421	1.10432
$746$	−6.52691	−0.238967
$747$	−15.0711	−0.551422
$748$	−48.6102	−1.77736
$749$	−20.4853	−0.748516
$750$	6.62742	0.241999
$751$	−18.6985	−0.682317	−0.341159	−	0.940006i	$-0.610819\pi$
$751$	−18.6985	−0.682317	−0.341159	+	0.940006i	$0.610819\pi$
$752$	28.7574	1.04867
$753$	1.17157	0.0426945
$754$	−12.2132	−0.444779
$755$	−26.8284	−0.976386
$756$	−20.6863	−0.752353
$757$	18.4558	0.670789	0.335395	−	0.942078i	$-0.391130\pi$
$757$	18.4558	0.670789	0.335395	+	0.942078i	$0.391130\pi$
$758$	7.54416	0.274016
$759$	−46.0000	−1.66969
$760$	−7.27208	−0.263786
$761$	−35.8284	−1.29878	−0.649390	−	0.760456i	$-0.724976\pi$
$761$	−35.8284	−1.29878	−0.649390	+	0.760456i	$0.724976\pi$
$762$	−9.21320	−0.333759
$763$	3.51472	0.127241
$764$	−39.7279	−1.43731
$765$	10.4853	0.379096
$766$	−4.31371	−0.155861
$767$	−5.27208	−0.190364
$768$	−5.61522	−0.202622
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	−4.20101	−0.151394
$771$	−27.5147	−0.990918
$772$	37.4558	1.34807
$773$	−36.3848	−1.30867	−0.654335	−	0.756205i	$-0.727051\pi$
$773$	−36.3848	−1.30867	−0.654335	+	0.756205i	$0.727051\pi$
$774$	3.31371	0.119109
$775$	2.48528	0.0892739
$776$	−0.816234	−0.0293011
$777$	1.65685	0.0594393
$778$	−6.24264	−0.223810
$779$	−9.95837	−0.356796
$780$	10.9706	0.392809
$781$	−5.07107	−0.181457
$782$	27.8579	0.996195
$783$	−55.5980	−1.98691
$784$	−9.00000	−0.321429
$785$	2.34315	0.0836305
$786$	6.38478	0.227737
$787$	−52.9706	−1.88820	−0.944098	−	0.329664i	$-0.893065\pi$
$787$	−52.9706	−1.88820	−0.944098	+	0.329664i	$0.893065\pi$
$788$	20.0589	0.714568
$789$	2.92893	0.104273
$790$	1.79899	0.0640052
$791$	−41.3137	−1.46895
$792$	−5.68629	−0.202054
$793$	1.75736	0.0624057
$794$	−4.24264	−0.150566
$795$	2.68629	0.0952729
$796$	−22.0711	−0.782288
$797$	−0.857864	−0.0303871	−0.0151936	−	0.999885i	$-0.504836\pi$
$797$	−0.857864	−0.0303871	−0.0151936	+	0.999885i	$0.504836\pi$
$798$	−3.79899	−0.134483
$799$	−71.0711	−2.51431
$800$	−13.2426	−0.468198
$801$	4.24264	0.149906
$802$	−3.18377	−0.112423
$803$	−16.6985	−0.589277
$804$	12.6690	0.446803
$805$	−25.6569	−0.904285
$806$	−1.02944	−0.0362604
$807$	12.0000	0.422420
$808$	−24.8284	−0.873461
$809$	−23.7990	−0.836728	−0.418364	−	0.908279i	$-0.637396\pi$
$809$	−23.7990	−0.836728	−0.418364	+	0.908279i	$0.637396\pi$
$810$	−2.92893	−0.102912
$811$	8.75736	0.307512	0.153756	−	0.988109i	$-0.450863\pi$
$811$	8.75736	0.307512	0.153756	+	0.988109i	$0.450863\pi$
$812$	35.9411	1.26129
$813$	−34.2843	−1.20240
$814$	0.870058	0.0304955
$815$	−25.6569	−0.898720
$816$	31.4558	1.10117
$817$	−25.9411	−0.907565
$818$	15.3137	0.535431
$819$	−6.00000	−0.209657
$820$	7.94113	0.277316
$821$	−10.5858	−0.369446	−0.184723	−	0.982791i	$-0.559139\pi$
$821$	−10.5858	−0.369446	−0.184723	+	0.982791i	$0.559139\pi$
$822$	−0.970563	−0.0338523
$823$	−2.27208	−0.0791997	−0.0395998	−	0.999216i	$-0.512608\pi$
$823$	−2.27208	−0.0791997	−0.0395998	+	0.999216i	$0.512608\pi$
$824$	−22.5858	−0.786813
$825$	−15.2132	−0.529656
$826$	−1.45584	−0.0506553
$827$	−2.20101	−0.0765366	−0.0382683	−	0.999267i	$-0.512184\pi$
$827$	−2.20101	−0.0765366	−0.0382683	+	0.999267i	$0.512184\pi$
$828$	−16.5858	−0.576396
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	8.82843	0.306439
$831$	19.5563	0.678402
$832$	−12.5147	−0.433870
$833$	22.2426	0.770662
$834$	2.34315	0.0811365
$835$	30.1421	1.04311
$836$	21.2599	0.735288
$837$	−4.68629	−0.161982
$838$	5.76955	0.199306
$839$	−6.14214	−0.212050	−0.106025	−	0.994363i	$-0.533812\pi$
$839$	−6.14214	−0.212050	−0.106025	+	0.994363i	$0.533812\pi$
$840$	6.34315	0.218859
$841$	67.5980	2.33096
$842$	−5.01219	−0.172732
$843$	23.7990	0.819681
$844$	20.0589	0.690455
$845$	−5.65685	−0.194602
$846$	−3.97056	−0.136511
$847$	3.71573	0.127674
$848$	−4.02944	−0.138371
$849$	−8.48528	−0.291214
$850$	9.21320	0.316010
$851$	5.31371	0.182152
$852$	3.65685	0.125282
$853$	−46.4558	−1.59062	−0.795309	−	0.606204i	$-0.792691\pi$
$853$	−46.4558	−1.59062	−0.795309	+	0.606204i	$0.792691\pi$
$854$	0.485281	0.0166060
$855$	−4.58579	−0.156831
$856$	16.2426	0.555162
$857$	−44.1421	−1.50787	−0.753933	−	0.656951i	$-0.771846\pi$
$857$	−44.1421	−1.50787	−0.753933	+	0.656951i	$0.771846\pi$
$858$	6.30152	0.215130
$859$	−4.75736	−0.162319	−0.0811595	−	0.996701i	$-0.525862\pi$
$859$	−4.75736	−0.162319	−0.0811595	+	0.996701i	$0.525862\pi$
$860$	20.6863	0.705397
$861$	8.68629	0.296028
$862$	13.6985	0.466572
$863$	−19.8995	−0.677387	−0.338693	−	0.940897i	$-0.609985\pi$
$863$	−19.8995	−0.677387	−0.338693	+	0.940897i	$0.609985\pi$
$864$	24.9706	0.849516
$865$	−20.4853	−0.696520
$866$	7.27208	0.247115
$867$	−53.6985	−1.82370
$868$	3.02944	0.102826
$869$	−11.0122	−0.373563
$870$	8.14214	0.276044
$871$	14.6985	0.498039
$872$	−2.78680	−0.0943728
$873$	−0.514719	−0.0174206
$874$	−12.1838	−0.412122
$875$	−22.6274	−0.764946
$876$	12.0416	0.406849
$877$	47.7990	1.61406	0.807029	−	0.590512i	$-0.201074\pi$
$877$	47.7990	1.61406	0.807029	+	0.590512i	$0.201074\pi$
$878$	12.3137	0.415568
$879$	−3.31371	−0.111769
$880$	−15.2132	−0.512837
$881$	9.07107	0.305612	0.152806	−	0.988256i	$-0.451169\pi$
$881$	9.07107	0.305612	0.152806	+	0.988256i	$0.451169\pi$
$882$	1.24264	0.0418419
$883$	0.213203	0.00717487	0.00358743	−	0.999994i	$-0.498858\pi$
$883$	0.213203	0.00717487	0.00358743	+	0.999994i	$0.498858\pi$
$884$	40.6690	1.36785
$885$	3.51472	0.118146
$886$	3.51472	0.118079
$887$	−35.1838	−1.18136	−0.590678	−	0.806908i	$-0.701139\pi$
$887$	−35.1838	−1.18136	−0.590678	+	0.806908i	$0.701139\pi$
$888$	−1.31371	−0.0440852
$889$	31.4558	1.05500
$890$	−2.48528	−0.0833068
$891$	17.9289	0.600642
$892$	38.7868	1.29868
$893$	31.0833	1.04016
$894$	−12.4853	−0.417570
$895$	−10.8284	−0.361954
$896$	−21.1127	−0.705326
$897$	38.4853	1.28499
$898$	−0.129942	−0.00433623
$899$	8.14214	0.271555
$900$	−5.48528	−0.182843
$901$	9.95837	0.331761
$902$	4.56140	0.151878
$903$	22.6274	0.752993
$904$	32.7574	1.08949
$905$	−7.07107	−0.235050
$906$	11.1127	0.369195
$907$	−28.2132	−0.936804	−0.468402	−	0.883515i	$-0.655170\pi$
$907$	−28.2132	−0.936804	−0.468402	+	0.883515i	$0.655170\pi$
$908$	16.7696	0.556517
$909$	−15.6569	−0.519305
$910$	3.51472	0.116512
$911$	37.3137	1.23626	0.618129	−	0.786077i	$-0.287891\pi$
$911$	37.3137	1.23626	0.618129	+	0.786077i	$0.287891\pi$
$912$	−13.7574	−0.455552
$913$	−54.0416	−1.78852
$914$	−16.2426	−0.537259
$915$	−1.17157	−0.0387310
$916$	−37.4558	−1.23758
$917$	−21.7990	−0.719866
$918$	−17.3726	−0.573381
$919$	16.5563	0.546144	0.273072	−	0.961994i	$-0.411960\pi$
$919$	16.5563	0.546144	0.273072	+	0.961994i	$0.411960\pi$
$920$	20.3431	0.670694
$921$	29.1716	0.961236
$922$	−1.02944	−0.0339027
$923$	4.24264	0.139648
$924$	−18.5442	−0.610058
$925$	1.75736	0.0577816
$926$	−13.6985	−0.450160
$927$	−14.2426	−0.467790
$928$	−43.3848	−1.42418
$929$	−34.5858	−1.13472	−0.567361	−	0.823469i	$-0.692036\pi$
$929$	−34.5858	−1.13472	−0.567361	+	0.823469i	$0.692036\pi$
$930$	0.686292	0.0225044
$931$	−9.72792	−0.318820
$932$	−31.3431	−1.02668
$933$	−17.5563	−0.574769
$934$	−13.4264	−0.439325
$935$	37.5980	1.22959
$936$	4.75736	0.155499
$937$	26.9706	0.881090	0.440545	−	0.897731i	$-0.354785\pi$
$937$	26.9706	0.881090	0.440545	+	0.897731i	$0.354785\pi$
$938$	4.05887	0.132527
$939$	43.5563	1.42141
$940$	−24.7868	−0.808456
$941$	30.5980	0.997466	0.498733	−	0.866756i	$-0.333799\pi$
$941$	30.5980	0.997466	0.498733	+	0.866756i	$0.333799\pi$
$942$	−0.970563	−0.0316226
$943$	27.8579	0.907177
$944$	−5.27208	−0.171592
$945$	16.0000	0.520480
$946$	11.8823	0.386325
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	7.94113	0.257916
$949$	13.9706	0.453504
$950$	−4.02944	−0.130732
$951$	−33.4142	−1.08353
$952$	23.5147	0.762116
$953$	2.65685	0.0860640	0.0430320	−	0.999074i	$-0.486298\pi$
$953$	2.65685	0.0860640	0.0430320	+	0.999074i	$0.486298\pi$
$954$	0.556349	0.0180125
$955$	30.7279	0.994332
$956$	−17.7868	−0.575266
$957$	−49.8406	−1.61112
$958$	−6.34315	−0.204938
$959$	3.31371	0.107005
$960$	8.34315	0.269274
$961$	−30.3137	−0.977862
$962$	−0.727922	−0.0234691
$963$	10.2426	0.330064
$964$	10.0833	0.324760
$965$	−28.9706	−0.932595
$966$	10.6274	0.341932
$967$	20.6274	0.663333	0.331667	−	0.943397i	$-0.392389\pi$
$967$	20.6274	0.663333	0.331667	+	0.943397i	$0.392389\pi$
$968$	−2.94618	−0.0946937
$969$	34.0000	1.09224
$970$	0.301515	0.00968107
$971$	17.8995	0.574422	0.287211	−	0.957867i	$-0.407272\pi$
$971$	17.8995	0.574422	0.287211	+	0.957867i	$0.407272\pi$
$972$	18.1005	0.580574
$973$	−8.00000	−0.256468
$974$	8.68629	0.278327
$975$	12.7279	0.407620
$976$	1.75736	0.0562517
$977$	−51.5563	−1.64943	−0.824717	−	0.565546i	$-0.808666\pi$
$977$	−51.5563	−1.64943	−0.824717	+	0.565546i	$0.808666\pi$
$978$	10.6274	0.339827
$979$	15.2132	0.486216
$980$	7.75736	0.247800
$981$	−1.75736	−0.0561082
$982$	−15.5147	−0.495095
$983$	42.3848	1.35186	0.675932	−	0.736964i	$-0.263741\pi$
$983$	42.3848	1.35186	0.675932	+	0.736964i	$0.263741\pi$
$984$	−6.88730	−0.219559
$985$	−15.5147	−0.494340
$986$	30.1838	0.961247
$987$	−27.1127	−0.863007
$988$	−17.7868	−0.565873
$989$	72.5685	2.30755
$990$	2.10051	0.0667584
$991$	30.7574	0.977040	0.488520	−	0.872553i	$-0.337537\pi$
$991$	30.7574	0.977040	0.488520	+	0.872553i	$0.337537\pi$
$992$	−3.65685	−0.116105
$993$	−42.0000	−1.33283
$994$	1.17157	0.0371600
$995$	17.0711	0.541189
$996$	38.9706	1.23483
$997$	37.9411	1.20161	0.600804	−	0.799397i	$-0.294847\pi$
$997$	37.9411	1.20161	0.600804	+	0.799397i	$0.294847\pi$
$998$	−6.94113	−0.219717
$999$	−3.31371	−0.104841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.a.a.1.2	✓	2
3.2	odd	2		4923.2.a.h.1.1		2
4.3	odd	2		8752.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.a.1.2	✓	2	1.1	even	1	trivial
4923.2.a.h.1.1		2	3.2	odd	2
8752.2.a.n.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.41421 q^{3} -1.82843 q^{4} +1.41421 q^{5} -0.585786 q^{6} +2.00000 q^{7} -1.58579 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} -1.41421 q^{3} -1.82843 q^{4} +1.41421 q^{5} -0.585786 q^{6} +2.00000 q^{7} -1.58579 q^{8} -1.00000 q^{9} +0.585786 q^{10} -3.58579 q^{11} +2.58579 q^{12} +3.00000 q^{13} +0.828427 q^{14} -2.00000 q^{15} +3.00000 q^{16} -7.41421 q^{17} -0.414214 q^{18} +3.24264 q^{19} -2.58579 q^{20} -2.82843 q^{21} -1.48528 q^{22} -9.07107 q^{23} +2.24264 q^{24} -3.00000 q^{25} +1.24264 q^{26} +5.65685 q^{27} -3.65685 q^{28} -9.82843 q^{29} -0.828427 q^{30} -0.828427 q^{31} +4.41421 q^{32} +5.07107 q^{33} -3.07107 q^{34} +2.82843 q^{35} +1.82843 q^{36} -0.585786 q^{37} +1.34315 q^{38} -4.24264 q^{39} -2.24264 q^{40} -3.07107 q^{41} -1.17157 q^{42} -8.00000 q^{43} +6.55635 q^{44} -1.41421 q^{45} -3.75736 q^{46} +9.58579 q^{47} -4.24264 q^{48} -3.00000 q^{49} -1.24264 q^{50} +10.4853 q^{51} -5.48528 q^{52} -1.34315 q^{53} +2.34315 q^{54} -5.07107 q^{55} -3.17157 q^{56} -4.58579 q^{57} -4.07107 q^{58} -1.75736 q^{59} +3.65685 q^{60} +0.585786 q^{61} -0.343146 q^{62} -2.00000 q^{63} -4.17157 q^{64} +4.24264 q^{65} +2.10051 q^{66} +4.89949 q^{67} +13.5563 q^{68} +12.8284 q^{69} +1.17157 q^{70} +1.41421 q^{71} +1.58579 q^{72} +4.65685 q^{73} -0.242641 q^{74} +4.24264 q^{75} -5.92893 q^{76} -7.17157 q^{77} -1.75736 q^{78} +3.07107 q^{79} +4.24264 q^{80} -5.00000 q^{81} -1.27208 q^{82} +15.0711 q^{83} +5.17157 q^{84} -10.4853 q^{85} -3.31371 q^{86} +13.8995 q^{87} +5.68629 q^{88} -4.24264 q^{89} -0.585786 q^{90} +6.00000 q^{91} +16.5858 q^{92} +1.17157 q^{93} +3.97056 q^{94} +4.58579 q^{95} -6.24264 q^{96} +0.514719 q^{97} -1.24264 q^{98} +3.58579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} + 4 q^{10} - 10 q^{11} + 8 q^{12} + 6 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 12 q^{17} + 2 q^{18} - 2 q^{19} - 8 q^{20} + 14 q^{22} - 4 q^{23} - 4 q^{24} - 6 q^{25} - 6 q^{26} + 4 q^{28} - 14 q^{29} + 4 q^{30} + 4 q^{31} + 6 q^{32} - 4 q^{33} + 8 q^{34} - 2 q^{36} - 4 q^{37} + 14 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{42} - 16 q^{43} - 18 q^{44} - 16 q^{46} + 22 q^{47} - 6 q^{49} + 6 q^{50} + 4 q^{51} + 6 q^{52} - 14 q^{53} + 16 q^{54} + 4 q^{55} - 12 q^{56} - 12 q^{57} + 6 q^{58} - 12 q^{59} - 4 q^{60} + 4 q^{61} - 12 q^{62} - 4 q^{63} - 14 q^{64} + 24 q^{66} - 10 q^{67} - 4 q^{68} + 20 q^{69} + 8 q^{70} + 6 q^{72} - 2 q^{73} + 8 q^{74} - 26 q^{76} - 20 q^{77} - 12 q^{78} - 8 q^{79} - 10 q^{81} - 28 q^{82} + 16 q^{83} + 16 q^{84} - 4 q^{85} + 16 q^{86} + 8 q^{87} + 34 q^{88} - 4 q^{90} + 12 q^{91} + 36 q^{92} + 8 q^{93} - 26 q^{94} + 12 q^{95} - 4 q^{96} + 18 q^{97} + 6 q^{98} + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^6 + 4 * q^7 - 6 * q^8 - 2 * q^9 + 4 * q^10 - 10 * q^11 + 8 * q^12 + 6 * q^13 - 4 * q^14 - 4 * q^15 + 6 * q^16 - 12 * q^17 + 2 * q^18 - 2 * q^19 - 8 * q^20 + 14 * q^22 - 4 * q^23 - 4 * q^24 - 6 * q^25 - 6 * q^26 + 4 * q^28 - 14 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 - 4 * q^33 + 8 * q^34 - 2 * q^36 - 4 * q^37 + 14 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^42 - 16 * q^43 - 18 * q^44 - 16 * q^46 + 22 * q^47 - 6 * q^49 + 6 * q^50 + 4 * q^51 + 6 * q^52 - 14 * q^53 + 16 * q^54 + 4 * q^55 - 12 * q^56 - 12 * q^57 + 6 * q^58 - 12 * q^59 - 4 * q^60 + 4 * q^61 - 12 * q^62 - 4 * q^63 - 14 * q^64 + 24 * q^66 - 10 * q^67 - 4 * q^68 + 20 * q^69 + 8 * q^70 + 6 * q^72 - 2 * q^73 + 8 * q^74 - 26 * q^76 - 20 * q^77 - 12 * q^78 - 8 * q^79 - 10 * q^81 - 28 * q^82 + 16 * q^83 + 16 * q^84 - 4 * q^85 + 16 * q^86 + 8 * q^87 + 34 * q^88 - 4 * q^90 + 12 * q^91 + 36 * q^92 + 8 * q^93 - 26 * q^94 + 12 * q^95 - 4 * q^96 + 18 * q^97 + 6 * q^98 + 10 * q^99

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