LMFDB - Embedded newform 538.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, 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("538.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.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:	$4.29595162874$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.38197 q^{5} -1.61803 q^{6} +0.236068 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.38197 q^{5} -1.61803 q^{6} +0.236068 q^{7} +1.00000 q^{8} -0.381966 q^{9} -1.38197 q^{10} -4.23607 q^{11} -1.61803 q^{12} -1.00000 q^{13} +0.236068 q^{14} +2.23607 q^{15} +1.00000 q^{16} -4.23607 q^{17} -0.381966 q^{18} +2.00000 q^{19} -1.38197 q^{20} -0.381966 q^{21} -4.23607 q^{22} -3.76393 q^{23} -1.61803 q^{24} -3.09017 q^{25} -1.00000 q^{26} +5.47214 q^{27} +0.236068 q^{28} -5.85410 q^{29} +2.23607 q^{30} -1.47214 q^{31} +1.00000 q^{32} +6.85410 q^{33} -4.23607 q^{34} -0.326238 q^{35} -0.381966 q^{36} +4.70820 q^{37} +2.00000 q^{38} +1.61803 q^{39} -1.38197 q^{40} -5.70820 q^{41} -0.381966 q^{42} -2.23607 q^{43} -4.23607 q^{44} +0.527864 q^{45} -3.76393 q^{46} -0.145898 q^{47} -1.61803 q^{48} -6.94427 q^{49} -3.09017 q^{50} +6.85410 q^{51} -1.00000 q^{52} +5.47214 q^{54} +5.85410 q^{55} +0.236068 q^{56} -3.23607 q^{57} -5.85410 q^{58} +13.5623 q^{59} +2.23607 q^{60} +3.47214 q^{61} -1.47214 q^{62} -0.0901699 q^{63} +1.00000 q^{64} +1.38197 q^{65} +6.85410 q^{66} +6.23607 q^{67} -4.23607 q^{68} +6.09017 q^{69} -0.326238 q^{70} -3.70820 q^{71} -0.381966 q^{72} +2.70820 q^{73} +4.70820 q^{74} +5.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} +1.61803 q^{78} +10.7984 q^{79} -1.38197 q^{80} -7.70820 q^{81} -5.70820 q^{82} -0.763932 q^{83} -0.381966 q^{84} +5.85410 q^{85} -2.23607 q^{86} +9.47214 q^{87} -4.23607 q^{88} +14.7984 q^{89} +0.527864 q^{90} -0.236068 q^{91} -3.76393 q^{92} +2.38197 q^{93} -0.145898 q^{94} -2.76393 q^{95} -1.61803 q^{96} -18.6525 q^{97} -6.94427 q^{98} +1.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{22} - 12 q^{23} - q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{33} - 4 q^{34} + 15 q^{35} - 3 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} - 5 q^{40} + 2 q^{41} - 3 q^{42} - 4 q^{44} + 10 q^{45} - 12 q^{46} - 7 q^{47} - q^{48} + 4 q^{49} + 5 q^{50} + 7 q^{51} - 2 q^{52} + 2 q^{54} + 5 q^{55} - 4 q^{56} - 2 q^{57} - 5 q^{58} + 7 q^{59} - 2 q^{61} + 6 q^{62} + 11 q^{63} + 2 q^{64} + 5 q^{65} + 7 q^{66} + 8 q^{67} - 4 q^{68} + q^{69} + 15 q^{70} + 6 q^{71} - 3 q^{72} - 8 q^{73} - 4 q^{74} + 10 q^{75} + 4 q^{76} - 2 q^{77} + q^{78} - 3 q^{79} - 5 q^{80} - 2 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} + 5 q^{85} + 10 q^{87} - 4 q^{88} + 5 q^{89} + 10 q^{90} + 4 q^{91} - 12 q^{92} + 7 q^{93} - 7 q^{94} - 10 q^{95} - q^{96} - 6 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9 - 5 * q^10 - 4 * q^11 - q^12 - 2 * q^13 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 3 * q^18 + 4 * q^19 - 5 * q^20 - 3 * q^21 - 4 * q^22 - 12 * q^23 - q^24 + 5 * q^25 - 2 * q^26 + 2 * q^27 - 4 * q^28 - 5 * q^29 + 6 * q^31 + 2 * q^32 + 7 * q^33 - 4 * q^34 + 15 * q^35 - 3 * q^36 - 4 * q^37 + 4 * q^38 + q^39 - 5 * q^40 + 2 * q^41 - 3 * q^42 - 4 * q^44 + 10 * q^45 - 12 * q^46 - 7 * q^47 - q^48 + 4 * q^49 + 5 * q^50 + 7 * q^51 - 2 * q^52 + 2 * q^54 + 5 * q^55 - 4 * q^56 - 2 * q^57 - 5 * q^58 + 7 * q^59 - 2 * q^61 + 6 * q^62 + 11 * q^63 + 2 * q^64 + 5 * q^65 + 7 * q^66 + 8 * q^67 - 4 * q^68 + q^69 + 15 * q^70 + 6 * q^71 - 3 * q^72 - 8 * q^73 - 4 * q^74 + 10 * q^75 + 4 * q^76 - 2 * q^77 + q^78 - 3 * q^79 - 5 * q^80 - 2 * q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 + 5 * q^85 + 10 * q^87 - 4 * q^88 + 5 * q^89 + 10 * q^90 + 4 * q^91 - 12 * q^92 + 7 * q^93 - 7 * q^94 - 10 * q^95 - q^96 - 6 * q^97 + 4 * q^98 + 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	1.00000	0.500000
$5$	−1.38197	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−1.38197	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	−1.61803	−0.660560
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	1.00000	0.353553
$9$	−0.381966	−0.127322
$10$	−1.38197	−0.437016
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	−1.61803	−0.467086
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0.236068	0.0630918
$15$	2.23607	0.577350
$16$	1.00000	0.250000
$17$	−4.23607	−1.02740	−0.513699	−	0.857971i	$-0.671725\pi$
$17$	−4.23607	−1.02740	−0.513699	+	0.857971i	$0.671725\pi$
$18$	−0.381966	−0.0900303
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.38197	−0.309017
$21$	−0.381966	−0.0833518
$22$	−4.23607	−0.903133
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	−1.61803	−0.330280
$25$	−3.09017	−0.618034
$26$	−1.00000	−0.196116
$27$	5.47214	1.05311
$28$	0.236068	0.0446127
$29$	−5.85410	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$29$	−5.85410	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$30$	2.23607	0.408248
$31$	−1.47214	−0.264403	−0.132202	−	0.991223i	$-0.542205\pi$
$31$	−1.47214	−0.264403	−0.132202	+	0.991223i	$0.542205\pi$
$32$	1.00000	0.176777
$33$	6.85410	1.19315
$34$	−4.23607	−0.726480
$35$	−0.326238	−0.0551443
$36$	−0.381966	−0.0636610
$37$	4.70820	0.774024	0.387012	−	0.922075i	$-0.373507\pi$
$37$	4.70820	0.774024	0.387012	+	0.922075i	$0.373507\pi$
$38$	2.00000	0.324443
$39$	1.61803	0.259093
$40$	−1.38197	−0.218508
$41$	−5.70820	−0.891472	−0.445736	−	0.895165i	$-0.647058\pi$
$41$	−5.70820	−0.891472	−0.445736	+	0.895165i	$0.647058\pi$
$42$	−0.381966	−0.0589386
$43$	−2.23607	−0.340997	−0.170499	−	0.985358i	$-0.554538\pi$
$43$	−2.23607	−0.340997	−0.170499	+	0.985358i	$0.554538\pi$
$44$	−4.23607	−0.638611
$45$	0.527864	0.0786893
$46$	−3.76393	−0.554962
$47$	−0.145898	−0.0212814	−0.0106407	−	0.999943i	$-0.503387\pi$
$47$	−0.145898	−0.0212814	−0.0106407	+	0.999943i	$0.503387\pi$
$48$	−1.61803	−0.233543
$49$	−6.94427	−0.992039
$50$	−3.09017	−0.437016
$51$	6.85410	0.959766
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	5.47214	0.744663
$55$	5.85410	0.789367
$56$	0.236068	0.0315459
$57$	−3.23607	−0.428628
$58$	−5.85410	−0.768681
$59$	13.5623	1.76566	0.882831	−	0.469691i	$-0.155635\pi$
$59$	13.5623	1.76566	0.882831	+	0.469691i	$0.155635\pi$
$60$	2.23607	0.288675
$61$	3.47214	0.444561	0.222281	−	0.974983i	$-0.428650\pi$
$61$	3.47214	0.444561	0.222281	+	0.974983i	$0.428650\pi$
$62$	−1.47214	−0.186961
$63$	−0.0901699	−0.0113603
$64$	1.00000	0.125000
$65$	1.38197	0.171412
$66$	6.85410	0.843682
$67$	6.23607	0.761857	0.380928	−	0.924605i	$-0.375604\pi$
$67$	6.23607	0.761857	0.380928	+	0.924605i	$0.375604\pi$
$68$	−4.23607	−0.513699
$69$	6.09017	0.733170
$70$	−0.326238	−0.0389929
$71$	−3.70820	−0.440083	−0.220041	−	0.975491i	$-0.570619\pi$
$71$	−3.70820	−0.440083	−0.220041	+	0.975491i	$0.570619\pi$
$72$	−0.381966	−0.0450151
$73$	2.70820	0.316971	0.158486	−	0.987361i	$-0.449339\pi$
$73$	2.70820	0.316971	0.158486	+	0.987361i	$0.449339\pi$
$74$	4.70820	0.547318
$75$	5.00000	0.577350
$76$	2.00000	0.229416
$77$	−1.00000	−0.113961
$78$	1.61803	0.183206
$79$	10.7984	1.21491	0.607456	−	0.794353i	$-0.292190\pi$
$79$	10.7984	1.21491	0.607456	+	0.794353i	$0.292190\pi$
$80$	−1.38197	−0.154508
$81$	−7.70820	−0.856467
$82$	−5.70820	−0.630366
$83$	−0.763932	−0.0838524	−0.0419262	−	0.999121i	$-0.513349\pi$
$83$	−0.763932	−0.0838524	−0.0419262	+	0.999121i	$0.513349\pi$
$84$	−0.381966	−0.0416759
$85$	5.85410	0.634967
$86$	−2.23607	−0.241121
$87$	9.47214	1.01552
$88$	−4.23607	−0.451566
$89$	14.7984	1.56862	0.784312	−	0.620366i	$-0.213016\pi$
$89$	14.7984	1.56862	0.784312	+	0.620366i	$0.213016\pi$
$90$	0.527864	0.0556418
$91$	−0.236068	−0.0247466
$92$	−3.76393	−0.392417
$93$	2.38197	0.246998
$94$	−0.145898	−0.0150482
$95$	−2.76393	−0.283573
$96$	−1.61803	−0.165140
$97$	−18.6525	−1.89387	−0.946936	−	0.321422i	$-0.895839\pi$
$97$	−18.6525	−1.89387	−0.946936	+	0.321422i	$0.895839\pi$
$98$	−6.94427	−0.701477
$99$	1.61803	0.162619
$100$	−3.09017	−0.309017
$101$	6.09017	0.605995	0.302997	−	0.952991i	$-0.402013\pi$
$101$	6.09017	0.605995	0.302997	+	0.952991i	$0.402013\pi$
$102$	6.85410	0.678657
$103$	4.14590	0.408507	0.204254	−	0.978918i	$-0.434523\pi$
$103$	4.14590	0.408507	0.204254	+	0.978918i	$0.434523\pi$
$104$	−1.00000	−0.0980581
$105$	0.527864	0.0515143
$106$	0	0
$107$	−7.76393	−0.750568	−0.375284	−	0.926910i	$-0.622455\pi$
$107$	−7.76393	−0.750568	−0.375284	+	0.926910i	$0.622455\pi$
$108$	5.47214	0.526557
$109$	8.70820	0.834095	0.417047	−	0.908885i	$-0.363065\pi$
$109$	8.70820	0.834095	0.417047	+	0.908885i	$0.363065\pi$
$110$	5.85410	0.558167
$111$	−7.61803	−0.723072
$112$	0.236068	0.0223063
$113$	−17.6180	−1.65737	−0.828683	−	0.559719i	$-0.810909\pi$
$113$	−17.6180	−1.65737	−0.828683	+	0.559719i	$0.810909\pi$
$114$	−3.23607	−0.303086
$115$	5.20163	0.485054
$116$	−5.85410	−0.543540
$117$	0.381966	0.0353128
$118$	13.5623	1.24851
$119$	−1.00000	−0.0916698
$120$	2.23607	0.204124
$121$	6.94427	0.631297
$122$	3.47214	0.314352
$123$	9.23607	0.832788
$124$	−1.47214	−0.132202
$125$	11.1803	1.00000
$126$	−0.0901699	−0.00803298
$127$	−3.79837	−0.337051	−0.168526	−	0.985697i	$-0.553901\pi$
$127$	−3.79837	−0.337051	−0.168526	+	0.985697i	$0.553901\pi$
$128$	1.00000	0.0883883
$129$	3.61803	0.318550
$130$	1.38197	0.121206
$131$	7.23607	0.632218	0.316109	−	0.948723i	$-0.397623\pi$
$131$	7.23607	0.632218	0.316109	+	0.948723i	$0.397623\pi$
$132$	6.85410	0.596573
$133$	0.472136	0.0409394
$134$	6.23607	0.538714
$135$	−7.56231	−0.650860
$136$	−4.23607	−0.363240
$137$	−13.3820	−1.14330	−0.571649	−	0.820498i	$-0.693696\pi$
$137$	−13.3820	−1.14330	−0.571649	+	0.820498i	$0.693696\pi$
$138$	6.09017	0.518430
$139$	−9.56231	−0.811064	−0.405532	−	0.914081i	$-0.632914\pi$
$139$	−9.56231	−0.811064	−0.405532	+	0.914081i	$0.632914\pi$
$140$	−0.326238	−0.0275721
$141$	0.236068	0.0198805
$142$	−3.70820	−0.311186
$143$	4.23607	0.354238
$144$	−0.381966	−0.0318305
$145$	8.09017	0.671852
$146$	2.70820	0.224133
$147$	11.2361	0.926735
$148$	4.70820	0.387012
$149$	−15.3262	−1.25557	−0.627787	−	0.778385i	$-0.716039\pi$
$149$	−15.3262	−1.25557	−0.627787	+	0.778385i	$0.716039\pi$
$150$	5.00000	0.408248
$151$	7.94427	0.646496	0.323248	−	0.946314i	$-0.395225\pi$
$151$	7.94427	0.646496	0.323248	+	0.946314i	$0.395225\pi$
$152$	2.00000	0.162221
$153$	1.61803	0.130810
$154$	−1.00000	−0.0805823
$155$	2.03444	0.163410
$156$	1.61803	0.129546
$157$	−8.65248	−0.690543	−0.345271	−	0.938503i	$-0.612213\pi$
$157$	−8.65248	−0.690543	−0.345271	+	0.938503i	$0.612213\pi$
$158$	10.7984	0.859072
$159$	0	0
$160$	−1.38197	−0.109254
$161$	−0.888544	−0.0700271
$162$	−7.70820	−0.605614
$163$	1.67376	0.131099	0.0655496	−	0.997849i	$-0.479120\pi$
$163$	1.67376	0.131099	0.0655496	+	0.997849i	$0.479120\pi$
$164$	−5.70820	−0.445736
$165$	−9.47214	−0.737405
$166$	−0.763932	−0.0592926
$167$	1.38197	0.106940	0.0534699	−	0.998569i	$-0.482972\pi$
$167$	1.38197	0.106940	0.0534699	+	0.998569i	$0.482972\pi$
$168$	−0.381966	−0.0294693
$169$	−12.0000	−0.923077
$170$	5.85410	0.448989
$171$	−0.763932	−0.0584193
$172$	−2.23607	−0.170499
$173$	1.47214	0.111924	0.0559622	−	0.998433i	$-0.482177\pi$
$173$	1.47214	0.111924	0.0559622	+	0.998433i	$0.482177\pi$
$174$	9.47214	0.718081
$175$	−0.729490	−0.0551443
$176$	−4.23607	−0.319306
$177$	−21.9443	−1.64943
$178$	14.7984	1.10919
$179$	5.29180	0.395527	0.197764	−	0.980250i	$-0.436632\pi$
$179$	5.29180	0.395527	0.197764	+	0.980250i	$0.436632\pi$
$180$	0.527864	0.0393447
$181$	−7.41641	−0.551257	−0.275629	−	0.961264i	$-0.588886\pi$
$181$	−7.41641	−0.551257	−0.275629	+	0.961264i	$0.588886\pi$
$182$	−0.236068	−0.0174985
$183$	−5.61803	−0.415297
$184$	−3.76393	−0.277481
$185$	−6.50658	−0.478373
$186$	2.38197	0.174654
$187$	17.9443	1.31222
$188$	−0.145898	−0.0106407
$189$	1.29180	0.0939643
$190$	−2.76393	−0.200517
$191$	−22.9443	−1.66019	−0.830095	−	0.557623i	$-0.811714\pi$
$191$	−22.9443	−1.66019	−0.830095	+	0.557623i	$0.811714\pi$
$192$	−1.61803	−0.116772
$193$	−3.76393	−0.270934	−0.135467	−	0.990782i	$-0.543253\pi$
$193$	−3.76393	−0.270934	−0.135467	+	0.990782i	$0.543253\pi$
$194$	−18.6525	−1.33917
$195$	−2.23607	−0.160128
$196$	−6.94427	−0.496019
$197$	−24.6180	−1.75396	−0.876981	−	0.480525i	$-0.840446\pi$
$197$	−24.6180	−1.75396	−0.876981	+	0.480525i	$0.840446\pi$
$198$	1.61803	0.114989
$199$	−7.61803	−0.540028	−0.270014	−	0.962856i	$-0.587028\pi$
$199$	−7.61803	−0.540028	−0.270014	+	0.962856i	$0.587028\pi$
$200$	−3.09017	−0.218508
$201$	−10.0902	−0.711706
$202$	6.09017	0.428503
$203$	−1.38197	−0.0969950
$204$	6.85410	0.479883
$205$	7.88854	0.550960
$206$	4.14590	0.288858
$207$	1.43769	0.0999266
$208$	−1.00000	−0.0693375
$209$	−8.47214	−0.586030
$210$	0.527864	0.0364261
$211$	−5.85410	−0.403013	−0.201506	−	0.979487i	$-0.564584\pi$
$211$	−5.85410	−0.403013	−0.201506	+	0.979487i	$0.564584\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	−7.76393	−0.530731
$215$	3.09017	0.210748
$216$	5.47214	0.372332
$217$	−0.347524	−0.0235915
$218$	8.70820	0.589794
$219$	−4.38197	−0.296106
$220$	5.85410	0.394683
$221$	4.23607	0.284949
$222$	−7.61803	−0.511289
$223$	10.3262	0.691496	0.345748	−	0.938327i	$-0.387625\pi$
$223$	10.3262	0.691496	0.345748	+	0.938327i	$0.387625\pi$
$224$	0.236068	0.0157730
$225$	1.18034	0.0786893
$226$	−17.6180	−1.17193
$227$	5.03444	0.334148	0.167074	−	0.985944i	$-0.446568\pi$
$227$	5.03444	0.334148	0.167074	+	0.985944i	$0.446568\pi$
$228$	−3.23607	−0.214314
$229$	2.70820	0.178963	0.0894816	−	0.995988i	$-0.471479\pi$
$229$	2.70820	0.178963	0.0894816	+	0.995988i	$0.471479\pi$
$230$	5.20163	0.342985
$231$	1.61803	0.106459
$232$	−5.85410	−0.384341
$233$	−0.236068	−0.0154653	−0.00773266	−	0.999970i	$-0.502461\pi$
$233$	−0.236068	−0.0154653	−0.00773266	+	0.999970i	$0.502461\pi$
$234$	0.381966	0.0249699
$235$	0.201626	0.0131526
$236$	13.5623	0.882831
$237$	−17.4721	−1.13494
$238$	−1.00000	−0.0648204
$239$	14.2705	0.923083	0.461541	−	0.887119i	$-0.347297\pi$
$239$	14.2705	0.923083	0.461541	+	0.887119i	$0.347297\pi$
$240$	2.23607	0.144338
$241$	8.70820	0.560945	0.280472	−	0.959862i	$-0.409509\pi$
$241$	8.70820	0.560945	0.280472	+	0.959862i	$0.409509\pi$
$242$	6.94427	0.446395
$243$	−3.94427	−0.253025
$244$	3.47214	0.222281
$245$	9.59675	0.613114
$246$	9.23607	0.588870
$247$	−2.00000	−0.127257
$248$	−1.47214	−0.0934807
$249$	1.23607	0.0783326
$250$	11.1803	0.707107
$251$	12.9098	0.814861	0.407431	−	0.913236i	$-0.366425\pi$
$251$	12.9098	0.814861	0.407431	+	0.913236i	$0.366425\pi$
$252$	−0.0901699	−0.00568017
$253$	15.9443	1.00241
$254$	−3.79837	−0.238331
$255$	−9.47214	−0.593168
$256$	1.00000	0.0625000
$257$	3.47214	0.216586	0.108293	−	0.994119i	$-0.465462\pi$
$257$	3.47214	0.216586	0.108293	+	0.994119i	$0.465462\pi$
$258$	3.61803	0.225249
$259$	1.11146	0.0690625
$260$	1.38197	0.0857059
$261$	2.23607	0.138409
$262$	7.23607	0.447046
$263$	−10.2705	−0.633307	−0.316653	−	0.948541i	$-0.602559\pi$
$263$	−10.2705	−0.633307	−0.316653	+	0.948541i	$0.602559\pi$
$264$	6.85410	0.421841
$265$	0	0
$266$	0.472136	0.0289485
$267$	−23.9443	−1.46537
$268$	6.23607	0.380928
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	−7.56231	−0.460227
$271$	−16.1246	−0.979500	−0.489750	−	0.871863i	$-0.662912\pi$
$271$	−16.1246	−0.979500	−0.489750	+	0.871863i	$0.662912\pi$
$272$	−4.23607	−0.256849
$273$	0.381966	0.0231176
$274$	−13.3820	−0.808434
$275$	13.0902	0.789367
$276$	6.09017	0.366585
$277$	−23.6180	−1.41907	−0.709535	−	0.704670i	$-0.751095\pi$
$277$	−23.6180	−1.41907	−0.709535	+	0.704670i	$0.751095\pi$
$278$	−9.56231	−0.573509
$279$	0.562306	0.0336644
$280$	−0.326238	−0.0194964
$281$	−6.81966	−0.406827	−0.203413	−	0.979093i	$-0.565204\pi$
$281$	−6.81966	−0.406827	−0.203413	+	0.979093i	$0.565204\pi$
$282$	0.236068	0.0140576
$283$	10.2361	0.608471	0.304236	−	0.952597i	$-0.401599\pi$
$283$	10.2361	0.608471	0.304236	+	0.952597i	$0.401599\pi$
$284$	−3.70820	−0.220041
$285$	4.47214	0.264906
$286$	4.23607	0.250484
$287$	−1.34752	−0.0795418
$288$	−0.381966	−0.0225076
$289$	0.944272	0.0555454
$290$	8.09017	0.475071
$291$	30.1803	1.76920
$292$	2.70820	0.158486
$293$	16.4721	0.962312	0.481156	−	0.876635i	$-0.340217\pi$
$293$	16.4721	0.962312	0.481156	+	0.876635i	$0.340217\pi$
$294$	11.2361	0.655301
$295$	−18.7426	−1.09124
$296$	4.70820	0.273659
$297$	−23.1803	−1.34506
$298$	−15.3262	−0.887825
$299$	3.76393	0.217674
$300$	5.00000	0.288675
$301$	−0.527864	−0.0304256
$302$	7.94427	0.457141
$303$	−9.85410	−0.566103
$304$	2.00000	0.114708
$305$	−4.79837	−0.274754
$306$	1.61803	0.0924968
$307$	−8.12461	−0.463696	−0.231848	−	0.972752i	$-0.574477\pi$
$307$	−8.12461	−0.463696	−0.231848	+	0.972752i	$0.574477\pi$
$308$	−1.00000	−0.0569803
$309$	−6.70820	−0.381616
$310$	2.03444	0.115549
$311$	−12.7639	−0.723776	−0.361888	−	0.932222i	$-0.617868\pi$
$311$	−12.7639	−0.723776	−0.361888	+	0.932222i	$0.617868\pi$
$312$	1.61803	0.0916031
$313$	19.8541	1.12222	0.561110	−	0.827741i	$-0.310374\pi$
$313$	19.8541	1.12222	0.561110	+	0.827741i	$0.310374\pi$
$314$	−8.65248	−0.488287
$315$	0.124612	0.00702108
$316$	10.7984	0.607456
$317$	−25.7639	−1.44705	−0.723523	−	0.690300i	$-0.757479\pi$
$317$	−25.7639	−1.44705	−0.723523	+	0.690300i	$0.757479\pi$
$318$	0	0
$319$	24.7984	1.38844
$320$	−1.38197	−0.0772542
$321$	12.5623	0.701160
$322$	−0.888544	−0.0495166
$323$	−8.47214	−0.471402
$324$	−7.70820	−0.428234
$325$	3.09017	0.171412
$326$	1.67376	0.0927011
$327$	−14.0902	−0.779188
$328$	−5.70820	−0.315183
$329$	−0.0344419	−0.00189884
$330$	−9.47214	−0.521424
$331$	23.1803	1.27411	0.637053	−	0.770820i	$-0.280153\pi$
$331$	23.1803	1.27411	0.637053	+	0.770820i	$0.280153\pi$
$332$	−0.763932	−0.0419262
$333$	−1.79837	−0.0985503
$334$	1.38197	0.0756178
$335$	−8.61803	−0.470853
$336$	−0.381966	−0.0208380
$337$	18.9787	1.03384	0.516918	−	0.856035i	$-0.327079\pi$
$337$	18.9787	1.03384	0.516918	+	0.856035i	$0.327079\pi$
$338$	−12.0000	−0.652714
$339$	28.5066	1.54826
$340$	5.85410	0.317483
$341$	6.23607	0.337702
$342$	−0.763932	−0.0413087
$343$	−3.29180	−0.177740
$344$	−2.23607	−0.120561
$345$	−8.41641	−0.453124
$346$	1.47214	0.0791425
$347$	−31.3607	−1.68353	−0.841765	−	0.539845i	$-0.818483\pi$
$347$	−31.3607	−1.68353	−0.841765	+	0.539845i	$0.818483\pi$
$348$	9.47214	0.507760
$349$	−12.2705	−0.656825	−0.328413	−	0.944534i	$-0.606514\pi$
$349$	−12.2705	−0.656825	−0.328413	+	0.944534i	$0.606514\pi$
$350$	−0.729490	−0.0389929
$351$	−5.47214	−0.292081
$352$	−4.23607	−0.225783
$353$	−3.14590	−0.167439	−0.0837196	−	0.996489i	$-0.526680\pi$
$353$	−3.14590	−0.167439	−0.0837196	+	0.996489i	$0.526680\pi$
$354$	−21.9443	−1.16632
$355$	5.12461	0.271986
$356$	14.7984	0.784312
$357$	1.61803	0.0856354
$358$	5.29180	0.279680
$359$	0.944272	0.0498368	0.0249184	−	0.999689i	$-0.492067\pi$
$359$	0.944272	0.0498368	0.0249184	+	0.999689i	$0.492067\pi$
$360$	0.527864	0.0278209
$361$	−15.0000	−0.789474
$362$	−7.41641	−0.389798
$363$	−11.2361	−0.589741
$364$	−0.236068	−0.0123733
$365$	−3.74265	−0.195899
$366$	−5.61803	−0.293659
$367$	−33.7984	−1.76426	−0.882130	−	0.471005i	$-0.843891\pi$
$367$	−33.7984	−1.76426	−0.882130	+	0.471005i	$0.843891\pi$
$368$	−3.76393	−0.196209
$369$	2.18034	0.113504
$370$	−6.50658	−0.338261
$371$	0	0
$372$	2.38197	0.123499
$373$	−10.9443	−0.566673	−0.283336	−	0.959021i	$-0.591441\pi$
$373$	−10.9443	−0.566673	−0.283336	+	0.959021i	$0.591441\pi$
$374$	17.9443	0.927876
$375$	−18.0902	−0.934172
$376$	−0.145898	−0.00752412
$377$	5.85410	0.301502
$378$	1.29180	0.0664428
$379$	12.8541	0.660271	0.330135	−	0.943934i	$-0.392906\pi$
$379$	12.8541	0.660271	0.330135	+	0.943934i	$0.392906\pi$
$380$	−2.76393	−0.141787
$381$	6.14590	0.314864
$382$	−22.9443	−1.17393
$383$	−19.6180	−1.00243	−0.501217	−	0.865321i	$-0.667114\pi$
$383$	−19.6180	−1.00243	−0.501217	+	0.865321i	$0.667114\pi$
$384$	−1.61803	−0.0825700
$385$	1.38197	0.0704315
$386$	−3.76393	−0.191579
$387$	0.854102	0.0434164
$388$	−18.6525	−0.946936
$389$	11.0557	0.560548	0.280274	−	0.959920i	$-0.409575\pi$
$389$	11.0557	0.560548	0.280274	+	0.959920i	$0.409575\pi$
$390$	−2.23607	−0.113228
$391$	15.9443	0.806336
$392$	−6.94427	−0.350739
$393$	−11.7082	−0.590601
$394$	−24.6180	−1.24024
$395$	−14.9230	−0.750857
$396$	1.61803	0.0813093
$397$	−0.729490	−0.0366121	−0.0183060	−	0.999832i	$-0.505827\pi$
$397$	−0.729490	−0.0366121	−0.0183060	+	0.999832i	$0.505827\pi$
$398$	−7.61803	−0.381858
$399$	−0.763932	−0.0382444
$400$	−3.09017	−0.154508
$401$	29.2705	1.46170	0.730850	−	0.682538i	$-0.239124\pi$
$401$	29.2705	1.46170	0.730850	+	0.682538i	$0.239124\pi$
$402$	−10.0902	−0.503252
$403$	1.47214	0.0733323
$404$	6.09017	0.302997
$405$	10.6525	0.529326
$406$	−1.38197	−0.0685858
$407$	−19.9443	−0.988601
$408$	6.85410	0.339329
$409$	19.6525	0.971752	0.485876	−	0.874028i	$-0.338501\pi$
$409$	19.6525	0.971752	0.485876	+	0.874028i	$0.338501\pi$
$410$	7.88854	0.389587
$411$	21.6525	1.06804
$412$	4.14590	0.204254
$413$	3.20163	0.157542
$414$	1.43769	0.0706588
$415$	1.05573	0.0518237
$416$	−1.00000	−0.0490290
$417$	15.4721	0.757674
$418$	−8.47214	−0.414386
$419$	−37.5623	−1.83504	−0.917519	−	0.397691i	$-0.869812\pi$
$419$	−37.5623	−1.83504	−0.917519	+	0.397691i	$0.869812\pi$
$420$	0.527864	0.0257571
$421$	2.52786	0.123201	0.0616003	−	0.998101i	$-0.480380\pi$
$421$	2.52786	0.123201	0.0616003	+	0.998101i	$0.480380\pi$
$422$	−5.85410	−0.284973
$423$	0.0557281	0.00270959
$424$	0	0
$425$	13.0902	0.634967
$426$	6.00000	0.290701
$427$	0.819660	0.0396661
$428$	−7.76393	−0.375284
$429$	−6.85410	−0.330919
$430$	3.09017	0.149021
$431$	−6.52786	−0.314436	−0.157218	−	0.987564i	$-0.550253\pi$
$431$	−6.52786	−0.314436	−0.157218	+	0.987564i	$0.550253\pi$
$432$	5.47214	0.263278
$433$	36.7984	1.76842	0.884208	−	0.467092i	$-0.154698\pi$
$433$	36.7984	1.76842	0.884208	+	0.467092i	$0.154698\pi$
$434$	−0.347524	−0.0166817
$435$	−13.0902	−0.627626
$436$	8.70820	0.417047
$437$	−7.52786	−0.360107
$438$	−4.38197	−0.209378
$439$	25.8541	1.23395	0.616974	−	0.786983i	$-0.288358\pi$
$439$	25.8541	1.23395	0.616974	+	0.786983i	$0.288358\pi$
$440$	5.85410	0.279083
$441$	2.65248	0.126308
$442$	4.23607	0.201489
$443$	6.58359	0.312796	0.156398	−	0.987694i	$-0.450012\pi$
$443$	6.58359	0.312796	0.156398	+	0.987694i	$0.450012\pi$
$444$	−7.61803	−0.361536
$445$	−20.4508	−0.969463
$446$	10.3262	0.488962
$447$	24.7984	1.17292
$448$	0.236068	0.0111532
$449$	14.0902	0.664956	0.332478	−	0.943111i	$-0.392115\pi$
$449$	14.0902	0.664956	0.332478	+	0.943111i	$0.392115\pi$
$450$	1.18034	0.0556418
$451$	24.1803	1.13861
$452$	−17.6180	−0.828683
$453$	−12.8541	−0.603938
$454$	5.03444	0.236278
$455$	0.326238	0.0152943
$456$	−3.23607	−0.151543
$457$	2.20163	0.102988	0.0514939	−	0.998673i	$-0.483602\pi$
$457$	2.20163	0.102988	0.0514939	+	0.998673i	$0.483602\pi$
$458$	2.70820	0.126546
$459$	−23.1803	−1.08197
$460$	5.20163	0.242527
$461$	−19.1246	−0.890722	−0.445361	−	0.895351i	$-0.646925\pi$
$461$	−19.1246	−0.890722	−0.445361	+	0.895351i	$0.646925\pi$
$462$	1.61803	0.0752778
$463$	−8.90983	−0.414075	−0.207037	−	0.978333i	$-0.566382\pi$
$463$	−8.90983	−0.414075	−0.207037	+	0.978333i	$0.566382\pi$
$464$	−5.85410	−0.271770
$465$	−3.29180	−0.152653
$466$	−0.236068	−0.0109356
$467$	−33.4164	−1.54633	−0.773163	−	0.634207i	$-0.781327\pi$
$467$	−33.4164	−1.54633	−0.773163	+	0.634207i	$0.781327\pi$
$468$	0.381966	0.0176564
$469$	1.47214	0.0679769
$470$	0.201626	0.00930032
$471$	14.0000	0.645086
$472$	13.5623	0.624256
$473$	9.47214	0.435529
$474$	−17.4721	−0.802522
$475$	−6.18034	−0.283573
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	14.2705	0.652718
$479$	12.5967	0.575560	0.287780	−	0.957697i	$-0.407083\pi$
$479$	12.5967	0.575560	0.287780	+	0.957697i	$0.407083\pi$
$480$	2.23607	0.102062
$481$	−4.70820	−0.214676
$482$	8.70820	0.396648
$483$	1.43769	0.0654173
$484$	6.94427	0.315649
$485$	25.7771	1.17048
$486$	−3.94427	−0.178916
$487$	38.8328	1.75968	0.879841	−	0.475267i	$-0.157649\pi$
$487$	38.8328	1.75968	0.879841	+	0.475267i	$0.157649\pi$
$488$	3.47214	0.157176
$489$	−2.70820	−0.122469
$490$	9.59675	0.433537
$491$	11.2361	0.507077	0.253538	−	0.967325i	$-0.418406\pi$
$491$	11.2361	0.507077	0.253538	+	0.967325i	$0.418406\pi$
$492$	9.23607	0.416394
$493$	24.7984	1.11686
$494$	−2.00000	−0.0899843
$495$	−2.23607	−0.100504
$496$	−1.47214	−0.0661009
$497$	−0.875388	−0.0392665
$498$	1.23607	0.0553895
$499$	−5.94427	−0.266102	−0.133051	−	0.991109i	$-0.542477\pi$
$499$	−5.94427	−0.266102	−0.133051	+	0.991109i	$0.542477\pi$
$500$	11.1803	0.500000
$501$	−2.23607	−0.0999001
$502$	12.9098	0.576194
$503$	−10.3607	−0.461960	−0.230980	−	0.972959i	$-0.574193\pi$
$503$	−10.3607	−0.461960	−0.230980	+	0.972959i	$0.574193\pi$
$504$	−0.0901699	−0.00401649
$505$	−8.41641	−0.374525
$506$	15.9443	0.708809
$507$	19.4164	0.862313
$508$	−3.79837	−0.168526
$509$	42.3262	1.87608	0.938039	−	0.346530i	$-0.112640\pi$
$509$	42.3262	1.87608	0.938039	+	0.346530i	$0.112640\pi$
$510$	−9.47214	−0.419433
$511$	0.639320	0.0282819
$512$	1.00000	0.0441942
$513$	10.9443	0.483201
$514$	3.47214	0.153149
$515$	−5.72949	−0.252472
$516$	3.61803	0.159275
$517$	0.618034	0.0271811
$518$	1.11146	0.0488346
$519$	−2.38197	−0.104557
$520$	1.38197	0.0606032
$521$	3.58359	0.157000	0.0785000	−	0.996914i	$-0.474987\pi$
$521$	3.58359	0.157000	0.0785000	+	0.996914i	$0.474987\pi$
$522$	2.23607	0.0978700
$523$	25.3820	1.10988	0.554938	−	0.831892i	$-0.312742\pi$
$523$	25.3820	1.10988	0.554938	+	0.831892i	$0.312742\pi$
$524$	7.23607	0.316109
$525$	1.18034	0.0515143
$526$	−10.2705	−0.447816
$527$	6.23607	0.271647
$528$	6.85410	0.298287
$529$	−8.83282	−0.384035
$530$	0	0
$531$	−5.18034	−0.224808
$532$	0.472136	0.0204697
$533$	5.70820	0.247250
$534$	−23.9443	−1.03617
$535$	10.7295	0.463876
$536$	6.23607	0.269357
$537$	−8.56231	−0.369491
$538$	1.00000	0.0431131
$539$	29.4164	1.26705
$540$	−7.56231	−0.325430
$541$	11.5279	0.495622	0.247811	−	0.968808i	$-0.420289\pi$
$541$	11.5279	0.495622	0.247811	+	0.968808i	$0.420289\pi$
$542$	−16.1246	−0.692611
$543$	12.0000	0.514969
$544$	−4.23607	−0.181620
$545$	−12.0344	−0.515499
$546$	0.381966	0.0163466
$547$	27.7984	1.18857	0.594286	−	0.804254i	$-0.297435\pi$
$547$	27.7984	1.18857	0.594286	+	0.804254i	$0.297435\pi$
$548$	−13.3820	−0.571649
$549$	−1.32624	−0.0566025
$550$	13.0902	0.558167
$551$	−11.7082	−0.498786
$552$	6.09017	0.259215
$553$	2.54915	0.108401
$554$	−23.6180	−1.00343
$555$	10.5279	0.446883
$556$	−9.56231	−0.405532
$557$	41.7082	1.76723	0.883617	−	0.468211i	$-0.155101\pi$
$557$	41.7082	1.76723	0.883617	+	0.468211i	$0.155101\pi$
$558$	0.562306	0.0238043
$559$	2.23607	0.0945756
$560$	−0.326238	−0.0137861
$561$	−29.0344	−1.22584
$562$	−6.81966	−0.287670
$563$	20.7082	0.872747	0.436373	−	0.899766i	$-0.356263\pi$
$563$	20.7082	0.872747	0.436373	+	0.899766i	$0.356263\pi$
$564$	0.236068	0.00994026
$565$	24.3475	1.02431
$566$	10.2361	0.430254
$567$	−1.81966	−0.0764185
$568$	−3.70820	−0.155593
$569$	−17.0902	−0.716457	−0.358229	−	0.933634i	$-0.616619\pi$
$569$	−17.0902	−0.716457	−0.358229	+	0.933634i	$0.616619\pi$
$570$	4.47214	0.187317
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	4.23607	0.177119
$573$	37.1246	1.55090
$574$	−1.34752	−0.0562446
$575$	11.6312	0.485054
$576$	−0.381966	−0.0159153
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	0.944272	0.0392765
$579$	6.09017	0.253099
$580$	8.09017	0.335926
$581$	−0.180340	−0.00748176
$582$	30.1803	1.25102
$583$	0	0
$584$	2.70820	0.112066
$585$	−0.527864	−0.0218245
$586$	16.4721	0.680458
$587$	−26.3262	−1.08660	−0.543300	−	0.839539i	$-0.682825\pi$
$587$	−26.3262	−1.08660	−0.543300	+	0.839539i	$0.682825\pi$
$588$	11.2361	0.463368
$589$	−2.94427	−0.121317
$590$	−18.7426	−0.771623
$591$	39.8328	1.63850
$592$	4.70820	0.193506
$593$	−27.4164	−1.12586	−0.562928	−	0.826506i	$-0.690325\pi$
$593$	−27.4164	−1.12586	−0.562928	+	0.826506i	$0.690325\pi$
$594$	−23.1803	−0.951101
$595$	1.38197	0.0566551
$596$	−15.3262	−0.627787
$597$	12.3262	0.504479
$598$	3.76393	0.153919
$599$	33.5967	1.37273	0.686363	−	0.727259i	$-0.259206\pi$
$599$	33.5967	1.37273	0.686363	+	0.727259i	$0.259206\pi$
$600$	5.00000	0.204124
$601$	34.6869	1.41491	0.707454	−	0.706759i	$-0.249843\pi$
$601$	34.6869	1.41491	0.707454	+	0.706759i	$0.249843\pi$
$602$	−0.527864	−0.0215141
$603$	−2.38197	−0.0970012
$604$	7.94427	0.323248
$605$	−9.59675	−0.390163
$606$	−9.85410	−0.400296
$607$	−48.5755	−1.97162	−0.985809	−	0.167873i	$-0.946310\pi$
$607$	−48.5755	−1.97162	−0.985809	+	0.167873i	$0.946310\pi$
$608$	2.00000	0.0811107
$609$	2.23607	0.0906100
$610$	−4.79837	−0.194280
$611$	0.145898	0.00590240
$612$	1.61803	0.0654051
$613$	−35.4721	−1.43271	−0.716353	−	0.697738i	$-0.754190\pi$
$613$	−35.4721	−1.43271	−0.716353	+	0.697738i	$0.754190\pi$
$614$	−8.12461	−0.327883
$615$	−12.7639	−0.514691
$616$	−1.00000	−0.0402911
$617$	−33.1591	−1.33493	−0.667467	−	0.744640i	$-0.732621\pi$
$617$	−33.1591	−1.33493	−0.667467	+	0.744640i	$0.732621\pi$
$618$	−6.70820	−0.269844
$619$	12.8328	0.515794	0.257897	−	0.966172i	$-0.416970\pi$
$619$	12.8328	0.515794	0.257897	+	0.966172i	$0.416970\pi$
$620$	2.03444	0.0817052
$621$	−20.5967	−0.826519
$622$	−12.7639	−0.511787
$623$	3.49342	0.139961
$624$	1.61803	0.0647732
$625$	0	0
$626$	19.8541	0.793530
$627$	13.7082	0.547453
$628$	−8.65248	−0.345271
$629$	−19.9443	−0.795230
$630$	0.124612	0.00496465
$631$	29.7984	1.18625	0.593127	−	0.805109i	$-0.297893\pi$
$631$	29.7984	1.18625	0.593127	+	0.805109i	$0.297893\pi$
$632$	10.7984	0.429536
$633$	9.47214	0.376484
$634$	−25.7639	−1.02322
$635$	5.24922	0.208309
$636$	0	0
$637$	6.94427	0.275142
$638$	24.7984	0.981777
$639$	1.41641	0.0560322
$640$	−1.38197	−0.0546270
$641$	−19.1803	−0.757578	−0.378789	−	0.925483i	$-0.623659\pi$
$641$	−19.1803	−0.757578	−0.378789	+	0.925483i	$0.623659\pi$
$642$	12.5623	0.495795
$643$	−10.8541	−0.428044	−0.214022	−	0.976829i	$-0.568656\pi$
$643$	−10.8541	−0.428044	−0.214022	+	0.976829i	$0.568656\pi$
$644$	−0.888544	−0.0350135
$645$	−5.00000	−0.196875
$646$	−8.47214	−0.333332
$647$	−25.1803	−0.989941	−0.494971	−	0.868910i	$-0.664821\pi$
$647$	−25.1803	−0.989941	−0.494971	+	0.868910i	$0.664821\pi$
$648$	−7.70820	−0.302807
$649$	−57.4508	−2.25514
$650$	3.09017	0.121206
$651$	0.562306	0.0220385
$652$	1.67376	0.0655496
$653$	39.5410	1.54736	0.773680	−	0.633577i	$-0.218414\pi$
$653$	39.5410	1.54736	0.773680	+	0.633577i	$0.218414\pi$
$654$	−14.0902	−0.550969
$655$	−10.0000	−0.390732
$656$	−5.70820	−0.222868
$657$	−1.03444	−0.0403574
$658$	−0.0344419	−0.00134268
$659$	−15.6738	−0.610563	−0.305282	−	0.952262i	$-0.598751\pi$
$659$	−15.6738	−0.610563	−0.305282	+	0.952262i	$0.598751\pi$
$660$	−9.47214	−0.368702
$661$	−14.1459	−0.550212	−0.275106	−	0.961414i	$-0.588713\pi$
$661$	−14.1459	−0.550212	−0.275106	+	0.961414i	$0.588713\pi$
$662$	23.1803	0.900929
$663$	−6.85410	−0.266191
$664$	−0.763932	−0.0296463
$665$	−0.652476	−0.0253019
$666$	−1.79837	−0.0696856
$667$	22.0344	0.853177
$668$	1.38197	0.0534699
$669$	−16.7082	−0.645976
$670$	−8.61803	−0.332944
$671$	−14.7082	−0.567804
$672$	−0.381966	−0.0147347
$673$	−2.70820	−0.104394	−0.0521968	−	0.998637i	$-0.516622\pi$
$673$	−2.70820	−0.104394	−0.0521968	+	0.998637i	$0.516622\pi$
$674$	18.9787	0.731033
$675$	−16.9098	−0.650860
$676$	−12.0000	−0.461538
$677$	−37.5623	−1.44364	−0.721818	−	0.692083i	$-0.756693\pi$
$677$	−37.5623	−1.44364	−0.721818	+	0.692083i	$0.756693\pi$
$678$	28.5066	1.09479
$679$	−4.40325	−0.168981
$680$	5.85410	0.224495
$681$	−8.14590	−0.312151
$682$	6.23607	0.238791
$683$	−16.7639	−0.641454	−0.320727	−	0.947172i	$-0.603927\pi$
$683$	−16.7639	−0.641454	−0.320727	+	0.947172i	$0.603927\pi$
$684$	−0.763932	−0.0292097
$685$	18.4934	0.706597
$686$	−3.29180	−0.125681
$687$	−4.38197	−0.167182
$688$	−2.23607	−0.0852493
$689$	0	0
$690$	−8.41641	−0.320407
$691$	25.1246	0.955785	0.477893	−	0.878418i	$-0.341401\pi$
$691$	25.1246	0.955785	0.477893	+	0.878418i	$0.341401\pi$
$692$	1.47214	0.0559622
$693$	0.381966	0.0145097
$694$	−31.3607	−1.19044
$695$	13.2148	0.501265
$696$	9.47214	0.359040
$697$	24.1803	0.915896
$698$	−12.2705	−0.464446
$699$	0.381966	0.0144473
$700$	−0.729490	−0.0275721
$701$	35.9787	1.35890	0.679449	−	0.733723i	$-0.262219\pi$
$701$	35.9787	1.35890	0.679449	+	0.733723i	$0.262219\pi$
$702$	−5.47214	−0.206532
$703$	9.41641	0.355147
$704$	−4.23607	−0.159653
$705$	−0.326238	−0.0122868
$706$	−3.14590	−0.118397
$707$	1.43769	0.0540701
$708$	−21.9443	−0.824716
$709$	−20.0557	−0.753209	−0.376604	−	0.926374i	$-0.622908\pi$
$709$	−20.0557	−0.753209	−0.376604	+	0.926374i	$0.622908\pi$
$710$	5.12461	0.192323
$711$	−4.12461	−0.154685
$712$	14.7984	0.554593
$713$	5.54102	0.207513
$714$	1.61803	0.0605534
$715$	−5.85410	−0.218931
$716$	5.29180	0.197764
$717$	−23.0902	−0.862318
$718$	0.944272	0.0352399
$719$	19.8197	0.739149	0.369574	−	0.929201i	$-0.379504\pi$
$719$	19.8197	0.739149	0.369574	+	0.929201i	$0.379504\pi$
$720$	0.527864	0.0196723
$721$	0.978714	0.0364492
$722$	−15.0000	−0.558242
$723$	−14.0902	−0.524019
$724$	−7.41641	−0.275629
$725$	18.0902	0.671852
$726$	−11.2361	−0.417010
$727$	−31.2918	−1.16055	−0.580274	−	0.814421i	$-0.697055\pi$
$727$	−31.2918	−1.16055	−0.580274	+	0.814421i	$0.697055\pi$
$728$	−0.236068	−0.00874926
$729$	29.5066	1.09284
$730$	−3.74265	−0.138522
$731$	9.47214	0.350340
$732$	−5.61803	−0.207649
$733$	11.3607	0.419616	0.209808	−	0.977743i	$-0.432716\pi$
$733$	11.3607	0.419616	0.209808	+	0.977743i	$0.432716\pi$
$734$	−33.7984	−1.24752
$735$	−15.5279	−0.572754
$736$	−3.76393	−0.138740
$737$	−26.4164	−0.973061
$738$	2.18034	0.0802594
$739$	5.96556	0.219447	0.109723	−	0.993962i	$-0.465004\pi$
$739$	5.96556	0.219447	0.109723	+	0.993962i	$0.465004\pi$
$740$	−6.50658	−0.239187
$741$	3.23607	0.118880
$742$	0	0
$743$	−33.7984	−1.23994	−0.619971	−	0.784625i	$-0.712856\pi$
$743$	−33.7984	−1.23994	−0.619971	+	0.784625i	$0.712856\pi$
$744$	2.38197	0.0873271
$745$	21.1803	0.775988
$746$	−10.9443	−0.400698
$747$	0.291796	0.0106763
$748$	17.9443	0.656108
$749$	−1.83282	−0.0669696
$750$	−18.0902	−0.660560
$751$	−42.8673	−1.56425	−0.782124	−	0.623123i	$-0.785864\pi$
$751$	−42.8673	−1.56425	−0.782124	+	0.623123i	$0.785864\pi$
$752$	−0.145898	−0.00532035
$753$	−20.8885	−0.761221
$754$	5.85410	0.213194
$755$	−10.9787	−0.399556
$756$	1.29180	0.0469822
$757$	−21.0557	−0.765283	−0.382642	−	0.923897i	$-0.624986\pi$
$757$	−21.0557	−0.765283	−0.382642	+	0.923897i	$0.624986\pi$
$758$	12.8541	0.466882
$759$	−25.7984	−0.936422
$760$	−2.76393	−0.100258
$761$	25.4164	0.921344	0.460672	−	0.887570i	$-0.347608\pi$
$761$	25.4164	0.921344	0.460672	+	0.887570i	$0.347608\pi$
$762$	6.14590	0.222642
$763$	2.05573	0.0744224
$764$	−22.9443	−0.830095
$765$	−2.23607	−0.0808452
$766$	−19.6180	−0.708828
$767$	−13.5623	−0.489706
$768$	−1.61803	−0.0583858
$769$	−17.6738	−0.637332	−0.318666	−	0.947867i	$-0.603235\pi$
$769$	−17.6738	−0.637332	−0.318666	+	0.947867i	$0.603235\pi$
$770$	1.38197	0.0498026
$771$	−5.61803	−0.202329
$772$	−3.76393	−0.135467
$773$	−1.59675	−0.0574310	−0.0287155	−	0.999588i	$-0.509142\pi$
$773$	−1.59675	−0.0574310	−0.0287155	+	0.999588i	$0.509142\pi$
$774$	0.854102	0.0307001
$775$	4.54915	0.163410
$776$	−18.6525	−0.669585
$777$	−1.79837	−0.0645163
$778$	11.0557	0.396367
$779$	−11.4164	−0.409035
$780$	−2.23607	−0.0800641
$781$	15.7082	0.562084
$782$	15.9443	0.570166
$783$	−32.0344	−1.14482
$784$	−6.94427	−0.248010
$785$	11.9574	0.426779
$786$	−11.7082	−0.417618
$787$	−33.6525	−1.19958	−0.599791	−	0.800157i	$-0.704749\pi$
$787$	−33.6525	−1.19958	−0.599791	+	0.800157i	$0.704749\pi$
$788$	−24.6180	−0.876981
$789$	16.6180	0.591618
$790$	−14.9230	−0.530936
$791$	−4.15905	−0.147879
$792$	1.61803	0.0574943
$793$	−3.47214	−0.123299
$794$	−0.729490	−0.0258886
$795$	0	0
$796$	−7.61803	−0.270014
$797$	37.7984	1.33889	0.669444	−	0.742863i	$-0.266533\pi$
$797$	37.7984	1.33889	0.669444	+	0.742863i	$0.266533\pi$
$798$	−0.763932	−0.0270429
$799$	0.618034	0.0218645
$800$	−3.09017	−0.109254
$801$	−5.65248	−0.199720
$802$	29.2705	1.03358
$803$	−11.4721	−0.404843
$804$	−10.0902	−0.355853
$805$	1.22794	0.0432791
$806$	1.47214	0.0518538
$807$	−1.61803	−0.0569575
$808$	6.09017	0.214251
$809$	24.4721	0.860394	0.430197	−	0.902735i	$-0.358444\pi$
$809$	24.4721	0.860394	0.430197	+	0.902735i	$0.358444\pi$
$810$	10.6525	0.374290
$811$	1.59675	0.0560694	0.0280347	−	0.999607i	$-0.491075\pi$
$811$	1.59675	0.0560694	0.0280347	+	0.999607i	$0.491075\pi$
$812$	−1.38197	−0.0484975
$813$	26.0902	0.915022
$814$	−19.9443	−0.699046
$815$	−2.31308	−0.0810237
$816$	6.85410	0.239942
$817$	−4.47214	−0.156460
$818$	19.6525	0.687133
$819$	0.0901699	0.00315079
$820$	7.88854	0.275480
$821$	−6.36068	−0.221989	−0.110995	−	0.993821i	$-0.535404\pi$
$821$	−6.36068	−0.221989	−0.110995	+	0.993821i	$0.535404\pi$
$822$	21.6525	0.755217
$823$	−15.1115	−0.526752	−0.263376	−	0.964693i	$-0.584836\pi$
$823$	−15.1115	−0.526752	−0.263376	+	0.964693i	$0.584836\pi$
$824$	4.14590	0.144429
$825$	−21.1803	−0.737405
$826$	3.20163	0.111399
$827$	4.74265	0.164918	0.0824590	−	0.996594i	$-0.473723\pi$
$827$	4.74265	0.164918	0.0824590	+	0.996594i	$0.473723\pi$
$828$	1.43769	0.0499633
$829$	19.2016	0.666900	0.333450	−	0.942768i	$-0.391787\pi$
$829$	19.2016	0.666900	0.333450	+	0.942768i	$0.391787\pi$
$830$	1.05573	0.0366449
$831$	38.2148	1.32566
$832$	−1.00000	−0.0346688
$833$	29.4164	1.01922
$834$	15.4721	0.535756
$835$	−1.90983	−0.0660924
$836$	−8.47214	−0.293015
$837$	−8.05573	−0.278447
$838$	−37.5623	−1.29757
$839$	−41.4508	−1.43104	−0.715521	−	0.698591i	$-0.753811\pi$
$839$	−41.4508	−1.43104	−0.715521	+	0.698591i	$0.753811\pi$
$840$	0.527864	0.0182130
$841$	5.27051	0.181742
$842$	2.52786	0.0871159
$843$	11.0344	0.380046
$844$	−5.85410	−0.201506
$845$	16.5836	0.570493
$846$	0.0557281	0.00191597
$847$	1.63932	0.0563277
$848$	0	0
$849$	−16.5623	−0.568417
$850$	13.0902	0.448989
$851$	−17.7214	−0.607480
$852$	6.00000	0.205557
$853$	−48.8115	−1.67127	−0.835637	−	0.549281i	$-0.814902\pi$
$853$	−48.8115	−1.67127	−0.835637	+	0.549281i	$0.814902\pi$
$854$	0.819660	0.0280482
$855$	1.05573	0.0361051
$856$	−7.76393	−0.265366
$857$	16.0689	0.548903	0.274451	−	0.961601i	$-0.411504\pi$
$857$	16.0689	0.548903	0.274451	+	0.961601i	$0.411504\pi$
$858$	−6.85410	−0.233995
$859$	29.7426	1.01481	0.507403	−	0.861709i	$-0.330606\pi$
$859$	29.7426	1.01481	0.507403	+	0.861709i	$0.330606\pi$
$860$	3.09017	0.105374
$861$	2.18034	0.0743058
$862$	−6.52786	−0.222340
$863$	1.67376	0.0569755	0.0284878	−	0.999594i	$-0.490931\pi$
$863$	1.67376	0.0569755	0.0284878	+	0.999594i	$0.490931\pi$
$864$	5.47214	0.186166
$865$	−2.03444	−0.0691731
$866$	36.7984	1.25046
$867$	−1.52786	−0.0518890
$868$	−0.347524	−0.0117957
$869$	−45.7426	−1.55171
$870$	−13.0902	−0.443798
$871$	−6.23607	−0.211301
$872$	8.70820	0.294897
$873$	7.12461	0.241132
$874$	−7.52786	−0.254634
$875$	2.63932	0.0892253
$876$	−4.38197	−0.148053
$877$	−53.5755	−1.80911	−0.904557	−	0.426352i	$-0.859799\pi$
$877$	−53.5755	−1.80911	−0.904557	+	0.426352i	$0.859799\pi$
$878$	25.8541	0.872534
$879$	−26.6525	−0.898966
$880$	5.85410	0.197342
$881$	−6.49342	−0.218769	−0.109384	−	0.994000i	$-0.534888\pi$
$881$	−6.49342	−0.218769	−0.109384	+	0.994000i	$0.534888\pi$
$882$	2.65248	0.0893135
$883$	48.1935	1.62184	0.810920	−	0.585157i	$-0.198967\pi$
$883$	48.1935	1.62184	0.810920	+	0.585157i	$0.198967\pi$
$884$	4.23607	0.142474
$885$	30.3262	1.01941
$886$	6.58359	0.221180
$887$	39.8328	1.33746	0.668728	−	0.743508i	$-0.266839\pi$
$887$	39.8328	1.33746	0.668728	+	0.743508i	$0.266839\pi$
$888$	−7.61803	−0.255644
$889$	−0.896674	−0.0300735
$890$	−20.4508	−0.685514
$891$	32.6525	1.09390
$892$	10.3262	0.345748
$893$	−0.291796	−0.00976458
$894$	24.7984	0.829382
$895$	−7.31308	−0.244449
$896$	0.236068	0.00788648
$897$	−6.09017	−0.203345
$898$	14.0902	0.470195
$899$	8.61803	0.287428
$900$	1.18034	0.0393447
$901$	0	0
$902$	24.1803	0.805117
$903$	0.854102	0.0284227
$904$	−17.6180	−0.585967
$905$	10.2492	0.340696
$906$	−12.8541	−0.427049
$907$	26.1246	0.867453	0.433727	−	0.901044i	$-0.357198\pi$
$907$	26.1246	0.867453	0.433727	+	0.901044i	$0.357198\pi$
$908$	5.03444	0.167074
$909$	−2.32624	−0.0771564
$910$	0.326238	0.0108147
$911$	−44.3394	−1.46903	−0.734515	−	0.678593i	$-0.762590\pi$
$911$	−44.3394	−1.46903	−0.734515	+	0.678593i	$0.762590\pi$
$912$	−3.23607	−0.107157
$913$	3.23607	0.107098
$914$	2.20163	0.0728233
$915$	7.76393	0.256668
$916$	2.70820	0.0894816
$917$	1.70820	0.0564099
$918$	−23.1803	−0.765065
$919$	−36.6525	−1.20905	−0.604527	−	0.796585i	$-0.706638\pi$
$919$	−36.6525	−1.20905	−0.604527	+	0.796585i	$0.706638\pi$
$920$	5.20163	0.171493
$921$	13.1459	0.433172
$922$	−19.1246	−0.629836
$923$	3.70820	0.122057
$924$	1.61803	0.0532294
$925$	−14.5492	−0.478373
$926$	−8.90983	−0.292795
$927$	−1.58359	−0.0520120
$928$	−5.85410	−0.192170
$929$	40.1803	1.31827	0.659137	−	0.752023i	$-0.270922\pi$
$929$	40.1803	1.31827	0.659137	+	0.752023i	$0.270922\pi$
$930$	−3.29180	−0.107942
$931$	−13.8885	−0.455179
$932$	−0.236068	−0.00773266
$933$	20.6525	0.676132
$934$	−33.4164	−1.09342
$935$	−24.7984	−0.810994
$936$	0.381966	0.0124849
$937$	−21.9098	−0.715763	−0.357881	−	0.933767i	$-0.616501\pi$
$937$	−21.9098	−0.715763	−0.357881	+	0.933767i	$0.616501\pi$
$938$	1.47214	0.0480669
$939$	−32.1246	−1.04835
$940$	0.201626	0.00657632
$941$	−0.0212862	−0.000693911 0	−0.000346956	−	1.00000i	$-0.500110\pi$
$941$	−0.0212862	−0.000693911 0	−0.000346956		1.00000i	$0.500110\pi$
$942$	14.0000	0.456145
$943$	21.4853	0.699657
$944$	13.5623	0.441415
$945$	−1.78522	−0.0580732
$946$	9.47214	0.307966
$947$	1.21478	0.0394751	0.0197376	−	0.999805i	$-0.493717\pi$
$947$	1.21478	0.0394751	0.0197376	+	0.999805i	$0.493717\pi$
$948$	−17.4721	−0.567469
$949$	−2.70820	−0.0879120
$950$	−6.18034	−0.200517
$951$	41.6869	1.35179
$952$	−1.00000	−0.0324102
$953$	−37.3050	−1.20843	−0.604213	−	0.796823i	$-0.706512\pi$
$953$	−37.3050	−1.20843	−0.604213	+	0.796823i	$0.706512\pi$
$954$	0	0
$955$	31.7082	1.02605
$956$	14.2705	0.461541
$957$	−40.1246	−1.29704
$958$	12.5967	0.406982
$959$	−3.15905	−0.102011
$960$	2.23607	0.0721688
$961$	−28.8328	−0.930091
$962$	−4.70820	−0.151799
$963$	2.96556	0.0955638
$964$	8.70820	0.280472
$965$	5.20163	0.167446
$966$	1.43769	0.0462570
$967$	−4.85410	−0.156097	−0.0780487	−	0.996950i	$-0.524869\pi$
$967$	−4.85410	−0.156097	−0.0780487	+	0.996950i	$0.524869\pi$
$968$	6.94427	0.223197
$969$	13.7082	0.440371
$970$	25.7771	0.827652
$971$	30.6525	0.983685	0.491842	−	0.870684i	$-0.336324\pi$
$971$	30.6525	0.983685	0.491842	+	0.870684i	$0.336324\pi$
$972$	−3.94427	−0.126513
$973$	−2.25735	−0.0723675
$974$	38.8328	1.24428
$975$	−5.00000	−0.160128
$976$	3.47214	0.111140
$977$	37.8541	1.21106	0.605530	−	0.795822i	$-0.292961\pi$
$977$	37.8541	1.21106	0.605530	+	0.795822i	$0.292961\pi$
$978$	−2.70820	−0.0865988
$979$	−62.6869	−2.00348
$980$	9.59675	0.306557
$981$	−3.32624	−0.106199
$982$	11.2361	0.358557
$983$	−51.5623	−1.64458	−0.822291	−	0.569067i	$-0.807304\pi$
$983$	−51.5623	−1.64458	−0.822291	+	0.569067i	$0.807304\pi$
$984$	9.23607	0.294435
$985$	34.0213	1.08401
$986$	24.7984	0.789741
$987$	0.0557281	0.00177384
$988$	−2.00000	−0.0636285
$989$	8.41641	0.267626
$990$	−2.23607	−0.0710669
$991$	−18.6180	−0.591421	−0.295711	−	0.955278i	$-0.595556\pi$
$991$	−18.6180	−0.591421	−0.295711	+	0.955278i	$0.595556\pi$
$992$	−1.47214	−0.0467404
$993$	−37.5066	−1.19024
$994$	−0.875388	−0.0277656
$995$	10.5279	0.333756
$996$	1.23607	0.0391663
$997$	−44.4164	−1.40668	−0.703341	−	0.710853i	$-0.748309\pi$
$997$	−44.4164	−1.40668	−0.703341	+	0.710853i	$0.748309\pi$
$998$	−5.94427	−0.188163
$999$	25.7639	0.815135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.a.1.1	✓	2
3.2	odd	2		4842.2.a.f.1.1		2
4.3	odd	2		4304.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.a.1.1	✓	2	1.1	even	1	trivial
4304.2.a.d.1.2		2	4.3	odd	2
4842.2.a.f.1.1		2	3.2	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.38197 q^{5} -1.61803 q^{6} +0.236068 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -1.38197 q^{5} -1.61803 q^{6} +0.236068 q^{7} +1.00000 q^{8} -0.381966 q^{9} -1.38197 q^{10} -4.23607 q^{11} -1.61803 q^{12} -1.00000 q^{13} +0.236068 q^{14} +2.23607 q^{15} +1.00000 q^{16} -4.23607 q^{17} -0.381966 q^{18} +2.00000 q^{19} -1.38197 q^{20} -0.381966 q^{21} -4.23607 q^{22} -3.76393 q^{23} -1.61803 q^{24} -3.09017 q^{25} -1.00000 q^{26} +5.47214 q^{27} +0.236068 q^{28} -5.85410 q^{29} +2.23607 q^{30} -1.47214 q^{31} +1.00000 q^{32} +6.85410 q^{33} -4.23607 q^{34} -0.326238 q^{35} -0.381966 q^{36} +4.70820 q^{37} +2.00000 q^{38} +1.61803 q^{39} -1.38197 q^{40} -5.70820 q^{41} -0.381966 q^{42} -2.23607 q^{43} -4.23607 q^{44} +0.527864 q^{45} -3.76393 q^{46} -0.145898 q^{47} -1.61803 q^{48} -6.94427 q^{49} -3.09017 q^{50} +6.85410 q^{51} -1.00000 q^{52} +5.47214 q^{54} +5.85410 q^{55} +0.236068 q^{56} -3.23607 q^{57} -5.85410 q^{58} +13.5623 q^{59} +2.23607 q^{60} +3.47214 q^{61} -1.47214 q^{62} -0.0901699 q^{63} +1.00000 q^{64} +1.38197 q^{65} +6.85410 q^{66} +6.23607 q^{67} -4.23607 q^{68} +6.09017 q^{69} -0.326238 q^{70} -3.70820 q^{71} -0.381966 q^{72} +2.70820 q^{73} +4.70820 q^{74} +5.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} +1.61803 q^{78} +10.7984 q^{79} -1.38197 q^{80} -7.70820 q^{81} -5.70820 q^{82} -0.763932 q^{83} -0.381966 q^{84} +5.85410 q^{85} -2.23607 q^{86} +9.47214 q^{87} -4.23607 q^{88} +14.7984 q^{89} +0.527864 q^{90} -0.236068 q^{91} -3.76393 q^{92} +2.38197 q^{93} -0.145898 q^{94} -2.76393 q^{95} -1.61803 q^{96} -18.6525 q^{97} -6.94427 q^{98} +1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{22} - 12 q^{23} - q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{33} - 4 q^{34} + 15 q^{35} - 3 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} - 5 q^{40} + 2 q^{41} - 3 q^{42} - 4 q^{44} + 10 q^{45} - 12 q^{46} - 7 q^{47} - q^{48} + 4 q^{49} + 5 q^{50} + 7 q^{51} - 2 q^{52} + 2 q^{54} + 5 q^{55} - 4 q^{56} - 2 q^{57} - 5 q^{58} + 7 q^{59} - 2 q^{61} + 6 q^{62} + 11 q^{63} + 2 q^{64} + 5 q^{65} + 7 q^{66} + 8 q^{67} - 4 q^{68} + q^{69} + 15 q^{70} + 6 q^{71} - 3 q^{72} - 8 q^{73} - 4 q^{74} + 10 q^{75} + 4 q^{76} - 2 q^{77} + q^{78} - 3 q^{79} - 5 q^{80} - 2 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} + 5 q^{85} + 10 q^{87} - 4 q^{88} + 5 q^{89} + 10 q^{90} + 4 q^{91} - 12 q^{92} + 7 q^{93} - 7 q^{94} - 10 q^{95} - q^{96} - 6 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9 - 5 * q^10 - 4 * q^11 - q^12 - 2 * q^13 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 3 * q^18 + 4 * q^19 - 5 * q^20 - 3 * q^21 - 4 * q^22 - 12 * q^23 - q^24 + 5 * q^25 - 2 * q^26 + 2 * q^27 - 4 * q^28 - 5 * q^29 + 6 * q^31 + 2 * q^32 + 7 * q^33 - 4 * q^34 + 15 * q^35 - 3 * q^36 - 4 * q^37 + 4 * q^38 + q^39 - 5 * q^40 + 2 * q^41 - 3 * q^42 - 4 * q^44 + 10 * q^45 - 12 * q^46 - 7 * q^47 - q^48 + 4 * q^49 + 5 * q^50 + 7 * q^51 - 2 * q^52 + 2 * q^54 + 5 * q^55 - 4 * q^56 - 2 * q^57 - 5 * q^58 + 7 * q^59 - 2 * q^61 + 6 * q^62 + 11 * q^63 + 2 * q^64 + 5 * q^65 + 7 * q^66 + 8 * q^67 - 4 * q^68 + q^69 + 15 * q^70 + 6 * q^71 - 3 * q^72 - 8 * q^73 - 4 * q^74 + 10 * q^75 + 4 * q^76 - 2 * q^77 + q^78 - 3 * q^79 - 5 * q^80 - 2 * q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 + 5 * q^85 + 10 * q^87 - 4 * q^88 + 5 * q^89 + 10 * q^90 + 4 * q^91 - 12 * q^92 + 7 * q^93 - 7 * q^94 - 10 * q^95 - q^96 - 6 * q^97 + 4 * q^98 + q^99

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