LMFDB - Embedded newform 159.1.d.a.158.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [159,1,Mod(158,159)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(159, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("159.158");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 159 = 3 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	159.d (of order $2$, degree $1$, minimal)

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:	$0.0793513370086$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.25281.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.25281.1

Embedding invariants

Embedding label			158.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	159.158

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} +1.00000 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.618034 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +1.00000 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.618034 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.618034 q^{12} -1.61803 q^{13} +0.381966 q^{14} -1.61803 q^{15} +0.618034 q^{18} +1.00000 q^{20} +0.618034 q^{21} +0.618034 q^{23} -1.00000 q^{24} +1.61803 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.381966 q^{28} -1.00000 q^{30} +1.00000 q^{32} -1.00000 q^{35} -0.618034 q^{36} +0.618034 q^{37} -1.61803 q^{39} +1.61803 q^{40} +0.618034 q^{41} +0.381966 q^{42} -1.61803 q^{43} -1.61803 q^{45} +0.381966 q^{46} -0.618034 q^{49} +1.00000 q^{50} +1.00000 q^{52} +1.00000 q^{53} +0.618034 q^{54} -0.618034 q^{56} +1.00000 q^{60} +0.618034 q^{63} +0.618034 q^{64} +2.61803 q^{65} +0.618034 q^{69} -0.618034 q^{70} -1.61803 q^{71} -1.00000 q^{72} +0.381966 q^{74} +1.61803 q^{75} -1.00000 q^{78} +1.00000 q^{81} +0.381966 q^{82} -1.61803 q^{83} -0.381966 q^{84} -1.00000 q^{86} -1.00000 q^{90} -1.00000 q^{91} -0.381966 q^{92} +1.00000 q^{96} -1.61803 q^{97} -0.381966 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 + q^4 - q^5 - q^6 - q^7 - 2 * q^8 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + q^{12} - q^{13} + 3 q^{14} - q^{15} - q^{18} + 2 q^{20} - q^{21} - q^{23} - 2 q^{24} + q^{25} - 2 q^{26} + 2 q^{27} - 3 q^{28} - 2 q^{30} + 2 q^{32} - 2 q^{35} + q^{36} - q^{37} - q^{39} + q^{40} - q^{41} + 3 q^{42} - q^{43} - q^{45} + 3 q^{46} + q^{49} + 2 q^{50} + 2 q^{52} + 2 q^{53} - q^{54} + q^{56} + 2 q^{60} - q^{63} - q^{64} + 3 q^{65} - q^{69} + q^{70} - q^{71} - 2 q^{72} + 3 q^{74} + q^{75} - 2 q^{78} + 2 q^{81} + 3 q^{82} - q^{83} - 3 q^{84} - 2 q^{86} - 2 q^{90} - 2 q^{91} - 3 q^{92} + 2 q^{96} - q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 + q^4 - q^5 - q^6 - q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + q^12 - q^13 + 3 * q^14 - q^15 - q^18 + 2 * q^20 - q^21 - q^23 - 2 * q^24 + q^25 - 2 * q^26 + 2 * q^27 - 3 * q^28 - 2 * q^30 + 2 * q^32 - 2 * q^35 + q^36 - q^37 - q^39 + q^40 - q^41 + 3 * q^42 - q^43 - q^45 + 3 * q^46 + q^49 + 2 * q^50 + 2 * q^52 + 2 * q^53 - q^54 + q^56 + 2 * q^60 - q^63 - q^64 + 3 * q^65 - q^69 + q^70 - q^71 - 2 * q^72 + 3 * q^74 + q^75 - 2 * q^78 + 2 * q^81 + 3 * q^82 - q^83 - 3 * q^84 - 2 * q^86 - 2 * q^90 - 2 * q^91 - 3 * q^92 + 2 * q^96 - q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/159\mathbb{Z}\right)^\times$.

$n$	$55$	$107$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$2$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$3$	1.00000	1.00000
$3$	1.00000	1.00000
$4$	−0.618034	−0.618034
$5$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0.618034	0.618034
$7$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$7$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$8$	−1.00000	−1.00000
$9$	1.00000	1.00000
$10$	−1.00000	−1.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.618034	−0.618034
$13$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$13$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$14$	0.381966	0.381966
$15$	−1.61803	−1.61803
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.618034	0.618034
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.00000	1.00000
$21$	0.618034	0.618034
$22$	0	0
$23$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$23$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$24$	−1.00000	−1.00000
$25$	1.61803	1.61803
$26$	−1.00000	−1.00000
$27$	1.00000	1.00000
$28$	−0.381966	−0.381966
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1.00000	−1.00000
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.00000	1.00000
$33$	0	0
$34$	0	0
$35$	−1.00000	−1.00000
$36$	−0.618034	−0.618034
$37$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$37$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$38$	0	0
$39$	−1.61803	−1.61803
$40$	1.61803	1.61803
$41$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$41$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$42$	0.381966	0.381966
$43$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$43$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$44$	0	0
$45$	−1.61803	−1.61803
$46$	0.381966	0.381966
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−0.618034	−0.618034
$50$	1.00000	1.00000
$51$	0	0
$52$	1.00000	1.00000
$53$	1.00000	1.00000
$53$	1.00000	1.00000
$54$	0.618034	0.618034
$55$	0	0
$56$	−0.618034	−0.618034
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	1.00000	1.00000
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0.618034	0.618034
$64$	0.618034	0.618034
$65$	2.61803	2.61803
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0.618034	0.618034
$70$	−0.618034	−0.618034
$71$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$72$	−1.00000	−1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.381966	0.381966
$75$	1.61803	1.61803
$76$	0	0
$77$	0	0
$78$	−1.00000	−1.00000
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.00000	1.00000
$82$	0.381966	0.381966
$83$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$83$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$84$	−0.381966	−0.381966
$85$	0	0
$86$	−1.00000	−1.00000
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−1.00000	−1.00000
$91$	−1.00000	−1.00000
$92$	−0.381966	−0.381966
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.00000	1.00000
$97$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$97$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$98$	−0.381966	−0.381966
$99$	0	0
$100$	−1.00000	−1.00000
$101$	2.00000	2.00000	1.00000			$0$
$101$	2.00000	2.00000	1.00000			$0$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.61803	1.61803
$105$	−1.00000	−1.00000
$106$	0.618034	0.618034
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.618034	−0.618034
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0.618034	0.618034
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−1.00000	−1.00000
$116$	0	0
$117$	−1.61803	−1.61803
$118$	0	0
$119$	0	0
$120$	1.61803	1.61803
$121$	1.00000	1.00000
$122$	0	0
$123$	0.618034	0.618034
$124$	0	0
$125$	−1.00000	−1.00000
$126$	0.381966	0.381966
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.618034	−0.618034
$129$	−1.61803	−1.61803
$130$	1.61803	1.61803
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.61803	−1.61803
$136$	0	0
$137$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$137$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$138$	0.381966	0.381966
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0.618034	0.618034
$141$	0	0
$142$	−1.00000	−1.00000
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.618034	−0.618034
$148$	−0.381966	−0.381966
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.00000	1.00000
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	1.00000	1.00000
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	1.00000	1.00000
$160$	−1.61803	−1.61803
$161$	0.381966	0.381966
$162$	0.618034	0.618034
$163$	2.00000	2.00000	1.00000			$0$
$163$	2.00000	2.00000	1.00000			$0$
$164$	−0.381966	−0.381966
$165$	0	0
$166$	−1.00000	−1.00000
$167$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$167$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$168$	−0.618034	−0.618034
$169$	1.61803	1.61803
$170$	0	0
$171$	0	0
$172$	1.00000	1.00000
$173$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$173$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$174$	0	0
$175$	1.00000	1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$179$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$180$	1.00000	1.00000
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−0.618034	−0.618034
$183$	0	0
$184$	−0.618034	−0.618034
$185$	−1.00000	−1.00000
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.618034	0.618034
$190$	0	0
$191$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$191$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$192$	0.618034	0.618034
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	−1.00000	−1.00000
$195$	2.61803	2.61803
$196$	0.381966	0.381966
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$199$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$200$	−1.61803	−1.61803
$201$	0	0
$202$	1.23607	1.23607
$203$	0	0
$204$	0	0
$205$	−1.00000	−1.00000
$206$	0	0
$207$	0.618034	0.618034
$208$	0	0
$209$	0	0
$210$	−0.618034	−0.618034
$211$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$211$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$212$	−0.618034	−0.618034
$213$	−1.61803	−1.61803
$214$	0	0
$215$	2.61803	2.61803
$216$	−1.00000	−1.00000
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0.381966	0.381966
$223$	2.00000	2.00000	1.00000			$0$
$223$	2.00000	2.00000	1.00000			$0$
$224$	0.618034	0.618034
$225$	1.61803	1.61803
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$229$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$230$	−0.618034	−0.618034
$231$	0	0
$232$	0	0
$233$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$233$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$234$	−1.00000	−1.00000
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.00000	2.00000	1.00000			$0$
$239$	2.00000	2.00000	1.00000			$0$
$240$	0	0
$241$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$241$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$242$	0.618034	0.618034
$243$	1.00000	1.00000
$244$	0	0
$245$	1.00000	1.00000
$246$	0.381966	0.381966
$247$	0	0
$248$	0	0
$249$	−1.61803	−1.61803
$250$	−0.618034	−0.618034
$251$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$251$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$252$	−0.381966	−0.381966
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.00000	−1.00000
$257$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$257$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$258$	−1.00000	−1.00000
$259$	0.381966	0.381966
$260$	−1.61803	−1.61803
$261$	0	0
$262$	0	0
$263$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$263$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$264$	0	0
$265$	−1.61803	−1.61803
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−1.00000	−1.00000
$271$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$271$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$272$	0	0
$273$	−1.00000	−1.00000
$274$	−1.00000	−1.00000
$275$	0	0
$276$	−0.381966	−0.381966
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	1.00000	1.00000
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.00000	1.00000
$285$	0	0
$286$	0	0
$287$	0.381966	0.381966
$288$	1.00000	1.00000
$289$	1.00000	1.00000
$290$	0	0
$291$	−1.61803	−1.61803
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−0.381966	−0.381966
$295$	0	0
$296$	−0.618034	−0.618034
$297$	0	0
$298$	0	0
$299$	−1.00000	−1.00000
$300$	−1.00000	−1.00000
$301$	−1.00000	−1.00000
$302$	0	0
$303$	2.00000	2.00000
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$307$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	1.61803	1.61803
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−1.00000	−1.00000
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.618034	0.618034
$319$	0	0
$320$	−1.00000	−1.00000
$321$	0	0
$322$	0.236068	0.236068
$323$	0	0
$324$	−0.618034	−0.618034
$325$	−2.61803	−2.61803
$326$	1.23607	1.23607
$327$	0	0
$328$	−0.618034	−0.618034
$329$	0	0
$330$	0	0
$331$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$331$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$332$	1.00000	1.00000
$333$	0.618034	0.618034
$334$	−1.00000	−1.00000
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.00000	1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−1.00000
$344$	1.61803	1.61803
$345$	−1.00000	−1.00000
$346$	0.381966	0.381966
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0.618034	0.618034
$351$	−1.61803	−1.61803
$352$	0	0
$353$	2.00000	2.00000	1.00000			$0$
$353$	2.00000	2.00000	1.00000			$0$
$354$	0	0
$355$	2.61803	2.61803
$356$	0	0
$357$	0	0
$358$	0.381966	0.381966
$359$	2.00000	2.00000	1.00000			$0$
$359$	2.00000	2.00000	1.00000			$0$
$360$	1.61803	1.61803
$361$	1.00000	1.00000
$362$	0	0
$363$	1.00000	1.00000
$364$	0.618034	0.618034
$365$	0	0
$366$	0	0
$367$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$367$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$368$	0	0
$369$	0.618034	0.618034
$370$	−0.618034	−0.618034
$371$	0.618034	0.618034
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.00000	−1.00000
$376$	0	0
$377$	0	0
$378$	0.381966	0.381966
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0.381966	0.381966
$383$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$383$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$384$	−0.618034	−0.618034
$385$	0	0
$386$	0	0
$387$	−1.61803	−1.61803
$388$	1.00000	1.00000
$389$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$389$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$390$	1.61803	1.61803
$391$	0	0
$392$	0.618034	0.618034
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−1.00000	−1.00000
$399$	0	0
$400$	0	0
$401$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$401$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$402$	0	0
$403$	0	0
$404$	−1.23607	−1.23607
$405$	−1.61803	−1.61803
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$409$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$410$	−0.618034	−0.618034
$411$	−1.61803	−1.61803
$412$	0	0
$413$	0	0
$414$	0.381966	0.381966
$415$	2.61803	2.61803
$416$	−1.61803	−1.61803
$417$	0	0
$418$	0	0
$419$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$419$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$420$	0.618034	0.618034
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0.381966	0.381966
$423$	0	0
$424$	−1.00000	−1.00000
$425$	0	0
$426$	−1.00000	−1.00000
$427$	0	0
$428$	0	0
$429$	0	0
$430$	1.61803	1.61803
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$433$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$439$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$440$	0	0
$441$	−0.618034	−0.618034
$442$	0	0
$443$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$443$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$444$	−0.381966	−0.381966
$445$	0	0
$446$	1.23607	1.23607
$447$	0	0
$448$	0.381966	0.381966
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1.00000	1.00000
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.61803	1.61803
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0.381966	0.381966
$459$	0	0
$460$	0.618034	0.618034
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−1.00000	−1.00000
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.00000	1.00000
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.00000	1.00000
$478$	1.23607	1.23607
$479$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$479$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$480$	−1.61803	−1.61803
$481$	−1.00000	−1.00000
$482$	0.381966	0.381966
$483$	0.381966	0.381966
$484$	−0.618034	−0.618034
$485$	2.61803	2.61803
$486$	0.618034	0.618034
$487$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$487$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$488$	0	0
$489$	2.00000	2.00000
$490$	0.618034	0.618034
$491$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$491$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$492$	−0.381966	−0.381966
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.00000	−1.00000
$498$	−1.00000	−1.00000
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.618034	0.618034
$501$	−1.61803	−1.61803
$502$	0.381966	0.381966
$503$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$503$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$504$	−0.618034	−0.618034
$505$	−3.23607	−3.23607
$506$	0	0
$507$	1.61803	1.61803
$508$	0	0
$509$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$509$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0.381966	0.381966
$515$	0	0
$516$	1.00000	1.00000
$517$	0	0
$518$	0.236068	0.236068
$519$	0.618034	0.618034
$520$	−2.61803	−2.61803
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$523$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$524$	0	0
$525$	1.00000	1.00000
$526$	−1.00000	−1.00000
$527$	0	0
$528$	0	0
$529$	−0.618034	−0.618034
$530$	−1.00000	−1.00000
$531$	0	0
$532$	0	0
$533$	−1.00000	−1.00000
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.618034	0.618034
$538$	0	0
$539$	0	0
$540$	1.00000	1.00000
$541$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$542$	−1.00000	−1.00000
$543$	0	0
$544$	0	0
$545$	0	0
$546$	−0.618034	−0.618034
$547$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$547$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$548$	1.00000	1.00000
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−0.618034	−0.618034
$553$	0	0
$554$	0	0
$555$	−1.00000	−1.00000
$556$	0	0
$557$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$557$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$558$	0	0
$559$	2.61803	2.61803
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$563$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.618034	0.618034
$568$	1.61803	1.61803
$569$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$569$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0.618034	0.618034
$574$	0.236068	0.236068
$575$	1.00000	1.00000
$576$	0.618034	0.618034
$577$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$577$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$578$	0.618034	0.618034
$579$	0	0
$580$	0	0
$581$	−1.00000	−1.00000
$582$	−1.00000	−1.00000
$583$	0	0
$584$	0	0
$585$	2.61803	2.61803
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.381966	0.381966
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.61803	−1.61803
$598$	−0.618034	−0.618034
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.61803	−1.61803
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.618034	−0.618034
$603$	0	0
$604$	0	0
$605$	−1.61803	−1.61803
$606$	1.23607	1.23607
$607$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$607$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.00000	−1.00000
$615$	−1.00000	−1.00000
$616$	0	0
$617$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$617$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$618$	0	0
$619$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$619$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$620$	0	0
$621$	0.618034	0.618034
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−0.618034	−0.618034
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0.618034	0.618034
$634$	0	0
$635$	0	0
$636$	−0.618034	−0.618034
$637$	1.00000	1.00000
$638$	0	0
$639$	−1.61803	−1.61803
$640$	1.00000	1.00000
$641$	2.00000	2.00000	1.00000			$0$
$641$	2.00000	2.00000	1.00000			$0$
$642$	0	0
$643$	2.00000	2.00000	1.00000			$0$
$643$	2.00000	2.00000	1.00000			$0$
$644$	−0.236068	−0.236068
$645$	2.61803	2.61803
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.00000	−1.00000
$649$	0	0
$650$	−1.61803	−1.61803
$651$	0	0
$652$	−1.23607	−1.23607
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$659$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$660$	0	0
$661$	2.00000	2.00000	1.00000			$0$
$661$	2.00000	2.00000	1.00000			$0$
$662$	0.381966	0.381966
$663$	0	0
$664$	1.61803	1.61803
$665$	0	0
$666$	0.381966	0.381966
$667$	0	0
$668$	1.00000	1.00000
$669$	2.00000	2.00000
$670$	0	0
$671$	0	0
$672$	0.618034	0.618034
$673$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$673$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$674$	0	0
$675$	1.61803	1.61803
$676$	−1.00000	−1.00000
$677$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$677$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$678$	0	0
$679$	−1.00000	−1.00000
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	2.61803	2.61803
$686$	−0.618034	−0.618034
$687$	0.618034	0.618034
$688$	0	0
$689$	−1.61803	−1.61803
$690$	−0.618034	−0.618034
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−0.381966	−0.381966
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−1.61803	−1.61803
$700$	−0.618034	−0.618034
$701$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$701$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$702$	−1.00000	−1.00000
$703$	0	0
$704$	0	0
$705$	0	0
$706$	1.23607	1.23607
$707$	1.23607	1.23607
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	1.61803	1.61803
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−0.381966	−0.381966
$717$	2.00000	2.00000
$718$	1.23607	1.23607
$719$	2.00000	2.00000	1.00000			$0$
$719$	2.00000	2.00000	1.00000			$0$
$720$	0	0
$721$	0	0
$722$	0.618034	0.618034
$723$	0.618034	0.618034
$724$	0	0
$725$	0	0
$726$	0.618034	0.618034
$727$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$727$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$728$	1.00000	1.00000
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$733$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$734$	0.381966	0.381966
$735$	1.00000	1.00000
$736$	0.618034	0.618034
$737$	0	0
$738$	0.381966	0.381966
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0.618034	0.618034
$741$	0	0
$742$	0.381966	0.381966
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.61803	−1.61803
$748$	0	0
$749$	0	0
$750$	−0.618034	−0.618034
$751$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$751$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$752$	0	0
$753$	0.618034	0.618034
$754$	0	0
$755$	0	0
$756$	−0.381966	−0.381966
$757$	2.00000	2.00000	1.00000			$0$
$757$	2.00000	2.00000	1.00000			$0$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$761$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$762$	0	0
$763$	0	0
$764$	−0.381966	−0.381966
$765$	0	0
$766$	−1.00000	−1.00000
$767$	0	0
$768$	−1.00000	−1.00000
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0.618034	0.618034
$772$	0	0
$773$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$773$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$774$	−1.00000	−1.00000
$775$	0	0
$776$	1.61803	1.61803
$777$	0.381966	0.381966
$778$	0.381966	0.381966
$779$	0	0
$780$	−1.61803	−1.61803
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	−1.61803	−1.61803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.61803	−1.61803
$796$	1.00000	1.00000
$797$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$797$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$798$	0	0
$799$	0	0
$800$	1.61803	1.61803
$801$	0	0
$802$	−1.00000	−1.00000
$803$	0	0
$804$	0	0
$805$	−0.618034	−0.618034
$806$	0	0
$807$	0	0
$808$	−2.00000	−2.00000
$809$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$809$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$810$	−1.00000	−1.00000
$811$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$811$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$812$	0	0
$813$	−1.61803	−1.61803
$814$	0	0
$815$	−3.23607	−3.23607
$816$	0	0
$817$	0	0
$818$	0.381966	0.381966
$819$	−1.00000	−1.00000
$820$	0.618034	0.618034
$821$	2.00000	2.00000	1.00000			$0$
$821$	2.00000	2.00000	1.00000			$0$
$822$	−1.00000	−1.00000
$823$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$823$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$827$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$828$	−0.381966	−0.381966
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	1.61803	1.61803
$831$	0	0
$832$	−1.00000	−1.00000
$833$	0	0
$834$	0	0
$835$	2.61803	2.61803
$836$	0	0
$837$	0	0
$838$	−1.00000	−1.00000
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	1.00000	1.00000
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	−0.381966	−0.381966
$845$	−2.61803	−2.61803
$846$	0	0
$847$	0.618034	0.618034
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.381966	0.381966
$852$	1.00000	1.00000
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$859$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$860$	−1.61803	−1.61803
$861$	0.381966	0.381966
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000	1.00000
$865$	−1.00000	−1.00000
$866$	0.381966	0.381966
$867$	1.00000	1.00000
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.61803	−1.61803
$874$	0	0
$875$	−0.618034	−0.618034
$876$	0	0
$877$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$877$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$878$	0.381966	0.381966
$879$	0	0
$880$	0	0
$881$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$881$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$882$	−0.381966	−0.381966
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−1.00000	−1.00000
$887$	2.00000	2.00000	1.00000			$0$
$887$	2.00000	2.00000	1.00000			$0$
$888$	−0.618034	−0.618034
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−1.23607	−1.23607
$893$	0	0
$894$	0	0
$895$	−1.00000	−1.00000
$896$	−0.381966	−0.381966
$897$	−1.00000	−1.00000
$898$	0	0
$899$	0	0
$900$	−1.00000	−1.00000
$901$	0	0
$902$	0	0
$903$	−1.00000	−1.00000
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$907$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$908$	0	0
$909$	2.00000	2.00000
$910$	1.00000	1.00000
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−0.381966	−0.381966
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	1.00000	1.00000
$921$	−1.61803	−1.61803
$922$	0	0
$923$	2.61803	2.61803
$924$	0	0
$925$	1.00000	1.00000
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	1.00000	1.00000
$933$	0	0
$934$	0	0
$935$	0	0
$936$	1.61803	1.61803
$937$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$937$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0.381966	0.381966
$944$	0	0
$945$	−1.00000	−1.00000
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.618034	0.618034
$955$	−1.00000	−1.00000
$956$	−1.23607	−1.23607
$957$	0	0
$958$	0.381966	0.381966
$959$	−1.00000	−1.00000
$960$	−1.00000	−1.00000
$961$	1.00000	1.00000
$962$	−0.618034	−0.618034
$963$	0	0
$964$	−0.381966	−0.381966
$965$	0	0
$966$	0.236068	0.236068
$967$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$968$	−1.00000	−1.00000
$969$	0	0
$970$	1.61803	1.61803
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.618034	−0.618034
$973$	0	0
$974$	−1.00000	−1.00000
$975$	−2.61803	−2.61803
$976$	0	0
$977$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$977$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$978$	1.23607	1.23607
$979$	0	0
$980$	−0.618034	−0.618034
$981$	0	0
$982$	−1.00000	−1.00000
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−0.618034	−0.618034
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.00000	−1.00000
$990$	0	0
$991$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$991$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$992$	0	0
$993$	0.618034	0.618034
$994$	−0.618034	−0.618034
$995$	2.61803	2.61803
$996$	1.00000	1.00000
$997$	2.00000	2.00000	1.00000			$0$
$997$	2.00000	2.00000	1.00000			$0$
$998$	0	0
$999$	0.618034	0.618034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	159.1.d.a.158.2	✓	2
3.2	odd	2		159.1.d.b.158.1	yes	2
4.3	odd	2		2544.1.o.a.2225.1		2
5.2	odd	4		3975.1.b.a.3974.3		4
5.3	odd	4		3975.1.b.a.3974.2		4
5.4	even	2		3975.1.h.d.476.1		2
12.11	even	2		2544.1.o.b.2225.2		2
15.2	even	4		3975.1.b.b.3974.2		4
15.8	even	4		3975.1.b.b.3974.3		4
15.14	odd	2		3975.1.h.c.476.2		2
53.52	even	2		159.1.d.b.158.1	yes	2
159.158	odd	2	CM	159.1.d.a.158.2	✓	2
212.211	odd	2		2544.1.o.b.2225.2		2
265.52	odd	4		3975.1.b.b.3974.2		4
265.158	odd	4		3975.1.b.b.3974.3		4
265.264	even	2		3975.1.h.c.476.2		2
636.635	even	2		2544.1.o.a.2225.1		2
795.158	even	4		3975.1.b.a.3974.2		4
795.317	even	4		3975.1.b.a.3974.3		4
795.794	odd	2		3975.1.h.d.476.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
159.1.d.a.158.2	✓	2	1.1	even	1	trivial
159.1.d.a.158.2	✓	2	159.158	odd	2	CM
159.1.d.b.158.1	yes	2	3.2	odd	2
159.1.d.b.158.1	yes	2	53.52	even	2
2544.1.o.a.2225.1		2	4.3	odd	2
2544.1.o.a.2225.1		2	636.635	even	2
2544.1.o.b.2225.2		2	12.11	even	2
2544.1.o.b.2225.2		2	212.211	odd	2
3975.1.b.a.3974.2		4	5.3	odd	4
3975.1.b.a.3974.2		4	795.158	even	4
3975.1.b.a.3974.3		4	5.2	odd	4
3975.1.b.a.3974.3		4	795.317	even	4
3975.1.b.b.3974.2		4	15.2	even	4
3975.1.b.b.3974.2		4	265.52	odd	4
3975.1.b.b.3974.3		4	15.8	even	4
3975.1.b.b.3974.3		4	265.158	odd	4
3975.1.h.c.476.2		2	15.14	odd	2
3975.1.h.c.476.2		2	265.264	even	2
3975.1.h.d.476.1		2	5.4	even	2
3975.1.h.d.476.1		2	795.794	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:	\( N \)	\(=\)	\( 159 = 3 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	159.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +1.00000 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.618034 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +1.00000 q^{3} -0.618034 q^{4} -1.61803 q^{5} +0.618034 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.618034 q^{12} -1.61803 q^{13} +0.381966 q^{14} -1.61803 q^{15} +0.618034 q^{18} +1.00000 q^{20} +0.618034 q^{21} +0.618034 q^{23} -1.00000 q^{24} +1.61803 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.381966 q^{28} -1.00000 q^{30} +1.00000 q^{32} -1.00000 q^{35} -0.618034 q^{36} +0.618034 q^{37} -1.61803 q^{39} +1.61803 q^{40} +0.618034 q^{41} +0.381966 q^{42} -1.61803 q^{43} -1.61803 q^{45} +0.381966 q^{46} -0.618034 q^{49} +1.00000 q^{50} +1.00000 q^{52} +1.00000 q^{53} +0.618034 q^{54} -0.618034 q^{56} +1.00000 q^{60} +0.618034 q^{63} +0.618034 q^{64} +2.61803 q^{65} +0.618034 q^{69} -0.618034 q^{70} -1.61803 q^{71} -1.00000 q^{72} +0.381966 q^{74} +1.61803 q^{75} -1.00000 q^{78} +1.00000 q^{81} +0.381966 q^{82} -1.61803 q^{83} -0.381966 q^{84} -1.00000 q^{86} -1.00000 q^{90} -1.00000 q^{91} -0.381966 q^{92} +1.00000 q^{96} -1.61803 q^{97} -0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 + q^4 - q^5 - q^6 - q^7 - 2 * q^8 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + q^{12} - q^{13} + 3 q^{14} - q^{15} - q^{18} + 2 q^{20} - q^{21} - q^{23} - 2 q^{24} + q^{25} - 2 q^{26} + 2 q^{27} - 3 q^{28} - 2 q^{30} + 2 q^{32} - 2 q^{35} + q^{36} - q^{37} - q^{39} + q^{40} - q^{41} + 3 q^{42} - q^{43} - q^{45} + 3 q^{46} + q^{49} + 2 q^{50} + 2 q^{52} + 2 q^{53} - q^{54} + q^{56} + 2 q^{60} - q^{63} - q^{64} + 3 q^{65} - q^{69} + q^{70} - q^{71} - 2 q^{72} + 3 q^{74} + q^{75} - 2 q^{78} + 2 q^{81} + 3 q^{82} - q^{83} - 3 q^{84} - 2 q^{86} - 2 q^{90} - 2 q^{91} - 3 q^{92} + 2 q^{96} - q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 + q^4 - q^5 - q^6 - q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + q^12 - q^13 + 3 * q^14 - q^15 - q^18 + 2 * q^20 - q^21 - q^23 - 2 * q^24 + q^25 - 2 * q^26 + 2 * q^27 - 3 * q^28 - 2 * q^30 + 2 * q^32 - 2 * q^35 + q^36 - q^37 - q^39 + q^40 - q^41 + 3 * q^42 - q^43 - q^45 + 3 * q^46 + q^49 + 2 * q^50 + 2 * q^52 + 2 * q^53 - q^54 + q^56 + 2 * q^60 - q^63 - q^64 + 3 * q^65 - q^69 + q^70 - q^71 - 2 * q^72 + 3 * q^74 + q^75 - 2 * q^78 + 2 * q^81 + 3 * q^82 - q^83 - 3 * q^84 - 2 * q^86 - 2 * q^90 - 2 * q^91 - 3 * q^92 + 2 * q^96 - q^97 - 3 * q^98

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