LMFDB - Embedded newform 920.1.p.c.229.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,1,Mod(229,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("920.229");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	920.p (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:	$0.459139811622$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.846400.1

Embedding invariants

Embedding label			229.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	920.229

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.61803 q^{7} +1.00000 q^{8} -0.618034 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.61803 q^{7} +1.00000 q^{8} -0.618034 q^{9} -1.00000 q^{10} -0.618034 q^{11} +0.618034 q^{12} -1.61803 q^{13} +1.61803 q^{14} -0.618034 q^{15} +1.00000 q^{16} -0.618034 q^{17} -0.618034 q^{18} +1.61803 q^{19} -1.00000 q^{20} +1.00000 q^{21} -0.618034 q^{22} -1.00000 q^{23} +0.618034 q^{24} +1.00000 q^{25} -1.61803 q^{26} -1.00000 q^{27} +1.61803 q^{28} -0.618034 q^{30} -1.61803 q^{31} +1.00000 q^{32} -0.381966 q^{33} -0.618034 q^{34} -1.61803 q^{35} -0.618034 q^{36} +1.61803 q^{38} -1.00000 q^{39} -1.00000 q^{40} +0.618034 q^{41} +1.00000 q^{42} -0.618034 q^{44} +0.618034 q^{45} -1.00000 q^{46} +0.618034 q^{48} +1.61803 q^{49} +1.00000 q^{50} -0.381966 q^{51} -1.61803 q^{52} -1.00000 q^{54} +0.618034 q^{55} +1.61803 q^{56} +1.00000 q^{57} -0.618034 q^{60} -0.618034 q^{61} -1.61803 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.61803 q^{65} -0.381966 q^{66} -0.618034 q^{68} -0.618034 q^{69} -1.61803 q^{70} +0.618034 q^{71} -0.618034 q^{72} +0.618034 q^{75} +1.61803 q^{76} -1.00000 q^{77} -1.00000 q^{78} -1.00000 q^{80} +0.618034 q^{82} +1.00000 q^{84} +0.618034 q^{85} -0.618034 q^{88} +0.618034 q^{90} -2.61803 q^{91} -1.00000 q^{92} -1.00000 q^{93} -1.61803 q^{95} +0.618034 q^{96} -0.618034 q^{97} +1.61803 q^{98} +0.381966 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 + q^7 + 2 * q^8 + q^9	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} - q^{13} + q^{14} + q^{15} + 2 q^{16} + q^{17} + q^{18} + q^{19} - 2 q^{20} + 2 q^{21} + q^{22} - 2 q^{23} - q^{24} + 2 q^{25} - q^{26} - 2 q^{27} + q^{28} + q^{30} - q^{31} + 2 q^{32} - 3 q^{33} + q^{34} - q^{35} + q^{36} + q^{38} - 2 q^{39} - 2 q^{40} - q^{41} + 2 q^{42} + q^{44} - q^{45} - 2 q^{46} - q^{48} + q^{49} + 2 q^{50} - 3 q^{51} - q^{52} - 2 q^{54} - q^{55} + q^{56} + 2 q^{57} + q^{60} + q^{61} - q^{62} - 2 q^{63} + 2 q^{64} + q^{65} - 3 q^{66} + q^{68} + q^{69} - q^{70} - q^{71} + q^{72} - q^{75} + q^{76} - 2 q^{77} - 2 q^{78} - 2 q^{80} - q^{82} + 2 q^{84} - q^{85} + q^{88} - q^{90} - 3 q^{91} - 2 q^{92} - 2 q^{93} - q^{95} - q^{96} + q^{97} + q^{98} + 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 + q^7 + 2 * q^8 + q^9 - 2 * q^10 + q^11 - q^12 - q^13 + q^14 + q^15 + 2 * q^16 + q^17 + q^18 + q^19 - 2 * q^20 + 2 * q^21 + q^22 - 2 * q^23 - q^24 + 2 * q^25 - q^26 - 2 * q^27 + q^28 + q^30 - q^31 + 2 * q^32 - 3 * q^33 + q^34 - q^35 + q^36 + q^38 - 2 * q^39 - 2 * q^40 - q^41 + 2 * q^42 + q^44 - q^45 - 2 * q^46 - q^48 + q^49 + 2 * q^50 - 3 * q^51 - q^52 - 2 * q^54 - q^55 + q^56 + 2 * q^57 + q^60 + q^61 - q^62 - 2 * q^63 + 2 * q^64 + q^65 - 3 * q^66 + q^68 + q^69 - q^70 - q^71 + q^72 - q^75 + q^76 - 2 * q^77 - 2 * q^78 - 2 * q^80 - q^82 + 2 * q^84 - q^85 + q^88 - q^90 - 3 * q^91 - 2 * q^92 - 2 * q^93 - q^95 - q^96 + q^97 + q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$	1.00000	1.00000
$2$	1.00000	1.00000
$3$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$3$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$4$	1.00000	1.00000
$5$	−1.00000	−1.00000
$5$	−1.00000	−1.00000
$6$	0.618034	0.618034
$7$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$7$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$8$	1.00000	1.00000
$9$	−0.618034	−0.618034
$10$	−1.00000	−1.00000
$11$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$11$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$12$	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	$0.800000\pi$
$14$	1.61803	1.61803
$15$	−0.618034	−0.618034
$16$	1.00000	1.00000
$17$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$17$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$18$	−0.618034	−0.618034
$19$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$19$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$20$	−1.00000	−1.00000
$21$	1.00000	1.00000
$22$	−0.618034	−0.618034
$23$	−1.00000	−1.00000
$23$	−1.00000	−1.00000
$24$	0.618034	0.618034
$25$	1.00000	1.00000
$26$	−1.61803	−1.61803
$27$	−1.00000	−1.00000
$28$	1.61803	1.61803
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.618034	−0.618034
$31$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$31$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$32$	1.00000	1.00000
$33$	−0.381966	−0.381966
$34$	−0.618034	−0.618034
$35$	−1.61803	−1.61803
$36$	−0.618034	−0.618034
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.61803	1.61803
$39$	−1.00000	−1.00000
$40$	−1.00000	−1.00000
$41$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$41$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$42$	1.00000	1.00000
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−0.618034	−0.618034
$45$	0.618034	0.618034
$46$	−1.00000	−1.00000
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0.618034	0.618034
$49$	1.61803	1.61803
$50$	1.00000	1.00000
$51$	−0.381966	−0.381966
$52$	−1.61803	−1.61803
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.00000	−1.00000
$55$	0.618034	0.618034
$56$	1.61803	1.61803
$57$	1.00000	1.00000
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−0.618034	−0.618034
$61$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$61$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$62$	−1.61803	−1.61803
$63$	−1.00000	−1.00000
$64$	1.00000	1.00000
$65$	1.61803	1.61803
$66$	−0.381966	−0.381966
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−0.618034	−0.618034
$69$	−0.618034	−0.618034
$70$	−1.61803	−1.61803
$71$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$71$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$72$	−0.618034	−0.618034
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0.618034	0.618034
$76$	1.61803	1.61803
$77$	−1.00000	−1.00000
$78$	−1.00000	−1.00000
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.00000	−1.00000
$81$	0	0
$82$	0.618034	0.618034
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	1.00000	1.00000
$85$	0.618034	0.618034
$86$	0	0
$87$	0	0
$88$	−0.618034	−0.618034
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.618034	0.618034
$91$	−2.61803	−2.61803
$92$	−1.00000	−1.00000
$93$	−1.00000	−1.00000
$94$	0	0
$95$	−1.61803	−1.61803
$96$	0.618034	0.618034
$97$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$97$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$98$	1.61803	1.61803
$99$	0.381966	0.381966
$100$	1.00000	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−0.381966	−0.381966
$103$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$103$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$104$	−1.61803	−1.61803
$105$	−1.00000	−1.00000
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1.00000	−1.00000
$109$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$109$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$110$	0.618034	0.618034
$111$	0	0
$112$	1.61803	1.61803
$113$	−2.00000	−2.00000	−1.00000			$\pi$
$113$	−2.00000	−2.00000	−1.00000			$\pi$
$114$	1.00000	1.00000
$115$	1.00000	1.00000
$116$	0	0
$117$	1.00000	1.00000
$118$	0	0
$119$	−1.00000	−1.00000
$120$	−0.618034	−0.618034
$121$	−0.618034	−0.618034
$122$	−0.618034	−0.618034
$123$	0.381966	0.381966
$124$	−1.61803	−1.61803
$125$	−1.00000	−1.00000
$126$	−1.00000	−1.00000
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	0	0
$130$	1.61803	1.61803
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−0.381966	−0.381966
$133$	2.61803	2.61803
$134$	0	0
$135$	1.00000	1.00000
$136$	−0.618034	−0.618034
$137$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$137$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$138$	−0.618034	−0.618034
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−1.61803	−1.61803
$141$	0	0
$142$	0.618034	0.618034
$143$	1.00000	1.00000
$144$	−0.618034	−0.618034
$145$	0	0
$146$	0	0
$147$	1.00000	1.00000
$148$	0	0
$149$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$149$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$150$	0.618034	0.618034
$151$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$151$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$152$	1.61803	1.61803
$153$	0.381966	0.381966
$154$	−1.00000	−1.00000
$155$	1.61803	1.61803
$156$	−1.00000	−1.00000
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−1.00000	−1.00000
$161$	−1.61803	−1.61803
$162$	0	0
$163$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$163$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$164$	0.618034	0.618034
$165$	0.381966	0.381966
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.00000	1.00000
$169$	1.61803	1.61803
$170$	0.618034	0.618034
$171$	−1.00000	−1.00000
$172$	0	0
$173$	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
$175$	1.61803	1.61803
$176$	−0.618034	−0.618034
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0.618034	0.618034
$181$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$181$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$182$	−2.61803	−2.61803
$183$	−0.381966	−0.381966
$184$	−1.00000	−1.00000
$185$	0	0
$186$	−1.00000	−1.00000
$187$	0.381966	0.381966
$188$	0	0
$189$	−1.61803	−1.61803
$190$	−1.61803	−1.61803
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0.618034	0.618034
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	−0.618034	−0.618034
$195$	1.00000	1.00000
$196$	1.61803	1.61803
$197$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$197$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$198$	0.381966	0.381966
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.00000	1.00000
$201$	0	0
$202$	0	0
$203$	0	0
$204$	−0.381966	−0.381966
$205$	−0.618034	−0.618034
$206$	−0.618034	−0.618034
$207$	0.618034	0.618034
$208$	−1.61803	−1.61803
$209$	−1.00000	−1.00000
$210$	−1.00000	−1.00000
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0.381966	0.381966
$214$	0	0
$215$	0	0
$216$	−1.00000	−1.00000
$217$	−2.61803	−2.61803
$218$	1.61803	1.61803
$219$	0	0
$220$	0.618034	0.618034
$221$	1.00000	1.00000
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.61803	1.61803
$225$	−0.618034	−0.618034
$226$	−2.00000	−2.00000
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.00000	1.00000
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$230$	1.00000	1.00000
$231$	−0.618034	−0.618034
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	1.00000	1.00000
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−1.00000	−1.00000
$239$	2.00000	2.00000	1.00000			$0$
$239$	2.00000	2.00000	1.00000			$0$
$240$	−0.618034	−0.618034
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−0.618034	−0.618034
$243$	1.00000	1.00000
$244$	−0.618034	−0.618034
$245$	−1.61803	−1.61803
$246$	0.381966	0.381966
$247$	−2.61803	−2.61803
$248$	−1.61803	−1.61803
$249$	0	0
$250$	−1.00000	−1.00000
$251$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$251$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$252$	−1.00000	−1.00000
$253$	0.618034	0.618034
$254$	0	0
$255$	0.381966	0.381966
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	1.61803	1.61803
$261$	0	0
$262$	0	0
$263$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$263$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$264$	−0.381966	−0.381966
$265$	0	0
$266$	2.61803	2.61803
$267$	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	$0.800000\pi$
$272$	−0.618034	−0.618034
$273$	−1.61803	−1.61803
$274$	1.61803	1.61803
$275$	−0.618034	−0.618034
$276$	−0.618034	−0.618034
$277$	2.00000	2.00000	1.00000			$0$
$277$	2.00000	2.00000	1.00000			$0$
$278$	0	0
$279$	1.00000	1.00000
$280$	−1.61803	−1.61803
$281$	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			$\pi$
$284$	0.618034	0.618034
$285$	−1.00000	−1.00000
$286$	1.00000	1.00000
$287$	1.00000	1.00000
$288$	−0.618034	−0.618034
$289$	−0.618034	−0.618034
$290$	0	0
$291$	−0.381966	−0.381966
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	1.00000	1.00000
$295$	0	0
$296$	0	0
$297$	0.618034	0.618034
$298$	1.61803	1.61803
$299$	1.61803	1.61803
$300$	0.618034	0.618034
$301$	0	0
$302$	0.618034	0.618034
$303$	0	0
$304$	1.61803	1.61803
$305$	0.618034	0.618034
$306$	0.381966	0.381966
$307$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$308$	−1.00000	−1.00000
$309$	−0.381966	−0.381966
$310$	1.61803	1.61803
$311$	2.00000	2.00000	1.00000			$0$
$311$	2.00000	2.00000	1.00000			$0$
$312$	−1.00000	−1.00000
$313$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$313$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$314$	0	0
$315$	1.00000	1.00000
$316$	0	0
$317$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$317$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−1.00000
$321$	0	0
$322$	−1.61803	−1.61803
$323$	−1.00000	−1.00000
$324$	0	0
$325$	−1.61803	−1.61803
$326$	−1.61803	−1.61803
$327$	1.00000	1.00000
$328$	0.618034	0.618034
$329$	0	0
$330$	0.381966	0.381966
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	1.00000	1.00000
$337$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$337$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$338$	1.61803	1.61803
$339$	−1.23607	−1.23607
$340$	0.618034	0.618034
$341$	1.00000	1.00000
$342$	−1.00000	−1.00000
$343$	1.00000	1.00000
$344$	0	0
$345$	0.618034	0.618034
$346$	0.618034	0.618034
$347$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$347$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.61803	1.61803
$351$	1.61803	1.61803
$352$	−0.618034	−0.618034
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−0.618034	−0.618034
$356$	0	0
$357$	−0.618034	−0.618034
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.618034	0.618034
$361$	1.61803	1.61803
$362$	1.61803	1.61803
$363$	−0.381966	−0.381966
$364$	−2.61803	−2.61803
$365$	0	0
$366$	−0.381966	−0.381966
$367$	−2.00000	−2.00000	−1.00000			$\pi$
$367$	−2.00000	−2.00000	−1.00000			$\pi$
$368$	−1.00000	−1.00000
$369$	−0.381966	−0.381966
$370$	0	0
$371$	0	0
$372$	−1.00000	−1.00000
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0.381966	0.381966
$375$	−0.618034	−0.618034
$376$	0	0
$377$	0	0
$378$	−1.61803	−1.61803
$379$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$379$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$380$	−1.61803	−1.61803
$381$	0	0
$382$	0	0
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$384$	0.618034	0.618034
$385$	1.00000	1.00000
$386$	0	0
$387$	0	0
$388$	−0.618034	−0.618034
$389$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$389$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$390$	1.00000	1.00000
$391$	0.618034	0.618034
$392$	1.61803	1.61803
$393$	0	0
$394$	0.618034	0.618034
$395$	0	0
$396$	0.381966	0.381966
$397$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$397$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$398$	0	0
$399$	1.61803	1.61803
$400$	1.00000	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	2.61803	2.61803
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−0.381966	−0.381966
$409$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$409$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$410$	−0.618034	−0.618034
$411$	1.00000	1.00000
$412$	−0.618034	−0.618034
$413$	0	0
$414$	0.618034	0.618034
$415$	0	0
$416$	−1.61803	−1.61803
$417$	0	0
$418$	−1.00000	−1.00000
$419$	−2.00000	−2.00000	−1.00000			$\pi$
$419$	−2.00000	−2.00000	−1.00000			$\pi$
$420$	−1.00000	−1.00000
$421$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$421$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.618034	−0.618034
$426$	0.381966	0.381966
$427$	−1.00000	−1.00000
$428$	0	0
$429$	0.618034	0.618034
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.00000	−1.00000
$433$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$433$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$434$	−2.61803	−2.61803
$435$	0	0
$436$	1.61803	1.61803
$437$	−1.61803	−1.61803
$438$	0	0
$439$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$439$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$440$	0.618034	0.618034
$441$	−1.00000	−1.00000
$442$	1.00000	1.00000
$443$	−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	0
$445$	0	0
$446$	0	0
$447$	1.00000	1.00000
$448$	1.61803	1.61803
$449$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$449$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$450$	−0.618034	−0.618034
$451$	−0.381966	−0.381966
$452$	−2.00000	−2.00000
$453$	0.381966	0.381966
$454$	0	0
$455$	2.61803	2.61803
$456$	1.00000	1.00000
$457$	−2.00000	−2.00000	−1.00000			$\pi$
$457$	−2.00000	−2.00000	−1.00000			$\pi$
$458$	−2.00000	−2.00000
$459$	0.618034	0.618034
$460$	1.00000	1.00000
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	−0.618034	−0.618034
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	1.00000	1.00000
$466$	0	0
$467$	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
$475$	1.61803	1.61803
$476$	−1.00000	−1.00000
$477$	0	0
$478$	2.00000	2.00000
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.618034	−0.618034
$481$	0	0
$482$	0	0
$483$	−1.00000	−1.00000
$484$	−0.618034	−0.618034
$485$	0.618034	0.618034
$486$	1.00000	1.00000
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.618034	−0.618034
$489$	−1.00000	−1.00000
$490$	−1.61803	−1.61803
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0.381966	0.381966
$493$	0	0
$494$	−2.61803	−2.61803
$495$	−0.381966	−0.381966
$496$	−1.61803	−1.61803
$497$	1.00000	1.00000
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.00000	−1.00000
$501$	0	0
$502$	1.61803	1.61803
$503$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$503$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$504$	−1.00000	−1.00000
$505$	0	0
$506$	0.618034	0.618034
$507$	1.00000	1.00000
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0.381966	0.381966
$511$	0	0
$512$	1.00000	1.00000
$513$	−1.61803	−1.61803
$514$	0	0
$515$	0.618034	0.618034
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0.381966	0.381966
$520$	1.61803	1.61803
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	1.00000	1.00000
$526$	−0.618034	−0.618034
$527$	1.00000	1.00000
$528$	−0.381966	−0.381966
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	2.61803	2.61803
$533$	−1.00000	−1.00000
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.00000	−1.00000
$540$	1.00000	1.00000
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.61803	−1.61803
$543$	1.00000	1.00000
$544$	−0.618034	−0.618034
$545$	−1.61803	−1.61803
$546$	−1.61803	−1.61803
$547$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$547$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$548$	1.61803	1.61803
$549$	0.381966	0.381966
$550$	−0.618034	−0.618034
$551$	0	0
$552$	−0.618034	−0.618034
$553$	0	0
$554$	2.00000	2.00000
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.00000	1.00000
$559$	0	0
$560$	−1.61803	−1.61803
$561$	0.236068	0.236068
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	2.00000	2.00000
$566$	0	0
$567$	0	0
$568$	0.618034	0.618034
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	−1.00000	−1.00000
$571$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$571$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$572$	1.00000	1.00000
$573$	0	0
$574$	1.00000	1.00000
$575$	−1.00000	−1.00000
$576$	−0.618034	−0.618034
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−0.618034	−0.618034
$579$	0	0
$580$	0	0
$581$	0	0
$582$	−0.381966	−0.381966
$583$	0	0
$584$	0	0
$585$	−1.00000	−1.00000
$586$	0	0
$587$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$587$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$588$	1.00000	1.00000
$589$	−2.61803	−2.61803
$590$	0	0
$591$	0.381966	0.381966
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0.618034	0.618034
$595$	1.00000	1.00000
$596$	1.61803	1.61803
$597$	0	0
$598$	1.61803	1.61803
$599$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$599$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$600$	0.618034	0.618034
$601$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$601$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$602$	0	0
$603$	0	0
$604$	0.618034	0.618034
$605$	0.618034	0.618034
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.61803	1.61803
$609$	0	0
$610$	0.618034	0.618034
$611$	0	0
$612$	0.381966	0.381966
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0.618034	0.618034
$615$	−0.381966	−0.381966
$616$	−1.00000	−1.00000
$617$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$617$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$618$	−0.381966	−0.381966
$619$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$619$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$620$	1.61803	1.61803
$621$	1.00000	1.00000
$622$	2.00000	2.00000
$623$	0	0
$624$	−1.00000	−1.00000
$625$	1.00000	1.00000
$626$	1.61803	1.61803
$627$	−0.618034	−0.618034
$628$	0	0
$629$	0	0
$630$	1.00000	1.00000
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−1.61803	−1.61803
$635$	0	0
$636$	0	0
$637$	−2.61803	−2.61803
$638$	0	0
$639$	−0.381966	−0.381966
$640$	−1.00000	−1.00000
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−1.61803	−1.61803
$645$	0	0
$646$	−1.00000	−1.00000
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	−1.61803	−1.61803
$651$	−1.61803	−1.61803
$652$	−1.61803	−1.61803
$653$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$653$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$654$	1.00000	1.00000
$655$	0	0
$656$	0.618034	0.618034
$657$	0	0
$658$	0	0
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$659$	−2.00000	−2.00000	−1.00000			$\pi$
$660$	0.381966	0.381966
$661$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$661$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$662$	0	0
$663$	0.618034	0.618034
$664$	0	0
$665$	−2.61803	−2.61803
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.381966	0.381966
$672$	1.00000	1.00000
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	1.61803	1.61803
$675$	−1.00000	−1.00000
$676$	1.61803	1.61803
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−1.23607	−1.23607
$679$	−1.00000	−1.00000
$680$	0.618034	0.618034
$681$	0	0
$682$	1.00000	1.00000
$683$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$683$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$684$	−1.00000	−1.00000
$685$	−1.61803	−1.61803
$686$	1.00000	1.00000
$687$	−1.23607	−1.23607
$688$	0	0
$689$	0	0
$690$	0.618034	0.618034
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0.618034	0.618034
$693$	0.618034	0.618034
$694$	0.618034	0.618034
$695$	0	0
$696$	0	0
$697$	−0.381966	−0.381966
$698$	0	0
$699$	0	0
$700$	1.61803	1.61803
$701$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$701$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$702$	1.61803	1.61803
$703$	0	0
$704$	−0.618034	−0.618034
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$709$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$710$	−0.618034	−0.618034
$711$	0	0
$712$	0	0
$713$	1.61803	1.61803
$714$	−0.618034	−0.618034
$715$	−1.00000	−1.00000
$716$	0	0
$717$	1.23607	1.23607
$718$	0	0
$719$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$719$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$720$	0.618034	0.618034
$721$	−1.00000	−1.00000
$722$	1.61803	1.61803
$723$	0	0
$724$	1.61803	1.61803
$725$	0	0
$726$	−0.381966	−0.381966
$727$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$727$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$728$	−2.61803	−2.61803
$729$	0.618034	0.618034
$730$	0	0
$731$	0	0
$732$	−0.381966	−0.381966
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−2.00000	−2.00000
$735$	−1.00000	−1.00000
$736$	−1.00000	−1.00000
$737$	0	0
$738$	−0.381966	−0.381966
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	−1.61803	−1.61803
$742$	0	0
$743$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$743$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$744$	−1.00000	−1.00000
$745$	−1.61803	−1.61803
$746$	0	0
$747$	0	0
$748$	0.381966	0.381966
$749$	0	0
$750$	−0.618034	−0.618034
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1.00000	1.00000
$754$	0	0
$755$	−0.618034	−0.618034
$756$	−1.61803	−1.61803
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−0.618034	−0.618034
$759$	0.381966	0.381966
$760$	−1.61803	−1.61803
$761$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$761$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$762$	0	0
$763$	2.61803	2.61803
$764$	0	0
$765$	−0.381966	−0.381966
$766$	−2.00000	−2.00000
$767$	0	0
$768$	0.618034	0.618034
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	1.00000	1.00000
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−1.61803	−1.61803
$776$	−0.618034	−0.618034
$777$	0	0
$778$	−0.618034	−0.618034
$779$	1.00000	1.00000
$780$	1.00000	1.00000
$781$	−0.381966	−0.381966
$782$	0.618034	0.618034
$783$	0	0
$784$	1.61803	1.61803
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0.618034	0.618034
$789$	−0.381966	−0.381966
$790$	0	0
$791$	−3.23607	−3.23607
$792$	0.381966	0.381966
$793$	1.00000	1.00000
$794$	0.618034	0.618034
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	1.61803	1.61803
$799$	0	0
$800$	1.00000	1.00000
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.61803	1.61803
$806$	2.61803	2.61803
$807$	0	0
$808$	0	0
$809$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$809$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−1.00000	−1.00000
$814$	0	0
$815$	1.61803	1.61803
$816$	−0.381966	−0.381966
$817$	0	0
$818$	−1.61803	−1.61803
$819$	1.61803	1.61803
$820$	−0.618034	−0.618034
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	1.00000	1.00000
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	−0.618034	−0.618034
$825$	−0.381966	−0.381966
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0.618034	0.618034
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	1.23607	1.23607
$832$	−1.61803	−1.61803
$833$	−1.00000	−1.00000
$834$	0	0
$835$	0	0
$836$	−1.00000	−1.00000
$837$	1.61803	1.61803
$838$	−2.00000	−2.00000
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−1.00000	−1.00000
$841$	1.00000	1.00000
$842$	1.61803	1.61803
$843$	0	0
$844$	0	0
$845$	−1.61803	−1.61803
$846$	0	0
$847$	−1.00000	−1.00000
$848$	0	0
$849$	0	0
$850$	−0.618034	−0.618034
$851$	0	0
$852$	0.381966	0.381966
$853$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$853$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$854$	−1.00000	−1.00000
$855$	1.00000	1.00000
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0.618034	0.618034
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0.618034	0.618034
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.00000	−1.00000
$865$	−0.618034	−0.618034
$866$	1.61803	1.61803
$867$	−0.381966	−0.381966
$868$	−2.61803	−2.61803
$869$	0	0
$870$	0	0
$871$	0	0
$872$	1.61803	1.61803
$873$	0.381966	0.381966
$874$	−1.61803	−1.61803
$875$	−1.61803	−1.61803
$876$	0	0
$877$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$877$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$878$	−1.61803	−1.61803
$879$	0	0
$880$	0.618034	0.618034
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1.00000	−1.00000
$883$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$883$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$884$	1.00000	1.00000
$885$	0	0
$886$	−1.61803	−1.61803
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	1.00000	1.00000
$895$	0	0
$896$	1.61803	1.61803
$897$	1.00000	1.00000
$898$	−1.61803	−1.61803
$899$	0	0
$900$	−0.618034	−0.618034
$901$	0	0
$902$	−0.381966	−0.381966
$903$	0	0
$904$	−2.00000	−2.00000
$905$	−1.61803	−1.61803
$906$	0.381966	0.381966
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	2.61803	2.61803
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.00000	1.00000
$913$	0	0
$914$	−2.00000	−2.00000
$915$	0.381966	0.381966
$916$	−2.00000	−2.00000
$917$	0	0
$918$	0.618034	0.618034
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	1.00000	1.00000
$921$	0.381966	0.381966
$922$	0	0
$923$	−1.00000	−1.00000
$924$	−0.618034	−0.618034
$925$	0	0
$926$	0	0
$927$	0.381966	0.381966
$928$	0	0
$929$	2.00000	2.00000	1.00000			$0$
$929$	2.00000	2.00000	1.00000			$0$
$930$	1.00000	1.00000
$931$	2.61803	2.61803
$932$	0	0
$933$	1.23607	1.23607
$934$	0	0
$935$	−0.381966	−0.381966
$936$	1.00000	1.00000
$937$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$937$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$938$	0	0
$939$	1.00000	1.00000
$940$	0	0
$941$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$941$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$942$	0	0
$943$	−0.618034	−0.618034
$944$	0	0
$945$	1.61803	1.61803
$946$	0	0
$947$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$947$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$948$	0	0
$949$	0	0
$950$	1.61803	1.61803
$951$	−1.00000	−1.00000
$952$	−1.00000	−1.00000
$953$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$953$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$954$	0	0
$955$	0	0
$956$	2.00000	2.00000
$957$	0	0
$958$	0	0
$959$	2.61803	2.61803
$960$	−0.618034	−0.618034
$961$	1.61803	1.61803
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	−1.00000	−1.00000
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.618034	−0.618034
$969$	−0.618034	−0.618034
$970$	0.618034	0.618034
$971$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$971$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$972$	1.00000	1.00000
$973$	0	0
$974$	0	0
$975$	−1.00000	−1.00000
$976$	−0.618034	−0.618034
$977$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$977$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$978$	−1.00000	−1.00000
$979$	0	0
$980$	−1.61803	−1.61803
$981$	−1.00000	−1.00000
$982$	0	0
$983$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$983$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$984$	0.381966	0.381966
$985$	−0.618034	−0.618034
$986$	0	0
$987$	0	0
$988$	−2.61803	−2.61803
$989$	0	0
$990$	−0.381966	−0.381966
$991$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$991$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$992$	−1.61803	−1.61803
$993$	0	0
$994$	1.00000	1.00000
$995$	0	0
$996$	0	0
$997$	2.00000	2.00000	1.00000			$0$
$997$	2.00000	2.00000	1.00000			$0$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.1.p.c.229.2	yes	2
4.3	odd	2		3680.1.p.c.689.1		2
5.4	even	2		920.1.p.a.229.1	✓	2
8.3	odd	2		3680.1.p.b.689.2		2
8.5	even	2		920.1.p.b.229.1	yes	2
20.19	odd	2		3680.1.p.a.689.2		2
23.22	odd	2		920.1.p.d.229.2	yes	2
40.19	odd	2		3680.1.p.d.689.1		2
40.29	even	2		920.1.p.d.229.2	yes	2
92.91	even	2		3680.1.p.d.689.1		2
115.114	odd	2		920.1.p.b.229.1	yes	2
184.45	odd	2		920.1.p.a.229.1	✓	2
184.91	even	2		3680.1.p.a.689.2		2
460.459	even	2		3680.1.p.b.689.2		2
920.229	odd	2	CM	920.1.p.c.229.2	yes	2
920.459	even	2		3680.1.p.c.689.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.1.p.a.229.1	✓	2	5.4	even	2
920.1.p.a.229.1	✓	2	184.45	odd	2
920.1.p.b.229.1	yes	2	8.5	even	2
920.1.p.b.229.1	yes	2	115.114	odd	2
920.1.p.c.229.2	yes	2	1.1	even	1	trivial
920.1.p.c.229.2	yes	2	920.229	odd	2	CM
920.1.p.d.229.2	yes	2	23.22	odd	2
920.1.p.d.229.2	yes	2	40.29	even	2
3680.1.p.a.689.2		2	20.19	odd	2
3680.1.p.a.689.2		2	184.91	even	2
3680.1.p.b.689.2		2	8.3	odd	2
3680.1.p.b.689.2		2	460.459	even	2
3680.1.p.c.689.1		2	4.3	odd	2
3680.1.p.c.689.1		2	920.459	even	2
3680.1.p.d.689.1		2	40.19	odd	2
3680.1.p.d.689.1		2	92.91	even	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	920.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.61803 q^{7} +1.00000 q^{8} -0.618034 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.61803 q^{7} +1.00000 q^{8} -0.618034 q^{9} -1.00000 q^{10} -0.618034 q^{11} +0.618034 q^{12} -1.61803 q^{13} +1.61803 q^{14} -0.618034 q^{15} +1.00000 q^{16} -0.618034 q^{17} -0.618034 q^{18} +1.61803 q^{19} -1.00000 q^{20} +1.00000 q^{21} -0.618034 q^{22} -1.00000 q^{23} +0.618034 q^{24} +1.00000 q^{25} -1.61803 q^{26} -1.00000 q^{27} +1.61803 q^{28} -0.618034 q^{30} -1.61803 q^{31} +1.00000 q^{32} -0.381966 q^{33} -0.618034 q^{34} -1.61803 q^{35} -0.618034 q^{36} +1.61803 q^{38} -1.00000 q^{39} -1.00000 q^{40} +0.618034 q^{41} +1.00000 q^{42} -0.618034 q^{44} +0.618034 q^{45} -1.00000 q^{46} +0.618034 q^{48} +1.61803 q^{49} +1.00000 q^{50} -0.381966 q^{51} -1.61803 q^{52} -1.00000 q^{54} +0.618034 q^{55} +1.61803 q^{56} +1.00000 q^{57} -0.618034 q^{60} -0.618034 q^{61} -1.61803 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.61803 q^{65} -0.381966 q^{66} -0.618034 q^{68} -0.618034 q^{69} -1.61803 q^{70} +0.618034 q^{71} -0.618034 q^{72} +0.618034 q^{75} +1.61803 q^{76} -1.00000 q^{77} -1.00000 q^{78} -1.00000 q^{80} +0.618034 q^{82} +1.00000 q^{84} +0.618034 q^{85} -0.618034 q^{88} +0.618034 q^{90} -2.61803 q^{91} -1.00000 q^{92} -1.00000 q^{93} -1.61803 q^{95} +0.618034 q^{96} -0.618034 q^{97} +1.61803 q^{98} +0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 + q^7 + 2 * q^8 + q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} - q^{13} + q^{14} + q^{15} + 2 q^{16} + q^{17} + q^{18} + q^{19} - 2 q^{20} + 2 q^{21} + q^{22} - 2 q^{23} - q^{24} + 2 q^{25} - q^{26} - 2 q^{27} + q^{28} + q^{30} - q^{31} + 2 q^{32} - 3 q^{33} + q^{34} - q^{35} + q^{36} + q^{38} - 2 q^{39} - 2 q^{40} - q^{41} + 2 q^{42} + q^{44} - q^{45} - 2 q^{46} - q^{48} + q^{49} + 2 q^{50} - 3 q^{51} - q^{52} - 2 q^{54} - q^{55} + q^{56} + 2 q^{57} + q^{60} + q^{61} - q^{62} - 2 q^{63} + 2 q^{64} + q^{65} - 3 q^{66} + q^{68} + q^{69} - q^{70} - q^{71} + q^{72} - q^{75} + q^{76} - 2 q^{77} - 2 q^{78} - 2 q^{80} - q^{82} + 2 q^{84} - q^{85} + q^{88} - q^{90} - 3 q^{91} - 2 q^{92} - 2 q^{93} - q^{95} - q^{96} + q^{97} + q^{98} + 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 2 * q^5 - q^6 + q^7 + 2 * q^8 + q^9 - 2 * q^10 + q^11 - q^12 - q^13 + q^14 + q^15 + 2 * q^16 + q^17 + q^18 + q^19 - 2 * q^20 + 2 * q^21 + q^22 - 2 * q^23 - q^24 + 2 * q^25 - q^26 - 2 * q^27 + q^28 + q^30 - q^31 + 2 * q^32 - 3 * q^33 + q^34 - q^35 + q^36 + q^38 - 2 * q^39 - 2 * q^40 - q^41 + 2 * q^42 + q^44 - q^45 - 2 * q^46 - q^48 + q^49 + 2 * q^50 - 3 * q^51 - q^52 - 2 * q^54 - q^55 + q^56 + 2 * q^57 + q^60 + q^61 - q^62 - 2 * q^63 + 2 * q^64 + q^65 - 3 * q^66 + q^68 + q^69 - q^70 - q^71 + q^72 - q^75 + q^76 - 2 * q^77 - 2 * q^78 - 2 * q^80 - q^82 + 2 * q^84 - q^85 + q^88 - q^90 - 3 * q^91 - 2 * q^92 - 2 * q^93 - q^95 - q^96 + q^97 + q^98 + 3 * q^99

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