LMFDB - Embedded newform 296.1.h.b.147.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [296,1,Mod(147,296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("296.147"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 296 = 2^{3} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	296.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.147723243739$
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 5.1.87616.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.87616.1

Embedding invariants

Embedding label			147.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	296.147

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.61803 q^{5} +0.618034 q^{6} +1.00000 q^{8} -0.618034 q^{9} -1.61803 q^{10} -1.61803 q^{11} +0.618034 q^{12} +0.618034 q^{13} -1.00000 q^{15} +1.00000 q^{16} -0.618034 q^{18} -1.61803 q^{20} -1.61803 q^{22} +0.618034 q^{23} +0.618034 q^{24} +1.61803 q^{25} +0.618034 q^{26} -1.00000 q^{27} +0.618034 q^{29} -1.00000 q^{30} -1.61803 q^{31} +1.00000 q^{32} -1.00000 q^{33} -0.618034 q^{36} +1.00000 q^{37} +0.381966 q^{39} -1.61803 q^{40} +0.618034 q^{41} -1.61803 q^{44} +1.00000 q^{45} +0.618034 q^{46} +0.618034 q^{48} +1.00000 q^{49} +1.61803 q^{50} +0.618034 q^{52} -1.00000 q^{54} +2.61803 q^{55} +0.618034 q^{58} -1.00000 q^{60} -1.61803 q^{61} -1.61803 q^{62} +1.00000 q^{64} -1.00000 q^{65} -1.00000 q^{66} -1.61803 q^{67} +0.381966 q^{69} -0.618034 q^{72} -1.61803 q^{73} +1.00000 q^{74} +1.00000 q^{75} +0.381966 q^{78} +0.618034 q^{79} -1.61803 q^{80} +0.618034 q^{82} +2.00000 q^{83} +0.381966 q^{87} -1.61803 q^{88} +1.00000 q^{90} +0.618034 q^{92} -1.00000 q^{93} +0.618034 q^{96} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + 2 q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 2 q^{16} + q^{18} - q^{20} - q^{22} - q^{23} - q^{24} + q^{25} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^5 - q^6 + 2 * q^8 + q^9 - q^10 - q^11 - q^12 - q^13 - 2 * q^15 + 2 * q^16 + q^18 - q^20 - q^22 - q^23 - q^24 + q^25 - q^26 - 2 * q^27 - q^29 - 2 * q^30 - q^31 + 2 * q^32 - 2 * q^33 + q^36 + 2 * q^37 + 3 * q^39 - q^40 - q^41 - q^44 + 2 * q^45 - q^46 - q^48 + 2 * q^49 + q^50 - q^52 - 2 * q^54 + 3 * q^55 - q^58 - 2 * q^60 - q^61 - q^62 + 2 * q^64 - 2 * q^65 - 2 * q^66 - q^67 + 3 * q^69 + q^72 - q^73 + 2 * q^74 + 2 * q^75 + 3 * q^78 - q^79 - q^80 - q^82 + 4 * q^83 + 3 * q^87 - q^88 + 2 * q^90 - q^92 - 2 * q^93 - q^96 + 2 * q^98 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/296\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$-1$	$-1$	$-1$

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	296.1.h.b.147.2	yes	2
3.2	odd	2		2664.1.m.a.739.2		2
4.3	odd	2		1184.1.h.a.591.1		2
8.3	odd	2		296.1.h.a.147.2	✓	2
8.5	even	2		1184.1.h.b.591.1		2
24.11	even	2		2664.1.m.b.739.1		2
37.36	even	2		296.1.h.a.147.2	✓	2
111.110	odd	2		2664.1.m.b.739.1		2
148.147	odd	2		1184.1.h.b.591.1		2
296.147	odd	2	CM	296.1.h.b.147.2	yes	2
296.221	even	2		1184.1.h.a.591.1		2
888.443	even	2		2664.1.m.a.739.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.1.h.a.147.2	✓	2	8.3	odd	2
296.1.h.a.147.2	✓	2	37.36	even	2
296.1.h.b.147.2	yes	2	1.1	even	1	trivial
296.1.h.b.147.2	yes	2	296.147	odd	2	CM
1184.1.h.a.591.1		2	4.3	odd	2
1184.1.h.a.591.1		2	296.221	even	2
1184.1.h.b.591.1		2	8.5	even	2
1184.1.h.b.591.1		2	148.147	odd	2
2664.1.m.a.739.2		2	3.2	odd	2
2664.1.m.a.739.2		2	888.443	even	2
2664.1.m.b.739.1		2	24.11	even	2
2664.1.m.b.739.1		2	111.110	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 296 = 2^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	296.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -1.61803 q^{5} +0.618034 q^{6} +1.00000 q^{8} -0.618034 q^{9} -1.61803 q^{10} -1.61803 q^{11} +0.618034 q^{12} +0.618034 q^{13} -1.00000 q^{15} +1.00000 q^{16} -0.618034 q^{18} -1.61803 q^{20} -1.61803 q^{22} +0.618034 q^{23} +0.618034 q^{24} +1.61803 q^{25} +0.618034 q^{26} -1.00000 q^{27} +0.618034 q^{29} -1.00000 q^{30} -1.61803 q^{31} +1.00000 q^{32} -1.00000 q^{33} -0.618034 q^{36} +1.00000 q^{37} +0.381966 q^{39} -1.61803 q^{40} +0.618034 q^{41} -1.61803 q^{44} +1.00000 q^{45} +0.618034 q^{46} +0.618034 q^{48} +1.00000 q^{49} +1.61803 q^{50} +0.618034 q^{52} -1.00000 q^{54} +2.61803 q^{55} +0.618034 q^{58} -1.00000 q^{60} -1.61803 q^{61} -1.61803 q^{62} +1.00000 q^{64} -1.00000 q^{65} -1.00000 q^{66} -1.61803 q^{67} +0.381966 q^{69} -0.618034 q^{72} -1.61803 q^{73} +1.00000 q^{74} +1.00000 q^{75} +0.381966 q^{78} +0.618034 q^{79} -1.61803 q^{80} +0.618034 q^{82} +2.00000 q^{83} +0.381966 q^{87} -1.61803 q^{88} +1.00000 q^{90} +0.618034 q^{92} -1.00000 q^{93} +0.618034 q^{96} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + 2 q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 2 q^{16} + q^{18} - q^{20} - q^{22} - q^{23} - q^{24} + q^{25} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^5 - q^6 + 2 * q^8 + q^9 - q^10 - q^11 - q^12 - q^13 - 2 * q^15 + 2 * q^16 + q^18 - q^20 - q^22 - q^23 - q^24 + q^25 - q^26 - 2 * q^27 - q^29 - 2 * q^30 - q^31 + 2 * q^32 - 2 * q^33 + q^36 + 2 * q^37 + 3 * q^39 - q^40 - q^41 - q^44 + 2 * q^45 - q^46 - q^48 + 2 * q^49 + q^50 - q^52 - 2 * q^54 + 3 * q^55 - q^58 - 2 * q^60 - q^61 - q^62 + 2 * q^64 - 2 * q^65 - 2 * q^66 - q^67 + 3 * q^69 + q^72 - q^73 + 2 * q^74 + 2 * q^75 + 3 * q^78 - q^79 - q^80 - q^82 + 4 * q^83 + 3 * q^87 - q^88 + 2 * q^90 - q^92 - 2 * q^93 - q^96 + 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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