LMFDB - Embedded newform 2904.1.n.a.485.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2904,1,Mod(485,2904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2904, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))

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

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

chi := DirichletCharacter("2904.485");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2904 = 2^{3} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2904.n (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:	$1.44928479669$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.8433216.2

Embedding invariants

Embedding label			485.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2904.485

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.00000 q^{6} -1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.00000 q^{6} -1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.618034 q^{10} -1.00000 q^{12} +1.61803 q^{14} +0.618034 q^{15} +1.00000 q^{16} -1.00000 q^{18} -0.618034 q^{20} +1.61803 q^{21} +1.00000 q^{24} -0.618034 q^{25} -1.00000 q^{27} -1.61803 q^{28} -0.618034 q^{29} -0.618034 q^{30} -1.61803 q^{31} -1.00000 q^{32} +1.00000 q^{35} +1.00000 q^{36} +0.618034 q^{40} -1.61803 q^{42} -0.618034 q^{45} -1.00000 q^{48} +1.61803 q^{49} +0.618034 q^{50} +1.61803 q^{53} +1.00000 q^{54} +1.61803 q^{56} +0.618034 q^{58} +1.61803 q^{59} +0.618034 q^{60} +1.61803 q^{62} -1.61803 q^{63} +1.00000 q^{64} -1.00000 q^{70} -1.00000 q^{72} +0.618034 q^{73} +0.618034 q^{75} +0.618034 q^{79} -0.618034 q^{80} +1.00000 q^{81} -0.618034 q^{83} +1.61803 q^{84} +0.618034 q^{87} +0.618034 q^{90} +1.61803 q^{93} +1.00000 q^{96} +0.618034 q^{97} -1.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - 2 q^{12} + q^{14} - q^{15} + 2 q^{16} - 2 q^{18} + q^{20} + q^{21} + 2 q^{24} + q^{25} - 2 q^{27} - q^{28} + q^{29} + q^{30} - q^{31} - 2 q^{32} + 2 q^{35} + 2 q^{36} - q^{40} - q^{42} + q^{45} - 2 q^{48} + q^{49} - q^{50} + q^{53} + 2 q^{54} + q^{56} - q^{58} + q^{59} - q^{60} + q^{62} - q^{63} + 2 q^{64} - 2 q^{70} - 2 q^{72} - q^{73} - q^{75} - q^{79} + q^{80} + 2 q^{81} + q^{83} + q^{84} - q^{87} - q^{90} + q^{93} + 2 q^{96} - q^{97} - q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 - q^10 - 2 * q^12 + q^14 - q^15 + 2 * q^16 - 2 * q^18 + q^20 + q^21 + 2 * q^24 + q^25 - 2 * q^27 - q^28 + q^29 + q^30 - q^31 - 2 * q^32 + 2 * q^35 + 2 * q^36 - q^40 - q^42 + q^45 - 2 * q^48 + q^49 - q^50 + q^53 + 2 * q^54 + q^56 - q^58 + q^59 - q^60 + q^62 - q^63 + 2 * q^64 - 2 * q^70 - 2 * q^72 - q^73 - q^75 - q^79 + q^80 + 2 * q^81 + q^83 + q^84 - q^87 - q^90 + q^93 + 2 * q^96 - q^97 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2904\mathbb{Z}\right)^\times$.

$n$	$727$	$1453$	$1937$	$2785$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.00000	−1.00000
$3$	−1.00000	−1.00000
$4$	1.00000	1.00000
$5$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	1.00000	1.00000
$7$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$7$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$8$	−1.00000	−1.00000
$9$	1.00000	1.00000
$10$	0.618034	0.618034
$11$	0	0
$11$	0	0
$12$	−1.00000	−1.00000
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	1.61803	1.61803
$15$	0.618034	0.618034
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.00000	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−0.618034	−0.618034
$21$	1.61803	1.61803
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	1.00000
$25$	−0.618034	−0.618034
$26$	0	0
$27$	−1.00000	−1.00000
$28$	−1.61803	−1.61803
$29$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$29$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$30$	−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	0
$34$	0	0
$35$	1.00000	1.00000
$36$	1.00000	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0.618034	0.618034
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.61803	−1.61803
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−0.618034	−0.618034
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.00000	−1.00000
$49$	1.61803	1.61803
$50$	0.618034	0.618034
$51$	0	0
$52$	0	0
$53$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$53$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$54$	1.00000	1.00000
$55$	0	0
$56$	1.61803	1.61803
$57$	0	0
$58$	0.618034	0.618034
$59$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$59$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$60$	0.618034	0.618034
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.61803	1.61803
$63$	−1.61803	−1.61803
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−1.00000	−1.00000
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.00000	−1.00000
$73$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$73$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$74$	0	0
$75$	0.618034	0.618034
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$79$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$80$	−0.618034	−0.618034
$81$	1.00000	1.00000
$82$	0	0
$83$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$83$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$84$	1.61803	1.61803
$85$	0	0
$86$	0	0
$87$	0.618034	0.618034
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.618034	0.618034
$91$	0	0
$92$	0	0
$93$	1.61803	1.61803
$94$	0	0
$95$	0	0
$96$	1.00000	1.00000
$97$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$97$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$98$	−1.61803	−1.61803
$99$	0	0
$100$	−0.618034	−0.618034
$101$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$101$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$102$	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	0
$105$	−1.00000	−1.00000
$106$	−1.61803	−1.61803
$107$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$107$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$108$	−1.00000	−1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−1.61803	−1.61803
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−0.618034	−0.618034
$117$	0	0
$118$	−1.61803	−1.61803
$119$	0	0
$120$	−0.618034	−0.618034
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−1.61803	−1.61803
$125$	1.00000	1.00000
$126$	1.61803	1.61803
$127$	2.00000	2.00000	1.00000			$0$
$127$	2.00000	2.00000	1.00000			$0$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	0	0
$131$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$131$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.618034	0.618034
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	1.00000	1.00000
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	1.00000
$145$	0.381966	0.381966
$146$	−0.618034	−0.618034
$147$	−1.61803	−1.61803
$148$	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	$0.400000\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	1.00000
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−0.618034	−0.618034
$159$	−1.61803	−1.61803
$160$	0.618034	0.618034
$161$	0	0
$162$	−1.00000	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0.618034	0.618034
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−1.61803	−1.61803
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$173$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$174$	−0.618034	−0.618034
$175$	1.00000	1.00000
$176$	0	0
$177$	−1.61803	−1.61803
$178$	0	0
$179$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$179$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$180$	−0.618034	−0.618034
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	−1.61803	−1.61803
$187$	0	0
$188$	0	0
$189$	1.61803	1.61803
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.00000	−1.00000
$193$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$193$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$194$	−0.618034	−0.618034
$195$	0	0
$196$	1.61803	1.61803
$197$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$197$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$198$	0	0
$199$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$199$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$200$	0.618034	0.618034
$201$	0	0
$202$	0.618034	0.618034
$203$	1.00000	1.00000
$204$	0	0
$205$	0	0
$206$	1.61803	1.61803
$207$	0	0
$208$	0	0
$209$	0	0
$210$	1.00000	1.00000
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	1.61803	1.61803
$213$	0	0
$214$	−1.61803	−1.61803
$215$	0	0
$216$	1.00000	1.00000
$217$	2.61803	2.61803
$218$	0	0
$219$	−0.618034	−0.618034
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$223$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$224$	1.61803	1.61803
$225$	−0.618034	−0.618034
$226$	0	0
$227$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$227$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0.618034	0.618034
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	1.61803	1.61803
$237$	−0.618034	−0.618034
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0.618034	0.618034
$241$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$242$	0	0
$243$	−1.00000	−1.00000
$244$	0	0
$245$	−1.00000	−1.00000
$246$	0	0
$247$	0	0
$248$	1.61803	1.61803
$249$	0.618034	0.618034
$250$	−1.00000	−1.00000
$251$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$251$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$252$	−1.61803	−1.61803
$253$	0	0
$254$	−2.00000	−2.00000
$255$	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
$260$	0	0
$261$	−0.618034	−0.618034
$262$	−1.61803	−1.61803
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−1.00000	−1.00000
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.00000	−2.00000	−1.00000			$\pi$
$269$	−2.00000	−2.00000	−1.00000			$\pi$
$270$	−0.618034	−0.618034
$271$	2.00000	2.00000	1.00000			$0$
$271$	2.00000	2.00000	1.00000			$0$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.61803	−1.61803
$280$	−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			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.00000	−1.00000
$289$	1.00000	1.00000
$290$	−0.381966	−0.381966
$291$	−0.618034	−0.618034
$292$	0.618034	0.618034
$293$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$293$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$294$	1.61803	1.61803
$295$	−1.00000	−1.00000
$296$	0	0
$297$	0	0
$298$	−1.61803	−1.61803
$299$	0	0
$300$	0.618034	0.618034
$301$	0	0
$302$	−0.618034	−0.618034
$303$	0.618034	0.618034
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	1.61803	1.61803
$310$	−1.00000	−1.00000
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$313$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$314$	0	0
$315$	1.00000	1.00000
$316$	0.618034	0.618034
$317$	−2.00000	−2.00000	−1.00000			$\pi$
$317$	−2.00000	−2.00000	−1.00000			$\pi$
$318$	1.61803	1.61803
$319$	0	0
$320$	−0.618034	−0.618034
$321$	−1.61803	−1.61803
$322$	0	0
$323$	0	0
$324$	1.00000	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−0.618034	−0.618034
$333$	0	0
$334$	0	0
$335$	0	0
$336$	1.61803	1.61803
$337$	2.00000	2.00000	1.00000			$0$
$337$	2.00000	2.00000	1.00000			$0$
$338$	−1.00000	−1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−1.00000
$344$	0	0
$345$	0	0
$346$	−1.61803	−1.61803
$347$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$347$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$348$	0.618034	0.618034
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.00000	−1.00000
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	1.61803	1.61803
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0.618034	0.618034
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.618034	0.618034
$361$	1.00000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.381966	−0.381966
$366$	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	0
$370$	0	0
$371$	−2.61803	−2.61803
$372$	1.61803	1.61803
$373$	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
$378$	−1.61803	−1.61803
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−2.00000	−2.00000
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	1.61803	1.61803
$387$	0	0
$388$	0.618034	0.618034
$389$	−2.00000	−2.00000	−1.00000			$\pi$
$389$	−2.00000	−2.00000	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.61803	−1.61803
$393$	−1.61803	−1.61803
$394$	0.618034	0.618034
$395$	−0.381966	−0.381966
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−0.618034	−0.618034
$399$	0	0
$400$	−0.618034	−0.618034
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−0.618034	−0.618034
$405$	−0.618034	−0.618034
$406$	−1.00000	−1.00000
$407$	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	0
$411$	0	0
$412$	−1.61803	−1.61803
$413$	−2.61803	−2.61803
$414$	0	0
$415$	0.381966	0.381966
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$419$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$420$	−1.00000	−1.00000
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−1.61803	−1.61803
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.61803	1.61803
$429$	0	0
$430$	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	$-0.800000\pi$
$433$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$434$	−2.61803	−2.61803
$435$	−0.381966	−0.381966
$436$	0	0
$437$	0	0
$438$	0.618034	0.618034
$439$	−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	0
$441$	1.61803	1.61803
$442$	0	0
$443$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$443$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$444$	0	0
$445$	0	0
$446$	−0.618034	−0.618034
$447$	−1.61803	−1.61803
$448$	−1.61803	−1.61803
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.618034	0.618034
$451$	0	0
$452$	0	0
$453$	−0.618034	−0.618034
$454$	0.618034	0.618034
$455$	0	0
$456$	0	0
$457$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$457$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.00000	−2.00000	−1.00000			$\pi$
$461$	−2.00000	−2.00000	−1.00000			$\pi$
$462$	0	0
$463$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$463$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$464$	−0.618034	−0.618034
$465$	−1.00000	−1.00000
$466$	0	0
$467$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$467$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.61803	−1.61803
$473$	0	0
$474$	0.618034	0.618034
$475$	0	0
$476$	0	0
$477$	1.61803	1.61803
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−0.618034	−0.618034
$481$	0	0
$482$	1.61803	1.61803
$483$	0	0
$484$	0	0
$485$	−0.381966	−0.381966
$486$	1.00000	1.00000
$487$	−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
$489$	0	0
$490$	1.00000	1.00000
$491$	−2.00000	−2.00000	−1.00000			$\pi$
$491$	−2.00000	−2.00000	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.61803	−1.61803
$497$	0	0
$498$	−0.618034	−0.618034
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1.00000	1.00000
$501$	0	0
$502$	0.618034	0.618034
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.61803	1.61803
$505$	0.381966	0.381966
$506$	0	0
$507$	−1.00000	−1.00000
$508$	2.00000	2.00000
$509$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$509$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$510$	0	0
$511$	−1.00000	−1.00000
$512$	−1.00000	−1.00000
$513$	0	0
$514$	0	0
$515$	1.00000	1.00000
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1.61803	−1.61803
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0.618034	0.618034
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.61803	1.61803
$525$	−1.00000	−1.00000
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	1.00000	1.00000
$531$	1.61803	1.61803
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.00000	−1.00000
$536$	0	0
$537$	0.618034	0.618034
$538$	2.00000	2.00000
$539$	0	0
$540$	0.618034	0.618034
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−2.00000	−2.00000
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1.00000	−1.00000
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$557$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$558$	1.61803	1.61803
$559$	0	0
$560$	1.00000	1.00000
$561$	0	0
$562$	0	0
$563$	−2.00000	−2.00000	−1.00000			$\pi$
$563$	−2.00000	−2.00000	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.61803	−1.61803
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	1.00000
$577$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$577$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$578$	−1.00000	−1.00000
$579$	1.61803	1.61803
$580$	0.381966	0.381966
$581$	1.00000	1.00000
$582$	0.618034	0.618034
$583$	0	0
$584$	−0.618034	−0.618034
$585$	0	0
$586$	−1.61803	−1.61803
$587$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$587$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$588$	−1.61803	−1.61803
$589$	0	0
$590$	1.00000	1.00000
$591$	0.618034	0.618034
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.61803	1.61803
$597$	−0.618034	−0.618034
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−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	0
$606$	−0.618034	−0.618034
$607$	2.00000	2.00000	1.00000			$0$
$607$	2.00000	2.00000	1.00000			$0$
$608$	0	0
$609$	−1.00000	−1.00000
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−1.61803	−1.61803
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	1.00000	1.00000
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−0.618034	−0.618034
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−1.00000	−1.00000
$631$	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
$633$	0	0
$634$	2.00000	2.00000
$635$	−1.23607	−1.23607
$636$	−1.61803	−1.61803
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0.618034	0.618034
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.61803	1.61803
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.00000	−1.00000
$649$	0	0
$650$	0	0
$651$	−2.61803	−2.61803
$652$	0	0
$653$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$653$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$654$	0	0
$655$	−1.00000	−1.00000
$656$	0	0
$657$	0.618034	0.618034
$658$	0	0
$659$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$659$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0.618034	0.618034
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−0.618034	−0.618034
$670$	0	0
$671$	0	0
$672$	−1.61803	−1.61803
$673$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$673$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$674$	−2.00000	−2.00000
$675$	0.618034	0.618034
$676$	1.00000	1.00000
$677$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$677$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$678$	0	0
$679$	−1.00000	−1.00000
$680$	0	0
$681$	0.618034	0.618034
$682$	0	0
$683$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$683$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	1.00000
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.61803	1.61803
$693$	0	0
$694$	−1.61803	−1.61803
$695$	0	0
$696$	−0.618034	−0.618034
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1.00000	1.00000
$701$	−2.00000	−2.00000	−1.00000			$\pi$
$701$	−2.00000	−2.00000	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.00000	1.00000
$708$	−1.61803	−1.61803
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0.618034	0.618034
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−0.618034	−0.618034
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.618034	−0.618034
$721$	2.61803	2.61803
$722$	−1.00000	−1.00000
$723$	1.61803	1.61803
$724$	0	0
$725$	0.381966	0.381966
$726$	0	0
$727$	2.00000	2.00000	1.00000			$0$
$727$	2.00000	2.00000	1.00000			$0$
$728$	0	0
$729$	1.00000	1.00000
$730$	0.381966	0.381966
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−0.618034	−0.618034
$735$	1.00000	1.00000
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	2.61803	2.61803
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−1.61803	−1.61803
$745$	−1.00000	−1.00000
$746$	0	0
$747$	−0.618034	−0.618034
$748$	0	0
$749$	−2.61803	−2.61803
$750$	1.00000	1.00000
$751$	2.00000	2.00000	1.00000			$0$
$751$	2.00000	2.00000	1.00000			$0$
$752$	0	0
$753$	0.618034	0.618034
$754$	0	0
$755$	−0.381966	−0.381966
$756$	1.61803	1.61803
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	2.00000	2.00000
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.00000	−1.00000
$769$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$769$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$770$	0	0
$771$	0	0
$772$	−1.61803	−1.61803
$773$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$773$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$774$	0	0
$775$	1.00000	1.00000
$776$	−0.618034	−0.618034
$777$	0	0
$778$	2.00000	2.00000
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.618034	0.618034
$784$	1.61803	1.61803
$785$	0	0
$786$	1.61803	1.61803
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.618034	−0.618034
$789$	0	0
$790$	0.381966	0.381966
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	1.00000	1.00000
$796$	0.618034	0.618034
$797$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$797$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$798$	0	0
$799$	0	0
$800$	0.618034	0.618034
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.00000	2.00000
$808$	0.618034	0.618034
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.618034	0.618034
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.00000	1.00000
$813$	−2.00000	−2.00000
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−0.618034	−0.618034
$819$	0	0
$820$	0	0
$821$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$821$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$822$	0	0
$823$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$823$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$824$	1.61803	1.61803
$825$	0	0
$826$	2.61803	2.61803
$827$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$827$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	−0.381966	−0.381966
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.61803	1.61803
$838$	0.618034	0.618034
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	1.00000	1.00000
$841$	−0.618034	−0.618034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.618034	−0.618034
$846$	0	0
$847$	0	0
$848$	1.61803	1.61803
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1.61803	−1.61803
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000	1.00000
$865$	−1.00000	−1.00000
$866$	1.61803	1.61803
$867$	−1.00000	−1.00000
$868$	2.61803	2.61803
$869$	0	0
$870$	0.381966	0.381966
$871$	0	0
$872$	0	0
$873$	0.618034	0.618034
$874$	0	0
$875$	−1.61803	−1.61803
$876$	−0.618034	−0.618034
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1.61803	1.61803
$879$	−1.61803	−1.61803
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1.61803	−1.61803
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	1.00000	1.00000
$886$	−1.61803	−1.61803
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−3.23607	−3.23607
$890$	0	0
$891$	0	0
$892$	0.618034	0.618034
$893$	0	0
$894$	1.61803	1.61803
$895$	0.381966	0.381966
$896$	1.61803	1.61803
$897$	0	0
$898$	0	0
$899$	1.00000	1.00000
$900$	−0.618034	−0.618034
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0.618034	0.618034
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−0.618034	−0.618034
$909$	−0.618034	−0.618034
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	−0.618034	−0.618034
$915$	0	0
$916$	0	0
$917$	−2.61803	−2.61803
$918$	0	0
$919$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$919$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$920$	0	0
$921$	0	0
$922$	2.00000	2.00000
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1.61803	1.61803
$927$	−1.61803	−1.61803
$928$	0.618034	0.618034
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	1.00000	1.00000
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−1.61803	−1.61803
$935$	0	0
$936$	0	0
$937$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$937$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$938$	0	0
$939$	−0.618034	−0.618034
$940$	0	0
$941$	−2.00000	−2.00000	−1.00000			$\pi$
$941$	−2.00000	−2.00000	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.61803	1.61803
$945$	−1.00000	−1.00000
$946$	0	0
$947$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$947$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$948$	−0.618034	−0.618034
$949$	0	0
$950$	0	0
$951$	2.00000	2.00000
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−1.61803	−1.61803
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0.618034	0.618034
$961$	1.61803	1.61803
$962$	0	0
$963$	1.61803	1.61803
$964$	−1.61803	−1.61803
$965$	1.00000	1.00000
$966$	0	0
$967$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$968$	0	0
$969$	0	0
$970$	0.381966	0.381966
$971$	−2.00000	−2.00000	−1.00000			$\pi$
$971$	−2.00000	−2.00000	−1.00000			$\pi$
$972$	−1.00000	−1.00000
$973$	0	0
$974$	1.61803	1.61803
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	−1.00000	−1.00000
$981$	0	0
$982$	2.00000	2.00000
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0.381966	0.381966
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	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
$995$	−0.381966	−0.381966
$996$	0.618034	0.618034
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2904.1.n.a.485.1		2
3.2	odd	2		2904.1.n.d.485.2		2
8.5	even	2		2904.1.n.d.485.2		2
11.2	odd	10		2904.1.t.c.2501.1		4
11.3	even	5		2904.1.t.f.2429.1		4
11.4	even	5		2904.1.t.f.269.1		4
11.5	even	5		264.1.t.b.245.1	yes	4
11.6	odd	10		2904.1.t.c.245.1		4
11.7	odd	10		2904.1.t.b.269.1		4
11.8	odd	10		2904.1.t.b.2429.1		4
11.9	even	5		264.1.t.b.125.1	yes	4
11.10	odd	2		2904.1.n.c.485.1		2
24.5	odd	2	CM	2904.1.n.a.485.1		2
33.2	even	10		2904.1.t.d.2501.1		4
33.5	odd	10		264.1.t.a.245.1	yes	4
33.8	even	10		2904.1.t.e.2429.1		4
33.14	odd	10		2904.1.t.a.2429.1		4
33.17	even	10		2904.1.t.d.245.1		4
33.20	odd	10		264.1.t.a.125.1	✓	4
33.26	odd	10		2904.1.t.a.269.1		4
33.29	even	10		2904.1.t.e.269.1		4
33.32	even	2		2904.1.n.b.485.2		2
44.27	odd	10		1056.1.bj.a.113.1		4
44.31	odd	10		1056.1.bj.a.785.1		4
88.5	even	10		264.1.t.a.245.1	yes	4
88.13	odd	10		2904.1.t.d.2501.1		4
88.21	odd	2		2904.1.n.b.485.2		2
88.27	odd	10		1056.1.bj.b.113.1		4
88.29	odd	10		2904.1.t.e.269.1		4
88.37	even	10		2904.1.t.a.269.1		4
88.53	even	10		264.1.t.a.125.1	✓	4
88.61	odd	10		2904.1.t.d.245.1		4
88.69	even	10		2904.1.t.a.2429.1		4
88.75	odd	10		1056.1.bj.b.785.1		4
88.85	odd	10		2904.1.t.e.2429.1		4
132.71	even	10		1056.1.bj.b.113.1		4
132.119	even	10		1056.1.bj.b.785.1		4
264.5	odd	10		264.1.t.b.245.1	yes	4
264.29	even	10		2904.1.t.b.269.1		4
264.53	odd	10		264.1.t.b.125.1	yes	4
264.101	even	10		2904.1.t.c.2501.1		4
264.125	odd	10		2904.1.t.f.269.1		4
264.149	even	10		2904.1.t.c.245.1		4
264.173	even	10		2904.1.t.b.2429.1		4
264.197	even	2		2904.1.n.c.485.1		2
264.203	even	10		1056.1.bj.a.113.1		4
264.245	odd	10		2904.1.t.f.2429.1		4
264.251	even	10		1056.1.bj.a.785.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.1.t.a.125.1	✓	4	33.20	odd	10
264.1.t.a.125.1	✓	4	88.53	even	10
264.1.t.a.245.1	yes	4	33.5	odd	10
264.1.t.a.245.1	yes	4	88.5	even	10
264.1.t.b.125.1	yes	4	11.9	even	5
264.1.t.b.125.1	yes	4	264.53	odd	10
264.1.t.b.245.1	yes	4	11.5	even	5
264.1.t.b.245.1	yes	4	264.5	odd	10
1056.1.bj.a.113.1		4	44.27	odd	10
1056.1.bj.a.113.1		4	264.203	even	10
1056.1.bj.a.785.1		4	44.31	odd	10
1056.1.bj.a.785.1		4	264.251	even	10
1056.1.bj.b.113.1		4	88.27	odd	10
1056.1.bj.b.113.1		4	132.71	even	10
1056.1.bj.b.785.1		4	88.75	odd	10
1056.1.bj.b.785.1		4	132.119	even	10
2904.1.n.a.485.1		2	1.1	even	1	trivial
2904.1.n.a.485.1		2	24.5	odd	2	CM
2904.1.n.b.485.2		2	33.32	even	2
2904.1.n.b.485.2		2	88.21	odd	2
2904.1.n.c.485.1		2	11.10	odd	2
2904.1.n.c.485.1		2	264.197	even	2
2904.1.n.d.485.2		2	3.2	odd	2
2904.1.n.d.485.2		2	8.5	even	2
2904.1.t.a.269.1		4	33.26	odd	10
2904.1.t.a.269.1		4	88.37	even	10
2904.1.t.a.2429.1		4	33.14	odd	10
2904.1.t.a.2429.1		4	88.69	even	10
2904.1.t.b.269.1		4	11.7	odd	10
2904.1.t.b.269.1		4	264.29	even	10
2904.1.t.b.2429.1		4	11.8	odd	10
2904.1.t.b.2429.1		4	264.173	even	10
2904.1.t.c.245.1		4	11.6	odd	10
2904.1.t.c.245.1		4	264.149	even	10
2904.1.t.c.2501.1		4	11.2	odd	10
2904.1.t.c.2501.1		4	264.101	even	10
2904.1.t.d.245.1		4	33.17	even	10
2904.1.t.d.245.1		4	88.61	odd	10
2904.1.t.d.2501.1		4	33.2	even	10
2904.1.t.d.2501.1		4	88.13	odd	10
2904.1.t.e.269.1		4	33.29	even	10
2904.1.t.e.269.1		4	88.29	odd	10
2904.1.t.e.2429.1		4	33.8	even	10
2904.1.t.e.2429.1		4	88.85	odd	10
2904.1.t.f.269.1		4	11.4	even	5
2904.1.t.f.269.1		4	264.125	odd	10
2904.1.t.f.2429.1		4	11.3	even	5
2904.1.t.f.2429.1		4	264.245	odd	10

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:	\( N \)	\(=\)	\( 2904 = 2^{3} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2904.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.00000 q^{6} -1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.618034 q^{5} +1.00000 q^{6} -1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.618034 q^{10} -1.00000 q^{12} +1.61803 q^{14} +0.618034 q^{15} +1.00000 q^{16} -1.00000 q^{18} -0.618034 q^{20} +1.61803 q^{21} +1.00000 q^{24} -0.618034 q^{25} -1.00000 q^{27} -1.61803 q^{28} -0.618034 q^{29} -0.618034 q^{30} -1.61803 q^{31} -1.00000 q^{32} +1.00000 q^{35} +1.00000 q^{36} +0.618034 q^{40} -1.61803 q^{42} -0.618034 q^{45} -1.00000 q^{48} +1.61803 q^{49} +0.618034 q^{50} +1.61803 q^{53} +1.00000 q^{54} +1.61803 q^{56} +0.618034 q^{58} +1.61803 q^{59} +0.618034 q^{60} +1.61803 q^{62} -1.61803 q^{63} +1.00000 q^{64} -1.00000 q^{70} -1.00000 q^{72} +0.618034 q^{73} +0.618034 q^{75} +0.618034 q^{79} -0.618034 q^{80} +1.00000 q^{81} -0.618034 q^{83} +1.61803 q^{84} +0.618034 q^{87} +0.618034 q^{90} +1.61803 q^{93} +1.00000 q^{96} +0.618034 q^{97} -1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - 2 q^{12} + q^{14} - q^{15} + 2 q^{16} - 2 q^{18} + q^{20} + q^{21} + 2 q^{24} + q^{25} - 2 q^{27} - q^{28} + q^{29} + q^{30} - q^{31} - 2 q^{32} + 2 q^{35} + 2 q^{36} - q^{40} - q^{42} + q^{45} - 2 q^{48} + q^{49} - q^{50} + q^{53} + 2 q^{54} + q^{56} - q^{58} + q^{59} - q^{60} + q^{62} - q^{63} + 2 q^{64} - 2 q^{70} - 2 q^{72} - q^{73} - q^{75} - q^{79} + q^{80} + 2 q^{81} + q^{83} + q^{84} - q^{87} - q^{90} + q^{93} + 2 q^{96} - q^{97} - q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 - q^10 - 2 * q^12 + q^14 - q^15 + 2 * q^16 - 2 * q^18 + q^20 + q^21 + 2 * q^24 + q^25 - 2 * q^27 - q^28 + q^29 + q^30 - q^31 - 2 * q^32 + 2 * q^35 + 2 * q^36 - q^40 - q^42 + q^45 - 2 * q^48 + q^49 - q^50 + q^53 + 2 * q^54 + q^56 - q^58 + q^59 - q^60 + q^62 - q^63 + 2 * q^64 - 2 * q^70 - 2 * q^72 - q^73 - q^75 - q^79 + q^80 + 2 * q^81 + q^83 + q^84 - q^87 - q^90 + q^93 + 2 * q^96 - q^97 - q^98

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