LMFDB - Embedded newform 983.1.b.b.982.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [983,1,Mod(982,983)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(983, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("983.982");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 983 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	983.b (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:	$0.490580907418$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.933714431521.1
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.933714431521.1

Embedding invariants

Embedding label			982.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	983.982

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} -0.652704 q^{6} -1.87939 q^{7} -0.652704 q^{8} +2.53209 q^{9} +O(q^{10})$	$q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} -0.652704 q^{6} -1.87939 q^{7} -0.652704 q^{8} +2.53209 q^{9} +1.65270 q^{12} -0.652704 q^{14} +0.652704 q^{16} +0.879385 q^{18} +0.347296 q^{19} +3.53209 q^{21} +1.53209 q^{23} +1.22668 q^{24} +1.00000 q^{25} -2.87939 q^{27} +1.65270 q^{28} -1.00000 q^{31} +0.879385 q^{32} -2.22668 q^{36} -1.00000 q^{37} +0.120615 q^{38} -1.00000 q^{41} +1.22668 q^{42} +0.347296 q^{43} +0.532089 q^{46} +1.53209 q^{47} -1.22668 q^{48} +2.53209 q^{49} +0.347296 q^{50} +0.347296 q^{53} -1.00000 q^{54} +1.22668 q^{56} -0.652704 q^{57} +2.00000 q^{59} -0.347296 q^{62} -4.75877 q^{63} -0.347296 q^{64} -1.00000 q^{67} -2.87939 q^{69} -1.65270 q^{72} -0.347296 q^{74} -1.87939 q^{75} -0.305407 q^{76} +2.00000 q^{79} +2.87939 q^{81} -0.347296 q^{82} -3.10607 q^{84} +0.120615 q^{86} -1.87939 q^{89} -1.34730 q^{92} +1.87939 q^{93} +0.532089 q^{94} -1.65270 q^{96} +0.879385 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 6 q^{12} - 3 q^{14} + 3 q^{16} - 3 q^{18} + 6 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} - 3 q^{31} - 3 q^{32} - 3 q^{37} + 6 q^{38} - 3 q^{41} - 3 q^{42} - 3 q^{46} + 3 q^{48} + 3 q^{49} - 3 q^{54} - 3 q^{56} - 3 q^{57} + 6 q^{59} - 3 q^{63} - 3 q^{67} - 3 q^{69} - 6 q^{72} - 3 q^{76} + 6 q^{79} + 3 q^{81} + 3 q^{84} + 6 q^{86} - 3 q^{92} - 3 q^{94} - 6 q^{96} - 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 6 * q^12 - 3 * q^14 + 3 * q^16 - 3 * q^18 + 6 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 - 3 * q^31 - 3 * q^32 - 3 * q^37 + 6 * q^38 - 3 * q^41 - 3 * q^42 - 3 * q^46 + 3 * q^48 + 3 * q^49 - 3 * q^54 - 3 * q^56 - 3 * q^57 + 6 * q^59 - 3 * q^63 - 3 * q^67 - 3 * q^69 - 6 * q^72 - 3 * q^76 + 6 * q^79 + 3 * q^81 + 3 * q^84 + 6 * q^86 - 3 * q^92 - 3 * q^94 - 6 * q^96 - 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/983\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$2$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$3$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$3$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$4$	−0.879385	−0.879385
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−0.652704	−0.652704
$7$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$7$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$8$	−0.652704	−0.652704
$9$	2.53209	2.53209
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.65270	1.65270
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.652704	−0.652704
$15$	0	0
$16$	0.652704	0.652704
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.879385	0.879385
$19$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$19$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$20$	0	0
$21$	3.53209	3.53209
$22$	0	0
$23$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$23$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$24$	1.22668	1.22668
$25$	1.00000	1.00000
$26$	0	0
$27$	−2.87939	−2.87939
$28$	1.65270	1.65270
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$31$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$32$	0.879385	0.879385
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−2.22668	−2.22668
$37$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$38$	0.120615	0.120615
$39$	0	0
$40$	0	0
$41$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$42$	1.22668	1.22668
$43$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$43$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$44$	0	0
$45$	0	0
$46$	0.532089	0.532089
$47$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$47$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$48$	−1.22668	−1.22668
$49$	2.53209	2.53209
$50$	0.347296	0.347296
$51$	0	0
$52$	0	0
$53$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$53$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$54$	−1.00000	−1.00000
$55$	0	0
$56$	1.22668	1.22668
$57$	−0.652704	−0.652704
$58$	0	0
$59$	2.00000	2.00000	1.00000			$0$
$59$	2.00000	2.00000	1.00000			$0$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−0.347296	−0.347296
$63$	−4.75877	−4.75877
$64$	−0.347296	−0.347296
$65$	0	0
$66$	0	0
$67$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$67$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$68$	0	0
$69$	−2.87939	−2.87939
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.65270	−1.65270
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.347296	−0.347296
$75$	−1.87939	−1.87939
$76$	−0.305407	−0.305407
$77$	0	0
$78$	0	0
$79$	2.00000	2.00000	1.00000			$0$
$79$	2.00000	2.00000	1.00000			$0$
$80$	0	0
$81$	2.87939	2.87939
$82$	−0.347296	−0.347296
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−3.10607	−3.10607
$85$	0	0
$86$	0.120615	0.120615
$87$	0	0
$88$	0	0
$89$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$89$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$90$	0	0
$91$	0	0
$92$	−1.34730	−1.34730
$93$	1.87939	1.87939
$94$	0.532089	0.532089
$95$	0	0
$96$	−1.65270	−1.65270
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.879385	0.879385
$99$	0	0
$100$	−0.879385	−0.879385
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0.120615	0.120615
$107$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$107$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$108$	2.53209	2.53209
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	1.87939	1.87939
$112$	−1.22668	−1.22668
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−0.226682	−0.226682
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0.694593	0.694593
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	1.87939	1.87939
$124$	0.879385	0.879385
$125$	0	0
$126$	−1.65270	−1.65270
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	−0.652704	−0.652704
$130$	0	0
$131$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$131$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$132$	0	0
$133$	−0.652704	−0.652704
$134$	−0.347296	−0.347296
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.00000	−1.00000
$139$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$139$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$140$	0	0
$141$	−2.87939	−2.87939
$142$	0	0
$143$	0	0
$144$	1.65270	1.65270
$145$	0	0
$146$	0	0
$147$	−4.75877	−4.75877
$148$	0.879385	0.879385
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.652704	−0.652704
$151$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$151$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$152$	−0.226682	−0.226682
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0.694593	0.694593
$159$	−0.652704	−0.652704
$160$	0	0
$161$	−2.87939	−2.87939
$162$	1.00000	1.00000
$163$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$163$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$164$	0.879385	0.879385
$165$	0	0
$166$	0	0
$167$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$167$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$168$	−2.30541	−2.30541
$169$	1.00000	1.00000
$170$	0	0
$171$	0.879385	0.879385
$172$	−0.305407	−0.305407
$173$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$173$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$174$	0	0
$175$	−1.87939	−1.87939
$176$	0	0
$177$	−3.75877	−3.75877
$178$	−0.652704	−0.652704
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−1.00000
$185$	0	0
$186$	0.652704	0.652704
$187$	0	0
$188$	−1.34730	−1.34730
$189$	5.41147	5.41147
$190$	0	0
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.652704	0.652704
$193$	2.00000	2.00000	1.00000			$0$
$193$	2.00000	2.00000	1.00000			$0$
$194$	0	0
$195$	0	0
$196$	−2.22668	−2.22668
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.652704	−0.652704
$201$	1.87939	1.87939
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.87939	3.87939
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.305407	−0.305407
$213$	0	0
$214$	−0.652704	−0.652704
$215$	0	0
$216$	1.87939	1.87939
$217$	1.87939	1.87939
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0.652704	0.652704
$223$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$223$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$224$	−1.65270	−1.65270
$225$	2.53209	2.53209
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0.573978	0.573978
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00000	2.00000	1.00000			$0$
$233$	2.00000	2.00000	1.00000			$0$
$234$	0	0
$235$	0	0
$236$	−1.75877	−1.75877
$237$	−3.75877	−3.75877
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.347296	0.347296
$243$	−2.53209	−2.53209
$244$	0	0
$245$	0	0
$246$	0.652704	0.652704
$247$	0	0
$248$	0.652704	0.652704
$249$	0	0
$250$	0	0
$251$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$252$	4.18479	4.18479
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−0.226682	−0.226682
$259$	1.87939	1.87939
$260$	0	0
$261$	0	0
$262$	0.532089	0.532089
$263$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$263$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$264$	0	0
$265$	0	0
$266$	−0.226682	−0.226682
$267$	3.53209	3.53209
$268$	0.879385	0.879385
$269$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$269$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	2.53209	2.53209
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0.532089	0.532089
$279$	−2.53209	−2.53209
$280$	0	0
$281$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$281$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$282$	−1.00000	−1.00000
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.87939	1.87939
$288$	2.22668	2.22668
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$294$	−1.65270	−1.65270
$295$	0	0
$296$	0.652704	0.652704
$297$	0	0
$298$	0	0
$299$	0	0
$300$	1.65270	1.65270
$301$	−0.652704	−0.652704
$302$	0.120615	0.120615
$303$	0	0
$304$	0.226682	0.226682
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$313$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$314$	0	0
$315$	0	0
$316$	−1.75877	−1.75877
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.226682	−0.226682
$319$	0	0
$320$	0	0
$321$	3.53209	3.53209
$322$	−1.00000	−1.00000
$323$	0	0
$324$	−2.53209	−2.53209
$325$	0	0
$326$	0.532089	0.532089
$327$	0	0
$328$	0.652704	0.652704
$329$	−2.87939	−2.87939
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−2.53209	−2.53209
$334$	0.120615	0.120615
$335$	0	0
$336$	2.30541	2.30541
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	0.347296	0.347296
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0.305407	0.305407
$343$	−2.87939	−2.87939
$344$	−0.226682	−0.226682
$345$	0	0
$346$	0.120615	0.120615
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$349$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$350$	−0.652704	−0.652704
$351$	0	0
$352$	0	0
$353$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$353$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$354$	−1.30541	−1.30541
$355$	0	0
$356$	1.65270	1.65270
$357$	0	0
$358$	0	0
$359$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$359$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$360$	0	0
$361$	−0.879385	−0.879385
$362$	0	0
$363$	−1.87939	−1.87939
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$367$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$368$	1.00000	1.00000
$369$	−2.53209	−2.53209
$370$	0	0
$371$	−0.652704	−0.652704
$372$	−1.65270	−1.65270
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−1.00000	−1.00000
$377$	0	0
$378$	1.87939	1.87939
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−0.347296	−0.347296
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.87939	1.87939
$385$	0	0
$386$	0.694593	0.694593
$387$	0.879385	0.879385
$388$	0	0
$389$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$389$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$390$	0	0
$391$	0	0
$392$	−1.65270	−1.65270
$393$	−2.87939	−2.87939
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	1.22668	1.22668
$400$	0.652704	0.652704
$401$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$401$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$402$	0.652704	0.652704
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.75877	−3.75877
$414$	1.34730	1.34730
$415$	0	0
$416$	0	0
$417$	−2.87939	−2.87939
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	3.87939	3.87939
$424$	−0.226682	−0.226682
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.65270	1.65270
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.87939	−1.87939
$433$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$434$	0.652704	0.652704
$435$	0	0
$436$	0	0
$437$	0.532089	0.532089
$438$	0	0
$439$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$440$	0	0
$441$	6.41147	6.41147
$442$	0	0
$443$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.65270	−1.65270
$445$	0	0
$446$	0.532089	0.532089
$447$	0	0
$448$	0.652704	0.652704
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.879385	0.879385
$451$	0	0
$452$	0	0
$453$	−0.652704	−0.652704
$454$	0	0
$455$	0	0
$456$	0.426022	0.426022
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$461$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0.694593	0.694593
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	1.87939	1.87939
$470$	0	0
$471$	0	0
$472$	−1.30541	−1.30541
$473$	0	0
$474$	−1.30541	−1.30541
$475$	0.347296	0.347296
$476$	0	0
$477$	0.879385	0.879385
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	5.41147	5.41147
$484$	−0.879385	−0.879385
$485$	0	0
$486$	−0.879385	−0.879385
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−2.87939	−2.87939
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−1.65270	−1.65270
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−0.652704	−0.652704
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−0.652704	−0.652704
$502$	−0.347296	−0.347296
$503$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$503$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$504$	3.10607	3.10607
$505$	0	0
$506$	0	0
$507$	−1.87939	−1.87939
$508$	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
$513$	−1.00000	−1.00000
$514$	0	0
$515$	0	0
$516$	0.573978	0.573978
$517$	0	0
$518$	0.652704	0.652704
$519$	−0.652704	−0.652704
$520$	0	0
$521$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$521$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$522$	0	0
$523$	2.00000	2.00000	1.00000			$0$
$523$	2.00000	2.00000	1.00000			$0$
$524$	−1.34730	−1.34730
$525$	3.53209	3.53209
$526$	−0.347296	−0.347296
$527$	0	0
$528$	0	0
$529$	1.34730	1.34730
$530$	0	0
$531$	5.06418	5.06418
$532$	0.573978	0.573978
$533$	0	0
$534$	1.22668	1.22668
$535$	0	0
$536$	0.652704	0.652704
$537$	0	0
$538$	−0.652704	−0.652704
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$547$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	1.87939	1.87939
$553$	−3.75877	−3.75877
$554$	0	0
$555$	0	0
$556$	−1.34730	−1.34730
$557$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$557$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$558$	−0.879385	−0.879385
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0.120615	0.120615
$563$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$563$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$564$	2.53209	2.53209
$565$	0	0
$566$	0	0
$567$	−5.41147	−5.41147
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$571$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$572$	0	0
$573$	1.87939	1.87939
$574$	0.652704	0.652704
$575$	1.53209	1.53209
$576$	−0.879385	−0.879385
$577$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$577$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$578$	0.347296	0.347296
$579$	−3.75877	−3.75877
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−0.347296	−0.347296
$587$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$588$	4.18479	4.18479
$589$	−0.347296	−0.347296
$590$	0	0
$591$	0	0
$592$	−0.652704	−0.652704
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.22668	1.22668
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.226682	−0.226682
$603$	−2.53209	−2.53209
$604$	−0.305407	−0.305407
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0.305407	0.305407
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$613$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000	2.00000	1.00000			$0$
$617$	2.00000	2.00000	1.00000			$0$
$618$	0	0
$619$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$619$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$620$	0	0
$621$	−4.41147	−4.41147
$622$	0	0
$623$	3.53209	3.53209
$624$	0	0
$625$	1.00000	1.00000
$626$	0.120615	0.120615
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	−1.30541	−1.30541
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0.573978	0.573978
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.22668	1.22668
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	2.53209	2.53209
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.87939	−1.87939
$649$	0	0
$650$	0	0
$651$	−3.53209	−3.53209
$652$	−1.34730	−1.34730
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−0.652704	−0.652704
$657$	0	0
$658$	−1.00000	−1.00000
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−0.879385	−0.879385
$667$	0	0
$668$	−0.305407	−0.305407
$669$	−2.87939	−2.87939
$670$	0	0
$671$	0	0
$672$	3.10607	3.10607
$673$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$673$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$674$	−0.347296	−0.347296
$675$	−2.87939	−2.87939
$676$	−0.879385	−0.879385
$677$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$677$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.773318	−0.773318
$685$	0	0
$686$	−1.00000	−1.00000
$687$	0	0
$688$	0.226682	0.226682
$689$	0	0
$690$	0	0
$691$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$691$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$692$	−0.305407	−0.305407
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−0.652704	−0.652704
$699$	−3.75877	−3.75877
$700$	1.65270	1.65270
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−0.347296	−0.347296
$704$	0	0
$705$	0	0
$706$	0.532089	0.532089
$707$	0	0
$708$	3.30541	3.30541
$709$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$709$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$710$	0	0
$711$	5.06418	5.06418
$712$	1.22668	1.22668
$713$	−1.53209	−1.53209
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−0.652704	−0.652704
$719$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$720$	0	0
$721$	0	0
$722$	−0.305407	−0.305407
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−0.652704	−0.652704
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.87939	1.87939
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$733$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$734$	0.532089	0.532089
$735$	0	0
$736$	1.34730	1.34730
$737$	0	0
$738$	−0.879385	−0.879385
$739$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$740$	0	0
$741$	0	0
$742$	−0.226682	−0.226682
$743$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$744$	−1.22668	−1.22668
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.53209	3.53209
$750$	0	0
$751$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$751$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$752$	1.00000	1.00000
$753$	1.87939	1.87939
$754$	0	0
$755$	0	0
$756$	−4.75877	−4.75877
$757$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$757$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0.879385	0.879385
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−1.75877	−1.75877
$773$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$773$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$774$	0.305407	0.305407
$775$	−1.00000	−1.00000
$776$	0	0
$777$	−3.53209	−3.53209
$778$	0.532089	0.532089
$779$	−0.347296	−0.347296
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.65270	1.65270
$785$	0	0
$786$	−1.00000	−1.00000
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	1.87939	1.87939
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0.426022	0.426022
$799$	0	0
$800$	0.879385	0.879385
$801$	−4.75877	−4.75877
$802$	0.532089	0.532089
$803$	0	0
$804$	−1.65270	−1.65270
$805$	0	0
$806$	0	0
$807$	3.53209	3.53209
$808$	0	0
$809$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$809$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.120615	0.120615
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$823$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$824$	0	0
$825$	0	0
$826$	−1.30541	−1.30541
$827$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$827$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$828$	−3.41147	−3.41147
$829$	2.00000	2.00000	1.00000			$0$
$829$	2.00000	2.00000	1.00000			$0$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	−1.00000	−1.00000
$835$	0	0
$836$	0	0
$837$	2.87939	2.87939
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	−0.652704	−0.652704
$844$	0	0
$845$	0	0
$846$	1.34730	1.34730
$847$	−1.87939	−1.87939
$848$	0.226682	0.226682
$849$	0	0
$850$	0	0
$851$	−1.53209	−1.53209
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	1.22668	1.22668
$857$	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
$861$	−3.53209	−3.53209
$862$	0	0
$863$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$864$	−2.53209	−2.53209
$865$	0	0
$866$	−0.347296	−0.347296
$867$	−1.87939	−1.87939
$868$	−1.65270	−1.65270
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0.184793	0.184793
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.347296	−0.347296
$879$	1.87939	1.87939
$880$	0	0
$881$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$881$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$882$	2.22668	2.22668
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−0.347296	−0.347296
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−1.22668	−1.22668
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−1.34730	−1.34730
$893$	0.532089	0.532089
$894$	0	0
$895$	0	0
$896$	1.87939	1.87939
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−2.22668	−2.22668
$901$	0	0
$902$	0	0
$903$	1.22668	1.22668
$904$	0	0
$905$	0	0
$906$	−0.226682	−0.226682
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−0.426022	−0.426022
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.87939	−2.87939
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−0.652704	−0.652704
$923$	0	0
$924$	0	0
$925$	−1.00000	−1.00000
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0.879385	0.879385
$932$	−1.75877	−1.75877
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0.652704	0.652704
$939$	−0.652704	−0.652704
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−1.53209	−1.53209
$944$	1.30541	1.30541
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	3.30541	3.30541
$949$	0	0
$950$	0.120615	0.120615
$951$	0	0
$952$	0	0
$953$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$954$	0.305407	0.305407
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	−4.75877	−4.75877
$964$	0	0
$965$	0	0
$966$	1.87939	1.87939
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.652704	−0.652704
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2.22668	2.22668
$973$	−2.87939	−2.87939
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1.00000	−1.00000
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.00000	1.00000
$983$	1.00000	1.00000
$984$	−1.22668	−1.22668
$985$	0	0
$986$	0	0
$987$	5.41147	5.41147
$988$	0	0
$989$	0.532089	0.532089
$990$	0	0
$991$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$991$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$992$	−0.879385	−0.879385
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$997$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$998$	0	0
$999$	2.87939	2.87939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.1.b.b.982.2	✓	3
983.982	odd	2	CM	983.1.b.b.982.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.1.b.b.982.2	✓	3	1.1	even	1	trivial
983.1.b.b.982.2	✓	3	983.982	odd	2	CM

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:	\( N \)	\(=\)	\( 983 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	983.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} -0.652704 q^{6} -1.87939 q^{7} -0.652704 q^{8} +2.53209 q^{9} +O(q^{10})\)	\(q+0.347296 q^{2} -1.87939 q^{3} -0.879385 q^{4} -0.652704 q^{6} -1.87939 q^{7} -0.652704 q^{8} +2.53209 q^{9} +1.65270 q^{12} -0.652704 q^{14} +0.652704 q^{16} +0.879385 q^{18} +0.347296 q^{19} +3.53209 q^{21} +1.53209 q^{23} +1.22668 q^{24} +1.00000 q^{25} -2.87939 q^{27} +1.65270 q^{28} -1.00000 q^{31} +0.879385 q^{32} -2.22668 q^{36} -1.00000 q^{37} +0.120615 q^{38} -1.00000 q^{41} +1.22668 q^{42} +0.347296 q^{43} +0.532089 q^{46} +1.53209 q^{47} -1.22668 q^{48} +2.53209 q^{49} +0.347296 q^{50} +0.347296 q^{53} -1.00000 q^{54} +1.22668 q^{56} -0.652704 q^{57} +2.00000 q^{59} -0.347296 q^{62} -4.75877 q^{63} -0.347296 q^{64} -1.00000 q^{67} -2.87939 q^{69} -1.65270 q^{72} -0.347296 q^{74} -1.87939 q^{75} -0.305407 q^{76} +2.00000 q^{79} +2.87939 q^{81} -0.347296 q^{82} -3.10607 q^{84} +0.120615 q^{86} -1.87939 q^{89} -1.34730 q^{92} +1.87939 q^{93} +0.532089 q^{94} -1.65270 q^{96} +0.879385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 6 q^{12} - 3 q^{14} + 3 q^{16} - 3 q^{18} + 6 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} - 3 q^{31} - 3 q^{32} - 3 q^{37} + 6 q^{38} - 3 q^{41} - 3 q^{42} - 3 q^{46} + 3 q^{48} + 3 q^{49} - 3 q^{54} - 3 q^{56} - 3 q^{57} + 6 q^{59} - 3 q^{63} - 3 q^{67} - 3 q^{69} - 6 q^{72} - 3 q^{76} + 6 q^{79} + 3 q^{81} + 3 q^{84} + 6 q^{86} - 3 q^{92} - 3 q^{94} - 6 q^{96} - 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 6 * q^12 - 3 * q^14 + 3 * q^16 - 3 * q^18 + 6 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 - 3 * q^31 - 3 * q^32 - 3 * q^37 + 6 * q^38 - 3 * q^41 - 3 * q^42 - 3 * q^46 + 3 * q^48 + 3 * q^49 - 3 * q^54 - 3 * q^56 - 3 * q^57 + 6 * q^59 - 3 * q^63 - 3 * q^67 - 3 * q^69 - 6 * q^72 - 3 * q^76 + 6 * q^79 + 3 * q^81 + 3 * q^84 + 6 * q^86 - 3 * q^92 - 3 * q^94 - 6 * q^96 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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