LMFDB - Embedded newform 983.1.b.b.982.3

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			982.3
Root			$-1.53209$ 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)

$f(q)$	$=$	$q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +0.532089 q^{6} +0.347296 q^{7} +0.532089 q^{8} -0.879385 q^{9} +O(q^{10})$	$q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +0.532089 q^{6} +0.347296 q^{7} +0.532089 q^{8} -0.879385 q^{9} +0.467911 q^{12} +0.532089 q^{14} -0.532089 q^{16} -1.34730 q^{18} +1.53209 q^{19} +0.120615 q^{21} -1.87939 q^{23} +0.184793 q^{24} +1.00000 q^{25} -0.652704 q^{27} +0.467911 q^{28} -1.00000 q^{31} -1.34730 q^{32} -1.18479 q^{36} -1.00000 q^{37} +2.34730 q^{38} -1.00000 q^{41} +0.184793 q^{42} +1.53209 q^{43} -2.87939 q^{46} -1.87939 q^{47} -0.184793 q^{48} -0.879385 q^{49} +1.53209 q^{50} +1.53209 q^{53} -1.00000 q^{54} +0.184793 q^{56} +0.532089 q^{57} +2.00000 q^{59} -1.53209 q^{62} -0.305407 q^{63} -1.53209 q^{64} -1.00000 q^{67} -0.652704 q^{69} -0.467911 q^{72} -1.53209 q^{74} +0.347296 q^{75} +2.06418 q^{76} +2.00000 q^{79} +0.652704 q^{81} -1.53209 q^{82} +0.162504 q^{84} +2.34730 q^{86} +0.347296 q^{89} -2.53209 q^{92} -0.347296 q^{93} -2.87939 q^{94} -0.467911 q^{96} -1.34730 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$2$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$3$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$3$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$4$	1.34730	1.34730
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0.532089	0.532089
$7$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$7$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$8$	0.532089	0.532089
$9$	−0.879385	−0.879385
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0.467911	0.467911
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0.532089	0.532089
$15$	0	0
$16$	−0.532089	−0.532089
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.34730	−1.34730
$19$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$19$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$20$	0	0
$21$	0.120615	0.120615
$22$	0	0
$23$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$23$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$24$	0.184793	0.184793
$25$	1.00000	1.00000
$26$	0	0
$27$	−0.652704	−0.652704
$28$	0.467911	0.467911
$29$	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	$0.666667\pi$
$32$	−1.34730	−1.34730
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.18479	−1.18479
$37$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$38$	2.34730	2.34730
$39$	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	$0.666667\pi$
$42$	0.184793	0.184793
$43$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$43$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$44$	0	0
$45$	0	0
$46$	−2.87939	−2.87939
$47$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$47$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$48$	−0.184793	−0.184793
$49$	−0.879385	−0.879385
$50$	1.53209	1.53209
$51$	0	0
$52$	0	0
$53$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$53$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$54$	−1.00000	−1.00000
$55$	0	0
$56$	0.184793	0.184793
$57$	0.532089	0.532089
$58$	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			$\pi$
$62$	−1.53209	−1.53209
$63$	−0.305407	−0.305407
$64$	−1.53209	−1.53209
$65$	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
$69$	−0.652704	−0.652704
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.467911	−0.467911
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.53209	−1.53209
$75$	0.347296	0.347296
$76$	2.06418	2.06418
$77$	0	0
$78$	0	0
$79$	2.00000	2.00000	1.00000			$0$
$79$	2.00000	2.00000	1.00000			$0$
$80$	0	0
$81$	0.652704	0.652704
$82$	−1.53209	−1.53209
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.162504	0.162504
$85$	0	0
$86$	2.34730	2.34730
$87$	0	0
$88$	0	0
$89$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$89$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$90$	0	0
$91$	0	0
$92$	−2.53209	−2.53209
$93$	−0.347296	−0.347296
$94$	−2.87939	−2.87939
$95$	0	0
$96$	−0.467911	−0.467911
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.34730	−1.34730
$99$	0	0
$100$	1.34730	1.34730
$101$	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
$106$	2.34730	2.34730
$107$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$107$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$108$	−0.879385	−0.879385
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−0.347296	−0.347296
$112$	−0.184793	−0.184793
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.815207	0.815207
$115$	0	0
$116$	0	0
$117$	0	0
$118$	3.06418	3.06418
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	−0.347296	−0.347296
$124$	−1.34730	−1.34730
$125$	0	0
$126$	−0.467911	−0.467911
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.00000	−1.00000
$129$	0.532089	0.532089
$130$	0	0
$131$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$131$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$132$	0	0
$133$	0.532089	0.532089
$134$	−1.53209	−1.53209
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.00000	−1.00000
$139$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$139$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$140$	0	0
$141$	−0.652704	−0.652704
$142$	0	0
$143$	0	0
$144$	0.467911	0.467911
$145$	0	0
$146$	0	0
$147$	−0.305407	−0.305407
$148$	−1.34730	−1.34730
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.532089	0.532089
$151$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$151$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$152$	0.815207	0.815207
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	3.06418	3.06418
$159$	0.532089	0.532089
$160$	0	0
$161$	−0.652704	−0.652704
$162$	1.00000	1.00000
$163$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$163$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$164$	−1.34730	−1.34730
$165$	0	0
$166$	0	0
$167$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$167$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$168$	0.0641778	0.0641778
$169$	1.00000	1.00000
$170$	0	0
$171$	−1.34730	−1.34730
$172$	2.06418	2.06418
$173$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$173$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$174$	0	0
$175$	0.347296	0.347296
$176$	0	0
$177$	0.694593	0.694593
$178$	0.532089	0.532089
$179$	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
$186$	−0.532089	−0.532089
$187$	0	0
$188$	−2.53209	−2.53209
$189$	−0.226682	−0.226682
$190$	0	0
$191$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.532089	−0.532089
$193$	2.00000	2.00000	1.00000			$0$
$193$	2.00000	2.00000	1.00000			$0$
$194$	0	0
$195$	0	0
$196$	−1.18479	−1.18479
$197$	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			$\pi$
$200$	0.532089	0.532089
$201$	−0.347296	−0.347296
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.65270	1.65270
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	2.06418	2.06418
$213$	0	0
$214$	0.532089	0.532089
$215$	0	0
$216$	−0.347296	−0.347296
$217$	−0.347296	−0.347296
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−0.532089	−0.532089
$223$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$223$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$224$	−0.467911	−0.467911
$225$	−0.879385	−0.879385
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0.716881	0.716881
$229$	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
$236$	2.69459	2.69459
$237$	0.694593	0.694593
$238$	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			$\pi$
$242$	1.53209	1.53209
$243$	0.879385	0.879385
$244$	0	0
$245$	0	0
$246$	−0.532089	−0.532089
$247$	0	0
$248$	−0.532089	−0.532089
$249$	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	$0.666667\pi$
$252$	−0.411474	−0.411474
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0.815207	0.815207
$259$	−0.347296	−0.347296
$260$	0	0
$261$	0	0
$262$	−2.87939	−2.87939
$263$	−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
$266$	0.815207	0.815207
$267$	0.120615	0.120615
$268$	−1.34730	−1.34730
$269$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$269$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$270$	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
$276$	−0.879385	−0.879385
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−2.87939	−2.87939
$279$	0.879385	0.879385
$280$	0	0
$281$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$281$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$282$	−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
$287$	−0.347296	−0.347296
$288$	1.18479	1.18479
$289$	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	$0.666667\pi$
$294$	−0.467911	−0.467911
$295$	0	0
$296$	−0.532089	−0.532089
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0.467911	0.467911
$301$	0.532089	0.532089
$302$	2.34730	2.34730
$303$	0	0
$304$	−0.815207	−0.815207
$305$	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
$313$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$313$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$314$	0	0
$315$	0	0
$316$	2.69459	2.69459
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.815207	0.815207
$319$	0	0
$320$	0	0
$321$	0.120615	0.120615
$322$	−1.00000	−1.00000
$323$	0	0
$324$	0.879385	0.879385
$325$	0	0
$326$	−2.87939	−2.87939
$327$	0	0
$328$	−0.532089	−0.532089
$329$	−0.652704	−0.652704
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0.879385	0.879385
$334$	2.34730	2.34730
$335$	0	0
$336$	−0.0641778	−0.0641778
$337$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$338$	1.53209	1.53209
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−2.06418	−2.06418
$343$	−0.652704	−0.652704
$344$	0.815207	0.815207
$345$	0	0
$346$	2.34730	2.34730
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$349$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$350$	0.532089	0.532089
$351$	0	0
$352$	0	0
$353$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$353$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$354$	1.06418	1.06418
$355$	0	0
$356$	0.467911	0.467911
$357$	0	0
$358$	0	0
$359$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$359$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$360$	0	0
$361$	1.34730	1.34730
$362$	0	0
$363$	0.347296	0.347296
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$367$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$368$	1.00000	1.00000
$369$	0.879385	0.879385
$370$	0	0
$371$	0.532089	0.532089
$372$	−0.467911	−0.467911
$373$	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
$378$	−0.347296	−0.347296
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−1.53209	−1.53209
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.347296	−0.347296
$385$	0	0
$386$	3.06418	3.06418
$387$	−1.34730	−1.34730
$388$	0	0
$389$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$389$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$390$	0	0
$391$	0	0
$392$	−0.467911	−0.467911
$393$	−0.652704	−0.652704
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0.184793	0.184793
$400$	−0.532089	−0.532089
$401$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$401$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$402$	−0.532089	−0.532089
$403$	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
$413$	0.694593	0.694593
$414$	2.53209	2.53209
$415$	0	0
$416$	0	0
$417$	−0.652704	−0.652704
$418$	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
$423$	1.65270	1.65270
$424$	0.815207	0.815207
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0.467911	0.467911
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.347296	0.347296
$433$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$434$	−0.532089	−0.532089
$435$	0	0
$436$	0	0
$437$	−2.87939	−2.87939
$438$	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
$441$	0.773318	0.773318
$442$	0	0
$443$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$444$	−0.467911	−0.467911
$445$	0	0
$446$	−2.87939	−2.87939
$447$	0	0
$448$	−0.532089	−0.532089
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.34730	−1.34730
$451$	0	0
$452$	0	0
$453$	0.532089	0.532089
$454$	0	0
$455$	0	0
$456$	0.283119	0.283119
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$461$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	3.06418	3.06418
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	−0.347296	−0.347296
$470$	0	0
$471$	0	0
$472$	1.06418	1.06418
$473$	0	0
$474$	1.06418	1.06418
$475$	1.53209	1.53209
$476$	0	0
$477$	−1.34730	−1.34730
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.226682	−0.226682
$484$	1.34730	1.34730
$485$	0	0
$486$	1.34730	1.34730
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−0.652704	−0.652704
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−0.467911	−0.467911
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.532089	0.532089
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0.532089	0.532089
$502$	−1.53209	−1.53209
$503$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$503$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$504$	−0.162504	−0.162504
$505$	0	0
$506$	0	0
$507$	0.347296	0.347296
$508$	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
$516$	0.716881	0.716881
$517$	0	0
$518$	−0.532089	−0.532089
$519$	0.532089	0.532089
$520$	0	0
$521$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$521$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$522$	0	0
$523$	2.00000	2.00000	1.00000			$0$
$523$	2.00000	2.00000	1.00000			$0$
$524$	−2.53209	−2.53209
$525$	0.120615	0.120615
$526$	−1.53209	−1.53209
$527$	0	0
$528$	0	0
$529$	2.53209	2.53209
$530$	0	0
$531$	−1.75877	−1.75877
$532$	0.716881	0.716881
$533$	0	0
$534$	0.184793	0.184793
$535$	0	0
$536$	−0.532089	−0.532089
$537$	0	0
$538$	0.532089	0.532089
$539$	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
$547$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$547$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−0.347296	−0.347296
$553$	0.694593	0.694593
$554$	0	0
$555$	0	0
$556$	−2.53209	−2.53209
$557$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$557$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$558$	1.34730	1.34730
$559$	0	0
$560$	0	0
$561$	0	0
$562$	2.34730	2.34730
$563$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$563$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$564$	−0.879385	−0.879385
$565$	0	0
$566$	0	0
$567$	0.226682	0.226682
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$571$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$572$	0	0
$573$	−0.347296	−0.347296
$574$	−0.532089	−0.532089
$575$	−1.87939	−1.87939
$576$	1.34730	1.34730
$577$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$577$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$578$	1.53209	1.53209
$579$	0.694593	0.694593
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−1.53209	−1.53209
$587$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$588$	−0.411474	−0.411474
$589$	−1.53209	−1.53209
$590$	0	0
$591$	0	0
$592$	0.532089	0.532089
$593$	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			$\pi$
$600$	0.184793	0.184793
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0.815207	0.815207
$603$	0.879385	0.879385
$604$	2.06418	2.06418
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−2.06418	−2.06418
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$613$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$614$	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
$619$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$619$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$620$	0	0
$621$	1.22668	1.22668
$622$	0	0
$623$	0.120615	0.120615
$624$	0	0
$625$	1.00000	1.00000
$626$	2.34730	2.34730
$627$	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	$0.666667\pi$
$632$	1.06418	1.06418
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0.716881	0.716881
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0.184793	0.184793
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−0.879385	−0.879385
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.347296	0.347296
$649$	0	0
$650$	0	0
$651$	−0.120615	−0.120615
$652$	−2.53209	−2.53209
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0.532089	0.532089
$657$	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
$666$	1.34730	1.34730
$667$	0	0
$668$	2.06418	2.06418
$669$	−0.652704	−0.652704
$670$	0	0
$671$	0	0
$672$	−0.162504	−0.162504
$673$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$673$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$674$	−1.53209	−1.53209
$675$	−0.652704	−0.652704
$676$	1.34730	1.34730
$677$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$677$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$678$	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			$\pi$
$684$	−1.81521	−1.81521
$685$	0	0
$686$	−1.00000	−1.00000
$687$	0	0
$688$	−0.815207	−0.815207
$689$	0	0
$690$	0	0
$691$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$691$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$692$	2.06418	2.06418
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0.532089	0.532089
$699$	0.694593	0.694593
$700$	0.467911	0.467911
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−1.53209	−1.53209
$704$	0	0
$705$	0	0
$706$	−2.87939	−2.87939
$707$	0	0
$708$	0.935822	0.935822
$709$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$709$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$710$	0	0
$711$	−1.75877	−1.75877
$712$	0.184793	0.184793
$713$	1.87939	1.87939
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0.532089	0.532089
$719$	−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
$722$	2.06418	2.06418
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0.532089	0.532089
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.347296	−0.347296
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$733$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$734$	−2.87939	−2.87939
$735$	0	0
$736$	2.53209	2.53209
$737$	0	0
$738$	1.34730	1.34730
$739$	−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
$742$	0.815207	0.815207
$743$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$744$	−0.184793	−0.184793
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.120615	0.120615
$750$	0	0
$751$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$751$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$752$	1.00000	1.00000
$753$	−0.347296	−0.347296
$754$	0	0
$755$	0	0
$756$	−0.305407	−0.305407
$757$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$757$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$758$	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
$764$	−1.34730	−1.34730
$765$	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
$772$	2.69459	2.69459
$773$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$773$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$774$	−2.06418	−2.06418
$775$	−1.00000	−1.00000
$776$	0	0
$777$	−0.120615	−0.120615
$778$	−2.87939	−2.87939
$779$	−1.53209	−1.53209
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0.467911	0.467911
$785$	0	0
$786$	−1.00000	−1.00000
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	−0.347296	−0.347296
$790$	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			$\pi$
$798$	0.283119	0.283119
$799$	0	0
$800$	−1.34730	−1.34730
$801$	−0.305407	−0.305407
$802$	−2.87939	−2.87939
$803$	0	0
$804$	−0.467911	−0.467911
$805$	0	0
$806$	0	0
$807$	0.120615	0.120615
$808$	0	0
$809$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$809$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$810$	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
$817$	2.34730	2.34730
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$823$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$824$	0	0
$825$	0	0
$826$	1.06418	1.06418
$827$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$827$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$828$	2.22668	2.22668
$829$	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
$837$	0.652704	0.652704
$838$	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
$843$	0.532089	0.532089
$844$	0	0
$845$	0	0
$846$	2.53209	2.53209
$847$	0.347296	0.347296
$848$	−0.815207	−0.815207
$849$	0	0
$850$	0	0
$851$	1.87939	1.87939
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0.184793	0.184793
$857$	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
$861$	−0.120615	−0.120615
$862$	0	0
$863$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$864$	0.879385	0.879385
$865$	0	0
$866$	−1.53209	−1.53209
$867$	0.347296	0.347296
$868$	−0.467911	−0.467911
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	−4.41147	−4.41147
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−1.53209	−1.53209
$879$	−0.347296	−0.347296
$880$	0	0
$881$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$881$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$882$	1.18479	1.18479
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−1.53209	−1.53209
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.184793	−0.184793
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−2.53209	−2.53209
$893$	−2.87939	−2.87939
$894$	0	0
$895$	0	0
$896$	−0.347296	−0.347296
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.18479	−1.18479
$901$	0	0
$902$	0	0
$903$	0.184793	0.184793
$904$	0	0
$905$	0	0
$906$	0.815207	0.815207
$907$	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			$\pi$
$912$	−0.283119	−0.283119
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.652704	−0.652704
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0.532089	0.532089
$923$	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
$931$	−1.34730	−1.34730
$932$	2.69459	2.69459
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−0.532089	−0.532089
$939$	0.532089	0.532089
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	1.87939	1.87939
$944$	−1.06418	−1.06418
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0.935822	0.935822
$949$	0	0
$950$	2.34730	2.34730
$951$	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	$0.666667\pi$
$954$	−2.06418	−2.06418
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	−0.305407	−0.305407
$964$	0	0
$965$	0	0
$966$	−0.347296	−0.347296
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.532089	0.532089
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1.18479	1.18479
$973$	−0.652704	−0.652704
$974$	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
$984$	−0.184793	−0.184793
$985$	0	0
$986$	0	0
$987$	−0.226682	−0.226682
$988$	0	0
$989$	−2.87939	−2.87939
$990$	0	0
$991$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$991$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$992$	1.34730	1.34730
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$997$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$998$	0	0
$999$	0.652704	0.652704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.1.b.b.982.3	✓	3
983.982	odd	2	CM	983.1.b.b.982.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.1.b.b.982.3	✓	3	1.1	even	1	trivial
983.1.b.b.982.3	✓	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)

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +0.532089 q^{6} +0.347296 q^{7} +0.532089 q^{8} -0.879385 q^{9} +O(q^{10})\)	\(q+1.53209 q^{2} +0.347296 q^{3} +1.34730 q^{4} +0.532089 q^{6} +0.347296 q^{7} +0.532089 q^{8} -0.879385 q^{9} +0.467911 q^{12} +0.532089 q^{14} -0.532089 q^{16} -1.34730 q^{18} +1.53209 q^{19} +0.120615 q^{21} -1.87939 q^{23} +0.184793 q^{24} +1.00000 q^{25} -0.652704 q^{27} +0.467911 q^{28} -1.00000 q^{31} -1.34730 q^{32} -1.18479 q^{36} -1.00000 q^{37} +2.34730 q^{38} -1.00000 q^{41} +0.184793 q^{42} +1.53209 q^{43} -2.87939 q^{46} -1.87939 q^{47} -0.184793 q^{48} -0.879385 q^{49} +1.53209 q^{50} +1.53209 q^{53} -1.00000 q^{54} +0.184793 q^{56} +0.532089 q^{57} +2.00000 q^{59} -1.53209 q^{62} -0.305407 q^{63} -1.53209 q^{64} -1.00000 q^{67} -0.652704 q^{69} -0.467911 q^{72} -1.53209 q^{74} +0.347296 q^{75} +2.06418 q^{76} +2.00000 q^{79} +0.652704 q^{81} -1.53209 q^{82} +0.162504 q^{84} +2.34730 q^{86} +0.347296 q^{89} -2.53209 q^{92} -0.347296 q^{93} -2.87939 q^{94} -0.467911 q^{96} -1.34730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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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