LMFDB - Embedded newform 983.1.b.c.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:	$9$
Coefficient field:	$\Q(\zeta_{54})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1 $ x^9 - 9x^7 + 27x^5 - 30x^3 + 9x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} - \cdots)$

Embedding invariants

Embedding label			982.3
Root			$1.37248$ 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)

$f(q)$	$=$	$q-1.37248 q^{2} +1.78727 q^{3} +0.883710 q^{4} -2.45299 q^{6} -1.67098 q^{7} +0.159606 q^{8} +2.19432 q^{9} +O(q^{10})$	$q-1.37248 q^{2} +1.78727 q^{3} +0.883710 q^{4} -2.45299 q^{6} -1.67098 q^{7} +0.159606 q^{8} +2.19432 q^{9} +1.57942 q^{12} +2.29339 q^{14} -1.10277 q^{16} -3.01166 q^{18} +1.94609 q^{19} -2.98648 q^{21} +0.792160 q^{23} +0.285258 q^{24} +1.00000 q^{25} +2.13456 q^{27} -1.47666 q^{28} +1.53209 q^{31} +1.35392 q^{32} +1.93914 q^{36} +0.347296 q^{37} -2.67098 q^{38} -1.87939 q^{41} +4.09889 q^{42} -0.573606 q^{43} -1.08723 q^{46} -1.98648 q^{47} -1.97094 q^{48} +1.79216 q^{49} -1.37248 q^{50} -0.573606 q^{53} -2.92965 q^{54} -0.266697 q^{56} +3.47818 q^{57} -1.00000 q^{59} -2.10277 q^{62} -3.66665 q^{63} -0.755470 q^{64} -1.87939 q^{67} +1.41580 q^{69} +0.350225 q^{72} -0.476658 q^{74} +1.78727 q^{75} +1.71978 q^{76} -1.00000 q^{79} +1.62071 q^{81} +2.57942 q^{82} -2.63918 q^{84} +0.787265 q^{86} -0.116290 q^{89} +0.700040 q^{92} +2.73825 q^{93} +2.72641 q^{94} +2.41982 q^{96} -2.45971 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^4 + 9 * q^9	$ 9 q + 9 q^{4} + 9 q^{9} - 9 q^{12} + 9 q^{16} - 9 q^{21} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{48} + 9 q^{49} - 9 q^{59} + 9 q^{64} - 9 q^{79} + 9 q^{81} - 9 q^{84} - 9 q^{86}+O(q^{100}) $ 9 * q + 9 * q^4 + 9 * q^9 - 9 * q^12 + 9 * q^16 - 9 * q^21 + 9 * q^25 - 9 * q^28 + 9 * q^36 - 9 * q^38 - 9 * q^48 + 9 * q^49 - 9 * q^59 + 9 * q^64 - 9 * q^79 + 9 * q^81 - 9 * q^84 - 9 * q^86

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 $

$2$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$2$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$3$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$3$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$4$	0.883710	0.883710
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−2.45299	−2.45299
$7$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$7$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$8$	0.159606	0.159606
$9$	2.19432	2.19432
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.57942	1.57942
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	2.29339	2.29339
$15$	0	0
$16$	−1.10277	−1.10277
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−3.01166	−3.01166
$19$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$19$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$20$	0	0
$21$	−2.98648	−2.98648
$22$	0	0
$23$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$23$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$24$	0.285258	0.285258
$25$	1.00000	1.00000
$26$	0	0
$27$	2.13456	2.13456
$28$	−1.47666	−1.47666
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$31$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$32$	1.35392	1.35392
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.93914	1.93914
$37$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$37$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$38$	−2.67098	−2.67098
$39$	0	0
$40$	0	0
$41$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$41$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$42$	4.09889	4.09889
$43$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$43$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$44$	0	0
$45$	0	0
$46$	−1.08723	−1.08723
$47$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$47$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$48$	−1.97094	−1.97094
$49$	1.79216	1.79216
$50$	−1.37248	−1.37248
$51$	0	0
$52$	0	0
$53$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$53$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$54$	−2.92965	−2.92965
$55$	0	0
$56$	−0.266697	−0.266697
$57$	3.47818	3.47818
$58$	0	0
$59$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−2.10277	−2.10277
$63$	−3.66665	−3.66665
$64$	−0.755470	−0.755470
$65$	0	0
$66$	0	0
$67$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$67$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$68$	0	0
$69$	1.41580	1.41580
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.350225	0.350225
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.476658	−0.476658
$75$	1.78727	1.78727
$76$	1.71978	1.71978
$77$	0	0
$78$	0	0
$79$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$80$	0	0
$81$	1.62071	1.62071
$82$	2.57942	2.57942
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.63918	−2.63918
$85$	0	0
$86$	0.787265	0.787265
$87$	0	0
$88$	0	0
$89$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$89$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$90$	0	0
$91$	0	0
$92$	0.700040	0.700040
$93$	2.73825	2.73825
$94$	2.72641	2.72641
$95$	0	0
$96$	2.41982	2.41982
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−2.45971	−2.45971
$99$	0	0
$100$	0.883710	0.883710
$101$	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
$106$	0.787265	0.787265
$107$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$107$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$108$	1.88633	1.88633
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0.620711	0.620711
$112$	1.84270	1.84270
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−4.77374	−4.77374
$115$	0	0
$116$	0	0
$117$	0	0
$118$	1.37248	1.37248
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	−3.35896	−3.35896
$124$	1.35392	1.35392
$125$	0	0
$126$	5.03242	5.03242
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.317053	−0.317053
$129$	−1.02519	−1.02519
$130$	0	0
$131$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$131$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$132$	0	0
$133$	−3.25187	−3.25187
$134$	2.57942	2.57942
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.94316	−1.94316
$139$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$139$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$140$	0	0
$141$	−3.55036	−3.55036
$142$	0	0
$143$	0	0
$144$	−2.41982	−2.41982
$145$	0	0
$146$	0	0
$147$	3.20306	3.20306
$148$	0.306909	0.306909
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.45299	−2.45299
$151$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$151$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$152$	0.310607	0.310607
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	1.37248	1.37248
$159$	−1.02519	−1.02519
$160$	0	0
$161$	−1.32368	−1.32368
$162$	−2.22440	−2.22440
$163$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$163$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$164$	−1.66083	−1.66083
$165$	0	0
$166$	0	0
$167$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$167$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$168$	−0.476658	−0.476658
$169$	1.00000	1.00000
$170$	0	0
$171$	4.27034	4.27034
$172$	−0.506902	−0.506902
$173$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$173$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$174$	0	0
$175$	−1.67098	−1.67098
$176$	0	0
$177$	−1.78727	−1.78727
$178$	0.159606	0.159606
$179$	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
$184$	0.126433	0.126433
$185$	0	0
$186$	−3.75820	−3.75820
$187$	0	0
$188$	−1.75547	−1.75547
$189$	−3.56680	−3.56680
$190$	0	0
$191$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$191$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$192$	−1.35023	−1.35023
$193$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$194$	0	0
$195$	0	0
$196$	1.58375	1.58375
$197$	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			$\pi$
$200$	0.159606	0.159606
$201$	−3.35896	−3.35896
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.73825	1.73825
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.506902	−0.506902
$213$	0	0
$214$	0.159606	0.159606
$215$	0	0
$216$	0.340688	0.340688
$217$	−2.56008	−2.56008
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−0.851915	−0.851915
$223$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$223$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$224$	−2.26237	−2.26237
$225$	2.19432	2.19432
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	3.07370	3.07370
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$234$	0	0
$235$	0	0
$236$	−0.883710	−0.883710
$237$	−1.78727	−1.78727
$238$	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			$\pi$
$242$	−1.37248	−1.37248
$243$	0.762078	0.762078
$244$	0	0
$245$	0	0
$246$	4.61012	4.61012
$247$	0	0
$248$	0.244530	0.244530
$249$	0	0
$250$	0	0
$251$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$251$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$252$	−3.24026	−3.24026
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.19062	1.19062
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	1.40705	1.40705
$259$	−0.580324	−0.580324
$260$	0	0
$261$	0	0
$262$	−1.63918	−1.63918
$263$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$263$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$264$	0	0
$265$	0	0
$266$	4.46314	4.46314
$267$	−0.207840	−0.207840
$268$	−1.66083	−1.66083
$269$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$269$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$270$	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
$276$	1.25116	1.25116
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.72641	2.72641
$279$	3.36189	3.36189
$280$	0	0
$281$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$281$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$282$	4.87281	4.87281
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.14041	3.14041
$288$	2.97094	2.97094
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$293$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$294$	−4.39615	−4.39615
$295$	0	0
$296$	0.0554304	0.0554304
$297$	0	0
$298$	0	0
$299$	0	0
$300$	1.57942	1.57942
$301$	0.958482	0.958482
$302$	1.88371	1.88371
$303$	0	0
$304$	−2.14608	−2.14608
$305$	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
$313$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$313$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$314$	0	0
$315$	0	0
$316$	−0.883710	−0.883710
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	1.40705	1.40705
$319$	0	0
$320$	0	0
$321$	−0.207840	−0.207840
$322$	1.81673	1.81673
$323$	0	0
$324$	1.43224	1.43224
$325$	0	0
$326$	−1.63918	−1.63918
$327$	0	0
$328$	−0.299960	−0.299960
$329$	3.31935	3.31935
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0.762078	0.762078
$334$	0.787265	0.787265
$335$	0	0
$336$	3.29339	3.29339
$337$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$337$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$338$	−1.37248	−1.37248
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−5.86097	−5.86097
$343$	−1.32368	−1.32368
$344$	−0.0915508	−0.0915508
$345$	0	0
$346$	−2.67098	−2.67098
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$349$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$350$	2.29339	2.29339
$351$	0	0
$352$	0	0
$353$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$353$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$354$	2.45299	2.45299
$355$	0	0
$356$	−0.102766	−0.102766
$357$	0	0
$358$	0	0
$359$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$359$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$360$	0	0
$361$	2.78727	2.78727
$362$	0	0
$363$	1.78727	1.78727
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$367$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$368$	−0.873567	−0.873567
$369$	−4.12397	−4.12397
$370$	0	0
$371$	0.958482	0.958482
$372$	2.41982	2.41982
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−0.317053	−0.317053
$377$	0	0
$378$	4.89537	4.89537
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−0.476658	−0.476658
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.566658	−0.566658
$385$	0	0
$386$	1.37248	1.37248
$387$	−1.25867	−1.25867
$388$	0	0
$389$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$389$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$390$	0	0
$391$	0	0
$392$	0.286039	0.286039
$393$	2.13456	2.13456
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	−5.81195	−5.81195
$400$	−1.10277	−1.10277
$401$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$401$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$402$	4.61012	4.61012
$403$	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
$413$	1.67098	1.67098
$414$	−2.38572	−2.38572
$415$	0	0
$416$	0	0
$417$	−3.55036	−3.55036
$418$	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
$423$	−4.35896	−4.35896
$424$	−0.0915508	−0.0915508
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−0.102766	−0.102766
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−2.35392	−2.35392
$433$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$433$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$434$	3.51367	3.51367
$435$	0	0
$436$	0	0
$437$	1.54161	1.54161
$438$	0	0
$439$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$439$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$440$	0	0
$441$	3.93257	3.93257
$442$	0	0
$443$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$443$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$444$	0.548528	0.548528
$445$	0	0
$446$	−1.63918	−1.63918
$447$	0	0
$448$	1.26237	1.26237
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−3.01166	−3.01166
$451$	0	0
$452$	0	0
$453$	−2.45299	−2.45299
$454$	0	0
$455$	0	0
$456$	0.555137	0.555137
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$461$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	1.37248	1.37248
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	3.14041	3.14041
$470$	0	0
$471$	0	0
$472$	−0.159606	−0.159606
$473$	0	0
$474$	2.45299	2.45299
$475$	1.94609	1.94609
$476$	0	0
$477$	−1.25867	−1.25867
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−2.36577	−2.36577
$484$	0.883710	0.883710
$485$	0	0
$486$	−1.04594	−1.04594
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	2.13456	2.13456
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−2.96835	−2.96835
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.68954	−1.68954
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−1.02519	−1.02519
$502$	2.57942	2.57942
$503$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$503$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$504$	−0.585218	−0.585218
$505$	0	0
$506$	0	0
$507$	1.78727	1.78727
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.31705	−1.31705
$513$	4.15405	4.15405
$514$	0	0
$515$	0	0
$516$	−0.905968	−0.905968
$517$	0	0
$518$	0.796485	0.796485
$519$	3.47818	3.47818
$520$	0	0
$521$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$521$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$522$	0	0
$523$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$524$	1.05543	1.05543
$525$	−2.98648	−2.98648
$526$	2.57942	2.57942
$527$	0	0
$528$	0	0
$529$	−0.372483	−0.372483
$530$	0	0
$531$	−2.19432	−2.19432
$532$	−2.87371	−2.87371
$533$	0	0
$534$	0.285258	0.285258
$535$	0	0
$536$	−0.299960	−0.299960
$537$	0	0
$538$	2.29339	2.29339
$539$	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
$547$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$547$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0.225970	0.225970
$553$	1.67098	1.67098
$554$	0	0
$555$	0	0
$556$	−1.75547	−1.75547
$557$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$557$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$558$	−4.61414	−4.61414
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−2.67098	−2.67098
$563$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$563$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$564$	−3.13749	−3.13749
$565$	0	0
$566$	0	0
$567$	−2.70817	−2.70817
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$571$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$572$	0	0
$573$	0.620711	0.620711
$574$	−4.31016	−4.31016
$575$	0.792160	0.792160
$576$	−1.65774	−1.65774
$577$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$577$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$578$	−1.37248	−1.37248
$579$	−1.78727	−1.78727
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−2.10277	−2.10277
$587$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$587$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$588$	2.83058	2.83058
$589$	2.98158	2.98158
$590$	0	0
$591$	0	0
$592$	−0.382987	−0.382987
$593$	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			$\pi$
$600$	0.285258	0.285258
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−1.31550	−1.31550
$603$	−4.12397	−4.12397
$604$	−1.21288	−1.21288
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	2.63486	2.63486
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$613$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$618$	0	0
$619$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$619$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$620$	0	0
$621$	1.69091	1.69091
$622$	0	0
$623$	0.194317	0.194317
$624$	0	0
$625$	1.00000	1.00000
$626$	1.88371	1.88371
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$631$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$632$	−0.159606	−0.159606
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.905968	−0.905968
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0.285258	0.285258
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−1.16975	−1.16975
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.258675	0.258675
$649$	0	0
$650$	0	0
$651$	−4.57555	−4.57555
$652$	1.05543	1.05543
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	2.07252	2.07252
$657$	0	0
$658$	−4.55576	−4.55576
$659$	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
$666$	−1.04594	−1.04594
$667$	0	0
$668$	−0.506902	−0.506902
$669$	2.13456	2.13456
$670$	0	0
$671$	0	0
$672$	−4.04346	−4.04346
$673$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$673$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$674$	−2.10277	−2.10277
$675$	2.13456	2.13456
$676$	0.883710	0.883710
$677$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$677$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$678$	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			$\pi$
$684$	3.77374	3.77374
$685$	0	0
$686$	1.81673	1.81673
$687$	0	0
$688$	0.632554	0.632554
$689$	0	0
$690$	0	0
$691$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$691$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$692$	1.71978	1.71978
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−2.45299	−2.45299
$699$	−1.78727	−1.78727
$700$	−1.47666	−1.47666
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0.675870	0.675870
$704$	0	0
$705$	0	0
$706$	2.72641	2.72641
$707$	0	0
$708$	−1.57942	−1.57942
$709$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$709$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$710$	0	0
$711$	−2.19432	−2.19432
$712$	−0.0185605	−0.0185605
$713$	1.21366	1.21366
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0.159606	0.159606
$719$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$719$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$720$	0	0
$721$	0	0
$722$	−3.82547	−3.82547
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−2.45299	−2.45299
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.258675	−0.258675
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$733$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$734$	2.72641	2.72641
$735$	0	0
$736$	1.07252	1.07252
$737$	0	0
$738$	5.66008	5.66008
$739$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$740$	0	0
$741$	0	0
$742$	−1.31550	−1.31550
$743$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$743$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$744$	0.437040	0.437040
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.194317	0.194317
$750$	0	0
$751$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$751$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$752$	2.19062	2.19062
$753$	−3.35896	−3.35896
$754$	0	0
$755$	0	0
$756$	−3.15202	−3.15202
$757$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$757$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$758$	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
$764$	0.306909	0.306909
$765$	0	0
$766$	0	0
$767$	0	0
$768$	2.12795	2.12795
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−0.883710	−0.883710
$773$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$773$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$774$	1.72751	1.72751
$775$	1.53209	1.53209
$776$	0	0
$777$	−1.03719	−1.03719
$778$	−1.08723	−1.08723
$779$	−3.65745	−3.65745
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.97633	−1.97633
$785$	0	0
$786$	−2.92965	−2.92965
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	−3.35896	−3.35896
$790$	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			$\pi$
$798$	7.97681	7.97681
$799$	0	0
$800$	1.35392	1.35392
$801$	−0.255176	−0.255176
$802$	2.72641	2.72641
$803$	0	0
$804$	−2.96835	−2.96835
$805$	0	0
$806$	0	0
$807$	−2.98648	−2.98648
$808$	0	0
$809$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$809$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$810$	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
$817$	−1.11629	−1.11629
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$823$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$824$	0	0
$825$	0	0
$826$	−2.29339	−2.29339
$827$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$827$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$828$	1.53611	1.53611
$829$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	4.87281	4.87281
$835$	0	0
$836$	0	0
$837$	3.27034	3.27034
$838$	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
$843$	3.47818	3.47818
$844$	0	0
$845$	0	0
$846$	5.98260	5.98260
$847$	−1.67098	−1.67098
$848$	0.632554	0.632554
$849$	0	0
$850$	0	0
$851$	0.275114	0.275114
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−0.0185605	−0.0185605
$857$	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
$861$	5.61274	5.61274
$862$	0	0
$863$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$863$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$864$	2.89003	2.89003
$865$	0	0
$866$	2.57942	2.57942
$867$	1.78727	1.78727
$868$	−2.26237	−2.26237
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	−2.11584	−2.11584
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.476658	−0.476658
$879$	2.73825	2.73825
$880$	0	0
$881$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$881$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$882$	−5.39738	−5.39738
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−2.10277	−2.10277
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0.0990689	0.0990689
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1.05543	1.05543
$893$	−3.86586	−3.86586
$894$	0	0
$895$	0	0
$896$	0.529788	0.529788
$897$	0	0
$898$	0	0
$899$	0	0
$900$	1.93914	1.93914
$901$	0	0
$902$	0	0
$903$	1.71306	1.71306
$904$	0	0
$905$	0	0
$906$	3.36669	3.36669
$907$	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			$\pi$
$912$	−3.83562	−3.83562
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.99567	−1.99567
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−2.45299	−2.45299
$923$	0	0
$924$	0	0
$925$	0.347296	0.347296
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	3.48770	3.48770
$932$	−0.883710	−0.883710
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−4.31016	−4.31016
$939$	−2.45299	−2.45299
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−1.48877	−1.48877
$944$	1.10277	1.10277
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1.57942	−1.57942
$949$	0	0
$950$	−2.67098	−2.67098
$951$	0	0
$952$	0	0
$953$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$953$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$954$	1.72751	1.72751
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.34730	1.34730
$962$	0	0
$963$	−0.255176	−0.255176
$964$	0	0
$965$	0	0
$966$	3.24697	3.24697
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.159606	0.159606
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.673457	0.673457
$973$	3.31935	3.31935
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−2.92965	−2.92965
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.00000	1.00000
$983$	1.00000	1.00000
$984$	−0.536109	−0.536109
$985$	0	0
$986$	0	0
$987$	5.93257	5.93257
$988$	0	0
$989$	−0.454388	−0.454388
$990$	0	0
$991$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$991$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$992$	2.07433	2.07433
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$997$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$998$	0	0
$999$	0.741325	0.741325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.1.b.c.982.3	✓	9
983.982	odd	2	CM	983.1.b.c.982.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.1.b.c.982.3	✓	9	1.1	even	1	trivial
983.1.b.c.982.3	✓	9	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)

\(f(q)\)	\(=\)	\(q-1.37248 q^{2} +1.78727 q^{3} +0.883710 q^{4} -2.45299 q^{6} -1.67098 q^{7} +0.159606 q^{8} +2.19432 q^{9} +O(q^{10})\)	\(q-1.37248 q^{2} +1.78727 q^{3} +0.883710 q^{4} -2.45299 q^{6} -1.67098 q^{7} +0.159606 q^{8} +2.19432 q^{9} +1.57942 q^{12} +2.29339 q^{14} -1.10277 q^{16} -3.01166 q^{18} +1.94609 q^{19} -2.98648 q^{21} +0.792160 q^{23} +0.285258 q^{24} +1.00000 q^{25} +2.13456 q^{27} -1.47666 q^{28} +1.53209 q^{31} +1.35392 q^{32} +1.93914 q^{36} +0.347296 q^{37} -2.67098 q^{38} -1.87939 q^{41} +4.09889 q^{42} -0.573606 q^{43} -1.08723 q^{46} -1.98648 q^{47} -1.97094 q^{48} +1.79216 q^{49} -1.37248 q^{50} -0.573606 q^{53} -2.92965 q^{54} -0.266697 q^{56} +3.47818 q^{57} -1.00000 q^{59} -2.10277 q^{62} -3.66665 q^{63} -0.755470 q^{64} -1.87939 q^{67} +1.41580 q^{69} +0.350225 q^{72} -0.476658 q^{74} +1.78727 q^{75} +1.71978 q^{76} -1.00000 q^{79} +1.62071 q^{81} +2.57942 q^{82} -2.63918 q^{84} +0.787265 q^{86} -0.116290 q^{89} +0.700040 q^{92} +2.73825 q^{93} +2.72641 q^{94} +2.41982 q^{96} -2.45971 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^4 + 9 * q^9	\( 9 q + 9 q^{4} + 9 q^{9} - 9 q^{12} + 9 q^{16} - 9 q^{21} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{48} + 9 q^{49} - 9 q^{59} + 9 q^{64} - 9 q^{79} + 9 q^{81} - 9 q^{84} - 9 q^{86}+O(q^{100}) \) 9 * q + 9 * q^4 + 9 * q^9 - 9 * q^12 + 9 * q^16 - 9 * q^21 + 9 * q^25 - 9 * q^28 + 9 * q^36 - 9 * q^38 - 9 * q^48 + 9 * q^49 - 9 * q^59 + 9 * q^64 - 9 * q^79 + 9 * q^81 - 9 * q^84 - 9 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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