LMFDB - Embedded newform 983.1.b.c.982.8

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			982.8
Root			$-1.78727$ 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)

$f(q)$	$=$	$q+1.78727 q^{2} +0.792160 q^{3} +2.19432 q^{4} +1.41580 q^{6} -1.98648 q^{7} +2.13456 q^{8} -0.372483 q^{9} +O(q^{10})$	$q+1.78727 q^{2} +0.792160 q^{3} +2.19432 q^{4} +1.41580 q^{6} -1.98648 q^{7} +2.13456 q^{8} -0.372483 q^{9} +1.73825 q^{12} -3.55036 q^{14} +1.62071 q^{16} -0.665726 q^{18} -1.67098 q^{19} -1.57361 q^{21} +1.94609 q^{23} +1.69091 q^{24} +1.00000 q^{25} -1.08723 q^{27} -4.35896 q^{28} +0.347296 q^{31} +0.762078 q^{32} -0.817346 q^{36} -1.87939 q^{37} -2.98648 q^{38} +1.53209 q^{41} -2.81245 q^{42} -0.116290 q^{43} +3.47818 q^{46} -0.573606 q^{47} +1.28386 q^{48} +2.94609 q^{49} +1.78727 q^{50} -0.116290 q^{53} -1.94316 q^{54} -4.24026 q^{56} -1.32368 q^{57} -1.00000 q^{59} +0.620711 q^{62} +0.739929 q^{63} -0.258675 q^{64} +1.53209 q^{67} +1.54161 q^{69} -0.795089 q^{72} -3.35896 q^{74} +0.792160 q^{75} -3.66665 q^{76} -1.00000 q^{79} -0.488773 q^{81} +2.73825 q^{82} -3.45299 q^{84} -0.207840 q^{86} +1.19432 q^{89} +4.27034 q^{92} +0.275114 q^{93} -1.02519 q^{94} +0.603688 q^{96} +5.26544 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$2$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$3$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$3$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$4$	2.19432	2.19432
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	1.41580	1.41580
$7$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$7$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$8$	2.13456	2.13456
$9$	−0.372483	−0.372483
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.73825	1.73825
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−3.55036	−3.55036
$15$	0	0
$16$	1.62071	1.62071
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−0.665726	−0.665726
$19$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$19$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$20$	0	0
$21$	−1.57361	−1.57361
$22$	0	0
$23$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$23$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$24$	1.69091	1.69091
$25$	1.00000	1.00000
$26$	0	0
$27$	−1.08723	−1.08723
$28$	−4.35896	−4.35896
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$31$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$32$	0.762078	0.762078
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−0.817346	−0.817346
$37$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$37$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$38$	−2.98648	−2.98648
$39$	0	0
$40$	0	0
$41$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$41$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$42$	−2.81245	−2.81245
$43$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$43$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$44$	0	0
$45$	0	0
$46$	3.47818	3.47818
$47$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$47$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$48$	1.28386	1.28386
$49$	2.94609	2.94609
$50$	1.78727	1.78727
$51$	0	0
$52$	0	0
$53$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$53$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$54$	−1.94316	−1.94316
$55$	0	0
$56$	−4.24026	−4.24026
$57$	−1.32368	−1.32368
$58$	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			$\pi$
$62$	0.620711	0.620711
$63$	0.739929	0.739929
$64$	−0.258675	−0.258675
$65$	0	0
$66$	0	0
$67$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$67$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$68$	0	0
$69$	1.54161	1.54161
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.795089	−0.795089
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−3.35896	−3.35896
$75$	0.792160	0.792160
$76$	−3.66665	−3.66665
$77$	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
$81$	−0.488773	−0.488773
$82$	2.73825	2.73825
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−3.45299	−3.45299
$85$	0	0
$86$	−0.207840	−0.207840
$87$	0	0
$88$	0	0
$89$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$89$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$90$	0	0
$91$	0	0
$92$	4.27034	4.27034
$93$	0.275114	0.275114
$94$	−1.02519	−1.02519
$95$	0	0
$96$	0.603688	0.603688
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	5.26544	5.26544
$99$	0	0
$100$	2.19432	2.19432
$101$	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
$106$	−0.207840	−0.207840
$107$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$107$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$108$	−2.38572	−2.38572
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−1.48877	−1.48877
$112$	−3.21950	−3.21950
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−2.36577	−2.36577
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.78727	−1.78727
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	1.21366	1.21366
$124$	0.762078	0.762078
$125$	0	0
$126$	1.32245	1.32245
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−1.22440	−1.22440
$129$	−0.0921200	−0.0921200
$130$	0	0
$131$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$131$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$132$	0	0
$133$	3.31935	3.31935
$134$	2.73825	2.73825
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	2.75527	2.75527
$139$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$139$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$140$	0	0
$141$	−0.454388	−0.454388
$142$	0	0
$143$	0	0
$144$	−0.603688	−0.603688
$145$	0	0
$146$	0	0
$147$	2.33377	2.33377
$148$	−4.12397	−4.12397
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.41580	1.41580
$151$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$151$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$152$	−3.56680	−3.56680
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−1.78727	−1.78727
$159$	−0.0921200	−0.0921200
$160$	0	0
$161$	−3.86586	−3.86586
$162$	−0.873567	−0.873567
$163$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$163$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$164$	3.36189	3.36189
$165$	0	0
$166$	0	0
$167$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$167$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$168$	−3.35896	−3.35896
$169$	1.00000	1.00000
$170$	0	0
$171$	0.622410	0.622410
$172$	−0.255176	−0.255176
$173$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$173$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$174$	0	0
$175$	−1.98648	−1.98648
$176$	0	0
$177$	−0.792160	−0.792160
$178$	2.13456	2.13456
$179$	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
$184$	4.15405	4.15405
$185$	0	0
$186$	0.491702	0.491702
$187$	0	0
$188$	−1.25867	−1.25867
$189$	2.15975	2.15975
$190$	0	0
$191$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$191$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$192$	−0.204911	−0.204911
$193$	−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
$196$	6.46466	6.46466
$197$	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			$\pi$
$200$	2.13456	2.13456
$201$	1.21366	1.21366
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.724886	−0.724886
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.255176	−0.255176
$213$	0	0
$214$	2.13456	2.13456
$215$	0	0
$216$	−2.32075	−2.32075
$217$	−0.689896	−0.689896
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−2.66083	−2.66083
$223$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$223$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$224$	−1.51385	−1.51385
$225$	−0.372483	−0.372483
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−2.90457	−2.90457
$229$	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
$236$	−2.19432	−2.19432
$237$	−0.792160	−0.792160
$238$	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			$\pi$
$242$	1.78727	1.78727
$243$	0.700040	0.700040
$244$	0	0
$245$	0	0
$246$	2.16913	2.16913
$247$	0	0
$248$	0.741325	0.741325
$249$	0	0
$250$	0	0
$251$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$251$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$252$	1.62364	1.62364
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−1.92965	−1.92965
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−0.164643	−0.164643
$259$	3.73336	3.73336
$260$	0	0
$261$	0	0
$262$	−2.45299	−2.45299
$263$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$263$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$264$	0	0
$265$	0	0
$266$	5.93257	5.93257
$267$	0.946090	0.946090
$268$	3.36189	3.36189
$269$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$269$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$270$	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
$276$	3.38279	3.38279
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.02519	−1.02519
$279$	−0.129362	−0.129362
$280$	0	0
$281$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$281$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$282$	−0.812112	−0.812112
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.04346	−3.04346
$288$	−0.283861	−0.283861
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$293$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$294$	4.17107	4.17107
$295$	0	0
$296$	−4.01166	−4.01166
$297$	0	0
$298$	0	0
$299$	0	0
$300$	1.73825	1.73825
$301$	0.231007	0.231007
$302$	3.19432	3.19432
$303$	0	0
$304$	−2.70817	−2.70817
$305$	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
$313$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$313$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$314$	0	0
$315$	0	0
$316$	−2.19432	−2.19432
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.164643	−0.164643
$319$	0	0
$320$	0	0
$321$	0.946090	0.946090
$322$	−6.90932	−6.90932
$323$	0	0
$324$	−1.07252	−1.07252
$325$	0	0
$326$	−2.45299	−2.45299
$327$	0	0
$328$	3.27034	3.27034
$329$	1.13946	1.13946
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0.700040	0.700040
$334$	−0.207840	−0.207840
$335$	0	0
$336$	−2.55036	−2.55036
$337$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$337$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$338$	1.78727	1.78727
$339$	0	0
$340$	0	0
$341$	0	0
$342$	1.11241	1.11241
$343$	−3.86586	−3.86586
$344$	−0.248227	−0.248227
$345$	0	0
$346$	−2.98648	−2.98648
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$349$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$350$	−3.55036	−3.55036
$351$	0	0
$352$	0	0
$353$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$353$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$354$	−1.41580	−1.41580
$355$	0	0
$356$	2.62071	2.62071
$357$	0	0
$358$	0	0
$359$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$359$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$360$	0	0
$361$	1.79216	1.79216
$362$	0	0
$363$	0.792160	0.792160
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$367$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$368$	3.15405	3.15405
$369$	−0.570677	−0.570677
$370$	0	0
$371$	0.231007	0.231007
$372$	0.603688	0.603688
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−1.22440	−1.22440
$377$	0	0
$378$	3.86004	3.86004
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−3.35896	−3.35896
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.969919	−0.969919
$385$	0	0
$386$	−1.78727	−1.78727
$387$	0.0433160	0.0433160
$388$	0	0
$389$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$389$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$390$	0	0
$391$	0	0
$392$	6.28861	6.28861
$393$	−1.08723	−1.08723
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	2.62946	2.62946
$400$	1.62071	1.62071
$401$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$401$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$402$	2.16913	2.16913
$403$	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
$413$	1.98648	1.98648
$414$	−1.29556	−1.29556
$415$	0	0
$416$	0	0
$417$	−0.454388	−0.454388
$418$	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
$423$	0.213659	0.213659
$424$	−0.248227	−0.248227
$425$	0	0
$426$	0	0
$427$	0	0
$428$	2.62071	2.62071
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.76208	−1.76208
$433$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$433$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$434$	−1.23303	−1.23303
$435$	0	0
$436$	0	0
$437$	−3.25187	−3.25187
$438$	0	0
$439$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$439$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$440$	0	0
$441$	−1.09737	−1.09737
$442$	0	0
$443$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$443$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$444$	−3.26684	−3.26684
$445$	0	0
$446$	−2.45299	−2.45299
$447$	0	0
$448$	0.513851	0.513851
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.665726	−0.665726
$451$	0	0
$452$	0	0
$453$	1.41580	1.41580
$454$	0	0
$455$	0	0
$456$	−2.82547	−2.82547
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$461$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−1.78727	−1.78727
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	−3.04346	−3.04346
$470$	0	0
$471$	0	0
$472$	−2.13456	−2.13456
$473$	0	0
$474$	−1.41580	−1.41580
$475$	−1.67098	−1.67098
$476$	0	0
$477$	0.0433160	0.0433160
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.06238	−3.06238
$484$	2.19432	2.19432
$485$	0	0
$486$	1.25116	1.25116
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−1.08723	−1.08723
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	2.66315	2.66315
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.562867	0.562867
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−0.0921200	−0.0921200
$502$	2.73825	2.73825
$503$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$503$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$504$	1.57942	1.57942
$505$	0	0
$506$	0	0
$507$	0.792160	0.792160
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−2.22440	−2.22440
$513$	1.81673	1.81673
$514$	0	0
$515$	0	0
$516$	−0.202140	−0.202140
$517$	0	0
$518$	6.67250	6.67250
$519$	−1.32368	−1.32368
$520$	0	0
$521$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$521$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$522$	0	0
$523$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$524$	−3.01166	−3.01166
$525$	−1.57361	−1.57361
$526$	2.73825	2.73825
$527$	0	0
$528$	0	0
$529$	2.78727	2.78727
$530$	0	0
$531$	0.372483	0.372483
$532$	7.28372	7.28372
$533$	0	0
$534$	1.69091	1.69091
$535$	0	0
$536$	3.27034	3.27034
$537$	0	0
$538$	−3.55036	−3.55036
$539$	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
$547$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$547$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	3.29067	3.29067
$553$	1.98648	1.98648
$554$	0	0
$555$	0	0
$556$	−1.25867	−1.25867
$557$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$557$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$558$	−0.231204	−0.231204
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−2.98648	−2.98648
$563$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$563$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$564$	−0.997071	−0.997071
$565$	0	0
$566$	0	0
$567$	0.970936	0.970936
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$571$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$572$	0	0
$573$	−1.48877	−1.48877
$574$	−5.43947	−5.43947
$575$	1.94609	1.94609
$576$	0.0963519	0.0963519
$577$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$577$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$578$	1.78727	1.78727
$579$	−0.792160	−0.792160
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0.620711	0.620711
$587$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$587$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$588$	5.12104	5.12104
$589$	−0.580324	−0.580324
$590$	0	0
$591$	0	0
$592$	−3.04594	−3.04594
$593$	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			$\pi$
$600$	1.69091	1.69091
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0.412870	0.412870
$603$	−0.570677	−0.570677
$604$	3.92183	3.92183
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−1.27341	−1.27341
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$613$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$614$	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
$619$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$619$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$620$	0	0
$621$	−2.11584	−2.11584
$622$	0	0
$623$	−2.37248	−2.37248
$624$	0	0
$625$	1.00000	1.00000
$626$	3.19432	3.19432
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$631$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$632$	−2.13456	−2.13456
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.202140	−0.202140
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.69091	1.69091
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−8.48293	−8.48293
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.04332	−1.04332
$649$	0	0
$650$	0	0
$651$	−0.546508	−0.546508
$652$	−3.01166	−3.01166
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	2.48307	2.48307
$657$	0	0
$658$	2.03651	2.03651
$659$	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
$666$	1.25116	1.25116
$667$	0	0
$668$	−0.255176	−0.255176
$669$	−1.08723	−1.08723
$670$	0	0
$671$	0	0
$672$	−1.19921	−1.19921
$673$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$673$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$674$	0.620711	0.620711
$675$	−1.08723	−1.08723
$676$	2.19432	2.19432
$677$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$677$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$678$	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			$\pi$
$684$	1.36577	1.36577
$685$	0	0
$686$	−6.90932	−6.90932
$687$	0	0
$688$	−0.188472	−0.188472
$689$	0	0
$690$	0	0
$691$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$691$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$692$	−3.66665	−3.66665
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.41580	1.41580
$699$	−0.792160	−0.792160
$700$	−4.35896	−4.35896
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	3.14041	3.14041
$704$	0	0
$705$	0	0
$706$	−1.02519	−1.02519
$707$	0	0
$708$	−1.73825	−1.73825
$709$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$709$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$710$	0	0
$711$	0.372483	0.372483
$712$	2.54934	2.54934
$713$	0.675870	0.675870
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	2.13456	2.13456
$719$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$719$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$720$	0	0
$721$	0	0
$722$	3.20306	3.20306
$723$	0	0
$724$	0	0
$725$	0	0
$726$	1.41580	1.41580
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.04332	1.04332
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$733$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$734$	−1.02519	−1.02519
$735$	0	0
$736$	1.48307	1.48307
$737$	0	0
$738$	−1.01995	−1.01995
$739$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$739$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$740$	0	0
$741$	0	0
$742$	0.412870	0.412870
$743$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$743$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$744$	0.587248	0.587248
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.37248	−2.37248
$750$	0	0
$751$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$751$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$752$	−0.929650	−0.929650
$753$	1.21366	1.21366
$754$	0	0
$755$	0	0
$756$	4.73917	4.73917
$757$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$757$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$758$	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
$764$	−4.12397	−4.12397
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.52859	−1.52859
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−2.19432	−2.19432
$773$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$773$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$774$	0.0774171	0.0774171
$775$	0.347296	0.347296
$776$	0	0
$777$	2.95741	2.95741
$778$	3.47818	3.47818
$779$	−2.56008	−2.56008
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	4.77476	4.77476
$785$	0	0
$786$	−1.94316	−1.94316
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	1.21366	1.21366
$790$	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			$\pi$
$798$	4.69954	4.69954
$799$	0	0
$800$	0.762078	0.762078
$801$	−0.444863	−0.444863
$802$	−1.02519	−1.02519
$803$	0	0
$804$	2.66315	2.66315
$805$	0	0
$806$	0	0
$807$	−1.57361	−1.57361
$808$	0	0
$809$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$809$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$810$	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
$817$	0.194317	0.194317
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$823$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$824$	0	0
$825$	0	0
$826$	3.55036	3.55036
$827$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$827$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$828$	−1.59063	−1.59063
$829$	−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
$834$	−0.812112	−0.812112
$835$	0	0
$836$	0	0
$837$	−0.377590	−0.377590
$838$	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
$843$	−1.32368	−1.32368
$844$	0	0
$845$	0	0
$846$	0.381865	0.381865
$847$	−1.98648	−1.98648
$848$	−0.188472	−0.188472
$849$	0	0
$850$	0	0
$851$	−3.65745	−3.65745
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	2.54934	2.54934
$857$	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
$861$	−2.41090	−2.41090
$862$	0	0
$863$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$863$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$864$	−0.828551	−0.828551
$865$	0	0
$866$	2.73825	2.73825
$867$	0.792160	0.792160
$868$	−1.51385	−1.51385
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	−5.81195	−5.81195
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−3.35896	−3.35896
$879$	0.275114	0.275114
$880$	0	0
$881$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$881$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$882$	−1.96129	−1.96129
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0.620711	0.620711
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−3.17788	−3.17788
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−3.01166	−3.01166
$893$	0.958482	0.958482
$894$	0	0
$895$	0	0
$896$	2.43224	2.43224
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−0.817346	−0.817346
$901$	0	0
$902$	0	0
$903$	0.182994	0.182994
$904$	0	0
$905$	0	0
$906$	2.53041	2.53041
$907$	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			$\pi$
$912$	−2.14530	−2.14530
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.72641	2.72641
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1.41580	1.41580
$923$	0	0
$924$	0	0
$925$	−1.87939	−1.87939
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−4.92284	−4.92284
$932$	−2.19432	−2.19432
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−5.43947	−5.43947
$939$	1.41580	1.41580
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	2.98158	2.98158
$944$	−1.62071	−1.62071
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1.73825	−1.73825
$949$	0	0
$950$	−2.98648	−2.98648
$951$	0	0
$952$	0	0
$953$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$953$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$954$	0.0774171	0.0774171
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−0.879385	−0.879385
$962$	0	0
$963$	−0.444863	−0.444863
$964$	0	0
$965$	0	0
$966$	−5.47328	−5.47328
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	2.13456	2.13456
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1.53611	1.53611
$973$	1.13946	1.13946
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1.94316	−1.94316
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.00000	1.00000
$983$	1.00000	1.00000
$984$	2.59063	2.59063
$985$	0	0
$986$	0	0
$987$	0.902631	0.902631
$988$	0	0
$989$	−0.226310	−0.226310
$990$	0	0
$991$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$991$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$992$	0.264667	0.264667
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$997$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$998$	0	0
$999$	2.04332	2.04332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.1.b.c.982.8	✓	9
983.982	odd	2	CM	983.1.b.c.982.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.1.b.c.982.8	✓	9	1.1	even	1	trivial
983.1.b.c.982.8	✓	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)

\(f(q)\)	\(=\)	\(q+1.78727 q^{2} +0.792160 q^{3} +2.19432 q^{4} +1.41580 q^{6} -1.98648 q^{7} +2.13456 q^{8} -0.372483 q^{9} +O(q^{10})\)	\(q+1.78727 q^{2} +0.792160 q^{3} +2.19432 q^{4} +1.41580 q^{6} -1.98648 q^{7} +2.13456 q^{8} -0.372483 q^{9} +1.73825 q^{12} -3.55036 q^{14} +1.62071 q^{16} -0.665726 q^{18} -1.67098 q^{19} -1.57361 q^{21} +1.94609 q^{23} +1.69091 q^{24} +1.00000 q^{25} -1.08723 q^{27} -4.35896 q^{28} +0.347296 q^{31} +0.762078 q^{32} -0.817346 q^{36} -1.87939 q^{37} -2.98648 q^{38} +1.53209 q^{41} -2.81245 q^{42} -0.116290 q^{43} +3.47818 q^{46} -0.573606 q^{47} +1.28386 q^{48} +2.94609 q^{49} +1.78727 q^{50} -0.116290 q^{53} -1.94316 q^{54} -4.24026 q^{56} -1.32368 q^{57} -1.00000 q^{59} +0.620711 q^{62} +0.739929 q^{63} -0.258675 q^{64} +1.53209 q^{67} +1.54161 q^{69} -0.795089 q^{72} -3.35896 q^{74} +0.792160 q^{75} -3.66665 q^{76} -1.00000 q^{79} -0.488773 q^{81} +2.73825 q^{82} -3.45299 q^{84} -0.207840 q^{86} +1.19432 q^{89} +4.27034 q^{92} +0.275114 q^{93} -1.02519 q^{94} +0.603688 q^{96} +5.26544 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) \) 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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