LMFDB - Embedded newform 983.1.b.c.982.4

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			982.4
Root			$0.573606$ of defining polynomial
Character	$\chi$	$=$	983.982

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.573606 q^{2} -0.116290 q^{3} -0.670976 q^{4} +0.0667045 q^{6} +1.78727 q^{7} +0.958482 q^{8} -0.986477 q^{9} +O(q^{10})$	$q-0.573606 q^{2} -0.116290 q^{3} -0.670976 q^{4} +0.0667045 q^{6} +1.78727 q^{7} +0.958482 q^{8} -0.986477 q^{9} +0.0780275 q^{12} -1.02519 q^{14} +0.121184 q^{16} +0.565849 q^{18} -1.37248 q^{19} -0.207840 q^{21} +1.19432 q^{23} -0.111462 q^{24} +1.00000 q^{25} +0.231007 q^{27} -1.19921 q^{28} +1.53209 q^{31} -1.02799 q^{32} +0.661902 q^{36} +0.347296 q^{37} +0.787265 q^{38} -1.87939 q^{41} +0.119219 q^{42} +1.94609 q^{43} -0.685068 q^{46} +0.792160 q^{47} -0.0140924 q^{48} +2.19432 q^{49} -0.573606 q^{50} +1.94609 q^{53} -0.132507 q^{54} +1.71306 q^{56} +0.159606 q^{57} -1.00000 q^{59} -0.878816 q^{62} -1.76310 q^{63} +0.468480 q^{64} -1.87939 q^{67} -0.138887 q^{69} -0.945521 q^{72} -0.199211 q^{74} -0.116290 q^{75} +0.920903 q^{76} -1.00000 q^{79} +0.959613 q^{81} +1.07803 q^{82} +0.139456 q^{84} -1.11629 q^{86} -1.67098 q^{89} -0.801358 q^{92} -0.178166 q^{93} -0.454388 q^{94} +0.119545 q^{96} -1.25867 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$2$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$3$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$3$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$4$	−0.670976	−0.670976
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0.0667045	0.0667045
$7$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$7$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$8$	0.958482	0.958482
$9$	−0.986477	−0.986477
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0.0780275	0.0780275
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−1.02519	−1.02519
$15$	0	0
$16$	0.121184	0.121184
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.565849	0.565849
$19$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$19$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$20$	0	0
$21$	−0.207840	−0.207840
$22$	0	0
$23$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$23$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$24$	−0.111462	−0.111462
$25$	1.00000	1.00000
$26$	0	0
$27$	0.231007	0.231007
$28$	−1.19921	−1.19921
$29$	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	$0.222222\pi$
$32$	−1.02799	−1.02799
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0.661902	0.661902
$37$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$37$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$38$	0.787265	0.787265
$39$	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	$0.888889\pi$
$42$	0.119219	0.119219
$43$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$43$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$44$	0	0
$45$	0	0
$46$	−0.685068	−0.685068
$47$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$47$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$48$	−0.0140924	−0.0140924
$49$	2.19432	2.19432
$50$	−0.573606	−0.573606
$51$	0	0
$52$	0	0
$53$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$53$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$54$	−0.132507	−0.132507
$55$	0	0
$56$	1.71306	1.71306
$57$	0.159606	0.159606
$58$	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			$\pi$
$62$	−0.878816	−0.878816
$63$	−1.76310	−1.76310
$64$	0.468480	0.468480
$65$	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
$69$	−0.138887	−0.138887
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.945521	−0.945521
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−0.199211	−0.199211
$75$	−0.116290	−0.116290
$76$	0.920903	0.920903
$77$	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
$81$	0.959613	0.959613
$82$	1.07803	1.07803
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.139456	0.139456
$85$	0	0
$86$	−1.11629	−1.11629
$87$	0	0
$88$	0	0
$89$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$89$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$90$	0	0
$91$	0	0
$92$	−0.801358	−0.801358
$93$	−0.178166	−0.178166
$94$	−0.454388	−0.454388
$95$	0	0
$96$	0.119545	0.119545
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.25867	−1.25867
$99$	0	0
$100$	−0.670976	−0.670976
$101$	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
$106$	−1.11629	−1.11629
$107$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$107$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$108$	−0.155000	−0.155000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−0.0403870	−0.0403870
$112$	0.216588	0.216588
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−0.0915508	−0.0915508
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0.573606	0.573606
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0.218553	0.218553
$124$	−1.02799	−1.02799
$125$	0	0
$126$	1.01132	1.01132
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.759271	0.759271
$129$	−0.226310	−0.226310
$130$	0	0
$131$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$131$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$132$	0	0
$133$	−2.45299	−2.45299
$134$	1.07803	1.07803
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.0796663	0.0796663
$139$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$139$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$140$	0	0
$141$	−0.0921200	−0.0921200
$142$	0	0
$143$	0	0
$144$	−0.119545	−0.119545
$145$	0	0
$146$	0	0
$147$	−0.255176	−0.255176
$148$	−0.233027	−0.233027
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.0667045	0.0667045
$151$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$151$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$152$	−1.31550	−1.31550
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0.573606	0.573606
$159$	−0.226310	−0.226310
$160$	0	0
$161$	2.13456	2.13456
$162$	−0.550440	−0.550440
$163$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$163$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$164$	1.26102	1.26102
$165$	0	0
$166$	0	0
$167$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$167$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$168$	−0.199211	−0.199211
$169$	1.00000	1.00000
$170$	0	0
$171$	1.35392	1.35392
$172$	−1.30578	−1.30578
$173$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$173$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$174$	0	0
$175$	1.78727	1.78727
$176$	0	0
$177$	0.116290	0.116290
$178$	0.958482	0.958482
$179$	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
$184$	1.14473	1.14473
$185$	0	0
$186$	0.102197	0.102197
$187$	0	0
$188$	−0.531520	−0.531520
$189$	0.412870	0.412870
$190$	0	0
$191$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$191$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$192$	−0.0544794	−0.0544794
$193$	−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
$196$	−1.47233	−1.47233
$197$	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			$\pi$
$200$	0.958482	0.958482
$201$	0.218553	0.218553
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.17817	−1.17817
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−1.30578	−1.30578
$213$	0	0
$214$	0.958482	0.958482
$215$	0	0
$216$	0.221416	0.221416
$217$	2.73825	2.73825
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0.0231662	0.0231662
$223$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$223$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$224$	−1.83730	−1.83730
$225$	−0.986477	−0.986477
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−0.107091	−0.107091
$229$	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
$236$	0.670976	0.670976
$237$	0.116290	0.116290
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−0.573606	−0.573606
$243$	−0.342600	−0.342600
$244$	0	0
$245$	0	0
$246$	−0.125363	−0.125363
$247$	0	0
$248$	1.46848	1.46848
$249$	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	$0.888889\pi$
$252$	1.18299	1.18299
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.904003	−0.904003
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0.129813	0.129813
$259$	0.620711	0.620711
$260$	0	0
$261$	0	0
$262$	1.13946	1.13946
$263$	−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
$266$	1.40705	1.40705
$267$	0.194317	0.194317
$268$	1.26102	1.26102
$269$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$269$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$270$	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
$276$	0.0931896	0.0931896
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−0.454388	−0.454388
$279$	−1.51137	−1.51137
$280$	0	0
$281$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$281$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$282$	0.0528406	0.0528406
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.35896	−3.35896
$288$	1.01409	1.01409
$289$	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	$0.222222\pi$
$294$	0.146371	0.146371
$295$	0	0
$296$	0.332877	0.332877
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0.0780275	0.0780275
$301$	3.47818	3.47818
$302$	0.329024	0.329024
$303$	0	0
$304$	−0.166323	−0.166323
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$313$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$314$	0	0
$315$	0	0
$316$	0.670976	0.670976
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.129813	0.129813
$319$	0	0
$320$	0	0
$321$	0.194317	0.194317
$322$	−1.22440	−1.22440
$323$	0	0
$324$	−0.643877	−0.643877
$325$	0	0
$326$	1.13946	1.13946
$327$	0	0
$328$	−1.80136	−1.80136
$329$	1.41580	1.41580
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−0.342600	−0.342600
$334$	−1.11629	−1.11629
$335$	0	0
$336$	−0.0251869	−0.0251869
$337$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$337$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$338$	−0.573606	−0.573606
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−0.776619	−0.776619
$343$	2.13456	2.13456
$344$	1.86529	1.86529
$345$	0	0
$346$	0.787265	0.787265
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$349$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$350$	−1.02519	−1.02519
$351$	0	0
$352$	0	0
$353$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$353$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$354$	−0.0667045	−0.0667045
$355$	0	0
$356$	1.12118	1.12118
$357$	0	0
$358$	0	0
$359$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$359$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$360$	0	0
$361$	0.883710	0.883710
$362$	0	0
$363$	−0.116290	−0.116290
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$367$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$368$	0.144732	0.144732
$369$	1.85397	1.85397
$370$	0	0
$371$	3.47818	3.47818
$372$	0.119545	0.119545
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0.759271	0.759271
$377$	0	0
$378$	−0.236825	−0.236825
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−0.199211	−0.199211
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.0882954	−0.0882954
$385$	0	0
$386$	0.573606	0.573606
$387$	−1.91977	−1.91977
$388$	0	0
$389$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$389$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$390$	0	0
$391$	0	0
$392$	2.10321	2.10321
$393$	0.231007	0.231007
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0.285258	0.285258
$400$	0.121184	0.121184
$401$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$401$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$402$	−0.125363	−0.125363
$403$	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
$413$	−1.78727	−1.78727
$414$	0.675804	0.675804
$415$	0	0
$416$	0	0
$417$	−0.0921200	−0.0921200
$418$	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
$423$	−0.781447	−0.781447
$424$	1.86529	1.86529
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.12118	1.12118
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.0279943	0.0279943
$433$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$433$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$434$	−1.57068	−1.57068
$435$	0	0
$436$	0	0
$437$	−1.63918	−1.63918
$438$	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
$441$	−2.16464	−2.16464
$442$	0	0
$443$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$443$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$444$	0.0270987	0.0270987
$445$	0	0
$446$	1.13946	1.13946
$447$	0	0
$448$	0.837299	0.837299
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.565849	0.565849
$451$	0	0
$452$	0	0
$453$	0.0667045	0.0667045
$454$	0	0
$455$	0	0
$456$	0.152979	0.152979
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$461$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0.573606	0.573606
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	−3.35896	−3.35896
$470$	0	0
$471$	0	0
$472$	−0.958482	−0.958482
$473$	0	0
$474$	−0.0667045	−0.0667045
$475$	−1.37248	−1.37248
$476$	0	0
$477$	−1.91977	−1.91977
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.248227	−0.248227
$484$	−0.670976	−0.670976
$485$	0	0
$486$	0.196517	0.196517
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0.231007	0.231007
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−0.146644	−0.146644
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.185665	0.185665
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−0.226310	−0.226310
$502$	1.07803	1.07803
$503$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$503$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$504$	−1.68990	−1.68990
$505$	0	0
$506$	0	0
$507$	−0.116290	−0.116290
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−0.240729	−0.240729
$513$	−0.317053	−0.317053
$514$	0	0
$515$	0	0
$516$	0.151849	0.151849
$517$	0	0
$518$	−0.356044	−0.356044
$519$	0.159606	0.159606
$520$	0	0
$521$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$521$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$522$	0	0
$523$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$524$	1.33288	1.33288
$525$	−0.207840	−0.207840
$526$	1.07803	1.07803
$527$	0	0
$528$	0	0
$529$	0.426394	0.426394
$530$	0	0
$531$	0.986477	0.986477
$532$	1.64590	1.64590
$533$	0	0
$534$	−0.111462	−0.111462
$535$	0	0
$536$	−1.80136	−1.80136
$537$	0	0
$538$	−1.02519	−1.02519
$539$	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
$547$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$547$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−0.133120	−0.133120
$553$	−1.78727	−1.78727
$554$	0	0
$555$	0	0
$556$	−0.531520	−0.531520
$557$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$557$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$558$	0.866932	0.866932
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0.787265	0.787265
$563$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$563$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$564$	0.0618102	0.0618102
$565$	0	0
$566$	0	0
$567$	1.71508	1.71508
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$571$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$572$	0	0
$573$	−0.0403870	−0.0403870
$574$	1.92672	1.92672
$575$	1.19432	1.19432
$576$	−0.462145	−0.462145
$577$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$577$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$578$	−0.573606	−0.573606
$579$	0.116290	0.116290
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−0.878816	−0.878816
$587$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$587$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$588$	0.171217	0.171217
$589$	−2.10277	−2.10277
$590$	0	0
$591$	0	0
$592$	0.0420867	0.0420867
$593$	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			$\pi$
$600$	−0.111462	−0.111462
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−1.99511	−1.99511
$603$	1.85397	1.85397
$604$	0.384876	0.384876
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.41090	1.41090
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$613$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$614$	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
$619$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$619$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$620$	0	0
$621$	0.275895	0.275895
$622$	0	0
$623$	−2.98648	−2.98648
$624$	0	0
$625$	1.00000	1.00000
$626$	0.329024	0.329024
$627$	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	$0.888889\pi$
$632$	−0.958482	−0.958482
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0.151849	0.151849
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−0.111462	−0.111462
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−1.43224	−1.43224
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.919772	0.919772
$649$	0	0
$650$	0	0
$651$	−0.318430	−0.318430
$652$	1.33288	1.33288
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−0.227751	−0.227751
$657$	0	0
$658$	−0.812112	−0.812112
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0.196517	0.196517
$667$	0	0
$668$	−1.30578	−1.30578
$669$	0.231007	0.231007
$670$	0	0
$671$	0	0
$672$	0.213659	0.213659
$673$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$673$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$674$	−0.878816	−0.878816
$675$	0.231007	0.231007
$676$	−0.670976	−0.670976
$677$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$677$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.908449	−0.908449
$685$	0	0
$686$	−1.22440	−1.22440
$687$	0	0
$688$	0.235835	0.235835
$689$	0	0
$690$	0	0
$691$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$691$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$692$	0.920903	0.920903
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0.0667045	0.0667045
$699$	0.116290	0.116290
$700$	−1.19921	−1.19921
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−0.476658	−0.476658
$704$	0	0
$705$	0	0
$706$	−0.454388	−0.454388
$707$	0	0
$708$	−0.0780275	−0.0780275
$709$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$709$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$710$	0	0
$711$	0.986477	0.986477
$712$	−1.60160	−1.60160
$713$	1.82980	1.82980
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0.958482	0.958482
$719$	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
$722$	−0.506902	−0.506902
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0.0667045	0.0667045
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.919772	−0.919772
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$733$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$734$	−0.454388	−0.454388
$735$	0	0
$736$	−1.22775	−1.22775
$737$	0	0
$738$	−1.06345	−1.06345
$739$	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
$742$	−1.99511	−1.99511
$743$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$743$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$744$	−0.170769	−0.170769
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.98648	−2.98648
$750$	0	0
$751$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$751$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$752$	0.0959970	0.0959970
$753$	0.218553	0.218553
$754$	0	0
$755$	0	0
$756$	−0.277026	−0.277026
$757$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$757$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$758$	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
$764$	−0.233027	−0.233027
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.105126	0.105126
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0.670976	0.670976
$773$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$773$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$774$	1.10119	1.10119
$775$	1.53209	1.53209
$776$	0	0
$777$	−0.0721822	−0.0721822
$778$	−0.685068	−0.685068
$779$	2.57942	2.57942
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0.265916	0.265916
$785$	0	0
$786$	−0.132507	−0.132507
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0.218553	0.218553
$790$	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			$\pi$
$798$	−0.163626	−0.163626
$799$	0	0
$800$	−1.02799	−1.02799
$801$	1.64838	1.64838
$802$	−0.454388	−0.454388
$803$	0	0
$804$	−0.146644	−0.146644
$805$	0	0
$806$	0	0
$807$	−0.207840	−0.207840
$808$	0	0
$809$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$809$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$810$	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
$817$	−2.67098	−2.67098
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$823$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$824$	0	0
$825$	0	0
$826$	1.02519	1.02519
$827$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$827$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$828$	0.790521	0.790521
$829$	−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
$834$	0.0528406	0.0528406
$835$	0	0
$836$	0	0
$837$	0.353923	0.353923
$838$	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
$843$	0.159606	0.159606
$844$	0	0
$845$	0	0
$846$	0.448243	0.448243
$847$	1.78727	1.78727
$848$	0.235835	0.235835
$849$	0	0
$850$	0	0
$851$	0.414782	0.414782
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1.60160	−1.60160
$857$	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
$861$	0.390612	0.390612
$862$	0	0
$863$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$863$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$864$	−0.237474	−0.237474
$865$	0	0
$866$	1.07803	1.07803
$867$	−0.116290	−0.116290
$868$	−1.83730	−1.83730
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0.940244	0.940244
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.199211	−0.199211
$879$	−0.178166	−0.178166
$880$	0	0
$881$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$881$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$882$	1.24165	1.24165
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−0.878816	−0.878816
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−0.0387102	−0.0387102
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1.33288	1.33288
$893$	−1.08723	−1.08723
$894$	0	0
$895$	0	0
$896$	1.35702	1.35702
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0.661902	0.661902
$901$	0	0
$902$	0	0
$903$	−0.404476	−0.404476
$904$	0	0
$905$	0	0
$906$	−0.0382621	−0.0382621
$907$	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			$\pi$
$912$	0.0193416	0.0193416
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.55036	−3.55036
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0.0667045	0.0667045
$923$	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
$931$	−3.01166	−3.01166
$932$	0.670976	0.670976
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	1.92672	1.92672
$939$	0.0667045	0.0667045
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−2.24458	−2.24458
$944$	−0.121184	−0.121184
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−0.0780275	−0.0780275
$949$	0	0
$950$	0.787265	0.787265
$951$	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	$0.444444\pi$
$954$	1.10119	1.10119
$955$	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
$963$	1.64838	1.64838
$964$	0	0
$965$	0	0
$966$	0.142385	0.142385
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.958482	0.958482
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.229876	0.229876
$973$	1.41580	1.41580
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−0.132507	−0.132507
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.00000	1.00000
$983$	1.00000	1.00000
$984$	0.209479	0.209479
$985$	0	0
$986$	0	0
$987$	−0.164643	−0.164643
$988$	0	0
$989$	2.32425	2.32425
$990$	0	0
$991$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$991$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$992$	−1.57498	−1.57498
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$997$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$998$	0	0
$999$	0.0802278	0.0802278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.1.b.c.982.4	✓	9
983.982	odd	2	CM	983.1.b.c.982.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.1.b.c.982.4	✓	9	1.1	even	1	trivial
983.1.b.c.982.4	✓	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)

\(f(q)\)	\(=\)	\(q-0.573606 q^{2} -0.116290 q^{3} -0.670976 q^{4} +0.0667045 q^{6} +1.78727 q^{7} +0.958482 q^{8} -0.986477 q^{9} +O(q^{10})\)	\(q-0.573606 q^{2} -0.116290 q^{3} -0.670976 q^{4} +0.0667045 q^{6} +1.78727 q^{7} +0.958482 q^{8} -0.986477 q^{9} +0.0780275 q^{12} -1.02519 q^{14} +0.121184 q^{16} +0.565849 q^{18} -1.37248 q^{19} -0.207840 q^{21} +1.19432 q^{23} -0.111462 q^{24} +1.00000 q^{25} +0.231007 q^{27} -1.19921 q^{28} +1.53209 q^{31} -1.02799 q^{32} +0.661902 q^{36} +0.347296 q^{37} +0.787265 q^{38} -1.87939 q^{41} +0.119219 q^{42} +1.94609 q^{43} -0.685068 q^{46} +0.792160 q^{47} -0.0140924 q^{48} +2.19432 q^{49} -0.573606 q^{50} +1.94609 q^{53} -0.132507 q^{54} +1.71306 q^{56} +0.159606 q^{57} -1.00000 q^{59} -0.878816 q^{62} -1.76310 q^{63} +0.468480 q^{64} -1.87939 q^{67} -0.138887 q^{69} -0.945521 q^{72} -0.199211 q^{74} -0.116290 q^{75} +0.920903 q^{76} -1.00000 q^{79} +0.959613 q^{81} +1.07803 q^{82} +0.139456 q^{84} -1.11629 q^{86} -1.67098 q^{89} -0.801358 q^{92} -0.178166 q^{93} -0.454388 q^{94} +0.119545 q^{96} -1.25867 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) \) 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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