LMFDB - Embedded newform 983.1.b.c.982.6

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			982.6
Root			$-0.792160$ 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)

$f(q)$	$=$	$q+0.792160 q^{2} +1.94609 q^{3} -0.372483 q^{4} +1.54161 q^{6} -0.573606 q^{7} -1.08723 q^{8} +2.78727 q^{9} +O(q^{10})$	$q+0.792160 q^{2} +1.94609 q^{3} -0.372483 q^{4} +1.54161 q^{6} -0.573606 q^{7} -1.08723 q^{8} +2.78727 q^{9} -0.724886 q^{12} -0.454388 q^{14} -0.488773 q^{16} +2.20796 q^{18} -1.98648 q^{19} -1.11629 q^{21} -1.67098 q^{23} -2.11584 q^{24} +1.00000 q^{25} +3.47818 q^{27} +0.213659 q^{28} -1.87939 q^{31} +0.700040 q^{32} -1.03821 q^{36} +1.53209 q^{37} -1.57361 q^{38} +0.347296 q^{41} -0.884279 q^{42} +1.19432 q^{43} -1.32368 q^{46} -0.116290 q^{47} -0.951196 q^{48} -0.670976 q^{49} +0.792160 q^{50} +1.19432 q^{53} +2.75527 q^{54} +0.623640 q^{56} -3.86586 q^{57} -1.00000 q^{59} -1.48877 q^{62} -1.59879 q^{63} +1.04332 q^{64} +0.347296 q^{67} -3.25187 q^{69} -3.03039 q^{72} +1.21366 q^{74} +1.94609 q^{75} +0.739929 q^{76} -1.00000 q^{79} +3.98158 q^{81} +0.275114 q^{82} +0.415799 q^{84} +0.946090 q^{86} -1.37248 q^{89} +0.622410 q^{92} -3.65745 q^{93} -0.0921200 q^{94} +1.36234 q^{96} -0.531520 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$2$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$3$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$3$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$4$	−0.372483	−0.372483
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	1.54161	1.54161
$7$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$7$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$8$	−1.08723	−1.08723
$9$	2.78727	2.78727
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.724886	−0.724886
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−0.454388	−0.454388
$15$	0	0
$16$	−0.488773	−0.488773
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	2.20796	2.20796
$19$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$19$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$20$	0	0
$21$	−1.11629	−1.11629
$22$	0	0
$23$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$23$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$24$	−2.11584	−2.11584
$25$	1.00000	1.00000
$26$	0	0
$27$	3.47818	3.47818
$28$	0.213659	0.213659
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$31$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$32$	0.700040	0.700040
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.03821	−1.03821
$37$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$37$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$38$	−1.57361	−1.57361
$39$	0	0
$40$	0	0
$41$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$41$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$42$	−0.884279	−0.884279
$43$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$43$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$44$	0	0
$45$	0	0
$46$	−1.32368	−1.32368
$47$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$47$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$48$	−0.951196	−0.951196
$49$	−0.670976	−0.670976
$50$	0.792160	0.792160
$51$	0	0
$52$	0	0
$53$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$53$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$54$	2.75527	2.75527
$55$	0	0
$56$	0.623640	0.623640
$57$	−3.86586	−3.86586
$58$	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			$\pi$
$62$	−1.48877	−1.48877
$63$	−1.59879	−1.59879
$64$	1.04332	1.04332
$65$	0	0
$66$	0	0
$67$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$67$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$68$	0	0
$69$	−3.25187	−3.25187
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.03039	−3.03039
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.21366	1.21366
$75$	1.94609	1.94609
$76$	0.739929	0.739929
$77$	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
$81$	3.98158	3.98158
$82$	0.275114	0.275114
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.415799	0.415799
$85$	0	0
$86$	0.946090	0.946090
$87$	0	0
$88$	0	0
$89$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$89$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$90$	0	0
$91$	0	0
$92$	0.622410	0.622410
$93$	−3.65745	−3.65745
$94$	−0.0921200	−0.0921200
$95$	0	0
$96$	1.36234	1.36234
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.531520	−0.531520
$99$	0	0
$100$	−0.372483	−0.372483
$101$	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
$106$	0.946090	0.946090
$107$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$107$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$108$	−1.29556	−1.29556
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	2.98158	2.98158
$112$	0.280363	0.280363
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−3.06238	−3.06238
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−0.792160	−0.792160
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0.675870	0.675870
$124$	0.700040	0.700040
$125$	0	0
$126$	−1.26650	−1.26650
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.126433	0.126433
$129$	2.32425	2.32425
$130$	0	0
$131$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$131$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$132$	0	0
$133$	1.13946	1.13946
$134$	0.275114	0.275114
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−2.57600	−2.57600
$139$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$139$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$140$	0	0
$141$	−0.226310	−0.226310
$142$	0	0
$143$	0	0
$144$	−1.36234	−1.36234
$145$	0	0
$146$	0	0
$147$	−1.30578	−1.30578
$148$	−0.570677	−0.570677
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.54161	1.54161
$151$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$151$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$152$	2.15975	2.15975
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−0.792160	−0.792160
$159$	2.32425	2.32425
$160$	0	0
$161$	0.958482	0.958482
$162$	3.15405	3.15405
$163$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$163$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$164$	−0.129362	−0.129362
$165$	0	0
$166$	0	0
$167$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$167$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$168$	1.21366	1.21366
$169$	1.00000	1.00000
$170$	0	0
$171$	−5.53684	−5.53684
$172$	−0.444863	−0.444863
$173$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$173$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$174$	0	0
$175$	−0.573606	−0.573606
$176$	0	0
$177$	−1.94609	−1.94609
$178$	−1.08723	−1.08723
$179$	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
$184$	1.81673	1.81673
$185$	0	0
$186$	−2.89729	−2.89729
$187$	0	0
$188$	0.0433160	0.0433160
$189$	−1.99511	−1.99511
$190$	0	0
$191$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$191$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$192$	2.03039	2.03039
$193$	−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
$196$	0.249927	0.249927
$197$	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			$\pi$
$200$	−1.08723	−1.08723
$201$	0.675870	0.675870
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.65745	−4.65745
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.444863	−0.444863
$213$	0	0
$214$	−1.08723	−1.08723
$215$	0	0
$216$	−3.78157	−3.78157
$217$	1.07803	1.07803
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	2.36189	2.36189
$223$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$223$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$224$	−0.401547	−0.401547
$225$	2.78727	2.78727
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.43997	1.43997
$229$	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
$236$	0.372483	0.372483
$237$	−1.94609	−1.94609
$238$	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			$\pi$
$242$	0.792160	0.792160
$243$	4.27034	4.27034
$244$	0	0
$245$	0	0
$246$	0.535397	0.535397
$247$	0	0
$248$	2.04332	2.04332
$249$	0	0
$250$	0	0
$251$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$251$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$252$	0.595524	0.595524
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.943161	−0.943161
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	1.84118	1.84118
$259$	−0.878816	−0.878816
$260$	0	0
$261$	0	0
$262$	1.41580	1.41580
$263$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$263$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$264$	0	0
$265$	0	0
$266$	0.902631	0.902631
$267$	−2.67098	−2.67098
$268$	−0.129362	−0.129362
$269$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$269$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$270$	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
$276$	1.21127	1.21127
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−0.0921200	−0.0921200
$279$	−5.23835	−5.23835
$280$	0	0
$281$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$281$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$282$	−0.179274	−0.179274
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.199211	−0.199211
$288$	1.95120	1.95120
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$293$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$294$	−1.03439	−1.03439
$295$	0	0
$296$	−1.66573	−1.66573
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.724886	−0.724886
$301$	−0.685068	−0.685068
$302$	0.627517	0.627517
$303$	0	0
$304$	0.970936	0.970936
$305$	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
$313$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$313$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$314$	0	0
$315$	0	0
$316$	0.372483	0.372483
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	1.84118	1.84118
$319$	0	0
$320$	0	0
$321$	−2.67098	−2.67098
$322$	0.759271	0.759271
$323$	0	0
$324$	−1.48307	−1.48307
$325$	0	0
$326$	1.41580	1.41580
$327$	0	0
$328$	−0.377590	−0.377590
$329$	0.0667045	0.0667045
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	4.27034	4.27034
$334$	0.946090	0.946090
$335$	0	0
$336$	0.545612	0.545612
$337$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$337$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$338$	0.792160	0.792160
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−4.38606	−4.38606
$343$	0.958482	0.958482
$344$	−1.29849	−1.29849
$345$	0	0
$346$	−1.57361	−1.57361
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$349$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$350$	−0.454388	−0.454388
$351$	0	0
$352$	0	0
$353$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$353$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$354$	−1.54161	−1.54161
$355$	0	0
$356$	0.511227	0.511227
$357$	0	0
$358$	0	0
$359$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$359$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$360$	0	0
$361$	2.94609	2.94609
$362$	0	0
$363$	1.94609	1.94609
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$367$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$368$	0.816728	0.816728
$369$	0.968007	0.968007
$370$	0	0
$371$	−0.685068	−0.685068
$372$	1.36234	1.36234
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0.126433	0.126433
$377$	0	0
$378$	−1.58044	−1.58044
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	1.21366	1.21366
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.246050	0.246050
$385$	0	0
$386$	−0.792160	−0.792160
$387$	3.32888	3.32888
$388$	0	0
$389$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$389$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$390$	0	0
$391$	0	0
$392$	0.729502	0.729502
$393$	3.47818	3.47818
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	2.21748	2.21748
$400$	−0.488773	−0.488773
$401$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$401$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$402$	0.535397	0.535397
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.573606	0.573606
$414$	−3.68945	−3.68945
$415$	0	0
$416$	0	0
$417$	−0.226310	−0.226310
$418$	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
$423$	−0.324130	−0.324130
$424$	−1.29849	−1.29849
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0.511227	0.511227
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.70004	−1.70004
$433$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$433$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$434$	0.853970	0.853970
$435$	0	0
$436$	0	0
$437$	3.31935	3.31935
$438$	0	0
$439$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$439$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$440$	0	0
$441$	−1.87019	−1.87019
$442$	0	0
$443$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$443$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$444$	−1.11059	−1.11059
$445$	0	0
$446$	1.41580	1.41580
$447$	0	0
$448$	−0.598453	−0.598453
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	2.20796	2.20796
$451$	0	0
$452$	0	0
$453$	1.54161	1.54161
$454$	0	0
$455$	0	0
$456$	4.20306	4.20306
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$461$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−0.792160	−0.792160
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	−0.199211	−0.199211
$470$	0	0
$471$	0	0
$472$	1.08723	1.08723
$473$	0	0
$474$	−1.54161	−1.54161
$475$	−1.98648	−1.98648
$476$	0	0
$477$	3.32888	3.32888
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.86529	1.86529
$484$	−0.372483	−0.372483
$485$	0	0
$486$	3.38279	3.38279
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	3.47818	3.47818
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−0.251750	−0.251750
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.918593	0.918593
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	2.32425	2.32425
$502$	0.275114	0.275114
$503$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$503$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$504$	1.73825	1.73825
$505$	0	0
$506$	0	0
$507$	1.94609	1.94609
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−0.873567	−0.873567
$513$	−6.90932	−6.90932
$514$	0	0
$515$	0	0
$516$	−0.865744	−0.865744
$517$	0	0
$518$	−0.696163	−0.696163
$519$	−3.86586	−3.86586
$520$	0	0
$521$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$521$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$522$	0	0
$523$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$524$	−0.665726	−0.665726
$525$	−1.11629	−1.11629
$526$	0.275114	0.275114
$527$	0	0
$528$	0	0
$529$	1.79216	1.79216
$530$	0	0
$531$	−2.78727	−2.78727
$532$	−0.424428	−0.424428
$533$	0	0
$534$	−2.11584	−2.11584
$535$	0	0
$536$	−0.377590	−0.377590
$537$	0	0
$538$	−0.454388	−0.454388
$539$	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
$547$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$547$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	3.53552	3.53552
$553$	0.573606	0.573606
$554$	0	0
$555$	0	0
$556$	0.0433160	0.0433160
$557$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$557$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$558$	−4.14961	−4.14961
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−1.57361	−1.57361
$563$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$563$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$564$	0.0842967	0.0842967
$565$	0	0
$566$	0	0
$567$	−2.28386	−2.28386
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$571$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$572$	0	0
$573$	2.98158	2.98158
$574$	−0.157807	−0.157807
$575$	−1.67098	−1.67098
$576$	2.90800	2.90800
$577$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$577$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$578$	0.792160	0.792160
$579$	−1.94609	−1.94609
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−1.48877	−1.48877
$587$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$587$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$588$	0.486381	0.486381
$589$	3.73336	3.73336
$590$	0	0
$591$	0	0
$592$	−0.748844	−0.748844
$593$	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			$\pi$
$600$	−2.11584	−2.11584
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−0.542683	−0.542683
$603$	0.968007	0.968007
$604$	−0.295066	−0.295066
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−1.39061	−1.39061
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$613$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$614$	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
$619$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$619$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$620$	0	0
$621$	−5.81195	−5.81195
$622$	0	0
$623$	0.787265	0.787265
$624$	0	0
$625$	1.00000	1.00000
$626$	0.627517	0.627517
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$631$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$632$	1.08723	1.08723
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.865744	−0.865744
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−2.11584	−2.11584
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	−0.357019	−0.357019
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−4.32888	−4.32888
$649$	0	0
$650$	0	0
$651$	2.09794	2.09794
$652$	−0.665726	−0.665726
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−0.169749	−0.169749
$657$	0	0
$658$	0.0528406	0.0528406
$659$	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
$666$	3.38279	3.38279
$667$	0	0
$668$	−0.444863	−0.444863
$669$	3.47818	3.47818
$670$	0	0
$671$	0	0
$672$	−0.781447	−0.781447
$673$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$673$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$674$	−1.48877	−1.48877
$675$	3.47818	3.47818
$676$	−0.372483	−0.372483
$677$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$677$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$678$	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			$\pi$
$684$	2.06238	2.06238
$685$	0	0
$686$	0.759271	0.759271
$687$	0	0
$688$	−0.583750	−0.583750
$689$	0	0
$690$	0	0
$691$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$691$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$692$	0.739929	0.739929
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.54161	1.54161
$699$	−1.94609	−1.94609
$700$	0.213659	0.213659
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−3.04346	−3.04346
$704$	0	0
$705$	0	0
$706$	−0.0921200	−0.0921200
$707$	0	0
$708$	0.724886	0.724886
$709$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$709$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$710$	0	0
$711$	−2.78727	−2.78727
$712$	1.49220	1.49220
$713$	3.14041	3.14041
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−1.08723	−1.08723
$719$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$719$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$720$	0	0
$721$	0	0
$722$	2.33377	2.33377
$723$	0	0
$724$	0	0
$725$	0	0
$726$	1.54161	1.54161
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	4.32888	4.32888
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$733$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$734$	−0.0921200	−0.0921200
$735$	0	0
$736$	−1.16975	−1.16975
$737$	0	0
$738$	0.766816	0.766816
$739$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$739$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$740$	0	0
$741$	0	0
$742$	−0.542683	−0.542683
$743$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$743$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$744$	3.97648	3.97648
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.787265	0.787265
$750$	0	0
$751$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$751$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$752$	0.0568392	0.0568392
$753$	0.675870	0.675870
$754$	0	0
$755$	0	0
$756$	0.743144	0.743144
$757$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$757$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$758$	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
$764$	−0.570677	−0.570677
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.83548	−1.83548
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0.372483	0.372483
$773$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$773$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$774$	2.63700	2.63700
$775$	−1.87939	−1.87939
$776$	0	0
$777$	−1.71025	−1.71025
$778$	−1.32368	−1.32368
$779$	−0.689896	−0.689896
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0.327955	0.327955
$785$	0	0
$786$	2.75527	2.75527
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0.675870	0.675870
$790$	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			$\pi$
$798$	1.75660	1.75660
$799$	0	0
$800$	0.700040	0.700040
$801$	−3.82547	−3.82547
$802$	−0.0921200	−0.0921200
$803$	0	0
$804$	−0.251750	−0.251750
$805$	0	0
$806$	0	0
$807$	−1.11629	−1.11629
$808$	0	0
$809$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$809$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$810$	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
$817$	−2.37248	−2.37248
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$823$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$824$	0	0
$825$	0	0
$826$	0.454388	0.454388
$827$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$827$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$828$	1.73482	1.73482
$829$	−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
$834$	−0.179274	−0.179274
$835$	0	0
$836$	0	0
$837$	−6.53684	−6.53684
$838$	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
$843$	−3.86586	−3.86586
$844$	0	0
$845$	0	0
$846$	−0.256763	−0.256763
$847$	−0.573606	−0.573606
$848$	−0.583750	−0.583750
$849$	0	0
$850$	0	0
$851$	−2.56008	−2.56008
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	1.49220	1.49220
$857$	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
$861$	−0.387683	−0.387683
$862$	0	0
$863$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$863$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$864$	2.43486	2.43486
$865$	0	0
$866$	0.275114	0.275114
$867$	1.94609	1.94609
$868$	−0.401547	−0.401547
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	2.62946	2.62946
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1.21366	1.21366
$879$	−3.65745	−3.65745
$880$	0	0
$881$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$881$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$882$	−1.48149	−1.48149
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−1.48877	−1.48877
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−3.24165	−3.24165
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−0.665726	−0.665726
$893$	0.231007	0.231007
$894$	0	0
$895$	0	0
$896$	−0.0725228	−0.0725228
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.03821	−1.03821
$901$	0	0
$902$	0	0
$903$	−1.33320	−1.33320
$904$	0	0
$905$	0	0
$906$	1.22120	1.22120
$907$	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			$\pi$
$912$	1.88953	1.88953
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.02519	−1.02519
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1.54161	1.54161
$923$	0	0
$924$	0	0
$925$	1.53209	1.53209
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1.33288	1.33288
$932$	0.372483	0.372483
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−0.157807	−0.157807
$939$	1.54161	1.54161
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−0.580324	−0.580324
$944$	0.488773	0.488773
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0.724886	0.724886
$949$	0	0
$950$	−1.57361	−1.57361
$951$	0	0
$952$	0	0
$953$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$953$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$954$	2.63700	2.63700
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.53209	2.53209
$962$	0	0
$963$	−3.82547	−3.82547
$964$	0	0
$965$	0	0
$966$	1.47761	1.47761
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−1.08723	−1.08723
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.59063	−1.59063
$973$	0.0667045	0.0667045
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	2.75527	2.75527
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.00000	1.00000
$983$	1.00000	1.00000
$984$	−0.734823	−0.734823
$985$	0	0
$986$	0	0
$987$	0.129813	0.129813
$988$	0	0
$989$	−1.99567	−1.99567
$990$	0	0
$991$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$991$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$992$	−1.31564	−1.31564
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$997$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$998$	0	0
$999$	5.32888	5.32888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.1.b.c.982.6	✓	9
983.982	odd	2	CM	983.1.b.c.982.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.1.b.c.982.6	✓	9	1.1	even	1	trivial
983.1.b.c.982.6	✓	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)

\(f(q)\)	\(=\)	\(q+0.792160 q^{2} +1.94609 q^{3} -0.372483 q^{4} +1.54161 q^{6} -0.573606 q^{7} -1.08723 q^{8} +2.78727 q^{9} +O(q^{10})\)	\(q+0.792160 q^{2} +1.94609 q^{3} -0.372483 q^{4} +1.54161 q^{6} -0.573606 q^{7} -1.08723 q^{8} +2.78727 q^{9} -0.724886 q^{12} -0.454388 q^{14} -0.488773 q^{16} +2.20796 q^{18} -1.98648 q^{19} -1.11629 q^{21} -1.67098 q^{23} -2.11584 q^{24} +1.00000 q^{25} +3.47818 q^{27} +0.213659 q^{28} -1.87939 q^{31} +0.700040 q^{32} -1.03821 q^{36} +1.53209 q^{37} -1.57361 q^{38} +0.347296 q^{41} -0.884279 q^{42} +1.19432 q^{43} -1.32368 q^{46} -0.116290 q^{47} -0.951196 q^{48} -0.670976 q^{49} +0.792160 q^{50} +1.19432 q^{53} +2.75527 q^{54} +0.623640 q^{56} -3.86586 q^{57} -1.00000 q^{59} -1.48877 q^{62} -1.59879 q^{63} +1.04332 q^{64} +0.347296 q^{67} -3.25187 q^{69} -3.03039 q^{72} +1.21366 q^{74} +1.94609 q^{75} +0.739929 q^{76} -1.00000 q^{79} +3.98158 q^{81} +0.275114 q^{82} +0.415799 q^{84} +0.946090 q^{86} -1.37248 q^{89} +0.622410 q^{92} -3.65745 q^{93} -0.0921200 q^{94} +1.36234 q^{96} -0.531520 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} + 9 q^{9}+O(q^{10}) \) 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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