LMFDB - Embedded newform 1231.1.b.c.1230.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1231,1,Mod(1230,1231)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1231, 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("1231.1230");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1231 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1231.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:	$0.614349030551$
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			1230.1
Root			$1.98648$ of defining polynomial
Character	$\chi$	$=$	1231.1230

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.98648 q^{2} +2.94609 q^{4} +1.53209 q^{5} -0.573606 q^{7} -3.86586 q^{8} +1.00000 q^{9} -3.04346 q^{10} -1.00000 q^{11} -0.116290 q^{13} +1.13946 q^{14} +4.73336 q^{16} -1.98648 q^{18} +1.19432 q^{19} +4.51367 q^{20} +1.98648 q^{22} +1.34730 q^{25} +0.231007 q^{26} -1.68990 q^{28} +1.78727 q^{29} -1.37248 q^{31} -5.53684 q^{32} -0.878816 q^{35} +2.94609 q^{36} +0.792160 q^{37} -2.37248 q^{38} -5.92284 q^{40} -1.87939 q^{41} +1.78727 q^{43} -2.94609 q^{44} +1.53209 q^{45} -0.670976 q^{49} -2.67637 q^{50} -0.342600 q^{52} -1.53209 q^{55} +2.21748 q^{56} -3.55036 q^{58} +2.72641 q^{62} -0.573606 q^{63} +6.26544 q^{64} -0.178166 q^{65} +1.74575 q^{70} -3.86586 q^{72} -1.57361 q^{74} +3.51857 q^{76} +0.573606 q^{77} +7.25192 q^{80} +1.00000 q^{81} +3.73336 q^{82} -3.55036 q^{86} +3.86586 q^{88} -3.04346 q^{90} +0.0667045 q^{91} +1.82980 q^{95} +1.33288 q^{98} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{4} + 9 q^{9} - 9 q^{11} + 9 q^{16} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{40} - 9 q^{44} + 9 q^{49} + 9 q^{64} - 9 q^{70} - 9 q^{74} + 9 q^{81} - 9 q^{99}+O(q^{100}) $ 9 * q + 9 * q^4 + 9 * q^9 - 9 * q^11 + 9 * q^16 + 9 * q^25 - 9 * q^28 + 9 * q^36 - 9 * q^38 - 9 * q^40 - 9 * q^44 + 9 * q^49 + 9 * q^64 - 9 * q^70 - 9 * q^74 + 9 * q^81 - 9 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1231\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$2$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	2.94609	2.94609
$5$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$5$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$6$	0	0
$7$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$7$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$8$	−3.86586	−3.86586
$9$	1.00000	1.00000
$10$	−3.04346	−3.04346
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	0	0
$13$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$13$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$14$	1.13946	1.13946
$15$	0	0
$16$	4.73336	4.73336
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.98648	−1.98648
$19$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$19$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$20$	4.51367	4.51367
$21$	0	0
$22$	1.98648	1.98648
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.34730	1.34730
$26$	0.231007	0.231007
$27$	0	0
$28$	−1.68990	−1.68990
$29$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$29$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$30$	0	0
$31$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$31$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$32$	−5.53684	−5.53684
$33$	0	0
$34$	0	0
$35$	−0.878816	−0.878816
$36$	2.94609	2.94609
$37$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$37$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$38$	−2.37248	−2.37248
$39$	0	0
$40$	−5.92284	−5.92284
$41$	−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	0
$43$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$43$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$44$	−2.94609	−2.94609
$45$	1.53209	1.53209
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−0.670976	−0.670976
$50$	−2.67637	−2.67637
$51$	0	0
$52$	−0.342600	−0.342600
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−1.53209	−1.53209
$56$	2.21748	2.21748
$57$	0	0
$58$	−3.55036	−3.55036
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	2.72641	2.72641
$63$	−0.573606	−0.573606
$64$	6.26544	6.26544
$65$	−0.178166	−0.178166
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	1.74575	1.74575
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.86586	−3.86586
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.57361	−1.57361
$75$	0	0
$76$	3.51857	3.51857
$77$	0.573606	0.573606
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	7.25192	7.25192
$81$	1.00000	1.00000
$82$	3.73336	3.73336
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−3.55036	−3.55036
$87$	0	0
$88$	3.86586	3.86586
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.04346	−3.04346
$91$	0.0667045	0.0667045
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.82980	1.82980
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.33288	1.33288
$99$	−1.00000	−1.00000
$100$	3.96926	3.96926
$101$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$101$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0.449560	0.449560
$105$	0	0
$106$	0	0
$107$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$107$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	3.04346	3.04346
$111$	0	0
$112$	−2.71508	−2.71508
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	5.26544	5.26544
$117$	−0.116290	−0.116290
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−4.04346	−4.04346
$125$	0.532089	0.532089
$126$	1.13946	1.13946
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−6.90932	−6.90932
$129$	0	0
$130$	0.353923	0.353923
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	−0.685068	−0.685068
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$137$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$138$	0	0
$139$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$139$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$140$	−2.58907	−2.58907
$141$	0	0
$142$	0	0
$143$	0.116290	0.116290
$144$	4.73336	4.73336
$145$	2.73825	2.73825
$146$	0	0
$147$	0	0
$148$	2.33377	2.33377
$149$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$149$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$150$	0	0
$151$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$151$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$152$	−4.61707	−4.61707
$153$	0	0
$154$	−1.13946	−1.13946
$155$	−2.10277	−2.10277
$156$	0	0
$157$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$157$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$158$	0	0
$159$	0	0
$160$	−8.48293	−8.48293
$161$	0	0
$162$	−1.98648	−1.98648
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−5.53684	−5.53684
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−0.986477	−0.986477
$170$	0	0
$171$	1.19432	1.19432
$172$	5.26544	5.26544
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−0.772818	−0.772818
$176$	−4.73336	−4.73336
$177$	0	0
$178$	0	0
$179$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$179$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$180$	4.51367	4.51367
$181$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$181$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$182$	−0.132507	−0.132507
$183$	0	0
$184$	0	0
$185$	1.21366	1.21366
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−3.63486	−3.63486
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−1.97675	−1.97675
$197$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$197$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$198$	1.98648	1.98648
$199$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$199$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$200$	−5.20846	−5.20846
$201$	0	0
$202$	3.31935	3.31935
$203$	−1.02519	−1.02519
$204$	0	0
$205$	−2.87939	−2.87939
$206$	0	0
$207$	0	0
$208$	−0.550440	−0.550440
$209$	−1.19432	−1.19432
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	−0.689896	−0.689896
$215$	2.73825	2.73825
$216$	0	0
$217$	0.787265	0.787265
$218$	0	0
$219$	0	0
$220$	−4.51367	−4.51367
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	3.17597	3.17597
$225$	1.34730	1.34730
$226$	0	0
$227$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$227$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−6.90932	−6.90932
$233$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$233$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$234$	0.231007	0.231007
$235$	0	0
$236$	0	0
$237$	0	0
$238$	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	0
$243$	0	0
$244$	0	0
$245$	−1.02799	−1.02799
$246$	0	0
$247$	−0.138887	−0.138887
$248$	5.30583	5.30583
$249$	0	0
$250$	−1.05698	−1.05698
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.68990	−1.68990
$253$	0	0
$254$	0	0
$255$	0	0
$256$	7.45976	7.45976
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−0.454388	−0.454388
$260$	−0.524893	−0.524893
$261$	1.78727	1.78727
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	1.36087	1.36087
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$271$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$272$	0	0
$273$	0	0
$274$	−3.86586	−3.86586
$275$	−1.34730	−1.34730
$276$	0	0
$277$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$277$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$278$	0.231007	0.231007
$279$	−1.37248	−1.37248
$280$	3.39738	3.39738
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	−0.231007	−0.231007
$287$	1.07803	1.07803
$288$	−5.53684	−5.53684
$289$	1.00000	1.00000
$290$	−5.43947	−5.43947
$291$	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	$0.444444\pi$
$294$	0	0
$295$	0	0
$296$	−3.06238	−3.06238
$297$	0	0
$298$	−2.37248	−2.37248
$299$	0	0
$300$	0	0
$301$	−1.02519	−1.02519
$302$	−1.57361	−1.57361
$303$	0	0
$304$	5.65313	5.65313
$305$	0	0
$306$	0	0
$307$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$307$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$308$	1.68990	1.68990
$309$	0	0
$310$	4.17710	4.17710
$311$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$311$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	3.94609	3.94609
$315$	−0.878816	−0.878816
$316$	0	0
$317$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$317$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$318$	0	0
$319$	−1.78727	−1.78727
$320$	9.59922	9.59922
$321$	0	0
$322$	0	0
$323$	0	0
$324$	2.94609	2.94609
$325$	−0.156677	−0.156677
$326$	0	0
$327$	0	0
$328$	7.26544	7.26544
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0.792160	0.792160
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$337$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$338$	1.95961	1.95961
$339$	0	0
$340$	0	0
$341$	1.37248	1.37248
$342$	−2.37248	−2.37248
$343$	0.958482	0.958482
$344$	−6.90932	−6.90932
$345$	0	0
$346$	0	0
$347$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$347$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.53518	1.53518
$351$	0	0
$352$	5.53684	5.53684
$353$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$353$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	2.72641	2.72641
$359$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$359$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$360$	−5.92284	−5.92284
$361$	0.426394	0.426394
$362$	1.13946	1.13946
$363$	0	0
$364$	0.196517	0.196517
$365$	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	0
$369$	−1.87939	−1.87939
$370$	−2.41090	−2.41090
$371$	0	0
$372$	0	0
$373$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$373$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.207840	−0.207840
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	5.39076	5.39076
$381$	0	0
$382$	0	0
$383$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$383$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$384$	0	0
$385$	0.878816	0.878816
$386$	0	0
$387$	1.78727	1.78727
$388$	0	0
$389$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$389$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$390$	0	0
$391$	0	0
$392$	2.59390	2.59390
$393$	0	0
$394$	−0.689896	−0.689896
$395$	0	0
$396$	−2.94609	−2.94609
$397$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$398$	3.94609	3.94609
$399$	0	0
$400$	6.37723	6.37723
$401$	−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	0
$403$	0.159606	0.159606
$404$	−4.92284	−4.92284
$405$	1.53209	1.53209
$406$	2.03651	2.03651
$407$	−0.792160	−0.792160
$408$	0	0
$409$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$409$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$410$	5.71983	5.71983
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0.643877	0.643877
$417$	0	0
$418$	2.37248	2.37248
$419$	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	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.02317	1.02317
$429$	0	0
$430$	−5.43947	−5.43947
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$433$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$434$	−1.56388	−1.56388
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	5.92284	5.92284
$441$	−0.670976	−0.670976
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−3.59390	−3.59390
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−2.67637	−2.67637
$451$	1.87939	1.87939
$452$	0	0
$453$	0	0
$454$	−3.86586	−3.86586
$455$	0.102197	0.102197
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$464$	8.45976	8.45976
$465$	0	0
$466$	1.13946	1.13946
$467$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$467$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$468$	−0.342600	−0.342600
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.78727	−1.78727
$474$	0	0
$475$	1.60910	1.60910
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$479$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$480$	0	0
$481$	−0.0921200	−0.0921200
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$487$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$488$	0	0
$489$	0	0
$490$	2.04209	2.04209
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0.275895	0.275895
$495$	−1.53209	−1.53209
$496$	−6.49645	−6.49645
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1.56758	1.56758
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	2.21748	2.21748
$505$	−2.56008	−2.56008
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−7.90932	−7.90932
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0.902631	0.902631
$519$	0	0
$520$	0.688766	0.688766
$521$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$521$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$522$	−3.55036	−3.55036
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	−2.01827	−2.01827
$533$	0.218553	0.218553
$534$	0	0
$535$	0.532089	0.532089
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.670976	0.670976
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−2.37248	−2.37248
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	5.73336	5.73336
$549$	0	0
$550$	2.67637	2.67637
$551$	2.13456	2.13456
$552$	0	0
$553$	0	0
$554$	3.31935	3.31935
$555$	0	0
$556$	−0.342600	−0.342600
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	2.72641	2.72641
$559$	−0.207840	−0.207840
$560$	−4.15975	−4.15975
$561$	0	0
$562$	0	0
$563$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$563$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.573606	−0.573606
$568$	0	0
$569$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$569$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$570$	0	0
$571$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$571$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$572$	0.342600	0.342600
$573$	0	0
$574$	−2.14148	−2.14148
$575$	0	0
$576$	6.26544	6.26544
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.98648	−1.98648
$579$	0	0
$580$	8.06713	8.06713
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.178166	−0.178166
$586$	−0.689896	−0.689896
$587$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$587$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$588$	0	0
$589$	−1.63918	−1.63918
$590$	0	0
$591$	0	0
$592$	3.74957	3.74957
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	3.51857	3.51857
$597$	0	0
$598$	0	0
$599$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$599$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	2.03651	2.03651
$603$	0	0
$604$	2.33377	2.33377
$605$	0	0
$606$	0	0
$607$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$607$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$608$	−6.61274	−6.61274
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	3.94609	3.94609
$615$	0	0
$616$	−2.21748	−2.21748
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$619$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$620$	−6.19494	−6.19494
$621$	0	0
$622$	−3.04346	−3.04346
$623$	0	0
$624$	0	0
$625$	−0.532089	−0.532089
$626$	0	0
$627$	0	0
$628$	−5.85234	−5.85234
$629$	0	0
$630$	1.74575	1.74575
$631$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$631$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$632$	0	0
$633$	0	0
$634$	3.31935	3.31935
$635$	0	0
$636$	0	0
$637$	0.0780275	0.0780275
$638$	3.55036	3.55036
$639$	0	0
$640$	−10.5857	−10.5857
$641$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$641$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$642$	0	0
$643$	−1.67098	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$643$	−1.67098	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$647$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$648$	−3.86586	−3.86586
$649$	0	0
$650$	0.311234	0.311234
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−8.89580	−8.89580
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$661$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.04959	−1.04959
$666$	−1.57361	−1.57361
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	2.72641	2.72641
$675$	0	0
$676$	−2.90625	−2.90625
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−2.72641	−2.72641
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	3.51857	3.51857
$685$	2.98158	2.98158
$686$	−1.90400	−1.90400
$687$	0	0
$688$	8.45976	8.45976
$689$	0	0
$690$	0	0
$691$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$691$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$692$	0	0
$693$	0.573606	0.573606
$694$	3.73336	3.73336
$695$	−0.178166	−0.178166
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−2.27679	−2.27679
$701$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$701$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$702$	0	0
$703$	0.946090	0.946090
$704$	−6.26544	−6.26544
$705$	0	0
$706$	−2.37248	−2.37248
$707$	0.958482	0.958482
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.178166	0.178166
$716$	−4.04346	−4.04346
$717$	0	0
$718$	−0.689896	−0.689896
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	7.25192	7.25192
$721$	0	0
$722$	−0.847021	−0.847021
$723$	0	0
$724$	−1.68990	−1.68990
$725$	2.40798	2.40798
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−0.257870	−0.257870
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1.57361	−1.57361
$735$	0	0
$736$	0	0
$737$	0	0
$738$	3.73336	3.73336
$739$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$740$	3.57555	3.57555
$741$	0	0
$742$	0	0
$743$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0	0
$745$	1.82980	1.82980
$746$	2.72641	2.72641
$747$	0	0
$748$	0	0
$749$	−0.199211	−0.199211
$750$	0	0
$751$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$751$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$752$	0	0
$753$	0	0
$754$	0.412870	0.412870
$755$	1.21366	1.21366
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	−7.07375	−7.07375
$761$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$761$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	3.73336	3.73336
$767$	0	0
$768$	0	0
$769$	1.94609	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$769$	1.94609	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$770$	−1.74575	−1.74575
$771$	0	0
$772$	0	0
$773$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$773$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$774$	−3.55036	−3.55036
$775$	−1.84914	−1.84914
$776$	0	0
$777$	0	0
$778$	3.73336	3.73336
$779$	−2.24458	−2.24458
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3.17597	−3.17597
$785$	−3.04346	−3.04346
$786$	0	0
$787$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$787$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$788$	1.02317	1.02317
$789$	0	0
$790$	0	0
$791$	0	0
$792$	3.86586	3.86586
$793$	0	0
$794$	1.98648	1.98648
$795$	0	0
$796$	−5.85234	−5.85234
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−7.45976	−7.45976
$801$	0	0
$802$	0.231007	0.231007
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−0.317053	−0.317053
$807$	0	0
$808$	6.45976	6.45976
$809$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$809$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$810$	−3.04346	−3.04346
$811$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$811$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$812$	−3.02029	−3.02029
$813$	0	0
$814$	1.57361	1.57361
$815$	0	0
$816$	0	0
$817$	2.13456	2.13456
$818$	2.72641	2.72641
$819$	0.0667045	0.0667045
$820$	−8.48293	−8.48293
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−0.728606	−0.728606
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−3.51857	−3.51857
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.19432	2.19432
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.51137	−1.51137
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$853$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$854$	0	0
$855$	1.82980	1.82980
$856$	−1.34260	−1.34260
$857$	−1.98648	−1.98648	−0.993238	−	0.116093i	$-0.962963\pi$
$857$	−1.98648	−1.98648	−0.993238	+	0.116093i	$0.962963\pi$
$858$	0	0
$859$	−0.573606	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$859$	−0.573606	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$860$	8.06713	8.06713
$861$	0	0
$862$	0	0
$863$	0.792160	0.792160	0.396080	−	0.918216i	$-0.370370\pi$
$863$	0.792160	0.792160	0.396080	+	0.918216i	$0.370370\pi$
$864$	0	0
$865$	0	0
$866$	3.31935	3.31935
$867$	0	0
$868$	2.31935	2.31935
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.305210	−0.305210
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	−7.25192	−7.25192
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.33288	1.33288
$883$	−0.116290	−0.116290	−0.0581448	−	0.998308i	$-0.518519\pi$
$883$	−0.116290	−0.116290	−0.0581448	+	0.998308i	$0.518519\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−1.00000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.10277	−2.10277
$896$	3.96323	3.96323
$897$	0	0
$898$	0	0
$899$	−2.45299	−2.45299
$900$	3.96926	3.96926
$901$	0	0
$902$	−3.73336	−3.73336
$903$	0	0
$904$	0	0
$905$	−0.878816	−0.878816
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	5.73336	5.73336
$909$	−1.67098	−1.67098
$910$	−0.203012	−0.203012
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$919$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.06727	1.06727
$926$	1.98648	1.98648
$927$	0	0
$928$	−9.89580	−9.89580
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.801358	−0.801358
$932$	−1.68990	−1.68990
$933$	0	0
$934$	−3.86586	−3.86586
$935$	0	0
$936$	0.449560	0.449560
$937$	1.19432	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$937$	1.19432	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	3.55036	3.55036
$947$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$947$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$948$	0	0
$949$	0	0
$950$	−3.19644	−3.19644
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	3.94609	3.94609
$959$	−1.11629	−1.11629
$960$	0	0
$961$	0.883710	0.883710
$962$	0.182994	0.182994
$963$	0.347296	0.347296
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$967$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0.0667045	0.0667045
$974$	−3.86586	−3.86586
$975$	0	0
$976$	0	0
$977$	1.78727	1.78727	0.893633	−	0.448799i	$-0.148148\pi$
$977$	1.78727	1.78727	0.893633	+	0.448799i	$0.148148\pi$
$978$	0	0
$979$	0	0
$980$	−3.02856	−3.02856
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0.532089	0.532089
$986$	0	0
$987$	0	0
$988$	−0.409173	−0.409173
$989$	0	0
$990$	3.04346	3.04346
$991$	−1.37248	−1.37248	−0.686242	−	0.727374i	$-0.740741\pi$
$991$	−1.37248	−1.37248	−0.686242	+	0.727374i	$0.740741\pi$
$992$	7.59922	7.59922
$993$	0	0
$994$	0	0
$995$	−3.04346	−3.04346
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1231.1.b.c.1230.1	✓	9
1231.1230	odd	2	CM	1231.1.b.c.1230.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1231.1.b.c.1230.1	✓	9	1.1	even	1	trivial
1231.1.b.c.1230.1	✓	9	1231.1230	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:	\( N \)	\(=\)	\( 1231 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1231.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.98648 q^{2} +2.94609 q^{4} +1.53209 q^{5} -0.573606 q^{7} -3.86586 q^{8} +1.00000 q^{9} -3.04346 q^{10} -1.00000 q^{11} -0.116290 q^{13} +1.13946 q^{14} +4.73336 q^{16} -1.98648 q^{18} +1.19432 q^{19} +4.51367 q^{20} +1.98648 q^{22} +1.34730 q^{25} +0.231007 q^{26} -1.68990 q^{28} +1.78727 q^{29} -1.37248 q^{31} -5.53684 q^{32} -0.878816 q^{35} +2.94609 q^{36} +0.792160 q^{37} -2.37248 q^{38} -5.92284 q^{40} -1.87939 q^{41} +1.78727 q^{43} -2.94609 q^{44} +1.53209 q^{45} -0.670976 q^{49} -2.67637 q^{50} -0.342600 q^{52} -1.53209 q^{55} +2.21748 q^{56} -3.55036 q^{58} +2.72641 q^{62} -0.573606 q^{63} +6.26544 q^{64} -0.178166 q^{65} +1.74575 q^{70} -3.86586 q^{72} -1.57361 q^{74} +3.51857 q^{76} +0.573606 q^{77} +7.25192 q^{80} +1.00000 q^{81} +3.73336 q^{82} -3.55036 q^{86} +3.86586 q^{88} -3.04346 q^{90} +0.0667045 q^{91} +1.82980 q^{95} +1.33288 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{4} + 9 q^{9} - 9 q^{11} + 9 q^{16} + 9 q^{25} - 9 q^{28} + 9 q^{36} - 9 q^{38} - 9 q^{40} - 9 q^{44} + 9 q^{49} + 9 q^{64} - 9 q^{70} - 9 q^{74} + 9 q^{81} - 9 q^{99}+O(q^{100}) \) 9 * q + 9 * q^4 + 9 * q^9 - 9 * q^11 + 9 * q^16 + 9 * q^25 - 9 * q^28 + 9 * q^36 - 9 * q^38 - 9 * q^40 - 9 * q^44 + 9 * q^49 + 9 * q^64 - 9 * q^70 - 9 * q^74 + 9 * q^81 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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