LMFDB - Embedded newform 3971.1.c.b.362.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3971,1,Mod(362,3971)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3971.362"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3971, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3971 = 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3971.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.98178716517$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.43681.1

Embedding invariants

Embedding label			362.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3971.362
Dual form			3971.1.c.b.362.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} +1.00000 q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000i q^{10} -1.00000i q^{11} -1.00000i q^{13} -1.00000 q^{15} -1.00000 q^{16} +1.00000i q^{17} +1.00000 q^{22} +1.00000 q^{23} -1.00000i q^{24} +1.00000 q^{26} +1.00000 q^{27} +1.00000i q^{29} -1.00000i q^{30} +1.00000i q^{33} -1.00000 q^{34} +1.00000i q^{39} +1.00000i q^{40} +1.00000i q^{41} +1.00000i q^{43} +1.00000i q^{46} +1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000i q^{51} +1.00000 q^{53} +1.00000i q^{54} -1.00000i q^{55} -1.00000 q^{58} -1.00000 q^{59} -1.00000i q^{61} -1.00000 q^{64} -1.00000i q^{65} -1.00000 q^{66} +1.00000 q^{67} -1.00000 q^{69} +1.00000 q^{71} +1.00000i q^{73} -1.00000 q^{78} -1.00000i q^{79} -1.00000 q^{80} -1.00000 q^{81} -1.00000 q^{82} +1.00000i q^{85} -1.00000 q^{86} -1.00000i q^{87} +1.00000 q^{88} +1.00000 q^{89} +1.00000i q^{94} +1.00000 q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{15} - 2 q^{16} + 2 q^{22} + 2 q^{23} + 2 q^{26} + 2 q^{27} - 2 q^{34} + 2 q^{47} + 2 q^{48} + 2 q^{49} + 2 q^{53} - 2 q^{58} - 2 q^{59} - 2 q^{64} - 2 q^{66} + 2 q^{67} - 2 q^{69}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^15 - 2 * q^16 + 2 * q^22 + 2 * q^23 + 2 * q^26 + 2 * q^27 - 2 * q^34 + 2 * q^47 + 2 * q^48 + 2 * q^49 + 2 * q^53 - 2 * q^58 - 2 * q^59 - 2 * q^64 - 2 * q^66 + 2 * q^67 - 2 * q^69 + 2 * q^71 - 2 * q^78 - 2 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3971\mathbb{Z}\right)^\times$.

$n$	$1806$	$2168$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$2$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	− 1.00000i	− 1.00000i
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	1.00000i
$9$	0	0
$10$	1.00000i	1.00000i
$11$	− 1.00000i	− 1.00000i
$11$	− 1.00000i	− 1.00000i
$12$	0	0
$13$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$14$	0	0
$15$	−1.00000	−1.00000
$16$	−1.00000	−1.00000
$17$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$17$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0	0
$22$	1.00000	1.00000
$23$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$23$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$24$	− 1.00000i	− 1.00000i
$25$	0	0
$26$	1.00000	1.00000
$27$	1.00000	1.00000
$28$	0	0
$29$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$29$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$30$	− 1.00000i	− 1.00000i
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.00000i	1.00000i
$34$	−1.00000	−1.00000
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	1.00000i	1.00000i
$40$	1.00000i	1.00000i
$41$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$41$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$42$	0	0
$43$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$43$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$44$	0	0
$45$	0	0
$46$	1.00000i	1.00000i
$47$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$47$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$48$	1.00000	1.00000
$49$	1.00000	1.00000
$50$	0	0
$51$	− 1.00000i	− 1.00000i
$52$	0	0
$53$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$53$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$54$	1.00000i	1.00000i
$55$	− 1.00000i	− 1.00000i
$56$	0	0
$57$	0	0
$58$	−1.00000	−1.00000
$59$	−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
$61$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$61$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−1.00000
$65$	− 1.00000i	− 1.00000i
$66$	−1.00000	−1.00000
$67$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$67$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$68$	0	0
$69$	−1.00000	−1.00000
$70$	0	0
$71$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$72$	0	0
$73$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$73$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	−1.00000	−1.00000
$79$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$79$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$80$	−1.00000	−1.00000
$81$	−1.00000	−1.00000
$82$	−1.00000	−1.00000
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	1.00000i	1.00000i
$86$	−1.00000	−1.00000
$87$	− 1.00000i	− 1.00000i
$88$	1.00000	1.00000
$89$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$89$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.00000i	1.00000i
$95$	0	0
$96$	0	0
$97$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$98$	1.00000i	1.00000i
$99$	0	0
$100$	0	0
$101$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$101$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$102$	1.00000	1.00000
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	1.00000i	1.00000i
$107$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$107$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$108$	0	0
$109$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$109$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$110$	1.00000	1.00000
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	1.00000	1.00000
$116$	0	0
$117$	0	0
$118$	− 1.00000i	− 1.00000i
$119$	0	0
$120$	− 1.00000i	− 1.00000i
$121$	−1.00000	−1.00000
$122$	1.00000	1.00000
$123$	− 1.00000i	− 1.00000i
$124$	0	0
$125$	−1.00000	−1.00000
$126$	0	0
$127$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$127$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$128$	− 1.00000i	− 1.00000i
$129$	− 1.00000i	− 1.00000i
$130$	1.00000	1.00000
$131$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$131$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$132$	0	0
$133$	0	0
$134$	1.00000i	1.00000i
$135$	1.00000	1.00000
$136$	−1.00000	−1.00000
$137$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$138$	− 1.00000i	− 1.00000i
$139$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$139$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$140$	0	0
$141$	−1.00000	−1.00000
$142$	1.00000i	1.00000i
$143$	−1.00000	−1.00000
$144$	0	0
$145$	1.00000i	1.00000i
$146$	−1.00000	−1.00000
$147$	−1.00000	−1.00000
$148$	0	0
$149$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$149$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$158$	1.00000	1.00000
$159$	−1.00000	−1.00000
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 1.00000i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	1.00000i	1.00000i
$166$	0	0
$167$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$167$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$168$	0	0
$169$	0	0
$170$	−1.00000	−1.00000
$171$	0	0
$172$	0	0
$173$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$173$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$174$	1.00000	1.00000
$175$	0	0
$176$	1.00000i	1.00000i
$177$	1.00000	1.00000
$178$	1.00000i	1.00000i
$179$	−2.00000	−2.00000	−1.00000			$\pi$
$179$	−2.00000	−2.00000	−1.00000			$\pi$
$180$	0	0
$181$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$182$	0	0
$183$	1.00000i	1.00000i
$184$	1.00000i	1.00000i
$185$	0	0
$186$	0	0
$187$	1.00000	1.00000
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	1.00000
$193$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$193$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$194$	1.00000i	1.00000i
$195$	1.00000i	1.00000i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$200$	0	0
$201$	−1.00000	−1.00000
$202$	−1.00000	−1.00000
$203$	0	0
$204$	0	0
$205$	1.00000i	1.00000i
$206$	0	0
$207$	0	0
$208$	1.00000i	1.00000i
$209$	0	0
$210$	0	0
$211$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$211$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$212$	0	0
$213$	−1.00000	−1.00000
$214$	2.00000	2.00000
$215$	1.00000i	1.00000i
$216$	1.00000i	1.00000i
$217$	0	0
$218$	−1.00000	−1.00000
$219$	− 1.00000i	− 1.00000i
$220$	0	0
$221$	1.00000	1.00000
$222$	0	0
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$229$	−2.00000	−2.00000	−1.00000			$\pi$
$230$	1.00000i	1.00000i
$231$	0	0
$232$	−1.00000	−1.00000
$233$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$233$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$234$	0	0
$235$	1.00000	1.00000
$236$	0	0
$237$	1.00000i	1.00000i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.00000	1.00000
$241$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$241$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$242$	− 1.00000i	− 1.00000i
$243$	0	0
$244$	0	0
$245$	1.00000	1.00000
$246$	1.00000	1.00000
$247$	0	0
$248$	0	0
$249$	0	0
$250$	− 1.00000i	− 1.00000i
$251$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$252$	0	0
$253$	− 1.00000i	− 1.00000i
$254$	1.00000	1.00000
$255$	− 1.00000i	− 1.00000i
$256$	0	0
$257$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$257$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$258$	1.00000	1.00000
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.00000	1.00000
$263$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$263$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$264$	−1.00000	−1.00000
$265$	1.00000	1.00000
$266$	0	0
$267$	−1.00000	−1.00000
$268$	0	0
$269$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$269$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$270$	1.00000i	1.00000i
$271$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$271$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$272$	− 1.00000i	− 1.00000i
$273$	0	0
$274$	1.00000i	1.00000i
$275$	0	0
$276$	0	0
$277$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$277$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$278$	−1.00000	−1.00000
$279$	0	0
$280$	0	0
$281$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$281$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$282$	− 1.00000i	− 1.00000i
$283$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$284$	0	0
$285$	0	0
$286$	− 1.00000i	− 1.00000i
$287$	0	0
$288$	0	0
$289$	0	0
$290$	−1.00000	−1.00000
$291$	−1.00000	−1.00000
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	− 1.00000i	− 1.00000i
$295$	−1.00000	−1.00000
$296$	0	0
$297$	− 1.00000i	− 1.00000i
$298$	−1.00000	−1.00000
$299$	− 1.00000i	− 1.00000i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 1.00000i	− 1.00000i
$304$	0	0
$305$	− 1.00000i	− 1.00000i
$306$	0	0
$307$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$307$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−1.00000	−1.00000
$313$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$314$	− 1.00000i	− 1.00000i
$315$	0	0
$316$	0	0
$317$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$318$	− 1.00000i	− 1.00000i
$319$	1.00000	1.00000
$320$	−1.00000	−1.00000
$321$	2.00000i	2.00000i
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.00000i	− 1.00000i
$328$	−1.00000	−1.00000
$329$	0	0
$330$	−1.00000	−1.00000
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	−1.00000	−1.00000
$335$	1.00000	1.00000
$336$	0	0
$337$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$337$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−1.00000	−1.00000
$345$	−1.00000	−1.00000
$346$	1.00000	1.00000
$347$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$347$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$	0	0
$349$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$349$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$350$	0	0
$351$	− 1.00000i	− 1.00000i
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	1.00000i	1.00000i
$355$	1.00000	1.00000
$356$	0	0
$357$	0	0
$358$	− 2.00000i	− 2.00000i
$359$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$359$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$360$	0	0
$361$	0	0
$362$	− 1.00000i	− 1.00000i
$363$	1.00000	1.00000
$364$	0	0
$365$	1.00000i	1.00000i
$366$	−1.00000	−1.00000
$367$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$368$	−1.00000	−1.00000
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	1.00000i	1.00000i
$375$	1.00000	1.00000
$376$	1.00000i	1.00000i
$377$	1.00000	1.00000
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	1.00000i	1.00000i
$382$	0	0
$383$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$383$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$384$	1.00000i	1.00000i
$385$	0	0
$386$	−1.00000	−1.00000
$387$	0	0
$388$	0	0
$389$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$390$	−1.00000	−1.00000
$391$	1.00000i	1.00000i
$392$	1.00000i	1.00000i
$393$	1.00000i	1.00000i
$394$	0	0
$395$	− 1.00000i	− 1.00000i
$396$	0	0
$397$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$398$	− 1.00000i	− 1.00000i
$399$	0	0
$400$	0	0
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	− 1.00000i	− 1.00000i
$403$	0	0
$404$	0	0
$405$	−1.00000	−1.00000
$406$	0	0
$407$	0	0
$408$	1.00000	1.00000
$409$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$409$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$410$	−1.00000	−1.00000
$411$	−1.00000	−1.00000
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.00000i	− 1.00000i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$422$	1.00000	1.00000
$423$	0	0
$424$	1.00000i	1.00000i
$425$	0	0
$426$	− 1.00000i	− 1.00000i
$427$	0	0
$428$	0	0
$429$	1.00000	1.00000
$430$	−1.00000	−1.00000
$431$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$431$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$432$	−1.00000	−1.00000
$433$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$434$	0	0
$435$	− 1.00000i	− 1.00000i
$436$	0	0
$437$	0	0
$438$	1.00000	1.00000
$439$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$439$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$440$	1.00000	1.00000
$441$	0	0
$442$	1.00000i	1.00000i
$443$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$443$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$444$	0	0
$445$	1.00000	1.00000
$446$	− 1.00000i	− 1.00000i
$447$	− 1.00000i	− 1.00000i
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	1.00000	1.00000
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$457$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$458$	− 2.00000i	− 2.00000i
$459$	1.00000i	1.00000i
$460$	0	0
$461$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$461$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	− 1.00000i	− 1.00000i
$465$	0	0
$466$	1.00000	1.00000
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	1.00000i	1.00000i
$471$	1.00000	1.00000
$472$	− 1.00000i	− 1.00000i
$473$	1.00000	1.00000
$474$	−1.00000	−1.00000
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$479$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$480$	0	0
$481$	0	0
$482$	−1.00000	−1.00000
$483$	0	0
$484$	0	0
$485$	1.00000	1.00000
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	1.00000	1.00000
$489$	0	0
$490$	1.00000i	1.00000i
$491$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$491$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$492$	0	0
$493$	−1.00000	−1.00000
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$500$	0	0
$501$	− 1.00000i	− 1.00000i
$502$	− 1.00000i	− 1.00000i
$503$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$503$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$504$	0	0
$505$	1.00000i	1.00000i
$506$	1.00000	1.00000
$507$	0	0
$508$	0	0
$509$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$510$	1.00000	1.00000
$511$	0	0
$512$	− 1.00000i	− 1.00000i
$513$	0	0
$514$	1.00000i	1.00000i
$515$	0	0
$516$	0	0
$517$	− 1.00000i	− 1.00000i
$518$	0	0
$519$	1.00000i	1.00000i
$520$	1.00000	1.00000
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$523$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$	0	0
$525$	0	0
$526$	1.00000	1.00000
$527$	0	0
$528$	− 1.00000i	− 1.00000i
$529$	0	0
$530$	1.00000i	1.00000i
$531$	0	0
$532$	0	0
$533$	1.00000	1.00000
$534$	− 1.00000i	− 1.00000i
$535$	− 2.00000i	− 2.00000i
$536$	1.00000i	1.00000i
$537$	2.00000	2.00000
$538$	1.00000i	1.00000i
$539$	− 1.00000i	− 1.00000i
$540$	0	0
$541$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$541$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$542$	1.00000	1.00000
$543$	1.00000	1.00000
$544$	0	0
$545$	1.00000i	1.00000i
$546$	0	0
$547$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$547$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	− 1.00000i	− 1.00000i
$553$	0	0
$554$	−2.00000	−2.00000
$555$	0	0
$556$	0	0
$557$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$558$	0	0
$559$	1.00000	1.00000
$560$	0	0
$561$	−1.00000	−1.00000
$562$	1.00000	1.00000
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	1.00000	1.00000
$567$	0	0
$568$	1.00000i	1.00000i
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	− 1.00000i	− 1.00000i
$580$	0	0
$581$	0	0
$582$	− 1.00000i	− 1.00000i
$583$	− 1.00000i	− 1.00000i
$584$	−1.00000	−1.00000
$585$	0	0
$586$	0	0
$587$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$587$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$588$	0	0
$589$	0	0
$590$	− 1.00000i	− 1.00000i
$591$	0	0
$592$	0	0
$593$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$593$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$594$	1.00000	1.00000
$595$	0	0
$596$	0	0
$597$	1.00000	1.00000
$598$	1.00000	1.00000
$599$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−1.00000
$606$	1.00000	1.00000
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	1.00000	1.00000
$611$	− 1.00000i	− 1.00000i
$612$	0	0
$613$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$613$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$614$	1.00000	1.00000
$615$	− 1.00000i	− 1.00000i
$616$	0	0
$617$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$617$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	1.00000	1.00000
$622$	0	0
$623$	0	0
$624$	− 1.00000i	− 1.00000i
$625$	−1.00000	−1.00000
$626$	− 1.00000i	− 1.00000i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$632$	1.00000	1.00000
$633$	1.00000i	1.00000i
$634$	− 1.00000i	− 1.00000i
$635$	− 1.00000i	− 1.00000i
$636$	0	0
$637$	− 1.00000i	− 1.00000i
$638$	1.00000i	1.00000i
$639$	0	0
$640$	− 1.00000i	− 1.00000i
$641$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$641$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$642$	−2.00000	−2.00000
$643$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$644$	0	0
$645$	− 1.00000i	− 1.00000i
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 1.00000i	− 1.00000i
$649$	1.00000i	1.00000i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.00000	2.00000	1.00000			$0$
$653$	2.00000	2.00000	1.00000			$0$
$654$	1.00000	1.00000
$655$	− 1.00000i	− 1.00000i
$656$	− 1.00000i	− 1.00000i
$657$	0	0
$658$	0	0
$659$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$659$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$660$	0	0
$661$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$662$	0	0
$663$	−1.00000	−1.00000
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.00000i	1.00000i
$668$	0	0
$669$	1.00000	1.00000
$670$	1.00000i	1.00000i
$671$	−1.00000	−1.00000
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−1.00000	−1.00000
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	−1.00000	−1.00000
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	1.00000	1.00000
$686$	0	0
$687$	2.00000	2.00000
$688$	− 1.00000i	− 1.00000i
$689$	− 1.00000i	− 1.00000i
$690$	− 1.00000i	− 1.00000i
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	−1.00000	−1.00000
$695$	1.00000i	1.00000i
$696$	1.00000	1.00000
$697$	−1.00000	−1.00000
$698$	2.00000	2.00000
$699$	1.00000i	1.00000i
$700$	0	0
$701$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$701$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$702$	1.00000	1.00000
$703$	0	0
$704$	1.00000i	1.00000i
$705$	−1.00000	−1.00000
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$710$	1.00000i	1.00000i
$711$	0	0
$712$	1.00000i	1.00000i
$713$	0	0
$714$	0	0
$715$	−1.00000	−1.00000
$716$	0	0
$717$	0	0
$718$	−1.00000	−1.00000
$719$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 1.00000i	− 1.00000i
$724$	0	0
$725$	0	0
$726$	1.00000i	1.00000i
$727$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	−1.00000	−1.00000
$731$	−1.00000	−1.00000
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.00000i	1.00000i
$735$	−1.00000	−1.00000
$736$	0	0
$737$	− 1.00000i	− 1.00000i
$738$	0	0
$739$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$739$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$743$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$744$	0	0
$745$	1.00000i	1.00000i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	1.00000i	1.00000i
$751$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$751$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$752$	−1.00000	−1.00000
$753$	1.00000	1.00000
$754$	1.00000i	1.00000i
$755$	0	0
$756$	0	0
$757$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$757$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$758$	0	0
$759$	1.00000i	1.00000i
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−1.00000	−1.00000
$763$	0	0
$764$	0	0
$765$	0	0
$766$	1.00000i	1.00000i
$767$	1.00000i	1.00000i
$768$	0	0
$769$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$769$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$770$	0	0
$771$	−1.00000	−1.00000
$772$	0	0
$773$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$774$	0	0
$775$	0	0
$776$	1.00000i	1.00000i
$777$	0	0
$778$	1.00000i	1.00000i
$779$	0	0
$780$	0	0
$781$	− 1.00000i	− 1.00000i
$782$	−1.00000	−1.00000
$783$	1.00000i	1.00000i
$784$	−1.00000	−1.00000
$785$	−1.00000	−1.00000
$786$	−1.00000	−1.00000
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	1.00000i	1.00000i
$790$	1.00000	1.00000
$791$	0	0
$792$	0	0
$793$	−1.00000	−1.00000
$794$	− 1.00000i	− 1.00000i
$795$	−1.00000	−1.00000
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	1.00000i	1.00000i
$800$	0	0
$801$	0	0
$802$	− 1.00000i	− 1.00000i
$803$	1.00000	1.00000
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.00000	−1.00000
$808$	−1.00000	−1.00000
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	− 1.00000i	− 1.00000i
$811$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$811$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$812$	0	0
$813$	1.00000i	1.00000i
$814$	0	0
$815$	0	0
$816$	1.00000i	1.00000i
$817$	0	0
$818$	−1.00000	−1.00000
$819$	0	0
$820$	0	0
$821$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$821$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$822$	− 1.00000i	− 1.00000i
$823$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$827$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	− 2.00000i	− 2.00000i
$832$	1.00000i	1.00000i
$833$	1.00000i	1.00000i
$834$	1.00000	1.00000
$835$	1.00000i	1.00000i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$839$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$840$	0	0
$841$	0	0
$842$	− 1.00000i	− 1.00000i
$843$	1.00000i	1.00000i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−1.00000	−1.00000
$849$	1.00000i	1.00000i
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$853$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$854$	0	0
$855$	0	0
$856$	2.00000	2.00000
$857$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$858$	1.00000i	1.00000i
$859$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$859$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$860$	0	0
$861$	0	0
$862$	1.00000	1.00000
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	− 1.00000i	− 1.00000i
$866$	− 1.00000i	− 1.00000i
$867$	0	0
$868$	0	0
$869$	−1.00000	−1.00000
$870$	1.00000	1.00000
$871$	− 1.00000i	− 1.00000i
$872$	−1.00000	−1.00000
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$877$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$878$	1.00000	1.00000
$879$	0	0
$880$	1.00000i	1.00000i
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$883$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$884$	0	0
$885$	1.00000	1.00000
$886$	1.00000i	1.00000i
$887$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$887$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$888$	0	0
$889$	0	0
$890$	1.00000i	1.00000i
$891$	1.00000i	1.00000i
$892$	0	0
$893$	0	0
$894$	1.00000	1.00000
$895$	−2.00000	−2.00000
$896$	0	0
$897$	1.00000i	1.00000i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	1.00000i	1.00000i
$902$	1.00000i	1.00000i
$903$	0	0
$904$	0	0
$905$	−1.00000	−1.00000
$906$	0	0
$907$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	2.00000	2.00000
$915$	1.00000i	1.00000i
$916$	0	0
$917$	0	0
$918$	−1.00000	−1.00000
$919$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$919$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$920$	1.00000i	1.00000i
$921$	1.00000i	1.00000i
$922$	1.00000	1.00000
$923$	− 1.00000i	− 1.00000i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.00000	1.00000
$936$	0	0
$937$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$937$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$938$	0	0
$939$	1.00000	1.00000
$940$	0	0
$941$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$941$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$942$	1.00000i	1.00000i
$943$	1.00000i	1.00000i
$944$	1.00000	1.00000
$945$	0	0
$946$	1.00000i	1.00000i
$947$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$948$	0	0
$949$	1.00000	1.00000
$950$	0	0
$951$	1.00000	1.00000
$952$	0	0
$953$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$953$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.00000	−1.00000
$958$	−1.00000	−1.00000
$959$	0	0
$960$	1.00000	1.00000
$961$	−1.00000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.00000i	1.00000i
$966$	0	0
$967$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$967$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$968$	− 1.00000i	− 1.00000i
$969$	0	0
$970$	1.00000i	1.00000i
$971$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	1.00000i	1.00000i
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	− 1.00000i	− 1.00000i
$980$	0	0
$981$	0	0
$982$	1.00000	1.00000
$983$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$984$	1.00000	1.00000
$985$	0	0
$986$	− 1.00000i	− 1.00000i
$987$	0	0
$988$	0	0
$989$	1.00000i	1.00000i
$990$	0	0
$991$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.00000	−1.00000
$996$	0	0
$997$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$997$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$998$	− 1.00000i	− 1.00000i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3971.1.c.b.362.2		2
11.10	odd	2	inner	3971.1.c.b.362.1		2
19.2	odd	18		3971.1.q.e.2265.1		12
19.3	odd	18		3971.1.q.e.3277.2		12
19.4	even	9		3971.1.q.d.54.1		12
19.5	even	9		3971.1.q.d.956.2		12
19.6	even	9		3971.1.q.d.967.2		12
19.7	even	3		3971.1.h.d.2595.1		4
19.8	odd	6		209.1.h.a.197.1	yes	4
19.9	even	9		3971.1.q.d.1506.1		12
19.10	odd	18		3971.1.q.e.1506.2		12
19.11	even	3		3971.1.h.d.3541.2		4
19.12	odd	6		209.1.h.a.87.2	yes	4
19.13	odd	18		3971.1.q.e.967.1		12
19.14	odd	18		3971.1.q.e.956.1		12
19.15	odd	18		3971.1.q.e.54.2		12
19.16	even	9		3971.1.q.d.3277.1		12
19.17	even	9		3971.1.q.d.2265.2		12
19.18	odd	2		3971.1.c.e.362.1		2
57.8	even	6		1881.1.bc.a.406.2		4
57.50	even	6		1881.1.bc.a.505.1		4
76.27	even	6		3344.1.bb.a.2705.2		4
76.31	even	6		3344.1.bb.a.2177.2		4
209.8	even	30		2299.1.w.a.596.1		16
209.10	even	18		3971.1.q.e.1506.1		12
209.21	even	18		3971.1.q.e.2265.2		12
209.27	odd	30		2299.1.w.a.1812.2		16
209.31	odd	30		2299.1.w.a.524.1		16
209.32	even	18		3971.1.q.e.967.2		12
209.43	odd	18		3971.1.q.d.956.1		12
209.46	even	30		2299.1.w.a.2097.1		16
209.50	even	30		2299.1.w.a.239.2		16
209.54	odd	18		3971.1.q.d.3277.2		12
209.65	even	6		209.1.h.a.197.2	yes	4
209.69	odd	30		2299.1.w.a.1322.2		16
209.84	even	30		2299.1.w.a.1546.2		16
209.87	odd	6		3971.1.h.d.3541.1		4
209.98	even	18		3971.1.q.e.3277.1		12
209.103	odd	30		2299.1.w.a.1546.1		16
209.107	even	30		2299.1.w.a.1322.1		16
209.109	even	18		3971.1.q.e.956.2		12
209.120	odd	18		3971.1.q.d.967.1		12
209.126	odd	30		2299.1.w.a.239.1		16
209.131	odd	18		3971.1.q.d.2265.1		12
209.141	odd	30		2299.1.w.a.2097.2		16
209.142	odd	18		3971.1.q.d.1506.2		12
209.145	even	30		2299.1.w.a.524.2		16
209.160	even	30		2299.1.w.a.1812.1		16
209.164	even	6		209.1.h.a.87.1	✓	4
209.175	odd	18		3971.1.q.d.54.2		12
209.179	odd	30		2299.1.w.a.596.2		16
209.183	even	30		2299.1.w.a.2272.2		16
209.186	even	18		3971.1.q.e.54.1		12
209.197	odd	6		3971.1.h.d.2595.2		4
209.202	odd	30		2299.1.w.a.2272.1		16
209.208	even	2		3971.1.c.e.362.2		2
627.65	odd	6		1881.1.bc.a.406.1		4
627.164	odd	6		1881.1.bc.a.505.2		4
836.483	odd	6		3344.1.bb.a.2705.1		4
836.791	odd	6		3344.1.bb.a.2177.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.1.h.a.87.1	✓	4	209.164	even	6
209.1.h.a.87.2	yes	4	19.12	odd	6
209.1.h.a.197.1	yes	4	19.8	odd	6
209.1.h.a.197.2	yes	4	209.65	even	6
1881.1.bc.a.406.1		4	627.65	odd	6
1881.1.bc.a.406.2		4	57.8	even	6
1881.1.bc.a.505.1		4	57.50	even	6
1881.1.bc.a.505.2		4	627.164	odd	6
2299.1.w.a.239.1		16	209.126	odd	30
2299.1.w.a.239.2		16	209.50	even	30
2299.1.w.a.524.1		16	209.31	odd	30
2299.1.w.a.524.2		16	209.145	even	30
2299.1.w.a.596.1		16	209.8	even	30
2299.1.w.a.596.2		16	209.179	odd	30
2299.1.w.a.1322.1		16	209.107	even	30
2299.1.w.a.1322.2		16	209.69	odd	30
2299.1.w.a.1546.1		16	209.103	odd	30
2299.1.w.a.1546.2		16	209.84	even	30
2299.1.w.a.1812.1		16	209.160	even	30
2299.1.w.a.1812.2		16	209.27	odd	30
2299.1.w.a.2097.1		16	209.46	even	30
2299.1.w.a.2097.2		16	209.141	odd	30
2299.1.w.a.2272.1		16	209.202	odd	30
2299.1.w.a.2272.2		16	209.183	even	30
3344.1.bb.a.2177.1		4	836.791	odd	6
3344.1.bb.a.2177.2		4	76.31	even	6
3344.1.bb.a.2705.1		4	836.483	odd	6
3344.1.bb.a.2705.2		4	76.27	even	6
3971.1.c.b.362.1		2	11.10	odd	2	inner
3971.1.c.b.362.2		2	1.1	even	1	trivial
3971.1.c.e.362.1		2	19.18	odd	2
3971.1.c.e.362.2		2	209.208	even	2
3971.1.h.d.2595.1		4	19.7	even	3
3971.1.h.d.2595.2		4	209.197	odd	6
3971.1.h.d.3541.1		4	209.87	odd	6
3971.1.h.d.3541.2		4	19.11	even	3
3971.1.q.d.54.1		12	19.4	even	9
3971.1.q.d.54.2		12	209.175	odd	18
3971.1.q.d.956.1		12	209.43	odd	18
3971.1.q.d.956.2		12	19.5	even	9
3971.1.q.d.967.1		12	209.120	odd	18
3971.1.q.d.967.2		12	19.6	even	9
3971.1.q.d.1506.1		12	19.9	even	9
3971.1.q.d.1506.2		12	209.142	odd	18
3971.1.q.d.2265.1		12	209.131	odd	18
3971.1.q.d.2265.2		12	19.17	even	9
3971.1.q.d.3277.1		12	19.16	even	9
3971.1.q.d.3277.2		12	209.54	odd	18
3971.1.q.e.54.1		12	209.186	even	18
3971.1.q.e.54.2		12	19.15	odd	18
3971.1.q.e.956.1		12	19.14	odd	18
3971.1.q.e.956.2		12	209.109	even	18
3971.1.q.e.967.1		12	19.13	odd	18
3971.1.q.e.967.2		12	209.32	even	18
3971.1.q.e.1506.1		12	209.10	even	18
3971.1.q.e.1506.2		12	19.10	odd	18
3971.1.q.e.2265.1		12	19.2	odd	18
3971.1.q.e.2265.2		12	209.21	even	18
3971.1.q.e.3277.1		12	209.98	even	18
3971.1.q.e.3277.2		12	19.3	odd	18

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3971 = 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3971.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} +1.00000 q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000i q^{10} -1.00000i q^{11} -1.00000i q^{13} -1.00000 q^{15} -1.00000 q^{16} +1.00000i q^{17} +1.00000 q^{22} +1.00000 q^{23} -1.00000i q^{24} +1.00000 q^{26} +1.00000 q^{27} +1.00000i q^{29} -1.00000i q^{30} +1.00000i q^{33} -1.00000 q^{34} +1.00000i q^{39} +1.00000i q^{40} +1.00000i q^{41} +1.00000i q^{43} +1.00000i q^{46} +1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000i q^{51} +1.00000 q^{53} +1.00000i q^{54} -1.00000i q^{55} -1.00000 q^{58} -1.00000 q^{59} -1.00000i q^{61} -1.00000 q^{64} -1.00000i q^{65} -1.00000 q^{66} +1.00000 q^{67} -1.00000 q^{69} +1.00000 q^{71} +1.00000i q^{73} -1.00000 q^{78} -1.00000i q^{79} -1.00000 q^{80} -1.00000 q^{81} -1.00000 q^{82} +1.00000i q^{85} -1.00000 q^{86} -1.00000i q^{87} +1.00000 q^{88} +1.00000 q^{89} +1.00000i q^{94} +1.00000 q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{15} - 2 q^{16} + 2 q^{22} + 2 q^{23} + 2 q^{26} + 2 q^{27} - 2 q^{34} + 2 q^{47} + 2 q^{48} + 2 q^{49} + 2 q^{53} - 2 q^{58} - 2 q^{59} - 2 q^{64} - 2 q^{66} + 2 q^{67} - 2 q^{69}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^15 - 2 * q^16 + 2 * q^22 + 2 * q^23 + 2 * q^26 + 2 * q^27 - 2 * q^34 + 2 * q^47 + 2 * q^48 + 2 * q^49 + 2 * q^53 - 2 * q^58 - 2 * q^59 - 2 * q^64 - 2 * q^66 + 2 * q^67 - 2 * q^69 + 2 * q^71 - 2 * q^78 - 2 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^97

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