LMFDB - Embedded newform 2016.1.cm.a.655.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,1,Mod(655,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4, 4]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.655");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2016.cm (of order $6$, degree $2$, not 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:	no
Analytic conductor:	$1.00611506547$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.254016.3

Embedding invariants

Embedding label			655.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2016.655
Dual form			2016.1.cm.a.751.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{3} +(0.866025 + 0.500000i) q^{5} -1.00000i q^{7} +1.00000 q^{9} +(-0.500000 - 0.866025i) q^{11} +(-0.866025 + 0.500000i) q^{13} +(0.866025 + 0.500000i) q^{15} +(0.500000 - 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{19} -1.00000i q^{21} +(-0.866025 - 0.500000i) q^{23} +1.00000 q^{27} +(0.866025 + 0.500000i) q^{29} +(-0.500000 - 0.866025i) q^{33} +(0.500000 - 0.866025i) q^{35} +(-0.866025 + 0.500000i) q^{37} +(-0.866025 + 0.500000i) q^{39} +(0.500000 + 0.866025i) q^{41} +(-0.500000 + 0.866025i) q^{43} +(0.866025 + 0.500000i) q^{45} -1.00000 q^{49} +(0.500000 - 0.866025i) q^{51} +(-0.866025 - 0.500000i) q^{53} -1.00000i q^{55} +(0.500000 + 0.866025i) q^{57} -1.00000i q^{63} -1.00000 q^{65} +(-0.866025 - 0.500000i) q^{69} +(-0.500000 + 0.866025i) q^{73} +(-0.866025 + 0.500000i) q^{77} +2.00000i q^{79} +1.00000 q^{81} +(-0.500000 + 0.866025i) q^{83} +(0.866025 - 0.500000i) q^{85} +(0.866025 + 0.500000i) q^{87} +(-0.500000 - 0.866025i) q^{89} +(0.500000 + 0.866025i) q^{91} +1.00000i q^{95} +(-0.500000 + 0.866025i) q^{97} +(-0.500000 - 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9} - 2 q^{11} + 2 q^{17} + 2 q^{19} + 4 q^{27} - 2 q^{33} + 2 q^{35} + 2 q^{41} - 2 q^{43} - 4 q^{49} + 2 q^{51} + 2 q^{57} - 4 q^{65} - 2 q^{73} + 4 q^{81} - 2 q^{83} - 2 q^{89} + 2 q^{91}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 - 2 * q^11 + 2 * q^17 + 2 * q^19 + 4 * q^27 - 2 * q^33 + 2 * q^35 + 2 * q^41 - 2 * q^43 - 4 * q^49 + 2 * q^51 + 2 * q^57 - 4 * q^65 - 2 * q^73 + 4 * q^81 - 2 * q^83 - 2 * q^89 + 2 * q^91 - 2 * q^97 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{2}{3}\right)$

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			0
$2$		0			0
$3$	1.00000			1.00000
$3$	1.00000			1.00000
$4$		0			0
$5$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$5$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$6$		0			0
$7$		−	1.00000i		−	1.00000i
$7$		−	1.00000i		−	1.00000i
$8$		0			0
$9$	1.00000			1.00000
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$12$		0			0
$13$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$14$		0			0
$15$	0.866025	+	0.500000i	0.866025	+	0.500000i
$16$		0			0
$17$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$18$		0			0
$19$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$19$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		−	1.00000i		−	1.00000i
$22$		0			0
$23$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$23$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$29$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$34$		0			0
$35$	0.500000	−	0.866025i	0.500000	−	0.866025i
$36$		0			0
$37$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$37$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$38$		0			0
$39$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$40$		0			0
$41$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$41$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$43$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$44$		0			0
$45$	0.866025	+	0.500000i	0.866025	+	0.500000i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$		0			0
$51$	0.500000	−	0.866025i	0.500000	−	0.866025i
$52$		0			0
$53$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$53$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		−	1.00000i		−	1.00000i
$56$		0			0
$57$	0.500000	+	0.866025i	0.500000	+	0.866025i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		−	1.00000i		−	1.00000i
$64$		0			0
$65$	−1.00000			−1.00000
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$73$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$78$		0			0
$79$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$79$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			1.00000
$82$		0			0
$83$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$83$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$84$		0			0
$85$	0.866025	−	0.500000i	0.866025	−	0.500000i
$86$		0			0
$87$	0.866025	+	0.500000i	0.866025	+	0.500000i
$88$		0			0
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$90$		0			0
$91$	0.500000	+	0.866025i	0.500000	+	0.866025i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1.00000i			1.00000i
$96$		0			0
$97$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$97$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$98$		0			0
$99$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$100$		0			0
$101$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$101$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$103$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$104$		0			0
$105$	0.500000	−	0.866025i	0.500000	−	0.866025i
$106$		0			0
$107$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$107$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$108$		0			0
$109$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$109$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$110$		0			0
$111$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$112$		0			0
$113$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$113$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$114$		0			0
$115$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$116$		0			0
$117$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$118$		0			0
$119$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	0.500000	+	0.866025i	0.500000	+	0.866025i
$124$		0			0
$125$		−	1.00000i		−	1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$130$		0			0
$131$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$131$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$132$		0			0
$133$	0.866025	−	0.500000i	0.866025	−	0.500000i
$134$		0			0
$135$	0.866025	+	0.500000i	0.866025	+	0.500000i
$136$		0			0
$137$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$138$		0			0
$139$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.866025	+	0.500000i	0.866025	+	0.500000i
$144$		0			0
$145$	0.500000	+	0.866025i	0.500000	+	0.866025i
$146$		0			0
$147$	−1.00000			−1.00000
$148$		0			0
$149$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$149$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$151$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$152$		0			0
$153$	0.500000	−	0.866025i	0.500000	−	0.866025i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$160$		0			0
$161$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$162$		0			0
$163$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$163$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		−	1.00000i		−	1.00000i
$166$		0			0
$167$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$167$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	0.500000	+	0.866025i	0.500000	+	0.866025i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$179$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$180$		0			0
$181$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$181$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000			−1.00000
$186$		0			0
$187$	−1.00000			−1.00000
$188$		0			0
$189$		−	1.00000i		−	1.00000i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$	−1.00000			−1.00000
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$199$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.500000	−	0.866025i	0.500000	−	0.866025i
$204$		0			0
$205$			1.00000i			1.00000i
$206$		0			0
$207$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$208$		0			0
$209$	0.500000	−	0.866025i	0.500000	−	0.866025i
$210$		0			0
$211$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$211$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$220$		0			0
$221$			1.00000i			1.00000i
$222$		0			0
$223$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$223$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$227$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$228$		0			0
$229$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$229$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$232$		0			0
$233$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$233$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			2.00000i			2.00000i
$238$		0			0
$239$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$239$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	1.00000			1.00000
$244$		0			0
$245$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$246$		0			0
$247$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$248$		0			0
$249$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			1.00000i			1.00000i
$254$		0			0
$255$	0.866025	−	0.500000i	0.866025	−	0.500000i
$256$		0			0
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$258$		0			0
$259$	0.500000	+	0.866025i	0.500000	+	0.866025i
$260$		0			0
$261$	0.866025	+	0.500000i	0.866025	+	0.500000i
$262$		0			0
$263$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$263$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$264$		0			0
$265$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$266$		0			0
$267$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$268$		0			0
$269$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$269$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$270$		0			0
$271$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$271$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$272$		0			0
$273$	0.500000	+	0.866025i	0.500000	+	0.866025i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$277$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$281$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$			1.00000i			1.00000i
$286$		0			0
$287$	0.866025	−	0.500000i	0.866025	−	0.500000i
$288$		0			0
$289$		0			0
$290$		0			0
$291$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$292$		0			0
$293$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$293$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$298$		0			0
$299$	1.00000			1.00000
$300$		0			0
$301$	0.866025	+	0.500000i	0.866025	+	0.500000i
$302$		0			0
$303$	0.866025	−	0.500000i	0.866025	−	0.500000i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	0.866025	+	0.500000i	0.866025	+	0.500000i
$310$		0			0
$311$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$311$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	0.500000	−	0.866025i	0.500000	−	0.866025i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		−	1.00000i		−	1.00000i
$320$		0			0
$321$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$322$		0			0
$323$	1.00000			1.00000
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.866025	+	0.500000i	0.866025	+	0.500000i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$337$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$338$		0			0
$339$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			1.00000i
$344$		0			0
$345$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$349$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$350$		0			0
$351$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$352$		0			0
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$358$		0			0
$359$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$359$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$366$		0			0
$367$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$367$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$368$		0			0
$369$	0.500000	+	0.866025i	0.500000	+	0.866025i
$370$		0			0
$371$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$372$		0			0
$373$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$373$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$374$		0			0
$375$		−	1.00000i		−	1.00000i
$376$		0			0
$377$	−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
$381$		0			0
$382$		0			0
$383$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$383$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$384$		0			0
$385$	−1.00000			−1.00000
$386$		0			0
$387$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$388$		0			0
$389$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$389$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$390$		0			0
$391$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$392$		0			0
$393$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$394$		0			0
$395$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$396$		0			0
$397$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$397$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$	0.866025	−	0.500000i	0.866025	−	0.500000i
$400$		0			0
$401$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$401$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.866025	+	0.500000i	0.866025	+	0.500000i
$406$		0			0
$407$	0.866025	+	0.500000i	0.866025	+	0.500000i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$416$		0			0
$417$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$418$		0			0
$419$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$419$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$420$		0			0
$421$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$421$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	0.866025	+	0.500000i	0.866025	+	0.500000i
$430$		0			0
$431$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$431$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	0.500000	+	0.866025i	0.500000	+	0.866025i
$436$		0			0
$437$		−	1.00000i		−	1.00000i
$438$		0			0
$439$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$439$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		−	1.00000i		−	1.00000i
$446$		0			0
$447$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	0.500000	−	0.866025i	0.500000	−	0.866025i
$452$		0			0
$453$	0.866025	−	0.500000i	0.866025	−	0.500000i
$454$		0			0
$455$			1.00000i			1.00000i
$456$		0			0
$457$	−2.00000			−2.00000			−1.00000			$\pi$
$457$	−2.00000			−2.00000			−1.00000			$\pi$
$458$		0			0
$459$	0.500000	−	0.866025i	0.500000	−	0.866025i
$460$		0			0
$461$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$461$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$462$		0			0
$463$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$463$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$467$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.00000			1.00000
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$478$		0			0
$479$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$479$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	0.500000	−	0.866025i	0.500000	−	0.866025i
$482$		0			0
$483$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$484$		0			0
$485$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$486$		0			0
$487$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$487$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$488$		0			0
$489$	0.500000	+	0.866025i	0.500000	+	0.866025i
$490$		0			0
$491$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$491$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	0.866025	−	0.500000i	0.866025	−	0.500000i
$494$		0			0
$495$		−	1.00000i		−	1.00000i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$500$		0			0
$501$	0.866025	−	0.500000i	0.866025	−	0.500000i
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	1.00000			1.00000
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$509$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	0.866025	+	0.500000i	0.866025	+	0.500000i
$512$		0			0
$513$	0.500000	+	0.866025i	0.500000	+	0.866025i
$514$		0			0
$515$	0.500000	+	0.866025i	0.500000	+	0.866025i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$522$		0			0
$523$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$523$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$534$		0			0
$535$		−	1.00000i		−	1.00000i
$536$		0			0
$537$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$538$		0			0
$539$	0.500000	+	0.866025i	0.500000	+	0.866025i
$540$		0			0
$541$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$541$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$542$		0			0
$543$		−	2.00000i		−	2.00000i
$544$		0			0
$545$	0.500000	+	0.866025i	0.500000	+	0.866025i
$546$		0			0
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1.00000i			1.00000i
$552$		0			0
$553$	2.00000			2.00000
$554$		0			0
$555$	−1.00000			−1.00000
$556$		0			0
$557$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$557$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$558$		0			0
$559$		−	1.00000i		−	1.00000i
$560$		0			0
$561$	−1.00000			−1.00000
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		−	1.00000i		−	1.00000i
$566$		0			0
$567$		−	1.00000i		−	1.00000i
$568$		0			0
$569$	2.00000			2.00000			1.00000			$0$
$569$	2.00000			2.00000			1.00000			$0$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.866025	+	0.500000i	0.866025	+	0.500000i
$582$		0			0
$583$			1.00000i			1.00000i
$584$		0			0
$585$	−1.00000			−1.00000
$586$		0			0
$587$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$593$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$596$		0			0
$597$	0.866025	+	0.500000i	0.866025	+	0.500000i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$607$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$608$		0			0
$609$	0.500000	−	0.866025i	0.500000	−	0.866025i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$613$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$614$		0			0
$615$			1.00000i			1.00000i
$616$		0			0
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$619$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$622$		0			0
$623$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$		0			0
$627$	0.500000	−	0.866025i	0.500000	−	0.866025i
$628$		0			0
$629$			1.00000i			1.00000i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.866025	−	0.500000i	0.866025	−	0.500000i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$646$		0			0
$647$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$647$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$653$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$656$		0			0
$657$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$658$		0			0
$659$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$			1.00000i			1.00000i
$664$		0			0
$665$	1.00000			1.00000
$666$		0			0
$667$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$668$		0			0
$669$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	0.866025	+	0.500000i	0.866025	+	0.500000i
$680$		0			0
$681$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$682$		0			0
$683$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$684$		0			0
$685$		−	1.00000i		−	1.00000i
$686$		0			0
$687$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$688$		0			0
$689$	1.00000			1.00000
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$694$		0			0
$695$		−	1.00000i		−	1.00000i
$696$		0			0
$697$	1.00000			1.00000
$698$		0			0
$699$	0.500000	+	0.866025i	0.500000	+	0.866025i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$708$		0			0
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$709$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$710$		0			0
$711$			2.00000i			2.00000i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	0.500000	+	0.866025i	0.500000	+	0.866025i
$716$		0			0
$717$	0.866025	−	0.500000i	0.866025	−	0.500000i
$718$		0			0
$719$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$719$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	0.500000	−	0.866025i	0.500000	−	0.866025i
$722$		0			0
$723$	0.500000	+	0.866025i	0.500000	+	0.866025i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$727$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$	0.500000	+	0.866025i	0.500000	+	0.866025i
$732$		0			0
$733$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$733$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$734$		0			0
$735$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$740$		0			0
$741$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$742$		0			0
$743$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$744$		0			0
$745$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$746$		0			0
$747$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$748$		0			0
$749$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$750$		0			0
$751$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$751$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.00000			1.00000
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$			1.00000i			1.00000i
$760$		0			0
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$	0.500000	−	0.866025i	0.500000	−	0.866025i
$764$		0			0
$765$	0.866025	−	0.500000i	0.866025	−	0.500000i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	0.500000	−	0.866025i	0.500000	−	0.866025i
$772$		0			0
$773$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$773$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.500000	+	0.866025i	0.500000	+	0.866025i
$778$		0			0
$779$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.866025	+	0.500000i	0.866025	+	0.500000i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$790$		0			0
$791$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$796$		0			0
$797$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$797$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$802$		0			0
$803$	1.00000			1.00000
$804$		0			0
$805$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$806$		0			0
$807$	0.866025	+	0.500000i	0.866025	+	0.500000i
$808$		0			0
$809$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$809$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$810$		0			0
$811$	2.00000			2.00000			1.00000			$0$
$811$	2.00000			2.00000			1.00000			$0$
$812$		0			0
$813$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$814$		0			0
$815$			1.00000i			1.00000i
$816$		0			0
$817$	−1.00000			−1.00000
$818$		0			0
$819$	0.500000	+	0.866025i	0.500000	+	0.866025i
$820$		0			0
$821$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$821$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$822$		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			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$829$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$832$		0			0
$833$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$834$		0			0
$835$	1.00000			1.00000
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$839$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$840$		0			0
$841$		0			0
$842$		0			0
$843$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.00000			1.00000
$852$		0			0
$853$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$853$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$854$		0			0
$855$			1.00000i			1.00000i
$856$		0			0
$857$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$857$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$860$		0			0
$861$	0.866025	−	0.500000i	0.866025	−	0.500000i
$862$		0			0
$863$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$863$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.73205	−	1.00000i	1.73205	−	1.00000i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$874$		0			0
$875$	−1.00000			−1.00000
$876$		0			0
$877$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$877$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$878$		0			0
$879$	0.866025	−	0.500000i	0.866025	−	0.500000i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$887$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$896$		0			0
$897$	1.00000			1.00000
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$902$		0			0
$903$	0.866025	+	0.500000i	0.866025	+	0.500000i
$904$		0			0
$905$	1.00000	−	1.73205i	1.00000	−	1.73205i
$906$		0			0
$907$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$907$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$908$		0			0
$909$	0.866025	−	0.500000i	0.866025	−	0.500000i
$910$		0			0
$911$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$911$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$	1.00000			1.00000
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.866025	+	0.500000i	0.866025	+	0.500000i
$918$		0			0
$919$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$919$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	0.866025	+	0.500000i	0.866025	+	0.500000i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$932$		0			0
$933$		−	2.00000i		−	2.00000i
$934$		0			0
$935$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		−	1.00000i		−	1.00000i
$944$		0			0
$945$	0.500000	−	0.866025i	0.500000	−	0.866025i
$946$		0			0
$947$	2.00000			2.00000			1.00000			$0$
$947$	2.00000			2.00000			1.00000			$0$
$948$		0			0
$949$		−	1.00000i		−	1.00000i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	1.00000i		−	1.00000i
$958$		0			0
$959$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$967$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$968$		0			0
$969$	1.00000			1.00000
$970$		0			0
$971$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$971$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$980$		0			0
$981$	0.866025	+	0.500000i	0.866025	+	0.500000i
$982$		0			0
$983$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$983$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.866025	−	0.500000i	0.866025	−	0.500000i
$990$		0			0
$991$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$991$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.500000	+	0.866025i	0.500000	+	0.866025i
$996$		0			0
$997$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$997$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$998$		0			0
$999$	−0.866025	+	0.500000i	−0.866025	+	0.500000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.1.cm.a.655.2		4
4.3	odd	2		504.1.ce.a.403.1	yes	4
7.2	even	3		2016.1.bi.a.79.1		4
8.3	odd	2	inner	2016.1.cm.a.655.1		4
8.5	even	2		504.1.ce.a.403.2	yes	4
9.4	even	3		2016.1.bi.a.1327.1		4
12.11	even	2		1512.1.ce.a.235.2		4
24.5	odd	2		1512.1.ce.a.235.1		4
28.3	even	6		3528.1.cg.c.2059.2		4
28.11	odd	6		3528.1.cg.d.2059.2		4
28.19	even	6		3528.1.ba.d.1843.1		4
28.23	odd	6		504.1.ba.a.331.1	yes	4
28.27	even	2		3528.1.ce.c.2419.1		4
36.23	even	6		1512.1.ba.a.739.1		4
36.31	odd	6		504.1.ba.a.67.2	yes	4
56.5	odd	6		3528.1.ba.d.1843.2		4
56.13	odd	2		3528.1.ce.c.2419.2		4
56.37	even	6		504.1.ba.a.331.2	yes	4
56.45	odd	6		3528.1.cg.c.2059.1		4
56.51	odd	6		2016.1.bi.a.79.2		4
56.53	even	6		3528.1.cg.d.2059.1		4
63.58	even	3	inner	2016.1.cm.a.751.1		4
72.5	odd	6		1512.1.ba.a.739.2		4
72.13	even	6		504.1.ba.a.67.1	✓	4
72.67	odd	6		2016.1.bi.a.1327.2		4
84.23	even	6		1512.1.ba.a.667.2		4
168.149	odd	6		1512.1.ba.a.667.1		4
252.23	even	6		1512.1.ce.a.1171.2		4
252.31	even	6		3528.1.cg.c.3235.1		4
252.67	odd	6		3528.1.cg.d.3235.1		4
252.103	even	6		3528.1.ce.c.3019.1		4
252.139	even	6		3528.1.ba.d.67.2		4
252.247	odd	6		504.1.ce.a.499.1	yes	4
504.13	odd	6		3528.1.ba.d.67.1		4
504.149	odd	6		1512.1.ce.a.1171.1		4
504.157	odd	6		3528.1.cg.c.3235.2		4
504.229	odd	6		3528.1.ce.c.3019.2		4
504.373	even	6		504.1.ce.a.499.2	yes	4
504.445	even	6		3528.1.cg.d.3235.2		4
504.499	odd	6	inner	2016.1.cm.a.751.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.1.ba.a.67.1	✓	4	72.13	even	6
504.1.ba.a.67.2	yes	4	36.31	odd	6
504.1.ba.a.331.1	yes	4	28.23	odd	6
504.1.ba.a.331.2	yes	4	56.37	even	6
504.1.ce.a.403.1	yes	4	4.3	odd	2
504.1.ce.a.403.2	yes	4	8.5	even	2
504.1.ce.a.499.1	yes	4	252.247	odd	6
504.1.ce.a.499.2	yes	4	504.373	even	6
1512.1.ba.a.667.1		4	168.149	odd	6
1512.1.ba.a.667.2		4	84.23	even	6
1512.1.ba.a.739.1		4	36.23	even	6
1512.1.ba.a.739.2		4	72.5	odd	6
1512.1.ce.a.235.1		4	24.5	odd	2
1512.1.ce.a.235.2		4	12.11	even	2
1512.1.ce.a.1171.1		4	504.149	odd	6
1512.1.ce.a.1171.2		4	252.23	even	6
2016.1.bi.a.79.1		4	7.2	even	3
2016.1.bi.a.79.2		4	56.51	odd	6
2016.1.bi.a.1327.1		4	9.4	even	3
2016.1.bi.a.1327.2		4	72.67	odd	6
2016.1.cm.a.655.1		4	8.3	odd	2	inner
2016.1.cm.a.655.2		4	1.1	even	1	trivial
2016.1.cm.a.751.1		4	63.58	even	3	inner
2016.1.cm.a.751.2		4	504.499	odd	6	inner
3528.1.ba.d.67.1		4	504.13	odd	6
3528.1.ba.d.67.2		4	252.139	even	6
3528.1.ba.d.1843.1		4	28.19	even	6
3528.1.ba.d.1843.2		4	56.5	odd	6
3528.1.ce.c.2419.1		4	28.27	even	2
3528.1.ce.c.2419.2		4	56.13	odd	2
3528.1.ce.c.3019.1		4	252.103	even	6
3528.1.ce.c.3019.2		4	504.229	odd	6
3528.1.cg.c.2059.1		4	56.45	odd	6
3528.1.cg.c.2059.2		4	28.3	even	6
3528.1.cg.c.3235.1		4	252.31	even	6
3528.1.cg.c.3235.2		4	504.157	odd	6
3528.1.cg.d.2059.1		4	56.53	even	6
3528.1.cg.d.2059.2		4	28.11	odd	6
3528.1.cg.d.3235.1		4	252.67	odd	6
3528.1.cg.d.3235.2		4	504.445	even	6

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2016.cm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(0.866025 + 0.500000i) q^{5} -1.00000i q^{7} +1.00000 q^{9} +(-0.500000 - 0.866025i) q^{11} +(-0.866025 + 0.500000i) q^{13} +(0.866025 + 0.500000i) q^{15} +(0.500000 - 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{19} -1.00000i q^{21} +(-0.866025 - 0.500000i) q^{23} +1.00000 q^{27} +(0.866025 + 0.500000i) q^{29} +(-0.500000 - 0.866025i) q^{33} +(0.500000 - 0.866025i) q^{35} +(-0.866025 + 0.500000i) q^{37} +(-0.866025 + 0.500000i) q^{39} +(0.500000 + 0.866025i) q^{41} +(-0.500000 + 0.866025i) q^{43} +(0.866025 + 0.500000i) q^{45} -1.00000 q^{49} +(0.500000 - 0.866025i) q^{51} +(-0.866025 - 0.500000i) q^{53} -1.00000i q^{55} +(0.500000 + 0.866025i) q^{57} -1.00000i q^{63} -1.00000 q^{65} +(-0.866025 - 0.500000i) q^{69} +(-0.500000 + 0.866025i) q^{73} +(-0.866025 + 0.500000i) q^{77} +2.00000i q^{79} +1.00000 q^{81} +(-0.500000 + 0.866025i) q^{83} +(0.866025 - 0.500000i) q^{85} +(0.866025 + 0.500000i) q^{87} +(-0.500000 - 0.866025i) q^{89} +(0.500000 + 0.866025i) q^{91} +1.00000i q^{95} +(-0.500000 + 0.866025i) q^{97} +(-0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9} - 2 q^{11} + 2 q^{17} + 2 q^{19} + 4 q^{27} - 2 q^{33} + 2 q^{35} + 2 q^{41} - 2 q^{43} - 4 q^{49} + 2 q^{51} + 2 q^{57} - 4 q^{65} - 2 q^{73} + 4 q^{81} - 2 q^{83} - 2 q^{89} + 2 q^{91}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 - 2 * q^11 + 2 * q^17 + 2 * q^19 + 4 * q^27 - 2 * q^33 + 2 * q^35 + 2 * q^41 - 2 * q^43 - 4 * q^49 + 2 * q^51 + 2 * q^57 - 4 * q^65 - 2 * q^73 + 4 * q^81 - 2 * q^83 - 2 * q^89 + 2 * q^91 - 2 * q^97 - 2 * q^99

Introduction

L-functions

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