LMFDB - Embedded newform 1800.1.bk.d.1051.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(1051,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4, 0]))

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

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

chi := DirichletCharacter("1800.1051");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1800.bk (of order $6$ , degree $2$ , 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:	$0.898317022739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 72)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.648.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			1051.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1800.1051
Dual form			1800.1.bk.d.1651.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{11} -1.00000 q^{12} +(-0.500000 + 0.866025i) q^{16} +1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +(-0.500000 - 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{24} -1.00000 q^{27} +(0.500000 + 0.866025i) q^{32} +(-0.500000 - 0.866025i) q^{33} +(0.500000 - 0.866025i) q^{34} +(-0.500000 + 0.866025i) q^{36} +(-0.500000 + 0.866025i) q^{38} +(0.500000 + 0.866025i) q^{41} +(-0.500000 + 0.866025i) q^{43} -1.00000 q^{44} +(0.500000 + 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{51} +(-0.500000 + 0.866025i) q^{54} +(-0.500000 + 0.866025i) q^{57} +(0.500000 + 0.866025i) q^{59} +1.00000 q^{64} -1.00000 q^{66} +(-0.500000 - 0.866025i) q^{67} +(-0.500000 - 0.866025i) q^{68} +(0.500000 + 0.866025i) q^{72} +1.00000 q^{73} +(0.500000 + 0.866025i) q^{76} +(-0.500000 + 0.866025i) q^{81} +1.00000 q^{82} +(1.00000 - 1.73205i) q^{83} +(0.500000 + 0.866025i) q^{86} +(-0.500000 + 0.866025i) q^{88} +2.00000 q^{89} +1.00000 q^{96} +(-0.500000 + 0.866025i) q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} + q^{3} - q^{4} - q^{6} - 2 q^{8} - q^{9} + q^{11} - 2 q^{12} - q^{16} + 2 q^{17} - 2 q^{18} - 2 q^{19} - q^{22} - q^{24} - 2 q^{27} + q^{32} - q^{33} + q^{34} - q^{36} - q^{38}+ \cdots - 2 q^{99}+O(q^{100})$ 2 * q + q^2 + q^3 - q^4 - q^6 - 2 * q^8 - q^9 + q^11 - 2 * q^12 - q^16 + 2 * q^17 - 2 * q^18 - 2 * q^19 - q^22 - q^24 - 2 * q^27 + q^32 - q^33 + q^34 - q^36 - q^38 + q^41 - q^43 - 2 * q^44 + q^48 - q^49 + q^51 - q^54 - q^57 + q^59 + 2 * q^64 - 2 * q^66 - q^67 - q^68 + q^72 + 2 * q^73 + q^76 - q^81 + 2 * q^82 + 2 * q^83 + q^86 - q^88 + 4 * q^89 + 2 * q^96 - q^97 - 2 * q^98 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$ .

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$1$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−1.00000			−1.00000
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$		0			0
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$12$	−1.00000			−1.00000
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$18$	−1.00000			−1.00000
$19$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$25$		0			0
$26$		0			0
$27$	−1.00000			−1.00000
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$34$	0.500000	−	0.866025i	0.500000	−	0.866025i
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$39$		0			0
$40$		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			$\pi$
$44$	−1.00000			−1.00000
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.500000	+	0.866025i	0.500000	+	0.866025i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$		0			0
$51$	0.500000	−	0.866025i	0.500000	−	0.866025i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$55$		0			0
$56$		0			0
$57$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$58$		0			0
$59$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$59$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$	−1.00000			−1.00000
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$68$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.500000	+	0.866025i	0.500000	+	0.866025i
$73$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$73$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$		0			0
$76$	0.500000	+	0.866025i	0.500000	+	0.866025i
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$	1.00000			1.00000
$83$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$83$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$84$		0			0
$85$		0			0
$86$	0.500000	+	0.866025i	0.500000	+	0.866025i
$87$		0			0
$88$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$89$	2.00000			2.00000			1.00000			$0$
$89$	2.00000			2.00000			1.00000			$0$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.00000			1.00000
$97$	−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			$\pi$
$98$	−1.00000			−1.00000
$99$	−1.00000			−1.00000
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$107$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$113$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$114$	0.500000	+	0.866025i	0.500000	+	0.866025i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.00000			1.00000
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	1.00000			1.00000
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$	0.500000	+	0.866025i	0.500000	+	0.866025i
$130$		0			0
$131$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$132$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$133$		0			0
$134$	−1.00000			−1.00000
$135$		0			0
$136$	−1.00000			−1.00000
$137$	−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	1.00000			$0$
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$	0.500000	−	0.866025i	0.500000	−	0.866025i
$147$	−1.00000			−1.00000
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	1.00000			1.00000
$153$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.500000	+	0.866025i	0.500000	+	0.866025i
$163$	−2.00000			−2.00000			−1.00000			$\pi$
$163$	−2.00000			−2.00000			−1.00000			$\pi$
$164$	0.500000	−	0.866025i	0.500000	−	0.866025i
$165$		0			0
$166$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	0.500000	+	0.866025i	0.500000	+	0.866025i
$172$	1.00000			1.00000
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	0.500000	+	0.866025i	0.500000	+	0.866025i
$177$	1.00000			1.00000
$178$	1.00000	−	1.73205i	1.00000	−	1.73205i
$179$	2.00000			2.00000			1.00000			$0$
$179$	2.00000			2.00000			1.00000			$0$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.500000	−	0.866025i	0.500000	−	0.866025i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.500000	−	0.866025i	0.500000	−	0.866025i
$193$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$193$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$194$	0.500000	+	0.866025i	0.500000	+	0.866025i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−1.00000			−1.00000
$202$		0			0
$203$		0			0
$204$	−1.00000			−1.00000
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$210$		0			0
$211$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$212$		0			0
$213$		0			0
$214$	0.500000	−	0.866025i	0.500000	−	0.866025i
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$		0			0
$219$	0.500000	−	0.866025i	0.500000	−	0.866025i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$	2.00000			2.00000
$227$	−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			$\pi$
$228$	1.00000			1.00000
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$	0.500000	−	0.866025i	0.500000	−	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$		0			0
$243$	0.500000	+	0.866025i	0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$	0.500000	−	0.866025i	0.500000	−	0.866025i
$247$		0			0
$248$		0			0
$249$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$250$		0			0
$251$	−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
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$257$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$258$	1.00000			1.00000
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−2.00000			−2.00000
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	0.500000	+	0.866025i	0.500000	+	0.866025i
$265$		0			0
$266$		0			0
$267$	1.00000	−	1.73205i	1.00000	−	1.73205i
$268$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$273$		0			0
$274$	0.500000	+	0.866025i	0.500000	+	0.866025i
$275$		0			0
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	1.00000			1.00000
$279$		0			0
$280$		0			0
$281$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$281$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$283$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.500000	−	0.866025i	0.500000	−	0.866025i
$289$		0			0
$290$		0			0
$291$	0.500000	+	0.866025i	0.500000	+	0.866025i
$292$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$295$		0			0
$296$		0			0
$297$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.500000	−	0.866025i	0.500000	−	0.866025i
$305$		0			0
$306$	−1.00000			−1.00000
$307$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$313$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.500000	−	0.866025i	0.500000	−	0.866025i
$322$		0			0
$323$	−1.00000			−1.00000
$324$	1.00000			1.00000
$325$		0			0
$326$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$327$		0			0
$328$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$329$		0			0
$330$		0			0
$331$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$331$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$332$	−2.00000			−2.00000
$333$		0			0
$334$		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			$\pi$
$338$	0.500000	+	0.866025i	0.500000	+	0.866025i
$339$	2.00000			2.00000
$340$		0			0
$341$		0			0
$342$	1.00000			1.00000
$343$		0			0
$344$	0.500000	−	0.866025i	0.500000	−	0.866025i
$345$		0			0
$346$		0			0
$347$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$347$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$	1.00000			1.00000
$353$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$353$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$354$	0.500000	−	0.866025i	0.500000	−	0.866025i
$355$		0			0
$356$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$357$		0			0
$358$	1.00000	−	1.73205i	1.00000	−	1.73205i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	0.500000	−	0.866025i	0.500000	−	0.866025i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$385$		0			0
$386$	−1.00000			−1.00000
$387$	1.00000			1.00000
$388$	1.00000			1.00000
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$	−2.00000			−2.00000
$394$		0			0
$395$		0			0
$396$	0.500000	+	0.866025i	0.500000	+	0.866025i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$401$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$402$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	0.500000	+	0.866025i	0.500000	+	0.866025i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.00000			1.00000
$418$	0.500000	+	0.866025i	0.500000	+	0.866025i
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$	−2.00000			−2.00000
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.500000	−	0.866025i	0.500000	−	0.866025i
$433$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$433$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		0			0
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$	1.00000			1.00000
$452$	1.00000	−	1.73205i	1.00000	−	1.73205i
$453$		0			0
$454$	0.500000	+	0.866025i	0.500000	+	0.866025i
$455$		0			0
$456$	0.500000	−	0.866025i	0.500000	−	0.866025i
$457$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$457$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$458$		0			0
$459$	−1.00000			−1.00000
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$	0.500000	−	0.866025i	0.500000	−	0.866025i
$467$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$467$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$473$	0.500000	+	0.866025i	0.500000	+	0.866025i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	1.00000			1.00000
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$490$		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	$0.666667\pi$
$492$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−2.00000			−2.00000
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.500000	+	0.866025i	0.500000	+	0.866025i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	1.00000			1.00000
$514$	−1.00000			−1.00000
$515$		0			0
$516$	0.500000	−	0.866025i	0.500000	−	0.866025i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−2.00000			−2.00000			−1.00000			$\pi$
$523$	−2.00000			−2.00000			−1.00000			$\pi$
$524$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.00000			1.00000
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$	0.500000	−	0.866025i	0.500000	−	0.866025i
$532$		0			0
$533$		0			0
$534$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$535$		0			0
$536$	0.500000	+	0.866025i	0.500000	+	0.866025i
$537$	1.00000	−	1.73205i	1.00000	−	1.73205i
$538$		0			0
$539$	−1.00000			−1.00000
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	0.500000	+	0.866025i	0.500000	+	0.866025i
$545$		0			0
$546$		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			$\pi$
$548$	1.00000			1.00000
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.500000	−	0.866025i	0.500000	−	0.866025i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$562$	1.00000	+	1.73205i	1.00000	+	1.73205i
$563$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$563$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$	2.00000			2.00000
$567$		0			0
$568$		0			0
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$570$		0			0
$571$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$571$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$577$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$578$		0			0
$579$	−1.00000			−1.00000
$580$		0			0
$581$		0			0
$582$	1.00000			1.00000
$583$		0			0
$584$	−1.00000			−1.00000
$585$		0			0
$586$		0			0
$587$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$587$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$588$	0.500000	+	0.866025i	0.500000	+	0.866025i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.00000			−2.00000			−1.00000			$\pi$
$593$	−2.00000			−2.00000			−1.00000			$\pi$
$594$	0.500000	+	0.866025i	0.500000	+	0.866025i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		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
$603$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	0.500000	−	0.866025i	0.500000	−	0.866025i
$615$		0			0
$616$		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	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	0.500000	+	0.866025i	0.500000	+	0.866025i
$627$	0.500000	+	0.866025i	0.500000	+	0.866025i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−2.00000			−2.00000
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$642$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$	1.00000			1.00000
$650$		0			0
$651$		0			0
$652$	1.00000	+	1.73205i	1.00000	+	1.73205i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$	−1.00000			−1.00000
$657$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$658$		0			0
$659$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$659$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	1.00000	+	1.73205i	1.00000	+	1.73205i
$663$		0			0
$664$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$673$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$674$	−1.00000			−1.00000
$675$		0			0
$676$	1.00000			1.00000
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$	1.00000	−	1.73205i	1.00000	−	1.73205i
$679$		0			0
$680$		0			0
$681$	0.500000	+	0.866025i	0.500000	+	0.866025i
$682$		0			0
$683$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$684$	0.500000	−	0.866025i	0.500000	−	0.866025i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$689$		0			0
$690$		0			0
$691$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$691$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−1.00000			−1.00000
$695$		0			0
$696$		0			0
$697$	0.500000	+	0.866025i	0.500000	+	0.866025i
$698$		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			0
$704$	0.500000	−	0.866025i	0.500000	−	0.866025i
$705$		0			0
$706$	0.500000	+	0.866025i	0.500000	+	0.866025i
$707$		0			0
$708$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−2.00000			−2.00000
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.00000			−1.00000
$738$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$739$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.00000			−2.00000
$748$	−1.00000			−1.00000
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$759$		0			0
$760$		0			0
$761$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.00000			−1.00000
$769$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$770$		0			0
$771$	−1.00000			−1.00000
$772$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	0.500000	−	0.866025i	0.500000	−	0.866025i
$775$		0			0
$776$	0.500000	−	0.866025i	0.500000	−	0.866025i
$777$		0			0
$778$		0			0
$779$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$787$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$787$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.00000			1.00000
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$802$	1.00000			1.00000
$803$	0.500000	−	0.866025i	0.500000	−	0.866025i
$804$	0.500000	+	0.866025i	0.500000	+	0.866025i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.500000	+	0.866025i	0.500000	+	0.866025i
$817$	0.500000	−	0.866025i	0.500000	−	0.866025i
$818$	1.00000			1.00000
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	1.00000			1.00000
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.00000			−2.00000			−1.00000			$\pi$
$827$	−2.00000			−2.00000			−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			0
$833$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$834$	0.500000	−	0.866025i	0.500000	−	0.866025i
$835$		0			0
$836$	1.00000			1.00000
$837$		0			0
$838$	−2.00000			−2.00000
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$	1.00000	+	1.73205i	1.00000	+	1.73205i
$844$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.00000			2.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	−1.00000			−1.00000
$857$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$857$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$858$		0			0
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$		0			0
$866$	0.500000	−	0.866025i	0.500000	−	0.866025i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.00000			1.00000
$874$		0			0
$875$		0			0
$876$	−1.00000			−1.00000
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			2.00000			1.00000			$0$
$881$	2.00000			2.00000			1.00000			$0$
$882$	0.500000	+	0.866025i	0.500000	+	0.866025i
$883$	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
$885$		0			0
$886$	0.500000	+	0.866025i	0.500000	+	0.866025i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		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			0
$896$		0			0
$897$		0			0
$898$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	0.500000	−	0.866025i	0.500000	−	0.866025i
$903$		0			0
$904$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$905$		0			0
$906$		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			$\pi$
$908$	1.00000			1.00000
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$913$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$914$	0.500000	+	0.866025i	0.500000	+	0.866025i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	0.500000	−	0.866025i	0.500000	−	0.866025i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$929$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	0.500000	+	0.866025i	0.500000	+	0.866025i
$932$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$933$		0			0
$934$	0.500000	−	0.866025i	0.500000	−	0.866025i
$935$		0			0
$936$		0			0
$937$	−2.00000			−2.00000			−1.00000			$\pi$
$937$	−2.00000			−2.00000			−1.00000			$\pi$
$938$		0			0
$939$	0.500000	+	0.866025i	0.500000	+	0.866025i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$	−1.00000			−1.00000
$945$		0			0
$946$	1.00000			1.00000
$947$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$947$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$953$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$964$	−1.00000			−1.00000
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$970$		0			0
$971$	2.00000			2.00000			1.00000			$0$
$971$	2.00000			2.00000			1.00000			$0$
$972$	0.500000	−	0.866025i	0.500000	−	0.866025i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$978$	1.00000	+	1.73205i	1.00000	+	1.73205i
$979$	1.00000	−	1.73205i	1.00000	−	1.73205i
$980$		0			0
$981$		0			0
$982$	1.00000			1.00000
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$	−1.00000			−1.00000
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	1.00000	+	1.73205i	1.00000	+	1.73205i
$994$		0			0
$995$		0			0
$996$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	1.00000			1.00000
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.1.bk.d.1051.1		2
5.2	odd	4		1800.1.ba.b.1699.2		4
5.3	odd	4		1800.1.ba.b.1699.1		4
5.4	even	2		72.1.p.a.43.1	✓	2
8.3	odd	2	CM	1800.1.bk.d.1051.1		2
9.4	even	3	inner	1800.1.bk.d.1651.1		2
15.14	odd	2		216.1.p.a.19.1		2
20.19	odd	2		288.1.t.a.79.1		2
35.4	even	6		3528.1.ba.b.1843.1		2
35.9	even	6		3528.1.ce.a.2419.1		2
35.19	odd	6		3528.1.ce.b.2419.1		2
35.24	odd	6		3528.1.ba.a.1843.1		2
35.34	odd	2		3528.1.cg.a.2059.1		2
40.3	even	4		1800.1.ba.b.1699.1		4
40.19	odd	2		72.1.p.a.43.1	✓	2
40.27	even	4		1800.1.ba.b.1699.2		4
40.29	even	2		288.1.t.a.79.1		2
45.4	even	6		72.1.p.a.67.1	yes	2
45.13	odd	12		1800.1.ba.b.499.2		4
45.14	odd	6		216.1.p.a.91.1		2
45.22	odd	12		1800.1.ba.b.499.1		4
45.29	odd	6		648.1.b.a.163.1		1
45.34	even	6		648.1.b.b.163.1		1
60.59	even	2		864.1.t.a.559.1		2
72.67	odd	6	inner	1800.1.bk.d.1651.1		2
80.19	odd	4		2304.1.o.c.511.2		4
80.29	even	4		2304.1.o.c.511.1		4
80.59	odd	4		2304.1.o.c.511.1		4
80.69	even	4		2304.1.o.c.511.2		4
120.29	odd	2		864.1.t.a.559.1		2
120.59	even	2		216.1.p.a.19.1		2
180.59	even	6		864.1.t.a.847.1		2
180.79	odd	6		2592.1.b.b.1135.1		1
180.119	even	6		2592.1.b.a.1135.1		1
180.139	odd	6		288.1.t.a.175.1		2
280.19	even	6		3528.1.ce.b.2419.1		2
280.59	even	6		3528.1.ba.a.1843.1		2
280.139	even	2		3528.1.cg.a.2059.1		2
280.179	odd	6		3528.1.ba.b.1843.1		2
280.219	odd	6		3528.1.ce.a.2419.1		2
315.4	even	6		3528.1.ce.a.3019.1		2
315.94	odd	6		3528.1.ce.b.3019.1		2
315.139	odd	6		3528.1.cg.a.3235.1		2
315.184	even	6		3528.1.ba.b.67.1		2
315.229	odd	6		3528.1.ba.a.67.1		2
360.29	odd	6		2592.1.b.a.1135.1		1
360.59	even	6		216.1.p.a.91.1		2
360.67	even	12		1800.1.ba.b.499.1		4
360.139	odd	6		72.1.p.a.67.1	yes	2
360.149	odd	6		864.1.t.a.847.1		2
360.229	even	6		288.1.t.a.175.1		2
360.259	odd	6		648.1.b.b.163.1		1
360.283	even	12		1800.1.ba.b.499.2		4
360.299	even	6		648.1.b.a.163.1		1
360.349	even	6		2592.1.b.b.1135.1		1
720.139	odd	12		2304.1.o.c.2047.2		4
720.229	even	12		2304.1.o.c.2047.1		4
720.499	odd	12		2304.1.o.c.2047.1		4
720.589	even	12		2304.1.o.c.2047.2		4
2520.139	even	6		3528.1.cg.a.3235.1		2
2520.499	odd	6		3528.1.ba.b.67.1		2
2520.859	even	6		3528.1.ba.a.67.1		2
2520.1579	odd	6		3528.1.ce.a.3019.1		2
2520.2299	even	6		3528.1.ce.b.3019.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.1.p.a.43.1	✓	2	5.4	even	2
72.1.p.a.43.1	✓	2	40.19	odd	2
72.1.p.a.67.1	yes	2	45.4	even	6
72.1.p.a.67.1	yes	2	360.139	odd	6
216.1.p.a.19.1		2	15.14	odd	2
216.1.p.a.19.1		2	120.59	even	2
216.1.p.a.91.1		2	45.14	odd	6
216.1.p.a.91.1		2	360.59	even	6
288.1.t.a.79.1		2	20.19	odd	2
288.1.t.a.79.1		2	40.29	even	2
288.1.t.a.175.1		2	180.139	odd	6
288.1.t.a.175.1		2	360.229	even	6
648.1.b.a.163.1		1	45.29	odd	6
648.1.b.a.163.1		1	360.299	even	6
648.1.b.b.163.1		1	45.34	even	6
648.1.b.b.163.1		1	360.259	odd	6
864.1.t.a.559.1		2	60.59	even	2
864.1.t.a.559.1		2	120.29	odd	2
864.1.t.a.847.1		2	180.59	even	6
864.1.t.a.847.1		2	360.149	odd	6
1800.1.ba.b.499.1		4	45.22	odd	12
1800.1.ba.b.499.1		4	360.67	even	12
1800.1.ba.b.499.2		4	45.13	odd	12
1800.1.ba.b.499.2		4	360.283	even	12
1800.1.ba.b.1699.1		4	5.3	odd	4
1800.1.ba.b.1699.1		4	40.3	even	4
1800.1.ba.b.1699.2		4	5.2	odd	4
1800.1.ba.b.1699.2		4	40.27	even	4
1800.1.bk.d.1051.1		2	1.1	even	1	trivial
1800.1.bk.d.1051.1		2	8.3	odd	2	CM
1800.1.bk.d.1651.1		2	9.4	even	3	inner
1800.1.bk.d.1651.1		2	72.67	odd	6	inner
2304.1.o.c.511.1		4	80.29	even	4
2304.1.o.c.511.1		4	80.59	odd	4
2304.1.o.c.511.2		4	80.19	odd	4
2304.1.o.c.511.2		4	80.69	even	4
2304.1.o.c.2047.1		4	720.229	even	12
2304.1.o.c.2047.1		4	720.499	odd	12
2304.1.o.c.2047.2		4	720.139	odd	12
2304.1.o.c.2047.2		4	720.589	even	12
2592.1.b.a.1135.1		1	180.119	even	6
2592.1.b.a.1135.1		1	360.29	odd	6
2592.1.b.b.1135.1		1	180.79	odd	6
2592.1.b.b.1135.1		1	360.349	even	6
3528.1.ba.a.67.1		2	315.229	odd	6
3528.1.ba.a.67.1		2	2520.859	even	6
3528.1.ba.a.1843.1		2	35.24	odd	6
3528.1.ba.a.1843.1		2	280.59	even	6
3528.1.ba.b.67.1		2	315.184	even	6
3528.1.ba.b.67.1		2	2520.499	odd	6
3528.1.ba.b.1843.1		2	35.4	even	6
3528.1.ba.b.1843.1		2	280.179	odd	6
3528.1.ce.a.2419.1		2	35.9	even	6
3528.1.ce.a.2419.1		2	280.219	odd	6
3528.1.ce.a.3019.1		2	315.4	even	6
3528.1.ce.a.3019.1		2	2520.1579	odd	6
3528.1.ce.b.2419.1		2	35.19	odd	6
3528.1.ce.b.2419.1		2	280.19	even	6
3528.1.ce.b.3019.1		2	315.94	odd	6
3528.1.ce.b.3019.1		2	2520.2299	even	6
3528.1.cg.a.2059.1		2	35.34	odd	2
3528.1.cg.a.2059.1		2	280.139	even	2
3528.1.cg.a.3235.1		2	315.139	odd	6
3528.1.cg.a.3235.1		2	2520.139	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1800.1.bk.d.1051.1

Level	$1800$
Weight	$1$
Character	1800.1051
Analytic conductor	$0.898$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-8
Inner twists	$4$