LMFDB - Embedded newform 1368.1.bo.b.227.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(227,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1, 3]))

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

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

chi := DirichletCharacter("1368.227");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1368 = 2^{3} \cdot 3^{2} \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1368.bo (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:	$0.682720937282$
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:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.8640364608.3

Embedding invariants

Embedding label			227.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1368.227
Dual form			1368.1.bo.b.1139.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 - 0.866025i) q^{11} +1.00000 q^{12} +(-0.500000 + 0.866025i) q^{16} -1.73205i q^{17} -1.00000 q^{18} +(0.500000 - 0.866025i) q^{19} +(-1.50000 + 0.866025i) q^{22} +(0.500000 - 0.866025i) q^{24} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(0.500000 + 0.866025i) q^{32} +(1.50000 - 0.866025i) q^{33} +(-1.50000 - 0.866025i) q^{34} +(-0.500000 + 0.866025i) q^{36} +(-0.500000 - 0.866025i) q^{38} +(0.500000 + 0.866025i) q^{41} +(-0.500000 + 0.866025i) q^{43} +1.73205i q^{44} +(-0.500000 - 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(1.50000 + 0.866025i) q^{51} +(0.500000 - 0.866025i) q^{54} +(0.500000 + 0.866025i) q^{57} +(-0.500000 - 0.866025i) q^{59} +1.00000 q^{64} -1.73205i q^{66} +(-1.50000 + 0.866025i) q^{67} +(-1.50000 + 0.866025i) q^{68} +(0.500000 + 0.866025i) q^{72} +1.00000 q^{73} +(0.500000 + 0.866025i) q^{75} -1.00000 q^{76} +(-0.500000 + 0.866025i) q^{81} +1.00000 q^{82} +(0.500000 + 0.866025i) q^{86} +(1.50000 + 0.866025i) q^{88} -2.00000 q^{89} -1.00000 q^{96} +(1.50000 + 0.866025i) q^{97} -1.00000 q^{98} +1.73205i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{8} - q^{9} - 3 q^{11} + 2 q^{12} - q^{16} - 2 q^{18} + q^{19} - 3 q^{22} + q^{24} + q^{25} + 2 q^{27} + q^{32} + 3 q^{33} - 3 q^{34} - q^{36} - q^{38}+ \cdots - 2 q^{98}+O(q^{100})$ 2 * q + q^2 - q^3 - q^4 + q^6 - 2 * q^8 - q^9 - 3 * q^11 + 2 * q^12 - q^16 - 2 * q^18 + q^19 - 3 * q^22 + q^24 + q^25 + 2 * q^27 + q^32 + 3 * q^33 - 3 * q^34 - q^36 - q^38 + q^41 - q^43 - q^48 - q^49 - q^50 + 3 * q^51 + q^54 + q^57 - q^59 + 2 * q^64 - 3 * q^67 - 3 * q^68 + q^72 + 2 * q^73 + q^75 - 2 * q^76 - q^81 + 2 * q^82 + q^86 + 3 * q^88 - 4 * q^89 - 2 * q^96 + 3 * q^97 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$ .

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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

)

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

$2$	0.500000	−	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		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$	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
$11$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$12$	1.00000			1.00000
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$17$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$18$	−1.00000			−1.00000
$19$	0.500000	−	0.866025i	0.500000	−	0.866025i
$19$	0.500000	−	0.866025i	0.500000	−	0.866025i
$20$		0			0
$21$		0			0
$22$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	0.500000	−	0.866025i	0.500000	−	0.866025i
$25$	0.500000	−	0.866025i	0.500000	−	0.866025i
$26$		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		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$	1.50000	−	0.866025i	1.50000	−	0.866025i
$34$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−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			$\pi$
$44$			1.73205i			1.73205i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$	1.50000	+	0.866025i	1.50000	+	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	0.500000	−	0.866025i	$-0.333333\pi$
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$60$		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
$66$		−	1.73205i		−	1.73205i
$67$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$67$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$68$	−1.50000	+	0.866025i	−1.50000	+	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
$75$	0.500000	+	0.866025i	0.500000	+	0.866025i
$76$	−1.00000			−1.00000
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$	1.00000			1.00000
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$	0.500000	+	0.866025i	0.500000	+	0.866025i
$87$		0			0
$88$	1.50000	+	0.866025i	1.50000	+	0.866025i
$89$	−2.00000			−2.00000			−1.00000			$\pi$
$89$	−2.00000			−2.00000			−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.00000			−1.00000
$97$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$97$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$98$	−1.00000			−1.00000
$99$			1.73205i			1.73205i
$100$	−1.00000			−1.00000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$	1.50000	−	0.866025i	1.50000	−	0.866025i
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		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			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$114$	1.00000			1.00000
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.00000			−1.00000
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	1.00000	+	1.73205i
$122$		0			0
$123$	−1.00000			−1.00000
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$133$		0			0
$134$			1.73205i			1.73205i
$135$		0			0
$136$			1.73205i			1.73205i
$137$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$137$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$138$		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		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$	1.00000			1.00000
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$153$	−1.50000	+	0.866025i	−1.50000	+	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			$0$
$163$	2.00000			2.00000			1.00000			$0$
$164$	0.500000	−	0.866025i	0.500000	−	0.866025i
$165$		0			0
$166$		0			0
$167$		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
$171$	−1.00000			−1.00000
$172$	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
$176$	1.50000	−	0.866025i	1.50000	−	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			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$193$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$193$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$194$	1.50000	−	0.866025i	1.50000	−	0.866025i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	1.50000	+	0.866025i	1.50000	+	0.866025i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$201$		−	1.73205i		−	1.73205i
$202$		0			0
$203$		0			0
$204$		−	1.73205i		−	1.73205i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$	−1.00000			−1.00000
$226$	−2.00000			−2.00000
$227$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$228$	0.500000	−	0.866025i	0.500000	−	0.866025i
$229$		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
$233$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$233$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$242$	2.00000			2.00000
$243$	−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
$249$		0			0
$250$		0			0
$251$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$251$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$252$		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
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$265$		0			0
$266$		0			0
$267$	1.00000	−	1.73205i	1.00000	−	1.73205i
$268$	1.50000	+	0.866025i	1.50000	+	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			$\pi$
$272$	1.50000	+	0.866025i	1.50000	+	0.866025i
$273$		0			0
$274$	1.50000	−	0.866025i	1.50000	−	0.866025i
$275$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$276$		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	$-0.666667\pi$
$283$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.500000	−	0.866025i	0.500000	−	0.866025i
$289$	−2.00000			−2.00000
$290$		0			0
$291$	−1.50000	+	0.866025i	−1.50000	+	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
$297$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$298$		0			0
$299$		0			0
$300$	0.500000	−	0.866025i	0.500000	−	0.866025i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.500000	+	0.866025i	0.500000	+	0.866025i
$305$		0			0
$306$			1.73205i			1.73205i
$307$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$307$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		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
$323$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$324$	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
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$338$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$339$	2.00000			2.00000
$340$		0			0
$341$		0			0
$342$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$343$		0			0
$344$	0.500000	−	0.866025i	0.500000	−	0.866025i
$345$		0			0
$346$		0			0
$347$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$347$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$348$		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
$352$		−	1.73205i		−	1.73205i
$353$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$353$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$354$	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			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	−2.00000			−2.00000
$364$		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		0.866025	−	0.500000i	$-0.166667\pi$
$373$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$374$	1.50000	+	2.59808i	1.50000	+	2.59808i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$379$			1.73205i			1.73205i	−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
$386$		−	1.73205i		−	1.73205i
$387$	1.00000			1.00000
$388$		−	1.73205i		−	1.73205i
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	1.50000	−	0.866025i	1.50000	−	0.866025i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	0.500000	+	0.866025i	0.500000	+	0.866025i
$401$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$401$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$402$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$409$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$409$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$410$		0			0
$411$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.00000			−1.00000
$418$			1.73205i			1.73205i
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$421$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$426$		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
$433$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$433$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.500000	+	0.866025i	0.500000	+	0.866025i
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		0			0
$443$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$443$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$451$		−	1.73205i		−	1.73205i
$452$	−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	$-0.666667\pi$
$457$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$458$		0			0
$459$		−	1.73205i		−	1.73205i
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		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
$466$	1.50000	+	0.866025i	1.50000	+	0.866025i
$467$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$467$		−	1.73205i		−	1.73205i	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
$473$	1.50000	−	0.866025i	1.50000	−	0.866025i
$474$		0			0
$475$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$483$		0			0
$484$	1.00000	−	1.73205i	1.00000	−	1.73205i
$485$		0			0
$486$	−1.00000			−1.00000
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$490$		0			0
$491$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$491$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$492$	0.500000	+	0.866025i	0.500000	+	0.866025i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$499$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		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
$513$	0.500000	−	0.866025i	0.500000	−	0.866025i
$514$	−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	$-0.333333\pi$
$521$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$			1.73205i			1.73205i
$529$	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
$536$	1.50000	−	0.866025i	1.50000	−	0.866025i
$537$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$538$		0			0
$539$			1.73205i			1.73205i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	1.50000	−	0.866025i	1.50000	−	0.866025i
$545$		0			0
$546$		0			0
$547$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$547$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$548$		−	1.73205i		−	1.73205i
$549$		0			0
$550$			1.73205i			1.73205i
$551$		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			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$562$	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	$0.333333\pi$
$569$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$570$		0			0
$571$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$571$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$572$		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	$-0.666667\pi$
$577$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$578$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$579$			1.73205i			1.73205i
$580$		0			0
$581$		0			0
$582$			1.73205i			1.73205i
$583$		0			0
$584$	−1.00000			−1.00000
$585$		0			0
$586$		0			0
$587$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$587$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$588$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−1.50000	+	0.866025i	−1.50000	+	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	$0.333333\pi$
$600$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$601$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$601$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	1.50000	+	0.866025i	1.50000	+	0.866025i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$	1.00000			1.00000
$609$		0			0
$610$		0			0
$611$		0			0
$612$	1.50000	+	0.866025i	1.50000	+	0.866025i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$615$		0			0
$616$		0			0
$617$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$617$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$619$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$	0.500000	+	0.866025i	0.500000	+	0.866025i
$627$		−	1.73205i		−	1.73205i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		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	$0.333333\pi$
$641$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$642$	0.500000	+	0.866025i	0.500000	+	0.866025i
$643$	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			0
$646$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$			1.73205i			1.73205i
$650$		0			0
$651$		0			0
$652$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		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		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		−	1.73205i		−	1.73205i
$675$	0.500000	−	0.866025i	0.500000	−	0.866025i
$676$	−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	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$	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	$-0.333333\pi$
$691$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$692$		0			0
$693$		0			0
$694$		−	1.73205i		−	1.73205i
$695$		0			0
$696$		0			0
$697$	1.50000	−	0.866025i	1.50000	−	0.866025i
$698$		0			0
$699$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$705$		0			0
$706$	1.50000	−	0.866025i	1.50000	−	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			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−1.00000			−1.00000
$723$	1.50000	−	0.866025i	1.50000	−	0.866025i
$724$		0			0
$725$		0			0
$726$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$727$		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
$731$	1.50000	+	0.866025i	1.50000	+	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
$737$	3.00000			3.00000
$738$	−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
$747$		0			0
$748$	3.00000			3.00000
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	1.50000	+	0.866025i	1.50000	+	0.866025i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1.50000	+	0.866025i	1.50000	+	0.866025i
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		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	$0.333333\pi$
$769$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	1.00000			1.00000
$772$	−1.50000	−	0.866025i	−1.50000	−	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
$776$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$777$		0			0
$778$		0			0
$779$	1.00000			1.00000
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		−	1.73205i		−	1.73205i
$793$		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
$800$	1.00000			1.00000
$801$	1.00000	+	1.73205i	1.00000	+	1.73205i
$802$	1.00000			1.00000
$803$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$804$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$809$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$811$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$817$	0.500000	+	0.866025i	0.500000	+	0.866025i
$818$			1.73205i			1.73205i
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$			1.73205i			1.73205i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		−	1.73205i		−	1.73205i
$826$		0			0
$827$	2.00000			2.00000			1.00000			$0$
$827$	2.00000			2.00000			1.00000			$0$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$834$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$835$		0			0
$836$	1.50000	+	0.866025i	1.50000	+	0.866025i
$837$		0			0
$838$		0			0
$839$		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
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.00000			2.00000
$850$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$851$		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	0.500000	−	0.866025i	$-0.333333\pi$
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$860$		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
$866$	1.50000	+	0.866025i	1.50000	+	0.866025i
$867$	1.00000	−	1.73205i	1.00000	−	1.73205i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	1.73205i		−	1.73205i
$874$		0			0
$875$		0			0
$876$	1.00000			1.00000
$877$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$877$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	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
$886$	1.50000	−	0.866025i	1.50000	−	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
$891$	1.50000	−	0.866025i	1.50000	−	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
$900$	0.500000	+	0.866025i	0.500000	+	0.866025i
$901$		0			0
$902$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$903$		0			0
$904$	1.00000	+	1.73205i	1.00000	+	1.73205i
$905$		0			0
$906$		0			0
$907$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$907$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$908$	−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	$0.666667\pi$
$912$	−1.00000			−1.00000
$913$		0			0
$914$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	1.50000	+	0.866025i	1.50000	+	0.866025i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	−1.00000			−1.00000
$932$	1.50000	−	0.866025i	1.50000	−	0.866025i
$933$		0			0
$934$	−1.50000	−	0.866025i	−1.50000	−	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
$946$		−	1.73205i		−	1.73205i
$947$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$	−1.00000			−1.00000
$951$		0			0
$952$		0			0
$953$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$954$		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
$964$			1.73205i			1.73205i
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$969$	1.50000	−	0.866025i	1.50000	−	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	1.00000			$0$
$977$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$978$	1.00000	+	1.73205i	1.00000	+	1.73205i
$979$	3.00000	+	1.73205i	3.00000	+	1.73205i
$980$		0			0
$981$		0			0
$982$			1.73205i			1.73205i
$983$		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			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		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

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.1.bo.b.227.1	yes	2
8.3	odd	2	CM	1368.1.bo.b.227.1	yes	2
9.5	odd	6		1368.1.bo.a.1139.1	yes	2
19.18	odd	2		1368.1.bo.a.227.1	✓	2
72.59	even	6		1368.1.bo.a.1139.1	yes	2
152.75	even	2		1368.1.bo.a.227.1	✓	2
171.113	even	6	inner	1368.1.bo.b.1139.1	yes	2
1368.1139	odd	6	inner	1368.1.bo.b.1139.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.1.bo.a.227.1	✓	2	19.18	odd	2
1368.1.bo.a.227.1	✓	2	152.75	even	2
1368.1.bo.a.1139.1	yes	2	9.5	odd	6
1368.1.bo.a.1139.1	yes	2	72.59	even	6
1368.1.bo.b.227.1	yes	2	1.1	even	1	trivial
1368.1.bo.b.227.1	yes	2	8.3	odd	2	CM
1368.1.bo.b.1139.1	yes	2	171.113	even	6	inner
1368.1.bo.b.1139.1	yes	2	1368.1139	odd	6	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1368.1.bo.b.227.1

Level	$1368$
Weight	$1$
Character	1368.227
Analytic conductor	$0.683$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{6}$
CM discriminant	-8
Inner twists	$4$