LMFDB - Embedded newform 2268.1.s.f.1511.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,1,Mod(755,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("2268.755");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2268 = 2^{2} \cdot 3^{4} \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2268.s (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:	$1.13187944865$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{4} + 1$ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			1511.3
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2268.1511
Dual form			2268.1.s.f.755.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(0.866025 - 0.500000i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.258819 + 0.448288i) q^{11} +(0.707107 + 0.707107i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-0.366025 + 0.366025i) q^{22} +(-0.707107 + 1.22474i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-0.500000 + 0.866025i) q^{28} +(1.22474 - 0.707107i) q^{29} +(0.965926 + 0.258819i) q^{32} +1.73205 q^{37} +(0.866025 - 0.500000i) q^{43} +(-0.448288 - 0.258819i) q^{44} +(-1.36603 - 0.366025i) q^{46} +(0.500000 - 0.866025i) q^{49} +(-0.707107 + 0.707107i) q^{50} +1.93185i q^{53} +(-0.965926 - 0.258819i) q^{56} +(1.00000 + 1.00000i) q^{58} +1.00000i q^{64} +(-1.50000 - 0.866025i) q^{67} -1.93185 q^{71} +(0.448288 + 1.67303i) q^{74} +(0.448288 + 0.258819i) q^{77} +(-1.50000 + 0.866025i) q^{79} +(0.707107 + 0.707107i) q^{86} +(0.133975 - 0.500000i) q^{88} -1.41421i q^{92} +(0.965926 + 0.258819i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{16} + 4 q^{22} + 4 q^{25} - 4 q^{28} - 4 q^{46} + 4 q^{49} + 8 q^{58} - 12 q^{67} - 12 q^{79} + 8 q^{88}+O(q^{100})$ 8 * q + 4 * q^16 + 4 * q^22 + 4 * q^25 - 4 * q^28 - 4 * q^46 + 4 * q^49 + 8 * q^58 - 12 * q^67 - 12 * q^79 + 8 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$ .

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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

)

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

$2$	0.258819	+	0.965926i	0.258819	+	0.965926i
$2$	0.258819	+	0.965926i	0.258819	+	0.965926i
$3$		0			0
$3$		0			0
$4$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$8$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$9$		0			0
$10$		0			0
$11$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$11$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$	0.707107	+	0.707107i	0.707107	+	0.707107i
$15$		0			0
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$23$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$23$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$29$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$29$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	0.965926	+	0.258819i	0.965926	+	0.258819i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$37$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$43$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$44$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$45$		0			0
$46$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$51$		0			0
$52$		0			0
$53$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$53$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$54$		0			0
$55$		0			0
$56$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$57$		0			0
$58$	1.00000	+	1.00000i	1.00000	+	1.00000i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$			1.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−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			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$71$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	0.448288	+	1.67303i	0.448288	+	1.67303i
$75$		0			0
$76$		0			0
$77$	0.448288	+	0.258819i	0.448288	+	0.258819i
$78$		0			0
$79$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$79$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$	0.707107	+	0.707107i	0.707107	+	0.707107i
$87$		0			0
$88$	0.133975	−	0.500000i	0.133975	−	0.500000i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		−	1.41421i		−	1.41421i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	0.965926	+	0.258819i	0.965926	+	0.258819i
$99$		0			0
$100$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$107$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$107$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		−	1.00000i		−	1.00000i
$113$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$113$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$114$		0			0
$115$		0			0
$116$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.366025	−	0.633975i	0.366025	−	0.633975i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$127$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$128$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	0.448288	−	1.67303i	0.448288	−	1.67303i
$135$		0			0
$136$		0			0
$137$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$137$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$139$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$		0			0
$142$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$149$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$149$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$151$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$	−0.133975	+	0.500000i	−0.133975	+	0.500000i
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$159$		0			0
$160$		0			0
$161$			1.41421i			1.41421i
$162$		0			0
$163$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$163$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	0.866025	+	0.500000i	0.866025	+	0.500000i
$176$	0.517638			0.517638
$177$		0			0
$178$		0			0
$179$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$179$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	1.36603	−	0.366025i	1.36603	−	0.366025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$191$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$192$		0			0
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$			1.00000i			1.00000i
$197$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$197$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	0.258819	−	0.965926i	0.258819	−	0.965926i
$201$		0			0
$202$		0			0
$203$	0.707107	−	1.22474i	0.707107	−	1.22474i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$211$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$212$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$213$		0			0
$214$	0.500000	+	1.86603i	0.500000	+	1.86603i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$	0.965926	−	0.258819i	0.965926	−	0.258819i
$225$		0			0
$226$	0.500000	−	1.86603i	0.500000	−	1.86603i
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$233$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$233$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.707107	+	0.189469i	0.707107	+	0.189469i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−0.732051			−0.732051
$254$	1.67303	−	0.448288i	1.67303	−	0.448288i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	1.50000	−	0.866025i	1.50000	−	0.866025i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$263$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.73205			1.73205
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	0.366025	+	0.366025i	0.366025	+	0.366025i
$275$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$276$		0			0
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$281$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	1.67303	−	0.965926i	1.67303	−	0.965926i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$297$		0			0
$298$	0.133975	−	0.500000i	0.133975	−	0.500000i
$299$		0			0
$300$		0			0
$301$	0.500000	−	0.866025i	0.500000	−	0.866025i
$302$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−0.517638			−0.517638
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	0.866025	−	1.50000i	0.866025	−	1.50000i
$317$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$317$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$318$		0			0
$319$	0.633975	+	0.366025i	0.633975	+	0.366025i
$320$		0			0
$321$		0			0
$322$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$331$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$337$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$338$	0.707107	−	0.707107i	0.707107	−	0.707107i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$345$		0			0
$346$		0			0
$347$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$347$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$351$		0			0
$352$	0.133975	+	0.500000i	0.133975	+	0.500000i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.366025	+	1.36603i	0.366025	+	1.36603i
$359$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$359$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	0.707107	+	1.22474i	0.707107	+	1.22474i
$369$		0			0
$370$		0			0
$371$	0.965926	+	1.67303i	0.965926	+	1.67303i
$372$		0			0
$373$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$373$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$379$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$380$		0			0
$381$		0			0
$382$	1.36603	−	1.36603i	1.36603	−	1.36603i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$389$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$390$		0			0
$391$		0			0
$392$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$393$		0			0
$394$	1.86603	−	0.500000i	1.86603	−	0.500000i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$401$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	1.36603	+	0.366025i	1.36603	+	0.366025i
$407$	0.448288	+	0.776457i	0.448288	+	0.776457i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$421$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$422$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$423$		0			0
$424$	1.36603	−	1.36603i	1.36603	−	1.36603i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$429$		0			0
$430$		0			0
$431$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$431$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$443$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$449$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$450$		0			0
$451$		0			0
$452$	1.93185			1.93185
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$457$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$463$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$464$		−	1.41421i		−	1.41421i
$465$		0			0
$466$	1.36603	−	0.366025i	1.36603	−	0.366025i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−1.73205			−1.73205
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.448288	+	0.258819i	0.448288	+	0.258819i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.500000	+	0.133975i	0.500000	+	0.133975i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$			0.732051i			0.732051i
$485$		0			0
$486$		0			0
$487$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$487$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$491$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$498$		0			0
$499$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$499$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$	−0.189469	−	0.707107i	−0.189469	−	0.707107i
$507$		0			0
$508$	0.866025	+	1.50000i	0.866025	+	1.50000i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$	0.707107	−	0.707107i	0.707107	−	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.22474	+	1.22474i	1.22474	+	1.22474i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	0.366025	−	0.366025i	0.366025	−	0.366025i
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.448288	+	1.67303i	0.448288	+	1.67303i
$537$		0			0
$538$		0			0
$539$	0.517638			0.517638
$540$		0			0
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$547$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$548$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$549$		0			0
$550$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$551$		0			0
$552$		0			0
$553$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$554$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$555$		0			0
$556$		0			0
$557$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$557$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.36603	+	1.36603i	1.36603	+	1.36603i
$569$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$569$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.41421			−1.41421
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.866025	−	1.50000i	0.866025	−	1.50000i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	0.517638			0.517638
$597$		0			0
$598$		0			0
$599$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$599$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$	0.965926	+	0.258819i	0.965926	+	0.258819i
$603$		0			0
$604$	0.500000	−	0.866025i	0.500000	−	0.866025i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$613$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$617$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$617$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		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			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	1.67303	+	0.448288i	1.67303	+	0.448288i
$633$		0			0
$634$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.189469	+	0.707107i	−0.189469	+	0.707107i
$639$		0			0
$640$		0			0
$641$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$642$		0			0
$643$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$643$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$644$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$653$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$653$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$659$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			2.00000i			2.00000i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$673$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$674$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$675$		0			0
$676$	0.866025	+	0.500000i	0.866025	+	0.500000i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$683$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$		0			0
$686$	0.965926	−	0.258819i	0.965926	−	0.258819i
$687$		0			0
$688$		−	1.00000i		−	1.00000i
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$	1.86603	+	0.500000i	1.86603	+	0.500000i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.00000			−1.00000
$701$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$701$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$702$		0			0
$703$		0			0
$704$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$709$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$717$		0			0
$718$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$723$		0			0
$724$		0			0
$725$	1.22474	+	0.707107i	1.22474	+	0.707107i
$726$		0			0
$727$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$727$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$737$		−	0.896575i		−	0.896575i
$738$		0			0
$739$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$739$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$743$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$743$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$747$		0			0
$748$		0			0
$749$	1.67303	−	0.965926i	1.67303	−	0.965926i
$750$		0			0
$751$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$751$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	0.965926	−	0.258819i	0.965926	−	0.258819i
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$	1.67303	+	0.965926i	1.67303	+	0.965926i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$779$		0			0
$780$		0			0
$781$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$782$		0			0
$783$		0			0
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	0.965926	+	1.67303i	0.965926	+	1.67303i
$789$		0			0
$790$		0			0
$791$	−1.93185			−1.93185
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$	0.258819	+	0.965926i	0.258819	+	0.965926i
$801$		0			0
$802$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$809$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$			1.41421i			1.41421i
$813$		0			0
$814$	−0.633975	+	0.633975i	−0.633975	+	0.633975i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$821$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$827$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$828$		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			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	0.500000	−	0.866025i	0.500000	−	0.866025i
$842$	0.707107	−	0.707107i	0.707107	−	0.707107i
$843$		0			0
$844$	−1.00000			−1.00000
$845$		0			0
$846$		0			0
$847$		−	0.732051i		−	0.732051i
$848$	1.67303	+	0.965926i	1.67303	+	0.965926i
$849$		0			0
$850$		0			0
$851$	−1.22474	+	2.12132i	−1.22474	+	2.12132i
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$863$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$863$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.776457	−	0.448288i	−0.776457	−	0.448288i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$877$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$883$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$884$		0			0
$885$		0			0
$886$	1.00000	−	1.00000i	1.00000	−	1.00000i
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$897$		0			0
$898$	0.500000	−	0.133975i	0.500000	−	0.133975i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.500000	+	1.86603i	0.500000	+	1.86603i
$905$		0			0
$906$		0			0
$907$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$907$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$911$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$912$		0			0
$913$		0			0
$914$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$919$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.866025	+	1.50000i	0.866025	+	1.50000i
$926$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$927$		0			0
$928$	1.36603	−	0.366025i	1.36603	−	0.366025i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$	0.707107	+	1.22474i	0.707107	+	1.22474i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.133975	+	0.500000i	−0.133975	+	0.500000i
$947$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$947$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$953$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$			0.517638i			0.517638i
$957$		0			0
$958$		0			0
$959$	0.258819	−	0.448288i	0.258819	−	0.448288i
$960$		0			0
$961$	0.500000	+	0.866025i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	−0.707107	+	0.189469i	−0.707107	+	0.189469i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$975$		0			0
$976$		0			0
$977$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$977$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.36603	+	0.366025i	1.36603	+	0.366025i
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.41421i			1.41421i
$990$		0			0
$991$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$991$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$992$		0			0
$993$		0			0
$994$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	0.517638	−	1.93185i	0.517638	−	1.93185i
$999$		0			0

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.1.s.f.1511.3		8
3.2	odd	2	inner	2268.1.s.f.1511.2		8
4.3	odd	2		2268.1.s.e.1511.1		8
7.6	odd	2	CM	2268.1.s.f.1511.3		8
9.2	odd	6		2268.1.s.e.755.2		8
9.4	even	3		2268.1.h.a.2267.1	✓	8
9.5	odd	6		2268.1.h.a.2267.8	yes	8
9.7	even	3		2268.1.s.e.755.3		8
12.11	even	2		2268.1.s.e.1511.4		8
21.20	even	2	inner	2268.1.s.f.1511.2		8
28.27	even	2		2268.1.s.e.1511.1		8
36.7	odd	6	inner	2268.1.s.f.755.2		8
36.11	even	6	inner	2268.1.s.f.755.3		8
36.23	even	6		2268.1.h.a.2267.2	yes	8
36.31	odd	6		2268.1.h.a.2267.7	yes	8
63.13	odd	6		2268.1.h.a.2267.1	✓	8
63.20	even	6		2268.1.s.e.755.2		8
63.34	odd	6		2268.1.s.e.755.3		8
63.41	even	6		2268.1.h.a.2267.8	yes	8
84.83	odd	2		2268.1.s.e.1511.4		8
252.83	odd	6	inner	2268.1.s.f.755.3		8
252.139	even	6		2268.1.h.a.2267.7	yes	8
252.167	odd	6		2268.1.h.a.2267.2	yes	8
252.223	even	6	inner	2268.1.s.f.755.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.1.h.a.2267.1	✓	8	9.4	even	3
2268.1.h.a.2267.1	✓	8	63.13	odd	6
2268.1.h.a.2267.2	yes	8	36.23	even	6
2268.1.h.a.2267.2	yes	8	252.167	odd	6
2268.1.h.a.2267.7	yes	8	36.31	odd	6
2268.1.h.a.2267.7	yes	8	252.139	even	6
2268.1.h.a.2267.8	yes	8	9.5	odd	6
2268.1.h.a.2267.8	yes	8	63.41	even	6
2268.1.s.e.755.2		8	9.2	odd	6
2268.1.s.e.755.2		8	63.20	even	6
2268.1.s.e.755.3		8	9.7	even	3
2268.1.s.e.755.3		8	63.34	odd	6
2268.1.s.e.1511.1		8	4.3	odd	2
2268.1.s.e.1511.1		8	28.27	even	2
2268.1.s.e.1511.4		8	12.11	even	2
2268.1.s.e.1511.4		8	84.83	odd	2
2268.1.s.f.755.2		8	36.7	odd	6	inner
2268.1.s.f.755.2		8	252.223	even	6	inner
2268.1.s.f.755.3		8	36.11	even	6	inner
2268.1.s.f.755.3		8	252.83	odd	6	inner
2268.1.s.f.1511.2		8	3.2	odd	2	inner
2268.1.s.f.1511.2		8	21.20	even	2	inner
2268.1.s.f.1511.3		8	1.1	even	1	trivial
2268.1.s.f.1511.3		8	7.6	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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	2268.1.s.f.1511.3

Level	$2268$
Weight	$1$
Character	2268.1511
Analytic conductor	$1.132$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{12}$
CM discriminant	-7
Inner twists	$8$