LMFDB - Embedded newform 1800.1.cj.b.1643.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(443,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 6, 2, 9]))

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

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

chi := DirichletCharacter("1800.443");

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.cj (of order $12$ , degree $4$ , 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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
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:	$D_{6}$
Projective field:	Galois closure of 6.0.157464000.2

Embedding invariants

Embedding label			1643.2
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1800.1643
Dual form			1800.1.cj.b.1307.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 - 0.965926i) q^{2} +(-0.965926 - 0.258819i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(-0.500000 + 0.866025i) q^{6} +(-0.707107 + 0.707107i) q^{8} +(0.866025 + 0.500000i) q^{9} +(1.50000 - 0.866025i) q^{11} +(0.707107 + 0.707107i) q^{12} +(0.500000 + 0.866025i) q^{16} +(0.707107 + 0.707107i) q^{17} +(0.707107 - 0.707107i) q^{18} -1.00000i q^{19} +(-0.448288 - 1.67303i) q^{22} +(0.866025 - 0.500000i) q^{24} +(-0.707107 - 0.707107i) q^{27} +(0.965926 - 0.258819i) q^{32} +(-1.67303 + 0.448288i) q^{33} +(0.866025 - 0.500000i) q^{34} +(-0.500000 - 0.866025i) q^{36} +(-0.965926 - 0.258819i) q^{38} +(-1.50000 - 0.866025i) q^{41} +(0.448288 - 1.67303i) q^{43} -1.73205 q^{44} +(-0.258819 - 0.965926i) q^{48} +(0.866025 + 0.500000i) q^{49} +(-0.500000 - 0.866025i) q^{51} +(-0.866025 + 0.500000i) q^{54} +(-0.258819 + 0.965926i) q^{57} +(-0.866025 + 1.50000i) q^{59} -1.00000i q^{64} +1.73205i q^{66} +(-0.448288 - 1.67303i) q^{67} +(-0.258819 - 0.965926i) q^{68} +(-0.965926 + 0.258819i) q^{72} +(-1.22474 - 1.22474i) q^{73} +(-0.500000 + 0.866025i) q^{76} +(0.500000 + 0.866025i) q^{81} +(-1.22474 + 1.22474i) q^{82} +(1.93185 + 0.517638i) q^{83} +(-1.50000 - 0.866025i) q^{86} +(-0.448288 + 1.67303i) q^{88} -1.00000 q^{96} +(1.67303 + 0.448288i) q^{97} +(0.707107 - 0.707107i) q^{98} +1.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{6} + 12 q^{11} + 4 q^{16} - 4 q^{36} - 12 q^{41} - 4 q^{51} - 4 q^{76} + 4 q^{81} - 12 q^{86} - 8 q^{96}+O(q^{100})$ 8 * q - 4 * q^6 + 12 * q^11 + 4 * q^16 - 4 * q^36 - 12 * q^41 - 4 * q^51 - 4 * q^76 + 4 * q^81 - 12 * q^86 - 8 * q^96

Character values

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

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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

)

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

$2$	0.258819	−	0.965926i	0.258819	−	0.965926i
$2$	0.258819	−	0.965926i	0.258819	−	0.965926i
$3$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$3$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$4$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$5$		0			0
$5$		0			0
$6$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$7$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$7$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$8$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$9$	0.866025	+	0.500000i	0.866025	+	0.500000i
$10$		0			0
$11$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$11$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$12$	0.707107	+	0.707107i	0.707107	+	0.707107i
$13$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$13$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$17$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$18$	0.707107	−	0.707107i	0.707107	−	0.707107i
$19$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$19$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$20$		0			0
$21$		0			0
$22$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$23$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$23$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$24$	0.866025	−	0.500000i	0.866025	−	0.500000i
$25$		0			0
$26$		0			0
$27$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	0.965926	−	0.258819i	0.965926	−	0.258819i
$33$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$34$	0.866025	−	0.500000i	0.866025	−	0.500000i
$35$		0			0
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$39$		0			0
$40$		0			0
$41$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$42$		0			0
$43$	0.448288	−	1.67303i	0.448288	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$43$	0.448288	−	1.67303i	0.448288	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$44$	−1.73205			−1.73205
$45$		0			0
$46$		0			0
$47$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$47$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$48$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$49$	0.866025	+	0.500000i	0.866025	+	0.500000i
$50$		0			0
$51$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$55$		0			0
$56$		0			0
$57$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$58$		0			0
$59$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$59$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$60$		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
$66$			1.73205i			1.73205i
$67$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$67$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$68$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$73$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$73$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$74$		0			0
$75$		0			0
$76$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	0.500000	+	0.866025i	0.500000	+	0.866025i
$82$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$83$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$83$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$84$		0			0
$85$		0			0
$86$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$87$		0			0
$88$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.00000			−1.00000
$97$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$97$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$98$	0.707107	−	0.707107i	0.707107	−	0.707107i
$99$	1.73205			1.73205
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$103$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$103$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$107$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$108$	0.258819	+	0.965926i	0.258819	+	0.965926i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$113$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$114$	0.866025	+	0.500000i	0.866025	+	0.500000i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.22474	+	1.22474i	1.22474	+	1.22474i
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	1.00000	−	1.73205i
$122$		0			0
$123$	1.22474	+	1.22474i	1.22474	+	1.22474i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$129$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	1.67303	+	0.448288i	1.67303	+	0.448288i
$133$		0			0
$134$	−1.73205			−1.73205
$135$		0			0
$136$	−1.00000			−1.00000
$137$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$137$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$138$		0			0
$139$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$139$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			1.00000i			1.00000i
$145$		0			0
$146$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$147$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	0.707107	+	0.707107i	0.707107	+	0.707107i
$153$	0.258819	+	0.965926i	0.258819	+	0.965926i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$157$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.965926	−	0.258819i	0.965926	−	0.258819i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$	0.866025	+	1.50000i	0.866025	+	1.50000i
$165$		0			0
$166$	1.00000	−	1.73205i	1.00000	−	1.73205i
$167$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$167$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$168$		0			0
$169$	0.866025	−	0.500000i	0.866025	−	0.500000i
$170$		0			0
$171$	0.500000	−	0.866025i	0.500000	−	0.866025i
$172$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$173$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$173$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$174$		0			0
$175$		0			0
$176$	1.50000	+	0.866025i	1.50000	+	0.866025i
$177$	1.22474	−	1.22474i	1.22474	−	1.22474i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		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
$187$	1.67303	+	0.448288i	1.67303	+	0.448288i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$193$	1.67303	−	0.448288i	1.67303	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$193$	1.67303	−	0.448288i	1.67303	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$194$	0.866025	−	1.50000i	0.866025	−	1.50000i
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$	0.448288	−	1.67303i	0.448288	−	1.67303i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			1.73205i			1.73205i
$202$		0			0
$203$		0			0
$204$			1.00000i			1.00000i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$210$		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
$214$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$		0			0
$219$	0.866025	+	1.50000i	0.866025	+	1.50000i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$223$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$224$		0			0
$225$		0			0
$226$	2.00000			2.00000
$227$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$227$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$228$	0.707107	−	0.707107i	0.707107	−	0.707107i
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$233$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$234$		0			0
$235$		0			0
$236$	1.50000	−	0.866025i	1.50000	−	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$242$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$243$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$244$		0			0
$245$		0			0
$246$	1.50000	−	0.866025i	1.50000	−	0.866025i
$247$		0			0
$248$		0			0
$249$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$250$		0			0
$251$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$251$			1.73205i			1.73205i	−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
$257$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$257$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$258$	1.22474	+	1.22474i	1.22474	+	1.22474i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$263$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$264$	0.866025	−	1.50000i	0.866025	−	1.50000i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$269$		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			$\pi$
$272$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$273$		0			0
$274$	0.866025	+	0.500000i	0.866025	+	0.500000i
$275$		0			0
$276$		0			0
$277$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$277$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$278$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$283$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.965926	+	0.258819i	0.965926	+	0.258819i
$289$		0			0
$290$		0			0
$291$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$292$	0.448288	+	1.67303i	0.448288	+	1.67303i
$293$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$293$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$294$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$295$		0			0
$296$		0			0
$297$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.866025	−	0.500000i	0.866025	−	0.500000i
$305$		0			0
$306$	1.00000			1.00000
$307$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$307$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$313$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$317$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.500000	+	0.866025i	0.500000	+	0.866025i
$322$		0			0
$323$	0.707107	−	0.707107i	0.707107	−	0.707107i
$324$		−	1.00000i		−	1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	1.67303	−	0.448288i	1.67303	−	0.448288i
$329$		0			0
$330$		0			0
$331$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$331$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$332$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.448288	+	1.67303i	0.448288	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$337$	0.448288	+	1.67303i	0.448288	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$338$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$339$		−	2.00000i		−	2.00000i
$340$		0			0
$341$		0			0
$342$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$343$		0			0
$344$	0.866025	+	1.50000i	0.866025	+	1.50000i
$345$		0			0
$346$		0			0
$347$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$347$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$348$		0			0
$349$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$349$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$350$		0			0
$351$		0			0
$352$	1.22474	−	1.22474i	1.22474	−	1.22474i
$353$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$353$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$354$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$367$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$368$		0			0
$369$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$373$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$374$	0.866025	−	1.50000i	0.866025	−	1.50000i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$379$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$383$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$384$	0.866025	+	0.500000i	0.866025	+	0.500000i
$385$		0			0
$386$		−	1.73205i		−	1.73205i
$387$	1.22474	−	1.22474i	1.22474	−	1.22474i
$388$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$402$	1.67303	+	0.448288i	1.67303	+	0.448288i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.965926	+	0.258819i	0.965926	+	0.258819i
$409$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$409$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$410$		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
$417$	0.707107	+	0.707107i	0.707107	+	0.707107i
$418$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$	1.41421	+	1.41421i	1.41421	+	1.41421i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.258819	+	0.965926i	0.258819	+	0.965926i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	0.258819	−	0.965926i	0.258819	−	0.965926i
$433$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$433$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.67303	−	0.448288i	1.67303	−	0.448288i
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	0.500000	+	0.866025i	0.500000	+	0.866025i
$442$		0			0
$443$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$443$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$449$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$450$		0			0
$451$	−3.00000			−3.00000
$452$	0.517638	−	1.93185i	0.517638	−	1.93185i
$453$		0			0
$454$	0.866025	+	0.500000i	0.866025	+	0.500000i
$455$		0			0
$456$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$457$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$457$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$458$		0			0
$459$		−	1.00000i		−	1.00000i
$460$		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
$463$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$463$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$464$		0			0
$465$		0			0
$466$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$467$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$467$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$473$	−0.776457	−	2.89778i	−0.776457	−	2.89778i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		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
$482$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$483$		0			0
$484$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$485$		0			0
$486$	−1.00000			−1.00000
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$491$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$492$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$		0			0
$502$	1.67303	+	0.448288i	1.67303	+	0.448288i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	0.707107	+	0.707107i	0.707107	+	0.707107i
$513$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$514$			1.00000i			1.00000i
$515$		0			0
$516$	1.50000	−	0.866025i	1.50000	−	0.866025i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$521$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$529$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$530$		0			0
$531$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.50000	+	0.866025i	1.50000	+	0.866025i
$537$		0			0
$538$		0			0
$539$	1.73205			1.73205
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	0.866025	+	0.500000i	0.866025	+	0.500000i
$545$		0			0
$546$		0			0
$547$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$547$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$548$	0.707107	−	0.707107i	0.707107	−	0.707107i
$549$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$562$		0			0
$563$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$563$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$570$		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
$577$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$577$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$578$		0			0
$579$	−1.73205			−1.73205
$580$		0			0
$581$		0			0
$582$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$583$		0			0
$584$	1.73205			1.73205
$585$		0			0
$586$		0			0
$587$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$587$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$588$	0.258819	+	0.965926i	0.258819	+	0.965926i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$593$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$594$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	0.448288	−	1.67303i	0.448288	−	1.67303i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$607$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$608$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.258819	−	0.965926i	0.258819	−	0.965926i
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$	0.866025	+	1.50000i	0.866025	+	1.50000i
$615$		0			0
$616$		0			0
$617$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$617$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$619$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	1.50000	+	0.866025i	1.50000	+	0.866025i
$627$	0.448288	+	1.67303i	0.448288	+	1.67303i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	1.41421	−	1.41421i	1.41421	−	1.41421i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$641$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$642$	0.965926	−	0.258819i	0.965926	−	0.258819i
$643$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$643$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$649$			3.00000i			3.00000i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$653$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$654$		0			0
$655$		0			0
$656$		−	1.73205i		−	1.73205i
$657$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	1.93185	−	0.517638i	1.93185	−	0.517638i
$663$		0			0
$664$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$665$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$673$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$674$	1.73205			1.73205
$675$		0			0
$676$	−1.00000			−1.00000
$677$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$677$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$678$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$679$		0			0
$680$		0			0
$681$	0.500000	−	0.866025i	0.500000	−	0.866025i
$682$		0			0
$683$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$683$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$684$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.67303	−	0.448288i	1.67303	−	0.448288i
$689$		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
$694$			1.00000i			1.00000i
$695$		0			0
$696$		0			0
$697$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$698$		0			0
$699$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$705$		0			0
$706$	0.500000	−	0.866025i	0.500000	−	0.866025i
$707$		0			0
$708$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$709$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$709$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		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
$723$	0.258819	+	0.965926i	0.258819	+	0.965926i
$724$		0			0
$725$		0			0
$726$	1.00000	+	1.73205i	1.00000	+	1.73205i
$727$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$727$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$	1.50000	−	0.866025i	1.50000	−	0.866025i
$732$		0			0
$733$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$733$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.12132	−	2.12132i	−2.12132	−	2.12132i
$738$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$739$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$739$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$743$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.41421	+	1.41421i	1.41421	+	1.41421i
$748$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	0.448288	−	1.67303i	0.448288	−	1.67303i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	0.965926	+	0.258819i	0.965926	+	0.258819i
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−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
$768$	0.707107	−	0.707107i	0.707107	−	0.707107i
$769$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$769$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$	1.00000			1.00000
$772$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$775$		0			0
$776$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$777$		0			0
$778$		0			0
$779$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			1.00000i			1.00000i
$785$		0			0
$786$		0			0
$787$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$787$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$797$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$803$	−2.89778	−	0.776457i	−2.89778	−	0.776457i
$804$	0.866025	−	1.50000i	0.866025	−	1.50000i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$809$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$811$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.500000	−	0.866025i	0.500000	−	0.866025i
$817$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$818$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$823$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$823$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$827$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$828$		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
$833$	0.258819	+	0.965926i	0.258819	+	0.965926i
$834$	0.866025	−	0.500000i	0.866025	−	0.500000i
$835$		0			0
$836$			1.73205i			1.73205i
$837$		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			0
$843$		0			0
$844$	1.73205	−	1.00000i	1.73205	−	1.00000i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$853$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$854$		0			0
$855$		0			0
$856$	1.00000			1.00000
$857$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$857$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$858$		0			0
$859$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$859$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$865$		0			0
$866$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.22474	+	1.22474i	1.22474	+	1.22474i
$874$		0			0
$875$		0			0
$876$		−	1.73205i		−	1.73205i
$877$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$877$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.965926	−	0.258819i	0.965926	−	0.258819i
$883$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$883$	1.22474	+	1.22474i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$884$		0			0
$885$		0			0
$886$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$887$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$887$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$888$		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
$898$	0.448288	−	1.67303i	0.448288	−	1.67303i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.776457	+	2.89778i	−0.776457	+	2.89778i
$903$		0			0
$904$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$905$		0			0
$906$		0			0
$907$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$907$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$908$	0.707107	−	0.707107i	0.707107	−	0.707107i
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$913$	3.34607	−	0.896575i	3.34607	−	0.896575i
$914$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		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		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	0.500000	−	0.866025i	0.500000	−	0.866025i
$932$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$933$		0			0
$934$	0.866025	−	0.500000i	0.866025	−	0.500000i
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$	0.866025	−	1.50000i	0.866025	−	1.50000i
$940$		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
$944$	−1.73205			−1.73205
$945$		0			0
$946$	−3.00000			−3.00000
$947$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$953$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$954$		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
$963$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$964$			1.00000i			1.00000i
$965$		0			0
$966$		0			0
$967$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$967$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$968$	0.517638	+	1.93185i	0.517638	+	1.93185i
$969$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$977$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	1.22474	−	1.22474i	1.22474	−	1.22474i
$983$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$983$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$984$	−1.73205			−1.73205
$985$		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
$993$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$994$		0			0
$995$		0			0
$996$	1.00000	+	1.73205i	1.00000	+	1.73205i
$997$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$997$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$998$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$999$		0			0

Display

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

with

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up to: 50 250 1000 (See only

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.1.cj.b.1643.2	yes	8
5.2	odd	4	inner	1800.1.cj.b.707.2	yes	8
5.3	odd	4	inner	1800.1.cj.b.707.1	yes	8
5.4	even	2	inner	1800.1.cj.b.1643.1	yes	8
8.3	odd	2	CM	1800.1.cj.b.1643.2	yes	8
9.2	odd	6	inner	1800.1.cj.b.443.2	yes	8
40.3	even	4	inner	1800.1.cj.b.707.1	yes	8
40.19	odd	2	inner	1800.1.cj.b.1643.1	yes	8
40.27	even	4	inner	1800.1.cj.b.707.2	yes	8
45.2	even	12	inner	1800.1.cj.b.1307.2	yes	8
45.29	odd	6	inner	1800.1.cj.b.443.1	✓	8
45.38	even	12	inner	1800.1.cj.b.1307.1	yes	8
72.11	even	6	inner	1800.1.cj.b.443.2	yes	8
360.83	odd	12	inner	1800.1.cj.b.1307.1	yes	8
360.227	odd	12	inner	1800.1.cj.b.1307.2	yes	8
360.299	even	6	inner	1800.1.cj.b.443.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.1.cj.b.443.1	✓	8	45.29	odd	6	inner
1800.1.cj.b.443.1	✓	8	360.299	even	6	inner
1800.1.cj.b.443.2	yes	8	9.2	odd	6	inner
1800.1.cj.b.443.2	yes	8	72.11	even	6	inner
1800.1.cj.b.707.1	yes	8	5.3	odd	4	inner
1800.1.cj.b.707.1	yes	8	40.3	even	4	inner
1800.1.cj.b.707.2	yes	8	5.2	odd	4	inner
1800.1.cj.b.707.2	yes	8	40.27	even	4	inner
1800.1.cj.b.1307.1	yes	8	45.38	even	12	inner
1800.1.cj.b.1307.1	yes	8	360.83	odd	12	inner
1800.1.cj.b.1307.2	yes	8	45.2	even	12	inner
1800.1.cj.b.1307.2	yes	8	360.227	odd	12	inner
1800.1.cj.b.1643.1	yes	8	5.4	even	2	inner
1800.1.cj.b.1643.1	yes	8	40.19	odd	2	inner
1800.1.cj.b.1643.2	yes	8	1.1	even	1	trivial
1800.1.cj.b.1643.2	yes	8	8.3	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	1800.1.cj.b.1643.2

Level	$1800$
Weight	$1$
Character	1800.1643
Analytic conductor	$0.898$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{6}$
CM discriminant	-8
Inner twists	$16$