LMFDB - Embedded newform 1764.1.y.b.667.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,1,Mod(667,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

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

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

chi := DirichletCharacter("1764.667");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1764.y (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:	$0.880350682285$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.49392.1
Artin image:	$C_6\times D_8$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{48} + \cdots)$

Embedding invariants

Embedding label			667.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1764.667
Dual form			1764.1.y.b.1243.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.707107 - 1.22474i) q^{5} +1.00000 q^{8} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.707107 - 1.22474i) q^{5} +1.00000 q^{8} +(0.707107 + 1.22474i) q^{10} +1.41421 q^{13} +(-0.500000 + 0.866025i) q^{16} +(-0.707107 - 1.22474i) q^{17} -1.41421 q^{20} +(-0.500000 - 0.866025i) q^{25} +(-0.707107 + 1.22474i) q^{26} -2.00000 q^{29} +(-0.500000 - 0.866025i) q^{32} +1.41421 q^{34} +(0.707107 - 1.22474i) q^{40} +1.41421 q^{41} +1.00000 q^{50} +(-0.707107 - 1.22474i) q^{52} +(1.00000 - 1.73205i) q^{58} +(0.707107 - 1.22474i) q^{61} +1.00000 q^{64} +(1.00000 - 1.73205i) q^{65} +(-0.707107 + 1.22474i) q^{68} +(0.707107 + 1.22474i) q^{73} +(0.707107 + 1.22474i) q^{80} +(-0.707107 + 1.22474i) q^{82} -2.00000 q^{85} +(0.707107 - 1.22474i) q^{89} -1.41421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 2 q^{16} - 2 q^{25} - 8 q^{29} - 2 q^{32} + 4 q^{50} + 4 q^{58} + 4 q^{64} + 4 q^{65} - 8 q^{85}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 - 2 * q^16 - 2 * q^25 - 8 * q^29 - 2 * q^32 + 4 * q^50 + 4 * q^58 + 4 * q^64 + 4 * q^65 - 8 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$5$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			1.00000
$9$		0			0
$10$	0.707107	+	1.22474i	0.707107	+	1.22474i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$13$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$17$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	−1.41421			−1.41421
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$27$		0			0
$28$		0			0
$29$	−2.00000			−2.00000			−1.00000			$\pi$
$29$	−2.00000			−2.00000			−1.00000			$\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$33$		0			0
$34$	1.41421			1.41421
$35$		0			0
$36$		0			0
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$		0			0
$40$	0.707107	−	1.22474i	0.707107	−	1.22474i
$41$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$41$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$		0			0
$50$	1.00000			1.00000
$51$		0			0
$52$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.00000	−	1.73205i	1.00000	−	1.73205i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$61$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$	1.00000	−	1.73205i	1.00000	−	1.73205i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$73$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	0.707107	+	1.22474i	0.707107	+	1.22474i
$81$		0			0
$82$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−2.00000			−2.00000
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$89$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$97$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$101$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	1.41421			1.41421
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$109$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	1.00000	+	1.73205i	1.00000	+	1.73205i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$	0.707107	+	1.22474i	0.707107	+	1.22474i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$	1.00000	+	1.73205i	1.00000	+	1.73205i
$131$		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			0
$135$		0			0
$136$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$137$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$137$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$146$	−1.41421			−1.41421
$147$		0			0
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$157$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$158$		0			0
$159$		0			0
$160$	−1.41421			−1.41421
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$	1.00000	−	1.73205i	1.00000	−	1.73205i
$171$		0			0
$172$		0			0
$173$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$173$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.707107	+	1.22474i	0.707107	+	1.22474i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$181$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$193$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$194$	0.707107	−	1.22474i	0.707107	−	1.22474i
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$201$		0			0
$202$	−1.41421			−1.41421
$203$		0			0
$204$		0			0
$205$	1.00000	−	1.73205i	1.00000	−	1.73205i
$206$		0			0
$207$		0			0
$208$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$229$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$230$		0			0
$231$		0			0
$232$	−2.00000			−2.00000
$233$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$233$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$241$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$242$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$243$		0			0
$244$	−1.41421			−1.41421
$245$		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			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$257$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$258$		0			0
$259$		0			0
$260$	−2.00000			−2.00000
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$269$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	1.41421			1.41421
$273$		0			0
$274$	−2.00000			−2.00000
$275$		0			0
$276$		0			0
$277$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$277$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.00000			2.00000			1.00000			$0$
$281$	2.00000			2.00000			1.00000			$0$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$291$		0			0
$292$	0.707107	−	1.22474i	0.707107	−	1.22474i
$293$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$313$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$314$	1.41421			1.41421
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$	0.707107	−	1.22474i	0.707107	−	1.22474i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$326$		0			0
$327$		0			0
$328$	1.41421			1.41421
$329$		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
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$339$		0			0
$340$	1.00000	+	1.73205i	1.00000	+	1.73205i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$349$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$353$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$354$		0			0
$355$		0			0
$356$	−1.41421			−1.41421
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	0.707107	−	1.22474i	0.707107	−	1.22474i
$363$		0			0
$364$		0			0
$365$	2.00000			2.00000
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.82843			−2.82843
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		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
$386$	−2.00000			−2.00000
$387$		0			0
$388$	0.707107	+	1.22474i	0.707107	+	1.22474i
$389$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$389$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$397$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$401$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$	0.707107	−	1.22474i	0.707107	−	1.22474i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$409$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$410$	1.00000	+	1.73205i	1.00000	+	1.73205i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−2.00000			−2.00000			−1.00000			$\pi$
$421$	−2.00000			−2.00000			−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$433$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$	2.00000			2.00000
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$457$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$458$	0.707107	+	1.22474i	0.707107	+	1.22474i
$459$		0			0
$460$		0			0
$461$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$461$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	1.00000	−	1.73205i	1.00000	−	1.73205i
$465$		0			0
$466$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		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
$482$	1.41421			1.41421
$483$		0			0
$484$	1.00000			1.00000
$485$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	0.707107	−	1.22474i	0.707107	−	1.22474i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	1.41421	+	2.44949i	1.41421	+	2.44949i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	2.00000			2.00000
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$509$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.00000	−	1.73205i	1.00000	−	1.73205i
$521$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$521$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.00000			2.00000
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.41421			1.41421
$539$		0			0
$540$		0			0
$541$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$541$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	1.00000	−	1.73205i	1.00000	−	1.73205i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−2.00000			−2.00000
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$569$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$577$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$578$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$579$		0			0
$580$	2.82843			2.82843
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.707107	+	1.22474i	0.707107	+	1.22474i
$585$		0			0
$586$	0.707107	−	1.22474i	0.707107	−	1.22474i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$593$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$601$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.707107	+	1.22474i	0.707107	+	1.22474i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$	2.00000			2.00000
$611$		0			0
$612$		0			0
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.00000			−2.00000			−1.00000			$\pi$
$617$	−2.00000			−2.00000			−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$627$		0			0
$628$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$629$		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
$640$	0.707107	+	1.22474i	0.707107	+	1.22474i
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$	1.41421			1.41421
$651$		0			0
$652$		0			0
$653$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$653$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$654$		0			0
$655$		0			0
$656$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$661$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$662$		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			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$677$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$678$		0			0
$679$		0			0
$680$	−2.00000			−2.00000
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$	2.82843			2.82843
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	1.41421			1.41421
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$698$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$699$		0			0
$700$		0			0
$701$	2.00000			2.00000			1.00000			$0$
$701$	2.00000			2.00000			1.00000			$0$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.41421			−1.41421
$707$		0			0
$708$		0			0
$709$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$709$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$710$		0			0
$711$		0			0
$712$	0.707107	−	1.22474i	0.707107	−	1.22474i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$	1.00000			1.00000
$723$		0			0
$724$	0.707107	+	1.22474i	0.707107	+	1.22474i
$725$	1.00000	+	1.73205i	1.00000	+	1.73205i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$731$		0			0
$732$		0			0
$733$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$733$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	1.00000	+	1.73205i	1.00000	+	1.73205i
$747$		0			0
$748$		0			0
$749$		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			0
$754$	1.41421	−	2.44949i	1.41421	−	2.44949i
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$761$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$769$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$		0			0
$772$	1.00000	−	1.73205i	1.00000	−	1.73205i
$773$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$773$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$774$		0			0
$775$		0			0
$776$	−1.41421			−1.41421
$777$		0			0
$778$	−2.00000			−2.00000
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.00000			−2.00000
$786$		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
$792$		0			0
$793$	1.00000	−	1.73205i	1.00000	−	1.73205i
$794$	0.707107	+	1.22474i	0.707107	+	1.22474i
$795$		0			0
$796$		0			0
$797$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$797$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$801$		0			0
$802$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.707107	+	1.22474i	0.707107	+	1.22474i
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.41421			−1.41421
$819$		0			0
$820$	−2.00000			−2.00000
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$829$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$830$		0			0
$831$		0			0
$832$	1.41421			1.41421
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	3.00000			3.00000
$842$	1.00000	−	1.73205i	1.00000	−	1.73205i
$843$		0			0
$844$		0			0
$845$	0.707107	−	1.22474i	0.707107	−	1.22474i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$851$		0			0
$852$		0			0
$853$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$853$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$857$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	1.00000	+	1.73205i	1.00000	+	1.73205i
$866$	0.707107	−	1.22474i	0.707107	−	1.22474i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$881$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$	2.00000			2.00000
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$915$		0			0
$916$	−1.41421			−1.41421
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	1.00000	+	1.73205i	1.00000	+	1.73205i
$929$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$929$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$930$		0			0
$931$		0			0
$932$	2.00000			2.00000
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$937$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$941$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	1.00000	+	1.73205i	1.00000	+	1.73205i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		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			0
$964$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$965$	2.82843			2.82843
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$969$		0			0
$970$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.707107	+	1.22474i	0.707107	+	1.22474i
$977$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$977$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		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
$986$	−2.82843			−2.82843
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$997$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.1.y.b.667.2		4
3.2	odd	2		1764.1.y.d.667.1		4
4.3	odd	2	CM	1764.1.y.b.667.2		4
7.2	even	3		1764.1.g.d.883.1	yes	2
7.3	odd	6	inner	1764.1.y.b.1243.1		4
7.4	even	3	inner	1764.1.y.b.1243.2		4
7.5	odd	6		1764.1.g.d.883.2	yes	2
7.6	odd	2	inner	1764.1.y.b.667.1		4
12.11	even	2		1764.1.y.d.667.1		4
21.2	odd	6		1764.1.g.b.883.2	yes	2
21.5	even	6		1764.1.g.b.883.1	✓	2
21.11	odd	6		1764.1.y.d.1243.1		4
21.17	even	6		1764.1.y.d.1243.2		4
21.20	even	2		1764.1.y.d.667.2		4
28.3	even	6	inner	1764.1.y.b.1243.1		4
28.11	odd	6	inner	1764.1.y.b.1243.2		4
28.19	even	6		1764.1.g.d.883.2	yes	2
28.23	odd	6		1764.1.g.d.883.1	yes	2
28.27	even	2	inner	1764.1.y.b.667.1		4
84.11	even	6		1764.1.y.d.1243.1		4
84.23	even	6		1764.1.g.b.883.2	yes	2
84.47	odd	6		1764.1.g.b.883.1	✓	2
84.59	odd	6		1764.1.y.d.1243.2		4
84.83	odd	2		1764.1.y.d.667.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.1.g.b.883.1	✓	2	21.5	even	6
1764.1.g.b.883.1	✓	2	84.47	odd	6
1764.1.g.b.883.2	yes	2	21.2	odd	6
1764.1.g.b.883.2	yes	2	84.23	even	6
1764.1.g.d.883.1	yes	2	7.2	even	3
1764.1.g.d.883.1	yes	2	28.23	odd	6
1764.1.g.d.883.2	yes	2	7.5	odd	6
1764.1.g.d.883.2	yes	2	28.19	even	6
1764.1.y.b.667.1		4	7.6	odd	2	inner
1764.1.y.b.667.1		4	28.27	even	2	inner
1764.1.y.b.667.2		4	1.1	even	1	trivial
1764.1.y.b.667.2		4	4.3	odd	2	CM
1764.1.y.b.1243.1		4	7.3	odd	6	inner
1764.1.y.b.1243.1		4	28.3	even	6	inner
1764.1.y.b.1243.2		4	7.4	even	3	inner
1764.1.y.b.1243.2		4	28.11	odd	6	inner
1764.1.y.d.667.1		4	3.2	odd	2
1764.1.y.d.667.1		4	12.11	even	2
1764.1.y.d.667.2		4	21.20	even	2
1764.1.y.d.667.2		4	84.83	odd	2
1764.1.y.d.1243.1		4	21.11	odd	6
1764.1.y.d.1243.1		4	84.11	even	6
1764.1.y.d.1243.2		4	21.17	even	6
1764.1.y.d.1243.2		4	84.59	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1764.y (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.707107 - 1.22474i) q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.707107 - 1.22474i) q^{5} +1.00000 q^{8} +(0.707107 + 1.22474i) q^{10} +1.41421 q^{13} +(-0.500000 + 0.866025i) q^{16} +(-0.707107 - 1.22474i) q^{17} -1.41421 q^{20} +(-0.500000 - 0.866025i) q^{25} +(-0.707107 + 1.22474i) q^{26} -2.00000 q^{29} +(-0.500000 - 0.866025i) q^{32} +1.41421 q^{34} +(0.707107 - 1.22474i) q^{40} +1.41421 q^{41} +1.00000 q^{50} +(-0.707107 - 1.22474i) q^{52} +(1.00000 - 1.73205i) q^{58} +(0.707107 - 1.22474i) q^{61} +1.00000 q^{64} +(1.00000 - 1.73205i) q^{65} +(-0.707107 + 1.22474i) q^{68} +(0.707107 + 1.22474i) q^{73} +(0.707107 + 1.22474i) q^{80} +(-0.707107 + 1.22474i) q^{82} -2.00000 q^{85} +(0.707107 - 1.22474i) q^{89} -1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 2 q^{16} - 2 q^{25} - 8 q^{29} - 2 q^{32} + 4 q^{50} + 4 q^{58} + 4 q^{64} + 4 q^{65} - 8 q^{85}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 + 4 * q^8 - 2 * q^16 - 2 * q^25 - 8 * q^29 - 2 * q^32 + 4 * q^50 + 4 * q^58 + 4 * q^64 + 4 * q^65 - 8 * q^85

Introduction

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