LMFDB - Embedded newform 3528.1.cm.a.3155.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,1,Mod(227,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3528.227");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3528.cm (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:	$1.76070136457$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{24}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{24} + \cdots)$

Embedding invariants

Embedding label			3155.4
Root			$0.991445 - 0.130526i$ of defining polynomial
Character	$\chi$	$=$	3528.3155
Dual form			3528.1.cm.a.227.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +(0.793353 - 0.608761i) q^{3} -1.00000 q^{4} +(-0.608761 - 0.793353i) q^{6} +1.00000i q^{8} +(0.258819 - 0.965926i) q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +(0.793353 - 0.608761i) q^{3} -1.00000 q^{4} +(-0.608761 - 0.793353i) q^{6} +1.00000i q^{8} +(0.258819 - 0.965926i) q^{9} +(1.67303 + 0.965926i) q^{11} +(-0.793353 + 0.608761i) q^{12} +1.00000 q^{16} +(0.608761 + 1.05441i) q^{17} +(-0.965926 - 0.258819i) q^{18} +(0.226078 + 0.130526i) q^{19} +(0.965926 - 1.67303i) q^{22} +(0.608761 + 0.793353i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.382683 - 0.923880i) q^{27} -1.00000i q^{32} +(1.91532 - 0.252157i) q^{33} +(1.05441 - 0.608761i) q^{34} +(-0.258819 + 0.965926i) q^{36} +(0.130526 - 0.226078i) q^{38} +(0.793353 - 1.37413i) q^{41} +(0.258819 + 0.448288i) q^{43} +(-1.67303 - 0.965926i) q^{44} +(0.793353 - 0.608761i) q^{48} +(0.866025 + 0.500000i) q^{50} +(1.12484 + 0.465926i) q^{51} +(-0.923880 + 0.382683i) q^{54} +(0.258819 - 0.0340742i) q^{57} -1.98289 q^{59} -1.00000 q^{64} +(-0.252157 - 1.91532i) q^{66} +1.73205 q^{67} +(-0.608761 - 1.05441i) q^{68} +(0.965926 + 0.258819i) q^{72} +(-1.71723 + 0.991445i) q^{73} +(0.130526 + 0.991445i) q^{75} +(-0.226078 - 0.130526i) q^{76} +(-0.866025 - 0.500000i) q^{81} +(-1.37413 - 0.793353i) q^{82} +(-0.382683 - 0.662827i) q^{83} +(0.448288 - 0.258819i) q^{86} +(-0.965926 + 1.67303i) q^{88} +(0.923880 - 1.60021i) q^{89} +(-0.608761 - 0.793353i) q^{96} +(-1.37413 + 0.793353i) q^{97} +(1.36603 - 1.36603i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 8 q^{25} - 16 q^{64} + 8 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 8 * q^25 - 16 * q^64 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	1.00000i
$2$		−	1.00000i		−	1.00000i
$3$	0.793353	−	0.608761i	0.793353	−	0.608761i
$3$	0.793353	−	0.608761i	0.793353	−	0.608761i
$4$	−1.00000			−1.00000
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$7$		0			0
$7$		0			0
$8$			1.00000i			1.00000i
$9$	0.258819	−	0.965926i	0.258819	−	0.965926i
$10$		0			0
$11$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$11$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$12$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			1.00000
$17$	0.608761	+	1.05441i	0.608761	+	1.05441i	0.991445	+	0.130526i	$0.0416667\pi$
$17$	0.608761	+	1.05441i	0.608761	+	1.05441i	−0.382683	+	0.923880i	$0.625000\pi$
$18$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$19$	0.226078	+	0.130526i	0.226078	+	0.130526i	0.608761	−	0.793353i	$-0.291667\pi$
$19$	0.226078	+	0.130526i	0.226078	+	0.130526i	−0.382683	+	0.923880i	$0.625000\pi$
$20$		0			0
$21$		0			0
$22$	0.965926	−	1.67303i	0.965926	−	1.67303i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	0.608761	+	0.793353i	0.608761	+	0.793353i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		−	1.00000i		−	1.00000i
$33$	1.91532	−	0.252157i	1.91532	−	0.252157i
$34$	1.05441	−	0.608761i	1.05441	−	0.608761i
$35$		0			0
$36$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	0.130526	−	0.226078i	0.130526	−	0.226078i
$39$		0			0
$40$		0			0
$41$	0.793353	−	1.37413i	0.793353	−	1.37413i	−0.130526	−	0.991445i	$-0.541667\pi$
$41$	0.793353	−	1.37413i	0.793353	−	1.37413i	0.923880	−	0.382683i	$-0.125000\pi$
$42$		0			0
$43$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$43$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$44$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	0.793353	−	0.608761i	0.793353	−	0.608761i
$49$		0			0
$50$	0.866025	+	0.500000i	0.866025	+	0.500000i
$51$	1.12484	+	0.465926i	1.12484	+	0.465926i
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$55$		0			0
$56$		0			0
$57$	0.258819	−	0.0340742i	0.258819	−	0.0340742i
$58$		0			0
$59$	−1.98289			−1.98289			−0.991445	−	0.130526i	$-0.958333\pi$
$59$	−1.98289			−1.98289			−0.991445	+	0.130526i	$0.958333\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$	−0.252157	−	1.91532i	−0.252157	−	1.91532i
$67$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$67$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$68$	−0.608761	−	1.05441i	−0.608761	−	1.05441i
$69$		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
$73$	−1.71723	+	0.991445i	−1.71723	+	0.991445i	−0.793353	+	0.608761i	$0.791667\pi$
$73$	−1.71723	+	0.991445i	−1.71723	+	0.991445i	−0.923880	+	0.382683i	$0.875000\pi$
$74$		0			0
$75$	0.130526	+	0.991445i	0.130526	+	0.991445i
$76$	−0.226078	−	0.130526i	−0.226078	−	0.130526i
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$82$	−1.37413	−	0.793353i	−1.37413	−	0.793353i
$83$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$83$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$84$		0			0
$85$		0			0
$86$	0.448288	−	0.258819i	0.448288	−	0.258819i
$87$		0			0
$88$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$89$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$89$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$97$	−1.37413	+	0.793353i	−1.37413	+	0.793353i	−0.991445	−	0.130526i	$-0.958333\pi$
$97$	−1.37413	+	0.793353i	−1.37413	+	0.793353i	−0.382683	+	0.923880i	$0.625000\pi$
$98$		0			0
$99$	1.36603	−	1.36603i	1.36603	−	1.36603i
$100$	0.500000	−	0.866025i	0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	0.465926	−	1.12484i	0.465926	−	1.12484i
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$108$	0.382683	+	0.923880i	0.382683	+	0.923880i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$113$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$114$	−0.0340742	−	0.258819i	−0.0340742	−	0.258819i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			1.98289i			1.98289i
$119$		0			0
$120$		0			0
$121$	1.36603	+	2.36603i	1.36603	+	2.36603i
$122$		0			0
$123$	−0.207107	−	1.57313i	−0.207107	−	1.57313i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$			1.00000i			1.00000i
$129$	0.478235	+	0.198092i	0.478235	+	0.198092i
$130$		0			0
$131$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$131$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$132$	−1.91532	+	0.252157i	−1.91532	+	0.252157i
$133$		0			0
$134$		−	1.73205i		−	1.73205i
$135$		0			0
$136$	−1.05441	+	0.608761i	−1.05441	+	0.608761i
$137$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$137$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$138$		0			0
$139$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.923880	−	0.382683i	$-0.875000\pi$
$139$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.793353	−	0.608761i	$-0.791667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.258819	−	0.965926i	0.258819	−	0.965926i
$145$		0			0
$146$	0.991445	+	1.71723i	0.991445	+	1.71723i
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$	0.991445	−	0.130526i	0.991445	−	0.130526i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−0.130526	+	0.226078i	−0.130526	+	0.226078i
$153$	1.17604	−	0.315118i	1.17604	−	0.315118i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$163$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$163$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$164$	−0.793353	+	1.37413i	−0.793353	+	1.37413i
$165$		0			0
$166$	−0.662827	+	0.382683i	−0.662827	+	0.382683i
$167$		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
$171$	0.184592	−	0.184592i	0.184592	−	0.184592i
$172$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	1.67303	+	0.965926i	1.67303	+	0.965926i
$177$	−1.57313	+	1.20711i	−1.57313	+	1.20711i
$178$	−1.60021	−	0.923880i	−1.60021	−	0.923880i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			2.35207i			2.35207i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$193$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$193$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$194$	0.793353	+	1.37413i	0.793353	+	1.37413i
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$201$	1.37413	−	1.05441i	1.37413	−	1.05441i
$202$		0			0
$203$		0			0
$204$	−1.12484	−	0.465926i	−1.12484	−	0.465926i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.252157	+	0.436749i	0.252157	+	0.436749i
$210$		0			0
$211$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$211$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$		0			0
$214$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$215$		0			0
$216$	0.923880	−	0.382683i	0.923880	−	0.382683i
$217$		0			0
$218$		0			0
$219$	−0.758819	+	1.83195i	−0.758819	+	1.83195i
$220$		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			0
$225$	0.707107	+	0.707107i	0.707107	+	0.707107i
$226$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$227$	0.793353	−	1.37413i	0.793353	−	1.37413i	−0.130526	−	0.991445i	$-0.541667\pi$
$227$	0.793353	−	1.37413i	0.793353	−	1.37413i	0.923880	−	0.382683i	$-0.125000\pi$
$228$	−0.258819	+	0.0340742i	−0.258819	+	0.0340742i
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$233$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$234$		0			0
$235$		0			0
$236$	1.98289			1.98289
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	1.05441	+	0.608761i	1.05441	+	0.608761i	0.923880	−	0.382683i	$-0.125000\pi$
$241$	1.05441	+	0.608761i	1.05441	+	0.608761i	0.130526	+	0.991445i	$0.458333\pi$
$242$	2.36603	−	1.36603i	2.36603	−	1.36603i
$243$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$244$		0			0
$245$		0			0
$246$	−1.57313	+	0.207107i	−1.57313	+	0.207107i
$247$		0			0
$248$		0			0
$249$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$250$		0			0
$251$	0.261052			0.261052			0.130526	−	0.991445i	$-0.458333\pi$
$251$	0.261052			0.261052			0.130526	+	0.991445i	$0.458333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.608761	+	0.793353i	$0.291667\pi$
$257$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.382683	+	0.923880i	$0.375000\pi$
$258$	0.198092	−	0.478235i	0.198092	−	0.478235i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.60021	+	0.923880i	−1.60021	+	0.923880i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$	0.252157	+	1.91532i	0.252157	+	1.91532i
$265$		0			0
$266$		0			0
$267$	−0.241181	−	1.83195i	−0.241181	−	1.83195i
$268$	−1.73205			−1.73205
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$	0.608761	+	1.05441i	0.608761	+	1.05441i
$273$		0			0
$274$	0.258819	−	0.448288i	0.258819	−	0.448288i
$275$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	−0.991445	+	1.71723i	−0.991445	+	1.71723i
$279$		0			0
$280$		0			0
$281$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$281$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$282$		0			0
$283$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$283$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$289$	−0.241181	+	0.417738i	−0.241181	+	0.417738i
$290$		0			0
$291$	−0.607206	+	1.46593i	−0.607206	+	1.46593i
$292$	1.71723	−	0.991445i	1.71723	−	0.991445i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.252157	−	1.91532i	0.252157	−	1.91532i
$298$		0			0
$299$		0			0
$300$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.226078	+	0.130526i	0.226078	+	0.130526i
$305$		0			0
$306$	−0.315118	−	1.17604i	−0.315118	−	1.17604i
$307$			1.21752i			1.21752i	0.793353	+	0.608761i	$0.208333\pi$
$307$			1.21752i			1.21752i	−0.793353	+	0.608761i	$0.791667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			1.21752i			1.21752i	0.793353	+	0.608761i	$0.208333\pi$
$313$			1.21752i			1.21752i	−0.793353	+	0.608761i	$0.791667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−1.71723	+	0.226078i	−1.71723	+	0.226078i
$322$		0			0
$323$			0.317837i			0.317837i
$324$	0.866025	+	0.500000i	0.866025	+	0.500000i
$325$		0			0
$326$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$327$		0			0
$328$	1.37413	+	0.793353i	1.37413	+	0.793353i
$329$		0			0
$330$		0			0
$331$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$331$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$332$	0.382683	+	0.662827i	0.382683	+	0.662827i
$333$		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	$0.833333\pi$
$338$	0.866025	−	0.500000i	0.866025	−	0.500000i
$339$	−1.40211	+	0.184592i	−1.40211	+	0.184592i
$340$		0			0
$341$		0			0
$342$	−0.184592	−	0.184592i	−0.184592	−	0.184592i
$343$		0			0
$344$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$345$		0			0
$346$		0			0
$347$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$347$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$348$		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
$352$	0.965926	−	1.67303i	0.965926	−	1.67303i
$353$	−0.130526	+	0.226078i	−0.130526	+	0.226078i	−0.923880	−	0.382683i	$-0.875000\pi$
$353$	−0.130526	+	0.226078i	−0.130526	+	0.226078i	0.793353	+	0.608761i	$0.208333\pi$
$354$	1.20711	+	1.57313i	1.20711	+	1.57313i
$355$		0			0
$356$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−0.465926	−	0.807007i	−0.465926	−	0.807007i
$362$		0			0
$363$	2.52409	+	1.04551i	2.52409	+	1.04551i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	−1.12197	−	1.12197i	−1.12197	−	1.12197i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	2.35207			2.35207
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$379$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	0.608761	+	0.793353i	0.608761	+	0.793353i
$385$		0			0
$386$		−	0.517638i		−	0.517638i
$387$	0.500000	−	0.133975i	0.500000	−	0.133975i
$388$	1.37413	−	0.793353i	1.37413	−	0.793353i
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$394$		0			0
$395$		0			0
$396$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$402$	−1.05441	−	1.37413i	−1.05441	−	1.37413i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.465926	+	1.12484i	−0.465926	+	1.12484i
$409$			1.58671i			1.58671i	0.608761	+	0.793353i	$0.291667\pi$
$409$			1.58671i			1.58671i	−0.608761	+	0.793353i	$0.708333\pi$
$410$		0			0
$411$	0.513210	−	0.0675653i	0.513210	−	0.0675653i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.96593	+	0.258819i	−1.96593	+	0.258819i
$418$	0.436749	−	0.252157i	0.436749	−	0.252157i
$419$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$419$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.21752			−1.21752
$426$		0			0
$427$		0			0
$428$	1.50000	+	0.866025i	1.50000	+	0.866025i
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$433$			0.261052i			0.261052i	0.991445	+	0.130526i	$0.0416667\pi$
$433$			0.261052i			0.261052i	−0.991445	+	0.130526i	$0.958333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.83195	+	0.758819i	1.83195	+	0.758819i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$443$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$449$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$450$	0.707107	−	0.707107i	0.707107	−	0.707107i
$451$	2.65461	−	1.53264i	2.65461	−	1.53264i
$452$	1.22474	+	0.707107i	1.22474	+	0.707107i
$453$		0			0
$454$	−1.37413	−	0.793353i	−1.37413	−	0.793353i
$455$		0			0
$456$	0.0340742	+	0.258819i	0.0340742	+	0.258819i
$457$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$457$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$458$		0			0
$459$	0.741181	−	0.965926i	0.741181	−	0.965926i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$	0.258819	−	0.448288i	0.258819	−	0.448288i
$467$	0.608761	−	1.05441i	0.608761	−	1.05441i	−0.382683	−	0.923880i	$-0.625000\pi$
$467$	0.608761	−	1.05441i	0.608761	−	1.05441i	0.991445	−	0.130526i	$-0.0416667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	1.98289i		−	1.98289i
$473$			1.00000i			1.00000i
$474$		0			0
$475$	−0.226078	+	0.130526i	−0.226078	+	0.130526i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	0.608761	−	1.05441i	0.608761	−	1.05441i
$483$		0			0
$484$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$485$		0			0
$486$	0.130526	+	0.991445i	0.130526	+	0.991445i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$	−0.184592	−	1.40211i	−0.184592	−	1.40211i
$490$		0			0
$491$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$491$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$492$	0.207107	+	1.57313i	0.207107	+	1.57313i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$499$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$499$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$500$		0			0
$501$		0			0
$502$		−	0.261052i		−	0.261052i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.923880	+	0.382683i	0.923880	+	0.382683i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$	0.0340742	−	0.258819i	0.0340742	−	0.258819i
$514$	1.71723	−	0.991445i	1.71723	−	0.991445i
$515$		0			0
$516$	−0.478235	−	0.198092i	−0.478235	−	0.198092i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.608761	+	0.793353i	$0.291667\pi$
$521$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.382683	+	0.923880i	$0.375000\pi$
$522$		0			0
$523$	0.662827	+	0.382683i	0.662827	+	0.382683i	0.793353	−	0.608761i	$-0.208333\pi$
$523$	0.662827	+	0.382683i	0.662827	+	0.382683i	−0.130526	+	0.991445i	$0.541667\pi$
$524$	0.923880	+	1.60021i	0.923880	+	1.60021i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.91532	−	0.252157i	1.91532	−	0.252157i
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		0			0
$531$	−0.513210	+	1.91532i	−0.513210	+	1.91532i
$532$		0			0
$533$		0			0
$534$	−1.83195	+	0.241181i	−1.83195	+	0.241181i
$535$		0			0
$536$			1.73205i			1.73205i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$	1.05441	−	0.608761i	1.05441	−	0.608761i
$545$		0			0
$546$		0			0
$547$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$547$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$548$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$549$		0			0
$550$	0.965926	+	1.67303i	0.965926	+	1.67303i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.71723	+	0.991445i	1.71723	+	0.991445i
$557$		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
$561$	1.43185	+	1.86603i	1.43185	+	1.86603i
$562$	0.707107	+	1.22474i	0.707107	+	1.22474i
$563$	0.261052			0.261052			0.130526	−	0.991445i	$-0.458333\pi$
$563$	0.261052			0.261052			0.130526	+	0.991445i	$0.458333\pi$
$564$		0			0
$565$		0			0
$566$	−1.84776			−1.84776
$567$		0			0
$568$		0			0
$569$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$570$		0			0
$571$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$571$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$577$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	−0.608761	−	0.793353i	$-0.708333\pi$
$577$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	0.382683	+	0.923880i	$0.375000\pi$
$578$	0.417738	+	0.241181i	0.417738	+	0.241181i
$579$	0.410670	−	0.315118i	0.410670	−	0.315118i
$580$		0			0
$581$		0			0
$582$	1.46593	+	0.607206i	1.46593	+	0.607206i
$583$		0			0
$584$	−0.991445	−	1.71723i	−0.991445	−	1.71723i
$585$		0			0
$586$		0			0
$587$	0.793353	+	1.37413i	0.793353	+	1.37413i	0.923880	+	0.382683i	$0.125000\pi$
$587$	0.793353	+	1.37413i	0.793353	+	1.37413i	−0.130526	+	0.991445i	$0.541667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$593$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$594$	−1.91532	−	0.252157i	−1.91532	−	0.252157i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$601$	0.226078	−	0.130526i	0.226078	−	0.130526i	−0.382683	−	0.923880i	$-0.625000\pi$
$601$	0.226078	−	0.130526i	0.226078	−	0.130526i	0.608761	+	0.793353i	$0.291667\pi$
$602$		0			0
$603$	0.448288	−	1.67303i	0.448288	−	1.67303i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$	0.130526	−	0.226078i	0.130526	−	0.226078i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.17604	+	0.315118i	−1.17604	+	0.315118i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	1.21752			1.21752
$615$		0			0
$616$		0			0
$617$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.923880	−	0.382683i	$-0.875000\pi$
$619$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.793353	−	0.608761i	$-0.791667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$	1.21752			1.21752
$627$	0.465926	+	0.192993i	0.465926	+	0.192993i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$641$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$642$	0.226078	+	1.71723i	0.226078	+	1.71723i
$643$	−1.05441	−	0.608761i	−1.05441	−	0.608761i	−0.130526	−	0.991445i	$-0.541667\pi$
$643$	−1.05441	−	0.608761i	−1.05441	−	0.608761i	−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$	0.317837			0.317837
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$	−3.31744	−	1.91532i	−3.31744	−	1.91532i
$650$		0			0
$651$		0			0
$652$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$	0.793353	−	1.37413i	0.793353	−	1.37413i
$657$	0.513210	+	1.91532i	0.513210	+	1.91532i
$658$		0			0
$659$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$659$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		−	1.41421i		−	1.41421i
$663$		0			0
$664$	0.662827	−	0.382683i	0.662827	−	0.382683i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$673$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$674$	1.50000	+	0.866025i	1.50000	+	0.866025i
$675$	0.991445	+	0.130526i	0.991445	+	0.130526i
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	0.184592	+	1.40211i	0.184592	+	1.40211i
$679$		0			0
$680$		0			0
$681$	−0.207107	−	1.57313i	−0.207107	−	1.57313i
$682$		0			0
$683$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$683$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$684$	−0.184592	+	0.184592i	−0.184592	+	0.184592i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.258819	+	0.448288i	0.258819	+	0.448288i
$689$		0			0
$690$		0			0
$691$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$691$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$692$		0			0
$693$		0			0
$694$	−0.517638			−0.517638
$695$		0			0
$696$		0			0
$697$	1.93185			1.93185
$698$		0			0
$699$	0.513210	−	0.0675653i	0.513210	−	0.0675653i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$705$		0			0
$706$	0.226078	+	0.130526i	0.226078	+	0.130526i
$707$		0			0
$708$	1.57313	−	1.20711i	1.57313	−	1.20711i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	1.60021	+	0.923880i	1.60021	+	0.923880i
$713$		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
$722$	−0.807007	+	0.465926i	−0.807007	+	0.465926i
$723$	1.20711	−	0.158919i	1.20711	−	0.158919i
$724$		0			0
$725$		0			0
$726$	1.04551	−	2.52409i	1.04551	−	2.52409i
$727$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$727$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$730$		0			0
$731$	−0.315118	+	0.545801i	−0.315118	+	0.545801i
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.89778	+	1.67303i	2.89778	+	1.67303i
$738$	−1.12197	+	1.12197i	−1.12197	+	1.12197i
$739$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$739$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.739288	+	0.198092i	−0.739288	+	0.198092i
$748$		−	2.35207i		−	2.35207i
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	0.207107	−	0.158919i	0.207107	−	0.158919i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$			1.93185i			1.93185i
$759$		0			0
$760$		0			0
$761$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$761$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.793353	−	0.608761i	0.793353	−	0.608761i
$769$	0.662827	+	0.382683i	0.662827	+	0.382683i	0.793353	−	0.608761i	$-0.208333\pi$
$769$	0.662827	+	0.382683i	0.662827	+	0.382683i	−0.130526	+	0.991445i	$0.541667\pi$
$770$		0			0
$771$	1.83195	+	0.758819i	1.83195	+	0.758819i
$772$	−0.517638			−0.517638
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$775$		0			0
$776$	−0.793353	−	1.37413i	−0.793353	−	1.37413i
$777$		0			0
$778$		0			0
$779$	0.358719	−	0.207107i	0.358719	−	0.207107i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$787$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$787$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.36603	+	1.36603i	1.36603	+	1.36603i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	0.866025	+	0.500000i	0.866025	+	0.500000i
$801$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$802$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$803$	−3.83065			−3.83065
$804$	−1.37413	+	1.05441i	−1.37413	+	1.05441i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$809$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$810$		0			0
$811$			1.58671i			1.58671i	0.608761	+	0.793353i	$0.291667\pi$
$811$			1.58671i			1.58671i	−0.608761	+	0.793353i	$0.708333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	1.12484	+	0.465926i	1.12484	+	0.465926i
$817$			0.135131i			0.135131i
$818$	1.58671			1.58671
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	−0.0675653	−	0.513210i	−0.0675653	−	0.513210i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−0.739288	+	1.78480i	−0.739288	+	1.78480i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$	0.258819	+	1.96593i	0.258819	+	1.96593i
$835$		0			0
$836$	−0.252157	−	0.436749i	−0.252157	−	0.436749i
$837$		0			0
$838$	−0.662827	−	0.382683i	−0.662827	−	0.382683i
$839$		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
$843$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.12484	−	1.46593i	−1.12484	−	1.46593i
$850$			1.21752i			1.21752i
$851$		0			0
$852$		0			0
$853$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$853$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$854$		0			0
$855$		0			0
$856$	0.866025	−	1.50000i	0.866025	−	1.50000i
$857$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$857$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$858$		0			0
$859$	1.37413	−	0.793353i	1.37413	−	0.793353i	0.382683	−	0.923880i	$-0.375000\pi$
$859$	1.37413	−	0.793353i	1.37413	−	0.793353i	0.991445	+	0.130526i	$0.0416667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$865$		0			0
$866$	0.261052			0.261052
$867$	0.0629609	+	0.478235i	0.0629609	+	0.478235i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.410670	+	1.53264i	0.410670	+	1.53264i
$874$		0			0
$875$		0			0
$876$	0.758819	−	1.83195i	0.758819	−	1.83195i
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$881$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$882$		0			0
$883$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$		0			0
$886$	1.00000			1.00000
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.00000			−1.00000
$899$		0			0
$900$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$901$		0			0
$902$	−1.53264	−	2.65461i	−1.53264	−	2.65461i
$903$		0			0
$904$	0.707107	−	1.22474i	0.707107	−	1.22474i
$905$		0			0
$906$		0			0
$907$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$908$	−0.793353	+	1.37413i	−0.793353	+	1.37413i
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$	0.258819	−	0.0340742i	0.258819	−	0.0340742i
$913$		−	1.47858i		−	1.47858i
$914$			1.93185i			1.93185i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.965926	−	0.741181i	−0.965926	−	0.741181i
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	0.741181	+	0.965926i	0.741181	+	0.965926i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$929$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$930$		0			0
$931$		0			0
$932$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$933$		0			0
$934$	−1.05441	−	0.608761i	−1.05441	−	0.608761i
$935$		0			0
$936$		0			0
$937$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$937$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$938$		0			0
$939$	0.741181	+	0.965926i	0.741181	+	0.965926i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	−1.98289			−1.98289
$945$		0			0
$946$	1.00000			1.00000
$947$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$947$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$948$		0			0
$949$		0			0
$950$	0.130526	+	0.226078i	0.130526	+	0.226078i
$951$		0			0
$952$		0			0
$953$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$953$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.00000			−1.00000
$962$		0			0
$963$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$964$	−1.05441	−	0.608761i	−1.05441	−	0.608761i
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$969$	0.193487	+	0.252157i	0.193487	+	0.252157i
$970$		0			0
$971$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$971$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$972$	0.991445	−	0.130526i	0.991445	−	0.130526i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$977$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$978$	−1.40211	+	0.184592i	−1.40211	+	0.184592i
$979$	3.09136	−	1.78480i	3.09136	−	1.78480i
$980$		0			0
$981$		0			0
$982$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$	1.57313	−	0.207107i	1.57313	−	0.207107i
$985$		0			0
$986$		0			0
$987$		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
$993$	1.12197	−	0.860919i	1.12197	−	0.860919i
$994$		0			0
$995$		0			0
$996$	0.707107	+	0.292893i	0.707107	+	0.292893i
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.1.cm.a.3155.4		16
7.2	even	3		3528.1.co.a.2939.3	yes	16
7.3	odd	6		3528.1.u.a.2579.8		16
7.4	even	3		3528.1.u.a.2579.5		16
7.5	odd	6		3528.1.co.a.2939.2	yes	16
7.6	odd	2	inner	3528.1.cm.a.3155.1		16
8.3	odd	2	CM	3528.1.cm.a.3155.4		16
9.2	odd	6		3528.1.u.a.803.8		16
56.3	even	6		3528.1.u.a.2579.8		16
56.11	odd	6		3528.1.u.a.2579.5		16
56.19	even	6		3528.1.co.a.2939.2	yes	16
56.27	even	2	inner	3528.1.cm.a.3155.1		16
56.51	odd	6		3528.1.co.a.2939.3	yes	16
63.2	odd	6		3528.1.co.a.587.2	✓	16
63.11	odd	6	inner	3528.1.cm.a.227.5		16
63.20	even	6		3528.1.u.a.803.5		16
63.38	even	6	inner	3528.1.cm.a.227.8		16
63.47	even	6		3528.1.co.a.587.3	yes	16
72.11	even	6		3528.1.u.a.803.8		16
504.11	even	6	inner	3528.1.cm.a.227.5		16
504.83	odd	6		3528.1.u.a.803.5		16
504.227	odd	6	inner	3528.1.cm.a.227.8		16
504.299	odd	6		3528.1.co.a.587.3	yes	16
504.443	even	6		3528.1.co.a.587.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.1.u.a.803.5		16	63.20	even	6
3528.1.u.a.803.5		16	504.83	odd	6
3528.1.u.a.803.8		16	9.2	odd	6
3528.1.u.a.803.8		16	72.11	even	6
3528.1.u.a.2579.5		16	7.4	even	3
3528.1.u.a.2579.5		16	56.11	odd	6
3528.1.u.a.2579.8		16	7.3	odd	6
3528.1.u.a.2579.8		16	56.3	even	6
3528.1.cm.a.227.5		16	63.11	odd	6	inner
3528.1.cm.a.227.5		16	504.11	even	6	inner
3528.1.cm.a.227.8		16	63.38	even	6	inner
3528.1.cm.a.227.8		16	504.227	odd	6	inner
3528.1.cm.a.3155.1		16	7.6	odd	2	inner
3528.1.cm.a.3155.1		16	56.27	even	2	inner
3528.1.cm.a.3155.4		16	1.1	even	1	trivial
3528.1.cm.a.3155.4		16	8.3	odd	2	CM
3528.1.co.a.587.2	✓	16	63.2	odd	6
3528.1.co.a.587.2	✓	16	504.443	even	6
3528.1.co.a.587.3	yes	16	63.47	even	6
3528.1.co.a.587.3	yes	16	504.299	odd	6
3528.1.co.a.2939.2	yes	16	7.5	odd	6
3528.1.co.a.2939.2	yes	16	56.19	even	6
3528.1.co.a.2939.3	yes	16	7.2	even	3
3528.1.co.a.2939.3	yes	16	56.51	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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3528.cm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(0.793353 - 0.608761i) q^{3} -1.00000 q^{4} +(-0.608761 - 0.793353i) q^{6} +1.00000i q^{8} +(0.258819 - 0.965926i) q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +(0.793353 - 0.608761i) q^{3} -1.00000 q^{4} +(-0.608761 - 0.793353i) q^{6} +1.00000i q^{8} +(0.258819 - 0.965926i) q^{9} +(1.67303 + 0.965926i) q^{11} +(-0.793353 + 0.608761i) q^{12} +1.00000 q^{16} +(0.608761 + 1.05441i) q^{17} +(-0.965926 - 0.258819i) q^{18} +(0.226078 + 0.130526i) q^{19} +(0.965926 - 1.67303i) q^{22} +(0.608761 + 0.793353i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.382683 - 0.923880i) q^{27} -1.00000i q^{32} +(1.91532 - 0.252157i) q^{33} +(1.05441 - 0.608761i) q^{34} +(-0.258819 + 0.965926i) q^{36} +(0.130526 - 0.226078i) q^{38} +(0.793353 - 1.37413i) q^{41} +(0.258819 + 0.448288i) q^{43} +(-1.67303 - 0.965926i) q^{44} +(0.793353 - 0.608761i) q^{48} +(0.866025 + 0.500000i) q^{50} +(1.12484 + 0.465926i) q^{51} +(-0.923880 + 0.382683i) q^{54} +(0.258819 - 0.0340742i) q^{57} -1.98289 q^{59} -1.00000 q^{64} +(-0.252157 - 1.91532i) q^{66} +1.73205 q^{67} +(-0.608761 - 1.05441i) q^{68} +(0.965926 + 0.258819i) q^{72} +(-1.71723 + 0.991445i) q^{73} +(0.130526 + 0.991445i) q^{75} +(-0.226078 - 0.130526i) q^{76} +(-0.866025 - 0.500000i) q^{81} +(-1.37413 - 0.793353i) q^{82} +(-0.382683 - 0.662827i) q^{83} +(0.448288 - 0.258819i) q^{86} +(-0.965926 + 1.67303i) q^{88} +(0.923880 - 1.60021i) q^{89} +(-0.608761 - 0.793353i) q^{96} +(-1.37413 + 0.793353i) q^{97} +(1.36603 - 1.36603i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 8 q^{25} - 16 q^{64} + 8 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 8 * q^25 - 16 * q^64 + 8 * q^99

Introduction

L-functions

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