LMFDB - Embedded newform 728.1.l.e.181.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [728,1,Mod(181,728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("728.181"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 728 = 2^{3} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	728.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.363319329197$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.75712.1

Embedding invariants

Embedding label			181.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	728.181
Dual form			728.1.l.e.181.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.41421 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.41421 q^{10} +1.41421 q^{12} +(-0.707107 + 0.707107i) q^{13} -1.00000 q^{14} -2.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -1.41421i q^{19} -1.41421i q^{20} -1.41421i q^{21} -2.00000 q^{23} +1.41421i q^{24} -1.00000 q^{25} +(-0.707107 - 0.707107i) q^{26} -1.00000i q^{28} +2.00000 q^{30} +1.00000i q^{32} -1.41421 q^{35} -1.00000 q^{36} +1.41421 q^{38} +(1.00000 - 1.00000i) q^{39} +1.41421 q^{40} +1.41421 q^{42} +1.41421i q^{45} -2.00000i q^{46} -1.41421 q^{48} -1.00000 q^{49} -1.00000i q^{50} +(0.707107 - 0.707107i) q^{52} +1.00000 q^{56} +2.00000i q^{57} +1.41421i q^{59} +2.00000i q^{60} +1.41421 q^{61} +1.00000i q^{63} -1.00000 q^{64} +(-1.00000 - 1.00000i) q^{65} +2.82843 q^{69} -1.41421i q^{70} +2.00000i q^{71} -1.00000i q^{72} +1.41421 q^{75} +1.41421i q^{76} +(1.00000 + 1.00000i) q^{78} +1.41421i q^{80} -1.00000 q^{81} +1.41421i q^{83} +1.41421i q^{84} -1.41421 q^{90} +(-0.707107 - 0.707107i) q^{91} +2.00000 q^{92} +2.00000 q^{95} -1.41421i q^{96} -1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{9} - 4 q^{14} + 4 q^{16} - 8 q^{23} - 4 q^{25} + 8 q^{30} - 4 q^{36} + 4 q^{39} - 4 q^{49} + 4 q^{56} - 4 q^{64} - 4 q^{65} + 4 q^{78} - 4 q^{81} + 8 q^{92} + 8 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^9 - 4 * q^14 + 4 * q^16 - 8 * q^23 - 4 * q^25 + 8 * q^30 - 4 * q^36 + 4 * q^39 - 4 * q^49 + 4 * q^56 - 4 * q^64 - 4 * q^65 + 4 * q^78 - 4 * q^81 + 8 * q^92 + 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/728\mathbb{Z}\right)^\times$.

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$4$	−1.00000			−1.00000
$5$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$5$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$6$		−	1.41421i		−	1.41421i
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$		−	1.00000i		−	1.00000i
$9$	1.00000			1.00000
$10$	−1.41421			−1.41421
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	1.41421			1.41421
$13$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$13$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$14$	−1.00000			−1.00000
$15$		−	2.00000i		−	2.00000i
$16$	1.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$			1.00000i			1.00000i
$19$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$19$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$20$		−	1.41421i		−	1.41421i
$21$		−	1.41421i		−	1.41421i
$22$		0			0
$23$	−2.00000			−2.00000			−1.00000			$\pi$
$23$	−2.00000			−2.00000			−1.00000			$\pi$
$24$			1.41421i			1.41421i
$25$	−1.00000			−1.00000
$26$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$27$		0			0
$28$		−	1.00000i		−	1.00000i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	2.00000			2.00000
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$			1.00000i			1.00000i
$33$		0			0
$34$		0			0
$35$	−1.41421			−1.41421
$36$	−1.00000			−1.00000
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	1.41421			1.41421
$39$	1.00000	−	1.00000i	1.00000	−	1.00000i
$40$	1.41421			1.41421
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	1.41421			1.41421
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$			1.41421i			1.41421i
$46$		−	2.00000i		−	2.00000i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−1.41421			−1.41421
$49$	−1.00000			−1.00000
$50$		−	1.00000i		−	1.00000i
$51$		0			0
$52$	0.707107	−	0.707107i	0.707107	−	0.707107i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$	1.00000			1.00000
$57$			2.00000i			2.00000i
$58$		0			0
$59$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$59$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$60$			2.00000i			2.00000i
$61$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$61$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$62$		0			0
$63$			1.00000i			1.00000i
$64$	−1.00000			−1.00000
$65$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	2.82843			2.82843
$70$		−	1.41421i		−	1.41421i
$71$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$71$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$72$		−	1.00000i		−	1.00000i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	1.41421			1.41421
$76$			1.41421i			1.41421i
$77$		0			0
$78$	1.00000	+	1.00000i	1.00000	+	1.00000i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$			1.41421i			1.41421i
$81$	−1.00000			−1.00000
$82$		0			0
$83$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$83$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$84$			1.41421i			1.41421i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−1.41421			−1.41421
$91$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$92$	2.00000			2.00000
$93$		0			0
$94$		0			0
$95$	2.00000			2.00000
$96$		−	1.41421i		−	1.41421i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		−	1.00000i		−	1.00000i
$99$		0			0
$100$	1.00000			1.00000
$101$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$101$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	0.707107	+	0.707107i	0.707107	+	0.707107i
$105$	2.00000			2.00000
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$			1.00000i			1.00000i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$	−2.00000			−2.00000
$115$		−	2.82843i		−	2.82843i
$116$		0			0
$117$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$118$	−1.41421			−1.41421
$119$		0			0
$120$	−2.00000			−2.00000
$121$	−1.00000			−1.00000
$122$			1.41421i			1.41421i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−1.00000			−1.00000
$127$	2.00000			2.00000			1.00000			$0$
$127$	2.00000			2.00000			1.00000			$0$
$128$		−	1.00000i		−	1.00000i
$129$		0			0
$130$	1.00000	−	1.00000i	1.00000	−	1.00000i
$131$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$131$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$	1.41421			1.41421
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$			2.82843i			2.82843i
$139$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$139$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$140$	1.41421			1.41421
$141$		0			0
$142$	−2.00000			−2.00000
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$	1.41421			1.41421
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$			1.41421i			1.41421i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−1.41421			−1.41421
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$157$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$157$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	−1.41421			−1.41421
$161$		−	2.00000i		−	2.00000i
$162$		−	1.00000i		−	1.00000i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−1.41421			−1.41421
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−1.41421			−1.41421
$169$		−	1.00000i		−	1.00000i
$170$		0			0
$171$		−	1.41421i		−	1.41421i
$172$		0			0
$173$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$173$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		−	1.00000i		−	1.00000i
$176$		0			0
$177$		−	2.00000i		−	2.00000i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		−	1.41421i		−	1.41421i
$181$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$181$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$182$	0.707107	−	0.707107i	0.707107	−	0.707107i
$183$	−2.00000			−2.00000
$184$			2.00000i			2.00000i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$			2.00000i			2.00000i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	1.41421			1.41421
$193$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$193$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$194$		0			0
$195$	1.41421	+	1.41421i	1.41421	+	1.41421i
$196$	1.00000			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			1.00000i			1.00000i
$201$		0			0
$202$		−	1.41421i		−	1.41421i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.00000			−2.00000
$208$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$209$		0			0
$210$			2.00000i			2.00000i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		−	2.82843i		−	2.82843i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−1.00000			−1.00000
$225$	−1.00000			−1.00000
$226$		0			0
$227$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$227$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$228$		−	2.00000i		−	2.00000i
$229$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$229$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$230$	2.82843			2.82843
$231$		0			0
$232$		0			0
$233$	2.00000			2.00000			1.00000			$0$
$233$	2.00000			2.00000			1.00000			$0$
$234$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$235$		0			0
$236$		−	1.41421i		−	1.41421i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		−	2.00000i		−	2.00000i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		−	1.00000i		−	1.00000i
$243$	1.41421			1.41421
$244$	−1.41421			−1.41421
$245$		−	1.41421i		−	1.41421i
$246$		0			0
$247$	1.00000	+	1.00000i	1.00000	+	1.00000i
$248$		0			0
$249$		−	2.00000i		−	2.00000i
$250$		0			0
$251$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$251$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$252$		−	1.00000i		−	1.00000i
$253$		0			0
$254$			2.00000i			2.00000i
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	1.00000	+	1.00000i	1.00000	+	1.00000i
$261$		0			0
$262$			1.41421i			1.41421i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$			1.41421i			1.41421i
$267$		0			0
$268$		0			0
$269$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$269$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$	1.00000	+	1.00000i	1.00000	+	1.00000i
$274$		0			0
$275$		0			0
$276$	−2.82843			−2.82843
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		−	1.41421i		−	1.41421i
$279$		0			0
$280$			1.41421i			1.41421i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$283$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$284$		−	2.00000i		−	2.00000i
$285$	−2.82843			−2.82843
$286$		0			0
$287$		0			0
$288$			1.00000i			1.00000i
$289$	1.00000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$293$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$294$			1.41421i			1.41421i
$295$	−2.00000			−2.00000
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.41421	−	1.41421i	1.41421	−	1.41421i
$300$	−1.41421			−1.41421
$301$		0			0
$302$		0			0
$303$	2.00000			2.00000
$304$		−	1.41421i		−	1.41421i
$305$			2.00000i			2.00000i
$306$		0			0
$307$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$307$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		−	1.41421i		−	1.41421i
$315$	−1.41421			−1.41421
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		−	1.41421i		−	1.41421i
$321$		0			0
$322$	2.00000			2.00000
$323$		0			0
$324$	1.00000			1.00000
$325$	0.707107	−	0.707107i	0.707107	−	0.707107i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		−	1.41421i		−	1.41421i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		−	1.41421i		−	1.41421i
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	1.00000			1.00000
$339$		0			0
$340$		0			0
$341$		0			0
$342$	1.41421			1.41421
$343$		−	1.00000i		−	1.00000i
$344$		0			0
$345$			4.00000i			4.00000i
$346$		−	1.41421i		−	1.41421i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$349$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$350$	1.00000			1.00000
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	2.00000			2.00000
$355$	−2.82843			−2.82843
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	1.41421			1.41421
$361$	−1.00000			−1.00000
$362$			1.41421i			1.41421i
$363$	1.41421			1.41421
$364$	0.707107	+	0.707107i	0.707107	+	0.707107i
$365$		0			0
$366$		−	2.00000i		−	2.00000i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−2.00000			−2.00000
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−2.00000			−2.00000
$381$	−2.82843			−2.82843
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$			1.41421i			1.41421i
$385$		0			0
$386$	−2.00000			−2.00000
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$391$		0			0
$392$			1.00000i			1.00000i
$393$	−2.00000			−2.00000
$394$		0			0
$395$		0			0
$396$		0			0
$397$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$397$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$	−2.00000			−2.00000
$400$	−1.00000			−1.00000
$401$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$401$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$	1.41421			1.41421
$405$		−	1.41421i		−	1.41421i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.41421			−1.41421
$414$		−	2.00000i		−	2.00000i
$415$	−2.00000			−2.00000
$416$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$417$	2.00000			2.00000
$418$		0			0
$419$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$419$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$420$	−2.00000			−2.00000
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	2.82843			2.82843
$427$			1.41421i			1.41421i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2.82843i			2.82843i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		−	1.00000i		−	1.00000i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−1.41421			−1.41421
$455$	1.00000	−	1.00000i	1.00000	−	1.00000i
$456$	2.00000			2.00000
$457$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$457$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$458$	1.41421			1.41421
$459$		0			0
$460$			2.82843i			2.82843i
$461$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$461$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$462$		0			0
$463$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$463$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$464$		0			0
$465$		0			0
$466$			2.00000i			2.00000i
$467$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$467$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$468$	0.707107	−	0.707107i	0.707107	−	0.707107i
$469$		0			0
$470$		0			0
$471$	2.00000			2.00000
$472$	1.41421			1.41421
$473$		0			0
$474$		0			0
$475$			1.41421i			1.41421i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	2.00000			2.00000
$481$		0			0
$482$		0			0
$483$			2.82843i			2.82843i
$484$	1.00000			1.00000
$485$		0			0
$486$			1.41421i			1.41421i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		−	1.41421i		−	1.41421i
$489$		0			0
$490$	1.41421			1.41421
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$495$		0			0
$496$		0			0
$497$	−2.00000			−2.00000
$498$	2.00000			2.00000
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		−	1.41421i		−	1.41421i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	1.00000			1.00000
$505$		−	2.00000i		−	2.00000i
$506$		0			0
$507$			1.41421i			1.41421i
$508$	−2.00000			−2.00000
$509$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$509$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.00000			2.00000
$520$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$523$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$524$	−1.41421			−1.41421
$525$			1.41421i			1.41421i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.00000			3.00000
$530$		0			0
$531$			1.41421i			1.41421i
$532$	−1.41421			−1.41421
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			1.41421i			1.41421i
$539$		0			0
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	−2.00000			−2.00000
$544$		0			0
$545$		0			0
$546$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	1.41421			1.41421
$550$		0			0
$551$		0			0
$552$		−	2.82843i		−	2.82843i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.41421			1.41421
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−1.41421			−1.41421
$561$		0			0
$562$		0			0
$563$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$563$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$			1.41421i			1.41421i
$567$		−	1.00000i		−	1.00000i
$568$	2.00000			2.00000
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		−	2.82843i		−	2.82843i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.00000			2.00000
$576$	−1.00000			−1.00000
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$			1.00000i			1.00000i
$579$		−	2.82843i		−	2.82843i
$580$		0			0
$581$	−1.41421			−1.41421
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$586$	1.41421			1.41421
$587$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$587$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$588$	−1.41421			−1.41421
$589$		0			0
$590$		−	2.00000i		−	2.00000i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.41421	+	1.41421i	1.41421	+	1.41421i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		−	1.41421i		−	1.41421i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	1.41421i		−	1.41421i
$606$			2.00000i			2.00000i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	1.41421			1.41421
$609$		0			0
$610$	−2.00000			−2.00000
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−1.41421			−1.41421
$615$		0			0
$616$		0			0
$617$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$617$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$618$		0			0
$619$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$619$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.00000	−	1.00000i	1.00000	−	1.00000i
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$	1.41421			1.41421
$629$		0			0
$630$		−	1.41421i		−	1.41421i
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			2.82843i			2.82843i
$636$		0			0
$637$	0.707107	−	0.707107i	0.707107	−	0.707107i
$638$		0			0
$639$			2.00000i			2.00000i
$640$	1.41421			1.41421
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$643$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$644$			2.00000i			2.00000i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$			1.00000i			1.00000i
$649$		0			0
$650$	0.707107	+	0.707107i	0.707107	+	0.707107i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$			2.00000i			2.00000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$661$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$662$		0			0
$663$		0			0
$664$	1.41421			1.41421
$665$			2.00000i			2.00000i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	1.41421			1.41421
$673$	2.00000			2.00000			1.00000			$0$
$673$	2.00000			2.00000			1.00000			$0$
$674$		0			0
$675$		0			0
$676$			1.00000i			1.00000i
$677$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$677$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	2.00000i		−	2.00000i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$			1.41421i			1.41421i
$685$		0			0
$686$	1.00000			1.00000
$687$			2.00000i			2.00000i
$688$		0			0
$689$		0			0
$690$	−4.00000			−4.00000
$691$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$691$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$692$	1.41421			1.41421
$693$		0			0
$694$		0			0
$695$		−	2.00000i		−	2.00000i
$696$		0			0
$697$		0			0
$698$	−1.41421			−1.41421
$699$	−2.82843			−2.82843
$700$			1.00000i			1.00000i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1.41421i		−	1.41421i
$708$			2.00000i			2.00000i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		−	2.82843i		−	2.82843i
$711$		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			$\pi$
$720$			1.41421i			1.41421i
$721$		0			0
$722$		−	1.00000i		−	1.00000i
$723$		0			0
$724$	−1.41421			−1.41421
$725$		0			0
$726$			1.41421i			1.41421i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$	2.00000			2.00000
$733$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$733$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$			2.00000i			2.00000i
$736$		−	2.00000i		−	2.00000i
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			1.41421i			1.41421i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.00000			2.00000			1.00000			$0$
$751$	2.00000			2.00000			1.00000			$0$
$752$		0			0
$753$	2.00000			2.00000
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		−	2.00000i		−	2.00000i
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		−	2.82843i		−	2.82843i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$768$	−1.41421			−1.41421
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		−	2.00000i		−	2.00000i
$773$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$773$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$		−	2.00000i		−	2.00000i
$786$		−	2.00000i		−	2.00000i
$787$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$787$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$794$	−1.41421			−1.41421
$795$		0			0
$796$		0			0
$797$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$797$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$798$		−	2.00000i		−	2.00000i
$799$		0			0
$800$		−	1.00000i		−	1.00000i
$801$		0			0
$802$	−2.00000			−2.00000
$803$		0			0
$804$		0			0
$805$	2.82843			2.82843
$806$		0			0
$807$	−2.00000			−2.00000
$808$			1.41421i			1.41421i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	1.41421			1.41421
$811$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$811$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		−	1.41421i		−	1.41421i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	2.00000			2.00000
$829$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$829$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$830$		−	2.00000i		−	2.00000i
$831$		0			0
$832$	0.707107	−	0.707107i	0.707107	−	0.707107i
$833$		0			0
$834$			2.00000i			2.00000i
$835$		0			0
$836$		0			0
$837$		0			0
$838$			1.41421i			1.41421i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		−	2.00000i		−	2.00000i
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.41421			1.41421
$846$		0			0
$847$		−	1.00000i		−	1.00000i
$848$		0			0
$849$	−2.00000			−2.00000
$850$		0			0
$851$		0			0
$852$			2.82843i			2.82843i
$853$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$853$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$854$	−1.41421			−1.41421
$855$	2.00000			2.00000
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$859$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$864$		0			0
$865$		−	2.00000i		−	2.00000i
$866$		0			0
$867$	−1.41421			−1.41421
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−2.82843			−2.82843
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$			2.00000i			2.00000i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		−	1.00000i		−	1.00000i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	2.82843			2.82843
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$			2.00000i			2.00000i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	1.00000			1.00000
$897$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$898$		0			0
$899$		0			0
$900$	1.00000			1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2.00000i			2.00000i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		−	1.41421i		−	1.41421i
$909$	−1.41421			−1.41421
$910$	1.00000	+	1.00000i	1.00000	+	1.00000i
$911$	−2.00000			−2.00000			−1.00000			$\pi$
$911$	−2.00000			−2.00000			−1.00000			$\pi$
$912$			2.00000i			2.00000i
$913$		0			0
$914$	2.00000			2.00000
$915$		−	2.82843i		−	2.82843i
$916$			1.41421i			1.41421i
$917$			1.41421i			1.41421i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−2.82843			−2.82843
$921$		−	2.00000i		−	2.00000i
$922$	1.41421			1.41421
$923$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$924$		0			0
$925$		0			0
$926$	2.00000			2.00000
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$			1.41421i			1.41421i
$932$	−2.00000			−2.00000
$933$		0			0
$934$		−	1.41421i		−	1.41421i
$935$		0			0
$936$	0.707107	+	0.707107i	0.707107	+	0.707107i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$941$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$942$			2.00000i			2.00000i
$943$		0			0
$944$			1.41421i			1.41421i
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$	−1.41421			−1.41421
$951$		0			0
$952$		0			0
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$			2.00000i			2.00000i
$961$	−1.00000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.82843			−2.82843
$966$	−2.82843			−2.82843
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$			1.00000i			1.00000i
$969$		0			0
$970$		0			0
$971$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$971$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$972$	−1.41421			−1.41421
$973$		−	1.41421i		−	1.41421i
$974$		0			0
$975$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$976$	1.41421			1.41421
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$			1.41421i			1.41421i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		−	2.00000i		−	2.00000i
$995$		0			0
$996$			2.00000i			2.00000i
$997$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$997$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	728.1.l.e.181.3	yes	4
4.3	odd	2		2912.1.l.e.2001.4		4
7.6	odd	2	inner	728.1.l.e.181.4	yes	4
8.3	odd	2		2912.1.l.e.2001.1		4
8.5	even	2	inner	728.1.l.e.181.4	yes	4
13.12	even	2	inner	728.1.l.e.181.1	✓	4
28.27	even	2		2912.1.l.e.2001.1		4
52.51	odd	2		2912.1.l.e.2001.3		4
56.13	odd	2	CM	728.1.l.e.181.3	yes	4
56.27	even	2		2912.1.l.e.2001.4		4
91.90	odd	2	inner	728.1.l.e.181.2	yes	4
104.51	odd	2		2912.1.l.e.2001.2		4
104.77	even	2	inner	728.1.l.e.181.2	yes	4
364.363	even	2		2912.1.l.e.2001.2		4
728.181	odd	2	inner	728.1.l.e.181.1	✓	4
728.363	even	2		2912.1.l.e.2001.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.1.l.e.181.1	✓	4	13.12	even	2	inner
728.1.l.e.181.1	✓	4	728.181	odd	2	inner
728.1.l.e.181.2	yes	4	91.90	odd	2	inner
728.1.l.e.181.2	yes	4	104.77	even	2	inner
728.1.l.e.181.3	yes	4	1.1	even	1	trivial
728.1.l.e.181.3	yes	4	56.13	odd	2	CM
728.1.l.e.181.4	yes	4	7.6	odd	2	inner
728.1.l.e.181.4	yes	4	8.5	even	2	inner
2912.1.l.e.2001.1		4	8.3	odd	2
2912.1.l.e.2001.1		4	28.27	even	2
2912.1.l.e.2001.2		4	104.51	odd	2
2912.1.l.e.2001.2		4	364.363	even	2
2912.1.l.e.2001.3		4	52.51	odd	2
2912.1.l.e.2001.3		4	728.363	even	2
2912.1.l.e.2001.4		4	4.3	odd	2
2912.1.l.e.2001.4		4	56.27	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	728.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.41421 q^{3} -1.00000 q^{4} +1.41421i q^{5} -1.41421i q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.41421 q^{10} +1.41421 q^{12} +(-0.707107 + 0.707107i) q^{13} -1.00000 q^{14} -2.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -1.41421i q^{19} -1.41421i q^{20} -1.41421i q^{21} -2.00000 q^{23} +1.41421i q^{24} -1.00000 q^{25} +(-0.707107 - 0.707107i) q^{26} -1.00000i q^{28} +2.00000 q^{30} +1.00000i q^{32} -1.41421 q^{35} -1.00000 q^{36} +1.41421 q^{38} +(1.00000 - 1.00000i) q^{39} +1.41421 q^{40} +1.41421 q^{42} +1.41421i q^{45} -2.00000i q^{46} -1.41421 q^{48} -1.00000 q^{49} -1.00000i q^{50} +(0.707107 - 0.707107i) q^{52} +1.00000 q^{56} +2.00000i q^{57} +1.41421i q^{59} +2.00000i q^{60} +1.41421 q^{61} +1.00000i q^{63} -1.00000 q^{64} +(-1.00000 - 1.00000i) q^{65} +2.82843 q^{69} -1.41421i q^{70} +2.00000i q^{71} -1.00000i q^{72} +1.41421 q^{75} +1.41421i q^{76} +(1.00000 + 1.00000i) q^{78} +1.41421i q^{80} -1.00000 q^{81} +1.41421i q^{83} +1.41421i q^{84} -1.41421 q^{90} +(-0.707107 - 0.707107i) q^{91} +2.00000 q^{92} +2.00000 q^{95} -1.41421i q^{96} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{9} - 4 q^{14} + 4 q^{16} - 8 q^{23} - 4 q^{25} + 8 q^{30} - 4 q^{36} + 4 q^{39} - 4 q^{49} + 4 q^{56} - 4 q^{64} - 4 q^{65} + 4 q^{78} - 4 q^{81} + 8 q^{92} + 8 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^9 - 4 * q^14 + 4 * q^16 - 8 * q^23 - 4 * q^25 + 8 * q^30 - 4 * q^36 + 4 * q^39 - 4 * q^49 + 4 * q^56 - 4 * q^64 - 4 * q^65 + 4 * q^78 - 4 * q^81 + 8 * q^92 + 8 * q^95

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