LMFDB - Embedded newform 1767.1.eh.a.737.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1767,1,Mod(86,1767)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1767.86"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1767, base_ring=CyclotomicField(90)) chi = DirichletCharacter(H, H._module([45, 85, 39])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1767 = 3 \cdot 19 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1767.eh (of order $90$, degree $24$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.881847877323$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{45})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{90}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{90} - \cdots)$

Embedding invariants

Embedding label			737.1
Root			$-0.241922 + 0.970296i$ of defining polynomial
Character	$\chi$	$=$	1767.737
Dual form			1767.1.eh.a.983.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.374607 + 0.927184i) q^{3} +(0.997564 + 0.0697565i) q^{4} +(-1.42864 + 1.03797i) q^{7} +(-0.719340 + 0.694658i) q^{9} +(0.309017 + 0.951057i) q^{12} +(-0.743520 + 0.464603i) q^{13} +(0.990268 + 0.139173i) q^{16} +(0.913545 - 0.406737i) q^{19} +(-1.49756 - 0.935782i) q^{21} +(-0.939693 + 0.342020i) q^{25} +(-0.913545 - 0.406737i) q^{27} +(-1.49756 + 0.935782i) q^{28} +(0.882948 + 0.469472i) q^{31} +(-0.766044 + 0.642788i) q^{36} +(0.374607 + 0.648838i) q^{37} +(-0.709299 - 0.515336i) q^{39} +(-0.278177 - 0.00971414i) q^{43} +(0.241922 + 0.970296i) q^{48} +(0.654617 - 2.01470i) q^{49} +(-0.774117 + 0.411606i) q^{52} +(0.719340 + 0.694658i) q^{57} +(0.0896772 - 0.106873i) q^{61} +(0.306644 - 1.73907i) q^{63} +(0.978148 + 0.207912i) q^{64} +(1.36821 + 0.241252i) q^{67} +(-0.258808 - 0.902570i) q^{73} +(-0.669131 - 0.743145i) q^{75} +(0.939693 - 0.342020i) q^{76} +(1.43518 - 0.100357i) q^{79} +(0.0348995 - 0.999391i) q^{81} +(-1.42864 - 1.03797i) q^{84} +(0.579979 - 1.43550i) q^{91} +(-0.104528 + 0.994522i) q^{93} +(-0.548255 - 1.91199i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{12} - 3 q^{13} + 3 q^{19} - 12 q^{21} - 3 q^{27} - 12 q^{28} + 3 q^{43} - 6 q^{49} - 6 q^{52} + 6 q^{61} + 6 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 3 q^{75} + 3 q^{79} + 3 q^{91} + 3 q^{93}+O(q^{100}) $ 24 * q - 6 * q^12 - 3 * q^13 + 3 * q^19 - 12 * q^21 - 3 * q^27 - 12 * q^28 + 3 * q^43 - 6 * q^49 - 6 * q^52 + 6 * q^61 + 6 * q^63 - 3 * q^64 + 6 * q^67 + 3 * q^73 - 3 * q^75 + 3 * q^79 + 3 * q^91 + 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1767\mathbb{Z}\right)^\times$.

$n$	$590$	$685$	$838$
$\chi(n)$	$-1$	$e\left(\frac{13}{30}\right)$	$e\left(\frac{11}{18}\right)$

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			0		−0.999391	−	0.0348995i	$-0.988889\pi$
$2$		0			0		0.999391	+	0.0348995i	$0.0111111\pi$
$3$	0.374607	+	0.927184i	0.374607	+	0.927184i
$3$	0.374607	+	0.927184i	0.374607	+	0.927184i
$4$	0.997564	+	0.0697565i	0.997564	+	0.0697565i
$5$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$5$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$6$		0			0
$7$	−1.42864	+	1.03797i	−1.42864	+	1.03797i	−0.438371	+	0.898794i	$0.644444\pi$
$7$	−1.42864	+	1.03797i	−1.42864	+	1.03797i	−0.990268	+	0.139173i	$0.955556\pi$
$8$		0			0
$9$	−0.719340	+	0.694658i	−0.719340	+	0.694658i
$10$		0			0
$11$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$11$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$12$	0.309017	+	0.951057i	0.309017	+	0.951057i
$13$	−0.743520	+	0.464603i	−0.743520	+	0.464603i	−0.848048	−	0.529919i	$-0.822222\pi$
$13$	−0.743520	+	0.464603i	−0.743520	+	0.464603i	0.104528	+	0.994522i	$0.466667\pi$
$14$		0			0
$15$		0			0
$16$	0.990268	+	0.139173i	0.990268	+	0.139173i
$17$		0			0		−0.898794	−	0.438371i	$-0.855556\pi$
$17$		0			0		0.898794	+	0.438371i	$0.144444\pi$
$18$		0			0
$19$	0.913545	−	0.406737i	0.913545	−	0.406737i
$19$	0.913545	−	0.406737i	0.913545	−	0.406737i
$20$		0			0
$21$	−1.49756	−	0.935782i	−1.49756	−	0.935782i
$22$		0			0
$23$		0			0		0.970296	−	0.241922i	$-0.0777778\pi$
$23$		0			0		−0.970296	+	0.241922i	$0.922222\pi$
$24$		0			0
$25$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$26$		0			0
$27$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$28$	−1.49756	+	0.935782i	−1.49756	+	0.935782i
$29$		0			0		−0.615661	−	0.788011i	$-0.711111\pi$
$29$		0			0		0.615661	+	0.788011i	$0.288889\pi$
$30$		0			0
$31$	0.882948	+	0.469472i	0.882948	+	0.469472i
$31$	0.882948	+	0.469472i	0.882948	+	0.469472i
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$37$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$37$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$38$		0			0
$39$	−0.709299	−	0.515336i	−0.709299	−	0.515336i
$40$		0			0
$41$		0			0		−0.999391	−	0.0348995i	$-0.988889\pi$
$41$		0			0		0.999391	+	0.0348995i	$0.0111111\pi$
$42$		0			0
$43$	−0.278177	−	0.00971414i	−0.278177	−	0.00971414i	−0.104528	−	0.994522i	$-0.533333\pi$
$43$	−0.278177	−	0.00971414i	−0.278177	−	0.00971414i	−0.173648	+	0.984808i	$0.555556\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.0348995	−	0.999391i	$-0.511111\pi$
$47$		0			0		0.0348995	+	0.999391i	$0.488889\pi$
$48$	0.241922	+	0.970296i	0.241922	+	0.970296i
$49$	0.654617	−	2.01470i	0.654617	−	2.01470i
$50$		0			0
$51$		0			0
$52$	−0.774117	+	0.411606i	−0.774117	+	0.411606i
$53$		0			0		−0.961262	−	0.275637i	$-0.911111\pi$
$53$		0			0		0.961262	+	0.275637i	$0.0888889\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.719340	+	0.694658i	0.719340	+	0.694658i
$58$		0			0
$59$		0			0		0.139173	−	0.990268i	$-0.455556\pi$
$59$		0			0		−0.139173	+	0.990268i	$0.544444\pi$
$60$		0			0
$61$	0.0896772	−	0.106873i	0.0896772	−	0.106873i	−0.719340	−	0.694658i	$-0.755556\pi$
$61$	0.0896772	−	0.106873i	0.0896772	−	0.106873i	0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$	0.306644	−	1.73907i	0.306644	−	1.73907i
$64$	0.978148	+	0.207912i	0.978148	+	0.207912i
$65$		0			0
$66$		0			0
$67$	1.36821	+	0.241252i	1.36821	+	0.241252i	0.809017	−	0.587785i	$-0.200000\pi$
$67$	1.36821	+	0.241252i	1.36821	+	0.241252i	0.559193	+	0.829038i	$0.311111\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.898794	−	0.438371i	$-0.855556\pi$
$71$		0			0		0.898794	+	0.438371i	$0.144444\pi$
$72$		0			0
$73$	−0.258808	−	0.902570i	−0.258808	−	0.902570i	−0.978148	−	0.207912i	$-0.933333\pi$
$73$	−0.258808	−	0.902570i	−0.258808	−	0.902570i	0.719340	−	0.694658i	$-0.244444\pi$
$74$		0			0
$75$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$76$	0.939693	−	0.342020i	0.939693	−	0.342020i
$77$		0			0
$78$		0			0
$79$	1.43518	−	0.100357i	1.43518	−	0.100357i	0.669131	−	0.743145i	$-0.266667\pi$
$79$	1.43518	−	0.100357i	1.43518	−	0.100357i	0.766044	+	0.642788i	$0.222222\pi$
$80$		0			0
$81$	0.0348995	−	0.999391i	0.0348995	−	0.999391i
$82$		0			0
$83$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$83$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$84$	−1.42864	−	1.03797i	−1.42864	−	1.03797i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.438371	−	0.898794i	$-0.644444\pi$
$89$		0			0		0.438371	+	0.898794i	$0.355556\pi$
$90$		0			0
$91$	0.579979	−	1.43550i	0.579979	−	1.43550i
$92$		0			0
$93$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.548255	−	1.91199i	−0.548255	−	1.91199i	−0.374607	−	0.927184i	$-0.622222\pi$
$97$	−0.548255	−	1.91199i	−0.548255	−	1.91199i	−0.173648	−	0.984808i	$-0.555556\pi$
$98$		0			0
$99$		0			0
$100$	−0.961262	+	0.275637i	−0.961262	+	0.275637i
$101$		0			0		0.719340	−	0.694658i	$-0.244444\pi$
$101$		0			0		−0.719340	+	0.694658i	$0.755556\pi$
$102$		0			0
$103$	1.49889	−	0.487017i	1.49889	−	0.487017i	0.559193	−	0.829038i	$-0.311111\pi$
$103$	1.49889	−	0.487017i	1.49889	−	0.487017i	0.939693	+	0.342020i	$0.111111\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$107$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$108$	−0.882948	−	0.469472i	−0.882948	−	0.469472i
$109$	−0.385545	+	0.155770i	−0.385545	+	0.155770i	−0.559193	−	0.829038i	$-0.688889\pi$
$109$	−0.385545	+	0.155770i	−0.385545	+	0.155770i	0.173648	+	0.984808i	$0.444444\pi$
$110$		0			0
$111$	−0.461262	+	0.590388i	−0.461262	+	0.590388i
$112$	−1.55919	+	0.829038i	−1.55919	+	0.829038i
$113$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$113$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.212103	−	0.850699i	0.212103	−	0.850699i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$122$		0			0
$123$		0			0
$124$	0.848048	+	0.529919i	0.848048	+	0.529919i
$125$		0			0
$126$		0			0
$127$	0.0637646	+	1.82598i	0.0637646	+	1.82598i	0.438371	+	0.898794i	$0.355556\pi$
$127$	0.0637646	+	1.82598i	0.0637646	+	1.82598i	−0.374607	+	0.927184i	$0.622222\pi$
$128$		0			0
$129$	−0.0952000	−	0.261560i	−0.0952000	−	0.261560i
$130$		0			0
$131$		0			0		−0.719340	−	0.694658i	$-0.755556\pi$
$131$		0			0		0.719340	+	0.694658i	$0.244444\pi$
$132$		0			0
$133$	−0.882948	+	1.52931i	−0.882948	+	1.52931i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.999391	−	0.0348995i	$-0.0111111\pi$
$137$		0			0		−0.999391	+	0.0348995i	$0.988889\pi$
$138$		0			0
$139$	0.870573	−	1.63731i	0.870573	−	1.63731i	0.104528	−	0.994522i	$-0.466667\pi$
$139$	0.870573	−	1.63731i	0.870573	−	1.63731i	0.766044	−	0.642788i	$-0.222222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$145$		0			0
$146$		0			0
$147$	2.11323	−	0.147771i	2.11323	−	0.147771i
$148$	0.328433	+	0.673388i	0.328433	+	0.673388i
$149$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$149$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$150$		0			0
$151$	−1.58268	+	1.14988i	−1.58268	+	1.14988i	−0.669131	+	0.743145i	$0.733333\pi$
$151$	−1.58268	+	1.14988i	−1.58268	+	1.14988i	−0.913545	+	0.406737i	$0.866667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.671624	−	0.563559i	−0.671624	−	0.563559i
$157$	1.63039	+	1.01878i	1.63039	+	1.01878i	0.961262	+	0.275637i	$0.0888889\pi$
$157$	1.63039	+	1.01878i	1.63039	+	1.01878i	0.669131	+	0.743145i	$0.266667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.207022	−	1.96969i	0.207022	−	1.96969i	−0.0348995	−	0.999391i	$-0.511111\pi$
$163$	0.207022	−	1.96969i	0.207022	−	1.96969i	0.241922	−	0.970296i	$-0.422222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.990268	−	0.139173i	$-0.0444444\pi$
$167$		0			0		−0.990268	+	0.139173i	$0.955556\pi$
$168$		0			0
$169$	−0.101405	+	0.207912i	−0.101405	+	0.207912i
$170$		0			0
$171$	−0.374607	+	0.927184i	−0.374607	+	0.927184i
$172$	−0.276821	−	0.0290951i	−0.276821	−	0.0290951i
$173$		0			0		0.927184	−	0.374607i	$-0.122222\pi$
$173$		0			0		−0.927184	+	0.374607i	$0.877778\pi$
$174$		0			0
$175$	0.987476	−	1.46399i	0.987476	−	1.46399i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$179$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$180$		0			0
$181$	−0.232387	+	1.31793i	−0.232387	+	1.31793i	0.615661	+	0.788011i	$0.288889\pi$
$181$	−0.232387	+	1.31793i	−0.232387	+	1.31793i	−0.848048	+	0.529919i	$0.822222\pi$
$182$		0			0
$183$	0.132685	+	0.0431119i	0.132685	+	0.0431119i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.72731	−	0.367150i	1.72731	−	0.367150i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.173648	+	0.984808i	0.173648	+	0.984808i
$193$	−1.88295	−	0.469472i	−1.88295	−	0.469472i	−0.882948	−	0.469472i	$-0.844444\pi$
$193$	−1.88295	−	0.469472i	−1.88295	−	0.469472i	−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	0.793561	−	1.96413i	0.793561	−	1.96413i
$197$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$197$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$198$		0			0
$199$	0.938371	−	1.76482i	0.938371	−	1.76482i	0.438371	−	0.898794i	$-0.355556\pi$
$199$	0.938371	−	1.76482i	0.938371	−	1.76482i	0.500000	−	0.866025i	$-0.333333\pi$
$200$		0			0
$201$	0.288855	+	1.35896i	0.288855	+	1.35896i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.800944	+	0.356603i	−0.800944	+	0.356603i
$209$		0			0
$210$		0			0
$211$	1.04374	−	0.184039i	1.04374	−	0.184039i	0.374607	−	0.927184i	$-0.377778\pi$
$211$	1.04374	−	0.184039i	1.04374	−	0.184039i	0.669131	+	0.743145i	$0.266667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.74871	+	0.245765i	−1.74871	+	0.245765i
$218$		0			0
$219$	0.739897	−	0.578071i	0.739897	−	0.578071i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.454664	+	0.165484i	0.454664	+	0.165484i	0.559193	−	0.829038i	$-0.311111\pi$
$223$	0.454664	+	0.165484i	0.454664	+	0.165484i	−0.104528	+	0.994522i	$0.533333\pi$
$224$		0			0
$225$	0.438371	−	0.898794i	0.438371	−	0.898794i
$226$		0			0
$227$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$227$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$228$	0.669131	+	0.743145i	0.669131	+	0.743145i
$229$	−1.33587	−	1.20282i	−1.33587	−	1.20282i	−0.961262	−	0.275637i	$-0.911111\pi$
$229$	−1.33587	−	1.20282i	−1.33587	−	1.20282i	−0.374607	−	0.927184i	$-0.622222\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.374607	−	0.927184i	$-0.622222\pi$
$233$		0			0		0.374607	+	0.927184i	$0.377778\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.630676	+	1.29308i	0.630676	+	1.29308i
$238$		0			0
$239$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$239$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$	−1.05094	+	1.55808i	−1.05094	+	1.55808i	−0.241922	+	0.970296i	$0.577778\pi$
$241$	−1.05094	+	1.55808i	−1.05094	+	1.55808i	−0.809017	+	0.587785i	$0.800000\pi$
$242$		0			0
$243$	0.939693	−	0.342020i	0.939693	−	0.342020i
$244$	0.0969138	−	0.100357i	0.0969138	−	0.100357i
$245$		0			0
$246$		0			0
$247$	−0.490268	+	0.726852i	−0.490268	+	0.726852i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.469472	−	0.882948i	$-0.344444\pi$
$251$		0			0		−0.469472	+	0.882948i	$0.655556\pi$
$252$	0.427209	−	1.71344i	0.427209	−	1.71344i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.961262	+	0.275637i	0.961262	+	0.275637i
$257$		0			0		−0.694658	−	0.719340i	$-0.744444\pi$
$257$		0			0		0.694658	+	0.719340i	$0.255556\pi$
$258$		0			0
$259$	−1.20865	−	0.538126i	−1.20865	−	0.538126i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.139173	−	0.990268i	$-0.455556\pi$
$263$		0			0		−0.139173	+	0.990268i	$0.544444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.34805	+	0.336106i	1.34805	+	0.336106i
$269$		0			0		0.0348995	−	0.999391i	$-0.488889\pi$
$269$		0			0		−0.0348995	+	0.999391i	$0.511111\pi$
$270$		0			0
$271$	−0.103678	−	1.48267i	−0.103678	−	1.48267i	−0.719340	−	0.694658i	$-0.755556\pi$
$271$	−0.103678	−	1.48267i	−0.103678	−	1.48267i	0.615661	−	0.788011i	$-0.288889\pi$
$272$		0			0
$273$	1.54823			1.54823
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.28716	−	1.15897i	−1.28716	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$277$	−1.28716	−	1.15897i	−1.28716	−	1.15897i	−0.309017	−	0.951057i	$-0.600000\pi$
$278$		0			0
$279$	−0.961262	+	0.275637i	−0.961262	+	0.275637i
$280$		0			0
$281$		0			0		−0.927184	−	0.374607i	$-0.877778\pi$
$281$		0			0		0.927184	+	0.374607i	$0.122222\pi$
$282$		0			0
$283$	0.594092	−	0.170353i	0.594092	−	0.170353i	0.0348995	−	0.999391i	$-0.488889\pi$
$283$	0.594092	−	0.170353i	0.594092	−	0.170353i	0.559193	+	0.829038i	$0.311111\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.615661	+	0.788011i	0.615661	+	0.788011i
$290$		0			0
$291$	1.56739	−	1.22458i	1.56739	−	1.22458i
$292$	−0.195217	−	0.918425i	−0.195217	−	0.918425i
$293$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$293$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.615661	−	0.788011i	−0.615661	−	0.788011i
$301$	0.407497	−	0.274860i	0.407497	−	0.274860i
$302$		0			0
$303$		0			0
$304$	0.961262	−	0.275637i	0.961262	−	0.275637i
$305$		0			0
$306$		0			0
$307$	1.08997	−	0.440376i	1.08997	−	0.440376i	0.241922	−	0.970296i	$-0.422222\pi$
$307$	1.08997	−	0.440376i	1.08997	−	0.440376i	0.848048	+	0.529919i	$0.177778\pi$
$308$		0			0
$309$	1.01305	+	1.20730i	1.01305	+	1.20730i
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−1.00797	−	1.61308i	−1.00797	−	1.61308i	−0.766044	−	0.642788i	$-0.777778\pi$
$313$	−1.00797	−	1.61308i	−1.00797	−	1.61308i	−0.241922	−	0.970296i	$-0.577778\pi$
$314$		0			0
$315$		0			0
$316$	1.43868			1.43868
$317$		0			0		−0.829038	−	0.559193i	$-0.811111\pi$
$317$		0			0		0.829038	+	0.559193i	$0.188889\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.104528	−	0.994522i	0.104528	−	0.994522i
$325$	0.539776	−	0.690882i	0.539776	−	0.690882i
$326$		0			0
$327$	−0.288855	−	0.299118i	−0.288855	−	0.299118i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.107320	+	0.330298i	0.107320	+	0.330298i	0.990268	−	0.139173i	$-0.0444444\pi$
$331$	0.107320	+	0.330298i	0.107320	+	0.330298i	−0.882948	+	0.469472i	$0.844444\pi$
$332$		0			0
$333$	−0.720190	−	0.206511i	−0.720190	−	0.206511i
$334$		0			0
$335$		0			0
$336$	−1.35275	−	1.13510i	−1.35275	−	1.13510i
$337$	0.482665	−	1.93586i	0.482665	−	1.93586i	0.173648	−	0.984808i	$-0.444444\pi$
$337$	0.482665	−	1.93586i	0.482665	−	1.93586i	0.309017	−	0.951057i	$-0.400000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.610294	+	1.87829i	0.610294	+	1.87829i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$347$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$348$		0			0
$349$	−1.54946	+	0.689864i	−1.54946	+	0.689864i	−0.990268	−	0.139173i	$-0.955556\pi$
$349$	−1.54946	+	0.689864i	−1.54946	+	0.689864i	−0.559193	+	0.829038i	$0.688889\pi$
$350$		0			0
$351$	0.868210	−	0.122019i	0.868210	−	0.122019i
$352$		0			0
$353$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$353$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.374607	−	0.927184i	$-0.622222\pi$
$359$		0			0		0.374607	+	0.927184i	$0.377778\pi$
$360$		0			0
$361$	0.669131	−	0.743145i	0.669131	−	0.743145i
$362$		0			0
$363$	−0.997564	+	0.0697565i	−0.997564	+	0.0697565i
$364$	0.678702	−	1.39154i	0.678702	−	1.39154i
$365$		0			0
$366$		0			0
$367$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$367$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$373$	0.704489	+	0.406737i	0.704489	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$373$	0.704489	+	0.406737i	0.704489	+	0.406737i	−0.104528	+	0.994522i	$0.533333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.731145	+	1.64218i	−0.731145	+	1.64218i	0.0348995	+	0.999391i	$0.488889\pi$
$379$	−0.731145	+	1.64218i	−0.731145	+	1.64218i	−0.766044	+	0.642788i	$0.777778\pi$
$380$		0			0
$381$	−1.66913	+	0.743145i	−1.66913	+	0.743145i
$382$		0			0
$383$		0			0		0.961262	−	0.275637i	$-0.0888889\pi$
$383$		0			0		−0.961262	+	0.275637i	$0.911111\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.206852	−	0.186250i	0.206852	−	0.186250i
$388$	−0.413545	−	1.94558i	−0.413545	−	1.94558i
$389$		0			0		0.970296	−	0.241922i	$-0.0777778\pi$
$389$		0			0		−0.970296	+	0.241922i	$0.922222\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.0840186	−	0.476493i	−0.0840186	−	0.476493i	−0.997564	−	0.0697565i	$-0.977778\pi$
$397$	−0.0840186	−	0.476493i	−0.0840186	−	0.476493i	0.913545	−	0.406737i	$-0.133333\pi$
$398$		0			0
$399$	−1.74871	−	0.245765i	−1.74871	−	0.245765i
$400$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$401$		0			0		0.0348995	−	0.999391i	$-0.488889\pi$
$401$		0			0		−0.0348995	+	0.999391i	$0.511111\pi$
$402$		0			0
$403$	−0.874607	+	0.0611585i	−0.874607	+	0.0611585i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$409$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$	1.52921	−	0.381274i	1.52921	−	0.381274i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.84421	+	0.193834i	1.84421	+	0.193834i
$418$		0			0
$419$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$419$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$420$		0			0
$421$	1.17121	+	0.915051i	1.17121	+	0.915051i	0.997564	−	0.0697565i	$-0.0222222\pi$
$421$	1.17121	+	0.915051i	1.17121	+	0.915051i	0.173648	+	0.984808i	$0.444444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.0171856	+	0.245765i	−0.0171856	+	0.245765i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.999391	−	0.0348995i	$-0.0111111\pi$
$431$		0			0		−0.999391	+	0.0348995i	$0.988889\pi$
$432$	−0.848048	−	0.529919i	−0.848048	−	0.529919i
$433$	1.29929	+	1.09023i	1.29929	+	1.09023i	0.990268	+	0.139173i	$0.0444444\pi$
$433$	1.29929	+	1.09023i	1.29929	+	1.09023i	0.309017	+	0.951057i	$0.400000\pi$
$434$		0			0
$435$		0			0
$436$	−0.395472	+	0.128496i	−0.395472	+	0.128496i
$437$		0			0
$438$		0			0
$439$	−1.28479	−	1.53116i	−1.28479	−	1.53116i	−0.669131	−	0.743145i	$-0.733333\pi$
$439$	−1.28479	−	1.53116i	−1.28479	−	1.53116i	−0.615661	−	0.788011i	$-0.711111\pi$
$440$		0			0
$441$	0.928639	+	1.90399i	0.928639	+	1.90399i
$442$		0			0
$443$		0			0		−0.241922	−	0.970296i	$-0.577778\pi$
$443$		0			0		0.241922	+	0.970296i	$0.422222\pi$
$444$	−0.501321	+	0.556774i	−0.501321	+	0.556774i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.61323	+	0.718254i	−1.61323	+	0.718254i
$449$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$449$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.65903	−	1.03668i	−1.65903	−	1.03668i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.755642	+	1.04005i	−0.755642	+	1.04005i	0.241922	+	0.970296i	$0.422222\pi$
$457$	−0.755642	+	1.04005i	−0.755642	+	1.04005i	−0.997564	+	0.0697565i	$0.977778\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.970296	−	0.241922i	$-0.922222\pi$
$461$		0			0		0.970296	+	0.241922i	$0.0777778\pi$
$462$		0			0
$463$	0.114616	−	0.539228i	0.114616	−	0.539228i	−0.882948	−	0.469472i	$-0.844444\pi$
$463$	0.114616	−	0.539228i	0.114616	−	0.539228i	0.997564	−	0.0697565i	$-0.0222222\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$467$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$468$	0.270928	−	0.833831i	0.270928	−	0.833831i
$469$	−2.20509	+	1.07549i	−2.20509	+	1.07549i
$470$		0			0
$471$	−0.333843	+	1.89332i	−0.333843	+	1.89332i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.719340	+	0.694658i	−0.719340	+	0.694658i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.719340	−	0.694658i	$-0.244444\pi$
$479$		0			0		−0.719340	+	0.694658i	$0.755556\pi$
$480$		0			0
$481$	−0.579979	−	0.308380i	−0.579979	−	0.308380i
$482$		0			0
$483$		0			0
$484$	−0.374607	+	0.927184i	−0.374607	+	0.927184i
$485$		0			0
$486$		0			0
$487$	−0.0646021	+	0.614648i	−0.0646021	+	0.614648i	0.913545	+	0.406737i	$0.133333\pi$
$487$	−0.0646021	+	0.614648i	−0.0646021	+	0.614648i	−0.978148	+	0.207912i	$0.933333\pi$
$488$		0			0
$489$	1.90381	−	0.545910i	1.90381	−	0.545910i
$490$		0			0
$491$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$491$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.809017	+	0.587785i	0.809017	+	0.587785i
$497$		0			0
$498$		0			0
$499$	−1.52921	−	1.19475i	−1.52921	−	1.19475i	−0.913545	−	0.406737i	$-0.866667\pi$
$499$	−1.52921	−	1.19475i	−1.52921	−	1.19475i	−0.615661	−	0.788011i	$-0.711111\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.997564	−	0.0697565i	$-0.977778\pi$
$503$		0			0		0.997564	+	0.0697565i	$0.0222222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.230759	−	0.0161363i	−0.230759	−	0.0161363i
$508$	−0.0637646	+	1.82598i	−0.0637646	+	1.82598i
$509$		0			0		−0.0348995	−	0.999391i	$-0.511111\pi$
$509$		0			0		0.0348995	+	0.999391i	$0.488889\pi$
$510$		0			0
$511$	1.30658	+	1.02081i	1.30658	+	1.02081i
$512$		0			0
$513$	−1.00000			−1.00000
$514$		0			0
$515$		0			0
$516$	−0.0767226	−	0.267564i	−0.0767226	−	0.267564i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−0.688547	+	1.02081i	−0.688547	+	1.02081i	0.309017	+	0.951057i	$0.400000\pi$
$523$	−0.688547	+	1.02081i	−0.688547	+	1.02081i	−0.997564	+	0.0697565i	$0.977778\pi$
$524$		0			0
$525$	1.72731	+	0.367150i	1.72731	+	0.367150i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.882948	−	0.469472i	0.882948	−	0.469472i
$530$		0			0
$531$		0			0
$532$	−0.987476	+	1.46399i	−0.987476	+	1.46399i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.482665	−	1.93586i	−0.482665	−	1.93586i	−0.309017	−	0.951057i	$-0.600000\pi$
$541$	−0.482665	−	1.93586i	−0.482665	−	1.93586i	−0.173648	−	0.984808i	$-0.555556\pi$
$542$		0			0
$543$	−1.30902	+	0.278240i	−1.30902	+	0.278240i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.37461	+	0.927184i	1.37461	+	0.927184i	1.00000			$0$
$547$	1.37461	+	0.927184i	1.37461	+	0.927184i	0.374607	+	0.927184i	$0.377778\pi$
$548$		0			0
$549$	0.00973193	+	0.139173i	0.00973193	+	0.139173i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.94618	+	1.63304i	−1.94618	+	1.63304i
$554$		0			0
$555$		0			0
$556$	0.982665	−	1.57259i	0.982665	−	1.57259i
$557$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$557$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$558$		0			0
$559$	0.211343	−	0.122019i	0.211343	−	0.122019i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.987476	+	1.46399i	0.987476	+	1.46399i
$568$		0			0
$569$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$569$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$570$		0			0
$571$	−0.955369	+	0.860218i	−0.955369	+	0.860218i	−0.990268	−	0.139173i	$-0.955556\pi$
$571$	−0.955369	+	0.860218i	−0.955369	+	0.860218i	0.0348995	+	0.999391i	$0.488889\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.848048	+	0.529919i	−0.848048	+	0.529919i
$577$	−0.604528	−	1.86055i	−0.604528	−	1.86055i	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−0.604528	−	1.86055i	−0.604528	−	1.86055i	−0.104528	−	0.994522i	$-0.533333\pi$
$578$		0			0
$579$	−0.270078	−	1.92171i	−0.270078	−	1.92171i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.999391	−	0.0348995i	$-0.988889\pi$
$587$		0			0		0.999391	+	0.0348995i	$0.0111111\pi$
$588$	2.11839			2.11839
$589$	0.997564	+	0.0697565i	0.997564	+	0.0697565i
$590$		0			0
$591$		0			0
$592$	0.280660	+	0.694658i	0.280660	+	0.694658i
$593$		0			0		−0.997564	−	0.0697565i	$-0.977778\pi$
$593$		0			0		0.997564	+	0.0697565i	$0.0222222\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.98783	+	0.208930i	1.98783	+	0.208930i
$598$		0			0
$599$		0			0		−0.139173	−	0.990268i	$-0.544444\pi$
$599$		0			0		0.139173	+	0.990268i	$0.455556\pi$
$600$		0			0
$601$	−0.524123	−	1.61308i	−0.524123	−	1.61308i	−0.766044	−	0.642788i	$-0.777778\pi$
$601$	−0.524123	−	1.61308i	−0.524123	−	1.61308i	0.241922	−	0.970296i	$-0.422222\pi$
$602$		0			0
$603$	−1.15180	+	0.776896i	−1.15180	+	0.776896i
$604$	−1.65903	+	1.03668i	−1.65903	+	1.03668i
$605$		0			0
$606$		0			0
$607$	0.409677	−	0.368875i	0.409677	−	0.368875i	−0.438371	−	0.898794i	$-0.644444\pi$
$607$	0.409677	−	0.368875i	0.409677	−	0.368875i	0.848048	+	0.529919i	$0.177778\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.224224	+	0.781961i	0.224224	+	0.781961i	0.990268	+	0.139173i	$0.0444444\pi$
$613$	0.224224	+	0.781961i	0.224224	+	0.781961i	−0.766044	+	0.642788i	$0.777778\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.848048	−	0.529919i	$-0.177778\pi$
$617$		0			0		−0.848048	+	0.529919i	$0.822222\pi$
$618$		0			0
$619$	0.477418	−	0.275637i	0.477418	−	0.275637i	−0.241922	−	0.970296i	$-0.577778\pi$
$619$	0.477418	−	0.275637i	0.477418	−	0.275637i	0.719340	+	0.694658i	$0.244444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.630676	−	0.609036i	−0.630676	−	0.609036i
$625$	0.766044	−	0.642788i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$	1.55535	+	1.13003i	1.55535	+	1.13003i
$629$		0			0
$630$		0			0
$631$	1.37461	+	0.927184i	1.37461	+	0.927184i	1.00000			$0$
$631$	1.37461	+	0.927184i	1.37461	+	0.927184i	0.374607	+	0.927184i	$0.377778\pi$
$632$		0			0
$633$	0.561629	+	0.898794i	0.561629	+	0.898794i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.449316	+	1.80211i	0.449316	+	1.80211i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.882948	−	0.469472i	$-0.155556\pi$
$641$		0			0		−0.882948	+	0.469472i	$0.844444\pi$
$642$		0			0
$643$	−1.49027	−	0.726852i	−1.49027	−	0.726852i	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	−1.49027	−	0.726852i	−1.49027	−	0.726852i	−0.990268	+	0.139173i	$0.955556\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$647$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.882948	−	1.52931i	−0.882948	−	1.52931i
$652$	0.343916	−	1.95045i	0.343916	−	1.95045i
$653$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$653$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.813149	+	0.469472i	0.813149	+	0.469472i
$658$		0			0
$659$		0			0		−0.694658	−	0.719340i	$-0.744444\pi$
$659$		0			0		0.694658	+	0.719340i	$0.255556\pi$
$660$		0			0
$661$	−1.17121	+	0.915051i	−1.17121	+	0.915051i	−0.997564	−	0.0697565i	$-0.977778\pi$
$661$	−1.17121	+	0.915051i	−1.17121	+	0.915051i	−0.173648	+	0.984808i	$0.555556\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.0168859	+	0.483549i	0.0168859	+	0.483549i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.60229	+	1.16413i	1.60229	+	1.16413i	0.882948	+	0.469472i	$0.155556\pi$
$673$	1.60229	+	1.16413i	1.60229	+	1.16413i	0.719340	+	0.694658i	$0.244444\pi$
$674$		0			0
$675$	0.997564	+	0.0697565i	0.997564	+	0.0697565i
$676$	−0.115661	+	0.200332i	−0.115661	+	0.200332i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	2.76784	+	2.16248i	2.76784	+	2.16248i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−0.438371	+	0.898794i	−0.438371	+	0.898794i
$685$		0			0
$686$		0			0
$687$	0.614811	−	1.68918i	0.614811	−	1.68918i
$688$	−0.274117	−	0.0483343i	−0.274117	−	0.0483343i
$689$		0			0
$690$		0			0
$691$	−0.190983	+	1.81708i	−0.190983	+	1.81708i	0.309017	+	0.951057i	$0.400000\pi$
$691$	−0.190983	+	1.81708i	−0.190983	+	1.81708i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.08719	−	1.39154i	1.08719	−	1.39154i
$701$		0			0		0.882948	−	0.469472i	$-0.155556\pi$
$701$		0			0		−0.882948	+	0.469472i	$0.844444\pi$
$702$		0			0
$703$	0.606126	+	0.440376i	0.606126	+	0.440376i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.77028	+	0.863423i	−1.77028	+	0.863423i	−0.809017	+	0.587785i	$0.800000\pi$
$709$	−1.77028	+	0.863423i	−1.77028	+	0.863423i	−0.961262	+	0.275637i	$0.911111\pi$
$710$		0			0
$711$	−0.962665	+	1.06915i	−0.962665	+	1.06915i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$719$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$720$		0			0
$721$	−1.63586	+	2.25157i	−1.63586	+	2.25157i
$722$		0			0
$723$	−1.83832	−	0.390746i	−1.83832	−	0.390746i
$724$	−0.323755	+	1.29851i	−0.323755	+	1.29851i
$725$		0			0
$726$		0			0
$727$	0.194206	+	0.287922i	0.194206	+	0.287922i	0.913545	−	0.406737i	$-0.133333\pi$
$727$	0.194206	+	0.287922i	0.194206	+	0.287922i	−0.719340	+	0.694658i	$0.755556\pi$
$728$		0			0
$729$	0.669131	+	0.743145i	0.669131	+	0.743145i
$730$		0			0
$731$		0			0
$732$	0.129354	+	0.0522625i	0.129354	+	0.0522625i
$733$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$733$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.24739	−	1.48658i	−1.24739	−	1.48658i	−0.809017	−	0.587785i	$-0.800000\pi$
$739$	−1.24739	−	1.48658i	−1.24739	−	1.48658i	−0.438371	−	0.898794i	$-0.644444\pi$
$740$		0			0
$741$	−0.857583	−	0.182285i	−0.857583	−	0.182285i
$742$		0			0
$743$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$743$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.0477162	−	0.682374i	0.0477162	−	0.682374i	−0.913545	−	0.406737i	$-0.866667\pi$
$751$	0.0477162	−	0.682374i	0.0477162	−	0.682374i	0.961262	−	0.275637i	$-0.0888889\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.74871	−	0.245765i	1.74871	−	0.245765i
$757$	−0.434410	−	0.339399i	−0.434410	−	0.339399i	0.374607	−	0.927184i	$-0.377778\pi$
$757$	−0.434410	−	0.339399i	−0.434410	−	0.339399i	−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$761$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$762$		0			0
$763$	0.389120	−	0.622722i	0.389120	−	0.622722i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.104528	+	0.994522i	0.104528	+	0.994522i
$769$	1.29929	+	1.09023i	1.29929	+	1.09023i	0.990268	+	0.139173i	$0.0444444\pi$
$769$	1.29929	+	1.09023i	1.29929	+	1.09023i	0.309017	+	0.951057i	$0.400000\pi$
$770$		0			0
$771$		0			0
$772$	−1.84561	−	0.599676i	−1.84561	−	0.599676i
$773$		0			0		0.990268	−	0.139173i	$-0.0444444\pi$
$773$		0			0		−0.990268	+	0.139173i	$0.955556\pi$
$774$		0			0
$775$	−0.990268	−	0.139173i	−0.990268	−	0.139173i
$776$		0			0
$777$	0.0461731	−	1.32223i	0.0461731	−	1.32223i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.928639	−	1.90399i	0.928639	−	1.90399i
$785$		0			0
$786$		0			0
$787$	0.339707	+	0.0722070i	0.339707	+	0.0722070i	0.374607	−	0.927184i	$-0.377778\pi$
$787$	0.339707	+	0.0722070i	0.339707	+	0.0722070i	−0.0348995	+	0.999391i	$0.511111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.0170232	+	0.121127i	−0.0170232	+	0.121127i
$794$		0			0
$795$		0			0
$796$	1.05919	−	1.69506i	1.05919	−	1.69506i
$797$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$797$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.193356	+	1.37580i	0.193356	+	1.37580i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$809$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$810$		0			0
$811$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$811$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$812$		0			0
$813$	1.33587	−	0.651546i	1.33587	−	0.651546i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.258078	+	0.104270i	−0.258078	+	0.104270i
$818$		0			0
$819$	0.579979	+	1.43550i	0.579979	+	1.43550i
$820$		0			0
$821$		0			0		−0.529919	−	0.848048i	$-0.677778\pi$
$821$		0			0		0.529919	+	0.848048i	$0.322222\pi$
$822$		0			0
$823$	−0.114616	+	0.399715i	−0.114616	+	0.399715i	−0.997564	−	0.0697565i	$-0.977778\pi$
$823$	−0.114616	+	0.399715i	−0.114616	+	0.399715i	0.882948	+	0.469472i	$0.155556\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.990268	−	0.139173i	$-0.0444444\pi$
$827$		0			0		−0.990268	+	0.139173i	$0.955556\pi$
$828$		0			0
$829$	−1.82264	+	0.811492i	−1.82264	+	0.811492i	−0.882948	+	0.469472i	$0.844444\pi$
$829$	−1.82264	+	0.811492i	−1.82264	+	0.811492i	−0.939693	+	0.342020i	$0.888889\pi$
$830$		0			0
$831$	0.592396	−	1.62760i	0.592396	−	1.62760i
$832$	−0.823868	+	0.299864i	−0.823868	+	0.299864i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.615661	−	0.788011i	−0.615661	−	0.788011i
$838$		0			0
$839$		0			0		−0.469472	−	0.882948i	$-0.655556\pi$
$839$		0			0		0.469472	+	0.882948i	$0.344444\pi$
$840$		0			0
$841$	−0.241922	+	0.970296i	−0.241922	+	0.970296i
$842$		0			0
$843$		0			0
$844$	1.05403	−	0.110783i	1.05403	−	0.110783i
$845$		0			0
$846$		0			0
$847$	−0.545692	−	1.67947i	−0.545692	−	1.67947i
$848$		0			0
$849$	0.380500	+	0.487017i	0.380500	+	0.487017i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.943248	−	1.20730i	0.943248	−	1.20730i	−0.0348995	−	0.999391i	$-0.511111\pi$
$853$	0.943248	−	1.20730i	0.943248	−	1.20730i	0.978148	−	0.207912i	$-0.0666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.970296	−	0.241922i	$-0.922222\pi$
$857$		0			0		0.970296	+	0.241922i	$0.0777778\pi$
$858$		0			0
$859$	0.292131	−	1.01878i	0.292131	−	1.01878i	−0.669131	−	0.743145i	$-0.733333\pi$
$859$	0.292131	−	1.01878i	0.292131	−	1.01878i	0.961262	−	0.275637i	$-0.0888889\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$868$	−1.76159	+	0.123183i	−1.76159	+	0.123183i
$869$		0			0
$870$		0			0
$871$	−1.12938	+	0.456298i	−1.12938	+	0.456298i
$872$		0			0
$873$	1.72256	+	0.994522i	1.72256	+	0.994522i
$874$		0			0
$875$		0			0
$876$	0.778419	−	0.525050i	0.778419	−	0.525050i
$877$	1.06579	−	0.718885i	1.06579	−	0.718885i	0.104528	−	0.994522i	$-0.466667\pi$
$877$	1.06579	−	0.718885i	1.06579	−	0.718885i	0.961262	+	0.275637i	$0.0888889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$881$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$882$		0			0
$883$	−1.57506	+	0.0550024i	−1.57506	+	0.0550024i	−0.809017	−	0.587785i	$-0.800000\pi$
$883$	−1.57506	+	0.0550024i	−1.57506	+	0.0550024i	−0.766044	+	0.642788i	$0.777778\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.788011	−	0.615661i	$-0.211111\pi$
$887$		0			0		−0.788011	+	0.615661i	$0.788889\pi$
$888$		0			0
$889$	−1.98640	−	2.54248i	−1.98640	−	2.54248i
$890$		0			0
$891$		0			0
$892$	0.442013	+	0.196797i	0.442013	+	0.196797i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.500000	−	0.866025i	0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	0.407497	+	0.274860i	0.407497	+	0.274860i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.0567450	−	0.811492i	−0.0567450	−	0.811492i	−0.939693	−	0.342020i	$-0.888889\pi$
$907$	−0.0567450	−	0.811492i	−0.0567450	−	0.811492i	0.882948	−	0.469472i	$-0.155556\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$911$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$912$	0.615661	+	0.788011i	0.615661	+	0.788011i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.24871	−	1.29308i	−1.24871	−	1.29308i
$917$		0			0
$918$		0			0
$919$	−0.442013	−	0.196797i	−0.442013	−	0.196797i	0.173648	−	0.984808i	$-0.444444\pi$
$919$	−0.442013	−	0.196797i	−0.442013	−	0.196797i	−0.615661	+	0.788011i	$0.711111\pi$
$920$		0			0
$921$	0.816620	+	0.845635i	0.816620	+	0.845635i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.573931	−	0.481585i	−0.573931	−	0.481585i
$926$		0			0
$927$	−0.739897	+	1.39154i	−0.739897	+	1.39154i
$928$		0			0
$929$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$929$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$930$		0			0
$931$	−0.221432	−	2.10678i	−0.221432	−	2.10678i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.948445	+	1.40613i	−0.948445	+	1.40613i	−0.0348995	+	0.999391i	$0.511111\pi$
$937$	−0.948445	+	1.40613i	−0.948445	+	1.40613i	−0.913545	+	0.406737i	$0.866667\pi$
$938$		0			0
$939$	1.11803	−	1.53884i	1.11803	−	1.53884i
$940$		0			0
$941$		0			0		−0.438371	−	0.898794i	$-0.644444\pi$
$941$		0			0		0.438371	+	0.898794i	$0.355556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0697565	−	0.997564i	$-0.522222\pi$
$947$		0			0		0.0697565	+	0.997564i	$0.477778\pi$
$948$	0.538939	+	1.33392i	0.538939	+	1.33392i
$949$	0.611765	+	0.550836i	0.611765	+	0.550836i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.438371	−	0.898794i	$-0.355556\pi$
$953$		0			0		−0.438371	+	0.898794i	$0.644444\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.559193	+	0.829038i	0.559193	+	0.829038i
$962$		0			0
$963$		0			0
$964$	−1.15707	+	1.48098i	−1.15707	+	1.48098i
$965$		0			0
$966$		0			0
$967$	−1.55208	+	0.273673i	−1.55208	+	0.273673i	−0.882948	−	0.469472i	$-0.844444\pi$
$967$	−1.55208	+	0.273673i	−1.55208	+	0.273673i	−0.669131	+	0.743145i	$0.733333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.529919	−	0.848048i	$-0.322222\pi$
$971$		0			0		−0.529919	+	0.848048i	$0.677778\pi$
$972$	0.961262	−	0.275637i	0.961262	−	0.275637i
$973$	0.455739	+	3.24275i	0.455739	+	3.24275i
$974$		0			0
$975$	0.842779	+	0.241663i	0.842779	+	0.241663i
$976$	0.103678	−	0.0933524i	0.103678	−	0.0933524i
$977$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$977$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.169131	−	0.379874i	0.169131	−	0.379874i
$982$		0			0
$983$		0			0		0.438371	−	0.898794i	$-0.355556\pi$
$983$		0			0		−0.438371	+	0.898794i	$0.644444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.539776	+	0.690882i	−0.539776	+	0.690882i
$989$		0			0
$990$		0			0
$991$	0.333843	+	1.89332i	0.333843	+	1.89332i	0.438371	+	0.898794i	$0.355556\pi$
$991$	0.333843	+	1.89332i	0.333843	+	1.89332i	−0.104528	+	0.994522i	$0.533333\pi$
$992$		0			0
$993$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0121205	+	0.0687386i	−0.0121205	+	0.0687386i	−0.990268	−	0.139173i	$-0.955556\pi$
$997$	−0.0121205	+	0.0687386i	−0.0121205	+	0.0687386i	0.978148	+	0.207912i	$0.0666667\pi$
$998$		0			0
$999$	−0.0783141	−	0.745109i	−0.0783141	−	0.745109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1767.1.eh.a.737.1	✓	24
3.2	odd	2	CM	1767.1.eh.a.737.1	✓	24
19.14	odd	18		1767.1.ew.a.1667.1	yes	24
31.22	odd	30		1767.1.ew.a.53.1	yes	24
57.14	even	18		1767.1.ew.a.1667.1	yes	24
93.53	even	30		1767.1.ew.a.53.1	yes	24
589.394	even	90	inner	1767.1.eh.a.983.1	yes	24
1767.983	odd	90	inner	1767.1.eh.a.983.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1767.1.eh.a.737.1	✓	24	1.1	even	1	trivial
1767.1.eh.a.737.1	✓	24	3.2	odd	2	CM
1767.1.eh.a.983.1	yes	24	589.394	even	90	inner
1767.1.eh.a.983.1	yes	24	1767.983	odd	90	inner
1767.1.ew.a.53.1	yes	24	31.22	odd	30
1767.1.ew.a.53.1	yes	24	93.53	even	30
1767.1.ew.a.1667.1	yes	24	19.14	odd	18
1767.1.ew.a.1667.1	yes	24	57.14	even	18

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:	\( N \)	\(=\)	\( 1767 = 3 \cdot 19 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1767.eh (of order \(90\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(0.374607 + 0.927184i) q^{3} +(0.997564 + 0.0697565i) q^{4} +(-1.42864 + 1.03797i) q^{7} +(-0.719340 + 0.694658i) q^{9} +(0.309017 + 0.951057i) q^{12} +(-0.743520 + 0.464603i) q^{13} +(0.990268 + 0.139173i) q^{16} +(0.913545 - 0.406737i) q^{19} +(-1.49756 - 0.935782i) q^{21} +(-0.939693 + 0.342020i) q^{25} +(-0.913545 - 0.406737i) q^{27} +(-1.49756 + 0.935782i) q^{28} +(0.882948 + 0.469472i) q^{31} +(-0.766044 + 0.642788i) q^{36} +(0.374607 + 0.648838i) q^{37} +(-0.709299 - 0.515336i) q^{39} +(-0.278177 - 0.00971414i) q^{43} +(0.241922 + 0.970296i) q^{48} +(0.654617 - 2.01470i) q^{49} +(-0.774117 + 0.411606i) q^{52} +(0.719340 + 0.694658i) q^{57} +(0.0896772 - 0.106873i) q^{61} +(0.306644 - 1.73907i) q^{63} +(0.978148 + 0.207912i) q^{64} +(1.36821 + 0.241252i) q^{67} +(-0.258808 - 0.902570i) q^{73} +(-0.669131 - 0.743145i) q^{75} +(0.939693 - 0.342020i) q^{76} +(1.43518 - 0.100357i) q^{79} +(0.0348995 - 0.999391i) q^{81} +(-1.42864 - 1.03797i) q^{84} +(0.579979 - 1.43550i) q^{91} +(-0.104528 + 0.994522i) q^{93} +(-0.548255 - 1.91199i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{12} - 3 q^{13} + 3 q^{19} - 12 q^{21} - 3 q^{27} - 12 q^{28} + 3 q^{43} - 6 q^{49} - 6 q^{52} + 6 q^{61} + 6 q^{63} - 3 q^{64} + 6 q^{67} + 3 q^{73} - 3 q^{75} + 3 q^{79} + 3 q^{91} + 3 q^{93}+O(q^{100}) \) 24 * q - 6 * q^12 - 3 * q^13 + 3 * q^19 - 12 * q^21 - 3 * q^27 - 12 * q^28 + 3 * q^43 - 6 * q^49 - 6 * q^52 + 6 * q^61 + 6 * q^63 - 3 * q^64 + 6 * q^67 + 3 * q^73 - 3 * q^75 + 3 * q^79 + 3 * q^91 + 3 * q^93

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