LMFDB - Embedded newform 1067.1.bg.a.967.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1067,1,Mod(32,1067)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1067, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 37]))

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

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

chi := DirichletCharacter("1067.32");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1067 = 11 \cdot 97 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1067.bg (of order $48$, degree $16$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.532502368479$
Analytic rank:	$0$
Dimension:	$16$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} - \cdots)$

Embedding invariants

Embedding label			967.1
Root			$-0.793353 + 0.608761i$ of defining polynomial
Character	$\chi$	$=$	1067.967
Dual form			1067.1.bg.a.32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.91532 + 0.252157i) q^{3} +(0.258819 - 0.965926i) q^{4} +(-1.34861 - 1.18270i) q^{5} +(2.63896 + 0.707107i) q^{9} +O(q^{10})$	\(q+(1.91532 + 0.252157i) q^{3} +(0.258819 - 0.965926i) q^{4} +(-1.34861 - 1.18270i) q^{5} +(2.63896 + 0.707107i) q^{9} +(-0.608761 + 0.793353i) q^{11} +(0.739288 - 1.78480i) q^{12} +(-2.28480 - 2.60531i) q^{15} +(-0.866025 - 0.500000i) q^{16} +(-1.49144 + 0.996552i) q^{20} +(-0.583242 + 1.18270i) q^{23} +(0.289445 + 2.19855i) q^{25} +(3.09136 + 1.28048i) q^{27} +(0.241181 - 1.83195i) q^{31} +(-1.36603 + 1.36603i) q^{33} +(1.36603 - 2.36603i) q^{36} +(-0.576581 + 0.284338i) q^{37} +(0.608761 + 0.793353i) q^{44} +(-2.72263 - 4.07470i) q^{45} +(0.707107 + 0.707107i) q^{47} +(-1.53264 - 1.17604i) q^{48} +(0.793353 + 0.608761i) q^{49} +(-0.158919 - 0.207107i) q^{53} +(1.75928 - 0.349942i) q^{55} +(-0.391239 - 0.793353i) q^{59} +(-3.10789 + 1.53264i) q^{60} +(-0.707107 + 0.707107i) q^{64} +(0.128293 + 0.0255190i) q^{67} +(-1.41532 + 2.11818i) q^{69} +(-0.123864 + 1.88981i) q^{71} +4.28393i q^{75} +(0.576581 + 1.69855i) q^{80} +(3.23205 + 1.86603i) q^{81} +(0.991445 + 0.869474i) q^{92} +(0.923880 - 3.44797i) q^{93} +(-0.793353 + 0.608761i) q^{97} +(-2.16748 + 1.66317i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{15} - 8 q^{20} + 8 q^{31} - 8 q^{33} + 8 q^{36} - 24 q^{45} - 16 q^{59} + 8 q^{69} + 24 q^{81}+O(q^{100}) $ 16 * q - 8 * q^15 - 8 * q^20 + 8 * q^31 - 8 * q^33 + 8 * q^36 - 24 * q^45 - 16 * q^59 + 8 * q^69 + 24 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1067\mathbb{Z}\right)^\times$.

$n$	$199$	$486$
$\chi(n)$	$e\left(\frac{11}{48}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$2$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$3$	1.91532	+	0.252157i	1.91532	+	0.252157i	0.991445	−	0.130526i	$-0.0416667\pi$
$3$	1.91532	+	0.252157i	1.91532	+	0.252157i	0.923880	+	0.382683i	$0.125000\pi$
$4$	0.258819	−	0.965926i	0.258819	−	0.965926i
$5$	−1.34861	−	1.18270i	−1.34861	−	1.18270i	−0.965926	−	0.258819i	$-0.916667\pi$
$5$	−1.34861	−	1.18270i	−1.34861	−	1.18270i	−0.382683	−	0.923880i	$-0.625000\pi$
$6$		0			0
$7$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$7$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$8$		0			0
$9$	2.63896	+	0.707107i	2.63896	+	0.707107i
$10$		0			0
$11$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$11$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$12$	0.739288	−	1.78480i	0.739288	−	1.78480i
$13$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$13$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$14$		0			0
$15$	−2.28480	−	2.60531i	−2.28480	−	2.60531i
$16$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$17$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$17$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$18$		0			0
$19$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$19$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$20$	−1.49144	+	0.996552i	−1.49144	+	0.996552i
$21$		0			0
$22$		0			0
$23$	−0.583242	+	1.18270i	−0.583242	+	1.18270i	0.382683	+	0.923880i	$0.375000\pi$
$23$	−0.583242	+	1.18270i	−0.583242	+	1.18270i	−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	0.289445	+	2.19855i	0.289445	+	2.19855i
$26$		0			0
$27$	3.09136	+	1.28048i	3.09136	+	1.28048i
$28$		0			0
$29$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$29$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$30$		0			0
$31$	0.241181	−	1.83195i	0.241181	−	1.83195i	−0.258819	−	0.965926i	$-0.583333\pi$
$31$	0.241181	−	1.83195i	0.241181	−	1.83195i	0.500000	−	0.866025i	$-0.333333\pi$
$32$		0			0
$33$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$34$		0			0
$35$		0			0
$36$	1.36603	−	2.36603i	1.36603	−	2.36603i
$37$	−0.576581	+	0.284338i	−0.576581	+	0.284338i	−0.707107	−	0.707107i	$-0.750000\pi$
$37$	−0.576581	+	0.284338i	−0.576581	+	0.284338i	0.130526	+	0.991445i	$0.458333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$41$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$42$		0			0
$43$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$43$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$44$	0.608761	+	0.793353i	0.608761	+	0.793353i
$45$	−2.72263	−	4.07470i	−2.72263	−	4.07470i
$46$		0			0
$47$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$47$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$48$	−1.53264	−	1.17604i	−1.53264	−	1.17604i
$49$	0.793353	+	0.608761i	0.793353	+	0.608761i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.158919	−	0.207107i	−0.158919	−	0.207107i	0.707107	−	0.707107i	$-0.250000\pi$
$53$	−0.158919	−	0.207107i	−0.158919	−	0.207107i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	1.75928	−	0.349942i	1.75928	−	0.349942i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.391239	−	0.793353i	−0.391239	−	0.793353i	0.608761	−	0.793353i	$-0.291667\pi$
$59$	−0.391239	−	0.793353i	−0.391239	−	0.793353i	−1.00000			$\pi$
$60$	−3.10789	+	1.53264i	−3.10789	+	1.53264i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$	0.128293	+	0.0255190i	0.128293	+	0.0255190i	0.258819	−	0.965926i	$-0.416667\pi$
$67$	0.128293	+	0.0255190i	0.128293	+	0.0255190i	−0.130526	+	0.991445i	$0.541667\pi$
$68$		0			0
$69$	−1.41532	+	2.11818i	−1.41532	+	2.11818i
$70$		0			0
$71$	−0.123864	+	1.88981i	−0.123864	+	1.88981i	0.258819	+	0.965926i	$0.416667\pi$
$71$	−0.123864	+	1.88981i	−0.123864	+	1.88981i	−0.382683	+	0.923880i	$0.625000\pi$
$72$		0			0
$73$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$73$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$74$		0			0
$75$			4.28393i			4.28393i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$79$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$80$	0.576581	+	1.69855i	0.576581	+	1.69855i
$81$	3.23205	+	1.86603i	3.23205	+	1.86603i
$82$		0			0
$83$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$83$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$89$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$90$		0			0
$91$		0			0
$92$	0.991445	+	0.869474i	0.991445	+	0.869474i
$93$	0.923880	−	3.44797i	0.923880	−	3.44797i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$97$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$98$		0			0
$99$	−2.16748	+	1.66317i	−2.16748	+	1.66317i
$100$	2.19855	+	0.289445i	2.19855	+	0.289445i
$101$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$101$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$102$		0			0
$103$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.923880	−	0.382683i	$-0.875000\pi$
$103$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.130526	+	0.991445i	$0.541667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$107$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$108$	2.03696	−	2.65461i	2.03696	−	2.65461i
$109$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$109$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$110$		0			0
$111$	−1.17604	+	0.399211i	−1.17604	+	0.399211i
$112$		0			0
$113$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.923880	−	0.382683i	$-0.875000\pi$
$113$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.793353	−	0.608761i	$-0.791667\pi$
$114$		0			0
$115$	2.18534	−	0.905198i	2.18534	−	0.905198i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$122$		0			0
$123$		0			0
$124$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$125$	1.21332	−	1.81587i	1.21332	−	1.81587i
$126$		0			0
$127$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$127$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$131$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$132$	0.965926	+	1.67303i	0.965926	+	1.67303i
$133$		0			0
$134$		0			0
$135$	−2.65461	−	5.38302i	−2.65461	−	5.38302i
$136$		0			0
$137$	0.534534	−	1.57469i	0.534534	−	1.57469i	−0.258819	−	0.965926i	$-0.583333\pi$
$137$	0.534534	−	1.57469i	0.534534	−	1.57469i	0.793353	−	0.608761i	$-0.208333\pi$
$138$		0			0
$139$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$139$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$140$		0			0
$141$	1.17604	+	1.53264i	1.17604	+	1.53264i
$142$		0			0
$143$		0			0
$144$	−1.93185	−	1.93185i	−1.93185	−	1.93185i
$145$		0			0
$146$		0			0
$147$	1.36603	+	1.36603i	1.36603	+	1.36603i
$148$	0.125419	+	0.630526i	0.125419	+	0.630526i
$149$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$149$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$150$		0			0
$151$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$151$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.49190	+	2.18534i	−2.49190	+	2.18534i
$156$		0			0
$157$	1.34861	−	0.665060i	1.34861	−	0.665060i	0.382683	−	0.923880i	$-0.375000\pi$
$157$	1.34861	−	0.665060i	1.34861	−	0.665060i	0.965926	+	0.258819i	$0.0833333\pi$
$158$		0			0
$159$	−0.252157	−	0.436749i	−0.252157	−	0.436749i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.0999004	+	0.758819i	−0.0999004	+	0.758819i	0.866025	+	0.500000i	$0.166667\pi$
$163$	−0.0999004	+	0.758819i	−0.0999004	+	0.758819i	−0.965926	+	0.258819i	$0.916667\pi$
$164$		0			0
$165$	3.45783	−	0.226638i	3.45783	−	0.226638i
$166$		0			0
$167$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$167$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$168$		0			0
$169$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$173$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$174$		0			0
$175$		0			0
$176$	0.923880	−	0.382683i	0.923880	−	0.382683i
$177$	−0.549299	−	1.61818i	−0.549299	−	1.61818i
$178$		0			0
$179$	−1.29335	−	1.47479i	−1.29335	−	1.47479i	−0.793353	−	0.608761i	$-0.791667\pi$
$179$	−1.29335	−	1.47479i	−1.29335	−	1.47479i	−0.500000	−	0.866025i	$-0.666667\pi$
$180$	−4.64053	+	1.57525i	−4.64053	+	1.57525i
$181$	−0.257264	+	0.293353i	−0.257264	+	0.293353i	−0.866025	−	0.500000i	$-0.833333\pi$
$181$	−0.257264	+	0.293353i	−0.257264	+	0.293353i	0.608761	+	0.793353i	$0.291667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.11387	+	0.298460i	1.11387	+	0.298460i
$186$		0			0
$187$		0			0
$188$	0.866025	−	0.500000i	0.866025	−	0.500000i
$189$		0			0
$190$		0			0
$191$	0.513210	+	0.0675653i	0.513210	+	0.0675653i	0.382683	−	0.923880i	$-0.375000\pi$
$191$	0.513210	+	0.0675653i	0.513210	+	0.0675653i	0.130526	+	0.991445i	$0.458333\pi$
$192$	−1.53264	+	1.17604i	−1.53264	+	1.17604i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	0.793353	−	0.608761i	0.793353	−	0.608761i
$197$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$197$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$198$		0			0
$199$	1.50046	+	1.31587i	1.50046	+	1.31587i	0.793353	+	0.608761i	$0.208333\pi$
$199$	1.50046	+	1.31587i	1.50046	+	1.31587i	0.707107	+	0.707107i	$0.250000\pi$
$200$		0			0
$201$	0.239288	+	0.0812272i	0.239288	+	0.0812272i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.37545	+	2.70868i	−2.37545	+	2.70868i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$211$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$212$	−0.241181	+	0.0999004i	−0.241181	+	0.0999004i
$213$	−0.713769	+	3.58836i	−0.713769	+	3.58836i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.117317	−	1.78990i	0.117317	−	1.78990i
$221$		0			0
$222$		0			0
$223$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.991445	−	0.130526i	$-0.958333\pi$
$223$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.965926	+	0.258819i	$0.916667\pi$
$224$		0			0
$225$	−0.790778	+	6.00655i	−0.790778	+	6.00655i
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	−0.991445	−	1.71723i	−0.991445	−	1.71723i	−0.608761	−	0.793353i	$-0.708333\pi$
$229$	−0.991445	−	1.71723i	−0.991445	−	1.71723i	−0.382683	−	0.923880i	$-0.625000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$233$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$234$		0			0
$235$	−0.117317	−	1.78990i	−0.117317	−	1.78990i
$236$	−0.867580	+	0.172572i	−0.867580	+	0.172572i
$237$		0			0
$238$		0			0
$239$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$239$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$240$	0.676037	+	3.39867i	0.676037	+	3.39867i
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$		0			0
$243$	3.06528	+	2.35207i	3.06528	+	2.35207i
$244$		0			0
$245$	−0.349942	−	1.75928i	−0.349942	−	1.75928i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.608761	−	1.79335i	0.608761	−	1.79335i		−	1.00000i	$-0.5\pi$
$251$	0.608761	−	1.79335i	0.608761	−	1.79335i	0.608761	−	0.793353i	$-0.291667\pi$
$252$		0			0
$253$	−0.583242	−	1.18270i	−0.583242	−	1.18270i
$254$		0			0
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$	−0.923880	−	0.617317i	−0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$257$	−0.923880	−	0.617317i	−0.923880	−	0.617317i	−0.923880	+	0.382683i	$0.875000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$263$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$264$		0			0
$265$	−0.0306258	+	0.467259i	−0.0306258	+	0.467259i
$266$		0			0
$267$		0			0
$268$	0.0578541	−	0.117317i	0.0578541	−	0.117317i
$269$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$269$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$270$		0			0
$271$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$271$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.92043	−	1.10876i	−1.92043	−	1.10876i
$276$	1.67969	+	1.91532i	1.67969	+	1.91532i
$277$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$277$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$278$		0			0
$279$	1.93185	−	4.66390i	1.93185	−	4.66390i
$280$		0			0
$281$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$281$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$282$		0			0
$283$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$283$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$284$	1.79335	+	0.608761i	1.79335	+	0.608761i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$290$		0			0
$291$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$292$		0			0
$293$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$293$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$294$		0			0
$295$	−0.410670	+	1.53264i	−0.410670	+	1.53264i
$296$		0			0
$297$	−2.89778	+	1.67303i	−2.89778	+	1.67303i
$298$		0			0
$299$		0			0
$300$	4.13795	+	1.10876i	4.13795	+	1.10876i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$307$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$308$		0			0
$309$	−2.17303	+	0.900100i	−2.17303	+	0.900100i
$310$		0			0
$311$	1.25026	−	0.835400i	1.25026	−	0.835400i	0.258819	−	0.965926i	$-0.416667\pi$
$311$	1.25026	−	0.835400i	1.25026	−	0.835400i	0.991445	+	0.130526i	$0.0416667\pi$
$312$		0			0
$313$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$313$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.00855514	−	0.130526i	0.00855514	−	0.130526i	−0.991445	−	0.130526i	$-0.958333\pi$
$317$	0.00855514	−	0.130526i	0.00855514	−	0.130526i	1.00000			$0$
$318$		0			0
$319$		0			0
$320$	1.78990	−	0.117317i	1.78990	−	0.117317i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	2.63896	−	2.63896i	2.63896	−	2.63896i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.125419	+	0.369474i	−0.125419	+	0.369474i	−0.991445	−	0.130526i	$-0.958333\pi$
$331$	−0.125419	+	0.369474i	−0.125419	+	0.369474i	0.866025	+	0.500000i	$0.166667\pi$
$332$		0			0
$333$	−1.72263	+	0.342652i	−1.72263	+	0.342652i
$334$		0			0
$335$	−0.142836	−	0.186147i	−0.142836	−	0.186147i
$336$		0			0
$337$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$337$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$338$		0			0
$339$	−3.03906	−	2.33195i	−3.03906	−	2.33195i
$340$		0			0
$341$	1.30656	+	1.30656i	1.30656	+	1.30656i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	4.41389	−	1.18270i	4.41389	−	1.18270i
$346$		0			0
$347$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$347$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$348$		0			0
$349$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$349$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$353$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$354$		0			0
$355$	2.40211	−	2.40211i	2.40211	−	2.40211i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$359$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$360$		0			0
$361$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$362$		0			0
$363$	−0.252157	−	1.91532i	−0.252157	−	1.91532i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.69855	+	0.837633i	1.69855	+	0.837633i	0.991445	+	0.130526i	$0.0416667\pi$
$367$	1.69855	+	0.837633i	1.69855	+	0.837633i	0.707107	+	0.707107i	$0.250000\pi$
$368$	1.09645	−	0.732626i	1.09645	−	0.732626i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.09136	−	1.78480i	−3.09136	−	1.78480i
$373$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$373$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$374$		0			0
$375$	2.78179	−	3.17203i	2.78179	−	3.17203i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.739288	−	0.198092i	−0.739288	−	0.198092i	−0.130526	−	0.991445i	$-0.541667\pi$
$379$	−0.739288	−	0.198092i	−0.739288	−	0.198092i	−0.608761	+	0.793353i	$0.708333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.25026	+	1.09645i	1.25026	+	1.09645i	0.991445	+	0.130526i	$0.0416667\pi$
$383$	1.25026	+	1.09645i	1.25026	+	1.09645i	0.258819	+	0.965926i	$0.416667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.382683	+	0.923880i	0.382683	+	0.923880i
$389$	−1.98289			−1.98289			−0.991445	−	0.130526i	$-0.958333\pi$
$389$	−1.98289			−1.98289			−0.991445	+	0.130526i	$0.958333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	1.04551	+	2.52409i	1.04551	+	2.52409i
$397$	−0.500000	−	0.133975i	−0.500000	−	0.133975i		−	1.00000i	$-0.5\pi$
$397$	−0.500000	−	0.133975i	−0.500000	−	0.133975i	−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	0.848609	−	2.04872i	0.848609	−	2.04872i
$401$	−0.0862466	+	0.0983454i	−0.0862466	+	0.0983454i	−0.793353	−	0.608761i	$-0.791667\pi$
$401$	−0.0862466	+	0.0983454i	−0.0862466	+	0.0983454i	0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.15183	−	6.33908i	−2.15183	−	6.33908i
$406$		0			0
$407$	0.125419	−	0.630526i	0.125419	−	0.630526i
$408$		0			0
$409$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$409$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$410$		0			0
$411$	1.42088	−	2.88125i	1.42088	−	2.88125i
$412$	0.315118	+	1.17604i	0.315118	+	1.17604i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.207107	+	1.57313i	−0.207107	+	1.57313i	0.500000	+	0.866025i	$0.333333\pi$
$419$	−0.207107	+	1.57313i	−0.207107	+	1.57313i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	1.12197	−	1.12197i	1.12197	−	1.12197i	0.130526	−	0.991445i	$-0.458333\pi$
$421$	1.12197	−	1.12197i	1.12197	−	1.12197i	0.991445	−	0.130526i	$-0.0416667\pi$
$422$		0			0
$423$	1.36603	+	2.36603i	1.36603	+	2.36603i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$431$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$432$	−2.03696	−	2.65461i	−2.03696	−	2.65461i
$433$	−1.05217	−	1.57469i	−1.05217	−	1.57469i	−0.793353	−	0.608761i	$-0.791667\pi$
$433$	−1.05217	−	1.57469i	−1.05217	−	1.57469i	−0.258819	−	0.965926i	$-0.583333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$439$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$440$		0			0
$441$	1.66317	+	2.16748i	1.66317	+	2.16748i
$442$		0			0
$443$	−0.867580	+	0.172572i	−0.867580	+	0.172572i	−0.608761	−	0.793353i	$-0.708333\pi$
$443$	−0.867580	+	0.172572i	−0.867580	+	0.172572i	−0.258819	+	0.965926i	$0.583333\pi$
$444$	0.0812272	+	1.23929i	0.0812272	+	1.23929i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$449$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$450$		0			0
$451$		0			0
$452$	−1.40211	+	1.40211i	−1.40211	+	1.40211i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$457$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$458$		0			0
$459$		0			0
$460$	−0.308746	−	2.34516i	−0.308746	−	2.34516i
$461$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$461$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$462$		0			0
$463$			0.261052i			0.261052i	0.991445	+	0.130526i	$0.0416667\pi$
$463$			0.261052i			0.261052i	−0.991445	+	0.130526i	$0.958333\pi$
$464$		0			0
$465$	−5.32386	+	3.55729i	−5.32386	+	3.55729i
$466$		0			0
$467$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.500000	−	0.866025i	$-0.333333\pi$
$467$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.965926	+	0.258819i	$0.0833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.75072	−	0.933745i	2.75072	−	0.933745i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.272933	−	0.658919i	−0.272933	−	0.658919i
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.00000			−1.00000
$485$	1.78990	+	0.117317i	1.78990	+	0.117317i
$486$		0			0
$487$	0.793353	−	0.608761i	0.793353	−	0.608761i	−0.130526	−	0.991445i	$-0.541667\pi$
$487$	0.793353	−	0.608761i	0.793353	−	0.608761i	0.923880	+	0.382683i	$0.125000\pi$
$488$		0			0
$489$	−0.382683	+	1.42819i	−0.382683	+	1.42819i
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	4.89011	+	0.320515i	4.89011	+	0.320515i
$496$	−1.12484	+	1.46593i	−1.12484	+	1.46593i
$497$		0			0
$498$		0			0
$499$	1.57469	−	0.534534i	1.57469	−	0.534534i	0.608761	−	0.793353i	$-0.291667\pi$
$499$	1.57469	−	0.534534i	1.57469	−	0.534534i	0.965926	+	0.258819i	$0.0833333\pi$
$500$	−1.43996	−	1.64196i	−1.43996	−	1.64196i
$501$		0			0
$502$		0			0
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	1.93185i		−	1.93185i
$508$		0			0
$509$	0.410670	+	1.53264i	0.410670	+	1.53264i	0.793353	+	0.608761i	$0.208333\pi$
$509$	0.410670	+	1.53264i	0.410670	+	1.53264i	−0.382683	+	0.923880i	$0.625000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.14196	+	0.426063i	2.14196	+	0.426063i
$516$		0			0
$517$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$521$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$522$		0			0
$523$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$523$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.86603	−	0.500000i	1.86603	−	0.500000i
$529$	−0.449843	−	0.586247i	−0.449843	−	0.586247i
$530$		0			0
$531$	−0.471477	−	2.37027i	−0.471477	−	2.37027i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.10531	−	3.15082i	−2.10531	−	3.15082i
$538$		0			0
$539$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$540$	−5.88666	+	1.17093i	−5.88666	+	1.17093i
$541$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$541$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$542$		0			0
$543$	−0.566715	+	0.496996i	−0.566715	+	0.496996i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	2.05816	+	0.852518i	2.05816	+	0.852518i
$556$		0			0
$557$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$557$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$563$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$564$	1.78480	−	0.739288i	1.78480	−	0.739288i
$565$	1.14330	+	3.36804i	1.14330	+	3.36804i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$569$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$570$		0			0
$571$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$571$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$572$		0			0
$573$	0.965926	+	0.258819i	0.965926	+	0.258819i
$574$		0			0
$575$	−2.76904	−	0.939963i	−2.76904	−	0.939963i
$576$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$577$	0.835400	+	0.732626i	0.835400	+	0.732626i	0.965926	−	0.258819i	$-0.0833333\pi$
$577$	0.835400	+	0.732626i	0.835400	+	0.732626i	−0.130526	+	0.991445i	$0.541667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.261052			0.261052
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.293353	+	0.257264i	0.293353	+	0.257264i	0.793353	−	0.608761i	$-0.208333\pi$
$587$	0.293353	+	0.257264i	0.293353	+	0.257264i	−0.500000	+	0.866025i	$0.666667\pi$
$588$	1.67303	−	0.965926i	1.67303	−	0.965926i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.641502	+	0.0420463i	0.641502	+	0.0420463i
$593$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$593$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.54206	+	2.89867i	2.54206	+	2.89867i
$598$		0			0
$599$	−0.206647	−	0.608761i	−0.206647	−	0.608761i	0.793353	−	0.608761i	$-0.208333\pi$
$599$	−0.206647	−	0.608761i	−0.206647	−	0.608761i	−1.00000			$\pi$
$600$		0			0
$601$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$601$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$602$		0			0
$603$	0.320515	+	0.158060i	0.320515	+	0.158060i
$604$		0			0
$605$	−0.793353	+	1.60876i	−0.793353	+	1.60876i
$606$		0			0
$607$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$607$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$613$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.608761	+	0.793353i	$0.291667\pi$
$617$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.382683	+	0.923880i	$0.375000\pi$
$618$		0			0
$619$	1.78990	−	0.882683i	1.78990	−	0.882683i	0.866025	−	0.500000i	$-0.166667\pi$
$619$	1.78990	−	0.882683i	1.78990	−	0.882683i	0.923880	−	0.382683i	$-0.125000\pi$
$620$	1.46593	+	2.97260i	1.46593	+	2.97260i
$621$	−3.31744	+	2.90932i	−3.31744	+	2.90932i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.64196	+	0.439963i	−1.64196	+	0.439963i
$626$		0			0
$627$		0			0
$628$	−0.293353	−	1.47479i	−0.293353	−	1.47479i
$629$		0			0
$630$		0			0
$631$	−0.793353	−	0.608761i	−0.793353	−	0.608761i	0.130526	−	0.991445i	$-0.458333\pi$
$631$	−0.793353	−	0.608761i	−0.793353	−	0.608761i	−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.487130	+	0.130526i	−0.487130	+	0.130526i
$637$		0			0
$638$		0			0
$639$	−1.66317	+	4.89953i	−1.66317	+	4.89953i
$640$		0			0
$641$	0.491445	+	0.996552i	0.491445	+	0.996552i	0.991445	+	0.130526i	$0.0416667\pi$
$641$	0.491445	+	0.996552i	0.491445	+	0.996552i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−0.793353	+	1.37413i	−0.793353	+	1.37413i	0.130526	+	0.991445i	$0.458333\pi$
$643$	−0.793353	+	1.37413i	−0.793353	+	1.37413i	−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.57313	−	0.207107i	1.57313	−	0.207107i	0.707107	−	0.707107i	$-0.250000\pi$
$647$	1.57313	−	0.207107i	1.57313	−	0.207107i	0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	0.867580	+	0.172572i	0.867580	+	0.172572i
$650$		0			0
$651$		0			0
$652$	0.707107	+	0.292893i	0.707107	+	0.292893i
$653$	0.0578541	−	0.882683i	0.0578541	−	0.882683i	−0.866025	−	0.500000i	$-0.833333\pi$
$653$	0.0578541	−	0.882683i	0.0578541	−	0.882683i	0.923880	−	0.382683i	$-0.125000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$659$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$660$	0.676037	−	3.39867i	0.676037	−	3.39867i
$661$	−0.241181	+	0.0999004i	−0.241181	+	0.0999004i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−0.241181	+	0.0999004i	−0.241181	+	0.0999004i	0.258819	+	0.965926i	$0.416667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.78135	−	0.247843i	−3.78135	−	0.247843i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$673$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$674$		0			0
$675$	−1.92043	+	7.16715i	−1.92043	+	7.16715i
$676$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$677$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$677$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.478235	−	1.78480i	0.478235	−	1.78480i	−0.130526	−	0.991445i	$-0.541667\pi$
$683$	0.478235	−	1.78480i	0.478235	−	1.78480i	0.608761	−	0.793353i	$-0.291667\pi$
$684$		0			0
$685$	−2.58326	+	1.49144i	−2.58326	+	1.49144i
$686$		0			0
$687$	−1.46593	−	3.53906i	−1.46593	−	3.53906i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$691$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$705$	0.226638	−	3.45783i	0.226638	−	3.45783i
$706$		0			0
$707$		0			0
$708$	−1.70521	+	0.111766i	−1.70521	+	0.111766i
$709$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$709$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i	−0.382683	+	0.923880i	$0.625000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.02598	+	1.35372i	2.02598	+	1.35372i
$714$		0			0
$715$		0			0
$716$	−1.75928	+	0.867580i	−1.75928	+	0.867580i
$717$		0			0
$718$		0			0
$719$	−0.357164	+	1.05217i	−0.357164	+	1.05217i	0.608761	+	0.793353i	$0.291667\pi$
$719$	−0.357164	+	1.05217i	−0.357164	+	1.05217i	−0.965926	+	0.258819i	$0.916667\pi$
$720$	0.320515	+	4.89011i	0.320515	+	4.89011i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	0.216773	+	0.324423i	0.216773	+	0.324423i
$725$		0			0
$726$		0			0
$727$	0.207107	+	0.158919i	0.207107	+	0.158919i	0.707107	−	0.707107i	$-0.250000\pi$
$727$	0.207107	+	0.158919i	0.207107	+	0.158919i	−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	2.63896	+	2.63896i	2.63896	+	2.63896i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$733$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$734$		0			0
$735$	−0.226638	−	3.45783i	−0.226638	−	3.45783i
$736$		0			0
$737$	−0.0983454	+	0.0862466i	−0.0983454	+	0.0862466i
$738$		0			0
$739$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$739$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$740$	0.576581	−	0.998667i	0.576581	−	0.998667i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.0999004	−	0.758819i	−0.0999004	−	0.758819i	−0.965926	−	0.258819i	$-0.916667\pi$
$751$	−0.0999004	−	0.758819i	−0.0999004	−	0.758819i	0.866025	−	0.500000i	$-0.166667\pi$
$752$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$753$	1.61818	−	3.28135i	1.61818	−	3.28135i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.389345	+	1.95737i	−0.389345	+	1.95737i	−0.130526	+	0.991445i	$0.541667\pi$
$757$	−0.389345	+	1.95737i	−0.389345	+	1.95737i	−0.258819	+	0.965926i	$0.583333\pi$
$758$		0			0
$759$	−0.818872	−	2.41232i	−0.818872	−	2.41232i
$760$		0			0
$761$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$761$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$762$		0			0
$763$		0			0
$764$	0.198092	−	0.478235i	0.198092	−	0.478235i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.739288	+	1.78480i	0.739288	+	1.78480i
$769$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$769$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$770$		0			0
$771$	−1.61387	−	1.41532i	−1.61387	−	1.41532i
$772$		0			0
$773$	0.758819	+	0.0999004i	0.758819	+	0.0999004i	0.500000	−	0.866025i	$-0.333333\pi$
$773$	0.758819	+	0.0999004i	0.758819	+	0.0999004i	0.258819	+	0.965926i	$0.416667\pi$
$774$		0			0
$775$	4.09745			4.09745
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−1.42388	−	1.24871i	−1.42388	−	1.24871i
$782$		0			0
$783$		0			0
$784$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$785$	−2.60531	−	0.698092i	−2.60531	−	0.698092i
$786$		0			0
$787$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$787$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.176481	+	0.887230i	−0.176481	+	0.887230i
$796$	1.65938	−	1.10876i	1.65938	−	1.10876i
$797$	0.576581	+	0.284338i	0.576581	+	0.284338i	0.707107	−	0.707107i	$-0.250000\pi$
$797$	0.576581	+	0.284338i	0.576581	+	0.284338i	−0.130526	+	0.991445i	$0.541667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.140392	−	0.210111i	0.140392	−	0.210111i
$805$		0			0
$806$		0			0
$807$	0.436749	−	3.31744i	0.436749	−	3.31744i
$808$		0			0
$809$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$809$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$811$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.03218	−	0.905198i	1.03218	−	0.905198i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$821$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$822$		0			0
$823$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$823$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$824$		0			0
$825$	−3.39867	−	2.60789i	−3.39867	−	2.60789i
$826$		0			0
$827$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$827$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$828$	2.00157	+	2.99556i	2.00157	+	2.99556i
$829$	−1.17604	−	1.53264i	−1.17604	−	1.53264i	−0.793353	−	0.608761i	$-0.791667\pi$
$829$	−1.17604	−	1.53264i	−1.17604	−	1.53264i	−0.382683	−	0.923880i	$-0.625000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.09136	−	5.35439i	3.09136	−	5.35439i
$838$		0			0
$839$	−1.25026	−	0.835400i	−1.25026	−	0.835400i	−0.258819	−	0.965926i	$-0.583333\pi$
$839$	−1.25026	−	0.835400i	−1.25026	−	0.835400i	−0.991445	+	0.130526i	$0.958333\pi$
$840$		0			0
$841$	0.991445	−	0.130526i	0.991445	−	0.130526i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.996552	+	1.49144i	−0.996552	+	1.49144i
$846$		0			0
$847$		0			0
$848$	0.0340742	+	0.258819i	0.0340742	+	0.258819i
$849$		0			0
$850$		0			0
$851$		−	0.847759i		−	0.847759i
$852$	3.28135	+	1.61818i	3.28135	+	1.61818i
$853$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$853$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	0.837633	−	0.284338i	0.837633	−	0.284338i	0.130526	−	0.991445i	$-0.458333\pi$
$859$	0.837633	−	0.284338i	0.837633	−	0.284338i	0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.866025	−	0.500000i	$-0.166667\pi$
$863$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.793353	+	0.608761i	$0.208333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−2.52409	+	1.04551i	−2.52409	+	1.04551i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$		0			0
$879$		0			0
$880$	−1.69855	−	0.576581i	−1.69855	−	0.576581i
$881$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$881$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$882$		0			0
$883$	−0.641502	−	0.0420463i	−0.641502	−	0.0420463i	−0.258819	−	0.965926i	$-0.583333\pi$
$883$	−0.641502	−	0.0420463i	−0.641502	−	0.0420463i	−0.382683	+	0.923880i	$0.625000\pi$
$884$		0			0
$885$	−1.17303	+	2.83195i	−1.17303	+	2.83195i
$886$		0			0
$887$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$887$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−3.44797	+	1.42819i	−3.44797	+	1.42819i
$892$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$893$		0			0
$894$		0			0
$895$			3.51856i			3.51856i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	5.59722	+	2.31844i	5.59722	+	2.31844i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.693897	−	0.0913533i	0.693897	−	0.0913533i
$906$		0			0
$907$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$907$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.284338	+	0.576581i	0.284338	+	0.576581i	0.991445	−	0.130526i	$-0.0416667\pi$
$911$	0.284338	+	0.576581i	0.284338	+	0.576581i	−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.91532	+	0.513210i	−1.91532	+	0.513210i
$917$		0			0
$918$		0			0
$919$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$919$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.792020	−	1.18534i	−0.792020	−	1.18534i
$926$		0			0
$927$	−3.21299	+	0.860919i	−3.21299	+	0.860919i
$928$		0			0
$929$	−0.123864	−	1.88981i	−0.123864	−	1.88981i	−0.382683	−	0.923880i	$-0.625000\pi$
$929$	−0.123864	−	1.88981i	−0.123864	−	1.88981i	0.258819	−	0.965926i	$-0.416667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	2.60531	−	1.28480i	2.60531	−	1.28480i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$	−0.436749	+	3.31744i	−0.436749	+	3.31744i
$940$	−1.75928	−	0.349942i	−1.75928	−	0.349942i
$941$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$941$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$942$		0			0
$943$		0			0
$944$	−0.0578541	+	0.882683i	−0.0578541	+	0.882683i
$945$		0			0
$946$		0			0
$947$	−0.491445	+	0.996552i	−0.491445	+	0.996552i	0.500000	+	0.866025i	$0.333333\pi$
$947$	−0.491445	+	0.996552i	−0.491445	+	0.996552i	−0.991445	+	0.130526i	$0.958333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.0492990	−	0.247843i	0.0492990	−	0.247843i
$952$		0			0
$953$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$953$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$954$		0			0
$955$	−0.612210	−	0.698092i	−0.612210	−	0.698092i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	3.45783	+	0.226638i	3.45783	+	0.226638i
$961$	−2.33195	−	0.624844i	−2.33195	−	0.624844i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$967$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$968$		0			0
$969$		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	$0.250000\pi$
$972$	3.06528	−	2.35207i	3.06528	−	2.35207i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.88981	+	0.641502i	1.88981	+	0.641502i	0.965926	+	0.258819i	$0.0833333\pi$
$977$	1.88981	+	0.641502i	1.88981	+	0.641502i	0.923880	+	0.382683i	$0.125000\pi$
$978$		0			0
$979$		0			0
$980$	−1.78990	−	0.117317i	−1.78990	−	0.117317i
$981$		0			0
$982$		0			0
$983$	−1.31587	+	1.50046i	−1.31587	+	1.50046i	−0.608761	+	0.793353i	$0.708333\pi$
$983$	−1.31587	+	1.50046i	−1.31587	+	1.50046i	−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.49144	−	0.735499i	−1.49144	−	0.735499i	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.49144	−	0.735499i	−1.49144	−	0.735499i	−0.991445	+	0.130526i	$0.958333\pi$
$992$		0			0
$993$	−0.333384	+	0.676037i	−0.333384	+	0.676037i
$994$		0			0
$995$	−0.467259	−	3.54918i	−0.467259	−	3.54918i
$996$		0			0
$997$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$997$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$998$		0			0
$999$	−2.14651	+	0.140690i	−2.14651	+	0.140690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1067.1.bg.a.967.1	yes	16
11.10	odd	2	CM	1067.1.bg.a.967.1	yes	16
97.32	even	48	inner	1067.1.bg.a.32.1	✓	16
1067.32	odd	48	inner	1067.1.bg.a.32.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1067.1.bg.a.32.1	✓	16	97.32	even	48	inner
1067.1.bg.a.32.1	✓	16	1067.32	odd	48	inner
1067.1.bg.a.967.1	yes	16	1.1	even	1	trivial
1067.1.bg.a.967.1	yes	16	11.10	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1067 = 11 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1067.bg (of order \(48\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(1.91532 + 0.252157i) q^{3} +(0.258819 - 0.965926i) q^{4} +(-1.34861 - 1.18270i) q^{5} +(2.63896 + 0.707107i) q^{9} +O(q^{10})\)	\(q+(1.91532 + 0.252157i) q^{3} +(0.258819 - 0.965926i) q^{4} +(-1.34861 - 1.18270i) q^{5} +(2.63896 + 0.707107i) q^{9} +(-0.608761 + 0.793353i) q^{11} +(0.739288 - 1.78480i) q^{12} +(-2.28480 - 2.60531i) q^{15} +(-0.866025 - 0.500000i) q^{16} +(-1.49144 + 0.996552i) q^{20} +(-0.583242 + 1.18270i) q^{23} +(0.289445 + 2.19855i) q^{25} +(3.09136 + 1.28048i) q^{27} +(0.241181 - 1.83195i) q^{31} +(-1.36603 + 1.36603i) q^{33} +(1.36603 - 2.36603i) q^{36} +(-0.576581 + 0.284338i) q^{37} +(0.608761 + 0.793353i) q^{44} +(-2.72263 - 4.07470i) q^{45} +(0.707107 + 0.707107i) q^{47} +(-1.53264 - 1.17604i) q^{48} +(0.793353 + 0.608761i) q^{49} +(-0.158919 - 0.207107i) q^{53} +(1.75928 - 0.349942i) q^{55} +(-0.391239 - 0.793353i) q^{59} +(-3.10789 + 1.53264i) q^{60} +(-0.707107 + 0.707107i) q^{64} +(0.128293 + 0.0255190i) q^{67} +(-1.41532 + 2.11818i) q^{69} +(-0.123864 + 1.88981i) q^{71} +4.28393i q^{75} +(0.576581 + 1.69855i) q^{80} +(3.23205 + 1.86603i) q^{81} +(0.991445 + 0.869474i) q^{92} +(0.923880 - 3.44797i) q^{93} +(-0.793353 + 0.608761i) q^{97} +(-2.16748 + 1.66317i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{15} - 8 q^{20} + 8 q^{31} - 8 q^{33} + 8 q^{36} - 24 q^{45} - 16 q^{59} + 8 q^{69} + 24 q^{81}+O(q^{100}) \) 16 * q - 8 * q^15 - 8 * q^20 + 8 * q^31 - 8 * q^33 + 8 * q^36 - 24 * q^45 - 16 * q^59 + 8 * q^69 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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