LMFDB - Embedded newform 1067.1.bg.a.516.1

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			516.1
Root			$-0.608761 - 0.793353i$ of defining polynomial
Character	$\chi$	$=$	1067.516
Dual form			1067.1.bg.a.945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.252157 + 1.91532i) q^{3} +(-0.258819 + 0.965926i) q^{4} +(0.0420463 + 0.641502i) q^{5} +(-2.63896 - 0.707107i) q^{9} +O(q^{10})$	\(q+(-0.252157 + 1.91532i) q^{3} +(-0.258819 + 0.965926i) q^{4} +(0.0420463 + 0.641502i) q^{5} +(-2.63896 - 0.707107i) q^{9} +(0.793353 + 0.608761i) q^{11} +(-1.78480 - 0.739288i) q^{12} +(-1.23929 - 0.0812272i) q^{15} +(-0.866025 - 0.500000i) q^{16} +(-0.630526 - 0.125419i) q^{20} +(1.88981 - 0.641502i) q^{23} +(0.581687 - 0.0765806i) q^{25} +(1.28048 - 3.09136i) q^{27} +(0.758819 + 0.0999004i) q^{31} +(-1.36603 + 1.36603i) q^{33} +(1.36603 - 2.36603i) q^{36} +(-0.284338 + 0.837633i) q^{37} +(-0.793353 + 0.608761i) q^{44} +(0.342652 - 1.72263i) q^{45} +(-0.707107 - 0.707107i) q^{47} +(1.17604 - 1.53264i) q^{48} +(0.608761 - 0.793353i) q^{49} +(-1.57313 + 1.20711i) q^{53} +(-0.357164 + 0.534534i) q^{55} +(-1.79335 - 0.608761i) q^{59} +(0.399211 - 1.17604i) q^{60} +(0.707107 - 0.707107i) q^{64} +(0.732626 + 1.09645i) q^{67} +(0.752157 + 3.78135i) q^{69} +(-1.18270 - 1.34861i) q^{71} +1.13343i q^{75} +(0.284338 - 0.576581i) q^{80} +(3.23205 + 1.86603i) q^{81} +(0.130526 + 1.99144i) q^{92} +(-0.382683 + 1.42819i) q^{93} +(-0.608761 - 0.793353i) q^{97} +(-1.66317 - 2.16748i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1067\mathbb{Z}\right)^\times$.

$n$	$199$	$486$
$\chi(n)$	$e\left(\frac{23}{48}\right)$	$-1$

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		−0.608761	−	0.793353i	$-0.708333\pi$
$2$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$3$	−0.252157	+	1.91532i	−0.252157	+	1.91532i	0.130526	+	0.991445i	$0.458333\pi$
$3$	−0.252157	+	1.91532i	−0.252157	+	1.91532i	−0.382683	+	0.923880i	$0.625000\pi$
$4$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$5$	0.0420463	+	0.641502i	0.0420463	+	0.641502i	0.965926	+	0.258819i	$0.0833333\pi$
$5$	0.0420463	+	0.641502i	0.0420463	+	0.641502i	−0.923880	+	0.382683i	$0.875000\pi$
$6$		0			0
$7$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$7$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$8$		0			0
$9$	−2.63896	−	0.707107i	−2.63896	−	0.707107i
$10$		0			0
$11$	0.793353	+	0.608761i	0.793353	+	0.608761i
$11$	0.793353	+	0.608761i	0.793353	+	0.608761i
$12$	−1.78480	−	0.739288i	−1.78480	−	0.739288i
$13$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$13$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$14$		0			0
$15$	−1.23929	−	0.0812272i	−1.23929	−	0.0812272i
$16$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$17$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$17$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$18$		0			0
$19$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$19$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$20$	−0.630526	−	0.125419i	−0.630526	−	0.125419i
$21$		0			0
$22$		0			0
$23$	1.88981	−	0.641502i	1.88981	−	0.641502i	0.923880	−	0.382683i	$-0.125000\pi$
$23$	1.88981	−	0.641502i	1.88981	−	0.641502i	0.965926	−	0.258819i	$-0.0833333\pi$
$24$		0			0
$25$	0.581687	−	0.0765806i	0.581687	−	0.0765806i
$26$		0			0
$27$	1.28048	−	3.09136i	1.28048	−	3.09136i
$28$		0			0
$29$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$29$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$30$		0			0
$31$	0.758819	+	0.0999004i	0.758819	+	0.0999004i	0.500000	−	0.866025i	$-0.333333\pi$
$31$	0.758819	+	0.0999004i	0.758819	+	0.0999004i	0.258819	+	0.965926i	$0.416667\pi$
$32$		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
$37$	−0.284338	+	0.837633i	−0.284338	+	0.837633i	0.707107	+	0.707107i	$0.250000\pi$
$37$	−0.284338	+	0.837633i	−0.284338	+	0.837633i	−0.991445	+	0.130526i	$0.958333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$41$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$42$		0			0
$43$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$43$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$44$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$45$	0.342652	−	1.72263i	0.342652	−	1.72263i
$46$		0			0
$47$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$47$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$48$	1.17604	−	1.53264i	1.17604	−	1.53264i
$49$	0.608761	−	0.793353i	0.608761	−	0.793353i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.57313	+	1.20711i	−1.57313	+	1.20711i	−0.707107	+	0.707107i	$0.750000\pi$
$53$	−1.57313	+	1.20711i	−1.57313	+	1.20711i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	−0.357164	+	0.534534i	−0.357164	+	0.534534i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.79335	−	0.608761i	−1.79335	−	0.608761i	−0.793353	−	0.608761i	$-0.791667\pi$
$59$	−1.79335	−	0.608761i	−1.79335	−	0.608761i	−1.00000			$\pi$
$60$	0.399211	−	1.17604i	0.399211	−	1.17604i
$61$		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
$67$	0.732626	+	1.09645i	0.732626	+	1.09645i	0.991445	+	0.130526i	$0.0416667\pi$
$67$	0.732626	+	1.09645i	0.732626	+	1.09645i	−0.258819	+	0.965926i	$0.583333\pi$
$68$		0			0
$69$	0.752157	+	3.78135i	0.752157	+	3.78135i
$70$		0			0
$71$	−1.18270	−	1.34861i	−1.18270	−	1.34861i	−0.923880	−	0.382683i	$-0.875000\pi$
$71$	−1.18270	−	1.34861i	−1.18270	−	1.34861i	−0.258819	−	0.965926i	$-0.583333\pi$
$72$		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
$75$			1.13343i			1.13343i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$79$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$80$	0.284338	−	0.576581i	0.284338	−	0.576581i
$81$	3.23205	+	1.86603i	3.23205	+	1.86603i
$82$		0			0
$83$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$83$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$89$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$90$		0			0
$91$		0			0
$92$	0.130526	+	1.99144i	0.130526	+	1.99144i
$93$	−0.382683	+	1.42819i	−0.382683	+	1.42819i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$97$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$98$		0			0
$99$	−1.66317	−	2.16748i	−1.66317	−	2.16748i
$100$	−0.0765806	+	0.581687i	−0.0765806	+	0.581687i
$101$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$101$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$102$		0			0
$103$	1.37413	−	0.793353i	1.37413	−	0.793353i	0.382683	−	0.923880i	$-0.375000\pi$
$103$	1.37413	−	0.793353i	1.37413	−	0.793353i	0.991445	+	0.130526i	$0.0416667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$107$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$108$	2.65461	+	2.03696i	2.65461	+	2.03696i
$109$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$109$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$	−1.53264	−	0.755815i	−1.53264	−	0.755815i
$112$		0			0
$113$	−0.226078	−	0.130526i	−0.226078	−	0.130526i	0.382683	−	0.923880i	$-0.375000\pi$
$113$	−0.226078	−	0.130526i	−0.226078	−	0.130526i	−0.608761	+	0.793353i	$0.708333\pi$
$114$		0			0
$115$	0.490985	+	1.18534i	0.490985	+	1.18534i
$116$		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
$124$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$125$	0.199004	+	1.00046i	0.199004	+	1.00046i
$126$		0			0
$127$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$127$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$131$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$132$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$133$		0			0
$134$		0			0
$135$	2.03696	+	0.691453i	2.03696	+	0.691453i
$136$		0			0
$137$	0.867580	+	1.75928i	0.867580	+	1.75928i	0.608761	+	0.793353i	$0.291667\pi$
$137$	0.867580	+	1.75928i	0.867580	+	1.75928i	0.258819	+	0.965926i	$0.416667\pi$
$138$		0			0
$139$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$139$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$140$		0			0
$141$	1.53264	−	1.17604i	1.53264	−	1.17604i
$142$		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
$148$	−0.735499	−	0.491445i	−0.735499	−	0.491445i
$149$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$149$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$150$		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
$155$	−0.0321808	+	0.490985i	−0.0321808	+	0.490985i
$156$		0			0
$157$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i	−0.965926	−	0.258819i	$-0.916667\pi$
$157$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i	0.923880	+	0.382683i	$0.125000\pi$
$158$		0			0
$159$	−1.91532	−	3.31744i	−1.91532	−	3.31744i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.83195	+	0.241181i	1.83195	+	0.241181i	0.965926	−	0.258819i	$-0.0833333\pi$
$163$	1.83195	+	0.241181i	1.83195	+	0.241181i	0.866025	+	0.500000i	$0.166667\pi$
$164$		0			0
$165$	−0.933745	−	0.818872i	−0.933745	−	0.818872i
$166$		0			0
$167$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$167$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	0.991445	−	0.130526i	0.991445	−	0.130526i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$173$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$174$		0			0
$175$		0			0
$176$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$177$	1.61818	−	3.28135i	1.61818	−	3.28135i
$178$		0			0
$179$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i	−0.608761	+	0.793353i	$0.708333\pi$
$180$	1.57525	+	0.776826i	1.57525	+	0.776826i
$181$	−1.65938	+	0.108761i	−1.65938	+	0.108761i	−0.866025	−	0.500000i	$-0.833333\pi$
$181$	−1.65938	+	0.108761i	−1.65938	+	0.108761i	−0.793353	+	0.608761i	$0.791667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.549299	−	0.147184i	−0.549299	−	0.147184i
$186$		0			0
$187$		0			0
$188$	0.866025	−	0.500000i	0.866025	−	0.500000i
$189$		0			0
$190$		0			0
$191$	−0.0675653	+	0.513210i	−0.0675653	+	0.513210i	0.923880	+	0.382683i	$0.125000\pi$
$191$	−0.0675653	+	0.513210i	−0.0675653	+	0.513210i	−0.991445	+	0.130526i	$0.958333\pi$
$192$	1.17604	+	1.53264i	1.17604	+	1.53264i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	0.608761	+	0.793353i	0.608761	+	0.793353i
$197$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$197$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$198$		0			0
$199$	−0.0983454	−	1.50046i	−0.0983454	−	1.50046i	−0.707107	−	0.707107i	$-0.750000\pi$
$199$	−0.0983454	−	1.50046i	−0.0983454	−	1.50046i	0.608761	−	0.793353i	$-0.291667\pi$
$200$		0			0
$201$	−2.28480	+	1.12674i	−2.28480	+	1.12674i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.44073	+	0.356604i	−5.44073	+	0.356604i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$211$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$212$	−0.758819	−	1.83195i	−0.758819	−	1.83195i
$213$	2.88125	−	1.92519i	2.88125	−	1.92519i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.423880	−	0.483342i	−0.423880	−	0.483342i
$221$		0			0
$222$		0			0
$223$	0.835400	+	0.732626i	0.835400	+	0.732626i	0.965926	−	0.258819i	$-0.0833333\pi$
$223$	0.835400	+	0.732626i	0.835400	+	0.732626i	−0.130526	+	0.991445i	$0.541667\pi$
$224$		0			0
$225$	−1.58920	−	0.209222i	−1.58920	−	0.209222i
$226$		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
$229$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	0.793353	−	0.608761i	$-0.208333\pi$
$229$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	−0.923880	+	0.382683i	$0.875000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$233$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$234$		0			0
$235$	0.423880	−	0.483342i	0.423880	−	0.483342i
$236$	1.05217	−	1.57469i	1.05217	−	1.57469i
$237$		0			0
$238$		0			0
$239$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$239$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$240$	1.03264	+	0.689989i	1.03264	+	0.689989i
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$		0			0
$243$	−2.35207	+	3.06528i	−2.35207	+	3.06528i
$244$		0			0
$245$	0.534534	+	0.357164i	0.534534	+	0.357164i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.793353	−	1.60876i	−0.793353	−	1.60876i	−0.793353	−	0.608761i	$-0.791667\pi$
$251$	−0.793353	−	1.60876i	−0.793353	−	1.60876i		−	1.00000i	$-0.5\pi$
$252$		0			0
$253$	1.88981	+	0.641502i	1.88981	+	0.641502i
$254$		0			0
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$257$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$263$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$264$		0			0
$265$	−0.840506	−	0.958414i	−0.840506	−	0.958414i
$266$		0			0
$267$		0			0
$268$	−1.24871	+	0.423880i	−1.24871	+	0.423880i
$269$		−	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		−0.980785	−	0.195090i	$-0.937500\pi$
$271$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.508103	+	0.293353i	0.508103	+	0.293353i
$276$	−3.84718	−	0.252157i	−3.84718	−	0.252157i
$277$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$277$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$278$		0			0
$279$	−1.93185	−	0.800199i	−1.93185	−	0.800199i
$280$		0			0
$281$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$281$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$282$		0			0
$283$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$283$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$284$	1.60876	−	0.793353i	1.60876	−	0.793353i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$290$		0			0
$291$	1.67303	−	0.965926i	1.67303	−	0.965926i
$292$		0			0
$293$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$293$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$294$		0			0
$295$	0.315118	−	1.17604i	0.315118	−	1.17604i
$296$		0			0
$297$	2.89778	−	1.67303i	2.89778	−	1.67303i
$298$		0			0
$299$		0			0
$300$	−1.09481	−	0.293353i	−1.09481	−	0.293353i
$301$		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
$309$	1.17303	+	2.83195i	1.17303	+	2.83195i
$310$		0			0
$311$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i	0.130526	−	0.991445i	$-0.458333\pi$
$311$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i	−0.258819	+	0.965926i	$0.583333\pi$
$312$		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
$317$	0.869474	+	0.991445i	0.869474	+	0.991445i	1.00000			$0$
$317$	0.869474	+	0.991445i	0.869474	+	0.991445i	−0.130526	+	0.991445i	$0.541667\pi$
$318$		0			0
$319$		0			0
$320$	0.483342	+	0.423880i	0.483342	+	0.423880i
$321$		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
$331$	0.735499	+	1.49144i	0.735499	+	1.49144i	0.866025	+	0.500000i	$0.166667\pi$
$331$	0.735499	+	1.49144i	0.735499	+	1.49144i	−0.130526	+	0.991445i	$0.541667\pi$
$332$		0			0
$333$	1.34265	−	2.00942i	1.34265	−	2.00942i
$334$		0			0
$335$	−0.672572	+	0.516083i	−0.672572	+	0.516083i
$336$		0			0
$337$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$337$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$338$		0			0
$339$	0.307007	−	0.400100i	0.307007	−	0.400100i
$340$		0			0
$341$	0.541196	+	0.541196i	0.541196	+	0.541196i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.39412	+	0.641502i	−2.39412	+	0.641502i
$346$		0			0
$347$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$347$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$348$		0			0
$349$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$349$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.793353	+	0.608761i	$0.208333\pi$
$353$	0.923880	+	1.60021i	0.923880	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$354$		0			0
$355$	0.815408	−	0.815408i	0.815408	−	0.815408i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$359$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$360$		0			0
$361$	0.382683	−	0.923880i	0.382683	−	0.923880i
$362$		0			0
$363$	−1.91532	+	0.252157i	−1.91532	+	0.252157i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.576581	−	1.69855i	−0.576581	−	1.69855i	−0.707107	−	0.707107i	$-0.750000\pi$
$367$	−0.576581	−	1.69855i	−0.576581	−	1.69855i	0.130526	−	0.991445i	$-0.458333\pi$
$368$	−1.95737	−	0.389345i	−1.95737	−	0.389345i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.28048	−	0.739288i	−1.28048	−	0.739288i
$373$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$373$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$374$		0			0
$375$	−1.96639	+	0.128884i	−1.96639	+	0.128884i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.78480	+	0.478235i	1.78480	+	0.478235i	0.991445	−	0.130526i	$-0.0416667\pi$
$379$	1.78480	+	0.478235i	1.78480	+	0.478235i	0.793353	+	0.608761i	$0.208333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.128293	−	1.95737i	−0.128293	−	1.95737i	−0.258819	−	0.965926i	$-0.583333\pi$
$383$	−0.128293	−	1.95737i	−0.128293	−	1.95737i	0.130526	−	0.991445i	$-0.458333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.923880	−	0.382683i	0.923880	−	0.382683i
$389$	−0.261052			−0.261052			−0.130526	−	0.991445i	$-0.541667\pi$
$389$	−0.261052			−0.261052			−0.130526	+	0.991445i	$0.541667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	2.52409	−	1.04551i	2.52409	−	1.04551i
$397$	−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
$400$	−0.542046	−	0.224523i	−0.542046	−	0.224523i
$401$	−1.31587	+	0.0862466i	−1.31587	+	0.0862466i	−0.707107	−	0.707107i	$-0.750000\pi$
$401$	−1.31587	+	0.0862466i	−1.31587	+	0.0862466i	−0.608761	+	0.793353i	$0.708333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.06116	+	2.15183i	−1.06116	+	2.15183i
$406$		0			0
$407$	−0.735499	+	0.491445i	−0.735499	+	0.491445i
$408$		0			0
$409$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$409$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$410$		0			0
$411$	−3.58836	+	1.21808i	−3.58836	+	1.21808i
$412$	0.410670	+	1.53264i	0.410670	+	1.53264i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.20711	+	0.158919i	1.20711	+	0.158919i	0.707107	−	0.707107i	$-0.250000\pi$
$419$	1.20711	+	0.158919i	1.20711	+	0.158919i	0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−0.860919	+	0.860919i	−0.860919	+	0.860919i	−0.991445	−	0.130526i	$-0.958333\pi$
$421$	−0.860919	+	0.860919i	−0.860919	+	0.860919i	0.130526	+	0.991445i	$0.458333\pi$
$422$		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	$0.916667\pi$
$432$	−2.65461	+	2.03696i	−2.65461	+	2.03696i
$433$	−0.349942	+	1.75928i	−0.349942	+	1.75928i	0.258819	+	0.965926i	$0.416667\pi$
$433$	−0.349942	+	1.75928i	−0.349942	+	1.75928i	−0.608761	+	0.793353i	$0.708333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$439$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$440$		0			0
$441$	−2.16748	+	1.66317i	−2.16748	+	1.66317i
$442$		0			0
$443$	1.05217	−	1.57469i	1.05217	−	1.57469i	0.258819	−	0.965926i	$-0.416667\pi$
$443$	1.05217	−	1.57469i	1.05217	−	1.57469i	0.793353	−	0.608761i	$-0.208333\pi$
$444$	1.12674	−	1.28480i	1.12674	−	1.28480i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$449$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$450$		0			0
$451$		0			0
$452$	0.184592	−	0.184592i	0.184592	−	0.184592i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$457$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$458$		0			0
$459$		0			0
$460$	−1.27203	+	0.167466i	−1.27203	+	0.167466i
$461$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$461$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$462$		0			0
$463$		−	1.98289i		−	1.98289i	−0.130526	−	0.991445i	$-0.541667\pi$
$463$		−	1.98289i		−	1.98289i	0.130526	−	0.991445i	$-0.458333\pi$
$464$		0			0
$465$	−0.932280	−	0.185442i	−0.932280	−	0.185442i
$466$		0			0
$467$	−0.465926	−	1.12484i	−0.465926	−	1.12484i	−0.965926	−	0.258819i	$-0.916667\pi$
$467$	−0.465926	−	1.12484i	−0.465926	−	1.12484i	0.500000	−	0.866025i	$-0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.226638	−	0.111766i	−0.226638	−	0.111766i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	5.00498	−	2.07313i	5.00498	−	2.07313i
$478$		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
$485$	0.483342	−	0.423880i	0.483342	−	0.423880i
$486$		0			0
$487$	0.608761	+	0.793353i	0.608761	+	0.793353i	0.991445	−	0.130526i	$-0.0416667\pi$
$487$	0.608761	+	0.793353i	0.608761	+	0.793353i	−0.382683	+	0.923880i	$0.625000\pi$
$488$		0			0
$489$	−0.923880	+	3.44797i	−0.923880	+	3.44797i
$490$		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
$495$	1.32051	−	1.15806i	1.32051	−	1.15806i
$496$	−0.607206	−	0.465926i	−0.607206	−	0.465926i
$497$		0			0
$498$		0			0
$499$	−1.75928	−	0.867580i	−1.75928	−	0.867580i	−0.965926	−	0.258819i	$-0.916667\pi$
$499$	−1.75928	−	0.867580i	−1.75928	−	0.867580i	−0.793353	−	0.608761i	$-0.791667\pi$
$500$	−1.01788	−	0.0667151i	−1.01788	−	0.0667151i
$501$		0			0
$502$		0			0
$503$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$503$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1.93185i			1.93185i
$508$		0			0
$509$	−0.315118	−	1.17604i	−0.315118	−	1.17604i	−0.923880	−	0.382683i	$-0.875000\pi$
$509$	−0.315118	−	1.17604i	−0.315118	−	1.17604i	0.608761	−	0.793353i	$-0.291667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.566715	+	0.848149i	0.566715	+	0.848149i
$516$		0			0
$517$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$521$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$522$		0			0
$523$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$523$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.86603	−	0.500000i	1.86603	−	0.500000i
$529$	2.36649	−	1.81587i	2.36649	−	1.81587i
$530$		0			0
$531$	4.30213	+	2.87459i	4.30213	+	2.87459i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.418773	−	2.10531i	0.418773	−	2.10531i
$538$		0			0
$539$	0.965926	−	0.258819i	0.965926	−	0.258819i
$540$	−1.19510	+	1.78859i	−1.19510	+	1.78859i
$541$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$541$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$542$		0			0
$543$	0.210111	−	3.20567i	0.210111	−	3.20567i
$544$		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	$0.333333\pi$
$548$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.420415	−	1.01497i	0.420415	−	1.01497i
$556$		0			0
$557$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$557$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$563$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$564$	0.739288	+	1.78480i	0.739288	+	1.78480i
$565$	0.0742271	−	0.150518i	0.0742271	−	0.150518i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$569$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$570$		0			0
$571$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$571$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$572$		0			0
$573$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$574$		0			0
$575$	1.05015	−	0.517876i	1.05015	−	0.517876i
$576$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$577$	0.0255190	+	0.389345i	0.0255190	+	0.389345i	0.991445	+	0.130526i	$0.0416667\pi$
$577$	0.0255190	+	0.389345i	0.0255190	+	0.389345i	−0.965926	+	0.258819i	$0.916667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.98289			−1.98289
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.108761	+	1.65938i	0.108761	+	1.65938i	0.608761	+	0.793353i	$0.291667\pi$
$587$	0.108761	+	1.65938i	0.108761	+	1.65938i	−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
$592$	0.665060	−	0.583242i	0.665060	−	0.583242i
$593$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$593$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.89867	+	0.189989i	2.89867	+	0.189989i
$598$		0			0
$599$	−0.391239	+	0.793353i	−0.391239	+	0.793353i	0.608761	+	0.793353i	$0.291667\pi$
$599$	−0.391239	+	0.793353i	−0.391239	+	0.793353i	−1.00000			$\pi$
$600$		0			0
$601$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$601$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$602$		0			0
$603$	−1.15806	−	3.41154i	−1.15806	−	3.41154i
$604$		0			0
$605$	−0.608761	+	0.206647i	−0.608761	+	0.206647i
$606$		0			0
$607$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$607$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$613$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.130526	+	0.226078i	0.130526	+	0.226078i	0.923880	−	0.382683i	$-0.125000\pi$
$617$	0.130526	+	0.226078i	0.130526	+	0.226078i	−0.793353	+	0.608761i	$0.791667\pi$
$618$		0			0
$619$	0.483342	−	1.42388i	0.483342	−	1.42388i	−0.382683	−	0.923880i	$-0.625000\pi$
$619$	0.483342	−	1.42388i	0.483342	−	1.42388i	0.866025	−	0.500000i	$-0.166667\pi$
$620$	−0.465926	−	0.158161i	−0.465926	−	0.158161i
$621$	0.436749	−	6.66350i	0.436749	−	6.66350i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0667151	+	0.0178763i	−0.0667151	+	0.0178763i
$626$		0			0
$627$		0			0
$628$	−0.108761	−	0.0726721i	−0.108761	−	0.0726721i
$629$		0			0
$630$		0			0
$631$	−0.608761	+	0.793353i	−0.608761	+	0.793353i	−0.991445	−	0.130526i	$-0.958333\pi$
$631$	−0.608761	+	0.793353i	−0.608761	+	0.793353i	0.382683	+	0.923880i	$0.375000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	3.70012	−	0.991445i	3.70012	−	0.991445i
$637$		0			0
$638$		0			0
$639$	2.16748	+	4.39522i	2.16748	+	4.39522i
$640$		0			0
$641$	−0.369474	−	0.125419i	−0.369474	−	0.125419i	0.130526	−	0.991445i	$-0.458333\pi$
$641$	−0.369474	−	0.125419i	−0.369474	−	0.125419i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	0.382683	+	0.923880i	$0.375000\pi$
$643$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	−0.991445	+	0.130526i	$0.958333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.158919	+	1.20711i	0.158919	+	1.20711i	0.866025	+	0.500000i	$0.166667\pi$
$647$	0.158919	+	1.20711i	0.158919	+	1.20711i	−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$	−1.05217	−	1.57469i	−1.05217	−	1.57469i
$650$		0			0
$651$		0			0
$652$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$653$	−1.24871	−	1.42388i	−1.24871	−	1.42388i	−0.866025	−	0.500000i	$-0.833333\pi$
$653$	−1.24871	−	1.42388i	−1.24871	−	1.42388i	−0.382683	−	0.923880i	$-0.625000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$659$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$660$	1.03264	−	0.689989i	1.03264	−	0.689989i
$661$	−0.758819	−	1.83195i	−0.758819	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−0.758819	−	1.83195i	−0.758819	−	1.83195i	−0.258819	−	0.965926i	$-0.583333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.61387	+	1.41532i	−1.61387	+	1.41532i
$670$		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
$675$	0.508103	−	1.89627i	0.508103	−	1.89627i
$676$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$677$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$677$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.198092	−	0.739288i	0.198092	−	0.739288i	−0.793353	−	0.608761i	$-0.791667\pi$
$683$	0.198092	−	0.739288i	0.198092	−	0.739288i	0.991445	−	0.130526i	$-0.0416667\pi$
$684$		0			0
$685$	−1.09210	+	0.630526i	−1.09210	+	0.630526i
$686$		0			0
$687$	0.465926	−	0.192993i	0.465926	−	0.192993i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$691$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$692$		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
$704$	0.991445	−	0.130526i	0.991445	−	0.130526i
$705$	0.818872	+	0.933745i	0.818872	+	0.933745i
$706$		0			0
$707$		0			0
$708$	2.75072	+	2.41232i	2.75072	+	2.41232i
$709$	−0.923880	−	1.38268i	−0.923880	−	1.38268i	−0.923880	−	0.382683i	$-0.875000\pi$
$709$	−0.923880	−	1.38268i	−0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.49811	−	0.297992i	1.49811	−	0.297992i
$714$		0			0
$715$		0			0
$716$	0.357164	−	1.05217i	0.357164	−	1.05217i
$717$		0			0
$718$		0			0
$719$	0.172572	+	0.349942i	0.172572	+	0.349942i	0.965926	−	0.258819i	$-0.0833333\pi$
$719$	0.172572	+	0.349942i	0.172572	+	0.349942i	−0.793353	+	0.608761i	$0.791667\pi$
$720$	−1.15806	+	1.32051i	−1.15806	+	1.32051i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	0.324423	−	1.63099i	0.324423	−	1.63099i
$725$		0			0
$726$		0			0
$727$	−1.20711	+	1.57313i	−1.20711	+	1.57313i	−0.500000	+	0.866025i	$0.666667\pi$
$727$	−1.20711	+	1.57313i	−1.20711	+	1.57313i	−0.707107	+	0.707107i	$0.750000\pi$
$728$		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
$735$	−0.818872	+	0.933745i	−0.818872	+	0.933745i
$736$		0			0
$737$	−0.0862466	+	1.31587i	−0.0862466	+	1.31587i
$738$		0			0
$739$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$739$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$740$	0.284338	−	0.492488i	0.284338	−	0.492488i
$741$		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
$751$	1.83195	−	0.241181i	1.83195	−	0.241181i	0.866025	−	0.500000i	$-0.166667\pi$
$751$	1.83195	−	0.241181i	1.83195	−	0.241181i	0.965926	+	0.258819i	$0.0833333\pi$
$752$	0.258819	+	0.965926i	0.258819	+	0.965926i
$753$	3.28135	−	1.11387i	3.28135	−	1.11387i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.25026	−	0.835400i	1.25026	−	0.835400i	0.258819	−	0.965926i	$-0.416667\pi$
$757$	1.25026	−	0.835400i	1.25026	−	0.835400i	0.991445	+	0.130526i	$0.0416667\pi$
$758$		0			0
$759$	−1.70521	+	3.45783i	−1.70521	+	3.45783i
$760$		0			0
$761$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$761$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$762$		0			0
$763$		0			0
$764$	−0.478235	−	0.198092i	−0.478235	−	0.198092i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.78480	+	0.739288i	−1.78480	+	0.739288i
$769$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$769$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$770$		0			0
$771$	0.0492990	+	0.752157i	0.0492990	+	0.752157i
$772$		0			0
$773$	0.241181	−	1.83195i	0.241181	−	1.83195i	−0.258819	−	0.965926i	$-0.583333\pi$
$773$	0.241181	−	1.83195i	0.241181	−	1.83195i	0.500000	−	0.866025i	$-0.333333\pi$
$774$		0			0
$775$	0.449046			0.449046
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.117317	−	1.78990i	−0.117317	−	1.78990i
$782$		0			0
$783$		0			0
$784$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$785$	−0.0812272	−	0.0217648i	−0.0812272	−	0.0217648i
$786$		0			0
$787$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$787$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	2.04761	−	1.36817i	2.04761	−	1.36817i
$796$	1.47479	+	0.293353i	1.47479	+	0.293353i
$797$	0.284338	+	0.837633i	0.284338	+	0.837633i	0.991445	+	0.130526i	$0.0416667\pi$
$797$	0.284338	+	0.837633i	0.284338	+	0.837633i	−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.496996	−	2.49857i	−0.496996	−	2.49857i
$805$		0			0
$806$		0			0
$807$	3.31744	+	0.436749i	3.31744	+	0.436749i
$808$		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
$815$	−0.0776914	+	1.18534i	−0.0776914	+	1.18534i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$821$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$822$		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
$825$	−0.689989	+	0.899211i	−0.689989	+	0.899211i
$826$		0			0
$827$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$827$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$828$	1.06371	−	5.34764i	1.06371	−	5.34764i
$829$	−1.53264	+	1.17604i	−1.53264	+	1.17604i	−0.608761	+	0.793353i	$0.708333\pi$
$829$	−1.53264	+	1.17604i	−1.53264	+	1.17604i	−0.923880	+	0.382683i	$0.875000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.28048	−	2.21786i	1.28048	−	2.21786i
$838$		0			0
$839$	0.128293	−	0.0255190i	0.128293	−	0.0255190i	−0.130526	−	0.991445i	$-0.541667\pi$
$839$	0.128293	−	0.0255190i	0.128293	−	0.0255190i	0.258819	+	0.965926i	$0.416667\pi$
$840$		0			0
$841$	0.130526	+	0.991445i	0.130526	+	0.991445i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.125419	+	0.630526i	0.125419	+	0.630526i
$846$		0			0
$847$		0			0
$848$	1.96593	−	0.258819i	1.96593	−	0.258819i
$849$		0			0
$850$		0			0
$851$			1.76537i			1.76537i
$852$	1.11387	+	3.28135i	1.11387	+	3.28135i
$853$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$853$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$854$		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
$859$	−1.69855	−	0.837633i	−1.69855	−	0.837633i	−0.991445	−	0.130526i	$-0.958333\pi$
$859$	−1.69855	−	0.837633i	−1.69855	−	0.837633i	−0.707107	−	0.707107i	$-0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.47479	−	1.29335i	1.47479	−	1.29335i	0.608761	−	0.793353i	$-0.291667\pi$
$863$	1.47479	−	1.29335i	1.47479	−	1.29335i	0.866025	−	0.500000i	$-0.166667\pi$
$864$		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
$873$	1.04551	+	2.52409i	1.04551	+	2.52409i
$874$		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
$880$	0.576581	−	0.284338i	0.576581	−	0.284338i
$881$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$881$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$882$		0			0
$883$	−0.665060	+	0.583242i	−0.665060	+	0.583242i	−0.923880	−	0.382683i	$-0.875000\pi$
$883$	−0.665060	+	0.583242i	−0.665060	+	0.583242i	0.258819	+	0.965926i	$0.416667\pi$
$884$		0			0
$885$	2.17303	+	0.900100i	2.17303	+	0.900100i
$886$		0			0
$887$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$887$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.42819	+	3.44797i	1.42819	+	3.44797i
$892$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$893$		0			0
$894$		0			0
$895$		−	0.714329i		−	0.714329i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.613408	−	1.48090i	0.613408	−	1.48090i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.139541	−	1.05992i	−0.139541	−	1.05992i
$906$		0			0
$907$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i	−0.382683	−	0.923880i	$-0.625000\pi$
$907$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i			1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.837633	+	0.284338i	0.837633	+	0.284338i	0.707107	−	0.707107i	$-0.250000\pi$
$911$	0.837633	+	0.284338i	0.837633	+	0.284338i	0.130526	+	0.991445i	$0.458333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.252157	−	0.0675653i	0.252157	−	0.0675653i
$917$		0			0
$918$		0			0
$919$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$919$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.101249	+	0.509015i	−0.101249	+	0.509015i
$926$		0			0
$927$	−4.18725	+	1.12197i	−4.18725	+	1.12197i
$928$		0			0
$929$	−1.18270	+	1.34861i	−1.18270	+	1.34861i	−0.258819	+	0.965926i	$0.583333\pi$
$929$	−1.18270	+	1.34861i	−1.18270	+	1.34861i	−0.923880	+	0.382683i	$0.875000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.0812272	−	0.239288i	0.0812272	−	0.239288i
$934$		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
$939$	−3.31744	−	0.436749i	−3.31744	−	0.436749i
$940$	0.357164	+	0.534534i	0.357164	+	0.534534i
$941$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$941$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$942$		0			0
$943$		0			0
$944$	1.24871	+	1.42388i	1.24871	+	1.42388i
$945$		0			0
$946$		0			0
$947$	0.369474	−	0.125419i	0.369474	−	0.125419i	−0.130526	−	0.991445i	$-0.541667\pi$
$947$	0.369474	−	0.125419i	0.369474	−	0.125419i	0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2.11818	+	1.41532i	−2.11818	+	1.41532i
$952$		0			0
$953$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$953$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$954$		0			0
$955$	−0.332066	−	0.0217648i	−0.332066	−	0.0217648i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.933745	+	0.818872i	−0.933745	+	0.818872i
$961$	−0.400100	−	0.107206i	−0.400100	−	0.107206i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$967$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$971$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$972$	−2.35207	−	3.06528i	−2.35207	−	3.06528i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.34861	+	0.665060i	−1.34861	+	0.665060i	−0.965926	−	0.258819i	$-0.916667\pi$
$977$	−1.34861	+	0.665060i	−1.34861	+	0.665060i	−0.382683	+	0.923880i	$0.625000\pi$
$978$		0			0
$979$		0			0
$980$	−0.483342	+	0.423880i	−0.483342	+	0.423880i
$981$		0			0
$982$		0			0
$983$	1.50046	−	0.0983454i	1.50046	−	0.0983454i	0.707107	−	0.707107i	$-0.250000\pi$
$983$	1.50046	−	0.0983454i	1.50046	−	0.0983454i	0.793353	+	0.608761i	$0.208333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.630526	−	1.85747i	−0.630526	−	1.85747i	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−0.630526	−	1.85747i	−0.630526	−	1.85747i	−0.130526	−	0.991445i	$-0.541667\pi$
$992$		0			0
$993$	−3.04206	+	1.03264i	−3.04206	+	1.03264i
$994$		0			0
$995$	0.958414	−	0.126178i	0.958414	−	0.126178i
$996$		0			0
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$		0			0
$999$	2.22534	+	1.95157i	2.22534	+	1.95157i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1067.1.bg.a.516.1	✓	16
11.10	odd	2	CM	1067.1.bg.a.516.1	✓	16
97.72	even	48	inner	1067.1.bg.a.945.1	yes	16
1067.945	odd	48	inner	1067.1.bg.a.945.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1067.1.bg.a.516.1	✓	16	1.1	even	1	trivial
1067.1.bg.a.516.1	✓	16	11.10	odd	2	CM
1067.1.bg.a.945.1	yes	16	97.72	even	48	inner
1067.1.bg.a.945.1	yes	16	1067.945	odd	48	inner

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)

\(f(q)\)	\(=\)	\(q+(-0.252157 + 1.91532i) q^{3} +(-0.258819 + 0.965926i) q^{4} +(0.0420463 + 0.641502i) q^{5} +(-2.63896 - 0.707107i) q^{9} +O(q^{10})\)	\(q+(-0.252157 + 1.91532i) q^{3} +(-0.258819 + 0.965926i) q^{4} +(0.0420463 + 0.641502i) q^{5} +(-2.63896 - 0.707107i) q^{9} +(0.793353 + 0.608761i) q^{11} +(-1.78480 - 0.739288i) q^{12} +(-1.23929 - 0.0812272i) q^{15} +(-0.866025 - 0.500000i) q^{16} +(-0.630526 - 0.125419i) q^{20} +(1.88981 - 0.641502i) q^{23} +(0.581687 - 0.0765806i) q^{25} +(1.28048 - 3.09136i) q^{27} +(0.758819 + 0.0999004i) q^{31} +(-1.36603 + 1.36603i) q^{33} +(1.36603 - 2.36603i) q^{36} +(-0.284338 + 0.837633i) q^{37} +(-0.793353 + 0.608761i) q^{44} +(0.342652 - 1.72263i) q^{45} +(-0.707107 - 0.707107i) q^{47} +(1.17604 - 1.53264i) q^{48} +(0.608761 - 0.793353i) q^{49} +(-1.57313 + 1.20711i) q^{53} +(-0.357164 + 0.534534i) q^{55} +(-1.79335 - 0.608761i) q^{59} +(0.399211 - 1.17604i) q^{60} +(0.707107 - 0.707107i) q^{64} +(0.732626 + 1.09645i) q^{67} +(0.752157 + 3.78135i) q^{69} +(-1.18270 - 1.34861i) q^{71} +1.13343i q^{75} +(0.284338 - 0.576581i) q^{80} +(3.23205 + 1.86603i) q^{81} +(0.130526 + 1.99144i) q^{92} +(-0.382683 + 1.42819i) q^{93} +(-0.608761 - 0.793353i) q^{97} +(-1.66317 - 2.16748i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

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