LMFDB - Embedded newform 1067.1.bg.a.593.1

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			593.1
Root			$-0.130526 - 0.991445i$ of defining polynomial
Character	$\chi$	$=$	1067.593
Dual form			1067.1.bg.a.538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.315118 - 0.410670i) q^{3} +(-0.965926 + 0.258819i) q^{4} +(-0.123864 + 0.0420463i) q^{5} +(0.189469 + 0.707107i) q^{9} +O(q^{10})$	\(q+(0.315118 - 0.410670i) q^{3} +(-0.965926 + 0.258819i) q^{4} +(-0.123864 + 0.0420463i) q^{5} +(0.189469 + 0.707107i) q^{9} +(0.991445 + 0.130526i) q^{11} +(-0.198092 + 0.478235i) q^{12} +(-0.0217648 + 0.0641169i) q^{15} +(0.866025 - 0.500000i) q^{16} +(0.108761 - 0.0726721i) q^{20} +(0.641502 - 0.0420463i) q^{23} +(-0.779779 + 0.598345i) q^{25} +(0.828328 + 0.343105i) q^{27} +(1.46593 + 1.12484i) q^{31} +(0.366025 - 0.366025i) q^{33} +(-0.366025 - 0.633975i) q^{36} +(0.0862466 - 1.31587i) q^{37} +(-0.991445 + 0.130526i) q^{44} +(-0.0531996 - 0.0796189i) q^{45} +(0.707107 + 0.707107i) q^{47} +(0.0675653 - 0.513210i) q^{48} +(0.130526 - 0.991445i) q^{49} +(1.57313 - 0.207107i) q^{53} +(-0.128293 + 0.0255190i) q^{55} +(-1.99144 - 0.130526i) q^{59} +(0.00442847 - 0.0675653i) q^{60} +(-0.707107 + 0.707107i) q^{64} +(-1.75928 - 0.349942i) q^{67} +(0.184882 - 0.276695i) q^{69} +(-1.34861 + 0.665060i) q^{71} +0.508781i q^{75} +(-0.0862466 + 0.0983454i) q^{80} +(-0.232051 + 0.133975i) q^{81} +(-0.608761 + 0.206647i) q^{92} +(0.923880 - 0.247553i) q^{93} +(-0.130526 - 0.991445i) q^{97} +(0.0955518 + 0.725788i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1067\mathbb{Z}\right)^\times$.

$n$	$199$	$486$
$\chi(n)$	$e\left(\frac{43}{48}\right)$	$-1$

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		−0.130526	−	0.991445i	$-0.541667\pi$
$2$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$3$	0.315118	−	0.410670i	0.315118	−	0.410670i	−0.608761	−	0.793353i	$-0.708333\pi$
$3$	0.315118	−	0.410670i	0.315118	−	0.410670i	0.923880	+	0.382683i	$0.125000\pi$
$4$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$5$	−0.123864	+	0.0420463i	−0.123864	+	0.0420463i	−0.382683	−	0.923880i	$-0.625000\pi$
$5$	−0.123864	+	0.0420463i	−0.123864	+	0.0420463i	0.258819	+	0.965926i	$0.416667\pi$
$6$		0			0
$7$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$7$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$8$		0			0
$9$	0.189469	+	0.707107i	0.189469	+	0.707107i
$10$		0			0
$11$	0.991445	+	0.130526i	0.991445	+	0.130526i
$11$	0.991445	+	0.130526i	0.991445	+	0.130526i
$12$	−0.198092	+	0.478235i	−0.198092	+	0.478235i
$13$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$13$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$14$		0			0
$15$	−0.0217648	+	0.0641169i	−0.0217648	+	0.0641169i
$16$	0.866025	−	0.500000i	0.866025	−	0.500000i
$17$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$17$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$18$		0			0
$19$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$19$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$20$	0.108761	−	0.0726721i	0.108761	−	0.0726721i
$21$		0			0
$22$		0			0
$23$	0.641502	−	0.0420463i	0.641502	−	0.0420463i	0.258819	−	0.965926i	$-0.416667\pi$
$23$	0.641502	−	0.0420463i	0.641502	−	0.0420463i	0.382683	+	0.923880i	$0.375000\pi$
$24$		0			0
$25$	−0.779779	+	0.598345i	−0.779779	+	0.598345i
$26$		0			0
$27$	0.828328	+	0.343105i	0.828328	+	0.343105i
$28$		0			0
$29$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$29$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$30$		0			0
$31$	1.46593	+	1.12484i	1.46593	+	1.12484i	0.965926	+	0.258819i	$0.0833333\pi$
$31$	1.46593	+	1.12484i	1.46593	+	1.12484i	0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$	0.366025	−	0.366025i	0.366025	−	0.366025i
$34$		0			0
$35$		0			0
$36$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$37$	0.0862466	−	1.31587i	0.0862466	−	1.31587i	−0.707107	−	0.707107i	$-0.750000\pi$
$37$	0.0862466	−	1.31587i	0.0862466	−	1.31587i	0.793353	−	0.608761i	$-0.208333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$41$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$42$		0			0
$43$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$43$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$44$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$45$	−0.0531996	−	0.0796189i	−0.0531996	−	0.0796189i
$46$		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	$0.583333\pi$
$48$	0.0675653	−	0.513210i	0.0675653	−	0.513210i
$49$	0.130526	−	0.991445i	0.130526	−	0.991445i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.57313	−	0.207107i	1.57313	−	0.207107i	0.707107	−	0.707107i	$-0.250000\pi$
$53$	1.57313	−	0.207107i	1.57313	−	0.207107i	0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$	−0.128293	+	0.0255190i	−0.128293	+	0.0255190i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.99144	−	0.130526i	−1.99144	−	0.130526i	−0.991445	−	0.130526i	$-0.958333\pi$
$59$	−1.99144	−	0.130526i	−1.99144	−	0.130526i	−1.00000			$\pi$
$60$	0.00442847	−	0.0675653i	0.00442847	−	0.0675653i
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$	−1.75928	−	0.349942i	−1.75928	−	0.349942i	−0.793353	−	0.608761i	$-0.791667\pi$
$67$	−1.75928	−	0.349942i	−1.75928	−	0.349942i	−0.965926	+	0.258819i	$0.916667\pi$
$68$		0			0
$69$	0.184882	−	0.276695i	0.184882	−	0.276695i
$70$		0			0
$71$	−1.34861	+	0.665060i	−1.34861	+	0.665060i	−0.965926	−	0.258819i	$-0.916667\pi$
$71$	−1.34861	+	0.665060i	−1.34861	+	0.665060i	−0.382683	+	0.923880i	$0.625000\pi$
$72$		0			0
$73$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$73$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$74$		0			0
$75$			0.508781i			0.508781i
$76$		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	$0.875000\pi$
$80$	−0.0862466	+	0.0983454i	−0.0862466	+	0.0983454i
$81$	−0.232051	+	0.133975i	−0.232051	+	0.133975i
$82$		0			0
$83$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$83$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$84$		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
$92$	−0.608761	+	0.206647i	−0.608761	+	0.206647i
$93$	0.923880	−	0.247553i	0.923880	−	0.247553i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$97$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$98$		0			0
$99$	0.0955518	+	0.725788i	0.0955518	+	0.725788i
$100$	0.598345	−	0.779779i	0.598345	−	0.779779i
$101$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$101$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$102$		0			0
$103$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.923880	−	0.382683i	$-0.875000\pi$
$103$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.793353	−	0.608761i	$-0.791667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$107$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$108$	−0.888905	−	0.117027i	−0.888905	−	0.117027i
$109$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$109$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$110$		0			0
$111$	−0.513210	−	0.450073i	−0.513210	−	0.450073i
$112$		0			0
$113$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.923880	−	0.382683i	$-0.875000\pi$
$113$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.130526	+	0.991445i	$0.541667\pi$
$114$		0			0
$115$	−0.0776914	+	0.0321808i	−0.0776914	+	0.0321808i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.965926	+	0.258819i	0.965926	+	0.258819i
$122$		0			0
$123$		0			0
$124$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$125$	0.144101	−	0.215662i	0.144101	−	0.215662i
$126$		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	$0.187500\pi$
$132$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$133$		0			0
$134$		0			0
$135$	−0.117027	−	0.00767033i	−0.117027	−	0.00767033i
$136$		0			0
$137$	1.09645	+	1.25026i	1.09645	+	1.25026i	0.965926	+	0.258819i	$0.0833333\pi$
$137$	1.09645	+	1.25026i	1.09645	+	1.25026i	0.130526	+	0.991445i	$0.458333\pi$
$138$		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
$141$	0.513210	−	0.0675653i	0.513210	−	0.0675653i
$142$		0			0
$143$		0			0
$144$	0.517638	+	0.517638i	0.517638	+	0.517638i
$145$		0			0
$146$		0			0
$147$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$148$	0.257264	+	1.29335i	0.257264	+	1.29335i
$149$		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		0.258819	−	0.965926i	$-0.416667\pi$
$151$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.228872	−	0.0776914i	−0.228872	−	0.0776914i
$156$		0			0
$157$	0.123864	−	1.88981i	0.123864	−	1.88981i	−0.258819	−	0.965926i	$-0.583333\pi$
$157$	0.123864	−	1.88981i	0.123864	−	1.88981i	0.382683	−	0.923880i	$-0.375000\pi$
$158$		0			0
$159$	0.410670	−	0.711301i	0.410670	−	0.711301i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.607206	−	0.465926i	−0.607206	−	0.465926i	0.258819	−	0.965926i	$-0.416667\pi$
$163$	−0.607206	−	0.465926i	−0.607206	−	0.465926i	−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$	−0.0299475	+	0.0607275i	−0.0299475	+	0.0607275i
$166$		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
$169$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$173$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$174$		0			0
$175$		0			0
$176$	0.923880	−	0.382683i	0.923880	−	0.382683i
$177$	−0.681144	+	0.776695i	−0.681144	+	0.776695i
$178$		0			0
$179$	−0.630526	+	1.85747i	−0.630526	+	1.85747i	−0.130526	+	0.991445i	$0.541667\pi$
$179$	−0.630526	+	1.85747i	−0.630526	+	1.85747i	−0.500000	+	0.866025i	$0.666667\pi$
$180$	0.0719938	+	0.0631369i	0.0719938	+	0.0631369i
$181$	−0.125419	−	0.369474i	−0.125419	−	0.369474i	0.866025	−	0.500000i	$-0.166667\pi$
$181$	−0.125419	−	0.369474i	−0.125419	−	0.369474i	−0.991445	+	0.130526i	$0.958333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0446445	+	0.166616i	0.0446445	+	0.166616i
$186$		0			0
$187$		0			0
$188$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$189$		0			0
$190$		0			0
$191$	1.17604	−	1.53264i	1.17604	−	1.53264i	0.382683	−	0.923880i	$-0.375000\pi$
$191$	1.17604	−	1.53264i	1.17604	−	1.53264i	0.793353	−	0.608761i	$-0.208333\pi$
$192$	0.0675653	+	0.513210i	0.0675653	+	0.513210i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	0.130526	+	0.991445i	0.130526	+	0.991445i
$197$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$197$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$198$		0			0
$199$	0.837633	−	0.284338i	0.837633	−	0.284338i	0.130526	−	0.991445i	$-0.458333\pi$
$199$	0.837633	−	0.284338i	0.837633	−	0.284338i	0.707107	+	0.707107i	$0.250000\pi$
$200$		0			0
$201$	−0.698092	+	0.612210i	−0.698092	+	0.612210i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.151276	+	0.445644i	0.151276	+	0.445644i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$211$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$212$	−1.46593	+	0.607206i	−1.46593	+	0.607206i
$213$	−0.151851	+	0.763406i	−0.151851	+	0.763406i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.117317	−	0.0578541i	0.117317	−	0.0578541i
$221$		0			0
$222$		0			0
$223$	0.867580	−	1.75928i	0.867580	−	1.75928i	0.258819	−	0.965926i	$-0.416667\pi$
$223$	0.867580	−	1.75928i	0.867580	−	1.75928i	0.608761	−	0.793353i	$-0.291667\pi$
$224$		0			0
$225$	−0.570838	−	0.438019i	−0.570838	−	0.438019i
$226$		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
$229$	0.608761	−	1.05441i	0.608761	−	1.05441i	−0.382683	−	0.923880i	$-0.625000\pi$
$229$	0.608761	−	1.05441i	0.608761	−	1.05441i	0.991445	−	0.130526i	$-0.0416667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$233$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$234$		0			0
$235$	−0.117317	−	0.0578541i	−0.117317	−	0.0578541i
$236$	1.95737	−	0.389345i	1.95737	−	0.389345i
$237$		0			0
$238$		0			0
$239$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$239$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$240$	0.0132096	+	0.0664093i	0.0132096	+	0.0664093i
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$		0			0
$243$	−0.135131	+	1.02642i	−0.135131	+	1.02642i
$244$		0			0
$245$	0.0255190	+	0.128293i	0.0255190	+	0.128293i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.991445	−	1.13053i	−0.991445	−	1.13053i	−0.991445	−	0.130526i	$-0.958333\pi$
$251$	−0.991445	−	1.13053i	−0.991445	−	1.13053i		−	1.00000i	$-0.5\pi$
$252$		0			0
$253$	0.641502	+	0.0420463i	0.641502	+	0.0420463i
$254$		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
$265$	−0.186147	+	0.0917975i	−0.186147	+	0.0917975i
$266$		0			0
$267$		0			0
$268$	1.78990	−	0.117317i	1.78990	−	0.117317i
$269$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$269$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$270$		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
$275$	−0.851207	+	0.491445i	−0.851207	+	0.491445i
$276$	−0.106968	+	0.315118i	−0.106968	+	0.315118i
$277$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$277$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$278$		0			0
$279$	−0.517638	+	1.24969i	−0.517638	+	1.24969i
$280$		0			0
$281$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$281$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$282$		0			0
$283$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$283$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$284$	1.13053	−	0.991445i	1.13053	−	0.991445i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$290$		0			0
$291$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$292$		0			0
$293$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$293$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$294$		0			0
$295$	0.252157	−	0.0675653i	0.252157	−	0.0675653i
$296$		0			0
$297$	0.776457	+	0.448288i	0.776457	+	0.448288i
$298$		0			0
$299$		0			0
$300$	−0.131682	−	0.491445i	−0.131682	−	0.491445i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$307$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$308$		0			0
$309$	−0.948288	+	0.392794i	−0.948288	+	0.392794i
$310$		0			0
$311$	−1.57469	+	1.05217i	−1.57469	+	1.05217i	−0.608761	+	0.793353i	$0.708333\pi$
$311$	−1.57469	+	1.05217i	−1.57469	+	1.05217i	−0.965926	+	0.258819i	$0.916667\pi$
$312$		0			0
$313$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$313$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.60876	−	0.793353i	1.60876	−	0.793353i	0.608761	−	0.793353i	$-0.291667\pi$
$317$	1.60876	−	0.793353i	1.60876	−	0.793353i	1.00000			$0$
$318$		0			0
$319$		0			0
$320$	0.0578541	−	0.117317i	0.0578541	−	0.117317i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.189469	−	0.189469i	0.189469	−	0.189469i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.257264	−	0.293353i	−0.257264	−	0.293353i	0.608761	−	0.793353i	$-0.291667\pi$
$331$	−0.257264	−	0.293353i	−0.257264	−	0.293353i	−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$	0.946800	−	0.188330i	0.946800	−	0.188330i
$334$		0			0
$335$	0.232626	−	0.0306258i	0.232626	−	0.0306258i
$336$		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
$339$	−0.0822623	+	0.624844i	−0.0822623	+	0.624844i
$340$		0			0
$341$	1.30656	+	1.30656i	1.30656	+	1.30656i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.0112663	+	0.0420463i	−0.0112663	+	0.0420463i
$346$		0			0
$347$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$347$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$348$		0			0
$349$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$349$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$353$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$354$		0			0
$355$	0.139081	−	0.139081i	0.139081	−	0.139081i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$359$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$360$		0			0
$361$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$362$		0			0
$363$	0.410670	−	0.315118i	0.410670	−	0.315118i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.0983454	+	1.50046i	0.0983454	+	1.50046i	0.707107	+	0.707107i	$0.250000\pi$
$367$	0.0983454	+	1.50046i	0.0983454	+	1.50046i	−0.608761	+	0.793353i	$0.708333\pi$
$368$	0.534534	−	0.357164i	0.534534	−	0.357164i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.828328	+	0.478235i	−0.828328	+	0.478235i
$373$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$373$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$374$		0			0
$375$	−0.0431571	−	0.127137i	−0.0431571	−	0.127137i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.198092	+	0.739288i	0.198092	+	0.739288i	0.991445	+	0.130526i	$0.0416667\pi$
$379$	0.198092	+	0.739288i	0.198092	+	0.739288i	−0.793353	+	0.608761i	$0.791667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.57469	+	0.534534i	−1.57469	+	0.534534i	−0.965926	−	0.258819i	$-0.916667\pi$
$383$	−1.57469	+	0.534534i	−1.57469	+	0.534534i	−0.608761	+	0.793353i	$0.708333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.382683	+	0.923880i	0.382683	+	0.923880i
$389$	1.21752			1.21752			0.608761	−	0.793353i	$-0.291667\pi$
$389$	1.21752			1.21752			0.608761	+	0.793353i	$0.291667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.280144	−	0.676327i	−0.280144	−	0.676327i
$397$	−0.500000	−	1.86603i	−0.500000	−	1.86603i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−0.500000	−	1.86603i	−0.500000	−	1.86603i		−	1.00000i	$-0.5\pi$
$398$		0			0
$399$		0			0
$400$	−0.376136	+	0.908072i	−0.376136	+	0.908072i
$401$	0.576581	+	1.69855i	0.576581	+	1.69855i	0.707107	+	0.707107i	$0.250000\pi$
$401$	0.576581	+	1.69855i	0.576581	+	1.69855i	−0.130526	+	0.991445i	$0.541667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.0231097	−	0.0263516i	0.0231097	−	0.0263516i
$406$		0			0
$407$	0.257264	−	1.29335i	0.257264	−	1.29335i
$408$		0			0
$409$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$409$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$410$		0			0
$411$	0.858958	−	0.0562991i	0.858958	−	0.0562991i
$412$	1.91532	+	0.513210i	1.91532	+	0.513210i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.207107	−	0.158919i	−0.207107	−	0.158919i	0.500000	−	0.866025i	$-0.333333\pi$
$419$	−0.207107	−	0.158919i	−0.207107	−	0.158919i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	0.184592	−	0.184592i	0.184592	−	0.184592i	−0.608761	−	0.793353i	$-0.708333\pi$
$421$	0.184592	−	0.184592i	0.184592	−	0.184592i	0.793353	+	0.608761i	$0.208333\pi$
$422$		0			0
$423$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$431$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$432$	0.888905	−	0.117027i	0.888905	−	0.117027i
$433$	0.835400	+	1.25026i	0.835400	+	1.25026i	0.965926	+	0.258819i	$0.0833333\pi$
$433$	0.835400	+	1.25026i	0.835400	+	1.25026i	−0.130526	+	0.991445i	$0.541667\pi$
$434$		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
$441$	0.725788	−	0.0955518i	0.725788	−	0.0955518i
$442$		0			0
$443$	1.95737	−	0.389345i	1.95737	−	0.389345i	0.965926	−	0.258819i	$-0.0833333\pi$
$443$	1.95737	−	0.389345i	1.95737	−	0.389345i	0.991445	−	0.130526i	$-0.0416667\pi$
$444$	0.612210	+	0.301908i	0.612210	+	0.301908i
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$449$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$450$		0			0
$451$		0			0
$452$	0.860919	−	0.860919i	0.860919	−	0.860919i
$453$		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
$460$	0.0667151	−	0.0511923i	0.0667151	−	0.0511923i
$461$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$461$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$462$		0			0
$463$			1.58671i			1.58671i	0.608761	+	0.793353i	$0.291667\pi$
$463$			1.58671i			1.58671i	−0.608761	+	0.793353i	$0.708333\pi$
$464$		0			0
$465$	−0.104027	+	0.0695087i	−0.104027	+	0.0695087i
$466$		0			0
$467$	0.241181	−	0.0999004i	0.241181	−	0.0999004i	−0.258819	−	0.965926i	$-0.583333\pi$
$467$	0.241181	−	0.0999004i	0.241181	−	0.0999004i	0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.737054	−	0.646379i	−0.737054	−	0.646379i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.444506	+	1.07313i	0.444506	+	1.07313i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.00000			−1.00000
$485$	0.0578541	+	0.117317i	0.0578541	+	0.117317i
$486$		0			0
$487$	0.130526	+	0.991445i	0.130526	+	0.991445i	0.923880	+	0.382683i	$0.125000\pi$
$487$	0.130526	+	0.991445i	0.130526	+	0.991445i	−0.793353	+	0.608761i	$0.791667\pi$
$488$		0			0
$489$	−0.382683	+	0.102540i	−0.382683	+	0.102540i
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.0423522	−	0.0858817i	−0.0423522	−	0.0858817i
$496$	1.83195	+	0.241181i	1.83195	+	0.241181i
$497$		0			0
$498$		0			0
$499$	−1.25026	−	1.09645i	−1.25026	−	1.09645i	−0.991445	−	0.130526i	$-0.958333\pi$
$499$	−1.25026	−	1.09645i	−1.25026	−	1.09645i	−0.258819	−	0.965926i	$-0.583333\pi$
$500$	−0.0833732	+	0.245609i	−0.0833732	+	0.245609i
$501$		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
$507$			0.517638i			0.517638i
$508$		0			0
$509$	−0.252157	−	0.0675653i	−0.252157	−	0.0675653i	0.130526	−	0.991445i	$-0.458333\pi$
$509$	−0.252157	−	0.0675653i	−0.252157	−	0.0675653i	−0.382683	+	0.923880i	$0.625000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.254391	+	0.0506014i	0.254391	+	0.0506014i
$516$		0			0
$517$	0.608761	+	0.793353i	0.608761	+	0.793353i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$521$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$522$		0			0
$523$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$523$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.133975	−	0.500000i	0.133975	−	0.500000i
$529$	−0.581687	+	0.0765806i	−0.581687	+	0.0765806i
$530$		0			0
$531$	−0.285020	−	1.43289i	−0.285020	−	1.43289i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.564117	+	0.844261i	0.564117	+	0.844261i
$538$		0			0
$539$	0.258819	−	0.965926i	0.258819	−	0.965926i
$540$	0.115024	−	0.0228797i	0.115024	−	0.0228797i
$541$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$541$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$542$		0			0
$543$	−0.191254	−	0.0649219i	−0.191254	−	0.0649219i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	−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
$555$	0.0824923	+	0.0341694i	0.0824923	+	0.0341694i
$556$		0			0
$557$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$557$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$558$		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	$0.562500\pi$
$564$	−0.478235	+	0.198092i	−0.478235	+	0.198092i
$565$	0.105007	−	0.119738i	0.105007	−	0.119738i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$569$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$570$		0			0
$571$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$571$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$572$		0			0
$573$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$574$		0			0
$575$	−0.475072	+	0.416627i	−0.475072	+	0.416627i
$576$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$577$	−1.05217	+	0.357164i	−1.05217	+	0.357164i	−0.793353	−	0.608761i	$-0.791667\pi$
$577$	−1.05217	+	0.357164i	−1.05217	+	0.357164i	−0.258819	+	0.965926i	$0.583333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.58671			1.58671
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.369474	+	0.125419i	−0.369474	+	0.125419i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−0.369474	+	0.125419i	−0.369474	+	0.125419i	0.130526	+	0.991445i	$0.458333\pi$
$588$	0.448288	+	0.258819i	0.448288	+	0.258819i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.583242	−	1.18270i	−0.583242	−	1.18270i
$593$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$593$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.147184	−	0.433591i	0.147184	−	0.433591i
$598$		0			0
$599$	−0.869474	+	0.991445i	−0.869474	+	0.991445i	0.130526	+	0.991445i	$0.458333\pi$
$599$	−0.869474	+	0.991445i	−0.869474	+	0.991445i	−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
$603$	−0.0858817	−	1.31030i	−0.0858817	−	1.31030i
$604$		0			0
$605$	−0.130526	+	0.00855514i	−0.130526	+	0.00855514i
$606$		0			0
$607$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$607$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$613$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	0.382683	+	0.923880i	$0.375000\pi$
$617$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	−0.991445	+	0.130526i	$0.958333\pi$
$618$		0			0
$619$	0.0578541	−	0.882683i	0.0578541	−	0.882683i	−0.866025	−	0.500000i	$-0.833333\pi$
$619$	0.0578541	−	0.882683i	0.0578541	−	0.882683i	0.923880	−	0.382683i	$-0.125000\pi$
$620$	0.241181	+	0.0158078i	0.241181	+	0.0158078i
$621$	0.545801	+	0.185274i	0.545801	+	0.185274i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.245609	−	0.916627i	0.245609	−	0.916627i
$626$		0			0
$627$		0			0
$628$	0.369474	+	1.85747i	0.369474	+	1.85747i
$629$		0			0
$630$		0			0
$631$	−0.130526	+	0.991445i	−0.130526	+	0.991445i	0.793353	+	0.608761i	$0.208333\pi$
$631$	−0.130526	+	0.991445i	−0.130526	+	0.991445i	−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.212578	+	0.793353i	−0.212578	+	0.793353i
$637$		0			0
$638$		0			0
$639$	−0.725788	−	0.827603i	−0.725788	−	0.827603i
$640$		0			0
$641$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i	−0.608761	+	0.793353i	$0.708333\pi$
$642$		0			0
$643$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	0.793353	−	0.608761i	$-0.208333\pi$
$643$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.158919	−	0.207107i	−0.158919	−	0.207107i	0.707107	−	0.707107i	$-0.250000\pi$
$647$	−0.158919	−	0.207107i	−0.158919	−	0.207107i	−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	−1.95737	−	0.389345i	−1.95737	−	0.389345i
$650$		0			0
$651$		0			0
$652$	0.707107	+	0.292893i	0.707107	+	0.292893i
$653$	1.78990	−	0.882683i	1.78990	−	0.882683i	0.866025	−	0.500000i	$-0.166667\pi$
$653$	1.78990	−	0.882683i	1.78990	−	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	$0.812500\pi$
$660$	0.0132096	−	0.0664093i	0.0132096	−	0.0664093i
$661$	−1.46593	+	0.607206i	−1.46593	+	0.607206i	−0.965926	−	0.258819i	$-0.916667\pi$
$661$	−1.46593	+	0.607206i	−1.46593	+	0.607206i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.449093	−	0.910670i	−0.449093	−	0.910670i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$673$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$674$		0			0
$675$	−0.851207	+	0.228080i	−0.851207	+	0.228080i
$676$	0.608761	−	0.793353i	0.608761	−	0.793353i
$677$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$677$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.78480	+	0.478235i	−1.78480	+	0.478235i	−0.991445	−	0.130526i	$-0.958333\pi$
$683$	−1.78480	+	0.478235i	−1.78480	+	0.478235i	−0.793353	+	0.608761i	$0.791667\pi$
$684$		0			0
$685$	−0.188380	−	0.108761i	−0.188380	−	0.108761i
$686$		0			0
$687$	−0.241181	−	0.582262i	−0.241181	−	0.582262i
$688$		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
$704$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$705$	−0.0607275	+	0.0299475i	−0.0607275	+	0.0299475i
$706$		0			0
$707$		0			0
$708$	0.456911	−	0.926523i	0.456911	−	0.926523i
$709$	−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
$713$	0.987691	+	0.659954i	0.987691	+	0.659954i
$714$		0			0
$715$		0			0
$716$	0.128293	−	1.95737i	0.128293	−	1.95737i
$717$		0			0
$718$		0			0
$719$	−0.732626	−	0.835400i	−0.732626	−	0.835400i	0.258819	−	0.965926i	$-0.416667\pi$
$719$	−0.732626	−	0.835400i	−0.732626	−	0.835400i	−0.991445	+	0.130526i	$0.958333\pi$
$720$	−0.0858817	−	0.0423522i	−0.0858817	−	0.0423522i
$721$		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	−	1.57313i	0.207107	−	1.57313i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	0.207107	−	1.57313i	0.207107	−	1.57313i	0.707107	−	0.707107i	$-0.250000\pi$
$728$		0			0
$729$	0.189469	+	0.189469i	0.189469	+	0.189469i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$733$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$734$		0			0
$735$	0.0607275	+	0.0299475i	0.0607275	+	0.0299475i
$736$		0			0
$737$	−1.69855	−	0.576581i	−1.69855	−	0.576581i
$738$		0			0
$739$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$739$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$740$	−0.0862466	−	0.149383i	−0.0862466	−	0.149383i
$741$		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
$751$	−0.607206	+	0.465926i	−0.607206	+	0.465926i	−0.866025	−	0.500000i	$-0.833333\pi$
$751$	−0.607206	+	0.465926i	−0.607206	+	0.465926i	0.258819	+	0.965926i	$0.416667\pi$
$752$	0.965926	+	0.258819i	0.965926	+	0.258819i
$753$	−0.776695	+	0.0509073i	−0.776695	+	0.0509073i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.172572	−	0.867580i	0.172572	−	0.867580i	−0.793353	−	0.608761i	$-0.791667\pi$
$757$	0.172572	−	0.867580i	0.172572	−	0.867580i	0.965926	−	0.258819i	$-0.0833333\pi$
$758$		0			0
$759$	0.219416	−	0.250196i	0.219416	−	0.250196i
$760$		0			0
$761$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$761$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$762$		0			0
$763$		0			0
$764$	−0.739288	+	1.78480i	−0.739288	+	1.78480i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.198092	−	0.478235i	−0.198092	−	0.478235i
$769$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$769$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$770$		0			0
$771$	−0.544645	+	0.184882i	−0.544645	+	0.184882i
$772$		0			0
$773$	−0.465926	+	0.607206i	−0.465926	+	0.607206i	−0.965926	−	0.258819i	$-0.916667\pi$
$773$	−0.465926	+	0.607206i	−0.465926	+	0.607206i	0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−1.81614			−1.81614
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−1.42388	+	0.483342i	−1.42388	+	0.483342i
$782$		0			0
$783$		0			0
$784$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$785$	0.0641169	+	0.239288i	0.0641169	+	0.239288i
$786$		0			0
$787$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$787$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.0209598	+	0.105372i	−0.0209598	+	0.105372i
$796$	−0.735499	+	0.491445i	−0.735499	+	0.491445i
$797$	−0.0862466	−	1.31587i	−0.0862466	−	1.31587i	−0.793353	−	0.608761i	$-0.791667\pi$
$797$	−0.0862466	−	1.31587i	−0.0862466	−	1.31587i	0.707107	−	0.707107i	$-0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.515853	−	0.772029i	0.515853	−	0.772029i
$805$		0			0
$806$		0			0
$807$	0.711301	+	0.545801i	0.711301	+	0.545801i
$808$		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	$-0.333333\pi$
$811$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.0948017	+	0.0321808i	0.0948017	+	0.0321808i
$816$		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
$825$	−0.0664093	+	0.504428i	−0.0664093	+	0.504428i
$826$		0			0
$827$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$827$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$828$	−0.261462	−	0.391306i	−0.261462	−	0.391306i
$829$	−0.513210	+	0.0675653i	−0.513210	+	0.0675653i	−0.382683	−	0.923880i	$-0.625000\pi$
$829$	−0.513210	+	0.0675653i	−0.513210	+	0.0675653i	−0.130526	+	0.991445i	$0.541667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.828328	+	1.43471i	0.828328	+	1.43471i
$838$		0			0
$839$	1.57469	+	1.05217i	1.57469	+	1.05217i	0.965926	+	0.258819i	$0.0833333\pi$
$839$	1.57469	+	1.05217i	1.57469	+	1.05217i	0.608761	+	0.793353i	$0.291667\pi$
$840$		0			0
$841$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.0726721	−	0.108761i	0.0726721	−	0.108761i
$846$		0			0
$847$		0			0
$848$	1.25882	−	0.965926i	1.25882	−	0.965926i
$849$		0			0
$850$		0			0
$851$		−	0.847759i		−	0.847759i
$852$	−0.0509073	−	0.776695i	−0.0509073	−	0.776695i
$853$		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	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	1.50046	+	1.31587i	1.50046	+	1.31587i	0.793353	+	0.608761i	$0.208333\pi$
$859$	1.50046	+	1.31587i	1.50046	+	1.31587i	0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.735499	−	1.49144i	−0.735499	−	1.49144i	−0.866025	−	0.500000i	$-0.833333\pi$
$863$	−0.735499	−	1.49144i	−0.735499	−	1.49144i	0.130526	−	0.991445i	$-0.458333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.676327	−	0.280144i	0.676327	−	0.280144i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$877$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$878$		0			0
$879$		0			0
$880$	−0.0983454	+	0.0862466i	−0.0983454	+	0.0862466i
$881$	−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
$883$	0.583242	+	1.18270i	0.583242	+	1.18270i	0.965926	+	0.258819i	$0.0833333\pi$
$883$	0.583242	+	1.18270i	0.583242	+	1.18270i	−0.382683	+	0.923880i	$0.625000\pi$
$884$		0			0
$885$	0.0517123	−	0.124844i	0.0517123	−	0.124844i
$886$		0			0
$887$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$887$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.247553	+	0.102540i	−0.247553	+	0.102540i
$892$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$893$		0			0
$894$		0			0
$895$		−	0.256586i		−	0.256586i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.664755	+	0.275350i	0.664755	+	0.275350i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.0310700	+	0.0404912i	0.0310700	+	0.0404912i
$906$		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
$911$	−1.31587	−	0.0862466i	−1.31587	−	0.0862466i	−0.608761	−	0.793353i	$-0.708333\pi$
$911$	−1.31587	−	0.0862466i	−1.31587	−	0.0862466i	−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.315118	+	1.17604i	−0.315118	+	1.17604i
$917$		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
$925$	0.720090	+	1.07769i	0.720090	+	1.07769i
$926$		0			0
$927$	0.375696	−	1.40211i	0.375696	−	1.40211i
$928$		0			0
$929$	−1.34861	−	0.665060i	−1.34861	−	0.665060i	−0.382683	−	0.923880i	$-0.625000\pi$
$929$	−1.34861	−	0.665060i	−1.34861	−	0.665060i	−0.965926	+	0.258819i	$0.916667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−0.0641169	+	0.978235i	−0.0641169	+	0.978235i
$934$		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
$939$	−0.711301	−	0.545801i	−0.711301	−	0.545801i
$940$	0.128293	+	0.0255190i	0.128293	+	0.0255190i
$941$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$941$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$942$		0			0
$943$		0			0
$944$	−1.78990	+	0.882683i	−1.78990	+	0.882683i
$945$		0			0
$946$		0			0
$947$	1.10876	−	0.0726721i	1.10876	−	0.0726721i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.10876	−	0.0726721i	1.10876	−	0.0726721i	0.608761	+	0.793353i	$0.291667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.181144	−	0.910670i	0.181144	−	0.910670i
$952$		0			0
$953$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$953$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$954$		0			0
$955$	−0.0812272	+	0.239288i	−0.0812272	+	0.239288i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.0299475	−	0.0607275i	−0.0299475	−	0.0607275i
$961$	0.624844	+	2.33195i	0.624844	+	2.33195i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$967$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$968$		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	$0.250000\pi$
$972$	−0.135131	−	1.02642i	−0.135131	−	1.02642i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.665060	−	0.583242i	0.665060	−	0.583242i	−0.258819	−	0.965926i	$-0.583333\pi$
$977$	0.665060	−	0.583242i	0.665060	−	0.583242i	0.923880	+	0.382683i	$0.125000\pi$
$978$		0			0
$979$		0			0
$980$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i
$981$		0			0
$982$		0			0
$983$	0.284338	+	0.837633i	0.284338	+	0.837633i	0.991445	+	0.130526i	$0.0416667\pi$
$983$	0.284338	+	0.837633i	0.284338	+	0.837633i	−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
$991$	0.108761	+	1.65938i	0.108761	+	1.65938i	0.608761	+	0.793353i	$0.291667\pi$
$991$	0.108761	+	1.65938i	0.108761	+	1.65938i	−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−0.201540	+	0.0132096i	−0.201540	+	0.0132096i
$994$		0			0
$995$	−0.0917975	+	0.0704387i	−0.0917975	+	0.0704387i
$996$		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
$999$	0.522921	−	1.06038i	0.522921	−	1.06038i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1067.1.bg.a.593.1	yes	16
11.10	odd	2	CM	1067.1.bg.a.593.1	yes	16
97.53	even	48	inner	1067.1.bg.a.538.1	✓	16
1067.538	odd	48	inner	1067.1.bg.a.538.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1067.1.bg.a.538.1	✓	16	97.53	even	48	inner
1067.1.bg.a.538.1	✓	16	1067.538	odd	48	inner
1067.1.bg.a.593.1	yes	16	1.1	even	1	trivial
1067.1.bg.a.593.1	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)

\(f(q)\)	\(=\)	\(q+(0.315118 - 0.410670i) q^{3} +(-0.965926 + 0.258819i) q^{4} +(-0.123864 + 0.0420463i) q^{5} +(0.189469 + 0.707107i) q^{9} +O(q^{10})\)	\(q+(0.315118 - 0.410670i) q^{3} +(-0.965926 + 0.258819i) q^{4} +(-0.123864 + 0.0420463i) q^{5} +(0.189469 + 0.707107i) q^{9} +(0.991445 + 0.130526i) q^{11} +(-0.198092 + 0.478235i) q^{12} +(-0.0217648 + 0.0641169i) q^{15} +(0.866025 - 0.500000i) q^{16} +(0.108761 - 0.0726721i) q^{20} +(0.641502 - 0.0420463i) q^{23} +(-0.779779 + 0.598345i) q^{25} +(0.828328 + 0.343105i) q^{27} +(1.46593 + 1.12484i) q^{31} +(0.366025 - 0.366025i) q^{33} +(-0.366025 - 0.633975i) q^{36} +(0.0862466 - 1.31587i) q^{37} +(-0.991445 + 0.130526i) q^{44} +(-0.0531996 - 0.0796189i) q^{45} +(0.707107 + 0.707107i) q^{47} +(0.0675653 - 0.513210i) q^{48} +(0.130526 - 0.991445i) q^{49} +(1.57313 - 0.207107i) q^{53} +(-0.128293 + 0.0255190i) q^{55} +(-1.99144 - 0.130526i) q^{59} +(0.00442847 - 0.0675653i) q^{60} +(-0.707107 + 0.707107i) q^{64} +(-1.75928 - 0.349942i) q^{67} +(0.184882 - 0.276695i) q^{69} +(-1.34861 + 0.665060i) q^{71} +0.508781i q^{75} +(-0.0862466 + 0.0983454i) q^{80} +(-0.232051 + 0.133975i) q^{81} +(-0.608761 + 0.206647i) q^{92} +(0.923880 - 0.247553i) q^{93} +(-0.130526 - 0.991445i) q^{97} +(0.0955518 + 0.725788i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

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