LMFDB - Embedded newform 2040.1.df.b.1589.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,1,Mod(389,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 4, 4, 4, 3])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.389"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2040.df (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)

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

Self dual:	no
Analytic conductor:	$1.01809262577$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{8} - \cdots)$

Embedding invariants

Embedding label			1589.2
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	2040.1589
Dual form			2040.1.df.b.389.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.707107 + 0.707107i) q^{2} +(0.382683 - 0.923880i) q^{3} -1.00000i q^{4} +(-0.923880 - 0.382683i) q^{5} +(0.382683 + 0.923880i) q^{6} +(0.707107 + 0.707107i) q^{8} +(-0.707107 - 0.707107i) q^{9} +(0.923880 - 0.382683i) q^{10} +(0.541196 + 1.30656i) q^{11} +(-0.923880 - 0.382683i) q^{12} +1.84776i q^{13} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} +(0.707107 + 0.707107i) q^{17} +1.00000 q^{18} +(-0.382683 + 0.923880i) q^{20} +(-1.30656 - 0.541196i) q^{22} +(0.292893 + 0.707107i) q^{23} +(0.923880 - 0.382683i) q^{24} +(0.707107 + 0.707107i) q^{25} +(-1.30656 - 1.30656i) q^{26} +(-0.923880 + 0.382683i) q^{27} +(-1.30656 - 0.541196i) q^{29} -1.00000i q^{30} +(0.292893 - 0.707107i) q^{31} +(0.707107 - 0.707107i) q^{32} +1.41421 q^{33} -1.00000 q^{34} +(-0.707107 + 0.707107i) q^{36} +(1.70711 + 0.707107i) q^{39} +(-0.382683 - 0.923880i) q^{40} +(1.30656 + 1.30656i) q^{43} +(1.30656 - 0.541196i) q^{44} +(0.382683 + 0.923880i) q^{45} +(-0.707107 - 0.292893i) q^{46} +(-0.382683 + 0.923880i) q^{48} +(0.707107 - 0.707107i) q^{49} -1.00000 q^{50} +(0.923880 - 0.382683i) q^{51} +1.84776 q^{52} +(0.382683 - 0.923880i) q^{54} -1.41421i q^{55} +(1.30656 - 0.541196i) q^{58} +(-0.541196 - 0.541196i) q^{59} +(0.707107 + 0.707107i) q^{60} +(0.292893 + 0.707107i) q^{62} +1.00000i q^{64} +(0.707107 - 1.70711i) q^{65} +(-1.00000 + 1.00000i) q^{66} -0.765367 q^{67} +(0.707107 - 0.707107i) q^{68} +0.765367 q^{69} -1.00000i q^{72} +(0.923880 - 0.382683i) q^{75} +(-1.70711 + 0.707107i) q^{78} +(0.707107 + 1.70711i) q^{79} +(0.923880 + 0.382683i) q^{80} +1.00000i q^{81} +(-0.382683 - 0.923880i) q^{85} -1.84776 q^{86} +(-1.00000 + 1.00000i) q^{87} +(-0.541196 + 1.30656i) q^{88} +(-0.923880 - 0.382683i) q^{90} +(0.707107 - 0.292893i) q^{92} +(-0.541196 - 0.541196i) q^{93} +(-0.382683 - 0.923880i) q^{96} +1.00000i q^{98} +(0.541196 - 1.30656i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{16} + 8 q^{18} + 8 q^{23} + 8 q^{31} - 8 q^{34} + 8 q^{39} - 8 q^{50} + 8 q^{62} - 8 q^{66} - 8 q^{78} - 8 q^{87}+O(q^{100}) $ 8 * q - 8 * q^16 + 8 * q^18 + 8 * q^23 + 8 * q^31 - 8 * q^34 + 8 * q^39 - 8 * q^50 + 8 * q^62 - 8 * q^66 - 8 * q^78 - 8 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$e\left(\frac{5}{8}\right)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$2$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	0.382683	−	0.923880i	0.382683	−	0.923880i
$3$	0.382683	−	0.923880i	0.382683	−	0.923880i
$4$		−	1.00000i		−	1.00000i
$5$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$5$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$6$	0.382683	+	0.923880i	0.382683	+	0.923880i
$7$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$7$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$8$	0.707107	+	0.707107i	0.707107	+	0.707107i
$9$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$10$	0.923880	−	0.382683i	0.923880	−	0.382683i
$11$	0.541196	+	1.30656i	0.541196	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$11$	0.541196	+	1.30656i	0.541196	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$12$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$13$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$13$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$14$		0			0
$15$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$16$	−1.00000			−1.00000
$17$	0.707107	+	0.707107i	0.707107	+	0.707107i
$17$	0.707107	+	0.707107i	0.707107	+	0.707107i
$18$	1.00000			1.00000
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$21$		0			0
$22$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$23$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$23$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$24$	0.923880	−	0.382683i	0.923880	−	0.382683i
$25$	0.707107	+	0.707107i	0.707107	+	0.707107i
$26$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$27$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$28$		0			0
$29$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$29$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$30$		−	1.00000i		−	1.00000i
$31$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$31$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$32$	0.707107	−	0.707107i	0.707107	−	0.707107i
$33$	1.41421			1.41421
$34$	−1.00000			−1.00000
$35$		0			0
$36$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$37$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$37$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$38$		0			0
$39$	1.70711	+	0.707107i	1.70711	+	0.707107i
$40$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$41$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$41$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$42$		0			0
$43$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$43$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$44$	1.30656	−	0.541196i	1.30656	−	0.541196i
$45$	0.382683	+	0.923880i	0.382683	+	0.923880i
$46$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$49$	0.707107	−	0.707107i	0.707107	−	0.707107i
$50$	−1.00000			−1.00000
$51$	0.923880	−	0.382683i	0.923880	−	0.382683i
$52$	1.84776			1.84776
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	0.382683	−	0.923880i	0.382683	−	0.923880i
$55$		−	1.41421i		−	1.41421i
$56$		0			0
$57$		0			0
$58$	1.30656	−	0.541196i	1.30656	−	0.541196i
$59$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$59$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$60$	0.707107	+	0.707107i	0.707107	+	0.707107i
$61$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$61$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$62$	0.292893	+	0.707107i	0.292893	+	0.707107i
$63$		0			0
$64$			1.00000i			1.00000i
$65$	0.707107	−	1.70711i	0.707107	−	1.70711i
$66$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$67$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$67$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$68$	0.707107	−	0.707107i	0.707107	−	0.707107i
$69$	0.765367			0.765367
$70$		0			0
$71$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$71$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$72$		−	1.00000i		−	1.00000i
$73$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$73$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$74$		0			0
$75$	0.923880	−	0.382683i	0.923880	−	0.382683i
$76$		0			0
$77$		0			0
$78$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$79$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$79$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$80$	0.923880	+	0.382683i	0.923880	+	0.382683i
$81$			1.00000i			1.00000i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$86$	−1.84776			−1.84776
$87$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$88$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$91$		0			0
$92$	0.707107	−	0.292893i	0.707107	−	0.292893i
$93$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$94$		0			0
$95$		0			0
$96$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$97$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$97$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$98$			1.00000i			1.00000i
$99$	0.541196	−	1.30656i	0.541196	−	1.30656i
$100$	0.707107	−	0.707107i	0.707107	−	0.707107i
$101$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$101$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$102$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$105$		0			0
$106$		0			0
$107$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$107$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$108$	0.382683	+	0.923880i	0.382683	+	0.923880i
$109$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$109$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$110$	1.00000	+	1.00000i	1.00000	+	1.00000i
$111$		0			0
$112$		0			0
$113$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$114$		0			0
$115$		−	0.765367i		−	0.765367i
$116$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$117$	1.30656	−	1.30656i	1.30656	−	1.30656i
$118$	0.765367			0.765367
$119$		0			0
$120$	−1.00000			−1.00000
$121$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$122$		0			0
$123$		0			0
$124$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$125$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$129$	1.70711	−	0.707107i	1.70711	−	0.707107i
$130$	0.707107	+	1.70711i	0.707107	+	1.70711i
$131$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$131$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$132$		−	1.41421i		−	1.41421i
$133$		0			0
$134$	0.541196	−	0.541196i	0.541196	−	0.541196i
$135$	1.00000			1.00000
$136$			1.00000i			1.00000i
$137$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$137$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$138$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$139$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$139$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.41421	+	1.00000i	−2.41421	+	1.00000i
$144$	0.707107	+	0.707107i	0.707107	+	0.707107i
$145$	1.00000	+	1.00000i	1.00000	+	1.00000i
$146$		0			0
$147$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$148$		0			0
$149$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$149$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$150$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$151$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$151$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$152$		0			0
$153$		−	1.00000i		−	1.00000i
$154$		0			0
$155$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$156$	0.707107	−	1.70711i	0.707107	−	1.70711i
$157$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$157$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$158$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$159$		0			0
$160$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$161$		0			0
$162$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$163$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$163$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$164$		0			0
$165$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$166$		0			0
$167$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$167$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	−2.41421			−2.41421
$170$	0.923880	+	0.382683i	0.923880	+	0.382683i
$171$		0			0
$172$	1.30656	−	1.30656i	1.30656	−	1.30656i
$173$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$173$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$174$		−	1.41421i		−	1.41421i
$175$		0			0
$176$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$177$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$178$		0			0
$179$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$179$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$180$	0.923880	−	0.382683i	0.923880	−	0.382683i
$181$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$181$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$182$		0			0
$183$		0			0
$184$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$185$		0			0
$186$	0.765367			0.765367
$187$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	0.923880	+	0.382683i	0.923880	+	0.382683i
$193$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$193$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$194$		0			0
$195$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$196$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$197$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$197$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$198$	0.541196	+	1.30656i	0.541196	+	1.30656i
$199$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$199$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$200$			1.00000i			1.00000i
$201$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$202$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$203$		0			0
$204$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$205$		0			0
$206$		0			0
$207$	0.292893	−	0.707107i	0.292893	−	0.707107i
$208$		−	1.84776i		−	1.84776i
$209$		0			0
$210$		0			0
$211$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$211$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$216$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−1.41421			−1.41421
$221$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		−	1.00000i		−	1.00000i
$226$	1.70711	+	0.707107i	1.70711	+	0.707107i
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$	0.541196	+	0.541196i	0.541196	+	0.541196i
$231$		0			0
$232$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$233$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$233$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$234$			1.84776i			1.84776i
$235$		0			0
$236$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$237$	1.84776			1.84776
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	0.707107	−	0.707107i	0.707107	−	0.707107i
$241$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$241$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$242$		−	1.00000i		−	1.00000i
$243$	0.923880	+	0.382683i	0.923880	+	0.382683i
$244$		0			0
$245$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$246$		0			0
$247$		0			0
$248$	0.707107	−	0.292893i	0.707107	−	0.292893i
$249$		0			0
$250$	0.923880	+	0.382683i	0.923880	+	0.382683i
$251$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$251$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$252$		0			0
$253$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$254$		0			0
$255$	−1.00000			−1.00000
$256$	1.00000			1.00000
$257$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$257$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$258$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$259$		0			0
$260$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$261$	0.541196	+	1.30656i	0.541196	+	1.30656i
$262$		0			0
$263$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$263$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$264$	1.00000	+	1.00000i	1.00000	+	1.00000i
$265$		0			0
$266$		0			0
$267$		0			0
$268$			0.765367i			0.765367i
$269$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$269$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$270$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$271$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$271$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$272$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$273$		0			0
$274$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$275$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$276$		−	0.765367i		−	0.765367i
$277$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$277$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$278$		0			0
$279$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$283$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$284$		0			0
$285$		0			0
$286$	1.00000	−	2.41421i	1.00000	−	2.41421i
$287$		0			0
$288$	−1.00000			−1.00000
$289$			1.00000i			1.00000i
$290$	−1.41421			−1.41421
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	0.923880	+	0.382683i	0.923880	+	0.382683i
$295$	0.292893	+	0.707107i	0.292893	+	0.707107i
$296$		0			0
$297$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$298$	0.541196	+	0.541196i	0.541196	+	0.541196i
$299$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$300$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$301$		0			0
$302$		−	1.41421i		−	1.41421i
$303$	0.707107	−	1.70711i	0.707107	−	1.70711i
$304$		0			0
$305$		0			0
$306$	0.707107	+	0.707107i	0.707107	+	0.707107i
$307$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$307$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$308$		0			0
$309$		0			0
$310$		−	0.765367i		−	0.765367i
$311$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$311$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$312$	0.707107	+	1.70711i	0.707107	+	1.70711i
$313$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$313$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$314$	0.541196	+	0.541196i	0.541196	+	0.541196i
$315$		0			0
$316$	1.70711	−	0.707107i	1.70711	−	0.707107i
$317$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$317$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$318$		0			0
$319$		−	2.00000i		−	2.00000i
$320$	0.382683	−	0.923880i	0.382683	−	0.923880i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$326$	0.541196	−	1.30656i	0.541196	−	1.30656i
$327$		0			0
$328$		0			0
$329$		0			0
$330$	1.30656	−	0.541196i	1.30656	−	0.541196i
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$		0			0
$334$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$335$	0.707107	+	0.292893i	0.707107	+	0.292893i
$336$		0			0
$337$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$337$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$338$	1.70711	−	1.70711i	1.70711	−	1.70711i
$339$	−1.84776			−1.84776
$340$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$341$	1.08239			1.08239
$342$		0			0
$343$		0			0
$344$			1.84776i			1.84776i
$345$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$346$		0			0
$347$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$347$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$348$	1.00000	+	1.00000i	1.00000	+	1.00000i
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$352$	1.30656	+	0.541196i	1.30656	+	0.541196i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$	0.292893	−	0.707107i	0.292893	−	0.707107i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.84776			−1.84776
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$361$		−	1.00000i		−	1.00000i
$362$		0			0
$363$	0.382683	+	0.923880i	0.382683	+	0.923880i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$367$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$368$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$373$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$373$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$374$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$375$	−1.00000			−1.00000
$376$		0			0
$377$	1.00000	−	2.41421i	1.00000	−	2.41421i
$378$		0			0
$379$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$379$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$383$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$384$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$385$		0			0
$386$		0			0
$387$		−	1.84776i		−	1.84776i
$388$		0			0
$389$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$389$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$390$	1.84776			1.84776
$391$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$392$	1.00000			1.00000
$393$		0			0
$394$		0			0
$395$		−	1.84776i		−	1.84776i
$396$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$397$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$397$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$398$	0.707107	−	0.292893i	0.707107	−	0.292893i
$399$		0			0
$400$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$401$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$401$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$402$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$403$	1.30656	+	0.541196i	1.30656	+	0.541196i
$404$		−	1.84776i		−	1.84776i
$405$	0.382683	−	0.923880i	0.382683	−	0.923880i
$406$		0			0
$407$		0			0
$408$	0.923880	+	0.382683i	0.923880	+	0.382683i
$409$	−2.00000			−2.00000			−1.00000			$\pi$
$409$	−2.00000			−2.00000			−1.00000			$\pi$
$410$		0			0
$411$	0.541196	−	1.30656i	0.541196	−	1.30656i
$412$		0			0
$413$		0			0
$414$	0.292893	+	0.707107i	0.292893	+	0.707107i
$415$		0			0
$416$	1.30656	+	1.30656i	1.30656	+	1.30656i
$417$		0			0
$418$		0			0
$419$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$419$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			1.00000i			1.00000i
$426$		0			0
$427$		0			0
$428$		0			0
$429$			2.61313i			2.61313i
$430$	1.70711	+	0.707107i	1.70711	+	0.707107i
$431$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$431$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$432$	0.923880	−	0.382683i	0.923880	−	0.382683i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$	1.30656	−	0.541196i	1.30656	−	0.541196i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$439$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$440$	1.00000	−	1.00000i	1.00000	−	1.00000i
$441$	−1.00000			−1.00000
$442$		−	1.84776i		−	1.84776i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$448$		0			0
$449$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$449$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$450$	0.707107	+	0.707107i	0.707107	+	0.707107i
$451$		0			0
$452$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$453$	0.541196	+	1.30656i	0.541196	+	1.30656i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$460$	−0.765367			−0.765367
$461$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$461$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	1.30656	+	0.541196i	1.30656	+	0.541196i
$465$	0.292893	+	0.707107i	0.292893	+	0.707107i
$466$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$469$		0			0
$470$		0			0
$471$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$472$		−	0.765367i		−	0.765367i
$473$	−1.00000	+	2.41421i	−1.00000	+	2.41421i
$474$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$479$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$480$			1.00000i			1.00000i
$481$		0			0
$482$	0.707107	+	1.70711i	0.707107	+	1.70711i
$483$		0			0
$484$	0.707107	+	0.707107i	0.707107	+	0.707107i
$485$		0			0
$486$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$487$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$487$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$488$		0			0
$489$			1.41421i			1.41421i
$490$	0.382683	−	0.923880i	0.382683	−	0.923880i
$491$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$491$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$492$		0			0
$493$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$494$		0			0
$495$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$496$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$497$		0			0
$498$		0			0
$499$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$499$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$500$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$501$	1.30656	+	1.30656i	1.30656	+	1.30656i
$502$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$503$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$503$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$504$		0			0
$505$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$506$		−	1.08239i		−	1.08239i
$507$	−0.923880	+	2.23044i	−0.923880	+	2.23044i
$508$		0			0
$509$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$509$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$510$	0.707107	−	0.707107i	0.707107	−	0.707107i
$511$		0			0
$512$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$513$		0			0
$514$		−	1.41421i		−	1.41421i
$515$		0			0
$516$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.70711	−	0.707107i	1.70711	−	0.707107i
$521$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$521$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$522$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$523$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$523$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$524$		0			0
$525$		0			0
$526$	1.41421			1.41421
$527$	0.707107	−	0.292893i	0.707107	−	0.292893i
$528$	−1.41421			−1.41421
$529$	0.292893	−	0.292893i	0.292893	−	0.292893i
$530$		0			0
$531$			0.765367i			0.765367i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$537$	1.70711	−	0.707107i	1.70711	−	0.707107i
$538$		0			0
$539$	1.30656	+	0.541196i	1.30656	+	0.541196i
$540$		−	1.00000i		−	1.00000i
$541$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$541$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$542$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$543$		0			0
$544$	1.00000			1.00000
$545$		0			0
$546$		0			0
$547$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$547$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$548$		−	1.41421i		−	1.41421i
$549$		0			0
$550$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$551$		0			0
$552$	0.541196	+	0.541196i	0.541196	+	0.541196i
$553$		0			0
$554$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	0.292893	−	0.707107i	0.292893	−	0.707107i
$559$	−2.41421	+	2.41421i	−2.41421	+	2.41421i
$560$		0			0
$561$	1.00000	+	1.00000i	1.00000	+	1.00000i
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$			1.84776i			1.84776i
$566$	1.30656	+	0.541196i	1.30656	+	0.541196i
$567$		0			0
$568$		0			0
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$571$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$572$	1.00000	+	2.41421i	1.00000	+	2.41421i
$573$		0			0
$574$		0			0
$575$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$576$	0.707107	−	0.707107i	0.707107	−	0.707107i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$579$		0			0
$580$	1.00000	−	1.00000i	1.00000	−	1.00000i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$589$		0			0
$590$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$591$		0			0
$592$		0			0
$593$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$593$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$594$	1.41421			1.41421
$595$		0			0
$596$	−0.765367			−0.765367
$597$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$598$	0.541196	−	1.30656i	0.541196	−	1.30656i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	0.923880	+	0.382683i	0.923880	+	0.382683i
$601$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$601$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$602$		0			0
$603$	0.541196	+	0.541196i	0.541196	+	0.541196i
$604$	1.00000	+	1.00000i	1.00000	+	1.00000i
$605$	0.923880	−	0.382683i	0.923880	−	0.382683i
$606$	0.707107	+	1.70711i	0.707107	+	1.70711i
$607$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$607$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.00000			−1.00000
$613$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$613$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$614$	0.541196	−	0.541196i	0.541196	−	0.541196i
$615$		0			0
$616$		0			0
$617$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$617$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$619$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$620$	0.541196	+	0.541196i	0.541196	+	0.541196i
$621$	−0.541196	−	0.541196i	−0.541196	−	0.541196i
$622$		0			0
$623$		0			0
$624$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$625$			1.00000i			1.00000i
$626$		0			0
$627$		0			0
$628$	−0.765367			−0.765367
$629$		0			0
$630$		0			0
$631$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$631$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$632$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.30656	+	1.30656i	1.30656	+	1.30656i
$638$	1.41421	+	1.41421i	1.41421	+	1.41421i
$639$		0			0
$640$	0.382683	+	0.923880i	0.382683	+	0.923880i
$641$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$641$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$642$		0			0
$643$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$643$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$644$		0			0
$645$	−1.84776			−1.84776
$646$		0			0
$647$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$647$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$648$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$649$	0.414214	−	1.00000i	0.414214	−	1.00000i
$650$		−	1.84776i		−	1.84776i
$651$		0			0
$652$	0.541196	+	1.30656i	0.541196	+	1.30656i
$653$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$653$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$659$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$660$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$661$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$661$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$662$		0			0
$663$	0.707107	+	1.70711i	0.707107	+	1.70711i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	1.08239i		−	1.08239i
$668$	1.70711	+	0.707107i	1.70711	+	0.707107i
$669$		0			0
$670$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$671$		0			0
$672$		0			0
$673$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$673$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$674$		0			0
$675$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$676$			2.41421i			2.41421i
$677$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$677$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$678$	1.30656	−	1.30656i	1.30656	−	1.30656i
$679$		0			0
$680$	0.382683	−	0.923880i	0.382683	−	0.923880i
$681$		0			0
$682$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$683$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$683$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$684$		0			0
$685$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$686$		0			0
$687$		0			0
$688$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$689$		0			0
$690$	0.707107	−	0.292893i	0.707107	−	0.292893i
$691$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$691$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$	−1.41421			−1.41421
$697$		0			0
$698$		0			0
$699$	0.541196	−	0.541196i	0.541196	−	0.541196i
$700$		0			0
$701$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$701$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$702$	1.70711	+	0.707107i	1.70711	+	0.707107i
$703$		0			0
$704$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$705$		0			0
$706$		0			0
$707$		0			0
$708$	0.292893	+	0.707107i	0.292893	+	0.707107i
$709$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$709$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$710$		0			0
$711$	0.707107	−	1.70711i	0.707107	−	1.70711i
$712$		0			0
$713$	0.585786			0.585786
$714$		0			0
$715$	2.61313			2.61313
$716$	1.30656	−	1.30656i	1.30656	−	1.30656i
$717$		0			0
$718$		0			0
$719$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$719$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$720$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$721$		0			0
$722$	0.707107	+	0.707107i	0.707107	+	0.707107i
$723$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$724$		0			0
$725$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$726$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	0.707107	−	0.707107i	0.707107	−	0.707107i
$730$		0			0
$731$			1.84776i			1.84776i
$732$		0			0
$733$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$733$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$734$		0			0
$735$			1.00000i			1.00000i
$736$	0.707107	+	0.292893i	0.707107	+	0.292893i
$737$	−0.414214	−	1.00000i	−0.414214	−	1.00000i
$738$		0			0
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$743$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$744$		−	0.765367i		−	0.765367i
$745$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$746$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$747$		0			0
$748$	1.30656	+	0.541196i	1.30656	+	0.541196i
$749$		0			0
$750$	0.707107	−	0.707107i	0.707107	−	0.707107i
$751$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$751$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$752$		0			0
$753$	0.707107	+	0.292893i	0.707107	+	0.292893i
$754$	1.00000	+	2.41421i	1.00000	+	2.41421i
$755$	1.30656	−	0.541196i	1.30656	−	0.541196i
$756$		0			0
$757$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$757$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$758$		0			0
$759$	0.414214	+	1.00000i	0.414214	+	1.00000i
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$766$	−1.41421			−1.41421
$767$	1.00000	−	1.00000i	1.00000	−	1.00000i
$768$	0.382683	−	0.923880i	0.382683	−	0.923880i
$769$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$769$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$770$		0			0
$771$	0.541196	+	1.30656i	0.541196	+	1.30656i
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$	1.30656	+	1.30656i	1.30656	+	1.30656i
$775$	0.707107	−	0.292893i	0.707107	−	0.292893i
$776$		0			0
$777$		0			0
$778$			1.84776i			1.84776i
$779$		0			0
$780$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$781$		0			0
$782$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$783$	1.41421			1.41421
$784$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$785$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$786$		0			0
$787$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$787$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$788$		0			0
$789$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$790$	1.30656	+	1.30656i	1.30656	+	1.30656i
$791$		0			0
$792$	1.30656	−	0.541196i	1.30656	−	0.541196i
$793$		0			0
$794$	1.30656	+	0.541196i	1.30656	+	0.541196i
$795$		0			0
$796$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			1.00000
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.707107	+	0.292893i	0.707107	+	0.292893i
$805$		0			0
$806$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$807$		0			0
$808$	1.30656	+	1.30656i	1.30656	+	1.30656i
$809$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$809$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$810$	0.382683	+	0.923880i	0.382683	+	0.923880i
$811$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$811$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$812$		0			0
$813$	0.541196	−	1.30656i	0.541196	−	1.30656i
$814$		0			0
$815$	1.41421			1.41421
$816$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$817$		0			0
$818$	1.41421	−	1.41421i	1.41421	−	1.41421i
$819$		0			0
$820$		0			0
$821$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$821$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$822$	0.541196	+	1.30656i	0.541196	+	1.30656i
$823$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$823$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$824$		0			0
$825$	1.00000	+	1.00000i	1.00000	+	1.00000i
$826$		0			0
$827$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$827$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$828$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	1.00000	−	1.00000i	1.00000	−	1.00000i
$832$	−1.84776			−1.84776
$833$	1.00000			1.00000
$834$		0			0
$835$	1.30656	−	1.30656i	1.30656	−	1.30656i
$836$		0			0
$837$			0.765367i			0.765367i
$838$	1.30656	+	0.541196i	1.30656	+	0.541196i
$839$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$839$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	0.707107	+	0.707107i	0.707107	+	0.707107i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.23044	+	0.923880i	2.23044	+	0.923880i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.41421			−1.41421
$850$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$851$		0			0
$852$		0			0
$853$	0.541196	−	1.30656i	0.541196	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$853$	0.541196	−	1.30656i	0.541196	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$857$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$858$	−1.84776	−	1.84776i	−1.84776	−	1.84776i
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$861$		0			0
$862$		0			0
$863$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$863$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$864$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$865$		0			0
$866$		0			0
$867$	0.923880	+	0.382683i	0.923880	+	0.382683i
$868$		0			0
$869$	−1.84776	+	1.84776i	−1.84776	+	1.84776i
$870$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$871$		−	1.41421i		−	1.41421i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.84776	−	0.765367i	1.84776	−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$877$	1.84776	−	0.765367i	1.84776	−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$878$	0.707107	+	1.70711i	0.707107	+	1.70711i
$879$		0			0
$880$			1.41421i			1.41421i
$881$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$881$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$882$	0.707107	−	0.707107i	0.707107	−	0.707107i
$883$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$883$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$884$	1.30656	+	1.30656i	1.30656	+	1.30656i
$885$	0.765367			0.765367
$886$		0			0
$887$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$887$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$892$		0			0
$893$		0			0
$894$	0.707107	−	0.292893i	0.707107	−	0.292893i
$895$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$896$		0			0
$897$			1.41421i			1.41421i
$898$		0			0
$899$	−0.765367	+	0.765367i	−0.765367	+	0.765367i
$900$	−1.00000			−1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.707107	−	1.70711i	0.707107	−	1.70711i
$905$		0			0
$906$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$907$	−0.765367	−	1.84776i	−0.765367	−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$907$	−0.765367	−	1.84776i	−0.765367	−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$908$		0			0
$909$	−1.30656	−	1.30656i	−1.30656	−	1.30656i
$910$		0			0
$911$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$911$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.923880	−	0.382683i	0.923880	−	0.382683i
$919$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$919$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$920$	0.541196	−	0.541196i	0.541196	−	0.541196i
$921$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$922$			1.84776i			1.84776i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$929$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$929$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$930$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$931$		0			0
$932$	0.292893	−	0.707107i	0.292893	−	0.707107i
$933$		0			0
$934$		0			0
$935$	1.00000	−	1.00000i	1.00000	−	1.00000i
$936$	1.84776			1.84776
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$941$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$942$	0.707107	−	0.292893i	0.707107	−	0.292893i
$943$		0			0
$944$	0.541196	+	0.541196i	0.541196	+	0.541196i
$945$		0			0
$946$	−1.00000	−	2.41421i	−1.00000	−	2.41421i
$947$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$947$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$948$		−	1.84776i		−	1.84776i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1.84776	−	0.765367i	−1.84776	−	0.765367i
$958$		0			0
$959$		0			0
$960$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$961$	0.292893	+	0.292893i	0.292893	+	0.292893i
$962$		0			0
$963$		0			0
$964$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−1.00000			−1.00000
$969$		0			0
$970$		0			0
$971$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$971$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$972$	0.382683	−	0.923880i	0.382683	−	0.923880i
$973$		0			0
$974$		0			0
$975$	0.707107	+	1.70711i	0.707107	+	1.70711i
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$979$		0			0
$980$	0.382683	+	0.923880i	0.382683	+	0.923880i
$981$		0			0
$982$			1.84776i			1.84776i
$983$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$983$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$984$		0			0
$985$		0			0
$986$	1.30656	+	0.541196i	1.30656	+	0.541196i
$987$		0			0
$988$		0			0
$989$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$990$		−	1.41421i		−	1.41421i
$991$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$991$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$992$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$993$		0			0
$994$		0			0
$995$	0.541196	+	0.541196i	0.541196	+	0.541196i
$996$		0			0
$997$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$997$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.1.df.b.1589.2	yes	8
3.2	odd	2		2040.1.df.a.1589.2	yes	8
5.4	even	2		2040.1.df.a.1589.1	yes	8
8.5	even	2	inner	2040.1.df.b.1589.1	yes	8
15.14	odd	2	inner	2040.1.df.b.1589.1	yes	8
17.15	even	8	inner	2040.1.df.b.389.2	yes	8
24.5	odd	2		2040.1.df.a.1589.1	yes	8
40.29	even	2		2040.1.df.a.1589.2	yes	8
51.32	odd	8		2040.1.df.a.389.2	yes	8
85.49	even	8		2040.1.df.a.389.1	✓	8
120.29	odd	2	CM	2040.1.df.b.1589.2	yes	8
136.117	even	8	inner	2040.1.df.b.389.1	yes	8
255.134	odd	8	inner	2040.1.df.b.389.1	yes	8
408.389	odd	8		2040.1.df.a.389.1	✓	8
680.389	even	8		2040.1.df.a.389.2	yes	8
2040.389	odd	8	inner	2040.1.df.b.389.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.1.df.a.389.1	✓	8	85.49	even	8
2040.1.df.a.389.1	✓	8	408.389	odd	8
2040.1.df.a.389.2	yes	8	51.32	odd	8
2040.1.df.a.389.2	yes	8	680.389	even	8
2040.1.df.a.1589.1	yes	8	5.4	even	2
2040.1.df.a.1589.1	yes	8	24.5	odd	2
2040.1.df.a.1589.2	yes	8	3.2	odd	2
2040.1.df.a.1589.2	yes	8	40.29	even	2
2040.1.df.b.389.1	yes	8	136.117	even	8	inner
2040.1.df.b.389.1	yes	8	255.134	odd	8	inner
2040.1.df.b.389.2	yes	8	17.15	even	8	inner
2040.1.df.b.389.2	yes	8	2040.389	odd	8	inner
2040.1.df.b.1589.1	yes	8	8.5	even	2	inner
2040.1.df.b.1589.1	yes	8	15.14	odd	2	inner
2040.1.df.b.1589.2	yes	8	1.1	even	1	trivial
2040.1.df.b.1589.2	yes	8	120.29	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2040.df (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{2} +(0.382683 - 0.923880i) q^{3} -1.00000i q^{4} +(-0.923880 - 0.382683i) q^{5} +(0.382683 + 0.923880i) q^{6} +(0.707107 + 0.707107i) q^{8} +(-0.707107 - 0.707107i) q^{9} +(0.923880 - 0.382683i) q^{10} +(0.541196 + 1.30656i) q^{11} +(-0.923880 - 0.382683i) q^{12} +1.84776i q^{13} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} +(0.707107 + 0.707107i) q^{17} +1.00000 q^{18} +(-0.382683 + 0.923880i) q^{20} +(-1.30656 - 0.541196i) q^{22} +(0.292893 + 0.707107i) q^{23} +(0.923880 - 0.382683i) q^{24} +(0.707107 + 0.707107i) q^{25} +(-1.30656 - 1.30656i) q^{26} +(-0.923880 + 0.382683i) q^{27} +(-1.30656 - 0.541196i) q^{29} -1.00000i q^{30} +(0.292893 - 0.707107i) q^{31} +(0.707107 - 0.707107i) q^{32} +1.41421 q^{33} -1.00000 q^{34} +(-0.707107 + 0.707107i) q^{36} +(1.70711 + 0.707107i) q^{39} +(-0.382683 - 0.923880i) q^{40} +(1.30656 + 1.30656i) q^{43} +(1.30656 - 0.541196i) q^{44} +(0.382683 + 0.923880i) q^{45} +(-0.707107 - 0.292893i) q^{46} +(-0.382683 + 0.923880i) q^{48} +(0.707107 - 0.707107i) q^{49} -1.00000 q^{50} +(0.923880 - 0.382683i) q^{51} +1.84776 q^{52} +(0.382683 - 0.923880i) q^{54} -1.41421i q^{55} +(1.30656 - 0.541196i) q^{58} +(-0.541196 - 0.541196i) q^{59} +(0.707107 + 0.707107i) q^{60} +(0.292893 + 0.707107i) q^{62} +1.00000i q^{64} +(0.707107 - 1.70711i) q^{65} +(-1.00000 + 1.00000i) q^{66} -0.765367 q^{67} +(0.707107 - 0.707107i) q^{68} +0.765367 q^{69} -1.00000i q^{72} +(0.923880 - 0.382683i) q^{75} +(-1.70711 + 0.707107i) q^{78} +(0.707107 + 1.70711i) q^{79} +(0.923880 + 0.382683i) q^{80} +1.00000i q^{81} +(-0.382683 - 0.923880i) q^{85} -1.84776 q^{86} +(-1.00000 + 1.00000i) q^{87} +(-0.541196 + 1.30656i) q^{88} +(-0.923880 - 0.382683i) q^{90} +(0.707107 - 0.292893i) q^{92} +(-0.541196 - 0.541196i) q^{93} +(-0.382683 - 0.923880i) q^{96} +1.00000i q^{98} +(0.541196 - 1.30656i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{16} + 8 q^{18} + 8 q^{23} + 8 q^{31} - 8 q^{34} + 8 q^{39} - 8 q^{50} + 8 q^{62} - 8 q^{66} - 8 q^{78} - 8 q^{87}+O(q^{100}) \) 8 * q - 8 * q^16 + 8 * q^18 + 8 * q^23 + 8 * q^31 - 8 * q^34 + 8 * q^39 - 8 * q^50 + 8 * q^62 - 8 * q^66 - 8 * q^78 - 8 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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