Properties

Label 1723.14
Modulus $1723$
Conductor $1723$
Order $287$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1723, base_ring=CyclotomicField(574))
 
M = H._module
 
chi = DirichletCharacter(H, M([478]))
 
pari: [g,chi] = znchar(Mod(14,1723))
 

Basic properties

Modulus: \(1723\)
Conductor: \(1723\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(287\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1723.m

\(\chi_{1723}(4,\cdot)\) \(\chi_{1723}(10,\cdot)\) \(\chi_{1723}(14,\cdot)\) \(\chi_{1723}(16,\cdot)\) \(\chi_{1723}(25,\cdot)\) \(\chi_{1723}(26,\cdot)\) \(\chi_{1723}(35,\cdot)\) \(\chi_{1723}(40,\cdot)\) \(\chi_{1723}(49,\cdot)\) \(\chi_{1723}(62,\cdot)\) \(\chi_{1723}(64,\cdot)\) \(\chi_{1723}(65,\cdot)\) \(\chi_{1723}(66,\cdot)\) \(\chi_{1723}(91,\cdot)\) \(\chi_{1723}(100,\cdot)\) \(\chi_{1723}(104,\cdot)\) \(\chi_{1723}(106,\cdot)\) \(\chi_{1723}(118,\cdot)\) \(\chi_{1723}(135,\cdot)\) \(\chi_{1723}(138,\cdot)\) \(\chi_{1723}(140,\cdot)\) \(\chi_{1723}(142,\cdot)\) \(\chi_{1723}(146,\cdot)\) \(\chi_{1723}(155,\cdot)\) \(\chi_{1723}(160,\cdot)\) \(\chi_{1723}(165,\cdot)\) \(\chi_{1723}(167,\cdot)\) \(\chi_{1723}(169,\cdot)\) \(\chi_{1723}(171,\cdot)\) \(\chi_{1723}(194,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{287})$
Fixed field: Number field defined by a degree 287 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{239}{287}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1723 }(14, a) \) \(1\)\(1\)\(e\left(\frac{109}{287}\right)\)\(e\left(\frac{239}{287}\right)\)\(e\left(\frac{218}{287}\right)\)\(e\left(\frac{223}{287}\right)\)\(e\left(\frac{61}{287}\right)\)\(e\left(\frac{226}{287}\right)\)\(e\left(\frac{40}{287}\right)\)\(e\left(\frac{191}{287}\right)\)\(e\left(\frac{45}{287}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1723 }(14,a) \;\) at \(\;a = \) e.g. 2