Properties

Label 1723.32
Modulus $1723$
Conductor $1723$
Order $574$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1723, base_ring=CyclotomicField(574))
 
M = H._module
 
chi = DirichletCharacter(H, M([345]))
 
pari: [g,chi] = znchar(Mod(32,1723))
 

Basic properties

Modulus: \(1723\)
Conductor: \(1723\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(574\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1723.n

\(\chi_{1723}(2,\cdot)\) \(\chi_{1723}(5,\cdot)\) \(\chi_{1723}(7,\cdot)\) \(\chi_{1723}(8,\cdot)\) \(\chi_{1723}(13,\cdot)\) \(\chi_{1723}(27,\cdot)\) \(\chi_{1723}(28,\cdot)\) \(\chi_{1723}(31,\cdot)\) \(\chi_{1723}(32,\cdot)\) \(\chi_{1723}(33,\cdot)\) \(\chi_{1723}(50,\cdot)\) \(\chi_{1723}(52,\cdot)\) \(\chi_{1723}(53,\cdot)\) \(\chi_{1723}(59,\cdot)\) \(\chi_{1723}(69,\cdot)\) \(\chi_{1723}(70,\cdot)\) \(\chi_{1723}(73,\cdot)\) \(\chi_{1723}(80,\cdot)\) \(\chi_{1723}(97,\cdot)\) \(\chi_{1723}(98,\cdot)\) \(\chi_{1723}(102,\cdot)\) \(\chi_{1723}(108,\cdot)\) \(\chi_{1723}(111,\cdot)\) \(\chi_{1723}(112,\cdot)\) \(\chi_{1723}(123,\cdot)\) \(\chi_{1723}(124,\cdot)\) \(\chi_{1723}(125,\cdot)\) \(\chi_{1723}(131,\cdot)\) \(\chi_{1723}(132,\cdot)\) \(\chi_{1723}(173,\cdot)\) ...

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 574 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{345}{574}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1723 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{239}{574}\right)\)\(e\left(\frac{345}{574}\right)\)\(e\left(\frac{239}{287}\right)\)\(e\left(\frac{173}{574}\right)\)\(e\left(\frac{5}{287}\right)\)\(e\left(\frac{277}{574}\right)\)\(e\left(\frac{143}{574}\right)\)\(e\left(\frac{58}{287}\right)\)\(e\left(\frac{206}{287}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1723 }(32,a) \;\) at \(\;a = \) e.g. 2