Properties

Label 1600.509
Modulus $1600$
Conductor $1600$
Order $80$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1600, base_ring=CyclotomicField(80))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,56]))
 
pari: [g,chi] = znchar(Mod(509,1600))
 

Basic properties

Modulus: \(1600\)
Conductor: \(1600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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 1600.cq

\(\chi_{1600}(29,\cdot)\) \(\chi_{1600}(69,\cdot)\) \(\chi_{1600}(109,\cdot)\) \(\chi_{1600}(189,\cdot)\) \(\chi_{1600}(229,\cdot)\) \(\chi_{1600}(269,\cdot)\) \(\chi_{1600}(309,\cdot)\) \(\chi_{1600}(389,\cdot)\) \(\chi_{1600}(429,\cdot)\) \(\chi_{1600}(469,\cdot)\) \(\chi_{1600}(509,\cdot)\) \(\chi_{1600}(589,\cdot)\) \(\chi_{1600}(629,\cdot)\) \(\chi_{1600}(669,\cdot)\) \(\chi_{1600}(709,\cdot)\) \(\chi_{1600}(789,\cdot)\) \(\chi_{1600}(829,\cdot)\) \(\chi_{1600}(869,\cdot)\) \(\chi_{1600}(909,\cdot)\) \(\chi_{1600}(989,\cdot)\) \(\chi_{1600}(1029,\cdot)\) \(\chi_{1600}(1069,\cdot)\) \(\chi_{1600}(1109,\cdot)\) \(\chi_{1600}(1189,\cdot)\) \(\chi_{1600}(1229,\cdot)\) \(\chi_{1600}(1269,\cdot)\) \(\chi_{1600}(1309,\cdot)\) \(\chi_{1600}(1389,\cdot)\) \(\chi_{1600}(1429,\cdot)\) \(\chi_{1600}(1469,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1151,901,577)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{7}{10}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{11}{80}\right)\)\(e\left(\frac{9}{80}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{73}{80}\right)\)\(e\left(\frac{67}{80}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{31}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1600 }(509,a) \;\) at \(\;a = \) e.g. 2