Properties

Label 1600.3
Modulus $1600$
Conductor $1600$
Order $80$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: 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([40,15,28]))
 
pari: [g,chi] = znchar(Mod(3,1600))
 

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.cm

\(\chi_{1600}(3,\cdot)\) \(\chi_{1600}(27,\cdot)\) \(\chi_{1600}(83,\cdot)\) \(\chi_{1600}(163,\cdot)\) \(\chi_{1600}(187,\cdot)\) \(\chi_{1600}(267,\cdot)\) \(\chi_{1600}(323,\cdot)\) \(\chi_{1600}(347,\cdot)\) \(\chi_{1600}(403,\cdot)\) \(\chi_{1600}(427,\cdot)\) \(\chi_{1600}(483,\cdot)\) \(\chi_{1600}(563,\cdot)\) \(\chi_{1600}(587,\cdot)\) \(\chi_{1600}(667,\cdot)\) \(\chi_{1600}(723,\cdot)\) \(\chi_{1600}(747,\cdot)\) \(\chi_{1600}(803,\cdot)\) \(\chi_{1600}(827,\cdot)\) \(\chi_{1600}(883,\cdot)\) \(\chi_{1600}(963,\cdot)\) \(\chi_{1600}(987,\cdot)\) \(\chi_{1600}(1067,\cdot)\) \(\chi_{1600}(1123,\cdot)\) \(\chi_{1600}(1147,\cdot)\) \(\chi_{1600}(1203,\cdot)\) \(\chi_{1600}(1227,\cdot)\) \(\chi_{1600}(1283,\cdot)\) \(\chi_{1600}(1363,\cdot)\) \(\chi_{1600}(1387,\cdot)\) \(\chi_{1600}(1467,\cdot)\) ...

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

\((1151,901,577)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{7}{20}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(e\left(\frac{41}{80}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{3}{80}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{80}\right)\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{43}{80}\right)\)
value at e.g. 2