Properties

Label 1600.303
Modulus $1600$
Conductor $400$
Order $20$
Real no
Primitive no
Minimal no
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(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,15,7]))
 
pari: [g,chi] = znchar(Mod(303,1600))
 

Basic properties

Modulus: \(1600\)
Conductor: \(400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{400}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1600.bt

\(\chi_{1600}(303,\cdot)\) \(\chi_{1600}(527,\cdot)\) \(\chi_{1600}(623,\cdot)\) \(\chi_{1600}(847,\cdot)\) \(\chi_{1600}(1167,\cdot)\) \(\chi_{1600}(1263,\cdot)\) \(\chi_{1600}(1487,\cdot)\) \(\chi_{1600}(1583,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.104857600000000000000000000000000000000000.2

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(-i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{5}\right)\)
value at e.g. 2