Properties

Label 1600.101
Modulus $1600$
Conductor $64$
Order $16$
Real no
Primitive no
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(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,0]))
 
pari: [g,chi] = znchar(Mod(101,1600))
 

Basic properties

Modulus: \(1600\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1600.bm

\(\chi_{1600}(101,\cdot)\) \(\chi_{1600}(301,\cdot)\) \(\chi_{1600}(501,\cdot)\) \(\chi_{1600}(701,\cdot)\) \(\chi_{1600}(901,\cdot)\) \(\chi_{1600}(1101,\cdot)\) \(\chi_{1600}(1301,\cdot)\) \(\chi_{1600}(1501,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{64})^+\)

Values on generators

\((1151,901,577)\) → \((1,e\left(\frac{9}{16}\right),1)\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(-i\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{16}\right)\)
value at e.g. 2