Properties

Label 1600.21
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([0,65,48]))
 
pari: [g,chi] = znchar(Mod(21,1600))
 

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.cr

\(\chi_{1600}(21,\cdot)\) \(\chi_{1600}(61,\cdot)\) \(\chi_{1600}(141,\cdot)\) \(\chi_{1600}(181,\cdot)\) \(\chi_{1600}(221,\cdot)\) \(\chi_{1600}(261,\cdot)\) \(\chi_{1600}(341,\cdot)\) \(\chi_{1600}(381,\cdot)\) \(\chi_{1600}(421,\cdot)\) \(\chi_{1600}(461,\cdot)\) \(\chi_{1600}(541,\cdot)\) \(\chi_{1600}(581,\cdot)\) \(\chi_{1600}(621,\cdot)\) \(\chi_{1600}(661,\cdot)\) \(\chi_{1600}(741,\cdot)\) \(\chi_{1600}(781,\cdot)\) \(\chi_{1600}(821,\cdot)\) \(\chi_{1600}(861,\cdot)\) \(\chi_{1600}(941,\cdot)\) \(\chi_{1600}(981,\cdot)\) \(\chi_{1600}(1021,\cdot)\) \(\chi_{1600}(1061,\cdot)\) \(\chi_{1600}(1141,\cdot)\) \(\chi_{1600}(1181,\cdot)\) \(\chi_{1600}(1221,\cdot)\) \(\chi_{1600}(1261,\cdot)\) \(\chi_{1600}(1341,\cdot)\) \(\chi_{1600}(1381,\cdot)\) \(\chi_{1600}(1421,\cdot)\) \(\chi_{1600}(1461,\cdot)\) ...

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

\((1151,901,577)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(1\)\(1\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{53}{80}\right)\)\(e\left(\frac{47}{80}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{39}{80}\right)\)\(e\left(\frac{61}{80}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{73}{80}\right)\)
value at e.g. 2