Properties

Label 1728.199
Modulus $1728$
Conductor $288$
Order $24$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1728, base_ring=CyclotomicField(24))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12,15,8]))
 
pari: [g,chi] = znchar(Mod(199,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(288\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{288}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.bp

\(\chi_{1728}(199,\cdot)\) \(\chi_{1728}(343,\cdot)\) \(\chi_{1728}(631,\cdot)\) \(\chi_{1728}(775,\cdot)\) \(\chi_{1728}(1063,\cdot)\) \(\chi_{1728}(1207,\cdot)\) \(\chi_{1728}(1495,\cdot)\) \(\chi_{1728}(1639,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.0.18351423083070806589199715754737431920771072.1

Values on generators

\((703,325,1217)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{24}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(199,a) \;\) at \(\;a = \) e.g. 2