Properties

Label 1400.243
Modulus $1400$
Conductor $280$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1400, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,6,9,10]))
 
pari: [g,chi] = znchar(Mod(243,1400))
 

Basic properties

Modulus: \(1400\)
Conductor: \(280\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{280}(243,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1400.cf

\(\chi_{1400}(243,\cdot)\) \(\chi_{1400}(507,\cdot)\) \(\chi_{1400}(843,\cdot)\) \(\chi_{1400}(1307,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.144627327488000000000.1

Values on generators

\((351,701,1177,801)\) → \((-1,-1,-i,e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(i\)\(1\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1400 }(243,a) \;\) at \(\;a = \) e.g. 2