Properties

Label 1375.31
Modulus $1375$
Conductor $1375$
Order $25$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1375, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,30]))
 
pari: [g,chi] = znchar(Mod(31,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25\)
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 1375.bu

\(\chi_{1375}(31,\cdot)\) \(\chi_{1375}(71,\cdot)\) \(\chi_{1375}(91,\cdot)\) \(\chi_{1375}(136,\cdot)\) \(\chi_{1375}(306,\cdot)\) \(\chi_{1375}(346,\cdot)\) \(\chi_{1375}(366,\cdot)\) \(\chi_{1375}(411,\cdot)\) \(\chi_{1375}(581,\cdot)\) \(\chi_{1375}(621,\cdot)\) \(\chi_{1375}(641,\cdot)\) \(\chi_{1375}(686,\cdot)\) \(\chi_{1375}(856,\cdot)\) \(\chi_{1375}(896,\cdot)\) \(\chi_{1375}(916,\cdot)\) \(\chi_{1375}(961,\cdot)\) \(\chi_{1375}(1131,\cdot)\) \(\chi_{1375}(1171,\cdot)\) \(\chi_{1375}(1191,\cdot)\) \(\chi_{1375}(1236,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: 25.25.227936564389216769066972959924266550757465665810741484165191650390625.2

Values on generators

\((1002,376)\) → \((e\left(\frac{12}{25}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{6}{25}\right)\)\(1\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{2}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(31,a) \;\) at \(\;a = \) e.g. 2