Properties

Label 1375.1132
Modulus $1375$
Conductor $275$
Order $20$
Real no
Primitive no
Minimal no
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(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([17,10]))
 
pari: [g,chi] = znchar(Mod(1132,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(197,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1375.bo

\(\chi_{1375}(32,\cdot)\) \(\chi_{1375}(43,\cdot)\) \(\chi_{1375}(582,\cdot)\) \(\chi_{1375}(593,\cdot)\) \(\chi_{1375}(857,\cdot)\) \(\chi_{1375}(868,\cdot)\) \(\chi_{1375}(1132,\cdot)\) \(\chi_{1375}(1143,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.75487840807181783020496368408203125.1

Values on generators

\((1002,376)\) → \((e\left(\frac{17}{20}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(1132,a) \;\) at \(\;a = \) e.g. 2