Properties

Label 1386.445
Modulus $1386$
Conductor $693$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1386)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,10,6]))
 
pari: [g,chi] = znchar(Mod(445,1386))
 

Basic properties

Modulus: \(1386\)
Conductor: \(693\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{693}(445,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1386.bw

\(\chi_{1386}(445,\cdot)\) \(\chi_{1386}(697,\cdot)\) \(\chi_{1386}(709,\cdot)\) \(\chi_{1386}(823,\cdot)\) \(\chi_{1386}(949,\cdot)\) \(\chi_{1386}(961,\cdot)\) \(\chi_{1386}(1087,\cdot)\) \(\chi_{1386}(1213,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((155,199,1135)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)

Values

\(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 15.15.3091133177133909578645502426129.1