Properties

Label 1386.13
Modulus $1386$
Conductor $693$
Order $30$
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([10,15,3]))
 
pari: [g,chi] = znchar(Mod(13,1386))
 

Basic properties

Modulus: \(1386\)
Conductor: \(693\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{693}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1386.cj

\(\chi_{1386}(13,\cdot)\) \(\chi_{1386}(139,\cdot)\) \(\chi_{1386}(349,\cdot)\) \(\chi_{1386}(391,\cdot)\) \(\chi_{1386}(475,\cdot)\) \(\chi_{1386}(601,\cdot)\) \(\chi_{1386}(853,\cdot)\) \(\chi_{1386}(1273,\cdot)\)

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

\((155,199,1135)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{1}{10}\right))\)

Values

\(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{15}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.30.756699223436275881084160546254906938181247229834490855635253.1