Properties

Label 1386.37
Modulus $1386$
Conductor $77$
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([0,5,3]))
 
pari: [g,chi] = znchar(Mod(37,1386))
 

Basic properties

Modulus: \(1386\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1386.bx

\(\chi_{1386}(37,\cdot)\) \(\chi_{1386}(163,\cdot)\) \(\chi_{1386}(235,\cdot)\) \(\chi_{1386}(289,\cdot)\) \(\chi_{1386}(361,\cdot)\) \(\chi_{1386}(487,\cdot)\) \(\chi_{1386}(1171,\cdot)\) \(\chi_{1386}(1369,\cdot)\)

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

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

Values

\(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{5}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 15.15.886528337182930278529.1