Properties

Label 3549.38
Modulus $3549$
Conductor $3549$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3549, base_ring=CyclotomicField(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([39,13,33]))
 
pari: [g,chi] = znchar(Mod(38,3549))
 

Basic properties

Modulus: \(3549\)
Conductor: \(3549\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 3549.dp

\(\chi_{3549}(38,\cdot)\) \(\chi_{3549}(194,\cdot)\) \(\chi_{3549}(311,\cdot)\) \(\chi_{3549}(467,\cdot)\) \(\chi_{3549}(584,\cdot)\) \(\chi_{3549}(740,\cdot)\) \(\chi_{3549}(857,\cdot)\) \(\chi_{3549}(1130,\cdot)\) \(\chi_{3549}(1286,\cdot)\) \(\chi_{3549}(1403,\cdot)\) \(\chi_{3549}(1559,\cdot)\) \(\chi_{3549}(1676,\cdot)\) \(\chi_{3549}(1832,\cdot)\) \(\chi_{3549}(1949,\cdot)\) \(\chi_{3549}(2105,\cdot)\) \(\chi_{3549}(2222,\cdot)\) \(\chi_{3549}(2378,\cdot)\) \(\chi_{3549}(2495,\cdot)\) \(\chi_{3549}(2651,\cdot)\) \(\chi_{3549}(2768,\cdot)\) \(\chi_{3549}(2924,\cdot)\) \(\chi_{3549}(3197,\cdot)\) \(\chi_{3549}(3314,\cdot)\) \(\chi_{3549}(3470,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((1184,1522,3382)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{11}{26}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{26}\right)\)
value at e.g. 2