Properties

Label 3724.75
Modulus $3724$
Conductor $3724$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,17,21]))
 
pari: [g,chi] = znchar(Mod(75,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.de

\(\chi_{3724}(75,\cdot)\) \(\chi_{3724}(759,\cdot)\) \(\chi_{3724}(1139,\cdot)\) \(\chi_{3724}(1291,\cdot)\) \(\chi_{3724}(1671,\cdot)\) \(\chi_{3724}(1823,\cdot)\) \(\chi_{3724}(2203,\cdot)\) \(\chi_{3724}(2355,\cdot)\) \(\chi_{3724}(2735,\cdot)\) \(\chi_{3724}(2887,\cdot)\) \(\chi_{3724}(3267,\cdot)\) \(\chi_{3724}(3419,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((1863,3041,3137)\) → \((-1,e\left(\frac{17}{42}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{3}{14}\right)\)
value at e.g. 2