Properties

Label 3920.39
Modulus $3920$
Conductor $1960$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3920, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,21,21,34]))
 
pari: [g,chi] = znchar(Mod(39,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(1960\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1960}(1019,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.ev

\(\chi_{3920}(39,\cdot)\) \(\chi_{3920}(359,\cdot)\) \(\chi_{3920}(599,\cdot)\) \(\chi_{3920}(919,\cdot)\) \(\chi_{3920}(1159,\cdot)\) \(\chi_{3920}(1479,\cdot)\) \(\chi_{3920}(1719,\cdot)\) \(\chi_{3920}(2279,\cdot)\) \(\chi_{3920}(2599,\cdot)\) \(\chi_{3920}(2839,\cdot)\) \(\chi_{3920}(3159,\cdot)\) \(\chi_{3920}(3719,\cdot)\)

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

\((1471,981,3137,3041)\) → \((-1,-1,-1,e\left(\frac{17}{21}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2