Properties

Label 3920.3
Modulus $3920$
Conductor $3920$
Order $84$
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(3920, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([42,63,63,2]))
 
pari: [g,chi] = znchar(Mod(3,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(3920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 3920.fg

\(\chi_{3920}(3,\cdot)\) \(\chi_{3920}(187,\cdot)\) \(\chi_{3920}(243,\cdot)\) \(\chi_{3920}(507,\cdot)\) \(\chi_{3920}(563,\cdot)\) \(\chi_{3920}(747,\cdot)\) \(\chi_{3920}(1067,\cdot)\) \(\chi_{3920}(1123,\cdot)\) \(\chi_{3920}(1307,\cdot)\) \(\chi_{3920}(1363,\cdot)\) \(\chi_{3920}(1627,\cdot)\) \(\chi_{3920}(1683,\cdot)\) \(\chi_{3920}(1867,\cdot)\) \(\chi_{3920}(1923,\cdot)\) \(\chi_{3920}(2243,\cdot)\) \(\chi_{3920}(2427,\cdot)\) \(\chi_{3920}(2483,\cdot)\) \(\chi_{3920}(2747,\cdot)\) \(\chi_{3920}(2803,\cdot)\) \(\chi_{3920}(2987,\cdot)\) \(\chi_{3920}(3043,\cdot)\) \(\chi_{3920}(3307,\cdot)\) \(\chi_{3920}(3603,\cdot)\) \(\chi_{3920}(3867,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1471,981,3137,3041)\) → \((-1,-i,-i,e\left(\frac{1}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2