Properties

Label 3724.123
Modulus $3724$
Conductor $3724$
Order $126$
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(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([63,48,56]))
 
pari: [g,chi] = znchar(Mod(123,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
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.ev

\(\chi_{3724}(123,\cdot)\) \(\chi_{3724}(291,\cdot)\) \(\chi_{3724}(359,\cdot)\) \(\chi_{3724}(403,\cdot)\) \(\chi_{3724}(415,\cdot)\) \(\chi_{3724}(499,\cdot)\) \(\chi_{3724}(823,\cdot)\) \(\chi_{3724}(891,\cdot)\) \(\chi_{3724}(935,\cdot)\) \(\chi_{3724}(947,\cdot)\) \(\chi_{3724}(1031,\cdot)\) \(\chi_{3724}(1187,\cdot)\) \(\chi_{3724}(1355,\cdot)\) \(\chi_{3724}(1423,\cdot)\) \(\chi_{3724}(1467,\cdot)\) \(\chi_{3724}(1479,\cdot)\) \(\chi_{3724}(1563,\cdot)\) \(\chi_{3724}(1719,\cdot)\) \(\chi_{3724}(1887,\cdot)\) \(\chi_{3724}(1955,\cdot)\) \(\chi_{3724}(1999,\cdot)\) \(\chi_{3724}(2011,\cdot)\) \(\chi_{3724}(2095,\cdot)\) \(\chi_{3724}(2251,\cdot)\) \(\chi_{3724}(2487,\cdot)\) \(\chi_{3724}(2531,\cdot)\) \(\chi_{3724}(2543,\cdot)\) \(\chi_{3724}(2783,\cdot)\) \(\chi_{3724}(2951,\cdot)\) \(\chi_{3724}(3063,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((1863,3041,3137)\) → \((-1,e\left(\frac{8}{21}\right),e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{83}{126}\right)\)\(e\left(\frac{10}{63}\right)\)\(e\left(\frac{20}{63}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{50}{63}\right)\)\(e\left(\frac{103}{126}\right)\)\(e\left(\frac{61}{63}\right)\)\(e\left(\frac{109}{126}\right)\)\(e\left(\frac{20}{63}\right)\)\(e\left(\frac{41}{42}\right)\)
value at e.g. 2