Properties

Label 3724.43
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([63,18,112]))
 
pari: [g,chi] = znchar(Mod(43,3724))
 

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.ei

\(\chi_{3724}(43,\cdot)\) \(\chi_{3724}(351,\cdot)\) \(\chi_{3724}(435,\cdot)\) \(\chi_{3724}(519,\cdot)\) \(\chi_{3724}(575,\cdot)\) \(\chi_{3724}(631,\cdot)\) \(\chi_{3724}(967,\cdot)\) \(\chi_{3724}(1023,\cdot)\) \(\chi_{3724}(1051,\cdot)\) \(\chi_{3724}(1107,\cdot)\) \(\chi_{3724}(1163,\cdot)\) \(\chi_{3724}(1415,\cdot)\) \(\chi_{3724}(1499,\cdot)\) \(\chi_{3724}(1555,\cdot)\) \(\chi_{3724}(1583,\cdot)\) \(\chi_{3724}(1639,\cdot)\) \(\chi_{3724}(1695,\cdot)\) \(\chi_{3724}(1947,\cdot)\) \(\chi_{3724}(2031,\cdot)\) \(\chi_{3724}(2087,\cdot)\) \(\chi_{3724}(2115,\cdot)\) \(\chi_{3724}(2171,\cdot)\) \(\chi_{3724}(2227,\cdot)\) \(\chi_{3724}(2479,\cdot)\) \(\chi_{3724}(2563,\cdot)\) \(\chi_{3724}(2619,\cdot)\) \(\chi_{3724}(2703,\cdot)\) \(\chi_{3724}(2759,\cdot)\) \(\chi_{3724}(3011,\cdot)\) \(\chi_{3724}(3095,\cdot)\) ...

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

\((1863,3041,3137)\) → \((-1,e\left(\frac{1}{7}\right),e\left(\frac{8}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{25}{126}\right)\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{10}{63}\right)\)\(e\left(\frac{71}{126}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{89}{126}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{25}{42}\right)\)
value at e.g. 2