Properties

Label 2415.509
Modulus $2415$
Conductor $2415$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2415, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,33,55,48]))
 
pari: [g,chi] = znchar(Mod(509,2415))
 

Basic properties

Modulus: \(2415\)
Conductor: \(2415\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2415.de

\(\chi_{2415}(59,\cdot)\) \(\chi_{2415}(164,\cdot)\) \(\chi_{2415}(269,\cdot)\) \(\chi_{2415}(374,\cdot)\) \(\chi_{2415}(404,\cdot)\) \(\chi_{2415}(509,\cdot)\) \(\chi_{2415}(584,\cdot)\) \(\chi_{2415}(614,\cdot)\) \(\chi_{2415}(719,\cdot)\) \(\chi_{2415}(794,\cdot)\) \(\chi_{2415}(899,\cdot)\) \(\chi_{2415}(929,\cdot)\) \(\chi_{2415}(1139,\cdot)\) \(\chi_{2415}(1214,\cdot)\) \(\chi_{2415}(1244,\cdot)\) \(\chi_{2415}(1319,\cdot)\) \(\chi_{2415}(1559,\cdot)\) \(\chi_{2415}(1664,\cdot)\) \(\chi_{2415}(1844,\cdot)\) \(\chi_{2415}(2189,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((806,967,346,1891)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{8}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(26\)
\(1\)\(1\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{5}{66}\right)\)\(-1\)\(e\left(\frac{10}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2415 }(509,a) \;\) at \(\;a = \) e.g. 2