Properties

Label 4015.49
Modulus $4015$
Conductor $4015$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,24,55]))
 
pari: [g,chi] = znchar(Mod(49,4015))
 

Basic properties

Modulus: \(4015\)
Conductor: \(4015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 4015.ej

\(\chi_{4015}(49,\cdot)\) \(\chi_{4015}(289,\cdot)\) \(\chi_{4015}(389,\cdot)\) \(\chi_{4015}(654,\cdot)\) \(\chi_{4015}(779,\cdot)\) \(\chi_{4015}(1384,\cdot)\) \(\chi_{4015}(1609,\cdot)\) \(\chi_{4015}(1874,\cdot)\) \(\chi_{4015}(1974,\cdot)\) \(\chi_{4015}(2214,\cdot)\) \(\chi_{4015}(2479,\cdot)\) \(\chi_{4015}(2579,\cdot)\) \(\chi_{4015}(2704,\cdot)\) \(\chi_{4015}(3309,\cdot)\) \(\chi_{4015}(3699,\cdot)\) \(\chi_{4015}(3799,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1607,2191,881)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 4015 }(49, a) \) \(1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{47}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4015 }(49,a) \;\) at \(\;a = \) e.g. 2