Properties

Label 1015.44
Modulus $1015$
Conductor $1015$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1015, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,28,81]))
 
pari: [g,chi] = znchar(Mod(44,1015))
 

Basic properties

Modulus: \(1015\)
Conductor: \(1015\)
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 1015.cx

\(\chi_{1015}(39,\cdot)\) \(\chi_{1015}(44,\cdot)\) \(\chi_{1015}(79,\cdot)\) \(\chi_{1015}(114,\cdot)\) \(\chi_{1015}(184,\cdot)\) \(\chi_{1015}(214,\cdot)\) \(\chi_{1015}(359,\cdot)\) \(\chi_{1015}(424,\cdot)\) \(\chi_{1015}(569,\cdot)\) \(\chi_{1015}(599,\cdot)\) \(\chi_{1015}(669,\cdot)\) \(\chi_{1015}(704,\cdot)\) \(\chi_{1015}(739,\cdot)\) \(\chi_{1015}(744,\cdot)\) \(\chi_{1015}(809,\cdot)\) \(\chi_{1015}(814,\cdot)\) \(\chi_{1015}(844,\cdot)\) \(\chi_{1015}(849,\cdot)\) \(\chi_{1015}(884,\cdot)\) \(\chi_{1015}(914,\cdot)\) \(\chi_{1015}(949,\cdot)\) \(\chi_{1015}(954,\cdot)\) \(\chi_{1015}(984,\cdot)\) \(\chi_{1015}(989,\cdot)\)

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

Values on generators

\((407,871,176)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{27}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1015 }(44, a) \) \(-1\)\(1\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1015 }(44,a) \;\) at \(\;a = \) e.g. 2