Properties

Label 4015.69
Modulus $4015$
Conductor $4015$
Order $90$
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(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,72,65]))
 
pari: [g,chi] = znchar(Mod(69,4015))
 

Basic properties

Modulus: \(4015\)
Conductor: \(4015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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.ff

\(\chi_{4015}(69,\cdot)\) \(\chi_{4015}(114,\cdot)\) \(\chi_{4015}(434,\cdot)\) \(\chi_{4015}(509,\cdot)\) \(\chi_{4015}(839,\cdot)\) \(\chi_{4015}(874,\cdot)\) \(\chi_{4015}(894,\cdot)\) \(\chi_{4015}(1164,\cdot)\) \(\chi_{4015}(1204,\cdot)\) \(\chi_{4015}(1259,\cdot)\) \(\chi_{4015}(1444,\cdot)\) \(\chi_{4015}(1604,\cdot)\) \(\chi_{4015}(1809,\cdot)\) \(\chi_{4015}(1934,\cdot)\) \(\chi_{4015}(1939,\cdot)\) \(\chi_{4015}(1989,\cdot)\) \(\chi_{4015}(2259,\cdot)\) \(\chi_{4015}(2304,\cdot)\) \(\chi_{4015}(2539,\cdot)\) \(\chi_{4015}(2699,\cdot)\) \(\chi_{4015}(3029,\cdot)\) \(\chi_{4015}(3034,\cdot)\) \(\chi_{4015}(3084,\cdot)\) \(\chi_{4015}(3634,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((1607,2191,881)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{13}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 4015 }(69, a) \) \(1\)\(1\)\(e\left(\frac{7}{90}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{1}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4015 }(69,a) \;\) at \(\;a = \) e.g. 2