Properties

Label 4015.81
Modulus $4015$
Conductor $803$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,6,10]))
 
pari: [g,chi] = znchar(Mod(81,4015))
 

Basic properties

Modulus: \(4015\)
Conductor: \(803\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{803}(81,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4015.cf

\(\chi_{4015}(81,\cdot)\) \(\chi_{4015}(356,\cdot)\) \(\chi_{4015}(1906,\cdot)\) \(\chi_{4015}(2181,\cdot)\) \(\chi_{4015}(2271,\cdot)\) \(\chi_{4015}(2546,\cdot)\) \(\chi_{4015}(3001,\cdot)\) \(\chi_{4015}(3276,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\((1607,2191,881)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 4015 }(81, a) \) \(1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4015 }(81,a) \;\) at \(\;a = \) e.g. 2