Properties

Label 4015.692
Modulus $4015$
Conductor $4015$
Order $36$
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(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,18,17]))
 
pari: [g,chi] = znchar(Mod(692,4015))
 

Basic properties

Modulus: \(4015\)
Conductor: \(4015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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.dr

\(\chi_{4015}(692,\cdot)\) \(\chi_{4015}(857,\cdot)\) \(\chi_{4015}(1143,\cdot)\) \(\chi_{4015}(1253,\cdot)\) \(\chi_{4015}(2397,\cdot)\) \(\chi_{4015}(2507,\cdot)\) \(\chi_{4015}(2793,\cdot)\) \(\chi_{4015}(2958,\cdot)\) \(\chi_{4015}(3673,\cdot)\) \(\chi_{4015}(3717,\cdot)\) \(\chi_{4015}(3948,\cdot)\) \(\chi_{4015}(3992,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((1607,2191,881)\) → \((i,-1,e\left(\frac{17}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 4015 }(692, a) \) \(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{31}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4015 }(692,a) \;\) at \(\;a = \) e.g. 2