Properties

Label 8015.39
Modulus $8015$
Conductor $8015$
Order $228$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, base_ring=CyclotomicField(228))
 
M = H._module
 
chi = DirichletCharacter(H, M([114,152,97]))
 
pari: [g,chi] = znchar(Mod(39,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(8015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(228\)
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 8015.hi

\(\chi_{8015}(39,\cdot)\) \(\chi_{8015}(74,\cdot)\) \(\chi_{8015}(79,\cdot)\) \(\chi_{8015}(179,\cdot)\) \(\chi_{8015}(219,\cdot)\) \(\chi_{8015}(319,\cdot)\) \(\chi_{8015}(499,\cdot)\) \(\chi_{8015}(954,\cdot)\) \(\chi_{8015}(1269,\cdot)\) \(\chi_{8015}(1479,\cdot)\) \(\chi_{8015}(1579,\cdot)\) \(\chi_{8015}(1719,\cdot)\) \(\chi_{8015}(1794,\cdot)\) \(\chi_{8015}(2174,\cdot)\) \(\chi_{8015}(2249,\cdot)\) \(\chi_{8015}(2314,\cdot)\) \(\chi_{8015}(2529,\cdot)\) \(\chi_{8015}(2669,\cdot)\) \(\chi_{8015}(2949,\cdot)\) \(\chi_{8015}(2984,\cdot)\) \(\chi_{8015}(3054,\cdot)\) \(\chi_{8015}(3089,\cdot)\) \(\chi_{8015}(3159,\cdot)\) \(\chi_{8015}(3404,\cdot)\) \(\chi_{8015}(3574,\cdot)\) \(\chi_{8015}(3714,\cdot)\) \(\chi_{8015}(3819,\cdot)\) \(\chi_{8015}(3824,\cdot)\) \(\chi_{8015}(3854,\cdot)\) \(\chi_{8015}(3959,\cdot)\) ...

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: $\Q(\zeta_{228})$
Fixed field: Number field defined by a degree 228 polynomial (not computed)

Values on generators

\((3207,4581,4586)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{97}{228}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(39, a) \) \(-1\)\(1\)\(e\left(\frac{175}{228}\right)\)\(e\left(\frac{25}{38}\right)\)\(e\left(\frac{61}{114}\right)\)\(e\left(\frac{97}{228}\right)\)\(e\left(\frac{23}{76}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{67}{114}\right)\)\(e\left(\frac{11}{57}\right)\)\(e\left(\frac{21}{76}\right)\)\(e\left(\frac{4}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(39,a) \;\) at \(\;a = \) e.g. 2