Properties

Label 1925.1061
Modulus $1925$
Conductor $1925$
Order $15$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,20,12]))
 
pari: [g,chi] = znchar(Mod(1061,1925))
 

Basic properties

Modulus: \(1925\)
Conductor: \(1925\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
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 1925.cz

\(\chi_{1925}(81,\cdot)\) \(\chi_{1925}(786,\cdot)\) \(\chi_{1925}(1061,\cdot)\) \(\chi_{1925}(1241,\cdot)\) \(\chi_{1925}(1516,\cdot)\) \(\chi_{1925}(1521,\cdot)\) \(\chi_{1925}(1731,\cdot)\) \(\chi_{1925}(1796,\cdot)\)

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

\((1002,276,1751)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 1925 }(1061, a) \) \(1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1925 }(1061,a) \;\) at \(\;a = \) e.g. 2