Properties

Label 1925.1047
Modulus $1925$
Conductor $1925$
Order $60$
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(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([51,40,6]))
 
pari: [g,chi] = znchar(Mod(1047,1925))
 

Basic properties

Modulus: \(1925\)
Conductor: \(1925\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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.gf

\(\chi_{1925}(513,\cdot)\) \(\chi_{1925}(723,\cdot)\) \(\chi_{1925}(772,\cdot)\) \(\chi_{1925}(788,\cdot)\) \(\chi_{1925}(842,\cdot)\) \(\chi_{1925}(877,\cdot)\) \(\chi_{1925}(998,\cdot)\) \(\chi_{1925}(1003,\cdot)\) \(\chi_{1925}(1047,\cdot)\) \(\chi_{1925}(1117,\cdot)\) \(\chi_{1925}(1152,\cdot)\) \(\chi_{1925}(1278,\cdot)\) \(\chi_{1925}(1437,\cdot)\) \(\chi_{1925}(1458,\cdot)\) \(\chi_{1925}(1712,\cdot)\) \(\chi_{1925}(1733,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1002,276,1751)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 1925 }(1047, a) \) \(1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{59}{60}\right)\)\(i\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{37}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1925 }(1047,a) \;\) at \(\;a = \) e.g. 2