Properties

Label 5225.734
Modulus $5225$
Conductor $5225$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,9,25]))
 
pari: [g,chi] = znchar(Mod(734,5225))
 

Basic properties

Modulus: \(5225\)
Conductor: \(5225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 5225.fu

\(\chi_{5225}(734,\cdot)\) \(\chi_{5225}(1304,\cdot)\) \(\chi_{5225}(2064,\cdot)\) \(\chi_{5225}(2444,\cdot)\) \(\chi_{5225}(2934,\cdot)\) \(\chi_{5225}(3504,\cdot)\) \(\chi_{5225}(4264,\cdot)\) \(\chi_{5225}(4644,\cdot)\)

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 30 polynomial

Values on generators

\((2927,2851,4676)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 5225 }(734, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(-1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{13}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5225 }(734,a) \;\) at \(\;a = \) e.g. 2