Properties

Label 5225.261
Modulus $5225$
Conductor $5225$
Order $90$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([72,27,35]))
 
pari: [g,chi] = znchar(Mod(261,5225))
 

Basic properties

Modulus: \(5225\)
Conductor: \(5225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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.hu

\(\chi_{5225}(261,\cdot)\) \(\chi_{5225}(281,\cdot)\) \(\chi_{5225}(811,\cdot)\) \(\chi_{5225}(831,\cdot)\) \(\chi_{5225}(1041,\cdot)\) \(\chi_{5225}(1086,\cdot)\) \(\chi_{5225}(1381,\cdot)\) \(\chi_{5225}(1421,\cdot)\) \(\chi_{5225}(1591,\cdot)\) \(\chi_{5225}(1636,\cdot)\) \(\chi_{5225}(1656,\cdot)\) \(\chi_{5225}(1971,\cdot)\) \(\chi_{5225}(2141,\cdot)\) \(\chi_{5225}(2206,\cdot)\) \(\chi_{5225}(2416,\cdot)\) \(\chi_{5225}(2461,\cdot)\) \(\chi_{5225}(2521,\cdot)\) \(\chi_{5225}(2796,\cdot)\) \(\chi_{5225}(2966,\cdot)\) \(\chi_{5225}(3031,\cdot)\) \(\chi_{5225}(3346,\cdot)\) \(\chi_{5225}(3791,\cdot)\) \(\chi_{5225}(4171,\cdot)\) \(\chi_{5225}(4936,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((2927,2851,4676)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{3}{10}\right),e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 5225 }(261, a) \) \(1\)\(1\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{83}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5225 }(261,a) \;\) at \(\;a = \) e.g. 2