Properties

Label 5225.53
Modulus $5225$
Conductor $5225$
Order $180$
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(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,108,110]))
 
pari: [g,chi] = znchar(Mod(53,5225))
 

Basic properties

Modulus: \(5225\)
Conductor: \(5225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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.ja

\(\chi_{5225}(53,\cdot)\) \(\chi_{5225}(97,\cdot)\) \(\chi_{5225}(192,\cdot)\) \(\chi_{5225}(223,\cdot)\) \(\chi_{5225}(488,\cdot)\) \(\chi_{5225}(603,\cdot)\) \(\chi_{5225}(763,\cdot)\) \(\chi_{5225}(773,\cdot)\) \(\chi_{5225}(808,\cdot)\) \(\chi_{5225}(922,\cdot)\) \(\chi_{5225}(927,\cdot)\) \(\chi_{5225}(1017,\cdot)\) \(\chi_{5225}(1048,\cdot)\) \(\chi_{5225}(1153,\cdot)\) \(\chi_{5225}(1313,\cdot)\) \(\chi_{5225}(1428,\cdot)\) \(\chi_{5225}(1477,\cdot)\) \(\chi_{5225}(1598,\cdot)\) \(\chi_{5225}(1687,\cdot)\) \(\chi_{5225}(1978,\cdot)\) \(\chi_{5225}(2027,\cdot)\) \(\chi_{5225}(2138,\cdot)\) \(\chi_{5225}(2237,\cdot)\) \(\chi_{5225}(2302,\cdot)\) \(\chi_{5225}(2423,\cdot)\) \(\chi_{5225}(2787,\cdot)\) \(\chi_{5225}(2803,\cdot)\) \(\chi_{5225}(2852,\cdot)\) \(\chi_{5225}(3062,\cdot)\) \(\chi_{5225}(3283,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((2927,2851,4676)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{3}{5}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 5225 }(53, a) \) \(1\)\(1\)\(e\left(\frac{101}{180}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{11}{90}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{8}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5225 }(53,a) \;\) at \(\;a = \) e.g. 2