Properties

Label 1225.486
Modulus $1225$
Conductor $1225$
Order $210$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,155]))
 
pari: [g,chi] = znchar(Mod(486,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.br

\(\chi_{1225}(61,\cdot)\) \(\chi_{1225}(66,\cdot)\) \(\chi_{1225}(96,\cdot)\) \(\chi_{1225}(131,\cdot)\) \(\chi_{1225}(136,\cdot)\) \(\chi_{1225}(171,\cdot)\) \(\chi_{1225}(206,\cdot)\) \(\chi_{1225}(236,\cdot)\) \(\chi_{1225}(241,\cdot)\) \(\chi_{1225}(271,\cdot)\) \(\chi_{1225}(306,\cdot)\) \(\chi_{1225}(311,\cdot)\) \(\chi_{1225}(341,\cdot)\) \(\chi_{1225}(346,\cdot)\) \(\chi_{1225}(381,\cdot)\) \(\chi_{1225}(416,\cdot)\) \(\chi_{1225}(446,\cdot)\) \(\chi_{1225}(481,\cdot)\) \(\chi_{1225}(486,\cdot)\) \(\chi_{1225}(516,\cdot)\) \(\chi_{1225}(556,\cdot)\) \(\chi_{1225}(586,\cdot)\) \(\chi_{1225}(591,\cdot)\) \(\chi_{1225}(621,\cdot)\) \(\chi_{1225}(661,\cdot)\) \(\chi_{1225}(691,\cdot)\) \(\chi_{1225}(696,\cdot)\) \(\chi_{1225}(731,\cdot)\) \(\chi_{1225}(761,\cdot)\) \(\chi_{1225}(796,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((1177,101)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(486, a) \) \(-1\)\(1\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{71}{210}\right)\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{23}{70}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{71}{105}\right)\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{67}{210}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{101}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(486,a) \;\) at \(\;a = \) e.g. 2