Properties

Label 1225.188
Modulus $1225$
Conductor $1225$
Order $140$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([133,50]))
 
pari: [g,chi] = znchar(Mod(188,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
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 1225.bp

\(\chi_{1225}(13,\cdot)\) \(\chi_{1225}(27,\cdot)\) \(\chi_{1225}(62,\cdot)\) \(\chi_{1225}(83,\cdot)\) \(\chi_{1225}(153,\cdot)\) \(\chi_{1225}(167,\cdot)\) \(\chi_{1225}(188,\cdot)\) \(\chi_{1225}(202,\cdot)\) \(\chi_{1225}(223,\cdot)\) \(\chi_{1225}(237,\cdot)\) \(\chi_{1225}(258,\cdot)\) \(\chi_{1225}(272,\cdot)\) \(\chi_{1225}(328,\cdot)\) \(\chi_{1225}(363,\cdot)\) \(\chi_{1225}(377,\cdot)\) \(\chi_{1225}(398,\cdot)\) \(\chi_{1225}(412,\cdot)\) \(\chi_{1225}(433,\cdot)\) \(\chi_{1225}(447,\cdot)\) \(\chi_{1225}(503,\cdot)\) \(\chi_{1225}(517,\cdot)\) \(\chi_{1225}(552,\cdot)\) \(\chi_{1225}(573,\cdot)\) \(\chi_{1225}(608,\cdot)\) \(\chi_{1225}(622,\cdot)\) \(\chi_{1225}(678,\cdot)\) \(\chi_{1225}(692,\cdot)\) \(\chi_{1225}(713,\cdot)\) \(\chi_{1225}(727,\cdot)\) \(\chi_{1225}(748,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((1177,101)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(188, a) \) \(1\)\(1\)\(e\left(\frac{33}{140}\right)\)\(e\left(\frac{1}{140}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{99}{140}\right)\)\(e\left(\frac{1}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{67}{140}\right)\)\(e\left(\frac{117}{140}\right)\)\(e\left(\frac{33}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(188,a) \;\) at \(\;a = \) e.g. 2