Properties

Label 2736.125
Modulus $2736$
Conductor $912$
Order $12$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,6,8]))
 
pari: [g,chi] = znchar(Mod(125,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(912\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{912}(125,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.dq

\(\chi_{2736}(125,\cdot)\) \(\chi_{2736}(197,\cdot)\) \(\chi_{2736}(1493,\cdot)\) \(\chi_{2736}(1565,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.106352130137086689804288.1

Values on generators

\((1711,2053,1217,1009)\) → \((1,-i,-1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2736 }(125, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(125,a) \;\) at \(\;a = \) e.g. 2