Properties

Label 1805.1512
Modulus $1805$
Conductor $95$
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(1805, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([3,8]))
 
pari: [g,chi] = znchar(Mod(1512,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{95}(87,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.m

\(\chi_{1805}(68,\cdot)\) \(\chi_{1805}(292,\cdot)\) \(\chi_{1805}(653,\cdot)\) \(\chi_{1805}(1512,\cdot)\)

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: 12.0.33171021564453125.1

Values on generators

\((362,1446)\) → \((i,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(1512, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{6}\right)\)\(1\)\(i\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(1512,a) \;\) at \(\;a = \) e.g. 2