Properties

Label 1232.45
Modulus $1232$
Conductor $112$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,2,0]))
 
pari: [g,chi] = znchar(Mod(45,1232))
 

Basic properties

Modulus: \(1232\)
Conductor: \(112\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{112}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1232.cl

\(\chi_{1232}(45,\cdot)\) \(\chi_{1232}(397,\cdot)\) \(\chi_{1232}(661,\cdot)\) \(\chi_{1232}(1013,\cdot)\)

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: 12.0.2426443912768913408.1

Values on generators

\((463,309,353,673)\) → \((1,-i,e\left(\frac{1}{6}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 1232 }(45, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1232 }(45,a) \;\) at \(\;a = \) e.g. 2