Properties

Label 9072.269
Modulus $9072$
Conductor $1008$
Order $12$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,2,2]))
 
pari: [g,chi] = znchar(Mod(269,9072))
 

Basic properties

Modulus: \(9072\)
Conductor: \(1008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1008}(605,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9072.du

\(\chi_{9072}(269,\cdot)\) \(\chi_{9072}(1349,\cdot)\) \(\chi_{9072}(4805,\cdot)\) \(\chi_{9072}(5885,\cdot)\)

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: 12.12.940054087216005776526016512.2

Values on generators

\((1135,6805,3809,2593)\) → \((1,-i,e\left(\frac{1}{6}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 9072 }(269, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9072 }(269,a) \;\) at \(\;a = \) e.g. 2