Properties

Label 2736.2627
Modulus $2736$
Conductor $912$
Order $36$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,27,18,32]))
 
pari: [g,chi] = znchar(Mod(2627,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(912\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{912}(803,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.gv

\(\chi_{2736}(35,\cdot)\) \(\chi_{2736}(251,\cdot)\) \(\chi_{2736}(899,\cdot)\) \(\chi_{2736}(1043,\cdot)\) \(\chi_{2736}(1187,\cdot)\) \(\chi_{2736}(1259,\cdot)\) \(\chi_{2736}(1403,\cdot)\) \(\chi_{2736}(1619,\cdot)\) \(\chi_{2736}(2267,\cdot)\) \(\chi_{2736}(2411,\cdot)\) \(\chi_{2736}(2555,\cdot)\) \(\chi_{2736}(2627,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((1711,2053,1217,1009)\) → \((-1,-i,-1,e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2736 }(2627, a) \) \(1\)\(1\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{29}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(2627,a) \;\) at \(\;a = \) e.g. 2