Properties

Label 2736.13
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,27,12,10]))
 
pari: [g,chi] = znchar(Mod(13,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(2736\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.gp

\(\chi_{2736}(13,\cdot)\) \(\chi_{2736}(421,\cdot)\) \(\chi_{2736}(565,\cdot)\) \(\chi_{2736}(661,\cdot)\) \(\chi_{2736}(781,\cdot)\) \(\chi_{2736}(1093,\cdot)\) \(\chi_{2736}(1381,\cdot)\) \(\chi_{2736}(1789,\cdot)\) \(\chi_{2736}(1933,\cdot)\) \(\chi_{2736}(2029,\cdot)\) \(\chi_{2736}(2149,\cdot)\) \(\chi_{2736}(2461,\cdot)\)

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: 36.0.1518491026041947944789864372537347836310504032267670560805344803268132977411739903962193307107328.2

Values on generators

\((1711,2053,1217,1009)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2736 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(-1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(13,a) \;\) at \(\;a = \) e.g. 2