Properties

Label 2736.59
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2736, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,9,30,2]))
 
pari: [g,chi] = znchar(Mod(59,2736))
 

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.gx

\(\chi_{2736}(59,\cdot)\) \(\chi_{2736}(371,\cdot)\) \(\chi_{2736}(659,\cdot)\) \(\chi_{2736}(1067,\cdot)\) \(\chi_{2736}(1211,\cdot)\) \(\chi_{2736}(1307,\cdot)\) \(\chi_{2736}(1427,\cdot)\) \(\chi_{2736}(1739,\cdot)\) \(\chi_{2736}(2027,\cdot)\) \(\chi_{2736}(2435,\cdot)\) \(\chi_{2736}(2579,\cdot)\) \(\chi_{2736}(2675,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.1106979957984580051751811127579726572670357439523131838827096361582468940533158389988438920881242112.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{1}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{19}{36}\right)\)\(1\)\(e\left(\frac{35}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(59,a) \;\) at \(\;a = \) e.g. 2