Properties

Label 2736.67
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: 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([18,27,12,34]))
 
pari: [g,chi] = znchar(Mod(67,2736))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.hh

\(\chi_{2736}(67,\cdot)\) \(\chi_{2736}(355,\cdot)\) \(\chi_{2736}(763,\cdot)\) \(\chi_{2736}(907,\cdot)\) \(\chi_{2736}(1003,\cdot)\) \(\chi_{2736}(1123,\cdot)\) \(\chi_{2736}(1435,\cdot)\) \(\chi_{2736}(1723,\cdot)\) \(\chi_{2736}(2131,\cdot)\) \(\chi_{2736}(2275,\cdot)\) \(\chi_{2736}(2371,\cdot)\) \(\chi_{2736}(2491,\cdot)\)

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: 36.36.1518491026041947944789864372537347836310504032267670560805344803268132977411739903962193307107328.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,-i,e\left(\frac{1}{3}\right),e\left(\frac{17}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{19}{36}\right)\)
value at e.g. 2