Properties

Label 2736.35
Modulus $2736$
Conductor $912$
Order $36$
Real no
Primitive no
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([18,27,18,8]))
 
pari: [g,chi] = znchar(Mod(35,2736))
 

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 \(\chi_{912}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
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()
 
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: 36.36.20429953877724548562115535825002213860630960981294226252926298027107484541386752.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,-i,-1,e\left(\frac{2}{9}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{36}\right)\)
value at e.g. 2