Properties

Label 2736.53
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([0,9,18,22]))
 
pari: [g,chi] = znchar(Mod(53,2736))
 

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 \(\chi_{912}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.hc

\(\chi_{2736}(53,\cdot)\) \(\chi_{2736}(269,\cdot)\) \(\chi_{2736}(413,\cdot)\) \(\chi_{2736}(485,\cdot)\) \(\chi_{2736}(629,\cdot)\) \(\chi_{2736}(773,\cdot)\) \(\chi_{2736}(1421,\cdot)\) \(\chi_{2736}(1637,\cdot)\) \(\chi_{2736}(1781,\cdot)\) \(\chi_{2736}(1853,\cdot)\) \(\chi_{2736}(1997,\cdot)\) \(\chi_{2736}(2141,\cdot)\)

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: 36.36.7375213349858562030923708432825799203687776914247215677306393587785801919440617472.1

Values on generators

\((1711,2053,1217,1009)\) → \((1,i,-1,e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{25}{36}\right)\)
value at e.g. 2