Properties

Label 1110.13
Modulus $1110$
Conductor $185$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1110, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,11]))
 
pari: [g,chi] = znchar(Mod(13,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(185\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{185}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1110.ch

\(\chi_{1110}(13,\cdot)\) \(\chi_{1110}(133,\cdot)\) \(\chi_{1110}(187,\cdot)\) \(\chi_{1110}(217,\cdot)\) \(\chi_{1110}(277,\cdot)\) \(\chi_{1110}(313,\cdot)\) \(\chi_{1110}(427,\cdot)\) \(\chi_{1110}(463,\cdot)\) \(\chi_{1110}(523,\cdot)\) \(\chi_{1110}(553,\cdot)\) \(\chi_{1110}(607,\cdot)\) \(\chi_{1110}(727,\cdot)\)

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

Values on generators

\((371,667,631)\) → \((1,-i,e\left(\frac{11}{36}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(-i\)\(e\left(\frac{11}{18}\right)\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.36.57444765302724909954814307473256133361395843470561362005770206451416015625.1