Properties

Label 1110.59
Modulus $1110$
Conductor $555$
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([18,18,31]))
 
pari: [g,chi] = znchar(Mod(59,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(555\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{555}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1110.by

\(\chi_{1110}(59,\cdot)\) \(\chi_{1110}(89,\cdot)\) \(\chi_{1110}(209,\cdot)\) \(\chi_{1110}(239,\cdot)\) \(\chi_{1110}(389,\cdot)\) \(\chi_{1110}(449,\cdot)\) \(\chi_{1110}(479,\cdot)\) \(\chi_{1110}(779,\cdot)\) \(\chi_{1110}(809,\cdot)\) \(\chi_{1110}(869,\cdot)\) \(\chi_{1110}(1019,\cdot)\) \(\chi_{1110}(1049,\cdot)\)

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

\((371,667,631)\) → \((-1,-1,e\left(\frac{31}{36}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(-i\)\(e\left(\frac{2}{9}\right)\)\(-i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.36.11394702880804821835353664975608445751870049996765358066064135272979736328125.1