Properties

Label 1110.61
Modulus $1110$
Conductor $37$
Order $36$
Real no
Primitive no
Minimal yes
Parity odd

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([0,0,29]))
 
pari: [g,chi] = znchar(Mod(61,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1110.ce

\(\chi_{1110}(61,\cdot)\) \(\chi_{1110}(91,\cdot)\) \(\chi_{1110}(241,\cdot)\) \(\chi_{1110}(301,\cdot)\) \(\chi_{1110}(331,\cdot)\) \(\chi_{1110}(631,\cdot)\) \(\chi_{1110}(661,\cdot)\) \(\chi_{1110}(721,\cdot)\) \(\chi_{1110}(871,\cdot)\) \(\chi_{1110}(901,\cdot)\) \(\chi_{1110}(1021,\cdot)\) \(\chi_{1110}(1051,\cdot)\)

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: \(\Q(\zeta_{37})\)

Values on generators

\((371,667,631)\) → \((1,1,e\left(\frac{29}{36}\right))\)

Values

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