Properties

Label 1110.7
Modulus $1110$
Conductor $185$
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([0,9,32]))
 
pari: [g,chi] = znchar(Mod(7,1110))
 

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 \(\chi_{185}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1110.ca

\(\chi_{1110}(7,\cdot)\) \(\chi_{1110}(127,\cdot)\) \(\chi_{1110}(157,\cdot)\) \(\chi_{1110}(367,\cdot)\) \(\chi_{1110}(403,\cdot)\) \(\chi_{1110}(493,\cdot)\) \(\chi_{1110}(673,\cdot)\) \(\chi_{1110}(793,\cdot)\) \(\chi_{1110}(823,\cdot)\) \(\chi_{1110}(847,\cdot)\) \(\chi_{1110}(937,\cdot)\) \(\chi_{1110}(1033,\cdot)\)

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: 36.0.1134084166835624937413663701523229292665702790961273014545440673828125.1

Values on generators

\((371,667,631)\) → \((1,i,e\left(\frac{8}{9}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\(-1\)\(1\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{7}{9}\right)\)\(-i\)
value at e.g. 2