Properties

Label 1089.10
Modulus $1089$
Conductor $121$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1089, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15]))
 
pari: [g,chi] = znchar(Mod(10,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(10,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.p

\(\chi_{1089}(10,\cdot)\) \(\chi_{1089}(109,\cdot)\) \(\chi_{1089}(208,\cdot)\) \(\chi_{1089}(307,\cdot)\) \(\chi_{1089}(406,\cdot)\) \(\chi_{1089}(505,\cdot)\) \(\chi_{1089}(703,\cdot)\) \(\chi_{1089}(802,\cdot)\) \(\chi_{1089}(901,\cdot)\) \(\chi_{1089}(1000,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.0.4978518112499354698647829163838661251242411.1

Values on generators

\((848,244)\) → \((1,e\left(\frac{15}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{9}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(10,a) \;\) at \(\;a = \) e.g. 2