Properties

Label 2300.51
Modulus $2300$
Conductor $92$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2300, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,0,1]))
 
pari: [g,chi] = znchar(Mod(51,2300))
 

Basic properties

Modulus: \(2300\)
Conductor: \(92\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{92}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2300.bc

\(\chi_{2300}(51,\cdot)\) \(\chi_{2300}(251,\cdot)\) \(\chi_{2300}(451,\cdot)\) \(\chi_{2300}(651,\cdot)\) \(\chi_{2300}(751,\cdot)\) \(\chi_{2300}(1351,\cdot)\) \(\chi_{2300}(1551,\cdot)\) \(\chi_{2300}(1851,\cdot)\) \(\chi_{2300}(1951,\cdot)\) \(\chi_{2300}(2251,\cdot)\)

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: \(\Q(\zeta_{92})^+\)

Values on generators

\((1151,277,1201)\) → \((-1,1,e\left(\frac{1}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2300 }(51,a) \;\) at \(\;a = \) e.g. 2