Properties

Label 2300.101
Modulus $2300$
Conductor $23$
Order $11$
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([0,0,10]))
 
pari: [g,chi] = znchar(Mod(101,2300))
 

Basic properties

Modulus: \(2300\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2300.u

\(\chi_{2300}(101,\cdot)\) \(\chi_{2300}(301,\cdot)\) \(\chi_{2300}(501,\cdot)\) \(\chi_{2300}(601,\cdot)\) \(\chi_{2300}(901,\cdot)\) \(\chi_{2300}(1001,\cdot)\) \(\chi_{2300}(1301,\cdot)\) \(\chi_{2300}(1501,\cdot)\) \(\chi_{2300}(2101,\cdot)\) \(\chi_{2300}(2201,\cdot)\)

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: \(\Q(\zeta_{23})^+\)

Values on generators

\((1151,277,1201)\) → \((1,1,e\left(\frac{5}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2300 }(101,a) \;\) at \(\;a = \) e.g. 2