Properties

Label 2300.921
Modulus $2300$
Conductor $25$
Order $5$
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(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,6,0]))
 
pari: [g,chi] = znchar(Mod(921,2300))
 

Basic properties

Modulus: \(2300\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(5\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2300.m

\(\chi_{2300}(461,\cdot)\) \(\chi_{2300}(921,\cdot)\) \(\chi_{2300}(1381,\cdot)\) \(\chi_{2300}(1841,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 5.5.390625.1

Values on generators

\((1151,277,1201)\) → \((1,e\left(\frac{3}{5}\right),1)\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2300 }(921,a) \;\) at \(\;a = \) e.g. 2