Properties

Label 1441.654
Modulus $1441$
Conductor $1441$
Order $10$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1441, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,5]))
 
pari: [g,chi] = znchar(Mod(654,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(1441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.u

\(\chi_{1441}(130,\cdot)\) \(\chi_{1441}(654,\cdot)\) \(\chi_{1441}(785,\cdot)\) \(\chi_{1441}(916,\cdot)\)

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: 10.0.8269856231139440531.2

Values on generators

\((1311,133)\) → \((e\left(\frac{2}{5}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(654,a) \;\) at \(\;a = \) e.g. 2