Properties

Label 1441.155
Modulus $1441$
Conductor $131$
Order $26$
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(1441, base_ring=CyclotomicField(26))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15]))
 
pari: [g,chi] = znchar(Mod(155,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(131\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{131}(24,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.be

\(\chi_{1441}(155,\cdot)\) \(\chi_{1441}(199,\cdot)\) \(\chi_{1441}(210,\cdot)\) \(\chi_{1441}(309,\cdot)\) \(\chi_{1441}(331,\cdot)\) \(\chi_{1441}(485,\cdot)\) \(\chi_{1441}(595,\cdot)\) \(\chi_{1441}(804,\cdot)\) \(\chi_{1441}(837,\cdot)\) \(\chi_{1441}(936,\cdot)\) \(\chi_{1441}(1134,\cdot)\) \(\chi_{1441}(1211,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 26.0.85463837848355618843993505004759251339989023133673251.1

Values on generators

\((1311,133)\) → \((1,e\left(\frac{15}{26}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{9}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(155,a) \;\) at \(\;a = \) e.g. 2