Properties

Label 1520.13
Modulus $1520$
Conductor $1520$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1520, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,27,10]))
 
pari: [g,chi] = znchar(Mod(13,1520))
 

Basic properties

Modulus: \(1520\)
Conductor: \(1520\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1520.ed

\(\chi_{1520}(13,\cdot)\) \(\chi_{1520}(117,\cdot)\) \(\chi_{1520}(173,\cdot)\) \(\chi_{1520}(333,\cdot)\) \(\chi_{1520}(357,\cdot)\) \(\chi_{1520}(413,\cdot)\) \(\chi_{1520}(573,\cdot)\) \(\chi_{1520}(813,\cdot)\) \(\chi_{1520}(1077,\cdot)\) \(\chi_{1520}(1237,\cdot)\) \(\chi_{1520}(1397,\cdot)\) \(\chi_{1520}(1477,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((191,1141,1217,401)\) → \((1,-i,-i,e\left(\frac{5}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1520 }(13,a) \;\) at \(\;a = \) e.g. 2