Properties

Label 4560.709
Modulus $4560$
Conductor $1520$
Order $36$
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(4560, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,0,18,28]))
 
pari: [g,chi] = znchar(Mod(709,4560))
 

Basic properties

Modulus: \(4560\)
Conductor: \(1520\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1520}(709,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4560.hk

\(\chi_{4560}(709,\cdot)\) \(\chi_{4560}(1069,\cdot)\) \(\chi_{4560}(1309,\cdot)\) \(\chi_{4560}(1429,\cdot)\) \(\chi_{4560}(1669,\cdot)\) \(\chi_{4560}(1909,\cdot)\) \(\chi_{4560}(2989,\cdot)\) \(\chi_{4560}(3349,\cdot)\) \(\chi_{4560}(3589,\cdot)\) \(\chi_{4560}(3709,\cdot)\) \(\chi_{4560}(3949,\cdot)\) \(\chi_{4560}(4189,\cdot)\)

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

\((1711,1141,3041,2737,1921)\) → \((1,i,1,-1,e\left(\frac{7}{9}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{7}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4560 }(709,a) \;\) at \(\;a = \) e.g. 2