Properties

Label 1480.1307
Modulus $1480$
Conductor $1480$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1480, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,18,9,28]))
 
pari: [g,chi] = znchar(Mod(1307,1480))
 

Basic properties

Modulus: \(1480\)
Conductor: \(1480\)
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 1480.dw

\(\chi_{1480}(83,\cdot)\) \(\chi_{1480}(107,\cdot)\) \(\chi_{1480}(123,\cdot)\) \(\chi_{1480}(403,\cdot)\) \(\chi_{1480}(747,\cdot)\) \(\chi_{1480}(867,\cdot)\) \(\chi_{1480}(1043,\cdot)\) \(\chi_{1480}(1107,\cdot)\) \(\chi_{1480}(1163,\cdot)\) \(\chi_{1480}(1267,\cdot)\) \(\chi_{1480}(1307,\cdot)\) \(\chi_{1480}(1403,\cdot)\)

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

\((1111,741,297,1001)\) → \((-1,-1,i,e\left(\frac{7}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1480 }(1307, a) \) \(1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1480 }(1307,a) \;\) at \(\;a = \) e.g. 2