Properties

Label 1480.557
Modulus $1480$
Conductor $1480$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1480, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,18,9,1]))
 
pari: [g,chi] = znchar(Mod(557,1480))
 

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.ed

\(\chi_{1480}(13,\cdot)\) \(\chi_{1480}(93,\cdot)\) \(\chi_{1480}(133,\cdot)\) \(\chi_{1480}(237,\cdot)\) \(\chi_{1480}(277,\cdot)\) \(\chi_{1480}(357,\cdot)\) \(\chi_{1480}(557,\cdot)\) \(\chi_{1480}(797,\cdot)\) \(\chi_{1480}(893,\cdot)\) \(\chi_{1480}(957,\cdot)\) \(\chi_{1480}(1053,\cdot)\) \(\chi_{1480}(1293,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((1111,741,297,1001)\) → \((1,-1,i,e\left(\frac{1}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1480 }(557, a) \) \(1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1480 }(557,a) \;\) at \(\;a = \) e.g. 2