Properties

Label 8880.503
Modulus $8880$
Conductor $4440$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8880, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,18,18,27,31]))
 
pari: [g,chi] = znchar(Mod(503,8880))
 

Basic properties

Modulus: \(8880\)
Conductor: \(4440\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4440}(2723,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8880.nc

\(\chi_{8880}(503,\cdot)\) \(\chi_{8880}(647,\cdot)\) \(\chi_{8880}(1943,\cdot)\) \(\chi_{8880}(2087,\cdot)\) \(\chi_{8880}(2807,\cdot)\) \(\chi_{8880}(2903,\cdot)\) \(\chi_{8880}(3143,\cdot)\) \(\chi_{8880}(4823,\cdot)\) \(\chi_{8880}(6647,\cdot)\) \(\chi_{8880}(8327,\cdot)\) \(\chi_{8880}(8567,\cdot)\) \(\chi_{8880}(8663,\cdot)\)

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

\((5551,6661,5921,1777,8401)\) → \((-1,-1,-1,-i,e\left(\frac{31}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 8880 }(503, a) \) \(1\)\(1\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(e\left(\frac{2}{9}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8880 }(503,a) \;\) at \(\;a = \) e.g. 2