Properties

Label 2280.2027
Modulus $2280$
Conductor $2280$
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(2280, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,18,18,9,10]))
 
pari: [g,chi] = znchar(Mod(2027,2280))
 

Basic properties

Modulus: \(2280\)
Conductor: \(2280\)
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 2280.fj

\(\chi_{2280}(203,\cdot)\) \(\chi_{2280}(827,\cdot)\) \(\chi_{2280}(1067,\cdot)\) \(\chi_{2280}(1283,\cdot)\) \(\chi_{2280}(1307,\cdot)\) \(\chi_{2280}(1427,\cdot)\) \(\chi_{2280}(1523,\cdot)\) \(\chi_{2280}(1667,\cdot)\) \(\chi_{2280}(1763,\cdot)\) \(\chi_{2280}(1883,\cdot)\) \(\chi_{2280}(2027,\cdot)\) \(\chi_{2280}(2123,\cdot)\)

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

\((1711,1141,761,457,1921)\) → \((-1,-1,-1,i,e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2280 }(2027, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2280 }(2027,a) \;\) at \(\;a = \) e.g. 2