Properties

Label 2280.107
Modulus $2280$
Conductor $2280$
Order $12$
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(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,6,3,10]))
 
pari: [g,chi] = znchar(Mod(107,2280))
 

Basic properties

Modulus: \(2280\)
Conductor: \(2280\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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.de

\(\chi_{2280}(107,\cdot)\) \(\chi_{2280}(563,\cdot)\) \(\chi_{2280}(1547,\cdot)\) \(\chi_{2280}(2003,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.2288408218591707648000000000.1

Values on generators

\((1711,1141,761,457,1921)\) → \((-1,-1,-1,i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2280 }(107, a) \) \(1\)\(1\)\(-i\)\(-1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2280 }(107,a) \;\) at \(\;a = \) e.g. 2