Properties

Label 5040.2413
Modulus $5040$
Conductor $560$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,0,9,10]))
 
pari: [g,chi] = znchar(Mod(2413,5040))
 

Basic properties

Modulus: \(5040\)
Conductor: \(560\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{560}(173,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5040.od

\(\chi_{5040}(2413,\cdot)\) \(\chi_{5040}(2917,\cdot)\) \(\chi_{5040}(3853,\cdot)\) \(\chi_{5040}(4357,\cdot)\)

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: 12.12.4739148267126784000000000.2

Values on generators

\((3151,3781,2801,2017,3601)\) → \((1,-i,1,-i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5040 }(2413, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5040 }(2413,a) \;\) at \(\;a = \) e.g. 2