Properties

Label 2640.53
Modulus $2640$
Conductor $2640$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2640, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,10,15,12]))
 
pari: [g,chi] = znchar(Mod(53,2640))
 

Basic properties

Modulus: \(2640\)
Conductor: \(2640\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 2640.eq

\(\chi_{2640}(53,\cdot)\) \(\chi_{2640}(317,\cdot)\) \(\chi_{2640}(533,\cdot)\) \(\chi_{2640}(773,\cdot)\) \(\chi_{2640}(797,\cdot)\) \(\chi_{2640}(1037,\cdot)\) \(\chi_{2640}(1973,\cdot)\) \(\chi_{2640}(2237,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((991,661,881,1057,1201)\) → \((1,i,-1,-i,e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2640 }(53, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(i\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2640 }(53,a) \;\) at \(\;a = \) e.g. 2