Properties

Label 7920.829
Modulus $7920$
Conductor $880$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7920, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,0,10,4]))
 
pari: [g,chi] = znchar(Mod(829,7920))
 

Basic properties

Modulus: \(7920\)
Conductor: \(880\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{880}(829,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7920.hr

\(\chi_{7920}(829,\cdot)\) \(\chi_{7920}(1549,\cdot)\) \(\chi_{7920}(2269,\cdot)\) \(\chi_{7920}(3349,\cdot)\) \(\chi_{7920}(4789,\cdot)\) \(\chi_{7920}(5509,\cdot)\) \(\chi_{7920}(6229,\cdot)\) \(\chi_{7920}(7309,\cdot)\)

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

\((991,5941,3521,6337,6481)\) → \((1,-i,1,-1,e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7920 }(829, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7920 }(829,a) \;\) at \(\;a = \) e.g. 2