Properties

Label 7920.193
Modulus $7920$
Conductor $495$
Order $60$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7920, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,20,45,54]))
 
pari: [g,chi] = znchar(Mod(193,7920))
 

Basic properties

Modulus: \(7920\)
Conductor: \(495\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{495}(193,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7920.li

\(\chi_{7920}(193,\cdot)\) \(\chi_{7920}(337,\cdot)\) \(\chi_{7920}(673,\cdot)\) \(\chi_{7920}(1393,\cdot)\) \(\chi_{7920}(1537,\cdot)\) \(\chi_{7920}(1777,\cdot)\) \(\chi_{7920}(2257,\cdot)\) \(\chi_{7920}(2833,\cdot)\) \(\chi_{7920}(2977,\cdot)\) \(\chi_{7920}(4417,\cdot)\) \(\chi_{7920}(5233,\cdot)\) \(\chi_{7920}(5953,\cdot)\) \(\chi_{7920}(6673,\cdot)\) \(\chi_{7920}(6817,\cdot)\) \(\chi_{7920}(7537,\cdot)\) \(\chi_{7920}(7873,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((991,5941,3521,6337,6481)\) → \((1,1,e\left(\frac{1}{3}\right),-i,e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7920 }(193, a) \) \(1\)\(1\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7920 }(193,a) \;\) at \(\;a = \) e.g. 2