Properties

Label 7440.343
Modulus $7440$
Conductor $1240$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7440, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,10,0,15,16]))
 
pari: [g,chi] = znchar(Mod(343,7440))
 

Basic properties

Modulus: \(7440\)
Conductor: \(1240\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1240}(963,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7440.hx

\(\chi_{7440}(343,\cdot)\) \(\chi_{7440}(1783,\cdot)\) \(\chi_{7440}(2023,\cdot)\) \(\chi_{7440}(2887,\cdot)\) \(\chi_{7440}(4807,\cdot)\) \(\chi_{7440}(5863,\cdot)\) \(\chi_{7440}(6247,\cdot)\) \(\chi_{7440}(6487,\cdot)\)

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

\((6511,1861,4961,2977,5521)\) → \((-1,-1,1,-i,e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(37\)\(41\)\(43\)
\( \chi_{ 7440 }(343, a) \) \(1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(i\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{9}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7440 }(343,a) \;\) at \(\;a = \) e.g. 2