Properties

Label 7942.653
Modulus $7942$
Conductor $209$
Order $15$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,10]))
 
pari: [g,chi] = znchar(Mod(653,7942))
 

Basic properties

Modulus: \(7942\)
Conductor: \(209\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{209}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7942.n

\(\chi_{7942}(653,\cdot)\) \(\chi_{7942}(1873,\cdot)\) \(\chi_{7942}(2819,\cdot)\) \(\chi_{7942}(4761,\cdot)\) \(\chi_{7942}(5483,\cdot)\) \(\chi_{7942}(5707,\cdot)\) \(\chi_{7942}(6429,\cdot)\) \(\chi_{7942}(7649,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\((5777,6139)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 7942 }(653, a) \) \(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7942 }(653,a) \;\) at \(\;a = \) e.g. 2