Properties

Label 7623.6131
Modulus $7623$
Conductor $693$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,15,6]))
 
pari: [g,chi] = znchar(Mod(6131,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(693\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{693}(587,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.cr

\(\chi_{7623}(608,\cdot)\) \(\chi_{7623}(1049,\cdot)\) \(\chi_{7623}(1721,\cdot)\) \(\chi_{7623}(2792,\cdot)\) \(\chi_{7623}(4262,\cdot)\) \(\chi_{7623}(5333,\cdot)\) \(\chi_{7623}(5690,\cdot)\) \(\chi_{7623}(6131,\cdot)\)

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 30 polynomial

Values on generators

\((848,4357,4600)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(6131, a) \) \(1\)\(1\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(-1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(6131,a) \;\) at \(\;a = \) e.g. 2