Properties

Label 7623.1250
Modulus $7623$
Conductor $231$
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([15,20,21]))
 
pari: [g,chi] = znchar(Mod(1250,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(231\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{231}(95,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.db

\(\chi_{7623}(233,\cdot)\) \(\chi_{7623}(1250,\cdot)\) \(\chi_{7623}(1304,\cdot)\) \(\chi_{7623}(1691,\cdot)\) \(\chi_{7623}(4517,\cdot)\) \(\chi_{7623}(4589,\cdot)\) \(\chi_{7623}(4958,\cdot)\) \(\chi_{7623}(5660,\cdot)\)

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

\((848,4357,4600)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1250, a) \) \(1\)\(1\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1250,a) \;\) at \(\;a = \) e.g. 2