Properties

Label 7623.5048
Modulus $7623$
Conductor $363$
Order $22$
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(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,0,21]))
 
pari: [g,chi] = znchar(Mod(5048,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(329,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.cc

\(\chi_{7623}(197,\cdot)\) \(\chi_{7623}(890,\cdot)\) \(\chi_{7623}(1583,\cdot)\) \(\chi_{7623}(2276,\cdot)\) \(\chi_{7623}(2969,\cdot)\) \(\chi_{7623}(3662,\cdot)\) \(\chi_{7623}(5048,\cdot)\) \(\chi_{7623}(5741,\cdot)\) \(\chi_{7623}(6434,\cdot)\) \(\chi_{7623}(7127,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((848,4357,4600)\) → \((-1,1,e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(5048, a) \) \(1\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{1}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(5048,a) \;\) at \(\;a = \) e.g. 2