Properties

Label 7623.4670
Modulus $7623$
Conductor $363$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,0,69]))
 
pari: [g,chi] = znchar(Mod(4670,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(314,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.ew

\(\chi_{7623}(8,\cdot)\) \(\chi_{7623}(134,\cdot)\) \(\chi_{7623}(260,\cdot)\) \(\chi_{7623}(512,\cdot)\) \(\chi_{7623}(701,\cdot)\) \(\chi_{7623}(827,\cdot)\) \(\chi_{7623}(953,\cdot)\) \(\chi_{7623}(1205,\cdot)\) \(\chi_{7623}(1394,\cdot)\) \(\chi_{7623}(1520,\cdot)\) \(\chi_{7623}(1646,\cdot)\) \(\chi_{7623}(1898,\cdot)\) \(\chi_{7623}(2087,\cdot)\) \(\chi_{7623}(2213,\cdot)\) \(\chi_{7623}(2591,\cdot)\) \(\chi_{7623}(2906,\cdot)\) \(\chi_{7623}(3032,\cdot)\) \(\chi_{7623}(3284,\cdot)\) \(\chi_{7623}(3473,\cdot)\) \(\chi_{7623}(3599,\cdot)\) \(\chi_{7623}(3725,\cdot)\) \(\chi_{7623}(3977,\cdot)\) \(\chi_{7623}(4166,\cdot)\) \(\chi_{7623}(4292,\cdot)\) \(\chi_{7623}(4418,\cdot)\) \(\chi_{7623}(4670,\cdot)\) \(\chi_{7623}(4859,\cdot)\) \(\chi_{7623}(4985,\cdot)\) \(\chi_{7623}(5111,\cdot)\) \(\chi_{7623}(5363,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((-1,1,e\left(\frac{69}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(4670, a) \) \(1\)\(1\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{21}{55}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{7}{110}\right)\)\(e\left(\frac{19}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(4670,a) \;\) at \(\;a = \) e.g. 2