Properties

Label 7623.247
Modulus $7623$
Conductor $7623$
Order $165$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,110,222]))
 
pari: [g,chi] = znchar(Mod(247,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(7623\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(165\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.fb

\(\chi_{7623}(25,\cdot)\) \(\chi_{7623}(58,\cdot)\) \(\chi_{7623}(214,\cdot)\) \(\chi_{7623}(247,\cdot)\) \(\chi_{7623}(466,\cdot)\) \(\chi_{7623}(499,\cdot)\) \(\chi_{7623}(592,\cdot)\) \(\chi_{7623}(625,\cdot)\) \(\chi_{7623}(718,\cdot)\) \(\chi_{7623}(751,\cdot)\) \(\chi_{7623}(907,\cdot)\) \(\chi_{7623}(940,\cdot)\) \(\chi_{7623}(1159,\cdot)\) \(\chi_{7623}(1192,\cdot)\) \(\chi_{7623}(1285,\cdot)\) \(\chi_{7623}(1318,\cdot)\) \(\chi_{7623}(1411,\cdot)\) \(\chi_{7623}(1444,\cdot)\) \(\chi_{7623}(1633,\cdot)\) \(\chi_{7623}(1852,\cdot)\) \(\chi_{7623}(1885,\cdot)\) \(\chi_{7623}(1978,\cdot)\) \(\chi_{7623}(2011,\cdot)\) \(\chi_{7623}(2104,\cdot)\) \(\chi_{7623}(2137,\cdot)\) \(\chi_{7623}(2293,\cdot)\) \(\chi_{7623}(2545,\cdot)\) \(\chi_{7623}(2578,\cdot)\) \(\chi_{7623}(2704,\cdot)\) \(\chi_{7623}(2797,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 165 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{3}\right),e\left(\frac{37}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(247, a) \) \(1\)\(1\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{19}{165}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{101}{165}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{49}{165}\right)\)\(e\left(\frac{83}{165}\right)\)\(e\left(\frac{76}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(247,a) \;\) at \(\;a = \) e.g. 2