Properties

Label 7623.740
Modulus $7623$
Conductor $7623$
Order $330$
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([55,275,24]))
 
pari: [g,chi] = znchar(Mod(740,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(7623\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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.fz

\(\chi_{7623}(47,\cdot)\) \(\chi_{7623}(59,\cdot)\) \(\chi_{7623}(185,\cdot)\) \(\chi_{7623}(236,\cdot)\) \(\chi_{7623}(311,\cdot)\) \(\chi_{7623}(488,\cdot)\) \(\chi_{7623}(500,\cdot)\) \(\chi_{7623}(740,\cdot)\) \(\chi_{7623}(752,\cdot)\) \(\chi_{7623}(878,\cdot)\) \(\chi_{7623}(929,\cdot)\) \(\chi_{7623}(1004,\cdot)\) \(\chi_{7623}(1181,\cdot)\) \(\chi_{7623}(1193,\cdot)\) \(\chi_{7623}(1307,\cdot)\) \(\chi_{7623}(1433,\cdot)\) \(\chi_{7623}(1445,\cdot)\) \(\chi_{7623}(1571,\cdot)\) \(\chi_{7623}(1622,\cdot)\) \(\chi_{7623}(1874,\cdot)\) \(\chi_{7623}(1886,\cdot)\) \(\chi_{7623}(2000,\cdot)\) \(\chi_{7623}(2126,\cdot)\) \(\chi_{7623}(2264,\cdot)\) \(\chi_{7623}(2315,\cdot)\) \(\chi_{7623}(2390,\cdot)\) \(\chi_{7623}(2567,\cdot)\) \(\chi_{7623}(2579,\cdot)\) \(\chi_{7623}(2693,\cdot)\) \(\chi_{7623}(2819,\cdot)\) ...

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 330 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{4}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(740, a) \) \(1\)\(1\)\(e\left(\frac{299}{330}\right)\)\(e\left(\frac{134}{165}\right)\)\(e\left(\frac{21}{55}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{59}{330}\right)\)\(e\left(\frac{103}{165}\right)\)\(e\left(\frac{148}{165}\right)\)\(e\left(\frac{67}{330}\right)\)\(e\left(\frac{32}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(740,a) \;\) at \(\;a = \) e.g. 2