Properties

Label 7623.1459
Modulus $7623$
Conductor $847$
Order $330$
Real no
Primitive no
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([0,55,21]))
 
pari: [g,chi] = znchar(Mod(1459,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(612,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.fx

\(\chi_{7623}(19,\cdot)\) \(\chi_{7623}(73,\cdot)\) \(\chi_{7623}(145,\cdot)\) \(\chi_{7623}(271,\cdot)\) \(\chi_{7623}(325,\cdot)\) \(\chi_{7623}(514,\cdot)\) \(\chi_{7623}(523,\cdot)\) \(\chi_{7623}(640,\cdot)\) \(\chi_{7623}(712,\cdot)\) \(\chi_{7623}(964,\cdot)\) \(\chi_{7623}(1018,\cdot)\) \(\chi_{7623}(1216,\cdot)\) \(\chi_{7623}(1333,\cdot)\) \(\chi_{7623}(1405,\cdot)\) \(\chi_{7623}(1459,\cdot)\) \(\chi_{7623}(1531,\cdot)\) \(\chi_{7623}(1657,\cdot)\) \(\chi_{7623}(1711,\cdot)\) \(\chi_{7623}(1900,\cdot)\) \(\chi_{7623}(2026,\cdot)\) \(\chi_{7623}(2098,\cdot)\) \(\chi_{7623}(2152,\cdot)\) \(\chi_{7623}(2224,\cdot)\) \(\chi_{7623}(2350,\cdot)\) \(\chi_{7623}(2404,\cdot)\) \(\chi_{7623}(2593,\cdot)\) \(\chi_{7623}(2602,\cdot)\) \(\chi_{7623}(2719,\cdot)\) \(\chi_{7623}(2791,\cdot)\) \(\chi_{7623}(2845,\cdot)\) ...

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

\((848,4357,4600)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{7}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1459, a) \) \(1\)\(1\)\(e\left(\frac{131}{330}\right)\)\(e\left(\frac{131}{165}\right)\)\(e\left(\frac{179}{330}\right)\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{51}{55}\right)\)\(e\left(\frac{97}{165}\right)\)\(e\left(\frac{47}{165}\right)\)\(e\left(\frac{19}{165}\right)\)\(e\left(\frac{37}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1459,a) \;\) at \(\;a = \) e.g. 2