Properties

Label 7623.13
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([110,165,303]))
 
pari: [g,chi] = znchar(Mod(13,7623))
 

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.fw

\(\chi_{7623}(13,\cdot)\) \(\chi_{7623}(139,\cdot)\) \(\chi_{7623}(160,\cdot)\) \(\chi_{7623}(349,\cdot)\) \(\chi_{7623}(391,\cdot)\) \(\chi_{7623}(580,\cdot)\) \(\chi_{7623}(601,\cdot)\) \(\chi_{7623}(706,\cdot)\) \(\chi_{7623}(832,\cdot)\) \(\chi_{7623}(853,\cdot)\) \(\chi_{7623}(1042,\cdot)\) \(\chi_{7623}(1084,\cdot)\) \(\chi_{7623}(1168,\cdot)\) \(\chi_{7623}(1273,\cdot)\) \(\chi_{7623}(1294,\cdot)\) \(\chi_{7623}(1399,\cdot)\) \(\chi_{7623}(1525,\cdot)\) \(\chi_{7623}(1735,\cdot)\) \(\chi_{7623}(1777,\cdot)\) \(\chi_{7623}(1861,\cdot)\) \(\chi_{7623}(1966,\cdot)\) \(\chi_{7623}(1987,\cdot)\) \(\chi_{7623}(2092,\cdot)\) \(\chi_{7623}(2239,\cdot)\) \(\chi_{7623}(2428,\cdot)\) \(\chi_{7623}(2470,\cdot)\) \(\chi_{7623}(2554,\cdot)\) \(\chi_{7623}(2680,\cdot)\) \(\chi_{7623}(2785,\cdot)\) \(\chi_{7623}(2911,\cdot)\) ...

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

\((848,4357,4600)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{101}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(13, a) \) \(1\)\(1\)\(e\left(\frac{83}{330}\right)\)\(e\left(\frac{83}{165}\right)\)\(e\left(\frac{37}{330}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{149}{165}\right)\)\(e\left(\frac{1}{165}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{203}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(13,a) \;\) at \(\;a = \) e.g. 2