Properties

Label 1323.530
Modulus $1323$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,23]))
 
pari: [g,chi] = znchar(Mod(530,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(383,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1323.bz

\(\chi_{1323}(143,\cdot)\) \(\chi_{1323}(152,\cdot)\) \(\chi_{1323}(332,\cdot)\) \(\chi_{1323}(341,\cdot)\) \(\chi_{1323}(530,\cdot)\) \(\chi_{1323}(710,\cdot)\) \(\chi_{1323}(719,\cdot)\) \(\chi_{1323}(899,\cdot)\) \(\chi_{1323}(908,\cdot)\) \(\chi_{1323}(1088,\cdot)\) \(\chi_{1323}(1277,\cdot)\) \(\chi_{1323}(1286,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((785,1081)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{23}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1323 }(530, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1323 }(530,a) \;\) at \(\;a = \) e.g. 2