Properties

Label 1323.8
Modulus $1323$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

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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([7,36]))
 
pari: [g,chi] = znchar(Mod(8,1323))
 

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 \(\chi_{441}(155,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1323.bq

\(\chi_{1323}(8,\cdot)\) \(\chi_{1323}(71,\cdot)\) \(\chi_{1323}(260,\cdot)\) \(\chi_{1323}(386,\cdot)\) \(\chi_{1323}(449,\cdot)\) \(\chi_{1323}(575,\cdot)\) \(\chi_{1323}(764,\cdot)\) \(\chi_{1323}(827,\cdot)\) \(\chi_{1323}(953,\cdot)\) \(\chi_{1323}(1016,\cdot)\) \(\chi_{1323}(1142,\cdot)\) \(\chi_{1323}(1205,\cdot)\)

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

\((785,1081)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1323 }(8, a) \) \(-1\)\(1\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1323 }(8,a) \;\) at \(\;a = \) e.g. 2