Properties

Label 1143.80
Modulus $1143$
Conductor $381$
Order $42$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,41]))
 
pari: [g,chi] = znchar(Mod(80,1143))
 

Basic properties

Modulus: \(1143\)
Conductor: \(381\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{381}(80,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1143.by

\(\chi_{1143}(80,\cdot)\) \(\chi_{1143}(89,\cdot)\) \(\chi_{1143}(287,\cdot)\) \(\chi_{1143}(305,\cdot)\) \(\chi_{1143}(386,\cdot)\) \(\chi_{1143}(458,\cdot)\) \(\chi_{1143}(548,\cdot)\) \(\chi_{1143}(701,\cdot)\) \(\chi_{1143}(737,\cdot)\) \(\chi_{1143}(899,\cdot)\) \(\chi_{1143}(1043,\cdot)\) \(\chi_{1143}(1070,\cdot)\)

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: 42.42.1885814229083612999756170685190350016159423006500366264928575446894255145089191907938237816767181.1

Values on generators

\((128,892)\) → \((-1,e\left(\frac{41}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1143 }(80, a) \) \(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1143 }(80,a) \;\) at \(\;a = \) e.g. 2