Properties

Label 8001.365
Modulus $8001$
Conductor $1143$
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(8001, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,0,33]))
 
pari: [g,chi] = znchar(Mod(365,8001))
 

Basic properties

Modulus: \(8001\)
Conductor: \(1143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1143}(365,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8001.jk

\(\chi_{8001}(365,\cdot)\) \(\chi_{8001}(2381,\cdot)\) \(\chi_{8001}(3746,\cdot)\) \(\chi_{8001}(3935,\cdot)\) \(\chi_{8001}(4187,\cdot)\) \(\chi_{8001}(4691,\cdot)\) \(\chi_{8001}(5699,\cdot)\) \(\chi_{8001}(6413,\cdot)\) \(\chi_{8001}(6602,\cdot)\) \(\chi_{8001}(6854,\cdot)\) \(\chi_{8001}(7358,\cdot)\) \(\chi_{8001}(7715,\cdot)\)

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

\((3557,1144,7750)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 8001 }(365, a) \) \(1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8001 }(365,a) \;\) at \(\;a = \) e.g. 2