Properties

Label 8041.375
Modulus $8041$
Conductor $43$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,34]))
 
pari: [g,chi] = znchar(Mod(375,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.cb

\(\chi_{8041}(375,\cdot)\) \(\chi_{8041}(2245,\cdot)\) \(\chi_{8041}(2432,\cdot)\) \(\chi_{8041}(3367,\cdot)\) \(\chi_{8041}(3928,\cdot)\) \(\chi_{8041}(4489,\cdot)\) \(\chi_{8041}(6172,\cdot)\) \(\chi_{8041}(6359,\cdot)\) \(\chi_{8041}(6546,\cdot)\) \(\chi_{8041}(6733,\cdot)\) \(\chi_{8041}(6920,\cdot)\) \(\chi_{8041}(7668,\cdot)\)

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 21 polynomial

Values on generators

\((6580,2366,562)\) → \((1,1,e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(375, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{11}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(375,a) \;\) at \(\;a = \) e.g. 2