Properties

Label 8624.7745
Modulus $8624$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1,0]))
 
pari: [g,chi] = znchar(Mod(7745,8624))
 

Basic properties

Modulus: \(8624\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8624.em

\(\chi_{8624}(353,\cdot)\) \(\chi_{8624}(1585,\cdot)\) \(\chi_{8624}(1937,\cdot)\) \(\chi_{8624}(2817,\cdot)\) \(\chi_{8624}(3169,\cdot)\) \(\chi_{8624}(4401,\cdot)\) \(\chi_{8624}(5281,\cdot)\) \(\chi_{8624}(5633,\cdot)\) \(\chi_{8624}(6513,\cdot)\) \(\chi_{8624}(6865,\cdot)\) \(\chi_{8624}(7745,\cdot)\) \(\chi_{8624}(8097,\cdot)\)

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

\((5391,6469,7745,3137)\) → \((1,1,e\left(\frac{1}{42}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 8624 }(7745, a) \) \(-1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8624 }(7745,a) \;\) at \(\;a = \) e.g. 2