Properties

Label 4950.287
Modulus $4950$
Conductor $75$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,9,0]))
 
pari: [g,chi] = znchar(Mod(287,4950))
 

Basic properties

Modulus: \(4950\)
Conductor: \(75\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{75}(62,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4950.dm

\(\chi_{4950}(287,\cdot)\) \(\chi_{4950}(683,\cdot)\) \(\chi_{4950}(1277,\cdot)\) \(\chi_{4950}(1673,\cdot)\) \(\chi_{4950}(2267,\cdot)\) \(\chi_{4950}(2663,\cdot)\) \(\chi_{4950}(3653,\cdot)\) \(\chi_{4950}(4247,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{75})^+\)

Values on generators

\((551,2377,4501)\) → \((-1,e\left(\frac{9}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4950 }(287, a) \) \(1\)\(1\)\(i\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4950 }(287,a) \;\) at \(\;a = \) e.g. 2