Properties

Label 4950.217
Modulus $4950$
Conductor $275$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13,6]))
 
pari: [g,chi] = znchar(Mod(217,4950))
 

Basic properties

Modulus: \(4950\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(217,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4950.dg

\(\chi_{4950}(217,\cdot)\) \(\chi_{4950}(1513,\cdot)\) \(\chi_{4950}(1927,\cdot)\) \(\chi_{4950}(2053,\cdot)\) \(\chi_{4950}(2323,\cdot)\) \(\chi_{4950}(2647,\cdot)\) \(\chi_{4950}(3583,\cdot)\) \(\chi_{4950}(4087,\cdot)\)

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: Number field defined by a degree 20 polynomial

Values on generators

\((551,2377,4501)\) → \((1,e\left(\frac{13}{20}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4950 }(217, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(1\)\(e\left(\frac{9}{20}\right)\)\(-1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4950 }(217,a) \;\) at \(\;a = \) e.g. 2