Properties

Label 4950.829
Modulus $4950$
Conductor $275$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,2]))
 
pari: [g,chi] = znchar(Mod(829,4950))
 

Basic properties

Modulus: \(4950\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4950.ch

\(\chi_{4950}(829,\cdot)\) \(\chi_{4950}(2269,\cdot)\) \(\chi_{4950}(2359,\cdot)\) \(\chi_{4950}(2539,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.163542847442626953125.2

Values on generators

\((551,2377,4501)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4950 }(829, a) \) \(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(1\)\(e\left(\frac{3}{10}\right)\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4950 }(829,a) \;\) at \(\;a = \) e.g. 2