Properties

Label 2550.409
Modulus $2550$
Conductor $25$
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(2550, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,0]))
 
pari: [g,chi] = znchar(Mod(409,2550))
 

Basic properties

Modulus: \(2550\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2550.be

\(\chi_{2550}(409,\cdot)\) \(\chi_{2550}(919,\cdot)\) \(\chi_{2550}(1429,\cdot)\) \(\chi_{2550}(1939,\cdot)\)

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: \(\Q(\zeta_{25})^+\)

Values on generators

\((851,1327,751)\) → \((1,e\left(\frac{7}{10}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2550 }(409, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2550 }(409,a) \;\) at \(\;a = \) e.g. 2