Properties

Label 1148.209
Modulus $1148$
Conductor $287$
Order $10$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,3]))
 
pari: [g,chi] = znchar(Mod(209,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(209,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.be

\(\chi_{1148}(209,\cdot)\) \(\chi_{1148}(433,\cdot)\) \(\chi_{1148}(517,\cdot)\) \(\chi_{1148}(769,\cdot)\)

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: 10.0.5502308171359302527.1

Values on generators

\((575,493,785)\) → \((1,-1,e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1148 }(209, a) \) \(-1\)\(1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1148 }(209,a) \;\) at \(\;a = \) e.g. 2