Properties

Label 1305.1304
Modulus $1305$
Conductor $435$
Order $2$
Real yes
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,1,1]))
 
pari: [g,chi] = znchar(Mod(1304,1305))
 

Basic properties

Modulus: \(1305\)
Conductor: \(435\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{435}(434,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1305.b

\(\chi_{1305}(1304,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{-435}) \)

Values on generators

\((146,262,901)\) → \((-1,-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1305 }(1304, a) \) \(-1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1305 }(1304,a) \;\) at \(\;a = \) e.g. 2