Properties

Label 2805.608
Modulus $2805$
Conductor $2805$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2805, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,15,16,5]))
 
pari: [g,chi] = znchar(Mod(608,2805))
 

Basic properties

Modulus: \(2805\)
Conductor: \(2805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2805.ea

\(\chi_{2805}(608,\cdot)\) \(\chi_{2805}(752,\cdot)\) \(\chi_{2805}(863,\cdot)\) \(\chi_{2805}(1373,\cdot)\) \(\chi_{2805}(2027,\cdot)\) \(\chi_{2805}(2138,\cdot)\) \(\chi_{2805}(2282,\cdot)\) \(\chi_{2805}(2792,\cdot)\)

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

\((1871,562,1531,496)\) → \((-1,-i,e\left(\frac{4}{5}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(19\)\(23\)\(26\)
\( \chi_{ 2805 }(608, a) \) \(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(-1\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2805 }(608,a) \;\) at \(\;a = \) e.g. 2