Properties

Label 2057.69
Modulus $2057$
Conductor $121$
Order $55$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2057, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([48,0]))
 
pari: [g,chi] = znchar(Mod(69,2057))
 

Basic properties

Modulus: \(2057\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(69,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2057.y

\(\chi_{2057}(69,\cdot)\) \(\chi_{2057}(86,\cdot)\) \(\chi_{2057}(103,\cdot)\) \(\chi_{2057}(137,\cdot)\) \(\chi_{2057}(256,\cdot)\) \(\chi_{2057}(273,\cdot)\) \(\chi_{2057}(290,\cdot)\) \(\chi_{2057}(324,\cdot)\) \(\chi_{2057}(443,\cdot)\) \(\chi_{2057}(460,\cdot)\) \(\chi_{2057}(477,\cdot)\) \(\chi_{2057}(630,\cdot)\) \(\chi_{2057}(647,\cdot)\) \(\chi_{2057}(664,\cdot)\) \(\chi_{2057}(698,\cdot)\) \(\chi_{2057}(817,\cdot)\) \(\chi_{2057}(834,\cdot)\) \(\chi_{2057}(851,\cdot)\) \(\chi_{2057}(885,\cdot)\) \(\chi_{2057}(1004,\cdot)\) \(\chi_{2057}(1021,\cdot)\) \(\chi_{2057}(1038,\cdot)\) \(\chi_{2057}(1072,\cdot)\) \(\chi_{2057}(1191,\cdot)\) \(\chi_{2057}(1208,\cdot)\) \(\chi_{2057}(1225,\cdot)\) \(\chi_{2057}(1259,\cdot)\) \(\chi_{2057}(1378,\cdot)\) \(\chi_{2057}(1395,\cdot)\) \(\chi_{2057}(1446,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial

Values on generators

\((970,122)\) → \((e\left(\frac{24}{55}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 2057 }(69, a) \) \(1\)\(1\)\(e\left(\frac{24}{55}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{3}{55}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2057 }(69,a) \;\) at \(\;a = \) e.g. 2