Properties

Label 1280.61
Modulus $1280$
Conductor $256$
Order $64$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,19,0]))
 
pari: [g,chi] = znchar(Mod(61,1280))
 

Basic properties

Modulus: \(1280\)
Conductor: \(256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{256}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1280.bv

\(\chi_{1280}(21,\cdot)\) \(\chi_{1280}(61,\cdot)\) \(\chi_{1280}(101,\cdot)\) \(\chi_{1280}(141,\cdot)\) \(\chi_{1280}(181,\cdot)\) \(\chi_{1280}(221,\cdot)\) \(\chi_{1280}(261,\cdot)\) \(\chi_{1280}(301,\cdot)\) \(\chi_{1280}(341,\cdot)\) \(\chi_{1280}(381,\cdot)\) \(\chi_{1280}(421,\cdot)\) \(\chi_{1280}(461,\cdot)\) \(\chi_{1280}(501,\cdot)\) \(\chi_{1280}(541,\cdot)\) \(\chi_{1280}(581,\cdot)\) \(\chi_{1280}(621,\cdot)\) \(\chi_{1280}(661,\cdot)\) \(\chi_{1280}(701,\cdot)\) \(\chi_{1280}(741,\cdot)\) \(\chi_{1280}(781,\cdot)\) \(\chi_{1280}(821,\cdot)\) \(\chi_{1280}(861,\cdot)\) \(\chi_{1280}(901,\cdot)\) \(\chi_{1280}(941,\cdot)\) \(\chi_{1280}(981,\cdot)\) \(\chi_{1280}(1021,\cdot)\) \(\chi_{1280}(1061,\cdot)\) \(\chi_{1280}(1101,\cdot)\) \(\chi_{1280}(1141,\cdot)\) \(\chi_{1280}(1181,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((511,261,257)\) → \((1,e\left(\frac{19}{64}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1280 }(61, a) \) \(1\)\(1\)\(e\left(\frac{25}{64}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{53}{64}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{11}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1280 }(61,a) \;\) at \(\;a = \) e.g. 2