Properties

Label 1280.37
Modulus $1280$
Conductor $1280$
Order $64$
Real no
Primitive yes
Minimal yes
Parity odd

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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([0,25,16]))
 
pari: [g,chi] = znchar(Mod(37,1280))
 

Basic properties

Modulus: \(1280\)
Conductor: \(1280\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1280.bz

\(\chi_{1280}(13,\cdot)\) \(\chi_{1280}(37,\cdot)\) \(\chi_{1280}(93,\cdot)\) \(\chi_{1280}(117,\cdot)\) \(\chi_{1280}(173,\cdot)\) \(\chi_{1280}(197,\cdot)\) \(\chi_{1280}(253,\cdot)\) \(\chi_{1280}(277,\cdot)\) \(\chi_{1280}(333,\cdot)\) \(\chi_{1280}(357,\cdot)\) \(\chi_{1280}(413,\cdot)\) \(\chi_{1280}(437,\cdot)\) \(\chi_{1280}(493,\cdot)\) \(\chi_{1280}(517,\cdot)\) \(\chi_{1280}(573,\cdot)\) \(\chi_{1280}(597,\cdot)\) \(\chi_{1280}(653,\cdot)\) \(\chi_{1280}(677,\cdot)\) \(\chi_{1280}(733,\cdot)\) \(\chi_{1280}(757,\cdot)\) \(\chi_{1280}(813,\cdot)\) \(\chi_{1280}(837,\cdot)\) \(\chi_{1280}(893,\cdot)\) \(\chi_{1280}(917,\cdot)\) \(\chi_{1280}(973,\cdot)\) \(\chi_{1280}(997,\cdot)\) \(\chi_{1280}(1053,\cdot)\) \(\chi_{1280}(1077,\cdot)\) \(\chi_{1280}(1133,\cdot)\) \(\chi_{1280}(1157,\cdot)\) ...

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

\((511,261,257)\) → \((1,e\left(\frac{25}{64}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1280 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{27}{64}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{7}{64}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{31}{64}\right)\)\(e\left(\frac{37}{64}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{17}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1280 }(37,a) \;\) at \(\;a = \) e.g. 2