Properties

Label 1007.748
Modulus $1007$
Conductor $1007$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1007, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,27]))
 
pari: [g,chi] = znchar(Mod(748,1007))
 

Basic properties

Modulus: \(1007\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
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 1007.z

\(\chi_{1007}(7,\cdot)\) \(\chi_{1007}(11,\cdot)\) \(\chi_{1007}(64,\cdot)\) \(\chi_{1007}(144,\cdot)\) \(\chi_{1007}(163,\cdot)\) \(\chi_{1007}(197,\cdot)\) \(\chi_{1007}(216,\cdot)\) \(\chi_{1007}(467,\cdot)\) \(\chi_{1007}(486,\cdot)\) \(\chi_{1007}(520,\cdot)\) \(\chi_{1007}(539,\cdot)\) \(\chi_{1007}(600,\cdot)\) \(\chi_{1007}(653,\cdot)\) \(\chi_{1007}(676,\cdot)\) \(\chi_{1007}(695,\cdot)\) \(\chi_{1007}(714,\cdot)\) \(\chi_{1007}(729,\cdot)\) \(\chi_{1007}(748,\cdot)\) \(\chi_{1007}(767,\cdot)\) \(\chi_{1007}(771,\cdot)\) \(\chi_{1007}(824,\cdot)\) \(\chi_{1007}(885,\cdot)\) \(\chi_{1007}(938,\cdot)\) \(\chi_{1007}(961,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((743,267)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1007 }(748, a) \) \(1\)\(1\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{17}{39}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{1}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1007 }(748,a) \;\) at \(\;a = \) e.g. 2