Properties

Label 1352.43
Modulus $1352$
Conductor $1352$
Order $78$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,39,61]))
 
pari: [g,chi] = znchar(Mod(43,1352))
 

Basic properties

Modulus: \(1352\)
Conductor: \(1352\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1352.bp

\(\chi_{1352}(43,\cdot)\) \(\chi_{1352}(75,\cdot)\) \(\chi_{1352}(179,\cdot)\) \(\chi_{1352}(251,\cdot)\) \(\chi_{1352}(283,\cdot)\) \(\chi_{1352}(355,\cdot)\) \(\chi_{1352}(387,\cdot)\) \(\chi_{1352}(459,\cdot)\) \(\chi_{1352}(491,\cdot)\) \(\chi_{1352}(563,\cdot)\) \(\chi_{1352}(595,\cdot)\) \(\chi_{1352}(667,\cdot)\) \(\chi_{1352}(771,\cdot)\) \(\chi_{1352}(803,\cdot)\) \(\chi_{1352}(875,\cdot)\) \(\chi_{1352}(907,\cdot)\) \(\chi_{1352}(979,\cdot)\) \(\chi_{1352}(1011,\cdot)\) \(\chi_{1352}(1083,\cdot)\) \(\chi_{1352}(1115,\cdot)\) \(\chi_{1352}(1187,\cdot)\) \(\chi_{1352}(1219,\cdot)\) \(\chi_{1352}(1291,\cdot)\) \(\chi_{1352}(1323,\cdot)\)

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

\((1015,677,1185)\) → \((-1,-1,e\left(\frac{61}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 1352 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1352 }(43,a) \;\) at \(\;a = \) e.g. 2