Properties

Label 1352.69
Modulus $1352$
Conductor $1352$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,39,49]))
 
pari: [g,chi] = znchar(Mod(69,1352))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1352.bm

\(\chi_{1352}(69,\cdot)\) \(\chi_{1352}(101,\cdot)\) \(\chi_{1352}(173,\cdot)\) \(\chi_{1352}(205,\cdot)\) \(\chi_{1352}(277,\cdot)\) \(\chi_{1352}(309,\cdot)\) \(\chi_{1352}(381,\cdot)\) \(\chi_{1352}(413,\cdot)\) \(\chi_{1352}(517,\cdot)\) \(\chi_{1352}(589,\cdot)\) \(\chi_{1352}(621,\cdot)\) \(\chi_{1352}(693,\cdot)\) \(\chi_{1352}(725,\cdot)\) \(\chi_{1352}(797,\cdot)\) \(\chi_{1352}(829,\cdot)\) \(\chi_{1352}(901,\cdot)\) \(\chi_{1352}(933,\cdot)\) \(\chi_{1352}(1005,\cdot)\) \(\chi_{1352}(1109,\cdot)\) \(\chi_{1352}(1141,\cdot)\) \(\chi_{1352}(1213,\cdot)\) \(\chi_{1352}(1245,\cdot)\) \(\chi_{1352}(1317,\cdot)\) \(\chi_{1352}(1349,\cdot)\)

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

\((1015,677,1185)\) → \((1,-1,e\left(\frac{49}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 1352 }(69, a) \) \(1\)\(1\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1352 }(69,a) \;\) at \(\;a = \) e.g. 2